staggered fermions and topological susceptibility in lattice qcd at β = 5.7

Volume 194, number 3 PHYSICS LETTERS B 13 August 1987

STAGGERED FERMIONS A N D TOPOLOGICAL SUSCEPTIBILITY IN LATTICE QCD AT p = 5.7

Jan SMIT and Jeroen C. VINK Institute of Theoretical Physics, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands

Received 21 April 1987; revised manuscript received 27 May 1987

In this pilot study we use staggered fermions to estimate the topological susceptibility of SU (3) lattice gauge theory at fl = 5.7.

1. Introduction. Recently we have presented a lattice derivation [ 1,2] of the Witten-Veneziano for- mula [3] for the masses of the neutral pseudoscalar mesons,

2 1 2 1 2 m n, - ~m n - ~m~o = (6/f~)~. (1)

In the continuum ;~ is the susceptibility of the topological charge. On the lattice this remains the case if the topological charge Q is defined as

O=xp(m/nr) Tr FsG, (2)

where G stands for the gauge field dependent quark propagator either in the staggered or in the Wilson formulation, F5 represents the Dirac matrix ~5, m is the quark mass parameter and xp is a finite renor- realization constant. In this letter we use staggered fermions with nf=4 flavours and a flavour degenerate mass term. The susceptibility is defined as

~ = ( Q 2 ) / V , (3)

where V is the spacetime volume and the brackets indicate the average over gauge fields.

The derivation of (1) on the lattice can be given under the assumption that the gauge coupling fl and the quark mass parameter m are in the scaling region. This means for example that the quenched flavour singlet pseudoscalar mass is degenerate with the nonsinglet mass, which is a nontrivial requirement for staggered fermions. The susceptibility is to be evaluated for rn~0. For nonzero m there may be perturbative contributions. Furthermore, xp (inde-

pendent of m) is such that the renormalized pseudoscalar density P = xpgFsZ and the scalar density S=;~Z are in the same chiral multiplet (Z is the staggered fermion field). It is important to include this factor xp, since it may deviate strongly from unity: only for f l = ~ , in the continuum limit, xp= 1.

In ref. [ 1 ] we have proposed a nonperturbative method to evaluate xp; the procedure is as follows. Each gauge field configuration U from an equilibrium ensemble is multiplied with a fixed smooth U(1 ) field V carrying a topological charge Q. Next the unrenormalized charge ~)(UV) of the combined field UV is evaluated, where

Q= (m/nf) Tr FsG=O/xv. (4)

Then in the quenched approximation xe is given by

xp ' = ( Q(UV) )/Qnc, (5)

where nc = 3 is the number of colours. Previously we have investigated O in two-dimensional models [4,5], here we extend this exploration to QCD with a pilot study on a lattice of 84 sites. In order not to run into large finite size effects we have used the rather low value fl= 5.7.

The evaluation of (2) presents considerable dif- ficulties since the values of (FsG)(x, x) are needed for all sites x. This prohibits the use of some iterative procedure to invert G--1 _]~_~_ r n as this only provides G(xo, x) for fixed Xo. Obviously the estimate Ex(FsG)(x ,x) = V(FsG)(xo, Xo) is not likely to give good results. We have chosen for the pseud0fermion method [6] to invert D + m . This provides us with

0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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the full inverse G(x, y) but the result is afflicted with a statistical error. This will be further discussed shortly.

2. Pseudofermion method. Denoting the pseudofermion field by O(x) it can be shown that

G(x,y)= ~ (-D+m)(x,z)<~(z)O*(y))~, (6)

m 2 (~= n--f- ~ <~*(x)Fs(x, y)f~(y) )0, (7)

where the brackets ( )~ indicate the pseudofermion average with weight e s, S=O*(DZ-m2)~. The her- mitian matrix /'5 connects sites that are four links apart.

The pseudofermion average is estimated using a Monte Carlo method. We have written a code that uses a heatbath updating and a "multi hit technique" [7] to reduce the variance of (7). For each value of m considered, an equilibrium ensemble with 2085 configurations was generated, obtained by perform- ing 9140 pseudofermion sweeps, discarding the first 800 and taking measurements every 4 sweeps. We checked that the configurations had reached equilibrium by following the evolution of (7) in MC time and also by varying the number of discarded initial measurements. An estimate of the error is obtained using a binning procedure.

