Structure of the Λ(1405) in the Skyrme...

2018/09/11 原子核中におけるハドロンの性質とカイラル対称性の役割@東北大ELPH 1

Structure of the Λ(1405) in the Skyrme model

RCNP, Osaka University ASRC, Japan Atomic Energy Agency

Takashi Ezoe

Atsushi Hosaka


1. Introduction 2. Method (Kaon-nucleon systems) 3. KN bound state T. Ezoe and A. Hosaka, Phys. Rev. D 94, no. 3, 034022 (2016). T. Ezoe and A. Hosaka, Phys. Rev. D 96, no. 5, 054002 (2017). 4. Λ(1405) resonance 5. Summary

Contents

_


1. Introduction


Introduction1. Strange quark in a nucleus・Strange baryon in a nucleus = Hypernucleus ・Strange meson in a nucleus = Kaonic nucleus2. Kaonic nucleus・Kaonic nucleus = few body unclear system with anti-kaon 　・KN strong attraction

　・Shrink and dense stateY. Akaishi and T. Yamazaki, Phys. Rev. C 65 (2002).

A. Dote, H. Horiuchi, Y. Akaishi and T. Yamazaki, Phys. Rev. C 70 (2004).

_

3. Λ(1405) (JP = 1/2-, I = 0)・Difficult to explain by a simple quark model ・KN bound state = Λ(1405)? 　→KN and πΣ resonance = Λ(1405)

T. Hyodo, and D. Jido, Prog. Part. Nucl. Phys. 67, 55 (2012).

__

T. Hyodo and W. Weise, PRC 77 (2008).T. Hyodo and W. Weise, PRC 77 (2008).

KN interaction is important to investigate the Λ(1405)_


Introduction1. Strange quark in a nucleus・Strange baryon in a nucleus = Hypernucleus ・Strange meson in a nucleus = Kaonic nucleus2. Kaonic nucleus・Kaonic nucleus = few body unclear system with anti-kaon 　・KN strong attraction

　・Shrink and dense stateY. Akaishi and T. Yamazaki, Phys. Rev. C 65 (2002).

A. Dote, H. Horiuchi, Y. Akaishi and T. Yamazaki, Phys. Rev. C 70 (2004).

_

3. Λ(1405) (JP = 1/2-, I = 0)・Difficult to explain by a simple quark model ・KN bound state = Λ(1405)? 　→KN and πΣ resonance = Λ(1405)

T. Hyodo, and D. Jido, Prog. Part. Nucl. Phys. 67, 55 (2012).

__

KN interaction is important to investigate the Λ(1405)_

T. Hyodo and W. Weise, PRC 77 (2008).T. Hyodo and W. Weise, PRC 77 (2008).

TETSUO HYODO AND WOLFRAM WEISE PHYSICAL REVIEW C 77, 035204 (2008)

-120

-100

-80

-60

-40

-20

0

20

Im z

[MeV

]

14401420140013801360Re z [MeV]

z2 (πΣ only)

z2(2)

z2(4)

z1(2)z1

(4)

z1 (KN only)

FIG. 5. (Color online) Pole positions of the KN (I = 0) scatteringamplitude resulting from the single-channel, two-channel, and full(four-channel) models.

symmetry dictates that their interaction strength grows linearly(in leading order) with their energy. Explicit chiral symmetrybreaking gives those mesons their masses and moves themeson-baryon threshold energies to their physical values.At the kaon-nucleon threshold, the leading KN interaction,V11 ∼ − 3mK

2f 2 , becomes sufficiently attractive to produce aweakly bound state. At the π" threshold the correspondingleading π" interaction, V22 ∼ − 2mπ

f 2 , is too weak to supporta π" bound state, but it still generates a resonance abovethreshold.

