Nuclear Physics B261 (1985) 273-284
© N o r t h - H o l l a n d P u b h s h l n g C o m p a n y
T H E SU(3) P H A S E T R A N S I T I O N S IN T H E P R E S E N C E O F L I G H T DYNAMICAL QUARKS
R V G A V A I
Department o f Phystcs, Brookhaven Natmnal Laboratory, Upton, Long Island, N Y 11973, USA
F K A R S C H
Department o f Physws, Umverstty o f llhnms at Urbana-Champmgn, 1110 West Green Street, Urbana, IL 61801, USA
Received 16 Aprd 1985
We mvesUgate the deconfinement and the chlral phase transitions m Q C D with 3 hght dynamical flavors, using the pseudo-fermlon m e t h o d Monte Carlo slmulaUons have been performed on a lattice o f size 83 x 4 with fermlons of m a s s 0 1 and 0 075 respectwely A rapid change from the low-temperature region o f h a d r o n s to the hlgh-temperature q u a r k - g l u o n p l a s m a is observed m all the physical quantlUes studied Our detailed, high-statistics results, however, do not show any signs o f a strong first-order transition In the zero mass h m l t we find evidence for a chlral p h a s e transition at Tc/A L ~- 183 + 10
1. Introduction
During the past few years considerable effort and computer time has been devoted to the study o f the thermodynamics o f quantum chromodynamics. The lattice simulations o f quenched Q C D [ 1 ] have reached a stage where the qualitative features like the thermodynamics in the low- and high-temperature phases and the order of the chtral and deconfinement phase transittons are well understood and numerically well u n d e r control. Quantitative results for the crlttcal temperature, latent heat and in part also critical exponents [2] are in a good shape and there relation to continuum parameters using the non-perturbattve features o f the S U ( N ) B-function [3] lead to results whtch are probably reliable on the 10% level.
In contrast to this the study of the influence of dynamical fermtons on the thermodynamics o f Q C D is still in an exploratory stage Although the results obtained so far [4-8] look very promising, they differ even on the quahtative level and are not able to predict continuum parameters with great confidence.
Probably one o f the most mterestmg questions, which consequently has been addressed first in the context of dynamtcal fermions, ts their influence on the deconfinement and chlral phase transttions. In the pure gauge sector these transitions are known to be first order [9] for SU(3). Theoretical considerations based on effecttve models in the strong coupling region suggest that dynamical fermions tend to weaken these phase transitions. This is what one observed m a MC simulatmn
273
2 7 4 R V Gavin, F Karsch / S U ( 3 ) phase transltmns
with very heavy quarks, where standard MC techniques are still a p p h c a b l e [10] In the light mass region, however, the results for these phase transitions were mconclus- we, ranging from a rapid crossover behavior [4] or a second-order phase t r a n s m o n [5, 6] to a strong first-order transition [7, 8] There has even been a claim of total disappearance [11] of the phase transition, although a lack of spontaneous break- down of the chlral symmetry at T = 0 in the model of ref [11] makes it difficult to compare w~th the above mentioned results. Recently most spectacular results have been presented in this context in ref. [8] where a strong first-order chiral and deconfinement transition has been reported to persist m the enUre mass range down to zero-mass fermions These results have been obtained by simulating the effect of 3 quark flavors using the pseudo-fermlon algorithm. They a p p e a r to be in disagree- ment with pseudo-fermlon results on a smaller lattice [6], and also w~th results obtained with Wilson fermions [5] Also mlcrocanomcal simulations gave no evidence for a strong first-order transltmn [4]
In this p a p e r we report the results of our detailed study of the t h e r m o d y n a m i c s of Q C D with 3 flavors of hght fermlons We use staggered fermions and employ the pseudo-fermlon algorithm to include the effect of hght quarks of mass ma = 0 075 and 0 1 on a 83 ×4 lattice A detailed analysis of the dependence of the results on the different parameters o f the pseudo-fermlon approxlmaUon scheme has been performed in order to clarify the discrepancies between the results o f different groups We will present evidence which suggests that the first-order signal observed m ref. [7] is most hkely due to lack o f convergence in the crossover region. We do not obtain any evidence for a strong dlscontmmty
The p a p e r is orgamzed as follows In sect. 2 we present the basic finite temperature f o r m a h s m and fix our notations. Sect. 3 reviews the pseudo-fermlon algorithm and discusses the various approximations introduced in order to make th~s method useful m an actual MC simulation. Sect. 4 contains our results and a comparison with earlier results of other groups and in sect 5 we present our conclusions
2. Lattice thermodynamics
The formalism of thermodynamxcs of euchdean lattices has been discussed exten- swely m the literature [12]. We will review here the basic features related to the introduction o f staggered fermlons m the formahsm.
