Anatomy of the Higgs Boson Decay into Two Photons in the Unitary Gauge

Abstract

We review and clarify computational issues about the 𝑊-gauge boson one-loop contribution to the 𝐻→𝛾𝛾 decay amplitude, in the unitary gauge and in the Standard Model. We find that highly divergent integrals depend upon the choice of shifting momenta with arbitrary vectors. One particular combination of these arbitrary vectors reduces the superficial divergency down to a logarithmic one. The remaining ambiguity is then fixed by exploiting gauge invariance and the Goldstone Boson Equivalence Theorem. Our method is strictly realised in four dimensions. The result for the amplitude agrees with the “famous” one obtained using dimensional regularisation (DR) in the limit 𝑑→4, where 𝑑 is the number of spatial dimensions in Euclidean space. At the exact equality 𝑑=4, a three-sphere surface term appears that renders the Ward Identities and the equivalence theorem inconsistent. We also examined a recently proposed four-dimensional regularisation scheme and found agreement with the DR outcome.

1. Introduction

Today one of the main focal points at the Large Hadron Collider (LHC) is to search for the Higgs boson (𝐻) [1–3] through its decay into two photons, 𝐻→𝛾𝛾 (for reviews see [4, 5]). Indeed, the recent [6, 7] observation by ATLAS and CMS experiments of a resonance, which could be the Standard Model (SM) Higgs boson, is based on data mainly driven by 𝐻→𝛾𝛾. In the SM [8–10], this particular decay process goes through loop-induced diagrams involving either charged fermions or 𝑊-gauge bosons. Their calculation was first performed in [11] in the limit of light Higgs mass 𝑚_𝐻≪𝑚_𝑊, using dimensional regularisation in the ’t Hooft-Feynman gauge. Since then, there are numerous works spent on improving this calculation including finite Higgs mass effects in linear and nonlinear gauges [12–14], different regularisation schemes [15–18], and/or different gauge choices [19].

The 𝐻→𝛾𝛾 amplitude is originated, in broken (unbroken) phase, by a dimension-5 (dimension-6) SM gauge invariant operator(s) and, therefore, its expression, within a renormalizable theory, must be finite, gauge invariant and independent of any gauge choice. The amplitude should also be consistent with the Goldstone Boson Equivalence Theorem (GBET) [20–22] since the SM is a renormalizable, spontaneously broken, gauge field theory.

A problem arises when the 𝑊-gauge boson contribution (see Figure 1) to 𝐻→𝛾𝛾 produces “infinite” results at intermediate steps. These problems are usually treated by using a gauge invariant regulator method, for example, dimensional regularization. In the unitary gauge [23], this indeterminacy is more pronounced and more difficult to handle with (however, using DR and unitary gauge with modern computer algorithms, this may not be a hard problem today [19]) due to the particular form of the 𝑊-gauge boson propagator. On the other hand it is much simpler to work with only few diagrams that involve physical particle masses, rather than many.

Figure 1 

𝑊-gauge boson contribution to the 𝐻→𝛾𝛾 amplitude. Momentum flow together with relevant shift vectors is indicated.

More specifically, in the unitary gauge, one encounters divergencies up to the sixth power. It is well known that, in four dimensions, shifting momenta in integrals that are more than logarithmically divergent is a “tricky business”—recall the calculation of linearly divergent fermion triangles in chiral anomalies [24, 25]—that requires keeping track of several “surface” terms for these integrals. There is also the situation we face here where apparent logarithmically divergent integrals turn out to be finite but discontinuous at 𝑑=4.

We would like to bypass those ambiguities and at the same time to present a “regularisation” method, by performing the calculation for the 𝐻→𝛾𝛾 amplitude strictly in 4 dimensions and in the physical unitary gauge. Our method is similar to the one used elsewhere for calculating triple gauge boson amplitudes [26, 27], or Lorentz noninvariant amplitudes [28], and consists of three steps:(1)We write down the most general Lorentz invariant 𝐻→𝛾𝛾 amplitude.(2)We introduce arbitrary vectors that account for the “shifting momentum” indeterminacy. We show that a particular choice of those “shifting vectors” cancels higher powers of infinities leaving still behind at most logarithmically divergent integrals that are treated as undetermined variables.(3)We exploit physics, that is, gauge invariance (Ward Identities) and the GBET, in order to fix the last undetermined variables.This method is quite general within a renormalizable theory and can be applied to other observables too. Following these steps we arrive at the same result for the 𝐻→𝛾𝛾 amplitude obtained by Ellis et al. [11] and by Shifman et al. [13] almost 35 years ago. Our analysis, among other issues, highlights that the recent observation [6, 7] of the 𝐻→𝛾𝛾 at the LHC signifies the validity of the Goldstone Boson Equivalence Theorem. As a further clarification we also make a remark on the direct calculation in the following three cases: we first perform the integrals in exactly 𝑑=4 (with no regularisation method beyond the one discussed in point 2 above), second, by exploiting Dimensional Regularisation (DR) as defined in [29, 30] and then taking the limit 𝑑→4, and finally third by using a four-dimensional regularization scheme introduced in [31].

Our calculation is complementary to, but somewhat different from, the two existing ones [19, 32, 33] performed in the unitary gauge. It is not our intend to redo the calculation in unitary gauge with DR as in [19]. On the contrary, we want to clarify subtle issues related to the amplitude in unitary gauge and 𝑑=4 raised in part by [32, 33]. We find, using arbitrary vectors, that divergencies (up to the 6th power) are reduced down to logarithmic ones. This is a new result that is not obvious when working in unitary gauge and cannot be seen when using dimensional regularisation. This fact was stated incorrectly in [32, 33].

The outline of the paper is as follows: in Section 2 we present the calculation of the 𝑊-loop contribution (note that the calculation of the fermion triangle contribution is well defined; that is, it is independent of arbitrary vectors and finite. We are not going to repeat this calculation here and refer the reader to the reviews in [4, 5]) to 𝐻→𝛾𝛾 amplitude, its ambiguities, and the resolution within physics arising from GBET. Next, in Section 3 we examine details of the amplitude calculation within an alternative and recently proposed four-dimensional regularization scheme [31], the one that resembles most closely the symmetry approach taken here. In Section 4 we discuss other possible physical setups plus experiment that may help to resolve inconsistencies. We conclude in Section 5. There are two appendices: Appendix A that contains some intermediate formulae and Appendix B where we present the details about the origin of surface terms in four dimensions.

