φ4 on F4: Numerical results

Nuclear Physics B353 (1991) 551-564 North-Holland

,~4

4:

NUMERICAL

SULT

Gyan BHANOT

Thinking Machines Corporation, 245, First Street, Cambridge, MA 02142, USA and

Institute for Advanced Study, Princeton, NJ 08540, USA Khalil BITAR and Urs M. HELLER

Supercomputer Computations Research Institute, Florida State University, Tallahassee, FL 32306, USA Herbert NEUBERGER

Department ofPhysics and Astronomy, Rutgers University, Piscataway, NJ 08855-0849, USA Received 6 August 1990 (Revised 9 October 1990)

In this paper we present results from numerically simulating the scalar sector of the minimal standard model on an F4 lattice. This lattice obeys Lorentz invariance to a higher degree than hypercubic lattices previously used in this context. The major result of our study is that if one is willing to tolerate cutoff effects up to about 3% in the WL-WL scattering cross section, the upper bound on the mass of the Higgs boson in the simplest F4 model is 590 GeV with an error of about 60 GeV.

1 . Introduction It is believed that the minimal standard model, although unambiguous to any finite order in the perturbative expansion, is not really a completely defined theory . In this respect it is different from other gauge theories, like for example QCD. The main reason for this lack of determinacy is the fact that the Higgs self-coupling becomes weaker as the postulated range of validity of the minimal standard model is extended to higher energies . One cannot keep this coupling at a finite non-vanishing value if the ultraviolet cutoff is taken to infinity . Alternatively, one cannot make the Higgs self-coupling arbitrarily strong . This reflects itself in two complementary ways : On the one hand the asymptotic series obtained from perturbation theory gives no useful estimate if the coupling is too large. On the other hand, the mass of the Higgs particle, which increases with the self-coupling, eventually becomes larger than the cutoff one needs to sustain the self-coupling and this means that the model has no real meaning . 0550-3213/91/$03 .50 U 1991 - Elsevier Science Publishers B.V . (North-Holland)

G. Bl:anot et al. /04 on F4 : Ncunerical results

jj?

e wish to find the maximal Higgs mass possible in an energy range within which we postulated that the standard model is providing an adequate description of nature [1]. If the top quark has a mass well below 1/2 TeV or so, it is reasonable to accept an estimate for this quantity obtained by studying the scalar sector independently of all other fields . By now there is a sizable amount of literature on this subject ; many of the relevant papers have been mentioned in our first paper in this series, ref. [2]. For updates and reviews see ref. [3]. The basic strategy is to use a lattice model that has built in enough flexibility to reproduce the complete class of 00/A2) corrections allowed by the general principles of relativistic field theory . In this way, any choice made by nature will be represented by some lattice action, up to and including O(1/A2) cutoff effects . A is an effective ultraviolet cutoff. One can quite easily convince oneself that one needs to investigate a class of lattice actions parametrized by four real parameters [2]. In this paper we shall only consider a one-dimensional submanifold of such a class. There are many different ways of choosing adequate families of lattice actions. Meaningful comparison between them is only possible by looking at physical quantities like particle masses, scattering amplitudes and so on. These cannot, in most cases, directly be measured on the lattice but, as long as the coupling is reasonably weak they can be estimated analytically . In the broken phase of the purely scalar theory we do perturbative calculations at fixed f,, ("pion" decay constant) and fixed m, (Higgs mass) . The expansion parameter is

e need one non-perturbative piece of information which is the value of the ultraviolet cutoff A in terms of m o. and f, Perturbative Renormalization Group leads to Mo,

A ~

C( gR/47r2 )13/24 exp[ -

(4 Tr2/9R)]

(1 + O( 9R))

as gR ---)'0 .

