Direct and indirect searches of heavy resonances in ... · 1 Chapter1 Introduction After the...

Direct and indirect searches of heavyresonances in strongly coupled physicsbeyond the Standard Model

Scuola di dottorato Vito VolterraDottorato di Ricerca in Fisica – XXVIII Ciclo

CandidateMatteo SalvarezzaID number 1168239

Thesis Advisor

Luca Silvestrini

Co-AdvisorRoberto Contino

A thesis submitted in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Physics2014 - 2015

Direct and indirect searches of heavy resonances in strongly coupledphysics beyond the Standard ModelPh.D. thesis. Sapienza – University of Rome

© 2015 Matteo Salvarezza. All rights reserved

This thesis has been typeset by LATEX and the Sapthesis class.

Author’s email: [email protected]

mailto:[email protected]

iii

Publication notes

• The content of chapter 2 is published in

Contino, R. and Salvarezza, M., One loop effects from spin-1 resonances in Composite Higgs models. JHEP 07 (2015) 065.arXiv:1504.02750, doi:10.1007/JHEP07(2015)065


Contino, R. and Salvarezza, M., Dispersion relations for Electroweakobservables in Composite Higgs models. Accepted for publication in Phys.Rev. D. arXiv:1511.00592.


Ghosh, D. Salvarezza, M., and Senia, F., Extending electroweakprecision analyses in Composite Higgs models. In preparation.


Azatov, A. and Salvarezza, M., Son, M., and Spannowsky, M.,Boosting top partnes searches in Composite Higgs models. Phys. Rev.D89 (2014) 065. arXiv:1308.6601, doi:10.1103/PhysRevD.89.075001

http://arxiv.org/abs/1504.02750

http://link.springer.com/article/10.1007%2FJHEP07%282015%29065



http://journals.aps.org/prd/abstract/10.1103/PhysRevD.89.075001

v

Contents

1 Introduction 1

2 One loop EWPT from Spin-1 resonances 52.1 Effective Lagrangian and its symmetries . . . . . . . . . . . . 5

2.1.1 Hidden local symmetry description . . . . . . . . . . . 92.1.2 Two-site model limit . . . . . . . . . . . . . . . . . . . 11

2.2 Electroweak parameters at 1 loop . . . . . . . . . . . . . . . . 152.2.1 Matching . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Fit to the EW observables . . . . . . . . . . . . . . . . . . . . 272.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3 A dispersion relation approach for the EWPO 393.1 Dispersion relation for ε3 . . . . . . . . . . . . . . . . . . . . . 40

3.1.1 Short- and long-distance contributions to ε3 . . . . . . 403.1.2 Dispersion relations for the short-distance contributions 43

3.2 Dispersive relation in the effective theory . . . . . . . . . . . 513.3 One-loop computation of ∆ε3 . . . . . . . . . . . . . . . . . . 543.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4 Extended analysis of EW constraints 654.1 Theoretical Framework . . . . . . . . . . . . . . . . . . . . . . 66

4.1.1 The model . . . . . . . . . . . . . . . . . . . . . . . . 664.1.2 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2 Calculation of Electroweak Precision Observables . . . . . . . 754.2.1 Contributions to the T parameter . . . . . . . . . . . 784.2.2 Contributions to the S parameter . . . . . . . . . . . . 82

vi Contents

4.2.3 Contributions to δg(b)L . . . . . . . . . . . . . . . . . . 85

4.3 Fit Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.3.1 Data and fit interpretation . . . . . . . . . . . . . . . 894.3.2 New physics parameters and physical masses . . . . . 92

4.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . 944.4.1 Fermionic sector . . . . . . . . . . . . . . . . . . . . . 954.4.2 Spin-1 sector . . . . . . . . . . . . . . . . . . . . . . . 1004.4.3 Combining spin-1/2 and spin-1 resonances . . . . . . . 1054.4.4 The two-site model limit . . . . . . . . . . . . . . . . . 107

4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5 Direct searches of top partners at the LHC 1135.1 Basic concepts and motivations . . . . . . . . . . . . . . . . . 1135.2 A Simplified Model . . . . . . . . . . . . . . . . . . . . . . . . 1185.3 Reinterpretation of existing searches . . . . . . . . . . . . . . 1215.4 Boosting searches using jet substructure . . . . . . . . . . . . 124

5.4.1 Cut and count analysis for l+jets final state . . . . . . 1255.4.2 Top partner mass reconstruction . . . . . . . . . . . . 129

5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

A Generators of SO(5)/SO(4) 139

B Useful formulas 141

C Two-site vector Lagrangian in the SO(5)× SO(5)H limit 149

D One-loop renormalization of the spin-1 Lagrangian 151

E Dispersion relations for a small breaking of SO(5) 155

F Recovering the asymptotic behavior at the cutoff scale 159

G Mass mixings 161

Bibliography 165

1

Chapter 1

Introduction

After the discovery of the Higgs boson [5, 42], one of the most importanttopics in today’s particle physics is to unravel the detailed dynamics of theElectroweak Symmetry Breaking (EWSB) mechanism of the Standard Model(SM). In absence of any sign of new physics, only one guiding theoretical prin-ciple remains: naturalness. More specifically, much effort has been put intoproviding solutions to stabilize the Higgs mass at the Electroweak (EW) scalewithout invoking the appearance of peculiar cancellations. Two approachesare commonly discussed; either the mass is protected by a symmetry (su-persymmetry), or the Higgs boson is a composite bound state from a newstrong dynamics, which protects his mass from virtual effects at very high en-ergy. In this thesis we will focus on the latter proposal. In order to accountfor the absence of additional bound states in the range of a few hundredsGeV, in this approach the Higgs boson is commonly introduced as a pseudoNambu-Goldstone (NG) Boson. In such picture, a natural mass gap betweenthe Higgs boson and the rest of the composite spectrum emerges naturally.This choice defines the Composite Higgs models [74, 57, 27, 75, 66, 65, 59],constructions featuring a spontaneous global symmetry breaking patternG → H (or, equivalently, a coset G /H ) in which the Higgs is an exactNG boson. Such breaking takes place at a scale f > v, where v ' 246GeV isthe Vacuum Expectation Value (VEV) of the Higgs boson field. A suitableexplicit breaking of the global symmetry G is then introduced, generating aHiggs potential accounting for both its mass and its VEV, thus triggering

2 1. Introduction

the EWSB. In order to generate a realistic EWSB, a tuning in the Higgspotential is required to take place. Such tuning is of order ξ ≡ (v/f)2, i.e.of the order of the separation between the NP and the EW scales.

Among the possible sources of experimental constraints for New Physics(NP) models, the Electroweak Precision Tests (EWPT) performed at LEP,SLD and Tevatron represent a serious challenge for Composite Higgs mod-els. A first important correction to the Electroweak Precision Observables(EWPO) in composite Higgs models arises as a consequence of the modifiedcouplings of the Higgs to the W and Z bosons [28], which receive shift of or-der ξ. This effect generates shifts to the oblique EWPO ε1, ε2 and ε3 [20, 21]and sets an upper bound ξ . 0.1, i.e. requires an amount of tuning roughlybelow 10%. A second important effect comes from threshold corrections atthe TeV scale and is due to the presence of the other composite resonances.Although their particular form being model dependent, a big effect is gener-ally expected to arise from tree-level propagations of spin-1 resonances. Aswe will see in detail, these effects are of order M2

W /M2ρ , implying a lower

limit on Mρ at the 2 − 3TeV level. Given this picture, it is interesting toinvestigate if additional contributions can improve or worsen the agreementwith the data. Such contributions arise at the one-loop level due to the spin-1and fermionic resonances.

While corrections to the EWPO can be naively estimated to be gener-ally large, their precise determination through the usual perturbative dia-grammatic calculations in the context of strongly-interacting dynamics is achallenge. A possible alternative, non-perturbative tool, consists in makinguse of dispersion relations to express an observable as the integral over thespectral functions of the strong dynamics. Extracting the spectral functionsfrom experimental data thus leads to a result which is, at least in principle,free from theoretical ambiguities. In the context of the EWPT, this possibil-ity was already discussed for Technicolor theories by Peskin and Takeuchi intheir seminal paper [89]. Although the most powerful use of dispersion rela-tions is in conjunction with experimental data, in the absence of the latterone can make models of the spectral functions based on theoretical consid-erations. Computing the spectral functions through a low-energy effectivetheory of resonances leads in fact to the same result obtained by a moreconventional diagrammatic technique, though the dispersive approach can

3

simplify the calculation and gives a different viewpoint.In order to produce a Higgs potential generating a realistic Higgs mass

without large tuning, composite fermionic resonances are required to liearound or below masses of 1 TeV. For this reason, in the context of Compos-ite Higgs model, great attention is put on the direct searches of these statesat the LHC. Since in these models SM fermions generally arise from mixingwith these composite states, they are called top partners. Up to now, no sig-nal of NP has been seen in the run 1 of the LHC, and the analysis of data hasput lower mass limits around 800GeV [1, 2, 3, 4, 43, 108], although the exactnumber is slightly model and species (i.e., electric charge) dependent [82].

This thesis is organized as follows: in chapter 2 we will introduce the mini-mal SO(5)/SO(4) Composite Higgs model and discuss a first EWPT analysisof one of its particular UV extensions, featuring the introduction of heavyspin-1 SU(2) triplets at the TeV scale. In chapter 3 we will discuss a differentapproach for the calculation of EWPO, based on dispersion relations. Thischapter is of purely theoretical interest rather than phenomenological. Onthe other hand, in chapter 4 we adopt a strong phenomenological approachand extensively discuss the EW fit of many different SO(5)/SO(4)-basedmodels, including various combinations of spin-1 and spin-1/2 resonances.Finally, chapter 5 is dedicated to a direct search strategy of heavy compositespin-1/2 states at the LHC. It is based on the combination of signals pro-duced by both single and pair production processes, and jet substructuretechniques used to resolve their boosted decay products.

5

Chapter 2

One loop EWPT fromSpin-1 resonances

In this chapter we will present and discuss a first analysis of EWPT in aComposite Higgs model: we will study the contributions from heavy spin-1 resonances. In section 2.1, we will introduce the minimal SO(5)/SO(4)Lagrangian describing the NGB Higgs doublet, followed by a detailed con-struction and discussion of an effective description for the resonances. Afterthis step, in section 2.2 we will proceed with the calculation of their effects,performed through a matching and running procedure. In addition we willpresent a detailed RG analysis of the whole spin-1 sector. Finally, in section2.3 we will present some example of comparison with data, and in section 2.4we will discuss the results obtained.

2.1 Effective Lagrangian and its symmetries

We will start discussing the minimal SO(5)/SO(4) Composite Higgs model,constructing the low-energy effective Lagrangian describing the NG bosonsand massive spin-1 resonances by using the formalism of Callan, Coleman,Wess and Zumino (CCWZ) [46, 41]. Nambu-Goldstone bosons are parametrizedin terms of the field

U(π) = exp(i√

2π(x)/f), (2.1)

6 2. One loop EWPT from Spin-1 resonances

where π(x) = πa(x)T a and f is the associated decay constant. 1 Underglobal rotations g ∈ SO(5), the NG fields transform as

U(π)→ U(g(π)) = g U(π)h†(g, π) , (2.2)

where h(g, π(x)) is an element of SO(4) which depends on g and π(x). TheCCWZ construction makes use of the covariant functions

dµ = daµTa,

E(L)µ =E(L) aT (L) a,

E(R)µ =E(R) aT (R) a,

(2.3)

defined byξµ ≡ −iU†DµU = dµ + Eµ = dµ + ELµ + ERµ . (2.4)

They transform as

dµ → h(g, π)dµh†(g, π),

Eµ → h(g, π)Eµh†(g, π)− ih(g, π)∂µh†(g, π) .(2.5)

In particular, Eµ = ELµ + ERµ transforms as a gauge field of SO(4) andcan be used to define a covariant derivative ∇µ = ∂µ + iEµ as well as afield strength Eµν = ∂µEν − ∂νEµ + i[Eµ, Eν ]. The SM electroweak vectorbosons weakly gauge a subgroup SU(2)L × U(1)Y ⊂ SO(4)′ contained inSO(5), where the SO(4)′ is misaligned by an angle θ with respect to theunbroken SO(4). Hypercharge is identified with Y = T 3R

θ , where T aLθ , T aRθare the generators of SO(4)′ (see again appendix A). The derivative appear-ing in Eq. (2.4) is thus covariant with respect to local transformations ofSU(2)L × U(1)Y : Dµ = ∂µ + iW aL

µ T aLθ + iBµY . Although the EW gaugingintroduces an explicit breaking of the global SO(5) symmetry, the low-energyLagrangian can still be expressed in an SO(5)-invariant fashion by introduc-ing suitable spurions that encode the breaking. We will be mainly interestedin custodially-breaking radiative effects induced by loops of the hyperchargefield, whileWµ will be treated as an external source. In this limit the explicitbreaking of SO(5) can be parametrized in terms of a single spurion

τ(π) = U†(π)g′T 3Rθ U(π) , (2.6)

1We denote with T a = {T (L) a, T (R) a} the generators of SO(4) ∼ SU(2)L×SU(2)Rand with T a those of SO(5)/SO(4), normalized such that Tr[TATB ] = δAB . Seeappendix A for more details.

2.1 Effective Lagrangian and its symmetries 7

whose formal transformation rule is

τ → h(g, π) τ h†(g, π) . (2.7)

The part of the Lagrangian which describes the interactions among NGbosons can be organized in a derivative expansion controlled by ∂/Λ:

L(π) = L(2)(π) + L(4)(π) + L(6)(π) + . . . (2.8)

where Λ . 4πf is the cutoff of the effective theory and L(n) indicates termswith n derivatives. Omitting for simplicity CP -violating operators, one has: 2

L(2)(π) = f2

4 Tr[dµdµ] + cT f2(Tr[dµτ ])2 + cτ f

2Tr[dµd

µτ2] (2.10)

L(4)(π) =∑i

ciOi + . . . (2.11)

where

O1 = Tr[dµdµ]2

O2 = Tr[dµdν ]Tr[dµdν ]

O±3 = Tr[(ELµν)2 ± (ERµν)2]

O±4 = Tr[(ELµν ± ERµν

)i[dµ, dν ]

] (2.12)

and the dots stand for higher-derivative terms and O(p4) operators involv-ing τ . We adopted the basis of four-derivative SO(5)-invariant operators ofRef. [24] (see also Ref. [49]) but dropped the operator O5 there appearingbecause it identically vanishes [19]. Among the terms with 6 derivatives weonly list two operators that are relevant for our analysis:

L(6)(π) = c2W(∇µELµν

)2 + c2B(∇µERµν

)2 + . . . (2.13)

The operators O−3 , O−4 are odd under the action of the parity PLR ex-changing the SU(2)L and SU(2)R groups inside the unbroken SO(4) [49]; allthe other operators in Eqs. (2.10),(2.12) are PLR even. In particular, underPLR the spurion τ transforms as

χ→ PLR U†(π)g′T 3L

θ U(π)PLR ≡ PLR τ PLR . (2.14)2Additional O(p2) operators with two powers of the spurion are not linearly inde-

pendent. Specifically, by using the identity ∇µτ = −i[dµ, τ ] it is easy to show that:

Tr[∇µτ∇µτ ] = 2Tr[dµd

µτ2]− (Tr[dµτ ])2

Tr[dµτdµτ ] =12

(Tr[dµτ ])2 .(2.9)


Considering that Tr[dµτ ] = −Tr[dµτ ] and τ2 = τ2, it easily follows that theoperators OT = f2(Tr[dµτ ])2 and Oτ = f2Tr

[dµd

µτ2] are even under PLR.While Oτ is also custodially symmetric, 3 the operator OT is the only onewhich breaks explicitly the custodial symmetry and thus contributes to the Tparameter. The S parameter instead gets a contribution from O+

3 [49, 24] 4.Spin-1 resonances will be described by vector fields ρLµ = ρ

(L) aµ T (L) a and

ρRµ = ρ(R) aµ T (R) a living in the adjoint of SO(4) ∼ SU(2)L × SU(2)R and

transforming non-homogeneously under SO(5) global rotations:

ρµ → h(g, π)ρµh†(g, π)− ih(g, π)∂µh†(g, π) . (2.17)

We will assume that the Lagrangian that describes their interactions can alsobe organized in a derivative expansion controlled by ∂/Λ, so that physicalquantities at E � Λ are saturated by the lowest terms [49]. In order to esti-mate the coefficients of the operators appearing in the effective Lagrangian,we adopt the criterion of Partial UV Completion (PUVC) [49]. This premisesthat the coupling strengths of the resonances to the NG bosons and to them-selves do not exceed, and preferably saturate, the σ-model coupling g∗ = Λ/fat the cutoff scale. Under this assumption, neglecting for simplicity CP -odd

3The operator Oτ breaks explicitly SO(5) down to the gauged SO(4)′. This canbe easily seen by rewriting Tr[dµdµτ2] = Tr[dµdµ] − (UdµdµU†)55, where the gaugedSO(4)′ acts on the first four components of SO(5). In the unitary gauge one hasOτ = (f2/16)[(W 1

µ)2 + (W 2µ)2 + (Bµ −W 3

µ)2] sin2(θ + h/f)(1 + sin2(θ + h/f)), whichis custodially symmetric.

4In the unitary gauge (with gauge kinetic terms normalized as −W aµνW

µν a/4g2,−BµνBµν/4g′2)

OT∣∣u.gauge

=g′2f2

4sin4(θ +

h

f

)(W 3µ −Bµ

)2

O+3

∣∣u.gauge

=12

sin2(θ +

h

f

)((W a

µν)2 + (Bµν)2 − 2W 3µνB

µν)

+ . . .

(2.15)

where the dots indicate terms with more than two gauge fields. By expanding in powersof the fields, at the level of dimension-6 operators, one has

OT =g′2

f2 |H†←→DνH|2 + . . .

O+3 = −

i

2f2DνW i

µν(H†σi←→DµH)−

i

2f2 ∂νBµν(H†

←→DµH) + . . .

(2.16)


operators, the leading terms in the Lagrangian are

L(ρ) =∑r=L,R

{− 1

4g2ρr

Tr(ρrµνρ

r µν)

+M2ρr

2g2ρr

Tr(ρrµ − Erµ

)2+ β1r Tr

[(ρrµ − Erµ)τ

]Tr(dµτ) + β2r

(Tr[(ρrµ − Erµ)τ

])2+ α1r Tr

(ρrµν i[dµ, dν ]

)+ α2r Tr

(ρr µνErµν

)}+ βLR Tr

[(ρLµ − ELµ )τ

]Tr[(ρRµ − ERµ )τ

].

(2.18)

Among the operators involving τ , we have kept only those relevant for thepresent analysis.

2.1.1 Hidden local symmetry description

The above construction relies on describing the resonances in terms of massivevector fields, which propagate three polarizations. At energiesMρ � E < Λ,however, the longitudinal and transverse polarizations behave differently(their interactions scale differently with the energy), and it is convenientto describe them in terms of distinct fields. Indeed, it is always possi-ble to parametrize the longitudinal polarizations of massive spin-1 fields interms of a set of eaten NG bosons 5. In the case of the Lagrangian (2.18)the corresponding coset is SO(5) × SO(4)H/SO(4)d, which leads to 10 NGbosons transforming under the unbroken diagonal SO(4)d as π = (2, 2),ηL = (3, 1) and ηR = (1, 3) [49]. Their σ-model Lagrangian can be ob-tained by taking the limit gρ → 0 with Mρ/gρ fixed; transverse polariza-tions are then reintroduced by gauging the SO(4)H subgroup with vec-tor fields ρµ. It is convenient to parametrize the NG bosons in terms ofU(π, η) = ei

√2π/feiη

L/fρL eiηR/fρR [49], where ηL(x) = ηaL(x)XaL , ηR(x) =

ηaR(x)XaR and, we recall, π(x) = πa(x)T a 6. It is thus straightforward to

5See for example Ref. [33].6We denote the SO(5) × SO(4)H/SO(4)d (broken) generators by T a, Xa = (T a −

T aH)/√

2, where T aH are those of SO(4)H , and the SO(4)d (unbroken) generators byY a = (T a + T aH)/

√2. We will consider their matrix representation on a 9 × 9 space,

so that T a, T a and T aH act respectively on 5 × 5 and 4 × 4 subspaces. All the traces


derive the CCWZ decomposition

− iU†DµU = dµ(π, η) + dLµ(π, η) + dRµ (π, η) + ELµ (π, η) + ERµ (π, η) (2.19)

where

dµ(π, η) = e−iηR/fρR e−iη

L/fρL dµ(π) eiηL/fρL eiη

R/fρR

drµ(π, η) + Erµ(π, η) = e−iηr/fρr

(−i∂µ + Erµ(π) + ρrµ

)eiη

r/fρr (r = L,R) ,(2.20)

where dµ(π, η), dµ(π, η) and Eµ(π, η) are obtained by projecting respectivelyalong the generators T a, Xa and Y a. Here dµ(π) and Eµ(π) denote the upliftof the corresponding SO(5)/SO(4) functions to the 9 × 9 space (they havenon-vanishing components in the 5×5 subspace). Notice that dµ(π, η) is justan (η-dependent) SO(4)d rotation of dµ(π). Since SO(5)× SO(4)H/SO(4)dis not a symmetric space, hence no grading of the algebra exists, the d and Esymbols will contain terms with both odd and even numbers of NG bosonsin their expansion. In particular (in the following formula ELµ = ELµ (π)),

(dLµ)aL = 1fρL

∂µηaL + 1√

2(ELµ − ρLµ

)aL − 12fρL

εaLbLcL(ELµ + ρLµ

)bLηcL

+ 14√

2fρL

[ηaL

(ELµ − ρLµ

)bLηbL −

(ELµ − ρLµ

)aLηbLηbL

]+ . . .

(2.21)

and similarly for dRµ . In the unitary gauge ηa = 0 one has (drµ)a = (Erµ(π)−ρrµ)a/

√2 (r = L,R). It is thus easy to see that the kinetic terms of the NG

bosons η are mapped into the ρ mass terms of Eq. (2.18),

f2ρr

2 Tr(drµ(π, η)dr µ(π, η)

)−→

f2ρr

4 Tr[(ρrµ − Erµ(π)

)2]5×5

, (2.22)

(where [ ]5×5 denotes a 5× 5 trace) with the identification

aρr ≡Mρr

gρrf= 1√

2fρrf

(r = L,R) . (2.23)

At the level of terms quadratic in the d symbols, other three operators withtwo powers of τ map into those with coefficients βi in Eq. (2.18), once eval-

in this section and in the next one (Sections 2.1.1 and 2.1.2) will be 9× 9 ones exceptwhere explicitly indicated.


uated in the unitary gauge:

Tr[drµ(π, η)τ(π, η)

]Tr[dµ(π, η)τ(π, η)

]→ −1

2Tr[ρrµ τ(π)

]Tr[dµ(π)τ(π)]

(Tr[drµ(π, η)τ(π, η)

])2 → 14(Tr[ρrµ τ(π)

])2Tr[dLµ(π, η)τ(π, η)

]Tr[dRµ (π, η)τ(π, η)

]→ 1

4Tr[ρLµ τ(π)

]Tr[ρRµ τ(π)

],

(2.24)

where the traces in the right hand side are understood being evaluatedin the 5 × 5 space. Here we defined ρrµ ≡ ρrµ − Erµ(π) and τ(π, η) ≡U†(π, η)T 3R

0 U(π, η).

2.1.2 Two-site model limit

While in general π, ηL, ηR form three irreducible representations of theunbroken group, in the gauge-less limit gρ = g = g′ = 0 and for the specialchoice fρL = fρR = f the O(p2) Lagrangian

f2

4 Tr(dµ(π)dµ(π)

)+f2ρL

2 Tr(dLµ(π, η)dLµ(π, η)

)+f2ρR

2 Tr(dRµ (π, η)dRµ(π, η)

)(2.25)

is invariant under a larger SO(5) × SO(5)H → SO(5)d global symmetry,under which the NG bosons transform as a single representation: a 10 ofSO(5)d. In this limit Eq. (2.25) describes an SO(5)×SO(5)H two-site model,where the EW vector bosons and the ρ gauge respectively the left and rightsite [88]. By virtue of Eqs. (2.22) and (2.23), the same two-site descriptionis obtained from a Lagrangian containing the kinetic and mass terms for πand ρ (first term of Eq. (2.10) and first two terms of Eq. (2.18)) for aρL =aρR = 1/

√2. Another, more convenient, parametrization of the Nambu-

Goldstone bosons is also possible in this case in terms of a 5 × 5 link field,U(π, η), transforming as a (5, 5) of SO(5) × SO(5)H , see Appendix C. Asdiscussed in detail in Ref. [88], the interest of the two-site model lies in thefact that the Higgs boson is doubly protected, and EWSB effects stem froma collective breaking of the global symmetry. There are indeed two sources ofexplicit breaking of SO(5)×SO(5)H : the EW gauging of an SU(2)L×U(1)Ysubgroup of SO(5) on the left site, and the gauging of SO(4)H by the ρ on the


right site. If either of these two gaugings is switched off, there is an unbrokenSO(5) symmetry which allows one to align the vacuum to θ = 0 without lossof generality. This means that for gρ → 0, with non-zero EW couplings, allEWSB effects must vanish in the two-site model. Indeed, the Higgs is a NGboson under both SO(5)’s, and both symmetries must be explicitly broken(hence the collective breaking) in order to generate any EWSB effect.

The authors of Ref. [88] also put forward a simple power counting argu-ment showing that collective breaking lowers the superficial degree of diver-gence of EWSB quantities. This is easy to see by working in a renormalizablegauge and noticing that the NG bosons η interact with strength E/fρ, whilethe gauge field ρµ has coupling gρ. In any 1PI diagram, replacing an internalη line with a ρ propagator lowers the degree of divergence by two unites.Indeed, if one focuses on the divergent part, the extra relative factor g2

ρf2ρ

of the new diagram can only be compensated by a factor 1/Λ2, where Λ isthe cutoff scale. Therefore, diagrams with loops of NG bosons alone (and notransverse gauge field ρ) carry the largest superficial degree of divergence.If they entail a breaking of the EW symmetry, then their sum will vanishin the two-site model, since one can set gρ = 0 in their evaluation and bythe previous argument the electroweak symmetry is exact in this limit. Theoriginal superficial degree of divergence is thus lowered. In particular, 1PIcontributions to EWSB observables will be finite in the SO(5)/SO(4) theory(with both ρL and ρR) for aρ = 1/

√2 7 if they are at most logarithmically

divergent in the general case.

This power counting argument was used in Ref. [88] to conclude that theS and T parameters are finite in the aρ = 1/

√2 limit. In the case of the S

parameter one can easily prove that for gρ = 0 there is no local countertermfor 1PI divergent contributions to the 〈W 3

µBν〉 Green function that can beconstructed in the two-site model compatibly with the SO(5) × SO(5)Hsymmetry, see Appendix C. Local operators built by including powers ofthe spurion gρ can be generated at the cutoff scale through loops where boththe heavier states and the ρ circulate. By power counting these effects arefinite at the 1-loop level, and lead to a contribution to the S parameter

7Here and in the following we use the notation aρ = 1/√

2 as a shorthand foraρL = aρR = 1/

√2. Similarly, gρ = 0 must be always interpreted as gρL = gρR = 0.


that is suppressed by an additional factor (gρf/Λ)2 = (gρ/g∗)2 comparedto the naive estimate. They are thus subleading and can be neglected ifgρ � g∗. As discussed in Section 2.2.1, our calculation confirms that the1PI divergence (hence the β-function of c+3 ) vanishes for aρ = 1/

√2. The S

parameter is thus calculable in terms of the renormalized gρ and α2, whichabsorb the divergences associated to subdiagrams. Things work differentlyfor the T parameter, however. It turns out that while the 1PI divergence tothe 〈W 1W 1〉−〈W 3W 3〉Green function vanishes according to the argument ofRef. [88], the β-function of cT does not vanish for aρ = 1/

√2 and there is still

a dependence on cT in the final result which enters through the cancellationof the subdivergences. This can be seen as follows.

First of all, we notice that in the theory aboveMρ it is possible to embedOT into the (SO(5)× SO(5)H)-invariant operator(

Tr[(dµ + 2dLµ + 2dRµ )τ(π, η)

])2 −→ (Tr[(dµ(π)− ρLµ − ρRµ )τ(π)

]5×5

)2,

(2.26)where the expression after the arrow is obtained by going to the unitary gaugeη = 0. The simplest way to show that this operator is SO(5) × SO(5)Hinvariant is through the link field U(π, η), see Appendix C. By expandingthe square in Eq (2.26) one obtains a linear combination of OT and otheroperators of the Lagrangian (2.18) with coefficients satisfying the relations

β1L = β1R = −βLR = −2cT , β2L = β2R = cT . (2.27)

These are the relations which must be imposed on the coefficients of theLagrangian (2.18) in order to recover the larger SO(5) × SO(5)H globalsymmetry at the level of terms quadratic in τ . This means that invarianceunder SO(5) × SO(5)H does not force cT to vanish, but simply to becomecorrelated with the coefficients of other operators in the Lagrangian.

But how a non-vanishing cT is compatible with the fact that no EWSBoccurs in the two-site model for gρ = 0 ? In this limit, there is an [SU(2)L×U(1)Y ] × SO(5)H → [SU(2)L × U(1)Y ]d invariance after the EW gaugingwhich gives 10 NG bosons. Four of these are eaten to give mass to theW aµ triplet and to the hypercharge, while the others remain massless and

transform as a 21/2 (the composite Higgs doublet), and a 1±1 of the unbroken[SU(2)L × U(1)Y ]d. 8. In particular, the unbroken global symmetry forces

8One can also describe the same particle content in terms of the NG bosons of


the W i to form a degenerate triplet. Compatibly with this, the operatorin Eq. (2.26) does not lead to any splitting between W 3 and W 1,2: theterm W 3

µW3µ contained in the expansion of OT is exactly canceled by a

similar contribution from the other operators in the Lagrangian (2.18) as aconsequence of the relations (2.27). One has:

Tr[(dµ + 2dLµ + 2dRµ )τ

]= g′

f

√2(∂µη3L

µ sin2(θ/2) + ∂µη3Rµ cos2(θ/2)

− sin θ2 ∂µπ

3)

+ g′(Bµ − ρ3L

µ sin2(θ/2)− ρ3Rµ cos2(θ/2)

)+ . . .

(2.28)

Since no corresponding counterterm is contained in Eq. (2.26), any 1PI con-tribution to the Green function 〈W 1W 1〉−〈W 3W 3〉 must be finite, in agree-ment with the power counting argument of Ref. [88]. This is however notsufficient to conclude that the T parameter is finite, since non-1PI diagramsalso contribute and can be divergent. 9 Our calculation in section 2.2.1indeed shows that a divergent contribution arises from subdiagrams throughthe 1-loop correction to the ρ propagator. The associated counterterm iscontained in the operator (2.26), whose coefficient cT thus enters in the ex-pression of the T parameter.

It is interesting to notice that the T parameter can also be extractedfrom the Green function 〈π3π3〉, as done in Section 2.2.1, for which a 1PIdivergent contribution does exist. The corresponding counterterm (π3)2 iscontained in Eq. (2.26), and it is not in clash with the argument of Ref. [88].This is because π3 appears in the linear combination of NG bosons, the onein parenthesis in the first line of Eq. (2.28), that is eaten to give mass to thehypercharge for gρ = 0. 10 The 〈π3π3〉 Green function thus does not breakthe [SU(2)L × U(1)Y ]d symmetry and can be divergent.

Although it depends on cT , the T parameter can still be regarded as acalculable quantity in the two-site limit, up to g2

ρ/g2∗ effects. This is because

the operator (2.26) gives a custodially-breaking shift to the mass of the neu-tral ρ’s, so that cT can be rewritten in terms of the difference of charged andSO(5)H/[SU(2)L × U(1)Y ] plus massive spin-1 resonances (Wµ and Bµ).

9We thank G. Panico and A. Wulzer for discussions on this point.10For θ = 0 the NG boson eaten by the hypercharge is η3R, while the ηaL are eaten

to give mass to the W triplet.

2.2 Electroweak parameters at 1 loop 15

neutral renormalized ρ masses. In this sense T , similarly to S, is calculablein terms of parameters related to the ρ, which can be fixed experimentallyby measuring its properties.

2.2 Electroweak parameters at 1 loop

Oblique corrections to the electroweak precision observables at the Z-poleare conveniently described by the three ε parameters [20, 21]

ε1 = e1 − e5

ε2 = e2 − s2e4 − c2e5

ε3 = e3 + c2e4 − c2e5

(2.29)

defined in terms of the following vector-boson self energies:

e1 = 1M2W

(A33(0)−AW+W−(0)) ,

e2 = FW+W−(M2W )− F33(M2

Z),

e3 = c

sF3B(M2

Z),

e4 = Fγγ(0)− Fγγ(M2Z),

e5 = M2ZF′ZZ(M2

Z) .

(2.30)

Here s (c) denotes the sine (cosine) of the Weinberg angle and, according tothe standard notation, the vacuum polarizations are decomposed as

Πµνij (q) = −iηµν

(Aij(0) + q2Fij(q2)

)+ qµqν terms . (2.31)

There are two kind of modifications to the self-energies (3.2) from new physicsin our model. The first is due to the virtual exchange of the spin-1 reso-nances, which at energies E ∼ MZ � Mρ can be parametrized in terms oflocal operators of the effective Lagrangian (2.8). The tree-level contributionof these local operators to physical observables is a pure short-distance ef-fect, while their insertion in 1-loop diagrams with light fields contains alsoa long-distance part. The second modification comes from the fact thatthe composite Higgs has non-standard couplings with the electroweak vectorbosons. The bulk of the correction in this case is given by a logarithmicallydivergent part that can be easily computed in the low-energy theory with


Figure 2.1. One-loop diagrams relative to the Higgs contribution to the epsilonparameters. Wavy, continuous and dashed lines denote respectively gaugefields (W± and Z), NG bosons of SO(4)/SO(3) (π1,2,3) and the Higgs boson.

light fields [28]. Extracting the finite part instead requires fully recomputingthe Higgs contribution to the vector boson self energies in Fig. 3.1, as pointedout in Ref. [86]. Since the Higgs boson is light, this is a long-distance effect.It is so even if the compositeness scale is large, f � v, so that the shifts ofthe Higgs couplings to vector bosons are parametrized by local operators atlow energies. Indeed, the insertion of these local operators into the 1-loopdiagrams of Fig. 3.1 contains both long- and short-distance contributions. 11

We have performed a calculation of the εi at the 1-loop level including allthe contributions mentioned above. We have used dimensional regularizationand performed a minimal subtraction of the divergences (MS scheme). Wechoose to work in the Landau gauge for the elementary gauge fields, ∂µW i

µ =0 = ∂µBµ, since it conveniently preserves the custodial invariance of thestrong sector and leads to massless (hence degenerate) NG bosons π1,2,3.The one-loop contribution from the spin-1 resonances is computed througha matching procedure. We integrate out the ρ at a scale µ ∼Mρ and matchwith a low-energy Lagrangian which has the same form of Eq. (2.8). Itscoefficients will be denoted by ci(µ), where the tilde distinguishes them fromthe corresponding quantities in the full theory. By working in such low-energy theory and defining the shifts to the epsilons to be ∆εi = εi − εSMi ,

11The divergent part of the diagrams corresponds to a renormalization of the localoperators of the effective Lagrangian, and it is thus a short-distance effect. The finitepart is instead genuinely long distance.


we find

∆ε1 = − 3g′2

32π2 sin2θ

[log µ

MZ+ f1

(M2h

M2Z

)]− 2g′2 sin2θ cT , (2.32)

∆ε2 = g2

192π2 sin2θ f2

(M2h

M2Z

)+ 2M2

W g2(c2W cos4 θ

2 + c2B sin4 θ

2

)

+ g4

24π2 sin2θ cos4 θ

2

[(c+3 + c−3

)− 1

2(c+4 + c−4

)]log µ

MZ

+ g4

24π2 sin2θ sin4 θ

2

[(c+3 − c

−3)− 1

2(c+4 − c

−4)]

log µ

MZ,

(2.33)

∆ε3 = g2

96π2 sin2θ

[log µ

MZ+ f3

(M2h

M2Z

)]− 2g2 sin2θ c+3 . (2.34)

The first term in each equation corresponds to the Higgs contribution ofFig. 3.1 12 and agrees with the results of Ref.[86]. The explicit expression ofthe functions f1,2,3 is given in Appendix B. The coefficients c+3 , cT , c2W , c2Bencode the short-distance contribution from the ρ and from cutoff states, andare in one-to-one correspondence with the parameters S, T,W, Y defined inRefs. [89, 29]. The latter are introduced through an expansion of the selfenergies (3.3) in powers of q2 and parametrize the contribution from newheavy physics. At the tree level one can identify

S = −2g2 sin2θ c+3 , W = −2M2W g

2(c2W cos4 θ

2 + c2B sin4 θ

2

)

T = −2g′2 sin2θ cT , Y = −2M2W g

2(c2W sin4 θ

2 + c2B cos4 θ

2

),

(2.35)

where S = (αem/4s2)S and T = αemT [29]. The naive estimate of W andY is suppressed by a factor g2/g2

ρ compared to that of S and T [29]. Wethus included their contribution (i.e. the contribution of c2W and c2B) onlyin ε2, where it gives the leading effect. At the 1-loop level, the expression ofS, T,W, Y includes the logµ terms of Eqs. (2.32)-(2.34). These arise from theshort-distance, logarithmically divergent part of the Higgs contribution, and

12It can be found from the Higgs contribution in the SM by considering that theHiggs couplings to vector bosons are rescaled by a factor cos θ, so that εi|Higgs =cos2θ εSMi |Higgs, hence ∆εi|Higgs = − sin2θ εSMi |Higgs.


