Diphoton decay of the higgs from the Epstein–Glaser viewpoint

Eur. Phys. J. C (2021) 81:131 https://doi.org/10.1140/epjc/s10052-021-08898-z Regular Article - Theoretical Physics Diphoton decay of the higgs from the Epstein–Glaser viewpoint Paweł Duch1,2, Michael Dütsch3, José M. Gracia-Bondía5,4,a 1Institut für Theoretische Physik, Universität Leipzig, 04103 Leipzig, Germany 2Max-Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany 3Institut für Theoretische Physik, Universität Göttingen, 37077 Göttingen, Germany 4CAPA and Departamento de Física Teórica, Universidad de Zaragoza, 50009 Zaragoza, Spain 5Laboratorio de Física Teórica y Computacional, Universidad de Costa Rica, San Pedro 11501, Costa Rica Received: 9 August 2020 / Accepted: 21 January 2021 © The Author(s) 2021 Abstract We revisit a nearly 10-year old controversy on the diphoton decay of the Higgs particle. To a large extent, the controversy turned around the respective merits of the regularization techniques employed. The novel aspect of our approach is that no regularization techniques are brought to bear: we work within the Bogoliubov–Epstein–Glaser scheme of renormalization by extension of distributions. Solving the problem actually required an expansion of this method’s toolkit, furnished in the paper. Die Eule der Minerva beginnt erst mit der einbrechenden Dämmerung ihren Flug – Georg Wilhelm Friedrich Hegel 1 Introduction: the controversy Due to its cleanness, it is hard to overstate the experimental importance of the decay of the Higgs particle into two photons. It goes mainly via virtual W-bosons, the heavier charged particles of flavourdynamics. The amplitude of this contribution was calculated to the first non-vanishing order (one-loop, cubic in the couplings) long ago in the light-higgs limit [1] – and then “exactly” in [2]. The accepted result was confirmed many times – see [3] for a particularly clever calculation. It does not vanish in the heavy-higgs limit – which seems to fly in the face of the “decoupling theorem” (DT) in [4], as often understood. Much more recently, those calculations were questioned in [5,6]. The ensuing debate highlights the theoretical relevance of this decay. The authors of these papers made the point that, since the higgs cannot couple directly to the phoTo the memory of Günter Scharf and Raymond Stora. ae-mail: [email protected] (corresponding author) tons, the one-loop contribution must be finite: there are no couplings requiring “renormalization”. The roundabout procedures through “renormalizable gauges”, they concluded, were unnecessary. Eschewing dimensional regularization, they recomputed the amplitude in the unitary gauge of electroweak (EW) theory. They did obtain a result differing from the standard one by an additive constant, which shows up for instance in the heavy-higgs limit – whereby their result is equal to zero. There was no shortage of rejoinders [7–14]to[5,6]. The authors of [9] are the ones of the original calculation [2]. Those papers made several points, some rather implausibly arguing that at a given point in the calculation in [6] electromagnetic gauge invariance is lost, and criticizing the interpretation of the DT made in [5,6]. There was in some of the the rejoinders an explanatory reliance on the heuristics of the Brout–Englert–Higgs mechanism, throwing back the so-called “equivalence theorem” (GBET). The criticisms received a rejoinder in turn in [15]. This later paper argues by the example that two computations of the same process in different gauges (Rξversus unitary gauge) may yield different results. This goes against the grain, although of course no theorem contradicts such an assertion. Meanwhile, a dispersion relation calculation carried out in [16] appeared to support the contentions of [5,6], and got in turn a – quite thoughtful – rejoinder in [17]. More recent papers dealing with the same or related issues are [18,19]. By and large, the majority’s opinion and the experimental results [20] support the first tally. On the other hand, from the theoretical point of view the situation is still obscure: it had to be so, since both parties draw strength from different casuistics of the calculations in perturbative quantum field theory. The debate about the uses and abuses of the unitary gauge and the role of the decoupling and equivalence “theorems” 0123456789().: V,-vol 123 131 Page 2 of 25 Eur. Phys. J. C (2021) 81:131 is to be saluted as salutary. And it is safe to admit that up to now we lack a full conceptual understanding of the problem. The cleanest way to address this lack is surely to renounce all the heuristics of mathematically ill-defined quantities, in favour of a method in which there can be no argument on the meaning of infinite terms. Such is the truly (perturbatively) stringent scheme by Bogoliubov, Epstein and Glaser (BEG) of “renormalization” without regularization, by extension of distributions. In the BEG construction, governed by causality, there is no such thing as a “divergent diagram”: one never encounters infinities. There may, however, remain in the extension procedures some additive ambiguity, that can be restricted (but not always completely removed) by physical principles. This is rather to be regarded as a strength of the BEG paradigm, because those ambiguities express precisely how, and to what extent, the theory is determined by the fundamental principles of perturbative QFT. A particular advantage of the inductive BEG construction [21] of the (functional) S-matrix is that in principle one is allowed to stay on configuration space, which makes more transparent the physics under examination. For examples of calculations within the BEG scheme explicitly carried out in configuration space, see [22]or[23, Sect. 3.5]. It is only for computational convenience that we switch at some moment to momentum space. Since we do not deal in infinities, we refer as normalization to the processes taking the place of regularization and renormalization in the BEG framework. For its relative paucity of diagrams, in our context the underlying argument is made clearer by working mostly in the unitary gauge – whereupon only the physical particles’ data are brought to bear.1 To summarize, so far: we were motivated to tackle this subject by wondering why most knowledgeable people, borrowing different (but all apparently sound) methods to work on such a basic process, were divided on the outcome. It all turns around a subtlety uncovered by use of the BEG normalization. That condenses the purpose of the present paper. 1.1 Main results and plan of the article In Appendix A we introduce our conventions and notations, recalling a few well-known formulae of QFT needed in the body of the paper, in particular the propagators for the EW theory in the unitary gauge. Let mhdenote the mass of the higgs h. The amplitude coming from the one-loop calculations may be quoted as [25–27]: 1The paper [24] dwells usefully on the subject of the Rξ-versus-unitary gauges, leaning to demonstrate the validity of the latter at the quantum level. A=gα 2πMF1(ρ)Pμν, with αthe fine structure constant, gthe EW coupling constant, Mthe mass of the intermediate W-boson and ρ:= m2 h/4M2. The polarization factor Pμν, reflecting electromagnetic gauge invariance (EGI) of A,2is written in this paper as Pμν := (k1k2)gμν −k1νk2μ;(P•νk1)=(Pμ•k2)=0, (1.1) with k1,k2the outgoing photons’ momenta. Finally, for the dimensionless factor: F1(ρ) := 2+3 ρ+3 ρ2−1 ρf(ρ). (1.2) Now that we are at that, we quote as well the comparable result for a charged scalar particle of mass Mat the place of the W-boson: F0(ρ) =1 ρ1−f(ρ) ρ;so that F1(ρ) =3F0(ρ) +6f(ρ) ρ+2.(1.3) For the benefit of the reader coming to the subject of this paper for the first time, Appendix B introduces the distribution f(ρ) appearing in both F1(1.2) and F0(1.3) – as well as in the amplitude of diphoton decay of hvia virtual fermions. The bone of contention is that the first summand 2 in (1.2) should not be there, according to [5,6,16]. Relations (B.3) and (B.6) tell us that, as ρ↓0: F1=2+3 ρ+6 ρ−3 ρ2ρ+ρ2 3+8ρ3 45 +··· =7+22 15 ρ+O(ρ2); so F1(0)=7 and F1(∞)=2from(1.2). Precisely the former figure is what was calculated in the paper [1]. The result argued by the “heretics” in the controversy is F1− 2, so their respective assertions are instead F1(0)=5 and F1(∞)=0. Also, from (1.3): F0(0)=−1/3 and F0(∞)= 0. Appendices A and B of this paper deal with conventions and mathematical prerequisites. The basics of the BEG scheme are recalled in Appendix C. Understanding of the BEG method is indispensable in what follows, and even readers familiar with it are advised not to miss our review. The relation between the normalization problem by extension of distributions (or by “distribution splitting”) and dispersion integrals is treated in its Sect. 1. New results in this respect are required, announced in the short Sect. 2and proved in Sects. 3.2 and 3.3 of this paper. So for aficionados of BEG 2That is, transversality of the outgoing photons. 