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Anomalous Gauge Boson Couplings†

arXiv:hep-ph/9209222v1 9 Sep 1992September 1992UdeM-LPN-TH-105McGill-92/39Loops, Cutoffs andAnomalous Gauge Boson Couplings†David Londona and C.P. Burgessba Laboratoire de Physique Nucl´eaire, Universit´e de Montr´ealC.P.

6128, Montr´eal, Qu´ebec, CANADA, H3C 3J7.b Physics Department, McGill University3600 University St., Montr´eal, Qu´ebec, CANADA, H3A 2T8.AbstractWe discuss several issues regarding analyses which use loop calculations to put constraintson anomalous trilinear gauge boson couplings (TGC’s). Many such analyses give far toostringent bounds.

This is independent of questions of gauge invariance, contrary to therecent claims of de R´ujula et. al., since the lagrangians used in these calculations are gaugeinvariant, but the SU(2)L × U(1)Y symmetry is nonlinearly realized.

The real source ofthe problem is the incorrect use of cutoffs – the cutoffdependence of a loop integral doesnot necessarily reflect the true dependence on the heavy physics scale M. If done carefully,one finds that the constraints on anomalous TGC’s are much weaker. We also compareeffective lagrangians in which SU(2)L × U(1)Y is realized linearly and nonlinearly, anddiscuss the role of custodial SU(2) in each formulation.†Invited talk presented by David London at the XXVI International Conference onHigh Energy Physics, Dallas, USA, August 19921

Although the standard model of the weak and electromagnetic interactions has beenextremely successful in explaining all experimental results to date, the gauge structure ofthe theory has not yet been tested. This task will be accomplished in the coming years insuch experiments as LEP200 and the TeVatron, where the three-gauge-boson self couplingswill be directly measured.

Of course, it is hoped that new physics will be observed. Withthis in mind, these experiments will search for, among other things, anomalous trilineargauge boson couplings (TGC’s) not found in the standard model.If one wants to parametrize new physics such as anomalous TGC’s, the easiest way todo so is to use an effective lagrangian.

In order to define the low-energy effective lagrangianwhich parametrizes the new physics, it is necessary only to specify the particle content andthe symmetries of the low-energy theory. In dealing with anomalous TGC’s, one basicallyhas three choices:1.

Linearly Realized SU(2)L×U(1)Y : Here, symmetry breaking is accomplished throughthe Higgs mechanism, and the low-energy theory includes the standard model Higgsdoublet.2. Nonlinearly Realized SU(2)L × U(1)Y : In these effective lagrangians, the symmetrybreaking mechanism is unspecified – the low-energy theory contains only those pseudo-Nambu-Goldstone bosons which are eaten to give mass to the W’s and Z’s.

Theseare also known as “chiral lagrangians”.3. Only U(1)em Gauge Invariance: In this case, the low-energy effective lagrangian isrequired only to obey electromagnetic gauge invariance.

Massive W’s and Z’s andtheir interactions are simply put in by hand.At this point we would like to make a comment regarding the first choice above. It is,in fact, a very strong assumption to assume that, even in the presence of new physics whichgives rise to anomalous TGC’s, SU(2)L×U(1)Y is still broken to U(1)em via a single Higgsdoublet.

After all, given that there is new physics, the method of symmetry breaking mightbe quite different from that of the standard model. Furthermore, anomalous TGC’s ofteninvolve the longitudinal components of the gauge bosons, which are intimately connectedto the symmetry breaking mechanism.Most conclusions based upon linearly realizedSU(2)L×U(1)Y could be altered if one changes, for example, the Higgs content.

Of course,this does not imply that it is incorrect to use an effective lagrangian based on linearlyrealized SU(2)L × U(1)Y . However, it is more conservative to use nonlinearly realizedSU(2)L × U(1)Y .

We will briefly return later to a comparison of the two formulations.Up to now, most analyses which concern themselves with anomalous TGC’s use thethird option – the effective lagrangian obeys only electromagnetic gauge invariance. That2

is,L ∼M 2W W †µW µ + 12M 2ZZµZµ + iκV W †µWνV µν+ iλVM 2W †λµW µνV νλ −gZ4 W †µWν (DµZν + DνZµ) + ...(1)Here, we have written only a subset of the possible terms, including explicit masses for theW and Z, and several triple gauge boson vertices [1]. In the above, V µ represents eitherthe photon or the Z, W µ is the W −field, Dµ is the electromagnetic covariant derivative,and Wµν = DµWν −DνWµ (and similarly for Vµν).

