# Application of coupled-channel Complex Scaling Method to Λ(1405)

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Application of coupled-channel Complex Scaling Method to (1405)IntroductionRecent status of theoretical study of K-pp

Application of ccCSM to (1405)Coupled-channel complex scaling method (ccCSM)Energy-independent KbarN potential

ccCSM with an energy-dependent KbarN potential for (1405)

Summary and Future planA. Dot (KEK Theory center)T. Inoue (Nihon univ.)T. Myo (Osaka Tech. univ.)International conference on the structure of baryons (BARYONS 10)10.12.10 (7-11) @ Convention center, Osaka univ., Japan1. Introduction1. IntroductionKbar nuclei = Exotic system !?I=0 KbarN potential very attractiveHighly dense state formed in a nucleusInteresting structures that we have never seen in normal nuclei Recently, ones have focused onK-pp= Prototye of Kbar nucleiK-Recent results of calculation of K-pp and related experimentsWidth (KbarNNYN) [MeV]- B.E. [MeV]Dote, Hyodo, Weise (Variational, Chiral SU(3))Akaishi, Yamazaki (Variational, Phenomenological)Shevchenko, Gal (Faddeev, Phenomenological)Ikeda, Sato(Faddeev, Chiral SU(3))Exp. : FINUDAif K-pp bound stateExp. : DISTOif K-pp bound stateConstrained by experimental data. KbarN scattering data, Kaonic hydrogen atom data, (1405) etc.1. Introduction4Recent results of calculation of K-pp and related experimentsWidth (KbarNNYN) [MeV]- B.E. [MeV]Dote, Hyodo, Weise (Variational, Chiral SU(3))Akaishi, Yamazaki (Variational, Phenomenological)Shevchenko, Gal (Faddeev, Phenomenological)Ikeda, Sato(Faddeev, Chiral SU(3))Exp. : FINUDAif K-pp bound stateExp. : DISTOif K-pp bound stateConstrained by experimental data. KbarN scattering data, Kaonic hydrogen atom data, (1405) etc.1. Introduction5Recent results of calculation of K-pp and related experimentsWidth (KbarNNYN) [MeV]- B.E. [MeV]Dote, Hyodo, Weise (Variational, Chiral SU(3))Akaishi, Yamazaki (Variational, Phenomenological)Shevchenko, Gal (Faddeev, Phenomenological)Ikeda, Sato(Faddeev, Chiral SU(3))Exp. : FINUDAif K-pp bound stateExp. : DISTOif K-pp bound stateConstrained by experimental data. KbarN scattering data, Kaonic hydrogen atom data, (1405) etc.1. IntroductionThree-body system calculated with the effective KbarN potentialKN+ = KNKNKNNNNNNN

conserved

N thee-body dynamics61. IntroductionKbar nuclei = Exotic system !?I=0 KbarN potential very attractiveHighly dense state formed in a nucleusInteresting structures that we have never seen in normal nuclei Recently, ones have focused onK-pp= Prototye of Kbar nucleiK- In the study of K-pp, it was pointed out that the N three-body dynamicsmight be important. Based on the variational approach, and explicitly treating the N channel, we try to investigate KbarNN-N resonant state with coupled-channel Complex Scaling MethodKbar + N + NKbar N N + + N(1405) : I=0 quasi-bound state of K-p two-body systemBefore K-pp, Kaonic nuclei sdtudied with Complex Scaling Method2. Application of CSM to (1405) Coupled-channel Complex Scaling Method (ccCSM)

Energy-independent KbarN potentialKbarN- coupled system with s-wave and isospin-0 state(1405) with c.c. Complex Scaling MethodKbar + N (1405) + 14351332 [MeV]B. E. (KbarN) = 27 MeV () 50 MeVJ = 1/2-I = 0Kbar(J=0-, T=1/2)N (J=1/2+, T=1/2)L=0 (J=0-, T=1)(J=1/2+, T=1) L=0

Schrdinger equation to be solved

: complex parameters to be determinedWave function expanded with Gaussian baseComplex-rotate , then diagonalize with Gaussian base.

(1405) with c.c. Complex Scaling MethodPhenomenological potentialY. Akaishi and T. Yamazaki, PRC 52 (2002) 044005= Energy independent potentialChiral SU(3) potentialN. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995)= Energy dependent potential

Complex scaling of coordinate

ABC theorem

The energy of bound and resonant states is independent of scaling angle .

