Nuclear Physics B (Proc. Suppl.) 9 (1989) 21-25 21 North-Holland, Amsterdam

U P G R A D E OF RESULTS F R O M M O N T E CARLO S T U D Y OF T H E O(4) I N V A R I A N T A# 4 MODEL IN T H E B R O K E N P H A S E

T H O M A S N E U H A U S

Fakult~t fiir Physik, Universit~t Bielefeld, D-4800 Bielefeld, F. 1%. Germany

A B S T R A C T : I summarize recent results from large scale Monte Carlo simulations of scalar 0(4) theory in the broken phase. The wave function renormalization constant Za, the mass of the scalar particle m~ and the scalar field expectation value ~ are determined. The effect of the finite lattice size on the scalar mass is estimated. The upper bound on the ratio rna/ fa is 2.6(1) at a value of the cut-off Acut ~ 2.5, corresponding to an triviality bound of 8.2(5) on the ratio of the Higgs mass over W mass. First results of an extensive calculation on various lattice sizes at finite small external source j and at a value of fG "" .4 are presented. Taking into account the effect of the Goldstone states via analytic formulae, the scalar field expectation value ~ is determined. The agreement of this value calculated at finite j and the calculation using rotated fields at j=0 is impressive.

1. I N T R O D U C T I O N

It is now understood that the four component 0(4) A# 4

model serves as the heart of the spontaneous symmetry break-

ing and vector boson mass generation mechanism of the Weinberg

Salam model defined at the Gaussian fixed point (GFP) [1]. While

it is almost rigorously proven that the renormalized quartic cou-

pling of the scalar model vanishes in D=4 in the infinite cut-off

limit, one can define an effective and interacting theory if one al-

lows for a finite and physical value of the cut-off Acu~. In such

an effective theory the ratio of the scalar mass mq over fa , the

Goldstone equivalent of the pion decay constant, is basically a free

parameter. It is expected however that R = ma/ fG is bounded

from above as one varies the bare quartic coupling at any fixed

value of the cut-off and this bound is saturated at a value oo for

the bare quartic coupling. Lowering the cut-off will increase the

bound. When A~,t is of the order of m~ quantities defined in the

lattice regularization will show significant deviations from a scal-

ing behavior characteristic of the Gaussian fixed point. Here the

effective theory loses its meaning and as a consequence a physi-

cally sensible value of R is bounded from above by R,na=, leading

to an upper bound of (rnH/rnw)rnaz = 2 X Rrnaz/gw [2], where gw denotes the renormalized SU(2) gauge field coupling constant

g~v - - . 4 .

Those considerations motivate the recent high precision

Monte Carlo (MC) calculations [3-5] of the scalar field expectation

value ~, the scalar mass ma, the wave function renormalization

constant of the scalar particle Z~, and of the Goldstone mode

z a = ~ l f~ . In analogy to QCD and its ~r meson is the low energy be-

havior of the 0(4) model governed by the Goldstone modes and

can be described by an 0(4) invariant non-linear o'-modeh Thus replacing the original theory by a low energy one with universal

properties [6], depending on its coupling constant fc , which is

the analog to f~ in the chiral model. Perturbation theory in the

finite box in ( f a x L) -1 then predicts formula for certain 0(4)

invariant correlation functions which allow e.g. the determination

of za [7,8]. In an alternative approach one can study the response of the

system to an external finite but small source j introducing an

explicit breaking of the 0(4) symmetry but still in a region of

couplings where a universal low energy model provides a sensible

description. The projection of the magnetisation to the direction

of the external source then becomes a well defined and positive

quantity even on the finite lattice. It can be used for the determi-

nation of the infinite volume ~ by comparing the measured data

with finite volume theorems for the magnetisation derived in the

framework of the low energy model at very small values of the

external source [9].

The work which I describe in these conference proceedings

was done in colloboration with A. Hasenfratz, K. Jansen, J. Jers~.k

, C. B. Lang and also in part with H. Yoneyama. Most of the

results at zero value of the external source have been published

already [3]. So I concentrate on the infinite volume extrapolation

of certain quantities and the comparison of our work with other

groups. The second part of this paper contains interesting new

results with a nonzero source, which will be published soon.

