# Diphoton decay of the higgs
from the Epstein–Glaser viewpoint

Paweł Duch1,2, Michael Dütsch3 and José M. Gracia-Bondía4,5111Email:
<EMAIL_ADDRESS>
1 Institut für Theoretische Physik, Universität Leipzig, 04103 Leipzig,
Germany
2 Max-Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany
3 Institut für Theoretische Physik, Universität Göttingen, Göttingen D-37077,
Germany
4 CAPA and Departamento de Física Teórica, Universidad de Zaragoza, Zaragoza
50009, Spain
5 Laboratorio de Física Teórica y Computacional, Universidad de Costa Rica,
San Pedro 11501, Costa Rica

###### Abstract

We revisit a nearly ten-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.

To the memory of Günter Scharf and Raymond Stora

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 [2] – and then
“exactly” in [3]. The accepted result was confirmed many times – see [4] 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 [5], as
often understood.

Much more recently, those calculations were questioned in [6, 7]. 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 photons, 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 [8, 9, 10, 11, 12, 13, 14, 15] to [6, 7].
The authors of [10] are the ones of the original calculation [3]. Those papers
made several points, some rather implausibly arguing that at a given point in
the calculation in [7] electromagnetic gauge invariance is lost, and
criticizing the interpretation of the DT made in [6, 7]. 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 [16]. This later paper argues
by the example that two computations of the same process in different gauges
($R_{\xi}$ versus 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 [17]
appeared to support the contentions of [6, 7], and got in turn a – quite
thoughtful – rejoinder in [18]. More recent papers dealing with the same or
related issues are [19, 20].

By and large, the majority’s opinion and the experimental results [21] 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” 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 [22] of the
(functional) $\mathbb{S}$-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 [23] or [24, 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.222The paper [25] dwells usefully
on the subject of the $R_{\xi}$-versus-unitary gauges, leaning to demonstrate
the validity of the latter at the quantum level.

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 App. 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 $m_{h}$ denote the
mass of the higgs $h$. The amplitude coming from the one-loop calculations may
be quoted as [26, 27, 28]:

$\mathcal{A}=\frac{g\alpha}{2\pi M}F_{1}(\rho)P_{\mu\nu},$

with $\alpha$ the fine structure constant, $g$ the EW coupling constant, $M$
the mass of the intermediate $W$-boson and $\rho:=m^{2}_{h}/4M^{2}$. The
polarization factor $P_{\mu\nu}$, reflecting electromagnetic gauge invariance
(EGI) of $\mathcal{A}$,333That is, transversality of the outgoing photons. is
written in this paper as

$P_{\mu\nu}:=(k_{1}k_{2})g_{\mu\nu}-k_{1\nu}k_{2\mu};\quad(P_{\bullet\nu}k_{1})=(P_{\mu\bullet}k_{2})=0,$
(1.1)

with $k_{1}$, $k_{2}$ the outgoing photons’ momenta. Finally, for the
dimensionless factor:

$F_{1}(\rho):=2+\frac{3}{\rho}+\frac{3}{\rho}\biggl{(}2-\frac{1}{\rho}\biggr{)}f(\rho).$
(1.2)

Now that we are at that, we quote as well the comparable result for a charged
scalar particle of mass $M$ at the place of the $W$-boson:

$F_{0}(\rho)=\frac{1}{\rho}\biggl{(}1-\frac{f(\rho)}{\rho}\biggr{)};\quad\text{so
that}\quad F_{1}(\rho)=3F_{0}(\rho)+\frac{6f(\rho)}{\rho}+2.$ (1.3)

For the benefit of the reader coming to the subject of this paper for the
first time, App. B introduces the distribution $f(\rho)$ appearing in both
$F_{1}$ (1.2) and $F_{0}$ (1.3) – as well as in the amplitude of diphoton
decay of $h$ via virtual fermions.

The bone of contention is that the first summand $2$ in (1.2) should not be
there, according to [6, 7, 17]. Relations (B.3) and (B.6) tell us that, as
$\rho\downarrow 0$:

$F_{1}=2+\frac{3}{\rho}+\biggl{(}\frac{6}{\rho}-\frac{3}{\rho^{2}}\biggr{)}\biggl{(}\rho+\frac{\rho^{2}}{3}+\frac{8\rho^{3}}{45}+\cdots\biggr{)}=7+\frac{22}{15}\,\rho+O(\rho^{2});$

so $F_{1}(0)=7$ and $F_{1}(\infty)=2$ from (1.2). Precisely the former figure
is what was calculated in the paper [2]. The result argued by the “heretics”
in the controversy is $F_{1}-2$, so their respective assertions are instead
$F_{1}(0)=5$ and $F_{1}(\infty)=0$. Also, from (1.3): $F_{0}(0)=-1/3$ and
$F_{0}(\infty)=0$.

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
subsection C.3. New results in this respect are required, announced in the
short Section 2 and proved in subsections 3.2 and 3.3 of this paper. So for
aficionados of BEG normalization there is novelty here – whose interest goes
beyond the particular problem that motivated it.

Sections 3 and 4 constitute the heart of the paper. The scalar model leading
to $F_{0}$ is worked out in Section 3. One is 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 [17].

Finally, in Section 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 $AAWW^{\dagger}$-vertex.

It is time to aver why the “no-renormalization” argument in [7] is not
watertight. A direct $h\gamma\gamma$ 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 $F_{1}(\rho)$.
Within the unitary gauge, a different argument to the same purpose is
discussed at the end of this Section 4. Section 5 is the conclusion.

##  2 The obstruction to distribution splitting for null momenta

Formula (C.16) in App. 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) are
in 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 $a^{c}(k_{1},k_{2})$ 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 $k_{j}^{2}=0$, it generally holds that $(k_{j}-v_{j})^{2}\neq 0$
because $v_{j}\in V_{+}$; it does not suffice to know the causal distribution
$d(k_{1},k_{2})$ only for $k_{1}^{2}=0=k_{2}^{2}$.

The next section solves this problem for models such that
$0<(k_{1}+k_{2})^{2}<4M^{2}$ and $k_{1}^{0}k_{2}^{0}>0$. The proof’s strategy
is as follows: starting from the dispersion integral (C.14) for $k_{1}^{2}>0$,
$k_{2}^{2}>0$ and $k_{1}^{0}k_{2}^{0}>0$, we intend to show that
$d(k_{1},k_{2})$ is regular enough that this integral commutes with the limit
$(k_{1}^{2}\downarrow 0\wedge k_{2}^{2}\downarrow 0)$. Therefore the
dispersion integrals (C.14) and (C.16) keep their usefulness for
$k_{1}^{2}=0=k_{2}^{2}$: indeed, for computing
$a^{c}(k_{1},k_{2})|_{k_{1}^{2}=0=k_{2}^{2}}$ it suffices to know
$d(k_{1},k_{2})$ only for $k_{1}^{2}=k_{2}^{2}=0$, because
$k_{1}^{2}=k_{2}^{2}=0$ implies $(tk_{1})^{2}=(tk_{2})^{2}=0$ for all $t$.

Crucially, in the resulting dispersion integrals (C.14) and (C.16) for
$k_{1}^{2}=0=k_{2}^{2}$, the parameter $\omega$ is the singular order of the
off-shell $d(k_{1},k_{2})$. As a consequence, the general solution (prior to
imposition of other invariance rules) of the distribution splitting is
obtained by adding to $a^{c}(k_{1},k_{2})|_{k_{1}^{2}=0=k_{2}^{2}}$ a
polynomial in $k_{1},k_{2}$, in principle arbitrary, whose degree is given by
the singular order of the off-shell amplitude $d(k_{1},k_{2})$. Now, it
frequently happens that the singular order of
$d(k_{1},k_{2})|_{k_{1}^{2}=0=k_{2}^{2}}$ has 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
$\omega$. Examples for this are the “toy model” in the next section and the EW
diphoton decay of the higgs in the unitary gauge (subsections 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.444Private communication, early 2013.

##  3 Higgs to diphoton decay via a charged scalar field

The scalar electrodynamics computation leading to $F_{0}$ works 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”
$e^{2}AA\varphi\varphi^{\dagger}$-vertex is derived by implementing EGI within
the construction rules of the method – as any other part of $T_{2}$ [29]. We
give full details on how this comes about for the quartic vertex in the EW
theory in subsection 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 subsection C.2.

###  3.1 A causal distribution on-shell

The starting point is given by the lower order time-ordered products (TOPs):

$\displaystyle
T_{1}(x_{3})=gM\,h(x_{3})\,\varphi(x_{3})\,\varphi^{\dagger}(x_{3});\quad
T_{1}(x_{j})=-ieA^{\lambda}(x_{j})\,\varphi^{\dagger}(x_{j})\overleftrightarrow{\partial_{\lambda}}\varphi(x_{j}),\;\;j=1,2;$
$\displaystyle T_{2}(x_{1},x_{2})$
$\displaystyle\quad=-e^{2}A^{\mu}(x_{1})A^{\nu}(x_{2})\bigl{[}\varphi^{\dagger}(x_{1})\,\partial_{\mu}\Delta^{F}(x_{1}-x_{2})\,\partial_{\nu}\varphi(x_{2})-\partial_{\mu}\varphi^{\dagger}(x_{1})\Delta^{F}(x_{1}-x_{2})\,\partial_{\nu}\varphi(x_{2})$
$\displaystyle\quad+\varphi^{\dagger}(x_{1})\,\bigl{(}\partial_{\nu}\partial_{\mu}\Delta^{F}(x_{1}-x_{2})+ig_{\mu\nu}\,\delta(x_{1}-x_{2})\bigr{)}\varphi(x_{2})-\partial_{\mu}\varphi^{\dagger}(x_{1})\,\partial_{\nu}\Delta^{F}(x_{1}-x_{2})\,\varphi(x_{2})\bigr{]}$
$\displaystyle\quad+(x_{1}\leftrightarrow
x_{2})+T_{1}(x_{1})\,T_{1}(x_{2})+[\text{irrelevant loop diagram terms}],$

where $\Delta^{F}$ denotes the Feynman propagator (A.3).

From our formulas (C.4) and (C.5):555The $D_{n}$ are always linear
combinations of commutators.

$D_{3}(x_{1},x_{2},x_{3})=-[\overline{T}_{1}(x_{1}),T_{2}(x_{2},x_{3})]-[\overline{T}_{1}(x_{2}),T_{2}(x_{1},x_{3})]+[\overline{T}_{2}(x_{1},x_{2}),T_{1}(x_{3})].$
(3.1)

Because the photons emitted at $x_{1},x_{2}$ are 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

$\overline{T}_{1}(x_{1})=T_{1}(x_{1});\quad\overline{T}_{2}(x_{1},x_{2})=-T_{2}(x_{1},x_{2})+T_{1}(x_{1})T_{1}(x_{2})+T_{1}(x_{2})T_{1}(x_{1}).$
(3.2)

For the same reasons just argued, only the connected tree diagram part of the
$T_{2}(x_{1},x_{2})$ summand in $\overline{T}_{2}(x_{1},x_{2})$ 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
[30, Sect. 3.8]. Going to the contractions, bringing in the vertices and the
propagators (A.2), (A.4), apart from a factor $4ge^{2}M$ we obtain:

$\displaystyle
A_{\mu}(x_{1})A_{\nu}(x_{2})h(x_{3})\bigl{[}\Delta^{-}(1)\,\partial^{\mu}\Delta^{F}(1-2)\,\partial^{\nu}\Delta^{-}(2)-\partial^{\mu}\Delta^{-}(1)\,\partial^{\nu}\Delta^{F}(1-2)\,\Delta^{-}(2)$
$\displaystyle\quad-\partial^{\mu}\Delta^{-}(1)\,\Delta^{F}(1-2)\,\partial^{\nu}\Delta^{-}(2)+\Delta^{-}(1)\bigl{(}\partial^{\mu}\partial^{\nu}\Delta^{F}(1-2)+ig^{\mu\nu}\,\delta(1-2)\bigr{)}\Delta^{-}(2)$
$\displaystyle\quad-[\text{the same four terms with $\Delta^{-}$ replaced by
$\Delta^{+}$}]+\cdots\bigr{]}$
$\displaystyle=:A_{\mu}(x_{1})A_{\nu}(x_{2})h(x_{3})\,d^{\mu\nu}(1,2),$

where $1\equiv y_{1}:=x_{1}-x_{3}$, $2\equiv y_{2}:=x_{2}-x_{3}$. Here and
further down, the dots stand for the terms coming from the other two cuts and
further terms not contributing to the on-shell amplitude. Note the advertised
additional $+ig^{\mu\nu}\,\delta$ to $\partial^{\mu}\partial^{\nu}\Delta^{F}$,
corresponding to the “closed seagull” or fish-like diagram contribution to the
$h\to 2\gamma$ decay 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:
$d^{\mu}(k_{1},k_{2})\equiv\hat{d}^{\mu}(k_{1},k_{2})$. We obtain

$\displaystyle d^{\mu\nu}(k_{1},k_{2})=$
$\displaystyle\frac{1}{(2\pi)^{2}}\bigl{[}4(I_{+}^{\mu\nu}-I_{-}^{\mu\nu})+2k_{2}^{\nu}(I_{+}^{\mu}-I_{-}^{\mu})-2k_{1}^{\mu}(I_{+}^{\nu}-I_{-}^{\nu})-k_{1}^{\mu}k_{2}^{\nu}(I_{+}-I_{-})$
$\displaystyle\quad-\frac{i}{(2\pi)^{2}}\,g^{\mu\nu}\,(J_{+}-J_{-})\bigr{]}+\cdots$
(3.3)

with the integrals

$\displaystyle I_{\pm}^{\\{\cdot|\mu|\mu\nu\\}}(k_{1},k_{2})$
$\displaystyle:=\int
d^{4}k\,\\{1|k^{\mu}|k^{\mu}k^{\nu}\\}\,\Delta^{\pm}(k_{1}-k)\Delta^{F}(k)\,\Delta^{\pm}(k+k_{2}),$
$\displaystyle J_{\pm}(k_{1},k_{2})$ $\displaystyle:=\int
d^{4}k\,\Delta^{\pm}(k_{1}-k)\,\Delta^{\pm}(k+k_{2}),$ (3.4)

where the $J_{\pm}$-term is the contribution of the fish-like diagram. Keep in
mind that the terms belonging to $A^{\prime}_{3}:=A_{3}-T_{3}$ are those
coming from the integrals $I^{\cdot|\mu|\mu\nu}_{-}$ and $J_{-}$, whereas the
contribution of $R^{\prime}_{3}:=R_{3}-T_{3}$ is given by the integrals
$I^{\cdot|\mu|\mu\nu}_{+}$ 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) to $1$:

$\displaystyle\int
dx_{1}\,dx_{2}\,dx_{3}\,A^{\mu}(x_{1})A^{\nu}(x_{2})h(x_{3})\,d^{\mu\nu}(x_{1}-x_{3},x_{2}-x_{3})$
$\displaystyle\enspace=(2\pi)^{2}\int
dk_{1}\,dk_{2}\,h(k_{1}+k_{2})A^{\mu}(-k_{1})A^{\nu}(-k_{2})\,d^{\mu\nu}(k_{1},k_{2}).$
(3.5)

In this limit the momenta $k_{1}$ and $k_{2}$ become the momenta of the
external photons: $k_{1}^{2}=k_{2}^{2}=0$.

From now on, we compute $d^{\mu\nu}(k_{1},k_{2})|_{k_{1}^{2}=0=k_{2}^{2}}$.
Were we to have included the other cuts in (3.1) or $T_{1}T_{1}T_{1}$-terms,
there would appear $\Delta^{\pm}$-type propagators at the place of the Feynman
propagators above. The former are $\sim\delta(k^{2}-M^{2})$, with $k$ denoting
the internal momentum variable in the loop: so to speak, in contrast with the
Feynman propagators, the $\Delta^{\pm}$ are “always on-shell”, even within
loops.666This point is made in [31, Sect. 6.4]. Thus no further internal
momenta can be on-shell: assuming $k^{2}=M^{2}$ one obtains
$(k_{1}-k)^{2}=M^{2}-2(k_{1}k)\neq M^{2}$; similarly for
$(k+k_{2})$.777Compare the discussion after [30, Eq. (3.8.24)].

#### Scalar integrals $I_{\pm}$.

We have to compute

$\displaystyle I_{\mp}(k_{1},k_{2}):=\frac{i}{(2\pi)^{4}}$ $\displaystyle\int
d^{4}k\,\theta(\mp(k_{1}^{0}-k^{0}))\,\delta((k_{1}-k)^{2}-M^{2})$
$\displaystyle\quad\times\frac{1}{k^{2}-M^{2}+i0}\,\theta(\mp(k^{0}+k_{2}^{0}))\,\delta((k+k_{2})^{2}-M^{2}).$

Let us make a change of variable $q:=k+k_{2}$, and introduce $P:=k_{1}+k_{2}$,
noting for later purposes that $P^{2}=2(k_{1}k_{2})$. One obtains the
integral:

$\displaystyle\int
d^{4}q\,\theta(\mp(P^{0}-q^{0}))\,\delta((P-q)^{2}-M^{2})\,\frac{1}{(q-k_{2})^{2}-M^{2}+i0}\,\theta(\mp
q^{0})\,\delta(q^{2}-M^{2}).$ (3.6)

It follows that $I_{\mp}(k_{1},k_{2})\propto\theta(\mp
P^{0})\,\theta(P^{2}-4M^{2})$, and that
$\operatorname{sgn}k_{1}^{0}=\operatorname{sgn}k_{2}^{0}$ for $P^{2}\geq
4M^{2}$.

