Quenched QCD at finite density

Volume 256, number 1 PHYSICS LETTERS B 28 February 1991

Quenched QCD at finite density

C.T .H . D a v i e s a n d E .G. K l e p f i s h Department of Physics and Astronomy, University of Glasgow, Glasgow G I 2 8QQ, UK

Received 13 November 1990

We extend earlier calculations in quenched QCD at finite density to large lattices and weak coupling. We continue to find the result that the critical chemical potential at which the chiral symmetry restoration transition is indicated is one half the pion mass. The conclusion is that it will not be possible to obtain a more physically reasonable answer in the continuum limit of the quenched approximation.

I. Introduction

Simple arguments suggest that chiral symmetry restorat ion should occur in QCD at high baryon density [ 1 ]. We would expect the crit ical baryon densi ty to be related to the mass of the lowest lying baryonic state, the proton, since at this densi ty it is no longer advantageous to have a dynamical ly generated effec- t ive mass.

The problem of f inding the critical densi ty can be tackled numerical ly with the techniques of latt ice QCD. A chemical potent ia l for quarks, #a is s imply in t roduced into the fermion matr ix as e ua mult iplying the forward links in the t ime direct ion and e -ua mult iplying the backward ones [2] . The K o g u t - Susskind fermion matr ix becomes

Mxy=madxy+½ ~ [U~(x)~(x)dy ,x+u Iz=Yc,p,2

- Utu(y)rlu(y)dy,x_u]

+ ½ [e ~a Ut-(x)q;(x)6y,x+;

- e - ~ a U}(Y)qi(Y)Oy,x-i] • (1.1)

The problem was first s tudied in the quenched approximat ion, for s implici ty [ 1,3 ]. Of course, there can be no phase t ransi t ion as a function o f # a in this approx imat ion since the gluon fields are generated independent ly o f#a . However, we might still hope to see rapid var ia t ions in the behav iour of the chiral condensate, 2 z ( m a ) , as a function o f # a which would be indicat ive of a phase t ransi t ion in the full theory.

Unfortunately, as we shall describe below, the quenched approximat ion fails to give sensible results. In this paper we assess whether this failure can be ascribed to latt ice artefacts.

A study of the eigenvalue spectrum of the K o g u t - Susskind fermion matrix, M, provides the clearest method of following the behaviour of the chiral condensate as a function of #a and the quark mass ma [3] . Since the mass term appears mult iplying the ident i ty [ eq. ( 1.1 ) ] we can calculate the spectrum of the der ivat ive part, D, and simply shift it backwards and forwards along the real axis to obta in the spec- t rum of M. In the quenched approx imat ion the spec- t rum at all values of ma is thus obta ined for one cal- culat ion of the spectrum of D at a given value o f#a . The chiral condensate, 2 z ( m a ) , given by ( T r M - ~ } in the quenched approximat ion , is then readily calculated.

The eigenvalues of D are purely imaginary at #a - - 0 but move out into the complex plane for #a C0, forming an ellipse. The chiral condensate changes rapidly as a function o f#a at the point where the half- width, A, of the eigenvalue dis t r ibut ion of D is equal to the quark mass, ma. If we call this value o f # the critical value #c, then

A ( # c a ) = m a . (1.2)

Early calculations [3 ] on small lattices and at low values of the gauge coupling, fl, showed a surprising result - #ca was equal to ½ m~a! This result is clearly unphysical since the point carries no baryon number.

68 0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )

Volume 256, number I PHYSICS LETTERS B 28 February 1991

The failure of the quenched approximation here is very unfortunate since calculations with dynamical fermions including a complex determinant are only now becoming possible [4] and remain very diffi- cult. These dynamical calculations give numbers for /tea at strong coupling which agree with mean field calculations and are quite different from those obtained in the quenched approximation. The dynamical calculations will be very expensive to extrapolate to the continuum limit so we must ask whether it might not be possible to extract useful numbers from the quenched approximation with a bit more effort than hitherto. Could it be that the unphysical behaviour on small lattices at strong coupling was an arte- fact of finite volume or lattice spacing? Recent work [ 5 ] has suggested that this latter possibility might be true for a two-dimensional U ( 1 ) model. There a device was used to study the system at high values offl which involved multiplying thermalised configurations by special smooth configurations of non-zero topological charge.

