Volume 149B, number 4,5 PHYSICS LETTERS 20 December 1984

BEHAVIOR OF QUARKS AND GLUONS AT FINITE TEMPERATURE AND DENSITY IN SU(2) QCD

Atsushi NAKAMURA 1

INFN, Laboratori Nazionali di Frascati, CP 13, 00044 Frasca ti, Rome, Italy

Received 9 August 1984

We have run a computer simulation in SU(2) lattice gauge theory on a 83 × 2 lattice including dynamical quark loops. No rapid variation is observed in the value of the Polyakov line, while the energy densities of quark and gluon show a strong indication of a second order phase transition around T ~ 250 MeV. In order to reduce finite size effects, the results are compared with those of a free gas on a lattice of the same size. The quark and gluon energy densities overshoot the free gas values at high temperature. The effect of the chemical potential is also studied. The behaviors of the energy densities and of the number density are fax from the free gas ease.

It has been conjectured that systems of quarks and gluons at high temperature and density show a com- pletely different behavior from those at zero temper- ature and normal density [1-3] . Above some temper- ature and/or chemical potential, quarks and gluons are expected to be liberated in a deconfined quark- gluon plasma.

Monte Carlo (MC) studies of SU(2) Yang-Mills theory in the absence of dynamical quarks by McLerran and Svetitsky [4] and by Kuti, Polonyi and Szlachanyi [5] have given the first numerical evi- dence for a second order transition from a conf'med phase to a deconf'med one. Groups at the University of Bielefeld and at the University of illinois have performed MC simulations of the gluon matter at finite temperature in detail; for SU(3) Yang-Mills theory, they have observed a first order phase tran- sition and ideal gas behavior ofgluons at high tem- perature * a

Such studies of QCD in unusual environments are done not only for a theorist's fun and amusement. We hope that in high energy heavy ion collisions high temperature and density matter might be produced in a controlled experimental environment. To under- stand the data which might arise from such experi-

! Fujukai Foundation fellow. ,1 See ref. [6] and references therein.

ments, we may develop and study models of the quark-gluon system. MC simulation of lattice QCD probably provides the most fundamental informa- tion for such an analysis. For the study of hadronic matter, it is important to include quark loops in the calculation since they play a crucial role in screening. The phase transition observed in the pure gauge cal- culation might be washed out by them [7,8]. In the presence of quark fields, the Polyakov line is no more a good order parameter for the confined and decon- freed phases, mathematically because the presence of quark fields breaks the symmetry under the center of the gauge group, or physically because isolated heavy quarks can survive due to the quark pair creation.

We will report here a MC study of the quark gluon system with dynamical quarks. We simulate the finite temperature and baryon number density plasma on an N t X N 3 lattice. The temperature of the system is given by T = 1/Nta t, where at(g ) is the lattice distance in the fourth direction. The action is composed of the kinetic term of gauge variables and the fermion part:

S = S C + S v, Sr: = ~ A qJ .

We employ the Wilson form for the action [9]. The matrix A has the form

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Volume 149B, number 4,5 PHYSICS LETTERS 20 December 1984

4

A = I - ~ [K~+)(1 -- 7#)Un,tJSn+lJ, m #=1

+ t o ( - ) ( 1 + ~/#) u+ ,#Sn ,m+~] •

The chemical potential, p, is introduced ~ la Hasenfratz and Karsch [10] :

t¢(4 ±) = K t exp(-+taat) •

This change of the hopping parameters results in re- placing P4 ~ P 4 + iv in the free propagators on a lat- tice. Following refs. [11,12] we assume that the hopping parameters depend upon the lattice distance and the gauge coupling g as

K t = ~ [as/(3at + as) ] ~(g).

_ , - K s - $[at/(3a t +as)]$(g ) (i= 1 ,2 ,3 ) .