Performing the $q^{0}$-integration and using the notation
$E_{q}:=\sqrt{|\bm{q}|^{2}+M^{2}}$, we extract

$\displaystyle I_{\mp}(k_{1},k_{2})$
$\displaystyle=\frac{i}{(2\pi)^{4}}\,\theta(\mp P^{0})\,\theta(P^{2}-4M^{2})$
$\displaystyle\quad\times\int\frac{d^{3}q}{2E_{q}}\,\theta(\mp(P^{0}-q^{0}))\,\delta((P-q)^{2}-M^{2})\,\frac{1}{(q-k_{2})^{2}-M^{2}+i0}\biggr{|}_{q^{0}=\mp
E_{q}}.$

Since $P^{2}>0$, one may choose a particular Lorentz frame such that

$P=(P^{0},\bm{0});\quad\text{hence}\quad\bm{k}_{1}=-\bm{k}_{2},\quad
k_{1}^{0}=\mp|\bm{k}_{1}|=\mp|\bm{k}_{2}|=k_{2}^{0}={\mathchoice{\tfrac{1}{2}}{\tfrac{1}{2}}{{\scriptstyle\frac{1}{2}}}{{\scriptstyle\frac{1}{2}}}}P^{0}.$
(3.7)

Taking into account $q^{2}=M^{2}$, we observe that
$(P-q)^{2}-M^{2}=2P^{0}({\mathchoice{\tfrac{1}{2}}{\tfrac{1}{2}}{{\scriptstyle\frac{1}{2}}}{{\scriptstyle\frac{1}{2}}}}P^{0}-q^{0})$,
which yields

$\delta((P-q)^{2}-M^{2})=\frac{\delta(q^{0}-{\mathchoice{\tfrac{1}{2}}{\tfrac{1}{2}}{{\scriptstyle\frac{1}{2}}}{{\scriptstyle\frac{1}{2}}}}P^{0})}{2|P^{0}|}=\frac{\delta(E_{q}-{\mathchoice{\tfrac{1}{2}}{\tfrac{1}{2}}{{\scriptstyle\frac{1}{2}}}{{\scriptstyle\frac{1}{2}}}}|P^{0}|)}{2|P^{0}|}\,,$

by using $q^{0}=\mp E_{q}$. For later aims, we point out that in the chosen
frame this distribution implies $q^{0}=k_{2}^{0}$; hence

$kP=(q-k_{2})P=(q^{0}-k_{2}^{0})P^{0}=0.$ (3.8)

From $\mp
q^{0}=\mp{\mathchoice{\tfrac{1}{2}}{\tfrac{1}{2}}{{\scriptstyle\frac{1}{2}}}{{\scriptstyle\frac{1}{2}}}}P^{0}$
comes
$\mp(P^{0}-q^{0})=\mp{\mathchoice{\tfrac{1}{2}}{\tfrac{1}{2}}{{\scriptstyle\frac{1}{2}}}{{\scriptstyle\frac{1}{2}}}}P^{0}>0$.
Therefore the factor $\theta(\mp(P^{0}-q^{0}))$ is redundant. Changing the
integration variables,

$\int d^{3}q\cdots=\int_{M}^{\infty}dE_{q}\,E_{q}\sqrt{E_{q}^{2}-M^{2}}\int
d\Omega_{q}\cdots\,,$

the $E_{q}$-integration can trivially be done, and we are left with:

$\displaystyle I_{\mp}(k_{1},k_{2})=i\,\theta(\mp
P^{0})\,\theta(P^{2}-4M^{2})\,\frac{\sqrt{(P^{0})^{2}-4M^{2}}}{(2\pi)^{4}\,8|P^{0}|}\int\\!\frac{d\Omega_{q}}{(q-k_{2})^{2}-M^{2}+i0}\biggr{|}_{q^{0}=P^{0}/2}.$
(3.9)

Let $\alpha$ be the angle between $\bm{k}_{2}$ and $\bm{q}$, and let
$z:=\cos\alpha$. Due to $q^{2}=M^{2}$, $k_{2}^{2}=0$,
$|\bm{q}|=\sqrt{E_{q}^{2}-M^{2}}={\mathchoice{\tfrac{1}{2}}{\tfrac{1}{2}}{{\scriptstyle\frac{1}{2}}}{{\scriptstyle\frac{1}{2}}}}\,\sqrt{P_{0}^{2}-4M^{2}}$
and relations (3.7) and (3.8), we obtain

$(q-k_{2})^{2}-M^{2}=-2(k_{2}q)=-2(k_{2}^{0}q^{0}-|\bm{q}|\cdot|\bm{k}_{2}|\,z)=\tfrac{a}{2}(-a+bz),$
(3.10)

where

$a:=|P^{0}|>0,\quad 0\leq b:=\sqrt{(P^{0})^{2}-4M^{2}}<a.$

We point out that $(-a+bz)<0$ for all $z\in[-1,1]$: there is no infrared
problem in our triangle graph. The remaining $\Omega_{q}$-integral can be
easily computed:

$\frac{4\pi}{a}\int_{-1}^{1}\frac{dz}{-a+bz}=\frac{4\pi}{|P^{0}|\,\sqrt{(P^{0})^{2}-4M^{2}}}\,\log\frac{(P^{0})^{2}-|P^{0}|\,\sqrt{(P^{0})^{2}-4M^{2}}-2M^{2}}{2M^{2}}\,.$
(3.11)

To obtain the result in a generic Lorentz frame, replace $(P^{0})^{2}$ by
$s:=P^{2}=2(k_{1}k_{2})$, so

$\displaystyle I_{\mp}(k_{1},k_{2})$ $\displaystyle=\frac{i\theta(\mp
P^{0})\,\theta(s-4M^{2})}{4(2\pi)^{3}\,s}\log\biggl{[}\frac{s-\sqrt{s(s-4M^{2})}}{2M^{2}}-1\biggr{]}$
$\displaystyle=:\theta(\mp P^{0})\,\theta(s-4M^{2})F(s).$ (3.12)

The result for $J_{\pm}(k_{1},k_{2})$ can be read off from (3.9) by omitting
the Feynman propagator $i(2\pi)^{-2}\,((q-k_{2})^{2}-M^{2}+i0)^{-1}$. One
obtains for the contribution of the $J$-integrals:

$\displaystyle J_{\pm}(k_{1},k_{2})=\frac{1}{8\pi}\,\theta(\pm
P^{0})\,\theta(s-4M^{2})\,\sqrt{1-4M^{2}/s}\,.$

#### Vector integrals $I^{\mu}_{\mp}\,$.

For the same reasons as for the scalar integral, it must hold that
$I^{\mu}_{\mp}(k_{1},k_{2})\propto\theta(\mp P^{0})\,\theta(s-4M^{2})$. From
Lorentz covariance and
$I^{\mu}_{\pm}(k_{1},k_{2})=-I^{\mu}_{\pm}(k_{2},k_{1})$ it follows

$I^{\mu}_{\mp}(k_{1},k_{2})=\theta(\mp
P^{0})\,\theta(s-4M^{2})\,(k_{1}^{\mu}-k_{2}^{\mu})\,G(s)$

for appropriate $G(s)$. An immediate consequence is $I^{\mu}P_{\mu}=0$. To
procure $G(s)$, compute

$k_{2,\mu}I^{\mu}_{\mp}(k_{1},k_{2})=\frac{1}{2}\theta(\mp
P^{0})\,\theta(s-4M^{2})\,s\,G(s)=\bigl{(}-i/8\,(2\pi)^{3}\bigr{)}\,\theta(\mp
P^{0})\,\theta(s-4M^{2})\,\sqrt{1-4M^{2}/s}$

The second equality is obtained by comparing with the scalar integral: there
is an extra factor $(k_{2}k)=(k_{2}q)=-a(-a+bz)/4$, where (3.10) is used. Then
the $\Omega_{q}$-integral becomes trivial. Thus we glean

$\displaystyle G(s)=\frac{-i}{32\,\pi^{3}\,s}\sqrt{1-4M^{2}/s}\,.$ (3.13)

#### Tensor integrals $I^{\mu\nu}_{\mp}$.

Proceeding analogously to the vector integrals, one argues that

$I^{\mu\nu}_{\mp}(k_{1},k_{2})=\theta(\mp
P^{0})\,\theta(s-4M^{2})\,\bigl{[}(k_{1}^{\mu}k_{1}^{\nu}+k_{2}^{\mu}k_{2}^{\nu})\,A(s)+(k_{1}^{\mu}k_{2}^{\nu}+k_{2}^{\mu}k_{1}^{\nu})\,B(s)+g^{\mu\nu}\,C(s)\bigr{]}.$

We need three independent identities to compute $A(s)$, $B(s)$ and $C(s)$. A
first one is:

$\displaystyle I^{\mu\nu}_{\mp}k_{2\mu}k_{2\nu}$ $\displaystyle=\theta(\mp
P^{0})\,\theta(s-4M^{2})\,A(s)\,s^{2}/4$ $\displaystyle=\theta(\mp
P^{0})\,\theta(s-4M^{2})\,\frac{-i}{2^{5}\,(2\pi)^{3}}\,s\,\sqrt{1-4M^{2}/s}\,.$
(3.14)

The second equality is obtained by a modification of the computation of the
scalar integral: there is the extra factor $(k_{2}k)^{2}=a^{2}(-a+bz)^{2}/16$.
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
$k^{2}=(q-k_{2})^{2}=M^{2}-2(k_{2}q)=M^{2}-2(kk_{2})$, hence

$I^{\mu}_{\mp,\mu}=\theta(\mp
P^{0})\,\theta(s-4M^{2})\,(sB+4C)=M^{2}I_{\mp}-2k_{2,\mu}I^{\mu}_{\mp}.$

A third identity following from (3.8) reads:

$I^{\mu\nu}_{\mp}P_{\nu}=\theta(\mp
P^{0})\,\theta(s-4M^{2})\,P^{\mu}\bigl{(}(A+B)s/2+C\bigr{)}=0.$

Pulling together these results, one arrives at

$B(s)=-M^{2}F(s)/s\quad\text{and}\quad C(s)=M^{2}F(s)/2-s\,G(s)/4.$

$\blacktriangleright$ 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:

$\displaystyle
d^{\mu\nu}(k_{1},k_{2})\biggr{|}_{k_{1}^{2}=0=k_{2}^{2}}=\frac{\operatorname{sgn}(P^{0})\,\theta(s-4M^{2})}{(2\pi)^{2}}\,\bigl{[}k_{1}^{\mu}k_{2}^{\nu}[4G(s)-(1+4M^{2}/s)F(s)]$
$\displaystyle\quad+2M^{2}g^{\mu\nu}F(s)-k_{1}^{\nu}k_{2}^{\mu}\frac{4M^{2}}{s}\,F(s)\bigr{]}=\operatorname{sgn}(P^{0})\,\theta(s-4M^{2})\frac{4M^{2}}{(2\pi)^{2}}\,P^{\mu\nu}\,\frac{F(s)}{s}.$
(3.15)

The $k_{1}^{\mu}k_{2}^{\nu}$-terms have been dropped in the last identity, due
to $k_{\mu}A^{\mu}(-k)=0$. The remainder is electromagnetically gauge-
invariant. Introducing the dimensionless variable

$\tilde{\rho}:=\frac{s}{4M^{2}}=\frac{P^{2}}{4M^{2}}\,,$

keeping in mind formula (3.12), and on use of (B.5), equation (3.15) can be
rewritten as

$\displaystyle
d^{\mu\nu}_{\mathrm{gi}}(k_{1},k_{2})\biggr{|}_{k_{1}^{2}=0=k_{2}^{2}}:=\frac{i\,\operatorname{sgn}(P^{0})\,\theta(\tilde{\rho}-1)}{(2\pi)^{5}}\,P^{\mu\nu}\,b(\tilde{\rho})$
(3.16) with $\displaystyle
b(\tilde{\rho}):=\frac{1}{16\,M^{2}\,\tilde{\rho}^{2}}\,\log\bigl{(}2\tilde{\rho}-2\sqrt{\tilde{\rho}(\tilde{\rho}-1)}-1\bigr{)}=-\frac{1}{16\,M^{2}\,\tilde{\rho}^{2}}\,\log\frac{1+\sqrt{1-\tilde{\rho}^{-1}}}{1-\sqrt{1-\tilde{\rho}^{-1}}}\,,$

where ‘gi’ stands for the gauge invariant part. The singular order of
$d^{\mu\nu}_{\mathrm{gi}}\bigr{|}_{k_{1}^{2}=0=k_{2}^{2}}$ is $\omega=-2$ by
power counting; whereas for the off-shell $d^{\mu\nu}(k_{1},k_{2})$ the value
is $\omega=0$.

###  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 $d^{\mu\nu}(k_{1},k_{2})$, for
$(k_{1},k_{2})\in\mathcal{V}:=\bigl{(}\overline{V}_{+}\setminus\\{0\\}\bigr{)}^{\times
2}\cup\bigl{(}\overline{V}_{-}\setminus\\{0\\}\bigr{)}^{\times 2}$. We look at
the terms coming from (3.3) by means of (3.4). Introducing the new integration
variable
$q:=-k+{\mathchoice{\tfrac{1}{2}}{\tfrac{1}{2}}{{\scriptstyle\frac{1}{2}}}{{\scriptstyle\frac{1}{2}}}}(k_{1}-k_{2})$,
the internal lines’ momenta are

$q_{1}=q+{\mathchoice{\tfrac{1}{2}}{\tfrac{1}{2}}{{\scriptstyle\frac{1}{2}}}{{\scriptstyle\frac{1}{2}}}}P,\quad
q_{2}=q-{\mathchoice{\tfrac{1}{2}}{\tfrac{1}{2}}{{\scriptstyle\frac{1}{2}}}{{\scriptstyle\frac{1}{2}}}}P,\quad
q_{3}=q-{\mathchoice{\tfrac{1}{2}}{\tfrac{1}{2}}{{\scriptstyle\frac{1}{2}}}{{\scriptstyle\frac{1}{2}}}}(k_{1}-k_{2}),$
(3.17)

and one sees that the considered terms are all of the type

$\displaystyle H^{\mu\nu}(k_{1},k_{2})$ $\displaystyle:=\int
d^{4}q\,\bigl{(}\theta(q_{1}^{0})\,\theta(-q_{2}^{0})-\theta(-q_{1}^{0})\,\theta(q_{2}^{0})\bigr{)}\,\delta(q_{1}^{2}-M^{2})\,\delta(q_{2}^{2}-M^{2})\,\frac{h^{\mu\nu}(k_{1},k_{2},q)}{M^{2}-q_{3}^{2}}$
(3.18)

for
$(k_{1},k_{2})\in\mathcal{V}_{1}:=\bigl{(}\overline{V}\setminus\\{0\\}\bigr{)}^{\times
2}$ with $\overline{V}:=\overline{V}_{+}\cup\overline{V}_{-}$, and where
$h^{\mu\nu}\colon{\mathbb{R}}^{4\times 3}\to\mathbb{C}\,$ is a polynomial of
degree $2$. We have used that for $(k_{1},k_{2})\in\mathcal{V}_{1}$ it holds
true that

$\int
d^{4}q\,\bigl{(}\theta(q_{1}^{0})\,\theta(-q_{2}^{0})-\theta(-q_{1}^{0})\,\theta(q_{2}^{0})\bigr{)}\,\delta(q_{1}^{2}-M^{2})\,\delta(q_{2}^{2}-M^{2})\,\delta(q_{3}^{2}-M^{2})=0.$
(3.19)

This last relation can be argued as follows:888We borrow the standard notation
for the mass shell: $H_{M}^{\pm}:=\\{\,p\in{\mathbb{R}}^{4}:p^{2}=M^{2},\ \pm
p_{0}>0\,\\}$. the various $\theta$\- and $\delta$-distributions yield the
restrictions $(q_{1},q_{2})\in(H_{M}^{+}\times H_{M}^{-})\cup(H_{M}^{-}\times
H_{M}^{+})$ and $q_{3}\in H_{M}^{+}\cup H_{M}^{-}$; taking moreover into
account that $q_{3}=q_{2}+k_{2}$ and $q_{3}=q_{1}-k_{1}$, it ensues that the
various restrictions on $q_{3}$ are not compatible.

The same identity implies that terms of the kind $T_{1}(x_{\pi
1})\,T_{1}(x_{\pi 2})\,T_{1}(x_{\pi 3})$ do not contribute to the third
commutator in formula (3.1) for $D_{3}$ when
$(k_{1},k_{2})\in\mathcal{V}_{1}$, for all permutations $\pi$: the whole
contribution to $d^{\mu\nu}(k_{1},k_{2})|_{(k_{1},k_{2})\in\mathcal{V}_{1}}$
coming from this commutator is of the kind (3.18).

The contributions to $d^{\mu\nu}(k_{1},k_{2})|_{(k_{1},k_{2})\in\mathcal{V}}$
coming from the other two commutators in (3.1) are of the same form up to
cyclic permutations $k_{1}\mapsto k_{2}\mapsto-(k_{1}+k_{2})\mapsto k_{1}$ of
the external momenta. Here we use that $(k_{1},k_{2})\in\mathcal{V}$ implies
$(k_{2},-k_{1}-k_{2})\in\mathcal{V}_{1}$ and
$(-k_{1}-k_{2},k_{1})\in\mathcal{V}_{1}$, hence we may apply the identity
(3.19) also for the permuted momenta. However, note that the polynomials
$h_{j}^{\mu\nu}$, $j=2,3$, belonging to these other two cuts are not obtained
by cyclic permutations of the external momenta in the original polynomial
$h_{1}^{\mu\nu}$, 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

$d^{\mu\nu}(k_{1},k_{2})\bigr{|}_{(k_{1},k_{2})\in\mathcal{V}}=H^{\mu\nu}_{1}(k_{1},k_{2})+H^{\mu\nu}_{2}(k_{2},-k_{1}-k_{2})+H^{\mu\nu}_{3}(-k_{1}-k_{2},k_{1}),$
(3.20)

for some $H^{\mu\nu}_{j}$ $(j=1,2,3)$ of the form (3.18), the pertinent
polynomials $h_{j}^{\mu\nu}$ being of degree $2$.

######  Lemma 1.

Let $q_{1},q_{2},q_{3}$ and $\mathcal{V}_{1}$ be defined as above in (3.17)
and after (3.18), and let $H^{\mu\nu}\colon{\mathbb{R}}^{4\times
2}\to\mathbb{C}$ be given in terms of a generic polynomial
$h^{\mu\nu}\colon{\mathbb{R}}^{4\times 3}\to\mathbb{C}$ of degree
$\zeta\in\mathbb{N}_{0}$, as in (3.18). Then for all
$(k_{1},k_{2})\in\mathcal{V}_{1}$ and for some $C>0$ the function $H^{\mu\nu}$
is continuous in the region $\mathcal{V}_{1}$, and can be bounded as follows:

$|H^{\mu\nu}(k_{1},k_{2})|\leq
C\,\frac{(1+|(k_{1},k_{2})|)^{\zeta}}{|(k_{1}k_{2})|}\,\theta((k_{1}+k_{2})^{2}-4M^{2})\,\log((k_{1}+k_{2})^{2}/M^{2}).$
(3.21)

Note that $|(k_{1}k_{2})|>0$ if $(k_{1},k_{2})\in\mathcal{V}_{1}$ and
$(k_{1}+k_{2})^{2}\geq 4M^{2}$.

###### Proof.

Let $P:=k_{1}+k_{2}$ and $k:=k_{1}-k_{2}$. We first observe, on the strength
of

$\displaystyle q_{1}^{2}-q_{2}^{2}=2(Pq),\quad
q_{1}^{2}+q_{2}^{2}-2M^{2}=2(q^{2}+\tfrac{1}{4}P^{2}-M^{2})$
$\displaystyle\quad\text{and of\\!\\!}\quad
M^{2}-q^{3}=(M^{2}-q^{2}-\tfrac{1}{4}P^{2})+\tfrac{1}{4}P^{2}-\tfrac{1}{4}k^{2}+(kq)\quad\text{\\!\\!that}\quad$
$\displaystyle H^{\mu\nu}(k_{1},k_{2})\sim\operatorname{sgn}(P^{0})\int
d^{4}q\,\delta(q^{2}+\tfrac{1}{4}P^{2}-M^{2})\,\delta((Pq))\,\frac{h^{\mu\nu}(k_{1},k_{2},q)}{P^{2}/4-k^{2}/4+(kq)}\,,$

omitting irrelevant prefactors. Since $q_{1}-q_{2}=P$ and
$(q_{1},q_{2})\in(H_{M}^{+}\times H_{M}^{-})\cup(H_{M}^{-}\times H_{M}^{+})$,
we know that $H^{\mu\nu}(k_{1},k_{2})$ vanishes for $P^{2}<4M^{2}$. Hence, to
perform the integrals in $q^{0}$ and $|\bm{q}|$ using the Dirac deltas, we may
work in the frame in which $\bm{P}=0$. There the $\delta$-distributions yield
$q^{0}=0$ and $|\bm{q}|=\sqrt{P_{0}^{2}/4-M^{2}}$.

