# Generalized stability conditions for binary bose mixtures

M. Benarous, A. Hocine and A. Mehedi
Laboratory for Theoretical Physics and Material Physics
Department of Physics
Faculty of Exact Sciences and Informatics
Hassiba Benbouali University of Chlef (Algeria)

###### Abstract

Considering both interspecies and intraspecies fluctuations in ultracold bose-
bose mixtures, we derive generalized stability conditions against collapse and
phase separation. Furthermore, by examining the energy density of the system,
we show that new LHY-like effects may enforce or destroy the stability
according to the excitation branch, therefore, permitting or prohibiting the
existence of droplet states.

PACS: 05.30.Jp, 11.15.Tk, 32.80.Pj

##  1 Introduction

Ultracold bose mixtures are raising a great deal of interest owing to their
unique properties of the combination between intra and interspecies
interactions. Experimentally, one can tune these interactions by means of
Feschbach resonances leading to a rich ground state phase diagram composed of
either mixed or separated phases and allowing structure formation[1, 2, 3].

Binary mixtures are generally studied within the mean field approximation by
adopting the famous (coupled) Gross-Pitaevskii equations (CGPE). The latter
are quite successful in describing various equilibrium properties of ultracold
gases. However, as soon as the mean field energy becomes small enough, they
become less accurate as quantum fluctuations show up. Since this regime can
readily be reached experimentally by finely tuning the interspecies
interactions, it becomes crucial to analyze the effects of quantum
fluctuations on the stability properties of the mixture. In the field of
ultracold gases, Lee-Huang-Yang (LHY) corrections[4] are among the widely
studied fluctuations. Indeed, they have been shown to play a major role in the
stabilization of a collapsing mixture[5]. Most importantly, such a phenomenon
is accompanied with the existence of previously unknown stable states named
droplets[2, 6].

In the simplest beyond-Gross-Pitaevskii models for a single condensate, only
one-body local fluctuations, denoted generally by $\tilde{n}$ and $\tilde{m}$,
are retained. The most widely known are the Time-Dependent Hartree-Fock-
Bogoliubov (TDHFB) approximation ([7]-[13]), the (G)eneralized HFB (taking
into account both $\tilde{n}$ and $\tilde{m}$)([14]-[17]), the HFB-Popov
(retaining only $\tilde{n}$)([9, 11, 19]-[25]), and the second order Beliaev
approximation[26] which introduces $\tilde{m}$ in a quite consistent way. More
involved techniques, such as the Bethe Ansatz method, associated with the
functional integral method[27] and the stochastic projected Gross-Pitaevskii
equation[28] have also been successfully used. For a relatively small number
of particles, the exact many-body problem has also been solved numerically
using the Diffusion Monte-Carlo Method[29]. The extended Gross-Pitaevskii
equation was also shown in [30] to yield predictions that differ from the
Diffusion Monte-Carlo Method indicating that correlations in the system are
significant.

Moreover, for multi-species mixtures, going beyond CGPE along the same lines
as the methods cited above, requires taking into account correlations between
atoms of the same species, but also between atoms of different species. In
this context, although one (and some many-body) correlation functions between
atoms or molecules of the same species have been extensively studied both
experimentally ([31]-[39]) and theoretically ([40]-[54]), little has been done
for interspecies correlations in mixtures[55, 56]. It is indeed only recently
that they were recognized to play important roles in the characterization of
quantum phase transitions in mixtures thus relating them to real many-body
effects (see [57] for an illuminating discussion.) For instance, Wang et
al.[58] studied bose-bose mixtures in optical lattices using a perturbative
approach, and showed first that interspecies coupling is a necessary condition
for interspecies entanglement. Second, by computing the entanglement entropy
(von Neumann entropy of the reduced density operator), they deduced that it is
a clear measure of interspecies entanglement. The latter plays an important
role in the characterization of the quantum-phase transitions of mixtures,
specifically in the Mott Insulator to superfluid transition considered there.
The entanglement entropy for a two-species Bose-Hubbard dimer in a double well
was also computed in [59] with the intuitive results that a non-zero hopping
causes a non-zero entanglement between spatial modes and that the intraspecies
interaction contributes to the entanglement. Links et al.[60] computed the
exact ground-state correlation functions of an atomic-molecular Bose-Einstein
condensate (BEC) and found that the interspecies correlations and variances
are highly non-trivial in certain regions of the parameter space. Ota et
al.[61] extended the LHY description in a perturbative way by computing the
ground state energy of a self-bound Bose mixture and Zin et al.[62, 63]
elaborated a self consistent description of bose-bose droplets. Moreover,
Sarkar et al.[64] analyzed the interspecies entanglement with impurity atoms
in a lattice gas.

On the other hand, focusing on the density matrix, Alon[65] found for a
harmonic interaction, that the interspecies reduced density matrix per
particle is separable and given by the product of the intraspecies reduced
density matrices per particle. The results were compared to [66] who found for
attractive mixtures vanishing bipartite entanglement between the two species
and between a single particle (of either kind) and the remaining particles in
the mixture. Moreover, Klaiman et al. [67] showed in the case of an exactly
solvable model for a BEC mixture that, while the interspecies two-body density
factors to a product of two one-body densities in the infinite-particle limit,
and the many-body energy and densities coincide with their mean field values,
a clear departure from the mean field result is observed for the variance of
many-particle operators, witnessing that many-body correlations exist in the
ground state of a trapped mixture whatever the number of particles.

Using a quite different approach, namely the Multi-layer Multi-Configuration
Time Dependent Hartree Bogoliubov (ML-MCTDHB) ([68]-[70]), Mistakidis et
al.[71] studied the quench dynamics of a binary BEC of 87Rb atoms prepared in
different internal states. The correlation functions, beyond the mean field,
were analyzed up to the one and two-body level. Owing to the maximum numbers
of orbitals to be used, the authors computed the one and two body inter and
intraspecies correlation functions for up to 50 atoms. A major challenge to
avoid finite size effects and explore the thermodynamic limit, is to
generalize these results to high enough numbers.

The previous results clearly depict the importance of entanglement in binary
mixtures. Indeed, except for very special situations such as [72], there is no
a priori argument to claim that interspecies correlations are negligible. In
order to account for these correlations, we propose in the present work, to
use a gaussian density operator for a bose-bose mixture that does not simply
factorize to a product of density operators for the individual species. The
interspecies correlations will result from this non-factorazibility. In this
spirit, the present approach generalizes the correlated gaussian wavefunctions
method ([73]-[76]). To proceed further, the dynamics is generated by a
variational method ([77]-[79]). We will henceforth consider a density operator
for the entire system that is the exponential of the most general quadratic
form in the bose field operators. To generate interspecies correlations, this
form includes products of fields of different species.

The paper is organized as follows. In section 2, using the previously
described density operator, we derive the energy density corresponding to a
general two-body Hamiltonian for a homogeneous binary bose mixture at zero
temperature. By minimizing the grand canonical energy density, we derive
generalized expressions for the chemical potentials and the stability
conditions for the mixture. In order to get more quantitative results, we
compute in section 3, the intraspecies and interspecies quantum fluctuations
for a quasi-one dimensional mixture. While we recover for the former the well-
known (negative) LHY terms, we show that interspecies fluctuations have the
same magnitude and thus confirm that their omission is a quite hazardous
approximation in the general case. We discuss at the end the significance of
these results and their implications on the stability of quasi-1D droplets.
Some perspectives for future works are also given.

##  2 Stability of the mixture

Consider a homogeneous mixture composed of two Bose-Einstein condensates at
zero temperature. For contact interactions, the generally adopted two-body
second quantized hamiltonian writes

$\begin{array}[]{rl}H&=\sum_{i=1}^{2}\int_{r}\psi_{i}^{+}(r)h_{i}\psi_{i}(r)+{\displaystyle
1\over\displaystyle
2}\sum_{i=1}^{2}g_{ii}\int_{r}\,\psi_{i}^{+}(r)\psi_{i}^{+}(r)\psi_{i}(r)\psi_{i}(r)\\\
&+g_{12}\int_{r}\,\psi_{1}^{+}(r)\psi_{2}^{+}(r)\psi_{2}(r)\psi_{1}(r),\end{array}$
(2.1)

where $\psi_{1}^{+}(r)$ and $\psi_{2}^{+}(r)$ are the creation bose field
operators corresponding to the two species satisfying the usual commutation
rules
$[\psi_{i}(r),\psi_{j}^{+}(r^{{}^{\prime}})]=\delta_{ij}\delta(r-r^{{}^{\prime}})$,
$[\psi_{i}(r),\psi_{j}(r^{{}^{\prime}})]=[\psi_{i}^{+}(r),\psi_{j}^{+}(r^{{}^{\prime}})]=0$.
The constants $g_{ii}$ and $g_{12}$ are respectively the intra- and
interspecies couplings, $m_{i}$ the masses of the particles and
$h_{i}=-(\hbar^{2}/2m_{i})\Delta$.

The energy density then writes

$\begin{array}[]{rl}{\cal E}&={\displaystyle 1\over\displaystyle
2}g_{11}n_{1}^{2}+{\displaystyle 1\over\displaystyle
2}g_{22}n_{2}^{2}+g_{12}n_{1}n_{2}\\\ &+{\displaystyle 1\over\displaystyle
2}g_{11}n_{{c}_{1}}^{2}\left(2(\tilde{N}_{11}+\tilde{M}_{11})+\tilde{N}_{11}^{2}+\tilde{M}_{11}^{2}\right)+\big{(}1\to
2\big{)}\\\
&+g_{12}n_{{c}_{1}}n_{{c}_{2}}\left(2(\tilde{N}_{12}+\tilde{M}_{12})+\tilde{N}_{12}^{2}+\tilde{M}_{12}^{2}\right)\end{array}$
(2.2)

where $n_{1}$, $n_{2}$ are the total densities and $n_{{c}_{i}}$ the
condensate densities. For the sake of clarity, we have introduced the reduced
”densities” $\tilde{N}_{ij}=\tilde{n}_{ij}/\sqrt{n_{{c}_{i}}n_{{c}_{j}}}$ and
$\tilde{M}_{ij}=\tilde{m}_{ij}/\sqrt{n_{{c}_{i}}n_{{c}_{j}}}$, where
$\tilde{n}_{ij}=\langle\psi_{i}^{+}\psi_{j}\rangle-\langle\psi_{i}^{+}\rangle\langle\psi_{j}\rangle$
and
$\tilde{m}_{ij}=\langle\psi_{i}\psi_{j}\rangle-\langle\psi_{i}\rangle\langle\psi_{j}\rangle$.
Obviously, $\tilde{n}_{ii}$ and $\tilde{m}_{ii}$ are the noncondensate and
anomalous densities respectively and the second line in (2.2) (without the
quadratic terms) refers, as we will show below, to the well-known LHY
contributions. Most importantly, the new terms $\tilde{N}_{12}$ and
$\tilde{M}_{12}$ appearing in the third line of (2.2) are related to the
entanglement between the two species. It is evident that interspecies
interactions are a necessary condition for interspecies correlations [58, 59].

