# Soft graviton exchange and the information paradox

Nava Gaddam and  Nico Groenenboom

###### Abstract

We show that there is a remarkable phase in quantum gravity where
gravitational scattering amplitudes mediated by virtual gravitons can be
calculated explicitly in effective field theory, when the impact parameter $b$
satisfies $L_{Pl}\ll b\lesssim R_{S}$, with $R_{S}$ being the Schwarzschild
radius. This phase captures collisions with energies satisfying
$\sqrt{s}\gg\gamma M_{Pl}$ (with $\gamma\sim M_{Pl}/M_{BH}$) near the horizon.
We call this the black hole eikonal phase, in contrast to its flat space
analogue where collisions are trans-Planckian. Hawking’s geometric optics
approximation neglects gravitational interactions near the horizon, and
results in thermal occupation numbers in the Bogoliubov coefficients. We show
that these interactions are mediated by graviton exchange in $2\rightarrow 2$
scattering near the horizon, and explicitly calculate the S-matrix non-
perturbatively in $M_{Pl}/M_{BH}$ and $\hbar$. This involves a re-summation of
infinitely many ladder diagrams near the horizon, all mediated by virtual soft
gravitons. The S-matrix turns out to be a pure phase only upon this re-
summation and is agnostic of Planckian physics.

Our calculation shows that non-renormalisability of gravity is irrelevant for
a resolution of the information problem, and is agnostic of any specific
ultraviolet completion. In contrast to the flat space eikonal limit, the black
hole eikonal phase captures collisions of extremely low energy near the
horizon. Given that the physics is dominated by low momentum transfer effects,
it appears that $2\rightarrow 2$ collisions avoid firewalls for black holes
much larger than Planck size.

##  1 Introduction and a summary of results

Hawking argued that semi-classical black hole physics is well approximated by
the propagation of free quantum fields in the presence of a fixed classical
background, and therefore that they reliably describe the state of radiation
[1, 2]. The supposition is therefore that the gravitational effect of quantum
radiation is described by an adiabatic change of the background. In stark
contrast to this picture, ’t Hooft has long argued that strong gravitational
interactions between the infalling matter and outgoing Hawking quanta
dramatically change the observations of the external observer [3, 4, 5]; the
claim therefore being that the radiation is not sufficiently described by
quantum radiation on a fixed background and that quantum coherence is in fact
preserved. If this claim is to be true, the natural question to then ask is,
how must Hawking’s calculation be modified to include these strong
gravitational interactions? An educated guess might be that this must be via
the inclusion of graviton exchange between the in and out going quanta near
the horizon; after all, gravitational interactions dominate all others in this
region. In this paper, we explicitly compute an infinite number of such
graviton exchange diagrams near the horizon, non-perturbatively re-sum them in
a controlled approximation, and show that the result provides for compelling
evidence in support of the said claim.

Consider a spherical shell of matter that is energetic enough to collapse into
a black hole. For as long as the matter fields are far away from the eventual
Schwarzschild radius of the horizon that is to be formed, they may safely be
taken to be propagating on flat space. At such large impact parameters,
gravitational interactions can largely be ignored if the energy of collisions
is small. However, if the energy of collisions becomes Planckian, graviton
exchange between the modes starts to dominate [6]. In the so-called eikonal
approximation, these processes can be reliably calculated non-perturbatively
by summing an infinite number of ladder diagrams [7]; this is the regime of
negligible momentum transfer. Considerable effort has gone into studying
trans-Planckian scattering (when impact parameters $b\gg G_{N}\sqrt{s}$, where
$\sqrt{s}$ measures the centre of mass energy of the collision), both within
semi-classical gravity and string theory [8, 9, 10, 11, 12]. However, when the
impact parameters reach a certain critical value $b\sim G_{N}\sqrt{s}$, the
eikonal result diverges [13, 14]. This suggests the production of an
intermediate state. In equal measure, it also prevents a study of scattering
at impact parameters smaller than the said critical scale. Notwithstanding
this difficulty, various attempts have been made to capture black hole
formation and evaporation [15, 16, 17, 18, 19, 20, 21]. Nevertheless, it is
fair to say that no universally accepted picture of information retrieval has
emerged so far. Any hope to repair this situation seems to be hidden in
physics near the only intrinsic ultra-violet scale available, when impact
parameters are comparable to Planck length; a regime where large momentum
transfers and microscopic black hole formation dominates. A regime where
little to nothing is known.

Gravity has the remarkable property that strong quantum effects arise not only
in scenarios with large momentum transfer, but also via the emergence of
strong gravitational effects at emergent scales (much larger than Planck
length). A case in point being the Schwarzschild radius. In fact, in the
collapsing spherical shell scenario under consideration, an apparent horizon
opens up long before the shell has fallen past the Schwarzschild radius of the
eventual horizon to be formed. This is depicted in Fig. 1.

Figure 1: In this figure, a schematic picture of black hole evolution is
shown. It is formed by a collapsing shell of null matter of width
$v_{1}-v_{0}$. Already at a classical level, an apparent horizon forms long
before the entire collapse has occurred. For most of the lifetime of the black
hole, however, the apparent horizon is essentially indistinguishable from the
event horizon. In fact, already at the final stages of the collapse, the
apparent horizon begins to resemble the event horizon very closely. Therefore,
except during the very dynamical phase of the collapse, all infalliing matter
essentially appears to fall into an already formed horizon. The infalling mode
is drawn in red, the outgoing one in green, while the symbolic exchange of
gravitons is shown by the blue wavy line. The central conclusion of this paper
is that the radiation receives information about the infalling matter from the
mediated virtual soft gravitons; this is manifest after a re-summation of
infinitely many ladder diagrams. Our calculations are valid so long as
$\gamma=\frac{M_{Pl}}{M_{BH}}$ is small. Consequently, the radiation imprints
this information on the Cauchy slice drawn in yellow, allowing the external
observer to retrieve it.

Of course, in the very early stages of the opening up of the horizon, it is
Planckian in size and all processes are dominated by large momentum transfers.
However, as the horizon grows to be larger than Planck size, impact parameters
of collisions on the horizon are of the order of the Schwarzschild radius or
less, but larger than Planck size: $L_{Pl}\ll b\leq R_{S}$. Consequently,
momentum transfer effects are suppressed. Nevertheless, in this regime, the
physics is necessarily dominated by scattering in the presence of the horizon
that has opened up due to the collapse [22]. The primary difficulty with
carrying out this calculation is that unlike in flat space, the graviton
propagator on such a background is analytically intractable, rendering the
scattering process difficult to study.

When there is a separation of scales of the kind $L_{Pl}\ll b\leq R_{S}$, it
is known that the transverse directions of the horizon can be integrated out.
Such an expectation arose from arguments due to the Verlindes [23, 24]. It was
anticipated in their work that single graviton exchange in this approximation
would already modify the state of Hawking radiation. Carrying this integration
out explicitly allows for a partial wave basis, in which the scattering
problem becomes analytically tractable near the horizon.

In this article, we calculate a four point correlator of matter fields in the
said partial wave basis, near the black hole horizon. Owing to a sub-dominant
transverse momentum transfer, we may choose the external momentum to be
ingoing to the black hole for the infalling modes and outgoing from the
horizon for the Hawking quanta. These modes exchange soft gravitons on the
horizon. The interactions are governed by the universal three-point vertex
coupling the graviton with the stress tensor of the matter fields. Upon
integrating the transverse sphere out, the strength of the interactions is
dictated by the emergent dimensionless parameter: $\gamma=\kappa/R_{S}\sim
M_{Pl}/M_{BH}$. The main result of this article is that, for every partial
wave $\ell$, the four point function is given by

$\displaystyle\langle\phi\phi\phi\phi\rangle~{}$
$\displaystyle=~{}4p_{\text{in}}p_{\text{out}}\left[\exp\left(i\dfrac{\gamma^{2}R^{2}_{S}}{\hbar\left(\ell^{2}+\ell+2\right)}p_{\text{in}}p_{\text{out}}\right)-1\right]$
$\displaystyle=~{}4p_{\text{in}}p_{\text{out}}\left[\exp\left(i\dfrac{8\pi
G_{N}}{\hbar\left(\ell^{2}+\ell+2\right)}p_{\text{in}}p_{\text{out}}\right)-1\right]\,,$
(1.1)

where $p_{in}$ and $p_{out}$ are the in and out going momenta of the infalling
matter and Hawking quanta respectively. This is a result of a resummation of
an infinite number of ladder diagrams on the horizon. Therefore, it is non-
perturbative111It is non-perturbative in the sense that, for small $\gamma$,
effects that are exponentially suppressed in $\gamma$ are captured. But it may
also be called ‘perturbatively exact’ in the sense that Planckian (large
momentum transfer) effects that arise from the $\gamma\sim 1$ regime are not
captured. in $\gamma$ and $\hbar$. Exponential behaviour in such four point
functions has received renewed interest owing to their close connection to
quantum chaos [25, 26]. Scattering near the horizon has also been argued to be
related to chaos [27, 28]. This may be seen to arise from (1), when one moves
to exponentiated coordinates near the horizon. In fact, the two-dimensional
Dray-’t Hooft scattering amplitude was shown to exactly agree with the semi-
classical limit of the four point function of the Schwarzian quantum mechanics
that describes the collective infra-red mode of the SYK model [29, 30, 31, 32,
33]222In contrast to the expression (1.1) in reference [33], we find the extra
factor arising from the partial wave $\ell$. This factor is of importance, as
we explain below (1.3) and discuss further in Section 5.2..

However, as far as four dimensional black holes are concerned, the importance
of the emergent scale and the new dimensionless coupling $\gamma$ cannot be
overstated. Based on intuition from the eikonal approximation in flat space,
one may have feared that near-horizon scattering may result in firewalls [34,
35, 36, 28]; it is indeed true that soft graviton exchange in flat space
requires collision energies that are trans-Planckian. So, an infalling
observer may worry about encountering highly energetic outgoing modes.
However, as we will show in the present article, this amplitude (1) actually
arises at leading order in the approximation

$\sqrt{s}~{}\gg~{}\gamma\,M_{Pl}\,.$ (1.2)

We call this the black hole eikonal phase of quantum gravity. This is in
contrast to the eikonal approximation in flat space for which $s\gg
M^{2}_{Pl}$. For an earth mass black hole (with $R_{s}\sim 1cm\gg L_{Pl}$),
where our calculation is valid, we see that (1.2) implies that $s\gg
10^{-64}M^{2}_{Pl}$. The calculation captures collisions of extremely low
energy, and one may ‘safely’ dive into a black hole fearing no drama at the
horizon because the physics is captured by zero momentum transfer effects for
the $2\rightarrow 2$ amplitudes under consideration.333We expect that momentum
transfer effects will play a significant role in general $2\rightarrow N$
amplitudes. Moreover, the time scale associated to those scattering processes
is also expected to be longer. Therefore, all external legs are better
interpreted as asymptotic states at future and past null infinity; therefore
conclusions about firewalls would require further care in general. One may
have arrived at a different conclusion had one ignored the consequences of
this emergent new scale in collapse. Moreover, in contrast to scattering in
flat space in a partial wave basis (which for a fixed impact parameter and
collision energy is dominated by large $\ell$ modes [23]), scattering near the
horizon is evidently dominated by the low $\ell$ modes, as can be seen from
(1). The importance of metric perturbations for a resolution of the
information paradox has previously been emphasised [37, 38, 39, 40].

A rather important consequence of (1) is the fact that upon a Fourier
transform of the right hand side of that equation, we see that the four point
function is non-zero only when

$y_{out}~{}=~{}\dfrac{8\pi G_{N}}{\ell^{2}+\ell+2}p_{in}\,.$ (1.3)

This relation is very close to the Shapiro delay derived from the backreaction
calculation of [41] in a first quantised formalism [42, 43, 44]. The curious
discrepancy is in the extra factor in $\ell^{2}+\ell+2$ instead of the
$\ell^{2}+\ell+1$ that was found in those references. As we will show in
Section 5.2, both results are correct in their own right and the present one
must be seen as a second quantised generalisation that includes arbitrary off-
shell fluctuations in the path integral. It is worth noting that our results
are consistent with the expectation that quantum chaos is non-perturbatively
realised; our calculation shows which parameters are required to be non-
perturbatively treated (namely $\gamma$ and $\hbar$), and which perturbatively
($\sqrt{s}/\gamma M_{Pl}$), for this realisation. Our effective field theory
setup allows us to see that while the eikonal expectation [25] that ladder
diagrams yield the shock wave geometry is indeed largely true, the
‘classicalisation’ also allows for off-shell gravitons to contribute to the
black hole eikonal amplitude, as we show in Section 5.2. Moreover, our
effective field theory is a natural framework that can capture exponentially
suppressed amplitudes, and exponentially many of them, in the form of general
$2\rightarrow N$ amplitudes in addition to the $2\rightarrow 2$ amplitudes we
study in this paper.

That scattering amplitudes capture non-linearities in Einstein’s equations is
increasingly being appreciated [45, 46, 47, 48, 49, 50, 51, 52, 53]. In fact,
the classical tree-level three point vertex is sufficient [14, 54] to
completely determine the Aichelberg-Sexl solution [55, 56, 57]. In the case of
black holes, under consideration in the present article, off-shell
fluctuations do contribute to the non-perturbative amplitudes444See [58] for a
recent perspective on classical scattering of a pair of black holes.. It is
curious and remarkable that their effect is to merely renormalise555The change
in prefactor is compared to the what appears in the non-linear Dray-’t Hooft
solution; see eq. (2.9) of [43] for instance. We discuss this further in
Section 5.2. the prefactor in (1.3). This may implore one to ask whether
quantum chaotic behaviour can also be seen to emerge in the first quantised
formalism of [42, 43, 44]. As it turns out, appropriate boundary conditions
that respect the scattering algebra near the central causal diamond do indeed
generate a rich and chaotic spectrum; those of the zeroes of the Riemann zeta
and Dirichlet beta functions [59]. This boundary condition can be motivated to
arise from the expectation that CPT is a gauge symmetry in a putative UV
complete theory of quantum gravity, as was also argued in [59]. It is to be
seen as a generalisation of the antipodal identification that has gained some
traction in the context of black holes [60, 43, 44, 61, 62, 63, 64] and
wormholes [65, 66].

##### Organisation of the paper

We begin with a quick review of the eikonal regime in flat space in Section 2.
In Section 3, we set up the theory of quantum gravity coupled to a massless
scalar field in a certain soft limit, in the background field method. We first
begin with arbitrary backgrounds, and then specialise to spherically symmetric
ones, integrate the sphere out, and formulate an effective two-dimensional
theory. Thereafter, in Section 4, we move to physics near the horizon and
derive the propagators and Feynman rules governing the interactions. In
Section 5, we compute the advertised four point function. We end with a
discussion on the implications of our results for various existing proposals
for a resolution of the information paradox and an outlook, in Section 6. The
sections and subsections have been so titled to allow for an understanding of
the flow of the paper, merely by reading the table of contents.

##  2 A review of eikonal physics in flat space

Certain features of Planckian scattering in flat space are dominated by
virtual gravitons. The significance of soft gravitons in quantum gravity was
perhaps first noted in this classic example. In the eikonal limit, elastic
forward scattering of massive scalar particles can be calculated exactly [6,
7]. It is instructive to review this example. We begin with the flat space
metric in Cartesian coordinates in four dimensions

$\displaystyle\text{d}s^{2}~{}=~{}-\text{d}t^{2}+\text{d}x^{2}+\text{d}y^{2}+\text{d}z^{2}\,.$
(2.1)

Of the four external particles, the two ingoing ones are taken to carry
momenta $p_{1}$ and $p_{2}$ while the outgoing momenta are labelled by $p_{3}$
and $p_{4}$. The Mandelstam variables of interest are

$s~{}\coloneqq~{}-\left(p_{1}+p_{2}\right)^{2}\qquad\text{and}\qquad
t~{}\coloneqq~{}-\left(p_{1}-p_{3}\right)^{2}\,,$ (2.2)

and we focus on the limit $s\gg t$. The eikonal limit consists of discarding
effects that are sub-dominant in $s/m^{2}$, where $m$ is the mass of the
scalar field. Moreover, to avoid large transverse momentum transfer, the
impact parameter is taken to be large; in flat space, the only availables
scales to compare the impact parameter with are the Planck length, i.e.
$b\gg\ell_{\text{Pl}}$, and the scale associated to the centre of mass energy
of the collisions, i.e. $b\gg G_{N}\sqrt{s}$. Therefore, the two scattering
particles maintain most of their momentum in the scattering direction which we
call longitudinal, i.e. $p_{1}^{\parallel}\approx p_{2}^{\parallel}$. The two
particles do however exchange momentum in the remaining two directions; the
transverse directions, such that $p_{1}^{\perp}\neq p_{2}^{\perp}$.
Nevertheless, for all particles, we take $p_{i}^{\parallel}\gg p_{i}^{\perp}$.
This will later allow us to consider diagrams where there is no exchange in
four momentum, all virtual gravitons will be ‘soft’.

The Feynman rules of interest include the familiar flat space scalar and
graviton propagators that are well known and relatively straight forward to
calculate
feyn_minkpropagators

and the interactions are governed by a three point vertex that arises from the
coupling of the stress tensor to the metric fluctuations. It is of the form:
feyn_minkvertex

$\displaystyle\fmfgraph*(100,100)\fmfleft{i8,i7}\fmfright{o2}\fmf{gluon}{i7,v1}\fmf{fermion}{i8,v1}\fmf{fermion}{v1,o2}\fmflabel{$h_{\mu\nu}$}{i7}\fmflabel{$p_{2}$}{i8}\fmflabel{$p_{1}~{}.$}{o2}\fmfdot{v1}\fmfv{label=$i~{}p^{1}_{\mu}~{}p^{2}_{\nu}$,label.angle=60,label.dist=20}{v1}\,$

The scalar propagator is the familiar Klein-Gordon propagator and the graviton
propagator is written in the harmonic gauge with $\kappa^{2}=8\pi G_{N}$. The
vertex in principle contains another term

$p_{\mu}p_{\nu}-\dfrac{1}{2}\eta_{\mu\nu}\left(p^{2}+m^{2}\right)\,,$ (2.3)

but the second recoil term of the matter field can be neglected for eikonal
scattering. Similarly, for large external momentum $p$, internal scalar
propagators can be approximated as:

$\dfrac{1}{\left(p+k\right)^{2}+m^{2}-i\epsilon}~{}\approx~{}\dfrac{1}{2p\cdot
k-i\epsilon}\,.$ (2.4)

The tree level contribution to the four point function
$\langle\phi\left(p_{1}\right)\phi\left(p_{2}\right)\phi\left(p_{3}\right)\phi\left(p_{4}\right)\rangle$
arises from the Feynman diagram shown in Fig. 2.

feyn_minkTree (130,130) i1,i2 o1,o2 $\phi\left(p_{2}\right)$i1
$\phi\left(p_{4}\right)$o1 $\phi\left(p_{1}\right)$i2
$\phi\left(p_{3}\right)$o2 fermioni1,v1 fermionv1,o1 fermioni2,v2 fermionv2,o2
gluonv1,v2

Figure 2: Tree level diagram with virtual graviton exchange, built out of two
three-point vertices.

