Nuclear Physics B256 (1985) 670-686
© North-Holland Publishing Company
F I N I T E - T E M P E R A T U R E L A T T I C E Q C D W I T H W I L S O N F E R M I O N S T. ~ELIK t, J. ENGELS and H. SATZ
Fakultllt fiir Physik, Universitiit Bielefeld, D-4800 Bielefeld, West Germany Received 20 November 1984
The thermodynamics of QCD with dynamical Wilson fermions is studied in a low-order hopping parameter expansion, using Monte Carlo simulation on 8 ~ x 3 to 103 × 5 lattices. We observe a clear deconfinement transition at Tc/A~ ~12 ~ 150; chiral symmetry restoration occurs at the same point. Within our approximation, both transitions are continuous. In the confinement regime, we find the global centre Z 3 symmetry only very weakly broken, in accord with a picture relating string breaking in QCD with ionization in insulating solids.
1. Introduction
T h e t h e r m o d y n a m i c s o f p u r e S U ( N ) g a u g e fields p r e d i c t s the o c c u r r e n c e o f a d e c o n f i n i n g p h a s e t r a n s i t i o n [ 1 ]. At s o m e t e m p e r a t u r e To, c o l o u r s c r e e n i n g d i s s o l v e s the c o n f i n i n g b o n d a n d t u r n s g l u o n i u m m a t t e r into a c h r o m o p l a s m a [2]. O n e o f the m o s t i n t e r e s t i n g p r o b l e m s in c u r r e n t s t a t i s t i c a l Q C D is w h a t effect the i n t r o d u c t i o n o f d y n a m i c a l q u a r k s has on this d e c o n f i n e m e n t .
O n a p h e n o m e n o l o g i c a l level, c o l o u r s c r e e n i n g c o n s i d e r a t i o n s a p p e a r to r e m a i n v a l i d ; t h e p r e s e n c e o f m a n y o t h e r c o l o u r c h a r g e s s h o u l d d i s s o l v e b o u n d q u a r k states j u s t as well as g l u o n i u m states. O n the o t h e r h a n d , the g l o b a l s y m m e t r y u n d e r the c e n t r e ZN o f t h e S U ( N ) g a u g e g r o u p is b r o k e n b y the f e r m i o n t e r m in the l a g r a n g i a n , a n d h e n c e d e c o n f i n e m e n t c a n no l o n g e r b e strictly c h a r a c t e r i z e d in t e r m s o f s p o n - t a n e o u s ZN s y m m e t r y b r e a k i n g . T h e e x p e c t a t i o n v a l u e /[ o f the t h e r m a l W i l s o n l o o p d o e s not a n y m o r e c o n s t i t u t e an o r d e r p a r a m e t e r for the d e c o n f i n e m e n t t r a n s i t i o n , s i n c e it n o w d o e s not v a n i s h in the c o n f i n e m e n t region. H o w e v e r , it is u n c l e a r h o w m u c h /7, differs from zero t h e r e , i.e. h o w s t r o n g l y t h e ZN s y m m e t r y is a c t u a l l y b r o k e n . W e m u s t t h e r e f o r e ask b y w h a t m e c h a n i s m /7, is d e t e r m i n e d in the c o n f i n e m e n t z o n e .
T h e d e c o n f i n e m e n t t r a n s i t i o n is the c h r o m o d y n a m i c a n a l o g o f the m e t a l - i n s u l a t o r t r a n s i t i o n [3] in s y s t e m s with e l e c t r o m a g n e t i c forces. In i n s u l a t i n g s o l i d s , the e l e c t r i c c o n d u c t i v i t y tr is not strictly zero, b u t o n l y e x p o n e n t i a l l y s m a l l [3]:
~r ~ e x p { - E i / T } , ( 1 )
w h e r e E, d e n o t e s the i o n i z a t i o n energy. A b o v e the t r a n s i t i o n p o i n t to a m e t a l , cr is n o n - z e r o b e c a u s e D e b y e s c r e e n i n g has g l o b a l l y d i s s o l v e d the C o u l o m b b i n d i n g
A.v. Humboldt Fellow. now at Hacettepe University, Ankara, Turkey.
