Many experiments have recently been proposed to test whether non-relativistic
gravitational interactions can generate entanglement. In this note, I consider
the extent to which these experiments can test if the graviton exists.
Assuming unitarity and Lorentz invariance of the $S$-matrix, I demonstrate
that this “Newtonian entanglement” requires the existence of massless bosons,
universally coupled to mass, in the Hilbert space of low-energy scattering
states. These bosons could be the usual spin-2 gravitons, but in principle
there are other possibilities like spin-0 scalar gravitons. I suggest a
concept for a more refined experiment to rule these out. The special role of
$d=3+1$ spacetime dimensions and the possibility that unitarity is violated by
gravity are highlighted.††Dedicated to the memory of Steve Weinberg
(1933-2021).

# Newton, entanglement, and the graviton

Daniel Carney
Physics Division  Lawrence Berkeley National Laboratory
Berkeley  CA

<EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 Unitary Newtonian gravity requires quantized radiation
    1. 2.1 Lorentzian bootstrap strategy
    2. 2.2 Tree-level unitarity
  3. 3 Implications and interpretation
    1. 3.1 Models which can explain the observation of entanglement
    2. 3.2 Measuring the mediator spin
  4. 4 Conclusions
  5. A Classical and semiclassical gravity models
  6. B Details of amplitude calculations
  7. C Unitarity violation in particle-antiparticle scattering

##  1 Introduction

Dyson and others have pointed out that detection of individual gravitons is
likely to be impossible [1, 2, 3]. Experimental proof of the existence of the
graviton may thus require a more subtle methodology.

Bell’s theorem provides a method to prove that a state of nature does not
admit a local, classical description [4, 5, 6]. There have been a plethora of
experimental proposals to determine if gravitational interactions can generate
such a non-classical, entangled state [7, 8, 9, 10, 11, 12, 13, 14, 15, 16,
17, 18, 19, 20, 21, 22, 23, 24]. A minimal realization is depicted in Fig. 1.

These are non-relativistic tabletop experiments. If we assume that gravity
generates a unitary channel on the objects, then observation of entanglement
generation consistent with a $1/r$ law would tell us that the Hamiltonian must
be

$H=H_{1}+H_{2}+V_{N},\ \ \
V_{N}=-\frac{G_{N}m_{1}m_{2}}{|\mathbf{x}_{1}-\mathbf{x}_{2}|}.$ (1)

It is natural to ask [25, 26, 27, 28, 29, 30] what we learn about any
gravitational degrees of freedom themselves, which do not appear in (1). Of
course, quantizing metric fluctuations into gravitons produces a perfectly
good effective quantum field theory, and this model reproduces the Newton
potential operator [31, 32, 33]. The question these experiments probe is
whether this is the correct model of nature.

In this paper, I study the implications if we further assume that (1) is the
non-relativistic limit of some Lorentz-invariant model. A minimal $S$-matrix
theory framework is sufficient to encode unitarity and Lorentz invariance
without assuming that the fundamental degrees of freedom are quantum fields
[34, 35, 36, 37]. These assumptions are enough to prove that one needs
massless bosonic degrees of freedom in the Hilbert space of scattering states
to be consistent with (1). These bosons must have a universal coupling to mass
with strength $\sqrt{G_{N}}m$. They can however have any integer spin, not
only $s=2$ like the usual graviton. Ruling out these other possibilities
requires a more refined experiment, as discussed in section 3.

Perhaps the more interesting possibility is that gravity does not generate a
unitary interaction. For example, gravity could emerge from interactions
mediated by unobserved microscopic degrees of freedom [38, 39, 40, 41, 42, 43,
44]. Models of this type in which gravity emerges in a semi-classical fashion
like $G_{\mu\nu}=8\pi G_{N}\braket{T_{\mu\nu}}$ would be ruled out, since
there gravity cannot entangle objects [7]. However, it may be possible that
gravity could form an open system in a different manner, in which it can
produce entanglement observables consistent with (1). Making a precise
statement about gravitons in this context is a difficult problem left to
future work.

Before moving on, we note some previous results in this direction [26, 27]. In
particular, Belenchia et al. [26] study a gedankanexperiment in which
Newtonian entanglement enables superluminal signaling, and resolve the paradox
by introducing quantized metric fluctuations. The arguments presented here are
related, but precise enough to demonstrate an exhaustive list of
possibilities: the _only_ way to resolve these types of paradoxes within a
unitary and Lorentz-invariant model is to include radiative graviton, or very
graviton-like, degrees of freedom.

$\ket{\psi_{1}}$$\ket{\psi_{2}}$$e^{-iV_{N}\Delta t}$(a) | $m_{1}$$m_{2}$$V_{N}$laserlaser(b)
---|---
Figure 1: (a) Circuit diagram of the simplest possible experimental
implementation. Two masses $m_{1},m_{2}$ are prepared in an initial product
state, interact gravitationally for a time $\Delta t$, and are read out to
check for entanglement. (b) Realization with free-falling masses. The Newton
interaction would squeeze the relative position $x_{-}$ while preserving total
momentum $p_{+}$, leading to a violation of the Duan inequality [45, 46] for
separable states $\braket{\Delta x_{-}^{2}\Delta p_{+}^{2}}\geq\hbar^{2}$.
This can be read out with local interferometers.

##  2 Unitary Newtonian gravity requires quantized radiation

Our core assumption is unitarity. Specifically, we assume that the
gravitational interaction between massive objects operates as a closed system.
Within a unitary framework, a demonstration of entanglement between masses,
with the right parametric dependences, means that we must have a two-body
potential of the form

$V_{N}=-\frac{G_{N}m^{2}e^{-\mu|\mathbf{x}_{1}-\mathbf{x}_{2}|}}{|\mathbf{x}_{1}-\mathbf{x}_{2}|}.$
(2)

Here $\mu$ is a small parameter with dimensions of inverse length which is
useful as a regulator; we will take the $\mu\to 0$ limit to recover the Newton
potential.111The experiments are done within some finite length scale,
typically a laboratory scale $10^{-6}~{}{\rm m}\lesssim\ell\lesssim 1~{}{\rm
m}$. Thus beyond its regulatory benefits, this kind of potential would be a
perfectly viable explanation for the observed entanglement as long as we
assume a sufficiently small value $\mu\ll\ell^{-1}$. For extensive discussion
on the $S$-matrix in the $\mu\to 0$ limit, see for example [47, 48, 49, 50,
51]. The real input from the experiments is that this is a two-body operator
in the full sense of the term: the $\mathbf{x}_{i}$ are local operators on a
bipartite Hilbert space. This is different from, for example, models where
gravity acts “semiclassically” through expectation values (see appendix A).

We will consider scattering processes with the Hamiltonian (2). Using the non-
relativistic matrix elements as input to Lorentz-invariant extensions of this
model, we will find violations of unitarity in a variety of simple scattering
processes. Moreover, the precise form of this unitarity violation is enough to
conclude that the only solution is to add radiative states of bosons with mass
$\mu$ which couple to massive matter with strength $\sqrt{G_{N}}m$.

Unitarity and Lorentz invariance can be defined in a manner which is
independent of the way we realize the interactions, i.e., does not depend on
the use of field theory. Consider scattering processes described by $S$-matrix
elements $S_{\alpha\to\beta}=\braket{\beta}{S}{\alpha}$. The $S$-matrix
elements are the transition amplitudes for an initial state $\ket{\alpha}$
prepared in the asymptotic past $t\to-\infty$ to evolve to a definite state
$\ket{\beta}$ in the asymptotic future $t\to+\infty$. To begin, we will only
need the minimal postulates:

  1. (A)

Unitarity. $S$ is a unitary operator, $S^{\dagger}S=1$.

  2. (B)

Lorentz invariance. For $\Lambda$ an element of the Lorentz group, the
scattering states transform in a unitary representation $U(\Lambda)$.
Furthermore, the $S$-matrix is invariant: $U(\Lambda)SU^{\dagger}(\Lambda)=S$.

Our implementation of (2) to compute $S$-matrix elements non-relativistically
will be based on a Schrödinger equation with past boundary conditions, and
thus automatically assumes the usual non-relativistic notion of causality.

Gravity is a weak interaction, so that we can expand $S=1+iT$, where the $1$
reflects the possibility of no scattering occurring. Assumption (B) means in
particular that a spinless particle of momentum $\mathbf{p}$ transforms like
$U(\Lambda)\ket{\mathbf{p}}=\sqrt{(\Lambda
p)^{0}/p^{0}}\ket{\Lambda\mathbf{p}}$ under a boost.222This is our
_definition_ of a single-particle state; it could be a field quanta, or a
closed string mode, or whatever else, as long as it transforms correctly.
Following [52], we define single-particle states to satisfy the non-
relativistic normalization
$\braket{\mathbf{p}^{\prime}}{\mathbf{p}}=\delta^{3}(\mathbf{p}^{\prime}-\mathbf{p})$,
and use $(-,+,+,+)$ signature. Multiple-particle states
$\ket{\alpha}=\ket{\mathbf{p}_{1}\mathbf{p}_{2}\cdots}$ are described by lists
of such momenta (as well as spins and any other internal quantum numbers).
Furthermore, spacetime translation invariance implies that total four-momentum
is conserved in every process. Thus we will define the usual “Feynman
amplitudes” $M$ by

$S_{\alpha\to\beta}=\delta_{\alpha\beta}-i(2\pi)^{4}\delta^{4}(p_{\alpha}-p_{\beta})B_{\alpha}B_{\beta}M_{\alpha\to\beta},$
(3)

where $p_{\alpha},p_{\beta}$ are the total incoming and outgoing four-momenta,
respectively. We will only deal with spinless massive objects, and so have
factored out the Lorentz-transformation factors
$B_{\alpha}=\prod_{i\in\alpha}[2(2\pi)^{3}E_{i}]^{-1/2}$, with
$E_{i}=p_{i}^{0}$ the energy of the $i$th particle. Defined this way,
$M_{\alpha\to\beta}$ should be invariant under the Lorentz group.

The main workhorse in what follows will be the unitarity condition on the $S$
matrix. We have $S^{\dagger}S=1$, which implies
$i(T-T^{\dagger})=T^{\dagger}T$. Inserting a complete set of final states
$\ket{X}$ and comparing to (3), we have

$i\left(M_{\alpha\to\beta}-M^{*}_{\beta\to\alpha}\right)=(2\pi)^{4}\sum_{X}B_{X}^{2}M_{\alpha\to
X}M^{*}_{\beta\to X}\delta^{4}(p_{\alpha}-p_{X}).$ (4)

This is known as the optical theorem. In the special case of forward
scattering $\alpha=\beta$, this reduces to the usual optical theorem ${\rm
Im}~{}f(0)=\sigma_{\rm total}$. This is a reflection of the fact that the
scattered wavefunction must have a specific interference pattern with the
unscattered wavefunction. For a model to be unitary, equation (4) must be
satisfied for all initial and final states $\ket{\alpha}$, $\ket{\beta}$.