The finite temperature partiUon function can be regularized by introducing a lattice of size N 3 x N , with lattice spacing a such that the volume and temperature of the system are given by (N~a) 3 and T -1=- N~a respectively. For a S U ( N ) gauge theory with staggered fermlons the p a r t m o n function then reads
Z(/3, V) = f H d U~.~, H dxx d)~ e -s(u'~'x) (2.1)
J x,~ x
R V Gavin, F Karsch / SU(3) phase transmons 275 with the euclidean action S given by
S( U, ,f, X) = S o ( U ) + SF( U, )?, X), (2 2) SG ='-~T ,~< ~ 1 - R e T r ( U x , uUx+~.~U;+~,~,U+~), (2.3) S v = m Z 2 x X x + ½ Z gxrl,(x)[U,~#,Xx+, - Uo,-,,~,Xx-~,], + (2.4)
x x,/~
being the gluonlc and fermlonic contributions to the action The fermiomc fields, )?, X are single component Grassmann fields defined on the sites of the lattice They also carry a flavor index whmh has been suppressed m eq (2 4) The phase factors
*l~(x) are defined as r h , ( x ) = ( - 1 ) x'+ +x , The action depends on the bare quark mass rn and the gauge coupling / 3 - - 2 N / g 2. After integrating over the fermionlc fields one obtains a partition function in terms o f the bosomc fields Ux,,, alone However, m a d d m o n one gets a highly non-local contribution from the fermmn determinant
Z = f [I dUx#,[det (m2-D2)]~j/Se -sG , (2.5)
d x,/z
where D --- ~ , D " and
D~xy = l'ot, ( x ) [ Ux, j.d~y,x+~ - Uy+,la. ~y,x-la. ] (26) In eq. (2.5) we have introduced the posmve defimte operator QQ+ where
Q = - m + D , (27)
whose determinant is equal to the square o f det (m + D) n s denotes the number o f flavors and is reqmred to be a multiple o f 4 for staggered fermlons. Following ref.
[13] we will use eq. (2.5) to simulate an arbitrary number of nf continuum flavors The thermodynamics and phase structure of the quark-gluon system can now be analyzed b y either looking at thermodynamic observables, which are expected to show singular behavior at the phase transmon temperature, or by looking at order parameter for various global symmetries o f the system.
In the following we will concentrate on an analysis of the energy density e = T Z v - I o In Z / O T
where
---- e c + eF, (2.8)
eG = 3/3 ((P,,) - (P,))
with P,.(.) = 1 - ( l / N ) Re Tr U~(.) denoting the space-space plaquettes. The "fermionic part" o f the energy density, eF, IS given by
e F = ]nf(tr D 4 ( D + m) -1) -- { l N n f -~m(~x)r=o}
(2 9) (space-time) like
(2.10)
276 R V Gavin, F Karsch / S U ( 3 ) phase transmons
The term m curly brackets in eq (2.10) comes from renormalizing the energy density by subtracting the zero temperature contributions. In the defimtion of the "gluonic part", eG, of the energy density we neglected contributions which result from the derivatives of the coupling with respect to the temperature [12] These contributions are m general expected to be small, of the order of a few percent
In the absence o f fermlons the gluonic part o f the action, SG(U), has in addition to its local gauge symmetry a global Z ( N ) symmetry, due to the finiteness of the euclidean lattice m time direction and the periodic b o u n d a r y conditions imposed in this direction An order p a r a m e t e r for the reahzation of this symmetry is the Polyakov line
L=-~-75-3 ~ R e T r U(x, x4L4 • (2 11)
N o - x 1
As the Polyakov hne is related to the excess free energy, F, of a static color source m the gluonic environment, (L) - exp { - F ~ T}, a non-vanishing value would indicate the a p p e a r a n c e of a deconfined phase. In the presence of dynamical fermions the Z ( N ) symmetry of the pure gauge action is exphcltly broken and thus ( L ) ~ 0 for all temperatures The Polyakov line is thus an order p a r a m e t e r for a deconfinement transitmn only in the pure gauge sector (or equwalently for infinitely heavy fermlons).