This tree level diagram Fig. 2 is easily shown to evaluate to

$i\mathcal{M}~{}=~{}\dfrac{2i\kappa^{2}j(s)}{-t}\qquad\text{with}\qquad
j(s)~{}\coloneqq~{}\dfrac{1}{2}\left(\left(s-2m^{2}\right)^{2}-2m^{4}\right)\,.$
(2.5)

In the eikonal limit, the Feynman diagrams that dominate are the so-called
ladder diagrams displayed in Figure 3. All other loop diagrams (self-energy,
vertex corrections) are insignificant in the eikonal regime, that is to say
that their contributions are sub-leading in $s/m^{2}$.

feyn_minkOneLoop1 (150,100) i1,i2 o1,o2 $\phi\left(p_{2}\right)$i1
$\phi\left(p_{4}\right)$o1 $\phi\left(p_{1}\right)$i2
$\phi\left(p_{3}\right)$o2 fermioni1,v1 fermionv3,o1 fermioni2,v2 fermionv1,v3
fermionv2,v4 fermionv4,o2 gluon, tension=0v1,v2 gluon, tension=0v3,v4

feyn_minkOneLoop2 (150,100) i1,i2 o1,o2 $\phi\left(p_{2}\right)$i1
$\phi\left(p_{4}\right)$o1 $\phi\left(p_{1}\right)$i2
$\phi\left(p_{3}\right)$o2 fermioni1,v1 fermionv3,o1 fermioni2,v2 fermionv1,v3
fermionv2,v4 fermionv4,o2 gluon, tension=0v1,v4 gluon, tension=0v2,v3

Figure 3: The one-loop ladder diagrams contributing to the eikonal scattering
amplitude. The two graviton lines crossing each other in the diagram on the
right do not interact, as graviton self-interactions are sub-leading in the
eikonal limit. The ladder diagrams give all leading contributions in
$s/m^{2}$.

At loop level, it is important to average over the various ways to attach the
internal graviton legs, and the corresponding conserved momentum. For the one-
loop case, as can be seen in Fig. 3, there are two possibilities arising from
fixing internal graviton momentum $k$ on either external leg in a direction of
choice. The total amplitude at one-loop results from summing over both of
these choices, and inserting a factor of a half. It is then straightforward to
proceed with the calculation for the amplitude, as outlined in [7]. The final
expression at one-loop becomes

$\displaystyle i\mathcal{M}~{}$
$\displaystyle=~{}\kappa^{2}j\left(s\right)\int\text{d}^{4}x\text{
}e^{-i\left(p_{1}-p_{3}\right)\cdot
x}\Delta\left(x\right)\chi\left(x\right)\,,$ (2.6) $\displaystyle\chi~{}$
$\displaystyle\coloneqq~{}-2\kappa^{2}j\left(s\right)\int\dfrac{\text{d}^{4}k}{\left(2\pi\right)^{4}}\text{
}e^{ik\cdot x}\dfrac{1}{k^{2}-i\epsilon}\times\left[\dfrac{1}{-2p_{1}\cdot
k-i\epsilon}\dfrac{1}{2p_{2}\cdot k-i\epsilon}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+~{}\dfrac{1}{-2p_{1}\cdot
k-i\epsilon}\dfrac{1}{-2p_{4}\cdot k-i\epsilon}+\dfrac{1}{2p_{3}\cdot
k-i\epsilon}\dfrac{1}{2p_{2}\cdot k-i\epsilon}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\left.+\dfrac{1}{2p_{3}\cdot
k-i\epsilon}\dfrac{1}{-2p_{4}\cdot k-i\epsilon}\right]\,.$ (2.7)

Here, we have defined a momentum space massless Klein-Gordon propagator

$\displaystyle\dfrac{-1}{k^{2}-i\epsilon}~{}=~{}\int\text{d}^{4}x\text{
}e^{-ik\cdot x}\Delta\left(x\right)\,.$ (2.8)

Furthermore, in arriving at the above expression for the amplitude, we
approximated $(p_{i}+k)_{\mu}\approx p_{i\mu}$ in the vertex since $p_{1}$ is
very large. While $k$ is integrated over, the eikonal approximation is such
that the leading order contributions to the integrand are for $k\ll p_{i}$
allowing us to safely take only $p_{i}$. This approximation notwithstanding,
the expression for $\chi$ does not contain any UV-divergences in the eikonal
approximation. The additional $k$ from the vertex would then be countered by a
$k^{2}$ in the matter propagator. Remarkably, the UV-divergences are all
embedded in different diagrams, whose effects are subdominant in eikonal
scattering. This is crucial as the theory is not renormalisable. While it may
seem counter-intuitive at first glance, the divergent diagrams when
dimensionally regularized are indeed lower order in the eikonal approximation.
Gravity does not allow us to do any better at this stage.

The next step is to sum over all higher loop diagrams. Since any diagram at
order $n$ would contain $n$ extra graviton legs (in comparison to the tree
level diagram), there is an $n!$ that arises from symmetry. This also holds
for diagrams where the graviton legs cross each other. Therefore, we find a
$1/n!$ factor for each order in perturbation theory. So, non-perturbatively an
exponential series ensues; this is a typical feature of eikonal scattering.
The total scattering amplitude is thus written as

$\displaystyle i\mathcal{M}=-2i\kappa^{2}j(s)\int\text{d}^{4}x\text{
}e^{-i(p_{1}-p_{3})\cdot x}\Delta(x)\frac{e^{i\chi}-1}{\chi}.$ (2.9)

A calculation of the non-perturbative amplitude from such an exponentiation of
the one-loop result is possible owing to the fact that vertex corrections and
self energy diagrams are sub-dominant, as argued in [67]. What remains now is
the evaluation of $\chi$ by solving the integral. To this end, we now
approximate $p_{1}\approx p_{3},p_{2}\approx p_{4}$. This may interpreted as a
soft limit as it restricts the virtual graviton momenta. Inserting
$p_{1}=p_{3},p_{2}=p_{4}$ into (2) results in:

$\displaystyle\chi~{}$
$\displaystyle=~{}-2\kappa^{2}j\left(s\right)\int\dfrac{\text{d}^{4}k}{(2\pi)^{4}}\text{
}e^{ik\cdot x}\dfrac{1}{k^{2}-i\epsilon}\left[\dfrac{1}{2p_{1}\cdot
k+i\epsilon}-\dfrac{1}{2p_{1}\cdot k-i\epsilon}\right]$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\left[\dfrac{1}{2p_{2}\cdot
k+i\epsilon}-\dfrac{1}{2p_{2}\cdot k-i\epsilon}\right]\,.$ (2.10)

Since $\epsilon$ is an infinitesimal regulator, we can now use the following
delta identity:

$\displaystyle\dfrac{1}{x+i\epsilon}-\dfrac{1}{x-i\epsilon}~{}=~{}-2\pi
i\delta(x)\,.$ (2.11)

Using this, two of the integrals can be removed from (2), allowing us to
rewrite it as

$\displaystyle\chi~{}$
$\displaystyle=~{}-2\kappa^{2}j(s)\int\dfrac{\text{d}^{4}k}{\left(2\pi\right)^{4}}\text{
}e^{ik\cdot x}\dfrac{1}{k^{2}-i\epsilon}\left(-2\pi
i\right)^{2}\delta\left(2p_{1}\cdot k\right)\delta\left(2p_{2}\cdot k\right)$
(2.12)
$\displaystyle=~{}\dfrac{\kappa^{2}j(s)}{4Ep}\int\dfrac{\text{d}^{2}k_{\perp}}{\left(2\pi\right)^{2}}\text{
}e^{ik_{\perp}\cdot
x_{\perp}}\dfrac{1}{k_{\perp}^{2}+\tilde{\mu}^{2}-i\epsilon}\,,$ (2.13)

where we switched to the center of mass frame defined by $p_{1}=(E,0,0,p)$ and
$p_{2}=(E,0,0,-p)$. The new parameter $\tilde{\mu}$ is an infrared regulator
corresponding to the graviton mass and $x_{\perp}$ are the two remaining
coordinates in the transverse directions. The solution to the integral in
(2.13) can be shown to be

$\displaystyle\chi~{}=~{}\dfrac{G_{N}j(s)}{Ep}K_{0}\left(\tilde{\mu}x_{\perp}\right)~{}\approx~{}-\dfrac{G_{N}j(s)}{Ep}\log\left(\tilde{\mu}x_{\perp}\right)\,,$
(2.14)

where the last approximation holds for $\tilde{\mu}x_{\perp}\ll 1$ since
$\tilde{\mu}$ is a regulator. In this expression, a numerical constant has
been absorbed into $\tilde{\mu}$. In contrast to this result, we will see in
this paper that the black hole provides for a natural infrared regulator that
is not put in by hand.

We now return to solving (2.9); now that $\chi$ is known, only the integral
remains. First we pay attention to $\Delta(x)$. In the eikonal approximation,
$p_{1}-p_{3}$ only contains transverse components, i.e. only the transverse
components of $q\equiv p_{1}-p_{3}$ are dominant. This allows us to write
$e^{-iq\cdot x}\approx e^{-iq_{\perp}\cdot x_{\perp}}$. We can then isolate

$\displaystyle\int\text{d}t\text{d}z\text{
}\Delta(x)~{}=~{}\int\dfrac{\text{d}^{2}q_{\perp}}{\left(2\pi\right)^{2}}e^{iq_{\perp}\cdot
x_{\perp}}\dfrac{-1}{q_{\perp}^{2}-i\epsilon}~{}=~{}\dfrac{-Ep}{2\pi
G_{N}j(s)}\,\chi\,.$ (2.15)

Since $\chi$ only depends on $x_{\perp}$, we can insert this expression into
(2.9). The integral to be solved simply becomes

$\displaystyle i\mathcal{M}~{}=~{}8Ep\int\text{d}^{2}x_{\perp}\text{
}e^{-iq_{\perp}\cdot x_{\perp}}\bigr{(}e^{i\chi}-1\bigr{)}\,.$ (2.16)

Solving this integral yields the final result for the scalar four point
function in the eikonal regime

$\displaystyle
i\mathcal{M}~{}=~{}\dfrac{2i\kappa^{2}j(s)}{-t}\dfrac{\Gamma\left(1-i\alpha(s)\right)}{\Gamma\left(1+i\alpha(s)\right)}\left(\dfrac{4\tilde{\mu}^{2}}{-t}\right)^{-i\alpha(s)}\,,$
(2.17)

where we defined

$\displaystyle\alpha(s)~{}=~{}G_{N}\dfrac{\left(s-2m^{2}\right)^{2}-2m^{4}}{\sqrt{s\left(s-4m^{2}\right)}}\,.$
(2.18)

This amplitude is non-perturbative in the coupling constant $\kappa$, but is
valid only to leading order in $s/m^{2}$. Therefore, the approximation gets
better with ultra-high energy scattering. For large $s$, we notice that
$\alpha(s)$ is real and the complete amplitude reduces to the tree-level
amplitude in (2.5) up to a phase. In the limit $m\to 0$ and for
$\tilde{\mu}=1$ the scattering amplitude matches the semi-classical scattering
matrix derived by ’t Hooft in [6] based on a first quantised description of
shockwaves on an Aichelburg-Sexl metric [56]. This corroborates the naive
notion that while certain aspects of ultra-high energy scattering can be well
understand, quantum gravity is generically very difficult and poorly
understood because regimes of large momentum transfer (which are ignored in
the eikonal approximation) are plagued with problems like UV divergences,
potential non-locality at the Planck scale, etc. In what we have calculated so
far, it is not clear how one may even proceed to account for these in
principle.

Nevertheless, this notion is indeed naive in that quantum gravity is not only
sensitive to large momentum transfers and Planckian physics, but also to
strong gravitational backgrounds. Indeed, remarkable puzzles arise in strong
gravitational backgrounds that are classical solutions to General Relativity.
The most celebrated of them is of course the paradox of black hole information
loss.

In this paper, we will address the problem of quantum gravity in a tractable
limit (that in which the virtual gravitons do not transfer momenta) in the
presence of strong gravitational backgrounds where new large scales emerge
dynamically. An example is the Schwarzschild solution where a new physical
scale of the Schwarzschild radius emerges well before a collapsing shell has
fallen past that radius. The graviton propagator is difficult to determine in
generic backgrounds, and as we will see, it will take us some effort to derive
the necessary Feynman rules before we are able to carry out a calculation, of
the kind reviewed in this section, on the horizon of a Schwarzschild black
hole.

##  3 Quantum gravity in the soft limit

We begin by setting the scene for a putative quantum gravity path integral

$\mathcal{Z}~{}=~{}\int\mathcal{D}\bar{g}_{\mu\nu}\mathcal{D}\phi\exp\left(S\left[\bar{g}_{\mu\nu},\phi\right]\right)\,,$
(3.1)

where we take the matter action to be that of a minimally coupled scalar field
for simplicity:

$\displaystyle S\left[\bar{g}_{\mu\nu,\phi}\right]~{}$
$\displaystyle=~{}S_{EH}\left[\bar{g}_{\mu\nu}\right]+S_{M}\left[\bar{g}_{\mu\nu},\phi\right]$
$\displaystyle=~{}\dfrac{1}{2\kappa^{2}}\int\text{d}^{4}x\sqrt{-\bar{g}}\,\bar{R}-\dfrac{1}{2}\int\text{d}^{4}x\sqrt{-\bar{g}}\,\bar{\nabla}_{\mu}\phi\bar{\nabla}^{\mu}\phi\,,$
(3.2)

with $\kappa^{2}=8\pi G_{N}$. The covariant derivatives with the bar are
associated to $\bar{g}_{\mu\nu}$ and in what follows, those without will be
associated to $g_{\mu\nu}$. We will work in the background field method with a
vanishing matter background,

$\bar{g}_{\mu\nu}~{}=~{}g_{\mu\nu}+\kappa h_{\mu\nu}\,,$ (3.3)

and assume that $g_{\mu\nu}$ solves the vacuum Einstein’s equations. This
implies that (3.1) becomes

$\mathcal{Z}~{}=~{}\int\mathcal{D}h_{\mu\nu}\mathcal{D}\phi\exp\left(S_{EH}\left[h_{\mu\nu}\right]+S_{M}\left[h_{\mu\nu},\phi\right]\right)\,.$
(3.4)

In principle, both terms in the exponent come with infinitely many powers of
$h_{\mu\nu}$ owing to the fluctuations of the inverse metric. The matter
action begins with:

$\displaystyle S_{M}\left[h_{\mu\nu},\phi\right]~{}$
$\displaystyle=~{}\dfrac{1}{2}\int\text{d}^{4}\sqrt{-g}\phi\Box\phi-\kappa\int\text{d}^{4}x\dfrac{\delta
S_{M}\left[\bar{g}_{\rho\sigma},\phi\right]}{\delta\bar{g}^{\mu\nu}(x)}\biggr{\rvert}_{\bar{g}=g}h^{\mu\nu}(x)$
$\displaystyle=~{}\dfrac{1}{2}\int\text{d}^{4}\sqrt{-g}\phi\Box\phi+\dfrac{1}{2}\kappa\int\text{d}^{4}x\sqrt{-g}h^{\mu\nu}T_{\mu\nu}$
(3.5)

where we defined the stress tensor

$T_{\mu\nu}~{}=~{}\dfrac{-2}{\sqrt{-g}}\dfrac{\delta S_{M}}{\delta
g^{\mu\nu}}\,.$ (3.6)

For most of this paper, we will restrict our attention to the quadratic
kinetic terms and the cubic interaction in (3). The pure gravity action is
(derived in Appendix A.1):

$S_{EH}\left[h_{\mu\nu}\right]~{}=~{}\dfrac{1}{4}\int\text{d}^{4}x\sqrt{-g}\left(h^{\mu\nu}-\dfrac{1}{2}g^{\mu\nu}h\right)\bigr{(}2\nabla^{\sigma}\nabla_{(\mu}h_{\nu)\sigma}-\Box
h_{\mu\nu}-\nabla_{\mu}\nabla_{\nu}h\bigr{)}\,,$ (3.7)

where we defined the trace of the graviton $h=g^{\mu\nu}h_{\mu\nu}$.

Higher order fluctuations give rise to quartic (and higher order) interaction
vertices. In the Einstein-Hilbert part of the action, some of these contain
terms with derivatives on $h_{\mu\nu}$. In momentum space, these contribute to
graviton momenta in the Feynman diagrams. Integration over these momenta
results in the familiar non-renormalisable features of gravity [68, 69]. These
arise from large momenta in the ultraviolet. As we will show in this paper,
extremely interesting physics arises in the soft limit where internal graviton
momenta are taken to vanish. Although it is premature at this stage, we will
argue in Section 4 and Section 5 that contributions from large virtual momenta
are vanishing for the four-point function of interest; the soft limit is
essentially enforced upon us when we restrict ourselves to three-point
vertices. There are additional higher order interaction terms that do not
contain derivatives (for instance, those coming from the expansion of the
inverse metric to higher orders), which we hope to address in future work.

Therefore the complete action (upto cubic vertices) for fluctuations about a
generic solution of vacuum Einstein’s equations of motion is given by the sum
of (3.7) and (3):

$S\left[h_{\mu\nu},\phi\right]~{}=~{}S_{EH}\left[h_{\mu\nu}\right]+S_{M}\left[h_{\mu\nu},\phi\right]\,.$
(3.8)

A recent discussion emphasising the role and importance of interactions in the
context of black holes can be found in [40, 70].

###  3.1 Spherically symmetric backgrounds

In what follows, we will focus on backgrounds with spherical symmetry. Of
course, Birkhoff’s theorem limits the class of such allowed vacuum solutions.
Nevertheless, large parts of our calculation is aimed at generic spherical
backgrounds supported by matter; we will only specify the Schwarzschild metric
in Section 4. The general metric we will concern ourselves with is of the
form:

$\text{d}s^{2}~{}=~{}-2A(x,y)\text{d}x\text{d}y+r(x,y)^{2}\text{d}\Omega^{2}\,.$
(3.9)

With $A=e^{1-r/R}R/r$ and $r$ implicitly defined by
$xy=2R^{2}(1-r/R)e^{r/R-1}$, this reduces to the Schwarzschild solution in
Kruskal-Szekeres coordinates. Perturbations around the Schwarzschild metric
have long been studied [71, 72, 73, 74, 75] with a predominant focus often
laid on a study of gravitational waves, cosmic censorship, and quasi normal
modes.

##### Gauge symmetry

Not all components of the fluctuations $h_{\mu\nu}$ are physical; there are
gauge redundancies due to infinitesimal diffeomorphisms $x\rightarrow
x+\xi(x)$, under which the graviton transforms as

$h_{\mu\nu}(x)~{}\rightarrow~{}h_{\mu\nu}(x)+\nabla_{\mu}\xi_{\nu}+\nabla_{\nu}\xi_{\mu}\,.$
(3.10)

A choice of the vector $\xi_{\mu}$ fixes a choice of gauge. This choice is of
course, not to be confused with the choice of a system of coordinates for the
background. Infinitesimal diffeomorphisms acting on the graviton fluctuations
are different from finite coordinate transformations of the background; as we
will see, $\xi_{\mu}$ will be linear in the fluctuations and therefore shall
not interfere with the choice of coordinates for the background. An
alternative way to think about this is to note that a covariant action can be
written for the fluctuations as in (3.8). A choice of background coordinates
would not break that covariance, while a choice of $\xi_{\mu}$ would make
unphysical components of the graviton $h_{\mu\nu}$ redundant.

The oft employed gauge in the literature is the harmonic (or covariant or De
Donder) gauge defined by

$\nabla_{\mu}\left(h^{\mu\nu}-\dfrac{1}{2}g^{\mu\nu}h\right)~{}=~{}0\,.$
(3.11)

In this gauge, the first order variation of the Ricci tensor is of the form

$R^{(1)}_{\mu\nu}~{}=~{}-\dfrac{1}{2}\left(g_{\mu\rho}g_{\nu\sigma}\Box-2R_{\mu\rho\nu\sigma}\right)h^{\rho\sigma}\,.$
(3.12)

The operator $g_{\mu\rho}g_{\nu\sigma}\Box+2R_{\mu\rho\nu\sigma}$ is the
familiar Lichnerowicz operator. It has been studied on the Schwarzschild
background in [76, 77]. An important difficulty with the choice of the
harmonic gauge is that the Lichnerowicz operator couples various spherical
harmonics of the graviton. Instead, we will use an alternative gauge; the
Regge-Wheeler gauge first proposed in [71]. This choice exploits the spherical
symmetry of the background and allows us to reduce the action (3.8) to an
infinite tower of decoupled666In principle interaction terms do couple the
partial waves, but the coupling between the partial waves can be ignored when
transverse momenta are ignored. effective two-dimensional actions; one for
each partial wave. Conveniently enough, the reduced effective action can be
written covariantly in two dimensions.

