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C.P. Burgessa and David Londonb

arXiv:hep-ph/9203215v2 25 Mar 1992March 1992McGill-92/04UdeM-LPN-TH-83Light Spin-One ParticlesImply Gauge InvarianceC.P. Burgessa and David Londonba Physics Department, McGill University3600 University St., Montr´eal, Qu´ebec, CANADA, H3A 2T8.b Laboratoire de Physique Nucl´eaire, Universit´e de Montr´ealC.P.

6128, Montr´eal, Qu´ebec, CANADA, H3C 3J7.AbstractRecently, calculations which consider the implications of anomalous trilinear gauge-bosoncouplings, both at tree-level and in loop-induced processes, have been criticized on thegrounds that the lagrangians employed are not SUL(2) × UY (1)gauge invariant. We provethat, in fact, the general Lorentz-invariant and U(1)em invariant but not SUL(2) × UY (1)invariant action is equivalent to the general lagrangian in which SUL(2) × UY (1) appearsbut is nonlinearly realized.

We demonstrate this equivalence in an explicit calculation, andshow how it is reconciled with loop calculations in which the different formulations can(superficially) appear to give different answers. In this sense any effective theory containinglight spin-one particles is seen to be automatically gauge invariant.1.

IntroductionPerhaps the biggest pothole in the otherwise reasonably well-maintained surface thatemail: cliff@physics.mcgill.ca; london@lps.umontreal.ca1

is high-energy theory is our ignorance of the origin of particle masses. This ignorance ispatched over in the standard model through the introduction of the Higgs couplings, but abetter understanding is expected once shorter distance scales have been probed.

One wayin which the physics underlying the Higgs sector might make itself known in acceleratorcollisions is through the deviations from standard-model predictions it can produce in thecouplings of particles to gauge-boson probes. The couplings of the massive W ± and Z0bosons themselves are particularly interesting in this regard since they directly involve thesymmetry-breaking physics through their longitudinal modes.This type of reasoning has led to considerable effort in outlining the potential formthat the anomalous couplings of these particles to the photon and the Z0 might take, sincethese are the probes that are currently the most cleanly available in collider experiments.Since the experimental success of the standard model up to and beyond the Z0 mass can beinterpreted as saying that the energy scale appropriate to any new physics must be large,the analysis of potential anomalous couplings has focused on the lowest electromagneticand electroweak moments of fermions [1], [2], [3], [4] and gauge bosons [5], [6], [7], [8],[9] that would dominate interactions at low energies.

The natural theoretical frameworkfor this type of analysis is an effective-lagrangian approach [10] in which the influenceof any at-present-unknown new heavy particles is parameterized through the effectivenonrenormalizable interactions that they generate among the lighter particles.Essentially only two ingredients are required to specify such a low-energy effectivelagrangian: the low-energy particle content and the symmetries that their interactionspreserve. Although by and large there is agreement on the low-energy particle content,there are currently two main choices that are made concerning the symmetries that shouldbe required of the low-energy lagrangian.

One school [1], [3], [6], [7], [11] imposes onlythe very minimal conditions of Lorentz-invariance, SO(3, 1), and electromagnetic gaugeinvariance, Uem(1). The alternative procedure [2], [12] is to require invariance with respectto the full electroweak gauge group, SUL(2)×UY (1), but with all but the unbroken Uem(1)subgroup being nonlinearly realized.† In this second framework the unknown symmetry-breaking sector is assumed at low energies to contain only the three Nambu-Goldstonebosons which are eaten by the massive W ± and Z0 particles.

The transformation propertiesof all fields are then determined by general arguments [13] that were developed within theframework of chiral perturbation theory many years ago.The principal goal of this article is to demonstrate the equivalence of these twoschemes. We show that each may be obtained from the other via field redefinitions.

This† A third choice [4], [8], [9] is to linearly realize SUL(2) × UY (1)-invariance by explicitly includingthe standard-model Higgs doublet in the low-energy theory. We do not pursue this option further here.2

demonstration is given in section (2) below. A practical implication of this equivalence is topermit the application of renormalizable gauges to loop calculations in what is nominallynot a gauge-invariant theory.

At a conceptual level it illustrates that spontaneously brokengauge invariance is automatic for any effective theory containing light spin-one particles.Recently, de R´ujula and coworkers have criticized most analyses involving anomaloustrilinear gauge-boson couplings, saying that the lagrangians used are not gauge invariant[9]. Although many of their conclusions are basically correct, the equivalence theoremestablished in this paper shows that the supposed non-gauge invariance of such effectivelagrangians is actually a red herring.

Incorrect conclusions that are based on the non-gauge invariant effective lagrangian really arise from other abuses of the effective-lagrangianformalism. We pursue these related issues in a separate publication [14].Of course, the claimed equivalence only makes sense within the domain of applica-bility of both formulations of the effective theory.