At a more quantitative level, the driving attractive(Weinberg-Tomozawa) interaction terms in the S = −1meson-baryon channels, with their characteristic energy de-pendence,

V11 ≡ VKN ≃ − 32f 2

!√s − MN

"

and

V22 ≡ Vπ" ≃ − 2f 2

!√s − M"

",

generate a KN bound state and a π" resonance already inthe absence of channel couplings (i.e., for V12 = V21 = 0). Inthe typical energy range of

√s ∼ 1410–1420 MeV of interest

here, the ratio of these driving interactions,

VKN

Vπ"

∼ −34

#√s − MN√s − M"

$,

is about 1.6. The KN interaction is effectively stronger, but theπ" interaction is sizable and cannot be ignored even thoughthe π" resonance pole is located at a considerable distancefrom the real axis in the complex energy plane. The isolatedKN bound state, in contrast, has a binding energy of only5 MeV at this stage. By turning on the nondiagonal KN ↔ π"couplings, V12 = V21, both the π" resonance and KN boundstate poles move to their final positions as shown in Fig. 5.The KN bound state turns into a quasibound state embeddedin the π" continuum, with a decay width of about 35 MeV. At

the same time, by its coupling to the (virtual) KN channel, thepreviously isolated π" resonance shifts and reduces its widthby about 20%, from #π" ≃ 190 MeV to #π" ≃ 150 MeV.

These considerations imply a subtlety in assigning a mass(or a KN binding energy) to the $(1405). Empirically, theonly information at hand is the π" mass spectrum givenby the imaginary part of T22 ≡ Tπ" [see Fig. 4 (right)]. Thismass spectrum has its maximum indeed at

√s ≃ 1405 MeV.

Although its spectral shape is far from that of a Breit-Wignerresonance form, one may nevertheless read off a width ofabout 50 MeV. This is the mass and width assignment givento the $(1405) in the Particle Data Group table. However,the amplitude T eff = T11 ≡ TKN in the KN channel shown inFig. 4 (left) in the form of FKN has evidently quite differentfeatures. This is the amplitude relevant for subthresholdextrapolations of the KN interaction. The KN quasiboundstate, signaled by the zero of Re FKN , is seen to be locatedat

√s ≃ 1420 MeV, not at 1405 MeV, and almost coincides

with the maximum of Im FKN . The actual KN → π" decaywidth is about 20% smaller than the one naively identifiedwith the breadth of the π" mass spectrum. One must thereforeconclude that the KN quasibound state, commonly associatedwith the $(1405), has a binding energy of only about 12 MeV(and not the 27 MeV often used to tune phenomenologicalKN potentials).2

Given the obvious relevance of this discussion to theexistence (or nonexistence) of deeply bound K-nuclear states,it is now important to estimate theoretical uncertainties andexamine possible ambiguities.

C. Theoretical uncertainties and I = 1 amplitude

Detailed investigations have been performed concerningthe position of the second pole z2, especially its sensitivityto higher order terms in the chiral effective Lagrangian [28,29,31–33]. This section presents a conservative assessment ofsuch uncertainties, examining different chiral coupled-channelcalculations in comparison.

Several variants of the chiral unitary approach will beused in this test, all of which start from a leading Weinberg-Tomozawa (WT) term in the interaction kernel but differin their detailed treatment of subtraction constants. Thedifferences among these models have their origin in the fittingprocedures to experimental data, primarily through ambigui-ties of the π" mass spectrum with its limited data quality.

Oset-Ramos-Bennhold (ORB) [35] determine the subtrac-tion constants by matching the loop function with that obtainedin the three-momentum cutoff, with which they successfullyreproduce the observables [23]. Hyodo-Nam-Jido-Hosaka(HNJH) [34] used one single subtraction constant in allchannels to fit the data. A systematic χ2 fit was performed byBorasoy-Nißler-Weise (BNW) [29] and by Borasoy-Meißner-Nißler [33], where all the subtraction constants were used to

2In our present calculations, the isospin-averaged K mass has beenused in practice, so that the KN threshold actually appears at

√s =

1435 MeV, shifted by 3 MeV from the K−p threshold. The value ofthe binding energies just mentioned are thus understood to be shiftedby the same amount.

035204-6

Pole position of KN(I=0)_

T. Hyodo and W. Weise, PRC 77 (2008).


Introduction

Investigate the KN interaction and theΛ(1405) in the Skyrme model,

where the nucleon is described as a soliton.

_

・Phenomenological approach 　Y. Akaishi and T. Yamazaki, Phys. Rev. C 65 (2002). ・Chiral theory: based on Weinberg-Tomozawa interaction 　T. Hyodo and W. Weise, Phys. Rev. C 77 (2008). 　K. Miyahara and T. Hyodo, Phys. Rev. C 93 (2016).

