aainstitutetext:  Mani L. Bhaumik Institute for Theoretical Physics
Department of Physics and Astronomy
University of California, Los Angeles, CA 90095, USA

# Consistent actions for massive particles interacting with electromagnetism
and gravity

Lukas W. Lindwasser<EMAIL_ADDRESS>
###### Abstract

Consistent interactions with electromagnetism and gravity for mass $m$
particles of any spin are obtained. This is done by finding interactions which
preserve the covariantized massive gauge symmetry present in recently
constructed massive particle actions. This gauge principle is sufficient for
finding consistent completions of minimal as well as non-minimal couplings of
any type. For spins $s\geq 3/2$, consistency requires infinitely many
interaction terms in the action, including arbitrarily high order derivatives
of electromagnetic and gravitational curvatures, with correspondingly high
powers of $1/m$. These interactions may be formally resummed and expressed in
terms of non-local operators. The inherent non-locality is a manifestation of
the known causality problems present in interacting massive particles with
spin $s\geq 3/2$.

##  1 Introduction

There exists in nature composite massive particles with spins higher than 1,
the highest spin particle observed at the moment being the $\Delta(2950)$
baryon with spin 15/2 ParticleDataGroup:2022pth . The basic problem of
modelling such particles interacting with the most relevant forces they
experience during their lifetime, electromagnetism and gravity, has a long
history, first considered by Fierz and Pauli in 1939 Fierz:1939ix . In that
work, they noted that introducing interactions generically does not preserve
the massive higher spin particle’s degrees of freedom, making the model
inconsistent with unitarity.

Another challenge to the consistent modelling of interacting massive higher
spin particles is that interactions violate causality. For instance, it was
realized by Velo and Zwanziger in 1969 PhysRev.186.1337  that minimally
coupling spin 3/2 particles to an electromagnetic background allows for
superluminal propagation. Subsequently, arguments Camanho:2014apa ; Afkhami-
Jeddi:2018apj  were made that any theory with a finite collection of
interacting massive particles of spin higher than 2 cannot avoid causality
violations at sufficiently high energies. For spin 3/2 particles, a solution
to this and the degree of freedom problem was found by realizing the particle
within the framework of supergravity Deser:1977uq . For higher spins, the only
known solution to both is to realize them within string theory Porrati:2010hm
, which has an infinite collection of higher spin particle excitations.

Barring these special solutions, these results are usually taken to mean that
massive higher spin particles cannot be elementary Arkani-Hamed:2017jhn ;
Afkhami-Jeddi:2018apj . This is related to the fact that tree level scattering
amplitudes involving massive higher spin particles have bad high energy
behavior which violate unitarity bounds. It is known that there are unique
non-minimal couplings with electromagnetism and gravity that improves the bad
high energy behavior in some, but importantly not all, scattering processes
Ferrara:1992yc ; Cucchieri:1994tx . Furthermore, loop diagrams involving
particles with spin 3/2 and higher result in nonrenormalizeable divergences.
At energies where unitarity starts to break down, the particle description
must be replaced with some yet more fundamental description. Indeed, all
massive particles/resonances with spin 3/2 and higher we have observed are
understood to be composites of low spin particles. For these reasons,
modelling interacting massive higher spin particles directly is firmly within
the regime of effective field theory.

At the moment, there is an active program of research modelling the inspiral
of two spinning black holes or neutron stars using effective field theory
techniques, treating them as point particles with very high spin
Guevara:2017csg ; Chung:2018kqs ; Guevara:2018wpp ; Arkani-Hamed:2019ymq ;
Guevara:2019fsj ; Bern:2020buy ; Bern:2020uwk ; Bern:2022kto ;
Chiodaroli:2021eug ; FebresCordero:2022jts ; Aoude:2022thd ; Cangemi:2022bew ,
in order to explain recently detected gravitational waves
LIGOScientific:2016aoc ; LIGOScientific:2017vwq ; LIGOScientific:2021djp .
This has reinvigorated research in the consistent modelling of interacting
massive higher spin particles.