2. The 𝑊-Loop Contribution to 𝐻→𝛾𝛾 in SM

The most general, Lorentz and CP-invariant, form of the of-shell 𝐻→𝛾𝛾 amplitude is, ℳ₁𝑔^𝜇𝜈+ℳ₂𝑘^𝜈₁𝑘^𝜇₂+ℳ₃𝑘^𝜇₁𝑘^𝜈₂+ℳ₄𝑘^𝜇₁𝑘^𝜈₁+ℳ₅𝑘^𝜇₂𝑘^𝜈₂ (1) where 𝑘₁ and 𝑘₂ are the outgoing photon momenta shown in Figure 1 and the coefficients ℳ_𝑖=1⋯5≡ℳ_𝑖=1⋯5(𝑘₁,𝑘₂) are scalar functions of 𝑘²₁,𝑘²₂, and 𝑘₁⋅𝑘₂. By considering that all particles are on-mass-shell, that is, 𝑘²₁=𝑘²₂=0,𝑘₁⋅𝑘₂=𝑚²_𝐻/2,𝑘₁⋅𝜖^∗(𝑘₁)=0,𝑘₂⋅𝜖^∗(𝑘₂)=0, we obtain an amplitude ℳ=ℳ^𝜇𝜈𝜖^∗_𝜇(𝑘₁)𝜖^∗_𝜈(𝑘₂) with only two undetermined (for the time being) coefficients ℳ^𝜇𝜈=ℳ₁𝑔^𝜇𝜈+ℳ₂𝑘^𝜈₁𝑘^𝜇₂. (2) In unitary gauge, the Feynman diagrams that contribute to ℳ₁ and ℳ₂ are displayed in Figure 1. In order to calculate them, we introduce three arbitrary four-vectors 𝑎,𝑏, and 𝑐, one for each diagram. These vectors shift the integration momentum, that is, 𝑝→𝑝+𝑎 for the first diagram, 𝑝→𝑝+𝑏 for the second diagram, and 𝑝→𝑝+𝑐 for the third diagram. As we will see, these arbitrary vectors operate as regulators capable to handle highly divergent integrals related to unitary gauge choice. Furthermore, the vectors 𝑎, 𝑏, and 𝑐 linearly depend upon the external momenta 𝑘₁ and 𝑘₂. Hence 𝑎, 𝑏, and 𝑐 are not linearly independent (cf. (4)). This is an important fact leading to the cancellation of infinities.

We first calculate the less divergent part of ℳ^𝜇𝜈 in (2) which is the ℳ₂ coefficient. (The coefficient ℳ₁ will be fixed later on by the requirement of gauge invariance.) By naive power counting, we see that ℳ₂ diverges by at most four powers. Then we perform the Feynman integral calculations strictly in 4 dimensions. For reasons that will become clear later, we shall keep the number of dimensions general in all intermediate steps of the calculation, that is, 𝑔^𝜇𝜈𝑔_𝜇𝜈=𝑑. As we will see, 𝑑 contributes only to finite pieces of ℳ₂ (cf. (9)). (On the contrary, we shall see that there are nontrivial 𝑑-contributions into ℳ₁-coefficient.)

With all the above definitions, we can write down the total amplitude in the form ℳ^𝜇𝜈𝑑∼⁴𝑝(2𝜋)⁴𝒜₁₁𝑔^𝜇𝜈+𝒜₂₁𝑝+𝑎^𝜇𝑝+𝑎^𝜈+𝒜₂₂𝑝+𝑏^𝜇𝑝+𝑏^𝜈+𝒜₂₃𝑝+𝑐^𝜇𝑝+𝑐^𝜈+𝒜₃₁𝑝+𝑎^𝜇𝑘^𝜈₁+𝒜₃₂𝑝+𝑏^𝜇𝑘^𝜈₁+𝒜₃₃𝑝+𝑐^𝜇𝑘^𝜈₁+𝒜₄₁𝑝+𝑎^𝜈𝑘^𝜇₂+𝒜₄₂𝑝+𝑏^𝜈𝑘^𝜇₂+𝒜₄₃𝑝+𝑐^𝜈𝑘^𝜇₂+𝒜₅₁𝑘^𝜇₂𝑘^𝜈₁, (3) where the coefficients 𝒜_𝑖𝑗=𝒜_𝑖𝑗(𝑝^𝑛;𝑘₁,𝑘₂;𝑎;𝑏;𝑐) with −6≤𝑛≤0 are given in Appendix A, and the ∼ sign is the proportionality factor: −2𝑖𝑒²/𝑣. Note that ℳ^𝜇𝜈 is a (superficially) 6th power divergent amplitude in the unitary gauge. 𝒜₁₁ in (3) solely contributes to ℳ₁ while all other 𝒜-elements contribute to both ℳ₁ and/or ℳ₂ in (2).

First we focus on the calculation of the “less divergent” coefficient ℳ₂ of (2). Based on naive power counting, we observe that the 𝒜₂₁,𝒜₂₂,𝒜₂₃ terms in (3) lead to and the most quartic divergent integrals. However, when adding all these pieces together, we find that quartic divergent integrals vanish for every arbitrary vectors 𝑎,𝑏, and 𝑐 leaving behind an expression with integrals of third power (in momenta) plus integrals with smaller divergencies. Then the cubically divergent integrals are proportional to all possible Lorentz invariant combinations like [(𝑎+𝑏−2𝑐)⋅𝑝]𝑝^𝜇𝑝^𝜈, [(𝑎+𝑏−2𝑐)^𝜈𝑝^𝜇]𝑝², and [(𝑎+𝑏−2𝑐)^𝜇𝑝^𝜈]𝑝². Therefore, choosing 𝑎+𝑏−2𝑐=0, (4) we ensure that third-order divergent integrals related to 𝒜₂₁,𝒜₂₂,and𝒜₂₃ terms vanish identically. In the same way, by naive power counting, 𝒜₃₁ and 𝒜₃₃ terms—these terms in (3) together with 𝒜₃₂ contribute solely to ℳ₂—lead again to at most third-order divergent integrals. However, in the sum of 𝒜₃₁ and 𝒜₃₃ terms in (3), third-order divergent integrals vanish for arbitrary 𝑎,𝑏, and 𝑐, leading to an expression that when added to 𝒜₃₂ term consists of at most quadratically divergent integrals, proportional to [(𝑐−𝑎)⋅𝑝]𝑝^𝜇𝑘^𝜈₁ and [(𝑐−𝑎)^𝜇𝑘^𝜈₁]𝑝². We choose 𝑐−𝑎=0, (5) for the quadratically divergent integrals to vanish. Likewise, when we add 𝒜₄₂ and 𝒜₄₃ terms—these terms, together with 𝒜₄₁, solely contribute to ℳ₂ in (2)—the third-order divergent integrals vanish for every choice of 𝑎,𝑏,and𝑐 leading to an expression that when added to 𝒜₄₁ consists of at most quadratically divergent integrals proportional to [(𝑐−𝑏)⋅𝑝]𝑝^𝜈𝑘^𝜇₂ and [(𝑐−𝑏)^𝜈𝑘^𝜇₂]𝑝². Therefore, we choose 𝑐−𝑏=0, (6) for infinities to vanish identically. From (4), (5), and (6) we arrive at the final relation among the three introduced vectors: 𝑎=𝑏=𝑐. (7) Equation (7) suggests that the rest of the divergent integrals depend on, at most, one arbitrary vector, say the 𝑎-vector. Note that 𝒜₅₁ contributes only to the finite part of ℳ₂. Now, if we impose conditions (7) onto the remaining expressions for 𝒜₂₁,𝒜₂₂,…,𝒜₅₁ terms of (3), we find that all quadratically and linearly divergent integrals vanish, independently of the direction of the 𝑎-vector. We stress here on the fact that the cancellation of divergencies down to logarithmic ones is a highly non-trivial, almost “miraculous” result. These cancellations only take place for a particular choice of the momentum-variable shift vectors (7). (As a corollary, if for instance, we had split the 𝑊𝑊𝛾-vertex into three pieces, each one associated with three different arbitrary vectors, then the generalised condition (7) would again downgrade the divergency of the amplitude to a logarithmic one.) Of course this is an expected outcome for an observable in a renormalizable theory.