Figure 2.2. One-loop diagram with one insertion of O±3 and O±4 (crossed vertex)contributing to the running of c2W and c2B in the low-energy theory. Wavyand continuous lines denote respectively gauge fields (W and B) and NGbosons of SO(5)/SO(4) (πa).

exactly compensate the dependence of the ci on µ to give an RG-invariantresult. The finite part of the Higgs contribution is a genuinely long-distancecorrection to the SM, and it is not encoded by S, T,W, Y , although it iscaptured by the ∆εi. These latter are also independent of µ, being observ-able quantities: the variation of the ci(µ) is canceled by the logarithms inEqs. (2.32)-(2.34). We find that the evolution of the ci is described by theRG equations

µd

dµc+3 (µ) = 1

192π2

µd

dµcT (µ) = − 3

64π2

µd

dµc2W (µ) = − g2

48π2sin2θ

M2W

[(c+3 + c−3 )− 1

2(c+4 + c−4 )]

µd

dµc2B(µ) = − g2

48π2sin2θ

M2W

[(c+3 − c

−3 )− 1

2(c+4 − c−4 )].

(2.36)

Notice that the β-function of c2W , c2B is proportional to c±3 and c±4 , since therunning of these coefficients arises from the 1-loop insertion of the operatorsO±3 and O±4 defined in Eq. (2.12), see Fig. 2.2. Since c±3 and c±4 are generatedat tree level at the matching scale, they should be included at 1-loop in thecalculation of ε2. The last two terms in Eq. (2.33) account for the divergentpart of the diagram of Fig. 2.2 and cancel the µ dependence due to therunning of c2W , c2B . An additional finite contribution from of the 1-loopinsertion of O±3 and O±4 has been omitted for simplicity. It is subleading bya factor logµ/MZ and its computation would require evaluating additionaldiagrams with gauge fields circulating in the loop.


2.2.1 Matching

The explicit contribution of the spin-1 resonances to the ci can be obtainedby integrating them out and matching to the low-energy Lagrangian. Weperform this matching at the 1-loop level. This requires working out atthe same time the renormalization of the Lagrangian for the ρ, in order toderive the RG evolution of its parameters. We considered two choices tofix the gauge invariance associated with the ρ field and checked that theyboth lead to the same result for physical quantities: the first is the unitarygauge, where the ρ is described by the Lagrangian (2.18); the second isthe Landau gauge ∂µρaµ = 0, obtained by introducing the NG bosons η asdiscussed in Section 2.1.1. In the following we will report results for theunitary gauge, and collect formulas for the Landau gauge in Appendix D.Particularly relevant for our calculation is the running of gρ and α2, sincethese parameters enter at tree level in the expression of the εi. In the unitarygauge we find

µd

dµgρ(µ) ≡ βgρ =

g3ρ

16π22a4ρ − 8512 (2.37)

µd

dµα2(µ) ≡ βα2 =

a2ρ(1− a2

ρ)96π2 , (2.38)

for both ρL and ρR (there is no mixed renormalization of left and rightparameters at the 1-loop level). Other details on the renormalization of theρ Lagrangian can be found Appendix D.

A few remarks should be made about our calculation. First of all, wewill compute the Green functions relevant for the matching by neglectingthe masses of the Higgs and of the vector bosons. This implies a relativeerror of order M2

h/M2ρ , which is of the same size of the error due to the

truncation of the effective Lagrangian to the leading derivative operators (ofO(p4) in the case of ε1,3 and O(p6) for ε2). Infrared divergences are regulatedby introducing a small common (hence custodially-preserving) mass λ forthe NG bosons. The dependence on λ cancels out when matching the fulland low-energy theories. Second, the expressions for the ci reported in thissection are obtained by including the contribution of α1,2 only at the treelevel. This is justified if these coefficients are generated at the 1-loop level atthe cutoff scale Λ. The additional contribution from α2 at 1-loop is reported


Figure 2.3. Tree-level diagram contributing to the 〈W 3µBν〉 Green function.

Single and double wavy lines denote respectively the elementary gauge fields(W and B) and the ρ.

Figure 2.4. Diagram with a loop of NG bosons contributing to the 〈W 3µBν〉

Green function. Wavy and continuous lines denote respectively the elementarygauge fields (W and B) and the NG bosons (πa and η).

in Appendix B. Finally, our formulas will include the contribution of both theρL and the ρR. In case only one resonance is present in the theory, c+3 andcT have the same expression for both ρL and ρR, whereas ρL only generatesc2W , and ρR only c2B . This follows from a simple symmetry argument.Given a theory with a ρL, the case with a ρR is obtained by performinga PLR transformation on the strong dynamics. The equality of c+3 and cT

then follows from the invariance of the operators O+3 and OT under such

transformation. On the other hand, acting with PLR interchanges O2W withO2B , so that the expression of c2W in a theory with a ρL equals that of c2Bin a theory with ρR. We report the corresponding expressions in Appendix Bfor convenience.

Let us start discussing the matching for c+3 . We make use of the two-pointGreen function 〈W 3

µBν〉, in particular its derivative evaluated at q2 = 0, andmatch its expression in the full and effective theories. We focus on the leadingcontribution in g2, thus considering diagrams where only the ρ and the NGbosons (i.e. no elementary gauge field) circulate in the loop. These are thediagrams of Figs. 2.3, 2.4 and 2.5 for the full theory (ρ + NG bosons) andof Fig. 2.4 for the effective theory (only NG bosons). Neglecting diagramswith EW vector bosons circulating in the loop implies a relative error of orderg2/g2

ρ. Divergences from subdiagrams in the full theory are canceled by theaddition of suitable counterterms. The remaining divergence is associated


Figure 2.5. One-loop diagrams with ρ exchange contributing to the 〈W 3µBν〉

Green function. Single and double wavy lines denote respectively the elemen-tary gauge fields (W and B) and the ρ; continuous and dashed lines denoterespectively the NG bosons (πa and η) and the ghosts associated to the gaugefixing of the ρ field. The diagrams obtained by crossing those in the secondline are not shown for simplicity.


with the running of c+3 betweenMρ and Λ due to loops of ρ’s and NG bosons.We find

µd

dµc+3 (µ) ≡ βc+

3= 1

192π2

[32 + 1

4a2ρL(2a2

ρL − 7) + 14a

2ρR(2a2

ρR − 7)].

(2.39)Notice that βc+

3(hence the associated divergence) vanishes for aρL = aρR =

1/√

2, in agreement with the symmetry argument of Section 2.1.2. By match-ing the full and low-energy theories at a scale µ ∼Mρ, we obtain

c+3 (µ) = c+3 (µ)− 12

(1

4g2ρL

− α2L + 14g2ρR

− α2R

)+ 1

192π2

[34(a2

ρL + 28) log µ

MρL

+ 34(a2

ρR + 28) log µ

MρR

+ 2 + 4116a

2ρL + 41

16a2ρR

].

(2.40)

Obviously, since c+3 contributes to an observable such as ∆ε3 (see Eq. (2.34)),its expression (2.40) is the same in any gauge. In fact, it turns out that eventhe β-function of c+3 , Eq. (2.39), is gauge invariant at one loop. The argumentgoes as follows. When working at the 1-loop level, the logarithms that appearin the expression of an observable determine the running of the combinationof the parameters giving the tree-level contribution. Since the expression ofthe observable is gauge invariant, also the RG evolution of such combinationwill be invariant. In the case of ∆ε3, the tree-level contribution is given bythe terms in the first line of Eq. (2.40). Furthermore, (1/2gρ − α2gρ)2 (foreach ρ species) also has a gauge invariant running, since it gives the tree-level contribution to another observable: the pole residue of the ρ two-pointfunction (see chapter 3). Working in the approximation in which 1-loopeffects from α1,2 are neglected, this in turn implies that (1/4g2

ρ − α2) hasan invariant RG evolution, 13 hence the same follows for c+3 . Clearly, whenincluding the 1-loop contribution of α2 or going to two loops, the running ofc+3 acquires a gauge-dependent part.

Let us now turn to cT . In order to extract it, we make use of the two-point Green function of the π field, in particular we consider the custodiallybreaking combination 〈π1π1〉 − 〈π3π3〉 and compute its derivative at q2 = 0.This gives access to the coefficient of the operator OT , as it follows from the

13The running of the α22 term is of the same order of the neglected terms.


Figure 2.6. One-loop diagram with NG bosons contributing to the 〈π1π1〉 −〈π3π3〉 Green function. Wavy and continuous lines denote respectively thehypercharge gauge field B and the NG bosons (πa and η).

Figure 2.7. One-loop diagrams with ρ exchange contributing to the 〈π1π1〉 −〈π3π3〉 Green function. Single and double wavy lines denote respectively thehypercharge gauge field B and the ρ, while continuous lines denote the NGbosons (πa and η). The diagram obtained by crossing the first one is notshown for simplicity.

expansion Tr[dµτ ] = g′ sin2θ (W 3µ−Bµ)−g′ sin θ (∂µπ3/f)+. . . . The relevant

1-loop diagrams are shown in Figs. 2.6 and 2.7 for the full theory (ρ + NGbosons), and in Fig. 2.6 for the low-energy theory of NG bosons. Onlydiagrams where an elementary Bµ circulates contribute, as this latter givesthe required breaking of custodial symmetry. As for c+3 , we neglect diagramswith further insertions of EW vector bosons, since they are of higher orderin g2. The corresponding relative error is of order g2/g2

ρ. Since there are nodivergent subdiagrams, the overall divergence in the full theory is associatedwith the running of cT between the scales Λ and Mρ. We find:

µd

dµcT (µ) ≡ βcT = − 3

64π2

(1− 3

4a2ρL −

34a

2ρR + a2

ρLa2ρR

). (2.41)

In alternative, one can extract cT by considering the combination 〈W 1W 1〉−〈W 3W 3〉: the relevant 1-loop diagrams are shown in Figs. 2.8 and 2.9 for thefull theory (ρ + NG bosons), and in Fig. 2.8 for the low-energy theory ofNG bosons. Some of the diagrams have subdivergences associated with therenormalization of the ρ propagator and of the ρ −W mixing. The corre-sponding counterterms in the unitary gauge are (Tr[ρrµχ])2, Tr[dµχ]Tr[ρrµχ]


Figure 2.8. One-loop diagram with NG bosons contributing to the 〈W 1W 1〉 −〈W 3W 3〉 Green function. External (internal) wavy lines denote the elemen-tary W (B) field, while continuous lines stand for the NG bosons (πa andη).

Figure 2.9. One-loop diagrams with ρ exchange contributing to the 〈W 1W 1〉 −〈W 3W 3〉 Green function. External (internal) wavy lines denote the elemen-tary W (B) field, while continuous lines stand for the NG bosons (πa and η).The diagrams obtained by crossing the second, third and fourth one are notshown for simplicity.

and Tr[ρLµχ]Tr[ρRµχ], where r = L,R and ρrµ ≡ ρrµ − Erµ. The contributionof these counterterms, however, cancels out when summing all the diagrams.The overall divergence of the 〈W 1W 1〉 − 〈W 3W 3〉 Green function is thusremoved by the single counterterm (Tr[dµχ])2, as required to reproduce thecalculation of cT through 〈π1π1〉 − 〈π3π3〉. By matching the low-energy andfull theories one obtains Eq. (2.43). A further check of the calculation fol-lows from the fact that in the limit aρL = aρR = 1/

√2 the counterterms

combine into the (SO(5)×SO(5)H)-invariant operator of Eq. (2.26). In thislimit the 1PI divergence vanishes, and the only divergent contribution to


〈W 1W 1〉 − 〈W 3W 3〉 comes from subdiagrams.

Since cT gives the only tree-level contribution to ∆ε1 (see Eq. (2.43)below), its RG evolution is gauge invariant. One can see that βcT does notvanish for aρL = aρR = 1/

√2. This confirms the argument of Section 2.1.2,

where it was noticed that a counterterm exists also in the SO(5) × SO(5)symmetric limit (see Eq. (2.26)), and no cancellation of the 1PI divergenceof the 〈π1π1〉 − 〈π3π3〉 Green function is expected in this case. There is infact a limit in which the divergence partly cancels, as already discussed inRef. [85] for a Higgsless model. Indeed, the diagram of Fig. 2.6 and the firsttwo diagrams in Fig. 2.7 can be combined into one where Bµ couples to theNG bosons through the effective vertex

where the Bπaπb form factor denoted by the gray blob is equal to

{12

[1− a2

ρL sin2 θ

2

(1 +

M2ρL

q2 −M2ρL

)− a2

ρR cos2 θ

2

(1 +

M2ρR

q2 −M2ρR

)]ε3ab

− 14

[cos θ + a2

ρL sin2 θ

2

(1 +

M2ρL

q2 −M2ρL

)− a2

ρR cos2 θ

2

(1 +

M2ρR

q2 −M2ρR

)]

×(δa4δb3 + δa3δb4

)}(k + k′)µ + qµ terms .

(2.42)

In the limit aρL = aρR = 1 one obtains Vector Meson Dominance (VMD)for any value of θ, i.e. the form factor goes to 0 in the limit q2 → ∞.Consequently, the diagram built with the effective vertex (i.e. the sum of thediagram in Fig. 2.6 and the first two of Fig. 2.7) is finite. This does not imply,however, that the β-function of cT vanishes, since the last diagram of Fig. 2.7is still divergent. One can explicitly check, indeed, that the coefficient of thelogarithm in Eq. (2.41) does not vanish for aρL = aρR = 1. By matching the


full and low-energy theory at the scale µ we finally obtain

cT (µ) = cT (µ)− 9256π2

[a2ρL

(1− 4

3a2ρR

M2ρL

M2ρL −M2

ρR

)log µ

MρL

+ a2ρR

(1− 4

3a2ρL

M2ρR

M2ρR −M2

ρL

)log µ

MρR

+ 34a

2ρL + 3

4a2ρR −

59a

2ρLa

2ρR

].

(2.43)

Since cT contributes to the observable ∆ε1, this expression is gauge invariant.Finally, we discuss the matching to extract c2W and c2B . We make use

of the 〈WµWν〉 and 〈BµBν〉 Green functions, in particular we compute theirsecond derivative evaluated at q2 = 0. Working at leading order in g2, thediagrams in the full and effective theories are the same as in Figs. 2.3, 2.4and 2.5, where now the external gauge fields are either two W ’s (to extractc2W ) or two B’s (to extract c2B). There is in fact one additional diagram,shown in Fig. 2.2, which has to be included in the effective theory. It containsone insertion of the operators O±3 and O±4 defined in Eq. (2.12). As previouslynoticed, this contribution is relevant in the effective theory below Mρ sincec±3 and c±4 are generated at the tree-level by the exchange of the ρ. InsertingO±3 and O±4 in a 1-loop diagram thus gives a contribution to c2W and c2B

which is formally of the same order as that of the diagrams in Figs. 2.3-2.5. In fact, such contribution is required in order to properly match theIR divergence of the full and low-energy theories. The cancellation occursif c±3 and c±4 are set to the value they have at tree-level for αi = 0 (thatis: c±3 = −1/8g2

ρL ∓ 1/8g2ρR and c±4 = 0) when evaluating the diagram of

Fig. 2.2; we will thus adopt this choice. 14 There are no divergences left afterremoving those from subdiagrams through the renormalization of the ρ massand kinetic terms. This implies that the running of the coefficients c2W andc2B vanishes in the full theory between Mρ and Λ:

µd

dµc2W (µ) ≡ βc2W = 0 , µ

d

dµc2B(µ) ≡ βc2B = 0 . (2.44)

14When including the contribution of α2 at the 1-loop level, as done in Appendix B,one should instead set c±3 = (−1/4g2

ρL+α2L)/2± (−1/4g2

ρR+α2R)/2, while including

α1 at the 1-loop level requires setting c±4 = (α1L ± α1R)/2.

2.3 Fit to the EW observables 27

This result is independent of the choice of gauge. Indeed, by matching thefull and low-energy theories we obtain

c2W (µ) = c2W (µ)− 12g2ρLM

2ρL

(1− 2α2Lg2ρL)2

+ 196π2M2

ρL

[77 log µ

MρL

+ 465 −

2732a

2ρL

sin2θ

1 + cos2θ

(1 +

g2ρL

g2ρR

)]

(2.45)

c2B(µ) = c2B(µ)− 12g2ρRM

2ρR

(1− 2α2Rg2ρR)2

+ 196π2M2

ρR

[77 log µ

MρR

+ 465 −

2732a

2ρR

sin2θ

1 + cos2θ

(1 +

g2ρR

g2ρL

)].

(2.46)

The tree-level contribution to ∆ε2 comes from the combination of terms inthe first line of the above equations. We already noticed that (1/gρ−2α2gρ)2

has an invariant RG evolution at the 1-loop level; the same holds true forMρ, since it gives the tree-level contribution to the pole mass. It thus followsthat the RG evolution of c2W and c2B is also gauge invariant at one loop.

2.3 Fit to the EW observables

The results of the previous section can be used to perform a fit to the εi. It isconvenient to express the corrections ∆εi in terms of the parameters gρ, α2

andMρ evaluated at the physical mass scale of the resonancesMpoleρ . 15 This

removes all the logarithms originating from subdivergences leaving only thoseassociated with the running of O+

3 , OT , O2W and O2B . We will consider twobenchmark scenarios: in the first (Scenario 1) both ρL and ρR are presentwith equal masses and couplings (as implied for example by PLR invariance);in the second (Scenario 2) only a ρL is included. In either case the ∆εi can

15 For this evaluation we approximate Mpoleρ ' Mρ, the difference being of higher

order.


be written as

∆ε1 = − 2g′2 sin2θ cT (Λ)

− 3g′2

32π2 sin2θ

[f1

(M2h

M2Z

)+ log Mρ

MZ+ β1 log Λ

Mρ+ ζ1

](2.47)

∆ε2 = 2M2W g

2(c2W (Λ) cos4 θ

2 + c2B(Λ) sin4 θ

2

)− γ2

g2

g2ρ

M2W

M2ρ

(1− 2α2g

2ρ

)2+ g2

192π2 sin2θ

[f2

(M2h

M2Z

)+ β2

g2

g2ρ

log Mρ

MZ+ β2

g2

g2ρ

log ΛMρ

+ ζ2g2

g2ρ

](2.48)

∆ε3 = − 2g2 sin2θ c+3 (Λ) + γ3g2

g2ρ

sin2θ(1− 4α2g

2ρ

)+ g2

96π2 sin2θ

[f3

(M2h

M2Z

)+ log Mρ

MZ+ β3 log Λ

Mρ+ ζ3

],

(2.49)

where gρ, α2 and Mρ are evaluated at µ = Mρ and the O(1) coefficientsβi, βi, ζi, γi are reported in Table 2.1 in the simplified limit where 1-loopcontributions from α1,2 are neglected.

Let us analyze Eqs. (2.47)-(2.49) and discuss the various terms. For each∆εi one can identify: a tree-level contribution from the exchange of spin-1 resonances (second term of Eqs.(2.48) and (2.49)); a threshold correctiondue to Higgs compositeness (first term in square parenthesis); the IR run-ning from Mρ down to MZ , controlled by the low-energy β-function (secondterm in square parenthesis); the running from the cutoff Λ to Mρ, computedincluding the spin-1 resonances (third term in square parenthesis); a finitepart from the 1-loop ρ exchange (last term in square parenthesis). Finally,each ∆εi receives a short-distance correction from physics at the cutoff scale,encoded by the coefficients ci(Λ) (first term in Eqs.(2.47)-(2.49)).

In the case of ε3, the leading corrections come from the tree-level contri-bution (of orderM2

W /M2ρ ) and the IR running. Compared to the former, the

latter effect is suppressed by a factor g2ρ/16π2 but enhanced by log(Mρ/MZ).

The 1-loop ρ contribution is subleading because also suppressed by g2ρ/16π2

and enhanced by the smaller logarithm associated with the running betweenΛ andMρ. The contribution from cutoff physics encoded by c+3 (Λ) can be es-timated through Naive Dimensional Analysis (NDA) [80]. If the dynamics at


Scenario 1 (ρL + ρR) Scenario 2 (ρL)

β1 1− 32a

2ρ + a4

ρ 1− 34a

2ρ

ζ1 −98a

2ρ −

112a

4ρ − 9

16a2ρ

β2 0 0

β2 −(1 + cos2θ) −2 cos4 θ

2

ζ2 (1 + cos2θ)(

235a2ρ

− 2732 tan2 θ

2

)cos4 θ

2

(465a2ρ

− 2732 tan2 θ

2

)γ2

12(1 + cos2θ) cos4 θ

2

β332 +

a2ρ

2 (2a2ρ − 7) 5

4 +a2ρ

4 (2a2ρ − 7)

ζ3 −2− 418 a

2ρ −1− 41

16a2ρ

γ312

14

Table 2.1. Expression of the coefficients βi, ζi and γi, defined in Eqs. (2.47)-(2.49), in the limit where 1-loop contributions from α1,2 are neglected. Sce-narios 1 includes ρL and ρR with equal masses and couplings, while only ρLis included in Scenario 2.

the scale Λ is maximally strongly coupled one expects c+3 (Λ) ∼ 1/16π2, whichleads to a correction of the same size of the finite part and thus subleadingcompared to the 1-loop ρ contribution by a factor log(Λ/Mρ). Although thislogarithm is not large, since one does not expect a very large separation ofscales, it gives a parametric justification for including the 1-loop effect of theρ. In general, if the cutoff dynamics is characterized by a coupling strengthg∗, one naively expects c+3 (Λ) ∼ 1/g2

∗. For gρ < g∗ < 4π this implies a correc-tion larger than the 1-loop ρ contribution, though smaller than the tree-levelone. Interestingly, in the two-site limit (Scenario 1 with aρ = 1/

√2) the

SO(5) × SO(5)H global invariance of the theory below the cutoff ensures


c+3 (Λ) = 0, since the corresponding operator vanishes. Notice that βc+3van-

ishes also in Scenario 2 for aρL = 1, although in that case no larger symmetryis realized that can enforce c+3 (Λ) = 0. Similarly, no symmetry protectionfollows from the vanishing of βcT , βc2W and βc2B for specific values of theparameters.

Similar estimates of the various terms hold for ∆ε1, except there is notree-level correction due to custodial invariance, so that the largest effectcomes from the IR running. In the case of ∆ε2, the contribution from theρ exchange (both at tree and loop level) is suppressed by a factor (g2/g2

ρ)compared to the one entering ∆ε1 and ∆ε3. This is because the leading short-distance contribution in the low-energy theory arises at O(p6) through theoperators O2W , O2B [67]. The RG evolution of these latter in turn proceedsthrough the 1-loop insertion of O(p4) operators, as discussed in the previoussection, implying that the IR running contribution to ∆ε2 is also suppressedby a factor (g2/g2

ρ). The only unsuppressed effect is the finite term fromHiggs compositeness, which is however numerically small. The overall shiftto ε2 thus tends to be small and plays a minor role in the fit.

Besides the direct contributions to the ∆εi described above there is alsoan indirect one from the evolution of gρ, Mρ and α2 from the cutoff Λ downto the scale Mρ. This is a numerically large effect if the ∆εi are expressedin terms of the values of these parameters at the scale Λ. The running of gρ,in particular, proceeds through a sizable and negative (for aρ not too large)β-function, growing quickly in the IR. This implies that for moderately largevalues of gρ at the cutoff scale, the gap Λ/Mρ cannot be too large otherwisegρ would hit a Landau pole for µ > Mρ. For example, gρ(Λ) = 3 gives aLandau pole at µ ' Λ/3.6 in the unitary gauge. Although the evolution ofgρ is gauge dependent, it gives a rough indication on how strongly coupledthe theory of spin-1 resonances is. A more refined estimate could make usefor example of the combination λ ≡ (1/gρ − 2α2gρ)−1 with gauge-invariantrunning. Notice also that βgρ will in general receive contributions also fromother resonances lighter than the cutoff, like for example the top partners,which could slow down the growth of gρ in the IR and allow larger gaps.

In the following we analyze the constraints from the current electroweakdata by constructing a χ2 function using the fit of Refs. [45, 44] to the ∆εi


and their theoretical predictions in Eqs. (2.47)-(2.49). 16 These latter willbe evaluated in terms of the values of the parameters gρ, Mρ and f at thescale µ = Mρ. In particular we use the identity gρ = Mρ/(aρf) (Eq. (2.23))to rewrite gρ in terms of aρ and fix

f(Mρ) = v√ξ, (2.50)

where ξ ≡ sin2θ and v = 246GeV is the electroweak scale. This relationfollows from the minimization of the Higgs potential generated by loops ofheavy resonances. 17 The value of the remaining parameters c+3 , cT , c2W ,c2B is set to vanish at the scale Λ. For the case of c+3 , whose β-functions isgauge dependent when including the contribution from α1,2 at one loop, thiscondition is imposed in the unitary gauge. 18

Our results are expressed as 95% CL exclusion regions in the plane(Mρ(Mρ), ξ). The left and right plots in Figure 2.10 show the limits re-spectively for Scenario 1 with aρ(Mρ) = 1/

√2 (two-site limit) and Scenario

2 with aρ(Mρ) = 1. Notice that the tree-level shift to ε3 is the same inthe two cases: ∆ε3|tree = (M2

W /M2ρ )(1 − 4α2g

2ρ) (see Eq. (2.49)). In both

cases we fix Λ = 3Mρ(Mρ) and set α2(Mρ) = a2ρ(1− a2

ρ)/(96π2) log(Mρ/Λ).This one-loop value is chosen so that α2 vanishes at the scale µ = Λ in theunitary gauge. The orange area represents the region allowed at 95% CLfollowing from the full 1-loop results of Eqs. (2.47)-(2.49). The dashed lineshows instead the corresponding limit obtained by including the effect of theρ at the tree level. The dotted blue lines are isocurves of constant gρ(Mρ),and the blue area corresponds to the region with gρ(Mρ) > 4π. As expected,

16We perform a 3-parameters fit by using Table 4 of Ref. [44] fixing εb = εSMb . Wederive the limits by determining the isocurves of ∆τ2 corresponding to 3 degrees offreedom. Considering that ε2 does not vary much in our model (the new physics cor-rections is small), one could adopt a more conservative choice and derive the isocurveswith 2 degrees of freedom. This would lead to slightly stronger constraints, withoutqualitatively affecting our conclusions.

17If electroweak symmetry breaking is triggered by the contribution of a lighterset of resonances with mass MΨ, for instance the top partners, the relation becomesf(MΨ) = v/

√ξ. In this case f(Mρ) can be derived by running from MΨ. Notice that

βf is gauge invariant at one loop, since f gives the tree-level correction to physicalobservables like the on-shell ππ → ππ scattering amplitude and the W mass.

18Equivalently, one can fix c+3 (Mρ) so that c+3 vanishes at µ = Λ in the unitarygauge. The condition formulated in this way at µ = Mρ is gauge independent.


0.5

1

2

4

8

0 2 4 6 8 10

0.005

0.01

0.02

0.05

0.1

0.2

0.5

1

mρ [TeV ]

ξ

0.5

1

2

4

8

0 2 4 6 8 10

0.005

0.01

0.02

0.05

0.1

0.2

0.5

1

mρ [TeV ]

ξ

Figure 2.10. Limits in the plane (Mρ(Mρ), ξ) from a fit to the εi. The parameterξ controls the degree of vacuum misalignment and is related to the decayconstant f as in Eq. (2.50): ξ ≡ sin2 θ = (v/f)2. On the left: Scenario 1 withaρ(Mρ) = 1/

√2; On the right: Scenario 2 with aρ(Mρ) = 1. Both plots are

done fixing Λ = 3Mρ(Mρ). The orange area denotes the region allowed at95% CL from the 1-loop results of Eqs. (2.47)-(2.49). The dashed line showsthe corresponding limit obtained by including the effect of the ρ at the treelevel. The dotted blue lines are isocurves of constant gρ(Mρ), and the blueregion corresponds to gρ(Mρ) > 4π.

the 1-loop ρ contribution is more important for larger values of gρ, for whichthe tree-level shift to ε3 is smaller. It gives a negative shift to ε3 and a smallcorrection to ε1, thus enlarging the allowed region. The numerical valuesare reported in Table 2.2 and compared to the shifts from the IR runningand Higgs compositeness. The effect of including the new physics correctionto ε2 is small, except for gρ . 1.5 where it makes the bound on Mρ lessstrong (tail of the orange region at smaller values of Mρ and ξ). For smallgρ the 1-loop ρ contribution becomes less important and the limit almostcoincides with the tree-level one. The interpretation of our results for verylarge values of gρ requires some caution: naively the perturbative expansionbreaks down for gρ & 4π (blue region), but in practice higher-loop effects canbecome sizable earlier, invalidating our approximate result. For example, wefind that the 1-loop correction to gρ and to the pole mass mpole

ρ becomes aslarge as the tree-level term already for gρ ∼ 5 − 6. 19 Also notice that, as

19It is because of the premature loss of perturbativity in the pole mass that we prefer


1-loop ρ IR HiggsScenario 1 Scenario 2 running comp.

103 0.1ξ

∆ε1 +0.0041 +0.035 −0.43 +0.057[−0.057,+0.097] [−0.091,+0.25]

103 0.1ξ

∆ε3 −0.21 −0.16 +0.16 +0.032[−0.67,−0.14] [−0.31,−0.032]

Table 2.2. Corrections to ε1 and ε3 in units 103(0.1/ξ) from different 1-loopeffects: 1-loop ρ contribution in Scenario 1 with aρ = 1/

√2 and Scenario 2

with aρ = 1 obtained by fixing Λ/Mρ = 3 and neglecting the effect of α1,2;IR running from Mρ = 3TeV to MZ ; long-distance contribution from Higgscompositeness. The values in squared parentheses indicate the range of the1-loop ρ contribution obtained by varying 0.5 < aρ < 1.5 in Scenario 1 and 2.

a consequence of fixing Λ/Mρ = 3, values gρ > 4π/(3aρ) correspond to acutoff scale Λ larger than its naive upper limit 4πf . The latter should notbe interpreted as a sharp bound but rather as an indicative values suggestedby NDA. Yet, the above estimate also suggests that perturbativity might belost for gρ somewhat smaller than 4π.

The plots of Figure 2.10 shows the limits for a benchmark choice ofparameters. When these latter are varied, the results can change even sig-nificantly. Increasing the value of the gap Λ/Mρ amplifies the logarithmicterm in the 1-loop ρ contribution. For values of the other parameters as inFig. 2.10, the effect turns out to be small and tends to reduce the allowed re-gion. Varying aρ has a larger impact on the fit, since this parameter controlsthe size of the tree-level correction to ε3: smaller values of aρ imply smaller∆ε3|tree, hence weaker bounds on Mρ. The value of aρ also controls the sizeand the sign of the 1-loop ρ contribution. Table 2.2 shows for example howthis changes when varying 0.5 < aρ < 1.5. We find that in general the finitepart is numerically comparable, if not larger, than the log term. For illus-tration we show in Figure 2.11 the limits obtained in Scenario 2 for aρ = 0.5(left plot) and aρ = 1.5 (right plot). Finally, one could consider a scenario

to show the plots of Fig. 2.10 in terms of the running mass Mρ rather than in terms


0.5

1

2

4

8

0 2 4 6 8 10

0.005

0.01

0.02

0.05

0.1

0.2

0.5

1

mρ [TeV ]

ξ

0.5

1

2

4

8

0 2 4 6 8 10

0.005

0.01

0.02

0.05

0.1

0.2

0.5

1

mρ [TeV ]

ξ

Figure 2.11. Limits in the plane (Mρ(Mρ), ξ) for Scenario 2 with aρ = 0.5 (leftplot) and aρ = 1.5 (right plot). The parameter ξ controls the degree of vacuummisalignment and is related to the decay constant f as in Eq. (2.50): ξ ≡sin2 θ = (v/f)2. Both plots are done fixing Λ = 3Mρ(Mρ). The interpretationof the various curves and regions is the same as in Fig. 2.10.

where α2 is of order 1/g2ρ, leading to a cancellation in the tree-level contribu-

tion to ε3. 20 A proper calculation of the ∆εi in this case requires includingthe 1-loop contribution from α2 through the formulas of Appendix B, thusre-summing all powers of α2g

2ρ. As an illustration, Figure 2.12 shows the

limits obtained for α2g2ρ = 1/8 and 1/4 at the scale µ = Mρ, corresponding

respectively to a 50% and 100% cancellation of the tree-level contribution toε3. In the (extreme) case of a complete cancellation, the tail of the allowedregion at large ξ and small Mρ is a result of the new physics contribution toε2. It is indeed possible to compensate the positive (negative) shift to ε3 (ε1)from the IR running with a sizable and negative ∆ε2, due to the correlationin the 3-dimensional τ2 function. For small gρ such large and negative ∆ε2is provided by the tree-level ρ exchange, thus leading to the narrow regionextending up to ξ ∼ 0.5 and Mρ ∼ 500GeV.

The bounds that follow on Mρ and ξ from our analysis are quite severe.As already pointed out in previous studies, this is because the tree-levelexchange of the ρ generally implies a large and positive ∆ε3, while the IR

of mpoleρ .

20A scenario of this kind, with α1 � α2 ∼ 1/g2ρ, does not satisfy the PUVC criterion,

since the latter requires α1 − α2 . 1/(g∗gρ).

