# Gravity Mediated Entanglement between Oscillators as Quantum Superposition
of Geometries

Ofek Bengyat  0000-0001-5547-9176<EMAIL_ADDRESS>Andrea Di Biagio
0000-0001-9646-8457 Markus Aspelmeyer  0000-0003-4499-7335 Marios
Christodoulou  0000-0001-6818-2478 Institute for Quantum Optics and Quantum
Information (IQOQI) Vienna, Austrian Academy of Sciences, Boltzmanngasse 3,
A-1090 Vienna, Austria  Vienna Center for Quantum Science and Technology
(VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090
Vienna, Austria

(August 28, 2024)

###### Abstract

Protocols for observing gravity induced entanglement typically comprise the
interaction of two particles prepared either in a superposition of two
discrete paths, or in a continuously delocalized (harmonic oscillator) state
of motion. An important open question has been whether these two different
approaches allow to draw the same conclusions on the quantum nature of
gravity. To answer this question, we analyse using the path-integral approach
a setup that contains both features: a superposition of two highly delocalized
center of mass states. We conclude that the two usual protocols are of similar
epistemological relevance. In both cases the appearance of entanglement,
within linearised quantum gravity, is due to gravity being in a highly non-
classical state: a superposition of distinct geometries.

Feynman argued that detecting the gravitational pull of a mass in
superposition would be direct evidence of the quantum nature of gravity [1],
as the description of such an experiment would require the assignment of
quantum mechanical probability amplitudes to different configurations of the
gravitational field. Due to the weakness of gravity this has remained a
gedankenexperiment since the last 65 years. Partly motivated by experimental
progress there has been a revival of this fundamental question and several
proposals have been put forward that analyze Feynman’s gravity-mediated
entanglement (GME) in more detail [2, 3, 4]. Given the increase in quantum
control of mesoscopic solid masses [5, 6, 7] and advances in gravity
measurements at small scales [8, 9, 10, 11], these experiments are expected to
become feasible in our times [12, 13].

The literature is split into two kinds of experimental protocols for observing
gravity induced entanglement. We refer to them as the _path protocol_ and the
_oscillator protocol_. The path protocol [2, 3] envisages witnessing
entanglement production due to the gravitational interaction between two
particles, each in a quantum superposition of centre of mass location. The
quantum delocalization around each position is neglected. The oscillator
protocol [4, 14, 15] takes a different route: entanglement arises due to
gravitational interaction between two continuously delocalized particles,
initially prepared as quantum ground states of harmonic oscillators. Both
protocols would serve as a test for the predictions of (linearized) quantum
gravity.

The question remains whether for both protocols one can draw the same
conclusion about the quantum nature of gravity. Specifically, that neither
case can be described by our existing (classical) theory of gravity, but
requires linearized quantum gravity.

In [16, 17, 18, 19] it was argued that during the path protocol the
entanglement is caused by spacetime existing in a superposition of causally
evolving, diffeomorphically inequivalent geometries. In this paper, we extend
this analysis to the oscillator protocol. We assume linearised gravity and
compute the evolution with the path integral, which has the advantage of
keeping spacetime locality explicit. We then consider a state that generalises
both the path and oscillator protocols: a superposition of two center of mass
states with significant quantum spread of each particle. We calculate the
generalised concurrence for this state—an entanglement measure for continuous
variable systems proposed in [20]— in the limit of an instantaneous
interaction (zeroth order in $1/c$). We recover results from the literature:
in the limit where the width of the wavepackets is small with respect to the
size of the path superposition, we recover the entanglement rate for the path
protocol. In the converse limit, we recover the entanglement rate for the
oscillator protocol.

We then discuss the interesting case of an initially very localised state,
which, perhaps counterintuitively, results in much _faster_ entanglement
generation. We calculate the leading order relativistic correction for this
case, which is a quantitatively larger correction than the relativistic
corrections discussed in [17] for the path protocol, as well as qualitatively
different as it arises solely due to the fast spreading of the wavepacket. We
conclude that while the physics of the oscillator protocol can in fact be
richer, observing gravity mediated entanglement from either the path or
oscillator protocol is due to superposition of spacetime geometries.

The computations are analytic but lengthy, involving thousands of terms at
times, and so require non-trivial use of a symbolic computation software. The
code used is provided as Supplementary Material.

### Protocol —

The protocol we study is depicted in Fig. 1. The initial distance between the
center of mass of the particles is denoted $d$. At time $t_{1}$, each of the
two particles $A$ and $B$ is in a well-localized state. This can be achieved
by applying a strong trapping potential at times $t<t_{1}$, which is switched
off at time $t_{1}$. Local operations are applied to each particle to produce
some desired wavefunction $\psi_{2}(t_{2})$, which is assumed separable in $A$
and $B$. The particles are left to interact for a time $T=t_{3}-t_{2}$ that is
much larger than $t_{2}-t_{1}$ and $t_{4}-t_{3}$. At time $t_{3}$, the
particles are in some entangled state. Local operations are applied again to
the particles to make them well localised (and thus approximately disentangeld
from the field) by the time $t_{4}$. That is, entanglement produced at times
$t\notin[t_{2},t_{3}]$ is taken negligible. Entanglement is then measured by
measuring center of mass momentum and position correlations.

Figure 1: A sketch of the protocol. The particles start in a separable, well-
localised state at $t_{1}$. A more general separable state is prepared by
local operations ($t_{2}$). The particles evolve freely for a period $T$,
resulting in a possibly entangled state. Finally, local operations on the
particles make them localised again, without changing the entanglement.

### Action —

We assume linearized gravity. For brevity, the metric perturbation
$h_{\mu\nu}$ at a fixed time is denoted as $\varphi$ and a ‘path’
$h_{\mu\nu}(t)$ as $\mathcal{G}$. The action of the joint system of field and
the two particles $A$ and $B$ is of the form

$\displaystyle
S\left[x_{a},\mathcal{G}\right]=S_{\text{G}}\left[x_{a},\mathcal{G}\right]+\sum_{a\in\\{A,B\\}}S_{a}\left[x_{a}\right]$
(1)

where $a=A,B$, $S_{a}$ the action of a particle with mass $m_{a}$ in flat
spacetime, and $S_{\text{G}}$ the gravitational action which includes the
kinetic terms for $\varphi$ and its coupling to the energy momentum tensor,
which in turn describes point particles with trajectories
${x_{a}^{\mu}=(t,\mathbf{x}_{a})}$. $S_{a}$, $S_{\text{G}}$ and $T_{\mu\nu}$
are given in the Appendix. Note that without the term $S_{\text{G}}$ there can
be no entanglement generation since
$e^{i\sum_{a}S_{a}\left[x_{a}\right]/\hbar}=\prod_{a}e^{iS_{a}\left[x_{a}\right]/\hbar}$.

_Path integral —_ We model the system by a quantum state
$\ket{\Psi(t)}\in\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{\text{G}}$,
where $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$ are associated with the centre
of mass of the particles and $\mathcal{H}_{\text{G}}$ contains the states of
the gravitational field. The specific assignment of a state to the
gravitational field is a delicate point which we discuss further below.

