Interface tension in QCD matter

Nuclear Physics B333 (1990) 100-119 North-Holland

I N T E R F A C E T E N S I O N IN Q C D M A T T E R K. KAJANTIE* and L. IC~RKIC~INEN* Department of Theoretical Physics, Uniuersity of Helsinki, Siltavuorenpenger 20 C, 00170 Helsinki, Finland K. RUMMUKAINEN** Research Institute for Theoretical Physics, Universi(v of Helsinki, Siltavuorenpenger 20 C, 00170 ttelsinki, Finland Received 7 July 1989

In hot QCD matter with a colour confining-non-confining first order phase transition at T,., the two phases can coexist at T~.with a stable interface in between. We show how the structure of this interface and its tension a can be determined with lattice Monte Carlo methods. On an Nt = 2 lattice with an 8 × 8 × 40 spatial size the result is a/T~3 = 0.24 _+0.06, the interface extends over more than 2 fm.

1. Introduction Statistical systems can often exist in stable configurations in which two different b u l k phases coexist with an interface in between. Stability d e m a n d s that temperature T, chemical potential(s) # a n d pressure p are c o n s t a n t across the surface while the order parameter, energy e a n d e n t r o p y density s are discontinuous. The interface itself carries a positive free energy density or, integrated perpendicularly over the interface, a n interface tension a. The study of these interfaces is an old a n d central p a r t of statistical physics [1, 2]. Extensive work carried out d u r i n g the past ten years b y using the lattice M o n t e C a r l o t e c h n i q u e [3 10] has shown that bulk Q C D matter at /~ = 0 (zero net b a r y o n n u m b e r ) can exist in two phases: a hadronic disordered colour c o n f i n i n g phase at T < T~ ( h a d r o n phase H) and a q u a r k - g l u o n plasma-like ordered colour n o n - c o n f i n ing phase at T > T~ (quark phase Q). In a n y case the order p a r a m e t e r a n d the e n t r o p y a n d energy densities change rapidly near T~, b u t on a finite lattice it is very difficult to precisely determine whether the transition is of first or second order or even whether there is a phase transition at all. This was the case even for such a * Bitnet address: kajantie@finuhcb, karkkainen@finfun. ** Bitnet address: rummukainen@finuhcb. 0550-3213/90/$03.50 ©Elsevier Science Publishers B.V. (North-Holland)

K. Kajantie et al. / QCD matter

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a)

Fig. 1. (a) The configuration of the system on the lattice, (b) the configuration of the system in physical units. To one-loop order, the T in the hotter part Q is T~(1 + 1.2 Aft) and its dimension is decreased by 1 - 1.2 AB, where 1.2 = 1/4Nfl~o; in HA B ~ -Aft.

m u c h s i m p l e r m o d e l as the three-state Potts m o d e l in three d i m e n s i o n s [11]. It is c o m m o n l y a c c e p t e d that, at least for N v = 0, the transition is of first order, although a r g u m e n t s [9,10] and c o u n t e r a r g u m e n t s [12] against this even for N F = 0 have r e c e n t l y b e e n presented. I n this p a p e r , we shall use the lattice M o n t e Carlo m e t h o d to study the interface b e t w e e n the c o l o u r confining and the colour n o n - c o n f i n i n g phases in hot Q C D m a t t e r at F = 0 [13-16]. The configuration of the system thus is (fig. 1) one in which the two p h a s e s of the system coexist in thermal equilibrium at T = Tc a n d p = Pc with a stable p l a n a r interface in between. The e n t r o p y density s ( T ) = p ' ( T ) a n d energy d e n s i t y ~ ( T ) = T s ( T ) - p ( T ) are discontinuous across the interface, SQ > SH, EQ> Eu. W e shall show that the value of a a n d all other t h e r m o d y n a m i c q u a n t i t i e s can be numerically c o m p u t e d as expectation values of certain o p e r a t o r s over a lattice in the two-phase configuration of fig. 1. Because of tunnelling it is actually impossible to m a i n t a i n a finite lattice at T = Tc in a n y single one of the 1 + 3 possible phases, let alone in a two-phase configuration. W e shall force the system into the two-phase c o n f i g u r a t i o n b y giving T = Tc + AT, T = T ~ - A T to the two halves of the lattice. F o r A T < A T0 (which decreases with i n c r e a s i n g size) tunnelling destroys the configuration. However, the physical values c o r r e s p o n d i n g to A T = 0 can be o b t a i n e d b y e x t r a p o l a t i o n of the results for A T > A To. This m e t h o d has been tested for the much simpler 7-state 2 d Ports m o d e l in refs. [13-15, 17]. Here also we shall observe a rather reliably linear b e h a v i o u r as a f u n c t i o n of AT. Physically, for N v = 0 and F = 0 the b e h a v i o u r of Q C D m a t t e r d e p e n d s on two d i m e n s i o n f u l parameters, T and A QCD" As we know, at T = Tc there is only one a n d we c a n write for the interface tension a a = aoTc3 ,