670
T. ~elic et al. / Finite-temperature lattice Q C D 671
between ions and electrons; but even below this point, ionization can locally provide some free electrons, and thus make ~ > 0. The corresponding p h e n o m e n o n in Q C D with dynamical quarks is the production o f qcI pairs. In pure gauge theory below the deconfinement temperature, an infinite energy is needed to break up a static q u a r k - a n t i q u a r k pair; this corresponds to an infinite ionization energy in eq. (1).
With d y n a m i c quarks, such a break-up becomes possible when the separation yields a binding energy equal to that needed to produce a qcl pair, i.e. a hadron. The newly-formed quarks neutralize the static quarks and thus allow their separation.
We therefore expect that /T will now no longer vanish in the confinement zone, but that
L--exp{--mH/2T},
(2)where mH is the mass of the dominant qcl bound state. Letting mH ~ oc in eq. (2), we recover the pure gauge result.
Both the conventional Mott transition in solids and the deconfinement transition in hadronic matter thus lead from a regime, in which the binding can locally be broken (by ionization or qcl formation, respectively) to one, where it is globally removed by a collective screening of the binding force. Although this provides us with an intuitive picture o f how an /7,~ 0 arises in the confinement region, it does not allow us to estimate it quantitatively. Thus also the question of the sharpness of the transition, or o f its order, if it still is a genuine phase transition, remains open.
For colour SU(3), the deconfinement transition of the pure gauge theory is of first order [4]. Hence very heavy quarks will not totally remove the discontinuities in t h e r m o d y n a m i c observables. Massive constituents are thermodynamically sup- pressed, and only if the quark mass becomes small enough can the deconfinement pattern of the gluon system be changed.
However, quark mass considerations around the deconfinement point remain rather arbitrary as long as the relation between chiral symmetry restoration and deconfinement is not clarified. In a state of broken chiral symmetry, the effective quark mass is not zero, and if it is large enough, we may still have a first-order deconfinement transition [5].
On the other hand, the role of the quark term in the Q C D lagrangian is similar to that of an external magnetic field applied to a spin system [6-9]. This immediately shows why /S should cease to vanish in the confinement zone, although it does not directly relate its value there with any physical quantity. The discontinuity in /S decreases with decreasing quark mass (increasing external field), and based on strong coupling arguments it has been suggested that the deconfinement transition may be completely washed out in the light-quark limit [6-9]. However, all quantitative studies p e r f o r m e d so far [10-13] continue to show a very abrupt deconfinement transition even for the lightest quark masses considered.
To study this problem in more detail, we have performed numerical calculations with Wilson fermions [14] of two flavours in a low-order hopping parameter
672 T. ~'elic et at / Finite-temperature lattice QCD
expansion, on lattices ranging from 83X3 tO 103 ×5. We have chosen this fermion formulation because it allows in reasonable time calculations with rather good statistics on fairly large lattices. Moreover, the Wilson fermion scheme even in the lowest-order hopping p a r a m e t e r expansion contains all the essential features used to argue for the d i s a p p e a r a n c e of the deconfinement transition. Therefore, if we understand here why it persists to such an extent, then this will be applicable to other fermion formulations as well; and as we have already noted, so far the different fermion schemes do lead to very similar results on deconfinement and chiral symmetry restoration.
In the next section, we shall sketch the formalism of statistical Q C D on the lattice, using the Wilson fermion scheme. In sect. 3 we then present our numerical results for the relevant t h e r m o d y n a m i c observables.
2. Statistical QCD with Wilson fermions
Our starting point is the euclidean lattice form of the Q C D partition function:
ZE(/3)= f 1-I d U { d e t ( 1 - K M ) } Nr e x p [ - S G ( U ) ] , (3) J l i n k s
as it appears after integration of the quark spinor fields. Here
\ % / p~ ~ P,,~(l R e t r ( U U U U ) )
(4)
is the Wilson action for SU(3) gauge fields at finite temperature [15], obtained by s u m m i n g over s p a c e - t i m e (P~) and s p a c e - s p a c e plaquettes (Pu); a~, and a s are the spatial and temporal lattice spacings, g,, and g~ the corresponding couplings. The form (3) holds for N / f l a v o u r s of quarks with equal mass. The matrix MM~,..,. = (l - %.) U.m,~ .. . . ,~+(l + 3,~) U ~ . 8 . . . . ~ (5) describes the interaction for each flavour, corresponding to the Wilson form o f the fermion action
S F ( U ) = Z ~s( l - KM)¢y ; (6)
f
in eq. (5), /2 is a unit vector along a lattice link. In the finite temperature case, the hopping p a r a m e t e r K(g 2) depends on the link direction
3
KM-=-K~Mo+K~. ~ M . ; (7)
~ = l
however, for a . = a s = a, Ko = K,. = K(g2).
