$\displaystyle\kappa$ $\displaystyle h_{\mu\nu}^{(l+1,j)}(\vec{x})$ (5.11)
$\displaystyle=-16\,\pi^{l+1}\int\frac{d^{d}\vec{q}}{(2\pi)^{d}}\frac{e^{i\vec{q}\cdot\vec{x}}}{\vec{q}^{2}}\left(c_{1}^{(l,j)}(d)\left(\delta_{\mu}^{0}\delta_{\nu}^{0}-\frac{\eta_{\mu\nu}}{d-1}\right)+c_{2}^{(l,j)}(d)\left(-\frac{q_{\mu}q_{\nu}}{\vec{q}^{2}}+\frac{\eta_{\mu\nu}}{d-1}\right)\right)$
$\displaystyle\times(G_{N}m)^{l+1-j}(\alpha
G_{N})^{\frac{j}{2}}J_{(l)}(\vec{q}^{2})\ .$

Then employing the Fourier transform in eqs. (B.3) and (B.4), one can derive
the explicit post-Minkowskian expansion of the metric perturbation according
to the definition (4.11) as

$\displaystyle h_{0}^{(l+1,j)}(r)=$
$\displaystyle-\frac{16}{d-1}\left((d-2)c_{1}^{(l,j)}(d)+c_{2}^{(l,j)}(d)\right)\left(\frac{\rho}{4}\right)^{l+1}(G_{N}m)^{l+1-j}(\alpha
G_{N})^{\frac{j}{2}}$ (5.12) $\displaystyle h_{1}^{(l+1,j)}(r)=$
$\displaystyle\frac{16}{d-1}\left(c_{1}^{(l,j)}(d)-\left(1+\frac{d-1}{2-l(d-2)}\right)c_{2}^{(l,j)}(d)\right)$
$\displaystyle\times\left(\frac{\rho}{4}\right)^{l+1}(G_{N}m)^{l+1-j}(\alpha
G_{N})^{\frac{j}{2}}$ $\displaystyle h_{2}^{(l+1,j)}(r)=$ $\displaystyle
16\frac{(d-2)(l+1)}{2-l(d-2)}c_{2}^{(l,j)}(d)\left(\frac{\rho}{4}\right)^{l+1}(G_{N}m)^{l+1-j}(\alpha
G_{N})^{\frac{j}{2}}\ .$

These expressions impose an explicit relation between the metric and the
quantum amplitude calculation, in which the only object needed in order to
completely recover the observable are the form factors, which can be extracted
directly from amplitude computations.

Following [18], in order to work out a systematic procedure to extract the
form factors from amplitudes like (5.6), we decompose such processes in

$\displaystyle\mathcal{N}_{\mu_{1}\nu_{1},...,\mu_{l-j+1}\nu_{l-j+1},\beta_{1},...,\beta_{j}}(p_{1},p_{2},\ell_{1},...,\ell_{l})=\includegraphics[width=173.44534pt,valign={c}]{Diagrams/Chapter5/img1_2.png}$
(5.13)
$\displaystyle=\frac{(i)^{l}\left(\prod_{i=1}^{l+1-j}\left(\tau_{\phi^{2}h}\right)_{\mu_{i}\nu_{i}}\right)\left(\prod_{i=1}^{j}\left(\tau_{\phi^{2}A}\right)_{\beta_{i}}\right)}{\prod_{i=1}^{l}\left(\left(p_{1}-\sum_{k=1}^{i}\ell_{k}\right)^{2}-m^{2}+i\epsilon\right)}\
,$

where we take into account all the vertices attached to the massive line,
whose explicit expressions are reported in appendix A, and in

$\mathcal{M}_{\lambda_{1}\sigma_{1},...,\lambda_{l-j+1}\sigma_{l-j+1},\alpha_{1},...,\alpha_{j},\mu\nu}(p_{1},p_{2},\ell_{1},...,\ell_{l})=\includegraphics[width=173.44534pt,valign={c}]{Diagrams/Chapter5/img1_3.png}\
,$ (5.14)

where the explicit form of the tree internal amplitude depends on the specif
process. Considering the momentum conservation
$\ell_{1}+...+\ell_{l+1}=p_{1}-p_{2}=q$, putting all together it is possible
to compactly write down the amplitude in eq. (5.6) as

$\displaystyle-\frac{i\,\kappa}{2}\,\sqrt{4E_{1}E_{2}}\,T_{\mu\nu}^{(l,j)}(q^{2})$
(5.15)
$\displaystyle=\int\prod_{i=1}^{l}\frac{d^{d+1}\ell_{i}}{(2\pi)^{d+1}}\mathcal{N}_{\mu_{1}\nu_{1},...,\beta_{j}}\mathcal{M}_{\lambda_{1}\sigma_{1},...,\alpha_{j},\mu\nu}\frac{\Big{(}\prod_{i=1}^{l+1-j}i\,P^{\lambda_{i}\sigma_{i},\mu_{i}\nu_{i}}\Big{)}\Big{(}\prod_{i=1}^{j}-i\,\eta^{\alpha_{i},\beta_{i}\nu_{i}}\Big{)}}{\prod_{i=1}^{l+1}\Big{(}\ell_{i}^{2}+i\epsilon\Big{)}}\
.$

Up to now the expressions of the above amplitudes are completely generic, but
since we are interested only in those contributions that reconstruct the
classical phenomenology, we can impose the classical limit333From hereafter
with $"\text{classical limit}"$ we mean the whole set of considerations needed
to extract the $\hbar$-independent long-range contributions from a loop
amplitude, including the energy expansion of a certain diagram. in order to
simplify the integral. As usual this means expand the amplitude in an energy
expansion and pick only the leading-order contribution considering the limit
$|\vec{p_{i}}|\ll m$ and neglecting higher orders in the momentum whose lead
to quantum corrections. Then the vertices attached to the massive line take
the form

$\left(\tau_{\phi^{2}h}\right)_{\mu\nu}\simeq-i\,\kappa\,m^{2}\delta_{\mu}^{0}\delta_{\nu}^{0}\quad\text{and}\quad\left(\tau_{\phi^{2}A}\right)_{\beta}\simeq-2\,i\,Q\,m\,\delta_{\beta}^{0}\
.$ (5.16)

Moreover it is possible to simplify also the expression of the massive
propagator contained in eq. 5.13. In fact for a generic internal massive line
it is verified that

$\frac{i}{\left(p_{1}-\sum_{k=1}^{i}\ell_{k}\right)^{2}-m^{2}+i\epsilon}\simeq\frac{i}{-2m\left(\sum_{k=1}^{i}\ell_{k}^{0}\right)+i\epsilon}\
,$ (5.17)

from which it is possible to write

$\displaystyle\mathcal{N}_{\mu_{1}\nu_{1},...,\beta_{j}}$
$\displaystyle=(-i)^{l+1}\kappa^{l+1-j}m^{2+2l-j}2^{j}Q^{j}\left(\prod_{i=1}^{l+1-j}\delta_{\mu_{i}}^{0}\delta_{\nu_{i}}^{0}\right)\left(\prod_{i=1}^{j}\delta_{\beta_{i}}^{0}\right)$
(5.18)
$\displaystyle\times\left(\frac{i}{-2m}\right)^{l}\prod_{i=1}^{l}\frac{1}{\left(\sum_{k=1}^{i}\ell_{k}^{0}\right)-i\epsilon}\
.$

Now the above expression is invariant under the exchange of any internal
massless line, which means that it is possible to sum over all the
permutations of the internal momenta up to weight the expression with the
right combinatorial factor. In particular identifying with
$\mathfrak{S}_{l+1}$ the group of $l+1$ permutations, using the relation [18]

$\sum_{\sigma\in\mathfrak{S}_{l+1}}\prod_{i=1}^{l+1}\frac{1}{\sum_{k=1}^{i}x_{\sigma(k)}}=\prod_{i=1}^{l+1}\frac{1}{x_{i}}\
,$ (5.19)

exploiting the momentum conservation444It is verified that
$\ell_{1}^{0}+...+\ell_{l+1}^{0}=q^{0}=0$. in order to write

$\prod_{i=1}^{l}\frac{1}{\left(\sum_{k=1}^{i}\ell^{0}_{k}\right)-i\epsilon}=-i\epsilon\prod_{i=1}^{l+1}\frac{1}{\left(\sum_{k=1}^{i}\ell^{0}_{k}\right)-i\epsilon}\
,$ (5.20)

we can finally get

$\sum_{\sigma\in\mathfrak{S}_{l+1}}\prod_{i=1}^{l+1}\frac{1}{\left(\sum_{k=1}^{i}\ell^{0}_{\sigma(k)}\right)-i\epsilon}=\prod_{i=1}^{l+1}\frac{1}{\ell^{0}_{i}-i\epsilon}\
,$ (5.21)

from which one obtains

$\displaystyle\mathcal{N}_{\mu_{1}\nu_{1},...,\beta_{j}}=\frac{1}{(l+1)!}\sum_{\sigma\in\mathfrak{S}_{l+1}}\mathcal{N}_{\mu_{1}\nu_{1},...,\beta_{j}}$
(5.22)
$\displaystyle=\frac{1}{(l+1)!}(-i)^{l+1}\kappa^{l+1-j}m^{2+2l-j}2^{j}Q^{j}\left(\prod_{i=1}^{l+1-j}\delta_{\mu_{i}}^{0}\delta_{\nu_{i}}^{0}\right)\left(\prod_{i=1}^{j}\delta_{\beta_{i}}^{0}\right)\left(\frac{i}{-2m}\right)^{l}(-i\epsilon)\prod_{i=1}^{l+1}\frac{1}{\ell^{0}_{i}-i\epsilon}\
.$

Now inserting back this expression in eq. (LABEL:EMT_Amplitude_Complete),
following [13], we can integrate out the temporal component of the internal
momenta from the loop integral. In fact in the limit $\epsilon\rightarrow 0$,
eq. (5.22) vanishes unless some of the $\ell_{i}^{0}$ vanish at the same time,
and since eq. (LABEL:EMT_Amplitude_Complete) must be non-vanishing, one needs
to integrate over $\ell_{i}^{0}$ in the loop integral picking the residue
$\ell_{i}^{0}=i\epsilon$. This leads to the cancellation of the extra
$-i\epsilon$ since now $\ell^{0}_{l+1}=0$, and simplifies the amplitude
calculation as

$\displaystyle-\frac{i\,\kappa}{2}\,2m\,T_{\mu\nu}^{(l,j)}(q^{2})$ (5.23)
$\displaystyle=\int\prod_{i=1}^{l}\frac{d^{d}\ell_{i}}{(2\pi)^{d}}\mathcal{M}_{\lambda_{1}\sigma_{1},...,\lambda_{l+1-j}\sigma_{l+1-j},0,...,0,\mu\nu}\Big{|}_{\ell_{i}^{0}=0}\frac{\Big{(}\prod_{i=1}^{l+1-j}i\,P^{\lambda_{i}\sigma_{i},00}\Big{)}(-i)^{j}}{\prod_{i=1}^{l+1}\Big{(}-\vec{\ell_{i}}^{2}\Big{)}}$
$\displaystyle\times\frac{1}{(l+1)!}(-i)^{l+1}\kappa^{l+1-j}m^{2+2l-j}2^{j}Q^{j}\left(\frac{i}{-2m}\right)^{l}\
,$

from which one finally obtains

$\displaystyle T_{\mu\nu}^{(l,j)}(\vec{q}^{2})$
$\displaystyle=\frac{i}{(l+1)!}\int\prod_{i=1}^{l}\frac{d^{d}\vec{\ell_{i}}}{(2\pi)^{d}}(-1)^{j+l+1}2^{j-l}m^{l+1-j}\kappa^{l-j}Q^{j}\frac{\prod_{i=1}^{l-j+1}P^{00,\lambda_{i}\sigma_{i}}}{\prod_{i=1}^{l+1}\vec{\ell_{i}}^{2}}$
(5.24)
$\displaystyle\times\mathcal{M}_{\lambda_{1}\sigma_{1},...,\lambda_{l-j+1}\sigma_{l-j+1},0,...,0,\mu\nu}(p_{1},p_{2},\ell_{1},...,\ell_{l})\Big{|}_{\ell_{i}^{0}=0}\
,$

where

$P_{00,\mu\nu}=\delta_{\mu}^{0}\delta_{\nu}^{0}-\frac{\eta_{\mu\nu}}{d-1}\ .$
(5.25)

Then the strategy to extract the form factors needed for the computation of
eq. (5.12) out of the above expression, will be to choose a particular process
which fixes the expression of $\mathcal{M}$, considering the classical limit
of such amplitude and performing the replacement $\ell_{i}^{0}=0$ once the
right factors due to the residue integration are added. Then out of the whole
expression in (5.24) we will select only those contributions proportional to
the master integral, extracting in this way the explicit form of
$c_{1}^{(l,j)}(d)$ and $c_{2}^{(l,j)}(d)$ by comparison with eq. (5.10).
Moreover such analysis shows how eq. (5.24) is equivalent to a quantum tree-
graph in which the internal massive lines are cut and replaced by external
sources

$-i\,\kappa\,m\,T_{\mu\nu}^{(l,j)}(\vec{q}^{2})\Big{|}_{\text{classical}}=\includegraphics[width=195.12767pt,valign={c}]{Diagrams/Chapter5/img1_4.png}\Bigg{|}_{\text{leading-
order}}\ .$ (5.26)

This was first noticed by Duff in [17], in which the Schwarzschild metric was
recovered up to $O(G_{N}^{3})$ by the explicit calculation of amplitudes like
(5.26). This argument motivates the selection rule given in section 3.1, which
states that the low-energy limit imposes that massive particles cannot mediate
the interaction, and in massless emission processes this means that massive
states enter only as a single line that connects the two initial particles.
The reason why this is true, is that since the classical limit cuts internal
massive lines, the amplitude is reorganized in terms of the interaction of
$l+1$ external sources like in eq. (5.26), whose in [17] are proven to fully
reconstruct classical phenomenology. For example the amplitude in figure 5.1,
even having in principle a purely classical leading order behavior obtained
from the replacement rules of section 2.4, an explicit calculation will show
that such contribution does not exist and the leading-order is actually
$O(\vec{q}^{4})$, which correspond to quantum corrections. In the modern
language we can say that the leading contribution of the diagram is not
proportional to the master integral and therefore it is not classical.

Figure 5.1: 2-loop amplitude that contributes to a quantum correction of the
third post-Minkowskian order of the metric.

###  5.2 Metric from amplitude calculations

Now we are ready to employ the techniques elaborated in the previous section
in order to explicitly compute the metric out of amplitude calculations. As
anticipated earlier the strategy will be to consider processes like (5.6), and
extract out of them the classical part following eq. (5.24), picking up only
those contributions proportional to the master integral. The calculation will
be performed up to fourth post-Minkowskian order, which means up to 3 loops.
Since for each loop order there exist different contributions of the stress-
energy tensor associated with different values of $j$, for each pair of $l$
and $j$ will be evaluated the corresponding form factors and the associated
metric contribution.

#### Tree-level

We first consider the tree-level amplitude in figure 5.2, corresponding to eq.
(5.6) for $l=0$, $j=0$.

Figure 5.2: Tree-level diagram for graviton emission.

Considering on-shell particles which satisfy $p_{1}^{2}=p_{2}^{2}=m^{2}$, with
transferred momentum $q=p_{1}-p_{2}$, the stress-energy tensor arising at
tree-level is

$\frac{-i\kappa}{2}\sqrt{4E_{1}E_{2}}T_{\mu\nu}^{(0,0)}(q^{2})=(\tau_{\phi^{2}h})_{\mu\nu}\
.$ (5.27)

Using the relations (5.16) one gets

$T_{\mu\nu}^{(0,0)}(\vec{q}^{2})=m\,\delta_{\mu}^{0}\delta_{\nu}^{0}\ ,$
(5.28)

that compared with eq. (5.10) lead to

$\displaystyle c_{1}^{(0,0)}(d)$ $\displaystyle=1$ (5.29) $\displaystyle
c_{2}^{(0,0)}(d)$ $\displaystyle=0\ ,$

and from (5.12) one gets the metric components

$\displaystyle h_{0}^{(1,0)}(r)$ $\displaystyle=-4\frac{d-2}{d-1}G_{N}m\,\rho$
(5.30) $\displaystyle h_{1}^{(1,0)}(r)$
$\displaystyle=\frac{4}{d-1}G_{N}m\,\rho$ $\displaystyle h_{2}^{(1,0)}(r)$
$\displaystyle=0\ .$

This result is in perfect agreement with the classical post-Minkowskian metric
derived in eq. (4.38) at first order.

#### 1-loop

We now compute the contribution to the stress-energy tensor that arises at
1-loop order. There are two diagrams that contribute, i.e. $l=1,\ j=0$ and
$l=1,\ j=2$. The first one is obtained by evaluating the amplitude that
involves a 3-gravitons vertex, as in figure 5.3.

Figure 5.3: 1-loop diagram for graviton emission with 2 internal gravitons.

The amplitude associated to this diagram is

$\displaystyle\frac{-i\kappa}{2}\sqrt{4E_{1}E_{2}}$ $\displaystyle
T^{(1,0)}_{\mu\nu}(q^{2})$ (5.31)
$\displaystyle=\int\frac{d^{d+1}\ell}{(2\pi)^{d+1}}\frac{-iP^{\alpha\beta,\lambda\kappa}P^{\gamma\delta,\rho\sigma}\left(\tau_{\phi^{2}h}\right)_{\,\alpha\beta}\,\left(\tau_{\phi^{2}h}\right)_{\,\gamma\delta}\,\left(\tau_{h^{2}h}\right)_{\mu\nu,\rho\sigma,\lambda\kappa}(\ell,q)}{(\ell^{2}+i\epsilon)((\ell-q)^{2}+i\epsilon)((\ell-
p_{1})^{2}-m^{2}+i\epsilon)}\ ,$

where the 3 gravitons vertex $(\tau_{h^{2}h})$ is defined in eq. (A.10). In
particular here we are allowed to consider the vertex with an external
graviton which simplifies the calculation, although the vertex
$(\tau_{h^{3}})$ in eq. (A.9) would give the same result after more tedious
calculations. The stress-energy tensor form factors associated to this diagram
are computed in [18], and they read

$\displaystyle c_{1}^{(1,0)}(d)=-2\frac{4d^{2}-15d+10}{(d-1)^{2}}$ (5.32)
$\displaystyle c_{2}^{(1,0)}(d)=-\frac{2(d-2)(3d-2)}{(d-1)^{2}}\ ,$

from which by eq. (5.12) we get

$\displaystyle h_{0}^{(2,0)}(r)$
$\displaystyle=\frac{8(d-2)^{2}}{(d-1)^{2}}(G_{N}m)^{2}\rho^{2}$ (5.33)
$\displaystyle h_{1}^{(2,0)}(r)$
$\displaystyle=-\frac{4(2d^{2}-9d+14)}{(d-4)(d-1)^{2}}(G_{N}m)^{2}\rho^{2}$
$\displaystyle h_{2}^{(2,0)}(r)$
$\displaystyle=\frac{4(d-2)^{2}(3d-2)}{(d-4)(d-1)^{2}}(G_{N}m)^{2}\rho^{2}\ .$

Again the above metric is in perfect agreement with eq. 4.38 for the terms
proportional to $m^{2}G_{N}^{2}\rho^{2}$. In particular we notice some
divergences that arise in $d=4$ (five space-time dimensions), that have to be
renormalized in order to obtain finite results.