###  2.1 Lorentzian bootstrap strategy

time

$\ket{\beta}$$\ket{\alpha}=\ket{\mathbf{p}_{1}\mathbf{p}_{2}\cdots}$(a)$V_{N}(t)$$\mathbf{p}^{\prime}_{1}$$\mathbf{p}^{\prime}_{2}$$\mathbf{p}_{1}$$\mathbf{p}_{2}$(b)$\mathbf{p}^{\prime}_{1}$$\mathbf{p}^{\prime}_{2}$$\mathbf{p}_{1}$$\mathbf{p}_{2}$(c)
Figure 2: (a) Depiction of an $S$-matrix element for the process
$\ket{\alpha}\to\ket{\beta}$. The blob represents a sum over all possible
intermediate processes. (b) Diagram describing the gravitational, non-
relativistic scattering amplitude at lowest order in perturbation theory. The
interaction is given by the instantaneous potential matrix element
$\sim[(\mathbf{p}^{\prime}_{1}-\mathbf{p}_{1})^{2}+\mu^{2}]^{-1}$. (c)
Lorentzian bootstrap version of the same amplitude. The dashed line represents
a factor $[(p_{1}^{\prime}-p_{1})^{2}+\mu^{2}]^{-1}$, which at this stage in
the argument has no interpretation in terms of an intermediate particle.

The strategy we will follow is to begin with a non-relativistic expression for
the scattering amplitude, then bootstrap the answer into something Lorentz-
invariant.333The “bootstrap” terminology is inspired from the modern
$S$-matrix bootstrap program, where one derives constraints on low-energy
models by demanding that they are low-energy limits of some high-energy model
with certain restrictions [53]. Usually this is used to relate a pair of
relativistic field theories, but for a few examples with non-relativistic
models, see [54, 55, 56]. Scattering theory is the study of solutions to the
Schrödinger equation combined with boundary conditions as $t\to-\infty$. Our
starting point will thus be the time-dependent perturbation series solution
for this system. Although the Born series is perhaps more familiar, consider
instead the Dyson series [57]

$S_{\alpha\to\beta}=\braket{\beta}{S}{\alpha}=\sum_{n=0}^{\infty}\frac{(-i)^{n}}{n!}\int_{-\infty}^{\infty}dt_{1}\cdots
dt_{n}\braket{\beta}{\mathcal{T}V_{I}(t_{1})\cdots V_{I}(t_{n})}{\alpha}.$ (5)

Here $\mathcal{T}$ is the time-ordering operator, and the potential is written
in the interaction picture. To illustrate the Lorentzian bootstrap idea,
consider $2\to 2$ scattering, so
$\ket{\alpha}=\ket{\mathbf{p}_{1}\mathbf{p}_{2}}$ and
$\ket{\beta}=\ket{\mathbf{p}^{\prime}_{1}\mathbf{p}^{\prime}_{2}}$. We will
write the interaction picture potential operator (2) explicitly in two-body
notation, in the lab frame:

$\displaystyle\begin{split}V_{N}(t)=&\ -\frac{1}{(2\pi)^{3}}\int
d^{3}\mathbf{k}_{1}d^{3}\mathbf{k}_{2}d^{3}\mathbf{k}^{\prime}_{1}d^{3}\mathbf{k}^{\prime}_{2}\delta^{3}(\mathbf{k}_{1}+\mathbf{k}_{2}-\mathbf{k}^{\prime}_{1}-\mathbf{k}^{\prime}_{2})\\\
&\times
e^{-i(E_{\mathbf{k}^{\prime}_{1}}+E_{\mathbf{k}^{\prime}_{2}}-E_{\mathbf{k}_{1}}-E_{\mathbf{k}_{2}})t}V_{\mathbf{k}^{\prime}_{1}\mathbf{k}^{\prime}_{2},\mathbf{k}_{1}\mathbf{k}_{2}}\ket{\mathbf{k}^{\prime}_{1}\mathbf{k}^{\prime}_{2}}\bra{\mathbf{k}_{1}\mathbf{k}_{2}},\end{split}$
(6)

where the Schrödinger-picture matrix elements are

$V_{\mathbf{k}^{\prime}_{1}\mathbf{k}^{\prime}_{2},\mathbf{k}_{1}\mathbf{k}_{2}}=\frac{4\pi
G_{N}m^{2}}{\mathbf{q}^{2}+\mu^{2}},\ \ \
\mathbf{q}=\frac{\mathbf{k}_{1}^{\prime}-\mathbf{k}_{2}^{\prime}}{2}-\frac{\mathbf{k}_{1}-\mathbf{k}_{2}}{2},$
(7)

and $\mathbf{q}$ is the change in the relative momentum. The zeroth order term
$n=0$ in (5) is just the identity operator in the expansion $S=1+iT$. The
first-order term gives the same result as the first Born approximation:

$S^{(1)}_{\mathbf{p}_{1}\mathbf{p}_{2}\to\mathbf{p}^{\prime}_{1}\mathbf{p}^{\prime}_{2}}=\frac{2\pi
i}{(2\pi)^{3}}\delta(E_{\mathbf{p}_{1}}+E_{\mathbf{p}_{2}}-E_{\mathbf{p}^{\prime}_{1}}-E_{\mathbf{p}^{\prime}_{2}})\delta^{3}(\mathbf{p}_{1}+\mathbf{p}_{2}-\mathbf{p}^{\prime}_{1}-\mathbf{p}^{\prime}_{2})\frac{4\pi
G_{N}m^{2}}{\Delta\mathbf{p}^{2}+\mu^{2}},$ (8)

with $\Delta\mathbf{p}=\mathbf{p}_{1}^{\prime}-\mathbf{p}_{1}$ the momentum
transfer.

Now we impose the requirement that (8) is the non-relativistic limit of some
Lorentz-invariant model. Thus we need to write this as Lorentz-invariant
function of the external momenta
$\mathbf{p}_{1},\mathbf{p}_{2},\mathbf{p}^{\prime}_{1},\mathbf{p}^{\prime}_{2}$.
Specifically, we want a Feynman amplitude $M$ with the property that
$S^{(1)}=\lim_{p^{0}\to m}-i(2\pi)^{4}M\delta^{4}(\sum p)$. There are 10
Lorentz generators and 12 independent variables in these four momenta, leaving
only two free variables. A convenient choice for these are the invariant
momentum transfer $t=-(p_{1}^{\prime}-p_{1})^{2}$ and total incoming mass-
squared $s=-(p_{1}+p_{2})^{2}$. Clearly the only Lorentz-invariant option is
to replace the momentum transfer $\Delta\mathbf{p}^{2}$ with $t$, and combine
the two delta functions into a factor
$\delta^{4}(p_{1}+p_{2}-p_{1}^{\prime}-p_{2}^{\prime})$. Comparing to (3), we
have the bootstrapped amplitude

$M_{\mathbf{p}_{1}\mathbf{p}_{2}\to\mathbf{p}^{\prime}_{1}\mathbf{p}^{\prime}_{2}}=-\frac{16\pi
G_{N}m^{4}}{-t+\mu^{2}}.$ (9)

We had to rescale the coupling $G_{N}m^{2}\to G_{N}m^{4}$ in order to cancel
the relativistic wavefunction normalizations $\sim 1/\sqrt{p^{0}}\to
1/\sqrt{m}$ in (3). The two expressions for the amplitude are depicted in Fig.
2. We have assumed that the two massive particles are distinguishable so that
we can ignore the exchange channel
$\mathbf{p}^{\prime}_{1}\leftrightarrow\mathbf{p}^{\prime}_{2}$, and will
continue to do this in what follows.

Unitarity is trivially satisfied in (9). The amplitude is manifestly real, so
the left-hand side of (4) vanishes. Similarly, there is no amplitude at order
$\sqrt{G_{N}}$, so to order $G_{N}$ the right hand side also vanishes. Notice
that for physical momenta $p_{1}^{2}=p_{2}^{2}=p_{1}^{\prime 2}=p_{2}^{\prime
2}=-m^{2}$, we have $t\leq 0$, so the denominator is always non-zero.

To see how the unitarity condition can become non-trivial, we have to go to
higher order in perturbation theory. We next show an example of how this
works. We consider a $3\to 3$ process which includes two non-gravitational
interactions with coupling strength $\lambda$. This is modeled directly after
the kind of Alice and Bob experiments of [26]. In this example, the unitarity
violation shows up at order $G_{N}\lambda^{2}$. To emphasize the generality of
the core idea, we also show a purely gravitational example in appendix C.
There we consider a $2\to 2$ process with an incoming particle-antiparticle
pair, in which unitarity violations arise at order $G_{N}^{2}$.

###  2.2 Tree-level unitarity

Consider a process where two experimentalists Alice and Bob each have a
massive particle. At some early time Bob interacts with his particle, say by
hitting it with a photon. The two particles proceed to scatter via the Newton
interaction. Long after this scattering event, Alice then performs a
measurement of her particle, say again by hitting it with a photon. Let us
model the photon interactions with an interaction strength $\lambda$
(proportional to the charge of Alice and Bob’s particles). Following the logic
of the Lorentzian bootstrap given above, one obtains an amplitude

$M_{\mathbf{k}\mathbf{p}_{1}\mathbf{p}_{2}\to\mathbf{k}^{\prime}\mathbf{p}_{1}^{\prime}\mathbf{p}_{2}^{\prime}}=\left(\frac{\lambda}{(p_{1}+k)^{2}+m^{2}-i\epsilon}\right)\left(\frac{G_{N}m^{4}}{\tilde{k}^{2}+\mu^{2}-i\epsilon}\right)\left(\frac{\lambda}{(p_{2}^{\prime}+k^{\prime})^{2}+m^{2}-i\epsilon}\right).$
(10)

This is depicted diagrammatically in Fig. 3. The photon momenta are
$\mathbf{k},\mathbf{k}^{\prime}$ respectively. The four-momentum transfer
between Alice and Bob’s massive particles is now
$\tilde{k}=p_{1}^{\prime}-(p_{1}+k)$. See appendix B for some details of this
calculation.

$V_{N}(t)$$\mathbf{p}^{\prime}_{1}$$\mathbf{p}^{\prime}_{2}$$\mathbf{p}_{1}$$\mathbf{p}_{2}$$\mathbf{k}$$\mathbf{k}^{\prime}$(a)$\mathbf{p}^{\prime}_{1}$$\mathbf{p}^{\prime}_{2}$$\mathbf{p}_{1}$$\mathbf{p}_{2}$$\mathbf{k}$$\mathbf{k}^{\prime}$$\tilde{k}$$p_{1}+k$$p_{2}^{\prime}+k^{\prime}$(b)
Figure 3: Scattering amplitude for the $3\to 3$ process with external photons,
in the same notation as Fig. 2. (a) is the non-relativistic amplitude and (b)
is its relativistic extension.

Crucially, it is now possible that $\tilde{k}^{2}=-\mu^{2}$. This happens when
the photon momentum $|\mathbf{k}|\gtrsim\mu$. This is why we have inserted the
$i\epsilon$ factors in the denominators: the poles are now accessible to the
experimentalist. When we use the $S$-matrix to describe an actual scattering
experiment, it needs to be integrated against some momentum-space wavepackets
describing the incoming and outgoing particles. To make these integrals well-
defined, we have to say what happens on the poles. The specific prescription
used here (the “Feynman prescription”) is the only possibility consistent with
our unitarity and Lorentz-invariance assumptions, as explained in appendix B.