T. ~'elic et al. / Finite-temperature lattice QCD 673 We now expand the logarithm of det (1 - KM) in powers of the hopping parameter [16]
I
In det (1 - KM) = - T r -~ (8)
Only closed loops contribute to eq. (8). The inclusion of the fermion determinant thus corresponds to having in eq. (3) an effective action
Se.(U) = S ~ ( U ) - 4 N I ( 2 K ) N~ Y. Re L - 1 6 N s K 4 ~ R e T r ( U U U U ) + O ( K ' ) .
sites Pt3,P,,
(9)
Since the third term on the r.h.s of eq. (9) simply shifts the coupling by6 6
g--i ~ g-'~ + 48 NfK 4 , ( 1 O)
the main effect of the presence of fermions is contained in the thermal Wilson loop term of Sen, with
L(x)---Tr [I Ux: .... , . (11)
T = I
The hopping parameter K(g 2) is for massless quarks given at small g2 by [17]
K(g 2) = ~[1 + 0.1 lg 2 + O ( g ' ) ] . (12) At the
g2
value we have considered, we expect 10-20% deviations from the weak coupling behaviour (12) ; in fig. 1, we show Monte Carlo results obtained by requiring m,, = 0 , together with eq. (12) up to order g2. Since the form (12) falls below the critical Monte Carlo curve, it corresponds to quarks of non-vanishing mass. This keeps us within the radius of convergence of the hopping parameter expansion. For similar reasons, other fermion schemes also use finite quark masses in the actual calculation. We have thus used the form (12), as we also use the renormalization group relationa A L = e x p { 41r 2 ( 6 ~ 4 5 9 - 5 7 N f [ 8~_22 (g6__i)]} (13) - 3 3 _ 2Ny k-g~] + ( 3 3 _ 2Nf)21°g L 33_ 2Ns
obtained from the weak coupling expansion of the/3-function,
( 3 3 - 2 N / ) g3[ 1 _t.
(306- 38 Nf)
O(g4) ] (14)_fl (g2)
3-('4 ~-~ (4 ~)--~-~----2-Ny)g2+ .It appears reasonable to expect both eq. (12) and eq. (14) to apply well for g2<~ 1 (see for example [18] for a survey of recent results of scaling tests).
From the partition function (3) we now get the energy density e as a function of the temperature. With e -= eG + eF, we have for a,, = a~
eG/T 4 = 18 N~{g-2(P,, - PB) + c ' ( P - P,,) + c ~ ( P - Pa)} (15)
674 T. (~elic et al. / Finite-temperature lattice QCD
X
0.24 -
D
0 . 2 0 -
D
0 . 1 6 -
0 . 1 2 -
\ \
" , \
",
\ \ \ \ \
I I I
4.0 5.0 6.0 7.0
6 / g 2
Fig. 1. Hopping parameter K as function o f 6 / g 2, for the quenched case (the dashed curve, obtained by interpolation o f numerical data), for the case N r = 2 (the solid curve, obtained from the previous curve by use o f the renormalization group relation), and the weak coupling form (dash-dotted curve,
obtained from relation (12)).
for the gluon sector and
E F / T 4 = N ~ N f { 3 ( 2 K ) NB R e / S + 1 4 4 K 4 ( P ~ -- Pt3) + O ( K s ) } ( 1 6 ) for the fermion sector. The separation of e into two such terms corresponds to the two components of S,n; in the interaction regime, an actual separation into gluon and quark systems is of course impossible. In eq. (15), F'~, and Pv denote the space-space and space-temperature plaquette averages on the N~ × Nv lattice;
! .t
is the average on a large (N~) symmetric lattice. The constants c,, and c~ come from differentiating the couplings g, and gv with respect to a~; for colour SU(3) and Wilson fermions of two flavours, one has [ 19] c~, = 0.19366 and c v - ' - -0.132463.
In eq. (16), /S denotes the lattice average of the thermal Wilson loop.