The other contribution at 1-loop order is given by the diagram that contains
two photons drawn in figure 5.4.

Figure 5.4: 1-loop diagram for graviton emission with 2 internal photons.

The resulting contribution to the stress-energy tensor is

$\displaystyle\frac{-i\kappa}{2}\sqrt{4E_{1}E_{2}}$ $\displaystyle
T^{(1,2)}_{\mu\nu}(q^{2})$ (5.34)
$\displaystyle=\int\frac{d^{d+1}\ell}{(2\pi)^{d+1}}\frac{-i\left(\tau_{\phi^{2}A}\right)_{\alpha}\left(\tau_{\phi^{2}A}\right)_{\beta}\left(\tau_{A^{2}h}\right){}_{\mu\nu}{}^{\alpha,\beta}(\ell,\ell-q)}{(\ell+i\epsilon)^{2}((\ell-q)^{2}+i\epsilon)((p_{1}-\ell)^{2}-m^{2}+i\epsilon)}\
,$

where $(\tau_{\phi^{2}A})$ is the 2 scalars - 1 photon vertex in (A.5) and
$(\tau_{A^{2}h})$ is the 2 photons - 1 graviton vertex in (A.7). Following the
procedure of the previous section one gets

$T^{(1,2)}_{\mu\nu}(\vec{q}^{2})=i\frac{Q^{2}}{\kappa}\int\frac{d^{d}\ell}{(2\pi)^{d}}\frac{\left(\tau_{A^{2}h}\right)_{\mu\nu}{}^{0,0}(\ell,\ell-q)\Big{|}_{\ell^{0}=0}}{\vec{\ell}^{2}(\vec{q}-\vec{\ell})^{2}}\
,$ (5.35)

where the numerator is explicitly

$\left(\tau_{A^{2}h}\right)_{\mu\nu}{}^{0,0}(\ell,\ell-q)\Big{|}_{\ell^{0}=0}=i\kappa\left(\left(\delta_{\mu}^{0}\delta_{\nu}^{0}-\frac{\eta_{\mu\nu}}{2}\right)\ell\cdot(\ell-q)+\ell_{\mu}\ell_{\nu}-\frac{1}{2}(q_{\mu}\ell_{\nu}+q_{\nu}\ell_{\mu})\right)\
.$ (5.36)

Now that the complete loop integral reads

$T^{(1,2)}_{\mu\nu}(\vec{q}^{2})=-Q^{2}\int\frac{d^{d}\ell}{(2\pi)^{d}}\frac{\left(\delta_{\mu}^{0}\delta_{\nu}^{0}-\frac{\eta_{\mu\nu}}{2}\right)\ell\cdot(\ell-q)+\ell_{\mu}\ell_{\nu}-\frac{1}{2}(q_{\mu}\ell_{\nu}+q_{\nu}\ell_{\mu})}{\vec{\ell}^{2}(\vec{q}-\vec{\ell})^{2}}\
,$ (5.37)

we have to extract out of it the contributions proportional to the master
integral. In order to do that one has to reshape the expression in eq. (5.37)
using symmetry and integral properties to obtain the integral definition of
$J_{(l)}(\vec{q}^{2})$. The needed loop integral reductions are listed in
appendix C, whose lead to

$\begin{gathered}T^{(1,2)}_{00}(\vec{q}^{2})=-\frac{Q^{2}}{4}J_{(1)}(\vec{q}^{2})\\\
\delta^{ij}T^{(1,2)}_{ij}(\vec{q}^{2})=\frac{Q^{2}}{4}(2-d)J_{(1)}(\vec{q}^{2})\
.\end{gathered}$ (5.38)

The reason why we express the stress-energy tensor in the above form is
because it allows us to use scalar loop integral reduction formulas like the
one in appendix C, instead of more complicated relations that involve free
indices. Moreover the expression (5.38) it is sufficient to extract the form
factors. In fact expressing eq. (5.10) in an equivalent form as

$\begin{gathered}T^{(l,j)}_{00}(\vec{q}^{2})=m\left(c_{1}^{(l,j)}(d)-c_{2}^{(l,j)}(d)\right)(G_{N}m)^{l-j}(\alpha
G_{N})^{\frac{j}{2}}\,\pi^{l}\,J_{(l)}(\vec{q}^{2})\\\
\delta^{ij}T^{(l,j)}_{ij}(\vec{q}^{2})=m\,c_{2}^{(l,j)}(d)\left(d-1\right)(G_{N}m)^{l-j}(\alpha
G_{N})^{\frac{j}{2}}\,\pi^{l}\,J_{(l)}(\vec{q}^{2})\ ,\end{gathered}$ (5.39)

then in the present case we get

$\begin{gathered}T^{(1,2)}_{00}(\vec{q}^{2})=\left(c_{1}^{(1,2)}(d)-c_{2}^{(1,2)}(d)\right)\frac{Q^{2}}{4}J_{(1)}(\vec{q}^{2})\\\
\delta^{ij}T^{(1,2)}_{ij}(\vec{q}^{2})=c_{2}^{(1,2)}(d)\left(d-1\right)\frac{Q^{2}}{4}J_{(1)}(\vec{q}^{2})\
,\end{gathered}$ (5.40)

from which a direct comparison with eq. (5.38) leads to

$\displaystyle c_{1}^{(1,2)}(d)=\frac{-2d+3}{d-1}$ (5.41) $\displaystyle
c_{2}^{(1,2)}(d)=\frac{2-d}{d-1}\ .$

Finally replacing the above expression inside eq. (5.12) one obtains

$\displaystyle h_{0}^{(2,2)}(r)$
$\displaystyle=-\frac{(-2d+4)}{d-1}G_{N}\alpha\rho^{2}$ (5.42) $\displaystyle
h_{1}^{(2,2)}(r)$ $\displaystyle=\frac{-2d+6}{(d-1)(d-4)}G_{N}\alpha\rho^{2}$
$\displaystyle h_{2}^{(2,2)}(r)$
$\displaystyle=2\frac{(d-2)^{2}}{(d-1)(d-4)}G_{N}\alpha\rho^{2}\ .$

The obtained metric is in perfect agreement with eq. (4.38) for the terms
proportional to $\alpha G_{N}\rho^{2}$. Moreover as expected we encounter
divergences in $d=4$ that have to be renormalized.

#### 2-loop

The third post-Minkowskian order of the metric is given by the 2-loop diagrams
of the type in (5.6), which correspond to $l=2$ and are divided in $j=0$ and
$j=2$. The pure graviton case ($j=0$) corresponds to the diagrams in which
there are only graviton internal lines, which are given in figure 5.5.

Figure 5.5: 2-loop diagrams for graviton emission with only graviton internal
lines.

The computation of these diagrams was carried out in [18] and we report here
the corresponding form factors

$\displaystyle c_{1}^{(2,0)}(d)=$
$\displaystyle\frac{32}{3(d-4)(d-1)^{3}}\left(9d^{4}-70d^{3}+203d^{2}-254d+104\right)$
(5.43) $\displaystyle c_{2}^{(2,0)}(d)=$
$\displaystyle\frac{64(d-2)}{3(d-4)(d-1)^{3}}\left(2d^{3}-13d^{2}+25d-10\right)\
.$

From these, one computes the metric components

$\displaystyle h_{0}^{(3,0)}(r)=$
$\displaystyle\frac{8(7-3d)(d-2)^{3}}{(d-4)(d-1)^{3}}m^{3}G_{N}^{3}\rho^{3}$
(5.44) $\displaystyle h_{1}^{(3,0)}(r)=$
$\displaystyle\frac{8\left(7d^{4}-63d^{3}+214d^{2}-334d+212\right)}{3(d-4)(d-3)(d-1)^{3}}m^{3}G_{N}^{3}\rho^{3}$
$\displaystyle h_{2}^{(3,0)}(r)=$
$\displaystyle\frac{8(d-2)^{2}\left(-2d^{3}+13d^{2}-25d+10\right)}{(d-4)(d-3)(d-1)^{3}}m^{3}G_{N}^{3}\rho^{3}\
.$

This result is in agreement with the post-Minkowskian expansion of the metric
(4.38) for the terms proportional to $m^{3}G_{N}^{3}\rho^{3}$. In particular,
they are all divergent in five dimensions, while $h_{1}^{(3,0)}$ and
$h_{2}^{(3,0)}$ are also divergent in four dimensions.

In the case $j=2$ we have to consider diagrams with two photons and one
graviton emitted from the scalar line. At this loop order it is important to
consider the multiplicity factor of each diagram, in fact since the classical
limit of the amplitudes is independent of the exchange between photon and
graviton lines emitted by the massive scalar, we have to consider all the
permutations between them, and weight the amplitudes with the right factor.
Since the vertices defined in appendix A are all written with the right
multiplicity factor inside, this is the only combinatorial observation we have
to do. Finally there are in total three diagrams that we must take into
account, and we will compute the stress-energy tensor of each diagram and then
sum all of them.

Figure 5.6: 2-loop diagram for graviton emission with 1 internal graviton and
3 internal photons.

The first diagram is given in figure 5.6, and its contribution to the stress-
energy tensor is

$\displaystyle-\frac{i\kappa}{2}\sqrt{4E_{1}E_{2}}T^{(2,2)(\text{a})}_{\mu\nu}(q^{2})$
(5.45)
$\displaystyle=\,6\times\int\frac{d^{d+1}\ell_{1}}{(2\pi)^{d+1}}\frac{d^{d+1}\ell_{2}}{(2\pi)^{d+1}}\frac{\left(\tau_{\phi^{2}A}\right)^{\alpha}\left(\tau_{\phi^{2}A}\right)^{\beta}\left(\tau_{\phi^{2}h}\right)^{\rho\sigma}}{\left((p_{1}-\ell_{1})^{2}-m^{2}+i\epsilon\right)\left((p_{1}-\ell_{1}-\ell_{2})^{2}-m^{2}+i\epsilon\right)}$
$\displaystyle\times\frac{P_{\rho\sigma,\eta\chi}\left(\tau_{A^{2}h}\right)_{\mu\nu,\gamma,\alpha}(k,k^{\prime})\left(\tau_{A^{2}h}\right)^{\eta\chi}{}_{\beta}{}^{\gamma}(p,p^{\prime})}{\left(\ell_{1}^{2}+i\epsilon\right)\left(\ell_{2}^{2}+i\epsilon\right)\left((\ell_{1}+\ell_{2})^{2}+i\epsilon\right)\left((\ell_{1}+\ell_{2}-q)^{2}+i\epsilon\right)}\
,$

where we introduced the new momenta

$\displaystyle p$ $\displaystyle=\ell_{1}$ $\displaystyle k$
$\displaystyle=\ell_{1}+\ell_{2}$ (5.46) $\displaystyle p^{\prime}$
$\displaystyle=\ell_{1}+\ell_{2}$ $\displaystyle k^{\prime}$
$\displaystyle=\ell_{1}+\ell_{2}-q\ ,$

and where the factor multiplying the integral is the multiplicity of the
diagram. Then from the analysis of section 5.1 one obtains

$\displaystyle T^{(2,2)(\text{a})}_{\mu\nu}(\vec{q}^{2})=$ $\displaystyle
mQ^{2}\int\frac{d^{d}\vec{\ell}_{1}}{(2\pi)^{d}}\frac{d^{d}\vec{\ell}_{2}}{(2\pi)^{d}}\frac{\left(\tau_{A^{2}h}\right)_{\mu\nu,\gamma,0}(k,k^{\prime})P_{00,\eta\chi}\left(\tau_{A^{2}h}\right)^{\eta\chi}{}_{0}{}^{\gamma}(p,p^{\prime})\bigl{|}_{\ell_{i}^{0}=0}}{\vec{\ell}_{1}^{2}\vec{\ell}_{2}^{2}(\vec{\ell}_{1}+\vec{\ell}_{2}-\vec{q})^{2}(\vec{\ell}_{1}+\vec{\ell}_{2})^{2}}\
,$ (5.47)

whose numerator, using the explicit expressions of the vertices, becomes

$\displaystyle\left(\tau_{A^{2}h}\right)_{\mu\nu,\gamma,0}(k,k^{\prime})$
$\displaystyle
P_{00,\eta\chi}\left(\tau_{A^{2}h}\right)^{\eta\chi}{}_{0}{}^{\gamma}(p,p^{\prime})\bigl{|}_{\ell_{i}^{0}=0}$
(5.48) $\displaystyle=-\kappa^{2}\frac{d-2}{d-1}p\cdot
p^{\prime}\Biggl{(}k\cdot
k^{\prime}\left(\delta^{0}_{\mu}\delta^{0}_{\nu}-\frac{1}{2}\eta_{\mu\nu}\right)+\frac{1}{2}(k_{\mu}k^{\prime}_{\nu}+k_{\nu}k^{\prime}_{\mu})\Biggl{)}\
.$

Substituting this in eq. (5.47) and using the identities in appendix C,
rewriting the stress-energy tensor according to eq. (5.39), one gets

$\begin{gathered}T^{(2,2)(\text{a})}_{00}(\vec{q}^{2})=\frac{1}{12}\frac{d-2}{d-1}mQ^{2}\kappa^{2}J_{(2)}(\vec{q}^{2})\\\
\delta^{ij}T^{(2,2)(\text{a})}_{ij}(\vec{q}^{2})=\frac{1}{12}\frac{(d-2)^{2}}{d-1}mQ^{2}\kappa^{2}J_{(2)}(\vec{q}^{2})\
,\end{gathered}$ (5.49)

which we will add to the other 2-loop contributions in order to obtain the
final form factors to be replaced inside (5.12).

Figure 5.7: 2-loop diagram for graviton emission with 1 internal graviton and
2 internal photons.

The second diagram that we consider is the one in figure 5.7, which gives the
contribution to the stress-energy tensor

$\displaystyle-\frac{i\kappa}{2}\sqrt{4E_{1}E_{2}}T^{(2,2)(\text{b})}_{\mu\nu}(q^{2})$
(5.50)
$\displaystyle=3\times\int\frac{d^{d+1}\ell_{1}}{(2\pi)^{d+1}}\frac{d^{d+1}\ell_{2}}{(2\pi)^{d+1}}\frac{i\left(\tau_{\phi^{2}A}\right)^{\alpha}\left(\tau_{\phi^{2}A}\right)^{\beta}\left(\tau_{\phi^{2}h}\right)^{\rho\sigma}}{\left((p_{1}-\ell_{1})^{2}-m^{2}+i\epsilon\right)\left((p_{1}-\ell_{1}-\ell_{2})^{2}-m^{2}+i\epsilon\right)}$
$\displaystyle\times\frac{P_{\rho\sigma,\gamma\delta}\left(\tau_{A^{2}h^{2}}\right){}_{\mu\nu}{}^{\gamma\delta}{}_{\alpha,\beta}(p,p^{\prime})}{\left(\ell_{1}^{2}+i\epsilon\right)\left(\ell_{2}^{2}+i\epsilon\right)\left((\ell_{1}+\ell_{2}-q)^{2}+i\epsilon\right)}\
,$

where we defined the momenta $p=\ell_{2}$ and $p^{\prime}=\ell_{1}+\ell_{2}-q$
and where $(\tau_{A^{2}h^{2}})$ is the 2 photons - 2 gravitons vertex in
(A.8). Imposing the classical limit to the above equation one obtains

$\displaystyle T^{(2,2)(\text{b})}_{\mu\nu}(\vec{q}^{2})=$
$\displaystyle\,\frac{i}{2}\,m\,Q^{2}\int\frac{d^{d}\vec{\ell}_{1}}{(2\pi)^{d}}\frac{d^{d}\vec{\ell}_{2}}{(2\pi)^{d}}\frac{P_{00,\gamma\delta}\left(\tau_{A^{2}h^{2}}\right){}_{\mu\nu}{}^{\gamma\delta}{}_{0,0}(p,p^{\prime})\bigl{|}_{\ell_{i}^{0}=0}}{\vec{\ell}_{1}^{2}\vec{\ell}_{2}^{2}(\vec{\ell}_{1}+\vec{\ell}_{2}-\vec{q})^{2}}\
,$ (5.51)

from which the numerator explicitly gets the expression

$\displaystyle
P_{00,\gamma\delta}\left(\tau_{A^{2}h^{2}}\right){}_{\mu\nu}{}^{\gamma\delta}{}_{0,0}(p,p^{\prime})\bigl{|}_{\ell_{i}^{0}=0}$
(5.52)
$\displaystyle=-\frac{i\kappa^{2}}{4}\left(\frac{2(d-3)}{d-1}\left(p_{\mu}p^{\prime}_{\nu}+p_{\nu}p^{\prime}_{\mu}-p\cdot
p^{\prime}\eta_{\mu\nu}\right)+\frac{2(3d-7)}{d-1}p\cdot
p^{\prime}\delta^{0}_{\mu}\delta^{0}_{\nu}\right)\ .$

Finally, substituting the above equation in eq. (5.51), the resulting
components of the stress-energy tensor are

$\begin{gathered}T^{(2,2)(\text{b})}_{00}(\vec{q}^{2})=-\frac{1}{12}\frac{d-2}{d-1}Q^{2}m\kappa^{2}J_{(2)}(\vec{q}^{2})\\\
\delta^{ij}T^{(2,2)(\text{b})}_{ij}(\vec{q}^{2})=-\frac{1}{24}\frac{(d-3)(d-2)}{d-1}mQ^{2}\kappa^{2}J_{(2)}(\vec{q}^{2})\
.\end{gathered}$ (5.53)

Figure 5.8: 2-loop diagram for graviton emission with 2 internal gravitons and
2 internal photons.