The pseudofermion algorithm provides us with an estimate of the charge (~ of one particular gauge field. In order to compute the susceptibility, an average over gauge fields has to be taken. For this averaging we used 28 configurations separated by 1200 single hit Metropolis sweeps, which were generated in Edinburgh and kindly made available to us by K.C. Bowler.

3. Error analysis. The estimate for (~ is subject to a statistical error since the number of pseudofermion measurements is finite. The value after averaging N independent measurements can be written as

O_N:O_~+6~,, (8)

where O ~ = 0 and gN is a random fluctuation with an approximately normal distribution, provided N is sufficiently large. The size of this fluctuation can be estimated using the standard deviation of Q:

t7 2 = (m4/n~)( < ~t/~5¢ ~tF5 ~>¢, -- < ~tF5~>~), (9)

(g2)o =a2/N. (10)

The value of Q~ averaged over gauge fields is

(02)=((O_~+6N)2)=(O2)+(a2)/N, (11)

where 62 is replaced by its pseudofermion expec- tation value (10), and the cross term is zero because ( Q ~ S N ) - - ( Q ~ ) ( 8 ~ v ) . This shows that an unbiased estimator for the renormalized susceptibility is given by

)~-~ ( (/¢p 0N) 2 )/V-- ((/~pa) 2 )/NV. (12)

It is possible to give a crude estimate for the value of (~pa) 2. Relation (9) is rewritten using the eigenfunctions of IZ),

DL~ = i ~ , ~ , (13)

the index a distinguishes degenerate eigenfunctions. We now assume that XvF5 behaves as in the continuum:

xpf~,,~Fsf,a=~5,~a, 2,= - 2 , # 0 , (14a)

=Z,~d, a, 2t=2s=O. (14b)

The zero modes are chosen to have a definite chir- ality Z,~ = + 1. Then it follows that

m 4 1 (XPO')2 ~- ~ r 2f s,otZ (22 + m 2 ) 2 , (15)

which can be related to the quark condensate

(~z),

m 1 <ez>-- +m2), (16)

(/¢e0") 2 Vm4 d ( ~ ) - - nf dm 2

m V -- 2n-~ Co + (zero modes) + O ( m 2 ) , (17)

where the quark condensate is parametrized as

()~Z) =Co +cl m+c2m log m

+ (zero modes)/mV+O(m2). (18)

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Relation (17) shows that/(pO is reduced for small quark mass but it grows with increasing volume V. This is no problem for the susceptibility since the factor Vdrops out of (12) but it will become increas- ingly difficult to identify the (integer) charge of a configuration with the pseudofermion method when V increases.

Using the pseudofermion method we have measured the quark condensate and found 4co= 0.25 + 0.02, which is consistent with values reported in the literature [ 8]. I f the values nf--4 and m = 0.2 are supplied it follows that ( /¢pO ' )2= l .5x10 -3 × V/N, neglecting the contribution of the zero modes. Therefore, this crude estimate suggests that an error of 0.1 on O on a V= 84 lattice requires ~ 600 independent measurements, which is a feasible number.

4. Results for Q. First we turn to the mass dependence of(~. In fig. 1 we have plotted a Q(m, UV) for 7 values of m. The field U is multiplied with a background U(1 ) field Vwith Q= 1 but the plot will have a similar shape without a background field.

It is seen that Q approaches zero for large m and actually it behaves as 1/m 2 as can be inferred from (7). For small values of m it also drops to zero and this can be understood from the "zero-mode shift" effect [4]: the Dirac operator in the external field UV would have exact zero modes in the continuum limit but for finite fl these eigenvalues fluctuate away from zero since there is no topological obstruction on the lattice preventing this. We expect that these

0.i00

{ + +

0.075

O.OSO ¢

0 ,025

+

+

°'°°°o.oo o12o o14o o16o-

+

o~oo 11oo m

Fig. 1. Unrenormalized charge O(m, UV)/3 for a typical field U (indicated by ©) and the charge ()(rn, V)/30 of the smooth U(l ) field V (indicated by + ).

