---

Numerical analysis of the Higgs mass triviality bound

arXiv:hep-ph/9303215v1 3 Mar 1993FSU-SCRI-93-29CU-TP-590RU-93-06Numerical analysis of the Higgs mass triviality boundUrs M. Hellera, Markus Klomfassb,Herbert Neubergerc,Pavlos Vranasaa Supercomputer Computations Research InstituteThe Florida State UniversityTallahassee, FL 32306b Department of PhysicsColumbia UniversityNew York, NY 10027c Department of Physics and AstronomyRutgers UniversityPiscataway, NJ 08855–0849ABSTRACT: Previous large N calculations are combined with numerical work atN = 4 to show that the Minimal Standard Model will describe physics to an accuracyof a few percent up to energies of the order 2 to 4 times the Higgs mass, MH, only ifMH ≤710 ± 60 GeV . This bound is the result of a systematic search in the space ofdimension six operators and is expected to hold in the continuum.

Given that studyingthe scalar sector in isolation is already an approximation, we believe that our result issufficiently accurate and that further refinements would be of progressively diminishinginterest to particle physics.

1. INTRODUCTION AND CONCLUSION.Our goal is to obtain an estimate for the triviality bound on the Higgs mass in the mini-mal standard model.

Much work has preceded this paper (for examples consult [1,2,3,4,5,6]and the review [7]). We build on these results and generalize them.

Specifically, we dealmore systematically with the quantitative uncertainty resulting from the arbitrariness ofthe coefficients of higher dimensional operators in the scalar sector [8]. By doing this weaim to obtain a number that is meaningful beyond lattice field theory and directly relevantto particle physics.The overall framework of the approach has been reviewed before (e.g.

[8]) and will notbe repeated here. Its main simplifying feature is to treat the scalar sector of the minimalstandard model in isolation.

When considering further efforts in this framework the po-tential impact on particle physics should be evaluated against the accuracy of neglectingother interactions, e.g. with the top quark.

We believe that our result is reliable to areasonable degree, given that the whole framework is an approximation, and we feel thatfurther refinements would be of progressively diminishing interest to particle physics*.Our result is that the minimal standard model will describe physics to an accuracyof a few percent up to energies of the order 2 to 4 times the Higgs mass, MH, only ifMH ≤710 ± 60 GeV . The two major assumptions made are that ignoring all couplingsbut the scalar self–coupling is a good approximation and that any higher energy theoryinto which the standard model is embedded will not conspire to eliminate all dimensionsix scalar field operators in the low energy effective action.

The number we obtain is notsurprising because of its closeness to tree level bounds [1,3]; what has been achieved is tofinally show that in any reasonable situation higher orders in perturbation theory cannotchange it substantially although quite strong scalar self–interactions are possible.In the sequel we shall rely quite heavily on [9] but we also try to make the paperaccessible to readers who are not familiar with [9]. Our basic strategy was to first use 1Nexpansion techniques in the generalization of the O(4) symmetric scalar sector to O(N)to obtain an analytical non–perturbative estimate for the bound and then follow up withMonte Carlo simulations at the physical value N = 4.

In the next section we summarizeneeded information at N = ∞and add some new results in this limit. The followingsection presents our numerical work with some emphasis on the checks that were made toascertain control over systematic errors.

The last section explains our main conclusion.Appendix A gives a few more details on the new large N results, and finally in appendixB we collect the numbers obtained from our simulations in several tables matching thegraphs shown in the main text. * An exception would be further investigations of the Higgs width on the lattice.2

Our notational conventions are: The Higgs mass in physical units (GeV ), defined asthe real part of the resonance pole, is denoted by MH and the width by ΓH. The samequantities in lattice units are denoted by lower case letters, mH and γH.The matrixelement of conventionally normalized currents of broken symmetries between the vacuumand single Goldstone boson states (referred to as pions, π) is denoted by F in physicalunits and by f in lattice units.

The scalar self–coupling is defined by:g = 3M2HF 2. (1.1)In the N = ∞section everything is written in terms of the above N = 4 notation.

We useΛ to denote a generic cutoff.Most of our work is on the F4 lattice which can be thought of as embedded in ahypercubic lattice from which odd sites (i.e. sites whose integer coordinates add up to anodd sum) have been removed.

