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March 1992; revised June 1993

arXiv:hep-ph/9203216v3 7 Jun 1993March 1992; revised June 1993McGill-92/05UdeM-LPN-TH-84hepph/9203216Uses and Abuses ofEffective LagrangiansC.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.AbstractMotivated by past and recent analyses we critically re-examine the use of effective la-grangians in the literature to constrain new physics and to determine the ‘physics reach’of future experiments. We demonstrate that many calculations, such as those involvinganomalous trilinear gauge-boson couplings, either considerably overestimate loop-inducedeffects, or give ambiguous answers.

The source of these problems is the use of cutoffs toevaluate the size of such operators in loop diagrams. In contrast to other critics of theseloop estimates, we prove that the inclusion of nonlinearly-realized gauge invariance intothe low-energy lagrangian is irrelevant to this conclusion.We use an explicit exampleusing known multi-Higgs physics above the weak scale to underline these points.

We showhow to draw conclusions regarding the nature of the unknown high-energy physics withoutmaking reference to low-energy cutoffs.email: cliff@physics.mcgill.ca; london@lps.umontreal.ca1

1. IntroductionAs experimentally accessible energies have risen above the thresholds for producingelectroweak gauge bosons it has become more and more clear that the mass scale associatedwith any new physics is probably at significantly higher energies.

This is reflected by thegreat success of the standard model in predicting the results of these experiments in generaland the properties of these gauge bosons in particular.Given that the scale of physics beyond the standard model is well above the weak scale,the low-energy effects of such new physics may be parametrized in terms of an effectivelagrangian [1] in which the influence of any at-present-unknown new heavy particles isfelt through the effective nonrenormalizable interactions that they generate among thelighter particles. These nonstandard interactions may be organized according to increasingoperator dimension.

At a practical level this method is useful only to the extent that itis possible to consider just those few interactions which have the lowest dimension. Thiscan usually be justified by the suppression of higher-dimension operators by extra powersof the inverse of some heavy mass scale, M.This type of reasoning has led to considerable effort in using experimental data to con-strain the coefficients of the operators in such an effective lagrangian which parametrizedeviations from the standard model.

Of particular interest are those terms which corre-spond to anomalous couplings of the photon and the Z0, since these are the probes that arecurrently the most cleanly available in collider experiments. Analyses have focused on thelowest electromagnetic and electroweak moments of the light fermions [2], [3], [4] as well asgauge-boson self-couplings [5], [6], [7], [8], [9], [10], [11] that would dominate interactions atlow energies.

In this way it is possible to ascertain which interactions could have hithertoescaped detection and might yet be detectable at upcoming experiments. Proponents ofparticular experiments can turn this argument around and estimate the scale, M, of newphysics to which a particular proposal can be sensitive—its so-called ‘physics reach’.

Themost interesting proposals are naturally those that are potentially sensitive to the highestscales and so whose physics reach is the longest.So far so good. A complication arises, however, when loop effects in the low-energy the-ory are important for detecting the effective interaction under study.

This is because suchloops are typically divergent and so can depend on positive powers of a large high-energycutoff, Λ.This cutoffphysically describes the maximum energy to which the effectivelagrangian is expected to apply and so is frequently also taken to be of order of the newphysics scale, M. To the extent that this is true the most divergent contributions to agiven amplitude could be taken as indications of a strong dependence on new physics at2

scale M, potentially indicating a long physics reach.Our main point in this paper is to show that the above argument can be very mis-leading, and can even lead to conclusions which contradict general decoupling results [12].At best it gives [10] an ambiguous — and at worst, a false — indication of the scale ofnew physics to which a given experiment may be sensitive, often yielding overly stringentconstraints on parameters in the effective lagrangian. The weak link in the argumentsused is the assumed connection between what can be computed (the cutoffdependenceof amplitudes in the low-energy effective theory) and what is meant to be bounded (thedependence of low-energy amplitudes on physical high-energy physics scales such as heavyparticle masses).In this paper we refine and expand on our results in Ref.

[13], by exploring in de-tail this connection between low-energy cutoffdependence and heavy-mass dependence.We demonstrate our conclusions within the context of a multi-Higgs model in which theinfluence of the high-energy physics is known and calculable. We show that cutoffdepen-dence can be a very poor indicator of heavy-mass dependence, particularly where massivespin-one particles are involved.

We then indicate how to extract the dependence on high-frequency physics without resorting to arguments that rely on cutoffs.In the literature, the misidentification of heavy-mass and cutoffdependence arises mostfrequently in the context of anomalous three-gauge-boson vertices (TGV’s). There are tworeasons for this.

First, since TGV’s cannot yet be measured directly, the only availableinformation concerning them arises indirectly through their contributions to loops. Second,since problems with interpreting cutoffdependence arise most strikingly for loops involvingmassive spin-one particles, TGV-induced loops are very easy to mishandle [10].

This hasled to misleadingly stringent constraints on anomalous TGV’s, as well as to mistakenpredictions of large effects in future experiments – that is to say: long physics reach.In addition to this confusion between cutoffbehaviour and new-physics dependence,the waters have recently become even more muddied due to a parallel confusion thathas arisen within the specific context of TGV analyses. The authors of Ref.

[11] agreethat physics reach as regards anomalous TGV’s is overstated in places in the literature.However, they go on to identify the error as being the gauge invariance (or lack thereof)of the analysis. (An alternative phrasing of this line of thought is to object to the use ofunitary gauge in performing loop calculations.

)The key question is whether the light particles in the effective theory being consideredfill out a linear representation of the gauge group. They do not, for instance, if there is nolight Higgs boson to transform with the longitudinal W and Z bosons.

In this case, gaugeinvariance can only be realized nonlinearly. We contend here that, for gauge symmetries,3

such a nonlinear realization can be included, or not, simply by a change of variables, andso nothing physical can depend on this choice.That this confusion can arise at all serves to underline a more pervasive hazard thatunderlies the association of a physical interpretation to divergences within an effectivelagrangian: the lagrangians themselves, and so also the divergences they contain, are notinvariant under field redefinitions. Conclusions that are based on them are genericallymarked by the same flaw, unless it is specifically demonstrated otherwise (as can be donefor the S-matrix, for example).

Proposals which link cutoffdependence in the lagrangianto heavy-mass dependence are therefore at best ambiguous, unless they are specificallyreferred to a set of variables which are to be used. They are simply wrong if the variablesused are poorly chosen.Some of these points are undoubtedly familiar to some of the effective-lagrangiancognicenti.

They have not, however, been absorbed into the wider community which isnow finding applications for these techniques. We therefore feel that an examination of theissues is timely given the present debate over the accuracy of estimates of physics reach,and over the nature of the properties that should be built into low-energy lagrangians.We next expose all of these points in more detail, with reference to explicit underlyingmodels for which both heavy-mass dependence and cutoffdependence are separately calcu-lable.

