1793203 Laurea Magistrale in Fisica Facoltà di Scienze Matematiche, Fisiche e
Naturali 2021/2022 Prof. Fabio Riccioni<EMAIL_ADDRESS>2022 Master Thesis

# The Reissner-Nordström-Tangherlini solution from graviton and photon
emission processes

Claudio Gambino

###### Abstract

It is known that the canonical quantization of general relativity leads to a
non-renormalizable theory, which however with the modern tools of effective
field theories it is possible to make well-defined at low energies. What is
emerging nowadays is that recovering general relativity from the classical
limit of scattering amplitudes seems more efficient than actually solve the
Einstein equations. The thesis then aims to explore the state-of-the-art
strategies to recover classical gravitational observables from quantum
computations and apply them in order to reconstruct the Reissner-Nordström-
Tangherlini solution, which is the generalization to arbitrary dimensions of
the Reissner-Nordström one, from the classical limit of scattering amplitudes
of charged scalars. In the first part of the thesis, we recover the most
generic energy-ordered effective action which is compatible with the
symmetries of the theory, and then we discuss the classical limit from which
it is possible to obtain back classical gravitational observables out of
quantum computations. In order to show how such limit works, an explicit
example is made, and following the historical papers it is shown how to
recover stationary space-time solutions from the classical limit of graviton
emission processes in four dimensions. Then the attention is focused on the
Reissner-Nordström-Tangherlini solution and on computing it by using quantum
scattering amplitudes in the de Donder gauge. Since the main objective will be
to compare the quantum computation with the classical one, a procedure to
rewrite such solution in the de Donder gauge is carried out. Finally, the
state-of-the-art techniques to extract the classical contributions out of
graviton emission amplitudes are reviewed, and they are applied to the context
of a theory which is also coupled to the electromagnetic field. Since
divergences arise in four and five dimensions it is crucial to define non-
minimal couplings with the purpose of canceling such singularities and
obtaining finite results. In the end, the considerations employed in the
metric case are used to recover the electromagnetic potential associated to
the black hole solution through the classical limit of photon emission
processes. Also in this case there appear divergences that have to be
renormalized. One of the most important outcomes is that the counter-terms
that have to be fixed in order to eliminate the poles are the same throughout
the theory, showing an extraordinary coherence of the procedure.

The thesis project has produced the publication of a paper in which the
results obtained in this work have been presented [S. D’Onofrio, F. Fragomeno,
C. Gambino and F. Riccioni, The Reissner-Nordström-Tangherlini solution from
scattering amplitudes of charged scalars, JHEP 09 (2022) 013, 2207.05841].

###### Acknowledgements.

First I would like to thank my advisor Fabio Riccioni for the precious help
and inspiration during the thesis project. I would like also to thank Simone
D’Onofrio and Federica Fragomeno which with their thesis inspired the present
work, and whose collaboration was very pleasant and productive. A special
thank to the Physics Department of the University of Rome "La Sapienza" for
the years of formation which culminate with this thesis. Finally, I am very
grateful for the support I received during these years from my colleagues,
which without them none of this would have been possible, and from friends and
family, who helped me during this journey in many ways.

###### Contents

  1. 1 Introduction
  2. 2 Quantum gravity as an effective field theory
    1. 2.1 Scalar quantum electrodynamics
    2. 2.2 General relativity as a gauge theory
    3. 2.3 Quantization of gravity
    4. 2.4 Classical limit
  3. 3 Classical metrics from scattering amplitudes in four dimensions
    1. 3.1 Schwarzschild metric from graviton emission
    2. 3.2 Schwarzschild metric in harmonic gauge
    3. 3.3 Reissner-Nordström metric from graviton emission
  4. 4 Reissner-Nordström-Tangherlini solution in de Donder gauge
    1. 4.1 The coordinate change function
    2. 4.2 Metric in de Donder gauge
    3. 4.3 Electromagnetic potential in de Donder gauge
  5. 5 Graviton emission processes
    1. 5.1 Classical limit of graviton emission processes
    2. 5.2 Metric from amplitude calculations
    3. 5.3 Renormalization of the metric
      1. 5.3.1 Regularization
      2. 5.3.2 Non-minimal couplings
      3. 5.3.3 Complete metric expression
  6. 6 Photon emission processes
    1. 6.1 Classical limit of photon emission processes
    2. 6.2 Potential from amplitude calculations
    3. 6.3 Renormalization of the potential
  7. 7 Conclusions
  8. A Feynman rules
  9. B Master integral and Fourier transforms
  10. C Loop integral reduction

##  Chapter 1 Introduction

Since the formulation of quantum mechanics and general relativity, attempts
have been made to unify these two theories to provide a quantum description of
gravity. However, to date, the formulation of a quantum theory of gravity
remains one of the greatest challenges of modern physics. One of the reasons
lies in the fact that a canonical quantization of gravity, in which the
interaction is mediated by a massless spin-2 boson called graviton, gives rise
to a non-renormalizable theory, which has pushed the scientific community to
search for alternative approaches to field theories. However, by exploiting
the principles of effective field theories, a non-renormalizable theory can be
reformulated in such a way to have predictive power as long as the energy of
the processes considered is below a certain threshold, above which a more
fundamental theory is necessary. For the gravitational interaction, we believe
that the energy scale of the so called heavy degrees of freedom is identified
by the Planck scale. In the language of effective field theories therefore,
quantum gravity is described in terms of an action composed of an infinite
series of terms weighted by inverse powers of the Planck mass, in which the
Einstein-Hilbert action corresponds to the first order of this expansion [1,
2, 3, 4, 5]. This scheme was employed by Donoghue in 1993 in order to recover
the quantum corrections of the Newton gravitational potential, showing how the
effective field theory approach to gravity could actually lead to relevant
conclusions [6, 7].

Later, it was starting to be clear how the most important outcome of this
result was that besides the quantum corrections of gravitational
observables111In this thesis, for gravitational observables we mean the set of
objects related to gravitational phenomena which are associated to measurable
quantities. For example, the metric in general relativity is not a directly
measurable quantity, however, from it, experimental predictions can be
extracted, such as the attraction strength between two bodies, the light-
bending angle around some mass, the gravitational-wave emission amplitude
generated by a merger of a binary system, etc… Therefore, in this thesis, the
metric is considered a gravitational observable., also classical terms at
zeroth-order in $\hbar$ could be extracted through a loop quantum expansion
[7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. In particular the awareness that
classical physics could be efficiently obtained from the classical limit of
quantum loop scattering amplitudes was emerging. In fact since quantum gravity
as an effective field theory is well-defined at low energies, in that regime
the classical limit in which $\hbar\rightarrow 0$ must reproduce the original
classical theory, which in this case corresponds to general relativity. In [9]
it was shown how besides $\hbar$-dependent contributions, the first post-
Minkowskian term of the Schwarzschild metric could be derived from the tree-
level amplitude in which a massive scalar emits a graviton, while the second
post-Minkowskian order is obtained by considering a 1-loop amplitude which
contains a 3 graviton interaction vertex. In the same context it was shown
that it is possible to reconstruct the Kerr metric by considering graviton
emission from spinors, as well as the terms proportional to the electric
charge of the Reissner-Nordström and Kerr-Newman metrics by considering the
1-loop scattering of charged particles with photons running in the loop [8].
The knowledge on how the classical limit of quantum gravity occurs was taking
shape, and in particular that general relativity is reconstructed by multi-
loop scattering amplitudes. This phenomenology differs from the classical
limit of the other known long-range force, the electromagnetic interaction, in
which classical electromagnetism is recovered by the $\hbar\rightarrow 0$
limit of tree-level processes in QED. The reason why gravity needs loop
amplitudes to be reconstructed is traceable to its strong non-linear behavior,
which is translated in the quantum context by graviton self-interactions
terms, that make loop amplitudes necessary for recovering classical physics.

More recently, a systematic procedure to extract the classical contribution of
graviton emission loop amplitudes in any dimension222In this thesis we will
only consider space-time dimensions $D>2$. was given in [13], in which it is
shown how such terms correspond to the interaction of external sources through
a quantum-tree graph, originally observed by Duff in [17]. Applying these
techniques, in [18, 19] the Schwarzschild-Tangherlini metric [20], that
generalizes the Schwarzschild solution to arbitrary dimensions, is
reconstructed at fourth order in the post-Minkowskian expansion from amplitude
calculations up to 3-loops, showing the existence of a precise relation
between the loop order in the amplitude computation and the post-Minkowskian
one in the classical formulation. Such derivation is performed in de Donder
gauge, and in order to compare the results with the classical metric one has
to write down the latter in the same gauge, in which logarithmic terms appear
in the post-Minkowskian expansion starting from the second order in five
dimensions and third order in four dimensions. Correspondingly the amplitudes
develop divergences whose renormalization needs the addition of non-minimal
couplings in the original action, originally noticed by Goldberger and
Rothstein in the context of world-line formalism in [21], as it is suggested
by general principles of the effective field theories. Such renormalization
process gives rise to logarithmic terms that are exactly the ones that appear
in the classical metric in de Donder gauge. Moreover in [22, 23] it is shown
how to recover the Schwarzschild-Tangherlini metric in a generic gravitational
gauge up to 1 loop, in which the same divergences are found, but it is proved
that there exists a particular gauge choice in which the metric is well-
defined in all space-time dimensions.

In this thesis it is shown how the analysis of [18] applied to the scattering
of charged scalars, whose interact with both gravitons and photons,
systematically leads to the post-Minkowskian expansion of the Reissner-
Nordström-Tangherlini solution (the generalization to arbitrary dimensions of
the Reissner-Nordström one), recovering the metric from graviton emission and
the electromagnetic potential from photon emission. The analysis of the metric
is a review of the theses [24, 25], in which it is presented how following the
prescription of [13] it is possible to isolate the classical terms of
scattering amplitudes of graviton emission with photons running in the loops,
in order to reconstruct the classical Reissner-Nordström-Tangherlini metric.
The calculations are performed up to 2 loops and a perfect agreement with the
classical solution rewritten in the de Donder gauge is obtained. The
renormalization procedure follows directly from the one outlined in [18], in
which the same non-minimal coupling is considered but generalized in order to
satisfy the electromagnetic gauge symmetry. The addition of a new coupling
constant in the theory, namely the electric charge, makes possible to define
other counter-terms that are needed in order to cancel the divergences that
appear in the electromagnetic sector of the theory. In fact while the pure
gravitational diagrams in which only gravitons circulate in the loops are
renormalized in [18], processes in which also internal photons are present
need a mixed counter-term which contains both gravitational and
electromagnetic coupling constant. So then, it is proved that the graviton
emission amplitudes in which photons circulate in the loops are renormalized
by considering the same counter-term of [18] combined with a new one
proportional to the electric charge.

However in order to fully reconstruct the Reissner-Nordström-Tangherlini
solution, besides the metric, one has to specify also the electromagnetic
potential associated to the black hole. So then the original work which aims
to reconstruct the electromagnetic potential of the solution from scattering
amplitudes is presented. The idea is that if the metric is recovered from
graviton emission, which reconstructs the gravitational source, then the
electromagnetic potential is reconstructed from photon emission processes from
which the electromagnetic current is obtained. The analysis is again performed
up to 2 loops and leads to a perfect agreement with the classical potential
associated to the Reissner-Nordström-Tangherlini solution in de Donder gauge
at third post-Minkowskian order. As it was for the metric case, the amplitudes
develop divergences, which in particular arise in five dimensions at 2 loops,
and the most important outcome is that they are renormalized by the insertion
of the same counter-terms that are needed for the renormalization of graviton
emission processes. Then the logarithmic terms that arise are compared with
the ones in the classical solution, and again a perfect match is found. The
thesis project has produced the publication of a paper in which the results
obtained in this work have been presented [26].

The reason why in the recent years the scientific community is becoming more
and more interested in recovering general relativity from quantum scattering
amplitudes, is because of the idea that this scheme allows to recover
gravitational observables in a more efficient and precise way than directly
solve the Einstein equations, in addition to the fact that these methods are
completely analytic. This idea has been tested in the context of gravitational
waves emitted from compact binary systems, in which emission templates are
recovered from the Newton potential between the binary constituents, that is
calculated employing scattering amplitudes techniques [27, 28]. In fact the
precision demanded by the future generation of gravitational-wave detectors
[29, 30, 31, 32], is beyond our theoretical capabilities prediction, which
make use of standard techniques like effective-one-body methods and numerical
relativity [33, 34, 35, 36]. Depending on the source, gravitational-waves from
binary systems will be observed with a signal-to-noise ratio that is one or
two orders of magnitude higher (more precise) than with current detectors [37,
38, 39]. So then the accuracy of current waveform models, which are built from
combining analytic and numerical relativity, needs to be improved by two order
of magnitudes. Therefore the necessity of new techniques which will improve
theoretical predictions is urgent, and the scattering amplitudes program is
candidate to be one of them, as well as a very promising analytic alternative
to the set of all gravitational processes which can be predicted only by
numerical methods.

Up to now, the state-of-the-art predictions by means of scattering amplitudes
methods are able to compute the gravitational-wave emission template in the
inspiral phase through the classical limit of 2-to-2 scattering amplitudes
between scalar particles mediated by gravitons, up to fourth order in the
post-Minkowskian expansion, as shown in figure 1.1.

Figure 1.1: 2-to-2 scattering amplitudes taken from [27], which reconstruct
the binary dynamics by means of interaction between massive particles (thick
lines) and gravitons (wiggly lines). The amplitudes (A) reconstruct the
conservative sector of the dynamics, (B) encodes all the radiative effects due
to gravitational-wave emission represented by a five point scattering
amplitude in which an external graviton is produced, while (C), through
higher-dimensional vertices (solid circle), take into account all the tidal
effects deriving from the fact that the binary constituents are not point
particles.

Combining these techniques with the effective-one-body formalism, we expect
that the required precision would be reached with calculations at
$O(G_{N}^{7})$ order in the post-Minkowskian expansion, or at $O(v^{12})$ in
the post-Newtonian one333While the post-Minkowskian expansion is relativistic
order by order, namely it is true for all velocities, the post-Newtonian
series is an expansion in velocities, which is non-relativistic., as sketched
in figure 1.2.

Figure 1.2: Map of perturbative corrections to the Newton potential taken from
[27], where $G_{N}$ is the universal gravitational Newton constant and $v$ is
the relative velocity of the binary constituents. The red box contains the
state-of-the-art calculations in the post-Minkowskian order, while the dark
triangle the ones in the post-Newtonian expansion. The light triangle shows
the contributions required by future detectors [40, 41, 42, 43, 44].

So then, the work presented in this thesis aims to be a step forward in the
understanding of how the next generation of theoretical gravitational
predictions could be made employing scattering amplitudes techniques.

The thesis is organized as follows. In chapter 2 it is first reviewed how to
interpret general relativity as a gauge theory. Then an attempt in order to
quantize gravity is made, proving how such strategy leads to a non-
renormalizable theory. The modern tools of effective field theories are
employed and the most generic quantum effective action compatible with the
symmetries of the theory is written. Finally it is discussed how to impose the
classical limit in this theory, and replacement rules that state how to
reinsert back $\hbar$ factors in the theory in order to perform the limit
$\hbar\rightarrow 0$ are obtained. In chapter 3 we present the review of the
original papers [7, 8, 9] as an explicit example of how to employ the
classical limit to obtain classical physics out of quantum computations.
Moreover the chapter gives the opportunity to introduce some fundamental
concepts and ideas very important in the following of the thesis. Then the
attention is focused on the derivation of the Reissner-Nordström-Tangherlini
solution from scattering amplitudes. In chapter 4 we report the classical
derivation of the solution by solving the Einstein and Maxwell equations.
Since the scattering amplitudes calculation we have in mind will be performed
in the de Donder gauge, in order to make possible the comparison of the two
derivations, the procedure to rewrite the Reissner-Nordström-Tangherlini
solution in such gauge is reviewed. Finally in chapter 5 the state-of-the-art
strategy to extract the classical limit out of graviton emission loop
amplitudes in generic dimensions, with photons and gravitons running in the
loops, is discussed and then applied to recover the Reissner-Nordström-
Tangherlini metric. In particular in section 5.3 the non-minimal action that
has to be added to the original theory in order to renormalize the divergences
that appear is defined. Then in chapter 6 the same analysis is performed in
order to consider the classical limit of photon emission process with both
gravitons and photons running in the loops. It is shown how such processes
reconstruct the electromagnetic potential associated to the Reissner-
Nordström-Tangherlini solution, and we discuss the relevant outcome by which
the divergences that appear in this case are renormalized by the very same
counter-terms that enter in the metric case. In the end chapter 7 contains a
discussion of the obtained results and an overview of future possible
perspectives.

The thesis contains also three appendices. In appendix A all the expressions
for the propagators and vertices that are used in this work are given. In
appendix B all the Fourier transforms employed to determine the metric and the
potential from the amplitudes are listed. Finally, in appendix C we list all
the loop integrals reduction formulas needed to manage the expressions that
arise from amplitude computations.

Notice that, unless it is specified the contrary, all the conclusions of this
thesis are carried out in generic space-time dimensions.

##  Chapter 2 Quantum gravity as an effective field theory

In the first section of this chapter it is briefly reviewed how to construct
scalar quantum electrodynamics by gauging the $\text{U}(1)$ global symmetry of
the scalar action, then by analogy it is shown how to recover general
relativity as a non-abelian gauge theory by gauging the space-time global
translational symmetry. Once one has a classical field theory in which gravity
is coupled to other fields, an attempt is made in order to quantize
gravitational interaction in the old fashion way employing the path integral
formalism, which however leads to a non-renormalizable theory. Then the modern
interpretation of effective field theories can make the theory consistent at
low energies, and the most generic quantum gravity effective Lagrangian is
written. Finally it is shown how to recover the classical limit of such theory
by explicitly restoring the $\hbar$’s and then performing the limit
$\hbar\rightarrow 0$. In this context the low-energy limit implies specific
rules of how vertices and propagators depend on the Planck constant, and they
are applied in order to recover classical physics from quantum computations in
an efficient way.

