# Application of coupled-channel Complex Scaling Method to the K bar N -πY system

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Application of coupled-channel Complex Scaling Method to the KbarN-Y systemIntroduction

coupled-channel Complex Scaling MethodFor resonance statesFor scattering problem

Set up for the calculation of KbarN-Y systemKSW-type potentialKinematics / Non-rela. Approximation

ResultsI=0 channel: Scattering amplitude / Resonance poleI=1 channel: Scattering amplitude

Summary and future plansA. Dot (KEK Theory center / IPNS/ J-PARC branch)T. Inoue (Nihon univ.)T. Myo (Osaka Inst. Tech. univ.)International workshop Inelastic Reactions in Light Nuclei, 09.Oct.13 @ IIAS, The Hebrew univ., Jerusalem, IsraelA. Dote, T. Inoue, T. Myo,Nucl. Phys. A912, 66-101 (2013)1. IntroductionKaonic nuclei = Nuclear system with Anti-kaon Kbar Excited hyperon (1405) = a quasi-bound state of K- and protonAttractive KbarN interaction!Difficult to explain by naive 3-quark modelConsistent with repulsive nature indicated by KbarN scattering length and 1s level shift of kaonic hydron atom

ubarsuudsudT. Hyodo and D. Jido, Prog. Part. Nucl. Phys. 67, 55 (2012)KbarN threshold = 1435MeVKaonic nuclei = Nuclear system with Anti-kaon Kbar Excited hyperon (1405) = a quasi-bound state of K- and protonAttractive KbarN interaction!Difficult to explain by naive 3-quark modelConsistent with repulsive nature indicated by KbarN scattering length and 1s level shift of kaonic hydron atomAntisymmetrized Molecular Dynamics with a phenomenological KbarN potentialAnti-kaon can be deeply bound and/or form dense state?Anti-kaon bound in light nuclei with ~ 100MeV binding energyDense matter (average density = 24 0)Relativistic Mean Field applied to medium to heavy nuclei with anti-kaonsAMD studyA. D., H. Horiuchi, Y. Akaishi and T. Yamazaki, PLB 590 (2004) 51; PRC 70 (2004) 044313.

RMF studyD. Gazda, E. Friedman, A. Gal, J. Mares, PRC 76 (2007) 055204; PRC 77 (2008) 045206;A. Cieply, E. Friedman, A. Gal, D. Gazda, J. Mares, PRC 84 (2011) 045206.T. Muto, T. Maruyama, T. Tatsumi, PRC 79 (2009) 035207.Kaonic nuclei = Nuclear system with Anti-kaon Kbar Excited hyperon (1405) = a quasi-bound state of K- and protonAttractive KbarN interaction!Difficult to explain by naive 3-quark modelConsistent with repulsive nature indicated by KbarN scattering length and 1s level shift of kaonic hydron atomAntisymmetrized Molecular Dynamics with a phenomenological KbarN potentialAnti-kaon can be deeply bound and/or form dense state?Anti-kaon bound in light nuclei with ~ 100MeV binding energyDense matter (average density = 24 0)Relativistic Mean Field applied to medium to heavy nuclei with anti-kaonsPrototype of kaonic nuclei = K-ppVariational calculation with Gaussian baseAv18 NN potentialChiral SU(3)-based KbarN potentialB. E. ~ 20MeV, = 40 ~70MeVpp distance ~ 2.1 fmA. D., T. Hyodo, W. Weise, PRC79, 014003 (2009)Theoretical studies of K-ppDot, Hyodo, Weise PRC79, 014003(2009) Variational with a chiral SU(3)-based KbarN potentialAkaishi, Yamazaki PRC76, 045201(2007) ATMS with a phenomenological KbarN potentialIkeda, Sato PRC76, 035203(2007) Faddeev with a chiral SU(3)-derived KbarN potentialShevchenko, Gal, Mares PRC76, 044004(2007) Faddeev with a phenomenological KbarN potentialBarnea, Gal, Liverts PLB712, 132(2012) Hyperspherical harmonics with a chiral SU(3)-based KbarN potential Wycech, Green PRC79, 014001(2009) Variational with a phenomenological KbarN potential (with p-wave)

Arai, Yasui, Oka/ Uchino, Hyodo, Oka PTP119, 103(2008) * nuclei model /PTPS 186, 240(2010)Nishikawa, Kondo PRC77, 055202(2008) Skyrme modelK-pp = Prototype of Kbar nuclei(KbarNN, Jp=1/2-, T=1/2)All studies predict that K-pp can be bound!Theoretical studies of K-ppDot, Hyodo, Weise PRC79, 014003(2009) Variational with a chiral SU(3)-based KbarN potentialAkaishi, Yamazaki PRC76, 045201(2007) ATMS with a phenomenological KbarN potentialIkeda, Sato PRC76, 035203(2007) Faddeev with a chiral SU(3)-derived KbarN potentialShevchenko, Gal, Mares PRC76, 044004(2007) Faddeev with a phenomenological KbarN potentialBarnea, Gal, Liverts PLB712, 132(2012) Hyperspherical harmonics with a chiral SU(3)-based KbarN potential Wycech, Green PRC79, 014001(2009) Variational with a phenomenological KbarN potential (with p-wave)

Arai, Yasui, Oka/ Uchino, Hyodo, Oka PTP119, 103(2008) * nuclei model /PTPS 186, 240(2010)Nishikawa, Kondo PRC77, 055202(2008) Skyrme modelK-pp = Prototype of Kbar nuclei(KbarNN, Jp=1/2-, T=1/2)All studies predict that K-pp can be bound!Width (KbarNNYN) [MeV]- B.E. [MeV]Dot, Hyodo, Weise [1](Variational, Chiral SU(3))Akaishi, Yamazaki [2](Variational, Phenomenological)Exp. : FINUDA [6]if K-pp bound state.Exp. : DISTO [7]if K-pp bound state.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, Mares [3](Faddeev, Phenomenological)[1] PRC79, 014003 (2009) [5] PLB94, 712 (2012)[2] PRC76, 045201 (2007)[3] PRC76, 044004 (2007)[4] PRC76, 035203 (2007)[6] PRL94, 212303 (2005)[7] PRL104, 132502 (2010)Typical results of theoretical studies of K-pp Barnea, Gal, Liverts [5](HH, Chiral SU(3))-100 MeV8Width (KbarNNYN) [MeV]- B.E. [MeV]Dot, Hyodo, Weise [1](Variational, Chiral SU(3))Akaishi, Yamazaki [2](Variational, Phenomenological)Exp. : FINUDA [6]if K-pp bound state.Exp. : DISTO [7]if K-pp bound state.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, Mares [3](Faddeev, Phenomenological)[1] PRC79, 014003 (2009) [5] PLB94, 712 (2012)[2] PRC76, 045201 (2007)[3] PRC76, 044004 (2007)[4] PRC76, 035203 (2007)[6] PRL94, 212303 (2005)[7] PRL104, 132502 (2010)Typical results of theoretical studies of K-pp Barnea, Gal, Liverts [5](HH, Chiral SU(3))Kbar+N+N threshold++N thresholdFrom theoretical viewpoint, K-pp exists between Kbar-N-N and --N thresholds!9Key points to study kaonic nucleiSimilar to the bound-state calculation1. Consider a coupled-channel problemKbar + N + NKbar N N + + N2. Treat resonant states adequately.Coupling of KbarN and YBound below KbarN threshold, but a resonant state above Y threshold.Their nature?3. Obtain the wave function to help the analysis of the state.coupled-channel Complex Scaling Method

