On thermodynamic properties of the chromoplasma

Volume 215, number 2

PHYSICS LETTERS B

15 December 1988

ON T H E R M O D Y N A M I C PROPERTIES OF THE CHROMOPLASMA V.K. M I T R J U S H K I N a, A.M. Z A D O R O Z H N Y a a n d G.M. Z I N O V J E V b a Laboratory of Theoretical Physics, J1NR, Dubna, Head Post Office, P.O. Box 79, 101 000 Moscow, USSR b Institute of Theoretical Physics, 252 130 Kiev, USSR

Received 9 August 1988

We calculate the contributions of electric and magnetic modes to some thermodynamic functions in SU (2)-gauge theory on the lattice 4 X 123. It is shown that the behaviour of the chromoelectric part of the energy EE can be interpreted within the Ising-type model in agreement with the universality hypothesis. At the same time the behaviour of the magnetic parts of the internal energy and pressure (i.e. EM and PM) differs drastically from that of EE and PE. The character of the temperature dependence of EM and PM exhibited here testifies to the presence of highly nonidentical properties of electric and magnetic excitation modes of chromoplasma and may shed light on the role of unstable modes in gauge theories.

At present it is well established that there is a transition to the d e c o n f i n e m e n t phase at high enough t e m p e r a t u r e s in gauge theories. This transition, predicted in refs. [ 1,2 ], has been c o n f i r m e d by numerical calculations in the f r a m e w o r k o f the lattice a p p r o a c h (see, for instance, refs. [ 3 - 9 ] ). According to these calculations, at high enough t e m p e r a t u r e s c h r o m o p l a s m a behaves as a gas o f almost free gluons a n d quarks. At the same t i m e at t e m p e r a t u r e s just above the phase transition t e m p e r a t u r e 0c p l a s m a is a rather c o m p l i c a t e d and nontrivial object. N o t e that the t e m p e r a t u r e region close to 0c is o f great interest for experimentalists as just this t e m p e r a t u r e interval is most probable for obtaining c h r o m o p l a s m a in nucleus-nucleus collisions. At present, we have no unequivocal idea about the excitation m o d e s o f c h r o m o p l a s m a . It is not i m p r o b a b l e (for a review see ref. [10] and references therein) that at temperatures close to 0c an i m p o r t a n t role is played by collective excitation m o d e s o f c h r o m o p l a s m a which give rise to electric and magnetic masses. These collective excitations can be p h o n o n [ 11 ] and massive [ 12 ]. As a result, a qualitative picture m a y be thought to be the following [ 1 2 - 1 4 ] : In the large m o m e n t u m region the d o m i n a t i n g part is played by almost noninteracting gluons whereas at m o m e n t a o f the o r d e r o f mD and mM (the D e b y e and magnetic masses, respectively) the m a i n c o n t r i b u t i o n comes from collective excitations, both massive a n d massless. These 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division )

collective excitations are assumed to be singlet with respect to colour ( d y n a m i c a l confinement hypothesis) and the electric and magnetic massive modes give similar contributions [ 14 ]. The role a n d nature o f collective excitation m o d e s o f c h r o m o p l a s m a can be elucidated by investigating t h e r m o d y n a m i c functions within the lattice approach. The s t a n d a r d Wilson action is S=/~ Y~ ( l - ½ S p U p , s ) + P Z ( 1 - ½ S p G , , ) , Pls

(1)

Pit

where Upls a n d UpI, are plaquette variables corresponding to spacelike and timelike plaquettes, respectively, and f l = 4 / g ~ , .... The calculations were done on a 4 X 12 3 lattice with periodic b o u n d a r y conitions in the range o f f l values: 2.2 ~
1 NtaAL

" 11

" 51/121

e x p ( 3 a 2 p ) [1 + O ( g 2 ) ] ,

(2) 371

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PHYSICS LETTERS B

where a is the lattice spacing and Art = 4 is the n u m b e r of sites along the time axis. The energy density E, pressure P and interaction measure 5 were determined by the expressions [ 5 ]

UJ

15 December 1988

{

/..

Q

/®z"

E / 0 4 = 12N 4 [ g - 2 ( (P1s > - (Pit > )

+ C s ( ( P1sym> - (P1, > )--1-Ct ( ( Pl~ym > -- ) ] =EE/O4+EM/O 4 ,

(3a)

p / 0 4 = 4N4 [ g - 2 ( - )

-2

+C~( - ) + C , ( < Plsym > -- ) -& 20

dg-2 - a ~ ((Ply> + -2) =PE/O4 + pM/O 4 ,

1/~0

®/A L

Fig. 1. The solid line without points represents qualitatively the behaviour of E/O 4, full circles and open circles correspond to EE/ 04 and EM/O4, respectively. The dash-dotted line denotes the perturbative contribution. Some points represent a typical statistical error.