The lattice spacing of the hypercubic lattice is a. It is setto unity when mH, γH and f are used.

The F4 lattice is always fully symmetric having,when finite, L sites in each principal axis direction so that the total number of sites is L4.Usually, x, x′, x′′ denote sites, < x, x′ >, l, l′ links, ≪x, x′ ≫next–nearest–neighboringpairs, < l, l′ > pairs of links, and the field is constrained by ⃗Φ2(x) = 1.2. RESULTS AT LARGE N.Here we summarize the large N results of [9] relevant to our numerical work.

Theycontain predictions that can be directly compared to N = 4 Monte Carlo data and eval-uations of observable cutoffeffects on π −π scattering. The dependence of the boundon the magnitude of the observable cutoffeffects is relatively insensitive: a change by afactor of 3 induces a variation of 50 GeV in the bound in the worst case.

Thus, we do notworry about 1N corrections to the cutoffeffects. Cutoffeffects could also be calculated inperturbation theory but we have argued (see Appendix B of [9]) that the 1N computationis probably more reliable.

Hence we use the 1N results here.2.1. Relaxing the bound.The na¨ıve expectation that heavier Higgs masses are obtained when the bare scalarself–coupling is increased is upheld by nonperturbative calculations.

The search for thebound can therefore be restricted to nonlinear actions.Among the nonlinear actions the bound is further increased by reducing as much aspossible the attraction between low momentum pions in the I = J = 0 channel. Given3

a bare action the approximate combination of parameters achieving this is identified asfollows. Expand in slowly varying fields and use a field redefinition to bring the action,including terms up to fourth order in the momenta, to the formSc =Zx12⃗φ(−∂2)⃗φ −b12N (∂µ⃗φ · ∂µ⃗φ)2 −b22N (∂µ⃗φ · ∂ν ⃗φ −14δµ,ν∂σ⃗φ · ∂σ⃗φ)2.

(2.1.1)At N = ∞b2 has no effect and the bound depends monotonically on b1, increasing withdecreasing b1. Overall stability of the homogeneous broken phase restricts the range of b1.For example on the F4 lattice, at the optimal value of b1 the bound is increased by about100 GeV relative to the simplest non linear action.The rule in the above paragraph does not lead to an exactly universal bound.

Differentbare actions that give the same effective parameter b1 can give somewhat different boundsbecause the dependence of physical observables on the bare action is highly nonlinear.For example, at the optimal b1 value, Pauli–Villars regularizations lead to bounds higherby about 100 GeV than some lattice regularizations. This difference between the latticeand Pauli–Villars can be traced to the way the free massless inverse Euclidean propagatordeparts from the O(p2) behavior at low momenta.

For Pauli–Villars it bends upwards toenforce the needed suppression of higher modes in the functional integral, while on thelattice it typically bends downwards to reflect the eventual compactification of momentumspace.Because we desire to preserve Lorentz invariance to order 1/Λ2 we use the F4 latticeand, on the basis of the above observations, there are three stages of investigation. Thefirst stage is to investigate the na¨ıve nearest–neighbor model.

This should be viewed as thegeneric lattice case where no special effort to increase the bound is made. Since this case hasbeen investigated thoroughly in [5] we can proceed to more complicated actions with welltested methods of analysis.

The next stage is to write down the simplest action that has atunable parameter b1. The last stage is to add a term to eliminate the “wrong sign” orderp4 term in the free nearest–neighbor propagator, amounting to Symanzik improvement of4

the large N pion propagator. The three F4 actions, investigated at large N, are given byS′1 = −2Nβ0X⃗Φ(x) · ⃗Φ(x′)S′2 = S′1 −Nβ248XxXl∩x̸=∅l=⃗Φ(x) · ⃗Φ(x′)2S′3 = −Nβ0Xx"2Xx′ n.n.

to x⃗Φ(x) · ⃗Φ(x′) −12Xx′′ n.n.n. to x⃗Φ(x) · ⃗Φ(x′′)#−Nβ272Xx"2Xx′ n.n.

to x⃗Φ(x) · ⃗Φ(x′) −12Xx′′ n.n.n. to x⃗Φ(x) · ⃗Φ(x′′)#2.

(2.1.2)These actions were chosen because they have relatively simple large N limits and permita simultaneous study of both F4 and hypercubic lattices. In each case, at constant β2,β0 is varied tracing out a line in parameter space approaching a critical point from thebroken phase.