###  3.2 Decomposition in spherical harmonics

The decomposition of the graviton into spherical tensor harmonics needs care.
Owing to its tensorial nature, the corresponding spherical harmonic
decomposition differs from the usual spherical harmonic decomposition of
scalars. The angular components $h_{aA}$ and $h_{AB}$ have a nontrivial
spherical harmonic expansion which must be taken into account. The lowercase
indices run over the lightcone coordinates while the uppercase indices span
the sphere. As was pointed out in [71], this can be achieved by splitting the
graviton tensor harmonics into odd and even parity modes:

$h_{\mu\nu}~{}=~{}\sum\limits_{\ell,m}h^{-}_{\ell
m,\mu\nu}+\sum\limits_{\ell,m}h^{+}_{\ell m,\mu\nu}\,,$ (3.13)

where

$h^{-}_{\ell
m,\mu\nu}=\begin{pmatrix}0&0&-h^{-}_{x}\csc\theta\partial_{\phi}&h^{-}_{x}\sin\theta\partial_{\theta}\\\
0&0&-h^{-}_{y}\csc\theta\partial_{\phi}&h^{-}_{y}\sin\theta\partial_{\theta}\\\
&&h_{\Omega}\csc\theta\left(\partial_{\theta}\partial_{\phi}-\cot\theta\partial_{\phi}\right)&\tfrac{1}{2}h_{\Omega}\left(\csc\theta\partial_{\phi}^{2}+\cos\theta\partial_{\theta}-\sin\theta\partial_{\theta}^{2}\right)\\\
&&&-h_{\Omega}\sin\theta\left(\partial_{\theta}\partial_{\phi}-\cot\theta\partial_{\phi}\right)\end{pmatrix}Y_{\ell}^{m}$
(3.14)

is the odd parity mode and

$h^{+}_{\ell
m,\mu\nu}=\begin{pmatrix}H_{xx}&H_{xy}&h^{+}_{x}\partial_{\theta}&h^{+}_{x}\partial_{\phi}\\\
&H_{yy}&h^{+}_{y}\partial_{\theta}&h^{+}_{y}\partial_{\phi}\\\
&&r^{2}(K+G\partial_{\theta}^{2})&r^{2}G(\partial_{\theta}\partial_{\phi}-\cot\theta\partial_{\phi})\\\
&&&r^{2}(K\sin^{2}\theta+G(\partial_{\phi}^{2}+\sin\theta\cos\theta\partial_{\theta}))\end{pmatrix}Y_{\ell}^{m}$
(3.15)

is the even parity mode, with $Y_{\ell}^{m}$ being the familiar spherical
harmonics of the two-sphere. In these expressions, the missing entries are
determined by the symmetry of the matrices. In [71], this decomposition was
made for the Schwarzschild black hole, written in Schwarzschild coordinates.
While we shall still work with a generic spherically symmetric background at
this stage, for comparison to [71], the entries of the graviton modes above
are given in terms of those of [71] as:

$\displaystyle h^{\pm}_{x}(x,y)~{}$
$\displaystyle=~{}-\dfrac{R}{x}\left(h^{\pm}_{0}+\left(1-\dfrac{R}{r}\right)h^{\pm}_{1}\right)\,,$
(3.16) $\displaystyle h^{\pm}_{y}(x,y)~{}$
$\displaystyle=~{}-\dfrac{R}{y}\left(h^{\pm}_{0}-\left(1-\dfrac{R}{r}\right)h^{\pm}_{1}\right)\,,$
(3.17) $\displaystyle h_{\Omega}(x,y)~{}$ $\displaystyle=~{}h_{2}\,,$ (3.18)
$\displaystyle H_{xx}(x,y)~{}$
$\displaystyle=~{}-\dfrac{y}{2x}\bigr{(}H_{0}+H_{2}+2H_{1}\bigr{)}\,,$ (3.19)
$\displaystyle H_{xy}(x,y)~{}$
$\displaystyle=~{}\dfrac{1}{2}\bigr{(}H_{0}-H_{2}\bigr{)}\,,$ (3.20)
$\displaystyle H_{yy}(x,y)~{}$
$\displaystyle=~{}-\frac{x}{2y}\bigr{(}H_{0}+H_{2}-2H_{1}\bigr{)}\,.$ (3.21)

To arrive at these transformations, we made extensive use of the identity
$\left(1-\tfrac{R}{r}\right)=\frac{xyA}{2R^{2}}$. It is evident that the time
translation invariance of the Schwarzschild solution manifests itself as $x\to
ax,y\to\tfrac{1}{a}y$, and we observe that $h_{x}^{\pm},h_{y}^{\pm}$ and
$H_{xx},H_{yy}$ are not time translation invariant.

Further on in this paper, for the sake of convenience, we will work in light-
cone coordinates of the Kruskal-Szekeres kind. Nevertheless, as argued in
[78], the covariance of the effective two-dimensional theory we shall present
and the conformally flatness of the horizon indicate that the resulting
amplitude is gauge invariant.

The decomposition above is general and therefore, the graviton still contains
redundant gauge degrees of freedom. The $r^{2}$ appearing in the graviton
modes depends on the light-cone coordinates $x,y$ as was noted earlier. All
functions $h^{\pm}_{x},h^{\pm}_{y},h_{\Omega},H_{xx},H_{xy},H_{yy},K,G$ are
functions of only $x,y$ with no further constraints. All of them naturally
carry $\ell,m$ indices; we have dropped them to avoid clutter of notation.
These account for the ten degrees of freedom of the graviton.

#### The Regge-Wheeler gauge

We now perform a gauge fixing similar to Regge and Wheeler in [71] of the form

$\displaystyle\xi_{a}$ $\displaystyle=$ $\displaystyle\zeta_{a}Y_{\ell}^{m},$
(3.22) $\displaystyle\xi_{A}$ $\displaystyle=$
$\displaystyle-\tfrac{1}{2}r^{2}G\partial_{A}Y_{\ell}^{m}-\tfrac{1}{2}h_{\Omega}\epsilon_{A}^{\text{
}\text{ }B}\partial_{B}Y_{\ell}^{m}.$ (3.23)

Here the antisymmetric tensor is defined in Appendix B.1 and

$\displaystyle\zeta_{a}=\left(\tfrac{1}{2}r^{2}\partial_{a}G-h^{+}_{a}\right).$
(3.24)

As was noted earlier, infinitesimal gauge transformations do not interfere
with the choice of background Kruskal-Szekeres system of coordinates because
$\xi_{a},\xi_{A}$ are of order $h$. The above choice of gauge, along with the
redefinitions of $h_{a}^{-}$ , $K$, and $H_{ab}$ as

$\displaystyle h^{-}_{a}~{}$
$\displaystyle\to~{}h_{a}-\tfrac{1}{2}r^{2}\partial_{a}\left(\dfrac{1}{r^{2}}h_{\Omega}\right)\,,$
(3.25) $\displaystyle K~{}$
$\displaystyle\to~{}K-2g^{ab}\zeta_{a}\partial_{b}\log r\,,$ (3.26)
$\displaystyle H_{ab}~{}$
$\displaystyle\to~{}H_{ab}-\nabla_{a}\zeta_{b}-\nabla_{b}\zeta_{a}\,,$ (3.27)

kills $G$, $h^{+}_{a}$, and $h_{\Omega}$. We have dropped the minus
superscript on $h^{-}_{x}$ and $h^{-}_{y}$ since $h^{+}_{x}$ and $g^{+}_{y}$
are now zero. It can be checked that the gauge transformation above, together
with the field redefinitions results in the same general form for the odd and
even graviton components, with $G,h_{a}^{+},h_{\Omega}$ set to be vanishing.
The gauge fixing procedure, therefore, makes four of the components redundant
leaving six physical off-shell modes behind. After gauge fixing the graviton
components are given as

$h^{-}_{\ell
m,\mu\nu}~{}=~{}\begin{pmatrix}0&0&-h_{x}\csc\theta\partial_{\phi}&h_{x}\sin\theta\partial_{\theta}\\\
0&0&-h_{y}\csc\theta\partial_{\phi}&h_{y}\sin\theta\partial_{\theta}\\\
-h_{x}\csc\theta\partial_{\phi}&-h_{y}\csc\theta\partial_{\phi}&0&0\\\
h_{x}\sin\theta\partial_{\theta}&h_{y}\sin\theta\partial_{\theta}&0&0\end{pmatrix}Y_{\ell}^{m}\,,$
(3.28)

and

$h^{+}_{\ell m,\mu\nu}~{}=~{}\begin{pmatrix}H_{xx}&H_{xy}&0&0\\\
H_{xy}&H_{yy}&0&0\\\ 0&0&r^{2}K&0\\\
0&0&0&r^{2}K\sin^{2}\theta.\end{pmatrix}Y_{\ell}^{m}\,.$ (3.29)

The minus sign in the odd mode $h^{-}_{\mu\nu}$ naturally arises when written
in index notation

$h^{-}_{aA}~{}=~{}-h_{a}\epsilon_{A}^{\text{ }\text{
}B}\partial_{B}Y_{\ell}^{m}\,.$ (3.30)

Owing to this antisymmetric nature of the odd parity mode, the even and odd
modes naturally decouple as we shall see. Furthermore, for any diffeomorphism
that acts on the light-cone and angular coordinates separately, we may simply
transform $h_{a}\to h_{a^{\prime}}$ and $\epsilon_{A}^{\text{ }\text{
}B}\partial_{B}Y_{\ell}^{m}\to\epsilon_{A^{\prime}}^{\text{ }\text{
}B^{\prime}}\partial_{B^{\prime}}Y_{\ell}^{m}$ accordingly. In the new
coordinates then, $h^{-}_{\mu\nu}$ is still given by (3.30), so that the
spherical harmonics decomposition is coordinate independent for all such
diffeomorphisms. Similary, for the even modes, we may write

$h^{+}_{ab}~{}=~{}H_{ab}Y_{\ell}^{m}\,\quad\text{and}\quad
h^{+}_{AB}~{}=~{}Kg_{AB}Y_{\ell}^{m}\,,$ (3.31)

so that coordinate invariance for the decomposition is manifest when $H_{ab}$
is transformed accordingly; $g_{AB}$ naturally transforms appropriately.
Because the decomposition persists so long as we also transform the fields
appropriately, we now define $H_{ab}$ to be a 2-tensor on the light-cone,
$h_{a}$ to be a vector on the light-cone, and $K$ to be a scalar on the light-
cone. Therefore, these fields transform under coordinate transformations on
the light-cone but not under diffeomorphisms of the angular coordinates. In a
similar fashion all quantities with indices along $A,B$ transform under
angular diffeomorphisms but not under those on the light-cone. The spherical
harmonics decomposition persists under any transformation that keeps the
light-cone and two-sphere separate in this sense. Any coordinate
transformations that break the decomposition will therefore be forbidden
henceforth.

Finally, it is worth noting that the even parity graviton carries four of the
six off-shell degrees of freedom in the form of a two-tensor $H_{ab}$ and the
scalar $K$, while the other two are with the odd-parity mode in the form of
the two-vector $h_{a}$. These degrees of freedom are now defined covariantly
and for arbitrary $A(r)$ in the metric.

###  3.3 Decoupling of the odd and even parity graviton modes

In this section we will investigate the coupling between the odd and even
parity modes defined by the graviton decomposition

$\displaystyle h^{-}_{aA}~{}$ $\displaystyle=~{}-h_{a}\epsilon_{A}^{\text{
}\text{ }B}\partial_{B}Y_{\ell}^{m}\,,$ $\displaystyle h^{+}_{ab}~{}$
$\displaystyle=~{}H_{ab}Y_{\ell}^{m}\,,$ (3.32) $\displaystyle h^{+}_{AB}~{}$
$\displaystyle=~{}Kg_{AB}Y_{\ell}^{m}\,,$

with the remaining terms vanishing. We expect this coupling to fall away due
to spherical symmetry and will now prove this. First we see from. (3.8) that
the part of the Lagrangian that couples the different parity modes is of the
form

$\mathcal{L}_{parity-
coupling}~{}=~{}-\dfrac{1}{4}h_{+}^{\mu\nu}G^{(1),-}_{\mu\nu}-\dfrac{1}{4}h_{-}^{\mu\nu}G^{(1),+}_{\mu\nu}$
(3.33)

where

$\displaystyle G^{(1),\pm}_{\mu\nu}~{}$
$\displaystyle=~{}\dfrac{1}{2}g^{\rho\sigma}\bigr{(}\nabla_{\rho}\nabla_{\mu}h^{\pm}_{\nu\sigma}+\nabla_{\rho}\nabla_{\nu}h^{\pm}_{\mu\sigma}-\nabla_{\rho}\nabla_{\sigma}h^{\pm}_{\mu\nu}-\nabla_{\mu}\nabla_{\nu}h^{\pm}_{\rho\sigma}\bigr{)}$
$\displaystyle\quad-~{}\dfrac{1}{2}g_{\mu\nu}\biggr{(}\nabla^{\rho}\nabla^{\sigma}h^{\pm}_{\rho\sigma}-\Box
h^{\pm}\biggr{)}\,.$ (3.34)

Evidently, the $\pm$ symbols denote the variation of the Einstein tensor
corresponding to $h^{\pm}$ respectively. Exploiting the decomposition
structure of (3.32), we find

$\displaystyle G^{(1),-}_{ab}~{}$ $\displaystyle=~{}0\,,$ (3.35)
$\displaystyle G^{(1),-}_{AB}g^{AB}~{}$ $\displaystyle=~{}0\,,$ (3.36)
$\displaystyle G_{aA}^{(1),+}~{}$
$\displaystyle=~{}\biggr{(}\tfrac{1}{2}\partial^{b}H_{ab}+\tfrac{1}{A}\partial_{a}H_{xy}-\tfrac{1}{2A}\bigr{(}\partial_{a}\log(Ar^{2})\bigr{)}H_{xy}-\frac{1}{2}\partial_{a}K\biggr{)}\partial_{A}Y_{\ell}^{m}$
(3.37)

with all other components being unnecessary owing to either the vanishing of
$h^{-}_{ab}$, $h^{-}_{AB}$ and $h^{+}_{aA}$, or the fact that
$h^{+}_{AB}=Kg_{AB}$. The second equation $G_{AB}^{(1),-}g^{AB}=0$ results
from the fact that $g^{AB}$ is even in $A,B$, while $G^{(1),-}_{AB}$ contains
an $\epsilon_{AB}$ which is odd in $A,B$. We now immediately see that the
first term in the coupling Lagrangian (3.33) vanishes. Therefore, we are left
with

$\mathcal{L}_{parity-coupling}~{}=~{}-\dfrac{1}{2}h_{-}^{aA}G^{(1),+}_{aA}\,.$
(3.38)

Thus, to evaluate the coupling of the even and odd parity modes, we only need
to evaluate $h^{-}_{aA}G_{(1),+}^{aA}$. At this point, the presence of
$\ell,m$ indices that were omitted require attention and explicit re-insertion

$\displaystyle h^{-}_{aA}~{}$
$\displaystyle=~{}-\sum\limits_{\ell,m}h_{a}^{\ell m}\epsilon_{A}^{\text{
}\text{ }\text{ }\text{ }B}\partial_{B}Y_{\ell}^{m}\,,$ (3.39) $\displaystyle
G^{(1),+}_{aA}~{}$ $\displaystyle=~{}\sum\limits_{\ell,m}F_{a}^{\ell
m}\partial_{A}Y_{\ell}^{m}\,.$ (3.40)

Here we defined the new quantity

$F^{\ell m}_{a}~{}\coloneqq~{}\dfrac{1}{2}\partial^{b}H^{\ell
m}_{ab}+\dfrac{1}{A}\partial_{a}H^{\ell
m}_{xy}-\dfrac{1}{2A}\bigr{(}\partial_{a}\log(Ar^{2})\bigr{)}H^{\ell
m}_{xy}-\dfrac{1}{2}\partial_{a}K^{\ell m}.$ (3.41)

This quantity can be directly read off from (3.37). Inserting the summations
in (3.39) and (3.40), the coupling Lagrangian may be rewritten as

$\mathcal{L}_{parity-
coupling}~{}=~{}-\dfrac{1}{2}\sum\limits_{\ell,m}\sum\limits_{\ell^{\prime},m^{\prime}}h^{\ell
m}_{a}F^{a}_{\ell^{\prime}m^{\prime}}\epsilon^{AB}\partial_{B}Y_{\ell}^{m}\partial_{A}\bar{Y}_{\ell^{\prime}}^{m^{\prime}}\,.$
(3.42)

We inserted complex conjugation denoted by $\bar{Y}$ because the Lagrangian is
required to be real and we take the imaginary representation of the spherical
harmonics. The corresponding action is found by integrating 3.42 to give

$S_{parity-
coupling}~{}=~{}-\tfrac{1}{2}\sum\limits_{\ell,m}\sum\limits_{\ell^{\prime},m^{\prime}}\int\text{d}^{2}xA\left(r\right)h^{\ell
m}_{a}F^{a}_{\ell^{\prime}m^{\prime}}\underbrace{\int\text{d}\phi\text{d}\theta\sqrt{g_{S_{2}}}\epsilon^{AB}\partial_{B}Y_{\ell}^{m}\partial_{A}\bar{Y}_{\ell^{\prime}}^{m^{\prime}}}_{\eqqcolon~{}C_{\ell
m;\ell^{\prime}m^{\prime}}}\,.$ (3.43)

Here $\sqrt{g_{S_{2}}}$ is the volume element on $S_{2}$ of radius $r$, i.e.
$\sqrt{g_{S_{2}}}=r^{2}\sin\theta$. The first integral is generally non-
vanishing, and is the only light-cone dependent piece (the $\sqrt{g_{S_{2}}}$
cancels as we shall see). The second integral is only dependent on
$\theta,\phi$ and contains only spherical harmonics; functions that we know
exactly. This angular integrand contains contractions over $A,B$ and is
integrated over the sphere. So, the quantity $C_{\ell
m;\ell^{\prime}m^{\prime}}$ is independent of the chosen two-sphere
coordinates.

