# The QED four-photon amplitudes off-shell: part 2

Naser Ahmadiniaz<EMAIL_ADDRESS>Cristhiam Lopez-Arcos
<EMAIL_ADDRESS>Misha A. Lopez-Lopez<EMAIL_ADDRESS>Christian
Schubert<EMAIL_ADDRESS>Helmholtz-Zentrum Dresden-
Rossendorf, Bautzner Landstraße 400, 01328 Dresden, Germany  Escuela de
Matemáticas, Universidad Nacional de Colombia Sede Medellín, Carrera 65 $\\#$
59A–110, Medellín, Colombia  ELI Beamlines Centre, Institute of Physics, Czech
Academy of Sciences, Za Radnicki 835, 25241 Dolni Břežany, Czech Republic
After May 15, 2023: 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

###### Abstract

This is the second one of a series of four papers devoted to a first
calculation of the scalar and spinor QED four-photon amplitudes completely
off-shell. We use the worldline formalism which provides a gauge-invariant
decomposition for these amplitudes as well as compact integral
representations. It also makes it straightforward to integrate out any given
photon leg in the low-energy limit, and in the present sequel we do this with
two of the four photons. For the special case where the two unrestricted
photon momenta are equal and opposite the information on these amplitudes is
also contained in the constant-field vacuum polarisation tensors, which
provides a check on our results. Although these amplitudes are finite, for
possible use as higher-loop building blocks we evaluate all integrals in
dimensional regularisation. As an example, we use them to construct the two-
loop vacuum polarisation tensors in the low-energy approximation, rederive
from those the two-loop $\beta$-function coefficients and analyse their
anatomy with respect to the gauge-invariant decomposition. As an application
to an external-field problem, we provide a streamlined calculation of the
Delbrück scattering amplitudes in the low-energy limit. All calculations are
done in parallel for scalar and spinor QED.

††journal: .

###### Contents

  1. 1 Introduction
  2. 2 Worldline representation of the four-photon amplitudes
    1. 2.1 Case distinction for the low energy limit of photons $3$ and $4$
    2. 2.2 Low-energy limit for two of the photons: computational examples
  3. 3 Results: low-energy limit for two of the photons
    1. 3.1 Dimensionally regularized amplitudes
      1. 3.1.1 Scalar QED
      2. 3.1.2 Spinor QED
    2. 3.2 Four-dimensional amplitudes
      1. 3.2.1 Scalar QED
      2. 3.2.2 Spinor QED
  4. 4 Check: the photon propagator in a constant field
    1. 4.1 Scalar QED
    2. 4.2 Spinor QED
  5. 5 Gauge-invariant decomposition of the two-loop $\beta$-functions
    1. 5.1 Scalar QED $\beta$-function
    2. 5.2 Spinor QED $\beta$-function
    3. 5.3 Four-dimensional computation of the Spinor QED $\beta$-function
  6. 6 Delbrück scattering at low energies
  7. 7 Summary and outlook
  8. A Collection of integral formulas
    1. A.1 Integrating out a low-energy leg
    2. A.2 The functions $Y_{nl}$ and tensor reduction
  9. B Explicit results for $Q_{{\rm scal}(34)}$ and $Q_{{\rm spin}(34)}$
    1. B.1 Scalar QED
    2. B.2 Spinor QED

##  1 Introduction

The present paper is the second one in a series of four devoted to a first
calculation of the scalar and spinor QED one-loop four-photon amplitudes fully
off-shell, using the worldline representation of these amplitudes. In part I
[1] we already derived these representations, discussed their properties, and
calculated them for the simple limiting case where all four photons are taken
in the low-energy (or “Euler-Heisenberg” limit). In the present sequel, we
calculate them explicitly for the case where only photons 3 and 4 are taken in
that limit, but 1 and 2 have arbitrary off-shell momenta, see Fig. 1.

Figure 1: Four-photon box diagram with two low-energy legs $k_{4}$ and $k_{3}$
indicated by empty bullets at their ends. A sum over permutations is
understood, as well as the inclusion of seagull diagrams in the scalar QED
case.

The results are given in compact form in terms of the hypergeometric function
${}_{2}F_{1}$ for the dimensionally continued case, and in trigonometric form
for $D=4$.

Although to our knowledge this special case of the four-photon amplitudes has
not been considered by other authors, for $k_{1}=-k_{2},k_{3}=-k_{4}$ it is
straightforward to extract them from the well-studied photon polarization
tensors in a constant field [2, 3, 4, 5, 6, 7, 8, 9], and we use this fact for
a check on our calculations.

Off-shell legs can be used for creating internal propagators by sewing, or for
connecting them to external fields. As an example for the former, we use our
results to construct, by sewing of the legs $1$ and $2$ that carry full
momentum, the two-loop scalar and spinor QED vacuum polarization tensors in
the low-energy limit, see Fig. 2.

Figure 2:  Construction of the two-loop photon propagator from the four-photon
amplitude by sewing.

In this limit the vacuum polarization tensors just reduce to the induced
Maxwell terms, from which we can rederive the two-loop $\beta$-function
coefficients. Apart from providing another check, this also improves on
previous work [10] where the worldline formalism had already been applied to
the calculation of these coefficients. Although these coefficients were
already known for decades [11, 12], that work was motivated by the fact that
the formalism allows one to obtain parameter-integral representations that
unify all the Feynman diagrams contributing to the quenched (single fermion
loop) photon propagator at any loop order, see., e.g., Fig. 3 for the three-
loop case.

Figure 3: The three-loop quenched photon propagator.

The interesting feature of the worldline formalism is that the sewing results
in parameter integrals that represent not some particular Feynman diagram, but
the whole set of Feynman diagrams shown in Fig. 3. And it is precisely this
type of sums of diagrams that are known for particularly extensive
cancellations between diagrams. In 1967 Johnson, Willey and Baker showed [13]
that the quenched vacuum polarisation has, at any loop order, only a single
overall UV divergence. And the coefficient of this pole, which is essentially
the $\beta$-function, turned out to be rational up to the four-loop level (see
[14] and refs. therein). The role of gauge invariance in these cancellations
is not transparent, and remains an object of active investigation even at the
two-loop level [15]. The worldline formalism was applied to the recalculation
of the two-loop scalar and spinor QED $\beta$-functions by M.G. Schmidt and
one of the authors in [10] (see also [16]) to achieve significant
cancellations already at the integrand level. However, those calculations were
done using a different approach based on two-loop worldline Green’s functions
[17], which provides a shortcut but obscures the role of gauge invariance.
Only after that work a refinement of the usual homogeneising Bern-Kosower
integration-by-parts procedure was found that led to a decomposition of the
four-photon amplitudes into sixteen individually gauge invariant contributions
(see [18, 19] and part I). This gives us a chance to improve on [10] by asking
the following two questions: first, can we identify parts of that
decomposition that drop out in the construction of the two-loop $\beta$
functions? Second, are there cancellations in their computation beyond what is
expected from gauge invariance, that is, not only inside each gauge invariant
structure, but also between them? Thus we will study here the distribution of
the $\beta$-function coefficients, as well as the cancellation of the double
pole in the $\frac{1}{\epsilon}$ expansion, in the gauge-invariant
decomposition derived in part I.

As an application to an external-field problem, among the many processes
related to the four-photon amplitudes we have chosen here to reanalyse low-
energy Delbrück scattering, the deflection of photons in the Coulomb field of
nuclei due to the vacuum polarization (partially motivated also by the fact
that the worldline formalism has already been applied extensively to
calculations involving constant [20, 21, 7, 16, 22, 23], plane-wave [24, 25]
and combinations of the two types of fields [26], but not as yet to Coulomb
fields). Delbrück introduced this scattering in 1933 in order to explain the
discrepancies Meitner and Kösters had found in their experiment for the
Compton scattering on heavy atoms [27]. Later Bethe and Rohrlich computed the
angular distribution for small angles and the total cross section for Delbrück
scattering [28]. In 1973 DESY reported a first observation of this scattering
in the high-energy, small-angle limit [29] which was in agreement with the
prediction made in a series of papers by Cheng and Wu [30, 31, 32, 33] and
later confirmed in [34, 35]. In 1975 the Göttingen group performed another
experiment which was the first one where exact predictions based on Feynman
diagrams were confirmed with high precision after considering other background
phenomena like atomic and nuclear Rayleigh scattering [36]. The most accurate
high energy experiment done so far on Delbrück scattering is the one carried
out at BINP [37, 38]. Here we will use our results for a short and efficient
calculation of the Delbrück scattering cross section in the low-energy limit.

This paper is organized as follows. In Section 2 we shortly review the
worldline representation of the scalar and spinor QED four-photon amplitudes,
and discuss our general computational strategy in this paper. In particular,
we provide all the integral formulas required to integrate out low-energy
photons without having to split these multi-photon integrals into ordered
sectors. The central Section is 3, where we list our explicit final results
for the amplitudes, in $D$ as well as in four dimensions, and for both scalar
and spinor loops. Section 4 contains the above-mentioned comparison with the
amplitudes extracted from the vacuum polarization in a constant field. In
Section 5 we recalculate the two-loop $\beta$-functions, while Section 6 is
devoted to low-energy Delbrück scattering. Finally, Section 7 gives a summary
and outline of future work. Supplementary information and formulas are
provided in the appendices.

##  2 Worldline representation of the four-photon amplitudes

In part I, we presented in detail the derivation and structure of the
worldline representation of the $N$-photon amplitudes. Due to the inherent
freedom in the integration-by-parts procedure, this representation comes in
various slightly different versions. For easy reference, let us summarize here
the representation of the four-photon amplitudes that we will actually use in
the present sequel 111Since henceforth we are concerned exclusively with the
four-photon case we will now omit the subscript $N$ on the $Q$’s.:

$\Gamma_{\rm
scal}(k_{1},\varepsilon_{1};\cdots;k_{4},\varepsilon_{4})=\frac{(-ie)^{4}}{(4\pi)^{\frac{D}{2}}}\int_{0}^{\infty}\frac{dT}{T}\,T^{4-\frac{D}{2}}\,{\rm
e}^{-m^{2}T}\,\int_{0}^{1}\prod_{i=1}^{4}du_{i}\,Q_{{\rm scal}}\,\,{\rm
e}^{(\cdot)}\,.$ (2.1)

Here we have already done the usual rescaling $\tau_{i}=Tu_{i}$ such that the
exponential part is

$\,{\rm e}^{(\cdot)}\equiv\,{\rm e}^{T\sum_{i<j=1}^{4}G_{ij}k_{i}\cdot
k_{j}}\,,$ (2.2)

the bosonic Green’s functions are

$G_{ij}\equiv G(u_{i},u_{j})=|u_{i}-u_{j}|-(u_{i}-u_{j})^{2}\,,$ (2.3)

and the polynomial $Q_{{\rm scal}}$ is given by 222When comparing with [16]
note that there a different basis was used for the four-cycle component
$Q^{4}$. The two bases are related by cyclicity and inversion.

$\displaystyle Q_{{\rm scal}}$ $\displaystyle=$ $\displaystyle Q^{4}_{{\rm
scal}}+Q^{3}_{{\rm scal}}+Q^{2}_{{\rm scal}}+Q^{22}_{{\rm scal}}\,,$
$\displaystyle Q^{4}_{{\rm scal}}$ $\displaystyle=$
$\displaystyle\dot{G}(1234)+\dot{G}(2314)+\dot{G}(3124)\,,$ $\displaystyle
Q^{3}_{{\rm scal}}$ $\displaystyle=$
$\displaystyle\dot{G}(123)T(4)+\dot{G}(234)T(1)+\dot{G}(341)T(2)+\dot{G}(412)T(3)\,,$
$\displaystyle Q^{2}_{{\rm scal}}$ $\displaystyle=$
$\displaystyle\dot{G}(12)T_{sh}(34)+\dot{G}(13)T_{sh}(24)+\dot{G}(14)T_{sh}(23)+\dot{G}(23)T_{sh}(14)+\dot{G}(24)T_{sh}(13)+\dot{G}(34)T_{sh}(12)\,,$
$\displaystyle Q^{22}_{{\rm scal}}$ $\displaystyle=$
$\displaystyle\dot{G}(12)\dot{G}(34)+\dot{G}(13)\dot{G}(24)+\dot{G}(14)\dot{G}(23)\,.$

The extraordinary compactness of this representation is made possible through
the introduction of the “Lorentz-cycle” $Z_{n}(i_{1}i_{2}\ldots i_{n})$ as

$\displaystyle\begin{split}Z_{2}(ij)&\equiv{1\over 2}{\rm
tr}\bigl{(}f_{i}f_{j}\bigr{)}=\varepsilon_{i}\cdot k_{j}\varepsilon_{j}\cdot
k_{i}-\varepsilon_{i}\cdot\varepsilon_{j}k_{i}\cdot k_{j}\,,\\\
Z_{n}(i_{1}i_{2}\ldots i_{n})&\equiv{\rm
tr}\Bigl{(}\prod_{j=1}^{n}f_{i_{j}}\Bigr{)}\,,\quad(n\geq 3)\,,\end{split}$
(2.5)

where
$f_{i}^{\mu\nu}=k_{i}^{\mu}\varepsilon_{i}^{\nu}-k_{i}^{\nu}\varepsilon_{i}^{\mu}$
is the field strength tensor of photon $i$, and the “bicycle”

$\displaystyle\dot{G}(i_{1}i_{2}\cdots i_{n})$ $\displaystyle\equiv$
$\displaystyle\dot{G}_{i_{1}i_{2}}\dot{G}_{i_{2}i_{3}}\cdots\dot{G}_{i_{n}i_{1}}Z_{n}(i_{1}\cdots
i_{n})\,.$ (2.6)

It is the “tails” that exist in various versions. For the present computation,
we use the one-photon tail $T(i)$ of the original $Q$-representation and the
“short tail” $T_{sh}(ij)$, introduced in part I, as the two-photon tail:

$\displaystyle T(i)$ $\displaystyle\equiv$ $\displaystyle\sum_{r\neq
i}\dot{G}_{ir}\varepsilon_{i}\cdot k_{r}\,,$ (2.7) $\displaystyle T_{sh}(ij)$
$\displaystyle\equiv$ $\displaystyle\sum_{r,s\neq
i,j}\dot{G}_{ri}\dot{G}_{js}~{}\frac{k_{r}\cdot f_{i}\cdot f_{j}\cdot
k_{s}}{k_{i}\cdot k_{j}}\,.$ (2.8)

The spinor-loop result is obtained by employing the _Bern-Kosower replacement
rule_ , i.e. replacing simultaneously every closed (full) cycle as
$\dot{G}_{i_{1}i_{2}}\dot{G}_{i_{2}i_{3}}\cdots\dot{G}_{i_{n}i_{1}}$ appearing
in the integrand of the scalar-loop with

$\displaystyle\dot{G}_{i_{1}i_{2}}\dot{G}_{i_{2}i_{3}}\cdots\dot{G}_{i_{n}i_{1}}-G_{Fi_{1}i_{2}}G_{Fi_{2}i_{3}}\cdots
G_{Fi_{n}i_{1}}$ (2.9)

where $G_{Fij}={\rm sgn}(u_{i}-u_{j})$ is the fermionic Green function (here
it is understood that $\dot{G}_{ij}=-\dot{G}_{ji}$ may have to be used to
achieve the cycle form). We write the spinor-loop amplitude as

$\displaystyle\Gamma_{\rm
spin}(k_{1},\varepsilon_{1};\cdots;k_{4},\varepsilon_{4})$ $\displaystyle=$
$\displaystyle-2\frac{(-ie)^{4}}{(4\pi)^{\frac{D}{2}}}\int_{0}^{\infty}\frac{dT}{T}\,T^{4-\frac{D}{2}}\,{\rm
e}^{-m^{2}T}\,\int_{0}^{1}\prod_{i=1}^{4}du_{i}\,Q_{{\rm spin}}\,\,{\rm
e}^{(\cdot)}\,.$ (2.10)

Thus, apart from a global factor of $-2$, the only difference to the scalar
QED formula (2.1) is the replacement of $Q_{\rm scal}$ by $Q_{\rm spin}$
according to the rule (2.9). Let us also emphasize once more that equations
(2.1), (2.10) are valid off-shell, and that the right-hand sides are
manifestly finite term-by-term. The well-known spurious UV-divergences of the
four-photon diagrams that usually cancel only in the sum of diagrams would
show up here as logarithmic divergences of the $T$-integration at $T=0$, but
have been eliminated already at the beginning by the IBP procedure that led
from the P-representation to the Q-representation, see part I.

To avoid carrying common prefactors we define

$\hat{\Gamma}_{\left\\{{\rm scal}\atop{\rm
spin}\right\\}}\equiv\int_{0}^{\infty}\frac{dT}{T}\,T^{4-\frac{D}{2}}\,{\rm
e}^{-m^{2}T}\,\int_{0}^{1}\prod_{i=1}^{4}du_{i}\,Q_{\left\\{{\rm
scal}\atop{\rm spin}\right\\}}\,\,{\rm e}^{(\cdot)}\,.$ (2.11)

###  2.1 Case distinction for the low energy limit of photons $3$ and $4$

As has been mentioned above, in the present part II the photons number 3 and 4
are taken in low-energy limit which means that in the expressions $Q_{\rm
scal/spin}\,e^{(\cdot)}$ we only consider those contributions which are linear
in both $k_{3}$ and $k_{4}$ (for more details about the low-energy limit see
part I). For the sake of clarity, in this subsection we list the different
cases that appear in our calculations:

  * 1.

Case 1: Terms in $Q_{\rm scal}$ which are already linear in both $k_{3}$ and
$k_{4}$ do not need any further factors of $k_{3,4}$ from the exponential, so
that we can simply replace $\,{\rm e}^{(\cdot)}\rightarrow\,{\rm
e}^{(\cdot)|_{k_{3},k_{4}\rightarrow 0}}=e^{TG_{12}k_{1}\cdot k_{2}}$. This
includes all the “pure cycle” terms, defined by the absence of tails. For
instance, the four-cycle $Q_{\rm scal}^{4}$

$\dot{G}(1234)\,\,{\rm e}^{(\cdot)}\rightarrow\dot{G}(1234)\,{\rm
e}^{TG_{12}k_{1}\cdot k_{2}}.$ (2.12)

Or the two-two cycles $Q_{\rm scal}^{22}$, where it is possible to have both
low-energy photons in the same cycle

$\dot{G}(12)\dot{G}(34)\,\,{\rm
e}^{(\cdot)}\rightarrow\dot{G}(12)\dot{G}(34)\,\,{\rm e}^{TG_{12}k_{1}\cdot
k_{2}},$ (2.13)

or distributed among different cycles

$\displaystyle\dot{G}(13)\dot{G}(24)\,\,{\rm
e}^{(\cdot)}\rightarrow\dot{G}(13)\dot{G}(24)\,\,{\rm e}^{TG_{12}k_{1}\cdot
k_{2}}.$ (2.14)

  * 2.