We start in section (2) by discussing the relevance of gauge symmetries for effectivelagrangians. In so doing we (re)demonstrate the equivalence between nonlinearly-realizedgauge symmetries and no gauge symmetries at all.

This is followed in section (3) by somegeneral observations about how cutoffdependence arises in low-energy effective theories.Section (4) contains the guts of our criticism. We first present the arguments for thinkingthat cutoffs might track heavy masses, and then criticize these arguments.

We provideseveral examples which indicate how field redefinitions can alter cutoffdependence, andargue which variables are most likely to allow cutoffs to mimic heavy-mass dependence inobservables. In section (5) we outline how to infer heavy-mass dependence without havingto rely on the cutoffdependence of low-energy graphs.

This permits the retention of mostapplications of cutoffmethods, but with the conceptual advantage of relying on a moresolid foundation. Section (6) then presents an explicit multi-Higgs model for underlyingphysics in which these ideas are explicitly worked out.

Our conclusions are summarized insection (7).2. The Pertinence of Gauge SymmetriesEssentially two ingredients are required to specify a low-energy effective lagrangian:the low-energy particle content and the symmetries that their interactions preserve.

Once4

these have been specified, all possible interactions of successively higher dimensions maygenerically be written down.When considering the interactions of W and Z bosons, the most important distinctionto be made concerns where the scale of the unknown new physics, M, lies in relation tothe electroweak scale, v ≃246 GeV. If M is much greater than roughly 4πv, then theperturbative unitarity of the low-energy theory requires that it must linearly realize theelectroweak gauge symmetries [14], [15].

In this case the low-energy theory must containmore particles than have presently been discovered (such as the standard-model Higgsboson and top quark) in order for the known particles to fill out a linear representation ofthe gauge group. This is the choice that has been pursued in Refs.

[4], [9], and [11].We are mostly concerned in what follows with the other alternative in which theunderlying physics we are groping for is the electroweak-breaking physics itself. In thiscase the particle content need not fall into linear representations of the gauge group, andso could in particular consist only of those particles that have already been discovered.Since perturbative unitarity fails in this type of effective theory at energies of order 4πv ≃8πMW/g, we are guaranteed that the effective theory must fail at or before this point.Below this scale, agreement exists in the literature as to the appropriate low-energy particlecontent that is to be chosen, but practictioners divide according to their choices for thesymmetries that these particles should respect:• No Gauge Invariance:In the first approach [2], [5], [6], [7], [8], [10], only electromagnetic gauge invariance isimposed, and all spontaneously broken gauge symmetries are simply ignored.• Nonlinearly-Realized Gauge Invariance: In the alternative framework [3], [16] invariancewith respect to the full electroweak gauge group is required, but with all but the unbrokenUem(1) subgroup being nonlinearly realized.

The physical motivation that underlies thissecond approach is the assumption that the low-energy degrees of freedom of the unknownsymmetry-breaking sector contain only the three Nambu-Goldstone bosons which are eatenby the massive W and Z particles. Given this assumption, the transformation propertiesof all fields are determined by general arguments [17], [18] that were developed within theframework of chiral perturbation theory many years ago.It is the point of this section to (re)demonstrate the equivalence of these last twoschemes.This result is not new, appearing as it does in Refs.

[14] and [18], but thereminder is worthwhile in order to put to rest more recent concerns as to the legitimacy ofignoring the broken electroweak symmetries in the effective lagrangian. The equivalenceis established by explicitly finding a change of variables that relates the two alternatives.5

Although our arguments can be made quite generally, we restrict ourselves here to twospecific cases: a simplified toy model involving a single massive spin-one particle, as wellas the realistic case appropriate to the couplings of the electroweak gauge bosons, W ±, Z0and the photon, γ.2.1) The 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)• 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 [17], which becomes in this caseϕ →ϕ + fω. (4)6

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—as opposed to being covariant—with respect to gauge transformations.

As a result, anyLorentz-invariant lagrangian, such as L2, that is built from these fields becomes gaugeinvariant automatically.• 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.7

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 further con-strained only by unbroken Uem(1)-invariance. All derivatives are taken to be the Uem(1)gauge-covariant derivative, Dµ, which for fermions takes the form Dµψ = ∂µψ −ieQAµψ.Q here denotes the diagonal matrix of fermion electric charges.The quantum theory is given in terms of a functional integral of the formZ1 =Z[dWµ] [dW ∗µ] [dZµ] [dAµ] [dψ] 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 the low-energy interactions of the Nambu-Goldstone bosons for the global symmetry-breaking pattern SUL(2) × UY (1) →Uem(1) [17]. We then promote the symmetry to localgauge transformations.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,[TaY ] = 0 and tr[Y 2] = 12. Finally define the matrix-valued scalar field containing theNambu-Goldstone bosons by ξ(x) = exp[2iXaϕa(x)/v], in which the three Xa’s representthe spontaneously broken generators X1 = T1, X2 = T2 and X3 = aT3 −bY .Herea2 + b2 = 1, and a/b is chosen to ensure that tr[X3Q] = 0, where Q is the unbrokengenerator: Q = bT3 + aY .8

The action of the gauge group SUL(2) × UY (1) on ξ and ψ may be written in thestandard form: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 = gW aµ Ta + g′Bµ YbyDµ(ξ) ≡ξ†∂µξ −iξ†Wµξ.

(11)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 SUL(2)×UY (1) transformation rulesthese are,e Aµ ≡2i tr[QDµ(ξ)],eAµ →eAµ + ∂µu;qg′2 + g2 Zµ ≡2i tr[X3Dµ(ξ)],Zµ →Zµ;(12)g W±µ ≡i√2 tr[T∓Dµ(ξ)],W±µ →e±iuQ W±µ .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′. (13)The main point to be appreciated here is that eqs.

(12) imply that all of the fieldsψ′, Dµψ′, Aµ(ξ), Zµ(ξ) and W±µ (ξ) transform purely electromagnetically under arbitrary9

SUL(2) × UY (1) transformations. This ensures that once the lagrangian is constructed tobe invariant under the unbroken group, Uem(1), it is automatically invariant with respectto the full nonlinearly-realized group SUL(2) × UY (1).With these transformation rules the most general SUL(2)×UY (1)-invariant lagrangianbecomesL2 = L2(Aµ, Wµ, Zµ, ψ′)(14)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. (15)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 (15) 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. (16)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. (16) above simply becomeequalities.More formally, using the unitary-gauge condition to perform the functional integralover ξ in eq.

(15), gives the resultZ2 =Z[dWµ] [dψ] expiZd4x L2δ [Gem] DetδGemδωemDetδϕaδωbϕ=0. (17)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′). (18)10

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.

(17). In practice theFadeev-Popov determinant does not in any case arise until at least two-loop order.The practical benefit of this equivalence is that it allows the use of the most conve-nient gauge for any particular application.