The action in (3.43) explicitly splits the light-cone part depending on
$x^{a}$, and the spherical part depending on $x^{A}$ in a manner that is
covariant under light-cone and angular diffeomorphisms separately, without
mixing. We will now explicitly calculate this coordinate independent quantity
$C_{\ell m;\ell^{\prime}m^{\prime}}$; clearly, it can be determined exactly
without the knowledge of the graviton fields. The spherical harmonics are
defined by

$\displaystyle Y_{\ell}^{m}=\mathcal{N}_{\ell
m}e^{im\phi}P^{m}_{\ell}(\cos\theta)$ (3.44)

where $\mathcal{N}_{\ell m}$ is a normalization constant and $P^{m}_{\ell}(x)$
the associated Legendre polynomial which are real. The antisymmetric tensor
$\epsilon^{AB}$ as given in (B.33) has a prefactor $1/(r^{2}\sin\theta)$, so
that it cancels the $\sqrt{g_{S_{2}}}$ in (3.43). Inserting this into the
definition of $C_{\ell m;\ell^{\prime}m^{\prime}}$ gives

$\displaystyle C_{\ell m;\ell^{\prime}m^{\prime}}~{}$
$\displaystyle=~{}-i\mathcal{N}_{\ell
m}\mathcal{N}_{\ell^{\prime}m^{\prime}}\int\limits_{0}^{\pi}\text{d}\theta\int\limits_{0}^{2\pi}\text{d}\phi
e^{i(m-m^{\prime})\phi}\bigr{[}m^{\prime}P^{m^{\prime}}_{\ell^{\prime}}(\cos\theta)\partial_{\theta}P^{m}_{\ell}(\cos\theta)\bigr{.}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\bigr{.}+mP^{m}_{\ell}(\cos\theta)\partial_{\theta}P^{m^{\prime}}_{\ell^{\prime}}(\cos\theta)\bigr{]}\,.$
(3.45)

Here, the $\epsilon^{AB}$ introduces a minus sign, which is then cancelled by
the different signs in $\partial_{\phi}e^{im\phi}$ and
$\partial_{\phi}e^{-im^{\prime}\phi}$. Now using the orthogonality of the
complex exponents

$\int\limits_{0}^{2\pi}\text{d}\phi
e^{i(m-m^{\prime})\phi}~{}=~{}2\pi\delta_{mm^{\prime}}\,,$

we may immediately write down

$\displaystyle C_{\ell m;\ell^{\prime}m^{\prime}}~{}$ $\displaystyle=~{}-2\pi
im\mathcal{N}_{\ell
m}\mathcal{N}_{\ell^{\prime}m}\delta_{mm^{\prime}}\int\limits_{0}^{\pi}\text{d}\theta\bigr{[}P^{m}_{\ell^{\prime}}(\cos\theta)\partial_{\theta}P^{m}_{\ell}(\cos\theta)\bigr{.}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\bigr{.}+P^{m}_{\ell}(\cos\theta)\partial_{\theta}P^{m}_{\ell^{\prime}}(\cos\theta)\bigr{]}\,,$
(3.46)

where we used that
$F_{m}\delta_{mm^{\prime}}=F_{m^{\prime}}\delta_{mm^{\prime}}$ for arbitrary
$F_{m}$. Finally this can be recognized as the product rule on derivatives of
the Associated Legendre polynomials. Thus the integral becomes a total
derivative, which is trivially integrated, yielding

$C_{\ell m;\ell^{\prime}m^{\prime}}~{}=~{}-2\pi im\mathcal{N}_{\ell
m}\mathcal{N}_{\ell^{\prime}m}\delta_{mm^{\prime}}\bigr{(}P^{m}_{\ell}(1)P^{m}_{\ell^{\prime}}(1)-P^{m}_{\ell}(-1)P^{m}_{\ell^{\prime}}(-1)\bigr{)}\,.$
(3.47)

This gives the result for $C_{\ell m;\ell^{\prime}m^{\prime}}$ for all
$\ell,m,\ell^{\prime},m^{\prime}$. Clearly it trivially vanishes for $m=0$, so
we only need to know the values $P^{m}_{\ell}(\pm 1)$ for $m>0$. The
definition [79]

$P_{\ell}^{m}(x)~{}=~{}\dfrac{(-1)^{m}}{2^{\ell}\text{
}\ell!}\left(1-x^{2}\right)^{m/2}\left(\dfrac{\text{d}}{\text{d}x}\right)^{\ell+m}\left(x^{2}-1\right)^{\ell}.$
(3.48)

shows that for $m>0$ this quantity vanishes at $x=\pm 1$. Therefore, we
finally find that

$C_{\ell m;\ell^{\prime}m^{\prime}}~{}=~{}0$ (3.49)

in all cases. This completes the proof of the decoupling of the odd and even
graviton modes. The antisymmetry arising from $\epsilon^{AB}$ is crucial. The
minus sign introduced by the antisymmetry of the odd parity mode is canceled
by the minus sign of the complex conjugate777However, complex conjugation is
not crucial. Had we inserted two $Y_{\ell}^{m}$’s without complex conjugation
instead, there would be no minus sign in the derivative. Nevertheless, a new
minus sign would be introduced in the orthogonality relations. This allows the
use of the product rule in the end., allowing for the use of the product rule.

In this decoupling we find that the choice of light-cone coordinates plays no
role, i.e. $r,t$ or $x,y$ yield the same result. The same holds for the choice
of angular coordinates $A,B$. Both the light-cone coordinates and the angular
coordinates are summed over separately. This proof also holds for any metric
where $A\left(x,y\right),r\left(x,y\right)$ only depend on the light-cone
coordinates, i.e. for any spherically symmetric metric of the form (3.9). In
particular, of course, it is valid for the Schwarzschild metric.

###  3.4 The odd parity graviton does not contribute

We have seen that the even and odd parity gravitons decouple in the quadratic
action. In what follows, we will see from studying the structure of the three-
point vertex that the odd parity graviton does not contribute to the four-
point function of interest.

####  3.4.1 The interaction vertex

The interaction vertex can be inferred from (3)

$S_{vertex}~{}=~{}\dfrac{\kappa}{2}\int\text{d}^{4}x\,\sqrt{-g}\,h^{\mu\nu}\left(\partial_{\mu}\phi\partial_{\nu}\phi-\dfrac{1}{2}g_{\mu\nu}g^{\rho\sigma}\partial_{\rho}\phi\partial_{\sigma}\phi\right)\,.$
(3.50)

As discussed in the introduction, the external momenta of interest for the
scalars are in-going to or out-going from the black hole horizon. We are also
interested in scattering very close to the horizon. Naively, one may expect
the out-going mode to carry transverse momentum after scattering in addition
to the longitudinal shifts (caused by gravitational backreaction [56, 41], for
instance). However, as one may have inferred from the brick-wall model [80] or
the Bekenstein-Hawking entropy formula, there is roughly one degree of freedom
per Planck area on the horizon. This has been more recently emphasised in the
work of Dvali et.al. [81, 18] by studying graviton scattering amplitudes.
Therefore, loosely speaking, it is an unlikely event that particles scatter at
impact parameters comparable to Planck length. This observation validates the
approximation of impact parameters larger than Planck length. It is well known
that transverse momentum transfer is a Planckian effect888Of course, when
collision energies are extremely low for very massive bodies at large impact
parameters, classical non-linear effects do not distinguish between
longitudinal and transverse effects. [41]. Therefore, transverse momentum
transfer may safely be considered small: $p_{A}=0$. Piecing this together with
the Regge-Wheeler gauge, we find

$\displaystyle S_{vertex}~{}$
$\displaystyle=~{}\dfrac{\kappa}{2}\int\text{d}\Omega\int\text{d}^{2}x\,A\left(r\right)\,r^{2}\left[h^{ab}\left(\partial_{a}\phi\partial_{b}\phi-\dfrac{1}{2}g_{ab}g^{cd}\partial_{c}\phi\partial_{d}\phi\right)+h^{AB}T_{AB}\right]\,,$
(3.51)

for the interaction vertex, where $A\left(r\right)$ and $r$ are implicitly
functions of the two-dimensional coordinates $x,y$ as before:
$A\left(r\right)=A\left(x,y\right)$ and $r=r\left(x,y\right)$. We see that the
odd parity graviton mode $h_{aA}$ drops out of the vertex. Therefore, all
scattering processes are entirely governed by the even parity graviton.

###  3.5 Action for the even parity graviton

Owing to the complete decoupling of the odd and even parity modes of the
graviton, and the subsequent dropping out of the odd mode from all
interactions, the Lagrangian arising from the Einstein-Hilbert action reduces
to

$\mathcal{L}_{even}~{}=~{}-\dfrac{1}{4}h^{+}_{\mu\nu}G^{\mu\nu}_{(1),+}\,,$
(3.52)

where $G^{\mu\nu}_{(1),+}$ is defined in (3.3). This Lagrangian can be brought
into a more convenient form, exploiting the spherical symmetry of the
background. To do so, we first note that the covariant tensor $h_{AB}=Kg_{AB}$
and therefore depends on the angular variables. Therefore, we will now first
use a raised index on the graviton to extract this angular dependence out as
${h_{a}}^{b}={H_{a}}^{b}$ and ${h_{A}}^{B}=K\delta_{A}^{B}$. We will maintain
covariance so that the indices can later be lowered or raised. This
intermediate step allows us to write down a spherical harmonic expansion for
the graviton as:

${h_{\nu}}^{\mu}~{}=~{}\sum_{\ell,m}{\left(h_{lm}\right)_{\nu}}^{\mu}Y_{\ell}^{m}\,,$
(3.53)

where $h_{lm}$ are now only dependent on the two-dimensional coordinates
$x,y$. Therefore, we also have

$\displaystyle{h_{a}}^{b}~{}$ $\displaystyle=~{}{H_{a}}^{b}Y_{\ell}^{m}$
(3.54) $\displaystyle{h_{A}}^{B}~{}$
$\displaystyle=~{}K\delta_{A}^{B}Y_{\ell}^{m}\,.$ (3.55)

The spherical harmonic expansion enables us to write the action (3.52) as

$\mathcal{L}_{even}~{}=~{}-\dfrac{1}{8}{h^{\mu}}_{\nu}G^{\nu\text{ }\text{
}\text{ }\text{ }\sigma}_{\text{ }\text{ }\mu\rho\text{ }\text{
}}{h_{\sigma}}^{\rho}\,,$ (3.56)

where we defined

$\displaystyle G^{\nu\text{ }\text{ }\text{ }\text{ }\sigma}_{\text{ }\text{
}\mu\rho\text{ }\text{ }}~{}$
$\displaystyle\coloneqq~{}\delta_{\mu}^{\sigma}\nabla_{\rho}\nabla^{\nu}+\delta^{\nu}_{\rho}\nabla^{\sigma}\nabla_{\mu}-\delta^{\nu}_{\mu}\nabla^{\sigma}\nabla_{\rho}-\delta^{\sigma}_{\rho}\nabla^{\nu}\nabla_{\mu}+2\bar{\delta}^{\nu\sigma}_{\mu\rho}\Box\,,$
(3.57) $\displaystyle\bar{\delta}^{\nu\sigma}_{\mu\rho}~{}$
$\displaystyle\coloneqq~{}\delta^{\nu}_{[\mu}\delta^{\rho}_{\sigma]}\,.$
(3.58)

###  3.6 Decoupling of the partial waves

Given that we are working on a spherically symmetric background, it is
expected that the partial waves are decoupled entirely. To see this, we first
evaluate the action of the operator (3.57) on the graviton:

$\displaystyle G^{\nu\text{ }\text{ }\text{ }\text{ }\sigma}_{\text{ }\text{
}\mu\rho\text{ }\text{ }}{h_{\sigma}}^{\rho}~{}$
$\displaystyle=~{}\sum_{\ell,m}G^{\nu\text{ }\text{ }\text{ }\text{
}\sigma}_{\text{ }\text{ }\mu\rho\text{ }\text{ }}(h_{\ell
m})^{\rho}_{\sigma}+\sum\limits_{\ell,m}(h_{\ell
m})^{\rho}_{\sigma}G^{\nu\text{ }\text{ }\text{ }\text{ }\sigma}_{\text{
}\text{ }\mu\rho\text{ }\text{ }}Y_{\ell}^{m}$
$\displaystyle\qquad+~{}\sum\limits_{\ell,m}\biggr{(}-4\bar{\delta}^{\nu\sigma}_{\mu\rho}(\partial_{\kappa}Y_{\ell}^{m})\nabla^{\kappa}+\mathcal{U}^{\sigma\nu}_{\mu\rho}+\mathcal{U}^{\nu\sigma}_{\rho\mu}-\mathcal{U}^{\nu\sigma}_{\mu\rho}-\mathcal{U}^{\sigma\nu}_{\rho\mu}\biggr{)}(h_{\ell
m})^{\rho}_{\sigma}\,,$ (3.59)

where we defined

$\mathcal{U}^{\nu\sigma}_{\mu\rho}~{}=~{}\delta^{\nu}_{\mu}\biggr{[}\left(\partial^{\sigma}Y_{\ell}^{m}\right)\nabla_{\rho}+\left(\partial_{\rho}Y_{\ell}^{m}\right)\nabla^{\sigma}\biggr{]}\,.$
(3.60)

The first line of (3.6) is the action of all the operators in (3.57) on either
the modes $h_{\ell m}$ or on the spherical harmonics, whereas the second line
arises from all the mixed terms. While these terms appear complicated, noting
that the spherical harmonics $Y_{\ell}^{m}$ only dependent on the angles,
several indices in the second line of (3.6) do not contribute. Moreover, using
the tensor structure in (3.54) and (3.55), together with the derivatives
listed in Appendix B.1, we find

$\biggr{(}-4\bar{\delta}^{\nu\sigma}_{\mu\rho}\left(\partial_{\kappa}Y_{\ell}^{m}\right)\nabla^{\kappa}+\mathcal{U}^{\sigma\nu}_{\mu\rho}+\mathcal{U}^{\nu\sigma}_{\rho\mu}-\mathcal{U}^{\nu\sigma}_{\mu\rho}-\mathcal{U}^{\sigma\nu}_{\rho\mu}\biggr{)}\left(h_{\ell
m}\right)^{\rho}_{\sigma}~{}\sim~{}\partial_{A}\left(h_{\ell
m}\right)^{\rho}_{\sigma}~{}=~{}0\,.$ (3.61)

Therefore, the mixed terms in (3.6) vanish, and we are left with

$\displaystyle G^{\nu\text{ }\text{ }\text{ }\text{ }\sigma}_{\text{ }\text{
}\mu\rho\text{ }\text{ }}{h_{\sigma}}^{\rho}~{}$
$\displaystyle=~{}\sum_{\ell,m}G^{\nu\text{ }\text{ }\text{ }\text{
}\sigma}_{\text{ }\text{ }\mu\rho\text{ }\text{ }}(h_{\ell
m})^{\rho}_{\sigma}+\sum\limits_{\ell,m}(h_{\ell
m})^{\rho}_{\sigma}G^{\nu\text{ }\text{ }\text{ }\text{ }\sigma}_{\text{
}\text{ }\mu\rho\text{ }\text{ }}Y_{\ell}^{m}\,.$ (3.62)

The action of the operator (3.57) on the spherical harmonics is given by

$\displaystyle G^{\nu\text{ }\text{ }\text{ }\text{ }\sigma}_{\text{ }\text{
}\mu\rho\text{ }\text{ }}Y_{\ell}^{m}~{}$
$\displaystyle=~{}\dfrac{\left(\Delta_{\Omega}Y_{\ell}^{m}\right)}{2r^{2}}\biggr{(}\delta^{\nu}_{\mu}\delta_{\rho}^{\sigma}+\delta^{\nu}_{a}\delta^{a}_{\mu}\delta^{\sigma}_{b}\delta^{b}_{\rho}-2\delta^{\nu}_{\rho}\delta^{\mu}_{\sigma}\biggr{)}$
$\displaystyle=~{}-Y_{\ell}^{m}\dfrac{\ell\left(\ell+1\right)}{2r^{2}}\biggr{(}\delta^{\nu}_{\mu}\delta_{\rho}^{\sigma}+\delta^{\nu}_{a}\delta^{a}_{\mu}\delta^{\sigma}_{b}\delta^{b}_{\rho}-2\delta^{\nu}_{\rho}\delta^{\mu}_{\sigma}\biggr{)}\,,$
(3.63)

where in the second line, we have used the property that the spherical
harmonics are the eigenfunctions of the spherical Laplacian
$\Delta_{\Omega}Y_{\ell}^{m}=\ell\left(\ell+1\right)Y_{\ell}^{m}$. That there
are no further derivatives on the $Y_{\ell}^{m}$ helps the decoupling of the
partial waves. Equation (3.6) can now be written as

$G^{\nu\text{ }\text{ }\text{ }\text{ }\sigma}_{\text{ }\text{ }\mu\rho\text{
}\text{
}}{h_{\sigma}}^{\rho}~{}=~{}\sum_{\ell,m}Y_{\ell}^{m}{{{\mathcal{G}_{\ell}}^{\nu}}_{\mu\rho}}^{\sigma}\left(h_{\ell
m}\right)_{\sigma}^{\text{ }\rho}\,,$ (3.64)

where we defined the modified operator

$\mathcal{G}^{\nu\text{ }\text{ }\text{ }\text{ }\sigma}_{\ell\text{
}\mu\rho\text{ }\text{ }}~{}\coloneqq~{}G^{\nu\text{ }\text{ }\text{ }\text{
}\sigma}_{\text{ }\text{ }\mu\rho\text{ }\text{
}}-\dfrac{\ell(\ell+1)}{2r^{2}}\bigr{(}\delta^{\mu}_{\nu}\delta_{\rho}^{\sigma}+\delta^{\mu}_{a}\delta^{a}_{\nu}\delta^{\sigma}_{b}\delta^{b}_{\rho}-2\delta^{\mu}_{\rho}\delta^{\sigma}_{\nu}\bigr{)}\,.$
(3.65)

As one many have expected, the angular parts contribute to a $1/r^{2}$
potential in this operation. The second order action now results in

$\displaystyle S_{even}~{}$
$\displaystyle=~{}-\dfrac{1}{8}\sum\limits_{\ell,m}\sum\limits_{\ell^{\prime},m^{\prime}}\int\text{d}\Omega\text{
}\bar{Y}_{\ell^{\prime}}^{m^{\prime}}Y_{\ell}^{m}\int\text{d}^{2}x\sqrt{-g}{\left(h_{\ell^{\prime}m^{\prime}}\right)^{\mu}}_{\nu}\mathcal{G}^{\nu\text{
}\text{ }\text{ }\text{ }\sigma}_{\text{ }\text{ }\mu\rho\text{ }\text{
}}\left(h_{\ell m}\right)^{\text{ }\rho}_{\sigma}\,.$ (3.66)

We now have an infinite tower of decoupled actions, one for each partial wave.
The decoupling is a result of spherical symmetry of the background and may be
seen as the conservation of angular momentum.

###  3.7 An effective two-dimensional theory

Noting that the graviton fields $h_{\ell m}$ are only functions of the light-
cone coordinates since the angular pieces have been extracted out, and using.
the orthogonality relations for the spherical harmonics

$\int\text{d}\Omega\,\bar{Y}_{\ell^{\prime}}^{m^{\prime}}\,Y_{\ell}^{m}~{}=~{}\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}\,,$
(3.67)

the action (3.66) reduces to

$S_{even}~{}=~{}-\dfrac{1}{8}\sum\limits_{\ell,m}\int\text{d}^{2}xA\left(r\right)\,r^{2}\,{\left(h_{\ell
m}\right)^{\mu}}_{\nu}\mathcal{G}^{\nu\text{ }\text{ }\text{ }\text{
}\sigma}_{\text{ }\text{ }\mu\rho\text{ }\text{ }}\left(h_{\ell
m}\right)^{\text{ }\rho}_{\sigma}\,.$ (3.68)

While this may appear to be a two-dimensional theory, the indices on the
graviton are still Greek and run over all four spacetime dimensions. Moreover,
owing to the implicit presence of the angular indices, a separation of light-
cone and spherical fields is no longer obvious. To appreciate this problem of
separation of variables, it is instructive to evaluate the quantity $\Box
h^{a}_{\text{ }b}$. Despite the presence of only the light-cone coordinates,
as show in Appendix B.3, we find a term of the form
$\left(\partial^{a}r\right)\left(\partial_{b}r\right)K$. This arises from the
two-sphere coordinates in the $\Box$ operator. The second, and perhaps more
serious, problem is seen when indices are raised and lowered with the metric.
Defining, for instance, $h_{\phi\phi}$ is problematic because while
$h^{\phi}_{\phi}$ is independent of the angular variables, lowering the index
would introduce it because $g_{\phi\phi}=r^{2}\sin^{2}\theta$. Therefore,
separation of variables along the light-cone and the two-sphere, and the
spherical harmonic expansion appear to be in conflict with each other. While
the two-dimensional propagator and equations of motion appear problematic,
these nevertheless naturally pose no problem in the complete four-dimensional
action. It is therefore evident that the price we pay for covariance on the
light-cone is an inherent mixing between the fields $H_{ab}$ and $K$.

In what follows, we define differential operators with a tilde to represent
those on the light-cone. So, $\tilde{\nabla}$ is the covariant derivative on
the light-cone, and $\tilde{\Box}=\tilde{\nabla}^{a}\tilde{\nabla}_{a}$ is the
’d Alembertian, and the indices are raised and lowered with the light-cone
metric $g_{ab}$. For consistency, we will also write the light-cone metric
with a tilde as $\tilde{g}_{ab}$. In this formulation, clearly
$\tilde{\Box}h^{a}_{\text{ }b}$ will never give rise to the scalar field $K$
as the lowercase Latin indices never sum over the angular coordinates. This
allows us to separate variables and maintain covariance on the light-cone all
the same. Notwithstanding which, the tensor $H_{ab}$ and $K$ will remain
coupled.