Of course, the question of whether it is the finite lattice spacing which yields the unphysical results of the quenched approximation must eventually be re- solved without this device. We have attempted to do this in QCD by calculating the eigenvalue spectrum of D on plain thermalised configurations with lattice sizes up to 1 6 4 and gauge couplings, fl, up to 6.2. The results are presented in section 2. We find that the behaviour #ca= ½m~a continues to hold and we conclude that this will be the continuum behaviour in the quenched approximation. A simple non-rigorous argument is presented for why this happens. In section 3 we discuss the eigenvalue spectra of smooth configurations with non-zero topological charge at finite/~. We find that the spectra for four-dimensional models differ from the two-dimensional example of ref. [5] in having eigenvalues with small imaginary part and relatively large real part. Section 4 contains our conclusions.

2. Studies of the spectrum on large lattices at weak coupling

The eigenvalue spectrum of D can be used to give a direct determination of Tr M-~ on a finite lattice and then, by averaging over configurations, 2z(ma)

can be calculated. More useful than this is the picture that the spectrum gives of the infinite lattice behaviour of2z(ma) . We can look at the density of eigenvalues from several configurations and imagine the spectrum "filling in" with increasing volume. The behaviour of the condensate resulting from this smooth eigenvalue distribution can be obtained by a useful analogy with 2D electrostatics [ 3 ]. The chiral condensate is the electric field produced at the posi- tion of the mass on the real axis by a distribution of positive charge which is that of the eigenvalue distribution. Viewed in these terms, we can hope to gain information from the eigenvalue spectrum which is free of large finite volume effects, unless the spectrum on small lattices is grossly distorted.

From the early calculations [3], the notable fea- tures of the eigenvalue spectrum of D at finite l t in an SU (3) gauge theory were:

( 1 ) The 6 function distribution of eigenvalues on the imaginary axis obtained for / t=0 disappears completely as soon as/~> 0, to give an ellipse popu- lated with almost uniform density.

(2) The half-width of the ellipse, J, grows with/ta according to

Aoc (/ta) 2 . (2.1)

A consequence of point ( 1 ) is that chiral symmetry is restored, 2Z(0 ) = 0, for all/t > 0, despite being broken at ~t= 0. From the electrostatic analogy it is clear that chiral symmetry breaking can only occur if there is an accumulation of eigenvalues on a line through the origin,, such as the imaginary axis. Point (2) has consequences for the behaviour of the chiral condensate at finite mass and leads to a prediction of how the value of/~c at which the high density phase is en- tered, depends on the quark mass, ma. 27 . (ma ) is non- zero at small ,u for m a non-zero. As/t is increased, nothing changes until 2z(ma) suddenly starts to fall at the point where the half-width of the distribution, A, is equal to ma. Using the electrostatic analogy this is equivalent to the fact that the magnitude of the electric field falls as we enter the charge distribution. If this point is taken as/Zca, we find

(t~ca ) Z= Crna . (2.2)

The constant of proportionality, C, was found to be such that #ca = ½ m~a at fl= 0.

These results from the quenched approximation

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make no physical sense but the calculations were performed far from the continuum limit. I f we go towards this limit, with larger lattices and weaker couplings, and find that the above result continues to hold in the scaling region i.e./~c= ½m~, then we must conclude that the quenched approximation is wrong. On the other hand, the situation may change as the lattice spacing is reduced to give physically reasonable numbers in the continuum limit.

