Nuclear Physics B (Proc. Suppl.) 4 (1988) 505-509 505 North-Holland, Amsterdam

UPPER BOUND ESTIMATE FOR THE HIGG$ MASS FROM THE LATTICE REGULARIZED WEINBERG-SALAM MODEL

Thomas Neuhaus

Ins t i tu te for Physics, University Bielefeld, D-4800 Bielefeld, F. R. Germany

There are indications that , as a consequence of the t r i v i a l i t y of the scalar ~' theory, the Weinberg-Salam model is a massive free f ie ld theory. However, con- sidering the W-S model as an effective f i e ld theory, an upper bound for the Higgs mass exists. We have studied th is phenomenon by simulating the SU(2) gauge Higgs model on a l a t t i ce . The resul ts indicate that ~ .~ = mR/row [~,~= 9.3 ± 1. However, serious f in i t e size effects make i t unfeasible to explore the region m~ _< A~,t-oll with Monte Carlo simulation.

1. INTRODUCTION In these conference proceedings I will de-

scribe work which was done together with A. Hasenfratz at the Supercomputer Computations Ins t i tu te in Tallahaseee [1].

The Weinberg-Salm model, the unified the- ory of electroweak interact ions describes mny perturbative phenomena successfully. There are indications however that the theory de- fined around the Gaussian (perturbative) fixed point is a t r i v i a l free f i e ld theory. Never- theless one can consider i t as an effect ive theory with a f i n i t e cut-off in the range of the parameter space where the cut-off is large and the renormalized couplings are non-zero. If we f ix the physical scale via the W-boson mass, changing the cut-off , Ac,t will change the Higgs-boson mass. Perturbative and approximate uon-perturbative analytic calculations sug- gest that (at fixed quartic coupling) decreas- ing L,t means increasing m,. Where ms is of the same order as L,t, the theory becomes mean- ingless which determines an upper bound for the Higgs mass. I t is expected that the bound in- creases with increasing quartic coupling giv- ing an absolute upper bound at in f in i t e scalar coupling.

In the present work we performed a Monte- Carlo calculation of the scalar sector of the Weinberg-Salam model at in f in i t e (bare) scalar quartic coupling and fixed bare gauge coupling (8 = 8), in a wide range of the hopping parameter x in the broken phase. The resul ts support the above picture giving Rm,, = ms/row Ira,,= 9.S + I. However, from studying the behavior of the W mass as a function of x on various l a t t i ce sizes, we argue that serious f in i t e size ef- fects make i t unfeasible to study the region where mw × N, < 2 (N, denotes the spacelike, Nt the timalike l a t t i ce size) which is the region mB <_ ~e,t at ~ = ec i f N8 ~ 12. 1. DEFINITIONS

We have considered the l a t t i ce regularized SU(2) gauge model coupled to a 2-coeponent com- plex scalar f ie ld O:

4

where ~ = 4[o 5 is the gauge coupling, ~ is the hopping parameter and A is the scalar quartic coupling. We took the l i m t ~ -~ ~ where the

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506 T. Neuhaus / Upper bound estimate for the Higgs mass

length of the scalar f ie ld is frozen to unity. In this limit the action reduces to

4

where the angle variable ~, is an SU(2) gauge matrix. By a suitable gauge transformation the

variables can be transformed to unity. ~e have computed the scalar and vector bosun

masses from the exponential decay of connected correlation functions at momentum zero. Ye have chosen the same operators used extensively in other Monte Carlo calculations already [2]. Specifically, for the Higga boson

s

o.(.,: (,,

while for the W bosun

i

. : L2,s;i--1,2,3, (4)

operators were used (n : pault matrices). We have measured the expectation values of planar Wilson loops to calculate the potential and to extract the renormalized gauge coupling. $. THE MONTE-CARLO CALCULATION