The state at time $t_{1}$ is assumed to be of the form

$\ket{\Psi(t_{1})}=\ket{\psi^{A}_{1}}\ket{\psi^{B}_{1}}\ket{\varphi_{1}},$ (2)

that is, a separable state of the two particles and the field. We also assume
that the state at time $t_{4}$ is

$\ket{\Psi(t_{4})}=U\ket{\Psi(t_{1})}=\ket{\psi_{4}}\ket{\varphi_{1}},$ (3)

where $U$ is the unitary evolution operator and now $\ket{\psi_{4}}$ is a
possibly non-separable state for the two particle, allowing for gravity
mediated entanglement.

Note that we have assumed that the state of the gravitational field is the
same at the beginning and at the end of the experiment, for two reasons.
First, because the particles are light and slow-moving, we assume that the
amplitude to excite radiation is negligible; in other words, we approximate
the final state as its zero graviton component. Secondly, we assume that the
particles start and end in well-localised states at the same position. We take
$\ket{\varphi_{1}}$ to be the classical-like state peaked on the classical
field $\varphi_{1}$: the newtonian field111We take the particles to be
localised at the same position $\mathbf{Y}_{a}$ at the beginning and end of
the experiment. If they would end up in a different place, the final
gravitational state would be peaked around a different newtonian field.
Nothing but convenience rests on this. of two localised particles at
$\mathbf{Y}_{A}$ and $\mathbf{Y}_{B}$.

We introduce the path integral by positing that

$\bra{\mathbf{y}_{4}^{a},\varphi_{1}}{U}\ket{\mathbf{y}_{1}^{a},\varphi_{1}}=\int_{\mathbf{y}_{1}^{a},\varphi_{1}}^{\mathbf{y}_{4}^{a},\varphi_{1}}\text{D}x^{\prime}_{a}\text{D}\mathcal{G^{\prime}}e^{iS[x^{\prime}_{a},\mathcal{G^{\prime}}]},$
(4)

where $\ket{\mathbf{y}^{a}}$ is a position eigenstate for particle $a.$
Therefore the wavefunction for the final state
${\psi_{4}(\mathbf{y}_{4}^{a}):=\braket{\mathbf{y}^{a}_{4}}{\psi_{4}}}$ is
given in terms of the initial wavefunctions
${\psi^{a}_{1}(\mathbf{y}^{a}_{1}):=\braket{\mathbf{y}^{a}_{1}}{\psi^{a}_{1}}}$
as

$\displaystyle\\!\\!\\!\\!\
\psi_{4}(\mathbf{y}_{4}^{a})=\int\mathrm{d}\mathbf{y}_{1}^{a}\psi_{1}^{a}(\mathbf{y}_{1}^{a})\int_{\mathbf{y}_{1}^{a},\varphi_{1}}^{\mathbf{y}_{4}^{a},\varphi_{1}}\text{D}x^{\prime}_{a}\text{D}\mathcal{G^{\prime}}e^{iS[x^{\prime}_{a},\mathcal{G^{\prime}}]}.$
(5)

We now take the stationary phase approximation for $\mathcal{G}$. The action
is to be evaluated on the classical gravitational field $\mathcal{G}[x_{a}]$
sourced by a particle moving as $x_{a}(t)$.

$\displaystyle\psi_{4}(\mathbf{y}_{4}^{a})\approx\int\mathrm{d}\mathbf{y}_{1}^{a}\psi_{1}(\mathbf{y}_{1}^{a})\int_{\mathbf{y}_{1}^{a}}^{\mathbf{y}_{4}^{a}}\text{D}x^{\prime}_{a}e^{iS[\mathcal{G}[x^{\prime}_{a}]]}.$
(6)

We have introduced the notation
${S[\mathcal{G}[x^{\prime}_{a}]]=S[x^{\prime}_{a},\mathcal{G}[x^{\prime}_{a}]]}$
for brevity. Note that, in general,
${\mathcal{G}[x^{\prime}_{a}](t_{1})\neq\varphi_{1}\neq\mathcal{G}[x^{\prime}_{a}](t_{4})}$,
therefore the path $\mathcal{G}[x^{\prime}_{a}]$ is _not_ one of the paths
integrated over in (5). However, we assume for the moment that the error is
not too significant, because the initial state will be very localized. The
validity of this assumption will be corroborated by our results.

A second stationary phase approximation for the trajectories is taken by
keeping only the contribution on the solution of the classical equations of
motion for the particles for initial and final positions $\mathbf{y}_{1}^{a}$
and $\mathbf{y}_{4}^{a}$, denoted as $x_{a}$. We obtain that

$\displaystyle\psi_{4}(\mathbf{y}_{4}^{a})\approx\int\mathrm{d}\mathbf{y}_{1}^{a}\psi_{1}(\mathbf{y}_{1}^{a})e^{iS[\mathcal{G}[x_{a}]]}.$
(7)

Since $\psi_{1}(\mathbf{y}_{1}^{a})$ is separable in $A$ and $B$, entanglement
generation will arise from the $S_{G}$ part of $S$. The gravitational
contribution at each $\mathbf{y}_{4}^{a}$ corresponds to a phase
${S_{G}[\mathcal{G}[x_{a}]]}$ with ${x_{a}(t_{1})=\mathbf{y}_{1}^{a}}$ and
${x_{a}(t_{4})=\mathbf{y}_{4}^{a}}$. Each such pair of trajectories sources a
gravitational field, and integrating over $\mathbf{y}_{1}^{a}$ corresponds to
‘summing’ the contributions from the gravitational interaction of all these
pairs of trajectories for $A$ and $B$.

We note two points that will be further discussed later. First, (7) indicates
that the gravitational field is not in a classical-like state during the
experiment. Second, the trajectories $x_{a}$ can be taken to be straight lines
even when going to the leading order in $1/c$, which indicates no radiation
due to no acceleration. This justifies our original assumption of no radiation
in $\ket{\varphi_{1}}$.

### State evolution —

We have yet to use the assumptions that the ‘splitting’ and ‘recombination’
phases are fast with respect to $T$, which imply we only need to consider the
free evolution for $t\in[t_{2},t_{3}]$. It can be shown that, to leading order
in $1/c$,

$\displaystyle\psi_{4}(\mathbf{y}_{4}^{a})\approx$
$\displaystyle\int\mathrm{d}\mathbf{y}_{3}^{a}e^{iS_{3}[\mathcal{G}[x_{3}^{a}]]}\cdot\int\mathrm{d}\mathbf{y}_{2}^{a}e^{iS_{2}[\mathcal{G}[x_{2}^{a}]]}$
$\displaystyle\cdot\int\mathrm{d}\mathbf{y}_{1}^{a}\psi_{1}(\mathbf{y}_{1}^{a})e^{iS_{1}[\mathcal{G}[x_{1}^{a}]]}.$
(8)

$x_{i}^{a}$ are the trajectories of the particles at the time interval
$[t_{i},t_{i+1}]$. Above, the actions are approximated up to leading order in
$1/c$ (which is $1/c^{2}$). We caution that due to retardation it is not
straightforward to arrive at (State evolution —), see the Appendix.

