格子上のベーテ・サルピータ波動関数に基づくクォーク間ポテンシャル
飯田英明（京大理）共同研究者：池田陽一（東工大）
Refs.) Y.Ikeda and H.Iida, arXiv:1102.2097.
Y.Ikeda and H.Iida, PoS LATTICE2010 (2010) 143.
「素核宇融合による計算基礎物理学の進展」− ミクロとマクロのかけ橋の構築 −2011年12月3日（土）−5日（月） @合歓の郷

Contents• Introduction• Our approach to define interquark
potentials for finite quark mass on lattice• Results• Summary and future plans

Heavy quark system
BaBar Collaboration• Quark potential models well describe mass
spectra below open charm threshold
Godfrey, Isgur, PRD 32 (1985).
Barnes, Godfrey, Swanson, PRD 72 (2005).

• Exotic states (X, Y, Z) can be expected as
non-standard cbar-c mesons

Important information is T-matrix elements based on QCDInteraction part is determined from QCD… First principle calculation is importantQbar-Q interquark potential
Qbar-Q potentials can be expected having the following form:

Spin-independent

Spin-dependent

Effective field theory approach (pNRQCD) for charmonium spectra :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).

Our approach through Nambu-Bethe-Salpeter (NBS) amplitude :We define effective inter-quark potential with finite quark mass.
Lin et al., NPB 619 (2001).
Aoki, Hatsuda, Ishii, PTP 123 (2010).

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

How to define quark model on the lattice
Y.Ikeda., H.I., arXiv:1102.2097[hep-lat](2011).
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 system
K(p,p’;P) : irreducible kernel
G(p;P) : product of quark propagator
w/ assumption of constant quark mass mQ

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

Nambu-Bethe-Salpeter wave functions satisfy Schrödinger-type equation :Note :
Potential is non-local but energy-independent below open charm thresholdbecause of Krolikowski-Rsewuski relation
Krolikowski, Rzewuski, Nuovo Cimento, 4 (1956).

qbar-q interquark potential on the lattice
Aoki, Hatsuda, Ishii, PTP 123 (2010).
Ikeda, Iida, arXiv:1102.2097[hep-lat](2011).

1. Measure equal-time Nambu-Bethe-Salpeter wave function
Spatial correlation of 4-point function

2. Define potential through Schrödinger-type equation

3. Velocity expansion of non-local potential

Leading orderNLO

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

LQCD setup
Y.I., Iida, arXiv:1102.2097[hep-lat](2011).
Quench QCD simulation

Plaquette gauge action & Standard Wilson quark action
β=6.0 (a=0.104 fm, a-1=1.9GeV)
Box size : 323 x 48 -> L=3.3 (fm)• Four different hopping parameters (κ=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

qbar-q wave function
NBS 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)

Pseudoscalar channels

mq

Vector channels

mq

Channel dependence appears in light quark sector-> spin-spin interaction is enhanced
Size of wave function becomes smaller as increasing mq

qbar-q potential from NBS wave functions
Interquark potentials with various finite quark masses
MPS=2.53, 1.77, 1.27, 0.94 (GeV), MV=2.55, 1.81,1.35, 1.04 (GeV)

mqmq

Coulomb + linear confinement forces are reproduced with finite quark masses(solid curves representing Coulomb + linear functions)Fitting results of qbar-q potential

fit function:
MV (GeV) σ (MeV/fm) A (MeV fm)
2.55
822 (49)200 (7)1.87
766 (38)228 (6)
1.35
726 (39)269 (7)1.04
699 (57)324 (12)String tension σ has moderate mq dependences
σ for the heaviest quark mass gives comparable value to that from Wilson loop
Coulomb coefficients increase as decreasing mq
see also, Kawanai and Sasaki, PRL 107 (2011).

Operator dependence of qbar-q potential
Operator dependence of inter-quark potential is studied
by using gauge invariant smearing operator

Comparison with Coulomb gauge potentials
Vector channel

Pseudoscalar channel

The potentials obtained from gauge invariant smearing operators
are comparable with Coulomb gauge potentialRelativistic kinematics
Inter-quark potentials with relativistic kinematics are studied

κ=0.1320 : mq ~ mc

κ=0.1520 : mq ~ ms

Even for relativistic kinematics, Coulomb + linear potentials are obtainedLong range parts of relativistic potentials are consistent with those of N.R. potentials
For charmonium, non-relativistic kinematics is good enoughIn strangeness sector, non-locality of potentials gets to large, if one employs nonrelativistic kinematics

Some other systems (in progress)• Lattice QCD result of effective 2q potential…potential in baryons

integrated out•qbar-q potential at finite temperature…related to charmonium dissociation
Potential in Euclidean time is defined bySummary
We study inter-quark interactions with finite quark massin 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 comparable with that of
Wilson loop analysis with large mq limit

Future plans
Studies of tensor, LS, non-locality of inter-quark potential
Three-quark potentials : ccc, ccs
Coupled channel analysis toward above open charm threshold
Investigation of exotic states (X, Y, Z)
qbar-q potential at finite temperatures … charmonium dissociation