"zero modes" are still characterized by a relatively large value of f tFs f such that they dominate in the spectral representation 4(~= ~f*Fsfm2/(22 + rn2). As a result of the zero-mode shifts, Q(m) ocm 2 for m--.0. The position of the maximum of [Q(m) l gives an indication of the size of this zero-mode shift. I Q(rn) I reaches its maximum at m E 0.2. (We would like to stress that the eigenvalues of the "zero modes" are not necessarily the smallest eigenvalues; smaller ones may occur with a small value of the corresponding f tFs f We observed this phenomenon in two-dimensional QED [ 5 ]. Note the difference between ~ and <2Z >: the latter depends on the spectral density of all small eigenvalues.)

In fig. 1 we have also plotted Q(m, V) for the background U(1) field V, scaled down to make it comparable with Q(m, UV). There is no zero-mode- shift effect here and O(m, V) is approximately constant for small m, O(m 2) effects are small for m < 0.1. Comparing the two plots it is seen that the region where Q(m, UV) should be approximately constant (m < 0.1 ) is inaccessible due to the zero-mode-shift effect. Apart from this effect and an overall scale factor the mass dependence is almost the same. There- fore the mass dependence in the ratio O(m, UV)/O_(m, V) is reduced for values of m outside the zero-mode-shift region.

This motivates the choice for the mass values for which Q(m, U) is measured on the equilibrium ensemble. We have taken m = 0 . 2 and 0.7. The smaller m value is just outside the zero-mode-shift region and [(~(m, U) I is maximal; for the larger m value, the O(m 2) terms in Q(m, U) are certainly important but they may be largely cancelled in ratios such as O(m, UV)/Q_(m, V). This happens in the U(1 ) model [5]. We are interested to see if the same cancellation occurs here as well.

5. Renormalization. Relation (5) provides a method to evaluate tcp in the scaling region but for gauge fields a t f l= 5.7, the RHS of (5) has the typical m dependence shown in fig. 1. Hence, scaling is not present yet. However, we may still try to exploit the m dependence in

x~-~(m, V)= ( Q ( m , uv)>/Qnc, (19)

to reduce the m dependence of the renormalized charge Q=xp(m, V)O.(m, U). In general xp(m, V)

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depends also on the background field Fused in (19). We have not investigated this dependence here but we expect it to be small for the smooth constant field strength gauge fields V that we consider [ 4 ]. We use Q = 1 in the sequel. In the scaling region the m and V dependence should disappear, xp(m, F ) = Xp. To get an estimate of Xp, we consider the ratio

x~l(rn, V) (~)(rn, UV)) (20) R i Q.(m, V)/Qnc - O.(m, v)

for values of rn outside the zero-mode-shift region, and extrapolate to m = 0 . The O ( m 2) and even the finite size effects are to some extent cancelled in this ratio and R should be equal to Xv if this cancellation were exact.

Since the renormalization is caused by the short distance fluctuations common to all configurations U, it should be possible to estimate xv with a rea- sonable accuracy, using only a few configurations, provided that the average charge (~ of these configurations is zero. After measuring (~ for all 28 configurations we have selected 9 configurations with average charge zero and obtained xb -1 using these configurations as indicated in (19). The error is estimated by dividing the set into 3 bins with approximately zero average (~. The results are in table 1. For rn=0.2 and 0.7 we find R~8 .8 and 7.6, which values are indeed rather close to each other. A linear extrapolation to rn=0 gives the estimate /~p~-~- 9.3+1.2.

Having determined rp (m, V), the renormalized charges O of each configuration can be evaluated. For sufficiently smooth configurations they will be close to integer values and their distribution will show sharp peaks but for finite fl these peaks are broad- ened [ 5 ]. The charge will not be integer either if very many zero modes contribute to O (when the volume becomes very large at a given value offl), but we do not expect this problem here. In fig. 2 we have plotted a histogram of Q(rn = 0.2). Since the bin width

10

I o -3 -2 0 1 2 2

Fig. 2. Histogram of the distribution of the renormalized charges Oa t m--0.2 .

is 0.5, the pseudofermion noise on Q, which is ~<0.2, will not have much effect. There is not much indication of peaking at this value of ft.