Case 2: Terms in $Q_{\rm scal}^{3}$ which contain one of the low-energy
momenta, say, $k_{3}$, in the three-cycle, but lack $k_{4}$. Those require an
expansion of the exponential factor to linear order in $k_{4}$. For instance,
the one-tail term $\dot{G}(123)T(4)\,\,{\rm e}^{(\cdot)}$ gives

$\displaystyle\dot{G}(123)T(4)\,\,{\rm
e}^{(\cdot)}\rightarrow\dot{G}(123)\Big{[}\dot{G}_{41}\varepsilon_{4}\cdot
k_{1}+\dot{G}_{42}\varepsilon_{4}\cdot k_{2}\Big{]}\Big{[}TG_{14}k_{1}\cdot
k_{4}+TG_{24}k_{2}\cdot k_{4}\Big{]}\,\,{\rm e}^{TG_{12}k_{1}\cdot k_{2}}.$
(2.15)

  * 3.

Case 3: Terms in $Q_{\rm scal}^{3}$ which have two low-energy momenta in the
three-cycle, for those we neglect terms in the one-tail that are linear in any
of the two low-energy momenta. For instance, the one-tail term
$\dot{G}(234)T(1)\,\,{\rm e}^{(\cdot)}$ gives

$\displaystyle\dot{G}(234)T(1)\,\,{\rm
e}^{(\cdot)}\rightarrow\dot{G}(234)\dot{G}_{12}\varepsilon_{1}\cdot
k_{2}\,{\rm e}^{TG_{12}k_{1}\cdot k_{2}}.$ (2.16)

However, it turns out that the contributions of these particular terms vanish
after integration over $u_{3}$ and $u_{4}$.

  * 4.

Case 4: Terms in $Q_{\rm scal}^{2}$ which lack both $k_{3}$ and $k_{4}$. Such
terms occur only in the two-tail term $\dot{G}(12)T_{\rm sh}(34)\,\,{\rm
e}^{(\cdot)}$. Here the exponential factor must be expanded to linear order in
both $k_{3}$ and $k_{4}$,

$\displaystyle\dot{G}(12)T_{\rm sh}(34)\,\,{\rm
e}^{(\cdot)}\rightarrow\dot{G}(12)T_{\rm sh}(34)\left(TG_{34}k_{3}\cdot
k_{4}+T^{2}\sum_{i,j=1}^{2}G_{i4}k_{i}\cdot k_{4}G_{j3}k_{j}\cdot
k_{3}\right)\,\,{\rm e}^{TG_{12}k_{1}\cdot k_{2}}.$ (2.17)

  * 5.

Case 5: Terms in $Q_{\rm scal}^{2}$ which have only one of the low-energy
momenta in the two-cycle. For instance, the two-tail term $\dot{G}(13)T_{\rm
sh}(24)\,\,{\rm e}^{(\cdot)}$ gives

$\displaystyle\dot{G}(13)T_{\rm sh}(24)\,\,{\rm
e}^{(\cdot)}\rightarrow\dot{G}(13)\dot{G}_{12}\dot{G}_{41}\frac{k_{1}\cdot
f_{2}\cdot f_{4}\cdot k_{1}}{k_{2}\cdot k_{4}}\Big{[}TG_{14}k_{1}\cdot
k_{4}+TG_{24}k_{2}\cdot k_{4}\Big{]}\,\,{\rm e}^{TG_{12}k_{1}\cdot k_{2}}$
(2.18)

where we took the terms linear in $k_{4}$ from the exponential.

  * 6.

Case 6: Terms in $Q_{\rm scal}^{2}$ which are already quadratic in at least
one of the low-energy momenta can be discarded. For instance, in the two-tail
term $\dot{G}(34)T_{\rm sh}(12)\,\,{\rm e}^{(\cdot)}$ we have terms like

$\displaystyle\dot{G}(34)\dot{G}_{31}\dot{G}_{24}\frac{k_{3}\cdot f_{1}\cdot
f_{2}\cdot k_{4}}{k_{1}\cdot k_{2}}\,\,{\rm e}^{(\cdot)}\rightarrow 0\,.$
(2.19)

In particular, all terms in $\dot{G}(34)T_{\rm sh}(12)\,{\rm e}^{(\cdot)}$ can
be neglected.

Thus we see that, in the limit of two low-energy photons which is our object
of interest in this paper, there are many terms that drop out of the amplitude
at the integrand level, and others that drop out after integration. The
transition to spinor QED does not lead to any new considerations.

###  2.2 Low-energy limit for two of the photons: computational examples

Next, let us explain our strategy for computing the four-photon amplitude with
two low-energy photons, using some sample terms from both $Q_{{\rm scal}}$ and
$Q_{{\rm spin}}$.

Example 1: Let us consider the following one-tail term:

$\begin{split}Q_{{\rm scal}}^{3}(123;4)=&\textbf{
}\dot{G}_{12}\dot{G}_{23}\dot{G}_{31}Z_{3}(123)\left(\dot{G}_{41}\varepsilon_{4}\cdot
k_{1}+\dot{G}_{42}\varepsilon_{4}\cdot k_{2}+\dot{G}_{43}\varepsilon_{4}\cdot
k_{3}\right).\end{split}$ (2.20)

In order to take the low-energy limit for leg 4, we define

$Q^{3}_{{\rm scal}(4)}(123;4)\equiv\left.\int_{0}^{1}du_{4}~{}Q_{{\rm
scal}}^{3}(123;4)~{}\,{\rm e}^{(\cdot 4)}\right|_{{\rm lin}~{}k_{4}}\,,$
(2.21)

where in the full exponent we single out those terms which have a $k_{4}$

$\displaystyle\,{\rm e}^{(\cdot 4)}\equiv\,{\rm e}^{T(G_{14}k_{1}\cdot
k_{4}+G_{24}k_{2}\cdot k_{4}+G_{34}k_{3}\cdot k_{4})}\,,$ (2.22)

such that

$\displaystyle\,{\rm e}^{(\cdot)}=\,{\rm e}^{(\cdot 4)}\,\,{\rm
e}^{T(G_{12}k_{1}\cdot k_{2}+G_{13}k_{1}\cdot k_{3}+G_{23}k_{2}\cdot k_{3})}.$
(2.23)

Here, for convenience, we set the following convention for low-energy legs: on
the left-hand side of expression (2.21) the subscript ‘$(4)$’ indicates that
the leg number 4 with momentum $k_{4}$ is to be taken in the low energy limit
and integrated out. In A we give a list of all the occurring integrals (up to
permutations). It allows us to perform the integral over $u_{4}$ without
fixing an ordering for the remaining photon legs (for efficient techniques to
derive such formulas see, e.g., [39]).

Since in this example we do not have linear terms in $k_{4}$, we take them
from the exponential part and we use Eq. (A.5) to integrate

$\begin{split}Q_{{\rm
scal}(4)}^{3}(123;4)&=\frac{T}{3}Z_{3}(123)\,\dot{G}_{12}\dot{G}_{23}\dot{G}_{31}\left(\sum_{i=1}^{3}\dot{G}_{1i}G_{i1}\,k_{i}\cdot
k_{4}\varepsilon_{4}\cdot k_{1}\right.\\\
&\quad+\sum_{i=1}^{3}\left.\dot{G}_{2i}G_{i2}\,k_{i}\cdot
k_{4}\varepsilon_{4}\cdot k_{2}+\sum_{i=1}^{3}\dot{G}_{3i}G_{i3}\,k_{i}\cdot
k_{4}\varepsilon_{4}\cdot k_{3}\right)\,.\end{split}$ (2.24)

Combining terms, we can write the result in a manifestly gauge invariant form

$\begin{split}Q_{{\rm
scal}(4)}^{3}(123;4)&=\frac{T}{3}Z_{3}(123)\dot{G}_{12}\dot{G}_{23}\dot{G}_{31}\Big{(}\dot{G}_{12}G_{21}\,k_{2}\cdot
f_{4}\cdot k_{1}\\\ &\quad+\dot{G}_{23}G_{32}\,k_{3}\cdot f_{4}\cdot
k_{2}+\dot{G}_{31}G_{13}\,k_{1}\cdot f_{4}\cdot k_{3}\Big{)}\,.\\\
\end{split}$ (2.25)

Next we repeat all this with photon number 3. We define

$\begin{split}Q_{{\rm
scal}(34)}^{3}(123;4)&\equiv\left.\int_{0}^{1}du_{3}\int_{0}^{1}du_{4}~{}Q_{{\rm
scal}}^{3}(123;4)~{}\,{\rm e}^{(\cdot 3)}\,{\rm e}^{(\cdot 4)}\right|_{{\rm
lin}~{}k_{4},k_{3}}\\\ &=\left.\int_{0}^{1}du_{3}~{}Q_{{\rm
scal}(4)}^{3}(123;4)~{}\,{\rm e}^{(\cdot 3)}\right|_{{\rm
lin}~{}k_{3}}\,,\end{split}$ (2.26)

where

$\,{\rm e}^{(\cdot 3)}\equiv\,{\rm e}^{T(G_{13}k_{1}\cdot
k_{3}+G_{23}k_{2}\cdot k_{3})}\,.$ (2.27)

Similarly to the above, the subscript ‘$(34)$’ in equation (2.26) indicates
that legs number 3 and 4 are both projected on their low energy limits. Note
that some terms in (2.25) are quadratic in $k_{3}$ and can be neglected. We
use (LABEL:intGdots) to perform the integral over $u_{3}$ and, with the aid of
the identity $\dot{G}_{ij}^{2}=1-4G_{ij}$, write the result as

$Q_{{\rm scal}(34)}^{3}(123;4)=-\frac{T}{9}Z_{3}(123)k_{2}\cdot f_{4}\cdot
k_{1}\left(G_{12}-10G_{12}^{2}+24G_{12}^{3}\right).$ (2.28)

Therefore at this stage the contribution of $Q_{{\rm scal}}^{3}(123;4)$ to the
four-photon amplitude with legs 3 and 4 at low energy is given by

$\hat{\Gamma}_{{\rm
scal}(34)}^{3}(123;4)=\int_{0}^{\infty}\frac{dT}{T}\,T^{4-\frac{D}{2}}\,{\rm
e}^{-m^{2}T}\,\int_{0}^{1}du_{1}du_{2}\,Q_{{\rm scal}(34)}^{3}(123;4)\,\,{\rm
e}^{TG_{12}k_{1}\cdot k_{2}}\,.$ (2.29)

This leads us to define

$\begin{split}Y_{nl}\equiv\int_{0}^{\infty}\frac{dT}{T}\,T^{n-D/2}~{}\int_{0}^{1}du_{1}\int_{0}^{1}du_{2}~{}G_{12}^{l}\,{\rm
e}^{-T[m^{2}-G_{12}k_{1}\cdot k_{2}]}\,.\end{split}$ (2.30)

The proper-time integral $T$ is elementary and, due to the unbroken
translation invariance along the loop, one of the two parameters $u_{1},u_{2}$
can be fixed at some arbitrary value, such as $u_{2}=0$ and $u_{1}=u$. To the
remaining $u$ integral we apply a tensor reduction procedure, explained in
A.2, to arrive at the following expression,

$\begin{split}Y_{nl}=\frac{\Gamma\left(n-\frac{D}{2}-l\right)}{m^{2n-D}}~{}\frac{d^{l}}{d\hat{k}_{12}^{l}}~{}{}_{2}F_{1}\left(1,n-l-\frac{D}{2};\frac{3}{2};\frac{\hat{k}_{12}}{4}\right),\end{split}$
(2.31)

where $\hat{k}_{12}\equiv\frac{k_{1}\cdot k_{2}}{m^{2}}$ and
${}_{2}F_{1}(a,b;c;x)$ is the Gauss hypergeometric function. With these
definitions, (2.28) and (2.29) transform into

$\hat{\Gamma}_{{\rm scal}(34)}^{3}(123;4)=-\frac{1}{9}Z_{3}(123)\,k_{2}\cdot
f_{4}\cdot k_{1}\left(Y_{51}-10Y_{52}+24Y_{53}\right)\,.$ (2.32)

Proceeding to the spinor QED case, since for the term at hand the integration
over $u_{4}$ does not interfere with the application of the Bern-Kosower
replacement rule (2.9), the term corresponding to (2.25) reads

$\displaystyle\begin{split}Q_{{\rm
spin}(4)}^{3}(123;4)&=\frac{T}{3}\Big{(}\dot{G}_{12}\dot{G}_{23}\dot{G}_{31}-G_{F12}G_{F23}G_{F31}\Big{)}Z_{3}(123)\\\
&\times\Big{(}\dot{G}_{12}G_{21}k_{2}\cdot f_{4}\cdot
k_{1}+\dot{G}_{23}G_{32}k_{3}\cdot f_{4}\cdot
k_{2}+\dot{G}_{31}G_{13}k_{1}\cdot f_{4}\cdot k_{3}\Big{)}\,.\end{split}$
(2.33)

It would seem that we now have to extend our table of integrals in A to
integrals involving both $\dot{G}_{ij}$ and $G_{Fij}$, but we can avoid this
by eliminating the latter in favor of the former. This can be done using the
basic identities

$\displaystyle G_{Fij}^{2}=1,\quad
G_{Fij}G_{Fjk}G_{Fki}=-\dot{G}_{ij}-\dot{G}_{jk}-\dot{G}_{ki}$ (2.34)

which transforms (2.33) into

$\displaystyle\begin{split}Q_{{\rm
spin}(4)}^{3}(123;4)&=\frac{T}{3}\Big{(}\dot{G}_{12}\dot{G}_{23}\dot{G}_{31}+\dot{G}_{12}+\dot{G}_{23}+\dot{G}_{31}\Big{)}Z_{3}(123)\\\
&\times\Big{(}\dot{G}_{12}G_{21}k_{2}\cdot f_{4}\cdot
k_{1}+\dot{G}_{23}G_{32}k_{3}\cdot f_{4}\cdot
k_{2}+\dot{G}_{31}G_{13}k_{1}\cdot f_{4}\cdot k_{3}\Big{)}\,.\end{split}$
(2.35)

Proceeding as in the scalar case, one finds that the spinor-QED equivalent of
(2.32) becomes

$\displaystyle{\hat{\Gamma}_{{\rm spin}(34)}}^{3}(123;4)$ $\displaystyle=$
$\displaystyle\frac{2}{9}Z_{3}(123)k_{2}\cdot f_{4}\cdot
k_{1}\left(Y_{51}-Y_{52}-12Y_{53}\right)\,,$ (2.36)

as the reader can easily verify using formulas of A.

Example 2: Let us consider another one-tail term,

$\begin{split}Q_{{\rm
scal}}^{3}(234;1)&=Z_{3}(234)\,\dot{G}_{23}\dot{G}_{34}\dot{G}_{42}\left(\dot{G}_{12}\varepsilon_{1}\cdot
k_{2}+\dot{G}_{13}\varepsilon_{1}\cdot k_{3}+\dot{G}_{14}\varepsilon_{1}\cdot
k_{4}\right)\,.\end{split}$ (2.37)

Notice that it has a quadratic term in $k_{4}$ and two linear terms in
$k_{4}$. Omitting the former, and integration the latter over $u_{4}$ using
(LABEL:intGdots), we obtain

$Q_{{\rm
scal}(4)}^{3}(234;1)=Z_{3}(234)\,\dot{G}_{23}\left(\frac{1}{6}-\frac{1}{2}\dot{G}_{32}^{2}\right)\left(\dot{G}_{12}\varepsilon_{1}\cdot
k_{2}+\dot{G}_{13}\varepsilon_{1}\cdot k_{3}\right)\,.$ (2.38)

Thus we have a term quadratic in $k_{3}$ and a term linear in $k_{3}$, and for
the latter the integral over $u_{3}$ turns out to vanish. Therefore we get
nothing,

${\hat{\Gamma}_{{\rm scal}(34)}}^{3}(234;1)=0\,.$ (2.39)

We leave it to the reader to check that the additional terms generated by
applying the Bern-Kosower replacement rule to the $(234)$ cycle vanish, too.
Therefore

${\hat{\Gamma}_{{\rm spin}(34)}}^{3}(234;1)=0\,.$ (2.40)

Example 3: As our final example, we consider the following 4-cycle term in the
spinor QED integrand,

$Q_{{\rm
spin}}^{4}(1234)=Z_{4}(1234)\Big{(}\dot{G}_{12}\dot{G}_{23}\dot{G}_{34}\dot{G}_{41}-G_{F12}G_{F23}G_{F34}G_{F41}\Big{)}\,.$
(2.41)

To reexpress the spin contribution $G_{F12}G_{F23}G_{F34}G_{F41}$, we insert a
factor $1=-G_{F13}G_{F31}$, and once more use the identity
$G_{Fij}G_{Fjk}G_{Fki}=-\dot{G}_{ij}-\dot{G}_{jk}-\dot{G}_{ki}$. This leads to

$\begin{split}Q_{{\rm
spin}(4)}^{4}(1234)&=Z_{4}(1234)\int_{0}^{1}du_{4}\Big{[}\dot{G}_{12}\dot{G}_{23}\dot{G}_{34}\dot{G}_{41}+\left(\dot{G}_{12}+\dot{G}_{23}+\dot{G}_{31}\right)\left(\dot{G}_{13}+\dot{G}_{34}+\dot{G}_{41}\right)\Big{]}\,.\end{split}$
(2.42)

Upon integration we find, after some cancellations,

$\begin{split}Q_{{\rm
spin}(4)}^{4}(1234)=Z_{4}(1234)\left[\dot{G}_{12}\dot{G}_{23}\left(\frac{1}{6}-\frac{1}{2}\dot{G}_{31}^{2}\right)+\left(\dot{G}_{12}+\dot{G}_{23}+\dot{G}_{31}\right)\dot{G}_{13}\right]\,.\end{split}$
(2.43)

Integrating over $u_{3}$ we obtain

$Q_{{\rm
spin}(34)}^{4}(1234)=-\frac{4}{3}Z_{4}(1234)\left(G_{12}+2G_{12}^{2}\right)\,.$
(2.44)

Thus with the notation (2.30), the contribution to the amplitude of this term
reads

${\hat{\Gamma}_{{\rm
spin}(34)}}^{4}(1234)=-\frac{4}{3}Z_{4}(1234)\left(Y_{41}+2Y_{42}\right)\,.$
(2.45)

##  3 Results: low-energy limit for two of the photons

In this section, we present our results for the complete four-photon
amplitudes in both scalar and spinor QED, off-shell but under the restriction
that photons 3 and 4 are taken in the low-energy limit i. e., we consider only
contributions that are linear in $k_{3}$ and $k_{4}$ (see part I). These
amplitudes represent the main result of the present article. In the worldline
formalism, they appear naturally decomposed as

$\hat{\Gamma}_{\left\\{{\rm scal}\atop{\rm
spin}\right\\}(34)}=\hat{\Gamma}_{\left\\{{\rm scal}\atop{\rm
spin}\right\\}(34)}^{4}+\hat{\Gamma}_{\left\\{{\rm scal}\atop{\rm
spin}\right\\}(34)}^{3}+\hat{\Gamma}_{\left\\{{\rm scal}\atop{\rm
spin}\right\\}(34)}^{2}+\hat{\Gamma}_{\left\\{{\rm scal}\atop{\rm
spin}\right\\}(34)}^{22}$ (3.1)

with

$\hat{\Gamma}_{\left\\{{\rm scal}\atop{\rm
spin}\right\\}(34)}^{i}=\int_{0}^{1}du_{1}\int_{0}^{1}du_{2}\int_{0}^{\infty}\frac{dT}{T}\,T^{4-\frac{D}{2}}\,{\rm
e}^{-T(m^{2}-G_{12}k_{1}\cdot k_{2})}Q_{\left\\{{\rm scal}\atop{\rm
spin}\right\\}(34)}^{i}~{}~{},~{}~{}i=4,3,2,22\,.~{}~{}~{}$ (3.2)

Recall that the subscript $(34)$ in $\hat{\Gamma}_{(34)}^{i}$ indicates that
the integrals over $u_{3}$ and $u_{4}$ were performed taking the low-energy
limit for the corresponding photons. The absolute normalizations of the
amplitudes are given by

$\displaystyle\Gamma_{\rm
scal(34)}(k_{1},\varepsilon_{1};\cdots;k_{4},\varepsilon_{4})$
$\displaystyle=$
$\displaystyle\frac{e^{4}}{(4\pi)^{\frac{D}{2}}}\hat{\Gamma}_{\rm
scal(34)}\,,$ (3.3) $\displaystyle\Gamma_{\rm
spin(34)}(k_{1},\varepsilon_{1};\cdots;k_{4},\varepsilon_{4})$
$\displaystyle=$
$\displaystyle-2\frac{e^{4}}{(4\pi)^{\frac{D}{2}}}\hat{\Gamma}_{\rm
spin(34)}\,.$ (3.4)

We give them first in dimensionally regularized form, and then in $D=4$. Since
the worldline representation of these amplitudes is manifestly finite, taking
this limit is trivial and will not involve a cancellation of spurious
$1/\epsilon$ poles as would be the case in a Feynman diagram calculation (in
our conventions $\epsilon=D-4$).