2.4 Discussion 35

0.5

1

2

4

8

0 2 4 6 8 10

0.005

0.01

0.02

0.05

0.1

0.2

0.5

1

mρ [TeV ]

ξ

Figure 2.12. Limits in the plane (Mρ(Mρ), ξ) for Scenario 2 with aρ = 1 andΛ = 3Mρ(Mρ). The parameter ξ controls the degree of vacuum misalignmentand is related to the decay constant f as in Eq. (2.50): ξ ≡ sin2 θ = (v/f)2.The brown and orange curves are obtained by fixing respectively α2g

2ρ = 1/8

and 1/4 at the scale µ = Mρ; the black curve refers to the case α2(Λ) = 0and corresponds to the limit shown in the right plot of Fig. 2.10. The regionbelow each curve is allowed at 95% CL. The dotted blue lines are isocurves ofconstant gρ(Mρ), and the blue region corresponds to gρ(Mρ) > 4π.

running gives a positive ∆ε3 and a negative ∆ε1. The combination of theseeffects brings the theoretical prediction far outside the 95% CL contour inthe plane (ε3, ε1) unless ξ (Mρ) is very small (large). This is illustrated byFigure 2.13, where the region spanned by varying Mρ and ξ is shown in redfor aρ = 0.5, 1, 1.5 in the case of Scenario 2. It is evident that an additionalnegative contribution to ε3 or positive contribution to ε1, as for examplecoming from loops of fermionic resonances, can relax even significantly thebounds (see for example Refs. [69, 24])

2.4 Discussion

In this chapter we have computed the 1-loop contribution to the electroweakparameters ε1,2,3 arising from spin-1 resonances in a class of SO(5)/SO(4)composite Higgs theories. We performed our analysis by giving a low-energy


!

!"#

!"$%

$"%&"%

#! ! ! '() "

# 2 # 1 0 1 2 3

# 2

# 1

0

1

2

10 33

10 31

*+(,-./0 $ 1 " $# % &#1 $ &!

! &! $ ##! " '() 1 &! ! 1!"2 "

34 ' 1 5% ' 67

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!"#

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#!! ! &'( "

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# 2

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10 33

10 31

)*'+,-./ $ 0 " $# % &!"% 0 $ &!

! &! #"% ##! " &'( 0 &! ! 0!"1 "

23 ' 0 4% ' 56

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2

10 33

10 31

*+(,-./0 $ 1 " $# % &#"% 1 $ &!

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34 ' 1 5% ' 67

Figure 2.13. Comparison between the experimental determination of ∆ε1, ∆ε3(blue ellipses at 68% and 95% CL) and the theoretical prediction in our model(red area). This latter is obtained for the case of Scenario 2 by fixing aρ andvarying ξ and Mρ as follows: aρ = 1, ξ = 0 − 0.4, Mρ = 2 − 10TeV (upperplot); aρ = 0.5, ξ = 0 − 0.4, Mρ = 1.5 − 10TeV (lower left plot); aρ = 1.5,ξ = 0−0.4, Mρ = 2.5−10TeV (lower right plot). The black dot indicates theSM point. All plots have been obtained by fixing Deltaε2 = 0.

2.4 Discussion 37

effective description of the strong dynamics in terms of Nambu-Goldstonebosons and lowest-lying spin-1 resonances (ρL and ρR), these latter trans-forming as an adjoint representation of the unbroken SO(4). We provided aclassification of the relevant operators by including the custodially-breakingeffects arising from the external gauging of hypercharge. A detailed discus-sion was given of the so-called ‘hidden local symmetry’ description of thespin-1 resonances, where their longitudinal polarizations are parametrized interms of the NG bosons from a larger coset. This was useful to analyze aparticular limit, noticed by Ref. [88], in which the theory acquires a largerSO(5)×SO(5)/SO(5) global symmetry and has a collective breaking mecha-nism. In particular, we reviewed the argument that shows how certain EWSBquantities enjoy an improved convergence in this limit, clarifying the role ofdivergent subdiagrams in the calculation of S and T .

The contribution of the ρ to the electroweak parameters was computedby performing a 1-loop matching to the low-energy theory of NG bosons. Weused dimensional regularization and analyzed in detail the renormalization ofthe spin-1 Lagrangian and the RG evolution of its coefficients. We estimatea relative uncertainty in our calculation of orderM2

h/M2ρ from neglecting the

EW and Higgs boson masses in the matching and truncating the effectiveLagrangian at leading order in the derivative expansion, and of order g2/g2

ρ

from neglecting diagrams with additional insertions of the elementary vectorbosons. Our results extend previous studies where the contribution fromspin-1 resonances was included only at the tree level. They represent astarting point for a complete 1-loop analysis including all the lowest-lyingresonances, in particular the top partners.

By including only the spin-1 resonances, a fit to the current electroweakdata gives rather strong bounds. We find that typical 95% probability limitson the ρ mass and the degree of Higgs compositeness are in the range Mρ &

3 − 4TeV and ξ . 0.1 − 0.05, although choices of parameters exist whichlead to less stringent constraints. The 1-loop contribution from the ρ canbe most easily evaluated by expressing the ∆εi in terms of the parametersof the spin-1 Lagrangian renormalized at the scale Mρ (Eqs. (2.47-2.49)).Although parametrically subdominant compared to the IR running and thetree-level contribution, we find it to be numerically important in a significantfraction of the parameter space, where the coupling strength gρ is moderately


large. Its effect is that of enlarging the allowed region providing a negativeshift to ε3 (see Table 2.2 and Figs. 2.10-2.12). The relative importance ofthe 1-loop contribution grows with gρ. Although one would naively expectperturbativity to remain valid until gρ ∼ 4π, the 1-loop correction becomesas important as the tree-level term already for gρ ∼ 5−6 in several quantities,as for example the running of gρ or the pole mass Mpole

ρ . This suggests thatany limit extending to such large values of gρ should be interpreted withcaution. The contribution from cutoff states to the electroweak observablesmight also be important. Its naive estimate in the case of a fully stronglycoupled dynamics at the scale Λ suggests that it is subleading compared tothe 1-loop ρ contribution only by a factor log(Λ/Mρ), which is not expectedto be very large. In fact, the very existence of a gap Λ/Mρ � 1 shouldbe considered as a working hypothesis of our study, not necessarily realizedby the underlying strong dynamics. In this sense our calculation shouldbe regarded as a way, more refined than a simple estimate, to assess thecontribution of the spectrum of resonances lying at the compositeness scaleto the oblique parameters.

39

Chapter 3

A dispersion relationapproach for the EWPO

An alternative strategy for the calculation of EWPO consists in making useof dispersion relations to express an observable as the integral over the spec-tral functions of the strong dynamics. Extracting the spectral functions fromexperimental data thus leads to a result which is, at least in principle, freefrom possible theoretical ambiguities. Although the most powerful use ofdispersion relations is in conjunction with experimental data, in absence ofthese latter one can build models of the spectral functions based on theoret-ical considerations. Computing the spectral functions through a low-energyeffective theory of resonances leads in fact to the same result obtained by themore conventional diagrammatic technique, though the dispersive approachcan simplify the calculation and gives a different viewpoint.

In section 3.1 we will state the problem in an clear way and develop ageneral master formula for the dispersive calculation of ∆ε3 in SO(5)/SO(4)-based composite Higgs models. Such master formula makes use of the spec-tral densities of the conserved current of a full, UV-complete theory, whichis not considered in current practical realizations. Hence, in section 3.2, wewill show how the dispersive approach can be modified in order to be usedin the context of a low energy effective theory. In section 3.3 we will usethe developed machinery to perform an actual calculation of ∆ε3 using the

40 3. A dispersion relation approach for the EWPO

same model introduced in chapter 2. Despite the result being, as expected,the same previously obtained, some interesting comments can be made, asdiscussed in section 3.4.

3.1 Dispersion relation for ε3We start by deriving the dispersion relation for the ε3 parameter in the con-text of SO(5)/SO(4) composite Higgs theories. Our analysis will be similarto that of Ref. [86], although it differs in the way in which short- and long-distance contributions from new physics are parametrized. In this respect ourapproach is closer to the original work of Peskin and Takeuchi [89], where theS parameter is defined to include only short-distance effects from the newdynamics.

3.1.1 Short- and long-distance contributions to ε3

It is well known that universal corrections to the electroweak precision ob-servables at the Z-pole can be described by three ε parameters [20]. In thispaper we are mainly interested in the ε3 parameter, which can be expressedas

ε3 = e3 + c2W e4 − c2W e5 + (non-oblique corrections) (3.1)

in terms of the vector-boson self energies

e3 = cWsW

F3B(M2Z), e4 = Fγγ(0)− Fγγ(M2

Z), e5 = M2ZF′ZZ(M2

Z) .(3.2)

Here sW (cW ) denotes the sine (cosine) of the Weinberg angle and we havefollowed the standard convention decomposing the self energies (for canoni-cally normalized gauge fields) as

Πµνij (q) = −iηµν

(Aij(0) + q2Fij(q2)

)+ qµqν terms . (3.3)

We consider scenarios in which the new physics modifies only the self energies,i.e. its effects are oblique. The form of the non-oblique vertex and boxcorrections in Eq. (3.1) is thus irrelevant to our analysis, since these cancelout when considering the new physics correction ∆ε3 ≡ ε3 − εSM3 . It isuseful to distinguish between a short- and a long-distance contribution to

3.1 Dispersion relation for ε3 41

∆ε3. Heavy states with mass M∗ �MZ affect only the short-distance part.This latter can can be expressed as the contribution of local operators, andis generated also by loops of light (i.e. Standard Model (SM)) particles. Wedefine it to be

∆ε3|SD = ∆e3 + c2W∆e4 − c2W∆e5 , (3.4)

where ∆ei ≡ ei − eSMi and

e3 = cWsW

(F3B(0) +M2

ZF′3B(0)

), e4 = −M2

ZF′γγ(0), e5 = M2

ZF′ZZ(0) .

(3.5)It is convenient to express ∆ε3|SD in terms of the parameters S, W , Y andX defined in Ref. [29]:

∆ε3|SD = S −W − Y + X

sW cW, (3.6)

where

S = cWsW

(F3B(0)− FSM3B (0))

X = M2W (F ′3B(0)− F ′SM3B (0)) ,

W = M2W (F ′WW (0)− F ′SMWW (0))

Y = M2W (F ′BB(0)− F ′SMBB (0)) .

(3.7)

The S parameter originally introduced by Peskin and Takeuchi in [89] isrelated to S by S = (αem/4s2

W )S.The long-distance correction to ε3 arises from loops of light particles only,

as a consequence of their non-standard couplings. We define

∆ε3|LD =[∆e3 −∆e3 + c2W (∆e4 −∆e4)− c2W (∆e5 −∆e5)

]light , (3.8)

where ∆ei ≡ ei − eSMi and the expression in square brackets is computedby including only the contribution of light particles. In the scenario underconsideration the dominant long-distance contribution arises from the com-posite Higgs, as a consequence of its modified couplings to vector bosons. At1-loop it is given by the diagrams in Fig. 3.1. Working in the Landau gaugefor the elementary gauge fields (∂µW i

µ = 0 = ∂µBµ), we find 1

∆ε3|LD = g2

96π2 sin2θ

[f3(h)− 5h4 + 7h3 + 21h2 + 151h+ 68

12(1− h)4

− h

2(1− h)5

(h4 − 5h3 + 19h2 − 9h+ 36

)log h

],

(3.9)

1The same formula holds in a generic theory with Higgs coupling to vector bosonscV provided one replaces the factor sin2θ with (1− c2V ).


Figure 3.1. One-loop diagrams relative to the Higgs contribution to ∆ε3. Wavy,continuous and dashed lines denote respectively gauge fields (W± and Z),Nambu-Goldstone bosons of SO(4)/SO(3) (π1,2,3) and the Higgs boson.

where h = M2h/M

2Z and the function f3 is given in appendix B.

Additional long-distance effects arise from the top quark which are fur-ther suppressed by at least a factor ζ2

t , where ζt is the degree of compositenessof the top quark. They will be neglected in the following.

From Eqs. (3.4), (3.6) and (3.8) we find

ε3 = εSM3 + ∆ε3|LD + S −W − Y + X

sW cW+ . . . . (3.10)

Together with Eq. (3.8), this is our master formula for the calculation ofε3. 2 It is accurate up to corrections (denoted by the dots) of relativeorder (M2

Z/M2∗ ), which are not captured by our definition of short- and

long-distance contributions in Eqs. (3.4), (3.8). We will assume the massscale of the new resonances to be much higher than the electroweak scale,M∗ �MZ ,Mh, and neglect these corrections.

As a consequence of the gap betweenM∗ andMZ , the contribution of thenew heavy states to ε3 is local and encoded by the S,W, Y,X parameters.Loops of light SM particles, in particular the Higgs boson, lead to an ad-ditional new physics correction through their modified couplings which is ofboth short- and long-distance type. In the composite Higgs theories under ex-amination the shifts to the Higgs couplings are of order (v/f)2, where f is theHiggs decay constant. Since f is related toM∗ through the coupling strengthof the resonances,M∗ ∼ g∗f , one could in principle get large modifications tothe Higgs couplings for f ∼ v while still having a mass gap provided g∗ � g.In fact, current experimental data on Higgs production at the LHC disfavorlarge shifts and constrain (v/f)2 . 0.2−0.3 [25, 61, 31]. In the limit of large

2An analogous formula was given in Eq. (6c) of Ref. [29], where however the long-distance term ∆ε3|LD is omitted.


compositeness scale, f � v, all the new physics contributions to low-energyobservables can be conveniently computed by matching the UV theory to aneffective Lagrangian built with SM fields (including the Higgs doublet) atthe scaleM∗. The leading contribution of light fields to ∆ε3 then arises from1-loop diagrams with one insertion of a dimension-6 operator. The divergentpart of these diagrams is associated with the RG running of the operators’coefficients, while the finite part is interpreted as a long-distance thresholdcorrection at the scale MZ . This shows that the contributions from heavymodes and light modes are not individually RG invariant, as only their sumis independent of the renormalization scale at the one-loop level. Clearly,no issue with the RG invariance arises if one works at the tree level, and inthat case it makes perfect sense to define the S,W, Y and X parameters toinclude only the contribution of heavy particles. When 1-loop corrections areconsidered, however, any RG-invariant definition of the short-distance con-tribution must include at least the divergent correction from loops of lightfields. According to our definition of Eq. (3.4), S,W, Y and X include suchdivergent part as well as a finite one.

3.1.2 Dispersion relations for the short-distance contri-butions

We are now ready to derive the dispersion relations for S, W and Y in termsof the spectral functions of the strongly-interacting dynamics. We start byconsidering S.

The strong dynamics is assumed to have a global SO(5) invariance spon-taneously broken to SO(4) ∼ SU(2)L × SU(2)R. The elementary Wµ andBµ fields gauge an SU(2)L×U(1)Y subgroup contained into an SO(4)′ mis-aligned by an angle θ with respect to the unbroken SO(4) (see Chapter 2 for


details). They couple to the following linear combinations of SO(5) currents 3

Lint = W aµJa [W ]µ +BµJ [B]

µ (3.11)

Ja [W ]µ = Tr

[T aL(0)TA(θ)

]JAµ

J [B]µ = Tr

[T 3R(0)TA(θ)

]JAµ ,

(3.12)

where TA(θ) are the SO(5) generators, while T a(0) are the generators of thegauged SO(4)′. Using the expressions for the generators given in AppendixA, we find

J3 [W ]µ =

(1 + cos θ

2

)J3Lµ +

(1− cos θ

2

)J3Rµ + sin θ√

2J 3µ

J [B]µ =

(1− cos θ

2

)J3Lµ +

(1 + cos θ

2

)J3Rµ −

sin θ√2J 3µ ,

(3.13)

where JaLµ , JaRµ are the SO(4) ∼ SU(2)L×SU(2)R currents (aL, aR = 1, 2, 3)and J ıµ the SO(5)/SO(4) ones (ı = 1, 2, 3, 4). We assume that these currentsare conserved in the limit in which the strong dynamics is taken in isolation,i.e. when the couplings to the elementary fields are switched off. This is forexample the case of holographic composite Higgs models [50, 11, 47]. Thegeneralization to the case in which the strong dynamics itself contains a smallsource of explicit SO(5) breaking is discussed in Appendix E. By working atsecond order in the interactions (3.11) (i.e. at second order in the weak cou-plings), the vector-boson self energies in Eq. (3.7) can be expressed in termsof two-point current correlators. The corresponding contribution to S andto the other oblique parameters W,Y,X is gauge invariant (see the detaileddiscussion in Ref. [89]). The S parameter, in particular, gets a naive contribu-tion of O(M2

Z/M2∗ ) from the exchange of the heavy resonances of the strong

dynamics, while loops of Nambu-Goldstone (NG) bosons are responsible forthe IR running of orderM2

Z/(16π2f2) log(M∗/Mh). Corrections from higher-order terms in the weak coupling expansion cannot be expressed as two-point

3We assume that the one in Eq. (3.11) is the only interaction between elementarygauge fields and strong sector, i.e. that the gauge fields couple linearly to the strongdynamics through its conserved currents. If the UV degrees of freedom of the strongdynamics include elementary scalar fields, then an interaction quadratic in the gaugefields is also present, as dictated by gauge invariance.


+ + · · ·

Figure 3.2. Contribution of the strong dynamics to the vector boson self energiesexpanded in powers of the weak gauge couplings. The gray blob in the firstdiagram corresponds to the correlator of two conserved currents of the strongdynamics.

current correlators and are not gauge invariant in general. A graphical rep-resentation of the various terms in the expansion is given in Fig. 3.2, wherea typical O(g4) contribution is exemplified by the second diagram. A naiveestimate shows that corrections at quartic order in the weak couplings fromthe exchange of heavy resonances are of order M2

Z/(16π2f2)(g2/g2∗). They

are subdominant if g � g∗, and we will neglect them in the following. Inthe case of corrections involving loops of light fields only, on the other hand,the additional g2 suppression can be compensated by inverse powers of thelight masses. The only such unsuppressed contribution to S comes from thediagram on the right in Fig. 3.1, featuring a Higgs boson and a Z in the loop.It is gauge invariant 4 and gives a correction

δSZh = g2

96π2sin2θ

(xh − 1)2

(9xh + 1

2(1− xh) log xh + 2xh + 3), (3.14)

which we will retain in our calculation. Notice that since this term is not ofthe form of a two-point current correlator of the strong dynamics in isola-tion, it was not included by Peskin and Takeuchi in their estimate of S inRef. [89]. 5

In the limit in which the strong sector is taken in isolation, i.e. forunbroken SO(5) symmetry, the Fourier transform of the Green functions oftwo conserved currents can be decomposed as follows:

〈JaLµ (q)JbLν (−q)〉 = − iδaLbL(PT )µν ΠLL(q2)

〈JaRµ (q)JbRν (−q)〉 = − iδaRbR(PT )µν ΠRR(q2)

〈J aµ(q)J bν(−q)〉 = − iδab(PT )µν ΠBB(q2) ,

(3.15)

4See the discussion in Ref. [86].5For Technicolor one must set sin θ = 1 in Eq. (3.14).


where (PT )µν ≡ (ηµν−qµqν/q2). Any other two-point current Green functionvanishes by SO(5) invariance. By using its definition in Eq. (3.7), togetherwith Eqs. (3.11), (3.13) and (3.15), the parameter S can be expressed interms of the correlators Πij as follows:

S = g2 (Π′3B(0)−ΠhSM ′3B (0)

)+ δSZh , (3.16)

where

Π3B(q2) ≡ 14 sin2θ

(ΠLL(q2) + ΠRR(q2)− 2ΠBB(q2)

), (3.17)

and ΠhSM3B denotes the expression of Π3B obtained by replacing the strong

dynamics with the Higgs sector of the SM. Equation (3.16) is still a prelim-inary expression, however. The correlators Πij(q2) are singular at q2 = 0due to the presence of the four massless NG bosons (including the Higgsboson), since they are computed by considering the strong dynamics in iso-lation. A similar IR divergence is also present in the SM Higgs sector, butonly originating from the three SO(4)/SO(3) NG bosons. Subtracting theSM contribution in Eq. (3.16) thus removes only partly the IR divergence. 6

There is however a simple way solve this problem and write a general for-mula for S in terms of two-point current correlators of the strong dynamicsin isolation. 7 Let us add and subtract in Eq. (3.16) the contribution from alinear SO(5)/SO(4) model defined in terms of the four NG bosons plus anadditional scalar field η which unitarizes the scattering amplitudes in the UV(see Appendix G of Ref. [49] for a definition). This model coincides with theSO(5)/SO(4) strong dynamics in the infrared and is renormalizable. Thus,we have:

S = g2 (Π′3B(0)−ΠLSO5 ′3B (0)

)+ δSLSO5 + δSZh , (3.18)

6The IR divergence is completely removed if the strong dynamics contains a smallbreaking of the SO(5) symmetry giving the Higgs boson a mass. It is shown in Ap-pendix E that even in this case it is useful to rewrite Eq. (3.16) as discussed below toexplicitly extract the Higgs chiral logarithm.

7A possible alternative strategy is to define the correlators Πij by including the ex-plicit breaking of SO(5) due to the coupling of the strong dynamics to the elementaryfermions, in particular to the top quark. The resulting formula, however, is less con-venient to compute S by means of non-perturbative tools such as lattice field theory.We thank Slava Rychkov for drawing our attention on the importance of working withtwo-point current correlators defined in terms of the strong sector in isolation.


where ΠLSO53B denotes the expression of Π3B obtained by replacing the strong

dynamics with the linear SO(5)/SO(4) model and

δSLSO5 ≡ g2 (ΠLSO5 ′3B (0)−ΠhSM ′

3B (0))

= g2

96π2 sin2θ log Mη

Mh(3.19)

is computed for a non-vanishing Higgs mass. The mass of the scalar η is anarbitrary parameter which can be taken to be of the order of the mass ofthe heavy resonances of the strong sector, Mη ∼ M∗. In this way the Higgschiral logarithm is fully captured by δSLSO5, and the first term in parenthesisin Eq. (3.18) can be evaluated setting the Higgs mass to zero (the relativeerror that follows is of order M2

h/M2∗ and can be thus neglected). The IR

singularities exactly cancel out in the difference of correlators in parenthesis,since the linear model by construction coincides with the strong dynamics inthe infrared. Equation (3.18), together with Eq. (3.17), is a generalizationto SO(5)/SO(4) composite Higgs theories of the analogous result derived inRef. [89] by Peskin and Takeuchi for Technicolor.

At this point we can make use of the dispersive representation of thecorrelators Πij . This is obtained by inserting a complete set of states in theT -product of the two currents and defining∑

n

δ(4)(q − qn)〈0|J iµ(0)|n〉〈n|Jjν(0)|0〉 = θ(q0)(2π)3

(−gµνq2ρij + qµqν ρij

).

(3.20)The spectral functions ρij(q2) and (ρij(q2)−ρij(q2)) encode respectively thecontribution of spin-1 and spin-0 intermediate states; they are real and pos-itive definite. Current conservation implies ρij = ρij , while from analyticityand unitarity it follows

ρij(s) = 1πIm[

Πij(s)s

]. (3.21)

The n-subtracted dispersive representation thus reads (for a given q20)

Πij(q2) = Pn(q2) + q2 (q2 − q20)n ∫ ∞

0ds

1(s− q2

0)nρij(s)

s− q2 + iε, (3.22)

where Pn(q2) is a polynomial of degree n. 8 It holds provided Πij(q2) ∼(q2)1+n−ε for |q2| → ∞, with ε > 0. In the full theory of strong dynamics,

8One has P0(q2) = Πij(0) and

Pn(q2) = Πij(0)(

1−q2

q20

)n+ q2

n−1∑k=0

(q2 − q20)k

k!dk

dsk

(Πij(s)s

)∣∣∣∣s=q2

0

(3.23)


the asymptotic behavior of the linear combination

Π1 ≡ ΠLL + ΠRR − 2ΠBB (3.24)

is controlled by the scaling dimension, ∆ ≥ 1, of the first scalar operatorentering its OPE (see discussion in Ref. [86]): Π1(s) ∼ s1−∆/2. One canthus write a dispersion representation for Π1 without subtractions (i.e. set-ting n = 0 in Eq. (3.22)). Using the explicit expression of ΠLSO5 ′

3B (0), it isstraightforward to derive our final dispersion relation for S:

S = g2

4 sin2θ

∫ ∞0

ds

s

{(ρLL(s) + ρRR(s)− 2ρBB(s))

− 148π2

1−(

1−M2η

s

)3

θ(s−M2η )

}

+ δSLSO5 + δSZh .

(3.25)

This result generalizes the dispersion formula derived by Peskin and Takeuchiin Ref. [89] for Technicolor to the case of SO(5)/SO(4) composite Higgstheories. The dispersive integral accounts for the contribution from heavystates (of O(M2

Z/M2∗ )), while the chiral logarithm due to Higgs compositeness

is encoded by δSLSO5. The dependence on Mη cancels out when summingthis latter term with the dispersive integral.

Let us now turn toW , Y and X. In our class of theories the contributionof heavy particles to X is of O(M4

Z/M4∗ ) and will be neglected (it is of the

same order as the uncertainty due to our definition of short- and long-distanceparts in ∆ε3). The contribution of heavy particles to W and Y is insteadof O[(M2

Z/M2∗ )(g2/g2

∗)] and will be retained. Finally, the contribution to W ,Y and X from the diagrams of Fig. 3.1 involving light particles only is notsuppressed and must be fully included. For X we find:

X = − g2

64π2 sW cW sin2θ

[3x2

h + 4xh(xh − 1)5 log xh −

x3h + x2

h + 73xh + 912(xh − 1)4

]+ . . .

(3.26)where the dots indicate O(M4

Z/M4∗ ) terms generated by the exchange of

heavy particles. In the case of W and Y , it is straightforward to derive a

For n ≥ 1. Notice that ΠLL(0) and ΠRR(0) vanish if the strong dynamics is consideredin isolation.


dispersion relation by following a procedure analogous to that discussed forS. 9 By neglecting terms of order O(M4

W /M4∗ ), we obtain 10:

W = M2W g

2∫ ∞

0

ds

s2

(ρLL(s)− 1

96π2

)− g2

96π2c2W8xh

sin2θ + δWZh, (3.28)

Y = M2W g′2∫ ∞

0

ds

s2

(ρRR(s)− 1

96π2

)− g′2

96π2c2W8xh

sin2θ + δYZh . (3.29)

The first term in each equation encodes the contribution from the heavyresonances and is of O[(M2

Z/M2∗ )(g2/g2

∗)]. In particular, the integral inEq. (3.28) equals (Π′′LL(0) − ΠLSO5 ′′

LL (0)), while that in Eq. (3.29) equals(Π′′RR(0) − ΠLSO5 ′′

RR (0)). The second terms come from the difference be-tween the linear SO(5)/SO(4) model and the SM (they are the analogous ofEq. (3.19)), while δWZh and δYZh are the contributions from the Zh loop inFig. 3.1:

δWZh = g2

g′2δYZh = g2

64π2 c2W sin2θ

[3h2 + 4h(h− 1)5 log h

− 5h3 + 67h2 + 13h− 112h(h− 1)4

].

(3.30)

By putting together the expressions of S, W , Y , X, and of the long-

9The dispersive representation of ΠLL and ΠRR in this case requires one subtraction,since ΠLL(q2) ∼ ΠRR(q2) ∼ q2 for |q2| → ∞.

10The O(M4W /M

4∗ ) neglected terms give a contribution to W which can be written

as follows:

δW = M2W g

2

{−

sin2 θ

4

∫ ∞0

ds

s2

[(ρLL(s) + ρRR(s)− 2ρBB(s))

−1

48π2

(1−(

1−M2η

s

)3

θ(s−M2η )

)]

− sin2 θ

2

∫ ∞0

ds

s2 (ρLL(s)− ρRR(s))}.

(3.27)

The additional contribution to Y has the same form provided one exchanges LL↔ RR

and g ↔ g′.


distance part Eq. (3.9), we obtain a dispersive formula for ∆ε3:

∆ε3 = g2

96π2 sin2θ

(f3(xh) + log xh

2 − 512 + log Mη

Mh

)+ g2

4 sin2θ

∫ ∞0

ds

s

{ρLL(s) + ρRR(s)− 2ρBB(s)

− 148π2

1−(

1−M2η

s

)3

θ(s−M2η )

}

+M2W

∫ ∞0

ds

s2

(g2ρLL(s) + g′2ρRR(s)− g2 + g′2

96π2

).

(3.31)

The second and third terms encode the contribution from the heavy reso-nances and are, respectively, of O(M2

Z/M2∗ ) and O[(M2

Z/M2∗ )(g2/g2

∗)]. Whenmodeling the spectral functions –as we will do in the next section– in termsof the lowest-lying resonances of the strong dynamics, these contributionsarise from the tree-level exchange of massive spin-1 states. We neglectedterms of O(M4

Z/M4∗ ) (arising in particular from our definition of short- and

long-distance contributions) and of O[(M2W /16π2f2)(g2/g2

∗)] (arising fromthe expansion in powers of the weak couplings required to obtain a formulain terms of current correlators).

Equation (3.31) should be compared to the analogous result previouslyderived by Rychkov and Orgogozo in Ref. [86]. The expression there givenalso relies on an expansion in g2, and does not include the heavy-particlecontribution to W and Y (the last term of our Eq. (3.31)). Rychkov andOrgogozo also define the dispersive integral to comprise the contribution ofthe heavy states only, but do not perform any subtraction to remove theNG boson contribution. Rather, the integration over light modes is doneexplicitly and in an approximate way. Their procedure implies a relativeuncertainty of order Mh/M∗, which follows in particular from neglecting theHiggs mass and the contribution of the heavy states in the evaluation of thelow-energy part of the dispersive integral. In our case the relative uncertaintyimplied by our definition of short- and long-distance parts is smaller and oforder (MZ/M∗)2. Within their accuracy, the two results coincide.

3.2 Dispersive relation in the effective theory 51

3.2 Dispersive relation in the effective theory

The dispersive integrals in Eq. (3.31), as well as those in Eqs. (3.25), (3.28)and (3.29), are convergent and well defined if the spectral functions are com-puted in the full theory of the strong dynamics. Here we want to provide anapproximate calculation of ∆ε3 which makes use of an effective description ofthe strong dynamics in terms of its lowest-lying resonances and NG bosons.We focus in particular on the contribution of a spin-1 resonance (ρL) trans-forming as a (3, 1) of the SO(4) ∼ SU(2)L × SU(2)R global symmetry. Wewill thus compute the spectral functions in the effective theory and integratethem to obtain S, W and Y , hence ∆ε3, through their dispersion relations.In this case, the spectral integrals are generically divergent in the ultravio-let, since the effective description is approximately valid at low energy butnot adequate for momenta larger than the cutoff scale. In other words, thedispersion relations derived in the previous section need to be modified inorder to be used in the effective theory. Let us see how.

By considering the gauge fields Aµ as external sources for the currents,any two-point current correlator can be expressed as the second derivative ofan effective action W [A] with respect to the source:

〈Jµ(x)Jν(y)〉 = (−i)2 δ2

δAµ(x)δAν(y)W [A]∣∣∣∣A=0

, (3.32)

whereW [A] = log

∫dϕ exp

(iS[ϕ] + i

∫d4xJµA

µ

)(3.33)

and ϕ denotes the UV degrees of freedom of the strong dynamics. In absenceof a description of the theory in terms of these fields, we can compute W [A]approximately as the integral over the IR degrees of freedom ϕIR:

W [A] ' log∫dϕIR exp (iSIR[ϕIR, A]) . (3.34)

Notice however that the low-energy action SIR will not depend on the sourceonly through its coupling to the low-energy conserved current JIRµ , but willcontain non-minimal interactions. At quadratic order in the source, we canwrite

SIR[ϕIR, A] = SIR[ϕIR] +∫d4x

(JIRµ Aµ +OµνA

µν + c02 AµA

µ

−c14 AµνAµν + . . .

),

(3.35)


where c0 and c1 are constants, Aµν is the field strength constructed withthe source and Oµν is an operator antisymmetric in its Lorentz indices. Thesecond term in the parenthesis is a non-minimal interaction that is generatedwhen flowing to the infrared. The last two terms in parenthesis depend onlyon the source and generate contact contributions upon differentiation; pure-source higher-derivative terms are denoted by the dots. By using Eqs. (3.35)and (3.34) to compute (3.32 one finds

〈Jµ(x)Jν(y)〉 = 〈Jµ(x)Jν(y)〉+ c0 ηµνδ(4)(x− y)

+ c1 (ηµν�− ∂µ∂ν) δ(4)(x− y) + . . . ,(3.36)

where Jµ ≡ JIRµ − 2 ∂ρOρµ is also a conserved current, and the dots standfor higher-derivative local terms. The Green functions 〈JµJν〉 can thus becomputed in terms of the two-point functions of the effective currents Jµ.The coefficients ci are arbitrary in the effective theory and can be chosento cancel the UV divergences arising in 〈JµJν〉. 11 Performing a Fouriertransformation one has

Πij(q2) = Πij(q2) + ∆ij(q2) , (3.37)

where Πij is the two-point current correlator in the effective theory and∆(q2) =

∑k(q2)kck denotes the local counterterms.

It is always possible to express Πij(q2) as an integral over a contour inthe complex plane that runs below and above its branch cut on the real axis(where the imaginary part of Πij is discontinuous) and then describes a circleof radius M2 counterclockwise. We thus obtain

Πij(q2) = Πij(0) + q2∫ M2

0ds

ρij(s)s− q2 + q2

2πi

∫CM2

dzΠij(z)z(z − q2) + ∆ij(q2) ,

(3.38)where CM2 denotes the part of the contour over the circle, and ρij(q2) =(1/π)Im[Πij(q2)/q2] is the spectral function of the currents Jµ. Since the

11The value of c0 can be adjusted to ensure that the contributions to the two-point correlator from the tree-level exchange of, respectively, one NG boson and onespin-1 resonance are transverse. A simple way to enforce the Ward identity is infact demanding that the effective action SIR[ϕIR, A] be invariant under local SO(5)transformations under which the source Aµ transforms as a gauge field. We thankMassimo Testa for a discussion on this point. Notice also that adding the pure sourceterms in Eq. (3.35) corresponds to a redefinition of the T ∗ product of two currents.

3.2 Dispersive relation in the effective theory 53

value of M is arbitrary (as long as q2 is inside the contour), the dependenceonM2 cancels out in Eq. (3.38). If Πij(q2)/q2 → 0 for |q2| → ∞, it is possibleto take the limit M2 →∞ so that the integral on the circle vanishes. In thiscase one obtains a dispersion relation for Πij(q2) in terms of ρij similar to theone valid in the full theory, except for the appearance of the local term. Ingeneral, however, the correlator Πij is not sufficiently well behaved at infinity,andM must be kept finite. If Π(q2) ∼ (q2)1+k at large q2, both the dispersiveintegral and the integral over the circle scale as (M2/M2

∗ )k, where M∗ is themass of the resonances included in the low-energy theory. Also, Πij generallyrequires a regularization to be defined and contains divergences which areremoved by the counterterm ∆ij . The dispersive integral, on the other hand,is convergent since ρij is finite (after subdivergences are removed).

A particularly convenient way to define Πij(q2) is through dimensionalregularization. Upon extending the theory to D dimension, indeed, itsasymptotic q2 behavior arising at the radiative level can be arbitrarily soft-ened. For example, the 1-loop contribution to Πij scales like (q2)1+n−ε/2

at large q2, where n is some integer and ε ≡ 4 − D. It is thus possible tochoose ε sufficiently large and positive (ε > 2n), such that the contributionto the integral on the circle from 1-loop effects vanishes when taking thelimit M2 →∞. In doing so, the dispersive integral (now with its upper limitextended to infinity) becomes singular for ε → 0. The divergence is thustransferred from the integral over the circle to the dispersive integral, andthe 1/ε poles are still removed by the counterterm ∆ij . The same argumentgoes through after including higher-loop contributions. The large-q2 behaviorof the tree-level part of Πij , on the other hand, can not be softened throughdimensional continuation. If thus Πij scales like (q2)1+n at tree level, withn > 0, it is not possible to take the M2 →∞ limit in Eq. (3.38) (unless oneperforms n subtractions). The case with n = 0 is special, in that M2 can besent to infinity but the integral over the circle tends to a constant and doesnot vanish. Assuming that Πij(q2) grows no faster than q2 in D dimension,one can thus derive the following dispersion relation:

Πij(q2) = Πij(0) + q2∫ ∞

0ds

ρij(s)s− q2 + ∆ij(q2) + q2Cij , (3.39)


where

Cij ≡ lim|q2|→∞

[Πij(q2)q2

]. (3.40)

This is the formula that we will use in the next section to compute S, W andY .