This line can also be parameterized by mH or g. For actions S′2 and S′3, β2is chosen so that on this line the bound on MH is expected to be largest. A simulationproduces a graph showing mHfas a function of mH along this line.

The y-axis is turnedinto an axis for MH by MH = mHf× 246 GeV . The large N predictions for these graphsare shown in Figure 2.1.

Since action S′3 was not treated in [9] we include a brief accountin appendix A.2.2. CutoffEffects.At N = ∞the cutoffeffects to order 1/Λ2 on the Higgs width and π −π scatteringare parameterizable by:Seff = SR(g) + c exp[−96π2/g]O(g) .

(2.2.1)SR contains only universal information and so does O. All the non–universal informationis in the g independent parameter c. The form of Seff reflects the factorization of cutoffeffects into a universal g dependent function and a g independent non–universal amplitude.The bound is increased by first varying the non–universal part so that c decreases atconstant g and then going along the selected line to higher g.¯δ|A|2 denotes the fractional deviation of the square of the πa−πa (a identifies one of theN −1 directions transverse to the order parameter in internal space) scattering amplitudeat 90 degrees in the CM frame at energy W from its large N universal value.

A plot of ¯δ|A|25

Figure 2.1The Higgs mass MH = mHf× 246 GeV in physical units vs. the Higgs massmH in lattice units for the three actions, eq. (2.1.2).as a function of MH(mH) for the three actions is presented in Figure 2.2.

If one considersonly the magnitude of cutoffeffects as a function of mH = MHΛ , one might conclude thatthe bound obtained with action S′1 would be larger than the bound obtained with S′2. Thisconclusion proves to be wrong when the mass in physical units is considered.

The values ofMH in GeV , determined from MH = mHf× 246 GeV , are put on three horizontal lines inFigure 2.2 at ¯δ|A|2 = 0.005, 0.01, 0.02. At large N the bound increases when going fromS′1 to S′2 and then to S′3 by a little over 10% at each step.

For example, for ¯δ|A|2 = .01 weget bounds on MH of 680, 764, 863 GeV for S′1, S′2 and S′3 respectively.2.3. Width.The width ΓH is important for phenomenology and for lattice work (see sect.

3.4).With massless pions perturbation theory seriously underestimates ΓH when MH is large[9].6

Figure 2.2Leading order cutoffeffects in the invariant π −π scattering amplitude at900 at center of mass energy W = 2MH vs.the Higgs mass in latticeunits for the three actions.The values of MH in GeV determined fromMH = mHf× 246 GeV are put on the three horizontal lines at ¯δ|A|2 =0.005, 0.01, 0.02.It would therefore be desirable to determine the width non–perturbatively. Up to datethe only methods known for measuring the width within a numerical simulation require anon–zero pion mass [10].

To study the effects of a non–zero pion mass we computed thewidth in this case at large N. The mass for the pions was induced by an external magneticfield that breaks the symmetry explicitly [11]. Because the cutoffeffects on the width arevery small we show in Figure 2.3 only the universal part and compare it to the leadingorder perturbative values.

Shown are the two results for Mπ = 0, 100, 200, and 300 GeV .We see that the deviations from perturbation theory decrease when the pion massincreases. Therefore to detect nonperturbative effects on the width at N = 4 one wouldhave to deal either with the massless case directly or, at least, with quite light pions, sayMπ ∼< 16MH.

This is one point we believe deserves further study.7

Figure 2.3The universal part of the width vs. MH. The solid line displays the large Nresult scaled to N = 4 and the dotted line shows the leading order term inperturbation theory.

From left to right the lines correspond to pion massesMπ = 0, 100, 200, and 300 GeV .3. THE PHYSICAL CASE N = 4.The primary aim of the numerical work is to produce at N = 4 the analogues of thegraphs in Figure 2.1.

The actions simulated are slightly different than those investigatedat large N.Firstly, the factors N accompanying the couplings in (2.1.2) are omitted.Secondly, we take advantage of the fact that on the F4 lattice, unlike on the hypercubiclattice, the simplest action with a tunable parameter b1 can be constructed in a way thatmaintains the nearest–neighbor character of the action. This is done by coupling fieldssited at the vertices of elementary bond–triangles.