###  2.1 Scalar quantum electrodynamics

The action that describes a scalar complex massive field has the form

$S=\int
d^{D}x\Bigl{(}\partial_{\mu}\phi^{*}\partial^{\mu}\phi-m^{2}\phi^{*}\phi\Bigl{)}\
,$ (2.1)

where we neglect the self-interaction term and where we consider from here on
a $D$-dimensional flat background space-time with signature $(+,-,...,-)$ and
natural units in which $\hbar=c=1$. This theory has an abelian $\text{U}(1)$
global symmetry, which means that the action is invariant under the
transformation

$\phi(x)\rightarrow e^{-iQ}\phi(x)\ ,$ (2.2)

with $Q=const$. Since this is a continuous symmetry, due to the Noether
theorem there exists a conserved current, whose expression is

$j^{\mu}(x)=iQ\Big{(}\phi^{*}(x)\partial^{\mu}\phi(x)-\phi(x)\partial^{\mu}\phi^{*}(x)\Big{)}\
,$ (2.3)

and the relation

$\partial_{\mu}j^{\mu}(x)=0$ (2.4)

is satisfied. Now we want gauge the theory, which means make the original
global symmetry local. However the local field transformation

$\phi(x)\rightarrow e^{-iQ\alpha(x)}\phi(x)\ ,$ (2.5)

is not a symmetry of the theory since the space-time dependency in $\alpha(x)$
spoils it. In order to make the theory symmetric under this transformation, we
need to define a new vector field, which in turn it makes possible to define
the so called covariant derivative as

$D_{\mu}=\partial_{\mu}+iQA_{\mu}(x)\ ,$ (2.6)

where $A_{\mu}(x)$ is the gauge field of the theory, called electromagnetic
field, and $Q$ assumes the meaning of electric charge. Imposing that under the
symmetry transformation we are looking for, the gauge field transforms as

$A_{\mu}(x)\rightarrow A_{\mu}(x)+\partial_{\mu}\alpha(x)\ ,$ (2.7)

the theory described by the action

$S=\int
d^{D}x\Bigl{(}(D_{\mu}\phi)^{*}(D^{\mu}\phi)-m^{2}\phi^{*}\phi\Bigl{)}$ (2.8)

is now symmetric under the local gauge transformation.

Since we have introduced a new field, we have to consider its dynamics. The
kinetic term can be constructed from the so called strength tensor, defined
starting from the commutator between the covariant derivatives. Considering
$F_{\mu\nu}$ as the strength tensor of the theory, here called electromagnetic
tensor, schematically we can state

$\left[D_{\mu},D_{\nu}\right]\thicksim F_{\mu\nu}\ ,$ (2.9)

where the actual definition is

$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\ .$ (2.10)

Finally, in order to construct the kinetic term of the electromagnetic field,
we have to consider all the gauge invariant Lorentz scalars that can be made
by contractions of $F_{\mu\nu}$ and that give rise to terms quadratic in the
first derivative of the gauge field. From the symmetries of the
electromagnetic tensor the only contraction that one must take into account is
$F_{\mu\nu}F^{\mu\nu}$, from which the complete scalar electrodynamics action
reads [45]

$S=\int
d^{D}x\Biggl{(}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+(D_{\mu}\phi)^{*}(D^{\mu}\phi)-m^{2}\phi^{*}\phi\Biggl{)}\
,$ (2.11)

leading to the Maxwell equations for the electromagnetic potential.

From the classical action recovered in (2.11) we want to move to a quantum
field theory. The standard approach to do this, is to define a generating
functional through the path integral formalism. In order to quantize the
scalar electrodynamics then, we need to define

$Z=\int[d\phi][d\phi^{*}][dA_{\mu}]\,e^{iS}\ .$ (2.12)

However it is known, that apply this approach to a gauge invariant theory is
problematic, in fact since we are integrating even over those configurations
of the electromagnetic field which are related through a gauge transformation,
we are considering infinite copies of the same physical system, which will
give a divergent generating functional. The way to properly consider gauge
theories in this context was originally outlined by Faddeev and Popov [46], in
which they formalized the gauge fixing procedure in order to not integrate
over gauge equivalent orbits. The outcome is that the generating functional
can be rewritten as

$Z=\int[d\phi][d\phi^{*}][dA_{\mu}]\,\Delta[A_{\mu}]\,e^{iS-\frac{i}{2\xi}\int
d^{D}x\,f[A_{\mu}]^{2}}\ ,$ (2.13)

where $f[A_{\mu}]$ is the gauge fixing function that eliminates the gauge
freedom in the theory and $\Delta[A_{\mu}]$ is the so called Faddeev-Popov
determinant, that in terms of the gauge transformation in eq. (2.7) has the
explicit form

$\Delta[A_{\mu}]=\det(\frac{\partial
f^{\prime}}{\partial\alpha})=\det(\frac{\partial
f[A_{\mu}+\partial_{\mu}\alpha]}{\partial\alpha})\ .$ (2.14)

The above expression can be seen as an integral over Grassman variables and
takes the form

$\Delta[A_{\mu}]=\int[dc][d\bar{c}]\,e^{i\int d^{D}x\,\bar{c}\frac{\partial
f^{\prime}}{\partial\alpha}c}\ ,$ (2.15)

where $c$ is called ghost field. It does not correspond to any physical field
since it violates the spin-statistic theorem, however in order to preserve the
gauge invariance in the quantized version of the theory, it is necessary to
consider the ghost Lagrangian implicitly defined in (2.15) in addition to the
complete theory. Although we considered explicitly the electromagnetic field
as an example, this procedure holds even for more complicated non-abelian
gauge theories.

In the case of electromagnetism, the standard choice for the gauge fixing
function is $f[A_{\mu}]=\partial_{\mu}A^{\mu}$, which corresponds to the
Lorentz gauge and it is motivated by the fact that the field equations take a
particularly simple form. This choice leads to the Faddeev-Popov determinant

$\Delta[A_{\mu}]=\det(\frac{\partial}{\partial\alpha(y)}(\partial_{\mu}A^{\mu}(x)+\Box\,\alpha(x)))=\det(\Box\,\delta^{(D)}(x-y))=const\
,$ (2.16)

which shows the well-known result that an abelian theory does not require
ghost fields since they do not interact with the gauge field. Notice that in
our conventions $\Box=\partial_{t}^{2}-\nabla^{2}$, where
$\nabla^{2}=\partial_{i}\partial_{i}$. Then fixing the so called Feynman gauge
by making the choice $\xi=1$, which leads to the gauge fixing Lagrangian term

$\mathcal{\mathcal{L}}^{GF}_{A}=-\frac{1}{2}(\partial_{\mu}A^{\mu})^{2}\ ,$
(2.17)

considering its explicit addition in the action in (2.11), one finally obtains

$S=\int
d^{D}x\Biggl{(}\frac{1}{2}\partial_{\mu}A_{\nu}\partial^{\mu}A^{\nu}+\partial_{\mu}\phi^{*}\partial^{\mu}\phi-m^{2}\phi^{*}\phi+Q^{2}A_{\mu}A^{\mu}\phi^{*}\phi-j^{\mu}A_{\mu}\Biggl{)}\
,$ (2.18)

where $j^{\mu}(x)$ is exactly the Noether current associated to the global
symmetry in eq. (2.3). The action in (2.18) is the very well-known scalar
quantum electrodynamics action, which is renormalizable up to the addition of
the self-interaction term of the scalar field. From eq. (2.18) it is then
possible to recover the electromagnetic vertices in our conventions, as the
photon propagator and the 2 scalars - 1 photon vertex, reported in appendix A.
It is also possible to write down the electromagnetic field equations in the
Feynman gauge as

$\Box A^{\mu}(x)=j^{\mu}_{cov}(x)\ ,$ (2.19)

where $j^{\mu}_{cov}(x)$ is exactly the covariantized version of the global
Noether current as

$j^{\mu}_{cov}(x)=iQ\Big{(}\phi^{*}(x)D^{\mu}\phi(x)-\phi(x)D^{\mu}\phi^{*}(x)\Big{)}\
.$ (2.20)

From now on the subscript $"cov"$ will be omitted and it will be clear from
the context whether or not an expression is meant to be covariant.

Even if the procedure reviewed is performed considering the electromagnetic
field, it can be easily generalized to more complicated theories. For example
by gauging other symmetries of the theory it is possible to carry out
different gauge fields and interactions vertices.

###  2.2 General relativity as a gauge theory

Following a procedure similar to the one discussed above, it is possible to
recover general relativity by gauging the space-time translations, which are a
global symmetry of any field theory which is Poincaré invariant. This approach
is presented in [47], and differs from other approaches, like the one in [48].
Starting from the free action in (2.1), the theory is invariant under global
space-time translations

$x^{\mu}\rightarrow x^{\mu}+a^{\mu}\ ,$ (2.21)

where $a^{\mu}=const$. Due to the Noether theorem there exists a conserved
current associated to the continuous symmetry, defined up to a divergenceless
object, which has the expression

$T_{\phi}^{\mu\nu}=\partial^{\mu}\phi^{*}\partial^{\nu}\phi+\partial^{\nu}\phi^{*}\partial^{\mu}\phi-\eta^{\mu\nu}\left(\partial_{\alpha}\phi^{*}\partial^{\alpha}\phi-m^{2}\phi^{*}\phi\right)\
,$ (2.22)

and satisfies

$\partial_{\mu}T_{\phi}^{\mu\nu}(x)=0\ .$ (2.23)

The current in (2.22) is exactly the stress-energy tensor of the scalar field,
in fact one can check that $T_{\phi}^{00}=\mathcal{H}_{\phi}$, where
$\mathcal{H}_{\phi}$ is the Hamiltonian of the massive field.

Now following the same steps as in the electromagnetic case, we can gauge the
translations making the transformation local. This corresponds to considering
a coordinate transformation like

$x^{\mu}\rightarrow x^{\mu}+a^{\mu}(x)\ ,$ (2.24)

which since $a^{\mu}(x)$ is a generic function of space-time coordinates, it
can be rewritten as

$x^{\mu}\rightarrow\Lambda^{\mu}{}_{\nu}(x)x^{\nu}\ ,$ (2.25)

where one recognizes the structure of general coordinate transformations. So
then, imposing the symmetry under the gauge transformation, means considering
a theory invariant under general coordinate transformations, which is exactly
what the machinery of general relativity does. In fact introducing a new gauge
field in terms of a rank-2 symmetric tensor $h_{\mu\nu}(x)$ called graviton,
and defining the object

$g_{\mu\nu}(x)=\eta_{\mu\nu}+\kappa h_{\mu\nu}(x)\ ,$ (2.26)

which corresponds to the metric, we can build a covariantizing procedure as
usual in gauge theories. Imposing that under the local symmetry transformation
the gauge field implicitly transforms as

$g_{\mu\nu}(x)\rightarrow\Lambda^{\alpha}{}_{\mu}(x)\Lambda^{\beta}{}_{\nu}(x)g_{\alpha\beta}(x)\
,$ (2.27)

by defining the usual Christoffel symbols as

$\Gamma^{\rho}_{\mu\nu}=\frac{1}{2}g^{\rho\sigma}\left(\partial_{\mu}g_{\sigma\nu}+\partial_{\nu}g_{\sigma\mu}-\partial_{\sigma}g_{\mu\nu}\right)\
,$ (2.28)

and the covariant derivative

$D_{\mu}V^{\nu}=\partial_{\mu}V^{\nu}+\Gamma^{\nu}_{\mu\sigma}V^{\sigma}\ ,$
(2.29)

the covariantization of the theory is now straightforward, and it is done by
the following replacements [49]

$\displaystyle\eta_{\mu\nu}$ $\displaystyle\rightarrow g_{\mu\nu}$
$\displaystyle\partial_{\mu}$ $\displaystyle\rightarrow D_{\mu}$
$\displaystyle d^{D}x$ $\displaystyle\rightarrow d^{D}x\sqrt{-g}\ ,$ (2.30)

which leads to the matter action

$S=\int d^{D}x\sqrt{-g}\,\Biggl{(}(D_{\mu}\phi)^{*}(D_{\nu}\phi)\
g^{\mu\nu}-m^{2}\phi^{*}\phi\Biggl{)}\ .$ (2.31)

Now since we introduced a new gauge field, we have to make it dynamical. As
before we can define the gravitational strength tensor starting from the
commutator between covariant derivatives, which leads to the definition

$[D_{\alpha},D_{\beta}]V^{\mu}=R^{\mu}_{\ \nu\alpha\beta}V^{\nu}\ ,$ (2.32)

where $R^{\mu}_{\ \nu\alpha\beta}$ is exactly the Riemann tensor. From it, it
is possible to define the Ricci tensor and the Ricci scalar as

$R_{\mu\nu}=R^{\alpha}{}_{\mu\alpha\nu}\quad\text{and}\quad
R=g_{\mu\nu}R^{\mu\nu}\ .$ (2.33)

Now in order to construct the kinetic term of the gravitational field we have
to consider all the possible scalars of the theory made by contractions of the
Riemann tensor which are quadratic in the first derivative of the field. The
term we are looking for is just the Ricci scalar, and the kinematic term is
given by the well-known Einstein-Hilbert action

$S=-\frac{2}{\kappa^{2}}\int d^{D}x\sqrt{-g}\,R\ ,$ (2.34)

where

$\kappa^{2}=32\,\pi\,G_{N}\ ,$ (2.35)

with $G_{N}$ the Newton universal gravitational constant. We then reconstruct
general relativity as a gauge theory, and in particular since the general
coordinate transformations that play the role of gauge transformations are
non-abelian, gravity is placed in the category of non-abelian gauge theories.

Finally we can consider a classical action in which a complex scalar field
transforms under both electromagnetic and gravitational gauge transformations.
In such a theory a covariant derivative is meant to take into account all the
symmetries111For example if we consider a charged vector field $W^{\nu}$, then
$D_{\mu}W^{\nu}=\partial_{\mu}W^{\nu}+iQA_{\mu}W^{\nu}+\Gamma^{\nu}_{\mu\sigma}W^{\sigma}\
.$ and the gravitational covariantization procedure in eq. (2.30) has to be
carried out also in the electromagnetic sector of the theory. This gives rise
to interaction terms between the electromagnetic and gravitational field, as
it is clear from the complete action

$S=\int
d^{D}x\,\sqrt{-g}\,\Biggl{(}-\frac{2}{\kappa^{2}}R-\frac{1}{4}F_{\mu\nu}F_{\alpha\beta}\,g^{\alpha\mu}g^{\beta\nu}+(D_{\mu}\phi)^{*}(D_{\nu}\phi)g^{\mu\nu}-m^{2}\phi^{*}\phi\Biggl{)}\
.$ (2.36)

In this theory the graviton field is considered electrically neutral, while
for the electromagnetic tensor holds its original expression since

$F_{\mu\nu}=D_{\mu}A_{\nu}-D_{\nu}A_{\mu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}-\Gamma_{\mu\nu}^{\sigma}A_{\sigma}+\Gamma_{\mu\nu}^{\sigma}A_{\sigma}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\
.$ (2.37)

From the action (2.36) one derives the Einstein equations as the field
equations of the gravitational field

$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\frac{\kappa^{2}}{4}T_{\mu\nu}\
,$ (2.38)

where $T_{\mu\nu}$ is the stress-energy tensor of the complete system, which
takes into account both scalar and electromagnetic fields. In fact we can
state that

$T^{\mu\nu}=T_{\phi}^{\mu\nu}+T_{A}^{\mu\nu}+T_{\phi A}^{\mu\nu}\ ,$ (2.39)

where $T_{A}^{\mu\nu}$ is the electromagnetic stress-energy tensor, which
corresponds exactly to the Noether current associated to the global
translational symmetry of the electromagnetic field, and takes the
expression222The tensor in (2.40) corresponds to the Noether conserved current
up to a symmetrization and a divergenceless object, since these operations do
not spoil the conservation condition. Moreover, since the Einstein equations
have to be symmetric under gauge transformations, the Noether current has to
be covariantized, and $T_{A}^{\mu\nu}$ must be written in terms of the metric
$g_{\mu\nu}$, exactly what we already commented in eq. (2.20).

$T_{A}^{\mu\nu}=F^{\mu\alpha}F_{\alpha}{}^{\nu}+\frac{1}{4}g^{\mu\nu}F^{\alpha\beta}F_{\alpha\beta}\
,$ (2.40)

often called in literature Belinfante–Rosenfeld stress-energy tensor, while
$T_{\phi A}^{\mu\nu}$ is the mixed term between scalar and electromagnetic
field. Equations (2.38) respect the general behavior of gauge theories, in
which the covariantized version of the current associated to the original
global symmetry enters in the field equations of the gauge field as a source.
Moreover eq. (2.38) shows the characteristic non-linearity of gravity, which
becomes relevant in the quantization process. In the end we notice that in a
similar way, everything we have done considering a scalar field can be
generalized to massive spinors [47].