4. Theoretical studies for the KN interaction_


2. Method (Kaon-nucleon systems)


Method・Describe the interaction between mesons and baryons by mesons ・Baryon emerges as a soliton of meson fields

・Skyrme model

L =F 2⇡

16tr�@µU@µU †�+ 1

32e2tr⇥(@µU)U †, (@⌫U)U†⇤2 + LSB + LWZ

・Ansatz

(U⇡ ! A (t)U⇡A† (t)

UK = UK

U⇡ =

✓UH 00 1

◆

Hedgehog soliton (2×2 matrix)

A(t): Isospin rotation matrix

T. Ezoe and A. Hosaka Phys. Rev. D 94, 034022 (2016).[1] C. G. Callan and I. Klebanov, Nucl. Phys. B 262 (1985). C. G. Callan, K .Hornbostel and I. Klebanov, Phys. Lett. B 202 (1988).

[1]

34 第 4章束縛法

2. 質量による明示的なカイラル対称性の破れは以下のLagrangianにより表される [40,

41, 43]。

Lmass =1

48F 2π

!m2

π + 2m2K

"tr#U + U † − 2

$+

√3

24

!m2

π −m2K

"tr#λ8!U + U †"$

(4.3)

ここで、mπとmKはそれぞれπ中間子とK中間子の質量を表す。本論文ではCallan

とKlebanovの論文に従い、uクォークと dクォークの質量をゼロにとり、sクォークの質量を有限の大きさにとる [28, 29]。これにより、mπ = 0となる。

K中間子に質量を与えることで対称性が明示的に破れた場合の場の配位として、K中間子が励起し、なおかつ純粋なπ中間子のソリトンに近い配位を選ぶ。解としてはカイラル対称性の非線形表現 [37, 38]を用いて以下のものを考える [28, 29]。

U =√NUK

√N (4.4)

ここで、N は SU(3)に埋め込んだπ中間子のソリトン解であり以下のように定義する。

N =

%Uπ 0

0 1

&(4.5)

Uπ ≡ ξ2 = exp

'i2

Fπτ · π

((4.6)

UK = exp

'i2

FπλaKa

(, a = 3, 4, 5, 6 (4.7)

τ はアイソスピン行列であり、その値は pauliのスピン行列 (2.6)式と同じ値をとる。そして、λa(a = 4, 5, 6, 7)は (2.36)式のGell-Mann行列である。(2.36)式を用いて λaKaを以下のように書く。

λaKa =√2

%02×2 K

K† 0

&(4.8)

ただし、Kは 2成分アイソスピナー

K =

%K+

K0

&(4.9)

である。(4.1)式、(4.3)式に (4.4)式を代入し、Kの 2次まで展開すると

L = LSU(2) + LK (4.10)

K† =

✓K0

K�

◆K =

✓K+

K0

◆

UCK = (3 ⇥ 3 matrix) ! A (t)p

U⇡UK

pU⇡A

†(t)

UEH = A (t)pU⇡A

† (t)UKA (t)pU⇡A

† (t)

J. K. Perring and T. H. R. Skyrme, Nucl. Phys. 31, 550 (1962). T. H. R. Skyrme, Proc. Roy. Soc. Lond. A 260, 127 (1961).

Parameters: Fπ, e, Mπ = 0, MK ≠ 0

UK = exp

i2

F⇡�aKa

�, a = 4, 5, 6, 7

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Difference in CK and EH(1) Callan-Klebanov Ansatz UCK = A (t)

pU⇡UK

pU⇡A

† (t)

(2) Ezoe-Hosaka Ansatz UEH = A (t)pU⇡A


† (t)

K

K Hedgehog soliton

the kaon around the hedgehog soliton

the kaon around the “rotating” hedgehog solitonHedgehog

soliton


Difference in CK and EH(1) Callan-Klebanov Ansatz UCK = A (t)

pU⇡UK

pU⇡A

† (t)

(2) Ezoe-Hosaka Ansatz UEH = A (t)pU⇡A


† (t)

K

K Hedgehog soliton

Hedgehog soliton

K ! KCK = A (t)K<latexit sha1_base64="py863BVyi7VmHAkPihY1me0vvYc=">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</latexit><latexit sha1_base64="py863BVyi7VmHAkPihY1me0vvYc=">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</latexit><latexit sha1_base64="py863BVyi7VmHAkPihY1me0vvYc=">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</latexit><latexit sha1_base64="py863BVyi7VmHAkPihY1me0vvYc=">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</latexit>

