# Amplitude of $H\to\gamma Z$ process via one W loop in unitary gauge
(I. Details of calculation with Dyson scheme)

Shi-Yuan Li  Institute of Theoretical Physics, Shandong University, Jinan
250100, P. R. China

###### Abstract

Decay amplitude of $H\to\gamma Z$ process via one W loop in the unitary gauge
is presented. The divergent integrals including those of high divergence
orders typical of unitary gauge are arranged to cancel to get the
electromagnetic $U(1)$ gauge invariant finite result, hence no contribution to
the renormalization constant of $Z\gamma$-mixing in this 1-loop subprocess.
For the calculation of the Feynman diagrams employing the Feynman rules, all
the integrations of the propagator momenta and all the $\delta$-functions
representing the 4-momentum conservation of every vertex are retained in the
beginning. Therefore, the ambiguity of setting independent loop momentum for
divergences worse than logarithmic does not exist, and shift of integrated
variable in such divergent integrals is eschewed. The calculation are done in
4-dimension Minkowski momentum space without the aid of any regularization.
The correct treatment on the surface terms for the quadratic and logarithmic
tensor integral is one of the key points.

This part I is devoted to the calculation details and the indications from the
key surface terms. Comparing with other gauge(s) and complete results for
$H\to\gamma Z$ are left for part II.

##  1 Introduction

The Glashow-Weinberg-Salam electroweak (EW) theory is a SU(2)$\times$U(1)
Yang-Mills gauge field theory, with the gauge symmetry ’broken’ by a scalar
field via the Englert-Brout-Higgs-Guralnik-Hagen-Kibble Mechanism and the
scalar field coupling the fermion field in Yukawa style provides the mass term
which distinguishes various SU(2)-doublet fermions. This theory has been
confirmed from experiment in the sense that the massive $W^{\pm}$ $Z$
particles v.s. the massless photon, and a ’remaining’ neutral scalar particle
which is generally referred to as the Higgs or the ’God’ particle, all are
well measured. In general, a realistic calculation of the S-matrix or
scattering/decay amplitude employing the quantized field theory of the
standard model need to fix a specific gauge and it is adopted that the
physical result should be independent from the choice (artificial rather than
by nature) of the gauge. However, recently, careful revisit on the
$H\to\gamma\gamma$ decay width in the unitary gauge and the $R_{\xi}$ gauge
Wu:2017rxt ; Wu:2016nqf ; Gastmans:2015vyh ; Gastmans:2011ks ; Gastmans:2011wh
would like to imply some paradox. For a review and remarks on the
uncertainties in this paradox, see e.g., Duch:2020was . In all ways this
paradox calls for calculations in the unitary gauge for loop diagrams to be
extensively studied. Many topics have been suggested Wu:2017rxt , and one of
them is the $H\to\gamma Z$ process via one W loop. Though less experimentally
significant Aad:2014fia ; Chatrchyan:2013vaa , it can also be an important
example to investigate in the unitary gauge and the $R_{\xi}$ gauge to gain
insights for the paradox.

It is well known that unitary gauge can be taken as a limit of the general
$R_{\xi}$ gauge (but can be defined independently wein73 ) that does not
commute with the loop integrations Wu:2017rxt , so that a lot of care has to
be taken when applied to loop calculations. In the above mentioned gauge
(non)invariance paradox, several uncertainties could arise from the (maybe)
non-commutability of various limitations Duch:2020was . Besides, high
divergence order integrals are one of the difficulties. Similar as
$H\to\gamma\gamma$ process via W loop, $H\to\gamma Z$ process via one W loop
in the unitary gauge also has many high divergence order integrals for each
single diagrams and should properly cancel, or else one can not get the
correct result, either not possible to make the comparison with results from
other gauges. Terms proportional to $M_{Z}^{2}$, not encountered in
$H\to\gamma\gamma$ process, cause new difficulties. The purpose of this paper
is to apply the experiences obtained from the investigation on the
$H\to\gamma\gamma$ process via one W loop Li:2017hnv , i.e., without the
setting of independent integrated loop momentum in the beginning and eschewing
the shift of integrated variable for high order divergences, to provide the
systematic framework to give the finite and electromagnetic U(1) gauge
invariant amplitude of the $H\to\gamma Z$ process via one W loop for further
study.

For any diagram whose divergence order higher than logarithmic, to shift the
integral momentum can lead to extra terms with lower divergence (or finite).
In such case, the proper set of diagrams with correct inter-relations of the
loop momenta must be treated together to get the correct result, as pointed by
Gastmans:2011ks ; Gastmans:2011wh  (in the following we refer to these two
papers and works therein as GWW). Only a part of diagrams of the set shifting
the momenta will change the result. This problem can be solved by the the
original Dyson formulation, ’Dyson scheme’ as called in Li:2017hnv , without
the ambiguity of setting independent loop momentum in the beginning, and shift
of integrated variable in high divergence order integrals can be, and is,
eschewed. Correspondingly, our cancellation of divergences are all at integral
level rather than integrand level. However, a wise setting of independent
integrated momentum as GWW or employing Dyson scheme Li:2017hnv  is inadequate
to definitely determine the final result in the unitary gauge because of the
presence of the surface term for the reduction of the divergent tensor
integral. In $H\to\gamma Z$ process, the same logarithmic divergent tensor
integral appears as in $H\to\gamma\gamma$ Li:2017hnv ; but there is also the
new quadratic ones, especially for terms proportional to $M_{Z}^{2}$, which
will be investigated in this paper.

The considerations and key points of employing Dyson scheme are as following:

To think that a high divergence order ($>$ logartithmic) integral from the
Feynman diagram is changed when shifting the integrated momentum, one
inevitably raises the question what is the ’original’ expression/value to be
changed? There may not exist the ’original’ one for a single diagram, once
considering that different Feynman diagrams are related and hence the loop
momenta (à la GWW). However, one can have the definiteness starting from the
original form derived from the perturbative expansion of the S-matrix
according to the standard Dyson-Wick procedure Dyson:1949ha , which is
integrations on space-time at each perturbative order. Once taking these
space-time integrations 111leaving out the integrations on momenta from each
propagators; this in fact exchanging order of integration, between the phase
space and configuration space., we get $\delta$ functions, one for each
vertex, relating all the momenta of the propagators with energy-momentum
conservation Dyson:1949ha . If we start from such a form for each diagram,
without integrating the propagator momenta and $\delta$ functions, there will
be no indefiniteness, or ambiguity to set independent integral momentum. This
corresponds to that the momentum space Feynman rules are slightly modified
222in fact ’recovered’, see the classical paper of Dyson Dyson:1949ha ,
especially its Eq. (20) and discussions before and after it. as: Any
propagator with momentum $q$ has an extra $[\int\frac{d^{4}q}{(2\pi)^{4}}]$
’operator’, i.e., should this integration on $q$ to be done in the calculation
of the Feynman diagram; any vertex has an extra factor
$(2\pi)^{4}\delta(\sum_{i}q_{i})$, with $q_{i}$, each momentum of all the
propagators attaching the vertex, incoming. In this paper we adopt this way to
write the amplitude corresponding to each Feynman diagram for calculation. The
method has been proved to be valid and feasible based on our investigation on
the $H\to\gamma\gamma$ process via one W loop in the unitary gauge Li:2017hnv
. Calculation in this way can also help us to eschew the shift of integrated
variables in high order divergences. For the present $\gamma Z$ case in
unitary gauge this seems the only practical way. In Wu:2017rxt , to eliminate
uncertainties, the authors suggested to calculate the difference between
unitary gauge and $R_{\xi}$ gauge by first to calculate the difference of the
integrand and then to do the loop integration. Since the difference of the
integrand still leads to high order divergences, the choice of integrated
momenta has also to be treated in the above suggested way to eliminated
ambiguity. There is another advantage in applying the Dyson scheme when
treating the surface term for the divergent tensor integral reduction. Since
the integrated momentum is just the one in the originally-defined Feynman
propagator, its natural physical boundary condition can be used to determine
the surface term (to be zero).

In the following section 2, we illustrate the details of the calculations in
the way mentioned above. The result is finite, $U_{EM}(1)$ invariant, without
the need of the Dyson subtraction with the correct treatment on the
logarithmic and quadratic surface terms. No regularization is introduced and
all calculations are done in the 4-dimension Minkowski momentum space. During
the calculation, only real convergence or real logarithmic divergence need not
to eschew shift. As is also drawn attention by Li:2017hnv , especially for
terms with odd power of momentum (fermion propagator is another example), it
has long been noticed that the real divergence can be worse than simple power
counting once all possible ways for momentum to infinity considered—this is
just the condition when we do not beg for regularization.

The physical implication of the finite terms proportional to $M_{Z}^{2}/M^{4}$
after all divergences from quadratic to logarithmic cancelled is very
interesting as well. In $\gamma\gamma$ case, the $M^{-2}$ term not zero is
explained because of the goldstone effects, which is equivalently exposed in
unitary gauge by the corresponding terms in W propagator (and going to the
final result once the surface term of logarithmic tensor reduction and
cancellation is properly treated). In the $\gamma Z$ case, the Z particle
includes transverse as well as longitudinal components, which can expose more
longitudinal (goldstone) effects from the W propagator. This is true because
of the extra effects proportional to $M^{2}_{Z}$.

Besides the success of calculating these H decay channels, this way also shows
power to discuss other important issues. For example, in Bao:2021byx  we
demonstrate that it is very easy to obtain both the vector current and axial
vector current conservations at the same time via this Dyson scheme 333It has
been long recognized, both current conservations can be obtained at the same
time via setting ’the most symmetric loop momentum’ as in Wu:2017rxt ;
Wu:2016nqf ; Gastmans:2015vyh ; Gastmans:2011ks ; Gastmans:2011wh . The author
thanks Prof. T.T. Wu for informing this fact., contrary to the Bell-Jackiw
claim. From such investigations, one also recognizes the important rôle of the
infinite momentum surface integrals. So we make some discussions on the
results, the Dyson scheme, the surface terms, and the physical implication on
probing the structures and properties of space-time in section 3.

This part I is devoted to the calculation details to get the electromagnetic
U(1) gauge invariant result and physical indication from this calculation
procedure, especially the the divergence cancelation from the surface term.
Comparing with other gauge(s) and complete results for $H\to\gamma Z$ are left
for part II.