The last diagram for the emission of a graviton at 2-loop order is the one in
figure 5.8. The corresponding stress-energy tensor is

$\displaystyle-\frac{i\kappa}{2}\sqrt{4E_{1}E_{2}}T^{(2,2)(\text{c})}_{\mu\nu}(q^{2})$
(5.54)
$\displaystyle=3\times\int\frac{d^{d+1}\ell_{1}}{(2\pi)^{d+1}}\frac{d^{d+1}\ell_{2}}{(2\pi)^{d+1}}\frac{-\left(\tau_{\phi^{2}A}\right)^{\alpha}\left(\tau_{\phi^{2}A}\right)^{\beta}\left(\tau_{\phi^{2}h}\right)^{\rho\sigma}}{\left((p_{1}-\ell_{1})^{2}-m^{2}+i\epsilon\right)\left((p_{1}-\ell_{1}-\ell_{2})^{2}-m^{2}+i\epsilon\right)}$
$\displaystyle\times\frac{P_{\rho\sigma}{}^{\eta\chi}P_{\theta\xi}{}^{\gamma\delta}\left(\tau_{A^{2}h}\right)^{\theta\xi}{}_{\alpha,\beta}(p,p^{\prime})\left(\tau_{h^{2}h}\right)_{\mu\nu,\gamma\delta,\eta\chi}(k,k^{\prime})}{\left(\ell_{1}^{2}+i\epsilon\right)\left(\ell_{2}^{2}+i\epsilon\right)\left((\ell_{1}+\ell_{2})^{2}+i\epsilon\right)\left((\ell_{1}+\ell_{2}-q)^{2}+i\epsilon\right)}\
,$

where we have introduced the new momenta

$\displaystyle p$ $\displaystyle=\ell_{1}$ $\displaystyle k$
$\displaystyle=\ell_{1}+\ell_{2}$ (5.55) $\displaystyle p^{\prime}$
$\displaystyle=-\ell_{2}$ $\displaystyle k^{\prime}$ $\displaystyle=q\ .$

Employing the considerations of section 5.1 then one obtains

$\displaystyle T^{(2,2)(\text{c})}_{\mu\nu}(\vec{q}^{2})=$
$\displaystyle-\frac{1}{2}mQ^{2}\int\frac{d^{d}\vec{\ell}_{1}}{(2\pi)^{d}}\frac{d^{d}\vec{\ell}_{2}}{(2\pi)^{d}}\frac{\left(\tau_{A^{2}h}\right)^{\alpha\beta}{}_{0,0}(p,p^{\prime})P_{\alpha\beta}{}^{\gamma\delta}P_{00}{}^{\rho\sigma}\left(\tau_{h^{2}h}\right)_{\mu\nu,\gamma\delta,\rho\sigma}(k,k^{\prime})\bigl{|}_{\ell_{i}^{0}=0}}{\vec{\ell}_{1}^{2}\vec{\ell}_{2}^{2}(\vec{\ell}_{1}+\vec{\ell}_{2}-\vec{q})^{2}(\vec{\ell}_{1}+\vec{\ell}_{2})^{2}}\
.$ (5.56)

Introducing the vertex $\left(\tilde{\tau}_{h^{2}h}\right)$ defined as

$\left(\tilde{\tau}_{h^{2}h}\right){}^{\mu\nu}{}_{\alpha\beta,\gamma\delta}={{P}_{\alpha\beta}}^{\rho\sigma}{{P}_{\gamma\delta}}^{\eta\xi}\left(\tau_{h^{2}h}\right){}^{\mu\nu}{}_{\rho\sigma,\eta\xi}\
,$ (5.57)

where detailed properties are given in [18, 25], the numerator of the
integrand can be written as

$\displaystyle\left(\tau_{A^{2}h}\right)^{\alpha\beta}{}_{0,0}(p,p^{\prime})P_{\alpha\beta}{}^{\gamma\delta}P_{00}{}^{\rho\sigma}\left(\tau_{h^{2}h}\right)_{\mu\nu,\gamma\delta,\rho\sigma}(k,k^{\prime})=\left(\tau_{A^{2}h}\right)^{\alpha\beta}{}_{0,0}(p,p^{\prime})\left(\tilde{\tau}_{h^{2}h}\right)_{\mu\nu,\alpha\beta,00}(k,k^{\prime})\
,$ (5.58)

from which an explicit calculation leads to

$\displaystyle\left(\tau_{A^{2}h}\right)^{\alpha\beta}{}_{0,0}(p,p^{\prime})\left(\tilde{\tau}_{h^{2}h}\right)_{00,\alpha\beta,00}(k,k^{\prime})\bigl{|}_{\ell_{i}^{0}=0}$
(5.59) $\displaystyle=\frac{\kappa^{2}}{4}\frac{1}{d-1}\left(p\cdot
p^{\prime}\left(-dk^{\prime
2}+\frac{(3d-3)(d-2)}{d-1}k^{2}+(d-2)(k-k^{\prime})^{2}\right)+(4d-8)(k^{\prime}\cdot
p)(k^{\prime}\cdot p^{\prime})\right)$

for the $00$-component, and to

$\displaystyle\delta^{ij}$
$\displaystyle\left(\tau_{A^{2}h}\right)^{\alpha\beta}{}_{0,0}(p,p^{\prime})\left(\tilde{\tau}_{h^{2}h}\right)_{ij,\alpha\beta,00}(k,k^{\prime})\bigl{|}_{\ell_{i}^{0}=0}$
(5.60) $\displaystyle=-\frac{\kappa^{2}}{4}\frac{d-2}{d-1}\left(p\cdot
p^{\prime}\left((d-2)k^{2}+(d-4)(k-k^{\prime})^{2}+3dk^{\prime
2}\right)+4(k^{\prime}\cdot p)(k^{\prime}\cdot p^{\prime})\right)$

for the trace of the spatial components. Substituting the above expressions in
(5.56), inserting the momenta (5.55) and using the loop reduction identities
in appendix C (in particular (C.12)), we have

$\begin{gathered}T^{(2,2)(\text{c})}_{00}(\vec{q}^{2})=\frac{(3d^{2}-19d+32)}{24(d-4)(d-1)}mQ^{2}\kappa^{2}J_{(2)}(\vec{q}^{2})\\\
\delta^{ij}T^{(2,2)(\text{c})}_{ij}(\vec{q}^{2})=\frac{(d-2)(5d^{2}-27d+30)}{24(d-4)(d-1)}mQ^{2}\kappa^{2}J_{(2)}(\vec{q}^{2})\
.\end{gathered}$ (5.61)

Finally, summing all the contributions, we obtain the total stress-energy
tensor

$T^{(2,2)}_{\mu\nu}(\vec{q}^{2})=\sum_{i=\text{a},\text{b},\text{c}}T^{(2,2)(i)}_{\mu\nu}(\vec{q}^{2})\
,$ (5.62)

that explicitly it reads

$\begin{gathered}T_{00}^{(2,2)}(\vec{q}^{2})=\frac{(3d^{2}-19d+32)}{24(d-4)(d-1)}mQ^{2}\kappa^{2}J_{(2)}(\vec{q}^{2})\\\
\delta^{ij}T^{(2,2)}_{ij}(\vec{q}^{2})=\frac{(d-2)(3d^{2}-16d+17)}{12(d-4)(d-1)}{mQ^{2}\kappa^{2}}J_{(2)}(\vec{q}^{2})\
.\end{gathered}$ (5.63)

Then from a direct comparison with eq. (5.39) we can easily read the
coefficients $c^{(l,j)}_{i}(d)$, obtaining

$\displaystyle c_{1}^{(2,2)}(d)$
$\displaystyle=\frac{16\left(9d^{3}-66d^{2}+149d-100\right)}{3(d-4)(d-1)^{2}}$
(5.64) $\displaystyle c_{2}^{(2,2)}(d)$
$\displaystyle=\frac{32(d-2)\left(3d^{2}-16d+17\right)}{3(d-4)(d-1)^{2}}\ .$

From these coefficients, using eq. (5.12) we finally obtain the metric
components as

$\displaystyle h_{0}^{(3,2)}(r)=$
$\displaystyle-\frac{4(d-2)^{2}(3d-11)}{(d-4)(d-1)^{2}}m\alpha
G_{N}^{2}{\rho}^{3}$ (5.65) $\displaystyle h_{1}^{(3,2)}(r)=$
$\displaystyle\frac{8(3d^{3}-25d^{2}+69d-65)}{3(d-4)(d-3)(d-1)^{2}}m\alpha
G_{N}^{2}{\rho}^{3}$ $\displaystyle h_{2}^{(3,2)}(r)=$
$\displaystyle-\frac{4(d-2)^{2}(3d^{3}-19d^{2}+33d-17)}{(d-4)(d-3)(d-1)^{3}}m\alpha
G_{N}^{2}{\rho}^{3}\ .$

We can compare this result with the part proportional to $m\alpha G_{N}^{2}$
of (4.38) and observe that they match perfectly. In particular, all these
terms diverge in $d=4$, and $h_{1}^{(3,2)}$ and $h_{2}^{(3,2)}$ also diverge
in $d=3$.

As a summary, we have shown that the loop computations give exactly the
expression for the metric of the Reissner-Nordström-Tangherlini solution in de
Donder gauge given in eq. (4.38). However at third post-Minkowskian order in
$d=3$ arise divergences in $h_{1}$ and $h_{2}$, while in $d=4$ all the
components diverge. To obtain the values of the metrics component in these
dimensions we need to perform a renormalization procedure which will be
described in the next section.

###  5.3 Renormalization of the metric

In the previous section we noticed the appearance of divergences in $d=3$ and
$d=4$, and now we show how such divergences are renormalized by non-minimal
couplings. After regularizing the singularities that appear in the metric
explicitating the poles structure, firstly we analyze the non-minimal
couplings that are needed to be added to the original theory and the
consequent variation of the metric, then we use these results to renormalize
the divergences.

####  5.3.1 Regularization

The regularization procedure consists only in isolating the poles through
dimensional regularization. Since the observable is already expressed in
generic space-time dimension we only need to expand the parameter $d$ around
the problematic values, as is usual in the regularization process of any field
theory.

####  $\bm{d=3}$ case

In this dimension the only divergences that appear are at 2-loop order. From
the explicit expression of the 2-loop contribution of the metric in $d=3$, one
can notice that $h_{0}^{(d=3)}\bigl{|}_{2-\text{loop}}$ is non divergent and
can be already compared with the (4.39) with a perfect match. After the
regularization process, the other two components turn out to be equal up to a
multiplicative constant, in particular we have
$h_{2}^{(d=3)}\bigl{|}_{2-\text{loop}}=-3h_{1}^{(d=3)}\bigl{|}_{2-\text{loop}}$,
so we will renormalize only the $h_{1}^{(d=3)}\bigl{|}_{2-\text{loop}}$
component and then obtain the other one using this relation. We perform a
dimensional regularization using $d=3+\epsilon$, where $\epsilon$ is a small
parameter. Neglecting terms that vanish when $\epsilon\rightarrow 0$, the
result is

$\displaystyle h_{1}^{(d=3)}\bigl{|}_{2-\text{loop}}$
$\displaystyle=\frac{1}{\epsilon}\left(\frac{4}{3}\frac{m\alpha
G_{N}^{2}}{r^{3}}-\frac{2}{3}\frac{m^{3}G_{N}^{3}}{r^{3}}\right)+\frac{m^{3}G_{N}^{3}}{r^{3}}\left(2\log\left(2C_{E}r\right)-\frac{4}{3}\right)$
(5.66) $\displaystyle-4\frac{m\alpha
G_{N}^{2}}{r^{3}}\log\left(2C_{E}r\right)\ ,$

where following [18] we defined the constant $C_{E}^{2}=\pi e^{\gamma_{EM}}$,
with $\gamma_{EM}$ the Euler-Mascheroni constant.

####  $\bm{d=4}$ case

In $d=4$ the metric is divergent at both 1-loop and 2-loop order. Again, from
the explicit form of the metric, we can see that at 1-loop order the
$h_{0}^{(d=4)}\bigl{|}_{1-\text{loop}}$ component is non-divergent and no
regularization procedure has to be carried on. The
$h_{1}^{(d=4)}\bigl{|}_{1-\text{loop}}$ and
$h_{2}^{(d=4)}\bigl{|}_{1-\text{loop}}$ components are related by the identity
$h_{2}^{(d=4)}\bigl{|}_{1-\text{loop}}=-4h_{1}^{(d=4)}\bigl{|}_{1-\text{loop}}$
and so we can focus just on one component. Regularizing in $d=4+\epsilon$, and
neglecting again terms that vanish when $\epsilon\rightarrow 0$, the result at
1-loop is

$\displaystyle h_{1}^{(d=4)}\bigl{|}_{1-\text{loop}}$
$\displaystyle=\frac{1}{\epsilon}\left(-\frac{2}{3}\frac{\alpha
G_{N}}{\pi^{2}r^{4}}-\frac{40}{9}\frac{G_{N}^{2}m^{2}}{\pi^{2}r^{4}}\right)+\frac{m^{2}G_{N}^{2}}{\pi^{2}r^{4}}\left(-\frac{4}{27}+\frac{40}{9}\log\left(C_{E}^{2}r^{2}\right)\right)$
(5.67) $\displaystyle+\frac{\alpha
G_{N}}{\pi^{2}r^{4}}\left(-\frac{4}{9}+\frac{2}{3}\log\left(C_{E}^{2}r^{2}\right)\right)\
.$

For what concerns the 2-loop order, we can focus just on the
$h_{1}^{(d=4)}\bigl{|}_{2-\text{loop}}$ component since the relation
$h_{0}^{(d=4)}\bigl{|}_{2-\text{loop}}=h_{2}^{(d=4)}\bigl{|}_{2-\text{loop}}=-2h_{1}^{(d=4)}\bigl{|}_{2-\text{loop}}$
holds, and after the regularization we obtain

$\displaystyle h_{1}^{(d=4)}\bigl{|}_{2-\text{loop}}$
$\displaystyle=\frac{1}{\epsilon}\left(\frac{8}{9}\frac{m\alpha
G_{N}^{2}}{\pi^{3}r^{6}}+\frac{160}{27}\frac{m^{3}G_{N}^{3}}{\pi^{3}r^{6}}\right)+\frac{m^{3}G_{N}^{3}}{\pi^{3}r^{6}}\left(\frac{208}{81}-\frac{80}{9}\log\left(C_{E}^{2}r^{2}\right)\right)$
(5.68) $\displaystyle+\frac{\alpha
mG_{N}^{2}}{\pi^{3}r^{6}}\left(\frac{64}{27}-\frac{4}{3}\log\left(C_{E}^{2}r^{2}\right)\right)\
.$

####  5.3.2 Non-minimal couplings

Now that the poles are isolated we need to cancel the singularities. As in any
other quantum field theory this is done by the so called renormalization
procedure, in which exploiting the fact that the couplings of the theory that
appear in the original action are not the physical ones, it is possible to
define some counter-terms that introduce new Feynman rules by which one can
eliminate the divergences. However, since we are treating an effective field
theory which is non-renormalizable, we expect that it is not possible to
perform such procedure only considering the original Lagrangian, and in fact
do not exist such counter-terms that eliminate the poles. Nevertheless,
according to the principles of effective field theories, it is possible to add
some non-minimal couplings that are higher-order in energy with the purpose of
canceling such divergences. As shown in [18], such non-minimal coupling terms
are linear in the Riemann tensor and quadratic in the first derivative of the
scalar field, whose explicit form is

$\begin{split}\delta^{(\mathfrak{n})}S^{ct}=&\sum_{k=0}^{+\infty}(G_{N}m)^{\frac{2(\mathfrak{n}-k)}{d-2}}(\alpha
G_{N})^{\frac{k}{d-2}}\int
d^{d+1}x\sqrt{-g}\Bigl{(}\alpha^{(\mathfrak{n},k)}(d)(D^{2})^{\mathfrak{n}-1}RD_{\mu}\phi
D^{\mu}\phi\\\
&+\left(\beta_{0}^{(\mathfrak{n},k)}(d)D_{\mu}D_{\nu}(D^{2})^{\mathfrak{n}-2}R+\beta_{1}^{(\mathfrak{n},k)}(d)(D^{2})^{\mathfrak{n}-1}R_{\mu\nu}\right)D^{\mu}\phi
D^{\nu}\phi\Bigr{)}\ ,\end{split}$ (5.69)

where $D_{\mu}$ is the covariant derivative, and where we consider only
positive integer powers of the gravitational and electromagnetic coupling. It
is important to notice that the above action is not the most generic one. In
fact, here and in the following, we will only consider the addition of the
non-minimal terms useful to the renormalization of the theory, since every
other operator belongs to high-energy physics.

Due to the properties of the Fourier transforms, terms with $\mathfrak{n}\geq
2$ and terms proportional to $\beta_{0}^{(\mathfrak{n},k)}$ and
$\beta_{1}^{(\mathfrak{n},k)}$ do not contribute to the classical limit of the
metric [18], and this statement holds even for the present electromagnetically
coupled theory. Defining

$\alpha^{(\mathfrak{n})}(d)=\sum_{k=0}^{+\infty}(G_{N}m)^{\frac{-2k}{d-2}}(\alpha
G_{N})^{\frac{k}{d-2}}\alpha^{(\mathfrak{n},k)}(d)\ $ (5.70)

for the sake of simplicity, and $\beta_{0}^{(\mathfrak{n})}$ and
$\beta_{1}^{(\mathfrak{n})}$ in the same way, thus the only contribution to
the non-minimal coupling action that will renormalize the metric is

$\delta^{(1)}S^{ct}=(G_{N}m)^{\frac{2}{d-2}}\alpha^{(1)}(d)\int
d^{d+1}x\sqrt{-g}\,R\,D_{\mu}\phi D^{\mu}\phi\ .$ (5.71)

In the following, we compute the Feynman rules associated to the counter-terms
only in the case $\mathfrak{n}=1$, but for completeness we give the complete
relations considering also the term proportional to $\beta_{1}^{(1)}$.
Considering the interaction between two scalars and one graviton that arise
from the action in eq. (5.69), for $\mathfrak{n}=1$ one gets

$\displaystyle\includegraphics[width=108.405pt,valign={c}]{Diagrams/Chapter5/img3_1.png}=\left(\tau_{\phi^{2}h}^{ct}\right)_{\mu\nu}(q)$
(5.72)
$\displaystyle=(G_{N}m)^{\frac{2}{d-2}}i\kappa\left(\alpha^{(1)}(d)\left(-q_{\mu}q_{\nu}+\eta_{\mu\nu}q^{2}\right)p_{1}\cdot
p_{2}+\beta^{(1)}_{1}(d)\frac{q^{2}}{2}p_{1\,\mu}p_{2\,\nu}\right)\ ,$

whose classical limit is

$\left(\tau_{\phi^{2}h}^{ct}\right)_{\mu\nu}(q)\simeq-(G_{N}m)^{\frac{2}{d-2}}i\kappa
m^{2}\left(\alpha^{(1)}(d)\left(q_{\mu}q_{\nu}+\eta_{\mu\nu}\vec{q}^{2}\right)+\beta^{(1)}_{1}(d)\frac{\vec{q}^{2}}{2}\delta_{\mu}^{0}\delta_{\nu}^{0}\right)\
.$ (5.73)

Another possible vertex is constituted by two photons and one graviton. It
arises since we must take into account the electromagnetic gauge symmetry into
the covariant derivative. Its contribution is then

$\displaystyle\includegraphics[width=108.405pt,valign={c}]{Diagrams/Chapter5/img3_2.png}=\left(\tau_{A^{2}h}^{ct}\right)_{\alpha\beta,\mu,\nu}(q)=$
$\displaystyle-(G_{N}m)^{\frac{2}{d-2}}Q^{2}i\kappa\Bigl{(}\alpha^{(1)}(d)(-q_{\alpha}q_{\beta}+\eta_{\alpha\beta}q^{2})2\eta_{\mu\nu}$
(5.74)
$\displaystyle+\beta^{(1)}(d)\frac{q^{2}}{2}(\eta_{\mu\alpha}\eta_{\nu\beta}+\eta_{\nu\alpha}\eta_{\mu\beta})\Bigr{)}\
.$

Now, we can use these counter-terms to compute the diagrams that will be
necessary for the renormalization process.