We will analyze the following quantities:

Nt

L 2N 3 Tr i=I-I1 Ut(ti' X ) ,

e = V - 1 [ - a / a 3 + (p//~)a/ap] logZ - e 0 ,

n = (.3v)-lalogZ/Ou, where

a/a~ = (Nt)-I a/aa t , V = (Nsa~)3 ,

z= f e-S,

and e 0 stands for the energy density at T = 0 and p = 0. The energy density, e, may be divided into two parts,

r "l" t e = e G + e q ( -S G) ( - S F ) ,

where

s~ = v - l [ _ a / a ~ + (u/~)a/a~,] s ~ ,

We have run the simulation on a 2 X 8 3 lattice with two flavors using the pseudo-fermion method. The gauge group SU(2) is replaced by its icosahedral subgroup. The reliability o f the program and effects of the quark vacuum polarization on ordinary ha-

dronic quantities are discussed in refs. [13,14]. The hopping parameters, ~t and Ks, are fixed to be 0.17 which would correspond to a very small quark mass in the quenched approximation. The number of gauge MC iterations is 100 including 20 thermalization ex- cept for 4[g 2 = 2.0, 2.2 and # = 0 where it is 84. The number of pseudo-fermion iterations is 45 including 15 thermalization for each sweep of gauge fields. For the calculation of e, we need ag- 2/aa t. We employed the formula in ref. [15] where the quark loops are not included. We used the data on a 44 lattice to evaluate e 0 except the pure gauge case for which the data on an 84 lattice is employed. This procedure introduces some systematic error. However the total contribution from these terms is small.

In fig. 1, we show the expectation value o f the Polyakov line as a function of 4/g 2 for p = 0. At high temperature (small g2) the Polyakov line has a large expectation value which implies the existence of iso- lated heavy quarks with only small suppression due to finite temperature effects, while its value is small at low temperature. The characters o f the system at high and low temperature are therefore different but heavy quarks are nevertheless deconfined in both

I.C

~) With Quarks

Without Quarks o ° !

o !

t t T

t .a =0 8Nz

, t [ ,o ,'.5 z:o i.s

4 / g ~

Q5 A

___d V

Fig. 1. Expectation value of the Polyakov line as a function of 4/g 2 for zero chemical potential. The circles are the data obtained in the presence of quark loops. The crossses corre- spond to the pure gluon matter.

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Volume 149B, number 4,5 PHYSICS LETTERS 20 December 1984

(.9

With Quarks

Without Quarks t

,.o ;5 2'.o 2'5 4/g

T=~'30+- 20 345±q5 MeV 250*-55

~r 8 F-

6

14

b po =0

83x 2

I ,,o r.s

4/g Fig. 2. (a) Gluon energy density, eG/T 4, as a function of 4[g 2 for zero chemical potential. The dashed line gives the free field limit on the same lattice. (b) Quark energy density, eq/T 4, as a function of 4/g ~ for zero chemical potential. (Two flavor.) The dashed line gives the free field limit on the same lattice.

phases. Its change is quite smooth, and no indication o f a second order transition can be seen.

On the other hand, the gluon and quark densities, which are plotted in figs. 2a and 2b show rapid varia- tion. Moreover their values at high temperature are much larger than that of the pure gluon system which agrees quite well with the "free gas" behavior at high temperature. Here "free gas" refers to the massless free Bose or Fermi fields on a lattice of the same size [ 16]. The energy density o f gluons is as large as 3/2 o f the "free gas" (the dashed line in the figure). This suggests more degrees o f freedom than those of sim- ple ideal quark gluon gas. The system of quarks and gluons at these temperature regions looks rich in structure.

The rapid change in the energy densities occurs at a smaller value o f 4/g 2 than in the pure gauge case. This does not mean that the transition temperature is lower than that o f pure gauge theory because the lattice distance a at the same value o f 4/g 2 is different between the two theories. In refs. [13,14] it is shown

that, once we switch on the quark loops, the gauge configuration is pushed into a more ordered state and that the f'mite size effects become more serious. This may be interpreted as the shrinkage of the lattice dis- tance driven by quark loops.