We conclude by noticing that another approach is also possible to derivea dispersion relation in the effective theory. One could use Eq. (3.37) andapproximate Im[Πij(q2)] ' Im[Πij(q2)] for q2 � Λ2. Substituting ρij(s) =ρij(s)+O(s/Λ2) in the dispersion relation of the full theory, one thus obtains

Πij(q2) = Πij(0) + q2∫ M2

0ds

ρij(s)s− q2 + q2

∫ ∞M2ds

ρij(s)s− q2 +O

(M2

Λ2

). (3.41)

The value ofM can be conveniently chosen to be much larger than the mass ofthe resonances M∗, so as to fully include their contribution to the dispersiveintegral, and much smaller than the cutoff scale Λ, as required for ρij to givea good approximation of the full spectral function. With this choice, the lasttwo terms in Eq. (3.41) encode the contribution from the cutoff dynamics.Comparing with Eq. (3.38), it follows that

q2∫ ∞M2ds

ρij(s)s− q2 = q2

2πi

∫CM2

dzΠij(z)z(z − q2) + ∆ij(q2) +O

(M2

Λ2

). (3.42)

3.3 One-loop computation of ∆ε3

Having discussed how the dispersion relations are modified in the effectivetheory, we now put them to work and perform an explicit calculation of ∆ε3.Our goal is thus computing the spectral functions ρij of the currents Jµ in theeffective theory with NG bosons and a spin-1 resonance ρL. The dynamics ofthe spin-1 resonance will be described by the effective Lagrangian of Chapter2.

The SU(2)L, SU(2)R and SO(5)/SO(4) components of Jµ read, respec-

3.3 One-loop computation of ∆ε3 55

tively:

JaLµ =(1− a2

ρ)2

(εaLbc∂µπ

bπc + ∂µπaLπ4 − ∂µπ4πaL

)−M2ρ

gρρaµ − 2α2gρ∂

αραµ + . . . (3.43)

JaRµ = 12(εaRbc∂µπ

bπc + ∂µπaRπ4 − ∂µπ4πaR

)+ . . . (3.44)

J aµ = f√2∂µπ

a − f√2a2ρgρ(εabcρbµπ

c + δa4ρbµπb − ρaµπ4)+ . . . (3.45)

where gρ is the resonance’s coupling strength, aρ ≡Mρ/(gρf) and the ellipsesdenote terms with higher powers of the fields or terms that are not relevantfor the present calculation. The last term in Eq. (3.43) proportional to α2

originates from the non-minimal coupling to the external source induced bythe operator Q2 = Tr[ρµνL fLµν ]. 12

In order to compute the spectral functions, we use the definition (3.20)in terms of a sum over intermediate states. The resonance ρL can decay totwo NG bosons and is not an asymptotic state. The intermediate states tobe considered are thus multi-NGB states: 13 ππ, 3π, 4π, . . . . It is howeverpossible to simplify the calculation by noticing the following. We want toderive an expression for the S parameter at order g0

ρ, by expanding for gρ/4πsmall. Since the contribution from the tree-level exchange of the ρL is oforder 1/g2

ρ, our result will include terms that appear at the 1-loop level in adiagrammatic calculation of S. The role of tree- and loop-level effects in thedispersive computation, on the other hand, is subtler. Consider for examplethe contribution to the ππ state coming from the exchange of a ρL, i.e. that ofthe second diagram in the first row of Fig. 3.3. The vertex with the currentis of order 1/gρ, while that with the two NG bosons is of order gρ. Thediagram, and thus its contribution to the parameter S, is naively of O(g0

ρ).There is however an enhanced contribution of O(1/g2

ρ) that comes from thekinematic region s ∼ m2

ρ in the dispersive integral (3.25), where mρ is thepole mass of the ρL. To see this, notice that the small gρ limit coincides witha narrow-width expansion. The Breit-Wigner function that follows from the

12Notice that a different basis was used in Chapter 2 where Q2 = Tr[ρµνL ELµν ]. Thedefinition adopted in this paper is more convenient for our discussion.

13The exchange of one NG boson contributes only to the spectral function ρBB andis thus irrelevant to our calculation.


ρLL:

ρRR:

ρBB :

Figure 3.3. Feynman diagrams contributing to the spectral functions in theeffective theory at O(g0

ρ). Continuous lines denote NG bosons of SO(5)/SO(4)(πa), while doubly wavy lines denote a ρL. The cross stands for the insertionof a current, while the blue blobs and box indicate respectively the 1-loopcorrected vertices and propagator.

square of the ρL propagator can be thus expanded as

Γρ(s−m2

ρ)2 +m2ρΓ2

ρ

= π

mρδ(s−m2

ρ) +O(g2ρ) , (3.46)

where Γρ is the decay width of the ρL. The left-hand side is of O(g2ρ) for s

away from m2ρ, but the delta-function term in the right-hand side is of O(g0

ρ).The contribution to the dispersive integral at the ρL peak is thus enhancedcompared to the naive counting. As a consequence, the leading contributionto the S parameter from the ππ final state is of order 1/g2

ρ, and in factcorresponds to the tree-level correction of the diagrammatic calculation.

Loosely speaking, we can say that whenever the ρL goes “on-shell”, theorder in powers of gρ is lowered by two units. This has two consequences.The first is that the leading contribution from the 3π and 4π states canbe captured by replacing them, respectively, with the states πρL and ρLρLobtained by treating the ρL as an asymptotic state. This approximation issufficient to extract S at O(g0

ρ) and simplifies considerably the calculation.


The second consequence is that, in the calculation of the ππ contribution, 1-loop corrections to the vertices and to the ρL propagator should be includedfor s ' m2

ρ, as they contribute at O(g0ρ). In other words, 1-loop corrections

to the spectral functions need to be retained (only) near the ρL peak.

The Feynman diagrams relative to the calculation of the spectral func-tions ρLL, ρRR and ρBB are shown in Fig. 3.3 in terms of the relevant finalstates ππ, ρLρL and πρL. We work in the unitary gauge for ρL, choosing, asin Chapter 2, dimensional regularization and an on-shell minimal subtrac-tion scheme to remove the divergences of the 1-loop contributions. Whilethe calculation of ρRR and ρBB is straightforward, it is worth discussing insome detail how the 1-loop corrections have been included in ρLL. As alreadystressed, we need to consider 1-loop effects only at the ρL peak, for s ∼ m2

ρ.The first and third diagram in the first row of Fig. 3.3 can thus be evaluatedat tree level. The second diagram gets 1-loop corrections in the vertex withthe current (light blue blob with a cross), the ρL propagator (dark blue box)and the ρLππ vertex (light blue blob). By decomposing each of these threeterms into a longitudinal and a transverse part, the contribution of the dia-gram to the matrix element of the current between the vacuum and two NGbosons can be written as:

〈0|JaLµ |πk(p1)πl(p2)〉∣∣ρ

= δaLi(ΠJρ(q2)PT µα + ΠJρ(q2)PLµα

)× δij

(G(q2)PαβT + G(q2)PαβL

)× 1

2εjkl[(p1 − p2)βV (q2) + qβV (q2)

],

(3.47)

where PµνT = (ηµν − qµqν/q2), PµνL = qµqν/q2 and q = p1 + p2. The spectralfunction ρLL is extracted by squaring this matrix element, integrating overthe two-particle phase space and finally projecting over the transverse part(see Eq. (3.20)). The expression of the longitudinal terms in Eq. (3.47) isthus not relevant, as they do not enter the final result. For the transverse


terms we use the following approximate expressions,

ΠJρ(q2) =M2ρ

gρ− 2α2gρq

2 + gρΠ(1L)Jρ , (3.48)

G(q2) = Zρq2 −m2

ρ + imρΓρ, (3.49)

V (q2) = Z−1/2ρ

(96π Γρ

mρ

)1/2, (3.50)

where the 1-loop parts have been evaluated at q2 = m2ρ. The quantity Π(1L)

Jρ

encodes the pure 1-loop correction from NG bosons to the current-ρL mixing.For the propagatorG(q2) we make use of its resummed expression near the ρLpole in terms of the pole mass mρ, total decay width Γρ and the pole residueZρ. Finally the vertex V (q2) is expressed in terms of the decay width Γρ. Wereport the analytic formulas for Π(1L)

Jρ , m2ρ, Zρ and Γρ in Appendix B. Notice

that a tree-level expression for Γρ is sufficient to reach the O(g0ρ) precision we

are aiming for in the spectral function. Adding the contribution of the firstdiagram in the first row of Fig. 3.3 and inserting the total matrix element inEq. (3.20), one finds the following result for the spectral function

ρ(ππ)LL (q2) = ρRR(q2)×

∣∣1− a2ρ + ΠJρ(q2)G(q2)V (q2)

∣∣2 , (3.51)

where ρRR is given in Eq. (B.23). Away from the ρL peak the 1-loop correc-tions can be neglected, and the second term in the absolute value in Eq. (3.51)is of order g0

ρ, like the first one. At the peak, on the other hand, this sec-ond term develops an O(1/g2

ρ) contribution. This can be identified by usingEq. (3.46) to expand ρ(ππ)

LL (s) as a distribution. One has:

ρ(ππ)LL (s) = ZLm

2ρ δ(s−m2

ρ) + fLL(s) . (3.52)

Here ZL is the pole residue of the two-point current correlator:

ZL =(

1gρ− 2α2gρ

)2−

2a4ρ − 4a2

ρ + 8596π2 log µ

Mρ

−10a4

ρ − 32a2ρ + 1289− 231π

√3

576π2 .

(3.53)

It is of order 1/g2ρ and, being an observable, is RG invariant. The function

fLL denotes instead the O(g0ρ) continuum (which receives a contribution from

both the NG bosons and the ρL).


ρ(s)

MH2 Mρ

2 4Mρ2

s

Figure 3.4. Plot of the spectral functions ρLL (continuous green curve), ρRR(dot-dashed blue curve) and ρBB (dashed orange curve), computed at O(g0

ρ)for the following choice of parameters: Mρ(Mρ) = 2TeV, gρ(Mρ) = 3, aρ =1, α2(Mρ) = 0. The kink of ρLL at s = 4m2

ρ is due to the onset of thecontribution of the ρLρL intermediate state. The scale is logarithmic on bothaxes.

The analytic expressions of the spectral functions are reported in Ap-pendix B. Their plot (in D = 4 dimensions) is shown in Fig. 3.4 for thefollowing benchmark choice of parameters: Mρ(Mρ) = 2TeV, gρ(Mρ) = 3,aρ = 1, α2(Mρ) = 0 (here Mρ(µ), gρ(µ) and α2(µ) are the running parame-ters, see Chapter 2). 14 One can notice the following. The functions ρLL(s)and ρRR(s) become constant and equal for s→ 0 (in D = 4). This constanttail corresponds to the NG boson contribution to the spectral functions; itgives rise to the IR logarithmic singularity in the S parameter that is even-tually canceled by the subtraction in Eq. (3.25). Having set α2 = 0, thespectral functions tend to a constant also for s→∞. This gives rise to a UVlogarithmic divergence in the spectral integral for S which can be regulatedby extending the theory to D dimensions (Notice that one should consis-tently extend both the spectral functions and also the subtraction term inEq. (3.25)). The divergence is canceled by the local counterterm generated

14We have checked that setting α2 to a value of order 1/16π2 at the scale Mρ, asobtained if α2 = 0 at the cutoff scale, does not change qualitatively the plot. Noticethat the running of aρ arises at the two-loop level and can be thus neglected.


by the operator O+3 = Tr[(ELµν)2 + (ERµν)2]. The correlator Π1 thus obeys a

dispersion relation of the form (3.39),

Π1(q2) = Π1(0) + q2∫ ∞

0ds

ρLL(s) + ρRR(s)− 2ρBB(s)s− q2

+ q2(C1 − 8c+3 ) + . . . ,

(3.54)

where C1 ≡ limq2→∞

[Π1(q2)/q2], c+3 is the coefficient of O+3 , and the dots indi-

cate local terms with higher powers of q2. For α2 = 0 the contribution fromthe integral on the circle vanishes, C1 = 0, when extending the theory to Ddimensions. For non-vanishing α2, on the other hand, Π1(q2) grows like q2

in any dimension (as a consequence of its tree level behavior) and one findsC1 = −4α2

2g2ρ.

Using the expressions of the spectral functions we can derive our finalexpression for S. We find:

S = g2

4g2ρ

sin2θ(1− 2α2g

2ρ

)2 + g2

96π2 sin2θ

(log µ

Mh+ 5

12

)− g2

96π2 sin2θ

[34(a2ρ + 28

)log µ

Mρ+ 1 + 41

16a2ρ

]+ g2 sin2θ

(−2c+3 (µ) + C1

4

).

(3.55)

Notice that the term proportional to C1 cancels the α22 part in the first term.

The parameters W and Y obey the same dispersion relations of the fulltheory, Eqs. (3.28) and (3.29), with ρij replaced by the spectral functions ofthe effective theory ρij . All contributions from the integrals on the circle, inthis case, can be made vanish through dimensional continuation. The con-tact terms to be added in the effective theory are generated by the operatorsO2W = (∇µELµν)2 and O2B = (∇µERµν)2. Their contribution is naively ofO[(M2

W /M2ρ )(g2/16π2)], i.e. of higher order in our approximation, and will

be thus neglected. Furthermore, since we are interested in the leading cor-rection of O[(M2

W /M2ρ )(g2/g2

ρ)] from the ρL, the integral in Eq. (3.28) canbe computed by retaining only the delta function in the expansion of ρLL in

3.4 Discussion 61

Eq. (3.52) (while that in Eq. (3.29) is negligible). We thus find: 15

W = g2

96π2 c2W sin2θ

[9x2

h + 12xh2(xh − 1)5 log xh −

x3h + x2

h + 73xh + 98(xh − 1)4

]+ M2

W

M2ρ

g2

g2ρ

(1− 2α2g

2ρ

)2,

(3.57)

Y = g′2

96π2 c2W sin2θ

[9x2

h + 12xh2(xh − 1)5 log xh −

x3h + x2

h + 73xh + 98(xh − 1)4

]. (3.58)

Using Eqs. (3.55), (3.57) and (3.58), together with Eq. (3.9), we obtainour final formula for ∆ε3:

∆ε3 = g2

4g2ρ

sin2θ(1− 4α2g

2ρ

)+ M2

W

M2ρ

g2

g2ρ

(1− 2α2g

2ρ

)2− 2g2 sin2θ c+3 (µ) + g2

96π2 sin2θ

(log µ

MZ+ f3(h)

)− g2

96π2 sin2θ

[34(a2ρ + 28

)log µ

Mρ+ 1 + 41

16a2ρ

].

(3.59)

3.4 Discussion

Equation (3.59) coincides with the result that we obtained in Chapter 2through a 1-loop diagrammatic calculation of ∆ε3. 16 It shows that at treelevel (i.e. at O(1/g2

ρ)) the sign of ∆ε3, as well as that of S in Eq. (3.55),is controlled by α2 and is not necessarily positive. This was considered asproblematic by Rychkov and Orgogozo in their analysis of Ref. [86], basedon the expectation that S should be positive if obtained through a dis-persion relation where the leading contribution arises from the (positivedefinite) spectral function ρLL. They suggested that the positivity of Sis in fact restored once the correct asymptotic behavior in the deep Eu-clidean (q2 → −∞) implied by the OPE is enforced on the expressions

15The O(M4W /M

4ρ ) terms of footnote 10 give the additional corrections

δW =M2W

M2ρ

g2

g2ρ

(1− 2α2g

2ρ

)2(

cos4 θ

2− 1), δY =

M2W

M2ρ

g′2

g2ρ

(1− 2α2g

2ρ

)2 sin4 θ

2,

(3.56)which also come from the delta function in the expansion of ρLL.

16The O[(M2W /M

2ρ )(g2/g2

ρ)] contribution from W and Y was neglected in Chapter 2see Eq. (2.49).


of the two-point current correlators computed in the effective theory. Inparticular, one expects that Π1(q2) ∼ (−q2)1−∆1/2 for q2 → −∞, where∆1 ≥ 1 is the scaling dimension of the first scalar operator contributing toits OPE. If this condition is enforced on Eq. (3.54) by neglecting the higher-derivative terms denoted by the dots, one obtains c+3 = C1/8 = −α2

2g2ρ/2,

where from now on we focus on the tree-level contribution neglecting theO(1/16π2) radiative corrections. This relation implies that the last term ofEq. (3.55) identically vanishes, giving the positive definite expression derivedin Ref. [86]: S = (g2 sin2θ/4g2

ρ)(1−2α2g2ρ)2. Now, the higher-derivative terms

in Eq. (3.54) are suppressed by corresponding powers of the cutoff scale Λ.As such they become important at energies E ∼ Λ. Neglecting them whenenforcing the asymptotic behavior is in fact equivalent to requiring that thislatter is attained at energies E ∼ mρ through the exchange of the ρL, whilethe cutoff states have no effect. In this sense, the correction coming fromc+3 should be regarded as characterizing part of the ρL contribution ratherthan encoding the effect of the cutoff states. Requiring that the asymptoticbehavior be obtained at the scale mρ, as effectively done in Ref. [86], thusleads to a positive S.

There is, on the other hand, the possibility that the correct asymptoticbehavior is recovered only at energies E ∼ Λ as the effect of the higher-derivative terms. That is to say, it can be enforced by the exchange ofthe cutoff states rather than by the lighter resonance ρL. In this case it isreasonable to assume c+3 < 1/g2

ρ, as suggested by its naive estimate, so thatS = (g2 sin2θ/4g2

ρ)(1 − 4α2g2ρ) up to smaller corrections. This expression

is not definite positive, as previously noticed. It is a result consistent withthe properties of the underlying strong dynamics and in fact plausible tosome degree. Indeed, the behavior of the correlators in the deep Euclideancould be determined by the dynamics at or beyond the cutoff scale, whilethe S parameter is saturated in the infrared and as such gets its leadingcontribution from the lightest modes. A simple model with three spin-1resonances is discussed in Appendix F which illustrates this possibility withan explicit example.

The tree-level value of the S parameter can then be tuned to be smallor may even become negative for α2 of order 1/g2

ρ. While such large valuesare not expected from a naive estimate if α2 is generated by the physics at

3.4 Discussion 63

the cutoff scale (in this case one would expect α2 ∼ f2/Λ2 or smaller), theyare consistent with the request of the absence of a ghost in the low-energytheory (see Chapter 2). Having α2 ∼ 1/g2

ρ, on the other hand, affects thenaive estimate of c+3 . For non-vanishing α2, the 1-loop correction to Π′1(0)is quadratically divergent, which implies c+3 (Λ) ∼ (Λ2/M2

ρ )(α22g

4ρ)/16π2. For

α2 ∼ 1/g2ρ and setting Λ = g∗f one has c+3 (Λ) ∼ g2

∗/(16π2g2ρ). This can be

as large as the tree-level contribution from the ρL exchange if g∗ ∼ 4π. Suchenhancement of the 1-loop contribution from the cutoff dynamics originatesfrom the increased coupling strength through which the transverse gaugefields interact with the composite states. In particular, the ππWρL vertexgets an energy-growing contribution of order ggρ(α2g

2ρ)E2/M2

ρ . For α2 ∼1/g2

ρ, this translates into a coupling strength squared of order gg∗(g∗/gρ) atthe cutoff scale, which is a factor (g∗/gρ) stronger than the naive estimatebased on the PUVC criterion [49]. This is precisely the enhancement factorappearing in the estimate of c+3 . We thus conclude that while for α2 ∼ 1/g2

ρ

it is possible to make the tree-level value of S small or even negative, this isat the price of increasing the naive size of the unknown contribution from thecutoff states. Such contribution becomes of order 1/g2

ρ if g∗ ∼ 4π, makingthe S parameter in practice incalculable in the effective theory.

As a final remark we notice that when including the 1-loop corrections,the asymptotic behavior of the full theory is not attained at mρ even forα2 = 0. In fact, one has Π1(q2) ∼ q2 log(−q2)(1−a2

ρ)(5/2−a2ρ) for q2 → −∞

(in D = 4). Setting a2ρ equal to 1 or 5/2 (and α2 = 0) thus gives a model

of the strong dynamics where the asymptotic behavior of Π1 is enforced bythe exchange of the ρL, and the dispersive integral of the S parameter in theeffective theory is convergent in D = 4. In a low-energy theory with both ρLand ρR, one has that Π1(q2)/q2 vanishes at infinity for a2

ρL = a2ρR = 1/2 or

3 (and α2L = α2R = 0). The choice a2ρL = a2

ρL = 1/2, in particular, corre-sponds to a two-site model limit in which the global symmetry is enhancedto SO(5)× SO(5)→ SO(5) (see Chapter 2). Finiteness of the S parameterin this case follows as a consequence of the larger symmetry [88].

In this chapter we have derived dispersion relations for the electroweakoblique parameters in the context of SO(5)/SO(4) composite Higgs theories.We have distinguished between long- and short-distance contributions to ε3,and obtained a dispersion relation for each of the parameters S, W and Y


characterizing the short distance part (Eqs.(3.25), (3.28) and (3.29)). Ouranalysis generalizes the dispersion relation written by Peskin and Takeuchifor the S parameter in the case of Technicolor [89]. We thus derived adispersion relation for ε3 (Eq. (3.31)), extending the work of Rychkov andOrgogozo [86]. Our formula (3.31) agrees with their result and further re-duces the relative theoretical uncertainty to order M2

h/M2∗ , where M∗ is the

mass scale of the resonances of the strong sector. This is to be comparedwith the O(Mh/M∗) relative uncertainty of Ref. [86]. We also discussed howthe dispersion relations can be used and get modified in the context of alow-energy effective description of the strong dynamics. Making use of di-mensional regularization we provided a definition of the otherwise divergentspectral integrals, pointing out the importance of the contribution from theintegral on the circle in the case in which the two-point correlators of the ef-fective theory do not die off fast enough at infinity. We utilized our formulato perform the dispersive calculation of ε3 at the 1-loop level in a theory witha spin-1 resonance ρL. We pointed out that 1-loop corrections need to beretained only at the ρL peak to obtain ε3 at the O(g0

ρ) level. This reproducedthe result of the diagrammatic computation that we performed in Chapter2. The dispersive approach is particularly suitable to clarify the connectionbetween the positivity of the S parameter and the UV behavior of two-pointcurrent correlators, as first suggested by Ref. [86]. We argued that if thebehavior dictated by the OPE in the deep Euclidean is enforced at the scalemρ through the exchange of the light resonances, then the S parameter ispositive definite in agreement with the expectation of Ref. [86]. It is possible,on the other hand, that the UV behavior is recovered only at the cutoff scaleas an effect of the heavier resonances, while the leading contribution to the Sparameter is still saturated by the lowest lying modes. In this case S can benegative if the ρL dynamics is characterized by a large kinetic mixing withthe gauge fields of order α2 ∼ 1/g2

ρ.

65

Chapter 4

Extended analysis of EWconstraints

The aim of this chapter is to perform an extensive one-loop analysis of elec-troweak precision constraints in the minimal SO(5)/SO(4) model, includingboth the lowest-lying fermionic and spin-1 resonances. Differently from theanalysis already carried out in chapter 2, this time we will perform a com-plete and refined fit of the various models, featuring a bayesian approachand full systematic explorations of the parameter spaces. We will include,in general, constraints coming from the parameters ∆ε1,3, the shift to theZbLbL coupling δg(b)

L and from the Z, top and Higgs masses. This chapteris organised as follows: in section 4.1 we will present the theoretical frame-work of the various models involved, as well as some remarks on symmetriesrelevant for our purpose. In section 4.2 we will outline our calculation ap-proach for the various contributions the EWPO, followed by a presentationand discussion of the results. Part of these contribution have been firstlycalculated in chapter 2 and in ref. [69]. In section 4.3 we will introduceour method for comparison with data, presenting a general BSM fit tailoredfor our assumptions and summarizing the parameters involved. Finally, insection 4.4, we will show and discuss the results obtained for various choicesof particular models.

66 4. Extended analysis of EW constraints

4.1 Theoretical Framework

4.1.1 The model

In this section we will present the basic structure of the composite Higgsmodels of our interest and show the various relevant parts of the Lagrangian.These models consist of two distinct sectors interacting with each other: acomposite, strongly-interacting sector is formed at the TeV scale by a strongUltraviolet (UV) dynamics and couples to an elementary, weakly-interactingsector represented by the SM fields.

For the composite sector we will adopt the choice of the minimal com-posite Higgs model, based on the global symmetry breaking pattern SO(5)⊗U(1)X → SO(4) ⊗ U(1)X , where the U(1)X factor is needed to reproducethe weak hypercharge of fermions. The breaking of the global symmetrySO(5) → SO(4) at some scale f gives rise to 4 Nambu-Goldstone-Bosons(NGBs) denoted by πa , a = 1, 2, 3, 4. These NGBs form a 4-dimensional vec-torial representation of the unbroken group SO(4). As SO(4) is isomorphic(at the Lie algebra level) to SU(2) ⊗ SU(2) (usually called as SU(2)L andSU(2)R respectively), the NGBs can also be thought as a bi-doublet (i.e.,(2,2)) of SU(2)L ⊗ SU(2)R. One of the basic ingredients of the compositeHiggs scenario is the identification of these NGBs with the Higgs doublet.Hence, it is clear that SO(5)/SO(4) is the minimal coset that has only oneHiggs doublet and also contains the custodial symmetry protecting the Tparameter. In this construction, the rescaling factor a is linked to the sepa-ration of scales ξ by,

a =√

1− ξ, ξ =(v

f

)2, (4.1)

where, v ≈ 246 GeV is the Vacuum Expectation Value (VEV) of the Higgsdoublet. We will adopt the formalism of Callan, Coleman, Wess and Zu-mino (CCWZ) [46, 41] to build a simplified low-energy effective Lagrangianwhich describes the phenomenology of the lowest-lying fermionic and spin-1composite resonances. Such a model has a natural cutoff Λ, which shouldbe identified as the mass scale of the heavier states. The masses of the SMfermions are assumed to be generated by their linear coupling to the com-posite sector, i.e., the so-called partial compositeness mechanism [73, 48]. In

4.1 Theoretical Framework 67

this paper we will only consider the top quark to be massive and all the otherquarks will be assumed to be massless. Partial compositeness will be imple-mented choosing heavy fermions into 4 and 1 dimensional representations ofthe unbroken SO(4), and both the tL and the tR fields will be assumed to bepartially composite. On the other hand, spin-1 resonances are introduced astriplets of SU(2)L and SU(2)R, namely (3,1) and (1,3) of SU(2)L×SU(2)R.

We will use the criterion of Partial Ultra Violet Completion (PUVC) [49]as our rule of thumb in order to estimate the size of the various operators inthe effective Lagrangian. According to PUVC, all couplings in the compositesector must not exceed, and preferably saturate, the sigma model couplingg∗ ≡ Λ/f at the cut-off scale Λ. The only exception to this prescription isapplied to the coupling strength gX associated (by naive dimensional analy-sis) to the masses of the various resonances MX ∼ gXf . For these couplingswe demand gX < g∗. These exceptions will also fix the order of magnitudeof the masses of the resonances to the scale MX < Λ. On the other hand,the elementary sector is not constrained by PUVC, thus for the operatorsinvolving elementary fields we assume a power-counting rule in which fieldsand derivative expansions are controlled by powers of 1/Λ. Couplings involv-ing the elementary sector are generally expected to be smaller than the onespurely related the composite strongly-interacting sector.

We will now discuss the construction of the model in detail. We willclosely follow Refs. [49, 69] to which we refer for more details. In the standardCCWZ language, one constructs the Goldstone matrix U out of the 4 NGBsas

U = exp[i√

2πa(x)f

T a], (4.2)

where T a are the SO(5)/SO(4) broken generators and f denotes the scaleof the global symmetry breaking SO(5) → SO(4) (note that the NGBs areneutral under U(1)X). Starting from U one can define the CCWZ symbolsdµ ∈ SO(5)/SO(4) and Eµ ∈ SO(4), where the Eµ structure can be furtherdecomposed into ELµ ∈ SU(2)L and ERµ ∈ SU(2)R. The SM gauge groupis contained in a different SO(4)′ ∼ SU(2)′L ⊗ SU(2)′R which is rotated byan angle θ (degree of misalignment) with respect to the unbroken SO(4). Itturns out that with this construction we have the relation[49],

ξ = sin2(θ). (4.3)


Using the dµ and the Eµ structures one can only form one invariantoperator at the level of two derivatives,

L(2)π = f2

4 Tr [dµdµ] , (4.4)

which contains the mass terms for the electroweak gauge bosons:

L(2)π ⊃

18f

2ξ(gelW

aµ − g′elδ

a3Bµ)2. (4.5)

Notice that, since the elementary EW gauge bosons can mix with heavyspin-1 resonances, their gauge couplings (gel, g

′el) must be in general different

from the couplings (g, g′) of the physical states.Fermionic resonances in the 1 and 4 dimensional representations of SO(4)

will be denoted by Ψ1 and Ψ4 respectively. As mentioned before, the elemen-tary fermions couple linearly to the composite sector following the paradigmof partial compositeness. In order to give mass to the top quark we mustassign a U(1)X charge equal to 2/3 to the composite fermions.1 Hence, Ψ4

and Ψ1 transform as 42/3 and 12/3 of the unbroken SO(4)⊗ U(1)X respec-tively. Given these quantum numbers, the Ψ4 and Ψ1 fields can be writtenin terms of the electric charge eigenstates as

Ψ4 = 1√2

iB − iX5/3

B +X5/3

iT + iX2/3

−T +X2/3

0

, Ψ1 =

0000T

. (4.6)

In the above notation, both the fields ψ4 and ψ1 symbolically transformas Ψ1,4 → hΨ1,4 (note that hΨ1 = Ψ1) where h is an element of SO(4).

When both Ψ4 and Ψ1 are present, the Lagrangian of the compositefermions at leading order in the derivative expansion can be written as

LΨ = Ψ4

[i

(/∇+ i

23/B

)−M4

]Ψ4+Ψ1

[i

(/∂ + i

23/B

)−M1

]Ψ1

+(cdΨ1/dΨ4 + h.c

),

(4.7)

1On the other hand, in order to generate the bottom mass one needs a X = −1/3representation.


where ∇µ = ∂µ + iEµ is the standard CCWZ covariant derivative. We willassume cd to be real in our analysis. For future convenience, it will be usefulto define the couplings

gi ≡Mi

f, i = 1, 4. (4.8)

Our working assumption requires that gi < g∗ while PUVC sets |cd| . 1.The elementary quark doublet qL = (tL, bL)T is a 22/3 of the rotated

SU(2)′L⊗U(1)X having T (R) 3θ = −1/2. It is thus embedded in an incomplete

42/3 of SO(4)′ ⊗ U(1)X as

q5L = ∆LqL = 1√

2

ibL

bL

itL

− cos (θ) tL− sin (θ) tL

, ∆L = 1√

2

0 i

0 1i 0

− cos (θ) 0− sin (θ) 0

. (4.9)

Similarly, tR transforms as 12/3 and is written as

t5R = ∆RtR =

000

− sin (θ) tRcos (θ) tR

, ∆R =

000

− sin (θ)cos (θ)

. (4.10)

As mentioned earlier, we only generate a mass for the top quark andthe bottom quark remains exactly massless. The mixing Lagrangian for thetop quark for the general case in which both ψ4 and ψ1 are present can bewritten as

Lmix = yL4fq5LUΨ4+yL1fq

5LUΨ1+yR4ft

5RUΨ4+yR1ft

5RUΨ1+h.c. . (4.11)

In order to show the various representations involved, the above operatorsare often written as

Lmix = yL4f (qL)α(

∆†L)αIUIj (Ψ4)j + yL1f (qL)α

(∆†L)αIUI5T

+ yR4f tR

(∆†R)IUIj (Ψ4)j + yR1f tR

(∆†R)IUI5T + h.c. ,

(4.12)

where I = 1 . . . 5, j = 1 . . . 4 and α = 1, 2 are respectively SO(5), SO(4) andSU(2)′L indices. Notice that PUVC does not provide any information aboutthe yi couplings.


The spin-1 resonances are introduced as two triplets i.e., (1,3) and (3,1)of the unbroken SU(2)L ⊗ SU(2)R. They are neutral under U(1)X and areintroduced as vector fields transforming like a gauge field. When both of themare assumed to be present, their leading order Lagrangian in the derivativeexpansion can be written as

Lρ =∑r=L,R

− 14g2ρ r

Tr[ρr µνρrµν

]+ 1

2M2ρ r

g2ρ r

Tr[(ρrµ − Erµ

)2]+α2 rTr[ρr µνfrµν

],

(4.13)where frµν is another CCWZ structure obtained from the field strength of theEW gauge bosons (see eq.(10) of Ref. [49]). The above Lagrangian generatesboth kinetic and mass mixings between composite vectors and elementaryEW gauge bosons. We define the following parameters for future convenience,

aρ r ≡Mρ r

gρ rf, β2 r ≡ α2 rg

2ρ r , (r = L,R). (4.14)

While PUVC requires aρ r . 1, the coefficients α2 r are not constrained bysuch criterion as they couple the elementary and the composite sectors. How-ever, as discussed in [49], large values of α2 r can result in wrong-sign kineticterms in the physical basis of spin-1 fields. For example, if only one vector res-onance is present, this gives a consistency condition |α2 r| . 1/gelgρ r. More-over, as pointed out in chapter 3, when α2 r are included the estimate of thecontributions to the S parameter coming from the states above the cutoff getsan additional correction which is increased by a factor α2

2 rg4ρ r(Λ2/M2

ρ ). Suchenhancement is avoided by the choice |α2 r| . aρ r/gρ rg∗ which is strongerthan the consistency condition mentioned above.

One may also consider the four-derivative operators

Q1 r = Tr (ρr µνi [dµ, dν ]) , (4.15)

whose coefficients, α1 r, are constrained by PUVC with a similar bound,|α1 r| . 1/g∗gρ r. However, their impact on EWPO is purely at the one-looplevel, whereas the operators in (4.13) contribute at the tree level to the Sparameter. Thus, these operators are not considered in our calculations.2

2The authors of ref. [24] used a different basis for the four-derivative operators,namely Q′1 r = Tr (ρr µνi [dµ, dν ]) and Q′2 r = Tr

(ρr µνErµν

). Their basis is related to

our basis by the relation Eµν = f+µν− i [dµ, dν ]. For our purpose we find it inconvenient


If the fermion and the vector resonances are considered together, one canwrite new operators which couple directly the two sectors. The leading onesin the derivative expansion read,

LΨρ =∑r=L,R

crΨ4(/ρr − /E

r)Ψ4. (4.16)

Again using PUVC we have |cr| . 1. Just for completeness, below wealso give the kinetic terms for the elementary fields,

Lel = − 14g2

elTr [WµνWµν ]− 1

4g′ 2elTr [BµνBµν ] + qLi /DqL + tRi /DtR . (4.17)

4.1.2 Symmetries

In this section we will discuss some approximate symmetries and symmetry-related issues which are relevant for our purpose.

The so-called PLR parity plays an important role in the EWPO as it isrelated to corrections to the ZbLbL coupling [9]. The action of this discretesymmetry is to exchange the generators of the unbroken SU(2)L and SU(2)Rsubgroups, and change the sign of the first three broken generators. It canbe defined by the following action on the fields3

πa → ηaπa, B → −B, T ↔ −X2/3,

ρL aµ ↔ ρR aµ , X5/3 → −X5/3, T → T , (4.18)

where, ηa = (−1,−1,−1,+1). Using the above rules one can derive

daµ → ηadaµ, EL aµ ↔ ER aµ . (4.19)

Following the above definition of PLR one can check that it is an exactsymmetry of the composite sector if masses and couplings of ρLµ are equal

to use their basis because in that case the PUVC bounds on α′1 r would be correlatedwith the bound on α′2 r. Hence, even at the price of enhancing the contribution from thecut-off, it will not be possible to explore a large α′2 r with much smaller α′1 r. Moreover,the operators Q1 r give rise to tree-level contributions to the decay h → Zγ, hence itis expected to be one loop suppressed under the requirement of minimal coupling [24].

