Semi-relativistic study of K-pp with a fully

coupled-channel complex scaling method

1. Introduction (Motivation)

2. Formalism

• Semi-relativistic treatment

• Chiral SU(3)-based potential

3. Result• Check: Complex eigenvalue distribution in Semi-rela. case

• Result w/ Martin constraint

• Result w/ SIDDHARTA constraint

4. A concern ...

5. Summary and future plan

Akinobu Doté (KEK Theory Center, IPNS / J-PARC branch)

Takashi Inoue (Nihon university)

Takayuki Myo (Osaka Institute of Technology)

ELPH 研究会 C023

「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」,

11. Sept. ’18

@ Research center for

ELectron Photon science (ELPH),

Tohoku University of Science, Sendai, Japan

1. Introduction

Why semi-relativistic?→ Of course, pion is light.

It should be treated in a relativistic way.

In addition, ...

P PK-

“K-pp” = KbarNN – πΣN – πΛN

K-

Proton

Excited hyperon Λ(1405) ... KbarN quasi-bound state

= KbarN two-body system

Kaonic nuclei = Nuclear many-body system

with antikaons

P PK-

K- pp = the simplest kaonic nucleus

... bridge from Λ(1405) to general kaonic nuclei

Doorway to dense matter?→ Chiral symmetry restoration in dense matter??

→ Strong KbarN attractionT. Hyodo, D. Jido, Prog. Part. Nucl. Phys. 67, 55 (2012)

Experimental search for K-pp• Deeply bound region (near πΣN threshold = 103 MeV below KbarNN threshold)

FINUDA (2005), DISTO (2010), J-PARC E27 (2013)

• Shallowly bound region (near KbarNN threshold)

J-PARC E15 1st run (2013)

• No signal in bound region

LEPS/SPring8 (2015) and more ...

Clear evidence of K-pp bound state

J-PARC E15 (2nd run):

Exclusive exp. 3He(K-, Λp)nmissing

⇒ “Fully coupled-channel

Complex Scaling Method”A. Dote, T. Inoue, T. Myo, PRC 95, 062201(R) (2017)

According to early theoretical studies of “K-pp” ...

Resonant state of

KbarNN-πΣN-πΛN coupled channel

three-body system

“K-pp” (Jπ=0-, T=1/2) • Doté, Hyodo, Weise, PRC79, 014003(2009).

• Akaishi, Yamazaki, PRC76, 045201(2007)

• Ikeda, Sato, PRC76, 035203(2007).

• Shevchenko, Gal, Mares, PRC76, 044004(2007)

• Barnea, Gal, Liverts, PLB712, 132(2012)

→ Summarized in

A. Gal, E. V. Hungerford, D. J. Millener,

Rev. Mod. Phys. 88, 035004 (2016).

Resonance & Channel coupling

Results of “K-pp”

J-PARC E15-1st

J-PARC E27

DISTO

FINUDA

Faddeev-AGS

Pheno. pot. (E-indep.)

Variational (Gauss)

Pheno. pot. (E-indep.)

Variational (Gauss)

Chial. pot. (E-dep.)

Faddeev-AGS

Chiral pot. (E-dep.)

K-pp binding energy BK-pp [MeV]KbarNN threshold πΣN threshold

SGM

AYDHW

BGL

IKS

Full ccCSM

w/ Chiral SU(3) pot. (SIDDHARTA)PLB 704, 405 (2018)

J-PARC E15 2nd

arXiv: 1805.12275 FINUDA

DISTO

J-PARC E27

Results of “K-pp”

Full ccCSM

w/ AY pot. (Coupled)PRC 95, 062201(R) (2017)

SGM

AYDHW

BGL

IKS

Full ccCSM

w/ Chiral SU(3) pot. (SIDDHARTA)PLB 704, 405 (2018)

J-PARC E15 2nd

arXiv: 1805.12275 FINUDA

DISTO

J-PARC E27

Results of “K-pp”

Full ccCSM

w/ AY pot. (Coupled)PRC 95, 062201(R) (2017)

Too small decay width obtained with Full ccCSM???

2. Relativistic effect??

... Phase volume for decay is underestimated

in non-relativistic kinematics.

Suggested by Prof. S. Shinmura

1. Contribution of non-mesonic decay (ΓNM)?

• Width calculated in theory = Mesonic decay width (ΓM)

• Width measured in exp’t = Total decay width (ΓM+ ΓNM)

2. Formalism

• Semi-relativistic treatment

• Chiral SU(3)-based KbarN potential

Summary of methodology

Employ Chiral SU(3)-based KbarN potential

in relativistic kinematicsNo non-rela. approximation

A. Dote, T. Inoue, T. Myo,

NPA 912, 66 (2013)

Use Semi-Relativistic Hamiltonian

... Kinetic energy is given in relativistic form.

