# One-loop fermion-photon vertex in arbitrary gauge and dimensions: a novel
approach

Victor Miguel Banda Guzmán<EMAIL_ADDRESS>Adnan Bashir
adnan.bashir<EMAIL_ADDRESS>Instituto de Física y Matemáticas
Universidad Michoacana de San Nicolás de Hidalgo Edificio C-3, Apdo. Postal
2-82 C.P. 58040, Morelia, Michoacán, México  Theory Center, Jefferson Lab,
Newport News, VA 23606, USA

###### Abstract

We compute one-loop electron-photon vertex with fully off-shell external
momenta in an arbitrary covariant gauge and space-time dimension. There exist
numerous efforts in literature where one-loop off-shell vertex is calculated
by employing the standard first order Feynman rules in different covariant
gauges and space-time dimensions of interest. The tensor structure which
decomposes this three-point vertex into the components transverse and
longitudinal to the photon momentum gets intertwined in this first order
formalism. The Ward-Takahashi identity is explicitly invoked to untangle the
two pieces and the results are expressed in a preferred basis of twelve spin-
amplitudes. We propose a novel approach based upon an efficient combination of
the first and second order formalisms of quantum electrodynamics to compute
this one-loop vertex. Among some conspicuous advantages is the fact that this
less known second order formalism separates the spin and scalar degrees of
freedom of an electron interacting electromagnetically. More noticeably, the
longitudinal and transverse contributions naturally disentangle from the onset
in our approach. Moreover, this decomposition leads to identities between one-
loop scalar Feynman integrals with higher powers in the propagators and
shifted space-time dimensions that can be used to prove the Ward-Takahashi
identity at one-loop order without the need to evaluate any Feynman integral.
Additionally, this natural decomposition allows us to establish the gauge-
independence of the Pauli form factor through explicit cancellations of scalar
Feynman integrals that depend on the gauge parameter. These cancellations
naturally lead to a compact expression for the Pauli form factor in arbitrary
dimensions. Wherever necessary and insightful, we make comparisons with
earlier works.

##  I Introduction

Solving any quantum field theory (QFT) is equivalent to computing its Green
functions. Three and four-point interaction vertices are the defining Green
functions of any QFT within the standard model of particle physics. These
vertices appear at the level of the Lagrangian. The simplest three-point
vertex in quantum electrodynamics (QED) involves a photon interacting with a
fermion, a charged lepton (like electron) or a quark. Therefore, the fermion-
photon vertex not only orchestrates the dance of events when purely
electromagnetic interactions are involved but it also plays a crucial role in
unravelling the internal structure of hadrons. Though this structure and
dynamics of hadrons are predominantly determined by quantum chromodynamics
(QCD), it is generally probed through electromagnetic interactions of photons
with electrically charged quarks which compose all hadrons. One such example
is provided by hadron electromagnetic form factors. The quark-photon vertex
not only ensures charge conservation at zero photon momentum-transfer through
the vector Ward-Takahashi identity (WTI) but its labyrinthine details also
explain how the asymptotic limit of these form factors is faithfully
approached for large momentum-transfer of the probing photons.

A popular approach to study non-perturbative effects in a QFT in continuum is
through the infinite set of Schwinger-Dyson equations (SDEs) which constitute
the defining equations of motion of a QFT. Their intricate mathematical
structure is such that the SDE of the $2$-point Green function is coupled to
that of the $3$-point Green function through an integral equation, the one of
the 3-point function is in turn entangled with that of the 4-point function ad
infinitum. Any practical attempt to look for a non-perturbative solution is
generally made by proposing a reliable ansatz for the relevant interaction
vertices to truncate the infinite tower of these coupled equations to a finite
solvable number. It is only natural to assume that any non-perturbative
construction, say that of the fermion-photon vertex, must agree with its
perturbative expansion in the weak coupling regime. Thus the perturbative
knowledge of this vertex can be used as a guiding tool for constructing such
an ansatz. In literature, several such studies can be found, spanned over the
last four decades. For example,

  * •

In [1], one-loop QED vertex was computed in the Feynman gauge and four space-
time dimensions, i.e., $D=4$. A convenient choice of the basis vectors was
adopted therein such that the corresponding coefficients are free of kinematic
singularities.

  * •

In Ref. [2], one-loop vertex was reported in the asymptotic limit of momenta
where momentum squared in one of the fermion legs is much greater than in the
other.

  * •

The work of Ball and Chiu in Ref. [1] was extended to an arbitrary covariant
gauge in Ref. [3]. The appearance of unwanted kinematic singularities for this
general choice of an arbitrary covariant gauge was observed. However, this
shortcoming was easily cured by redefining two of the basis vectors as a
superposition of the previous set of these vectors.

  * •

In three space-time dimensions, $D=3$, this vertex for massless and massive
fermions in an arbitrary covariant gauge was calculated in Refs. [4] and [5],
respectively.

  * •

For scalar QED, the one-loop vertex in arbitrary covariant gauge and
dimensions was reported in Ref. [6].

  * •

A complete calculation of the quark-gluon vertex in an arbitrary covariant
gauge and dimensions was carried out in detail in [7]. All spinor QED results
can be derived from the general expressions present therein through an
appropriate choice of the color factors and selecting $D$ as desired.

  * •

In Ref. [8], numerical and analytical expressions for the transverse and
longitudinal form factors of the quark-gluon vertex in different kinematical
regimes are presented. Again, one can deduce QED results from there.

Following Ref. [1], subsequent works commence with the WTI which relates the
full fermion-photon vertex $\Gamma^{\mu}(p,p^{\prime},k)$ with the full
fermion propagator $S(p)$ not only at every order of perturbation theory but
even more generally, non-perturbatively:

$\displaystyle k\cdot\Gamma(p,p^{\prime},k)=S^{-1}(p^{\prime})-S^{-1}(p)\;,$
(1)

where $k$ is the incoming momentum of the photon, $p$ is that of the fermion
and $p^{\prime}$ is the outgoing momentum of the latter. This identity readily
allows us to decompose the vertex into two components, the so called
longitudinal piece $\Gamma^{\mu}_{L}(p,p^{\prime},k)$ and the remaining part
$\Gamma^{\mu}_{T}(p,p^{\prime},k)$ which is transverse to the photon
4-momentum $k^{\mu}$ 111Our metric in the Minkowski space is
$\eta^{\mu\nu}=\text{diag}(-,+,+,+)$, while the Dirac gamma matrices satisfy
the anticommutation relation
$\\{\gamma^{\mu},\gamma^{\nu}\\}=-2\eta^{\mu\nu}$.:

$\displaystyle\Gamma^{\mu}(p,p^{\prime},k)=\Gamma^{\mu}_{L}(p,p^{\prime},k)+\Gamma^{\mu}_{T}(p,p^{\prime},k)\;,$
(2)

where $\Gamma^{\mu}_{T}(p,p^{\prime},k)$ satisfies the following restrictions:

$\displaystyle
k\cdot\Gamma_{T}(p,p^{\prime},k)=0\;,\quad\Gamma_{T}(p,p,0)=0\;.$ (3)

The longitudinal part $\Gamma^{\mu}_{L}(p,p^{\prime},k)$ alone satisfies the
WTI. Starting from the limiting form of the WTI, namely, the Ward identity

$\displaystyle\Gamma^{\mu}(p,p,0)=\frac{\partial}{\partial
p_{\mu}}S^{-1}(p)\;,$ (4)

Ball and Chiu proposed how to construct the longitudinal component, see [1]
for the details and the explicit form of the vertex. We choose to call it the
Ball-Chiu vertex and adopt the following notation

$\displaystyle\Gamma^{\mu}_{L}(p,p^{\prime},k)=\Gamma^{\mu}_{BC}(p,p^{\prime},k)\;.$
(5)

We would like to emphasize that neither the method of its construction nor the
final form of the vertex is unique. However, it has become a standard practice
to decompose the vertex in this manner:

$\displaystyle\Gamma^{\mu}(p,p^{\prime},k)=\Gamma^{\mu}_{BC}(p,p^{\prime},k)+\Gamma^{\mu}_{T}(p,p^{\prime},k)\;.$
(6)

All one-loop results for the vertex are then presented by ensuing the
following strategy:

  * •

The vertex in a particular kinematic configuration, gauge and dimensions is
calculated at the one-loop level.

  * •

Accordingly, one-loop fermion propagator is also computed within the same set
of requirements.

  * •

Using the last result, $\Gamma^{\mu}_{BC}(p,p^{\prime},k)$ is evaluated at the
one-loop level: $\Gamma^{\mu\,1-{\rm loop}}_{BC}(p,p^{\prime},k)$.

  * •

One-loop transverse vertex is obtained by simply subtracting the longitudinal
vertex from the full vertex:

$\displaystyle\Gamma^{\mu\,1-{\rm loop}}_{T}=\Gamma^{\mu\,1-{\rm
loop}}-\Gamma^{\mu\,1-{\rm loop}}_{BC}\;.$

  * •

This result is finally projected onto the transverse basis proposed in [1] or
[3] and the transverse form factors are identified for each basis vector.

The procedure outlined above is cumbersome though straightforward. There is a
plethora of available research dedicated to constructing increasingly refined,
reliable and physically meaningful ans$\ddot{a}$tze, taking perturbation
theory as a guide where key elements of a QFT, such as gauge invariance and
renormalization, are satisfied order by order in a systematic manner. A
comprehensive list of articles is hard to be provided without intentionally
overlooking some of the several efforts in the literature. However, a selected
list of references akin to the work we present here is [9, 10, 11, 12, 13, 14,
15, 16, 17, 18, 19].

All the one-loop results we have discussed so far are obtained solely through
employing the standard first order formalism of QED. In this article, we weave
this approach with a relatively less scrutinized second order formalism, Refs.
[20, 21, 22, 23]. Explicit calculations presented here convincingly reveal
that an adequate merger of both the first and the second order formalisms is
particularly advantageous and efficient in the following aspects:

1\. Vertex decomposition: We shall observe in the next section that the
electron-photon vertex $V^{\mu}(p,p^{\prime},k)$ decomposes naturally into its
longitudinal and transverse pieces at one-loop order without requiring a Ball-
Chiu like decomposition222From now on, we shall adopt the notation $V^{\mu}$
for the one-loop vertex instead of $\Gamma^{\mu}$ which we will reserve for
the full vertex.:

$\displaystyle
V^{\mu}(p,p^{\prime},k)=V_{L}^{\mu}(p,p^{\prime},k)+V_{T}^{\mu}(p,p^{\prime},k)\;,$
(7)

where $V_{T}^{\mu}$ satisfies the transversality condition

$\displaystyle k\cdot V_{T}(p,p^{\prime},k)=0\;,$ (8)

while the longitudinal piece $V_{L}^{\mu}$ satisfies the WTI at the same level
of approximation:

$\displaystyle k\cdot
V_{L}(p,p^{\prime},k)=e\left[\Sigma(\not{p})-\Sigma(\not{p}^{\prime})\right],$
(9)

where $\Sigma(\not{p})$ is the fermion self-energy and $e$ is the usual QED
coupling. We believe that this feature of a natural decomposition of the full
vertex into its longitudinal and transverse components would continue to
persist at higher orders of the electromagnetic coupling. However, only an
explicit calculation will be able to confirm this statement.

2\. Expeditious computation: By employing an astute concoction of the first
and the second order formalisms, we are able to select efficient routes to not
only expedite the complete computation of one-loop vertex in arbitrary gauge
and dimensions but also write it in a much more compact and concise form.
However, in order to make a swift comparison with other results in the
literature, we can always project our result onto the Ball-Chiu [1] or
Kızelers${\rm\ddot{u}}$-Pennington [3] basis.

An added advantage of using this combined analysis is to keep track of those
terms which identically vanish when external momenta go on-shell at all
intermediate stages of this calculation. Both in the longitudinal and the
transverse components of the vertex the terms which are explicitly irrelevant
on-shell are ordered conveniently, e.g., the operator $(\not{p}+m)$ remains on
the far right whereas $(\not{p}-m)$ is present on the left. Therefore, when
on-shell conditions $(\not{p}+m)u_{s}(p)=0$ and
$\overline{u}_{s}(p)(\not{p}+m)=0$ are imposed, they vanish. After eliminating
these terms and using the on-shell symmetries of the Feynman integrals, we
explicitly see the instantaneous cancellations of the integrals that depend
explicitly on the covariant gauge parameter $\xi$, e.g., in the evaluation of
the on-shell Pauli form factor. These cancellations naturally lead to a
compact expression for it in an arbitrary space-time dimension in terms of
scalar integrals with: (i) higher powers of scalar propagators and (ii)
shifted dimensions. It hints towards a possible analogous simplification in
the evaluation of the anomalous magnetic moment of the charged fermion at
higher orders of perturbation theory.

