Inter-quark Potential from Nambu-Bethe-Salpeter Amplitude

Yoichi IKEDA(Tokyo Institute of Technology)

in collaboration with

Hideaki IIDA (Kyoto Univ.)

based on Ikeda, Iida, arXiv:1102.2097[hep-lat](2011).

Inter-quark potentials Constituent quark model from Lattice QCD Results on Qbar-Q potentials with finite masses Summary

Contents

Mass spectrum of charmonium system Quark potential models well describe

mass spectra below open charm threshold

Charmonium-like (X, Y, Z) states can be expected as candidates of exotic hadrons

All NEW states reveal as resonances above open charm threshold

BaBar Collaboration

Important information is T-matrix elements based on QCD

T1(s) =

i

RisWi + a(s0) +

s s02pi

s+

ds(s)

(s s)(s s0)(General form of amplitudes from N/D method)

Interaction parts cannot be determined within the scattering theory

Key observations are QQbar and/or meson-meson interactions

Godfrey, Isgur, PRD 32 (1985).Barnes, Godfrey, Swanson, PRD 72 (2005).

Qbar-Q interquark potential Qbar-Q potential is expected having the following form :

Spin-independent Spin-dependent

Effective field theory approach (pNRQCD) : Wilson loop + relativistic correction (1/mQ, v (velocity), 1/mQv expansion)

Bali, Phys. Rept. 343 (2001).Brambilla, Pineda, Soto, Vairo, NPB 566 (2000); Rev. Mod. Phys. 77 (2005). Koma et al., PRL 97 (2006).Koma et al., NPB 769 (2007).

Reliable input based on QCD for quark potential models can be extracted

Our approach through Nambu-Bethe-Salpeter (NBS) amplitude : We can define inter-quark potential with finite quark mass which becomes faithful to QCD T-matrix

Lin et al., NPB 619 (2001).Aoki, Hatsuda, Ishii, PTP 123 (2010).

VQQ(r) = r 43s

r+ Vspin(r)#SQ #SQ + VT(r)S12 + VLS(r)#L #S +

Confinement + Coulomb-like + spin-dependent force

How to define quark model on the lattice

Homogenous Nambu-Bethe-Salpeter equation :

P : meson 4-momentum P = (E, 0) = (Mmeson, 0) at meson-rest framep, p : relative 4-momentum of Qbar-Q systemK(p,p;P) : irreducible kernelG(p;P) : product of free quark propagator w/ assumption of constant quark mass mQ

E(p0, p) = G(p;P )d4pK(p, p;P )E(p0, p)

Note :Potential is non-local but energy-independent below open charm thresholdbecause of Krolikowski-Rsewuski relation

Nambu-Bethe-Salpeter wave functions satisfy Shrdinger-type equation :

(E 2!(p))E(p) =dpU(p, p)E(p)

Krolikowski, Rzewuski, Nuovo Cimento, 4 (1956).

!(p) =m2Q + p2

Y.I., Iida, arXiv:1102.2097[hep-lat](2011).

Non-relativistic reduction through Levy-Klein-Macke (LKM) method :Reviewed in Klein, Lee, PRD 10 (1974).

-> Equal-time Nambu-Bethe-Salpeter amplitude = Nambu-Bethe-Salpeter wave function

E(p) =12pii

dp0

[(p0 P 0/2 + #(p) i)1 + (p0 p0)1

]E(p0, p)

Qbar-Q interquark potential on the lattice

1. Measure equal-time Nambu-Bethe-Salpeter wave function

G(2)(r, t tsrc) =

x,X,Y

0|q(x, t)q(x + r, t)(q(X, tsrc)q(Y, tsrc)

)|0=x

En

AEn0|q(x)q(x + r)|EneEn(ttsrc)

(E0 = M, t! tsrc) AE0E0(r)eE0(ttsrc)

Spacial correlation of 4-point functionE(r) =x

0|q(x)q(x + r)|E; JPC

3. Velocity expansion of non-local potential

U(r, r) = (VC(r) + Vspin(r)!SQ !SQ + VT(r)S12 + VLS(r)!L !S + )(r r)Leading order NLO

We examine s-wave effective LO potentials in pseudo-scalar and vector channels

Aoki, Hatsuda, Ishii, PTP 123 (2010).Ikeda, Iida, arXiv:1102.2097[hep-lat](2011).

2. Define potential through Schrdinger-type equation

(E H0)E(r) =drU(r, r)E(r)

LQCD setup

Quench QCD simulation

Plaquette gauge action & Standard Wilson quark action =6.0 (a=0.104 fm, a-1=1.9GeV)

Box size : 323 x 48 -> L=3.3 (fm) Four different hopping parameters (=0.1320, 0.1420, 0.1480, 0.1520)

-> MPS=2.53, 1.77, 1.27, 0.94 (GeV), MV=2.55, 1.81,1.35, 1.04 (GeV) Nconf=100

Wall source

Coulomb gauge fixing

Y.I., Iida, arXiv:1102.2097[hep-lat](2011).

