Flavour degrees of freedom and the transition temperature in QCD

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Volume 241, number 4 PHYSICS LETTERS B 24 May 1990

F L A V O U R D E G R E E S O F F R E E D O M A N D T H E T R A N S I T I O N T E M P E R A T U R E I N

QCD

MT~ C o l l a b o r a t i o n

R.V. GAVAI a, S o u r e n d u G U P T A b, A. IRB~,CK b, F. K A R S C H c, S. M E Y E R d, B. P E T E R S S O N b, H. S A T Z c a n d H.W. W Y L D ~

a TIFR, Homi Bhabha Road, Bombay 400 005, India

b Fakultfity~#r Physik. Universitfit Bielefeld, D-4800 Bielefeld l, FRG c Theory Division, CERN, CH-1211 Geneva 23, Switzerland

Fachbereich Physik, Universitfit Kaiserslautern, D-6 750 Kaiserslautern. FRG e Department o f Physics, University oflllinois, Urbana, IL 61801, USA

Received 1 March 1990

We study the finite temperature transition in QCD with four flavours of dynamical quarks. Our simulation is performed on an 8 × 163 lattice for quarks of mass ma = 0.01. At the coupling tic = 5.15, we find long-lived metastable states, as well as an abrupt change in the entropy density, suggesting a first order phase transition. Using presently available results from corresponding calculations of the hadron masses and the chiral condensate, this gives us a transition temperature Tc-~ 100 MeV, which is a factor two lower than the transition temperature found in quenched QCD.

The most striking feature in strong i n t e r a c t i o n t h e r m o d y n a m i c s is the p r e d i c t e d t r a n s i t i o n from h a d r o n i c m a t t e r at low t e m p e r a t u r e to a p l a s m a o f d e c o n f i n e d quarks a n d gluons at high t e m p e r a t u r e . In the case o f pure S U ( 3 ) gauge field t h e r m o d y - namics, this t r a n s i t i o n has been extensively s t u d i e d a n d a p p e a r s to be quite well u n d e r s t o o d : there is a first o r d e r phase t r a n s i t i o n from the Z3-symmetric confined phase to a d e c o n f i n e d phase with sponta- neously b r o k e n Z3 s y m m e t r y . Using results from string tension calculations [ 1 ], one finds for the transition t e m p e r a t u r e T c / x / a = 0.58 _+ 0.04. W i t h a string tension o f x / a - ~ 4 4 0 M e V [ 2 ] , this i m p l i e s a decon- f i n e m e n t t e m p e r a t u r e T o - ~ 250 MeV. W h a t h a p p e n s in full Q C D , when q u a r k s are included? The presence o f d y n a m i c a l q u a r k s m a k e s the d i s s o l u t i o n o f b o u n d states " e a s i e r " , since they can directly screen the col- our charge o f the b i n d i n g partners. However, their presence breaks the Z3 s y m m e t r y o f the lagrangian and hence could weaken the a b r u p t n e s s o f deconfinement. On the other hand, for v a n i s h i n g quark mass the lagrangian b e c o m e s chirally s y m m e t r i c , so that in this l i m i t we can have a phase t r a n s i t i o n restoring the chiral s y m m e t r y , which is s p o n t a n e o u s l y b r o k e n in the low t e m p e r a t u r e phase.

First studies with d y n a m i c a l quarks on relatively small lattices and with fairly large quark masses show an astonishingly strong flavour dependence. There are hints that the transition temperature, if converted into physical units by utilising h a d r o n mass calculations at the same coupling, decreases as the n u m b e r o f f l a - vours is increased, from a b o u t 200 MeV for N f = 0 to values a r o u n d 100 MeV for N~-= 4 [ 3 - 6 ]. This has a d r a m a t i c effect on the critical energy d e n s i t y n e e d e d for the transition. F o r an ideal gas o f quarks and gluons, the energy density is given by

¢ = ~ r ~ 2 ( 8 + ~ N f ) T 4 .