##  2 Calculation

### 2.1

There are totally 5 Feynman diagrams. Three are similar as those of the
$H\to\gamma\gamma$ process, via the direct coupling of Higgs and W-boson loop.
We will demonstrate the calculations in details. The other two are via the
direct 3-point coupling of Higgs and ZZ, while one Z transiting to gamma via
the W bubble (3-point vertices) and W tadpole (4-point vertex). However, these
latter two diagrams sum to be zero Wu:2017rxt , and the details of the
calculation will not be presented in this paper. In this calculation the same
surface term of the quadratic tensor integral is encountered, which will be
investigated in details for the former three diagrams in the following 444As a
matter of fact, these diagrams are self-energy like diagrams, and by Lorentz
and $U(1)_{EM}$ gauge invariant arguments should be proportional to
$k_{1}^{2}g_{\mu\alpha}-k_{1\mu}k_{1\alpha}$ ($\mu$ the on shell photon index,
$\alpha$ some dummy index), so are zero. But just the same as the QED photon
self-energy diagram, direct calculation is non-trivial. The superficial
quadratic latter also need the correct quadratic surface term to get the
’proper’ form, which was regarded only available via a gauge invariant
regularization. With the same argument, and also can be shown explicitly by
the help of the same quadratic surface term, the fermion bubble of $\gamma Z$
also zero since $k_{1}$ on shell, i.e., $k_{1}^{2}=0,k_{1}\cdot\epsilon=0$.
The application to the fermion case in fact is even more useful since here we
can freely work in 4 dimension Minkowski momentum space without the ambiguity
of $\gamma^{5}$.. This zero result together with the finite result of the
former three diagrams show in this 1-loop subprocess no contribution to the
$Z\gamma$ mixing renormalization constant.

Figure 1 showes the three diagrams to be calculated in the following, $k_{1}$
is the 4-momentum of $\gamma$. $k_{2}$ is the 4-momentum of the Z particle,
$k_{2}^{2}=M_{Z}^{2}$, with the corresponding polarization vectors
$\epsilon_{Z\lambda}^{\nu}$, $\lambda=1,2,3$, since Z is massive. We still
have $k_{2\nu}\epsilon_{Z\lambda}^{\nu}=0,~{}\forall\lambda$ (see Eq. 87).
There is also an extra $\cot\theta_{W}$ factor for the WWZ vertex, where
$\theta_{W}$ is the Weinberg angle.

[0.3] | [0.3] |
---|---|---
($T_{1}$) | ($T_{2}$) |
[0.33] |  |
($T_{3}$) |  |
Figure 1: The one W loop Feynman diagrams with non-zero contribution to the
process $H\to\gamma Z$. Taking the waving line with momentum $k_{2}$
represents the Z particle with polarization vectors
$\epsilon_{\lambda}^{\nu}$. In general, the inner integrated momenta should be
considered as not correlated between different diagrams, so here we mark those
of $T_{2}$ and $T_{3}$ with prime or double primes. In the manuscript, this is
also implied though they are written as the same in the concrete step of
derivation. Since the cancellation is determined by the whole integral (with
the integrand including the $\delta$ functions), this way of writing dummy
variables does not lead to ambiguity and is easily to be tracked.

As convention, The S-matrix and T-matrix have the relation $S=I+iT$, and the
matrix element between initial and final states
$iT_{fi}=i(2\pi)^{4}\delta(P_{f}-P_{i})\mathfrak{M}_{fi}$ for the space-time
displacement invariant case. Here we keep all the momenta respectively
corresponding to each propagator and hence all $\delta$ functions respectively
corresponding to each vertex. The one corresponding to the initial-final state
energy momentum conservation is contained in these $\delta$ functions. After
integrating over them, one will get the above form of T-matrix element with
the $\mathfrak{M}_{fi}$ is the integration of the independent loop momenta
only, without the $\delta$ functions attached to the vertices. This is the
standard procedure in developing the Dyson-Wick perturbation theory in the
interaction picture Dyson:1949ha . The four-momentum conservation $\delta$
function attached to each of the vertices is the result of integration of
space-time variables in the perturbative expansion of the S-matrix, and is the
manifestation of space-time displacement invariance 555This requires not only
the boundary of space-time at infinity trivial but also no point or structure
singular to ’block’ the displacement. We conjecture this may be the condition
to exchange the integration order of configuration space and momentum space,
see footnote 1. We also would like to point out ”no point or structure
singular to ’block’ the displacement” guarantees the investigation on the
surface term at infinity in momentum space Li:2017hnv .. In the following, we
do not integrate out the $\delta$ functions of each vertex until have to and
is allowed to integrate out some of the momenta with corresponding $\delta$
functions (only for ’hidden’ case, see the following). So here we deal with
the matrix elements $T_{fi}$ rather than $\mathfrak{M}_{fi}$:

$\displaystyle T_{1}$ $\displaystyle=$
$\displaystyle\frac{-ie^{2}gM\cot\theta_{W}}{(2\pi)^{4}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
(1) $\displaystyle\times$
$\displaystyle(g_{\alpha}~{}^{\beta}-\frac{q_{1\alpha}q_{1}^{\beta}}{M^{2}})(g^{\rho\sigma}-\frac{q_{3}^{\rho}q_{3}^{\sigma}}{M^{2}})(g^{\alpha\gamma}-\frac{q_{2}^{\alpha}q_{2}^{\gamma}}{M^{2}})\frac{V_{\beta\mu\rho}(q_{1},-k_{1},-q_{3})~{}V_{\sigma\nu\gamma}(q_{3},-k_{2},-q_{2})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})},$
$\displaystyle T_{2}$ $\displaystyle=$
$\displaystyle\frac{ie^{2}gM\cot\theta_{W}}{(2\pi)^{4}}\int
d^{4}q_{1}d^{4}q_{2}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-q_{2}-k_{1}-k_{2})$
(2) $\displaystyle\times$
$\displaystyle(g_{\alpha}~{}^{\beta}-\frac{q_{1\alpha}q_{1}^{\beta}}{M^{2}})(g^{\alpha\gamma}-\frac{q_{2}^{\alpha}q_{2}^{\gamma}}{M^{2}})\frac{2g_{\mu\nu}g_{\beta\gamma}-g_{\mu\beta}g_{\nu\gamma}-g_{\mu\gamma}g_{\nu\beta}}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})},$
$\displaystyle T_{3}$ $\displaystyle=$
$\displaystyle\frac{-ie^{2}gM\cot\theta_{W}}{(2\pi)^{4}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{2}-q_{3})\delta(q_{3}-k_{1}-q_{2})$
(3) $\displaystyle\times$
$\displaystyle(g_{\alpha}~{}^{\beta}-\frac{q_{1\alpha}q_{1}^{\beta}}{M^{2}})(g^{\rho\sigma}-\frac{q_{3}^{\rho}q_{3}^{\sigma}}{M^{2}})(g^{\alpha\gamma}-\frac{q_{2}^{\alpha}q_{2}^{\gamma}}{M^{2}})\frac{V_{\beta\nu\rho}(q_{1},-k_{2},-q_{3})~{}V_{\sigma\mu\gamma}(q_{3},-k_{1},-q_{2})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})}.$

Here we do not explicitly write the matrix element subscript$~{}_{fi}$, and
all the $\delta$ fuctions are understood as four-dimensional one, i.e.,
$\delta(P_{1}-P_{2}):=\delta^{4}(P_{1}-P_{2})$. As GWW, we also omit the
polarization vector, so, e.g., $T_{1}$ should be understood as $T_{1\mu\nu}$.
In all this paper, we use M to represent the W mass. The relation between
$T_{1}$ and $T_{3}$, i.e., $\mu\leftrightarrow\nu$, and $k_{1}\leftrightarrow
k_{2}$ is clear to be read out.

To better illustrate the calculation procedure, we list the formulae we use
repeatedly in Appendix A. They correspond to Eqs. (2.5-2.12) of GWW
Gastmans:2011wh  and can go to GWW (2.5-2.12) by simply taking $M_{Z}=0$ since
there all for photons. Similarly as the $H\to\gamma\gamma$ process, the
property of 3-particle vertex, WW$\gamma$ or WWZ, which is named as Ward
Identity (W.I.) in GWW, is the key role in the evaluation, because the extra
terms $q^{\alpha}q^{\beta}/M^{2}$ in W propagator product with the vertex is
typical of the unitary gauge. There is more subtle elements in considering
these W.I.’s, since they demonstrate the special relations of the various
propagator momenta provided by the concrete dynamics in the standard mode.
Furthermore, the equations (A9), (A10), and first and third terms of (A7),
(A8) are independent of the integrated momentum choice (the
$(q_{3}^{2}-M^{2})$ term reduces with one from the denominator, so in fact
corresponding to a $g_{\mu\nu}$ term).

In the following, we investigate the terms according to their minus power of
M. The $\frac{-ieg^{2}M\cos\theta_{W}}{(2\pi)^{4}}$ factor will not explicitly
written, and all terms should multiply with this factor to get the proper
terms in the corresponding T amplitude of Eqs. (1-3). So when we mention a
term, we do not take into account the overall M factor coming from the HWW
coupling except explicitly addressed. Since there is still a WW$\gamma$
vertex, i.e., $V_{\beta\mu\rho}(q_{1},-k_{1},-q_{3})$ of $T_{1}$,
$V_{\sigma\mu\gamma}(q_{3},-k_{1},-q_{2})$ of $T_{3}$, respectively, the
$M^{-6}$ terms in $T_{1}$ and $T_{3}$ are again zero as the $H\to\gamma\gamma$
process, according to Eq. (95).

### 2.2

For the $M^{-4}$ terms, in $T_{1}$ and $T_{3}$, respectively, there are 3 ways
of the combination of two of three $q_{i}^{\alpha_{i}}q_{i}^{\beta_{i}}/M^{2}$
($i=1,2,3$) from the W propagators. One combination is zero because of the
W.I. of the WW$\gamma$ vertex, Eq. (95). Another gives a $M_{Z}^{2}/M^{4}$
terms because of the W.I. of the WWZ vertex, Eq. (96). The third corresponds
to those of the $H\to\gamma\gamma$ process, gives likely terms besides extra
$M_{Z}^{2}/M^{4}$ terms.

Those from $T_{1}$:

$\displaystyle T^{\prime\prime}_{11}$ $\displaystyle=$
$\displaystyle\frac{1}{M^{4}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
(4) $\displaystyle\times$ $\displaystyle
q_{1\alpha}q_{1}^{\beta}q_{3}^{\rho}q_{3}^{\sigma}g^{\alpha\gamma}\frac{V_{\beta\mu\rho}(q_{1},-k_{1},-q_{3})~{}V_{\sigma\nu\gamma}(q_{3},-k_{2},-q_{2})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})}=0,$
$\displaystyle T^{\prime}_{11}$ $\displaystyle=$
$\displaystyle\frac{1}{M^{4}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
(5) $\displaystyle\times$ $\displaystyle
g_{\alpha}~{}^{\beta}q_{3}^{\rho}q_{3}^{\sigma}q_{2}^{\alpha}q_{2}^{\gamma}\frac{V_{\beta\mu\rho}(q_{1},-k_{1},-q_{3})~{}V_{\sigma\nu\gamma}(q_{3},-k_{2},-q_{2})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})}$
$\displaystyle=$ $\displaystyle\frac{M_{Z}^{2}}{M^{4}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
$\displaystyle\times$
$\displaystyle\frac{(-q_{1}^{2}q_{2\mu}q_{3\nu}+q_{1}\cdot
q_{2}q_{1\mu}q_{3\nu})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})},$

employing the corresponding W.I.’s of Appendix A. Obviously,
$T^{\prime}_{11}=0$ if $M_{Z}=0$, which is consistent with the
$H\to\gamma\gamma$ case.