Finally we add a term to eliminatethe “wrong sign” order p4 term in the free nearest–neighbor propagator, leaving, unlikein S′3, the term with the tunable b1 coupling unchanged. This new term couples next–8

nearest–neighbors and amounts to tree level Symanzik improvement. * The three F4 actionssimulated (with Si having the same expansion in slowly varying fields as S′i investigatedat large N) areS1 = −2β0X⃗Φ(x) · ⃗Φ(x′)S2 = S1 −β28XxXl,l′∩x̸=∅, l∩x′̸=∅, l′∩x′′̸=∅x,x′,x′′ all n.n.h⃗Φ(x) · ⃗Φ(x′) ⃗Φ(x) · ⃗Φ(x′′)iS3 = −2(2β0 + β2)X⃗Φ(x) · ⃗Φ(x′) + (β0 + β2)X≪x,x′≫⃗Φ(x) · ⃗Φ(x′)−β28XxXl,l′∩x̸=∅, l∩x′̸=∅, l′∩x′′̸=∅x,x′,x′′ all n.n.h⃗Φ(x) · ⃗Φ(x′) ⃗Φ(x) · ⃗Φ(x′′)i.

(3.1)The data for action S1 can be found in [5]. Preliminary data for action S2 were presentedin [8,12] and references therein.

Our main task is to finalize the results for action S2 andcarry out some new measurements for action S3.Each action is investigated in two main steps. First, the phase diagram is established;next, a particular line is chosen in the broken phase, which for actions S2 and S3, amountsto picking a value for β2.

On this line we make several measurements at different values ofβ0 approaching the critical point β0c(β2). We chose β2 based on the large N criteria andon the measured structure of the phase diagram.

We may therefore be missing the “best”action, but, from our experience at large N, we expect at most a 20 −30 GeV additionalincrease in the bound. Because we object to excessive fine–tuning and since one may viewwhat we are doing already as some amount of fine tuning, the 20 −30 GeV might go ineither direction and represents the systematic uncertainty that we assign to the questionof fine tuning.3.1.

Methods in General.We follow closely the approach of [5]. We use a Metropolis algorithm to map out thephase diagram and a single cluster spin reflection algorithm, tested against the Metropolisalgorithm, for the actual measurements.

Typically we use 10, 000 −100, 000 lattice passes,depending on lattice size and couplings, and simulate systems of increasing sizes with even* On the hypercubic lattice tree level improvement of the simplest action may also helpreduce Lorentz violation effects at order 1/Λ2 and was investigated in [6].9

L (to avoid frustration effects), L = 6, · · ·, 16. The spin update speeds on a single processorCRAY Y–MP are about 20µsecsite for action S1, 50µsecsite for S2 and 55µsecsite for S3.The total computer time invested is 400 hrs for action S1, 500 hrs for S2 and 400 hrsfor S3.

The approximate constancy of the total amount of time spent for each action, inspite of the significant increase in complexity indicated by the rise of time per spin update,reflects the diminishing need for checks as confidence in the numerical methods builds up.The total amount of time spent, approximately 1300 hrs, is quite modest and shows thatcareful preparation, continuous support by analytical work and an incremental approachpay off.Statistical errors are always treated by the Jackknife method and in least χ2–fits cor-relations between measurements are taken into account.3.2. Phase Diagrams.For actions S2 and S3 we determine critical points β0c(β2) at several fixed β2’s usingBinder cumulants of the magnetization M =Px ⃗Φ(x)L4.

The critical points are obtainedfrom the intersection points of the Binder cumulant graphs for different volumes, producedby reweighting and patching [13] the measurements from several couplings β0. The resultsare shown in Figures 3.1 and 3.2 and the actual numbers are listed in Tables 1 and 2.They compare well with our large N results which were helpful in deciding where to scanin the first place.

At large N the critical lines are straight, a feature that seems to persistat N = 4.Based on our large N work we know that we want extremal values of β2 [9].Bythis we mean the most negative value of β2 for which the leading term in the long wave–length expansion, eq. (2.1.1), still has the usual ferromagnetic sign on the critical line (i.e.β0c + β2 ∼> 0).

We therefore select β2 = −0.11 for actions S2 and S3. The critical pointsat β2 = −0.11 are β0c = 0.1118(3) and β0c = 0.1113(2) respectively.