Evolution from $\psi_{2}$ to $\psi_{3}$ is done by the second integral in
(State evolution —). The assumptions made previously imply that the evolution
done with the actions $S_{1}$ and $S_{3}$ will not contribute to the
entanglement measured for $\psi_{4}$. We thus assume that the state
$\psi_{2}=\int\mathrm{d}\mathbf{y}_{1}^{a}\psi_{1}e^{iS_{1}}$ has been
prepared separable in $A$ and $B$ and write

$\psi_{3}(\mathbf{y}_{3}^{a})\approx\int\mathrm{d}\mathbf{y}_{2}^{a}e^{iS_{2}[\mathcal{G}[x_{2}^{a}]]}\psi_{2}(\mathbf{y}_{2}^{a}).$
(9)

Once $\psi_{3}(\mathbf{y}_{3}^{a})$ is calculated, we proceed to compute the
entanglement in $\psi_{3}$, which in the approximation we work in will be the
same as that in $\psi_{4}$.

Figure 2:  The initial state of matter. We take the convention that the two
particles have opposite positive directions to reduce the number of sign
changes in what follows. Note that $\omega\equiv\hbar/m\alpha^{2}$.

### Two gaussians in path superposition —

We now fix the states $\psi_{2}^{a}$ and apply the above to the generalised
case shown in Fig. 2 to arrive at an analytical expression for the
entanglement at zeroth order in $1/c$. That is, in this (and the next) section
the gravitational interaction is approximated as an instantaneous interaction
through a newtonian potential. The state $\psi_{2}=\otimes_{a}\psi_{2}^{a}$ is
separable in $A$ and $B$ and describes two quantum particles of mass
$m_{a}=m$, each prepared initially in a spatial superposition of two
gaussians. The initial wavefunction of each particle in the position basis is

$\displaystyle\\!\\!\\!\psi^{a}_{2}(\mathbf{y}^{a})\propto\exp\left(-\frac{(\mathbf{y}^{a}-\beta/2)^{2}}{2\alpha^{2}}\right)+\exp\left(-\frac{(\mathbf{y}^{a}+\beta/2)^{2}}{2\alpha^{2}}\right),$
(10)

where $\alpha$ is the delocalization of the position of each path and $\beta$
is the separation between the two paths. The normalisation will drop out from
the entanglement. Thinking of the gaussians as ground states of a harmonic
potential (which was turned off at $t=t_{2}$), we introduce the (nominal)
frequency $\omega=\hbar/m\alpha^{2}$.

We can calculate $\psi_{3}$ using (9). To simplify the difficult calculation,
we assume $d\gg\beta,\alpha,\alpha\omega t$, and expand to second order in the
dimensionless parameters $\alpha/d$ and $\beta/d$. This is a realistic
assumption for most realistic experimental scenarios. We also expand to zeroth
order in the dimensionless parameter $\omega d/c$. The above make the
integrals over $\text{d}\mathbf{y}_{2}^{a}$ in (9) computationally tractable.
The integrands become sums of gaussians multiplied by polynomials, a case for
which closed formulas can be found. The code used for the long analytical
calculation is provided as Supplementary Material.

To quantify the amount of entanglement generated by gravity in $\psi_{3}$, we
compute the generalised entanglement measure $\mathcal{E}$ recently introduced
in [20] and given in the Appendix. This is a generalization to the case of
continuous sets of states of the concurrence, a well known entanglement
measure for discrete sets of states. We find

$\displaystyle\mathcal{E}=$ $\displaystyle\frac{2Gm^{2}t}{\hbar
d}\frac{1}{d^{2}}\Biggl{[}\frac{1}{4}\beta^{4}+\alpha^{2}\beta^{2}\left(1+\frac{1}{4}(\omega
t)^{2}\right)$ (11)
$\displaystyle+\alpha^{4}\biggl{[}f_{0}\left(\frac{\beta}{\alpha}\right)+f_{2}\left(\frac{\beta}{\alpha}\right)\frac{(\omega
t)^{2}}{3}+f_{4}\left(\frac{\beta}{\alpha}\right)\frac{(\omega
t)^{4}}{9}\biggr{]}\Biggr{]}^{1/2}.$

The functions $f_{0},f_{2}$, and $f_{4}$ are of order unity and given in the
appendix. As is shown below, the faster than linear scaling with time is due
to the growth in uncertainty.

### Path and oscillator protocols —

When $\beta\gg\alpha$ each of the particles begins in a superposition of two
gaussians with negligible overlap. This will remain true during the experiment
so long as $\beta\gg\alpha\omega T$. Then, (11) reduces to the path protocol
entanglement [13]

$\displaystyle\mathcal{E}_{\beta\gg\alpha}=\frac{Gm^{2}T}{\hbar
d}\left(\frac{\beta}{d}\right)^{2}.$ (12)

When $\beta\ll\alpha$ each particle state is approximately a single gaussian
and (11) reduces to

$\displaystyle\mathcal{E}_{\beta\ll\alpha}=\frac{Gm^{2}T}{\hbar
d}\left(\frac{\sqrt{2}\alpha}{d}\right)^{2}\sqrt{1+\frac{(\omega
T)^{2}}{3}+\frac{(\omega T)^{4}}{9}}.$ (13)

We see that the distance $\beta$ between the paths in the path protocol is
replaced by the initial gaussian width $\alpha$. The oscillator protocol has
an additional amplification term that depends on $\omega T$. When $\omega T$
is large, we obtain

$\displaystyle\mathcal{E}_{\beta\ll\alpha}^{\omega T\gg
1}=\frac{2}{3}\frac{Gm^{2}T}{\hbar d}\left(\frac{\alpha\omega
T}{d}\right)^{2},$ (14)

which is in accordance with previous literature on the oscillator protocol
[4]. Recall that $\omega\propto\alpha^{-2}.$ This means that, while in the
small $\omega T$ limit, larger $\alpha$ leads to larger entanglement, in the
large $\omega T$ limit, smaller spread leads to larger entanglement. This was
already remarked in [4].

### Relativistic correction —

We now report the leading correction in $1/c$ for the oscillator protocol when
$\omega T\gg 1$. The newtonian potential is now replaced by a Liénard-Wiechert
type [17], approximated to leading order in $1/c$. The tedious calculation is
given in the Supplementary Material. As in the previous section, we expand the
action to second order in $\alpha/d$, but, this time keep the leading order in
$\omega d/c$. The state at $t_{2}$ is now

$\displaystyle\psi_{2}(\mathbf{y}_{a})=\otimes_{a}\psi_{2}^{a}(\mathbf{y}_{a})\propto\otimes_{a}\exp\left(-\frac{{\mathbf{y}^{a}}^{2}}{2\alpha^{2}}\right).$
(15)

The entangled state $\psi_{3}$ is computed again through (9) and is given in
the Supplementary Material. The entanglement is
$\mathcal{E}_{\beta\ll\alpha}^{\omega T\gg
1}+\Delta\mathcal{E}_{\beta\ll\alpha}^{\omega T\gg 1}$ with

$\Delta\mathcal{E}_{\beta\ll\alpha}^{\omega T\gg
1}=-\frac{3}{2}\left(\frac{d}{ct}\right)^{2}\mathcal{E}_{\beta\ll\alpha}^{\omega
T\gg 1}.$ (16)

Using (14), $\Delta\mathcal{E}_{\beta\ll\alpha}^{\omega t\gg 1}$ scales with
$(\omega d/c)^{2}$. Note that there will be _less_ entanglement due to this
relativistic effect. The correction for general values of $\omega T$ is given
in the Appendix.