2. LATTICE A C T I O N AND MC C A L C U L A T I O N For the lattice action we choose the form

4

z6AD=I

+~ ~(+~¢~ - 1)= + ~ +~¢~ + ~ ~ ~:, (1) zEA zEA zEA

0920-5632/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

22 T. Neuhaus / R e s u l t s from Monte Carlo s t u d y

where to, ,~ > 0 are the hare coupling parameters. The fields

¢~ are real and we use the convention tha t summat ion over the

index a = 1 . . . 4 is implied, the • fields should be resealed to

the fields ~ = x / ~ . Here the emphasis of the presentation is on

results for bare quart ic coupling ,~ = oo and various values of x in

the broken phase, Results for non-zero j are presented at ~ = o¢

and ~ = .355. We have studied ensembles of configurations for

hypercnbic lat t ices A of size 84 up to 164 with direct MC simulat ion

and periodic boundary conditions. The updat ing method was a

2-hit Metropolis with more than .3 x 106 MC-sweeps per point

considered. The calculations at finite j required typically up to

106 sweeps. The calculations were performed on vector computers

CYBER 205, ETA10 and CRAY X-MP/48.

3. R E S U L T S A T J = 0

3.1 ~ E X P E C T A T I O N V A L U E

We measured the magnet isa t ion

1 M a = 7~'.4 Z ~ " (2)

x

and i ts length averaged over the Monte Carlo run ((MaMa')½1 is

close to the scalar field expectat ion value. Following Neuberger

[8] we have at hand the finite size formula

42 ( ( M t ' M a ) ~ ) 2 = ~2(1 + 2~-~) + O(L-41og'rL) (3)

~, I I ' ' ' ' I '

.4

.2

.0

I I I t I I

. 3 . 3 5

Figure 1. ~ as a function of ~ at )~ = co, triangles up are

124, crosses 144 and triangles down 164 lattices. We only plot the

data which were fitted in .305 < x < .34. The curve corresponds

to the fit .

valid for large and periodic lattices. We extract values for Y],

by using eq. (3) to order O(1/L ~) and then f i t our da ta on 124,

144 and 164 lat t ices from Table 1 with MINUIT to the sealing

form

= At, x t x/2 x (lnt) 1/4

t = ~ - ~---~ (4)

assuming exponents and logari thmic corrections of the GFP. The

results from the fit are A= = .822(1) and ge = .304(1), agreeing

very well with [4,5].

3 . 2 S C A L A R M A S S ma

We determine the scalar mass from the exponential decay of

the propagator of the "rotated" scalar field

~ M ~

~o,~ = 1[ M II (5)

a t zero momentum. We find tha t our correlation functions are well

measurable and consistent with the propagat ion of one massive

particle. The effects of resonant behavior due to the coupling of

Goldstone modes can be controlled. For the details of the analysis

I refer to [3]. The obtained values of ma are given in Table 1

showing a pronounced finite size dependence as can be seen in

Figure 2.

ma

0

I i I I I J I

O

O ~ / ~ l l i , i i o / 3~ ~(L~ I ' ' ' _ _ I''1

0 0 ~ " 2 L ~ ~

4 8 z I , , , , I

. 3 . 3 5 f

Figure 2. m,, as function of ~ at ,~ = ~ . Circles are 84,

triangles up 124 , crosses 144 and triangles down 164 lattices. The

curve corresponds to a infinite volume estimate. The inlet shows a

plot of the ratio mo(L) /mo(oo) against the variable z = L x m o ( L )

The data form a common curve supporting the validity of the

infinite volume estimate eq. (7).

The observed large finite size effects make a infinite volume

extrapolat ion for the scalar mass neccessary and we present here

an es t imate of the infinite volume mass. Guided by per turbat ive

arguments [13] we assume tha t the leading contribution to the

finite size dependence of the scalar mass from the Goldstone states

is given by A

m , ( L ) ~-- m,,(oo) + - ~ (6)

T. Neuhaus / Resul ts from Monte Carlo s tudy 23

at fixed values of the hopping parameter r . Here we have however

only few ~¢ values where such an extrapolat ion is leasable and we

are left wi th 3 points where we estimate the infinite volume scalar

mass. F i t t ing our da ta with eq. (6) we obtain m ¢ = .29(1) a t ~ =

.3075, ma = .40(1) at a = .31 and ma -- .62(2) a t ~ = .3175. We

note however tha t the determinat ion of the finite size dependence

of a resonance mass is far from tr ivial and should be investigated

more precisely.