The obvious fact is that the contributions of $\tilde{N}_{12}$ and
$\tilde{M}_{12}$ are formally comparable to $\tilde{N}_{ii}$ and
$\tilde{M}_{ii}$. However, for unknown reasons, they are generally omitted in
the existing literature [5, 80, 81, 82, 83, 84]. Such assumptions are well
justified when the mean field energy dominates. On the contrary, when the
latter happens to be vanishingly small, as would be the case when one enters
the droplet region, these assumptions completely break down. Indeed, the
finding made by Petrov[5] that a collapsing mixture
($g_{12}+\sqrt{g_{11}g_{22}}\to 0$) may be stabilized by quantum fluctuations
becomes questionable as the new terms may have dramatic effects on the
stability of the droplets. It is worth mentioning that these terms have been
considered in two recent papers. In [18], they were written explicitly, but
never considered for further calculations. On the contrary, the authors of
[56] considered seriously interspecies fluctuations (which they named mixed
anomalous densities), but were mainly interested in the Popov approximation at
finite temperature. Hence, real quantum fluctuations were hidden by thermal
effects.

One of the objectives of the present work is to show that the contributions of
quantum interspecies fluctuations are not only qualitatively, but also
quantitatively comparable to the intraspecies fluctuations, even at zero
temperature. To this end, we will compute $\tilde{N}_{12}$ and
$\tilde{M}_{12}$ using the Bogoliubov approach.

Before proceeding further, we begin by writing the general stability
conditions of the mixture. The latter are ensured when the grand canonical
energy density ${\cal E}-\mu_{1}n_{1}-\mu_{2}n_{2}$ reaches a minimum. This
requires that

${\displaystyle\partial{\cal E}\over\partial n_{{c}_{i}}}=0$ (2.3)

and that the Hessian matrix

$A_{ij}={\displaystyle\partial^{2}{\cal E}\over\partial n_{{c}_{i}}\partial
n_{{c}_{j}}}$

be positive definite, which means ${\rm Tr}\,A>0$ and ${\rm Det}\,A>0$. The
first condition leads to the following expressions for the chemical potentials

$\begin{array}[]{rl}\mu_{1}&=g_{11}n_{1}+g_{12}n_{2}+g_{11}n_{{c}_{1}}\left(\tilde{N}_{11}+\tilde{M}_{11}\right)+g_{12}n_{{c}_{2}}\left(\tilde{N}_{12}+\tilde{M}_{12}\right)\\\
\mu_{2}&=g_{22}n_{2}+g_{12}n_{1}+g_{22}n_{{c}_{2}}\left(\tilde{N}_{22}+\tilde{M}_{22}\right)+g_{12}n_{{c}_{1}}\left(\tilde{N}_{12}+\tilde{M}_{12}\right)\\\
\end{array}$ (2.4)

which are clearly a generalization, taking into account interspecies
fluctuations, of the results obtained from the extensions of the stationary
Gross-Pitaevskii equation [7, 13, 56, 85, 86]. The second requirements yield
the stability conditions against collapse $G_{1}+G_{2}>0$ and against phase
separation $G_{1}G_{2}-G_{12}^{2}>0$, where the generalized coupling constants
are defined as follows:

$\begin{array}[]{rl}G_{1}&=g_{11}-g_{12}{\displaystyle\tilde{N}_{12}+\tilde{M}_{12}\over\displaystyle
2}{\displaystyle n_{{c}_{2}}\over\displaystyle n_{{c}_{1}}}\\\
G_{2}&=g_{22}-g_{12}{\displaystyle\tilde{N}_{12}+\tilde{M}_{12}\over\displaystyle
2}{\displaystyle n_{{c}_{1}}\over\displaystyle n_{{c}_{2}}}\\\
G_{12}&=g_{12}\left(1+{\displaystyle\tilde{N}_{12}+\tilde{M}_{12}\over\displaystyle
2}\right)\end{array}$ (2.5)

Obviously, the previous conditions generalize in a quite natural way the well-
known stability conditions $g_{ii}>0$, $g_{11}g_{22}-g_{12}^{2}>0$ for a non-
correlated homogeneous bose mixture[87]. Notice that this generalization
involves the interspecies fluctuations and not the intraspecies ones, which is
a quite new result that may have great implications on the stability
properties of the mixture. For instance, having repulsive intraspecies forces
does not prevent in general the mixture from collapse. In parameter space, the
stability windows against collapse and phase separation are strongly affected
by the interspecies fluctuations. It is worthwhile noticing that these effects
may be quite small when one addresses only the question of stability. However,
from the expression (2.2) of the energy, we see that they may have dramatic
effects in situations where the mean field energy happens to vanish. This is
at the heart of the observation made by Petrov[5].

In order to make these remarks more quantitative, we will compute in the next
section the fluctuations $\tilde{N}_{ij}$ and $\tilde{M}_{ij}$ using the
Bogoliubov approximation.

##  3 Single Particle Excitations and Stability of Droplets in Quasi-One
Dimension

The single particle excitations may be analyzed by examining the small
amplitude motion around static equilibrium. Decomposing the shifted operators
$\bar{\psi_{i}}=\psi_{i}-\langle\psi_{i}\rangle$ according to Bogoliubov’s
prescription, we may write in general

$\begin{array}[]{rl}\tilde{n}_{ij}&=\phantom{-}\sum_{k}\left[f_{k}U_{k}^{(i)}U_{k}^{(j)}+(1+f_{k})V_{k}^{(i)}V_{k}^{(j)}\right]\\\
\tilde{m}_{ij}&=-\sum_{k}\left[(1+2f_{k})U_{k}^{(i)}V_{k}^{(j)}\right]\end{array}$
(3.1)

where $U_{k}^{(i)}$ and $V_{k}^{(i)}$ are the quasi-particles amplitudes, and
$f_{k}=(e^{\hbar\omega_{k}/k_{B}T}-1)^{-1}$ is the Bose-Einstein distribution
function for the energy mode $\hbar\omega_{k}$. Evidently, $f_{k}=0$ at zero
temperature, which we shall consider from now on.

Using the variational method cited above[77, 78, 79], we finally obtain the
Hartree-Fock-Bogoliubov-De Gennes (HFB-BdG) equations [16, 17, 18] in their
generalized form

$\begin{array}[]{rl}\hbar\omega_{k}U_{k}^{(i)}&=\phantom{-}e_{k_{i}}U_{k}^{(i)}-g_{ii}\kappa_{ii}V_{k}^{(i)}+g_{12}\left(\eta_{12}U_{k}^{(3-i)}-\kappa_{12}V_{k}^{(3-i)}\right)\\\
\hbar\omega_{k}V_{k}^{(i)}&=-e_{k_{i}}V_{k}^{(i)}+g_{ii}\kappa_{ii}U_{k}^{(i)}-g_{12}\left(\eta_{12}V_{k}^{(3-i)}-\kappa_{12}U_{k}^{(3-i)}\right)\\\
\end{array}$ (3.2)

where
$e_{k_{i}}=\hbar^{2}k^{2}/2m_{i}+g_{ii}n_{c_{i}}\left[1-\tilde{M}_{ii}-\lambda_{i}\frac{n_{c_{3-i}}}{n_{c_{i}}}\left(\tilde{N}_{12}+\tilde{M}_{12}\right)\right]$,
$\lambda_{i}=g_{12}/g_{ii}$, $\eta_{ij}=\sqrt{n_{{c}_{i}}n_{{c}_{j}}}\\\
(1+\tilde{N}_{ij})$ and
$\kappa_{ij}=\sqrt{n_{{c}_{i}}n_{{c}_{j}}}(1+\tilde{M}_{ij})$, $i,j=1,2$. For
the sake of clarity, we will consider from now on a two-component balanced
system, for which $m_{1}=m_{2}=m$, $g_{11}=g_{22}=g$ and $n_{1}=n_{2}=n$.
Then, $n_{{c}_{1}}=n_{{c}_{2}}=n_{c}$,
$\tilde{N}_{11}=\tilde{N}_{22}=\tilde{N}$,
$\tilde{M}_{11}=\tilde{M}_{22}=\tilde{M}$ and
$\lambda_{1}=\lambda_{2}=\lambda$. The Eqs. (3.2) yield two branches
($\tilde{\omega}=\hbar\omega_{k}/gn_{c}$):

$\begin{array}[]{rl}\tilde{\omega}_{-}&=\sqrt{\epsilon_{k}\left(\epsilon_{k}+2(1-\lambda)-2\lambda(\tilde{N}_{12}+\tilde{M}_{12})\right)}\\\
\tilde{\omega}_{+}&=\sqrt{\epsilon_{k}\left(\epsilon_{k}+2(1+\lambda)\right)}\\\
\end{array}$ (3.3)

with $\epsilon_{k}=\frac{\hbar^{2}k^{2}}{2mgn_{c}}$. Notice that
$\tilde{\omega}_{+}$ does not depend on the interspecies correlations and that
the symmetry $\tilde{\omega}_{-}(-\lambda)=\tilde{\omega}_{+}(\lambda)$ (valid
when $\tilde{N}_{12}$ and $\tilde{M}_{12}$ vanish) is broken. The quasi-
particle amplitudes read now $U_{k}^{(2)}=-U_{k}^{(1)}$ and
$V_{k}^{(2)}=-V_{k}^{(1)}$ for the ($-$) branch, and
$U_{k}^{(2)}=+U_{k}^{(1)}$ and $V_{k}^{(2)}=+V_{k}^{(1)}$ for the ($+$)
branch. This yields the following remarkable solutions:
$\tilde{N}_{12}=-\tilde{N}$, $\tilde{M}_{12}=-\tilde{M}$ for
$\tilde{\omega}_{-}$ and $\tilde{N}_{12}=+\tilde{N}$,
$\tilde{M}_{12}=+\tilde{M}$ for $\tilde{\omega}_{+}$. Plugging these solutions
into the energy density (2.2), one gets

${\displaystyle{\cal E}_{\mp}\over\displaystyle
gn^{2}}=1+\lambda+2(1\mp\lambda)\left(\tilde{N}+\tilde{M}\right)+(1+\lambda)\left(\tilde{N}^{2}+\tilde{M}^{2}\right).$
(3.4)

One may get a better insight by comparing (3.4) to its analog in the absence
of interspecies fluctuations (called for brevity uncorrelated):

${\displaystyle{\cal E}\over\displaystyle
gn^{2}}=1+\lambda+2\left(\tilde{N}+\tilde{M}\right)+\tilde{N}^{2}+\tilde{M}^{2}.$
(3.5)

The key point at this level is that, for $\lambda\simeq-1$ (droplet regime),
the LHY term is doubled in the ($-$) sector whereas it is canceled by
interspecies fluctuations in the ($+$) sector. Since it is assumed that this
(positive in 3D) correction stabilizes a collapsing condensate[5], we may
infer from these considerations that, in the ($+$) sector, it will no longer
be the case in the presence of interspecies fluctuations.

