Recent progress on anti-kaon--nucleon interactions and dibaryon - - PowerPoint PPT Presentation

Recent progress on anti-kaon--nucleon interactions and dibaryon resonances

Yoichi IKEDA (RIKEN, Nishina Center)

“Recent progress in hadron physics” --From hadrons to quark and gluon-- @Yonsei Univ., Korea, Feb.20, 2013.

Outline of lectures

• Chiral symmetry in QCD
• Chiral effective field theory (chiral perturbation theory)
• Antikaon - nucleon interactions and nature of Λ(1405)

✦ Chiral symmetry and chiral dynamics ✦ Antikaon in nuclei

• Antikaon-nucleon potentials model based on chiral dynamics
• Faddeev equations to handle three-body dynamics (KbarNN - πYN coupled system)
• Possible spectrum of KbarNN - πYN system

✦ Hadron interactions from lattice QCD

• Introduction to lattice QCD
• Hadron scattering on the lattice
• Application to meson-baryon scattering

(3) Hadron interactions from LQCD

Kbar-N and Kbar-nuclear physics

✓ KbarN (I=0) interaction is...

• coupled with πΣ channel
• strongly attractive

➡ quasi-bound state of Λ(1405)

• Λ(1405) is the lowest state among negative parity baryons

✓ Quest for quasi-bound Kbar-NN systems

• FINUDA, DISTO, J-PARC (E15, E27), ...

√s[MeV]

πΣ : 1330[MeV] ¯ KN : 1435[MeV]

Λ(1405)

Phenomenological construction of KbarN interaction leads to dense Kbar-nuclei Central density is much larger than normal nuclei <- Λ(1405) doorway process to dense matter

Akaishi, Yamazaki, PRC65 (2002). Dote, Horiuchi, Akaishi, Yamazaki, PRC70 (2004).

✓ Many theoretical studies have been motivated..., but...

Coupled-channel Faddeev approach

B [MeV] Γ two-body input: KbarN interaction

Overview

Variational approach

B [MeV] Γ two-body input: KbarN interaction 48 17-23 40-80 16 61 40-70 40-85 42 Phenomenological optical potential [1] Effective chiral SU(3) potential [2] Phenomenological potential [3] Effective chiral SU(3) potential [4] 50-70 45-80 9-16 90-110 45-75 34-46 Phenomenological potential [5] Chiral SU(3) potential (E-indep.) [6] Chiral SU(3) potential (E-dep.) [7]

[1] Akaishi, Yamazaki, PLB535, 70 (2002); PRC76, 045201 (2007). [2] Dote, Hyodo, Weise, NPA804, 197 (2008); PRC79, 014003 (2009). [3] Wycech, Green, PRC79, 014001 (2009). [4] Barnea, Gal, Liverts, PLB712, 132 (2012).

• Binding energy and width of quasi-bound [Kbar[NN]I=1]

[5] Shevchenko, Gal, Mares, (+Revay,) PRL98, 082301 (2007); PRC76, 044004 (2007). [6] Ikeda, Sato, PRC76, 035203 (2007); PRC79, 035201 (2009). [6] Ikeda, Kamano, Sato, PTP124, 533 (2010).

Coupled-channel Faddeev approach

B [MeV] Γ two-body input: KbarN interaction

Overview

Variational approach

B [MeV] Γ two-body input: KbarN interaction

[2] Dote, Hyodo, Weise, NPA804, 197 (2008); PRC79, 014003 (2009). [3] Wycech, Green, PRC79, 014001 (2009). [4] Barnea, Gal, Liverts, PLB712, 132 (2012).

• Binding energy and width of quasi-bound [Kbar[NN]I=1]

[5] Shevchenko, Gal, Mares, (+Revay,) PRL98, 082301 (2007); PRC76, 044004 (2007). [6] Ikeda, Sato, PRC76, 035203 (2007); PRC79, 035201 (2009). [6] Ikeda, Kamano, Sato, PTP124, 533 (2010).

Energy independent

[1] Akaishi, Yamazaki, PLB535, 70 (2002); PRC76, 045201 (2007).

48 17-23 40-80 16 61 40-70 40-85 42 Phenomenological optical potential [1] Effective chiral SU(3) potential [2] Phenomenological potential [3] Effective chiral SU(3) potential [4] 50-70 45-80 9-16 90-110 45-75 34-46 Phenomenological potential [5] Chiral SU(3) potential (E-indep.) [6] Chiral SU(3) potential (E-dep.) [7]

[6] Ikeda, Sato, PRC76, 035203 (2007); PRC79, 035201 (2009). [5] Shevchenko, Gal, Mares, (+Revay,) PRL98, 082301 (2007); PRC76, 044004 (2007). [3] Wycech, Green, PRC79, 014001 (2009).