(2)

The number that cannot be obtained from perturbation theory alone is C. The definition of the UV cutoff A has a certain element of arbitrariness. For example, we could take as A -1 the distance between nearest neighbors on the F4 lattice or, equally justifiable, the distance between nearest neighbors on the "parent" hypercubic lattice . To give the cutoff an unambiguous physical meaning it has to be fixed from a physically observable cutoff effect . Such cutoff effects can be calculated perturbatively . In view of the fact that we want to avoid introducing lattice artefacts at 00/A 2 ) it is necessary to maintain Lorentz invariance to this order. This can be done exactly if one works with an action on an F4 lattice [4] that obeys the larger discrete symmetry group associated with this lattice . The necessary perturbative calculations for this case have been worked out in ref. [2].

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In the present paper we shall present first results coming from non-perturbative calculations on the F4 lattice . An estimate of the Higgs mass bound will be given within a restricted class of actions and with well-defined criteria for limiting the cutoff effects . Once a more complete search of the full four-dimensional manifold of actions is carried out a much more reliable estimate will be obtained. Our bound compares well with that obtained previously on hypercubic lattices [5,6]. To relate our results to the hypercubic ones the perturbative formulas from refs. [2,5] will be needed . Our results indicate that the presence of O(1 /A2) Lorentz breaking terms may have a numerical effect of the order of 10% on the bound within the class of models investigated and within the criteria adopted . There is a dependence of the bound on the precise form of the criterion adopted which can be as large as 10% ; in particular, the Lorentz breaking term affects more a direction-dependent quantity, like the 7r-7r scattering cross section at 90 degrees in the center-of-mass frame, than one which includes an average over all directions, like the total width of the o-particle . 2. Phase structure There exists a natural way of embedding the F4 lattice in the hypercubic one by removing all sites that have E,_, x,_, =odd. One can imagine integrating out the fields on the unwanted sites and being left with a local but somewhat longer range interaction on the F4 lattice . Although the latter is not invariant under the full symmetry group of the F4 lattice it is close enough to behaving as if it were, when probed at distances of the order of the correlation length once the system is in the critical regime. The results of our simulations also add to the above considerations a quantitative evaluation of the actual magnitude of the Lorentz symmetry violating effects on the hypercubic lattice (see also ref. [7]). The four-dimensional set of actions that allows independent variation of the physical effects of all possible renormalizable and 00/A2 ) terms is chosen to be given by 112 d +2( X ) + A .d [ +2( X S= -2K nn

x

x

] +n1: [tb2( X) 3+n'~ 4) 2(x) e2(x ? ) .

Here, the x's are sites on an F4 lattice and nn stands for nearest neighbor. This action has enough symmetry to preclude the appearance of the quantity E,,a4 which is not forbidden by the hypercubic lattice symmetry group . This quantity contributes to 00/A2 ) and violates Lorentz invariance . It is impractical to exhaustively search the whole range Of K, A, -q and 'q'; one needs to develop some technique for identifying regions in this parameter space

/04 on F4: Numerical results G. Blianot et al.

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where the Higgs mass can be made relatively large . As a first step we have to get a

feeling for the kind of numbers that are typical of the generic F4 lattice action . The simplest choice, analogous to the one in refs . [5,6], is to take

= ?7' = 0. The perturbative calculations of ref. [2] apply directly only in this case . Within this subclass it is quite plausible, and known to be true in the hypercubic case, that the maximal Higgs mass can be obtained when

A = oo . In the present paper we

therefore focus on a model with the action S= -2K F,+(X)+(X'), nn where

4~ = (OP 02 ,

03,04) and

4

(4)

is constrained by

We are interested in the critical regime and therefore the first step in our analysis was to accurately determine the critical point of the model with the action (4). This was done by simulating the theory on lattices of size 44 to 144 by standard Monte Carlo methods using Metropolis and heat bath algorithms . The results on the larger lattices are based on typically 10 5 to 4 X 10 5 heat bath sweeps after sufficient thermalization . Preliminary results were presented in ref. [8]. In all the results reported in this paper, the error estimates take correlations between successive measurements and between different observables into account . Autocorrelations are eliminated by bunching the data into sets of sufficient length . Correlations are accounted for by using the jackknife method [9] or by doing X2-fits correlated [10] . The critical point can quite accurately and efficiently be determined as rc c =

lim

rc c ( L,

L'- 0C

(6)

where K,(L, L') is the coupling at which the Binder cumulant

BL

__ 3 _ (

(.