Covariant gauges, such as Feynman gauge, areparticularly useful for making power-counting arguments, since all propagators explicitlyvary like 1/p2 for large four-momenta, and the pathologies of the unitary-gauge propagatorare put into derivative couplings. For instance, this is the simplest way to understand whyQED remains renormalizable once a photon mass term is added, while the same is not truefor a nonabelian gauge theory.

This distinction is most easily seen from the form of theNambu-Goldstone boson couplings. While an invariant renormalizable lagrangian existsfor a U(1) Nambu-Goldstone boson — i.e.

it is simply its kinetic term −12 DµϕDµϕ — thesame is not true for a nonabelian symmetry group. This is because the kinetic terms are inthis case not by themselves invariant with respect to the nonlinearly-realized symmetries.Conversely, unitary gauge has the simplicity of just involving physical particles, allowinga direct identification of the physical significance of the effective interactions.2.3) Derivative vs. Yukawa CouplingsIn this section we wish to make the previous arguments concrete by considering anexplicit one-loop example.

Besides having applications later in the paper, the example alsoserves to bring out three general, but not-so-widely appreciated, features of the equivalencewe have described. These general points are listed at the end of the section.It is a basic feature of the chiral lagrangian described above that all of the would-be Nambu-Goldstone bosons (WBGB’s) couple derivatively to all other fields (and tothemselves).

This expresses a completely generic feature of any Nambu-Goldstone-bosoninteraction, and is easily seen from the expansion of e.g. Wµ(ξ) (c.f.

eqs. (11) and (12))in powers of fields:W±µ = W ±µ −1MW∂µϕ± + · · · .

(19)The second term in this expansion gives a very simple Feynman rule for WBGBcouplings: simply contract the result for the corresponding W ± coupling by ikµ/MW,11

where kµ is the WBGB four-momentum. This is equally true regardless of whether theparticle to which the W ± or ϕ± couples is a scalar, fermion or a gauge boson.Notice that this type of coupling is not the same as what is obtained for the WBGB’sin a covariant gauge in the standard model.

In the Standard Model, for example, theWBGB–fermion interactions do not involve any derivatives at all, since they come fromthe Yukawa couplings to the Higgs multiplet. In the standard model these two formulationsare physically equivalent, since it is possible to pass from one to the other by performingan appropriate field redefinition.

As we shall now see, however, they can and do give riseto different types of divergences in off-shell quantities like the effective lagrangian. Thispoint will become important once we begin trying to track the cutoffdependence of loopsin later sections.Consider then, the following effective interaction:1La = −a ZµW +ν W −µν + W −ν W +µν.

(20)The couplings that this interaction induces for the WBGB’s are found by substitutingW →W and Z →Z from eqs. (12), and expanding the result in powers of fields.

Wechoose to compute the following CP-violating Z–τ–τ vertex (or Zdm):Lzdm = −iz2 τ γ5σµντ Zµν. (21)that this coupling induces at one loop.In the unitary-gauge formulation we must evaluate the graph of Fig.

(1). The Wpropagator that appears in each of the two internal boson lines is:GµνU (k) =−ik2 −M 2Wgµν −kµkνM 2W.

(22)On the other hand, working with the chiral lagrangian in a general covariant gaugeleads not only to to Fig. (1), but also to the three other graphs that are obtained from thisone by replacing each W line by the corresponding WBGB propagator.

In the standardone-parameter-family of covariant gauges, the two types of boson propagators that appear1 This interaction happens to violate CP and corresponds on shell to the interaction denoted gZ4 inRef. [5].12

are:G(η)(k) =ik2 −ηM 2WandGµν(η)(k) =−ik2 −M 2Wgµν + (η −1)kµkνk2 −ηM 2W. (23)The equivalence theorem of the previous sections argues that the unitary-gauge resultequals the sum of the four covariant-gauge graphs.

This is easy to see by using the followingidentity in the unitary-gauge result:GµνU (k) = Gµν(η)(k) + kµkνM 2WG(η)(k). (24)The two factors (ikµ/MW)(−ikν/MW) are just what is required to reproduce the Feynmanrules for the WBGB couplings as given in eq.

(19). (The relative sign arises becausemomentum in at one end of the boson line corresponds to momentum out at the other.

)Thus, the diagrams with WBGB’s simply cancel the η-dependence of the W propagatorsin Fig. (1).

Notice that it is crucial for this result to use the derivative WBGB couplingsfor both the WWZ vertex, and the W-fermion vertices.Since the integrands in the two formulations are equal, they give the same result forthe τ weak dipole moment, regardless of how the graphs are regularized. We choose hereto regulate the graph by inserting the form factor, F(p, Λ) = −Λ2/(p2 −Λ2), into eachinternal line.2 (This may be viewed as a higher-derivative regularization, for which a higherderivative kinetic term has been added to the unperturbed lagrangian for each field.) Theresult for the most-divergent part becomes:zmost−div = −ag22304π2Λ2M 4Wmτ,(25)where g is the SUL(2) coupling constant.It is instructive to compare this result with what would have been obtained if we hadused the chiral lagrangian only for the WWZ vertex in Fig.

(1), and had simply usedStandard-Model Yukawa-type Feynman rules for the WBGB-fermion vertices. In this case2 The momentum flowing through each line must be regulated separately, or else the result will dependon how momentum is routed through the graph.13

the WBGB graphs are less divergent, since there are fewer powers of momentum associatedwith each vertex. The most divergent part of the result becomeszmost−div = −ag2384π2mτm2τ −m2ντM 4Wln Λ2M 2W.

(26)There are three lessons to be learned from this section and from this example:• 1: First, we explicitly verify the equivalence between the chiral lagrangian and thelagrangian which ignores all but Uem(1). This equivalence relies crucially on the derivativecouplings of the WBGB’s in chiral perturbation theory.

Criticizing the apparent non-gaugeinvariance of the TGV lagrangians that are used in loop calculations (or, equivalently, ofunitary gauge) in favour of chiral perturbation theory clearly misses the point. If there areproblems with the large loop estimates that have been obtained, then the reason must befound elsewhere.

We point this reason out in the following section.• 2: Next, we see explicitly that even the dominant cutoffdependence of off-shell quantities,such as couplings in the effective lagrangian, depend strongly on the choice of field variablesused. In particular, the two kinds of Feynman rules for the fermion–WBGB vertex may beobtained from one another by performing a WBGB-dependent nonlinear field redefinitionon the fermion fields of the form ψ →f(ϕ)ψ.

(In fact, the answer would have remainedunchanged if the higher-derivative terms which implement the cutoffwere also transformed,since this transformation introduces new cutoff-dependent fermion–WBGB interactions. )This is part of the occupational hazard of trading in off-shell divergences: they depend indetail on which field variables are regulated.• 3: But the last, and most important, point is this: without knowing the underlyingphysics, which of these two answers is correct?

If one interprets Λ to agree, in order ofmagnitude, with the new physics scale, they have very different physical implications. Thedifference between them could well be the difference between detecting z at LEP or not.We shall argue in the following sections that in this case it is eq.

