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Inelastic Channels in the Electroweak Symmetry-Breaking Sector

arXiv:hep-ph/9206249v1 25 Jun 1992JHU-TIPAC-920012June, 1992Inelastic Channels in the Electroweak Symmetry-Breaking SectorS. G. Naculichand C.–P.

YuanDepartment of Physics and AstronomyThe Johns Hopkins UniversityBaltimore, MD 21218ABSTRACTIt has been argued that if light Higgs bosons do not exist then the self–interactions of W’s become strong in the TeV region and can be observed inlongitudinal WW scattering. We present a model with many inelastic channelsin the WW scattering process, corresponding to the creation of heavy fermionpairs.

The presence of these heavy fermions affects the elastic scattering of W’sby propagating in loops, greatly reducing the amplitudes in some charge channels.Consequently, the symmetry–breaking sector cannot be fully explored by using,for example, the W +W + mode alone; all WW →WW scattering modes must bemeasured.

If light Higgs bosons do not exist, it is believed that elastic longitudinal WWscattering will be enhanced, indicating the presence of new strong interactions at orabove 1 TeV. (We use W to denote either the W ± or Z0 boson.) It has been claimedthat the energy and luminosity of the SSC are large enough that, whatever formthe new interactions take, they would be observable in WW two–body interactionsvia leptonic decays of W’s.

[1] This is called the “no–lose theorem.” Study of theW +W + mode has been particularly promoted[2] in the context of observing thesestrong interactions because the standard model background for this mode is small.Recently, Chivukula and Golden have emphasized the possible existence ofinelastic channels in the WW scattering process. [3] They studied an O(4) × O(n)model in which the WW interactions are almost entirely inelastic (to n species ofpseudo–Goldstone bosons).

Consequently, the elastic WW amplitude is reduced,the more so for a larger number n of inelastic channels. (The total event ratefor elastic scattering, however, does not decrease as n increases.

[4]) In that model,inelastic scattering leads to a very broad low–energy resonance in the elastic WWscattering amplitudes.In this letter, we present a model with many inelasticchannels in the electroweak sector and which has no resonances.In general, the presence of inelastic channels corresponds to additional particlesin the theory. Even if the production of these particles is not directly observed,they affect the elastic scattering of W’s by propagating in loops.These loopsnecessarily contribute to the imaginary part of the elastic scattering amplitude,which is related by the optical theorem to the total cross section.

The loops alsocontribute to the real part of the elastic amplitude, interfering with the Borncontribution. This interference may dramatically reduce the signal in some chargechannels, e.g.

the W +W + channel. Other channels may be enhanced, however,both by real and imaginary loop corrections.

The model we present below hasprecisely this behavior. The moral is that to be certain of detecting the symmetry–breaking sector it will be necessary to measure scattering in all the final state WWmodes.2

The no–lose theorem, with its prediction of strong WW scattering in the ab-sence of a light Higgs resonance, is based on the low–energy theorems for a theorywith spontaneously–broken symmetry. The pattern of symmetry breaking for theelectroweak sector is SU(2)L × SU(2)R −→SU(2)V if we assume that it respectsa custodial SU(2) symmetry.

At energies s ≫M2W, the longitudinal vector bosonsW correspond, via the equivalence theorem, to the three Goldstone bosons φa re-sulting from this broken symmetry. The interactions of these Goldstone bosons aredescribed by low–energy theorems, which emerge automatically when we describethis broken symmetry using chiral Lagrangians.

In this approach, the Goldstonebosons are parametrized by the matrixΣ = expiτaφaf,a = 1, 2, 3,(1)where τa are the Pauli matrices. The lowest energy term of the chiral LagrangianisL = f24 Tr∂µΣ∂µΣ†.