The eigenvalue distribution must change radically to give sensible answers. For small/~ we must have an accumulation of small eigenvalues on a line through the origin, probably the imaginary axis, so that chiral symmetry is broken as a t /~= 0. At/~ equal to the smallest baryonic mass these eigenvalues can then leave the axis, moving either upward, as in the finite temperature transition, or outward. In a U (N) gauge theory, with no baryons, there should be no phase transition as a function of~t. One might hope in that case that the small eigenvalues (at least) would stay on the imaginary axis independently of/~. This is the picture envisaged in ref. [ 5 ].

The simplest possibility that could give the correct answer would be for the whole eigenvalue spectrum to collapse on the imaginary axis as the lattice spacing, a, is made smaller at fixed physical /~ ( < mba~yon). The half-width, ~f, must then go to zero faster than a physical mass in lattice units vanishes. If, on the other hand, A behaves like a mass in lattice units then there is a physical relationship, albeit wrong, between/tc and m. IfA remains fixed or grows as a is reduced than the quenched approximation is giving really nonsensical results. Later we will discuss the possibility that only a part of the spectrum col- lapses on the imaginary axis. First we describe results on the variation of A as a function of ft.

The eigenvalues of the matrix D were calculated on various lattice sizes for various couplings, fl, using the complex Lanczos algorithm [ 6 ]. We generally used antiperiodic boundary conditions in all directions on the fermion field, although using periodic conditions gives essentially identical results on these configurations. On large lattices it is not possible to calculate all the eigenvalues. The ones that converge first are luckily the ones on the outer edge of the spectrum. Since we are only interested at this stage in the width of the spectrum it is sufficient to run the Lanczos algorithm until the largest real or almost real eigenval-

ues have converged. This generally takes a few thou- sand iterations, growing with the lattice volume. Fig. 1 shows a typical result for the first 2392 eigenvalues to converge in the spectrum of D from one configuration at fl= 5.7 and p.a = 0.5 on a n 8 4 lattice. The half- width, A, in this case is 0.163, assessing where the edge of the distribution is by eye. Our error estimates for zl take into account odd eigenvalues lying by fluctua- tion outside the main distribution. We do not find large variations in the width from configuration to configuration.

Consider first the strong coupling limit, fl= 0.0. Fig. 2 shows a plot of the half-width of the distribution, 3, against (2#a) 2. Also shown is the straight line corresponding to m a = ( 1 / 4 . 6 )(m,~a )2 which is the next- to-leading order mean field prediction [7] for the pion mass at fl= 0.0, calculated from the standard Kogut-Susskind fermion matrix [equation (1.1) with/~a = 0.0 ]. The agreement between the curves is very good, implying that given a value for m a and a corresponding m,~a, the eigenvalue distribution in the presence of a chemical potential will have a width equal to m a when #a = ½ m~a. As explained above, this value of ira is the critical one at which the chiral condensate will start to fall from its value at high mass and where the quenched approximation believes that chiral symmetry is restored. Data on smaller vol- umes give identical results, showing that increasing

2.5

O. 5 ""~!? ~~ '~'~ <

0.0

• , ,-,~ "'L~".~ -0. 5 S~ :.~ ' ~,%'.

~..~. ~,,,..

-~.o .'~. :~.

-2. o ' ~ , ~ , ~ . _ . ~ f

- 2 . 5 I I I I I I I I I I

-0.25-0.20-0. 15-0. I0-0.05 0.00 0.05 0.10 0.15 0.20 0.25

ReX

Fig. I. The outer eigenvalues o f the spectrum o f D fo r one conf ig-

u ra t ion on an 84 latt ice at # = 5.7 and I ~ = 0.5.

70


0 . 3 "

0.2.

0 .1

/ !

/

/ /

/ T

/

/

/

/ . . . . t .

0 . 5 0

0 . 0 • . . i . . . .