KbarN (-Y) interaction??4. Confirmed that CSM works well on many-body systems.Chiral SU(3)-based potential Anti-kaon = Nambu-Goldstone boson 2. Complex Scaling Method(1405) = a building block of kaonic nucleiK-Proton For Resonance states

For Scattering problem

KbarN- coupled system with s-wave and isospin-0 state(1405) with c.c. Complex Scaling MethodKbar(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 Kbar + N (1405) + 14351332 [MeV]Complex Scaling Method for ResonanceBy Complex scaling, Complex rotation of coordinate (Complex scaling)

Resonant state

, : real, >0DivergentDamping under some condition

Complex Scaling Method for ResonanceComplex rotation of coordinate (Complex scaling)

By Complex scaling, Resonance wave function: divergent function damping functionBoundary condition is the same as that for a bound state. Resonance energy (pole position) doesnt change. ABC theoremThe energy of bound and resonant states is independent of scaling angle . J. Aguilar and J. M. Combes, Commun. Math. Phys. 22 (1971),269. E. Balslev and J. M. Combes, Commun. Math. Phys. 22 (1971),280 Diagonalize H with Gaussian base,we can obtain resonant states, in the same way as bound states!Complex Scaling Method for Resonance

KbarNpS continuumKbarN continuum[MeV]q =30 deg.B. E. (KbarN) = 28.2 MeV = 40.0 MeV Nominal position of (1405)Akaishi-Yamazaki potentialLocal Gaussian formEnergy independent

KbarN

Y. Akaishi and T. Yamazaki, PRC65, 044005 (2002)Test calculation with a phenomenological potential2. Complex Scaling Method For Resonance states

For Scattering problem

Calculation of KbarN scattering amplitudeCalc. of scattering amplitude with CSM

With help of the CSM, all problems for bound, resonant and scattering states can be treated with Gaussian base!Calc. of scattering amplitude with CSMA. T. Kruppa, R. Suzuki and K. Kat, PRC 75, 044602 (2007)1. Separate incoming wave UnknownNon square-integrable

square-integrable for 0 < < 2. Complex scaling r rei3. Calculate scattering amplitude with help of Cauchy theorem

Born term is OK!Scattered part is unknown.

Scattered wave function along rei is known!By Cauchy theorem, we can obtain asExpanding with square-integrable basis function (ex: Gaussian basis)r0r ei

Cauchy theoremCalc. of scattering amplitude with CSMA. T. Kruppa, R. Suzuki and K. Kat, PRC 75, 044602 (2007)

square-integrable for 0 < < can be expanded with Gaussian basisScattering problem can be solved as bound-state problem by matrix calculation!Equation to be solved:

Linear equation to be solved with matrix calculation!3. Set up for the calculation of KbarN-Y system KSW-type potential Chiral SU(3)-based

Kinematics

Non-rela. approximation of KSW-type potential

KSW-type potential chiral-SU(3) based N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995)Effective Chiral Lagrangian

PseudopotentialDelta-function type ( Yukawa / separable type)Up to order q2

Local Gaussian form in r-space

Weinberg-Tomozawa term

Energy dependenceKSW-type potential

a(I)ij : range parameter [fm] Easy to handle in many-body calculation with Gaussian baseRelative strength between channels determined by the SU(3) algebra Chiral SU(3) theory

KbarNConstrained by KbarN scattering length (Martins value)

aKN(I=0) = -1.70+i0.67fm, aKN(I=1) = 0.37+i0.60fm A.D.Martin, NPB179, 33(1979)Kinematics1. Non-relativistic2. Semi-relativistic

Reduced massReduced energy22Non-rela. approximation of KSW potential- Original

Normalized GaussianRange parameter for coupling potential (NRv1) Non-rela. approx. version 1

Comparison of the flux factor for differential cross sectionbetween non-rela. and rela.

(NRv2) Non-rela. approx. version 2

@ non-rela. limit (small p2)23Kinematics- Original(NRv1) Non-rela. approx. ver. 1(NRv2) Non-rela. approx. ver. 2

Non-rela.Semi-rela.Potential244. ResultI=0 channel

I=0 KbarN scattering length Range parameters of KSW-type potential

Scattering amplitude

Resonance pole wave function and size

Using Chiral SU(3) potential KSW-type r-space, Gaussian form, Energy-dependentI=0 KbarN scattering lengthRange parameters of KSW-type potential

Range parameters:

Pion decay constant f as a parameterf = 90 ~ 120 MeVFound sets of range parameters (dKN,KN, d,) to reproduce the Martins value. Two sets found in semi-rela. case. (SR-A, SR-B)

dKN,KNd,Re aI=0KNIm aI=0KNf=110 MeV26Scattering amplitude Non-rela. / KSW-type NRv1f=110 MeV

KbarNKbarNResonance structure appears in KbarN and channels.

ReImKbarN KbarN Resonance structure at 1413 MeVResonance structure at 1405 MeV27Scattering amplitude Non-rela. / KSW-type NRv2f=110 MeVKbarNKbarN

Essentially same as NRv1 case.

ReImKbarN KbarN Resonance structure at 1416 MeVResonance structure at 1408 MeV28Scattering amplitude Semi-rela. / KSW-type SR-Af=110 MeVKbarNKbarNQualitatively similar to non-rela. amplitudes.

ReImKbarN KbarN Resonance structure at 1410 MeVResonance structure at 1398 MeV29Scattering amplitude Semi-rela. / KSW-type SR-Bf=110 MeVKbarNKbarNVery different behavior of amplitude from other cases.