The gZ4 term is a CP violating TGCcalled (for obscure reasons) the anapole coupling, which we will use in our examples below.Now, given that these anomalous three-gauge-boson vertices will not be measured forseveral years, it is reasonable to ask whether limits can be placed on them using currentdata.Obviously the only constraints can come through contributions to loop-inducedprocesses, and a number of papers have considered this possibility [2]. Let us illustrate atypical such calculation.Consider the CP violating anapole coupling gZ4 defined in Eq.

(1) above. This willcontribute at one loop to the W- and Z-masses.

However, because it is CP violating, therewill be a nonzero result only if this anomalous TGC appears at both vertices in the loopdiagrams, as in Fig. 2.

Now, the anapole coupling is nonrenormalizable, and hence thediagrams diverge. The standard way to regularize the divergent loop integrals is to simplyinsert a cutoff.

If this is done, then one finds (keeping only the highest divergence)δπW Wq2= −gZ426π2Λ6M 2W M 2ZδπZZq2= −gZ42144π2Λ4M 4Wq2. (2)Since the values of δπW W and δπZZ at q2 = 0 are unequal, there will be a nonzerocontribution to the deviation of the ρ-parameter from unity, often parametrized using theT-parameter of Peskin and Takeuchi [3].

Taking Λ to be the scale of new physics, say 1TeV, and using the present limit of |T| < 0.8 [4], one obtains a very stringent constrainton gZ4 :gZ4 < 3.5 × 10−41 TeVΛ3. (3)Intuitively, this result seems suspect.

After all, the anomalous coupling contributesonly at one loop, and the quantity which is used to constrain gZ4 , the ρ-parameter, has3

only been measured to a precision of a couple of percent or so. It’s not as if this limitcomes from an extremely well-measured process such as µ →eγ, for example.In fact, there is a recent paper by de R´ujula et.

al. [5], in which they claim that thisconstraint is a considerable overestimate (and similarly for calculations which predict largemeasurable effects in loop-induced processes involving anomalous TGC’s).

The reason,they say, is that the lagrangian used (Eq. (1)) is not SU(2)L × U(1)Y gauge invariant.This claim is only partly correct.

It is true that the result in Eq. (3) is an overestimate,for reasons we will explain later.However, the reason has nothing whatsoever to dowith gauge invariance.The point is that the lagrangian in Eq.

(1), which obeys onlyelectromagnetic gauge invariance, is equivalent, term by term, to a chiral lagrangian inwhich SU(2)L × U(1)Y is present but nonlinearly realized [6].Briefly, the proof goes as follows. To construct a lagrangian which contains a nonlinearrealization of SU(2)L × U(1)Y , broken to U(1)em, one introduces the matrix-valued scalarfield ξ(x) ≡exp [iXaφa/f], in which the φa are the Nambu-Goldstone bosons, and the Xaare the broken generators.

(The afficionados may remark that, as we have written it, ξ(x)respects custodial SU(2), since there is a common decay constant f for each of the φa’s.However, we could equally well have ignored this symmetry by writing fa – the proof isindependent of the existence of a custodial SU(2).) With ξ(x), one can define a nonlinear“covariant derivative”Dµ(ξ) ≡ξ†∂µξ −iξ†Wµξ,(4)in which Wµ = g2W aµ Ta + g1Bµ Y .

In terms of Dµ(ξ), one constructs the three fieldse Aµ ≡i tr[QDµ(ξ)],qg21 + g22 Zµ ≡2i tr[(T3 −Y )Dµ(ξ)],g2 W±µ ≡i√2 tr[T∓Dµ(ξ)]. (5)The significance of these three quantities, Aµ, Zµ and Wµ is that, under arbitrary SU(2)L×U(1)Y transformations, they transform purely electromagnetically.

Thus, any lagrangianwhich is constructed using these quantities and which is required to obey electromagneticgauge invariance will automatically obey the full SU(2)L × U(1)Y symmetry. In unitarygauge, these fields reduce to the standard photon, W and Z fields:Aµ ↔Aµ,Zµ ↔Zµ,W±µ ↔W ±µ .

(6)With this construction one sees explicitly that SU(2)L × U(1)Y gauge invariance is auto-matic for any lagrangian which obeys U(1)em gauge invariance. Therefore, the claim that4

Eq. (1) is not gauge invariant is simply a red herring.The real reason that Eq.