At points in phase space where $\tilde{k}^{2}=-\mu^{2}$, the pole in the
middle term has an imaginary residue, because

$\lim_{\epsilon\to 0}\frac{1}{x-i\epsilon}=\frac{1}{x}-i\pi\delta(x)$ (11)

in the sense of distributions (for $x$ real). Therefore, unitarity is no
longer trivial: the left-hand side of the optical theorem (4) is now non-zero.
Specifically, it has a pole singularity at $\tilde{k}^{2}=-\mu^{2}$ with
residue of order $\lambda^{2}G_{N}$. Therefore, if unitarity were to hold, we
would need to have an amplitude $M\sim\mathcal{O}(\lambda\sqrt{G_{N}})$ to use
in the right-hand side of (4). But if the full set of scattering states is
only massive particles and photons, it is easy to see that there simply is no
such amplitude!444One could distribute the coupling factors differently
between the two amplitudes on the right-hand side of (4). In fact there are
some disconnected diagrams like this with the correct scalings of the
couplings, but they have the wrong detailed momentum dependence. Therefore the
optical theorem fails and we have a violation of unitarity.

What happened? The unitarity violation means that an incoming 3-body state
$\ket{\psi}=\ket{\mathbf{k}\mathbf{p}_{1}\mathbf{p}_{2}}$ will evolve to some
state $\ket{\psi^{\prime}}=S\ket{\psi}$ with a deficit in its norm
$|\braket{\psi^{\prime}}{\psi^{\prime}}|<1$. Inspecting the form of (4), and
remembering that the dynamics are fixed by assumption of (2), we see that
there is only one possible resolution: modify the sum over final states
$\ket{X}$. In more detail, notice that near the pole $\tilde{k}^{2}=-\mu^{2}$,
the imaginary part of the amplitude behaves like

${\rm
Im}~{}M_{\mathbf{k}\mathbf{p}_{1}\mathbf{p}_{2}\to\mathbf{k}^{\prime}\mathbf{p}_{1}^{\prime}\mathbf{p}_{2}^{\prime}}\to
G_{N}m^{4}\left(\frac{\lambda}{(p_{1}+k)^{2}+m^{2}}\right)\delta(\tilde{k}^{2}+\mu^{2})\left(\frac{\lambda}{(p_{2}^{\prime}+k^{\prime})^{2}+m^{2}}\right).$
(12)

But this is essentially just the product of two amplitudes with final state
$\ket{X}=\ket{\tilde{\mathbf{k}}\mathbf{p}_{1}^{\prime}\mathbf{p}_{2}}$, where
the $\tilde{\mathbf{k}}$ represents a radiated particle of mass $\mu$ coupled
with strength $\sqrt{G_{N}}m$. See Fig. 4. To be precise, if we include such
final states, the optical theorem (4) will be satisfied

$i(M_{\mathbf{k}\mathbf{p}_{1}\mathbf{p}_{2}\to\mathbf{k}^{\prime}\mathbf{p}_{1}^{\prime}\mathbf{p}_{2}^{\prime}}-M^{*}_{\mathbf{k}^{\prime}\mathbf{p}_{1}^{\prime}\mathbf{p}_{2}^{\prime}\to\mathbf{k}\mathbf{p}_{1}\mathbf{p}_{2}})\sim
M_{\mathbf{k}\mathbf{p}_{1}\mathbf{p}_{2}\to\tilde{k}\mathbf{p}_{1}^{\prime}\mathbf{p}_{2}}M^{*}_{\mathbf{k}^{\prime}\mathbf{p}_{1}^{\prime}\mathbf{p}_{2}^{\prime}\to\tilde{k}\mathbf{p}_{1}^{\prime}\mathbf{p}_{2}},$
(13)

where the sum on final states $\ket{X}$ has collapsed into a single outgoing
state which includes this gravitationally-coupled radiation. (The $\sim$
represents some awkward factors arising from the disconnected propagators; see
appendix B for the detailed equality). Thus we conclude that we need to
include such states in the Hilbert space of scattering states. In the limit
$\mu\to 0$ this radiated particle is essentially a graviton in terms of its
masslessness and $\sqrt{G_{N}}$ coupling to matter. It must be a boson,
otherwise the diagrams on the right-hand side would violate angular momentum
conservation. However, nothing in this argument is sensitive to which integer
spin this boson carries.

Im()$\mathbf{p}^{\prime}_{1}$$\mathbf{p}^{\prime}_{2}$$\mathbf{p}_{1}$$\mathbf{p}_{2}$$\mathbf{k}$$\mathbf{k}^{\prime}$$\tilde{k}$$\approx$$\tilde{k}^{2}\to-\mu^{2}$()$\mathbf{p}^{\prime}_{1}$$\mathbf{p}_{2}$$\mathbf{p}_{1}$$\mathbf{p}_{2}$$\mathbf{k}$$\tilde{\mathbf{k}}$$\times$()$*$$\mathbf{p}_{2}$$\mathbf{p}_{1}^{\prime}$$\mathbf{p}_{2}^{\prime}$$\mathbf{p}_{1}^{\prime}$$\mathbf{k}^{\prime}$$\tilde{\mathbf{k}}$
Figure 4: Unitarity in the tree-level, six-point amplitude. The pole at
$\tilde{k}^{2}=-\mu^{2}$ has an imaginary residue. The form of this residue is
exactly equal to the product of a pair of amplitudes where the “graviton” line
(dashed) is emitted into the final state. The disconnected lines represent
trivial propagators.

The fact that we get a unitarity violation precisely when
$\tilde{k}^{2}=-\mu^{2}$ has a simple physical interpretation. In the language
of field theory, $\tilde{k}^{2}=-\mu^{2}$ occurs when the momentum transfer
$\tilde{k}=p_{1}^{\prime}-(p_{1}+k)$ is tuned so that the “virtual graviton”
mediating the interaction satisfies its relativistic dispersion relation,
i.e., “goes on shell”. This is why we need the external photon to see the
effect: if $k=0$ then $\tilde{k}^{2}\geq 0$, but including the photon allows
us to reach the pole at $\tilde{k}^{2}=-\mu^{2}$.

This example bears some important similarities to the Alice and Bob
gedankenexperiment of [26]. There, causality (or rather faster-than-light
signaling) was used as a primary diagnosis of the issues arising if one
neglects to include final-state graviton radiation. Here we have instead
focused on a scattering calculation, in which boundary conditions are imposed
on both the past and future, which obscures the causal properties of the
process. We found a unitarity violation instead of a causality violation. This
reflects the fact that unitarity and causality are intimately linked in a
relativistic model. Very similar considerations have long been discussed in
the context of a classic paradox of Fermi [58], who incorrectly argued that
perturbation theory predicted superluminal communications. As is now well-
known, the solution is that Fermi forgot to include final-state radiation (see
[59] for a review).

It is interesting to note that the arguments given here for the necessity of a
graviton-like particle depend strongly on the dimensionality of space-time.
Consider general relativity in $d=2+1$ dimensions, defined as usual by the
action

$S=\int d^{3}x\sqrt{-g}\left[\frac{R}{16\pi G_{N}}+\mathcal{L}_{\rm
matter}\right].$ (14)

This model is “topological” in the sense that is has no propagating
gravitational waves [60]. However, particles can pick up braiding phases while
scattering [61, 62], and thus can become entangled. The discrepancy with the
argument above is that the Newton potential, or rather its logarithmic cousin
in two spatial dimensions, is not the non-relativistic limit $d=2+1$ Einstein
gravity [60]. This exemplifies the fact that entanglement generation alone
does not require a mediator—the key is the local form of the non-relativistic
potential.

##  3 Implications and interpretation

The bottom-up argument given above says that Newtonian entanglement can only
be explained within a unitary, Lorentz-invariant model if that model includes
radiative graviton-like degrees of freedom. Here I emphasize the fact that
this does not uniquely pick out the spin-2 graviton by constructing some top-
down counterexamples. A refined, non-Newtonian experiment is then outlined
which could distinguish the spin of the gravitational mediator.

###  3.1 Models which can explain the observation of entanglement

We can certainly show that the graviton reproduces the necessary entangling
operation on the masses. To see this, assume the metric is perturbatively
expanded $g_{\mu\nu}=\eta_{\mu\nu}+\sqrt{32\pi G_{N}}h_{\mu\nu}$ where
$\eta_{\mu\nu}$ is flat spacetime. We have scaled out a factor of the Planck
mass $m_{\rm pl}\sim 1/\sqrt{G_{N}}$ to give $h$ dimensions of mass. The
graviton $h_{\mu\nu}$ couples to matter in the usual way

$\mathcal{L}_{\rm int}=\sqrt{8\pi
G_{N}}h^{\mu\nu}T_{\mu\nu}+\mathcal{O}(h^{2})$ (15)

with $T$ the stress-energy tensor. We will not need the terms quadratic and
higher order in the gravitons. The Feynman rules for calculating amplitudes in
this model are given in appendix B. One finds the lowest-order $S$-matrix
element [31, 32, 33]

$M_{\mathbf{p}_{1}^{\prime}\mathbf{p}_{2}^{\prime},\mathbf{p}_{1}\mathbf{p}_{2}}=\vbox{\hbox{
\leavevmode\hbox to128.83pt{\vbox
to87.67pt{\pgfpicture\makeatletter\hbox{\hskip
35.96355pt\lower-43.44666pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{17.84377pt}{-13.33925pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$p_{1}^{\prime}-p_{1}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}
{}{}{}\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@lineto{85.35828pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{81.1805pt}{35.03033pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\mathbf{p}_{2}^{\prime}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}
{}{}{}\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@lineto{85.35828pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{81.1805pt}{-38.16922pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\mathbf{p}_{2}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}=4\pi
G_{N}\frac{N_{2}}{(p_{1}^{\prime}-p_{1})^{2}-i\epsilon}$ (16)

where the numerator

$N_{2}=4\left[(p_{1}\cdot p_{2}^{\prime})(p_{1}^{\prime}\cdot
p_{2})-m^{2}(p_{1}\cdot p_{1}^{\prime})-m^{2}(p_{2}\cdot
p_{2}^{\prime})-2m^{4}\right]\to 4m^{4}$ (17)

in the non-relativistic limit $p^{0}\to m,\mathbf{p}\to 0$ (so
$p^{2}\to-m^{2}$). This recovers the Newtonian result (9), in the $\mu\to 0$
limit. In other words, virtual graviton exchange leads to the Newtonian
potential operator.

It is tempting to conclude from the above that observing Newtonian
entanglement experimentally would imply that the graviton exists. But this is
not correct. Nothing in any of the proposed experiments to date can detect the
spin of the boson.

To make this clear, let us study a simple model of scalar gravity [63].
Variants on this model go back to the days before general relativity [64].
Consider a scalar field $\phi$ coupled to matter through the trace of the
stress tensor:

$\mathcal{L}_{\rm int}=\sqrt{8\pi G_{N}}\phi T_{\mu}^{\mu}.$ (18)

This coupling is Lorentz invariant and has the same mass dimensions as the
usual graviton coupling. One way to obtain this would be to write the usual
metric interaction but constrain the metric to be of the form
$g_{\mu\nu}=\phi\eta_{\mu\nu}$ (“conformally flat”). This model is equivalent
to Einstein gravity in the non-relativistic limit. To see this, consider the
same $2\to 2$ scattering matrix element:

$M_{\mathbf{p}_{1}^{\prime}\mathbf{p}_{2}^{\prime},\mathbf{p}_{1}\mathbf{p}_{2}}=\vbox{\hbox{
\leavevmode\hbox to128.31pt{\vbox
to87.67pt{\pgfpicture\makeatletter\hbox{\hskip
35.70139pt\lower-43.44666pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{}{}\pgfsys@moveto{-28.45276pt}{-28.45276pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p_{1}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}{}\pgfsys@moveto{-28.45276pt}{28.45276pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p_{1}^{\prime}$}}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{56.90552pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{17.84377pt}{-13.33925pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$p_{1}^{\prime}-p_{1}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}
{}{}{}\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@lineto{85.35828pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{81.44266pt}{35.03033pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p_{2}^{\prime}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}{}\pgfsys@moveto{56.90552pt}{0.0pt}\pgfsys@lineto{85.35828pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{81.44266pt}{-38.16922pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p_{2}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}=4\pi
G_{N}\frac{N_{0}}{(p_{1}^{\prime}-p_{1})^{2}-i\epsilon}.$ (19)

The difference is the numerator,

$N_{0}=4(p_{1}\cdot p_{1}^{\prime}+2m^{2})(p_{2}\cdot
p_{2}^{\prime}+2m^{2})\to 4m^{4}$ (20)

which is in exact agreement with the spin-2 model in the non-relativistic
limit. In other words, this spin-0 model produces the exact same effective
Newtonian potential as the spin-2 model.