3

1

22

NN MB

i

iiH Vm V

p1 2 3 p p p 0 (CM sys.)

Y. Suzuki and K. Varga, “Stochastic Variational

Approach to Quantum-Mechanical Few-Body

Problems” (Springer, Berlin, 1998).

, where

Investigate “K-pp” with Full ccCSM as our previous studies

• “K-pp” = KbarNN-πΣN-πΛN (Jπ=0-, T=1/2)

• Search for resonance pole on complex-energy plane

• Correlated Gaussian basis function

• Self-consistent calculation for energy-dependent potential

A. Dote, T. Inoue, T. Myo,

PRC 95, 062201(R) (2017);

PLB 704, 405 (2018)

Complex Scaling Method… Powerful tool for resonance study of many-body system

S. Aoyama, T. Myo, K. Kato, K. Ikeda, PTP116, 1 (2006)

T. Myo, Y. Kikuchi, H. Masui, K. Kato, PPNP79, 1 (2014)

!!! Caution !!!

This is Non-rela. case.

Chiral SU(3)-based KbarN potential

Coupled-channel chiral dynamics

(Chiral Unitary model)N. Kaiser, P.B. Siegel, W. Weise, NPA 594 (1995) 325

E. Oset, A. Ramos, NPA 635 (1998) 99

Weinberg-Tomozawa term

of effective chiral Lagrangian

Based on Chiral SU(3) theory

→ Energy dependence

• Anti-kaon, Pion = Nambu-Goldstone boson

... governed by chiral dynamics

Constrained by KbarN scattering length

• Old data: aKN(I=0) = -1.70 + i0.67 fm, aKN(I=1) = 0.37 + i0.60 fm A. D. Martin, NPB179, 33(1979)

• SIDDHARTA K-p data with a coupled-channel chiral dynamics:

aK-p = -0.70 + i0.89 fm, aK-n = 0.57 + i0.72 fm M. Bazzi et al., NPA 881, 88 (2012)

Y. Ikeda, T. Hyodo, W. Weise, NPA 881, 98 (2012)

( 0,1)

( 0,1)

28

I

ij i jI

ij ijji

i j

C M MV r g r

f s

3

2

/ 2 3ex

1p

ijij

ij

g rd

r d

Potential in relativistic kinematics

: Gaussian form

ωi: meson energy

A. Dote, T. Inoue, T. Myo, NPA 912, 66 (2013)

Chiral SU(3)-based KbarN potential

Two types of the potentials...

Λ* resonance pole

Broad Narrow

Martin constraint

fπ=110 MeV

A. Dote, T. Inoue, T. Myo, NPA 912, 66 (2013)

3. Result

• Check: Complex eigenvalue distribution

in Semi-rela. case

• Result w/ Martin constraint

• Result w/ SIDDHARTA constraint

Check: Complex eigenvalue distribution in Semi-rela. case

22 2

22

2

i ii i

i

i

i

pp e m m

e

m

In low energy region, eigenvalues of three-body continuum

states are on 2θ lines, similarly to Non-rela. case.

πΛN πΣN KbarNNComplex energy plane

(Units are MeV.)

Case:

• SIDDHARTA / Field

• fπ=110 MeV

• θ=24°, 6400 dim.

“K-pp” “Λ*”

Check: Complex eigenvalue distribution in Semi-rela. case

In high energy region, eigenvalues of three-body continuum

states are on 1θ lines, differently from Non-rela. case.

Complex energy plane

(Units are MeV.)

Case:

• SIDDHARTA / Field

• fπ=110 MeV

• θ=24°, 6400 dim.

πΛN

2 2 2 2 2 2i i

i i i

i

ip m p e p ee

Start at –(mπ+MΛ+MN),

since the origin is taken

to be the πΛN threshold.

Result w/ Martin constraint

• Field

• Particle

－62.6 － i16.8

－50.5 － i22.2

－20.4 － i11.3

－21.7 － i11.7

“K-pp”－ BK-pp － i Γ/2 [MeV]

SR-A SR-B

Λ* 1419.5 － i 25.0 1420.0 － i 12.7

M － i Γ/2 [MeV]

( fπ=110 )

An interacting MB pair carries

100% of B(M) = “Field picture”

50% of B(M) = “Particle picture”

* B(M) ... “Meson’s binding energy”

NN potential = Av18 potential (R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, PRC 51, 38 (1995))

Result w/ Martin constraint

• Field

• Particle

－62.6 － i16.8

－50.5 － i22.2

－20.4 － i11.3

－21.7 － i11.7

“K-pp”－ BK-pp － i Γ/2 [MeV]

SR-A SR-B

Λ* 1419.5 － i 25.0 1420.0 － i 12.7

M － i Γ/2 [MeV]

( fπ=110 )

SR-A: Large binding energy as well as large decay width

Very attractive in bound region!