123 Eur. Phys. J. C (2021) 81:131 Page 3 of 25 131 normalization there is novelty here – whose interest goes beyond the particular problem that motivated it. Sections 3and 4constitute the heart of the paper. The scalar model leading to F0is worked out in Sect. 3.Oneis able to perform the “adiabatic limit” of Epstein and Glaser at an intermediate step, which simplifies computations – this is rigorously justified. This “toy model” allows the reader to familiarize with the BEG construction of time-ordered products in a relatively simple case. For it, the ambiguity in the Epstein–Glaser result can be disposed of, and the unique outcome happens to coincide with the result of a “naive” on-shell calculation, of the kind performed in [16]. Finally, in Sect. 4, we compute the EW amplitude, working first in the unitary gauge. We start in earnest by illustrating in this relevant instance the machinery of the BEG formalism in constructing time-ordered products, at the lowest non-trivial order: from cubic interaction vertices, identified to time-ordered products at first order in the couplings, we derive the quartic, second-order AAWW†-vertex. It is time to aver why the “no-renormalization” argument in [6] is not watertight. A direct hγγ coupling in flavourdynamics is forbidden also because of EGI. Thus to obtain the general amplitude, which lives off-shell, one must add to the naive calculations a polynomial in the external momenta, of degree given by the singular order of that amplitude. Computing the 1-loop contribution in the unitary gauge by the Epstein–Glaser method, we ratify this fact. To find the coefficients of that polynomial, beyond EGI here we call upon gauge-fixing independence of the on-shell amplitude. This locks in the indetermination; and in the end we do obtain F1(ρ). Within the unitary gauge, a different argument to the same purpose is discussed at the end of this Sect. 4. Section 5is the conclusion. 2 The obstruction to distribution splitting for null momenta Formula (C.16) in Appendix C is our main workhorse: in momentum space the Epstein–Glaser distribution splitting amounts to a dispersion integral. But it pertains to remark that, by construction, prescriptions (C.14) and (C.16)arein principle valid only for timelike k. Thus, in order to solve the problem in this paper, one has to run an extra mile. The explicit splitting procedure introduced here exhibits relevant novel features: we have to compute the central solution ac(k1,k2)for null momenta. Hence, one cannot immediately use the dispersion integrals (C.14)or(C.16). On trying to work instead with the convolution integral (C.13), there appears the problem that, in spite of k2 j=0, it generally holds that (kj−vj)2= 0 because vj∈V+;itdoes not suffice to know the causal distribution d(k1,k2)only for k2 1=0=k2 2. The next section solves this problem for models such that 0<(k1+k2)2<4M2and k0 1k0 2>0. The proof’s strategy is as follows: starting from the dispersion integral (C.14) for k2 1>0, k2 2>0 and k0 1k0 2>0, we intend to show that d(k1,k2)is regular enough that this integral commutes with the limit (k2 1↓0∧k2 2↓0). Therefore the dispersion integrals (C.14) and (C.16) keep their usefulness for k2 1=0=k2 2: indeed, for computing ac(k1,k2)|k2 1=0=k2 2it suffices to know d(k1,k2)only for k2 1=k2 2=0, because k2 1=k2 2=0 implies (tk1)2=(tk2)2=0 for all t. Crucially, in the resulting dispersion integrals (C.14) and (C.16)fork2 1=0=k2 2, the parameter ωis the singular order of the off-shell d(k1,k2). As a consequence, the general solution (prior to imposition of other invariance rules) of the distribution splitting is obtained by adding to ac(k1,k2)|k2 1=0=k2 2 a polynomial in k1,k2, in principle arbitrary, whose degree is given by the singular order of the off-shell amplitude d(k1,k2). Now, it frequently happens that the singular order of d(k1,k2)|k2 1=0=k2 2has a smaller value. Consequently, it may happen that the required dispersion integral appears to be “oversubtracted” – i.e., it would be convergent also for a smaller value of ω. Examples for this are the “toy model” in the next section and the EW diphoton decay of the higgs in the unitary gauge (Sects. 3.3 and 4.3, respectively). These issues were realized by Raymond Stora, who, referring to the very subject process of this paper, pointed out to one of us that the good behaviour of the absorptive part of the form factor involving Compton scattering of the W-bosons should not make one forget that BEG-generated dispersion integrals, just as perturbative renormalization theory in general, applies off-shell. 3 3 Higgs to diphoton decay via a charged scalar field The scalar electrodynamics computation leading to F0works like a kind of toy model, allowing the reader to familiarize with our methods in a less complicated, although non-trivial case. We develop it in the present section. Notice the following: in the Epstein–Glaser scheme the “seagull” e2AAϕϕ†- vertex is derived by implementing EGI within the construction rules of the method – as any other part of T2[28]. We give full details on how this comes about for the quartic vertex in the EW theory in Sect. 4.1. The game here would be similar, only simpler. The reader is advised to keep in mind the methods and standard notations recalled in Sect. C.2. 3Private communication, early 2013. 123 131 Page 4 of 25 Eur. Phys. J. C (2021) 81:131 3.1 A causal distribution on-shell The starting point is given by the lower order time-ordered products (TOPs): T1(x3)=gM h(x3)ϕ(x3)ϕ†(x3); T1(xj)=−ieAλ(xj)ϕ†(xj)←→ ∂λϕ(xj), j=1,2; T2(x1,x2) =−e2Aμ(x1)Aν(x2)ϕ†(x1)∂ μF(x1−x2)∂ νϕ(x2) −∂μϕ†(x1)F(x1−x2)∂ νϕ(x2) +ϕ†(x1)∂ν∂μF(x1−x2)+igμν δ(x1−x2)ϕ(x2) −∂μϕ†(x1)∂ νF(x1−x2)ϕ(x2) +(x1↔x2)+T1(x1)T1(x2) +[irrelevant loop diagram terms], where Fdenotes the Feynman propagator (A.3). From our formulas (C.4) and (C.5):4 D3(x1,x2,x3)=−[T1(x1), T2(x2,x3)] −[T1(x2), T2(x1,x3)]+[T2(x1,x2), T1(x3)].(3.1) Because the photons emitted at x1,x2are on-shell, only the third commutator is relevant here – in the language of Cutkosky rules, one needs only the triangle cut separating the higgs vertex from the propagator connecting the photons. We give the explanation further on. From the general formula for the antichronological product (C.3), we particularly know that T1(x1)=T1(x1);T2(x1,x2)=−T2(x1,x2) +T1(x1)T1(x2)+T1(x2)T1(x1). (3.2) For the same reasons just argued, only the connected tree diagram part of the T2(x1,x2)summand in T2(x1,x2)contributes. A most convenient parallel for the coming calculation is the treatment of the vertex function in QED in the first edition of the finite QED book by Scharf [29, Sect. 3.8]. Going to the contractions, bringing in the vertices and the propagators (A.2), (A.4), apart from a factor 4ge2Mwe obtain: Aμ(x1)Aν(x2)h(x3)−(1)∂μF(1−2)∂ν−(2) −∂μ−(1)∂νF(1−2) −(2) −∂μ−(1)F(1−2)∂ν−(2)+−(1)∂μ∂νF(1−2) +igμν δ(1−2)−(2) −[the same four terms with −replaced by +]+··· =: Aμ(x1)Aν(x2)h(x3)dμν(1,2), 4The Dnare always linear combinations of commutators. where 1 ≡y1:= x1−x3,2≡y2:= x2−x3. Here and further down, the dots stand for the terms coming from the other two cuts and further terms not contributing to the onshell amplitude. Note the advertised additional +igμν δto ∂μ∂νF, corresponding to the “closed seagull” or fish-like diagram contribution to the h→2γdecay in this model. We now proceed to momentum space, where computations are carried out more simply. For Fourier transformations, consult the convention (C.7). In this section and the next, in keeping with physicists’ notation, we indicate the transforms by just exhibiting the variables, namely: dμ(k1,k2)≡ˆ dμ(k1,k2). We obtain dμν(k1,k2)=1 (2π)24(Iμν +−Iμν −)+2kν 2(Iμ +−Iμ −) −2kμ 1(Iν +−Iν −)−kμ 1kν 2(I+−I−) −i (2π)2gμν (J+−J−)+··· (3.3) with the integrals I{·|μ|μν} ±(k1,k2):= d4k{1|kμ|kμkν}±(k1−k) F(k) ±(k+k2), J±(k1,k2):= d4k±(k1−k) ±(k+k2), (3.4) where the J±-term is the contribution of the fish-like diagram. Keep in mind that the terms belonging to A 3:= A3−T3are those coming from the integrals I·|μ|μν −and J−, whereas the contribution of R 3:= R3−T3is given by the integrals I·|μ|μν + and J+. For our purposes one may perform the adiabatic limit already at this stage. Since all internal lines of the diagrams correspond to massive fields, this limit can be done here in the naive way by just setting the switching function g(x) in (C.1)to1: dx1dx2dx3Aμ(x1)Aν(x2)h(x3)dμν (x1−x3,x2−x3) =(2π)2dk1dk2h(k1+k2)Aμ(−k1)Aν(−k2)dμν (k1,k2). (3.5) In this limit the momenta k1and k2become the momenta of the external photons: k2 1=k2 2=0. From now on, we compute dμν(k1,k2)|k2 1=0=k2 2.Werewe to have included the other cuts in (3.1)orT1T1T1-terms, there would appear ±-type propagators at the place of the Feynman propagators above. The former are ∼δ(k2−M2), with kdenoting the internal momentum variable in the loop: so to speak, in contrast with the Feynman propagators, the ±are 123 Eur. Phys. J. C (2021) 81:131 Page 5 of 25 131 “always on-shell”, even within loops.5Thus no further internal momenta can be on-shell: assuming k2=M2one obtains (k1−k)2=M2−2(k1k)= M2; similarly for (k+k2).6 Scalar integrals I±. We have to compute I∓(k1,k2) := i (2π)4d4kθ(∓(k0 1−k0)) δ((k1−k)2−M2) ×1 k2−M2+i0θ(∓(k0+k0 2)) δ((k+k2)2−M2). Let us make a change of variable q:= k+k2, and introduce P:= k1+k2, noting for later purposes that P2=2(k1k2). One obtains the integral: d4qθ(∓(P0−q0)) δ((P−q)2−M2) 1 (q−k2)2−M2+i0θ(∓q0)δ(q2−M2). (3.6) It follows that I∓(k1,k2)∝θ(∓P0)θ(P2−4M2), and that sgn k0 1=sgn k0 2for P2≥4M2. Performing the q0-integration and using the notation Eq:= |q|2+M2, we extract I∓(k1,k2)=i (2π)4θ(∓P0)θ(P2−4M2) ×d3q 2Eq θ(∓(P0−q0)) δ((P−q)2−M2) 1 (q−k2)2−M2+i0q0=∓Eq . Since P2>0, one may choose a particular Lorentz frame such that P=(P0,0);hence k1=−k2,k0 1=∓|k1|=∓|k2|=k0 2=1 2P0.