###  2.3 Quantization of gravity

The complete classical action in (2.36) is constructed in terms of the metric,
which in turn contains the graviton field in its definition. In order to make
the physical content of the theory explicit in terms of graviton field
interactions, one has to consider the expansions

$g^{\mu\nu}=\eta^{\mu\nu}-\kappa
h^{\mu\nu}+\kappa^{2}h^{\mu}{}_{\alpha}h^{\alpha\nu}+O(\kappa^{3})$ (2.41)

and

$\sqrt{-g}=1+\frac{\kappa}{2}h+\frac{\kappa^{2}}{4}\left(\frac{1}{2}h^{2}-h^{\alpha}{}_{\beta}h^{\beta}{}_{\alpha}\right)+O(\kappa^{3})\
,$ (2.42)

by which it is possible to expand the theory perturbatively around the
coupling $\kappa$, which makes the strongly non-linearity of gravity manifest.
Considering the expansion of the Christoffel symbols in powers of coupling
constant, explicitly derived by the insertion of (2.41) and (2.42) in its
definition, schematically one obtains

$\Gamma^{\mu}_{\alpha\beta}=\Gamma^{\mu\ (1)}_{\alpha\beta}+\Gamma^{\mu\
(2)}_{\alpha\beta}+O(\kappa^{3})\ ,$ (2.43)

where explicitly

$\begin{gathered}\Gamma^{\mu\
(1)}_{\alpha\beta}=\frac{\kappa}{2}\eta^{\mu\nu}\Bigl{(}\partial_{\beta}h_{\nu\alpha}+\partial_{\alpha}h_{\nu\beta}-\partial_{\nu}h_{\alpha\beta}\Bigl{)}\\\
\Gamma^{\mu\
(2)}_{\alpha\beta}=-\frac{\kappa^{2}}{2}h^{\mu\nu}\Bigl{(}\partial_{\beta}h_{\nu\alpha}+\partial_{\alpha}h_{\nu\beta}-\partial_{\nu}h_{\alpha\beta}\Bigl{)}\
.\end{gathered}$ (2.44)

From this expansion it is possible to consider also

$R_{\mu\nu}=R_{\mu\nu}^{(1)}+R_{\mu\nu}^{(2)}+O(\kappa^{3})\quad\text{and}\quad
R=R^{(1)}+R^{(2)}+O(\kappa^{3})\ ,$ (2.45)

where for example at first order they explicitly read

$\begin{gathered}R_{\mu\nu}^{(1)}=\frac{\kappa}{2}\Bigl{(}\partial_{\mu}\partial_{\nu}h+\Box
h_{\mu\nu}-\partial_{\mu}\partial_{\lambda}h^{\lambda}{}_{\nu}-\partial_{\nu}\partial_{\lambda}h^{\lambda}{}_{\mu}\Bigl{)}\\\
R^{(1)}=\kappa\Bigl{(}\Box h-\partial_{\mu}\partial_{\nu}h^{\mu\nu}\Bigl{)}\
,\end{gathered}$ (2.46)

where $h=\eta^{\mu\nu}h_{\mu\nu}$.

This expansion can be performed for each term in the action (2.36). Rewriting
it as

$S=\int
d^{D}x\,\sqrt{-g}\,\Biggl{(}\mathcal{L}_{h}+\mathcal{L}_{A}+\mathcal{L}_{\phi}+\mathcal{L}_{\phi^{2}A}+\mathcal{L}_{\phi^{2}A^{2}}\Biggl{)}\
,$ (2.47)

where respectively

$\begin{gathered}\mathcal{L}_{h}=-\frac{2}{\kappa^{2}}R\\\
\mathcal{L}_{A}=-\frac{1}{4}F_{\mu\nu}F_{\alpha\beta}\,g^{\alpha\mu}g^{\beta\nu}\\\
\mathcal{L}_{\phi}=(\partial_{\mu}\phi)^{*}(\partial_{\nu}\phi)g^{\mu\nu}-m^{2}\phi^{*}\phi\\\
\mathcal{L}_{\phi^{2}A}=-j_{\mu}A_{\nu}g^{\mu\nu}\\\
\mathcal{L}_{\phi^{2}A^{2}}=Q^{2}A_{\mu}A_{\nu}g^{\mu\nu}\phi^{*}\phi\
,\end{gathered}$ (2.48)

one gets the expansions

$\begin{gathered}\sqrt{-g}\mathcal{L}_{h}=-\frac{2}{\kappa^{2}}\Bigl{(}R^{(1)}+R^{(2)}+\frac{\kappa}{2}hR^{(1)}\Bigl{)}+O(\kappa)\\\
\sqrt{-g}\mathcal{L}_{A}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{\kappa}{2}h_{\mu\nu}T^{\mu\nu}_{A}+O(\kappa^{2})\\\
\sqrt{-g}\mathcal{L}_{\phi}=(\partial_{\mu}\phi)^{*}(\partial^{\mu}\phi)-m^{2}\phi^{*}\phi-\frac{\kappa}{2}h_{\mu\nu}T^{\mu\nu}_{\phi}+O(\kappa^{2})\\\
\sqrt{-g}\mathcal{L}_{\phi^{2}A}=-j_{\mu}A^{\mu}+\kappa
j_{\mu}A_{\nu}\Bigl{(}h^{\mu\nu}-\frac{1}{2}h\,\eta^{\mu\nu}\Bigl{)}+O(\kappa^{2})\\\
\sqrt{-g}\mathcal{L}_{\phi^{2}A^{2}}=Q^{2}A_{\mu}A^{\mu}\phi^{*}\phi-\kappa
Q^{2}A_{\mu}A_{\nu}\phi^{*}\phi\Bigl{(}h^{\mu\nu}-\frac{1}{2}h\,\eta^{\mu\nu}\Bigl{)}+O(\kappa^{2})\
,\end{gathered}$ (2.49)

where the indices are raised and lowered by the flat Minkowskian metric
$\eta_{\mu\nu}$. The above expansion reproduces the terms at zeroth order in
the graviton as well as terms that describe the interaction between
electromagnetic, scalar and gravitational fields. For what concerns the pure
gravity sector, the term linear in $h_{\mu\nu}$ in the expansion of the pure
gravitational Lagrangian does not contribute since it is a total derivative,
then it will be no longer considered in this thesis. The term quadratic in the
graviton represents the kinetic term of the field, while higher terms in the
expansion of $\mathcal{L}_{h}$ represent the self-interaction of the graviton.
Then the pure scalar and electromagnetic sector, as usual in any gauge theory,
give rise to an interaction term linear in the graviton which couples the
gauge field directly with the original global Noether current. Again, due to
the non-linearity of gravity, there are infinitely many other interaction
terms that are higher order in the graviton field. Finally there exist terms
which involve the simultaneous interaction of all the fields, with an
arbitrary number of graviton fields involved, that correspond to the classical
mixed contributions to the stress-energy tensor.

Now that the field content of the theory is manifest, it is possible to
quantize the gravitational field [1, 2, 3, 4]. Following the procedure
outlined in section 2.1, since general relativity is a gauge theory, in order
to quantize it we have to work out the Faddeev-Popov procedure for the gauge
fixing, where now the quantum gauge field that mediates the interaction is the
graviton field $h_{\mu\nu}$ [5]. For simplicity we first consider gravity only
coupled to the scalar field, and then after the quantization procedure we put
back the electromagnetic interaction. The generating functional associated to
the action in eq. (2.36), neglecting the electromagnetic interaction, is
expected to be

$Z=\int[d\phi][d\phi^{*}][dh_{\mu\nu}]\,e^{iS}\ ,$ (2.50)

where explicitly the kinetic term of the graviton field reads

$\mathcal{L}_{h}=\frac{1}{2}\partial_{\sigma}h_{\mu\nu}\partial^{\sigma}h^{\mu\nu}-\partial^{\nu}h_{\mu\nu}\partial_{\sigma}h^{\mu\sigma}+\partial^{\nu}h_{\mu\nu}\partial_{\mu}h-\frac{1}{2}\partial_{\mu}h\partial^{\mu}h+O(\kappa)\
.$ (2.51)

As already noticed in the electromagnetic case, eq. (2.50) diverges since we
are integrating over infinite equivalent copies of the same physical systems
due to gauge invariance. In order to remove such redundancy, we must choose a
gauge, rewriting the generating functional in the form

$Z=\int[d\phi][d\phi^{*}][dh_{\mu\nu}]\,\Delta[h_{\mu\nu}]e^{iS+i\xi\int
d^{D}x\,F_{\lambda}F^{\lambda}}\ ,$ (2.52)

where $F_{\lambda}[h_{\mu\nu}]$ is the gauge fixing functional and
$\Delta[h_{\mu\nu}]$ is the Faddeev-Popov determinant. Differently from
electromagnetism, gravity is non-abelian, this means that the ghost Lagrangian
that arises by the Faddeev-Popov determinant becomes relevant for the gauge
invariance of the theory. Considering that under an infinitesimal gauge
transformation333In literature one usually refers to this kind of
transformation as the Lie derivative of the metric. the transformed graviton
field reads [50]

$\kappa\,h^{\prime}_{\mu\nu}=\kappa\,h_{\mu\nu}-\partial_{\mu}a_{\nu}-\partial_{\nu}a_{\mu}-\kappa\,h_{\lambda\nu}\partial_{\mu}a^{\lambda}-\kappa\,h_{\lambda\mu}\partial_{\nu}a^{\lambda}-\kappa\,a^{\lambda}\partial_{\lambda}h_{\mu\nu}+O(a^{2})\
,$ (2.53)

one can consequently obtain

$\Delta[h_{\sigma\rho}]=\det(\frac{\partial
F^{\mu}[h^{\prime}_{\sigma\rho}]}{\partial
a^{\nu}})=\int[dc_{\mu}][d\bar{c}^{\nu}]e^{i\int
d^{D}x\,\bar{c}^{\nu}\frac{\partial F^{\prime\mu}}{\partial a^{\nu}}c_{\mu}}\
,$ (2.54)

where $c_{\mu}$ and $\bar{c}^{\nu}$ are the anticommuting $"\text{fermionic
vector}"$ ghost fields.

In the context of quantization a particular convenient gauge choice is the de
Donder gauge, whose condition is defined by

$\eta^{\mu\nu}\Gamma_{\mu\nu}^{\alpha}=0\ ,$ (2.55)

from which in terms of the graviton field can be rewritten as

$\partial^{\mu}h_{\mu\alpha}-\frac{1}{2}\partial_{\alpha}h=0\ .$ (2.56)

Now we can consider the gauge fixing function associated to the de Donder
gauge, which is exactly

$F_{\lambda}=\partial^{\mu}h_{\mu\lambda}-\frac{1}{2}\partial_{\lambda}h\ ,$
(2.57)

from which it is possible to compute explicitly the ghost Lagrangian.
Considering the infinitesimal gauge transformation in eq. (2.53), then one
gets

$F^{\prime}_{\lambda}=F_{\lambda}-\Box a_{\lambda}+O(\kappa)\ ,$ (2.58)

that leads to

$\mathcal{L}_{ghost}=\partial_{\mu}\bar{c}_{\nu}\partial^{\mu}c^{\nu}+O(\kappa)\
.$ (2.59)

From the above Lagrangian one can derive the Feynman rules that involve ghosts
in the old fashion way, in particular from the kinetic term one computes the
propagator

$\includegraphics[width=78.04842pt,valign={c}]{Diagrams/Ghost_Propagator.png}=-\frac{i\eta^{\mu\nu}}{q^{2}+i\epsilon}\
,$ (2.60)

while an explicit calculation of the interaction term will give rise to the
vertex (for the explicit derivation see [50]).

Now it is possible to exploit the gauge fixing procedure to recover the
graviton propagator. In fact considering the addition of

$\mathcal{L}_{h}^{GF}=F_{\lambda}F^{\lambda}$ (2.61)

to the Lagrangian in eq. (2.51), choosing $\xi=1$, then the kinematical term
of the graviton reads

$S=\int d^{D}x\,\Bigl{(}\mathcal{L}_{h}+\mathcal{L}_{h}^{GF}\Bigl{)}=\int
d^{D}x\,\Biggl{(}\frac{1}{2}\partial_{\sigma}h_{\mu\nu}\partial^{\sigma}h^{\mu\nu}-\frac{1}{4}\partial_{\mu}h\partial^{\mu}h+O(\kappa)\Biggl{)}\
,$ (2.62)

by which after an integration by parts one gets

$S=\int
d^{D}x\,\Biggl{(}h_{\mu\nu}V^{\mu\nu\alpha\beta}h_{\alpha\beta}+O(\kappa)\Biggl{)}\
,$ (2.63)

where

$V^{\mu\nu\alpha\beta}=\left(\frac{1}{4}\eta^{\mu\nu}\eta^{\alpha\beta}-\frac{1}{2}\eta^{\mu\alpha}\eta^{\nu\beta}\right)\Box\
.$ (2.64)

The operator above can be inverted in order to determine the graviton
propagator in the de Donder gauge, which in generic space-time dimension reads
[47]

$\includegraphics[width=86.72267pt,valign={c}]{Diagrams/Graviton_Propagator.png}=i\frac{P_{\alpha\beta,\mu\nu}}{q^{2}+i\epsilon}\
,$ (2.65)

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

$P_{\alpha\beta,\mu\nu}=\frac{1}{2}\Big{(}\eta_{\mu\alpha}\eta_{\nu\beta}+\eta_{\mu\beta}\eta_{\nu\alpha}-\frac{2}{D-2}\eta_{\mu\nu}\eta_{\alpha\beta}\Big{)}\
.$ (2.66)

Notice that here and in the following the double wiggly lines stand for
gravitons. In the same gauge one can find the field equations of the graviton
field, which read

$\Box\Bigl{(}h_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}h\Bigl{)}=-\frac{\kappa}{2}\Bigl{(}T_{\mu\nu}+t_{\mu\nu}\Bigl{)}\
,$ (2.67)

where $t_{\mu\nu}$ is the stress-energy tensor associated to the gravitational
field itself. In fact the non-linearity of general relativity gives rise to
gravitational self-interaction terms, that can be computed from the higher
order terms in $\sqrt{-g}\mathcal{L}_{h}$. From here on we will write only
$T_{\mu\nu}$ as a unique stress-energy tensor, and it will be clear from the
context if a gravitational contribution is implied. Finally, by tracing the
field equations, one obtains

$\Box
h_{\mu\nu}=-\frac{\kappa}{2}\Biggl{(}T_{\mu\nu}-\eta_{\mu\nu}\frac{1}{D-2}T\Biggl{)}\
.$ (2.68)

Now putting back the electromagnetic interaction, since the gauge we choose is
second order in the first derivative of the field, it does not affect the
expressions of the other terms in eq. (2.49) rather than the kinetic one. So
it is possible to write down all the Feynman rules of the theory, keeping in
mind that the interaction terms in (2.49) are just the first pieces of an
infinite expansion in the gravitational coupling constant, since the non-
linearity of gravity gives rise to vertices with an arbitrary number of
gravitons attached to them [51]. Some of these, in particular the ones useful
for further analysis in this thesis, are reported in appendix A. It is worth
to comment the 3 graviton vertex that arises from the $O(\kappa)$ contribution
of the Einstein-Hilbert action expansion. The explicit expression of it was
originally provided by de Witt [4], in which the vertex is recovered in the
old fashion way where the expression is symmetric under the exchange of any
leg, and is listed in (A.9). Since such object can be painful to handle due to
the proliferation of Lorentz indices, one can derive a simplified expression
that describes the interaction of two quantum fields and an external field.
The expression was firstly given by Donoghue [7], which used the background
field method and obtained the vertex reported in eq. (A.10). The idea is to
expand the metric rather than on a flat background space-time, around a
generic $\bar{g}_{\mu\nu}$ metric, such as

$g_{\mu\nu}=\bar{g}_{\mu\nu}+\kappa\,h_{\mu\nu}\ .$ (2.69)

Imposing than $\bar{g}_{\mu\nu}=\eta_{\mu\nu}+\kappa\,H_{\mu\nu}$, obtaining
the relation

$g_{\mu\nu}=\eta_{\mu\nu}+\kappa\,H_{\mu\nu}+\kappa\,h_{\mu\nu}\ ,$ (2.70)

where $H_{\mu\nu}$ is an external gravitational field, by replacing this
expansion inside the Einstein-Hilbert action, the $O(\kappa)$ piece with two
quantum fields and an external one gives rise to the aforementioned vertex.
Notice that by construction such vertex is not symmetric under the exchange of
all the legs, but only between the internal ones, therefore it is not possible
to use such expression as an internal vertex.

Once the quantization procedure is done and the Feynman rules are carried out,
the natural question is whether this theory is renormalizable or not. The
gravitational Einstein-Hilbert action, which gives rise to the propagator and
the self-interaction terms of the graviton, is quadratic in the first
derivatives of the metric. This means that taking $q$ as a characteristic
momentum, the propagator goes like $q^{-2}$ and each vertex goes like $q^{2}$.
Moreover in a $D$-dimensional space-time the measure of a loop integral is
$d^{D}q$, from which considering a diagram with $V$ vertices, $I$ internal
lines and $L$ loops, the superficial degree of divergence of the theory reads

$\mathfrak{D}=DL+2V-2I\ .$ (2.71)

Replacing the topological relation

$L=1+I-V\ ,$ (2.72)

one obtains the relation

$\mathfrak{D}=2+(D-2)L\ .$ (2.73)

It is easy to see that for dimensions $D>2$ the superficial degree of
divergence of the theory is positive and increases with the number of loops,
therefore the theory is non-renormalizable since an infinite number of
processes produce divergent amplitudes. This can be seen also from the
dimension of gravitational coupling. In fact in terms of the canonical mass
dimension in which $[M]=1$, the dimension of the gravitational Newton constant
is

$[G_{N}]=2-D\ ,$ (2.74)

which is negative for $D>2$, and then leads to a non-renormalizable theory.

What happens when a theory is non-renormalizable, is that since each physical
process is divergent, one should in principle add to the original Lagrangian
an infinite number of terms in order to cancel the infinite amount of
divergences that appear, for this reason such kind of theories are classified
as non-fundamental ones. However in the modern way of thinking, non-
renormalizable theories can be interpreted as effective field theories, which
can be used to extrapolate predictions below some energy cut-off, in which
above of it the effective theory is not useful anymore. The idea is that at
high energies the unknown fundamental theory possesses some heavy degrees of
freedom, which however at low energy are expected to not contribute, and their
effect can be parametrized by the addition in the Lagrangian of all the
possible higher-derivative non-minimal couplings compatible with the symmetry
of the theory, whose ordered in an energy expansion mimic the fundamental
theory and separates high-energy effects from the low-energy ones.