The specific pointswe choose to study in detail are shown in Figures 3.1 and 3.2.3.3. Coupling Constant.To obtain the coupling constant g, eq.

(1.1), we need f and mH. f is obtained frommeasuring the magnetization, M, and the pion wave function renormalization constant,Zπ, defined from the residue of the pole at zero momentum in the pion–pion propagator.The pion field, ⃗π, is defined as the component of ⃗Φ transverse to M, and for low momentawe have < |⃗π(p)|2 >= Z2πp2 +regular terms.

The magnetization is obtained by extrapolating10

Figure 3.1Phase diagram for action S2. The solid line is a least square straight line fitto the data that are denoted by diamonds.

The squares indicate the pointswhere we made simulations to determine mH and fπ. They all lie on thevertical line β2 = −0.11.to L = ∞the quantities < M2 >L with an O(1/L2) correction, using the methods in [14].f is then obtained from f = M/Zπ.

This method is safe and the reliability of the numbersone obtains for f is high for our purposes. The error never exceeds 1.5% in the region ofhigher Higgs masses which we are interested in.

The estimate of f by analytical methods[3] and [15] has an error of order 5% in the same region; thus f is better determined byMonte Carlo. This is due mainly to the good theoretical control one has over finite volumeeffects [5,6,14].

Unfortunately, for the determination of mH we are not so lucky.To obtain mH we measure the correlations of zero total three–momentum sigma statesand low relative momentum two–pion states at different time separations.The sigmafield, σ, is defined as the component of ⃗Φ parallel to M. mH is the energy of one of thelightest states created by superpositions of the σ field and two–pion composite operatorsat zero total three–momentum and is obtained from the eigenvalues of the measured timecorrelation matrix.The evaluation of mH is not as clean as that of f and the next11

Figure 3.2Same as Figure 3.1 but for action S3.subsection discusses the determination of the numbers in greater detail.The main result of this paper is in Figure 3.3 which shows MH = 246pg/3 GeV asa function of mH for all three actions. The actual numbers are given in Tables 3, 4 and5.

One clearly sees the progressive increase of the bound. A glance at Figure 2.2 showsthat in all cases the cutoffeffects on the pion–pion scattering are below a few percenteven at the maximal MH of each curve.

Thus we can take the largest of these maxima asour bound. The ordering of the points and their relative positions are in agreement withFigure 2.1, while the differences in overall scale, reflecting the difference between N = ∞and N = 4, come out as expected [9].3.4.

More about mass determination.The main source of systematic errors is in the evaluation of mH. In an infinite volumethe Higgs particle would decay into two pions and extracting mH from the fall–offof theσ–σ correlation function would yield nonsense.

In a finite volume the decay is prohibited12

Figure 3.3The Higgs mass MH = mHf× 246 GeV in physical units vs.the Higgsmass mH in lattice units from the numerical simulations. The diamondscorrespond to action S1 [5], the squares to action S2 and the crosses toaction S3.or severely restricted by the rather large minimal amount of energy even the softest pionshave due to momentum quantization.

This makes the measurement possible but leavesone with the difficult task of estimating the accuracy of the so determined real part of theresonance pole.In [5] we dealt with action S1. As a check of the results obtained in the broken phasewe also measured the coupling g in the symmetric phase (as defined from the zero four–momentum four–point correlation) and used perturbation theory to predict g in the brokenphase.

This method is free of finite width contamination and gave results consistent withthe broken phase analysis. However, the determination of the coupling in the symmetricphase had a large statistical error.

This problem has been eliminated by one of us [15] whocarried out a complete analysis modeled on the work of [3]. The numbers of [5] survivethis test reasonably.

Of course, the analytical method suffers from systematic errors ofa different type.It is difficult to know how much of the discrepancies at larger mass13

values in the broken phase between the analytical and the Monte Carlo results are due toeither approach. Still, the comparison gives a worst–case scenario for the systematics inboth cases, since the methods are so different that systematics are unlikely to conspire towork in the same direction.

Indeed, while the discrepancies on the F4 lattice and on thehypercubic lattice [6] are roughly of similar magnitude, the sign is opposite. We cannotbe very precise, but the general conclusion we draw is that for mHfas a function of mHthe overall systematic error in the Monte Carlo results cannot exceed a few percent for thehighest Higgs masses measured.For S2 and S3 an analysis similar to [15] would be very demanding and has not beencarried out.