ReImKbarN KbarN (Another set of SR)???Resonance structure at 1419 MeVResonance structure at 1421 MeV30Pole position of the resonancef=90 - 120MeV

- / 2 [MeV]

901209012090120M [MeV]12090- / 2 [MeV]Complex energy plane31Pole position of the resonance- / 2 [MeV]f=90 - 120MeV

90120901209012012090M [MeV]Non-rela.(M, /2) = (1418.2 1.6, 21.4 4.7) NRv1 (1418.9 1.1, 18.6 4.6) NRv2- / 2 [MeV]32Pole position of the resonance- / 2 [MeV]f=90 - 120MeV

901209012090120M [MeV]12090Semi-rela.(M, /2) = (1420.5 3, 24.5 2) SR-A- / 2 [MeV]33Pole position of the resonance- / 2 [MeV]f=90 - 120MeV

90120901209012012090M [MeV]Semi-rela.(Another set)(M, /2) = (1419.5 0.5, 13.0 1.4) SR-BDifferent behavior from other cases- / 2 [MeV]34Wave function of the resonance polef=110 MeV=30

Non-rela. (NRv2)

Semi rela. (SR-A) component also localizeddue to Complex scaling.Mean distance between meson and baryon

BmNR: 1.3 - 0.3 i [fm]SR: 1.2 - 0.5 i [fm] ? Somehow small. cf) 1.9 fm @ M = 1423 MeV (B = 12 MeV)

- Difference of binding energy? - Different definition of pole? Gamow state or bound state

A. D., T. Hyodo, W. Weise, PRC79, 014003 (2009)354. Result I=1 channel

Range parameters of the KSW-type potential

Scattering amplitude

Using Chiral SU(3) potential (KSW-type) r-space, Gaussian form, Energy-dependentI=1 channel KbarN - - Data : aI=1KbarN = 0.37 + i 0.60 fm (A. D. Martin)

Range parameters: A. D. Martin, Nucl. Phys. B 179, 33 (1981)

Potential Weinberg-Tomozawa term only37I=1 channel KbarN - -

Range parameters:

SU(3) C.G. coefficients for I=1 channel

Data : aI=1KbarN = 0.37 + i 0.60 fm (A. D. Martin)

Potential Weinberg-Tomozawa term only38I=1 channel KbarN - -

Range parameters:

SU(3) C.G. coefficients for I=1 channel

Data : aI=1KbarN = 0.37 + i 0.60 fm (A. D. Martin)Potential Weinberg-Tomozawa term only39I=1 channel KbarN - -

Range parameters:

SU(3) C.G. coefficients for I=1 channelIsospin symmetric choice

Ref.) chiral unitary model

Data : aI=1KbarN = 0.37 + i 0.60 fm (A. D. Martin)

Potential Weinberg-Tomozawa term only40I=1 channel KbarN - - A. D. Martin : aI=1KbarN = 0.37 + i 0.60 fm Iso-symmetric choice : {dKN,KN, d,} fixed to I=0 channel onesSearch dKN, to reproduce Re or Im aI=1KbarN of Matins value.Difficult to reproduce Re aI=1KbarN Martins value within our model.When Re aI=1KbarN is reproduced, Im aI=1KbarN deviates largely from Martins value.

(fit)(fit)(fit)(fit)(I=0 one)(Search)41I=1 channel KbarN - - NRv2 (c): aI=1KbarN = 0.657 + i 0.599 fmf=110 MeV

Search dKN, to reproduce Im aI=1KbarN of Matins value.

KbarNKbarNSR-A (c): aI=1KbarN = 0.659 + i 0.600 fm42I=1 channel KbarN - - Only dKN,KN fixed to I=0 channel onesIn a study with separable potential, the cutoff parameter for KbarN is not so different between I=0 and 1 channels.Search {d,, dKN,} to reproduce Re and Im aI=1KbarN of Matins value.

When the only dKN,KN is fixed iso-symmetrically, we can find a set of {d,, dKN,}

to reproduce simultaneously Re and Im of Martins value(fit)(fit)(fit)(fit) Y. Ikeda and T. Sato, PRC 76, 035203 (2007)(I=0 one)(Search)43I=1 channel KbarN - - NRv2 (a): aI=1KbarN = 0.376 + i 0.606 fmf=110 MeVSearch {d,, dKN,} to reproduce Re and Im aI=1KbarN of Matins value.SR-A (a): aI=1KbarN = 0.375 + i 0.605 fm

KbarN

KbarNA narrow resonance exists at a few MeV below threshold repulsive ???445. Summary and Future plan5. SummaryScattering and resonant states of KbarN-Y system is studied with a coupled-channel Complex Scaling Method using a chiral SU(3) potentialA Chiral SU(3) potential KSW-type r-space, Gaussian form, energy dependenceCalculated scattering amplitude with help of CSMScattering states as well as resonant and bound states are treated with Gaussian base.Non-rela. / Semi-rela. kinematics and two types of non-rela. approximation of KSW-type potential are tried. Determined by the KbarN scattering length obtained by Martins analysis of old data. f dependence (f = 90 120MeV) Found two sets of range parameters in SR case.KbarN-Y system is essential for the study of Kbar nuclear system which is expected to be an exotic nuclear system with strangeness. 5. Summary and future plansFuture plansThree-body system (KbarNN-YN); Updated data of K-p scattering length by SHIDDARTA Pole position Size of the I=0 pole stateNR: 1.3 0.3i fmSR: 1.2 0.5i fm (M, /2) NRv1 : (1418.2 1.6, 21.4 4.7) NRv2 : (1418.9 1.1, 18.6 4.6) SR-A : (1420.5 0.5, 24.5 2) SR-B : (1419 1, 13.0 2)I=0 channel (KbarN-)I=1 channel (KbarN--) Difficult to reproduce Re aI=1KbarN of Martins value within our model, in case of iso-symmetric choice of range parameter. Another self-consistent solution NR: (~1360, 40~90), SR: (1350~1390, 30~100) Lower pole of double pole? Thank you very much!A. D. is thankful to Prof. Kat for his advice on the scattering-amplitude calculation in CSM, and to Dr. Hyodo for useful discussion.Backup slidesRough estimation of charge radiusK- pDistance CM

Calculated from electric form factor with chiral unitary model T. Sekihara, T. Hyodo and D. Jido, PLB 669, 133 (2008)Pole position [MeV] [fm2]NRv21417.8 i 16.6-0.49 + i 0.24SR-A1419.5 i 25.0-0.37 + i 0.37(Sekihara et al)1426.11 i 16.54-0.131 + i 0.303Charge radiusChiral (HW-HNJH): B 12 MeV, Distance = 1.86 fm = -1.07 fm2