(1.1)

w h e r e a 0 is a dimensionless c o n s t a n t to be d e t e r m i n e d . F o r the e q u a t i o n of state

K. Kajantie et al. / QCD matter

102

(EOS) of bulk matter we have the ideal gas EOS to set the scale of corresponding dimensionless constants, but there is no similar model for the interface tension. We expect the constant to be of the order of one, but it could as well be, say, 0.01. The larger it is, the more significant are surface effects. We shall describe in sect. 2 how a 0 affects one important quantity, the average distance of the nucleation centers of h a d r o n i c matter in the quark---, hadron phase transition in the early universe [18-20]. O u r result for % is a 0 = 0.24 + 0.06. The magnitude of a can be characterized by the observation that a = p i T c. The result is obtained with reasonably large statistics on a N t = 2 lattice with a 8 × 8 × 40 spatial part. The interface is of the size 8 × 8 (or a b o u t (4 fm) 2) and the length of the system is effectively 20 (or about 10 fm); because of periodic b o u n d a r y conditions there actually are two interfaces in the system. Present results with N t = 4 are compatible with the same physical value but they have a larger statistical error so that they are also compatible with zero. In bulk t h e r m o d y n a m i c s scaling of To is observed for N t = 2 - 4 , is violated for N t = 6 - 8 and seems to set in again for N t = 10 14. It is clear that similar non-scaling effects on the level of several tens of % may be observed for a 0, too. Obtaining a nonzero value for the interface tension is equivalent to proving that the phase transition is of first order. As discussed above, one has failed to do this in a completely convincing manner even when studying the bulk system only. Our value for a 0 for N t = 2 is about four standard deviations away from zero. In any case one has a rather reliable upper limit which is useful for restricting cosmological scenarios. While we are here studying the case of large T, /~ = 0, one should also emphasize that the concept of surface (rather than interface) tension has played an important role in bag-type models at T = 0, large/~ [21]. Because of the difficulty of applying the lattice M o n t e Carlo method at /z :~ 0, only model calculations of the magnitude of the surface tension exist [22-24].

2. Thermodynamics and phenomenology In the g r a n d canonical ensemble with b¢= 0 the general t h e r m o d y n a m i c relations for a system with an interface are, in standard notation, as follows:

F= F(T, V, A) = - T l n

Z=

- p ( T ) V + a ( T ) A = E(T, V, A) - TS(T, V, A), (2.1)

dF=

-SdT-pdV+adA,

(2.2)

K. Kajantieet aL / QCDmatter

103

so that S=p'(T)V-a'(T)A,

V+ [a(T)-

E = [Tp'(T)-p(T)]

(2.3)

Ta'(T)IA.

From these one finds the surface free energy (or tension), entropy and internal energy: F~ = a ( T ) A ,

E~ = [ a ( T ) - T a ' ( T ) ] A .

S~ = - a ' ( T ) A ,

(2.4)

Because the tension or free energy is the first while the entropy and internal energy are the second derivatives of the partition function, the interface tension is far easier to determine with lattice Monte Carlo techniques. Phenomenologically, the lattice results for the bulk equation of state can be described either by the bag EOS

pq(T)=aq T4- B ph(T) = ahT4

IEq(T) =3aqT4+B Eh(T ) =3ah T4

/xq(T)=4aqT 3 and

~Sh(T )

4ah T3

(2.5)

or even better by the K~illman EOS [25, 26]

{ pq(T)=aqT4- AT ph(T) = OhT4

{ 'q(T)= 3aqT4 •h(T) = 3ah T4

sq(T) =4aqT 3 - A and

Sh(T ) = 4ah T3 (2.6)

Both of these contain three parameters, the transition temperature Tc and the numbers of degrees of freedom in the quark and hadron phases gq and gh- These are related to the parameters in eqs. (2.5) and (2.6) by the equilibrium condition (2.7)

Pc ~ pq(Tc) =ph(Tc), which leads to B=(aq-ah)T

4

or

A=(aq-ah)T~

3,

(2.8)

= 0.33.

(2.9)

and by ( N v = O)

,//.2 aq = g q - ~ = 1.75,

71.2 ah = g h ~

K. Kajantie et al. / QCD matter

104

For the latent heat we have E q ( T c ) -- ( h ( T c )

4B = 4(aq - a h ) T c 4

-- £Q -- ~H =

= 3ATc = 3(aq - a h ) T 4

bag

EOS,

K~illman EOS.