$\displaystyle T_{11}$ $\displaystyle=$ $\displaystyle\frac{1}{M^{4}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
(6) $\displaystyle\times$ $\displaystyle
q_{1\alpha}q_{1}^{\beta}g^{\rho\sigma}q_{2}^{\alpha}q_{2}^{\gamma}\frac{V_{\beta\mu\rho}(q_{1},-k_{1},-q_{3})~{}V_{\sigma\nu\gamma}(q_{3},-k_{2},-q_{2})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})}.$

Now since
$q_{1}^{\beta}V_{\beta\mu\rho}(q_{1},-k_{1},-q_{3})=(q_{3}^{2}-M^{2})g_{\mu\rho}-q_{3\mu}q_{3\rho}+M^{2}g_{\mu\rho}$,
according to (93), $T_{11}=T_{111}+T_{112}+T_{113}$. For this kind of step we
employ the W.I. for WW$\gamma$ (A7) in priority than WWZ vertex (A8) because
it straightforwardly gives the proper minus power of M for each term.

$\displaystyle T_{113}$ $\displaystyle=$ $\displaystyle\frac{M^{2}}{M^{4}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
(7) $\displaystyle\times$ $\displaystyle q_{1}\cdot
q_{2}\frac{V_{\mu\nu\gamma}(q_{3},-k_{2},-q_{2})q_{2}^{\gamma}}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})}$

is obviously $M^{-2}$ term and to be discussed with other $M^{-2}$ terms
later.

$\displaystyle T_{112}$ $\displaystyle=$ $\displaystyle\frac{1}{M^{4}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
(8) $\displaystyle\times$ $\displaystyle q_{1}\cdot
q_{2}q_{3\mu}\frac{q_{3}^{\sigma}V_{\sigma\nu\gamma}(q_{3},-k_{2},-q_{2})(-q_{2}^{\gamma})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})}$
$\displaystyle=$ $\displaystyle\frac{M_{Z}^{2}}{M^{4}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
$\displaystyle\times$ $\displaystyle\frac{q_{1}\cdot
q_{2}q_{3\mu}q_{3\nu}}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})},$
$\displaystyle T_{111}$ $\displaystyle=$ $\displaystyle\frac{1}{M^{4}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
(9) $\displaystyle\times$ $\displaystyle q_{1}\cdot
q_{2}\frac{V_{\mu\nu\gamma}(q_{3},-k_{2},-q_{2})q_{2}^{\gamma}}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})}$
$\displaystyle=$ $\displaystyle\frac{1}{M^{4}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
$\displaystyle\times$ $\displaystyle q_{1}\cdot q_{2}\frac{q_{1}\cdot
q_{2}g_{\mu\nu}-q_{1\mu}q_{2\nu}+(q_{1}\cdot k_{2}-q_{2}\cdot k_{1}-k_{1}\cdot
k_{2})g_{\mu\nu}-M_{Z}^{2}g_{\mu\nu}}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})}.$

Again to employ the corresponding W.I. of these two kinds of 3-particle
vertices. In the last step of Eq. (9), we also take into account that, $\int
dxf(x)\delta(x-a)=\int dxf(a)\delta(x-a)$, to use the relation
$q_{3}=q_{1}-k_{1}=q_{2}+k_{2}$, to get

$(q_{3}^{2}g_{\mu\nu}-q_{3\mu}q_{3\nu})=q_{1}\cdot
q_{2}g_{\mu\nu}-q_{1\mu}q_{2\nu}+(q_{1}\cdot k_{2}-q_{2}\cdot k_{1}-k_{1}\cdot
k_{2})g_{\mu\nu}.$

In fact, all the Ward identities for the 3-boson vertex we use here also have
employed the energy momentum conservation at the vertex.

We write

$\displaystyle T_{11B}$ $\displaystyle=$ $\displaystyle
T^{\prime}_{11}+T_{112}$ $\displaystyle=$
$\displaystyle\frac{M_{Z}^{2}}{M^{4}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
$\displaystyle\times$
$\displaystyle\frac{(-q_{1}^{2}q_{2\mu}q_{2\nu}+2q_{1}\cdot
q_{2}q_{3\mu}q_{3\nu})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})},$

so all the $M^{-4}$ term from $T_{1}$ are now $T_{11B}$ and $T_{111}$, with
the 4th and 3th order divergences only in $T_{111}$ (formally similar as those
in $H\to\gamma\gamma$).

Now we come to $T_{3}$, from external variables, its relation with $T_{1}$ is
$\mu\leftrightarrow\nu$, and $k_{1}\leftrightarrow k_{2}$; from internal
momenta, it is $q_{1}\leftrightarrow-q_{2}$ and $q_{3}\leftrightarrow-q_{3}$.
For easy to investigate, we now deal with it separately, according to the same
thread of $T_{1}$. Similarly for the three ways of combination,
$T^{\prime\prime}_{31}=0$, and

$\displaystyle T^{\prime}_{31}$ $\displaystyle=$
$\displaystyle\frac{M_{Z}^{2}}{M^{4}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{2}-q_{3})\delta(q_{3}-k_{1}-q_{2})$
(11) $\displaystyle\times$
$\displaystyle\frac{(-q_{2}^{2}q_{1\mu}q_{1\nu}+q_{1}\cdot
q_{2}q_{3\mu}q_{3\nu})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})}.$

For $T_{31}$, we also have $T_{31}=T_{311}+T_{312}+T_{313}$, with

$\displaystyle T_{313}$ $\displaystyle=$ $\displaystyle\frac{M^{2}}{M^{4}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{2}-q_{3})\delta(q_{3}-k_{1}-q_{2})$
(12) $\displaystyle\times$ $\displaystyle q_{1}\cdot
q_{2}\frac{q_{1}^{\beta}V_{\beta\nu\mu}(q_{1},-k_{2},-q_{3})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})}$

is obviously $M^{-2}$ term and to be discussed with other $M^{-2}$ terms
later.

$\displaystyle T_{312}$ $\displaystyle=$
$\displaystyle\frac{M_{Z}^{2}}{M^{4}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{2}-q_{3})\delta(q_{3}-k_{1}-q_{2})$
(13) $\displaystyle\times$ $\displaystyle\frac{q_{1}\cdot
q_{2}q_{3\mu}q_{3\nu}}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})},$
$\displaystyle T_{311}$ $\displaystyle=$ $\displaystyle\frac{1}{M^{4}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{2}-q_{3})\delta(q_{3}-k_{1}-q_{2})$
(14) $\displaystyle\times$ $\displaystyle q_{1}\cdot q_{2}\frac{q_{1}\cdot
q_{2}g_{\mu\nu}-q_{1\nu}q_{2\mu}+(q_{1}\cdot k_{1}-q_{2}\cdot k_{2}-k_{1}\cdot
k_{2})g_{\mu\nu}-M_{Z}^{2}g_{\mu\nu}}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})}.$

We write

$\displaystyle T_{31B}$ $\displaystyle=$ $\displaystyle
T^{\prime}_{31}+T_{312}$ $\displaystyle=$
$\displaystyle\frac{M_{Z}^{2}}{M^{4}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{2}-q_{3})\delta(q_{3}-k_{1}-q_{2})$
$\displaystyle\times$
$\displaystyle\frac{(-q_{2}^{2}q_{1\mu}q_{1\nu}+2q_{1}\cdot
q_{2}q_{3\mu}q_{3\nu})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})},$

again all the $M^{-4}$ term from $T_{3}$ are now $T_{31B}$ and $T_{311}$, with
the 4th and 3th order divergences only in $T_{311}$. $T_{111}$ and $T_{311}$
can be combined, since for both of them $q_{3}$ only appear in the $\delta$
functions and can be integrated out. Then the 4th and 3th order divergences
cancel after summing with those of $T_{2}$ and only quadratic one proportional
to $M_{Z}^{2}/M^{4}$ left (zero for $M_{Z}=0$).

$\displaystyle T_{111}+T_{311}$ $\displaystyle=$
$\displaystyle\frac{1}{M^{4}}\int
d^{4}q_{1}d^{4}q_{2}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-q_{2}-k_{1}-k_{2})$
(16) $\displaystyle\times$ $\displaystyle q_{1}\cdot
q_{2}~{}~{}\frac{2q_{1}\cdot
q_{2}g_{\mu\nu}-q_{1\mu}q_{2\nu}-q_{2\mu}q_{1\nu}-M_{Z}^{2}g_{\mu\nu}}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})},$

since $(q_{1}\cdot k_{2}-q_{2}\cdot k_{1}-k_{1}\cdot
k_{2})g_{\mu\nu}+(q_{1}\cdot k_{1}-q_{2}\cdot k_{2}-k_{1}\cdot
k_{2})g_{\mu\nu}=k_{2}^{2}g_{\mu\nu}=M_{Z}^{2}g_{\mu\nu}$, takeing now
$q_{1}-q_{2}=k_{1}+k_{2}$ from the $\delta$ function. The $M^{-4}$ term in
$T_{2}$ is

$\displaystyle T_{21}$ $\displaystyle=$ $\displaystyle\frac{-1}{M^{4}}\int
d^{4}q_{1}d^{4}q_{2}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-q_{2}-k_{1}-k_{2})$
(17) $\displaystyle\times$ $\displaystyle q_{1}\cdot
q_{2}~{}~{}\frac{2q_{1}\cdot
q_{2}g_{\mu\nu}-q_{1\mu}q_{2\nu}-q_{2\mu}q_{1\nu}}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})},$

so $T_{111}+T_{311}+T_{21}=T_{A}$:

$\displaystyle T_{A}$ $\displaystyle=$ $\displaystyle\frac{1}{M^{4}}\int
d^{4}q_{1}d^{4}q_{2}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-q_{2}-k_{1}-k_{2})\frac{-M_{Z}^{2}}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})}$
(18) $\displaystyle\times$ $\displaystyle q_{1}\cdot q_{2}g_{\mu\nu},$
$\displaystyle=$ $\displaystyle\frac{M_{Z}^{2}}{M^{4}}\int
d^{4}q_{1}d^{4}q_{2}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-q_{2}-k_{1}-k_{2})\frac{1}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})}$
$\displaystyle\times$
$\displaystyle(\frac{(k_{1}+k_{2})^{2}}{2}+(-\frac{(q_{1}^{2}-M^{2})+(q_{2}^{2}-M^{2})}{2}-M^{2}))g_{\mu\nu}.$

All the uncancelled terms of $M^{-4}$ order, $T_{A}$, $T_{11B}$, $T_{31B}$,
are proportional to $M_{Z}^{2}$, so are zero when Z mass goes to zero. Since
$M_{Z}$ is a constant parameter in the dynamics, the result is not because of
the in-proper choice of the integrated variables; as a matter of fact, all the
above derivation is independent from any special choice of the integrated
variable. In the integral $T_{A}$, the $q_{3}$ does not appear (or only
appearing in the $\delta$ functions and is integrated out). This case, which
has appeared above, is called ’hidden’ in the following.