Our final result contains at most logarithmically divergent integrals. (This result is different from the one obtained in [32, 33], where there are remaining quadratically and linearly divergent terms. Following (11) in Gastmans et al. paper [32] or (3.36) in their sequel paper [33], and unless there is a typo in both their formulae, we find that there is a missing quadratically divergent term proportional to 𝑘²𝑘_𝜇𝑘_𝜈. Even in the case that this is a typo, their formulae contain a linearly divergent term that is also referred to by the authors claiming that this last term reduces to a logarithmically divergent integral by changing the internal momentum 𝑘→−𝑘 and further manipulating the integral. Note here that in our calculation all divergent integrals are reduced to at most logarithmically divergent ones without any further manipulation nor any assumption other than (7) for every arbitrary vector: 𝑎, 𝑏, and 𝑐. Finally, the authors in [32, 33] have made a specific choice for routing the internal loop momentum corresponding to 𝑎=𝑏=𝑐=(1/2)(𝑘₁+𝑘₂), and therefore their result should be the same as ours.) Despite of the fact that the resulting expressions so far contain the shift 𝑝+𝑎 instead of 𝑝 with an arbitrary vector 𝑎, its presence is irrelevant since logarithmically divergent integrals are momentum-variable shift independent [34]. Summing up all the above contributions to ℳ₂, we find a particularly nice and symmetric form for ℳ^𝜇𝜈:ℳ^𝜇𝜈∼𝑑⁴𝑝(2𝜋)⁴𝑝^𝜇𝑝^𝜈4(𝑑−1)𝑚²_𝑊+2𝑚²_𝐻𝑝²−𝑚²_𝑊𝑝−𝑘₁²−𝑚²_𝑊𝑝−𝑘₁−𝑘₂²−𝑚²_𝑊+4(𝑑−1)𝑚²_𝑊+2𝑚²_𝐻𝑝²−𝑚²_𝑊𝑝−𝑘₂²−𝑚²_𝑊𝑝−𝑘₁−𝑘₂²−𝑚²_𝑊+𝑑⁴𝑝(2𝜋)⁴𝑝^𝜇𝑘^𝜈₁−4(𝑑−1)𝑚²_𝑊−4𝑝⋅𝑘₂𝑝²−𝑚²_𝑊𝑝−𝑘₁²−𝑚²_𝑊𝑝−𝑘₁−𝑘₂²−𝑚²_𝑊+−4𝑝⋅𝑘₂𝑝²−𝑚²_𝑊𝑝−𝑘₂²−𝑚²_𝑊𝑝−𝑘₁−𝑘₂²−𝑚²_𝑊𝑑+⁴𝑝(2𝜋)⁴𝑝^𝜈𝑘^𝜇₂−4𝑝⋅𝑘₁𝑝²−𝑚²_𝑊𝑝−𝑘₁²−𝑚²_𝑊𝑝−𝑘₁−𝑘₂²−𝑚²_𝑊+−4(𝑑−1)𝑚²_𝑊−4𝑝⋅𝑘₁𝑝²−𝑚²_𝑊𝑝−𝑘₂²−𝑚²_𝑊𝑝−𝑘₁−𝑘₂²−𝑚²_𝑊+𝑑⁴𝑝(2𝜋)⁴𝑘^𝜈₁𝑘^𝜇₂6𝑚²_𝑊+2𝑝²𝑝²−𝑚²_𝑊𝑝−𝑘₁²−𝑚²_𝑊𝑝−𝑘₁−𝑘₂²−𝑚²_𝑊+6𝑚²_𝑊+2𝑝²𝑝²−𝑚²_𝑊𝑝−𝑘₂²−𝑚²_𝑊𝑝−𝑘₁−𝑘₂²−𝑚²_𝑊. (8)Introducing Feynman parameters, shifting momentum variable from 𝑝 to ℓ, and ignoring all terms (these terms will be used later in arriving at (24)) that contribute to ℳ₁, we find that the contribution to ℳ₂ in (2) arises solely from the term ℳ₂𝑘^𝜈₁𝑘^𝜇₂∼8¹₀𝑑𝑥₀^1−𝑥𝑑𝑑𝑦⁴ℓ(2𝜋)⁴×ℓ²𝑘^𝜈₁𝑘^𝜇₂−2ℓ⋅𝑘₂ℓ^𝜇𝑘^𝜈₁−2ℓ⋅𝑘₁ℓ^𝜈𝑘^𝜇₂ℓ²−Δ³+8𝑚²_𝑊¹₀𝑑𝑥₀^1−𝑥×𝑑𝑑𝑦₄ℓ(2𝜋)⁴3−2(𝑑−1)𝑥1−𝑥−𝑦ℓ²−Δ³𝑘^𝜈₁𝑘^𝜇₂, (9) with Δ=𝑥(𝑥+𝑦−1)𝑚²_𝐻+𝑚²_𝑊. Obviously, the first integral in (9) is (superficially) logarithmically divergent while the second one is finite. The number of dimensions (𝑑) appears only at the finite integral, and therefore we can fearlessly set 𝑑=4 everywhere. This means that we do not use dimensional regularisation in what follows (see however the discussion below). We state here few additional remarks to be exploited later on: (a) we observe that the top line in the integrand of (9) does not vanish in the limit 𝑚²_𝑊→0 and (b) despite of the appearances in (8), there is no 𝑚²_𝐻 in the numerators of the subsequent expression equation (9). The whole 𝑚²_𝐻 contribution arises from the denominator’s Δ-term.