(2.10)

The K~illman EOS (2.6) is slightly better since it reproduces the 0-function-like behaviour of ~ / T 4 ; the data do not exhibit the overshoot for T>~ Tc implied by the bag EOS. The clearest signal for the phase transition comes from the large j u m p in E or s and, surprisingly, the observed values of the number of degrees of freedom correspond to those of a massless ideal gluon and (to greater accuracy) pion gas, even for T near Tc. This also implies that the critical pressure Pc = E Q / 1 5 is small so that its observation requires an increase in accuracy of one order of magnitude relative to the latent heat. The interface tension being a free energy, this also sets the scale of accuracy for the observation of interface phenomena. For the bulk EOS we can use the ideal gas behaviour to set the scale of the dimensionless constants in the EOS. For the interface tension there is no corresponding model, the constant a 0 in eq. (1.1) is undetermined. To illustrate the effects of various magnitudes, consider a situation in which the interface is in the x, y plane at z = 0 and write eqs. (2.1) and (2.3) in terms of intensive quantities in the form

f=f(z)

= -~+a(z),

E=~(z),

Ls=((z)-f(z),

(2.11)

where O0

a=

f

dza(z).

To have a very simple model, choose o~(z)/Tc3=O~o/2cosh2(Tcz), ~ ( z ) / T 4 = ~ ( a q + a h ) + ~ ( a q - ah)tanh(Tcz ) and obtain, for c~ = 1.5To3 as an example, the result shown in fig. 2. One clearly sees that this value of a is quite large in the sense that it implies values of interface free energy density much larger than Pc- In fact, the observed value will be close to pJT~ ~ 0.3To3. The main application of the interface tension is to the quark ~ hadron phase transition in early cosmology. This is a phenomenological application; there is no known first-principle method of studying time-dependent non-equilibrium phenomena in finite-T QCD. To begin with, there are two vastly different scales in the problem: the local QCD scale 1/T~ = 1 fm and the Hubble distance

1/X ~ 1/Tc2~/G - 1/GvFGB = 10 k m ,

K. Kajantie et al. / QCD matter

.-

-I

-3

105

.,.

i

i

i

-2

-1

0

1

3

Fig. 2. Schematic variation of the free energy (dotted line), energy (solid line) and entropy (dashed line) densities across the interface if a 0 = !,.5. All quantities are scaled with appropriate powers of T~,,

so that

L=ln(T4/x4)= 175. One now assumes that the transition proceeds like in classical nucleation theory so that the rate of bubble nucleation is

p(t)

= poT4 exp

w(v(t)) ] T

'

(2.12)

where W(T) is the work done to create a bubble of critical radius (for r > rc = 2o~/( p h - Pq) bubbles expand instead of shrinking): 167r~ 3

W(T)-

3(ph--Pq)

16¢rao3T~ 2 -- 3 ( a q _ a h ) 2 (

1 _ T4/T4)

(2.13)

and where T and t are related by the Einstein equation T2t = (45/16~r3Gg) 1/2, g = 51.25 (or an equation slightly modified by the effects of the bag constant B in the EOS). As the universe cools to To and below, the probability of nucleating bubbles of hadronic matter is, according to eqs. (2.12) and (2.13), exponentially damped. A short calculation shows that the fraction of the universe affected by the

K. Kajantie et al. / Q C D matter

106

nucleation process is simply

f(t)

= exp[L-

W(T(t))T(t)]"

(2.14)

neglecting logarithmic terms containing the P o m eq. (2.12), velocities of bubble expansion, etc. Setting 1 gives the temperature and the time at which this first nucleation stage is over. It is quite essential that the universe is not yet entirely filled with hadronic matter: the hadronic bubbles grow with a very small deflagration velocity, while this deflagration front is preceded by a shock wave in the quark phase moving with Ushock > Usound = 1/~/3. These shock waves from different hadronic bubbles start colliding and reheat the quark phase into T--~ Tc. It is the termination of this phase that gives = 1 in eq. (2.14). Beyond that the universe continues expanding at T = T~ and decreases its entropy density by converting matter from the quark phase at Se into hadron phase at s H by slowly expanding the hadronic bubbles. An essential new scale [27] now is the separation of these hadronic bubbles. This one obtains by expanding the exponent in eq, (2.14) near the time of completion tf of the nucleation stage as i. The result is [19]

f(t)=

f(t)

L-W(T(tf))/T(tf)-/)shock(If--t)/R

Ri

0.1 1 - - c ~ 3/2 L3/2

0

X "

(2.15)

The relation of this scale to the horizon distance 1/X is thus given by the magnitude of the logarithm L and the interface tension a in units of Tc3. The estimate given in ref. [20] is the same (noting that X - Tc2) but for the numerical constant, which is by about a factor 50 too large (the constant 1.4 in eq. (26) should be 0.37 there). The significance of the above calculation is that R i gives the scale of the hadronic inhomogeneities after the quark--* hadron phase transition. These, on the other hand, may affect light element nucleosynthesis [28 30] and have important cosmological consequences. For any quantitative results a precise knowledge of the value of a 0 is essential.