0.00 1 . 0 0 1 . 5 0

2 (2~a)

Fig. 2. A plot of the half-width of the spectrum of D, A, against (2p~) 2 on an 8 4 lattice. We use the following symbols: + fl=0.0; [] fl= 5.7; @ data for the pion mass at various values of the quark mass at fl=5.7 [8], plotted as ma versus (m,~a) 2. The straight lines are (m~a)2=Bma with dotted lines: B-- 4.6, the mean field prediction at fl=0, and full lines: B=7.5, a fit to the pion mass data at fl= 5.7.

the volume has simply filled in the spectrum in the expected manner. Results with the U (3) gauge group lie on the same curve despite the fact that no phase transition as a function of/~a should occur at all (and cannot in the full theory). Strong coupling calculations thus indicate something is wrong with the quenched approximation here and it is not a finite volume effect.

We now turn to calculations at weaker gauge coupling. We have calculated the outer eigenvalues in the spectrum of five 84 configurations at f l= 5.7 and one

16 4 configuration at f l= 6.2. These couplings and lattice sizes were chosen because data were available for these lattices on the pion mass and its (Kogut - Susskind) quark mass dependence in the quenched approximation. Large lattices are necessary because there are finite temperature effects associated with the deconfinement phase transition which cause the eigenvalues to leave the real axis and behave more like the free spectrum. If this happens, as it did in the early work on 44 lattices [ 3 ], a calculation of the half-width of the distribution is unreliable. We find that it is eigenvalues close to the real axis which are most sensi- tive to changes in ft. In figs. 2 and 3 we show our re- suits on a plot of A against (2#a) 2 as before. Also shown are the pion mass data from the Edinburgh and Los Alamos groups [ 8-10 ] and the lines corresponding to their fits of the form (m~a) 2 = B m a for small m a . Since physically B must be proportional to the lattice spacing, it will vanish in the a - , 0 limit. How- ever, B actually increases from f l=0 to a maximum at f l= 5.7 before falling with increasing fl in the (weak) scaling region above fl= 6.0. Our curves agree with the data on the pion mass and follow the fl dependence of the constant B exactly. In fact our data seem to represent no more than a m e a s u r e m e n t of the pion mass at various quark masses. At f l= 6.2, we even re- produce the quark mass dependence of the pion mass when it deviates from the line (m~a)2= B m a at large m.

We conclude that the half-width of the eigenvalue distribution of D will scale towards the cont inuum limit as the mass m a which corresponds to a pion mass, m ~ a = 2 z t a . A simple argument can be presented to show why this happens, or, equivalently, why we expect the matrix M to have its first zero mode as a function of/z at #ca = ~ m~a. The argument is very similar to that o f Gibbs [ 11 ] but is perhaps easier to follow.

Consider the following positive semi-definite quantity, H:

H = ~ T r c o l o u r M o x l ( M * ) 2 o ~ . (2.3) x

At/~a = 0, H represents the pion propagator. There- fore we expect

H ( / l a = 0 ) larger e . . . . t. (2.4)

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0 . 3 -

< 1

0 . 2

0 . 1

a -

0 . 0 • , . i . , • i • t .

o.o 0.2 0.4 0.6 0.8 1,o ~.2

(21aa) 2

Fig. 3. A plot of the half-width of the spectrum of D, LI, against (2/~a) 2 on a 164 lattice, where [] fl= 6.2. Also plotted are data for the pion mass at various values of the quark mass as ma versus (m~a)2, where • fl=6.2 [10], and • fl=6.15 [9]. The straight line is (m,,a)2=Bma with B=4.2, a fit to the pion mass data at fl=6.2.