(3), and other results like it, are incorrect is due to theimproper use of cutoffs [7]. This can be seen as follows.

Suppose we knew what the fulltheory was at the new physics scale, M, which is much larger than m, the scale of thelight physics. Now let us calculate the effect of the heavy physics on a light particle masssuch as MW , for example.

The contribution one obtains after integrating out the heavyparticles has the following form:δµ2(m, M) = aM 2 + bm2 + cm4M 2 + · · ·(7)The dots represent terms which are of higher order in the small mass ratio m2/M 2, andthe coefficients a, b, c, ... may depend at most logarithmically on this ratio. Note thatthere is no term of the form M 4/m2.

There is an old paper of Weinberg [8] which explainsthat such terms are not allowed since only logarithmic infrared divergences are allowed atzero temperature in four spacetime dimensions.Now suppose we split this calculation up into a “high-energy” piece and a “low-energy”piece by choosing a cutoffΛ such that M ≫Λ ≫m. In this case, the two contributionscan have the formδµ2he(m, Λ, M) = a′M 2 + b′Λ2 + c′Λ4m2 + · · ·δµ2le(m, Λ, M) = b′′Λ2 + c′′Λ4m2 + · · ·(8)Obviously, this is just a reorganization of the full calculation (Eq.

(7)), so we must haveδµ2(m, M) = δµ2le(m, Λ, M) + δµ2he(m, Λ, M). (9)Furthermore, the full result is independent of the cutoffΛ, which implies thata = a′ ,b′ = −b′′ ,c′ = −c′′ ,· · ·(10)In other words, all quadratic and higher dependence on Λ in the low-energy piece of thecalculation is simply cancelled by counterterms coming from the high-energy piece of thecalculation!

Note also that the coefficient b′′ is unrelated to the coefficient a. That is, acalculation of the quadratic divergence in the low-energy theory does not tell you how thefull calculation depends on M 2.5

The point is that there is no physical significance to terms containing the cutoffΛ.Another way of saying this, perhaps more to the point, is that cutoffs do not accuratelytrack the true heavy mass dependence of the full theory.There is one exception to this statement – the case of a logarithmic divergence. Sup-pose that, in the full theory, there were a term of the formδµ2 ∼dm2 logM 2m2.

(11)If one used a cutoff, the high-energy and low-energy contributions would beδµ2he ∼d′m2 logM 2Λ2,δµ2le ∼d′′m2 logΛ2m2. (12)Clearly, cancellation of the Λ-dependence requires a = a′ = a′′.

This is the only casein which the low-energy cutoffdependence accurately reflects the true dependence on theheavy mass M.Before returning to the example of the anapole, let us explicitly demonstrate in a toymodel the fact that the cutoffdependence found in a low-energy loop calculation is ingeneral unrelated to the true heavy mass dependence. Consider a renormalizable modelwith only two scalars: ψ, which has mass M, and φ, of mass m. The potential describingthe interactions of these two scalars can be split up into two pieces, depending on whetherthe potential is odd or even under separate reflections of the fields.

We have U = U+ +U−,withU+ = 12m2φ2 + 12M 2ψ2 + λφ4 + λ′ψ4 + λ′′φ2ψ2,U−= 13hψ3 + 13gψφ3. (13)Note that, in principle, other terms could have been included in U−, but we assume thatthese are the only terms which appear in the lagrangian at some scale, µ0.Now consider integrating out the field ψ, so that the potential is a function of φ only,V (φ), and consider just the V−piece of the potential.

At tree and one-loop levels, thelowest order terms which appear are (see Fig. 2)tree level:V−(φ) = hg381M 6φ9,one-loop level:V−(φ) = −hg92π2φ31 + 2 logM 2µ20.

(14)6

What is important to note here is that the dependence of the φ3 term on the heavy massM is logarithmic.Now suppose that the heavy physics, characterized by scale M, were unknown. Inthis case one assumes the most general form for V−,V−(φ) = aMφ3 + bMφ5 + cM 3φ7 + dM 5φ9 + ...(15)In order to make the connection with the anapole calculation we showed you earlier,consider the contribution of the d-term to the a-term.

In other words, aMφ3 plays therole of the ρ-parameter, while (d/M 5)φ9 acts like an anomalous WWV coupling. Here onefinds a contribution at 3 loops (Fig.

3),δa ∼Λ216π2M 23d. (16)The upshot is that one finds that the cutoffdependence of the φ3 term goes like Λ6.