Our next step is to parametrize the logarithmically divergent integral in (9) by an unknown, dimensionless parameter 𝜆 to be determined later by a physical argument. So we define¹₀𝑑𝑥₀^1−𝑥𝑑𝑑𝑦⁴ℓ(2𝜋)⁴×ℓ²𝑘^𝜈₁𝑘^𝜇₂−2ℓ⋅𝑘₂ℓ^𝜇𝑘^𝜈₁−2ℓ⋅𝑘₁ℓ^𝜈𝑘^𝜇₂ℓ²−Δ³≡𝑖𝜆4(4𝜋)²𝑘^𝜈₁𝑘^𝜇₂. (10)A parenthesis must be added at this point. We could of course promote 𝑑⁴ℓ→𝑑^𝑑ℓ and use dimensional regularisation [29] by exploiting symmetric integration ℓ^𝜇ℓ^𝜈→(1/𝑑)ℓ²𝑔^𝜇𝜈 in 𝑑-dimensions. In this case, and after taking the limit 𝑑→4, one finds 𝜆=−1 which is finite and nonzero and agrees with the one we find below in (20) after imposing the GBET condition. This is also the result found in the original references [11–13]. However, according to [32, 33], the integral in (10) is discontinuous at 𝑑=4; in fact, when symmetric integration ℓ^𝜇ℓ^𝜈→(1/4)ℓ²𝑔^𝜇𝜈 in 𝑑=4 is used, one finds instead 𝜆=0. This is also understood in a slightly different context. It has long been known [34–36] that shifts of integration variables in linearly (and above) divergent integrals are accompanied by “surface” terms that appear only in four dimensions, a famous example being the integrals in chiral anomaly triangle graphs. For our purpose here let us start with the following shift of variables in a linearly divergent integral that has been generalised [36] to work in 2𝜔-dimensions following the expression 𝑑^2𝜔ℓℓ_𝜇(ℓ−𝑘)²−Δ²−𝑑^2𝜔ℓ(ℓ+𝑘)_𝜇ℓ²−Δ²=−𝑖𝜋²2𝑘_𝜇𝛿_𝜔,2 (11) that is valid for 𝜔<5/2 and Δ constant, possibly dependent on Feynman parameters, like the one given below (9), and 𝑘_𝜇 is an arbitrary constant four-vector. By taking the derivative, 𝜕/𝜕𝑘^𝜈 of both sides in (11) shifting the integration variable for the logarithmically divergent integral encountered, and evaluating the finite one, we easily arrive at 𝑑^2𝜔ℓℓ²𝑔_𝜇𝜈−4ℓ_𝜇ℓ_𝜈ℓ²−Δ³=−𝑖𝜋²2𝑔_𝜇𝜈𝜋^𝜔−2Γ(3−𝜔)Δ^2−𝜔−𝛿_𝜔,2. (12) For an alternative and detailed proof of (12), see Appendix B. (The same result is obtained by standard algebraic tricks. We would like to thank R. Jackiw for communicating his calculation to us.) Applying (12) to ℓ^𝜎ℓ^𝜌 terms of (10) with 𝑑⁴ℓ/(2𝜋)⁴→𝑑^2𝜔ℓ/(2𝜋)^2𝜔, we find ().𝜆=0,𝜔=2−1,𝜔=2−𝜖DR (13) This is consistent with the symmetric integration in 4 dimensions (𝜔=2) but, is also consistent with the usual tabulated textbook result [37] from dimensional regularisation in 4−2𝜖-dimensions (𝜔=2−𝜖). Equation (13) shows that 𝜆 is discontinuous at 𝑑=2𝜔=4. Then the question arises: which 𝜆 to believe in? Answer: the one that is indicated by well defined, calculable, boundary conditions and symmetries of the underlying theory.

The above parenthesis to our calculation motivates us to avoid the direct calculation of integral (10), but set 𝑑=4 everywhere and treat 𝜆 as an unknown parameter to be defined later within a physical context or experiment. Substituting (10) into (9), we arrive at ℳ₂∼𝑖8𝜋²𝜆−6𝑚²_𝑊×¹₀𝑑𝑥₀^1−𝑥𝑑𝑦1−2𝑥1−𝑥−𝑦Δ. (14) Evaluating the double finite integral in (14) and restoring the proportionality factor given below (3), we obtain ℳ₂𝑒=−²𝑔(4𝜋)²𝑚_𝑊−2𝜆+3𝛽+3𝛽2−𝛽𝑓𝛽, (15) where 𝛽=4𝑚²_𝑊𝑚²_𝐻,𝑓𝛽=arctan²1−1𝛽−1,𝛽≥14ln1+1−𝛽1−1−𝛽−𝑖𝜋²,𝛽<1. (16)

Our final step is to determine the unknown parameter 𝜆 in (15). For this we need physics that reproduces ℳ₂ in a different and unambiguous way. One choice, probably not the only one, is to adopt the Goldstone Boson Equivalence Theorem (GBET) [20–22] which states that the amplitude for emission or absorption of a longitudinally polarised 𝑊 at high energy becomes equivalent to the emission or absorption of the Goldstone boson that was eaten. Mathematically, this is written by an [38] 𝑆𝑊^±_𝐿,physical=𝑖^𝑛𝑆𝑠^±,physical, (17) which says that the 𝑆-matrix elements for the scattering of the physical longitudinal vector bosons 𝑊_𝐿 with other physical particles are the same as the 𝑆-matrix elements of the theory where the 𝑊_𝐿’s have been replaced by physical Goldstone bosons (𝑠^±). We are not going to get into details here; apart from the original literature the reader is also referred to the articles [19, 38–40]. Following [38], within perturbation theory and in the limit of high energies, 𝑚²_𝑊/𝑠→0, GBET can be expressed with physics in two different limits of the theory: (a) 𝑔²/𝜆_𝐻→0 or (b) 𝑚²_𝐻/𝑠→0.

The limit (b) is irrelevant (the limit (b) simply says that matrix elements for the theory which contains the physical 𝑊_𝐿’s and zero v.e.v are equal to those produced by scattering of massless physical Goldstone bosons (instead of 𝑊_𝐿's) at high energies. We have checked that (17) is satisfied in this limit) for defining 𝜆 in (15) so we completely focus on the limit (a). It is very easy to see that, in the unitary gauge, the 𝑊_𝐿’s do not decouple (this is another advantage of calculating in the unitary gauge. Note that here the word “decoupling” is stated in a different context than is usually quoted following the Appelquist-Carazzone [41] theorem, where particle masses circulating in loops are much heavier than the external ones) for vanishing gauge coupling 𝑔. Consider for example the diagrams in Figure 1: there is always an 𝑚²_𝑊 from the 𝐻𝑊𝑊-vertex that cancels another 𝑚²_𝑊 sitting in the denominator of the longitudinal part for the internal 𝑊-boson propagator expression written in the unitary gauge. So, as it was already noted in the paragraph below (9), in the limit 𝑔→0, there are remaining nondecoupled terms. Unfortunately, these effects may be obscured or misjudged by the regularisation method needed to handle divergent, intermediate loop integrals. This is exactly what happens here when trying to calculate 𝜆 directly from its ambiguous form (10). On the other hand, however, at the exact 𝑔=0, with fixed v.e.v 𝑣 and Higgs quartic coupling 𝜆_𝐻, (17) suggests that the Goldstone bosons (𝑠^±) should reappear at the physical spectrum of the theory while the longitudinal components of 𝑊’s become unphysical. At this limit, the SM is a spontaneously broken global 𝑆𝑈(2)_𝐿×𝑈(1)_𝑌-symmetry that couples, minimally, to electromagnetism. The interactions between the Higgs and photon with the Goldstone bosons are simply those of a spontaneously broken scalar QED with 𝑈(1)_em: 𝐻𝑠⁺𝑠⁻∶−𝑖𝑚²_𝐻𝑣,𝛾𝑠⁺𝑝₁𝑠⁻𝑝₂∶−𝑖𝑒𝑝₁+𝑝₂^𝜇,𝛾𝛾𝑠⁺𝑠⁻∶2𝑖𝑒²𝑔^𝜇𝜈. (18)