(26) which is correct. Itis clearly important to be able to decide which is right in advance!3.

Cutoffs – General ArgumentsThe main point of this paper is to critically reassess the common habit of inferringheavy-mass dependence from the cutoffdependence obtained purely within low-energyloops. In this section we make our main points.

In order to do so, we start by presenting14

the arguments in favour of using cutoffs in this way, followed by our criticisms of thesearguments. We then provide a few explicit examples to illustrate the relevant points.3.1) Why Might One Think That Cutoffs Track New Physics?Associating a physical interpretation with the cutoffis an almost irresistable impulsewhen dealing with effective lagrangians.

After all, the effective theory is from the startonly meant to describe physics below some scale, Λ, above which we cannot probe. Sinceeffective theories are not renormalizable in the traditional sense, the insertion of effectivevertices into loop graphs can produce very divergent results.It is natural to supposethat these divergences indicate that the amplitude in question gets its most importantcontributions from the highest frequencies: those just below the cutoff.

Presumably thisstrong sensitivity is removed once all of the heavy degrees of freedom of mass M > Λare included, such as would happen if this underlying theory were renormalizable. As aresult, so the argument goes, the strong short-distance contributions should saturate at M,leaving a result whose size is set by replacing Λ with M, at least up to order of magnitude.This reasoning can be made considerably more precise by rephrasing it as the followingprinciple [19]:If there is a divergent graph in the low energy theory, cutting it offat the scale wherethe theory breaks down due to new physics gives a lower bound to the actual value of thegraph in the full theory (in the absence of fine tuning).What could possibly be wrong with such a physically appealing argument?Theanswer is that, in certain circumstances, nothing goes wrong with it.

Unfortunately, it cansometimes also happen that it is completely false, and it fails because it does not take intoaccount cancellations that are automatically built into any effective theory. We describehere what these cancellations are, and return in following sections to the question of howto tell when the above reasoning will fail.3.2) The Curse of CancellationsConsider, then, a theory which involves two very different mass scales M ≫m.

(Wehave in mind that m represents the weak scale — say m ∼MW — while M representsthe scale of unknown new physics.) Suppose that within this theory we wish to computea physical low-energy observable, such as a calculable low-energy mass shift, δµ2, as afunction of these two mass scales.

An example of this type of observable in the electroweak15

interactions would be the deviation from unity of the ρ-parameter, which is related to thecomparative strength of the low-energy charged- and neutral-current weak interactions.We are interested in the form taken by δµ2 in the limit where m/M is taken to beasymptotically small, with all dimensionless couplings held fixed. It is possible to makefairly general statements as to the result in this limit (in four spacetime dimensions) ifthe renormalizable part of the low-energy theory is perturbative, so that all fields scaleapproximately as the noninteracting lagrangian would indicate.

Typically the answer inthis case takes the following general formδµ2(m, M) = c0M 2 + c1m2 + c2m4M 2 + · · ·(27)in which the dots represent terms that are suppressed by more than two powers of m/M.The dimensionless coefficients are functions of the other (renormalized) dimensionless pa-rameters of the theory, and they may also depend at most logarithmically on the large massratio M/m. Notice that the largest power of M here is just set by dimensional analysis.For applications to the electroweak interactions m ∼MW, it is important to be awarethat the above form strictly applies only asymptotically for MW/M →0.

It may thereforebe expected to hold when the new physics can be at very high scales compared to theweak scale, such as if the underlying physics were a Grand Unified Theory of some kind.Its application is less straightforward when the new physics is associated with electroweaksymmetry breaking, since in this case M cannot be larger than of order 4πv, and soM/MW <∼8π/g. In this case eq.

(27) must be interpreted as applying to the g →0 limitrather than for MW/M →0 with g fixed.Imagine now performing the same calculation, but this time dividing the contributionsinto a ‘low-energy’ part and a ‘high-energy’ part. To this end choose a cutoff, Λ, whichsatisfies m ≪Λ ≪M.

First integrating out the high energy part of the spectrum producesa low-energy effective lagrangian that is applicable at scales below Λ. Next compute thephysical mass shift in this low-energy effective theory.

Since this simply corresponds to aparticular way of organizing the calculation in the full theory it must produce the correctanswer of eq. (27) above.

The full expression may therefore be broken up as followsδµ2(m, M) = δµ2le(m, Λ, M) + δµ2he(m, Λ, M)(28)in which the first (second) term here respectively contains only the low-energy (high-energy) contributions. (This split between low and high frequencies may be conveniently16

formulated in euclidean signature according to whether the four-momentum, p, for a parti-cle of mass mi in each internal line of a Feynman graph satisfies the condition p2+m2i > Λ2).The low-energy and high-energy contributions to δµ2 in general take the followingform:δµ2he = c0M 2 + b1Λ2 + · · ·δµ2le = b′1Λ2 + · · ·(29)In both of these equations the ellipses represent terms that depend differently on the smallmass ratios m/Λ, Λ/M or m/M than the terms that are explicitly written. Examples inlater sections include, for example, such quartically-divergent terms as Λ4/m2.

Clearlythe condition that the two contributions sum up to the full result of eq. (27), whose Λ-independence is manifest, requires that the coefficients satisfy b1 + b′1 = 0, etc.Now comes the main point.

In order to calculate the scale of new physics that may beprobed by a detailed measurement of a quantity like δµ2 we require the accurate knowledgeof the coefficient c0 in eq. (27).

If we only have access to the low-energy effective lagrangianbelow scale Λ then it is impossible to precisely compute c0. In particular, knowledge of thecoefficient, b′1, of the low-energy quadratic divergence gives no a priori information regard-ing c0, since it is completely cancelled by the high-energy contribution (or counterterm) b1.There is nothing miraculous about this cancellation; it simply reflects how physics cannotdepend on the intermediate steps in a calculation.There are occasions, however, when knowledge of the coefficient of a particular di-vergence in the low-energy theory can be parlayed into reliable information about theheavy-mass dependence of the full result [20].

A logarithmic divergence furnishes perhapsthe simplest example. Here the full and partial results for a dimensionless observable, callit A, can take the following formA = Ale + Ahe = a0 logM 2m2+ · · ·whileAhe = a′0 logM 2Λ2+ · · ·(30)andAle = a′′0 logΛ2m2+ · · ·In this case the condition that the cutoffdependence cancel requires that a0 = a′0 = a′′0 andso the coefficient of the large logarithm within the full theory may be determined simply17

by identifying the coefficient of the logarithmic divergence within the low-energy theory.It is important to realize that this property is not generically shared by other types ofdivergences.3.3) “Good” vs. “Bad” VariablesWith the general concepts regarding cutoffs now firmly in hand, we can now demon-strate the flaw in the principle enunciated earlier (Section 3.1), which states that cutoffsfurnish lower bounds for the contributions of new physics. A brief example here is instruc-tive.Consider the case where the standard model itself — Higgs and all — is the low-energy theory, as might be appropriate to a Grand Unified Theory.In this case thebounds of eq.