We have

$\displaystyle T_{11B}$ $\displaystyle=$
$\displaystyle\frac{M_{Z}^{2}}{M^{4}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
(19) $\displaystyle\times$
$\displaystyle\frac{1}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})(q_{3}^{2}-M^{2})}$
$\displaystyle\times$
$\displaystyle(-(k_{1}+k_{2})^{2}q_{3\mu}q_{3\nu}+q_{2}^{2}q_{3\mu}q_{3\nu}+(q_{3}^{2}-M^{2})k_{2\mu}q_{2\nu}+M^{2}k_{2\mu}q_{2\nu}+2k_{1}\cdot
q_{3}k_{2\mu}q_{3\nu});$ $\displaystyle T_{31B}$ $\displaystyle=$
$\displaystyle\frac{M_{Z}^{2}}{M^{4}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{2}-q_{3})\delta(q_{3}-k_{1}-q_{2})$
(20) $\displaystyle\times$
$\displaystyle\frac{1}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})(q_{3}^{2}-M^{2})}$
$\displaystyle\times$
$\displaystyle(-(k_{1}+k_{2})^{2}q_{3\mu}q_{3\nu}+q_{1}^{2}q_{3\mu}q_{3\nu}-(q_{3}^{2}-M^{2})k_{2\mu}q_{1\nu}-M^{2}k_{2\mu}q_{1\nu}+2k_{1}\cdot
q_{3}k_{2\mu}q_{3\nu}).$

In the above equations, $(q_{1}-q_{2})=(k_{1}+k_{2})$ is used. In the last
line of each of Eqs. (18,19, 20), we have decomposed them into various terms
with various orders of divergences.

Before discussing the $M_{Z}^{2}/M^{4}$ terms, we first collect the
uncancelled finite $M_{Z}^{2}/M^{2}$ terms appearing in above equations for
further investigation. The ones directly read from the Eqs. (19) and (20)
respectively are the fourth term in last line of each

$\displaystyle T_{11B1}$ $\displaystyle=$
$\displaystyle\frac{M_{Z}^{2}}{M^{2}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
(21) $\displaystyle\times$
$\displaystyle\frac{k_{2\mu}q_{2\nu}}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})(q_{3}^{2}-M^{2})},$
$\displaystyle T_{31B1}$ $\displaystyle=$
$\displaystyle\frac{M_{Z}^{2}}{M^{2}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{2}-q_{3})\delta(q_{3}-k_{1}-q_{2})$
(22) $\displaystyle\times$
$\displaystyle\frac{-k_{2\mu}q_{1\nu}}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})(q_{3}^{2}-M^{2})}.$

The denominators are not the same, since different kinematic configuration.
Similar attention should be paid in the following. These two terms (21, 22)
are latter in need to be combined to get the $U_{EM}(1)$ gauge invariant term.
This very subtle fact is a signal of the self-consistency of the standard
model.

The linear divergent terms in Eqs. (19) and (20) respectively are the third
term in last line of each. The ($q_{3}^{2}-M^{2}$) reduces the corresponding
factor in denominator, and then they are independent on $q_{3}$. Integrating
on $q_{3}$ in both and they can combine and give

$\frac{M_{Z}^{2}}{M^{4}}\int
d^{4}q_{1}d^{4}q_{2}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-q_{2}-k_{1}-k_{2})\frac{-1}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})}k_{2\mu}k_{1\nu}.$
(23)

Again $(q_{1}-q_{2})=(k_{1}+k_{2})$ is used.

Further, dividing Eq. (23) by 2, each respectively recover a
$(q_{3}^{2}-M^{2})$ factor in the numerator and denominator, recover a third
$\delta$ function and integration on $q_{3}$ corresponding to $T_{1}$ and
$T_{3}$. The $q_{3}^{2}$ term of numerator cancels the logarithmic divergence
of fifth (last) term in last line respectively of Eqs. (19) and (20) (the
remaining finite terms are in the following). The remaining $M_{Z}^{2}/M^{2}$
terms are

$\displaystyle T_{11B2}$ $\displaystyle=$
$\displaystyle\frac{M_{Z}^{2}}{M^{2}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
(24) $\displaystyle\times$
$\displaystyle\frac{k_{2\mu}k_{1\nu}/2}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})(q_{3}^{2}-M^{2})},$
$\displaystyle T_{31B2}$ $\displaystyle=$
$\displaystyle\frac{M_{Z}^{2}}{M^{2}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{2}-q_{3})\delta(q_{3}-k_{1}-q_{2})$
(25) $\displaystyle\times$
$\displaystyle\frac{k_{2\mu}k_{1\nu}/2}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})(q_{3}^{2}-M^{2})}.$

Similar cancellation as above is also done for first term of last line
respectively of Eqs. (19) and (20) with the first term of last line of the
equation of $T_{A}$ (the remaining finite terms are in the following), and the
remaining $M_{Z}^{2}/M^{2}$ terms from $T_{A}$ (coming from the
$q_{3}^{2}-M^{2}$ factor for both numerator and denominator, recovering the
$\int dq_{3}$ integration) are

$\displaystyle T_{A11}$ $\displaystyle=$
$\displaystyle\frac{M_{Z}^{2}}{M^{2}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
(26) $\displaystyle\times$
$\displaystyle\frac{-(k_{1}+k_{2})^{2}g_{\mu\nu}/4}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})(q_{3}^{2}-M^{2})},$
$\displaystyle T_{A31}$ $\displaystyle=$
$\displaystyle\frac{M_{Z}^{2}}{M^{2}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{2}-q_{3})\delta(q_{3}-k_{1}-q_{2})$
(27) $\displaystyle\times$
$\displaystyle\frac{-(k_{1}+k_{2})^{2}g_{\mu\nu}/4}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})(q_{3}^{2}-M^{2})}.$

In the above two logarithmic divergence cancellations, the nonzero finite
$M_{Z}^{2}/M^{4}$ terms are

$\displaystyle T_{11BF}$ $\displaystyle=$
$\displaystyle\frac{M_{Z}^{2}}{M^{4}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
(28) $\displaystyle\times$
$\displaystyle\frac{\frac{1}{4}(k_{1}+k_{2})^{2}q_{3}^{2}g_{\mu\nu}-(k_{1}+k_{2})^{2}q_{3\mu}q_{3\nu}-\frac{1}{2}q_{3}^{2}k_{2\mu}k_{1\nu}+2k_{1}\cdot
q_{3}k_{2\mu}q_{3\nu}}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})(q_{3}^{2}-M^{2})},$
$\displaystyle T_{31BF}$ $\displaystyle=$
$\displaystyle\frac{M_{Z}^{2}}{M^{4}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{2}-q_{3})\delta(q_{3}-k_{1}-q_{2})$
(29) $\displaystyle\times$
$\displaystyle\frac{\frac{1}{4}(k_{1}+k_{2})^{2}q_{3}^{2}g_{\mu\nu}-(k_{1}+k_{2})^{2}q_{3\mu}q_{3\nu}-\frac{1}{2}q_{3}^{2}k_{2\mu}k_{1\nu}+2k_{1}\cdot
q_{3}k_{2\mu}q_{3\nu}}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})(q_{3}^{2}-M^{2})}.$

For the above six remaining non zero finite terms, contrary to (21, 22), it
can be shown that Eqs. (24) to (27) are cancelled by those terms in $T_{11BF}$
and $T_{31BF}$ proportional to an extra $M^{2}$ factor. Those remaining in
$T_{11BF}$ and $T_{31BF}$ all proportional to $(k_{1}\cdot
k_{2}g^{\mu\nu}-k_{2}^{\mu}k_{1}^{\nu})M_{Z}^{2}/M^{4}$. This in fact only can
be approved after the integration on Feynman-Schwinger parameters
$x_{1},x_{2}$, by which extra terms other than the gauge invariant one
integrated to be zero. The coefficient of $(k_{1}\cdot
k_{2}g^{\mu\nu}-k_{2}^{\mu}k_{1}^{\nu})M_{Z}^{2}/M^{4}$ is then

$\frac{i}{(4\pi)^{2}}\int
dx_{1}dx_{2}dx_{3}\delta(x_{1}+x_{2}+x_{3}-1)\frac{x_{1}x_{2}(2k_{1}\cdot
k_{2}+M_{Z}^{2})}{x_{1}x_{2}2k_{1}\cdot
k_{2}-x_{2}^{2}M_{Z}^{2}+x_{2}M_{Z}^{2}-M^{2}}.$ (30)

The result is of the form

$\frac{1}{2}+O(\frac{M_{Z}^{2}}{2k_{1}\cdot k_{2}})\cdot\cdot\cdot),$ (31)

which guarantee the non-zero $M_{Z}^{2}/M^{4}$ terms in the unitary gauge.

This term is the most important part of the $U_{EM}(1)$ gauge invariant final
result, in the sense crucial to comparing with the result from other gauges
(e.g., $R_{\xi}$ gauge). For feasibility to collect for the whole result, we
mark it as R1.

Now the terms including quadratically divergence to be considered are:

$\displaystyle\frac{1}{M^{4}}\int
d^{4}q_{1}d^{4}q_{2}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-q_{2}-k_{1}-k_{2})\frac{M_{Z}^{2}}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})}$
(32) $\displaystyle\times$
$\displaystyle(-\frac{(q_{1}^{2}-M^{2})+(q_{2}^{2}-M^{2})}{2}-M^{2})g_{\mu\nu},$
$\displaystyle\frac{1}{M^{4}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
(33) $\displaystyle\times$
$\displaystyle\frac{M_{Z}^{2}((q_{2}^{2}-M^{2})+M^{2})}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})(q_{3}^{2}-M^{2})}q_{3\mu}q_{3\nu},$
$\displaystyle\frac{1}{M^{4}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{2}-q_{3})\delta(q_{3}-k_{1}-q_{2})$
(34) $\displaystyle\times$
$\displaystyle\frac{M_{Z}^{2}((q_{1}^{2}-M^{2})+M^{2})}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})(q_{3}^{2}-M^{2})}q_{3\mu}q_{3\nu}.$

But to get a clear and definite cancellation of the quadratic divergence, from
the above three equations we still have to again take out the logarithmic
divergent ($\frac{M_{Z}^{2}}{M^{2}}$) terms to be considered in next sections
(which is indeed necessary to cancel divergence there, that also guarantees
this derivation):

$\frac{M_{Z}^{2}}{M^{2}}\int
d^{4}q_{1}d^{4}q_{2}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(k_{1}+k_{2}-q_{1}+q_{2})\frac{-g_{\mu\nu}}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})},$
(35)

and

$\displaystyle\frac{M_{Z}^{2}}{M^{2}}[\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
(36) $\displaystyle+$ $\displaystyle\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{2}-q_{3})\delta(q_{3}-k_{1}-q_{2})]$
$\displaystyle\times$
$\displaystyle\frac{q_{3\mu}q_{3\nu}}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})(q_{3}^{2}-M^{2})}.$

So the quadratic divergence from Eq. (32) is

$T_{Aq}=\frac{M_{Z}^{2}}{M^{4}}(2\pi)^{4}\delta(P-k_{1}-k_{2})\frac{-g_{\mu\nu}}{2}(\int
d^{4}q_{1}\frac{1}{(q_{1}^{2}-M^{2})}+d^{4}q_{2}\frac{1}{(q_{2}^{2}-M^{2})}),$
(38)

after cancelling the similar factor in the numerator and denominator, and
integrating the variable only appearing in the $\delta$ function.

