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Can the Electroweak Symmetry-breaking Sector Be Hidden?

arXiv:hep-ph/9206254v1 26 Jun 1992JHU-TIPAC-920017June, 1992Can the Electroweak Symmetry-breaking Sector Be Hidden?S. G. Naculichand C.–P.

YuanDepartment of Physics and AstronomyThe Johns Hopkins UniversityBaltimore, MD 21218ABSTRACTIn a recent paper, Chivukula and Golden claimed that the electroweak symmetry–breaking sector could be hidden if there were many inelastic channels in the longi-tudinal WW scattering process. They presented a model in which the W’s coupleto pseudo–Goldstone bosons, which may be difficult to detect experimentally.

Be-cause of these inelastic channels, the WW interactions do not become strong inthe TeV region. We demonstrate that, despite the reduced WW elastic amplitudesin this model, the total event rate (∼5000 extra longitudinal W +W −pairs pro-duced in one standard SSC year) does not decrease with an increasing number ofinelastic channels, and is roughly the same as in a model with a broad high–energyresonance and no inelastic channels.

The “no–lose theorem” states that if light Higgs bosons do not exist, elasticlongitudinal W scattering becomes strong at or above 1 TeV, and that the newstrong interactions can be detected by observing WW scattering via leptonic de-cays of W’s. [1] (We use W to denote either the W ± or Z0 boson.) In a recentpaper,[2] Chivukula and Golden have argued that the “no–lose theorem” breaksdown if there are many inelastic channels into which the W’s can scatter.

Theypresented a toy model in which the W’s couple to a large number of pseudo–Goldstone bosons, which may be difficult to detect experimentally. Because of thelarge number of inelastic channels, there are light resonances in the elastic WWscattering amplitudes, which are too broad to be discernible as peaks.

Moreover,the growth of the elastic scattering amplitudes is cut offat the scale of the lightresonances, so the WW interactions do not become strong in the TeV region. Theyconclude that, unless the pseudo–Goldstone bosons themselves can be observed,the electroweak–symmetry breaking sector will remain hidden.In this note, we point out that the total event rate for elastic WW scattering inthe model of ref.

2 does not decrease as the number of inelastic channels increases.The elastic amplitude is smaller, but the resonance is at a lower energy, where theparton luminosity is greater. We give a simple scaling argument to show that thetotal rate (∼5000 W +W −pairs and ∼2500 Z0Z0 pairs produced in one standardSSC year) is roughly independent of the number of inelastic channels, and is aboutthe same as that in the standard O(4) model[3,4] with no inelastic channels and abroad TeV scale resonance.

We also briefly comment on possible methods to detectthe signal.The model presented in ref. 2 to demonstrate the possible effects of inelasticchannels in the electroweak sector contains both exact Goldstone bosons and alarge number of pseudo–Goldstone bosons.

This model possesses an approximateO(j + n) symmetry which is explicitly broken to O(j) × O(n). The exact O(j)symmetry is spontaneously broken to O(j −1), yielding j −1 massless Goldstonebosons, φ, and one massive scalar boson.

The O(n) symmetry remains unbroken,and there are n degenerate pseudo–Goldstone bosons, ψ, with mass mψ.2

To use this model to describe the scattering of longitudinal W’s, one appliesthe equivalence theorem, replacing the longitudinal W with its corresponding Gold-stone boson φ in the S–matrix. This equivalence holds only when EW ≫MW ,where EW is the energy of the W boson in the WW center–of–mass frame.

There-fore, strictly speaking, one should not use this model to describe WW scatteringwhen the invariant mass MW W is less than a couple of times the mass threshold2MW. The amplitudes for the scattering of longitudinal W’s are given by[5]M(Z0Z0 →W −W +) = A(s, t, u),M(W −W + →Z0Z0) = A(s, t, u),M(W −W + →W −W +) = A(s, t, u) + A(t, s, u),M(Z0Z0 →Z0Z0) = A(s, t, u) + A(t, s, u) + A(u, t, s),M(W ±Z0 →W ±Z0) = A(t, s, u),M(W ±W ± →W ±W ±) = A(t, s, u) + A(u, t, s).

(1)In the models we consider in this note, A(s, t, u) = A(s) depends only on s.Before turning to the model of ref. 2, we recall some relevant features of theO(N) model.

[3,4] To leading order in 1/N, with the parameters v and Λc held fixedas N →∞, the amplitude A(s) in the O(N) model is given byA(s) = sNv2 −s32π2lneΛ2c|s|+ iπΘ(s)−1. (2)Here Θ(s > 0) = 1 and Θ(s < 0) = 0, and Λc is the cut–offscale of the theory,[6]related to the tachyon mass µt throughΛc = µt√e exp−16π2v2µ2t.

(3)(For µ2t ≫v2, the tachyon and cut–offscales are roughly the same, Λc ≃µt/√e. )Because of the presence of the tachyon, the O(N) model must be regarded as aneffective theory, valid only at energy scales well below µt.