With the notation $\hat{p}:=\bm{p}/|\bm{p}|$ for $p\in\\{\,\\!q,k\\!\,\\}$, it
follows that

$\tfrac{1}{4}P^{2}-\tfrac{1}{4}k^{2}+(kq)=(k_{1}k_{2})\biggl{(}1-(\hat{q}\,\hat{k})\sqrt{1-4M^{2}/P^{2}}\,\frac{|P^{0}|\,|\bm{k}|}{2(k_{1}k_{2})}\biggr{)},$

and one verifies that

$0\leq P_{0}^{2}\,|\bm{k}/2|^{2}=(k_{1}k_{2})^{2}-k_{1}^{2}k_{2}^{2},$ (3.22)

with $k_{1}=(k_{1}^{0},\bm{k}_{1})$ and $k_{2}=(P^{0}-k_{1}^{0},-\bm{k}_{1})$.
With the help of these results we obtain

$\displaystyle(k_{1}k_{2})\,H^{\mu\nu}(k_{1},k_{2})$
$\displaystyle\sim\operatorname{sgn}(P^{0})\,\theta(P^{2}-4M^{2})\,\sqrt{1-4M^{2}/P^{2}}$
$\displaystyle\enspace\times\int_{\mathbb{S}^{2}}d\Omega(\hat{q})\,\frac{h^{\mu\nu}\bigl{(}k_{1},k_{2},(0,\sqrt{P^{2}/4-M^{2}}\,\hat{q})\bigr{)}}{1-(\hat{q}\,\hat{k})\sqrt{(1-4M^{2}/P^{2})\,(1-k_{1}^{2}k_{2}^{2}/(k_{1}k_{2})^{2})}}\,,$
(3.23)

valid in the frame in which $\bm{P}=\bm{0}$. Let moreover
$\mathcal{V}_{1}^{M}:=\\{\,(k_{1},k_{2})\in\mathcal{V}_{1}:(k_{1}+k_{2})^{2}\geq
4M^{2}\,\\}$. We know that

$\displaystyle 4M^{2}/P^{2}\in(0,1]\quad\text{and}\quad
k_{1}^{2}k_{2}^{2}/(k_{1}k_{2})^{2}\in[0,1]\quad\text{for}\quad(k_{1},k_{2})\in\mathcal{V}_{1}^{M};$
$\displaystyle\quad\text{hence}\quad
a:=\sqrt{(1-4M^{2}/P^{2})\,(1-k_{1}^{2}k_{2}^{2}/(k_{1}k_{2})^{2})}\in[0,1).$

In particular, the denominator in the integrand of (3.23) does not vanish for
$(k_{1},k_{2})\in\mathcal{V}_{1}^{M}$. Since
$\theta(P^{2}-4M^{2})\,\sqrt{1-4M^{2}/P^{2}}$ is continuous, $H^{\mu\nu}$ is
continuous on $\mathcal{V}_{1}$.

Observe now that for all $\hat{q}\in\mathbb{S}^{2}$ the inequality

$\bigl{|}h^{\mu\nu}\bigl{(}k_{1},k_{2},(0,\sqrt{P^{2}/4-M^{2}}\hat{q})\bigr{)}\bigr{|}\leq\mathrm{const}(1+|(k_{1},k_{2})|)^{\zeta}$

holds, with $|(k_{1},k_{2})|^{2}:=\sum_{j=0}^{3}(k_{1j}^{2}+k_{2j}^{2})$.

Setting $z:=\bigl{(}\hat{q}\hat{k}\bigr{)}$, the remaining integral is of the
type

$\int_{-1}^{1}\frac{dz}{1-az}=\frac{1}{a}\log\biggl{(}\frac{1+a}{1-a}\biggr{)}\leq
2(1-\log(1-a)),$

valid for $a\in[0,1)$. Using that
$a\leq\sqrt{1-4M^{2}/P^{2}}\leq(1-2M^{2}/P^{2})$ and monotonicity of the
logarithm, we see that

$-\log(1-a)=\log\frac{1}{1-a}\leq\log\frac{P^{2}}{2M^{2}}\,.$

Putting together the estimates, we end up with

$|(k_{1}k_{2})\,H^{\mu\nu}(k_{1},k_{2})|\leq\mathrm{const}\cdot\theta(P^{2}-4M^{2})\,(1+|(k_{1},k_{2})|)^{\zeta}\bigl{(}1+\log(P^{2}/2M^{2})\bigr{)},$
(3.24)

impliying (3.21), since $1+\log(P^{2}/2M^{2})<2\,\log(P^{2}/M^{2})$ for
$P^{2}\geq 4M^{2}$. ∎

The reader should keep in mind that $d^{\mu\nu}(k_{1},k_{2})$ 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 $d^{\mu\nu}(k_{1},k_{2})$ given in (3.20) is
continuous on $\mathcal{V}$ and fulfills the bound:

$|d^{\mu\nu}(k_{1},k_{2})|\leq\mathrm{const}\,\frac{(1+|(k_{1},k_{2})|)^{\omega+2}}{|(k_{1}k_{2})|}\,\log(2+|(k_{1},k_{2})|/M)\text{
for all }(k_{1},k_{2})\in\mathcal{V}.$ (3.25)

###### Proof.

Continuity follows immediately from Lemma 1. For the bound (3.25) we have
substituted $\omega+2\equiv\omega(d)+2$ for $\zeta$ of the Lemma, since the
singular order of $H_{j}^{\mu\nu}$ ($j=1,2,3$) is $\zeta-2$ by power counting
in (3.18). In addition, for $H^{\mu\nu}_{1}(k_{1},k_{2})$ we have used that
$(k_{1}+k_{2})^{2}\leq 4\,|(k_{1},k_{2})|^{2}$, and in order to omit the
$\theta$-distribution we have replaced $\log(2\,|(k_{1},k_{2})|/M)$ by
$2\log(2+|(k_{1},k_{2})|/M)$. One deals analogously with
$H^{\mu\nu}_{2}(k_{2},-k_{1}-k_{2})$ and $H^{\mu\nu}_{3}(-k_{1}-k_{2},k_{1})$.
∎

###  3.3 Distribution splitting by the dispersion integral for null momenta

Recall that for $(k_{1},k_{2})\in V_{\eta}\times V_{\eta}$ the advanced part
$a^{\mu\nu}$ of $d^{\mu\nu}$ can be computed by the dispersion integral
(C.14). Using the regularity properties of $d^{\mu\nu}$ given in Corollary 2,
we finally aim to show that the limit $k_{1}^{2}\downarrow 0$,
$k_{2}^{2}\downarrow 0$ in (C.14) commutes with integration; that is, the
dispersion integral is also valid for $k_{1}^{2}=0=k_{2}^{2}$. To formulate
the assertion, let

$\mathcal{K}:=\\{\,(k_{1},k_{2})\in({\mathbb{R}}^{4})^{\times
2}:k_{1}^{2},k_{2}^{2}<4M^{2},\ (k_{1}+k_{2})^{2}<4M^{2},\ (k_{1}k_{2})\neq
0\,\\}.$ (3.26)

Bearing in mind the factors $\theta(q^{2}-4M^{2})$ for
$q\in\\{k_{1},k_{2},k_{1}+k_{2}\\}$ appearing in each term of
$d^{\mu\nu}(k_{1},k_{2})$, we see that for $(k_{1},k_{2})\in(V_{\eta}\times
V_{\eta})\cap\mathcal{K}$, formula (C.14) can be rewritten as:

$a^{\mu\nu}(k_{1},k_{2})=\frac{i\eta}{2\pi}\int_{|t|\geq
t_{\min}}dt\,\frac{d^{\mu\nu}(tk_{1},tk_{2})}{t^{\omega+1}(1-t)},$ (3.27)

for some $t_{\min}>1$ depending on $k_{1},k_{2}$. Now, as discussed in
subsection C.3, one knows $a^{\mu\nu}(k_{1},k_{2})$ to be analytic on the
region $\mathcal{K}$. The Lebesgue dominated convergence theorem [32, Th.
4.6.3] with the bound (3.25) allows us conclude that (3.27) is a valid
identity for $(k_{1},k_{2})\in\mathcal{V}\cap\mathcal{K}$. Indeed, introducing
the set of limit points

$\mathcal{M}:=\mathcal{V}\cap\mathcal{K}\cap\\{(k_{1},k_{2})\in{\mathbb{R}}^{8}:k_{i}^{2}=0\\}=\\{(k_{1},k_{2})\in{\mathbb{R}}^{8}:k_{i}^{2}=0,\
0<(k_{1}+k_{2})^{2}<4M^{2}\\},$

it is enough to observe that for any
$(\tilde{k}_{1},\tilde{k}_{2})\in\mathcal{M}$ – implying
$(\tilde{k}_{1}\,\tilde{k}_{2})>0$ and $\tilde{k}_{1}^{0}\tilde{k}_{2}^{0}>0$
– there is a neighbourhood $\mathcal{U}_{(\tilde{k}_{1},\tilde{k}_{2})}$ such
that

$\displaystyle\biggl{|}\theta(|t|-t_{\min})\frac{d(tk_{1},tk_{2})}{t^{\omega+1}(1-t)}\biggr{|}$
$\displaystyle\leq\mathrm{const}\cdot\frac{\theta(|t|-t_{\min})}{|t(1-t)|}\,\frac{\bigl{(}1+|(k_{1},k_{2})|\bigr{)}^{\omega+2}}{|(k_{1}\,k_{2})|}\,\log(2+|t|\,|(k_{1},k_{2})|/M)$
$\displaystyle\leq C\,\frac{\theta(|t|-t_{1})}{|t(1-t)|}\,\log(2+C_{1}\,|t|),$

for all $(k_{1},k_{2})\in(V_{\eta}\times
V_{\eta})\cap\mathcal{K}\cap\mathcal{U}_{(\tilde{k}_{1},\tilde{k}_{2})}\,$,
for some $C,C_{1}>0$ and some $t_{1}>1$ independent of $(k_{1},k_{2})$. The
function on the right hand side is absolutely integrable in $t$ – here we see
the reason for the condition $(k_{1}k_{2})\neq 0$ in (3.26).

###  3.4 Normalization of the scalar model by distribution splitting

We must finally compute the gauge invariant part
$t^{\mu\nu}_{\mathrm{gi}}(k_{1},k_{2})$ for momenta lying on the set
$\mathcal{M}$. Considering the formula $T_{3}=A_{3}-A^{\prime}_{3}$ and
reckoning that $a^{\prime\mu\nu}(k_{1},k_{2})|_{k_{1}^{2}=0=k_{2}^{2}}$
contains the factor $\theta(P^{2}-4M^{2})$ where $P:=k_{1}+k_{2}$, we see that
on $\mathcal{M}$ its contribution vanishes, that is $t^{\mu\nu}=a^{\mu\nu}$
there.

The upshot of the preceding two subsections is that we may compute valid terms
of the central solution $a^{\mu\nu}|_{\mathcal{M}}\equiv
a^{c\,\mu\nu}|_{\mathcal{M}}$ by inserting the on-shell amplitude (3.15) into
the dispersion integral, with $\omega$ the singular order of the off-shell
$d^{\mu\nu}$, equal to $0$ in the present case.

Looking at (3.15), observe that a $k_{r}^{\mu}k_{s}^{\nu}$\- or
$g^{\mu\nu}$-term of $d^{\mu\nu}$ goes over to a $k_{r}^{\mu}k_{s}^{\nu}$\- or
$g^{\mu\nu}$-term of $a^{\mu\nu}$, respectively. Therefore, such factors may
be taken out of the dispersion integral. Since moreover
$P^{\mu\nu}(tk)=t^{2}\,P^{\mu\nu}(k)$, we see that the gauge invariant part
$a^{\mu\nu}_{\mathrm{gi}}$ can be obtained by inserting just the gauge
invariant part $d^{\mu\nu}_{\mathrm{gi}}$ in (3.16) into the dispersion
integral. The latter is of the form (C.15). So we may use the version (C.16)
of the dispersion integral.

Lastly,
$t^{\mu\nu}_{\mathrm{gi}}|_{\mathcal{M}}=a^{\mu\nu}_{\mathrm{gi}}|_{\mathcal{M}}$
is obtained from (C.16) by setting $\omega=0$ and substituting there
$b(u\tilde{\rho})$ as given in (3.16) for
$f\bigl{(}u(k_{1}^{2},k_{2}^{2},(k_{1}+k_{2})^{2})\bigr{)}$ – in our case only
$(k_{1}+k_{2})^{2}=s$ is present. Allowing for the dilation factor in
$P^{\mu\nu}$ this leads, for $(k_{1},k_{2})\in\mathcal{M}$, to

$\displaystyle t^{\mu\nu}_{\mathrm{gi}}(k_{1},k_{2})$
$\displaystyle=-\frac{P^{\mu\nu}}{(2\pi)^{6}}\int_{\tilde{\rho}^{-1}}^{\infty}du\,\frac{u\,b(u\tilde{\rho})}{u(1-u)}$
(3.28)
$\displaystyle=\frac{P^{\mu\nu}}{16M^{2}\,(2\pi)^{6}}\int_{\tilde{\rho}^{-1}}^{\infty}\frac{du}{\tilde{\rho}^{2}u^{2}(1-u)}\,\log\frac{1+\sqrt{1-u^{-1}\tilde{\rho}^{-1}}}{1-\sqrt{1-u^{-1}\tilde{\rho}^{-1}}}=-\frac{P^{\mu\nu}}{8\,(2\pi)^{6}}\,\frac{J_{2}(\tilde{\rho})}{M^{2}}\,,$
where $\displaystyle 2\,J_{2}(\tilde{\rho})$
$\displaystyle:=\int_{1}^{\infty}dv\,\frac{1}{(v-\tilde{\rho})v^{2}}\,\log\frac{1+\sqrt{1-v^{-1}}}{1-\sqrt{1-v^{-1}}},$

after the change of integration variable $v:=u\tilde{\rho}$. Integrals like
$J_{2}$ have been computed in [20]. From Appendix C of that reference:

$J_{1}(\tilde{\rho},a):=\frac{1}{2}\int_{1}^{\infty}dv\,\frac{1}{(v-\tilde{\rho})(v-a)}\,\log\frac{1+\sqrt{1-v^{-1}}}{1-\sqrt{1-v^{-1}}}=\frac{f(\tilde{\rho})-f(a)}{\tilde{\rho}-a}$
(3.29)

for $0\leq\tilde{\rho}\leq 1$, $0\leq a\leq 1$, where $f$ is the distribution
(B.3). We infer that

$\displaystyle J_{2}(\tilde{\rho})=\frac{\partial}{\partial
a}\biggr{|}_{a=0}\,J_{1}(\tilde{\rho},a)=\frac{f(\tilde{\rho})}{\tilde{\rho}^{2}}-\frac{1}{\tilde{\rho}}\quad\text{for}\quad
0\leq\tilde{\rho}\leq 1,$ (3.30)

by bringing in the values $f(0)=0$ and $f^{\prime}(0)=1$, which can be read
off from (B.6).

Summing up, the final result reads, as expected:

$\displaystyle
t^{\mu\nu}_{\mathrm{gi}}(k_{1},k_{2})=\frac{P^{\mu\nu}}{8\,(2\pi)^{6}}\,\frac{1}{M^{2}}\biggl{(}\frac{1}{\tilde{\rho}}-\frac{f(\tilde{\rho})}{\tilde{\rho}^{2}}\biggr{)}=\frac{P^{\mu\nu}}{8\,(2\pi)^{6}}\,\frac{F_{0}(\tilde{\rho})}{M^{2}}\quad\text{for
}(k_{1},k_{2})\in\mathcal{M},$ (3.31)

where $F_{0}$ was given in (1.3). We conjecture that this formula holds true
for all $(k_{1},k_{2})$ satisfying $k_{1}^{2}=0=k_{2}^{2}$ and
$(k_{1}+k_{2})^{2}>0$.

The reader should remember that (3.31) stands in principle for just a member
of a solution set. Since $\omega=0$, the general Lorentz-invariant
Epstein–Glaser solution is obtained by adding to expression (3.31) a term of
the type $Cg^{\mu\nu}$ with $C\in\mathbb{C}$ arbitrary. But such a term with
$C\neq 0$ would violate EGI. Therefore we regard the above result as unique.

Recovering formula (3.5) and the factor $4ge^{2}M$, one ends up with

$\displaystyle\int
dx_{1}\,dx_{2}\,dx_{3}\,T_{3}(x_{1},x_{2},x_{3})=\frac{g\alpha}{(2\pi)^{3}M}\int
dk_{1}\,dk_{2}\,h(k_{1}+k_{2})A^{\mu}(-k_{1})A^{\rho}(-k_{2})\,P_{\mu\nu}\,F_{0}(\tilde{\rho}),$

which, on substituting $\rho$ for $\tilde{\rho}$, that is, $m_{h}^{2}$ for
$s\equiv(k_{1}+k_{2})^{2}$, agrees with the literature [26].

######  Remark 1.

In the occasion an (unsubtracted) dispersion integral applied to $b(u)$,
performed in [17, 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 $AAWW^{\dagger}$-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 $T_{1}$ – that is the sole “empirical” input. Here in going
from $T_{1}$ to $T_{2}$ we derive the $AAWW^{\dagger}$-vertex which
contributes by a “fish-like” diagram to the amplitude to be computed, see
Figure 4.1.

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
$\mathbb{S}$-matrix under the variations $A^{\mu}(x)\to
A^{\mu}(x)+\partial^{\mu}\Lambda(x)$: interaction dictates symmetry. The
method is similar to the derivation of the $AA\varphi\varphi^{\dagger}$
“seagull” vertex from the cubic coupling in scalar QED, first performed in
this way in [29].

The concept works on configuration space, as follows. Recall the pertinent
Hermitian vertex – see for instance [33, Sect. 7.2.2], explicitly referring to
the unitary gauge. With
$G_{\mu\nu}:=\partial_{\mu}W_{\nu}-\partial_{\nu}W_{\mu}$, one has:

$\displaystyle
T_{1}(x_{1})=ie[(W^{\mu}G^{\dagger}_{\mu\nu}-W^{\dagger\mu}G_{\mu\nu})A^{\nu}-W^{\mu}W_{\nu}^{\dagger}F_{\mu}^{\,\nu}](x_{1}).$
(4.1)

All indicated operator products are Wick products. We copy a second vertex
similar to (4.1):

$\displaystyle
T_{1}(x_{2})=ie[(W^{\rho}G^{\dagger}_{\rho\lambda}-W^{\dagger\rho}G_{\rho\lambda})A^{\lambda}-W^{\rho}W_{\lambda}^{\dagger}F_{\rho}^{\,\lambda}](x_{2}),$
(4.2)

and proceed to make the contractions, according to the BEG method to construct
(the relevant sector of) the second-order $T_{2}(x_{1},x_{2})$ out of
$T_{1}(x_{1}),T_{1}(x_{2})$. Now $T_{2}(x_{1},x_{2})=T_{1}(x_{1})T_{1}(x_{2})$
for $x_{1}$ not to the past of $x_{2}$, and
$T_{2}(x_{1},x_{2})=T_{1}(x_{2})T_{1}(x_{1})$ for $x_{2}$ not to the past of
$x_{1}$.

In view of the triangle diagram given in Figure 4.1, we are only interested in
contractions yielding a connected tree diagram with the photons uncontracted.
Once they are made, we analyze the terms for which the resulting distributions
are not uniquely defined on the diagonal $x_{1}=x_{2}$. The ambiguities in
this extension are eliminated by EGI as sole selection criterion.

From the last pair of terms in (4.1) and (4.2), and with $\Delta^{\mu\beta}$
standing for the Feynman–Proca propagator (A.8) for the $W$-bosons, there
comes:

$\displaystyle-e^{2}\,\bigl{(}F_{\mu\nu}(x_{1})W^{\dagger\nu}(x_{1})\langle\\!\langle\operatorname{T}W^{\mu}(x_{1})\,W^{\dagger\beta}(x_{2})\rangle\\!\rangle
W^{\alpha}(x_{2})F_{\alpha\beta}(x_{2})$
$\displaystyle\quad+F_{\mu\nu}(x_{1})W^{\mu}(x_{1})\langle\\!\langle\operatorname{T}W^{\dagger\nu}(x_{1})\,W^{\alpha}(x_{2})\rangle\\!\rangle
W^{\dagger\beta}(x_{2})F_{\alpha\beta}(x_{2})\bigr{)}$
$\displaystyle=-e^{2}\,F_{\mu\nu}(x_{1})F_{\alpha\beta}(x_{2})W^{\dagger\nu}(x_{1})\,\Delta^{\mu\beta}(x_{1}-x_{2})W^{\alpha}(x_{2})+(x_{1}\leftrightarrow
x_{2}),$ denoted $T_{2}^{D}(x_{1},x_{2})$ for later use; (4.3)

where explicitly
$\Delta_{\beta}^{\mu}=-\bigl{(}g_{\beta}^{\mu}+\partial^{\mu}\partial_{\beta}/M^{2}\bigr{)}\Delta^{F}$
with $\Delta^{F}$ the scalar Feynman propagator (A.2). Electromagnetic gauge
variations obviously do not affect (4.3).