For the quadratics in Eqs. (33) and (34), respectively, after cancelling the
similar factor in the numerator and denominator, and integrating the variable
only appearing in the $\delta$ function, they become,

$T_{11Bq}=\frac{M_{Z}^{2}}{M^{4}}\int
d^{4}q_{1}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{3}-k_{2})\delta(q_{1}-k_{1}-q_{3})\frac{q_{3\mu}q_{3\nu}}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})},$
(39) $T_{31Bq}=\frac{M_{Z}^{2}}{M^{4}}\int
d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P+q_{2}-q_{3}-k_{2})\delta(q_{3}-k_{1}-q_{2})\frac{q_{3\mu}q_{3\nu}}{(q_{2}^{2}-M^{2})(q_{3}^{2}-M^{2})}.$
(40)

In the above derivations, we take many ’petty’ steps to extract various terms
which is finite and definite or divergent only logarithmically, and at last
arrive at the above three quadratic terms to sum and cancel. The reason is
just that we eschew any shift of the integrated variable in the integral of
high divergence order. The subtle need of them (except the above U(1)
invariant $M_{Z}^{2}/M^{4}$ term) to cancel divergence and to get U(1)
invariant form is of course a nontrivial guarantee the derivation.

The above three terms sum to be zero, because of the quadratic surface term.
Here we come to investigate: Similar as the relation of logarithmic tensor
integral (here we only write the integrand)

$\partial_{(l)}^{\mu}\frac{l^{\nu}}{(l^{2}-\Delta)^{2}}=\frac{g^{\mu\nu}}{(l^{2}-\Delta)^{2}}-\frac{4l^{\mu}l^{\nu}}{(l^{2}-\Delta)^{3}}=\frac{l^{2}g^{\mu\nu}-4l^{\mu}l^{\nu}}{(l^{2}-\Delta)^{3}}-\frac{\Delta
g^{\mu\nu}}{(l^{2}-\Delta)^{3}},$ (41)

we have the quadratic one

$\partial_{(l)}^{\mu}\frac{l^{\nu}}{(l^{2}-\Delta)}=\frac{g^{\mu\nu}}{(l^{2}-\Delta)}-\frac{2l^{\mu}l^{\nu}}{(l^{2}-\Delta)^{2}}=\frac{l^{2}g^{\mu\nu}-2l^{\mu}l^{\nu}}{(l^{2}-\Delta)^{2}}-\frac{\Delta
g^{\mu\nu}}{(l^{2}-\Delta)^{2}}.$ (42)

When we take the surface term to be integrated zero, we obtain the tensor
reduction formula we need. Why the surface term can be zero considering the
boundary condition at infinity inherent of the phase space free Feynman
propagator Li:2017hnv , and the indication that this momentum space divergence
as sensitive probe on local property of space time et vice verse, are to be
investigated in the discussion section. This quadratic form is also appear in
QED photon self energy. It is easy for the above formula to give the
electromagnetic gauge invariant form (W.I.), proportional only to a log pole
which only affect the residue of the photon propagator and is absorbed by the
coupling constant renormalization Bao:2021byx .

By changing the dummy integrated variables according the relations set by the
$\delta$ functions, Eq.(39) and Eq.(40) are

$q_{1}\to l_{1},q_{3}\to l_{2}$:

$T_{11Bq}=\frac{M_{Z}^{2}}{M^{4}}\int
d^{4}l_{1}d^{4}l_{2}(2\pi)^{4}\delta(P-(l_{1}-l_{2})-k_{2})\delta(l_{1}-l_{2}-k_{1})\frac{l_{2\mu}l_{2\nu}}{(l_{1}^{2}-M^{2})(l_{2}^{2}-M^{2})},$
(43)

$q_{2}\to l_{2},q_{3}\to l_{1}$:

$T_{31Bq}=\frac{M_{Z}^{2}}{M^{4}}\int
d^{4}l_{1}d^{4}l_{2}(2\pi)^{4}\delta(P-(l_{1}-l_{2})-k_{2})\delta(l_{1}-l_{2}-k_{1})\frac{l_{1\mu}l_{1\nu}}{(l_{1}^{2}-M^{2})(l_{2}^{2}-M^{2})}.$
(44)

Effectively these two integrals equals to the following two integrals

$\displaystyle T^{\prime}_{11Bq}$ $\displaystyle=$
$\displaystyle\frac{M_{Z}^{2}}{M^{4}}\int
d^{4}l_{1}d^{4}l_{2}(2\pi)^{4}\delta(P-(l_{1}-l_{2})-k_{2})\delta(l_{1}-l_{2}-k_{1})\frac{l_{2\mu}l_{2\nu}}{(l_{2}^{2}-M^{2})^{2}}$
(45) $\displaystyle=$ $\displaystyle\frac{M_{Z}^{2}}{M^{4}}\int
d^{4}l_{2}(2\pi)^{4}\delta(P-k_{1}-k_{2})\frac{l_{2\mu}l_{2\nu}}{(l_{2}^{2}-M^{2})^{2}},$
(46) $\displaystyle T^{\prime}_{31Bq}$ $\displaystyle=$
$\displaystyle\frac{M_{Z}^{2}}{M^{4}}\int
d^{4}l_{1}d^{4}l_{2}(2\pi)^{4}\delta(P-(l_{1}-l_{2})-k_{2})\delta(l_{1}-l_{2}-k_{1})\frac{l_{1\mu}l_{1\nu}}{(l_{1}^{2}-M^{2})^{2}}$
(47) $\displaystyle=$ $\displaystyle\frac{M_{Z}^{2}}{M^{4}}\int
d^{4}l_{1}(2\pi)^{4}\delta(P-k_{1}-k_{2})\frac{l_{1\mu}l_{1\nu}}{(l_{1}^{2}-M^{2})^{2}}.$
(48)

In the second line of each of the above two equations, a dummy $\delta$
function is integrated out. It is obviously that (38) + (46) + (48)=0
according to the above quadratic surface integral formula, so that (38) + (39)
+ (40)=0, i.e., $T_{Aq}+T_{11Bq}+T_{31Bq}=0$.

In the following we show that ((43) + (44))$-$ ((45) + (47))=0, i.e.,

$(T_{11Bq}+T_{31Bq})-(T^{\prime}_{11Bq}+T^{\prime}_{31Bq})=0$:

(43)$-$ (45) is

$T_{11Bq}-T^{\prime}_{11Bq}=-\frac{M_{Z}^{2}}{M^{4}}\int
d^{4}l_{1}d^{4}l_{2}(2\pi)^{4}\delta(P-(l_{1}-l_{2})-k_{2})\delta(l_{1}-l_{2}-k_{1})\frac{l_{2\mu}l_{2\nu}2l_{2}\cdot
k_{1}}{(l_{1}^{2}-M^{2})(l_{2}^{2}-M^{2})^{2}},$ (49)

(44)$-$ (47) is

$T_{31Bq}-T^{\prime}_{31Bq}=\frac{M_{Z}^{2}}{M^{4}}\int
d^{4}l_{1}d^{4}l_{2}(2\pi)^{4}\delta(P-(l_{1}-l_{2})-k_{2})\delta(l_{1}-l_{2}-k_{1})\frac{l_{1\mu}l_{1\nu}2l_{1}\cdot
k_{1}}{(l_{1}^{2}-M^{2})^{2}(l_{2}^{2}-M^{2})}.$ (50)

The above two terms equal to, again by the quadratic surface formula,

$-\frac{M_{Z}^{2}}{M^{4}}\int
d^{4}l_{1}d^{4}l_{2}(2\pi)^{4}\delta(P-(l_{1}-l_{2})-k_{2})\delta(l_{1}-l_{2}-k_{1})\frac{g_{\mu\nu}l_{2}\cdot
k_{1}}{(l_{1}^{2}-M^{2})(l_{2}^{2}-M^{2})}+S_{1},$ (51)
$\frac{M_{Z}^{2}}{M^{4}}\int
d^{4}l_{1}d^{4}l_{2}(2\pi)^{4}\delta(P-(l_{1}-l_{2})-k_{2})\delta(l_{1}-l_{2}-k_{1})\frac{g_{\mu\nu}l_{1}\cdot
k_{1}}{(l_{1}^{2}-M^{2})(l_{2}^{2}-M^{2})}+S_{2}.$ (52)

The fact expressed in the $\delta$ function is that
$l_{1}-l_{2}=k_{1},k_{1}^{2}=0,l_{1}\cdot k_{1}=l_{2}\cdot k_{1}$. So two
terms explicitly written above cancel, leaving the surface related terms
$S_{1},S_{2}$.

$S_{1}+S_{2}=\frac{l_{2}\cdot
k_{1}}{(l_{1}^{2}-M^{2})}\partial_{\mu}\frac{l_{2\nu}}{(l_{2}^{2}-M^{2})}-\frac{l_{1}\cdot
k_{1}}{(l_{2}^{2}-M^{2})}\partial_{\mu}\frac{l_{1\nu}}{(l_{1}^{2}-M^{2})}.$
(53)

(Here we neglect the integral symbol and the $\delta$ functions, and employ
the fact $\partial_{(l_{1})\mu}=\partial_{(l_{2})\mu}=\partial_{\mu}$.)

$S_{1}+S_{2}=-\frac{2l_{2}\cdot
k_{1}l_{1\mu}}{(l_{1}^{2}-M^{2})^{2}}\frac{l_{2\nu}}{(l_{2}^{2}-M^{2})}+\frac{2l_{1}\cdot
k_{1}l_{2\mu}}{(l_{2}^{2}-M^{2})^{2}}\frac{l_{1\nu}}{(l_{1}^{2}-M^{2})}+surface~{}term.$
(54)

Here we use $\partial_{\mu}l\cdot k_{1}=k_{1\mu}=0$. The surface term comes
from the total derivative

$\partial_{\mu}[\frac{l_{2}\cdot
k_{1}}{(l_{1}^{2}-M^{2})}\frac{l_{2\nu}}{(l_{2}^{2}-M^{2})}-\frac{l_{1}\cdot
k_{1}}{(l_{2}^{2}-M^{2})}\frac{l_{1\nu}}{(l_{1}^{2}-M^{2})}],$ (55)

and is zero (integrated).

Again with $l_{1}\cdot k_{1}=l_{2}\cdot k_{1}$, but
$l_{1\nu}-l_{2\nu}=k_{1\nu}$,

$\displaystyle
T_{11Bq}-T^{\prime}_{11Bq}+T_{31Bq}-T^{\prime}_{31Bq}=S_{1}+S_{2}=-(T_{11Bq}-T^{\prime}_{11Bq}+T_{31Bq}-T^{\prime}_{31Bq})$
(56) $\displaystyle+\frac{2l_{1}\cdot
k_{1}l_{1\mu}}{(l_{1}^{2}-M^{2})^{2}}\frac{k_{1\nu}}{(l_{2}^{2}-M^{2})}+\frac{2l_{2}\cdot
k_{1}l_{2\mu}}{(l_{2}^{2}-M^{2})^{2}}\frac{k_{1\nu}}{(l_{1}^{2}-M^{2})}.$

The second line is now logarithmic, so we can employ Feynman-Schwinger
parameterization to directly calculate it. It will lead to the factor
$k_{1\mu}$ and is zero, and then
$T_{11Bq}-T^{\prime}_{11Bq}+T_{31Bq}-T^{\prime}_{31Bq}$ is solved to be zero.
q.e.d.

