Monte Carlo measurement of the topological susceptibility in SU (3) lattice gauge theory

Volume 127B, number 3,4

PHYSICS LETTERS

28 July 1983

MONTE CARLO MEASUREMENT OF THE TOPOLOGICAL SUSCEPTIBILITY IN SU (3) LATTICE GAUGE THEORY K. FABRICIUS

Physics Department, University of Wuppertal, Wuppertal, Fed. Rep. Germany and

G.C. ROSSI

Istituto di Fisica, Universitd di Roma I Dipartimento di Fisica, Universitgldi Roma H INFN, Sezione di Roma, Italy Received 11 April 1983

A Monte Carlo measurement of the topological susceptibility A = f d4x(OIT(Q (x) Q (0))10), where Q (x) is the topological charge density, is performed in the SU (3) case on a 44 lattice. We find in string tension unitsA 1 / 4 - (0.12 +_ 0.02)x/~ as in the SU (2) case. For x/rK-= 420 MeV this corresponds to A 1/4 = (50 -+ 10) MeV.

An elegant and physically appealing solution of the UA (1) [1] problem based on 1/N and/or topological arguments has been proposed already some years ago in refs. [2] and [3] and further elaborated in ref. [4]. It requires

A = f d 4 x < O l r ( Q ( x ) Q(O))lO)noquarkloops,

A L = ~ 4 n ~ / \' q~ L L) a~ A , n o~0

(1)

with

v

-- 1

with the definition

+-4 QL _ 24 . 32 • 7r2 ta,v,p,o=+-I euvP~r X tr (Un,u Un+u, v Un+u+v,p Un+u+v+p,o

Q = (g2/32zr2) F~vff'~v , ]~a

(3)

~a

- ~-euvpoJ'~a,

× Utn+v+o+o,uU?n+p+o,vU~fn+o,,U?n,a), (2)

to be different from zero despite the fact that Q(x) can be formally written as a total divergence. In this letter we present the results of a Monte Carlo study of A on a 44 periodic lattice for the pure gauge SU(3) theory with Wilson's action. The SU(2) case has been discussed in refs. [5] and [6], hereafter referred to as I and II , t . As in I we write ,1 We use here the same notations as in I and II. 0 0 3 1 - 9 1 6 3 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 03.00 © 1983 North-Holland

e1234 = --e2134 = --e-1234-- 1 .

(4) (5)

We have measured A L in the range 5.0 ~
6/g 2 <~ 10.0 at 16 distinct values ofl5 with a m i n i m u m of about 2000 sweeps per point. The total number of sweeps was 66 700 corresponding to about 60 hours of CDC 7600 CPU time. The results are presented in table 1 where simply the statistical error is attributed to each point ,2. We ,2 For footnote see next page. 229

Volume 127B, number 3,4

PHYSICS LETTERS

Table 1 Monte Carlo data for n42aSa4A~c [see eqs. (3) and (4)].

28 July 1983

T

a

6.0

13

7r4218a4ALc

Width

Number of sweeps

5.00 5.25 5.50 5.55 5.58 5.63 5.70 5.75 6.00 6.25 6.50 7.00 7.50 8.00 9.00 10.00

1876 1688 1532 1408 1351 1250 1121 1090 943 798 692 567 466 370 265 187

59 54 35 31 44 22 17 18 19 16 21 19 15 15 10 6

2900 2700 4600 4900 2000 7400 9400 8900 5300 5400 2900 2300 1900 1900 1800 2400

A

~m

=

4.0

3.0

5.0

_

7r4218aaAL.T" = C3//33 + C4//34 + C5//35 .

(i)

5.8

6.2

I

5,0

4.0

3.0

2.0

'"I

....

J ....

50

J ....

' ....

i ....

~ . . . . . . . . .

x ....

5.4

i . . . . . . . . .

5.8

~ ....

, ....

....

i

8.2

c

(6)

As for the other coefficients we tried to determine the first few of them b y fitting our data for/3 > 6.50. Three forms for A L in p e r t u r b a t i o n theory were considered:

5.4

I

6.0

also report in table 1 the n u m b e r of sweeps taken at each temperature. The procedure to extract from the raw data o f table 1 the physical value o f A has been described in detail in I and II and it will n o t be repeated here. We only record the value of the coefficient of the computed first n o n trivial term in the perturbative (large /3) expansion of 7r4218a4AL. For a 44 lattice one finds C3 = 14321 • 3 3 / 2 .

