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Higgs Mass Bound in the Minimal

arXiv:hep-ph/9303260v1 15 Mar 1993Higgs Mass Bound in the MinimalStandard Model∗Urs M. HellerSupercomputer Computations Research InstituteFlorida State UniversityTallahassee, FL 32306U.S.A.FSU-SCRI-93-40March 1993AbstractA brief review of the role of the Higgs mechanism and the ensuing Higgs par-ticle in the Minimal Standard Model is given. Then the property of trivialityof the scalar sector in the Minimal Standard Model and the upper bound onthe Higgs mass that follows is discussed.

It is emphasized that the bound isobtained by limiting cutoffeffects on physical processes. Actions that allowa parameterization and tuning of the leading cutoffeffects are studied bothanalytically, in the large N limit of the generalization of the O(4) symmetryof the scalar sector to O(N), and numerically for the physical case N = 4.Combining those results we show that the Minimal Standard Model will de-scribe physics to an accuracy of a few percent up to energies of the order 2to 4 times the Higgs mass, MH, only if MH ≤710 ± 60 GeV .

This bound isthe result of a systematic search in the space of dimension six operators andis expected to hold in the continuum.∗Invited talk given at the 4th Hellenic School on Elementary Particle Physics, Corfu, Greece,Sept. 2–20, 1992.

To appear in the proceedings.

1. IntroductionThe elementary particles and their interactions are described in a highly eco-nomical and successful way in the Minimal Standard Model.

The symmetry amongthe elementary fermions is described by the internal symmetry groupG = SU(3)color × SU(2)L × U(1)Y(1)and we have three known families of such fermions. Indeed, precision measurementsof the width of the Z vector boson at LEP have yielded for the number of masslessneutrinos (and thus of families, since we have one massless lefthanded neutrino perfamily) Nν = 3.04 ± 0.04 [1].The interactions among the elementary fermions are introduced by making theglobal symmetry group G into a local symmetry group with the help of 8 + 3 + 1 =12 massless vector bosons (8 gluons for the SU(3)color factor, 3 vector bosons forSU(2)L and 1 for the U(1) hypercharge).

The resulting theory is perturbativelyrenormalizable, which means in particular that it is applicable for energies rangingover many orders of magnitude and that it has a lot of predictive power.However, so far we do not describe nature appropriately. We know that explicitlyrealized is only the subgroup SU(3)color × U(1)em of G, where here the U(1) factordescribes electromagnetism.

Furthermore the theory does not allow for masses. Inthe case of fermions, mass terms are forbidden by the chiral nature of the symmetrygroup G, and in the case of the vector bosons, mass terms are forbidden by thegauge symmetry.

But in nature 14 out of the 15 fermions per family are massive,as are the weak vector bosons W ± and Z. In the case of the three vector bosonswe don’t even have enough degrees of freedom, since massless vector boson havetwo (transverse) polarizations, while massive ones need in addition a state withlongitudinal polarization.In the Minimal Standard Model we correct these shortcomings by adding an el-ementary complex scalar field, transforming as a doublet under SU(2)L.Grouptheory determines that the scalar self-interactions have an enhanced symmetry,SU(2)L × SU(2)custodial ≃O(4).This symmetry is then arranged to be brokenspontaneously to O(3) by giving the scalar field a non-vanishing vacuum expecta-tion value, F. (This notation is chosen from the analogy to the current algebradescribing the soft pions of QCD, in which the analog of the vacuum expectationvalue is the pion decay constant, fπ).

This spontaneous symmetry breaking turnsout to do all the tricks we need. It breaks the symmetry group from G to the sub-group SU(3)color × U(1)em, explicitly realized.

The three Goldstone bosons of thebreaking O(4) →O(3) provide the longitudinal degrees of freedom of the W ± and Z,making those massive, and finally the spontaneous symmetry breaking gives massesto the fermions via gauge invariant Yukawa couplings. And all these good thingshappen while the theory remains perturbatively renormalizable and maintains itspredictive power.

After the spontaneous symmetry breaking, one out of the 4 scalardegrees of freedom that we added is left, the so far elusive Higgs boson.While the Minimal Standard Model, briefly outlined above, has had many spec-tacular successes, relatively little is known experimentally about the Higgs sector.One exception is the value of the vacuum expectation value F. From its relation to

the W boson mass and the latter’s to the four-fermion coupling GF one can easilydeduce that F = 246 GeV . Furthermore, recent experiments at LEP led to a lowerbound on the Higgs mass of about 60 GeV [2].In the remainder of this seminar I will describe what we know about the massof this Higgs particle on purely theoretical grounds.In particular I will presentthe arguments leading to an upper bound on the mass of the Higgs particle, the socalled triviality bound.