By pairing the last term in (4.2) with $A$-type terms in (4.1) and the last
term in (4.1) with $A$-type terms in (4.2) we obtain eight terms, that we
organize as follows:

$\displaystyle
e^{2}\,A^{\mu}(x_{1})F_{\rho\beta}(x_{2})\bigl{[}G^{\dagger}_{\alpha\mu}(x_{1})\,\Delta^{\alpha\beta}(x_{1}-x_{2})\,W^{\rho}(x_{2})-G_{\alpha\mu}(x_{1})\,\Delta^{\alpha\rho}(x_{1}-x_{2})\,W^{\beta\dagger}(x_{2})+{}$
$\displaystyle
W^{\alpha}(x_{1})(-g^{\rho}_{\mu}\partial_{\alpha}+g^{\rho}_{\alpha}\partial_{\mu})\Delta^{F}(x_{1}{-}x_{2})W^{\beta\dagger}(x_{2})-W^{\alpha\dagger}(x_{1})(-g^{\beta}_{\mu}\partial_{\alpha}+g^{\beta}_{\alpha}\partial_{\mu})\Delta^{F}(x_{1}{-}x_{2})W^{\rho}(x_{2})\bigr{]}$
$\displaystyle+(x_{1}\leftrightarrow
x_{2})=:T_{2}^{B}(x_{1},x_{2})+T_{2}^{C}(x_{1},x_{2}),$ (4.4)

where $T_{2}^{C}$ denotes the $(x_{1}\leftrightarrow x_{2})$-term. As noted in
App. A, third-order derivatives of $\Delta^{F}$ cancel here of their own
accord. In order to verify EGI in the above expression, note first that it can
only be violated on the diagonal $x_{1}=x_{2}$, that is by contact terms – see
the discussion on this in Sect. C.2. Therefore in computing the divergence of
expressions like the above one selects only such terms that under
$\partial_{1}^{\mu}$ produce local expressions. For instance, the third term
in (4.4) brings forth:

$W^{\alpha}(x_{1})\,g^{\rho}_{\alpha}\,\partial_{1}^{\mu}\partial_{\mu}\,\Delta^{F}(x_{1}-x_{2})\,W^{\beta\,\dagger}(x_{2})=-iW^{\rho}(x_{1})W^{\dagger\beta}(x_{1})\,\delta(x_{1}-x_{2})+\cdots$

where the dots stand for a term ${}\sim M^{2}W\Delta^{F}W^{\dagger}$; but it
is not hard to see that it is cancelled by a similar one coming from the next
term. In conclusion: $T_{2}^{B}$ is individually electromagnetically gauge-
invariant, and $T_{2}^{D}$, $T_{2}^{B}$, $T_{2}^{C}$ calculated up to now have
no bearing on the generation of the $AAWW^{\dagger}$-vertex in constructing
$T_{2}$.

Still, we are left with the most interesting contractions to calculate. By
pairing the two first terms in (4.1) with the two first in (4.2) we get:

$\displaystyle-e^{2}\,A^{\mu}(x_{1})A^{\rho}(x_{2})\bigl{[}-W^{\alpha}(x_{1})\widetilde{D}_{\alpha\mu\;\beta\rho}(x_{1}-x_{2})W^{\beta\dagger}(x_{2})-G^{\dagger}_{\alpha\mu}(x_{1})\,\Delta^{\alpha\beta}(x_{1}-x_{2})\,G_{\beta\rho}(x_{2})+{}$
$\displaystyle
G^{\dagger}_{\alpha\mu}(x_{1})(g^{\alpha}_{\rho}\partial_{\beta}-g^{\alpha}_{\beta}\partial_{\rho})\Delta^{F}(x_{1}-x_{2})W^{\beta}(x_{2})+W^{\alpha\dagger}(x_{1})(-g^{\beta}_{\mu}\partial_{\alpha}+g^{\beta}_{\alpha}\partial_{\mu})\Delta^{F}(x_{1}-x_{2})G_{\beta\rho}(x_{2})\bigr{]}$
$\displaystyle+(x_{1}\leftrightarrow x_{2})=:T_{2}^{A}(x_{1},x_{2}).$ (4.5)

Outside the diagonal the distribution $\widetilde{D}_{\alpha\mu\;\beta\rho}$
necessarily coincides with $D_{\alpha\mu\;\beta\rho}$, defined in Eq. (A.9) as
the propagator for the $G$-fields. Following the Epstein–Glaser method, we now
look for the most general Lorentz-covariant extension of this distribution
having the same scaling degree. The solution reads:

$\widetilde{D}_{\alpha\mu,\beta\rho}(y)=D_{\alpha\mu,\beta\rho}(y)+i(a\,g_{\alpha\beta}g_{\mu\rho}-b\,g_{\alpha\rho}g_{\mu\beta})\,\delta(y)$
(4.6)

with as yet unknown numbers $a,b\in\mathbb{C}$. Note that the second term in
the above propagator generates a contact term eventually yielding the
$AAWW^{\dagger}$-vertex

$T_{F}(x_{1},x_{2}):=e^{2}\,A^{\mu}(x_{1})A^{\rho}(x_{2})W^{\alpha}(x_{1})W^{\dagger\beta}(x_{2})\,i(a\,g_{\alpha\beta}g_{\mu\rho}-b\,g_{\alpha\rho}g_{\mu\beta})\,\delta(x_{1}-x_{2}).$
(4.7)

A third Lorentz tensor might appear in the Ansatz (4.6), namely
$g_{\alpha\mu}g_{\rho\beta}\,\delta(y)$. However, on insertion into (4.5), one
obtains the same contribution as $g_{\alpha\rho}g_{\mu\beta}\,\delta(y)$,
namely

$e^{2}\,A^{\mu}(x_{1})A^{\rho}(x_{1})W_{\mu}(x_{1})W^{\dagger}_{\rho}(x_{1})\,\delta(x_{1}-x_{2}).$

Since EGI of $T_{2}$ can be violated only by local terms, it will suffice to
select the terms which are $\sim(\partial)\delta(x_{1}-x_{2})$ after taking
the divergence $\partial_{1}^{\mu}$. Those can be of two types:

  1. (a)

Either the contact term already contains a $\delta(x_{1}-x_{2})$; or

  2. (b)

Such terms are generated when the $\partial^{\mu}_{x_{1}}$ acts on
$\partial_{\mu}\Delta^{F}(x_{1}-x_{2})$ or on
$\partial\partial_{\mu}\Delta^{F}(x_{1}-x_{2})$, due to
$(\square+m^{2})\Delta^{F}=-i\delta$.

From the $W(x_{1})\,W^{\dagger}(x_{2})$ part in (4.5), we find:

  * $\star$

The type (a) contributions:

$\displaystyle
i\partial^{\mu}W^{\alpha}(x_{1})\,(-a\,g_{\alpha\beta}g_{\mu\rho}+b\,g_{\alpha\rho}g_{\mu\beta})\,\delta(x_{1}-x_{2})\,W^{\beta\,\dagger}(x_{2})$
$\displaystyle\quad+iW^{\alpha}(x_{1})\,\bigl{(}(-a\,g_{\alpha\beta}\partial_{\rho}+b\,g_{\alpha\rho}\partial_{\beta})\delta\bigr{)}(x_{1}-x_{2})\,W^{\beta\,\dagger}(x_{2})$
$\displaystyle=i\bigl{(}-a\,\partial_{\rho}W^{\alpha}(x_{1})W^{\dagger}_{\alpha}(x_{1})+b\,\partial_{\beta}W_{\rho}(x_{1})W^{\beta\dagger}(x_{1})\bigr{)}\delta(x_{1}-x_{2})$
$\displaystyle\quad-
ia\,W^{\alpha}(x_{1})W^{\dagger}_{\alpha}(x_{2})\,\partial_{\rho}\delta(x_{1}-x_{2})+ib\,W_{\rho}(x_{1})W^{\beta\dagger}(x_{2})\,\partial_{\beta}\delta(x_{1}-x_{2}),$
(4.8)

and the type (b) contribution:

$\displaystyle
iW^{\alpha}(x_{1})\,\bigl{(}(g_{\alpha\beta}\partial_{\rho}-g_{\alpha\rho}\partial_{\beta})\delta\bigr{)}(x_{1}-x_{2})\,W^{\beta\,\dagger}(x_{2})$
$\displaystyle=iW^{\alpha}(x_{1})W^{\dagger}_{\alpha}(x_{2})\,\partial_{\rho}\delta(x_{1}-x_{2})-iW_{\rho}(x_{1})W^{\beta\dagger}(x_{2})\,\partial_{\beta}\delta(x_{1}-x_{2}).$
(4.9)

  * $\star$

From the $W^{\dagger}(x_{1})\,W(x_{2})$-part of the exchange term, we obtain
terms (4.8) and (4.9) with $W\leftrightarrow W^{\dagger}$ interchanged.

  * $\star$

From the $W^{\dagger}(x_{1})\,G(x_{2})$-part of the original term, we get the
type (b) contribution:

$\displaystyle
iW^{\alpha\dagger}(x_{1})\,g^{\beta}_{\alpha}\,\delta(x_{1}-x_{2})\,\bigl{(}-\partial_{\beta}W_{\rho}(x_{2})+\partial_{\rho}W_{\beta}(x_{2})\bigr{)}$
$\displaystyle=i\bigl{(}-W^{\alpha\dagger}(x_{1})\,\partial_{\alpha}W_{\rho}(x_{1})+W^{\alpha\dagger}(x_{1})\,\partial_{\rho}W_{\alpha}(x_{1})\bigr{)}\,\delta(x_{1}-x_{2}).$
(4.10)

  * $\star$

From the $W(x_{1})\,G^{\dagger}(x_{2})$-part of the exchange term, we obtain
the term (4.10) with $W\leftrightarrow W^{\dagger}$ interchanged.

Summing up, the requirement is:

$\displaystyle 0$
$\displaystyle\overset{!}{=}W^{\alpha}(x_{1})W^{\dagger}_{\alpha}(x_{2})\,\partial_{\rho}\delta(x_{1}-x_{2})\,(1-a)+W_{\rho}(x_{1})W^{\beta\dagger}(x_{2})\,\partial_{\beta}\delta(x_{1}-x_{2})\,(b-1)$
$\displaystyle\quad+\partial_{\beta}W_{\rho}(x_{1})W^{\beta\dagger}(x_{1})\,\delta(x_{1}-x_{2})\,(b-1)+\partial_{\rho}W^{\alpha}(x_{1})W^{\dagger}_{\alpha}(x_{1})\,\delta(x_{1}-x_{2})\,(1-a)$
$\displaystyle\quad+[W\leftrightarrow W^{\dagger}].$ (4.11)

Generally, for two free fields $B(x)$, $C(x)$ it holds that the three terms

$\partial B(x_{1})\,C(x_{1})\,\delta(x_{1}-x_{2}),\quad B(x_{1})\,\partial
C(x_{1})\,\delta(x_{1}-x_{2})\quad\text{and}\quad
B(x_{1})\,C(x_{2})\,\partial\delta(x_{1}-x_{2})$

are linearly independent. We conclude that condition (4.11) has a unique
solution, namely $a=1=b$. In fine, the resulting contact term may be written
as

$\widetilde{D}_{\alpha\mu,\beta\rho}(y)-D_{\alpha\mu,\beta\rho}(y)=\tfrac{i}{2}(2g_{\alpha\beta}g_{\mu\rho}-g_{\alpha\rho}g_{\mu\beta}-g_{\mu\alpha}g_{\rho\beta})\,\delta(y),$
(4.12)

which reproduces the seagull $e^{2}AAWW^{\dagger}$-vertex in the Feynman rules
for the EW interaction.

$H$$x_{3}$$\gamma$$\gamma$$W^{+}$$x_{1}$$W^{+}$$x_{2}$$W^{+}$$H$$\gamma$$\gamma$$W^{+}$$W^{+}$
Figure 4.1: Diagrams contributing to the amplitude to be computed

$\blacktriangleright$ Recapitulation: assembling $T_{3}$ from $T_{1}$ and
$T_{2}$ by the Epstein–Glaser method, the above derived
$AAWW^{\dagger}$-vertex (which is part of $T_{2}$) generates the fish-like
diagram in Figure 4.1. In the Feynman gauge, the $AAWW^{\dagger}$-vertex was
already derived in [34] by the same method. The fact that we had to modify
only $D^{\bullet\bullet}_{\bullet\bullet}$ (i.e., the $G$-propagator), and not
$\Delta^{\bullet}_{\bullet}$ (i.e., the $W$-propagator), although here both
are of the same singular order $\omega=0$, is in accordance with the
corresponding modification in the Feynman gauge or, more generally, in a
$R_{\xi}$ gauge: in such gauges the $AAWW^{\dagger}$ term can be derived in
the same way, but only the $G$-propagator may be modified, because only this
propagator has $\omega=0$. The $WG$ and $GW$-propagators have singular order
$\omega=-1$ (see Eqs. (A.11) and (A.12)) and are not to be modified.

$\blacktriangleright$ In conclusion: almost the first thing to learn when
working in the BEG scheme is that what is usually taken from a top-down,
mysterious prevalence in particle physics of classical non-Abelian gauge
theories, with their quartic, second-order couplings, is here derived by pure
quantum field theory operations. To wit, the inductive construction of the
time-ordered products respecting “gauge invariance”, in the sense of [29, 35,
34, 36, 37], order by order in the couplings. In fact, all of the reductive
Lie algebra structure of the Standard Model interactions, up to including at
least one scalar particle, comes automatically in the BEG formalism from
respecting first principles of quantum field theory in building the
$\mathbb{S}$-matrix999The derivation of the reductive Lie algebra structure
was annunciated by Stora [38]. His kind of principled bottom-up approach has
long been neglected in textbooks. A refreshing exception is [39, Problem 9.3].
– without invoking unobservable mechanisms. To go further into this subject
here would take us too far afield. We have merely dealt with the example
necessary for our purposes.

###  4.2 Computation of the causal distribution at third order

We wish to mention here that triangular graphs in Epstein–Glaser theory have
been examined, for instance, in [40, Chap. 3.8] – the vertex correction in
QED; in [41] – anomalies; as well as in [42] and [24, Sect. 3.2].101010Some of
these works are very instructive, in that they show that cherished invariance
properties cannot always be preserved under distribution splitting. Of course,
here we shall have more terms and a forest of indices.

Proceeding similarly to Sect. 3, we perform the adiabatic limit already at
this stage. Since all the propagators in the loop are massive, we may use
(3.5); hence, we only have to compute the restricted distribution
$d^{\mu\nu}(k_{1},k_{2})|_{k_{1}^{2}=0=k_{2}^{2}}$. Again, because of the
kinematic constraints, only the first cut in formula (3.1) for $D_{3}$ counts
(including the “fish graph” as derived in the previous subsection). Moreover,
the discussions and general conclusions of Sects. 2, 3.2 and 3.3 hold here,
and will be assumed without further ado.

The two $WWA$-vertices are given in (4.1) and (4.2), and the $WWh$-vertex
reads

$T_{1}(x_{3})=gM\,h(x_{3})W^{\nu}(x_{3})W^{\dagger}_{\nu}(x_{3}).$

The first step is to compute $R^{\prime
A|B|C|D|F}_{3}(x_{1},x_{2};x_{3}):=T_{2}^{A|B|C|D|F}(x_{1},x_{2})\,T_{1}(x_{3})\bigr{|}_{\triangle}$,
where the lower-order TOPs $T_{2}^{\bullet}$ have been respectively given by
lexicographical order in (4.5), (4.4), (4.3), and finally (4.7) together with
(4.12). As it happens, their sum is symmetric in $(x_{1},x_{2})$, so we just
introduce a general factor $2$ and need not mention exchange anymore.

We start with

$\displaystyle R^{\prime D}_{3}(x_{1},x_{2};x_{3})$
$\displaystyle=-2e^{2}gM\,h(x_{3})F^{\mu\nu}(x_{1})F^{\alpha\beta}(x_{2})\,r^{\prime
D}_{\mu\nu,\alpha\beta}(y_{1},y_{2}),\quad\text{where}\quad$ $\displaystyle
r^{\prime D}_{\mu\nu,\alpha\beta}(y_{1},y_{2})$
$\displaystyle:=\Delta^{\gamma\,+}_{\nu}(y_{1})\,\Delta_{\mu\beta}(y_{1}-y_{2})\,\Delta^{+}_{\alpha\gamma}(y_{2}),\quad\text{and}\quad
y_{k}:=x_{k}-x_{3}.$ (4.13)

The “fish” diagram contribution reads:

$\displaystyle R^{\prime F}_{3}(x_{1},x_{2};x_{3})$
$\displaystyle=-2e^{2}gM\,h(x_{3})A^{\mu}(x_{1})A^{\nu}(x_{2})\,r^{\prime
F}_{\mu\nu}(y_{1},y_{2})\quad\text{with}\quad$ $\displaystyle r^{\prime
F}_{\mu\nu}(y_{1},y_{2})$
$\displaystyle:=i\,\delta(y_{1}-y_{2})\,\bigl{[}-g_{\mu\nu}\,\Delta^{\gamma\,+}_{\beta}(y_{1})\,\Delta^{\beta\,+}_{\gamma}(y_{1})+\Delta^{\gamma\,+}_{\nu}(y_{1})\,\Delta^{+}_{\mu\gamma}(y_{1})\bigr{]}.$
(4.14)

Next we compute $R^{\prime B}_{3}$:

$\displaystyle R^{\prime B}_{3}(x_{1},x_{2};x_{3})$
$\displaystyle=-2e^{2}gM\,h(x_{3})A^{\mu}(x_{1})F_{\nu\beta}(x_{2})\,{r^{\prime
B}}_{\mu}^{\nu\beta}(y_{1},y_{2})$ $\displaystyle\text{with}\quad{r^{\prime
B}}_{\mu}^{\nu\beta}(y_{1},y_{2})$
$\displaystyle:=(g_{\mu}^{\gamma}\partial_{\alpha}-g^{\gamma}_{\alpha}\partial_{\mu})\Delta^{+}(y_{1})\,\Delta^{\alpha\beta}(y_{1}-y_{2})\,\Delta^{\nu\,+}_{\gamma}(y_{2})$
$\displaystyle\quad+\Delta^{\alpha\gamma\,+}(y_{1})\,(g_{\mu}^{\nu}\partial_{\alpha}-g^{\nu}_{\alpha}\partial_{\mu})\Delta^{F}(y_{1}-y_{2})\,\Delta^{\beta\,+}_{\gamma}(y_{2}).$
(4.15)

The relation $R^{\prime C}_{3}(x_{1},x_{2};x_{3}):=R^{\prime
B}_{3}(x_{2},x_{1};x_{3})$ obviously holds. Finally, for $R^{\prime A}_{3}$,
we collect

$\displaystyle R^{\prime A}_{3}(x_{1},x_{2};x_{3})$
$\displaystyle=-2e^{2}gM\,h(x_{3})A^{\mu}(x_{1})A^{\nu}(x_{2})\,r^{\prime
A}_{\mu\nu}(y_{1},y_{2}),\quad\text{with}\quad$ $\displaystyle r^{\prime
A}_{\mu\nu}(y_{1},y_{2})$
$\displaystyle:=-(g_{\mu}^{\gamma}\partial_{\alpha}-g^{\gamma}_{\alpha}\partial_{\mu})\Delta^{+}(y_{1})\,\Delta^{\alpha\beta}(y_{1}-y_{2})\,(g_{\nu\gamma}\partial_{\beta}-g_{\beta\gamma}\partial_{\nu})\Delta^{+}(y_{2})$
$\displaystyle\quad-\Delta^{\alpha\gamma\,+}(y_{1})\,D_{\alpha\mu,\beta\nu}(y_{1}-y_{2})\,\Delta^{\beta\,+}_{\gamma}(y_{2})$
$\displaystyle\quad-(g_{\mu}^{\gamma}\partial_{\alpha}-g^{\gamma}_{\alpha}\partial_{\mu})\Delta^{+}(y_{1})\,(g^{\alpha}_{\nu}\partial_{\beta}-g^{\alpha}_{\beta}\partial_{\nu})\Delta^{F}(y_{1}-y_{2})\,\Delta^{\beta\,+}_{\gamma}(y_{2})$
$\displaystyle\quad+\Delta^{\alpha\gamma\,+}(y_{1})\,(g_{\mu\beta}\partial_{\alpha}-g_{\alpha\beta}\partial_{\mu})\Delta^{F}(y_{1}-y_{2})\,(g_{\gamma\nu}\partial^{\beta}-g^{\beta}_{\gamma}\partial_{\nu})\Delta^{+}(y_{2}).$
(4.16)

