Anatomy of the Higgs boson decay into two photons in the unitary gauge

In this work, we review and clarify computational issues about the W-gauge boson one-loop contribution to the H ->gamma gamma 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.


Introduction
Today one of the main focal points at the Large Hadron Collider (LHC) is to search for the Higgs boson (H) [1][2][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, that could be the Standard Model (SM) Higgs boson, is based on data mainly driven by H → γγ. In the (SM) [8][9][10], this particular decay process goes through loop induced diagrams involving either charged fermions or W -gauge bosons. Their calculation was first performed in ref. [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 non-linear gauges [12][13][14], different regularisation schemes [15][16][17][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][21][22] since the SM is a renormalizable, spontaneously broken, gauge field theory.
A problem arises when the W -gauge boson contribution (see Fig. 1) to H → γγ produces "infinite" results at intermediate steps. These problems are usually treated by using a gauge invariant regulator method, e.g., dimensional regularization. In the unitary gauge [23], this indeterminacy is more pronounced and more difficult to handle with 1 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.
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 non-invariant 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" cancel higher powers of infinities leaving still behind at most logarithmically divergent integrals that are treated as undetermined variables.
3. We exploit physics, i.e., 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 J. Ellis et.al [11] and by M. 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 refs. [29,30] and then taking the limit d → 4, and finally third by using a four-dimensional regularization scheme introduced in ref. [31].
Our calculation is complementary to, but somewhat different than, 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 ref. [19]. On the contrary, we want to clarify subtle issues related to the amplitude in unitary gauge and d = 4 raised in part by refs. [32,33]. We find, using arbitrary vectors, that divergencies (up to 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 refs. [32,33].
The outline of the paper is as following: in section 2 we present the calculation of the W -loop contribution 2 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 : A that contains some intermediate formulae and 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, where k 1 and k 2 are the outgoing photon momenta shown in Fig. 1, and the coefficients M i=1..5 ≡ M i=1..5 (k 1 , k 2 ) are scalar functions of k 2 1 , k 2 2 , and k 1 · k 2 . By considering that all particles are on-mass-shell, that is k 2 with only two, undetermined (for the time being), coefficients, In unitary gauge, the Feynman diagrams that contribute to M 1 and M 2 are displayed in Fig. 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, i.e., p → p + a for the first diagram, p → p + b for the second diagram and p → p + c for the third diagram. As we shall 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 k 1 and k 2 . Hence a, b and c are not linearly independent [c.f. eq. (2.4)]. This is an important fact leading to the cancellation of infinities.
We first calculate the less divergent part of M µν in eq. (2.2) which is the M 2 coefficient 3 . By naive power counting, we see that M 2 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 i.e., g µν g µν = d. As we will see, d contributes only in finite pieces of M 2 [c.f. eq. (2.9)] 4 . With all the above definitions, we can write down the total amplitude in the form The coefficient M1 will be fixed later on by the requirement of gauge invariance. 4 On the contrary, we shall see that there are non-trivial d-contributions into M1-coefficient.
where the coefficients A ij = A ij (p n ; k 1 , k 2 ; a; b; c) with −6 ≤ n ≤ 0, are given in Appendix A, and the ∼ sign is the proportionality factor: − 2ie 2 v . Note that M µν is a (superficially) 6th power divergent amplitude in the unitary gauge. A 11 in eq. (2.3) solely contributes to M 1 while all other A-elements contribute to both M 1 and/or M 2 in eq. (2.2).
First we focus on the calculation of the "less divergent" coefficient M 2 of eq. (2.2). Based on naive power counting, we observe that the A 21 , A 22 , A 23 -terms in eq. (2.3), lead to at 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: for the quadratically divergent integrals to vanish. Likewise, when we add A 42 and A 43 -termsthese terms, together with A 41 , solely contribute to M 2 in eq. (2.2) -the third order divergent integrals vanish for every choice of a, b, c leading to an expression, that when added to A 41 , consists of at most quadratically divergent integrals proportional to 3), we find that all quadratically and linearly divergent integrals vanish, independently of the direction of the a-vector. We stress here 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, [eq. (2.7)] 5 . Of course this is an expected outcome for an observable in a renormalizable theory.
Our final result contains at most logarithmically divergent integrals. 6 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 M 2 , we find a particularly nice and symmetric form for M µν , . (2.8) Introducing Feynman parameters, shifting momentum variable from p to and ignoring all terms 7 that contribute to M 1 we find that the contribution to M 2 in eq. (2.2) arises solely from the term, Obviously, the first integral in eq. (2.9) 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 eq. (2.9) does not vanish in the limit m 2 W → 0 and, b) despite of appearances in eq. (2.8), there is no m 2 H in the numerators of the subsequent expression eq. (2.9). The whole m 2 H contribution arises from the denominator's ∆-term. Our next step is to parametrize the logarithmically divergent integral in eq. (2.9) by an unknown, dimensionless, parameter λ to be determined later by a physical argument. So we define, (2.10) An important parenthesis here. We could of course promote d 4 → d d and use dimensional regularisation [29] by exploiting symmetric integration µ ν → 1 d 2 g µν in d-dimensions. In this case, and after taking the limit d → 4, one finds λ = −1 which is finite and non-zero, and, agrees with the one we find below in eq. (2.20) after imposing the GBET condition. This is also the result found in the original refs. [11][12][13]. However, according to refs. [32,33], the integral in eq. (2.10) is discontinuous at d = 4; in fact, when symmetric integration, µ ν → 1 4 2 g 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][35][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 lets 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, that is valid for ω < 5/2 and ∆ constant, possibly dependent on Feynman parameters, like the one given below eq. (2.9), and k µ is an arbitrary constant four vector. By taking the derivative, ∂ ∂k ν , of both sides in eq. (2.11) and shifting the integration variable for the logarithmically divergent integral encountered, and evaluating the finite one, we easily arrive at For an alternative and detailed proof of eq. (2.12), see Appendix B. 8 Applying eq. (2.12) to σ ρ terms of eq. (2.10) with d 4 (2π) 4 → d 2ω (2π) 2ω , we find, .