Charge radius derived from electric form factorT. Sekihara, T. Hyodo and D. Jido, PLB 669, 133 (2008)Rough estimationKbar 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 nuclei

3He0.0 0.15

K- ppn0.0 2.0

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

A. Dote, H. Horiuchi, Y. Akaishi and T. Yamazaki, PRC70, 044313 (2004)K-Relate to various interesting physics such as Restoration of chiral symmetry in dense matterInteresting structureNeutron starExcited hyperon

I=0 Proton-K- bound state with 30MeV binding energy

Not 3 quark state,cant be explained with a simple quark model

But rather a molecular state

ubarsuudsudT. Hyodo and D. Jido, Prog. Part. Nucl. Phys. 67, 55 (2012)Today, K-pp has been focused in theor. and exp. studies!As the first trial of our study with ccCSM, consider (1405).K-PPPrototype of Kaonic nuclei3HeK-, pppK-, 4HeK-, pppnK-, 8BeK-,Complicated nuclear system with K-K-Proton(1405)=Building block of Kaonic nuclei54Asymptotic behavior of the wave function before/after complex scaling

Bound state Resonant state Continuum state

B : real, >0, : real, >0k : real, >0DampingDamping under some conditionOscillatingOscillatingDivergingDamping under some condition55Complex Scaling Method for ResonanceGreat success in nuclear physics in particular, unstable nuclei.Well applied to usual calculation of bound states diagonalize with Gaussian base (Gaussian Expansion Method, Correlated Gaussian)Tiny modification of matrix elements

Possible to study many-body systems up to 5-body system (4He+n+n+n+n)

Cluster-Orbital Shell ModelT. Myo, R. Ando, K. Kato, PLB691, 150 (2010)

Calc. of scattering amplitude with CSMA. T. Kruppa, R. Suzuki and K. Kat, PRC 75, 044602 (2007)

Complex scalingr r eiUnknownNon square-integrable

Separate the incoming wave !

Point 1

Point 2square-integrable for 0 < < Expanding with square-integrable basis function (ex: Gaussian basis)Calculation of scattering amplitude

We dont have which is a solution along r,

but we have which is a solution along rei.

r0r ei

Cauchy theoremPoint 3

Calc. of scattering amplitude with CSMis independent of .

expressed with Gaussian baseA. T. Kruppa, R. Suzuki and K. Kat, PRC 75, 044602 (2007)Test calculation with AY potentialResonance pole

KbarNpS continuumKbarN continuum[MeV]q =30 deg.B. E. (KbarN) = 28.2 MeV = 40.0 MeV Nominal position of (1405)Akaishi-Yamazaki potentialPhenomenological potentialLocal Gaussian formEnergy independent

KbarN

Y. Akaishi and T. Yamazaki, PRC65, 044005 (2002)59Complex Scaling Method for ResonanceStudy of unstable nuclei Resonance of 8He

Cluster-Orbital Shell ModelT. Myo, R. Ando, K. Kato, PLB691, 150 (2010)Bound stateResonanceContinuumon 2 lineTest calculation with AY potentialUnitarity violation of S-matrixPhase shift sum

| |Det S| -1 |EKbarN [MeV]EKbarN [MeV]Checked by Continuum Level Density method (R. Suzuki, A.T. Kruppa, B.G. Giraud, and K. Kat, PTP119, 949(2008))

KbarN + [deg.]61Test calculation with AY potentialKbarN (I=0) scattering amplitude

Scattering length (Scatt. amp. @ EKbarN=0) = -1.77 + i 0.47 fmY. Akaishi and T. Yamazaki, PRC65, 044005 (2002)Scattering length = -1.76 + i 0.46 fm

EKbarN [MeV]62Scattering amplitude Non-rela. / KSW-type SRf=90 MeVKbarNKbarN

Kinematics-potential mismatched case

ReImKbarN KbarN A virtual state exists. (a, re)=(61,-6.3) fm satisfies -a /2 < re. Y. Ikeda et al, PTP 125, 1205 (2011)Singular behavior at threshold63I=0 KbarN scattering lengthRange parameters of KSW-type potentialf=110 MeVData : aI=0KbarN = -1.70 + i 0.68 fm (A. D. Martin)Found sets of range parameters (dKN,KN, d,) to reproduce the Martins value. Two sets found in semi-rela. case. (SR-A, SR-B)

dKN,KNd,Re aI=0KNIm aI=0KN64Resonant structure in the scattering amplitudeEnergy @ Re fKbarN KbarN = 0Energy @ Re f = 0f=90 - 120MeV

KSW - Semi rela.KbarN KbarNf = 90MeV65Resonant structure in the scattering amplitudeEnergy @ Re fKbarN KbarN = 0Energy @ Re f = 0f=90 - 120MeV

Strange behavior?SR-BSR-A66Wave function of the resonance polef=110 MeV=30 component also localizeddue to Complex scaling.* The wave functions shown above are multiplied by a phase factor so that the KbarN wfn. becomes real at r=0.

Non-rela. (NRv2)

Semi rela. (SR-A)

Semi rela. (SR-B)67Size of the resonance pole state

Mean distance between meson and baryonReImf=90 - 120MeVNR: 1.3 - 0.3 i [fm]SR: 1.2 - 0.5 i [fm] BM? Somehow small. cf) 1.9 fm @ M = 1423 MeV (B = 12 MeV)

- Difference of binding energy? - Different definition of pole? Gamow state or bound state

A. D., T. Hyodo, W. Weise, PRC79, 014003 (2009)68Strange nuclear physicsNucleusHypernucleiudsHyperonbaryon = qqqStrangeness is introduced through baryons.69Strange nuclear physicsNucleusKaonic nuclei !ubarsK- meson(anti-kaon, Kbar)Strangeness is introduced through mesons meson = qqbar70Key points to study kaonic nucleiSimilar to the bound-state calculation1. Consider a coupled-channel problemKbar + N + NKbar N N + + N2. Treat resonant states adequately.KbarN couples with Y.Bound state for KbarN channelbut a resonant state for Y channelTheir nature?3. Obtain the wave function to help the analysis of the state.coupled-channel Complex Scaling Method

KbarN (-Y) interaction??Chiral SU(3)-based potential Excited hyperon

939p,n11151190p + K-1435(1405)14051250 + + 1325Energy [MeV]sudMysterious state; (1405)

Quark model prediction calculated as 3-quark stateN. Isgar and G. Karl, Phys. Rev. D18, 4187 (1978)qqq(1405) cant be well reproduced as a 3-quark state!observed (1405)calculated (1405)73

flux factor

KSWBornki: meson momentumi: reduced energy

KSWBorn

fij : vi : kij : i :

