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PHYSICS ON LATTICE

© 2009 M. I. Polikarpov*

Institute for Theoretical and Experimental Physics, Moscow, Russia Received September 18, 2008; in final form, January 19, 2009

A short review of physical results obtained recently in lattice gluodynamics and lattice QCD is given. The topics are: formation and breaking of the confining string, spectrum of hadrons, QCD at finite temperature, monopoles and vortices in Abelian and non-Abelian gauge theories.

PACS:11.15.Ha, 13.38.Ge

1. INTRODUCTION

The numerical simulations of QCD become possible if we pass from continuous Minkowski theory to discrete Euclidean lattice formulation. After the change of the time to the imaginary time, t ^ it, the partition function of a field theory becomes analogous to the statistical sum:

Z = j eiSMMVy ^J e-SE[v]Vy, (1)

where y denotes the collection of all fields in the considered theory. The similarity with the statistical physics becomes exact after the discretization of the space—time; we consider finite Euclidean space, 0 < x1,x2,x3,x4 < R, with periodic boundary conditions, the coordinates have discrete values. Thus we get the four-dimensional lattice with sites at the points x = (xi,x2, x3,x4), 1 < xk < L = R/a, where a is the lattice spacing and L is the size of the lattice (for calculations at the finite temperature we consider asymmetric lattices L x L x L x Nt (Nt < L), and the temperature of the system is defined as T = = (Nta)-1). After these changes the partition function of the theory is reduced to the finite-dimensional integral,

Z = / n Ms)e-S^, (2)

■' s

which can be evaluated numerically using Monte Carlo method. The continuum limit of the theory corresponds to the limits L a ^ 0, while

numerical calculations are performed at finite values of L and a. The systematic errors corresponding to the finite volume and finite lattice spacing can be estimated using standard methods by varying L and a.

E-mail: polykarp@itep.ru

In lattice QCD the numerical integration is possible only over the gauge fields, integration over the quark (fermionic) fields can be performed analytically:

J V^V^W = det M. (3)

Since matrix M is the function of the gauge field, M = M(AM), after such integration the gauge action becomes effectively nonlocal: S(AM) = SqF(Am) + + lnM(AM); here, Sqf(Am) is the lattice analog of the gauge action f TrF%d4x. The lattice discretization of the fermionic matrix M is not unique, there are infinitely many versions of M which correspond to (D + m) in the continuum limit. The different discretizations are usually called different types of lattice fermions, which have different properties. A part of them is simpler to use in numerical calculations, the others have better analytical properties (see also Section 3).

Due to the term induced by quarks, ln M(A^), the gauge-field action is highly nonlocal and the calculations in lattice QCD with dynamical fermions are very time consuming. The multiplicity of the integrals in (1) for typical lattice size, L4 = 484, is 32L4 = 169 869 312; the dimension of matrix M in Eq. (3) is 12L4 x 12L4 = 63 700 992 x 63 700 992. Due to large number of degrees of freedom we need large-scale calculations on supercomputers to simulate QCD. Only recently there appear calculations in QCD with two light and one massive (strange) quark. The results of the hadron-spectrum calculation in such lattice QCD, called (2 + 1) QCD, can be found in [1] (see Section 3).

Calculations in lattice QCD with two light dynamical quarks are more popular, in Section 2 we discuss some results of the DIK (DESY—ITEP— Kanazawa) Collaboration, and when we refer to "full

1616

0

x

16 -

16 -8

Fig. 1. The dimensionless action density pA (s)r^ of the Abelian flux tube in (left) full and (right) quenched QCD. x and y axes are in dimensionless lattice units.

8

Fig. 2. Distribution of the color electric field in (left) full and (right) quenched QCD.

QCD" we mean Nf = 2 lattice QCD with nonper-turbatively improved Wilson fermions (review of the results of the DIK Collaboration is given in [2]). SU(2) and SU(3) lattice gluodynamics is much simpler to study in computer simulations than QCD with dynamical quarks, but, as we show in Section 2, even from these simple models we can get useful information about confining strings. In Section 3 we briefly discuss the lattice QCD phenomenology. In Section 4 we present two examples of confinement mechanisms in Abelian gauge models. In Section 5 we discuss monopoles and vortices in lattice gluodynamics as gauge-field fluctuations responsible for confinement of color.

