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The Phenomenology of a Hidden

arXiv:hep-ph/9206255v2 21 Jul 1992BUHEP-92-23hep-ph/9206255The Phenomenology of a HiddenSymmetry Breaking SectorR. Sekhar Chivukula, Mitchell Golden, Dimitrios Kominis, and M. V. Ramana∗Boston UniversityDept.

of Physics590 Commonwealth AvenueBoston, MA 02215ABSTRACTWe calculate the production rate of gauge-boson pairs at the SSC in a model with a“hidden” electroweak symmetry breaking sector. We show that the signal of electroweaksymmetry breaking is lower than the background and that we cannot necessarily rely ongauge boson pairs as a signal of the dynamics of symmetry breaking.06/92∗sekhar@weyl.bu.edu, golden@weyl.bu.edu, kominis@budoe.bu.edu, mani@ryan.bu.edu

It is generally assumed that in the elastic scattering of longitudinally polarized Wsand Zs either there will be resonances with masses of less than 1 TeV (as in the weaklycoupled one doublet Higgs model) or the scattering amplitudes will become large, indicatingthe presence of new strong interactions at or above 1 TeV (see, for example [1]). In arecent paper [2], two of us argued that there is another possibility: If the electroweaksymmetry breaking sector has a large numbers of particles other than the longitudinalcomponents of the W and Z, then the elastic W and Z scattering amplitudes can be smalland structureless, i.e.

lacking any discernible resonances, at all energies. It was furtherargued that in such a model it may not be possible to rely on the two-gauge-boson eventsas a signal of the symmetry breaking sector.

A toy model with these properties, based onan O(N) scalar field theory solved in the limit of large N, was discussed.These arguments have been recently been disputed by Kane, Naculich, and Yuan [3]who argue that, independent of N, the total number of WLWL, ZLZL →ZLZL scatteringevents in the O(N) model is comparable to the number of events due to a standard-model“TeV Higgs” boson, and may therefore be large enough to be observed.However, forlarge N the would-be Higgs resonance is [2] both light and very broad. In this note, wecompute both the gauge boson pair signal and background for the toy model presented inref.

[2]. We show that, while the number of gauge boson scattering events is approximatelyindependent of N, the background is much larger for a light resonance and this signal isnot observable.We also estimate the ZLZL signal which arises from gluon fusion through a top quarkloop [4].

Because the would-be Higgs of this model is light and broad, even this contributionto the signal is smaller than the background by a factor of four or more making detectionproblematic, at best.We begin by reviewing the toy model of the electroweak symmetry sector constructedin [2]. This model has both exact Goldstone bosons (which will represent the longitudinalcomponents of the W and Z [5]) and pseudo-Goldstone bosons.

The Lagrangian densityisL = 12(∂⃗φ)2 + 12(∂⃗ψ)2 −12µ20φ⃗φ2 −12µ20ψ ⃗ψ2 −λ08N (⃗φ2 + ⃗ψ2)2,(1)where ⃗φ and ⃗ψ are j- and n-component real vector fields. This theory has an approximateO(j + n) symmetry (i.e.N = j + n) which is softly broken to O(j) × O(n) so longas µ20φ ̸= µ20ψ.

If µ20φ is negative and less than µ20ψ, one of the components of ⃗φ gets avacuum expectation value (VEV), breaking the approximate O(N) symmetry to O(N −1).1

With this choice of parameters, the exact O(j) symmetry is broken to O(j −1) and thetheory has j −1 massless Goldstone bosons and one massive Higgs boson.The O(n)symmetry is unbroken, and there are n degenerate pseudo-Goldstone bosons of mass mψ(m2ψ = µ20ψ −µ20φ). This model is particularly interesting since it can be solved (even forstrong coupling) in the limit of large N [6].

We will consider this model in the limit thatj, n →∞with j/n held fixed.The scalar sector of the standard one-doublet Higgs model has a global O(4) ≈SU(2) × SU(2) symmetry, where the 4 of O(4) transforms as one complex scalar dou-blet of the SU(2)W × U(1)Y electroweak gauge interactions. It is this symmetry whichis enlarged in the O(N) model: we will model the scattering amplitudes of longitudinalgauge bosons by the corresponding O(j) Goldstone boson scattering amplitudes in theO(j + n) model solved in the large j and n limit.

Of course, j = 4 is not particularlylarge. Nonetheless, the resulting model will have all of the correct qualitative features, theGoldstone boson scattering amplitudes will be unitary (to the appropriate order in 1/jand 1/n), and we can investigate the theory at moderate to strong coupling [7].