2

agrees on lattices of size L and

1g

2)2)L

2 2 ~~ > L

'

L' [11]. BL

is shown in fig. 1 from which we infer

the estimate k C = 0.0917(2) . We have done simulations both in the symmetric and the broken phase of the model. ®f course, only the results in the broken phase are of direct physical

relevance . However, the work of Li1scher and Weisz [5] has shown that with the help of the Renormalization Group one can make use also of data obtained in the symmetric phase . The numbers we obtained in the symmetric phase are of a

G.

hanot et al. /,04 on F4 : Numerical results

555

0.6

0 .4

n

n v

0.2

v

0.0 0 .09

0 .092

0 .094

k

0.096

Fig . 1 . Binder cumulant for L = 4, 6, 8, 10, 12 and 14. The intersection of the curves gives estimates for K C.

quality inferior to the ones we got in the broken phase . Their main role was to provide a means of independently checking the validity of the main results in the broken phase . Such a check is needed because there are conceptual difficulties in intrinsically estimating the systematic errors of the values obtained for m," in the broken phase . The check is highly nontrivial as it involves going through the transition region with the help of a quite extensive analysis. The necessary Renormalization Group functions have been calculated in ref. [2]. The quality of the check suffers mainly from finite-size effects in the evaluation of the coupling constant g R in the symmetric phase. This problem could be completely eliminated if a high-temperature expansion to sufficiently high order were carried out in the symmetric phase on the F4 lattice . This has not been done yet - it is likely that, due to the fact that the F4 lattice is not bipartite, such an analysis would be even more demanding than the one worked out for the hypercubic case .

3. Symmetric phase We first discuss our results for the symmetric phase. Here it is relatively easy to compute the mass of the O(4) scalar particle and we did this in two ways . In the first we measured the momentum space propagator G(p) = (
- p)%

for ten different small momenta ranging in magnitude from 27r/L to Err/L, all of them unrelated under the F4 symmetry, as well as for zero momentu~i~. The mass and wave function renormalization Z R was then extracted from fits to the form of

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/04 on F4: Numerical results

1 .0 0.8 0.6 0.4 e

0.2 0.0

0.07

Fig. 2. mR as a function of

K

e

0.08

0.09

in the symmetric phase from lattices of size 14`x.

a free massive propagator [6],

G( p) -

(4 /6a)Z R g( P) + mR

where the factor of 4 counts the multiplicity, the factor of 1/6K converts the fields

from lattice to continuum normalization [2], and g(p) is the kinetic energy term on the F4 lattice, g(h)

=6

~ LZ

jt, *

v

- COS(pla+pv) - COS (pl~ `pv)] .

(10)

Z We carried out correlated X -fits of our data to eq . (9) for various ranges of momenta. For each fit we obtain errors from varying the X a by 1 . The spread of the results from different fits provides an estimate of the systematic uncertainty.

ZR always came out close to one and the rnass :~i R is shown in fig. 2 for our largest lattice, of size 14 4, as a function of tc . The errors shown reflect both the statistical and the systematic uncertainties.

The second way of computing the mass is from the exponential fall-off of the correlation function in coordinate space. As explained in ref. [2) the time direction

is along the euclidean direction e 1 and we computed correlation functions at zero spatial momentum . The mass computed in this way comes from the pole at

p1

= ï1V1 and

p2, p3, p4

= 0, whereas the mass obtained previously from the momentum space propagator comes from the behavior near zero four-momentum . The two masses are related in ordinary perturbation theory by mR=4sinh 2 (2M)+O(g R ) .

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G. Bhanot et al. / q4 on F4: Numerical results

55 7

Fig. 3. The finite-size behavior of M at rc = 0.09 near K, in the symmetric phase. The curve is a fit to the leading finite-size correction, eq. (13).

