Λ-parameters for asymmetric lattice gauge theories with fermions

Nuclear Physics B227 (1983) 50-60 O North-Holland Publishing Company

A-PARAMETERS FOR ASYMMETRIC LATTICE THEORIES WITH FERMIONS

GAUGE

P. D. SACKETT

Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pa 15260, USA Received 13 October 1982 Dimensional quantities obtained from Monte Carlo simulations on the lattice depend on the lattice mass parameter, AL. To make a connection with continuum physics, a relationship is needed between AL and the A-parameters of the continuum theory. This has been done for the euclidean symmetric lattice by others. However, in order to incorporate finite temperature into Monte Carlo studies, or to study the transition from the euclidean formulation to the hamiltonian formulation of gauge theories, asymmetric lattices (as # at) may be used. In this paper, the symmetric calculations are extended and the ratio Amin/AL, where Aminis the continuum mass parameter in the minimal subtraction scheme, is given to one loop for nf flavors of Wilson and Susskind massless fermions on an asymmetric four-dimensional lattice for two different asymmetric lattice actions.

1. Introduction T h e use of lattice r e g u l a r i z a t i o n t o g e t h e r with M o n t e Carlo s i m u l a t i o n s have b e e n a powerful m e a n s of s t u d y i n g the b e h a v i o r of z e r o - t e m p e r a t u r e gauge theories [1]. H o w e v e r , to i n c o r p o r a t e finite t e m p e r a t u r e in a M o n t e Carlo study requires lattices in which the spatial e x t e n t of the lattice differs from the t e m p o r a l extent. T o extract i n f o r m a t i o n from these studies a n d relate it to c o n t i n u u m t h e o r y q u a n t i t i e s at finite t e m p e r a t u r e , the ratio of lattice a n d c o n t i n u u m mass scales must be calculated as a f u n c t i o n of the a s y m m e t r y of the lattice. T h e l a m b d a p a r a m e t e r d e p e n d s on the choice of action a n d the s c h e m e used in i n t r o d u c i n g a s y m m e t r y into the lattice. A s y m m e t r i c lattice r e g u l a r i z a t i o n s (for which the spatial lattice spacing, as, is r e l a t e d to the t e m p o r a l lattice spacing, at, by at = eas) have a n o t h e r i n t e r e s t i n g feature. T h e e = 1 limit r e p r e s e n t s a s y m m e t r i c e u c l i d e a n f o r m u l a t i o n of the theory, while the e - - 0 limit r e p r e s e n t s the h a m i l t o n i a n f o r m u l a t i o n [2]. C o n t i n u o u s l y v a r y i n g the a s y m m e t r y factor, e, describes a t r a n s i t i o n from o n e to the other. A m e t h o d of t r e a t i n g a s y m m e t r i c lattices was first i n t r o d u c e d [2] to calculate the ratio of l a m b d a p a r a m e t e r s for these two f o r m u l a t i o n s , a n d later the e l e g a n t lattice b a c k g r o u n d field m e t h o d [3] was d e v e l o p e d which simplified calculations. Pure gauge theory a n d f e r m i o n i c c o n t r i b u t i o n s to the ratio of lattice a n d c o n t i n u u m l a m b d a p a r a m e t e r s have b e e n c o m p u t e d in p r e v i o u s work to the o n e - l o o p level for s y m m e t r i c lattice actions [2-7]. D i f f e r e n t lattice actions have b e e n t r e a t e d [8, 9], 50

P. Sackett / Lattice gauge theories with ferrnions

51

and for the pure gauge case, the contribution to one loop has been computed for an asymmetric lattice [10]. This p a p e r will consider two schemes for introducing an asymmetry in the temporal and spatial directions which keep euclidean invariance of the effective action intact to one loop. For each scheme, the ratio of the continuum mass p a r a m e t e r , A m i n , to the lattice parameter, A L, is determined to one loop, using the background field method [3]. Results are summarized as a function of the a s y m m e t r y factor, e, in tabular and graphical form for lattice theories with nf flavors of Wilson or Susskind fermions interacting with SU(2) or SU(3) gauge bosons. A discussion of these results and the relative merits of the two schemes for use in Monte Carlo simulations concludes the paper.

