# K - pp studied with Coupled-channel Complex Scaling method

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K-pp studied with Coupled-channel Complex Scaling methodWorkshop on Hadron and Nuclear Physics (HNP09) 09.11.19 @ Arata hall, Osaka univ., Ibaraki, Osaka, JapanIntroduction

Question --- Variational calculation and Faddeev ---

Complex Scaling Method

Result

Summary and Future planA. Dot (KEK) and T. Inoue (Tsukuba univ.)Two-body K-p system (1405)studied with AY potentialNot new calculation but confirmation of the past work

Kbar nuclei (Nuclei with anti-koan) = Exotic system !?I=0 KbarN potential very attractiveDeeply bound (Total B.E. 100MeV)Highly dense state formed in a nucleusInteresting structures that we have never seen in normal nucleiTo make the situation more clear K-pp = Prototype of Kbar nuclei(KbarNN, Jp=1/2-, T=1/2)K-

3He0.0 0.15

K- ppn0.0 2.0

K- ppp3 x 3 fm20.0 2.0 [fm-3]Calculated with Antisymmetrized Molecular Dynamics method employing a phenomenological KbarN potential

A. Dot, H. Horiuchi, Y. Akaishi and T. Yamazaki, PRC70, 044313 (2004)1. Introduction1. IntroductionVariational calculation of K-pp with a chiral SU(3)-based KbarN potential Av18 NN potential a realistic NN potential with strong repulsive core. Effective KbarN potential based on Chiral SU(3) theory Variational method Investigate various properties with the obtained wave function. reproduce the original KbarN scattering amplitude obtained with coupled channel chiral dynamics.

Single channel, Energy dependent, Complex, Gaussian-shape potentialFour variants of chiral unitary models

Total B. E. : 20 3 MeVGKbarNpY : 40 70 MeVShallow binding and large decay widthNN distance = 2.2 fmKbarN distance = 2.0 fmNN distance in normal nucleiA. Dot, T. Hyodo and W. Weise,Nucl. Phys. A804, 197 (2008)Phys. Rev. C79, 014003 (2009)

I=0 KbarN resonance (1405)appears at 1420 MeV, not 1405 MeVI=0 KbarN resonance (1405)appears at 1420 MeV, not 1405 MeV3Width (KbarNNYN) [MeV]- B.E. [MeV]Dot, Hyodo, Weise [1](Variational, Chiral SU(3))Akaishi, Yamazaki [2](Variational, Phenomenological)Exp. : FNUDA [5]Exp. : DISTO [6](Preliminary)Using S-wave KbarN potentialconstrained by experimental data. KbarN scattering data, Kaonic hydrogen atom data, (1405) etc.Ikeda, Sato [4](Faddeev, Chiral SU(3))Shevchenko, Gal [3](Faddeev, Phenomenological)[1] PRC79, 014003 (2009)[2] PRC76, 045201 (2007)[3] PRC76, 044004 (2007)[4] PRC76, 035203 (2007)[5] PRL94, 212303 (2005)[6] arXiv:0810.5182 [nucl-ex]Recent results of calculation of K-pp Recent results of calculation of K-pp and related experiments4Width (KbarNNYN) [MeV]- B.E. [MeV]Dot, Hyodo, Weise [1](Variational, Chiral SU(3))Akaishi, Yamazaki [2](Variational, Phenomenological)Exp. : FNUDA [5]Exp. : DISTO [6](Preliminary)Recent results of calculation of K-pp and related experimentsUsing S-wave KbarN potentialconstrained by experimental data. KbarN scattering data, Kaonic hydrogen atom data, (1405) etc.Ikeda, Sato [4](Faddeev, Chiral SU(3))Shevchenko, Gal [3](Faddeev, Phenomenological)[1] PRC79, 014003 (2009)[2] PRC76, 045201 (2007)[3] PRC76, 044004 (2007)[4] PRC76, 035203 (2007)[5] PRL94, 212303 (2005)[6] arXiv:0810.5182 [nucl-ex]52. Question --- Variational calc. and Faddeev ---The KbarN potentials used in both calculations are constrained with Chiral SU(3) theory, but Total B. E. = 203 MeV, Decay width = 4070 MeVA. Dot, T. Hyodo and W. Weise,Phys. Rev. C79, 014003 (2009)Variational calculation(DHW)Faddeev calculation(IS)Total B. E. = 6095 MeV, Decay width = 4580 MeVY. Ikeda, and T. Sato,Phys. Rev. C76, 035203 (2007)??? Discrepancy between Variational calc. and Faddeev calc. Separable potential used in Faddeev calculation? Non-relativistic (semi-relativistic) vs relativistic? Energy dependence of two-body system (KbarN) in the three-body system (KbarNN)? ???

Why ?A possible reason is N thee-body dynamicsY. Ikeda and T. Sato, Phys. Rev. C79, 035201(2009) This is correct for the two-body system. The effective KbarN potential has energy dependence due to the elimination of channel. It reproduces the original KbarN scattering amplitude calculated with KbarN- coupled channel model.Two-body systemIn the variational calculation (DHW), channel is eliminated and incorporated into the effective KbarN potential.KN+ + + = KNKNKNKNKNV11V1iVj1

2. Question --- Variational calc. and Faddeev ---Three-body system calculated with the effective KbarN potentialKN+ + + = KNKNKNKNKNNNNNNNThe energy of the intermediate state is fixed to the initial KbarN energy.Not true!