The energy density thus determined will certainly display finite-size effects - both because o f the lattice regularization [20] and because of the truncation of the hopping parameter expansion. To compare our results with those of a non-interacting quark- gluon system, we therefore calculate the ideal gas forms e sB and e s" on lattices of the same size and in the same order of hopping parameter expansion. For e~;, the
T. (?elic et al. / Finite-temperature lattice QCD 675
resulting values are given in ref. [20]; for e sB we have from eq. (16) by setting all U = 1
e s a / 7 `4 = 9 N ~ Nr(4)- N~, (17)
where we have taken K sB =4 and neglected fifth and higher order contributions.
If the q u a r k - g l u o n system undergoes a sudden transition from the hadronic to the plasma phase, we expect this to reflect strongly in the behaviour of e. To study the restoration o f chiral symmetry, we consider the order p a r a m e t e r [16]
( ~ b ) a 3 ] p e , fl . . . =2K ~ KITr M I = 2 K { 1 2 - - 4 ( 2 K ) N" Re/S
1=o
- 1 ! 52K4(1 - ½(P,, + Pt~)) + O(KS)} • (18) Since in the Wilson formulation (4~q') never vanishes on a finite lattice, the relevant quantity to consider as chiral symmetry order p a r a m e t e r is
A (t~qj) -= ( ~ 0 ) - ( ~ 9 ) w c , (19) where in ( ~ 0 ) w c the weak coupling limits* for Re £ and the plaquette averages of ref. [21] are used. Chiral symmetry restoration is a non-perturbative effect, and by subtracting the perturbative form with our K (i.e. for finite mass), we can obtain an idea of where the restoration occurs.
The aim o f our study will now be the numerical determination of the energy density (15)/(16), the average of the thermal Wilson loop (11), and the chiral symmetry measure (19).
3. Numerical results
The m a j o r part of our results is obtained on an 83x3 lattice, with about 2000 lattice sweeps (iterations) per value of the coupling; of these, the first 500 are generally discarded.
In fig. 2 we show the behaviour of the gluon sector energy density (15) as a function o f
6/g 2,
together with the Stefan-Boltzmann result on an 83 x 3 lattice [20].Also shown is the corresponding result for the pure gauge theory [22]. We note that the transition now occurs at a lower value of 6/g2; this is to be expected because of the modified renormalization group relation (13), if Tc is not significantly changed.
In contrast to the pure gauge case, the transition now also has b e c o m e continuous.
Finally we note a rather sizeable overshoot in the transition region. This is, as we shall see more clearly in a moment, due to the Re/S part of the effective action (9) ; it causes a more complete alignment of the U than would be the case in the pure gauge theory.
The energy density o f the quark sector with Nj = 2, shown in fig. 3, still falls short of the limit (17); the reason for this is that Re/S attains its asymptotic limit only
* The results of ref. [21] correspond to the quenched case; we do not expect significant changes in eq.
(19) when the determinant is included.
676
eG/T 4
15
10
T. (~elic et at. / Finite-temperature lattice Q C D
SB
{ ,} {
¢0 } s.s 6!0 61s
6 / g 2
Fig. 2. T h e e n e r g y d e n s i t y e G for the g l u o n sector, c a l c u l a t e d on a n 8 3 x 3 lattice in the f o u r t h - o r d e r h o p p i n g p a r a m e t e r e x p a n s i o n ( 9 ) . A l s o s h o w n are t h e p u r e g a u g e t h e o r y r e s u l t ( x ) a n d the i d e a l g a s
l i m i t ( d a s h e d line) o n t h e s a m e size lattice.
rather slowly, as is seen in fig. 4. We might note at this point that within o u r error bars I m / ~ is essentially zero, even below the d e c o n f i n e m e n t point, where it is a b o u t r6 or less o f Re/7.. 1
C o m b i n i n g eG a n d eF, we obtain the b e h a v i o u r o f the overall e n e r g y density. It is s h o w n in fig. 5 as f u n c t i o n o f the t e m p e r a t u r e T / A L , o b t a i n e d with the help o f the r e n o r m a l i z a t i o n g r o u p relation (13). T h e S t e f a n - B o l t z m a n n limit in fig. 5 is the sum o f the c o r r e s p o n d i n g gluon a n d q u a r k sector forms. It thus includes the effect both o f finite lattice size a n d o f f o u r t h - o r d e r h o p p i n g p a r a m e t e r truncation. The latter only results in a 5% r e d u c t i o n o f the c o m p l e t e series.
