- Email: [email protected]
On the chiral phase transition in two-flavor lattice QCD
Volume 200, number 1,2
PHYSICS LETTERS B
7 January 1988
O N THE CHIRAL P H A S E TRANSITION IN TWO-FLAVOR LATTICE QCD R.V. GAVAI Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400 005, India
J. P O T V I N and S. SANIELEVICI Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA
Received 31 July 1987
We use a Langevin algorithm to simulate QCD with two light flavors on an 83X4 lattice. We find metastability signals at a quark mass of 0.025 in lattice units, but not at a quark mass of 0.1. These results are discussed in the light of other recent findings on the phase diagram of finite-temperature QCD and of the special significance of the two-flavor problem.
The phase diagram o f finite-temperature Q C D has m a n y interesting implications for the physics o f heavy-ion collisions, of the early universe and o f astrophysical matter under extreme conditions [ 1 ]. A first-order transition from the hadron phase to a quark-gluon phase could yield spectacular signatures in heavy-ion collisions, such as explosions or eruptions as the plasma supercools back into the hadron phase [ 1 ]. It seems reasonable to d e m a n d that the quark species which contribute dynamically to this transition satisfy the condition m << Tc (the transition temperature Tc is currently estimated to lie between 100 and 200 MeV [ 1 ]). Indeed, any species with m ~ T~ or heavier ought to freeze out o f the dynamics. Thus, the question of the phase transition in the real world should be posed for Q C D with n r = 2 almost massless quark flavors, corresponding to u and d current quarks o f about 10 MeV, perhaps in the presence of a 150 MeV strange quark. For nf>~ 3 degenerate flavors o f massless quarks, studies of effective a-models predict a first-order chiral symmetry restoration transition at finite temperature [ 2,3 ]. For n~= 2, this transition is predicted to be first-order iff the effects o f the U(1 ) axial anomaly are much weaker at finite temperature than at zero temperature, and second-order iff the anomaly is temperature independent [ 2 ]. Therefore, a firstorder chiral transition in the physical world might have other interesting signatures, such as a dramatic
reduction o f the q ' - n mass difference in superhot matter. In recent months, there has been very intense activity in the effort to understand the phase diagram of finite-temperature Q C D with dynamical quarks by means o f numerical simulations on the lattice, using the staggered fermion scheme [ 4 - 1 0 ] . The emerging picture seems to be clearest in the case of nf= 4 continuum flavors of degenerate quarks. There is a first-order phase transition for sufficiently large quark masses; it becomes a smooth crossover for masses of the order m a = 0.2 in lattice units but there is again a first-order transition for m a = 0 . 0 2 5 . It is believed that the transition at high masses is the remnant of the deconfinement transition o f pure S U ( 3 ) theory [ 11 ] whereas the low-mass transition is the remnant of the chiral transition o f Q C D with massless quarks [ 2 ]. There is some controversy concerning the value o f m a at which the chiral transition begins to be felt: refs. [4,5] maintain that the transition is already first order at m a = O . 1 whereas [ 6,7] hold that even m a = 0 . 0 5 is still a rapid crossover. For three flavors, metastabilities have been seen at m a = 0 . 0 2 5 [10] but the transition is at best very weakly first order for m a = O. 1 [ 1,12 ]. A similar controversy exists in the case of two quark flavors. The K y o t o - T s u k u b a group claims a first-order transition at m a = O . 1 [8], whereas Gottlieb et al. [ 7] do not see unambiguous metasta-
0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
137
Volume 200, number 1,2
PHYSICS LETTERS B
bility signals even at ma=0.025. To be precise, the latter authors do find thermodynamic behavior consistent with a first-order transition (a sharp jump in internal energy in a very narrow interval of lattice temperature, over which the pressure is constant) but they cannot see any two-state signal in their time histories of observables. One must bear in mind that these various simulations were carried out using diverse algorithms and that different strategies were used to search for signals of metastability. Each computation has its own sources of systematic and statistical errors, so that it is not rigorously possible to compare the results directly. It is clear, however, that each has its own shortcomings. Gupta et al. [5] use an algorithm which is biasfree except for the effects of the finite residual in the conjugate-gradient inversion of the fermion matrix (which seem to be under control in all simulations). In particular, there are no renormalizations of the QCD coupling fl, of the mass ma or of the number of flavors nf. Therefore, they are able in principle to determine the transition coupling tic exactly. On