Armed with these Feynman rules we calculate the diagrams in Figure 2. By doing so, we introduce again three momentum variable shift vectors, one for each diagram, exactly in the same way we did for the calculation of the diagrams in Figure 1. The Lorentz structure of the amplitude is completely analogous to (2) with ℳ_1,2→ℳ_1,2(GBET), but now due to the scalar propagators, the superficial degree of divergence, for diagrams contributing to ℳ_2(GBET), is 𝐷=−2. Hence, all integrals involved in ℳ_2(GBET) are finite and in addition, they are independent of any momentum integration shift vector variable. As a consequence, ℳ_2(GBET) is well defined, calculable, independent of any regularisation method, and at the limit of 𝑔→0 (or 𝛽=4𝑚²_𝑊/𝑚²_𝐻→0) is ℳ_2(GBET)=−2𝑒²𝑔(4𝜋)²𝑚_𝑊,𝛽⟶0. (19) By equating (15) (in the limit 𝛽→0) and (19) which represent the l.h.s and r.h.s of the GBET condition (17), respectively, we find 𝜆=−1. (20) This value agrees with dimensional regularization [29, 30] in the limit 𝑑→4 (see (13)). The final form of the ℳ₂ in (2) is ℳ₂𝑒=−²𝑔(4𝜋)²𝑚_𝑊2+3𝛽+3𝛽2−𝛽𝑓𝛽, (21) with 𝛽 and 𝑓(𝛽) defined in (16).

Figure 2 

Charged Goldstone boson contributions to 𝐻→𝛾𝛾 in the limit of 𝑔→0.

To complete the picture there is still the coefficient ℳ₁ in (2) to be calculated. Naive power counting says that this is by two powers more divergent than ℳ₂ and, in general, undetermined. It can be fixed however by using quantum gauge invariance, that is, conservation of charge, for the 𝑈(1)_em: 𝑘_1𝜇ℳ^𝜇𝜈𝑘=0,_2𝜈ℳ^𝜇𝜈𝑘=0,₂₁=𝑘²₂=0, (22) and thus from (2), ℳ₁=−𝑘₁⋅𝑘₂ℳ₂. (23) Equation (23) is substituted to (2) with ℳ₂ read by (21). This is exactly the same result for the 𝑊-boson contribution to 𝐻→𝛾𝛾 amplitude that has been obtained in [11–14, 19] using dimensional regularisation in 𝑅_𝜉-gauges.

It is interesting here to note the result from the explicit algebraic manipulation of ℳ₁ in the unitary gauge and check the validity of gauge invariance (see (23)). Exactly as for ℳ₂, the condition 𝑎=𝑏=𝑐 for the arbitrary vectors given in (7) is crucial in reducing the divergence of ℳ₁ down to a logarithmic one (see expression equation (A.2)). In 𝑑-dimensions the expression for ℳ₁ is finally independent of any arbitrary vector and, up to a proportionality factor, readsℳ₁∼4¹₀𝑑𝑥₀^1−𝑥𝑑𝑑𝑦^𝑑ℓ(2𝜋)^𝑑4ℓ⋅𝑘₁ℓ⋅𝑘₂+2(2/𝑑−1)ℓ²𝑘₁⋅𝑘₂+((𝑑−1)/𝑑)(4−𝑑)ℓ²𝑚²_𝑊ℓ²−Δ³++(𝑑−1)𝑚⁴_𝑊−3𝑚²_𝑊𝑚²_𝐻+(1−𝑑)𝑥𝑥+𝑦−1𝑚²_𝑊𝑚²_𝐻ℓ²−Δ³. (24)Clearly the first integral in (24) is ill-defined in four dimensions. If, however, we insist on doing the calculation of (24) in 𝑑=4 with symmetric integration, like in [32, 33], we find that gauge invariance (see (23)) is not satisfied. This is of course unacceptable. By going a little bit deeper, gauge invariance is lost because of the term proportional to 4−𝑑 in (24) which vanishes when 𝑑=4. Quite the contrary in DR, this term results in a non-zero contribution when 𝑚_𝑊≠0 since the (log divergent) integral in front of (4−𝑑) contains a simple pole at 𝑑=4. This changes the final result and renders (21), (24), and (23) consistent only if 𝜆=−1. This outcome is in agreement with [19].

Few remarks are worth mentioning here. Had we started first calculating ℳ₁, there would be no possibility of defining unambiguously 𝜆 without using a gauge invariant regulator: the 𝑔_𝜇𝜈 part of the amplitude at 𝑔→0 involving Goldstone bosons (see diagrams Figure 2) is not well defined—an integral as the one in (12) appears again. Another remark is that the same expressions for the coefficients 𝒜_𝑖𝑗 displayed in Appendix A in the unitary gauge appear also when one exploits the 𝑅_𝜉-gauge. In the latter there are in addition 𝜉-dependent terms [19] that vanish in the end from unphysical scalar contributions. Therefore, the logarithmic ambiguity in (12), found here in the unitary gauge, is similar in every other gauge.

The ambiguity of the integral in (12) has been discussed by many articles in the recent literature. References [15, 16] have used gauge-invariant regulators alá Pauli-Villars. Most notably, Piccinini et al. [18] showed that the unitary gauge with a cut-off regularisation scheme turns out to be nonpredictive: new physics input, along the lines of [28] also followed in our article, is needed. As already noted, Marciano et al. [19] were the first to make the calculation of ℎ→𝛾𝛾 in unitary gauge with DR. Furthermore, [15, 18, 19, 39, 40] showed that the “decoupling limit” 𝑚_𝑊/𝑚_𝐻→0  must hold because of the GBET. All these articles together with the one at hand conclude that the result in [32, 33] is incorrect. Furthermore, the aim of our paper is not to redo the calculation in the unitary gauge with DR as in [19]; this is only a by-product of our analysis. On the contrary, we want to clarify subtle issues related to this calculation in the unitary gauge and in 𝑑=4 raised in part by [32, 33]. We find by working strictly in 𝑑=4, using arbitrary vectors, that divergencies (up to the 6th power) are reduced down to logarithmic ones. This is a new result that is not obvious when working in the unitary gauge and cannot be seen when using dimensional regularisation.

3. Four-Dimensional Regularization (FDR)

So far we have proposed a regularization scheme which is four-dimensional and uses the basic symmetries and underlying physics of the SM. However, in more complicated models or observables with more parameters to adjust, such a scheme can become cumbersome. For example, it is not always obvious which physics argument will fix undefined integrals.

Very recently, Pittau [31] proposed a scheme that is fairly easy to handle and, to the best of our knowledge, is the closest to four-dimensional calculations and thereby coined four-dimensional regularisation/renormalization scheme or just FDR. According to this scheme, infinite bubble graph contributions, that is, large loop momenta contributions that do not depend upon external momenta, are absorbed into the shift of the vacuum while the remaining finite corrections are calculable in four dimensions in addition to being Lorentz and gauge invariant.