3. SU(3) gauge field thermodynamics with an interface on the lattice

The usual relations [31] for the bulk thermodynamics of pure gluon matter are based on writing the partition function in the form

Z = Z(T, V)

=

f~Uexp[-s(u)l,

(3.1)

107

K. Kajantie et al. / QCD matter with the action

--(Pol+Po2+P03)+

S(U)=

ao

g)---~(P12+P23+P31)

,

(3.2)

where the sum goes over lattice sites n.

P,,=- P.~(n) = T r [ 2 - V . . ( n ) -

(3.3)

V~(n)]

is the ~v plaquette associated with site n,

1

an 1

(a)

?g--3-=2/3°ln~-+c t ---1 ao

-;- = 2rio l n - - 7 + Cs m - 1 , a0

,

(3.4)

and (N~ = 3, N F = 0) [32] c~ = 0.20162,

ct = -0.13194,

ct + Cs = fi0 = 0.06966.

(3.5)

As the system is at temperature T = 1/aoN t in a volume V = Ns3a3, the thermodynamical partial derivatives can be explicitly computed with the result (a 0 = a at the end)

S = , ~ [ 4 ~ + ~(c't2 __ Cs)] (el2 _1_ev3. ~~- e31 v PT =

1 1 ~g . + c~ . + 2c .t .

-

(1)1

-ct-2

.

eo, - e(p.- e03) .

(3.6)

P,s + Psi )

]

V

G 5 ( P()I nt- P(}2q- P03) --Pvac(g2) T ,

V I~--=

T

--

g-

Cs ( P I z + P ~ ~ - { - P s I )

-

-

---}-ct

g-

(/:)Ol

+ a,: + a,3)

- {v..(g2)T, (3.7)

where Evac(,g2)a4=

_ P v a c ( g 2 ) a 4 = _ 3~o{ p ( g 2 ) )

(3.8)

is the average plaquette on a symmetric lattice. Observe, in particular, that the O ( g 2) corrections factorize for the entropy and it is unaffected by the vacuum contributions.

K. Kajantieet al. / QCD matter

108

In the s i t u a t i o n with an interface in the 1,2 plane there are three variables T, V a n d A, a n d the t h e r m o d y n a m i c p a r t i a l derivatives require keeping two of these fixed. This can b e i m p l e m e n t e d by choosing three different lattice spacings a 0, a 1 =

a 2 = a r a n d a 3 so that T-

1

,

V = Ns3a~a3,

A = Ns2a2r.

(3.9)

Ntao T h e a c t i o n n o w is 1 S(U)=

~ 2 [ 1 a3 l"o(p13+P23)+ T,,[7-~l-aoo(Pm+P°2)+g~a3

a2

g32 aoa3

1 P03 +

g42

]

a°a3

7

P12 ,] (3.10)

w h e r e the g2 = g~(ao, at, a3). F o r the leading terms the a j d e p e n d e n c e of the gff c a n be neglected but for the O ( g 2) corrections, which c o n t a i n essential d y n a m i c s , we also n e e d some of the derivatives Og~/O In a j n e a r a i all equal. It first seems t h a t the relations (3.4), which require a r = a3, are not sufficient. However, it is now p o s s i b l e to r e d u c e the c o m p u t a t i o n of the partial derivatives to a c o m b i n a t i o n of the two cases a o, a~ = a 2 = a 3 a n d a 0 = a x = a2, a 3 a n d express the O ( g 2) result e n t i r e l y in terms of the coefficients c t, c S in eq. (3.4). T o see this c o n s i d e r explicitly the c o m p u t a t i o n of

.

T

.

.

.

.

I

A -~

T OA T,V

W e write the action (3.10) in the form S

.

= S(a 3, ~0, ~3),

a3 ~o = - - ,

aT

T h e n , expressing T, A and V in terms of a3, finds that

(3.11)

T,V

where

aT & =

- - .

ao

~0 and ~3 a n d

c o m p u t i n g OS/O~ o one

OS r, 1 OS I r O~v r,,4 1 0S v, A~ v = 2 ~ ° ~ o ,,, & + -~ V ~ A

(3.12)

O n the l e f t - h a n d side we have to keep ao:/:ar:/: a> O n the r i g h t - h a n d side, however, we can m a i n t a i n the required c o n s t a n t variables b y taking a o = a r :# a 3 in the first a n d second terms and a 0 :~ a r = a 3 in the third one. W h e n c o m p u t i n g the derivatives o f the 1 / g 2 terms in S we can then use eq. (3.4) as it stands for the third

109

K. Kajantie et aL / Q C D matter

term, and with the change a t ~ a 3, a ---, a 0 = a r, 1/g2t ~ 1/g~ appearing with a o / a 3, for the first two. After the derivation one, of course, sets all a i equal. Note that physically the first term in eq. (3.12) is related to stretching the interface and changing the T while keeping the length of the system constant; the expectation value of the second term gives p V / T and that of the last one - E / 2 T . Without going through the computation it is hard to see why the relative weights are those in eq. (3.12). Using eq. (3.12) together with (3.10) and (3.4) and doing the same for all the thermodynamic variables, one can derive the following set of equations (in which