A low energy effective field theory with an energy cutoff $\Lambda$ never sees
the aforementioned bad high energy behavior. Causality violations meanwhile
are suppressed but still detectable if a superluminal signal with speed
difference $\Delta v=v-c>0$ propagates for a sufficiently long time $T$ such
that $\Delta vT\gtrsim 1/\Lambda$, or if $\Lambda$ is parametrically larger
than the higher spin particle’s mass $m$. The original degree of freedom
problem on the other hand persists in an effective field theory and is still
of fundamental importance. The generic solution to this problem is understood
to be one which has been employed successfully for spin $1,3/2,2$ particles,
i.e. gauge invariance. For free particles, gauge symmetry is an invaluable
tool for ensuring that the field theory description used has the correct
physical degrees of freedom while keeping Lorentz invariance manifest. There
is a reliable perturbative procedure for introducing interactions consistent
with the gauge symmetry, while also possibly deforming the gauge
transformations, described for instance in Berends:1984rq  and references
therein. In the case of spin 1 particles, gauge invariance in interacting
theories implies generalized Ward identities which guarantee that all Feynman
diagrams with external legs with unphysical polarizations vanish
tHooft:1971akt ; tHooft:1971qjg . Thus whenever perturbation theory is valid,
the gauge invariant interacting theory can continue describing a spin 1
particle at the level of the $S$-matrix. A natural expectation is that this
general result will continue to hold for higher spins, although an independent
proof of this is necessary. One complication of such a proof is that when
interactions are introduced, one generally needs to correspondingly deform the
gauge symmetry, as is what happens for instance in Yang-Mills theory and
general relativity. The exact form of the generalized Ward identities in turn
depends on the gauge symmetry. One should therefore first find a consistent
set of interactions and corresponding deformations of the gauge symmetry.

Progress in this direction has been made in the past, including work at the
level of equations of motion Porrati:2010hm ; Cortese:2013lda ; Rahman:2016tqc
; Rahman:2020qal , and at the Lagrangian level for constrained external
backgrounds Argyres:1989cu ; Porrati:2009bs ; Buchbinder:2000fy ;
Zinoviev:2006im ; Zinoviev:2008ck ; Zinoviev:2009hu ; Buchbinder:2012iz . An
important challenge is to find consistent actions of massive higher spin
fields interacting in an arbitrary electromagnetic and gravitational field, so
that they too can be treated as dynamical. The general expectation is that an
exactly gauge invariant theory of massive higher spins with electromagnetism
and gravity will require infinitely many interactions, including all orders of
derivatives of curvatures, and correspondingly all orders in $1/m$. In this
paper, a prescription for writing down all such interactions is found for any
spin.

To facilitate this, it was important to find an action principle for massive
higher spins, recently detailed in a previous paper Lindwasser:2023zwo , which
is simple enough to determine the set of consistent interactions. The original
Singh-Hagen actions Singh:1974qz ; Singh:1974rc  are not suited for this task
for at least two reasons. First, although they successfully describe a free
massive spin $s$ particle with the correct degrees of freedom, they achieve
this without an underlying gauge symmetry, and so there is no guiding
principle for adding interactions which preserve the degrees of freedom.
Second, these actions are only consistent in four spacetime dimensions, making
it unclear how to dimensionally regulate. Actions for massive higher spins
which address both of these points were constructed in Zinoviev:2001dt ;
Metsaev:2006zy . Introducing consistent interactions to these actions is
possible, but technically challenging.

In contrast, the actions $S_{n}$ and $S_{n+1/2}$ constructed in
Lindwasser:2023zwo  are built from gauge invariant field strengths
$\mathcal{F}_{n-i}$, $i=0,1,2,3$ for integer spins and $\mathcal{S}_{n-i}$,
$i=0,1,2$ for half integers, respectively, which satisfy massive Bianchi
identities111The name “massive Bianchi identity” comes from the fact that they
descend from massless higher spin identities, explained in Lindwasser:2023zwo
, which are analogous to the linearized second contracted Bianchi identity in
Riemannian geometry when the spin $s=n=2$. (2.15), (2.16), (2.23), enabling a
simple prescription for finding exactly gauge invariant interactions. While
the general procedure for adding consistent interactions Berends:1984rq
entails also deforming the free theory’s gauge transformations, possibly
including terms proportional to arbitrary powers of electromagnetic and
gravitational curvatures $F_{\mu\nu}$ and $R_{\omega\sigma\mu\nu}$, we find it
sufficient to minimally deform the gauge transformations by simply replacing
spacetime derivatives present in them with covariant derivatives, so that it
is at least consistent with the $U(1)$ electromagnetic gauge and general
coordinate invariance. Given this gauge principle, the gauge variation of the
actions $\delta S_{n}$ and $\delta S_{n+1/2}$ depends only on the
aforementioned properties of the field strengths.