Qbar-Q wave functionsNBS wave functions (channel & quark mass dependence)MPS=2.53, 1.77, 1.27, 0.94 (GeV), MV=2.55, 1.81,1.35, 1.04 (GeV)

Channel dependence appears in light quark sector -> spin-spin interaction is enhanced at light quark mass (one-gluon exchange predicts Vspin proportional to 1/mq2)

Size of wave function becomes smaller as increasing mq

E(r) =x

0|q(x)q(x + r)|E; JPC

mq mq

Pseudoscalar channels Vector channels

Qbar-Q potentials

V effspinindep.(r) E =1mq

[142PS(r)PS(r)

+342V(r)V(r)

]Effective spin-independent & dependent forces are constructed by linear combination of PS & V channel potentials

Spin-independent forces reveal Coulomb + linear behavior

Spin-dependent forces have strong quark mass dependence Repulsive spin-dependent forces as expected by mass spectrum

V effspindep.(r) E =1mq

(

2PS(r)PS(r)

+2V(r)V(r)

)

Fit results of Qbar-Q potentialsV effspinindep.(r) E =

1mq

[142PS(r)PS(r)

+342V(r)V(r)

]

MV (GeV) (MeV/fm) A (MeV fm)2.551.871.351.04

822 (49)766 (38)726 (39)699 (57)

200 (7)228 (6)269 (7)324 (12)

String tension has moderate mq dependences Naive extrapolation to infinite mass gives comparable value from Wilson loop Coulomb coefficients increase as decreasing mq

V (r) = r Ar+ Cfit function:

see also, Kawanai and Sasaki, PRL 107 (2011).

Operator dependence of Qbar-Q potentialsOperator dependence of inter-quark potential is studiedby using gauge invariant smearing operator

smr.E (r) =x

0|q(x)L(x, r)q(x + r)|E; JPC

mq(V effC (r) E) =2smr.E (r)smr.E (r)

Comparison with Coulomb gauge potentials

The potentials obtained from gauge invariant smearing operators are consistent with the Coulomb gauge potentials

Vector channel

Pseudoscalar channel

Relativistic kinematicsInter-quark potentials with relativistic kinematics are studied

Even for relativistic kinematics, Coulomb + linear potentials are obtained Long range parts of relativistic potentials are consistent with those of N.R. potentials For charmonium, non-relativistic kinematics is good enough In strangeness sector, non-locality of potentials gets to large, if one employes non-

relativistic kinematics

=0.1320 : mq ~ mc

=0.1520 : mq ~ ms

(V effC (r) E) =H0VE (r)VE (r)

H0E(r) =dr( dp

(2pi)22m2q + p2e

ip(rr))E(r)

Summary We study inter-quark interactions with finite quark mass in

quenched QCD simulation

Effective central Qbar-Q potentials from NBS amplitudes reveal Coulomb + linear forms

Coulomb coefficients become smaller and smaller as increasing mq String tension also has mq dependence and is consistent with that

of Wilson loop analysis at large mq limit Spin-spin interactions are repulsive and strongly dependent on

quark masses

Studies of tensor, LS, non-locality of inter-quark potential Interquark potential for baryon (talk by H. Iida) Coupled channel analysis toward above open charm threshold Investigation of exotic states (X, Y, Z)

Future plans : Full QCD

Thank you very much for your attention

Small difference!

We compare Standard Wilson quark action with O(a) improved action (clover action)!

! We study cutoff dependence of the qbar-q potentail by adopting O(a)-improved Wilson-clover quark action

Check (I) : O(a) improvement

Check (II) : Volume dependence

L=4.5fm (!=5.8, mPS=2.47GeV, clover): red L=3.2fm (!=6.0, mPS=2.58GeV, standard): green!

! Small difference between them volume is enough

! We study volume dependence of the qbar-q potentail by varying lattice spacing for O(a)-improved Wilson-clover quark action

(Lat

tice

unit)!

Pseudoscalar channel!

Our setup (!=6.0, a=0.1fm, standard Wilson, (3.2fm)3) seems sufficient for the calculation of qbar-q potential (in quark mass region calculated here)!

Qbar-Q potential from NBS wave functionInter-quark potential with various finite quark massesMPS=2.53, 1.77, 1.27, 0.94 (GeV), MV=2.55, 1.81,1.35, 1.04 (GeV)

V effC (r) E =1mq

2E(r)E(r) mq =MV/2

Coulomb + linear confinement forces are reproduced with finite quark masses(solid curves representing Coulomb + linear functions)

Pseudoscalar channels Vector channels

Vc(r

)-E (M

eV)

Vc(r

)-E (M

eV)

Relativistic kinematicsInter-quark potentials with relativistic kinematics are studied

Even for relativistic kinematics, Coulomb + linear potentials are obtained Long range parts of relativistic potentials are consistent with those of N.R. potentials For charmonium, non-relativistic kinematics is good enough For strangeness sector, non-locality of potentials may get to large, if one employes

non-relativistic kinematics

=0.1320 (MV=2.55 GeV)

=0.1520 (MV=1.04 GeV)

=0.1420 (MV=1.77 GeV)

=0.1480 (MV=1.27 GeV)

cbar-c potential from full QCD

Spin-independent force shows Coulomb + linear form

Lattice QCD potential is consistent with NRp model

Kawanai, Sasaki, arXiv:1110.0888[hep-lat].