(27) should apply since the low-energy theory is perturbative in the regimeof interest, and we may take MW/M to be extremely small. Suppose we choose to computethe cutoffdependence in this theory of the coefficient of the effective operator FµνF µν,which contributes to the vacuum polarization of the photon.

Since the standard modelis renormalizable, this result in a manifestly renormalizable gauge is finite — varyinglike 1/Λ2 for Λ ≫MW. If the same result is computed in unitary gauge, however, (or,equivalently, in any gauge using the derivative WBGB couplings of chiral perturbationtheory) then it diverges quadratically: ∼Λ2/M 4W.

If taken seriously, this example woulddrastically overestimate the heavy-mass dependence of the underlying theory, which cannotbe larger than O(1/M 2).Once again, just as in our earlier example involving the weak dipole moment of theτ, a change of variables has dramatically altered the cutoffdependence of the effectivelagrangian. These two examples illustrate the difference between what might be called“good” and “bad” variables.

To see the distinction between these variables, notice that forboth examples the divergences of the S-matrix are the same in both sets of variables, sincethe S-matrix is unchanged by field redefinitions. “Bad” variables are therefore character-ized by large cancellations in physical quantities, such as the S-matrix, between enormousterms in the effective lagrangian.

As a result, these are variables for which the couplingsin the lagrangian do not follow the couplings that would be defined in terms of scatteringamplitudes.With this in mind, one can propose a modification of the above principle [19]:If there is a divergent graph in the low energy theory, cutting it offat the scale wherethe theory breaks down due to new physics gives a lower bound to the actual value of thegraph in the full theory (in the absence of fine tuning), so long as “good” variables are18

used in the calculation.It appears that this principle holds in all known examples. However, its utility relieson the existence of an practical algorithm for determining in advance whether the variablesof interest are “good” or “bad”.This is our main criticism of the papers in Ref.

[10].In using cutoffs to regulatedivergent loops involving anomalous TGV’s, they obtain limits on the coefficients of theseoperators which depend on their choice of variables.Without knowing whether thesevariables are “good” or “bad”, one cannot ascertain if the bounds obtained are reasonable.A second criticism of some of these papers, and indeed of some of those in Ref. [8], is thatthe scale of new physics, M, is often allowed to be greater than 4πv, which is not permittedif the symmetry is realized nonlinearly.

This typically leads to overly stringent bounds onnew operators.4. Banishing CutoffsRather than searching for a practical algorithm for “good” and “bad” variables, weprefer to recast the above principle in a way which does not refer to cutoffs at all.

Itamounts, in essence, to the judicious use of dimensional analysis, together with any otherinformation that may also be available purely within the low-energy theory. This informa-tion is all that is really required of any analysis of low-energy graphs, and in applicationswhere cutoffdependence happens to track the underlying masses, produces identical an-swers.

It has the conceptual advantage, however, of being insensitive to field redefinitions,and so of never leading one badly astray through the mistaken use of “bad” variables.In the remainder of this section, we describe this procedure, followed immediately by adetailed calculation using a known model of underlying physics with which both cutoffandour results can be compared.Suppose, then, that some physics that is associated with a heavy mass scale, M,(which might, for example, denote the mass of the lightest unkown particle) is integratedout to produce a low-energy effective lagrangian, Leff:δLeff= cnOn. (31)We are interested in the M–dependence of the coupling for an effective operator of scalingdimension (mass)dn which appears in this effective lagrangian.

In general this is an ill-defined question, since the dependence of cn on heavy physics requires a proper definitionof the composite operator it multiplies, On. We therefore pause here to make a brief asideconcerning a particularly convenient formalism for these purposes.19

4.1) A Regularizational AsideA particularly clean and convenient scheme with which to work in an effective theory isdimensional regularization supplemented by the ‘decoupling subtraction’ renormalizationscheme [21]. This scheme consists of minimal subtraction supplemented by the explicitremoval of heavy degrees of freedom as the renormalization point is lowered below thecorresponding mass thresholds.

This ‘integrating out’ of the heavy particles is in practiceimplemented as a set of matching conditions for the appropriate effective couplings atthese threshholds. The resulting couplings may then be used as initial conditions for therenormalization group equations that define the scale-dependence of such couplings in thetheory below the threshhold.

With this scheme a logarithmic dependence on the massesof the problem (including M) is introduced into the coefficients cn as the various effectiveoperators are evolved between particles threshholds.The beauty of using dimensional regularization in this way is that no confusion ispossible between the cutoffand the heavy-physics scale, since within this framework nocutoff, Λ, arises at all. As a result only the physical masses ever arise in effective couplings.Furthermore, more and more divergent graphs in the effective theory, which involves onlylight particles, simply introduce higher and higher powers of the light mass, m, rather thansome higher scale such as Λ or M. As a result it becomes possible (and convenient) toinclude within the loops of the low-energy theory all of the momenta of the light fields,right up to infinity.

This leads to a real distinction in the nature of the matching betweenthe underlying theory and the effective theory when using cutoffs and dimensional regular-ization. When using a cutoff, all frequencies above the scale Λ are integrated out, includingall of the modes of the heavy particles as well as the high frequency components of the lightparticles.

In dimensional regularization, one instead integrates out only the heavy-particlecontributions; leaving all of the momenta of the light fields in the low-energy theory. Thisallows the matching between the effective and the underlying theories to be made at theheavy mass threshold itself, and so the only mass which appears due to this matching istypically this threshold mass, M.There is another practical benefit in using dimensional regularization.

Dimensionally-regularized graphs are much less sensitive to the field redefinitions that relate the “good”and “bad” variables of the earlier examples. For instance, if dimensional regularization isused to regularize the contribution of the WWZ interaction to the Zdm, the divergentpiece is found to bezpole =ag2384π2mτm2τ −m2ντM 4W1ǫ.

(32)20

where n = 4−2ǫ is the dimension of spacetime. This result holds using either Yukawa-typeor derivative couplings for the WBGB’s to the fermions.Within minimal subtraction, we find therefore that the renormalized parameter zmixes with the renormalized parameter a in the following way:z(µ) = z(M) −g2384π2mτm2τ −m2ντM 4Wa(M) lnM 2µ2.

(33)Notice the similarity between the logarithmic dependence here, and the previous resultsof eq. (26).Both terms in eq.

(33) have a clear interpretation. The logarithmic dependence corre-sponds to the explicit operator mixing that can be unambiguously computed purely withinthe low-energy effective theory.

The initial conditions, z(M) and a(M), however are de-termined by matching to the underlying theory and so cannot be known until this theoryis specified. At best we can only try to estimate the size of these initial conditions, andthis is the goal of the remainder of this section.4.2) The Generic EstimateWith this definition in mind, we wish now to estimate how the couplings cn of eq.