Minimally coupling the massive higher spin fields $\Phi_{n-i},\,\Psi_{n-i}$ to
electromagnetism and gravity modifies the field strengths so that they are no
longer gauge invariant $\delta\mathcal{F}_{n-i}$, $\delta\mathcal{S}_{n-i}\neq
0$, and they no longer satisfy massive Bianchi identities. This in turn breaks
the gauge invariance of the actions $\delta S_{n},\,\delta S_{n+1/2}\neq 0$.
Because of the way in which the gauge symmetry is broken, the gauge symmetry
can be restored by further modifying the field strengths
$\mathcal{F}_{n-i}\to\mathcal{F}_{n-i}+\Delta\mathcal{F}_{n-i}$ and
$\mathcal{S}_{n-i}\to\mathcal{S}_{n-i}+\Delta\mathcal{S}_{n-i}$ so that they
again are gauge invariant
$\delta(\mathcal{F}_{n-i}+\Delta\mathcal{F}_{n-i}),\,\delta(\mathcal{S}_{n-i}+\Delta\mathcal{S}_{n-i})=0$,
and satisfy massive Bianchi identities. $\Delta\mathcal{F}_{n-i}$ and
$\Delta\mathcal{S}_{n-i}$ remain linear in $\Phi_{n-i}$ and $\Psi_{n-i}$,
respectively, but include powers of $F_{\mu\nu}$ and $R_{\omega\sigma\mu\nu}$
and all powers of their derivatives. More general interactions, corresponding
for instance to multipole and tidal couplings classified in Bern:2020buy ;
Bern:2020uwk , can be made gauge invariant in the same way.

###  1.1 Conventions

Throughout this paper, we use the metric signature
$\eta=\text{diag}(-1,+1,\cdots,+1)$. In terms of the $U(1)$ gauge field
$A_{\mu}$ and Levi-Civita connection $\Gamma^{\lambda}_{\;\;\mu\nu}$, our
convention for electromagnetic and gravitational curvatures are

$\displaystyle F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ (1.1)
$\displaystyle
R^{\lambda}_{\;\;\sigma\mu\nu}=\partial_{\mu}\Gamma^{\lambda}_{\;\;\nu\sigma}-\partial_{\nu}\Gamma^{\lambda}_{\;\;\mu\sigma}+\Gamma^{\lambda}_{\;\;\mu\kappa}\Gamma^{\kappa}_{\;\;\nu\sigma}-\Gamma^{\lambda}_{\;\;\nu\kappa}\Gamma^{\kappa}_{\;\;\mu\sigma}$
(1.2)

Finally, we will on occasion write
$\gamma_{\mu\nu}\equiv\frac{1}{2}[\gamma_{\mu},\gamma_{\nu}]$ for shorthand,
where $\gamma_{\mu}$ are Dirac matrices satisfying a Clifford algebra for
either the flat $\eta_{\mu\nu}$ or curved $g_{\mu\nu}$ metric depending on the
context.

###  1.2 Outline

In section 2, we review the hyperfield formalism that will help simplify the
subsequent analysis, and the free actions for massive integer and half integer
particles constructed in Lindwasser:2023zwo . In section 3, we present exactly
gauge invariant interactions between massive integer spin particles and
electromagnetism and gravity in subsection 3.1, and then repeat the analysis
for half integer spin particles in subsection 3.2. Finally in section 4, we
discuss our results and conclude with some directions for future research.

##  2 Free theory

In this section, we briefly review the formulation of free massive spinning
particles Lindwasser:2023zwo  that will be used as our starting point for
incorporating interactions.