Nonetheless it is clearly of interest to study its b e h a w o r m the presence of dynamical ferm~ons also to contrast from the corresponding behavior in the pure gauge theory
In the zero mass limit the action, eq (2 2), has a flavor nonsmglet axial chlral symmetry for all values of lattice spacing It can be shown to be U ( n s) x U(ny) The order p a r a m e t e r to check whether this symmetry is spontaneously broken is given by
( ~ ) =- ()(X) = -~ny(tr ( D + m) -~) (2.12) 3. Simulation of dynamical fermions
The G r a s s m a n n nature of the fermlon fields reflects Itself in a highly non-local determinant once these fields have been integrated out In the past different approxl-
• I
matlon schemes have been suggested in order to deal w~th this determinant Presently the pseudo-fermlon algorithm [13-15] and the mlcrocanomcal method [16] are widely used and seem to be most promising In the following we wdl discuss m some detail the pseudo-fermion algorithm and the approxtmations revolved when implementing it in a M C simulation in order to make this method useful•
After integrating over the fermlon fields X, )? the partition function reads Z --- [ H d Ux,~, det Q e - S G (3.1)
d x,/~
with
det Q -= det (m + D ) = (det (m 2 - D2)) 1/2 . (3 2) The basic idea of the pseudo-fermlon method is to think of this determinant not as
R V Gavat, F Karsch / SU(3) phase transmons 277 resulting f r o m an integration over G r a s s m a n n fields but resulting from an integration over scalar fields. Actually what is needed in a MC simulation, using the Metropolis method, is not the whole determinant but its change under a change o f one link variable, Ux,~ --> Ux,~ + 6Ux,~ For small e n o u g h changes 8 U we find
det ( Q + 6Q )
= det (1 + Q-16Q) det Q
= 1 + T r Q-13Q+O(~U2) • (3.3) The matrix elements Qxy ~ can then be obtained from a MC simulation with scalar fields ~b, Q~y = ( Q+ Q);) Q=y -1 +
( Q ÷ Q ) = ~ : z f H d ~ x d d ~ x ~ y ~ x e x p { - ~ @ ( Q + Q ) t m ~ b m } l (3.4)
The task of evaluating the determinant o f Q for every hnk change is thus reduced to evaluating Q-1. A further i m p r o v e m e n t in the time taken per hnk update can be brought a b o u t by noting that all the relevant matrix elements of Q-~ may be calculated before a given sweep o f all the link variables and used for the entire sweep. For the errors induced by this procedure can be shown to be O(6U 2) and thus negligible m the a p p r o x i m a t i o n used m eq. (3.3) I f Npf denotes the n u m b e r of Monte Carlo iterattons over &-fields to obtain (Q+Q)~ty using eq. (3 4) then it is obvlous that the algorithm (and the procedure above) becomes exact in the hmlt N p f - ~ and 6 U ~ O One can thus easily recognize potential sources o f statlstmal a n d / o r systematic errors in any practical application of this method. Choosing too small Nor m a y lead to intolerable statistical errors in Q-1 which will be earned over m subsequent link updates whereas too large a change 6U m a y invalidate either the e x p a n s m n , Eq (3.3), or the procedure o f calculating Q-1 only once per update o f all links or even both. C o m p u t e r time requirements clearly prevent one from going to the other extremes where, in fact, the results will necessarily be more reliable. O f course, w~th too small a 6U one has to be cautious again The integration in eq. (3.1) over link variables runs over the entire group space A reliable estimate of averages for any observable m a y thus need increasingly large n u m b e r of iterations over the link variables as 6U~O.