KSWBorn

(KSW)

iiKSW KSWpseudo-potential

i 1 / sETotKSWComparison with other studies of I=0 channelComparison with other studies of I=0 KbarN-Kaiser-Siegel-WeiseHyodo-WeiseIkeda-Hyodo-Jido-Kamano-Sato-YazakiDote-Inoue-MyoBaseChargeIsospinIsospinIsopinTermsAll terms up to q2WT termWT termWT termRegularizationYukawa / Separable formDimensional regularizationDimensional regularization/ Separable formGaussian formConstraintLow energy scattering data, Branching ratio at KbarN thresholdLow energy scattering data, Branching ratio at KbarN threshold KbarN scattering length/ (1405) poleKbarN scattering lengthRef.NPA 594, 325 (1995)PRC 77, 035204 (2008)PTP 125, 1205 (2011)NPA 912, 66 (2013)Comparison with other studies of I=0 KbarN-IHJKSY, HW:

DIM:

Different energy dependence of interaction kernelProportional to just meson energyNot proportional to meson energy including relativistic q2 correction and non-static effect of baryonsComparison with other studies of I=0 KbarN-IHJKSYHW

Comparison with other studies of I=0 KbarN-

Use the same interaction kernel as that of IHJKSY and HW in our ansatz (Flux factor and Gaussian form factor)

Suppressed the enhancement near the threshold, but still different from the amplitudes of IHJKSY.Different interaction kernelUse of flux factorDifferent regularization scheme (Gaussian form factor vs Dimensional reg.)* Gaussian form factor enhances the amplitude far below the threshold as |E| exp [c|E|] with E -. (c>0) 1. Introduction

Neutron star in universeRadius 10kmMass MsunHigh dense at core > 2 0Maybe Involving strangenessKbar nuclear system on the Earth Self-bound Kbar-nuclear systemK- light nucleus with K- mesonsInvolving strangeness via K- meson1. Introduction

Neutron star in universeRadius 10kmMass MsunHigh dense at core > 2 0Maybe Involving strangenessKbar nuclear system on the Earth

Density [fm-3]0.00 0.143He

Density [fm-3]0.0 0.75 1.53HeK-44 fm2 light nucleus with K- mesonsInvolving strangeness via K- mesonStrong attraction by K- meson Dense nuclear system A. D., H. Horiuchi, Y. Akaishi and T. Yamazaki, PLB 590 (2004) 51; PRC 70 (2004) 044313.A phenomenological KbarN potential B(K) 100MeV, 240 FINUDA collaboration (DANE, Frascati) K- absorption at rest on various nuclei (6Li, 7Li, 12C, 27Al, 51V) Invariant-mass method

Strong correlation between emitted p and (back-to-back)

Invariant mass of p and K-pppIf it is K-pp, Total binding energy = 115 MeVDecay width = 67 MeVPRL 94, 212303 (2005)

Experimental studies of K-pp Re-analysis of KEK-PS E549 DISTO collaboration p + p -> K+ + + p @ 2.85GeV p invariant mass Comparison with simulation data T. Yamazaki et al. (DISTIO collaboration), PRL104, 132502 (2010) K- stopped on 4He target p invariant mass

T. Suzuki et al (KEK-PS E549 collaboration), arXiv:0711.4943v1[nucl-ex]K- pp???B. E.= 103 3 5 MeV = 118 8 10 MeVStrong p back-to-back correlation is confirmed.Unknown strength is there in the same energy region as FINUDA.

Experimental studies of K-pp

E15: A search for deeply bound kaonic nuclear states by 3He(inflight K-, n) reaction (M. Iwasaki (RIKEN), T. Nagae (Kyoto))

E17: Precision spectroscopy of kaonic 3He atom 3d2p X-rays (R. Hayano (Tokyo), H. Outa (Riken))

E27: Search for a nuclear Kbar bound state K-pp in the d(+, K+) reaction (T. Nagae (Kyoto))

E31: Spectroscopic study of hyperon resonances below KN threshold via the (K-, n) reaction on deuteron (H. Noumi (Osaka))

as of 2011 Experiments of kaonic nuclei at J-PARC87

E15: A search for deeply bound kaonic nuclear states by 3He(inflight K-, n) reaction (M. Iwasaki (RIKEN), T. Nagae (Kyoto))

E17: Precision spectroscopy of kaonic 3He atom 3d2p X-rays (R. Hayano (Tokyo), H. Outa (Riken))

E27: Search for a nuclear Kbar bound state K-pp in the d(+, K+) reaction (T. Nagae (Kyoto))

E31: Spectroscopic study of hyperon resonances below KN threshold via the (K-, n) reaction on deuteron (H. Noumi (Osaka))

as of 2011 Experiments of kaonic nuclei at J-PARC

Dr. Fujiokas talk (KEK workshop, 7-9. Aug. 08)at K1.8BR beam line1.8GeV/c88

E15: A search for deeply bound kaonic nuclear states by 3He(inflight K-, n) reaction (M. Iwasaki (RIKEN), T. Nagae (Kyoto))

E17: Precision spectroscopy of kaonic 3He atom 3d2p X-rays (R. Hayano (Tokyo), H. Outa (Riken))

E27: Search for a nuclear Kbar bound state K-pp in the d(+, K+) reaction (T. Nagae (Kyoto))

E31: Spectroscopic study of hyperon resonances below KN threshold via the (K-, n) reaction on deuteron (H. Noumi (Osaka))

as of 2011 Experiments of kaonic nuclei at J-PARC

Missing mass spectroscopyat K1.8BR beam line1.8GeV/cInvariant mass spectroscopyDr. Fujiokas talk (KEK workshop, 7-9. Aug. 08)All emitted particles will be measured.89

E17: Precision spectroscopy of kaonic 3He atom 3d2p X-rays (R. Hayano (Tokyo), H. Outa (Riken))

E27: Search for a nuclear Kbar bound state K-pp in the d(+, K+) reaction (T. Nagae (Kyoto))

as of 2011 Experiments of kaonic nuclei at J-PARCd+K+(1405)npppK-Preceding E15, E27 experiment was performed in June. Analysis is now undergoing! Use lots of pion Missing mass90

E17: Precision spectroscopy of kaonic 3He atom 3d2p X-rays (R. Hayano (Tokyo), H. Outa (Riken))

E27: Search for a nuclear Kbar bound state K-pp in the d(+, K+) reaction (T. Nagae (Kyoto))

as of 2011 Experiments of kaonic nuclei at J-PARCKbarN interaction(1405)K-ppK-pp91These days, K-pp has been focused in both of theoretical and experimental studies!K-Proton(1405)3HeK-, pppK-, 4HeK-, pppnK-, 8BeK-,Complicated nuclear system with K-=K-PPMost essential kaonic nucleus!92Coupled Channel Chiral Dynamics (Chiral Unitary model)We are thinking about KbarN interaction.S=-1 meson-baryon system is constrained byThe leading couplings between Low mass pseudo-scalar meson octet (Nambu-Goldston bosons) and Baryon octetare determined bySpontaneous breaking ofSU(3)SU(3) Chiral symmetryChiral SU(3) dynamics !Chiral low-energy theorem tells us

KbarNKFor I=0 channel,Remark There are no free parameters as for coupling. There is an attractive interaction in - channel, while AY potential doesnt have it.