For both approaches this is necessarilyrestricted to energy scales that are not too large compared to the spin-one boson masses,M. For a weak spin-one coupling, g, the maximum applicable scale may be estimatedto be ≃4πM/g in order of magnitude.

At higher energies pathologies such as the fail-ure of perturbative unitarity may be expected, indicating a breakdown of the low-energyapproximation and the appearance of some sort of ‘new physics’.In order to bring out some of the peripheral issues which can confuse this equivalencewe compute the one-loop-induced weak fermion dipole moment that would be generatedby a particular (anapole) anomalous moment in the WWZ interaction. We show thatalthough the equivalence is manifest within a gauge-invariant regularization—such as di-mensional regularization—it is hidden when a cutoffis used (as is frequently done in theliterature).

In this case the induced weak dipole moment can be quadratically or log-arithmically divergent depending on the gauge, or even on the field variables that areemployed.Although some of these points are undoubtedly known to the effective-lagrangiancognicenti, it is evident that they have not percolated out into the wider community whichis now finding applications for these techniques. (This is particularly clear in criticisms [9]of the ‘non-gauge invariant’ formulation discussed above.) For this reason we feel that are-examination of these issues is appropriate here.2.

The Equivalence ResultThere are two natural ways to incorporate spontaneously broken gauge symmetrieswithin a low-energy effective lagrangian:3

• No Gauge Invariance: In the first formulation massive spin-one bosons are representedby vector fields and the lagrangian is only required to be Lorentz invariant. Only invariancewith respect to unbroken gauge symmetries is imposed and all broken gauge symmetriesare simply ignored.• Nonlinearly-Realized Gauge Invariance: The alternative is the second approach in whichboth Lorentz and gauge symmetries are built in from the beginning.

Spontaneous symme-try breaking is incorporated by coupling all fields to a symmetry-breaking sector. All thatis assumed about this sector is that its only light degrees of freedom are the appropriateset of Nambu-Goldstone bosons that are required on general grounds by Goldstone’s theo-rem.

These are, of course, ultimately ‘eaten’ by the gauge bosons via the Higgs mechanismonce the nonlinearly-realized action of the broken-symmetry transformations amongst theNambu-Goldstone bosons is ‘gauged’.We demonstrate in this section a precise form for the equivalence of these two formula-tions for the low-energy lagrangian. Although the arguments can be made quite generally,we restrict ourselves here to establishing this equivalence for two specific cases: a simpli-fied toy model involving a single massive spin-one particle, as well as the realistic caseappropriate to the couplings of the electroweak gauge bosons, W ±, Z0 and the photon, γ.2.1) A Toy ExampleIn order to describe the argument within its simplest context, consider first the cou-pling of a single massive spin-one particle, Vµ, coupled to various forms of spinless orspin-half matter, ψ.

We first state the two alternative forms for the effective lagrangianand then demonstrate their equivalence.• No Gauge Invariance: The lagrangian in the first formulation then takes the form:L1 = L1(Vµ, ψ),(1)in which L1 is a priori an arbitrary local Lorentz-invariant function of the fields Vµ, ψand their spacetime derivatives. Since ψ and Vµ are independent degrees of freedom thequantum theory could be defined in this case by a functional integral of the form:Z1 =Z[dψ] [dVµ] expiZd4x L1(Vµ, ψ).

(2)4

• Nonlinearly Realized Gauge Invariance: The alternative formulation is to consider aU(1) gauge theory with matter fields, χi, carrying U(1) charges qi. The gauge symmetrytransformations acting on these fields and on the gauge potential, Aµ, are the usual ones:χi →eiqiω χi;gAµ →gAµ + ∂µω.

(3)g here is the gauge coupling constant.Symmetry breaking is incorporated by coupling these matter and gauge fields in acompletely general way to a single Nambu-Goldstone boson, ϕ, for a spontaneously brokenU(1). The action of the U(1) on the Nambu-Goldstone bosons may always be chosen totake a standard form [13], which becomes in this caseϕ →ϕ + fω.

(4)f here is the Nambu-Goldstone boson’s decay constant which is of the order of the scale atwhich the U(1) symmetry is spontaneously broken. It is related to the mass of the gaugeboson by the relation M = gf.The most general gauge-invariant low-energy lagrangian may then be written in thefollowing form:L2 = L2(Dµϕ, χ′),(5)in which the redefined field is χ′i ≡e−iqiϕ/f χi and the gauge-covariant derivative for ϕis given by Dµϕ ≡∂µϕ −gfAµ.