(3.7) Taking into account q2=M2, we observe that (P−q)2− M2=2P0(1 2P0−q0), which yields δ((P−q)2−M2)=δ(q0−1 2P0) 2|P0|=δ(Eq−1 2|P0|) 2|P0|, by using q0=∓Eq. For later aims, we point out that in the chosen frame this distribution implies q0=k0 2; hence kP =(q−k2)P=(q0−k0 2)P0=0.(3.8) 5This point is made in [30,Sect.6.4]. 6Compare the discussion after [29, Eq. (3.8.24)]. From ∓q0=∓ 1 2P0comes ∓(P0−q0)=∓ 1 2P0>0. Therefore the factor θ(∓(P0−q0)) is redundant. Changing the integration variables, d3q···=∞ M dEqEqE2 q−M2dq···, the Eq-integration can trivially be done, and we are left with: I∓(k1,k2)=iθ(∓P0)θ(P2−4M2) (P0)2−4M2 (2π)48|P0|dq (q−k2)2−M2+i0q0=P0/2 .(3.9) Let αbe the angle between k2and q, and let z:= cos α.Duetoq2=M2,k2 2=0, |q|=E2 q−M2= 1 2P2 0−4M2and relations (3.7) and (3.8), we obtain (q−k2)2−M2=−2(k2q)=−2(k0 2q0−|q|·|k2|z) =a 2(−a+bz), (3.10) where a:= |P0|>0,0≤b:= (P0)2−4M2<a. We point out that (−a+bz)<0 for all z∈[−1,1]: there is no infrared problem in our triangle graph. The remaining q-integral can be easily computed: 4π a1 −1 dz −a+bz =4π |P0|(P0)2−4M2log (P0)2−|P0|(P0)2−4M2−2M2 2M2. (3.11) To obtain the result in a generic Lorentz frame, replace (P0)2 by s:= P2=2(k1k2),so I∓(k1,k2)=iθ(∓P0)θ(s−4M2) 4(2π)3slogs−s(s−4M2) 2M2−1 =: θ(∓P0)θ(s−4M2)F(s). (3.12) The result for J±(k1,k2)can be read off from (3.9)by omitting the Feynman propagator i(2π)−2((q−k2)2−M2+ i0)−1. One obtains for the contribution of the J-integrals: J±(k1,k2)=1 8πθ(±P0)θ(s−4M2)1−4M2/s. Vector integrals Iμ ∓. For the same reasons as for the scalar integral, it must hold that Iμ ∓(k1,k2)∝θ(∓P0)θ(s−4M2). From Lorentz covariance and Iμ ±(k1,k2)=−Iμ ±(k2,k1)it follows Iμ ∓(k1,k2)=θ(∓P0)θ(s−4M2)(kμ 1−kμ 2)G(s) 123 131 Page 6 of 25 Eur. Phys. J. C (2021) 81:131 for appropriate G(s). An immediate consequence is IμPμ= 0. To procure G(s), compute k2,μ Iμ ∓(k1,k2)=1 2θ(∓P0)θ(s−4M2)sG(s) =−i/8(2π)3θ(∓P0)θ(s−4M2)1−4M2/s The second equality is obtained by comparing with the scalar integral: there is an extra factor (k2k)=(k2q)=−a(−a+ bz)/4, where (3.10) is used. Then the q-integral becomes trivial. Thus we glean G(s)=−i 32 π3s1−4M2/s.(3.13) Tensor integrals I μν ∓. Proceeding analogously to the vector integrals, one argues that Iμν ∓(k1,k2)=θ(∓P0)θ(s−4M2)(kμ 1kν 1+kμ 2kν 2)A(s) +(kμ 1kν 2+kμ 2kν 1)B(s)+gμν C(s). We need three independent identities to compute A(s),B(s) and C(s). A first one is: Iμν ∓k2μk2ν=θ(∓P0)θ(s−4M2)A(s)s2/4 =θ(∓P0)θ(s−4M2)−i 25(2π)3s1−4M2/s. (3.14) The second equality is obtained by a modification of the computation of the scalar integral: there is the extra factor (k2k)2=a2(−a+bz)2/16. This yields A(s)=G(s)/2. A second identity is given by the trace. The result is again obtained by comparing with the computation of the scalar integral: there is an additional factor k2=(q−k2)2= M2−2(k2q)=M2−2(kk2), hence Iμ ∓,μ =θ(∓P0)θ(s−4M2)(sB +4C)=M2I∓−2k2,μ Iμ ∓. A third identity following from (3.8) reads: Iμν ∓Pν=θ(∓P0)θ(s−4M2)Pμ(A+B)s/2+C=0. Pulling together these results, one arrives at B(s)=−M2F(s)/sand C(s)=M2F(s)/2−sG(s)/4. •At this point we are able to show that the triangle plus fish-like parts constitute a gauge-invariant quantity. For that, insert the results already known for the integrals into (3.3), obtaining: dμν(k1,k2)k2 1=0=k2 2=sgn(P0)θ(s−4M2) (2π)2kμ 1kν 2[4G(s) −(1+4M2/s)F(s)] +2M2gμν F(s)−kν 1kμ 2 4M2 sF(s) =sgn(P0)θ(s−4M2)4M2 (2π)2Pμν F(s) s.(3.15) The kμ 1kν 2-terms have been dropped in the last identity, due to kμAμ(−k)=0. The remainder is electromagnetically gauge-invariant. Introducing the dimensionless variable ˜ρ:= s 4M2=P2 4M2, keeping in mind formula (3.12), and on use of (B.5), equation (3.15) can be rewritten as dμν gi (k1,k2)k2 1=0=k2 2:= isgn(P0)θ(˜ρ−1) (2π)5Pμν b(˜ρ) (3.16) with b(˜ρ) := 1 16 M2˜ρ2log2˜ρ−2˜ρ( ˜ρ−1)−1 =− 1 16 M2˜ρ2log 1+1−˜ρ−1 1−1−˜ρ−1, where ‘gi’ stands for the gauge invariant part. The singular order of dμν gi k2 1=0=k2 2 is ω=−2 by power counting; whereas for the off-shell dμν(k1,k2)the value is ω=0. 3.2 Regularity of absorptive parts in momentum space This subsection is devoted to prove essential regularity properties of the off-shell d-distribution, more precisely of dμν(k1,k2),for(k1,k2)∈V:= V+\{0}×2∪V−\{0}×2. We look at the terms coming from (3.3) by means of (3.4). Introducing the new integration variable q:= −k+1 2(k1− k2), the internal lines’ momenta are q1=q+1 2P,q2=q−1 2P,q3=q−1 2(k1−k2), (3.17) and one sees that the considered terms are all of the type Hμν(k1,k2) := d4qθ(q0 1)θ(−q0 2) −θ(−q0 1)θ(q0 2)δ(q2 1−M2)δ(q2 2−M2)hμν(k1,k2,q) M2−q2 3 (3.18) 123 Eur. Phys. J. C (2021) 81:131 Page 7 of 25 131 for (k1,k2)∈V1:= V\{0}×2with V:= V+∪V−, and where hμν :R4×3→Cis a polynomial of degree 2. We have used that for (k1,k2)∈V1it holds true that d4qθ(q0 1)θ(−q0 2) −θ(−q0 1)θ(q0 2)δ(q2 1−M2)δ(q2 2−M2)δ(q2 3−M2)=0. (3.19) This last relation can be argued as follows:7the various θand δ-distributions yield the restrictions (q1,q2)∈(H+ M× H− M)∪(H− M×H+ M)and q3∈H+ M∪H− M; taking moreover into account that q3=q2+k2and q3=q1−k1, it ensues that the various restrictions on q3are not compatible. The same identity implies that terms of the kind T1(xπ1) T1(xπ2)T1(xπ3)do not contribute to the third commutator in formula (3.1)forD3when (k1,k2)∈V1, for all permutations π:thewhole contribution to dμν(k1,k2)|(k1,k2)∈V1 coming from this commutator is of the kind (3.18). The contributions to dμν(k1,k2)|(k1,k2)∈Vcoming from the other two commutators in (3.1) are of the same form up to cyclic permutations k1→ k2→−(k1+k2)→ k1of the external momenta. Here we use that (k1,k2)∈Vimplies (k2,−k1−k2)∈V1and (−k1−k2,k1)∈V1, hence we may apply the identity (3.19) also for the permuted momenta. However, note that the polynomials hμν j,j=2,3, belonging to these other two cuts are not obtained by cyclic permutations of the external momenta in the original polynomial hμν 1, meant in (3.18). This is due to the difference between the higgs vertex and the photon vertices; in particular, these other two cuts contain no term giving rise to a fish-like diagram. Summing up, it holds that dμν(k1,k2)(k1,k2)∈V =Hμν 1(k1,k2)+Hμν 2(k2,−k1−k2)+Hμν 3(−k1−k2,k1), (3.20) for some Hμν j(j=1,2,3)of the form (3.18), the pertinent polynomials hμν jbeing of degree 2. Lemma 1 Let q1,q2,q3and V1be defined as above in (3.17) and after (3.18), and let Hμν :R4×2→Cbe given in terms of a generic polynomial hμν :R4×3→Cof degree ζ∈N0, as in (3.18). Then for all (k1,k2)∈V1and for some C >0 the function Hμν is continuous in the region V1, and can be bounded as follows: |Hμν (k1,k2)| ≤C(1+|(k1,k2)|)ζ |(k1k2)|θ((k1+k2)2−4M2)log((k1+k2)2/M2). (3.21) 7We borrow the standard notation for the mass shell: H± M:= { p∈ R4:p2=M2,±p0>0}. Note that |(k1k2)|>0if (k1,k2)∈V1and (k1+k2)2≥4M2. Proof Let P:= k1+k2and k:= k1−k2. We first observe, on the strength of q2 1−q2 2=2(Pq), q2 1+q2 2−2M2=2(q2+1 4P2−M2) and of M2−q3=(M2−q2−1 4P2)+1 4P2−1 4k2+(kq)that Hμν (k1,k2)∼sgn(P0)d4qδ(q2+1 4P2−M2) δ((Pq)) hμν (k1,k2,q) P2/4−k2/4+(kq), omitting irrelevant prefactors. Since q1−q2=Pand (q1,q2)∈(H+ M×H− M)∪(H− M×H+ M), we know that Hμν(k1,k2)vanishes for P2<4M2. Hence, to perform the integrals in q0and |q|using the Dirac deltas, we may work in the frame in which P=0. There the δ-distributions yield q0=0 and |q|=P2 0/4−M2. With the notation ˆp:= p/|p|for p∈{q,k}, it follows that 1 4P2−1 4k2+(kq)=(k1k2)1−(ˆqˆ k)1−4M2/P2|P0||k| 2(k1k2), and one verifies that 0≤P2 0|k/2|2=(k1k2)2−k2 1k2 2,(3.22) with k1=(k0 1,k1)and k2=(P0−k0 1,−k1). With the help of these results we obtain (k1k2)Hμν(k1,k2)∼sgn(P0)θ(P2−4M2)1−4M2/P2 ×S2d(ˆq)hμνk1,k2,(0,P2/4−M2ˆq) 1−(ˆqˆ k)(1−4M2/P2)(1−k2 1k2 2/(k1k2)2) , (3.23) valid in the frame in which P=0. Let moreover VM 1:= {(k1,k2)∈V1:(k1+k2)2≥4M2}. We know that 4M2/P2∈(0,1]and k2 1k2 2/(k1k2)2∈[0,1]for (k1,k2)∈VM 1; hence a:= (1−4M2/P2)(1−k2 1k2 2/(k1k2)2)∈[0,1). In particular, the denominator in the integrand of (3.23) does not vanish for (k1,k2)∈VM 1. Since θ(P2−4M2) 1−4M2/P2is continuous, Hμν is continuous on V1. Observe now that for all ˆq∈S2the inequality hμνk1,k2,(0,P2/4−M2ˆq)≤const(1+|(k1,k2)|)ζ holds, with |(k1,k2)|2:= 3 j=0(k2 1j+k2 2j). Setting z:= ˆqˆ k, the remaining integral is of the type 1 −1 dz 1−az =1 alog1+a 1−a≤2(1−log(1−a)), 123 131 Page 8 of 25 Eur. Phys. J. C (2021) 81:131 valid for a∈[0,1). Using that a≤1−4M2/P2≤(1− 2M2/P2)and monotonicity of the logarithm, we see that −log(1−a)=log 1 1−a≤log P2 2M2. Putting together the estimates, we end up with |(k1k2)Hμν(k1,k2)| ≤const ·θ(P2−4M2)(1+|(k1,k2)|)ζ1+log(P2/2M2), (3.24) impliying (3.21), since 1 +log(P2/2M2)<2log(P2/M2) for P2≥4M2. The reader should keep in mind that dμν(k1,k2)is supported outside a certain neighbourhood of the origin on momentum space – have a look back at Eq. (3.23). Corollary 2 The off-shell d-distribution dμν (k1,k2)given in (3.20) is continuous on Vand fulfills the bound: |dμν(k1,k2)|≤const (1+|(k1,k2)|)ω+2 |(k1k2)| log(2+|(k1,k2)|/M)for all (k1,k2)∈V.