Apply the effective KbarN potential to the three-body system

2. Question --- Variational calc. and Faddeev ---Three-body system calculated with the effective KbarN potentialKN+ + + = KNKNKNKNKNNNNNNNDue to the third nucleon, the intermediate energy can differ from the initial KbarN energy.NN

Apply the effective KbarN potential to the three-body system

2. Question --- Variational calc. and Faddeev ---Three-body system calculated with the effective KbarN potentialKN+ = KNKNKNNNNNDue to the third nucleon, the intermediate energy can differ from the initial KbarN energy.NNOnly the total energy of the three-body system should be conserved. The intermediate energy of the state can be variable, due to the third nucleon.

conservedN thee-body dynamics

Apply the effective KbarN potential to the three-body system

2. Question --- Variational calc. and Faddeev ---N three-body dynamics is taken into account in the Faddeev calculation, because the channel is directly treated in their calculation.

Dr. Ikedas talk in KEK theory center workshop Nuclear and Hadron physics(KEK, 11-13 Aug. 09)2. Question --- Variational calc. and Faddeev ---Coupled channel calculationDirect treatment of the degree of freedom The obtained state is possibly to appear below KbarNN threshold, but above N threshold. The actual calculation is done in multi channels such as KbarN and . A bound state for KbarNN channel, Kbar + N + NKbar N N + + Nbut a resonant state for N channelResonant state cant be treated with a variational method. Here, we employ

Complex Scaling method

to deal with resonant state.3. Complex Scaling method3. Complex Scaling method

The resonant state can be obtained by diagonalizing with Gaussian basis, quite in the same way as calculating bound state.

When, the wave function of a resonant state changes to a dumping function.

Wave function of a resonant state (two-body system)Negative!Boundary condition is the same as that for a bound state.Complex rotation of coordinateAdvised by Dr. Myo (RCNP)The energy of bound and resonant states is independent of scaling angle . (ABC theorem) Two-body system: KbarN- coupled system with s-wave and isospin zero state (1405)Here, we consider just two-channel problem; KbarN and channels

Employ Akaishi-Yamazaki potential

Y. Akaishi and T. Yamazaki, PRC 52 (2002) 044005Schrdinger equation to be solved

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

4. Result4. Resultq trajectory # Gauss base (n) = 30 Max range (b) = 10 [fm]E-G / 2[MeV]

q = 0 deg.2q 4. ResultE-G / 2[MeV]

q = 5 deg.q trajectory # Gauss base (n) = 30 Max range (b) = 10 [fm]2q 4. ResultE-G / 2[MeV]

q =10 deg.q trajectory # Gauss base (n) = 30 Max range (b) = 10 [fm]2q 4. ResultE-G / 2[MeV]

q =15 deg.q trajectory # Gauss base (n) = 30 Max range (b) = 10 [fm]2q 4. ResultE-G / 2[MeV]

q =20 deg.q trajectory # Gauss base (n) = 30 Max range (b) = 10 [fm]2q 4. ResultE-G / 2[MeV]

q =25 deg.q trajectory # Gauss base (n) = 30 Max range (b) = 10 [fm]4. ResultE-G / 2[MeV]

q =30 deg.q trajectory # Gauss base (n) = 30 Max range (b) = 10 [fm]2q 4. ResultE-G / 2[MeV]

q =35 deg.q trajectory # Gauss base (n) = 30 Max range (b) = 10 [fm]2q 4. ResultE-G / 2[MeV]

q =40 deg.q trajectory # Gauss base (n) = 30 Max range (b) = 10 [fm]2q 4. ResultE-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) !4. Result

b trajectory # Gauss base (n) = 30 Rotation angle (q) = 30 [deg]

n trajectory Max range (b) = 10 [fm] Rotation angle (q) = 30 [deg]

b = 50 [fm]Stability of the resonance for other parameters # Gauss base (n) = 30 Max range (b) = 10 [fm]Analysis of trajectory withResonance found at

B. E. (KbarN) = 28.2 MeV = 40.0 MeV5. Summary and Future plan No other resonant state (no interaction between and ) Solve the Coupled Channel problem = KbarN + Complex Scaling Method applying to (1405)Resonance found at

B. E. (KbarN) = 28.2 MeV = 40.0 MeV In case of AY potential, Comment In case of only VKN-KN, a bound state appears at B. E. = 3.2 MeV. Coupling with continuum state increases drastically the binding energy. Also, it gives the decay width.

KbarN thr.VKN-VKN-KN5. Summary and Future planFuture planKeep doing calculations!2. Calculate three-body system KbarNN-N system corresponding to K-pp1. Calculate two-body system KbarN- system corresponding to (1405) See what happen if we use a chiral SU(3)-based KbarN potential (HW potential, energy-dependent). T. Hyodo and W. Weise, PRC77, 035204(2008) For a test calculation, a phenomenological KbarN potential (AY potential, energy-independent) will be employed. Y. Akaishi and T. Yamazaki, PRC 52 (2002) 044005 Done Investigate the property of the obtained wave functionand thinking

J-PARC will give us lots of interesting data!E15: A search for deeply bound kaonic nuclear states by 3He(inflight K-, n) reaction --- Spokespersons: M. Iwasaki (RIKEN), T. Nagae (Kyoto)

E17: Precision spectroscopy of kaonic 3He atom 3d2p X-rays --- Spokespersons: R. Hayano (Tokyo), H. Outa (Riken)Thank you very much!Acknowledgement:Dr. Myo (RCNP) taught me CSM and supplied a subroutine for diagonalization of a complex matrix.