There is a physical explanation for this, namely, the fact that when $\omega
t\gg 1$ the entanglement generation is not due to the initial spread per se,
but, due to the fast growth of the spread with free evolution. The particles
see each other with retardation and so they see a smaller spread of the other
particle, which results in less entanglement. The amount of entanglement
reduction is controlled by the ratio of the distance of the center of mass and
the light crossing time.

### Gravitational attraction —

Feynman imagined detecting the gravitational field of a mass in superposition
by measuring the displacement of another mass. However, surprisingly, gravity
mediated entanglement can arise with no measurable displacement. This has been
the main insight from the path protocol [2, 3]. Owing to our path integral
formalism, we can conclude that this is the case for the oscillator protocol
too.

To see this, we compare the results of using (i) straight lines trajectories
with (ii) the accelerated trajectories of the Kepler problem for the
particles. One gets the exact same results at leading order in $1/c$ both for
the entanglement (11) and all derived results, as well as for the relativistic
correction (16). Therefore at this level of approximation, the entanglement in
the oscillator protocol, like that in the path-protocol, is not due to the
gravity induced displacement between the particles.

### Superposition of geometries —

Let us now return to (7) and discuss the state of the field. For the scope of
this computation, we assume that the state of the field is the same at the
beginning and at the end of the experiment. In the evaluation of the path
integral, we take both the particles and the fields to be on-shell. The
picture that arises, to this approximation, is that of a superposition of
point particles on constant-velocity trajectories sourcing a corresponding
gravitational field. Different trajectories of the particles will source
diffeomorphically inequivalent geometries. Additionally, since the
trajectories of the particles can be taken to not be accelerated, these
geometries do not contain radiation. Therefore we may say that during the
experiment, spacetime is in a superposition of a continuum of different,
radiation-free, geometries. We may approximate the state of the particles and
field at times $t\in[t_{2},t_{3}]$ as

$\ket{\Psi(t)}\approx\int\mathrm{d}\mathbf{y}^{a}\,\psi(\mathbf{y}^{a})\ket{\mathbf{y}^{a}}\ket{\varphi(\mathbf{y}^{a})},$
(17)

where $\psi(\mathbf{y}^{a})$ is given by (9), and
$\ket{\varphi(\mathbf{y}^{a})}$ are semiclassical states of the fields, peaked
on classical fields $\varphi(\mathbf{y}^{a})$ sourced by particles at
positions $\mathbf{y}^{a}$.

### Conclusions —

We considered a protocol (see Figs. 1 and 2) that generalises the path and
oscillator protocols for gravity mediated entanglement. We calculated the
newtonian limit entanglement for this protocol in (11) and saw that the path
and oscillator protocol entanglement are recovered as physical subregimes in
(12) and (13). This is the first indication that entanglement in the
oscillator and path protocols is due to superposition of spacetime geometries.

The oscillator and path protocols amount to specific choices for the
$\psi^{a}_{2}$ wavefunctions for particles $a=A,B$ in (9) (a sum of Dirac
deltas for the path protocol and gaussians for the oscillator protocol). Note
that (9) and the general setup and procedure used is applicable to more
general (continuous or discrete) prepared wavefunctions $\psi^{a}_{2}$. We see
that the final matter state will be entangled in $A$ and $B$ because different
trajectories $x_{a}$ of the particles source a different classical field and
give rise to a different phase in (9).

Our computation relies on two approximations typical for GME experiments. One
of them is $\alpha/d\ll 1$, meaning that the size of each wavefunction is much
smaller than the distance between the two. Another approximation is that
$\omega d/c\ll 1$, which means that the experiment’s time scales $T,1/\omega$
are long enough relative to the distance scales $d,\alpha$ such that
relativistic effects are negligible. The saddle-point approximation is
expected to be valid as long as typical values of the action are much larger
than $\hbar$.

Another assumption in our computation is that the initial and final states of
the particles are very localised, and that therefore they are disentangled
from the field. Of course, even though much harder to identify experimentally,
detecting gravitationally induced decoherence due to entanglement with the
field [21] would in itself be a great feat.

Using the path integral formalism, rather than treating the problem as a
direct Hamiltonian interaction term as in [4, 15, 13], allows us to work with
covariant quantities: the actions that feature in the entangling phases in (7)
and (9). Thus, in this setting, entanglement is generated in a spacetime-local
manner. It follows that if the entire protocol lasts for about time $T$ and
$T\ll d/c$ the particles will not become entangled. This was shown by some of
us for the path protocol in [17]. In this work, we applied a similar
computational technique to more general wavefunctions.

We reported in (16) the first correction in $1/c$ for the oscillator
entanglement (13) for the case $\omega t\gg 1$. In this case the entanglement
is generated due to the fast rate of growth of the spread and scales inversely
with the initial spread $\alpha$, as noticed in [4]. The relativistic
correction we calculated here intuitively corresponds to the particles seeing
each other in the past due to retardation and thus interacting with a gaussian
with smaller spread. This gives rise to a negative correction with respect to
the newtonian calculation, that is, less entanglement. The relative error of
the correction versus the newtonian entanglement is
$-\frac{3}{2}\left(d/cT\right)^{2}$. This relativistic correction is very
difficult to detect for gravity, but, it could potentially be observed in an
electromagnetic version of the experiment.

Our results for the oscillator protocol without relativistic corrections match
those of [4]. Relativistic corrections to the entanglement in the oscillator
protocol were also computed in [15], but for a slightly different setup. While
we compute the entanglement of an initially separable state as a function of
interaction time, they compute the entanglement in the joint ground state of
gravity and optically-trapped massive particles. As a result, their
entanglement does not depend on time. The time-dependence shown here could be
exploited in an experimental configuration that starts in two separable
oscillator states and introduces the wanted interaction at a later time222This
also fits better the scope of the discussions of a theory independent argument
for the non-classicality of gravity [22, 23, 24]., e.g. by bringing the
particles more closely together.

It has been previously argued by some of us in [16, 17, 25, 18] that the path
protocol would show that the gravitational field has been indirectly detected
in a quantum superposition of macroscopically distinct geometries. An
analogous understanding for continuous matter wavefunctions sourcing the
gravitational field has thus far been missing. We have shown here that the
entanglement in the oscillator protocol can also be attributed to a
superposition of geometries, specifically a superposition of a continuum of
different, radiation-free geometries, and the different phases these accrue.