(2)It follows that the low–energy amplitude for Z0Z0 →W +W −, for example, isM(Z0Z0 →W −W +) = sf2,(3)where f = 250 GeV. This amplitude grows with increasing center–of–mass energy,becoming strong at around 1 TeV.At higher energies, the amplitude (3) is modified by corrections of order s2/16π2f4coming from higher–dimension operators induced by physics at a higher scale,and from Goldstone boson loop corrections.

[5] Indeed, we know that the ampli-tude (3) must eventually break down, because it violates partial–wave unitarity(|Re a00| > 12) at about 2√2πf ∼1.2 TeV. Nevertheless, below the scale of unitar-ity breakdown, the scattering of W’s will be roughly governed by eq.

(3), as longas there are no “light” particles (M <∼1 TeV) in the symmetry–breaking sectorother than the Goldstone bosons. On the other hand, if there do exist such lightparticles, the amplitude (3) will only be valid for s ≪M2.3

If such light particles can be exchanged by the W’s, as in the case of a lightHiggs boson for example, there will be narrow resonances in the WW scatteringamplitudes at s ∼M2. We consider another possibility, particles with mass well be-low 1 TeV which can only be produced in pairs; in particular we have in mind heavyfermions.

These fermions correspond to inelastic channels in the WW scatteringprocess. Unlike exchange particles, they will not necessarily produce resonances inthe elastic WW scattering amplitudes.

They alter these amplitudes through loopeffects, however, and lead to behavior markedly differently from that predicted bylow–energy theorems at scales above the threshold for fermion pair production butbelow 1 TeV. (New physics must still enter at around 1 TeV, because the fermions,unlike the Higgs boson, do not unitarize the amplitudes.

)In this letter, we present a chiral Lagrangian model coupled to heavy fermiondoublets. This model has no resonances in the elastic WW scattering amplitudes,therefore we must study these amplitudes in the TeV region where they becomestrong.

We find that the presence of the fermions dramatically reduces the scat-tering rates for the W +W + and W +Z0 modes relative to the predictions of thelow–energy theorems. On the other hand, the rates for scattering into the modesW +W −and Z0Z0 are enhanced.This enhancement partially results from thelarge imaginary part of the loop amplitude in the forward direction, which via theoptical theorem is due to the large total cross section.

Here we will only presentour results; details of the calculation will be given elsewhere. [6]The Lagrangian for the model isL = Nv24 Tr∂µΣ∂µΣ†+NXj=1ψj i/∂ψj −gvψjLΣψjR + ψjRΣ†ψjL ,Σ = expiτaφa√Nv,ψjL = 12 (1 −γ5) ψj ,ψjR = 12 (1 + γ5) ψj .

(4)The fields ψj represent N degenerate heavy fermion doublets with mass m = gv.The effects of the fermions on the WW scattering amplitudes will be significantwhen the Yukawa coupling g is large. To capture this, we will not calculate the4

amplitudes perturbatively in g, but rather in a 1/N expansion, holding the pa-rameters g and v fixed as N →∞. The results will be valid for arbitrary Yukawacoupling g, i.e., for all values of the fermion mass m.Were we to calculate perturbatively in the Yukawa coupling, the real partof the loop correction would contribute through interference with the tree–levelamplitude, but the imaginary part would be higher order.

In the large–N approach,the imaginary part of the loop correction contributes in leading order. Because itis related to the total cross section via the optical theorem, this contribution isimportant when there are many inelastic channels.To leading order in 1/N, the only corrections come from fermion loops.

Thesecontribute a divergence to the Goldstone boson self-energy. Accordingly, we add acountertermδL = δZNv24Tr∂µΣ∂µΣ†,(5)with δZ chosen so that the residue of the Goldstone boson propagator at p2 = 0 isunity.

This prescription will ensure that the low–energy theorems for the scatteringamplitudes are satisfied.The Lagrangian (4) is not the most general one with global SU(2)L × SU(2)Rchiral symmetry; we have omitted a possible derivative coupling of the form κLψL(Σi/∂Σ†)ψL+κRψR(Σ†i/∂Σ)ψR. (If parity is conserved, then κL = κR.) We also have not in-cluded any four-derivative terms involving Σ.