Next we express the resulting integrals by momentum space integrals. By using
$F^{\mu\nu}(k)=-i\bigl{(}k^{\mu}A^{\nu}(k)-k^{\nu}A^{\mu}(k)\bigr{)}$ and
$R^{\prime}_{3}:=R^{\prime A}_{3}+R^{\prime B}_{3}+R^{\prime C}_{3}+R^{\prime
D}_{3}+R^{\prime F}_{3}$ we gather

$\displaystyle\int dx_{1}\,dx_{2}\,dx_{3}\,R^{\prime}_{3}(x_{1},x_{2};x_{3})$
(4.17) $\displaystyle\quad=-2e^{2}gM(2\pi)^{2}\int
dk_{1}\,dk_{2}\,h(k_{1}+k_{2})A^{\mu}(-k_{1}){A^{\nu}}(-k_{2})\,r^{\prime}_{\mu\nu}(k_{1},k_{2}),$
$\displaystyle\quad\text{where}\quad
r^{\prime}_{\mu\nu}(k_{1},k_{2}):=r^{\prime A}_{\mu\nu}(k_{1},k_{2})+r^{\prime
F}_{\mu\nu}(k_{1},k_{2})$ (4.18)
$\displaystyle+\bigl{[}ik_{2}^{\beta}(r^{\prime
B}_{\mu,\beta\nu}(k_{1},k_{2})-r^{\prime
B}_{\mu,\nu\beta}(k_{1},k_{2}))+(k_{1},\mu)\leftrightarrow(k_{2},\nu)\bigr{]}$
$\displaystyle-k_{1}^{\beta}k_{2}^{\alpha}\bigl{[}r^{\prime
D}_{\beta\mu,\alpha\nu}(k_{1},k_{2})-r^{\prime
D}_{\mu\beta,\alpha\nu}(k_{1},k_{2})-r^{\prime
D}_{\beta\mu,\nu\alpha}(k_{1},k_{2})+r^{\prime
D}_{\mu\beta,\nu\alpha}(k_{1},k_{2})\bigr{]}.$

To compute the (combinations of) Fourier transformed
${r^{\prime}}^{\dots}_{\dots}$-distributions appearing on the right hand side
of (4.18), we bring in $k_{1}^{2}=0=k_{2}^{2}$, and omit all terms containing
a factor $k_{1\mu}$ or $k_{2\nu}$: this is allowed in view of
$k^{\lambda}A_{\lambda}(-k)=0$. Similarly to the analogous computation for the
scalar model in the previous section, let us work with the integration
variable $q:=k+k_{2}$, and introduce $P:=k_{1}+k_{2}$, hence $k_{1}-k=P-q$ and
$s:=P^{2}=2k_{1}k_{2}$ and $Pk_{1}=k_{1}k_{2}=Pk_{2}$. Due to the factors
$\Delta^{+}(q)$ and $\Delta^{+}(P-q)$, we may use the relations

$\displaystyle q^{2}=M^{2},\quad(P-q)^{2}=M^{2},\quad\text{hence}\quad
2Pq=s=2(k_{1}k_{2}),\quad\text{implying}\quad$
$\displaystyle(P-q)q=(k_{1}k_{2})-M^{2},\quad
0=(P(q-k_{2}))=(Pk),\quad(q-k_{2})^{2}=M^{2}-2(qk_{2}),$

and $((P-q)\,k_{1})=(qk_{2})$, that is, $(qk_{1})+(qk_{2})=(k_{1}k_{2})$.
Remember also that we may replace $P_{\mu}\to k_{2\mu}$ and $P_{\nu}\to
k_{1\nu}$.

To tally the fish diagram contribution, we introduce the integrals

$J^{\\{\cdot|\mu|\mu\nu\\}}(P):=\int
d^{4}q\,\\{1|q^{\mu}|q^{\mu}q^{\nu}\\}\,\Delta^{+}(P-q)\,\Delta^{+}(q).$
(4.19)

These symbols $J^{\bullet}(P)$ generalize that of the scalar $J(P)$, defined
in (3.4). We obtain:

$\displaystyle r^{\prime F}_{\mu\nu}(k_{1},k_{2})$
$\displaystyle=\frac{i}{(2\pi)^{4}}\biggl{(}g_{\mu\nu}\Bigl{(}-2+\frac{s}{M^{2}}-\frac{s^{2}}{4M^{4}}\Bigr{)}J(P)-\frac{k_{2\mu}k_{1\nu}}{M^{2}}J(P)$
$\displaystyle\hskip
30.00005pt+\frac{1}{M^{2}}\Bigl{(}k_{2\mu}J_{\nu}(P)+k_{1\nu}J_{\mu}(P)\,\frac{s}{2M^{2}}\Bigr{)}-J_{\mu\nu}(P)\Bigl{(}\frac{1}{M^{2}}+\frac{s}{2M^{4}}\Bigr{)}\biggr{)}.$
(4.20)

To figure out $J^{\mu}$ and $J^{\mu\nu}$ one first argues that

$\displaystyle J^{\mu}(P)$
$\displaystyle=\theta(P_{0})\,\theta(s-4M^{2})\,P^{\mu}\,g(s),$ $\displaystyle
J^{\mu\nu}(P)$
$\displaystyle=\theta(P_{0})\,\theta(s-4M^{2})\,(P^{\mu}P^{\nu}\,a(s)+g^{\mu\nu}c(s)),$
(4.21)

for appropriate $g(s),\,a(s)$ and $c(s)$. The latter can be obtained from the
identites

$J^{\mu}(P)P_{\mu}={\mathchoice{\tfrac{1}{2}}{\tfrac{1}{2}}{{\scriptstyle\frac{1}{2}}}{{\scriptstyle\frac{1}{2}}}}sJ(P),\quad
J^{\mu}_{\mu}(P)=M^{2}J(P),\quad
J^{\mu\nu}(P)P_{\mu}P_{\nu}=\tfrac{1}{4}s^{2}J(P).$

and from $J(P)=\theta(P^{0})\,\theta(s-4M^{2})\,4i\pi^{2}\,s\,G(s)$ – see
(3.13). So we arrive at

$g(s)=2i\pi^{2}s\,G(s);\quad a(s)=\frac{4i\pi^{2}}{3}\,(s-M^{2})G(s);\quad
c(s)=\frac{4i\pi^{2}}{3}\,(M^{2}-s/4)s\,G(s),$

yielding

$\displaystyle r^{\prime F}_{\mu\nu}(k_{1},k_{2})$
$\displaystyle=\frac{1}{4\pi^{2}}\,\theta(P^{0})\,\theta(s-4M^{2})\,G(s)\biggl{[}\Bigl{(}\frac{14}{3}-\frac{11}{6}\,\frac{s}{M^{2}}+\frac{5}{12}\,\frac{s^{2}}{M^{4}}\Bigr{)}g_{\mu\nu}(k_{1}k_{2})$
$\displaystyle\quad+k_{1\nu}k_{2\mu}\Bigl{(}-\frac{1}{3}+\frac{2}{3}\,\frac{s}{M^{2}}-\frac{1}{12}\,\frac{s^{2}}{M^{4}}\Bigr{)}\biggr{]}.$
(4.22)

#### Other new integrals.

We introduce already all the new integrals required in this section. By means
of $(q-k_{2})^{2}-M^{2}=-2(qk_{2})$, implying
$(qk_{2})\,\Delta^{F}(q-k_{2})=-i/8\pi^{2}$, some of the integrals to appear
are calculated:

$\displaystyle K^{\\{\cdot|\mu|\mu\nu\\}}(P)$
$\displaystyle:=\int\\!d^{4}q\,\\{1|q^{\mu}|q^{\mu}q^{\nu}\\}\,(qk_{2})\,\Delta^{+}(P-q)\Delta^{F}(q-k_{2})\,\Delta^{+}(q)=-\frac{i}{8\pi^{2}}\,J^{\\{\cdot|\mu|\mu\nu\\}}(P),$
$\displaystyle L(P)$
$\displaystyle:=\int\\!d^{4}q\,(qk_{2})^{2}\,\Delta^{+}(P-q)\Delta^{F}(q-k_{2})\,\Delta^{+}(q)=k_{2\mu}\,K^{\mu}.$

The following integrals will also be needed:

$\displaystyle N^{\\{\cdot|\mu|\mu\nu\\}}(P)$ $\displaystyle:=\int
d^{4}q\,\\{1|q^{\mu}|q^{\mu}q^{\nu}\\}\,\Delta^{+}(P-q)\Delta^{F}(q-k_{2})\,\Delta^{+}(q).$

By using $k_{2}^{\nu}=0$, they are easily be expressed in terms of the
integrals computed in Sect. 3.1:

$N(P)=I(P)\,,\qquad N^{\mu}(P)=I^{\mu}(P)+k_{2}^{\mu}I(P)\,,\qquad
N^{\mu\nu}(P)=I^{\mu\nu}(P)+k_{2}^{\mu}I^{\nu}(P)\,,$

#### Electromagnetic gauge invariance.

The computation of the individual terms in (4.18) is lengthy, but
straightforward. For the $r^{\prime B}$-terms we reap

$\displaystyle ik_{2}^{\beta}\bigl{(}r^{\prime
B}_{\mu,\beta\nu}(k_{1},k_{2})-r^{\prime
B}_{\mu,\nu\beta}(k_{1},k_{2})\bigr{)}=\frac{1}{(2\pi)^{2}}\biggl{\\{}-2N(P)(g_{\mu\nu}(k_{1}k_{2})-k_{1\nu}k_{2\mu})$
$\displaystyle\quad+g_{\mu\nu}(k_{1}k_{2})\biggl{(}2\frac{K(P)}{M^{2}}-\frac{L(P)}{M^{4}}\biggr{)}-k_{1\nu}k_{2\mu}(k_{1}k_{2})\,\frac{K(P)}{M^{4}}+N_{\mu\nu}(P)\biggl{(}2\frac{(k_{1}k_{2})}{M^{2}}-\frac{(k_{1}k_{2})^{2}}{M^{4}}\biggr{)}$
$\displaystyle\quad+k_{2\mu}\biggl{(}N_{\nu}(P)\Bigl{(}-2\frac{(k_{1}k_{2})}{M^{2}}+\frac{(k_{1}k_{2})^{2}}{M^{4}}\,\Bigr{)}+K_{\nu}(P)\,\frac{(k_{1}k_{2})}{M^{4}}\biggr{)}+k_{1\nu}\,K_{\mu}(P)\biggl{(}-\frac{2}{M^{2}}+\frac{k_{1}k_{2}}{M^{4}}\biggr{)}\biggr{\\}}.$

Inserting the integrals calculated above, one reaches for
$ik_{2}^{\beta}\bigl{(}r^{\prime B}_{\mu,\beta\nu}(k_{1},k_{2})-r^{\prime
B}_{\mu,\nu\beta}(k_{1},k_{2})\bigr{)}+(k_{1},\mu)\leftrightarrow(k_{2},\nu)$
the following result:

$\displaystyle\frac{\theta(P^{0})\,\theta(s-4M^{2})}{(2\pi)^{2}}\,\bigl{(}g_{\mu\nu}(k_{1}k_{2})-k_{1\nu}k_{2\mu}\bigr{)}\,\biggl{(}F(s)\Bigl{(}-2-\frac{s}{2M^{2}}\Bigr{)}+G(s)\,\frac{s}{M^{2}}\biggr{)}.$
(4.23)

Note that $ik_{2}^{\beta}\bigl{(}r^{\prime
B}_{\mu,\beta\nu}(k_{1},k_{2})-r^{\prime
B}_{\mu,\nu\beta}(k_{1},k_{2})\bigr{)}$ is individually gauge invariant. This
reflects the known fact that $T_{2}^{B}$ given by (4.4) is individually gauge
invariant.

Analogously, since $T_{2}^{D}$ (4.3) is trivially gauge invariant, we expect
the combination

$-k_{1}^{\beta}k_{2}^{\alpha}\bigl{(}r^{\prime
D}_{\beta\mu,\alpha\nu}(k_{1},k_{2})-r^{\prime
D}_{\mu\beta,\alpha\nu}(k_{1},k_{2})-r^{\prime
D}_{\beta\mu,\nu\alpha}(k_{1},k_{2})+r^{\prime
D}_{\mu\beta,\nu\alpha}(k_{1},k_{2})\bigr{)},$

to contain the factor $g_{\mu\nu}(k_{1}k_{2})-k_{1\nu}k_{2\mu}$. Indeed, we
obtain

$\displaystyle\frac{1}{(2\pi)^{2}}\biggl{\\{}-2(g_{\mu\nu}(k_{1}k_{2})-k_{1\nu}k_{2\mu})\,N(P)+g_{\mu\nu}\biggl{(}-2\frac{L(P)}{M^{2}}+(k_{1}k_{2})\Bigl{(}2\frac{K(P)}{M^{2}}-\frac{L(P)}{M^{4}}\Bigr{)}\biggr{)}$
$\displaystyle\quad+k_{1\nu}k_{2\mu}\,(k_{1}k_{2})\,\frac{K(P)}{M^{4}}+N_{\mu\nu}\biggl{(}2\frac{(k_{1}k_{2})}{M^{2}}+\frac{(k_{1}k_{2})^{2}}{M^{4}}\biggr{)}+k_{1\nu}\biggl{(}-2\frac{K_{1\mu}(P)}{M^{2}}-(k_{1}k_{2})\,\frac{K_{\mu}(P)}{M^{4}}\biggr{)}$
$\displaystyle\quad+k_{2\mu}\biggl{(}2\frac{K_{\nu}(P)}{M^{2}}-2\frac{(k_{1}k_{2})}{M^{2}}\,N_{\nu}(P)-\frac{(k_{1}k_{2})^{2}}{M^{4}}\,N_{\nu}(P)+(k_{1}k_{2})\,\frac{K_{1\nu}(P)}{M^{4}}\biggr{)}\biggr{\\}}$
$\displaystyle=\frac{\theta(P^{0})\,\theta(s-4M^{2})}{(2\pi)^{2}}\,(g_{\mu\nu}(k_{1}k_{2})-k_{1\nu}k_{2\mu})\biggl{[}\Bigl{(}-1+\frac{s}{4M^{2}}\Bigr{)}F(s)-\frac{s^{2}}{4M^{4}}\,G(s)\biggr{]}.$
(4.24)

Finally, for $r^{\prime A}$ we get

$\displaystyle r^{\prime A}_{\mu\nu}(k_{1},k_{2})$
$\displaystyle=\frac{1}{(2\pi)^{2}}\biggl{\\{}-2(g_{\mu\nu}(k_{1}k_{2})-k_{1\nu}k_{2\mu})\,N(P)$
$\displaystyle\quad+g_{\mu\nu}\biggl{(}\Bigl{(}2+\frac{2(k_{1}k_{2})}{M^{2}}\Bigr{)}\,K(P)-\Bigl{(}\frac{2}{M^{2}}+\frac{(k_{1}k_{2})}{M^{4}}\Bigr{)}\,L(P)\biggr{)}$
$\displaystyle\quad+k_{1\nu}k_{2\mu}\Bigl{(}-\frac{(k_{1}k_{2})}{M^{4}}\Bigr{)}\,K(P)+k_{1\nu}\Bigl{(}-\frac{2}{M^{2}}+\frac{(k_{1}k_{2})}{M^{4}}\Bigr{)}\,K_{\mu}(P)$
$\displaystyle\quad+k_{2\mu}\biggl{(}\Bigl{(}-12+2\frac{(k_{1}k_{2})}{M^{2}}-\frac{(k_{1}k_{2})^{2}}{M^{4}}\Bigr{)}\,N_{\nu}(P)+\Bigl{(}\frac{4}{M^{2}}+\frac{(k_{1}k_{2})}{M^{4}}\Bigr{)}\,K_{\nu}(P)\biggr{)}$
$\displaystyle\quad+\Bigl{(}12-2\frac{k_{1}k_{2}}{M^{2}}+\frac{(k_{1}k_{2})^{2}}{M^{4}}\Bigr{)}\,N_{\mu\nu}(P)-\Bigl{(}\frac{2}{M^{2}}+2\frac{(k_{1}k_{2})}{M^{4}}\Bigr{)}\,K_{\mu\nu}(P)\biggr{\\}}.$

After insertion of the integrals, this is equal to

$\displaystyle\frac{\theta(P^{0})\,\theta(s-4M^{2})}{(2\pi)^{2}}\biggl{\\{}(g_{\mu\nu}(k_{1}k_{2})-k_{1\nu}k_{2\mu})\,F(s)\,\Bigl{(}-3+12\,\frac{M^{2}}{s}+\frac{s}{4M^{2}}\Bigr{)}$
$\displaystyle+G(s)\biggl{[}g_{\mu\nu}(k_{1}k_{2})\Bigl{(}-\frac{14}{3}+\frac{5}{6}\,\frac{s}{M^{2}}-\frac{1}{6}\,\frac{s^{2}}{M^{4}}\Bigr{)}+k_{1\nu}k_{2\mu}\Bigl{(}\frac{1}{3}+\frac{s}{3M^{2}}-\frac{s^{2}}{6M^{4}}\Bigr{)}\biggr{]}\biggr{\\}}.$

The sum $r^{\prime A}+r^{\prime F}$ is indeed gauge invariant:

$\displaystyle r^{\prime A}_{\mu\nu}(k_{1},k_{2})+r^{\prime
F}_{\mu\nu}(k_{1},k_{2})$
$\displaystyle=\frac{\theta(P^{0})\,\theta(s-4M^{2})}{(2\pi)^{2}}\,(g_{\mu\nu}(k_{1}k_{2})-k_{1\nu}k_{2\mu})$
$\displaystyle\quad\times\biggl{(}F(s)\,\Bigl{(}-3+12\,\frac{M^{2}}{s}+\frac{s}{4M^{2}}\Bigr{)}+G(s)\,\Bigl{(}-\frac{s}{M^{2}}+\frac{s^{2}}{4M^{4}}\Bigr{)}\biggr{)},$

as expected from the outcome of the discussion in subsection 4.1.

The $G(s)$-terms are seen to cancel out, and for the total $r^{\prime}$ we
obtain the following gauge-invariant result:

$r^{\prime}_{\mu\nu}(k_{1},k_{2})=\frac{\theta(P^{0})\,\theta(s-4M^{2})}{(2\pi)^{2}}\,(g_{\mu\nu}(k_{1}k_{2})-k_{1\nu}k_{2\mu})\,6F(s)\,(-1+2M^{2}/s).$
(4.25)

Like for the scalar model, $a^{\prime}_{\mu\nu}(k_{1},k_{2})$ is obtained from
$r^{\prime}_{\mu\nu}(k_{1},k_{2})$ in (4.25) by replacing $\theta(P^{0})$ by
$\theta(-P^{0})$, and hence
$d_{\mu\nu}=a^{\prime}_{\mu\nu}-r^{\prime}_{\mu\nu}$ by replacing
$\theta(P^{0})$ by $-\operatorname{sgn}(P^{0})$.

