Chiral phase transition in lattice QCD with dynamical quarks

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PHYSICS LETTERS

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CHIRAL PHASE TRANSITION IN LATTICE QCD WITH DYNAMICAL QUARKS R.V. GAVAI Zentrum jTtr inrerdiszipliniire Forschung and Fakultiit flir Physik, Universit~t BielefeM, Bielefeld, Germany

and M. LEV and B. PETERSSON Fakultiit flit Physik, Universitiit Bielefeld, Bielefeld, Germany Received 31 July 1984

The effect of virtual light quark loops on the chiral phase transition in lattice QCD with staggered fermions is investigated by employing the pseudo-fermion Monte Carlo method on a 63 × 2 lattice. The variation in the order parameter ( ~ ) is found to become less sharp than the quenched case, indicating a second order chiral phase transition in the full theory.

One way to learn more about a theory is to investigate its consequences in unusual environments such as at Finite temperatures or finite densities. The advent of Monte Carlo simulation techniques for lattice gauge theories has enabled one to obtain interesting results this way [1 ]. In the approximation o f neglecting virtual quark loops (quenched approximation) it has thus been shown that the spontaneously broken [2] chiral symmetry at low temperatures is restored at high temperatures [3] and that for SU(3) a first order chiral phase transition separates these two regimes. At almost the same value o f the temperature (or the coupling constant o f the lattice theory), there is also a first order deconFmement phase transition in the SU(3) Yang-Mills theory, as has been shown by studying the order parameter (L), the average thermal Wilson loop. In view o f the importance o f such results for the early universe or ultra-relativistic heavy ion collisions, it appears desirable to study how dependent they are on the quenched approximation. Indeed, arguments have been presented [4] to suggest that the deconfinement phase transition may be washed away due to the inclusion o f the virtual quark loops. Such arguments may, however, depend on the approximations such as strong coupling, heavy quarks, mean field theory etc. On the other hand, no similar argu492

ments have been presented for the chiral phase transition so far. To study QCD in the presence o f light dynamical quarks, we, therefore, undertook a numerical study o f the SU(3) lattice gauge theory with dynamical staggered fermions. In this letter, we report our results on ( ~ ) , the total energy density and the average thermal Wilson loop, obtained by using the pseudofermion method [5,6] to include the virtual quark loops *x . Let us begin by defining our action: S = SG + S F ,

(1)

with SG=/~ ~

(1 - ~ R 1 etrU~U It, x +V h U xut_ ~ U Xvt )

X,t.t
=/3

~

X,/.t
Px~v ,

(2)

,1 Result s obtained at m = 0.1,but with different boundary conditions, have been published elsewhere, where more details of the method may b.e found (ref. [7 ]). While the importance Of these boundary conditions is unclear, the ones used here are usually more favoured, and although using the other boundary conditions on our lattice seems to sharpen the variation in (~-~0), no significant difference is expected to be found on bigger lattices. 0370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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Xx(Dxx,+m~xx,)Xx, = 2 ( D + m ) x ,

L(x) = tr ( temporal links U~x ) .

(4)

As mentioned above, we have used the pseudofermion method to incorporate the virtual quark loops. Details of this method can be found elsewhere [ 5 - 7 ] . It may suffice here to recall briefly that for lattices with even number of sites in all directions, eq. (5) can be rewritten as

4

Dxx, = 1 ~ r u ( x ) (U~x6x,x+i~ _ UUx? ' 6x ' x ~) 2 ~=1

' -

"

Here U~x is the gauge variable associated with the link connecting sites x and x +/~,/3 = 6/g 2, Xx and Xx are single component fermionic fields associated with site x in a four-dimensional hypercubic lattice and Pu(x) = ( _ 1)xl+-.-+xu-1. Unless explicitly stated, the colour indices will be suppressed and the lattice spacing a set to unity. The euclidean form of the partition function is then given by

z = f l-I dU~x I-I d X x d ~ x e x p ( - S ) , "links

sites

(5)

with (anti-) periodic boundary conditions for (fermionic) gauge variables in the temperature direction. To reduce the surface effects periodic boundary conditions were used in the space directions. The action defined above has a continuous (flavour non-singlet) axial symmetry [8] which is a remnant of the chiral symmetries o f the corresponding continuum theory. The order parameter for the investigation o f its spontaneous breakdown is given by lim

~(~xiXxi)/Ns~, m ~0 Ns-,'~ \ x,i [ ] (6)

tim ( ~ k ) = lira lira

m~ 0 Ns~ ~,

where i is the colour i n d e x , N s is the number of sites and ( ) denotes expectation value with respect to Z in eq. (5). In addition to (~ff), we also studied the total energy density e = eG + e F, where e G and e F (for nf flavours) are given by [1]

eG=fl(( ~ PUxV- ~ P~xV;t/Ns,

(7)

e F = nf ~

(S)

\ \ space

X~X

temp

///

(trD(4)x,x'[D+m]x,1)/4Ns - 3nf/16

H

(3)

and

4 - ~ r)(u) ~=1 ~xx'

27 December 1984

'

and the average thermal Wilson loop (L)with L defined

(9)

Z = f I-I dUx~ exp [ - S G ( U ) + I nf Tr In ( - D 2 + m2)] a links

=-fliI-lsdU~xexp(-Seff).