the other hand, the algorithm is so expensive that they can only run on a 44 lattice, which is known to induce appreciable finite-size fluctuations [ 10]. If the first-order transition is strong enough, as in the case of four flavors at rna=0.025, these fluctuations are not large enough to obscure the metastability signal. For large ma or for two flavors, however, the transition is weaker, so that the finite-size noise can make the determination of its order very difficult [ 10] (see also fig. 1 below). The use of larger lattices (such as 8 3 X 4 ) may thus be necessary for such cases, and the advantages of the exact algorithm may be lost. Refs. [4,6-8,10] all use approximate algorithms based on the numerical integration of differential equations with finite step size. This enables them to run on larger lattices, to conduct systematic searches for the transition fl and to gather considerable statistics. On the other hand, the finiteness of the step size induces systematic errors which become more and more severe as the quark mass is reduced [ 13-15 ]. One immediate effect is that all the couplings in the theory (fl, ma, nf) are shifted away from their true values. For instance, to first order in the Langevin step size e, fl is shifted to fl(1 + ~ e C A - eCv), nf to nf(1 +~2eCA) and the quark mass to 138
7 January1988
m / ( l - leCv) ~. The justification for using such algorithms despite the systematic errors is that one hopes that the effective actions of the simulations are in the same universality class as QCD for the step sizes one uses. However, it is not known at present how to relate the step size in the Langevin algorithms used in refs. [4,8,10] to the step size of the hybrid algorithms used in refs. [6,7]. Therefore, it is possible that these simulations correspond to QCD at slightly different values of r, nf and ma. The main focus of our work is to study the flavordependence of the low-mass phase transition [ 16]. Given the confusing situation described above and the special physical importance of the two-flavor case, we wish to report our findings on nf= 2, ma = 0.025 and 0.1 in the present letter. Our method of investigation of the order of the phase transition [10] is consistently based on the comparison of the time histories of two separate runs: one starting from a completely ordered configuration and one which starts from a completely random configuration. This is done for each value offl which we wish to study. This procedure is designed to minimize the effect of systematic errors and of time correlations, since it does not depend on the accurate measurement of averages. We use the discrete first-order Langevin algorithm described in ref. f 13 ]. In this algorithm the first-order renormalizations of r, ma and nf are explicitly compensated. To reduce the effects of residual systematic errors while keeping the algorithm as efficient as possible, we have run at e = 0.01. To simulate nf continuum flavors by means of staggered lattice fermions, we raise the fermion determinant in the QCD action to the power nf/4 [17]. The boundary conditions are periodic in the space directions and antiperiodic in the time direction. The inversion of the fermion matrix was done by the conjugate gradient method. We used a stopping residue r = 0.2 (in the normalization of ref. [13]) for the runs at m a = 0 . 0 2 5 and r=0.01 at m a = 0 . 1 and m a = 0 . 2 . By means of preliminary runs on a 4 4 lattice, we had found that the transition region for nf=2, m a = 0 . 0 2 5 is between fl = 5.3 and r = 5.4 [ 10] (see fig. la). To look for possible metastabilities with better resolution, we ran on an 83× 4 lattice in that in-
~ Ref. [ 10] contains misprints in the shift formulae forfl and nf.
Volume 200, number 1,2
PHYSICS LETTERS B
1.4
0.8
nf=2
I.••
•
j
1.4
~
/3= 5 . 3 3
1.2~
/3=5.30
1.0 d
1.4, A
I
(o) 4 4 ' m a = 0 . 0 2 5 ,
1,2
7 January 1988
1.0F 3.8~
A••I'•I.
I
/3 = 5.35
1.0 AA•
••
3.6~
o 0.6
I
1.2
•
•
•A•
••• • •
•
0.8
•
•••
0.6
•
• •
•
•
,L•
• AAA
0,4
•
:3.4 F
A•
•" •
• •
A• •
0.4 ,i
0.2 0.0
3.2F
•A IIi
I • ii
iiii
ii
•
• i t £ IIi
3.0 ~
i
i I
-0.20
r
1000
2000
# o.6
1.4},,
'
, 2000
/3=,5.37
300 '
do.2b
""
i
I
1000
2000
I
I
1"41"
,5000
/3 = 5.40
/3=5.35
I.O 0.8
LA •
••
AA•
•
1.0 • ••
•A A
""
A •
A
Q2 II
i
1000
2000
iiii~
•
.
301, ." -",J
I000
,",• •
.
"," "t" ' 2000
3.8
•
"
•
I
500C • 0
•
""
9.4 ~-
0.4
0.0 0
, I000
300C
1.0 0.8
•m e
•'++
(b) 441mo=0.10, I nf--2
j
0.0 m|me
I
1.4 1.2
0.2
.• ..- "'."
t
--
3+I
••
•• • •A
"
•A
i 3.4
30 3000 4000 '
• "I•IT
|1" •
1000
," "1 •
• I
2000
3000
ITERATIONS
Fig. 1. Time histories for the thermal Wilson line Re L on a 44 lattice with two continuum flavors of quarks. (a) refers to ma= 0.025 and shows fl=5.30, 5.33, 5.35 from left to right. (b): ma=0.1, fl= 5.35, 5.37, 5.40 from left to right. Squares describe the runs from disordered starts and triangles describe the runs from ordered starts. Each point represents the average over 100 iterations. Re L is normalized to 3.