Hence $d^{\mu\nu}_{\mathrm{gi}}(k_{1},k_{2})|_{k_{1}^{2}=0=k_{2}^{2}}$ is of
the form

$\displaystyle
d^{\mu\nu}_{\mathrm{gi}}(k_{1},k_{2})\bigr{|}_{k_{1}^{2}=0=k_{2}^{2}}=P^{\mu\nu}(k_{1},k_{2})\,d_{0}(P),\quad\text{where}\quad$
(4.26) $\displaystyle
d_{0}(P):=\frac{i\,\operatorname{sgn}(P^{0})\,\theta(\tilde{\rho}-1)}{(2\pi)^{5}}\,b_{1}(\tilde{\rho}),\quad
b_{1}(\tilde{\rho}):=-\frac{3}{16M^{2}\,\tilde{\rho}}\,\biggl{(}\frac{1}{\tilde{\rho}}-2\biggr{)}\,\log\frac{1+\sqrt{1-\tilde{\rho}^{-1}}}{1-\sqrt{1-\tilde{\rho}^{-1}}}\,.$

The factor $3$ above was to be expected, since the Proca field has three
components. The singular order of the on-shell distribution
$d^{\mu\nu}_{\mathrm{gi}}(k_{1},k_{2})|_{k_{1}^{2}=0=k_{2}^{2}}$ is clearly
equal to zero. It is also easy to show that power counting rules imply that
the singular order of the off-shell $d$-distribution111111This notation is
badly abused in this paper. But that is hardly avoidable. $d^{\mu\nu}$
satisfies the bounds $6\geq\omega:=\omega(d^{\mu\nu})\geq 2$.

###  4.3 Distribution splitting in EW theory

The off-shell $d^{\mu\nu}$-distribution for the diphoton decay of the higgs
via EW vector bosons is also of the genre (3.20), the difference being that
the pertinent polynomials $h_{j}^{\mu\nu}$ are of a higher degree, i.e.,
$\omega:=\omega(d^{\mu\nu})$ has a greater value. Therefore, the distribution
splitting method for null momenta developed in Section 2 does apply:
$t^{\mu\nu}_{\mathrm{gi}}|_{\mathcal{M}}=a^{c\,\mu\nu}_{\mathrm{gi}}|_{\mathcal{M}}$
can be computed by inserting the light-cone restriction of
$d^{\mu\nu}_{\mathrm{gi}}(k_{1},k_{2})$ into the dispersion integral (3.27).
Again, this restricted $d$-distribution is of the genre (3.16) for the given
$b_{1}$. Hence, the central solution compatible with EGI is obtained by:

$\displaystyle
t^{c\,\mu\nu}_{\mathrm{gi}}(k_{1},k_{2})=P^{\mu\nu}(k_{1},k_{2})\,t^{c}_{\mathrm{gi}}(P),\quad\text{where}\quad$
$\displaystyle
t^{c}_{\mathrm{gi}}(P)=\frac{i\eta}{2\pi}\int_{|t|\geq\sqrt{1/\tilde{\rho}}}\,\frac{t^{2}\,d_{0}(tP)\,dt}{(1-t)\,t^{\max\\{\omega+1,\,0\\}}}=-\frac{1}{(2\pi)^{6}}\int_{1/\tilde{\rho}}^{\infty}\frac{u\,b_{1}(u\tilde{\rho})\,du}{u^{\max\\{\lfloor\omega/2\rfloor+1,\,0\\}}(1-u)},$
(4.27)

for $(k_{1},k_{2})\in\mathcal{M}$. Let us first assume that $\omega=2$. Before
computing, we recall that the ambiguity of the result for
$t^{\mu\nu}_{\mathrm{gi}}$ will be given in that case by a polynomial of
second degree in $(k_{1},k_{2})$ containing the factor $P^{\mu\nu}$; that is,
in the occasion, a constant multiple of $P^{\mu\nu}$ itself.

Besides a global factor $-3(16M^{2}(2\pi)^{6})^{-1}$, in view of (4.26) and
making the customary change of variable $v=u\tilde{\rho}$, to obtain
$t^{c}_{\mathrm{gi}}(\tilde{\rho})$ we ought to compute:

$\displaystyle\tilde{\rho}\int_{1}^{\infty}\\!dv\,\Biggl{[}\frac{\log\bigl{(}1+\sqrt{1-v^{-1}}\bigr{)}/\bigl{(}1-\sqrt{1-v^{-1}}\bigr{)}}{v^{3}(\tilde{\rho}-v)}-\frac{2\log\bigl{(}1+\sqrt{1-v^{-1}}\bigr{)}/\bigl{(}1-\sqrt{1-v^{-1}}\bigr{)}}{v^{2}(\tilde{\rho}-v)}\Biggr{]}$
$\displaystyle=\int_{1}^{\infty}\\!dv\,\Biggl{[}\frac{\log\bigl{(}1+\sqrt{1-v^{-1}}\bigr{)}/\bigl{(}1-\sqrt{1-v^{-1}}\bigr{)}}{v^{2}(\tilde{\rho}-v)}-\frac{2\log\bigl{(}1+\sqrt{1-v^{-1}}\bigr{)}/\bigl{(}1-\sqrt{1-v^{-1}}\bigr{)}}{v(\tilde{\rho}-v)}\Biggr{]}$
$\displaystyle+\int_{1}^{\infty}\\!dv\,\Biggl{[}\frac{\log\bigl{(}1+\sqrt{1-v^{-1}}\bigr{)}/\bigl{(}1-\sqrt{1-v^{-1}}\bigr{)}}{v^{3}}-\frac{2\log\bigl{(}1+\sqrt{1-v^{-1}}\bigr{)}/\bigl{(}1-\sqrt{1-v^{-1}}\bigr{)}}{v^{2}}\Biggr{]}.$

Notice that the last two integral terms just yield a number, equal to minus
the sum of the values of the two previous ones at $\tilde{\rho}=0$. The first
integral term is already known from the discussion in subsection 3.4, and in
all yields

$\frac{3}{8M^{2}(2\pi)^{6}}J_{2}(\tilde{\rho})=\frac{-3}{8M^{2}(2\pi)^{6}}\biggl{(}\frac{1}{\tilde{\rho}}-\frac{f(\tilde{\rho})}{\tilde{\rho}^{2}}\biggr{)}.$

This is essentially $3F_{0}(\tilde{\rho})$, that clearly contributes to the
expected final result for $F_{1}(\tilde{\rho})$ – recall (1.3). According to
Eq. (3.29) – also in subsection 3.4 – in the case $a=0$, the second integral
term yields

$-\frac{2\cdot
3}{8M^{2}(2\pi)^{6}}\,J_{1}(\tilde{\rho},0)=-\frac{6}{8M^{2}(2\pi)^{6}}\,\frac{f(\tilde{\rho})}{\tilde{\rho}}\,.$

Including the last two integrals, we recover
$t^{c}_{\mathrm{gi}}(\tilde{\rho})\sim F_{1}(\tilde{\rho})-7$. Taking into
account the Epstein–Glaser ambiguity as limited by EGI, and bringing in
constant prefactor appearing in (4.17), we end up with the amplitude

$-2e^{2}gM(2\pi)^{2}\,t^{\mu\nu}_{\mathrm{gi}}(k_{1},k_{2})=\frac{e^{2}g}{4(2\pi)^{4}\,M}\,P^{\mu\nu}(k_{1},k_{2})\,\bigl{(}F_{1}(\tilde{\rho})+C\bigr{)},$
(4.28)

with $C$ an arbitrary constant.

If $\omega>2$, the central solution within the unitary gauge is obtained by
adding the following expression to the above obtained result, more precisely,
to $-(8M^{2}(2\pi)^{6})^{-1}\bigl{(}F_{1}(\tilde{\rho})-7\bigr{)}$:

$\displaystyle-\frac{1}{(2\pi)^{6}}\int_{\tilde{\rho}^{-1}}^{\infty}du\,\frac{b_{1}(u\tilde{\rho})}{1-u}\biggl{(}\frac{1}{u^{\lfloor\omega/2\rfloor}}-\frac{1}{u}\biggr{)}=-\frac{1}{(2\pi)^{6}}\int_{\tilde{\rho}^{-1}}^{\infty}du\,b_{1}(u\tilde{\rho})\sum_{k=0}^{\lfloor\omega/2\rfloor-2}\frac{1}{u^{\lfloor\omega/2\rfloor-k}}$
$\displaystyle\quad=\sum_{k=0}^{\lfloor\omega/2\rfloor-2}\tilde{c}_{k}\,\tilde{\rho}^{\lfloor\omega/2\rfloor-1-k};\quad\text{where
the}\quad\tilde{c}_{k}:=-\frac{1}{2\pi}\int_{1}^{\infty}dv\,\frac{b_{1}(v)}{v^{\lfloor\omega/2\rfloor-k}}$
(4.29)

do not depend on $\tilde{\rho}$; in the first step we have used the relation

$\frac{1}{u^{\lfloor\omega/2\rfloor}}-\frac{1}{u}=(1-u)\sum_{k=0}^{\lfloor\omega/2\rfloor-2}\frac{1}{u^{\lfloor\omega/2\rfloor-k}}\,,$

and in the last step we have made again the substitution $v:=u\tilde{\rho}$.
The point is that (4.29) is a polynomial in $\tilde{\rho}$ allowed by the
surviving Epstein–Glaser ambiguity. Therefore, without knowing the precise
value of $\omega$, the general Epstein–Glaser solution respecting EGI can be
written as

$-2e^{2}gM(2\pi)^{2}\,t^{\mu\nu}_{\mathrm{gi}}(k_{1},k_{2})=\frac{e^{2}g}{4(2\pi)^{4}\,M}\,P^{\mu\nu}(k_{1},k_{2})\,\biggl{(}F_{1}(\tilde{\rho})+\sum_{k=0}^{\lfloor\omega/2\rfloor-1}C_{k}\tilde{\rho}^{k}\biggr{)},$
(4.30)

where the constants $C_{k}$ are all arbitrary. However, terms corresponding to
$\omega\geq 4$ can be discarded on grounds of perturbative unitarity.

Let us now to come back to reference [17]. It is argued there that the
convergent integral

$t_{0}(\tilde{\rho}):=\int_{\tilde{\rho}^{-1}}^{\infty}du\,\frac{b_{1}(u\tilde{\rho})}{1-u}$

leads to the correct result. From the standpoint of this reference, formula
(4.27) is “oversubtracted”. One obtains there, yet again

$\displaystyle P^{\mu\nu}t_{0}(\tilde{\rho})$
$\displaystyle=P^{\mu\nu}\\!\int_{\tilde{\rho}^{-1}}^{\infty}du\,\frac{b_{1}(u\tilde{\rho})}{1-u}=-\frac{3\,P^{\mu\nu}}{8M^{2}(2\pi)^{6}}\,(2J_{1}(\tilde{\rho},0)-J_{2}(\tilde{\rho}))$
$\displaystyle=-\frac{P^{\mu\nu}}{8M^{2}(2\pi)^{6}}\,\biggl{[}\frac{3}{\tilde{\rho}}+\frac{6f(\tilde{\rho})}{\tilde{\rho}}-\frac{3f(\tilde{\rho})}{\tilde{\rho}^{2}}\biggr{]}=-\frac{P^{\mu\nu}}{8M^{2}(2\pi)^{6}}\,(F_{1}(\tilde{\rho})-2).$
(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
$CP^{\mu\nu}$ for $C$ an 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) is the unique Epstein–Glaser
solution respecting EGI, corresponding to the following causal
$d$-distribution: let the result (4.26) for
$d^{\mu\nu}_{\mathrm{gi}}(k_{1},k_{2})$ (obtained by light-cone restriction of
the photon momenta) be interpreted as an unrestricted element of
$\mathcal{S}^{\prime}({\mathbb{R}}^{8})$, that is, all values
$(k_{1},k_{2})\in{\mathbb{R}}^{8}$ are 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.121212Actually, in
the first edition of the book by Scharf on quantum electrodynamics (i.e., [30]
rather than [40]), the vertex function in QED at third order was computed by
such a method.

###  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 $h\to\gamma\gamma$ decay in the Feynman gauge and gauge-
fixing independence, namely, the requirement that observable quantities should
not depend on the choice of gauge.131313The equivalence or inequivalence of
calculations performed in different gauges was a nagging worry of Raymond
Stora in his last years. The classic paper [43] illustrates the difficulties
lurking here. Motivated by results of [44],141414This reference works with a
formulation of gauge invariance suitable for the BEG scheme. In that framework
it was shown for the various $R_{\xi}$-gauges that the $T$-products can be
normalized in such a way that the physical $S$-matrix (i.e., for in- and out-
states being on-shell) does not depend on the gauge-fixing parameter $\xi$ in
the formal adiabatic limit; and that this normalization is compatible with
gauge invariance in the mentioned sense. 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, $\tilde{\rho}=\rho:=m_{h}^{2}/4M^{2}$.

Denote the Epstein–Glaser result for the $d$-distribution in the Feynman gauge
by $d^{1}_{\mu\nu}$. In contrast with the unitary gauge, there additionally
contribute diagrams with Stückelberg fields and Faddeev–Popov ghosts (as inner
lines) to $d^{1}_{\mu\nu}$, see e.g. [16]. We spare the reader the details of
the construction of the TOPs, and in particular the derivation of the
$AAWW^{\dagger}$-vertex in this context. For photons on-shell with physical
polarizations (setting $k_{1}^{2}=0=k_{2}^{2}$ and omitting pure gauge terms
$\sim k_{1\mu}$ or $\sim k_{2\nu}$), our result reads:

$\displaystyle
d_{\mu\nu}^{1}(k_{1},k_{2})=-\frac{1}{2^{3}\,(2\pi)^{6}\,M^{2}}\times$ (4.32)
$\displaystyle\bigg{[}(k_{1}k_{2})g_{\mu\nu}\left(-\frac{3}{\tilde{\rho}^{2}}+\frac{7}{\tilde{\rho}}-\frac{\rho}{\tilde{\rho}^{2}}\right)-k_{2\mu}k_{1\nu}\left(-\frac{3}{\tilde{\rho}^{2}}+\frac{8}{\tilde{\rho}}-\frac{2\rho}{\tilde{\rho}^{2}}\right)\bigg{]}\Im
f(\tilde{\rho}).$

The tedious computation of the above absorptive part was done with the aid of
the Mathematica package FeynCalc [45]. 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 Sec. 3.1 and 4.2, respectively. As before, all terms contributing to
the distribution $d_{\mu\nu}^{1}$ can be represented by Feynman diagrams with
cuts – for the complete list see e.g. [16]. Just like in subsections. 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.

An important feature of electroweak theory in the $R_{\xi}$-gauges is the fact
that all interaction vertices have dimensions lower or equal to four (because
$\mathrm{dim}\,W^{\mu}=1$, in contrast to the value $\mathrm{dim}\,W^{\mu}=2$
for the unitary gauge). In particular, a straightforward power counting
argument gives the upper bound $\omega\bigl{(}d^{1}_{\mu\nu}\bigr{)}\leq 0$
for the singular order of the off-shell distribution $d_{\mu\nu}^{1}$. Noting
that $\tilde{\rho}=(k_{1}k_{2})/2M^{2}$ and
$f(\tilde{\rho})=O(\log\tilde{\rho})$ we see that the on-shell restriction of
$d^{1}_{\mu\nu}(k_{1},k_{2})$ in Eq. (4.32) grows logarithmically for big
values of $\tilde{\rho}$. For the off-shell $d_{\mu\nu}^{1}$, this implies the
equality $\omega\bigl{(}d^{1}_{\mu\nu}\bigr{)}=0$. This should be contrasted
with the bounds $6\geq\omega\bigl{(}d_{\mu\nu}\bigr{)}\geq 2$ in the case of
the absorptive part computed in the unitary gauge.

The off-shell distribution $d_{\mu\nu}^{1}$ is again of the type considered in
Sec. 3.2. In particular, the method of distribution splitting developed in
subsection 3.3 is applicable. For photons on-shell with physical
polarizations, the central solution reads

$\displaystyle
t^{1\,c}_{\mu\nu}(k_{1},k_{2})=-\frac{1}{2^{3}\,(2\pi)^{6}\,M^{2}}\,\times$
(4.33)
$\displaystyle\biggl{\\{}g_{\mu\nu}(k_{1}k_{2})\biggl{[}\Bigl{(}-\frac{3}{\tilde{\rho}^{2}}+\frac{7}{\tilde{\rho}}-\frac{\rho}{\tilde{\rho}^{2}}\Bigr{)}f(\tilde{\rho})+\frac{3}{\tilde{\rho}}+\frac{2\rho}{\tilde{\rho}}\biggr{]}-k_{1\nu}k_{2\mu}\biggl{[}\Bigl{(}-\frac{3}{\tilde{\rho}^{2}}+\frac{8}{\tilde{\rho}}-\frac{2\rho}{\tilde{\rho}^{2}}\Bigr{)}f(\tilde{\rho})+\frac{3}{\tilde{\rho}}+\frac{2\rho}{\tilde{\rho}}\biggr{]}\biggr{\\}}.$

According to the postulate ‘Divergence degree’ (in Sect. C.1), we have to
demand for the off-shell $t^{1}_{\mu\nu}$ that

$\omega\bigl{(}t^{1}_{\mu\nu}\bigr{)}=\omega\bigl{(}d^{1}_{\mu\nu}\bigr{)}=0.$

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

$t^{1}_{\mu\nu}(k_{1},k_{2})=t^{1\,c}_{\mu\nu}(k_{1},k_{2})+g_{\mu\nu}D,$
(4.34)

where $D$ is 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 $R_{\xi}$-gauges.
However, entirely on-shell EGI can be satisfied: setting $\tilde{\rho}:=\rho$
in (4.33), we plainly get

$t^{1}_{\mu\nu}(k_{1},k_{2})\bigr{|}_{\tilde{\rho}=\rho}=-\frac{1}{2^{3}\,(2\pi)^{6}\,M^{2}}\,\bigl{(}P_{\mu\nu}(k_{1},k_{2})\,F_{1}(\rho)+D\,g_{\mu\nu}\bigr{)},$
(4.35)

and one sees that $D$ must be put equal to zero. This fixes completely the
normalization freedom in the construction of $t_{\mu\nu}^{1}$ in the Feynman
gauge. At this level there is of course coincidence with the result in [46],
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 $\omega\equiv\omega(d_{\mu\nu})$. We stress that the constants $C_{k}$
appearing 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 $C_{k}$ of the polynomial in the expression (4.30)
for $t_{\mathrm{gi},\mu\nu}$ in the unitary gauge in such a way that the
following equality

$t_{\mathrm{gi},\mu\nu}(k_{1},k_{2})\bigr{|}_{\tilde{\rho}=\rho}=t^{1}_{\mu\nu}(k_{1},k_{2})\bigr{|}_{\tilde{\rho}=\rho}.$
(4.36)

holds entirely on-shell, i.e. for $\tilde{\rho}=\rho$. In fact, we must set
$C_{0}:=2$ and $C_{k}:=0$ for all $k\geq 1$ in (4.30), which fixes completely
the normalization freedom of $t_{\mathrm{gi},\mu\nu}$. 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 $t_{\mathrm{gi},\mu\nu}$ in the expression (4.30).

In summary, our final result for the entirely on-shell EW $h\to\gamma\gamma$
decay reads:

$t_{\mu\nu}(k_{1},k_{2})\bigr{|}_{\tilde{\rho}=\rho}=-\frac{1}{2^{3}\,(2\pi)^{6}\,M^{2}}\,P_{\mu\nu}(k_{1},k_{2})\,F_{1}(\rho),$

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 $M\to 0$) and “light-higgs” (or
$M\to\infty$) arguments to bolster the case of $F_{1}(\rho)$ versus
$F_{1}(\rho)-2$, 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
$\beta$-function series associated to electric charge
renormalization.151515$F_{0}(0)=-1/3$, which has been calculated in this
paper, means precisely this. It was a fortunate historical fact that a
calculation of the effective Lagrangian for charged Proca particles [47] was
already available when the first “exact” computation of the higgs to digamma
process that we are aware of was performed [3] – 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 [42, Sect. 7] and
[48] – and expected to yield the required value $F_{1}(0)=7$. 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 à la Bogoliubov–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” [7], nor adopt the “loop regularization
method” [11], 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 $h\to\gamma\gamma$ 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
$h\to\gamma\gamma$ 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 [40, Chap. 3.8] and in [49].