Notice that all of the dependence on Aµ in L2 arisesthrough this gauge-covariant derivative. For example, the gauge field strength is given bygfFµν = ∂µDνϕ −∂νDµϕ.The corresponding functional integral defining the quantum theory then has the stan-dard form:Z2 =Z[dχ′i] [dAµ] [dϕ] expiZd4x L2(Dµϕ, χ′)δ[G] DetδGδω,(6)in which the second-to-last term is the functional delta function, δ[G], which enforces thegauge condition G = 0, and the last term is the associated Fadeev-Popov-DeWitt—orghost—functional determinant.It is crucial for the remainder of the argument that both χ′i and Dµϕ are invariant—asopposed to being covariant—with respect to gauge transformations.

As a result even if5

the lagrangian, L2, is only required to be Lorentz invariant it becomes automatically alsogauge invariant.• Equivalence: Now comes the main point. The two lagrangians, L1 and L2, are identicalto one another.

There is a one-to-one correspondence between the terms in each given bythe replacement ψ ↔χ′i and Dµϕ ↔−gf Vµ. This is only possible because both L1 andL2 are constrained only by Lorentz invariance and so any interaction which is allowed forone is equally allowed for the other.More formally, the functional integral of eq.

(2) may be obtained from that of eq. (6) by simply choosing unitary gauge, defined by the condition G ≡ϕ(x), and using thefunctional delta function to perform the integration over ϕ.

The ghost ‘operator’ is in thiscase δG(x)/δω(x′) = f δ4(x −x′) and so the ghost determinant contributes just a trivialfield-independent normalization factor.The integration over the ‘extra’ Nambu-Goldstone degree of freedom of the gauge-invariant theory is thereby seen to be precisely compensated by the freedom to choose agauge.2.2) Applications to the Electroweak BosonsThe argument as applied to a more complicated symmetry-breaking pattern, suchas appears in the electroweak interactions, has essentially the same logic although thetechnical details are slightly more intricate.• No Gauge Invariance: We take for the purposes of illustration the degrees of freedom inthe low-energy effective lagrangian for the electroweak interactions of leptons and quarks.These are: the massless photon, Aµ, the massive weak vector bosons, Wµ and Zµ, andthe usual fermions, ψ. Although other particles such as gluons may also be very simplyincluded we do not do so here for simplicity of notation.

The general lagrangian for thesefields may be written:L1 = L1(Aµ, Wµ, Zµ, ψ),(7)in which L1 is a general local and Lorentz-invariant function whose form is constrainedonly by the requirement of invariance with respect to the unbroken electromagnetic gaugetransformations, Uem(1). All derivatives are taken to be the Uem(1) gauge-covariant deriva-tive, Dµ, which for fermions takes the form Dµψ = ∂µψ −ieQAµψ.

Q here denotes thediagonal matrix of fermion electric charges.6

The quantum theory is given in terms of a functional integral of the formZ1 =Z[dWµ] [dW ∗µ] [dZµ] [dAµ] [dℓi] expiZd4x L1δ [Gem] DetδGemδωem. (8)We next outline the nonlinear realization of SUL(2) × UY (1).• Nonlinearly Realized Gauge Invariance: The first step is to briefly review the formulationfor realizing the symmetry-breaking pattern SUL(2) × UY (1) →Uem(1) nonlinearly [13].Consider, therefore, a collection of matter fields, ψ, on which SUL(2) × UY (1) isrepresented (usually reducibly) by the matrices G = exp[iωa2Ta + iω1Y ].

We choose herea slightly unconventional normalization for the generators Ta and Y , viz tr[TaTb] = 12 δab,tr[TaY ] = 0 and tr[Y 2] = 12. Finally define the matrix-valued scalar field containing theNambu-Goldstone bosons by ξ(x) = exp[iXaϕa(x)/f], in which the three Xa’s representthe spontaneously broken generators X1 = T1, X2 = T2 and X3 = T3 −Y .

X3 here ischosen to be orthogonal to the unbroken generator of Uem(1): Q = T3 + Y .The action of the gauge group SUL(2) × UY (1) on ξ and ψ may be written in thestandard form [13]:ψ →Gψandξ →ξ′,whereG ξ = ξ′ H†. (9)Here H = exp[iQ u(ξ, ξ′, G)] and u = u(ξ, ξ′, G) is implicitly defined by the condition thatξ′ on the right-hand-side of eq.

(9) involves only the broken generators.As was the case for the toy example, for the purposes of constructing the lagrangianit is convenient to define new matter fields, ψ′, according to ψ′ ≡ξ† ψ since this has theSUL(2) × UY (1) transformation rule:ψ′ →ξ′† G ψ= H ψ′. (10)Notice that even for global UY (1) rotations, for which ω1 is constant, u(ξ, ξ′, G) is spacetimedependent because of its dependence on the scalar field ξ(x).The next step is the construction of the general locally SUL(2) × UY (1) invarianteffective lagrangian.To this end consider the auxiliary quantity Dµ(ξ) which may bedefined in terms of ξ and the SUL(2) × UY (1) gauge potentials Wµ = g2W aµ Ta + g1Bµ YbyDµ(ξ) ≡ξ†∂µξ −iξ†Wµξ.