We extracted the mass from the time slice correlation functions by correlated X'-fits to the formula for a single particle intermediate state in a periodic box, -(t) =Acosh(M(t - 1L)),

(12)

for various ranges of t. The results, when converted by eq. (11), are consistent with the "propagator masses" shown in fig. 2. On finite lattices the masses are contaminated by finite-size effects . In ref. [2] we have shown that on the F4 lattice the leading finite-size effect is parametrized by A

M( L) = M(oo) + L3 /2exp [ - JM(oo) L]

(13)

as L -+ oo. Fig. 3 shows the mass at K = 0.09 as a function of L together with a fit to eq. (13) from the data for L = 8, 10, 12 and 14. The fit gives M(00) = 0.2054(12) and A = 7.8(4) which is in good agreement with leading-order perturbation theory [2]. This implies that we have good control over the finite-size effects on the mass in the symmetric phase . To be able to make use of these numbers in the broken phase we need some additional information . The most useful and natural quantity is the coupling constant 9R . A possible definition is extracted from the connected four-point function at zero external four-momenta . Our analytical work in ref. [2] was carried out with this particular definition . Defining the magnetization 4'= (1 /V )E x +(x), the coupling constant is given by gR

-

2 -Z R

32 (6K

)2

(C~~~4>-

L

4

C

(

.~2i2) . ~, 2)4

14

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558

The major difficulty in using eq. (14) in practice is that the statistical error grows 1:_. . ..~--' "~ vv ith the x,nlume . This can be seen by replacing the quantities in eq. (14) by lineally their values for a free theory and calculating their intrinsic dispersions . It is impractical to overcome this by increasing simulation times. Unfortunately, we could not find a range of volumes in which the finite-size effects are already asymptotic and, at the same time, the statistical error has not grown beyond the limit of usefulness. At only one point, close to the transition, we obtained some consistent numbers, for several volumes but, they were still not sufficiently accurate to carry out a credible finite-size analysis. In summary, the direct evaluation of the coupling from its definition (14) proved impossible. It seems that in the hypercubic case one can partially circumvent this problem by computing the renormalized coupling by fitting measured moments of the magnetiJ .'f/ 1, compatible zation to those obtained from a probability distribution for with the Û(4) symmetry, of the form ILit

a .c. ..

P(,/1) a .~1 3 exp -L4 m 2 (6K)~~Z~ 2 +

2 4~

9R( 6 K)

2 ~~ 4

.

(15)

Here the factors of 6K again convert lattice to continuum normalizations and we neglected the wave function renormalizations, which are, as we have seen, close to l . One cannot get meaningful estimates from our numbers for g R using this method unless one inputs in the fit a fixed value for the mass. Such a calculation is not really justifiable and we decided against using results obtained in this way. A third way of trying to avoid the problems with direct usage of eq. (14) is to consider an alternative definition of the coupling, where the external momenta are not all set to zero . A simple way is to define the coupling from the truncated four-point one-particle irreducible vertex function with two of the momenta set to zero and the other two at p = (im, 0, 0, 0) and p = - (im, 0, 0, 0). For small masses one expects the numerical difference between this coupling and the one of eq. (14) to be small . The reason for this definition is that, in a euclidean rotational invariant system, the leading finite-size correction to ( ./z?. 2 ) is given by a formula similar to (13) (see ref. [121). If one replaces the exponent by a particular modified Bessel function the further subleading corrections are exponentially suppressed . The amplitude of the leading correction is proportional to the above defined coupling . Fitting to the exponential correction only and comparing the results to those obtained by a fit to the formula with the modified Bessel function one can form an estimation of the magnitude of the systematic error. This method proved to work somewhat better close to the transition . We decided to take the measurement at K = 0.09 as providing the initial data for integrating the '8-function across the transition into the broken phase . Still, g R had a relatively large error of about 10 %, gR = 1_5.5(1 .5), while the mass is known accurately, MR = 0.2075(15).