To leading order in 1/N, no suchterms are needed to absorb divergences; the counterterm (5) suffices to cancel thedivergences of the fermion loops in the WW →WW amplitudes.The WW scattering amplitudes are given by5

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). (6)Including only diagrams which contribute to leading order in 1/N, we findA(s, t, u) = 1N sv2 −m24π2v4sF2(s) −m44π2v4hF4(s, t) + F4(s, u) −F4(t, u)i, (7)whereF2(s) =1Z0dx lnh1 −sm2x(1 −x) −iεi,F4(s, t) =1Z0dxx2 −x + m2(s + t)st−1×lnh1 −sm2x(1 −x) −iεi+ lnh1 −tm2x(1 −x) −iεi.

(8)The integral F2(s) is given byF2(s < 0) = −2 +r1 −4m2sln √4m2 −s + √−s√4m2 −s −√−s!,F2(0 < s < 4m2) = −2 + 2r−1 + 4m2sarctanrs4m2 −s,F2(s > 4m2) = −2 +r1 −4m2s"ln √s +√s −4m2√s −√s −4m2!−iπ#,(9)6

and the integral F4(s, t) can be written[7] in terms of Spence functions asF4(s, t) = 21 −4m2(s + t)st−12 Spx+x+ −y+(s) + iσsε+ Spx+x+ −y−(s) −iσsε−Sp−x−x+ −y+(s) + iσsε−Sp−x−x+ −y−(s) −iσsε+ Spx+x+ −y+(t) + iσtε+ Spx+x+ −y−(t) −iσtε−Sp−x−x+ −y+(t) + iσtε−Sp−x−x+ −y−(t) −iσtε+ 2πiΘ(st) ln x+−x− ,(10)where σs denotes the sign of s, andx± = 12 ±r14 −m2(s + t)st,y±(s) = 12 ±r14 −m2s. (11)Now we fix N to a finite value, the number of fermion doublets in the model.

Thenwe set the scale v equal to f/√N, where f = 250 GeV characterizes the scale ofsymmetry breaking, to obtainA(s, t, u) = sf2 −Nm24π2f4sF2(s) −Nm44π2f4hF4(s, t) + F4(s, u) −F4(t, u)i. (12)In the limit s ≪m2, A(s, t, u) approaches s/f2, in accord with low–energy theo-rems.Our model depends on only two parameters: the number of fermion doubletsN and the fermion mass m. Experimental bounds require N to be ≤15.

Thisconstraint derives from the contribution of additional heavy degenerate fermionsto the shift in MW . [8]We consider a model with m = 250 GeV and N = 15, corresponding to 5additional generations of degenerate quark doublets, due to the color degeneracy.For these parameters, the partial waves obey unitarity (|aJI | ≤1) for the energyregion which we will consider.

[9] In table 1, we show the event rates and angular7

distributions of the W’s produced in various modes at the SSC (with integratedluminosity 104 pb−1) for this model, comparing them with results[10] from the“low–energy theorem model,” with Lagrangian (2). We do not include W pairsproduced by either quark or gluon fusion, restricting our consideration to WWscattering.

The invariant mass of the W pair is required to be within 850 GeV and1350 GeV. The branching ratio of the W boson decay has not been included.

Thetransverse momentum of each decay product of the W’s is required to be at least20 GeV. No rapidity cut has been imposed on the final–state particles.

The eventrate is calculated using the effective–W approximation. The parton distributionfunction used is the leading order set, Fit SL, of Morfin and Tung.