(3.25) Proof Continuity follows immediately from Lemma 1.For the bound (3.25) we have substituted ω+2≡ω(d)+2forζ of the Lemma, since the singular order of Hμν j(j=1,2,3) is ζ−2 by power counting in (3.18). In addition, for Hμν 1(k1,k2)we have used that (k1+k2)2≤4|(k1,k2)|2, and in order to omit the θ-distribution we have replaced log(2|(k1,k2)|/M)by 2 log(2+|(k1,k2)|/M). One deals analogously with Hμν 2(k2,−k1−k2)and Hμν 3(−k1−k2,k1).  3.3 Distribution splitting by the dispersion integral for null momenta Recall that for (k1,k2)∈Vη×Vηthe advanced part aμν of dμν can be computed by the dispersion integral (C.14). Using the regularity properties of dμν given in Corollary 2, we finally aim to show that the limit k2 1↓0, k2 2↓0in(C.14) commutes with integration; that is, the dispersion integral is also valid for k2 1=0=k2 2. To formulate the assertion, let K:= {(k1,k2)∈(R4)×2:k2 1,k2 2<4M2, (k1+k2)2<4M2,(k1k2)= 0}.(3.26) Bearing in mind the factors θ(q2−4M2)for q∈ {k1,k2,k1+k2}appearing in each term of dμν(k1,k2),we see that for (k1,k2)∈(Vη×Vη)∩K,formula(C.14) can be rewritten as: aμν(k1,k2)=iη 2π|t|≥tmin dt dμν(tk1,tk2) tω+1(1−t),(3.27) for some tmin >1 depending on k1,k2. Now, as discussed in Sect. 1, one knows aμν(k1,k2)to be analytic on the region K. The Lebesgue dominated convergence theorem [31, Th. 4.6.3] with the bound (3.25) allows us conclude that (3.27) is a valid identity for (k1,k2)∈V∩K. Indeed, introducing the set of limit points M:= V∩K∩{(k1,k2)∈R8:k2 i=0} ={(k1,k2)∈R8:k2 i=0,0<(k1+k2)2<4M2}, it is enough to observe that for any (˜ k1,˜ k2)∈M– implying (˜ k1˜ k2)>0 and ˜ k0 1˜ k0 2>0 – there is a neighbourhood U(˜ k1,˜ k2) such that θ(|t|−tmin)d(tk1,tk2) tω+1(1−t) ≤const ·θ(|t|−tmin) |t(1−t)|1+|(k1,k2)|ω+2 |(k1k2)| log(2+|t||(k1,k2)|/M) ≤Cθ(|t|−t1) |t(1−t)|log(2+C1|t|), for all (k1,k2)∈(Vη×Vη)∩K∩U(˜ k1,˜ k2),forsomeC,C1>0 and some t1>1 independent of (k1,k2). The function on the right hand side is absolutely integrable in t– here we see the reason for the condition (k1k2)= 0in(3.26). 3.4 Normalization of the scalar model by distribution splitting We must finally compute the gauge invariant part tμν gi (k1,k2) for momenta lying on the set M. Considering the formula T3=A3−A 3and reckoning that aμν(k1,k2)|k2 1=0=k2 2contains the factor θ(P2−4M2)where P:= k1+k2,wesee that on Mits contribution vanishes, that is tμν =aμν there. The upshot of the preceding two subsections is that we may compute valid terms of the central solution aμν|M≡ acμν|Mby inserting the on-shell amplitude (3.15) into the dispersion integral, with ωthe singular order of the off-shell dμν, equal to 0 in the present case. Looking at (3.15), observe that a kμ rkν s-orgμν-term of dμν goes over to a kμ rkν s-orgμν-term of aμν , respectively. Therefore, such factors may be taken out of the dispersion integral. Since moreover Pμν (tk)=t2Pμν(k), we see that the gauge invariant part aμν gi can be obtained by inserting just the gauge invariant part dμν gi in (3.16) into the dispersion integral. The latterisoftheform(C.15). So we may use the version (C.16) of the dispersion integral. Lastly, tμν gi |M=aμν gi |Mis obtained from (C.16) by setting ω=0 and substituting there b(u˜ρ) as given in (3.16)for fu(k2 1,k2 2,(k1+k2)2)– in our case only (k1+k2)2=s is present. Allowing for the dilation factor in Pμν this leads, 123 Eur. Phys. J. C (2021) 81:131 Page 9 of 25 131 for (k1,k2)∈M,to tμν gi (k1,k2) =−Pμν (2π)6∞ ˜ρ−1 du ub(u˜ρ) u(1−u) =Pμν 16M2(2π)6∞ ˜ρ−1 du ˜ρ2u2(1−u)log 1+1−u−1˜ρ−1 1−1−u−1˜ρ−1 =− Pμν 8(2π)6 J2(˜ρ) M2,(3.28) where 2J2(˜ρ) := ∞ 1 dv1 (v −˜ρ)v2log 1+√1−v−1 1−√1−v−1, after the change of integration variable v:= u˜ρ.Integrals like J2have been computed in [19]. From Appendix C of that reference: J1(˜ρ,a):= 1 2∞ 1 dv1 (v −˜ρ)(v −a)log 1+√1−v−1 1−√1−v−1 =f(˜ρ) −f(a) ˜ρ−a(3.29) for 0 ≤˜ρ≤1, 0 ≤a≤1, where fis the distribution (B.3). We infer that J2(˜ρ) =∂ ∂aa=0 J1(˜ρ,a) =f(˜ρ) ˜ρ2−1 ˜ρfor 0 ≤˜ρ≤1,(3.30) by bringing in the values f(0)=0 and f(0)=1, which can be read off from (B.6). Summing up, the final result reads, as expected: tμν gi (k1,k2)=Pμν 8(2π)6 1 M21 ˜ρ−f(˜ρ) ˜ρ2 =Pμν 8(2π)6 F0(˜ρ) M2for (k1,k2)∈M,(3.31) where F0was given in (1.3). We conjecture that this formula holds true for all (k1,k2)satisfying k2 1=0=k2 2and (k1+ k2)2>0. The reader should remember that (3.31) stands in principle for just a member of a solution set. Since ω=0, the general Lorentz-invariant Epstein–Glaser solution is obtained by adding to expression (3.31) a term of the type Cgμν with C∈Carbitrary. But such a term with C= 0 would violate EGI. Therefore we regard the above result as unique. Recovering formula (3.5) and the factor 4ge2M, one ends up with dx1dx2dx3T3(x1,x2,x3) =gα (2π)3Mdk1dk2h(k1+k2)Aμ(−k1)Aρ(−k2)Pμν F0(˜ρ), which, on substituting ρfor ˜ρ, that is, m2 hfor s≡(k1+k2)2, agrees with the literature [25]. Remark 1 In the occasion an (unsubtracted) dispersion integral applied to b(u), performed in [16, Eq. (3.2)], leads to the same integral (3.28) and so the same correct result. As the next section shows, this does not hold for the higgs to diphoton decay via EW vector bosons. 4 Higgs to diphoton decay via EW vector bosons 4.1 Derivation of the quartic AAWW†-vertex in the unitary gauge The amplitude in question in this paper describes an EW decay process at third order in the coupling constant. Its structure is given by the cubic vertices in the first TOP T1 – that is the sole “empirical” input. Here in going from T1 to T2we derive the AAWW†-vertex which contributes by a “fish-like” diagram to the amplitude to be computed, see Fig. 1. The general idea is to examine the propagator which is to become the internal line linking the di-photon in the one-loop, three-vertex graph, and to obtain the one-loop, two-vertex graph from a modification of that propagator, demanded by EGI – by which here we precisely understand invariance of the S-matrix under the variations Aμ(x)→ Aμ(x)+∂μ(x): interaction dictates symmetry. The method is similar to the derivation of the AAϕϕ†“seagull” vertex from the cubic coupling in scalar QED, first performed in this way in [28]. The concept works on configuration space, as follows. Recall the pertinent Hermitian vertex – see for instance [32, Sect. 7.2.2], explicitly referring to the unitary gauge. With Gμν := ∂μWν−∂νWμ, one has: T1(x1)=ie[(WμG† μν −W†μGμν)Aν−WμW† νFν μ](x1). (4.1) All indicated operator products are Wick products. We copy a second vertex similar to (4.1): T1(x2)=ie[(WρG† ρλ −W†ρGρλ)Aλ−WρW† λFλ ρ](x2), (4.2) 123 131 Page 16 of 25 Eur. Phys. J. C (2021) 81:131 Let us now to come back to reference [16]. It is argued there that the convergent integral t0(˜ρ) := ∞ ˜ρ−1 du b1(u˜ρ) 1−u leads to the correct result. From the standpoint of this reference, formula (4.27) is “oversubtracted”. One obtains there, yet again Pμνt0(˜ρ) =Pμν∞ ˜ρ−1 du b1(u˜ρ) 1−u =− 3Pμν 8M2(2π)6(2J1(˜ρ,0)−J2(˜ρ)) =− Pμν 8M2(2π)63 ˜ρ+6f(˜ρ) ˜ρ−3f(˜ρ) ˜ρ2 =− Pμν 8M2(2π)6(F1(˜ρ) −2). (4.31) So the naive on-shell computation yields a particular Epstein–Glaser solution. In the present case, however, equation (4.30) tells us that we are forced to add (at least) a polynomial of degree two respecting EGI, that is, a term CPμν for Can indeterminate constant – with which our result for the amplitude is compatible with the generally accepted one. Remark 2 From our viewpoint, the expression in (4.31)isthe unique Epstein–Glaser solution respecting EGI, corresponding to the following causal d-distribution: let the result (4.26) for dμν gi (k1,k2)(obtained by light-cone restriction of the photon momenta) be interpreted as an unrestricted element of S(R8), that is, all values (k1,k2)∈R8are admitted. One easily verifies that this d-distribution has causal support, so the splitting problem is well defined, and since its singular order is zero, the EGI requirement selects a unique splitting solution. Writing the latter suitably as a dispersion integral in momentum space, one verifies the claim. This procedure strongly simplifies explicit computations, but it is not conceptually correct.11 4.4 Fixing the normalization polynomial by agreement with the Feynman gauge In order to determine the normalization polynomial we may as well invoke the computation of the h→γγ decay in the Feynman gauge and gauge-fixing independence, namely, the requirement that observable quantities should not depend 11 Actually, in the first edition of the book by Scharf on quantum electrodynamics (i.e., [29] rather than [39]), the vertex function in QED at third order was computed by such a method. on the choice of gauge.12 Motivated by results of [43],13 we contend that the “entirely on-shell” amplitude coming out of our previous computation should coincide with that of an Epstein–Glaser computation in the Feynman gauge. By “entirely on-shell” we mean that not only the photons, but also the higgs is on-shell, that is, ˜ρ=ρ:= m2 h/4M2. Denote the Epstein–Glaser result for the d-distribution in the Feynman gauge by d1 μν. In contrast with the unitary gauge, there additionally contribute diagrams with Stückelberg fields and Faddeev–Popov ghosts (as inner lines) to d1 μν, see e.g. [15]. We spare the reader the details of the construction of the TOPs, and in particular the derivation of the AAWW†-vertex in this context. For photons on-shell with physical polarizations (setting k2 1=0=k2 2and omitting pure gauge terms ∼k1μor ∼k2ν), our result reads: d1 μν(k1,k2)=− 1 23(2π)6M2 ×(k1k2)gμν −3 ˜ρ2+7 ˜ρ−ρ ˜ρ2 −k2μk1ν−3 ˜ρ2+8 ˜ρ−2ρ ˜ρ2f(˜ρ). (4.32) The tedious computation of the above absorptive part was done with the aid of the Mathematica package FeynCalc [44]. The computation proceeds along the lines of the computations of related absorptive parts in scalar electrodynamics and electroweak theory in the unitary gauge presented in full detail in Sects. 