G. Bhanot et al. / p4 on F4: Numerical results

55 9

1.00

0.95

Za

0.90

0.85

0.80 0.09

0.095

0.1

0.105

0.11

Fig . 4. Z,,, and Z,T as a function of a in the broker, phase from lattices of size 14 4 .

4. Broken phase We now turn to the results in the broken phase . Here we need to measure the pion wave function renormalization constant, the pion decay constant and the Higgs mass . First one has to settle on some definition for what one calls the longitudinal and transverse fields in a finite volume, where the symmetry does not break . We chose the component of + along the "magnetization" // (the longitudinal part) to represent the Higgs field a(x ) The transverse part represents the three "pions" which would be eaten by the gauge fields when weak SUM x U(1) is turned on, and is denoted by ir. The measurement of the pion wave function renormalization and its decay constant are relatively easy. The estimation of m, however, is a nontrivial matter . We measured momentum space propagators and time slice correlation functions for the rr- and a-fields. The numbers were fitted to the appropriate analogues of ey. (9). We allowed for a free mass parameter even for the pions because we are working in a finite volume . As expected, this parameter came out very small numerically and consistent with going to zero in the infinite-volume limit. Fig. 4 shows Z,r and for comparison, Z., as a function of K for our largest, 144, lattice . The physical scale is set by the analogue of the pion decay constant in this model, f, We extracted it using the method developed in ref. [13] taking into account the numerical changes in the finite-size formula that we worked out in ref. [2]. Measuring several moments // 2 "~ we determined the parameters in the following assumed form for the probability distribution of the "magnetization" .,//: P( ..//) a~l/~exp [- b(L)L4(6K~//2-a(L))2] .

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/04 on F4: Numerical results G. Bhanot et al.

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0.20

0.15

0 .10

0.05

0.00

0

0.005

0.01

0.015

0.02

Integral - Sum

Fig. 5. Finite-size fits of a(L) versus the difference between a finite-volume momentum sum and its limiting integral. The data are, from top to bottom, for K = 0.110, 0.105, 0.100, 0.0975, 0.095 and 0.0925.

The finite-size formula we used is

y 3a + ® log (f,L) a = 0 .105346 . . . . (17) a( L) = f 2 Z11 1 + ~ (f, L)2 Like in ref.

[131 we found that our fits improved when we replaced the 11L 2 term

in eq . (17) by the appropriate difference between a momentum

sum and a 2

momentum integral where an effective pion mass of the order of 11L included in the propagator . We left the coefficient of the

was also

11L term as a free 2

= f,2

/L2 parameter in the fits, i.e . we made fits of the form a(L) Z, + c 1 and P a(L) f,~Z,~ + c'( j l/(g( p) + mi ff ) - (1/2L4)ß'1/g(p)) . Using the previously

=

measured value of

Z,rr

we found c =

3ZTra and c' = 3Z. . This analysis also gave us

an estimate for the magnitude of the subleading finite-size effects that have been left unaccounted for. The finite-size systematic error of

fr

turned out to be larger

by almost an order of magnitude than the statistical uncertainty . Mill our numbers for

f,

are satisfactorily accurate for our purposes .

a(L) against the difference between /L2 term as the momentum sum and its continuous integral limit that replace the 1 In fig. 5, we plat the extracted values for

explained above. The fits work quite well and the extrapolated values yield, with the Z7r's obtained before, good estimates of function of

K.

f,

which are shown in fig. 6 as a

b(L) should behave as 1 /log L for large L because of the pion cut. The values of L we worked at are too small to see the logarithm. However, we did detect a stronger volume sensitivity in

b(L) than in other quantities, like a(L) for example.

G. Bhanot et al. / 0' on F4 : Numerical results

56 1

0.20

0.15

0.10 49

0.05

0.00 0.09

0.095

0.1

0 .105

0.11

Fig. 6. f;, versus rc in the broken phase.