Particles with spin are covariantly described by totally symmetric tensors
$\phi_{\mu_{1}\cdots\mu_{n}}(x)$ for spin $n$, or spinor tensors
$\psi_{\mu_{1}\cdots\mu_{n}}(x)$ for spin $n+1/2$. The formulation of these
particles dramatically simplifies after the introduction of an auxiliary
vector coordinate $s^{\mu}$, and instead writing the theory in terms of a
“hyperfield”
$\Phi_{n}(X,s)=\frac{1}{n!}i^{-n/2}\phi_{\mu_{1}\cdots\mu_{n}}(X)s^{\mu_{1}}\cdots
s^{\mu_{n}}$, where the factor $i^{-n/2}$ is added for later convenience. All
of the basic operations needed for constructing the covariant actions for
massive spinning particles can be performed at the level of the hyperfield.
For instance, one often has to take traces of the fields
$\phi^{\lambda}_{\;\;\lambda\mu_{3}\cdots\mu_{n}}(X)$. This can be achieved at
the level of $\Phi_{n}(X,s)$ by taking the Laplacian with respect to $s^{\mu}$

$\displaystyle
i\,\partial_{s}^{2}\Phi_{n}(X,s)=\frac{1}{(n-2)!}i^{-(n-2)/2}\phi^{\lambda}_{\;\;\lambda\mu_{1}\cdots\mu_{n-2}}(X)s^{\mu_{1}}\cdots
s^{\mu_{n-2}}$ (2.1) The divergence
$\partial^{\lambda}\phi_{\lambda\mu_{2}\cdots\mu_{n}}(X)$ may also be written
in terms of $\Phi_{n}(X,s)$ $\displaystyle
i^{1/2}\,\partial_{s}\cdot\partial_{X}\Phi_{n}(X,s)=\frac{1}{(n-1)!}i^{-(n-1)/2}\partial^{\lambda}\phi_{\lambda\mu_{1}\cdots\mu_{n-1}}(X)s^{\mu_{1}}\cdots
s^{\mu_{n-1}}$ (2.2) The symmetric derivative
$\partial_{(\mu_{1}}\phi_{\mu_{2}\cdots\mu_{n+1})}(X)$ is written in terms of
$\Phi_{n}(X,s)$ via $\displaystyle
i^{-1/2}s\cdot\partial_{X}\Phi_{n}(X,s)=\frac{1}{n!}i^{-(n+1)/2}\partial_{(\mu_{1}}\phi_{\mu_{2}\cdots\mu_{n+1})}(X)s^{\mu_{1}}\cdots
s^{\mu_{n+1}}$ (2.3) One can also contract indices between two equal rank $n$
hyperfields $A_{n}(X,s)$ and $B_{n}(X,s)$
$\displaystyle\int\frac{d^{d}sd^{d}s^{\prime}}{(2\pi)^{d}}e^{is\cdot
s^{\prime}}A_{n}(X,s)B_{n}(X,s^{\prime})=\frac{1}{n!}a_{\mu_{1}\cdots\mu_{n}}b^{\mu_{1}\cdots\mu_{n}}$
(2.4) These four operations are sufficient for the formulation of integer spin
fields. For half integer spin fields, contraction with gamma matrices
$\gamma^{\mu}$ will also be necessary. For a Dirac hyperfield $\Psi_{n}(X,s)$,
we have $\displaystyle
i^{1/2}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\Psi_{n}(X,s)=\frac{1}{(n-1)!}i^{-(n-1)/2}\gamma^{\lambda}\psi_{\lambda\mu_{1}\cdots\mu_{n}}(X)s^{\mu_{1}}\cdots
s^{\mu_{n}}$ (2.5)

The formula (2.4) is particularly interesting, as it suggests introducing a
(pseudo) inner product on the space of hyperfields

$\displaystyle(A_{n},B_{n})=\int
d^{d}X\frac{d^{d}sd^{d}s^{\prime}}{(2\pi)^{d}}e^{is\cdot
s^{\prime}}\tilde{A}_{n}(X,s)B_{n}(X,s^{\prime})=\int
d^{d}X\frac{1}{n!}a^{*}_{\mu_{1}\cdots\mu_{n}}b^{\mu_{1}\cdots\mu_{n}}$ (2.6)
$\displaystyle\text{where
}\tilde{A}_{n}(X,s)=\frac{1}{n!}i^{-n/2}a^{*}_{\mu_{1}\cdots\mu_{n}}(X)s^{\mu_{1}}\cdots
s^{\mu_{n}}$ (2.7)

and we may consider the (pseudo) Hilbert space of hyperfields $\Phi(X,s)$ with
finite (pseudo) norm $(\Phi,\Phi)<\infty$222$(A,B)$ is only positive definite
in Euclidean signature. Bargmann:1977gy . Note that
$(i^{-1/2}s^{\mu}A,B)=(A,i^{1/2}\partial_{s}^{\mu}B)$, and so for instance,
the divergence and symmetric derivative are anti-Hermitian adjoints of each
other
$(i^{1/2}\partial_{s}\cdot\partial_{X})^{\dagger}=-i^{-1/2}s\cdot\partial_{X}$
in this space.