In quasi-1D geometries, the situation is different as the LHY term is
negative[80, 81, 82] and hence, an uncorrelated mixture may in principle
undergo instability as soon as the mean field energy vanishes. However, when
the mean field is sufficiently close to zero, there will be a possibility for
a self bound stable state.

Let us see this more concretely by computing the various fluctuating terms in
1D (or quasi-1D). Plugging the solutions of (3.2) into (3.1), we get:

$\begin{array}[]{rl}\tilde{N}&=\frac{1}{2\pi}\left(1-\frac{{\rm
arcsinh}(1)}{\sqrt{2}}\right)\gamma_{1D}\sqrt{1\mp\lambda}\\\
\tilde{M}&=-\frac{\sqrt{2}}{8}\gamma_{1D}\sqrt{1\mp\lambda}\\\ \end{array}$
(3.6)

where $\gamma_{1D}=\sqrt{mg/\hbar^{2}n}$ is the Lieb-Liniger gas parameter.
Hence, as mentioned above, the LHY term
$\tilde{N}+\tilde{M}\simeq-0.117\,\gamma_{1D}\sqrt{1\mp\lambda}$ is
effectively negative[80, 82, 83] and falls in the stability region defined in
section 2. Moreover, one may easily check that $\tilde{\omega}_{-}$
corresponds to the lowest branch for any $\lambda>0$ while
$\tilde{\omega}_{+}$ corresponds to the lowest branch for any $\lambda<0$.

When approaching the droplet regime by taking $1+\lambda=\delta g/g<<1$, the
energy densities (3.4-3.5) write, up to first order in the fluctuations,

$\begin{array}[]{rl}{\cal E}_{+}&\simeq\delta gn^{2}-\gamma n^{3/2}\delta
g^{3/2}\\\ {\cal E}&\simeq\delta gn^{2}-\gamma n^{3/2}g\delta g^{1/2}\\\
\end{array}$ (3.7)

where $\gamma=0.234\sqrt{m/\hbar^{2}}$. Hence, although the two expressions
have the same functional form in density characterizing the existence of a
stable self-bound state[80, 81, 82], they differ by their structure in the
coupling constants. Indeed, we can easily show that the equilibrium density
and the minimum energy for the correlated case are smaller by a factor of
$(\delta g/g)^{2}$ and $(\delta g/g)^{4}$ respectively with respect to their
uncorrelated values. Hence, interspecies fluctuations shift the equilibrium
density and the minimum energy to almost zero, forbiding the existence of a
self-bound droplet. This is clearly depicted on Fig.1, where we have set
$m=\hbar=1$, $\delta g=0.01$ and $g=1$ for the sake of a better visualization.

Figure 1: Energy (arbitrary units) vs. density in the ($+$) sector (3.7) for
the correlated (${\cal E}_{+}$: red) and uncorrelated ($\cal E$: black) case.
$m=\hbar=1$, $\delta g=0.01$, $g=1$.

On the other hand, the ($-$) branch reveals a more interesting behavior, since
the energy densities (3.4-3.5) write in this case

$\begin{array}[]{rl}{\cal E}_{-}&\simeq\delta
gn^{2}-\sqrt{8}\left(1-{\displaystyle 3\over\displaystyle
4}{\displaystyle\delta g\over\displaystyle g}\right)\gamma(gn)^{3/2}\\\ {\cal
E}&\simeq\delta gn^{2}-\sqrt{2}\left(1-{\displaystyle 1\over\displaystyle
4}{\displaystyle\delta g\over\displaystyle g}\right)\gamma(gn)^{3/2}\\\
\end{array}$ (3.8)

The minimum of energy writes ${\cal E}_{-}^{(min)}=-1/3\,n_{-}^{2}\delta g$
where the liquid phase equilibrium density is given by

$n_{-}\simeq\frac{9\gamma^{2}}{2}\frac{g^{3}}{\delta g^{2}}.$ (3.9)

One easily shows that ${\cal E}_{-}^{(min)}=16\,{\cal E}^{(min)}$ and
$n_{-}=4\,n_{0}$, where $n_{0}$ is the equilibrium density for the
uncorrelated case. As can be inferred from Fig.2, this minimum is drastically
shifted downward leading to a more stable self-bound droplet state in the
liquid phase.

Figure 2: Energy (arbitrary units) vs. density in the ($-$) sector (3.8) for
the correlated (${\cal E}_{-}$: red) and uncorrelated ($\cal E$: black) case.
$m=\hbar=1$, $\delta g=0.01$, $g=1$.

##  4 Conclusions

In conclusion, contrary to the abundant scientific literature on binary bose
mixtures, where beyond mean field effects consist only in intraspecies quantum
fluctuations, the so-called LHY corrections, we demonstrate in this work that
completely different behaviors may show up when one takes into account
interspecies fluctuations. In a first step, we derive generalized stability
conditions against collapse and phase separation taking account of the
interspecies fluctuations. These conditions nicely generalize the well known
conditions in the absence of interspecies fluctuations. Moreover, in the
framework of Bogoliubov’s approach, the interspecies fluctuations are shown to
be of the same order of magnitude than the LHY terms. Thus, omitting them, as
it is generally assumed, is no longer justified in the general case. Most
importantly, depending on the excitation branch in the vicinity of the droplet
regime, the interspecies fluctuations are shown to either enforce the LHY
effects or completely cancel them which may lead to the destabilization of the
mixture.

In order to better apprehend the impact of these novel behaviors, the previous
calculations will be generalized in a forthcoming work to the unbalanced non-
symmetric case, where one should observe the effects associated with unequal
populations and interactions. Furthermore, the effects of geometry of the
system should also be examined carefully as the LHY corrections are not at all
the same in 2 or 3 dimensions. Some other interesting perspectives are the
presence of traps and the effects of temperature. These are currently under
investigation and are postponed for future publications.

Finally, as all these effects occur in very narrow windows of energy and
density, their experimental checks are quite challenging. One may imagine
special setups to finely adapt the measurements to such situations. In
particular, the comparison of the excitation spectra reveals that one may
select the phonon region where the frequencies are well separated.

M. B. and A. H. wish to thank G. Shlyapnikov for fruitful discussions on the
subject of correlations in BEC.

This work has been supported by The Faculty of Exact Sciences and Informatics
and the Laboratory for Theoretical Physics and Material Physics, Hassiba
Benbouali University of Chlef, Ministry of Higher Education and Scientific
Research, through the project grants B00L02UN020120180002.

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††institutetext:  Instituto de Física, Universidade de São Paulo,
Rua do Matão, 1371, 05508-090 São Paulo, SP, Brazil

# Virtual Exchange of Continuous Spin Fields

Victor O. Rivelles<EMAIL_ADDRESS>
###### Abstract

We discuss the long range interactions mediated by continuous spin fields. We
start by deriving the propagator for a continuous spin field using the
antifield BRST formalism. Then we couple the continuous spin field to a
conserved current to find the interaction energy due to static disturbances of
the vacuum. For sources having charges of the same sign we find an attractive
force at small distances and a repulsive force at large distances. We also
discuss the conserved current degrees of freedom.

##  1 Introduction

All known elementary particles are classified according to the unitary
irreducible representations of the Poincaré group and the interactions among
them are successfully described by quantum field theories in Minkowski
spacetime. However there is a gap which needs to be fulfilled. The continuous
spin particle is a massless unitary irreducible representation of the Poincaré
group which is characterized by a non-vanishing quartic Casimir operator
having eigenvalue $\rho^{2}$ with $\rho$ having mass dimension one
Wigner:1939cj . In the context of quantum field theory it presents serious
troubles Yngvason:1970fy ; Iverson:1971hq ; Chakrabarti:1971rz ; Abbott:1976bb
; Hirata:1977ss . Only recently a classical field theory for continuous spin
particles was proposed not in spacetime but on a cotangent bundle over
Minkowski spacetime where the field depends not only on the spacetime
coordinates $x^{\mu}$ but also on an extra coordinate $\eta^{\mu}$
Schuster:2014hca . Like the other massless particles the continuous spin field
has a gauge symmetry but now on the cotangent bundle not in spacetime. The
fact that we have to work on a cotangent bundle brings new issues which are
absent in conventional quantum field theory. In this work we present the first
steps towards a systematic quantization of bosonic continuous spin fields by
deriving the static energy generated by the virtual exchange of continuous
spin fields.

In order to do that we must make sure that continuous spin particles can
interact with other particles. As argued in Rivelles:2016rwo  we can couple an
abelian gauge field to a complex continuous spin field preserving the abelian
gauge symmetry. However the local symmetry of the continuous spin field is
completely lost. This was expected because when the value of continuous spin
goes to zero the continuous spin field action reduces to a sum of actions for
all integer values of the spin which, of course, do not propagate in flat
spacetime. If we restrict to cubic couplings involving one continuous spin
field and two massive scalar fields Bekaert:2017xin ; Rivelles:2018tpt  or one
continuous spin tachyon and two massive scalar fields Rivelles:2018tpt  then
conserved currents can be found on the cotangent bundle. More general cubic
vertices involving massless or massive fields of any spin and continuous spin
fields or continuous spin tachyons also have been built in a light-cone gauge
formulation Metsaev:2017cuz ; Metsaev:2018moa . More recently a worldline
coupling of spinless matter particle to continuous spin fields was also
analyzed Schuster:2023xqa .