The article is organized as follows: Sec. II contains a summary of the second
order formalism. In Sec. III, we apply a convenient combination of first and
second order formalisms to the fermion-photon vertex to decompose it into its
longitudinal and transverse components. In Sec. IV, we provide explicit
confirmation of the fact that the longitudinal part of the vertex is indeed
longitudinal, i.e., it satisfies the WTI. We express it in terms of Feynman
integrals. Then in Sec. V, transverse contributions of the vertex which were
previously obtained in Sec. III, are written in terms of Feynman scalar
integrals. Employing these expressions, and the results obtained in Sec. IV
for the longitudinal part of the vertex, we also provide the coefficients of
the Ball-Chiu basis in terms of scalar integrals. In Sec. VI we calculate the
Pauli form factor in $D$ dimensions, explicitly showing its independence of
the covariant gauge parameter $\xi$ by cancellations of the scalar integrals
that depend on this parameter in the longitudinal and transverse contributions
of the vertex. A compact expression for the Pauli form factor in terms of
scalar integrals is obtained. Making use of this general expression, special
cases, i.e., $D=4$ and $D=3$ are reviewed, recovering the known results in the
literature. Concluding remarks with prospects for future endeavors are
provided in Sec. VII. The manuscript is complemented with three appendices
with supplemental information. Appendix A provides explicit difference between
the Ball-Chiu and our construction of the longitudinal vertex in perturbation
theory. Naturally one is related to the other through the addition of a
transverse piece. Appendices B and C detail expressions of the transverse and
longitudinal vertices, respectively, at one-loop order, and the scalar
integrals involved with relevant identities relating them.

##  II Second order formalism for QED

The second order formalism for QED is based on Feynman rules displayed in Fig.
1 which were formally derived in [21]. It provides an efficacious
correspondence between spinor and scalar QED. This formalism has been used to
compute fermion loops and amplitudes involving external fermions, see [21] and
references therein. In this article, we apply this strategy to achieve similar
success in the off-shell evaluation of the QED vertex. The only difference is
the additional spin factor
$\sigma^{\mu\nu}=\frac{1}{2}[\gamma^{\mu},\gamma^{\nu}]$ that appears in the
QED vertex. It disentangles explicit contributions of the electromagnetic
interaction of the fermionic particle into scalar and spin degrees of freedom
with appropriate usage of Gordon-like identities. In addition to the rules of
Fig. 1, whenever a fermionic loop appears in a Feynman diagram, an extra
factor of $-1/2$ must be included in the amplitude. Though the structure of
the first order and the second order rules are different, the computed
amplitude of a QED process following both formalisms is equivalent.

Figure 1: Second order rules for spinor QED

As mentioned before, one can derive the second order rules from the first
order [21], by formally rewriting the product of the fermion propagator
$S(p+k)$ with the first order vertex $e\gamma^{\mu}$ as,

$\displaystyle e\,S(p+k)\,\gamma^{\mu}$ $\displaystyle=$ $\displaystyle
e\,\dfrac{-(\not{p}+\not{k})+m}{(p+k)^{2}+m^{2}}\,\gamma^{\mu}=\dfrac{A^{\mu}_{p,k}}{D_{p+k}},$
(10)

where

$\displaystyle A^{\mu}_{p,k}$ $\displaystyle=$ $\displaystyle
B^{\mu}_{p,k}+C_{p}^{\mu},$ $\displaystyle B^{\mu}_{p,k}$ $\displaystyle=$
$\displaystyle e(2p+k)^{\mu}+e\sigma^{\mu\alpha}k_{\alpha},$ $\displaystyle
C_{p}^{\mu}$ $\displaystyle=$ $\displaystyle e\gamma^{\mu}(\not{p}+m),$
$\displaystyle D_{q}$ $\displaystyle=$ $\displaystyle q^{2}+m^{2}.$ (11)

Here, $D_{q}^{-1}$ represents the scalar propagator, and $B_{p,k}$ is the
three point vertex of the second order rules which naturally separates the
scalar and spin electromagnetic interactions. Thus a product of a propagator
and a vertex in a first order Feynman diagram can be decomposed into a second
order contribution, and a left over first order term given by the operator
$C_{p}^{\mu}$.

Now, the four-point scalar vertex that appears in the second order rules
arises when there are two consecutive pairs of a fermion propagator and a
first order vertex in a Feynman diagram, as shown in Fig. 2. An amplitude
constructed from a first order Feynman diagram that contains a subdiagram as
the one shown in Fig. 2 is proportional to
$e^{2}S(p+k_{1}+k_{2})\gamma^{\nu}S(p+k_{1})\gamma^{\mu}$. According to Eq.
(10), it can be decomposed as,

$\displaystyle
e^{2}S(p+k_{1}+k_{2})\gamma^{\nu}S(p+k_{1})\gamma^{\mu}=\dfrac{A^{\nu}_{p+k_{1},k_{2}}}{D_{p+k_{1}+k_{2}}}\dfrac{A^{\mu}_{p+k_{1}}}{D_{p+k_{1}}}.$
(12)

Since

$\displaystyle C_{p+k}^{\mu}\;\dfrac{A_{p,k}^{\nu}}{D_{p+k}}$ $\displaystyle=$
$\displaystyle
e^{2}\gamma^{\mu}\gamma^{\nu}-e^{2}\left(\eta^{\mu\nu}-\sigma^{\mu\nu}\right),$
(13)

Eq. (12) acquires the following form

$\displaystyle e^{2}S(p+k_{1}+k_{2})\gamma^{\nu}S(p+k_{1})\gamma^{\mu}=$
$\displaystyle\hskip
40.00006pt\dfrac{B^{\nu}_{p+k_{1},k_{2}}}{D_{p+k_{1}+k_{2}}}\dfrac{A^{\mu}_{p+k_{1}}}{D_{p+k_{1}}}-e^{2}\dfrac{\eta^{\mu\nu}+\sigma^{\mu\nu}}{D_{p+k_{1}+k_{2}}}.$
(14)

Thus, the multiplication of fermion propagators and first order vertices on
the left-hand side of this equation generates a term that is proportional to
$\eta^{\mu\nu}$ and has one less power of the scalar propagator. It readily
gets identified as a four-point scalar vertex.

Following the procedure where fermion propagators and first order vertices are
written according to Eq. (10) and applying identity (13), any amplitude
constructed from the first order rules can be obtained equivalently and
conveniently from the second order rules given in Fig. (1). The proof of this
equivalence can be found in Ref. [21]. We now proceed to show how this
systematic procedure can generate a natural decomposition of the QED vertex
into its longitudinal and transverse pieces.

Figure 2: Subdiagram that produces the contribution to the four-point scalar
vertex of the second order formalism.

##  III The QED vertex at one-loop from the combination of the first and
second order formalisms

Fig. 3 depicts the Feynman diagram for the vertex function $V^{\mu}$ at one-
loop within the first order formalism. Following the standard Feynman rules,
its mathematical expression is:

$V^{\mu}(p^{\prime},p)=\dfrac{e^{3}}{i}\int\dfrac{d^{D}l}{(2\pi)^{D}}\left[\gamma^{\rho}S(p^{\prime}+l)\gamma^{\mu}S(p+l)\gamma^{\nu}\right]\Delta_{\nu\rho}(l),$
(15)

where the photon propagator $\Delta_{\mu\nu}(l)$ in an arbitrary covariant
gauge $\xi$ is given by:

$\Delta_{\mu\nu}(l)=\dfrac{1}{l^{2}}\left(\eta_{\mu\nu}-(1-\xi)\dfrac{l_{\mu}l_{\nu}}{l^{2}}\right).$
(16)

Using Eq. (10), the vertex $V^{\mu}$ of Eq. (15) becomes:

$\displaystyle
V^{\mu}(p^{\prime},p)=e\int\dfrac{d^{D}l}{i(2\pi)^{D}}\gamma^{\rho}\left(\dfrac{A_{p+l,k}^{\mu}}{D_{p^{\prime}+l}}\right)\left(\dfrac{A^{\nu}_{p,l}}{D_{p+l}}\right)\Delta_{\rho\nu}(l),$
(17)

Figure 3: One-loop vertex $V^{\mu}(p^{\prime},p)$ in the first order
formalism.

which can be written as:

$\displaystyle V^{\mu}(p^{\prime},p)$ $\displaystyle=$ $\displaystyle
e\,\int\dfrac{d^{D}l}{i(2\pi)^{D}}\,\gamma^{\rho}\Bigg{(}\dfrac{B^{\mu}_{p+l,k}\,B^{\nu}_{p,l}}{D_{p^{\prime}+l}D_{p+l}}+\dfrac{B^{\mu}_{p+l,k}\,C^{\nu}_{p}}{D_{p^{\prime}+l}D_{p+l}}$
(18)
$\displaystyle+e^{2}\dfrac{\gamma^{\mu}\,\gamma^{\nu}}{D_{p^{\prime}+l}}\Bigg{)}\Delta_{\rho\nu}(l),$

after application of identity (13), and the definitions given in Eqs. (11).
Here $V^{\mu}$ is expressed as a combination of first and second order
vertices, with the corresponding scalar propagators of the Feynman diagram of
Fig. (3). The numerator of the third term on the right-hand side of Eq. (18)
can be rearranged as

$\displaystyle{l^{4}}\gamma^{\rho}\,\gamma^{\mu}\,\gamma^{\nu}\,\Delta_{\rho\nu}(l)={l^{2}(D-3+\xi)\gamma^{\mu}+2(1-\xi)l^{\mu}\not{l}},$
(19)

where we have used the following identity:

$\displaystyle{l^{2}}\gamma^{\rho}\,\gamma^{\nu}\,\Delta_{\rho\nu}={{1-D-\xi}}.$
(20)

The result that stems from expression (19) will form part of the longitudinal
vertex as we show later. On the other hand, the second term on the right-hand
side of Eq. (18) can be split into longitudinal and transverse pieces by
commuting $\gamma^{\rho}$ with $B^{\mu}_{p+l,k}$. Using the identity

$\displaystyle\gamma^{\mu}\,B^{\nu}_{p,k}=B^{\nu}_{p,k}\,\gamma^{\mu}-2e\,\eta^{\mu\nu}\not{k}+2e\,k^{\mu}\gamma^{\nu},$
(21)

the numerator of the second term acquires the form:

$\displaystyle\gamma^{\rho}\,B^{\mu}_{p+l,k}\,C^{\nu}_{p}\,\Delta_{\rho\nu}$
$\displaystyle=e\Big{(}B^{\mu}_{p+l,k}\,\gamma^{\rho}\,\gamma^{\nu}-2e\,\eta^{\rho\mu}\not{k}\,\gamma^{\nu}+2e\,k^{\rho}\,\gamma^{\mu}\,\gamma^{\nu}\Big{)}$
$\displaystyle\hskip 8.0pt\times\Delta_{\rho\nu}\left(\not{p}+m\right)$
$\displaystyle=\dfrac{e^{2}}{l^{2}}\Big{[}(1-D-\xi)(p+p^{\prime}+2l)^{\mu}+\dfrac{2(1-\xi)}{l^{2}}\Big{(}l^{\mu}\not{k}\not{l}$
$\displaystyle\hskip 8.0pt-k\cdot
l\,\gamma^{\mu}\not{l}\Big{)}+(5-D-\xi)\sigma^{\mu\alpha}k_{\alpha}\Big{]}(\not{p}+m),$
(22)

where we have made use of identity (20). It can be readily observed that only
the term proportional to $(2p+k+2l)^{\mu}$ pertains to the longitudinal
vertex; the other terms are transverse. Similar procedure allows us to re-
write the numerator of the first term on the right-hand side of Eq. (18) as
follows:

$\displaystyle\gamma^{\rho}\,B^{\mu}_{p+l,k}\,B^{\nu}_{p,l}\,\Delta_{\rho\nu}$
$\displaystyle=$ $\displaystyle
B^{\mu}_{p+l,k}\,\gamma^{\rho}\,B^{\nu}_{p,l}\,\Delta_{\rho\nu}-2e\,\not{k}\,B^{\nu}_{p,l}\,\Delta_{\mu\nu}$
(23)
$\displaystyle+2e\,k^{\rho}\,\gamma^{\mu}\,B^{\nu}_{p,l}\,\Delta_{\rho\nu}.$

Since

$\displaystyle\gamma^{\rho}\,B^{\nu}_{p,l}\,\Delta_{\rho\nu}$ $\displaystyle=$
$\displaystyle\dfrac{e}{l^{2}}\Bigg{[}2\not{p}+\Bigg{(}1-D+\xi$
$\displaystyle-(1-\xi)\dfrac{2p\cdot l}{l^{2}}\Bigg{)}\not{l}\Bigg{]},$
$\displaystyle\not{k}\,B^{\nu}_{p,l}\,\Delta_{\mu\nu}$ $\displaystyle=$
$\displaystyle\dfrac{1}{l^{2}}\Big{[}B^{\mu}_{p,l}\,\not{k}+2e(k\cdot
l\gamma^{\mu}-k^{\mu}\not{l})$ $\displaystyle-e(1-\xi)\dfrac{2p\cdot
l+l^{2}}{l^{2}}l^{\mu}\,\not{k}\Big{]},$ $\displaystyle
k^{\rho}\,B^{\nu}_{p,l}\,\Delta_{\rho\nu}$ $\displaystyle=$
$\displaystyle\dfrac{e}{l^{2}}\Bigg{[}k\cdot(2p+l)+\sigma^{\beta\alpha}l_{\alpha}k_{\beta}$
(24) $\displaystyle-(1-\xi)k\cdot l\left(2\dfrac{p\cdot
l}{l^{2}}+1\right)\Bigg{]},$

we can now make use of these detailed identities to cast Eq. (23) in the
following form