To o b t a i n s o m e feeling for the size o f e in the c o n f i n e m e n t region, we also include in fig. 5 the e n e r g y density o f an ideal gas o f at, p a n d to mesons, using for illustration p u r p o s e s A IN..' ~2 = 1.5 M e V for the lattice scale in physical units. We see that such an ideal m e s o n gas i n d e e d leads to an e o f the same order o f m a g n i t u d e as we obtain f r o m o u r Q C D evaluation.
We have already s h o w n in fig. 4 the average thermal Wilson loop Re/7.; at the end o f the d e c o n f i n e m e n t regime, it falls rather rapidly to a very small value. T h e b e h a v i o u r o f the c o r r e s p o n d i n g fluctuation is given by the susceptibility
X -= { L 2 - ~
L}N,,;
3 (20)and is s h o w n in fig. 6. T h e a p p a r e n t strong increase at the critical point, together
~.F/T 4
T. ~?elic et al. / Finite-temperature lattice Q C D
SB
15
10-
5
0 I I I
5.0 6.0 7.0
61g 2
20
677
Fig. 3. The energy density e G for the fermion sector, calculated on an 8 3 x3 lattice in the fourth-order hopping parameter expansion (O), together with the ideal gas limit on a lattice of the same size (dashed
line).
with the r a p i d b u t s m o o t h v a r i a t i o n o f b o t h /_7, a n d e suggests a s e c o n d - o r d e r t r a n s i t i o n . C o m p a r i n g t h e results o f hot a n d c o l d starts a r o u n d t h e d e c o n f i n e m e n t p o i n t , we h a v e in fact n o t o b s e r v e d a n y t w o - s t a t e signal.
F i n a l l y we preoent in fig. 7 the b e h a v i o u r o f the c h i r a l s y m m e t r y m e a s u r e (19), t o g e t h e r w i t h t h a t o f /S. D e c o n f i n e m e n t a n d c h i r a l s y m m e t r y r e s t o r a t i o n are t h u s seen to o c c u r at the s a m e p o i n t .
Before we s t u d y the effect o f h i g h e r o r d e r s a n d l a r g e r lattices, let us n o t e here t h a t the b e h a v i o u r we h a v e o b s e r v e d for W i l s o n f e r m i o n s in f o u r t h - o r d e r h o p p i n g p a r a m e t e r e x p a n s i o n a g r e e s in all p o i n t s with t h a t o b s e r v e d for K o g u t - S u s s k i n d f e r m i o n s b o t h in the m i c r o c a n o n i c a l [ 13] a n d in the c a n o n i c a l [ 11, 12] f o r m u l a t i o n . T h e t r u n c a t i o n o f the h o p p i n g p a r a m e t e r e x p a n s i o n (8) at ! = 4 is c l e a r l y a r a t h e r d r a s t i c m e a s u r e . To o b t a i n at least s o m e i d e a o f the effect o f h i g h e r - o r d e r t e r m s , we have r e p e a t e d o u r c a l c u l a t i o n s i n c l u d i n g the n e x t (K 5) o r d e r , a g a i n on an 8 3 x 3 lattice. In fig. 8 we s h o w the r e s u l t i n g c h a n g e b o t h in e a n d in i ; it d o e s not s e e m
678
1.5
T. ('elic et al. / Finite-temperature lattice Q C D
1.0
0.5
0 I I 1 I
5.0 5 5 6.0 6.5
6 / g 2
Fig. 4. T h e a v e r a g e o f t h e t h e r m a l W i l s o n l o o p o n an 8 ~ x 3 lattice as a f u n c t i o n o f 6 / g 2.
I~/ESB
1.0
0.8
0.6
0.4
0.2
o ¢ o
0 0
- 0
_ O f
J
I I I I I I I I
80 100 150
I I I I I I
200 250 300 400 500 600
T/AN, :z
Fig. 5. T h e t o t a l e n e r g y d e n s i t y e, n o r m a l i z e d to the i d e a l g a s l i m i t esa, as a f u n c t i o n o f t h e t e m p e r a t u r e , c a l c u l a t e d o n a n 83 x 3 l a t t i c e in t h e f o u r t h - o r d e r h o p p i n g p a r a m e t e r e x p a n s i o n . A l s o s h o w n are the i d e a l g a s l i m i t s for the q u a r k - g l u o n p l a s m a ( d a s h e d l i n e ) a n d for a s y s t e m o f ~r, p a n d to m e s o n s ( s o l i d
line).