###  3.1 Dimensionally regularized amplitudes

The integrals appearing in (3.2) are all of the form of (2.30) so that we
choose to express them in terms of the functions $Y_{nl}$ that have been given
in terms of ${}_{2}F_{1}$ in (2.31). We note that this implies some redundancy
since, as shown in A.2, there exist recursion relations between the $Y_{nl}$
functions that would also allow us to rewrite all our results entirely in
terms of the $Y_{n0}$. However, the resulting representation would be less
compact than the one given in the following.

####  3.1.1 Scalar QED

For $Q_{{\rm scal}}^{4}$

$\displaystyle{\hat{\Gamma}_{{\rm scal}(34)}}^{4}(1234)$ $\displaystyle=$
$\displaystyle\frac{2}{3}Z_{4}(1234)\left(Y_{41}-4Y_{42}\right)\,,$ (3.5)
$\displaystyle{\hat{\Gamma}_{{\rm scal}(34)}}^{4}(2314)$ $\displaystyle=$
$\displaystyle\frac{1}{9}Z_{4}(2314)\left(Y_{40}-12Y_{41}+36Y_{42}\right)\,,$
(3.6) $\displaystyle{\hat{\Gamma}_{{\rm scal}(34)}}^{4}(3124)$
$\displaystyle=$
$\displaystyle\frac{2}{3}Z_{4}(3124)\left(Y_{41}-4Y_{42}\right)\,.$ (3.7)

For $Q_{{\rm scal}}^{3}$

$\displaystyle{\hat{\Gamma}_{{\rm scal}(34)}}^{3}(123;4)$ $\displaystyle=$
$\displaystyle-\frac{1}{9}Z_{3}(123)k_{2}\cdot f_{4}\cdot
k_{1}\left(Y_{51}-10Y_{52}+24Y_{53}\right)\,,$ (3.8)
$\displaystyle{\hat{\Gamma}_{{\rm scal}(34)}}^{3}(234;1)$ $\displaystyle=$
$\displaystyle 0\,,$ (3.9) $\displaystyle{\hat{\Gamma}_{{\rm
scal}(34)}}^{3}(341;2)$ $\displaystyle=$ $\displaystyle 0\,,$ (3.10)
$\displaystyle{\hat{\Gamma}_{{\rm scal}(34)}}^{3}(412;3)$ $\displaystyle=$
$\displaystyle-\frac{1}{9}Z_{3}(412)k_{2}\cdot f_{3}\cdot
k_{1}\left(Y_{51}-10Y_{52}+24Y_{53}\right)\,.$ (3.11)

For $Q_{{\rm scal}}^{2}$

$\begin{split}{\hat{\Gamma}_{{\rm
scal}(34)}}^{2}(12;34)&=-\frac{1}{18}Z_{2}(12)\Bigg{\\{}\left[\frac{1}{5}\left(Y_{50}-4Y_{51}\right)-6\left(Y_{52}-4Y_{53}\right)\right]k_{1}\cdot
f_{3}\cdot f_{4}\cdot k_{2}\\\
&\quad+\frac{1}{5}\left(Y_{50}-4Y_{51}\right)k_{1}\cdot f_{3}\cdot f_{4}\cdot
k_{1}+(1\leftrightarrow 2)\Bigg{\\}}\\\
&\quad-\frac{1}{9}Z_{2}(12)\left(Y_{62}-8Y_{63}+16Y_{64}\right)k_{1}\cdot
f_{3}\cdot k_{2}k_{1}\cdot f_{4}\cdot k_{2}\,,\end{split}$ (3.12)
${\hat{\Gamma}_{{\rm scal}(34)}}^{2}(13;24)+{\hat{\Gamma}_{{\rm
scal}(34)}}^{2}(23;14)=-\frac{1}{9}Z_{2}(13)k_{1}\cdot f_{2}\cdot f_{4}\cdot
k_{1}\,\left(Y_{51}-4Y_{52}\right)+(1\leftrightarrow 2)\,,$ (3.13)
${\hat{\Gamma}_{{\rm scal}(34)}}^{2}(14;23)+{\hat{\Gamma}_{{\rm
scal}(34)}}^{2}(24;13)=-\frac{1}{9}Z_{2}(14)k_{1}\cdot f_{2}\cdot f_{3}\cdot
k_{1}\,\left(Y_{51}-4Y_{52}\right)+(1\leftrightarrow 2)\,,$ (3.14)
${\hat{\Gamma}_{{\rm scal}(34)}}^{2}(34;12)=0\,.$ (3.15)

And finally for $Q_{{\rm scal}}^{22}$

$\displaystyle{\hat{\Gamma}_{{\rm scal}(34)}}^{22}(12,34)$ $\displaystyle=$
$\displaystyle\frac{1}{3}Z_{2}(12)Z_{2}(34)\left(Y_{40}-4Y_{41}\right)\,,$
(3.16) $\displaystyle{\hat{\Gamma}_{{\rm scal}(34)}}^{22}(13,24)$
$\displaystyle=$ $\displaystyle\frac{1}{9}Z_{2}(13)Z_{2}(24)Y_{40}\,,$ (3.17)
$\displaystyle{\hat{\Gamma}_{{\rm scal}(34)}}^{22}(14,23)$ $\displaystyle=$
$\displaystyle\frac{1}{9}Z_{2}(14)Z_{2}(23)Y_{40}\,.$ (3.18)

####  3.1.2 Spinor QED

For $Q_{{\rm spin}}^{4}$

$\displaystyle{\hat{\Gamma}_{{\rm spin}(34)}}^{4}(1234)$ $\displaystyle=$
$\displaystyle-\frac{4}{3}Z_{4}(1234)\left(Y_{41}+2Y_{42}\right)\,,$ (3.19)
$\displaystyle{\hat{\Gamma}_{{\rm spin}(34)}}^{4}(2314)$ $\displaystyle=$
$\displaystyle-\frac{4}{9}Z_{4}(2314)\left(2Y_{40}-6Y_{41}-9Y_{42}\right)\,,$
(3.20) $\displaystyle{\hat{\Gamma}_{{\rm spin}(34)}}^{4}(3124)$
$\displaystyle=$
$\displaystyle-\frac{4}{3}Z_{4}(3124)\left(Y_{41}+2Y_{42}\right)\,.$ (3.21)

For $Q_{{\rm spin}}^{3}$

$\displaystyle{\hat{\Gamma}_{{\rm spin}(34)}}^{3}(123;4)$ $\displaystyle=$
$\displaystyle\frac{2}{9}Z_{3}(123)k_{2}\cdot f_{4}\cdot
k_{1}\left(Y_{51}-Y_{52}-12Y_{53}\right)\,,$ (3.22)
$\displaystyle{\hat{\Gamma}_{{\rm spin}(34)}}^{3}(234;1)$ $\displaystyle=$
$\displaystyle 0\,,$ (3.23) $\displaystyle{\hat{\Gamma}_{{\rm
spin}(34)}}^{3}(341;2)$ $\displaystyle=$ $\displaystyle 0\,,$ (3.24)
$\displaystyle{\hat{\Gamma}_{{\rm spin}(34)}}^{3}(412;3)$ $\displaystyle=$
$\displaystyle\frac{2}{9}Z_{3}(412)k_{2}\cdot f_{3}\cdot
k_{1}\left(Y_{51}-Y_{52}-12Y_{53}\right)\,.$ (3.25)

For $Q_{{\rm spin}}^{2}$

$\begin{split}{\hat{\Gamma}_{{\rm
spin}(34)}}^{2}(12;34)&=\frac{2}{9}Z_{2}(12)\,\Bigg{\\{}\left[\frac{1}{5}Y_{51}k_{1}\cdot
f_{3}\cdot f_{4}\cdot k_{1}+\left(\frac{1}{5}Y_{51}-6Y_{53}\right)k_{1}\cdot
f_{3}\cdot f_{4}\cdot k_{2}+(1\leftrightarrow 2)\right]\\\
&\quad+2\,\left(Y_{63}-4Y_{64}\right)k_{1}\cdot f_{3}\cdot k_{2}k_{1}\cdot
f_{4}\cdot k_{2}\Bigg{\\}}\,,\end{split}$ (3.26) ${\hat{\Gamma}_{{\rm
spin}(34)}}^{2}(13;24)+{\hat{\Gamma}_{{\rm
spin}(34)}}^{2}(23;14)=\frac{2}{9}Z_{2}(13)k_{1}\cdot f_{2}\cdot f_{4}\cdot
k_{1}\,\left(Y_{51}-4Y_{52}\right)+(1\leftrightarrow 2)\,,$ (3.27)
${\hat{\Gamma}_{{\rm spin}(34)}}^{2}(14;23)+{\hat{\Gamma}_{{\rm
spin}(34)}}^{2}(24;13)=\frac{2}{9}Z_{2}(14)k_{1}\cdot f_{2}\cdot f_{3}\cdot
k_{1}\,\left(Y_{51}-4Y_{52}\right)+(1\leftrightarrow 2)\,,$ (3.28)
${\hat{\Gamma}_{{\rm spin}(34)}}^{2}(34;12)=0\,.$ (3.29)

And finally for $Q_{{\rm spin}}^{22}$

${\hat{\Gamma}_{{\rm
spin}(34)}}^{22}(12,34)=\frac{8}{3}Z_{2}(12)Z_{2}(34)Y_{41}\,,$ (3.30)
${\hat{\Gamma}_{{\rm
spin}(34)}}^{22}(13,24)=\frac{4}{9}Z_{2}(13)Z_{2}(24)Y_{40}\,,$ (3.31)
${\hat{\Gamma}_{{\rm
spin}(34)}}^{22}(14,23)=\frac{4}{9}Z_{2}(14)Z_{2}(23)Y_{40}\,.$ (3.32)

###  3.2 Four-dimensional amplitudes

In this subsection we specialize the previous results to four space-time
dimensions ($D=4$), after which it becomes possible to write the amplitudes
completely in terms of elementary and trigonometric functions. In the interest
of compactness, it will now be useful to introduce the following dimensionless
variables,

$\displaystyle{\hat{k}_{12}}=\frac{k_{1}\cdot
k_{2}}{m^{2}}\,,~{}~{}~{}~{}~{}p_{0}\equiv\frac{{\rm
arcsinh}\Bigl{(}\frac{\sqrt{-{\hat{k}_{12}}}}{2}\Bigr{)}}{\sqrt{(4-{\hat{k}_{12}})(-{\hat{k}_{12}})}}\,.$
(3.33)

####  3.2.1 Scalar QED

For $Q_{{\rm scal}}^{4}$

$\begin{split}{\hat{\Gamma}_{{\rm
scal}(34)}}^{4}(1234)&=-\frac{12+8({\hat{k}_{12}}-6)p_{0}}{3m^{4}{\hat{k}_{12}}^{2}}Z_{4}(1234)\,,\\\
{\hat{\Gamma}_{{\rm
scal}(34)}}^{4}(2314)&=-\frac{2({\hat{k}_{12}}^{2}-30{\hat{k}_{12}}+108)-8p_{0}(5{\hat{k}_{12}}^{2}-48{\hat{k}_{12}}+108)}{9m^{4}({\hat{k}_{12}}-4){\hat{k}_{12}}^{2}}Z_{4}(2314)\,,\\\
{\hat{\Gamma}_{{\rm
scal}(34)}}^{4}(3124)&=-\frac{12+8({\hat{k}_{12}}-6)p_{0}}{3m^{4}{\hat{k}_{12}}^{2}}Z_{4}(3124)\,.\end{split}$
(3.34)

For $Q_{{\rm scal}}^{3}$

$\begin{split}{\hat{\Gamma}_{{\rm
scal}(34)}}^{3}(123;4)&=-\frac{2({\hat{k}_{12}}^{2}-48{\hat{k}_{12}}+180)-16p_{0}(4{\hat{k}_{12}}^{2}-39{\hat{k}_{12}}+90)}{9m^{6}({\hat{k}_{12}}-4){\hat{k}_{12}}^{3}}Z_{3}(123)k_{2}\cdot
f_{4}\cdot k_{1}\,,\\\ {\hat{\Gamma}_{{\rm
scal}(34)}}^{3}(412;3)&=-\frac{2({\hat{k}_{12}}^{2}-48{\hat{k}_{12}}+180)-16p_{0}(4{\hat{k}_{12}}^{2}-39{\hat{k}_{12}}+90)}{9m^{6}({\hat{k}_{12}}-4){\hat{k}_{12}}^{3}}Z_{3}(412)k_{2}\cdot
f_{3}\cdot k_{1}\,.\end{split}$ (3.35)

For $Q_{{\rm scal}}^{2}$

$\begin{split}{\hat{\Gamma}_{{\rm
scal}(34)}}^{2}(12;34)&=\frac{-1}{90}Z_{2}(12)\Bigg{\\{}\Bigg{[}\frac{2(-{\hat{k}_{12}}+2)-16p_{0}}{m^{6}({\hat{k}_{12}}-4){\hat{k}_{12}}}\,k_{1}\cdot
f_{3}\cdot f_{4}\cdot k_{1}\\\ &+P_{1}\,k_{1}\cdot f_{3}\cdot f_{4}\cdot
k_{2}+(1\leftrightarrow 2)\Bigg{]}+10P_{2}~{}k_{2}\cdot f_{4}\cdot
k_{1}k_{2}\cdot f_{3}\cdot k_{1}\Bigg{\\}}\,,\\\ {\hat{\Gamma}_{{\rm
scal}(34)}}^{2}(13;24)&=-\frac{2({\hat{k}_{12}}-6)-16p_{0}({\hat{k}_{12}}-3)}{9m^{6}({\hat{k}_{12}}-4){\hat{k}_{12}}^{2}}Z_{2}(13)k_{1}\cdot
f_{2}\cdot f_{4}\cdot k_{1}\,,\\\ {\hat{\Gamma}_{{\rm
scal}(34)}}^{2}(23;14)&=-\frac{2({\hat{k}_{12}}-6)-16p_{0}({\hat{k}_{12}}-3)}{9m^{6}({\hat{k}_{12}}-4){\hat{k}_{12}}^{2}}Z_{2}(23)k_{2}\cdot
f_{1}\cdot f_{4}\cdot k_{2}\,,\\\ {\hat{\Gamma}_{{\rm
scal}(34)}}^{2}(14;23)&=-\frac{2({\hat{k}_{12}}-6)-16p_{0}({\hat{k}_{12}}-3)}{9m^{6}({\hat{k}_{12}}-4){\hat{k}_{12}}^{2}}Z_{2}(14)k_{1}\cdot
f_{2}\cdot f_{3}\cdot k_{1}\,,\\\ {\hat{\Gamma}_{{\rm
scal}(34)}}^{2}(24;31)&=-\frac{2({\hat{k}_{12}}-6)-16p_{0}({\hat{k}_{12}}-3)}{9m^{6}({\hat{k}_{12}}-4){\hat{k}_{12}}^{2}}Z_{2}(24)k_{2}\cdot
f_{1}\cdot f_{3}\cdot k_{2}\,.\end{split}$ (3.36)

And finally, for $Q_{{\rm scal}}^{22}$

$\begin{split}{\hat{\Gamma}_{{\rm
scal}(34)}}^{22}(12,34)&=-\frac{2-8p_{0}}{3m^{4}{\hat{k}_{12}}}Z_{2}(12)Z_{2}(34)\,,\\\
{\hat{\Gamma}_{{\rm
scal}(34)}}^{22}(13,24)&=-\frac{2+8p_{0}}{9m^{4}({\hat{k}_{12}}-4)}Z_{2}(13)Z_{2}(24)\,,\\\
{\hat{\Gamma}_{{\rm
scal}(34)}}^{22}(14,23)&=-\frac{2+8p_{0}}{9m^{4}({\hat{k}_{12}}-4)}Z_{2}(14)Z_{2}(23)\,.\end{split}$
(3.37)

Here we have introduced two more functions, $P_{i}=P_{i}({\hat{k}_{12}})$ with
$i=1,2$, which are defined as

$P_{1}\equiv\frac{2(-{\hat{k}_{12}}^{3}+2{\hat{k}_{12}}^{2}-210{\hat{k}_{12}}+900)-32p_{0}(8{\hat{k}_{12}}^{2}-90{\hat{k}_{12}}+225)}{m^{6}({\hat{k}_{12}}-4){\hat{k}_{12}}^{3}}\,,$
(3.38)
$P_{2}\equiv\frac{4(-{\hat{k}_{12}}^{2}+55{\hat{k}_{12}}-210)+48p_{0}(3{\hat{k}_{12}}^{2}-30{\hat{k}_{12}}+70)}{m^{8}({\hat{k}_{12}}-4){\hat{k}_{12}}^{4}}\,.$
(3.39)

####  3.2.2 Spinor QED

For $Q_{{\rm spin}}^{4}$

$\begin{split}{\hat{\Gamma}_{{\rm
spin}(34)}}^{4}(1234)&=\frac{16[3+p_{0}({\hat{k}_{12}}^{2}+2{\hat{k}_{12}}-12)]}{3m^{4}({\hat{k}_{12}}-4){\hat{k}_{12}}^{2}}Z_{4}(1234)\,,\\\
{\hat{\Gamma}_{{\rm
spin}(34)}}^{4}(2314)&=\frac{4[4{\hat{k}_{12}}^{2}-3{\hat{k}_{12}}-54-8p_{0}({\hat{k}_{12}}^{2}+3{\hat{k}_{12}}-27)]}{9m^{4}({\hat{k}_{12}}-4){\hat{k}_{12}}^{2}}Z_{4}(2314)\,,\\\
{\hat{\Gamma}_{{\rm
spin}(34)}}^{4}(3124)&=\frac{16[3+p_{0}({\hat{k}_{12}}^{2}+2{\hat{k}_{12}}-12)]}{3m^{4}({\hat{k}_{12}}-4){\hat{k}_{12}}^{2}}Z_{4}(3124)\,.\end{split}$
(3.40)

