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HOW TO PUT A HEAVIER HIGGS ON THE LATTICE§

arXiv:hep-lat/9107001v1 8 Apr 1992RU–91–31FSU-SCRI-91-94HOW TO PUT A HEAVIER HIGGS ON THE LATTICE§byUrs M. Heller†, Herbert Neuberger‡, and Pavlos Vranas†† Supercomputer Computations Research InstituteThe Florida State UniversityTallahassee, FL 32306‡ Department of Physics and AstronomyRutgers UniversityPiscataway, NJ 08855–0849AbstractLattice work, exploring the Higgs mass triviality bound, seems to indicate that a strongly interactingscalar sector in the minimal standard model cannot exist while low energy QCD phenomenology seems toindicate that it could. We attack this puzzle using the 1/N expansion and discover a simple criterion forselecting a lattice action that is more likely to produce a heavy Higgs particle.

Our large N calculationsuggests that the Higgs mass bound might be around 850 GeV , which is about 30% higher than previouslyobtained.§ Submitted to Physical Review Letters1

The recent quantitative results that emerged from the investigation of triviality in the λ(⃗φ2)2 theoryon the lattice [1] have given an upper bound on the ratio of the Higgs mass, mH, to the weak scale,fπ = 246 GeV , of about 2.5–2.7. In the hypothetical world of QCD with just the up-down doublet ofquarks with zero masses, which is described by the same effective theory at low energies, one believes, onphenomenological grounds, that the above ratio would be significantly larger, equal to about 6–8.

This isadmittedly an extreme situation because the sigma enhancement in π π scattering doesn’t quite qualify tobe called a resonance, and, it comes accompanied by a host of other resonances, representing all kinds ofnontrivial “cutoffeffects”. Still, there is a large gap between 2.7 and 6 and we are left with essentially twoalternatives: either the QCD example provides a totally unreasonable estimation for the bound, or, we admitthat the lattice result, while solidly established within a subclass of actions, is relatively far from a realisticestimate for the bound.

We would have much less cause for worry if the lattice bound for the ratio came outto be about 4 (this would correspond to mH ≃1 T eV ) or more [2].This issue could be addressed in principle by an extensive search of all possible lattice actions.Inpractice one needs some theoretical indication as to what kind of regularized action is likely to produce ahigher bound. If the physical picture behind the particular form of the suggested action is reasonable, aninvestigation of its consequences would help resolve the above puzzle by either refuting the existing latticebound or by strengthening the contention that the QCD example is grossly exaggerated.The purpose of the present note is to make a proposal for such a generalized action and sketch thetheoretical basis for it.

The basic point of the proposal is that one should put on the lattice nonlinearO(4) models that have a naive continuum limit with freely adjustable four derivative couplings and studythem nonperturbatively. We solve the model in the large N limit of O(N) and find the region in couplingspace where the highest bound is expected; we test the extent to which the results are dependent on theregularization and extract the basic features that are generic.

The results can be explained by a roughphysical picture as follows: When the regularized model is nonlinear one has to think about the Higgsresonance as a loosely bound state of two pions in an I = 0, J = 0 state. Pions in such a state attractbecause superposing the field configurations corresponding to individual pions makes the state look more likethe vacuum and hence lowers the energy.

The four derivative terms in the action can add to this attraction;if the attraction is increased to a certain level the bound state will become massless and stable. This occurswhile the system is still relatively deep in the broken phase; hence the weak scale fπ is nonvanishing.

Theappearance of this massless bound state signals a new kind of phase transition; the ordinary phase transitionoccurs when both fπ and mH vanish and leads to a scale invariant theory at the transition - here we stillhave a scale and therefore scale invariance seems to be spontaneously broken, with the massless boundstate,the Higgs particle, playing the role of the expected dilaton. This phenomenon is reminiscent of the resultsobtained in three dimensions by Bardeen, Moshe and Bander [3].

As far as the Higgs mass bound goes, itis clear that we have to try to be as far away as possible from this new transition point; we wish to induceenough repulsion between the pions to delay the creation of the bound state as much as possible. Of course,the interaction is still attractive overall, because only the order p4 term in the interaction can be tamperedwith; the order p2 and p4 log(p2) terms are fixed by current algebra and are attractive [4].