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Hence ~ d r o p s at the critical p o i n t by m o r e than a factor 4 between N f = 0 and N f = 4. It thus seems that the results for critical t e m p e r a t u r e o b t a i n e d for N f = 0 are not directly a p p l i c a b l e to the "real w o r l d " s t u d i e d in h e a v y ion collisions; for an e x p e r i m e n t a l q u a r k m a t t e r search, a clarification o f the f l a v o u r d e p e n - dence o f Tc is clearly very i m p o r t a n t .

At the t r a n s i t i o n point, a further flavour d e p e n - dence appears: for Nf>~ 2, the chiral t r a n s i t i o n seems to be o f first o r d e r for q u a r k masses below a critical value m c ( N f ) . This critical q u a r k mass seems to in-

0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland ) 567

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V o l u m e 2 4 I, n u m b e r 4 P H Y S I C S L E T T E R S B 24 M a y 1990

crease as the number of flavours is increased.

In contrast to this, the ratio

Tc/A~rs

seems to vary little or not at all when Nf is increased [ 7 ].

For an understanding of deconfinement and chiral symmetry restoration in full QCD, the flavour dependence of the transition thus becomes of essential importance. Most of the calculations of the transition temperature carried out so far were performed at couplings well below the asymptotic scaling regime, with comparatively large quark masses, and on lattices of volume

V=N, XN 3

with No<~ 10, N~=4-6 [3]. It is therefore necessary to push simulations closer to the continuum regime, keeping at the same time the physical quark mass in units of the temperature as small as possible; however, already the ex- tension to N~=8 leads to enormous computational effort, if one attempts to keep the quark mass small in units of the temperature [ 4 - 6 ] . Most useful are calculations at several small quark mass values, al- lowing an extrapolation to vanishing quark mass.

We have therefore carried out simulations with N f = 4 on an 8×163 lattice, for quarks of mass rna=0.01. In a preliminary study [5], we bad con- sidered the corresponding case for m a = 0 . 0 2 5 on an 8 × 123 lattice, so that we can combine the results for the two investigations in order to estimate the zero quark mass limit. The quark mass m a = 0 . 0 1 corre- sponds to

m/T=O.08;

previous calculations [8] at NT=4 for

m/T=O.1

have shown a first order transition. Finally, the choice of N f = 4 allows the use of an exact algorithm for the simulation, thus eliminating systematic errors in the determination of the critical coupling.

The partition function for four flavour QCD can be written in terms of the usual gluonic link matrices

Uand the Wilson action S ( U ) as

Z= f dUdetQ(U)

e x p [ - S ( U ) ] , (2) where

Q ( U)xy =

madly

• , - Ux_,,,dx,y+~) (3)

kt

is the staggered fermion matrix with phase factors eL,,= ( - 1 )x~ +...+x,_,. We have simulated this partition function with the hybrid Monte Carlo algorithm [9]. Our implementation of this algorithm was dis-

cussed in ref. [ 5 ]. Even for the large lattice and small quark mass used in this study, we found that a rea- sonable acceptance rate (of about 70%) could be achieved without making the step size impractically small. We have used Ar=0.004-0.0125, and ad- justed the number of steps in each trajectory so that its length r=0.3. The stopping criterion in the con- jugate gradient inversion of the matrix Q was that the norm of the residual vector be less than 1.25 × 10- ~ V.

Further details about the performance of the algorithm, including a formula giving the dependence of the acceptance on the coupling, quark mass, volume and step size, are given in ref. [ 10].

In order to look for critical behaviour, we studied the evolution of the Polyakov loop R e ( L ) and the chiral condensate,Of at the couplings fl= 5.1, 5.15, 5.2 and 5.3. Here

1 Nr

L=~7~-~3 ~ T r l~

U(,o,,),o,

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N a n o = l

and

77,Z= ~- r Q - ' 1 T (5)

Fig. 1 shows the run time histories for R e ( L ) . From an ordered start at fl= 5.1, the system attained equi- librium in a disordered state with Re (L) close to zero.