Barnes, Godfrey, Swanson, PRD 72 (2005).

Spin-dependent force shows short range but not point-like repulsion

cbar-c potential from 2+1 flavor FULL QCD simulation at almost PHYSICAL POINT generated by PACS-CS Coll. (m=156(7), mK=553(2) MeV)

Iwasaki gauge action (=1.9, a=0.091 fm) + RHQ action -> Mave.(1S) = 3.069(2) GeV, Mhyp.=111(2) MeV

Spin-independent force Spin-dependent force

see also, Kawanai and Sasaki, PRL 107 (2011).

How to define quark model on the lattice

How to define quark model on the lattice We start with NBS equation for invariant amplitudes at meson rest frame :

P : meson 4-momentum P=(M, 0) at center-of-mass framep, p, k : relative 4-momentum of Qbar-Q systemK(p,p;P) : irreducible kernelG(k;P) : product of free quark propagator

Note :We do not require instantaneous NBS kernel K(p,p;P) in LKM method

Non-relativistic reduction through Levy-Klein-Macke (LKM) methodReviewed in Klein, Lee, PRD 10 (1974).

Concept of LKM method :Replacement of free-propagator G(k;P) to non-relativistic one GN.R.(k;P) leads to rearrangement of interaction kernel of original NBS equation

I(p,p;P) is new kernel and satisfying

w/ assumption of constant quark mass mQ

M(p, p;P ) = K(p, p;P ) +d4kK(p, k;P )G(k;P )M(k, p;P )

M(p, p;P ) = I(p, p;P ) +d4kI(p, k;P )F(k;P )GN.R.(!k, P )M(k, p;P )

I = K +K(G FGN.R.)IF(p;P ) 1

2pii

[(p0 P 0/2 + "(#p) i)1 + (p0 p0)1

]

3-dimensional LKM equation for NBS invariant amplitude:

Summary of LKM method : Relativistic 3-dimensional equation are extracted from equal-time NBS wave

function -> suitable for LQCD simulation Obtained 3-dimensional equation is Schrdinger-type equation (LKM equation) Effective potential of LKM equation is related to irreducible kernel K(p.p;P) of

NBS equation -> potential model based on QCD can be constructed

Reviewed in Klein, Lee, PRD 10 (1974).

L.H.S. of LKM equation is found as equal-time NBS wave function (positive energy)-> Schrdinger-type equation for NBS wave function is easily derived :

with non-local, energy-dependent potential U(p,p;E) satisfying

Schrdinger-type equation for NBS wave function in r-space

E = P 0 =Mmeson

P UHP

dp0F(p;P )

PM(p, p;P ) = P I(p, p;P ) +Pd4kI(p, k;P )F(k;P )GN.R.(!k;P )M(k, p;P )

(E H0)E("r) =d3rU("r,"r;E)E("r)

U(!p, !p;E) =P I(p, p;E)P

How to define quark model on the lattice

(E 2!("p))E("p) =d3pU("p, "p;E)E("p)

Interquark potentials for baryons Three-body Schrdinger-type equation :

Effective two-quark potentials : integrate out spectator particleDoi et al, (HAL QCD Coll.), arXiv:1106,2276 [hep-lat] (2011).

(

2r

2

2

2+ V (r, )

)E(r, ) = EE(r, )

2qE (r)

dE(r, )(

2r

2+ V 2qeff .(r)

)2qE (r) = E

2qE (r)

Effective 2q potential v.s. qbar-q potential (spin independent parts)

Vqq(r) = r 43s

r+ V2q(r) = r 23

s

r+

String tension would be comparable (Wilson loop analyses)Coulomb part is different by factor two with one-gluon exchange

Three-quark potential Valence light quark effect for 2Q

T.T.Takahashi et al., PRD70, 074506 (2004) A.Yamamoto et al., PLB664, 129 (2008)

Interquark potentials for baryons Lattice QCD result of effective 2q potential:

V 2qeff .(r) =122r2qE (r)2qE (r)

+ E

V fit2q (r) = r A

r+ C

2/Ndf = 0.7 [ 0.2 < r < 0.8 fm ] = 797 (8) [MeV / fm]A = 78 (2) [MeV fm]

2qE (r) =

0|"abc(qTa (r/2)C5qb(r/2))qc,()|B, JP = 1/2+

Effective 2q potential shows Coulomb + linear form

String tension of effective 2q potential is comparable with that of qbar-q potential(c.f., qbar-q =822 [MeV / fm], Aqbar-q =200 [MeV fm])