Moreover, inthe critical regime the model still is governed by the λ(⃗φ2)2 trivial fixed point and is expected therefore togenerate a one particle state corresponding to the σ field component. While it is quite possible that theabove described scale invariance breaking transition cannot actually materialize at N = 4 (we shall see thatat N = ∞it does appear in all regularizations that we tested), the basic trend for the magnitude of thetriviality bound must be the same.Assuming global O(4) symmetry and that the leading cutoffeffects are represented by dimension sixoperators, one arrives, after eliminating couplings that won’t lead to observable effects in the S-matrix toleading order in the inverse cutoff, to a space of actions that depends on 4 parameters [5].

One may try toinduce the dependence on the associated renormalized operators by simply writing identical forms for termsin the bare action. However, we are looking for a bound on the physical four point coupling and it is likelythat it obtains in the limit that the bare four point coupling is taken to infinity.

When the bare couplingbecomes infinite the bare model is reduced to a nonlinear model; this has the effect that the naive forms ofthe dimension six operators in terms of bare linear fields become trivial. The parameterization in terms oflinear fields and operators of naive dimension six breaks down completely at the point where the bare theorybecomes nonlinear (we have tested this explicitly in the linear model, within 1/N, and with a particular form2

of a bare dimension-six term in the Lagrangian). So we were expecting to be left with three freely adjustableparameters (within some range) but we ended up with only one.

This is the case that has been investigatednonperturbatively until now [1]. To include the effects of dimension six operators at physical scales we needto put in operators of higher order in derivatives in the bare, nonlinear, theory that describes the physics atcutoffscales.

There are exactly two additional terms that one can write down if one restricts the number ofderivatives to four [4]; thus the right number of parameters is obtained. This time, unlike in the linear case,the effect of the bare couplings has no reason to disappear.It is well known that the 1/N expansion, at least to leading order, doesn’t have to work well quantita-tively simply because there is a factor of N + 8 in the coefficients of the β function and therefore N = 4 isbadly approximated by N = ∞[6-8].

We shall say more about this later; for the moment keep in mind thatwe are interested mainly in relative effects, that is, in how the bound varies with the action, and less in theabsolute magnitude of the bound. To get the latter one would have to resort to numerical means.If a strongly interacting scalar sector could be produced on the lattice its investigation would requiresome nontrivial extension of the tools that were used until now.

However, if we only wish to establish theexistence of a strongly interacting regime, we can do that indirectly, while restricting our attention just toweak couplings. For example, a strongly coupled Higgs sector was excluded on lattices with a pure nearest-neighbor coupling, by showing that, for a ratio mH/fπ of only 2.5, one already had mH/Λ equal to 0.5,leaving no “room” below the cutoffΛ for further growth in the coupling; if we could show that a differentaction would, at the same value of mH/fπ, give mH/Λ equal to 1/10 say, our case would be made.

Therefore,our subsequent study is restricted to the weakly coupled sector. Generically, we expect there thatmHΛ≈C(b) exp[−16π2f 2π/(Nm2H)],(1)where C(b) is a function of the additional parameters we introduced in the action.

We aim to show thatby varying b C(b) can be significantly lowered below the value it had when no four derivative terms wereincluded (this is the only case that has been investigated to date). Our finding is best summarized in aqualitative self-explanatory diagram that we checked for a class of Pauli-Villars regularizations and for theF4 lattice regularization (the generalization of the calculations to the hypercubic lattice should be a simpleexercise).We first look at the models with Pauli-Villars, or higher derivative, regularization.

We choose to replacethe ordinary propagator by 1/[p2(1 + (p2/M 2)n)] where M is a “bare” cutoff(as opposed to Λ the physicalcutoffto be defined below) and n ≥3 is an integer. The limit n = ∞corresponds to a sharp momentumcutoffand is expected to be singular.

For any finite n the bare propagator has n “ghost” poles all situatedon the unit circle in the complex s/M 2 plane. We shall define the physical cutoff, Λ, as the square root ofthe modulus of the location of the closest complex pole in s in the full pion propagator (at leading order in1/N).

The action, written in Euclidean space, is:S =Zx12⃗φ g(−∂2)⃗φ −b12N (∂µ⃗φ · ∂µ⃗φ)2 −b22N (∂µ⃗φ · ∂ν ⃗φ −14δµ,ν∂σ⃗φ · ∂σ⃗φ)2(2)where g(−∂2) = (−∂2)[1 + (−∂2/M 2)n], and the field is subjected to the constraint ⃗φ2 = Nβ. The partitionfunction is Z =R[d⃗φ] exp[−S].Using the technique employed in ref.

