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ANOMALOUS GAUGE-BOSON COUPLINGS

arXiv:hep-ph/9209237v1 15 Sep 1992FERMILAB–CONF–92/246–TSeptember 1992ANOMALOUS GAUGE-BOSON COUPLINGSAT HADRON SUPERCOLLIDERSG. Valencia∗Fermi National Accelerator LaboratoryP.O.

Box 500, Batavia, IL 60510AbstractWe discuss anomalous gauge boson couplings at hadron supercolliders. Wereview the usual description of these couplings, as well as the studies of astrongly interacting electroweak symmetry breaking sector.

We present an ef-fective field theory formulation of the problem that relates the two subjects,and that allows a consistent and systematic analysis. We end with some phe-nomenology.

[*] Talk presented at the XXVI International Conference on High Energy Physics. Dallas, 1992.

1IntroductionThe main reason for building hadron supercolliders like the SSC or the LHC is to studythe mechanism that breaks electroweak symmetry. There are many possibilities forthis, and each one has distinctive signals.

The simplest one is the minimal standardmodel with a Higgs boson, in which case the goal would be to find this particle.Typical technicolor models contain a large number of particles that should be foundby these colliders; the case of the technirho has been studied most. Similarly, otherpossibilities like supersymmetry should have several particles within reach.An interesting question is whether any of these new particles will be within thereach of the SSC/LHC.

In anticipation that all new particles could lie just out of reach,many studies have been undertaken to extract information on electroweak symmetrybreaking from its indirect effects. There are two fields in the literature that addressthis issue.

One goes by the name of “anomalous gauge-boson couplings”, and theother one is known as “strong longitudinal gauge-boson scattering”. In this talk wewill present an effective field theory formulation of the problem, that relates the twosubjects and that allows a consistent and systematic study of anomalous couplings.2Anomalous CouplingsConventional studies of anomalous couplings start from the Lagrangian1L=−igVgV1 (W †µνW µ −W †µWµν)V ν + κV W +µ W −ν V µν + λVM2WW †λµW µν V νλ+ac(W +µ W −µ )2 + · · ·(1)where V is either a photon or a Z-boson, and · · · stands for all other terms that wehave not written.There are several questions that arise when this Lagrangian is used.

The first oneis that it is not obviously SUL(2) ×UY (1) gauge invariant.2 Although this has causedsome confusion in the literature, it is not a problem, as was recently emphasized byBurgess and London.3 The reason why this is not a problem is that this Lagrangianis the unitary gauge version of an explicitly gauge invariant equivalent Lagrangian.For practical applications to supercolliders, it turns out to be useful to work with theexplicitly gauge-invariant version. In particular, because this permits the use of theequivalence theorem to simplify the calculations.1

Another issue that is not clear in Eq. (1), is the question of how many independentanomalous couplings there are (the · · ·), and if there is a hierarchy amongst them.The use of Eq.

(1) also creates problems at the one-loop level. For example, thestandard model at one-loop generates some of the “anomalous” couplings and oneneeds a procedure to separate these contributions from others due to electroweaksymmetry breaking.

Also, the Lagrangian of Eq. 1 is not renormalizable, and noprocedure has been specified to treat the divergences that arise beyond tree-level.Some authors have addressed this problem at the practical level by introducing formfactors.1 Doing so, further complicates the issue of counting and classifying the inde-pendent couplings.We answer all these questions by using an effective (“chiral”) Lagrangian supple-mented with the rules of chiral perturbation theory.4 This gives us an effective fieldtheory formalism that allows a systematic and consistent study of these issues.3Strongly Interacting VLMuch work has been done under the heading of strongly interacting longitudinalW’s and Z’s (≡VL).5,6 Perhaps the most important concept in this field is that of“enhanced electroweak strength”.

In the standard model, the amplitude for WWscattering is proportional to g2 multiplied by polarization vectors for the W’s. If onelooks at the longitudinal polarization (in a frame where E/MW ≫1), one finds thatsome of the g2 terms are now multiplied by M2H/M2W.

For a very heavy Higgs boson,these terms are thus much larger than the usual g2 terms, they are of “enhanced”electroweak strength.5,6 In the standard model without a Higgs boson the Higgs massis replaced by the energy of the W’s. One then has terms of order g2, terms of orderg2s/M2W ∼s/v2, v ≈250 GeV, and also terms of order (s/v2)(s/Λ2)n where Λ is thescale at which the effective theory breaks down.

The terms that grow with s are ofenhanced electroweak strength at high energies. It is these terms that are of interestat hadron supercolliders.

When one is only interested in extracting these terms, onecan resort to the equivalence theorem and replace the gauge bosons W, Z with theirwould-be Goldstone bosons w, z even inside loops.5b,5c,7By computing VL scattering in the standard model, one is able to place “unitarity”bounds on the Higgs mass.5a,5b At high energies, the partial waves are proportional toM2H, and thus if MH is too large, they violate the unitarity condition |a| ≤1. In theabsence of the Higgs boson, these partial wave amplitudes grow like s, violating the2

unitarity bound at about 2 TeV. One expects the physics associated with electroweaksymmetry breaking to come in at this scale (or below).