Here we emphasize again this petty derivation just to show the importance of
the surface term and the possibility to eschew any shift for high order
divergence.

Now we see that the total result on the terms proportional to
$M_{Z}^{2}/M^{4}$ is finite, non-zero, and U(1) gauge invariant. And the
$M_{Z}^{2}/M^{2}$ terms left after all the above cancellation are to be shown
necessary for the following cancellation.

From the above, we learn that the calculation in the way introduced in this
paper, especially not to integrate the $\delta$ functions before have to and
ia allowed to (once a variable only appearing in the delta function, it is a
dummy and the integrand is independent from it), provides the exact definition
of the Feynman diagram. One may suspect that for the most general case of the
calculation of the Feynman diagrams, the proper way of setting the independent
integrated variables at beginning as done by GWW Gastmans:2011ks , may not be
available. So calculation without integration on the $\delta$ functions until
have to is a more proper or maybe necessary way of the employment of the
Feynman rules. As for the surface term, it is consistent with the naïve
symmetric integration for convergent integral. The surface integral formula is
also consistent with a D dimension regularization calculation (which is always
considered as convergent), and share the similar spirit of the ’IBP’. But it
is unclear All the above calculations done in D dimension can get the same
result.

### 2.3

Now here we investigate the $M^{-2}$ terms ($T_{113}$ (7), $T_{313}$ (12),
$T_{11B1}$ (21), $T_{31B1}$ (22), and Eqs. (35), (36), (2.2) included).

Besides the above listed $M^{-2}$ terms in bracket, we need to investigate the
following: From $T_{1}$

$\displaystyle T_{12}$ $\displaystyle=$ $\displaystyle\frac{-1}{M^{2}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
(57) $\displaystyle\times$ $\displaystyle
g_{\alpha}~{}^{\beta}g^{\rho\sigma}q_{2}^{\alpha}q_{2}^{\gamma}\frac{V_{\beta\mu\rho}(q_{1},-k_{1},-q_{3})~{}V_{\sigma\nu\gamma}(q_{3},-k_{2},-q_{2})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})},$
$\displaystyle T_{13}$ $\displaystyle=$ $\displaystyle\frac{-1}{M^{2}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
(58) $\displaystyle\times$ $\displaystyle
g_{\alpha}~{}^{\beta}q_{3}^{\rho}q_{3}^{\sigma}g^{\alpha\gamma}\frac{V_{\beta\mu\rho}(q_{1},-k_{1},-q_{3})~{}V_{\sigma\nu\gamma}(q_{3},-k_{2},-q_{2})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})},$
$\displaystyle T_{14}$ $\displaystyle=$ $\displaystyle\frac{-1}{M^{2}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
(59) $\displaystyle\times$ $\displaystyle
q_{1\alpha}q_{1}^{\beta}g^{\rho\sigma}g^{\alpha\gamma}\frac{V_{\beta\mu\rho}(q_{1},-k_{1},-q_{3})~{}V_{\sigma\nu\gamma}(q_{3},-k_{2},-q_{2})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})}.$

$T_{12}$ and $T_{14}$ both give $q_{3}^{2}-M^{2}$ term in numerator when
applying the Ward identity directly, which could reduce the corresponding
factor in denominator and combine with $T_{22+23}$,

$\displaystyle T_{22+23}$ $\displaystyle=$ $\displaystyle\frac{1}{M^{2}}\int
d^{4}q_{1}d^{4}q_{2}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-q_{2}-k_{1}-k_{2})$
(60) $\displaystyle\times$
$\displaystyle\frac{2q_{1}^{2}g_{\mu\nu}+2q_{2}^{2}g_{\mu\nu}-2q_{1\mu}q_{1\nu}-2q_{2\mu}q_{2\nu}}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})}.$

$T_{12}=T_{121}+T_{122}+T_{123}+T_{124}$:

$\displaystyle T_{121}$ $\displaystyle=$ $\displaystyle\frac{-1}{M^{2}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
(61) $\displaystyle\times$ $\displaystyle
q_{2}^{\beta}\frac{V_{\beta\mu\nu}(q_{1},-k_{1},-q_{3})}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})};$
$\displaystyle T_{122}$ $\displaystyle=$ $\displaystyle\frac{-1}{M^{2}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
(62) $\displaystyle\times$ $\displaystyle
q_{2}^{\beta}\frac{V_{\beta\mu\sigma}(q_{1},-k_{1},-q_{3})(-q_{3}^{\sigma})q_{3\nu}}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})};$
$\displaystyle T_{123}$ $\displaystyle=$ $\displaystyle\frac{-1}{M^{2}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
(63) $\displaystyle\times$ $\displaystyle
q_{2}^{\beta}\frac{V_{\beta\mu\nu}(q_{1},-k_{1},-q_{3})M^{2}}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})}$

is $M^{0}$ term;

$\displaystyle T_{124}$ $\displaystyle=$ $\displaystyle\frac{-1}{M^{2}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
(64) $\displaystyle\times$ $\displaystyle
q_{2}^{\beta}\frac{V_{\beta\mu\nu}(q_{1},-k_{1},-q_{3})(-M_{Z}^{2})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})}.$

$T_{14}=T_{141}+T_{142}+T_{143}$:

$\displaystyle T_{141}$ $\displaystyle=$ $\displaystyle\frac{-1}{M^{2}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
(65) $\displaystyle\times$
$\displaystyle\frac{V_{\mu\nu\gamma}(q_{3},-k_{2},-q_{2})}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})}q_{1}^{\gamma};$
$\displaystyle T_{142}$ $\displaystyle=$ $\displaystyle\frac{1}{M^{2}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
(66) $\displaystyle\times$
$\displaystyle\frac{q_{3\mu}q_{3}^{\sigma}V_{\sigma\nu\gamma}(q_{3},-k_{2},-q_{2})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})}q_{1}^{\gamma};$
$\displaystyle T_{143}$ $\displaystyle=$ $\displaystyle\frac{-1}{M^{2}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
(67) $\displaystyle\times$
$\displaystyle\frac{M^{2}V_{\mu\nu\gamma}(q_{3},-k_{2},-q_{2})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})}q_{1}^{\gamma}$

is $M^{0}$ term.

This discussion also applies to $T_{3}$, so the following is to investigate
$T_{121}+T_{141}+T_{321}+T_{341}+T_{22+23}$, and we use $q_{2}=q_{1}-P$, so
can use W.I. again, while the extra term
$-P^{\beta}V_{\beta\mu\nu}(q_{1},-k_{1},-q_{3})$ will combine with the
corresponding term from $T_{141}$. We use $q_{1}=q_{2}+P$, and the extra term
is $V_{\mu\nu\gamma}(q_{3},-k_{2},-q_{2})P^{\gamma}$ there. We found that the
extra $M_{Z}^{2}$ term from the W.I. and from $(k_{1}+k_{2})^{2}$ just
cancelled, so all the other similar as two photon case Li:2017hnv , i.e.,

$T_{121}+T_{141}+T_{321}+T_{341}+T_{22+23}=0.$

Now the remaining $M^{-2}$ terms are all from $T_{1}$ and $T_{3}$, as well as
the remained $M_{Z}^{2}$ terms from the above subsection. As the case of the
two photons, cancellation for the linear divergence will be taken by the
summation of the corresponding terms from $T_{1}$ and $T_{3}$ respectively.
Those directly from $T_{1}$ are

$T_{113}+T_{13}+T_{122}+T_{124}+T_{142},$ (68)

which must be considered together with $M_{Z}^{2}$ terms half of Eq. (35),
then $T_{11B1}$ (21) and Eq. (36). It is easy to find that the terms without
the $M_{Z}^{2}$ factor are quite similar as two photon case, but extra terms
from $k_{2}^{2}=M_{Z}^{2}$ are subtle. They emerge from various equations, and
cancel in various ways leading to the U(1) invariant final result, which is a
manifest of the self-consistent of the standard model.

So we separate (68) as two parts: A, those explicitly without $M^{2}_{Z}$
factor in the beginning; B, those with. Then we calculate A, to see which
canceled, which giving extra $M_{Z}^{2}$ terms to be canceled with extra terms
from last section, or to be arranged with part B—together with all extra terms
from last subsection, leading to the final U(1) invariant results.

We here again apply the W.I. and combine them together to obtain a simple form
of part A:

$\displaystyle(T_{113}+T_{13}+T_{122}+T_{142})_{A}$ (69) $\displaystyle=$
$\displaystyle\frac{1}{M^{2}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}\frac{(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})}$
$\displaystyle\times$
$\displaystyle(2q_{1}^{2}q_{2\mu}q_{2\nu}+2q_{2}^{2}q_{1\mu}q_{1\nu}-4q_{1}\cdot
q_{2}q_{1\mu}q_{2\nu}+q_{1}\cdot
q_{2}q_{3}^{2}g_{\mu\nu}-q_{1}^{2}q_{2}^{2}g_{\mu\nu}).$

It looks as quadratic, but easy to see in fact to the most linear, quite
similar as the two photon case, since

$2q_{1}\cdot q_{2}=-(q_{1}-q_{2})^{2}+q_{1}^{2}+q_{2}^{2},$ then
$2q_{1}^{2}q_{2\mu}q_{2\nu}+2q_{2}^{2}q_{1\mu}q_{1\nu}-4q_{1}\cdot
q_{2}q_{1\mu}q_{2\nu}$ equals to

$2q_{1}^{2}(q_{2\mu}-q_{1\mu})q_{2\nu}+2q_{2}^{2}q_{1\mu}(q_{1\nu}-q_{2\nu})+2(q_{1}-q_{2})^{2}q_{1\mu}q_{2\nu}=2q_{1}^{2}(-k_{2\mu})q_{2\nu}+2q_{2}^{2}q_{1\mu}k_{1\nu}+2(k_{1}+k_{2})^{2}q_{1\mu}q_{2\nu},$

and $q_{1}\cdot
q_{2}q_{3}^{2}g_{\mu\nu}-q_{1}^{2}q_{2}^{2}g_{\mu\nu}=-\frac{(k_{1}+k_{2})^{2}}{2}q_{3}^{2}g_{\mu\nu}+\frac{q_{1}^{2}+q_{2}^{2}}{2}q_{3}^{3}g_{\mu\nu}-q_{1}^{2}q_{2}^{2}g_{\mu\nu}.$

However,

$\frac{q_{1}^{2}+q_{2}^{2}}{2}q_{3}^{3}g_{\mu\nu}-q_{1}^{2}q_{2}^{2}g_{\mu\nu}=(\frac{q_{1}^{2}}{2}(q_{3}+q_{2})\cdot
k_{2}-\frac{q_{2}^{2}}{2}(q_{3}+q_{1})\cdot k_{1})g_{\mu\nu}$

hence is also linear ($q_{3}$ and $q_{2}$ can not combine since
$k_{2}^{2}=M^{2}_{Z}\neq 0$).