We have performed Monte Carlo (WC) simula- tion at ~ = 8,A = oo varying x in the broken phase in the range of x E (0.~,0.475). The simulations were performed on la t t ices of size 6 s ×16, 83 ×16, 10 ~ × 20 and 1~ × 24. We used periodic boundary conditions. Most of the calculations were per- formed on 83 × 16 and on the l~ × 24 la t t ices . The other la t t i ce sizes were used only at few x val- ues, where we wanted to study f in i t e l a t t i ce size effects in more detai l . We have carefully thermalized our configurations by discarding 1~ to 5.1o 4 seeeps before s tar t ing measurements, for a detailed account of our s t a t i s t i c s see

[iS. The computations were performed on a CYBER 205 using approximately 500 hours of cpu time. $.1 THEORETICAL CONSIDERATIONS

The t r i v i a l free f ie ld behavior of the Weinberg-Salam model is the consequence of the t r i v i a l i t y of the scalar ~' theory. It is almost rigorously proven that the only fixed point of a 2-component complex scalar model where a renormalized theory can be defined is the Gaussian fixed point with vanishing renor- malized couplings [3]. It can then be shown that at the Gaussian fixed point the inclu- sion of the gauge f ie ld can not generate a f i - nite quartic coupling, L = 0 in the gauge-Higgs systems as well [4]. It means that the rat io R = rex/row is an undefined parameter of the theory. However, perturbation theory already indicates that the Higgs mass is bounded from above [5]. If we f ix the scale by the W mass, decreasing cut-off will result in increasing Higgs mass. Where the cut-off is of the same or- der as the Higgs mass, the theory breaks down. As we increase the bare quartic coupling ~, this break-down happens at higher and higher ener- gies. However, i t does not increase without bound but saturates at values ~ > 50. Of course this coupling value is outside the range of per- turbation theory. Recently, a non-perturbative (approximate) renormalization group study was performed for the scalar 0(4) model [6]. The assumption there is that the effect of the gauge interaction can be taken into account perturba- t ively (~ ~ 0.4), only the scalar sector needs non-perturbative treatment. The results sup- ported the above picture giving R I~a~ 9.

Here we address the above questions using a method which includes the gauge interaction non-perturbatively as well, namely we study the SU(2) gauge Higgs model on a lattice. 4. RESULTS 4.1 THE RENOILMALIZED GAUGE COUPLING

We have measured space-time planar Wilson loop operators WR,T with R _< 6 and T <_ 12 on the 12 S x 24 lattice each I0 sweeps. We then adopted

T. Neuhaus / Upper bound estimate for the Higgs mass 507

the standard procedure for the calculation of the potential V(R) between two s ta t ic sources by determining the slope of the linear decay of - l , <WR,r > at fixed values of R and for T > 5 . At the considered values of n in the broken phase we found extremely well measurable Wil- son loop operators up to the largest calculated extensions.

While we expect a coulombic behavior of the potential at short distances there will be no l inear r ise of the potential at large distances as screening of the charges sets in. Such a be- havior can be described bya Yukawa type poten- t i a l . Here i t is then reasonable to take the exchange part icle equal to mw. We therefore assumed that the form of the potential is given by a Yukawa potential evaluated on a f in i t e la t - t ice and we deters/ned the renormalized fine structure constant ~,and thus the renormalized gauge coupling cons tan tS= , , x4r from a f i t to the potential values. For a detailed analysis see [1].

We find that the renormalized gauge cou- pling constant ~ shows analmost constant be- havior over a wide range of x values at a value of approx, o.65 in the broken phase of the SU(2) gauge-Higgs system at ~ = S. We note that the estimated values for g~ are quite close to the bare value ~ =0.5. 4,2 THE WMASS

The calculation of mw from the e~onential decay of the correlation functions is straight- forward. Figure i is the collection of all our W masses at different x values and lattice sizes. A few co~ents are in order here:

I) The correlation functions for the W are stable and well measurable even up to the largest distance. The l-mass fit gives a good quality fit starting from distance r = 3 and the obtained mass values are in general consis- tent with values determined from fits with r = 4 or 5. In short, it is possible and technically feasible to determine the W mass on the given size lattices and with our statistics within an (statistical) error of about 5 ~.