In [5] and for the lighter Higgs masses of actions S2 and S3 the Higgs mass wasextracted using only the σ field correlation function. Since now, for the larger masses, thewidth is suspected to be larger some additional checks are needed.

The most direct test isto write down reasonable operators that would create predominantly two pion states andcheck for mixing effects. We did this for action S1 at β0 = 0.10, the coupling from whichthe bound quoted in [5] was extracted, and also for actions S2 and S3 at the couplingswhere we extract the bound.

The large N results of section 2.3 show that more ambitiousattempts to actually measure the width on the lattice by making the pions massive [11]may not be directly relevant to the massless case.The mass is now estimated by measuring the correlation matrix C(t)Cij(t) =< Oi(t)Oj(0) > −< Oi(t) >< Oj(0) >between several operators, Oi, described in the previous subsection, evaluated on two timeslices separated by t lattice spacings. The eigenvalues are determined by diagonalizingC(t0)−1/2 C(t) C(t0)−1/2 as in [16] with t0 = 1 except for the largest lattice with actionS1 where t0 = 0 was used.

After some trials it was decided to diagonalize a 3 × 3 matrix.For each lattice size we evaluate the lowest eigenvalue with this method. We also computea “trial mass” from the σ field correlation function alone.

When the two numbers aredifferent within errors we expect to have some level repulsion between the two lowesteigenvalues.We then identify the eigenvalue closer to the free two pion energy as theenergy of the two pion state and the other as the resonance energy. To approximatelycorrect for the repulsion we adjust the resonance energy by the amount that the two pionenergy differs from its free value.

This technique should eliminate leading 1/L correctionswhen they are significant numerically. The so obtained Higgs masses are extrapolatedto L = ∞by assuming an1L2 behavior.

The quality of these fits is acceptable but themotivation for the method of analysis is somewhat empirical and therefore we allow for anorder 3% systematic error in the mass determination.The low lying spectrum for selected high mass values for each action are shown inFigures 3.4, 3.5 and 3.6 with the actual numbers given in Tables 6, 7 and 8. The new14

Figure 3.4The low lying spectrum for action S1 for various lattice sizes as measured ina numerical simulation at β0 = 0.10. The dotted lines correspond to the twolowest energies of two free pions.method of analysis confirms that the older results for action S1 are acceptable, which inturn increases our confidence when the method is applied to actions S2 and S3.

We seeno indications for strong mixing because the largest level repulsions observed are smallcompared to the eigenvalues themselves. The two–pion states have a spectrum well de-scribed by the lattice free particle dispersion and show only small amounts of sensitivityto the resonance.

Taking into account statistical errors we settle on an error estimate onthe mass determinations of 5%. This results in an error of about 6% on the determinationof the physical MH.4.

MAIN RESULT.A realistic and not overly conservative value for the Higgs mass triviality bound is710 GeV . At 710 GeV the Higgs particle is expected to have a width of about 210 GeV15

Figure 3.5Same as Figure 3.4 but for action S2 at β0 = 0.12, β2 = −0.11.and is therefore quite strongly interacting. We estimate the overall accuracy of our boundas ±60 GeV .

This includes the statistical error and systematic uncertainty of the measure-ments of MH (see section 3.4) as well as the systematic uncertainty assigned to fine tuning.The latter should also allow for effects of the hereto neglected b2 coupling in eq. (2.1.1).The meaning of the error is that it would be quite surprising if evidence were producedfor a reasonable generic model of the scalar sector with observable cutoffeffects bound bya few percent, but a Higgs mass larger than 770 GeV .

It would be even more surprising ifa future analysis ended up concluding that the bound is some number less than 650 GeV .The first estimates for the nearest–neighbor hypercubic actions gave a bound of 640 GeV[3,4] which is about 10% below the new number whereas the old F4 bound was 590 GeV[5]. Thus, the older results have proven to be quite robust and this is an indication thatmore search in the space of actions is unlikely to yield surprises.16

Figure 3.6Same as Figure 3.4 but for action S3 at β0 = 0.1175, β2 = −0.11.ACKNOWLEDGMENTS.The simulations for obtaining the phase diagram and measurements on some of thesmaller lattices were done on the cluster of IBM workstations at SCRI. The other mea-surements were done on the CRAY Y-MP at FSU.