$\displaystyle\gamma^{\rho}\,B^{\mu}_{p+l,k}\,B^{\nu}_{p,l}\,\Delta_{\rho\nu}=$
$\displaystyle\dfrac{e^{2}}{l^{2}}\Big{[}2\left[(p+p^{\prime}+2l)^{\mu}+\sigma^{\mu\alpha}k_{\alpha}\right]\not{p}+4(p\cdot
k\,\gamma^{\mu}-p^{\mu}\not{k})$ $\displaystyle+2\xi\left(l\cdot
k\,\gamma^{\mu}-l^{\mu}\,\not{k}\right)+4\left(\gamma^{\mu}\not{k}\not{l}+k^{\mu}\,\not{l}\right)$
$\displaystyle+2\left(k\cdot
l\,\gamma^{\mu}-l^{\mu}\,\not{k}\right)+(1-D+\xi)\big{[}(p+p^{\prime}+2l)^{\mu}$
$\displaystyle+\sigma^{\mu\alpha}k_{\alpha}\big{]}\not{l}-2\,\dfrac{1-\xi}{l^{2}}\Big{\\{}p\cdot
l\left[(p+p^{\prime}+2l)^{\mu}+\sigma^{\mu\alpha}k_{\alpha}\right]\not{l}$
$\displaystyle+2\left[l\cdot kp\cdot l\gamma^{\mu}-p\cdot
ll^{\mu}\,\not{k}\right]\Big{\\}}\Big{]}.$ (25)

Again we have an unambiguous separation between the longitudinal and
transverse components of the vertex. The former correspond to the terms
proportional to $(p+p^{\prime})\not{p}$, $l^{\mu}\not{p}$,
$(p+p^{\prime})\not{l}$ and $l^{\mu}\not{l}$, while the transverse parts form
the remaining terms. According to Eqs. (19), (22), (25), the longitudinal and
transverse components, $V^{\mu}_{L}$ and $V^{\mu}_{T}$, respectively, defined
in Eq. (7), read as,

$\displaystyle V^{\mu}_{L}$ $\displaystyle=$ $\displaystyle
e^{3}\int\dfrac{d^{D}l}{i(2\pi)^{D}}\,\Bigg{\\{}\dfrac{1}{D_{p^{\prime}+l}D_{p+l}l^{2}}\Bigg{[}2(p+p^{\prime}+2l)^{\mu}\not{p}+(1-D+\xi)(p+p^{\prime})^{\mu}\not{p}+2(1-D+\xi)l^{\mu}\not{l}$
(26) $\displaystyle-$ $\displaystyle
2(1-\xi)(p+p^{\prime}+2l)^{\mu}\dfrac{p\cdot
l\not{l}}{l^{2}}+(1-D-\xi)\left[(p+p^{\prime})^{\mu}+2l^{\mu}\right](\not{p}+m)\Bigg{]}+\dfrac{D-3+\xi}{D_{p^{\prime}+l}\,l^{2}}\gamma^{\mu}+2\dfrac{(1-\xi)l^{\mu}\,\not{l}}{D_{p^{\prime}+l}\,l^{4}}\Bigg{\\}},$
$\displaystyle V^{\mu}_{T}$ $\displaystyle=$ $\displaystyle
e^{3}\int\dfrac{d^{D}l}{i(2\pi)^{D}}\,\dfrac{1}{D_{p^{\prime}+l}D_{p+l}l^{2}}\Bigg{\\{}\left[(5-D-\xi)\sigma^{\mu\alpha}k_{\alpha}+\dfrac{2(1-\xi)}{l^{2}}\left(l^{\mu}\not{k}\not{l}-k\cdot
l\gamma^{\mu}\not{l}\right)\right](\not{p}+m)+2\sigma^{\mu\alpha}k_{\alpha}\not{p}$
(27) $\displaystyle+$ $\displaystyle 4(p\cdot
k\gamma^{\mu}-p^{\mu}\not{k})+2\xi\left(k\cdot
l\gamma^{\mu}-l^{\mu}\not{k}\right)+(1-D+\xi)\sigma^{\mu\alpha}k_{\alpha}\not{l}+4\left(\gamma^{\mu}\not{k}\not{l}+k^{\mu}\not{l}\right)+2(k\cdot
l\gamma^{\mu}-l^{\mu}\not{k})$ $\displaystyle-$ $\displaystyle
2(1-\xi)\dfrac{p\cdot
l}{l^{2}}\sigma^{\mu\alpha}k_{\alpha}\not{l}+4\dfrac{1-\xi}{l^{2}}\left(p\cdot
l\not{k}l^{\mu}-l\cdot kp\cdot l\gamma^{\mu}\right)\Bigg{\\}}.$

From the expression above for $V_{T}^{\mu}$, one can readily infer that it
satisfies the transversality condition $k\cdot V_{T}=0$. However, it is less
obviously discernible that $V_{L}^{\mu}$ satisfies the WTI given in Eq. (9).
However, we set out to show this explicitly in the next section.

##  IV $V_{L}^{\mu}$ and the Ward-Takashi identity

In order to demonstrate that the longitudinal vertex $V_{L}^{\mu}$ given by
Eq. (26) indeed satisfies the WTI at the one-loop level, a natural starting
point is to contract it with the photon momentum $k_{\mu}$:

$\displaystyle k\cdot V_{L}$ $\displaystyle=$ $\displaystyle
e^{3}\int\dfrac{d^{D}l}{i(2\pi)^{D}}\,\Bigg{\\{}\dfrac{1}{D_{p^{\prime}+l}D_{p+l}\,l^{2}}\Big{[}(1-D-\xi)$
(28) $\displaystyle\times[k\cdot(p+p^{\prime})+2k\cdot
l](m+\not{l})+(3-D-\xi)$ $\displaystyle\times[k\cdot(p+p^{\prime})+2k\cdot
l]\,\not{p}\Big{]}+\dfrac{D-3+\xi}{D_{p^{\prime}+l}\,l^{2}}\,\not{k}$
$\displaystyle+\dfrac{1-\xi}{D_{p^{\prime}+l}D_{p+l}\,l^{4}}\Big{[}2(p^{2}+m^{2})p^{\prime}\cdot
l+2l\cdot p^{\prime}l^{2}$ $\displaystyle-(p^{\prime 2}+m^{2})p\cdot l-2l\cdot
pl^{2}\Big{]}\not{l}\Bigg{\\}}.$

Perhaps the most efficient way to proceed is to reduce the vector and tensor
integrals involved into scalar integrals. We can then make use of the known
identities connecting these scalar integrals to simplify the resulting
expression and rearrange it in a convenient manner so that the final terms can
readily be identified with the expressions which define electron self energy.
One way to achieve this goal is to employ the tensor reduction algorithm
described first in Ref. [24] and later also used in Refs. [25]-[26]. It allows
us to write any multi-loop tensor integrals in terms of scalar integrals:

$\displaystyle\int\dfrac{d^{D}l}{i\pi^{D/2}}\dfrac{l^{\mu}}{D_{p^{\prime}+l}\,D_{p+l}\;l^{2\,a}}$
$\displaystyle=$
$\displaystyle-p^{\mu}\;J_{1,2,a}^{D+2}-p^{\prime\mu}\;J_{2,1,a}^{D+2},$
$\displaystyle\int\dfrac{d^{D}l}{i\pi^{D/2}}\dfrac{l^{\mu}l^{\nu}}{D_{p^{\prime}+l}\,D_{p+l}\;l^{2\,a}}$
$\displaystyle=$
$\displaystyle\dfrac{1}{2}\eta^{\mu\nu}\;J_{1,1,a}^{D+2}+2p^{\prime\mu}p^{\prime\nu}\;J_{3,1,a}^{D+4}$
(29)
$\displaystyle+2p^{\mu}p^{\nu}\;J_{1,3,a}^{D+4}+(p^{\mu}p^{\prime\nu}+p^{\nu}p^{\prime\mu})\;J_{2,2,a}^{D+4},$

where

$\displaystyle
J_{a,b,c}^{\tilde{D}}(p,p^{\prime})=\int\dfrac{d^{\tilde{D}}l}{i\pi^{\tilde{D}/2}}\dfrac{1}{D_{p^{\prime}+l}^{a}\,D_{p+l}^{b}\;l^{2\,c}}.$
(30)

Using this tensor reduction algorithm multiple times, Eq. (28) can be
rearranged in the following form:

$\displaystyle k\cdot V_{L}=$
$\displaystyle\dfrac{e^{3}}{(4\pi)^{\frac{D}{2}}}\Bigg{\\{}m(1-D-\xi)\Big{[}(p^{\prime
2}-p^{2})J_{1,1,1}^{D}-2k\cdot pJ_{1,2,1}^{D+2}$ $\displaystyle-2k\cdot
p^{\prime}J_{2,1,1}^{D+2}\Big{]}+\Big{(}(3-D-\xi)\Big{[}(p^{\prime
2}-p^{2})J_{1,1,1}^{D}$ $\displaystyle-2k\cdot pJ_{1,2,1}^{D+2}-2k\cdot
p^{\prime}J_{2,1,1}^{D+2}+J_{1,0,1}^{D}\Big{]}$
$\displaystyle+(2-D)\Big{[}(p^{2}-p^{\prime
2})J_{1,2,1}^{D+2}-J_{1,1,1}^{D+2}+2k\cdot p^{\prime}J_{2,2,1}^{D+4}$
$\displaystyle+4k\cdot pJ_{1,3,1}^{D+4}\Big{]}+(1-\xi)\Big{[}(p^{\prime
2}-p^{2})J_{1,2,1}^{D+2}$ $\displaystyle+(p^{2}+m^{2})\left(2p^{\prime
2}J_{2,2,2}^{D+4}+4p\cdot p^{\prime}J_{1,3,2}^{D+4}\right)$
$\displaystyle-(p^{\prime 2}+m^{2})\Big{(}J_{1,1,2}^{D+2}+2p\cdot
p^{\prime}J_{2,2,2}^{D+4}+4p^{2}J_{1,3,2}^{D+4}\Big{)}\,\Big{]}\;\Big{)}\;\not{p}$
$\displaystyle+\Bigg{(}(2-D)\Big{[}(p^{2}-p^{\prime
2})J_{2,1,1}^{D+2}+J_{1,1,1}^{D+2}+2k\cdot pJ_{2,2,1}^{D+4}$
$\displaystyle+4k\cdot
p^{\prime}J_{3,1,1}^{D+4}\Big{]}+(1-\xi)\Big{[}(p^{\prime
2}-p^{2})J_{2,1,1}^{D+2}$
$\displaystyle+(p^{2}+m^{2})\left(J_{1,1,2}^{D+2}+2p\cdot
p^{\prime}J_{2,2,2}^{D+4}+4p^{\prime 2}J_{3,1,2}^{D+4}\right)$
$\displaystyle-(p^{\prime 2}+m^{2})\left(2p^{2}J_{2,2,2}^{D+4}+4p\cdot
p^{\prime}J_{3,1,2}^{D+4}\right)\Big{]}$
$\displaystyle+(D-3+\xi)J_{1,0,1}^{D}\Bigg{)}\not{p}^{\prime}\Bigg{\\}}.$ (31)

All the scalar integrals appearing in Eq. (31) can be expressed as a linear
combination of the master integrals $J_{1,1,1}^{D}$, $J_{0,1,0}^{D}$,
$J_{0,1,1}^{D}$, $J_{1,0,1}^{D}$ and $J_{1,1,0}^{D}$ by implementing the well-
established and widely employed Integration by Parts Technique (IBP) [27, 28]
as well as dimensional recurrence relations [29, 30, 31]. These methods, aided
with the symbolic programming package LiteRed [32, 33], yield the following
practically useful identities:

$\displaystyle J_{0,1,1}^{D}-J_{1,0,1}^{D}$ $\displaystyle=$
$\displaystyle(p^{\prime 2}-p^{2})J_{1,1,1}^{D}-2k\cdot pJ_{1,2,1}^{D+2}$ (32)
$\displaystyle-2k\cdot p^{\prime}J_{2,1,1}^{D+2},$ $\displaystyle
J_{0,1,0}^{D}+(m^{2}+p^{2})J_{0,1,1}^{D}$ $\displaystyle=$
$\displaystyle-2p^{2}\Big{[}(p^{2}-p^{\prime 2})J_{1,2,1}^{D+2}$ (33)
$\displaystyle-J_{1,1,1}^{D+2}+2k\cdot p^{\prime}J_{2,2,1}^{D+4}$
$\displaystyle+4k\cdot pJ_{1,3,1}^{D+4}\Big{]},$ $\displaystyle
J_{0,1,0}^{D}+(p^{\prime 2}+m^{2})J_{1,0,1}^{D}$ $\displaystyle=$
$\displaystyle 2p^{\prime 2}\Big{[}(p^{2}-p^{\prime 2})J_{2,1,1}^{D+2}$ (34)
$\displaystyle+J_{1,1,1}^{D+2}+2k\cdot pJ_{2,2,1}^{D+4}$
$\displaystyle+4k\cdot p^{\prime}J_{3,1,1}^{D+4}\Big{]},$

and

$\displaystyle(2-D)J_{0,1,0}^{D}$ $\displaystyle=$
$\displaystyle\left[(4-D)p^{2}+(D-2)m^{2}\right]J_{0,1,1}^{D}$ (35)
$\displaystyle+2p^{2}\Big{[}(p^{\prime 2}-p^{2})J_{1,2,1}^{D+2}+(p^{2}+m^{2})$
$\displaystyle\times\left(2p^{\prime 2}J_{2,2,2}^{D+4}+4p\cdot
p^{\prime}J_{1,3,2}^{D+4}\right)$ $\displaystyle-(p^{\prime
2}+m^{2})(J_{1,1,2}^{D+2}+2p\cdot p^{\prime}J_{2,2,2}^{D+4}$
$\displaystyle+4p^{2}J_{1,3,2}^{D+4})\Big{]},$
$\displaystyle(2-D)J_{0,1,0}^{D}$ $\displaystyle=$
$\displaystyle\left[(4-D)p^{\prime\,2}+(D-2)m^{2}\right]J_{1,0,1}^{D}$ (36)
$\displaystyle+2p^{\prime\,2}\Big{[}(p^{\prime
2}-p^{2})J_{2,1,1}^{D+2}-(p^{\prime 2}+m^{2})$
$\displaystyle\times\left(2p^{2}J_{2,2,2}^{D+4}+4p\cdot
p^{\prime}J_{3,1,2}^{D+4}\right)$
$\displaystyle+(p^{2}+m^{2})(J_{1,1,2}^{D+2}+2p\cdot
p^{\prime}J_{2,2,2}^{D+4}$ $\displaystyle+4p^{\prime
2}J_{3,1,2}^{D+4})\Big{]}.$

Using these identities, Eq. (31) simplifies to

$\displaystyle k\cdot V_{L}$ $\displaystyle=$
$\displaystyle\dfrac{e^{3}}{(4\pi)^{\frac{D}{2}}}\Big{\\{}m(1-D-\xi)\left(J_{0,1,1}^{D}-J_{1,0,1}^{D}\right)$
(37) $\displaystyle+$
$\displaystyle\dfrac{D-2}{2p^{2}}\xi\left[J_{0,1,0}^{D}+(m^{2}-p^{2})J_{0,1,1}^{D}\right]\not{p}$
$\displaystyle-$ $\displaystyle\dfrac{D-2}{2p^{\prime
2}}\xi\left[J_{0,1,0}^{D}+(m^{2}-p^{\prime
2})J_{1,0,1}^{D}\right]\not{p}^{\prime}\Big{\\}}.$

Since fermion self-energy $\Sigma(\not{p})$ at one-loop in an arbitrary
covariant gauge $\xi$ and dimensions $D$ is given by [22]:

$\displaystyle\Sigma(\not{p})$ $\displaystyle=$
$\displaystyle\dfrac{e^{2}}{(4\pi)^{\frac{D}{2}}}\Big{\\{}m(1-D-\xi)J_{0,1,1}^{D}$
(38) $\displaystyle+$
$\displaystyle\dfrac{D-2}{2p^{2}}\xi\left[J_{0,1,0}^{D}+(m^{2}-p^{2})J_{0,1,1}^{D}\right]\not{p}\Big{\\}},$

it can be readily inferred that $V_{L}^{\mu}$ satisfies the WTI, Eq. (9), as
expected.

To wind up this section, let us write down the expression for $V_{L}^{\mu}$ in
terms of Feynman scalar integrals. Using Eq. (26) and the tensor reduction
algorithm, $V_{L}^{\mu}$ reads as:

$\displaystyle V_{L}^{\mu}$ $\displaystyle=$
$\displaystyle\dfrac{e^{3}}{(4\pi)^{\frac{D}{2}}}\Big{\\{}(D-1-\xi)(p+p^{\prime})^{\mu}\left(J_{1,2,1}^{D+2}\not{p}+J_{2,1,1}^{D+2}\not{p}^{\prime}\right)$
(39)
$\displaystyle+2J_{1,1,1}^{D}(p+p^{\prime})^{\mu}\not{p}-4\left(J_{1,2,1}^{D+2}p^{\mu}+J_{2,1,1}^{D+2}p^{\prime\mu}\right)\not{p}$
$\displaystyle+(2-D)\Big{[}J_{1,1,1}^{D+2}\,\gamma^{\mu}+2J_{2,2,1}^{D+4}(p^{\prime\mu}\not{p}+p^{\mu}\not{p}^{\prime})$
$\displaystyle+4J_{1,3,1}^{D+4}p^{\mu}\not{p}+4J_{3,1,1}^{D+4}p^{\prime\mu}\not{p}^{\prime}\Big{]}$
$\displaystyle+(1-D-\xi)\Big{[}J_{1,1,1}^{D}(p+p^{\prime})^{\mu}-2J_{1,2,1}^{D+2}p^{\mu}$
$\displaystyle-2J_{2,1,1}^{D+2}p^{\prime\mu}\Big{]}(\not{p}+m)+(D-3+\xi)J_{1,0,1}^{D}\,\gamma^{\mu}$
$\displaystyle+(1-\xi)(p^{2}+m^{2})\Big{[}J_{1,1,2}^{D+2}\,\gamma^{\mu}+4J_{1,3,2}^{D+4}p^{\mu}\not{p}$
$\displaystyle+4J_{3,1,2}^{D+4}p^{\prime\mu}\not{p}^{\prime}+2J_{2,2,2}^{D+4}(p^{\prime\mu}\not{p}+p^{\mu}\not{p}^{\prime})\Big{]}$
$\displaystyle-(1-\xi)(p+p^{\prime})^{\mu}\Big{[}J_{1,1,2}^{D+2}\not{p}+4p\cdot
p^{\prime}J_{3,1,2}^{D+4}\not{p}^{\prime}$
$\displaystyle+4p^{2}J_{1,3,2}^{D+4}\not{p}+2J_{2,2,2}^{D+4}(p\cdot
p^{\prime}\not{p}+p^{2}\not{p}^{\prime})\Big{]}\Big{\\}}.$

In the basis
$\\{p^{\mu},p^{\prime\mu},\gamma^{\mu},p^{\mu}\not{p},p^{\mu}\not{p}^{\prime},p^{\prime\mu}\not{p},p^{\prime\mu}\not{p}^{\prime}\\}$
it can be rewritten as follows:

$\displaystyle
V_{L}^{\mu}=\dfrac{e^{3}}{(4\pi)^{\frac{D}{2}}}\Big{\\{}m(1-\xi-D)(J_{1,1,1}^{D}-2J_{1,2,1}^{D+2})p^{\mu}$
$\displaystyle+m(1-\xi-D)(J_{1,1,1}^{D}-2J_{2,1,1}^{D+2})p^{\prime\mu}$
$\displaystyle+\Big{[}(2-D)J_{1,1,1}^{D+2}+(D-3+\xi)J_{1,0,1}^{D}$
$\displaystyle+(m^{2}+p^{2})(1-\xi)J_{1,1,2}^{D+2}\Big{]}\gamma^{\mu}$
$\displaystyle+\Big{[}(2-D)\left(J_{1,1,1}^{D}-3J_{1,2,1}^{D+2}+4J_{1,3,1}^{D+4}\right)$
$\displaystyle+(1-\xi)\big{(}J_{1,1,1}^{D}-J_{1,1,2}^{D+2}-J_{1,2,1}^{D+2}$
$\displaystyle+4m^{2}J_{1,3,2}^{D+4}-2p\cdot
p^{\prime}J_{2,2,2}^{D+4}\big{)}\Big{]}p^{\mu}\not{p}$
$\displaystyle+\Big{[}(D-2)\left(J_{2,1,1}^{D+2}-2J_{2,2,1}^{D+4}\right)+(1-\xi)\big{(}J_{2,1,1}^{D+2}$
$\displaystyle+2m^{2}J_{2,2,2}^{D+4}-4p\cdot
p^{\prime}J_{3,1,2}^{D+4}\big{)}\Big{]}p^{\mu}\not{p}^{\prime}$
$\displaystyle+\Big{[}(2-D)\big{(}J_{1,1,1}^{D}-J_{1,2,1}^{D+2}-2J_{2,1,1}^{D+2}+2J_{2,2,1}^{D+4}\big{)}$
$\displaystyle+(1-\xi)\Big{(}J_{1,1,1}^{D}-J_{1,1,2}^{D+2}+J_{1,2,1}^{D+2}-4p^{2}J_{1,3,2}^{D+4}$
$\displaystyle-2J_{2,1,1}^{D+2}+2(m^{2}+p^{2}-p\cdot
p^{\prime})J_{2,2,2}^{D+4}\big{)}\Big{]}p^{\prime\mu}\not{p}$
$\displaystyle+\Big{[}(D-2)\left(J_{2,1,1}^{D+2}-4J_{3,1,1}^{D+4}\right)+(1-\xi)\Big{(}J_{2,1,1}^{D+2}$
$\displaystyle-2p^{2}J_{2,2,2}^{D+4}+4(m^{2}+p^{2}-p\cdot
p^{\prime})J_{3,1,2}^{D+4}\Big{)}\Big{]}p^{\prime\mu}\not{p}^{\prime}\Big{\\}}.$
(40)

The scalar integrals sprinkled all over Eq. (40) contain higher powers of
propagators and have shifted space-time dimensions. These are explicitly
detailed in appendix B as a linear combination of elementary master integrals
whose solutions are well-known in literature in various space-time dimensions
$D$. We can now proceed to carry out a similar analysis for the transverse
vertex in the next section. As the details have already been outlined, we will
merely present the results.

##  V $V_{T}^{\mu}$ and the Ball-Chiu Basis

One-loop transverse vertex $V_{T}^{\mu}$ is given by Eq. (27). On applying
tensor reduction algorithm it can be written in terms of scalar Feynman
integrals as follows:

$\displaystyle V_{T}^{\mu}=$
$\displaystyle\dfrac{e^{3}}{(4\pi)^{\frac{D}{2}}}\Big{\\{}\Big{(}2(1-\xi)\Big{[}J_{2,2,2}^{D+4}\left(p^{\mu}\not{k}-p\cdot
k\gamma^{\mu}\right)\not{p}^{\prime}$
$\displaystyle+J_{2,2,2}^{D+4}\left(p^{\prime\mu}\not{k}-p^{\prime}\cdot
k\gamma^{\mu}\right)\not{p}+2J_{1,3,2}^{D+4}\left(p^{\mu}\not{k}-p\cdot
k\gamma^{\mu}\right)\not{p}$
$\displaystyle+2J_{3,1,2}^{D+4}\left(p^{\prime\mu}\not{k}-p^{\prime}\cdot
k\gamma^{\mu}\right)\not{p}^{\prime}\Big{]}+\Big{[}(4-D)J_{1,1,1}^{D}$
$\displaystyle+(1-\xi)(J_{1,1,1}^{D}-2J_{1,1,2}^{D+2})\Big{]}\sigma^{\mu\alpha}k_{\alpha}\Big{)}(\not{p}+m)$
$\displaystyle+\Big{[}2J_{1,1,1}^{D}+(D-6)J_{1,2,1}^{D+2}-(1-\xi)\Big{(}J_{1,1,2}^{D+2}$
$\displaystyle+2p\cdot
p^{\prime}J_{2,2,2}^{D+4}+4p^{2}J_{1,3,2}^{D+4}-J_{1,2,1}^{D+2}\Big{)}\Big{]}\sigma^{\mu\alpha}k_{\alpha}\not{p}$
$\displaystyle-\Big{[}(6-D)J_{2,1,1}^{D+2}+(1-\xi)\Big{(}2p^{2}J_{2,2,2}^{D+4}$
$\displaystyle+4p\cdot
p^{\prime}J_{3,1,2}^{D+4}-J_{2,1,1}^{D+2}\Big{)}\Big{]}\sigma^{\mu\alpha}k_{\alpha}\not{p}^{\prime}$
$\displaystyle+2\Big{[}-2J_{1,1,1}^{D}+2J_{1,2,1}^{D+2}+(1-\xi)\Big{(}J_{1,1,2}^{D+2}$
$\displaystyle+2p\cdot
p^{\prime}J_{2,2,2}^{D+4}+4p^{2}J_{1,3,2}^{D+4}-J_{1,2,1}^{D+2}\Big{)}\Big{]}(p^{\mu}\not{k}-k\cdot
p\gamma^{\mu})$
$\displaystyle+2\Big{[}2J_{2,1,1}^{D+2}+(1-\xi)\Big{(}2p^{2}J_{2,2,2}^{D+4}+4p\cdot
p^{\prime}J_{3,1,2}^{D+4}$ $\displaystyle-
J_{2,1,1}^{D+2}\Big{)}\Big{]}(p^{\prime\mu}\not{k}-p^{\prime}\cdot
k\gamma^{\mu})\Big{\\}}.$ (41)