[9] we find the phase diagram of the model paying particularattention to the competition between various saddles. We set⃗φ =√N⃗v + ˆvH(x) + ⃗π(x)(3)with ˆv = ⃗v/v, v = |⃗v|, ˆv·⃗π = 0,Rx H(x) = 0,Rx ⃗π(x) = 0.

We define the pion wave function renormalizationconstant Zπ for the field ⃗π as usual. The pion decay constant fπ can be easily gotten from v and Zπ.It turns out that it is more convenient, for the broken phase, to parameterize the theory by Zπ:b1 = 4πM 22 n sin(2π/n)2πZπ −1Z2(n−1)/nπ(4)3

Pion Wave Function Renormalization Constant Pion Decay Constant 123456789012345678901234567123456789012345678901234567123456789012345678901234567123456789012345678901234567123456789012345678901234567123456789012345678901234567Cutoff Effects will Become Significant for Energies in this Range. Second Order Transition Line 123456712345671234567123456712345671234567Tricritical Point Z=1 Line of Fixed Higgs Mass to Pion Decay Constant RatioSYMMETRIC PHASE BROKEN PHASE First Order Transition Line.

Frustrated phase 2345678901234567890123423456789012345678901234234567890123456789012342345678901234567890123423456789012345678901234234567890123456789012342345678901234567890123423456789012345678901234234567890123456789012342345678901234567890123423456789012345678901234234567890123456789012342345678901234567890123423456789012345678901234234567890123456789012342345678901234567890123423456789012345678901234234567890123456789012342345678901234567890123423456789012345678901234234567890123456789012342345678901234567890123423456789012345678901234234567890123456789012342345678901234567890123423456789012345678901234234567890123456789012342345678901234567890123423456789012345678901234234567890123456789012342345678901234567890123423456789012345678901234234567890123456789012342345678901234567890123423456789012345678901234234567890123456789012342345678901234567890123423456789012345678901234234567890123456789012342345678901234567890123423456789012345678901234234567890123456789012342345678901234567890123423456789012345678901234234567890123456789012342345678901234567890123423456789012345678901234234567890123456789012342345678901234567890123423456789012345678901234234567890123456789012342345678901234567890123423456789012345678901234234567890123456789012342345678901234567890123423456789012345678901234234567890123456789012342345678901234567890123423456789012345678901234234567890123456789012342345678901234567890123423456789012345678901234234567890123456789012342345678901234567890123423456789012345678901234234567890123456789012342345678901234567890123423456789012345678901234234567890123456789012342345678901234567890123423456789012345678901234234567890123456789012342345678901234567890123423456789012345678901234Nearest Neighbor Action Fig. 1 : Summary Diagram Qualitative Features Common to all Models Investigated where 0 ≤Zπ ≤2(n −1)/(n −2).

In this range, b1 increases monotonically with Zπ and at the largerend-point we have ∂b1/∂Zπ = 0. The portion of the phase diagram we are interested in does not dependon b2 because giving an expectation value to the O(N) singlets associated with b2 would break Lorentzinvariance.

The other coupling is best replaced by the unitless weak scale, F = fπ/Λ. The physical cutoff,Λ, is given by Λ = M/Z1/(2n)π.

From the large N saddle point equations we obtain the expectation value vwhich is related to fπ by v2 = Zπf 2π resulting in:(4πF)2N=4πM2βZπ(1−n)/n −πn sin(π/n). (5)There is a second order transition line at F = 0 and, on it, at the end of the allowed range of Zπ, thereis a tricritical point where the second order line turns into first order.

At the tricritical point one obtains amassless pole in the two point correlation function of the trace of the energy-momentum tensor. We shall notbe concerned with the region beyond the tricritical point.

The calculation of mH proceeds in the standardmanner; it turns out that one has to find the complex zeros of the determinant of a four by four matrix.Since we restrict our attention to the perturbative regime, i.e. F, p << 1 we can replace the quantity mHby the technically more accessible real quantity mR, defined from the smallest positive zero in s = −p2 ofthe real part of the determinant by mR = √s.