How-ever, we have already determined the true dependence on the heavy mass to be logarithmic.This demonstrates explicitly that cutoffs do not accurately track the true dependence ofthe full theory on the heavy mass scale. This also demonstrates that the issue of gaugeinvariance is indeed a red herring, since it is clear that same type of problems arise inmodels which contain only scalars.Let us now return to our original example of the anapole contribution to the W- andZ-masses.

Given the problems with cutoffs, how can one extract physically meaningfulbounds on the anapole coupling, given that there is a nonzero contribution to the ρ-parameter? The easiest way to do this (but by no means the only way) is not to use cutoffsat all to regularize the divergent integrals.

Instead, one uses dimensional regularization,supplemented by the decoupling subtraction renormalization scheme.Using dimensional regularization, the divergent pieces of the diagrams in Fig. 1 areδπW Wq2|q2=0 = −gZ424π23M 2W21 + M 2ZM 2W−M 4ZM 4W 2ǫ,δπZZq2|q2=0 = 0,(17)where ǫ = n−4 in n spacetime dimensions.

The key point now is the following. In the mostgeneral effective lagrangian, there will be a term contributing directly to the T-parameter7

(∆ρ). The contribution of Eq.

(17) to ∆ρ renormalizes this direct contribution:αT(µ2) = αTµ′2−38π2gZ4µ′22 1 + M 2ZM 2W−M 4ZM 4Wln µ′2µ2!. (18)This shows how the two operators in the effective lagrangian, T and gZ4 , mix as thelagrangian is renormalized and evolved down from scale µ′ to scale µ (in the absence ofthresholds).

This is a point which seems to have been overlooked in many of the analyseswhich deal with anomalous TGC’s. Even in an effective (nonrenormalizable) lagrangian,the parameters must be renormalized.In general, this requires an infinite number ofcounterterms, but this is of no consequence since the effective lagrangian already containsan infinite number of terms.Note also that there is, in general, a contribution whichdepends quadratically on the heavy mass scale M. This is contained in the initial conditionTM 2, which is, however, incalculable if one does not know the underlying theory.If one assumes no accidental cancellation between the two terms on the right handside of Eq.

(18), one can now use the experimental limit on |T| to constrain the anapolecoupling:gZ4 < 0.24(@ 1 TeV). (19)This is 3 orders of magnitude weaker than the bound found using cutoffs (Eq.

(3))!Before concluding, we would like to return to a subject we briefly touched upon at thebeginning, namely the comparison of conclusions based upon effective lagrangians withlinearly and nonlinearly realized SU(2)L × U(1)Y .One of the terms in Eq. (1) is theelectric quadrupole moment of the W,iλVM 2W †λµW µνV νλ.

(20)In the literature, one often sees statements to the effect that λγ = λZ (modulo cot θW ).The reasons given vary – occasionally gauge invariance or custodial SU(2) symmetry areinvoked, and sometimes this relation is required in order to avoid too large contributionsto well-measured quantities which are in agreement with the standard model. The fact is,none of these reasons is valid – there is no reason, in general, to require λγ = λZ.There are basically two sources of confusion.

First of all, if one calculates the contri-bution to ∆ρ using a cutoff, one finds∆ρ ∼(λγ −λZ) Λ4M 4W. (21)8

This has led some authors to require λγ = λZ in order to avoid large contributions to ∆ρfor large values of the cutoff. However, as we have argued above, this cutoffbehaviour isnot physically meaningful – this type of term is cancelled by a counterterm coming fromthe high-energy theory, and no conclusions as to the relative size of λγ and λZ should bedrawn.A more important, and subtle, source of confusion is that the relation λγ = λZ istrue, to a good approximation, if one uses an effective lagrangian with a linearly realizedSU(2)L × U(1)Y gauge symmetry.

If there is only one Higgs doublet, then it necessarilyfollows thatW3µ = Zµ cos θW + Aµ sin θW . (22)Since the standard model Higgs sector possesses an approximate custodial SU(2) symme-try, one is led quite naturally in this context to λγ = λZ.

Even if one were to add moreHiggs doublets, for example, this relation would continue to be approximately true.On the other hand, and this is where the subtlety arises, if one realizes the symmetrynonlinearly, then it does not necessarily follow that λγ = λZ. Although the symmetrybreaking sector continues to possess an approximate custodial SU(2) symmetry, the rela-tion among W3µ, Zµ and Aµ need not be that found in Eq.

(22) above [9]. In general, thereis no reason, neither gauge invariance nor custodial SU(2), for λγ and λZ to be related.This is the point of this discussion.