To see how H is modified in the presence of a chemical potential we can consider its expansion in terms of loops of gauge fields. The simplest loops which join o and x with only minor deviations in the t ime direction will pick up a factor ofe 2~'. This is because gauge links point ing upwards in time in M and those pointing downwards in M t are multiplied by e ~. The gauge loops themselves will have a value independent ofl ta in the quenched approximation. Thus if we neglect loops that go right around the lattice in the time direction,

H(lza) large, e_<m.a 2~a)t. (2.5)

This form for H becomes obvious if we note that, at /mve 0, H represents the propagator of a #~q2 pion in a (quenched) theory where #1 = - #2 ~

From eq. (2.5) we see that as/ta is increased from zero the exponent in the exponential decay becomes

smaller and vanishes at /~ca= ½m,,a (the (12q~ pion gives solution #ca = - ½ m,~a). If we sum over t and consider the infinite volume limit, T r ( M * M ) - l will start from a finite value at small/za and diverge as/m approaches ½ rn,~a, signalling the appearance of a zero eigenvalue. This will also be a zero eigenvalue of M, or, equivalently, a real eigenvalue of D at A= ma. This is what we find, as described above.

The conclusions we draw from the change in width of the spectrum with fl would be invalidated if the density of eigenvalues in the spectrum were to change markedly. If the density increased on the imaginary axis, this could become a 6 function in the a-+0 limit and give the correct answer even if the overall width of the distribution behaved as above. We cannot check this by calculating the eigenvalues of D on a large lattice since the eigenvalues near the imaginary axis take too long to converge. What we can do is study the density of small eigenvalues of M t M as a function of ma. Since M t M and M have common zero modes, this should give some informat ion about the density of eigenvalues of D at different points on the real axis. We can then see whether the density is decreasing rapidly as we move away from the imaginary axis.

We have done this for the 164 configuration at

f l= 6.2 with ma = 0.0 and m a = 0.05. We calculate all eigenvalues o f M t M less than 0.00025 ( ~ equivalent to eigenvalues of M < 0.02 ). We find that the density of small eigenvalues actually increases with the mass. It therefore seems unlikely that a ~ function is devel- oping on the imaginary axis in the spectrum of D at weak coupling, Our conclusion #c=½m~ remains valid.

3. Smooth configurations with topological charge

In studying the eigenvalue spectrum for D for the

~J We thank Jan Smit for pointing this out [ 12 ].

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U ( 1 ) gauge theory in 2D, Vink [ 5 ] made use of a smooth configuration, V, of non-zero topological charge which he explicitly constructed. Surprisingly this gave an eigenvalue distribution which was on the imaginary axis for all small eigenvalues.

In ref. [ 5 ] configurations at finite fl, U, are multiplied by V and the eigenvalue spectrum of the resulting configuration calculated. The resulting spectrum grows rapidly narrower in physical units as fl is increased. It is argued that this demonstrates that the finite width of the spectrum on the original configuration, U, is a spurious effect induced by the finite lattice spacing. We believe our results show that the finite width is a real physical prediction, unfortunately wrong, from the quenched approximation. Fi- nite volume effects could account for the results o f ref. [5]. Small lattices at large values off l will have essentially free gauge fields. When multiplied by V the spectrum will become close to that of Vwith zero width. In the same way, if no multiplication by Vwere done the spectrum would become like that of the free case, with width A3c#a. Neither of these results re- flects the true physical result on a lattice large enough to allow for a low mass pion.

For interest's sake we have studied the spectrum of D from the smooth configurations of non-zero topological charge defined in ref. [ 5 ] but now in 4D. The first 1396 eigenvalues to converge f o r / m = 1.0 on an 84 lattice are shown in fig. 4. The results from SU (2), SU (3) and U (3) gauge groups are identical. There are clear signs of eigenvalues close to the real axis which are not purely imaginary and which move out- wards with/ta. This is in clear contrast to the results in 2D. We have also calculated spectra from instanton-like configurations placed on the lattice and smoothed by cooling [13]. These spectra are very similar to the one shown in fig. 4, although not as regular.