A clear takeaway from our analysis is that if linearised quantum gravity is
assumed, whatever the field state is taken to be, it cannot be a classical-
like state that approximately solves Einstein’s equations.

## acknowledgments

We acknowledge support of the ID# 61466 and ID# 62312 grants from the John
Templeton Foundation, as part of the “Quantum Information Structure of
Spacetime (QISS)” project (qiss.fr), and support from the Research Network
Quantum Aspects of Spacetime (TURIS). OB acknowledges support from the
Blaumann foundation. We thank Časlav Brukner, Anton Zasedatelev, Sougato Bose,
Richard Howl, Anupam Mazumdar and Carlo Rovelli for useful discussions.

## Appendix

_Action —_ The metric tensor is approximated as
${{g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}}}$ where $h_{\mu\nu}$ is the metric
perturbation that satisfies $|h_{\mu\nu}|\ll 1$ and $\eta_{\mu\nu}$ is the
flat metric. The signature is $(-,+,+,+)$ and we have: $g=\det g_{\mu\nu}$,
$t$ the coordinate time, $R$ the Ricci scalar, $G$ Newton’s gravitational
constant, $c$ the speed of light, and $x_{a}^{\mu}=(ct,\mathbf{x}_{a}(t))$ the
trajectory of particle $a$. We have

$S_{a}[x_{a}]=-m_{a}c\int\text{d}t\sqrt{-\eta_{\mu\nu}\dot{x}_{a}^{\mu}\dot{x}_{a}^{\nu}},$
(18) $\displaystyle S_{\text{G}}\left[x_{a},g\right]=$
$\displaystyle\int\text{d}^{4}x\frac{c^{3}}{16\pi
G}[\sqrt{-g}R]_{\mathcal{O}(h_{\mu\nu}^{2})}$ (19)
$\displaystyle-\int\text{d}^{4}x\frac{1}{2c}h_{\mu\nu}T^{\mu\nu}\big{|}_{g_{\mu\nu}=\eta_{\mu\nu}}+O(h_{\mu\nu}^{3}),$

and

$T^{\mu\nu}\big{|}_{g=\eta}=\sum_{a}\gamma_{a}(t)m_{a}\dot{x}_{a}^{\mu}\dot{x}_{a}^{\nu}\delta^{(3)}(\mathbf{x}-\mathbf{x}_{a}).$
(20)

Note that taking only the interaction term $\sim T_{\mu\nu}h^{\mu\nu}$ of
$S_{\text{G}}$ would give a wrong numerical result by a factor of two as the
contribution of the kinetic terms of $h_{\mu\nu}$ also need to be taken into
account. The on-shell action $\tilde{S}_{\text{G}}$ reads [17]

$\displaystyle S_{\text{G}}[\mathcal{G}[x^{\prime}_{a}]]=$
$\displaystyle\sum_{a\neq
b}\int_{0}^{t}\text{d}t^{\prime}\frac{Gm_{a}m_{b}\bar{V}_{a}^{\mu\nu}(t_{ab})V_{b\mu\nu}/c^{4}}{d_{ab}^{\mu}\dot{x}^{\prime}_{a\mu}(t_{ab})/c},$
(21)

with

$\displaystyle V_{a}^{\mu\nu}$
$\displaystyle=\dot{x}_{a}^{\mu}\dot{x}_{a}^{\nu}/\sqrt{\dot{x}_{a}^{\mu}\dot{x}_{a}^{\nu}\eta_{\mu\nu}/c^{2}},$
(22) $\displaystyle\bar{V}^{\mu\nu}$
$\displaystyle=V^{\mu\nu}-\frac{1}{2}\eta^{\mu\nu}\eta_{\alpha\beta}V^{\alpha\beta},$
(23) $\displaystyle d_{ab}^{\mu}\equiv x_{b}^{\mu}-x_{a}^{\mu}($
$\displaystyle t_{ab})=(ct-ct_{ab},\mathbf{x}_{b}-\mathbf{x}_{a}(t_{ab})),$
(24)

where $t_{ab}$ is the retarded time, defined implicitly as ${d_{ab}^{2}=0}$.
The omitted time argument in $V_{a\mu\nu},d_{ab},\mathbf{x}_{a}$ and
$x_{a}^{\mu}$ implies dependence on the non-retarded time $t^{\prime}$.

_Path integral at leading order in $1/c$ —_ We split the path integral at
times $t_{2},t_{3}$ as

$\displaystyle\psi_{4}(\mathbf{y}_{4}^{a})=$
$\displaystyle\int\mathrm{d}\mathbf{y}_{3}^{a}\int\mathrm{d}\mathbf{y}_{2}^{a}\int\mathrm{d}\mathbf{y}_{1}^{a}\psi_{1}(\mathbf{y}_{1}^{a})$
(25)
$\displaystyle\int_{\mathbf{y}_{1}^{a}}^{\mathbf{y}_{2}^{a}}\text{D}x_{1}^{a}e^{iS_{1}[\mathcal{G}[x_{1}^{a}]]}\int_{\mathbf{y}_{2}^{a}}^{\mathbf{y}_{3}^{a}}\text{D}x_{2}^{a}e^{iS_{2}[\mathcal{G}[x_{1}^{a},x_{2}^{a}]]}$
$\displaystyle\int_{\mathbf{y}_{3}^{a}}^{\mathbf{y}_{4}^{a}}\text{D}x_{3}^{a}e^{iS_{3}[\mathcal{G}[x_{1}^{a},x_{2}^{a},x_{3}^{a},]]}.$

where the path $x_{i}$ connects $y_{i}$ and $y_{i+1}$. Note that, because of
the retardation of the field, $S_{2}$ and $S_{3}$ depend on the path in the
previous time interval. This dependence can be disregarded to leading order in
$1/c$, since the time interval influenced by the previous trajectory is then
of order $d/c$, which implies that the contribution of the previous
trajectories will be one order higher in $1/c$.

Schematically, we approximate the action in interval $i$ as

$\displaystyle S_{i}$
$\displaystyle=\int_{t_{i}}^{t_{i+1}}L_{[-\infty,t^{\prime}]}\mathrm{d}t^{\prime}\approx\int_{t_{i}}^{t_{i+1}}L_{[t^{\prime}-d/c,t^{\prime}]}\mathrm{d}t^{\prime}$
(26)
$\displaystyle=\int_{t_{i}}^{t_{i}+d/c}L_{[t^{\prime}-d/c,t^{\prime}]}\mathrm{d}t^{\prime}+\int_{t_{i}+d/c}^{t_{i+1}}L_{[t^{\prime}-d/c,t^{\prime}]},\mathrm{d}t^{\prime}$

where $L_{[t^{\prime},t^{\prime\prime}]}$ depends on the particle trajectories
between times $t^{\prime}$ and $t^{\prime\prime}$. Now, notice that in the
first term the integration is over a time interval of length $d/c$. This
implies that the contribution from that term will be one order higher in $1/c$
than the second term, meaning we can neglect it. Therefore, after a simple
change of variables, we may write

$\displaystyle S_{i}$
$\displaystyle\approx\int_{t_{i}}^{t_{i+1}-d/c}L_{[t^{\prime},t^{\prime}+d/c]}.$
(27)

Thus $S_{i}$ only depends on the particle trajectories between
$t_{i},t_{i+1}$. This way, we can neglect the dependence of $S_{2}$ and
$S_{3}$ on the past of the trajectories and write
$S_{3}[\mathcal{G}[x_{1}^{a},x_{2}^{a},x_{3}^{a}]]\approx
S[\mathcal{G}[x_{3}^{a}]]$ and $S_{2}[\mathcal{G}[x_{1}^{a},x_{2}^{a}]]\approx
S[\mathcal{G}[x_{2}^{a}]]$.