X
2 B
B
0 4.5
I I
5.0 5.5
6 / g 2
T. (~elic et al. / Finite-temperature lattice QCD 679
I 6.0
Fig. 6. The susceptibility X in the fifth-order hopping parameter expansion, calculated on an 8 ~ x 3 lattice.
to effect o u r c o n c l u s i o n s . W e e x p e c t the role o f h i g h e r - o r d e r t e r m s to b e m o s t crucial in the d e t e r m i n a t i o n o f the a c t u a l v a l u e o f the critical c o u p l i n g ; since the i n c l u s i o n o f h i g h e r o r d e r s results in a shift to s m a l l e r 6/g2o o u r f o u r t h - o r d e r results p r o v i d e an u p p e r b o u n d for the critical t e m p e r a t u r e . W e shall see s h o r t l y t h a t this b o u n d is not m u c h a b o v e the c o r r e s p o n d i n g Tc v a l u e s in o t h e r f e r m i o n s c h e m e s [11-13].
zt( >a s -E
1.5
B
0.06 -
0.04 -
0.02
0 I
5.0
1.0
0.5
I I 0
6.0 7.0
6 1 g 2
Fig. 7. Chiral symmetry measure (O) and the average thermal Wilson loop (O), calculated on an 83 ×3 lattice in the fourlh-order hopping parameter expansion.
/ESB
1.0
0 5
E
1.5
1.0
0.5
0
T. ~elic et al. / Finite-temperature lattice Q C D o )
• • 0
° 0 0
0 o
0 6.0 | 7.0 !
6 / g 2 0
O o O
1 5.0
b) 680
1
5.0 6 0 7 0
6 / g 2
Fig. 8. C o m p a r i s o n o f results in the fourth ( 0 ) and fifth ( 0 ) order h o p p i n g parameter expansion on an 8 3 x 3 lattice, for the overall energy density e/esB (a) and for the average thermal Wilson loop (b).
T. ~elic et al. / Finite-temperature lattice QCD 681 In fig. 9, we t h e n s h o w t h e b e h a v i o u r o f eG, as c a l c u l a t e d o n lattices o f size 83 x 3, 8 3 x 4 a n d
103×5,
a n d n o r m a l i z e d in e a c h c a s e to the c o r r e s p o n d i n g S t e f a n - B o l t z m a n n form. In fig. 10, the results for/7, on the s a m e set o f lattices a r e p r e s e n t e d . T h e shift in the critical c o u p l i n g with i n c r e a s i n g N o is c l e a r ; the v a l u e s o f 6/g:c, t o g e t h e r with the r e s u l t i n g critical t e m p e r a t u r e s , are listed in t a b l e 1. W i t h i n o u r e r r o r m a r g i n s , we t h u s d o n o t find a n y n o t a b l e d e v i a t i o n f r o m scaling.The rise o f Ecs a b o v e the S t e f a n - B o l t z m a n n limit, w h i c h we h a d a l r e a d y n o t e d b e f o r e , is n o w seen to d e c r e a s e with g r o w i n g N o . T h e t e c h n i c a l r e a s o n for the o v e r s h o o t t h u s b e c o m e s e v i d e n t : the p r e s e n c e o f R e / ~ in the effective a c t i o n c a u s e s a m o r e c o m p l e t e o r d e r i n g on t e m p o r a l t h a n on s p a t i a l links. As a result, g'0 is r e d u c e d in c o m p a r i s o n to the q u e n c h e d case, Po not so m u c h . W i t h i n c r e a s i n g 6 / g 2, all links a c q u i r e o r d e r e d U, a n d eG r e t u r n s to its S t e f a n - B o l t z m a n n limit. W h e n N o is i n c r e a s e d , the effect o f Re [ in the effective a c t i o n is d o u b l y r e d u c e d : the f a c t o r
(~/ESB)G
1.5
1.0
0.5
-0.5
t t ...
I I
T
I I5.0 5.5 6.0 6.5
6 / g 2
Fig. 9. Comparison of the gluon sector energy density values calculated on an 83 ×3 (O), an 83 x4 ( × ) and a 103 x5 (A) lattice, normalized in each case to the ideal gas result on a lattice of the same size.