We have applied FDR to the calculation of the 𝐻→𝛾𝛾 amplitude and found agreement with our physics approach and with DR results. In FDR one introduces an arbitrary scale 𝜇 which is considered to be much smaller than internal momenta and particle masses in loops. Self-contracted loop momenta quantities like ℓ² become ℓ²=ℓ²−𝜇², while, for gauge invariance to hold, vector momenta, 𝑝^𝜇, remain untouched. For example the integral of (12) becomes 𝑑⁴ℓℓ²𝑔_𝜇𝜈−4ℓ_𝜇ℓ_𝜈𝐷³=𝑑⁴ℓ−𝜇²𝐷³𝑔_𝜇𝜈, (25) where 𝐷=(ℓ²−Δ) and [𝑑⁴ℓ] stands for integration over 𝑑⁴ℓ, dropping all divergent terms from the integrand (see below) and taking the limit 𝜇→0. In going from l.h.s to r.h.s of (25) the symmetry property ℓ_𝜇ℓ_𝜈=𝑔_𝜇𝜈ℓ²/4 has been used in four dimensions. Then, using the partial fractions identity, 1𝐷³=1ℓ⁶1+Δ𝐷³ℓ²+1𝐷²ℓ⁴+1𝐷ℓ⁶, (26) the term in square bracket is recognised as divergent and therefore removed, and integrating the r.h.s of (25) over [𝑑⁴ℓ] one obtains 𝑑⁴ℓ−𝜇²𝐷³≡−Δlim_𝜇→0𝜇²𝑑⁴ℓ1𝐷³ℓ²+1𝐷²ℓ⁴+1𝐷ℓ⁶=−𝑖𝜋²2, (27) that is, exactly the same result as in DR which eventually leads to 𝜆=−1 consistent with gauge invariance and GBET. What in fact FDR scheme does is to restate the correct DR answer through the regulator 𝜇² keeping (12) correct in 𝑑=4. We therefore understand that the constant (𝛽-independent) term of (21) in FDR arises from the fact that the arbitrary scale, 𝜇², must disappear from physical observables.

4. Discussion

It is evident that our calculation for the amplitude incorporates two physical inputs: one is the conservation of charge and the other is the equivalence theorem. They are both direct consequences of the gauge invariance of the underlying physical theory. The first one is experimentally indisputable while the second one is theoretical (this is not entirely correct. There is of course the high energy behaviour of 𝑒⁺𝑒⁻→𝑊⁺𝑊⁻ found at LEP [42] consistent with the GBET.) and has been proven in [43], which is valid in any spontaneously broken renormalizable theory, like, for example, the SM. One may think however that there is a loophole in our use of this second argument: so far, and, to our knowledge, the replacement of the 𝑊-bosons with Goldstone bosons at high energies has been proven to be valid only for external 𝑊-bosons [38, 44, 45] and not for internal ones which is the case exploited here. Although it has been tested in several phenomenological examples [46], a formal, to all orders, proof is still missing. Although this may be true, it is difficult to argue against the validity of decoupling limit 𝑔→0 (with fixed v.e.v and Higgs quartic coupling) discussed in the paragraph above (18).

Is there another physics context from which one can define 𝜆? One possibility is to exploit the low-energy Higgs theorem [11, 47–49] instead. Although this may serve as a consistency check and indeed is compatible with 𝜆=−1, we cannot use it to define 𝜆. The reason here is threefold: first, when treating the Higgs field as an external background field with zero momentum one needs to take partial derivative w.r.t 𝑚_𝑊 of the 2-point photon vacuum polarization amplitude, Π_𝛾𝛾(𝑞²). The latter is notoriously difficult, if meaningful, to be calculated in the unitary gauge. Second, according to [11], we know that to the lowest order in weak coupling, the amplitude for the process ⟨𝛾𝛾|𝐻⟩ is proportional to ⟨𝛾𝛾|Θ^𝜇_𝜇|0⟩, where Θ^𝜇_𝜇=2𝑚²_𝑊𝑊⁺𝑊⁻+⋯ is the improved energy momentum tensor [50]. However, the calculation of ⟨𝛾𝛾|Θ^𝜇_𝜇|0⟩ goes through the same steps as for the calculation for the 𝐻→𝛾𝛾 amplitude and therefore involves the same ambiguity for calculating 𝜆. Third, one could examine the 𝑊-contribution to 𝐻→𝛾𝛾 within the dispersion relation approach. It can be shown [51] that the nonvanishing limit at 𝑔_𝑊→0 is due to a finite subtraction induced by the corresponding trace anomaly [52]. However, in order to calculate unambiguously this finite piece, one has to make full use of a (physical) boundary condition of the theory.

As a final remark, suppose that we did not know DR and wanted to calculate a certain observable in 4 dimensions. In this observable we encounter singularities, that is, undefined and undetermined integrals. Then we use physics arguments to fix these ambiguities. However, we can always question whether we are using the right physics setup or not. In that sense the final judgement should come from the experiment. Therefore it may be not academic to ask whether LHC could see the difference between 𝜆=−1 and 𝜆=0. Setting the SM Higgs mass 𝑚_𝐻=125 GeV and including the top-loop contribution, we find Br𝐻⟶𝛾𝛾,𝜆=0Br𝐻⟶𝛾𝛾,𝜆=−1≈0.46. (28) This is certainly within LHC’s sensitivity for 14 TeV c.m energy and luminosity of 30 fb⁻¹ (see, e.g., Figure 3 in [53]). In fact, the recent observation by LHC experiments [6, 7] indicates a value Br(𝐻→𝛾𝛾,(exp))/Br(𝐻→𝛾𝛾,𝜆=−1)=1.6±0.3 [54] which highly disfavours the case 𝜆=0 by almost four standard deviations. We can turn this around and state that this is an indirect hint towards the validity of the equivalence theorem.

5. Conclusions

In this work we review the 𝑊-gauge boson loop contribution to the 𝐻→𝛾𝛾 amplitude in the unitary gauge. Our objective is to fix intermediate step indeterminacies arising from divergent diagrams by making full use of physics at 𝑑=4 much in the same way as in the calculation of the chiral anomaly triangle.

We anticipate a finite result for the loop-induced 𝐻→𝛾𝛾-amplitude in the renormalizable SM. Therefore the amplitude has to be independent of any shifting momentum variables we have originally introduced. But finite or even log divergent integrals are independent of these vectors, so the vectors have to be accompanied only by infinite contributions, if at all. Therefore, infinities and arbitrary vectors are eliminated altogether by a certain combination among them (see (7)).

The whole calculation in the unitary gauge boils down to a logarithmically divergent integral (10). We find that this integral results in two different values depending on whether 𝑑→4 or 𝑑=4. This is due to a surface term remaining at the exact 𝑑=4 case after the part-by-part integration in 𝑑-dimensions (see Appendix B). To proceed, we identify this integral with an undefined parameter 𝜆 (see (12)). This parameter is then fixed unambiguously by assuming the validity of the Goldstone Boson Equivalence Theorem (GBET). Its value is consistent with DR in the limit 𝑑→4.

In our calculation we are very careful not to perform shifting of integration variables for highly divergent integrals by introducing three arbitrary momentum variable shift vectors straight from the beginning. Divergencies and arbitrariness from these unknown vectors are altogether removed, leaving behind a log-like divergent integral in ℳ₂ of (2). This is defined by a physical input taken from the GBET and is connected to ℳ₁ by electromagnetic charge conservation.

As noted many times in the text, the key point towards deriving an unambiguous amplitude for 𝐻→𝛾𝛾 in the unitary gauge is the limit of vanishing gauge couplings; this is an aspect of GBET (see (17)). In this limit, the Goldstone boson loop contributions to the coefficient ℳ₂ finite, that is, independent of any regularisation scheme.