[5

s=E o~--= ,~T

+

7

]

~(~.~- cO (2e~ +

Pig + P23 - 2Po3 - eol - No2),

+½(ct-Cs) ( 2 P ° 3 - P m - P ° 2 - 2 P 1 2 + P 1 3 + P 2 3 ) '

(3.13)

(3.14)

v[(1)

7+e, (Po,+P1,+P23)- 7 - e ~ (Po1+Po2+P12) --Pvac(g2)¥,

pf=E

(3.15) is given by the same equation (3.7) as for bulk thermodynamics. From these one can further derive S+o~ F T

-pV+ --

-~

A

T

}

n ~5 + ~ ( c t - Cs) 2(Pt3 +/)23 -- POI -- P02), (3.16)

T

c~A

T

1

E+pV

E ~ - - -- S

~--Cs

T

P03-

7

+¢t

P12

2

o(Pol+Po2+P13+P23)

-'vac(g2)~,

(3.17) and, finally, the interaction measure E-

3 p V + 2c~A T

V = --2/~0 E ( P o l tl

q" P02 "}- P03 "~ P12 "}- P23 -Jr-P13) - 41[ vac ( g 2 ) ..~ .

(3.18)

110

K. Kajantie et al. / QCD matter

All these equations are written in the one-loop approximation so that, for instance, a

d 1 d a g2 - -2130 = - 2 ( c t + cs).

This is anyway the accuracy to which numerical work is possible, at present. We shall, in the following, study the above quantities as functions of the longitudinal coordinate z. Then it is convenient to use V / T = Ns3Nta 4 or A / T = N,2Nt a3 to write, for instance, 1

°~a3= E NtNs2 ~ ( . . . ) Z

.~c , . ' /

Y'~a(z) a3

(3.19)

Z

and give the results in terms of ce(z)a 3, similarly for the other thermodynamic quantities. However, because the results (3.13)-(3.18) are given in terms of plaquettes, the association of a value c~(z) for a site z (averaged over x and y ) is actually ambiguous. Note that for both the entropy and the interface tension the O(g 2) corrections factorize entirely. Thus, even including these corrections, the expression for c~ vanishes identically if there is no interface (all the directions 1, 2 and 3 are equivalent) at finite T (the 0 direction is different) while the expression for S vanishes if there is no thermal matter (all the directions are equivalent). Physically one sees how the free energy of the interface is measured by probing the interface with plaquettes _l_ and ]] to the interface and looking at the difference. The expression is thus very suggestive and has also been given a rigorous derivation. A corresponding suggestive expression for the tension in spin models was only derived heuristically [17]. To evaluate the pressure and energy, free energy and enthalpy densities one also needs to know the values of the vacuum symmetric plaquettes (eq. (3.8)). It may also be of some interest to compare the above expressions with the classic expression for the interface tension [1]

.= f dz tplz)-pT(Z)3, where p and PT are the "normal" and "transverse" pressures. This is nothing other than the relation F = - p V + eeA, modified by writing F - - - p T V and written differentially in z. A different method proposed [14,15] to determine the interface tension starts from the following general relation, satisfied by the free energy of QCD matter:

c3fl T -

2N c

( Pm + Po2 + Po3 + P12 + P13 + P23) ,

(3.20)

K. Kajantieet al. / QCD matter

111

where fi = 2 N J g 2 and our P,~ are normalized to 2N c (eq. (3.3)). Note that eq. (3.18) relates the right-hand side of eq. (3.20) to the interaction measure. By starting the system from a configuration with both halves at /3 =/3+ and cooling one of the halves to 13 -- fi , one can create the interface. By integrating eq. (3.20) over/3 one can then compute the associated z~(F/T) and by choosing the path properly one m a y separate the z I ( F / T ) associated with the interface and from this infer Zl F~ = aA. The method has already been applied to a spin model, for which eq. (3.20) reads

0

F

o(1/r)

T

- (H(1/T)).

The expressions (3.13)-(3.18) are the main formal result of this paper. With their aid the entire thermodynamics of a QCD system with an interface can, in principle, be computed in terms of four different types of plaquette expectation values, Pol + Po2, Po3, P13 + ~ 3 and P12- However, the numerical evaluation is, in practice, highly nontrivial, since on a finite lattice the interface is not stable.

4. Numerical results

In this section, we present results for lattices with N t = 2, for which/3c = 5.09. If T~ = 200 MeV, the lattice spacing in physical units is a = 1/(T~Nt) = 0.5 fro.