Coupled-channel Faddeev approach

B [MeV] Γ two-body input: KbarN interaction

Overview

Variational approach

B [MeV] Γ two-body input: KbarN interaction

[2] Dote, Hyodo, Weise, NPA804, 197 (2008); PRC79, 014003 (2009). [4] Barnea, Gal, Liverts, PLB712, 132 (2012).

• Binding energy and width of quasi-bound [Kbar[NN]I=1]

[6] Ikeda, Kamano, Sato, PTP124, 533 (2010).

Energy dependent

[1] Akaishi, Yamazaki, PLB535, 70 (2002); PRC76, 045201 (2007).

48 17-23 40-80 16 61 40-70 40-85 42 Phenomenological optical potential [1] Effective chiral SU(3) potential [2] Phenomenological potential [3] Effective chiral SU(3) potential [4] 50-70 45-80 9-16 90-110 45-75 34-46 Phenomenological potential [5] Chiral SU(3) potential (E-indep.) [6] Chiral SU(3) potential (E-dep.) [7]

πΣ pole position is relevant to determine...

• Nature of Λ(1405) : single resonance pole v.s. double resonance pole
• Strange dibaryon energy, fate of “kaonic nuclei”

How can we determine πΣ pole position? --> scattering parameter is key

Importance of πΣ scattering

Energy of strange dibaryon:

• Phenomenological potential / E-indep models
• -> deeply quasi-bound KbarNN state
• Chiral SU(3) potential models
• -> shallow quasi-bound KbarNN state

πΣ resonance pole Chiral SU(3) model

X

πΣ virtual pole

X

E-indep. model Difference: πΣ pole position

Demonstration of classification input: KN scattering length --> πΣ scattering parameters Scattering amplitude near threshold Low-energy scattering parameters such as length, effective range...

• > nature of πΣ pole position

f(k) = 1 kcotδ(k) − ik

Threshold behavior of πΣ scattering

Y.I., Hyodo, Jido, Kamano, Sato, Yazaki, PTP125 (2011).

⇣ kcotδ(k) = 1 a + 1 2rek2 + · · · ⌘

πΣ scattering length from Λc decay

Hyodo & Oka, PRC84 (2011).

Two independent equations & three unknown valiables

• -> One of πΣ scattering lengths is important input

S-wave I=2 πΣ scattering on the lattice ( Clear signal in I=2 πΣ scattering is expected from LQCD ) Scattering amplitude near threshold Low-energy scattering parameters such as length, effective range...

• > nature of πΣ pole position

f(k) = 1 kcotδ(k) − ik

Threshold behavior of πΣ scattering

Y.I., Hyodo, Jido, Kamano, Sato, Yazaki, PTP125 (2011).

⇣ kcotδ(k) = 1 a + 1 2rek2 + · · · ⌘

Path integral formalism in Euclidean space-time

LQCD = −1 2trFµνF µν + ¯ ψ(iγµDµ − m)ψ

Well defined statistical field theory Gauge invariant formalism Non-perturbative method

Monte Calro calculation L a space Euclidean time X X

: Quenched QCD : Full QCD

detD(U, m) = 1

detD(U, m) 6= 1 hO(τ)i = 1 Z Z DUe−Sg(U)detD(U, m)O(τ)

Un,ˆ

µ

ψ(n)

• Quarks : ψ(n)
• Gluons : Un,µ

Lattice QCD

Quark mass ( mq --> mphys. ) Lattice spacing (a --> 0) Lattice volume ( 1/L, T --> 0 )

Lattice QCD simulation with physical quark masses, finer lattice, large volume is desirable

Important limit in LQCD

• Physical quark (pion) mass (mphys.)
• Continuum limit (a --> 0)

PACS-CS Coll., PRD81 (2010). BMW Coll., Science (2008). BMW Coll., Science (2008).

• Thermodynamic limit (1/L, T --> 0): on-going

10fm3, phys. point, full QCD gauge configuration by PACS-CS Coll.

Scattering on the lattice

ψ(r > R): phase shift (Luscher’s formula) (asymptotic region) [R : interaction range] (interacting region) ψ(r < R): potential --> observable

• M. Lüscher, NPB354, 531 (1991).

CP-PACS Coll., PRD71, 094504(2005). Ishii, Aoki, Hatsuda, PRL99, 02201 (2007).

• -> Ozaki’s talk(Tue.)