Atfl= 5.2, the runs from ordered and disordered starts converged to the same value Re(L) > 0. We then used

.2 _I _I _I

.z I L I

500 1 Of 30 I DO0 g(YO0

ntraj

Fig. 1. T h e r u n t i m e h i s t o r i e s f o r R e ( L ) at t h e i n d i c a t e d v a l u e s o f ft.

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Volume 241, number 4 PHYSICS LETTERS B 24 May 1990 e q u i l i b r i u m c o n f i g u r a t i o n s from runs at 5.1 a n d 5.2

as " d i s o r d e r e d " and " o r d e r e d " starts at f l = 5.15. Even after n e a r l y 2000 t r a j e c t o r i e s d i d they not converge to a c o m m o n value. We take this as an i n d i c a t i o n for long lived m e t a s t a b l e states at f l = 5.15, suggestive o f a first o r d e r phase transition. A similar b e h a v i o u r was seen in the run t i m e histories for 2Z (fig. 2). In the s i m u l a t i o n s at the mass m a = 0 . 0 2 5 r e p o r t e d in ref.

[5] we d i d not observe such m e t a s t a b i l i t i e s . F o r N ~ = 4 , in contrast, there are m e t a s t a b l e states for

m a = 0.025, a n d a s m o o t h t r a n s i t i o n for m a = 0.05.

This is consistent with the end o f the first o r d e r transition line scaling with m c / T c . It should be noted, however, that only a m u c h m o r e extensive investiga- tion can d e f i n i t i v e l y establish the o r d e r o f the transition.

In fig. 3 we show the b e h a v i o u r o f the averages,

< Re ( L ) > a n d <Z:(), as o b t a i n e d after d i s c a r d i n g the t r a n s i e n t parts o f the runs. F o r those cases in which the system was started in the phase in which it even- tually c a m e to e q u i l i b r i u m , this m e a n t d i s c a r d i n g ap- p r o x i m a t e l y the first 1000 trajectories; when crossing f r o m one phase to a n o t h e r was involved, a b o u t 1500 t r a j e c t o r i e s were d i s c a r d e d . Also shown in fig. 3 are the results o f our earlier work at m a = 0.025.

At f l = 5.1, a linear e x t r a p o l a t i o n o f the values o f (ZZ) o b t a i n e d at the two masses (fig. 4) gives a finite value at m a = O , whereas for f l = 5.2 a n d 5.3 the e x t r a p o l a t i o n s lead to vanishing, or even negative, values. This shows that for m a = 0.01, the point fl= 5.1

4

. 2

.4

, 2

.4

. 2

P I T

F

8=5 15

I I I

~ = 5 . 2

5 0 0 1OOO 1 5 0 0 2 0 0 0

n t r a j

Fig. 2. The run time histories f o r ~ at the indicated values of ft.

O, 20

0 , 1 5

/k

v

O. O:

(a)

[ I I I I I

o

~ .~o

~ 1 I I I I

5 , 0 5 . 1 5 . 2 5 . 3 5 . 4 5 . 5 5 . 6 P

0.6

0,5

0 . 3

0 . 2

0 . 1

] 1 1 1 1 1 1 1

(b) o

o

{

o

• o

o I

I I I I ] I I

5 . 0 5 . 1 5 . 2 5 . 3 5 . 4 5 . 5 5 . 6

Fig. 3. The averages (a) <Re(L)> and (b) <ZZ). The filled circles are for data obtained on the 8X 163 lattice with ma=0.01, whereas the open circles are for data with ma = 0.025 on an 8 X 123 lattice.