3.1 and 4.2 , respectively. As before, all terms contributing to the distribution d1 μν can be represented by Feynman diagrams with cuts – for the complete list see e.g. [15]. Just like in Sects. 3.1 and 4.2 , because of the kinematic constraints one needs to consider only the cut separating the higgs vertex from the photon vertices. All the appearing expressions have a very similar structure to those that have been already considered in the above-mentioned parts. Thanks to the presence of the cut, each integral over the four-momentum flowing in the loop can be converted into an integral over a sphere, which can be evaluated explicitly. We stress the fact that, due to compactness of the region of integration, the computation of the absorptive part does not involve any regularization. 12 The equivalence or inequivalence of calculations performed in different gauges was a nagging worry of Raymond Stora in his last years. The classic paper [42] illustrates the difficulties lurking here. 13 This reference works with a formulation of gauge invariance suitable for the BEG scheme. In that framework it was shown for the various Rξ-gauges that the T-products can be normalized in such a way that the physical S-matrix (i.e., for inand out-states being on-shell) does not depend on the gauge-fixing parameter ξin the formal adiabatic limit; and that this normalization is compatible with gauge invariance in the mentioned sense. 123 Eur. Phys. J. C (2021) 81:131 Page 17 of 25 131 An important feature of electroweak theory in the Rξgauges is the fact that all interaction vertices have dimensions lower or equal to four (because dim Wμ=1, in contrast to the value dim Wμ=2 for the unitary gauge). In particular, a straightforward power counting argument gives the upper bound ωd1 μν≤0 for the singular order of the off-shell distribution d1 μν. Noting that ˜ρ=(k1k2)/2M2and f(˜ρ) = O(log ˜ρ) we see that the on-shell restriction of d1 μν(k1,k2) in Eq. (4.32) grows logarithmically for big values of ˜ρ.For the off-shell d1 μν, this implies the equality ωd1 μν=0. This should be contrasted with the bounds 6 ≥ωdμν≥2inthe case of the absorptive part computed in the unitary gauge. The off-shell distribution d1 μν is again of the type considered in Sects. 3.2. In particular, the method of distribution splitting developed in Sect. 3.3 is applicable. For photons onshell with physical polarizations, the central solution reads t1c μν (k1,k2)=− 1 23(2π)6M2 ×gμν(k1k2)−3 ˜ρ2+7 ˜ρ−ρ ˜ρ2f(˜ρ) +3 ˜ρ+2ρ ˜ρ −k1νk2μ−3 ˜ρ2+8 ˜ρ−2ρ ˜ρ2f(˜ρ) +3 ˜ρ+2ρ ˜ρ. (4.33) According to the postulate ‘Divergence degree’ (in Sect. C.1), we have to demand for the off-shell t1 μν that ωt1 μν=ωd1 μν=0. This implies that the pertaining normalization freedom consists of a constant term which is a tensor with two indices. By the Lorentz invariance such a term has to be proportional to the metric tensor. Consequently, the general off-shell solution of the splitting problem is of the form t1 μν(k1,k2)=t1c μν (k1,k2)+gμν D,(4.34) where Dis an arbitrary constant; note that this relation holds also after restriction to on-shell photons with physical polarizations. Observe that, in contrast to the unitary gauge, as long as the higgs is off-shell, the distributions (4.32) and (4.33) are not electromagnetically gauge-invariant. This was to be expected and is related to the presence of unphysical degrees of freedom in electroweak theory in the Rξ-gauges. However, entirely on-shell EGI can be satisfied: setting ˜ρ:= ρ in (4.33), we plainly get t1 μν(k1,k2)˜ρ=ρ =− 1 23(2π)6M2Pμν(k1,k2)F1(ρ) +Dg μν,(4.35) and one sees that Dmust be put equal to zero. This fixes completely the normalization freedom in the construction of t1 μν in the Feynman gauge. At this level there is of course coincidence with the result in [45], despite different game rules. Recall that in the unitary gauge, for on-shell photons with physical polarizations, the general normalization freedom fulfilling electromagnetic gauge invariance and Lorentz covariance is given by the last term in (4.30), where ω≡ ω(dμν). We stress that the constants Ckappearing in that term cannot be fixed without imposing some further normalization conditions. To address this problem, observe that it is possible to adjust the coefficients Ckof the polynomial in the expression (4.30)fortgi,μν in the unitary gauge in such a way that the following equality tgi,μν(k1,k2)˜ρ=ρ=t1 μν(k1,k2)˜ρ=ρ.(4.36) holds entirely on-shell, i.e. for ˜ρ=ρ. In fact, we must set C0:= 2 and Ck:= 0 for all k≥1in(4.30), which fixes completely the normalization freedom of tgi,μν .Eq.(4.36) expresses the independence of the physical amplitude of the diphoton decay of the higgs of the choice of the gauge. We regard (4.36) as a normalization condition of time-ordered products. We have shown that this condition can be satisfied in the case at hand and determines uniquely the indeterminate normalization polynomial of tgi,μν in the expression (4.30). In summary, our final result for the entirely on-shell EW h→γγ decay reads: tμν(k1,k2)˜ρ=ρ=− 1 23(2π)6M2Pμν(k1,k2)F1(ρ), in agreement with the majority of the literature. 4.5 On settling the controversy Should one infer that by computing in the “physical” unitary gauge there is no way to entirely settle the controversy that motivates this work, by removing the remaining ambiguity in determining the amplitude in question? Not without at least pondering credible “heavy-higgs” (or M→0) and “lighthiggs” (or M→∞) arguments to bolster the case of F1(ρ) versus F1(ρ) −2, that have been made in the literature. Now, for the present authors the question is not whether either class of arguments is compelling enough. Instead, the question is whether they can be made within the BEG prescriptions, and at the level of rigour of this paper. The arguments in the first-named class involve plays with field transformations, power counting rules and the adiabatic limit that we find hard to countenance in the BEG formalism. However, those of the second class are persuasive within our purview. Note that F(0), for both scalar and vector boson charged fields, as well as for Dirac fermions, must coincide with (the first coefficient of) the β-function series associated 123 131 Page 18 of 25 Eur. Phys. J. C (2021) 81:131 to electric charge renormalization.14 It was a fortunate historical fact that a calculation of the effective Lagrangian for charged Proca particles [46] was already available when the first “exact” computation of the higgs to digamma process that we are aware of was performed [2] – thus making possible a dependable “light-higgs” argument. A computation of the renormalization of the electric charge of massive vector bosons in the unitary gauge by means of BEG technology is in principle feasible – cf. in this respect [41, Sect. 7] and [47] – and expected to yield the required value F1(0)=7. That would complete the analysis of this paper, without going beyond the unitary gauge framework. 5 Conclusion Contrary to custom, we begin this section by declaring what we have not done in the paper. Finite QFT àlaBogoliubov– Epstein–Glaser is mathematically a rigorous method. So, referring to what is found in the literature – like that cited in the Introduction – we have not employed dimensional regularization, deemed an “artifact” by some. Nor do we borrow Pauli–Villars’, nor cutoff regularizations, for that matter. We did not have to practice “judicious routings of the external momenta” [6], nor adopt the “loop regularization method” [10], or any of the techniques to handle divergent integrals, resulting from the blind application of Feynman graph technology on momentum space. We do not pore over divergent integrals, at all. Each and every one of the integrals appearing in this paper produces an unambiguous result; each amplitude is finite. We expected the BEG procedure to yield a conceptually clear understanding of the EW h→γγ decay in the unitary gauge. We have succeeded in this – at a price. According to Epstein and Glaser, the adiabatic limit is to be performed after distribution splitting. Such an off-shell procedure for the h→ γγ decay in the unitary gauge demands computations more than one order of magnitude greater than the ones performed in this paper – compare the computation of the QED vertex function in [39, Chap. 3.8] and in [48]. We were not disposed to inflict this on ourselves, nor our surviving readers. Thus we were forced to innovate on the method, generalizing the splitting dispersion integral to production