This research was supported in part bythe DOE under grant # DE-FG05-85ER250000 (UMH, MK and PV), grant # DE-FG05-92ER40742 (UMH and PV), and under grant # DE-FG05-90ER40559 (HN).17

APPENDIX A.For the interested reader we sketch in this appendix the large N analysis for action S′3that was not treated in [9]. We can write the action S′3 as in eq.

(5.1) of [9]S′3 = ηN(β0 + β2)12Zx,y⃗Φ(x)gx,y⃗Φ(y) −Nβ2η28ǫZxZy⃗Φ(x)gx,y⃗Φ(y)2(A.1)withη = 6 ,ǫ = 18 ,Zx= 2Xx,Zp=ZB∗d4p(2π)4gx,y = 6˜δx,y −13Xx′ n.n. to x˜δy,x′ + 112Xx′′ n.n.n.

to x˜δy,x′′g(p) = 13Xµ̸=ν2 −cos(pµ + pν) −cos(pµ −pν)−112Xµ2 −2 cos(2pµ)+X{ǫµ=±1}"1 −cos Xµǫµpµ!#= p2 + O(p6) . (A.2)Here we used ˜δx,y = 1/2δx,y for an F4 lattice and B∗is its Brillouin zone.With the above notations the computations of the phase diagram, Higgs mass andcutoffeffects are just as in sections 5 and 6 of [9].

We only need a few numerical constants.They are defined in [9] and for the present case take the valuesr0 = 0.09932603c1 = 0.0281844 ,c2 = 2.2639 · 10−4γ = 0.01089861ζ =0 . (A.3)The resulting large N plots for the Higgs mass and cutoffeffects, analogous to those in [9]for the actions considered there, are shown in Figure A.1.APPENDIX B.In this appendix we collect the numbers obtained from our simulations in several tablesmatching the graphs shown in the main text.18

Figure A.1(a) mH/fπ vs. mH: The solid line corresponds to β2 = 0 and the dottedline to the optimal value β2 = −β2,t.c.. (b) Leading order cutoffeffect in thewidth to mass ratio. (c) Leading order cutoffeffects in the invariant π −πscattering amplitude at 900.

Here the dotted line represents center of massenergy W = 2MH, the dashed line W = 3MH and the solid line W = 4MH.19

β2β0c0.0000.0917( 2)−0.0400.0995( 5)−0.0800.1070(10)−0.1100.1118( 3)Table 1Some points of the critical line for action S2.β2β0c0.0000.0653(3)−0.0400.0825(5)−0.0800.0990(4)−0.1100.1113(2)Table 2Some points of the critical line for action S3.β0mHMH (GeV )0.09250.198(37)438(84)0.09500.414(25)524(32)0.09750.550(27)546(27)0.10000.702(30)593(25)0.10500.830(41)576(30)0.11000.946(50)568(30)Table 3The Higgs mass mH in lattice units and the Higgs mass MH = mHf ×246 GeVin physical units from the numerical simulations of action S1. The errorsquoted are statistical errors only.β0mHMH (GeV )0.11200.126(15)529(62)0.11300.239(13)571(30)0.11500.391( 8)627(12)0.11750.534( 8)672(10)0.12000.646( 9)696(10)0.12250.721(10)694(10)Table 4Same as in Table 3 but for the action S2 at β2 = −0.11.β0mHMH (GeV )0.11250.302( 7)634(15)0.11500.528(12)699(16)0.11750.663(11)708(12)20