(31)depend on the new-physics scale, M. We are specifically interested here in the powers ofM that arise at the threshhold, M, rather than any logarithmic dependence.Simpledimensional analysis would indicate:cn = ˆcn M 4−dn. (34)Without any additional information about the nature of the new physics that is responsiblefor this effective lagrangian, all that can be said about the dimensionless coupling, ˆcn, isthat it is O(1) or smaller.With more assumptions concerning the physics at M, more information can be ex-tracted about the cn.

We next illustrate how different kinds of physics can differ in theirimplications for cn by contrasting two plausible alternatives for electroweak symmetry-breaking physics at M.21

4.3) Strong Coupling: Naive Dimensional AnalysisSuppose first that the symmetry-breaking sector is strongly coupled, with only theWBGB’s appearing at energies much less than M. In this case chiral perturbation theoryorganizes their couplings according to the numbers of derivatives which appear in thelagrangian. For applications to energies that are much less than the electroweak scale, v,simple dimensional analysis with M ∼v properly describes the size of each interaction.Of more practical interest, however, is the application of this lagrangian to electroweakenergies, E ≃v ≪M.

In this case higher-derivative interactions should be suppressed bypowers of M rather than v, and it becomes important to keep track of the powers of v/Mwhich can appear in the coefficients cn. A set of self-consistent statements for the sizes thatcan be expected for any given term in the chiral lagrangian is called “Naive DimensionalAnalysis” (NDA) [22].

It states that a term having b WBGB fields, f weakly-interactingfermions fields, d derivatives and w gauge fields has a coefficient whose size is:cn(M) ∼v2M 21vb 1M 3/2f 1Md gMw,(35)with M <∼4πv. (If the fermions are strongly interacting, then the appropriate factor is1/v√M for each fermion.

)Some examples of this counting are instructive, particularly when these are com-pared with the alternative estimates of the next section. For instance, according to theabove estimate, the mass terms for the W and Z bosons are both of order g2v2.

Thisindicates that the small size of the deviation from unity of the rho parameter cannot beunderstood in this picture as being simply the result of a suppression by powers of v/M.Additional approximate symmetries are required in order to explain the small size of δρ.Also, typical corrections to the charged- and neutral-current interactions for fermions arehere of the order of gv2/M 2. Finally, triple-gauge boson operators such as κW ∗µWνZµνand λW ∗µνWνλZλµ are respectively of order κ ∼g3v2/M 2 and λ ∼g3v2/M 4.

We nextcompare these estimates with the implications of an alternative scenario.4.4) Weak Coupling: Linearly-Realized LagrangianAn alternative perspective arises if the low-energy theory fills out a linear realizationof the electroweak group. In this case M need not be small compared to 4πv, and theWBGB’s fall into some linear representation of this group.Again operators can havecoefficients that are suppressed by powers of v/M once the low-energy Higgs fields are22

given their v.e.v.s, and so the power that arises depends on the representation in whichthe symmetry-breaking order parameter transforms. Much the most plausible choice forsuch a linearly-realized Higgs representation is one or more doublets, with the standardhypercharge assignment.

In this case the dependence on v/M of any non-Higgs interactionsmay be found by taking cn = ˆcnM 4−dn, as before, and then replacing any Higgs multipletsin the effective operator by their v.e.v.s. In this case the linearly-realized gauge symmetryenforces relations amongst the coefficients of operators of a given type, depending on howthese operators fall into linearly-realized multiplets.This is best illustrated with a few examples.

Consider the W and Z boson mass terms:OW = W ∗µW µ and OZ = 12 ZµZµ. The lowest-dimension operator which contains theseterms is simply the dimension-four Higgs kinetic term, (Dµφ)†(Dµφ).

Just as for the stan-dard model, replacement of φ by its expectation value in this operator generates the par-ticular combination cos2 θw OW + OZ with a coefficient that is of order g2v2. More generalcombinations arise at dimension six, such as through the operator (φ†Dµφ) (φ†Dµφ)/M 2.This and similar operators ruin the mass relation MW = MZ cos θw, by amounts thatare of order g2v4/M 2.

In contrast with the NDA estimate, δρ is automatically small ifv2/M 2 ≪1.As we shall see in a later section, the smallness of the present estimate in comparisonwith the NDA result has a simple explanation within the context of an underlying multi-Higgs model. In this case contributions such as those to δρ typically arise at one loopand are proportional to gλ2H/16π2, where λH ≃gmH/MW is a Higgs self-coupling.

If thisself-coupling is weak, then the suppression by 1/(4π)2 corresponds to a factor of v2/M 2.Once λH is of order 4π, however, for which mH ∼4πv, this suppression is lost and weobtain the NDA result.It is not always true that NDA gives a larger estimate for effective couplings thanwould a linearly-realized underlying theory, however. For example, both predict devia-tions from the standard model charged- and neutral-current couplings that are of ordergv2/M 2.

Similarly, both estimates for the coupling κW ∗µWνZµν are of order g3v2/M 2.Furthermore, for the coupling λ (which premultiplies the interaction W ∗µνWνλZλµ), theNDA estimate is actually smaller than that for a linearly realized model. NDA wouldpredict λ ∼g3v2/M 4 while the linearly-realized estimate is λ ∼g3/M 2, since this inter-action can be embedded into the linearly-realized operator Tr[W µνWνλW λµ] without thenecessity for Higgs doublets.4.5) Using Loops to Infer Further InformationThese estimates that are simply based on dimensional analysis can be sharpened23

using additional assumptions. A dimensional estimate for cn(M) can be obtained by usingloops in the low-energy theory to estimate factors of dimensionless coupling constants and1/(16π2) which arise from the low-energy contribution to cn at lower scales.

If these areassumed to not cancel with the high-frequency contribution, then these factors may be usedto place a lower bound on cn, just as in the principle that was enunciated in the earliersection to describe the potential relevance of cutoffdependence. The main difference inthe present formulation is the use of these loops purely to determine the dependence ondimensionless combinations of couplings, with only dimensional analysis being used to fixthe dependence on M.For loops which involve WBGB’s there is one dimensionless coupling that is of partic-ular interest.

This is the dimensionless coupling which describes the interactions betweenWBGB’s and the other particles of the theory. It is always possible to choose variables suchthat these are proportional to the ratio of the particle’s mass to v, rather than using thederivative coupling of Section (2).

For instance: λϕff ∼gmf/MW, and λϕww ∼gM 2W/v,etc.. Including these couplings is important if the corresponding particle masses are large,in that they can produce what appears to be a positive power of a heavy mass.

We illus-trate this in more detail in the following section. Use of this coupling strength amounts tousing our freedom to use field redefinitions to remove as many derivatives as possible fromWBGB couplings [23].

This is where the use of ‘good’ variables enters our rules [19].This procedure is clearly operationally very similar to what is usually done when usingcutoffs to estimate effective interactions. In particular, it reproduces the many successfulestimates that are often argued from using cutoffs.