(10)

Here Tr denotes the sum over both x and i. If in the upgrading U-~ U' in the Metropolis algorithm, 6 U = U ' - U is sufficiently small, then one obtains

6Seff=6SG-nf~_lRetr Jx6Ux " ", 4 x,~

(11)

where all the "currents" J x~, defined by # _1 J:~;i! - ~ Pu(x)[(D+m) -1 +(D-m)-l]x+u,x;i] , (12)

may be evaluated before a full sweep of the gauge variables U is initiated [6]. The estimation O f J xu (and other quantities depending on D) proceeds via another Monte Carlo simulation over bosonic (pseudofermion) Variables ~0x, ~x with the measure exp [ - ~ ( - D 2 + m 2) ~0]. Our calculations were made on a 63 X 2 lattice with nf = 2 , 2 and for quark masses m (in lattice units) = 0.1,0.15,0.2. The numerical procedure consisted o f the following. Starting from a random configuration of gauge variables a quenched configuration was generated by performing 400 MC iterations on the pure gauge system at each/3. Effects of dynamical fermions were then introduced using the method mentioned above. Also for the pseudo-fermion Monte Carlo we used the Metropolis algorithm, essentially due to its simplicity. A full iteration then comprised of 60 pseudo-fermion MC iterations, over last 40 o f which average values o f all the currents J xu were formed, followed by a sweep o f all gauge variables with 6Seff givenby eq. (11). After every 80 such full iterations (ffff) and e F were measured by performing 5000 pseudo-fermion MC iterations and dis-

as

:1:2Ideally our equations are valid for nf = 4n, where n is an integer; see, however, ref. [6]. 493

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Fig. 1. (~-~) as a function of~ = 6 / g 2 . The points indicated by full dots are for the full theory with two massless flavours while the open dots denote the points obtained by extrapolating the data of ref. [10]. carding the first 1000. To judge whether equilibrium had been reached, the whole procedurewas repeated at each/3, starting from a quenched, thermalised configuration at/3 = 5 25 (ordered start). Averages o f all physical quantities were formed only when the results from these two runs were statistically consistent; the first 160 iterations (approximately twice in the critical region) had to be thus always discarded. Typically 7 - 8 measurements of ( t ~ ) w e r e obtained at each/3. Configurations thus generated were then frequently used as the starting point for the calculations at a different quark mass value. The values o f ( ~ ) thus measured at m = 0.1,0.15, 0.2 (in lattice units)were extrapolated linearly ,a to obtain (~qJ) for massless quarks at each/3. The results so obtained are displayed in fig. 1 ; the errors shown are purely statistical. Also shown in fig. 1 are the (~qJ) values obtained also on a 63 X 2 lattice but in the quenched approximation. We have obtained these latter points from the data o f ref. [10] by again a linear fit. One sees clearly that the virtual quark loops push the transition towards lower/3 (to/3 ~- 4.6), as ,3 The linear extrapolation procedure together with our choice of mass values is expected to yield typically a 20% larger value for (~-~0)than the other procedures favoured by other authors, as one can see from the analysis in ref. [9]. 494