682 T. ~elic et al. / Finite-temperature lattice Q C D
-E
1.5
1.0
0.5
0
0 0 0
o o
x x O
x
o •
o
x
x •
0 I x i • I I I
5.0 5.5 6.0 6.5 7.0
6 / g 2
Fig. 10. C o m p a r i s o n o f the average thermal Wilson l o o p values on 83 x 3 (O), 83 x 4 ( x ) a n d 103 x 5 ( & ) lattices.
K N~ b e c o m e s smaller, and - as seen from fig. 10 - the value o f £ in the deconfinement z o n e decreases. This decrease o f / S with N~ is k n o w n from pure gauge theory [23];
it is due to the fact that asymptotically divergent point-source contributions for the colour charge still have to be removed to obtain a physically meaningful observable.
In pure gauge theory, this is possible in the weak coupling limit [21]. Here, with a finite number o f terms in the hopping parameter expansion, it is not clear h o w this could be carried out ir~the action. Hence, in the transition region the energy densities calculated at ditterent N~ will in general not coincide as functions o f T. It would be interesting to see if other fermion schemes can avoid this problem, or if it is a general normalization difficulty for the quark-quark interaction on the lattice.
Having seen that the critical temperature is rather insensitive to the temporal lattice size, let us return to its dependence on the order o f the hopping parameter expansion. We had already observed that higher order implied smaller
6/g2c.
InTABt.E 1
Critical couplings and temperatures
3 5.300+ 0.050 152 ± 10
4 5.575 ± 0.025 162 ± 5
5 5.725 ± 0.025 157 ± 5
T. ~elic et al. / Finite-temperature lattice Q C D
TABI,E 2
Critical parameters in different fermion schemes
683
Scheme N o N r 6/g~ Tc/,t ° Ref.
Wilson K 4 3 2 5.30+0.05 1 3 1 + 8 ]
Wilson K s 3 2 5.25 ±0.05 123 ± 8~ this paper
KS, canonical 4 3 5.3 100 11
KS, canonical 2 2 4.6 89 12
KS, micro-canonical 4 4 5.1 106 13
table 2 we list our results on the 83 ×3 lattice for fourth and fifth order of the hopping p a r a m e t e r expansion, together with the results from other fermion schemes. To allow a comparison, we convert all critical couplings to temperature values Tc/A °, where A ° is the lattice scale for N/-= 0; its relation to the A ~, with Nj ~ 0 is known perturbatively [23], and we have used this in the conversion. It is seen that our values tend to lie about 20% above the average value obtained in other schemes.
This gives us some indication of the effect of the truncation in K t. It must be kept in mind, however, that the perturbative relations between A ~, and A ° are at present coupling values very likely not reliable; hence we do not believe that the TffA ° values in table 2 can be used to obtain Tc in physical units by using pure gauge theory values for A °.
Finally, we want to return to the size o f the deconfinement measure /.T. in the region below To. As we have seen in figs. 4 and 10, /T is indeed very small there, so that the Z3 symmetry a p p e a r s to be only weakly broken. To test this in more detail, we have studied the distribution of L(x) values over the lattice in different equili- brium configurations; the results on an 83 ×4 lattice are shown in fig. 11. We see in fig. 1 la that in the deconfined state, the symmetry is clearly broken, while in the confinement regime, as shown in fig. 1 lb, the symmetry is only slightly perturbed.
Note that the lattice average of It(x)l is practically the same in the two cases: it is really the near-restoration of Zs symmetry which leads to the small L v a l u e below To.
4. Conclusions
We thus find that in the presence of dynamical quarks deconfinement persists as a transition p h e n o m e n o n between two distinct regimes - and this is as clearly evident here as it is in pure gauge theory. The transition brings the system from a temperature region in which the global Z3 symmetry is almost unbroken to one where it is completely absent. At the same point, chiral symmetry is essentially restored. In the Wilson scheme, the functional connection between deconfinement and chiral sym- metry restoration is particularly transparent; as can be seen in eq. (18), it is the rapid change of E which drives the chiral symmetry restoration in addition to signalling the onset o f deconfinement.
684 T. ~elic et al. / Finite-temperature lattice Q C D r-] 1 - 9
~- [ ] 10-19
_ _ • 2 0 - 3 0
b)
= R e L
[ ] 1-14 [ ] 15-29 30-50
R e L
F 1
o)Fig. 11. The distribution of thermal Wilson Loop values in the complex L plane for one equilibrium configuration on an 83 x4 lattice. Part (a) is above To, at 6 / g 2 =6.0, where I[I= 1.08: part (b) is below
To, at 6 / g 2 = 5.4, with ILl = 0.89.