The main difference is that the powerof M that contributes here is explicitly argued purely on dimensional grounds, therebyremoving the uncertainty that is associated with the choice of “good” and “bad” variables.5. Known New Physics: An ExampleWe now wish to apply this reasoning to a model for which all of the heavy-massdependence is known and calculable.

This permits a comparison of the above argumentswith the known correct dependence on M, as well as with the cutoffdependence of thelow-energy effective theory.5.1) An Explicit CalculationWe consider a two-Higgs doublet model with soft CP-breaking terms in the Higgspotential, and where we imagine that the physical Higgs particles all have masses thatare as large as is possible: mH <∼4πv.Larger masses are not possible here without24

disbelieving the perturbative analysis, since the Higgs masses can be made larger thanv only by increasing their self couplings. In this model the anomalous WWZ couplingof eq.

(20) arises at one loop, with a calculable coefficient. We may therefore computethe contributions which this operator makes to the Zdm of section (2), as well as to theρ-parameter, and contrast this with an estimate of the corresponding higher-loop graphsthat are obtained within the underlying theory when the effective WWZ vertex is resolved.Following Ref.

[24], we consider a two-Higgs doublet model in which CP is sponta-neously broken. This occurs when there is a relative phase between the vacuum expectationvalues (vevs) of the two Higgs doublets.

In such a scenario, tree-level flavour changing neu-tral currents (FCNC’s) are usually generated, but these can naturally be made small ifCP violation is generated via soft CP breaking terms in the Higgs potential. The twoHiggs doublets can then be written φTi = (φ+i , φ0i + vieiθi), i = 1, 2, in which vieiθi arethe vevs.

For calculational purposes, it is useful to change bases such that the WBGBfields (ϕ0Z, ϕ+W) are decoupled from the physical Higgs fields (H+, H01,2, I02). The new basisis φ′T1 = (ϕ+W, H01 + iϕ0Z +pv21 + v22), φ′T2 = (H+, H02 + iI02), in which only the vev of φ′1 isnonzero.

Although H+ is a mass eigenstate, the neutral states H01,2 and I02 are not. Theyare related to the mass eigenstates by an orthogonal matrix dij:H01H02I02=d11d12d13d21d22d23d31d32d33φm1φm2φm3.

(36)In the absence of CP violation, d13 = d23 = d31 = d32 = 0.• The WWZ Effective Operator: There are two graphs which contribute at one loop to theCP violating WWZ vertex in eq. (20).

These are shown in Fig. (2).

In fact, since we areonly interested in getting an idea of the dependence on the Higgs’ masses, we concentrateonly on diagram (a) in Fig. (2).If we had no knowledge of the underlying theory, we could estimate the dependenceof a, the coefficient of the WWZ vertex, on the heavy mass scale M ∼mH by using thedimensional analysis of the previous section.

In order to determine the suppression by vthat is appropriate, we use the estimate of NDA, since this is appropriate to the case of astrongly-coupled Higgs bosons that we are considering. Since a is dimensionless, we expect(after inserting a factor of g for each vector boson):adim ∼g3 v2M 2∼g g4π2.

(37)25

Direct calculation, on the other hand, givesamodel =Xijig332π2 cos θW(d3id2j −d3jd2i) (d2id2j + d3id3j)m2i −m2jI1,(38)whereI1 =Z 10dx1Z x10dx2x2(x1 −x2)1x2(m2i −m2j) + x1(m2j −m2c) + m2c + M 2Wx1(x1 −1). (39)In the above equations, the sum is over the physical neutral Higgs bosons with mi and mcbeing the neutral and charged Higgs masses, respectively.

From the above expression, itis clear thatamodel ≃g316π2 lnm2HM 2W,(40)where mH is a generic Higgs mass.This agrees with the estimates from dimensionalanalysis, for mH ∼M <∼4πv.For our later purposes we wish to embed the anomalous WWZ interaction into loopsin order to estimate their implications for other effective interactions. Since the strongestdependence on heavy masses comes from the longitudinal W particles, ϕW, in these loops,we pause here to present an estimate for the size of the coefficient of the anomalous ZϕWϕWvertex.

There is only one difference from the previous case: the ϕW bosons couple with astrength that is proportional to gM/MW rather than simply to g. On dimensional groundsthe largest contributions to the ZϕWϕW coupling should be proportional to:aϕdim ∼g v2M 2 gMMW2∼g. (41)In this model, the lowest-dimension anomalous ZϕWϕW coupling arises at one loopfrom the graphs of Fig.

(2). Keeping only terms linear in q, the four-momentum of theexternal Z, we find that the ZϕWϕW vertex isXij(−g34 cos θW(d3id2j −d3jd2i) (d2id2j + d3id3j)m2i −m2j m2i −m2c m2j −m2cM 2WIµ2),(42)withIµ2 =Zd4l(2π)4(2l)µ(2q · l)[(l + K/2)2 −m2c] (l2 −m2i )2(l2 −m2j)2 ,(43)26

where K = k −k′. It is not necessary to solve this integral exactly – what is importantis that for the external momenta of interest (i.e.

those that are <∼mH) it is dominated bymomenta of order mH, giving an integral that is of order m−4H . This gives the resultaϕmodel ≃g316π2mHMW2lnm2HM 2W,(44)which is larger by an additional factor of m2H/M 2W in comparison with the result for trans-verse W’s: eq.

(40). This enhancement corresponds, in the underlying theory, to the re-placement of two gauge couplings, g, with two WBGB-Higgs couplings, λHHϕ ∼gmH/MW.It agrees with estimate (41) when mH ∼M ∼4πv.• The Weak Dipole Moment: Next consider the Zdm of eq.

(21) in this effective theory.Using only naive dimensional analysis, we can therefore only conclude:zdim ∼gv2M 3. (45)Any further information is more model specific.In order to sharpen our estimate we next consider the size of the Zdm that is inducedin the low-energy theory from the effective WWZ operator considered previously, via theloop of Fig.

(1). The dominant short-distance behaviour comes from the contributions oflongitudinal W’s to this graph.

In this case there is now an additional factor of mτ fromthe required helicity flip, as well as two factors of the longitudinal W couplings to thefermion line, λϕττ ∼g mτ/MW. Taking our estimate for this graph as a lower bound, wetherefore expect:zdim >∼aϕdim16π2g mτMW2 mτM 2∼g5(4π)2 v2M 2W m2τM 2W mτM 2,>∼g5(4π)4 m3τM 4W.

(46)We have used v2/M 2 >∼1/(4π)2 in this last equation.In the underlying theory, the Zdm appears at two loops as in Fig. (3).

The strongestdependence on mH again comes when both W’s in the loop are longitudinal — in a covariantgauge they are WBGB’s. Although the full 2-loop diagram is difficult to solve completely,27

it is sufficient for our purposes to estimate the integrals using dimensional analysis. Again,the important region in the loop integration comes from momenta ∼mH, since mH is thelargest scale in the problem.