Table 5Same as in Table 3 but for the action S3 at β2 = −0.11.LThree lowest energy levelsTwo lowest free π −π80.802(24)1.202(54)1.818(77)1.3621.571100.801(25)1.005(47)1.315(67)1.0891.257120.754(13)0.894(18)1.063(19)0.9071.047140.739(16)0.804(23)0.916(27)0.7770.898160.721(33)0.679(23)0.839(23)0.6800.785Table 6The low lying spectrum for action S1 for various lattice sizes as measured ina numerical simulation at β0 = 0.10. The two lowest energy levels of a freetwo pion state of zero total three–momentum are also given.LThree lowest energy levelsTwo lowest free π −π80.792(24)1.277(102)1.711(223)1.3621.571100.775(14)1.115( 36)1.171( 39)1.0891.257120.686(16)0.889( 32)1.034( 38)0.9071.047140.673(15)0.794( 24)0.914( 25)0.7770.898160.656(20)0.728( 21)0.863( 27)0.6800.785Table 7Same as in Table 6 but for action S2 at β0 = 0.12, β2 = −0.11.LThree lowest energy levelsTwo lowest free π −π80.817(15)1.250(146)1.660(196)1.3691.571100.759( 9)1.108( 53)1.267( 64)1.0911.257120.740( 8)0.964( 16)1.050( 15)0.9081.047140.684( 8)0.817( 14)0.913( 15)0.7780.898160.659( 8)0.719( 10)0.811( 11)0.6800.785Table 8Same as in Table 6 but for action S3 at β0 = 0.1175, β2 = −0.11.REFERENCES1)B. W. Lee, C. Quigg, H. B. Thacker, Phys.

Rev. D16 (1977) 1519.2)N. Cabbibo, L. Maiani, G. Parisi, R. Petronzio,Nucl.

Phys. B158 (1979) 295;B. Freedman, P. Smolensky, D. Weingarten,Phys.

Lett. 113B (1982) 209; R.21

Dashen and H. Neuberger, Phys. Rev.

Lett. 50 (1983) 1897; H. Neuberger, Phys.Lett.

B199 (1987) 536.3)M. L¨uscher and P. Weisz, Nucl. Phys.

B318 (1989) 705.4)J. Kuti, L. Lin and Y. Shen,Phys. Rev.

Lett. 61 (1988) 678; A. Hasenfratz, K.Jansen, J. Jersak, C. B. Lang, T. Neuhaus, H. Yoneyama Nucl.

Phys. B317 (1989)81; G. Bhanot, K. Bitar, Phys.

Rev. Lett.

61 (1988) 798; U. M. Heller, H. Neubergerand P. Vranas Phys. Lett.

B283 (1992) 335; U. M. Heller, M. Klomfass, H. Neubergerand P. Vranas, Nucl. Phys.

B (Proc. Suppl) 26 (1992) 522.5)G. Bhanot, K. Bitar, U. M. Heller and H. Neuberger, Nucl.

Phys. B353 (1991) 551(Erratum Nucl.

Phys. B375 (1992) 503).6)M. G¨ockeler, H. A. Kastrup, T. Neuhaus and F. Zimmermann, preprint HLRZ 92–35/PITHA 92-21.7)“The Standard Model Higgs Boson”, Vol.

8, Current Physics Sources and Comments,edited by M. B. Einhorn, North Holland, 1991.8)H. Neuberger, U. M. Heller, M. Klomfass, P. Vranas, Talk delivered at the XXVIInternational Conference on High Energy Physics, August 6-12, 1992, Dallas, TX,USA; FSU–SCRI–92C–114, to appear in the proceedings.9)U. M. Heller, H. Neuberger, P. Vranas, FSU–SCRI–92–99/RU–92–18, to appear inNucl. Phys.

B.10)M. L¨uscher, Nucl. Phys.

B354 (1991) 531; Nucl. Phys.

B364 (1991) 237.11)F. Zimmermann, J. Westphalen, M. G¨ockeler, H. A. Kastrup, talk presented by F.Z. at the International Symposium Lattice ’92, Amsterdam, Sept. 15–19, 1992; toappear in the proceedings.12)U. M. Heller, M. Klomfass, H. Neuberger and P. Vranas, FSU–SCRI–92C–150/RU–92–43, to appear in the proceedings of the LATTICE ’92 conference in Amsterdam.13)A.M. Ferrenberg and R.H. Swendsen, Phys.

Rev. Lett.

61 (1988) 2635, [erratum 63,(1989), 1658]; Phys. Rev.

Lett. 63 (1989) 1196.14)U. M. Heller, H. Neuberger,Phys.

Lett. 207B (1988) 189; H. Neuberger,Phys.Rev.

Lett. 60 (1988) 889.15)M. Klomfass, to appear in the proceedings of the LATTICE ’92 conference in Ams-terdam, and in preparation.

See also Ph.D. thesis, FSU–SCRI–92T-117 (1992).16)M. L¨uscher, U. Wolff, Nucl. Phys.

B339 (1990) 222.22

---

출처: arXiv:9303.215 • 원문 보기