3The transformation properties of fermionic resonances under PLR can be obtainedby expressing the (2,2) multiplet in a matrix form (in analogy with NG bosons),Ψ4 = T a (Ψ4)a where, a = 1 . . . 4.


to those for ρRµ .4 When the composite sector couples to the elementarysector, this symmetry gets explicitly broken by the gauge couplings (g, g′)and the four yL/Ri (i = 1, . . . 4) mass mixings whose insertions are needed togenerate a nonzero δgL. This property can lower the degree of divergence ofthe various contributions in our model as we will discuss in subsection 4.2.3.

The Custodial symmetry is represented by the subgroup SU(2)′L+R,spanned by the generators TL aθ +TR a

θ .5 It is explicitly broken by the gaugecoupling g′el and the left mixings yLi. The insertion of powers of those mix-ings makes the pure fermionic contribution to the T parameter finite (seesubsection 4.2.1 below).

One interesting case, the two-site model [88, 81], is obtained by includingthe full 5-plet of fermionic resonances as well as both the vector triplets andenforcing the relations,

yL1 = yL4, yR1 = yR4, cd = 0,

gρL = gρR , aρL = aρR = 1√2, (4.20)

β2L = β2R = 0, cL = cR = −1.

Note that the above constraints imply MρL = MρR . The Lagrangianobtained after imposing the above relations is equivalent to a model char-acterised by the global symmetry breaking pattern SO(5)L ⊗ SO(5)R →SO(5)V in which the EW gauge bosons and the vectorial resonances appearas gauge fields arising from gauging part of the SO(5)L and the SO(5)Rgroups. As a characteristic feature of this construction, the EWSB can onlybe achieved if the global symmetry in both the left and the right sites areexplicitly broken. As shown in Ref. [88], a consequence of this fact is that thedegree of divergence of the 1PI contributions to any EWSB-related quantityis lowered. In particular, the fermionic contribution to the Higgs potentialis changed from quadratically to logarithmically divergent, and the contri-butions to the S and T parameters become finite. Non-1PI divergent con-tributions can still be present because of the presence of spin-1 resonances.

4The symmetry is lost when operators beyond the leading chiral order in the NGbosons Lagrangian are introduced. It is possible to enforce the symmetry at all ordersin the chiral expansion by promoting the global symmetry group to O(5) [49].

5As the custodial group also belongs to the unbroken subgroup, such combinationis equal to TLa + TRa.


As pointed out in Ref. [51], for the S parameter such divergences can bereabsorbed by the renormalization of the parameters of Lρ, making it fullycalculable at one-loop, while this is not the case for the T parameter becausean additional operator is needed to remove the non-1PI subdivergences. Thecoefficient of such operator gives an additional contribution to the mass split-ting between neutral and charged vector resonances, hence the T parameterbecomes calculable after fixing the splitting. Similarly, it is found that theHiggs potential is calculable by fixing a single counterterm to reproduce theHiggs VEV.

Another special case is obtained by cd = ±1. In this case, as discussedin ref. [69], in the composite fermion Lagrangian the goldstone fields can beshown to appear only in the mass terms. Hence the degree of divergencein any EWSB related quantity is decreased. As a result of this, fermioniccontributions to the S parameters become finite in the case cd = ±1.

The global SO(5) symmetry is explicitly broken by two different sources:the EW gauging and the fermion mass mixings. The breaking can be pa-rameterised by the couplings gel, g′el and the four yL/R couplings. The EWobservables T and δgL are sensitive to these symmetry-breaking couplings.Hence it is useful to analyse their effect with a spurion method. In particu-lar, it is relevant to analyse the number of multiplicative symmetry breakingcouplings that is required to generate T and δgL.

As mentioned earlier, the breaking of SO(5) proceeds through the EWgauging and the fermion mass mixing Lagrangian (4.11). In particular, theterms involving the yL couplings break SU(2)′R, while the yR couplings pre-serve the full SO(4)′. The terms involving the gauge couplings gel and g′el

respectively preserve SO(4)′ and the gauge group SU(2)′L ⊗ U(1)Y . Hencethe custodial symmetry is broken by the couplings yL and g′el. On the otherhand, as tR is an SO(4) singlet the yR couplings do not break the PLR parity,which is only broken by the yL and the gauge couplings. Moreover, since inour calculation we neglect O (gel, g

′el) contributions to δg(b)

L , we will be onlyinterested in the yL mediated contributions to δg(b)

L and T , and g′el medi-ated contributions to T . As far as these couplings are concerned, the globalSO(5) can be restored by promoting them to fields with suitable transfor-mation rules. However, for our purpose we find it more suitable to promotethe associated matrices ∆L and TR 3

θ . This is because in the fermionic sector


there are four mass mixing couplings but only two possible mixing-relatedstructures. For the EW gauging part we promote TR 3

θ to the spurion TR 3θ

transforming as

TR 3θ → gTR 3

θ g†, (4.21)

while for the fermion mixing, we introduce the ∆L spurions transforming as

∆L → g∆Lk†, (4.22)

where k = exp(i αLi σ

i/2)is a 2× 2 SU(2)′L representation.

Using TR 3θ we can form the SO(4)-covariant matrix

τ = U†TR 3θ U, τ → hτh†. (4.23)

Two structures, a matrix and a 5-component vector, can also be con-structed by ∆L:

χL = U†∆LqL, χL → hχL, (4.24)

ηL = U†∆L∆†LU, ηL → hηLh†. (4.25)

The structures τ , χL and ηL can be further deconstructed as fourpletsand singlets of SO(4) as

(τ)IJ = τij + τi5 + τ5j + τ55, (4.26)

(χL)I = χL i + χL 5, (4.27)

(ηL)IJ = ηL ij + ηL i5 + ηL 5j + ηL 55. (4.28)

where, the range of the indices are given by I, J = 1 . . . 5 and i, j = 1 . . . 4.Using only the τ field one can form a custodial symmetry breaking op-

erator which contributes to the T parameter [51],

O(τ)T = Tr [dµτ ]2 . (4.29)

Similarly, using the η field we can build

O(η)T = Tr [dµη]2 . (4.30)

It is now clear that in order to generate the T parameter one either needstwo powers of the coupling g′el or four powers of yL(1/4).6

6Other invariant operators with two powers of τ or η are not independent and differonly by custodial symmetry invariant terms.

4.2 Calculation of Electroweak Precision Observables 75

As shown in ref. [69], in the effective theory with only SM particles andat the lowest order in the mass mixing spurions, one can form the followinglocal operator contributing to δg(b)

L ,

O(qq)δ = χLγ

µχLTr [ηLdµ] . (4.31)

The above operator contains four powers of the spurion ∆L, and henceat least four powers of yL1/4 are needed to generate δg(b)

L . Notice that thevectorial sector also introduces a genuine breaking of PLR without breakingSO(5)× U(1).

4.2 Calculation of Electroweak Precision Ob-servables

In this section we will discuss the details of our computation of the elec-troweak precision observables. Some of our results were computed for thefirst time in ref. [69, 51]. The ε parameters [20, 21] that are relevant for usare

ε1 = e1 − e5 + non-oblique leptonic terms, (4.32)

ε3 = e3 + c2e4 − c2e5 + non-oblique leptonic terms , (4.33)

where, the ei are related to gauge bosons vacuum polarisation amplitudes inthe following way,

e1 = 1M2W

(A33(0)−A11(0)) , e3 = c

sF3B

(M2Z

),

e4 = Fγγ(0)− Fγγ(M2Z

), e5 = M2

ZF′ZZ

(M2Z

).

(4.34)

The Aij and the Fij functions are defined through the vacuum polarizationamplitudes of the EW gauge bosons,

Πµνij (q) = −igµν

(Aij(0) + Fij(q2)q2)+ qµqν terms . (4.35)

As we have neglected couplings between the composite sector and leptons,non-oblique leptonic terms vanish at one-loop. For our analysis we will needto compute the deviations from the SM predictions, namely ∆εi = εi−ε(SM)

i .


As discussed in reference [51], the calculation of the oblique observablescan be interpreted as a matching procedure at the scale of the heavy reso-nances followed by renormalisation group (RG) evolution down to the weakscale and an additional calculation for the light physics contributions (i.e.threshold corrections from the composite Higgs and the top quark at theEW scale). The first step corresponds to the calculation of the effects comingfrom the heavy resonances, and by neglecting terms of O

(M2Z/M

2∗)(where

∗ stands for any resonance) this becomes equivalent to the calculation of Sand T parameters,

S = c

sΠ′3B(0) , (4.36)

T = 1M2W

(Π33(0)−Π11(0)) . (4.37)

On the other hand, both the RG evolution and the threshold correctionsarise from the non-standard dynamics of the Higgs boson and also the topquark (if it mixes with the heavy fermion resonances).

Another quantity which is relevant for us is the parameter g(b)L which is

defined through the non-universal corrections to the effective coupling

gZ bLbLeff ≡ e

scΓ(b)L (q2) bL(p1) γµ bL(p2)Zµ(q = p1 + p2) . (4.38)

The quantity δg(b)L is defined as the NP contribution to Γ(b)

L (q2) at q2 = 0.We could in principle also consider the parameters ε2 and g(b)

R . In general,in composite Higgs models the resonance contributions to the first one issuppressed by powers of g2/g2

∗ (where g∗ is any strong coupling) and the IRcontribution from the composite Higgs is finite and small. These suppressionsare often strong enough to make the impact of ε2 negligible in any stronglycoupled model and can be safely neglected (see, for example [51]). On theother hand, the coupling g

(b)R is in general expected to be produced with

similar size as g(b)L . However, it is much less constrained than the left coupling

(see Fig. 4.7), hence it will not have much effect on our fit and we will neglectit for simplicity.

We will use dimensional regularisation with the MS scheme and fix therenormalisation scale to the cutoff Λ, which will be fixed to the somewhat ar-bitrary value of three times the mass of the heaviest particle in the particularmodel.


The various contributions to the EWPO can be divided into four cate-gories:

• The purely spin-1 resonance contributions where heavy vectors propa-gate alone or together with light particles (NGBs, Higgs or EW gaugebosons).

• The purely heavy fermion contributions arising from diagrams whereat least one heavy fermionic resonance propagates in the loop.

• The mixed heavy fermion-vector contribution comprising of diagramswhere both type of resonances propagate.

• The light physics contribution from the loops of only light particles.Since we are interested in the deviations with respect to the SM predic-tions, only light particles with non standard dynamics will contribute,i.e. the Higgs boson and the top quark.

Logarithmically divergent terms in all four categories account for RGevolutions, while the finite terms of the first three categories represent theshort-distance effects at the M∗ threshold. This is the part for which we canneglectO

(M2Z/M

2∗)terms and calculate contributions to ∆T and ∆S instead

of ∆ε1,3. On the other hand, when calculating the light physics contributions(which must also undergo a subtraction of the SM result), we have insteadto consider the full ε parameters, thus probing the full IR structure of themodels: for this reason, the contribution coming from this last part to theoblique ε1,3 parameters will contain a complicated finite part. In all ourcalculations we neglect the effects coming from the propagation of the EWgauge bosons inside loops except for the pure vectorial contribution to the Tparameter, as it is the leading contribution of the purely vectorial part.

We stress that in the class of models we are interested in (or in general,in any effective theory), EWPO also get additional unknown contributionsfrom local operators that are generated from physics above the cut-off. Sincethey depend on the details of the UV theory, we are completely ignorantabout these contributions. We will assume them to vanish at the cut-offscale (at lower scales they will in general be generated by loops of resonancesand SM particles). It is important to stress that these contributions can benon-negligible in particular, if non-decoupling effects are present.


Both the fermionic and the spin-1 resonances mix with each other, hencethe interaction eigenstates are in general not the mass eigenstates. In thefermion sector, when both the 4-plet and the singlet are present, in the limitξ = 0 there are two degenerate SU(2)L doublets (T,B) and

(X5/3, X2/3)

with masses√M2

4 + y2L4f

2 and M4 respectively and a singlet T . The massdegeneracy is lifted by non-zero ξ. The same happens for the SU(2)L tripletof vector resonances, while the mass degeneracy of the SU(2)R triplet res-onances is already lifted at ξ = 0 by the coupling g′el. Elementary statesare also involved in the mixings: the top quark and the EW vector bosonsacquire mass for ξ 6= 0, while the bottom quark remains exactly massless,as its compositness has been neglected. See appendix G for more details onmass mixings.

Mixings in the vector boson sector are regulated by powers of (gel, g′el) /gρr ,

and as long as β2 r . 1 they can be safely treated in perturbation theory.Things are different for fermionic mixings, as they are associated with powersof yiMj/f = yi/gj which in general may not be a suitable expansion param-eter. Hence calculations have been carried out by resumming power of massinsertions in the fermionic sector by numerically diagonalizing fermionic massmatrices.

In the previous few paragraphs we sketched the details of our calculationfor the EWPO that will be used in the numerical fits of section 4.4. We willnow present simplified analytic expressions for the various contributions sothat (at least) some broad features of the results can be understood analyt-ically. In order to do so, at first we will neglect the threshold correctionsat the EW scale coming from top quark loops. Moreover, for the fermionicresonance contributions we will only keep the terms that are leading orderin ξ and also in the mass mixings yi. In the fermionic 5-plet scenario, ad-ditionally we will set yL1 = yL4 = yL and yR1 = yR4 = yR. Finally, themixed fermion-vector contributions will be expanded in M2

4 /M2ρr and only

the leading order terms will be kept.

4.2.1 Contributions to the T parameter

Every contribution to the T parameter depends on sources of breaking ofthe custodial symmetry. In the absence of EW gauging, the sector of vec-


Figure 4.1. Pure vectorial contributions to the T parameter. Smooth singlelines represent all the NGBs including the Higgs. External (internal) singlewavy lines stand for a W i

µ (Bµ) field. Double wavy lines represent eithera ρL or a ρR. Although in the first diagram there is no actual resonancepropagation, it gets contributions from the (Eµ)2 term of Eq. (4.13). Henceit must be included in the resonance contribution. Note that, due to a mistakein ref. [51], the set of diagrams listed there (Figs. 12 and 13 therein) is slightlydifferent from the above set. However, the mistake was only done in showingthe list of diagrams, and in both calculations all the correct contributions havebeen included.

torial composite resonances is SO(5) symmetric, hence insertions of the hy-percharge coupling (at least two insertions, see section 4.1.2) are needed togenerate a contribution as shown in figure 4.1. Pure vector contributions tothe T parameter have been firstly calculated in [51]. As pointed out in thesame reference, the contribution from a single ρL has the same form as thatcoming from a single ρR. Below, we report the expressions for the single


resonance case as well as for the scenario when both ρL and ρR are present.

T∣∣ρL/R

= 3g′ 2el32π2

34ξa

2ρL/R

[(1− 8

3β22L/R

)log(

ΛMρL/R

)(4.39)

+34 −

43β2L/R + 2

9β22L/R

], (4.40)

T∣∣ρL+ρR

= T∣∣ρL

+ T∣∣ρR

+ T∣∣ρLR

, (4.41)

where

T∣∣ρLR

= − g′ 2el32π2

32ξa

2ρLa

2ρR (1− 2β2L − 2β2R + 4β2Lβ2R)×[

log(

ΛMρL

)+ log

(Λ

MρR

)−M2ρL +M2

ρR

M2ρL −M2

ρR

log(MρL

MρR

)+ 5

6

].

(4.42)

Notice that the contributions coming from β2 are different from the onespresented in ref. [51]. This is because we used different basis for the four-derivative operators of the spin-1 resonances, see eq. (4.13) and the textfollowing it.

If |β2r | . 1, it can be easily seen that the contribution coming froma single resonance can compensate for the negative shift coming from thecomposite Higgs boson (see Eqs. B.1 and B.3). When both the resonancesare present, the contribution can still be positive however, the interplay ismuch more complicated and involves also the values of aρr .

The fermionic sector provides instead its own source of breaking, i.e.the mass mixings with elementary fermions, thus fermionic self-energies willcontribute to the T parameter. The contributions have been firstly calcuatedin ref. [69]. As already reported in the same reference, a spurion analysis (seesection 4.1.2) shows that the contributions must appear at least at O

(y4L

),

and are therefore finite by power counting. By including the top contributionsand subtracting the SM result we find the following approximate resultsfor ∆T = T − T (SM) in the singlet, fouplet and fiveplet cases (which aredefined as the scenarios in which we only include a 1, 4 and 4⊕ 1 of SO(4)


respectively):

∆T∣∣Ψ1≈ 3y4

L164π2g2

1ξ, (4.43)

∆T∣∣Ψ4≈ − y4

L432π2g2

4ξ, (4.44)

∆T∣∣Ψ5≈ y4

L

32π2g21ξ

x2

(1− x2)2

[− 3c2d

(2x3 − 5x2 + 18x− 5

)+ 3cd

(x3 + 7x2 − x− 5 + 2

x

)− x4 − 5

2x2 − 4 + 3

2x2

+ 6 log(x)1− x2

(c2d(x4 − 12x3 + 8x2 − 8x+ 1

)+ (4.45)

cd(x4 + 2x2 + x− 2

)− 2x4 + x2 − 1

) ]. (4.46)

In the above and also the rest of this section x ≡M1/M4 = g1/g4.The signs for the singlet and fourplet cases are fixed in the above expres-

sions. This is tipically true also in the full numerical calculations, where allpowers of ξ and yi are resummed. On the other hand, in the fiveplet casethere is no preference for positive or negative sign. Due to the finitenessof the above contributions, inverse powers of g1,4 arise, resulting in gooddecoupling properties for both the fourplet and the singlet.

If one introduces the operators in eq. (4.16), new diagrams with the ex-ternal propagation of heavy spin-1 resonances appear (as shown in Fig. 4.3),yielding however no contribution to T . This can be easily seen by the factthat substituting the equations of motion, namely

ρr aµ = Er aµ −crg

2ρr

M2ρr

Ψ4γµT(r) aΨ4 −

1M2ρr

∇αEr aαµ +O(p5). (4.47)

in the UV Lagrangian of Eqs. (4.13) and (4.16) does not produce any lo-cal operator contributing to T . In other words, the tree level exchanges ofvector resonances cannot generate shifts to the couplings of gauge bosons tofermions at q2 = 0, where qµ is the momentum of the gauge boson. Froma diagrammatic point of view, the absence of these contributions is due toa cancellation between two diagrams. The operators in Eq. (4.16) generateadditional tree level contributions to the vertices between a gauge boson andtwo fermions proportional to cr, which cancels out (at q2 = 0) with thediagram containing a gauge boson-ρ mixing.


Figure 4.2. Pure vectorial contribution to the S parameter. Smooth single linesrepresent all the NGBs including the Higgs. Single wavy lines on the left(right) represent a W 3

µ (Bµ) field. Double wavy lines represent either a ρL ora ρR. Although in the first diagram there is no actual resonance propagation,it gets contributions from the (Eµ)2 term of Eq. (4.13). Hence it must beincluded in the resonance contribution.

4.2.2 Contributions to the S parameter

The S parameter gets an important contribution at tree level from the ex-change of vector resonances. The pure vectorial contribution at one-loopwas calculated in ref. [51] and is shown in figure 4.2. In analogy with theT parameter, the single resonance case is L/R symmetric. We report the


contributions:7

S∣∣ρL/R

= g2

4g2ρL/R

(Λ)ξ(1− 4β2L/R (Λ)

)− g2

96π2 ξ

[34

(a2ρL/R

+ 28 + 24β2L/R

(a2ρL/R

β2L/R − a2ρL/R

− 2))×

log(

ΛMρL/R

)+ 1 + 41

16a2ρL/R

− 32β2L/R

(9a2ρL/R

− 4)

+ 32β

22L/R

(9a2ρL/R

− 8)]

,

(4.48)

S∣∣ρL+ρR

= S∣∣ρR

+ S∣∣ρR. (4.49)

The inclusion of β2r can partly or even fully cancel the tree level con-tribution within its considered range (see Eq. (4.14) and the text followingit), and in principle can also make it negative.8 On the other hand, theone-loop term can be negative even when when β2r is small, hence reducingthe total contribution to S. Note that the explicit dependence on the RGscale (here, fixed to Λ) in the tree level term (first line of Eq. 4.48) ensuresthe RG independence of the full result till one loop order. See again ref. [51]for a detailed discussion about the running of the parameters in the vectorialsector.

Fermionic self-energies will also contribute to the S parameter. For thefourplet and fiveplet case the contributions start at O

(y0), while in the

singlet case the leading order isO(y2), as it must provide a finite contribution

to S (see ref. [69]). By including the top contribution and subtracting the

7In this case the change of operator basis from the one used in ref [51] has no effect(see the beginning of sec. 4.2.1).

8As pointed out in 3, the enforcement of a good UV behaviour of the SU(2)Land SU(2)R spectral functions within the low-energy effective theory would produce anon-negative tree level contribution to S. See also ref. [86]


Figure 4.3. Mixed fermion-vector contribution to the S parameter. Fermioniclines in the above diagrams stand for any fermionic resonance or light quark.Double wavy lines represent either a ρL or a ρR. Although in the first diagramthere is no actual ρ propagation, it gets contributions from the Ψ/EΨ term ofLagrangian (4.16), which must be included in the mixed fermion-vector part.

corresponding SM result, we find the approximate expressions:

∆S∣∣Ψ1≈ − g2

96π2y2L1g2

1ξ

[log(f4ξy2

L1y2R1

2M41

)+ 5

2

], (4.50)

∆S∣∣Ψ4≈ g2

4π2 ξ log(

ΛM4

), (4.51)

∆S∣∣Ψ5≈ g2

4π2 ξ

[ (1− c2d

)log(

ΛM4

)+ log(x)c2d

x3(x3 − 3x+ 3)(1− x2)3 −

c2d2x4 + 9x3 − 16x2 + 9x+ 2

12(1− x2)2

]. (4.52)

Part of these contributions were previously computed in [69, 24]. The con-tributions in the fourplet and fiveplet cases are tipically big also in the fullnumerical calculation. In the fourplet and singlet cases the full contributionsare mostly positive, while in the fiveplet it can have both signs.

With the introduction of the operator in Eq. (4.16) vector resonances cancouple to fermions and thus contribute to the S parameter forming vector-fermion diagrams shown in figure 4.3. Such contributions exist only if acomposite fourplet is present in the spectrum. They start at O

(y2) and,

more interestingly, at order O(M2

4 /M2ρ

). Their approximate expressions


yield:

S∣∣ρL/R+Ψ4

≈ 3g2

32π2y2L4 − 2y2

R4g2

4

M24

M2ρL/R

ξ(1− 2β2L/R

)cL/R

[log(

ΛM4

)+ 1

4

](4.53)

S∣∣ρL/R+Ψ5

≈ 3g2

32π2y2L − 2y2

R

g24

M24

M2ρL/R

ξ(1− β2L/R

)cL/R×[

(1 + cd) log(

ΛM4

)+ cd log(x) (2− x)x2

(1− x)2(1 + x)

+14cd

3− x1− x + 1

4

], (4.54)

S∣∣ρL+ρR+Ψ4

≈ S∣∣ρL+Ψ4

+ S∣∣ρR+Ψ4

, (4.55)

S∣∣ρL+ρR+Ψ5

≈ S∣∣ρL+Ψ5

+ S∣∣ρR+Ψ5

. (4.56)

Since lower mass bounds on vectors are sizeably higher than the ones onfermions (both from direct and indirect indications), a factor M2

4 /M2ρ can

be a strong suppression. Just like the absence of analogous terms in the Tparameter, this suppression can be also explained using the solution (4.47)to the equations of motion of vectorial resonances. Using such solution itis found that an interaction of the form cr

(q2/M2

ρr

)ΨγµΨAµ is generated,

which contributes to S when used to form a fermionic loop together with astandard ΨγµΨAµ gauge-like interaction. The explicit q2 appearing in thevertex forces us to extract a mass term proportional to M2

4 from the loop,thus generating a factor M2

4 /M2ρr .

4.2.3 Contributions to δg(b)L

Tree level contributions δg(b)L from the fermionic sector are proportional to

the degree of compositeness of the bottom quark, and although in some mod-els they can be comparable with the ones coming from fermion loops (see ref.[69]) they are generally expected to be small. As we neglect the bottomquark mass, this contribution is absent in our case. Moreover, tree level con-tributions proportional to cr due to the presence of spin-1 resonances vanishfor the same reason discussed at the end of sec. 4.2.1 for the T parameter.Hence, the contributions to δg(b)

L start at the one loop level.In our models the shift to the ZbLbL coupling is linked to the breaking

of the PLR parity. The spurion analysis of section 4.1.2 shows that the


Figure 4.4. Pure fermionic contribution to δg(b)L . Fermionic lines in the above

diagrams stand for any resonance or light quark, while smooth continouslines denote any NG boson including the Higgs. Note that the wave functionrenormalization of the Z leg must not be included in the calculation.

total pure fermionic contributions must appear at least at O(y4L

), see the

operator O(qq)δ in equation (4.31). In the singlet and the fourplet cases the

insertions of four mass mixings is enough to generate finite pure fermioniccontributions. In the fiveplet case we may also make use of the couplingsproportional to cd, which contain a derivative and can therefore increase thedegree of divergence. However, as explained in ref. [69], the contributionsare finite also in this case. Purely spin-1 resonance contributions to δg(b)

L areabsent, as their effects only appear when coupled to the composite fermionsthrough the operators in Eq. 4.16.

On the pure fermionic side, the various contributions (firstly calculated


in ref. [69]) are shown in figure 4.4, and the approximate expressions read:

δg(b)L

∣∣Ψ1+t ≈

y4L1

64π2g21ξ, (4.57)

δg(b)L

∣∣Ψ4+t ≈

y4L4

32π2g24

y2R4g2

4ξ

[log(

2g44

ξy2L4y

2R4

)+ 2], (4.58)

δg(b)L

∣∣Ψ5+t ≈ −

y4L

64π2g21ξ

x2

1− x2×[log(x)1− x2

(4(1− x2)+ cd

(2− 4x3)− 4c2d

(1− x+ x2)

−2c3dx (2− x))

+ 1− 1x2 + cd

(1 + 2x− 4

x

)+ c2d

(x2 + 2x− 5

)+ c3dx (x− 2)

]. (4.59)

Differently from the oblique parameters, δg(b)L is loosely correlated with

the other EW observables and almost equally accepts positive and negativevalues (see section 4.3.1), so the sign of these contributions is not a relevantinformation.

The vectorial sector is also able to break the PLR parity. The vectors cou-ple to the fermionic sector through the parameters cr introduced in Eq. (4.16).As discussed in the beginning of this section, their effect starts at the one-loop level. Even in this case, the same cancellation discussed for the tree levelcontributions takes place, so that one-loop contributions are only generatedby diagrams with the propagation of spin-1 resonances inside loops.

δg(b)L

∣∣ρL+Ψ4+t ≈

y2L4y

2R4

16π2g24ξcL

[log(

ΛM4

)+ 1

2

], (4.60)

δg(b)L

∣∣ρR+Ψ4+t ≈ −

y2L4y

2R4

16π2g24ξcR

[log(

ΛM4

)+ 1

2

], (4.61)

δg(b)L

∣∣ρL+Ψ5+t ≈

3y2Ly

2R

64π2g24cLξ (1 + cd)

[log(

ΛMρL

)+ 5

12

], (4.62)

δg(b)L

∣∣ρR+Ψ5+t ≈ −

3y2Ly

2R

64π2g24cRξ (1 + cd)

[log(

ΛMρR

)+ 5

12

], (4.63)

δg(b)L

∣∣ρL+ρR+Ψ4+t ≈ δg

(b)L

∣∣ρL+Ψ4+t + δg

(b)L

∣∣ρR+Ψ4+t , (4.64)

δg(b)L

∣∣ρL+ρR+Ψ5+t ≈ δg

(b)L

∣∣ρL+Ψ5+t + δg

(b)L

∣∣ρR+Ψ5+t . (4.65)


Figure 4.5. Mixed vector-fermion contributions to δg(b)L . Fermionic lines stand

for any resonance or light quark, while smooth continous lines denote anyNG boson including the Higgs. Double wavy lines represent either a ρL or aρR. Note that the wave function renormalization of the Z leg must not beincluded, and mixings on the same leg followed by vertices proportional to crdo not contribute (see text).

The fact that the leading contribution does not scale as y4L is a signal that

the breaking of PLR is not coming from the fermionic sector. Despite theappearance of four mass mixings the above results are divergent as the mas-sive spin-1 fields change the power counting. Notice that, for each particularmodel the ρL and ρR contributions are obtained from each other through asign flip and the replacement L↔ R. This happens because a spin-1 sectorwith equal ρL and ρR parameters is PLR symmetric, hence in this case anycontribution to δg(b)

L with less than four powers of yL should cancel.

4.3 Fit Procedure

In order to fit the composite Higgs model parameters to the electroweakprecision data, we follow a two step procedure. In the first step, we performa relatively model independent fit that can be used for a broader class ofmodels. The details of this step is described in the following subsection. Inthe second step we apply the results of the model independent fit to our

4.3 Fit Procedure 89

composite Higgs scenarios. More details about the second step will be givenin section 4.3.2.

4.3.1 Data and fit interpretation

As discussed at length in section 4.2, the fit to the EWPO in the compositeHiggs models of our interest can be performed by only using the two epsilonparameters ∆ε1 ,3 and the modification to the Z bL bL vertex, namely δg(b)

L .Therefore, in our specific NP scenarios one can perform a complete EW fitusing ∆ε1, ∆ε3, δg(b)

L , αs(MZ), ∆α5had(MZ), MZ , mpole

t and mh as the inputparameters 9 and keeping ∆ε2 and δg(b)

R fixed to zero. The result of this fit,namely the central values, uncertainties and the correlation matrix for the 8input parameters (we will call them pseudo-observables below) are shown intables 4.1 and 4.2. The marginalised 68% and 95% probability regions in the∆ε3 −∆ε1 plane and for δg(b)

L are also shown in Fig. 4.6. The numerical fithas been done using the HEPfit package10 (formerly SUSYfit).

Here we would like to emphasise that δg(b)L = 0 is fully consistent with

our fit (it is almost at the centre of the 68% allowed region) which is slightlyinconsistent with the results obtained in [30, 70] (which was also used in [69]).In order to directly compare their results with our fit, in Fig. 4.7 we showthe 68% and 95% marginalised probability regions in the δg(b)

R − δg(b)L plane

when δg(b)R is also included in the input parameters of our fit. It can be seen

that δg(b)R = δg

(b)L = 0 is now marginally consistent at the 95% CL while in

[30, 70] it was slightly outside the 95% CL region. Our results are consistentwith ref. [44], and we believe that the difference with the earlier literatures isdue to the new SM calculation of Rb [64, 63, 62] and possibly also due to thedifferent methodologies used to obtain the 68% and 95% probability regions.

The results of table 4.1 and 4.2 will be used as constraints in the fits ofour NP models discussed in section 4.4. As αs(MZ) and ∆α5

had(MZ) do notget any NP contributions and their uncertainties are also very small, we fixthem to their central values and remove them from the fit. Moreover, as the

9For more details of the EWPO and the experimental data used for the fit, see [45].10The HEPfit is available under the GNU General Public License (GPL) from https/

github.com/silvest/HEPfit.

https/github.com/silvest/HEPfit

https/github.com/silvest/HEPfit


Quantity Central value Uncertainty

∆ε3 0.62× 10−3 0.74× 10−3

∆ε1 0.71× 10−3 0.52× 10−3

δg(b)L −0.13× 10−3 0.61× 10−3

αs(MZ) 0.11850 0.00050

∆α5had(MZ) 0.02750 0.00033

MZ 91.1900 2.1× 10−3

mt 173.30 0.76mh 125.50 0.30

Table 4.1. Central values and the uncertainties of the pseudo-observables.

∆ε3 ∆ε1 δg(b)L αs ∆α5

had MZ mt mh

∆ε3 1 0.864 0.060 −0.013 −0.400 −0.039 −0.004 −0.003∆ε1 0.864 1 0.123 −0.011 −0.116 −0.124 −0.146 0.002

δg(b)L 0.060 0.123 1 0.110 −0.033 −0.003 −0.064 0.001αs −0.013 −0.011 0.110 1 0.003 0.001 0.001 0.001

∆α5had −0.400 −0.116 −0.033 0.003 1 0.004 −0.002 −0.001

MZ −0.039 −0.124 −0.003 0.001 0.004 1 0.000 0.000mt −0.004 −0.146 −0.064 0.001 −0.002 0.000 1 0.000mh −0.003 0.002 0.001 0.001 −0.001 0.000 0.000 1

Table 4.2. Full correlation matrix of the 8 pseudo-observables. The quantitiesαs and ∆α5

had are evaluated at the scale MZ .

Higgs mass is not calculable in all the cases we consider in this paper (exceptthe two-site model), it drops out from the list of constraints. Hence, we endup with only five pseudo-observables, namely ∆ε1,3, δg(b)

L , MZ and mt anda simpler correlation matrix of dimension 5. As the Higgs mass is calculablein the two-site model, we include it in that case. However, as the Higgs massis very weakly correlated to all the other pseudo-observables, we treat it asan independent constraint.

In addition to the constraints coming from the pseudo-observables listedin table 4.1, we also take into account lower mass bounds on the masses ofspin-1/2 and spin-1 resonances from direct searches. Including these directsearch limits in a proper way is beyond the scope of this work, so we have


3∈∆ emα

2W4 s

0.3− 0.2− 0.1− 0 0.1 0.2 0.3 0.4

1∈∆ emα1

0.2−

0.1−

0

0.1

0.2

0.3

0.4

b

Lgδ

0.005− 0 0.005

)b LgδP

(

0

200

400

600

Figure 4.6. The 68% and 95% marginalised probability regions of the epsilonparameters (left) and δg

(b)L (right). Since δg(b)

L is very weakly correlated tothe other pseudo-observables (see table 4.2), we show only its one dimensionalprobability distribution.

chosen to impose approximate limits, namely of 800 GeV and 1.5 TeV on thelightest fermionic and vectorial resonances respectively. However, as we willsee in the next section, the constraints from the EW fit are often strongerthan the direct search limits, hence these lower limits do not have noticeableimpact on our results.

In order to perform the numerical fit, we use the data given in tables 4.1and 4.2 and also the lower bounds. We follow a Bayesian statistical approachusing the BAT library [40] and present the 68% and 95% probability regionsof the posterior distributions of the relevant model parameters. We will notmake any effort to quantify the goodness-of-fit for the various compositeHiggs scenarios. Hence, in general the fit results of two different scenarioscan not be compared. However, as we know that in the limit ξ → 0 one getsback the SM, for very small ξ all the models are equally acceptable so that theposterior probability distributions of ξ give a good qualitative indication ofthe goodness-of-fit. In particular, models that allow larger values of ξ providebetter fit to the EWPO. In other words, the posterior distributions actuallycarry the information of ∆χ2 and not the absolute χ2, however, as all ourmodels have a smooth SM limit (i.e., ξ → 0) and the SM provides a very


b

Lgδ

0.002− 0 0.002 0.004 0.006 0.008

b Rgδ

0.01−

0

0.01

0.02

0.03

0.04

0.05

Figure 4.7. The 68% and 95% marginalised probability regions in the δg(b)L −δg

(b)R

plane.

good fit to the EWPO, the shape and range of the probability distributionof ξ can be taken as a qualitative measure of the goodness-of-fit.