Including the factor of mτ due to the required helicity-flip onthe fermion line, and two factors of the WBGB-τ Yukawa coupling: λτ ≃gmτ/MW, wearrive at the following estimate for z:zmodel ∼g5 116π22 m3τM 4Wln2m2HM 2W. (47)This result agrees both with the our current estimate of eq.

(46), as well as with the earliercutoff-based estimate of eq. (26), but not with the ‘bad-variable’ result of eq.

(25).• The Vacuum Polarization: It is instructive to also consider the contributions towardsthe Z vacuum polarization that are induced by the WWZ operator in this model. Besidesproviding another comparison with the estimates, it furnishes an example for which there is(superficially) an enhancement by powers of M/MW, and for which a simple cutoffanalysisin unitary gauge proves to be correct.The required contribution to the Z vacuum polarization comes from the three-loopgraph of Fig.

(4). Again, in the underlying model the largest contribution comes whenboth W’s in the inside loop are longitudinal.

Then each ZϕWϕW vertex contributes afactor of orderaϕmodel q,(48)where aϕmodel is given in eq. (44), and q is the four-momentum which flows through theexternal Z line.

The middle loop gives just the loop factor 1/16π2 times a logarithm.Therefore we find that the contribution to the Z vacuum polarization in this model hasthe form[δΠZZ]model ∼ g216π23 m4HM 4Wq2 ln3m2HM 2W. (49)Notice the large power of mH/MW.

This result agrees with the most-divergent part ofthe unitary gauge cutoffdependence that is obtained by inserting two effective WWZinteractions into a one-loop vacuum polarization diagram:[δΠZZ]most−divq2= −a2576π2Λ4M 4Wq2. (50)if we take our earlier estimate for the WWZ interaction: adim ∼g3v2/m2H >∼g3/(4π)2.28

Our dimensional estimate for this quantity, on the other hand, is[δΠZZ]dim ∼1(4π)2 (aϕdim)2∼g6(4π)2 v4M 4 M 4M 4W,(51)which also agrees with the result of the underlying model once we use v/M >∼1/4π.Notice that, keeping in mind gmH <∼8πMW, what appears to be an enhancement offour powers of mH/MW in eq. (49) is really more than compensated for by the suppressionby six powers of g/4π, as it must be in order for the result to be sensible.

Thus, it ismisleading in this case to use the corresponding enhancement in eq. (50) without alsoincluding the accompanying suppression that is implicit in the coefficient a.6.

ConclusionsEffective lagrangians are the natural way to parametrize the effects of the new physicswhich must lie beyond the standard model. The next generation of experiments will havethe ability to probe a number of these new effective operators.

Quite naturally, then, onewants to have an idea of how big these new effects might be.Much work has gone into constraining the new operators, particularly those corre-sponding to trilinear gauge boson vertices, through their loop contributions to quantitieswhich are measured at lower energies. We have argued here that these estimates [10] aretypically misleading, and often give bounds which are overly stringent.Other authors [11] have made the same criticisms.

However, they trace the cause ofthe problem to the apparent non-gauge invariance of the operators that are widely used inthe literature. We argue instead that in this instance gauge invariance is a complete redherring and is not the source of the problem.If one does not wish to explicitly include a Higgs scalar in the low-energy theory,there are two principal candidates for such an effective lagrangian – one which requires onlyUem(1) gauge invariance, but not SUL(2)×UY (1) gauge invariance, and one which imposesthe full 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) gauge29

invariance. Alternatively, one can say that at low energies there is little to choose betweena spontaneously-broken gauge invariance and no gauge invariance at all.The real source of the problems is the widespread use of cutoffs to regulate diver-gent graphs in the low-energy effective lagrangian.

Both the effective lagrangian and itsdivergences, being off-shell quantities, are not invariant under field redefinitions. As a con-sequence, the result of a loop calculation will generically depend on the choice of variables,if cutoffs are used to regulate the divergences.It is in principle possible to use “good” variables in such loop calculations, in whichcase the cutoffbehaviour of the final answer accurately reflects the true dependence ofthe operator on the heavy mass scale M. However, it is equally possible to choose “bad”variables, characterized by cancellations in the S-matrix between large terms in the effectivelagrangian, in which case the cutoffdoes not properly track the dependence on M. If “bad”variables are used, the bounds on effective operators inferred from such calculations aretypically much too strong, and completely unreliable.In the absence of an algorithmto distinguish “good” and “bad” variables, the constraints obtained from such cutoff-regulated calculations are ambiguous at best.A separate mistake that has also been made when bounding effective interactions hasbeen to take the scale of new physics M to be 10 TeV, or higher [8], [10], even whenthe effective theory does not linearly realize the electroweak gauge group.

In this casethe effective lagrangian is simply being applied beyond its domain of applicability, sinceperturbative unitarity typically fails for such models when M >∼4πv.If one wants to estimate the size of the new operators, we advocate dispensing withcutoffs completely. A simpler method is to just use simple dimensional arguments, supple-mented by any additional information concerning dependence on coupling-constants and(4π)’s that can be gleaned by inspecting underlying or low-energy graphs.

These rulescoincide in practice with currently-used lore when this lore is sufficiently well spelled out.It has the conceptual advantage of not relying on the cutoffdependence of low-energydiagrams.One quantity which is accurately calculable within the low-energy effective lagrangian(as opposed to being an order-of-magnitude estimate) is the mixing among operators asthe effective lagrangian is evolved down from the heavy mass scale M to low energies. Thismixing, which is always logarithmic, is most easily computed using dimensional regulariza-tion, along with the decoupling-subtraction renormalization scheme.

Among the beautiesof dimensional regularization is that it is comparatively insensitive to the choice of “good”or “bad” variables.30

By using dimensional regularization to calculate the mixing of operators, and dimen-sional analysis to estimate the size of the initial conditions, i.e. the effective operators atscale M, one sees that it is never necessary to deal with cutoffs in a low-energy effectivelagrangian.Note Added:After this paper was released, we have become aware of Ref.

[25], whose authorspresent a point of view more similar to our own.AcknowledgmentsWe would like to thank Joe Polchinski for numerous enlightening discussions on thesubject of cutoffs and effective lagrangians. D.L.

thanks F. del Aguila for the hospitality ofthe University of Granada, where part of this work was done. Many thanks also to FawziBoudjema, Steven Godfrey, Markus Luty, Ivan Maksymyk, Yossi Nir, Santi Peris, XerxesTata and German Valencia for helpful criticism.

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.31

Figure Captions• Figure (1): The Feynman graph through which the anomalous gauge-boson vertex con-tributes to fermion weak dipole moments.• Figure (2): The Feynman graphs which generate the CP-violating anomalous gauge-boson vertex in the two-Higgs model.• Figure (3): A Feynman graph which generates the CP-violating τ Zdm in the two-Higgsmodel.• Figure (4): The 3-loop contribution to the Z-boson vacuum polarization. The blobsindicate the 1-loop anomalous gauge-boson vertices whose structure is shown in Fig.

(2).32

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