(4.1)

The spatial size of the lattice is 8 × 8 × 40. Because of periodic boundary conditions the interface appears twice and in order to decouple the two interfaces it is important to have a rather long lattice in the z direction. To impose the required two-phase configuration on the lattice, we set /3 = / 3 = B - , ~ B on the links between z = l , 2 .... , N J 2 , B = B + = B + z I B on the links between z = N : / 2 + 1 . . . . . ~ and /3 =/3c on the links parallel to the z direction between N J 2 and NJ 2 + 1. The interface thus effectively appears between z = N J 2 and N J 2 + 1 and between z = Nz and z = 1. To one-loop accuracy in the beta function we have for one half of the total temperature j u m p 1 AT AT Aft = - = 1.2-4N~/30 T T

(4.2)

The values of A/3 used are less than 0.3 so that the temperature is at most 30% above Tc on the hot side of the interface and 30% below on the cold one. On the lattices we use, the interface cannot be maintained for A/3 below about 0.04. The

112

K. Kajantie et al. / QCD matter

°Tt.

!

0.65

0.6 ~

F

-

-

= 5.~9

0.55 N=2

0.45 ~- ' [

o4L-~ " 4.4 4.6

4.8

5 . 0 5.2

55.4

5.6

5.8

6.0

Fig. 3. Expectation value of the vacuum plaquette = (P~,~)/6, where P~,~ is given in eq. (3.3), calculated on a symmetric 84 lattice (so that it is independent of/~ and ~,) as a function o f / 3 = 6 / g 2. A fit to the curve is given in eq. (4.4). The critical values of 17 for N t = 2 and 4 are marked on the curve.

physical value then is obtained by first taking the limit V, A ~ 0 and then Aft ~ 0: a=

lim

lira

Afl~O V , A ~ O

a(AB).

(4.3)

This limit is analogous to the one encountered in the study [33] of spontaneous chiral symmetry breaking: there ( ~ ( m f ) ) is computed and first the limit V ~ 0 and then the limit m f ~ 0 is taken. One also has an analytic control over the leading finite-size effects [34,35], which is not the case here. Note that in eq. (4.3) the finite-size effects depend on both V and A. For the energy density and pressure we also need the vacuum plaquettes. These have been computed on a 84 lattice and the result is shown in fig. 3. A sixth-order fit to the points within the interval 4.8 < fl < 6.0 is f i ( f l ) = -5691.1705 + 6532.3383fl - 3114.6373fl 2 + 789.67446fl 3 - 112.27329/74 + 8.4864872fl 5 - 0.26641652fl 6 .

(4.4)

Because of the imposed discontinuity in fl it is important to include correctly the fl dependence of the vacuum plaquettes to reproduce correctly the discontinuities in energy density and pressure between the two halves of the lattice. When presented as functions of z the results in eqs. (3.13)-(3.18) are differences of (differences of) plaquette expectation values averaged over x and y, varying rapidly with z (see fig. 4) but known only in a few points within the range of rapid variation. This causes a discretization ambiguity since the z-values for plaquettes entirely in the transverse plane, P12(z) and Pot(Z)+ Po2(Z), and the z-values for those perpendicular to the interface, Po3(Z) and Pz3(Z) + P13(z), differ by ½. This ambiguity vanishes when summed over z.

K. Kajantie et aL / QCD matter

113

a :o.o5 0.60

0.55

Pm + Po2

/

I \\

0.50

10

20

30

Fig. 4. Behaviour of the fundamental plaquette expectation values (multiplied by 1/6 so that the maximum value for a single plaquette is 1) (Pro + Po2)/2, (P12 + P23)/2, P03 and P12 averaged over x, y as a function of the longitudinal coordinate z for a 2 × 8 × 8 × 40 lattice in the two-phase configuration of fig. 1 at Aft = 0.05. For clarity, only the region of + 10 lattice spacings around the interface in the middle is shown.

The results for the N t = 2, 2 × 8 × 8 × 40 lattices in the two-phase configuration of fig. 1 are shown in figs. 4-8. They are based on 50,000 iterations at Aft = 0.30, 0.20, 0.16, 0.12 and 100,000 iterations at Af = 0 . 1 0 , 0.07, 0.05 and the total computer time used was 220 hours CPU of one processor of CRAY X-MP. The main output of the computer runs were the four types of the plaquette expectation values averaged over x, y, P12(z), Pol(Z) + Po2(Z), P03(z) and P12(z) + P13(z). All the other relevant quantities can be computed in terms of these using eqs. (3.13)-(3.18). Fig. 4 shows an example of the general pattern for the smallest value of Aft, Aft = 0.05. Only the region around the interface in the middle at z ~ 20 is shown; the interface near z = 40 is similar. The bulk hadron phase is to the left and the bulk quark phase to the right. Because of the symmetry we know that P12(z) = P23(z)= P13(z) and Pol(Z)= Po2(Z)= P03(z) in the bulk phase and from the expression of the energy density in eq. (3.7) we know that the space-space plaquettes must be larger than the space-time plaquettes. Also, because the energy density is much larger in the quark phase, the difference between space-space and space-time plaquettes must be much larger in the quark phase. All the features are clearly visible in fig. 4. As far as the interface is concerned, all the thermodynamics is determined by how the four types of plaquettes vary between their bulk values. Fundamentally, the aim of the lattice computations is to find out this and to extrapolate to the physical configuration Aft = O.