Symbolically we want to find an action of the following form [82]

$\displaystyle
S_{even}~{}=~{}\frac{1}{4}\int\text{d}^{2}x\sqrt{-\tilde{g}}\biggr{(}\tilde{H}^{ab}\tilde{\Delta}^{-1}_{abcd}\tilde{H}^{cd}+\tilde{H}^{ab}\tilde{\Delta}^{-1}_{L,ab}\tilde{K}+\tilde{K}\tilde{\Delta}^{-1}_{R,ab}\tilde{H}^{ab}+\tilde{K}\tilde{\Delta}^{-1}\tilde{K}\biggr{)}.$
(3.69)

Here we defined all of the new $\tilde{\Delta}$ operators listed below. We
have also defined the new lightcone fields
$\tilde{H}_{ab}=rH_{ab},\tilde{K}=rK$. These definitions have been made to
absorb the residual $r^{2}$ of the two-sphere Jacobian in the action in (3.68)
into the fields. This gives rise to the light-cone Jacobian
$\sqrt{-\tilde{g}}$, and consequently results in the canonical form of the
action (3.69). A detailed derivation of these operators is given in Appendix
B.4. The result is:

$\displaystyle\tilde{\Delta}^{-1}~{}$
$\displaystyle=~{}-\tilde{\Box}+F^{a}_{a}\,,$ (3.70)
$\displaystyle\tilde{\Delta}^{-1}_{R,ab}~{}$
$\displaystyle=~{}-g_{ab}\left(\tilde{\Box}-\dfrac{1}{2}V_{c}\tilde{\nabla}^{c}+\dfrac{1}{4}V_{c}V^{c}-F_{c}^{c}-\dfrac{\ell(\ell+1)}{2r^{2}}\right)+\tilde{\nabla}_{a}\tilde{\nabla}_{b}-F_{ab}\,,$
(3.71) $\displaystyle\tilde{\Delta}^{-1}_{L,ab}~{}$
$\displaystyle=~{}-g_{ab}\left(\tilde{\Box}+\dfrac{1}{2}V_{c}\tilde{\nabla}^{c}-\dfrac{\ell(\ell+1)}{2r^{2}}\right)+\tilde{\nabla}_{a}\tilde{\nabla}_{b}-F_{ab}\,,$
(3.72) $\displaystyle\tilde{\Delta}^{-1}_{abcd}~{}$
$\displaystyle=~{}\dfrac{1}{2}g_{ac}V_{[b}\tilde{\nabla}_{d]}+\dfrac{1}{2}g_{bd}V_{[a}\tilde{\nabla}_{c]}+\dfrac{1}{2}g_{ab}\left(V_{(c}\tilde{\nabla}_{d)}+2F_{cd}\right)$
$\displaystyle\qquad+\dfrac{1}{2}g_{cd}\left(-V_{(a}\tilde{\nabla}_{b)}+\dfrac{1}{2}V_{a}V_{b}\right)+g_{ab}g_{cd}\left(\dfrac{1}{4}R_{2d}+\dfrac{\ell(\ell+1)}{2r^{2}}\right)$
$\displaystyle\qquad-
g_{ac}g_{bd}\left(\dfrac{1}{2}R_{2d}+\dfrac{\ell(\ell+1)}{2r^{2}}\right)\,,$
(3.73)

where we defined the following tensors:

$\displaystyle V_{a}~{}$ $\displaystyle=~{}2\partial_{a}\log r\,,$ (3.74)
$\displaystyle F_{ab}~{}$
$\displaystyle=~{}\dfrac{1}{r}\tilde{\nabla}_{a}\tilde{\nabla}_{b}r~{}=~{}\dfrac{1}{2}\tilde{\nabla}_{(a}V_{b)}+\dfrac{1}{4}V_{a}V_{b}\,,$
(3.75) $\displaystyle R_{2d}~{}$
$\displaystyle=~{}-\dfrac{1}{A}\tilde{\Box}\log A\,.$ (3.76)

Here $V_{a}$ is a residual curvature potential arising from the two-sphere,
$F_{ab}$ its corresponding field strength, and $R_{2d}$ is the Ricci scalar of
the two-dimensional background metric $\tilde{g}_{ab}$. We note that all
operators are in fact symmetric in the fields; in particular,
$\tilde{\Delta}^{-1}_{R,ab}$ equals $\tilde{\Delta}^{-1}_{L,ab}$ up to total
derivatives. We now have a covariant description of the graviton modes on the
lightcone, where the degrees of freedom $\tilde{H}_{ab}$ and $\tilde{K}$ are
explicitly separated, albeit coupled. The two-sphere is now integrated out
entirely; it leaves a residual $\ell,m$ dependence and the curvature potential
$V_{a}$. The lightcone metric is at this stage still general
$\tilde{g}_{ab}=A(r)\eta_{ab}$ parametrised by an arbitrary function.

####  3.7.1 Weyl rescaling

Before attempting to insert the Schwarzschild metric to find the propagators,
we first observe that the effective two-dimensional metric is conformally
flat:

$\displaystyle\tilde{g}_{ab}~{}=~{}A\left(r\right)\eta_{ab}\,,$ (3.77)

with $\eta_{ab}$ being the two-dimensional Minkowski metric:

$\eta_{ab}~{}=~{}\begin{pmatrix}0&-1\\\ -1&0\end{pmatrix}\,.$ (3.78)

This allows us to perform a Weyl transformation to reduce our theory to a two-
dimensional flat theory, with modified potentials. This will allow us to trade
the subtleties of curved space, and quantum field theory on the said curved
space to a flat theory for additional potential terms. An important
consequence is that despite being in the region within the gravitational
potential, where flat space kinematics is invalid, the Weyl transformation
will allow us to define centre of mass energies.

In order to migrate to a flat two-dimensional background we first make a
transformation that explicitly removes the function $A(r)$ from the metric.
The transformations are given by

$\displaystyle\tilde{g}_{ab}~{}$
$\displaystyle\to~{}A\left(r\right)\eta_{ab}\,,$ (3.79)
$\displaystyle\tilde{H}_{ab}~{}$
$\displaystyle\to~{}A\left(r\right)\mathfrak{h}_{ab}\,,$ (3.80)
$\displaystyle\tilde{K}~{}$ $\displaystyle\to~{}\mathcal{K}\,,$ (3.81)

where $\mathfrak{h}^{ab}$ is raised and lowered with the new flat Minkowski
metric. Conveniently enough, partial derivatives are now covariant
derivatives, so covariance of the physical quantities is trivially achieved.

The Weyl transformation brings about a few noteworthy points: $\tilde{K}$ does
not transform (and yet we give it a new name $\mathcal{K}$) and neither does
$V_{b}=2\partial_{b}\log r$ (however,
$V^{b}=\tilde{g}^{ab}V_{b}=\tfrac{1}{A}\eta^{ab}V_{b}$ does transform).
Moreover, $F_{ab}$ also does not transform.

Inserting the Weyl rescaling, we may rewrite the action quadratic in
$\mathcal{K}$, for instance, as

$\dfrac{1}{4}\int\text{d}^{2}xA\left(r\right)\mathcal{K}\left((-\Box+F_{a}^{a}\right)\mathcal{K}~{}=~{}\dfrac{1}{4}\int\text{d}^{2}x\mathcal{K}\left(-\partial^{2}+\eta^{ab}F_{ab}\right)\mathcal{K}\,,$
(3.82)

where we included the $A\left(r\right)$ that comes from the $\sqrt{-g}$ and
defined $\partial^{2}=\eta^{ab}\partial_{a}\partial_{b}$. For consistency we
redefine

$\displaystyle
F_{ab}~{}=~{}\dfrac{1}{r}\tilde{\nabla}_{a}\tilde{\nabla}_{b}r~{}\eqqcolon~{}\dfrac{1}{r}\partial_{a}\partial_{b}r-\dfrac{1}{2}U_{(a}V_{b)}+\dfrac{1}{4}\eta_{ab}U^{c}V_{c}\,,$
(3.83)

where all index manipulations are now done with the flat metric, i.e.
$U^{c}V_{c}=\eta^{ac}U_{a}V_{c}$ and $U^{a}=\eta^{ab}U_{b}$. This redefinition
is in principle necessary, since the previous lightcone covariant derivative
does not hold after the Weyl transformation. Here we defined a new potential
that will embed the curvature remnants of $A\left(r\right)$:

$\displaystyle U_{a}~{}=~{}\partial_{a}\log A\left(r\right)\,.$ (3.84)

Then the action quadratic in $\mathcal{K}$ becomes

$\displaystyle\dfrac{1}{4}\int\text{d}^{2}x\mathcal{K}\left(-\partial^{2}+F_{a}^{a}\right)\mathcal{K}\,,$
(3.85)

where $F_{a}^{a}=\eta^{ab}F_{ab}$. The Weyl transformation essentially allows
us a change in viewpoint but no change in physics; we may either study free
fields in a curved spacetime, or Weyl rescaled fields bounded by potentials in
flat spacetime. Both descriptions are equivalent for conformally flat
spacetimes. The fact that the spacetime is conformally flat is important
because all curvature effects are then isotropic, allowing for an embedding
into a scalar function $A\left(r\right)$. For anisotropic curvatures we
naturally need a tensor potential, losing the simplicity of a Weyl
transformation.

We now proceed to a Weyl rescaling of all fields and operators, with due
regard for which fields transform, and which fields do not. For the following,
all Christoffel symbols need to be worked out in order to safely pull the
$\tfrac{1}{A}$ in $\tilde{H}^{cd}$ through. Essentially, we need to calculate:

$\displaystyle\Delta^{-1}_{R,ab}~{}$
$\displaystyle\coloneqq~{}A\left(r\right)\tilde{\Delta}^{-1}_{R,ab}\dfrac{1}{A\left(r\right)}\,,$
(3.86) $\displaystyle\Delta^{-1}_{L,ab}~{}$
$\displaystyle\coloneqq~{}\tilde{\Delta}^{-1}_{L,ab}\,,$ (3.87)
$\displaystyle\Delta^{-1}_{abcd}~{}$
$\displaystyle\coloneqq~{}\tilde{\Delta}^{-1}_{abcd}\dfrac{1}{A\left(r\right)}\,.$
(3.88)

The action is now given by

$\displaystyle
S~{}=~{}\dfrac{1}{4}\int\text{d}^{2}x\biggr{(}\mathfrak{h}^{ab}\Delta^{-1}_{abcd}\mathfrak{h}^{cd}+\mathfrak{h}^{ab}\Delta^{-1}_{L,ab}\mathcal{K}+\mathcal{K}\Delta^{-1}_{R,ab}\mathfrak{h}^{ab}+\mathcal{K}\Delta^{-1}\mathcal{K}\biggr{)}\,.$
(3.89)

An explicit calculation of all $\Delta$ operators shows that

$\displaystyle\Delta^{-1}~{}$ $\displaystyle=~{}-\partial^{2}+F_{a}^{a}\,,$
(3.90a) $\displaystyle\Delta^{-1}_{R,ab}~{}$
$\displaystyle=~{}-\eta_{ab}\left(\partial^{2}+\dfrac{1}{2}\left(U^{c}-V^{c}\right)\partial_{c}+\dfrac{1}{2}\eta^{cd}\left(\mathcal{W}^{A}_{cd}-\mathcal{W}^{r}_{cd}\right)\right.$
$\displaystyle\qquad\qquad\qquad-\left.A\left(r\right)\dfrac{\ell(\ell+1)}{2r^{2}}\right)+\partial_{a}\partial_{b}+U_{(a}\partial_{b)}+\mathcal{W}^{A}_{ab}-F_{ab}\,,$
(3.90b) $\displaystyle\Delta^{-1}_{L,ab}~{}$
$\displaystyle=~{}-\eta_{ab}\left(\partial^{2}-\dfrac{1}{2}\left(U^{c}-V^{c}\right)\partial_{c}-A\left(r\right)\dfrac{\ell(\ell+1)}{2r^{2}}\right)+\partial_{a}\partial_{b}-U_{(a}\partial_{b)}-F_{ab}\,,$
(3.90c) $\displaystyle\Delta^{-1}_{abcd}~{}$
$\displaystyle=~{}\dfrac{1}{2}\eta_{ac}V_{[b}\partial_{d]}+\dfrac{1}{2}\eta_{bd}V_{[a}\partial_{c]}+\dfrac{1}{2}\eta_{ab}\left(V_{(c}\partial_{d)}+\mathcal{W}^{r}_{cd}+\dfrac{1}{2}V_{c}V_{d}\right)$
$\displaystyle\qquad+\dfrac{1}{2}\eta_{cd}\left(-V_{(a}\partial_{b)}+\dfrac{1}{2}V_{a}V_{b}\right)+\eta_{ab}\eta_{cd}\left(\dfrac{1}{4}A\left(r\right)R_{2d}-\dfrac{1}{4}V^{e}U_{e}+A\left(r\right)\dfrac{\ell(\ell+1)}{2r^{2}}\right)$
$\displaystyle\qquad-\eta_{ac}\eta_{bd}\left(\dfrac{1}{2}A\left(r\right)R_{2d}-\dfrac{1}{2}V^{e}U_{e}+A\left(r\right)\dfrac{\ell(\ell+1)}{2r^{2}}\right)\,,$
(3.90d)

where repeated application of the product rule on any relevant $\tfrac{1}{A}$
was necessary. We defined two new field tensors

$\displaystyle\mathcal{W}^{r}_{ab}~{}\coloneqq~{}\partial_{(a}V_{b)}\qquad\text{and}\qquad\mathcal{W}^{A}_{ab}~{}\coloneqq~{}\partial_{(a}U_{b)}\,.$
(3.91)

While the $\Delta$ operators now appear more complicated, the background is
flat, making all covariant derivatives trivial. Moreover, while coordinate
transformations need care, any coordinate system may be chosen by an
appropriate choice of the function $A(r)$.

This completes the calculation of the even action. The four-dimensional
spherically symmetric spacetime is now reduced to a flat two dimensional
Minkowski spacetime. This is largely owed to the spherical harmonics expansion
and a careful exploitation of the background spherical symmetry. Curvature is
embedded in two potentials $V_{a},U_{a}$ and their respective field strengths.
While we have focused our attention on vacuum solutions to Einstein’s
equations, extension to more general spacetimes should be straightforward. The
first order term in (A.2) will be nontrivial but can be accommodated for, in
this formalism.

To complete the effective action, we also need to reduce the scalar action in
(3) and the interaction vertex (3.51) on to two-dimensional flat spacetime. We
will incorporate these in the following section as we move near the horizon.
Once equipped with the complete effective action, we will then proceed to
determine the graviton and scalar propagators and the interaction vertex, on
the Schwarzschild horizon, as advertised.

##  4 Feynman rules on the horizon

While the soft limit of the path integral about spherically symmetric
backgrounds conveniently split into effective two-dimensional theories, it is
still difficult to invert the quadratic operators (3.90) to find the graviton
propagator for general backgrounds. In what follows, we will evaluate the
action (3.89) on the Schwarzschild background specified by

$A\left(r\right)~{}=~{}\dfrac{R}{r}e^{1-r/R}\quad\text{and}\quad
xy~{}=~{}2R^{2}\left(1-\dfrac{r}{R}\right)e^{r/R-1}\,.$ (4.1)

Therefore, the potentials defining the operators (3.90) can now be written in
the $x,y$ coordinates:

$\displaystyle V_{a}~{}$ $\displaystyle=~{}\dfrac{A}{rR}x_{a}\,,$ (4.2a)
$\displaystyle U_{a}~{}$
$\displaystyle=~{}-\dfrac{A}{2rR}\left(1+\tfrac{r}{R}\right)x_{a}\,,$ (4.2b)
$\displaystyle\mathcal{W}^{r}_{ab}~{}$
$\displaystyle=~{}\dfrac{A}{rR}\eta_{ab}-\dfrac{A^{2}}{2R^{2}r^{2}}\left(2+\dfrac{r}{R}\right)x_{a}x_{b}\,,$
(4.2c) $\displaystyle\mathcal{W}^{A}_{ab}~{}$
$\displaystyle=~{}-\dfrac{A}{2rR}\left(1+\dfrac{r}{R}\right)\eta_{ab}+\dfrac{A^{2}}{4R^{2}r^{2}}\left(2+2\dfrac{r}{R}+\dfrac{r^{2}}{R^{2}}\right)x_{a}x_{b}\,,$
(4.2d) $\displaystyle F_{ab}~{}$
$\displaystyle=~{}\dfrac{AR}{2r^{3}}\eta_{ab}\,,$ (4.2e) $\displaystyle
R_{2d}~{}$ $\displaystyle=~{}\dfrac{2R}{r^{3}}\,.$ (4.2f)

The choice of background (4.1) is evidently that of an eternal Schwarzschild
solution in Kruskal-Szekeres coordinates. This retains time-translational
invariance in the effective two-dimensional theory: $x\to ax,y\to a^{-1}y$. In
what is to follow, however, we will make a near-horizon approximation. As it
turns out, the physically relevant information for the derivation of the
graviton propagator is the conformal flatness of the bifurcation sphere. In a
collapsing scenario, the nature of the apparent horizon may be rather
different. However, in the soft limit, we will argue towards the end of the
paper that the general lessons drawn may well hold. There are of course more
detailed questions further away from the horizon where the near horizon
propagator needs to be modified.

As an aside, the relation between the trace of the residual curvature field
strength of the two-sphere $F_{ab}$ and the scalar curvature of the two-
dimensional spacetime is striking: $A\left(r\right)R_{2d}=2F_{a}^{a}$. They
may be seen to conspire to make up the vacuum Schwarzschild solution in four
dimensions.

We may now insert all curvature potentials into the $\Delta$ operators. Our
interest is however, in studying scattering near the horizon. Therefore, we
will first define the near-horizon approximation before writing down the
simplified potentials.

###  4.1 The near horizon approximation

The naive near-horizon approximation of the Scwharzschild solution yields
Rindler spacetime. However, we would like to keep the spherical nature of the
horizon intact. Moreover, as mentioned earlier, all wave fronts received on
future null infinity appear to emanate from the central causal diamond in a
collapsing scenario. Therefore, the natural approximation of interest is such
that the two-dimensional light-cone coordinates $x,y\ll R$; the reference
scale for the coordinates is the Schwarzschild radius $R$ instead of the
Planck length and the near-horizon region effectively measures $1-r/R$. In
terms of the light-cone coordinates, the horizon is of course defined by
$xy=0$. So, we have that

$\displaystyle r~{}=~{}R+R\mathcal{O}\left(\dfrac{xy}{R^{2}}\right)\,,$ (4.3)

in this approximation. Therefore, to linear order, we simply find $r=R$ and
$A\left(r\right)=1$. There are of course other linear contributions in the
form of the couplings $x_{a}\partial_{b}$, while all potentials have now
simply become mass terms. The $\Delta$ operators (3.90) now simplify to

$\displaystyle\Delta^{-1}~{}$ $\displaystyle=~{}-\partial^{2}+\mu^{2}\,,$
(4.4a) $\displaystyle\Delta^{-1}_{R,ab}~{}$
$\displaystyle=~{}-\eta_{ab}\left(\partial^{2}-\mu^{2}x^{c}\partial_{c}-\dfrac{1}{2}\mu^{2}\lambda\right)+\partial_{a}\partial_{b}-\mu^{2}x_{(a}\partial_{b)}\,,$
(4.4b) $\displaystyle\Delta^{-1}_{L,ab}~{}$
$\displaystyle=~{}-\eta_{ab}\left(\partial^{2}+\mu^{2}x^{c}\partial_{c}-\dfrac{1}{2}\mu^{2}\left(\lambda-2\right)\right)+\partial_{a}\partial_{b}+\mu^{2}x_{(a}\partial_{b)}\,,$
(4.4c) $\displaystyle\Delta^{-1}_{abcd}~{}$
$\displaystyle=~{}\dfrac{1}{2}\mu^{2}\left(\eta_{ac}x_{[b}\partial_{d]}+\eta_{bd}x_{[a}\partial_{c]}+\eta_{ab}x_{(c}\partial_{d)}-\eta_{cd}x_{(a}\partial_{b)}\right)$
$\displaystyle\qquad+\dfrac{\mu^{2}\left(\lambda+1\right)}{2}\left(\eta_{ab}\eta_{cd}-\eta_{ac}\eta_{bd}\right)\,,$
(4.4d)

where we defined

$\displaystyle\lambda~{}\coloneqq~{}\ell^{2}+\ell+1\,,$ (4.5)

and used the inverse Schwarzschild radius $\mu=1/R$ again, which can now be
understood as effective mass due to the positive curvature of the two-sphere.