The first step to quantize the continuous spin field is to find its
propagator. Adding a source term to the action and removing pure gauge terms
from the equations of motion provides a quick way to find the propagator
Bekaert:2017xin . But since the action for the continuous spin field has a
gauge symmetry which is reducible in the sense that its gauge parameters also
have a gauge symmetry means that we must be more careful. One way of doing
this is by using the BRST approach. Several approaches of BRST symmetry were
developed and applied to study properties of the continuous spin case.
Treating the Wigner equations for continuous spin particles as a constrained
system allowed a formulation of the BRST symmetry developed in
Bengtsson:2013vra ; Buchbinder:2018yoo ; Buchbinder:2020nxn ;
Buchbinder:2022msd . Considering the continuous spin field by itself as a
constrained system leads to the BRST formulation developed in Alkalaev:2017hvj
; Alkalaev:2018bqe  and to the BRST-BFV approach of Burdik:2019tzg ;
Metsaev:2018lth . In this paper we will take into account the fact that the
classical gauge theory is reducible and apply the antifield BRST
formalism111For a review of the antifield BRST formalism and original
references see Fuster:2005eg ; Gomis:1994he ; Henneaux:1989jq  to the original
classical action defined on a cotangent bundle Schuster:2014hca  leading to a
gauge fixed action with a manifest nilpotent BRST symmetry. We then find that
for a simple gauge choice the propagator goes like $1/k^{2}$ as for other
massless particles and also that it does not depend on $\rho$.

We then set up the path integral formalism in the cotangent bundle over
Minkowski spacetime to get the generating functional for connected diagrams.
To this end we add external sources minimally coupled to the continuous spin
field. They will be used to find out the interaction energy due to localized
static disturbances of the vacuum. To do that we need to consider a conserved
current in the cotangent bundle. We find that the current is not point like as
in conventional gauge theories but it is spread around a point in space and
has a constant flux on the cotangent bundle. We then find the classical
potential energy between two charged particles. Charges of the same sign go
like $-1/r^{D-3}$ at small distances and it is independent of $\rho$. At large
distances we find $\alpha^{(D-2)/2}\exp(-\alpha r)/r^{(D-4)/2}$ with $\alpha$
proportional to $\rho$. So at small distances particles with charges of the
same sign attract each other while at large distances they repel each other.
Quite interestingly this has a flavor of the dark side of the universe where
at small distances we find attraction and at large distances we find repulsion
of ordinary matter.

Next we consider the exchange of continuous spin fields between the two
sources. Firstly we solve the current conservation equation to identify the
independent components of the current when it is expanded in $\eta^{\mu}$
finding that the components are traceless in their transversal $SO(D-2)$
indices as expected. However when we compute the generating functional for
connected diagrams we get the product of three summations involving the
components of the two currents instead of a single summation which is expected
if each helicity appears just once. However, taking the first few terms of the
three summations we find that they remarkably simplifies to a single summation
only. We then construct a new generating functional with a single summation of
the product of the two traceless $SO(D-2)$ currents leaving the coefficients
of the summation arbitrary. Comparing the first few terms of each functional
suggests a general pattern for the coefficients of the new functional.
Unfortunately it is not possible to compare both functionals analytically but
only in a case by case situation. In all tested cases we found the same result
for both functionals giving a very strong support that they are really the
same.

The contents of the paper is as follows. In Section 2 we get the propagator
for the continuous spin field using the antifield BRST formalism. Then we
obtain the continuous spin propagator in Section 3 and derive the potential
energy due to static forces generated by the exchange of virtual continuous
spin fields in Section 4. In Section 5 we discuss the degrees of freedom of
the conserved current and in Section 6 we present some final comments.

##  2 Antifield BRST Formalism for Continuous Spin Fields

The field for a continuous spin particle is defined on a $D$-dimensional
cotangent bundle with coordinates $x^{\mu}$ and $\eta^{\mu}$,
$\mu=0,\dots,D-1$ and its action is given by Schuster:2013vpr

$S=\frac{1}{2}\int
dx\,d\eta\,\delta^{\prime}(\eta^{2}+\mu^{2})\left((\partial_{x}\Psi(\eta,x))^{2}-\frac{1}{2}(\eta^{2}+\mu^{2})\left(\Delta\Psi(\eta,x)\right)^{2}\right),$
(1)

where $\Delta=\partial_{\eta}\cdot\partial_{x}+\rho$, $\rho$ is the real
continuous spin parameter and $\delta^{\prime}$ is the derivative of the delta
function with respect to its argument. The metric is mostly minus.
Integrations over $\eta^{\mu}$ are divergent so they must be regulated. As in
Schuster:2014hca  we use a Wick rotation to perform all $\eta^{\mu}$
integrals. The factor $\mu^{2}$ is introduced to track dimensions of
$\eta^{\mu}$ and to make dimensional analysis easier. It can always be set
equal to 1 by rescaling $\eta^{\mu}$. Notice also that $\mu\rho$ has dimension
of inverse length and it is the only parameter in the free theory. The
derivative of the delta function constrains the dynamics to the hyperboloid
$\eta^{2}+\mu^{2}=0$ and its first neighborhood.

The action is invariant under the following global transformations: spacetime
translations, Lorentz transformations and a $\eta^{\mu}$ dependent translation
along $x^{\mu}$ given by $\delta x^{\mu}=\omega^{\mu\nu}\eta_{\nu}$, with
$\omega^{\mu\nu}$ antisymmetric. This last symmetry does not preserve the
natural symplectic structure of the cotangent bundle Rivelles:2016rwo . The
action is also invariant under the following local transformations

$\delta\Psi(\eta,x)=\left(\eta\cdot\partial_{x}-\frac{1}{2}(\eta^{2}+\mu^{2})\Delta\right)\epsilon(\eta,x)+\frac{1}{4}(\eta^{2}+\mu^{2})^{2}\chi(\eta,x),$
(2)

with $\epsilon(\eta,x)$ and $\chi(\eta,x)$ being the local parameters. Fields
defined in the cotangent bundle can be expanded around the hyperboloid and the
role of the $\chi$ symmetry is to remove all components of such an expansion
except for the first two so that it restricts the propagation of $\Psi$ to the
$\eta^{2}+\mu^{2}$ hyperboloid and its first neighborhood. On the other side,
the $\epsilon$ symmetry is an usual gauge symmetry removing gauge degrees of
freedom. These local symmetries are reducible Rivelles:2014fsa  since

$\displaystyle\delta\epsilon(\eta,x)$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\eta^{2}+\mu^{2})\Lambda(\eta,x),$ (3)
$\displaystyle\delta\chi(\eta,x)$ $\displaystyle=$
$\displaystyle\Delta\Lambda(\eta,x),$ (4)

leave (2) invariant. This symmetry mimics the $\chi$ symmetry for $\Psi$ and
can be used to limit the expansion of $\epsilon$ around the hyperboloid
$\eta^{2}+\mu^{2}=0$ to just the first term of the expansion. More details can
be found in Rivelles:2018tpt .

Since we have local symmetries which are reducible we have to introduce ghost
fields in such a way that the reducibility is taken into account. We will use
the BRST field-antifield formalism where the original local symmetries are
replaced by a rigid nilpotent BRST symmetry Fuster:2005eg ; Gomis:1994he ;
Henneaux:1989jq . This is achieved in two steps. Firstly we introduce ghost
fields and antifields on the cotangent bundle in such a way that a BRST
symmetry is manifest and the action is still invariant under the local
transformations (2). Then we introduce a gauge fixing fermion which will
require additional fields and antifields. These additional fields and
antifields are added keeping the nilpotency of the BRST transformation. Once
this is done we can make a choice for the gauge fixing fermion to obtain the
propagator.

We start by introducing ghost fields for each local symmetry of
$\Psi(\eta,x)$. We need two fermionic ghost fields $c_{\epsilon}(\eta,x)$ and
$c_{\chi}(\eta,x)$ both with ghost number 1, and one additional bosonic ghost
field $c_{\Lambda}(\eta,x)$ with ghost number 2, all of them required by the
reducible structure (3) and (4). Then we introduce the corresponding
antifields, a fermionic antifield $\Psi^{*}(\eta,x)$ with ghost number -1, two
bosonic antifields $c^{*}_{\epsilon}(\eta,x)$ and $c^{*}_{\chi}(\eta,x)$ both
with ghost number -2, and a fermionic antifield $c^{*}_{\Lambda}(\eta,x)$ with
ghost number -3. They are listed in Table 1 and Table 2.

| statistics | ghost number
---|---|---
$\Psi(\eta,x)$ | bosonic | 0
$c_{\epsilon}(\eta,x)$ | fermionic | 1
$c_{\chi}(\eta,x)$ | fermionic | 1
$c_{\Lambda}(\eta,x)$ | bosonic | 2

Table 1: Fields

| statistics | ghost number
---|---|---
$\Psi^{*}(\eta,x)$ | fermionic | -1
$c^{*}_{\epsilon}(\eta,x)$ | bosonic | -2
$c^{*}_{\chi}(\eta,x)$ | bosonic | -2
$c^{*}_{\Lambda}(\eta,x)$ | fermionic | -3

Table 2: Antifields

Then we can build the action

$\displaystyle S_{1}=\int
dx\,d\eta\left\\{\frac{1}{2}\delta^{\prime}(\eta^{2}+\mu^{2})\left((\partial_{x}\Psi(\eta,x))^{2}-\frac{1}{2}(\eta^{2}+\mu^{2})\left(\Delta\Psi(\eta,x)\right)^{2}\right)\right.$
$\displaystyle\left.+\Psi^{*}\left[\left(\eta\cdot\partial_{x}-\frac{1}{2}(\eta^{2}+\mu^{2})\Delta\right)c_{\epsilon}+\frac{1}{4}(\eta^{2}+\mu^{2})^{2}c_{\chi}\right]+\left(c^{*}_{\chi}\Delta+\frac{1}{2}(\eta^{2}+\mu^{2})c^{*}_{\epsilon}\right)c_{\Lambda}\right\\},$
(5)

which is invariant under the nilpotent BRST transformations222Our convention
for the BRST transformation for the product of two fields is
$\delta(XY)=X\delta Y+(-1)^{\epsilon_{Y}}(\delta X)Y$, where $\epsilon_{Y}$ is
the grading of $Y$.