###  2.1 Integer spins

As detailed in Lindwasser:2023zwo , the covariant formalism of a complex
massive spin $n$ particle we work with not only uses a rank $n$ hyperfield
$\Phi_{n}(X,s)$, but also three auxiliary hyperfields $\Phi_{n-1}(X,s)$,
$\Phi_{n-2}(X,s)$, and $\Phi_{n-3}(X,s)$. These auxiliary hyperfields couple
to $\Phi_{n}(X,s)$ in such a way so that after imposing the equations of
motion, the auxiliary hyperfields may be set to zero, and $\Phi_{n}(X,s)$ is a
Fierz-Pauli system Fierz:1939ix  in hyperspace

$\displaystyle(\partial_{X}^{2}-m^{2})\Phi_{n}$ $\displaystyle=0$
$\displaystyle\partial_{s}\cdot\partial_{X}\Phi_{n}$ $\displaystyle=0$
$\displaystyle\partial_{s}^{2}\Phi_{n}$ $\displaystyle=0$ (2.8)

$\Phi_{n}(X,s)$ therefore has the correct degrees of freedom for the
description of a massive spin $n$ particle, with mass $m$. This special
coupling is facilitated by demanding that the theory be invariant under the
gauge transformations

$\displaystyle\delta\Phi_{n}=i^{-1/2}s\cdot\partial_{X}\epsilon_{n-1}$
$\displaystyle\delta\Phi_{n-1}=i^{-1/2}s\cdot\partial_{X}\epsilon_{n-2}+im\,\epsilon_{n-1}$
$\displaystyle\delta\Phi_{n-2}=-i^{1/2}s\cdot\partial_{X}\partial_{s}^{2}\epsilon_{n-1}+2im\,\epsilon_{n-2}$
$\displaystyle\delta\Phi_{n-3}=-i^{1/2}s\cdot\partial_{X}\partial_{s}^{2}\epsilon_{n-2}+3m\partial_{s}^{2}\epsilon_{n-1}$
(2.9)

where the gauge parameters $\epsilon_{n-1}(X,s)$ and $\epsilon_{n-2}(X,s)$ are
arbitrary rank $n-1$ and rank $n-2$ hyperfields, respectively.

This theory includes gauge invariant field strengths $\mathcal{F}_{n}$,
$\mathcal{F}_{n-1}$, $\mathcal{F}_{n-2}$, and $\mathcal{F}_{n-3}$, which are
linear in the $\Phi_{n-i}$’s and quadratic in derivatives

$\displaystyle\mathcal{F}_{n}=$
$\displaystyle\big{(}\partial_{X}^{2}-m^{2}-s\cdot\partial_{X}\partial_{s}\cdot\partial_{X}+\frac{1}{2}(s\cdot\partial_{X})^{2}\partial_{s}^{2}\big{)}\Phi_{n}-\frac{i}{2}(s\cdot\partial_{X})^{2}\Phi_{n-2}-i^{1/2}m\,s\cdot\partial_{X}\Phi_{n-1}$
(2.10) $\displaystyle\mathcal{F}_{n-1}=$
$\displaystyle\big{(}\partial_{X}^{2}-s\cdot\partial_{X}\partial_{s}\cdot\partial_{X}+\frac{1}{2}(s\cdot\partial_{X})^{2}\partial_{s}^{2}\big{)}\Phi_{n-1}-\frac{i}{2}(s\cdot\partial_{X})^{2}\Phi_{n-3}$
$\displaystyle+i^{-1/2}m\partial_{s}\cdot\partial_{X}\Phi_{n}-i^{-1/2}m\,s\cdot\partial_{X}\partial_{s}^{2}\Phi_{n}$
(2.11) $\displaystyle\mathcal{F}_{n-2}=$
$\displaystyle\big{(}\partial_{X}^{2}-s\cdot\partial_{X}\partial_{s}\cdot\partial_{X}-\frac{1}{2}(s\cdot\partial_{X})^{2}\partial_{s}^{2}\big{)}\Phi_{n-2}-\frac{i}{2}(s\cdot\partial_{X})^{2}\partial_{s}^{4}\Phi_{n}-im^{2}\partial_{s}^{2}\Phi_{n}$
$\displaystyle+2i^{-1/2}m\partial_{s}\cdot\partial_{X}\Phi_{n-1}-2i^{-1/2}m\,s\cdot\partial_{X}\partial_{s}^{2}\Phi_{n-1}+i^{1/2}m\,s\cdot\partial_{X}\Phi_{n-3}$
(2.12) $\displaystyle\mathcal{F}_{n-3}=$
$\displaystyle\big{(}\partial_{X}^{2}-m^{2}-s\cdot\partial_{X}\partial_{s}\cdot\partial_{X}-\frac{1}{2}(s\cdot\partial_{X})^{2}\partial_{s}^{2}\big{)}\Phi_{n-3}-\frac{i}{2}(s\cdot\partial_{X})^{2}\partial_{s}^{4}\Phi_{n-1}-3im^{2}\partial_{s}^{2}\Phi_{n-1}$
$\displaystyle+2i^{1/2}m\,s\cdot\partial_{X}\partial_{s}^{4}\Phi_{n}+3i^{-1/2}m\partial_{s}\cdot\partial_{X}\Phi_{n-2}+i^{-1/2}m\,s\cdot\partial_{X}\partial_{s}^{2}\Phi_{n-2}$
(2.13)