terval. As in the three-flavor case, we find that the signal is considerably e n h a n c e d by going to the larger lattice. Fig. 2 shows that t i c = 5.33 presents clear metastability signals in the P o l y a k o v loop a n d in the chiral o r d e r parameter. There a p p e a r to be two distinct coexisting phases; we also see clear tunneling p h e n o m e n a from both the " c o l d " a n d the " h o t " time histories. Thus our simulation strongly suggests that the transition is indeed first o r d e r at m a = 0 . 0 2 5 . While the d i s c o n t i n u i t y in the P o l y a k o v loop and in the chiral o r d e r p a r a m e t e r is o f the same o r d e r as in the n r = 4 a n d nf= 3 simulations, we find the j u m p in the plaquette to be m u c h smaller. This, together with the fact that the transition region is narrower than in those cases, tends to c o n f i r m that the transition weakens as nf is decreased. The picture we o b t a i n for m a = 0.1 is quite different. The p r e l i m i n a r y runs on the 44 lattice pin down
the transition interval to fie [5.37, 5.38], in agreem e n t with the one r e p o r t e d by ref. [8] (fig. l b ) . Based on the runs at fl = 5.37 and fl = 5.38 one might draw the conclusion that tunneling is taking place. However, going to the 83X 4 lattice reveals that the hot a n d cold runs meet after a relatively long evolution a n d evolve together thereafter (fig. 3). This suggests a smooth crossover, over a narrow interval o f fl, from the d i s o r d e r e d to the ordered phase. In any case, we see no signs o f metastability c o m p a rable to fig. 2 or to the ones we r e p o r t e d in ref. [ 10]. We also simulated the theory at m a = 0.2, where refs. [7,8] agree that there is no first-order transition. Here, the crossover b e h a v i o r is a p p a r e n t even on the 44 lattice. O u r findings thus a p p e a r to differ from ref. [ 7 ], in that we do see metastability at m a = 0 . 0 2 5 , a n d from ref. [8], in that we do not see any at m a = O. 1. 139
Volume 200, number 1,2
1,5
I
A (o)
5
PHYSICS LETTERS B
I
8 x4
I
I
mo=O.025
I
I
nf=2
7 January 1988
]
/3=5.554
5
I ,OC
/~
q
I
:
0.9 J
!
0.7 0.70 0.5 0.3 0.1 -0.1
....'.j. ,,~_~2-"
..,,...~,.
ml~ I IOOO 2000
0
•4
....
I 5000
I 4000
I 5000
•
I 6000
i
| 7000
O
IOO
200
500
ITERATIONS Fig. 2. Metastability signals in two-flavor QCD on an 83× 4 lattice at m a = 0.025. Re L is shown in (a) and the chiral order parameter (normalized to 3) is shown in (b). The symbols are as explained in the caption to fig. 1.
This is perhaps not surprising, since, as we pointed out above, small differences in the simulation procedures can lead to misunderstandings in the inter-
1.5
i--
(0)
8'3X4
t
mo =0.10
i
nf=2
1.5
~ = 5.565
1.5
o.7
I
I
t
~
I
(b} -
t.I 0.9
pretation of the results, especially in the case of weak first-order transitions. We believe, however, that our procedure is superior to those based on the conca-
I
,B= 5.575
1.5 I.I
•
0.9 •
••
0.7
• "---..
.
0,5;
"'"~''7.:::,7:"t o.~~
O.5
,,
O.I -O.I
....
,.,,,,.,,,"
"'7"" ,ooo
,
~ooo
4oE,
~-~ooo
~," "~-;.~.::~ .~.1 ,--"'""
5 - o . , - ' ' ~ ' " ~oooo ,ooo
~ooo
'
~ooo
ITERATIONS Fig. 3. Time histories for Re L for nf--2,
140
ma
= 0.1 on an 83X 4 lattice• (a) is for fl = 5.365 and (b) is for fl = 5•375.
~ooo
Volume 200, number 1,2
PHYSICS LETTERS B
tenation of runs done at various couplings [ 7,8 ] and step sizes [ 8 ]. This can lead to a buildup of systematic and statistical errors, thereby raising the probability that the order of the transition will be misjudged. The monitoring of only one run for "tunneling" events and the use of histograms as indicators of metastability also seem more error-prone than our approach. In this context, it is interesting to compare the oscillations in R e L shown in fig. 4 of ref. [8] (which they interpret as a metastability signal for m a = O . 1 ) with the tunneling events in our fig. 2 above. In their case, it is striking that the time the system spends with "high" or "low" Re L is comparable to the time it spends transiting between the two ranges of values. Also, the gap between the two ranges is not much larger than the amplitude of the fluctuations shown in their figure. Thus their signal could well be interpreted as being due to long-wavelength fluctuations within a single phase. As we remarked above, the absence of such oscillations in our time histories shown in fig. 3 is probably due to the very different preparation of the system; note that our total run time in fig. 3b corresponds to T = 70 in the notation of ref.