$\displaystyle\delta\Psi=\left(\eta\cdot\partial_{x}-\frac{1}{2}(\eta^{2}+\mu^{2})\Delta\right)c_{\epsilon}+\frac{1}{4}(\eta^{2}+\mu^{2})^{2}c_{\chi},$
(6) $\displaystyle\delta
c_{\epsilon}=\frac{1}{2}(\eta^{2}+\mu^{2})c_{\Lambda},$ (7)
$\displaystyle\delta c_{\chi}=\Delta c_{\Lambda},$ (8)
$\displaystyle\delta\Psi^{*}=\delta^{\prime}(\eta^{2}+\mu^{2})\left(\Box_{x}-\eta\cdot\partial_{x}\Delta+\frac{1}{2}(\eta^{2}+\mu^{2})\Delta^{2}\right)\Psi,$
(9) $\displaystyle\delta
c^{*}_{\epsilon}=-2\left(\eta\cdot\partial_{x}+\frac{1}{4}(\eta^{2}+\mu^{2})\Delta\right)\Psi^{*},$
(10) $\displaystyle\delta
c^{*}_{\chi}=\frac{1}{4}(\eta^{2}+\mu^{2})^{2}\Psi^{*},$ (11)
$\displaystyle\delta c^{*}_{\Lambda}=-\Delta
c^{*}_{\chi}-\frac{1}{2}(\eta^{2}+\mu^{2})c^{*}_{\epsilon}.$ (12)

Notice that the terms in the second line of (2) are not restricted to
propagate on the hyperboloid and its first neighbourhood. On the other side,
the BRST transformation of $\Psi^{*}$ (9) is restricted to the hyperboloid and
its first neighbourhood. This structure is crucial, for instance, to show that
the BRST transformation of $c^{*}_{\epsilon}$ is indeed nilpotent. Notice also
that (2) is still invariant under the local transformations (2) so that no
gauge fixing was done yet.

The next step is to gauge fix (2). In order to do that we need the
introduction of further fields and antifields as follows. Due to the
$\epsilon$ symmetry we add the pair $\overline{c}_{\epsilon}(\eta,x)$ and
$b_{\epsilon}(\eta,x)$, the first a fermionic and the second a bosonic field,
with ghost number -1 and zero, respectively. In a similar way, for the $\chi$
symmetry we add $\overline{c}_{\chi}(\eta,x)$ and $b_{\chi}(\eta,x)$,
fermionic and bosonic fields respectively, also with ghost number -1 and zero.
We also need a pair $\overline{c}_{\Lambda}(\eta,x)$ and
$b_{\Lambda}(\eta,x)$, bosonic and fermionic fields respectively, with ghost
number -2, and -1. Finally the last pair $\sigma(\eta,x)$ and $\pi(\eta,x)$,
bosonic and fermionic fields respectively, have ghost number zero and 1. Then
we have to add the corresponding antifields
$\overline{c}^{*}_{\epsilon},b^{*}_{\epsilon},\overline{c}_{\chi},b^{*}_{\chi},\overline{c}^{*}_{\Lambda},b^{*}_{\Lambda},\sigma^{*}$
and $\pi^{*}$ with the opposite statistics to the former ones and ghost number
0, -1, 0, -1, 1, 0 -1 and -2, respectively. They are all listed in Table 3 and
Table 4.

| statistics | ghost number
---|---|---
$\overline{c}_{\epsilon}(\eta,x)$ | fermionic | -1
${b}_{\epsilon}(\eta,x)$ | bosonic | 0
$\overline{c}_{\chi}(\eta,x)$ | fermionic | -1
${b}_{\chi}(\eta,x)$ | bosonic | 0
$\overline{c}_{\Lambda}(\eta,x)$ | bosonic | -2
${b}_{\Lambda}(\eta,x)$ | fermionic | -1
$\sigma(\eta,x)$ | bosonic | 0
$\pi(\eta.x)$ | fermionic | 1

Table 3: Fields

| statistics | ghost number
---|---|---
$\overline{c}^{*}_{\epsilon}(\eta,x)$ | bosonic | 0
${b}^{*}_{\epsilon}(\eta,x)$ | fermionic | -1
$\overline{c}^{*}_{\chi}(\eta,x)$ | bosonic | 0
${b}^{*}_{\chi}(\eta,x)$ | fermionic | -1
$\overline{c}^{*}_{\Lambda}(\eta,x)$ | fermionic | 1
${b}^{*}_{\Lambda}(\eta,x)$ | bosonic | 0
$\sigma^{*}(\eta,x)$ | fermionic | -1
$\pi^{*}(\eta.x)$ | bosonic | -2

Table 4: Antifields

They are used to write the action

$\displaystyle S_{2}=\int
dx\,d\eta\left(\overline{c}^{*}_{\epsilon}b_{\epsilon}+\overline{c}^{*}_{\chi}b_{\chi}+\overline{c}^{*}_{\Lambda}b_{\Lambda}+\sigma^{*}\pi\right),{\color[rgb]{1,0,0}}$
(13)

which is trivially invariant under the BRST transformations

$\displaystyle\delta\overline{c}_{\epsilon}$
$\displaystyle=b_{\epsilon},\qquad\qquad\delta\overline{c}_{\chi}=b_{\chi},$
(14) $\displaystyle\delta\overline{c}_{\Lambda}$
$\displaystyle=b_{\Lambda},\qquad\qquad\delta\sigma=\pi,$ (15)
$\displaystyle\delta b^{*}_{\epsilon}$
$\displaystyle=-\overline{c}^{*}_{\epsilon},\qquad\quad\,\delta
b^{*}_{\chi}=-\overline{c}^{*}_{\chi},$ (16) $\displaystyle\delta
b^{*}_{\Lambda}$
$\displaystyle=\overline{c}^{*}_{\Lambda},\qquad\qquad\delta\pi^{*}=\sigma^{*},$
(17)

so that $S_{1}+S_{2}$ is also BRST invariant. The action $S_{2}$ has no
dependence on $\delta^{\prime}$.

Now we introduce the gauge fixing fermion ${\bf\Omega}$ which is a quadratic
function of fields, antifields and their derivatives, has ghost number -1 and
is fermionic. It will be used to eliminate all the antifields by imposing the
condition

$\displaystyle\Phi^{*}_{A}=\frac{\partial\bf\Omega}{\partial\Phi_{A}},$ (18)

where $\Phi_{A}$ is the set of all fields and $\Phi^{*}_{A}$ the set of all
antifields. A suitable choice for the gauge fixing fermion depends on two real
parameters $\zeta$ and $\xi$ and is given by

$\displaystyle{\bf\Omega}=\int
dx\,d\eta\,\delta(\eta^{2}+\mu^{2})\left[\overline{c}_{\epsilon}(\Delta\Psi+\sigma)+\overline{c}_{\chi}\left(\Psi+\frac{1}{\zeta}b_{\chi}+\frac{1}{\sqrt{\zeta\xi}}\sigma\right)+b_{\Lambda}\sigma+\overline{c}_{\Lambda}c_{\chi}\right].$
(19)

Using (18) we can eliminate all antifields so that the $S_{1}+S_{2}$ reduces
to the gauge fixed action

$\displaystyle S_{gf}=\int
dx\,d\eta\,\delta^{\prime}(\eta^{2}+\mu^{2})\times\hskip 270.30118pt$
$\displaystyle\times\left\\{\frac{1}{2}(\partial_{x}\Psi(\eta,x))^{2}-(\eta^{2}+\mu^{2})\left[\frac{1}{4}(\Delta\Psi)^{2}+(\Delta\Psi+\sigma)b_{\epsilon}+(\Psi+\frac{1}{\zeta}b_{\chi}+\frac{1}{\sqrt{\zeta\xi}}\sigma)b_{\chi}\right.\right.\hskip
14.22636pt$
$\displaystyle\left.\left.+(c_{\chi}-\pi)b_{\Lambda}+\overline{c}_{\epsilon}(\Box_{x}c_{\epsilon}+\pi)+\overline{c}_{\chi}(\eta\cdot\partial_{x}c_{\epsilon}+\frac{1}{\sqrt{\zeta\xi}}\pi)+\overline{c}_{\Lambda}\Delta
c_{\Lambda}\right]\right\\}.$ (20)

Notice that the two first terms are those of the classical action (1) while
the remaining terms involve all ghost fields. The gauge fixed action (2) is
invariant under the nilpotent BRST transformations (6), (7), (8), (14) and
(15).

##  3 Continuous Spin Field Propagator

In order to find the propagator we have to consider the path integral
formulation on a cotangent bundle. So let us consider a path integral
involving a field $\Xi(\eta,x)$ on a cotangent bundle, an operator $\cal{O}$
involving spacetime derivatives, $\eta^{\mu}$ and its derivatives, and also a
delta function $\delta(\eta^{2}+\mu^{2})$. We then get

$\displaystyle\int\,{\cal D}\Xi\,\,e^{i\int
dx\,d\eta\,\delta(\eta^{2}+\mu^{2})\,\Xi(\eta,x)\,{\cal
O}\,\Xi(\eta,x)}=det^{\mp 1/2}\left(\delta(\eta^{2}+\mu^{2})\cal{O}\right),$
(21)

where the minus (plus) sign is for a bosonic (fermionic) field $\Xi(\eta,x)$.
Using the gauge fixed action (2) we find that the integration over
$c_{\Lambda}$ and $\overline{c}_{\Lambda}$ gives a factor of
$det^{-1}\left(\delta(\eta^{2}+\mu^{2})\,\Delta\right)$ in the path integral
measure. Integration over $\pi$ and $\eta$ gives rise to two delta functions
which can be used to solve for $b_{\epsilon}$ and $b_{\Lambda}$. Then
integration over $b_{\chi}$ gives a factor of
$det^{-1/2}\left(\frac{2}{\zeta}\delta(\eta^{2}+\mu^{2})\right)$ and a
contribution to the action of the form
$\frac{\zeta}{2}\left((1-\frac{\Delta}{\sqrt{\zeta}})\Psi\right)^{2}$. Finally
integration over $\overline{c}_{\epsilon}$ and $\overline{c}_{\chi}$ gives
rise to delta functions which can be solved for $c_{\chi}$ and $c_{\epsilon}$
contributing with a factor of
$det\left(\delta(\eta^{2}+\mu^{2})(\eta\cdot\partial_{x}-\frac{\Box}{\sqrt{\zeta}})\right)$.
All together gives a factor of

$\displaystyle
det^{-1}\left(\delta(\eta^{2}+\mu^{2})\Delta\right)\,\,det^{-1/2}\left(\frac{2}{\zeta}\delta(\eta^{2}+\mu^{2})\right)\,\,det\left(\delta(\eta^{2}+\mu^{2})\left(\eta\cdot\partial_{x}-\frac{\Box_{x}}{\sqrt{\zeta}}\right)\right),$
(22)

in the path integral measure and the gauge fixed action (2) is reduced to

$\displaystyle S_{gf}=\int
dx\,d\eta\,\,\frac{1}{2}\delta^{\prime}(\eta^{2}+\mu^{2})\times\hskip
213.39566pt$
$\displaystyle\times\left\\{(\partial_{x}\Psi(\eta,x))^{2}-\frac{1}{2}(\eta^{2}+\mu^{2})\left((1-\frac{1}{\xi})(\Delta\Psi)^{2}+2\sqrt{\frac{\zeta}{\xi}}\Psi\Delta\Psi-\zeta\Psi^{2}\right)\right\\}.$
(23)

This gauge fixed action is no longer invariant under the $\epsilon$ gauge
transformation in (2) but it is still invariant under the $\chi$
transformation as expected.