4.3.2 New physics parameters and physical masses

In the presence of both, the fermionic 4-plet and the singlet as well as boththe spin-1 resonances, the ρL and ρR, there are in total 20 free parametersin our fit which are listed below,

{ξ, f, gel, g

′el, gρL/R , aρL/R , β2L/R , g1/4, y(L/R)(1/4), cd, cL/R, Λ

}. (4.66)

The global symmetry breaking scale f can be related to ξ using the HiggsVEV. In particular, we use the following tree-level relation,

GF = 1√2f2ξ

, (4.67)

where we take GF = 1.166371× 10−5 GeV−2. Moreover, one of the vectorialcouplings can be fixed using the tree-level relation

14πα

EM

= 1g′2el

+ 1g2

el+ 1g2ρL

+ 1g2ρR

, (4.68)


Parameter Range Parameter Range

ξ [0, 1] g1/4 [0.1, 7]

g′el [0.34, 0.4] y(L/R)(1/4) [0, 5.5]

gρL/R [1.5, 5.5] |cd| [−4, 4]

aρL/R [0.1, 6] |cL/R| [−3, 3]

|β2L/R | [−1, 1.5]

Table 4.3. Ranges of the model parameters used in our fit. We use flat priorsfor all the parameters.

where αEM = 1/128.96. We will use the above equation to fix gel. Thecut-off scale Λ, which only enters logarithmically in our formulas, will be setto three times the mass of the heaviest resonance. Thus, in the most generalcase, we end up with 17 free parameters:{

ξ, g′el, gρL/R , aρL/R , β2L/R , g1/4, y(L/R)(1/4), cd, cL/R}. (4.69)

Note that this choice is preferable because the decoupling limit of thecomposite sector (i.e., the SM limit) is achieved by setting ξ → 0 with allthe other parameters fixed to any value. In table 4.3 we summarise theapproximate ranges of the input parameters that were used in our fit. Weuse flat priors for all the parameters.

Since in the one-loop expressions for the EWPO we fixed the renormal-ization scale to the cutoff Λ, all our input parameters are understood as MSparameters at the same scale. Hence, also the tree-level masses obtained fromthese model parameters are understood as MS masses. In order to comparewith the top quark pole mass, we will use the SM QCD contribution to therunning of the top mass from the cutoff to the EW scale which is given by

mMSt (mt) = mMS

t (Λ)(αs(Λ)αs(mt)

) γ(0)m

2β0, (4.70)

where, γ(0)m = 7, β0 = 8, αs(mt) = 0.1083 and

αs(Λ) = αs(mt)

1 + β0αs(mt)

2π log (Λ/mt). (4.71)


In order to obtain the pole mass from the MS mass we use the one loopconversion formula,

mpolet = mMS

t (mt)(

1 + 43αs(mt)π

). (4.72)

The EW contributions to the top mass running and also the contributionsdue to the resonances will be neglected.

4.4 Results and discussion

In this section we present the results of our fits and discuss their qualitativefeatures. As discussed in the previous section, we construct the likelihoodfunction from the results of our tables 4.1 and 4.2 removing the entries cor-responding to αs, ∆α5

had and mh. Depending on the particular model understudy, we also remove the entries related to MZ and/or mt. For instance, ifvectorial resonances are not considered, MZ does not receive NP contribu-tions (at the tree level) and is automatically reproduced by setting g′el = g′

(and gel = g, as it would naturally follow from the constraint (4.68)). For thesame reason, in these cases we will also remove the parameter g′el from thefit. On the other hand, in the absence of fermionic resonances the particularmechanism which generates the top mass is not specified. In this case, we willassume it to have the correct value and remove the corresponding constraintfrom the fit.

In all the different scenarios, the effect of the nonstandard Higgs dynamicson the ∆ε1,3 parameters will always be present. These effects are completelydictated by the SO(5)/SO(4) symmetry breaking pattern, thus being thesame in all the cases. They were first computed in ref. [86] and are reportedin Eq. (B.1)-(B.3).

As a warm-up and to compare with the literature, we first perform thefit taking into account only the composite Higgs contributions. In this casewe set δg(b)

L = 0 and g′el = g′. Once the cutoff is fixed to a value Λ = 3TeV,the only remaining free parameter is the separation of scales ξ. Its posterioris presented in Fig. 4.8, which clearly shows the tight constraint, namelyξ . 0.075 at 95% CL.

In order to ease the comparison with the literature, in Fig. 4.9 we alsoplot the absolute χ2 and the ∆χ2 as a function of ξ. In the left panel the

4.4 Results and discussion 95

ξ0.05 0.1 0.15

)ξP

(

0

10

20

30

Figure 4.8. Posterior probability for ξ including only the effects of the nonlinearhiggs dynamics. The blue and the grey colors respectively indicate the allowedregions at 68% and 95% probability.

blue, grey and transparent regions satisfy ∆χ2 < (2.3, 6.18, 11.83), whichcorrespond to the 1-, 2- and 3-σ regions assuming 2 degrees of freedom andgaussian distributions of the observables as a function of ξ. Similarly, in theright panel we plot the absolute χ2 function.

4.4.1 Fermionic sector

In this section we investigate the constraints on the fermionic resonances.We start with the discussion of fermionic singlet for which a full analyticalcalculation is possible. The masses of the top quark and the heavy toppartner T are given by

m2T ,t

= 14f

2[2g2

1 + ξy2L1 + 2 (1− ξ) y2

R1

±√

4 (g21 + (1− ξ) y2

R1)2 + 4ξy2L1 (g2

1 − (1− ξ) y2R1)].

(4.73)

The contributions to the EWPO δg(b)L , T and S coming from loops of T


0.00 0.05 0.10 0.15 0.200

2

4

6

8

10

12

ξ

Δχ2

0.00 0.05 0.10 0.15 0.200

2

4

6

8

10

12

ξ

χ2

Figure 4.9. The ∆χ2 and the absolute χ2 as a function of ξ in the only compositeHiggs scenario. See text for more details.

and the top quark are given by

δg(b)L

∣∣Ψ1+t = 1

3 T∣∣Ψ1+t , (4.74)

∆T∣∣Ψ1+t = 3m2

t

16π2v2 sin2(2φ)[m2T

∆m2 log(mT

mt

)

+ 14

(tan2(φ)

m2T

m2t

− 1− 1cos2(φ)

)],

(4.75)

∆S∣∣Ψ1+t = g2

96π2 sin2(2φ)[

12

(6m4

T

m2T− 3m2

t

∆m6 − 3− 1cos2(φ)

)log(mT

mt

)

+3m2Tm2t

∆m4 −54

].

(4.76)

Where ∆m2 = m2T−m2

t , and the angle φ is related to the rotation matrix

which defines the mass eigenstates(tL, TL

)cos2(φ) =

2m2T− v2y2

L1

2∆m2 . (4.77)

In order to obtain the contributions to ∆ε1,3 parameters, the above con-tributions to ∆T and ∆S must be supplemented with the additional con-


tributions coming from top loops. It is found that the fermionic resonancecontributions to ∆S and ∆T are generically dominant in the models underconsideration, and the additional terms usually contribute with a smallercorrection. In fact, we have checked that removing these terms from thefit does not produce sizeably different results. They can be obtained fromequations (B.33), (B.34) using the following expressions for the top couplingswith EW gauge bosons (which, except for gRZ , deviate from the SM valuesby a shift of O(ξ)):

gLZ = 23g

c

(s2 − 3

4 cos2(φ)), gα = 1

3g2 s

ccos2(φ)

(1− 3

4 cos2(φ)),

(4.78)

gRZ = 23g

cs2, gβ = 1

3g2 s

ccos2(φ). (4.79)

The singlet contribution to the T parameter is often positive and big,while S can have both signs. The contribution to δg(b)

L is, on the other hand,always positive. Combination of these properties help the singlet scenario im-prove the overall fit to the data dramatically compared to the only compositeHiggs scenario, as can be seen from the posterior of ξ shown in Fig. 4.10. Itcan be seen that the 95% CL allowed region in ξ has now increased to ∼ 0.4as compared to ∼ 0.08 in the only composite Higgs case (see Fig. 4.8). Theposterior probability distributions for a few other relevant quantities are alsoshown in Fig. 4.10.

By expanding equations (4.73) at leading order in ξ, it is found thatscenarios in which the mass mixings y(L/R)1 are both too big or too small areforbidden by the mass constraints. On the other hand, EWPO forbid them tohave too different values. Hence the posteriors for both the couplings yL1 andyR1 are peaked around ∼ 2. We are not showing the posterior distributionfor the coupling g1 as it is loosely constrained by the fit. As a consequence,the mass of the top partner T can also vary in a big range, as shown in thebottom right plot of Fig. 4.10.

We now move to the other cases, namely the 4 and the 4⊕ 1 of SO(4),for which fully analytical calculations are not possible and a numerical ap-proach is in order. We find that the fermionic 4-plet scenario is stronglydisfavoured by the EWPO. This is because the 4-plet contributions to theoblique observables are rather large and have the same sign as the composite


ξ0.2 0.4 0.6

)ξP

(

0

5

10

L1y

0 2 4

)L1

P(y

0

0.2

0.4

R1y

0 2 4

)R

1P

(y

0

0.2

0.4

0.6

(GeV)Tm0 5000 10000 15000 20000

)T

P(m

0

0.05

0.1

0.15

3−10×

Figure 4.10. Posterior probability density functions for the case of a fermionicsinglet of SO(4) (see main text). The sharp drop for small values of yL1 isdue to the top mass constraint.

Higgs contributions. Our findings are qualitatively in agreement with [69].From the results of our fit we find a rather stringent 95% CL upper boundon ξ, ξ . 0.02. Consequently, all the fermionic resonances are constrained tobe rather heavy; with masses above 4 TeV at 95% CL.

In the presence of both the fermionic singlet and the 4-plet (i.e., a fullSO(5) vector), the fit shows a mild improvement over the only Composite


ξ0.05 0.1 0.15 0.2

)ξP

(

0

20

40

60

80

dc4− 2− 0 2 4

) dP

(c

0

0.1

0.2

0.3

(GeV)Bm5000 10000 15000 20000

(GeV

)5/

3X

m

5000

10000

15000

20000

Figure 4.11. Posterior probability density functions for the fermionic fiveplet (a4⊕ 1 of SO(4)) case (see main text).

Higgs case, as shown in figure 4.11. The interplay between the 4-plet and thesinglet contributions to the observables is quite non-trivial. The parametersy(L/R)4, g1/4 are loosely constrained, while there is a preference for y(L/R)1 >

1, mainly in order to avoid the T parameter being too negative. On theother hand, the S parameter contains a big logarithmically divergent termproportional to

(1− c2d

). In order to keep this term not so large, |cd| ∼ O(1)


is preferred, as can be seen from Fig. 4.11. All resonances are generallyconstrained to be rather heavy (& 2 TeV) with no clear mass hierarchybetween the 4-plet and the singlet.

As discussed in Ref. [81], the amount of tuning coming from the require-ment of correct EWSB for models with our fermionic sectors is estimated asO ((yi/gj) ξ), i.e. the naive estimate of O(ξ) is corrected by a factor (yi/gj).From naive theoretical expectations (see for example, sec. 4.1.1) such factoris expected to be smaller than 1, thus increasing the amount of fine tuning.However, in our analysis we did not use any theoretical prejudice for thechoice of the ranges of the input parameters, and from the results of our fitwe do not see any preference for (yi/gj) to populate values either lower orlarger than 1.

4.4.2 Spin-1 sector

In this section we will analyse models in which only vectorial resonances arepresent. In contrast to the contributions coming from fermionic resonances,in this case both a tree level and a one-loop term are present, which allows usto have more control on perturbation theory. As explained in section 4.3.2,our input parameters are taken at the scale Λ, but since important runningeffects of the tree level term arise, we have to ensure that the perturbativeseries controlled by the couplings gρr holds also at the scale Mρ, at whichspin-1 resonances are integrated out: formally, this is the scale at whichthe effects of resonances is computed. Hence, in addition to the constraintsmentioned in section 4.3.1, we also impose an upper bound on gρr (Mρ). Inaddition, couplings at both scales will be bounded from below, followingthe paradigm that the composite sector should remain reasonably stronglycoupled.11 Thus, in order to be conservative, we have chosen a range of[1.5, 5.5] for the couplings gρr at both the scalesMρr and Λ. Their β functionsare calculated in ref. [51] and are given by:

βgρr = g3ρr

[2a4ρr − 85192π2 − β2r

a4ρr − a

2ρr − 3

24π2 − β22ra4ρr

24π2

], r = L, R.

(4.80)11This choice is also driven by the fact that we have neglected terms of O

(g2

el/g2ρr

)in our calculations.


They are generally negative for aρr inside their PUVC bound, hence thecouplings will increase in the Infra Red. However, as we will see in the nextsection, things can easily change when also including fermionic resonances.

Even after imposing these bounds, it is found that, at the mass scale ofcomposite resonances, the one-loop contribution can still be of the same orderof the tree level one. This happens in particular for big values of aρr , whichenters quadratically in the one-loop expression of S (see eq. (4.48)). Thus, inorder to be as safe as possible, we also explicitly constrain the ratio betweenthe one loop term and the tree level one to values smaller than 0.5 at themass scale of resonances. However, this constraint is only implemented usingβ2r = 0, otherwise it would completely rule out the possibility of cancelingthe tree level term with β2r = 0.25. We are interested in exploring thiscase, as such possibility is only related to an accidental cancellation of treelevel term and has nothing to do with the higher-order terms of perturbationtheory.

It is relevant to note that, throughout this paper, we did not considerthe corrections to ∆ε3 coming from the parameters W and Y [29]. In factthey are suppressed by a factor g2/g2

ρr with respect to S, but since they alsoappear at tree level, their relative parametrical suppression with respect tothe one loop contribution to S is only a factor 16π2g2/g4

ρr ' (2.9/gρr )4. This

factor can be easily 1, hence tree level contributions toW and Y must also betaken into account. They have been calculated in chapters 2, 3, and amountto:

W∣∣ρL

= g4

4g4ρL

ξcos4(θ/2)a2ρL

(1−2β2L)2, W∣∣ρR

= tan4(θ

2

)W∣∣ρL, (4.81)

and the contributions to Y are obtained by exchanging cos4(θ/2)↔ sin4(θ/2)(we recall that ξ = sin2(θ)). These sin4(θ/2) trigonometrical factors producean additional ξ suppression to W

∣∣ρR

and Y∣∣ρL, which can be neglected.

As one could expect, the mass of the Z boson exhibits a strong sensitivityto the coupling g′el, and in addition, aside from a mild dependence on β2r ,no other parameter is sizeably relevant. As a consequence, the coupling g′el

always sits in a very narrow region centred around the physical coupling g′,and the MZ constraint can be satisfied with a slight tuning of g′el for anyvalue of the other parameters. For this reason, it does not play an importantrole in the fit.


We will start considering the case for which a single resonance is present.As far as the EWPO are concerned, it makes no difference wether we choose aρL or a ρR vector, as their final expressions are exacty the same at our level ofapproximation (i.e., gel = g′el = 0). On the other hand, since our calculationsfor the masses of resonances are done by numerical diagonalization, theyresum terms of all orders in gel and g′el making (as already observed, again,in the beginning of section 4.2) the mass splittings between neutral andcharged states slightly different for the left and the right vectors. However,the difference between the two scenarios is still very small, hence we will onlydiscuss one single-resonance case: for instance, we will focus on a ρL.

The expressions for the oblique parameters are in this case given byequations (4.40), (4.48), (4.81) and the text following Eq. (4.81). Since weare not including any fermionic resonance, we will assume the quark sectorto be exactly SM-like, so that no contributions to δg(b)

L are generated andthe top quark is assumed to have the correct mass.

As already commented, the positive tree level contribution to S can be re-duced by tuning the parameter β2L , while the one-loop contribution is mostlynegative. On the other hand, the contribution to T can be positive, and cansizeably compensate, or also overcome, the negative shift from the Higgs foraρ ∼ 1.5 or more. Remarkably, all these effects can appear simultaneously,making a single vectorial resonance a good candidate to accomodate EWPO.

Indeed, as shown in Fig. 4.12 the bound on ξ is weakened considerablyand values up to ∼ 0.4 are allowed at 95% probability. The parametersgρL and aρL are mainly constrained, respectively, by the requests on the RG-evolved gρL (MρL) and the ratio between the one-loop term and the tree-levelterm of S at the same scale. The parameter β2L , as expected, sits in a range∼ ±1 peaked around β2L = 0.25 12. Despite being parametrically sizeable,the contribution of W to ∆ε3 plays a minor role in the fit: this is due to aslight suppression 1/a2

ρL and also accidental numerical factors. Resonanceshave mass in the multi-TeV range with a small splitting between neutral andcharged states. Just like all the other cases, experimental lower mass boundsonly play a small role.

12Notice that (according to our assumption Λ = 3Mρ) in the central part of thisrange the estimate of the contributions coming from the local operators does not getthe enhancement mentioned in sec. 4.1.1.


ξ0.1 0.2 0.3 0.4 0.5

)ξP

(

0

5

10

Lρg

2 3 4

) LρP

(g0

0.2

0.4

0.6

Lρa1 2 3 4 5

) LρP

(a

0

0.2

0.4

0.6

(GeV)±Lρm0 5000 10000 15000 20000

) ± LρP

(m

0

0.1

0.2

3−10×

Figure 4.12. Posterior probability density functions for the case of an SU(2)Lvectorial triplet (see main text).

We will now move to the case in which both resonances are present. Inthis scenario the observable δg(b)

L is still set to zero, and the contributions tothe oblique observables are given by eqs. (4.41), (4.49). Since the contribu-tion to S is just the sum of the two single-resonance contributions, the samequalitative statements of the single-resonance case also apply here, while onthe quantitative side the contributions are of course typically bigger. On theother hand, in the T parameter an extra term (mostly negative and growing


ξ0.1 0.2 0.3

)ξP

(

0

20

40

Lρg

2 3 4 5

) LρP

(g

0

0.2

0.4

0.6

Lρa2 4 6

) LρP

(a

0

0.2

0.4

(L)0ρm

5000 10000 15000

(nL) 0 ρ

m

5000

10000

15000

20000

Figure 4.13. The posterior probability distributions for a few parameters andmasses in the scenario when both the SU(2)L and SU(2)R vectors are present.The superscripts (L) and (nL) correspond to the lightest and the next to thelightest resonances respectively.

like a4ρr ) arises, and the behaviour changes completely: in particular, while

in the single-resonance case values of aρr around ∼ 2 provided an overcom-pensation of the Higgs contribution, in this case the extra term grows quicklyand brings the overall resonance contribution to negative values.

Given these remarks, it turns out that the double-resonances case offers


less room to accommodate the EWPT. Still, as shown in figure 4.13, animprovement of the picture offered by the Higgs is still obtained, with asmall 95% probability tail extending up to ∼ 0.2.

4.4.3 Combining spin-1/2 and spin-1 resonances

In this section we will discuss a scenario where both the fermionic and vec-torial resonances are present. In particular, we will investigate whether thepresence of a single vectorial resonance can help relax the strong constraintson the fermionic 4-plet case. For definiteness we will consider the SU(2)Ltriplet of vectors. The fit results for the SU(2)R triplet do not show anysizeable difference compared to the SU(2)L one, except that the posterior ofcR is similar to that of −cL. This difference is driven by δg(b)

L , as can be seenby the sign difference between Eq. (4.60) and Eq. (4.61) (see also the textbelow).

In this case we still apply all the additional requests on the spin-1 sectordiscussed in section 4.4.2, namely the requests on gρL (MρL) and the ratiobetween the one loop and the tree level term of the pure vectorial contribu-tion. In this case the beta function of gρL gets a big positive contribution byfermion loops proportional to c2L:

βgρL = g3ρL

[2a4ρL + 48c2L − 85

192π2 − β2La4ρL − a

2ρL − 3

24π2 − β22L

a4ρL

24π2

], (4.82)

Notice that, differently from the typical situation in the absence of fermionicresonances, the inclusion of such contribution can easily make the β functionpositive.

The results presented in Fig. 4.14 show that a sizeable improvement isindeed present, with a 95% probability upper bound relaxed from ξ . 0.02to ξ . 0.2. We did not find any significant change in the posterior for theparameters yL4, yR4 and g4 with respect to the pure fermionic case. Afterall, the purely fermionic contributions to the oblique observables cannot betoo big, so the EWPO still require values of the fermionic parameteres thatminimise them: in particular the large negative T can be partially tamedby choosing yL4 < g4. On the other hand, the top mass constraint strictlyforbids values of yR4 . 1, just like yL1 for the singlet case (compare with fig.4.10).


ξ0.1 0.2

)ξP

(

0

20

40

60

Lρg

2 3 4 5

) LρP

(g

0.2

0.3

Lc2− 0 2

) LP

(c

0

0.1

0.2

0.3

±ρm0 10000 20000

(H) 2/3

ψm

0

10000

20000

Figure 4.14

The request gρL (MρL) ∈ [1.5, 5.5] forbids the beta function of gρL frombeing too big, which is reflected in a posterior distribution showing |cL| .2. The presence of cL increases the possibility for some cancellation in thebeta function, which results in a broader allowed range of gρL comparedto the purely vectorial case. For values of cL of O(1), the mixed fermion-vector contribution to S is typically subleading with respect to the purevectorial and fermionic terms. On the other hand, the contribution to δg(b)

L

proportional to cL is typically dominant over the fermionic term. We have


checked that for cL ∼ −1 δg(b)L is typically smaller (and negative) than for

cL ∼ 1, so that the former is slightly preferred by the fit.Differently from the pure vectorial fit, in this case the posterior distri-

bution for β2L is peaked at the slightly negative value β2L ∼ −0.1 insteadof 0.25. With this choice it is possible to increase the positive shift to theT parameter from the pure vectorial contribution. On the other hand, thepositive tree level contribution to the S parameter in Eq. 4.49 is increased bythis choice. However, the one-loop contribution is also bigger and negative,and its relative effect to the tree level one is increased by the fact that biggervalues of the coupling gρL are now allowed by the fit. We stress the factthat gρL is still a perturbative coupling in the window [MρL ,Λ] and we stillconstrain the ratio of the tree level to the one loop term of the pure vectorialcontribution at the scale MρL to be smaller than 0.5.

4.4.4 The two-site model limit

As discussed in sec. 4.1.2, the two-site model limits is obtained by enforcingthe relations,

yL1 = yL4, yR1 = yR4, cd = 0,

gρL = gρR , aρL = aρR = 1√2, (4.83)

β2L = β2R = 0, cL = cR = −1.

The two-site model is of phenomenological interest because the Higgspotential becomes calculable in terms of the Higgs VEV. The dominant con-tribution to the Higgs mass arises from the fermionic resonances, and aninteresting approximate relation holds [81]:

m2h

m2t

' Ncπ2

m2Tm

2T

f2(m2T −m2

T

) log(m2T

m2T

). (4.84)

In the above formula mT and mT

are understood as the top partnermasses before the EWSB, i.e. for ξ = 0. In the two-site model they are givenby:

mT = f√y2L + g2

4 , mT

= f√y2R + g2

1 . (4.85)


0 2 4 6 80

2

4

6

8

MT /f

MT∼ /f

ξ=0.05

0 2 4 6 80

2

4

6

8

MT /f

MT∼ /f

ξ=0.15

Figure 4.15. Scattered plots showing the points allowed at 95% probability bymt (in red) and mh/mt (in blue) for two values of ξ, ξ = 0.05 (left panel) andξ = 0.15 (right panel). The grey bands correspond to the area ruled out bylower bound on top partner masses from direct searches. See text for moredetails.

Equation (4.84) is a very good approximation of the exact numericalcalculation of the fermionic contribution to the Higgs mass from the potential.Notice that, on substituting the explicit expressions for the masses in theright hand side of Eq. (4.84) it becomes independent of ξ. We also remindthe reader that, according to our procedure, the top mass in the left handside of Eq. (4.84) should be identified with the MS top mass at the cut-off.

Differently from the benchmark models already discussed, in this case a(sizeable) tension with data already comes from mass constraints alone. Itis already well known that sub-TeV top partners are needed to generate alight Higgs without large tuning of ξ, and a big part of the sub-TeV range isalready ruled out by direct searches at the LHC.

In order to get a feeling of the different mass constraints, we throw a largenumber of random points in the (yL, yR, g1, g4) space and check how manyof them satisfy the individual constraint from mt and mt/mh. The allowedpoints are plotted in Fig. 4.15 in the plane of

(y2L + g2

4)1/2 and

(y2R + g2

1)1/2

(which correspond tomT /f andmT/f at ξ = 0 respectively) for two different

values of ξ.13 In this plane the points passing the mh/mt constraint sit in13For the purpose of this analysis we have used the O (ξ2) approximate expression


a localised hyperbola-shaped region; the bigger the Higgs mass the moredoes this region move towards the upper-right corner of the plot. The red(blue) points refer to the points that are allowed at 95% probability by themt (mh/mt) constraint. The grey area correspond to the region ruled outby requiring the lightest top partner to have a mass greater than 800 GeV(for simplicity, here we neglected the ξ dependent terms in the top partnermasse). It can be seen from Fig. 4.15 that already for ξ ∼ 0.15 the lowerbound on the top partner mass rules out almost all the points that satisfythe constraint from mh/mt.

We now show the results from our numerical fit in Fig. 4.16. In the upperleft panel we show the posterior of ξ when only the mass constraints (mt,mh/mt and direct search bounds on the top partner masses) are taken intoaccount. It is again clear from this figure that the mass constraints aloneare enough to provide an upper bound on ξ. We would like to remark atthis point that the Eq. (4.84) above only takes into account the fermioniccontributions and ignores the subdominant gauge contribution which hasbeen neglected in our analysis. Moreover, the running of the Higgs mass fromthe cut-off to the EW scale has also been neglected. In order to take intoaccount these missing contributions in a conservative way, we also performa fit by naively adding a 30 GeV additional uncertainty to the Higgs mass.This would grant the fermionic sector the freedom to produce a Higgs masssizeably bigger than the central value, thus relaxing the bound coming fromthe masses. The result of this fit is presented in the upper right panel ofFig. 4.16, which shows a clear improvement over the left plot. In the lowerpanel we show the same fits when the EWPO are also included. The requestsfrom masses and EWPO appear to be scarcely compatible, combining in atightly constrained final result. If no additional uncertainty on the Higgsmass is added (lower left panel), the 95% probability region extends up toξ ∼ 0.07, with a distribution shape strongly peaked around zero. When theadditional 30 GeV uncertainty to the Higgs mass is included the distributionshape is slightly relaxed, but still constrains ξ very tightly (the lower rightpanel).

for the top mass and fixed, for simplicity, mt (Λ) = 150GeV.


ξ0.05 0.1 0.15 0.2 0.25

)ξP

(

0

5

10

ξ0.05 0.1 0.15 0.2 0.25

)ξP

(

0

5

10

ξ0.05 0.1 0.15 0.2 0.25

)ξP

(

0

50

100

150

200

ξ0.05 0.1 0.15 0.2 0.25

)ξP

(

0

20

40

60

80

Figure 4.16. Fit results for the 2-site model. In the upper panel the posteriorprobability distributions for ξ are shown when only the mass constraints (mt,mh/mt and direct searches) are considered. In the right panel an additional±30 GeV uncertainty on the Higgs mass is included (see text for more details).The plots in the lower panel are the same as the corresponding plot in theupper panel once the constraints from EWPO are also considered.

4.5 Discussion 111

4.5 Discussion

In this chapter we have presented an extensive analysis of electroweak preci-sion constraints in specific composite Higgs models. We have considered theSO(5)/SO(4) coset including various combinations of fermionic and vecto-rial resonances. In particular, we have considered combinations of fermionicresonances living in (2,2) and (1,1), and vectorial resonances in (3,1) and(1,3) representations of SU(2)L⊗SU(2)R ∼ SO(4). We have used a simpli-fied effective Lagrangian describing the dynamics of such resonances at lowenergy and discussed some of its features following the approach of references[69] and [49].

We have calculated the one-loop contributions to the EWPO ∆ε1, ∆ε3(thus including the light physics contributions from the non-standard topquark and Higgs couplings) and δg

(b)L coming from the various resonances.

While the contributions coming from the vectorial resonances have been cal-culated at leading order in the elementary-composite mixings, the mixingeffects in the fermionic sector have been resummed by numerical diagonal-ization. We have also presented approximate analytical expressions and dis-cussed some general features of the results.

We have carried out a general NP fit to the EWPO applicable to themodels under consideration and, in general, to a broader class of modelsproducing negligible contributions to ∆ε2 and δg

(b)R . The results of this fit

(shown in tables 4.1 and 4.2) have been used to constrain the various NPmodels. We have adopted a bayesian statistical approach and, contrary tothe previous literature, performed a complete and systematic exploration ofthe parameter space using all the EWPO and the mass constraints simulta-neously.

Following our approach, at first we have studied the scenario in whichonly the nonlinear Higgs dynamics is considered and found a strong 95%probability upper bound ξ . 0.075 (or equivalently f & 900GeV). Althougha different statistical approach has been followed, our result is generally con-sistent with the previous studies. Going further, we analysed the effect ofthe various resonances alone and combinations thereof. We have shown thata scenario with a fermionic SO(4) singlet or a single spin-1 triplet (of eitherSU(2)L or SU(2)R) can considerably improve the agreement with data, re-


laxing the 95% probability bound on ξ to ∼ 0.4 (f & 400GeV). For thecases in which we include either both the spin-1 triplets or the combinationof the fermionic 4-plet and a single spin-1 (again either SU(2)L or SU(2)R),we find the 95% probability upper bound on ξ to be ∼ 0.2 (f & 550GeV).Finally, the two scenarios with only a fermionic 5-plet and only a fermionic4-plet turn out to be very tightly constrained with the 95% probability upperbounds on ξ to be ξ . 0.1 and 0.02 (f & 780 and 1700GeV) respectively.

We have also analysed the interesting case of the two-site model wherethe Higgs mass and the S parameter becomes fully calculable. We find that,in this case, it is rather difficult to obtain the correct Higgs mass once theexperimental lower bounds on fermionic resonances are taken into account.In fact, a fit with only the mass constraints (i.e., mt, mh/mt and the directsearch bounds) already constrains ξ to be ξ . 0.15 at 95% probability. In-clusion of the EWPO worsens the fit furthermore and the 95% upper boundon ξ reduces to ξ . 0.075.

As far as the masses of the fermionic and spin-1 resonances are con-cerned, we find that the EWPO do not constrain them severely, generallyallowing resonance masses below lower bounds from direct searches. Hence,fermionic top partners of mass around or below 1 TeV and spin-1 resonancesof mass around 2 − 3 TeV are consistent with our fits. As the constraintsfrom EWPO are not expected to improve considerably in the near future,the direct searches of resonances at the LHC [100] and the Higgs couplingmeasurements [7] will provide new constraints on these models.

113

Chapter 5

Direct searches of toppartners at the LHC

In this chapter we will present a direct search strategy for composite spin-1/2 resonances (top partners) at the LHC. It is based on the combinationof signals produced by both single and pair production processes for theresonances, and jet substructure techniques used to resolve their boosteddecay products. In Section 5.1 we will introduce the problem and motivateour proposals, while in Section 5.2 we begin with a brief description of asimplified model. In Section 5.3 we recast the constraint of a recent CMSsearch on X5/3 in the same-sign dilepton (SSDL) channel by considering thecontributions from both production processes and also the B resonance. InSection 5.4 we develop a search strategy focused on the one-lepton channel,considering both production processes and making use of jet substructuretechniques. We quantify the exclusion reach for the top partners with LHC8data. Finally we conclude with a discussion of the implications of the resultsfrom Sections 5.3 and 5.4.

5.1 Basic concepts and motivations

While a lot of effort has already been dedicated to setting limits on top part-ners in a model-independent way, we take a rather different approach. We

114 5. Direct searches of top partners at the LHC

customise the search strategy specifically for the lightest exotic top partnerX5/3. At the LHC the top partners can be either pair produced, throughQCD interactions alone, or singly produced, involving a model dependentcoupling. We show the relevant diagrams in Fig. 5.1. The relevant couplingfor the single production process is 1

gX X /V t , (5.1)

where V denotes an electroweak gauge boson (W±, Z) and X can be any toppartner. The model dependence of the single production process is encoded inthe couplings gX which are functions of the model parameters that determinethe mass spectrum and the particles’ interactions. However, we emphasisethat the model-dependence we are exploiting is rather minimal, and as suchrequired in a large class of composite Higgs models. The relative size ofthe two production processes entirely depends on the top partners’ massesand the couplings gX probed in the single production process. Compared tothe pair production process the single production cross section suffers fromthe exchange of a virtual electroweak gauge boson and a gluon splitting toheavy quarks. This results in a larger pair production cross section if the toppartner is light. However, for heavier top partners the mainly gluon-inducedpair production cross section drops quickly and the single production modebegins to take over.

While the prospects for the single production process at the LHC arediscussed in [83, 36, 110, 109, 78, 55, 14] (see [12, 52, 13, 83, 72, 68] forthe double production process), most experimental top partner searches arebased on the pair production process because of its dominant signal rates atlower masses and promising kinematic features to overcome the large Stan-dard Model backgrounds (see [101, 104, 103, 102, 106, 108, 107, 105] forrecent analyses on top partners using LHC8 data). Based on those searches,currently the limit on top partner masses is close to or has already passed thecrossing point of the single vs double production cross section. For massesmuch heavier than the top quark the top partner’s decay products are neces-sarily boosted. While this is a general feature when a heavy resonance decays

1There are also interactions involving the bottom quark, X /V b. However, we willnot consider those interactions in this chapter.

5.1 Basic concepts and motivations 115

into two much lighter resonances, the impact of the related kinematics on thetop partner searches is dramatic.2

In the mass range of interest, the increasing relevance of the single pro-duction process extends to an increased sensitivity on the model parameters3.This fact and its implications on the minimal composite Higgs scenario wererecently discussed in detail in Ref. [55] where the same-sign dilepton andtrilepton searches performed by CMS and ATLAS were recast. The authorsof [55] propose a tailored search for the single production process exploitingforward tagging jets. While this search can enhance the sensitivity to specificparts of the parameter space, we take a different path in this work. We keepour search as inclusive as possible to both the single- and pair-productionprocesses. By combining both signal processes we enhance the total signalrate while maintaining a good statistical significance. In other words, we cor-relate the rates of several production mechanisms to increase the sensitivityon the model parameters (process correlation). Another promising approachwe advertise to constrain the model further is to correlate the contributionsof several resonances to the same search (particle correlation). The numberof combined contributions to the same final state from different top partnerscrucially depends on their mass spectra and branching fractions which aredetermined by the same set of model parameters that determine the cou-plings gX of the single production process. If interference is negligible, thetotal rate of the two types of processes (single and double production) foreach top partner (if multiple candidates exist) can be expressed as,

Ntotal =∑X

(NX

pair + g2X N

Xsingle(gX = 1)

). (5.2)

N denotes the number of events for a given integrated luminosity and theindex X runs over the top partners. To facilitate a scan over the model de-pendent parameter gX we explicitly factor it from the single production rate.We will argue in this work that our proposed search strategy can significantly

2A top partner search exploiting boosted techniques in the single production processwas studied in [78].

3In the pair production process the model parameters can be constrained by mea-suring the branching fractions of top partner decays. In case the top partner decaysto certain final states with unit coupling, it becomes rather insensitive to the modelparameters that define the structure of the composite Higgs model.


Figure 5.1. The diagrams of the double production and single production pro-cesses. We show only the dominant diagrams. t is a collective symbol to referto either top or anti-top. Similarly X denotes any top partner of interest, andit collectively refers to either top partner or anti-top partner.

improve the limit setting on top partner masses while simultaneously probinga large part of the model’s parameter space.

All signal final states with sufficient overlap are of potential interest forsuch a cross correlation. More precisely, after the decay of the two resonancesX5/3 or B to tW , in both the single and double production processes, a ttWsystem will emerge, see Fig. 5.1. Naturally, a signature that can be used todisentangle the signal from the SM backgrounds is the clean final state withsame-sign dileptons and jets. However, particularly at the LHC with 8 TeVcenter-of-mass energy an economical use of the signal rate is crucial. We willdemonstrate that the final state with one lepton and jets has the potentialto exceed the exclusion limits set by the same-sign dilepton search.