In order to know the value o f the temperature in physical units, we should evaluate the lattice distance a. Note that we cannot use the values in the literature which were obtained without quark loops. We ran a simulation on an N t X N x X Ny X N z = 8 X 8 X 4 X 4lattice at 4/g 2 = 1.6, 1.8 and 2.0 and measured Wilson loops on the t - x plane. The heavy quark po- tential is estimated by the Stack method [17]. We fit the results to a Martin phenomenological potential [18] with the lattice distance and the constant part of the potential as parameters. The detailed calcula- tion of the potential will be published elsewhere. The obtained temperatures are shown in fig. 2a.

We can expect the rapid change of the quark gluon energy density between T ~ 200 and 350 MeV, which may be more drastic in case of SU(3).

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Volume 149B, number 4,5 PHYSICS LETTERS 20 December 1984

0.6

0.4 A

V 0.2 4/g z= 1.4

83x2

o & ,'o ,'s z'o pa

Fig. 3. Expectation value of the Polyakov line as a function of the chemical potential for fixed temperature.

This picture is consistent with the recent analysys by Celik et al. [19], by Fucito and Solomon [20], and by Gavai et al. [21].

Next we shall study the effects of the chemical potential. We plot the MC data in figs. 3, 4 and 5 as a function of the chemical potential for 4[g 2 = 1.4.

k.O 2

6 a

4/g z =1.4

83x2

. . . . . . . . . . . . . . . . . . . . .

f

60

0

0

/ z /

/ /

/ /

/ 4O /

/ / /

0 / C / /

/ / /

20 / / /

/ i"

i" 1 1

I I 1. 1.

0.5 1.0 15 20

pa Fig. 5. Quark baryon number density, niT 3 , as a function of the chemical potential for fixed temperature. (Two flavor.) The dashed line gives the free field limit on the same lattice.

As the chemical potential increases, the value of the Polyakov line increases very slowly and monotonous- ly, while the thermodynamic quantities show peculiar behavior. At large chemical potential, isolated heavy quarks can survive longer than at zero chemical po-

80

60

I - - 4¢

2(3

b ,

4 igZ= 1.4

83x 2

/ 0 / i

/ J J

,s// / /

OpJ" 7

0

0

/ / jJ

i i

JJ

I I 0 015 t I.,5 1.0 ~0 015 IO 115 210

pa pa

Fig. 4. (a) Gluon energy density, eG/T 4, as a function of the chemical potential for fixed temperature. The dashed line gives the free field limit on the same lattice. (b) Quark energy density, eq/T 4, as a function of the chemical potential for fixed temperature. (Two flavor.) The dashed line gives the free field limit on the same lattice.

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Volume 149B, number 4,5 PHYSICS LETTERS 20 December 1984

tential. The gluon energy density shown in fig. 4a in- creases quickly when we increase the chemical po- tential, i.e., the gluons are not independent o f quark matter density and exhibit behavior far from that of a "free gas". However it falls suddenly at large chemical potential . The quark energy density in fig. 4b increases l ike a "free gas" but the value is much higher. At these chemical potential regions, the free quark gluon picture is not correct. There might be other degrees of freedom. The number density, n, shown in fig. 5 also overshoots the "free gas" values at large chemical potential in a similar manner to the quark energy density. To obtain the system with large chemical potential , a higher density is required than that estimated from the ideal gas equation.

This calculation was done at CERN and Frascati. I am grateful to the theory divisions there for their hospitali ty and to N. Oshima for his advise in the numerical computat ion. I am indebted to the parti- cipants and organizers of the Warsaw symposium and of Zacopane summer school, 1984, for constructive criticisfn, especially A. Bialas and L.D. McLerran for valuable discussions and careful reading o f the manu- script.

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