2. Symmetric lattice actions Consider a symmetric euclidean lattice with spacing a in each direction. The partition function is given by

Z ( N , a) = I [dU][d47d0] e ~s~u~+sf(e;,,,u~l,

(2.1)

where the U are link variables and 47, 0 are the Grassman spinor variables (or components thereof in the Susskind fermion action) which live on the lattice sites. If U.,, is the link from site n to a site one lattice spacing away in the/~ direction, then the gauge piece of the action is given by [11] 2N

S~(U) = ~ g

Z Y. [1 - ( l / N ) Re [Tr (U.,,•

U.+,...U

+ * , . . . . ~ U.,~)]]

(2.2)

n /x~-v

where here N specifies the S U ( N ) gauge group and n is the set of integers labelling a particular site. A naive discretization of the Dirac continuum action leads to the fermion doubling problem. To circumvent this difficulty the two most popular choices are the Wilson [11] and Susskind [12] fermion lattice actions. The Wilson approach inserts an extra piece into the naive fermion action which breaks chiral symmetry even for massless fermions, but which disappears in the continuum limit. In the Susskind method the components of the Fermi spinor each live on different, but adjacent, lattice sites. This approach requires the n u m b e r of fermion flavors to satisfy nf = 2 a/2, where d is the dimension of the lattice. The contribution to the action l[rom rtf flavors of massless fermions in each of these approaches is given below. The Wilson lattice action for I'tf fermion flavors is

Swv(O, O, U ) = Y [O.(~+~,.)U...O.+. +47. ( ~ - v . I U . * - , . . . O . - . -247. ~ 0 . ] . (2.3)

52

P. Sackett / Lattice gauge theories with fermions

For Susskind fermions, SSF(~, ~b, U ) = Y~f,,,.~.(U.,~,~b.+,,- U*._~,,~,~b._.)

(2.4)

n,~,

(In both cases, there is a hidden sum over flavor indices.) Here, the ~b are the anti-commuting components of the fermion spinor and f is a complex vector function of n which plays the role of the y-matrices on the lattice, namely [13] f . , . =)¢~,( - 1 ) "~,="~ ,

n=

if00!)0 (it 1

0

1

1

f=

"

(2.5)

(2.6)

The finite temperature theory [14] is realized by imposing periodic (anti-periodic) boundary conditions on Bose (Fermi) fields in the euclidean time direction, and associating the inverse physical temperature, /3, with the temporal extent of the lattice via /3 = 1 / k T = N t a t . (2.7) The physical volume of the system is then V = (Nsas) 3 .

(2.8)

Here, Nt(Ns) -- number of lattice sites in the time (spatial) direction(s) at(as) = lattice spacing in the time (spatial) direction(s).

(2.9)

(Physical inverse temperature,/3, should be distinguished from the inverse coupling constant,/3g = 2 N / g 2 for SU(N), also seen in the literature.) Clearly, in order to vary temperature without varying the spatial volume of the box, one needs an asymmetric lattice with either Nt # Ns, or at # as, or both. In Monte Carlo simulations, computer storage limits the realizable values of Art and ]Vs. To examine any long-distance behavior of the theory, one would like to make Nt and Ns large (indeed, the necessity of this choice has been emphasized by Engels, et al. [15]), so we are naturally led to fixing Nt and Ns as large as possible and allowing at # as. Throughout the remainder of this paper, a~, = e~,a, where e~, is the asymmetry factor in the /z-direction and lattices with all e,, = 1 are considered symmetric lattices.

3. Introducing the asymmetry For simplicity we shall choose 8i

=

1,

for i = 1, 2, 3 ,

84

=

e

.