For $Q_{{\rm spin}}^{3}$

$\begin{split}{\hat{\Gamma}_{{\rm
spin}(34)}}^{3}(123;4)&=\frac{4({\hat{k}_{12}}^{2}+15{\hat{k}_{12}}-90)+16p_{0}({\hat{k}_{12}}^{2}-30{\hat{k}_{12}}+90)}{9m^{6}({\hat{k}_{12}}-4){\hat{k}_{12}}^{3}}Z_{3}(123)k_{2}\cdot
f_{4}\cdot k_{1}\,,\\\ {\hat{\Gamma}_{{\rm
spin}(34)}}^{3}(412;3)&=\frac{4({\hat{k}_{12}}^{2}+15{\hat{k}_{12}}-90)+16p_{0}({\hat{k}_{12}}^{2}-30{\hat{k}_{12}}+90)}{9m^{6}({\hat{k}_{12}}-4){\hat{k}_{12}}^{3}}Z_{3}(412)k_{2}\cdot
f_{3}\cdot k_{1}\,.\end{split}$ (3.41)

For $Q_{{\rm spin}}^{2}$

$\begin{split}{\hat{\Gamma}_{{\rm
spin}(34)}}^{2}(12;34)&=\frac{1}{3}Z_{2}(12)\Bigg{\\{}\Bigg{[}\frac{4({\hat{k}_{12}}+2)+32p_{0}({\hat{k}_{12}}-1)}{15m^{6}(4-{\hat{k}_{12}})^{2}{\hat{k}_{12}}}\,k_{1}\cdot
f_{3}\cdot f_{4}\cdot k_{1}\\\ &+\frac{2}{15}\tilde{P}_{1}\,k_{1}\cdot
f_{3}\cdot f_{4}\cdot k_{2}+(1\leftrightarrow
2)\Bigg{]}+\frac{4}{3}\tilde{P}_{2}\,k_{2}\cdot f_{4}\cdot k_{1}k_{2}\cdot
f_{3}\cdot k_{1}\Bigg{\\}}\,,\\\ {\hat{\Gamma}_{{\rm
spin}(34)}}^{2}(13;24)&=\frac{4({\hat{k}_{12}}-6)-32p_{0}({\hat{k}_{12}}-3)}{9m^{6}({\hat{k}_{12}}-4){\hat{k}_{12}}^{2}}Z_{2}(13)k_{1}\cdot
f_{2}\cdot f_{4}\cdot k_{1}\,,\\\ {\hat{\Gamma}_{{\rm
spin}(34)}}^{2}(23;14)&=\frac{4({\hat{k}_{12}}-6)-32p_{0}({\hat{k}_{12}}-3)}{9m^{6}({\hat{k}_{12}}-4){\hat{k}_{12}}^{2}}Z_{2}(23)k_{2}\cdot
f_{1}\cdot f_{4}\cdot k_{2}\,,\\\ {\hat{\Gamma}_{{\rm
spin}(34)}}^{2}(14;23)&=\frac{4({\hat{k}_{12}}-6)-32p_{0}({\hat{k}_{12}}-3)}{9m^{6}({\hat{k}_{12}}-4){\hat{k}_{12}}^{2}}Z_{2}(14)k_{1}\cdot
f_{2}\cdot f_{3}\cdot k_{1}\,,\\\ {\hat{\Gamma}_{{\rm
spin}(34)}}^{2}(24;31)&=\frac{4({\hat{k}_{12}}-6)-32p_{0}({\hat{k}_{12}}-3)}{9m^{6}({\hat{k}_{12}}-4){\hat{k}_{12}}^{2}}Z_{2}(24)k_{2}\cdot
f_{1}\cdot f_{3}\cdot k_{2}\,.\end{split}$ (3.42)

And finally, for $Q_{{\rm spin}}^{22}$

$\begin{split}{\hat{\Gamma}_{{\rm
spin}(34)}}^{22}(12,34)&=-\frac{16[1+2p_{0}({\hat{k}_{12}}-2)]}{3m^{4}({\hat{k}_{12}}-4){\hat{k}_{12}}}Z_{2}(12)Z_{2}(34)\,,\\\
{\hat{\Gamma}_{{\rm
spin}(34)}}^{22}(13,24)&=-\frac{8(1+4p_{0})}{9m^{4}({\hat{k}_{12}}-4)}Z_{2}(13)Z_{2}(24)\,,\\\
{\hat{\Gamma}_{{\rm
spin}(34)}}^{22}(14,23)&=-\frac{8(1+4p_{0})}{9m^{4}({\hat{k}_{12}}-4)}Z_{2}(14)Z_{2}(23)\,,\end{split}$
(3.43)

Here, we have defined $\tilde{P}_{i}=\tilde{P}_{i}({\hat{k}_{12}})$ as

$\displaystyle\tilde{P}_{1}\equiv\frac{2({\hat{k}_{12}}^{3}+32{\hat{k}_{12}}^{2}-390{\hat{k}_{12}}+900)+16p_{0}({\hat{k}_{12}}^{3}-46{\hat{k}_{12}}^{2}+270{\hat{k}_{12}}-450)}{m^{6}({\hat{k}_{12}}-4)^{2}{\hat{k}_{12}}^{3}},$
(3.44)
$\displaystyle\tilde{P}_{2}\equiv\frac{2(-23{\hat{k}_{12}}^{2}+200{\hat{k}_{12}}-420)-24p_{0}({\hat{k}_{12}}^{3}-18{\hat{k}_{12}}^{2}+90{\hat{k}_{12}}-140)}{m^{8}({\hat{k}_{12}}-4)^{2}{\hat{k}_{12}}^{4}}.$
(3.45)

##  4 Check: the photon propagator in a constant field

The kinematical regime of the four-photon amplitudes that we are studying in
this paper has, to the best of our knowledge, not been treated in the
literature before. However, for the special case where $k_{1}=-k_{2}$ (which
then also implies $k_{3}=-k_{4}$) they can be easily extracted from the photon
polarization tensor in a constant background field, a quantity that has been
intensely studied in spinor [2, 4, 5, 6, 7, 8, 9] and to a lesser degree in
scalar QED [3, 7]. The field can be treated non-perturbatively using the exact
Dirac or Klein-Gordon propagator in the field, indicated by the double line on
the left-hand side of Fig. 4 as is customary.

Figure 4:  The first three terms in the diagrammatic expansion of the photon
polarization tensor in powers of the constant background field. The second
diagram on the right-hand side represents the four-photon amplitude with two
low-energy photons and $k_{1}=-k_{2}$.

Expanding out the full non-perturbative polarization tensor in powers of the
background field one obtains the series depicted on the right-hand side of
Fig. 4, containing any even number of interactions with the field. Since a
constant field cannot inject energy or momentum into the loop, these
interactions are mediated by zero-momentum photons. Thus the second diagram on
the right-hand side of Fig. 4 corresponds to our case of two arbitrary and two
low-energy photons, the two low-energy legs corresponding to the two photons
taken from the background field, however under the restriction
$k_{1}=-k_{2}=k$ imposed by the energy-momentum conservation in the constant
field.

We would now like to use this correspondence for a check on our above results.
While such a comparison could be made using the results of any of the
references cited above, it will be convenient to compare with [7], where the
worldline formalism was already used to compute the constant-field photon
polarization tensors for both scalar and spinor QED. Although that calculation
is still quite different from our present one (mainly because the integrating
out of the low-energy photon legs there was effectively done already at the
level of the construction of the generalized worldline Green’s functions,
shown below in (4.22), (4.44)), it is still close enough to our present one to
make it possible to verify the equivalence at an early stage, without the need
to perform all the integrals. It will be sufficient to work with the
intermediate results of our above calculations after integrating out $u_{3}$
and $u_{4}$, which we collect in B. Using suitable integrations-by-part, we
identify the resulting set of parameter integrals with the ones obtained by
expanding the integral representations given in [7] for the scalar and spinor
QED photon propagators in a constant background field to second order in the
field strength tensor $F_{\mu\nu}$.

###  4.1 Scalar QED

The above replacement for the momenta $(k_{1}=-k_{2}=k)$ actually makes some
terms in $Q_{{\rm scal}}$ vanish. The remaining terms in the scalar loop case
can be compactly written as:

$Q_{{\rm
scal}(34)}^{4}(1234)=\frac{2}{3}Z_{4}(1234)\left(G_{12}-4G_{12}^{2}\right),$
(4.1) $Q_{{\rm
scal}(34)}^{4}(2314)=\frac{1}{9}Z_{4}(2314)\left(1-12G_{12}+36G_{12}^{2}\right),$
(4.2) $Q_{{\rm
scal}(34)}^{4}(3124)=\frac{2}{3}Z_{4}(3124)\left(G_{12}-4G_{12}^{2}\right),$
(4.3) $\displaystyle Q_{{\rm scal}(34)}^{3}(123;4)=0,$ (4.4) $\displaystyle
Q_{{\rm scal}(34)}^{3}(234;1)=0,$ (4.5) $\displaystyle Q_{{\rm
scal}(34)}^{3}(314;2)=0,$ (4.6) $\displaystyle Q_{{\rm
scal}(34)}^{3}(124;3)=0,$ (4.7) $Q_{{\rm
scal}(34)}^{2}(12;34)=-\frac{2T}{3}Z_{2}(12)k\cdot f_{4}\cdot f_{3}\cdot
k\left(G_{12}^{2}-4G_{12}^{3}\right),$ (4.8) $Q_{{\rm
scal}(34)}^{2}(13;24)=-\frac{T}{9}Z_{2}(13)k\cdot f_{4}\cdot f_{2}\cdot
k\left(G_{12}-4G_{12}^{2}\right),$ (4.9) $Q_{{\rm
scal}(34)}^{2}(23;14)=-\frac{T}{9}Z_{2}(23)k\cdot f_{4}\cdot f_{1}\cdot
k\left(G_{12}-4G_{12}^{2}\right),$ (4.10) $Q_{{\rm
scal}(34)}^{2}(14;23)=-\frac{T}{9}Z_{2}(14)k\cdot f_{2}\cdot f_{3}\cdot
k\left(G_{12}-4G_{12}^{2}\right),$ (4.11) $Q_{{\rm
scal}(34)}^{2}(24;13)=-\frac{T}{9}Z_{2}(24)k\cdot f_{1}\cdot f_{3}\cdot
k\left(G_{12}-4G_{12}^{2}\right),$ (4.12) $Q_{{\rm scal}(34)}^{2}(34;12)=0,$
(4.13) $Q_{{\rm
scal}(34)}^{22}(12,34)=\frac{1}{3}Z_{2}(12)Z_{2}(34)\left(1-4G_{12}\right),$
(4.14) $Q_{{\rm scal}(34)}^{22}(13,24)=\frac{1}{9}Z_{2}(13)Z_{2}(24),$ (4.15)
$Q_{{\rm scal}(34)}^{22}(14,23)=\frac{1}{9}Z_{2}(14)Z_{2}(23).$ (4.16)

Note that with the above replacement
$f_{1}^{\mu\nu}=k^{\mu}\varepsilon_{1}^{\nu}-k^{\nu}\varepsilon_{1}^{\mu}$ and
$f_{2}^{\mu\nu}=-k^{\mu}\varepsilon_{2}^{\nu}+k^{\nu}\varepsilon_{2}^{\mu}$.
Rewriting the remaining integrals we have

$\displaystyle\Gamma_{\rm
scal}(k,\varepsilon_{1};-k,\varepsilon_{2};k_{3},\varepsilon_{3};k_{4},\varepsilon_{4})=\frac{e^{4}}{(4\pi)^{\frac{D}{2}}}\int_{0}^{\infty}\frac{dT}{T}\,T^{4-\frac{D}{2}}\,{\rm
e}^{-m^{2}T}\int_{0}^{1}du\,Q_{{\rm scal}(34)}\,{\rm e}^{-Tk^{2}G(u,0)}\,,$
(4.17)

where we have also set $u_{2}=0$ and $u_{1}=u$ so that now
$G_{12}=G(u,0)=u(1-u)$. In order to be able to make the comparison at the
integrand level is important to use IBP or equivalently the identity

$T\left(G_{12}-4G_{12}^{2}\right)k^{2}\,\,{\rm
e}^{-TG_{12}k^{2}}=\left(1-6G_{12}\right)\,{\rm
e}^{-TG_{12}k^{2}}-\partial_{u}\left(\dot{G}_{12}G_{12}\,{\rm
e}^{-TG_{12}k^{2}}\right)$ (4.18)

for many of the terms in $Q_{{\rm scal}(34)}^{2}$. Note that this identity can
also be expressed in terms of the $Y_{nl}$ as

$\displaystyle\left(Y_{51}-4Y_{52}\right)k^{2}=Y_{40}-6Y_{41}$ (4.19)

which will be useful in Section 5.

Turning our attention now to the photon polarization tensors in a constant
background field, for scalar QED the worldline calculation of [7] yielded the
following presentation:

$\Pi^{\mu\nu}_{\rm
scal}=-\frac{e^{2}}{(4\pi)^{\frac{D}{2}}}\int_{0}^{\infty}dT\,T^{1-\frac{D}{2}}\,{\rm
e}^{-m^{2}T}\,{\rm det}^{-{1\over
2}}\left(\frac{\sin\mathcal{Z}}{\mathcal{Z}}\right)\int_{0}^{1}du\,I^{\mu\nu}_{\rm
scal}\,{\rm e}^{-Tk\cdot\Phi_{12}\cdot k}\,,$ (4.20)

where

$I_{\rm
scal}^{\mu\nu}=\dot{\mathcal{G}}_{B12}^{\mu\nu}\,k\cdot\dot{\mathcal{G}}_{B12}\cdot
k-\Big{(}\dot{\mathcal{G}}_{B11}-\dot{\mathcal{G}}_{B12}\Big{)}^{\mu\lambda}\Big{(}\dot{\mathcal{G}}_{B21}-\dot{\mathcal{G}}_{B22}\Big{)}^{\nu\kappa}k^{\lambda}k^{\kappa}\,,$
(4.21)

and ${\mathcal{G}}_{Bij}$ is the bosonic worldline Green’s function in a
constant field background [21],

$\begin{split}\mathcal{G}_{Bij}&=\frac{T}{2\mathcal{Z}^{2}}\Big{(}\frac{\mathcal{Z}}{\sin\mathcal{Z}}\,{\rm
e}^{-i\mathcal{Z}\dot{G}_{ij}}+i\mathcal{Z}\dot{G}_{ij}-1\Big{)}\,,\\\
\mathcal{G}_{Bii}&=\frac{T}{2\mathcal{Z}^{2}}\Big{(}\mathcal{Z}\cot\mathcal{Z}-1\Big{)}\,,\end{split}$
(4.22)

with $\mathcal{Z}^{\mu\nu}=eTF^{\mu\nu}$. For the exponent of (4.20), we have

$-Tk\cdot\Phi_{ij}\cdot
k=-k\cdot\left[\frac{T}{2\mathcal{Z}}\Big{(}\frac{\,{\rm
e}^{-i\mathcal{Z}\dot{G}_{ij}}-\cos\mathcal{Z}}{\sin\mathcal{Z}}+i\dot{G}_{ij}\Big{)}\right]\cdot
k\,.$ (4.23)

In order to compare with the results in the present work, we choose
$F^{\mu\nu}=i(f_{3}^{\mu\nu}+f_{4}^{\mu\nu})$ and expand the whole
polarization tensor to second order in $F^{\mu\nu}$ so as to obtain the first
two diagrams on the right-hand side of Fig. 4. After some straightforward
algebra, we can express the result as

$\begin{split}\varepsilon_{1}\cdot\Pi_{\rm
scal}\cdot\varepsilon_{2}&=-\frac{e^{4}}{(4\pi)^{D/2}}\int_{0}^{\infty}dT\,T^{3-D/2}\,{\rm
e}^{-m^{2}T}\int_{0}^{1}du\,\,{\rm e}^{-TG_{12}k^{2}}\\\
&\times\bigg{\\{}(1-4G_{12})\Big{(}\varepsilon_{1}\cdot\varepsilon_{2}k^{2}-\varepsilon_{1}\cdot
k\varepsilon_{2}\cdot k\Big{)}\Big{[}(eT)^{-2}-\frac{2T}{3}G_{12}^{2}k\cdot
f_{3}\cdot f_{4}\cdot k+\frac{1}{6}tr(f_{3}f_{4})\Big{]}\\\
&+\frac{2}{3}\Big{(}G_{12}-4G_{12}^{2}\Big{)}\Big{[}2\varepsilon_{1}\cdot\varepsilon_{2}k\cdot
f_{3}\cdot f_{4}\cdot k+k^{2}\left(\varepsilon_{1}\cdot f_{3}\cdot
f_{4}\cdot\varepsilon_{2}+\varepsilon_{1}\cdot f_{4}\cdot
f_{3}\cdot\varepsilon_{2}\right)\\\ &-\varepsilon_{2}\cdot
k\Big{(}\varepsilon_{1}\cdot f_{3}\cdot f_{4}\cdot k+\varepsilon_{1}\cdot
f_{4}\cdot f_{3}\cdot k\Big{)}-\varepsilon_{1}\cdot
k\Big{(}\varepsilon_{2}\cdot f_{3}\cdot f_{4}\cdot k+\varepsilon_{2}\cdot
f_{4}\cdot f_{3}\cdot k\Big{)}\Big{]}\\\
&-4G_{12}^{2}\Big{(}\varepsilon_{1}\cdot f_{3}\cdot k\varepsilon_{2}\cdot
f_{4}\cdot k+\varepsilon_{1}\cdot f_{4}\cdot k\varepsilon_{2}\cdot f_{3}\cdot
k\Big{)}\bigg{\\}}\,,\end{split}$ (4.24)

Note that the term proportional to $(eT)^{-2}$ corresponds to the two-point
diagram or the vacuum polarization diagram in the absence of the background
field. The remaining terms give the sought after four-photon amplitude, and
are found to be in complete agreement with (4.17).