[8]. Our results appear to fit into the general picture of the flavor-number dependence of the chiral transition on the lattice [ 9 ]. Since the first-order character weakens progressively as nr decreases and the order of the transition at m a = 0.1 is already controversial for nf=4 and nr=3, it would seem surprising if m a = 0 . 1 , nr=2 were first order (the crossover at m a = 0.2 means that this transition cannot be driven by the "deconfinement" mechanism). Our conclusion that the time history at m a = 0 . 0 2 5 does show metastability agrees nicely with the findings of ref. [ 7 ] on the behavior of the internal energy and of the pressure in the transition region. Our estimate for the critical coupling, fl,.= 5.33, is consistent with the Langevin results of ref. [8] but it differs from the hybrid estimate, ~c=5.2875. It is interesting to compare this to the situation for m a = O . 0 2 5 , nr=4, where the hybrid studies [6,7] yield tic = 4.96 _+0.03 while the Langevin simulations find flc~ 5.03 and the bias-free simulation [5] sees metastability signals at both fl=4.90 and fl=4.95. Part of these differences could be due to the fact that metastabilities always occur over a finite range offl
7 January 1988
and to the imprecisions of the localization procedures, but the recurrence of the same trend at two different values of nf does suggest a systematic effect. This favors our earlier conjecture that Langevin and hybrid algorithms correspond to QCD with different fl, nf and ma. The magnitude of the shifts quoted above is larger than expected on the basis of the firstorder formulae, which is not surprising since higherorder systematic errors are known to be appreciable at low quark masses: the effective step size grows as the mass decreases [13]. From a physical point of view, two questions come immediately to mind concerning the first-order character of the nf= 2 chiral transition as seen on a lattice with staggered quarks. Firstly, is the transition we see really due to the chiral mechanism of refs. [ 2,3 ], or does the deconfinement mechanism extend to lower masses as the number of flavors is decreased? The latter statement should be true if the quenched approximation could be reached as an analytical limit nf ~ 0 . The results of the present work tend to support the existence of a gap between the high-mass and the low-mass first-order transition regions, so that the low-mass transition should not be caused by the deconfinement mechanism. Therefore, the question of the limit nf--,O, ma-+O must be addressed for nr< 2 [ 16 ]. Secondly, is it really possible to use a lattice at a typical fl of ~ 5.33 and staggered quarks in order to study a problem so intimately related to the axial U(1 ) anomaly? For, if the effects of the anomaly are suppressed by our simulation, the reasoning of ref. [ 2 ] implies that our first-order transition might not survive the passage to the continuum limit. This question must obviously be answered before we can definitely predict the partial restoration of axial U(1 ) symmetry as one of the experimental signatures which could be obtained in heavy-ion collisions. We would like to thank M. Creutz, R. Gupta, I.H. Lee, H. Satz and D. Toussaint for interesting discussions. The computations were done on the CRAY X-MP at NMFECC, Livermore. This work was supported by the US Department of energy under contract number DE-AC02-76CH00016.
141
Volume 200, number 1,2
PHYSICS LETTERS B
References [ 1 ] J. Cleymans, R.V. Gavai and E. Suhonen, Phys. Rep. 130 (1986) 217; R.V. Gavai, in: Proc. Lake Louise Conf. (1986), ed. D.F. Geesaman, AlP Conf. Proc. No. 150 (ALP, New York) p. 848. [2] R.D. Pisarski and F. Wilczek, Phys. Rev. D 29 (1984) 338. [3] H. Goldberg, Phys. Lett. B 131 (1983) 133; A. Margaritis and A. Patkos, Phys. Lett. B 178 (1986) 272. [4] M. Fukugita and A. Ukawa, Phys. Rev. Lett. 57 (1986) 503. [5] R, Gupta, G. Guralnik, G.W. Kilcup, A. Patel and S.R. Sharpe, Phys. Rev. Lett. 57 (1986) 2621; R. Gupta, More on the first-order chiral symmetry transition in QCD, in: Proc. NATO ARW 641/86 Workshop, eds. H. Satz, I. Harrity and J. Potvin (Plenum, New York, 1987), to be published. [6] J.B. Kogut and D.K. Sinclair, Nucl. Phys. B 280 [FSI8] (1987) 625; J.B. Kogut, H.W. Wyld, F. Karsch and D.K. Sinclair, preprint ILL-(TH)-87-6 (1987); E.V.E. Kovacs, D.K. Sinclair and J.B. Kogut, Phys. Rev. Lett. 58 (1987) 751; and preprint ILL-(TH)-87-29 (1987). [7] S. Gottlieb, W. Liu, D. Toussaint, R.L. Renken and R.L. Sugar, Phys. Rev. D 35 (1987) 3972.
142
7 January 1988
[ 8 ] M. Fukugita, S. Ohta, Y. Oyanagi and A. Ukawa, Phys. Rev. Lett. 58 (1987) 2515. [9] N. Attig, B. Petersson and M. Wolff, Phys. Lett. B 190 (1987) 143. [ 10] R.V. Gavai, J. Potvin and S. Sanielevici, Phys. Rev. Lett. 58 (1987) 2519. [ 11 ] B. Svetitsky, Phys. Rep. 132 (1986) 1. [ 12 ] F. Fucito, C. Rebbi and S. Solomon, Nucl. Phys. B 180 [ FS2 ] (1984) 615; Phys. Rev. D31 (1985) 1461; F. Fucito and S. Solomon, Phys. Rev. Lett. 55 (1985 ) 2641; R.V. Gavai and F. Karsch, Nucl. Phys. B 261 (1985) 273; R.V. Gavai, Nucl. Phys. B 269 (1986) 530, and references therein. [ 13] R.V. Gavai, J. Potvin and S. Sanielevici, Phys. Lett. B 183 (1987) 86; preprint BNL-39021/TIFR/TH/86-38 (1986), Phys. Rev. D, to be published. [14] G.G. Batrouni, G.R. Katz, A.S. Kronfeld, G.P. Lepage, B. Svetitsky and K.G. Wilson, Phys. Rev. D 32 (1985) 2736. [ 15] S. Gottlieb, W. Liu, D. Toussaint, R.L. Renken and R.L. Sugar, Phys. Rev. D 35 (1987) 2531. [ 16] R.V. Gavai, J. Potvin and S. Sanielevici, in preparation. [ 17 ] H.W. Hamber, E. Marinari, G. Parisi and C. Rebbi, Phys. Lett. B 124 (1983) 99.