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 [6, 7] and [17]. We have
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.161616Variants of the “light-higgs” or “low energy” argument besides
[3, 10] and [18] are found for instance in [28, Ch. 24.8], in [50] and in
[51]. 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, ITP-Göttingen and ZiF-Bielefeld is gratefully
acknowledged.

##  Appendix A Notations and prerequisites

Our Minkowski metric is mostly-negative. The Minkowski inner product of two
vectors $x\equiv x^{\mu}$, $p\equiv p^{\nu}$ is denoted with parentheses:
$(xp)=x^{\mu}p_{\mu}$. When (we hope) it does not cause confusion, we often
denote $p^{2}=(pp)$.

We signal the standard formula for time-ordered $2$-point function:

$\langle\\!\langle\operatorname{T}\varphi(x)\,\chi(x^{\prime})\rangle\\!\rangle:=\frac{i}{(2\pi)^{4}}\int
d^{4}p\,\frac{e^{-i(p(x-x^{\prime}))}}{p^{2}-M^{2}+i0}\,M^{\varphi\chi}(p),$
(A.1)

where $M^{\varphi\chi}$ is the multiplier appearing in the corresponding
$2$-point function for the fields $\varphi,\chi$ with the same mass $M$.

Propagators for a (complex) scalar field. Clearly, for (say, complex) scalar
fields the Feynman propagator

$\displaystyle\Delta^{F}(x-x^{\prime})$
$\displaystyle:=\langle\\!\langle\operatorname{T}\varphi(x)\,\varphi^{\dagger}(x^{\prime})\rangle\\!\rangle$
(A.2) fulfils $\displaystyle\square\Delta^{F}(x-x^{\prime})$
$\displaystyle=-M^{2}\Delta^{F}(x-x^{\prime})-i\delta(x-x^{\prime}),$ (A.3)

where $M$ is the mass of the $\varphi$-field. Also, with $\theta$ denoting the
Heaviside function, the Wightman functions

$\displaystyle\Delta^{+}(x-x^{\prime}):=\langle\\!\langle\varphi(x)\varphi^{\dagger}(x^{\prime})\rangle\\!\rangle=\langle\\!\langle\varphi^{\dagger}(x)\varphi(x^{\prime})\rangle\\!\rangle=\frac{1}{(2\pi)^{3}}\int
d^{4}p\,\theta(p^{0})\,\delta(p^{2}-M^{2})e^{-i(p(x-x^{\prime}))}$
$\displaystyle\quad\text{so
that}\quad(\square+M^{2})\Delta^{+}(x)=0,\qquad\Delta^{-}(x):=-\Delta^{+}(-x),$
(A.4)

are used in our calculations.

Massive vector fields. A dreibein $e_{r}(p)$ on Minkowski momentum space, with
the properties:

$\bigl{(}e_{r}(p)\,e_{s}(p)\bigr{)}=-\delta_{rs}\quad\text{for}\quad
r,s=1,2,3;\qquad\bigl{(}p\,e_{r}(p)\bigr{)}=0,$

describes polarization states for particles with squared mass $M^{2}=p^{2}>0$
and spin $j=1$. From the above identities, one derives the projector formula:

$\sum_{r=1}^{3}e_{r}^{\mu}(p)e_{r}^{\nu}(p)=-g^{\mu\nu}+\frac{p^{\mu}p^{\nu}}{M^{2}}.$
(A.5)

The set $e$ is regarded as an intertwiner matrix mapping the natural
representation space of the Lorentz group onto the representation space
$\mathbb{C}^{3}$ for spin $1$ objects. Let $a_{r}^{\dagger}(p)$ and $a_{r}(p)$
be respectively the creation and annihilation operators on the boson Fock
space for such particles – whose $1$-particle subspace is the corresponding
Wigner unirrep space; and $b_{r}^{\dagger}(p)$ and $b_{r}(p)$ for their
antiparticles.

There is a quantum vector field acting on that space given by the formula

$W^{\mu}(x):=\sum_{r}\int
d\mu(p)\,\bigl{[}e^{i(px)}e_{r}^{\mu}(p)\,b^{\dagger}_{r}(p)+e^{-i(px)}e_{r}^{\mu*}(p)\,a_{r}(p)\bigr{]};$
(A.6)

In (A.6) and in other formulas $d\mu(p)$ denotes the usual invariant measure
$d^{3}\bm{p}/2E(p)=d^{3}\bm{p}/\sqrt{m^{2}+|\bm{p}|^{2}}$ over the mass
hyperboloid $H_{M}^{\pm}:=\\{p\in{\mathbb{M}}\,|\,p^{2}=M^{2}\wedge\pm
p^{n}>0\\}$. By its definition, the charged Proca field $W$ is divergenceless:
$(\partial W)=0$. Its equations of motion can be variously written as

$(\square+M^{2})W^{\mu}=(\square+M^{2})W^{\mu}-\partial^{\mu}(\partial
W)=\partial_{\nu}G^{\nu\mu}(x)+M^{2}W^{\mu}=0,$ (A.7)

where $G^{\mu\nu}:=\partial^{\mu}W^{\nu}-\partial^{\nu}W^{\mu}$.

The theory of massive vector fields is a gauge theory [52, 53], its Proca
version being a “unitary gauge” for it. It has been analyzed, in terms
parallel to Maxwell field theory, in [54]; wherein the associated BRST
machinery is “deconstructed” in terms of Koszul cohomology.

The high-energy limit of $(-g^{\mu\nu}+p^{\mu}p^{\nu}/M^{2})/(p^{2}-M^{2})$
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 $e_{r}$, 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

$\displaystyle\Delta^{\alpha}_{\beta}(x-x^{\prime}):=\langle\\!\langle\operatorname{T}W^{\alpha}(x)\,W_{\beta}^{\dagger}(x^{\prime})\rangle\\!\rangle=-(g^{\alpha}_{\beta}+\partial^{\alpha}\partial_{\beta}/M^{2})\Delta^{F}(x-x^{\prime}),$
(A.8)

where $M$ is the mass of the $W$-field, and its properties:

$\square\Delta^{\alpha}_{\beta}=-M^{2}\Delta^{\alpha}_{\beta}+i(g^{\alpha}_{\beta}+\partial^{\alpha}\partial_{\beta}/M^{2})\delta;\quad\partial_{\mu}\Delta^{\mu}_{\nu}=i\partial_{\nu}\delta/M^{2}.$

The corresponding formulas for the Wightman functions respectively read:

$\displaystyle\Delta^{\alpha\,+}_{\beta}(x-x^{\prime}):=\langle\\!\langle
W^{\alpha}(x)\,W_{\beta}^{\dagger}(x^{\prime})\rangle\\!\rangle=\langle\\!\langle
W^{\alpha\,\dagger}(x)\,W_{\beta}(x^{\prime})\rangle\\!\rangle=-(g^{\alpha}_{\beta}+\partial^{\alpha}\partial_{\beta}/M^{2})\Delta^{+}(x-x^{\prime})$
$\displaystyle\quad\text{and}\quad\square\Delta^{\alpha\,+}_{\beta}=-M^{2}\Delta^{\alpha\,+}_{\beta},\quad\partial_{\mu}\Delta^{\mu\,+}_{\nu}=0.$

We will invoke also the Maxwell-like fields, where $\\#=\dagger$ or naught,

$F^{\mu\nu}:=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu};\qquad
G^{\\#}_{\mu\nu}:=\partial_{\mu}W^{\\#}_{\nu}-\partial_{\nu}W^{\\#}_{\mu},$

and introduce the propagator

$\displaystyle D^{\alpha\mu}_{\beta\rho}(x-x^{\prime})$
$\displaystyle:=\langle\\!\langle\operatorname{T}G^{\alpha\mu}(x)\,G^{\dagger}_{\beta\rho}(x^{\prime})\rangle\\!\rangle=\langle\\!\langle\operatorname{T}G^{\alpha\mu\,\dagger}(x)\,G_{\beta\rho}(x^{\prime})\rangle\\!\rangle$
$\displaystyle=\langle\\!\langle\operatorname{T}(\partial^{\alpha}W^{\mu}(x)-\partial^{\mu}W^{\alpha}(x))(\partial_{\beta}W^{\dagger}_{\rho}(x^{\prime})-\partial_{\rho}W^{\dagger}_{\beta}(x^{\prime}))\rangle\\!\rangle$
$\displaystyle=-\partial^{\mu}\bigl{(}\partial_{\rho}\Delta^{\alpha}_{\beta}(x-x^{\prime})-\partial_{\beta}\Delta^{\alpha}_{\rho}(x-x^{\prime})\bigr{)}+\partial^{\alpha}\bigl{(}\partial_{\rho}\Delta^{\mu}_{\beta}(x-x^{\prime})-\partial_{\beta}\Delta^{\mu}_{\rho}(x-x^{\prime})\bigr{)}$
$\displaystyle=(g^{\alpha}_{\beta}\partial^{\mu}_{\rho}-g^{\alpha}_{\rho}\partial^{\mu}_{\beta}-g^{\mu}_{\beta}\partial^{\alpha}_{\rho}+g^{\mu}_{\rho}\partial^{\alpha}_{\beta})\Delta^{F}(x-x^{\prime}),\quad\partial^{\alpha}_{\rho}:=\partial^{\alpha}\partial_{\rho},$
(A.9)

since in

$\partial_{\rho}\Delta^{\alpha}_{\beta}-\partial_{\beta}\Delta^{\alpha}_{\rho}=(-g^{\alpha}_{\beta}\partial_{\rho}+g^{\alpha}_{\rho}\partial_{\beta})\Delta^{F}$
(A.10)

the terms with three derivatives cancel out, due to the antisymmetry of
$G_{\mu\nu}^{\\#}$. That fact is relevant in this paper. Analogously we obtain

$\displaystyle D^{\alpha\mu\,+}_{\beta\rho}(x-x^{\prime})$
$\displaystyle:=\langle\\!\langle
G^{\alpha\mu}(x)\,G_{\beta\rho}^{\dagger}(x^{\prime})\rangle\\!\rangle=\langle\\!\langle
G^{\alpha\mu\,\dagger}(x)\,G_{\beta\rho}(x^{\prime})\rangle\\!\rangle$
$\displaystyle=(g^{\alpha}_{\beta}\partial^{\mu}_{\rho}-g^{\alpha}_{\rho}\partial^{\mu}_{\beta}-g^{\mu}_{\beta}\partial^{\alpha}_{\rho}+g^{\mu}_{\rho}\partial^{\alpha}_{\beta})\Delta^{+}(x-x^{\prime}),$

without third-order derivatives. We also note that

$\partial_{\mu}D^{\alpha\mu}_{\beta\rho}=(g^{\alpha}_{\beta}\partial_{\rho}-g^{\alpha}_{\rho}\partial_{\beta})\square\Delta^{F}=(-g^{\alpha}_{\beta}\partial_{\rho}+g^{\alpha}_{\rho}\partial_{\beta})(M^{2}\Delta^{F}+i\delta),$

since third-order derivatives appear only in the form
$\partial\square\Delta^{F}$, 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

$\displaystyle\langle\\!\langle\operatorname{T}W^{\mu}(x)\,G^{\dagger}_{\alpha\nu}(x^{\prime})\rangle\\!\rangle=\langle\\!\langle\operatorname{T}W^{\mu\,\dagger}(x)\,G_{\alpha\nu}(x^{\prime})\rangle\\!\rangle$
(A.11)
$\displaystyle=-(\partial_{\alpha}\Delta^{\mu}_{\nu}(x-x^{\prime})-\partial_{\nu}\Delta^{\mu}_{\alpha}(x-x^{\prime}))=(g^{\mu}_{\nu}\partial_{\alpha}-g^{\mu}_{\alpha}\partial_{\nu})\Delta^{F}(x-x^{\prime}),$
$\displaystyle\langle\\!\langle\operatorname{T}G^{\dagger}_{\alpha\nu}(x)\,W^{\mu}(x^{\prime})\rangle\\!\rangle=\langle\\!\langle\operatorname{T}G_{\alpha\nu}(x)\,W^{\mu\,\dagger}(x^{\prime})\rangle\\!\rangle=(-g^{\mu}_{\nu}\partial_{\alpha}+g^{\mu}_{\alpha}\partial_{\nu})\Delta^{F}(x-x^{\prime}).$

With the parallel Wightman functions we proceed similarly:

$\displaystyle\langle\\!\langle
W^{\mu}(x)\,G^{\dagger}_{\alpha\nu}(x^{\prime})\rangle\\!\rangle$
$\displaystyle=\langle\\!\langle
W^{\mu\,\dagger}(x)\,G_{\alpha\nu}(x^{\prime})\rangle\\!\rangle=(g^{\mu}_{\nu}\partial_{\alpha}-g^{\mu}_{\alpha}\partial_{\nu})\Delta^{+}(x-x^{\prime}),$
$\displaystyle\langle\\!\langle
G^{\dagger}_{\alpha\nu}(x)\,W^{\mu}(x^{\prime})\rangle\\!\rangle$
$\displaystyle=\langle\\!\langle
G_{\alpha\nu}(x)\,W^{\mu\,\dagger}(x^{\prime})\rangle\\!\rangle=(-g^{\mu}_{\nu}\partial_{\alpha}+g^{\mu}_{\alpha}\partial_{\nu})\Delta^{+}(x-x^{\prime}).$
(A.12)

Comparing with the Feynman gauge, in which the $W^{\\#}$ two-point functions
$\Delta^{\alpha}_{\beta}$ and $\Delta^{\alpha\,+}_{\beta}$ are replaced by
$-g^{\alpha}_{\beta}\Delta^{F}$ (A.13) and $-g^{\alpha}_{\beta}\Delta^{+}$
(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

$\displaystyle\langle\\!\langle\operatorname{T}W^{\alpha}(x)\,W_{\beta}^{\dagger}(x^{\prime})\rangle\\!\rangle$
$\displaystyle=-g^{\alpha}_{\beta}\Delta^{F}(x-x^{\prime}),$ (A.13)
$\displaystyle\langle\\!\langle
W^{\alpha}(x)\,W_{\beta}^{\dagger}(x^{\prime})\rangle\\!\rangle=\langle\\!\langle
W^{\alpha\,\dagger}(x)\,W_{\beta}(x^{\prime})\rangle\\!\rangle$
$\displaystyle=-g^{\alpha}_{\beta}\Delta^{+}(x-x^{\prime}).$ (A.14)

Besides the $W$-field the computation from Sec. 4.4 involves the Stückelberg
fields $\varphi^{\pm}$ and the ghost and anti-ghost fields
$C^{\pm},\bar{C}^{\pm}$ – where $\phi^{\pm}:=\frac{1}{\sqrt{2}}(\phi_{1}\pm
i\phi_{2}\bigr{)}$ for $\phi=\varphi,\,C,\,\bar{C}$. Below we list the non-
vanishing Feynman propagators and two-point functions for these fields:

$\displaystyle\langle\\!\langle\operatorname{T}\varphi^{+}(x)\,\varphi^{-}(x^{\prime})\rangle\\!\rangle$
$\displaystyle=\Delta^{F}(x-x^{\prime}),$ (A.15)
$\displaystyle\langle\\!\langle\varphi^{+}(x)\,\varphi^{-}(x^{\prime})\rangle\\!\rangle=\langle\\!\langle\varphi^{-}(x)\,\varphi^{+}(x^{\prime})\rangle\\!\rangle$
$\displaystyle=\Delta^{+}(x-x^{\prime}),$ (A.16)
$\displaystyle\langle\\!\langle\operatorname{T}C^{+}(x)\,\bar{C}^{-}(x^{\prime})\rangle\\!\rangle=\langle\\!\langle\operatorname{T}C^{-}(x)\,\bar{C}^{+}(x^{\prime})\rangle\\!\rangle$
$\displaystyle=\Delta^{F}(x-x^{\prime}),$ (A.17)
$\displaystyle\langle\\!\langle
C^{+}(x)\,\bar{C}^{-}(x^{\prime})\rangle\\!\rangle=-\langle\\!\langle\bar{C}^{-}(x)\,C^{+}(x^{\prime})\rangle\\!\rangle$
$\displaystyle=\Delta^{+}(x-x^{\prime}),$ (A.18)
$\displaystyle\langle\\!\langle
C^{-}(x)\,\bar{C}^{+}(x^{\prime})\rangle\\!\rangle=-\langle\\!\langle\bar{C}^{+}(x)\,C^{-}(x^{\prime})\rangle\\!\rangle$
$\displaystyle=\Delta^{+}(x-x^{\prime}).$ (A.19)

##  Appendix B An interesting distribution

In this appendix we study the distribution $f(\rho)$ appearing in the
amplitude of the $h\to\gamma\gamma$ decay via both scalar QED and
flavourdynamics.

To define $\sqrt{\cdot}\colon\mathbb{C}\to\mathbb{C}$ and
$\log\colon\mathbb{C}\to\mathbb{C}$ one uses a cut on the negative real axis:

$\sqrt{r\,e^{i\varphi}}=\sqrt{r}\,e^{i\varphi/2},\quad\log{re^{i\varphi}}=\log{r}+i\varphi,\quad\text{both
with }\varphi\in(-\pi,\pi].$

The complex function

$\tilde{f}:\begin{cases}\mathbb{C}\setminus\bigl{(}(-\infty,0)\cup(1,\infty)\bigr{)}\longrightarrow\mathbb{C}\\\
z\longmapsto-\bigl{(}\log(\sqrt{1-z}+i\sqrt{z})\bigr{)}^{2}\end{cases}$ (B.1)

is analytic, in view of the two cuts on the real axis. The distribution
$f(\rho)$ is defined by

$f:[0,\infty)\longrightarrow\mathbb{C}:\rho\longmapsto
f(\rho):=\tilde{f}(\rho+i0).$ (B.2)

We claim that

$\displaystyle f(\rho)$
$\displaystyle=(\arcsin\sqrt{\rho}\bigr{)}^{2}=\biggl{[}\arctan\frac{\rho}{\sqrt{1-\rho^{2}}}\biggr{]}^{2}\quad\text{for}\quad
0\leq\rho\leq 1,$ (B.3) $\displaystyle f(\rho)$
$\displaystyle=-\frac{1}{4}\,\Biggl{[}\log\frac{1+\sqrt{1-\rho^{-1}}}{1-\sqrt{1-\rho^{-1}}}-i\pi\Biggr{]}^{2}\quad\text{for}\quad\rho\geq
1,$ (B.4)

from which one easily obtains the following formula for the imaginary part:

$\Im
f(\rho)=\theta(\rho-1)\,\frac{\pi}{2}\log\frac{\sqrt{\rho}+\sqrt{\rho-1}}{\sqrt{\rho}-\sqrt{\rho-1}}=-\theta(\rho-1)\,\frac{\pi}{2}\log\bigl{(}2\rho-2\sqrt{\rho(\rho-1)}-1\bigr{)}.$
(B.5)

The first claim (B.3) follows immediately from the identity

$\arcsin\sqrt{\rho}=-i\log\bigl{(}\sqrt{1-\rho}+i\sqrt{\rho}\bigr{)}\quad\text{for}\quad\rho\in[0,1],$

which is obvious from $\exp(i\arcsin x)=\sqrt{1-x^{2}}+ix,\quad|x|\leq 1$.