In the following we will analyze in detail the dependence of the results obtained for the t h e r m o d y n a m i c s o f full Q C D on these parameters entering in the pseudo- fermion algorithm In particular we will show that they have to be handled especially carefully in a regmn of large correlatmn length in order to get conclusive results on the order of phase transitions m the presence of dynamical fermmns.
4. Results
In the following we will present our results for a MC simulation o f SU(3) gauge theory with 3 flavors o f staggered fermlons of mass ma = 0.1 and 0 075 on a lattice
278 R V Gavaz, F Karseh / SU(3) phase transmons
50
20
I0 E/T 4
{
{
O(g z) _o(gO)
SU(5) nf =5 mo=OI
I ~1 I I =_
51 ~ 55 5 4 B
) I ' I ' T/AL
180 200
Fig 1 Energy density versus couphng /3 for SU(3) with 3 flavors of mass rna =0 1 on a 83 x4 latttce Also shown are the lowest order ( - - - ) and O(g 2) (-- - - ) weak coupling perturbatlve results The temperature scale has been obtained by assuming the vahdlty of the asymptotic scahng relatmn eq (4 1) o f s i z e 8 3 × 4 T h e m a i n results are b a s e d on a p s e u d o - f e r m i o n (pf) s l m u l a t a o n with N p f = 50 i t e r a t i o n s m t h e p f u p d a t e , n e g l e c t i n g the first 25 for e q u i l i b r a t i o n . W e u s e d a h e a t - b a t h a l g o r i t h m to u p d a t e t h e p s e u d o - f e r m l o n s a n d a M e t r o p o l i s a l g o r i t h m w i t h 8 hits p e r h n k for t h e g a u g e fields. T h e m a x i m a l c h a n g e in t h e g a u g e fields a l l o w e d in a u p d a t e has b e e n ad}usted s u c h t h a t a n o v e r a l l a c c e p t a n c e rate o f - 6 3 % h a s b e e n a c h i e v e d . W e will c o m e b a c k l a t e r to t h e q u e s t i o n o f h o w o p t i m a l t h e s e c h o i c e s are a n d w h a t effect t h e y h a v e o n t h e final results.
I n figs 1 a n d 2 we s h o w o u r results f o r t h e e n e r g y d e n s i t y e a n d the P o l y a k o v h n e <L) at m a s s m a = 0,1 m the entire t e m p e r a t u r e r a n g e c o n s i d e r e d b y us As can b e s e e n b o t h q u a n t i t i e s c h a n g e r a p i d l y b u t s e e m i n g l y c o n t i n u o u s l y o v e r a s m a l l c o u p h n g r a n g e Al3 ~ 0 . 1 ( A T / A L = 3 0 ) , F o r l a r g e r c o u p l i n g s / 3 - - - 6 / g 2 the e n e r g y d e n s i t y a g r e e s well wtth w e a k c o u p h n g results [17] A t all c o u p h n g s o r d e r e d a n d r a n d o m start c o n f i g u r a t m n s h a v e b e e n a n a l y z e d to l o o k for m e t a s t a b l e states N o n e h a v e b e e n o b s e r v e d F i g 3 d i s p l a y s e v o l u t m n o f the real p a r t o f t h e P o l y a k o v line f r o m a r a n d o m (/3 = 5 2 q u e n c h e d , t h e r m a l i z e d c o n f i g u r a t i o n ) start a n d an o r d e r e d start (Ux,~, = 1, Vx, ~ ) a t / 3 = 5 3 O n e sees t h a t after = 8 0 0 i t e r a t i o n s the two starts c o m e t o g e t h e r a n d t h e r e a f t e r y i e l d the s a m e v a l u e a p a r t f r o m s t a t t s t l c a l f l u c t u a t i o n s . At all t h e c o u p l i n g s we s t u d i e d , we o b s e r v e d s i m i l a r b e h a v i o r , the o n l y difference b e i n g the n u m b e r o f l t e r a t t o n s r e q m r e d to c o n v e r g e t o g e t h e r , a w a y f r o m the critical r e g i o n t h e y d e c r e a s e d *