PHYSICS LETTERS B
7 January 1988
O N THE CHIRAL P H A S E TRANSITION IN TWO-FLAVOR LATTICE QCD R.V. GAVAI Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400 005, India
J. P O T V I N and S. SANIELEVICI Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA
Received 31 July 1987
We use a Langevin algorithm to simulate QCD with two light flavors on an 83X4 lattice. We find metastability signals at a quark mass of 0.025 in lattice units, but not at a quark mass of 0.1. These results are discussed in the light of other recent findings on the phase diagram of finite-temperature QCD and of the special significance of the two-flavor problem.
The phase diagram o f finite-temperature Q C D has m a n y interesting implications for the physics o f heavy-ion collisions, of the early universe and o f astrophysical matter under extreme conditions [ 1 ]. A first-order transition from the hadron phase to a quark-gluon phase could yield spectacular signatures in heavy-ion collisions, such as explosions or eruptions as the plasma supercools back into the hadron phase [ 1 ]. It seems reasonable to d e m a n d that the quark species which contribute dynamically to this transition satisfy the condition m << Tc (the transition temperature Tc is currently estimated to lie between 100 and 200 MeV [ 1 ]). Indeed, any species with m ~ T~ or heavier ought to freeze out o f the dynamics. Thus, the question of the phase transition in the real world should be posed for Q C D with n r = 2 almost massless quark flavors, corresponding to u and d current quarks o f about 10 MeV, perhaps in the presence of a 150 MeV strange quark. For nf>~ 3 degenerate flavors o f massless quarks, studies of effective a-models predict a first-order chiral symmetry restoration transition at finite temperature [ 2,3 ]. For n~= 2, this transition is predicted to be first-order iff the effects o f the U(1 ) axial anomaly are much weaker at finite temperature than at zero temperature, and second-order iff the anomaly is temperature independent [ 2 ]. Therefore, a firstorder chiral transition in the physical world might have other interesting signatures, such as a dramatic
reduction o f the q ' - n mass difference in superhot matter. In recent months, there has been very intense activity in the effort to understand the phase diagram of finite-temperature Q C D with dynamical quarks by means o f numerical simulations on the lattice, using the staggered fermion scheme [ 4 - 1 0 ] . The emerging picture seems to be clearest in the case of nf= 4 continuum flavors of degenerate quarks. There is a first-order phase transition for sufficiently large quark masses; it becomes a smooth crossover for masses of the order m a = 0.2 in lattice units but there is again a first-order transition for m a = 0 . 0 2 5 . It is believed that the transition at high masses is the remnant of the deconfinement transition o f pure S U ( 3 ) theory [ 11 ] whereas the low-mass transition is the remnant of the chiral transition o f Q C D with massless quarks [ 2 ]. There is some controversy concerning the value o f m a at which the chiral transition begins to be felt: refs. [4,5] maintain that the transition is already first order at m a = O . 1 whereas [ 6,7] hold that even m a = 0 . 0 5 is still a rapid crossover. For three flavors, metastabilities have been seen at m a = 0 . 0 2 5 [10] but the transition is at best very weakly first order for m a = O. 1 [ 1,12 ]. A similar controversy exists in the case of two quark flavors. The K y o t o - T s u k u b a group claims a first-order transition at m a = O . 1 [8], whereas Gottlieb et al. [ 7] do not see unambiguous metasta-
0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
137
Volume 200, number 1,2
PHYSICS LETTERS B
bility signals even at ma=0.025. To be precise, the latter authors do find thermodynamic behavior consistent with a first-order transition (a sharp jump in internal energy in a very narrow interval of lattice temperature, over which the pressure is constant) but they cannot see any two-state signal in their time histories of observables. One must bear in mind that these various simulations were carried out using diverse algorithms and that different strategies were used to search for signals of metastability. Each computation has its own sources of systematic and statistical errors, so that it is not rigorously possible to compare the results directly. It is clear, however, that each has its own shortcomings. Gupta et al. [5] use an algorithm which is biasfree except for the effects of the finite residual in the conjugate-gradient inversion of the fermion matrix (which seem to be under control in all simulations). In particular, there are no renormalizations of the QCD coupling fl, of the mass ma or of the number of flavors nf. Therefore, they are able in principle to determine the transition coupling tic exactly. On the other hand, the algorithm is so expensive