4. C o n c l u s i o n s

We have demonstrated that/to for the chiral symmetry restoration transition in QCD as provided by the quenched approximation follows the behaviour ~tc = ~ ms as we change the gauge coupling from f l= 0.0 to f l= 6.2. There is no sign o f a different and more desirable answer as we go to smaller lattice spacing.

2 . 5

2 . 0

1 , 5

1 . 0

0 . 5

0 , 0

- 0 , 5

- 1 , 0

- 1 . 5

. . . . i / i i i i i \ \ . . . 1 / . / . • . - \ . . \ . ~

• ' /~J" . .. .

" .-..:...

. / . . .

',.~..~. : : , . . / . ( : . - 2 . 0 . ', ~ ~ i i i I /

- 1 . 2 - 1 . 0 - 0 . 6 - 0 . 6 - 0 . Z+-0 .2 0 . 0 0 . 2 O.Z~ 0 . 6 0 . 6 1 . 0 1 . 2

Re X

Fig. 4. The outer eigenvalues of the spectrum of D for an explicit smooth configuration with topological charge 2 [ 5 ] on an 8 4 lattice at#a= 1.0.

We must conclude that the quenched approximation will give the same wrong result all the way to the con- t inuum limit. The study of QCD at finite chemical potential is one of the few examples of a calculation which must clearly be performed in the full theory.

A c k n o w l e d g e m e n t

We are grateful to Ian Barbour, Jeroen Vink and David Sutherland for useful discussions and to Mike Teper for giving us the programme to generate smooth instanton configurations. The calculations described here were performed on a number of different com- puters. Preliminary work was done on the CRAY- XMP at the Supercomputing Facility of Ohio State University, Columbus, Ohio. C.T.H.D. thanks John Wilkins and the Physics Department at Columbus for hospitality during that stage of the work. Later work was done on the IBM 3090/VF machines at Glasgow University and the Atlas Centre, Rutherford Labo- ratory. Both machines are supported by IBM under study contracts. C.T.H.D. thanks the UK SERC for an Advanced Fellowship and E.G.K. is supported under SERC grant G R / F 1374.4.

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Refe rences

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[ 2 ] P. Hasenfratz and F. Karsch, Phys. Lett. B 125 ( 1983 ) 308. [ 3 ] I. Barbour, N. Behilil, E. Dagotto, F. Karsch, A. Moreo, M.

Stone and H. Wyld, Nucl. Phys. B 275 (1986) 296. [4] I.M. Barbour, C.T.H. Davies and Z. Sabeur, Phys. Lett. B

215 (1988) 567; I.M. Barbour and Z.A. Sabeur, Nucl. Phys. B 342 (1990) 269; F. Karsch and K. Miitter, Nucl. Phys. B 313 (1989) 541; A. Gocksch, Phys. Rev. Lett. 61 (1988) 2054.

[5] J.C. Vink, Nucl. Phys. B 323 (1989) 399.

[ 6 ] J. Cullum and R.A. Willoughby, Lanczos algorithms for large symmetric eigenvalue computations (Birkhaiiser, Basel, 1985); C.T.H. Davies and I.M. Barbour, The eigenvalues of the fermion matrix in lattice gauge theories, Glasgow University preprint GUTPA/90/5-3, in preparation.

[7 ] T. Jolicoeur et al., Nucl. Phys. B 235 (1984) 455. [ 8] K. Bowler et al., Phys. Lett. B 162 (1985 ) 354. [9] K. Bowler et al., Nucl. Phys. B 296 (1988) 732.

[ 10] R. Gupta et al., Phys. Rev. D 36 (1987) 2813. [ 11 ] P.E. Gibbs, Phys. Lett. B 172 (1986) 53. [ 12] See also A. Gocksch, Phys. Rev. D 37 (1988) 1014. [ 13 ] M. Teper, Large instantons in lattice gauge theory and their

stability under "cooling", Oxford University preprint Oxford-TP-59/88.

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Quenched QCD at finite density