belongs to a phase with s p o n t a n e o u s l y b r o k e n chiral s y m m e t r y , whereas for fl>~ 5.2 there is no such sym- m e t r y breaking. The slightly negative zero-mass ex- trapolations at fl= 5.2 and 5.15 ( d i s o r d e r e d start) are readily u n d e r s t o o d from the d e p e n d e n c e of/~c on the q u a r k mass: for m a = 0 . 0 2 5 , / ~ = 5.2 is in the b r o k e n s y m m e t r y phase, whereas for m a = 0 . 0 1 it is in the s y m m e t r i c phase. F o r the s a m e reason, the extrapo- 569

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.6

.4

/

.2

/

o /

o

i i 5 .

f f

• 0 1 . 0 2

g l a

03

Fig. 4. (;?30 obtained at ma=0.025 [5] and ma=0.01, plotted against ma. The values offl are as marked.

lation o f the d a t a at f l = 5.1 do not necessarily give us the actual value at r n a = O .

T u r n i n g now to ( R e ( L ) > , we note that the r a p i d change across the transition region seen on the 8 × 123 lattice persists also for 8 X 16 ~. T h e curves for the two lattice sizes are similar, < R e ( L ) ) being small a n d nearly zero at small fl, but clearly n o n - z e r o at larger coupling. A p a r t from this change o f scale, there is also a shift o f the whole curve towards lowerfl for the larger lattice. This b e h a v i o u r is entirely consistent with t h a t o f the phase t r a n s i t i o n between fl = 5.1 a n d 5.2 seen f r o m ( Z Z ) .

We have also m e a s u r e d the e n t r o p y density, which, for four flavours, is given by

s = 4 f i N 4 1 g2 [<p~>_<p~)]

T 3 2

4 4

+ 5 N , ( 1 + ¢ k g 2 )

(' 4+ )

× ~ < T r D o Q - ) + m a < T r Q - ~ ) - 3 , ( 6 ) where ( P ~ ) and <PT> are the e x p e c t a t i o n values o f the s p a c e - s p a c e and s p a c e - t i m e plaquettes, Do is the t i m e c o m p o n e n t o f the D i r a c o p e r a t o r on the lattice, a n d the c o n s t a n t s c~,, c'~ a n d c~ are o b t a i n e d f r o m a weak coupling c o m p u t a t i o n [ l 1 ] that yields the for- m u l a for the e n t r o p y d e n s i t y given above. We note that s / T 3 involves the difference o f two p l a q u e t t e av-

erages. T h i s m a k e s it a difficult m e a s u r e m e n t , as re- flected in the e s t i m a t e s o f errors. Nevertheless, the a b r u p t change at f l = 5 . 1 5 (fig. 5) is also consistent with a first o r d e r transition. In the high t e m p e r a t u r e phase the results are in a g r e e m e n t with an ideal gas o f q u a r k s a n d gluons.

In s u m m a r y , the present d a t a at fl = 5.1 a n d 5.2 establish lower a n d u p p e r b o u n d s for tic at m a = 0 . 0 1 . F o r the p u r p o s e o f the following discussion o f the critical t e m p e r a t u r e we therefore take

f l c ( m a = 0 . 0 1 ) = 5.15_+ 0.05 ( 7 ) as the critical coupling on the 8X 163 lattice. C o m - b i n i n g this result with that for m a = 0 . 0 2 5 , / 3 c ( m a = 0 . 0 2 5 ) = 5 . 2 5 0 + _ 0 . 0 2 5 [ 5 ] , we can extract the critical coupling in the zero mass limit. A l i n e a r e x t r a p o l a t i o n gives

flc( m a = O ) = 5 . 0 8 + O . 0 8 . ( 8 )

U s i n g the a s y m p t o t i c scaling f o r m u l a at the t w o - l o o p level, this gives us

T c ( m a = O ) _ 1.8+_0.2. ( 9 )

A~rs

T h e c o r r e s p o n d i n g value for SU ( 3 ) pure gauge the-

60

40

)

I q _ I I I I I

5,0 541 5 . 2 5.3 5.4 5.5 5.6

#

Fig. 5. The values of s~ T 3 as a function of ft. The filled circles are for data obtained on the 8 × 163 lattice with ma = 0.01, whereas the open circles are for data with ma = 0.025 on an 8 × 12 3 lattice.

