KEK理論センター研究会「ストレンジネス核物理2010」@KEK, 2-4 Dec 2010.

Study of KN-πΣ interactions with accurate
experimental data- based on coupled-channel chiral SU(3) dynamics -Y. Ikeda (RIKEN, Nishina Center)in collaboration with
T. Hyodo(Tokyo Tech.), W. Weise(TUM)
andHAL QCD Collaboration

Plan of this talkPart 1 : Coupled-channel chiral dynamics with
SIDDHARTA K--hydrogen energy shift data
Y.I., T.Hyodo, W.Weise(2010).

✓ Introduction to KbarN - πΣ interactions✓ Coupled-Channel Chiral SU(3) dynamics✓ χ2 analysis of leading order terms of chiral EFT withSIDDHARTA (preliminary) data

Part 2 : πΣ scattering on lattices
Y.I. for HAL QCD Coll.(2010)✓ Inﬂuence on dibaryon energy by I=0 πΣ pole positions✓ LQCD potential for I=2 πΣ system

Why KbarN - !" interactions?
KbarN interaction is strongly attractive

Energy-shift of K- atoms

Presence of Λ(1405)

Two poles in amplitudes : Double pole structure of Λ(1405)
Jido et al., NPA723, 205 (2003).
Hyodo, Weise, PRC77, 035204 (2008).

Chiral dynamics
KbarN-KbarN interaction :strong enough to form bound stateπΣ-πΣ interaction :strong correlation & energy-dependence
to generate resonanceFrom #(1405) to strange dibaryon
Phenomenological or Separable interactions
Akaishi, Yamazaki-> (B,Γ)=(48,61) MeV [PRC65(2002).]
Shevchenko, Gal, Mares -> (B,Γ)=(50-70,90-110) MeV [PRL98(2007).]
Ikeda, Sato-> (B,Γ)=(45-80,45-75) MeV [PRC76(2007).]
Wycech, Green-> (B,Γ)=(40-80,40-85) MeV [PRC79(2009).]-> Deeply bound dibaryon state

Chiral interactions (leading order Weinberg-Tomozawa interactions)
Dote, Hyodo, Weise-> (B,Γ)=(17-23,40-70) MeV [NPA804(2008)., PRC79(2009).]
Ikeda, Kamano, Sato-> (B,Γ)=(9-16,34-46) MeV [PTP124(2010).]-> Weakly bound dibaryon state

Issues to be studied in next step

✓ More quantitative investigation of the energy of strange dibaryon✓ Comparison with experimental data such as DISTO
Constructing realistic KbarN - πΣ interactions
based on low-energy QCD

Coupled-channel chiral SU(3) dynamics
Low-energy interactions of Nambu-Goldstone bosons

couple to baryons are described byLeﬀ (Φ, B) = LM (Φ) + LB (Φ, B)“Systematic” low-energy expansion in mπ, momenta(Chiral Perturbation Theory)

+ΦΦ

Φ

+ ...

+ΒΦΒ

$: Meson field, B : Baryon field

Vertices at leading order :Derivative coupling -> Energy-dependent interactions
If systems contain a bound state... (e.g., KbarN-πΣ system)
Non-perturbative Coupled-channel dynamics withinteractions based on chiral SU(3) symmetry is necessary
Kaiser, Siegel, Weise, NPA594(1995).
Oset, Ramos, NPA635(1998).
Oller, Meissner, PLB500(2001).
many others...

Coupled-channel chiral SU(3) dynamics
Coupled-channel chiral SU(3) dynamics
Non-perturbative dynamics

Ti,j = Ki,j +�Ki,n Gn Tn,jn

T=+T

Interaction kernels from chiral EFT-> Couplings are fully determined by chiral SU(3) symmetry
with low-energy constants
Loop integrals with the use of dimensional regularization
�dd qi2G(p ) =2(2π)d [(p − q)2 − MB+ i�][q 2 − m2φ + i�]-> Finite parts include subtraction constant : a(μ)

Toward realistic KbarN-!" interactions
Construct more realistic KbarN-πΣ interactions
Framework : Chiral SU(3) dynamics withWT + Born + NLO terms-> χ2 analysis of experimental data with chiral SU(3) dynamics
Experimental data we input are...
Accurate K-p scattering length <- SIDDHARTA
Threshold branching ratios
K-p reaction cross sections
Necessary conditions
What we will predict in the future are...

πΣ invariant mass spectra -> HADES, CLAS, LEPS,...
Mass spectra of strange dibaryons -> DISTO, J-PARC,..