(3C)

enough temperatures the behaviour of the energy E and pressure P is in the agreement with the StefanBoltzman law (taking into account one-loop corrections). At temperatures close to the critical one in the deconfinement phase the values o f P / O 4 are strongly suppressed in comparison with the values of E / O 4. This is quite natural since for the second order phase transition in the vicinity of the critical temperature

= 12N4a d g - 2 ( ( P 1 s ) + < P l t ) - 2 < PIsym ) )

q2-a

where < Pls > and < Pit > are the spacelike and timelike plaquettes, respectively, and < Plsym > is the plaquette average calculated on the hypercubic lattice. We have used values of the coefficients C~ and Ct calculated within the one-loop approximation [ 15 ]: Cs=0.1140, C , = - 0 . 0 6 7 5 8 . The electric and magnetic contributions to the thermodynamic functions are determined in a standard way through a lattice generalisation o f the strength tensor F~,,. Correspondingly, in formulae ( 3 a ) - ( 3 c ) the electric (magnetic) components o f the thermodynamic functions are composed o f spacelike (timelike) plaquettes and - It is worth noting that all thermodynamic functions listed above are normalised in such a way that their values at zero temperature are equal to zero by defnition. Our values for E(O), P(O) and 5(0) are their electric and magnetic components are shown in figs. 13 ~l. The dash-dotted line in fig. 1 denotes the behaviour o f the thermodynamic quantity E / O 4 calculated by perturbation theory within the one-loop approximation [ 16 ]. It is seen from these figures that at large

~' We are indebted to Professor H. Satz for providing us with the values of (PIsym> on the 124 lattice. 372

I00

(3b)

5/10 4= ( E - 3P) /O 4

= t~E/04 "]-5M/04 ,

6'0

{ 3PIe4

2 %, 0

-2

-& 20

! /°

60

I00

l&O

8/At"

Fig. 2. The solid line without points represents qualitatively the behaviour of 3P/O4, full circles and open circles correspond to 3PE/O4 and 3PM/O4, respectively. Some points represent a typical statistical error.

Volume 215, number 2

PHYSICS LETTERS B

15 December 1988

x=exp(-

~/0 ~

8-ga0),

(5 cont'd)

the thermodynamic functions should have the following behaviour:

i.e., we have developed the three-dimensional "spin" model of the Ising type, with the difference that the integration in ( 5 ) is performed over the Haar measures dTi. The role of spins is played by representations ½Xl/2(7i)=s, (A consistent procedure of reducing (4) to the partition function of the "spin" system is demonstrated in ref. [ 19 ]. ) A model of this type is known to predict a second order phase transition. For a qualitative investigation of the behaviour of thermodynamic functions we use the meanfield approximation. Using the standard technique of the mean-field approximation (see, for instance, ref. [ 20 ] ) we get the equation for the mean "magnetization" M = ( s , ) :

E~_constlO-O~ l ~-" ,

M = 12( 2 4 x M ) / I ,

P--~const I 0 - 0c [~--",

where/2 and 11 are the modified Bessel functions. This equation at x~< xc= 1/6 has only zero solutions for the mean magnetization M whereas at x~> Kc the magnetization changes continuously from zero at K= Kc, i.e. at x = x~ the second order phase transition occurs. The effective action W ( M ) = - I n Z in this approximation is

1

0.2

Bdo ~

:o

t

O

8M / ¢4

-0.5 20

60

100

1L0

0/A L

Fig. 3. The quantities 6]/04, ~E/04 and ~M/04 with the same notation as in figs. 1, 2.

where a is the relevant critical exponent for the specific heat (the notion of universality classes [ 17,18 ] suggested us that a-~ 0.12 as in the 3D-Ising model). It is possible to reach some understanding of the behaviour of the chromoelectric part of the energy using the hamiltonian approach to SU(2)-gluodynamics in the strong coupling limit. In this regime the partition function is of the form

z = f ]~i.dPi

linksI~D(yk, ~,) ,

(4a)

D(7,, ~,)

(4b)

1

Xk(y,)X~(7,),

where 7i is the S U ( 2 ) group determined in site i, 0 is the temperature, a = as is the lattice spacing in spatial directions and X/~(7) is the character for the kth representation of the SU(2) group ( k = 1/2, 1, 3/2 .... ). At temperatures close to the critical one the expression for the partition function can be exposed in the form [ 1,2 ] Z = f [I. d?, e x p ( - S e f f ) , S e f f ( ~ ) = / ¢ Z Xl/2(~t)Xl/2(~)J), links

(5)

= 12KmZ--ln[l~ (24KM)/12tcM],

(6)

(7)

where N~ is the number of sites in the three-dimensional space. Using (7) we get for the internal energy E 04-

= 1 + ~ , exp - ~ k ( k + l )

W(M)