An alternative and more commonly adopted approach is to express the vertex
$V^{\mu}$ by expanding it out in the Ball-Chiu basis, Eq. [3, 1, 7]:

$\displaystyle
V^{\mu}=\sum_{i=1}^{4}\lambda_{i}L_{i}^{\mu}+\sum_{i=1}^{8}\tau_{i}T_{i}^{\mu},$
(42)

where $L_{i}^{\mu}$ and $T_{i}^{\mu}$ are the longitudinal and the transverse
basis vectors. $L_{i}^{\mu}$ are:

$\displaystyle L_{1}^{\mu}$ $\displaystyle=$ $\displaystyle\gamma^{\mu},$
$\displaystyle L_{2}^{\mu}$ $\displaystyle=$
$\displaystyle(p+p^{\prime})^{\mu}(\not{p}+\not{p}^{\prime}),$ $\displaystyle
L_{3}^{\mu}$ $\displaystyle=$ $\displaystyle(p+p^{\prime})^{\mu},$
$\displaystyle L_{4}^{\mu}$ $\displaystyle=$
$\displaystyle\sigma^{\mu\alpha}(p+p^{\prime})^{\alpha},$ (43)

and

$\displaystyle T_{1}^{\mu}$ $\displaystyle=$ $\displaystyle p^{\prime}\cdot
kp^{\mu}-p\cdot kp^{\prime\mu},$ $\displaystyle T_{2}^{\mu}$ $\displaystyle=$
$\displaystyle\left(p^{\prime}\cdot kp^{\mu}-p\cdot
kp^{\prime\mu}\right)\left(\not{p}^{\prime}+\not{p}\right)$ $\displaystyle
T_{3}^{\mu}$ $\displaystyle=$ $\displaystyle
k^{2}\gamma^{\mu}-k^{\mu}\not{k},$ $\displaystyle T_{4}^{\mu}$
$\displaystyle=$ $\displaystyle-\left(p^{\prime}\cdot kp^{\mu}-p\cdot
kp^{\prime\mu}\right)\sigma^{\alpha\beta}p^{\prime}_{\alpha}p_{\beta},$
$\displaystyle T_{5}^{\mu}$ $\displaystyle=$
$\displaystyle\sigma^{\mu\alpha}k_{\alpha},$ $\displaystyle T_{6}^{\mu}$
$\displaystyle=$ $\displaystyle\left(p^{\prime
2}-p^{2}\right)\gamma^{\mu}-(p+p^{\prime})^{\mu}\not{k},$ $\displaystyle
T_{7}^{\mu}$ $\displaystyle=$ $\displaystyle-\dfrac{1}{2}\left(p^{\prime
2}-p^{2}\right)\left[\gamma^{\mu}(\not{p}^{\prime}+\not{p})+(p^{\prime}+p)^{\mu}\right]$
$\displaystyle+(p^{\prime}+p)^{\mu}\sigma^{\alpha\beta}p^{\prime}_{\alpha}p_{\beta},$
$\displaystyle T_{8}^{\mu}$ $\displaystyle=$
$\displaystyle\gamma^{\mu}\sigma^{\alpha\beta}p^{\prime}_{\alpha}p_{\beta}+p^{\prime\mu}\not{p}-p^{\mu}\not{p}^{\prime}.$
(44)

In order to provide a direct verification of our readily available and compact
results, we can compare our findings with those reported in [7] by projecting
onto the above basis. Starting from the expressions for the longitudinal
$V_{L}^{\mu}$ and transverse $V_{T}^{\mu}$ vertices given in Eqs. (40) and
(41), the coefficients $\lambda_{i}$ and $\tau_{i}$ of Eq. (42) can be
identified in terms of scalar integrals as detailed in appendix C. These
results for the complete one-loop vertex in arbitrary gauge and dimensions are
completely equivalent to the ones evaluated in [7].

##  VI The Pauli form factor $F_{2}(k^{2})$

The Pauli form factor $F_{2}(k^{2})$ is defined through the on-shell matrix
element of the vertex as follows:

$\displaystyle\bar{u}_{s^{\prime}}(p^{\prime})V^{\mu}(p^{\prime},p)u_{s}(p)=$
$\displaystyle
e\bar{u}_{s^{\prime}}(p^{\prime})\Big{[}F_{1}(k^{2})\gamma^{\mu}+\dfrac{1}{2m}F_{2}(k^{2})\sigma^{\mu\nu}k_{\nu}\Big{]}u_{s}(p),$
(45)

where $F_{1}(k^{2})$ represents the Dirac form factor, and the Dirac spinors
$u_{s}(p)$ and $\bar{u}_{s^{\prime}}(p^{\prime})$ satisfy

$\displaystyle\bar{u}_{s^{\prime}}(p^{\prime})(\not{p}^{\prime}+m)=(\not{p}+m)u_{s}(p)=0.$
(46)

Using Eq. (39), the matrix element of the longitudinal vertex $V_{L}^{\mu}$
reads as

$\displaystyle\bar{u}_{s^{\prime}}(p^{\prime})V^{\mu}_{L}(p^{\prime},p)u_{s}(p)=$
$\displaystyle\dfrac{e^{3}}{(4\pi)^{\frac{D}{2}}}\bar{u}_{s^{\prime}}(p^{\prime})\Bigg{\\{}2m(p+p^{\prime})^{\mu}(2J_{2,1,1}^{D+2}-J_{1,1,1}^{D})$
$\displaystyle+2m(1-D+\xi)(p+p^{\prime})^{\mu}J_{2,1,1}^{D+2}$
$\displaystyle+(2-D)\Big{[}J_{1,1,1}^{D+2}\gamma^{\mu}-2m(p+p^{\prime})^{\mu}(2J_{3,1,1}^{D+4}+J_{2,2,1}^{D+4})\Big{]}$
$\displaystyle+(D-3+\xi)J_{1,0,1}^{D}\gamma^{\mu}-(p+p^{\prime})^{\mu}\Big{[}2m(m^{2}-p\cdot
p^{\prime})J_{2,2,2}^{D+4}$ $\displaystyle+4m(m^{2}-p\cdot
p^{\prime})J_{3,1,2}^{D+4}-mJ_{1,1,2}^{D+2}\Big{]}\Bigg{\\}}u_{s}(p),$ (47)

where Eq. (46) has been used together with the relation
$J_{\alpha,\beta,\gamma}^{\tilde{D}}=J_{\beta,\alpha,\gamma}^{\tilde{D}}$,
valid under on-shell condition $p^{2}=p^{\prime 2}=-m^{2}$. In this section
all the scalar integrals are taken on-shell, which means that the relation
$p^{2}=p^{\prime 2}=-m^{2}$ is satisfied. After applying Gordon identity,

$\displaystyle\bar{u}_{s^{\prime}}(p^{\prime})\left[(p+p^{\prime})^{\mu}+\sigma^{\mu\alpha}k_{\alpha}\right]u_{s}(p)=2m\bar{u}_{s^{\prime}}(p^{\prime})\gamma^{\mu}u_{s}(p),$
(48)

Eq. (47) becomes

$\displaystyle\bar{u}_{s^{\prime}}(p^{\prime})V^{\mu}_{L}(p^{\prime},p)u_{s}(p)=\dfrac{e^{3}}{(4\pi)^{\frac{D}{2}}}\times$
$\displaystyle\bar{u}_{s^{\prime}}(p^{\prime})\Big{\\{}\left[(2-D)J_{1,1,1}^{D+2}+(D-3+\xi)J_{1,0,1}^{D}+f_{2\,L}\right]\gamma^{\mu}$
$\displaystyle\hskip 17.07164pt-
f_{2\,L}\;\sigma^{\mu\alpha}k_{\alpha}\Big{\\}}u_{s}(p),$ (49)

where

$\displaystyle f_{2\,L}$ $\displaystyle=$
$\displaystyle(1-\xi)m\Big{[}J_{1,1,2}^{D+2}-2(m^{2}-p\cdot
p^{\prime})J_{2,2,2}^{D+4}$ (50) $\displaystyle-4(m^{2}-p\cdot
p^{\prime})J_{3,1,2}^{D+4}-2J_{2,1,1}^{D+2}\Big{]}-2mJ_{1,1,1}^{D}$
$\displaystyle+4m(D-2)J_{3,1,1}^{D+4}+2(4-D)J_{2,1,1}^{D+2}$
$\displaystyle-2m(2-D)J_{2,2,1}^{D+4}.$

Similarly, we can evaluate the on-shell matrix element of the transverse
vertex $V_{T}$ using Eq. (41):

$\displaystyle\bar{u}_{s^{\prime}}(p^{\prime})V^{\mu}_{T}(p^{\prime},p)u_{s}(p)=\dfrac{e^{3}}{(4\pi)^{\frac{D}{2}}}\times$
$\displaystyle\hskip
48.36958pt\bar{u}_{s^{\prime}}(p^{\prime})\left[f_{1\,T}\gamma^{\mu}+f_{2\,T}\sigma^{\mu\alpha}k_{\alpha}\right]u_{s}(p),$
(51)

where

$\displaystyle f_{2\,T}$ $\displaystyle=$
$\displaystyle(1-\xi)m\Big{[}J_{1,1,2}^{D+2}-2(m^{2}-p\cdot
p^{\prime})J_{2,2,2}^{D+4}$ $\displaystyle-$ $\displaystyle 4(m^{2}-p\cdot
p^{\prime})J_{3,1,2}^{D+4}-2J_{2,1,1}^{D+2}\Big{]}-2mJ_{1,1,1}^{D}$
$\displaystyle+$ $\displaystyle 2(6-D)mJ_{2,1,1}^{D+2},$ $\displaystyle
f_{1\,T}$ $\displaystyle=$ $\displaystyle 4k\cdot pJ_{1,1,1}^{D}-4k\cdot
pJ_{1,2,1}^{D+2}+2(4-D)k\cdot p^{\prime}J_{2,1,1}^{D+2}$ (52) $\displaystyle-$
$\displaystyle 2k\cdot p(1-\xi)\big{(}J_{1,1,2}^{D+2}+2p\cdot
p^{\prime}J_{2,2,2}^{D+4}+4p^{2}J_{1,3,2}^{D+4}$ $\displaystyle-$
$\displaystyle J_{1,2,1}^{D+2}\big{)}.$

According to Eq. (45), supplemented with the results of Eqs. (49) and (51),
the Pauli form factor $F_{2}(k^{2})$ in any space-time dimension $D$ reads as,

$\displaystyle
F_{2}(k^{2})=\dfrac{4e^{2}m^{2}}{(4\pi)^{\frac{D}{2}}}\Big{[}2J_{2,1,1}^{D+2}+(2-D)\left(2J_{3,1,1}^{D+4}+J_{2,2,1}^{D+4}\right)\Big{]},$
(53)

which agrees with Eq. (4.30) of Ref. [7] after all the scalar integrals are
represented as linear combination of on-shell master integrals.

It is important to notice the cancellation of the scalar integrals that are
weighted by the factor $1-\xi$ in the Pauli form factor, implying its explicit
gauge independence.

###  VI.1 $F_{2}(k^{2})$ in $D=4$ dimensions

For $D=4$, Eq. (53) for the Pauli form factor yields

$\displaystyle
F_{2}(k^{2})=-\dfrac{2m^{2}\alpha}{\pi}\left[2J_{3,1,1}^{8}+J_{2,2,1}^{8}-J_{2,1,1}^{6}\right],$
(54)

where $\alpha=e^{2}/(4\pi)$. The combination of scalar integrals above can be
computed easily by a direct application of Feynman parametrization without the
need to use the expressions given in the appendix B to transform it in terms
of master integrals. Thus it can be shown that,

$\displaystyle
F_{2}(k^{2})=-\dfrac{2\alpha}{\pi}\int_{0}^{1}dx\int_{0}^{1-x}dy\dfrac{x^{2}+xy-x}{x^{2}+y^{2}+(2+c)xy},$
(55)

where $c={k^{2}}/{m^{2}}$. After a convenient change of variables
$y\xrightarrow{}x\left({1}/{y}-1\right)$ in the second integral, and
interchanging the order of integration, the equation above becomes:

$\displaystyle F_{2}(k^{2})$ $\displaystyle=$
$\displaystyle\dfrac{\alpha}{\pi}\int_{0}^{1}dy\,\dfrac{y}{1+cy(1-y)}.$ (56)

Since

$\int_{0}^{1}dy\,\dfrac{y}{1+cy(1-y)}=\dfrac{1}{2}\int_{0}^{1}dy\,\dfrac{1}{1+cy(1-y)},$
(57)

Eq. 56 for the Pauli form factor in $D=4$ dimensions becomes

$\displaystyle
F_{2}(k^{2})=\dfrac{\alpha}{2\pi}\int_{0}^{1}\dfrac{dy}{1+cy(1-y)},$ (58)

which agrees with the standard textbook result [34].