We are really interested in MR = mR/Λ. If we also takeMR << 1 of the same order (up to logs) as F, the fourth row and column of the matrix decouple and the4

dependence on b2 disappears. We need the expression for the “bubble” diagramBP V (p) =Zd4k(2π)41g(k −p/2)g(k + p/2)= −116π2log(p2/M 2) −n −1n+ O(p2/M 2)(6)which holds for any n < ∞(for n = ∞the correction term goes aspp2/M 2 and represents a non-local termin the effective action as expected when employing a sharp momentum cutoff).

One has then:M 2R ≈exp"−2N4πFMR2#C2(Zπ),C(Zπ) = expn −12n1 −πn tan(π/n)1 −Zπ1 −Zπ(n −2)/(2(n −1)). (7)There is no four derivative term when Zπ = 1; the minimal C obtains at Zπ = 0 which is as far away fromthe tricritical point as possible:C(Zπ = 1)C(Zπ = 0) = expn −12nπn tan (π/n)≤exp(0.5) ≈1.65.

(8)We have obtained some increase in the cutoffbut not a factor of about 5 as we would have liked.However, we know from the work of Einhorn [7] that even at b1 = 0 this particular large N model will havea strongly interacting sector; therefore it may be that there is no room left to dramatically increase thecutoffeven further. It is also clear that if it turned out that the lattice actions that were so far investigatedreally correspond to an action with b1 > 0 in Pauli-Villars regularization, a much larger variation could beobtained because C diverges at some finite, positive b1.

b1 > 0 means that the extra term has induced moreattraction in the I, J = 0 channel and the divergence in C occurs at the tricritical point where the resonantbound state becomes massless and stable.We turn now to the lattice. Since our objective here is to provide a useful indication for what kind ofactions would be instructive to study by nonperturbative means, we pick the lattice regularization that isbest suited for this purpose, namely we put the theory on an F4 lattice.

This lattice has a larger symmetrygroup than the more commonly used hypercubic lattice which ensures that no Lorentz invariance breakingfour derivative terms appear in the (naive) continuum limit of its lattice actions. The F4 lattice can beviewed as a hypercubic lattice (Z4) with unit lattice spacing from which all the sites with an odd sum ofcoordinates have been erased [5].

We shall denote sites on F4 by x, x′, x′′, and links by < x, x′ >, l, l′. Theaction is chosen asS = −2Nβ0X⃗Φ(x) · ⃗Φ(x′) −Nβ1Xh⃗Φ(x) · ⃗Φ(x′)i2−N β248XxXl∩x̸=∅l=⃗Φ(x) · ⃗Φ(x′)2.

(9)Here the field is constrained by ⃗Φ2(x) = 1. To express the action in a form closer to the one we employedbefore we rescale the field ⃗φ =p6N(β0 + β1 + β2)⃗Φ (we only consider the region β0 + β1 + β2 > 0).

Theaction has the same form as above (eq. (2)), up to terms that contain fourth order derivatives, if oneidentifies:g(kµ) =16Xµ̸=ν[2 −cos(kµ + kν) −cos(kµ −kν)] ≈k2 −(k2)212+ O(k6µ),b1 = 148β1 + β2(β0 + β1 + β2)2 ,b2 = 136β1(β0 + β1 + β2)2 .

(10)5

Again b2 plays no role and one may set β1 to 0. The situation is essentially the same as in the Pauli-Villars case.

b1 can be traded for the pion wave function renormalization constant,b1 = 1Zπ−1Z2π,(11)where Zπ varies between 0 and 2. The standard non-linear action corresponds again to Zπ = 1 and, on theline where the pion decay constant F vanishes, Zπ = 2 corresponds to the tricritical point.

F is given by:F 2N = 6(β0 + β1 + β2)Zπ−r0,r0 =Zk1g(k) = 0.13823. (12)The momentum integral is over the F4 Brillouin zone and a factor of 1/(2π)4 has been absorbed in thedefinition ofRk; the numerical value is taken from ref.

[5].We proceed now to find MR defined exactly as before; for this we need the lattice “bubble” which canbe easily evaluated using the results of ref. [5] (but note a missing minus sign in the value of r1 in equation(4.12) there):BF4(p) = −116π2 log(p2) + c1 + c2p2 + O(p4 log p2),c1 = 0.0466316,c2 = 0.0005497.