With v, Λc, and s heldfixed, the amplitude (2) evidently scales as 1/N.3

To extract physical predictions from the O(N) model, one must set N equalto some finite value; N = 4 corresponds to the electroweak sector with its threeGoldstone bosons and one massive Higgs boson. To ensure that low–energy the-orems for the scattering amplitudes are satisfied, one must then set v = f/√N,where f = 250 GeV characterizes the symmetry–breaking scale.

The amplitude isthen given byA(s) = sf2 −sN32π2lneΛ2c|s|+ iπΘ(s)−1. (4)With f held fixed, the scaling property of the amplitude differs slightly from thatdescribed above; A(s) scales as 1/N, but only if we simultaneously scale s with1/N and Λc with 1/√N.

In other words,A(s) = 1N F(˜s, ˜Λc),˜s = Ns,˜Λc =√NΛc,(5)where F(˜s, ˜Λc) only depends on N through ˜s and ˜Λc.To locate resonances in the scattering amplitudes (1), we look for the positionof the (complex) pole of A(s) as a function of the parameters of the theory. Theposition of the pole s can be parametrized by its “mass” m and “width” Γ throughthe relation s = (m −i2Γ)2, though we should not take these terms literally whenΓ is comparable to m. The pole traces out a curve in the s–plane as µt is varied.When µt is very large, the real and imaginary parts of the pole are both small, cor-responding to a light, narrow resonance.

As µt decreases, Re(s) increases, reachesa maximum and then begins to decrease, while Im(s) continues to increase. Werefer to the pole position with maximum Re(s) as the “heaviest” resonance.

Thisresonance is very broad, with Γ roughly equal to m. In the O(4) model, the “heav-iest” resonance is found[4]to have “mass” m = 845 GeV and “width” Γ = 640 GeV,and corresponds to a cut–offΛc = 4.9 TeV and tachyon mass µt = 8.4 TeV. Fromthe scaling property of eq.

(4), the values of m and Γ corresponding to the heaviestresonance for the O(N) model areq4N times those for the O(4) model; the cut–offand tachyon mass scale in the same way.4

We now turn to the O(j)×O(n) model of ref. 2.

The amplitudes are calculatedin the limit j, n →∞with the ratio j/n fixed; only diagrams which contribute toleading order in 1/(j + n) are included. The amplitude A(s) is given byA(s) = s(f2 −sj32π2lneΛ2c|s|+ iπΘ(s)−sn32π2"ln Λ2cem2ψ!−F2(s, mψ)#)−1,(6)whereF2(s, m) = −2 +r1 −4m2sln √4m2 −s + √−s√4m2 −s −√−s!fors < 0,F2(s, m) = −2 + 2r−1 + 4m2sarctanrs4m2 −sfor0 < s < 4m2,F2(s, m) = −2 +r1 −4m2s"ln √s +√s −4m2√s −√s −4m2!−iπ#fors > 4m2,(7)and the cut–offΛc (equal to M of ref.

2) is related to the tachyon scale byΛc = µt√e exp−16π2f2(j + n)µ2t+n2(j + n)ln m2ψµ2t!−s1 +4m2ψµ2tlnqµ2t + 4m2ψ −µtqµ2t + 4m2ψ + µt. (8)Qualitatively speaking, for center–of–mass energies well below the ψ mass thresh-old, s ≪4m2ψ, the O(j) × O(n) model behaves like the O(j) model; the pseudo–Goldstone bosons play little role.

On the other hand, well above the threshold,s ≫4m2ψ, the O(j) × O(n) model behaves like the O(j + n) model.Having obtained the amplitude (6), one sets f = 250 GeV and j = 4; theexact Goldstone bosons in this model correspond to the longitudinal W’s. Threeindependent parameters now specify the model: the number of pseudo–Goldstonebosons n, their mass mψ, and the tachyon mass µt.

(Again, the model is only validat energy scales well below the tachyon mass. )We now compare the total event rates in the O(4) × O(n) model for differentvalues of n. As in the O(N) model, the amplitude (6) has a complex pole, whose5

real part increases and then decreases as µt varies. We choose the parameter mψ sothat the resonance is well above the the pseudo–Goldstone mass threshold, wherethe model essentially behaves like the O(4 + n) model.

Thus, using the scalingbehavior described earlier, the “heaviest” resonance of the O(4) × O(n) model hasm ≃q44+n × 845 GeV and Γ ≃q44+n × 640 GeV; the corresponding tachyonmass and cut–offalso scale asq44+n relative to the O(4) model. We choose thetachyon mass µt for each value of n to correspond to the “heaviest” resonance ofthat model.