that they can only run on a 44 lattice, which is known to induce appreciable finite-size fluctuations [ 10]. If the first-order transition is strong enough, as in the case of four flavors at rna=0.025, these fluctuations are not large enough to obscure the metastability signal. For large ma or for two flavors, however, the transition is weaker, so that the finite-size noise can make the determination of its order very difficult [ 10] (see also fig. 1 below). The use of larger lattices (such as 8 3 X 4 ) may thus be necessary for such cases, and the advantages of the exact algorithm may be lost. Refs. [4,6-8,10] all use approximate algorithms based on the numerical integration of differential equations with finite step size. This enables them to run on larger lattices, to conduct systematic searches for the transition fl and to gather considerable statistics. On the other hand, the finiteness of the step size induces systematic errors which become more and more severe as the quark mass is reduced [ 13-15 ]. One immediate effect is that all the couplings in the theory (fl, ma, nf) are shifted away from their true values. For instance, to first order in the Langevin step size e, fl is shifted to fl(1 + ~ e C A - eCv), nf to nf(1 +~2eCA) and the quark mass to 138
7 January1988
m / ( l - leCv) ~. The justification for using such algorithms despite the systematic errors is that one hopes that the effective actions of the simulations are in the same universality class as QCD for the step sizes one uses. However, it is not known at present how to relate the step size in the Langevin algorithms used in refs. [4,8,10] to the step size of the hybrid algorithms used in refs. [6,7]. Therefore, it is possible that these simulations correspond to QCD at slightly different values of r, nf and ma. The main focus of our work is to study the flavordependence of the low-mass phase transition [ 16]. Given the confusing situation described above and the special physical importance of the two-flavor case, we wish to report our findings on nf= 2, ma = 0.025 and 0.1 in the present letter. Our method of investigation of the order of the phase transition [10] is consistently based on the comparison of the time histories of two separate runs: one starting from a completely ordered configuration and one which starts from a completely random configuration. This is done for each value offl which we wish to study. This procedure is designed to minimize the effect of systematic errors and of time correlations, since it does not depend on the accurate measurement of averages. We use the discrete first-order Langevin algorithm described in ref. f 13 ]. In this algorithm the first-order renormalizations of r, ma and nf are explicitly compensated. To reduce the effects of residual systematic errors while keeping the algorithm as efficient as possible, we have run at e = 0.01. To simulate nf continuum flavors by means of staggered lattice fermions, we raise the fermion determinant in the QCD action to the power nf/4 [17]. The boundary conditions are periodic in the space directions and antiperiodic in the time direction. The inversion of the fermion matrix was done by the conjugate gradient method. We used a stopping residue r = 0.2 (in the normalization of ref. [13]) for the runs at m a = 0 . 0 2 5 and r=0.01 at m a = 0 . 1 and m a = 0 . 2 . By means of preliminary runs on a 4 4 lattice, we had found that the transition region for nf=2, m a = 0 . 0 2 5 is between fl = 5.3 and r = 5.4 [ 10] (see fig. la). To look for possible metastabilities with better resolution, we ran on an 83× 4 lattice in that in-
~ Ref. [ 10] contains misprints in the shift formulae forfl and nf.
Volume 200, number 1,2
PHYSICS LETTERS B
1.4
0.8
nf=2
I.••
•
j
1.4
~
/3= 5 . 3 3
1.2~
/3=5.30
1.0 d
1.4, A
I
(o) 4 4 ' m a = 0 . 0 2 5 ,
1,2
7 January 1988
1.0F 3.8~
A••I'•I.
I
/3 = 5.35
1.0 AA•
••
3.6~
o 0.6
I
1.2
•
•
•A•
••• • •
•
0.8
•
•••
0.6
•
• •
•
•
,L•
• AAA
0,4
•
:3.4 F
A•
•" •
• •
A• •
0.4 ,i
0.2 0.0
3.2F
•A IIi
I • ii
iiii
ii
•
• i t £ IIi
3.0 ~
i
i I
-0.20
r
1000
2000
# o.6
1.4},,
'
, 2000
/3=,5.37
300 '
do.2b
""
i
I
1000
2000
I
I
1"41"
,5000
/3 = 5.40
/3=5.35
I.O 0.8
LA •
••
AA•
•
1.0 • ••
•A A
""
A •
A
Q2 II
i
1000
2000
iiii~
•
.
301, ." -",J
I000
,",• •
.
"," "t" ' 2000
3.8
•
"
•
I
500C • 0
•
""
9.4 ~-
0.4
0.0 0
, I000
300C
1.0 0.8
•m e
•'++
(b) 441mo=0.10, I nf--2
j
0.0 m|me
I
1.4 1.2
0.2
.• ..- "'."
t
--
3+I
••
•• • •A
"
•A
i 3.4
30 3000 4000 '
• "I•IT
|1" •
1000
," "1 •
• I
2000
3000
ITERATIONS
Fig. 1. Time histories for the thermal Wilson line Re L on a 44 lattice with two continuum flavors of quarks. (a) refers to ma= 0.025 and shows fl=5.30, 5.33, 5.35 from left to right. (b): ma=0.1, fl= 5.35, 5.37, 5.40 from left to right. Squares describe the runs from disordered starts and triangles describe the runs from ordered starts. Each point represents the average over 100 iterations. Re L is normalized to 3.