5.0

6.0

5.0 ¸

"%

49

(7a) 311

(ii)

rra218aaA P.T. L = C3//33 + C4/~ 4

(7b)

(iii)

7ra218a4AL.T" = C3//33 .

(7c)

2J~

The resulting values of log [a 4 ( A L C A L.T.) ] are shown in fig. l a , b , c , respectively in the range 5.0 ~< /3 ~< 6.25. Above/3 = 6.00 the signal a4(ALMc -- ALpT ) • L " " IS smaller t h a n the statistical error on a 4 AMC. In any case for large/3 one should see finite volume effects. The straight line in fig. l a , b , c , represents the small

5.0

As it is well known, the statistical error is certainly an underestimate of the effect of thermal fluctuations.

230

5,8

6.2

a (large/3) scaling behaviour expected for a physical q u a n t i t y o f d i m e n s i o n (mass)4:

a4(ALc_AL.T.) :~2

fi .4

L --AD L ,r )] as a function of j3, Fig. 1. Plot of log[It4 21 8 4a (A~r, assuming 7r4 2 1 8 4a Ap T of the form (a) C3/[33 + C4/[34 + Cs/t35 (b) C3/~ 3 + C4/~ 4, (ci C3/t33.

c~ exp ( - - 1~6 7r2/3).

The data seem to show such a behaviour for t3/>

(8)

Volume 127B, number 3,4

PHYSICS LETTERS

Table 2 Coefficients of the perturbative tail for the cases (7a), (7b) and (7e). C3 (7a) (7b) (7c)

1.93 X 10 s 1.93 X 105 1.93 X 105

C4 2.40 X 104 -3.47 x 104 0

(50 -+ 10) MeV.

1601) I

I

1200

-3.86 X 105 0 0

4O0 I BOO

(9)

a numerical value which is the same we have found for the SU(2) case in I and II. It is interesting to note that the value (9) satisfies the theoretical b o u n d found in ref. [8] ,4 Taking x/K ~- 420 MeV one obtains:

A 1/4 ~

2001]

C5

5.5 and are remarkably stable against the change of the form of the perturbative tail [eqs. (7)], due to the high accuracy of our points in the perturbative region (t3 > 6.5). The fitted values of C4 and C5 in the three cases examined are reported in table 2. One notes that in the cases (7a) and (7b) the fitted perturbative term C4//34 + C5//35 is very small as compared to the computed one C3//33 in the interesting region 5.5 ~
28 July 1983

5

6

7

9

8

10

Fig. 2. Plot of the raw Monte Carlo data. The solid curve shows the computed perturbative t e r m C3/f33.

The value of Aphen has been obtained [3,4] by fitting the mass matrix of the flavourless pseudoscalars after a proper inclusion of the contribution due to the UA (1) anomaly. A delicate point in this approach is perhaps the extrapolation of the Ward-Takahashi identities from qZ= 0 to the values of the actual masses of the pseudoscalar nonet. In fact, noticing ,6 that the form factor A(q 2)

=fd4xexp(iqx)(OIT(Q(x)

Q(0))I0>

(11)

has dimension (mass)4 we may argue that A (q2) obeys a doubly subtracted dispersion relation A (q2) = A (0) + q2A'(O)

(10)

This number is phenomenologically too small by a factor of about 3.5 (although definitively not zero), as compared to the expected one Al~4n ~ (180 + 30) MeV * s ,3 We also have few runs on a 64 lattice. The indication is that one finds a similar number forA. ,4 For the SU(2) case the measured valueA ~- (1.5 ± 0.5) X 10-4K 2 is a bit too small, but essentially of the same order of magnitude as the theoretical lower bound (A >~ 7 X 10-4K 2) computed in ref. [8]. This calculation assumes thatA = /V is a monotonically increasing function of the space-time volume V and uses multi-instantonsolutions to evaluate its small V value. ,5 Recent instanton type calculations [9] seem also to indicate a somewhat small value ofA.

+ (q2)2 /. imA(q,2) ,2 7r j (q,Z)2(q,2 _ q2) dq ,

(12)

where A (0) = A .