of massless particles, and showing that in the present situation the adiabatic limit may be performed before distribution splitting. Only, then one may have to add to the result so obtained an a priori indeterminate polynomial in the external momenta, of a degree given by the singular order of the amplitude off-shell. It is precisely the addition of this polynomial that is missing in references [5,6] and [16]. We have 14 F0(0)=−1/3, which has been calculated in this paper, means precisely this. resolved the ambiguity by recourse to gauge-fixing independence of the entirely on-shell amplitude. Alternatively, the ambiguity could be resolved within the unitary gauge in the BEG scheme, by invoking the low-energy argument.15 We have not attempted here a rigorous proof of this argument, nor computed the relevant coefficient of the beta function, leaving the task for a separate analysis in future work. Acknowledgements We are grateful to E. Alvarez, L. Alvarez-Gaumé, M. Herrero, C. P. Martín, J. C. Várilly and T. T. Wu for comments, discussions and helpful remarks. We particularly thank I. T. Todorov for keen help in the beginning, and his continued and thought-provoking, if contrarian, interest in this work. As well we thank an anonymous referee for knowledgeable reporting, definitely contributing to improve the paper. During the inception and writing of this article, PD received funding from the National Science Center, Poland, under the Grant UMO-2017/25/N/ST2/01012. He also gratefully acknowledges the hospitality of the University of Zaragoza. JMG-B received funding from the European Union’s Horizon 2020 research programme under the Marie Skłodowska-Curie Grant agreement RISE 690575; from Project FPA2015–65745–P of MINECO/Feder; from CERN; from the COST actions MP1405 and CA18108. Hospitality of CERN, IFT-Madrid, ITPGöttingen and ZiF-Bielefeld is gratefully acknowledged. Data Availability Statement This manuscript has no associated data or the data will not be deposited. [Authors’ comment: This manuscript is a theoretical work and no databases were generated.] Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/. Funded by SCOAP3. Appendix A: Notations and prerequisites Our Minkowski metric is mostly-negative. The Minkowski inner product of two vectors x≡xμ,p≡pνis denoted with parentheses: (xp)=xμpμ. When (we hope) it does not cause confusion, we often denote p2=(pp). We signal the standard formula for time-ordered 2-point function: Tϕ(x)χ(x) := i (2π)4d4pe−i(p(x−x)) p2−M2+i0Mϕχ(p), (A.1) 15 Variants of the “light-higgs” or “low energy” argument besides [2, 9,17] are found for instance in [27, Ch. 24.8], in [49]andin[50]. 123 Eur. Phys. J. C (2021) 81:131 Page 19 of 25 131 where Mϕχ is the multiplier appearing in the corresponding 2-point function for the fields ϕ,χ with the same mass M. Propagators for a (complex) scalar field. Clearly, for (say, complex) scalar fields the Feynman propagator F(x−x):= Tϕ(x)ϕ†(x) (A.2) fulfils F(x−x)=−M2F(x−x)−iδ(x−x), (A.3) where Mis the mass of the ϕ-field. Also, with θdenoting the Heaviside function, the Wightman functions +(x−x):= ϕ(x)ϕ†(x) =ϕ†(x)ϕ(x) = 1 (2π)3d4pθ(p0)δ(p2−M2)e−i(p(x−x)) so that (+M2)+(x)=0, −(x):= −+(−x), (A.4) are used in our calculations. Massive vector fields. A dreibein er(p)on Minkowski momentum space, with the properties: er(p)es(p)=−δrs for r,s=1,2,3;pe r(p)=0, describes polarization states for particles with squared mass M2=p2>0 and spin j=1. From the above identities, one derives the projector formula: 3  r=1 eμ r(p)eν r(p)=−gμν +pμpν M2.(A.5) The set eis regarded as an intertwiner matrix mapping the natural representation space of the Lorentz group onto the representation space C3for spin 1 objects. Let a† r(p)and ar(p)be respectively the creation and annihilation operators on the boson Fock space for such particles – whose 1particle subspace is the corresponding Wigner unirrep space; and b† r(p)and br(p)for their antiparticles. There is a quantum vector field acting on that space given by the formula Wμ(x):=  rdμ(p)ei(px)eμ r(p)b† r(p) +e−i(px)eμ∗ r(p)ar(p);(A.6) In (A.6) and in other formulas dμ(p)denotes the usual invariant measure d3p/2E(p)=d3p/m2+|p|2over the mass hyperboloid H± M:= {p∈M|p2=M2∧±pn>0}.By its definition, the charged Proca field Wis divergenceless: (∂W)=0. Its equations of motion can be variously written as (+M2)Wμ=(+M2)Wμ−∂μ(∂W) =∂νGνμ(x)+M2Wμ=0,(A.7) where Gμν := ∂μWν−∂νWμ. The theory of massive vector fields is a gauge theory [51,52], its Proca version being a “unitary gauge” for it. It has been analyzed, in terms parallel to Maxwell field theory, in [53]; wherein the associated BRST machinery is “deconstructed” in terms of Koszul cohomology. The high-energy limit of (−gμν +pμpν/M2)/(p2− M2)apparently signals quadratic divergences and trouble with unitarity of the scattering matrix: cross-sections would appear to grow without bound due to the longitudinal momentum states. The difficulty lies with the closure relation (A.5) of the intertwiners er, whose dimension does not allow the standard sufficiency criterion for renormalizability. This is usually “cured” nowadays by the cohomological extension of the Wigner representation space for massive spin-1 particles into spaces populated by Faddeev-Popov ghosts and anti-ghosts and Stückelberg fields. In this paper we work mainly with the Proca field (i.e., we use the unitary gauge), where these additional unphysical fields do not appear; the apparently bad UV-behaviour of the propagators is under control, as we verify, thanks to amazing cancellations in the amplitudes. Propagators for the EW theory in the unitary gauge. We will make frequent use of α β(x−x):= TWα(x)W† β(x) = −(gα β+∂α∂β/M2) ×F(x−x), (A.8) where Mis the mass of the W-field, and its properties: α β=−M2α β+i(gα β+∂α∂β/M2)δ;∂μμ ν=i∂νδ/M2. The corresponding formulas for the Wightman functions respectively read: α+ β(x−x):= Wα(x)W† β(x) = Wα†(x)Wβ(x) =−(gα β+∂α∂β/M2)+(x−x) and α+ β=−M2α+ β,∂ μμ+ ν=0. We will invoke also the Maxwell-like fields, where # =†or naught, Fμν := ∂μAν−∂νAμ;G# μν := ∂μW# ν−∂νW# μ, and introduce the propagator Dαμ βρ (x−x):= TGαμ(x)G† βρ (x) = TGαμ †(x)Gβρ (x) =T(∂αWμ(x)−∂μWα(x))(∂βW† ρ(x)−∂ρW† β(x)) =−∂μ∂ρα β(x−x)−∂βα ρ(x−x) +∂α∂ρμ β(x−x)−∂βμ ρ(x−x) =(gα β∂μ ρ−gα ρ∂μ β−gμ β∂α ρ+gμ ρ∂α β)F(x−x), ∂α ρ:= ∂α∂ρ,(A.9) 123 131 Page 20 of 25 Eur. Phys. J. C (2021) 81:131 since in ∂ρα β−∂βα ρ=(−gα β∂ρ+gα ρ∂β)F(A.10) the terms with three derivatives cancel out, due to the antisymmetry of G# μν. That fact is relevant in this paper. Analogously we obtain Dαμ+ βρ (x−x):= Gαμ(x)G† βρ (x) = Gαμ †(x)Gβρ (x) =(gα β∂μ ρ−gα ρ∂μ β−gμ β∂α ρ+gμ ρ∂α β)+(x−x), without third-order derivatives. We also note that ∂μDαμ βρ =(gα β∂ρ−gα ρ∂β)F=(−gα β∂ρ+gα ρ∂β)(M2F+iδ), since third-order derivatives appear only in the form ∂F, removable with the help of (A.3). For the 2-point functions with one G#plus one W#,we use (A.10) to get rid of the terms with three derivatives. For time-ordered ones we obtain TWμ(x)G† αν(x) = TWμ†(x)Gαν(x) =−(∂αμ ν(x−x)−∂νμ α(x−x)) =(gμ ν∂α−gμ α∂ν) ×F(x−x), TG† αν(x)Wμ(x) =TGαν(x)Wμ†(x) =(−gμ ν∂α+gμ α∂ν)F(x−x). (A.11) With the parallel Wightman functions we proceed similarly: Wμ(x)G† αν(x) = Wμ†(x)Gαν(x) =(gμ ν∂α−gμ α∂ν)+(x−x), G† αν(x)Wμ(x) = Gαν(x)Wμ†(x) =(−gμ ν∂α+gμ α∂ν)+(x−x). (A.12) Comparing with the Feynman gauge, in which the W#twopoint functions α βand α+ βare replaced by −gα βF(A.13) and −gα β+(A.14), respectively, we find the W#G#,G#W# and G#G#two-point functions to be the same, thanks to the cancellations in (A.10). Propagators for the EW theory in the Feynman gauge. The Feynman propagator and the Wightman two-point function for the W-field in the Feynman gauge read TWα(x)W† β(x) = −gα βF(x−x), (A.13) Wα(x)W† β(x) = Wα†(x)Wβ(x) = −gα β+(x−x). (A.14) Besides the W-field the computation from Sects. 4.4 involves the Stückelberg fields ϕ±and the ghost and anti-ghost fields C±,¯ C±– where φ±:= 1 √2(φ1±iφ2for φ=ϕ, C,¯ C. Below we list the non-vanishing Feynman propagators and two-point functions for these fields: Tϕ+(x)ϕ−(x) = F(x−x), (A.15) ϕ+(x)ϕ−(x) = ϕ−(x)ϕ+(x) = +(x−x), (A.16) TC+(x)¯ C−(x) = TC−(x)¯ C+(x) = F(x−x), (A.17) C+(x)¯ C−(x) = − ¯ C−(x)C+(x) = +(x−x), (A.18) C−(x)¯ C+(x) = − ¯ C+(x)C−(x) = +(x−x). (A.19) Appendix B: An interesting distribution In this appendix we study the distribution f(ρ) appearing in the amplitude of the h→γγ decay via both scalar QED and flavourdynamics. To define √·:C→Cand log:C→Cone uses a cut on the negative real axis: re iϕ=√re iϕ/2,log reiϕ=log r+iϕ, both with ϕ∈(−π,π]. The complex function ˜ f:C\(−∞,0)∪(1,∞)−→ C z−→ −log(√1−z+i√z)2(B.1) is analytic, in view of the two cuts on the real axis. The distribution f(ρ) is defined by f:[0,∞)−→ C:ρ−→ f(ρ) := ˜ f(ρ +i0). (B.2) We claim that f(ρ) =(arcsin √ρ2=arctan ρ 1−ρ22 for 0 ≤ρ≤1, (B.3) f(ρ) =−1 4log 1+1−ρ−1 1−1−ρ−1−iπ2 for ρ≥1,(B.4) from which one easily obtains the following formula for the imaginary part: f(ρ) =θ(ρ −1)π 2log √ρ+√ρ−1 √ρ−√ρ−1 =−θ(ρ −1)π 2log2ρ−2ρ(ρ −1)−1.