* The slowest convergence we observed was at/3 = 5 25 where the ordered start required -- 1500 iterations to catch up with the random start which was eqmhbrated after ~700 lteratmns
R V Gaval, F Karsch / SU(3) phase transitions 279 (L>
0.5
0.I
( ~ ~ ' ) m = o
' t I
l I
5.1 5 2 5 3
' I ' l
180 200
SU(3) nf =3
I 5 4
T/A L
Fig 2 The Polyakov hne expectation value (0) versus/~ for SU(3) with 3 flavors of mass ma = 0 1 on a 83 ×4 lattice and the zero mass extrapolated chlral order parameter ( I ) (~O)m=o has been obtained
from a linear extrapolation of data at ma = 0 075 and 0 1
In fig 2 we also s h o w the chiral o r d e r p a r a m e t e r ( ~ ¢ ) e x t r a p o l a t e d to zero mass A linear e x t r a p o l a t i o n f r o m o u r data for m a = 0.1 a n d 0 075 has b e e n m a d e to obtain these results. Clearly ( ~ 0 ) vanishes a r o u n d / 3 = 5 25 A s s u m i n g the validity o f the a s y m p t o t i c s c a h n g relation
f 4zr2/3 4 5 9 - 5 7 n y 87r2/3 ~ (4.1)
a A L = exp [ 3 3 - 2 n y (33 - 2 n y ) ~ In 33 - 2 n y J we find f o r the chiral transition t e m p e r a t u r e
T ~ , / A L = 1 8 3 + 10. (4.2)
As m the q u e n c h e d a p p r o x i m a t i o n , all the physical quantities we considered, n a m e l y e, (L) a n d (q~q,), exhibit a rapid c h a n g e in b e h a v i o r m a small interval o f A/3 This has b e e n a feature o f previous calculations [7] too. T h e difference w h i c h we find is the a p p a r e n t lack o f discontinuity m all o f them. I n this respect o u r findings are at least qualitatively m a g r e e m e n t with those o f ref. [6] where 63 × 2 lattice with ny = 2 was used a n d ref. [5] where a 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 has b e e n used. O u r results seem to indicate that the chiral p h a s e transition is c o n t i n u o u s c o n t r a r y to w h a t one w o u l d have expected f o r n s = 3 (and larger) by considering effective chiral m o d e l s [18]. It m a y be e m p h a s i z e d t h o u g h that all m e t h o d s to obtain
280 R V Gavin, F Karsch / S U ( 3 ) phase transmons
<L)
A _
- k / v , , y
/1 Npf = 50 (-25) /
^ f~ o r d e r e d s t a r t / v - - - - r a n d o m start I
p.~. jhv",, v _ I
O I " " t l
r-•. / /
i L i i I i i i i i , , -
5 0 0 I 0 0 0 # ,ter-
Fig 3 The Polyakov line versus number o f M C iteratlons Shown is the evo|utmn o f ( L ) from r a n d o m (- - -) and ordered ( ) start configurations at/3 = 5 3 The data have been averaged over 20 s u b s e q u e n t
iterations
(~70),,=o f r o m simulations on finite lattices necessarily involve extrapolations and a weak first order chiral phase transition could easily be buried in the errors of these extrapolations. Thus we certainly cannot rule out a weak fluctuation induced first-order phase t r a n s m o n [18].