(3.1)

P. S a c k e t t / Lattice g a u g e theories with f e r m i o n s

53

Naively, one can introduce the asymmetry e into the previously described actions by replacing 1

~e ~ t..t,v

2 2,

inSG,

la.,v e ~ e v

y~, ~ Y__e~, e~

~~ - - ,

in SWF,

Et~

in Ssv •

f~,.. -~ f"'", e~

(3.2)

However, care must be taken to ensure that the effective action maintains invariance under the discrete rotation subgroup for all choices of e. This may be accomplished by introducing two independent gauge couplings, gs and gt, for the space-space plaquette traces and the space-time plaquette traces, respectively [2, 10]. Alternatively, one can insert an extra e-dependent term, _F~,~(e), into the gauge action as done by Duncan, Roskies and Vaidya (DRV) [13]. The lattice A-parameter will then depend on the choice of asymmetric lattice action, as well as on the asymmetry factor e. The modified gauge action is S~oc _

2Ne 2N ~ ~ Pii + 2 ~ ~ P4i, g2(e,a) - i>j egt(e,a) i~4

(3.3)

_ _

i¢:4

in the scheme utilizing two independent gauge couplings (IGC), and sDRV .

2Ne . . . {(I~__P~) +NF,~(e)(P,~) 2} g2(e,a) ~ ~>~ e~,e~

(3.4)

in the D R V approach, where P,, •~ = [ ( l / N ) R e T r ( u . , , , u , +,,,~ u , +*v , ~ u . ,*~ ) ] .

(3.5)

If two couplings are introduced, they are e-dependent but each tends to gE, the euclidean (symmetric lattice) coupling, as e ~ 1. The one-loop quantum corrections are responsible for the e-dependence of the couplings, and in the weak-coupling limit [2]:

gs-2 (e, a) =gEZ(a)+cs(e)+Order (g2), gt- 2 (e, a) =gE2(a)+ct(e)+Order (g2E).

(3.6)

The cs(e) and ct(e) are adjusted in such a way as to restore invariance of the effective action under rotations which take one lattice axis into another. In the D R V method, the particular choice of e~, (eq. (3.1)) allows the freedom to choose IF(e), for/z = 1, 2, 3 and v = 4 F~,~ (e) = ~[0, otherwise. (3.7)

54

P. Sackett/ Lattice gauge theories with[ermions

In both cases, eqs. (2.7) and (2.8) still hold and the continuum thermodynamic limit is obtained by letting Nt-~ oo, Ns-~ c~, at -~ 0, as + 0, and bare couplings g2 -+ 0 in such a way that the physical temperature remains fixed and A L tends to a well-defined non-zero limit set by the one-loop renormalization group/3-function:

8~"2 l l N - 2 n f l n g2N = 6N

1 (aAL) 2"

(3.8)

4. The background field method The background field method is used here to evaluate Amin/AL. More detail on this approach can be found in earlier work [3, 10, 16]. Briefly, a weak, slowly varying, background field is assumed to be a solution of the classical equations of motion on the lattice in question. The action is expanded about this solution and quadratic quantum fluctuations corresponding to a one-loop approximation are computed. A similar calculation is done in the continuum theory using some regularization scheme to control the divergences. Here, minimal subtractio~q is used. If the lattice theory is to be a good approximation of the continuum theory, their vacuum persistence amplitudes (partition functions) must be equal as the lattice spacing a -~ 0 [3]. Therefore, as a -+ 0,

ln-~----= ~ Lmi, b . , . ~

d(ma, e)

- - + - 4gE(e,a) 4g20(m)

d4x(Fb,~)2-~0

(4.1)

where g(a, e) is the lattice coupling, go(m) is the continuum coupling at mass m and d(ma, e) is the term resulting from the calculation of the one-loop quantum corrections. In the D R V approach, F(e) is adjusted so that d(ma, e) is independent of tz, v. Then, together with the condition above one is led to

1 4gZ(e,a)

1 4go2(m)

d(ma, e ) .