(B.5) 123 Eur. Phys. J. C (2021) 81:131 Page 21 of 25 131 The first claim (B.3) follows immediately from the identity arcsin √ρ=−ilog1−ρ+i√ρfor ρ∈[0,1], which is obvious from exp(iarcsin x)=√1−x2+ ix,|x|≤1. To prove the second claim (B.4), first note that one has √1−(ρ +i0)=−i√ρ−1forρ≥1. Hence, there holds: log1−(ρ +i0)+i√ρ=log√ρ−ρ−1+iπ/2 =1 2log(√ρ−ρ−1)2+iπ =1 2log √ρ−√ρ−1 √ρ+√ρ−1+iπ−1 2log 1+1−ρ−1 1−1−ρ−1−iπ, from which assertion (B.4) follows. We point out that, for ρ∈[0,1], in the distribution F0(ρ) =ρ−11−ρ−1f(ρ)in Eq. (1.3)theterms∼ρ−1 cancel. We bring in the power series expansion arcsin x=x+x3 2·3+3x5 2·4·5 +3·5x7 2·4·6·7+··· for |x|≤1,yielding f(ρ) =(arcsin √ρ) 2=ρ+ρ2 3 +8ρ3 45 +··· so that F0(ρ) =−1 3−8 45 ρ+···. (B.6) Appendix C: Bogoliubov–Epstein–Glaser normalization Epstein and Glaser [21,54] started from Bogoliubov’s functional S[g]-matrix [55, Sect. 21], based on [56] and on previous work by Stückelberg and Rivier [57]. That is an expansion of operator-valued distributions (OVD) on configuration space, of the form S[g]=1 +∞  n=1 in n!d4x1···d4xnTn(x1,...,xn)g(x1)···g(xn), g∈S(R4,R). (C.1) We have taken ¯ h=1. The g’s are multiplets of coupling functions which work as adiabatic cutoffs. The Tn,symmetric in their arguments, are identified with chronological or time-ordered n-products. This is Bogoliubov’s version of the summands in the formal Dyson expansion for the scattering matrix in the interaction picture. One tries to recursively build the Tnfrom natural postulates: the ultraviolet problem is solved in that construction. In the “adiabatic limit” g↑1 the functional scattering matrix (C.1) is expected to converge to the physical Sin suitable senses [58]. C.1: The Epstein–Glaser postulates Beginning of induction: The procedure is perturbative, the basic building blocks being finite sets of quantum free fields on their corresponding Fock spaces. Precisely, T1(x)is a Wick polynomial in those and their derivatives – a well-defined OVD.16 The coupling constants of the model are included in the Tn, the expansion being a power series on them. The other postulates shall enable us to construct the Tnfrom T1by induction on n. Causality: This is the key requirement, for which the Epstein–Glaser manufacturing of TOPs is also called “causal perturbation theory”. Let V±and V±respectively denote the open forward and backward lightcones and their closures. If g1,g2are such that supp g2∩supp g1+V−=∅,then S[g1+g2]=S[g2]S[g1]; equivalently, Tn(x1,...,xn) =Tr(x1,...,xr)Tn−r(xr+1,...,xn)whenever {x1,...,xr}∩{xr+1,...,xn}+V−=∅, for all rand nwith1 ≤r≤n−1. This is a powerful postulate, called causal factorization. It means that on large open sets of the n-point Minkowski space (M4)×n≡Mnthe TOP Tncan be built up from its lower-order counterparts. In the inductive step of the Epstein–Glaser method, this requirement uniquely determines Tnon the set of Schwartz functions S(Mn\n), in terms of the given Tkat lower orders k≤n−1, where nis the “thin” diagonal n:= {(x1,...,xn): x1=x2= ··· = xn}. Perturbative normalization is the extension of the operator-valued distribution Tnfrom S(Mn\n)to S(Mn). The gist of BEG normalization is that in local quantum field theory this problem finds a solution, the induction process going through. So there is no need to deal with infinities. The solution of the extension problem is non-unique: in principle one may add any OVD which is supported on n. All further postulates of Epstein–Glaser have the purpose of giving guidance for this problem; hence they may be called “normalization conditions”. Causal Wick expansion: The TOPs are required to satisfy the Wick expansion formula. We display the latter in terms of the interaction T1(x)=ϕk(x),forϕa real 16 One can think of T1as an “interaction Lagrangian”. However, the Lagrangian mindset is inessential here. 123 131 Page 22 of 25 Eur. Phys. J. C (2021) 81:131 scalar field: Tnϕk(x1),...,ϕk(xn) = k  l1,...,ln=0k l1···k lnTn(ϕk−l1(x1), . . . , ×ϕk−ln(xn)) ϕl1(x1)···ϕln(xn) with · · ·  denoting vacuum expectation value. This postulate reduces the extension problem for the OVD Tn(···) to one of numerical distributions – a simpler task. Poincaré Covariance: Let there be given the standard lifting U(a,) to Fock space of the Poincaré unitary irreducible representations (unirreps) on 1-particle subspaces. Then U(a,)S[g]U†(a,)=S(a,)·g, where ((a,) ·g)(x)=g(−1(x−a)). In particular, translation invariance implies that the coefficients in the causal Wick expansion depend only on the relative coordinates. Therefore, the extension problem for the numerical distributions is step by step simplified to an extension to one point, namely from S(R4(n−1)\{0}) to S(R4(n−1)). Unitarity (conservation of probability): S[g]S†[g]=S†[g]S[g]=1;here we denote: S−1[g] =: 1+∞  n=1 (−i)n n!d4x1···d4xnTn(x1,...,xn) ×g(x1)···g(xn). Divergence degree: Heuristically, this is the requirement that normalization does not make the T-product “more singular” (in the UV-region). This is expressed in terms of the scaling degree of the coefficients (i.e., the numerical distributions) in the causal Wick expansion of the T-product: that degree may not be increased by the extension. The standard definitions of the scaling degree sd(t) and the singular order ω(t)of a distribution t∈S(Rk) or t∈S(Rk\{0})– see, e.g., [23, Sect. 3.2.2] – are as follows: sd(t):= inf{r∈R:lim λ↓0λrt(λx)=0},ω(t):= sd(t)−k, (C.2) where inf ∅:=∞and inf R:= −∞. For instance, for a translation-invariant distribution d(x1−x3,x2−x3)∈ S(R8)fulfilling sd(d)=8, equivalently ω(d)=0, we say that the amplitude superficially is “logarithmically divergent”. Other invariance rules and physical requirements: Discrete symmetries can be accomodated in the Epstein– Glaser construction [59]. A Ward identity playing a paramount role in this paper corresponds to EGI – see Sects. 3.1 and 4.2 for this. For different types of requirements, consult Sects. 4.4 and 4.5. C.2: Iterative building of the time-ordered products To assemble the Tnoutside of the thin diagonal nfrom the inductively known (Tk)1≤k≤n−1directly by causal factorization, one would need a partition of unity subordinate to an open cover of Mn\n–see[60] and [23, Sect. 3.3]. This is problematic for practical computations. For this reason the original Epstein–Glaser construction [21,39]isless direct: it introduces an intermediate Dn-distribution having causal support; and the crucial step is the splitting of Dninto its advanced and retarded parts. This splitting corresponds precisely to the above-mentioned extension problem, that is, to perturbative normalization. A decisive advantage of the method is that the problem is solved in momentum space by a dispersion integral. To explain the construction, we first express the antichronological product Tnin terms of the TOPs (Tk)1≤k≤n.Let N={x1,...,xn}and I⊆Nwith |I| = 0 elements. Define T|I|(I)=T|I|(xi:xi∈I). By the standard inversion of a formal power series with noncommuting terms in terms of set compositions, we obtain T|N|(N)= n  k=1 (−)n+k I1$···$Ik=N T|I1|(I1)···T|Ik|(Ik), (C.3) where the disjoint union is over nonempty blocks I j.Theterminology of antichronological products is appropriate, since if I∩(J+V−)=∅, then T(I∪J)=T(J)T(I). Retarded and advanced products, denoted by Rnand An respectively, are the coefficients in the perturbative expansion of the respective retarded and advanced interacting fields. For them we follow the convention in the book [23], identical to that of [21] except that Rnand Anhave an extra factor in−1. In general, Bogoliubov’s definitions read: Rn+1(x1,...,xn+1):= in I⊂{1,...,n} (−1)|I|T|I|(I)T|Ic|+1(Ic,xn+1), (C.4) An+1(x1,...,xn+1):= in I⊂{1,...,n} (−1)|I|T|Ic|+1(Ic,xn+1)T|I|(I), (C.5) 123 Eur. Phys. J. C (2021) 81:131 Page 23 of 25 131 where Ic:= {1,...,n}\I. Epstein and Glaser [21] prove that An+1,Rn+1have advanced or retarded support, respectively: supp An+1⊆{x∈Mn+1:xj−xn+1∈V+∀j}; supp Rn+1⊆{x∈Mn+1:xj−xn+1∈V−∀j}. In the induction step n→n+1 neither the Tn+1nor the Rn+1 nor the An+1are known. But by the induction hypothesis the difference Dn+1, defined by Dn+1:= An+1−Rn+1, only depends on known quantities. For instance, in D3the unknown T3has dropped out – and T1,T2are uniquely given in terms of T1and T2.ItfollowsthatDn+1has causal support: supp Dn+1⊆{x∈Mn+1:xj−xn+1∈V+∀j} ∪{x∈Mn+1:xj−xn+1∈V−∀j}. If one finds a way to extract the advanced part An+1of Dn+1, that is, to split the OVD Dn+1into An+1and −Rn+1in such a way that the latter two satisfy the just given support properties, then one can construct a candidate for Tn+1.17 For the sake of normalization conditions, at this stage we may add to Tn+1any OVD supported on n+1which is symmetric in x1,...,xn+1.TheDn+1fulfils all the normalization conditions, in particular the ‘Causal Wick expansion’ and ‘Translation invariance’, because of the validity of those for the inductively given (Tk)1≤k≤n. Therefore, the splitting problem for Dn+1translates into a consonant problem for the coefficients d(x1−xn+1,...,xn−xn+1)∈S(R4n,C)in the Wick expansion of Dn+1, yielding a,r(x1−xn+1,...,xn− xn+1)∈S(R4n,C), which are the coefficients in the Wick expansion of An+1and Rn+1, respectively. In fine, by the induction process, one specifies the ambiguity in the vacuum expectation value of each Tn+1by adding to it a contact term, that is, t(x1−xn+1,...,xn−xn+1) + |a|≤ω ca∂aδ(x1−xn+1,...,xn−xn+1), (C.6) where ωis the singular order of the pertinent