(11)7

In terms of this quantity it is possible to construct fields which transform in a simple waywith respect to SUL(2) × UY (1). Together with their transformation rules these are,e Aµ ≡i tr[QDµ(ξ)],eAµ →eAµ + ∂µu;(12)qg21 + g22 Zµ ≡2i tr[(T3 −Y )Dµ(ξ)],Zµ →Zµ;(13)g2 W±µ ≡i√2 tr[T∓Dµ(ξ)],W±µ →e±iuQ W±µ .

(14)T± is defined as usual to be T1 ± iT2. The first of these fields, Aµ(ξ), transforms in sucha way as to permit the construction of a covariant derivative for the local transformationsas realized on ψ′:Dµψ′ ≡(∂µ −ieAµ Q) ψ′.

(15)The main point to be appreciated here is that all of the fields ψ′, Dµψ′, Aµ(ξ),Zµ(ξ) and W±µ (ξ) transform purely electromagnetically under arbitrary SUL(2) × UY (1)transformations.This ensures that once the lagrangian is constructed to be invariantunder the unbroken group, Uem(1), it is automatically invariant with respect to the fullnonlinearly-realized group SUL(2) × UY (1).With these transformation rules the most general SUL(2)×UY (1)-invariant lagrangianbecomesL2 = L2(Aµ, Wµ, Zµ, ψ′)(16)with L2 restricted only by the unbroken Uem(1) gauge invariance. The functional integralwhich defines the quantum theory may then be writtenZ2 =Z[dWµ] [dξ] [dψ′] expiZd4x L2δ [Ga] DetδGaδωb.

(17)Four gauge conditions, Ga = 0, a = 1, ...4, are required—one for each generator of SUL(2)×UY (1).• Equivalence: The demonstration of the equivalence between eqs. (8) and (17) proceedsalong lines that are similar to those used in the abelian toy example presented previ-ously.

As was the case in this earlier example, the equivalence works term-by-term in thelagrangian. The correspondence between the field variables isAµ ↔Aµ,Zµ ↔Zµ,W±µ ↔W ±µ ,ψ′ ↔ψ.

(18)8

The equivalence is explicit in unitary gauge, which is defined in this case by thecondition ϕa(x) ≡0, or equivalently ξ(x) ≡1, throughout spacetime. As is seen fromthe transformation rules of eq.

(9) this condition does not completely fix the gauge. It ispreserved by the unbroken electromagnetic transformations which satisfy G = H = eiωem.In this gauge the relations for Zµ, Wµ and ψ indicated in eqs.

(18) above simply becomeequalities.More formally, using the unitary gauge-condition to perform the functional integralover ξ in eq. (17), gives the resultZ2 =Z[dWµ] [dψ] expiZd4x L2δ [Gem] DetδGemδωemDetδϕaδωbϕ=0.

(19)Since L2(ξ = 1) = L1 this clearly agrees with eq. (8) apart from the final Fadeev-Popov-DeWitt ghost determinant that is associated with the choice of unitary gaugeδϕa(x)/δωb(x′) ≡∆ab(x) δ4(x −x′).

(20)The final point is that the identity Det ≡exp Tr Log may be used to rewrite thisdeterminant as the exponential of a local, Lorentz- and Uem(1)-invariant function. As suchit may be considered as a shift in the parameters appearing in the original lagrangian, L2.Furthermore, since its contribution to L2 is proportional to δ4(x = 0) its coefficients areultraviolet divergent and so their contribution may be absorbed into the renormalizationsthat are anyhow required in defining the functional integral of eq.

(19). At a practicallevel, the Fadeev-Popov determinant does not in any case arise until at least two-looporder.3.

An Illustrative CalculationIn order to illustrate explicitly the equivalence of the two formulations, we will computethe CP-violating ‘weak dipole moment’ [1] (which we denote by Zdm) of the τ lepton,Lzdm = −iz τ γ5σµντ ∂µZν,(21)that is induced at one loop by an anomalous WWZ vertex. We consider for these purposesthe following CP-violating anomalous anapole coupling such as appears in the non-gauge9

invariant formulation of Hagiwara et.al. ref.

[5]:†La = −a W ∗µWν (∂µZν + ∂νZµ) . (22)We may translate this effective interaction into a form in which the gauge invarianceis nonlinearly realized using the general correspondence of the previous section.

The resultis to simply make the substitutions of eqs. (18) in eq.