L

a

τ

~ x

X X X X

(~ r, t) = X

~ x, ~ X,~ Y

h0|M(~ x + ~ r, t)B(~ x, t)M( ~ X, t = 0)†B(~ Y , t = 0)†|0i

• Scattering parameters:

low-energy expansion of phase shift

kcotδ(k) = 1 a − 1 2rek2 + · · ·

• Relevant to resonance property

Potential method (recipe)

Nambu-Bethe-Saltpeter wave function Many applications:

nuclei, exotics, ... astrophysics input...

• -> phase shift, T-matrix

Potential defined on the lattice

Baryon-Baryon, Baryon-Baryon-Baryon, Meson-Baryon, Meson-Meson, ...

• S. Aoki, B. Charron, T. Doi, T. Hatsuda,
• Y. Ikeda, T. Inoue, N. Ishii, K. Murano,
• H. Nemura, K. Sasaki

HAL QCD Collaboration

Full details, see, Aoki, Hatsuda, Ishii, PTP123, 89 (2010).

Nambu-Bethe-Salpeter wave function

NBS wave function in quantum field theory is the best analogue to wave function in quantum mechanics At large distance r, NBS amplitudes satisfy free Klein-Gordon equations:

(@2

t r2 i + m2 π)Ψ(~

x1, t; ~ x2, t) = 0 (i = 1, 2) (r2

r + ~

k2) ππ(~ r; W ) = 0

Equal-time Nambu-Bethe-Salpeter(NBS) amplitudes (e.g., ππ scattering)

Ψ(~ x1, t; ~ x2, t) ⌘ h0|⇡(x1)⇡(x2)|⇡(~ k)⇡(~ k); ini

W = 2 q m2

π + ~

k2

= ππ(~ r; W )e−iW t

Spatial correlation ψ(r) is NBS wave function and satisfies Helmholtz equation Asymptotic form of NBS wave function:

• -> faithful to scattering phase shift

ψ(l)

~ k (r) ∼ eil(k)

kr sin

• kr + δl(k) − lπ/2

Full details, see, Aoki, Hatsuda, Ishii, PTP123, 89 (2010).

Lattice QCD potential

Derive potential: non-local, energy-independent potential by construction

U(~ r, ~ r0) = Z dW 2⇡ KW (~ r) ⇤

MB(~

r0; W )

Non-local potential satisfies time-independent Schrödinger-type equation:

(E − H0) MB(~ r; W ) = Z d~ r0U(~ r, ~ r0) MB(~ r0; W )

Possible improvement: We do NOT need ground (single) state saturation in LQCD simulation to extract potentials, because they are energy-independent. Define half off-shell T-matrix in interacting region: Plane wave components are projected out

(E − H0) MB(~ r; W ) = KW (~ r) (r < R)

E = ~ k2 2µ, H0 = r2

r

2µ

Helmholtz equation of NBS wave function:

(r2

r + ~

k2) MB(~ r; W ) = 0 (r > R)

W = q m2 + ~ k2 + q M 2 + ~ k2

Ishii et al,(HAL QCD Coll.), PLB712,437(2012).

Improved lattice QCD potential

1) Start with normalized meson-baryon correlation functions (R-correlators) Advantage of this method: We obtain potentials even with excited state contaminations --> check in NN system 3) Velocity expansion: (LO)

U(⇥ r, ⇥ r0) = V (⇥ r, r)(⇥ r ⇥ r0)

(NLO) VMB(~ r, r) = VC(~ r) + ~ L · ~ SVLS(~ r) + O(r2)

= X

~ k

A~

kexp

h −∆W (~ k)t i ~

k(~

r)

Interaction energy: ∆W (~

k) = q m2

M + ~

k2 + q m2

B + ~

k2 − (mM + mB) ψ(l)

~ k (r) ∼ eil(k)

kr sin

• kr + δl(k) − lπ/2
• NBS wave func.:

R(~ r, t) = e(mM +mB)t X

~ x, ~ X,~ Y

h0|M(~ x + ~ r, t)B(~ x, t)M( ~ X, t = 0)†B(~ Y , t = 0)†|0i

X

~ X,~ Y

h~ k; B = 1, J⇡|M( ~ X, t = 0)†B(~ Y , t = 0)†|0i

Overlap:

2) Time derivative of R-correlators --> time-dependent Schrödinger-type equation

(− @ @t − H0 + · · · )R(~ r, t) = Z d~ r0U(~ r, ~ r0)R(~ r0, t)

H0 = r2

r

2µ

Relativistic correction: W (~ k)rel = −

• ∆W (~

k) − ~ k2/2µ

1st term Total central force Operator dependence disappear Mid-range attraction becomes large (2nd term is crucial)

NN potential from HAL QCD method

Central force of singlet NN system (Leading order of velocity expansion) VC(~ r) = −H0R(~ r, t) R(~ r, t) − @ @t log R(~ r, t) + 1 4MN (@/@t)2R(~ r, t) R(~ r, t) Relativistic correction: W (~

k)rel = −∆W (~ k)2/4MN

Source operator dependence: possible contaminations are different

f(x, y, z) = 1 + α

• cos(2πx/L) + cos(2πy/L) + cos(2πz/L)
• Full QCD, mπ~700 MeV

Ishii et al,(HAL QCD Coll.), PLB712,437(2012).