K. Kajantie et al. / QCD matter

114

A~= 0.20

zx~ 0.30 =

A]3 = 0.16

2.0

!IUI

. . . . .

1,5i 1.0

i. . . . . . . . . . ......................

0.5

iilil!iiiii!iI.iiiiiii!ililiii!i+

0£

i 21

i

21

40

A[3 = 0.10

1.5

1.0

i .................

ZXl3= 0.07

21

40

40

A[~= o.o5

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

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

0.5

0o

j(

Fig. 5. The o r d e r p a r a m e t e r ( T r L ( z ) )

,

2,

for various k fl for a 2 × 8 x 8 X 4 0 configuration.

i

21

40

lattice in the two-phase

One quantity not calculable from the plaquettes in fig. 4 is the standard order parameter, Tr(Wilson line), trace of the string of N t SU(3) matrices in the periodic time direction. How this behaves is shown in fig. 5. The two-phase structure with two finite but narrow interfaces separating two extended regions of bulk matter is observed. The statistical errors are marked on the figure but are hardly visible. One estimates that the interface extends over 4 - 6 lattice spacings or over 2 - 3 fm. The two-phase structure is also clearly visible in fig. 6, which shows the entropy, energy density and pressure, evaluated from eqs. (3.13), (3.7) and (3.15). The correct inclusion of the vacuum energy density is essential for these curves. The magnitudes of S and c in the T > Tc phase are in rough agreement with the same quantities for an ideal gluon gas, with proper inclusion of finite-size effects [36], but we do not enter in this discussion here. Note, however, that the existence of the temperature gradient A T / T c = 0.Sail implies that there is a discontinuity even in the pressure; in the physical limit Aft = 0 the pressure will be the same in the two bulk phases. The main results are shown in figs. 7 and 8. The former shows a ( z ) a 3 evaluated from eqs. (3.14) and (3.19), and the latter shows a as a function of Aft and the extrapolation to Aft = 0. Again the two bulk phases separated by two interfaces are clearly visible in fig. 7. The negative values of a ( z ) observed on the hot side of the interface are not physical but are due to the ambiguous discretization effect referred to above. Since the difference between the two halves decreases with Aft, the discretization effect also disappears. The sum over z shown in fig. 8 is unaffected by the ambiguity.

115

K. Kajantie et al. / QCD matter

aB = 0.20

A[3 = 0.30

A~=0.16

2.0 S

1'5i H .....

0.0 ' 1

2'1

2.0

40

1

40

21

1

40

a~ = o.o5

A~ = 0.07

AI~ = O. 10

21

1.5 1.0 0.5

~-"--'----'~,

o.o ~

:

1

21

-

40

21

40

1

21

40

Fig. 6. The e n t r o p y Sa 3 (eq. (3.13)), energy density ca 4 (eq. (3.7)) and pressure pa 4 (eq. (3.15)) for various values of J/3 for a 2 × 8 X 8 × 40 lattice in the two-phase configuration.

A~ = 0.30

3,~ = 0.16

A[3 = 0.20

0.15 ~

IZIIII221ZI211iI2ZIZIIZII£1

IZ2ZZZII ZZZZ~ .................

o oool

0.15

i .....................

i

a~ = 0.05

a~ = 0.07

al~=0.t0

0.10 0.05 0.00 -0.05 -0.10

L....................................

i .................................

21

40

1

21

40

21

40

Fig. 7. The free energy density a(z) of the interface (eqs. (3.14) and (3.19)) for various values of A/3 for a 2 X 8 × 8 × 40 lattice in the two-phase configuration.

K. Kajantie et al. / QCD matter

116 0.4

0,3

06 0.2

0.1

0.0 0.0

0.1

0.2

0.3

0.4

Fig. 8. The interface tension s u m m e d over the two interfaces o n the lattice ( = sum over z of the curves in fig. 7) for various values of zl]~ for a 2 × 8 × 8 X 40 lattice in the two-phase configuration. The intercept at Aft = 0 is 0.060 _+ 0.014.

The errors of the points shown in fig. 8 are estimated by dividing the data (50,000 or 100,000 iterations) in five blocks and computing the standard deviation. The extrapolation to A/~ = 0 gives a a 3 = 0.060 _+ 0.014. Converting this to physical units and dividing by two to obtain the result for one interface, we find that

a 0 -- Tc-X = 0.24 _+ 0.06.