Note that as n increases, the resonance moves to smaller mass m; thewidth to mass ratio of the resonance is of course independent of n.Above the pseudo–Goldstone mass threshold, where the model behaves like theO(4 + n) model, the amplitude (6) has the scaling propertyA(s) =14 + nF(˜s),˜s = (4 + n)s,(9)This follows from eq. (5) because the cut–offΛc for the heaviest resonance scalesas 1/√4 + n, and so ˜Λc is independent of n. Since the amplitude (9) scales as1/(4 + n), it would seem that the scattering rate becomes smaller as n increases.On the other hand, for larger n, the resonance occurs at lower invariant mass,where the WW parton luminosity LW W is higher.We now show that the twoeffects cancel each other.In the effective–W approximation[7,8] we haveσpp→W W →W W(S) =1Zτmindτ dLW WdτσW W →W W(τS),σW W →W W(τS) =ZdΩ12τS |M(τS)|2 ,(10)where√S is the center–of–mass energy of the pp collider, √s =√τS is the in-variant mass of the WW pair, τmin = 4M2W /S, and dΩintegrates over the di-rection of the out–going W in the WW center–of–mass frame.

The parton lu-minosity (dLW W/dτ) scales[9] approximately as 1/τ2 for MW W <∼1 TeV at the6

SSC. By rewriting eq.

(10) in terms of ˜τ = (4 + n)τ, using eq. (9), we find thatσpp→W W →W W(S) is actually independent of n. Thus we conclude that, although theamplitude decreases as n increases, the total elastic event rate stays the same.To see how large the event rate actually is, we choose n = 8 and mψ = 125GeV.

We expect from our scaling arguments that the heaviest resonance for n = 8will have m ≃q412 ×845 = 490 GeV and Γ ≃q412 ×640 = 370 GeV. Indeed, usingeq.

(6) explicitly, we find that the heaviest resonance occurs for m = 485 GeV andΓ = 350 GeV, corresponding to tachyon mass µt = 4.3 TeV. Because EW ≫MW ,use of the equivalence theorem is probably justified for these parameters.To obtain the event rate for the O(4)×O(8) model with parameters mψ = 125GeV and µt = 4.3 TeV, we fold the amplitudes with the parton luminosities.

Wefind that the elastic W −W + event rate for MW W ≥350 GeV at the SSC (with√S = 40 TeV and integrated luminosity 104 pb−1) is about 0.5 pb. This rateis about the same as the total rate (0.6 pb) for the O(4) model (i.e., the n = 0limit) with a resonance with m = 845 GeV and Γ = 640 GeV, as expected fromthe scaling argument given above.

Moreover, this rate is not much smaller thanthe rate (3.4 pb) for a 500 GeV standard model Higgs boson, with width 64 GeV,produced via the W–fusion process. The Z0Z0 event rate in the O(4)×O(8) modelis about half the W +W −event rate.

We have not included in these rates W pairsproduced by either quark or gluon fusion, restricting our consideration to WWscattering.Many studies have been performed on detecting Higgs bosons at the SSC. It hasbeen shown that a ∼500 GeV standard model Higgs boson can be detected usingthe “gold–plated” mode alone, and does not require the application of techniquessuch as jet–tagging[10] and/or jet–vetoing[11] used in studying TeV WW interac-tions.However, these techniques, together with others, such as measuring thecharged particle multiplicity of the event[12] or testing the fraction of longitudinalW’s,[5] could be used to further improve the signal–to–background ratio to study a∼500 GeV standard model Higgs boson produced via WW fusion processes.

We7

think it is clear that similar strategies could be applied to detect the ∼500 GeVresonance in the O(4) × O(8) model discussed above.For the O(4) × O(32) model of ref. 2, with parameters mψ = 125 GeV andµt = 2.5 TeV, the resonance (m = 275 GeV and Γ = 120 GeV) is in a regionwhere the energy EW of the longitudinal W in the WW center–of–mass frame isless than twice MW .

[13] To see whether such a resonance could be detected wouldrequire a detailed Monte Carlo study, which we will not perform in this paper.We would argue, however, that with ∼5000 extra longitudinal W +W −pairs and∼2500 extra longitudinal Z0Z0 pairs produced in one standard SSC year, thissignal could probably be observed with appropriate detectors.In this note, we have considered the O(4)×O(n) model presented in ref. 2 con-taining many inelastic channels in the WW scattering process.

We demonstratedthrough a simple scaling argument that, although the amplitude for elastic WWscattering in this model decreases as the number n of inelastic channels increases,the total elastic event rate remains more or less the same. (We choose the pa-rameters for each model to give the “heaviest” possible resonance.) This rate isabout the same as that for the O(4) model, with no inelastic channels and a heavyresonance.Acknowledgements:We would like to thank G. L. Kane for asking questions which stimulatedthis work, and for insisting that the total elastic cross–section could not decreasewith many inelastic channels.

We are also grateful to J. Bagger, Gordon Feldman,C. Im, G. Ladinsky, S. Meshkov, F. Paige, E. Poppitz, and E. Wang for discussions.This work has been supported by the National Science Foundation under grantno.

PHY-90-96198.8

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13. The equivalence theorem, which allows us to identify the Goldstone bosonsφ with the longitudinal gauge bosons W in the WW scattering processes,requires EW ≫MW .10

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