(22).In order to illustrate the equivalence of these two formulations we next compute theZdm using the anapole vertex as derived from interaction (22) before and after makingthe substitution (18).3.1) Unitary Gauge CalculationIn the non-gauge-invariant formulation the anapole vertex of Fig. 1 is represented bythe following Feynman ruleakβgµα + kαgµβ,(23)and the gauge bosons propagate with the usual massive vector-boson propagatorGµνU (k) = −i P µν(k)k2 −M 2WwithP µν(k) = gµν −kµkνM 2W.

(24)The expression for z may then be read from the amplitude (see Fig. 2)T µ = −ag2w2Zdnq(2π)n1Dkβgµα + kαgµβPαρ(q + p2) Pβσ(q −p1)uτ (p2) γρ/qγσγL vτ (p1) ,(25)D here represents the denominators of the propagators that appear in the graphD = (q2 −m2ντ )(q + p2)2 −M 2W (q −p1)2 −M 2W.

(26)†The coefficient ‘a’ in this equation corresponds to gZ4 of ref. [5].10

Since this amplitude diverges we regularize the integral by working in n ̸= 4 dimensions.We will return to the issue of regularization later in this section. The divergent part maybe explicitly evaluated to beT µ = −ag2w384π2mτm2τ −m2ντM 4Wuτ (p2) σµνkνγ5vτ (p1)24 −n,(27)and may be absorbed by renormalizing the coefficient z of the Zdm operator of eq.

(21).This determines how these operators mix due to renormalization. In the minimal subtrac-tion scheme we therefore find:z(µ) = z(µ′) +g2w384π2mτm2τ −m2ντM 4Wa(µ′) logµ2µ′2.

(28)3.2) Renormalizable-Gauge CalculationThe same calculation may be performed in a general gauge using the Feynman rulesappropriate to the effective lagrangian with nonlinearly-realized gauge invariance. Theprincipal difference here is that there are now four diagrams – that of Fig.

2, and thosein which one or both of the W ±’s is replaced by the corresponding would-be-Goldstoneboson (WBGB), ϕ±.In the standard family of covariant renormalizable gauges parameterized by the vari-able α the ϕ±-scalar and W ±-boson propagators are respectively given byG(α)(k) =ik2 −αM 2W(29)andGµν(α)(k) = −i1k2 −M 2gµν + (α −1)kµkνk2 −αM 2W= GµνU (k) −kµkνM 2WG(α)(k). (30)As is clear from the expansion of Wµ(ξ) in terms of powers of fields:W±µ = g2W ±µ +1MW∂µϕ± + · · ·,(31)the Feynman rule for the emission of a WBGB, w, of four-momentum kµ from the anapolevertex is found by simply contracting the result for the emission of the corresponding11

gauge particle—i.e. that of eq.

(23)—by kµ/MW . The same is true for the emission of aWBGB by a fermion line.

As may be easily verified these are precisely the vertices thatare required to preserve the α-independence of tree level amplitudes.From these Feynman rules it is immediately clear that the sum of the four graphs thatcontribute in the renormalizable gauges precisely corresponds to the four terms that wouldbe obtained by substituting eq. (30) into the unitary-gauge result of eq.

(25). This demon-strates the equivalence of the induced Zdm as computed with the non-gauge-invariant andthe nonlinearly-realized gauge-invariant formulations.Notice that this equivalence has relied on the WBGB’s having derivative couplings tofermions as well as to the anapole vertex.

Such couplings are an automatic consequence ofthe replacement (18) in the nonlinearly-realized effective lagrangian. They differ superfi-cially from those that appear in the standard model, however, where the WBGB’s coupleto fermions via renormalizable Yukawa couplings.This difference is irrelevant becauseone set of couplings may be changed into the other by performing an appropriate fieldredefinition, which cannot alter any scattering amplitudes.

It is in fact straightforward tocheck that use of these Yukawa couplings in the previous calculation does not at all alterour conclusions.3.3) Related Red HerringsThis equivalence as outlined appears to be so simple as to be almost trivial. It is there-fore worth outlining some circumstances which can act, and have acted in the literature,to obscure this conclusion.The main obstacle to understanding this equivalence is the widespread use of cutoffs toregularize the divergent integrals that arise in loop-level effective-lagrangian applications.For the present purposes an uncritical use of cutoffs can cause confusion in two distinctways.

At a purely technical level they can hide the transformation properties of the theoryunder field redefinitions in general, and gauge transformations in particular, and so cangive the impression of obtaining differing results in different gauges. Cutoffs also introducea more conceptual difficulty once an attempt is made to associate a physical interpretationwith the cutoff-dependence of a given amplitude.

We speak briefly to each of these issuesin the following paragraphs.At the technical level, it is notoriously easy to inadvertently break gauge-invariancewith a cutoffregularization. One way to see this is to implement the cutoffin the effectivetheory by adding higher-derivative kinetic terms to the lagrangian.