(13)

Especially i f A (0) is very small, the effect of the term q2A'(O), even if one can neglect the dispersive integral, may be substantial. Then the whole question of the location of the poles of the selfenergy matrix of the pseudoscalar flavour singlets should be reconsidered. It would clearly be useful to have even a crude esti,6 We are indebted to G. Veneziano for calling our attention on this point. 231

Volume 127B, number 3,4

PHYSICS LETTERS

mate o f A ' ( 0 ) or at least an indication of its sign ,7. Notice that since

A'(O) - f d4x x2(OIT(a (x) a (0)) I0 ) ,

(14)

it would be possible to compute it b y Monte Carlo methods. A weak point o f the approach we have been pursuing in I, II, and here is admittedly the necessity o f the perturbative subtraction. In view o f the topological meaning of Q(x), one may hope to find a perturbation theory-free lattice definition o f it. This indeed has been constructed b y Lfischer [10]. His definition, however, is rather complicated and has not yet been used for actual lattice calculations in four dimensions ,8 Another possibility would be to use the definition of Q(x) suggested by the lattice version o f the anomalous Ward-Takahashi identities [ 12]. A self-consistent measurement of the ~7'-mass along these lines has been recently performed by Hamber and Parisi [13]. They essentially measured both F~r and the quantity

(m2/V) (a4 tr (~/5G) a4 tr (~5G))no quark loops,

(15)

which in the naive a ~ 0 limit tends to A [12]. Expressing their result for m n' in terms o f A , one finds A~.4p. = (1 l 0 + 30) M e V , a value in between the phenomenological one and our Monte Carlo result. The question o f the existence o f a possible perturbative tail in ( 1 5 ) h a s not been considered in ref. [13]. This is clearly a crucial issue for the actual determination o f the *2'-mass and we plan to investigate it in the future. ,7 A gluebaU pole would give a negative contribution to A '(0). ¢8 For a discussion of Liischer's definition in the case of the two-dimensional Schwinger model see ref. [ 11].

232

28 July 1983

We wish to thank E. Marinari and C. Rebbi for providing us their very efficient Monte Carlo program for the updating of the lattice in the SU(3) case. We also thank P. Di Vecchia, G. Veneziano and K. Yoshida for many useful discussions. One o f us (K.F.) acknowledges the warm hospitality o f the INFN in Rome, where part o f this work was done.

References [1] G. 't Hooft, Phys. Rev. Lett. 37 (1976) 8; Phys. Rev. D14 (1976) 3432. [2] E. Witten, Nucl. Phys. B156 (1979) 269. [31 G. Veneziano, Nucl. Phys. B159 (1979) 213. [4] C. Rosenzweig, J. Schechter and G. Trahern, Phys. Rev. D21 (1980) 3388; P. Di Vecchia and G. Veneziano, Nucl. Phys. B171 (1980) 253; E. Witten, Ann. Phys. (NY) 128 (1980) 363; P. Nath and R. Arnowitt, Phys. Rev. D23 (1981) 473; P. Di Vecchia, F. Nicodemi, R. Pettorino and G. Veneziano, Nucl. Phys. B181 (1981) 318; K. Kawarabayashi and N. Ohta, Nucl. Phys. B175 (1980) 477; D.I. Dyakanov and M.J. Eides, Leningrad Nucl. Phys. Inst. preprint 639 (1981). [5 ] P. Di Vecchia, K. Fabricius, G.C. Rossi and G. Veneziano, Nucl. Phys. B192 (1981) 392. [6] P. Di Vecchia, K. Fabricius, G.C. Rossi and G. Veneziano, Phys. Lett. 108B (1982) 323. [7] M. Creutz, Phys. Rev. Lett. 45 (1980) 313; H. Harnber and G. Parisi, Phys. Rev. Lett. 47 (1981) 1792; E. Pietarinen, Nucl. Phys. B190 (1981) 349. [8] M. Liischer, Nucl. Phys. B205 (1982) 483. [9] M. MuUer-Preussker, CERN preprint TH 3431 (1982), and references quoted therein. [10] M. L/ischer, Commun. Math. Phys. 85 (1982) 39. [11] R. Flume and D. Wyler, Phys. Lett. 108B (1981) 317. [12] E. Seller and I.O. Stamatescu, Phys. Rev. D25 (1982) 2177; G. Immirzi and K. Yoshida, Nucl. Phys. B210 (1982) 499; W. Kerler, Phys. Rev. D23 (1981) 2384. [13] H. Hamber and G. Parisi, Brookhaven preprint BNL 31322 (1982).