The value of s I T 3 obtained for a free gas of quarks and gluons on a lattice of this size is marked by the line.

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Volume 241, n u m b e r 4 PHYSICS LETTERS B 24 May 1990

ory is 1.87_+0.04 [12]. The observation made at smaller N~, that T~/Ayes for the theories with N f = 0, 2 and 4 are nearly equal [ 7 ], thus seems to carry over to N~= 8. One should note, however, that higher loop corrections to the Tc/Asvs value are not yet under complete control. Even in the pure gauge theory, where data have been taken up to N~= 14, the functional behaviour o f T~/ATvrs in fl is as well described by the one-loop/?-function as it is by the two-loop function. Using the one-loop renormalisation group equation, one finds that Tc/Ayrs decreases by (12_+5)% when going from N~=8 to 14, while the two-loop formula gives a decrease o f ( 15 _+ 5 )%. For the two different forms, the value for Tc/A~rs, however, differs by a factor three. Existing data therefore, do not provide a sensitive test of the functional form b e y o n d the one-loop level. In particular, the exis- tence o f higher loop contributions which vary little over the presently probed region, but affect the con- t i n u u m value o f Tc/A~rs significantly, cannot be ruled out. Furthermore, before we can c o m p a r e the value o f Tc in pure gauge theory with that for four flavour Q C D , we must know the flavour dependence o f A Ms.

If we want to extract Tc in physical units without invoking the R G relation, and hence the assumption o f asymptotic scaling, then we should c o m p a r e Tc directly to another observable calculated on the lattice at T = 0 with the same quark mass and at the same coupling. Using hadron masses, we can construct the dimensionless ratio

T~ 1

m ~ ptl - - N~mHa " ( 1 0 )

At the critical coupling determined by us [eq. (7) ], such a calculation does not exist at present. For the m o m e n t we have to rely on an extrapolation based on calculations performed at nearby couplings with the same bare quark masses [4,13,14]. In particular the hadron mass calculations at t = 5 . 2 have been checked with different fermion algorithms, and seem to be rather insensitive to the particular integration scheme used. We obtain an estimate o f the critical temperature by a linear extrapolation o f the hadron masses, measured for m a = 0 . 0 1 at B = 5 . 2 and 5.35 [ 14 ], to our critical coupling. This yields at m a = 0.01,

r~ T~

= 0 . 1 2 _ + 0 . 0 2 , - - = 0 . 0 9 _ + 0 . 0 1 , ( l l )

m o m N

which is consistent with our earlier estimates for m a = 0.025 [ 5 ]. In contrast, the corresponding ratios for the quenched theory are more than a factor two bigger:

T c = 0 . 3 0 + 0 . 0 5 , Tc - 0 . 2 1 + 0 . 0 2 . (12)

m p m N

If the lattice spacing a is determined by inserting the experimental hadron masses in eqs. ( 11 ) and (12), then Tc = 1/N~a implies approximately the same lattice spacing for N f = 4 at N~= 8 as in the quenched theory for N~=4. It has, however, been observed in the quenched theory that these ratios do not vary significantly as N~ is changed from 4 to 14. We take this to be an indication that the difference between eqs.

( 11 ) and ( 12 ) is a general feature.

Inserting the experimental values of the hadron masses into eq. (11) we find that for four flavour Q C D at N~ = 8 and m a = 0.01,

T c = 9 3 + 1 5 M e V ( f r o m m p ) ,

= 8 5 _ + 1 1 M e V ( f r o m m N ) . (13) The error estimates given here take into account the uncertainties in our determination o f to, as well as the propagation o f the statistical errors in the masses on extrapolating to tic. In contrast, using the experimental hadron masses in eq. (12) results in a transition temperature around 200 MeV. We thus find that in four flavour Q C D the finite temperature phase transition is about a factor of two lower than in the quenched theory.