We also saw that DR (FDR), a regularisation scheme introduced to maintain Ward Identities at intermediate steps of a calculation, supports the GBET in the limit 𝑑→4 (𝑑=4). On the contrary, we find that performing the integrals in 𝑑=4 with symmetric integration is not a good choice because it leads to the violation of gauge invariance (see (23) and the discussion below). The main reason is due to surface terms that are developed in exactly 𝑑=4 dimensions (see discussion below (10) and Appendix B). The latter are axiomatically discarded in DR [29, 30]. Another reason is the appearance of the (𝑑−4)-term in the numerator of (24).

In conclusion, the four-dimensional calculation of 𝐻→𝛾𝛾 amplitude in the unitary gauge is ambiguous without introduction of a physics input beyond gauge invariance. As we have demonstrated, this physics, which uniquely defines the amplitude, may arise from the Goldstone Boson Equivalence Theorem (GBET). This effectively proves that GBET comprises an additional important pillar of the Standard Model dynamics.

Appendices

A. The Coefficients 𝒜_𝑖𝑗

We append here the integrand expressions for the coefficients 𝒜_𝑖𝑗 in (3). The corresponding formula for 𝒜₁₁ is quite long and is not included here. It can be provided by the authors upon request. Note that the number of dimensions 𝑑 has been kept arbitrary throughout, and on-shell conditions for the external particles have been imposed: 𝒜₂₁=1×𝑝+𝑎²−𝑚²_𝑊𝑝+𝑎−𝑘₁²−𝑚²_𝑊×𝑝+𝑎−𝑘₁−𝑘₂²−𝑚²_𝑊⁻¹×(4𝑑−6)𝑚²_𝑊+3𝑝+𝑎⋅𝑝+𝑎−5𝑝+𝑎⋅𝑘₁−𝑝+𝑎⋅𝑘₂+2𝑚²_𝐻+1𝑚²_𝑊−𝑝+𝑎⋅𝑝+𝑎²+3𝑝+𝑎⋅𝑘₁×𝑝+𝑎⋅𝑝+𝑎−2𝑝+𝑎⋅𝑘₁²−2𝑝+𝑎⋅𝑘₁𝑝+𝑎⋅𝑘₂+𝑝+𝑎⋅𝑝+𝑎𝑝+𝑎⋅𝑘₂,𝒜₂₂=1×𝑝+𝑏²−𝑚²_𝑊𝑝+𝑏−𝑘₂²−𝑚²_𝑊×𝑝+𝑏−𝑘₁−𝑘₂²−𝑚²_𝑊⁻¹×(4𝑑−6)𝑚²_𝑊+3𝑝+𝑏⋅𝑝+𝑏−5𝑝+𝑏⋅𝑘₂−𝑝+𝑏⋅𝑘₁+2𝑚²_𝐻+1𝑚²_𝑊−𝑝+𝑏⋅𝑝+𝑏²+3𝑝+𝑏⋅𝑘₂×𝑝+𝑏⋅𝑝+𝑏−2𝑝+𝑏⋅𝑘₂²−2𝑝+𝑏⋅𝑘₁𝑝+𝑎⋅𝑘₂+𝑝+𝑏⋅𝑝+𝑏𝑝+𝑏⋅𝑘₁,𝒜₂₃=−1𝑝+𝑐²−𝑚²_𝑊𝑝+𝑐−𝑘₁−𝑘₂²−𝑚²_𝑊×24+𝑚²_𝑊−⋅+𝑝+𝑐𝑝+𝑐𝑝+𝑐⋅𝑘₁+𝑝+𝑐⋅𝑘₂,𝒜₃₁=1×𝑝+𝑎²−𝑚²_𝑊𝑝+𝑎−𝑘₁²−𝑚²_𝑊×𝑝+𝑎−𝑘₁−𝑘₂²−𝑚²_𝑊⁻¹×(7−4𝑑)𝑚²_𝑊−4𝑝+𝑎⋅𝑝+𝑎−7𝑝+𝑎⋅𝑘₁+3𝑝+𝑎⋅𝑘₂+1𝑚²_𝑊𝑝+𝑎⋅𝑝+𝑎²−3𝑝+𝑎⋅𝑘₁𝑝+𝑎⋅𝑝+𝑎+2𝑝+𝑎⋅𝑘₁²+2𝑝+𝑎⋅𝑘₁×𝑝+𝑎⋅𝑘₂−⋅×𝑝+𝑎𝑝+𝑎𝑝+𝑎⋅𝑘₂,𝒜₃₂=−1×𝑝+𝑏²−𝑚²_𝑊𝑝+𝑏−𝑘₂²−𝑚²_𝑊×𝑝+𝑏−𝑘₁−𝑘₂²−𝑚²_𝑊⁻¹×𝑚²_𝑊+−𝑝+𝑏⋅𝑝+𝑏+6𝑝+𝑏⋅𝑘₂,𝒜₃₃=1𝑝+𝑐²−𝑚²_𝑊𝑝+𝑐−𝑘₁−𝑘₂²−𝑚²_𝑊×12−𝑚²_𝑊⋅−𝑝+𝑐𝑝+𝑐𝑝+𝑐⋅𝑘₁−𝑝+𝑐⋅𝑘₂,𝒜₄₁=−1×𝑝+𝑎²−𝑚²_𝑊𝑝+𝑎−𝑘₁²−𝑚²_𝑊×𝑝+𝑎−𝑘₁−𝑘₂²−𝑚²_𝑊⁻¹×𝑚²_𝑊+−𝑝+𝑎⋅𝑝+𝑎+6𝑝+𝑎⋅𝑘₁,𝒜₄₂=1×𝑝+𝑏²−𝑚²_𝑊𝑝+𝑏−𝑘₂²−𝑚²_𝑊×𝑝+𝑏−𝑘₁−𝑘₂²−𝑚²_𝑊⁻¹×(7−4𝑑)𝑚²_𝑊−4𝑝+𝑏⋅𝑝+𝑏−7𝑝+𝑏⋅𝑘₂+3𝑝+𝑏⋅𝑘₁+1𝑚²_𝑊𝑝+𝑏⋅𝑝+𝑏²−3𝑝+𝑏⋅𝑘₂𝑝+𝑏⋅𝑝+𝑏+2𝑝+𝑏⋅𝑘₂²+2𝑝+𝑏⋅𝑘₁×𝑝+𝑏⋅𝑘₂−⋅×𝑝+𝑏𝑝+𝑏𝑝+𝑏⋅𝑘₁,𝒜₄₃=𝒜₃₃,𝒜₅₁=1×𝑝+𝑎²−𝑚²_𝑊𝑝+𝑎−𝑘₁²−𝑚²_𝑊×𝑝+𝑎−𝑘₁−𝑘₂²−𝑚²_𝑊⁻¹×5𝑚²_𝑊+3𝑝+𝑎⋅𝑝+𝑎−2𝑝+𝑎⋅𝑘₁+1×𝑝+𝑏²−𝑚²_𝑊𝑝+𝑏−𝑘₂²−𝑚²_𝑊×𝑝+𝑏−𝑘₁−𝑘₂²−𝑚²_𝑊⁻¹×5𝑚²_𝑊+3𝑝+𝑏⋅𝑝+𝑏−2𝑝+𝑏⋅𝑘₂−2×𝑝+𝑐²−𝑚²_𝑊×𝑝+𝑐−𝑘₁−𝑘₂²−𝑚²_𝑊⁻¹. (A.1) It is straightforward, but long and tedious, to show that after implementing the condition (7) to coefficients in (A.1) we arrive at (8) which is at most logarithmically divergent.