The equations of motion are equivalent to setting all four field strengths to
zero $\mathcal{F}_{n-i}=0$. In terms of these ingredients, the gauge invariant
free action for a massive spin $n$ particle may be written as

$\displaystyle S_{n}=\frac{1}{2}$ $\displaystyle n!\int
d^{d}X\frac{d^{d}sd^{d}s^{\prime}}{(2\pi)^{d}}e^{is\cdot s^{\prime}}\times$
$\displaystyle\Bigg{\\{}$ $\displaystyle\sum_{k=0}^{\cramped{\left\lfloor
n/2\right\rfloor}}\frac{(-1)^{k}}{(2k)!}\Bigg{(}$
$\displaystyle\Big{(}1-\frac{3k}{2}\Big{)}\partial_{s}^{2k}\tilde{\Phi}_{n}\partial_{s^{\prime}}^{2k}\mathcal{F}_{n}+i\frac{k}{2}\partial_{s}^{2k}\tilde{\Phi}_{n}\partial_{s^{\prime}}^{2(k-1)}\mathcal{F}_{n-2}$
$\displaystyle+i\frac{k}{2}\partial_{s}^{2(k-1)}\tilde{\Phi}_{n-2}\partial_{s^{\prime}}^{2k}\mathcal{F}_{n}-\frac{k}{2}\partial_{s}^{2(k-1)}\tilde{\Phi}_{n-2}\partial_{s^{\prime}}^{2(k-1)}\mathcal{F}_{n-2}\Bigg{)}$
$\displaystyle+\sum_{k=0}^{\cramped{\left\lfloor(n-1)/2\right\rfloor}}\frac{(-1)^{k}}{(2k+1)!}\Bigg{(}$
$\displaystyle\Big{(}1-\frac{5k}{2}\Big{)}\partial_{s}^{2k}\tilde{\Phi}_{n-1}\partial_{s^{\prime}}^{2k}\mathcal{F}_{n-1}+i\frac{3k}{2}\partial_{s}^{2k}\tilde{\Phi}_{n-1}\partial_{s^{\prime}}^{2(k-1)}\mathcal{F}_{n-3}$
$\displaystyle+i\frac{3k}{2}\partial_{s}^{2(k-1)}\tilde{\Phi}_{n-3}\partial_{s^{\prime}}^{2k}\mathcal{F}_{n-1}+\frac{k}{2}\partial_{s}^{2(k-1)}\tilde{\Phi}_{n-3}\partial_{s^{\prime}}^{2(k-1)}\mathcal{F}_{n-3}\Bigg{)}\Bigg{\\}}$
$\displaystyle\hskip 199.16928pt+\text{c.c.}$ (2.14)

That this action is gauge invariant is guaranteed by the fact that the field
strengths are themselves gauge invariant $\delta\mathcal{F}_{n-i}=0$, and that
they satisfy what we call massive Bianchi identities