Embracing this idea and considering for the moment only the purely
gravitational action, one can then rewrite it in terms of an infinite series
of all the possible Riemann tensor contractions as

$S=\int
d^{D}x\,\sqrt{-g}\Biggl{(}-\Lambda-\frac{2}{\kappa^{2}}R+c_{1}R^{2}+c_{2}R_{\mu\nu}R^{\mu\nu}+c_{3}R_{\alpha\beta\sigma\rho}R^{\alpha\beta\sigma\rho}+\cdots\Biggl{)}\
,$ (2.75)

where $\Lambda,c_{1},c_{2},c_{3}...$ are effective couplings that have to be
fixed experimentally. Since we are going to talk about their experimental
values, it is useful to switch to $D=4$ until the end of this section, where
in particular in four space-time dimensions the term proportional to $c_{3}$
disappears due to the Gauss-Bonnet identity. From the fact that the Riemann
tensor is quadratic in the derivative of the metric, schematically written as
$R_{\mu\nu\alpha\beta}\thicksim\partial^{2}$, in momentum space it is
considered to be quadratic in energy. Therefore the energy expansion is then
organized as follows

$\Lambda\thicksim O(1)\quad R\thicksim O(E^{2})\quad
R^{2},R_{\mu\nu}R^{\mu\nu}\thicksim O(E^{4})\ ,$ (2.76)

meaning that the contribution of higher-order terms in energy becomes
negligible when low-energy processes are considered. In fact schematically the
coefficients $\Lambda,c_{1},c_{2},...$ are not dimensionless in $D$ dimension,
and they can be rewritten making explicit the cut-off energy scale of the
gravitational interaction. We believe that such a scale is identified by the
Planck scale, above which a fundamental theory of quantum gravity is needed.
Including the Planck mass $M_{p}\simeq 10^{19}\ \text{GeV}$ and the
dimensionless coefficient $\lambda,d_{1},d_{2},d_{3},...$, then it is possible
to rewrite the effective action (2.78) in a more manifest way as

$S=\int
d^{D}x\,\sqrt{-g}\Biggl{(}-M_{p}^{D}\lambda-\frac{2}{\kappa^{2}}R+\frac{d_{1}}{M_{p}^{4-D}}R^{2}+\frac{d_{2}}{M_{p}^{4-D}}R_{\mu\nu}R^{\mu\nu}+\frac{d_{3}}{M_{p}^{4-D}}R_{\alpha\beta\sigma\rho}R^{\alpha\beta\sigma\rho}+\cdots\Biggl{)}\
,$ (2.77)

which in $D=4$ reads

$S=\int
d^{4}x\,\sqrt{-g}\Biggl{(}-M_{p}^{4}\lambda-\frac{2}{\kappa^{2}}R+d_{1}R^{2}+d_{2}R_{\mu\nu}R^{\mu\nu}+\cdots\Biggl{)}\
.$ (2.78)

Once the most generic effective Lagrangian is defined, one can think about
fixing the effective coupling constants.

  * •

The constant $\Lambda$ is called cosmological constant, and it is believed to
be responsible for the current accelerated expansion of the Universe. By
observations we inferred that its value should approximately be $\Lambda\simeq
10^{-47}\ \text{GeV}^{4}$. The energy of such object is very tiny compared to
ordinary scales in which gravity occurs, so then we will neglect this term any
further in this thesis.

  * •

The parameter $\kappa^{2}$ defines the strength of the gravitational field at
long distances and its value is fixed by matching the classical non-
relativistic potential. In fact the contribution to the Newton potential only
due to the Einstein-Hilbert action is

$V(r)=-\frac{\kappa^{2}}{32\pi}\frac{m_{1}m_{2}}{r}=-G_{N}\frac{m_{1}m_{2}}{r}\
.$ (2.79)

  * •

The constants $d_{1},d_{2},...$ produce Yukawa–type corrections to the
gravitational potential which become relevant at short distances. Indeed by
measuring the classical contribution to the gravitational potential of such
pieces [52], by experiments one gets $|d_{i}|<10^{-56}$. The length scale at
which these terms are relevant is well below the astrophysical distances we
are interested in, so any further in this thesis we will neglect any high-
energy Riemann coupling.

Finally we can consider the non-minimal coupling between the gravitational
field, the massive scalar and the photon. Considering the terms in eq. (2.48),
one should consider also the addition of terms like

$\begin{gathered}\mathcal{L}_{\phi}\thicksim
m^{2}R\phi^{*}\phi,\,R^{\mu\nu}\partial_{\mu}\phi^{*}\partial_{\nu}\phi,\,Rg^{\mu\nu}\partial_{\mu}\phi^{*}\partial_{\nu}\phi,\,...\\\
\mathcal{L}_{A}\thicksim
F_{\mu\nu}F^{\mu\nu}R,\,F_{\mu\nu}R^{\mu\nu},\,F_{\mu\alpha}F^{\mu}{}_{\beta}R^{\alpha\beta},\,...\end{gathered}$
(2.80)

In order to organize an energy expansion of these terms, one has to notice
that in this effective field theory approach, the energy of the processes,
namely the typical transferred momentum between interacting particles, must be
much smaller than every energy scale of the theory. This means that since in
our theory also a massive field is present, this gives rise to another energy
scale rather than the Planck one, and in the effective approach the typical
transferred momentum must be $q\ll M_{p},\,m$, which means that the derivative
of massive fields is grater in energy than the derivative of massless fields,
since in momentum space $m\thicksim\partial\phi$. In virtue of this
consideration it is crucial to differentiate the quantum effects of massless
particles from the ones which have mass. While massless particles can
propagate to long distances at low energies, massive particles cannot. In fact
considering schematically a massive propagator in momentum space, since
$q^{2}\ll m^{2}$ one can expand it as

$\frac{1}{q^{2}-m^{2}+i\epsilon}=-\frac{1}{m^{2}}-\frac{q^{2}}{m^{4}}-\frac{q^{4}}{m^{6}}+O(q^{6})\
.$ (2.81)

Then going back to coordinate space, one gets

$-\frac{1}{m^{2}}\thicksim\delta^{(D)}(x)\quad-\frac{q^{2}}{m^{4}}\thicksim\partial^{2}\quad-\frac{q^{4}}{m^{6}}\thicksim\partial^{4}\
,$ (2.82)

meaning that the leading order in the expansion of the massive propagator
gives a local contribution, which in turn generates an effective interaction
vertex, while higher order terms can be just reabsorbed in high-derivative
couplings in the effective action. Obviously such expansion cannot be
performed on massless propagators which go like $1/q^{2}$, and whose
contribution cannot be integrated out.

$\includegraphics[width=108.405pt,valign={c}]{Diagrams/Local1.png}\underrightarrow{q^{2}\ll
m^{2}}\includegraphics[width=108.405pt,valign={c}]{Diagrams/Local2.png}$
Figure 2.1: In the left a 2-to-2 graviton scattering process which is mediated
by a massive loop, which in the low-energy limit degenerate in an effective 4
graviton vertex on the right.

Then the final outcome is that a process in which the interaction is mediated
by massive lines will collapse in an effective vertex as in figure 2.1, as
usually happens when the heavy degrees of freedom are integrated out in an
effective field theory.

So in the very end, quantum gravity as an effective field theory is a well-
defined quantum theory in the low-energy regime, in which at any fixed loop
order it is possible to obtain finite predictions, as it is shown in [53],
where the 2-loop effective pure gravitational Lagrangian is obtained. Since
this approach only enables us to make predictions when the exchanged momentum
between virtual particles is well below all the energy scales of the theory,
in coordinate space this is translated to the fact that we can only infer the
long-range physics of the systems.

###  2.4 Classical limit

In the last section we reviewed how the effective field theory approach to
quantum gravity gives a perfectly well-defined quantum theory, able to make
predictions in the low-energy regime. Then as any other quantum theory, the
classical limit, in which $\hbar\rightarrow 0$, must reconstruct the original
classical phenomenology, that in this case corresponds to general relativity.
In order to do that, we have to restore the explicit presence of $\hbar$ in
the theory. Generalizing to arbitrary space-time dimensions the procedure
outlined in [54], we still consider relativistic natural units in which $c=1$,
while the Planck constant is considered to have dimensions

$[\hbar]=L\cdot M\ ,$ (2.83)

where $L$ is a length dimension and $M$ is a mass dimension (which is
equivalent to an energy). In order to recover the dimension of both fields and
couplings, now the dimensional analysis must be performed keeping in mind that
the dimensionless object is $S/\hbar$ and therefore the generating functional
of the theory becomes

$Z=\int[d\phi][d\phi^{*}][dA_{\mu}][dh_{\alpha\beta}]e^{iS/\hbar}\ .$ (2.84)

From this consideration in a $D$-dimensional space-time the Lagrangian
satisfies

$[\mathcal{L}]=[\hbar]\,L^{-D}\ .$ (2.85)

Studying the kinetic term of the fields, each boson of the theory has the same
dimension, which is given by

$[\phi]=[A_{\mu}]=[h_{\mu\nu}]=[\hbar]^{\frac{1}{2}}\,L^{-\frac{D}{2}+1}\ .$
(2.86)

Then considering the interaction terms of the theory and imposing eq. (2.85),
it is possible to recover whether it is needed or not to add an extra $\hbar$
in the Lagrangian by studying the dimension of the couplings.

  * •

For what concerns the mass term, adding an unknown dimensional constant, the
equation to solve is

$[x\,m^{2}\phi^{2}]=[\hbar]\,L^{-D}\ ,$ (2.87)

which leads to

$[x]=[\hbar]^{-2}\ .$ (2.88)

This means that whenever $m$ appears in the theory, we have to consider the
replacement

$m\rightarrow\frac{m}{\hbar}\ .$ (2.89)

  * •

Considering the electromagnetic interaction term which involves the charge
$Q$, the equation to solve is

$[x\,QA_{\mu}\phi\partial^{\mu}\phi]=[\hbar]\,L^{-D}\ .$ (2.90)

However one has also to keep in mind that the fine structure constant defined
by

$\alpha=\frac{Q^{2}}{4\pi\,\hbar}\ ,$ (2.91)

has to be dimensionless in $D=4$. So then the only solution that satisfies
both relations is

$[Q]=[\hbar]^{\frac{1}{2}}\,L^{\frac{D}{2}-2}\quad\text{and}\quad[x]=[\hbar]^{-1}\
,$ (2.92)

which in turn implies the replacement

$Q\rightarrow\frac{Q}{\hbar}\ ,$ (2.93)

whenever a charge is present.

  * •

The last coupling to take into account is the gravitational one. As it was for
the charge we need to satisfy two dimensional relations. The first is due by
considering the dimension of a coupling that involves $\kappa$. For example
taking the 2 scalars - 1 graviton vertex the dimensional equation is

$[x\,\kappa h_{\mu\nu}T_{\phi}^{\mu\nu}]=[\hbar]\,L^{-D}\ ,$ (2.94)

with

$[T_{\phi}^{\mu\nu}]=[\hbar]\,L^{-D}\ .$ (2.95)

The other relation to be satisfied is the one that defines the Planck mass,
which is

$M_{p}=\sqrt{\frac{\hbar}{G_{N}}}\ ,$ (2.96)

and in $D=4$ imposes

$[G_{N}]=[\hbar]^{-1}\,L^{2}\ .$ (2.97)

Keeping in mind that $[\kappa]=[G_{N}]^{\frac{1}{2}}$, than the solution is

$[G_{N}]=[\hbar]^{-1}L^{D-2}\quad\text{and}\quad[x]=1\ ,$ (2.98)

which means that terms involving $\kappa$ do not need an explicit insertion of
$\hbar$ factors.

Now that the explicit presence of $\hbar$’s is restored in the theory, the
idea is to study how Feynman rules change, in order to consider the explicit
dependence of the Planck constant in the diagrams. As in any other approach
like this one, the fact that the ration $S/\hbar$ enters in the generating
functional, means that for each vertex we have to add an extra $\hbar^{-1}$,
and for each propagator we have to consider a $\hbar$ factor in front.
Moreover, whenever one encounters a momentum, this is implied to be a wave-
number $\tilde{q}$ instead of a physical one, where the relation that links
each other is $q=\tilde{q}\,\hbar$, with $[\tilde{q}]=L^{-1}$. Making this
explicit, for example the scalar propagator in coordinate space reads

$i\Delta(x-y)=\int
d^{D}\tilde{q}\,e^{i\tilde{q}\cdot(x-y)}\frac{i\,\hbar}{\tilde{q}^{2}-\frac{m^{2}}{\hbar^{2}}+i\epsilon}\
.$ (2.99)

In this context it is important to remind the considerations of the section
before, in which we underlined the differences between massive and massless
particles since we constantly consider the low-energy limit. Even though we
said that the only processes which give long-distance contribution are the
ones with no massive particles circulating in the loops, an internal massive
line may be attached to an external massive state, since the constraint of
low-energy transferred momentum would be still valid. Consider then a massive
propagator with momentum $q=p-k$, where $p$ is the momentum of the external
particle, which satisfies the mass-shell relation $p^{2}=m^{2}$, and $k$ is an
exchanged momentum that respects $k^{2}\ll m^{2}$. In this case the massive
propagator can be expanded as

$\frac{\hbar}{(\tilde{p}-\tilde{k})^{2}-\frac{m^{2}}{\hbar^{2}}+i\epsilon}=\frac{\hbar}{\tilde{k}^{2}-2\,\tilde{p}\cdot\tilde{k}+i\epsilon}\
,$ (2.100)

where we used $\tilde{p}^{2}=m^{2}/\hbar^{2}$. Now considering
$\tilde{p}\thicksim\frac{m}{\hbar}\gg\tilde{k}$, one gets schematically

$\frac{\hbar}{\tilde{k}^{2}-2\,\tilde{p}\cdot\tilde{k}+i\epsilon}=-\frac{\hbar^{2}}{2m\,\tilde{k}}\propto\hbar^{2}\
.$ (2.101)

It is important to underline that this is just the leading order behavior, and
a complete expansion of the propagator would lead to terms $O(\hbar^{3})$.
Finally one can consider a massless propagator which simply goes like

$\frac{\hbar}{\tilde{q}^{2}+i\epsilon}\propto\hbar\ .$ (2.102)

Similar arguments hold for the study of $\hbar$ dependency of vertices, which
has a different behavior whether the interaction involves a massive particle
or not. Vertices that involve only massless lines do not depend on the charge
or the mass, and they simply go like

$\tau_{massless}\propto\hbar^{-1}\ .$ (2.103)

To this category belong the 2 photons - 1 graviton vertex, 2 photons - 2
gravitons vertex, 3 gravitons vertex, and so on. On the other hand, vertices
that are attached to external massive lines behave differently. For the 2
scalars - 1 photon vertex, the expression in eq. (A.5) becomes

$\left(\tau_{\phi^{2}A}\right)^{\mu}(p,p^{\prime})=-i\frac{1}{\hbar}\frac{Q}{\hbar}\left(\tilde{p}+\tilde{p}\,^{\prime}\right)^{\mu}\
,$ (2.104)

where considering again $\tilde{p}\thicksim m/\hbar$ one gets

$\left(\tau_{\phi^{2}A}\right)^{\mu}(p,p^{\prime})\propto\hbar^{-3}\ .$
(2.105)

Analogously for the 2 scalars - 1 graviton vertex one firstly has

$\left(\tau_{\phi^{2}h}\right)^{\mu\nu}(p,p^{\prime})=-\frac{1}{\hbar}\frac{i\kappa}{2}\Big{(}\big{(}\tilde{p}^{\mu}\tilde{p}\,^{\prime}{}^{\nu}+\tilde{p}^{\nu}\tilde{p}\,^{\prime}{}^{\mu}\big{)}-\eta^{\mu\nu}\big{(}\tilde{p}\cdot\tilde{p}\,^{\prime}-\frac{m^{2}}{\hbar^{2}}\big{)}\Big{)}\
,$ (2.106)

which leads to

$\left(\tau_{\phi^{2}h}\right)^{\mu\nu}(p,p^{\prime})\propto\hbar^{-3}\ .$
(2.107)

From these two computations one can infer then that

$\tau_{massive}\propto\hbar^{-3}\ .$ (2.108)

Again this is just a leading order contribution, while a higher-order
expansion contains pieces $O(\hbar^{-2})$.

So then we obtain some replacement rules, which given a specific amplitude,
allow us to see immediately to what power of $\hbar$ the process is
proportional to. Then since a generic gravitational quantum observable can be
described perturbatively in terms of Feynman diagrams, exploiting these
considerations we are able to infer before calculations how each contribution
of the observables depends on the Planck constant. In fact, since the main
scope of this machinery is to recover classical physics from quantum
processes, if we know a priori that some contributions of a specific
observable depend on $\hbar$, then we can conclude that the Feynman diagrams
associated to that specific piece do not contribute to the classical limit,
and only reconstruct the so called quantum corrections that are vanishing in
the limit $\hbar\rightarrow 0$. This procedure is crucial in order to perform
actual calculations of classical gravitational observables from scattering
amplitudes, because it allows us to focus only on those diagrams we know will
contribute to the classical regime, avoiding all other processes which would
lead just to $\hbar$-dependent contributions. The idea then, is to define some
selection rules which make the choice of classical processes systematical. To
see explicitly how this machinery can be exploited, in the next chapter we
pick a specific classical gravitational observable in order to see how it can
be recovered from quantum calculations.

##  Chapter 3 Classical metrics from scattering amplitudes in four dimensions

In this section we want to exploit the procedure outlined so far in order to
recover classical metrics from the $\hbar\rightarrow 0$ limit of quantum
gravity as an effective field theory [8, 9]. In first place it will be
reconstructed the Schwarzschild metric. The strategy will be firstly identify
the quantum counterpart of the classical gravitational observable, and then
organize a perturbative expansion in terms of Feynman diagrams, picking up
only those pieces we know will contribute to the classical limit, following
the original work of [7]. Such observable, obtained from scattering
amplitudes, will be compared with the one computed by directly solving the
Einstein equations, whose however is expressed in the harmonic gauge. Then we
will outline a procedure to systematically move from one gauge to the other,
originally discussed in [9], and we will prove the coherence of the procedure
by matching the two derivations. Finally it will be computed the
Reissner–Nordström metric from graviton emission processes with photons
circulating in the loops, and it will be commented the procedure to recover
the Kerr-Newman space-time from spinors-scattering in a completely analogous
way. The chapter follows the historical path of how we learned to employ the
quantum field theory formalism in order to recover classical physics in
gravity context, and gives the chance to introduce some important concepts and
ideas later useful in this thesis.