Result w/ Martin constraint

• fπ and ansatz dependence

SR-A SR-B

fπ=100

120

120

100fπ=90

120X

• No self-consistent solution at fπ=100 MeV

• Large binding energy and decay width

• Strongly depends on fπ and ansatz

• Small binding energy and decay width

• Small fπ dependence

• Similar solutions obtained by both ansatzes

(BK-pp, Γ/2)

• Field: (55,17) ～ (63,17)

• Particle: (43,20) ～ (62,26)

(BK-pp, Γ/2)

• Field: (19,10) ～ (28,16)

• Particle: (21,11) ～ (29,16)

Result w/ SIDDHARTA constraint

a(I=0) [fm] -1.974 + i1.058

a(I=1) [fm] 0.570 + i0.731

KbarN scat. leng. SIDDHARTA+IHW

-1.97 + i1.05

0.57 + i0.73

• Field

• Particle

－46.4 － i 23.9

－33.8 － i 23.7

1.62 － i0.19

1.66 － i0.06

“K-pp”－ BK-pp － i Γ/2 [MeV] RNN [fm]

fπ=110 MeV

Λ*1429.1 － i 16.2

M － i Γ/2 [MeV]

SR-A ( fπ=110 MeV)

Result w/ SIDDHARTA constraint

• fπ and ansatz dependence

fπ=100

120120

100SR-A

Binding energy (BKpp)

and half decay width (ΓπYN/2)

• Field picture

BKpp = 39 ～ 59 MeV

ΓπYN/2 = 21 ～ 26 MeV

• Particle picture

BKpp = 29 ～ 41 MeV

ΓπYN/2 = 19 ～ 33 MeVNN potential = Av18 potential

Result w/ SIDDHARTA constraint

• Comparison with Martin case

SR-A

• Weaker binding (10~20 MeV)

• Larger decay width

... as expected from the scattering length

Matin case

• Comparison with Non-rela. case

Non-rela.

A. Dote, T. Inoue, T. Myo,

PLB 704, 405 (2018)

• Larger decay width, as expected

• Larger binding energy

4. A concern ...

Consistency between

NN potential and kinematics?

Av18 NN potential was constructed under non-relativistic kinematics.

Can it be used in semi-rela. kinematics? (By Shinmura, Harada, Akaishi)

← NN kinematics might not give an influence to the result,

because nucleon mass is much larger, compared with its momentum. (Dote)

NN phase shift(1E)

Av18 potential

Phase shift calculated in SR is slightly more attractive

than that calculated in NR.

A test calculation:

Baryons are again treated non-relativisitcally.

• Meson ... Semi-relativistic kinematics• Baryon ... Non-relativistic kinematics

22

( )

( )

1 ( )

22

2

Baryon i

Baryon i

i Bary

NN MBMesonMe

o

so

i

n

n

H m V Vmm

p

p

NN kinematics modifies the result more than my expectation...

Result

• Binding energy decreases

by 5~16 MeV.

• Decay width increases.

• Case: SIDDHARTA/ Field picture

SR+NR SRfπ -B.E. (K-pp) -Γ /2 -B.E. (K-pp) -Γ /2

100 -52.3 -32.8 -58.5 -25.8

110 -30.2 -29.3 -46.4 -23.9120 -23.4 -22.3 -39.4 -21.4

5. Summary

and Future plan

Future plan• NN potential for semi-relativistic kinematics

• More conditions to constrain the KbarN potential

• Non-mesonic decay mode (KbarNN→YN)

• Spectrum calculation using Green function method in ccCSM

Focusing on the decay width of K-pp, we examine the semi-relativistic kinematics (SR)

in a fully coupled-channel complex scaling method.

It is confirmed that the decay width of K-pp can be larger in SR kinematics

than in non-relativistic kinematics (NR). (ΓπYN/2 is less than 20 MeV in NR.)

However, there is a concern on consistency between NN potential and kinematics.

Since Av18 NN potential is constructed under the NR kinematics, it might be inconsistent

to use it in SR kinematics ...

SIDDHARTA constraint case,

• (BK-pp, ΓπYN/2) = (39,21) ～ (59,26) w/ Field picture

• (BK-pp, ΓπYN/2) = (29,19) ～ (41,33) w/ Particle picture

* Av18 potential is used as a realistic NN potential.

Summary

In addition, it is found that in SR kinematics K-pp could be deeply bound with

a chiral SU(3)-based KbarN potential, although it is obtained to be shallowly bound

in NR kinematics. (BK-pp is up to 40 MeV with Field picture in NR.)

References:

• A. Dote, T. Inoue, T. Myo, PLB 784, 405 (2018)• A. Dote, T. Inoue, T. Myo, PRC 95, 062201(R) (2017)

Thank you very much!