T h e c h a n g e in e n e r g y d e n s i t y we observe at d e c o n f i n e m e n t agrees well with that e x p e c t e d w h e n g o i n g f r o m a n ideal m e s o n gas to a n ideal c h r o m o p l a s m a . T h e c h a n g e o c c u r s over an i n t e r v a l o f 30 A ~J ~2 or less, i.e. for s o m e t h i n g like a 45 MeV increase in t e m p e r a t u r e * . All o u r results i n d i c a t e a c o n t i n u o u s t r a n s i t i o n - b u t in view o f the t r u n c a t e d h o p p i n g p a r a m e t e r e x p a n s i o n we c a n n o t really e x c l u d e a first-order t r a n s i t i o n .
N e i t h e r /S n o r A ( ~ / , ) r e p r e s e n t g e n u i n e o r d e r p a r a m e t e r s in o u r f o r m u l a t i o n ; /S b e c a u s e o f the Z3 s y m m e t r y b r e a k i n g b y d y n a m i c a l q u a r k s , A ( ~ 0 ) b e c a u s e we are n o t c a l c u l a t i n g at zero q u a r k mass. T h e s e q u a n t i t i e s s h o u l d t h e r e f o r e p r o v i d e us with s o m e i n f o r m a t i o n a b o u t the physics u n d e r l y i n g the respective s y m m e t r y b r e a k - ing. In the chirally s y m m e t r i c state, A ( ~ 0 ) s h o u l d give s o m e i n d i c a t i o n o f the effective q u a r k mass at the c o u p l i n g in q u e s t i o n . It is n o t clear to us h o w this c a n be e x t r a c t e d from o u r d a t a or from that o b t a i n e d in o t h e r f e r m i o n schemes. T o
* Here, as in figs. 5 and 12, we have to relate A~, -2 to physical units. For illustration purposes, we have taken A~ ",'2= 1.5 MeV.
T. (~elic et
al.
/ Finite-temperature lattice Q C D 685mH//1Nf
=2 tO00800
600
400
i
200 -
0 I I I I I I
50 80 120 160
T/AN~=Z
Fig. 12. Values ofthe effective hadron mass m R obtained from the average thermal Wilson loop, compared to the average hadron mass in an ideal resonance gas with 7r, p and to (solid line).
o b t a i n s o m e i d e a o f the q u a r k m a s s v a l u e s c o r r e s p o n d i n g to o u r v a l u e s o f K(g2), we t h e r e f o r e c o n s i d e r the a p p r o x i m a t i o n [25]
- = e x p ( m q a ) - 1 , (21)
w h e r e Kc is the ( n u m e r i c a l ) K-value o b t a i n e d at the p o i n t w h e r e t h e p i o n m a s s v a n i s h e s . U s i n g the results o f refs. [25, 26] as s h o w n in fig. l, we o b t a i n mqa---0.31 a n d t h u s mq/Tc = 0.94.
As i n d i c a t e d a b o v e , /S in the c o n f i n e d r e g i m e s h o u l d tell us s o m e t h i n g a b o u t the m a s s o f t h e d o m i n a n t h a d r o n state f o r m e d w h e n we try to b r e a k a string. In fig. 12, we s h o w t h e m a s s v a l u e s mR o b t a i n e d f r o m o u r /S d a t a using eq. (2), t o g e t h e r with the a v e r a g e m a s s in a m e s o n gas o f the zr, p a n d to. A c c o r d i n g to d u a l m o d e l s [27]
o r s t a t i s t i c a l b o o t s t r a p a r g u m e n t s [28], ff~H s h o u l d i n c r e a s e as we a p p r o a c h T~. Fig.
12 is s e e n to b e at least q u a l i t a t i v e l y in a c c o r d with o u r i n t e r p r e t a t i o n o f / S in the c o n f i n e m e n t r e g i o n .
It is a p l e a s u r e to t h a n k R.V. G a v a i , P. H a s e n f r a t z , F. K a r s c h , J. P o l 6 n y i a n d A.
U k a w a for s t i m u l a t i n g d i s c u s s i o n s , a n d t h e B o c h u m C o m p u t e r C e n t e r for p r o v i d i n g us with t h e f a c i l i t y ( C Y B E R 205) a n d c o m p u t e r time.
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