.4

.3

0 .01 1/L~ Figure 3. Values of mo as function of 1/L 2 at ~¢ = .3075

on lattices ranging from 84 to 164 . The straight line shows the fit

according to eq. (6) leading to ma = .29(1)

Assuming finally critical exponents and logari thmic scaling

corrections of the GFP we fit the infinite volume masses to the

scaling Ansatz

mo = ASM x (t) 112 x (lnt) -114 (z)

where t has the same meaning as in eq. (4) and ~:c = .304 is fixed.

From the fit we obtain ASM = 3.94(9). We have also considered

mff using the Ganssian model relat ion m f f = 2 x sinh(m~/2).

Here we find ARSM = 3.98(9). The last ampl i tude can be compared

with [5] and is consistent with the number given there.

3.3 W A V E F U N C T I O N R E N O R M A L I Z A T I O N

The wave function renormalizat ion of the scalar field Za is

determined by the residue of the propagator of the "rotated" field

eq. (5) at nonzero momentum. For the wave function renormal-

ization of the Goldstone mode Z c we s tudy the operator

1

arEA,

and its 0 (4) invariant zero momentum correlation function having

contributions from the Goldstone modes and the massive s ta te as

well. Lowest order per turbat ion theory in ( f c L ) -1 in the frame-

work of the low energy model predicts the following parabolic de-

pendence

< 0~(0)0~(~) >

= C + Z a x ~ x ( ~ - )2+... (i0)

of the 0 (4) invariant correlation function on r for large values of

the distance r allowing the determinat ion of Za . The results for

Zc are given in Table 1, for more details see [3].

3.4 RESULT FOR THE RATIO AND COMPARISON

In Figure 4. we present our final result for the the ratio

R = ~- x Z~ at a value of A = o0 as a function of the cut-off in

units of the scalar mass. The triangles correspond to 124 , crosses

to 144 and triangles down to 164 lattices. In a region offthe cut-off

1/m~, ranging from 2-3 the upper bound on R is ]?~na~ - 2.6(1). The values for R may however be little overestimated as can be

seen if we include our estimate of possible finite size effects. Here

the main effect comes from the finite size dependence of the scalar

mass. Using the fits to E and mo according to equations (4) and

(7) and alternatively replacing ma by rnff everywhere we arrive at

curves labeled 1 and 2 in the plot. We assumed Za to be .97 for the

graph. We did not include errors in the curves, but the statistical

error is basically given by the error of the scalar mass while the

systematic errors of the FS extrapolation is unknown. We learn

that the finite size corrections of R at values of the cut-off 2-3 may

be as large as 5 percent even on a 164 lattice. Here one should

keep in mind that for the moment we are only in the position to

give a reasonable estimate of the finite size effect. Work in this

direction is in progress.

We can now compare our data [3] with the high statistics

numerical simulations of the group around J. Kuti [4,5]. Our

infinite volume estimate for R and their infinite volume extrap-

olation which corresponds to curve 3 are very close indeed. In

addition one can state that the numerical simulations coincide in

their raw data obtained on finite lattices [10]. Finally we also

compare the analytic calculation of Liischer and Weisz [11] with

the various curves. We find impressive agreement showing that

numerical and analytical methods are consistent within a few per-

cent in their determination of R at the considered values of the

cutoff.

~ I I I I

3 . 5

3

2 I 1 2 3 4

A

Figure 4. The ratio R as function of the cut-off Curves I and 2 are ezplained in the tezt. Symbols have the same meaning as

24 T. Neuhaus / Results from Monte Carlo study

in Fig. ~. The 3 diamond shaped data points are from the analytic

calculation by Lf=scher and Weiss using g1¢ = 3 x R 2 [11]. Curve 3

is taken from the fits to mo and ~ in [5] where I took the values of

the amplitudes given there and used a common xe = .304 for the

scaling curves. Z was chooses as above. We add to our graph the

result from a calculation in the SU(~) coupled gauge field Higgs

model at a value of fl = 8. [1~] (fat circles ), assuming m~t = mo undrape= 2 2 2 9 z ~g f~ with = .5.