(4.2)

The d(ma, e) acts as a finite renormalization of the coupling on the asymmetric lattice. With the two independent couplings of the IGC approach, d(ma, e) depends on the choice of/z, v so that the space-space and space-time couplings are separately renormalized by

1 1 4gZ(e,a) - 4go2(m) ds(ma, e),

1

1

4 g t2( e , a )

4go2(m)

dt(ma, e)

(4.3)

55

P. Sackett / Lattice gauge theories with fermions

Asymptotically (i.e. for small a) we expect

d(ma, e) = ~bo[ln ma + 6 ( e ) ] , 1

1

492(e, a) - ~bo In ALa '

1 1 m 4g 2 (m) - ~bo In Am,,--'

(4.4)

where bo-

llN-2nf 48~.2

(4.5)

Combining eqs. (4.2)-(4.4) yields the relationship between the lattice and continuum A-parameters Amin e -a~') , AL

(4.6)

=

in the D R V approach or

Amin [(Cs +Ct) AL - - e x p / 4bo

6(1)]

(4.7)

J

in the IGC scheme. 5. The results

The work in any regularization scheme is to compute the quantum fluctuations, done here to the one-loop level. After a lengthy calculation, the gauge piece of the effective lattice may be written as SeG" = S G ( U o ) -~- E

b.,*v

(N2-1) 8e

2 2(G+G) 2N

l? lz E v

2 '}I N [A,~.+l(e.B,.)_~C.v]

(N2+l) 8e

4

4

(C~+Cv)V,.~

dax (F~.) 2

(5.1)

where the Uo are the background field links and D----~[1 - c o s (k.e~,a)],

A..=

I D -dk --~[l+cos(k.e.a)][l+cos(k~eva)],

dk By = I ~---~...cos (kveva), uotK)

(5.2) (5.3) (5.4)

P. Sackett / Lattice gauge theories with fermions

56

C.~ =

I DG(k) ~ cd, os(k.e.a)

(5.5)

cos (k~e~a)

4 _

dk.] ,

e a

(5.6)

J with

DG(k) = 2 ~ (1/e~)2[1 - c o s (k,~e~a)].

(5.7)

o-

The contribution to the effective action from nf flavors of Wilson fermions is eft Swv

--

nf

E

16e b,.,v

G~,vf d4x (Fbv)2

(5.8)

where

I d,

G,.~ = 8 Dewy (k) [23--sinE (kge.a)sin E (k~e~a)] 2

2

- -dk ) a ~,(DWF)O,,(DwF) + 2 e , e ~ f d-------~k DwF(k) cos (k~,e~,a) cos (k~eva)-~a1 4 f Dew~(k (5.9)

DwF(k)=y. sinz(k~e~a) 2 Eo-

o-

q-

[~[c°s(kxexa)-l]]

2

•

(5.10)

EA

For n~ flavors of Susskind fermions, the effective fermion lattice action is instead serfSF ~--- 48enf b.,~H"v

I d4x ( F b " ) 2 ,

(5.11)

where here

H.~ = I

dk

D_IF (k-----~ [c°s2 (k~.e . a) + cos e (kv e~ a ) - sin 2 (k~.e. a) sin e (kv e ~a)] d

x

(5.12) DsF(k) = 2 E sin2 (k~e~,a). o-

(5.13)

~'o-

In the above, Tr (1)Dirac = 4 is used. From the above, the DRV F~,~(e) function may now be computed to ensure the euclidean invariance of the complete lattice action. Alternatively, F,,~ (e) may be set to zero; the one-loop quantum corrections then split into ds(ma, e ) and dt (ma, e) from which the Cs(e) and ct(e) of the IGC scheme may be computed. Finally, the ratio of Amin/AL c a n be obtained from eqs. (4.6) and (4.7) as a function of e in both the D R V and IGC approaches.

57

P. Sackett / Lattice gauge theories with [errnions TABLE 1 Amin for a symmetric (e = 1) lattice

Gauge group

SU(2) SU(3)