Now we write $(T_{113}+T_{13}+T_{122}+T_{142})_{A}$ as the summation of two
parts:

$\displaystyle\frac{1}{M^{2}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}\frac{(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})}$
$\displaystyle\times$ $\displaystyle
2q_{1}^{2}(-k_{2\mu})q_{2\nu}+2q_{2}^{2}q_{1\mu}k_{1\nu}+(\frac{q_{1}^{2}}{2}(q_{3}+q_{2})\cdot
k_{2}-\frac{q_{2}^{2}}{2}(q_{3}+q_{1})\cdot k_{1})g_{\mu\nu},$ (70)
$\displaystyle\frac{1}{M^{2}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}\frac{(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})}$
$\displaystyle\times$ $\displaystyle
2(k_{1}+k_{2})^{2}q_{1\mu}q_{2\nu}-\frac{(k_{1}+k_{2})^{2}}{2}q_{3}^{2}g_{\mu\nu},$
(71)

i.e., the linear and logarithmic divergent terms respectively. Then the above
linear term, after further taking out logarithmic and finite terms from it,
should combine with the corresponding term from $T_{3}$, then is deduced to
get as two terms, one is logarithmic divergent, the other is finite.

Some details are:

$\displaystyle 2q_{1}^{2}(-k_{2\mu})q_{2\nu}$ $\displaystyle=$
$\displaystyle-2(q_{3}^{2}-M^{2})k_{2\mu}q_{2\nu}-4q_{3}\cdot
k_{1}k_{2\mu}q_{2\nu}-2M^{2}k_{2\mu}q_{2\nu}$ (72) $\displaystyle
2q_{2}^{2}q_{1\mu}k_{1\nu}$ $\displaystyle=$ $\displaystyle
2(q_{3}^{2}-M^{2})q_{1\mu}k_{1\nu}-4q_{3}\cdot
k_{2}q_{1\mu}k_{1\nu}+2M^{2}q_{1\mu}k_{1\nu}+2M_{Z}^{2}q_{1\mu}k_{1\nu}$ (73)
$\displaystyle\frac{q_{1}^{2}}{2}(q_{3}+q_{2})\cdot k_{2}g_{\mu\nu}$
$\displaystyle=$ $\displaystyle\frac{(q_{3}^{2}-M^{2})}{2}(q_{3}+q_{2})\cdot
k_{2}g_{\mu\nu}+q_{3}\cdot k_{1}(q_{3}+q_{2})\cdot k_{2}g_{\mu\nu}$ (74)
$\displaystyle+$ $\displaystyle\frac{M^{2}}{2}(q_{3}+q_{2})\cdot
k_{2}g_{\mu\nu}$ $\displaystyle-\frac{q_{2}^{2}}{2}(q_{3}+q_{1})\cdot
k_{1}g_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{-(q_{3}^{2}-M^{2})}{2}(q_{3}+q_{1})\cdot
k_{1}g_{\mu\nu}+q_{3}\cdot k_{2}(q_{3}+q_{1})\cdot k_{1}g_{\mu\nu}$ (75)
$\displaystyle-$ $\displaystyle\frac{M^{2}}{2}(q_{3}+q_{1})\cdot
k_{1}g_{\mu\nu}-\frac{M_{Z}^{2}}{2}(q_{3}+q_{1})\cdot k_{1}g_{\mu\nu}$

First of all we address that the $M_{Z}^{2}$ terms (some are not explicitly
written in the above) are all finite terms and are to be calculated later. So
it is the following linear term (with $(q_{3}^{2}-M^{2})$ factor reduced with
the common one in denominator, and $(q_{3}+q_{2})$ written as
$(2q_{2}+k_{2})$, $(q_{3}+q_{1})$ written as $(2q_{1}-k_{1})$, then $q_{3}$
integrated)

$\displaystyle\frac{1}{M^{2}}\int
d^{4}q_{1}d^{4}q_{2}\frac{(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-q_{2}-k_{1}-k_{2})}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})}(-2k_{2\mu}q_{2\nu}+2q_{1\mu}k_{1\nu}$
(76) $\displaystyle+$ $\displaystyle\frac{1}{2}((2q_{2}+k_{2})\cdot
k_{2}-(2q_{1}-k_{1})\cdot k_{1}))g_{\mu\nu})$

to be combined with that from $T_{3}$:

$\displaystyle\frac{1}{M^{2}}\int
d^{4}q_{1}d^{4}q_{2}\frac{(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-q_{2}-k_{1}-k_{2})}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})}(-2k_{1\nu}q_{2\mu}+2q_{1\nu}k_{2\mu}$
(77) $\displaystyle+$ $\displaystyle\frac{1}{2}((2q_{2}+k_{1})\cdot
k_{1}-(2q_{1}-k_{2})\cdot k_{2}))g_{\mu\nu}),$

and summed with Eq. (35) (!) then deduces to logarithmic term. Half of their
summation is then:

$\frac{1}{M^{2}}\int
d^{4}q_{1}d^{4}q_{2}\frac{(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-q_{2}-k_{1}-k_{2})}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})}(2k_{2\mu}k_{1\nu}-\frac{(k_{1}+k_{2})^{2}}{2}g_{\mu\nu})$
(78)

This term can again be separated into a logarithmic term and a finite term,

$\displaystyle\frac{1}{M^{2}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}\frac{(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})}q_{3}^{2}(2k_{2\mu}k_{1\nu}-\frac{(k_{1}+k_{2})^{2}}{2}g_{\mu\nu})$
$\displaystyle+\frac{1}{M^{2}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}\frac{(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})}(-M^{2})(2k_{2\mu}k_{1\nu}-\frac{(k_{1}+k_{2})^{2}}{2}g_{\mu\nu})$

Hence effectively
$(T_{113}+T_{13}+T_{122}+T_{142})_{A}+Eq.(\ref{the2})/2=T1LG_{A}+T1LG_{B}+T1F_{A}+T1F_{B}$,
with

$\displaystyle T1LG_{A}=\frac{1}{M^{2}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}\frac{(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})}$
$\displaystyle\times[(-2q_{3}^{2}k_{1}\cdot k_{2}+4k_{1}\cdot q_{3}k_{2}\cdot
q_{3})g_{\mu\nu}-4k_{1}\cdot q_{3}k_{2\mu}q_{3\nu}-4k_{2}\cdot
q_{3}q_{3\mu}k_{1\nu}+2q_{3}^{2}k_{2\mu}k_{1\nu}+4k_{1}\cdot
k_{2}q_{3\mu}q_{3\nu}]$ (79) $\displaystyle T1F_{A}$ $\displaystyle=$
$\displaystyle\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}\frac{(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})}$
$\displaystyle\times$
$\displaystyle[2q_{1\mu}k_{1\nu}-2k_{2\mu}q_{2\nu}-2k_{2\mu}k_{1\nu}+\frac{1}{2}((q_{3}+q_{2})\cdot
k_{2}-(q_{3}+q_{1})\cdot k_{1}+(k_{1}+k_{2})^{2})g_{\mu\nu}],$

($M^{0}$ term to be combined into the final result), and U(1) invariant finite

$\displaystyle T1F_{B}$ $\displaystyle=$
$\displaystyle\frac{M_{Z}^{2}}{M^{2}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
$\displaystyle\times$ $\displaystyle\frac{2q_{1\mu}k_{1\nu}-2q_{1}\cdot
k_{1}g_{\mu\nu}}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})}.$

But the $T1LG_{B}$ term is the logarithmic terms with $M_{Z}^{2}$ factor,
summing the $M_{Z}^{2}$ terms $(T_{113}+T_{13}+T_{124}+T_{142})_{B}$, Eq. (36)
, and $T_{11B1}$ (21). The result is

$\displaystyle T1F_{C}$ $\displaystyle=$
$\displaystyle\frac{M_{Z}^{2}}{M^{2}}\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
$\displaystyle\times$ $\displaystyle\frac{2k_{1}\cdot
k_{2}g_{\mu\nu}-2k_{2\mu}k_{1\nu}}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})}.$

The $T_{3}$ is in fact effectively discussed, and just make the above doubled.
We see that all the above are finite and U(1) invariant ($T1F_{A}$ is to be
combined in the following), and all the remaining ones in last subsection has
been cancelled and combined. This procedure is subtle, interesting and self-
consistent (due to standard model). From this subsection we obtain U(1) EM
gauge invariant terms proportional to $1/M^{2}$, $T1LG_{A}$ (the logarithmic
cancelled so finite) as the two photon case, we mark as R2. We also obtain the
extra finite U(1) EM invariant terms proportional to $M_{Z}^{2}/M^{2}$,
$T1F_{B}$ and $T1F_{C}$, we mark as R3, R4. These are also to be considered
for the investigation of the gauge invariant w.r.t. $R_{\xi}$ gauge result.

### 2.4

Now the $M^{0}$ terms.

The third part of $T_{12},~{}T_{14}$, i.e., $T_{123},~{}T_{143}$, as well as
those corresponding ones from $T_{3}$, are $M^{0}$ terms and to be
investigated here together with the corresponding terms from $T_{2}$ (Pay
attention that $T_{2}$ lack of a overall minus sign) and the other remaining
$M^{0}$ terms form $T_{1}$ and $T_{3}$:

$\displaystyle T_{15}$ $\displaystyle=$ $\displaystyle\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
(83) $\displaystyle\times$ $\displaystyle
g_{\alpha}~{}^{\beta}g^{\rho\sigma}g^{\alpha\gamma}\frac{V_{\beta\mu\rho}(q_{1},-k_{1},-q_{3})~{}V_{\sigma\nu\gamma}(q_{3},-k_{2},-q_{2})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})},$

and $T_{35}$ is not explicitly written here.

$\displaystyle T_{24}$ $\displaystyle=$ $\displaystyle-\int
d^{4}q_{1}d^{4}q_{2}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-q_{2}-k_{1}-k_{2})$
(84) $\displaystyle\times$ $\displaystyle
g_{\alpha}~{}^{\beta}g^{\alpha\gamma}\frac{2g_{\mu\nu}g_{\beta\gamma}-g_{\mu\beta}g_{\nu\gamma}-g_{\mu\gamma}g_{\nu\beta}}{(q_{1}^{2}-M^{2})(q_{2}^{2}-M^{2})}.$

These four terms (half of $T_{24}$) summed still give terms as logarithmic

$\displaystyle\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
$\displaystyle\times$
$\displaystyle\frac{(-3q_{3}^{2}g_{\mu\nu}+12q_{3\mu}q_{3\nu})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})},$
(85)

but all the divergences cancelled, and the total summation of the terms, half
of $T_{24}$, $T_{15}$, $T_{123},~{}T_{143}$, as well as $T1F_{A}$, give the
exactly similar form of Eq. (36) of Li:2017hnv  ($H\to\gamma\gamma$ process),
i.e.,

$\displaystyle\int
d^{4}q_{1}d^{4}q_{2}d^{4}q_{3}(2\pi)^{4}\delta(P-q_{1}+q_{2})\delta(q_{1}-k_{1}-q_{3})\delta(q_{3}-k_{2}-q_{2})$
$\displaystyle\times$ $\displaystyle\frac{-6k_{1}\cdot
k_{2}g_{\mu\nu}+6k_{2\mu}k_{1\nu}+3M^{2}g_{\mu\nu}+(-3q_{3}^{2}g_{\mu\nu}+12q_{3\mu}q_{3\nu})}{(q_{1}^{2}-M^{2})(q_{3}^{2}-M^{2})(q_{2}^{2}-M^{2})}.$
(86)

But pay attention since the denominator different because of
$k_{2}^{2}=M_{Z}^{2}\neq 0$, this is U(1) invariant just after some of the
terms integrated to be zero. However, same as two photon case, the Dyson
subtraction is not needed since the surface term formula for log divergence.
We mark this term as R5.