terval. As in the three-flavor case, we find that the signal is considerably e n h a n c e d by going to the larger lattice. Fig. 2 shows that t i c = 5.33 presents clear metastability signals in the P o l y a k o v loop a n d in the chiral o r d e r parameter. There a p p e a r to be two distinct coexisting phases; we also see clear tunneling p h e n o m e n a from both the " c o l d " a n d the " h o t " time histories. Thus our simulation strongly suggests that the transition is indeed first o r d e r at m a = 0 . 0 2 5 . While the d i s c o n t i n u i t y in the P o l y a k o v loop and in the chiral o r d e r p a r a m e t e r is o f the same o r d e r as in the n r = 4 a n d nf= 3 simulations, we find the j u m p in the plaquette to be m u c h smaller. This, together with the fact that the transition region is narrower than in those cases, tends to c o n f i r m that the transition weakens as nf is decreased. The picture we o b t a i n for m a = 0.1 is quite different. The p r e l i m i n a r y runs on the 44 lattice pin down
the transition interval to fie [5.37, 5.38], in agreem e n t with the one r e p o r t e d by ref. [8] (fig. l b ) . Based on the runs at fl = 5.37 and fl = 5.38 one might draw the conclusion that tunneling is taking place. However, going to the 83X 4 lattice reveals that the hot a n d cold runs meet after a relatively long evolution a n d evolve together thereafter (fig. 3). This suggests a smooth crossover, over a narrow interval o f fl, from the d i s o r d e r e d to the ordered phase. In any case, we see no signs o f metastability c o m p a rable to fig. 2 or to the ones we r e p o r t e d in ref. [ 10]. We also simulated the theory at m a = 0.2, where refs. [7,8] agree that there is no first-order transition. Here, the crossover b e h a v i o r is a p p a r e n t even on the 44 lattice. O u r findings thus a p p e a r to differ from ref. [ 7 ], in that we do see metastability at m a = 0 . 0 2 5 , a n d from ref. [8], in that we do not see any at m a = O. 1. 139
Volume 200, number 1,2
1,5
I
A (o)
5
PHYSICS LETTERS B
I
8 x4
I
I
mo=O.025
I
I
nf=2
7 January 1988
]
/3=5.554
5
I ,OC
/~
q
I
:
0.9 J
!
0.7 0.70 0.5 0.3 0.1 -0.1
....'.j. ,,~_~2-"
..,,...~,.
ml~ I IOOO 2000
0
•4
....
I 5000
I 4000
I 5000
•
I 6000
i
| 7000
O
IOO
200
500
ITERATIONS Fig. 2. Metastability signals in two-flavor QCD on an 83× 4 lattice at m a = 0.025. Re L is shown in (a) and the chiral order parameter (normalized to 3) is shown in (b). The symbols are as explained in the caption to fig. 1.
This is perhaps not surprising, since, as we pointed out above, small differences in the simulation procedures can lead to misunderstandings in the inter-
1.5
i--
(0)
8'3X4
t
mo =0.10
i
nf=2
1.5
~ = 5.565
1.5
o.7
I
I
t
~
I
(b} -
t.I 0.9
pretation of the results, especially in the case of weak first-order transitions. We believe, however, that our procedure is superior to those based on the conca-
I
,B= 5.575
1.5 I.I
•
0.9 •
••
0.7
• "---..
.
0,5;
"'"~''7.:::,7:"t o.~~
O.5
,,
O.I -O.I
....
,.,,,,.,,,"
"'7"" ,ooo
,
~ooo
4oE,
~-~ooo
~," "~-;.~.::~ .~.1 ,--"'""
5 - o . , - ' ' ~ ' " ~oooo ,ooo
~ooo
'
~ooo
ITERATIONS Fig. 3. Time histories for Re L for nf--2,
140
ma
= 0.1 on an 83X 4 lattice• (a) is for fl = 5.365 and (b) is for fl = 5•375.
~ooo
Volume 200, number 1,2
PHYSICS LETTERS B
tenation of runs done at various couplings [ 7,8 ] and step sizes [ 8 ]. This can lead to a buildup of systematic and statistical errors, thereby raising the probability that the order of the transition will be misjudged. The monitoring of only one run for "tunneling" events and the use of histograms as indicators of metastability also seem more error-prone than our approach. In this context, it is interesting to compare the oscillations in R e L shown in fig. 4 of ref. [8] (which they interpret as a metastability signal for m a = O . 1 ) with the tunneling events in our fig. 2 above. In their case, it is striking that the time the system spends with "high" or "low" Re L is comparable to the time it spends transiting between the two ranges of values. Also, the gap between the two ranges is not much larger than the amplitude of the fluctuations shown in their figure. Thus their signal could well be interpreted as being due to long-wavelength fluctuations within a single phase. As we remarked above, the absence of such oscillations in our time histories shown in fig. 3 is probably due to the very different preparation of the system; note that our total run time in fig. 3b corresponds to T = 70 in the notation of ref.