While this spin-0 mediator is a simple example, it is not the only
possibility. In fact, one could construct a model like this using any integer
spin. Even fermions are possible, in some sense. The force mediator could be
condensed fermion pairs (like supercurrent fluctuations in a BCS
superconductor), if one could think of a palatable way to violate the
assumptions of the Weinberg-Witten theorem [65, 66].

###  3.2 Measuring the mediator spin

This brings up the question: what would be needed to further pin down the
spin? Of course, one could try to appeal to various classical observations,
like the tensorial nature of gravitational waves detected by LIGO. It may be
hard to imagine that the world has spin-2 classical radiation at astrophysical
wavelengths but spin-0 radiation at tabletop scales. However, precisely this
kind of situation could arise in a modified gravity scenario [67, 68] or a
variety of dark matter models [69]. More to the point, we are discussing
experimental tests of quantum gravity, so one should be careful.

It is not too difficult, at least in principle, to extend these tabletop
experiments to be spin-sensitive. Consider for example generating entanglement
through the bending of light, as in Fig. 5. In general relativity, the
deflection angle of the light depends on both the $g_{00}$ and $g_{ij}$
components of the metric. In a scalar gravity model, one would typically get
an incorrect deflection angle (e.g., the famous “factor of 2” difference with
light bending in the $m\to 0$ limit of Newtonian gravity).555In the specific
dilaton-type model considered here where $g_{\mu\nu}=\phi\eta_{\mu\nu}$, light
actually doesn’t bend at all, because it is coupled to gravity through
$\mathcal{L}=\sqrt{-{\rm
det}~{}g}g_{\alpha\beta}g_{\gamma\delta}F^{\alpha\gamma}F^{\beta\delta}=\eta_{\alpha\beta}\eta_{\gamma\delta}F^{\alpha\gamma}F^{\beta\delta}$.
In other scalar gravity examples, for example Brans-Dicke gravity [70], the
answer may not be so simple but will still generically differ from general
relativity. More generally, different tensor structures in the interaction
will give rise to different dependence on the momenta in the numerators of
these scattering amplitudes, as seen in comparing $N_{2}$ with $N_{0}$.

$\ket{L}$$\ket{R}$$\ket{0}$$\ket{\ell}$$\ket{r}$ Figure 5: Entanglement from
the bending of light around a superposed mass. A heavy mass (shaded) is
prepared in a spatial superposition $\ket{L}+\ket{R}$, and a light beam
prepared in some initial wavepacket $\ket{0}$ is scattered. This produces an
entangled state through evolution of the form
$(\ket{L}+\ket{R})\otimes\ket{0}\to\ket{L\ell}+\ket{Rr}$.

Let us estimate the size of the entanglement in an experiment like that
depicted in Fig. 5. The deflection angle of the light is classically given by
$\theta=G_{N}M/c^{2}b$ where $b$ is the impact parameter and $M$ is the heavy
source mass. Thus the differential bending along the two paths shown in Fig. 5
is of order

$\Delta\theta=\frac{G_{N}M}{c^{2}b}\frac{\Delta b}{b}\approx 7.4\times
10^{-27}\times\left(\frac{M}{1~{}{\rm g}}\right)\left(\frac{\Delta
b}{10~{}{\rm\mu m}}\right)\left(\frac{100~{}{\rm\mu m}}{b}\right)^{2}$ (21)

where $\Delta b$ is the distance over which the source mass $M$ is superposed.
This would be impossible to detect. One could improve the situation by having
the light reflect back and forth within a cavity $N$ times. For a cavity of
length $L$, the differential movement of the two forward peaks during one
crossing would be $\Delta\ell=L\Delta\theta$. To get the two peaks separated
by of order one laser wavelength $\lambda$ then requires
$N\Delta\ell=NL\Delta\theta=\lambda$, meaning $N=\lambda/L\Delta\theta$, or a
total integration time

$T=N\frac{L}{c}=\frac{b^{2}\lambda c}{G_{N}M\Delta b}\approx 10^{16}~{}{\rm
s}$ (22)

with the same parameters given above. This can be improved by a factor
$\sqrt{n_{\gamma}}$ with $n_{\gamma}$ photons in the cavity. Amusingly,
allowing for a one second integration time, this works out to the requirement
of $\sim 10^{32}$ laser photons, each of energy $(1000~{}{\rm
nm})^{-1}=0.2~{}{\rm eV}$. In terms of mass, this is $m=10^{32}\times
0.2~{}{\rm eV}\approx 10^{4}~{}m_{\rm pl}$.

Beyond the many practical difficulties [71], the decay of this laser field
into $e^{+}e^{-}$ pairs would render this fundamentally impossible [72, 73].
However, the fact that the required effective mass works out to be in the
ballpark of $m_{\rm pl}\approx 0.2~{}{\rm\mu g}$ motivates some analogue of
this experiment with two composite masses, much like the Newtonian experiments
currently in development. One would need to engineer the coupling in a way
sensitive to the tensor structure of the $h_{\mu\nu}T^{\mu\nu}$ interaction.

##  4 Conclusions

Experiments in the reasonably near future will be able to test whether or not
gravity can entangle pairs of non-relativistic masses. Observation of this
Newtonian entanglement would provide compelling but incomplete evidence for
the existence of the graviton. These experiments would definitely rule out
models where gravity emerges in something like a semi-classical sense. Such a
model is a logical possibility consistent with all classical gravitational
phenomena observed to date.

However, to make a definitive statement about the graviton itself, two key
loopholes need to be closed. The simplest is that the Newtonian experiments
are insensitive to the spin of the graviton. A more sophisticated experiment,
like the one described in section 3.2, could resolve this.

The more difficult issue is that the specific unitarity assumptions used in
this paper could be too strong. It is important to understand that we have
imposed unitarity and Lorentz invariance as conditions on arbitrary scattering
processes. It may be that an emergent gravitational interaction can still
generate the appropriate entanglement signatures these experiments seek
without admitting a unitary, Hamiltonian description like (1). Construction of
or no-go statements about such a model would be extremely valuable. But even
if a graviton were not required in such a model, observation of Newtonian
entanglement would teach us something remarkable about the detailed way in
which gravity emerges: it would have to be able to coherently communicate
quantum information between spatially disjoint systems.

## Acknowledgements

I thank Daniele Alves, Niklas Beisert, Nikita Blinov, Sougato Bose, Colby
Delisle, Anson Hook, Matthew McCollough, Gordon Semenoff, David Simmons-
Duffin, Raman Sundrum, Jacob Taylor, Oleksandr Tomalek, Jessica Turner, Mark
van Raamsdonk, Junwu Wang, and Jordan Wilson-Gerow for discussions. I am
especially grateful to Simon Caron-Huot for the suggestion to consider the
six-point amplitudes of section 2.2, and pointing me to reference [59]. I also
gratefully acknowledge hospitality at the Cornell and University of Maryland
physics departments while this work was being completed. My work at LBNL is
supported by the US Department of Energy under contract DE-AC02-05CH11231 and
by the Quantum Information Science Enabled Discovery (QuantISED) for High
Energy Physics grant KA2401032.

##  Appendix A Classical and semiclassical gravity models

In this appendix we briefly review the idea that gravity could be “classical”.
The quotes represent the fact that this could mean a variety of things in
detail. There is a large literature on these ideas, see for example [7, 74]
for much more extensive discussion. Here I want to just highlight why it is
possible to formulate some kind of classical gravity coupled to quantum matter
in a way that can avoid the classic no-go statements of Weinberg and
Polchinski [75, 76].

The naive starting point for a classical gravity model would be to source the
Einstein equation with the expectation value of the stress tensor
$G_{\mu\nu}=8\pi G_{N}\braket{T_{\mu\nu}}$. This does not specify the full
dynamics because we need an equation of motion for the matter state
$\ket{\psi}$. Closing the system with a simple Schrödinger equation leads to
fundamental problems. The Newtonian limit is sufficient to see the basic
issue. Imagine trying to define a model like666In this section, and only this
section, operators are written with hats for clarity.

$i\partial_{t}\ket{\psi}=\left[\hat{H}_{\rm
matter}+\sum_{i}V_{i}(\hat{\mathbf{x}}_{i})\right]\ket{\psi}$ (23)

with the indices $i,j=1,\ldots,N$ running over some set of $N$ particles,
$\hat{H}_{\rm matter}$ describing any non-gravitational evolution, and a
“semiclassical” gravitational potential

$V_{i}(\hat{\mathbf{x}}_{i})=-\sum_{j\neq
i}\frac{G_{N}m}{|\hat{\mathbf{x}}_{i}-\braket{\hat{\mathbf{x}}_{j}}|}.$ (24)

In a limit where all the particle wavefunctions are strongly peaked on
classical trajectories, we can approximate
$\hat{\mathbf{x}}_{i}\approx\mathbf{x}_{i}$ as $c$-numbers, and the system
(23) exactly reproduces classical Newton gravity. However, the interaction in
general is a direct sum, and thus cannot generate entanglement. In this sense,
the model is “classical”.

The trouble is that this is a non-linear modification of the Schrödinger
dynamics, since $V$ depends quadratically on $\ket{\psi}$. It was noted long
ago that generic such modifications lead to fundamental problems when applied
to states where the classical approximation fails. For example, one can use
Schrödinger-cat type states to superluminally signal [76, 77, 78].

$S_{1}:\ \ \ket{\psi_{1}}$$A_{1}:\ \ \ket{\phi}$$A_{2}:\ \ \ket{\phi}$$S_{2}:\
\ \ket{\psi_{2}}$$U_{ent}$$U_{ent}$$U_{FB}$$U_{FB}$ Figure 6: Adversarial
measurement-and-feedback gravity, following [42]. The figure is taken from
[7].

However, we can construct a model designed to evade these difficulties [42].
The basic strategy is to realize the non-linearity in the Schrödinger equation
as arising from a normal quantum mechanical process. We might call this
approach “adversarial gravity”. We imagine that the universe is a quantum
simulation designed specifically to make it look like a classical
gravitational force exists. See Fig. 6 for a circuit diagram of one timestep
of the simulation, with two masses for simplicity.