_Entanglement measure —_ The definition of $\mathcal{E}$ for a normalized
bi–partite pure state $\ket{\psi}$ is [20]

$\displaystyle\mathcal{E}^{2}=2-2\int\text{d}y_{1}\text{d}y^{\prime}_{1}\left|\int\text{d}y_{2}\psi(y_{1},y_{2})\psi^{*}(y^{\prime}_{1},y_{2})\right|^{2}.$
(28)

_The functions $f_{0}$,$f_{2}$,$f_{4}$ —_ The functions from Eq. (11) are
order unit and given by

$\displaystyle f_{0}(x)$
$\displaystyle=\frac{-x^{4}-4x^{2}+4e^{x^{2}/2}-2e^{x^{2}/4}(x^{4}+2x^{2}-4)+4}{4(e^{x^{2}/4}+1)^{2}}$
(29) $\displaystyle f_{2}(x)$
$\displaystyle=\frac{-12x^{2}+8e^{x^{2}/2}+(-3x^{4}-12x^{2}+16)e^{x^{2}/4}+8}{8(e^{x^{2}/4}+1)^{2}}$
$\displaystyle f_{4}(x)$
$\displaystyle=\frac{(-x^{2}/2+e^{x^{2}/4}+1)^{2}}{(e^{x^{2}/4}+1)^{2}}$

They are plotted in the Supplementary Material.

_Relativistic correction —_ The relativistic correction to (13) for general
values of $\omega T$ was found to be

$\displaystyle\Delta\mathcal{E}_{\beta\ll\alpha}=$
$\displaystyle-\frac{1}{2}\left(\frac{\omega
d}{c}\right)^{2}\frac{Gm^{2}T}{\hbar
d}\left(\frac{\sqrt{2}\alpha}{d}\right)^{2}$ (30)
$\displaystyle\times\frac{\left(\omega T\right)^{2}/3-1}{\sqrt{1+\frac{(\omega
T)^{2}}{3}+\frac{(\omega T)^{4}}{9}}}.$

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##  I Supplementary Material

_Mathematica notebooks—_ We give an overview of the computation performed in
the Mathematica notebooks published with this work. When descriptions differ,
we will refer to the one used to achieve the relativistic corrections as rel
and to the one used to calculate the entanglement of paths with uncertainty as
path.The notebooks are available under
https://github.com/ofek-b/gme_superposition_geometries. We first define three
dimesionless quantities: $\phi=\frac{Gm^{2}}{\hbar\omega d}$, which is a
measure for the effect of gravity in the system; $\epsilon=\alpha/d$, which is
a measure of the quantum uncertainty of the particles’ paths; and
$\Omega=\omega d/c$, which is a measure for the effect of retardation in the
system. When we write that we _expand_ quantities, it is meant that we perform
a series expansion to second order in $\epsilon$ in both computations and to
zeroth (second) order in $\Omega$ in paths (rel). Note that in paths we define
$\xi=\beta/\alpha$ and so take second order in $\beta/d=\epsilon\xi$
automatically as well.

In rel, we find the retarded time $t_{ab}$ by solving a quadratic equation
resulting from expanding its implicit definition $d_{ab}^{2}=0$ with $d_{ab}$
defined in Eq. (24).

We then proceed to expand the action
$S[q_{1},q_{2}]=\int_{0}^{t}dt^{\prime}(L_{A}+L_{B}+L_{\text{G}})$, a
functional of general trajectories $q_{1}(t),q_{2}(t)$. We use the
relativistic kinetic energy for the uncoupled part $L_{A}+L_{B}$ and the on-
shell action Eq. (21) for $L_{\text{G}}$. In rel, we introduce the bookkeeping
variables ret and kin as follows to keep track of which relativistic
contributions result from retardation (i.e., the finite-speed propagation of
the change in spacetime’s geometry) and which from kinetic energy and momenta
gravitating (i.e., the correction to the sourcing of those changes predicted
by GR). Note that this distinction is frame dependent and is not addressed in
the main text, but will be used shortly.

$\displaystyle(L_{A}+L_{B})/\hbar\approx\frac{q_{1}^{\prime}(t)^{2}+q_{2}^{\prime}(t)^{2}}{2\omega}+\frac{\Omega^{2}\epsilon^{2}\left(q_{1}^{\prime}(t)^{4}+q_{2}^{\prime}(t)^{4}\right)}{8\omega^{3}}-\frac{2\omega}{\Omega^{2}\epsilon^{2}},$
$\displaystyle L_{\text{G}}/\hbar\approx$
$\displaystyle\omega\epsilon^{2}\phi(q_{1}(t)+q_{2}(t))^{2}-\omega\epsilon\phi(q_{1}(t)+q_{2}(t))+\omega\phi$
$\displaystyle+\text{kin}\left(-\frac{4\Omega^{2}\epsilon^{2}\phi
q_{1}^{\prime}(t)q_{2}^{\prime}(t)}{\omega}+\frac{3\Omega^{2}\epsilon^{2}\phi
q_{1}^{\prime}(t)^{2}}{2\omega}+\frac{3\Omega^{2}\epsilon^{2}\phi
q_{2}^{\prime}(t)^{2}}{2\omega}\right)$
$\displaystyle+\text{ret}^{2}\left(\frac{\Omega^{2}\epsilon\phi\left(q_{1}^{\prime\prime}(t)+q_{2}^{\prime\prime}(t)\right)}{2\omega}-\frac{2\Omega^{2}\epsilon^{2}\phi
q_{1}^{\prime}(t)q_{2}^{\prime}(t)}{\omega}+\frac{\Omega^{2}\epsilon^{2}\phi
q_{1}^{\prime}(t)^{2}}{\omega}+\frac{\Omega^{2}\epsilon^{2}\phi
q_{2}^{\prime}(t)^{2}}{\omega}\right)$
$\displaystyle+\text{ret}\left(-\frac{\Omega^{2}\epsilon\phi\left(q_{1}^{\prime\prime}(t)+q_{2}^{\prime\prime}(t)\right)}{4\omega}+\frac{\Omega^{2}\epsilon^{2}\phi
q_{1}^{\prime}(t)q_{2}^{\prime}(t)}{\omega}-\frac{\Omega^{2}\epsilon^{2}\phi
q_{1}^{\prime}(t)^{2}}{2\omega}-\frac{\Omega^{2}\epsilon^{2}\phi
q_{2}^{\prime}(t)^{2}}{2\omega}\right).$