[8]. Our results appear to fit into the general picture of the flavor-number dependence of the chiral transition on the lattice [ 9 ]. Since the first-order character weakens progressively as nr decreases and the order of the transition at m a = 0.1 is already controversial for nf=4 and nr=3, it would seem surprising if m a = 0 . 1 , nr=2 were first order (the crossover at m a = 0.2 means that this transition cannot be driven by the "deconfinement" mechanism). Our conclusion that the time history at m a = 0 . 0 2 5 does show metastability agrees nicely with the findings of ref. [ 7 ] on the behavior of the internal energy and of the pressure in the transition region. Our estimate for the critical coupling, fl,.= 5.33, is consistent with the Langevin results of ref. [8] but it differs from the hybrid estimate, ~c=5.2875. It is interesting to compare this to the situation for m a = O . 0 2 5 , nr=4, where the hybrid studies [6,7] yield tic = 4.96 _+0.03 while the Langevin simulations find flc~ 5.03 and the bias-free simulation [5] sees metastability signals at both fl=4.90 and fl=4.95. Part of these differences could be due to the fact that metastabilities always occur over a finite range offl
7 January 1988
and to the imprecisions of the localization procedures, but the recurrence of the same trend at two different values of nf does suggest a systematic effect. This favors our earlier conjecture that Langevin and hybrid algorithms correspond to QCD with different fl, nf and ma. The magnitude of the shifts quoted above is larger than expected on the basis of the firstorder formulae, which is not surprising since higherorder systematic errors are known to be appreciable at low quark masses: the effective step size grows as the mass decreases [13]. From a physical point of view, two questions come immediately to mind concerning the first-order character of the nf= 2 chiral transition as seen on a lattice with staggered quarks. Firstly, is the transition we see really due to the chiral mechanism of refs. [ 2,3 ], or does the deconfinement mechanism extend to lower masses as the number of flavors is decreased? The latter statement should be true if the quenched approximation could be reached as an analytical limit nf ~ 0 . The results of the present work tend to support the existence of a gap between the high-mass and the low-mass first-order transition regions, so that the low-mass transition should not be caused by the deconfinement mechanism. Therefore, the question of the limit nf--,O, ma-+O must be addressed for nr< 2 [ 16 ]. Secondly, is it really possible to use a lattice at a typical fl of ~ 5.33 and staggered quarks in order to study a problem so intimately related to the axial U(1 ) anomaly? For, if the effects of the anomaly are suppressed by our simulation, the reasoning of ref. [ 2 ] implies that our first-order transition might not survive the passage to the continuum limit. This question must obviously be answered before we can definitely predict the partial restoration of axial U(1 ) symmetry as one of the experimental signatures which could be obtained in heavy-ion collisions. We would like to thank M. Creutz, R. Gupta, I.H. Lee, H. Satz and D. Toussaint for interesting discussions. The computations were done on the CRAY X-MP at NMFECC, Livermore. This work was supported by the US Department of energy under contract number DE-AC02-76CH00016.
141
Volume 200, number 1,2
PHYSICS LETTERS B
References [ 1 ] J. Cleymans, R.V. Gavai and E. Suhonen, Phys. Rep. 130 (1986) 217; R.V. Gavai, in: Proc. Lake Louise Conf. (1986), ed. D.F. Geesaman, AlP Conf. Proc. No. 150 (ALP, New York) p. 848. [2] R.D. Pisarski and F. Wilczek, Phys. Rev. D 29 (1984) 338. [3] H. Goldberg, Phys. Lett. B 131 (1983) 133; A. Margaritis and A. Patkos, Phys. Lett. B 178 (1986) 272. [4] M. Fukugita and A. Ukawa, Phys. Rev. Lett. 57 (1986) 503. [5] R, Gupta, G. Guralnik, G.W. Kilcup, A. Patel and S.R. Sharpe, Phys. Rev. Lett. 57 (1986) 2621; R. Gupta, More on the first-order chiral symmetry transition in QCD, in: Proc. NATO ARW 641/86 Workshop, eds. H. Satz, I. Harrity and J. Potvin (Plenum, New York, 1987), to be published. [6] J.B. Kogut and D.K. Sinclair, Nucl. Phys. B 280 [FSI8] (1987) 625; J.B. Kogut, H.W. Wyld, F. Karsch and D.K. Sinclair, preprint ILL-(TH)-87-6 (1987); E.V.E. Kovacs, D.K. Sinclair and J.B. Kogut, Phys. Rev. Lett. 58 (1987) 751; and preprint ILL-(TH)-87-29 (1987). [7] S. Gottlieb, W. Liu, D. Toussaint, R.L. Renken and R.L. Sugar, Phys. Rev. D 35 (1987) 3972.
142
7 January 1988
[ 8 ] M. Fukugita, S. Ohta, Y. Oyanagi and A. Ukawa, Phys. Rev. Lett. 58 (1987) 2515. [9] N. Attig, B. Petersson and M. Wolff, Phys. Lett. B 190 (1987) 143. [ 10] R.V. Gavai, J. Potvin and S. Sanielevici, Phys. Rev. Lett. 58 (1987) 2519. [ 11 ] B. Svetitsky, Phys. Rep. 132 (1986) 1. [ 12 ] F. Fucito, C. Rebbi and S. Solomon, Nucl. Phys. B 180 [ FS2 ] (1984) 615; Phys. Rev. D31 (1985) 1461; F. Fucito and S. Solomon, Phys. Rev. Lett. 55 (1985 ) 2641; R.V. Gavai and F. Karsch, Nucl. Phys. B 261 (1985) 273; R.V. Gavai, Nucl. Phys. B 269 (1986) 530, and references therein. [ 13] R.V. Gavai, J. Potvin and S. Sanielevici, Phys. Lett. B 183 (1987) 86; preprint BNL-39021/TIFR/TH/86-38 (1986), Phys. Rev. D, to be published. [14] G.G. Batrouni, G.R. Katz, A.S. Kronfeld, G.P. Lepage, B. Svetitsky and K.G. Wilson, Phys. Rev. D 32 (1985) 2736. [ 15] S. Gottlieb, W. Liu, D. Toussaint, R.L. Renken and R.L. Sugar, Phys. Rev. D 35 (1987) 2531. [ 16] R.V. Gavai, J. Potvin and S. Sanielevici, in preparation. [ 17 ] H.W. Hamber, E. Marinari, G. Parisi and C. Rebbi, Phys. Lett. B 124 (1983) 99.