Role of Weinberg-Tomozawa term
χ2 analysis with coupled-channel chiral SU(3) dynamics( 10 channels : K-p, Kbar0n, πΛ, π0Σ0, π+Σ-, π-Σ+, ηΛ, ηΣ0, K+Ξ-, K0Ξ0 )Borasoy, Nissler, Weise, EPJA25(2005).
Brasoy, Meissner, Nissler, PRC74(2006).
Nissler, PhD thesis (2008)

Experimental data used in the ﬁt1) Threshold branching ratios
2) K-p reaction cross sections3) πΣ mass spectrum by Hemingway

Prediction of KbarN
scattering length<- Predicted energy shift ofkaonic hydrogen with
Weinberg-Tomozawa term
Weinberg-Tomozawa prediction
agrees with

SIDDHARTA?
Nissler, PhD thesis (2008)(preliminary) SIDDHARTA resultOur strategy in the fitting1) updated analysis with

Weinberg-Tomozawa term in coupled-channel( 10 channels : K-p, Kbar0n, πΛ, π0Σ0, π+Σ-, π-Σ+, ηΛ, ηΣ0, K+Ξ-, K0Ξ0 )Experimental data used in this study1) Accurate SIDDHARTA results (preliminary)2) Threshold branching ratios
fπ = 92.4(MeV)
3) K p reaction cross sections
fK = (1.22 ± 0.01)fπwith “physical meson decay constants”
fη = (1.30 ± 0.05)fπ
Predict πΣ mass spectrum & compare to data
Can Weinberg-Tomozawa term excellently
describe low-energy K-p scattering data?
2) Inclusion of Born + NLO terms
A study to construct realistic KbarN interactions

Results (Weinberg-Tomozawa term)
Best ﬁt results :kaonic hydrogen shift & width

TheoryEXP.

345

280 35488

520 80Γ(K − p→π + Σ− )Γ(K − p→π − Σ+ )2.35

2.36 0.04

Γ(K − p→π + Σ− , π − Σ+ )Γ(K − p→all inelastic channels)

0.66

0.66 0.01

Γ(K − p→π 0 Λ0 )
Γ(K − p→neutral channels)

0.21

0.19 0.02

K-p scattering lengthaK-p = -0.86+i0.78 (fm)Pole position of Λ(1405)

1422-i20.8, 1386-i74.8 (MeV)∆E (eV)Γ (eV)−2α3 µ2c aK − p [1 − 2αµc (lnα − 1)aK − p ]Threshold branching ratios

Shift of kaonic hydrogen is a little bit larger thanthat of SIDDHARTA (preliminary) value

Results (Weinberg-Tomozawa term)
Best ﬁt results : total cross sections

All cross sections of K-p reactions are described very wellResults (Weinberg-Tomozawa term)
Subtraction constants ai(μ) x 10-3 @ μ=1GeVaKbarN(μ)=-1.95aπΣ(μ)=1.93aπΛ(μ)=-58.0aηΛ(μ)=-0.21aηΣ(μ)=116.3
aKΞ(μ)=64.9

suitable value for ai(μ) : -20 < ai(1GeV) < 20

Some subtraction constants are “unnaturally” large-> Model may include (CDD) pole contributions
Hyodo, Jido, Hosaka, PRC79(2008).

Born term contributions are studied+Results (WT+Born terms)
Inclusion of Born terms : No additional parameterkaonic hydrogen shift & width

TheoryEXP.

345 -> 339

280 35488 -> 661

520 80Γ(K − p→π + Σ− )Γ(K − p→π − Σ+ )2.35 -> 2.35

2.36 0.04

Γ(K − p→π + Σ− , π − Σ+ )Γ(K − p→all inelastic channels)

0.66 -> 0.64

0.66 0.01

Γ(K − p→π 0 Λ0 )
Γ(K − p→neutral channels)

0.21 -> 0.23

0.19 0.02

K-p scattering lengthaK-p = -0.75+i1.02 (fm)Pole position of Λ(1405)

1427-i19.0, 1381-i75.7 (MeV)∆E (eV)Γ (eV)−2α3 µ2c aK − p [1 − 2αµc (lnα − 1)aK − p ]Threshold branching ratios

best-fit values from WT term by red

Results (Weinberg-Tomozawa term)
Subtraction constants ai(μ) x 10-3 @ μ=1GeVaKbarN(μ)=-0.52aπΣ(μ)=1.47aπΛ(μ)=-19.7aηΛ(μ)=-5.48aηΣ(μ)=15.2
aKΞ(μ)=-5.96

suitable value for ai(μ) : -20 < ai(1GeV) < 20All subtraction constants are “natural”
WT + Born terms well describe KbarN experimental data++Small deviations <- NLO terms in chiral EFT

Results (Weinberg-Tomozawa + Born terms)
Theory prediction : πΣ mass spectrum

dNdm

2∝ |tI=0πΣ−πΣ | pc.m.