###  3.1 Schwarzschild metric from graviton emission

The Schwarzschild metric describes the four-dimensional space-time geometry
associated to a spherically symmetric and static source, which is also
electrically neutral. The quantum counterpart of the observable is the
graviton field $h_{\mu\nu}(x)$, since by its evaluation it is possible to
reconstruct the metric as $g_{\mu\nu}=\eta_{\mu\nu}+\kappa h_{\mu\nu}$. Such a
solution of the Einstein equations is associated to an action that couples an
electrically neutral scalar field to gravity, and where such scalar field is
static, meaning that $\partial_{t}\phi(x)=0$. Consider then the graviton
field, whose field equations in de Donder gauge and $D=4$ read

$\Box
h_{\mu\nu}(x)=-\frac{\kappa}{2}\left(T_{\mu\nu}(x)-\frac{1}{2}\eta_{\mu\nu}T(x)\right)\
,$ (3.1)

with $T^{\mu\nu}(x)=T^{\mu\nu}_{\phi}(x)+t^{\mu\nu}(x)$, where
$T^{\mu\nu}_{\phi}(x)$ is simply the stress-energy tensor of the scalar field,
while $t^{\mu\nu}(x)$ contains inside all the self-interaction terms of the
gravitational field. Imposing the staticity of the field and an explicit
insertion of $\hbar$ factors, the solution of the above equation takes the
form

$h_{\mu\nu}(\vec{x})=-\frac{\kappa}{2}\int\frac{d^{3}\tilde{q}}{(2\pi)^{3}}\frac{e^{i\vec{\tilde{q}}\cdot\vec{x}}}{\vec{\tilde{q}}^{2}}\left(T_{\mu\nu}(\vec{\tilde{q}}^{2})-\frac{1}{2}\eta_{\mu\nu}T(\vec{\tilde{q}}^{2})\right)\
,$ (3.2)

where $T_{\mu\nu}(\vec{\tilde{q}}^{2})$ is the Fourier transform of the
stress-energy tensor and again $\tilde{q}=q/\hbar$ has the meaning of wave-
number. From the expression above we see that there are no explicit $\hbar$
inside the integral, meaning that eventual terms proportional to the Planck
constant come from the stress-energy tensor itself. By definition
$T_{\mu\nu}(q^{2})$ is associated to the matrix element

$T_{\mu\nu}(q^{2})=\bra{p^{\prime}}T_{\mu\nu}(x=0)\ket{p}\ ,$ (3.3)

with $q=p-p^{\prime}$ the transferred momentum, $p$ and $p^{\prime}$ the
external physical momenta associated to on-shell particles in which
$p^{2}=p^{\prime 2}=m^{2}$, and where the normalization of the particle states
is

$\bra{p^{\prime}}\ket{p}=(2\pi)^{3}\delta^{(3)}(\vec{p}\,^{\prime}-\vec{p})=(2\pi)^{3}\frac{1}{\hbar^{3}}\delta^{(3)}(\vec{\tilde{p}}\,^{\prime}-\vec{\tilde{p}})\
.$ (3.4)

From (3.3) it is clear how $T_{\mu\nu}(q^{2})$ can be expressed perturbatively
in terms of Feynman diagrams which reconstruct the generic interaction vertex
between two scalars with an emission of a graviton, as

$\includegraphics[width=130.08731pt,valign={c}]{Diagrams/Generic_Graviton_Emission_Plain.png}=-i\frac{1}{\hbar^{3}}\frac{\kappa}{2}\sqrt{4E\,E^{\prime}}\,T_{\mu\nu}(q^{2})\
,$ (3.5)

where the energy-dependent factor is just a normalization choice with $E$ and
$E^{\prime}$ the energy of the external particles, and where the power of
$\hbar$ is due to dimensional reasons. It is possible then to expand the
stress-energy tensor in terms of form factors, and from the convention chosen
the decomposition reads

$T_{\mu\nu}(q^{2})=\frac{1}{\sqrt{4E\,E^{\prime}}}\Biggl{(}2P_{\mu}P_{\nu}F_{1}(q^{2})+(q_{\mu}q_{\nu}-\eta_{\mu\nu}q^{2})F_{2}(q^{2})\Biggl{)}\
,$ (3.6)

with $P_{\mu}=\frac{1}{2}(p+p^{\prime})_{\mu}$ and where the normalization
condition imposes $F_{1}(0)=1$. It is easy to check the consistency of the
above form factor expansion with the normalization given. Considering
$\mathscr{P}^{\mu}$ as the momentum operator that acts on particle states and
defined by the Noether current as

$\mathscr{P}^{\mu}=\int d^{3}x\,T^{\mu 0}(x)\ ,$ (3.7)

then it follows

$\bra{p^{\prime}}\mathscr{P}^{\mu}\ket{p}=p^{\mu}\bra{p^{\prime}}\ket{p}=p^{\mu}(2\pi)^{3}\delta^{(3)}(\vec{p}\,^{\prime}-\vec{p})\
.$ (3.8)

On the other hand, it is possible to compute the same object considering

$\bra{p^{\prime}}\mathscr{P}^{\mu}\ket{p}=\int d^{3}x\bra{p^{\prime}}T^{\mu
0}(x)\ket{p}=(2\pi)^{3}\delta^{(3)}(\vec{p}\,^{\prime}-\vec{p})T^{\mu
0}(q^{2}=0)\ ,$ (3.9)

that using the normalization condition $F_{1}(0)=1$ leads to the same
conclusion of (3.8).

So then to compute the graviton field in eq. (3.2), it is sufficient to
evaluate the form factors defined in (3.6). The standard way to do that is to
expand perturbatively the resummed vertex in (3.5), and assign a value to
$F_{1}$ and $F_{2}$ order by order in the coupling constant. However we are
not interested in all the processes that reconstruct a given order in the
quantum computation, but we only care about the ones which are not vanishing
in the limit $\hbar\rightarrow 0$. Moreover, it is important to take into
account that from the low-energy limit imposed by the effective approach, each
diagram organizes itself in an energy expansion, and since the power of the
momentum is related to the power of $\hbar$, we conclude that even if the
leading contribution of a diagram contributes classically to an observable,
its next-to-leading-order does not [7]. Making this clear, in the limit in
which the momentum of the external particles satisfies
${m}^{2}\gg{\vec{p}}^{2}$ (imposed by staticity), the energy of the external
particles reads

$E=\sqrt{m^{2}-\vec{\tilde{p}}^{2}\hbar^{2}}=m\left(1-\frac{\vec{\tilde{p}}^{2}\hbar^{2}}{2m^{2}}+O(\vec{\tilde{p}}^{4}\hbar^{4})\right)\
.$ (3.10)

Exploiting this analysis, in the following we will only consider the leading
order, since terms $O(\vec{\tilde{p}}^{2}\hbar^{2})$ contribute to quantum
corrections. Finally considering the static limit in momentum space, which
reads $q^{0}=0$, by replacing (3.6) inside eq. (3.2) one gets

$\kappa\,h_{\mu\nu}(\vec{x})=-16G_{N}\pi\int\frac{d^{3}\tilde{q}}{(2\pi)^{3}}\frac{e^{i\vec{\tilde{q}}\cdot\vec{x}}}{\vec{\tilde{q}}^{2}}\frac{1}{2m}\Biggl{(}\left(\delta_{\mu}^{0}\delta_{\nu}^{0}-\frac{1}{2}\eta_{\mu\nu}\right)F_{1}(\vec{\tilde{q}}^{2})+\frac{\tilde{q}_{\mu}\tilde{q}_{\nu}-\frac{1}{2}\eta_{\mu\nu}\vec{\tilde{q}}^{2}}{2m^{2}}\hbar^{2}F_{2}(\vec{\tilde{q}}^{2})\Biggl{)}\
.$ (3.11)

##### Tree-level

The first order of the perturbative expansion is trivially the tree-level
process, reported in figure 3.1.

Figure 3.1: Tree-level of the 2 scalars - 1 graviton process.

From eq. (3.5) one obtains

$-i\frac{1}{\hbar^{3}}\frac{\kappa}{2}\sqrt{4E\,E^{\prime}}\,T^{tree}_{\mu\nu}(\tilde{q}^{2})=(\tau_{\phi^{2}h})_{\mu\nu}(\tilde{p},\tilde{p}^{\prime})\
,$ (3.12)

where $(\tau_{\phi^{2}h})_{\mu\nu}$ is the 2 scalars - 1 graviton vertex in
eq. (A.6). The leading order of the resulting stress-energy tensor,
remembering that $(\tau_{massive})\propto\hbar^{-3}$, is then

$T^{tree}_{\mu\nu}(\vec{\tilde{q}}^{2})=m\,\delta_{\mu}^{0}\delta_{\nu}^{0}+O(\hbar)\
,$ (3.13)

which leads to the form factors

$F_{1}^{tree}(\vec{\tilde{q}}^{2})=1+O(\hbar)\quad\text{and}\quad
F_{2}^{tree}(\vec{\tilde{q}}^{2})=O(\hbar)\ .$ (3.14)

From prior considerations, since (3.13) is classical, we can conclude that the
metric contribution that comes from the leading-order of the tree-level
amplitude is classical, and will be non-vanishing in the limit
$\hbar\rightarrow 0$ since it is independent of the Planck constant. Replacing
the form factors inside eq. (3.11) and using the Fourier transforms in
appendix B, in particular eq. (B.6), one finally obtains

$\displaystyle\kappa\,h_{00}^{tree}$
$\displaystyle=-2\frac{G_{N}\,m}{r}+O(\hbar)$ (3.15)
$\displaystyle\kappa\,h_{i0}^{tree}$ $\displaystyle=O(\hbar)$
$\displaystyle\kappa\,h_{ij}^{tree}$
$\displaystyle=-2\frac{G_{N}\,m}{r}\,\delta_{ij}+O(\hbar)\ ,$

where $r=\sqrt{x^{i}x^{i}}$.

##### 1-loop

For the 1-loop case the situation is more tricky. Since there exist many
1-loop diagrams it is not known a priori which process has a classical leading
behavior. Some of the diagrams that contribute to the process are reported in
figure 3.2.

Figure 3.2: Some of the processes that contribute to the 1-loop correction of
the 2 scalars - 1 graviton vertex.

Now it is not necessary to compute every single process and see at the end
whether or not they are classical, in fact exploiting the knowledge of how
vertices and propagators depend explicitly by $\hbar$, it can be understood
immediately which diagram has a purely quantum leading behavior. For what
concerns diagram (a), the behavior of the stress-energy tensor is

$\frac{1}{\hbar^{3}}T_{\mu\nu}^{(\text{a})1-loop}\propto\frac{\hbar\hbar}{\hbar^{3}\hbar}\Rightarrow
T_{\mu\nu}^{(\text{a})1-loop}\propto\hbar\ ,$ (3.16)

which shows that such contribution will vanish in the classical limit, and
therefore it is not needed if we are interested only in classical
phenomenology. For diagram (b) the same procedure reads

$\frac{1}{\hbar^{3}}T_{\mu\nu}^{(\text{b})1-loop}\propto\frac{\hbar\hbar\hbar}{\hbar^{3}\hbar\hbar}\Rightarrow
T_{\mu\nu}^{(\text{b})1-loop}\propto\hbar\ ,$ (3.17)

which lead to the same conclusions. The process (c) instead behaves like

$\frac{1}{\hbar^{3}}T_{\mu\nu}^{(\text{c})1-loop}\propto\frac{\hbar^{2}\hbar\hbar}{\hbar^{3}\hbar^{3}\hbar}\Rightarrow
T_{\mu\nu}^{(\text{c})1-loop}\propto\hbar^{0}\ ,$ (3.18)

which means that the leading contribution is entirely classical and it is
needed for reconstructing classical phenomenology. Different analysis for the
diagram (d), which since turns out to be independent from the transferred
momentum $\vec{\tilde{q}}$, gives rise to a local contribution [7, 9]. So then
this argument shows that among the processes in figure 3.2, only the type (c)
contributes to the classical limit of the observable, and this result can be
generalized to conclude that every other diagram gives either an $\hbar$
correction or a local one, including diagrams that involve ghosts.

This explicit example outlines the idea behind the formulation of selection
rules of Feynman diagrams, which only pick the contributions needed for
classical phenomenology. So then, we can state that:

  * •

The low-energy limit imposes that the only processes that contribute to the
classical behavior are the ones in which internal massive lines do not mediate
the interaction, meaning that such intermediate states cannot propagate the
external momentum to the final states. This idea is explained in details in
chapter 5.1.

  * •

The classical limit imposes that a classical gravitational observable is
recovered only from those diagrams which have an internal tree structure.

Therefore the selection rules lead to the conclusion that every classical
contribution to the metric is made of $l$-loop amplitudes whose structure is
composed by a single massive line which emits $l+1$ gravitons that interact in
a tree-level structure ending up with a graviton emission. One of the most
important outcomes of this analysis is that while classical electromagnetism
is recovered only by tree-level processes, general relativity is reconstructed
with multi-loop diagrams, from which we interpret the non-linearity of gravity
as self-interaction of the gravitons even at the classical level. In fact we
proved that the naive statement that says that a loop expansion is synonymous
of an $\hbar$ expansion is false, and that gravity is classically
reconstructed by quantum processes like loop amplitudes, which is something
not deeply understood yet.

The actual calculation for the 1-loop contribution to the metric then follows
the same steps as before. Considering then the low-energy limit of the
amplitude (c), an explicit calculation leads to the form factors [7]

$\displaystyle F_{1}^{1-loop}(\vec{\tilde{q}}^{2})=\frac{1}{16}\frac{G_{N}\pi
m\,\tilde{q}^{2}}{\sqrt{-\tilde{q}^{2}}}+O(\hbar)=-\frac{1}{16}G_{N}\pi\,m\,|\vec{\tilde{q}}|+O(\hbar)$
(3.19) $\displaystyle
F_{2}^{1-loop}(\vec{\tilde{q}}^{2})=\frac{7}{8}\frac{G_{N}\pi
m^{3}}{\hbar^{2}\sqrt{-\tilde{q}^{2}}}+O(1/\hbar)=\frac{7}{8}\frac{G_{N}\pi
m^{3}}{\hbar^{2}|\vec{\tilde{q}}|}+O(1/\hbar)\ ,$

where we used $\tilde{q}^{2}=-\vec{\tilde{q}}^{2}$. Replacing the above
coefficients inside eq. (3.11) and using the relations (B.8) and (B.9), it is
possible to extract the classical contributions of the metric perturbation as

$\displaystyle\kappa\,h_{00}^{1-loop}$
$\displaystyle=2\frac{G_{N}^{2}m^{2}}{r^{2}}+O(\hbar)$ (3.20)
$\displaystyle\kappa\,h_{i0}^{1-loop}$ $\displaystyle=O(\hbar)$
$\displaystyle\kappa\,h_{ij}^{1-loop}$
$\displaystyle=-\frac{G_{N}^{2}m^{2}}{r^{2}}\left(5\delta_{ij}-7\frac{x_{i}x_{j}}{r^{2}}\right)+O(\hbar)\
.$

So at the very end, considering that the metric is reconstructed by

$g_{\mu\nu}=\eta_{\mu\nu}+\kappa\,h_{\mu\nu}=\eta_{\mu\nu}+\kappa\Bigl{(}h_{\mu\nu}^{tree}+h_{\mu\nu}^{1-loop}+\cdots)\
,$ (3.21)

picking only the classical contributions one gets the Schwarzschild metric in
the de Donder gauge as

$\displaystyle g_{00}$
$\displaystyle=1-\frac{2G_{N}\,m}{r}+\frac{2G_{N}^{2}\,m^{2}}{r^{2}}+O(1/r^{3})$
(3.22) $\displaystyle g_{i0}$ $\displaystyle=0$ $\displaystyle g_{ij}$
$\displaystyle=-\delta_{ij}-\delta_{ij}\frac{2G_{N}\,m}{r}-\frac{G_{N}^{2}m^{2}}{r^{2}}\left(5\delta_{ij}-7\frac{x_{i}x_{j}}{r^{2}}\right)+O(1/r^{3})\
.$

Eq. (3.22) shows a very important result: the classical observable is
reconstructed in terms of a post-Minkowskian expansion, which is related to
the loop perturbative series of the quantum computation. In fact we can infer
that an $l$-loop amplitude contributes to the $l+1$-post-Minkowskian order of
the observable, which goes like $1/r^{l+1}$. In this expansion the first order
is recovered from the tree-level stress-energy tensor that corresponds to what
we called $T^{\mu\nu}_{\phi}$, which is zeroth-order in the post-Minkowskian
series, while the multi-loop diagrams reconstruct the self-interaction stress-
energy tensor $t^{\mu\nu}$. It is important to notice that in this case the
powers of the radial coordinate and the gravitational coupling constant are
the same, but this is just an artifact of the fact that our theory has one
coupling.

###  3.2 Schwarzschild metric in harmonic gauge

In the previous section we recovered the second post-Minkowskian order of the
Schwarzschild metric in the de Donder gauge performing the classical limit of
quantum gravity scattering amplitudes. Now we want to compare the solution in
eq. (3.22) with the classical derivation of it and prove the consistency of
the strategy outlined before. The crucial step is to take into account the
gauge in which the metric is computed, in fact while in the quantum context
the de Donder gauge is very useful, the classical space-time solutions in
literature are often given in spherical coordinates or harmonic ones. It turns
out that the de Donder gauge and the harmonic one are very similar, and it is
possible to link them and convert eq. (3.22) in harmonic coordinates, and
finally compare the derivation from scattering amplitudes with the completely
classical one.