(13)Unlike in the Pauli-Villars case, the cutoff(defined for example by a certain amount of violation of Euclideanrotational invariance in the full pion propagator) is independent of the couplings in the action.This time we shall be interested to go beyond the O(1) term and that’s why we evaluated the nextcorrection in p to the bubble (13). Ignoring this correction for the moment we get the usual formula for MR(eq.

(7)), but the coefficient C is now given by,C(Zπ) = exp8π2c1 −r201 −Zπ1 −Zπ/2,(14)leading to:C(Zπ = 1)C(Zπ = 0) = exp(8π2r20) ≈4.521. (15)We see that we have obtained a much larger variation.In practice, for vectorization purposes, it would be better to have an action that involves only nearestneighbors on the F4 lattice.

Nothing in what we have said until now changes if we replace the β2 term inthe action byN 3β28XxXl,l′∩x̸=∅, l∩x′̸=∅, l′∩x′′̸=∅, x,x′,x′′ all n.n.h⃗Φ(x) · ⃗Φ(x′) ⃗Φ(x) · ⃗Φ(x′′)i. (16)To check sensitivity to higher order terms in F 2 and/or M 2R, while still neglecting b2, we go to the nexthigher order in F, p, MR when we find the Higgs mass.

It turns out that the numerical correction is small (ofthe order of 1–2 % ) even for MR ≈1. For the usually studied action with b1 and b2 = 0 we can check how wellthe 1/N expansion works, on the quantitative level, in the region of main interest, namely mR/fπ ≈2.0−2.5;in this region we have numerical data at N = 4 (see last reference in [1]).

We find agreement to within25 percent for the MR/F ratio as a function of MR; the large N results are systematically larger than theN = 4 ones but the relative change in the ratio MR/F when MR is varied is more accurately predicted.This is enough to convince us that our large N results ought to be taken seriously as an indication for whatis going to happen in the real system. For example, we would predict that if we limit cutoffeffects by thereasonable [5] requirement MR ≤√2/2 we find, with N = 4:MRF≤2πqlog√2C(Zπ).

(17)6

This gives for the nearest neighbor case (Zπ = 1) MR/F =3.1 while for the extreme case (Zπ = 0) one findsMR/F = 4.0. For N = 4 the numerical result is smaller by about 20%.

We therefore are led to conjecturethat for the simplest case with β1 = 0 the bound would increase by 30% to about 850 GeV .We should comment here on the possible relationship between this note and the work based on effectiveLagrangians that was carried out in the context of the investigation of the scalar sector of the standard modelby several groups recently [10]. Although in both cases one employs a nonlinear Lagrangian of identical forms(at least for N = 4) there is a fundamental difference that ought to be stressed: In the effective Lagrangianapproach one uses renormalized perturbation theory as defined by Weinberg [4] to summarize all relationsbetween soft pion amplitudes to any finite order in the external momenta of the pions; one then tries toextrapolate [10] these results by several schemes, all attempting to approximately impose unitarity andcrossing, to the region of energies dominated by the tails of the lowest resonances.

At no point does thisapproach consider the question of whether the physics so defined can be consistently extended to the cutoff(see second reference in [4] for some discussion). Technically speaking, chiral renormalized perturbationtheory, very much like ordinary perturbation theory, does not provide the relation between the parametersin the bare Lagrangian (i.e.

cutoffscale physics) and the parameters that describe low energy properties;it only provides a very large number of constraints between all the low energy physical processes. Thus,questions having to do with triviality are completely outside the reach of these approaches.

In more intuitiveterms, we are taking here very seriously a model defined by a chiral Lagrangian, all the way to the highestenergies, and seek for constraints on the allowable low energy effects that result from the requirement thatthey occur below the cutoff. If the cutoffis very high relative to the lowest resonance we are bound, by thegeneral principles of the renormalization group, to view this as just another regularization of λ(⃗φ2)2 andhence a perfectly reasonable framework for investigating consequences of triviality.We have not discussed in any detail the term with coupling b2.

We have seen that it has no effect inthe perturbative regime; however, when terms beyond the equivalent of the O(p2) correction to the bubbleare taken into account, b2 comes into play and can become important. The structure of the term indicatesthat b2 is the right dial to tune if we wish to make our theory look more like QCD.