To prove the second claim (B.4), first note that one has
$\sqrt{1-(\rho+i0)}=-i\sqrt{\rho-1}$ for $\rho\geq 1$. Hence, there holds:

$\displaystyle\log\bigl{(}\sqrt{1-(\rho+i0)}+i\sqrt{\rho}\bigr{)}=\log\bigl{(}\sqrt{\rho}-\sqrt{\rho-1}\bigr{)}+i\pi/2=\frac{1}{2}\bigl{(}\log\bigl{(}(\sqrt{\rho}-\sqrt{\rho-1})^{2}\bigr{)}+i\pi\bigr{)}$
$\displaystyle=\frac{1}{2}\biggl{(}\log\frac{\sqrt{\rho}-\sqrt{\rho-1}}{\sqrt{\rho}+\sqrt{\rho-1}}+i\pi\biggr{)}-\frac{1}{2}\biggl{(}\log\frac{1+\sqrt{1-\rho^{-1}}}{1-\sqrt{1-\rho^{-1}}}-i\pi\biggr{)},$

from which assertion (B.4) follows.

We point out that, for $\rho\in[0,1]$, in the distribution
$F_{0}(\rho)=\rho^{-1}\bigl{(}1-\rho^{-1}f(\rho)\bigr{)}$ in Eq. (1.3) the
terms $\sim\rho^{-1}$ cancel. We bring in the power series expansion

$\displaystyle\arcsin x$ $\displaystyle=x+\frac{x^{3}}{2\cdot
3}+\frac{3\,x^{5}}{2\cdot 4\cdot 5}+\frac{3\cdot 5\,x^{7}}{2\cdot 4\cdot
6\cdot 7}+\cdots\quad\text{for}\quad|x|\leq 1,\quad\text{yielding}\quad$
$\displaystyle f(\rho)$
$\displaystyle=(\arcsin\sqrt{\rho}\,)^{2}=\rho+\frac{\rho^{2}}{3}+\frac{8\rho^{3}}{45}+\cdots\quad\text{so
that}\quad F_{0}(\rho)=-\frac{1}{3}-\frac{8}{45}\,\rho+\cdots\,.$ (B.6)

##  Appendix C Bogoliubov–Epstein–Glaser normalization

Epstein and Glaser [22, 55] started from Bogoliubov’s functional
$\mathbb{S}[g]$-matrix [56, Sect. 21], based on [57] and on previous work by
Stückelberg and Rivier [58]. That is an expansion of operator-valued
distributions (OVD) on configuration space, of the form

$\mathbb{S}[g]=1+\sum_{n=1}^{\infty}\frac{i^{n}}{n!}\int d^{4}x_{1}\cdots
d^{4}x_{n}\,T_{n}(x_{1},\dots,x_{n})\,g(x_{1})\cdots g(x_{n}),\quad
g\in\mathcal{S}({\mathbb{R}}^{4},{\mathbb{R}}).$ (C.1)

We have taken $\hbar=1$. The $g$’s are multiplets of coupling functions which
work as adiabatic cutoffs. The $T_{n}$, 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 $T_{n}$ from natural postulates: the ultraviolet problem is solved in that
construction. In the “adiabatic limit” $g\\!\uparrow\\!1$ the functional
scattering matrix (C.1) is expected to converge to the physical $\mathbb{S}$
in suitable senses [59].

###  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, $T_{1}(x)$
is a Wick polynomial in those and their derivatives – a well-defined
OVD.171717One can think of $T_{1}$ as an “interaction Lagrangian”. However,
the Lagrangian mindset is inessential here. The coupling constants of the
model are included in the $T_{n}$, the expansion being a power series on them.
The other postulates shall enable us to construct the $T_{n}$ from $T_{1}$ by
induction on $n$.

Causality:

This is the key requirement, for which the Epstein–Glaser manufacturing of
TOPs is also called “causal perturbation theory”. Let $V_{\pm}$ and
$\overline{V}_{\pm}$ respectively denote the open forward and backward
lightcones and their closures. If $g_{1},g_{2}$ are such that

$\displaystyle\operatorname{supp}g_{2}\cap\bigl{(}\operatorname{supp}g_{1}+\overline{V}_{-}\bigr{)}=\emptyset,\quad\text{then}\quad\mathbb{S}[g_{1}+g_{2}]=\mathbb{S}[g_{2}]\,\mathbb{S}[g_{1}];$
$\displaystyle\text{equivalently,}\quad
T_{n}(x_{1},\dots,x_{n})=T_{r}(x_{1},\dots,x_{r})\,T_{n-r}(x_{r+1},\dots,x_{n})\quad\text{whenever}$
$\displaystyle\\{x_{1},\dots,x_{r}\\}\cap\bigl{(}\\{x_{r+1},\dots,x_{n}\\}+\overline{V}_{-}\bigr{)}=\emptyset,\text{for
all $r$ and $n$ with $1\leq r\leq n-1$.}$

This is a powerful postulate, called causal factorization. It means that on
large open sets of the $n$-point Minkowski space $({\mathbb{M}}_{4})^{\times
n}\equiv{\mathbb{M}}^{n}$ the TOP $T_{n}$ can be built up from its lower-order
counterparts. In the inductive step of the Epstein–Glaser method, this
requirement uniquely determines $T_{n}$ on the set of Schwartz functions
$\mathcal{S}({\mathbb{M}}^{n}\setminus\Delta_{n})$, in terms of the given
$T_{k}$ at lower orders $k\leq n-1$, where $\Delta_{n}$ is the “thin” diagonal
$\Delta_{n}:=\\{\,(x_{1},\dots,x_{n}):x_{1}=x_{2}=\cdots=x_{n}\,\\}$.
Perturbative normalization is the extension of the operator-valued
distribution $T_{n}$ from
$\mathcal{S}^{\prime}({\mathbb{M}}^{n}\setminus\Delta_{n})$ to
$\mathcal{S}^{\prime}({\mathbb{M}}^{n})$. 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 $\Delta_{n}$. 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 $T_{1}(x)=\varphi^{k}(x)$, for $\varphi$ a
real scalar field:

$\displaystyle
T_{n}\bigl{(}\varphi^{k}(x_{1}),\dots,\varphi^{k}(x_{n})\bigr{)}$
$\displaystyle\enspace=\sum_{l_{1},\dots,l_{n}=0}^{k}\binom{k}{l_{1}}\cdots\binom{k}{l_{n}}\langle\\!\langle
T_{n}(\varphi^{k-l_{1}}(x_{1}),\dots,\varphi^{k-l_{n}}(x_{n}))\rangle\\!\rangle\,\varphi^{l_{1}}(x_{1})\cdots\varphi^{l_{n}}(x_{n})$

with $\langle\\!\langle\cdots\rangle\\!\rangle$ denoting vacuum expectation
value. This postulate reduces the extension problem for the OVD
$T_{n}(\cdots)$ to one of numerical distributions – a simpler task.

Poincaré Covariance:

Let there be given the standard lifting $U(a,\Lambda)$ to Fock space of the
Poincaré unitary irreducible representations (unirreps) on $1$-particle
subspaces. Then

$U(a,\Lambda)\,\mathbb{S}[g]\,U^{\dagger}(a,\Lambda)=\mathbb{S}\bigl{[}(a,\Lambda)\cdot
g\bigr{]},$

where $((a,\Lambda)\cdot g)(x)=g(\Lambda^{-1}(x-a))$. 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
$\mathcal{S}^{\prime}({\mathbb{R}}^{4(n-1)}\setminus\\{0\\})$ to
$\mathcal{S}^{\prime}({\mathbb{R}}^{4(n-1)})$.

Unitarity (conservation of probability):

$\displaystyle\mathbb{S}[g]\,\mathbb{S}^{\dagger}[g]=\mathbb{S}^{\dagger}[g]\,\mathbb{S}[g]=1;\quad\text{here
we denote:}$
$\displaystyle\mathbb{S}^{-1}[g]=:1+\sum_{n=1}^{\infty}\frac{(-i)^{n}}{n!}\int
d^{4}x_{1}\cdots
d^{4}x_{n}\,\overline{T}_{n}(x_{1},\dots,x_{n})\,g(x_{1})\cdots g(x_{n}).$

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
$\operatorname{sd}(t)$ and the singular order $\omega(t)$ of a distribution
$t\in\mathcal{S}^{\prime}({\mathbb{R}}^{k})$ or
$t\in\mathcal{S}^{\prime}({\mathbb{R}}^{k}\setminus\\{0\\})$ – see, e.g., [24,
Sect. 3.2.2] – are as follows:

$\operatorname{sd}(t):=\inf\\{\,r\in{\mathbb{R}}:\lim_{\lambda\downarrow
0}\lambda^{r}\,t(\lambda x)=0\,\\},\quad\omega(t):=\operatorname{sd}(t)-k,$
(C.2)

where $\inf\,\emptyset:=\infty$ and $\inf\,{\mathbb{R}}:=-\infty$. For
instance, for a translation-invariant distribution
$d(x_{1}-x_{3},x_{2}-x_{3})\in\mathcal{S}^{\prime}({\mathbb{R}}^{8})$
fulfilling $\operatorname{sd}(d)=8$, equivalently $\omega(d)=0$, 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
[60]. A Ward identity playing a paramount role in this paper corresponds to
EGI – see subsections 3.1 and 4.2 for this. For different types of
requirements, consult subsections 4.4 and 4.5.

###  C.2 Iterative building of the time-ordered products

To assemble the $T_{n}$ outside of the thin diagonal $\Delta_{n}$ from the
inductively known $(T_{k})_{1\leq k\leq n-1}$ directly by causal
factorization, one would need a partition of unity subordinate to an open
cover of ${\mathbb{M}}^{n}\setminus\Delta_{n}$ – see [61] and [24, Sect. 3.3].
This is problematic for practical computations. For this reason the original
Epstein–Glaser construction [22, 40] is less direct: it introduces an
intermediate $D_{n}$-distribution having causal support; and the crucial step
is the splitting of $D_{n}$ into 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
$\overline{T}_{n}$ in terms of the TOPs $(T_{k})_{1\leq k\leq n}$. Let
$N=\\{x_{1},\dots,x_{n}\\}$ and $I\subseteq N$ with $|I|\neq 0$ elements.
Define $T_{|I|}(I)=T_{|I|}(x_{i}:x_{i}\in I)$. By the standard inversion of a
formal power series with noncommuting terms in terms of set compositions, we
obtain

$\overline{T}_{|N|}(N)=\sum_{k=1}^{n}(-)^{n+k}\sum_{I_{1}\uplus\cdots\uplus
I_{k}=N}T_{|I_{1}|}(I_{1})\cdots T_{|I_{k}|}(I_{k}),$ (C.3)

where the disjoint union is over nonempty blocks $I_{j}$. The terminology of
antichronological products is appropriate, since if
$I\cap(J+\overline{V}_{-})=\emptyset$, then $\overline{T}(I\cup
J)=\overline{T}(J)\,\overline{T}(I)$.

Retarded and advanced products, denoted by $R_{n}$ and $A_{n}$ 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
[24], identical to that of [22] except that $R_{n}$ and $A_{n}$ have an extra
factor $i^{n-1}$. In general, Bogoliubov’s definitions read:

$\displaystyle R_{n+1}(x_{1},\dots,x_{n+1})$
$\displaystyle:=i^{n}\sum_{I\subset\\{1,\dots,n\\}}(-1)^{|I|}\,\overline{T}_{|I|}(I)\,T_{|I^{c}|+1}(I^{c},x_{n+1}),$
(C.4) $\displaystyle A_{n+1}(x_{1},\dots,x_{n+1})$
$\displaystyle:=i^{n}\sum_{I\subset\\{1,\dots,n\\}}(-1)^{|I|}\,T_{|I^{c}|+1}(I^{c},x_{n+1})\,\overline{T}_{|I|}(I),$
(C.5)

where $I^{c}:=\\{1,\dots,n\\}\setminus I$. Epstein and Glaser [22] prove that
$A_{n+1},\,R_{n+1}$ have advanced or retarded support, respectively:

$\displaystyle\operatorname{supp}A_{n+1}$
$\displaystyle\subseteq\\{\,x\in{\mathbb{M}}^{n+1}:x_{j}-x_{n+1}\in\overline{V}_{+}\
\forall j\,\\};$ $\displaystyle\operatorname{supp}R_{n+1}$
$\displaystyle\subseteq\\{\,x\in{\mathbb{M}}^{n+1}:x_{j}-x_{n+1}\in\overline{V}_{-}\
\forall j\,\\}.$

In the induction step $n\to n+1$ neither the $T_{n+1}$ nor the $R_{n+1}$ nor
the $A_{n+1}$ are known. But by the induction hypothesis the difference
$D_{n+1}$, defined by $D_{n+1}:=A_{n+1}-R_{n+1}$, only depends on known
quantities. For instance, in $D_{3}$ the unknown $T_{3}$ has dropped out – and
$\overline{T}_{1}$, $\overline{T}_{2}$ are uniquely given in terms of $T_{1}$
and $T_{2}$. It follows that $D_{n+1}$ has causal support:

$\operatorname{supp}D_{n+1}\subseteq\\{\,x\in{\mathbb{M}}^{n+1}:x_{j}-x_{n+1}\in\overline{V}_{+}\
\forall
j\,\\}\cup\\{\,x\in{\mathbb{M}}^{n+1}:x_{j}-x_{n+1}\in\overline{V}_{-}\
\forall j\,\\}.$

If one finds a way to extract the advanced part $A_{n+1}$ of $D_{n+1}$, that
is, to split the OVD $D_{n+1}$ into $A_{n+1}$ and $-R_{n+1}$ in such a way
that the latter two satisfy the just given support properties, then one can
construct a candidate for $T_{n+1}$.181818That sometimes needs to be
symmetrized, by adding a suitable OVD supported on $\Delta_{n+1}$.

For the sake of normalization conditions, at this stage we may add to
$T_{n+1}$ any OVD supported on $\Delta_{n+1}$ which is symmetric in
$x_{1},\dots,x_{n+1}$. The $D_{n+1}$ fulfils all the normalization conditions,
in particular the ‘Causal Wick expansion’ and ‘Translation invariance’,
because of the validity of those for the inductively given $(T_{k})_{1\leq
k\leq n}$. Therefore, the splitting problem for $D_{n+1}$ translates into a
consonant problem for the coefficients
$d(x_{1}-x_{n+1},\dots,x_{n}-x_{n+1})\in\mathcal{S}^{\prime}({\mathbb{R}}^{4n},\mathbb{C})$
in the Wick expansion of $D_{n+1}$, yielding
$a,r(x_{1}-x_{n+1},\dots,x_{n}-x_{n+1})\in\mathcal{S}^{\prime}({\mathbb{R}}^{4n},\mathbb{C})$,
which are the coefficients in the Wick expansion of $A_{n+1}$ and $R_{n+1}$,
respectively.

In fine, by the induction process, one specifies the ambiguity in the vacuum
expectation value of each $T_{n+1}$ by adding to it a contact term, that is,

$t(x_{1}-x_{n+1},\dots,x_{n}-x_{n+1})+\sum_{|a|\leq\omega}c_{a}\,\partial^{a}\delta(x_{1}-x_{n+1},\dots,x_{n}-x_{n+1}),$
(C.6)

where $\omega$ is the singular order of the pertinent $d(x_{1}-x_{n+1},\dots)$
and the coefficients $c_{a}\in\mathbb{C}$ depending on the multi-index $a$ are
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 transform of
$f\in\mathcal{S}\bigl{(}{\mathbb{R}}^{8}\bigr{)}$ we employ the following
convention:

$\displaystyle f(y_{1},y_{2})=(2\pi)^{-4}\int
dk_{1}\,dk_{2}\,e^{-i(k_{1}y_{1}+k_{2}y_{2})}\,\hat{f}(k_{1},k_{2}).$ (C.7)

Let $\Gamma_{\pm}:=\overline{V}_{\pm}\times\overline{V}_{\pm}$ henceforth.
Given a “causal distribution”, that is,
$d\in\mathcal{S}^{\prime}({\mathbb{R}}^{8})$ with

$\operatorname{supp}d\subseteq\Gamma_{+}\cup\Gamma_{-}\quad\text{and}\quad\operatorname{sd}(d)<\infty,$
(C.8)

by a splitting solution of $d$ we mean a distribution
$a\in\mathcal{S}^{\prime}({\mathbb{R}}^{8})$ with

$(a-d)\bigr{|}_{\mathcal{S}({\mathbb{R}}^{8}\setminus\Gamma_{-})}=0,\quad\operatorname{supp}a\subseteq\Gamma_{+}\quad\text{and}\quad\operatorname{sd}(a)\leq\operatorname{sd}(d).$
(C.9)

In what follows we assume that the Fourier transform $\hat{d}$ of the causal
$d$-distribution we wish to split vanishes in an open ball
$\mathcal{R}\subset{\mathbb{R}}^{8}$ centered at $k=0$. This holds if all
propagators contributing to $d$ are massive, as it is the case in this paper –
see [22, Sect. 5.2]. Also in [22] it is shown for any splitting solution $a$
that $\hat{d}|_{\mathcal{R}}=0$ entails analyticity of $\hat{a}(k)$ on
$\mathcal{R}$. In this case there exists a distinguished splitting solution,
the so-called central solution $a^{c}$, characterized by the conditions

$\partial^{a}\hat{a}^{c}(0)=0,\quad\text{for all}\quad|a|\leq\omega(d).$
(C.10)

As indicated in Eq. (C.6), for $\operatorname{sd}(d)\geq 8$ – i.e., for
$\omega(d)\geq 0$, as defined in Eq. (C.2) – the splitting solution of $d$ is
not uniquely determined. Any two solutions $a_{1}$ and $a_{2}$ differ by

$a_{1}(y)-a_{2}(y)=\sum_{|a|=0}^{\omega(d)}C_{a}\,\partial^{a}\delta(y)\quad\text{or
equivalently,}\quad\hat{a}_{1}(k)-\hat{a}_{2}(k)=\frac{1}{(2\pi)^{4}}\sum_{|a|=0}^{\omega(d)}C_{a}\,(-ik)^{a},$

with arbitrary constants $C_{a}\in\mathbb{C}$.

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.191919For further detail we refer to [40, Sect. 3.2], which relies on
[22, Sect. 6.5]. The naive way to extract the advanced part $a$ of $d$ is to
multiply the latter by a $\theta$-function:

$a_{\mathrm{naive}}(y_{1},y_{2}):=d(y_{1},y_{2})\,\chi(y_{1},y_{2})\quad\text{with}\quad\chi(y_{1},y_{2}):=\theta\bigl{(}(y_{1}v_{1})+(y_{2}v_{2})\bigr{)},$

where $v:=(v_{1},v_{2})\in V_{+}\times V_{+}$ is arbitrary. But for
$\operatorname{sd}(d)\geq 8$ the pointwise product $d\chi$ exists only as an
element of $\mathcal{S}^{\prime}({\mathbb{R}}^{8}\setminus\\{0\\})$.
Therefore, the splitting problem is an extension problem: we have to extend
$d\chi\in\mathcal{S}^{\prime}({\mathbb{R}}^{8}\setminus\\{0\\})$ to an
$a\in\mathcal{S}^{\prime}({\mathbb{R}}^{8})$ such that
$\operatorname{sd}(a)=\operatorname{sd}(d\chi)=\operatorname{sd}(d)$.202020A
priori, it might happen that $\operatorname{sd}(d\chi)<\operatorname{sd}(d)$;
but in the applications to Epstein–Glaser normalization known to us one always
finds $\operatorname{sd}(d\chi)=\operatorname{sd}(d)$. Hence we assume the
latter relation to hold true.

The problem is studied in its particulars in [24, Sect. 3.2.2]. Given $a$ with
singular order $\omega$, there exists an obvious extension $a^{\omega}$,
belonging in the dual space $\mathcal{S}_{\omega}^{\prime}({\mathbb{R}}^{8})$
of

$\mathcal{S}_{\omega}({\mathbb{R}}^{8}):=\\{\,f\in\mathcal{S}({\mathbb{R}}^{8}):\partial^{b}f(0)=0~{}~{}\text{for
all}~{}~{}|b|\leq\omega\,\\},$