We now turn to the discussion of the discrepanctes between our present work and that of ref. [7]. Those authors also used 83 x 4 lattice, n s = 3 and staggered fermlons but they chose to use Npr= 24, discarding 4 out of these to allow for equilibration and they adjusted the acceptance to be - 8 0 % . Since smaller the size of 8U the greater its probability of being accepted, their acceptance rate translates rata a smaller s~ze of ~ U than what we used. They presented evidence for strong first-order phase transmons: (0~) and (L) at m a = 0 1 showed d l s c o n t m u m e s and evolution o f (L) at fl = 5.3 (same c o u p h n g as the one used in our fig, 3) showed a two-state signal for 960 iterations Fucito et al., have recently extended th~s work to higher values of m a and found that for all o f them the first order character of the transition persists [8].
As we noted m the previous secuon a smaller step length 8 U is clearly better.
However, It m a y lead to problems with convergences, especially m the regions of large correlation length as m the wctmty o f / 3 = 5.3 in our case; one m a y simply need more iterations On the other hand, our choice m a y have been simply too large to be acceptable for eq. (3.3) to be still vahd. In table 1 we show that the latter is most likely not the case. We compare the average plaquette values (] Re Tr Up) at /3 = 5.2, 5.3 and 5.4 with those obtained by Fuclto, Rabbi and Solomon with 80%
acceptance. They agree extremely well Even at/3 = 5.3 our results are in agreement
R V Gavaz, F Karsch / SU(3) p h a s e t r a n s m o n s TABLE 1
Comparison of plaquette expectation values at different values of fl and m a = 0 1
281
~ReTr Up
fl nf = 3, acc= 80% n: = 3, acc = 63% nf = 0
5 2 0 48 l ± 0 002 0 4813 ± 0 0008 0 43173 ± 0 00040
5 3 0 528±0 001 0 5283±0 0008
5 4 0 545 ± 0 002 0 5475 ± 0 0010 0 47163 ± 0 00087
The first column shows the results of ref [7] obtained with an acceptance rate of 80% The second column g~ves our results obtained with 63% acceptance The last column shows pure gauge theory results
with thetr o r d e r e d start. We take this r e a s s u r i n g a g r e e m e n t o n the level o f 0.001 to m e a n that o u r choice o f 63% a c c e p t a n c e rate is at least as g o o d as theirs. Both the works mdicate" that the i n c l u s i o n o f d y n a m i c a l f e r m l o n s c h a n g e s the average p l a q u e t t e b y a p p r o x i m a t e l y 0.05 c o m p a r e d to the p u r e g a u g e values.
I n o r d e r to test w h e t h e r o u r first h y p o t h e s i s a b o u t the d i s c r e p a n c y is correct, we m a d e l o n g r u n s a t / 3 = 5 3, r n a = 0.1 s t a r t i n g f r o m the s a m e r a n d o m c o n f i g u r a t i o n b u t with a c c e p t a n c e m a i n t a i n e d at - 5 3 % , 63% a n d 79% Npr was c h o s e n to be 24 a n d 4 i t e r a t i o n s were d i s c a r d e d so as to b e able to c o m p a r e with ref. [7], Fig. 4 exhibits the results o f this study, Also s h o w n ~s the e q u i l i b r i u m v a l u e at /3 = 5.3 o b t a i n e d f r o m the r u n d i s p l a y e d in fig. 3. O n e sees a clear rising t r e n d i n all the three curves. T h o u g h o n e n e e d s m o r e t h a n 2000 i t e r a t i o n s to b e c o n v i n c e d that even with 79% a c c e p t a n c e the final result will be the s a m e This p e r h a p s e x p l a i n s w h y the a u t h o r s o f ref. [7] m t e r p r e t e d t h e i r results as signals for two state b e h a v i o r after
( L ) /~ =5.5
Npf = 2 4 ( - 4 )
0 5 ~ / ' ~ 5 5 %
_ / 6 3 % o
/ - / 79'/o
I 0 0 0 2 0 0 0 # )ter
Fig 4 The Polyakov hne versus number of MC iterations for various acceptance rates at fl = 53 The dashed hne indicates the eqmhbnum value obtained in the run shown m fig 3 The data have been
averaged over 50 subsequent lteraUons
282 R V Gava~, F Karsch / SU(3) phase transitions
I
( L ) /3 = 5.3 Npf = 5 0 (-25)acc 6 3 % A / ' \ / ~ v /
/
0"5 t- / Npf = 24. (-4)
/ /
x~"- t ~ t 1 : x ~ : I t I
,500 I000 # iter"-
Fig 5 The Polyakov hne versus number of MC iterations for different number of pseudo-fermlon Iterations (Npr) at/3 = 5 3 and fixed acceptance rate o f 63% The numbers m brackets denote the lteratmns
dlsgarded before taking averages The data have been averaged over 50 subsequent iterations
I000 iterations. In fig 5 we display the d e p e n d e n c e on Npf. We c o m p a r e at/3 = 5.3 and m a = 0.1 the two chmces of Nrr used by us and ref. [7]. One notices that equihbraUon time depends on Npr too.