This has the effectof multiplying each propagator by a form factor which separately implements the cutoff12

on each internal line of any graph and ensures, for example, that the cutoffresult isindependent of extraneous issues such as how momentum is routed through the graph.Considered this way, however, it is clear that higher-derivative terms cannot be gaugeinvariant unless the derivatives used are gauge covariant.Gauge covariant derivativesnecessarily imply additional cutoff-dependent interaction terms, however, whose effectsare easily missed if cutoffs are simply applied a posteriori to loop integrals.A related issue concerns the behaviour of cutoff-regulated amplitudes under field re-definitions. For instance, in the example considered above it is superficially possible tochange the divergent behaviour of the result simply by performing a field redefinition.This may be seen by comparing the result of evaluating the given graph using two kindsof WBGB–fermion couplings: on the one hand using the derivative WBGB–fermion cou-plings which come from the general substitution (18), and on the other hand using thestandard-model Yukawa-type couplings between these particles.In order to see these difficulties explicitly consider using the following form factorregularization in the one-loop-generated Zdm−Λ2q2 −Λ2−Λ2(q + p2)2 −Λ2−Λ2(q −p1)2 −Λ2.

(32)Using this regularization together with the derivatively-coupled fermion–WBGB vertexone finds the following quadratic divergenceT µ = −ag2w2304π2Λ2M 4Wmτ uτ (p2) σµνkνγ5vτ (p1) . (33)This result holds for both the unitary-gauge and the α-gauge calculations.Performing the same calculation using Yukawa-type WBGB–fermion vertices in α-gauge gives instead only linear and logarithmic divergences.

These arise only from thegraph in which both vector bosons in Fig. 2 are replaced by WBGB’s.

The result fromthis graph isT µ = −ag2w2M 4WZd4q(2π)41D [2qµq · k −q · k (p1 −p2)µ]uτ (p2)/qm2τγR + m2ντ γL−mτm2ντvτ (p1) . (34)Regularizing using eq.

(32) as before, we findT µ = −ag2w384π2mτm2τ −m2ντM 4Wln Λ2M 2Wuτ (p2) σµνkνγ5vτ (p1) ,(35)13

which is only logarithmically divergent, as advertised.The problem here is that these two kinds of Feynman rules for the fermion–WBGB ver-tex may be obtained from one another by performing a WBGB-dependent nonlinear fieldredefinition on the fermion fields. The answer would be unchanged if the higher-derivativeterm which implements the cutoffwere also transformed, since this transformation wouldintroduce new cutoff-dependent fermion–WBGB interactions.

Of course, this is not whatwas compared between eqs. (33) and (35).There are two lessons to be learned from this example.

The first is that it is very simpleto miss contributions when performing field redefinitions on cutoff-regulated quantities.More important, however, is the realization that the cutoffdependence of an amplitudein an effective theory is not necessarily simply related to its dependence on the heavymass scales that appear within whatever short-distance physics generates that effectivelagrangian. Since cutoffs are frequently used to estimate the scale of new physics whichmight be probed in proposed experiments, we will deal with this issue in more depth ina separate publication [14].

It suffices here to remark that the connection between cutoffsand the scale of new physics is completely unrelated to how gauge-invariance is realized inthe effective lagrangian. Furthermore, we repeat that superficial gauge-variance of cutoff-regulated results can usually be traced to the non-invariance of the regularization – andnot to the lagrangian itself.4.

ConclusionsEffective lagrangians are the natural way to parameterize the effects of the new physicsthat is ultimately responsible for the breaking of the electroweak gauge group.If onedoes not wish to explicitly include a Higgs scalar in the low-energy theory, there are twoprincipal candidates for such an effective lagrangian – one which requires only Uem(1)gauge invariance, but not SUL(2) × UY (1) gauge invariance, and one which imposes thefull SUL(2) × UY (1) gauge invariance, nonlinearly realized. We have demonstrated theequivalence of these two lagrangians.The same arguments as are used here may be similarly used to prove this equivalencefor more general symmetry-breaking patterns G →H.This shows that any effectivetheory containing light spin-one particles automatically has a (spontaneously broken) gaugeinvariance.

Alternatively, one can say that at low energies there is little to choose betweena spontaneously-broken gauge invariance and no gauge invariance at all. It also shows thatcriticisms of effective lagrangians based on the absence of gauge invariance are actuallyred herrings.

Problems with these lagrangians tend to arise for other reasons, such as thecareless use of cutoffs to regularize loop diagrams.14

At a practical level this equivalence has the advantage that it allows the use of thetechniques of renormalizable gauges for calculations in what is nominally not a gauge-invariant theory. This is useful when powercounting arguments are being used in thatall propagators explicitly vary like 1/p2 for large four-momenta.