The Lagrangian of paths (this of a Newtonian two-body problem) is achieved for
kin, ret, $\Omega\rightarrow 0$. The Lagrangian of GR up to leading order in
$1/c$ is achieved for taking both kin $=$ ret $=1$:

$\displaystyle L_{\text{G}}/\hbar\approx$
$\displaystyle\omega\epsilon^{2}\phi(q_{1}(t)+q_{2}(t))^{2}-\omega\epsilon\phi(q_{1}(t)+q_{2}(t))+\omega\phi+\frac{\Omega^{2}\epsilon\phi}{\omega}\left(\frac{q_{1}^{\prime\prime}(t)+q_{2}^{\prime\prime}(t)}{4}-5\epsilon
q_{1}^{\prime}(t)q_{2}^{\prime}(t)+8\epsilon q_{1}^{\prime}(t)^{2}+2\epsilon
q_{2}^{\prime}(t)^{2}\right)$

The classical trajectories beginning at $\mathbf{y}^{\prime}_{a}$ and ending
at $\mathbf{y}_{a}$ at time $t$ for the system are then calculated by solving
the Euler-Lagrange equations resulting from the action. In rel, we first solve
the non-relativistic equations of motion to achieve $\tilde{x}_{a}(t)$, and
then plug in $q_{a}(t)=\tilde{x}_{a}(t)+\Omega c_{a}(t)$ to the relativistic
equations of motion. We expand them and then solve for the correction
$c_{a}(t)$. As it turns out, at the order of our expansion there is no
relativistic correction to the classical trajectories. Note that as we have
checked, the final result for the entanglement remains the same for both
calculations if one takes the classical trajectories to be straight lines.
This could be expected on order-of-magnitude grounds.

The trajectories are then plugged into the action and the wavefunction under
the stationary phase approximation Eq. (9) is achieved after expanding the
integrand and solving the integral. The wavefunction is a sum of polynomials
multiplied by gaussians (in rel this sum has a single term, in paths two).
Using a closed formula for the moments of the gaussian distribution we are
able to perform the integral defining the generalized entanglement measure Eq.
(28) as a sum of expectation values of polynomials under a gaussian
distribution (one in rel, sixteen in paths). Before solving the integral we
expand the polynomials to an order double of the usual one, as ultimately a
square root will be taken. We then arrive at our results Eqs. (11), (16). In
the latter, we can set only one of kin and rel to one to achieve the different
corrections from retardation and kinetic energy and momenta gravitating. We
find that each time we get the correction reported in the main text multiplied
by a numerical factor: $4/5$ and $1/5$ respectively.

Figure 3: The functions from Eq. (11). They are of order unit. The horizontal
axis is $x=\beta/\alpha$.

in 1,…,8 See pages \x of paths.pdf in 1,…,11 See pages \x of rel.pdf


Institute of Physics, Jan Kochanowski University, ul. Uniwersytecka 7, 25-406,
Kielce, Poland

# Tensor glueball decay into nucleon-antinucleon

A. Vereijken ins:x ins:xins:x

###### Abstract

In the context of a chiral hadronic model, we compute the decay ratio of a
tensor glueball decaying into a nucleon and antinucleon compared to the decay
into 2 pions. Tensor meson dominance is assumed to also hold for the tensor
glueball in order to relate the coupling constants of the different decay
channels. We find that the decay width to nucleons is slightly larger than the
decay width to pions, but still in the same order of magnitude.

##  1 Introduction

Glueballs, bound states made of only gluons, are one of the oldest predictions
of Quantum Chromodynamics (QCD) [1]. Various theoretical (e.g. [2, 3, 4]) and
experimental [5] works have made progress, yet their experimental status is
not resolved [7, 6, 8, 9]. Different theoretical methods agree on the mass
hierarchy of the lowest lying glueball states, with the scalar
$(J^{PC}=0^{++})$ being the lightest and the tensor $(J^{PC}=2^{++})$ the
second lightest glueball. In this work, in the context of a chiral model
described in [10, 11], we will calculate the decay of the tensor glueball into
a nucleon-antinucleon pair, based on arguments used in tensor mesons dominance
models. Different glueballs have been studied before in hadronic models, such
as the scalar glueball in [12] and the pseudoscalar glueball in [13].

##  2 Decay Amplitude

The decays of the tensor glueball were studied in the extended Linear Sigma
Model in [10], where the $\rho\rho$ and $K^{*}\bar{K}^{*}$ channels were found
as the dominant decays. Here, we extend that model by coupling the tensor
glueball to nucleons with the following interaction term [14, 17]

$\mathcal{L}_{GNN}=\frac{g_{NN}}{m}G_{\mu\nu}\bar{N}(\gamma^{\mu}\overset{\leftrightarrow}{\partial^{\nu}}+\gamma^{\nu}\overset{\leftrightarrow}{\partial^{\mu}})N,$
(1)

where $G_{\mu\nu}$ is the tensor glueball, $\bar{N},N$ is the (anti-)nucleon
field containing the proton and neutron, i.e. $N^{T}=(p,n)$, $m$ is the
nucleon mass, and
$\overset{\leftrightarrow}{\partial^{\mu}}=\overset{\rightarrow}{\partial^{\mu}}-\overset{\leftarrow}{\partial^{\mu}}$.
Writing it out explicitly and using the symmetry of $G_{\mu\nu}$ we have the
Feynman diagram in figure 1 and associated matrix element:

Figure 1: Feynman diagram for tensor glueball decay

$\mathcal{M}(\alpha,r,s)=2\frac{g_{NN}}{m}\epsilon_{\mu\nu}(\vec{p},\alpha)\bar{u}(\vec{k}_{1},r)\gamma^{\mu}q^{\nu}v(\vec{k}_{2},s),$
(2)

with $\epsilon_{\mu\nu}(p,\alpha)$ the spin 2 polarization tensors,
$\bar{u}(\vec{k}_{1},r),v(\vec{k}_{2},s)$ Dirac spinors, and $q=k_{1}-k_{2}$
is the difference of outgoing momenta. Using the Casimir trick, the spin-
averaged modulus squared matrix element is

$\displaystyle\bar{|\mathcal{M}|}^{2}=\frac{4}{5}\left(\frac{g_{NN}}{m}\right)^{2}\sum_{\alpha}\epsilon_{\mu\nu}(p,\alpha)\epsilon_{\mu^{\prime}\nu^{\prime}}(p,\alpha)q^{\nu}q^{\nu^{\prime}}\text{Tr}\left[\gamma^{\mu}(\not{k}_{2}-m)\gamma^{\mu^{\prime}}(\not{k}_{1}+m)\right].$
(3)

The polarization tensors fulfill the completeness relation [15]