Following [18], the insertion of the non-minimal coupling at tree-level gives
the process in figure 5.9, and the associated contribution to the stress-
energy tensor is given by

$\frac{-i\kappa}{2}\,\sqrt{4E_{1}E_{2}}\,\delta^{(1)}T_{\mu\nu}^{(0)}(q^{2})=\left(\tau_{\phi^{2}h}^{ct}\right)_{\mu\nu}(q)\
.$ (5.75)

Figure 5.9: Insertion of non-minimal coupling at tree level.

Considering the relation between the stress-energy tensor and the metric
perturbation, we obtain the variations of the metric components

$\displaystyle\delta^{(1)}h_{0}^{(1)}(r)$ $\displaystyle=0$ (5.76)
$\displaystyle\delta^{(1)}h_{1}^{(1)}(r)$
$\displaystyle=\frac{16\alpha^{(1)}(d)\Gamma\left(\frac{d}{2}\right)}{\pi^{\frac{d-2}{2}}}\frac{(G_{N}m)^{\frac{d}{d-2}}}{r^{d}}$
$\displaystyle\delta^{(1)}h_{2}^{(1)}(r)$
$\displaystyle=\frac{-32\alpha^{(1)}(d)\Gamma\left(\frac{d+2}{2}\right)}{\pi^{\frac{d-2}{2}}}\frac{(G_{N}m)^{\frac{d}{d-2}}}{r^{d}}\
,$

where $\alpha^{(1)}(d)$ is defined in (5.70).

Figure 5.10: Insertion of non-minimal coupling in the 1-loop diagram for
graviton emission with 2 internal gravitons.

Then we can consider the insertion in the 1-loop diagram as in figure 5.10,
whose contribution to the stress-energy tensor is

$\displaystyle-\frac{i\,\kappa}{2}\,\sqrt{4E_{1}E_{2}}\,\delta^{(1)}T_{\mu\nu}^{(1)}(q^{2})=$
(5.77)
$\displaystyle=2\times\int\frac{d^{D}\ell}{(2\pi)^{D}}\frac{-i\,P_{\rho\sigma,\alpha\beta}P_{\eta\xi,\gamma\delta}\left(\tau_{\phi^{2}h}\right)_{\rho\sigma}\left(\tau_{\phi^{2}h}^{ct}\right)_{\eta\xi}(q-\ell)\left(\tau_{h^{2}h}\right){}^{\mu\nu}{}_{\alpha\beta,\gamma\delta}(l,q)}{((p_{1}-\ell)-m^{2}+i\epsilon)(\ell^{2}+i\epsilon)((q-\ell)^{2}+i\epsilon)}$

from which using the property in (B.5), we determine the variation of the
metric as

$\displaystyle\delta^{(1)}h_{0}^{(2)}(r)$
$\displaystyle=64\alpha^{(1)}(d)\frac{(d-2)\Gamma\left(\frac{d}{2}\right)^{2}}{(d-1)\pi^{d-2}}\left(\frac{\left(G_{N}m\right)^{\frac{1}{d-2}}}{r}\right)^{2(d-1)}$
(5.78) $\displaystyle\delta^{(1)}h_{1}^{(2)}(r)$
$\displaystyle=-64\alpha^{(1)}(d)\frac{(d-2)\Gamma\left(\frac{d}{2}\right)^{2}}{(d-1)\pi^{d-2}}\left(\frac{\left(G_{N}m\right)^{\frac{1}{d-2}}}{r}\right)^{2(d-1)}$
(5.79) $\displaystyle\delta^{(1)}h_{2}^{(2)}(r)$
$\displaystyle=128\alpha^{(1)}(d)\frac{(d-2)\Gamma\left(\frac{d}{2}\right)^{2}}{(d-1)\pi^{d-2}}\left(\frac{\left(G_{N}m\right)^{\frac{1}{d-2}}}{r}\right)^{2(d-1)}\
.$ (5.80)

In principle there exists another 1-loop insertion given by the amplitude in
figure 5.11.

Figure 5.11: Insertion of non-minimal coupling in the 1-loop diagram for
graviton emission with 2 internal photons.

However it turns out that the classical limit of the metric variation
associated to such process is vanishing, and then does not contribute to the
renormalization process of the classical observable.

####  5.3.3 Complete metric expression

Now that we have the variations of the metric associated to the insertion of
counter-terms in the amplitudes, we can use them to compute the complete
metric expression in which the divergences are canceled. In order to do that
we have to sum the right metric variation contribution to the regularized
metric by checking the power of the coupling constants. Finally we can fix the
parameter $\alpha^{(1)}(d)$, in which referred to eq. (5.70) do contribute
only specific allowed values of $k$, imposing that the complete summed metric
expression becomes finite.

#### Renormalization in $\bm{d=3}$

In four dimensions, the only values of $k$ that respect the constraints on the
power of the coupling constant are $k=0,1,2$. However, the case $k=2$ does not
have a match in the post-Minkowskian expansion and will not be considered. As
a consequence, eq. (5.70) becomes

$\alpha^{(1)}(3)=\alpha^{(1,0)}(3)+\frac{\alpha
G_{N}}{(G_{N}m)^{2}}\alpha^{(1,1)}(3)\,.$ (5.81)

In order to cure the divergences, $\alpha^{(1)}(3)$ must have the form

$\alpha^{(1)}(3)=\frac{\omega(3)}{d-3}+\Omega(3)\ ,$ (5.82)

where $\omega(3)$ is the coefficient that we need to fix to renormalize the
metric, while $\Omega(3)$ is a finite term. In $d=3$ the 2-loop corrections to
the metric are given by the diagram in figure 5.9 and, expanding the metric
contributions (5.76) in $d=3+\epsilon$, we find the renormalized metric
component [18, 25]

$\displaystyle h_{1}^{(d=3)}$
$\displaystyle\bigl{|}^{renorm}_{2-loop}=h_{1}^{(d=3)}\bigl{|}_{2-loop}+\delta^{(1)}h_{1}^{(1)}$
(5.83) $\displaystyle=\frac{8G_{N}^{3}m^{3}}{\epsilon
r^{3}}\left(\omega(3)-\frac{1}{12}+\frac{\alpha}{6G_{N}m^{2}}\right)-\frac{4G_{N}^{3}m^{3}}{3r^{3}}+\frac{8G_{N}^{3}m^{3}}{r^{3}}\omega(3)+\frac{8G_{N}^{3}m^{3}}{r^{3}}\Omega(3)$
$\displaystyle+\frac{2G_{N}^{3}m^{3}}{r^{3}}\log\left(2C_{E}r\right)-\frac{4\alpha
G_{N}^{2}m}{r^{3}}\log\left(2C_{E}r\right)-\frac{8G_{N}^{3}m^{3}}{r^{3}}\omega(3)\log\left(2C_{E}rG_{N}^{2}m^{2}\right)\
.$

Imposing the above expression to be finite, we can fix $\omega(3)$ in order to
cancel the divergent term in the metric component, obtaining for the divergent
contribution

$\omega(3)=\frac{1}{12}-\frac{\alpha G_{N}}{6(G_{N}m)^{2}}\ .$ (5.84)

In the above expression we recognize the term independent of the electric
charge to be equivalent to the one already found in [18], while the other is
the new piece due to electromagnetic interaction, which leads to

$\displaystyle h_{1}^{(d=3)}\bigl{|}^{renorm}_{2-loop}=$
$\displaystyle\frac{2G_{N}^{3}m^{3}}{3r^{3}}\left(2\log\left(\frac{2C_{E}r}{G_{N}m}\right)+12\,\Omega(3)-1\right)$
(5.85)
$\displaystyle-\frac{4G_{N}^{2}m\alpha}{3r^{3}}\left(1+2\log\left(\frac{2C_{E}r}{G_{N}m}\right)\right)\
.$

Now it is straightforward to see that this expression matches the metric in
(4.40) up to finite terms. In fact, although we cannot fix the finite part
$\Omega(3)$, since it belongs to the high-energy regime, we can keep trace of
it, and imposing a strict agreement between the renormalized metric and the
one computed classically, the condition

$\Omega(3)=2\,\omega(3)\log\left(\frac{c}{4C_{E}}\right)-\omega(3)+\frac{2}{3}\
$ (5.86)

must hold, which relates the freedom in the quantum computation with the gauge
freedom in the classical one.

#### Renormalization in $\bm{d=4}$

Referring to (5.69), in five dimensions the only allowed values of $k$ are
$k=0,2$. Then the expression in (5.70) reads

$\alpha^{(1)}(4)=\alpha^{(1,0)}(4)+\frac{\alpha
G_{N}}{(G_{N}m)^{2}}\alpha^{(1,2)}(4)\ .$ (5.87)

As we already did in the previous case, in order to cancel the divergences we
write

$\alpha^{(1)}(d)=\frac{\omega(4)}{d-4}+\Omega(4)\ ,$ (5.88)

where $\omega(4)$ is the coefficient we need to fix to renormalize the
divergences in $d=4$ and $\Omega(4)$ is a finite contribution. In
$d=4+\epsilon$ the metric is renormalized at 1 loop adding the contribution of
the diagram in figure 5.9. Then we can set the constant $\omega(4)$ in order
to cancel the divergence of the metric, whose value becomes

$\omega(4)=\frac{5}{18\pi}+\frac{G_{N}\alpha}{(G_{N}m)^{2}}\frac{1}{24\pi}\ ,$
(5.89)

and again we can match the purely gravitational result with [18]. As done for
$d=3$ we can obtain a perfect match with the classical expression 4.40 by
fixing the finite contribution $\Omega(4)$, finding

$\Omega(4)=\frac{\omega(4)}{2}\log\left(\frac{3\pi
c}{8C_{E}^{2}}\right)+\frac{\omega(4)}{6}-\frac{4}{27\pi}\ .$ (5.90)

Finally the 1-loop renormalized metric component with this choice of coupling
constants is [18, 24]

$\displaystyle
h_{1}^{(d=4)}\bigl{|}^{renorm}_{1-loop}=h_{1}^{(d=4)}\bigl{|}_{1-loop}$
$\displaystyle+\delta^{(1)}h_{1}^{(1)}=\frac{56G_{N}^{2}m^{2}}{27\pi^{2}r^{4}}-\frac{\alpha
G_{N}}{9\pi^{2}r^{4}}+\frac{16G_{N}^{2}m^{2}}{\pi r^{4}}\Omega(4)$ (5.91)
$\displaystyle+\frac{20G_{N}^{2}m^{2}}{9\pi^{2}r^{4}}\log\left(\frac{C_{E}^{2}r^{2}}{G_{N}m}\right)+\frac{\alpha
G_{N}}{3\pi^{2}r^{4}}\log\left(\frac{C_{E}^{2}r^{2}}{G_{N}m}\right)\ ,$

which perfectly match the corresponding order of eq. (4.40). For the 2-loop
computation the steps are the same as the above case, but using the 1-loop
diagram in figure 5.10. The computations are independent from the previous
case and lead to the same constants $\omega(4)$ and $\Omega(4)$ defined above.
At this order the renormalized metric component is found to be [18, 25]

$\displaystyle
h_{1}^{(d=4)}\bigl{|}^{renorm}_{2-loop}=h_{1}^{(d=4)}\bigl{|}_{2-loop}$
$\displaystyle+\delta^{(1)}h_{1}^{(2)}=-\frac{112G_{N}^{3}m^{3}}{81\pi^{3}r^{6}}+\frac{16\alpha
G_{N}^{2}m}{9\pi^{3}r^{6}}-\frac{64G_{N}^{3}m^{3}\Omega(4)}{3\pi^{2}r^{6}}$
(5.92)
$\displaystyle-\frac{80G_{N}^{3}m^{3}}{27\pi^{3}r^{6}}\log\left(\frac{C_{E}^{2}r^{2}}{G_{N}m}\right)-\frac{4\alpha
G_{N}^{2}m}{9\pi^{3}r^{6}}\log\left(\frac{C_{E}^{2}r^{2}}{G_{N}m}\right)\ ,$

which again shows a perfect agreement with the classical expression in eq.
(4.40).

Summarizing, we obtain complete renormalized expressions for the metric that
are in complete agreement with the classical calculation of chapter 4. We
showed that while the logarithmic terms coincide naturally, in order to get a
strict correspondence between the quantum and the classical derivation, which
involves also finite parts, one must impose a relation that links the freedom
in the quantum computation, parametrized by $\Omega(d)$, with the classical
gauge freedom contained in the parameter $c$, and this unique definition is
coherent with every independent object of the theory. Moreover it is important
to observe that in the quantum context it is possible to prove that the
divergences that we renormalized are the only ones that can appear,
reproducing the classical result of chapter 4. In fact for $\mathfrak{n}=1$,
$d=3$ and $d=4$ are the only dimensions in which the coupling constant
structures in eq. (5.69) have non-negative integer powers. This means that in
other dimensions it would not exist any counter-term that renormalizes the
observable, therefore cannot exist any divergence in those cases.

##  Chapter 6 Photon emission processes

In the previous chapters we reviewed the literature on the state-of-the-art
techniques for extracting classical physics from quantum gravity loop
amplitudes, which culminate with the reconstruction of the Reissner-Nordström-
Tangherlini metric from graviton emission processes. Since in chapter 4 it is
showed how the complete black hole solution requires also the knowledge of the
electromagnetic potential, we can ask ourselves if this can be recovered from
scattering amplitudes too.

In this chapter, where all the results have to be considered original, we will
first show how along the same picture of section 5.1, it is possible to
formalize a systematic procedure in order to obtain the electromagnetic
potential from photon emission scattering amplitudes. As in the metric case,
an explicit calculation of the potential shows the appearance of divergences
in five dimensions, which as in the metric case have to be renormalized. In
the end it is shown how the procedure to cancel the singularities leads to the
very same counter-terms of chapter 5, which underlines the extraordinary
coherence of the theory.

###  6.1 Classical limit of photon emission processes

Following the procedure outlined in section 5.1, considering the usual quantum
action defined in section 2.3, we want to solve the field equations

$\Box A_{\mu}(x)=j_{\mu}(x)$ (6.1)

of the electromagnetic potential in the Feynman gauge and space-time
dimensions $D=d+1$, perturbatively in the post-Minkowskian expansion, where
$j_{\mu}(x)$ is the electromagnetic source current. The idea then, is that the
photon field that appears in the above equation, is the quantum counterpart of
the classical observable we want to recover up to consider the classical limit
in which $\hbar\rightarrow 0$. Considering the post-Minkowskian expansion of
both the potential and the source, with the usual notation which employs the
indices $n$ and $j$, we can write

$j_{\mu}=\sum_{n=0}^{+\infty}\sum_{\begin{subarray}{c}j=0\\\ j\
\text{even}\end{subarray}}^{n}j_{\mu}^{(n,j)}\quad\text{and}\quad
A_{\mu}=\sum_{n=1}^{+\infty}\sum_{\begin{subarray}{c}j=0\\\ j\
\text{even}\end{subarray}}^{n}A_{\mu}^{(n,j)}\ ,$ (6.2)

and the field equations at each order become

$\Box A_{\mu}^{(n,j)}(x)=j_{\mu}^{(n-1,j)}(x)\ ,$ (6.3)

where again the $n$-th post-Minkowskian order of the potential is generated
from the $n-1$-th order of the source.

As usual we are interested in static solutions of eq. (6.3), from which
imposing such condition we obtain

$A_{\mu}^{(n,j)}(\vec{x})=\int\frac{d^{d}q}{(2\pi)^{d}}\frac{e^{i\vec{q}\cdot\vec{x}}}{\vec{q}^{2}}j_{\mu}^{(n-1,j)}(\vec{q}^{2})\
,$ (6.4)

where $j_{\mu}^{(n-1,j)}(\vec{q}^{2})$ is the Fourier transform of the
electromagnetic current that corresponds to the matrix element

$j_{\mu}(q^{2})=\bra{p_{2}}j_{\mu}(x=0)\ket{p_{1}}\ ,$ (6.5)

in which $p_{1}$ and $p_{2}$ are the momenta of two external on-shell massive
particles and where $q=p_{1}-p_{2}$ is the momentum transferred. So then, the
problem of calculating the electromagnetic potential is moved to the problem
of compute the matrix element of the source, and since we are interested in
classical phenomenology, we pick only those contributions we know will not
vanish in the classical limit, which are given exactly by the selection rules
of section 3.1. The outcome then, is that the classical contributions of
$j_{\mu}(\vec{q}^{2})$ are reconstructed by $l$-loop photon emission
amplitudes, in which a single massive line emits $n-j-1$ gravitons and $j+1$
photons, ending up in a tree internal interaction structure as

$\includegraphics[width=173.44534pt,valign={c}]{Diagrams/Chapter6/img1_1.png}=-i\,\sqrt{4E_{1}E_{2}}\,j_{\mu}{}^{(l,j)}(q^{2})\
,$ (6.6)

where the number of loops and the post-Minkowskian order are related through
the usual relation $n=l+1$. Now analogously to the metric case, since eq.
(6.4) is meant to reconstruct the classical electromagnetic potential, we are
interested only in those contributions of the electromagnetic source whose

$j_{\mu}(\vec{q}^{2})^{(l,j)}\propto J_{(l)}(\vec{q}^{2})\ ,$ (6.7)

from which using the Lorentz covariance and the fact that the current is
conserved, an expansion in terms of dimensionless form factors $c^{(l,j)}(d)$
leads to

$j_{\mu}^{(l)}(\vec{q}^{2})=\sum_{\begin{subarray}{c}j=0\\\ j\
\text{even}\end{subarray}}^{l+1}j_{\mu}^{(l,j)}(\vec{q}^{2})\simeq\sum_{\begin{subarray}{c}j=0\\\
j\
\text{even}\end{subarray}}^{l+1}Q\,\delta_{\mu}^{0}\,c^{(l,j)}(d)(G_{N}m)^{l-j}(\alpha
G_{N})^{\frac{j}{2}}J_{(l)}(\vec{q}^{2})\ .$ (6.8)

Then using the Fourier transform of the master integral in eq. (B.3), one
finally obtains

$A_{\mu}^{(l+1,j)}(r)=Q\,\delta_{\mu}^{0}c^{(l,j)}(d)(G_{N}m)^{l-j}(\alpha
G_{N})^{\frac{j}{2}}\left(\frac{\rho}{4\pi}\right)^{l+1}\ ,$ (6.9)

where we observe that only the temporal component of the potential is non-
vanishing. This analysis outlines the explicit relation between the classical
electromagnetic potential and the amplitude calculations, by which the form
factors111Notice that in the expansion in eq. (6.8) we miss a $\pi^{l}$ factor
which makes the form factors dependent by it. can be directly extracted.
However, with respect to the metric case, the above expressions are much
simpler to manage, therefore it is not needed to extract the form factors and
$j_{\mu}^{(l,j)}(\vec{q}^{2})$ can be directly computed. This means that the
form factor expansion in eq. (6.9) will be not used in actual computations,
nevertheless shows that the temporal component is the only non-vanishing one,
reproducing the classical result of chapter 4.