$\displaystyle\partial_{s}\cdot\partial_{X}\mathcal{F}_{n}-\frac{1}{2}s\cdot\partial_{X}\partial_{s}^{2}\mathcal{F}_{n}+\frac{i}{2}s\cdot\partial_{X}\mathcal{F}_{n-2}+i^{1/2}m\mathcal{F}_{n-1}=0$
(2.15)
$\displaystyle\partial_{s}\cdot\partial_{X}\mathcal{F}_{n-1}-\frac{1}{2}s\cdot\partial_{X}\partial_{s}^{2}\mathcal{F}_{n-1}+\frac{i}{2}s\cdot\partial_{X}\mathcal{F}_{n-3}+\frac{1}{2}i^{-1/2}m\partial_{s}^{2}\mathcal{F}_{n}+\frac{1}{2}i^{1/2}m\mathcal{F}_{n-2}=0$
(2.16)

Indeed, the gauge variation of the action $\delta S_{n}$ is a straightforward
linear combination of terms proportional to the left hand sides of (2.15),
(2.16), and the $\delta\mathcal{F}_{n-i}$’s, and their complex conjugates.
This fact will be very important for us when considering interactions in
section 3.

###  2.2 Half integer spins

The covariant formalism for Dirac spin $n+1/2$ particles Lindwasser:2023zwo
is very analogous to the integer spin case. It requires the use of not only a
rank $n$ Dirac spinor hyperfield $\Psi_{n}(X,s)$, but also two auxiliary Dirac
spinor hyperfields $\Psi_{n-1}(X,s)$ and $\Psi_{n-2}(X,s)$. As before, these
auxiliary hyperfields couple to $\Psi_{n}(X,s)$ in such a way so that after
imposing the equations of motion, the auxiliary hyperfields may be set to
zero, and $\Psi_{n}(X,s)$ is a Rarita-Schwinger system PhysRev.60.61  in
hyperspace

$\displaystyle(\mathrlap{\\!\not{\phantom{\partial}}}\partial_{X}+m)\Psi_{n}$
$\displaystyle=0$
$\displaystyle\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\Psi_{n}$
$\displaystyle=0$ (2.17)

$\Psi_{n}(X,s)$ therefore has the correct degrees of freedom for the
description of a massive spin $n+1/2$ particle, with mass $m$. This special
coupling is facilitated by demanding that the theory be invariant under the
gauge transformations

$\displaystyle\delta\Psi_{n}=i^{-1/2}s\cdot\partial_{X}\epsilon_{n-1}$
$\displaystyle\delta\Psi_{n-1}=-is\cdot\partial_{X}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\epsilon_{n-1}+im\epsilon_{n-1}$
$\displaystyle\delta\Psi_{n-2}=-i^{1/2}s\cdot\partial_{X}\partial_{s}^{2}\epsilon_{n-1}+2i^{1/2}m\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\epsilon_{n-1}$
(2.18)

where the gauge parameter $\epsilon_{n-1}(X,s)$ is an arbitrary rank $n-1$
hyperfield.

This theory includes gauge invariant field strengths $\mathcal{S}_{n}$,
$\mathcal{S}_{n-1}$, and $\mathcal{S}_{n-2}$, which are linear in the
$\Psi_{n-i}$’s and linear in derivatives

$\displaystyle\mathcal{S}_{n}$
$\displaystyle=(\mathrlap{\\!\not{\phantom{\partial}}}\partial_{X}+m-s\cdot\partial_{X}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s})\Psi_{n}+i^{1/2}s\cdot\partial_{X}\Psi_{n-1}$
(2.19) $\displaystyle\mathcal{S}_{n-1}$
$\displaystyle=(\mathrlap{\\!\not{\phantom{\partial}}}\partial_{X}-s\cdot\partial_{X}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s})\Psi_{n-1}+i^{1/2}s\cdot\partial_{X}\Psi_{n-2}+i^{-1/2}m\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\Psi_{n}$
(2.20) $\displaystyle\mathcal{S}_{n-2}$
$\displaystyle=(\mathrlap{\\!\not{\phantom{\partial}}}\partial_{X}-m)\Psi_{n-2}+is\cdot\partial_{X}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}^{3}\Psi_{n}+i^{-1/2}s\cdot\partial_{X}\partial_{s}^{2}\Psi_{n-1}+2i^{-1/2}m\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\Psi_{n-1}$
(2.21)