As a simple example,this equivalence provides an extremely easy way to see why QED remains renormalizableeven after it is supplemented by a photon mass term while a nonabelian gauge theory likethe standard model does not. The difference may be most easily seen in the version ofthese theories in which the WBGB’s are explicit.

It arises because although it is possibleto construct an invariant power-counting renormalizable lagrangian for a U(1) WBGB –simply its kinetic term −12 DµϕDµϕ – such a term is not possible for a nonabelian symmetrygroup. This is because the kinetic terms are in this case not by themselves invariant withrespect to the nonlinearly-realized symmetries.AcknowledgmentsD.L.

thanks F. del Aguila for the hospitality of the University of Granada, where partof this work was done. Many thanks also to Fawzi Boudjema, Steven Godfrey, Yossi Nir,Santi Peris, and Xerxes Tata and German Valencia for helpful criticism, and to MarkusLuty for pointing out an error in an earlier draft.

This research was partially funded byfunds from the N.S.E.R.C. of Canada and les Fonds F.C.A.R.

du Qu´ebec.15

Figure Captions• Figure 1: The Feynman rule for the CP-violating anomalous gauge-boson vertex dis-cussed in the text. All momenta are outgoing.• Figure 2: The Feynman graph through which the anomalous gauge-boson vertex con-tributes to fermion weak dipole moments.16

5. References[1] W. Bernreuther, U. L¨ow, J.P. Ma and O. Nachtmann, Zeit.

Phys. C43 (1989) 117.

[2] R.D. Peccei, S. Peris and X. Zhang, Nucl.

Phys. B349 (1991) 305.

[3] G. Valencia and A. Soni, Phys. Lett.

263B (1991) 517. [4] W. Buchm¨uller and D. Wyler, Nucl.

Phys. B268 (1986) 621; J. Anglin, C.P.

Burgess,H. de Guise, C. Mangin and J.A.

Robinson, Phys. Rev.

D43 (1991) 703; C.P. Burgessand J.A.

Robinson, Int. J. Mod.

Phys. A6 (1991) 2707; A.

De R´ujula, M.B. Gavela, O.P`ene and F.J. Vegas, Nucl.

Phys. B357 (1991) 311; M. Traseira and F.J. Vegas, Phys.Lett.

262B (1991) 12. [5] K.J.F.

Gaemers and G.J. Gounaris, Zeit.

Phys. C1 (1979) 259; K. Hagiwara, R.D.Peccei, D. Zeppenfeld and K. Hikasa, Nucl.

Phys. B282 (1987) 253.

[6] C.L. Bilchak and J.D.

Stroughair, Phys. Rev.

D30 (1984) 1881; J.A. Robinson andT.G.

Rizzo, Phys. Rev.

D33 (1986) 2608; S.-C. Lee and W.-C. Su, Phys. Rev.

D38(1988) 2305; Phys. Lett.

205B (1988) 569; Phys. Lett.

212B (1988) 113; Phys. Lett.214B (1988) 276; C.-H. Chang and S.-C. Lee, Phys.

Rev. D37 (1988) 101; D. Zeppenfeldand S. Willenbrock, Phys.

Rev. D37 (1988) 1775; U. Baur and D. Zeppenfeld, Nucl.Phys.

B308 (1988) 127; Phys. Lett.

201B (1988) 383; Nucl. Phys.

B325 (1989) 253;Phys. Rev.

D41 (1990) 1476; G. Couture, S. Godfrey and P. Kalyniak, Phys. Rev.

D39(1989) 3239; K. Hagiwara, J. Woodside and D. Zeppenfeld, Phys. Rev.

D41 (1990) 2113;V. Barger and T. Han, Phys. Lett.

241B (1990) 127; E.N. Argyres, O. Korakianitis, C.G.Papadopoulos and W.J.

Stirling, Phys. Lett.

259B (1991) 195; E.N. Argyres and C.G.Papadopoulos, Phys.

Lett. 263B (1991) 298; S. Godfrey, Carleton preprint OCIP-C-91-2;S. Godfrey and H. Konig, Carleton preprint OCIP-C-91-5; G. Couture, S. Godfrey and R.Lewis, Phys.

Rev. D45 (1992) 777.

[7] S.J. Brodsky and J.D.

Sullivan, Phys. Rev.

156 (1967) 1644; F. Herzog, Phys. Lett.148B (1984) 355; Phys.

Lett. 155B (1985) 468E; M. Suzuki, Phys.

Lett. 153B (1985)289; J.C. Wallet, Phys.

Rev. D32 (1985) 813; A. Grau and J.A.

Grifols, Phys. Lett.

154B(1985) 283; Phys. Lett.