Unphysical kaon (s-quark)

K ! KEH = K<latexit sha1_base64="79m8qniYYFwhnDWOIOU1WxXz3rw=">AAAD5niclVJNa9RQFL3T+FHHj051U3ATHFrExfDSjVIQCiIUZtOvaQudMrxkXmfCJC/x5c2UMcwfEHEnXXRjCy7EnVvdufEPuOhPEJcV3Ljw5E38ajHiDck777x7Tu59XDcO/EQzdlyasM6dv3Bx8lL58pWr16Yq09c3kqivPNHwoiBSWy5PROBL0dC+DsRWrAQP3UBsur0H2fnmQKjEj+S6HsZiJ+Qd6e/6HtegWpW5pit42UbU7abyO13NlYr27Horfbg0su/b9XJTCN6qVFmNmbDPAicHVcpjOZourVOT2hSRR30KSZAkDRwQpwTPNjnEKAa3Qyk4BeSbc0EjKkPbR5ZABgfbw7eD3XbOSuwzz8SoPfwlwKugtGmWfWSv2An7wF6zT+zbX71S45HVMsTqjrUibk09mVn7+k9ViFVT95eqQOEi2+QBZZW2C/vTtEv3TF8++owNk3XsjT0Gj/dP1hZWZ9M5dsQ+o9dDdszeo1s5+OK9XBGrB4WVKOAesDZuWabEbs/cYWi6kqgvBe+C58j6gTumnhTokbl/ZbyGdAc5o0In8ZuTyNn/d2mDD3KX+OfEcMMV6SQYaWZwdGqfdYWZwWw7pyf5LNiYrzms5qzMVxcX8imfpJt0i25jku/SIi3RMjXw/6f0ht7SO6trPbOeW/vj1IlSrrlBf4T14jubqdy3</latexit><latexit sha1_base64="79m8qniYYFwhnDWOIOU1WxXz3rw=">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</latexit><latexit sha1_base64="79m8qniYYFwhnDWOIOU1WxXz3rw=">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</latexit><latexit sha1_base64="79m8qniYYFwhnDWOIOU1WxXz3rw=">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</latexit>

Physical kaon


Derivation 11. Substitute our ansatz for the Lagrangian

Lagrangian

L =F 2⇡

16tr�@µU@µU†�+ 1

32e2tr⇥(@µU)U†, (@⌫U)U †⇤2

+LSB + LWZ

Ansatz

34 第 4章束縛法

2. 質量による明示的なカイラル対称性の破れは以下のLagrangianにより表される [40,

41, 43]。

Lmass =1

48F 2π

!m2

π + 2m2K

"tr#U + U † − 2

$+

√3

24

!m2

π −m2K

"tr#λ8!U + U †"$

(4.3)

ここで、mπとmKはそれぞれπ中間子とK中間子の質量を表す。本論文ではCallan

とKlebanovの論文に従い、uクォークと dクォークの質量をゼロにとり、sクォークの質量を有限の大きさにとる [28, 29]。これにより、mπ = 0となる。

K中間子に質量を与えることで対称性が明示的に破れた場合の場の配位として、K中間子が励起し、なおかつ純粋なπ中間子のソリトンに近い配位を選ぶ。解としてはカイラル対称性の非線形表現 [37, 38]を用いて以下のものを考える [28, 29]。

U =√NUK

√N (4.4)

ここで、N は SU(3)に埋め込んだπ中間子のソリトン解であり以下のように定義する。

N =

%Uπ 0

0 1

&(4.5)

Uπ ≡ ξ2 = exp

'i2

Fπτ · π

((4.6)

UK = exp

'i2

FπλaKa

(, a = 3, 4, 5, 6 (4.7)

τ はアイソスピン行列であり、その値は pauliのスピン行列 (2.6)式と同じ値をとる。そして、λa(a = 4, 5, 6, 7)は (2.36)式のGell-Mann行列である。(2.36)式を用いて λaKaを以下のように書く。

λaKa =√2

%02×2 K

K† 0

&(4.8)