(4.5)

In the simple pion-gluon gas model of the EOS described in eqs. (2.5)-(2.9) we had Pc = 0.33T4, so that c~ is numerically very close to p J T c. Although we do not have any model for the interface directly, the magnitude of its tension (which is a free e n e r g y / a r e a ) is thus seen to be close to (minus) the free e n e r g y / V of the bulk phases times 1/T~. We can also estimate the thickness of the interface: for N t = 2 it extends over 4 - 6 lattice spacings or 2 - 3 fm ~ (2-3)/T¢. At the position of the interface it thus maximally contributes an extra 30-50% increase to the ambient (minus) free energy density. Finally, the result (4.5) implies that the average distance between the hadronic nucleation centers in the early universe, as given by eq. (2.15), is about 5 cm.

5. Fermions

Although we shall not carry out any numerical work for fermions, we shall present the relevant formtflas to exhibit their structure. Placing the interface again in the 1,2 plane and choosing the three different lattice spacings ao, a r and a 3, the

K. Kajantieet al. / QCDmatter

117

fermion part of the total action S~ + S F is for Wilson fermions

S F = ~ Q t ~ - ~ b [ 1 - K o M o - K T ( M I + Mz)-K3M3]~/,

(5.1)

where, to leading order in g2 and for m e = 0, 1

1

m

K i - 2ai 1 / a o + 2 / a r + l / a 3

i = 0 , T, 3,

(5.2)

and M, ..... = (1 - ,/,)Un,,~,,,,_, + (1 + "/~)U2,6,,m+ ~. A simple calculation then gives for the fermionic parts of the thermodynamic quantities E F

1

V - 32 ( ~ ( 3 M ° -

M 1 - M 2 - M3)~b)'

V 1 P F -T- - - -32 - (~/(M° + M1 + M2- 3 M 3 ) q ' ) ' A aF T

1 32 ( ~ ( 2 M 1 -

2 M 2 - 4M3)~b) '

SF= ~---~(~b(4Mo- 2 M 1 - 2M2)+),

(5.3)

where

- f

Udet Q Tr(MQ -1) exp(f N U d e t Q exp( - S~)

(5.4)

The general structure of the expressions is again apparent: that for a F vanishes if there is no interface even at finite T, while that for S v vanishes at T = 0. As for gluons, the O ( g 2) corrections can also be computed for the interface case using the recent one-loop results in ref. [37] for the bulk thermodynamics case with a 0 :~ a I -0 2 = a 3.

6. Conclusions

In this paper we have presented a method for determining the interface tension of the interface at T = Tc between the two phases of QCD matter, the colour confining hadronic and the colour non-confining plasma-like phase. The method involves imposing the two-phase configuration on the lattice by giving T = Tc + AT to one half of the lattice and T = Tc - AT to the other and then extrapolating to the physical value AT = 0. We have applied the method to pure gluon matter without quarks using an N t = 2 lattice with the spatial size 8 x 8 × 40. The result in eq. (4.5)

118

K. Kajantie et al. / QCD matter

is a / T c 3 = 0.24 _+ 0.06. Quantitatively, a is a p p r o x i m a t e l y equal to p c / T o , where Pc is the e q u i l i b r i u m pressure of the two phases. P r e l i m i n a r y results with N t = 4 are c o m p a t i b l e with this but, due to a larger statistical error, they are also c o m p a t i b l e with zero. It is q u i t e o b v i o u s how the present study should be e x t e n d e d to larger lattices a n d to larger n u m b e r s of iterations p e r point. The total i n v e s t m e n t of c o m p u t e r p o w e r to the results in this p a p e r is a few h u n d r e d hours C P U time of a single p r o c e s s o r of C R A Y X - M P so that it is clear that m u c h m o r e can b e done. In this way one would also o b t a i n a m u c h better control over the finite V a n d A effects, which is n e e d e d to h a v e c o m p l e t e c o n f i d e n c e in the e x t r a p o l a t i o n to the physical value. A n analytic u n d e r s t a n d i n g of these finite-size effects w o u l d also be very useful. A s e c o n d d i r e c t i o n of generalization w o u l d be to include quarks a n d use eq. (5.3). This is c l e a r l y n e c e s s a r y to o b t a i n a n u m b e r directly a p p l i c a b l e to the cosmological situation. F i n a l l y , it will be very interesting to c o m p a r e with the n u m b e r o b t a i n e d with the a i d of eq. (3.20). W e t h a n k D o u g Toussaint for asking the question which lead to this investigation a n d T o m D e G r a n d , H a n n u K u r k i - S u o n i o , Jean Potvin, A k i r a U k a w a and, in p a r t i c u l a r , F r i t h j o f K a r s c h for several instructive discussions and c o r r e s p o n d e n c e . T h e c o m p u t i n g facilities have been p r o v i d e d b y the C e n t e r for Scientific C o m p u t ing, F i n n i s h State C o m p u t i n g Center a n d f u n d e d b y the U n i v e r s i t y of Helsinki.

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