166B (1986) 233; J.J. van der Bij, Phys. Rev.

D35 (1987) 1088;F. Hoogeveen, Max Planck Inst. preprint MPI.PAE/PTh 25/87 (1987), unpublished.

W.J.Marciano and A. Queijeiro, Phys. Rev.

D33 (1986) 3449; G.L. Kane, J. Vidal and C.-P.Yuan, Phys.

Rev. D39 (1989) 2617; G. B´elanger, F. Boudjema and D. London, Phys.Rev.

Lett. 665 (1990) 2943; F. Boudjema, K. Hagiwara, C. Hamzaoui and K. Numata,17

Phys. Rev.

D43 (1991) 2223. [8] D. Atwood, C.P.

Burgess, C. Hamzaoui, B. Irwin and J.A. Robinson, Phys.

Rev. D42(1990) 3770; F. Boudjema, C.P.

Burgess, C. Hamzaoui and J.A. Robinson, Phys.

Rev.D43 (1991) 3683. [9] A. de R´ujula, M.B.

Gavela, P. Hernandez and E. Mass´o, CERN preprint CERN-Th.6272/91, 1991. [10] S. Weinberg, Physica 96A (1979) 327; J. Polchinski, Nucl.

Phys. B231 (1984) 269;H. Georgi, Weak Interactions and Modern Particle Theory (Benjamin/Cummings MenloPark, 1984); C.P.

Burgess and J.A. Robinson, in BNL Summer Study on CP Violation S.Dawson and A. Soni editors, (World Scientific, Singapore, 1991).[11]R.

K¨ogerler and D. Schildknecht, CERN preprint CERN-TH.3231 (1982), unpub-lished; D. Schildknecht, in ”Electroweak effects at high energies”, ed.N.B. Newman(Plenum, New York, 1985) p. 551.; M. Kuroda, D. Schildknecht and K.-H. Schwarzer,Nucl.

Phys. B261 (1985) 432; J. Maalampi, D. Schildknecht, and K.-H. Schwarzer, Phys.Lett.

166B (1986) 361; H. Neufeld, J.D. Stroughair and D. Schildknecht, Phys.

Lett.198B (1987) 563; M. Kuroda, J. Maalampi, D. Schildknecht and K.-H. Schwarzer, Phys.Lett. 190B (1987) 217; Nucl.

Phys. B284 (1987) 271; M. Kuroda, F.M.

Renard and D.Schildknecht, Phys. Lett.

183B (1987) 366; C. Bilchak, M. Kuroda and D. Schildknecht,Nucl. Phys.

B299 (1988) 7; Y. Nir, Phys. Lett.

209B (1988) 523; J.A. Grifols, S. Perisand J. Sol`a, Phys.

Lett. 197B (1987) 437; Int.

J. Mod. Phys.

A3 (1988) 225; R. Alcorta,J.A. Grifols and S. Peris, Mod.

Phys. Lett.

A2, (1987) 23; C. Bilchak and J.D. Stroughair,Phys.

Rev. D41 (1990) 2233; H. Schlereth, Phys.

Lett. 256B (1991) 267.

[12] J.M. Cornwall, D.N.

Levin and G. Tiktopoulos, Phys. Rev.

D10 (1974) 1145; B.W.Lee, C. Quigg and H. Thacker, Phys. Rev.

D16 (1977) 1519; M. Veltman, Acta. Phys.Pol.

B8 (1977) 475; M.S. Chanowitz and M.K.

Gaillard, Nucl. Phys.

B261 (1985) 379;M.S. Chanowitz, M. Golden and H. Georgi, Phys.

Rev. D36 (1987) 1490; M.S.

Chanowitz,Ann. Rev.

Nucl. Part.

Sci. 38, (1988) 323; R.D.

Peccei and X. Zhang, Nucl. Phys.

B337(1990) 269; B. Holdom and J. Terning, Phys. Lett.

247B (1990) 88; J. Bagger, S. Dawsonand G. Valencia, preprint BNL-45782; M. Golden and L. Randall, Nucl. Phys.

B361(1991) 3; B. Holdom, Phys. Lett.

259B (1991) 329; A. Dobado, M.J. Herrero and D.Espriu, Phys. Lett.

255B (1991) 405; R.D. Peccei and S. Peris, Phys.

Rev. D44 (1991)809; A. Dobado and M.J. Herrero, preprint CERN-TH-6272/91.18

[13] C. Callan, S. Coleman, J. Wess and B. Zumino, Phys. Rev.

177 (1969) 2239; Phys.Rev. 177 (1969) 2247; J. Gasser and H. Leutwyler, Ann.

Phys. (NY) 158 (1984) 142.

[14] C.P. Burgess and David London, preprint McGill-92/05, UdeM-LPN-TH-84.19

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