(24~cM),

1 0W

1

9 9 • i~

1702 ~0 -- ( a 0 ) 4 ~g-Kwl-,

(8)

where V=Ns a3 is the three-dimensional volume. With relations ( 6 ) - ( 8 ) one can easily obtain the behaviour for the internal energy which coincides qualitatively with the one depicted in fig. 1 for EE/ 04. This fact is hardly surprising in the light of the universality hypothesis mentioned above [ 17,18 ]. (Similar calculations for P and ~ are more ambiguous because of the presence of derivative dg-2/da which is unknown in the strong-coupling limit. ) The problem of interpretation like this for the chromomagnetic contributions remains open. At temperatures just above the critical one the magnetic component EM and PM (both being the difference be373

Volume 215, number 2

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t w e e n the c o r r e s p o n d i n g n o n z e r o t e m p e r a t u r e a n d z e r o t e m p e r a t u r e v a l u e s ) take n e g a t i v e v a l u e s (figs. 1, 2 ) [ 21 ] ~2. T h i s m e a n s that the c o n t r i b u t i o n o f the c h r o m o m a g n e t i c e x c i t a t i o n m o d e s differs drastically f r o m t h a t o f the c h r o m o e l e c t r i c e x c i t a t i o n m o d e s (at least for the S U ( 2 ) g r o u p ) , a n d extra e x c i t a t i o n m e c h a n i s m s o f c h r o m o p l a s m a are to be s e a r c h e d for. We a s s u m e that the t e m p e r a t u r e d e p e n d e n c e o f magnetic c o m p o n e n t s o f energy a n d p r e s s u r e p r e s e n t e d h e r e is m a y b e an i n d i c a t i o n o f an i m p o r t a n t role p l a y e d by u n s t a b l e m o d e s [23,24] in the Q C D vacuum.

~2 It is worth noting that nonperturbative corrections [22], changing appreciably the values of coefficients Cs and C, outside the region of asymptotic scaling, do not influence this conclusion.

References [ 1 ] A.M. Polyakov, Phys. Lett. B 72 (1978) 477. [2] U Susskind, Phys. Rev. D 20 (1979) 2610. [3] L.D. McLerran and B. Svetitsky, Phys. Lett. B 98 ( 1981 ) 195; Phys. Rev. D 24 ( 1981 ) 450. [4]J. Kuti, J. Polonyi and K. Szlachanyi, Phys. Lett. B98 (1981) 199.

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[5] J. Engels, F. Karsch, J. Montvay and H. Satz, Phys. Lett. B 102 ( 1981 ) 332; Nucl. Phys. B 205 (1982) 545. [6] V.P. Gerdt and V.K. Mitrjushkin, Pis'ma Zh. Eksp. Teor. Fiz. 37 (1983) 400. [7] K. Kanaya and H. Satz, Phys. Rev. D 34 (1986) 3193. [8 ] V.K. Mitrjushkin and A.M. Zadorozhny, Phys. Lett. B 185 (1987) 377. [ 9 ] J. Engels, J. Jersak, K. Kanaya, E. Laermann, C.B. Lang, T. Neuhaus and H. Satz, Nucl. Phys. B 280 (1987) 577. [ 10] F. Karsh, CERN-TH.4851/87. [ 11 ] P. Carruthers, Phys. Rev. Lett. 50 (1983) 1179. [12] C.E. DeTar, Phys. Rev. D 32 (1985) 276. [ 13] T.A. DeGrand and C.E. DeTar, Phys. Rev. D 34 (1986) 2469. [14] T.A. DeGrand and C.E. DeTar, Phys. Rev. D 35 (1987) 742. [ 15 ] F. Karsh, Nucl. Phys. B 205 (1982) 239. [ 16 ] U. Heller and F. Karsh, Nucl. Phys. B 251 ( 1985 ) 254. [ 17 ] B. Svetitsky and L.G.Yaffe, Nucl. Phys. B 210 ( 1982 ) 423. [ 18] J. Polonyi and K. Szlachanyi, Phys. Lett. B 1l0 (1982) 395. [ 19] O.A.Borisenko, V.K. Petrov and G.M. Zinovjev, preprint ITP-87-154E (Kiev). [20] J.-M. Drouffe and J.-B. Zuber, Phys. Rep. 102 (1983) 1. [21] V.K. Mitrjushkin, A.M. Zadorozhny and G.M. Zinovjev, 19th Spring Symp. on High energy physics (Corobang, DDR, April 1988). [22] G. Burgers, F. Karsch, A. Nakamura and I.O. Stamatescu, CERN-TH.4843/87. [23] N.K. Nielsen and P. Olesen, Nucl. Phys. B 144 (1978) 376. [ 24 ] J. Amhjorn, N.K. Nielsen and P. Olesen, Nucl. Phys. B 152 (1979) 75.