###  VI.2 $F_{2}(k^{2})$ in $D=3$ dimensions

For $D=3$, Eq. (53) for the Pauli form factor reads as

$\displaystyle
F_{2}(k^{2})=\dfrac{4e^{2}m^{2}}{(4\pi)^{3/2}}\left(2J_{2,1,1}^{5}-2J_{3,1,1}^{7}-J_{2,2,1}^{7}\right).$
(59)

In this case, the first scalar integral is infrared divergent. To regularize
this divergence a fictitious photon mass $m_{\gamma}$ is introduced in the
photon propagator. Thus, after Feynman parametrization, we have

$\displaystyle F_{2}(k^{2})=\dfrac{e^{2}}{4\pi
m}\int_{0}^{1}dx\int_{0}^{1-x}dy\dfrac{x(2-x-y)}{A^{3/2}},$ (60)

where $c=k^{2}/m^{2}$ as before, and

$\displaystyle A$ $\displaystyle=$
$\displaystyle(x+y)^{2}+cxy+(1-x-y)\kappa^{2},$ $\displaystyle\kappa$
$\displaystyle=$ $\displaystyle m_{\gamma}/m.$ (61)

These integrals are easily evaluated when $c=0$, which defines the anomalous
magnetic moment of the electron in 3-dimensions. After computing the integrals
involved, Eq. (60) gives $F(k^{2}=0)\equiv F_{2}$:

$\displaystyle F_{2}=\dfrac{e^{2}}{8\pi
m}\Big{[}3(\kappa-1)+\left(2-\dfrac{3}{2}\kappa\right)\ln\left(\dfrac{2+\kappa}{\kappa}\right)\Big{]},$
(62)

which agrees with Eq. (16) of Ref. [35]. It reaffirms that in 3-dimensions
usual QED does not yield well-defined electron anomalous magnetic moment.
However, it can be cured by adding a Chern-Simon term to the original
Lagrangian as discussed in Ref. [35].

##  VII Summary and conclusions

The second order formalism of QED is based on the Feynman rules depicted in
Fig. 1. It allows conceptual and clear decoupling between scalar and spin
degrees of freedom in an electromagnetic interaction of charged fermions. Any
QED amplitude, constructed from these rules, is equivalent to the one obtained
from the standard first order formalism. This equivalence is established by
rewriting the product of the tree-level fermion propagator and the fermion-
photon vertex as illustrated in Eq. (10), and applying identity (13).

In this article, a hybrid perspective to study the Green functions in QED,
such as the vertex $V^{\mu}$, is proposed. As the name suggests, it blends
together the first and second order formalisms. It can be employed to obtain a
more instructive Dirac and tensor structure of the Green functions. For the
vertex, the combined analysis yields a natural separation between its
longitudinal and transverse components. Before any attempt to perform the
Feynman integrals is made, these components are given in Eqs. (26) and (27).
There, the transversality condition for $V_{T}^{\mu}$ is clearly observed
while the longitudinal part is demonstrated to satisfy the WTI.

After the tensor reduction method is applied to the tensor Feynman integrals
in Eq. (26) for $V_{L}^{\mu}$, and the contraction with the photon momenta
$k_{\mu}$ is done, useful identities between scalar integrals with higher
powers in the propagators and shifted dimension can be deduced. These
identities are displayed in Eqs. (32), (33), (34), (35) and (36). The
complicated scalar integrals that appear on its left hand side conspire in
such a way as to obtain simple relations of master integrals with at most two
propagators. Employing these identities one can demonstrate the WTI holds at
one-loop order for the component $V_{L}^{\mu}$ of the vertex given in either
Eqs. (26), (39) or (40).

The longitudinal and transverse separation that is obtained from the
combination of the first and second order formalism also allows us to show
explicitly, with minimum effort, the independence of the Pauli form factor
$F_{2}$ from the gauge parameter $\xi$. In fact, the form factor in any
dimensions has the compact expression obtained in Eq. (53) in terms of scalar
integrals. As shown in section VI, in dimensions $D=4$ and $D=3$, this
combination of the scalar integrals can be easily evaluated to recover known
results in the literature.

As a final remark the longitudinal and transverse decomposition obtained in
this work are in agreement with the expressions obtained in Ref. [7].

###### Acknowledgements.

A.B. acknowledges Coordinación de la Investigación Científica of the
Universidad Michoacana de San Nicolás de Hidalgo Grant No. 4.10., US
Department of Energy (DOE) under the Contract No. DE-AC05-6OR23177 and the
Fulbright-García Robles Scholarship. V.M.B.G is grateful to Consejo Nacional
de Ciencia y Tecnología (Mexico) for support.

##  Appendix A The difference between the Ball-Chiu vertex and $V_{L}^{\mu}$
at one-loop

The results of Eq. (40) suggest the following vector structure for the
expanded out longitudinal part $\Gamma_{L}^{\mu}$ of the full fermion-photon
vertex $\Gamma^{\mu}$,

$\displaystyle\Gamma_{L}^{\mu}$ $\displaystyle=$ $\displaystyle
a_{1}p^{\mu}+a_{2}p^{\prime\mu}+a_{3}\gamma^{\mu}+a_{4}p^{\mu}\not{p}+a_{5}p^{\mu}\not{p}^{\prime}$
(63)
$\displaystyle+a_{6}p^{\prime\mu}\not{p}+a_{7}p^{\prime\mu}\not{p}^{\prime},$

where the scalar coefficients $a_{i}$ at one-loop are given by Eq. (40). These
coefficients are not entirely independent. Since, according to the WTI,

$\displaystyle k\cdot\Gamma_{L}^{\mu}$ $\displaystyle=$ $\displaystyle
S^{-1}(p^{\prime})-S^{-1}(p),$ $\displaystyle S^{-1}(p)$ $\displaystyle=$
$\displaystyle F(p^{2})\,\not{p}+G(p^{2}),$ (64)

the following relations must be satisfied

$\displaystyle a_{1}k\cdot p+a_{2}k\cdot p^{\prime}$ $\displaystyle=$
$\displaystyle G(p^{\prime 2})-G(p^{2}),$ $\displaystyle a_{3}+a_{5}k\cdot
p+a_{7}k\cdot p^{\prime}$ $\displaystyle=$ $\displaystyle F(p^{\prime 2}),$
$\displaystyle-a_{3}+a_{4}k\cdot p+a_{6}k\cdot p^{\prime}$ $\displaystyle=$
$\displaystyle-F(p^{2}).$ (65)

On-shell, these relations are reduced to:

$\displaystyle a_{1}$ $\displaystyle=$ $\displaystyle a_{2},$ $\displaystyle
a_{4}+a_{5}$ $\displaystyle=$ $\displaystyle a_{6}+a_{7},$ $\displaystyle
a_{3}+k\cdot p(a_{5}-a_{7})$ $\displaystyle=$ $\displaystyle F(m^{2}).$ (66)

The definition of $\Gamma_{L}^{\mu}$ in (63) differs from the Ball-Chiu vertex
[1],

$\displaystyle\Gamma_{BC}^{\mu}$ $\displaystyle=$
$\displaystyle\dfrac{G(p^{\prime 2})-G(p^{2})}{p^{\prime
2}-p^{2}}(p+p^{\prime})^{\mu}$ (67) $\displaystyle+\dfrac{F(p^{\prime
2})+F(p^{2})}{2}\gamma^{\mu}$
$\displaystyle+\dfrac{\not{p}^{\prime}+\not{p}}{2}(p+p^{\prime})^{\mu}\dfrac{F(p^{\prime
2})-F(p^{2})}{p^{\prime 2}-p^{2}},$

by a transverse piece, which can be written in terms of the coefficients
$a_{i}$ as follows:

$\displaystyle\delta\Gamma^{\mu}_{L}$ $\displaystyle=$
$\displaystyle\Gamma_{L}^{\mu}-\Gamma_{BC}^{\mu}$ (68) $\displaystyle=$
$\displaystyle\dfrac{a_{1}-a_{2}}{p^{\prime
2}-p^{2}}\;\tilde{T}_{1}^{\mu}+\dfrac{a_{4}-a_{5}}{2}\;\tilde{T}_{2}^{\mu}+\dfrac{a_{6}-a_{7}}{2}\;\tilde{T}_{3}^{\mu}$
$\displaystyle+\dfrac{a_{4}+a_{5}-a_{6}-a_{7}}{p^{\prime
2}-p^{2}}\;\tilde{T}_{4}^{\mu},$

where

$\displaystyle\tilde{T}_{1}^{\mu}=k\cdot p^{\prime}p^{\mu}-k\cdot
pp^{\prime\,\mu},$ $\displaystyle\tilde{T}_{2}^{\mu}=k\cdot
p\gamma^{\mu}+p^{\mu}\not{p}-p^{\mu}\not{p}^{\prime},$
$\displaystyle\tilde{T}_{3}^{\mu}=k\cdot
p^{\prime}\gamma^{\mu}+p^{\prime\mu}\not{p}-p^{\prime\mu}\not{p}^{\prime},$
$\displaystyle\tilde{T}_{4}^{\mu}=\Big{[}k\cdot
p^{\prime}\left(p^{\mu}\not{p}+p^{\mu}\not{p}^{\prime}\right)-k\cdot
p\left(p^{\prime\mu}\not{p}+p^{\prime\mu}\not{p}^{\prime}\right)\Big{]}.$ (69)

In terms of the transverse Ball-Chiu basis defined earlier in Eq. (44), these
new vectors read as

$\displaystyle\tilde{T}_{1}^{\mu}$ $\displaystyle=$ $\displaystyle
T_{1}^{\mu},$ $\displaystyle\tilde{T}_{2}^{\mu}$ $\displaystyle=$
$\displaystyle-\dfrac{1}{2}\left(T_{3}^{\mu}-T_{6}^{\mu}\right),$
$\displaystyle\tilde{T}_{3}^{\mu}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\left(T_{3}^{\mu}+T_{6}^{\mu}\right),$
$\displaystyle\tilde{T}_{4}^{\mu}$ $\displaystyle=$ $\displaystyle
T_{2}^{\mu}.$ (70)

Eq. (68) can be rewritten as follows in terms of scalar Feynman integrals at
one-loop,

$\displaystyle\delta\Gamma_{L}^{\mu\,1-L}$ $\displaystyle=$
$\displaystyle\delta V_{L}^{\mu}=\sum_{i=1}^{4}\delta_{i}\tilde{T}_{i}^{\mu},$
(71)

where

$\displaystyle\delta_{1}$ $\displaystyle=$
$\displaystyle\dfrac{2m(D-1+\xi)}{p^{\prime
2}-p^{2}}\left(J_{1,2,1}^{D+2}-J_{2,1,1}^{D+2}\right),$ (72)
$\displaystyle\delta_{2}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\Big{[}(2-D)\big{(}J_{1,1,1}^{D}-3J_{1,2,1}^{D+2}+4J_{1,3,1}^{D+4}+J_{2,1,1}^{D+2}$
(73)
$\displaystyle-2J_{2,2,1}^{D+4}\big{)}+(1-\xi)\big{(}J_{1,1,1}^{D}-J_{1,1,2}^{D+2}-J_{1,2,1}^{D+2}$
$\displaystyle-J_{2,1,1}^{D+4}+4m^{2}J_{1,3,2}^{D+4}+4p\cdot
p^{\prime}J_{3,1,2}^{D+4}$ $\displaystyle-2(m^{2}+p\cdot
p^{\prime})J_{2,2,2}^{D+4}\big{)}\Big{]},$ $\displaystyle\delta_{3}$
$\displaystyle=$
$\displaystyle\dfrac{1}{2}\Big{[}(2-D)\big{(}J_{1,1,1}^{D}-J_{1,2,1}^{D+2}-J_{2,1,1}^{D+2}+2J_{2,2,1}^{D+4}$
(74)
$\displaystyle-4J_{3,1,1}^{D+4}\big{)}+(1-\xi)\big{(}J_{1,1,1}^{D}-J_{1,1,2}^{D+2}+J_{1,2,1}^{D+2}$
$\displaystyle-3J_{2,1,1}^{D+2}+2(m^{2}-p\cdot
p^{\prime}+2p^{2})J_{2,2,2}^{D+4}$ $\displaystyle-4(m^{2}-p\cdot
p^{\prime}+p^{2})J_{3,1,2}^{D+4}-4p^{2}J_{1,3,2}^{D+4}\big{)}\Big{]},$
$\displaystyle\delta_{4}$ $\displaystyle=$ $\displaystyle\dfrac{2}{p^{\prime
2}-p^{2}}\Big{\\{}(D-2)\big{(}J_{1,2,1}^{D+2}-2J_{1,3,1}^{D+4}-J_{2,1,1}^{D+2}$
(75)
$\displaystyle+2J_{3,1,1}^{D+4}\big{)}+(\xi-1)\big{[}J_{1,2,1}^{D+2}-J_{2,1,1}^{D+2}$
$\displaystyle-2(m^{2}+p^{2})\big{(}J_{1,3,2}^{D+4}-J_{3,1,2}^{D+4}\big{)}\big{]}\Big{\\}}.$