(2.13)
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 4−2 -dimensions (ω = 2 − ). Eq. (2.13) 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 (2.10) but set d = 4 everywhere and treat λ as an unknown parameter to be defined later within a physical context or experiment. Substituting eq. (2.10) into eq. (2.9) we arrive at (2.14) Evaluating the double finite integral in eq. (2.14), and restoring the proportionality factor given below eq. (2.3), we obtain, , and, f (β) = arctan 2 Our final step is to determine the unknown parameter λ in eq. (2.15). For this we need physics that reproduces M 2 in a different and unambiguous way. One choice, probably not the only one, is to adopt the Goldstone Boson Equivalence Theorem (GBET) [20][21][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 equation [38], 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][39][40]. Following ref. [38], within perturbation theory and in the limit of high energies, m 2 W /s → 0, Figure 2: Charged Goldstone boson contributions to H → γγ in the limit of g → 0.
GBET can be expressed with physics in two different limits of the theory: (a) g 2 /λ H → 0, or (b) m 2 H /s → 0. The limit (b) is irrelevant 9 for defining λ in eq. (2.15) 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 10 for vanishing gauge coupling g. Consider for example the diagrams in Fig. 1: there is always a m 2 W from the HW W -vertex that cancels another m 2 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 eq. (2.9), in the limit g → 0 there are remaining non-decoupled 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 (2.10). On the other hand however, at the exact g = 0, with fixed v.e.v v and Higgs quartic coupling λ H , eq. (2.17) 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 , Armed with these Feynman rules we calculate the diagrams in Fig. 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 Fig. 1. The Lorentz structure of the amplitude is completely analogous to eq. (2.2) 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 with β, and f (β) defined in eq. (2.16).
To complete the picture there is still the coefficient M 1 in eq. (2.2) to be calculated. Naive power counting says that this is by two powers more divergent than M 2 and, in general, undetermined. It can be fixed however by using quantum gauge invariance i.e., conservation of charge, for the U (1) em , and thus from eq. (2.2), Eq. (2.23) is substituted to eq. (2.2) with M 2 read by eq. (2.21). This is exactly the same result for the W -boson contribution to H → γγ amplitude, that has been obtained in refs. [11][12][13][14]19] using dimensional regularisation in R ξ -gauges.
It is interesting here to note the result from the explicit algebraic manipulation of M 1 in the unitary gauge and check the validity of gauge invariance [eq. (2.23)]. Exactly as for M 2 , the condition a = b = c for the arbitrary vectors given in eq. (2.7) is crucial in reducing the divergence of M 1 down to a logarithmic one [see expression eq. (A.11)]. In d-dimensions the expression for M 1 is finally independent of any arbitrary vector and, up to a proportionality factor, reads:  [19].
Few remarks are worth mentioning here. Had we started first calculating M 1 , 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 fig. 2] is not well defined -an integral as the one in eq. (2.12) 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 eq. (2.12), found here in the unitary gauge, is similar in every other gauge.
The ambiguity of the integral in eq. (2.12) has been discussed by many articles in the recent literature. Refs. [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 non-predictive: new physics input, along the lines of ref. [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, refs. [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 refs. [32,33] is incorrect. Furthermore, the aim of our paper is not to redo the calculation in the unitary gauge with DR as in ref. [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 refs. [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.