We can also perform a similar c o m p u t a t i o n using the chiral condensate (,?.Z) at T = 0 instead o f the hadron masses. Since this is a bulk quantity, finite size effects may influence it less than the correlation lengths 1/mH. For four flavour QCD, we can estimate {7~) at fl~(N~=8, m a = 0 ) = 5 . 0 8 _ + 0 . 0 8 from two different evaluations. One estimate comes from calculations at t = 5.2 and 5.35 [ 13,14 ]. Extrapolat- ing to zero quark mass and t = 5.08 gives (Z)f~)=

0.35_+0.06. The second estimate is based on results obtained at fl=5.1 on an 84 lattice at larger quark masses [ 15]. A linear extrapolation down to m a = O yields ( j j ) = 0.40_+ 0.06. Using the average o f these, we extract the renormalisation group invariant chiral condensate per quark flavour, (ZZ) R~[, as defined in ref. [ 16 ], to obtain

571

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(,~)I/3RGI

Tc = 0 . 4 + 0 . 1 . ( 1 4 )

C o m b i n i n g this with the estimate given in ref. [ 17 ], x/3 _ 1 9 0 + 2 0 MeV, we find that our c o m p u t a - tion at N~= 8 yields for rna=O in four flavour Q C D

Tc = 8 0 _ + 2 0 M e V . ( 1 5 )

In q u e n c h e d QCD, we can use the values of the chiral c o n d e n s a t e at f l = 5.7 a n d 5.9, corresponding to the critical couplings for N~= 4 a n d 6 respectively, from ref. [ 18 ]. Converting again to the R G i n v a r i a n t value, we get, instead o f e q . ( 1 4 ) , the ratio

Tc 0 . 8 2 + 0 . 0 3 ( 1 6 )

for N ~ = 4 a n d a 10% larger value for NT=6. We thus note that, like the ratios T c / x / ~ a n d T~/mH, this ratio too is fairly i n d e p e n d e n t o f N~. Hence we can com- pare this estimate with that given in eq. (14). We find, once more, a factor o f two between the values of Tc o b t a i n e d for four flavour Q C D a n d the q u e n c h e d theory.

In conclusion, we have d e t e r m i n e d the critical coupling tic = 5.15 4- 0.05 for the finite t e m p e r a t u r e transition in four flavour Q C D on an 8 × 163 lattice with d y n a m i c a l quarks of mass m a = 0.01. At this value of fl the q u a n t i t i e s ( R e ( L ) > , (Z;(> a n d s / T 3 change abruptly; within o u r statistics, their b e h a v i o u r is that expected of a first order transition. C o m b i n i n g our present results with those for larger quark mass, m a = 0 . 0 2 5 , we find that this t r a n s i t i o n separates phases of s p o n t a n e o u s l y b r o k e n a n d restored chiral symmetry.

U s i n g existing results from calculations of the hadron masses a n d the chiral c o n d e n s a t e in this region offl, we have two different ways of d e t e r m i n i n g Tc in physical units. Both result in Tc-~ 100 MeV for four flavour Q C D ; this value lies a factor two below that o b t a i n e d in the q u e n c h e d theory. Let us stress again, however, that n e i t h e r value can be applied directly to the physical world of two light a n d one or more heavy quark flavours.

The c o m p u t a t i o n s reported here were performed on the C R A Y - Y M P at H L R Z Jtilich. We are very grateful to Professor H. R o l l n i k for his k i n d support of the MTc project, a n d we would also like to t h a n k

the staff of the c o m p u t e r centre at Jtilich, in particular Professor F. Hossfeld a n d Dr. N. Attig, Dr. S.

K n e c h t a n d Dr. L. Wollschl~iger. F u r t h e r m o r e , we t h a n k our colleagues from the M part of the collabo- ration, in particular E. L a e r m a n n , for helpful discus- sions a n d support. We t h a n k the Deutsche For- schungsgerneinschaft for support u n d e r contract Pe 340/1-3.

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