The computer begins by using some ancillae degrees of freedom to weakly
measure the positions of the masses, allowing it to make estimates for
$\braket{\hat{\mathbf{x}}_{i}}$ with $i=1,2$. The computer uses this
information to compute a semiclassical Newton potential (24). This information
is then used to evolve the state of the masses through a feedback unitary

$U_{\rm fb}=\exp\left\\{-i\sum_{i}V_{i}(\hat{\mathbf{x}}_{i})dt\right\\}.$
(25)

It is clear that this unitary is a product $U=U_{1}\otimes U_{2}$, so it
cannot generate any entanglement between the two masses. This model produces a
Schrödinger-like evolution for the matter state of the form

$d\ket{\psi}=-iH_{\rm
mat}dt\ket{\psi}-i\sum_{i}V_{i}(\hat{\mathbf{x}}_{i})dt\ket{\psi}+\sum_{i}\sqrt{\gamma}\xi_{i}dW\ket{\psi}+\sum_{i,j}\xi_{i}\xi_{j}dt\ket{\psi}.$
(26)

This is like (23), except with the addition of the two final terms. These
represent noise due to the weak measurement step. The differential $dW$ is a
classical stochastic process satisfying the usual Itô rule $dW^{2}=dt$, and
$\gamma,\xi$ represent the strength of the noise. We have replaced the
deterministic equations (23), (24) with a stochastic evolution law. There are
still non-linear terms in (26), but their origin comes from quantum
measurements via ancilla degrees of freedom which are traced out [79]. Such
non-linear terms are hardly pathological; they arise for example in any
experiment which uses a measurement-and-feedback protocol. The description
here was non-relativistic, but similar relativistic models of non-linear
quantum mechanics also exist [80, 81, 82].

This model is a kind of emergent gravity scenario. In this example the
emergent gravitational force cannot entangle objects. It is not at all clear
that this is a general property of emergent interactions. Indeed, one can
construct models of entropic forces which produce quantum coherent
interactions [83]. Moreover, in AdS/CFT, the bulk graviton exists by
construction, at least in the traditional sense of AdS/CFT as a string duality
[84], since the closed string has a massless spin-2 excitation. This suggests
an interesting question as to whether one can realize emergent gravity in the
sense of Jacobson [38] or Verlinde [40] in such a way that gravity can
communicate quantum information but remains a fundamentally open system.

##  Appendix B Details of amplitude calculations

### External probe interaction

The $3\to 3$ process of section 2.2 used an external probe system (a “photon”)
to model the effect the experimentalists interacting with the massive,
gravitationally-coupled objects. Before moving to the detailed calculation of
the $3\to 3$ amplitude we thus need to provide some details about this
interaction. This will also provide an excellent warmup to the $3\to 3$
amplitude itself.

In our non-relativistic language, we will model the external probe as a
massless scalar (i.e., a photon in the limit that we can ignore the
polarizations). In abuse of language I will refer to these probe particles as
photons throughout. The coupling is, in the interaction picture,

$V_{e}(t)=\lambda\int
d^{3}\mathbf{k}d^{3}\mathbf{p}d^{3}\mathbf{p}^{\prime}\delta^{3}(\mathbf{k}+\mathbf{p}-\mathbf{p}^{\prime})\left[W_{\mathbf{k}\mathbf{p},\mathbf{p}^{\prime}}e^{i(E_{\mathbf{k}}+E_{\mathbf{p}}-E_{\mathbf{p}^{\prime}})t}\ket{\mathbf{k}\mathbf{p}}\bra{\mathbf{p}^{\prime}}+{\rm
h.c.}\right].$ (27)

The coupling $\lambda$ has mass dimension one and the matrix elements $W$ are
given by

$W_{\mathbf{k}\mathbf{p},\mathbf{p}^{\prime}}=\frac{1}{\sqrt{[2(2\pi)^{3}]^{3}E_{\mathbf{k}}E_{\mathbf{p}}E_{\mathbf{p}^{\prime}}}},$
(28)

thus carrying dimension ${\rm mass}^{-3/2}$. I will use $\mathbf{k}$ for the
probe momentum and $\mathbf{p},\mathbf{p}^{\prime}$ for the massive object.
The form of this matrix element can be derived for example by considering the
massive object as a scalar particle and quantizing the interaction
$V_{e}=\lambda\int d^{3}\mathbf{x}\chi^{2}\phi$ with $\phi$ the probe and
$\chi$ the massive object.

$\mathbf{p}$$\mathbf{k}$$\mathbf{p}^{\prime}$$\mathbf{k}^{\prime}$$\mathbf{p}$$\mathbf{k}$$\mathbf{p}^{\prime}$$\mathbf{k}^{\prime}$
Figure 7: Diagrams for the probe-system interactions, one for each term in
(30).

To get a feel for this interaction, consider a process with a single mass
which absorbs and emits a probe photon. This means we need to go to second
order in the Dyson series

$S^{(2)}=\frac{(-i)^{2}}{2!}\int_{-\infty}^{\infty}dt_{1}dt_{2}\braket{\beta}{\mathcal{T}V_{e}(t_{1})V_{e}(t_{2})}{\alpha},$
(29)

with initial state $\ket{\alpha}=\ket{\mathbf{k}\mathbf{p}}$ and final state
$\ket{\beta}=\ket{\mathbf{k}^{\prime}\mathbf{p}^{\prime}}$. There are two
contributions to this due to the time-ordering

$\mathcal{T}V_{e}(t_{1})V_{e}(t_{2})=V_{e}(t_{1})V_{e}(t_{2})\Theta(t_{1}-t_{2})+V_{e}(t_{2})V_{e}(t_{1})\Theta(t_{2}-t_{1}).$
(30)

Each of these terms will further contribute two terms, one for each ordering
of the emission and absorption [i.e. one each for either the $W$ or $W^{*}$
term in (27)]. Using the explicit matrix element (27), the two terms which
contribute to the first diagram in Fig. 7, representing absorption followed by
emission, are

$\displaystyle\begin{split}S_{A}&=-\frac{\lambda^{2}}{2}\int_{-\infty}^{\infty}dt_{1}dt_{2}d^{3}\mathbf{q}W_{\mathbf{k}^{\prime}\mathbf{p}^{\prime},\mathbf{q}}e^{i(E_{\mathbf{k}^{\prime}}+E_{\mathbf{p}^{\prime}}-E_{\mathbf{q}})t_{2}}\delta^{3}(\mathbf{k}^{\prime}+\mathbf{p}^{\prime}-\mathbf{q})\\\
&\times
W^{*}_{\mathbf{q},\mathbf{k}\mathbf{p}}e^{i(E_{\mathbf{q}}-E_{\mathbf{k}}-E_{\mathbf{p}})t_{1}}\delta^{3}(\mathbf{q}-\mathbf{k}-\mathbf{p})\Theta(t_{2}-t_{1})\\\
&+(t_{1}\leftrightarrow t_{2}),\end{split}$ (31)

where the last line just means a copy of the previous term but with the time
variables switched. To do the time integrals, we can re-write the step
function using the identity

$\Theta(t)e^{-iEt}=\frac{1}{2\pi
i}\int_{-\infty}^{\infty}d\omega\frac{e^{i\omega t}}{\omega-E+i\epsilon}$ (32)

which can be easily checked by contour integration. Here and throughout, the
limit $\epsilon\to 0$ is taken at the end of all computations. Using this on
the phase $e^{-iE_{\mathbf{q}}(t_{2}-t_{1})}$ and evaluating the time
integrals we obtain

$\displaystyle\begin{split}S_{A}&=-\frac{2(2\pi)^{2}\lambda^{2}}{4\pi i}\int
d\omega
d^{3}\mathbf{q}W_{\mathbf{k}^{\prime}\mathbf{p}^{\prime},\mathbf{q}}\delta(E_{\mathbf{k}^{\prime}}+E_{\mathbf{p}^{\prime}}-\omega)\delta^{3}(\mathbf{k}^{\prime}+\mathbf{p}^{\prime}-\mathbf{q})\\\
&\times\frac{1}{\omega-
E_{\mathbf{q}}+i\epsilon}W^{*}_{\mathbf{q},\mathbf{k}\mathbf{p}}\delta(\omega-
E_{\mathbf{k}}-E_{\mathbf{p}})\delta^{3}(\mathbf{q}-\mathbf{k}-\mathbf{p}).\end{split}$
(33)

The overall factor of 2 appears because the $(t_{1}\leftrightarrow t_{2})$
term contributes identically. The physical interpretation of the $i\epsilon$
here can be understood from (32). This says that the intermediate positive-
energy state of energy $E_{\mathbf{q}}$ has energy flowing forward in time.

Now we would like to apply the Lorentzian bootstrap idea to this expression.
In other words, we want this to be a Lorentz-invariant function times the
overall wavefunction re-scaling $\sim
1/\sqrt{E_{\mathbf{p}}E_{\mathbf{k}}E_{\mathbf{p}^{\prime}}E_{\mathbf{k}^{\prime}}}$
as in (9). This re-scaling is already taken care of in the matrix elements
(28), which also produce an additional factor $1/E_{\mathbf{q}}$. Identifying
$\omega=q^{0}$, the delta-functions can be combined in the obvious way to
produce a pair of four-dimensional delta functions.

The tricky part is the non-relativistic propagator $\sim 1/(\omega-
E_{\mathbf{q}}+i\epsilon)$. To get something Lorentz-invariant we will need to
make a quadratic function out of the denominator, and the only option is to
use $q^{2}$. The function $q^{2}+m^{2}$ has precisely the same pole
$\omega=q^{0}=+E_{\mathbf{q}}$ (once we move to the relativistic dispersion
relation $E_{\mathbf{q}}=\sqrt{m^{2}+\mathbf{q}^{2}}$). Specifically, we have

$\frac{1}{q^{2}+m^{2}}=\frac{-1}{2E_{\mathbf{q}}}\left[\frac{1}{q^{0}-E_{\mathbf{q}}}-\frac{1}{q^{0}+E_{\mathbf{q}}}\right].$
(34)

The remaining question is the prescription for handling integration of the two
poles $q^{0}=\pm E_{\mathbf{q}}$. In the non-relativistic propagator above, we
saw that the positive-energy pole was displaced $E_{\mathbf{q}}\to
E_{\mathbf{q}}-i\epsilon$, so the only issue is what to do with this new
negative-energy pole. We will see in the next section that the only answer
consistent with Lorentz invariance and the optical theorem is to displace this
pole $-E_{\mathbf{q}}\to-(E_{\mathbf{q}}-i\epsilon)$.777This is not quite
true. What is inconsistent would be to use the retarded or advanced
prescriptions, in which both poles are shifted above the real axis or both are
shifted below the real axis. One could, however, use the prescription
$q^{2}+m^{2}+i\epsilon$ instead of the usual Feynman $q^{2}+m^{2}-i\epsilon$.
This would correspond to an overall time-reversal, i.e., a model in which the
definition of positive and negative energies are reversed. One can make
interesting models in which both conventions are used [85, 86]. In what
follows we will use the standard Feynman prescription. We will thus identify

$\frac{1}{\omega-
E_{\mathbf{q}}+i\epsilon}\to-\frac{2E_{\mathbf{q}}}{q^{2}+m^{2}-i\epsilon}.$
(35)

This is not Lorentz-invariant, but the numerator cancels the extra factor of
$1/E_{\mathbf{q}}$ we picked up from the interaction vertices, leaving an
overall Lorentz-invariant expression.