. 4

.3

.2

W-mass I ' ' " l ' ' " J ' " ' l ' "

+

hopp ing parameter • 3 . 3 5 . 4 . 4 5

FIGURE l : mw as function of x. The black diamonds correspond to 1 ~ × ~ , the open ci rc les to 83 × 16, the open squares to t~ ×20and the open triangles t o ~ × 16 la t t ices .

2) We observe defini te f in i t e size effects as we approach the phase transi t ion. The data for the W mass show a 'bend-over' typical for a system with correlation length compara- ble or larger then the spatial extent of the la t t ice . We return to the interpretation of the f in i t e size effects in the next section. 4.$ THE HIGGS MASS In order to determine the Higgs mass we mea- sured correlation functions of operators de- fined in sect. 2. We performed high s t a t i s t i c s MC measurements in the broken phase at x = 0.32, 0.325, 0.34 and 0.355 on 83 x 16 la t t ices . Table 1 contains 'ef fect ive mass' values determined from ratios of correlation functions at consec- utive time distances. The mass values stabi- l ize af ter distance S for x = 0.32 and x = 0.325 giving a good quality overall single exponen- t i a l f i t . At x = 0.34 the errors of the corre- lation functions are increasing (although we

508 T. Neuhaus / Upper bound estimate for the Higgs mass

have very good statistics here), still a sin- gle exponential flt Is acceptable. At x = 0.355 the correlation function indicates the mixture of two separate masses. This is understandable because the t r i a l operator for the Htggsmass couples to a 2if s ta te as well [2]. Using a 2- mass exponential f i t with the second mass fixed at 2.mw, we get a good f i t for the Higgsmass.

TABLE1

"~+l,~ ~.320 .325 .340 .355

m2,! .62 .73 1.05 1.28

ms,~ .49 .60 .98 1.23

m~.s .41 .64 .05 .76 ms,, .40 .60 .O0 .79 m6,s .38 .67 1.04 noise m7,6 .40 .70 noise noise

4.4 EXTRAPOLATION AND ESTIMATES FOR R~a, From the discussion of the previous two sec- tions i t is clear that the region where the Higgs mass is measurable within reasonable er- rors and where the if mass is not badly distorted from f in i t e size effects is very narrow. I t is impossible to obtain a reasonable estimate for mw by a direct measurement on 1~ × 24 or similar size la t t i ces in the region m,. a _< 1, which is the physically interesting region. One needs either some type of extrapolation in • or a re- l iable f in i te size analysis or preferably both in order to extend the accessible mu. a range to lower values.

Here we use what one can learn from the 0(, -. co} model, namely the remarkable linear behavior of < ~ >2~ m~ in (x-~c,) [1]. We suppose that the effect of the gauge interaction on the scaling behavior is small. Using values of mw from our 12 ~ x 24 l a t t i ces with x _> .39 and ~c, = 0.317(5) the linear ~2-fit to m~ gives mw(~ = .355) = .15 and mv/(~ = .34) = .12. These f in i t e size effects are somewhat larger than we expected.

Are these f in i t e size ef fects consistent with the mass values obtained on different la t -

tices? In figure 2 we plot z = mw • N, versus mw(N,)/mw(co) for all our data including dif- ferent x values ~ > .34 and lattice sizes. For the infinite volume mass row(co) we used the extrapolation. The solid curve is the corre- sponding (universal) curve for the scalar mass in the 0(. -~ co) fixed length model. This curve is in good agreement with the measured data, in spite of the fact that It corresponds to the 0(. -~ co) scalar model (although it is probably very close to the 0(4) case) and, what Is a more severe problem, it describes the finite size dependence of the scalar mass while we compare it to the vector bosun. Nevertheless our data indicate the existence of a universal scaling curve which in addition turns out to be consis- tent with the finite slze scaling of the scalar mass in the 0(. -~ co) model. We think that these results provide additional support for our In- finite volume mass extrapolation.