The propagator can be found by adding a source term ${J}(\eta,x)$ to the
action (3)

$\displaystyle S_{J}=\int
d\eta\,dx\,\delta^{\prime}(\eta^{2}+\mu^{2})\,{J}(\eta,x)\,\Psi(\eta,x).$ (24)

The presence of the $\delta^{\prime}(\eta^{2}+\mu^{2})$ makes $S_{J}$
invariant under

$\displaystyle\delta
J(\eta,x)=\frac{1}{4}(\eta^{2}+\mu^{2})^{2}\,\,\Xi(\eta,x),$ (25)

so that $J(\eta,x)$ is restricted to the hyperboloid $\eta^{2}+\mu^{2}=0$ and
its first neighborhood. We can then find the field equation from (3) and (24),
multiply it by $\eta^{2}+\mu^{2}$ and apply $\Delta$ to get the generalization
of the current conservation equation in a cotangent space Rivelles:2016rwo

$\displaystyle\delta(\eta^{2}+\mu^{2})\Delta J(\eta,x)=0.$ (26)

The gauge choice which gives the simplest propagator is $\zeta=0,\xi=1$ so
that the field equation in momentum space becomes

$\delta^{\prime}(\eta^{2}+\mu^{2})k^{2}\tilde{\Psi}(\eta,k)=\delta^{\prime}(\eta^{2}+\mu^{2})\tilde{J}(\eta,k).$
(27)

The propagator in momentum space, $\tilde{D}(\eta,k)$, is taken as
$\tilde{\Psi}(\eta,k)=\tilde{D}(\eta,k)\,\tilde{J}(\eta,k)$ so that the
momentum space propagator is simply

$\displaystyle\tilde{D}(\eta,k)=\frac{1}{k^{2}}+\frac{1}{4}(\eta^{2}+\mu^{2})^{2}\tilde{\varepsilon}(\eta,k),$
(28)

with $\tilde{\varepsilon}(\eta,k)$ arbitrary. The propagator then lives only
on the $\eta^{2}+\mu^{2}=0$ hyperboloid and its first neighborhood but with no
contribution to the first order neighborhood. This is expected since in the
classical theory $\Psi(\eta,x)$ also does not propagate beyond the first
neighborhood of the $\eta^{2}+\mu^{2}=0$ hyperboloid.

##  4  Virtual Exchange of Continuous Spin Fields

As remarked before the path integral formalism for continuous spin fields has
to be formulated on a cotangent bundle in such a way that the dynamics is
constrained to the hyperboloid $\eta^{2}+\mu^{2}=0$ and its first
neighborhood. Apart from that the procedure is similar to the case of ordinary
fields in Minkowski spacetime. Then the generating functional for a classical
external source $J(\eta,x)$ is taken to be

$\displaystyle Z[J]=\int{\it
D}\Psi\,\,e^{iS_{gf}[\Psi]+i\int{dx\,d\eta\,\,\delta^{\prime}(\eta^{2}+\mu^{2})\,J(\eta,x)\Psi(\eta,x)}}=e^{i\it{W[J]}},$
(29)

where $S_{gf}$ is the gauge fixed classical action (3) and $W[J]$ is the
generating functional for connected diagrams which can be written in momentum
space as

$\displaystyle W[J]=-\frac{1}{2}\int
dk\,d\eta\,\,\delta^{\prime}(\eta^{2}+\mu^{2})\,\tilde{J}(\eta,k)\tilde{D}(\eta,k)\tilde{J}(\eta,-k),$
(30)

with $\tilde{D}(\eta,k)$ being the propagator for the continuous spin field in
the gauge $\zeta=0,\xi=1$ (28).

In general the static force among charged particles can be derived by the
analysis of virtual particles exchanged by its sources333See Zee:2003mt  for
instance. In the present case continuous spin fields are the gauge fields
carrying a force that intermediate interactions among matter particles charged
under this force. Then we can use the path integral formulation for continuous
spin fields found in the previous section to find the classical force that
acts on the matter particles. This can be achieved by computing the virtual
particle exchange of continuous spin fields. In order to do so we take the
source $J(\eta,x)$ as a matter current that interacts with the continuous spin
field.

So we assume that a virtual continuous spin field is created by a disturbance
of the vacuum and is then absorbed by another disturbance of vacuum. These
disturbances associated to the two charged particles are given by currents
which are assumed to be static and localized in space at $\vec{x}_{1}$ and
$\vec{x}_{2}$

$\displaystyle
J(\eta,\vec{x})=J_{1}(\eta,\vec{x}-\vec{x}_{1})+J_{2}(\eta,\vec{x}-\vec{x}_{2}),$
(31)

and must satisfy the conservation equation (26). We will also neglect current
self-interactions so that (30) becomes

$\displaystyle W_{12}[J]=-\frac{1}{2}\int
dk\,d\eta\,\,\delta^{\prime}(\eta^{2}+\mu^{2})\left(\tilde{J}_{1}(\eta,k)\tilde{D}(\eta,k)\tilde{J}_{2}(-\eta,k)+\tilde{J}_{2}(\eta,k)\tilde{D}(\eta,k)\tilde{J}_{1}(-\eta,k)\right).$
(32)

In this case $Z[J]$ gives the probability amplitude for the creation,
propagation and annihilation of the virtual continuous spin field which can be
written in terms of its Hamiltonian operator ${H}$ as
$<0|e^{-iHT}|0>=e^{-iET}$, where $E$ is the energy change caused by the
disturbance and $T$ the elapsed time. Then the interaction energy due to the
disturbances of the vacuum is given by $E=-W_{12}/T$.

In order to compute the interaction energy due to static disturbances of the
vacuum (32) we have to specify the currents $J_{a}(\eta,x),\,a=1,2$. The
currents must be static and assumed to be concentrated at two different
regions around $\vec{x}_{1}$ and $\vec{x}_{2}$. Since the total current
$J(\eta,x)$ must be conserved (26) we assume that each current
$J_{a}(\eta,\vec{x}_{a})$ is also conserved. For spins 0 and 1 the current is
simply proportional to $\delta(\vec{x}-\vec{x}_{a})$ but in the continuous
spin case this is non longer true and requires the addition of terms such that
the current is not strictly localized at $\vec{x}_{a}$ but spread around
$\vec{x}_{a}$. Besides that it should lead to a non vanishing exchange current
(32). The current dependence on $\eta$ is restricted by (25) so that it goes
up to first order in $\eta^{2}+\mu^{2}$. All this leads to a conserved current
satisfying (26) given by

$\displaystyle
J_{a}(\eta,\vec{x})=\frac{2}{D-2}\,q_{a}\,\biggl{[}\delta(\vec{x}-\vec{x}_{a})-\biggl{.}$
$\displaystyle\left.\frac{1}{(2\pi)^{\frac{D-1}{2}}}\left(\rho\vec{\eta}\cdot\vec{\partial}_{x}+\frac{(\mu\rho)^{2}}{D-2}-\frac{1}{2}\rho^{2}(\eta^{2}+\mu^{2})\right)\left(\frac{\mu\rho}{\sqrt{D-2}}\frac{1}{|\vec{x}-\vec{x}_{a}|}\right)^{\frac{D-3}{2}}K_{\frac{D-3}{2}}\left(\frac{\mu\rho}{\sqrt{D-2}}|\vec{x}-\vec{x}_{a}|\right)\right],$
(33)

where $K_{i}(x)$ is the modified Bessel function of the second kind. Since
$K_{i}(x)$ is complex for $x<0$ we take $\mu\rho$ to be positive in order for
the current to be real. We can also choose $\mu$ to be positive to eventually
set $\mu=1$ which means that $\rho$ is also positive. Notice that
$x^{i}K_{i}(x)$ (no summation over $i$) is divergent at $x=0$ only for $i=0$
which means that $D\geq 4$. Notice also that near $|\vec{x}_{a}|$ the current
goes like $|\vec{x}-\vec{x}_{a}|^{D-2}$ so its localized around
$|\vec{x}_{a}|$. We can also integrate the current over $\eta^{\mu}$ to obtain

$\displaystyle J_{a}(\vec{x})=\int
d\eta\,\delta^{\prime}(\eta^{2}+\mu^{2})J_{a}(\eta,\vec{x})=q_{a}\,\mu^{D-4}\,\delta(\vec{x}-\vec{x_{a}}),$
(34)

showing that from the space-time point of view it is localized at
$\vec{x}_{a}$. We can also compute the flux of this current up to the first
neighborhood of the cotangent bundle and we find that the flux is constant and
non vanishing

$\displaystyle\Phi_{a}=\int
d\vec{x}\,d\eta\,\delta^{\prime}(\eta^{2}+\mu^{2})\,J_{a}(\eta,\vec{x})=-q_{a}\mu^{D-4},$
(35)

and it is independent of $\rho$.