• Gauge coupling : β=1.90
• Iwasaki gauge & Wilson clover
• Lattice spacing : a=0.091 (fm)
• Box size : 323x64 -> L=2.9 (fm)
• Conf. # : 399

Full QCD configurations generated by PACS-CS Coll.

Hadron masses Mπ~704 (MeV) MK~784 (MeV) MN~1568 (MeV) MΣ~1637 (MeV)

Lattice QCD setup for MB system

PACS-CS Coll., S. Aoki et al., PRD79, 034503, (2009).

S-wave meson-baryon potentials@mπ=700MeV

I=2 πΣ = π+Σ+ -> (udbar)(uus) I=1 KN = K+p -> (usbar)(uud)

VC(~ r) = −H0R(~ r, t) R(~ r, t) + E

Potentials constructed from time-independent Schrödinger equation Strong repulsive core near origin + mid-range attraction?? Note that single state saturation is not sufficient @ t-tsrc=13

• Y. Ikeda et al.[HAL QCD Coll.], PoS Lattice2011 (2011) arXiv:1111.2663[hep-lat].
• S. Aoki et al.[HAL QC Coll.], PTEP 2012 (2012) 01A105

Potentials constructed from time-dependent Schrödinger equation Strong repulsive core near origin Mid-range attraction disappears

VC(~ r) = −H0R(~ r, t) R(~ r, t) − @ @t log R(~ r, t)

I=2 πΣ = π+Σ+ -> (udbar)(uus) I=1 KN = K+p -> (usbar)(uud) Relativistic correction is neglected

S-wave meson-baryon potentials@mπ=700MeV

• Y. Ikeda et al.[HAL QCD Coll.], PoS Lattice2011 (2011) arXiv:1111.2663[hep-lat].
• S. Aoki et al.[HAL QC Coll.], PTEP 2012 (2012) 01A105

Hard core does NOT appear in I=0 KN potential see also, Baryon-Baryon potentials in SU(3) limit,

Inoue et al [Hal QCD]., PTP124, 591 (2010).

I=0 KN ~ K+n -> (usbar)(udd) I=1 KN = K+p -> (usbar)(uud) I=1 KN state : one up-quark cannot be in s-state

Pauli blocking in MB systems

Pauli blocking meson-baryon system (Quark model expectation)

Machida, Namiki, PTP33 (1965).

Quark mass dependence

• Gauge coupling : β=1.90
• Iwasaki gauge & Wilson clover
• Lattice spacing : a=0.091 (fm)
• Box size : 323x64 -> L=2.9 (fm)
• Conf. # : 400

Full QCD configurations generated by PACS-CS Coll.

Hadron masses Mπ~567 (MeV) MK~714 (MeV) MN~1389 (MeV) MΣ~1518 (MeV) PACS-CS Coll., S. Aoki et al., PRD79, 034503, (2009).

Potentials constructed from time-dependent Schrödinger equation Strong repulsive core near origin (Pauli principle at work)

VC(~ r) = −H0R(~ r, t) R(~ r, t) − @ @t log R(~ r, t)

Relativistic correction is neglected

S-wave meson-baryon potentials@mπ=570MeV

I=2 πΣ = π+Σ+ -> (udbar)(uus) I=1 KN = K+p -> (usbar)(uud)

• Y. Ikeda et al.[HAL QCD Coll.], PoS Lattice2011 (2011) arXiv:1111.2663[hep-lat].
• S. Aoki et al.[HAL QCD Coll.], PTEP 2012 (2012) 01A105

Very weak quark mass dependence Short range repulsions become a little bit large as decreasing mq

VC(~ r) = −H0R(~ r, t) R(~ r, t) − @ @t log R(~ r, t)

Quark mass dependence of potentials

I=2 πΣ = π+Σ+ -> (udbar)(uus) I=1 KN = K+p -> (usbar)(uud)

• Y. Ikeda et al.[HAL QCD Coll.], PoS Lattice2011 (2011) arXiv:1111.2663[hep-lat].
• S. Aoki et al.[HAL QC Coll.], PTEP 2012 (2012) 01A105