To summarize then we find that the p s e u d o - f e r m l o n method works rather well with comparatively small acceptance rates also Average values of physical observ- ables tend to be quite m d e p e n d e n t of the parameters Npf and 8U, provided one makes sure that e q u d l b n u m is reached. The convergence rate a p p e a r s to depend strongly on b o t h these parameters and if one prefers to optimize for smaller 8 U then extra care needs to be taken to ensure that measurements are made m equili- brium only
5. Conclusions
We have studied the thermodynamics o f SU(3) with 3 hght quark flavors. A rapid change from the low-temperature phase to the high-temperature q u a r k - g l u o n plasma has been observed In the zero-mass hmlt we find evidence for a chlral phase trans~tton In wew of the present data st ~s suggestwe that the first-order phase transition present m the pure gauge sector of the theory weakens and m a y disappear at some critical mass value. MC simulations on a 63 ×2 lattice [19] indicate that this h a p p e n s around rnc/Tc<~ 2 4 This is considerably smaller than what has been estimated earher from a large mass a p p r o x i m a t i o n [10] However, m total we beheve a generic phase diagram hke the one shown in fig 6 m a y be emerging out of these MC simulations for SU(3) with 3 flavors To support this picture it certainly would be interesting to find out whether the second-order e n d p o m t (A) exists m this phase diagram also on larger latttces, to determine mc at that point and confirm the
R V Gavaz, F Karsch / SU(3) phase transtttons (20
m
O0 T x B co
283
Fig 6 Generic phase diagram for SU(3) with 3 flavors The circle on the m = ~ hne m&cates the first order phase transmon m the pure gauge sector from which a hne of first order transmons emerges ending in a second order transmon at the point A The point B indicates the second order choral transition at m = 0 u n i v e r s a l i t y o f t h e r a t i o me~ To. W h e t h e r t h e r a p i d c r o s s o v e r b e h a v i o r s e e n f o r h g h t q u a r k m a s s e s b e t w e e n mca a n d t h e c h i r a l t r a n s t t i o n at m a = 0 is j u s t a r e m n a n t o f t h e s e p h a s e t r a n s i t i o n s o r l n d ~ c a t e s a h n e o f s e c o n d - o r d e r t r a n s i t i o n s c o n n e c t i n g t h e p o i n t s at A a n d B r e m a i n s u n c l e a r o n t h e b a s i s o f t h e p r e s e n t d a t a O u r u n d e r s t a n d i n g o f t h e Q C D p h a s e d i a g r a m m t h e p r e s e n c e o f f e r m l o n s is, h o w e v e r , still i n c o m p l e t e . S u r p r i s e s m a y a l s o s h o w u p b y a n a l y z i n g in m o r e d e t a i l t h e f l a v o r d e p e n d e n c e o f t h e p h a s e d i a g r a m [20].
T h i s w o r k w a s s u p p o r t e d tn p a r t b y a g r a n t o f t h e N a t i o n a l S c t e n c e F o u n d a t i o n ( N S F - P H Y 8 2 - 0 1 9 4 8 ) a n d t h e U S D e p a r t m e n t o f E n e r g y u n d e r c o n t r a c t D E - A C 0 2 - 7 6 C H 0 0 0 1 6 .
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