For complementarity reasons, it is useful in deriving (24) to present the expression for the coefficient 𝒜₁₁ after the imposition of the arbitrary vector relation equation (7): 𝒜₁₁=1×𝑝+𝑎²−𝑚²_𝑊𝑝+𝑎−𝑘₁²−𝑚²_𝑊×𝑝+𝑎−𝑘₁−𝑘₂²−𝑚²_𝑊⁻¹×𝑝+𝑎−𝑘₁²−𝑚²_𝑊(1−𝑑)𝑚²_𝑊+4𝑝+𝑎⋅𝑘₁𝑝+𝑎⋅𝑘₂−3𝑚²_𝑊+𝑝+𝑎²𝑚²_𝐻+1×𝑝+𝑎²−𝑚²_𝑊𝑝+𝑎−𝑘₂²−𝑚²_𝑊×𝑝+𝑎−𝑘₁−𝑘₂²−𝑚²_𝑊⁻¹×𝑝+𝑎−𝑘₂²−𝑚²_𝑊(1−𝑑)𝑚²_𝑊+4𝑝+𝑎⋅𝑘₁𝑝+𝑎⋅𝑘₂−3𝑚²_𝑊+𝑝+𝑎²𝑚²_𝐻. (A.2) This integrand expression, under ∫𝑑⁴𝑝, is obviously at most logarithmically divergent.

B. Dimensional Regularization and the Surface Term

We would like to examine the surface terms arising in 𝑑=4 when calculating the integral on the l.h.s of (12). This integral after Wick rotation into Euclidean space reads 𝑖𝑑^2𝜔ℓℓ²𝑔_𝜇𝜈−4ℓ_𝜇ℓ_𝜈ℓ²+Δ³, (B.1) where ℓ≡ℓ_𝐸 and drop for clarity the subscript 𝐸 from now on. We follow very closely’t Hooft and Veltman’s seminal paper in [29]. In our calculation for a physical process we should notice first that ℓ_𝜇,ℓ_𝜈 are strictly 4-vectors since they are contracted with physical external momenta 𝑘^𝜇_1,2 or 𝑘^𝜈_1,2. On the other hand, the loop momentum ℓ in ℓ² has components in all, 𝑑=2𝜔, dimensions. We write ℓ as a sum of a vector ℓ_‖ which has non-zero components in dimensions 0, 1, 2, and 3 and a vector ℓ_⟂ which has nonzero components in (2𝜔−4)-dimensions: ℓ=ℓ_‖+ℓ_⟂. (B.2) With this definition, the integral (B.1) reduces to 𝑖𝑑^2𝜔ℓℓ²_⟂𝑔_𝜇𝜈ℓ²+Δ³, (B.3) where the ℓ_‖ components in the numerator of (B.1) vanish thanks to symmetric integration formula, ℓ^𝜇_‖ℓ^𝜈_‖→(1/4)ℓ²_‖𝑔^𝜇𝜈. In order not to carry the 𝑔_𝜇𝜈 in all formulae below we just concentrate on the integral ℐ≡𝑖𝑑^2𝜔ℓℓ²_⟂ℓ²+Δ³=𝑖𝑑⁴ℓ_‖𝑑^2𝜔−4ℓ_⟂ℓ²_⟂ℓ²_‖+ℓ²_⟂+Δ³. (B.4) Integrating over the extra-dimensional solid angle 𝑑Ω_2𝜔−4 we arrive at ℐ=2𝑖𝜋^𝜔−2Γ()𝑑𝜔−2⁴ℓ_‖^∞₀𝐿𝑑𝐿^2𝜔−3ℓ²_‖+𝐿²+Δ³, (B.5) where Γ(𝑥) is the Euler Γ-function and 𝐿 is the length of the ℓ_⟂ vector. This integral is UV divergent for 𝜔≥2 and IR divergent for 𝜔≤1. Therefore, the region of convergence, 1<𝜔<2, is finite, but it does not yet include the point 𝜔=2. In order to enlarge the region of convergence to include 𝜔=2 one has to analytically continue ℐ by inserting the identity 11=5𝜕ℓ_‖𝜇𝜕ℓ_‖𝜇+𝜕𝐿,𝜕𝐿 (B.6) in (B.5). After integrating by parts in the region of convergence, rewriting the r.h.s in terms of ℐ from (B.5), and keeping only, potentially, nonvanishing surface terms, we arrive at ℐ=𝑖𝜋^𝜔−2Γ(4−𝜔)4𝑑³𝑆^𝜇ℓ_‖𝜇ℓ²_‖+Δ^4−𝜔−6𝑖𝜋^𝜔−2ΔΓ()𝑑𝜔−1⁴ℓ_‖^∞₀𝐿𝑑𝐿^2𝜔−3ℓ²_‖+𝐿²+Δ⁴, (B.7) where the first integral is over the Euclidean spatial components of a 4-vector on a three-sphere. The surface integral converges in 1<𝜔<2 while the other in 1<𝜔<3. By taking the surface integral on a three-sphere with radius 𝑅 and eventually taking the limit 𝑅→∞ we find 𝑑³𝑆^𝜇ℓ_‖𝜇ℓ²_‖+Δ^4−𝜔=2𝜋²lim_𝑅→∞𝑅^2𝜔−4, (B.8) which now converges in the region 𝜔≤2 that is, it includes the point 𝜔=2. For 𝜔<2 this surface term vanishes while for 𝜔=2 there is a finite piece, 2𝜋², remaining. This is exactly the term that spoils gauge invariance and the equivalence theorem. In DR this term is axiomatically absent—allowing the shifting of integral momenta is among DR’s main properties.

Turning into the second integral of (B.7) we note first that the region of convergence includes now 𝜔=2. It gives 𝑑⁴ℓ_‖^∞₀𝐿𝑑𝐿^2𝜔−3ℓ²_‖+𝐿²+Δ⁴=𝜋²Γ()Γ()12𝜔−13−𝜔Δ^3−𝜔. (B.9) By placing (B.8) and (B.9) into (B.7) we finally arrive at (12).

Acknowledgments

The authors are grateful to Adrian Signer for illuminating discussions and critical comments on the manuscript. The authors would also like to thank R. Pittau, R. Jackiw, F. Piccinini, P. Kanti, S. Martin, and K. Tamvakis for discussions and comments. The authors also thank C. Coriano for bringing to our attention [51]. This research Project is cofinanced by the European Union—European Social Fund (ESF) and National Sources, in the framework of the program “ARISTEIA” of the “Operational Program Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) 2007–2013. Kristaq Suxho acknowledges full the financial support from the Greek State Scholarships Foundation (I.K.Y).