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, R. Pittau [31] proposed a scheme that is fairly easy to handle and, to the best of our knoweledge, is the closest to four dimensional calculations, thereby coined four-dimensional regularisation/renormalization scheme or just FDR. According to this scheme, infinite bubble graph contributions, i.e., 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 into 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 2 become¯ 2 = 2 − µ 2 , while for gauge invariance to hold, vector momenta, p µ , remain untouched. For example the integral of eq. (2.12) becomes, whereD = (¯ 2 − ∆) and [d 4 ] stands for integration over d 4 , 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 eq. (3.25) the symmetry property µ ν = g µν 2 /4 has been used in four dimensions. Then, using the partial fractions identity, the term in square bracket is recognised as divergent and therefore removed, and integrating the r.h.s of eq. (3.25) over [d 4 ] one obtains i.e., 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 µ 2 keeping eq. (2.12) correct in d = 4. We therefore understand that the constant (β-independent) term of eq. (2.21) in FDR arises from the fact that the arbitrary scale, µ 2 , must disappear from physical observables.

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 11 and has been proven in ref. [43] that 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 eq. (2.18).
Is there another physics context from which one can define λ? One possibility is to exploit the low energy Higgs theorem [11,[47][48][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 (q 2 ). The later, is notoriously difficult, if meaningful, to be calculated in the unitary gauge. Second, according to ref. [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 = 2m 2 W W + 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 non-vanishing limit at g W → 0 is due to a finite subtraction induced by the corresponding trace anomaly [52]. However, in order to calculate unambigiously 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 i.e., 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 set up 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  [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.

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 eq. (2.7)].
The whole calculation in the unitary gauge boils down to a logarithmically divergent integral (2.10). 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-bypart integration in d-dimensions [see Appendix B]. To proceed, we identify this integral with an undefined parameter λ [see eq. (2.12)]. 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 M 2 of eq. (2.2). This is defined by a physical input taken from the GBET and is connected to M 1 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 [eq. (2.17)]. In this limit, the Goldstone boson loop contributions to the coefficient M 2 is finite, i.e., 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 eq. (2.23) and the discussion below]. The main reason is due to surface terms that are developed in exactly d = 4 dimensions [see discussion below eq. (2.10) 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 eq. (2.24).
(A. 10) where the components in the numerator of (B.1) vanish thanks to symmetric integration formula, µ ν → 1 4 2 g µν . In order not to carry the g µν in all formulae below we just concentrate on the integral

(B.4)
Integrating over the extra dimensional solid angle dΩ 2ω−4 we arrive at where Γ(x) is the Euler Γ-function and L is the length of the ⊥ 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, in (B.5). After integrating by parts in the region of convergence, rewriting the r.h.s in terms of I from eq. (B.5) and keeping only, potentially, non-vanishing surface terms, we arrive at 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 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, 2π 2 , 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 eq. (B.7) we note first that the region of convergence includes now ω = 2. It gives,