And finally I’ll describe a non-perturbative computation ofthe bound, that we found to be 710 ± 60 GeV .2. Perturbative indications of trivialityAs indicated in the introduction we are interested in an upper bound on theHiggs mass.

It turns out, as we will see below, that the Higgs mass increases withincreasing scalar self coupling. At the energy scale of the upper bound of Higgsmass, all gauge interactions are relatively weak and can be treated perturbatively.The same holds true for the Yukawa interactions, including the top if it is not heavierthan about 200 GeV , as is favored by experiment [1].

Therefore we concentrate hereon the scalar sector of the Minimal Standard Model alone. It is described by anO(4) invariant scalar field theory with potentialV (⃗φ) = 12µ20⃗φ2 + g04!

(⃗φ2)2(2)To have spontaneous symmetry breaking we assume µ20 < 0. As usual the theoryso far described is ill defined.We need to introduce a cutoffΛ to regulate andthen renormalize the theory.

A two-loop perturbative computation together withapplication of the renormalization group leads to the relation between the cutoff, Λ,the physical Higgs mass, MH, and the renormalized coupling, gR = 3M2H/F 2MHΛ= C gR4π213/24exp(−4π2gR)[1 + O(gR)](3)Usually, at the end of the calculation one would like to remove the cutoffby takingthe limit Λ →∞. However, from eq.

(3) we see that the limit Λ →∞impliesgR →0, i.e. we are left with a non-interacting, trivial theory.

But we need aninteracting scalar sector for the Higgs mechanism to work. Therefore we need tokeep the cutofffinite: the Minimal Standard Model has to be viewed as an effectivetheory that describes physics at energies below the cutoffscale.Since we have to keep a finite cutoff, we may ask what happens if we try tomake the (renormalized) scalar self-interactions stronger.

Eq. (3) tells us that asgR increases, so does the ratio MH/Λ.But since the Higgs mass is one of thephysical quantities that the standard model is supposed to describe, we certainlyneed MH/Λ < 1.

Hence we have arrived at an upper bound on the Higgs mass.Since it comes from the triviality of the scalar sector, this bound is referred to asthe triviality bound.In the next three sections I will make the definition of the bound, namely themeaning of the “<” in MH/Λ < 1 more precise and give the results of a numericalcomputation of the bound.

3. Cutoffeffects and generalized actionsThe triviality of the scalar sector of the Minimal Standard Model, and thereforethe need to retain a finite cutoffΛ, are by now very well established [3, 4, 5, 6].

Asa consequence, all observable predictions have a weak cutoffdependence, of order1/Λ2. I will later on show explicit examples of such cutoffeffects, computed in thelarge N limit of the generalization of the O(4) symmetric scalar sector to O(N).This generalization is useful, because we can solve the model analytically in thelarge N limit, and therefore e.g.

compute cutoffeffects. The cutoffeffects becomelarger when the ratio MH/Λ, and hence gR, increases.

Therefore, by limiting thecutoffeffect on some physical observable – we shall use the square of the invariantscattering amplitude of Goldstone bosons at 90o in the center of mass frame – weobtain a more precise definition of the upper bound on the Higgs mass.The need for a finite cutoffin the Minimal Standard Model means, that it is onlyan effective theory, applicable for energies below the cutoff. There will be some, asyet unknown, embedding theory.

But we assume that for “small” energies – energiessmaller than a few times the Higgs mass – the scalar sector is representable by aneffective action (for a more detailed discussion see e.g. [7] and references therein)Leff= Lren + 1Λ2XAcAOA ,dimOA ≤6(4)with Lren the usual renormalized φ4 Lagrangian.

OA are operators with the correctsymmetry properties and dimension less than or equal to 6, and the coefficientscA depend on the embedding theory.Since we don’t know this theory, we justparameterize our ignorance with these cA’s, i.e., we consider reasonable bare cutoffmodels with enough free parameters to reproduce the effective action, eq. (4).

Thisallows us to tune the cutoffeffects. Eliminating redundant operators, which leave theS-matrix unchanged, we end up with two “measurable” cA’s.

The Higgs mass boundis now obtained as the maximal value MH can take, when varying the parameters cAwhile maintaining our requirement of limiting the cutoffeffects by some prescribedvalue, typically a few percent. However, since we do not know the embedding theory,we will in this maximization avoid excessive fine tuning that might eliminate leadingcutoffeffects, of order 1/Λ2.The most straightforward implementation is to start with an action of the formeq.

(4) on the level of bare fields and parameters. It turns out, however, that, asour intuition would tell us, the maximal renormalized coupling gR, and hence themaximal Higgs mass, is obtained at maximal bare φ4 self coupling, i.e., at g0 →∞(see e.g.