###  4.2 Spinor QED

For the spinor QED case we find, in complete analogy,

$Q_{{\rm
spin}(34)}^{4}(1234)=-\frac{4}{3}Z_{4}(1234)\left(G_{12}+2G_{12}^{2}\right),$
(4.25) $Q_{{\rm
spin}(34)}^{4}(2314)=-\frac{4}{9}Z_{4}(2314)\left(2-6G_{12}-9G_{12}^{2}\right),$
(4.26) $Q_{{\rm
spin}(34)}^{4}(3124)=-\frac{4}{3}Z_{4}(3124)\left(G_{12}+2G_{12}^{2}\right),$
(4.27) $\displaystyle Q_{{\rm spin}(34)}^{3}(123;4)=0,$ (4.28) $\displaystyle
Q_{{\rm spin}(34)}^{3}(234;1)=0,$ (4.29) $\displaystyle Q_{{\rm
spin}(34)}^{3}(314;2)=0,$ (4.30) $\displaystyle Q_{{\rm
spin}(34)}^{3}(124;3)=0,$ (4.31) $Q_{{\rm
spin}(34)}^{2}(12;34)=\frac{8T}{3}Z_{2}(12)k\cdot f_{4}\cdot f_{3}\cdot
k\,G_{12}^{3}\,,$ (4.32) $Q_{{\rm
spin}(34)}^{2}(13;24)=\frac{2T}{9}Z_{2}(13)k\cdot f_{4}\cdot f_{2}\cdot
k\left(G_{12}-4G_{12}^{2}\right),$ (4.33) $Q_{{\rm
spin}(34)}^{2}(23;14)=\frac{2T}{9}Z_{2}(23)k\cdot f_{4}\cdot f_{1}\cdot
k\left(G_{12}-4G_{12}^{2}\right),$ (4.34) $Q_{{\rm
spin}(34)}^{2}(14;23)=\frac{2T}{9}Z_{2}(14)k\cdot f_{2}\cdot f_{3}\cdot
k\left(G_{12}-4G_{12}^{2}\right),$ (4.35) $Q_{{\rm
spin}(34)}^{2}(24;13)=\frac{2T}{9}Z_{2}(24)k\cdot f_{1}\cdot f_{3}\cdot
k\left(G_{12}-4G_{12}^{2}\right),$ (4.36) $Q_{{\rm spin}(34)}^{2}(34;12)=0,$
(4.37) $Q_{{\rm spin}(34)}^{22}(12,34)=\frac{8}{3}Z_{2}(12)Z_{2}(34)G_{12}\,,$
(4.38) $Q_{{\rm spin}(34)}^{22}(13,24)=\frac{4}{9}Z_{2}(13)Z_{2}(24),$ (4.39)
$Q_{{\rm spin}(34)}^{22}(14,23)=\frac{4}{9}Z_{2}(14)Z_{2}(23),$ (4.40)

and the following integral representation

$\Gamma_{\rm
spin}(k,\varepsilon_{1};-k,\varepsilon_{2};k_{3},\varepsilon_{3};k_{4},\varepsilon_{4})=-2\frac{e^{4}}{(4\pi)^{\frac{D}{2}}}\int_{0}^{\infty}\frac{dT}{T}\,T^{4-\frac{D}{2}}\,{\rm
e}^{-m^{2}T}\int_{0}^{1}du\,Q_{{\rm spin}(34)}\,{\rm e}^{-Tk^{2}G(u,0)}\,.\\\
$ (4.41)

As in the scalar case, to be able to compare at the integrand level it is
important to make repeated use of (4.18).

The spinor-loop result of the vacuum polarization tensor (from [16, 7]) is

$\Pi^{\mu\nu}_{\rm
spin}=2\frac{e^{2}}{(4\pi)^{\frac{D}{2}}}\int_{0}^{\infty}dTT^{1-\frac{D}{2}}\,{\rm
e}^{-m^{2}T}\,{\rm det}^{-{1\over
2}}\left(\frac{\tan\mathcal{Z}}{\mathcal{Z}}\right)\int_{0}^{1}du\,I^{\mu\nu}_{\rm
spin}\,\,{\rm e}^{-Tk\cdot\Phi_{12}\cdot k}\,,$ (4.42)

where

$\begin{split}I_{\rm
spin}^{\mu\nu}&=\dot{\mathcal{G}}_{B12}^{\mu\nu}\,k\cdot\dot{\mathcal{G}}_{B12}\cdot
k-\mathcal{G}_{F12}^{\mu\nu}\,k\cdot\mathcal{G}_{F12}\cdot k\\\
&-\Big{[}\Big{(}\dot{\mathcal{G}}_{B11}-\mathcal{G}_{F11}-\dot{\mathcal{G}}_{B12}\Big{)}^{\mu\lambda}\Big{(}\dot{\mathcal{G}}_{B21}-\dot{\mathcal{G}}_{B22}+\mathcal{G}_{F22}\Big{)}^{\nu\kappa}+\mathcal{G}_{F12}^{\mu\lambda}\mathcal{G}_{F21}^{\nu\kappa}\Big{]}k^{\lambda}k^{\kappa}\,,\end{split}$
(4.43)

with the fermionic Green’s function in a constant field and its coincidence
limit [21]

$\begin{split}\mathcal{G}_{Fij}&=G_{Fij}\frac{\,{\rm
e}^{-i\mathcal{Z}\dot{G}_{ij}}}{\cos\mathcal{Z}}\,,\\\
\mathcal{G}_{Fii}&=-i\tan\mathcal{Z}\,.\end{split}$ (4.44)

After the expansion, choosing $\mathcal{Z}$ as in the scalar case, the spinor-
loop gives

$\begin{split}\varepsilon_{1}\cdot\Pi_{\rm
spin}\cdot\varepsilon_{2}&=\frac{2e^{4}}{(4\pi)^{\frac{D}{2}}}\int_{0}^{\infty}dT\,T^{3-\frac{D}{2}}\,{\rm
e}^{-m^{2}T}\int_{0}^{1}du\,\,{\rm e}^{-TG_{12}k^{2}}\\\
&\times\bigg{\\{}4G_{12}\Big{(}\varepsilon_{1}\cdot\varepsilon_{2}k^{2}-\varepsilon_{1}\cdot
k\varepsilon_{2}\cdot k\Big{)}\Big{[}-(eT)^{-2}+\frac{2T}{3}G_{12}^{2}k\cdot
f_{3}\cdot f_{4}\cdot k+\frac{1}{3}{\rm tr}(f_{3}f_{4})\Big{]}\\\
&-\frac{4}{3}\Big{(}G_{12}+2G_{12}^{2}\Big{)}\Big{[}2\varepsilon_{1}\cdot\varepsilon_{2}k\cdot
f_{3}\cdot f_{4}\cdot k+k^{2}(\varepsilon_{1}\cdot f_{3}\cdot
f_{4}\cdot\varepsilon_{2}+\varepsilon_{1}\cdot f_{4}\cdot
f_{3}\cdot\varepsilon_{2})\\\ &-\varepsilon_{2}\cdot
k\Big{(}\varepsilon_{1}\cdot f_{3}\cdot f_{4}\cdot k+\varepsilon_{1}\cdot
f_{4}\cdot f_{3}\cdot k\Big{)}-\varepsilon_{1}\cdot
k\Big{(}\varepsilon_{2}\cdot f_{3}\cdot f_{4}\cdot k+\varepsilon_{2}\cdot
f_{4}\cdot f_{3}\cdot k\Big{)}\Big{]}\\\
&-4G_{12}^{2}\Big{(}\varepsilon_{1}\cdot f_{3}\cdot k\varepsilon_{2}\cdot
f_{4}\cdot k+\varepsilon_{1}\cdot f_{4}\cdot k\varepsilon_{2}\cdot f_{3}\cdot
k\Big{)}\bigg{\\}}\,,\end{split}$ (4.45)

which again after removing the term proportional to $(eT)^{-2}$ is in perfect
agreement with (4.42). Despite of the restriction $k_{1}=-k_{2}$, this
provides an excellent check on many of the results presented in Section 3
above.

##  5 Gauge-invariant decomposition of the two-loop $\beta$-functions

In this section, we use the results presented in Section 3 to go to two loops
by sewing together the unrestricted photons 1 and 2 (Fig. 2). This yields the
two-loop photon propagators in the low-energy limit, that is, the induced
Maxwell terms, from whose $1/\epsilon$-poles we can (up to a contribution from
mass renormalization) compute the two-loop $\beta$-function coefficients for
scalar and spinor QED, recuperating the known results [11, 12, 40, 10, 41,
16]. As explained in the introduction, besides providing another check this is
motivated by the following two questions suggested by the highly structured
worldline integrand (LABEL:q4-scal): first, are there terms that drop out in
the sewing procedure? Second, does the fact that the sixteen terms in this
decomposition are (differently from the usual decomposition into Feynman
diagrams) individually gauge invariant, have a bearing on the issue of gauge
cancellations? Specifically, will the vanishing of the double poles involve
only cancellations inside each gauge-invariant structure, or between them?

We use Feynman gauge in the sewing, so that it is done by the replacement

$\varepsilon_{1}^{\mu}\varepsilon_{2}^{\nu}\rightarrow\frac{\eta^{\mu\nu}}{k^{2}}\,,$
(5.1)

($k_{1}=k=-k_{2}$) with $k$ to be integrated over. The induced Maxwell term in
our present set-up will appear as ${\rm tr}(f_{3}f_{4})$. The sewing will now
change the argument of $Y_{nl}$ functions, such that in this section it is
understood that $Y_{nl}=Y_{nl}(-k^{2},m^{2},D)$. Going back to the results in
$D$-dimensions in Section 3 for the scalar/spinor-loop, we have the following
expressions which will be used in both cases:

$\begin{split}Z_{4}(1234)|_{\rm sewing}=Z_{4}(3124)|_{\rm
sewing}&\Rightarrow\frac{2}{D}(D-1){\rm tr}(f_{3}f_{4})\,,\\\
Z_{4}(2314)|_{\rm sewing}&\Rightarrow\frac{2}{D}{\rm tr}(f_{3}f_{4})\,,\\\
Z_{3}(123)|_{\rm sewing}\,k\cdot f_{4}\cdot k&\Rightarrow 0\,,\\\
Z_{2}(12)|_{\rm sewing}\,k\cdot f_{3}\cdot f_{4}\cdot
k&\Rightarrow\frac{1}{D}(D-1)k^{2}{\rm tr}(f_{3}f_{4})\,,\\\ Z_{2}(13)(k\cdot
f_{4}\cdot f_{2}\cdot k)|_{\rm sewing}&\Rightarrow\frac{1}{D}{\rm
tr}(f_{3}f_{4})k^{2}\,,\\\ Z_{2}(12)Z_{2}(34)|_{\rm
sewing}&\Rightarrow\frac{(D-1)}{2}{\rm tr}(f_{3}f_{4})\,,\\\
Z_{2}(13)Z_{2}(24)|_{\rm sewing}=Z_{2}(14)Z_{2}(23)|_{\rm
sewing}&\Rightarrow\frac{1}{D}{\rm tr}(f_{3}f_{4})\,.\\\ \end{split}$ (5.2)

Here we have used the Passarino-Veltman type identity

$\int d^{D}k\,(k\cdot f_{3}\cdot f_{4}\cdot
k)\,F\left(k^{2}\right)\rightarrow\int d^{D}k\,\frac{k^{2}}{D}{\rm
tr}(f_{3}f_{4})\,F\left(k^{2}\right)\,,$ (5.3)

where $F$ is an arbitrary scalar function.

Note that the three-cycle $Z_{3}(123)$ drops out in the sewing, together with
its three permutations. Therefore $Q^{3}_{\rm scal/spin}$ does not contribute
here, and the total amplitudes can be written as (using a similar convention
as in (2.11))

$\displaystyle\begin{split}\Gamma^{(2)}_{\rm
scal}\big{(}f_{3},f_{4}\big{)}&=\frac{e^{4}}{2(4\pi)^{\frac{D}{2}}}\int\frac{d^{D}k}{(2\pi)^{D}}\left(\hat{\Gamma}^{(2)4}_{\rm
scal}+\hat{\Gamma}^{(2)2}_{\rm scal}+\hat{\Gamma}^{(2)22}_{\rm
scal}\right)\,,\\\ \Gamma^{(2)}_{\rm
spin}\big{(}f_{3},f_{4}\big{)}&=-\frac{e^{4}}{(4\pi)^{\frac{D}{2}}}\int\frac{d^{D}k}{(2\pi)^{D}}\left(\hat{\Gamma}^{(2)4}_{\rm
spin}+\hat{\Gamma}^{(2)2}_{\rm spin}+\hat{\Gamma}^{(2)22}_{\rm
spin}\right)\,.\end{split}$ (5.4)

(note that a symmetry factor of $\frac{1}{2}$ has to be included).

###  5.1 Scalar QED $\beta$-function

In this way, we find the following result for the Maxwell term of the two-loop
vacuum polarization tensor in scalar QED,

$\begin{split}\Gamma^{(2)}_{\rm
scal}\big{(}f_{3},f_{4}\big{)}&=\frac{e^{4}}{2(4\pi)^{\frac{D}{2}}}\int\frac{d^{D}k}{(2\pi)^{D}}\Bigg{\\{}\left(\frac{4}{9D}+\frac{D-1}{6}\right)Y_{40}-\frac{2}{3D}\big{[}(D-4)(D-1)+4\big{]}Y_{41}\\\
&-\frac{8}{3D}(4D-7)Y_{42}-\frac{4}{9D}(Y_{51}-4Y_{52})k^{2}-\frac{2(D-1)}{3D}(Y_{52}-4Y_{53})k^{2}\Bigg{\\}}{\rm
tr}(f_{3}f_{4})\,.\end{split}$ (5.5)

Using the identity (4.19) this can be somewhat simplified,

$\begin{split}\Gamma^{(2)}_{\rm
scal}(f_{3},f_{4})&=\frac{e^{4}}{2(4\pi)^{\frac{D}{2}}}\int\frac{d^{D}k}{(2\pi)^{D}}\bigg{[}\frac{D-1}{6}Y_{40}-\frac{2}{3D}(D-4)(D-1)Y_{41}\\\
&\hskip
28.45274pt-\frac{8}{3D}(4D-7)Y_{42}-\frac{2(D-1)}{3D}\,k^{2}(Y_{52}-4Y_{53})\bigg{]}{\rm
tr}(f_{3}f_{4})\,.\end{split}$ (5.6)

Next, we rewrite this as

$\displaystyle\Gamma^{(2)}_{\rm
scal}(f_{3},f_{4})\equiv\frac{e^{4}}{2(4\pi)^{4}}\gamma_{{\rm
scal}}(D,m^{2}){\rm tr}(f_{3}f_{4})\,.$ (5.7)

We then use (A.2) to eliminate the $k$ \- integration, which leads to

$\begin{split}\gamma_{{\rm
scal}}(D,m^{2})&=\frac{(4\pi)^{4}\Gamma(4-D)}{4(4\pi)^{D}m^{2(4-D)}}\,\int_{0}^{1}\,du\,\frac{1}{3D}\Big{\\{}2(D-1)D\Big{[}G(u,0)\Big{]}^{-\frac{D}{2}}\\\
&\quad-4(D-1)(3D-8)\Big{[}G(u,0)\Big{]}^{1-\frac{D}{2}}+16(D-7)(D-2)\Big{[}G(u,0)\Big{]}^{2-\frac{D}{2}}\Big{\\}}.\end{split}$
(5.8)

The remaining $u$-integral can be expressed in terms of the Euler Beta-
function as

$\int_{0}^{1}du\Big{[}G(u,0)\Big{]}^{x}=B(x+1,x+1),\qquad
B(x,y)\equiv\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}\,,$ (5.9)

which leads to our final result for the coefficient of the induced Maxwell
term,

$\gamma_{{\rm
scal}}(D,m^{2})=-\frac{(4\pi)^{4}\,\Gamma\left(4-D\right)}{(4\pi)^{D}\,m^{2(4-D)}}\,B\left(1-\frac{D}{2},1-\frac{D}{2}\right)\frac{(D-4)[(D-3)D+8]}{12(D-5)(D-3)D}\,.$
(5.10)

Note the global factor of $(D-4)$, which makes it clear that, as expected, the
$1/\epsilon$-expansion has only a single pole,

$\displaystyle\gamma_{{\rm scal}}(\epsilon,m^{2})$ $\displaystyle=$
$\displaystyle\frac{2}{\epsilon}+\mathcal{O}(\epsilon^{0})\,.$ (5.11)

After substituting (5.11) into (5.7) one arrives at

$\displaystyle\Gamma^{(2)}_{\rm
scal}(f_{3},f_{4})=\frac{e^{4}}{(4\pi)^{4}\epsilon}{\rm
tr}(f_{3}f_{4})+\mathcal{O}(\epsilon^{0})\,,$ (5.12)

which is in agreement with previous calculations of the two-loop dimensionally
regularized effective action in Scalar QED [10, 42] 333Note that there is an
overall sign difference between the above results and the one obtained in [16]
which comes from the fact that the two external photons in Fig. 2 have equal
and opposite momenta ($k_{3}=-k_{4}$) which leads to an extra sign in ${\rm
tr}(f_{3}f_{4})$.  .

For the full two-loop renormalization of the photon propagator, one still
needs to add an equal contribution $\Delta\Gamma^{(2)}_{\rm
scal}(f_{3},f_{4})$ from mass renormalization [10],

$\displaystyle\Gamma^{(2)}_{\rm scal}(f_{3},f_{4})+\Delta\Gamma^{(2)}_{\rm
scal}(f_{3},f_{4})\sim\frac{2e^{4}}{(4\pi)^{4}\epsilon}{\rm
tr}(f_{3}f_{4})\,.$ (5.13)

This then gives the two-loop photon wave-function renormalization factor and
leads to the standard result for the two-loop $\beta$-function coefficient in
Scalar QED [12, 10]

$\displaystyle\beta^{(2)}_{\rm
scal}(\alpha)=\frac{\alpha^{3}}{2\pi^{2}}~{}~{}~{}~{},~{}~{}~{}~{}\alpha=\frac{e^{2}}{4\pi}\,.$
(5.14)

More interesting is the distribution of the double and single - pole
coefficients $a_{\rm scal}$ and $b_{\rm scal}$ in the decomposition
(LABEL:q4-scal), shown in Table 1.

${\gamma_{{\rm scal}}}$ | $a_{\rm scal}$ | $b_{\rm scal}$ | ${\gamma_{{\rm scal}}}$ | $a_{\rm scal}$ | $b_{\rm scal}$ | ${\gamma_{{\rm scal}}}$ | $a_{\rm scal}$ | $b_{\rm scal}$
---|---|---|---|---|---|---|---|---
${\gamma_{{\rm scal}}^{4}}(1234)$ | $4$ | $\frac{13}{3}$ | ${\gamma_{{\rm scal}}^{2}}(13;24)$ | $\frac{4}{9}$ | $-\frac{2}{9}$ | ${\gamma_{{\rm scal}}^{22}}(12,34)$ | $-4$ | $\frac{2}{3}$
${\gamma_{{\rm scal}}^{4}}(1243)$ | $4$ | $\frac{13}{3}$ | ${\gamma_{{\rm scal}}^{2}}(14;23)$ | $\frac{4}{9}$ | $-\frac{2}{9}$ | ${\gamma_{{\rm scal}}^{22}}(13,24)$ | $\frac{2}{9}$ | $\frac{1}{18}$
${\gamma_{{\rm scal}}^{4}}(1324)$ | $-\frac{20}{9}$ | $-\frac{11}{9}$ | ${\gamma_{{\rm scal}}^{2}}(24;13)$ | $\frac{4}{9}$ | $-\frac{2}{9}$ | ${\gamma_{{\rm scal}}^{22}}(14,23)$ | $\frac{2}{9}$ | $\frac{1}{18}$
${\gamma_{{\rm scal}}^{2}}(12;34)$ | $-4$ | $-\frac{16}{3}$ | ${\gamma_{{\rm scal}}^{2}}(23;14)$ | $\frac{4}{9}$ | $-\frac{2}{9}$ |  |  |
Table 1: Contributions of individual terms in the worldline integrand to the
double and single poles of the two-loop Maxwell term in Scalar QED. The
$a_{\rm scal}$ sum to zero, the $b_{\rm scal}$ to $2$, see (5.11).

Although all terms here are already individually gauge invariant, the
vanishing of the double pole still happens through an intricate cancellation
between them. Similarly, apart from the vanishing of the three-cycle
contributions the other terms all contribute to the single pole, without any
clear pattern emerging as one might have hoped.