6. Susceptibility. Table 1 shows the various results that lead to the renormalized susceptibility. The first column shows xe(m, V) which is rn dependent; the second and third column show the unrenormalized average value of ()2 which is also m dependent and the contribution of the noise & The last column gives the renormalized unbiased susceptibility (12). The quoted error is mainly due to the small number of gauge field configurations used in the average. It is remarkable that the susceptibility comes out being almost m independent, within errors. This makes it straightforward to extrapolate to m - 0 , giving ( 0 2 ) = 1.1 with an error of about 30%.

I f relation (1) is supplied with the experimental values of the various masses, it follows that )? = (180 MeV) 4. In order to compare our result with this value, it is necessary to know the inverse lattice distance in MeV. Measurements of the string tension [9] lead to a-1=0.95_+0.03 GeV which gives )?=(122_+9 MeV) 4. The value a 1=1.14_+0.02 GeV, following from ref. [10], would lead to )~=(146+11 MeV) 4.

Table 1 Renormalization factor, unrenormalized average (~2, contribution of the pseudofermion fluctuations and the renormalized, unbiased average 0 2.

m Kp(m, F) <02> <62> <Q2>

0.2 10.5 +-0.9 1 .00+0 .24× 10 -2 0.03-+0.01N 10 -2 1.07+0.27 0.7 20.3-+1.1 0 .27+0 .07X 10 2 0 .04±0.01N 10-2 0.96+_0.27

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It is also possible to de te rmine the scale by measuring the p ion decay constant , using the pseudofermion method. We have measured quenched correla t ion funct ions for the pion using a zero-l ink operator and also using the four-link Fs operator. The corresponding masses should be degenerate in the scaling region. Wi th the a forement ioned statistics it was possible for the signal to drop by 10 -2 before getting lost in the noise. This enabled us to extract a p ion mass m ~ l . 1 at m = 0 . 2 . F rom the much weaker signal of the flavour singlet opera tor we could only infer that the corresponding mass is roughly 1.5-2 t imes heavier.

I f we use the relat ion 2 2 m~f~ = m ( ~ x ) , and sub- st i tute the measured value for m~ and ( Z Z ) , we obtain f~ ~ 0.10. Usingf~ = 93 MeV, we find a - ~ ~ 0.9 GeV, which is consistent with the string tension result [9] .

Since there is evident ly not yet scaling for fermions at f l = 5 . 7 (strong f lavour symmet ry break- ing), we do not wish to a t tach physical significance to the number ob ta ined for ;~. However , the reader may find it interest ing to compare our result with other values ob ta ined by different methods at the same value offl: 104a4 Z = 2 0 [ 11 ]; 25 [ 12]; 6.5 [ 13]; 2.7 [this work]; (lattice size 84, except for ref. [13] which uses a 54 lattice; ref. [13] also investigates "cuts on the m i n i m u m sizes o f the topological fluc- tuat ion to be inc luded" , which leads to a lower value l O'~ a4z = 2.2 ).

7. Conclusion. The lattice der ivat ion of (1) assumes that fl and m are in the scaling region. For fl = 5.7 we are forced to use large values for m which are evi- dently not in the saling region and for which ~ is strongly rn dependent . However , the renormal ized susceptibi l i ty comes out being almost independen t o f m. F rom our results it can be inferred, that the fluctuations in the gauge fields at fl = 5.7 do not allow the staggered fermions to behave like con t inuum fer- talons. The renormal ized charges Q are not cluster- ing a round integers; the mass o f the quenched four-

l ink pion is not degenerate with the zero-l ink pion. This lack o f f lavour symmet ry is also expressed in the large zero-mode-shif t effect. For exact f lavour symmet ry the almost zero eigenvalues must be degenerate and therefore zero. Restora t ion of flavour symmet ry in the spectrum of 12) on the scale of the quark mass implies that 52/m-~0 for f l ~ , where 82 is the difference between almost degenerate eigenvalues. Results should improve considerably at higher fl values.

We wish to thank K.C. Bowler very much for giving us gauge field configurations. Pseudofermion cal- culat ions were done on the Cyber 205 computer in Amsterdam. This work is suppor ted in part by the "St icht ing voor Fundamentee l Onderzoek der Mate- rie ( F O M ) " .

References

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staggered fermions and topological susceptibility in lattice qcd at β = 5.7

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