[4, 8]). But in this limit the model becomes nonlinear, with the field havinga fixed length, and all dimension six operators become trivial.

In a non-linear theoryit are terms with four derivatives that allow us to tune the cutoffeffects of order 1/Λ2.Maintaining O(N) invariance, there are three different terms with four derivatives,and we are led to consider actions of the formS =Zx"12⃗φ(−∂2 + 2b0∂4)⃗φ −b12N (∂µ⃗φ · ∂µ⃗φ)2 −b22N (∂µ⃗φ · ∂ν⃗φ −14δµ,ν∂σ⃗φ · ∂σ⃗φ)2#(5)

with φ2 = Nβ fixed. Up to terms with more derivatives, the parameter b0 can beeliminated with a field redefinition⃗φ →⃗φ + b0∂2⃗φq⃗φ2 + b20(∂2⃗φ)2 + 2b0⃗φ∂2⃗φqNβ.

(6)This leaves two free parameters to tune the cutoffeffects, exactly the number ofmeasurable coefficients cA in (4). Therefore (5) should give a good parameterizationto obtain the Higgs mass bound.4.

Solution at large NTo understand the effect of the four-derivative couplings in the action eq. (5) westudied these models first in the soluable large N limit [8].

We considered differentregularizations: a class of Pauli-Villars regularizations obtained by replacing theterm ⃗φ(−∂2)⃗φ by ⃗φ(−∂2(1 + (−∂2/Λ2)n)⃗φ with n ≥3 and b0 set to zero, and sometranscriptions of action (5) on a lattice, i.e. lattice actions such that eq.

(5) appearsin their expansion in slowly varying fields.The result of our investigations is that at N = ∞, after b0 has been eliminated,b2 has no effect and that the bound depends monotonically on b1, increasing withdecreasing b1. Overall stability of the homogeneous broken phase restricts the rangeof b1 and thus we find a finite optimal value for b1.

The physical picture that emergesis that among the nonlinear actions the bound is further increased by reducing asmuch as possible the attraction between low momentum pions in the I = J = 0channel.The rule in the above paragraph does not lead to an exactly universal bound.Different bare actions that give the same effective parameter b1 can give somewhatdifferent bounds because the dependence of physical observables on the bare actionis highly nonlinear. For example, at the optimal b1 value, Pauli–Villars regulariza-tions lead to bounds higher by about 100 GeV than some lattice regularizations.This difference between the lattice and Pauli–Villars can be traced to the way thefree massless inverse Euclidean propagator departs from the O(p2) behavior at lowmomenta.

For Pauli–Villars it bends upwards to enforce the needed suppressionof higher modes in the functional integral, while on the lattice it typically bendsdownwards to reflect the eventual compactification of momentum space.When considering lattice regularizations, because we desire to preserve Lorentzinvariance to order 1/Λ2, we use the F4 lattice. The F4 lattice can be thought ofas embedded in a hypercubic lattice from which odd sites (i.e.

sites whose integercoordinates add up to an odd sum) have been removed. The F4 lattice turns outto have a larger symmetry group than the hypercubic lattice, which forbids Lorentzinvariance breaking terms at order 1/Λ2.On the basis of the above observations, we went through three stages of inves-tigation.

The first stage was to investigate the na¨ıve nearest–neighbor model. Thisshould be viewed as the generic lattice case where no special effort to increase thebound is made.

The next stage is to write down the simplest action that has atunable parameter b1. We should emphasize that on the F4 lattice, unlike on the

Figure 1: Large N prediction of the Higgs mass MH = MH/F ×246 GeV in physicalunits vs. the Higgs mass mH = aMH in lattice units for the three actions on the F4latticehypercubic lattice, this can be done in a way that maintains the nearest–neighborcharacter of the action, namely by coupling fields sited at the vertices of elementarybond–triangles. The last stage is to add a term to eliminate the “wrong sign” orderp4 term in the free propagator, amounting to Symanzik improvement.

The three F4actions we investigated are given byS1 =−2β0P ⃗Φ(x) · ⃗Φ(x′)S2 =S1 −β28PxPl,l′∩x̸=∅, l∩x′̸=∅, l′∩x′′̸=∅x,x′,x′′ all n.n.h⃗Φ(x) · ⃗Φ(x′) ⃗Φ(x) · ⃗Φ(x′′)i(7)S3 =−2(2β0 + β2) P ⃗Φ(x) · ⃗Φ(x′) + (β0 + β2) P≪x,x′≫⃗Φ(x) · ⃗Φ(x′)−β28PxPl,l′∩x̸=∅, l∩x′̸=∅, l′∩x′′̸=∅x,x′,x′′ all n.n.h⃗Φ(x) · ⃗Φ(x′)⃗Φ(x) · ⃗Φ(x′′)i.The coupling β2 here plays the role of b1 in eq. (5).In the large N limit, to simplify the calculation somewhat, we actually consideredslight variants of actions S2 and S3 (see [8]), that have the same expansion forslowly varying fields and are expected to give the same physics.