This suggests that itmight help to make the Higgs mass bound even higher. But in the present note we were interested mainlyin estimating the Higgs-mass triviality bound when nothing special (technicolor-like) has been imposed onthe theory.

Further discussion of the b2 term would take us too far afield. We shall also omit any discussionof subleading terms in the 1/N expansion; particularly interesting would be the corrections to Zπ and tothe phase structure in the vicinity of the tricritical point.

These topics and a comprehensive analysis ofRenormalization Group flows in the models under consideration will hopefully be addressed in future work.We should also comment here on the fact that it is well known that once the scalar sector is stronglyinteracting, looking at mR rather than mH, is a bad approximation due to the large width [11]. This effectleads to a saturation of the growth of mH much before mR stops growing and can be best seen in the 1/Nexpansion, as shown by Einhorn [7].

It is important to realize that this sort of bound has nothing to dowith triviality; it appears even when the issue of triviality is totally ignored, for example, when the bare fourpoint coupling in the linear model is allowed to become negative and the theory has a tachyonic instability.Our calculations can all be done for a relatively narrow Higgs, and all they aim to show is that a stronglyinteracting scalar sector in the minimal standard model has not yet been convincingly ruled out. The relationbetween mR and mH alluded to above may very well hold in such a sector, if only we could show that oneexists.In summary, we propose that the scalar sector of the minimal Standard Model can be made to have astronger renormalized self coupling by introducing, at the cutofflevel, four-derivative couplings that inducesome repulsion between Goldstone Bosons.

Within the 1/N expansion this increases the Higgs mass bound,compared to the usual lattice result, by up to about 30%. Obviously it would be extremely interesting tostudy, by numerical means, whether this conclusion persists for the physically relevant case of N = 4.

Thiswould help resolve the puzzle presented at the beginning of this Letter. We hope to address this question inthe future.ACKNOWLEDGEMENTS.This research was supported in part by the Florida State University Supercomputer ComputationsResearch Institute which is partially funded by the U.S. Department of Energy through DOE grant #DE-FG05-85ER250000 (UMH and PV) and under DOE grant # DE-FG05-90ER40559 (HN).

UMH also7

acknowledges partial support by the NSF under grant # INT-8922411 and he would like to thank F. Karschand the other members of the Fakult¨at f¨ur Physik at the University of Bielefeld for the kind hospitality whilepart of this work was done. HN would like to thank F. David and M. Einhorn for useful discussions.8

REFERENCES[1] M. L¨uscher and P. Weisz, Phys. Lett., B212 (1988) 472; J. Kuti, L. Lin and Y. Shen, Phys.

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

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

Lett., 61 (1988) 427; G. Bhanot, K. Bitar, U. M.Heller, H. Neuberger, Nucl. Phys., B353 (1991) 551.

[2] see the The Standard Model Higgs Boson edited by M. B. Einhorn, CPSC Vol. 8, Elsevier Pub.

[3] W. Bardeen, M. Moshe, M. Bander, Phys. Rev.

Lett., 52 (1983) 1188. [4] S. Weinberg, Physica, 96A (1979) 327; H. Georgi, Weak Interactions and Modern Particle Theory (1984)Addison-Wesley Pub.

[5] G. Bhanot, K. Bitar, U. M. Heller, H. Neuberger, Nucl. Phys., B343, (1990) 467.

[6] R. Dashen, H. Neuberger, Phys. Rev.

Lett., 50 (1983) 1897. [7] M. Einhorn, Nucl.

Phys., B246 (1984) 75. [8] L. Lin, J. Kuti, Y. Shen, in Lattice Higgs Workshop, FSU, May 16–18 1988, Editors B.

Berg., et. al.

[9] F. David, D. Kessler, H. Neuberger, Nucl. Phys., B257 [FS14] (1985) 695.

[10] J. Bagger, S. Dawson, G. Valencia, BNL – 45782, Brookhaven Nat. Lab.

preprint; S. Dawson, G.Valencia, Nucl. Phys., B352(1991) 27; J. F. Donoghue, C. Ramirez, G. Valencia, Phys.

Rev., D39(1989) 1947; A. Dobado, M. Herrero, J. Terron, CERN–TH.5670/90, CERN preprint and referencestherein. [11] B. W. Lee, C. Quigg, H. B. Thacker, Phys.

Rev., D16 (1977) 1519.9

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