4. RESULTS AT F I N I T E J

4.1 M A G N E T I S A T I O N S PARALLEL C O M P O N E N T

The coupling of an external source to the 0(4) model provides

the laboratory of an alternative, more rigorous and even more aes-

thetical description of the models properties in the broken phase.

One immediate consequence is now that the expectation value of

the magnetisations parallel component

1 < ~ - - ~ > (11) < Mp > = ~-4 m

is a positive number even on the finite lattice. Only if the external

source is 0 its average value is 0 showing that there is no sponta-

neous symmetry breaking on the finite lattice. In the case of the

j=0 simulations this difficulty was circumvented by assuming that

the length of the magnetisation vector serves as an estimate of Z,

however strictly spoken assuming that the state prepared on the

periodic lattice is a pure state in the infinite volume limit. Intro-

ducing the external source, this assumption is no longer needed

and one can try to determine ~ in a different way. It is possible in the framework of the low energy effective

model and analytic tbrmulae have been worked out by Leutwyler

and Gasser [6,9]. For the very small values of the external source the lowest order large volume theorem for the parallel component

of the magnetisation gives

1 ~Xz(s) (12) < Mp > - - ~ x 2 x X2(s)

where X2 (s) = I1 (2s)/s is a Bessel function of the scaling variable

s = ½ x E x f. We introduced f = j /(2n) "s. The formula describes

how the magnetization < Mp > vanishes in the "thirst limit"

j --* 0 on the finite lattice. It assumes that the lattice is large

compared to the massive a-field

1 - - << L (13) IT~ a

and that mo '(~ me. Here we perform the calculation at a value

of ~ = .355 and A = ~ where the scalar mass takes a value of

ma ~- 1.2 and fG -~ .4. The condition of eq. (13) is therefore

easily fulfilled if we choose lattices ranging in size from L=8 to

L=12. We also decided to choose such small values of j, that the

lattices were finite compared to the Goldstone modes

1 - - >> L (14) m G

with

m~ = ~ x f (15) J a

allowing for large finite size effects and a genuine study of those

effects in the chiral limit. Typical values of j were smaller than

.005. Such small values of the external source require a high statis-

tics calculation of < NIp > on the already mentioned lattices

and we have typically used 106 sweeps on the 84 and 104 lattices,

.3 x 106 sweeps on the 124 lattices. We performed a careful error

analysis of the magnetisation by introducing blocks of data with a

typical block length of a few thousand sweeps. We observed sat-

uration of the error determined from the blocked data at a block

length of the order of 104 sweeps. The results for < Mp > are

given in Figure 4., nicely exhibiting the huge finite size effect in

vicinity of the j=0 point.

We fitted all the data using eq. (12) and determined :E by a

global one parameter MINUIT fit. Using the lowest order formula

eq. (12) we obtain the value E = .409(4) from the fit, which

one can compare with the value E = .403(1) coming from the

infinite volume extrapolation at j=0, eq. (3). In addition one also

knows the next higher order corrections to eq. (12) [6]. Using

these one obtains ~ = .404(4), for a detailed analysis I refer to a

forthcoming paper. The agreement of the fitted numbers coming

from the calculation at finite source and the calculation at j=0 is

impressive.

I ' I ' I

.4 •

(Mp)

. 2

0

I , I , 1 , 0 . 0 0 2 . 0 0 4 j

Figure 5. < Mp > as function of the external source j at

A = go and t¢ = .355. Circles are 84, triangles 124 and crosses

124 lattices. The 3 curves correspond to the 1 parameter fit using

eq. (1~}. The fat circle plotted at j=O corresponds to the infinite

volume ~ value.

5. C O N C L U S I O N S

We determined the triviality bound on the ratio 1% of the

scalar boson mass ma over the analog of the "pion decay constant"

fa at a value 1~ms = 2 - 3. We obtain an upper bound of 2.6(1),

which for a renormalized gauge coupling of g~v = 0.4 corresponds

T. Neuhaus / Resul ts from Monte Cado s tudy 25

to a maximal ratio m H / m w = 8.2(5). We estimated possible finite size effects and our calculation is in very good agreement with other groups [4,5,11].