In the case where no newparticles lie below this scale, the study of anomalous couplings consists of looking fordeviations of the leading s behavior of amplitudes at high energies. One can think ofthis as a general way of taking the infinite mass limit of the standard model Higgs.4Effective LagrangianTo construct the effective Lagrangian we introduce the Goldstone bosons w±, zthrough the matrix U = exp(i⃗τ · ⃗w/v), and the gauge fields through the covariantderivativeDµU = ∂µU + ig′2 Bµτ3U −ig2UWµWe also have the field strength tensorsWµν=12∂µWν −∂νWµ −ig2[Wµ, Wν]Bµν=12∂µBν −∂νBµτ3(2)The lowest order effective Lagrangian is:8L(2) = v24 TrDµU†DµU + · · ·where · · · stands for the usual gauge boson kinetic terms, couplings to fermions, andgauge fixing terms.

This is the term that gives the W and Z their mass as can beseen immediately in unitary gauge (U = 1). This is a non-renormalizable Lagrangian,and divergences that occur at the loop level are handled in the usual way by chiralperturbation theory.

At tree-level, this Lagrangian produces amplitudes of order E2in an energy expansion. At one-loop, the divergences that appear are of order E4, andare absorbed by renormalization of the next to leading O(E4) effective Lagrangian.By looking for the most general form consistent with a global SU(2) × U(1) sym-metry spontaneously broken to U(1) (but conserving CP), Longhitano found8 thatthe next to leading order Lagrangian contains 13 terms (and the leading order La-grangian contains an extra term, ∆ρ).

We will not use this general Lagrangian, butinstead we will introduce an additional assumption: that there is a custodial SU(2)in the physics of electroweak symmetry breaking. That is, that the global symmetryis SU(2) × SU(2) broken to SU(2).

This amounts to requiring that the additional“custodial” SU(2) be broken only by g′ and by the difference in fermion masses.3

This is a reasonable assumption, which is true both for the minimal standardmodel and for common extensions such as technicolor. It is also a consistent assump-tion for experiments at supercolliders.

The reason is that at the very high energiesof these machines we can concentrate only on those terms of enhanced electroweakstrength as explained before. The counterterms needed to renormalize the loop dia-grams of enhanced electroweak strength respect this custodial SU(2).

Note that thisis no longer valid for low energy experiments, such as the ones that will be carriedout at LEP2. In that case the distinction between electroweak strength and enhancedelectroweak strength is not meaningful, and for consistency one must keep the fullcounterterm structure that appears at one-loop.The next to leading order effective Lagrangian is:9L(4)=v2Λ2gg′L10TrU†BµνUW µν−igL9LTrWµνDµU†DνU−ig′L9RTrBµνDµUDνU†+L1TrDµU†DµU2+ L2TrDµU†DνU2(3)The scale Λ is determined by the mass of the lightest particle in the symmetry break-ing sector, and in any case it is Λ <∼4πv.

The next to leading order terms in theeffective Lagrangian have been normalized by v2/Λ2; this reflects the fact that theyappear as the heavy physics associated with the scale Λ is integrated out. With thisnormalization, the Li are all expected to be O(1).

By going to unitary gauge onecan easily see what are the contents of this Lagrangian. The first line contains acorrection to the Z self energy that has been thoroughly discussed:10Lr10(MZ) = −πSwhen we take Λ = 4πv.

The next line contains the lowest order anomalous threegauge boson couplings.11,12 The more common κγ, κZ and gZ1 are simply some linearcombinations of L9L, L9R and L10;12gZ1 −1 ∼κV −1 ∼Og2L9L,9R,10v2Λ2Finally, the last line contains the lowest order anomalous four gauge boson couplings.There are only two of them: L1 and L2.13The bare Li coupling constants that appear in the Lagrangian are used to ab-sorb the divergences generated by L(2) at one-loop. It is the renormalized running4

couplings Lri(µ) that are physical and can be related to observables. A convenientrenormalization scheme has been defined in the literature,9 and dimensional regular-ization is typically used.

Once again, since we are only interested in terms of enhancedelectroweak strength, there is no need to renormalize the electroweak gauge sector.This is the reason why we do not need custodial SU(2) breaking counterterms like∆ρ (or “T”).Some of the anomalous couplings in Ref. 1a, namely those with λV , are not presentin the effective Lagrangian at order E4.

These terms appear at the next order, E6,and are thus expected to produce much smaller effects (suppressed by ∼s/Λ2) thanthe κV terms. They are of the same order as the slope terms introduced when κV ismodified with a form factor.

Within our assumptions, these terms in Eq. (1), shouldhave been normalized by Λ2 instead of M2W.

The energy expansion breaks down atsome scale near 2 TeV, where all the terms become equally important.We have emphasized that the Lri (µ) couplings are naturally of order one. A valuemuch larger than 1 of one or more of these couplings, would indicate that the formal-ism is breaking down at much lower energies than it should.

This is associated withthe presence of some new, relatively light, particle beyond the standard model. In ourLagrangian, we have explicitly included all the known particles in the standard modeland we have assumed that any new particles associated with electroweak symmetrybreaking are heavy: of order a few TeV.