The Schwarzschild metric in spherical coordinates $(t,r,\theta,\varphi)$ has
the form

$ds^{2}=\left(1-\frac{2G_{N}m}{r}\right)dt^{2}-\left(1-\frac{2G_{N}m}{r}\right)^{-1}dr^{2}-r^{2}d\Omega_{2}\
,$ (3.23)

where $d\Omega_{2}=d\theta^{2}+\sin(\theta)^{2}d\varphi^{2}$ is the 2-sphere
metric and $m$ is the mass of the black hole. It is possible to construct
another set of coordinates $\left(t,\vec{x}\right)$, called harmonic, in which
following [55] the metric reads

$ds^{2}=\left(\frac{r-G_{N}m}{r+G_{N}m}\right)dt^{2}-\left(1+\frac{G_{N}m}{r}\right)^{2}d\vec{x}^{2}-\left(\frac{r+G_{N}m}{r-G_{N}m}\right)\frac{G_{N}^{2}m^{2}}{r^{4}}\left(\vec{x}\cdot
d\vec{x}\right)\ ,$ (3.24)

where $r^{2}=\vec{x}\cdot\vec{x}$ and where the metric satisfies the relation

$g^{\mu\nu}\Gamma_{\mu\nu}^{\alpha}=0\ ,$ (3.25)

which imposes the harmonic gauge. Finally, the metric can be expanded in a
post-Minkowskian series component by component, which reads

$\displaystyle g_{00}=$ $\displaystyle\frac{\displaystyle
1-\frac{G_{N}m}{r}}{\displaystyle
1+\frac{G_{N}m}{r}}=1-2\frac{G_{N}m}{r}+2\frac{G_{N}^{2}m^{2}}{r^{2}}+O(1/r^{3})$
(3.26) $\displaystyle g_{i0}=$ $\displaystyle 0$ $\displaystyle g_{ij}=$
$\displaystyle-\left(1+\frac{G_{N}m}{r}\right)^{2}\delta_{ij}-\left(\frac{\displaystyle
1+\frac{G_{N}m}{r}}{\displaystyle
1-\frac{G_{N}m}{r}}\right)\frac{G_{N}^{2}m^{2}}{r^{4}}x_{i}x_{j}$
$\displaystyle=-\delta_{ij}\left(1+2\frac{G_{N}m}{r}+\frac{G_{N}^{2}m^{2}}{r^{2}}\right)-\frac{x_{i}x_{j}}{r^{2}}\frac{G_{N}^{2}m^{2}}{r^{2}}+O(1/r^{3})\
.$

Now in order to do the comparison we are looking for, we need to rewrite eq.
(3.22) in the harmonic gauge, and compare the result with the expression
(3.26). In the last section we saw that a perturbative computation of the
metric in terms of scattering amplitudes organizes itself in a post-
Minkowskian series. Therefore it is useful to decouple the Einstein equations
from the beginning as

$G_{\mu\nu}^{tree}=\frac{\kappa^{2}}{4}T_{\mu\nu}^{tree}$ (3.27)
$G_{\mu\nu}^{1-loop}=\frac{\kappa^{2}}{4}T_{\mu\nu}^{1-loop}\ .$ (3.28)

Then imposing the metric decomposition in terms of a post-Minkowskian
expansion as well

$g_{\mu\nu}=\eta_{\mu\nu}+\kappa h_{\mu\nu}^{tree}+\kappa
h_{\mu\nu}^{1-loop}+\cdots\ ,$ (3.29)

the harmonic gauge condition, which is non-linear in the graviton field, reads

$\displaystyle 0=g^{\mu\nu}\Gamma_{\mu\nu}^{\alpha}=$
$\displaystyle\kappa\Bigl{(}\partial_{\beta}h^{tree\,\alpha\beta}-\frac{1}{2}\partial^{\alpha}h^{tree}\Bigl{)}+\kappa\Bigl{(}\partial_{\beta}h^{1-loop\,\alpha\beta}-\frac{1}{2}\partial^{\alpha}h^{1-loop}\Bigl{)}$
(3.30)
$\displaystyle-\kappa^{2}h^{tree\,\alpha\beta}\Bigl{(}\partial^{\mu}h^{tree}_{\beta\mu}-\frac{1}{2}\partial_{\beta}h^{tree}\Bigl{)}+\cdots\
.$

The above equation has to be satisfied order by order, and then for the first
two post-Minkowskian contributions it is verified

$\begin{gathered}\kappa\Bigl{(}\partial_{\beta}h^{tree\,\alpha\beta}-\frac{1}{2}\partial^{\alpha}h^{tree}\Bigl{)}=0\\\
\kappa\Bigl{(}\partial_{\beta}h^{1-loop\,\alpha\beta}-\frac{1}{2}\partial^{\alpha}h^{1-loop}\Bigl{)}-\kappa^{2}h^{tree\,\alpha\beta}\Bigl{(}\partial^{\mu}h^{tree}_{\beta\mu}-\frac{1}{2}\partial_{\beta}h^{tree}\Bigl{)}=0\
.\end{gathered}$ (3.31)

Now we have to replace these conditions in both (3.27) and (3.28) and obtain
the analogue of eq. (3.1) in the harmonic case. The tree-level harmonic gauge
condition corresponds exactly to the de Donder gauge111The fact that the
leading order of the harmonic gauge corresponds to the de Donder gauge, means
that the graviton propagator remains unchanged since the correction quadratic
in the first derivative of the field is the same., therefore the field
equations for the graviton field at first post-Minkowskian order coincide with
eq. (3.1) and the metric perturbation remains unchanged in this gauge

$\displaystyle\kappa\,h_{00}^{tree}$ $\displaystyle=-2\frac{G_{N}\,m}{r}$
(3.32) $\displaystyle\kappa\,h_{i0}^{tree}$ $\displaystyle=0$
$\displaystyle\kappa\,h_{ij}^{tree}$
$\displaystyle=-2\frac{G_{N}\,m}{r}\,\delta_{ij}\ ,$

in fact we can see that the expressions above perfectly coincide with the
metric in (3.26) at first post-Minkowskian order.

At the second post-Minkowskian order the two gauges differ, and we expect the
field equations to be modified, in fact they read

$\displaystyle\Box\left(h_{\mu\nu}^{1-loop}-\frac{1}{2}\eta_{\mu\nu}h^{1-loop}\right)=$
$\displaystyle\frac{\kappa}{4}T_{\mu\nu}^{1-loop}+\partial_{\mu}\left(h^{tree\,\lambda\sigma}\left(\partial_{\lambda}h_{\sigma\nu}^{tree}-\frac{1}{2}\partial_{\nu}h_{\lambda\sigma}^{tree}\right)\right)$
(3.33)
$\displaystyle+\partial_{\nu}\left(h^{tree\,\lambda\sigma}\left(\partial_{\lambda}h_{\sigma\mu}^{tree}-\frac{1}{2}\partial_{\mu}h_{\lambda\sigma}^{tree}\right)\right)$
$\displaystyle-\eta_{\mu\nu}\partial^{\alpha}\left(h^{tree\,\lambda\sigma}\left(\partial_{\lambda}h_{\alpha\sigma}^{tree}-\frac{1}{2}\partial_{\alpha}h_{\lambda\sigma}^{tree}\right)\right)\
.$

Now we know the explicit solution of $h^{tree}_{\mu\nu}$ that it is needed to
be replaced inside the above equation, which can be compactly rewritten as

$h_{\mu\nu}^{tree}=f(r)\delta_{\mu\nu}\ ,$ (3.34)

where

$f(r)=-2\frac{G_{N}m}{r}\ ,$ (3.35)

and finally eq. (3.33) can be traced and expressed as

$\Box
h_{\mu\nu}^{1-loop}=-\frac{\kappa}{2}\left(T_{\mu\nu}^{1-loop}-\frac{1}{2}\eta_{\mu\nu}T^{1-loop}\right)-\partial_{\mu}\Bigl{(}f(r)\partial_{\nu}f(r)\Bigl{)}-\partial_{\nu}\Bigl{(}f(r)\partial_{\mu}f(r)\Bigl{)}\
.$ (3.36)

Since it holds the static limit, which implies that the temporal derivative is
vanishing both on the metric and on the function $f(r)$, using the relation

$-\partial_{\mu}\Bigl{(}f(r)\partial_{\nu}f(r)\Bigl{)}-\partial_{\nu}\Bigl{(}f(r)\partial_{\mu}f(r)\Bigl{)}\\\
=\partial_{i}\Bigl{(}f(r)\partial_{j}f(r)\Bigl{)}+\partial_{j}\Bigl{(}f(r)\partial_{i}f(r)\Bigl{)}=4G_{N}^{2}m^{2}\nabla^{2}\left(\frac{\delta_{ij}}{r^{2}}-2\frac{x_{i}x_{j}}{r^{4}}\right)\
,$ (3.37)

where
$\Box=\partial_{\mu}\partial_{\nu}\eta^{\mu\nu}=\partial_{t}^{2}-\nabla^{2}$,
the field equations in (3.36) becomes

$\displaystyle\nabla^{2}h_{00}^{1-loop}=$
$\displaystyle\frac{\kappa}{2}\left(T_{00}^{1-loop}-\frac{1}{2}T^{1-loop}\right)$
(3.38) $\displaystyle\nabla^{2}h_{i0}^{1-loop}=$ $\displaystyle 0$
$\displaystyle\nabla^{2}h_{ij}^{1-loop}=$
$\displaystyle\frac{\kappa}{2}\left(T_{ij}^{1-loop}+\frac{1}{2}\delta_{\mu\nu}T^{1-loop}\right)+\nabla^{2}\left(4\frac{G_{N}^{2}m^{2}}{r^{2}}\left(\delta_{ij}-2\frac{x_{i}x_{j}}{r^{2}}\right)\right)\
.$

Comparing the above equations with the analogous expression in the de Donder
gauge in eq. (3.2), one concludes that in order to move to the harmonic gauge,
it is sufficient to add the extra term to the de Donder gauge expression in
eq. (3.20), obtaining

$\displaystyle\kappa\,h_{00}^{1-loop}=$ $\displaystyle
2\frac{G_{N}^{2}m^{2}}{r^{2}}$ (3.39) $\displaystyle\kappa\,h_{i0}^{1-loop}=$
$\displaystyle 0$ $\displaystyle\kappa\,h_{ij}^{1-loop}=$
$\displaystyle-\frac{G_{N}^{2}m^{2}}{r^{2}}\left(5\delta_{ij}-7\frac{x_{i}x_{j}}{r^{2}}\right)+4\frac{G_{N}^{2}m^{2}}{r^{2}}\left(\delta_{ij}-2\frac{x_{i}x_{j}}{r^{2}}\right)$
$\displaystyle=-\frac{G_{N}^{2}m^{2}}{r^{2}}\left(\delta_{ij}+\frac{x_{i}x_{j}}{r^{2}}\right)\
,$

which at this order matches exactly the Schwarzschild metric in harmonic gauge
in eq. (3.26).

In the end we proved that the approach of section 3.1, with the right gauge
choice, is perfectly coherent and leads to the expected classical metric. This
analysis also underlines how the de Donder gauge is much more suitable for
this approach, since the procedure outlined to convert the de Donder gauge
into the harmonic one can be very tedious at higher orders. To avoid this
problem, further in this thesis the harmonic gauge will be no longer used, and
the quantum computation will be compared directly with the observables in the
de Donder gauge, whose procedure to impose it is shown in chapter 4.

###  3.3 Reissner-Nordström metric from graviton emission

In the previous two sections we recovered the Schwarzschild metric from
graviton emission amplitudes in which the gravitational field interacts
exclusively with discharged scalars. However there exist classical cases in
which gravity is coupled to the electromagnetic field, and for a static and
spherically symmetrically mass distribution, such interaction gives rise to
the Reissner-Nordström metric [56, 57], which in spherical coordinates reads

$ds^{2}=\left(1-\frac{2G_{N}m}{r}+\frac{Q^{2}G_{N}}{4\pi\,r^{2}}\right)dt^{2}-\left(1-\frac{2G_{N}m}{r}+\frac{Q^{2}G_{N}}{4\pi\,r^{2}}\right)^{-1}dr^{2}-r^{2}d\Omega_{2}\
,$ (3.40)

where $Q$ is the electric charge of the black hole. As we did for the
Schwarzschild case, it is possible to express the above metric in harmonic
gauge, and the post-Minkowskian expansion, truncated at the first order in
$G_{N}$, leads to

$\displaystyle g_{00}$
$\displaystyle=1-\frac{2G_{N}m}{r}+\frac{Q^{2}G_{N}}{4\pi\,r^{2}}+O(G_{N}^{2})$
(3.41) $\displaystyle g_{i0}$ $\displaystyle=0$ $\displaystyle g_{ij}$
$\displaystyle=-\delta_{ij}-\delta_{ij}\frac{2G_{N}m}{r}+\frac{Q^{2}G_{N}}{4\pi\,r^{2}}\frac{x_{i}x_{j}}{r^{2}}+O(G_{N}^{2})\
.$

As we anticipated earlier, in this thesis we define the post-Minkowskian
expansion as an expansion in $O(1/r)$, which due to the presence of the charge
does not coincide with an expansion in $G_{N}$, conversely to what happened in
the Schwarzschild case. This means that the expansion in (3.41) is missing
some pieces in order to be second order in the post-Minkowskian expansion, and
those terms are exactly the ones already encountered in the Schwarzschild case
and computed in the harmonic gauge.

Now following the same steps outlined in section 3.1, it is possible to
recover such metric from scattering amplitudes in which both photons and
gravitons circulate in the loops [7]. In fact the theory associated to the
present system is exactly the one studied in chapter 2, and since the long-
range limit imposes that the interaction is mediated by massless particles,
one has to take into account all the massless particles present in the theory.
This means that the Reissner-Nordström metric is recovered from eq. (3.11) as
well, where to the 1-loop correction of the vertex in eq. (3.5) contributes
also the amplitude in figure 3.3, in which two photons are emitted from the
massive line and end up interacting in a 2 photons - 1 graviton vertex. Notice
that here and in the following the single wiggly lines stand for photons.

Figure 3.3: 1-loop correction of the 2 scalars - 1 graviton vertex with
photons circulating in the loop.

From the selection rules previously introduced, it is possible to show that up
to this loop order, this process is the only one that contributes to
reconstruct the term of the metric proportional to $Q^{2}$. From an explicit
computation it is possible to extract the classical contributions of the form
factors associated to such process [8], which explicitly read

$\displaystyle F_{1}^{RN}(\vec{\tilde{q}}^{2})$
$\displaystyle=\frac{3}{64}\frac{Q^{2}\,\tilde{q}^{2}}{m\sqrt{-\tilde{q}^{2}}}+O(\hbar)=-\frac{3}{64}\frac{Q^{2}\,|\vec{\tilde{q}}|}{m}+O(\hbar)$
(3.42) $\displaystyle F_{2}^{RN}(\vec{\tilde{q}}^{2})$
$\displaystyle=\frac{1}{32}\frac{Q^{2}m}{\hbar^{2}\sqrt{-\tilde{q}^{2}}}+O(1/\hbar)=\frac{1}{32}\frac{Q^{2}m}{\hbar^{2}|\vec{\tilde{q}}|}+O(1/\hbar)\
.$

Replacing eq. (3.42) inside (3.11) and using the usual Fourier transforms in
appendix B, the metric perturbation reads

$\displaystyle\kappa\,h_{00}^{RN}$ $\displaystyle=\frac{Q^{2}G_{N}}{4\pi
r^{2}}+O(\hbar)$ (3.43) $\displaystyle\kappa\,h_{i0}^{RN}$ $\displaystyle=0$
$\displaystyle\kappa\,h_{ij}^{RN}$ $\displaystyle=\frac{Q^{2}G_{N}}{4\pi
r^{2}}\frac{x_{i}x_{j}}{r^{2}}+O(\hbar)\ ,$

which exactly corresponds to the classical metric in eq. (3.41). Interestingly
in fact, in this case even if the classical calculation is done in the
harmonic gauge and the quantum one in de Donder gauge, the two expressions
coincide. This is traceable to the fact that the difference between the two
gauges enters only in graviton self-interaction vertices, and since in the
diagram in figure 3.3 they are not present, the calculation is independent by
this choice.

In the end we can comment that the same analysis can be performed in order to
obtain non-static space-time solutions, such as the Kerr or the Kerr-Newman
ones. Considering in fact fermionic matter instead of scalar fields, it is
possible to take into account the spinning dynamics of metric and describe
rotating black holes. The explicit calculation in $D=4$ is performed in [9]
where the Kerr metric is obtained up to 1-loop order from spinor-graviton
interactions, while in [8] it is computed the lowest order in the
electromagnetic coupling constant of the Kerr-Newman metric.

##  Chapter 4 Reissner-Nordström-Tangherlini solution in de Donder gauge

The picture that has emerged so far, is that it is possible to compute the
post-Minkowskian expansion of gravitational observables by exploiting the
classical limit of scattering amplitudes in quantum gravity as an effective
field theory. The rest of the thesis will aim to recover the
Reissner–Nordström-Tangherlini solution from scattering amplitudes in which
charged scalars, photons and gravitons interact with each other. In this
chapter we will firstly review such solution, originally obtained by
Tangherlini [20] from the generalization to arbitrary dimension of the
Reissner–Nordström metric, and then, following the procedure discussed in [18]
applied to an electromagnetic coupled theory in [24, 25], a systematic
procedure will be outlined in order to impose the de Donder gauge and express
the solution in a post-Minkowskian expansion. During the derivation it is
noticed how the expansion is affected by divergences in four and five space-
time dimensions, and a procedure to eliminate them and obtain finite
observables is carried out.