No fermions n~=0 7.463 10.85

Wilson fermions

Susskind fermions 4

1

2

3

4

34.44 28.78

9.294 12.80

12.15 15.46

16.99 19.19

26.15 24.67

The four-dimensional integrals in the gauge and Susskind fermion contributions may be reduced to equivalent one-dimensional integrals and are then easily done [3]. The contribution from the Wilson fermion action, however, must be done as numerical four-dimensional integrals and this reduces the accuracy of the results in that case. The numerical calculation of the pure gauge and Susskind fermion lambda ratios converged to two parts in 1 0 - 7 o v e r most of the range of epsilon, and never worse than to two parts in 1 0 - 4 e v e n at very small epsilon (e < 0.06). The lambda ratios in theories with Wilson fermions generally converged to two parts in 10 -5 or better, and never worse than four parts in 1 0 - 3 at very small epsilon. N u m e r o u s calculations have been done by others to compute the relationship between the symmetric lattice theories and the corresponding continuum theories [2-7, 16]. The symmetric limit (e = 1) of AmiJAe is of course the same in both of the asymmetric lattice regularization schemes considered here, and is summarized in table 1. Figs. 1-4 indicate h o w the ratio Amin/AL varies with epsilon in both the regularization schemes for pure SU(2) and SU(3) gauge theories, as well as for those with nf flavors of Susskind or Wilson fermions. To the accuracy quoted, the 40

30

zo

I0

O0

0 2

0 6

0.8

1,0

E

Fig. 1. Amln/A L as a function of e for SU(2) and SU(3) without fermions in the D R V ( ( - - - ) approaches.

) and I G C

58

P. Sackett / Lattice gauge theories with fermions

2 0 0 ~

. . . . . . .

%

O0

02

0.4

06

08

i.O

Fig. 2. Amin/AL as a function of e for SU(2) and SU(3) with four flavors of Susskind fermions in the DRV ( ) and IGC ( - - - ) approaches.

values in table 1 agree with those of Sharatchandra et al. [7] for a theory with four Susskind fermion flavors, and with those of Kawai et al. [6] for a theory with Wilson fermions (rtf=3,4) on a symmetric lattice. The lambda ratios Amin/AL in pure SU(2) and SU(3) gauge theories computed by Sharatchandra et al. [7] and Weisz [16] are also in agreement with table 1. Values in table 1 for Wilson fermions (nf = 1, 2, 3, 4) are slightly, but consistently, higher than those quoted by Weisz [16], but Kawai et al. [6] and Celmaster and Maloof [5] also quote ratios that are higher than those of Weisz. Furthermore, for the special case of pure S U ( N ) gauge LO0

f

f

i

i

6o

<~

4o

2o

o

0

0.2

0.4

0.6

0.8

1.0

Fig. 3. Amin/AL as a function of e for SU(2) with nf flavors of Wilson fermions in the D R V ( IGC ( - - - ) approaches.

) and

P. Sackett / Lattice gauge theories with fermions

59

I00

80

60 "
4.0

20

O0

I

I

0.2

I

I

0.4

I

i"

0 6

I

0.8

I

1.0

E

Fig. 4.

a m i n / A L as

a function of e for SU(3) with nf flavors of Wilson fermions in the DRV ( IGC ( - - - ) approaches.

) and

theories with I G C regularization, agreement is found with Karsch [10] over the full range of epsilon. (Karsch actually computes a different lambda ratio, A L ( e ) / A L ( 1 ) , normalized to the symmetric lattice lambda.)

6. Discussion It is striking that the I G C and D R V methods of introducing an asymmetry on the lattice lead to such drastically different epsilon dependent behavior in the lattice scale factors. This is of interest to those doing finite-temperature Monte Carlo simulations since the dimensionless quantity A L a controls the coupling g through eq. (3.8). On one hand, since AL varies m o r e drastically in the D R V approach than in the I G C approach, if the physical volume is to remain fixed during a Monte Carlo, the coupling g will vary over a wider range of values in the D R V approach as e is changed. However, to ensure the simultation is being done in the weak-coupling regime, the coupling g (and therefore A L a ) must remain small. Since most Monte Carlo simulations are done on small lattices (typically 44-84 lattice sites), only a small physical volume is being sampled unless a is fairly large. Thus, it is desirable to have A L as small as possible so that the simulation may be done in the weakcoupling regime but still sample a relatively large volume. In every case, for e < 1, the D R V approach yields lattice scale factors smaller than those in the I G C approach and varying over a wider range. This is actually a serious problem for both of these regularization schemes, and for Monte Carlo simulations in general. For example, for Monte Carlo simulations of SU(3) with AL'S ranging from 1-20 MeV, the spatial