2. VISUALIZATION OF CONFINING STRINGS

We cannot strictly prove the confinement of color in non-Abelian gauge theories, but we can clearly see the formation of gluonic cylinder-type object between quark and antiquark. This object called confining string was first time observed in SU(2) gluodynamics in [3]. A standard lap-top computer can reproduce the profile of confining string in gluodynamics after 10—20 h of calculations, the linear quark—antiquark potential can be obtained after 2—3 h of calculations; at the same time the color-confinement problem cannot be solved analytically already during more than 30 years. The proof of the existence of the mass gap in the spectrum of Yang-Mills theories is one of the "Millenium problems" (see http://www.claymath.org/millennium/Yang-Mills_Theory/).

The material of this section is based on papers of the DIK Collaboration [2, 4-6], we discuss the visualization of the gluon fields inside hadrons. The difference between confining string in gluodynamics (quenched QCD) and in QCD with dynamical quarks is not very large as it is seen from Fig. 1, the definition of Abelian action density is given in [4, 5], r0 is the parameter which defines the scale, r0 & 0.5 fm.

In Fig. 2 we show the color electric field in Abelian projection [4, 5]. One can see only small differences between distributions obtained in full and in quenched QCD. Figure 2 shows that the electric field is purely longitudinal in a region between the sources as we expect for the flux tube.

If in lattice QCD we separate the test quark and antiquark at sufficient large distance, the confining string should break due to the creation of the quark— antiquark pair from the vacuum. Thus, from the meson made from static quark—antiquark pair we get two heavy—light mesons. At zero temperature it is hard to observe the confining string breaking due to large statistical noise. At finite temperature below the phase transition we can observe the formation of the confining string in the static meson and also the string breaking.

The explanation is simple: the heavy-quark potential V(r,T) is determined from the Polyakov loop correlator:

y(r,T) = -^ln<L(s)Lt(s')>. (4)

Lt

When r

(L(s)L (S)) — \(L)\2, (5)

V(r)r0

8r 1/Tc 0.80 ■e—0.94 -¿—1.00 -: 1.25

6

_I_I_I_I_I_I

0 12 3

R/ro

Fig. 3. Static potential from monopole Polyakov loop correlators for 3 = 5.2.

r,

r,

Cl

Fig. 4. Abelian action density (vertical axis on top figure) in 3Q system in full QCD. The horizontal axes are spacial axes. The lower figure is the corresponding contour plot.

and \(L)\2 = 0 even below Tc, since global Z3 is broken by fermions. From (4) and (5) it follows that V(r, T) ^ const when r ^<x>. This flattening of the potential at T <Tc is due to the creation of a quark— antiquark pair from the vacuum which screens the test sources. The flattening (corresponding to the string breaking) is clearly seen in Fig. 3, the details

Fig. 5. Three-quark Wilson loop.

of calculations are given in [4, 5]. The mesonic flux tube breaks when the Q-Q distance R is increasing at a fixed temperature T. We measure profiles of the action density at T/Tc = 0.94 for various values of R/r0. Figure 4 of paper [5] shows the profiles of the action density of the mesonic system at T/Tc = 0.94 (below the deconfining temperature). It occurs that the flux tube persists for distances R/r0 = 0.98, 1.71, while for R/r0 = 2.42 it seems to disappear, leaving only lumps of the action density around the sources.

It is important to learn about the forces and the distribution of color electric flux in the 3Q system, a particularly interesting question is whether a three-body force exists and the confining flux tube is of Y shape, or whether the long-range potential is simply the sum of two-body potentials, resulting in a flux tube of A shape. Several lattice studies give evidence for a A-type long-range potential [7, 8], while others show the existence of a Y type potential [9, 10]. The latter result is also being supported by the field-correlators method [11]. The difference between A-and Y-shape potentials is rather small and it is difficult to detect it numerically. The recent results [6] obtained by DIK Collaboration show that the baryon flux tube in quenched lattice QCD and in full QCD has Y shape. The example of the density of baryon flux in the full QCD is shown in Fig. 4. The peak in the center of three-quark system supports the Y shape of the flux. Thus we see that in QCD there exist genuine three-body forces, that is the potential between quarks in baryon is not the sum of the pair potentials: V (ri ,r2, r) = V (ri2) + V (r23) + V (r3i).

To obtain Fig. 4 we used the baryon creation operator (the analog of the Wilson loop which creates the meson state)

WSQ = —SijkSi'j'k'U%% (rCl) x x (rC2)Ukk'(rC3),

4

2

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Fig. 6. The Abelian action density (vertical axis on left figure) of baryon in the deconfined phase at T/Tc = 1.28. The right figure is the corresponding contour plot.

where U(r^) = ns,^erc U^(s) is the ordered product of link matrices, along the path rCi , as shown in Fig. 5.

As in case of the meson, if we separate the source quarks by sufficient distance, the effect of baryon string breaking appears. This effect is shown in Fig. 6 obtained at finite temperature above the deconfinement temperat

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