We indicate thisby denoting the modified actions with a ′. In each case, at constant β2, β0 is varied

tracing out a line in parameter space approaching a critical point from the brokenphase. This line can also be parameterized by MH/Λ (on the lattice Λ = a−1) or gR.For actions S′2 and S′3, β2 is chosen so that on this line the bound on MH is expectedto be largest.

A simulation produces a graph showing MH/F as a function of MH/Λalong this line. The y-axis is turned into an axis for MH by MH = MH/F ×246 GeV .The large N predictions for these graphs are shown in Figure 1.To obtain a well defined bound on the Higgs mass from such graphs, as wealready explained, we need to compute the cutoffeffects on some physical quantity.We show the cutoffeffect in the square of the invariant π −π scattering amplitudeat 90o at several center of mass energies in Figure 2.Figure 2: Leading order cutoffeffects in the invariant π −π scattering amplitudeat 90o at center of mass energy W = 2MH vs.the Higgs mass mH = aMH inlattice units for the three actions.The values of MH in GeV determined fromMH = MH/F× GeV are put on the three horizontal lines at ¯δ|A|2 = 0.005, 0.01, 0.02.If one considers only the magnitude of cutoffeffects as a function of MH/Λ, onemight conclude that the bound obtained with action S1 would be larger than thebound obtained with S′2.

This conclusion proves to be wrong when the mass in phys-ical units is considered. The values of MH in GeV , determined from MH = MH/F ×246 GeV , are put on three horizontal lines in Figure 2 at ¯δ|A|2 = 0.005, 0.01, 0.02.At large N the bound increases when going from S1 to S′2 and then to S′3 by a little

over 10% at each step. For example, for ¯δ|A|2 = .01 we get bounds on MH of 680,764, 863 GeV for S1, S′2 and S′3 respectively.5.

The physical case N = 4.For the physical case, N = 4, we do not have at our disposal non-perturbativeanalytical methods of computation. We therefore resort to numerical simulations ofthe models in eq.

(7). The result of these simulations are show in Figure 3 whichshows MH = 246qgR/3 GeV as a function of aMH for all three actions.

One clearlysees the progressive increase of MH from action S1 to S2 and then S3, just as in thelarge N limit.Figure 3: The Higgs mass MH = MH/F × 246 GeV in physical units vs. the Higgsmass mH = aMH in lattice units from the numerical simulations. The diamondscorrespond to action S1 [6], the squares to action S2 and the crosses to action S3 [9].To obtain a bound on the Higgs mass we also need estimates of the cutoffeffects.We do not know how to compute cutoffeffects in numerical simulations.

Thereforewe take the estimates from the large N calculation, which should be accurate enoughfor our purpose here (observe that the cutoffeffects are relatively insensitive to theHiggs mass in lattice units, mH = aMH). A glance at Figure 2 shows that in all

cases the cutoffeffects on the pion–pion scattering are below a few percent even atthe 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 agreementwith Figure 1, while the differences in overall scale, reflecting the difference betweenN = ∞and N = 4, come out compatible with 1/N corrections, as expected [8].We conclude that the Minimal Standard Model will describe physics to an accu-racy of a few percent up to energies of the order 2 to 4 times the Higgs mass, MH,only if MH ≤710 ± 60 GeV . The error quoted accounts for the statistical errors,shown in Figure 3, as well as the systematic uncertainty associated with the remain-ing regularization dependence, e.g.

a not completely optimal choice of b1 in (5) andthe possible small dependence on b2 for N ̸= ∞. Since this bound is the result of asystematic search in the space of dimension six operators, we expect it to hold in thecontinuum.

A Higgs particle of mass 710 GeV is expected to have a width between180 GeV (the perturbative estimate) and 280 GeV (the large N non-perturbativeestimate). Thus, if the Higgs mass bound turns out to be saturated in nature, theHiggs would be quite strongly interacting.AcknowledgementsI would like to thank my collaborators, H. Neuberger, P. Vranas and M. Klomfassfor a most fruitful and stimulating collaboration, and the organizers of the School forinviting me to contribute and for organizing a very pleasant and interesting School.This work was supported in part by the DOE under grants # DE-FG05-85ER250000and # DE-FG05-92ER40742.

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