At this point we have to follow section 5.1 in order to extract the
contributions proportional to the master integral out of amplitudes like
(6.6). The procedure is exactly the same and it is not needed to repeat it.
Decouple the amplitude in a massive line emitting massless particles,
parametrized by $\mathcal{N}$, and in an internal tree structure that depends
on the specific process, parametrized by $\mathcal{M}$, then the classical
limit of the electromagnetic current reads

$\displaystyle
j_{\mu}^{(l,j)}(\vec{q}^{2})=\frac{i}{(l+1)!}\int\prod_{i=1}^{l}\frac{d^{d}\vec{\ell_{i}}}{(2\pi)^{d}}(-1)^{l+j}2^{j-l}m^{l-j}\kappa^{l-j}Q^{j+1}\frac{\prod_{i=1}^{l-j}P^{00,\lambda_{i}\sigma_{i}}}{\prod_{i=1}^{l+1}\vec{\ell_{i}}^{2}}$
(6.10)
$\displaystyle\times\mathcal{M}_{\lambda_{1}\sigma_{1},...,\lambda_{l-j+1}\sigma_{l-j+1},0,...,0,\mu}(p_{1},p_{2},\ell_{1},...,\ell_{l})\Bigl{|}_{\ell_{i}^{0}=0}\
.$

It is worth stressing again the fact that classical physics is recovered from
extracting out of the above expression only the contributions whose leading-
order in energy is proportional to the master integral, while all the other
pieces give rise to quantum corrections. This statement can be summarized by
rewriting eq. (6.10) in terms of quantum tree-graphs, adapting the formalism
developed by Duff in [17] to photon emission processes as

$-i\,2m\,j_{\mu}^{(l,j)}(\vec{q}^{2})\Big{|}_{\text{classical}}=\includegraphics[width=195.12767pt,valign={c}]{Diagrams/Chapter6/img1_2.png}\Bigg{|}_{\text{leading-
order}}\ .$ (6.11)

###  6.2 Potential from amplitude calculations

Now we can perform the actual computation of the photon emission process in
(6.6) up to 2-loop order, and explicitly recover the expression of the
electromagnetic potential by directly evaluating eq. (6.4). Then we will show
the appearance of divergences at 2 loops in $d=4$, which will be treated in
detail in the next section. As already noticed previously, only the time
component of the gauge potential is non-vanishing, so in the following it will
be the only component taken into account.

#### Tree level

As for the metric case, we start the analysis at tree level for completeness.
In the notation of section 6.1, the tree-level diagram corresponds to the term
$l=0,\ j=0$ in the expansion of eq. (6.6).

Figure 6.1: Tree-level diagram for photon emission.

Evaluating the matrix element of the current due to the diagram in figure 6.1,
and using the Feynman rules in section A, one obtains

$-i\,\sqrt{4E_{1}E_{2}}\,j_{\nu}^{(0,0)}(\vec{q}^{2})=\left(\tau_{\phi^{2}A}\right)_{\nu}\
.$ (6.12)

Considering the classical limit, the tree-level electromagnetic current is
simply the charge,

$j_{0}^{(0,0)}(\vec{q}^{2})=Q\ ,$ (6.13)

and replacing it inside eq. (6.4) and using the Fourier transform identities
of the master integral in appendix B, we get

$A_{0}^{(1,0)}(r)=\frac{Q}{4\pi}\rho\ ,$ (6.14)

which is exactly the first order of the expression in (4.45).

#### 1-loop

At this loop order we can only have $j=0$. So the 1-loop process is given by
the amplitude in figure 6.2.

Figure 6.2: 1-loop diagram for photon emission.

The electromagnetic current associated to this process reads

$-i\,\sqrt{4E_{1}E_{2}}\,j_{\nu}^{(1,0)}(q^{2})=2\times\int\frac{d^{d+1}\ell}{(2\pi)^{d+1}}\frac{i\,P_{\alpha\beta,\sigma\rho}\left(\tau_{\phi^{2}A}\right)^{\mu}\left(\tau_{\phi^{2}h}\right)^{\alpha\beta}\left(\tau_{A^{2}h}\right)^{\sigma\rho}{}_{\mu,\nu}\left(\ell,q\right)}{\left((p_{1}-\ell)^{2}-m^{2}+i\epsilon\right)\left(\ell^{2}+i\epsilon\right)\left((q-\ell)^{2}+i\epsilon\right)}\
,$ (6.15)

where a factor $"2"$ is implied in order to consider the multiplicity of the
diagram. Performing the classical limit to obtain an expression like eq.
(6.10), one gets

$j_{0}^{(1,0)}(\vec{q}^{2})=-\frac{i}{2}\kappa
Qm\int\frac{d^{d}\ell}{(2\pi)^{d}}\frac{P_{00,\alpha\beta}\left(\tau_{A^{2}h}\right)^{\alpha\beta}{}_{0,0}\left(\ell,q\right)\big{|}_{\ell^{0}=0}}{\vec{\ell}^{2}(\vec{q}-\vec{\ell})^{2}}\
,$ (6.16)

from which considering the tensorial contraction

$P_{00,\alpha\beta}\left(\tau_{A^{2}h}\right)^{\alpha\beta}{}_{0,0}\left(\ell,q\right)\big{|}_{\ell^{0}=0}=-i\kappa\frac{d-2}{d-1}\vec{\ell}\cdot\vec{q}\
,$ (6.17)

we can express the current in terms of the master integral through the
identities in appendix C. In particular using (C.2) one end up with the
compact expression

$j_{0}^{(1,0)}(\vec{q}^{2})=-\frac{1}{4}mQ\kappa^{2}\frac{d-2}{d-1}J_{(1)}(\vec{q}^{2})\
,$ (6.18)

recovering the structure of eq. (6.8). Replacing the above expression inside
the (6.4), exploiting the property (B.3), one obtains

$A_{0}^{(2,0)}(r)=-\frac{Q}{4\pi}\frac{2(d-2)}{d-1}mG_{N}\rho^{2}\ ,$ (6.19)

which shows, at this order, a perfect agreement with the expression (4.45).

#### 2-loop

At 2 loops we can have either $j=0$ or $j=2$. We first discuss the case $j=0$,
in which contribute three different diagrams that differ by the internal tree
structure. As we already did in the metric case, after computing the current
for each process we sum all the contributions in order to get the complete
electromagnetic potential. The first amplitude we consider is the one in
figure 6.3, which has 2 internal photons and 2 internal gravitons.

Figure 6.3: 2-loop diagram for photon emission with 2 internal photons and 2
internal gravitons.

The associated electromagnetic current to this process is then

$-i\,\sqrt{4E_{1}E_{2}}\,j_{\nu}^{(2,0)(\text{a})}(q^{2})=6\times\int\frac{d^{d+1}\ell_{1}}{(2\pi)^{d+1}}\frac{d^{d+1}\ell_{2}}{(2\pi)^{d+1}}\frac{-P_{\sigma\rho,\alpha\beta}P_{\gamma\delta,\eta\chi}\left(\tau_{\phi^{2}A}\right)_{\lambda}\left(\tau_{\phi^{2}h}\right)^{\eta\chi}\left(\tau_{\phi^{2}h}\right)^{\sigma\rho}}{\left(\ell_{1}^{2}+i\epsilon\right)\left(\ell_{2}^{2}+i\epsilon\right)\left((q-\ell_{1}-\ell_{2})^{2}+i\epsilon\right)}\\\
\times\frac{\left(\tau_{A^{2}h}\right)^{\gamma\delta,\mu,\lambda}(\ell_{1},\ell_{1}+\ell_{2})\left(\tau_{A^{2}h}\right)^{\alpha\beta}{}_{\mu,\nu}(\ell_{1}+\ell_{2},q)}{\left((\ell_{1}+\ell_{2})^{2}+i\epsilon\right)((p_{1}-\ell_{1})^{2}-m^{2}+i\epsilon)((p_{1}-\ell_{1}-\ell_{2})^{2}-m^{2}+i\epsilon)}\
.$ (6.20)

Exploiting the procedure outlined in section 6.1, one gets

$\displaystyle j_{0}^{(2,0)(\text{a})}(\vec{q}^{2})$
$\displaystyle=-\frac{\kappa^{2}Qm^{2}}{4}\int\frac{d^{d}\ell_{1}}{(2\pi)^{d}}\frac{d^{d}\ell_{2}}{(2\pi)^{d}}\frac{P^{\alpha\beta,00}P^{\sigma\rho,00}}{\vec{\ell_{1}}^{2}\vec{\ell_{2}}^{2}(\vec{q}-\vec{\ell_{1}}-\vec{\ell_{2}})^{2}(\vec{\ell_{1}}+\vec{\ell_{2}})^{2}}$
(6.21)
$\displaystyle\times(\tau_{A^{2}h})_{0\mu,\alpha\beta}(\ell_{1},\ell_{1}+\ell_{2})(\tau_{A^{2}h})^{0\mu}{}_{\sigma\rho}(\ell_{1}+\ell_{2},q)\bigl{|}_{\ell_{i}^{0}=0}\
,$

where the tensor contraction gives

$\displaystyle P^{\alpha\beta,00}$ $\displaystyle
P^{\sigma\rho,00}(\tau_{A^{2}h})_{0,\mu,\alpha\beta}(\ell_{1},\ell_{1}+\ell_{2})(\tau_{A^{2}h})^{0,\mu}{}_{\sigma\rho}(\ell_{1}+\ell_{2}q)\bigl{|}_{\ell_{i}^{0}=0}$
(6.22)
$\displaystyle=-\kappa^{2}\frac{(d-2)^{2}}{(d-1)^{2}}(\vec{\ell_{1}}+\vec{\ell_{2}})\cdot\vec{q}\,(\vec{\ell_{1}}+\vec{\ell_{2}})\cdot\vec{\ell_{1}}\
.$

Then using the reduction identities in appendix C, one gets the current in
terms of the master integral as

$j_{0}^{(2,0)(\text{a})}(\vec{q}^{2})=\frac{\kappa^{4}Qm^{2}}{12}\frac{(d-2)^{2}}{(d-1)^{2}}J_{(2)}(\vec{q}^{2})\
,$ (6.23)

from which, due to the usual relations of the Fourier transform of the master
integral, one recovers

$A_{0}^{(3,0)(\text{a})}(r)=\frac{Q}{4\pi}m^{2}G_{N}^{2}\frac{16}{3}\frac{(d-2)^{2}}{(d-1)^{2}}\rho^{3}\
.$ (6.24)

The second amplitude we consider is the one in figure 6.4, which has 1
internal photon and 2 internal gravitons.

Figure 6.4: 2-loop diagram for photon emission with 1 internal photon and 2
internal gravitons.

The electromagnetic current associated to the diagram is

$\displaystyle-i\,\sqrt{4E_{1}E_{2}}$ $\displaystyle
j_{\nu}^{(2,0)(\text{b})}(q^{2})=3\times\int\frac{d^{d+1}\ell_{1}}{(2\pi)^{d+1}}\frac{d^{d+1}\ell_{2}}{(2\pi)^{d+1}}\frac{-iP_{\sigma\rho,\alpha\beta}P_{\gamma\delta,\eta\chi}\left(\tau_{\phi^{2}A}\right)^{\mu}\left(\tau_{\phi^{2}h}\right)^{\eta\chi}\left(\tau_{\phi^{2}h}\right)^{\sigma\rho}}{\left(\ell_{1}^{2}+i\epsilon\right)\left(\ell_{2}^{2}+i\epsilon\right)\left((q-\ell_{1}-\ell_{2})^{2}+i\epsilon\right)}$
(6.25)
$\displaystyle\times\frac{\left(\tau_{A^{2}h^{2}}\right)^{\alpha\beta,\gamma\delta}{}_{\mu,\nu}(\ell_{1},q)}{((p_{1}-\ell_{1})^{2}-m^{2}+i\epsilon)((p_{1}-\ell_{1}-\ell_{2})^{2}-m^{2}+i\epsilon)}\
.$

Performing the classical limit, one gets

$j_{0}^{(2,0)(\text{b})}(\vec{q}^{2})=\frac{i}{8}\kappa^{2}m^{2}Q\int\frac{d^{d}\ell_{1}}{(2\pi)^{d}}\frac{d^{d}\ell_{2}}{(2\pi)^{d}}\frac{P_{00,\mu\nu}P_{00,\alpha\beta}(\tau_{A^{2}h^{2}})^{0,0,\mu\nu,\alpha\beta}(\ell_{1},q)\bigl{|}_{\ell_{i}^{0}=0}}{\vec{\ell_{1}}^{2}\vec{\ell_{2}}^{2}\left(\vec{q}-\vec{\ell_{1}}-\vec{\ell_{2}}\right)^{2}}\
,$ (6.26)

where the tensorial contraction of the numerator leads to

$P_{00,\mu\nu}P_{00,\alpha\beta}(\tau_{A^{2}h^{2}})^{0,0,\mu\nu,\alpha\beta}(\ell_{1},q)\bigl{|}_{\ell_{i}^{0}=0}=i\frac{\kappa^{2}}{2}\frac{(d-2)(3d-7)}{(d-1)^{2}}\vec{\ell_{1}}\cdot\vec{q}\
.$ (6.27)

Then using the relation (C.5), the current expressed in terms of the master
integral reads

$j_{0}^{(2,0)(\text{b})}(\vec{q}^{2})=-\frac{m^{2}Q\kappa^{4}}{48}\frac{(d-2)(3d-7)}{(d-1)^{2}}J_{(2)}(\vec{q}^{2})\
,$ (6.28)

from which considering the usual Fourier transform one gets

$A_{0}^{(3,0)(\text{b})}(r)=-\frac{Q}{4\pi}m^{2}G_{N}^{2}\frac{4}{3}\frac{(d-2)(3d-7)}{(d-1)^{2}}\rho^{3}\
.$ (6.29)

The last diagram that contributes to the process with $l=2,\ j=0$ is the one
with an internal 3 gravitons vertex, as shown in figure 6.5.

Figure 6.5: 2-loop diagram for photon emission with 1 internal photon and 3
internal gravitons.

Since the 3 gravitons vertex is internal, we must use the expression given in
(A.9). Then the current associated to this diagram is

$\displaystyle-i\,\sqrt{4E_{1}E_{2}}\,j_{\nu}^{(2,0)(\text{c})}(q^{2})=3\times\int\frac{d^{d+1}\ell_{1}}{(2\pi)^{d+1}}\frac{d^{d+1}\ell_{2}}{(2\pi)^{d+1}}\frac{P_{\alpha_{1}\beta_{1},\gamma\delta}P_{\alpha_{2}\beta_{2},\eta\chi}P_{\alpha_{3}\beta_{3},\sigma\rho}\left(\tau_{\phi^{2}A}\right)^{\mu}}{\left(\ell_{1}^{2}+i\epsilon\right)\left(\ell_{2}^{2}+i\epsilon\right)\left((q-\ell_{1}-\ell_{2})^{2}+i\epsilon\right)}$
(6.30)
$\displaystyle\times\frac{\left(\tau_{\phi^{2}h}\right)^{\gamma\delta}\left(\tau_{\phi^{2}h}\right)^{\eta\chi}\left(\tau_{h^{3}}\right)^{\alpha_{1}\beta_{1},\alpha_{2}\beta_{2},\alpha_{3}\beta_{3}}(\ell_{1},\ell_{2},-\ell_{1}-\ell_{2})\left(\tau_{A^{2}h}\right)^{\sigma\rho}{}_{\mu,\nu}(q-\ell_{1}-\ell_{2},q)}{\left((\ell_{1}+\ell_{2})^{2}+i\epsilon\right)((p_{1}-\ell_{1})^{2}-m^{2}+i\epsilon)((p_{1}-\ell_{1}-\ell_{2})^{2}-m^{2}+i\epsilon)}\
.$

Referring to eq. (6.10), one gets

$\displaystyle j_{0}^{(2,0)(\text{c})}(\vec{q}^{2})$
$\displaystyle=\frac{1}{8}\kappa^{2}Qm^{2}\int\frac{d^{d}\ell_{1}}{(2\pi)^{d}}\frac{d^{d}\ell_{2}}{(2\pi)^{d}}\frac{P_{00,\mu\nu}P_{00,\alpha\beta}P_{\gamma\delta,\sigma\rho}}{\vec{\ell_{1}}^{2}\vec{\ell_{2}}^{2}(\vec{q}-\vec{\ell_{1}}-\vec{\ell_{2}})^{2}(\vec{\ell_{1}}+\vec{\ell_{2}})^{2}}$
(6.31)
$\displaystyle\times\left(\tau_{h^{3}}\right)^{\mu\nu,\alpha\beta,\gamma\delta}(\ell_{1},\ell_{2},-\ell_{1}-\ell_{2})\left(\tau_{A^{2}h}\right)^{\sigma\rho}{}_{0,0}(q-\ell_{1}-\ell_{2},q)\bigl{|}_{\ell_{i}^{0}=0}\
,$

where considering from the beginning the fact that all the expressions that
contain internal momenta are symmetric under the exchange of
$\ell_{1}\leftrightarrow\ell_{2}$, the tensor contraction at the numerator
reads

$\displaystyle
P_{\gamma\delta,\sigma\rho}P_{\mu\nu,00}P_{\alpha\beta,00}\left(\tau_{h^{3}}\right)^{\mu\nu,\alpha\beta,\gamma\delta}(\ell_{1},\ell_{2},-\ell_{1}-\ell_{2})\left(\tau_{A^{2}h}\right)^{\sigma\rho}{}_{0,0}(q-\ell_{1}-\ell_{2},q)\bigl{|}_{\ell_{i}^{0}=0}$
(6.32)
$\displaystyle=-\frac{\kappa^{2}}{4}\frac{d-2}{(d-1)^{2}}\Biggl{(}4(d-1)\vec{\ell_{1}}\cdot(\vec{q}-\vec{\ell_{1}}-\vec{\ell_{2}})\
\vec{\ell_{1}}\cdot\vec{q}+2(d-1)\vec{\ell_{1}}\cdot(\vec{q}-\vec{\ell_{1}}-\vec{\ell_{2}})\
\vec{\ell_{2}}\cdot\vec{q}$
$\displaystyle+(d-5)\vec{q}\cdot(\vec{q}-\vec{\ell_{1}}-\vec{\ell_{2}})\
\vec{\ell_{1}}\cdot\vec{\ell_{2}}\Biggl{)}\ .$

Exploiting the expressions in appendix C, the current finally reads

$j_{0}^{(2,0)(\text{c})}(\vec{q}^{2})=\frac{\kappa^{4}m^{2}Q}{96}\frac{(d-2)(34-29d+7d^{2})}{(d-4)(d-1)^{2}}J_{(2)}(\vec{q}^{2})\
,$ (6.33)

from which we obtain

$A_{0}^{(3,0)(\text{c})}(r)=\frac{Q}{4\pi}m^{2}G_{N}^{2}\frac{2}{3}\frac{(d-2)(34-29d+7d^{2})}{(d-4)(d-1)^{2}}\rho^{3}\
.$ (6.34)

At the end we can sum up all the $j=0$ pieces to get

$A_{0}^{(3,0)}(r)=\sum_{i=\text{a},\text{b},\text{c}}A_{0}^{(3,0)(i)}(r)=\frac{Q}{4\pi}m^{2}G_{N}^{2}\frac{2(d-2)^{2}(3d-7)}{(d-1)^{2}(d-4)}\rho^{3}\
,$ (6.35)

which is exactly the contribution that appears in (4.45). In this last
expression a divergence in $d=4$ arises, and as we did in the metric case it
has to be renormalized.