d(x1− xn+1,...) and the coefficients ca∈Cdepending on the multi-index aare arbitrary, up to restrictions coming from the ‘Poincaré covariance’ and ‘Other invariance rules’ requirements. C.3: Dispersion integrals from splitting in BEG normalization: the central solution For simplicity, here we restrict ourselves to the case of two four-variables, relevant for this paper. For the Fourier trans17 That sometimes needs to be symmetrized, by adding a suitable OVD supported on n+1. form of f∈SR8we employ the following convention: f(y1,y2)=(2π)−4dk1dk2e−i(k1y1+k2y2)ˆ f(k1,k2). (C.7) Let ±:= V±×V±henceforth. Given a “causal distribution”, that is, d∈S(R8)with supp d⊆+∪−and sd(d)<∞,(C.8) byasplitting solution of dwe mean a distribution a∈S(R8) with (a−d)S(R8\−)=0,supp a⊆+and sd(a)≤sd(d). (C.9) In what follows we assume that the Fourier transform ˆ dof the causal d-distribution we wish to split vanishes in an open ball R⊂R8centered at k=0. This holds if all propagators contributing to dare massive, as it is the case in this paper – see [21, Sect. 5.2]. Also in [21] it is shown for any splitting solution athat ˆ d|R=0 entails analyticity of ˆa(k)on R. In this case there exists a distinguished splitting solution, the so-called central solution ac, characterized by the conditions ∂aˆac(0)=0,for all |a|≤ω(d). (C.10) As indicated in Eq. (C.6), for sd(d)≥8 – i.e., for ω(d)≥0, as defined in Eq. (C.2) – the splitting solution of dis not uniquely determined. Any two solutions a1and a2differ by a1(y)−a2(y)= ω(d)  |a|=0 Ca∂aδ(y)or equivalently, ˆa1(k)−ˆa2(k)=1 (2π)4 ω(d)  |a|=0 Ca(−ik)a, with arbitrary constants Ca∈C. Essential for dealing with our situation is that the central solution of the splitting problem in momentum space can be computed by a dispersion integral. Now we sketch the derivation of a few versions of this distinguished splitting integral. 18 The naive way to extract the advanced part aof dis to multiply the latter by a θ-function: anaive(y1,y2):= d(y1,y2)χ(y1,y2)with χ(y1,y2):= θ(y1v1)+(y2v2), where v:= (v1,v 2)∈V+×V+is arbitrary. But for sd(d)≥8 the pointwise product dχexists only as an element of S(R8\{0}). Therefore, the splitting problem is an 18 For further detail we refer to [39, Sect. 3.2], which relies on [21, Sect. 6.5]. 123 131 Page 24 of 25 Eur. Phys. J. C (2021) 81:131 extension problem: we have to extend dχ∈S(R8\{0})to an a∈S(R8)such that sd(a)=sd(dχ) =sd(d).19 The problem is studied in its particulars in [23, Sect. 3.2.2]. Given awith singular order ω, there exists an obvious extension aω, belonging in the dual space S ω(R8)of Sω(R8):= { f∈S(R8):∂bf(0)=0 for all |b|≤ω}, uniquely determined by the requirement that sd(aω)= sd(d). Next, a projection is introduced: Wω:S(R8)−→ Sω(R8); Wωf(y):= f(y)−w(y) ω  |b|=0 yb b!∂bf(0), (C.11) where the suitably decaying function wmust fulfil w(0)=1 and ∂bw(0)=0for1≤|b|≤ω. One verifies that a solution aw(depending on the choice of the function w) of the splitting problem (C.9) is obtained by setting aw|f:=aω|Wωf.(C.12) The aωinvolved here is dχwith enlarged domain. If furthermore assumption d|R=0 is satisfied, the infrared behaviour of d(y)is harmless. Hence, one may simply choose w(y)=1 for all y∈R4. Then the correspondent splitting solution aw=1is actually the central solution ac(C.10). Substituting dχfor aωand further using the convolution formula ˆ fˆχ(k)=(2π)4i 2πR dt t+i0ˆ f(k−tv) and  fg =(2π)−4ˆ fˆg, we see that ˆac(k)=i 2πR dt t+i0ˆ d(k−tv) − ω  |b|=0 kb b!∂bˆ d(−tv). (C.13) This splitting integral does not depend on the choice of v∈ V+×V+. Moreover, for k∈Vη×Vη, where η∈{+,−}, we may choose v:= ηk– that vvary with kis admissible. With some extra work [39, Prop. 3.4], this formula is then simplified into a convergent dispersion integral: ˆac(k) =iη 2πR dt ˆ d(tk) (t−ηi0)max{ω+1,0}(1−t+iη0)for k∈Vη×Vη. (C.14) In the applications treated in this paper, d(tk)is of the form ˆ d(tk) =ηsgn(t)θ(t2−t2 min)ft2k2 1,t2k2 2,t2(k1+k2)2for k∈Vη×Vη, (C.15) 19 A priori, it might happen that sd(dχ)<sd(d); but in the applications to Epstein–Glaser normalization known to us one always finds sd(dχ) =sd(d). Hence we assume the latter relation to hold true. for some f∈S(R3), where tmin >0 depends on the squares of the momenta. So finally, introducing the new integration variable u:= t2, the integral (C.14) goes over into ˆac(k) =i 2π∞ t2 min du f(uk2 1,uk2 2,u(k1+k2)2) umax{ω/2+1,0}(1−u+iη0)for k∈Vη×Vη, (C.16) where ·denotes the integer part. References 1. J.R. Ellis, M.K. Gaillard, D.V. Nanopoulos, Nucl. Phys. B 106, 292 (1976) 2. M.A. Shifman, A.I. Vainshtein, M.B. Voloshin, V.I. Zakharov, Sov. J. Nucl. Phys. 30, 711 (1979) 3. M.B. Gavela, G. Girardi, C. Malleville, P. Sorba, Nucl. Phys. B 193, 257 (1981) 4. T. Appelquist, J. Carazzone, Phys. Rev. D 11, 2856 (1975) 5. R. Gastmans, S.L. Wu, T.T. Wu, Higgs decay H→γγ through a Wloop: difficulty with dimensional regularization. arXiv:1108.5322 6. R. Gastmans, S.L. Wu, T.T. Wu, Int. J. Mod. Phys. A 30, 15502000 (2015). arXiv:1108.5872 7. H.-S. Shao, Y.-J. Zhang, K.-T. Chao, JHEP 2012–01, 053 (2012) 8. W.J. Marciano, C. Zhang, S. Willenbrock, Phys. Rev. D 85, 013002 (2012) 9. M.A. Shifman, A.I. Vainshtein, M.B. Voloshin, V.I. Zakharov, Phys.Rev.D85, 013015 (2012) 10. D. Huang, Y. Tang, Y.-L. Wu, Commun. Theor. Phys. 57, 427 (2012) 11. F. Jegerlehner, Comment on H→γγ and the role of the decoupling theorem and the equivalence theorem, arXiv:1110.0869 12. F. Piccinini, A. Pilloni, A.D. Polosa, Chin. Phys. C 37, 043102 (2013) 13. A. Dedes, K. Suxho, Adv. High Energy Phys. 2013, 631841 (2013) 14. S. Weinzierl, Mod. Phys. Lett. A 29, 1430015 (2014) 15. T.T. Wu, S.L. Wu, Nucl. Phys. B 914, 421 (2017) 16. E. Christova, I. Todorov, Bulg. J. Phys. 42, 296 (2015) 17. K. Melnikov, A. Vainshtein, Phys. Rev. D 93, 053015 (2016) 18. J. Gegelia, U.-G. Meissner, Nucl. Phys. B 934, 1 (2018) 19. I. Boradjiev, E. Christova, H. Eberl, Phys. Rev. D 97, 073008 (2018) 20. K. Jacobs, talk given at IFT, Madrid, December 2019 21. H. Epstein, V.J. Glaser, Ann. Inst. Henri Poincaré A 19, 211 (1973) 22. J.M. Gracia-Bondía, H. Gutiérrez, J.C. Várilly, Nucl. Phys. B 886, 824 (2014) 23. M. Dütsch, From Classical Field Theory to Perturbative Quantum Field Theory, Progress in Mathematical Physics 74 (Birkhäuser, Cham, 2019) 24. N. Irges, F. Koutroulis, Nucl. Phys. B 924, 178 (2017) 25. J.F. Gunion, H.E. Haber, G. Kane, S. Dawson, The higgs Hunter’s Guide (Addison-Wesley, Redwood City, 1990) 26. D. Bardin, G. Passarino, The Standard Model in the Making (Oxford University Press, Oxford, 1999) 27. L.B. Okun, Leptons and Quarks (World Scientific, Singapore, 2014) 28. M. Dütsch, F. Krahe, G. Scharf, Nuovo Cimento A 106, 277 (1993) 29. G. Scharf, Finite Quantum Electrodynamics (Springer, Berlin, 1989) 30. G. ’t Hooft, M. Veltman, Diagrammar, CERN 73-9 (1973) 123 Eur. Phys. J. C (2021) 81:131 Page 25 of 25 131 31. B. Simon, Real Analysis (American Mathematical Society, Providence, 2015) 32. P. Langacker, The Standard Model and Beyond (CRC Press, New York, 2010) 33. M. Dütsch, G. Scharf, Ann. Phys. (Leipzig) 8, 359 (1999) 34. M. Dütsch, T. Hurth, F. Krahe, G. Scharf, Nuovo Cimento A 107, 375 (1994) 35. A. Aste, M. Dütsch, G. Scharf, Ann. Phys. (Leipzig) 8, 389 (1999) 36. G. Scharf, Quantum Gauge Theories: A True Ghost Story (Wiley, New York, 2001) 37. R. Stora, Local gauge groups in quantum field theory: perturbative gauge theories. talk given at the workshop Local Quantum Physics, Erwin-Schrödinger Institut, Vienna (1997) 38. M.D. Schwartz, Quantum Field Theory and the Standard Model (Cambridge University Press, Cambridge, 2014) 39. G. Scharf, Finite Quantum Electrodynamics: The Causal Approach (Dover, New York, 2014) 40. M. Dütsch, F. Krahe, G. Scharf, Nuovo Cim. A 105, 399 (1992) 41. R. Brunetti, M. Dütsch, K. Fredenhagen, Adv. Theor. Math. Phys. 13, 1541 (2009) 42. N.H. Christ, T.D. Lee, Phys. Rev. D 22, 939 (1980) 43. A. Aste, G. Scharf, M. Dütsch, J. Phys. A 31, 1563 (1998) 44. R. Mertig, M. Böhm, A. Denner, Comput. Phys. Commun. 64, 345–359 (1991) 45. M. Herrero, R.A. Morales, Phys. Rev. D 102, 075040 (2020) 46. V.S. Vanyashin, M.V. Terent’ev, Sov. Phys. JETP 21, 375 (1965) 47. M. Dütsch, Rev. Math. Phys. 27, 1550024 (2015) 48. M. Dütsch, F. Krahe, G. Scharf, J. Phys. G 19, 485 (1993) 49. B.A. Kniehl, M. Spira, Z. Phys. C 69, 77 (1995) 50. J. Hoˇrejší, M. Stöehr, Phys. Lett. B 379, 159 (1996) 51. W. Pauli, Rev. Mod. Phys. 13, 203 (1941) 52. H. Ruegg, M. Ruiz-Altaba, Int. J. Mod. Phys. A 19, 3265 (2004) 53. R. Stora, From Koszul complexes to gauge fixing, in 50 Years of Yang-Mills Theory, ed. by G ’t Hooft (World Scientific, Singapore, 2005), pp. 137–167 54. H. Epstein, V.J. Glaser, Adiabatic limit in perturbation theory, in Renormalization Theory, ed. by G. Velo, A.S. Wightman (Springer, 1976), pp. 193–254 55. N.N. Bogoliubov, D.V. Shirkov, Introduction to the Theory of Quantized Fields, 3rd edn. (Wiley, New York, 1980) 56. N.N. Bogoliubov, Izv. Akad. Nauk SSSR Ser. Fiz. 19, 237 (1955) 57. E.C.G. Stückelberg, D. Rivier, Helv. Phys. Acta 23, 215 (1950) 58. P. Duch, Ann. Henri Poincaré 19, 875 (2018) 59. M. Dütsch, J.M. Gracia-Bondía, Phys. Lett. B 711, 428 (2012) 60. R. Brunetti, K. Fredenhagen, Commun. Math. Phys. 208, 623 (2000) 123