Upper bound estimate for the Higgs mass from the lattice regularized Weinberg-Salam model

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

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UPPER BOUND ESTIMATE FOR THE HIGG$ MASS FROM THE LATTICE REGULARIZED WEINBERG-SALAM MODEL Thomas Neuhaus Institute 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 field theory. However, considering the W-S model as an effective f i e l d theory, an upper bound for the Higgs mass exists. Wehave studied this phenomenon by simulating the SU(2) gauge Higgs model on a l a t t i c e . The results indicate that ~ . ~ = mR/row [~,~= 9.3 ± 1. However, serious f i n i t e size effects make i t unfeasible to explore the region m~ _
coupling. 0920 5632/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

In the present work we performed a MonteCarlo calculation of the scalar sector of the Weinberg-Salam model at i n f i n 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 results support the above picture giving Rm,, = ms/row Ira,,= 9.S + I. However, from studying the behavior of the Wmass as a function of x on various l a t t i c e sizes, we argue that serious f i n i t e size effects make i t unfeasible to study the region where mw × N, < 2 (N, denotes the spacelike, Nt the timalike l a t t i c e size) which is the region mB <_~e,t at ~ = ec if N8 ~ 12. 1. DEFINITIONS We have considered the l a t t i c e regularized SU(2) gauge model coupled to a 2-coeponent complex scalar field O: 4

where ~ = 4[o5 is the gauge coupling, ~ is the hopping parameter and A is the scalar quartic coupling. We took the l i m t ~ -~ ~ where the

T. Neuhaus / Upper bound estimate for the Higgs mass

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length of the scalar field 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 momentumzero. 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 Wbosun 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) simulation at ~ = 8,A = oo varying x in the broken phase in the range of x E (0.~,0.475). The simulations were performed on l a t t i c e s of size 6s ×16, 83 ×16, 10~ × 20 and 1~ × 24. Weused periodic boundary conditions. Most of the calculations were performed on 83 × 16 and on the l~ × 24 l a t t i c e s . The other l a t t i c e sizes were used only at few x values, where we wanted to study f i n i t e l a t t i c e size effects in more detail. Wehave carefully thermalized our configurations by discarding 1~ to 5.1o4 seeeps before starting 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 i e l d 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 renormalized couplings [3]. It can then be shown that at the Gaussian fixed point the inclusion of the gauge f i e l d can not generate a f i nite quartic coupling, L = 0 in the gauge-Higgs systems as well [4]. It means that the ratio 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 fix the scale by the Wmass, decreasing cut-off will result in increasing Higgs mass. Where the cut-off is of the same order as the Higgs mass, the theory breaks down. As we increase the bare quartic coupling ~, this break-down happens at higher and higher energies. However, i t does not increase without bound but saturates at values ~ > 50. Of course this coupling value is outside the range of perturbation 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 perturbatively (~ ~ 0.4), only the scalar sector needs non-perturbative treatment. The results supported 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 G A U G E COUPLING We have measured space-time planar Wilson loop operators WR,T with R _<6 and T <_ 12 on the 12S x 24 lattice each I0 sweeps. We then adopted

T. Neuhaus / Upper bound estimate for the Higgs mass

the standard procedure for the calculation of the potential V(R) between two static sources by determining the slope of the linear decay of -l, at fixed values of R and for T > 5 . At the considered values of n in the broken phase we found extremely well measurable Wilson loop operators up to the largest calculated extensions. While we expect a coulombic behavior of the potential at short distances there will be no linear rise of the potential at large distances as screening of the charges sets in. Such a behavior can be described bya Yukawatype potent i a l . Here it is then reasonable to take the exchange particle equal to mw. Wetherefore assumed that the form of the potential is given by a Yukawapotential evaluated on a f i n i t e lattice and we deters/ned the renormalized fine structure constant ~,and thus the renormalized gauge coupling c o n s t a n t S = ,, x4r from a f i t to the potential values. For a detailed analysis see [1]. Wefind that the renormalized gauge coupling constant ~ shows analmost constant behavior 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. Wenote 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 straightforward. 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 consistent 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 ~.