###  5.2 Spinor QED $\beta$-function

For the more relevant spinor QED case, let us also write down the individual
contributions in terms of the $Y_{nl}$:

$\begin{split}\hat{\Gamma}^{(2)4}_{\rm spin}(1234)=\hat{\Gamma}^{(2)4}_{\rm
spin}(3124)&=-\frac{8(D-1)}{3D}\left(Y_{41}+2Y_{42}\right){\rm
tr}(f_{3}f_{4})\,,\\\ \hat{\Gamma}^{(2)4}_{\rm
spin}(2314)&=-\frac{8}{9D}\left(2Y_{40}-6Y_{41}-9Y_{42}\right){\rm
tr}(f_{3}f_{4})\,,\\\ \hat{\Gamma}^{(2)2}_{\rm
spin}(12;34)&=\frac{8}{3D}(D-1)\,k^{2}\,Y_{53}\,{\rm tr}(f_{3}f_{4})\,,\\\
\hat{\Gamma}^{(2)2}_{\rm spin}(13;24)=\hat{\Gamma}^{(2)2}_{\rm
spin}(23;14)&=\frac{2}{9D}\left(Y_{40}-6Y_{41}\right){\rm
tr}(f_{3}f_{4})\,,\\\ \hat{\Gamma}^{(2)2}_{\rm
spin}(14;23)=\hat{\Gamma}^{(2)2}_{\rm
spin}(24;13)&=\frac{2}{9D}\left(Y_{40}-6Y_{41}\right){\rm
tr}(f_{3}f_{4})\,,\\\ \hat{\Gamma}^{(2)22}_{\rm
spin}(12,34)&=\frac{4}{3}(D-1)\,Y_{41}\,{\rm tr}(f_{3}f_{4})\,,\\\
\hat{\Gamma}^{(2)22}_{\rm spin}(13,24)=\hat{\Gamma}^{(2)22}_{\rm
spin}(14,23)&=\frac{4}{9D}\,Y_{40}\,{\rm tr}(f_{3}f_{4})\,.\end{split}$ (5.15)

Collecting all terms, we obtain

$\begin{split}\Gamma^{(2)}_{\rm
spin}(f_{3},f_{4})&=-\frac{e^{4}}{(4\pi)^{\frac{D}{2}}}\int\frac{d^{D}k}{(2\pi)^{D}}\bigg{[}\frac{4}{3D}(D-4)(D-1)Y_{41}-\frac{8}{3D}(4D-7)Y_{42}\\\
&\hskip 85.35826pt+\frac{8}{3D}(D-1)\,k^{2}\,Y_{53}\bigg{]}{\rm
tr}(f_{3}f_{4})\,.\end{split}$ (5.16)

As in the scalar case, we define

$\Gamma^{(2)}_{\rm
spin}(f_{3},f_{4})\equiv-\frac{e^{4}}{(4\pi)^{4}}\,\gamma_{{\rm
spin}}(D,m^{2})\,{\rm tr}(f_{3}f_{4})\,.$ (5.17)

Application of (A.2) leads to the following representation for $\gamma_{\rm
spin}(D,m^{2})$,

$\begin{split}\gamma_{\rm
spin}(D,m^{2})&=\frac{(4\pi)^{4}\Gamma(4-D)}{4(4\pi)^{D}m^{2(4-D)}}\,\int_{0}^{1}\,du\,\frac{16}{3D}\Big{\\{}(D-4)(D-1)\Big{[}G(u,0)\Big{]}^{1-\frac{D}{2}}\\\
&\quad+(D-7)(D-2)\Big{[}G(u,0)\Big{]}^{2-\frac{D}{2}}\Big{\\}}.\end{split}$
(5.18)

This result agrees already at the integrand level with Eq. (36) of [10],
providing another excellent check on many of the equations of the present
paper. Performing the $u$ \- integral brings us to our final result for the
induced Maxwell term,

$\begin{split}\gamma_{{\rm
spin}}(D,m^{2})&=\frac{(4\pi)^{4}\Gamma\left(4-D\right)}{(4\pi)^{D}\,m^{2(4-D)}}\,B\left(2-\frac{D}{2},2-\frac{D}{2}\right)\frac{(D-4)[34+D(5D-33)]}{3D(D-5)}\,.\end{split}$
(5.19)

Note once more the global factor of $(D-4)$, which reduces the pole in
dimensional regularization to a single one,

$\begin{split}\gamma_{{\rm
spin}}(\epsilon,m^{2})=\frac{6}{\epsilon}+\mathcal{O}(\epsilon^{0})\,.\end{split}$
(5.20)

Adding the appropriate term from mass renormalization [10]

$\displaystyle\Delta\Gamma^{(2)}_{\rm
spin}(f_{3},f_{4})\sim\frac{8e^{4}}{(4\pi)^{4}\epsilon}{\rm
tr}(f_{3}f_{4})\,,$ (5.21)

we get the total pole of the induced Maxwell term,

$\displaystyle\Gamma^{(2)}_{\rm spin}(f_{3},f_{4})+\Delta\Gamma^{(2)}_{\rm
spin}(f_{3},f_{4})\sim\frac{2e^{4}}{(4\pi)^{4}\epsilon}{\rm
tr}(f_{3}f_{4})\,,\hskip 25.6073pt$ (5.22)

which finally gives the correct two-loop spinor QED $\beta$-function [11, 40],

$\displaystyle\beta^{(2)}_{\rm spin}(\alpha)=\frac{\alpha^{3}}{2\pi^{2}}\,.$
(5.23)

Finally, in Table 2 we list the double and single pole contributions of the
individual terms in the worldline integrand. The picture that emerges is
similar to the scalar QED case above.

${\gamma_{{\rm spin}}}$ | $a_{\rm spin}$ | $b_{\rm spin}$ | ${\gamma_{{\rm spin}}}$ | $a_{\rm spin}$ | $b_{\rm spin}$ | ${\gamma_{{\rm spin}}}$ | $a_{\rm spin}$ | $b_{\rm spin}$
---|---|---|---|---|---|---|---|---
${\gamma_{{\rm spin}}^{4}}(1234)$ | $-8$ | $\frac{10}{3}$ | ${\gamma_{{\rm spin}}^{2}}(13;24)$ | $-\frac{8}{9}$ | $\frac{4}{9}$ | ${\gamma_{{\rm spin}}^{22}}(12,34)$ | $16$ | $\frac{16}{3}$
${\gamma_{{\rm spin}}^{4}}(1243)$ | $-8$ | $\frac{10}{3}$ | ${\gamma_{{\rm spin}}^{2}}(14;23)$ | $-\frac{8}{9}$ | $\frac{4}{9}$ | ${\gamma_{{\rm spin}}^{22}}(13,24)$ | $\frac{8}{9}$ | $\frac{2}{9}$
${\gamma_{{\rm spin}}^{4}}(1324)$ | $\frac{16}{9}$ | $-\frac{38}{9}$ | ${\gamma_{{\rm spin}}^{2}}(24;13)$ | $-\frac{8}{9}$ | $\frac{4}{9}$ | ${\gamma_{{\rm spin}}^{22}}(14,23)$ | $\frac{8}{9}$ | $\frac{2}{9}$
${\gamma_{{\rm spin}}^{2}}(12;34)$ | $0$ | $-4$ | ${\gamma_{{\rm spin}}^{2}}(23;14)$ | $-\frac{8}{9}$ | $\frac{4}{9}$ |  |  |
Table 2: Contributions of individual terms in the worldline integrand to the
double and single poles of the two-loop Maxwell term in Spinor QED. The
$a_{\rm spin}$ sum to zero, the $b_{\rm spin}$ to $6$, see (5.20).

###  5.3 Four-dimensional computation of the Spinor QED $\beta$-function

Although the calculation of corresponding quantities in Scalar and Spinor QED
in the worldline formalism usually proceeds with only minor differences, a
surprising finding of [10] had been that, in the case of the two-loop QED
$\beta$ functions, this holds true when employing dimensional regularization
but not for strictly four-dimensional regularization schemes such as the use
of a proper-time cutoff. In [16], this observation was explained as a
consequence of the above-mentioned theorem by Johnson et al. [13] that the
total induced effective action can have only a global divergence. Since,
unlike Scalar QED, Spinor QED does not possess true quadratic divergences, the
required cancellation of subdivergences here leads to two constraint
equations. And since, by the structure of the worldline representation of the
induced Maxwell term, the integrand before the final integration over $u$ in
$D=4$ must be of the form (compare (5.18))

$\left(\frac{A}{G_{12}^{2}}+\frac{B}{G_{12}}+C\right){\rm tr}(F^{2})\,,$
(5.24)

one can first conclude from the absence of a quadratic subdivergence that
$A=0$, and then from the absence of a logarithmic subdivergence that $B=0$,
leaving only the coefficient $C$ non-vanishing and making the final
$u$-integral trivial (this argument does not work in dimensional
regularization due to the principal suppression of quadratic subdivergences by
that scheme).

Thus one would expect that also the present recalculation should simplify in
the Spinor QED case when a four-dimensional scheme is used. And indeed, if we
return to (5.16) and put $D=4$ there, we find

$\displaystyle\gamma_{{\rm spin}}(D,m^{2})$ $\displaystyle\xrightarrow[]{D=4}$
$\displaystyle(4\pi)^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\bigg{[}-6Y_{42}+2Y_{53}k^{2}\bigg{]}\,,$
(5.25)

and then using any four-dimensional method for the regularization of the
global divergence that is still contained in the global $T$-integration will
lead to a trivial $u$-integration.

##  6 Delbrück scattering at low energies

In this section, we compute the differential cross section of Delbrück
scattering in scalar and spinor QED under the assumption that the photon that
interacts with the Coulomb field has low energy. For the spinor QED case, this
quantity was computed in detail in [43], therefore we will follow their
conventions for easy comparison.

Figure 5: Feynman diagram for low-energy Delbrück scattering.

Here we use our above results for the four-photon amplitudes with two low-
energy photons, and replace the two unrestricted legs $1$ and $2$ with two
Coulomb photons. Furthermore, we now take the two low-energy photons ($3$ and
$4$) on-shell.

The vector potential for the photons from the Coulomb field is given by

$A_{\mu}(x)=\left(-\frac{Ze}{4\pi r},\,{\bf 0}\right)\,,$ (6.1)

and has the following Fourier representation,

$A_{\mu}(x)=\varepsilon_{\mu}\int\frac{d^{4}k}{(2\pi)^{4}}\,\frac{Ze}{k^{2}}\,2\pi\,\delta(k_{0})\,\,{\rm
e}^{ik\cdot x}\,.$ (6.2)

From here the new vertex operator for photons from the Coulomb field read as

$\begin{split}V^{\gamma}_{\rm Nuc}[k,\varepsilon]&=\frac{Ze}{(2\pi)^{3}}\int
d^{4}k\frac{\delta(k_{0})}{k^{2}}\int_{0}^{T}d\tau\,\varepsilon\cdot\dot{x}\,{\rm
e}^{ik\cdot x}\\\ &=\frac{Ze}{(2\pi)^{3}}\int\frac{d^{3}{\bf k}}{{\bf
k}^{2}}\int_{0}^{T}d\tau\,\varepsilon\cdot\dot{x}\,\,{\rm e}^{i{\bf
k}\cdot{\bf x}}\,.\end{split}$ (6.3)

Thus in the present formalism, we have

$\Gamma_{\left\\{{\rm scal}(34)\atop{\rm
spin}(34)\right\\}}=\left\\{1\atop-2\right\\}\frac{1}{2}\frac{e^{4}(Ze)^{2}}{(4\pi)^{2}(2\pi)^{6}}\int\frac{(d^{3}{{\bf
k}_{1}})(d^{3}{{\bf k}_{2}})}{{\bf k}_{1}^{2}{\bf
k}_{2}^{2}}(2\pi)^{4}\delta^{4}(k_{1}+k_{2}+k_{3}+k_{4})\hat{\Gamma}_{\left\\{{\rm
scal}(34)\atop{\rm spin}(34)\right\\}}\,,$ (6.4)

with $\hat{\Gamma}_{\left\\{{\rm scal}(34)\atop{\rm spin}(34)\right\\}}$ as
defined in (3.1), (3.2). Note the factor of $\frac{1}{2}$ that takes the
symmetry between legs $1$ and $2$ into account 444In Feynman diagram
calculations of Delbrück scattering, this factor if usually implemented by
restricting the sum over diagrams to run only over three diagrams, instead of
the six light-by-light scattering ones (which are equal in pairs by CP
invariance). In the worldine formalism we do not have this option. . In order
to reproduce the result given in [43] for the differential cross section, we
employ the following kinematics

$\begin{split}k_{1}&=(0,{\bf q}-{\bf
q}^{\prime})~{},~{}~{}~{}~{}~{}~{}k_{2}=(0,-{\bf q}-{\bf q}^{\prime}),\\\
k_{3}&=(i\omega,{\bf k}+{\bf
q}^{\prime})~{},~{}~{}~{}~{}~{}k_{4}=(-i\omega,{\bf q}^{\prime}-{\bf
k}),\end{split}$ (6.5)

where

$\begin{split}{\bf q^{\prime}}&=(\omega\sin\theta/2,0,0),\\\ {\bf
k}&=(0,0,\omega\cos\theta/2),\\\ {\bf q}&=(q_{1},q_{2},q_{3}),\end{split}$
(6.6)

with $\omega$ as the energy and $\theta$ the scattering angle. The
polarizations are chosen as

$\begin{split}\varepsilon_{1}^{\mu}&=\varepsilon_{2}^{\mu}=(i,0,0,0),\\\
\varepsilon_{3}^{\mu}&=\frac{1}{\sqrt{2}}(0,-i\lambda_{3}\cos\theta/2,1,+i\lambda_{3}\sin\theta/2),\\\
\varepsilon_{4}^{\mu}&=\frac{1}{\sqrt{2}}(0,-i\lambda_{4}\cos\theta/2,1,-i\lambda_{4}\sin\theta/2),\end{split}$
(6.7)

where $\lambda_{i}=\pm 1$ for right-and left-handed circular polarization,
respectively. In the following it is understood that $\hat{\Gamma}_{{\rm
scal}(34)}^{+-}=\hat{\Gamma}_{{\rm
scal}(34)}|_{\lambda_{3}=1,\,\lambda_{4}=-1}$ etc. and we will also use the
abbreviations

$P_{0}\equiv\frac{{\rm
arcsinh}\left(\frac{q}{2m}\right)}{q\sqrt{4m^{2}+q^{2}}},\qquad
S\equiv\sin\frac{\theta}{2},\qquad C\equiv\cos\frac{\theta}{2}\,.$ (6.8)

With the kinematics of (6.5) and using conservation of momentum we can write
(6.4) as

$\Gamma_{\left\\{{\rm scal}(34)\atop{\rm
spin}(34)\right\\}}=\left\\{1\atop-2\right\\}\frac{1}{2}\frac{e^{4}(Ze)^{2}}{(4\pi)^{2}(2\pi)^{6}}(2\pi)^{4}\delta(k_{3}^{0}+k_{4}^{0})\int\frac{d^{3}{\bf
q}}{|{\bf q}-{\bf q^{\prime}}|^{2}|{\bf q}+{\bf
q}^{\prime}|^{2}}\hat{\Gamma}_{\left\\{{\rm scal}(34)\atop{\rm
spin}(34)\right\\}}\,.$ (6.9)

For convenience, let us further define

$\tilde{\Gamma}_{\left\\{{\rm scal}\atop{\rm
spin}\right\\}}\equiv\int\frac{d^{3}{\bf q}}{|{\bf q}-{\bf
q^{\prime}}|^{2}|{\bf q}+{\bf q}^{\prime}|^{2}}\hat{\Gamma}_{\left\\{{\rm
scal}(34)\atop{\rm spin}(34)\right\\}}\,.$ (6.10)

Since we are considering the low-energy case, $\omega\ll m$, we neglect
contributions of order superior to $\omega^{2}$. We notice that

$\begin{split}\int\frac{d^{3}{\bf q}}{|{\bf q}-{\bf q^{\prime}}|^{2}|{\bf
q}+{\bf
q}^{\prime}|^{2}}&=\int_{0}^{\infty}\int_{0}^{\pi}\int_{0}^{2\pi}\frac{q^{2}\sin\theta^{\prime}dq\,d\theta^{\prime}\,d\phi^{\prime}}{\left(q^{2}+\omega^{2}\sin^{2}\frac{\theta}{2}\right)^{2}-4q_{1}^{2}\omega^{2}\sin^{2}\frac{\theta}{2}}\\\
&=\int_{0}^{\infty}\int_{0}^{\pi}\int_{0}^{2\pi}\frac{\sin\theta^{\prime}dq\,d\theta^{\prime}\,d\phi^{\prime}}{q^{2}}+\mathcal{O}(\omega)\,.\end{split}$
(6.11)

Using the kinematics of (6.5), we find

$\begin{split}\hat{\Gamma}_{{\rm
scal}(34)}^{+-}&=\frac{4\omega^{2}}{3m^{2}q^{4}(4m^{2}+q^{2})}\Big{\\{}3m^{2}[(6m^{2}+q^{2})-8m^{2}(3m^{2}+q^{2})P_{0}](q_{2}^{2}-q_{1}^{2})\\\
&\quad+[q^{4}(2m^{2}+q^{2})-3m^{2}(6m^{2}+q^{2})q_{2}^{2}]S^{2}-8m^{4}[q^{4}-3(3m^{2}+q^{2})q_{2}^{2}]S^{2}P_{0}\Big{\\}}+\mathcal{O}(\omega^{3})\,,\end{split}$
(6.12) $\begin{split}\hat{\Gamma}_{{\rm
spin}(34)}^{+-}&=\frac{4\omega^{2}}{3q^{4}(4m^{2}+q^{2})^{2}}\Big{\\{}3(4m^{2}+q^{2})[(6m^{2}+q^{2})-8m^{2}(3m^{2}+q^{2})P_{0}](q_{2}^{2}-q_{1}^{2})\\\
&\quad+[q^{4}(2m^{2}-q^{2})-3(4m^{2}+q^{2})(6m^{2}+q^{2})q_{2}^{2}]S^{2}\\\
&\quad-8m^{2}[q^{4}(m^{2}+q^{2})-3(4m^{2}+q^{2})(3m^{2}+q^{2})q_{2}^{2}]S^{2}P_{0}\Big{\\}}+\mathcal{O}(\omega^{3})\,,\end{split}$
(6.13)

for the helicity non-conserving component, and

$\begin{split}\hat{\Gamma}_{{\rm
scal}(34)}^{++}&=\frac{4\omega^{2}}{3q^{4}(4m^{2}+q^{2})}\Big{\\{}4[(6m^{2}+q^{2})-8m^{2}(3m^{2}+q^{2})P_{0}](q_{2}^{2}-q_{3}^{2})\\\
&\quad-[3q^{2}(2m^{2}+q^{2})+4(6m^{2}+q^{2})q_{2}^{2}]C^{2}\\\
&\quad+4[q^{2}(6m^{4}+4m^{2}q^{2}+q^{4})+8m^{2}(3m^{2}+q^{2})q_{2}^{2}]C^{2}P_{0}\Big{\\}}+\mathcal{O}(\omega^{3})\,,\end{split}$
(6.14) $\begin{split}\hat{\Gamma}_{{\rm
spin}(34)}^{++}&=\frac{4\omega^{2}}{3q^{4}(4m^{2}+q^{2})^{2}}\Big{\\{}(6m^{2}+q^{2})(16m^{2}+7q^{2})(q_{2}^{2}-q_{3}^{2})\\\
&\quad+8[(3m^{2}+q^{2})(4m^{2}+q^{2})^{2}-3m^{4}q^{2}](q_{3}^{2}-q_{2}^{2})P_{0}\\\
&\quad-[3q^{2}(8m^{4}-2m^{2}q^{2}-q^{4})+(6m^{2}+q^{2})(16m^{2}+7q^{2})q_{2}^{2}]C^{2}\\\
&\quad+8q^{2}(12m^{6}-m^{4}q^{2}-5m^{2}q^{4}-q^{6})C^{2}P_{0}\\\
&\quad+8[(3m^{2}+q^{2})(4m^{2}+q^{2})^{2}-3m^{4}q^{2}]q_{2}^{2}C^{2}P_{0}\Big{\\}}+\mathcal{O}(\omega^{3})\,,\end{split}$
(6.15)

for the conserving one. To perform the integral over ${\bf q}$ we use
spherical coordinates:

$q_{1}=q\cos\theta^{\prime},\qquad
q_{2}=q\sin\theta^{\prime}\cos\phi^{\prime},\qquad
q_{3}=q\sin\theta^{\prime}\sin\phi^{\prime}\,.$ (6.16)

The integrals over $\theta^{\prime}$ and $\phi^{\prime}$ are trivial, and what
remains to be calculated is

$\tilde{\Gamma}_{{\rm scal}}^{+-}=\frac{16\pi
S^{2}\omega^{2}}{3}\int_{0}^{\infty}\frac{dq}{q^{2}}\,\frac{-6m^{4}+m^{2}q^{2}+q^{4}+24m^{6}P_{0}}{m^{2}q^{2}(4m^{2}+q^{2})}\,,$
(6.17) $\tilde{\Gamma}_{{\rm spin}}^{+-}=\frac{32\pi
S^{2}\omega^{2}}{3}\int_{0}^{\infty}\frac{dq}{q^{2}}\,\frac{-12m^{4}-4m^{2}q^{2}-q^{4}+24m^{4}(2m^{2}+q^{2})P_{0}}{q^{2}(4m^{2}+q^{2})^{2}}\,,$
(6.18) $\tilde{\Gamma}_{{\rm scal}}^{++}=\frac{16\pi
C^{2}\omega^{2}}{9}\int_{0}^{\infty}\frac{dq}{q^{2}}\,\frac{-42m^{2}-13q^{2}+4(42m^{4}+20m^{2}q^{2}+3q^{4})P_{0}}{q^{2}(4m^{2}+q^{2})}\,,$
(6.19) $\tilde{\Gamma}_{{\rm spin}}^{++}=\frac{32\pi
C^{2}\omega^{2}}{9}\int_{0}^{\infty}\frac{dq}{q^{2}}\,\frac{-84m^{4}-20m^{2}q^{2}+q^{4}+8(42m^{6}+17m^{4}q^{2}-2m^{2}q^{4}-q^{6})P_{0}}{q^{2}(4m^{2}+q^{2})^{2}}\,.$
(6.20)

Performing the integral over $q$, we get

$\tilde{\Gamma}_{{\rm
scal}}^{+-}=\frac{15\pi^{3}S^{2}\omega^{2}}{32m^{3}},\qquad\tilde{\Gamma}_{{\rm
spin}}^{+-}=-\frac{5\pi^{3}S^{2}\omega^{2}}{32m^{3}}\,,$ (6.21)
$\tilde{\Gamma}_{{\rm
scal}}^{++}=\frac{3\pi^{3}C^{2}\omega^{2}}{32m^{3}},\qquad\tilde{\Gamma}_{{\rm
spin}}^{++}=-\frac{73\pi^{3}C^{2}\omega^{2}}{288m^{3}}\,.$ (6.22)

Finally, the differential cross section is

$d\sigma_{{\rm
scal}(\lambda_{3}\lambda_{4})}=\frac{(Z\alpha)^{4}\alpha^{2}}{4(2\pi)^{6}}\,|\tilde{\Gamma}_{{\rm
scal}}^{\lambda_{3}\lambda_{4}}|^{2}\,d\Omega\,,$ (6.23) $d\sigma_{{\rm
spin}(\lambda_{3}\lambda_{4})}=\frac{(Z\alpha)^{4}\alpha^{2}}{(2\pi)^{6}}\,|\tilde{\Gamma}_{{\rm
spin}}^{\lambda_{3}\lambda_{4}}|^{2}\,d\Omega\,.$ (6.24)

For scalar QED, we have

$\displaystyle d\sigma_{{\rm scal}(++)}$ $\displaystyle=$ $\displaystyle
d\sigma_{{\rm
scal}(--)}=(Z\alpha)^{4}\left(\frac{\alpha}{m}\right)^{2}\left(\frac{3}{16}\right)^{2}\left(\frac{1}{32}\right)^{2}\left(\frac{\omega}{m}\right)^{4}\cos^{4}\frac{\theta}{2}d\Omega\,,$
(6.25) $\displaystyle d\sigma_{{\rm scal}(+-)}$ $\displaystyle=$
$\displaystyle d\sigma_{{\rm
scal}(-+)}=(Z\alpha)^{4}\left(\frac{\alpha}{m}\right)^{2}\left(\frac{15}{16}\right)^{2}\left(\frac{1}{32}\right)^{2}\left(\frac{\omega}{m}\right)^{4}\sin^{4}\frac{\theta}{2}d\Omega\,.$
(6.26)

For spinor QED, we find

$\displaystyle d\sigma_{{\rm spin}(++)}$ $\displaystyle=$ $\displaystyle
d\sigma_{{\rm
spin}(--)}=(Z\alpha)^{4}\left(\frac{\alpha}{m}\right)^{2}\left(\frac{73}{72}\right)^{2}\left(\frac{1}{32}\right)^{2}\left(\frac{\omega}{m}\right)^{4}\cos^{4}\frac{\theta}{2}d\Omega\,,$
(6.27) $\displaystyle d\sigma_{{\rm spin}(+-)}$ $\displaystyle=$
$\displaystyle d\sigma_{{\rm
spin}(-+)}=(Z\alpha)^{4}\left(\frac{\alpha}{m}\right)^{2}\left(\frac{5}{8}\right)^{2}\left(\frac{1}{32}\right)^{2}\left(\frac{\omega}{m}\right)^{4}\sin^{4}\frac{\theta}{2}d\Omega\,,$
(6.28)

in agreement with [43].

##  7 Summary and outlook

In this second part of our series of papers on the off-shell four-photon
amplitudes in scalar and spinor QED we have obtained these amplitudes for the
case where two of the photon momenta were taken in the low-energy limit, as
defined in part I. We have used the worldline representation of these
amplitudes which, as outlined in part I, allows one to treat the scalar and
spinor cases in parallel, and to arrive at a permutation and gauge-invariant
decomposition of these amplitudes that is closely related to the one
previously obtained by [43]. The coefficient functions in this decomposition
are written in terms of Feynman-Schwinger type parameter integrals, and as our
main result we have explicitly evaluated them both in four and in $D$
dimensions. Since in our formalism the four-photon amplitudes are manifestly
free of UV divergences, the former results are sufficient for one-loop
applications. The latter formulas serve the double purpose of making the off-
shell four-photon amplitudes useful as input for higher-loop calculations in
dimensional regularization, but also giving the correct results for them in
other physical dimensions (arbitrary $D$ for scalar QED, even $D$ for spinor
QED). After the integrating out of the two low-energy legs these integrals are
of two-point type, and can therefore be written in terms of ${}_{2}F_{1}$ for
general $D$, and for $D=4$ in terms of trigonometric functions.

To the best of our knowledge, for this momentum configuration the four-photon
amplitudes have not been obtained before. However, for the special case
$k_{1}+k_{2}=k_{3}+k_{4}=0$ they can be extracted from the low-energy limit of
the known vacuum polarization tensors in a constant field, and performing this
comparison provided an excellent check on our results for both scalar and
spinor QED.

From a technical point of view, probably the most important aspect of our
calculation is that it demonstrates how to integrate out a low-energy photon
leg without fixing an ordering for the remaining legs. As discussed in part I,
when using the light-by-light diagram as a subdiagram in higher-loop
calculation this property effectively allows one to unify the calculation of
Feynman diagrams of different topologies, and considering the proliferation of
diagrams in higher-loop QED calculations it is obviously of great interest to
explore what level of simplification can be achieved along these lines.

Off-shell photon legs can be used for creating internal photons by sewing, or
for connecting to external fields. As an example for sewing, we have used our
results for a construction of the two-loop vacuum polarisation tensors in the
low-energy limit, which allowed us to recover the two-loop $\beta$-function
coefficients for scalar and spinor QED. Although by present-days standards
this is easy enough to do using Feynman diagrams, the worldline calculation in
the version presented here has the advantage that it replaces the non-gauge
invariant decomposition into diagrams by a gauge-invariant decomposition into
sixteen substructures. This leads not only to a more compact parameter
integral representation, but also has given us an opportunity to probe into
the nature of the extensive cancellations that one generally finds in
perturbative QED calculations [13, 44, 14, 45]. If, as is sometimes assumed,
these were entirely due to gauge invariance, one might have expected the
cancellation of the double poles in the scalar and spinor QED $\beta$-function
calculations to have happened already at the level of each of the gauge-
invariant partial structures; instead, we have seen an intricate pattern of
cancellations between the various structures. Another interesting result of
this calculation has been that four of the sixteen partial structures, namely
the one involving a three-cycle, drop out in the sewing. It remains to be seen
whether this fact admits a generalization to larger numbers of external
photons and multiple sewing.

As an example of an external-field calculation, we have applied our formulas
to a calculation of the low-energy Delbrück scattering cross sections for both
scalar and spinor QED. For the spinor QED case this served just as another
check of efficiency and correctness, while the scalar QED result is new, to
the best of our knowledge. Both the scalar and spinor QED calculations have
been presented in a way that would be easy to adapt to other external fields.

The forthcoming third part of this series is devoted to the explicit,
$D$-dimensional calculation of the off-shell four-photon amplitudes with only
one leg taken in the low-energy limit and its applications.

##### Acknowledgments:

We would like to thank D. Broadhurst, F. Karbstein and D. Kreimer for
discussions and correspondence. C. Lopez-Arcos and M. A. Lopez-Lopez thank
CONACYT for financial support. N. Ahmadiniaz and M. A. Lopez-Lopez would like
to thank R. Schützhold for his support.

##  Appendix A Collection of integral formulas

Here we collect a number of results for the parameter integrals appearing in
one-loop worldline calculations, mostly taken from [16]. All Green’s functions
have been rescaled to the unit circle, $T=1$.

###  A.1 Integrating out a low-energy leg

All integrals encountered in the integrating out of a low-energy photon can,
using the identities (2.34), be reduced to integrals over polynomials in
$\dot{G}_{ij}$. For this type of integrals a closed-form master formula is
available even for the most general (abelian) case, integrating an arbitrary
monomial in the $\dot{G}_{ij}$’s involving an arbitrary number of variables in
one of the variables, and giving the result as a polynomial in the remaining
$\dot{G}_{ij}$’s [39]:

$\displaystyle\int_{0}^{1}du\,\dot{G}(u,u_{1})^{k_{1}}\dot{G}(u,u_{2})^{k_{2}}\cdots\dot{G}(u,u_{n})^{k_{n}}$
$\displaystyle=$ $\displaystyle{1\over 2n}\sum_{i=1}^{n}\,\prod_{j\neq
i}\sum_{l_{j}=0}^{k_{j}}{k_{j}\choose
l_{j}}\dot{G}_{ij}^{k_{j}-l_{j}}\sum_{l_{i}=0}^{k_{i}}{k_{i}\choose l_{i}}$
$\displaystyle\times{(-1)^{\sum_{j=1}^{n}l_{j}}\over(1+\sum_{j=1}^{n}l_{j})n^{\sum_{j=1}^{n}l_{j}}}\biggr{\\{}\Bigl{(}\sum_{j\neq
i}\dot{G}_{ij}+1\Bigr{)}^{1+\sum_{j=1}^{n}l_{j}}-(-1)^{k_{i}-l_{i}}\Bigl{(}\sum_{j\neq
i}\dot{G}_{ij}-1\Bigr{)}^{1+\sum_{j=1}^{n}l_{j}}\biggr{\\}}\,.$

At the four-photon level, apart from the single four-point integral

$\int_{0}^{1}du_{4}\,\dot{G}_{41}\dot{G}_{42}\dot{G}_{43}=-\frac{1}{6}(\dot{G}_{12}-\dot{G}_{23})(\dot{G}_{23}-\dot{G}_{31})(\dot{G}_{31}-\dot{G}_{12})\,,$
(A.2)

only three-point integrals appear, for which the master formula
(LABEL:intmaster) reduces to

$\int_{0}^{1}du_{3}\,\dot{G}_{13}^{k}\dot{G}_{32}^{l}=\frac{k!l!}{2}\sum_{m=0}^{l}\frac{(1-(-1)^{k+l-m+1})\dot{G}_{12}^{m}-(1-(-1)^{m})\dot{G}_{12}^{k+l-m+1}}{m!(k+l-m+1)!}\,.$
(A.3)

However, the latter formula generally does not provide the most compact way of
writing the result in terms of the $\dot{G}_{ij}$, therefore for easy
reference we provide below a list of optimized versions for all the integrals
that appeared in our calculations.

$\begin{split}&\int_{0}^{1}du_{3}\,\dot{G}_{13}=0\,,\\\
&\int_{0}^{1}du_{3}\,\dot{G}_{13}^{2}=\frac{1}{3}\,,\\\
&\int_{0}^{1}du_{3}\,\dot{G}_{13}^{4}=\frac{1}{5}\,,\\\
&\int_{0}^{1}du_{3}\dot{G}_{13}\dot{G}_{32}=\frac{1}{6}-\frac{1}{2}\dot{G}_{12}^{2}\,,\\\
&\int_{0}^{1}du_{3}\,\dot{G}_{13}\dot{G}_{32}^{2}=\frac{1}{3}(\dot{G}_{12}-\dot{G}_{12}^{3})\,,\\\
&\int_{0}^{1}du_{3}\,\dot{G}_{23}^{3}\dot{G}_{31}=\frac{3!}{5!}-\frac{\dot{G}_{12}^{4}}{4}\,,\\\
&\int_{0}^{1}du_{3}\,\dot{G}_{13}^{2}\dot{G}_{32}^{2}=\frac{4}{5!}-\frac{\dot{G}_{12}^{4}}{3!}+\frac{\dot{G}_{12}^{2}}{3!}\,,\\\
&\int_{0}^{1}du_{3}\,\dot{G}_{13}^{4}\dot{G}_{32}=\frac{1}{5}(\dot{G}_{12}-\dot{G}_{12}^{5})\,,\\\
&\int_{0}^{1}du_{3}\,\dot{G}_{13}^{3}\dot{G}_{32}^{2}=\frac{1}{10}(\dot{G}_{12}-\dot{G}_{12}^{5})\,,\end{split}$
(A.4) $\begin{split}&\int_{0}^{1}du_{3}\,G_{13}=\frac{1}{6}\,,\\\
&\int_{0}^{1}du_{3}\,G_{13}G_{32}=\frac{1}{30}-\frac{1}{6}G_{12}^{2}\,,\\\
&\int_{0}^{1}du_{3}\,G_{23}\dot{G}_{31}=\frac{1}{3}\dot{G}_{12}G_{12}\,.\end{split}$
(A.5)

###  A.2 The functions $Y_{nl}$ and tensor reduction

Here we study some properties of the functions $Y_{nl}$ defined in (2.30),

$Y_{nl}=\int_{0}^{\infty}\frac{dT}{T}T^{n-D/2}~{}\int_{0}^{1}du_{1}\int_{0}^{1}du_{2}~{}G_{12}^{l}\,{\rm
e}^{-T[m^{2}-G_{12}k_{1}\cdot k_{2}]}\,.$ (A.6)

In terms of $\hat{k}_{12}=\frac{k_{1}\cdot k_{2}}{m^{2}}$ and using the
translational invariance to fix $u_{2}=0$ and $u_{1}=u$, so that
$G_{12}=u(1-u)$, we can perform the following tensor reduction

$\displaystyle Y_{nl}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\frac{dT}{T}T^{n-D/2}~{}\int_{0}^{1}du~{}[u(1-u)]^{l}\,{\rm
e}^{-Tm^{2}[1-u(1-u)\hat{k}_{12}]}$ (A.7) $\displaystyle=$
$\displaystyle\frac{\Gamma\left(n-\frac{D}{2}\right)}{m^{2n-D}}~{}\int_{0}^{1}du~{}\frac{[u(1-u)]^{l}}{[1-u(1-u)\hat{k}_{12}]^{n-\frac{D}{2}}}$
$\displaystyle=$
$\displaystyle\frac{\Gamma\left(n-\frac{D}{2}-l\right)}{m^{2n-D}}~{}\frac{d^{l}}{d\hat{k}_{12}^{l}}\int_{0}^{1}du~{}\frac{1}{[1-u(1-u)\hat{k}_{12}]^{n-l-\frac{D}{2}}}\,.$

This leaves us with the single integral

$\begin{split}\int_{0}^{1}du~{}\frac{1}{[1-u(1-u)\hat{k}_{12}]^{n-l-\frac{D}{2}}}={}_{2}F_{1}\left(1,n-l-\frac{D}{2};\frac{3}{2};\frac{\hat{k}_{12}}{4}\right)\,.\end{split}$
(A.8)

So, in terms of the gaussian hypergeometric function ${}_{2}F_{1}$,

$\begin{split}Y_{nl}=\frac{\Gamma\left(n-\frac{D}{2}-l\right)}{m^{2n-D}}~{}\frac{d^{l}}{d\hat{k}_{12}^{l}}~{}{}_{2}F_{1}\left(1,n-l-\frac{D}{2};\frac{3}{2};\frac{\hat{k}_{12}}{4}\right)\,.\end{split}$
(A.9)

In Section 5 we need the elementary integral

$\displaystyle\int\frac{d^{D}k}{(2\pi)^{D}}\,(k^{2})^{\lambda}\,Y_{nl}$
$\displaystyle=$
$\displaystyle\,\frac{\Gamma\left(\frac{D}{2}+\lambda\right)\,\Gamma\left(n-\lambda-D\right)}{(4\pi)^{\frac{D}{2}}\Gamma\left(\frac{D}{2}\right)m^{2(n-\lambda-D)}}\,\int_{0}^{1}\,du\bigl{[}u(1-u)\bigr{]}^{l-\lambda-\frac{D}{2}}$
$\displaystyle=$
$\displaystyle\,\frac{\Gamma\left(\frac{D}{2}+\lambda\right)\,\Gamma\left(n-\lambda-D\right)}{(4\pi)^{\frac{D}{2}}\Gamma\left(\frac{D}{2}\right)m^{2(n-\lambda-D)}}\,\,B\Bigl{(}l-\lambda-\frac{D}{2}+1,l-\lambda-\frac{D}{2}+1\Bigr{)}.$

The following identities are not used in the present paper, but let us include
them here for whatever their worth might be:

$Y_{nl}=\Gamma\left(n-\frac{D}{2}\right)\hat{k}_{12}^{-l}\,\sum_{j=0}^{l}(-1)^{j}\binom{l}{j}\left[\Gamma\left(n-j-\frac{D}{2}\right)\right]^{-1}m^{-2j}\,Y_{(n-j)0}\,,$
(A.11)

which can be used to recursively eliminate all $Y_{nl}$’s with $l\neq 0$. And

$(2n-D)(2n-D-1)\,Y_{n0}=\left[2-(2n-D)\left(\hat{k}_{12}^{2}-8\right)\right]\,m^{2}\,Y_{(n+1)0}+\left(\hat{k}_{12}^{2}-4\right)m^{4}\,Y_{(n+2)0}\,,$
(A.12)

that follows from the following identity of contiguous functions [46]

$(c-b){}_{2}F_{1}(a,b-1;c;z)+(2b-c-
bz+az){}_{2}F_{1}(a,b;c;z)+b(z-1){}_{2}F_{1}(a,b+1;c;z)=0.$ (A.13)

##  Appendix B Explicit results for $Q_{{\rm scal}(34)}$ and $Q_{{\rm
spin}(34)}$

In this appendix, we explicitly write down the results for $Q_{{\rm scal}}$
and $Q_{{\rm spin}}$ after integration over $u_{4}$ and $u_{3}$, under the
assumption that photons $3$ and $4$ are taken in the low-energy limit.