###  4.1 The coordinate change function

The Reissner–Nordström-Tangherlini solution describes the metric and the
electromagnetic potential of a static, spherically symmetric and electrically
charged mass distribution in generic space-time dimension. The resulting
metric in $D=d+1$ dimensions, obtained by solving the Einstein equations in
spherical coordinates, has the expression

$ds^{2}=\bigg{(}1-\frac{\mu_{m}}{r^{d-2}}+\frac{\mu_{Q}}{r^{2(d-2)}}\bigg{)}dt^{2}-\frac{dr^{2}}{1-\frac{\mu_{m}}{r^{d-2}}+\frac{\mu_{Q}}{r^{2(d-2)}}}-r^{2}d\Omega_{d-1}\
,$ (4.1)

where $\mu_{m}$ and $\mu_{Q}$ are respectively related to the mass and the
electric charge of the black hole, by the relations

$\mu_{m}=\frac{16\pi
G_{N}m}{(d-1)\Omega_{d-1}}\quad\text{and}\quad\mu_{Q}=\frac{8\pi
G_{N}Q^{2}}{(d-2)(d-1)\Omega_{d-1}^{2}}\ ,$ (4.2)

with $d\Omega_{d-1}$ the metric of the $(d-1)$-sphere, whose surface is

$\Omega_{d-1}=\frac{2\pi^{d/2}}{\Gamma(d/2)}\ ,$ (4.3)

while the electromagnetic potential of the mass distribution, obtained by
solving the Maxwell equations in Lorentz gauge, has the form

$A_{\mu}(r)=\delta_{\mu}^{0}\frac{1}{(d-2)\Omega_{d-1}}\frac{Q}{r^{d-2}}\ .$
(4.4)

Following the analysis outlined in [18], we want to express such solution in
the de Donder gauge, imposed by the condition

$\eta^{\mu\nu}\Gamma^{\alpha}_{\mu\nu}=0\ .$ (4.5)

Given a generic metric of the form

$ds^{2}=C(r)dt^{2}-\frac{dr^{2}}{C(r)}-r^{2}d\Omega_{d-1}\ ,$ (4.6)

moving to cartesian coordinates $(t,\vec{x})$, in which
$r^{2}=\vec{x}\cdot\vec{x}$ and considering the relations

$dr=\frac{\vec{x}\cdot d\vec{x}}{r}\quad\text{and}\quad
d\vec{x}^{2}=dr^{2}+r^{2}d\Omega_{d-1}\ ,$ (4.7)

its expression becomes

$ds^{2}=C(r)dt^{2}-d\vec{x}^{2}-\frac{1-C(r)}{C(r)}\frac{(\vec{x}\cdot
d\vec{x})^{2}}{r^{2}}\ .$ (4.8)

Inspired by the fact that the metric in (4.8) depends only on $r$, in order to
find a reference frame that satisfies the condition (4.5), we look for a
transformation that involves only the radial coordinate, such as

$(t,\vec{x})\rightarrow(t,f(r)\vec{x})\quad\text{with}\quad r\rightarrow
f(r)r\ ,$ (4.9)

where $f(r)$ is the coordinate change function that completely determines the
transformation. Then keeping trace of

$d\vec{x}\rightarrow
d\Big{(}f(r)\vec{x}\Big{)}=f(r)d\vec{x}+f^{\prime}(r)\frac{\vec{x}\cdot
d\vec{x}}{r}\,\vec{x}\ ,$ (4.10)

the metric in the new coordinates becomes

$ds^{2}=h_{0}(r)dt^{2}-h_{1}(r)d\vec{x}^{2}-h_{2}(r)\frac{(\vec{x}\cdot
d\vec{x})^{2}}{r^{2}}\ ,$ (4.11)

where $h_{0}(r)$, $h_{1}(r)$ and $h_{2}(r)$ are determined in terms of $C(r)$
and $f(r)$ as

$\displaystyle h_{0}(r)$ $\displaystyle=C(f(r)r)$ (4.12) $\displaystyle
h_{1}(r)$ $\displaystyle=f(r)^{2}$ $\displaystyle h_{2}(r)$
$\displaystyle=-f(r)^{2}+\frac{(f(r)+rf^{\prime}(r))^{2}}{C(f(r)r)}\ .$

Expanding the condition (4.5) in terms of the metric, it reads

$-\partial_{j}g_{00}-2\partial_{i}g_{ij}+\partial_{j}g_{ii}=0\ ,$ (4.13)

where we used the fact that the metric is time-independent. Imposing the above
condition for

$\displaystyle g_{00}$ $\displaystyle=h_{0}(r)$ (4.14) $\displaystyle g_{i0}$
$\displaystyle=0$ $\displaystyle g_{ij}$
$\displaystyle=-h_{1}(r)\delta_{ij}-h_{2}(r)\frac{x_{i}x_{j}}{r^{2}}\ ,$

then the de Donder gauge condition leads to the equation

$r\frac{d}{dr}\Big{(}h_{0}(r)+(d-2)h_{1}(r)-h_{2}(r)\Big{)}=2(d-1)h_{2}(r)\ ,$
(4.15)

by which replacing in it eq. (4.12), it becomes a differential equation for
$f(r)$.

Since we are interested in the metric in (4.1), we want to solve eq. (4.15)
for

$C(r)=1-\frac{\mu_{m}}{r^{d-2}}+\frac{\mu_{Q}}{r^{2(d-2)}}\ .$ (4.16)

Following [18], we define the parameter

${\rho}(r)=\frac{\Gamma(\frac{d}{2}-1)\pi^{1-d/2}}{r^{d-2}}\ ,$ (4.17)

from which substituting eq. (4.16) in (4.12) one gets

$\displaystyle h_{0}(r)$
$\displaystyle=1-4mG_{N}\frac{d-2}{d-1}\frac{\rho(r)}{f(r)^{d-2}}\bigg{(}1-\frac{\alpha}{2m}\frac{\rho(r)}{f(r)^{d-2}}\bigg{)}$
(4.18) $\displaystyle h_{1}(r)$ $\displaystyle=f(r)^{2}$ $\displaystyle
h_{2}(r)$
$\displaystyle=-f(r)^{2}+\frac{\left(f(r)+(2-d)\rho(r)\frac{df(r)}{d\rho}\right)^{2}}{1-4mG_{N}\frac{d-2}{d-1}\frac{\rho(r)}{f(r)^{d-2}}\left(1-\frac{\alpha}{2m}\frac{\rho(r)}{f(r)^{d-2}}\right)}\
,$

where we have introduced for convenience the fine structure constant
$\alpha=\frac{Q^{2}}{4\pi}$ in natural units. Plugging this last expression
into eq. (4.15) and rewriting it as an equation for the parameter $\rho$,
using the relation

$r\frac{d}{dr}=(2-d)\rho(r)\frac{d}{d\rho(r)}$ (4.19)

we obtain

$(2-d)\rho(r)\frac{d}{d\rho(r)}\Big{(}h_{0}(r)+(d-2)h_{1}(r)-h_{2}(r)\Big{)}=2(d-1)h_{2}(r)\
.$ (4.20)

Now the idea is to solve this equation perturbatively for $f(r)$ by expanding
it as a power series in $\rho$

$f(r)=1+\sum_{n=1}^{\infty}\sum_{\begin{subarray}{c}j=0\\\ j\
\text{even}\end{subarray}}^{n}b_{n,j}(d)\,(mG_{N})^{n-j}(\alpha
G_{N})^{\frac{j}{2}}\,\rho(r)^{n}\ ,$ (4.21)

which corresponds to its post-Minkowskian expansion111In this context we
define the post-Minkowskian expansion as an expansion in $\rho(r)$.. The
structure of the above expression follows from dimensional reasons, in fact in
natural units the dimension of the objects involved is

$[\rho]=L^{2-d}\quad[G_{N}]=L^{d-1}\quad[\alpha]=L^{d-3}\quad[m]=L^{-1}\ ,$
(4.22)

which imply

$[mG_{N}]^{n-j}[\alpha G_{N}]^{\frac{j}{2}}[\rho]^{n}=1\ ,$ (4.23)

and therefore this structure makes the coefficients $b_{n,j}(d)$ dimensionless
and only dependent on the space-time dimension. Moreover for the sake of
simplicity it is useful to compact eq. (4.21) in

$f(r)=1+\sum_{n=1}^{\infty}a_{n}(d)\,\rho(r)^{n}\ ,$ (4.24)

in which we defined implicitly the coefficients

$a_{n}(d)=\sum_{\begin{subarray}{c}j=0\\\ j\
\text{even}\end{subarray}}^{n}b_{n,j}(d)\,(mG_{N})^{n-j}(\alpha
G_{N})^{\frac{j}{2}}\ ,$ (4.25)

that are still functions of the space-time dimension, but clearly not
dimensionless anymore. Finally replacing (4.24) inside eq. (4.20) and solving
the equation, up to third order in the post-Minkowskian expansion, the
coordinate change function reads

$\displaystyle f(r)$ $\displaystyle=1+\frac{2mG_{N}}{d-1}\rho-\frac{\alpha
G_{N}\left(d^{2}-4d+3\right)+4\left(d^{2}-4d+5\right)m^{2}G_{N}^{2}}{(d-4)(d-1)^{2}}\rho^{2}$
(4.26) $\displaystyle+\frac{2mG_{N}}{3(d-1)^{3}(d-4)(d-3)}\Big{(}\alpha
G_{N}\left(6d^{4}-53d^{3}+167d^{2}-223d+103\right)$
$\displaystyle+2\left(7d^{4}-57d^{3}+172d^{2}-232d+122\right)m^{2}G_{N}^{2}\Big{)}\rho^{3}+O\left(\rho^{4}\right)\
,$

where the explicit value of the coefficients can be easily read and the
$r$-dependence of $\rho$ left understood from here on.

Once the function $f(r)$ is known, it enables through the coordinate
transformation (4.9) to determine the expression that any gravitational
observable takes in the de Donder gauge. However in four and five space-time
dimensions, that is when $d=3$ and $d=4$, eq. (4.26) is affected by
divergences. This leads to the conclusion that in these particular cases the
ansatz in eq. (4.24) is no longer true and has to be changed, creating the
need to properly define some adding terms inside $f(r)$ in order to cancel the
divergences whenever they appear. This can be done according to the following
strategy: every time there is a divergent coefficient we have to add at that
order all the terms with a power of logarithm permitted by the order of the
polynomial, considering a new coefficient in the expansion for each newly
added term. In the end every coefficient will be fixed except for the ones
associated to the divergent terms, which will remain free parameters [18].

#####  $\bm{f(r)}$ in $\bm{d=3}$

In the case of $d=3$, looking at eq. (4.26) the only divergence that appears
is at third post-Minkowksian order, which corresponds to $\rho^{3}$ order.
Following the statement that we just gave, in the new ansatz the coordinate
change function becomes

$f(r)\bigl{|}_{d=3}=1+a_{1}(3)\rho+a_{2}(3)\rho^{2}+(a_{3,2}(3)\log(\rho)^{2}+a_{3,1}(3)\log(\rho)+a_{3}(3))\rho^{3}+O(\rho^{4})\
.$ (4.27)

Replacing it inside the differential equation (4.20), each coefficient of the
expansion reads

$\begin{gathered}a_{1}(3)=G_{N}m\qquad a_{2}(3)=2G_{N}^{2}m^{2}\\\
a_{3,2}(3)=0\qquad a_{3,1}(3)=\frac{2}{3}\Big{(}2G_{N}^{2}m\alpha-
G_{N}^{3}m^{3}\Big{)}\qquad a_{3}(3)=\text{not fixed}\ .\end{gathered}$ (4.28)

The fact that $a_{3}(3)$ is not fixed means that with this new ansatz the de
Donder gauge condition is satisfied for any $a_{3}(3)$, introducing a
redundant degree of freedom in the system. Physically this means that by
changing the value of $a_{3}(3)$ we move to a different reference frame still
preserving the de Donder condition. So then redefining the arbitrary constant
in a convenient form, such that the argument of the logarithms become
dimensionless, the function $f(r)\bigl{|}_{d=3}$ takes the expression

$\displaystyle f(r)\bigl{|}_{d=3}$
$\displaystyle=1+mG_{N}\rho+2m^{2}G_{N}^{2}\rho^{2}+\frac{2}{3}mG_{N}\left(2\alpha
G_{N}\log\left(\frac{2mG_{N}\rho}{c}\right)\right.$ (4.29)
$\displaystyle\left.-m^{2}G_{N}^{2}\log\left(\frac{2mG_{N}\rho}{c}\right)\right)\rho^{3}+O\left(\rho^{4}\right)\
,$

where the arbitrary constant is redefined in terms of $c$.

#####  $\bm{f(r)}$ in $\bm{d=4}$

The same arguments hold for the $d=4$ case, in which looking at eq. (4.26) we
have divergences both at second and third post-Minkowskian order. Then in this
case the function $f(r)$ takes the form

$\displaystyle f(r)\bigl{|}_{d=4}=$ $\displaystyle
1+a_{1}(4)\rho+(a_{2,1}(4)\log(\rho)+a_{2}(4))\rho^{2}$ (4.30)
$\displaystyle+(a_{3,2}(4)\log(\rho)^{2}+a_{3,1}(4)\log(\rho)+a_{3}(4))\rho^{3}+O(\rho^{4})\
.$

Replacing it inside (4.20) and solving the differential equation, the
coefficients read

$\begin{gathered}a_{1}(4)=\frac{2}{3}G_{N}m\qquad
a_{2,1}(4)=-\frac{1}{18}\bigl{(}20G_{N}^{2}m^{2}+3G_{N}\alpha\bigl{)}\qquad
a_{2}(4)=\text{not fixed}\\\
a_{3,1}(4)=\frac{1}{9}G_{N}^{2}m\left(3\alpha+20G_{N}m^{2}\right)\qquad
a_{3,2}(4)=0\\\
a_{3}(4)=\frac{2}{81}\left(-81\,a_{2}(4)G_{N}m+16G_{N}^{3}m^{3}+33\alpha
G_{N}^{2}m\right)\ .\end{gathered}$ (4.31)

Interestingly even if in this case the unfixed coefficients are two, namely
$a_{2}(4)$ and $a_{3}(4)$, they are related to each other, which means that
the free degree of freedom in the transformation is still one. Finally,
conveniently manipulating the free constant similarly as before, in this case
the coordinate change function reads

$\displaystyle f$
$\displaystyle(r)\bigl{|}_{d=4}=1+\frac{2}{3}mG_{N}\rho+\frac{1}{18}G_{N}\left(-20\log\left(\frac{8mG_{N}\rho}{3c}\right)G_{N}m^{2}-3\alpha\log\left(\frac{8mG_{N}\rho}{3c}\right)\right)\rho^{2}$
(4.32)
$\displaystyle+\frac{1}{81}mG_{N}^{2}\left(180\log\left(\frac{8mG_{N}\rho}{3c}\right)G_{N}m^{2}+32G_{N}m^{2}+66\alpha+27\alpha\log\left(\frac{8mG_{N}\rho}{3c}\right)\right)\rho^{3}$
$\displaystyle+O\left(\rho^{4}\right)\ ,$

where again the arbitrary constant becomes $c$.

The fact that in $d=3$ and $d=4$ the coordinate change function has
logarithmic terms and it is defined up to an arbitrary constant is not a
coincidence, and it is traceable to the properties of the gauge
transformations. Consider an infinitesimal coordinate transformation
$x^{\alpha}\rightarrow x^{\alpha}+\epsilon^{\alpha}(x)$, performed while the
de Donder gauge is satisfied. If the condition

$\Box\epsilon^{\alpha}(x)=0$ (4.33)

is verified up to orders $O(\epsilon^{2})$, then even the new reference frame
satisfies the de Donder gauge [49]. This means that the function
$\epsilon^{\alpha}(x)$ parametrizes a redundancy inside the gauge. Explicitly,
an harmonic function that satisfies eq. (4.33) and does not involve time
coordinates is

$\epsilon^{i}(\vec{x})=\beta(d)\frac{x^{i}}{r^{d}}\qquad\epsilon^{0}(\vec{x})=0\
,$ (4.34)

where $\beta(d)$ is a dimensional constant inserted in order to respect
$[\epsilon^{i}(\vec{x})]=L$. In $d=3$ this last condition imposes that
$[\beta(3)]=L^{3}$, whose using the coupling we have a disposal can only be
satisfied if

$\beta(3)\thicksim(G_{N}m)^{3},\alpha G_{N}^{3}m\ .$ (4.35)

This means that in $d=3$ it is possible to define a redundant gauge
transformation like

$\epsilon^{i}(\vec{x})\big{|}_{d=3}=\beta(3)\frac{x^{i}}{r^{3}}$ (4.36)

inside the de Donder gauge, and this redundancy, that appear exactly at third
post-Minkowskian order, is in fact reconstructed by the parameter $a_{3}(3)$
that we already discussed. Similarly, in $d=4$ in order to define such
coordinate transformation it must be true that $[\beta(4)]=L^{4}$, which
corresponds to

$\beta(4)\thicksim(G_{N}m)^{2},\alpha G_{N}\ .$ (4.37)

Again since the redundant transformation is possible, we expect a free
parameter in the coordinate change function already at second post-Minkowskian
order, which is exactly $a_{2}(4)$. Moreover this analysis shows that the fact
that the free degree of freedom is just one, and the two unfixed coefficients
are related, is not a coincidence but it is because the whole redundancy is
controlled just by one parameter, namely $\beta(4)$.

Finally in $d\geq 5$ there is no combination of the couplings with an integer
power that satisfy $[\beta(d)]=L^{d}$, therefore does not exist any harmonic
function like the one in (4.33). This means that for $d\geq 5$ no redundant
gauge transformation is possible, meaning that no free parameters can be
present in the coordinate change function. This argument not only shows that
we expected the divergences of $f(r)$ exactly where they are, but states also
that besides $d=3$ and $d=4$ no divergences can occur [22].