Energy and mass of baryon in channel i

andWeinberg-Tomozawa term derived from Chiral SU(3) effective Lagrangian

: Psuedo-scalar meson decay constantCoupled Channel Chiral Dynamics (Chiral Unitary model)

3. Solve coupled channel Bethe-Salpeter equation.T-matrix of coupled channel scattering

+ + + = 2. Using the WT-term as a building block,1. Weinberg-Tomozawa term derived from Chiral SU(3) effective Lagrangian:

K-p scattering length Threshold branching ratio Total cross section of K-p scatteringCoupled Channel Chiral Dynamics (Chiral Unitary model)

Threshold branching ratio Total cross section of K-p scatteringT. Hyodo, S. I. Nam, D. Jido, and A. Hosaka, Phys. Rev. C68, 018201 (2003)Coupled Channel Chiral Dynamics (Chiral Unitary model)

3. Solve coupled channel Bethe-Salpeter equation.T-matrix of coupled channel scattering

+ + + = 2. Using the WT-term as a building block,(1405) is dynamically generated as meson-baryon system.1. Weinberg-Tomozawa term derived from Chiral SU(3) effective Lagrangian:

K-p scattering length Threshold branching ratio Total cross section of K-p scatteringCoupled Channel Chiral Dynamics (Chiral Unitary model)

I=0 mass distributionT. Hyodo, S. I. Nam, D. Jido, and A. Hosaka, Phys. Rev. C68, 018201 (2003)Dynamical generation of (1405) !Remark:Calculated with - scattering amplitude.Coupled Channel Chiral Dynamics (Chiral Unitary model)Chiral SU(3) potential (KSW)N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995)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.

Flavor SU(3) symmetryOriginal: -function typePresent: Normalized Gaussian type

a: range parameter [fm]Energy dependence is determined by Chiral low energy theorem. Kaon: Nambu-Goldstone boson

Chiral 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

4. Experiments related to Kbar nuclear physics LEPS / SPring-8 invariant mass (Kaonic atom) (1405) CLAS / JLab Kaonic 4He atom, 2p (3d2p X) @ KEK (E570) S. Okada et. al., Phys. Lett. B653, 387 (2007) Kaonic hydrogen atom, 1s @ DEAR group, DANE, G. Beer et al., Phys. Rev. Lett. 94, 212302 (2005) Frascati National Laboratories Kaonic hydrogen, deuterium @ SIDDHARTA group Kaonic 3He atom, 2p (3d2p X) @ J-PARC (E17, DAY-1) + p K+ + (1405), (1405) K. Moriya and R. Schumacher, Nucl. Phys. A835, 325 (2010)J. K. Ahn, Nucl. Phys. A835, 329 (2010)-+, 00, +- M. Bazzi et al., Phys. Lett. B704, 113 (2011) LEPS / SPring-8 invariant mass (Kaonic atom) (1405) CLAS / JLab Kaonic 4He atom, 2p (3d2p X) @ KEK (E570) S. Okada et. al., Phys. Lett. B653, 387 (2007) Kaonic hydrogen atom, 1s @ DEAR group, DANE, G. Beer et al., Phys. Rev. Lett. 94, 212302 (2005) Frascati National Laboratories Kaonic hydrogen, deuterium @ SIDDHARTA group Kaonic 3He atom, 2p (3d2p X) @ J-PARC (E17, DAY-1) + p K+ + (1405), (1405) K. Moriya and R. Schumacher, Nucl. Phys. A835, 325 (2010)J. K. Ahn, Nucl. Phys. A835, 329 (2010)-+, 00, +- M. Bazzi et al., Phys. Lett. B704, 113 (2011)DEAR exp. for kaonic hydrogen atomKaonic hydrogen atom, 1s @ DEAR Collaboration, DANE, Frascati National LaboratoriesG. Beer et al., Phys. Rev. Lett. 94, 212302 (2005)

cf) KEK exp. M.Iwasaki et al., Phys. Rev. Lett. 78, 3067 (1997)

B. Borasoy et al., Phys. Rev. Lett. 94, 213401 (2005)KEK exp.DEARCoupled channel chiral dynamics (Chiral unitary model) DEARSHIDDARTA exp. for kaonic hydrogen atomKEKKaonic hydrogen atom, 1s @ SHIDDARTA Collaboration, DANE, Frascati National LaboratoriesM. Bazzi et al., Phys. Lett. B704, 113 (2011)

K-pKbarN subthreshold(1405)Y. Ikeda, T. Hyodo and W. Weise, Phys. Lett. B706, 63 (2011) LEPS / SPring-8 invariant mass (Kaonic atom) (1405) CLAS / JLab Kaonic 4He atom, 2p (3d2p X) @ KEK (E570) S. Okada et. al., Phys. Lett. B653, 387 (2007) Kaonic hydrogen atom, 1s @ DEAR group, DANE, G. Beer et al., Phys. Rev. Lett. 94, 212302 (2005) Frascati National Laboratories Kaonic hydrogen, deuterium @ SIDDHARTA group Kaonic 3He atom, 2p (3d2p X) @ J-PARC (E17, DAY-1) + p K+ + (1405), (1405) K. Moriya and R. Schumacher, Nucl. Phys. A835, 325 (2010)J. K. Ahn, Nucl. Phys. A835, 329 (2010)-+, 00, +- M. Bazzi et al., Phys. Lett. B704, 113 (2011)KEK E570 for kaonic 4He atom 0 eV consisitentKaonic 4He atom, 2p 3d2p X @ KEK, E5700eV-43 eVKaonic helium puzzleS. Okada et. al., Phys. Lett. B653, 387 (2007)

S. Hirenzaki et al., Phys. Rev. C61, 055205 (2000)J-PARC for kaonic 3He atomKaonic 3He atom, 2p 3d2p X @ J-PARC, E17 DAY-1S. Okada et. al., Phys. Lett. B653, 387 (2007)