60

P. Sackett / Lattice gauge theories with fermions

lattice s p a c i n g m u s t b e k e p t b e l o w 0 . 2 - 0 . 0 1 fm to e n s u r e t h a t g lies in the w e a k c o u p l i n g r e g i m e . T h e difficulty is w o r s e n e d by a f a c t o r of a b o u t t e n w h e n f o u r flavors of f e r m i o n s a r e i n t r o d u c e d . T h e e = 0 limit is e s p e c i a l l y i n t e r e s t i n g as it r e p r e s e n t s t h e h a m i l t o n i a n f o r m u l a tion of lattice t h e o r i e s . T h e I G C a n d D R V s c h e m e s for a s y m m e t r i c lattice h a v e different h a m i l t o n i a n limits; this d i f f e r e n c e is e v e n m o r e m a r k e d w h e n f e r m i o n s a r e i n t r o d u c e d . W o r k n o w in p r o g r e s s will e x t e n d t h e s e results to i n c l u d e calculations of A min/A L as a f u n c t i o n of e p s i l o n for m o r e g e n e r a l lattice a c t i o n s including: (i) m a s s i v e f e r m i o n s ; (ii) f e r m i o n i c c h e m i c a l p o t e n t i a l ; (iii) c h o o s i n g a m o r e g e n e r a l (C + y , ) r a t h e r t h a n (~+ y , ) in t h e W i l s o n ferm i o n i c lattice a c t i o n in eq. (2.3). T h e a u t h o r w o u l d like to t h a n k P r o f e s s o r s R a l p h Z. R o s k i e s a n d A n t h o n y D u n c a n for m a n y useful discussions a n d careful r e a d i n g of the m a n u s c r i p t .

N o t e added A f t e r s u b m i t t i n g this p a p e r for p u b l i c a t i o n , a p r e p r i n t b y R . C . T r i n c h e r o c a m e to the a t t e n t i o n of the a u t h o r . W h i l e t h e a n a l y t i c results c o n c e r n i n g t h e I G C r e g u l a r i z a t i o n a r e e s s e n t i a l l y t h e s a m e , o u r n u m e r i c a l results differ in the r e g i o n of small e.

References [1] J.M. Drouffe and C. Itzykson, Phys. Reports 38 (1978) 133; Proc. Int. Symp. on statistical mechanics of quarks and hadrons, Bielefeld, FR Germany, ed. H. Satz (North-Holland, Amsterdam, 1981), and references cited in both [2] A. Hasenfratz and P. Hasenfratz, NucL Phys. 193B (1981) 210 [3] R. Dashen and D. Gross, Phys. Rev. D23 (1981) 2340 [4] A. Hasenfratz and P. Hasenfratz, Phys. Lett. 93B (1980) 165 [5] W. Celmaster and D. Maloof, Phys. Rev. D24 (1981) 2730 [6] H. Kawai, R. Nakayama and K. Seo, Nucl. Phys. 189B (1981) 40 [7] H.S. Sharatchandra, H.J. Thun and P. Weisz, Nucl. Phys. 192B (1981) 205 [8] A. Gonzalez-Arroyo and C.P. Korthals-Altes, Nucl. Phys. 205B (1982) 46 [9] C.B. Lang, C. Rebbi, P. Salomonson and B.S. Skagerstam, Phys. Rev. D26 (1982) 2028 [10] F. Karsch, Nucl. Phys. 205B (1982) 285 [11] K. Wilson, Phys. Rev. D10 (1974) 2445 [12] L. Susskind, Phys. Rev. D16 (1977) 3031 [13] A. Duncan, R. Roskies and H. Vaidya, Phys. Lett. l14B (1982) 439 [14] S. Weinberg, Phys. Rev. D9 (1974) 3357 [15] J. Engels, F. Karsch and H. Satz, Nucl. Phys. 205B (1982) 239 [16] P. Weisz, Phys. Lett. 100B (1981) 331