$\sum_{\alpha}\epsilon_{\mu\nu}(p,\alpha)\epsilon_{\mu^{\prime}\nu^{\prime}}(p,\alpha)=\frac{1}{2}(A_{\mu\mu^{\prime}}A_{\nu\nu^{\prime}}+A_{\mu\nu^{\prime}}A_{\mu^{\prime}\nu})-\frac{1}{3}A_{\mu\nu}A_{\mu^{\prime}\nu^{\prime}},$
(4)

with the tensor $A_{\mu\nu}$ defined as

$A_{\mu\nu}=g_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{M^{2}},$ (5)

with $M$ being the mass of the tensor glueball, which we take from lattice QCD
to be 2369 MeV [4]. It is useful to note early on that $p\cdot
q=(k_{1}+k_{2})\cdot(k_{1}-k_{2})=k_{1}^{2}-k_{2}^{2}=0$ since it is the mass
difference of the two daughter particles. This simplifies things because any
$p_{\nu},p_{\nu^{\prime}}$ from the completeness relation automatically gives
0. The amplitude squared then becomes

$\displaystyle\bar{|\mathcal{M}|}^{2}=\frac{4}{5}\left(\frac{g_{NN}}{m}\right)^{2}$
$\displaystyle\left[\frac{1}{2}(g_{\mu\mu^{\prime}}-\frac{p_{\mu}p_{\mu^{\prime}}}{M^{2}})g_{\nu\nu^{\prime}}+\frac{1}{2}g_{\mu\nu^{\prime}}g_{\mu^{\prime}\nu}-\frac{1}{3}g_{\mu\nu}g_{\mu^{\prime}\nu^{\prime}}\right]q^{\nu}q^{\nu^{\prime}}$
(6)
$\displaystyle\left(k_{2}{}_{\alpha}k_{1}{}_{\beta}\text{Tr}[\gamma^{\mu}\gamma^{\alpha}\gamma^{\mu^{\prime}}\gamma^{\beta}]-m^{2}\text{Tr}[\gamma^{\mu}\gamma^{\mu^{\prime}}]\right)$

Using well-known trace identities for gamma matrices and contracting with $q$
we find

$\displaystyle\bar{|\mathcal{M}|}^{2}=$
$\displaystyle\frac{16}{5}\left(\frac{g_{NN}}{m}\right)^{2}\left[\frac{1}{2}(g_{\mu\mu^{\prime}}-\frac{p_{\mu}p_{\mu^{\prime}}}{M^{2}})q^{2}+\frac{1}{6}q_{\nu}q_{\nu^{\prime}}\right]$
(7)
$\displaystyle\left(k_{1}^{\mu}k_{2}^{\mu^{\prime}}+k_{1}^{\mu^{\prime}}k_{2}^{\mu}-(k_{1}\cdot
k_{2})g^{\mu\mu^{\prime}}-m^{2}g^{\mu\mu^{\prime}}\right),$

which simplifies to

$\displaystyle\bar{|\mathcal{M}|}^{2}=\frac{16}{5}\left(\frac{g_{NN}}{m}\right)^{2}\Bigg{[}$
$\displaystyle\frac{q^{2}}{2}\left(-(k_{1}\cdot k_{2})-3m^{2}-2\frac{(p\cdot
k_{1})(p\cdot k_{2})}{M^{2}}\right)$ (8)
$\displaystyle+\frac{1}{6}\Big{(}2(k_{1}\cdot q)(k_{2}\cdot q)-(k_{1}\cdot
k_{2})q^{2}-m^{2}q^{2}\Big{)}\Bigg{]}$

Evaluating all dot products in the rest-frame of the tensor glueball the
amplitude takes the form

$\displaystyle\frac{8}{15}\left(\frac{g_{NN}}{m}\right)^{2}\left(3M^{4}-4m^{2}M^{2}-32m^{4}\right).$
(9)

The decay width for a boson decaying into two nucleons is given by [16]

$\Gamma_{GN\bar{N}}=2\frac{\sqrt{\frac{M^{2}}{4}-m^{2}}}{8\pi
M^{2}}\bar{|\mathcal{M}|}^{2}=\frac{1}{30}\left(\frac{2g_{NN}}{m}\right)^{2}\frac{\sqrt{\frac{M^{2}}{4}-m^{2}}}{\pi
M^{2}}\left(3M^{4}-4m^{2}M^{2}-32m^{4}\right),$ (10)

where the factor 2 counts the $p\bar{p}$ and $n\bar{n}$ modes.

##  3 Tensor Meson Dominance and decay ratio

We cannot compute the decay width without knowing the value of $g_{NN}$.
Although this coupling constant is not known experimentally, assuming tensor
meson dominance [17] one has certain relations between couplings of different
channels for the tensor meson decays. We will assume these for the tensor
glueball as well. The Lagrangian for the decay of the tensor glueball to 2
pions is of the form [10, 11]

$\mathcal{L}_{G\pi\pi}=\frac{g_{\pi\pi}}{M}G_{\mu\nu}\partial^{\mu}\vec{\pi}\partial^{\nu}\vec{\pi},$
(11)

where $\vec{\pi}=(\pi^{1},\pi^{2},\pi^{3})$ refers to the isospin triplet.
This Lagrangian leads to decay width of the form

$\Gamma_{G\pi\pi}=6\left(\frac{g_{\pi\pi}}{M}\right)^{2}\frac{\left(\frac{M^{2}}{4}-m^{2}\right){}^{5/2}}{60\pi
M^{2}},$ (12)

where the factor 6 counts the isospin and identical particle factors. Tensor
meson dominance (TMD) states that the dominant contribution to the hadron
energy momentum tensor $\Theta^{\mu\nu}$ is the tensor mesons $T^{\mu\nu}$.
Then, assuming tensor meson dominance leads to the following identity [17]

$\frac{2g_{NN}}{m}=\frac{g_{\pi\pi}}{M},$ (13)

which we will assume is also a valid approximation for the tensor glueball.
This allows us to calculate the decay ratio of $G\to N\bar{N}/G\to\pi\pi$ as

$\frac{\Gamma_{GN\bar{N}}}{\Gamma_{G\pi\pi}}\approx 5.3.$ (14)

The decay ratio is large enough to be a relevant factor in the search for a
tensor glueball. In comparison to a chiral hadronic model [10] or a
holographic model [18], the decay width is lower than the 2-vector channel
widths, but larger than the other channels. For example compared to the
dominant $\rho\rho$ channel found in [10], the ratio is
$\Gamma_{G\rho\rho}/\Gamma_{GN\bar{N}}\approx 9.6$. The applicability of
tensor meson dominance to the tensor glueball is not completely clear, so it
is at best an approximate result. However, as an order of magnitude
estimation, the outcome can be useful.

##  4 Conclusion

In this note we have computed the tensor glueball decay ratio of the nucleon-
antinucleon and 2 pion channels. A speculative assumption of tensor meson
dominance relations applying also to the tensor glueball has been made. The
approximate result we obtained for the decay into nucleon-antinucleon is
larger than the 2 pion ratio but the ratio is still of order 1. Compared to
previous works it is not the largest decay channel found. Nevertheless it
could still be a fruitful process to investigate in glueball searches.

###### Acknowledgements.

The author acknowledges Francesco Giacosa and Shahriyar Jafarzade for useful
discussions. The author also acknowledges financial support from the Polish
National Science Centre (NCN) via the OPUS project 2019/33/B/ST2/00613.

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