Kaonic 4He atom, 2p 0 eV

KbarN potential

3HeKbarN potentialY. Akaishi, Proceedings of EXA05, Austrian Academy of Sciences press, Vienna, 2005, p.452007 LEPS / SPring-8 invariant mass (Kaonic atom) (1405) CLAS / JLab Kaonic 4He atom, 2p (3d2p X) @ KEK (E570) S. Okada et. al., Phys. Lett. B653, 387 (2007) Kaonic hydrogen atom, 1s @ DEAR group, DANE, G. Beer et al., Phys. Rev. Lett. 94, 212302 (2005) Frascati National Laboratories Kaonic hydrogen, deuterium @ SIDDHARTA group Kaonic 3He atom, 2p (3d2p X) @ J-PARC (E17, DAY-1) + p K+ + (1405), (1405) K. Moriya and R. Schumacher, Nucl. Phys. A835, 325 (2010)J. K. Ahn, Nucl. Phys. A835, 329 (2010)-+, 00, +- M. Bazzi et al., Phys. Lett. B704, 113 (2011)(1405) - invariant mass -LEPS / Spring-8K. Moriya and R. Schumacher, Nucl. Phys. A835, 325 (2010) CLAS / Jefferson Laboratory + p K+ + (1405), (1405)

(chiral unitary)

Highest peak+ -- +J. K. Ahn, Nucl. Phys. A835, 329 (2010)p (, K+ ) at E = 1.5-2.4GeVCharged

What is the object observed experimentally?A bound state of K-pp, or another object such as N ???Only what we can say from only this spectrum is thatThere is some object with B=2, S=-1, charge=+1

DISTO collaboration

112Self-consistent solutions for complex energy

: complex energySelf-consistent solution

Ex) KSW ; f = 100 MeV, a= 0.47 fm ( =35)

Self-consistency for complex energy should be considered!Chiral potential Energy-dependent potential

Self-consistency achieved after a few times iteration.A. D., T. Inoue and T. Myo, Proc. of BARYONS 10, AIP conf. proc. 1388, 549 (2010)113f = 90 MeVScattering amplitude 114Scattering amplitude KSW org. Non. rela.

Phase shift ()KbarN KbarNReIm KbarN KbarNf=90 MeV

KbarN115Scattering amplitude KSW NRv1 Non. rela.f=90 MeV

Phase shift ()ReImKbarN KbarNKbarN KbarN

KbarN116Scattering amplitude KSW NRv2 Non. rela.f=90 MeV

Phase shift ()ReImKbarN KbarNKbarN KbarN

KbarN117Scattering amplitude KSW org. Semi rela.f=90 MeV

Phase shift ()ReImKbarN KbarNKbarN KbarN

KbarN118

Succeeded in nuclear physics Especially used in the study of unstable nuclei.Resonant state of 6He4HennS. Aoyama, T. Myo, K. Kato and K. Ikeda, Prog. Theor. Phys. 116, 1 (2006)ContinuumResonanceComplex Scaling MethodResult Scattering amplitude

Position of resonant structure appeared in the scattering amplitude

Wave function of the resonance pole

Size of the resonance pole state

dependence of Size and Wave function

Comparison with Hyodo-Weises resultPotential: KSW original

Kinematics: Non. rela.

Scattering amplitude 121Scattering amplitude KSW org. Non. rela.

f=90 MeV

Phase shift ()ReImKbarN KbarN KbarN KbarN122Scattering amplitude KSW org. Non. rela.

f=100 MeV

Phase shift ()ReImKbarN KbarN KbarN KbarN123Scattering amplitude KSW org. Non. rela.

f=110 MeV

Phase shift ()

ReImKbarN KbarN KbarN KbarN124Scattering amplitude KSW org. Non. rela.

f=120 MeV

Phase shift ()ReImKbarN KbarN KbarN KbarN125Potential: KSW NRv1

Kinematics: Non. rela.Scattering amplitude

126Scattering amplitude KSW NRv1 Non. rela.

f=90 MeV

Phase shift ()

ReImKbarN KbarN KbarN KbarN127Scattering amplitude KSW NRv1 Non. rela.Phase shift ()

f=100 MeV

KbarN KbarN

ReImKbarN KbarN128Scattering amplitude KSW NRv1 Non. rela.Phase shift ()

f=110 MeV

KbarN KbarN

ReImKbarN KbarN129Scattering amplitude KSW NRv1 Non. rela.Phase shift ()

f=120 MeV

KbarN KbarN

ReImKbarN KbarN130Potential: KSW NRv2

Kinematics: Non. rela.Scattering amplitude

131Scattering amplitude KSW NRv2 Non. rela.Phase shift ()

f=90 MeV

ReImKbarN KbarN KbarN KbarN132Scattering amplitude KSW NRv2 Non. rela.Phase shift ()

f=100 MeV

ReImKbarN KbarN KbarN KbarN133Scattering amplitude KSW NRv2 Non. rela.Phase shift ()

f=110 MeV

ReImKbarN KbarN KbarN KbarN134Scattering amplitude KSW NRv2 Non. rela.Phase shift ()

f=120 MeV

KbarN KbarN

ReImKbarN KbarN135Potential: KSW original

Kinematics: Semi rela.

Scattering amplitude 136Scattering amplitude KSW org. Semi rela.Phase shift ()

f=90 MeV

ReImKbarN KbarN KbarN KbarN137Scattering amplitude KSW org. Semi rela.Phase shift ()

f=100 MeV

ImKbarN KbarN KbarN KbarNRe138Scattering amplitude KSW org. Semi rela.Phase shift ()

f=110 MeV

ImKbarN KbarN KbarN KbarNRe139Scattering amplitude KSW org. Semi rela.Phase shift ()

f=120 MeV

ImKbarN KbarN KbarN KbarNRe140Position of resonant structure appeared in the scattering amplitude141Position of resonant structure appeared in the scattering amplitudef=90 MeV

KbarN KbarN 142Position of resonant structure appeared in the scattering amplitudef=100 MeV

KbarN KbarN 143Position of resonant structure appeared in the scattering amplitudef=110 MeV

KbarN KbarN 144Position of resonant structure appeared in the scattering amplitudef=120 MeV

KbarN KbarN 145Position of resonant structure appeared in the scattering amplitude

Resonance position in KbarN KbarNResonance position in 146Wave function of the resonance pole147Wave function of the resonance pole

Org. Non.rela.NRv1 Non.rela.NRv2 Non.rela.Org. Semi rela.f=90 MeV=30148Wave function of the resonance pole

Org. Non.rela.Org. Semi rela.NRv1 Non.rela.NRv2 Non.rela.f=100 MeV=30149Wave function of the resonance pole

Org. Non.rela.NRv1 Non.rela.NRv2 Non.rela.Org. Semi rela.f=110 MeV=30150Wave function of the resonance pole

Org. Non.rela.NRv1 Non.rela.NRv2 Non.rela.