ただし、Kは 2成分アイソスピナー

K =

%K+

K0

&(4.9)

である。(4.1)式、(4.3)式に (4.4)式を代入し、Kの 2次まで展開すると

L = LSU(2) + LK (4.10)

K =

✓K+

K0

◆

UK = exp

i2

F⇡�aKa

�, a = 4, 5, 6, 7

U⇡ =

✓UH 00 1

◆UH: Hedgehog soliton (2×2 matrix)

UEH = A (t)pU⇡A

† (t)UKA (t)p

U⇡A† (t)


Obtained LagrangianL = LSU(2) + LKN

U = A(t)UHA†(t), ⇠ = A(t)pUHA†(t) DµK = @µK + vµK

vµ =1

2

⇣⇠†@µ⇠ + ⇠@µ⇠

†⌘

aµ =1

2

⇣⇠†@µ⇠ � ⇠@µ⇠

†⌘

G. S. Adkins, C. R. Nappi and E. Witten, Nucl. Phys. B 228 (1983)

LSU(2) =1

16F⇡

2trh@µU

†@µUi+

1

32e2trh@µU U†, @⌫U U†

i2

LKN = (DµK)† DµK �K†a†µaµK �m2

KK†K

+1

(eF⇡)2

⇢�K†Ktr

h@µU U†, @⌫U U†

i2� 2 (DµK)† D⌫Ktr (aµa⌫)

�1

2(DµK)† DµKtr

⇣@⌫U

†@⌫U⌘+ 6 (D⌫K)† [a⌫ , aµ]DµK

�

+3i

F 2⇡

Bµh(DµK)† K �K† (DµK)

i

Bµ = �"µ⌫↵�

24⇡2trh⇣

U †@⌫U⌘⇣

U†@↵U⌘⇣

U†@�U⌘i

2. Expand UK up to second order of the kaon field K


3. Decompose the kaon filedSpatial wave function

4. Expand the K(r) by the spherical harmonicsYlm(θ, φ): Spherical harmonics l：orbital angular momentum m：the 3rd component of l α：the other quantum numbers

K (r) =X

l,m

Clm↵Ylm (✓,�) k↵l (r)

Isospin wave function

5. Take a variation with respect to the kaon radial function ⇒Obtain the equation of motion for the kaon around the nucleon

✓K+

K0

◆= IK (t, r) ! I K (r) e�iEt

Derivation 2


3. KN bound state

T. Ezoe. and A. Hosaka Phys. Rev. D 94, 034022 (2016). T. Ezoe and A. Hosaka, Phys. Rev. D 96, 054002 (2017).

_


Result: E.o.M and potential1. Equation of motion(E.o.M)� 1

r2d

dr

✓r2h(r)

dk↵l (r)

dr

◆� E2f(r)k↵l (r) +

�m2

K + V (r)�k↵l (r) = 0 : Klein-Gordon like


Result: E.o.M and potential1. Equation of motion(E.o.M)

� 1

mK + E

1

r2d

dr

✓r2

dk↵l (r)

dr

◆+ U (r)k↵l (r) = "k↵l (r)

� 1

r2d

dr

✓r2h(r)

dk↵l (r)

dr

◆� E2f(r)k↵l (r) +

�m2

K + V (r)�k↵l (r) = 0

U (r) =U (r) k↵l (r)

k↵l (r)

2. Properties of resulting potential U 1. Nonlocal and depend on the kaon energy 2. Contain isospin dependent and independent central forces 　 and the similar spin-orbit(LS) forces 3. A repulsive component is proportional to 1/r2 at short distances

3. Equivalent local potential:

U (r) = � 1

mK + E

h(r)� 1

r2d

dr

✓r2

d

dr

◆+

dh(r)

dr

d

dr

�� (f(r)� 1)E2

mK + E+

V (r)

mK + E

= U c0 (r) + U c

⌧ (r) ⌧K · ⌧N +

�ULS0 (r) + ULS

⌧ (r) ⌧K · ⌧N�L · S

: Klein-Gordon like

: Schrödinger like(E = mK + ε)

2018/09/11 原子核中におけるハドロンの性質とカイラル対称性の役割@東北大ELPH

KN (JP = 1/2-, I = 0) Bound stateWave func

tion [1/fm

3/2 ]