The effect of these heavy particles is onlyfelt indirectly through the anomalous couplings. If there are some relatively lightparticles, for example a 300 GeV Higgs boson, then the formalism has to be modi-fied to include the light Higgs explicitly in the Lagrangian.

For this example, thereexists another formulation of the effective Lagrangian that one could use, namelythat in which the symmetry breaking is linearly realized.2 For studies at the SSC itis reasonable to assume that there are no such light particles, since they would bediscovered directly. For studies at lower energy machines like LEP2 this is not thecase.

A 300 GeV Higgs boson would still not be seen directly and the study of itsindirect effects remains interesting. Of course this is not the only possibility, therecould be, for example, a 300 GeV vector resonance.

To study that case at LEP2 onecould use an effective Lagrangian that contains this field explicitly.145

5PhenomenologyThe explicit gauge invariance of Eq. (3) allows us to use the equivalence theorem tosimplify the calculations.

As long as we are only interested in terms of enhanced elec-troweak strength, we can compute with the O(E4) terms presented here, replacingall the vector bosons with their corresponding would-be Goldstone bosons. The onlyexception is for vector bosons in the initial state since these couple to light fermions.For q¯q annihilation we must keep the “initial” vector boson.

For vector boson scatter-ing, the effective luminosity of transverse gauge bosons in the protons is much largerthan that of longitudinal gauge bosons.15 In practice, we find that for energies above∼500 GeV, the longitudinally polarized initial states completely dominate the crosssections.9There are three mechanisms to produce vector boson pairs at hadron colliders.Each of them is sensitive to different anomalous couplings. The largest source of vectorboson pairs is q¯q annihilation.16 This process is sensitive to anomalous three gaugeboson couplings L9L and L9R (also to L10 but not in terms of enhanced electroweakstrength).9,12 The vector boson fusion mechanism is sensitive to all the anomalouscouplings, but only to L1 and L2 at the enhanced electroweak strength level.9,13Finally, gluon fusion is not sensitive to any of the anomalous couplings we havediscussed (to O(E4)), but it is to anomalous couplings of the top-quark gA −1.17 Ithas been argued in the literature that the vector boson fusion process can be separatedexperimentally from the other two by tagging one forward jet.18For our numerical studies we will take Λ = 4πv, in accord with our assumptionthat there are no new particles below a few TeV.One of the couplings of the effective Lagrangian has already been measured.

Afit to all data by Altarelli19 translates into Lr10(µ) = 0.5 ± 1.6 at µ = 1500 GeV. Thisone doesn’t contribute to the processes of interest at the SSC (enhanced electroweakstrength production of VL pairs).

The UA2 collaboration has reported:20 −2.2 ≤κγ −1 ≤2.6. This translates into |L9| <∼900.

This is expected to improve21 by afactor of 2 at the Tevatron. Similar results are expected from LEP2.1 Within ourframework this means that there will not be any significant bounds on L9 beforethe SSC/LHC.

There are no present bounds on the anomalous four-gauge bosoncouplings.We have done a very crude phenomenological analysis, in which we assume thatit is possible to measure the polarization of the vector bosons. We have computed6

the contribution of the anomalous couplings to the integrated cross section for 0.5 ∼2.5.

If weassume that it is not possible to extract the longitudinal polarization, the change inthe rate is always less than a few percent.The W +L W −L channel will be sensitive to a combination of L9L and L9R if L9 <∼−4.0or L9 >∼3.0. Again, this is assuming that all backgrounds can be eliminated andpolarizations measured.The W +L W +L channel is sensitive to a combination of L1 and L2 if |L1| >∼1.

or|L2| >∼1.To understand the possible significance of these numbers it is instructive to com-pare with previous studies on anomalous three gauge boson couplings at the SSC thatconsidered all polarizations. Kane, Vidal and Yuan22 found that the SSC would besensitive to |L9| >∼25 and Falk, Falk and Simmons12 found that the SSC would besensitive to L9L <∼−16 or L9L >∼7 by looking at the WZ channel.

This result isconsistent with our estimate. They also found that the SSC would be sensitive toL9R <∼−119 or L9R >∼113 from the Wγ channel.

In this case the bound is weakerbecause the final state can have at most one longitudinal polarization, so the ampli-tude can only grow as √s. In the WZ channel, the leading term in the amplitudegrows like s.We have argued that if there are no new light particles, the Li should be of orderone and not significantly larger.

This implies that to obtain meaningful bounds onanomalous three gauge boson couplings at the SSC, an effort to separate the transversebackground is necessary. We have not studied the feasibility of this separation, andmore detailed phenomenology is clearly needed.On the other hand, the W +W +channel seems to be a very promising one to place significant bounds on anomalousfour gauge boson couplings.

This channel is particularly useful because it is the onewith the lowest backgrounds, as has been emphasized in the literature.23AcknowledgementsThis work was done in collaboration with J. Bagger and S. Dawson. I thank C.Burgess, S. Willenbrock and D. Zeppenfeld for useful discussions.7

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