Finally, the only process that contributes at $l=2,\ j=2$ is the one which has
3 internal photons and 1 internal graviton, shown in figure 6.6.

Figure 6.6: 2-loop diagram for photon emission with 3 internal photons and 1
internal graviton.

The electromagnetic current associated to this process is then

$\displaystyle-i\,\sqrt{4E_{1}E_{2}}$ $\displaystyle
j_{\nu}^{(2,2)}(q^{2})=3\times\int\frac{d^{d+1}\ell_{1}}{(2\pi)^{d+1}}\frac{d^{d+1}\ell_{2}}{(2\pi)^{d+1}}\frac{P_{\gamma\delta,\sigma\alpha}\left(\tau_{\phi^{2}A}\right)^{\beta}\left(\tau_{\phi^{2}A}\right)^{\sigma}\left(\tau_{\phi^{2}A}\right)^{\rho}}{\left(\ell_{1}^{2}+i\epsilon\right)\left(\ell_{2}^{2}+i\epsilon\right)\left((q-\ell_{1}-\ell_{2})^{2}+i\epsilon\right)}$
(6.36)
$\displaystyle\times\frac{\left(\tau_{A^{2}h}\right)^{\sigma\alpha}{}_{\beta,\sigma}(\ell_{1},-\ell_{2})\left(\tau_{A^{2}h}\right)^{\gamma\delta}{}_{\rho,\nu}(q-\ell_{1}-\ell_{2},q)}{\left((\ell_{1}+\ell_{2})^{2}+i\epsilon\right)((p_{1}-\ell_{1})^{2}-m^{2}+i\epsilon)((p_{1}-\ell_{1}-\ell_{2})^{2}-m^{2}+i\epsilon)}\
,$

which in the classical regime reads

$j_{0}^{(2,2)}(\vec{q}^{2})=\frac{1}{2}Q^{3}\int\frac{d^{d}\ell_{1}}{(2\pi)^{d}}\frac{d^{d}\ell_{2}}{(2\pi)^{d}}\frac{P_{\mu\nu,\alpha\beta}\left(\tau_{A^{2}h}\right)^{\mu\nu}{}_{00}(\ell_{1},-\ell_{2})\left(\tau_{A^{2}h}\right)^{\alpha\beta}{}_{00}(q-\ell_{1}-\ell_{2},q)\bigl{|}_{\ell_{i}^{0}=0}}{\vec{\ell_{1}}^{2}\vec{\ell_{2}}^{2}(\vec{q}-\vec{\ell_{1}}-\vec{\ell_{2}})^{2}(\vec{\ell_{1}}+\vec{\ell_{2}})^{2}}\
.$ (6.37)

Computing the tensor contraction in the numerator as

$\displaystyle P_{\mu\nu,\alpha\beta}$
$\displaystyle\left(\tau_{A^{2}h}\right)^{\mu\nu}{}_{0,0}(\ell_{1},-\ell_{2})\left(\tau_{A^{2}h}\right)^{\alpha\beta}{}_{0,0}(q-\ell_{1}-\ell_{2},q)\bigl{|}_{\ell_{i}^{0}=0}$
(6.38)
$\displaystyle=\frac{\kappa^{2}}{2}\left(\frac{d-3}{d-1}\vec{\ell_{1}}\cdot\vec{\ell_{2}}\
\vec{q}\cdot(\vec{q}-\vec{\ell_{1}}-\vec{\ell_{2}})+2\
\vec{q}\cdot\vec{\ell_{2}}\
\vec{\ell_{1}}\cdot(\vec{q}-\vec{\ell_{1}}-\vec{\ell_{2}})\right)\ ,$

in which again we exploited the symmetry $\ell_{1}\leftrightarrow\ell_{2}$,
and using the expressions in appendix C, we obtain

$j_{0}^{(2,2)}(\vec{q}^{2})=\frac{Q^{3}\kappa^{2}}{8}\frac{(d-3)(d-2)}{(d-4)(d-1)}J_{(2)}(\vec{q}^{2})\
.$ (6.39)

From this we get the contribution to the electromagnetic potential

$A_{0}^{(3,2)}(r)=\frac{Q}{4\pi}\alpha
G_{N}\frac{(d-3)(d-2)}{(d-4)(d-1)}\rho^{3}\ .$ (6.40)

This last expression matches perfectly the corresponding term of eq. (4.45),
and as before we notice a divergence for $d=4$ which must be renormalized.

To summarize, we have shown that from the loop computations one recovers
exactly the expression for the electromagnetic potential of the Reissner-
Nordström-Tangherlini solution in de Donder gauge given in eq. (4.45). However
a renormalization procedure has to be discussed since at third post-
Minkowskian order appear a divergence in $d=4$, as expected from the analysis
in chapter 4.

###  6.3 Renormalization of the potential

In this section we renormalize the electromagnetic potential following the
strategy already outlined in section 5.3. First we regularize the potential by
isolating the pole, and then, considering the non-minimal action defined in
eq. (5.69), we compute the potential variation that arises by the insertion of
counter-terms in photon emission amplitudes. In the end we sum the regularized
potential with its variation in order to obtain the complete renormalized
observable, which will be compared with the one in eq. (4.46).

In the present case the only divergence that appears is for $d=4$ at 2-loop
order. Then we just need to regularize the expression given by

$A_{0}\big{|}_{\text{2-loop}}=A_{0}^{(3,0)}+A_{0}^{(3,2)}\ ,$ (6.41)

from which expanding it around $d=4+\epsilon$, neglecting terms that vanish
for $\epsilon\rightarrow 0$, we obtain

$\displaystyle A_{0}^{(d=4)}\bigl{|}_{2-\text{loop}}$
$\displaystyle=\frac{1}{\epsilon}\left(\frac{10}{9}\frac{Q(G_{N}m)^{2}}{\pi^{4}r^{6}}+\frac{1}{6}\frac{QG_{N}\alpha}{\pi^{4}r^{6}}\right)-\frac{1}{27}\frac{Q(G_{N}m)^{2}}{\pi^{4}r^{6}}\left(-28+45\log(C_{E}^{2}r^{2})\right)$
(6.42)
$\displaystyle-\frac{1}{36}\frac{QG_{N}\alpha}{\pi^{4}r^{6}}\left(-7+9\log(C_{E}^{2}r^{2})\right)\
.$

Now we have to consider the potential variation that arises from the insertion
of the counter-terms vertices reported in eqs. (5.72) and (5.74) into photon
emission amplitudes. At tree-level no insertion can be performed since does
not exists a 2 scalars - 1 photon counter-term vertex. This means that the
first amplitude in which we can insert the counter-term is at 1-loop order,
leading to the fact that no divergences can arise at that order. So then, the
fact that eq. (6.19) is perfectly finite it is not a coincidence, but derives
from the consideration by which there would be no counter-term to cancel the
singularity, and this protects the 1-loop contribution of the potential to be
well-defined. Moreover from the same argument can be derived that we expect
divergences in $d=3$ only starting from the 3-loop order. For what concerns
the insertion in the 1-loop amplitude, the first process to take into account
is the one in figure 6.7, whose contribution to the electromagnetic current is

$\displaystyle-i\,\sqrt{4E_{1}E_{2}}\,\delta^{(1)}j_{\nu}^{(1)}(q^{2})$ (6.43)
$\displaystyle=2\times\int\frac{d^{d+1}\ell}{(2\pi)^{d+1}}\frac{iP_{\alpha\beta\gamma\delta}\
\left(\tau_{\phi^{2}A}\right)^{\mu}\left(\tau_{\phi^{2}h}^{ct}\right)^{\gamma\delta}\left(q-\ell\right)\left(\tau_{A^{2}h}\right)^{\alpha\beta}{}_{\mu\nu}(\ell,q)}{(\left(p_{1}-\ell\right)^{2}-m^{2}+i\epsilon)(\ell^{2}+i\epsilon)(\left(q-\ell\right)^{2}+i\epsilon)}\
.$

Figure 6.7: Insertion of an internal counter-term vertex at 1-loop with an
external photon.

Considering the classical limit we get

$-i\,2m\,\delta^{(1)}j_{0}^{(1)}(\vec{q}^{2})=-iQ\int\frac{d^{d}\ell}{(2\pi)^{d}}\frac{P_{\mu\nu,\alpha\beta}\left(\tau_{\phi^{2}h}^{ct}\right)^{\mu\nu}\left(q-\ell\right)\left(\tau_{A^{2}h}\right)^{\alpha\beta}{}_{00}(\ell,q)\bigl{|}_{\ell^{0}=0}}{\vec{\ell}^{2}(\vec{q}-\vec{\ell})^{2}}\
,$ (6.44)

from which computing

$\displaystyle P_{\mu\nu,\alpha\beta}$
$\displaystyle\left(\tau_{\phi^{2}h}^{ct}\right)^{\mu\nu}\left(q-\ell\right)\left(\tau_{A^{2}h}\right)^{\alpha\beta}{}_{00}(\ell,q)\bigl{|}_{\ell^{0}=0}$
(6.45)
$\displaystyle=-\frac{\kappa^{2}}{2(d-1)}(G_{N}m)^{\frac{2}{d-2}}\Bigl{(}\beta^{(1)}(d)(d-2)(\vec{\ell}^{2}\,\vec{\ell}\cdot\vec{q}-2\vec{\ell}\cdot\vec{q}\,\vec{\ell}\cdot\vec{q}+\vec{\ell}\cdot\vec{q}\,\vec{q}^{2})$
$\displaystyle-2\alpha^{(1)}(d)m^{2}\vec{\ell}\cdot\vec{q}(\vec{q}^{2}+\vec{\ell}^{2})-2\alpha^{(1)}(d)(d-3)m^{2}\vec{\ell}\cdot\vec{q}\,\vec{\ell}\cdot\vec{q}\Bigl{)}\
,$

and exploiting eqs. (C.3) and (C.4), one finally obtains the contribution to
the electromagnetic current

$\delta^{(1)}j_{0}^{(1)}(\vec{q}^{2})=\frac{1}{4}Q\kappa^{2}m\alpha^{(1)}(d)(G_{N}m)^{\frac{2}{d-2}}\vec{q}^{2}J_{(1)}(\vec{q}^{2})\
.$ (6.46)

Then using eq. (B.5), we obtain the variation to the electromagnetic potential
as

$\delta^{(1)}A_{0}^{(2)}(r)=-4G_{N}m(G_{N}m)^{\frac{2}{d-2}}Q\alpha^{(1)}(d)\frac{\Gamma\left(\frac{d}{2}\right)^{2}}{\pi^{d-1}r^{2(d-1)}}\
.$ (6.47)

Furthermore, likely to the metric case, there should be another counter-term
insertion given by the vertex in eq. (5.74), drawn in figure 6.8.

Figure 6.8: Insertion of an external counter-term vertex at 1-loop with an
external photon.

However also in this case an explicit calculation shows that the classical
limit of the electromagnetic potential associated to such process is
vanishing. We conclude then, that up to 2-loop order, the pure massless
counter-term (5.74) never contributes to the renormalization of the theory. It
is interesting to observe that there is no known reason by which this should
happen, so then cannot be excluded the possibility for it to be necessary for
higher loop-order calculations.

In the end it is possible to obtain the complete renormalized potential by
considering the addition of eq. (6.47), which must be expanded around
$d=4+\epsilon$, to the regularized expression in (6.42). One of the most
important outcome is that the counter-term that is needed to be fixed in order
to obtain finite results, is the same as the one already encountered in
section 5.3. In fact, reporting here for clarity

$\alpha^{(1)}(d)=\frac{\omega(4)}{d-4}+\Omega(4)\ ,$ (6.48)

where

$\omega(4)=\frac{5}{18\pi}+\frac{G_{N}\alpha}{(G_{N}m)^{2}}\frac{1}{24\pi}\ ,$
(6.49)

the 2-loop renormalized potential reads

$A_{0}^{(d=4)}\bigl{|}^{renorm}_{2-loop}=A_{0}^{(d=4)}\bigl{|}_{2-loop}+\delta^{(1)}A_{0}^{(2)}=\frac{Q}{4\pi}\Biggl{(}-\frac{8}{27}\frac{(G_{N}m)^{2}}{\pi^{3}r^{6}}+\frac{1}{9}\frac{G_{N}\alpha}{\pi^{3}r^{6}}\\\
-\frac{16(G_{N}m)^{2}}{\pi^{2}r^{6}}\Omega(4)+\frac{20}{9}\frac{(G_{N}m)^{2}}{\pi^{3}r^{6}}\log\left(\frac{G_{N}m}{C_{E}^{2}r^{2}}\right)+\frac{1}{3}\frac{G_{N}\alpha}{\pi^{3}r^{6}}\log\left(\frac{G_{N}m}{C_{E}^{2}r^{2}}\right)\Biggl{)}\
.$ (6.50)

Finally using the relation (5.90) the renormalized electromagnetic potential
perfectly matches the classical calculation in eq. (4.46).

In conclusion, we proved that from photon emission scattering amplitudes, in
which both photons and gravitons circulate in the loops, it is possible to
recover the electromagnetic potential associated to the Reissner-Nordström-
Tangherlini solution. We showed the divergences that arise in $d=4$ are
renormalized by the same counter-terms already employed in the metric case,
and that such procedure gives rise to logarithmic terms that perfectly
reproduce the classical solution.

##  Chapter 7 Conclusions

In this thesis we have shown that the Reissner-Nordström-Tangherlini solution,
describing the metric and the potential generated by a static and spherically
symmetric object of charge $Q$ and mass $m$ in any dimension, can be derived
from scattering amplitudes describing the emission of either a graviton or a
photon from a scalar field with the same charge and mass. This generalizes the
work of [18] in which the chargeless case was considered up to 3-loop order.
The present work extends the original papers [8, 9] where amplitude
computations were applied to derive the metric of all black hole solutions in
four dimensions up to second post-Minkowskian order.

In chapter 2 we reconstructed the scalar electrodynamics action by gauging the
global $\text{U}(1)$ symmetry, and we discussed the Faddeev-Popov procedure to
quantize it. Then we applied the same considerations in order to get general
relativity by gauging the global translational symmetry, recovering the
Einstein-Hilbert action as the kinetic term of such theory. Moreover we
concluded that since the general coordinate transformations are intrinsically
non-abelian, general relativity can be considered a non-abelian gauge theory.
Following the Faddeev-Popov procedure an attempt to quantize general
relativity has been made, but since the canonical mass dimension of the
gravitational coupling constant is negative for all the dimensions greater
than two, we recover the well-known result by which such theory is non-
renormalizable. Then, we applied the tools of effective field theories in
order to rewrite such theory by means of an infinite series of all the
possible energy ordered operators compatible with the symmetry of the original
action. Finally in section 2.4 we showed how systematically restore the
$\hbar$’s in the theory with the idea to make the classical limit in which
$\hbar\rightarrow 0$ more simple. An important outcomes is that the effective
theory allows us to separate the effects of massive and massless particles,
which contribute to the classical limit in different ways.

In chapter 3 we made an explicit example of a classical gravitational
observable recovered from the classical limit of quantum computations.
Reviewing the original work of [7, 8, 9], we showed how to compute static
metrics in four space-time dimensions from graviton emission scattering
amplitudes. Starting with the Schwarzschild metric, considering the tree-level
graviton emission from a scalar particle, in eq. (3.15) it is recovered the
first post-Minkowskian order of the metric in the de Donder gauge. In fact,
since in section 2.3 we chose to employ the de Donder gauge for the
gravitational sector (since the graviton field equations acquire a simple
expression), then all the quantum computations we have made in this thesis are
expressed in this gauge. For the 1-loop case the number of processes to take
into account proliferate, and it is necessary to state some selection rules
that pick only those contributions which are non-vanishing in the classical
limit. These rules then lead to eq. (3.20), and the comparison with the
classical Schwarzschild metric is done in section 3.2, in which a procedure to
convert the de Donder gauge in the harmonic one, which is more used in
literature, has been provided. Finally a perfect match between the two
computations is found. In the end we consider another static metric, namely
the Reissner-Nordström one, which describes a static electrically charged
black hole solution. In order to recover such metric, it is shown how the
addition of photons in the graviton emission loop amplitude is needed, and
from the 1-loop computation it is recovered the term proportional to the
electric charge of the metric in the harmonic gauge in eq. (3.43).

Then the attention is focused to recover the Reissner-Nordström-Tangherlini
solution from scattering amplitudes, and as a preliminary step, since we are
interested in compare classical and quantum derivations, in chapter 4 we
reviewed the classical solution provided in [20]. In particular, following the
strategy outlined in [18], we recovered the classical solution in the de
Donder gauge in terms of a post-Minkowskian expansion of both the metric and
the electromagnetic potential associated to the black hole. The core of the
procedure was find the coordinate change function in eq. (4.26), that
completely determines the reference frame transformation from spherical
coordinates to de Donder coordinates. Substituting this expression inside the
metric and the potential, in eqs. (4.38) and (4.45) we reach our claim.
However the procedure gives rise to divergences in four and five space-time
dimensions, that we classically proved to be traceable to a redundant gauge
freedom in those dimensions. Once modified the ansatz of the coordinate change
function, we found its well-defined form for $d=3$ and $d=4$, which
consequently lead to the expressions of the metric and the potential in those
dimensions. In the end we observed how in such expressions logarithmic terms
arise in the post-Minkowksian expansion, as well as free constants, which
parametrize the redundant gauge freedom.

In chapter 5 actual computations of graviton emission processes are carried
out with the aim to recover the Reissner-Nordström-Tangherlini metric. First
we outlined the general procedure to extract from the quantum amplitudes only
the terms which will reconstruct the classical phenomenology. Following [18],
this is accomplished by considering only the amplitudes in which a single
massive line emits $n-j$ gravitons and $j$ photons, whose internal-tree
structure ends up with a graviton emission, and extracting out from their
energy expansion the term proportional to the master integral
$J_{(l)}(\vec{q}^{2})$. In order to do that the strategy followed is the one
presented in [13], in which exploiting the low-energy and static limit it is
possible to integrate out the temporal component of the internal momenta, and
reduce the amplitudes in an interaction of external sources reproducing the
original work of [17]. In fact this result can be read backward, and state
that the only quantum processes that contribute to reconstruct classical
physics are the ones in which performed the classical limit, that corresponds
to cut the massive lines, reproduce the quantum tree-graphs of [17]. Then in
section 5.2 calculations are carried out, and we showed, as expected, the
appearance of divergences in four and five dimensions. These divergences are
renormalized by the insertion of the non-minimal action in eq. (5.69), in
which are taken into account both gravitational and electromagnetic coupling
constants. Then we commented how from Fourier transform properties, it is
possible to show that in the renormalization process of the metric only eq.
(5.71) contributes, and decoupling the counter-term (5.70) in a finite and a
divergent part, following [18, 24, 25], a renormalized expression of the
metric has obtained, showing that the same logarithmic terms of the classical
solution arise. We commented that this procedure only needs to fix the
divergent part of the counter-term, since the finite one belongs to high-
energy physics. Moreover, even if the logarithmic terms match naturally, it is
possible to impose a strict relation between the finite parts, which leads to
a relation between the constant that parametrize the redundant classical gauge
freedom and the unfixed high-energy parameter. We finally showed that the
exact same relation is coherent for both the first and second post-Minkowskian
order expansion of the metric.