Finally we can compute the interaction energy due to the disturbances of the
vacuum. Inserting (4) in (32) and integrating in $\eta^{\mu}$ we find

$\displaystyle
E=q_{1}\,q_{2}\frac{\mu^{D-4}}{(D-2)\,(2\pi)^{\frac{D-1}{2}}}\times\hskip
213.39566pt$
$\displaystyle\times\left[(D-2)\left(\frac{\mu\rho}{\sqrt{D-2}}\right)^{\frac{D-1}{2}}\frac{1}{|\Delta\vec{x}|^{\frac{D-5}{2}}}K_{\frac{D-1}{2}}\left(\frac{\mu\rho}{\sqrt{D-2}}|\Delta\vec{x}|\right)\right.\hskip
42.67912pt$
$\displaystyle\left.-\left(1+(D-2)(D-3)\right)\left(\frac{\mu\rho}{\sqrt{D-2}|\Delta\vec{x}|}\right)^{\frac{D-3}{2}}K_{\frac{D-3}{2}}\left(\frac{\mu\rho}{\sqrt{D-2}}|\Delta\vec{x}|\right)\right],$
(36)

where $|\Delta\vec{x}|=|\vec{x}_{2}-\vec{x}_{1}|$. At short distances
$|\Delta\vec{x}|<<(\mu\rho)^{-1}$ we find

$\displaystyle
E=-q_{1}\,q_{2}\frac{\mu^{D-4}\,\Gamma\left(\frac{D-3}{2}\right)}{16(D-2)\,\pi^{\frac{D-1}{2}}}\,\frac{1}{|\Delta\vec{x}|^{D-3}},$
(37)

so that the force is attractive for charges $q_{1}$ and $q_{2}$ of the same
sign and it is independent of $\rho$. For large distances
$|\Delta\vec{x}|>>(\mu\rho)^{-1},$

$\displaystyle
E=q_{1}\,q_{2}\frac{\mu^{D-4}}{2\,(2\pi)^{\frac{D-2}{2}}}\left(\frac{\mu\rho}{\sqrt{D-2}}\right)^{\frac{D-2}{2}}\frac{e^{-\frac{\mu\rho|\Delta\vec{x}|}{\sqrt{D-2}}}}{|\Delta\vec{x}|^{\frac{D-4}{2}}},$
(38)

the force is repulsive for charges of same sign and it depends on $\rho$.
Notice that at short distances we get the standard behavior for low spin
massless particles since $E$ goes like $-1/|\Delta\vec{x}|^{D-3}$. At large
distances however we get the surprising behavior that $E$ approaches zero as
${e^{-\frac{\mu\rho|\Delta\vec{x}|}{\sqrt{D-2}}}}/{|\Delta\vec{x}|^{\frac{D-4}{2}}}$.

In Fig.1 we plot $E$ for the case $D=4$ when $q_{1}$ and $q_{2}$ have the same
sign. Notice that for $D>4$ we get a similar behavior. There is just one point
of maximum for any $D$ which is the turning point where the attractive force
becomes repulsive as $|\Delta\vec{x}|$ grows. Taking the derivative of $E$
with respect to $|\Delta\vec{x}|$ we find

$\displaystyle
K_{\frac{D-1}{2}}\left(\frac{\mu\rho}{\sqrt{D-2}}|\Delta\vec{x}|\right)-\sqrt{D-2}\,\mu\rho\,|\Delta\vec{x}|\,K_{\frac{D-3}{2}}\left(\frac{\mu\rho}{\sqrt{D-2}}|\Delta\vec{x}|\right)=0,$
(39)

which can be used to determine the turning points of $E$. In general (39) has
several real and complex roots. It is quite remarkable that there is just one
positive real root. Its value depends on $D$ being
$|\Delta\vec{x}|=\sqrt{2}/(\mu\rho)$ for $D=4$ and growing like
$\sqrt{D}/(\mu\rho)$ for $D>>1$. We could not find a closed form for it. The
roots of $E$ can also be determined. Again, there are real and complex roots
but just one which is real and positive.

Figure 1: $E\times x$ graphics for $D=4$ with
$x=\frac{\mu\rho|\Delta\vec{x}|}{\sqrt{2}}$ dimensionless and charges $q_{1}$
and $q_{2}$ of the same sign. $E$ has a maximum at $x=1$ and vanishes for
finite $x$ at $x=1/2$. For other values of $D$ we have a similar behavior.

For $D=4$ (see Fig.1) our results are quite simple since

$\displaystyle
E=\frac{q_{1}\,q_{2}}{8\pi}\left(\sqrt{2}{\mu\rho}-\frac{1}{|\Delta\vec{x}|}\right)e^{-\frac{\mu\rho}{\sqrt{2}}|\Delta\vec{x}|}.$
(40)

For $|\Delta\vec{x}|<(\sqrt{2}\mu\rho)^{-1}$ the charged particles attract
each other if $q_{1}$ and $q_{2}$ have the same sign while for
$|\Delta\vec{x}|>(\sqrt{2}\mu\rho)^{-1}$ the charges repel each other. At
$|\Delta\vec{x}|=(\sqrt{2}\mu\rho)^{-1}$ there is no force between them.

Back to the general $D$ case we find that for $\rho=0$ both terms in (4) are
non vanishing and we get

$\displaystyle
E=-q_{1}\,q_{2}\,\frac{\mu^{D-4}\,\Gamma(\frac{D-3}{2})}{4\,(D-2)\,\pi^{\frac{D-1}{2}}}\frac{1}{|\Delta\vec{x}|^{D-3}},$
(41)

so that charged particles with the same sign always attract each other. For
$\rho=0$ the free continuous spin field reduces to a massless higher spin
theory with helicities going from zero to infinity Schuster:2014hca ;
Rivelles:2014fsa . Since the no-go theorems Weinberg:1964ew ; Weinberg:1965rz
; Porrati:2012rd  show that higher spin fields can not mediate long range
interactions we expect that for $\rho=0$ only the lower helicities survive in
(41).

##  5  Conserved Current Degrees of Freedom

Now we want to compute the generating functional (30) with the propagator (28)
to show that the exchanged continuous spin field carries all integer
helicities, each one of them appearing just once.

Taking into account that the current $J(\eta,x)$ will be expanded around the
hyperboloid $\eta^{2}+\mu^{2}=0$ let us introduce new coordinates
$(|\eta|,\hat{\eta}^{\mu}),\,\,\eta^{\mu}=|\eta|\hat{\eta}^{\mu}$, satisfying
the constraint $\hat{\eta}^{2}=-1$, so that $\hat{\eta}^{\mu}$ parametrizes
points on the hyperboloid while $|\eta|$ parametrizes the hyperboloids
Rivelles:2018tpt . It must be noticed that these new coordinates must be
handled carefully due to the constraint imposed on them. In the new
coordinates the current $J(\eta,x)$ can be expanded as

$\displaystyle
J(\eta,x)=\sum_{n=0}^{\infty}\frac{1}{n!}\,(\mu^{2}-|\eta|^{2})^{n}\,J^{T}_{n}(\hat{\eta},x),$
(42) $\displaystyle
J^{T}_{n}(\hat{\eta},x)=\sum_{p=0}^{\infty}\frac{1}{p!}\hat{\eta}^{\mu_{1}}\cdots\hat{\eta}^{\mu_{p}}J^{T}_{\mu_{1}\dots\mu_{p}}(x),$
(43)

where $J^{T}_{\mu_{1}\dots\mu_{p}}(x)$ is traceless due to the constraint
$\hat{\eta}^{2}=-1$. We can now use the $\Xi$ symmetry of (25) and expand its
parameter as in (42) so that all terms in the expansion (42) can be removed
leaving only the first two terms

$\displaystyle
J(\eta,x)=J_{0}^{T}(\hat{\eta},x)+(\mu^{2}-|\eta|^{2})\,J_{1}^{T}(\hat{\eta},x).$
(44)

The current conservation equation (26) can then be written in momentum space
as

$\displaystyle\delta(\mu^{2}-|\eta|^{2})\left[\left(\rho-\frac{1}{\mu}ik\cdot\partial_{\hat{\eta}}\right){\tilde{J}}_{0}^{T}(\hat{\eta},k)-2\mu\,ik\cdot\hat{\eta}\,{\tilde{J}}_{1}^{T}(\hat{\eta},k)\right]=0,$
(45)

showing that ${\tilde{J}}_{0}^{T}$ and ${\tilde{J}}_{1}^{T}$ are not
independent. Since we are on-shell we can choose $k_{+}$ as the only non-
vanishing component of the momentum and use (43) to solve (45) as

$\displaystyle{\tilde{J}^{T}}_{0,-\mu_{1}\dots\mu_{n}}(k_{+})=\frac{\mu\rho}{ik_{+}}\,{\tilde{J}^{T}}_{0,\mu_{1}\dots\mu_{n}}(k_{+}),$
(46)
$\displaystyle{\tilde{J}^{T}}_{1,\mu_{1}\dots\mu_{n}}(k_{+})=-\frac{n}{2\mu^{2}}\,{\tilde{J}^{T}}_{0,\mu_{1}\dots\mu_{n}}(k_{+}).$
(47)

Then $\tilde{J}^{T}_{1}$ can be expressed in terms of $\tilde{J}^{T}_{0}$ so
that

$\displaystyle{\tilde{J}^{T}}_{0,(-)^{p}\,\overline{\mu}_{1}\dots\overline{\mu}_{n}}(k_{+})=\left(\frac{\mu\rho}{ik_{+}}\right)^{p}\,{\tilde{J}^{T}}_{0,\overline{\mu}_{1}\dots\overline{\mu}_{n}}(k_{+}),$
(48)

where $\overline{\mu}=(+,i)$. This means that all minus components of
$\tilde{J}^{T}_{0}$ can be removed and only the $+$ and $i$ components of
$\tilde{J}^{T}_{0}$ are left. We can now consider the traceless condition of
$\tilde{J}^{T}_{0}$ which reads

$\displaystyle
2\tilde{J}^{T}_{0,+-\mu_{1}\dots\mu_{n}}(k_{+})=\tilde{J}^{T}_{0,ii\mu_{1}\dots\mu_{n}}(k_{+}),$
(49)

and use (48) in it to get

$\displaystyle\tilde{J}^{T}_{0,+\mu_{1}\dots\mu_{n}}(k_{+})=\frac{ik_{+}}{2\mu\rho}\,\tilde{J}^{T}_{0,ii\mu_{1}\dots\mu_{n}}(k_{+}).$
(50)

This last relation together with (46) allow us to write all $+$ and $-$
components of $\tilde{J}^{T}_{0}$ in terms of those with i components only.
Thus the independent components of the current are
$\tilde{J}^{T}_{0,i_{1}\dots i_{n}}(k_{+})$ which are traceless in their
$SO(D-2)$ indices.