It is now evident that this approximation is identical to expanding around
$\mu\to 0$, which holds for large black holes. Of course the approximation is
better defined in terms of the dimensionless quantity $\mu x_{a}$. The
equations of motion are also written in terms of the $\Delta$ operators as

$\displaystyle\Delta^{-1}_{abcd}\mathfrak{h}^{cd}+\Delta^{-1}_{L,ab}\mathcal{K}~{}$
$\displaystyle=~{}0\,,$ (4.6)
$\displaystyle\Delta^{-1}_{R,ab}\mathfrak{h}^{ab}+\Delta^{-1}\mathcal{K}~{}$
$\displaystyle=~{}0\,.$ (4.7)

An evident disconcerting feature of the operators (4.4) is the asymmetry in
the ‘left’ and ‘right’ operators. Removing first order terms naively with $\mu
x_{a}\rightarrow 0$ does not fix the problem either. The first derivative
terms contribute to the effective mass. A symmetric representation of these
operators can be achieved with a field redefinition as we will now show. We
first note the following identities:

$\displaystyle\partial_{a}\partial_{b}e^{-\tfrac{\mu^{2}x^{2}}{4}}~{}$
$\displaystyle=~{}e^{-\tfrac{\mu^{2}x^{2}}{4}}\left(\partial_{a}\partial_{b}-\mu^{2}x_{(a}\partial_{b)}-\dfrac{1}{2}\mu^{2}\eta_{ab}+\dfrac{1}{4}\mu^{4}x_{a}x_{b}\right)\,,$
(4.8) $\displaystyle\partial_{a}\partial_{b}e^{\tfrac{\mu^{2}x^{2}}{4}}~{}$
$\displaystyle=~{}e^{\tfrac{\mu^{2}x^{2}}{4}}\left(\partial_{a}\partial_{b}+\mu^{2}x_{(a}\partial_{b)}+\dfrac{1}{2}\mu^{2}\eta_{ab}+\dfrac{1}{4}\mu^{4}x_{a}x_{b}\right)\,.$
(4.9)

In what follows, we will ignore the last terms quadratic in $\mu x_{a}$. This
allows us to identify

$\displaystyle
e^{\tfrac{\mu^{2}x^{2}}{4}}\Delta^{-1}_{L,ab}e^{-\tfrac{\mu^{2}x^{2}}{4}}~{}=~{}e^{-\tfrac{\mu^{2}x^{2}}{4}}\Delta^{-1}_{R,ab}e^{\tfrac{\mu^{2}x^{2}}{4}}~{}=~{}-\eta_{ab}\bigr{(}\partial^{2}-\tfrac{1}{2}\mu^{2}(\lambda-1)\bigr{)}+\partial_{a}\partial_{b}\,.$
(4.10)

This gives a symmetric representation. Moreover, conveniently enough, it also
removes all first derivatives and first order terms in $\mu x_{a}$. Therefore,
the field re-definitions to achieve this symmetric representation are given by
the following transformations

$\displaystyle\mathcal{K}~{}$
$\displaystyle\to~{}e^{-\tfrac{\mu^{2}x^{2}}{4}}\mathcal{K}\,,$ (4.11)
$\displaystyle\mathfrak{h}_{ab}~{}$
$\displaystyle\to~{}e^{\tfrac{\mu^{2}x^{2}}{4}}\mathfrak{h}_{ab}\,.$ (4.12)

Under this field redefinition, it is worth noting that $\Delta^{-1}_{abcd}$
does not transform because of the anti-symmetry of the derivatives. What
remains is $\Delta^{-1}$. We first note the part of the action quadratic in
$\mathcal{K}$:

$\displaystyle
S_{\mathcal{K}}~{}=~{}\dfrac{1}{4}\int\text{d}^{2}x\mathcal{K}\Delta^{-1}\mathcal{K}~{}=~{}\dfrac{1}{4}\int\text{d}^{2}x\text{
}e^{-\tfrac{\mu^{2}x^{2}}{2}}\mathcal{K}\left(-\partial^{2}+\dfrac{1}{2}\mu^{2}x^{c}\partial_{c}+2\mu^{2}\right)\mathcal{K}\,.$
(4.13)

While there was no first derivative term to begin with, the field redefinition
has generated one. The first derivatives in the terms coupling $\mathcal{K}$
and $\mathfrak{h}$ have essentially been traded for a first derivative in the
pure $\mathcal{K}$ action. The exponent can be removed in the near-horizon
approximation

$e^{-\tfrac{\mu^{2}x^{2}}{2}}~{}=~{}1+\mathcal{O}(\mu^{2}xy)\,.$ (4.14)

The use of the exponential in the field redefinition may appear to be at odds
with the recurring use of the approximation $\mu x_{a}\ll 1$. A remark on this
is in order. But before that, we will first massage the said first derivative.
The integral with the first derivative can be rewritten as

$\displaystyle\int\text{d}^{2}x\mathcal{K}\mu^{2}x^{c}\partial_{c}\mathcal{K}~{}=~{}\dfrac{1}{2}\mu^{2}\int\text{d}^{2}xx^{c}\partial_{c}\left(\mathcal{K}^{2}\right)\,.$
(4.15)

Integration by parts with vanishing boundary conditions yields

$\displaystyle\dfrac{1}{2}\mu^{2}\int\text{d}^{2}xx^{c}\partial_{c}\left(\mathcal{K}^{2}\right)~{}=~{}-\dfrac{1}{2}\mu^{2}\int\text{d}^{2}x\mathcal{K}^{2}\partial_{c}x^{c}~{}=~{}-\mu^{2}\int\text{d}^{2}x\mathcal{K}^{2}\,.$
(4.16)

The first derivative is therefore traded for another mass term. The complete
even parity graviton action near the horizon is now given by

$\displaystyle
S~{}=~{}\dfrac{1}{4}\int\text{d}^{2}x\biggr{(}\mathfrak{h}^{ab}\Delta^{-1}_{abcd}\mathfrak{h}^{cd}+\mathfrak{h}^{ab}\Delta^{-1}_{ab}\mathcal{K}+\mathcal{K}\Delta^{-1}_{ab}\mathfrak{h}^{ab}+\mathcal{K}\Delta^{-1}\mathcal{K}\biggr{)}\,,$
(4.17)

with

$\displaystyle\Delta^{-1}~{}$ $\displaystyle=~{}-\partial^{2}+\mu^{2}\,,$
(4.18a) $\displaystyle\Delta^{-1}_{ab}~{}$
$\displaystyle=~{}-\eta_{ab}\left(\partial^{2}-\dfrac{1}{2}\mu^{2}\left(\lambda-1\right)\right)+\partial_{a}\partial_{b}\,,$
(4.18b) $\displaystyle\Delta^{-1}_{abcd}~{}$
$\displaystyle=~{}\dfrac{1}{2}\mu^{2}\left(\eta_{ac}x_{[b}\partial_{d]}+\eta_{bd}x_{[a}\partial_{c]}+\eta_{ab}x_{(c}\partial_{d)}-\eta_{cd}x_{(a}\partial_{b)}\right)$
$\displaystyle\qquad+\dfrac{\mu^{2}\left(\lambda+1\right)}{2}\bigr{(}\eta_{ab}\eta_{cd}-\eta_{ac}\eta_{bd}\bigr{)}\,.$
(4.18c)

What we see is that $\Delta^{-1}$ and $\Delta^{-1}_{abcd}$ remain unchanged.
It was important that we were able to remove the exponential
$\exp\left(\pm\tfrac{\mu^{2}x^{2}}{2}\right)$ to linear order in $\mu x_{a}$.
A word about the consistency of the approximation is now in order. In the
small $\mu x_{a}$ limit, the fields themselves do not transform as can be seen
from (4.11) and (4.12). The transformations are only nontrivial at higher
orders in $\mu x^{a}$. So we began with an infinitesimally small
transformation, and nevertheless transformed the operators. Therefore, the
difference between the first derivatives in $\Delta^{-1}_{L,ab}$ and
$\Delta^{-1}_{R,ab}$ and the effective mass in $\Delta^{-1}_{ab}$ is vanishing
to linear order in $\mu x^{a}$. The transformation we chose is merely one
consistent choice, but the underlying structure that
$\Delta^{-1}_{L,ab},\Delta^{-1}_{R,ab}$ equal $\Delta^{-1}_{ab}$ up to
infinitesimal differences holds regardless of the transformation chosen.

This quadratic action is now in a convenient enough form to allow for
inversion of the above operators to find the graviton propagator. All terms
linear in $\mu x_{a}$ have dropped out, except in $\Delta^{-1}_{abcd}$. We
shall see that in the following subsection that they do not contribute either.

###  4.2 The graviton propagator

In this subsection we derive the complete graviton propagator for the even
action on the Schwarzschild horizon near the horizon. The propagator is found
from the Green’s function of the equation of motion. The equations of motion
of the graviton are now

$\displaystyle\Delta^{-1}_{abcd}\mathfrak{h}^{cd}+\Delta^{-1}_{ab}\mathcal{K}~{}$
$\displaystyle=~{}0\,,$ (4.19a)
$\displaystyle\Delta^{-1}_{ab}\mathfrak{h}^{ab}+\Delta^{-1}\mathcal{K}~{}$
$\displaystyle=~{}0\,,$ (4.19b)

with the operators defined in (4.18). These can be written in the following
matrix form

$\displaystyle\begin{pmatrix}\Delta^{-1}_{abcd}&\Delta^{-1}_{ab}\\\
\Delta^{-1}_{cd}&\Delta^{-1}\end{pmatrix}\begin{pmatrix}\mathfrak{h}^{cd}\\\
\mathcal{K}\end{pmatrix}~{}=~{}0\,.$ (4.20)

Then the Green’s function of this matrix differential equation is defined by

$\displaystyle\begin{pmatrix}\Delta^{-1}_{abcd}&\Delta^{-1}_{ab}\\\
\Delta^{-1}_{cd}&\Delta^{-1}\end{pmatrix}\begin{pmatrix}\mathcal{P}^{cdef}&\mathcal{P}^{cd}\\\
\mathcal{P}^{ef}&\mathcal{P}\end{pmatrix}~{}=~{}\begin{pmatrix}\delta^{ef}_{ab}&0\\\
0&1\end{pmatrix}\delta^{(2)}\left(x-x^{\prime}\right)\,.$ (4.21)

The Green’s function matrix is symmetric owing to the symmetrised operators
(4.18). There are three propagators. The first is $\mathcal{P}^{cdef}$, which
is the propagator of $\mathfrak{h}_{cd}\to\mathfrak{h}_{ef}$. The function
$\mathcal{P}_{ab}$ corresponds to $\mathfrak{h}^{ab}\to\mathcal{K}$ whereas
$\mathcal{P}$ corresponds to $\mathcal{K}\to\mathcal{K}$. From equation (4.21)
we find that all individual propagators are defined by

$\displaystyle\Delta^{-1}_{abcd}\mathcal{P}^{cdef}+\Delta^{-1}_{ab}\mathcal{P}^{ef}~{}$
$\displaystyle=~{}\delta^{ef}_{ab}\delta^{(2)}\left(x-x^{\prime}\right)\,,$
(4.22a)
$\displaystyle\Delta^{-1}_{ab}\mathcal{P}^{abcd}+\Delta^{-1}\mathcal{P}^{cd}~{}$
$\displaystyle=~{}0\,,$ (4.22b)
$\displaystyle\Delta^{-1}_{abcd}\mathcal{P}^{cd}+\Delta^{-1}_{ab}\mathcal{P}~{}$
$\displaystyle=~{}0\,,$ (4.22c)
$\displaystyle\Delta^{-1}_{ab}\mathcal{P}^{ab}+\Delta^{-1}\mathcal{P}~{}$
$\displaystyle=~{}\delta^{(2)}\left(x-x^{\prime}\right)\,.$ (4.22d)

The coupling between the propagators is naturally owed to the coupling between
the tensorial and scalar fields in the quadratic action (4.17). To find the
propagators, we first need to find the inverses of $\Delta^{-1}$,
$\Delta^{-1}_{ab}$, and $\Delta^{-1}_{abcd}$. These inverses are defined as
usual:

$\displaystyle\Delta^{-1}_{abcd}\Delta^{cdef}~{}$
$\displaystyle=~{}\delta^{ef}_{ab}\delta^{(2)}\left(x-x^{\prime}\right)\,,$
(4.23) $\displaystyle\Delta^{-1}_{ab}\Delta^{bc}~{}$
$\displaystyle=~{}\delta^{c}_{a}\delta^{(2)}\left(x-x^{\prime}\right)\,,$
(4.24) $\displaystyle\Delta^{-1}\Delta~{}$
$\displaystyle=~{}\delta^{(2)}\left(x-x^{\prime}\right)\,.$ (4.25)

##### The inverse of $\Delta^{-1}$

This is the easiest of the lot, and requires us to solve

$\displaystyle(-\partial^{2}+\mu^{2})\Delta\left(x;x^{\prime}\right)~{}=~{}\delta^{(2)}\left(x-x^{\prime}\right)\,.$
(4.26)

Using the Fourier transforms

$\displaystyle\Delta\left(x;x^{\prime}\right)~{}$
$\displaystyle=~{}\dfrac{1}{(2\pi)^{2}}\int\text{d}^{2}k\text{
}e^{ik_{a}(x-x^{\prime})^{a}}\Delta(k)\,,$ (4.27)
$\displaystyle\delta^{(2)}\left(x-x^{\prime}\right)~{}$
$\displaystyle=~{}\dfrac{1}{(2\pi)^{2}}\int\text{d}^{2}p\text{
}e^{ik_{a}(x-x^{\prime})^{a}}\,,$ (4.28)

the solution for the inverse is

$\displaystyle\Delta(k)~{}=~{}\dfrac{1}{k^{2}+\mu^{2}}$ (4.29)

where $k^{2}=\eta^{ab}k_{a}k_{b}$. This is the Klein-Gordon propagator with an
effective mass $\mu^{2}$ arising from the two-sphere reduction of the
Schwarzschild metric. In contrast to the flat space eikonal approximation, we
see that the effective reduction to two dimensions near the horizon provides a
natural infrared regulator given by the Schwarzschild radius. In the soft
limit, this cures infrared divergences. However, the four dimensional graviton
of course remains massless.

##### The inverse of $\Delta^{-1}_{abcd}$

The operator is given in (4.18c). In order to find the inverse we start with
the following ansatz

$\displaystyle\Delta^{abcd}~{}=~{}T\left(x-x^{\prime}\right)\eta^{ab}\eta^{cd}+Q\left(x-x^{\prime}\right)\eta^{a(c}\eta^{d)b}\,.$
(4.30)

This is the most general form that respects the symmetries of the graviton
$\mathfrak{h}_{ab}=\mathfrak{h}_{ba}$, and time translation symmetry $x\to
ax,y\to y/a$ up to linear order in $\mu x_{a}$. Here we used time translation
symmetry to require that $\Delta^{abcd}$ transforms at most as a tensor under
time translations, so that the only allowed tensor structures are
$\eta_{ab}\eta_{cd},\mu^{2}\eta_{ab}x_{c}x_{d}$ and
$\mu^{4}x_{a}x_{b}x_{c}x_{d}$ with any index permutation. We included the
$\mu^{2}$ to have dimensionless tensor structures. Now the latter two can be
removed up to linear order in $\mu x_{a}$, so that only $\eta_{ab}\eta_{cd}$
remains. Then (4.30) gives the most general form that respects symmetries of
the problem. There is another allowed dimensionless tensor as well:
$\tfrac{x_{a}x_{b}}{x^{2}}$. This tensor is however not regular over the
entire manifold. The irregularities of usual momentum space poles determine
the mass-shell, whereas this tensor contains a pole that would be a coordinate
singularity. In Kruskal-Szekers coordinates there is no singularity at
$x^{2}=0$. This are not the usual $1/r$ singularity of spherical coordinates
either, as that would correspond to $1/(x^{2}+2R^{2})$ instead. Therefore, we
forbid this tensor, leaving behind (4.30) as the most general form.

Inserting (4.30) in the definition of the inverse (4.23) results in

$\displaystyle\dfrac{2}{\mu^{2}(\lambda+1)}\delta^{ef}_{ab}\delta^{(2)}(x-x^{\prime})~{}$
$\displaystyle=~{}-Q\delta^{ab}_{ef}+\eta_{ab}\eta^{ef}\bigr{(}x^{c}\partial_{c}T-\left(\lambda+1\right)\left(T+Q\right)\bigr{)}$
$\displaystyle\qquad-\eta^{ef}x_{(a}\partial_{b)}\left(2T+Q\right)+\eta_{ab}x^{(e}\partial^{f)}Q$
$\displaystyle\qquad+\dfrac{1}{2}\bigr{(}\delta^{e}_{a}x_{[b}\partial^{f]}+\delta^{f}_{a}x_{[b}\partial^{e]}+\delta^{f}_{b}x_{[a}\partial^{e]}+\delta^{e}_{b}x_{[a}\partial^{f]}\bigr{)}Q\,.$
(4.31)

Neither the anti-symmetric terms nor the terms containing the metric tensor
can reduce to the delta function as required by the left hand side of this
equation. Therefore, we find

$\displaystyle
Q~{}=~{}-\dfrac{2}{\mu^{2}(\lambda+1)}\delta^{(2)}\left(x-x^{\prime}\right)\,.$
(4.32)

This is a peculiar result; the inverse of the operator is instantaneous.
Nevertheless it is allowed. The two-dimensional operator on the light-cone
(4.18c) does not contain any second derivatives $\partial^{2}$. So it is not
expected to have a momentum dependence. Instantaneous operators are not
familiar in other gauge choices. Presumably this is a peculiarity of the
choice of gauge. This operator is not the physical propagator; we denoted that
by $\mathcal{P}^{abcd}$. That full propagator will of course have momentum
dependence. Notice that the inclusion of the irregular
$\tfrac{x_{a}x_{b}}{x^{2}}$ tensor would not have lead to a $\delta^{ab}_{ef}$
either and the inverse would still have been instantaneous. To proceed
further, we make a guess that $T=-Q$ in (4.30). Inserting this into the
definition of the inverse shows that (4.30) is only a valid inverse if the
following equation is satisfied:

$\displaystyle 0~{}$
$\displaystyle\stackrel{{\scriptstyle?}}{{=}}~{}\biggr{(}\eta_{ab}\eta^{ef}x^{c}\partial_{c}-\eta^{ef}x_{(a}\partial_{b)}-\eta_{ab}x^{(e}\partial^{f)}$
$\displaystyle\qquad-\dfrac{1}{2}\bigr{(}\delta^{e}_{a}x_{[b}\partial^{f]}+\delta^{f}_{a}x_{[b}\partial^{e]}+\delta^{f}_{b}x_{[a}\partial^{e]}+\delta^{e}_{b}x_{[a}\partial^{f]}\bigr{)}\biggr{)}\delta^{(2)}\left(x-x^{\prime}\right)\,.$
(4.33)

At first glance this does not seem to vanish. However should we interpret the
Dirac delta function as a distribution obeying

$\displaystyle
x_{a}\partial_{b}\delta^{(2)}\left(x-x^{\prime}\right)~{}=~{}-\eta_{ab}\delta^{(2)}\left(x-x^{\prime}\right)\,,$
(4.34)

and recognise that the right hand side of (4.2) appears inside a two-
dimensional integral, we find

$\displaystyle\eta_{ab}\eta^{ef}x^{c}\partial_{c}\delta^{(2)}\left(x-x^{\prime}\right)-\eta^{ef}x_{(a}\partial_{b)}\delta^{(2)}\left(x-x^{\prime}\right)-\eta_{ab}x^{(e}\partial^{f)}\delta^{(2)}\left(x-x^{\prime}\right)~{}=~{}$
$\displaystyle\qquad=~{}-2\eta_{ab}\eta^{ef}\delta^{(2)}\left(x-x^{\prime}\right)+\eta_{ab}\eta^{ef}\delta^{(2)}\left(x-x^{\prime}\right)+\eta_{ab}\eta^{ef}\delta^{(2)}\left(x-x^{\prime}\right)$
$\displaystyle\qquad=~{}0\,.$

A similar analysis for $x_{[b}\partial^{f]}$ results in

$\displaystyle x_{[b}\partial^{f]}\delta^{(2)}\left(x-x^{\prime}\right)~{}$
$\displaystyle=~{}\dfrac{1}{2}\left(x_{b}\partial^{f}-x^{f}\partial_{b}\right)\delta^{(2)}\left(x-x^{\prime}\right)$
$\displaystyle=~{}-\dfrac{1}{2}\left(\delta^{f}_{b}-\delta^{f}_{b}\right)\delta^{(2)}\left(x-x^{\prime}\right)$
$\displaystyle=~{}0\,.$

Of course, all these manipulations are legitimate only under the integration
over all $x,y$. Therefore, we now find that the inverse is given by

$\displaystyle\Delta_{abcd}~{}=~{}\dfrac{1}{\mu^{2}\left(\lambda+1\right)}\left(2\eta_{ab}\eta_{cd}-\eta_{ac}\eta_{bd}-\eta_{ad}\eta_{bc}\right)\delta^{(2)}\left(x-x^{\prime}\right)\,.$
(4.35)

In momentum space the $\delta^{(2)}\left(x-x^{\prime}\right)$ can simply be
replaced by unity, giving the final tensor

$\displaystyle\Delta_{abcd}~{}=~{}\dfrac{1}{\mu^{2}\left(\lambda+1\right)}\left(2\eta_{ab}\eta_{cd}-\eta_{ac}\eta_{bd}-\eta_{ad}\eta_{bc}\right)\,.$
(4.36)

With the inverses of $\Delta^{-1}$ and $\Delta^{-1}_{abcd}$, the propagators
can be determined from (4.22).