With this identification, we can do the integral in (33) using one set of the
delta-functions, and obtain

$S_{A}=\frac{1}{\sqrt{[2(2\pi)^{3}]^{4}E_{\mathbf{k}}E_{\mathbf{p}}E_{\mathbf{k}^{\prime}}E_{\mathbf{p}^{\prime}}}}\frac{i\lambda^{2}}{(2\pi)^{2}}\frac{1}{(p+k)^{2}+m^{2}-i\epsilon}\delta^{4}(p+k-p^{\prime}-k^{\prime}).$
(36)

By the exact same logic, the other two terms (absorption followed by emission,
the second part of Fig. 7) produce a contribution $S_{B}$ identical to $S_{A}$
except with the denominator $(p+k)^{2}\to(p-k^{\prime})^{2}$. In terms of our
Feynman amplitudes, this means we would identify

$M_{\mathbf{k}\mathbf{p}\to\mathbf{k}^{\prime}\mathbf{p}^{\prime}}=\frac{\lambda^{2}}{(2\pi)^{3}}\left[\frac{1}{(p+k)^{2}+m^{2}-i\epsilon}+\frac{1}{(p-k^{\prime})^{2}+m^{2}-i\epsilon}\right].$
(37)

This argument contains all the same pieces we will need to evaluate the $3\to
3$ amplitude of section 2.2, to which we now turn.

### Six-point amplitude derivation

Consider the process involving two gravitationally coupled masses
$m_{1},m_{2}$ and some external “photons” depicted in Fig. 3. For simplicity
we will assume that there are two distinguishable probes,

$V_{i}(t)=\lambda\int
d^{3}\mathbf{k}d^{3}\mathbf{p}_{i}d^{3}\mathbf{p}_{i}^{\prime}\delta^{3}(\mathbf{k}+\mathbf{p}_{i}-\mathbf{p}_{i}^{\prime})\left[W_{\mathbf{k}\mathbf{p}_{i},\mathbf{p}_{i}^{\prime}}e^{i(E_{\mathbf{k}}+E_{\mathbf{p}_{i}}-E_{\mathbf{p}_{i}^{\prime}})t}\ket{\mathbf{k}\mathbf{p}_{i}}\bra{\mathbf{p}_{i}^{\prime}}+{\rm
h.c.}\right]\otimes 1_{j\neq i}$ (38)

which will simplify some combinatoric factors.

Starting from the Dyson series (5) we need to go to third order in the
couplings. More specifically we need to go to order $\lambda^{2}G_{N}$ and
consider the $S$-matrix contribution

$S^{(3)}=\int_{-\infty}^{\infty}dt_{1}dt_{2}dt_{N}\braket{\beta}{\mathcal{T}V_{1}(t_{1})V_{2}(t_{2})V_{N}(t_{N})}{\alpha}.$
(39)

This gives six independent terms from the time-ordering symbol

$\displaystyle\mathcal{T}V_{1}V_{2}V_{N}=V_{1}V_{N}V_{2}\Theta(t_{1}-t_{N})\Theta(t_{N}-t_{2})+V_{2}V_{N}V_{1}\Theta(t_{2}-t_{N})\Theta(t_{N}-t_{1})+\cdots,$
(40)

which can be represented by a sum of four Feynman diagrams, as in Fig. 8. We
will now proceed to compute one of these in detail in terms of our Lorentzian
bootstrap. After we are done it will be clear that the others just add
essentially identical factors. Alternatively, one could integrate against
localized-in-time wavepackets for the probes, which would suppress the
contributions from the other diagrams.

$V_{N}(t)$$\mathbf{p}^{\prime}_{1}$$\mathbf{p}^{\prime}_{2}$$\mathbf{p}_{1}$$\mathbf{p}_{2}$$\mathbf{k}$$\mathbf{k}^{\prime}$$+$$+$$+$
Figure 8: Contributions to the scattering amplitude for the $3\to 3$ process
with external photons.

Let’s study a particular term corresponding to the first diagram of Fig. 8,
namely the $S$-matrix contribution

$S_{1N2}=\int_{-\infty}^{\infty}dt_{1}dt_{2}dt_{N}\braket{\beta}{V_{2}(t_{2})V_{N}(t_{N})V_{1}(t_{1})}{\alpha}\Theta(t_{2}-t_{N})\Theta(t_{N}-t_{1}).$
(41)

Inserting the explicit forms of the interactions (6), (38) and using the
states $\ket{\alpha}=\ket{\mathbf{k}\mathbf{p}_{1}\mathbf{p}_{2}}$,
$\ket{\beta}=\ket{\mathbf{k}^{\prime}\mathbf{p}_{1}^{\prime}\mathbf{p}_{2}^{\prime}}$,
we have

$\displaystyle\begin{split}S_{1N2}&=\lambda^{2}\int_{-\infty}^{\infty}dt_{2}dt_{N}dt_{1}d^{3}\underline{\mathbf{p}}_{1}d^{3}\underline{\mathbf{p}}_{2}d^{3}\underline{\underline{\mathbf{p}}}_{1}d^{3}\underline{\underline{\mathbf{p}}}_{2}\Theta(t_{2}-t_{N})\Theta(t_{N}-t_{1})\\\
&\times
W_{\mathbf{k}^{\prime}\mathbf{p}_{2}^{\prime},\underline{\underline{\mathbf{p}}}_{2}}e^{i(E_{\mathbf{k}^{\prime}}+E_{\mathbf{p}_{2}^{\prime}}-E_{\underline{\underline{\mathbf{p}}}_{2}})t_{2}}\delta^{3}(\mathbf{k}^{\prime}+\mathbf{p}_{2}^{\prime}-\underline{\underline{\mathbf{p}}}_{2})\delta^{3}(\mathbf{p}_{1}^{\prime}-\underline{\underline{\mathbf{p}}}_{1})\\\
&\times
V_{\underline{\underline{\mathbf{p}}}_{1}\underline{\underline{\mathbf{p}}}_{2},\underline{\mathbf{p}}_{1}\underline{\mathbf{p}}_{2}}e^{i(E_{\underline{\underline{\mathbf{p}}}_{2}}+E_{\underline{\underline{\mathbf{p}}}_{1}}-E_{\underline{\mathbf{p}}_{2}}-E_{\underline{\mathbf{p}}_{1}})t_{N}}\delta^{3}(\underline{\underline{\mathbf{p}}}_{1}+\underline{\underline{\mathbf{p}}}_{2}-\underline{\mathbf{p}}_{1}-\underline{\mathbf{p}}_{2})\\\
&\times
W_{\underline{\mathbf{p}}_{1},\mathbf{k}\mathbf{p}_{1}}e^{i(E_{\underline{\mathbf{p}}_{1}}-E_{\mathbf{k}}-E_{\mathbf{p}_{1}})t_{1}}\delta^{3}(\underline{\mathbf{p}}_{1}-\mathbf{k}-\mathbf{p}_{1})\delta^{3}(\underline{\mathbf{p}}_{2}-\mathbf{p}_{2}).\end{split}$
(42)

Using the step function identity (32) in much the same way as before, the time
integrals may be evaluated in terms of some frequency integrals,

$\displaystyle\begin{split}&\int_{-\infty}^{\infty}dt_{2}dt_{N}dt_{1}\Theta(t_{2}-t_{N})\Theta(t_{N}-t_{1})\\\
&\times
e^{i(E_{\mathbf{k}^{\prime}}+E_{\mathbf{p}_{2}^{\prime}}-E_{\underline{\underline{\mathbf{p}}}_{2}})t_{2}}e^{i(E_{\underline{\underline{\mathbf{p}}}_{2}}+E_{\underline{\underline{\mathbf{p}}}_{1}}-E_{\underline{\mathbf{p}}_{2}}-E_{\underline{\mathbf{p}}_{1}})t_{N}}e^{i(E_{\underline{\mathbf{p}}_{1}}-E_{\mathbf{k}}-E_{\mathbf{p}_{1}})t_{1}}\\\
&=\int\frac{d\omega_{2}}{\omega_{2}-E_{\underline{\underline{p}}_{2}}+i\epsilon}\frac{d\omega_{1}}{\omega_{1}-E_{\underline{p}_{1}}+i\epsilon}\\\
&\times\delta(E_{\mathbf{k}^{\prime}}+E_{\mathbf{p}_{2}^{\prime}}-\omega_{2})\delta(E_{\underline{\underline{\mathbf{p}}}_{1}}+\omega_{2}-\omega_{1}-E_{\underline{\mathbf{p}}_{2}})\delta(\omega_{1}-E_{\mathbf{p}_{1}}-E_{\mathbf{k}}).\end{split}$
(43)

Performing the
$d^{3}\underline{\underline{\mathbf{p}}}_{1}d^{3}\underline{\mathbf{p}}_{2}$
momentum integrals with the simple delta functions then leaves free integrals
over
$d\omega_{1}d\omega_{2}d^{3}\underline{\mathbf{p}}_{1}d^{3}\underline{\underline{\mathbf{p}}}_{2}$.
Identifying
$\omega_{1}=\underline{p}_{1}^{0},\omega_{2}=\underline{\underline{p}}_{2}^{0}$
and forming four-vectors we then obtain

$\displaystyle\begin{split}S_{1N2}&=\lambda^{2}\int
d^{4}\underline{p}_{1}d^{4}\underline{\underline{p}}_{2}W_{\mathbf{k}^{\prime}\mathbf{p}_{2}^{\prime},\underline{\underline{\mathbf{p}}}_{2}}\delta^{4}(k^{\prime}+p_{2}^{\prime}-\underline{\underline{p}}_{2})\frac{1}{\omega_{2}-E_{\underline{\underline{p}}_{2}}+i\epsilon}V_{\mathbf{p}_{1}^{\prime}\underline{\underline{\mathbf{p}}}_{2},\underline{\mathbf{p}}_{1}\mathbf{p}_{2}}\\\
&\times\delta^{4}(p_{1}^{\prime}+\underline{\underline{p}}_{2}-\underline{p}_{1}-p_{2})\frac{1}{\omega_{1}-E_{\underline{\mathbf{p}}_{1}}+i\epsilon}W_{\underline{\mathbf{p}}_{1},\mathbf{k}\mathbf{p}_{1}}\delta^{4}(\underline{p}_{1}-k-p_{1}).\end{split}$
(44)

Again, everything in the above is simply a formal manipulation of the non-
relativistic amplitude.

Finally, we now make the same “bootstrap” identifications as in earlier
sections. The energy denominators are replaced according to (35), while the
Newton interaction matrix elements are replaced in line with the discussion of
section 2.1. We include the various $1/\sqrt{E}$ factors appropriately.
Performing the two remaining momentum integrals, we then identify the
resulting $S$-matrix contribution:

$\displaystyle\begin{split}S_{1N2}&\to\frac{\lambda^{2}G_{N}m^{4}}{\sqrt{[2(2\pi)^{3}]^{6}E_{\mathbf{k}}E_{\mathbf{p}_{1}}E_{\mathbf{p}_{2}}E_{\mathbf{k}^{\prime}}E_{\mathbf{p}_{1}^{\prime}}E_{\mathbf{p}_{2}^{\prime}}}}\delta^{4}(k^{\prime}+p_{1}^{\prime}+p_{2}^{\prime}-k-p_{1}-p_{2})\\\
&\times\left(\frac{1}{(p_{1}+k)^{2}+m^{2}-i\epsilon}\right)\left(\frac{1}{\tilde{k}^{2}+\mu^{2}-i\epsilon}\right)\left(\frac{1}{(p_{2}^{\prime}+k^{\prime})^{2}+m^{2}-i\epsilon}\right),\end{split}$
(45)

where $\tilde{k}=p_{1}^{\prime}-(p_{1}+k)$. Comparing to our $S$-matrix
conventions (3) then leads to the identification of the Feynman amplitude

$\displaystyle
M=\lambda^{2}G_{N}m^{4}\left(\frac{1}{(p_{1}+k)^{2}+m^{2}-i\epsilon}\right)\left(\frac{1}{\tilde{k}^{2}+\mu^{2}-i\epsilon}\right)\left(\frac{1}{(p_{2}^{\prime}+k^{\prime})^{2}+m^{2}-i\epsilon}\right),$
(46)

precisely as in (10).