We makeno assumptions about the embedding of SU(2)W × U(1)Y in O(n), i.e. no assumptionsabout the electroweak quantum numbers of the pseudo-Goldstone bosons: we will assume,however, that the pseudo-Goldstone bosons are SU(3) color singlets1.One may compute Goldstone boson scattering to leading order in 1/N.

The detailsof the calculation may be found in [2]. The amplitude aij;kl(s, t, u) for the process φiφj →φkφl isaij;kl(s, t, u) = A(s)δijδkl + A(t)δikδjl + A(u)δilδjk(2)whereA(s) =sv2 −Ns1λ(M) + eB(s; mψ, M),(3)andeB(s; mψ, M) =n32Nπ21 +iqs/(4m2ψ −s)logi −qs/(4m2ψ −s)i +qs/(4m2ψ −s)−logm2ψM 2+j32Nπ21 + log M 2−s.

(4)1 Gauge boson pair production in models with colored pseudo-Goldstone bosons is discussedin detail in [8].2

Here s, t, and u are the usual Mandelstam variables, v is the weak scale (approximately 250GeV), and M is a renormalization point which we chose below such that the renormalizedcoupling, λ(M), satisfies 1/λ(M) = 0.The amplitude aij;kl of eqn. (2) may be used to derive partonic cross sections forWLWL, ZLZL →ZLZL which can then be folded with the appropriate gauge bosonstructure functions (using the “effective-W approximation” [9] and the EHLQ set II [10]structure functions) to yield the contribution of gauge boson scattering to the processpp →ZZ + X.

This contribution to the differential cross section for ZZ production asa function of ZZ invariant mass is shown in the dot-dash lines of fig. 1 for n = 32 andM = 1500 GeV [2].As in the standard model, gluon fusion through a top quark loop [4] provides a signalfor gauge boson pairs comparable to the signal from gauge boson scattering.

The correctcomputation in this model is somewhat nontrivial. We wish to work to lowest nonvanishingorder in αs (the QCD coupling constant) and the top quark Yukawa coupling.

To this order,there are three diagrams that contribute to the ZLZL final state. The first is the simpletop quark triangle diagram, the analogue of the Higgs production diagram in the standardmodel.

Next there is the top-quark box, which produces final state longitudinal Z’s exactlyas in the standard model [11]. Lastly, there is a two-loop diagram, in which a box of quarks(not all four sides of which are top) produces a pair of Goldstone bosons which rescatterthrough the Higgs boson into ZLZL.

This last diagram must also be computed to get acorrect, gauge invariant answer, since it is leading in 1/N.To get a rough estimate of the rate, we concentrate on the top-quark triangle, ignoringthe other two diagrams. The amplitude for this diagram isαs s δab2πhv2 −Ns1λ(M) + eB(s; mψ, M)igµν −2pµ2pν1sI(s, m2t).

(5)Here p1 and p2 are the momenta of the two incoming gluons with polarization vectorsassociated with µ and ν and colors with the a and b respectively, and the function I(s, m2t)is the Feynman parameter integralI(s, m2t) = m2tZ 10dxZ 1−x0dy(1 −4xy)m2t −xys −iǫ,(6)and mt is the mass of the top quark. The contribution of this process to ZZ productionis shown as the solid curve in fig.

1.3

The irreducible background to observing the Higgs boson in Z pairs comes from theprocess q¯q →ZZ and is shown as the dashed curves in fig. 1.

We see that the backgroundis more than an order of magnitude larger than the gauge boson scattering signal and thatthis signal is unobservable. Even the gluon fusion contribution to the ZLZL productioncross section is lower than the background by a factor of four or more, making detectionof such a broad resonance problematic.In [3] it was shown that the numbers of final state gauge boson pairs from gauge bosonscattering is roughly independent of N if√NM is held fixed.

This is because as N increasesfor fixed√NM, M and the mass and width of the Higgs boson decrease like 1/√N. Theincreased production of Higgs bosons due to their smaller mass2 is approximately cancelledby the Higgs boson’s smaller branching ratio into Ws and Zs.

The number of signal events,therefore, is approximately independent of N and is the same as the number which wouldbe present in the model with n = 0 described in ref. [7].

Since the signal for gauge bosonscattering in that model is (marginally) observable [5] and since the number of ZZ eventsis roughly independent of n, the authors of ref. [3] argue that the signal may be observablefor any n.We have calculated the signal and background for the parameters chosen in [3], n = 8and M = 2500 GeV (with a Higgs mass of approximately 485 GeV).

The results are plottedin fig. 2 and the results for n = 0, M = 4300 GeV are plotted in fig.

3. It is true thatthe total number of gauge boson scattering events is comparable in figs.

2,3, and even infig. 1.