####  4.2.1 The $\mathcal{K}$ propagator

We first address to find the easiest propagator: the scalar $\mathcal{K}$
propagator. From (4.22c) and (4.22d) we can find that in momentum space the
propagator is given by

$\displaystyle\mathcal{P}_{K}~{}=~{}\dfrac{1}{\Delta^{-1}-\Delta^{-1}_{ab}\Delta^{abcd}\Delta^{-1}_{cd}}\,.$
(4.37)

The second term in the denominator can be expanded as

$\displaystyle\Delta^{-1}_{ab}\Delta^{abcd}\Delta^{-1}_{cd}~{}=~{}\dfrac{\lambda-1}{\lambda+1}\left(\mu^{2}\left(\lambda-1\right)+2k^{2}\right)\,.$
(4.38)

Together with $\Delta^{-1}=k^{2}+\mu^{2}$, we have for the $\mathcal{K}$
propagator that

$\displaystyle\mathcal{P}_{K}~{}=~{}-\dfrac{\lambda+1}{\lambda-3}\dfrac{1}{k^{2}+\mu^{2}\lambda}\,.$
(4.39)

This obviously resembles the Klein-Gordon propagator with an effective mass
$\mu\sqrt{\lambda}$. We will have more to say about the effective graviton
mass at the end of this section. The prefactor, however, deserves attention.
It is not well defined for $\lambda=\ell^{2}+\ell+1=3$ which corresponds to
$\ell=1$. Moreover, there is a flip in sign for $\ell=0$, hinting at negative
norm states. That the $\ell=0$ and $\ell=1$ cases are special is familiar [72,
83]. The equations of motion are not invertible for $\ell=1$, so a gauge
redundancy is to be expected. The $\ell=0$ corresponds to a change in black
hole mass since it is spherically symmetric and Birkhoff’s theorem states that
the $\ell=0$ perturbation is also of a Schwarzschild form [72]. Therefore this
also suggests leftover gauge redundancies since at $\ell=0$ we have many more
degrees of freedom than just the mass perturbation. Similarly $\ell=1$ has has
more degrees of freedom than can be reduced to a form with only the odd-parity
graviton components $h_{aA}$ [72]. So the $\ell=1$ mode may be thought of as
describing a rotating gravitational field. It has to be said, however, that
this intuition of [72] is based on a classical analysis. Nevertheless, that
these modes require additional care and indicate gauge redundancies is clear.
For the four-point function of interest in this paper, fortunately, this
propagator will not contribute as we will see in further sections. Therefore,
we leave further analysis of the $\mathcal{K}$ propagator for future work.

####  4.2.2 The $\mathfrak{h}_{ab}$ propagator

For the purposes of the scalar four-point function of interest, the propagator
of $\mathfrak{h}_{ab}$ will turn out to be the most important one. To find
$\mathcal{P}^{abcd}$, we begin by using (4.22a) and (4.22b) to write

$\displaystyle\mathcal{P}^{abcd}~{}$ $\displaystyle=$
$\displaystyle~{}\left(\mathcal{L}^{ab}_{\text{ }\text{ }\text{ }\text{
}ef}\right)^{-1}\Delta^{efcd}\,,$ (4.40)

where we defined

$\displaystyle\mathcal{L}^{ab}_{\text{ }\text{ }\text{ }\text{ }ef}~{}$
$\displaystyle\coloneqq$
$\displaystyle~{}\delta^{ab}_{ef}-\Delta^{abcd}\Delta^{-1}_{cd}\Delta\Delta^{-1}_{ef}\,.$
(4.41)

We now write down $\Delta^{-1}_{ab}$ in momentum space:

$\displaystyle\Delta^{-1}_{ab}~{}=~{}\eta_{ab}\left(k^{2}+\dfrac{1}{2}\mu^{2}\left(\lambda-1\right)\right)-k_{a}k_{b}\,.$
(4.42)

Using this, $\mathcal{L}^{ab}_{\text{ }\text{ }\text{ }\text{ }ef}$ is given
by its definition in (4.41). Expanding the tensor structure results in

$\displaystyle\mathcal{L}^{ab}_{\text{ }\text{ }\text{ }\text{ }ef}~{}$
$\displaystyle=~{}\delta^{ab}_{ef}-\dfrac{\Delta\left(k\right)\left(k^{2}+\frac{1}{2}\mu^{2}\left(\lambda-1\right)\right)\left(\lambda-1\right)}{\lambda+1}\eta^{ab}\eta_{ef}-\dfrac{2\Delta\left(k\right)\left(k^{2}+\frac{1}{2}\mu^{2}\left(\lambda-1\right)\right)}{\mu^{2}\left(\lambda+1\right)}k^{a}k^{b}\eta_{ef}$
$\displaystyle\qquad+~{}\dfrac{\Delta\left(k\right)\left(\lambda-1\right)}{\left(\lambda+1\right)}\eta^{ab}k_{e}k_{f}+\dfrac{2\Delta\left(k\right)}{\mu^{2}\left(\lambda+1\right)}k^{a}k^{b}k_{e}k_{f}$
$\displaystyle\eqqcolon~{}\delta^{ab}_{ef}-L_{1}\eta^{ab}\eta_{ef}-L_{2}k^{a}k^{b}\eta_{ef}+L_{3}\eta^{ab}k_{e}k_{f}+L_{4}k^{a}k^{b}k_{e}k_{f}\,.$
(4.43)

where we defined four functions $L_{i}$ that can be seen by comparing the two
equalities above. To invert (4.43) we make the ansatz

$\displaystyle(\mathcal{L}^{ab}_{\text{ }\text{ }\text{ }\text{
}ef})^{-1}~{}=~{}\delta^{ab}_{ef}-Q_{1}\eta^{ab}\eta_{ef}-Q_{2}k^{a}k^{b}\eta_{ef}+Q_{3}\eta^{ab}k_{e}k_{f}+Q_{4}k^{a}k^{b}k_{e}k_{f}\,,$
(4.44)

such that

$\displaystyle\mathcal{L}^{ab}_{\text{ }\text{ }\text{ }\text{
}ef}(\mathcal{L}^{ef}_{\text{ }\text{ }\text{ }\text{
}cd})^{-1}~{}=~{}\delta^{ab}_{cd}\,.$ (4.45)

Using the definition of $\mathcal{L}$ in this equation yields a neat relation
between all $L_{i}$ and $Q_{i}$. This relation can be written in matrix form
as follows

$\displaystyle\begin{pmatrix}\mathcal{M}&0\\\
0&\mathcal{M}\end{pmatrix}\begin{pmatrix}Q_{1}\\\ Q_{2}\\\ Q_{3}\\\
Q_{4}\end{pmatrix}~{}=~{}\begin{pmatrix}L_{1}\\\ L_{2}\\\ L_{3}\\\
L_{4}\end{pmatrix}\,,$ (4.46)

where

$\displaystyle\mathcal{M}~{}$
$\displaystyle=~{}\begin{pmatrix}-1+2L_{1}-L_{3}k^{2}&L_{1}k^{2}-L_{3}k^{4}\\\
2L_{2}-L_{4}k^{2}&-1+L_{2}k^{2}-L_{4}k^{4}\end{pmatrix}$
$\displaystyle=~{}\frac{\Delta\left(k\right)}{\lambda+1}\begin{pmatrix}-2k^{2}+\mu^{2}\lambda(\lambda-3)&\frac{1}{2}\mu^{2}k^{2}(\lambda-1)^{2}\\\
\frac{2}{\mu^{2}}(k^{2}+\mu^{2}(\lambda-1))&-2k^{2}-\mu^{2}(\lambda+1)\end{pmatrix}\,.$
(4.47)

The matrix relation (4.46) is block diagonal and therefore, an inverse of the
the $2\times 2$ matrix $\mathcal{M}$ suffices to invert the matrix relation.
An explicit calculation shows that the matrix determinant is given by

$\displaystyle|\mathcal{M}|~{}$
$\displaystyle=~{}-\dfrac{\lambda-3}{\lambda+1}\left(k^{2}+\mu^{2}\lambda\right)\Delta\left(k\right)$
$\displaystyle=~{}\dfrac{\Delta\left(k\right)}{\mathcal{P}_{K}}\,.$ (4.48)

This determinant vanishes for $\lambda=3$; a problem similar to what was
evident in the inverse propagator of $\mathcal{K}$. This shows that the
equations of motion are indeed not orthogonal at $\lambda=3$ and that there is
residual gauge freedom. Furthermore, the observations of the zeros and poles
of the prefactor of $\mathcal{P}$ are again relevant. We can now easily write
down the inverse matrix to be

$\displaystyle\mathcal{M}^{-1}~{}=~{}\dfrac{\mathcal{P}_{K}}{\lambda+1}\begin{pmatrix}-2k^{2}-\mu^{2}\left(\lambda+1\right)&-\frac{1}{2}\mu^{2}k^{2}\left(\lambda-1\right)^{2}\\\
-\frac{2}{\mu^{2}}\left(k^{2}+\mu^{2}\left(\lambda-1\right)\right)&-2k^{2}+\mu^{2}\lambda\left(\lambda-3\right)\end{pmatrix}\,.$
(4.49)

We now recognize that $2L_{1}=\mu^{2}\left(\lambda-1\right)L_{2}$, and that
$2L_{3}=\mu^{2}\left(\lambda-1\right)L_{4}$. Thus we only need to calculate

$\displaystyle\mathcal{M}^{-1}\begin{pmatrix}\mu^{2}\left(\lambda-1\right)\\\
2\end{pmatrix}~{}=~{}-\dfrac{\mathcal{P}_{K}}{\Delta}\begin{pmatrix}\mu^{2}\left(\lambda-1\right)\\\
2\end{pmatrix}\,.$ (4.50)

What we see is in fact an eigenvalue equation, so that the $2\times 2$ matrix
$\mathcal{M}$ is simply reduced to only a scalar in this particular case. The
corresponding eigenvalue is exactly given by the negative of the reciprocal of
the determinant. This allows us to easily identify the solution to the inverse

$\displaystyle Q_{i}~{}=~{}-\dfrac{\mathcal{P}_{K}}{\Delta}L_{i}\,.$ (4.51)

Inserting $Q_{i}$ into (4.44) results in

$\displaystyle(\mathcal{L}^{ab}_{\text{ }\text{ }\text{ }\text{ }ef})^{-1}~{}$
$\displaystyle=~{}\delta^{ab}_{ef}+\dfrac{\mathcal{P}_{K}\left(k^{2}+\frac{1}{2}\mu^{2}\left(\lambda-1\right)\right)\left(\lambda-1\right)}{\lambda+1}\eta^{ab}\eta_{ef}+\dfrac{2\mathcal{P}_{K}\left(k^{2}+\frac{1}{2}\mu^{2}\left(\lambda-1\right)\right)}{\mu^{2}\left(\lambda+1\right)}k^{a}k^{b}\eta_{ef}$
$\displaystyle\qquad-~{}\dfrac{\mathcal{P}_{K}\left(\lambda-1\right)}{\left(\lambda+1\right)}\eta^{ab}k_{e}k_{f}-\dfrac{2\mathcal{P}_{K}}{\mu^{2}\left(\lambda+1\right)}k^{a}k^{b}k_{e}k_{f}\,.$
(4.52)

We can now insert the above solution for $(\mathcal{L}^{ab}_{\text{ }\text{
}\text{ }\text{ }ef})^{-1}$ into (4.40) to find the propagator. It is worth
noting that the detailed mathematical calculations result in what one may have
expected on physical grounds based on symmetry. The underlying structure
merely consists of eigentensors, elegantly displaying the symmetry structure
of the graviton.

Finally, inserting (4.2.2) into (4.40) results in

$\displaystyle\mathcal{P}^{abcd}~{}=~{}\Delta^{abcd}+\mathcal{P}_{K}\text{p}^{ab}\text{p}^{cd}\,,$
(4.53)

with

$\displaystyle\text{p}^{ab}~{}\coloneqq~{}\dfrac{\lambda-1}{\lambda+1}\eta^{ab}+\dfrac{2k^{a}k^{b}}{\mu^{2}\left(\lambda+1\right)}\,.$
(4.54)

Here $\Delta^{abcd}$ is the same quantity defined in (4.36). The
$\text{p}^{ab}$ operators are the projection operators for the massive
graviton. In fact, observing the scalar part $\mathcal{P}_{k}$, there is
clearly a pole at $-\mu^{2}\lambda$. Therefore, the tensor field
$\mathfrak{h}_{ab}$ has also become massive. Note that the four-dimensional
graviton is still massless, it is only the two-dimensional reduction that
gains an effective mass. This is an important result, one that provides for a
natural infrared regulator. This is in contrast to the flat space eikonal
limit where a regulator was necessarily inserted by hand (2.17).

A projection operator similar to $\text{p}^{ab}$ was observed for the massive
graviton in [84], where unlike in the present case the mass was owed to a
direct insertion of mass terms in the action. While the resemblance between
the propagator in [84] and (4.53) is notable, the most significant difference
is the $\Delta^{abcd}$ term; this quantity still has no momentum dependence in
the present case. It represents the effect of the mass of the spin-2 particle
in two dimensions. This mass generates a curvature and $\Delta^{abcd}$ embeds
this curvature into the propagator. This is also the reason that it has no
momentum dependence: the curvature contribution of the tensor modes is
constant in the Lagrangian, there is no $k^{2}$ contributing to the two-
dimensional light-cone action in (4.17). The second term involving the
$\mathcal{K}$ mode carries momentum dependence, and implicitly also gives it
to the tensor modes.

####  4.2.3 Propagator for $\mathfrak{h}_{ab}\to\mathcal{K}$

The propagator that couples the tensorial mode in the graviton to the scalar
one can be found from (4.22b) to be given by

$\displaystyle\mathcal{P}^{ab}~{}$
$\displaystyle=~{}-\Delta\Delta^{-1}_{cd}\mathcal{P}^{cdab}$
$\displaystyle=~{}-\mathcal{P}_{K}\text{p}^{ab}\,.$ (4.55)

The operator $\text{p}^{ab}$ is still the same projection operator as before,
defined in (4.54). This propagator resembles that of a massive spin-1 field;
given that this it couples a spin-2 field with a scalar field, this may not be
a surprise. For the purposes of this article, this plays no role as we will
see in the upcoming sections.

###  4.3 The scalar propagator

The propagator for the scalar field is relatively straight forward to find,
but first requires a reduction of the matter action on the two sphere. We
begin with the kinetic term in (3):

$\displaystyle
S~{}=~{}\dfrac{1}{2}\int\text{d}^{4}x\sqrt{-g}\,\phi\,\Box\,\phi\,.$ (4.56)

For the box operator, we write

$\displaystyle\Box~{}=~{}\tilde{\Box}+V^{a}\tilde{\nabla}_{a}+\dfrac{1}{r^{2}}\Delta_{\Omega}\,,$
(4.57)

where, as before, the tilde operators correspond to the light-cone.
Furthermore, we used the residual curvature tensor $V_{a}$, which is defined
to be $V_{a}=2\partial_{a}\log r$. We now expand the scalar field in spherical
harmonics:

$\displaystyle\phi~{}=~{}\sum\limits_{\ell,m}\phi_{\ell m}\,Y^{m}_{\ell}\,.$
(4.58)

We again make use of $\Delta_{\Omega}Y_{\ell}^{m}=-\ell(\ell+1)Y_{\ell}^{m}$
and the orthogonality relation (3.67) to write the action as

$\displaystyle
S_{m}~{}=~{}-\dfrac{1}{2}\sum\limits_{\ell,m}\int\text{d}^{2}x\,A\left(r\right)r^{2}\phi_{\ell
m}\left(-\tilde{\Box}-V^{a}\tilde{\nabla}_{a}+\dfrac{\ell(\ell+1)}{r^{2}}\right)\phi_{\ell,m}\,.$
(4.59)

We now perform the Weyl transformation as was done in Section 3.7.1, along
with a field redefinition for the scalar as $\phi\to r\tilde{\phi}$. This
removes the Jacobian, and yields

$\displaystyle
S_{m}~{}=~{}-\dfrac{1}{2}\sum\limits_{\ell,m}\int\text{d}^{2}x\text{
}\phi_{\ell
m}\left(-\partial^{2}+\dfrac{1}{r}\left(\partial^{2}r\right)+\dfrac{A\left(r\right)\ell\left(\ell+1\right)}{r^{2}}\right)\phi_{\ell,m}\,.$
(4.60)

The field redefinition also removes the first derivatives, and replaces them
with a potential $\sim\left(\partial^{2}r\right)$. Finally, we insert that on
the horizon $r\sim R$ and $\partial^{2}r\sim-\tfrac{1}{2R}$, as derived in
Appendix B.1 in the approximation $x,y\ll R$, to find the following:

$\displaystyle
S_{m}~{}=~{}-\dfrac{1}{2}\sum\limits_{\ell,m}\int\text{d}^{2}x\,\phi_{\ell
m}\bigr{(}-\partial^{2}+\mu^{2}\lambda\bigr{)}\phi_{\ell m}\,,$ (4.61)

where we defined $\lambda=\ell^{2}+\ell+1$ and $\mu=1/R$ as before. The
propagator is easily found by the Fourier transformation

$\displaystyle\phi(x)~{}$
$\displaystyle=~{}\dfrac{1}{\left(2\pi\right)^{2}}\int\text{d}^{2}p\,e^{ip_{a}x^{a}}\phi(p)\,,$
(4.62) $\displaystyle\phi(p)~{}$
$\displaystyle=~{}\int\text{d}^{2}x\,e^{-ip_{a}x^{a}}\phi(x)\,.$ (4.63)

The inverse propagator can be read off from (4.61) as
$\partial^{2}-\mu^{2}\lambda$. Under the chosen conventions for the Fourier
transform, the propagator is directly found by substituting
$\partial_{a}=ip_{a}$. This results in

$\displaystyle\mathcal{P}_{\phi}~{}=~{}-\dfrac{1}{p^{2}+\mu^{2}\lambda-i\epsilon}\,.$
(4.64)

This is the propagator for a fixed $(\ell,m)$ wave and spherical symmetry of
the background ensures that different partial waves do not interact below
Planck scale.