At this stage, we can close the loop and explain the need for the specific
$i\epsilon$ pole prescription used here. In the previous section, I claimed
that this (or its $-i\epsilon$ time-reversed version) is the only prescription
consistent with Lorentz invariance and unitarity. To see this, consider what
would happen if we used instead a prescription like

$\frac{1}{q^{0}-E_{\mathbf{q}}-i\epsilon}-\frac{1}{q^{0}+E_{\mathbf{q}}-i\epsilon},$
(47)

in the notation of (34), in which both poles are shifted above the real
$q^{0}$ axis. When we compute the left-hand side of the optical theorem (4),
we would pick up a term proportional to the imaginary part of this factor,
which is $\delta(q^{0}-E_{\mathbf{q}})-\delta(q^{0}+E_{\mathbf{q}})$. This is
not Lorentz invariant. In contrast, the usual Feynman prescription produces an
imaginary part proportional to
$\delta(q^{0}-E_{\mathbf{q}})+\delta(q^{0}+E_{\mathbf{q}})\propto\delta(q^{2}+m^{2})$,
as used in the main text. Thus the retarded prescription (47) or the advanced
prescription where both poles are shifted below the real axis would be
incapable of satisfying the optical theorem (4), since the right-hand side is
manifestly Lorentz-invariant.

### Optical theorem for the six-point amplitude

Finally, we spell out the details of the new amplitudes on the right-hand side
of (13). Everything is almost identical to the $2\to 2$ process of the
previous subsection. The only change is the $\sqrt{G_{N}}m$ coupling, and an
additional factor for the disconnected part:

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{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.7}{0.0}{0.0}{0.7}{-1.84721pt}{-38.2809pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\mathbf{k}$}}
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{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.7}{0.0}{0.0}{0.7}{-1.74998pt}{33.86317pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\underline{\mathbf{k}}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}=\frac{\sqrt{G_{N}}m^{2}\lambda}{(p_{1}+k)^{2}+m^{2}-i\epsilon}\delta^{3}(\underline{\mathbf{p}}_{2}-\mathbf{p}_{2})\times
2E_{\mathbf{p}_{2}}(2\pi)^{3}.$ (48)

The final factor $2E_{\mathbf{p}_{2}}(2\pi)^{3}$ is needed to cancel out the
Lorentzian phase-space factors of equation (3) for the disconnected particle.
To apply this amplitude in the optical theorem (4) to reproduce the unitarity
condition (13), the non-trivial part of the sum over $\ket{X}$ (the sum over
final three-body states) can be written explicitly

$\sum_{\ket{X}}B_{X}^{2}\delta^{4}(p_{X}-p_{\alpha})=\int\frac{d^{3}\underline{\mathbf{k}}d^{3}\underline{\mathbf{p}}_{1}d^{3}\underline{\mathbf{p}}_{2}}{[2(2\pi)^{3}]^{3}E_{\underline{\mathbf{k}}}E_{\underline{\mathbf{p}}_{1}}E_{\underline{\mathbf{p}}_{2}}}\delta^{4}(\underline{k}+\underline{p}_{1}+\underline{p}_{2}-k-p_{1}-p_{2}),$
(49)

where $\ket{\alpha}=\ket{\mathbf{k}\mathbf{p}_{1}\mathbf{p}_{2}}$. Using (48)
and (49) on the right-hand side of (4), with
$\ket{\beta}=\ket{\mathbf{k}^{\prime}\mathbf{p}^{\prime}_{1}\mathbf{p}^{\prime}_{2}}$,
one finds

$\displaystyle\begin{split}&\sum_{\ket{X}}B_{X}^{2}\delta^{4}(p_{X}-p_{\alpha})M_{\alpha\to
X}M^{*}_{\beta\to
X}=\int\frac{d^{3}\underline{\mathbf{k}}d^{3}\underline{\mathbf{p}}_{1}d^{3}\underline{\mathbf{p}}_{2}}{[2(2\pi)^{3}]^{3}E_{\underline{\mathbf{k}}}E_{\underline{\mathbf{p}}_{1}}E_{\underline{\mathbf{p}}_{2}}}\delta^{4}(\underline{k}+\underline{p}_{1}+\underline{p}_{2}-k-p_{1}-p_{2})\\\
&\times\left[\frac{\sqrt{G_{N}}m^{2}\lambda}{(p_{1}+k)^{2}+m^{2}-i\epsilon}\delta^{3}(\underline{\mathbf{p}}_{2}-\mathbf{p}_{2})\times
2E_{\underline{\mathbf{p}}_{2}}(2\pi)^{3}\right]\\\
&\times\left[\frac{\sqrt{G_{N}}m^{2}\lambda}{(p^{\prime}_{2}+k^{\prime})^{2}+m^{2}-i\epsilon}\delta^{3}(\underline{\mathbf{p}}_{1}-\mathbf{p}^{\prime}_{1})\times
2E_{\underline{\mathbf{p}}_{1}}(2\pi)^{3}\right]\\\
&=G_{N}m^{4}\lambda^{2}\int\frac{d^{3}\underline{\mathbf{k}}}{[2(2\pi)^{3}]E_{\underline{\mathbf{k}}}}\delta^{4}(\underline{k}+p^{\prime}_{1}-k-p_{1})\frac{1}{(p_{1}+k)^{2}+m^{2}-i\epsilon}\frac{1}{(p^{\prime}_{2}+k^{\prime})^{2}+m^{2}-i\epsilon}\\\
&=G_{N}m^{4}\lambda^{2}\delta(\underline{k}^{2}+\mu^{2})\frac{1}{(p_{1}+k)^{2}+m^{2}-i\epsilon}\frac{1}{(p^{\prime}_{2}+k^{\prime})^{2}+m^{2}-i\epsilon}\Big{|}_{\underline{k}=k+p_{1}-p^{\prime}_{1}}\\\
\end{split}$ (50)

where we used the usual Lorentz phase space identity $\int
d^{3}\mathbf{k}/((2\pi)^{3}2E_{\mathbf{k}})=\int
d^{4}k\delta^{2}(k^{2}+\mu^{2})\Theta(k^{0})$ before doing the integral to
obtain the last line. Identifying $\underline{k}=\tilde{k}$ in the notation of
the main text, this verifies the optical theorem (4), as described in (13).
The sum over final states has collapsed to a discrete point in phase space,
namely the specific final state
$\ket{X}=\ket{\tilde{\mathbf{k}}\mathbf{p}_{1}\mathbf{p}^{\prime}_{2}}$ with a
radiated “graviton”.

### Graviton and scalar graviton Feynman rules

I record here the Feynman rules used in section 3. We will use a scalar field
for the matter for simplicity. In transverse-traceless gauge, the graviton-
matter interactions lead to the Feynman rules [31, 32, 33]

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The Feynman rules for the scalar graviton model used in the main text can be
obtained by simply tracing over the Lorentz indices in the spin-2 model.
Specifically, one has

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##  Appendix C Unitarity violation in particle-antiparticle scattering

In the example of section 2, we stayed very close to the basic premise of the
actual experiments. The massive “particles” there can be viewed directly as
the massive, composite objects used to demonstrate Newtonian entanglement in
concrete experimental proposals. However, to emphasize the generality of this
argument, we now turn to a more sophisticated example, in which we use the
crossing symmetry property of relativistic $S$-matrices to study gravitational
particle-antiparticle scattering.

We thus add another postulate to our $S$-matrix rules:

  1. (C)

Crossing symmetry. The matrix element for a process $A+B\to C+D$ is given by
the same function as that for a process, like $A+B+\bar{C}\to D$, where an
incoming/outgoing particle is exchanged with an outgoing/incoming antiparticle
(and momenta switched accordingly).

For example, the amplitude for electron-electron scattering $e^{-}e^{-}\to
e^{-}e^{-}$ determines the amplitude for $e^{-}e^{+}\to e^{-}e^{+}$. In
quantum field theory, crossing is a consequence of micro-causality, the
requirement that field operators satisfy $[\phi(x),\phi(x^{\prime})]=0$ for
$x,x^{\prime}$ space-like separated [87]. It is rigorously satisfied in $2\to
2$ scattering [88, 89]. The same property holds in string theory despite the
absence of microscopic local fields [89]. Following the analytic $S$-matrix
program [37] we thus will simply assume it as a postulate of scattering
processes.

Consider the $2\to 2$, purely gravitational scattering amplitude for two
particles. I will refer to the massive particles as $\chi$ and antiparticles
as $\bar{\chi}$. From the second-order term of (5), we have

$\displaystyle\begin{split}S^{(2)}_{\mathbf{p}_{1}\mathbf{p}_{2}\to\mathbf{p}^{\prime}_{1}\mathbf{p}^{\prime}_{2}}&=-\int_{-\infty}^{\infty}dt_{1}dt_{2}d^{3}\mathbf{k}_{1}d^{3}\mathbf{k}_{2}V_{\mathbf{p}^{\prime}_{1}\mathbf{p}^{\prime}_{2},\mathbf{k}_{1}\mathbf{k}_{2}}\delta^{3}(\mathbf{p}_{1}^{\prime}+\mathbf{p}_{2}^{\prime}-\mathbf{k}_{1}-\mathbf{k}_{2})V_{\mathbf{k}_{1}\mathbf{k}_{2},\mathbf{p}_{1}\mathbf{p}_{2}}\\\
&\times\delta^{3}(\mathbf{k}_{1}+\mathbf{k}_{2}-\mathbf{p}_{1}-\mathbf{p}_{2})\Theta(t_{1}-t_{2})e^{-i(E_{\mathbf{p}^{\prime}_{1}}+E_{\mathbf{p}^{\prime}_{2}})t_{1}}e^{i(E_{\mathbf{k}_{1}}+E_{\mathbf{k}_{2}})(t_{1}-t_{2})}e^{-i(E_{\mathbf{p}_{1}}+E_{\mathbf{p}_{2}})t_{2}}\\\
&+(t_{1}\leftrightarrow t_{2}).\end{split}$ (53)

To get the second line, we inserted a complete set of states
$\ket{\mathbf{k}_{1}\mathbf{k}_{2}}$. Now we insert an additional step
function by noting that $\Theta(t)=\Theta^{2}(t)$, and use the same Fourier
transform $\Theta(t)e^{-iEt}=(2\pi i)^{-1}\int d\omega e^{-i\omega
t}/(\omega-E+i\epsilon)$ on each of the internal-line phase factors.
Performing the time integrals, we obtain