###  4.2 Metric in de Donder gauge

Now that the coordinate transformation is completely worked out, to get a
specific observable in the de Donder gauge it is sufficient to consider eq.
(4.9) associated with the explicit form of $f(r)$, which we computed up to
third post-Minkowskian order. In the metric case part of the work has already
been done, and it is sufficient to plug eq. (4.26) inside (4.18), which leads
to

$\displaystyle h_{0}$
$\displaystyle(r)=1-\frac{4(d-2)}{d-1}mG_{N}{\rho}+\left(\frac{2(d-2)}{d-1}\alpha
G_{N}+\frac{8(d-2)^{2}}{(d-1)^{2}}m^{2}G_{N}^{2}\right){{\rho}^{2}}$ (4.38)
$\displaystyle+\left(-\frac{4(d-2)^{2}(3d-11)}{(d-4)(d-1)^{2}}m\alpha
G_{N}^{2}+\frac{8(7-3d)(d-2)^{3}}{(d-4)(d-1)^{3}}m^{3}G_{N}^{3}\right){\rho}^{3}+O({\rho}^{4})$
$\displaystyle h_{1}$
$\displaystyle(r)=1+\frac{4}{d-1}mG_{N}{\rho}+\left(-\frac{2(d-3)}{(d-4)(d-1)}\alpha
G_{N}+\frac{4\left(-2d^{2}+9d-14\right)}{(d-4)(d-1)^{2}}m^{2}G_{N}^{2}\right){\rho}^{2}$
$\displaystyle+\left(\frac{8(3d^{3}-25d^{2}+69d-65)}{3(d-4)(d-3)(d-1)^{2}}m\alpha
G_{N}^{2}+\frac{8\left(7d^{4}-63d^{3}+214d^{2}-334d+212\right)}{3(d-4)(d-3)(d-1)^{3}}m^{3}G_{N}^{3}\right)\rho^{3}$
$\displaystyle+O(\rho^{4})$ $\displaystyle h_{2}$
$\displaystyle(r)=\left(\frac{2(d-2)^{2}}{(d-4)(d-1)}\alpha
G_{N}+\frac{4(d-2)^{2}(3d-2)}{(d-4)(d-1)^{2}}m^{2}G_{N}^{2}\right)\rho^{2}$
$\displaystyle+\left(\frac{4(d-2)^{2}\left(-3d^{3}+19d^{2}-33d+17\right)}{(d-4)(d-3)(d-1)^{3}}m\alpha
G_{N}^{2}\right.$
$\displaystyle\left.+\frac{8(d-2)^{2}\left(-2d^{3}+13d^{2}-25d+10\right)}{(d-4)(d-3)(d-1)^{3}}m^{3}G_{N}^{3}\right)\rho^{3}+O(\rho^{4})\
.$

The above expression has a structure that directly follows from eq. (4.21), in
which the coupling constants combination together with the associated power of
the $\rho$ are dimensionless. As expected eq. (4.38) is not well-defined in
$d=3$ and $d=4$, whose cases need the proper form of the coordinate change
function. In the $d=3$ case in fact, one need to consider the explicit form of
$f(r)\big{|}_{d=3}$ given in eq. (4.29), from which the metric reads

$\displaystyle h_{0}(r)\bigl{|}_{d=3}$
$\displaystyle=1-\frac{2mG_{N}}{r}+\left({\alpha
G_{N}}+2m^{2}G_{N}^{2}\right)\frac{1}{r^{2}}+\left(-2m\alpha
G_{N}^{2}+2m^{3}G_{N}^{3}\right)\frac{1}{r^{3}}+O\left(\frac{1}{r^{4}}\right)$
(4.39) $\displaystyle h_{1}(r)\bigl{|}_{d=3}$
$\displaystyle=1+\frac{2mG_{N}}{r}+\frac{5m^{2}G_{N}^{2}}{r^{2}}$
$\displaystyle+\left(\frac{8}{3}m\alpha
G_{N}^{2}\log\left(\frac{2mG_{N}}{cr}\right)+\frac{4}{3}m^{3}G_{N}^{3}\left(3-\log\left(\frac{2mG_{N}}{cr}\right)\right)\right)\frac{1}{r^{3}}+O\left(\frac{1}{r^{4}}\right)$
$\displaystyle h_{2}(r)\bigl{|}_{d=3}$ $\displaystyle=\left(-\alpha
G_{N}-7m^{2}G_{N}^{2}\right)\frac{1}{r^{2}}+\left(\frac{2}{3}m\alpha
G_{N}^{2}\left(-12\log\left(\frac{2mG_{N}}{cr}\right)-7\right)\right.$
$\displaystyle\left.+\frac{2}{3}m^{3}G_{N}^{3}\left(6\log\left(\frac{2mG_{N}}{cr}\right)-19\right)\right)\frac{1}{r^{3}}+O\left(\frac{1}{r^{4}}\right)\
.$

As we can see from the above metric, in $d=3$ logarithmic terms arise and the
free constant $c$ introduces a redundant gauge freedom in the system.
Similarly in the $d=4$ case, taking into account (4.32), the outcome is

$\displaystyle h_{0}$ $\displaystyle(r)\bigl{|}_{d=4}=1-\frac{8mG_{N}}{3\pi
r^{2}}+\left(\frac{4\alpha
G_{N}}{3\pi^{2}}+\frac{32m^{2}G_{N}^{2}}{9\pi^{2}}\right)\frac{1}{r^{4}}+\left(-\frac{8m\alpha
G_{N}^{2}}{9\pi^{3}}\left(\log\left(\frac{8mG_{N}}{3c\pi
r^{2}}\right)+4\right)\right.$ (4.40)
$\displaystyle\left.+\frac{32m^{3}G_{N}^{3}}{27\pi^{3}}\left(-5\log\left(\frac{8mG_{N}}{3c\pi
r^{2}}\right)-3\right)\right)\frac{1}{r^{6}}+O\left(\frac{1}{r^{8}}\right)$
$\displaystyle h_{1}$ $\displaystyle(r)\bigl{|}_{d=4}=1+\frac{4mG_{N}}{3\pi
r^{2}}+\left(-\frac{\alpha G_{N}}{3\pi^{2}}\log\left(\frac{8mG_{N}}{3c\pi
r^{2}}\right)+\frac{4m^{2}G_{N}^{2}}{9\pi^{2}}\left(1-5\log\left(\frac{8mG_{N}}{3c\pi
r^{2}}\right)\right)\right)\frac{1}{r^{4}}$
$\displaystyle+\left(\frac{4m\alpha
G_{N}^{2}}{27\pi^{3}}\left(3\log\left(\frac{8mG_{N}}{3c\pi
r^{2}}\right)+11\right)+\frac{16m^{3}G_{N}^{3}}{81\pi^{3}}\left(15\log\left(\frac{8mG_{N}}{3c\pi
r^{2}}\right)+4\right)\right)\frac{1}{r^{6}}+O\left(\frac{1}{r^{8}}\right)$
$\displaystyle h_{2}$ $\displaystyle(r)\bigl{|}_{d=4}=\left(\frac{2\alpha
G_{N}}{3\pi^{2}}\left(2\log\left(\frac{8mG_{N}}{3c\pi
r^{2}}\right)-1\right)+\frac{40m^{2}G_{N}^{2}}{9\pi^{2}}\left(2\log\left(\frac{8mG_{N}}{3c\pi
r^{2}}\right)+1\right)\right)\frac{1}{r^{4}}$
$\displaystyle+\left(-\frac{8m\alpha
G_{N}^{2}}{9\pi^{3}}\left(\log\left(\frac{8mG_{N}}{3c\pi
r^{2}}\right)+13\right)+\frac{32m^{3}G_{N}^{3}}{27\pi^{3}}\left(-5\log\left(\frac{8mG_{N}}{3c\pi
r^{2}}\right)-4\right)\right)\frac{1}{r^{6}}$
$\displaystyle+O\left(\frac{1}{r^{8}}\right)\ ,$

where again logarithmic terms are present as well as the free parameter $c$.

###  4.3 Electromagnetic potential in de Donder gauge

Having determined the post-Minkowskian expansion of the metric in the de
Donder gauge, we now proceed to compute the same expansion for the
electromagnetic potential, reported in eq. (4.4) in cartesian coordinates.
Considering the coordinate transformation (4.9), the potential transforms
according to

$A_{\mu}(r)\rightarrow A_{\mu}^{\text{dD}}(r)=\frac{\partial x^{\nu}}{\partial
x^{\prime\mu}}A_{\nu}(f(r)r)\ ,$ (4.41)

where $x^{\prime\mu}$ and $x^{\nu}$ are respectively the new and the old
coordinates. Since in the cartesian reference frame only the temporal
component of the electromagnetic potential is non-vanishing, then

$A_{\mu}^{\text{dD}}(r)=\frac{\partial x^{0}}{\partial
x^{\prime\mu}}A_{0}(f(r)r)\ ,$ (4.42)

but since all the $\vec{x}^{\prime}=f(r)\vec{x}$ are time-independent, even
for the transformed potential the temporal component is the only non-vanishing
one, and the transformation simply reads

$A_{\mu}^{\text{dD}}(r)=\delta_{\mu}^{0}A_{0}(f(r)r)=\delta_{\mu}^{0}\frac{1}{(d-2)\Omega_{d-1}}\frac{Q}{f(r)^{d-2}r^{d-2}}\
.$ (4.43)

Finally, analogously to the metric case we can express it in terms of the
parameter $\rho$ as

$A^{\text{dD}}_{\mu}(r)=\delta_{\mu}^{0}\frac{Q\,\rho}{4\pi}\frac{1}{f(r)^{d-2}}\
.$ (4.44)

Now in order to obtain an explicit expression it is sufficient to replace
(4.26) inside the above equation, which leads to

$\displaystyle A^{\rm dD}_{0}(r)$
$\displaystyle=\frac{Q}{4\pi}\Biggl{(}\rho-\frac{2(d-2)}{(d-1)}mG_{N}\rho^{2}$
(4.45)
$\displaystyle+\left(\frac{2(d-2)^{2}(3d-7)}{(d-4)(d-1)^{2}}m^{2}G_{N}^{2}+\frac{(d-3)(d-2)}{(d-4)(d-1)}\alpha
G_{N}\right)\rho^{3}\Biggl{)}+O(\rho^{4})\ .$

In this case the third post-Minkowskian order of the potential is only
divergent in $d=4$. This is just an artifact of the expression (4.44), in
which it is clear that for the computation of the $O(\rho^{3})$ terms of the
potential we only need the second post-Minkowskian order of $f(r)$, which is
well-defined in $d=3$. So then in five space-time dimensions the
electromagnetic field reads

$\displaystyle A^{\rm dD}_{0}$
$\displaystyle(r)\bigl{|}_{d=4}=\frac{Q}{4\pi}\Biggl{(}\frac{1}{\pi}\frac{1}{r^{2}}-\frac{4}{3\pi^{2}}G_{N}m\frac{1}{r^{4}}$
(4.46)
$\displaystyle+\left(\frac{4G_{N}^{2}m^{2}}{9\pi^{3}}\left(5\log\left(\frac{8G_{N}m}{3\pi
cr^{2}}\right)+3\right)+\frac{\alpha
G_{N}}{3\pi^{3}}\log\left(\frac{8G_{N}m}{3\pi
cr^{2}}\right)\right)\frac{1}{r^{6}}\Biggl{)}+O\left(\frac{1}{r^{8}}\right)\
,$

in which it is coherently considered the same redefinition of the unfixed
constant of the metric case.

##  Chapter 5 Graviton emission processes

The original work of [7, 8, 9], which gave birth to the idea that it is
possible to recover general relativity from the classical limit of quantum
gravity as an effective field theory, inspired a series of papers whose
objective was to formalize and make systematical the procedure for extracting
classical physics from quantum scattering amplitudes.

In this chapter we will firstly review the work presented in [18, 13], in
which the strategy to extract the classical limit from graviton emission
processes is improved, and then applied to the computation of the
Schwarzschild-Tangherlini metric in a completely systematical way. Then
following [24, 25] the procedure is generalized to an electromagnetically
coupled theory in order to obtain the Reissner-Nordström-Tangherlini metric
from scattering amplitudes with photons and gravitons circulating in the
loops. After the strategy review, explicit calculations of graviton emission
processes are carried out, and the perfect agreement with the classical
derivation of chapter 4 is shown. Then it is noticed how in the metric
divergences arise in four and five space-time dimensions, and a procedure to
renormalize them and obtain finite results has to be carried out. According to
[18], in section 5.3 in fact the non-minimal action compatible with the
principles of the effective field theories is defined with the intention to
use the new interaction vertices as counter-terms111In the context of the
world-line formalism, these higher-derivative couplings had already been
introduced in [21], and shown not to contribute to any physical observable.
The equivalence between the scattering amplitude and world-line approaches was
shown in [58]., in order to renormalize the observable [24, 25]. Finally it is
shown how such procedure gives results in complete agreement with the
classical derivation in which logarithmic terms appear, and it is proved that
the counter-terms needed to renormalize the electromagnetic sector of the
metric are exactly the ones already found in [18]. Notice that from here on it
is set again $\hbar=1$.

###  5.1 Classical limit of graviton emission processes

Inspired by chapter 3 we now want to recover the Reissner-Nordström-
Tangherlini metric from the classical limit of graviton emission scattering
amplitudes. Considering the quantum effective action defined in section 2.3 in
space-time dimension $D=d+1$, we look for a solution of the gravitational
field equation expanded in a post-Minkowskian series. Referring to the the
notation of chapter 4, in which to the $n$-th post-Minkowskian order are
associated powers of $\alpha$ up to the integer part of $j/2$, the expansion
of the gravitational source takes the form

$T_{\mu\nu}=\sum_{n=0}^{+\infty}\sum_{\begin{subarray}{c}j=0\\\ j\
\text{even}\end{subarray}}^{n}T_{\mu\nu}^{(n,j)}\ .$ (5.1)

In the same way we can expand from the beginning the graviton field in a post-
Minkowskian series as

$h_{\mu\nu}=\sum_{n=1}^{+\infty}\sum_{\begin{subarray}{c}j=0\\\ j\
\text{even}\end{subarray}}^{n}h_{\mu\nu}^{(n,j)}\ ,$ (5.2)

in which the de Donder gauge condition (4.5) now holds order by order as

$\partial_{\lambda}h^{\lambda}{}_{\nu}{}^{(n,j)}-\frac{1}{2}\partial_{\nu}h^{(n,j)}=0\
.$ (5.3)

Then it is possible to write down the graviton field equations in the de
Donder gauge, and for each order they read

$\Box
h_{\mu\nu}^{(n,j)}(x)=-\frac{\kappa}{2}\left(T_{\mu\nu}^{(n-1,j)}(x)-\frac{1}{d-1}\eta_{\mu\nu}T^{(n-1,j)}(x)\right)\
,$ (5.4)

which say that the $n-1$-th post-Minkowskian term of the stress-energy tensor
is the source of the $n$-th post-Minkowskian term of the metric. As we already
commented, this is due to the fact that the zeroth order of the stress-energy
tensor is reconstructed by the matter sector of the theory, which gives rise
to the first post-Minkowskian order of the metric, while higher orders
correspond to graviton self-interaction terms [9].

We are interested in static solutions of eq. (5.4), which means impose
$\partial_{t}h_{\mu\nu}(x)=0$. From this condition we can explicit the
graviton field as

$h_{\mu\nu}^{(n,j)}(\vec{x})=-\frac{\kappa}{2}\int\frac{d^{d}\vec{q}}{(2\pi)^{d}}\frac{e^{i\vec{q}\cdot\vec{x}}}{\vec{q}^{2}}\left(T^{(n-1,j)}_{\mu\nu}(\vec{q}^{2})-\frac{1}{d-1}\eta_{\mu\nu}T^{(n-1,j)}(\vec{q}^{2})\right)\
,$ (5.5)

where $T_{\mu\nu}(\vec{q}^{2})$ is the Fourier transform of the stress-energy
tensor defined in eq. (3.3) and in which $q^{0}=0$ is the usual momentum-space
version of the static limit. While the graviton field explicitated in the
above equation is a quantum object, we are interested only in its classical
contributions that survive at $\hbar\rightarrow 0$, such pieces are
reconstructed by picking the terms of the perturbative expansion of
$T_{\mu\nu}(\vec{q}^{2})$ which satisfy the selection rules stated in section
3.1. Reproducing the results of [18], the classical contributions of the
observable are reconstructed from $l$-loop amplitudes in which $n-j$ gravitons
and $j$ photons are emitted by a single massive line and whose internal tree
structure ends up with a graviton emission as

$\includegraphics[width=173.44534pt,valign={c}]{Diagrams/Chapter5/img1_1.png}=-\frac{i\,\kappa}{2}\,\sqrt{4E_{1}E_{2}}\,T_{\mu\nu}^{(l,j)}(\vec{q}^{2})\
,$ (5.6)

where, as already noticed in prior chapters, the post-Minkowskian expansion is
related to the perturbative loop series by the relation $n=l+1$. Now since eq.
(5.5) must reconstruct the post-Minkowskian expansion of the classical metric
perturbation, it has to be true that

$\int\frac{d^{d}\vec{q}}{(2\pi)^{d}}\frac{e^{i\vec{q}\cdot\vec{x}}}{\vec{q}^{2}}\,T^{(l,j)}_{\mu\nu}(\vec{q}^{2})\propto\rho^{l+1}\
,$ (5.7)

where $\rho$ is defined in eq. (4.17). The function that satisfy the relation
(5.7) is the so called $l$-loop massless sunset integral (or briefly master
integral), defined by

$J_{(l)}(\vec{q}^{2})=\int\prod_{i=1}^{l}\frac{d^{d}\vec{\ell_{i}}}{(2\pi)^{d}}\frac{\vec{q}^{2}}{\left(\prod_{i=1}^{l}\vec{\ell_{i}}^{2}\right)\left(\vec{q}-\vec{\ell_{1}}-...-\vec{\ell_{l}}\right)^{2}}\
,$ (5.8)

whose mathematical properties are listed in appendix B. Out of this analysis
turns out that222Since we are interested in recovering classical physics, eq.
(5.9) must be understood in the sense that the contributions of the stress-
energy tensor which reconstruct the classical metric must be proportional to
the master integral. In principle there would be also contributions not
proportional to $J_{(l)}(\vec{q}^{2})$, which however we neglect since do not
belong to classical phenomenology.

$T_{\mu\nu}^{(l,j)}(\vec{q}^{2})\propto J_{(l)}(\vec{q}^{2})\ ,$ (5.9)

from which exploiting the fact that the stress-energy tensor is conserved and
Lorentz covariant, from dimensional reasons already discussed in eq. (4.23),
in terms of dimensionless form factors $c_{1}^{(l,j)}(d)$ and
$c_{2}^{(l,j)}(d)$, the gravitational source in momentum space reads

$\displaystyle T_{\mu\nu}^{(l)}(\vec{q}^{2})=\sum_{\begin{subarray}{c}j=0\\\
j\ \text{even}\end{subarray}}^{l+1}T_{\mu\nu}^{(l,j)}(\vec{q}^{2})$ (5.10)
$\displaystyle\simeq\sum_{\begin{subarray}{c}j=0\\\ j\
\text{even}\end{subarray}}^{l+1}m\left(c_{1}^{(l,j)}(d)\delta_{\mu}^{0}\delta_{\nu}^{0}+c_{2}^{(l,j)}(d)\left(-\frac{q_{\mu}q_{\nu}}{\vec{q}^{2}}-\eta_{\mu\nu}\right)\right)(G_{N}m)^{l-j}(\alpha
G_{N})^{\frac{j}{2}}\,\pi^{l}\,J_{(l)}(\vec{q}^{2})\ ,$

in which from here on the symbol $"\simeq"$ means that we consider only the
leading-order in energy and the factor $\pi^{l}$ is inserted in order to make
the form factors independent by it. The reason why we consider only the
leading-order momentum expansion, is because since we are interested only in
classical physics, non-leading-order pieces lead to quantum corrections even
considering the amplitude in eq. (5.6), exactly as it was in section 3.1.

Now it is possible to replace eq. (5.10) inside (5.5) obtaining the classical
metric perturbation order by order in $l$ and $j$ as