Org. Semi rela.f=120 MeV=30151Size of the resonance pole state152

Size of the resonance pole statef=90 MeV=30153Size of the resonance pole statef=100 MeV=30

154Size of the resonance pole statef=110 MeV=30

155Size of the resonance pole statef=120 MeV=30

156Size of the resonance pole state=30

Mean distance between meson and baryonReIm157Pole position of the resonance

- / 2 [MeV]M [MeV]90100110120901001101209010011012090100110120158 dependence of Size and Wave function159 dependence of SizeNRv2-NRf=90 MeV

[deg][fm]

160 dependence of SizeOrg-SRf=90 MeV

[deg][fm]

161 dependence of Wave functionNRv2-NRf=90 MeV

KbarN bound state resonant stateKbarNr [fm]r [fm] =10 deg.162 dependence of Wave functionNRv2-NRf=90 MeVKbarN bound state resonant stateKbarN

r [fm]r [fm] =15 deg.163 dependence of Wave functionNRv2-NRf=90 MeVKbarN bound state resonant stateKbarN

r [fm]r [fm] =20 deg.164 dependence of Wave functionNRv2-NRf=90 MeVKbarN bound state resonant stateKbarN

r [fm]r [fm] =25 deg.165 dependence of Wave functionNRv2-NRf=90 MeVKbarN bound state resonant stateKbarN

r [fm]r [fm] =30 deg.166 dependence of Wave functionNRv2-NRf=90 MeVKbarN bound state resonant stateKbarN

r [fm]r [fm] =35 deg.167Comparison with Hyodo-Weises result168Comparison with Hyodo-Weises result

KbarN KbarN Scattering amplitudesDIMHWHW: PRC 77, 035204 (2008)Fig. 4 I=0 KbarN/ scattering amplitude in two channel model169Comparison with Hyodo-Weises resultPole positionHW: PRC 77, 035204 (2008)DIMHWFig. 5 Pole positions of I=0 scattering amplitude

(Upper pole in two channel model)z = 1419 14.4i MeV170

Schrdinger equation to be solved

: complex parameters to be determinedWave function expanded with Gaussian basec.c. CSM for resonant statesPhenomenological 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

Schrdinger equation to be solved

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

c.c. CSM for resonant statesPhenomenological 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 Chiral 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

1. Chiral SU(3) potentialr-space a f 2. A.D. Martin I=0 KbarN aKbarN(I=0) (a, f)Kruppa3. (a, f) (1405)KbarN-Chiral SU(3) potentialself consistency

N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995)

A. T. Kruppa, R. Suzuki and K. Kato, PRC 75, 044602 (2007)Martinf=90120MeV

f=75.5MeVself-consistentKbarN- (EKbarN, /2) = (-19, 27) (-5,8) MeV : (EKbarN, /2) = (-32-24, 2866) MeV PDG1405 (EKbarN, /2) = (-294, 251) MeV f=75.5: (EKbarN, /2) = (-23, 34) MeV Mysterious state; (1405)

Quark model prediction calculated as 3-quark stateN. Isgar and G. Karl, Phys. Rev. D18, 4187 (1978)qqq(1405) cant be well reproduced as a 3-quark state!observed (1405)calculated (1405)178KbarN- 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 Calc. of scattering amplitude with CSM Kruppa methodA. T. Kruppa, R. Suzuki and K. Kat, PRC 75, 044602 (2007)Radial Schroedinger eq.

Asymptotic behavior of wave func.

Equation to be solved

V(r) : short-range potential

Asymptotic behavior of scattering part of wave func.

: Ricatti-Bessel, Hankel func.is not square-integrable

Search the solution in the form

Separate the incoming wave !

Point 1Calc. of scattering amplitude with CSM Kruppa methodA. T. Kruppa, R. Suzuki and K. Kat, PRC 75, 044602 (2007)Complex scalingEquation to be solved

Expanding with square-integrable basis function (ex: Gaussian basis)

square-integrable for 0 < < Point 2

: square integrable functionLinear equation:Calc. of scattering amplitude with CSM Kruppa methodA. T. Kruppa, R. Suzuki and K. Kat, PRC 75, 044602 (2007)Calculation of scattering amplitudeWe dont have which is a solution along r,

but we have which is a solution along rei.

r0r ei

Cauchy theoremPoint 32. 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)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 complex energy

Search for such a solution that both of real and imaginary parts of energy are identical to assumed ones.

(B.E., )Calculated = (B.E., )Assumed

More reasonable?

Pole search of T-matrix is done on complex-energy plane.Z: complex energy

Self consistency for complex energyKSWf = 100 MeVa=0.47, =35-B [MeV]

1 step-/2 [MeV]

obtained by the self-consistency for the real energyKbarNSelf consistency for complex energyKSWf = 100 MeVa=0.47, =35-B [MeV]

2 steps-/2 [MeV]

KbarNSelf consistency for complex energyKSWf = 100 MeVa=0.47, =35-B [MeV]

3 steps-/2 [MeV]KbarNSelf consistency for complex energyKSWf = 100 MeVa=0.47, =35-B [MeV]

4 steps-/2 [MeV]KbarNSelf consistency for complex energyKSWf = 100 MeVa=0.47, =35

5 steps-/2 [MeV]-B [MeV]Self consistent!KbarNSelf consistency for complex energyKSWf = 100 MeV

AssumedCalc.AssumedCalc.

Self consistency for complex energyKSWf = 100 MeV-B [MeV]- / 2 [MeV]a=0.60a=0.50a=0.47a=0.45S.C. for real energyS.C. for complex energyKbarNRepulsively shifted!Mean distance between Kbar () and N ()Kbar ()N ()Distance Chiral (HW-HNJH): B 12 MeV, Distance = 1.86 fm

Rough estimation of charge radiusK- pDistance Chiral (HW-HNJH): B 12 MeV, Distance = 1.86 fm = -1.07 fm2CM

Charge radiusChiral (HW-HNJH): B 12 MeV, Distance = 1.86 fm = -1.07 fm2

Charge radius derived from electric form factorT. Sekihara, T. Hyodo and D. Jido, PLB 669, 133 (2008)Rough estimation