In general, an effective lagrangian has an infinitenumber of terms, the coefficients of which are arbitrary and independent. It is possibleto relate some of these coefficients by imposing certain symmetries, or to construct theeffective lagrangian in a special way.

However, the lagrangian thus obtained is less general.This is the case for an effective lagrangian with SU(2)L × U(1)Y realized linearly. Theassumption of the breaking of the symmetry via the Higgs mechanism results in certainrelationships among the parameters of the low-energy effective lagrangian.

These rela-tionships are not present if one makes no assumption about the mechanism of symmetrybreaking, i.e. if one realizes the symmetry nonlinearly.

Again, this is not to say that thelinearly realized effective lagrangian is incorrect; however, it is more constrained than aneffective lagrangian with nonlinearly realized SU(2)L × U(1)Y .To conclude,1. Bounds on anomalous trilinear gauge boson couplings which are obtained from theircontributions to loop diagrams are often significantly overestimated (and similarly forpredictions of large effects in loop-induced processes).2.

This is unrelated to any questions of gauge invariance. The fact is, any lagrangianwhich obeys Lorentz invariance and electromagnetic gauge invariance is equivalent,9

term by term, to a lagrangian in which the SU(2)L × U(1)Y symmetry is present,with the breaking SU(2)L × U(1)Y →U(1)em nonlinearly realized.3. The real source of the problem is the incorrect use of cutoffs in estimating the effect ofheavy masses in the loop diagrams.

In general, the cutoffbehaviour does not properlytrack the true dependence on the heavy mass scale M. (The one exception is the caseof a logarithmic divergence.)4. A more straightforward way to do the calculation is not to use cutoffs at all to reg-ularize the divergent integrals.

Instead one uses dimensional regularization, supple-mented by the decoupling subtraction renormalization scheme, to calculate the effectof anomalous triple-gauge-boson couplings in loops.5. In general, there is no reason to have relationships such as λγ = λZ among the param-eters of the low-energy effective lagrangian – neither gauge invariance nor custodialSU(2) symmetry requires this.

Such relations arise naturally when one uses an effec-tive lagrangian in which the breaking SU(2)L × U(1)Y →U(1)em is linearly realized.However, this is a strong assumption – it is more conservative to use the nonlinearlyrealized version of the effective lagrangian in calculations involving anomalous trilineargauge boson couplings.AcknowledgmentsThis work was supported in part by the Natural Sciences and Engineering ResearchCouncil of Canada, and by FCAR, Qu´ebec.10

1. References[1] The most general trilinear gauge boson couplings are discussed in K.J.F.

Gaemers andG.J. Gounaris, Zeit.

Phys. C1 (1979) 259; K. Hagiwara, R.D.

Peccei, D. Zeppenfeld andK. Hikasa, Nucl.

Phys. B282 (1987) 253.

[2] For references to the literature, see Refs. 5,7.

[3] M.E. Peskin and T. Takeuchi, Phys.

Rev. Lett.

65 (1990) 964; W.J. Marciano and J.L.Rosner, Phys.

Rev. Lett.

65 (1990) 2963; D.C. Kennedy and P. Langacker, Phys. Rev.Lett.

65 (1990) 2967; Phys. Rev.

D44 (1991) 1591. [4] P. Langacker, U. Penn preprint UPR-0492T (1991).

[5] A. de R´ujula, M.B. Gavela, P. Hernandez and E. Mass´o, CERN preprint CERN-Th.6272/91, 1991.

[6] M.S. Chanowitz, M. Golden and H. Georgi, Phys.

Rev. D36 (1987) 1490; C.P.

Burgessand David London, preprint McGill-92/04, UdeM-LPN-TH-83, 1992. [7] C.P.

Burgess and David London, preprint McGill-92/05, UdeM-LPN-TH-84, 1992. [8] S. Weinberg, Phys.

Rev. 140 (1965) B516; in Asymptotic Realms Of Physics (Cam-bridge, 1981).

[9] For an explicit example, see the appendix of Ref. 7.11

FiguresFigure 1. Contribution of the CP violating anapole TGC (blob) to W- and Z-boson propagators.Figure 2.

Diagrams which result in φ3 and φ9 couplings once the heavy field ψ is integratedout. The ψ field is denoted by lines in bold type, while the fine (external) lines representthe φ field.Figure 3.

3-loop contribution of the φ9 coupling to the φ3 coupling.12

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