507

W-mass

I''"l''"J'"'l'" .4

+

.3

.2

hopping parameter •3

.35

.4

.45

FIGUREl : mw as function of x. The black diamonds correspond to 1 ~ × ~ , the open circles to 83 × 16, the open squares to t~ ×20and the open triangles t o ~ × 16 l a t t i c e s .

2) We observe definite f i n i t e size effects as we approach the phase transition. The data for the Wmass show a 'bend-over' typical for a system with correlation length comparable or larger then the spatial extent of the l a t t i c e . We return to the interpretation of the f i n i t e size effects in the next section. 4.$ THE HIGGS MASS In order to determine the Higgs mass we measured correlation functions of operators defined in sect. 2. Weperformed high s t a t i s t i c s MCmeasurements in the broken phase at x = 0.32, 0.325, 0.34 and 0.355 on 83 x 16 lattices. Table 1 contains 'effective mass' values determined from ratios of correlation functions at consecutive time distances. The mass values stabilize after distance S for x = 0.32 and x = 0.325 giving a good quality overall single exponent i a l f i t . At x = 0.34 the errors of the correlation functions are increasing (although we

T. Neuhaus / Upper bound estimate for the Higgs mass

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have very good statistics here), still a single 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 state as well [2]. Using a 2mass 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,! ms,~ m~.s ms,, m6,s m7,6

.73 .60 .64 .60 .67 .70

.62 .49 .41 .40 .38 .40

1.05 .98 .05 .O0 1.04 noise

1.28 1.23 .76 .79 noise noise

4.4 EXTRAPOLATION AND ESTIMATES FOR R~a, From the discussion of the previous two sections i t is clear that the region where the Higgs mass is measurable within reasonable errors and where the if mass is not badly distorted from f i n i t e size effects is very narrow. It is impossible to obtain a reasonable estimate for mw by a direct measurement on 1~ × 24 or similar size l a t t i c e s in the region m,. a _<1, which is the physically interesting region. One needs either some type of extrapolation in • or a reliable f i n i t e 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 c e s with x _>.39 and ~c, = 0.317(5) the linear ~2-fit to m~ gives mw(~ = .355) = .15 and mv/(~ = .34) = .12. These f i n i t e size effects are somewhat larger than we expected. Are these f i n i t e size effects consistent with the mass values obtained on different l a t -

tices? In figure 2 we plot z = mw •N, versus mw(N,)/mw(co)for all our data including different x values ~ > .34 and lattice sizes. For the infinite volume mass row(co)we used the extrapolation. The solid curve is the corresponding (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 consistent with the finite slze scaling of the scalar mass in the 0(. -~ co) model. We think that these results provide additional support for our Infinite volume mass extrapolation. Combining values of the Higgs masses and W masses from our infinite volume extrapolation z I

I

[

[

I

[

(

I

W-mass 1

(

[

i

i

i

I

i

i

I

I

ratio

'1.5

2

FIGURE2 : z : row(N,) x N, as function of mw(N,)/mw(co). The symbols have the same meaning as in the previous figure.

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

~e estimate for the ratio: i = .97 at ~=.340

R= g.0

a.mS

R= 9.6

~'mH

I =.72 at ~=.355

Weestimate 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 varying 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 HtKKsmass. 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 maximumvalue at A = oo, we consider this as an absolute 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 renormalized gauge coupling is almost the same as the bare and depends on x only very weakly, simulating the 0(4) scalar model with Monte Carlo and adding the gauge interaction perturbatively should be a good approximation to the full model. Then the W mass is obtained from the scalar field expectation value through the formula m~ = ~ < #, >2. Sznce < # > zs a bulk 3

.

.

509

quantity, its f i n i t e size dependence is presumably much smaller then that of row, making the calculation possible. On the Seillac conference two groups reported ( 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 lattice calculation of the maximal R within the gauge-HiKEs model at large values of ~ [1,7] ~ g h t well be consistent 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 PITHA85/25. [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. Sirlln, Phys. Rev. L e tt., 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 87020 (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. LanK, T. Neuhaus and H. Yoneyama, preprint FSU-SCRI-87-52.