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

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Application of coupled-channel Complex Scaling Method to Λ(1405) 1. Introduction Recent status of theoretical study of K - pp 2. Application of ccCSM to Λ(1405) Coupled-channel complex scaling method (ccCSM) Energy-independent K bar N potential 3. ccCSM with an energy-dependent K bar N potential for Λ(1405) 4. Summary and Future plan A. Doté (KEK Theory center) T. Inoue (Nihon univ.) T. Myo (Osaka Tech. univ.) International conference on the structure of baryons (BARYONS ’10.12.10 (7-11) @ Convention center, Osaka univ.,
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Application of coupled-channel Complex Scaling Method to Λ(1405). A. Doté ( KEK Theory center) T. Inoue (Nihon univ .) T. Myo (Osaka Tech. univ .). Introduction Recent status of theoretical study of K - pp Application of ccCSM to Λ(1405) - PowerPoint PPT Presentation

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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