###  B.1 Scalar QED

The explicit form of $Q_{{\rm scal}(34)}$ is written as

$Q_{{\rm
scal}(34)}^{4}(1234)=\frac{2}{3}Z_{4}(1234)\left(G_{12}-4G_{12}^{2}\right),$
(B.1) $Q_{{\rm
scal}(34)}^{4}(2314)=\frac{1}{9}Z_{4}(2314)\left(1-12G_{12}+36G_{12}^{2}\right),$
(B.2) $Q_{{\rm
scal}(34)}^{4}(3124)=\frac{2}{3}Z_{4}(3124)\left(G_{12}-4G_{12}^{2}\right),$
(B.3) $Q_{{\rm scal}(34)}^{3}(123;4)=-\frac{T}{9}Z_{3}(123)k_{2}\cdot
f_{4}\cdot k_{1}\left(G_{12}-10G_{12}^{2}+24G_{12}^{3}\right),$ (B.4) $Q_{{\rm
scal}(34)}^{3}(234;1)=0,$ (B.5) $Q_{{\rm scal}(34)}^{3}(341;2)=0,$ (B.6)
$Q_{{\rm scal}(34)}^{3}(412;3)=-\frac{T}{9}Z_{3}(412)k_{2}\cdot f_{3}\cdot
k_{1}\left(G_{12}-10G_{12}^{2}+24G_{12}^{3}\right),$ (B.7)
$\begin{split}Q_{{\rm
scal}(34)}^{2}(12;34)&=-\frac{T}{18}Z_{2}(12)\left(1-4G_{12}\right)\Bigg{\\{}\Bigg{[}\frac{1}{5}k_{1}\cdot
f_{3}\cdot f_{4}\cdot k_{1}+\left(\frac{1}{5}-6G_{12}^{2}\right)k_{1}\cdot
f_{3}\cdot f_{4}\cdot k_{2}\\\ &\quad+(1\leftrightarrow
2)\Bigg{]}+2T\left(1-4G_{12}\right)G_{12}^{2}\,k_{1}\cdot f_{3}\cdot
k_{2}k_{1}\cdot f_{4}\cdot k_{2}\Bigg{\\}}\,,\end{split}$ (B.8) $Q_{{\rm
scal}(34)}^{2}(13;24)+Q_{{\rm
scal}(34)}^{2}(23;14)=-\frac{T}{9}Z_{2}(13)k_{1}\cdot f_{2}\cdot f_{4}\cdot
k_{1}\,\left(1-4G_{12}\right)G_{12}+(1\leftrightarrow 2)\,,$ (B.9) $Q_{{\rm
scal}(34)}^{2}(14;23)+Q_{{\rm
scal}(34)}^{2}(24;13)=-\frac{T}{9}Z_{2}(14)k_{1}\cdot f_{2}\cdot f_{3}\cdot
k_{1}\,\left(1-4G_{12}\right)G_{12}+(1\leftrightarrow 2)\,,$ (B.10) $Q_{{\rm
scal}(34)}^{2}(34;12)=0,$ (B.11) $Q_{{\rm
scal}(34)}^{22}(12,34)=\frac{1}{3}Z_{2}(12)Z_{2}(34)\left(1-4G_{12}\right),$
(B.12) $Q_{{\rm scal}(34)}^{22}(13,24)=\frac{1}{9}Z_{2}(13)Z_{2}(24),$ (B.13)
$Q_{{\rm scal}(34)}^{22}(14,23)=\frac{1}{9}Z_{2}(14)Z_{2}(23).$ (B.14)

###  B.2 Spinor QED

The explicit form of $Q_{{\rm spin}(34)}$ is written as

$Q_{{\rm
spin}(34)}^{4}(1234)=-\frac{4}{3}Z_{4}(1234)\left(G_{12}+2G_{12}^{2}\right),$
(B.15) $Q_{{\rm
spin}(34)}^{4}(2314)=-\frac{4}{9}Z_{4}(2314)\left(2-6G_{12}-9G_{12}^{2}\right),$
(B.16) $Q_{{\rm
spin}(34)}^{4}(3124)=-\frac{4}{3}Z_{4}(3124)\left(G_{12}+2G_{12}^{2}\right),$
(B.17) $Q_{{\rm spin}(34)}^{3}(123;4)=\frac{2T}{9}Z_{3}(123)k_{2}\cdot
f_{4}\cdot k_{1}\left(G_{12}-G_{12}^{2}-12G_{12}^{3}\right),$ (B.18) $Q_{{\rm
spin}(34)}^{3}(234;1)=0,$ (B.19) $Q_{{\rm spin}(34)}^{3}(341;2)=0,$ (B.20)
$Q_{{\rm spin}(34)}^{3}(412;3)=\frac{2T}{9}Z_{3}(412)k_{2}\cdot f_{3}\cdot
k_{1}\left(G_{12}-G_{12}^{2}-12G_{12}^{3}\right),$ (B.21)
$\begin{split}Q_{{\rm
spin}(34)}^{2}(12;34)&=\frac{2T}{9}Z_{2}(12)\,G_{12}\Bigg{\\{}\Bigg{[}\frac{1}{5}k_{1}\cdot
f_{3}\cdot f_{4}\cdot k_{1}+\left(\frac{1}{5}-6G_{12}^{2}\right)k_{1}\cdot
f_{3}\cdot f_{4}\cdot k_{2}+(1\leftrightarrow 2)\Bigg{]}\\\
&\quad+2T\,\left(1-4G_{12}\right)G_{12}^{2}\,k_{1}\cdot f_{3}\cdot
k_{2}k_{1}\cdot f_{4}\cdot k_{2}\Bigg{\\}}\,,\end{split}$ (B.22) $Q_{{\rm
spin}(34)}^{2}(13;24)+Q_{{\rm
spin}(34)}^{2}(23;14)=\frac{2T}{9}Z_{2}(13)k_{1}\cdot f_{2}\cdot f_{4}\cdot
k_{1}\,\left(1-4G_{12}\right)G_{12}+(1\leftrightarrow 2)\,,$ (B.23) $Q_{{\rm
spin}(34)}^{2}(14;23)+Q_{{\rm
spin}(34)}^{2}(24;13)=\frac{2T}{9}Z_{2}(14)k_{1}\cdot f_{2}\cdot f_{3}\cdot
k_{1}\,\left(1-4G_{12}\right)G_{12}+(1\leftrightarrow 2)\,,$ (B.24) $Q_{{\rm
spin}(34)}^{2}(34;12)=0,$ (B.25) $Q_{{\rm
spin}(34)}^{22}(12,34)=\frac{8}{3}Z_{2}(12)Z_{2}(34)G_{12}\,,$ (B.26) $Q_{{\rm
spin}(34)}^{22}(13,24)=\frac{4}{9}Z_{2}(13)Z_{2}(24),$ (B.27) $Q_{{\rm
spin}(34)}^{22}(14,23)=\frac{4}{9}Z_{2}(14)Z_{2}(23).$ (B.28)

## References

  * [1] N. Ahmadiniaz, C. Lopez-Arcos, M. A. Lopez-Lopez, C. Schubert, The QED four – photon amplitudes off-shell: part 1. arXiv:2012.11791.
  * [2] I. A. Batalin, A. E. Shabad, Photon green function in a stationary homogeneous field of the most general form, Zh. Eksp. Teor. Fiz. 60 (1971) 894–900.
  * [3] V. N. Baier, V. M. Katkov, V. M. Strakhovenko, An Operator Approach to Quantum Electrodynamics in External Field: Mass Operator, Zh. Eksp. Teor. Fiz. 67 (1974) 453.
  * [4] V. N. Baier, V. M. Katkov, V. M. Strakhovenko, An operator approach to quantum electrodynamics in external field 2. Electron loops, Zh. Eksp. Teor. Fiz. 68 (1975) 405.
  * [5] W.-y. Tsai, T. Erber, The Propagation of Photons in Homogeneous Magnetic Fields: Index of Refraction, Phys. Rev. D 12 (1975) 1132.  doi:10.1103/PhysRevD.12.1132.
  * [6] L. F. Urrutia, Leptonic and Radiative Meson Decays: A Phenomenologically Broken Symmetry Approach, Phys. Rev. D 18 (1978) 819.  doi:10.1103/PhysRevD.18.819.
  * [7] C. Schubert, Vacuum polarization tensors in constant electromagnetic fields. Part 1., Nucl. Phys. B 585 (2000) 407–428.  arXiv:hep-ph/0001288, doi:10.1016/S0550-3213(00)00423-5.
  * [8] W. Dittrich, H. Gies, Probing the quantum vacuum. Perturbative effective action approach in quantum electrodynamics and its application, Vol. 166, 2000\.  doi:10.1007/3-540-45585-X.
  * [9] F. Karbstein, R. Shaisultanov, Photon propagation in slowly varying inhomogeneous electromagnetic fields, Phys. Rev. D 91 (8) (2015) 085027.  arXiv:1503.00532, doi:10.1103/PhysRevD.91.085027.
  * [10] M. G. Schmidt, C. Schubert, Multiloop calculations in the string inspired formalism: The Single spinor loop in QED, Phys. Rev. D 53 (1996) 2150–2159.  arXiv:hep-th/9410100, doi:10.1103/PhysRevD.53.2150.
  * [11] R. Jost, J. M. Luttinger, Vacuumpolarisation und $e^{4}$-Ladungsrenormalisation für Elektronen, Helv. Phys. Act. I-II (1950) 2150–2159.  doi:10.5169/seals-112105.
  * [12] Z. Bialynicka-Birula, FOURTH ORDER RENORMALIZATION CONSTANTS IN QUANTUM ELECTRODYNAMICS, Bull. Acad. Polon. Sci., Ser. Sci., Math., Astron. Phys. 3.
  * [13] K. Johnson, R. Willey, M. Baker, Vacuum polarization in quantum electrodynamics, Phys. Rev. 163 (1967) 1699–1715.  doi:10.1103/PhysRev.163.1699.
  * [14] D. J. Broadhurst, R. Delbourgo, D. Kreimer, Unknotting the polarized vacuum of quenched QED, Phys. Lett. B 366 (1996) 421–428.  arXiv:hep-ph/9509296, doi:10.1016/0370-2693(95)01343-1.
  * [15] B. Grauel, $\beta$ function QED to two loops - traditionally and with Corolla polynomial, MSc thesis, Humboldt University Berlin (2014).
  * [16] C. Schubert, Perturbative quantum field theory in the string-inspired formalism, Physics Reports 355 (2) (2001) 73 – 234.  arXiv:hep-th/0101036, doi:https://doi.org/10.1016/S0370-1573(01)00013-8.
URL http://www.sciencedirect.com/science/article/pii/S0370157301000138

  * [17] M. G. Schmidt, C. Schubert, Worldline Green functions for multiloop diagrams, Phys. Lett. B 331 (1994) 69–76.  arXiv:hep-th/9403158, doi:10.1016/0370-2693(94)90944-X.
  * [18] C. Schubert, The Structure of the Bern-Kosower integrand for the N gluon amplitude, Eur. Phys. J. C 5 (1998) 693–699.  arXiv:hep-th/9710067, doi:10.1007/s100520050311.
  * [19] N. Ahmadiniaz, C. Schubert, V. M. Villanueva, String-inspired representations of photon/gluon amplitudes, JHEP 01 (2013) 132.  arXiv:1211.1821, doi:10.1007/JHEP01(2013)132.
  * [20] R. Shaisultanov, On the string-inspired approach to QED in external field, Physics Letters B 378 (1) (1996) 354 – 356.  doi:https://doi.org/10.1016/0370-2693(96)00359-0.
URL http://www.sciencedirect.com/science/article/pii/0370269396003590

  * [21] M. Reuter, M. G. Schmidt, C. Schubert, Constant external fields in gauge theory and the spin 0, $1/2$, 1 path integrals, Annals of Physics 259 (2) (1997) 313 – 365.  arXiv:hep-th/9610191, doi:https://doi.org/10.1006/aphy.1997.5716.
URL http://www.sciencedirect.com/science/article/pii/S000349169795716X

  * [22] N. Ahmadiniaz, J. P. Edwards, A. Ilderton, Reducible contributions to quantum electrodynamics in external fields, JHEP 05 (2019) 038.  arXiv:1901.09416, doi:10.1007/JHEP05(2019)038.
  * [23] N. Ahmadiniaz, F. Bastianelli, O. Corradini, J. P. Edwards, C. Schubert, One-particle reducible contribution to the one-loop spinor propagator in a constant field, Nucl. Phys. B 924 (2017) 377–386.  arXiv:1704.05040, doi:10.1016/j.nuclphysb.2017.09.012.
  * [24] A. Ilderton, G. Torgrimsson, Worldline approach to helicity flip in plane waves, Phys. Rev. D 93 (8) (2016) 085006.  arXiv:1601.05021, doi:10.1103/PhysRevD.93.085006.
  * [25] J. P. Edwards, C. Schubert, N-photon amplitudes in a plane-wave background, Phys. Lett. B 822 (2021) 136696.  arXiv:2105.08173, doi:10.1016/j.physletb.2021.136696.
  * [26] C. Schubert, R. Shaisultanov, Master formulas for photon amplitudes in a combined constant and plane-wave background fieldarXiv:2303.08907.
  * [27] L. Meitner, H. Kösters, Über die streuung kurzwelliger $\gamma$-strahlen, Zeitschrift für Physik 84 (3-4) (1933) 137–144.
  * [28] H. Bethe, F. Rohrlich, Small angle scattering of light by a coulomb field, Physical Review 86 (1) (1952) 10.
  * [29] G. Jarlskog, L. Jönsson, S. Prünster, H. Schulz, H. Willutzki, G. Winter, Measurement of delbrück scattering and observation of photon splitting at high energies, Physical Review D 8 (11) (1973) 3813.
  * [30] H. Cheng, T. T. Wu, High-energy collision processes in quantum electrodynamics. i, Physical Review 182 (5) (1969) 1852.
  * [31] H. Cheng, T. T. Wu, High-energy collision processes in quantum electrodynamics. ii, Physical Review 182 (5) (1969) 1868.
  * [32] H. Cheng, T. T. Wu, High-energy collision processes in quantum electrodynamics. iii, Physical Review 182 (5) (1969) 1873.
  * [33] H. Cheng, T. T. Wu, High-energy collision processes in quantum electrodynamics. iv, Physical Review 182 (5) (1969) 1899.
  * [34] A. Mil’shtein, V. Strakhovenko, Quasiclassical approach to the high-energy delbrück scattering, Physics Letters A 95 (3-4) (1983) 135–138.
  * [35] A. Mil’shtein, M. Strakhovenko, Coherent scattering of high energy photons in a coulomb field, Zh. Eksp. Teor. Fiz 85 (1983) 14–23.
  * [36] M. Schumacher, I. Borchert, F. Smend, P. Rullhusen, Delbrück scattering of $2.75$ MeV photons by lead, Physics Letters B 59 (2) (1975) 134 – 136.  doi:https://doi.org/10.1016/0370-2693(75)90685-1.
URL http://www.sciencedirect.com/science/article/pii/0370269375906851

  * [37] A. Milstein, M. Schumacher, Present status of delbrück scattering, Physics Reports 243 (4) (1994) 183 – 214.  doi:https://doi.org/10.1016/0370-1573(94)00058-1.
URL http://www.sciencedirect.com/science/article/pii/0370157394000581

  * [38] M. Schumacher, Delbrück scattering, Radiation Physics and Chemistry 56 (1) (1999) 101 – 111.  doi:https://doi.org/10.1016/S0969-806X(99)00289-3.
URL http://www.sciencedirect.com/science/article/pii/S0969806X99002893

  * [39] J. P. Edwards, C. M. Mata, U. Müller, C. Schubert, New Techniques for Worldline Integration, SIGMA 17 (2021) 065.  arXiv:2106.12071, doi:10.3842/SIGMA.2021.065.
  * [40] C. Itzykson, J. Zuber, Quantum Field Theory, International Series In Pure and Applied Physics, McGraw-Hill, New York, 1980.
  * [41] M. G. Schmidt, C. Schubert, Multiloop calculations in the string-inspired formalism: The single spinor loop in QED, Phys. Rev. D 53 (1996) 2150–2159.  doi:10.1103/PhysRevD.53.2150.
URL https://link.aps.org/doi/10.1103/PhysRevD.53.2150

  * [42] D. Fliegner, M. Reuter, M. Schmidt, C. Schubert, The Two loop Euler-Heisenberg Lagrangian in dimensional renormalization, Theor. Math. Phys. 113 (1997) 1442–1451.  arXiv:hep-th/9704194, doi:10.1007/BF02634170.
  * [43] V. Costantini, B. De Tollis, G. Pistoni, Nonlinear effects in quantum electrodynamics, Nuovo Cim. A 2 (3) (1971) 733–787.  doi:10.1007/BF02736745.
  * [44] P. Cvitanovic, Asymptotic Estimates and Gauge Invariance, Nucl. Phys. B 127 (1977) 176–188.  doi:10.1016/0550-3213(77)90357-1.
  * [45] I. Huet, M. Rausch de Traubenberg, C. Schubert, Asymptotic Behaviour of the QED Perturbation Series, Adv. High Energy Phys. 2017 (2017) 6214341.  arXiv:1707.07655, doi:10.1155/2017/6214341.
  * [46] M. Abramowitz, I. A. Stegun, R. H. Romer, Handbook of mathematical functions with formulas, graphs, and mathematical tables, American Journal of Physics 56 (10) (1988) 958–958.  arXiv:https://doi.org/10.1119/1.15378, doi:10.1119/1.15378.
URL https://doi.org/10.1119/1.15378

  * [47] Wolfram Research, Inc., Mathematica, Version 11.1, Champaign, IL, 2017.