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expected naively from the renormalization group equation. Moreover, the abrupt behaviour in the quenched case, characteristic o f a first order phase transition, is reduced substantially, although a sharp change is still indicated by the data. Thus it appears that the full theory (with nf = 2) most likely has a second order chiral phase transition. If the experience gained in the quenched calculations is any guide, we do not expect that calculations on bigger lattices or with different procedures to extract (~qJ) at m --- 0 will alter our conclusions drastically;they are certainly necessary for the question of scaling and an eventual precise determination of the temperature at which the chiral transition occurs. Although the recent resuits on bigger lattices [ 11 ] indicate that asymptotic scaling begins for 13/> 6, it is interesting to note that assuming it were good for/3 = 5.1 (the phase transition point in the quenched theory) and assuming perturbative A-ratios, one obtains the following ratio for T~I~ (quenched) and T~2H ) (with 2 dynamical flavours): = 1.15. Figs. 2 and 3 exhibit our data for the total energy density e (in lattice units) and the magnitude o f average thermal Wilson loop i(L)L respectively. Displayed are the data points for m = 0.1 and 0.2; the points for m = 0.15, not shown here for the sake o f clarity, lie in between the two. Also displayed are the linearly extrapolated values at m = 0.0. One sees that both these quantities have a sharp variation at approximately the same/3-value as the ( ~ ) i.e. at/3 ~ 4 . 5 - 4 . 6 . The changes in both of them due to the change in the quark mass are rather small; m = 0.2 data points appear to form a somewhat steeper curve. Although the limited statistics and the finite size effects may specially affect the lower/3-region of this data, it seems to indicate that the sharp variation is a continuation o f the first order phase transition observed in the quenched theory. Whether the data at fixed quark masses in figs. 2 and 3 or the extrapolations to m = 0 indicate the presence o f a separate deconfmement phase transition is hard to say. One would naively expect such a behaviour of the data in any case, if the sharp change in @~) in fig. 1 is due to a phase transition, and thus due to some nonanalyticity in the partition function. To conclude then, we fred that the inclusion o f virtual light (massless) quark loops ( o f 2 flavours) softens the discontinuous behaviour of ( t ~ ) and other physical

Volume 149B, number 6

PHYSICS LETTERS

27 December 1984

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Ea41

Ea 4

i

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t

0.4

i

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o.81-

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e m =0.1 o m =0.2

i

0.0 4.0

-

I 4.4

I

i 0.01

I 4.8

I

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4.0

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Fig. 2. The total energy density in the full theory, plotted in lattice units as a function of#. The dashed line is the energy density of an ideal gas of masstess quarks and gluons, placed on a 63 ×2 lattice.

I
L>II

1.2-

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Fig. 3. The thermal Wilson loop I(L>Ias a function off.

quantities, observed in the quenched approximation in QCD; a sharp variation, indicating a second order phase transition, is present in all o f them, however. Our results thus cherish the hopes o f seeing new physics in the ultra-relativistic heavy ion collisions.

It is a pleasure to thank Professor J. Engels and Professor H. Satz for stimulating discussions. We grate~lly acknowledge the support o f the Bochum University Computer Center, on whose CYBER 205 this investigation was carried out. 495

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Note added. After this paper was written we received two articles where the same problem is addressed. Polonyi et al. [12] using a 4 X 83 lattice, nf = 4 and the microcanonical method find a transition which looks somewhat sharper than ours, but they consider their results not precise enough to determine the order of the transition. Fucito et al. [13] using a 4 X 63 lattice and n f = 3 claim signs of metastability. Because o f the different value of the number of points in the temperature direction, and differe n t nf, the results in these two articles cannot be directly compared to ours. To determine conclusively the order of the physical transition as a function of nf, clearly more work is needed. References [1] See e.g.J. Kogut, Nucl. Phys. A418 (1984) 381c; H. Satz, Nucl. Phys. A418 (1984) 447c. [2] H. Hamber and G. Parisi, Phys. Rev. Lett. 47 (1981) 1792.

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[ 3 ] J. Kogut, M. St one, H.W.Wyld,W.R. Gibbs, J. Shigemitsu, S.H. Shenker and D.K. Sinclair, Phys. Rev. Lett. 50 (1983) 393. [4] T. Banks and A. Ukawa, Nucl. Phys. B225 [FS9] (1983) 145. [5 ] F. Fucito, E. Marinari, G. Parisi and C. Rebbi, Nucl. Phys. B180 [FS3] (1981) 369. [6] H.W. Hamber, E. Marinari, G. Parisi and C. Rebbi, Phys. Lett. 124B (1983) 99. [7] R.V. Gavai, M. Lev and B. Petersson,Phys. Lett. 140B (1984) 397. [ 8 ] H. Kluberg-Stern, A. Morel, O. Napoly and B. Petersson, Nucl. Phys. B220 [FS8] (1983) 447. [9] I.M. Barbour, P. Gibbs, J.R. Gilchrist, H. Schneider, G. Schierholz and M. Teper, Phys. Lett. 136B (1983) 80. [ 10] J. Kogut, H. Matsuoka, M. Stone, H.W. Wyld, S.H. Shenker, J. Shigemitsu and D.K. Sinclair, Nucl. Phys. B225 [FS9] (1983) 93. [ 11 ] D. Barkai, K.J.M. Moriarty and C. Rebbi, The force between static quarks, Brookhaven preprint BNL-34462. [12] J. Polonyi et al., Phys. Rev. Lett. 53 (1984) 644. [13] F. Fucito et al., Caltech preprint CALT-68-1124.