Relativistic correction is neglected

Quark mass dependence of phase shifts

I=2 πΣ phase shift is repulsive (No experimental data) I=1 KN phase shift is qualitatively consistent with experimental data Weak quark mass dependences are observed Fit potential data --> Solve Schrödinger equations --> phase shifts Scattering length (fm): Mπ=570MeV Mπ=700MeV aπΣ

• 0.229(25)
• 0.218(25)

aKN

• 0.241(33)
• 0.221(26)

Relativistic kinematics

R-correlator with

R(t, ⇥ x) = e(m1+m2)tCh1,h2(t, ⇥ x) =

• k

A

ke−∆W t k(⇥

x)

Definition of semi-relativistic potential Local semi-relativistic potential (leading order of derivative expansion)

∆W =

• m2

1 +

k2 +

• m2

2 +

k2 − (m1 + m2)

⌅ m2

1 + ⇥

k2 + ⌅ m2

2 + ⇥

k2 − H0 ⇥

• k(⇥

x) = ⇤ d⇥ xU S.R.(⇥ x, ⇥ x)

k(⇥

x)

V (⇥ x) = −H0R(t, ⇥ x) R(t, ⇥ x) − t ln R(t, ⇥ x) + (m1 + m2)

Semi-relativistic v.s. Non-relativistic Schrödinger equations (momentum space)

• In large momentum (short distance) region, potentials can be different
• Both potentials MUST give correct scattering phase shifts

interaction energy :

⌅ W − ⌃ m2

1 + ⇥

p2 + ⌃ m2

2 + ⇥

p2⇥⇧ ˜

• k(⇥

p) = ⇤ d⇥ pU S.R.(⇥ p, ⇥ p) ˜

• k(⇥

p) ⇥⇥ k2 2µ − ⇥ p2 2µ ⇤ ˜

• k(⇥

p) =

• d⇥

pU(⇥ p, ⇥ p) ˜

• k(⇥

p)

H0 ~

k(~

x) = F.T. hq m2

1 + ~

p2 + q m2

2 + ~

p2 ˜ ~

k(~

p) i

I=1 KN potential@mπ=570MeV (Semi-rela)

S.R. potential : short-range (strong) repulsion qualitatively consistent with NR potential short-range part Semi-relativistic KN potentials (I=1) I=1 phase shift from S.R. Schrödinger equation agrees with NR one at low-energy Relativistic correction is quite small

πΣ(I=2) and KN(I=1) phase shifts

Comparison with chiral perturbation theory: leading order Scattering length (fm): Mπ=570MeV Mπ=700MeV aπΣ

• 0.229(25)
• 0.218(25)

aKN

• 0.241(33)
• 0.221(26)

aI=1

KN =

1 4πf 2

K

µKN = −0.42 (fm)

aI=2

πΣ =

1 4πf 2

π

µπΣ = −0.23 (fm)

Empirical value KN(I=1):

aI=1

KN = −0.31(1) fm

πΣ(I=2) phase shift calculated by LQCD simulation is consistent with leading order chiral perturbation theory, although the quark mass is very large... For definite conclusion, LQCD calculation with physical quark mass, large volume and fine lattice is necessary.

I=2 πΣ & I=1 KN scattering on the lattice

• We study s-wave MB interactions in full QCD simulations
• I=2 πΣ potential reveals scattering phase shift as repulsive
• Calculated I=1 KN phase shift is qualitatively consistent with

experimental data

• Weak quark mass dependences on potentials and observable

Other systems:

• NN(central, tensor, LS), YN, YY potentials calculated
• Heavy MB and MM (DbarN, Tcc, Tcs, ...) data taken
• Non-locality of NN potential calculated
• Results of physical point simulation on K-computer coming soon!

Summary

Thanks to...

Sinya Aoki (Univ. Tsukuba), Bruno Charron (Univ. Tokyo), Takumi Doi (RIKEN), Tetsuo Hatsuda (RIKEN), Takashi Inoue (Nihon Univ.), Noriyoshi Ishii (Univ. Tsukuba), Keiko Murano (RIKEN), Hidekatsu Nemura (Univ. Tsukuba), Kenji Sasaki (Univ. Tsukuba)

HAL QCD Collaboration (Hadrons to Atomic nuclei from Lattice QCD)

Tetsuo Hyodo (Tokyo Inst. Tech.), Hiroyuki Kamano (RCNP), Shota Ohnishi (Tokyo Inst. Tech), Toru Sato (Osak Univ.), Wolfram Weise (ECT*, TUM)