In what is to follow, we will assume $\partial_{A}\to 0$ since the external
particles have insignificant transverse momenta when the black hole is larger
than Planck size. We will first analyse the consequence of this for the
vertex, before proceeding to compute the amplitudes of interest.

###  4.4 The interaction vertex

The scattering processes of interest in this article are mediated by three
point vertices as we noted before. This vertex written in the Regge Wheeler
gauge was given in (3.51). Inserting the light-cone fields taking the Weyl-
rescaling into account,

$h_{ab}~{}=~{}\dfrac{1}{A\left(r\right)r}\mathfrak{h}_{ab}\,,\qquad\phi~{}=~{}\dfrac{1}{r}\phi\,,\qquad
K~{}=~{}\dfrac{1}{r}\mathcal{K}\,,$ (4.65)

this vertex action becomes

$S_{\text{vertex}}~{}=~{}\dfrac{\kappa}{2R}\int\text{d}\Omega\int\text{d}^{2}x\text{
}\left[\mathfrak{h}^{ab}\left(\partial_{a}\phi\partial_{b}\phi-\dfrac{1}{2}\eta_{ab}\eta^{cd}\partial_{c}\phi\partial_{d}\phi\right)+\mathcal{K}\eta^{AB}T_{AB}\right]\,.$
(4.66)

in the $x^{a}\ll R$ approximation. As can be seen from the second term in
(3.50), the definition of the stress tensor shows that the second term of
(4.66) does not vanish with a mere assumption of vanishing transverse momenta
$p_{A}=0$. In fact, we see that

$\mathcal{K}\eta^{AB}T_{AB}~{}\sim~{}\mathcal{K}\eta^{ab}\partial_{a}\phi\partial_{b}\phi\,.$
(4.67)

Now, the scattering amplitude of interest has one particle, say $p_{1}$, with
a momentum exclusively going into the black hole while the other particle, say
$p_{2}$, is exclusively exiting the horizon. Therefore, we have that
$p_{1}=\left(p_{1,x},p_{1,y}\right)=\left(p_{1,x},0\right)$ and
$p_{2}=\left(p_{2,x},p_{2,y}\right)=\left(0,p_{2,y}\right)$. We now define the
two-dimensional Mandelstam variable

$s~{}=~{}-\dfrac{1}{2}\left(p_{1}+p_{2}\right)^{2}~{}=~{}-p_{1}\cdot
p_{2}~{}=~{}p_{1,x}p_{2,y}\,.$ (4.68)

The amplitudes arise from the interaction terms in (4.66). Let us first
consider the tree level diagram arising from (4.67) mediated by propagators
involving $\mathcal{K}$:

feyn_Ktree (130,130) i1,i2 o1,o2 $\phi\left(p_{2}\right)$i1
$\phi\left(p_{1}\right)$o1 $\phi\left(p_{1}\right)$i2
$\phi\left(p_{2}\right)$o2 fermioni1,v1 fermionv1,o1 fermioni2,v2 fermionv2,o2
gluon,label=$\quad\mathcal{P}_{\mathcal{K}}\left(p_{1}-p_{2}\right)$,label.d=-0.6wv1,v2
$p_{1}\cdot p_{2}$v1 $p_{1}\cdot p_{2}$v2

feyn_habToKtree (130,130) i1,i2 o1,o2 $\phi\left(p_{2}\right)$i1
$\phi\left(p_{1}\right)$o1 $\phi\left(p_{1}\right)$i2
$\phi\left(p_{2}\right)$o2 fermioni1,v1 fermionv1,o1 fermioni2,v2 fermionv2,o2
gluon,label=$\quad\mathcal{P}^{ab}\left(p_{1}-p_{2}\right)$,label.d=-0.6wv1,v2
$p_{1,a}p_{2,b}$v1 $p_{1}\cdot p_{2}$v2

Figure 4: The diagram on the left gives rise to an amplitude, say
$\mathcal{M}_{\mathcal{K}}$, and is mediated by the propagator
$\mathcal{P}_{\mathcal{K}}$ given in (4.39). The diagram on the right yields
an amplitude, say $\mathcal{M}_{\mathfrak{h}_{ab}\to\mathcal{K}}$, mediated by
the propagator $\mathcal{P}^{ab}$ given in (4.2.3).

The amplitudes in Fig. 4 can easily be calculated. The diagram on the left
mediated by $\mathcal{P}_{\mathcal{K}}$ gives

$\mathcal{M}_{\mathcal{K}}~{}\sim~{}\dfrac{s^{2}}{2s+\mu^{2}\lambda}\,,$
(4.69)

and the diagram on the right, mediated by $\mathcal{P}^{ab}$, gives

$\displaystyle\mathcal{M}_{\mathfrak{h}_{ab}\to\mathcal{K}}~{}$
$\displaystyle\sim~{}\left(p_{1}\cdot
p_{2}\right)\,\mathcal{P}^{ab}\left(p_{1,a}p_{2,b}-\dfrac{1}{2}\eta_{ab}\,\left(p_{1}\cdot
p_{2}\right)\right)$ $\displaystyle=~{}\left(p_{1}\cdot
p_{2}\right)\mathcal{P}^{xy}\left(p_{1,x}p_{2,y}-p_{1,x}p_{2,y}\right)$
$\displaystyle=~{}0\,,$ (4.70)

where we used that the light-cone metric is given by $\eta_{xy}=-1$, and that
$p_{1}=\left(p_{1,x},0\right)$ and $p_{2}=\left(0,p_{2,y}\right)$. The
remaining tree-level diagrams involve $\mathfrak{h}_{ab}$ exclusively, as
shown in Fig. 5.

feyn_habtreeSoft (130,130) i1,i2 o1,o2 $\phi\left(p_{2}\right)$i1
$\phi\left(p_{2}\right)$o1 $\phi\left(p_{1}\right)$i2
$\phi\left(p_{1}\right)$o2 fermioni1,v1 fermionv1,o1 fermioni2,v2 fermionv2,o2
gluon,label=$\quad\mathcal{P}^{xxyy}\left(0\right)$,label.d=-0.6wv1,v2
$p_{2,y}p_{2,y}$v1 $p_{1,x}p_{1,x}$v2 feyn_habtreeWithMomentum (130,130) i3,i4
o3,o4 $\phi\left(p_{2}\right)$i3 $\phi\left(p_{1}\right)$o3
$\phi\left(p_{1}\right)$i4 $\phi\left(p_{2}\right)$o4 fermioni3,v3
fermionv3,o3 fermioni4,v4 fermionv4,o4
gluon,label=$\quad\mathcal{P}^{abcd}\left(p_{1}-p_{2}\right)$,label.d=-0.7wv3,v4
$p_{1,a}p_{2,b}$v3 $p_{1,c}p_{2,d}$v4

Figure 5: The diagram on the left, say $\mathcal{M}_{soft}$, is the soft one
mediated by the propagator $\mathcal{P}^{abcd}$ given in (4.53), with no
momentum exchanged. The diagram on the right yields an amplitude, say
$\mathcal{M}_{hard}$, mediated by the same propagator $\mathcal{P}^{abcd}$.

Owing to the symmetry of the propagator $\mathcal{P}^{abcd}$ under
$a\leftrightarrow b$, and under $c\leftrightarrow d$, the amplitude
$\mathcal{M}_{hard}$ vanishes due to an argument similar to (4.4). The soft
amplitude on the other hand is again easily calculated to be

$\mathcal{M}_{soft}~{}\sim~{}\left(p_{1,x}p_{2,y}\right)^{2}\mathcal{P}^{xxyy}\left(0\right)~{}\sim~{}\dfrac{s^{2}}{\mu^{2}}\,.$
(4.71)

###  4.5 The black hole eikonal limit: $\sqrt{s}\gg\gamma M_{Pl}$

Using the parameters $\mu=1/R_{S}$ and $\kappa=\sqrt{8\pi G_{N}}$, we define

$\gamma~{}\coloneqq~{}\mu\kappa~{}\sim~{}\dfrac{M_{Pl}}{M_{BH}}\,.$ (4.72)

With this definition, we observe that

$\mu^{2}~{}\sim~{}\dfrac{M^{4}_{Pl}}{M^{2}_{BH}}~{}\sim~{}\gamma^{2}M^{2}_{Pl}\,.$
(4.73)

The only non-vanishing tree-level amplitudes were (4.69) and (4.71). When
written in terms of $\gamma$, in the limit $s\gg\gamma^{2}M^{2}_{Pl}$ for
every fixed $\ell$, these read

$\displaystyle\mathcal{M}_{\mathcal{K}}~{}$
$\displaystyle\sim~{}s\quad\text{and}\quad\mathcal{M}_{soft}~{}\sim~{}\dfrac{s^{2}}{\gamma^{2}M^{2}_{Pl}}\,.$
(4.74)

Clearly $\mathcal{M}_{\mathcal{K}}$ is sub-leading for black holes that are
bigger than Planck size. In the eikonal approximation in flat space, reviewed
in Section 2, we saw that the analogous approximation necessitated ultra-high
energy scattering. A black hole background provides for a new scale that
tempers this requirement dramatically. The requirement for the energy of
collision is merely

$\sqrt{s}~{}\gg~{}\gamma\,M_{Pl}\,.$ (4.75)

The factor $\gamma$ is of the order of $10^{-32}$ for a black hole with a
Schwarzschild radius of $1\,cm$ (this would be a black hole with the mass of
the earth). Therefore, the requirement $\sqrt{s}\gg 10^{-32}M_{Pl}$ is very
easily satisfied for all known particles in the Standard Model for any black
hole that is much heavier than Planck mass. In this approximation, we see that
we can ignore the amplitude for $\mathcal{K}$ in (4.74) and we are left with
$\mathcal{M}_{soft}$ to leading order. Therefore, this approximation
simplifies the vertex (4.66) because the field $\mathcal{K}$ drops out and we
are left with

$\mathfrak{h}^{ab}\left(\partial_{a}\phi\partial_{b}\phi-\dfrac{1}{2}\eta_{ab}\eta^{cd}\partial_{c}\phi\partial_{d}\phi\right)~{}=~{}\mathfrak{h}^{xx}\partial_{x}\phi\partial_{x}\phi+\mathfrak{h}^{yy}\partial_{y}\phi\partial_{y}\phi\,.$
(4.76)

That only $\mathfrak{h}_{xx}$ and $\mathfrak{h}_{yy}$ remain was argued to be
a feature of the eikonal approximation in flat space based on boundary
conditions. Near the horizon of the Schwarzschild solution in the
$s\gg\gamma^{2}M^{2}_{Pl}$ approximation, we see that the same result can be
explicitly derived. Therefore, with the implicit understanding that the off-
diagonal components $\mathfrak{h}_{xy}$ do not contribute

$\mathfrak{h}^{ab}~{}=~{}\begin{pmatrix}\mathfrak{h}^{xx}&0\\\
0&\mathfrak{h}^{yy}\end{pmatrix}\,,$ (4.77)

the vertex action (4.66) can be written as

$\displaystyle S_{\text{vertex}}~{}$
$\displaystyle=~{}\dfrac{\gamma}{2}\sum\limits_{\ell,m}\sum\limits_{\ell_{1},m_{1}}\sum\limits_{\ell_{2},m_{2}}\int\text{d}\Omega\text{
}Y_{\ell}^{m}Y_{\ell_{1}}^{m_{1}}Y_{\ell_{2}}^{m_{2}}\int\text{d}^{2}x\text{
}\mathfrak{h}^{ab}_{\ell
m}\partial_{a}\phi_{\ell_{1}m_{1}}\partial_{b}\phi_{\ell_{2}m_{2}}\,,$ (4.78)

where we inserted the spherical harmonic expansions

$\mathfrak{h}^{ab}~{}=~{}\sum\limits_{\ell,m}\mathfrak{h}^{ab}_{\ell
m}Y_{\ell}^{m}\qquad\text{and}\qquad\phi~{}=~{}\sum\limits_{\ell,m}\phi_{\ell
m}Y_{\ell}^{m}\,.$ (4.79)

Integrating over the sphere will result in a coupling between the various
partial waves. We saw in Section 3.6 that the various partial waves decouple
at quadratic order, owing to the spherical symmetry of the background. The
different partial waves are then expected to only be coupled by large
transverse momentum transfers [23]. This is subdominant for scattering
processes on the horizon when the black hole is greater than Planck size.
Therefore, we do not expect that angular momentum is distributed among
external legs. This implies that one of the external scalar legs is kept with
some fixed partial wave, say $\phi_{0}$. The vertex action therefore becomes

$S_{\text{vertex}}~{}=~{}\gamma\sum\limits_{\ell,m}\int\text{d}^{2}x\text{
}\mathfrak{h}^{ab}_{\ell m}\partial_{a}\phi_{0}\partial_{b}\phi_{\ell m}\,,$
(4.80)

where the additional factor of two is owed to the different possible
attachments of the fixed partial wave $\phi_{0}$.

Since the off-diagonal components of $\mathfrak{h}^{ab}$ play no role in
scattering with a massless scalar field, the graviton propagator (4.53)
simplifies significantly

$\mathcal{P}^{abcd}\left(k\right)~{}=~{}\dfrac{1}{4}f_{\ell}\left(\eta^{ac}\eta^{bd}+\eta^{ad}\eta^{bc}-f_{\ell}\left(\dfrac{\lambda+1}{\lambda+3}\right)\dfrac{k^{a}k^{b}k^{c}k^{d}}{4\left(k^{2}+\mu^{2}\lambda\right)}\right)\,.$
(4.81)

The choice of $1/4$ is arbitrary. From (4.53), the soft form factor $f_{\ell}$
can be found to be given by

$f_{\ell}~{}=~{}-\dfrac{4}{\mu^{2}(\lambda+1)}\,.$ (4.82)

For the scattering processes to be considered, we will see in the Section 5
that the soft limit $k\rightarrow 0$ is imposed on us. Observing (4.81), we
see familiar problems in the ultraviolet. In the high energy limit, we have
that $\mathcal{P}^{abcd}\sim k^{4}/\left(\mu^{2}k^{2}+\mu^{4}\right)$ which
diverges. This is a feature familiar from massive gravity [84]. It is a result
of the mass in the projection operator $\text{p}^{ab}$. For a massive spin-1
field, there is a similar feature in the UV-limit, which renders the theory
non-renormalizable by the power counting theorem [85]. Massive Quantum
Electrodynamics is renormalizable due to the introduction of additional scalar
fields [86]. A similar trick can be applied to the spin-2 case, introducing
vector fields to restore the power counting theorem for the graviton. This was
also done in [84], however renormalization remains a problem, as it was
already a problem for the massless graviton.

Fortunately enough, owing to the soft limit that is to be enforced upon us, we
may simply work with the propagator in (4.81). We shall see that the divergent
momentum-dependent tail is projected out in the ladder diagrams of interest.
And it will be far more important that $\mathcal{P}^{abcd}$ is well defined
when $k=0$. Unlike in the flat space eikonal calculation reviewed in Section
2, the emergent scale of the Schwarzschild radius provides for a natural
regulator, with no necessity for a regulator to be put in by hand.

As a final comment, if instead of writing down the honest propagator as we
have done above, had we postulated an ansatz of the form [78]

$\mathcal{P}^{abcd}\left(k\right)~{}=~{}\dfrac{1}{4}f_{\ell}\left(k^{2}\right)\left(\eta^{ac}\eta^{bd}+\eta^{ad}\eta^{bc}\right)\,,$
(4.83)

we find that using (4.53), the solution is given by

$f_{\ell}\left(k^{2}\right)~{}=~{}-\dfrac{4}{\mu^{2}\left(\lambda+1\right)}-\dfrac{2}{\left(\lambda+1\right)\left(\lambda-3\right)}\dfrac{1}{k^{2}+\mu^{2}\lambda}\dfrac{k^{4}}{\mu^{4}}\,.$
(4.84)

In the soft limit, which we will be projected on to in Section 5, either
choice results in the same answer since $f_{\ell}\left(0\right)=f_{\ell}$.

###  4.6 The Feynman rules

Gathering all the results we have obtained so far in (4.61), (4.80), and
(4.81), we find that the action near the horizon reduces to

$\displaystyle S_{\text{hor}}~{}$
$\displaystyle=~{}\dfrac{1}{4}\int\mathrm{d}^{2}k\,\mathfrak{h}^{ab}\,\mathcal{P}^{-1}_{abcd}\left(k\right)\,\mathfrak{h}^{cd}~{}-~{}\dfrac{1}{2}\int\mathrm{d}^{2}p\,\phi\left(p^{2}+\mu^{2}\lambda\right)\phi$
$\displaystyle\quad~{}+\gamma\int\mathrm{d}^{2}k\,\mathrm{d}^{2}p_{1}\,\mathrm{d}^{2}p_{2}\,\delta^{(2)}\left(k+p_{1}+p_{2}\right)\mathfrak{h}_{ab}\left(k\right)p^{a}_{1}p^{b}_{2}\phi_{0}\left(p_{1}\right)\phi\left(p_{2}\right)$
(4.85)

for each partial wave, in momentum space. This allows us to read off the
Feynman rules. For the propagators we find

feyn_propagators

$\displaystyle\fmfgraph*(130,0)\fmfleft{i1}\fmfright{i2}\fmf{phantom}{i1,i2}\fmfi{fermion,label=$\phi_{\ell
m}(p)$,label.side=left}{vpath(_{_{i}}1,_{_{i}}2)}~{}~{}~{}$
$\displaystyle=~{}\dfrac{-i}{p^{2}+\frac{\lambda}{R^{2}_{S}}-i\epsilon}$
$\displaystyle\fmfgraph*(130,0)\fmfleft{i3}\fmfright{i4}\fmf{phantom}{i3,i4}\fmfi{dashes,label=$\phi_{0}(p)$,label.side=left}{vpath(_{_{i}}3,_{_{i}}4)}~{}~{}$
$\displaystyle=~{}\dfrac{-i}{p^{2}+\frac{1}{R^{2}_{S}}-i\epsilon}$
$\displaystyle\fmfgraph*(100,0)\fmfleft{i5}\fmfright{i6}\fmf{phantom}{i5,i6}\fmflabel{$\mathfrak{h}^{ab}_{\ell
m}$}{i5}\fmflabel{$\mathfrak{h}^{cd}_{\ell
m}$}{i6}\fmfi{curly,label=$\mathfrak{h}_{\ell
m}(k)$,label.side=left}{vpath(_{_{i}}5,_{_{i}}6)}~{}\qquad$
$\displaystyle=~{}2i\mathcal{P}^{abcd}\left(k\right)\,,$

and the vertex is given by feyn_vertex

$\displaystyle\fmfgraph*(100,100)\fmfleft{i8,i7}\fmfright{o2}\fmf{gluon}{i7,v1}\fmf{dashes_{a}rrow}{i8,v1}\fmf{fermion}{v1,o2}\fmflabel{$\mathfrak{h}^{ab}_{\ell
m}$}{i7}\fmflabel{$p_{2}$}{i8}\fmflabel{$p_{1}~{},$}{o2}\fmfdot{v1}\fmfv{label=$i\gamma~{}p^{1}_{a}~{}p^{2}_{b}$,label.angle=60,label.dist=20}{v1}$

with a symmetry factor of 8 for the graviton propagator accounting for the