Combining values of the Higgs masses and W masses from our infinite volume extrapolation

z

I I [ [ I [ ( I ( [ i i i I i i I I

W - m a s s r a t i o

1 ' 1 . 5 2

FIGURE 2 : z : row(N,) x N, as function of mw(N,)/mw(co). The symbols have the same mean- ing as in the previous figure.

7". Neuhaus / Upper bound estimate for the Higgs mass 509

~e estimate for the ratio:

R= g.0 i = .97 at ~=.340 a.mS

R = 9.6 I =.72 at ~=.355 ~'mH

We estimate the error for those R values to be around to 10~. CONCLUSION AND OUTLOOK

We carried out a Monte Carlo simulation for the SU(2) gauge HiKEs system at ~= 8.,A = 0o vary- ing x in the broken phase.

Our goal was to investigate the region m, < 1 to study the scaling behavior of the scalar and vector boson masses and to obtain an estimate for the upper bound of the HtKKs mass.

Our results indicate that in the region where a. me ~ 1, mw is about a magnitude smaller then me, giving for the ratio R = mu/mw = 9.3±1 I a.mH = O(t). As R is expected to take its maxi- mum value at A = oo, we consider this as an ab- solute upper bound within the SU(2) gauge HiKEs model.

Considering the dlfficulties coming from the finite size dependence of the W-mass, we think that the direct Monte Carlo simulation of the gauge-Higgs model in the range a.mu = O(I) at large A is unfeasible. However, as the renor- malized gauge coupling is almost the same as the bare and depends on x only very weakly, simu- lating the 0(4) scalar model with Monte Carlo and adding the gauge interaction perturba- tively should be a good approximation to the full model. Then the W mass is obtained from the scalar field expectation value through the for-

3 . .

mula m~ = ~ < #, >2. Sznce < # > zs a bulk

quantity, i t s f in i te size dependence is pre- sumably much smaller then that of row, making the calculation possible.

On the Seillac conference two groups re- ported ( see contributions by K. Jansen and J. Kuti ) their results for the above mentioned quantities calculated In scalar theory. It seems to emerge that the la t t ice calculation of the maximal R within the gauge-HiKEs model at large values of ~ [1,7] ~ght well be consis- tent with the upper bounds derived there [8]. REFERENCES

[1] A. Hasenfratz and T. Neuhaus, preprint FSU-SCRI-87-29, to be published in Nucl. Phys B.

[2] for a sugary, see J. Jers~k, preprint PITHA 85/25.

[3] K.G. Wilson, Phys. Rev. 84 (1971) 3184. K.G. Wilson and J. Kogut, Phys. Rep., 12C (1974) 76.

[4] k. Hasenfratz and P. Hasenfratz Phys. Rev. D~ (1986) 3160.

[5] R. Dashen and H. Neuberger, Phys. Rev. Lett . , 50 (1983) 1897. M.A. Beg, C. Panagiotakopoulos and A. Sir l ln, Phys. Rev. Le t t . , 52 (1984) 883. D.J.E. Callaway, Nucl. Phys. 8223 (1984) 189.

[6] P. Hasenfratz and J. Nager, preprint BUTP-86/20

[7] I. Montvay, preprint DESY 86-143 (1986). W. Langguth and I. Montvay, preprint DESY 87- 020 (1987).

[8] M. LQsher and P. Welsz, preprlnt DESY 87-017 and DESY 87-rxx. J. Kuti and Y. Shen, preprint UCSD/PTH 87-14. A. Hasenfratz, K. Jansen, C. B. Lan K, T. Neuhaus and H. Yoneyama, preprint FSU-SCRI-87-52.