The absence of any excess in existing new physics searches leads us toconsider heavier top partners. Electroweak scale resonances, W/Z/h/top,decayed from those heavy top partners are necessarily boosted, and as a re-sult all the final state particles of those boosted tops and electroweak bosonsare collimated in the laboratory frame. As we are in a transition region be-tween the boosted and unboosted regime, many standard search strategiescease to work well. Relatively simple observables like the multiplicity of jetsproved to be a powerful discriminator between signal and SM backgrounds,yet when the decay products are collimated in a narrow opening angle over-lapping radiation will spoil the aimed-for jet-parton matching and jet count-ing becomes much less effective. A simple example is the reconstruction of aW boson. AW boson with small transverse momentum decays to two widelyseparated jets, counted as two, whereas for a highly boostedW boson, whose

5.1 Basic concepts and motivations 117

transverse momentum is much bigger than its mass (pT,W �MW ), the twojets merge into a single jet which would be counted as one. This obscuresthe two-prong nature of the W boson, as opposed to the one-prong QCD-jet.Here jet substructure techniques can be helpful to recover sensitivity in dis-criminating the W jet from QCD jets. Those techniques organise the energydistribution of jet constituents such that they correctly identify two hardobjects in a single jet, while efficiently rejecting QCD-like jets. By exploitingthe substructure of a jet the traditional observable jet multiplicity, Nj , canbe consistently extended to Ncon, the number of jet constituents4, defined as

Ncon =∑n=1

n ·Nn−prong. (5.3)

Nn−prong is the number of boosted n-prong jets, tagged by n-prong taggers,and N1−prong is just the number of traditional jets. In this classification,top taggers [32, 99, 76, 18, 17, 91, 60, 98, 97, 93] (see also [8, 22]) are 3-prong taggers and W/Z/h-taggers [94, 34, 35, 18, 17, 60, 77, 95, 53, 98,71, 96, 16, 54] are 2-prong taggers. A crucial advantage of this approach isthat it continuously interpolates between the boosted and unboosted regime.Therefore sensitivity for the signal is restored over a maximum range in phasespace.

2- and 3-prong taggers do not only count the number of subjets insidea jet but also impose kinematic requirements, e.g. the reconstruction of thecorrect top or W mass. Thus, we find that apart from Ncon the numberof reconstructed top quarks Ntop and W bosons NW using jet substructuretechniques provide a strong handle in disentangling the signal from the SMbackgrounds. In general top taggers do not make use of b-tagging. However,b-tagging can be applied in addition to a top tag [72], though we will notpursue this possibility.

We will investigate a cut-and-count analysis based on Ncon, HT , NW andNtop in the context of top partner searches.

4This variable is already being explored in a top partner search by CMS [108].


5.2 A Simplified Model

For the discussion of LHC phenomenology of top partners we consider a sim-plified model of the minimal composite Higgs scenario based on SO(5)/SO(4) 5.We assume that tR is a completely composite chiral state and only qL =(b, t)L is realised as partially composite. To avoid large corrections of theT -parameter and a large ZbLbL interaction we assume the custodial symme-try is preserved [10]. For simplicity we consider the case where top partnersbelong to the fourplet, Ψ ≡ 42/3, embedded in 52/3 of SO(5). The four com-ponents of the fourplet are T,B,X2/3, X5/3. Here 2/3 refers to the chargeof an extra U(1)X , introduced for the correct assignment of the SM hyper-charge Y = T 3

R + X. LHC phenomenology for those top partners has beendiscussed in [55]. The leading order Lagrangian in Callan-Coleman-Wess-Zumino (CCWZ) [46, 41] language is given by

L = Lkin − Ψ/eΨ−MΨΨΨ

+ i c1(ΨR)iγµdiµtR + y f (Q5L)IUI iΨi

R + y c2 f(Q5L)IUI 5tR + h.c. ,

(5.4)

where Lkin includes the covariant kinetic terms of qL, tR and Ψ (see [55]for details about the conventions used in Eq. 5.4). The pNGB Higgs isparameterised by the matrix U in Eq. 5.4, defined as U ≡ exp[i

√2/f ΠiT

i].T i=1···4 are broken generators parameterising the coset of SO(5)/SO(4) andf is the associated symmetry breaking scale. One can construct a tensorU†DµU which decomposes into two components, dµ = diµT

i and eµ = eaµTa

as used in Eq. 5.4, where T a are unbroken generators. The simplified modelin Eq. 5.4 leads to a non-tunable structure of the Higgs potential at leadingorder (see [87] for other possibilities). While top partners can be embeddedin a bigger representation of SO(5) [55] or be part of a less minimal model,the physics captured by the simplified model in Eq. 5.4 is likely to be a subsetof them. The model is defined by five parameters in addition to the ones ofthe SM. One of them is fixed by the top mass constraint, leaving eventuallyonly four free parameters

5While the simplified top partner models of generic composite Higgs scenarios havebeen discussed in [48, 52, 83], the interpretation of our result in those models is straight-forward as the couplings and mass spectrum take simple forms.

5.2 A Simplified Model 119

First of all, X5/3 can not mix with any other state due to its exoticcharge. Hence its leading-order physical mass is expressed by MΨ in Eq. 5.4.The bottom-type quark sector takes the form of a 2× 2 mass matrix in thebasis of (b, B), and the mass of the heavy eigenstate is derived from Eq. 5.4as

MB =√M2

Ψ + y2f2 . (5.5)

Eq. 5.5 holds even after electroweak symmetry breaking. The mass spectrumof the up-type quark sector can be obtained by diagonalising the 3× 3 massmatrix in the basis of (t, T, X2/3). It was pointed out in Ref. [55] that themass matrix can be made to be block-diagonal, by an appropriate field redef-inition of T and X2/3, such that the mass of X2/3 can be expressed by MΨ.In other words, in this model the masses of X2/3 and X5/3 are degenerate.The masses with non-trivial dependence on the model parameters are thoseof t and T . They are approximately given by 6

Mt ∼c2 yf√

2gΨ√g2

Ψ + y2

√ξ

[1 +O

(y2

g2Ψξ

)],

MT ∼√M2

Ψ + y2f2[1− y2(g2

Ψ + (1− c22)y2)4(g2

Ψ + y2)2 ξ + · · ·],

(5.6)

where gΨ ≡MΨ/f and ξ ≡ (v/f)2.The top partners X5/3 and B are in this model particularly interesting.

The X5/3 is the lightest top partner and it decays to tW with unit coupling7

due to its exotic charge. If this type of composite Higgs model is indeed re-sponsible for electroweak symmetry breaking and is kinematically accessibleat the LHC, X5/3 is the most promising candidate to be discovered first. Themass limit on X5/3 automatically extends to other top partners via the massrelations of Eqs. 5.5 and 5.6, constraining all top partners indirectly. Forinstance, while the search for X2/3 is more difficult due to its predominantdecay to tZ or th, and their subsequent mostly hadronic decays, one can geta stringent bound from the limit on X5/3. According to Eqs. 5.5 and 5.6 themass hierarchy between T , B and X5/3 is controlled by yf . By measuringthe masses of X5/3 and B the overall mass scale of the top partners MΨ and

6Note, in our numerical evaluations in Secs. 5.3 and 5.4, we use exact expressions.7In a more general setup, this property can be relaxed such that X5/3 also decays

to Wq (q = u, c) [37].


the symmetry breaking scale f can be constrained simultaneously. Remark-ably, both states decay predominantly into the same class of particles8: atop quark and a W boson. Therefore, the same search strategy can be ap-plied to both particles simultaneously irrespective of whether they are singlyor pair produced. When including contributions from both particles in onesearch, the reduced signal rate for the heavier B can be compensated by thecontribution from X5/3, assuming MX5/3 not being far from MB , such thata sizeable total signal rate is maintained. Therefore, in the following we willfocus on the two most accessible top partners9 X5/3 and B.

The single production process is directly sensitive to the details of thestrong dynamics in our simplified model. The coupling in Eq. 5.2 is derivedfrom the model’s free parameters,

gX = gX(f, y, c1, c2, MΨ) , (5.7)

where one of the parameters, for instance c2, can be removed using the topmass constraint. y controls the mixing between elementary and the compos-ite states, which leads to the partial compositeness of the left-handed topand bottom quarks. y explicitly breaks the SO(5) symmetry, generating aleading contribution to the Higgs potential. c2 is expected to be O(1). To-gether with other parameters it sets the top mass. c1 is also expected tobe O(1) and it constitutes the dominant interaction for the single produc-tion process with an associated top. While we use the exact expressions forthe coupling constants in Eq. 5.1 in the unitary gauge for numerical evalua-tion, the dominant contribution to the coupling in Eq. 5.7 can be estimatedby the Goldstone boson equivalence theorem. For instance, the vertices ofφ+X5/3LtR and φ−BLtR using mass eigenstates are [55]

gX5/3 ∼√

2 c1MΨ

f,

gB ∼√

2 c1√y2 + (MΨ/f)2 − c2

y2√y2 + (MΨ/f)2

.

(5.8)

8In the composite Higgs model, defined by Eq. 5.4, the branching fraction of B totW is dominant as th and tZ modes are forbidden [55].

9While there can be an extra contribution from X2/3 to the one-lepton channelwhen decaying to tZ, the branching fraction of tZtZ to one lepton is roughly twotimes smaller than for the tWtW system. Further, BR(X2/3 → tZ) ' 0.5 in thismodel.

5.3 Reinterpretation of existing searches 121

700 750 800 850 900

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

mX5�3HGeVL

gX

700 750 800 850 9000

50

100

150

200

250

300

mX5�3HGeVL

∆m=

mB-

mX

5�3HG

eVL

Pair production only

X5�3 and B

0.5 1

CMS SSDL

on X5�3

Figure 5.2. Recasted CMS SSDL search. Left: the single production processis added such that the limit on the mass extends into a two-dimensionalexclusion (MX , gX)-plane. Right: the contribution from the bottom-like toppartner B is added to those of X5/3. Plot is restricted to only pair productionprocesses of X5/3 and B for simplicity. BR(X5/3 → tW )=1 is assumed, andBR(B → tW )=0.5 (dashed black) and 1 (solid black) are plotted. Red linein the right panel indicates the bound on X5/3 set by CMS SSDL. The reddashed line in the left panel indicates O(φ+Xt) ∼ gX(MX/MW ) = 3.5 .

The φ± are the Goldstone bosons eaten byW±. The couplings in Eq. 5.8 arebasically Yukawa couplings of the composite states, implying that the singleproduction processes can be sizeable.

5.3 Reinterpretation of existing searches

Due to the importance of top partners for the naturalness problem, theirsearch is part of the core program of ATLAS and CMS. In this section,with Eqs. 5.5-5.7 in mind, we will discuss the impact of existing searches,performed by ATLAS and CMS, on MX5/3 and MB . We will demonstratethat recasting searches in terms of a combined measurement of X5/3 and Bin the single and double production channels, as outlined in Eq. 5.2, directlyresults in an improved limit on the simplified model’s parameters.

The four processes we exploit to perform this task are pp→ X5/3X5/3 →W+tW−t, pp → BB → W−tW+t, pp → X5/3t + h.c. → W+tt + h.c. and


X5/3 Mass (GeV) 2SS leptons m(ll) Veto Ncon ≥ 5 HT ≥ 900

Pair production of X5/3 (→ tW )700 55.7 49.5 29.4 20.6800 20.1 18.3 11.6 9.48900 7.88 7.22 4.59 4.081000 3.33 3.1 2.01 1.89

Single production of X5/3 (→ tW ) with an associated t700 579.5 545.8 146.9 45.4800 388.6 365.8 101.1 38.8900 250.9 234.4 65.1 30.41000 173.8 163.4 48.9 26.3

Table 5.1. Summary table of the expected signal events for the pair produc-tion (as the validation of our analysis) and the single production processes.Branching fraction of the pair production (single production) to same-sign(SS) leptons is 0.21 (0.11). The expected signal events of the single produc-tion process assumes gX = 1.

pp→ Bt+h.c.→W+tt + h.c. All of the processes can give rise to same-sign(SS) dilepton signatures.

CMS recently published a search [106] which explores many channelsthat include a varying number of leptons (i.e. one lepton, SS dileptons, twotypes of opposite-sign (OS) dileptons and trileptons), to derive the limitsfor various final states, e.g. bW , tZ, tH, in pair produced vector-like toppartner signals. While their SS dilepton search will pick up our signals, amore tailored SS dileption search targeted on X5/3, using full LHC8 data,has been shown in [108]. To date this search yields the strongest bound onthe top partner X5/3 using 19.6 fb−1 of LHC8 data. The limit is set usingthe pair-produced X5/3, each of them decaying to tW with unit branchingfraction. This search not only demands same sign dileptons, but also exploitsjet substructure techniques to efficiently capture boosted top quarks or Wbosons. The W can either originate from the top or directly from X5/3. Thesearch imposes the cuts HT > 900 GeV and Ncon ≥ 5, resulting in a limit ofMX5/3 > 770 GeV at 95% CL.

5.3 Reinterpretation of existing searches 123

Starting from the findings of [108] we can improve the limit and extractmore information on the model parameters. While the cuts in this searchare rather exclusive to the doubly produced X5/3, a significant fraction ofevents where X5/3 is produced in association with a top quark still passesthose cuts, hereby contributing to the total signal rate. We show the numberof signal events passing the different event selection cuts in Table 5.1. Bycombining the single and double top partner production according to Eq. 5.2,the separate limits on either the model dependent coupling gX or the toppartner mass MX5/3 can now be unfolded on a two-dimensional exclusionplane.

We run the same analysis on the signals of the single and pair produc-tion processes, generated by MadGraph5 v1.4.7, interfaced with our ownsimplified UFO [56] model, processed through PYTHIA8 and clustered usingFastJet v3.0.3 [38]. We normalise signal cross sections to their NNLO val-ues, derived by HATHOR [15]. We validate our procedure by comparing thepair production signals with the CMS results, see Table 5.1. In Fig. 5.2 weshow that the limit on MX5/3 can be improved to ∼ 830 GeV for gX = 0.35this way. As discussed in Sec. 5.2 the coupling gX is a function of the modelparameters which determine the mass spectrum of the top partners. There-fore, an extension of the exclusion region for MX5/3 in comparison with thelimit from pair production alone constitutes a strong discriminator betweenthe models.

So far we only studied the impact of the combination of the single anddouble X5/3 production processes on the exclusion plane. As mentioned inSec. 5.2, by including the bottom-like top partner B we can access orthogonalinformation on the model parameters. As long as the mass hierarchy betweenB and X5/3 is small, they can both contribute to the same final state with asizeable rate. We only consider the double production processes of X5/3 andB. The samples of B are simulated and processed in the same way as theones for X5/3. While the inclusion of B introduces two more free parameters,namely its mass MB and the branching ratio to tW , we demonstrate thiseffect on the limit of MX5/3 for a varying mass gap δm ≡MB −MX5/3 withtwo different choices of the branching ratio. We show the results in the rightpanel of Fig. 5.2.

One can further include the single production processes of X5/3 and B,


hereby introducing sensitivity on two more couplings. We will discuss thisoption in Section 5.5 in more detail.

In addition to the SS leptons, the one lepton channel is also very sensitiveto our signal. Multivariate techniques are used in [106] , i.e. a boosteddecision tree, to infer the limit from the one lepton final state. The reportedexpected (observed) limit for BR(tZ)=1 is 689 (644) GeV. If one applies thissearch to the pair produced X5/3 with subsequent decay to tW , the limit canbe even stronger due to the roughly two times larger signal rate. A potentialimpact on the limit for X5/3 using the one lepton analysis of [106] can beseen in the right panel of Fig. 5.2 (see Fig. 5.8 as well) where increase ofthe signal rate by a factor of two improves the bound on top partners byroughly 80 GeV. One would naively expect a similar effect when applied to[106], and this would make an expected limit comparable with limits of theSSDL analysis in [108]. Similar argument applies to the singly produced toppartners. While we do not attempt to recast the search channels in [106] togive a reinterpretation in terms of the (mX , gX) exclusion plane, performingsuch an analysis would be straightforward. Instead we will discuss the onelepton search in the context of a different search strategy in Section 5.4.

5.4 Boosting searches using jet substructure

While for our signal processes final states with same-sign dileptons and trilep-tons are certainly the cleanest to exploit, in this Section we will focus on theone lepton channel. By changing from the SS dileptons, discussed in Sec. 5.3,to the one lepton channel we increase the rate in the pair production processby a factor six and in the single production process by a factor nine. Conse-quently the relative sensitivity of the single production process with respectto the pair production process is increased as well. A lepton in the one lep-ton channel can be produced from either of the W s. The other two or threeW s will decay hadronically with accompanying b-jets if they originate fromtop quarks. The hadronic tops and W s from the heavy top partner’s decay,as opposed to those in SM backgrounds, are necessarily boosted, while theones produced in association with the top partner (in the single productionprocess) are not. There is only one source of missing transverse momentumin the final state. Therefore by requiring the lepton and the neutrino to

5.4 Boosting searches using jet substructure 125

reconstruct the W mass MW = Mνl one can reconstruct the leptonic top.This way we are able to reconstruct the entire ttW subsystem, from whichwe can reconstruct the top partner mass.

The single and double production processes, both for X5/3 and B, sharea common feature: they contain a ttW subsystem which can lead to a finalstate with one lepton and at least six partons. This ensures that the signalwill likely populate the high-valued Ncon region. The ttW subsystem of thesignal is produced from the decay of heavy top partners, as opposed to thosefrom non-resonant QCD processes, and thus the pT -summed HT is roughlyproportional to the heavy top partner masses. Thus, Ncon and HT are veryeffective observables to separate the signals from backgrounds.

The signal samples are simulated as in Sec. 5.3. The major backgroundsin our one lepton analysis include tt+jets matched up to two jets andW+jetsmatched up to four jets using MLM matching [23] with R = 0.4 and pT =30 GeV. Conservatively, we apply a K-factor of 2 to both backgrounds. Theevents are not further processed to take into account detector effects.10 Wecheck other irreducible backgrounds such as ttW+jets, and we find that thoseare subleading.

5.4.1 Cut and count analysis for l+jets final state

The events are triggered by one isolated lepton. A lepton is considered iso-lated if the surrounding hadronic activity within a cone of size R = 0.3 sat-isfies pT (l)/(pT (l) + pT (cone)) > 0.85. While a mini-isolation criterium [92],i.e. a pT dependent isolation cone-size, which is already being used in AT-LAS [102], can result in a higher reconstruction efficiency for leptons fromboosted resonances, we find that the standard isolation criterium we applyretains in the top partner mass range of interest a good efficiency of ∼ 85%.The hadronic activity is organised such that objects with more substructureare looked at first, followed by object with less substructure in the ttW+

10To our knowledge, no publicly available detector software has a validated smearingprofile for subjets or jet sub-structure observables. Therefore including a detectorsimulation can not ensure to improve on the precision of our analysis. However, inmany MC-data comparisons at LHC7 a good agreement for jet substructre observableswas found [6].


jN5 6 7 8 9 10

Norm

aliz

ed r

ate

0

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4)≥ j

> 800 GeV, NT

(HjN 4)≥ j

> 800 GeV, NT

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conN

5 6 7 8 9 10

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alized r

ate

0

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0.15

0.2

0.25

0.3

0.35

4)≥ j

> 800 GeV, NT

(HconN 4)≥ j

> 800 GeV, NT

(HconN

Figure 5.3. Normalized distributions of the number of traditional jets (left) andof the constituents (right) for signals and backgrounds in l + jets channel.Signals are the pair (black solid) and single (black dashed) production pro-cesses of 800 GeV X5/3, and the backgrounds are tt + jets (solid Red) andW + jets (solid blue). Events in the plots were restricted to those satisfyingHT > 800 GeV and Nj ≥ 4. The area of each curve over the full range ofNj , Ncon is normalized to 1. We only display 5 ≤ Nj , Ncon ≤ 10.

conN

4 5 6 7 8 9 10 11

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alized r

ate

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(HconN > 800 GeV)T

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(GeV)TH600 800 1000 1200 1400 1600 1800 2000

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ate

/100G

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0.22

7)≥ con

(NTH 7)≥ con

(NTH

Figure 5.4. Left: Number of reconstructed constituents for the X5/3 top partnerwith 800 (black) and 1000 (red) GeV from pair (solid) and single (dashed)production processes. Right: HT distributions of the X5/3 with 800 GeVfrom pair (solid black) and single (dashed black) production processes. Thebackground distributions are tt + jets (solid red) and W + jets (solid blue).


topN0.5 0 0.5 1 1.5 2 2.5 3 3.5

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ed r

ate

0

0.1

0.2

0.3

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> 1000 GeV)T

7, H≥ con

(NtopN > 1000 GeV)T

7, H≥ con

(NtopN

WN

0.5 0 0.5 1 1.5 2 2.5 3 3.5

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alized r

ate

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(NWN

top+NWN0.5 0 0.5 1 1.5 2 2.5 3 3.5

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aliz

ed r

ate

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0.2

0.3

0.4

0.5

> 1000 GeV)T

7, H≥ con

(Ntop+NWN > 1000 GeV)T

7, H≥ con

(Ntop+NWN

Figure 5.5. Normalized distributions of the boosted tops (left), boosted W s(middle), and boosted tops + W s (right) in the l + jets channel. Signals arepair (black solid) and single (black dashed) production of 800 GeV X5/3, andbackgrounds are tt + jets (solid red) and W + jets (solid blue). Events areselected requiring Ncon ≥ 7 and HT > 1000 GeV.


anything system. We cluster the event into R = 0.8 “fat-jets” using the Cam-bridge/Aachen algorithm [58, 111]. We apply the HEPTopTagger [90, 91] onevery fat-jet in the event to look for hadronic tops and remove them fromthe list of fat jets. Next we run the so-called BDRS tagger [35] over theremaining fat-jets to look for hadronic W s. If the invariant mass of threefiltered subjets meets the W mass requirement, mreco

W = (65, 95) GeV, weconsider the jet to be a hadronic W -jet and remove it from the event. Alltagged hadronic top-jets or W -jets are required to be within |η| < 2.5. Wecollect all constituents from the remaining fat-jets that were not top or Wtagged, and we recluster them into R = 0.5 anti-kT [39] jets. We accept onlyanti-kT jets with pT > 35 GeV and |η(j)| < 4.5. Eventually we count thenumbers of boosted tops and W s, and anti-kT jets and sum up the numberof constituents Ncon according to Eq. 5.3.

We show in Fig. 5.3 the distributions of the traditional jet multiplicity,Nj , and the number of constituents, Ncon, for X5/3 with 800 GeV and back-grounds respectively. The change of shapes between Ncon and Nj in thesignal clearly indicates that using jet substructure methods is necessary toresolve the decay products of the top partners. In the background this effectis much less pronounced.

In Fig. 5.4 we compare the Ncon distributions of X5/3 with 800 and 1000GeV. The distributions are rather insensitive to the top partner masses, andthey tend to have at least six (sub)jets. While single production is expected tohave fewer constituents than pair production in general, the two distributionswhen restricted to the high HT region become very similar. We find thatimposing Ncon ≥ 7 is very effective to achieve a good statistical significance.We choose the HT cut such that we keep ∼90% of the pair produced eventsafter Ncon ≥ 7 is imposed. The same HT cut keeps only ∼40-60% of thesingle production events for the top partner mass range of interest. However,its smaller efficiency is compensated by the higher initial cross section in thesingle production process for heavy top partners.

We present the distributions of the boosted W -jets and top-jets, taggedusing jet substructure techniques, in Fig. 5.5. Imposing cuts on the totalnumber of W -jets and top-jets is more effective than cutting on NW or Ntop

individually. The variable NW + Ntop is insensitive to whether a top istagged as top-jet or only the W -boson of the top is tagged as W -jet. Thus


we are less sensitive to the choice of the fat-jet’s cone size11. We demandNW + Ntop ≥ 2 which suppresses the backgrounds while keeping a handfulof signal events of both single and pair production processes. One mightwant to try NW + Ntop ≥ 3, as suggested by the rightmost plot in Fig. 5.5.While this more aggressive cut can significantly improve S/B, we lose thesensitivity to the single production process. We will not pursue this option,though we point out that at 14 TeV center-of-mass energy, with increasedsignal cross sections, this cut may improve the sensitivity of the search. Afterapplying the outlined cuts on both pair and single production events as wellas the backgrounds, we sum the expected signal events according to Eq. 5.2.The estimated exclusion plot we show on the left panel of Fig. 5.7.

5.4.2 Top partner mass reconstruction

X5/3 and B decay both into a boosted top and W boson. Sometimes thetop quark is not boosted enough to capture all decay products in a singlejet and the b-jet and W are reconstructed separately. Hence we iterativelypair each hadronic W -jet with any of the anti-kT jets within ∆R < 1.5 tolook for a top candidate. If the invariant mass of the pair falls into the topmass window mreco

top = (150, 200) GeV and it satisfies pT (Wj) > 200 GeVand |η(Wj)| < 2.5, we count the jet pair as a top candidate. While ourcut-and-count analysis does not care where the isolated lepton comes from,the reconstruction of the resonant top partner mass depends on its origin.Leptons are either produced from a W originated in a top partner decay ora W from an associated top quark in case of the single production process.In order to cover the maximum number of possibilities we reconstruct theleptonic W and top quark as well. The leptonic W is reconstructed byrequiring M2

νl = M2W . We resolve the twofold ambiguity in calculating the

longitudinal component of the neutrino momentum by choosing the solutionwhich reconstructs the top mass best. After reconstructing the leptonic Wwe treat them on equal footing with the hadronic W: the leptonicW is pairedwith any of the remaining anti-kT jets within ∆R < 1.5, and we considerthe pair a leptonic top if it satisfies the same pT and η requirements as the

11We explicitly studied the effect of the jet radius on NW + Ntop and find similarresults.


(GeV)Xrecom

200 400 600 800 1000 1200 1400 1600

Norm

aliz

ed r

ate

/200G

eV

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

= 800 GeVX

m = 800 GeVX

m

(GeV)Xrecom

200 400 600 800 1000 1200 1400 1600

1E

vents

/200G

eV

, 20 fb

0

5

10

15

20

25

30

35

40

Figure 5.6. The invariant mass of the reconstructed top partner. Left: normal-ized distributions of the signal’s pair (solid black) and single (dashed black)production processes, and backgrounds, tt + jets (solid red) and W + jets(solid blue). Right: stacked distributions of the signal’s pair (black) and single(grey) production processes, and backgrounds, tt + jets (red) and W + jets(blue), assuming 20 fb−1 of LHC8 data. For the single production process wechoose gX = 0.32. All events are required to satisfy Ncon ≥ 7, HT > 1000GeV and NW +Ntop ≥ 2.

5.5 Discussion 131

700 750 800 850 900 950 1000

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

mX5�3HGeVL

gX

CMS SSDL

0.35

0.3

0.25

700 750 800 850 900 950 1000

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

mX5�3HGeVL

gX

CMS SSDL

0.6

0.5

0.45

0.4

Figure 5.7. Left: the estimated exclusion plot from our “l + jets” cut-and-countanalysis, assuming 20 fb−1 of LHC8 data. Right: the corresponding exclusionplot after top partner mass reconstruction. The boundaries are defined byS/√S +B = 2. The solid red line displays the limit from the CMS SSDL

search. The solid blue line in the right panel represents the exclusion curvefrom our cut-and-count analysis in the left panel. Black dotted lines indicateS/B ratios. Red dashed lines indicate O(φ+Xt) ∼ gX (mX/mW ) = 3.5.

hadronic top, as well asmlj < 160 GeV. Once an event is organised in terms oftops,W s, and everything else, we choose the top-W pair with largest distancein azimuthal angle ∆φtW . The resulting reconstructed invariant masses ofthe top partners are shown in Fig. 5.6. The grey-coloured region in the rightplot is contributed by the single production process for the coupling constantgX ∼ 0.32, which roughly translates to O(φ+Xt) ∼ 3.2. We then count thenumber of signal and background events in the top partner mass windowmrecoX = (mX − 20%,mX + 20%) to estimate the sensitivity, assuming 20

fb−1 of LHC8 data.

5.5 Discussion

Our results are summarised in a series of plots in Figs. 5.7, 5.8. We demon-strate that our one lepton analysis, taking into account the boosted kinemat-ics, significantly improves the existing excluded region. Figs. 5.7 and 5.8 showthe relevant roles of two main ingredients, “the correlation of processes” and


700 750 800 850 900 950 10000

100

200

300

400

mX5�3HGeVL

∆m=

mB-

mX

5�3HG

eVL

Pair production only

X5�3 and B

CMS SSDL

Figure 5.8. The exclusion plot when the two contributions from X5/3 and B aresummed in the total signal rate. The plot is restricted to only pair productionprocesses of X5/3 and B for simplicity. BR(X5/3 → tW )=1 is assumed, andBR(B → tW )=0.5 (dashed) and 1 (solid) are plotted. Blue lines are obtainedby our “l + jets” style cut-and-count analysis, assuming 20 fb−1 of LHC8data. Black lines are after the top partner mass window is applied. Red linesindicates the recasted CMS SSDL.

5.5 Discussion 133

700 800 900 1000 11000

1

2

3

4

5

mX5�3HGeVL

c1

Ξ=0.4

Ξ=0.4

Ξ=0.2

Ξ=0.2

Ξ=0.1

Ξ=0.1

CMS SSDL

y = 3

700 800 900 1000 11000

1

2

3

4

5

mX5�3HGeVL

c1

0.40.4

0.20.2

0.10.1

CMS SSDL

y = 0.3

Figure 5.9. The excluded region of (c1, mX) for a fixed set of ξ and y. Twochoices of y-values are shown. Left: y = 3, corresponding to the case withmB � m5/3. Right: y = 0.3, corresponding to the case with mB ∼> m5/3.For each y-value, we plot the contours for three different values of ξ ≡ (v/f)2:ξ = 0.1 (dotted), ξ = 0.2 (solid), and ξ = 0.4 (dashed). Black lines areobtained by our “l + jets” style cut-and-count analysis, assuming 20 fb−1 ofLHC8 data. Red lines indicate the recast CMS SSDL analysis.


“correlation of particles”, respectively, in improving the limit setting. Forthe former, in case the coupling gX of the single production process is large,the limit on the top partner X5/3 can be increased up to ∼ 1 TeV by our “l+ jets” style cut-and-count analysis. The reconstructed top partner peak inFig. 5.6 appears on top of the backgrounds with a similar shape. This some-what unwanted feature is mainly due to pT -cuts on the reconstructed tops,W ’s and the wide angle between them. Hence exploiting the top partnermass reconstruction has little effect on the statistical significance. However,S/B is improved by roughly a factor 2. For the latter, when the two massesof X5/3 and B are degenerate and their branching ratios to tW are 1, theirmass limit can be improved to ∼ 930 GeV using the pair production processalone.

While Figs. 5.7 and 5.8 show the effect on the limits for the top partnermasses, their implication for the composite Higgs model’s input parametersis not transparent. Therefore we will rephrase our findings according toEqs. 5.5, 5.6 and 5.7 in terms of the free parameters of the theory. Forinstance we show in Fig. 5.9 the excluded region of c1 for varying top partnermasses, while keeping y and ξ fixed. For further illustration we choose twovalues of y, representing two different mass hierarchies between B and X5/3

(see Eq. 5.5). If y = 3, the contribution from B is negligible comparedto X5/3 due to mB � mX5/3 , and the limit is mainly set by X5/3 alone.For small y (e.g. y = 0.3 as in Fig. 5.9), the additional contribution from B

becomes very important. The way we defined the simplified model in Eq. 5.4,the bottom-like top partner B dominantly decays to tW as decays to bZ andbh are forbidden [55]. While the limit from direct searches for B is identicalto the limit on X5/3, the indirect bound set by Eq. 5.5 is higher. For y = 3and ξ = 0.1 − 0.4 in Fig. 5.9, the bounds on m5/3 indicate that B can beexcluded at masses below mB ∼> 1.5 − 2.5 TeV. For the squeezed spectrumwith y = 0.3, as shown in Fig. 5.9, the B mass is expected to be larger thanmB ∼> 930 − 940 GeV assuming ξ = 0.1 − 0.4. The mass splitting betweenB and T after EWSB is much smaller than the splitting between the SU(2)doublets (B, T ) and (X5/3, X2/3) (see Eq 5.6). The indirect bound on T ,derived from the direct limit on X5/3, is very similar to the one of B.12 Wepoint out that the bottom-like top partner, B, is likely to exists in most

12The parameters in our simplified model are also indirectly constrained by elec-

5.5 Discussion 135

realization of the composite Higgs model discussed in Sec. 5.2 and it decaysto tW whereas the existence of X5/3 is tied to the custodial symmetry. Thecorresponding limit would be similar to the limit on X5/3 assuming a masshierarchy of mX5/3 � mB .

While we show exclusion bounds in Figs. 5.7 and 5.8 only for a few bench-mark values of the parameters, from the set of contours one can extract otherchoices easily. The results shown in Figs. 5.7 and 5.8 are not restricted tothe simplified model as defined in Eq. 5.4 only. Their benefit extends toany heavy vector-like fermion model that shares the same decay topology.Therefore, the parameters of those models can be constrained by our results,shown in Figs. 5.7 and 5.8.

So far in this chapter we have only discussed results for 8 TeV center-of-mass energy. At the end of 2014 the LHC is going to restart with

√s = 13−

14 TeV. At this energy, due to the lower Bjorken-x needed to produce the toppartners, the production cross section for the gluon-induced pair productionprocess will be strongly increased. Thus, the cross over point where thesingle production process has the same production cross section as the pairproduction process will be shifted to larger top partner masses. Still, ourfinding that exploiting the correlation of different production processes andcontributions from different top partners to the ttW final state is beneficial inconstraining the free parameters of the composite Higgs model carries overstraightforwardly. As heavier top partners can be probed at 14 TeV theirdecay products will be more boosted and their radiation will be confined toa smaller area of the detector. Particularly for the reconstruction of isolatedleptons this can pose a severe challenge. However, already in searches at8 TeV mini-isolation criteria for the reconstruction of isolated leptons wereproposed and successfully applied [92]. In this kinematic regime boostedtechniques will be indispensable. In fact, some of the existing taggers mightneed further development to exploit the LHC’s energy reach to the fullest[93]. In any case, the observables and search strategies discussed in this workwill be directly applicable at 13 (14) TeV, hereby helping to discover TeV-

troweak precision measurements (EWPM) [69], e.g. ξ ≡ (v/f)2 < 0.2 from the singleHiggs boson fit at LHC8. Another possibility to constrain top partners indirectly isthe production of a Higgs boson recoiling against a hard jet [26].


scale top partners or constraining the parameter space of composite Higgsmodels.

5.5 Discussion 137

mX

700GeV

800GeV

900GeV

1000

GeV

1100

GeV

HTcu

t900GeV

1000

GeV

1110

GeV

1200

GeV

1300

GeV

simple

(Nco

n,HT

)pa

irprod

uctio

n6.4fb

2.37

fb0.93

fb0.38

fb0.16

fbN

con≥

7sin

gle-(g X

=1)

55.7

fb33.9

fb20.1

fb12.7

fb7.5fb

tt+jets

84.9

fb52.2

fb31

fb20.1

fb12.2

fbW

+jets

26.1

fb18.9

fb13.5

fb10.2

fb7.1fb

simple

(Nco

n,H

T)

pair

prod

uctio

n3.3fb

1.4fb

0.61

fb0.26

fb0.11

fbN

con≥

7sin

gle-(g X

=1)

14.9

fb10

fb6.9fb

4.6fb

3fb

+(N

W+N

top)≥

2tt+jets

7.1fb

4.8fb

3.2fb

2fb

1.3fb

W+jets

2.1fb

1.6fb

0.93

fb0.62

fb0.4fb

(Nj,H

T),N

con≥

7pa

irprod

uctio

n1.32

fb0.64

fb0.3fb

0.13

fb0.06

fb+

(NW

+N

top)≥

2sin

gle-(g X

=1)

6.5fb

5.1fb

3.6fb

2.6fb

1.73

fb+

toppa

rtnerreconstruc

tion

tt+jets

2.4fb

1.5fb

1fb

0.75

fb0.48

fbwith

0.8m

X<m

reco

X<

1.2m

XW

+jets

0.52

fb0.31

fb0.16

fb0.1fb

0.06

fb

Tab

le5.2.