##  Appendix B Scalar integrals

In this section the Feynman scalar integrals that appear in the longitudinal
and trasverse parts of the vertex, $V_{L}^{\mu}$ and $V_{T}^{\mu}$
respectively, in terms of master integrals are provided. Using the IBP
technique [27, 28] and the dimensional recurrence relations [29, 30, 31], with
the aid of the symbolic programming package LiteRed [32, 33], the scalar
integrals read as,

$\displaystyle J_{1,2,1}^{D+2}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2\beta_{1}}\Big{[}p\cdot
p^{\prime}J_{0,1,1}^{D}-p^{\prime 2}J_{1,0,1}^{D}+(p^{\prime 2}-p\cdot
p^{\prime})J_{1,1,0}^{D}+(p^{\prime 2}(m^{2}-k\cdot p)-m^{2}p\cdot
p^{\prime})J_{1,1,1}^{D}\Big{]},$ (76) $\displaystyle J_{2,1,1}^{D+2}$
$\displaystyle=$
$\displaystyle\dfrac{1}{2\beta_{1}}\Big{[}-p^{2}J_{0,1,1}^{D}+p\cdot
p^{\prime}J_{1,0,1}^{D}-k\cdot pJ_{1,1,0}^{D}+(k\cdot p^{\prime}\,p^{2}-k\cdot
p\,m^{2})J_{1,1,1}^{D}\Big{]},$ (77) $\displaystyle J_{1,1,2}^{D+2}$
$\displaystyle=$ $\displaystyle\dfrac{1}{2\beta_{1}}\Big{[}-k\cdot
pJ_{0,1,1}^{D}+k\cdot
p^{\prime}J_{1,0,1}^{D}-k^{2}J_{1,1,0}^{D}-k^{2}(m^{2}-p\cdot
p^{\prime})J_{1,1,1}^{D}\Big{]},$ (78) $\displaystyle J_{1,1,1}^{D+2}$
$\displaystyle=$ $\displaystyle\dfrac{1}{2(D-2)\beta_{1}}\Big{\\{}(p^{2}k\cdot
p^{\prime}-m^{2}k\cdot p)J_{0,1,1}^{D}+[p^{\prime 2}(m^{2}-k\cdot
p)-m^{2}p\cdot p^{\prime}]J_{1,0,1}^{D}$ (79)
$\displaystyle-[k^{2}m^{4}+k^{2}p^{2}p^{\prime 2}-2m^{2}(p^{\prime 2}p\cdot
p^{\prime}+p^{2}(p\cdot p^{\prime}-2p^{\prime
2}))]J_{1,1,1}^{D}-k^{2}(m^{2}-p\cdot p^{\prime})J_{1,1,0}^{D}\Big{\\}},$
$\displaystyle J_{2,2,1}^{D+4}$ $\displaystyle=$
$\displaystyle\dfrac{1}{4(D-2)\beta_{1}^{2}}\left(\alpha^{(1)}_{1}J_{0,1,0}^{D}+\alpha^{(1)}_{2}J_{0,1,1}^{D}+\alpha^{(1)}_{3}J_{1,0,1}^{D}+\alpha^{(1)}_{4}J_{1,1,0}^{D}+\alpha^{(1)}_{5}J_{1,1,1}^{D}\right),$
(80) $\displaystyle J_{2,2,2}^{D+4}$ $\displaystyle=$
$\displaystyle\dfrac{1}{4\beta_{1}^{2}\beta_{2}}\left(\alpha^{(2)}_{1}J_{0,1,0}^{D}+\alpha^{(2)}_{2}J_{0,1,1}^{D}+\alpha^{(2)}_{3}J_{1,0,1}^{D}+\alpha^{(2)}_{4}J_{1,1,0}^{D}+\alpha^{(2)}_{5}J_{1,1,1}^{D}\right),$
(81)

where

$\displaystyle\beta_{1}$ $\displaystyle=$ $\displaystyle p^{2}p^{\prime
2}-(p\cdot p^{\prime})^{2},\qquad\beta_{2}=(p^{\prime
2}+m^{2})(p^{2}+m^{2})k^{2}-m^{2}(p^{\prime 2}-p^{2})^{2},$ (82)

and

$\displaystyle\alpha_{1}^{(1)}$ $\displaystyle=$
$\displaystyle(D-2)\beta_{1},$ $\displaystyle\alpha_{2}^{(1)}$
$\displaystyle=$ $\displaystyle-p^{2}\left\\{(p\cdot
p^{\prime})^{2}+\left[(D-2)p^{2}+(1-D)p\cdot p^{\prime}\right]p^{\prime
2}\right\\}+m^{2}\left\\{p^{2}\left[(D-1)p\cdot p^{\prime}-(D-2)p^{\prime
2}\right]-(p\cdot p^{\prime})^{2}\right\\},$ $\displaystyle\alpha_{3}^{(1)}$
$\displaystyle=$ $\displaystyle-m^{2}\left\\{(p\cdot
p^{\prime})^{2}+(D-2)p^{2}p^{\prime 2}-(D-1)p\cdot p^{\prime}p^{\prime
2}\right\\}-p^{\prime 2}\left\\{(p\cdot p^{\prime})^{2}+p^{2}\left[(1-D)p\cdot
p^{\prime}+(D-2)p^{\prime 2}\right]\right\\},$ $\displaystyle\alpha_{4}^{(1)}$
$\displaystyle=$ $\displaystyle(D-2)p^{4}p^{\prime 2}+(p\cdot
p^{\prime})^{2}\left[(D-4)p\cdot p^{\prime}+p^{\prime
2}\right]+p^{2}\left[(p\cdot p^{\prime})^{2}-3(D-2)p\cdot p^{\prime}p^{\prime
2}+(D-2)p^{\prime 4}\right]$
$\displaystyle+m^{2}\left\\{p^{2}\left[(1-D)p\cdot p^{\prime}+(D-2)p^{\prime
2}\right]+p\cdot p^{\prime}\left[Dp\cdot p^{\prime}+(1-D)p^{\prime
2}\right]\right\\},$ $\displaystyle\alpha_{5}^{(1)}$ $\displaystyle=$
$\displaystyle m^{2}\Big{\\{}Dp^{2}(p\cdot
p^{\prime})^{2}+\left[(D-2)p^{4}-4(D-1)p^{2}p\cdot p^{\prime}+D(p\cdot
p^{\prime})^{2}\right]p^{\prime 2}+(D-2)p^{2}p^{\prime 4}\Big{\\}}$
$\displaystyle+m^{4}\left\\{p^{2}\left[(1-D)p\cdot p^{\prime}+(D-2)p^{\prime
2}\right]+p\cdot p^{\prime}\left[Dp\cdot p^{\prime}+(1-D)p^{\prime
2}\right]\right\\}$ $\displaystyle+p^{2}p^{\prime
2}\left\\{p^{2}\left[(1-D)p\cdot p^{\prime}+(D-2)p^{\prime 2}\right]+p\cdot
p^{\prime}\left[Dp\cdot p^{\prime}+(1-D)p^{\prime 2}\right]\right\\},$
$\displaystyle\alpha_{1}^{(2)}$ $\displaystyle=$
$\displaystyle(D-2)\beta_{1}\left[p\cdot p^{\prime}p^{\prime 2}+p^{2}(p\cdot
p^{\prime}-2p^{\prime 2})-m^{2}k^{2}\right],$ $\displaystyle\alpha_{2}^{(2)}$
$\displaystyle=$ $\displaystyle-p^{2}p^{\prime 2}k^{2}\left\\{(D-2)(p\cdot
p^{\prime})^{2}+p^{2}\left[(1-D)p\cdot p^{\prime}+p^{\prime
2}\right]\right\\}+m^{4}k^{2}\left\\{-(p\cdot
p^{\prime})^{2}+p^{2}\left[(D-1)p\cdot p^{\prime}-(D-2)p^{\prime
2}\right]\right\\}$ $\displaystyle+m^{2}\Big{\\{}(3-D)p^{6}p^{\prime
2}+2(p\cdot p^{\prime})^{3}p^{\prime 2}+p^{2}p\cdot
p^{\prime}\left[2(D-2)(p\cdot p^{\prime})^{2}+(7-5D)p\cdot p^{\prime}p^{\prime
2}+2(D-2)p^{\prime 4}\right]$ $\displaystyle-p^{4}\left[(D+1)(p\cdot
p^{\prime})^{2}+2(1-2D)p\cdot p^{\prime}p^{\prime 2}+(1+D)p^{\prime
4}\right]\Big{\\}},$ $\displaystyle\alpha_{3}^{(2)}$ $\displaystyle=$
$\displaystyle-p^{2}p^{\prime 2}k^{2}\left\\{(D-2)(p\cdot
p^{\prime})^{2}+p^{\prime 2}\left[p^{2}+(1-D)p\cdot
p^{\prime}\right]\right\\}-m^{4}k^{2}\left\\{(p\cdot
p^{\prime})^{2}+\left[(D-2)p^{2}+(1-D)p\cdot p^{\prime}\right]p^{\prime
2}\right\\}$ $\displaystyle+m^{2}\Big{\\{}2p^{2}(p\cdot p^{\prime})^{3}+p\cdot
p^{\prime}p^{\prime 2}\left[2(D-2)p^{4}+(7-5D)p^{2}p\cdot
p^{\prime}+2(D-2)(p\cdot p^{\prime})^{2}\right]$ $\displaystyle-p^{\prime
4}\left[(D+1)p^{4}+2(1-2D)p^{2}p\cdot p^{\prime}+(1+D)(p\cdot
p^{\prime})^{2}\right]-(D-3)p^{2}p^{\prime 6}\Big{\\}},$
$\displaystyle\alpha_{4}^{(2)}$ $\displaystyle=$
$\displaystyle-m^{4}k^{2}\left\\{p^{2}\left[(D-1)p\cdot
p^{\prime}-(D-2)p^{\prime 2}\right]+p\cdot p^{\prime}\left[-Dp\cdot
p^{\prime}+(D-1)p^{\prime 2}\right]\right\\}$ $\displaystyle+p^{2}p^{\prime
2}k^{2}\left\\{p^{2}\left[(1-D)p\cdot p^{\prime}+(D-2)p^{\prime
2}\right]+p\cdot p^{\prime}\left[Dp\cdot p^{\prime}+(1-D)p^{\prime
2}\right]\right\\}$ $\displaystyle+m^{2}\Big{\\{}(D-3)p^{6}p^{\prime
2}+(p\cdot p^{\prime})^{2}p^{\prime 2}\left[-2Dp\cdot
p^{\prime}+(1+D)p^{\prime 2}\right]$ $\displaystyle+p^{4}\left[(D+1)(p\cdot
p^{\prime})^{2}+2(4-3D)p\cdot p^{\prime}p^{\prime 2}+2(D-1)p^{\prime
4}\right]$ $\displaystyle+p^{2}\left[-2D(p\cdot p^{\prime})^{3}+10(D-1)(p\cdot
p^{\prime})^{2}p^{\prime 2}+2(4-3D)p\cdot p^{\prime}p^{\prime
4}+(D-3)p^{\prime 6}\right]\Big{\\}},$ $\displaystyle\alpha_{5}^{(2)}$
$\displaystyle=$ $\displaystyle p^{2}p^{\prime
2}k^{2}\left\\{(D-2)(p^{2}-p\cdot p^{\prime})(p\cdot p^{\prime})^{2}+p^{\prime
2}\left[p^{4}-Dp^{2}p\cdot p^{\prime}+(D-2)(p\cdot
p^{\prime})^{2}\right]+p^{2}p^{\prime 4}\right\\}$ (83)
$\displaystyle+m^{4}\Big{\\{}(p\cdot
p^{\prime})^{2}\left[2Dp^{4}+(6-5D)p^{2}p\cdot p^{\prime}+2(D-2)(p\cdot
p^{\prime})^{2}\right]$ $\displaystyle+p^{\prime
2}\left[(D-3)p^{6}+(6-7D)p^{4}p\cdot p^{\prime}+2(7D-8)p^{2}(p\cdot
p^{\prime})^{2}+(6-5D)(p\cdot p^{\prime})^{3}\right]$ $\displaystyle p^{\prime
4}\left[2(1+D)p^{4}+(6-7D)p^{2}p\cdot p^{\prime}+2D(p\cdot
p^{\prime})^{2}\right]+(D-3)p^{2}p^{\prime 6}\Big{\\}}$
$\displaystyle+m^{2}\Big{\\{}-p^{2}(p\cdot
p^{\prime})^{3}\left[D(p^{2}-2p\cdot p^{\prime})+4p\cdot p^{\prime}\right]$
$\displaystyle+p^{\prime 2}p\cdot
p^{\prime}\left[(3-2D)p^{6}+(9D-10)p^{4}p\cdot p^{\prime}-12(D-2)p^{2}(p\cdot
p^{\prime})^{2}+2(D-2)(p\cdot p^{\prime})^{3}\right]$ $\displaystyle+p^{\prime
4}\left[(D+2)p^{6}-6(D+1)p^{4}p\cdot p^{\prime}+(9D-10)p^{2}(p\cdot
p^{\prime})^{2}-D(p\cdot p^{\prime})^{3}\right]$ $\displaystyle+p^{2}p^{\prime
6}\left[(D+2)p^{2}+(3-2D)p\cdot p^{\prime}\right]\Big{\\}}$
$\displaystyle-m^{6}k^{2}\left\\{p^{2}\left[(D-1)p\cdot
p^{\prime}-(D-2)p^{\prime 2}\right]+p\cdot p^{\prime}\left[-Dp\cdot
p^{\prime}+(D-1)p^{\prime 2}\right]\right\\}.$