Using the propagator (28) we can write $W[J]$ (30) as

$\displaystyle W[J]=-\frac{1}{2}\int\frac{dk}{k^{2}}\,I(k_{+}),$ (51)

where

$\displaystyle I(k_{+})=I(k)\Big{|}_{k^{2}=0},\qquad I(k)=\int
d\eta\,\delta^{\prime}(\eta^{2}+\mu^{2})\,\tilde{J}(\eta,k)\,\tilde{J}(\eta,-k).$
(52)

The next step is to expand $\tilde{J}(\eta,k_{+})$ over $\eta$ and perform the
integral in (52). Using the results of Appendix A we find

$\displaystyle
I(k_{+})=\frac{1}{2}\mu^{D-4}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\frac{(D+2n-2)(D+2n)}{\prod\limits_{\ell=0}^{n}(D+2\ell)}\,{\tilde{J}}_{0}^{T\,\mu_{1}\dots\mu_{n}}(k_{+})\tilde{J}^{T}_{0\,\mu_{1}\dots\mu_{n}}(-k_{+}).$
(53)

We can now use (46) and (50) to eliminate the $+$ and $-$ components of
$\tilde{J}^{T}_{0}$ leaving only its $SO(D-2)$ indices. After some work (53)
can be rewritten as

$\displaystyle
I(k_{+})=\frac{1}{2}\mu^{D-4}\sum_{n=0}^{\infty}\frac{(D-2)!!}{(D+2n-4)!!}\sum_{p=0}^{n}\frac{(-1)^{p}}{2^{p}(n-p)!}\sum_{q=0}^{p}\frac{1}{q!(p-q)!}{\tilde{J}}^{T(q)}_{i_{1}\dots
i_{n-p}}(k_{+})\,{\tilde{J}}^{T(p-q)}_{i_{1}\dots i_{n-p}}(-k_{+}),$ (54)

where $q$ in ${\tilde{J}}^{T(q)}_{i_{1}\dots i_{n}}$ means $q$ traces of
${\tilde{J}}^{T}_{i_{1}\dots i_{n}}$ in its $SO(D-2)$ indices. It must also be
noticed that ${\tilde{J}}^{T(q)}_{i_{1}\dots i_{n}}$ is not $SO(D-2)$
traceless. Another important point is that a term like
${{\tilde{J}}^{T(p)}}_{i_{1}\dots i_{n}}(k_{+}){\tilde{J}}^{T(q)}_{i_{1}\dots
i_{n}}(-k_{+})$ appears just once in the summation.

If (54) is to describe the exchange of a continuous spin field it must be
possible to rewrite it as a sum of $SO(D-2)$ traceless symmetric tensors with
ranks from zero to infinity, each rank appearing just once. Let us denote this
tensor as $\tilde{\cal J}^{t}_{i_{1}\dots i_{n}}$ where $t$ means traceless
only in the $SO(D-2)$ indices. This tensor must be obtained by taking
traceless linear combinations of ${\tilde{J}}^{T}_{i_{1}\dots i_{n}}$ and also
of his traces ${\tilde{J}}^{T(p)}_{i_{1}\dots i_{n}}$

$\displaystyle{\tilde{\cal J}}^{t}_{i_{1}\dots
i_{n}}=\sum_{p=0}^{\infty}\alpha_{p}^{n}{\tilde{J}}^{T\,(p)}_{i_{1}\dots
i_{n}}.$ (55)

Then we must compute

$\displaystyle\frac{1}{2}\mu^{D-4}\sum_{n=0}^{\infty}\frac{1}{n!}\frac{(D-2)!!}{(D+2n-4)!!}{{\cal{\tilde{J}}}^{t}}_{i_{1}\dots
i_{n}}(k_{+}){{\cal{\tilde{J}}}^{t}}_{i_{1}\dots i_{n}}(-k_{+}),$ (56)

express it in terms of $\tilde{J}^{T(p)}_{i_{1}\dots i_{n}}$ and compare with
(54) to find the coefficients $\alpha^{n}_{p}$. As remarked before, in (54)
each tensor product ${\tilde{J}}^{T(p)}_{i_{1}\dots
i_{n}}(k_{+})\tilde{J}^{T(q)}_{i_{1}\dots i_{n}}(-k_{+})$ appears just once
but in (56) it may appear more than once. Comparing the first few terms of
both expansions we can find the coefficients $\alpha_{p}^{n}$ for low values
of $p$ and $n$ which suggests a simple pattern for the higher values of $p$
and $n$

$\displaystyle\alpha_{p}^{n}=\frac{(-1)^{p}}{(2p)!!}\frac{(D+2n-4)!!}{(D+2n+2p-4)!!}.$
(57)

Then (56) can be rewritten as

$\displaystyle\frac{1}{2}\mu^{D-4}(D-2)!!\sum_{n=0}^{\infty}(D+2n-4)\,\sum_{p=0}^{[n/2]}\left(-\frac{1}{2}\right)^{p}\frac{(D-2p+2n-6)!!}{p!(n-2p)!}\times$
$\displaystyle\times\sum_{a,b=0}^{\infty}\frac{(-1)^{a+b}}{(2a)!!(2b)!!}\frac{1}{(D+2n+2a-4)!!(D+2n+2b-4)!!}\tilde{J}^{T(a+p)}_{i_{1}\dots
i_{n-2p}}(k_{+})\tilde{J}^{T(b+p)}_{i_{1}\dots i_{n-2p}}(-k_{+}).$ (58)

It is not possible to compare (54) and (5) directly since the range of the
summations are quite different. However we can compare the coefficients for
the currents in both cases. So lets us consider a generic term
$\tilde{J}^{T(\ell)}_{i_{1}\dots i_{r}}(k_{+})\,\tilde{J}^{T(m)}_{i_{1}\dots
i_{r}}(-k_{+})$ with $\ell\geq m$. Its coefficient in (54) is

$\displaystyle\frac{(-1)^{l+m}(D-2)!!}{2^{D/2+2\ell+2m+r-2}(D/2+\ell+m+r-2)!\,\ell!\,m!\,r!},$
(59)

while in (5) it is

$\displaystyle\frac{(-1)^{\ell+m}(D-2)!!}{2^{D/2+2\ell+2m+r-2}r!}\sum_{p=0}^{m}\frac{(-1)^{p}(D/2+2p+r-2)(D/2+p+r-3)!}{p!(\ell-p)!(m-p)!(D/2+p+r+\ell-2)!(D/2+p+r+m-2)!},$
(60)

so if they are the same we must have

$\displaystyle\sum_{p=0}^{m}\frac{(-1)^{p}(D/2+2p+r-2)(D/2+p+r-3)!}{p!(\ell-p)!(m-p)!(D/2+p+r+\ell-2)!(D/2+p+r+m-2)!}=$
$\displaystyle\frac{1}{\ell!m!(D/2+\ell+m+r-2)!},\qquad\ell\geq m.$ (61)

This is an equation which depends on $\ell,m,$ and $r$. For low numeric values
of $m$ and any arbitrary value of the others two parameters we can show that
this equation holds using symbolic computation software. If we use only
numerical values for all parameters then (5) was satisfied for all choices
that were made.

It is also remarkable that (5) can be rewritten in terms of the generalized
hypergeometric function ${}_{3}F_{2}(a_{1},a_{2},a_{3};b_{1},b_{2};x)$ as

$\displaystyle(D/2+r-2)!\left[\frac{{}_{3}F_{2}(D/2+r-2,-\ell,-m,D/2+\ell+r-1,D/2+m+r-1;-1)!}{({(D/2+\ell+r-2)!(D/2+m+r-2)!})}\right.$
$\displaystyle\left.-\frac{2\,\ell\,m\,\,{}_{3}F_{2}(D/2+r-1,1-\ell,1-m,D/2+\ell+r,D/2+m+r;-1)}{(D/2+\ell+r-1)!(D/2+m+r-1)!}\right]$
$\displaystyle=$ $\displaystyle\frac{1}{(D/2+\ell+m+r-2)!},\qquad\ell\geq m.$
(62)

Again, giving numerical values for $\ell,m$ and $r$ we found that (5) is
always satisfied. Unfortunately we were not able to find any property of
hypergeometric functions that allows us to show that equation (5) holds for
all values of $\ell,m$ and $r$. All these results provide a very strong
evidence that the exchanged fields carries the right number of degrees of
freedom for a continuous spin particle.

##  6 Conclusions

We have shown that like the photon and the graviton the continuous spin
particle mediates long range interactions. We have found the interaction
energy between two static particles having charges of the same sign which are
coupled to a continuous spin field. This shows that continuous spin fields can
have consistent long range interactions with matter closing a gap in our
understanding of massless particles and long range physics. It would be
interesting to extended theses results to the other kinds of continuous spin
particles like the fermionic continuous spin particle Najafizadeh:2015uxa ,
the supersymmetric spin particle Najafizadeh:2019mun  and the tachyonic spin
particle Rivelles:2018tpt .

It also quite interesting that at small distances charges of the same sign are
attractive while at large distances they repel each other. Since continuous
spin fields can propagate in (A)dS spaces Metsaev:2016lhs ; Metsaev:2017ytk ;
Metsaev:2019opn ; Metsaev:2021zdg  it is quite important to extend our
formalism to (A)dS and search for a connection involving dark matter and dark
energy.

##  Appendix A Appendix

Since integrals over the hyperboloid like $\int
d\eta\,\,\delta(\eta^{2}+\mu^{2})\,\eta^{\mu_{1}}\cdots\eta^{\mu_{n}}$ are
divergent they must be regulated. As discussed in detail in Appendix A of
Schuster:2014hca  we can regulate these integrals by analytic continuation in
$\eta$ space taking Wick rotated coordinates
$[\eta]^{\mu}=(i\eta^{0},\eta^{1},\dots,\eta^{D-1})$. To keep (engineering)
dimension counting right we choose the normalization $\int
d^{D}[\eta]\,\delta(-[\eta]^{2}+\mu^{2})=\mu^{D-2}$. In Schuster:2014hca  just
the the cases with low values of $n$ were given. Here we present the results
for arbitrary values of $n\geq 2$

$\displaystyle\int
d[\eta]\,\delta^{\prime}(-[\eta]^{2}+\mu^{2})[\eta]^{\mu_{1}}\cdots[\eta]^{\mu_{2n}}=\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle\frac{1}{2}(-1)^{n}\mu^{D+2(n-2)}\frac{(D-2)(D-4)}{\prod\limits_{\ell=0}^{n}(D-4+2\ell)}(g^{\mu_{1}\mu_{2}}\cdots
g^{\mu_{2n-1}\mu_{2n}}+permutations),$ (63)

where the permutations have weight 1.

###### Acknowledgements.

This work was supported by FAPESP Grant 2019/21281-4.

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