Cross

sections

atLH

C8forthesign

alswith

diffe

rent

toppa

rtne

rmassesan

dthecorrespo

ndingba

ckgrou

nds,

afterthe

diffe

rent

analysiscuts,inthel+

jets

chan

nel.g X

isthecoup

lingconstant,inv

olvedin

theprod

uctio

nof

thesing

letoppa

rtne

r,an

dthenu

mbe

rsareforun

itcoup

ling.

139

Appendix A

Generators of SO(5)/SO(4)

In this appendix we summarize our definitions for the SO(5)/SO(4) algebra.The unbroken SO(5) generators are defined as :

T aIJ = i√2

(δaI δ5J − δaJδ5I) ,

TL/R aIJ =− i

2

(12εabc(δbIδ

cJ − δbJδcI

)± (δaI δ4J − δaJδ4I)

),

(A.1)

where I, J = 1, . . . , 5, a = 1, . . . , 4, a = 1, . . . 3.The generators of the rotated SO(5)′ are obtained through:

TAθ = R (θ)TAR (θ)−1, (A.2)

where R(θ) is defined as:

R (θ) =(I3x3 0

0 r(θ)

), r (θ) =

(cos(θ) sin(θ)− sin(θ) cos(θ)

). (A.3)

The two sets can be related through:

TL/R aθ =1

2(TL a + TR a

)± 1

2 cos(θ)(TL a − TR a

)∓ sin(θ)√

(2)T a,

T aθ =sin(θ)√(2)

(TL a − TR a

)+ cos(θ)T a (a = 1, 2, 3) ,

T 4θ =T 4.

(A.4)

141

Appendix B

Useful formulas

In this appendix we collect some useful formulas used throughout the maintext. As to begin, we report here the 1-loop Higgs contribution to the εi:

∆ε1∣∣H

= − 3g′ 2

32π2 ξ

[log(

ΛMZ

)+ f1(h)

], (B.1)

∆ε2∣∣H

= g2

192π2 ξf2(h), (B.2)

∆ε3∣∣H

= g2

96π2 ξ

[log(

ΛMZ

)+ f3(h)

], (B.3)

where h = M2H/M

2Z and

f1(h) = 1s2

(−5c2

12 + h2

6 −7h12 + 31

18

)−[(c2 + 5

)h3 −

(5c2 + 12

)h2 + 2

(9c2 + 2

)h− 4c2 − h4]×

log(h)12s2 (c2 − h) −

c4

s2 (h− c2) log(c)

+h(h3 − 7h2 + 20h− 28

)6s2√

(4− h)harctan

(√4h− 1),

(B.4)

142 B. Useful formulas

f2 (h) =(− 1c4− 2)h2 +

(9

2c2 + 6)h− 47

2

+ log(c)c6 (c2 − h)

(2c8 − 38c6h+ 24c4h2 − 7c2h3 + h4)

+ log(h)2c6 (c2 − h)

[− 12c8 −

(2c6 + 1

)h4 + 6

(3c2 + 8

)c6h

− 3(3c4 + 6c2 + 8

)c4h2 +

(2c6 + 9c4 + 7

)c2h3

]−(2h3 − 13h2 + 32h− 36

)h√

(4− h)harctan

(√4h− 1)

+(48c6h− 28c4h2 + 8c2h3 − h4)

c6√h (4c2 − h)

arctan(√

4c2h− 1),

(B.5)

f3 (h) =(−h2 + 3h− 31

6

)+ 1

4(2h3 − 9h2 + 18h− 12

)log(h)

−(2h3 − 13h2 + 32h− 36

)h

2√

(4− h)harctan

(√4h− 1).

(B.6)

They agree with the functions Hi of Ref. [86], see also Ref. [84].

Next, in order to complete the discussion for the effective theory matchingof chapter 2, we report the RG evolution and the matching conditions for c+3and cT in a theory with a single spin-1 resonance, either ρL or ρR (neglecting1-loop contributions from α1,2)

µd

dµc+3 (µ) = 1

192π2

[54 + 1

4a2ρ(2a2

ρ − 7)]

(B.7)

µd

dµcT (µ) = − 3

64π2

(1− 3

4a2ρ

)(B.8)

and

c+3 (µ) = c+3 (µ)− 12

(1

4g2ρ

− α2

)+ 1

192π2

[34(a2

ρ + 28) log µ

mρ+ 1 + 41

16a2ρ

](B.9)

cT (µ) = cT (µ)− 9256π2

[a2ρ log µ

mρ+ 3

4a2ρ

]. (B.10)

The β-functions of c2W and c2B vanish. In a theory with only ρL one has

143

the matching conditions

c2W (µ) = c2W (µ)− 12g2ρLm

2ρL

(1− 2α2Lg2ρL)2

+ 196π2m2

ρL

[77 log µ

mρL

+ 465 −

2732a

2ρL tan2 θ

2

] (B.11)

c2B(µ) = c2B(µ) , (B.12)

while only a ρR gives

c2W (µ) = c2W (µ) (B.13)

c2B(µ) = c2B(µ)− 12g2ρRm

2ρR

(1− 2α2Rg2ρR)2

+ 196π2m2

ρR

[77 log µ

mρR

+ 465 −

2732a

2ρR tan2 θ

2

].

(B.14)

When including the effect of α2 at the 1-loop level, there arise the fol-lowing additional contributions to the εi (in the following β2r = α2rg

2ρr ):

∆ε1∣∣α2

= − 9g′2

128π2 sin2θ

×

{83

a2ρLm

2ρL

m2ρL −m2

ρR

[8 (1− β2L)β2Lβ2R

(a2ρR − β2Ra

2ρL

)− (1− β2L)β2L

(2a2ρR +

m2ρR

m2ρL

− 1)

− 2β2R

(a2ρR − β2R

g2ρL

g2ρR

a2ρL

)]log µ

mρL

+ 29a

2ρLβ2L

[11− 10a2

ρR + 20β2Ra2ρR

+ 20β2Lβ2Ra2ρR

(1 +

m2ρL

m2ρR

+m2ρR

m2ρL

)

− 40β2Lβ2Ra2ρR

(1 +

m2ρR

m2ρL

)

− β2L

(11− 10a2

ρR

(1 +

m2ρR

m2ρL

))]}+ {L↔ R} ,

(B.15)


∆ε2∣∣α2

= g2

96π2g2

g2ρL

1a2ρL

sin2θ cos4 θ

2

×

{log µ

mρR

[116β2L − β2

2L

(74− 6a2

ρL tan2 θ

2

)]

+ β2L

(5− 6a2

ρL tan2 θ

2

)+ β2

2L

(7 + 17

2 a2ρL tan2 θ

2

)}

+ {L↔ R, θ → π − θ}

(B.16)

∆ε3∣∣α2

= g2

96π2 sin2θ

[32β2L

(9a2ρL − 4 + β2L

(9a2ρL − 8

))+ 18

(β2L

(a2ρL + 2

)− β2

2L)

log µ

mρL

]+ {L↔ R} .

(B.17)The renormalization of the various parameters is also affected, in particulareach β-function gets an additional contribution. We report the correspondingexpressions in the unitary gauge:

∆βc+3

=− α2Lg2ρL

2a4ρL − 20a2

ρL + 11192π2 + α2

2Lg4ρL

3a4ρL − 7a2

ρL + 696π2

− α32Lg

6ρL

a4ρL

12π2 + {L↔ R}(B.18)

∆βcT =− 332π2

a2ρLm

2ρL

m2ρL −m2

ρR

×[

8(1− α2Lg

2ρL

)α2Lα2Rg

2ρLg

2ρR

(a2ρR − α2Rg

2ρLa

2ρL

)− 2α2Rg

2ρR

(a2ρR − α2Rg

2ρLa

2ρL

)−(1− α2Lg

2ρL

)α2Lg

2ρL

(2a2ρR +

m2ρR

m2ρL

− 1)]

+ {L↔ R}

(B.19)

∆βc2W = − 1m2ρL

(α2Lg

2ρL

2a2ρL − 8548π2 + α2

2Lg4ρL

37− 3a2ρL tan2(θ/2)24π2

)(B.20)

∆βc2B = − 1m2ρR

(α2Rg

2ρR

2a2ρR − 8548π2 + α2

2Rg4ρR

37− 3a2ρR cot2(θ/2)24π2

)(B.21)

145

∆βgρ =−α2g

5ρ

24π2

(a4ρ − a2

ρ − 3 + α2g2ρa

4ρ

)∆βmρ =mρ α2g

4ρ

a4ρ

24π2

(−1 + α2g

2ρ

)∆βα2 =α2g

2ρ

4a4ρ − 4a2

ρ + 2596π2 .

(B.22)

We now move on to the calculation of the spectral functions of chapter3. We report here the expressions of the spectral functions computed in thelow-energy effective theory in D dimensions, which can be used to performthe dispersive integrals using dimensional regularization. For conveniencethey are given for a finite Higgs mass mh, so that one should set mh = 0in evaluating the integrals of Eqs. (3.25), (3.28), (3.29) and (3.31). TheLL and RR spectral functions are computed by introducing a small mass λfor the three SO(4)/SO(3) NG bosons which acts as an IR regulator whenconsidering their individual contribution to the dispersive integrals. Notice,on the other hand, that the linear combination of spectral functions appearingin Eqs. (3.25), (3.28), (3.29) and (3.31) is free from IR divergences, and thatone should set λ = 0 when evaluating them.

The function ρRR receives a contribution from the intermediate statesχχ and χh, where χ1,2,3 ≡ π1,2,3 and h = π4. We find:

ρRR(q2) = ρ(χχ)RR (q2) + ρ

(χh)RR (q2) , (B.23)

ρ(χχ)RR (q2) = µ4−D

π(D−1)/2 4D Γ(D+1

2) (1− 4λ

2

q2

)(D−1)/2

(q2)(D−4)/2 θ(q2 − 4λ2) ,

(B.24)

ρ(χh)RR (q2) = µ4−D

π(D−1)/2 4D Γ(D+1

2) (1− m2

h

q2

)D−1

(q2)(D−4)/2 θ(q2 −m2

h

).

(B.25)


The intermediates states contributing to ρLL are ππ and ρρ. We have:

ρLL(q2) = ρ(ππ)LL (q2) + ρ

(ρρ)LL (q2) , (B.26)

ρ(ρρ)LL (q2) = µ4−D

π(D−1)/2 4D Γ(D+1

2) q4 + 20q2m2

ρ + 12m4ρ(

q2 −m2ρ

)2×

(1− 4

m2ρ

q2

)3/2 (q2 − 4m2

ρ

)(D−4)/2θ(q2 − 4M2

ρ

).

(B.27)

where ρ(ππ)LL (q2) is given by Eq. (3.51). Finally, the only contribution to ρBB

is from the intermediate state ρπ:

ρBB(q2) = 3µ4−D

2π(D−1)/2 4D Γ(D+1

2) a2

ρ

(1 + 10

m2ρ

q2 +m4ρ

q4

)

×

(1−

m2ρ

q2

)D−3

(q2)(D−4)/2 θ(q2 −M2

ρ

).

(B.28)

Notice that for simplicity the contribution of α2 has been included only inρ

(ππ)LL , see Eq. (3.48), and omitted in ρ

(ρρ)LL and ρBB . This corresponds to

including α2 only at the tree level in the diagrammatic calculation.For completeness, we also report the expression for the ρL pole mass

squared M2ρ , the pole residue Zρ, the decay width Γρ (tree-level expression),

and the 1-loop vertex correction Π(1L)Jρ used in Section 3.3:

M2ρ = m2

ρ −m2ρ

g2ρ

96π2

[(2a4ρ − 69

)log µ

mρ+ 8

3a4ρ − 103 + 33

√3

2 π

],

(B.29)

Zρ = 1−g2ρ

96π2

[(2a4ρ − 53

)log µ

mρ+ 5

3a4ρ −

536 −

11√

32 π

], (B.30)

Γρ =g2ρa

4ρ

96π mρ , (B.31)

Π(1L)Jρ = − 1

48π2m2ρa

2ρ

(a2ρ − 1

)(log µ

mρ+ 4

3 + i

2π). (B.32)

We report now the contributions of the top quark to the parametersε1 and ε3, used for the numerical fits of chapter 4. They depend on the

147

particular choice of the composite fermionic sector through the top couplingsto EW gauge bosons. We have calculated its contributions as a function ofsuch couplings. Separating the contribution to T and S, they can be writtenas:

ε1∣∣top = T

∣∣t

+((1− 2t)

(g2LZ + g2

RZ

)− 2 (1− 6t) gLZgRZ

)×

38π2

t√4t− 1

arctan(√

4t− 11− 2t

)3

16π2

[(23 − 2t

)(g2LZ + g2

RZ

)+ 12tgLZgRZ

],

(B.33)

ε3∣∣top = S

∣∣t

+ 18π2

c

s

√4t− 1 arctan

(√4t− 1

1− 2t

)((1− t) gα + 3tgβ)

− 38π2 c

2 t√4t− 1

arctan(√

4t− 11− 2t

)×[

(1− 2t)(g2LZ + g2

RZ

)− 2 (1− 6t) gLZgRZ

]+ 1

8π2 c2 ((1− 3t)

(g2LZ + g2

RZ

)+ 18tgLZgRZ

)+ 1

48π2c

s((13− 12t) gα + 3(12t− 1)gβ) + c2e4

∣∣t,

(B.34)

where T∣∣top and S

∣∣top are given by

T∣∣top = 3t

16π2c2

[4 log

(Λmt

)(g2L1 + g2

R1 − (gL3 − gR3)2)

+ g2L1 + g2

R1

],

(B.35)

S∣∣top = 1

16π2c

s

(4 log

(Λmt

)gα − gα + gβ

). (B.36)

In the above equations t = m2t/M

2Z , g(L/R)i indicates the coupling of the

left (right) chirality top pair with the i-th EW gauge boson (for example, inthe SM gLZ = −g/c

(1/2− 2/3s2)). The quantities gα and gβ correspond to

the combinations:

gα = gL3gLB + gR3gRB (B.37)

gβ = gL3gRB + gR3gLB . (B.38)

The above formulae must undergo a subtraction of the same contributionsin which the SM couplings are used. For this reason we kept the dependance


from e4 implicit in ε3, as it will always cancel when subtracting the SMresut. This is of course because e4 obtained through photon self energies,and in order to respect the exact U(1)Q symmetry photon couplings arenever shifted.

149

Appendix C

Two-site vector Lagrangianin the SO(5)× SO(5)H limit

As discussed in Section 2.1, in the limit aρL = aρR = 1/√

2 the Lagrangian(2.25) enjoys a larger SO(5)×SO(5)H → SO(5)d global symmetry, partiallygauged by the EW and ρµ fields. The theory is in fact equivalent to a two-siteSO(5)×SO(5)H model whereWµ and Bµ gauge a subgroup SU(2)L×U(1)Yon the left site, while ρµ gauges an SO(4)H on the right site. The mostconvenient way to construct the Lagrangian, in this case, is in terms of a5×5 link field U(π, η) = ei

√2π(x)/fei

√2 η(x)/f , where π(x) = πa(x)T a, η(x) =

ηa(x)T a and T a, T a are the SO(5) generators. The link transforms as a (5, 5)under SO(5)× SO(5)H

U(π, η)→ g U(π, η) g†H , (C.1)

so that its covariant derivative is (we conveniently normalize gauge fields sothat gauge couplings appear in their kinetic terms)

DµU = ∂µU + iW aLµ T aLU + iBµT

3RU − iUρaµT a . (C.2)

Given the above transformation rules, it is possible to eat all the NG bosonsη by making an SO(4)H local transformation and go to a gauge in whichthe link field coincides with U(π) defined in Section 2.1: U(π, η = 0) =ei√

2π(x)/f = U(π). When acting on U from the left with a global rotation g ∈SO(5), the unitary gauge can be maintained by simultaneously performing

150 C. Two-site vector Lagrangian in the SO(5)× SO(5)H limit

a suitable, local SO(4)H transformation gH = h(g, π). The fields thus obeythe following transformation rules

U(π, 0)→ U(g(π), 0) = g U(g, 0)h†(g, π)

ρµ → h(g, π)ρµh†(g, π)− ih(g, π)∂µh†(g, π) ,(C.3)

which are the same as those in the SO(5)/SO(4) theory with massive spin-1resonance ρµ.

By working in the η = 0 gauge, it is easy to recast the kinetic term ofU in SO(5)/SO(4) CCWZ notation. Since −iU(π, 0)DµU(π, 0) = dµ(π) +Eµ(π)− ρµ, it simply follows

f2

4 Tr[(DµU)†(DµU)

]= f2

4 Tr[dµ(π)dµ(π)] + f2

4 Tr[(ρµ − Eµ(π))2] , (C.4)

which gives aρ = 1/√

2 upon comparison with Eq. (2.18).At the level of two derivatives and two powers of the hypercharge spu-

rion g′T 3R0 , there is one (SO(5)× SO(5)H)-invariant operator which can be

constructed:OT =

(Tr[U iDµU

†g′T 3R0])2

. (C.5)

Notice that the combination UDµU† transforms as UDµU

† → g(UDµU†)g†.

In the η = 0 gauge, by defining χ(π) = U†(π, 0)g′T 3R0 U(π, 0), one has

OT = (Tr[(dµ + Eµ(π)− ρµ)χ])2 (C.6)

which coincides with the right-hand side of Eq. (2.26). On the other hand, atorder g0

ρ there is no operator with two EW field strengths and no derivativeacting on U which can contribute to the S parameter. This is because thereis no way to saturate the SO(5)H index of U except in the trivial productU U† = 1.

151

Appendix D

One-loop renormalizationof the spin-1 Lagrangian

In this appendix we discuss the one-loop renormalization of the spin-1 La-grangian, deriving the results used in chapter 2. We first describe our pro-cedure for the unitary gauge and then give the results also for the Landaugauge. We will not specify the quantum numbers of the spin-1 resonanceunless necessary since the same expressions hold for both ρL and ρR, therebeing no mixed renormalization at one loop.

Starting from the bare Lagrangian, we define renormalized fields andparameters as follows

πa(0) = Z1/2π πa

ρa(0)µ = Z1/2

ρ ρaµ

W i(0)µ = Z

1/2W W i

µ

B(0)µ = Z

1/2B Bµ

f (0) = µ−ε/2Z1/2f f(µ)

m(0)ρ = Zmmρ(µ)

g(0)ρ = µε/2Zgρgρ(µ)

g(0) = µε/2Zgg(µ)

g′(0) = µε/2Zg′g′(µ) ,

(D.1)

where Zi are renormalization functions and we make use of dimensional reg-ularization in d = 4 − ε dimensions with a renormalization scale µ. Therenormalization of the elementary gauge fields and coupling constants arisesat O(g2, g′2) so we can set ZW , ZB , Zg and Zg′ to unity when working atleading order in an expansion in powers of the elementary couplings. The re-

152 D. One-loop renormalization of the spin-1 Lagrangian

maining functions Zπ, Zρ, Zm, Zf and Zgρ can be computed by renormalizingthe 2-point functions 〈ππ〉, 〈ρµρν〉, 〈AµAν〉 and 〈ρµAν〉, where Aµ = Wµ, Bµ.We adopt a subtraction scheme where the above Green functions (and theirderivatives) are evaluated at q2 = m2

ρ and made finite by removing their polesin 1/ε, where 2/ε ≡ 2/ε − γ − log(4π). This hybrid MS on-shell scheme isconvenient, as it requires the same number of counterterms as in the Landaugauge. Performing instead a minimal subtraction on off-shell Green functionswould require further counterterms to remove the q4 and q6 divergent termsin the ρ propagator. We thus obtain

Zρ = 1− g2ρ

2a4ρ − 5396π2

1ε

(D.2)

Zgρ = 1 + g2ρ

2a4ρ − 85

192π21ε

(D.3)

Zm = 1 + g2ρ

2a4ρ − 69

192π21ε

(D.4)

Zπ = 1 +(g2ρL

3a4ρL

16π2 + g2ρR

3a4ρR

16π2

)1ε

(D.5)

Zf = 1 +(g2ρL

9a4ρL

32π2 + g2ρR

9a4ρR

32π2

)1ε. (D.6)

From these expressions it follows Eq. (2.37) and

µ∂mρ

∂µ≡βmρ = g2

ρ

2a4ρ − 69

192π2 mρ (D.7)

µ∂f

∂µ≡βf = f

(g2ρL

9a4ρL

64π2 + g2ρR

9a4ρR

64π2

). (D.8)

The renormalized ci and α2 are instead defined by

c(0)i = µ−ε

(ci(µ) + 1

ε∆i

)' ci(µ) + ∆i

(1ε− logµ

)

α(0)2 = µ−ε

(α2(µ) + 1

ε∆α2

)' α2(µ) + ∆α2

(1ε− logµ

).

(D.9)

The value of the counterterm ∆α2 is obtained by renormalizing the 〈ρµAµ〉Green function. We find ∆α2 = a2

ρ(1− a2ρ)/96π2, which leads to Eq. (2.38).

The value of the counterterms ∆ci is instead found by renormalizing theGreen functions in Figs. 2.3-2.7 after canceling the divergences from subdi-agrams. The corresponding RG evolution of the coefficients ci is given inEqs. (2.39), (2.41) and (2.44).

153

A similar procedure also applies in the Landau gauge with a few differ-ences however. First, another field is present, that of the NG bosons η, whichneeds to be renormalized. Second, the ρ mass originates from the η kineticterm, and mρ is defined in terms of fρ according to Eq. (2.23). It is thusmore convenient to include fρ in the list of renormalized quantities and treatmρ as a derived parameter. By defining

ηa(0) = Z1/2η ηa , f (0)

ρ = µ−ε/2Z1/2fρ

fρ(µ) (D.10)

we find

Zρ = 1− g2ρ

2a4ρ − 5196π2

1ε

(D.11)

Zgρ = 1 + g2ρ

2a4ρ − 87

192π21ε

(D.12)

Zfρ = Zη = 1 + g2ρ

316π2

1ε

(D.13)

Zπ = 1 +(g2ρL

a4ρL

4π2 + g2ρR

a4ρR

4π2

)1ε

(D.14)

Zf = 1 +(g2ρL

9a4ρL

32π2 + g2ρR

9a4ρR

32π2

)1ε

(D.15)

and ∆α2 = (2a2ρ(1− a2

ρ))/192π2. The corresponding RG equations read

µ∂gρ∂µ

= g3ρ

2a4ρ − 87

192π2 ,

µ∂α2

∂µ=

2a2ρ(1− a2

ρ) + 1192π2 ,

µ∂fρ∂µ

= g2ρ

332π2 fρ ,

µ∂f

∂µ= f

(g2ρL

9a4ρL

64π2 + g2ρR

9a4ρR

64π2

).

(D.16)

155

Appendix E

Dispersion relations for asmall breaking of SO(5)

In deriving the dispersion relations of chapter 3 we have assumed that thestrong dynamics in isolation is SO(5) symmetric. It is conceivable, on theother hand, that the global symmetry is only approximate and that a smallexplicit breaking arises internal to the strong dynamics. This is for examplewhat happens in the Conformal Technicolor model of Ref. [79], where thesmall breaking arises from the techniquark mass terms. Generalizing ourprocedure to such a scenario is straightforward. We will assume that anSO(3) × PR subgroup of the strong dynamics is unbroken, where SO(3) isthe custodial isospin and PR is the grading of the SO(5) algebra under whichthe SO(5)/SO(4) generators are odd. This allows for a Higgs boson potential,hence a Higgs mass, ensuring a correct phenomenology. The definitions ofthe two-point correlators generalizing Eq. (3.15) thus read:

〈JaLµ (q)JbLν (−q)〉 = − iδaLbL(ηµνΠLL(q2)− qµqνΠLL(q2)

)〈JaRµ (q)JbRν (−q)〉 = − iδaRbR

(ηµνΠRR(q2)− qµqνΠRR(q2)

)〈JaLµ (q)JbRν (−q)〉 = − iδaLbR

(ηµνΠLR(q2)− qµqνΠLR(q2)

)〈J aµ(q)J bν(−q)〉 = − iδab

(ηµνΠBB(q2)− qµqνΠBB(q2)

)− iδa4δb4

(ηµνΠ(4)

BB(q2)− qµqνΠ(4)BB(q2)

).

(E.1)

156 E. Dispersion relations for a small breaking of SO(5)

Any two-point function with one SO(5)/SO(4) and one SO(4) current van-ishes due to PR invariance. As a consequence of the SO(5) breaking, inparticular, ΠLR does not vanish and must be included in the definition ofΠ3B when deriving Eq. (3.16):

Π3B(q2) ≡ 14 sin2θ

(ΠLL(q2) + ΠRR(q2)− 2ΠBB(q2)

)+ 1

2(1 + cos2θ

)ΠLR(q2) .

(E.2)

Since now the Higgs boson mass is non-vanishing, Eq. (3.16) is free fromIR singularities, which cancel when taking the difference with the SM. Itis still convenient, however, to add and subtract the contribution from theSO(5)/SO(4) linear model, as done in the text. A first motivation to do so isthat the SO(5) breaking internal to the strong dynamics only partly accountsfor the Higgs mass, an important (if not dominant) contribution comes fromthe coupling to the elementary top quark, which is not included. The secondmotivation is that subtracting the SO(5)/SO(4) linear model allows one toisolate the Higgs chiral logarithm, so that the final dispersive integral en-codes the contribution from the heavy resonances only. By performing thesubtraction as explained in the text, the result that follows coincides withthe massless case. That is: Eq. (3.18) is valid also in the massive case, withΠ3B defined as in Eq. (3.17). This is because the only unsuppressed contri-bution to ΠLR comes from the NG bosons and cancels out when subtractingthe SO(5)/SO(4) linear model. Although Eq. (3.18) is formally unchanged,ΠLSO5 ′

3B (0) in parenthesis must be evaluated by setting the Higgs mass to thesame value m0h generated by the strong dynamics. The dispersion relationgeneralizing Eq. (3.25) reads

S = g2

4 sin2θ

∫ ∞0

ds

s

{(ρLL(s) + ρRR(s)− 2ρBB(s))

− 148π2

[12 + 1

2

(1− m2

0hs

)3

θ(s−m20h)

−

(1−

m2η

s

)3

θ(s−m2η)]}

+ δSLSO5 + δSZh ,

(E.3)

157

where δSLSO5 is still defined by Eq. (3.19) and computed at the physicalHiggs mass. Similarly, the dispersion relations for W and Y are:

W =−m2W g

2∫ ∞

0

ds

s2

{ρLL(s)− 1

192π2

[1 +

(1− m2

0hs

)3

θ(s−m20h)]}

+ g2

96π2c2W8xh

sin2θ + δWZh

(E.4)

Y =−m2W g′2∫ ∞

0

ds

s2

{ρRR(s)− 1

192π2

[1 +

(1− m2

0hs

)3

θ(s−m20h)]}

+ g′2

96π2c2W8xh

sin2θ + δYZh .

(E.5)

The formula for ∆ε3 finally reads:

∆ε3 = g2

96π2 sin2θ

(f3(xh)− 1

8xh+ log xh

2 − 512 + log mη

mh

)+ g2

4 sin2θ

∫ ∞0

ds

s

{ρLL(s) + ρRR(s)− 2ρBB(s)

− 196π2

[12 + 1

2

(1− m2

0hs

)3

θ(s−m20h)

−

(1−

m2η

s

)3

θ(s−m2η)]}

+m2W

∫ ∞0

ds

s2

{g2ρLL(s) + g′2ρRR(s)

− g2 + g′2

192π2

[1 +

(1− m2

0hs

)3

θ(s−m20h)]}

.

(E.6)

Notice that the dependence on m0h in Eqs. (E.3)-(E.6) cancels out up tonegligible terms with relative suppression of order m2

0h/m2∗.

159

Appendix F

Recovering the asymptoticbehavior at the cutoff scale

In section 3.4 we mentioned the possibility of enforcing the correct asymp-totic behaviour of the correlators of the global SO(5) currents through theexchange of states at the scale Λ, while keeping the leading contribution tothe S parameter dominated by the lighter resonances. Here we discuss theconstruction of a simple model which implements this idea.

Consider a low-energy theory with three spin-1 resonances transforming,respectively, as a (3, 1) (the ρL), a (1, 3) (ρR) and a (2, 2) (ρB) of SU(2)L ×SU(2)R. We will assume for the moment that their masses are all of the sameorder and accidentally (much) lighter than the cutoff scale. The Lagrangiancharacterizing the ρL and the ρR is defined in Chapter 2. The ρB is insteaddescribed by

L(ρB) = − 14g2ρB

Tr[ρBµνρ

B µν]−m2ρB

2g2ρB

Tr[ρBµ ρ

B µ]+α2B Tr

[ρBµνf

−µν] , (F.1)where ρBµν ≡ ∇µρBν − ∇νρBµ and f−µν is the component of the dressed fieldstrength along the broken SO(5)/SO(4) generators (see again Chapter 2). Asimple calculation shows that in the deep Euclidean ΠLL(q2)/q2 ' 4α2

2Lg2ρL ,

ΠRR(q2)/q2 ' 4α22Rg

2ρR and ΠBB(q2)/q2 ' 4α2

2Bg2ρB , where the L,R,B

subindices are used to denote the parameters of the corresponding resonances.The asymptotic behavior ΠLL(q2) ∼ ΠRR(q2) ∼ ΠBB(q2) ∼ γ q2, where γ is

160 F. Recovering the asymptotic behavior at the cutoff scale

a constant proportional to the central charge of the OPE, is thus reproducedby the correlators in the effective theory if

α22Lg

2ρL = α2

2Rg2ρR = α2

2Bg2ρB . (F.2)

Under this condition, Π1(q2)/q2 → 0 for q2 → −∞, and the integral onthe circle vanishes (i.e. δ1 = 0 in this model). The contribution to S fromthe tree-level exchange of the resonances, as obtained through the dispersionintegral, thus reads

S = g2

4 sin2θ

[(1gρL− 2α2LgρL

)2+(

1gρR− 2α2RgρR

)2− 8α2

2Bg2ρB

]

= g2

4 sin2θ

[(1g2ρL

− 4α2L

)+(

1g2ρR

− 4α2R

)],

(F.3)

where the second equality follows from Eq. (F.2). The expression in the lastline coincides with the result of the diagrammatic calculation, where the tree-level exchange of the ρB gives no contribution to S. 1 Notice that althoughS is obtained through a dispersive integral it is not positive definite, becausethe contribution from the spectral function ρBB comes with a negative signin Eq. (3.25).

Now consider the limit in which the resonance ρB is much heavier thanthe other two and has a mass mρB ∼ g∗f � mρL ∼ mρR ∼ gρf . The scalemρB acts as a cutoff for the effective theory with just ρL and ρR. In suchlow-energy description the leading O(1/g2

ρ) contribution to the S parameteris fully accounted for by the exchange of the light resonances (last line ofEq. (F.3)), and no anomalously large coefficient for the dimension-6 opera-tors is generated by the cutoff dynamics. The result from the diagrammaticcalculation is reproduced by the dispersive approach only after adding thecontribution of the integral on the circle at infinity. While S is not positivedefinite, the correct asymptotic behavior of the two-point current correlatorsis recovered at the cutoff scale through the exchange of the ρB , as a conse-quence of Eq. (F.2). This latter can be satisfied for α2L ∼ α2R ∼ 1/g2

ρ andα2B ∼ 1/(gρg∗).

1This can be most easily seen by noticing that integrating out the ρB from theLagrangian (F.1) by using the equations of motions does not generate any O(p4) op-erator.

161

Appendix G

Mass mixings

In this appendix we will review some useful formulas about particle mixingsin the extended models discussed in chapter 4. We will present the mostgeneral case in which all fermionic and vectorial resonances are present.

Spin-1 states exhibit both mass and kinetic mixings. Defining V µ0 =(Wµ

3 , Bµ, ρµL 3, ρ

µR 3) and V µ± =

(Wµ±, ρ

µL±, ρ

µR±)their mixing Lagrangian is

given by

L(mix)V =− 1

4VµT0 [−2gµν�+ 2∂µ∂ν ]KV0V0 + 1

2VµT0 MV0V0

− 12V

µT+ [−2gµν�+ 2∂µ∂ν ]KV±V− + V µT+ MV±V−,

(G.1)

162 G. Mass mixings

where

KV0 =

1/g2

el 0 −2α2L cos2 ( θ2)−2α2R sin2 ( θ

2)

1/g′ 2el −2α2L sin2 ( θ2)−2α2R cos2 ( θ

2)

1/g2ρL 0

1/g2ρR

, (G.2)

KV± =

1/g2

el −2α2L cos2 ( θ2)−2α2R sin2 ( θ

2)

1/g2ρL 0

1/g2ρR

, (G.3)

MV0 =14f

2 sin2 (θ)

M11 M12 M13 M14

M22 M23 M24

M33 M34

M44

, (G.4)

MV0 =14f

2 sin2 (θ)

M11 M13 M14

M33 M34

M44

, (G.5)

and

M11 = g2el

(1 + a2

ρL cot2(θ

2

)+ a2

ρR tan2(θ

2

)),

M22 = g′2el

(1 +M2

ρL tan2(θ

2

)+M2

ρR cot2(θ

2

)),

M12 = −gelg′el(1− a2

ρL −M2ρR

), M13 = −gelgρL csc2

(θ

2

)a2ρL ,

M14 = −gelgρR sec2(θ

2

)a2ρR , M23 = −g′elgρL sec2

(θ

2

)a2ρL ,

M24 = −g′elgρR csc2(θ

2

)(a2ρL

), M33 = 4g2

ρL csc2 (θ) a2ρL

M34 = 0, M44 = 4g2ρR csc2 (θ) a2

ρR . (G.6)

On the fermionic side only mass mixings appear. Defining D = (b, B)and U =

(t, T,X2/3, T

)we have

L(mix)Ψ = −ULMUUR −DLMDDR + h.c., (G.7)

163

MU =

0 −yL4f sin2

(θ2

)−yL4f cos2

(θ2

)1√2yL1f sin (θ)

1√2yR4f sin (θ) M4 0 0

− 1√2yR4f sin (θ) 0 M4 0−yR1f cos (θ) 0 0 M1

,

(G.8)

MD =(

0 −yL4f

0 M4

). (G.9)

For simplicity, we report the expression for the physical masses at leadingorder in ξ and for α2r = 0:

M2Z = 1

4f2ξ

(g2

elg2ρL

g2el + g2

ρL

+g′ 2el g

2ρR

g′ 2el + g2ρR

)+O

(ξ2) ,

M2W = 1

4f2ξ

g2elg

2ρL

g2el + g2

ρL

+O(ξ2) ,

m2t = 1

2f2ξ

(g1yL4yR4 − g4yL1yR1)2

(g21 + y2

R1) (g24 + y2

L4) +O(ξ2) ,

M0ρL = M±ρL = M2

ρL

(1 + g2

elg2ρL

)+O (ξ) ,

M0ρR = M2

ρR

(1 + g′ 2el

g2ρR

)+O (ξ) , (G.10)

M±ρR = M2ρR +O (ξ) ,

m2T = m2

B = M24

(1 + y2

L4g2

4

)+O (ξ) ,

m2X2/3 = m2

X5/3 = M24 +O (ξ)

m2T

= M21

(1 + y2

R1g2

1

)+O (ξ) .

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