However, the background is much greater when n = 8 or 32 since the correspondingHiggs is much lighter. The gauge boson scattering signal with n = 8 or 32 is “hidden”because the Higgs boson is both light and broad.

In both cases, the gluon fusion signalis substantially larger than the gauge boson scattering signal. The signal, however, is stillsignificantly below the background, making detection of a broad resonance difficult, atbest.By way of comparison, the signal for a 485 GeV standard model Higgs is shown infig.

4. In this case, because the Higgs is relatively narrow, on the peak the gauge bosonscattering Higgs signal is comparable to the background and the gluon fusion signal is wellabove the background.There are two technical shortcomings in the calculation of the cross sections presentedabove.

Firstly, in computing the gauge boson scattering signal, we have used both the2 And therefore higher gauge boson partonic luminosity [9].4

equivalence theorem [5] and the effective-W approximation [9]. Strictly speaking, both ofthese approximations hold only at energies above a few times the W mass.

Even at 200GeV, however, the corrections should be of order one [9][12], whereas the background atthese energies when n = 32 is more than an order of magnitude larger. While we cannotprecisely determine the number of gauge boson scattering events, it is clear that they areswamped by the background.

Second, in computing the gluon fusion signal, we have onlycomputed the contribution from a top quark triangle diagram and have not included thecontributions from the other two of the three leading diagrams, as discussed above. Thesewill interfere with the contribution we have computed.

However, we do not expect this tochange any of our conclusions.In general, the two-gauge-boson scattering signal of the symmetry breaking sectoris not visible above the background unless the gauge boson elastic scattering amplitudesare big. This can happen either if the symmetry breaking sector is strongly coupled orat the peak of a narrow resonance.

We can see this just by counting coupling constants:the background (qq →ZZ) is order g2 and is a two body final state, while the signal(qq →qqZZ) is naively of order g4 and is a four body final state. When the symmetrybreaking sector is strongly interacting and the final state gauge bosons are longitudinal,this naive g4 gets replaced by g2aij;kl, and the signal may compete with the background.Since aij;kl in the O(4 + 32) model is never large, the signal rate never approaches thebackground rate.Moreover, while we have concentrated on the signal for the ZZ final state, the ar-guments given here should apply equally to all other two-gauge-boson signals as well.

Inthe O(4 + 32) model it is likely that none of the two-gauge-boson signals of the symmetrybreaking sector may be observed over the background. In this model even observing alltwo-gauge-boson modes will not be sufficient to detect the dynamics of electroweak sym-metry breaking – one will need to observe the pseudo-Goldstone bosons, and identify themwith symmetry breaking.In conclusion, we see that in models of electroweak symmetry breaking with a largenumber of pseudo-Goldstone bosons the longitudinal gauge boson scattering amplitudesmay be small and structureless at all energies.

In this case we cannot necessarily rely ongauge boson pairs as a signal of the dynamics of symmetry breaking.We would like to thank Kenneth Lane for useful conversations, Elizabeth Simmons forreading the manuscript, and Gordon Kane, Steven Naculich, and C.-P. Yuan for sending5

us ref. [3] prior to publication.

R.S.C. acknowledges the support of an Alfred P. SloanFoundation Fellowship, an NSF Presidential Young Investigator Award and DOE Out-standing Junior Investigator Award.

This work was supported in part under NSF contractPHY-9057173 and DOE contracts DE-AC02-89ER40509 and DE-FG02-91ER40676, and byfunds from the Texas National Research Laboratory Commission under grant RGFY91B6.6

Figure CaptionsFig. 1.Differential production cross section for pp →ZZ (at a pp center of mass energyof 40 TeV) as a function of invariant Z-pair mass for j = 4, n = 32, mψ = 125GeV and the renormalization point M = 1500 GeV.

A rapidity cut of |y| < 2.5has been imposed on the final state Zs. The gauge boson scattering signal isshown as the dot-dash curve and gluon fusion signal (with mt = 120 GeV) asthe solid curve.

The background from q¯q annihilation is shown as the dashedcurve. In all contributions, the rapidities of the Zs must satisfy |yZ| < 2.5.

Allcomputations use the EHLQ set II [10] structure functions with Q2 = M 2W in thegauge boson scattering curve, and Q2 = ˆs in the other two cases.Fig. 2.Same as fig.

1 with n = 8 and M = 2500 GeV, as in ref. [3].Fig.

3.Same as fig. 1 with n = 0 (ref.

[7]) and M = 4300 GeV.Fig. 4.Same as fig.

1 for a standard model Higgs boson with mass 485 GeV.7

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