According to the identities (33), (34), (35) and (36), the remaining scalar
integrals, $J_{3,1,1}^{D+4}$, $J_{1,3,1}^{D+4}$, $J_{3,1,2}^{D+4}$ and
$J_{1,3,2}^{D+4}$, can be expressed as a linear combination of the others as,

$\displaystyle J_{3,1,1}^{D+4}$ $\displaystyle=$
$\displaystyle\dfrac{1}{4k\cdot p^{\prime}}\left[\dfrac{1}{2p^{\prime
2}}J_{0,1,0}^{D}+\dfrac{p^{\prime 2}+m^{2}}{2p^{\prime
2}}J_{1,0,1}^{D}-2k\cdot pJ_{2,2,1}^{D+4}-J_{1,1,1}^{D+2}-(p^{2}-p^{\prime
2})J_{2,1,1}^{D+2}\right],$ $\displaystyle J_{1,3,1}^{D+4}$ $\displaystyle=$
$\displaystyle\dfrac{1}{4k\cdot p}\left[J_{1,1,1}^{D+2}-(p^{2}-p^{\prime
2})J_{1,2,1}^{D+2}-2k\cdot
p^{\prime}J_{2,2,1}^{D+4}-\dfrac{1}{2p^{2}}J_{0,1,0}^{D}-\dfrac{p^{2}+m^{2}}{p^{2}}J_{0,1,1}^{D}\right],$
$\displaystyle J_{3,1,2}^{D+4}$ $\displaystyle=$
$\displaystyle\dfrac{1}{4p^{\prime 2}(p^{2}+m^{2})-4p\cdot
p^{\prime}(p^{\prime 2}+m^{2})}\Bigg{[}\dfrac{1}{2p^{\prime
2}}\left\\{\left[(D-4)p^{\prime
2}-(D-2)m^{2}\right]J_{1,0,1}^{D}-(D-2)J_{0,1,0}^{D}\right\\}$
$\displaystyle-(p^{\prime
2}-p^{2})J_{2,1,1}^{D+2}-(p^{2}+m^{2})\left(J_{1,1,2}^{D+2}+2p\cdot
p^{\prime}J_{2,2,2}^{D+4}\right)+2p^{2}(p^{\prime
2}+m^{2})J_{2,2,2}^{D+4}\Bigg{]},$ $\displaystyle J_{1,3,2}^{D+4}$
$\displaystyle=$ $\displaystyle\dfrac{1}{4p\cdot
p^{\prime}(p^{2}+m^{2})-4p^{2}(p^{\prime
2}+m^{2})}\Bigg{[}\dfrac{1}{2p^{2}}\left\\{\left[(D-4)p^{2}-(D-2)m^{2}\right]J_{0,1,1}^{D}-(D-2)J_{0,1,0}^{D}\right\\}$
(84) $\displaystyle-(p^{\prime 2}-p^{2})J_{1,2,1}^{D+2}-2p^{\prime
2}(p^{2}+m^{2})J_{2,2,2}^{D+4}+(p^{\prime
2}+m^{2})\left(J_{1,1,2}^{D+2}+2p\cdot
p^{\prime}J_{2,2,2}^{D+4}\right)\Bigg{]}.$

##  Appendix C Ball-Chiu coefficients

Using the results given in Eqs. (40) and (41), the coefficients $\lambda_{i}$
in Eq. (42) reads as

$\displaystyle\lambda_{1}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}(D-2)\Big{[}2J_{1,0,1}^{D}+(p^{\prime
2}-p^{2})J_{1,1,1}^{D}-2J_{1,1,1}^{D+2}+(3p^{2}-2p\cdot p^{\prime}-p^{\prime
2})J_{1,2,1}^{D+2}+(4p\cdot p^{\prime}-4p^{2})J_{1,3,1}^{D+4}$
$\displaystyle+(2p\cdot p^{\prime}-p^{2}-p^{\prime
2})J_{2,1,1}^{D+2}+(2p^{2}-4p\cdot p^{\prime}+2p^{\prime
2})J_{2,2,1}^{D+4}+4(p\cdot p^{\prime}-p^{\prime 2})J_{3,1,1}^{D+4}\Big{]}$
$\displaystyle+\dfrac{1}{2}(1-\xi)\Big{\\{}-2J_{1,0,1}^{D}+(p^{2}-p^{\prime
2})J_{1,1,1}^{D}+(2m^{2}+p^{2}+p^{\prime 2})J_{1,1,2}^{D+2}+(2p\cdot
p^{\prime}-p^{2}-p^{\prime 2})J_{1,2,1}^{D+2}$
$\displaystyle+4(m^{2}p^{2}-m^{2}p\cdot p^{\prime}-p^{2}p\cdot
p^{\prime}+p^{2}p^{\prime 2})J_{1,3,2}^{D+4}+(3p^{\prime 2}-p^{2}-2p\cdot
p^{\prime})J_{2,1,1}^{D+2}$ $\displaystyle+2(2m^{2}p\cdot
p^{\prime}-m^{2}p^{2}+p^{2}p\cdot p^{\prime}-m^{2}p^{\prime 2}-2p^{2}p^{\prime
2}+p\cdot p^{\prime}p^{\prime 2})J_{2,2,2}^{D+4}$ $\displaystyle+4(p^{\prime
2}[m^{2}+p^{2}-p\cdot p^{\prime}]-m^{2}p\cdot
p^{\prime})J_{3,1,2}^{D+4}\Big{\\}},$ $\displaystyle\lambda_{2}$
$\displaystyle=$ $\displaystyle\dfrac{1}{2(p^{2}-p^{\prime
2})}\Bigg{\\{}(D-2)\Big{[}(p^{\prime 2}-p^{2})J_{1,1,1}^{D}+(3p^{2}-2p\cdot
p^{\prime}-p^{\prime 2})J_{1,2,1}^{D+2}+4(p\cdot
p^{\prime}-p^{2})J_{1,3,1}^{D+4}$ $\displaystyle+(p^{2}+2p\cdot
p^{\prime}-3p^{\prime 2})J_{2,1,1}^{D+2}+(2p^{\prime
2}-2p^{2})J_{2,2,1}^{D+4}+4(p^{\prime 2}-p\cdot
p^{\prime})J_{3,1,1}^{D+4}\Big{]}$
$\displaystyle(1-\xi)\Big{[}(p^{2}-p^{\prime 2})J_{1,1,1}^{D}+(p^{\prime
2}-p^{2})J_{1,1,2}^{D+2}+(2p\cdot p^{\prime}-p^{2}-p^{\prime
2})J_{1,2,1}^{D+2}$ $\displaystyle+4(m^{2}p^{2}-m^{2}p\cdot
p^{\prime}-p^{2}p\cdot p^{\prime}+p^{2}p^{\prime
2})J_{1,3,2}^{D+4}+(p^{2}-2p\cdot p^{\prime}+p^{\prime 2})J_{2,1,1}^{D+2}$
$\displaystyle+2(m^{2}p^{2}-p^{2}p\cdot p^{\prime}-m^{2}p^{\prime 2}+p\cdot
p^{\prime}p^{\prime 2})J_{2,2,2}^{D+4}+4\Big{(}m^{2}p\cdot
p^{\prime}-p^{\prime 2}(m^{2}+p^{2}-p\cdot
p^{\prime})\Big{)}J_{3,1,2}^{D+4}\Big{]}\Bigg{\\}},$
$\displaystyle\lambda_{3}$ $\displaystyle=$
$\displaystyle\dfrac{m(D-1+\xi)}{p^{\prime 2}-p^{2}}\left[(p^{2}-p^{\prime
2})J_{1,1,1}^{D}+2(p\cdot p^{\prime}-p^{2})J_{1,2,1}^{D+2}+2(p^{\prime
2}-p\cdot p^{\prime})J_{2,1,1}^{D+2}\right],$ $\displaystyle\lambda_{4}$
$\displaystyle=$ $\displaystyle 0,$ (85)

while the coefficients $\tau_{i}$ correspond to

$\displaystyle\tau_{1}$ $\displaystyle=$
$\displaystyle\dfrac{2m(D-1+\xi)}{p^{\prime
2}-p^{2}}\left(J_{1,2,1}^{D+2}-J_{2,1,1}^{D+2}\right),$
$\displaystyle\tau_{2}$ $\displaystyle=$
$\displaystyle\dfrac{1}{p^{2}-p^{\prime
2}}\Big{\\{}(2-D)\left(J_{1,2,1}^{D+2}-2J_{1,3,1}^{D+4}-J_{2,1,1}^{D+2}+2J_{3,1,1}^{D+4}\right)$
$\displaystyle+(1-\xi)\left[J_{1,2,1}^{D+2}-2(m^{2}+p^{2})J_{1,3,2}^{D+4}-J_{2,1,1}^{D+2}+(p^{\prime
2}-p^{2})J_{2,2,2}^{D+4}+2(m^{2}+p^{\prime
2})J_{3,1,2}^{D+4}\right]\Big{\\}},$ $\displaystyle\tau_{3}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\Big{\\{}(2-D)\left[J_{1,1,1}^{D}-2\left(J_{1,3,1}^{D+4}-J_{2,2,1}^{D+4}+J_{3,1,1}^{D+4}\right)\right]+(1-\xi)\Big{[}J_{1,1,1}^{D}-J_{1,1,2}^{D+2}$
$\displaystyle-2(m^{2}+p^{2})J_{1,3,2}^{D+4}+(2m^{2}+p^{2}+p^{\prime
2})J_{2,2,2}^{D+4}-2(m^{2}+p^{\prime 2})J_{3,1,2}^{D+4}\Big{]}\Big{\\}},$
$\displaystyle\tau_{4}$ $\displaystyle=$
$\displaystyle\dfrac{4m(1-\xi)}{p^{2}-p^{\prime
2}}\left(J_{1,3,2}^{D+4}-J_{3,1,2}^{D+4}\right),$ $\displaystyle\tau_{5}$
$\displaystyle=$
$\displaystyle(4-D)mJ_{1,1,1}^{D}+m(1-\xi)\Big{[}J_{1,1,1}^{D}-2J_{1,1,2}^{D+2}+2(p\cdot
p^{\prime}-p^{2})J_{1,3,2}^{D+4}$ $\displaystyle+(p^{2}-2p\cdot
p^{\prime}+p^{\prime 2})J_{2,2,2}^{D+4}+2(p\cdot p^{\prime}-p^{\prime
2})J_{3,1,2}^{D+4}\Big{]},$ $\displaystyle\tau_{6}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\Big{\\{}(D-2)\left(J_{1,2,1}^{D+2}-2J_{1,3,1}^{D+4}-J_{2,1,1}^{D+2}+2J_{3,1,1}^{D+4}\right)+(1-\xi)\Big{[}J_{1,2,1}^{D+2}+2(m^{2}-p^{2})J_{1,3,2}^{D+4}$
$\displaystyle-J_{2,1,1}^{D+2}+(p^{2}-p^{\prime 2})J_{2,2,2}^{D+4}+2(p^{\prime
2}-m^{2})J_{3,1,2}^{D+4}\Big{]}\Big{\\}},$ $\displaystyle\tau_{7}$
$\displaystyle=$ $\displaystyle\dfrac{2m(1-\xi)}{p^{\prime
2}-p^{2}}\left[2(p\cdot p^{\prime}-p^{2})J_{1,3,2}^{D+4}+(p^{\prime
2}-p^{2})J_{2,2,2}^{D+4}+2(p^{\prime 2}-p\cdot
p^{\prime})J_{3,1,2}^{D+4}\right],$ $\displaystyle\tau_{8}$ $\displaystyle=$
$\displaystyle(6-D)\left(J_{1,1,1}^{D}-J_{1,2,1}^{D+2}-J_{2,1,1}^{D+2}\right)+(1-\xi)\Big{(}J_{1,1,1}^{D}-3J_{1,1,2}^{D+2}+J_{1,2,1}^{D+2}-4p^{2}J_{1,3,2}^{D+4}$
(86) $\displaystyle+J_{2,1,1}^{D+2}-4p\cdot
p^{\prime}J_{2,2,2}^{D+4}-4p^{\prime 2}J_{3,1,2}^{D+4}\Big{)}.$

All these coefficients agree with the results presented in [7] after changing
to its conventions, replacing $1-\xi$ by $\xi$, and using the expressions of
the appendix B to rewrite the coefficients in terms of master integrals.

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