Finally the original calculations in which we proved that the electromagnetic
potential associated to the Reissner-Nordström-Tangherlini solution is
recovered from photon emission scattering amplitudes are presented. In
particular we use the exact same scheme we employed in the metric case, in
which the classical contributions of the electromagnetic current in momentum
space are given by the terms of the amplitudes proportional to the master
integral, in which a single massive line emits $n-j-1$ gravitons and $j+1$
photons, ending up in a photon emission through an internal tree-structure
interaction. Then we proved that such structure leads to the conclusion that
the classical limit is again reconstructed by quantum-tree graphs, which using
the formalism of Duff, are expressed in eq. (6.11). Exploiting again the
program outlined in [13], integrating out the temporal component of the
internal momenta, explicit calculations are provided and we showed a perfect
agreement with the classical potential up to the problematic case in which a
divergence appears, which in particular happens at third post-Minkowksian
order for $d=4$. One of the most important results then, is that such
divergence is canceled by the very same count-terms that we already defined in
the metric case, leading to eq. (6.50), from which using the relation (5.90) a
perfect match with the classical solution can be observed, also in the finite
pieces. The aforementioned observations gave rise to the publication of a
paper in which all the results obtained in this thesis are presented [26].

In the end, we conclude that the present strategy for extracting classical
gravitational physics from quantum scattering amplitudes is proved to be
perfectly coherent and self-consistent. In particular all the general results
that can be recovered classically have a quantum counter-part, which makes
this calculation method completely equivalent to a classical derivation. For
instance we showed that classically the reason why we expect divergences in
the de Donder gauge is because in $d=3$ and $d=4$ there exists a redundant
gauge transformation. The quantum approach to the problem shows that it is
possible to arrive at the same conclusion just by looking at the non-minimal
action that renormalizes the theory, and since it would not be possible to
define eq. (5.71) in those dimensions, there would be no divergences except
the ones we encounter. Similarly in section 4.3 we showed that only the
temporal component of the classical potential is non-vanishing, and in eq.
(6.9) we arrive to the same conclusion through a completely quantum approach.
This extraordinary correspondence is not present only for these observables,
but in every application of this approach. In fact many considerations we did,
were not specific for the metric or the electromagnetic potential, but hold
for every gravitational observable computed with the present program. For
example if we focus on the derivation of the Newton potential between two
massive objects, even non static ones, then it is possible to reconstruct it
in a post-Minkowskian expansion by means of 2-to-2 scattering amplitudes of
massive particles mediated by gravitons. In this case, even if the procedure
to extract the classical contributions from the amplitudes is more involved
than the one we discussed in this thesis, the general conclusions regarding
the phenomenology of the classical limit hold. In fact, performing the
classical limit of such amplitudes, it leads to cut the internal massive lines
and consider the interaction of external source like in figure 7.1.

Figure 7.1: Example of 2-loop diagrams that contributes to the classical limit
of the 2-to-2 scattering amplitudes of scalar massive particles taken from
[28]. In the picture it is shown how in the classical limit, that corresponds
to cut the internal massive lines, the diagrams correspond to external sources
(represented by the blobs) that interact through a quantum-tree graph.

Such quantum tree-graphs are identical to the ones that arise from the world-
line formalism [59, 60, 58, 61, 62, 63], and show how many results obtained in
this thesis are universal, since we got the same kind of diagrams for the
graviton and photon emission amplitudes.

There are many ways in which this work can be extended. As mentioned in
chapters 5 and 6, the counter-term vertex in eq. (5.74) does not contribute to
the renormalization of the theory up to 2-loop calculations. However, it is
interesting to observe that there are no known reasons why this should happen,
and therefore an higher order calculation could show whether this vertex
contributes or does not. If the vertex will contribute at higher-order, then
what happens in this work is proved to be a coincidence. Otherwise if it will
not contribute even beyond 2 loops, this would seem an insight of a general
behavior. It would be also very interesting to perform this higher loop
computation in a generalized hybrid gauge as in [22]. In fact, since in the
reference it is shown that up to 1-loop order there exists a particular gauge
choice in which the graviton emission amplitudes do not diverge, and
consequently the Schwarzschild-Tangherlini metric is completely finite, it
could be nice to see if this special gauge makes the amplitudes divergenceless
also in the electromagnetically coupled case, and if it is still true at
higher orders. Moreover, since the non-minimal couplings are gauge invariant,
they still contribute, and then we expect that in this gauge the variations of
the observables that come from the counter-terms disappear in the classical
limit. Finally one of the most exciting extension is the possibility to
generalize the present work to graviton and photon emission processes from
fermionic matter, extending the four-dimensional 1-loop analysis of [8, 9],
finding the post-Minkowskian expansion of a rotating black hole metric in
arbitrary dimension. The reason why this is important is because the
literature on spinning black holes in arbitrary dimensions is much less
abundant than the one that treat static space-time solutions, and this program
could give many insights in order to explore the richer structure of black
hole solutions in higher dimensions [64].

The techniques to extract classical contributions from quantum processes
discussed in thesis then, are not only interesting on their own, but have many
applications. In fact they have been widely applied in the last few years to
determine the dynamics of binary systems from two-body scattering amplitudes
[15, 65, 66, 67, 68, 69, 70, 58, 71, 72, 73, 74, 75, 76]. As we did our
analysis in generic space-time dimensions, also these computations have been
extended to higher-dimensions [77, 78] (for a review see [28]), for two
reasons mainly. First, it is known that gravity has a much richer
phenomenology in higher dimensions, which means that hidden theoretical
structures could emerge in $D\neq 4$, which would give clues for other
problems. Second, as it is clear from this thesis work, even if we calculate
classical physics out of quantum computations, the amplitudes are affected by
divergences that have to be renormalized. However, in every renormalization
process one has to impose a regularization procedure, which usually
corresponds to the dimensional one. So then, following this approach, it is
worth to impose a generic space-time dimension from the beginning.

The reason why the scientific community is so interested in the dynamics of
massive objects though, and in particular in bound binary systems, is because
it can be linked to the gravitational-wave emission of a coalescence process.
In fact, since this research program can be applied to make systematic
computations for the post-Minkowskian dynamics of binary bound systems, it
could lead to high-precision predictions in the context of gravitational-wave
emission [27]. Moreover one of the most relevant feature of the proposed
techniques, is that they allow the computations of gravitational observables
in a complete analytic form, which means that such program, not only has the
power to significantly improve the state-of-the-art precision of the
predictions of gravitational-wave emission templates, but it is candidate to
be an analytical alternative to numerical methods. In fact since up to now the
only way to theoretically model the merger phase of a binary coalescence is by
employing numerical relativity, it seems to be possible to extend these
techniques on a curved space-time in order to obtain analytic predictions of
such process [27]. This last research line, and all the other ones, are far
from being completely understood or worked out, and this is one of the reasons
why it is worth to continue to explore the methods to extract classical
gravitational observables from scattering amplitudes.

##  Appendix A Feynman rules

In this appendix are listed all the Feynman rules used in the thesis. In the
first part are given the propagators in the interested gauges and in generic
space-time dimension $D=d+1$, while in the second part are reported the
interaction vertices between photons, gravitons and massive scalars. Notice
that the single wiggly lines stand for photons, the double wiggly lines stand
for gravitons and the solid lines stand for charged scalars.

  * •

Scalar propagator of mass $m$:

$\includegraphics[width=65.04034pt,valign={c}]{Diagrams/Scalar_Propagator.png}=\frac{i}{q^{2}-m^{2}+i\epsilon}\
.$ (A.1)

  * •

Photon propagator in Feynman gauge:

$\includegraphics[width=78.04842pt,valign={c}]{Diagrams/Photon_Propagator.png}=-\frac{i\eta^{\mu\nu}}{q^{2}+i\epsilon}\
.$ (A.2)

  * •

Graviton propagator in de Donder gauge:

$\includegraphics[width=86.72267pt,valign={c}]{Diagrams/Graviton_Propagator.png}=i\frac{P_{\alpha\beta,\mu\nu}}{q^{2}+i\epsilon}\
,$ (A.3)

with $P_{\alpha\beta,\mu\nu}$ defined by

$P_{\alpha\beta,\mu\nu}=\frac{1}{2}\Big{(}\eta_{\mu\alpha}\eta_{\nu\beta}+\eta_{\mu\beta}\eta_{\nu\alpha}-\frac{2}{d-1}\eta_{\mu\nu}\eta_{\alpha\beta}\Big{)}\
.$ (A.4)

  * •

2 scalars - 1 photon vertex:

$\includegraphics[width=108.405pt,valign={c}]{Diagrams/2Scalars_1Photon.png}=\left(\tau_{\phi^{2}A}\right)^{\mu}(p,p^{\prime})=-iQ\left(p+p^{\prime}\right)^{\mu}\
.$ (A.5)

  * •

2 scalars - 1 graviton vertex [51]:

$\includegraphics[width=108.405pt,valign={c}]{Diagrams/2Scalars_1Graviton.png}=\left(\tau_{\phi^{2}h}\right)^{\mu\nu}(p,p^{\prime})=-\frac{i\kappa}{2}\Big{(}\big{(}p^{\mu}p^{\prime}{}^{\nu}+p^{\nu}p^{\prime}{}^{\mu}\big{)}-\eta^{\mu\nu}\big{(}p\cdot
p^{\prime}-m^{2}\big{)}\Big{)}\ .$ (A.6)

  * •

2 photons - 1 graviton vertex [51]:

$\includegraphics[width=108.405pt,valign={c}]{Diagrams/2Photons_1Graviton.png}=\left(\tau_{A^{2}h}\right)^{\mu\nu,\alpha,\beta}(p,p^{\prime})$
$\begin{split}=i\kappa&\left(P_{(4)}^{\mu\nu,\alpha\beta}(p\cdot
p^{\prime})+\frac{1}{2}\left(\frac{1}{2}\eta^{\mu\nu}(p^{\alpha}p^{\prime}{}^{\beta}+p^{\beta}p^{\prime}{}^{\alpha})+\eta^{\alpha\beta}(p^{\mu}p^{\prime}{}^{\nu}+p^{\nu}p^{\prime}{}^{\mu})\right.\right.\\\
&\left.\left.-\frac{1}{2}\left(\eta^{\beta\nu}(p^{\mu}p^{\prime}{}^{\alpha}+p^{\alpha}p^{\prime}{}^{\mu})+\eta^{\alpha\nu}(p^{\mu}p^{\prime}{}^{\beta}+p^{\beta}p^{\prime}{}^{\mu})+\eta^{\alpha\mu}(p^{\nu}p^{\prime}{}^{\beta}+p^{\beta}p^{\prime}{}^{\nu})\right.\right.\right.\\\
&\left.\left.\left.+\eta^{\beta\mu}(p^{\nu}p^{\prime}{}^{\alpha}+p^{\alpha}p^{\prime}{}^{\nu})\right)\right)\right)\
,\end{split}$ (A.7)

where $P_{(4)}^{\mu\nu,\alpha\beta}$ is (A.4) for $d=3$.

  * •

2 photons - 2 gravitons vertex [51]:

$\includegraphics[width=108.405pt,valign={c}]{Diagrams/2Photons_2Gravitons.png}=\left({\tau}_{A^{2}h^{2}}\right){}_{\mu\nu,}{}_{\rho\sigma,\alpha,\beta}(p,p^{\prime})$
$\displaystyle\quad=$
$\displaystyle-\frac{i\kappa^{2}}{4}\left(\left(p_{\beta}p^{\prime}_{\alpha}-\eta_{\alpha\beta}p\cdot
p^{\prime}\right)\left(\eta_{\mu\rho}\eta_{\nu\sigma}+\eta_{\mu\sigma}\eta_{\nu\rho}-\eta_{\mu\nu}\eta_{\rho\sigma}\right)+\eta_{\mu\rho}\left(\eta_{\alpha\beta}\left(p_{\nu}p^{\prime}_{\sigma}+p_{\sigma}p^{\prime}_{\nu}\right)\right.\right.$
(A.8)
$\displaystyle\left.-\eta_{\alpha\nu}p_{\beta}p^{\prime}_{\sigma}-\eta_{\beta\nu}p_{\sigma}p^{\prime}_{\alpha}-\eta_{\beta\sigma}p_{\nu}p^{\prime}_{\alpha}-\eta_{\alpha\sigma}p_{\beta}p^{\prime}_{\nu}+p\cdot
p^{\prime}\left(\eta_{\alpha\nu}\eta_{\beta\sigma}+\eta_{\alpha\sigma}\eta_{\beta\nu}\right)\right)$
$\displaystyle+\eta_{\mu\sigma}\left(\eta_{\alpha\beta}\left(p_{\nu}p^{\prime}_{\rho}+p_{\rho}p^{\prime}_{\nu}\right)-\eta_{\alpha\nu}p_{\beta}p^{\prime}_{\rho}-\eta_{\beta\nu}p_{\rho}p^{\prime}_{\alpha}-\eta_{\beta\rho}p_{\nu}p^{\prime}_{\alpha}-\eta_{\alpha\rho}p_{\beta}p^{\prime}_{\nu}\right.$
$\displaystyle\left.+p\cdot
p^{\prime}\left(\eta_{\alpha\nu}\eta_{\beta\rho}+\eta_{\alpha\rho}\eta_{\beta\nu}\right)\right)+\eta_{\nu\rho}\left(\eta_{\alpha\beta}\left(p_{\mu}p^{\prime}_{\sigma}+p_{\sigma}p^{\prime}_{\mu}\right)-\eta_{\alpha\mu}p_{\beta}p^{\prime}_{\sigma}-\eta_{\beta\mu}p_{\sigma}p^{\prime}_{\alpha}\right.$
$\displaystyle\left.-\eta_{\beta\sigma}p_{\mu}p^{\prime}_{\alpha}-\eta_{\alpha\sigma}p_{\beta}p^{\prime}_{\mu}+p\cdot
p^{\prime}\left(\eta_{\alpha\mu}\eta_{\beta\sigma}+\eta_{\alpha\sigma}\eta_{\beta\mu}\right)\right)+\eta_{\nu\sigma}\left(\eta_{\alpha\beta}\left(p_{\mu}p^{\prime}_{\rho}+p_{\rho}p^{\prime}_{\mu}\right)\right.$
$\displaystyle\left.-\eta_{\alpha\mu}p_{\beta}p^{\prime}_{\rho}-\eta_{\beta\mu}p_{\rho}p^{\prime}_{\alpha}\right.\left.-\eta_{\beta\rho}p_{\mu}p^{\prime}_{\alpha}-\eta_{\alpha\rho}p_{\beta}p^{\prime}_{\mu}+p\cdot
p^{\prime}\left(\eta_{\alpha\mu}\eta_{\beta\rho}+\eta_{\alpha\rho}\eta_{\beta\mu}\right)\right)$
$\displaystyle-\eta_{\mu\nu}\left(\eta_{\alpha\beta}\left(p_{\rho}p^{\prime}_{\sigma}+p_{\sigma}p^{\prime}_{\rho}\right)-\eta_{\alpha\rho}p_{\beta}p^{\prime}_{\sigma}-\eta_{\beta\rho}p_{\sigma}p^{\prime}_{\alpha}-\eta_{\beta\sigma}p_{\rho}p^{\prime}_{\alpha}-\eta_{\alpha\sigma}p_{\beta}p^{\prime}_{\rho}\right.$
$\displaystyle\left.+p\cdot
p^{\prime}\left(\eta_{\alpha\rho}\eta_{\beta\sigma}+\eta_{\beta\rho}\eta_{\alpha\sigma}\right)\right)-\eta_{\rho\sigma}\left(\eta_{\alpha\beta}\left(p_{\mu}p^{\prime}_{\nu}+p_{\nu}p^{\prime}_{\mu}\right)-\eta_{\alpha\mu}p_{\beta}p^{\prime}_{\nu}-\eta_{\beta\mu}p_{\nu}p^{\prime}_{\alpha}\right.$
$\displaystyle\left.-\eta_{\beta\nu}p_{\mu}p^{\prime}_{\alpha}-\eta_{\alpha\nu}p_{\beta}p^{\prime}_{\mu}+p\cdot
p^{\prime}\left(\eta_{\alpha\mu}\eta_{\beta\nu}+\eta_{\beta\mu}\eta_{\alpha\nu}\right)\right)$
$\displaystyle+\left(\eta_{\alpha\rho}p_{\mu}-\eta_{\alpha\mu}p_{\rho}\right)\left(\eta_{\beta\sigma}p^{\prime}_{\nu}-\eta_{\beta\nu}p^{\prime}_{\sigma}\right)+\left(\eta_{\alpha\sigma}p_{\nu}-\eta_{\alpha\nu}p_{\sigma}\right)\left(\eta_{\beta\rho}p^{\prime}_{\mu}-\eta_{\beta\mu}p^{\prime}_{\rho}\right)$
$\displaystyle+\left.\left(\eta_{\alpha\sigma}p_{\mu}-\eta_{\alpha\mu}p_{\sigma}\right)\left(\eta_{\beta\rho}p^{\prime}_{\nu}-\eta_{\beta\nu}p^{\prime}_{\sigma}\right)+\left(\eta_{\alpha\rho}p_{\nu}-\eta_{\alpha\nu}p_{\rho}\right)\left(\eta_{\beta\sigma}p^{\prime}_{\mu}-\eta_{\beta\mu}p^{\prime}_{\sigma}\right)\right)\
.$

  * •

3 gravitons vertex [4, 79]:

$\includegraphics[width=130.08731pt,valign={c}]{Diagrams/3Gravitons.png}=\left(\tau_{h^{3}}\right)^{\mu_{1}\nu_{1},\mu_{2}\nu_{2},\mu_{3}\nu_{3}}(p_{1},p_{2},p_{3})$
$\displaystyle=-2i\kappa\,\text{Sym}\ \Big{(}$
$\displaystyle-\frac{1}{4}P_{3}\big{(}p_{1}\cdot
p_{2}\eta^{\mu_{1}\nu_{1}}\eta^{\mu_{2}\nu_{2}}\eta^{\mu_{3}\nu_{3}}\big{)}-\frac{1}{4}P_{6}\big{(}p_{1}^{\mu_{2}}p_{1}^{\nu_{2}}\eta^{\mu_{1}\nu_{1}}\eta^{\mu_{3}\nu_{3}}\big{)}$
(A.9) $\displaystyle+\frac{1}{4}P_{3}\big{(}p_{1}\cdot
p_{2}\eta^{\mu_{1}\mu_{2}}\eta^{\nu_{1}\nu_{2}}\eta^{\mu_{3}\nu_{3}}\big{)}+\frac{1}{2}P_{6}\big{(}p_{1}\cdot
p_{2}\eta^{\mu_{1}\nu_{1}}\eta^{\mu_{2}\mu_{3}}\eta^{\nu_{2}\nu_{3}}\big{)}$
$\displaystyle+P_{3}\big{(}p_{1}^{\mu_{2}}p_{1}^{\nu_{3}}\eta^{\mu_{1}\nu_{1}}\eta^{\nu_{2}\mu_{3}}\big{)}-\frac{1}{2}P_{3}\big{(}p_{1}^{\nu_{2}}p_{2}^{\mu_{1}}\eta^{\nu_{1}\mu_{2}}\eta^{\mu_{3}\nu_{3}}\big{)}$
$\displaystyle+\frac{1}{2}P_{3}\big{(}p_{1}^{\mu_{3}}p_{2}^{\nu_{3}}\eta^{\mu_{1}\mu_{2}}\eta^{\nu_{1}\nu_{2}}\big{)}+\frac{1}{2}P_{6}\big{(}p_{1}^{\mu_{3}}p_{1}^{\nu_{3}}\eta^{\mu_{1}\mu_{2}}\eta^{\nu_{1}\nu_{2}}\big{)}$
$\displaystyle+P_{6}\big{(}p_{1}^{\mu_{2}}p_{2}^{\nu_{3}}\eta^{\nu_{2}\mu_{1}}\eta^{\nu_{1}\mu_{3}}\big{)}+P_{3}\big{(}p_{1}^{\mu_{2}}p_{2}^{\mu_{1}}\eta^{\nu_{2}\mu_{3}}\eta^{\nu_{3}\nu_{1}}\big{)}$
$\displaystyle-P_{3}\big{(}p_{1}\cdot
p_{2}\eta^{\nu_{1}\mu_{2}}\eta^{\nu_{2}\mu_{3}}\eta^{\nu_{3}\mu_{1}}\big{)}\Big{)}\
.$

Here $"\text{Sym}"$ indicates that a symmetrization has to be performed on
each index pair, while $"P"$ indicates that a summation has to be performed on
all the distinct permutations of the momentum-index triplets (the subscript
gives the number of terms).111For example:
$\text{Sym}\big{[}P_{3}\big{(}p_{1}^{\mu_{2}}p_{2}^{\mu_{1}}\eta^{\nu_{2}\mu_{3}}\eta^{\nu_{3}\nu_{1}}\big{)}\big{]}=p_{1}^{\mu_{2}}p_{2}^{\mu_{1}}\eta^{\nu_{2}\mu_{3}}\eta^{\nu_{3}\nu_{1}}+p_{1}^{\mu_{3}}p_{3}^{\mu_{1}}\eta^{\nu_{3}\mu_{2}}\eta^{\nu_{2}\nu_{1}}+p_{2}^{\mu_{3}}p_{3}^{\mu_{2}}\eta^{\nu_{3}\mu_{1}}\eta^{\nu_{1}\nu_{2}}\
.$

  * •

3 gravitons vertex with two quantum fields and an external field [7]:

$\includegraphics[width=108.405pt,valign={c}]{Diagrams/3Gravitons_Don.png}=\left(\tau_{h^{2}h}\right){}^{\mu\nu}{}_{\alpha\beta,\gamma\delta}(l,q)$
$\displaystyle=\frac{i\kappa}{2}$
$\displaystyle\bigg{(}P^{(4)}_{\alpha\beta,\gamma\delta}\left(l^{\mu}l^{\nu}+(l-q)^{\mu}(l-q)^{\nu}+q^{\mu}q^{\nu}-\frac{3}{2}\eta^{\mu\nu}q^{2}\right)$
(A.10)
$\displaystyle+2q_{\lambda}q_{\sigma}\left({I_{\alpha\beta}}^{\lambda\sigma}{I_{\gamma\delta}}^{\mu\nu}+{I_{\gamma\delta}}^{\lambda\sigma}{I_{\alpha\beta}}^{\mu\nu}-{I_{\alpha\beta}}^{\lambda\mu}{I^{\sigma\nu}}_{\gamma\delta}-{I_{\alpha\beta}}^{\sigma\nu}{I_{\gamma\delta}}^{\lambda\nu}\right)$
$\displaystyle+q_{\lambda}q^{\mu}\left(\eta_{\alpha\beta}I^{\lambda\nu}{}_{\gamma\delta}+\eta_{\gamma\delta}{I^{\lambda\nu}}_{\alpha\beta}\right)+q_{\lambda}q^{\nu}\left(\eta_{\alpha\beta}{I_{\gamma\delta}}^{\lambda\mu}+\eta_{\gamma\delta}{I_{\alpha\beta}}^{\lambda\mu}\right)$
$\displaystyle-q^{2}\left(\eta_{\alpha\beta}{I_{\gamma\delta}}^{\mu\nu}+\eta_{\gamma\delta}{I_{\alpha\beta}}^{\mu\nu}\right)-\eta^{\mu\nu}q^{\lambda}q^{\sigma}\left(\eta_{\alpha\beta}{I_{\gamma\delta\lambda\sigma}}+\eta_{\gamma\delta}I_{\alpha\beta\lambda\sigma}\right)$
$\displaystyle+2q^{\lambda}\left({I_{\alpha\beta}}^{\sigma\lambda}I_{\gamma\delta\sigma\nu}(l-q)^{\mu}+{I_{\alpha\beta}}^{\lambda\sigma}I_{\gamma\delta\sigma}{}^{\mu}(l-q)^{\nu}-{I_{\gamma\delta}}^{\lambda\sigma}I_{\alpha\beta\sigma}{}^{\nu}l^{\mu}-{I_{\gamma\delta}}^{\lambda\sigma}I_{\alpha\beta\sigma}{}^{\mu}l^{\nu}\right)$
$\displaystyle+q^{2}\left({I_{\alpha\beta}}^{\sigma\mu}{I_{\gamma\delta,\sigma}}^{\nu}+I_{\alpha\beta,\sigma}{}^{\nu}{I_{\alpha\delta}}^{\sigma\mu}\right)+\eta^{\mu\nu}q^{\lambda}q_{\sigma}\left({I_{\gamma\delta}}^{\rho\sigma}I_{\alpha\beta,\lambda\rho}+{I_{\alpha\beta}}^{\rho\sigma}I_{\gamma\delta,\lambda\rho}\right)$
$\displaystyle+\left(l^{2}+(l-q)^{2}\right)\left({I_{\alpha\beta}}^{\sigma\mu}{I_{\gamma\delta,\sigma}}^{\nu}+I^{\sigma\nu}{}_{\alpha\beta}{I_{\gamma\delta,\sigma}}^{\mu}-\frac{1}{2}\eta^{\mu\nu}P^{(4)}_{\alpha\beta,\gamma\delta}\right)$
$\displaystyle-l^{2}\eta_{\gamma\delta}{I_{\alpha\beta}}^{\mu\nu}-(l-q)^{2}\eta_{\alpha\beta}{I_{\gamma\delta}}^{\mu\nu}\bigg{)}\
,$

with
$I_{\mu\nu,\alpha\beta}=\frac{1}{2}(\eta_{\mu\alpha}\eta_{\nu\beta}+\eta_{\mu\beta}\eta_{\nu\alpha})$,
and where the $\mu\nu$ graviton correspond to the external one, meaning that
the expression is symmetric only under the exchange of the lines
$\alpha\beta\leftrightarrow\gamma\delta$.

##  Appendix B Master integral and Fourier transforms

Here we review the definition of the master integral introduced in section 5.1
and we list its $d$-dimensional Fourier transform identities. In a
$d$-dimensional Euclidean space in which
$\vec{q}^{2}=q_{1}^{2}+...+q_{d}^{2}$, the $l$-loop massless sunset integral
is defined by

$J_{(l)}(\vec{q}^{2})=\int\prod_{i=1}^{l}\frac{d^{d}\vec{\ell}_{i}}{(2\pi)^{d}}\frac{\vec{q}^{2}}{\left(\prod_{i=1}^{l}\vec{\ell}_{i}^{2}\right)\left(\vec{q}-\sum_{i=1}^{l}\vec{\ell}_{i}\right)^{2}}\
.$ (B.1)

It can be explicitly computed [80] as

$J_{(l)}(\vec{q}^{2})=\frac{\Gamma\left(l+1-\frac{ld}{2}\right)\Gamma\left(\frac{d-2}{2}\right)^{l+1}}{(4\pi)^{\frac{ld}{2}}\Gamma\left(\frac{(l+1)(d-2)}{2}\right)}(\vec{q}^{2})^{\frac{l(d-2)}{2}}\
,$ (B.2)

from which it is clear the presence of poles for
$\frac{(l+1)(d-2)}{2}=0,-1,-2$. Then computing the Fourier transforms111A
proof of such Fourier transforms can be found in [26]. needed for the
amplitude calculation of chapters 5 and 6, they read

$\int\frac{d^{d}\vec{q}}{(2\pi)^{d}}\frac{J_{(l)}(\vec{q}^{2})}{\vec{q}^{2}}e^{i\vec{q}\cdot\vec{x}}=\left(\frac{\rho}{4\pi}\right)^{l+1}$
(B.3)
$\int\frac{d^{d}\vec{q}}{(2\pi)^{d}}\frac{q_{i}q_{j}}{\vec{q}^{2}}\frac{J_{(l)}(\vec{q}^{2})}{\vec{q}^{2}}e^{i\vec{q}\cdot\vec{x}}=\frac{1}{2-l(d-2)}\left(\delta_{ij}+(l+1)(2-d)\frac{x_{i}x_{j}}{r}\right)\left(\frac{\rho}{4\pi}\right)^{l+1}\
,$ (B.4)

where the $\rho$ parameter is the one defined in (4.17). One can notice how
eqs. (B.3) and (B.4) are perfectly finite, which means that the divergences in
the explicit form of the master integral are canceled by the Fourier
transforms. Then it is possible to compute a specific relation needed for
sections 5.3 and 6.3, which reads [18]

$\int\frac{d^{d}\vec{q}}{(2\pi)^{d}}J_{(1)}(\vec{q}^{2})e^{i\vec{q}\cdot\vec{x}}=-\frac{\Gamma\left(\frac{d}{2}\right)^{2}}{2\pi^{d}r^{2(d-1)}}\
.$ (B.5)

Finally from the above identities it is possible to recover the particular
cases needed for the computations in chapter 3, recovering the Fourier
transforms listed in [8] and [9]. Since $J_{(0)}(\vec{q}^{2})=1$, in $d=3$ and
$l=0$ eq. (B.3) becomes

$\int\frac{d^{3}\vec{q}}{(2\pi)^{3}}\frac{1}{\vec{q}^{2}}e^{i\vec{q}\cdot\vec{x}}=\frac{1}{4\pi\,r}\
,$ (B.6)

while for $l=1$, the master integral explicitly reads

$J_{(1)}(\vec{q}^{2})\Big{|}_{d=3}=\frac{1}{8}|\vec{q}|\ ,$ (B.7)

whose replaced inside eqs. (B.3) and (B.4) gives

$\int\frac{d^{3}\vec{q}}{(2\pi)^{3}}\frac{1}{|\vec{q}|}e^{i\vec{q}\cdot\vec{x}}=\frac{1}{2\pi^{2}\,r^{2}}$
(B.8)
$\int\frac{d^{3}\vec{q}}{(2\pi)^{3}}\frac{q_{i}q_{j}}{\vec{q}^{3}}e^{i\vec{q}\cdot\vec{x}}=\frac{1}{2\pi^{2}\,r^{2}}\left(\delta_{ij}-2\frac{x_{i}x_{j}}{r^{2}}\right)\
.$ (B.9)

##  Appendix C Loop integral reduction

The classical limit of both the stress-energy tensor and the electromagnetic
current, arise by extracting from the loop integrals the contributions
proportional to the master integral and neglecting all the higher powers of
the momentum transferred in the energy expansion, which correspond to the
quantum corrections. In this appendix we list such contributions for all the
loop integrals that occur in the amplitude computations in sections 5.2 and
6.2. Making the notation clear, when an integral below is set to be vanish, it
means that no term in the energy expansion is proportional to the master
integral. Notice that for some of these relations we made use of the LiteRed
package of Mathematica [81].

##### 1-loop

$\int\frac{d^{d}\ell}{(2\pi)^{d}}\frac{\vec{\ell}^{2}}{\vec{\ell}^{2}(\vec{q}-\vec{\ell})^{2}}=0\
,$ (C.1)
$\int\frac{d^{d}\ell}{(2\pi)^{d}}\frac{\vec{\ell}\cdot\vec{q}}{\vec{\ell}^{2}(\vec{q}-\vec{\ell})^{2}}=\frac{1}{2}J_{(1)}(\vec{q}^{2})\
,$ (C.2)
$\int\frac{d^{d}\ell}{(2\pi)^{d}}\frac{\vec{\ell}\cdot\vec{q}\,\vec{\ell}^{2}}{\vec{\ell}^{2}(\vec{q}-\vec{\ell})^{2}}=0\
,$ (C.3)
$\int\frac{d^{d}\ell}{(2\pi)^{d}}\frac{\vec{\ell}\cdot\vec{q}\,\vec{\ell}\cdot\vec{q}}{\vec{\ell}^{2}(\vec{q}-\vec{\ell})^{2}}=\frac{1}{4}\vec{q}^{2}\,J_{(1)}(\vec{q}^{2})\
,$ (C.4)

##### 2-loop

$\int\frac{d^{d}\ell_{1}}{(2\pi)^{d}}\frac{d^{d}\ell_{2}}{(2\pi)^{d}}\frac{\vec{\ell_{1}}\cdot\vec{q}}{\vec{\ell_{1}}^{2}\vec{\ell_{2}}^{2}\left(\vec{q}-\vec{\ell_{1}}-\vec{\ell_{2}}\right)^{2}}=\frac{1}{3}J_{(2)}\left(\vec{q}^{2}\right)$
(C.5)

$\int\frac{d^{d}\ell_{1}}{(2\pi)^{d}}\frac{d^{d}\ell_{2}}{(2\pi)^{d}}\frac{\vec{\ell_{1}}\cdot\vec{\ell_{2}}}{\vec{\ell_{1}}^{2}\vec{\ell_{2}}^{2}\left(\vec{q}-\vec{\ell_{1}}-\vec{\ell_{2}}\right)^{2}}=\frac{1}{6}J_{(2)}\left(\vec{q}^{2}\right)$
(C.6)

$\int\frac{d^{d}\ell_{1}}{(2\pi)^{d}}\frac{d^{d}\ell_{2}}{(2\pi)^{d}}\frac{\vec{\ell_{1}}\cdot\vec{q}\
\vec{\ell_{1}}^{2}}{\vec{\ell_{1}}^{2}\vec{\ell_{2}}^{2}\left(\vec{q}-\vec{\ell_{1}}-\vec{\ell_{2}}\right)^{2}\left(\vec{\ell_{1}}+\vec{\ell_{2}}\right)^{2}}=0$
(C.7)

$\int\frac{d^{d}\ell_{1}}{(2\pi)^{d}}\frac{d^{d}\ell_{2}}{(2\pi)^{d}}\frac{\vec{\ell_{2}}\cdot\vec{q}\
\vec{\ell_{1}}^{2}}{\vec{\ell_{1}}^{2}\vec{\ell_{2}}^{2}\left(\vec{q}-\vec{\ell_{1}}-\vec{\ell_{2}}\right)^{2}\left(\vec{\ell_{1}}+\vec{\ell_{2}}\right)^{2}}=0$
(C.8)

$\int\frac{d^{d}\ell_{1}}{(2\pi)^{d}}\frac{d^{d}\ell_{2}}{(2\pi)^{d}}\frac{\vec{\ell_{1}}\cdot\vec{\ell_{2}}\
\vec{q}\cdot\vec{\ell_{1}}}{\vec{\ell_{1}}^{2}\vec{\ell_{2}}^{2}\left(\vec{q}-\vec{\ell_{1}}-\vec{\ell_{2}}\right)^{2}\left(\vec{\ell_{1}}+\vec{\ell_{2}}\right)^{2}}=\frac{1}{6}J_{(2)}\left(\vec{q}^{2}\right)$
(C.9)

$\int\frac{d^{d}\ell_{1}}{(2\pi)^{d}}\frac{d^{d}\ell_{2}}{(2\pi)^{d}}\frac{\vec{\ell_{1}}^{2}\
\vec{q}^{2}}{\vec{\ell_{1}}^{2}\vec{\ell_{2}}^{2}\left(\vec{q}-\vec{\ell_{1}}-\vec{\ell_{2}}\right)^{2}\left(\vec{\ell_{1}}+\vec{\ell_{2}}\right)^{2}}=0$
(C.10)

$\int\frac{d^{d}\ell_{1}}{(2\pi)^{d}}\frac{d^{d}\ell_{2}}{(2\pi)^{d}}\frac{\vec{\ell_{1}}\cdot\vec{\ell_{2}}\
\vec{q}^{2}}{\vec{\ell_{1}}^{2}\vec{\ell_{2}}^{2}\left(\vec{q}-\vec{\ell_{1}}-\vec{\ell_{2}}\right)^{2}\left(\vec{\ell_{1}}+\vec{\ell_{2}}\right)^{2}}=\frac{1}{2}J_{(2)}\left(\vec{q}^{2}\right)$
(C.11)

$\int\frac{d^{d}\ell_{1}}{(2\pi)^{d}}\frac{d^{d}\ell_{2}}{(2\pi)^{d}}\frac{\vec{\ell_{1}}\cdot\vec{q}\
\vec{\ell_{2}}\cdot\vec{q}}{\vec{\ell_{1}}^{2}\vec{\ell_{2}}^{2}\left(\vec{q}-\vec{\ell_{1}}-\vec{\ell_{2}}\right)^{2}\left(\vec{\ell_{1}}+\vec{\ell_{2}}\right)^{2}}=\frac{2d-7}{6(d-4)}J_{(2)}\left(\vec{q}^{2}\right)$
(C.12)

$\int\frac{d^{d}\ell_{1}}{(2\pi)^{d}}\frac{d^{d}\ell_{2}}{(2\pi)^{d}}\frac{\vec{\ell_{1}}\cdot\vec{q}\
\vec{\ell_{1}}\cdot\vec{q}}{\vec{\ell_{1}}^{2}\vec{\ell_{2}}^{2}\left(\vec{q}-\vec{\ell_{1}}-\vec{\ell_{2}}\right)^{2}\left(\vec{\ell_{1}}+\vec{\ell_{2}}\right)^{2}}=\frac{d-3}{3(d-4)}J_{(2)}\left(\vec{q}^{2}\right)\
.$ (C.13)

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