In this paper we do not intend to produce the final results. Investigation on
$SU(2)\times U(1)$ gauge invariance and combining fermion loops are to be done
in the following paper. In this paper we just give the original form we
obtained above, R1+R2+R3+R4+R5.

##  3 Discussion and speculation

In this paper we eliminate the paradoxes and ambiguities via the framework of
the original Dyson scheme, each propagator momentum integration kept and each
$\delta$ function at each vertex not integrated out: 1, the original loop
momentum definite in the beginning to write down the amplitude in momentum
space with Feynman rules; 2, any momentum shift for divergences worse than
logarithmic eschewed; 3, no regularization or dimension extrapolation. Some
speculation of new physics beyond the standard model can be based on some new
symmetry which introduces some correspondence to the known standard model
particles, and may simplify the renormalization dramatically via divergence
cancellation, especially high order divergence cancelation. So to properly
write down these divergences and then to properly cancel them become a very
crucial point.

To proceed the concrete calculation, the evaluation on the surface terms
Ferreira:2011cv  for the loop momentum going to infinite is very important to
eliminate any uncertainty. We refer to the physical boundary condition of the
scattering S matrix static problem to determine the surface term to be zero
Li:2017hnv  which is taken as undetermined in Ferreira:2011cv . In this paper
we encounter and determine the quadratic (42) and logarithmic (41) ones.
Besides the $H\to\gamma\gamma$, $H\to\gamma Z$, the photon and photon-Z self-
energy diagram Bao:2021byx , these also appear in other places, as for the
axial anomaly. The Dyson scheme and the ’most symmetric loop momentum’ (see
footnote 3) are both in contrary to the Bell-Jackiw claim. There, no necessity
referring to the surface term, but we can see the surface term determined same
as above via the physical boundary condition of the free propagator, give the
consistency Bao:2021byx .

However, one must adopt that in the most general consideration which is not
restricted in the physical boundary condition of the free propagator in
Minkowski momentum space (the scattering S matrix static problem), whether the
surface term equals to zero depending on the concrete condition of the
infinite momentum surface, hence the space-time structure, geometry or
topology, which is very crucial. For example, when the divergence of the
chiral current is calculated via the concrete QFT defined on a specified
space-time manifold, the value can be zero or not. This is in fact taken as an
input to generate many interesting physics which have been widely studied,
e.g., relation with the Atiyah-Singer index/Pontryagin class, ’t Hoot symmetry
breaking, etc. What lesson we learn here is that for a concrete physical
problem, the Atiyah-Singer index in no way must equal to nonzero integer, but
must be determined by the proper calculation of the QFT defined on the
manifold, as well as the physical condition, especially for special limits. As
a classical original example, the Polyakov solution in Euclidean space-time
Belavin:1975fg . Here we only want to point out that besides the index $q\neq
0$ case which gives many interesting speculations so far, the $q=0$ case also
exist and can be considered as special limit of the $q=1$ solution. One
trivial case is the group element g in the paper independent from the space-
time at the large sphere $S_{3}$, i.e., becoming a global gauge. This is more
consistent with confinement than pure gauge. Another case is when $x_{4}=-it$
can be neglected, index q again can be calculated to be zero. This may show a
triviality of the high energy behaviour with the axial current conservation,
sharing similar spirit as the divergence cancellation in our present paper
with Energy larger larger than 3-momentum on the infinite momentum surface.

So it is important to clarify the concrete condition for the divergence
(un)cancellation, the surface term ($=$ or $\neq$ 0). The concrete manifold of
space-time can make zero nonzero, make finite nonfinite, can make a
renormalizable field theory non renormalizable, or need more operators to
close, etc. Besides others the anomaly is very important considering the ’t
Hooft way to get charge non-conservation. This can explain the non-
conservation of U(1) number and CP violation of early universe, and
conservation restoration now, based the difference of the property of the
space-time. In other words, divergence integral (or cancellation, or the
concrete value of the surface term) in the calculation of the QFT in momentum
space is a sensitive probe on the topology or some other local structure
(holes…) of the space-time.

For the study of the cosmology, the above discussion on the (ultraviolet)
divergence cancellation possibly depending on the concrete (local) structure
of spacetime implies interesting speculation. The ’historical versions’ of
QFT’s vary with the universe evolution. QFT’s can be defined on various
spacetime math structure. In more details, different versions of QFT’s,
defined on different manifolds or more general math structures corresponding
to our universe—spacetime—at various special periods can be self -consistent
or -inconsistent. But in whatsoever cases, the divergences may cancel to get
finite (including zero) prediction or may not and can not give clear
prediction, it can always probe the math structure of the spacetime of the
universe. The ultraviolet probes the local, the infrared probes the global, or
just manifests the special information of the space time which leads to the
inconsistency between the spacetime and QFT. In the more concrete cases
addressed above in the paper, the spacetime in early universe manifold may
have singularity, defect, bubble wall or other structure to cause the anomaly
to give the CP violation and U(1) charge nonconservation for baryogenesis but
these structures of the spacetime manifold now evolute/expand so that locally
become Minkowski and no anomaly, no violations but only conservation of the
corresponding currents, which is consistent with experiments and has to ask
for cancellation in some theoretical framework without elimination inherently.
At the same time these early defects or structures can also play the role of
’seed’ of the curvature of spacetime (primordial curvature perturbation) via
the homogeneous Einstein equations, i.e., this curvature is not caused by
’matter’ (inhomogeneous term in Einstein equations) but is now observed as
’dark matter’ when the universe evolute to today.

## Acknowledgments

The authors greatly thank Prof. Tai Tsun Wu for stimulating the topic related
with the gauge paradox, for encouragement and many instructive discussions.

This work is supported in part by National Natural Science Foundation of China
(grant No. 12275157, 11775130, 11635009).

##  Appendix A Mathematics for $H\to\gamma Z$ corresponding to Eqs. (2.5-2.12)
of GWW Gastmans:2011wh

(These formulation will recover to the complemented GWW formulations which we
employed in the above for calculating the $H\to\gamma\gamma$ process once
taking $M_{Z}=0$.)

$k_{1}^{2}=0,~{}k_{2}^{2}=M_{Z}^{2};\hskip 85.35826ptk_{1\mu}=k_{2\nu}=0.$
(87) $(k_{1}+k_{2})^{2}=2k_{1}\cdot k_{2}+M_{Z}^{2}=M_{H}^{2}.$ (88)
$\displaystyle
V_{\alpha\beta\gamma}(p_{1},p_{2},p_{3})=(p_{2}-p_{3})_{\alpha}g_{\beta\gamma}+(p_{3}-p_{1})_{\beta}g_{\gamma\alpha}+(p_{1}-p_{2})_{\gamma}g_{\alpha\beta};$
(89) $\displaystyle p_{1}+p_{2}+p_{3}=0~{}~{}(incoming).$ $\displaystyle
p_{1}^{\alpha}V_{\alpha\beta\gamma}(p_{1},p_{2},p_{3})=(p_{3}^{2}g_{\beta\gamma}-p_{3\beta}p_{3\gamma})-(p_{2}^{2}g_{\beta\gamma}-p_{2\beta}p_{2\gamma})$
(90) $\displaystyle
V_{\alpha\beta\gamma}(p_{1},p_{2},p_{3})p_{3}^{\gamma}=-(p_{1}^{2}g_{\alpha\beta}-p_{1\alpha}p_{1\beta})+(p_{2}^{2}g_{\alpha\beta}-p_{2\alpha}p_{2\beta})$
$\displaystyle
p_{1}^{\alpha}V_{\alpha\mu\gamma}(p_{1},-k_{1},p_{3})=p_{3}^{2}g_{\mu\gamma}-p_{3\mu}p_{3\gamma}$
(91) $\displaystyle
V_{\alpha\mu\gamma}(p_{1},-k_{1},p_{3})p_{3}^{\gamma}=-(p_{1}^{2}g_{\alpha\mu}-p_{1\alpha}p_{1\mu})$
$\displaystyle
p_{1}^{\alpha}V_{\alpha\nu\gamma}(p_{1},-k_{2},p_{3})=p_{3}^{2}g_{\nu\gamma}-p_{3\nu}p_{3\gamma}-M_{Z}^{2}g_{\nu\gamma}$
(92) $\displaystyle
V_{\alpha\nu\gamma}(p_{1},-k_{2},p_{3})p_{3}^{\gamma}=-(p_{1}^{2}g_{\alpha\nu}-p_{1\alpha}p_{1\nu})+M_{Z}^{2}g_{\alpha\nu}$
$\displaystyle
p_{1}^{\alpha}V_{\alpha\mu\gamma}(p_{1},-k_{1},p_{3})=(p_{3}^{2}-M^{2})g_{\mu\gamma}-p_{3\mu}p_{3\gamma}+M^{2}g_{\mu\gamma}$
(93) $\displaystyle
V_{\alpha\mu\gamma}(p_{1},-k_{1},p_{3})p_{3}^{\gamma}=-[(p_{1}^{2}-M^{2})g_{\alpha\mu}-p_{1\alpha}p_{1\mu}]-M^{2}g_{\alpha\mu}$
$\displaystyle
p_{1}^{\alpha}V_{\alpha\nu\gamma}(p_{1},-k_{2},p_{3})=(p_{3}^{2}-M^{2})g_{\nu\gamma}-p_{3\nu}p_{3\gamma}+(M^{2}-M_{Z}^{2})g_{\nu\gamma}$
(94) $\displaystyle
V_{\alpha\nu\gamma}(p_{1},-k_{2},p_{3})p_{3}^{\gamma}=-[(p_{1}^{2}-M^{2})g_{\alpha\nu}-p_{1\alpha}p_{1\nu}]-(M^{2}-M_{Z}^{2})g_{\alpha\nu}$
$p_{1}^{\alpha}V_{\alpha\mu\gamma}(p_{1},-k_{1},p_{3})p_{3}^{\gamma}=0$ (95)
$p_{1}^{\alpha}V_{\alpha\nu\gamma}(p_{1},-k_{2},p_{3})p_{3}^{\gamma}=-M_{Z}^{2}p_{3\nu}=M_{Z}^{2}p_{1\nu}$
(96)

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