J. Aguilar and J. M. Combes, Commun. Math. Phys. 22 (1971),269E. Balslev and J. M. Combes, Commun. Math. Phys. 22 (1971),280 2. Application of CSM to (1405) Coupled-channel Complex Scaling Method (ccCSM)

Energy-independent KbarN potentialPhenomenological potential (AY)Y. Akaishi and T. Yamazaki, PRC 52 (2002) 044005Energy-independent potential

KbarNfree KbarN scattering data1s level shift of kaonic hydrogen atomBinding energy and width of (1405) = K- + protonThe result that I show hereafter is not new, because the same calculation was done by Akaishi-san, when he made AY potential. Remark !q trajectory # Gauss base (n) = 30 Max range (b) = 10 [fm]E-G / 2[MeV]

q = 0 deg.2q (1405) with c.c. Complex Scaling MethodE-G / 2[MeV]

q = 5 deg.q trajectory # Gauss base (n) = 30 Max range (b) = 10 [fm]2q (1405) with c.c. Complex Scaling MethodE-G / 2[MeV]

q =10 deg.q trajectory # Gauss base (n) = 30 Max range (b) = 10 [fm]2q (1405) with c.c. Complex Scaling MethodE-G / 2[MeV]

q =15 deg.q trajectory # Gauss base (n) = 30 Max range (b) = 10 [fm]2q (1405) with c.c. Complex Scaling MethodE-G / 2[MeV]

q =20 deg.q trajectory # Gauss base (n) = 30 Max range (b) = 10 [fm]2q (1405) with c.c. Complex Scaling MethodE-G / 2[MeV]

q =25 deg.q trajectory # Gauss base (n) = 30 Max range (b) = 10 [fm](1405) with c.c. Complex Scaling MethodE-G / 2[MeV]

q =30 deg.q trajectory # Gauss base (n) = 30 Max range (b) = 10 [fm]2q (1405) with c.c. Complex Scaling MethodE-G / 2[MeV]

q =35 deg.q trajectory # Gauss base (n) = 30 Max range (b) = 10 [fm]2q (1405) with c.c. Complex Scaling MethodE-G / 2[MeV]

q =40 deg.q trajectory # Gauss base (n) = 30 Max range (b) = 10 [fm]2q (1405) with c.c. Complex Scaling MethodE-G / 2[MeV]q trajectory2q

q =30 deg.pS KbarNpS continuumKbarN continuumResonance!(E, /2) = (75.8, 20.0)Measured from KbarN thr.,

B. E. (KbarN) = 28.2 MeV = 40.0 MeV L(1405) !(1405) with c.c. Complex Scaling Method3. ccCSM with an

energy-dependent potential

for (1405)Chiral SU(3) potential (KSW)N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995)Original: -function type

Energy dependence is determined by Chiral low energy theorem. Kaon: Nambu-Goldstone bosonChiral SU(3) potential (KSW)N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995)Original: -function typePresent: Normalized Gaussian type

a: range parameter [fm]

Energy dependence is determined by Chiral low energy theorem. Kaon: Nambu-Goldstone bosonChiral SU(3) potential (KSW)N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995)Original: -function typePresent: Normalized Gaussian type

a: range parameter [fm]Mi , mi : Baryon, Meson mass in channel iEi : Baryon energy, i : Meson energy

Reduced energy:

KbarNEnergy dependence of Vij is controlled by CM energy s.

Energy dependence is determined by Chiral low energy theorem. Kaon: Nambu-Goldstone bosonFlavor SU(3) symmetryChiral SU(3) potential (KSW)Energy dependence

s [MeV]KbarN-KbarN-KbarN-N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995)KbarN threshold thresholdChiral SU(3) potential = Energy-dependent potential Calculational procedurePerform the Complex Scaling method.Then, find a pole of resonance or bound state.

CheckFinished !If YesSelf consistency for the energy!Assume the values of the CM energy s.

If No

ResultRange parameter (a) and pion-decay constant f are ambiguous in this model. Various combinations (a,f) are tried. f = 95 105 MeVSelf consistency for real energy

-B (Assumed) [MeV]-B (Calculated) [MeV]a=0.60a=0.56a=0.54a=0.52a=0.51a=0.50a=0.49a=0.48a=0.44a=0.45f = 100 MeVNo resonance for a>0.60s [MeV]1435KbarN

Resonant state

Self consistency for real energy-B (Assumed) [MeV]-B (Calculated) [MeV]s [MeV]1435a=0.48a=0.45a=0.44f = 100 MeV bound state1331a=0.43No self-consistent solution for a