[11] The scale usedin evaluating the parton distribution function in conjunction with the effective–Wmethod is MW .From the results of table 1, we see that in the chiral Lagrangian model withheavy fermion doublets, the rates for the W +Z0 and W +W + modes are greatlyreduced relative to the low–energy theorem model. In fact, the W +W + event rateat MW W = 1.5 TeV is down by about a factor of 10 from the prediction of thelow–energy theorem model, and the W +Z0 mode is reduced by a similar factor.This is because the tree level amplitude is largely cancelled by the real part ofthe loop amplitude, and the imaginary part of the amplitude is zero for W +W +and small for W +Z0.

On the other hand, the heavy fermion model has a largerrate than the low–energy theorem model in both the W −W + and Z0Z0 modesbecause of enhancement from the loop contribution to the scattering amplitudes.This increase, however, is less than a factor of 2 even at MW W = 1.5 TeV. Theevent rate for the W −W + and Z0Z0 modes in the TeV region is almost entirelydue to the imaginary forward part of the amplitude (related by the optical theoremto the total cross section), the effect becoming stronger for higher MW W.In this letter, we have examined the effects of inelastic channels in the WWscattering process in one specific model.

In this model, containing heavy fermiondoublets, the rate of elastic WW scattering at energies above the threshold forfermion pair production differs significantly from low–energy theorem predictions.8

In particular, we found a large suppression of some elastic WW charge channelsand an enhancement of others.A lesson to be drawn from this model is thatall charge modes of the WW →WW process need to be observed to be sure ofdetecting the symmetry–breaking sector.Acknowledgements:It is a pleasure to thank G. L. Kane for asking the questions which stimu-lated this work, and for many fruitful discussions. We are also grateful to J. Bag-ger, J. Bjorken, Gordon Feldman, B. Grinstein, C. Im, G. Ladinsky, S. Meshkov,S.

Mrenna, F. Paige, and E. Poppitz for discussions. This work has been supportedby the National Science Foundation under grant no.

PHY-90-96198.9

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Phys. B261, 379 (1985);M. S. Chanowitz, Ann.

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38, 323 (1988).2. M. S. Chanowitz and M. Golden, Phys.

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61, 1053 (1988); 63,466(E) (1989); M. S. Berger and M. S. Chanowitz, LBL-30476 (1991).3. R. S. Chivukula and M. Golden, Phys.

Lett. B267, 233 (1991).4.

S. G. Naculich and C.–P. Yuan, preprint JHU–TIPAC–920017.5.

J. Bagger, S. Dawson and G. Valencia, preprint BNL-45782; preprint JHU-TIPAC-920009.6. S. G. Naculich and C.–P.

Yuan, in preparation.7. G. ’t Hooft and M. Veltman, Nucl.

Phys. B153, 365 (1979).8.

M. Veltman, Phys. Lett.

91B, 95 (1980);W. J. Marciano and A. Sirlin, Phys. Rev.

D22, 2695 (1980);S. Bertollini and A. Sirlin, Nucl. Phys.

B248, 589 (1984).9. The partial waves in both our model (with these parameters) and the low–energy theorem model satisfy |Re aJI | ≤12 up to about 1.2 TeV.10.

We use the scattering amplitudes given by M. S. Chanowitz and M. K.Gaillard, cited in ref. 1.11.

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TABLE 1(W +W +)LETHF0 < |η| < 194220 < |η| < 2246570 < |η| < 434380others31(W −W +)LETHF0 < |η| < 11993330 < |η| < 25308900 < |η| < 47411257others611(Z0Z0)LETHF0 < |η| < 11191670 < |η| < 23084440 < |η| < 4421622others36(W +Z0)LETHF0 < |η| < 188220 < |η| < 2253690 < |η| < 4373110others42TABLE 1. The event rates for W +W +, W −W +, Z0Z0, and W +Z0 production in one SSCyear for the low–energy theorem model and for the chiral Lagrangian model with heavy fermiondoublets.

The invariant mass of the W pair is required to satisfy 850 GeV < MWW < 1350 GeV.The branching ratio of the W boson decay is not included. 0 < |η| < 1 means that both Wbosons have pseudo–rapidity between 0 and 1, etc.11

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