We also present for the first time results at finite external source and a determination of E from those data using low energy and large volume theorems. The gain is twofold. Firstly we find a remarkable agreement of the determined value for Z with the calculation at zero source, implying that the assumptions made there are sound and justified, at least for the large values of fa . Secondly we find that those theorems provide a consistent way to describe some properties of the 0(4) scalar theory in the broken phase. Work in this direction is in progress. A C K N O W L E D G E M E N T

The support of the Computer Centers at Florida State Uni- versity and KFA Jiilich, where the necessary computations were performed, is acknowledged. 6. R E F E R E N C E S

[1] A. Hasenfratz and P. I:Iasenfratz, Phys. Rev. D34 (1986) 3160.

[2] 1%. Dashen and H. Neuberger, Phys. Rev. Lett. DS0 (1983) 1897.

[3] A. Hasenfratz, K. Jansen, C.B. Lang, T. Neuhaus and H. Yoneyama, Phys. Lett. 199B (1987) 531 ; A. tIasenfratz, K. Jansen, J. Jers~,k, C.B. Lang, T. Neuhaus and It. Yoneyama, preprint HLRZ Jiilieh 88-02 and UNIGRAZ-UTP-03-88 to appear in Nucl. Phys. B.

[4] J. Kuti, L. Lin, Y. Shen, Nucl. Phys. (proe. Suppl.) B4 (1988) 397; Phys. Rev. Lett. 61 (1988) 678; U.C. San Diego preprints, UCSD/PTIt 88-05; UCSD/PTH 88-07; UCSD/PTH 88-06 (with S. Meyer).

[5] J. Kuti, San Diego preprint UCSD/PTH 88-12 , October 1988, invited talk at Munich conference, August 4-10,1988.

[6] H. Leutwyler, private communication ; P. Hasenfratz, talk at this conference.

[7] H. Leutwyler, Nucl. Phys. B4 (Proc. Suppl.) (1988) 248. [8] H. Neuberger, Phys. Rev. Lett. 60 (1988) 889; Nuel. Phys.

B300 [FS22] (1988) 180. [9] J. Gasser and H. Leutwyler, Ann. of Phys. 158 (1984) 142; J.

Gasser and It. Leutwyler, Phys. Lett. B184 (1987) 83; Phys. Lett. B188 (1987) 477; Univ. Bern preprint BUTP-87/22.

[10] J. Kuti, private communication. [11] M. Liiseher and P. Weisz, preprint DESY 88-083 to appear

in Phys. Lett. B; preprint DESY 88-146 (October 1988). [12] A. Hasenfratz and T. Neuhaus, Nuel. Phys. B297 (1988)

205. [13] L. Lin, Talk at this conference.

7. TABLES

L x < ~ > mo Zo R

12 .3075 .178(1) .38(1) .96(1) 2.67(7) 12 .3100 .214(1) .47(1) .97(1) 2.75(6) 12 .3175 .292(1) .67(1) .96(1) 2.82(5) 12 .3250 .347(1) .80(1) .96(1) 2.80(4) 12 .3330 .392(1) .92(2) .97(1) 2.83(7) 12 .3550 .480(i) 1.22(4) .97(1) 2.97(9) 14 .3075 .169(2) .35(1) 2.64(8) 14 .3100 .209(1) .43(1) .96(1) 2.56(6) 14 .3175 .289(1) .64(1) 2.78(5) 14 .3200 .310(1) .71(1) .97(1) 2.82(4) 14 .3250 .345(1) .81(2) 2.91(7) 14 .3300 .374(1) .87(2) .97(1) 2.82(7) 14 .3350 .400(1) .92(2) 2.81(6) 16 .3060 .135(1) .27(1) .96(1) 2.50(9) 16 .3075 .165(1) .33(1) .96(1) 2.50(7) 16 .3300 .373(1) .85(5) .97(1) 2.76(15) 16 .3550 .478(1) 1.09(10) .96(1) 2.65(23)

Table 1 : High precision data in the 0(4) model at A =

and in the broken phase. L is the lattice size, x the hopping

paramter, < ~ > denotes < (MaMa)½ > /(2x) 's, R is R = 1 k

mo x - :-r- x Z~ . At few data points on the 144 lattice we <(M~Mo)~ > did not evaluate Za for technical reasons. We then assumed the value .97.