$\displaystyle\begin{split}S^{(2)}_{\mathbf{p}_{1}\mathbf{p}_{2}\to\mathbf{p}^{\prime}_{1}\mathbf{p}^{\prime}_{2}}&=\int
d\omega_{1}d^{3}\mathbf{k}_{1}d\omega_{2}d^{3}\mathbf{k}_{2}V_{\mathbf{p}^{\prime}_{1}\mathbf{p}^{\prime}_{2},\mathbf{k}_{1}\mathbf{k}_{2}}\delta(\omega_{1}+\omega_{2}-E_{\mathbf{p}^{\prime}_{1}}-E_{\mathbf{p}^{\prime}_{2}})\delta^{3}(\mathbf{p}_{1}^{\prime}+\mathbf{p}_{2}^{\prime}-\mathbf{k}_{1}-\mathbf{k}_{2})\\\
&\times\frac{1}{\omega_{1}-E_{\mathbf{k}_{1}}+i\epsilon}\frac{1}{\omega_{2}-E_{\mathbf{k}_{2}}+i\epsilon}\\\
&\times
V_{\mathbf{k}_{1}\mathbf{k}_{2},\mathbf{p}_{1}\mathbf{p}_{2}}\delta(\omega_{1}+\omega_{2}-E_{\mathbf{p}_{1}}-E_{\mathbf{p}_{2}})\delta^{3}(\mathbf{k}_{1}+\mathbf{k}_{2}-\mathbf{p}_{1}-\mathbf{p}_{2})\\\
&+{\mathbf{p}_{1}\mathbf{p}_{2}\leftrightarrow\mathbf{p}^{\prime}_{1}\mathbf{p}^{\prime}_{2}}.\end{split}$
(54)

One can interpret this as the amplitude for the initial state to transition to
an intermediate state of two particles with momenta
$\mathbf{k}_{1},\mathbf{k}_{2}$ while conserving total energy and momentum,
followed by free propagation of the two intermediate particles, followed by
the transition to the final state
$\ket{\mathbf{p}^{\prime}_{1}\mathbf{p}^{\prime}_{2}}$. See the first panel of
Fig. 9. The term labeled
${\mathbf{p}_{1}\mathbf{p}_{2}\leftrightarrow\mathbf{p}^{\prime}_{1}\mathbf{p}^{\prime}_{2}}$,
which comes from the $V(t_{2})V(t_{1})$ part of (53), is the same up to
complex-conjugating the $V$ matrix elements.

$V_{N}(t_{1})$$V_{N}(t_{2})$$\mathbf{k}_{1}$$\mathbf{k}_{2}$$\mathbf{p}^{\prime}_{1}$$\mathbf{p}^{\prime}_{2}$$\mathbf{p}_{1}$$\mathbf{p}_{2}$Lorentz$k_{1}$$k_{2}$$\mathbf{p}^{\prime}_{1}$$\mathbf{p}^{\prime}_{2}$$\mathbf{p}_{1}$$\mathbf{p}_{2}$crossing$k_{1}$$k_{2}$$\mathbf{p}^{\prime}_{1}$$\mathbf{p}^{\prime}_{2}$$\mathbf{p}_{1}$$\mathbf{p}_{2}$
Figure 9: Basic procedure to compute the elastic $\chi\bar{\chi}$ scattering
amplitude. The left diagram represents the non-relativistic
$\chi\chi\to\chi\chi$ scattering ampitude, computed to second order in
perturbation theory. Its relativistic extension is depicted in the middle
panel. Finally, by crossing symmetry, the same mathematical expression can be
used to compute the $\chi\bar{\chi}\to\chi\bar{\chi}$ amplitude to the same
order, depicted in the third panel.

Following the same logic as in previous sections, we promote this to a
Lorentz-invariant amplitude:

$\displaystyle\begin{split}M_{\mathbf{p}_{1}\mathbf{p}_{2}\to\mathbf{p}^{\prime}_{1}\mathbf{p}^{\prime}_{2}}&=\tilde{\alpha}^{4}\int
d^{4}k_{1}d^{4}k_{2}\delta^{4}(k_{1}+k_{2}-p_{1}-p_{2})\\\
&\times\frac{1}{k_{1}^{2}+m^{2}-i\epsilon}\frac{1}{k_{2}^{2}+m^{2}-i\epsilon}\frac{1}{(k_{1}-p_{1})^{2}+\mu^{2}-i\epsilon}\frac{1}{(p^{\prime}_{1}-k_{1})^{2}+\mu^{2}-i\epsilon}.\end{split}$
(55)

We could have done one of the integrals with the remaining delta-function to
leave a single unconstrained loop integral, but it will become clear shortly
why we write things this way. In the language of field theory, this is a box
diagram with four scalars running in the loop. See Fig. 9. This brings up an
important point: this expression is correct only in the very low-energy limit,
i.e., the momentum integrals should have a “UV” cutoff $\Lambda\lesssim m$.
Above $\Lambda$, we cannot know the detailed spectrum of the model (the
intermediate states, non-relativistically included by inserting
$1=\sum\ket{\mathbf{k}_{1}\mathbf{k}_{2}}\bra{\mathbf{k}_{1}\mathbf{k}_{2}}$
above). However, we will only need the infrared part of this diagram in what
follows, where the expression (55) is reliable.

It is easy to check that (55) satisfies the optical theorem. The imaginary
part of the amplitude, which is of order $G_{N}^{2}$, is determined precisely
by the square of the elastic tree-level $\chi\chi\to\chi\chi$ amplitude of
order $G_{N}$. However, there will be a problem when we now extend these
results to $\chi\bar{\chi}$ scattering. By crossing symmetry, the amplitude
for $\chi\bar{\chi}\to\chi\bar{\chi}$ should be given by the
$\chi\chi\to\chi\chi$ amplitude, but with the roles of some outgoing momenta
switched with incoming momenta (see Fig. 9). At lowest order in perturbation
theory, we simply get (9), but with the momentum transfer
$t=-(p^{\prime}_{1}-p_{1})^{2}$ replaced by the total incoming energy-momentum
$s=-(p_{1}+p_{2})^{2}$:

$M^{\chi\bar{\chi}\to\chi\bar{\chi}}_{\mathbf{p}_{1}\mathbf{p}_{2}\to\mathbf{p}^{\prime}_{1}\mathbf{p}^{\prime}_{2}}=\frac{4\pi\tilde{\alpha}^{2}}{-s+\mu^{2}}.$
(56)

For physical momenta $p_{1}^{2}=p_{2}^{2}=p_{1}^{\prime 2}=p_{2}^{\prime
2}=-m^{2}$, we have $s\geq 4m^{2}$. Thus the pole in the denominator cannot be
reached, so the amplitude is purely real. Much like the $t$-channel $\chi\chi$
amplitude, the optical theorem is therefore trivial at this order since ${\rm
Im}~{}M=0+\mathcal{O}(\tilde{\alpha}^{4})$.

Im()$\mathbf{p}^{\prime}_{1}=\mathbf{p}_{1}$$\mathbf{p}^{\prime}_{2}=\mathbf{p}_{2}$$\mathbf{p}_{1}$$\mathbf{p}_{2}$$=$$\int$$d^{3}\mathbf{k}_{1}d^{3}\mathbf{k}_{2}$$|$$|$$2$$\mathbf{k}_{1}$$\mathbf{k}_{2}$$\mathbf{p}_{1}$$\mathbf{p}_{2}$
Figure 10: Forward $\chi\bar{\chi}$ scattering at second order in the
gravitational interaction. The imaginary part of the forward amplitude has a
branch cut singularity, which precisely equals the total cross section for the
process $\chi\bar{\chi}\to{\rm gravitons}$.

However, at the next order in perturbation theory
$\mathcal{O}(\tilde{\alpha}^{4})$, we will have unitarity problems. In
particular, the amplitude will contain a term given by (55) but with external
momenta crossed:

$\displaystyle\begin{split}M^{\chi\bar{\chi}\to\chi\bar{\chi}}_{\mathbf{p}_{1}\mathbf{p}_{2}\to\mathbf{p}^{\prime}_{1}\mathbf{p}^{\prime}_{2}}&=\mathcal{O}(\tilde{\alpha}^{2})+\tilde{\alpha}^{4}\int
d^{4}k_{1}d^{4}k_{2}\delta^{4}(k_{1}+k_{2}-p_{1}-p_{2})\\\
&\times\frac{1}{k_{1}^{2}+\mu^{2}-i\epsilon}\frac{1}{k_{2}^{2}+\mu^{2}-i\epsilon}\frac{1}{(k_{1}-p_{1})^{2}+m^{2}-i\epsilon}\frac{1}{(p^{\prime}_{1}-k_{1})^{2}+\mu^{2}-i\epsilon}.\end{split}$
(57)

This looks almost identical to the part of the $\chi\chi\to\chi\chi$ amplitude
given in (55), except that the roles of the particle mass $m^{2}$ and Yukawa
scattering parameter $\mu^{2}$ are switched. To check unitarity, we can
compute the imaginary part of the forward limit
$\mathbf{p}_{1}^{\prime}\to\mathbf{p}_{1},\mathbf{p}_{2}^{\prime}\to\mathbf{p}_{2}$
of this amplitude. A classic result of Cutkosky [90] says that the poles at
$k_{i}^{2}=-\mu^{2}$ combine to form an overall branch cut in the complex
$s$-plane. The imaginary part of the diagram is computed as the complex
discontinuity across this cut, and is given by replacing the two factors
$1/(x+\mu^{2}-i\epsilon)\to i\pi\delta(x+\mu^{2})$. That is,

${\rm
Im}~{}M^{\chi\bar{\chi}\to\chi\bar{\chi}}_{f}(s)=-\pi^{2}\tilde{\alpha}^{4}\int\frac{d^{3}\mathbf{k}_{1}}{2E_{\mathbf{k}_{1}}}\frac{d^{3}\mathbf{k}_{2}}{2E_{\mathbf{k}_{2}}}\left|\frac{1}{(p_{1}-k_{1})^{2}+m^{2}-i\epsilon}\right|^{2}\delta^{4}(k_{1}+k_{2}-p_{1}-p_{2}).$
(58)

To satisfy the optical theorem, this would have to be given by something of
the form of the right-hand side of (4). But much like the example of section
2, if the outgoing states are made up of $\chi$ and $\bar{\chi}$ particles
alone, there is simply no set of diagrams that can satisfy this. In
particular, the total elastic $\chi\bar{\chi}$ cross-section is given by

$\sigma_{\chi\bar{\chi}\to\chi\bar{\chi}}=-\pi^{2}\tilde{\alpha}^{4}\int\frac{d^{3}\mathbf{k}_{1}}{2E_{\mathbf{k}_{1}}}\frac{d^{3}\mathbf{k}_{2}}{2E_{\mathbf{k}_{2}}}\left|\frac{1}{(p_{1}-k_{1})^{2}+\mu^{2}-i\epsilon}\right|^{2}\delta^{4}(k_{1}+k_{2}-p_{1}-p_{2}).$
(59)

While these look similar, the problem is the mass in the denominators are
different. Thus (4) is not satisfied and we have a unitarity violation.

The solution is the same as before. We recognize that on the right-hand side
of

${\rm Im}~{}M_{f}=\sum_{X}|\braket{X}{S}{\psi_{\rm in}}|^{2}$ (60)

we need to add some new states $\ket{X}$. In this case, what we need to add is
the process $\chi\bar{\chi}\to{\rm``2~{}gravitons^{\prime\prime}}$. See Fig.
10. Inserting the amplitude $M_{\chi\bar{\chi}\to gg}$ on the right hand side
of (60) makes the equation precisely correct. Unlike the tree-level example,
the sum is not over a single isolated point of phase space but rather a full
integral over all possible kinematically allowed annihilations to two
gravitons.

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