This behavior of b(L) indicates that fits of the a-propagator in momentum space to a free field propagator might give unrealistic estimates for the real part of the a-particle resonance location . Although we saw no numerical evidence for a logarithmic size dependence this does not ensure that there is no sizable contribution, effectively constant over the range of volumes under investigation, of the two pion cut, which contaminates any Higgs mass estimate one might try to obtain from the zero-momentum regime. Therefore we also extracted estimates for the Higgs mass from time slice correlations . At least, in this method, one is sure that what one gets is an estimate for the lightest state in the appropriate channel . However, for large enough volumes this is guaranteed to be a state consisting, essentially, of two pions of minimal nonvanishing momenta reflecting the instability of a to decay into multiple (even number of) pion states in an infinite volume . The finite volume cuts off the very soft pions and, since the minimal momenta are not really small numerically, essentially forbids the decays. It is hoped [14], but not really proven, that this stabilization does not contaminate to a large extent the value one obtains for the mass of the a-particle . The space lattice corresponding to the F4 lattice is an fcc lattice and the minimal nonvanishing three-momentum that is allowed has a magnitude C7r/L . Hence, when the Higgs masses get close to twice this value one should allow for an unknown systematic error of the order of the Higgs width there. With our range of masses and volumes we estimate that this effect can generate a systematic error of the order of one half width (about 5%). There is another source for finite-size effects coming in, and numerically it is even more significant than the mixing. Were we to compute the 0-propagator for the finite-volume a-field defined above at a generic finite and non-vanishing value of lattice momentum (this means at a p = 2-rrs/L with L and s going to infinity

562

/04 on F4: Numerical results G. Bl:anot et al.

0 .5

0.0

0

0.01

0.02

1/L2

0 .03

2 Fig. 7. Finite-size fits of M, (L) versus 11L . The data are, from top to bottom, for a = 0.110, 0.105, 0.100, 0.0975, 0.095 and 0.0925.

while p is held fixed and away from zero) we would expect the answer to converge to its infinite-volume limit with a leading correction that goes asymptotically as 1 /L2 . This follows from the same sort of analysis that led to eq. (17) order by order in perturbation theory . Since the evaluation of a mass from time slice correlations attempts to get numbers describing the o--propagator in some range of complex momenta close to the resonance it is reasonable to try to account for part of the finite-size effects on m, by fitting the results obtained at various volumes to an 1 /L2 behavior . These fits are shown in fig. 7 and seem to work reasonably well . Fig. 8 is the main result of the simulation in the broken phase. There we plot the ratio m,/f,, as a function of ma in lattice units. The masses used are the extrapolations to infinite volume as described above . The horizontal axis gives the ratio between the Higgs mass in physical units and the UV cutoff. The Higgs mass in physical units is obtained by multiplying the plotted ratio by 247 GeV, the experimentally known value of f, These numbers are now subjected to a global consistency check by comparing them with the integration of the 6-functions (one for the symmetric and the other for the broken phase) we calculated in ref. [2]. These functions included all scaling violations to tree level, scaling violations to order m R log(m R ) at one-loop order and all the universal terms to three loops . One constant of integration is needed to be able to fix the line on both sides of the transition . We worked out several cases taking as initial conditions either the K = 0.090 point in the symmetric phase or K = 0.095 or K = 0.105 in the broken phase . The errors are quite large but the numbers are self-consistent. In the broken phase the data seem to even reproduce the turn-over of the curve which originates from the tree level scaling violation term in the 6-function . In fig. 8 we also show the curve obtained from integrating

G. Bhanot et al. /,0 4 on F4 : Numerical results

563

3.0

2.5

2.0

b

r  I  I  I  I  I , ; 0.2 0 .4 0 .6 0 0.8 1 Mc MQ in lattice units. The line is the result from the integration of the

Fig. 8. The ratio M,,/f,, versus three-loop 6-function with one-loop scaling violation terms included with the initial value taken from the point at MQ = 0.414(25). The dashed lines represent the uncertainty propagated by the RG integration from the errors of the initial point.