Wave function Potential

⌦r2N

↵=

Z 1

0dr r2⇢B (r) , ⇢B (r) = � 2

⇡sin2 FF 0 G. S. Adkins, C. R. Nappi and E. Witten,

Nucl. Phys. B 228 (1983)⌦r2K

↵=

ZdV r2

⇥Y00 (r) k

00 (r)

⇤2=

Z 1

0dr r4k2 (r) Y00 =

1p4⇡

・Parameter sets and bound state properties

Repulsive core

Attractive pocket

mK = 495 MeV mK = 495 MeV

F⇡ [MeV] e B.E. [MeV]p

hr2N i [fm]p

hr2Ki [fm]Parameter set A 205 4.67 19.9 0.43 1.30Parameter set B 186 4.82 32.2 0.46 1.15

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_


4. Λ(1405) resonance


・Λ(1405) = KN and πΣ resonance

Λ(1405) resonance(|⇤ (1405)iH = aH |KNires + bH |⇡⌃ires|⇤ (1405)iL = aL |KNires + bL |⇡⌃ires

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h⇡⌃| L |⇤ (1405)Hi = aH h⇡⌃| L1 |KNires + bH h⇡⌃| L2 |⇡⌃ires<latexit sha1_base64="yL2Qfg8XFzTrzE6VLLYELgZYAME=">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</latexit><latexit sha1_base64="yL2Qfg8XFzTrzE6VLLYELgZYAME=">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</latexit><latexit sha1_base64="yL2Qfg8XFzTrzE6VLLYELgZYAME=">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</latexit><latexit sha1_base64="yL2Qfg8XFzTrzE6VLLYELgZYAME=">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</latexit>

Decay from Λ(1405)(higher) to πΣ


-120

-100

-80

-60

-40

-20

0

20

Im z

[MeV

]

14401420140013801360Re z [MeV]

z2 (πΣ only)

z2(2)

z2(4)

z1(2)z1

(4)

z1 (KN only)






V11 ≡ VKN ≃ − 32f 2

!√s − MN

"

and

V22 ≡ Vπ" ≃ − 2f 2

!√s − M"

",




VKN

Vπ"

∼ −34

#√s − MN√s − M"

$,




√s ≃ 1405 MeV.










√s =


035204-6



_







＿

NΣ

πK

di-quark

s-quarkAxial current: J5,aµ

L1 / @µ⇡aJ5,aµ

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-120

-100

-80

-60

-40

-20

0

20

Im z

[MeV

]

14401420140013801360Re z [MeV]

z2 (πΣ only)

z2(2)

z2(4)

z1(2)z1

(4)

z1 (KN only)






V11 ≡ VKN ≃ − 32f 2

!√s − MN

"

and

V22 ≡ Vπ" ≃ − 2f 2

!√s − M"

",




VKN

Vπ"

∼ −34

#√s − MN√s − M"

$,




√s ≃ 1405 MeV.










√s =


035204-6



_







Σ

π

di-quark

s-quarkVector current: Jaµ

L2 /�✏abc⇡b@µ⇡c

�Jaµ

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di-quark

s-quarkΣ

π


-120

-100

-80

-60

-40

-20

0

20

Im z

[MeV

]

14401420140013801360Re z [MeV]

z2 (πΣ only)

z2(2)

z2(4)

z1(2)z1

(4)

z1 (KN only)






V11 ≡ VKN ≃ − 32f 2

!√s − MN

"

and

V22 ≡ Vπ" ≃ − 2f 2

!√s − M"

",




VKN

Vπ"

∼ −34

#√s − MN√s − M"

$,




√s ≃ 1405 MeV.










√s =


035204-6



_


5. Summary


SummaryConstruct a new method to investigate 1. the kaon-nucleon systems 2. the Λ(1405) resonance in the Skyrme model

Future work ・coupling constant for the “πΣ-πΣ” vertex

Results 1. Properties of the obtained potential a. nonlocal and depends on the kaon energy b. contain central and LS terms

with and without isospin dependence 　 c. repulsion proportional to 1/r2 for small r 2. KN(I = 0) bound states exist with B.E. of order ten MeV 3. 1/r2 repulsion at short distances is related to the rotating Hedgehog soliton

_


Thank you for you attention!!

Structure of the Λ(1405) in the Skyrme...

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