Theory prediction agrees with πΣ(I=0) invariant massspectrum by Hemingway below KbarN threshold
Deviations above KbarN threshold is improved by NLO terms

Nissler, PhD thesis (2008)

Importance of !" interaction

Y.I.,Kamano,Sato, PTP124(2010).
arXiv:1004.4877[nucl-th].

Pole of Λ(1405)~12MeV~25MeV~10MeV

Pole ofstrange dibaryon!"N : 2267

KbarNN : 2370
E (MeV)

100200300
Γ (MeV)~15MeV~35MeV~55MeVImportance of !" interaction
Y.I., Hyodo, Jido, Yazaki et al., in preparation.

Scattering amplitude near thresholdf (k) =�1
kcotδ(k) − ik112

�− re k + · · ·a2
Scattering length & effective rangekcotδ(k) =-> nature of pole position

Challenging issue is extracting πΣ scattering length in I=0 channel-> Hyodoʼs talk-> Useful constraint on “realistic KbarN-πΣ interactions”<- Combined analysis using experimental data & Lattice QCD data
HAL QCD Collaboration
S. Aoki, T. Doi, K. Sasaki (Univ. Tsukuba)
T. Hatsuda, N. Ishii(Univ. Tokyo)
H. Nemura(Tohoku Univ.)
T. Inoue
K. Murano
Y. Ikeda(Nihon Univ.)(KEK)(RIKEN)LQCD approach to I=0 πΣ scattering
Direct calculations of πΣ scattering in I=0 channel
Luescherʼs formula -> scattering length, phase shift
Bethe-Salpeter wave func. -> potential -> observable
Quark annihilation diagram in I=0 channel-> LQCD data would be noisy (need high statistics)πΣ scattering length from Λc decay

Hyodo, Oka, in preparation

I=2 πΣ scattering length is important input( Clear signal can be expected for I=2 πΣ scattering from LQCD )We examine s-wave I=2 πΣ scattering from LQCD potential

LQCD “potentials”

4-point function

G4 (�r , t) =

�Aoki, Hatsuda, Ishii, PTP123, 89 (2010).

< φ1 (�x+�r , t)φ2 (�x, t)J1† (0)J2† (0) >�x→ A0 ψ(�r )e−Erel t
spacial correlation :ψ(�r ) is Bethe-Salpeter wave func.

Deﬁne potentials : energy independent, non-local
from half off-shell T-matrix

�rscattering length, effective range, phase shift,binding energy, ...

tNon-locality determined order by order in derivative expansion

0LQCD setup
Full QCD conﬁgurations generated by PACS-CS Coll.

•Gauge coupling : β=1.90•Iwasaki gauge & Wilson clover•Lattice spacing : a=0.091 (fm)•Box size : 323x64 -> L=2.9 (fm)•Conf. # : 399•Static wall source
Hadron masses
Mπ~710 (MeV)
MK~790 (MeV)
MN~1580 (MeV)MΣ~1650 (MeV)Wave functions & Potentials

Phase shifts

Primielyrna

I=2 πΣ phase shift is repulsive (No experimental data)
I=1 KN phase shift is qualitatively consistent withexperimental data
Deviations from experimental data for I=1 KN scattering
would attribute to heavy quark masses used in this simulation

Summary and Future targets
1. χ2 analysis of the experimental data with leading order terms ofchiral EFT is studied

We wait for SIDDHARTA final data
2. Constructing realistic KbarN-πΣ interactions including NLO termsof chiral EFT is now on-going
3. We will calculate πΣ invariant mass and 3-body scatteringobservables with “realistic KbarN-πΣ interactions”

We will compare theory results with
precisely measured experimental data
4. Constraint from πΣ observable(Decay of heavy particle & LQCD calc.)
5. Input for Λ*N interaction (Uchinoʼs talk)
6. Kbar-meson in nuclear medium (optical potentials)

7. .... so many applications with “realistic KbarN - πΣ interaction”

Thank you very much for your attention!!