the 8-function with initial conditions given by the point at K = 0.095 with a width generated by the error there. When extended into the symmetric phase with the help of the connection formula derived in ref. [5] one obtains reasonable agreement with the values measured there at K = 0.09. Using this curve one also can estimate the unknown constant in eq. (2): in the broken phase C = 13 with an asymmetric error of + 7 and - 3. 5. Conclusions To make a meaningful statement about the Higgs mass bound we need physical limiting criteria for UV cutoff effects [2,51. In ref. [2] we calculated, using perturbation theory, that, for c.m. energies smaller than twice the Higgs mass, pion-pion scattering has no more than a 3% UV cutoff effect, as long as ma in lattice units is smaller than 1 . If one goes to a cutoff twice as large, the unitless Higgs mass becomes 1/2 and the UV cutoff correction decreases dramatically to 0.3% . In ref. [2] we also calculated the cutoff corrections to the Higgs width . For the range of masses under consideration these relative corrections turned out to be smaller by more than a factor of 5 when compared to the ones mentioned above. A good representative of our result is summarized in the following : M11 < 590(60) GeV

3% cutoff effect on scattering,

Mf , < 530(60) GeV

0.3% cutoff effect on scattering .

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Bhanot et al. /04 on F4 : Numerical results

This should be compared to the results from hypercubic lattices, which, when presented in an analogous fashion, we estimate from refs. [5,61 as MH < 640(65) GeV

3% cutoff effect on scattering,

MH < 520(50) GeV

0 .3% cutoff effect on scattering .

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This work was partially supported by the US Department of Energy under contract DE-FC05-85ER250000 (K .B. and U.M.H.) and contract DF.-FG0590ER40559 (H.N.). The work of G .B. was partly supported by a New Jersey High Technology Grant 88-240090-2 . The numerical calculations were done on the ETA10-G at SCRI and, supported by the NSF under grant PHY-870041, on the Cray-YMP at the Pittsburgh Supercomputing Center. eferences [11 L. Maiani, G. Parisi and R. Petronzio, Nucl. Phys. B136 (1978) 115 ; R. Dashen and H. Neuberger, Phys. Rev. Lett. 50 (1983) 1897 [21 G. Bhanot, K. Bitar, U . M. Heller and H. Neuberger, Nuci. Phys. B343 (1990) 467 ; U.M. Heller, Nucl. Phys. B (froc . Suppl .) 17 (1990) 649 [3] J. Jersàk, in Lattice gauge theory - a challenge in large-scale computing, cut. r . Bunk and K. Schilling (Plenum Press, New York, 1986) ; P. Hasenfratz, Nucl. Phys. B (Proc. Suppl .) 9 (1989) 3; H . Neuberger, Nucl. Phys. B (Proc. Suppl .) 17 (1990) 17 [41 H . Neuberger, Phys. Lett. B199 (1987) 536 [5] M. Li1scher and P. Weisz, Nucl. Phys. B318 (1988) 705 [61 J. Kuti, L. Lin and Y. Shen, Phys. Rev. Lett . 61 (1988) 678 ; A. Hasenfratz, K. Jansen, J. Jersák, C.B. Lang, T. Neuhaus and H. Yoneyama, Nucl. Phys. B317 (1989) 81 [7) C. Lang, Nucl. Phys. B (Proc . Suppl.) 17 (1990) 665 [8] G. Bhanot, Nucl . Phys. B (Proc. Suppl .) 17 (1990) 653 [91 B. Efron, The jackknife, the bootstrap and other resampling plans, Soc . Ind . Appl. Math . (1982) [101 D. Toussaint, in From actions to answers, Proc. 1989 Theoretical Advanced Summer Institute in Particle physics, ed. T. DeGrand and D. Toussaint (World Scientific, Singapore, 1990) [111 K. Binder, Z. Phys. B43 (1981) 119; W. Bernreuther, M. G6ckler and M. Kremer, Nucl. Phys. B295 (1988) 211 [12] H. Neuberger, Phys. Lett. B233 (1989) 183 [131 U. M. Heller and H. Neuberger, Phys. Lett. B207 (1988) 189 [14] U. J. Wiese, Nucl. Phys. B (Proc. Suppl .) 9 (1989) 609