The equations of motion are equivalent to setting all three field strengths to
zero $\mathcal{S}_{n-i}=0$. In terms of these ingredients, the gauge invariant
free action for a massive spin $n+1/2$ particle in the Dirac representation
may be written as

$\displaystyle S_{n+1/2}$ $\displaystyle=-n!\int
d^{d}X\frac{d^{d}sd^{d}s^{\prime}}{(2\pi)^{d}}e^{is\cdot s^{\prime}}\times$
$\displaystyle\Bigg{\\{}$ $\displaystyle\sum_{k=0}^{\cramped{\left\lfloor
n/2\right\rfloor}}\frac{(-1)^{k}}{(2k)!}\Bigg{(}$
$\displaystyle\Big{(}1-\frac{3k}{2}\Big{)}\partial_{s}^{2k}\overline{\Psi}_{n}\partial_{s^{\prime}}^{2k}\mathcal{S}_{n}+i\frac{k}{2}\partial_{s}^{2k}\overline{\Psi}_{n}\partial_{s^{\prime}}^{2(k-1)}\mathcal{S}_{n-2}$
$\displaystyle+i\frac{k}{2}\partial_{s}^{2(k-1)}\overline{\Psi}_{n-2}\partial_{s^{\prime}}^{2k}\mathcal{S}_{n}+\frac{k}{2}\partial_{s}^{2(k-1)}\overline{\Psi}_{n-2}\partial_{s^{\prime}}^{2(k-1)}\mathcal{S}_{n-2}$
$\displaystyle-i^{-1/2}k\partial_{s}^{2(k-1)}\overline{\Psi}_{n-1}\overleftarrow{\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}}\partial_{s^{\prime}}^{2(k-1)}\mathcal{S}_{n-2}+i^{-1/2}k\partial_{s}^{2(k-1)}\overline{\Psi}_{n-2}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s^{\prime}}\partial_{s^{\prime}}^{2(k-1)}\mathcal{S}_{n-1}$
$\displaystyle\hskip
78.24507pt+ik\partial_{s}^{2(k-1)}\overline{\Psi}_{n-1}\overleftarrow{\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s^{\prime}}\partial_{s^{\prime}}^{2(k-1)}\mathcal{S}_{n-1}\Bigg{)}$
$\displaystyle+\sum_{k=0}^{\cramped{\left\lfloor(n-1)/2\right\rfloor}}\frac{(-1)^{k}}{(2k+1)!}\Bigg{(}$
$\displaystyle-i\Big{(}\frac{1}{2}+\frac{3k}{2}\Big{)}\partial_{s}^{2k}\overline{\Psi}_{n}\overleftarrow{\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s^{\prime}}\partial_{s^{\prime}}^{2k}\mathcal{S}_{n}-\frac{k}{2}\partial_{s}^{2k}\overline{\Psi}_{n}\overleftarrow{\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s^{\prime}}\partial_{s^{\prime}}^{2(k-1)}\mathcal{S}_{n-2}$
$\displaystyle-\frac{k}{2}\partial_{s}^{2(k-1)}\overline{\Psi}_{n-2}\overleftarrow{\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s^{\prime}}\partial_{s^{\prime}}^{2k}\mathcal{S}_{n}+i\frac{k}{2}\partial_{s}^{2(k-1)}\overline{\Psi}_{n-2}\overleftarrow{\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s^{\prime}}\partial_{s^{\prime}}^{2(k-1)}\mathcal{S}_{n-2}$
$\displaystyle+\frac{1}{2}i^{-1/2}\partial_{s}^{2k}\overline{\Psi}_{n}\overleftarrow{\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}}\partial_{s^{\prime}}^{2k}\mathcal{S}_{n-1}-\frac{1}{2}i^{-1/2}\partial_{s}^{2k}\overline{\Psi}_{n-1}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s^{\prime}}\partial_{s^{\prime}}^{2k}\mathcal{S}_{n}$
$\displaystyle+i^{1/2}k\partial_{s}^{2(k-1)}\overline{\Psi}_{n-2}\overleftarrow{\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}}\partial_{s^{\prime}}^{2k}\mathcal{S}_{n-1}-i^{1/2}k\partial_{s}^{2k}\overline{\Psi}_{n-1}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s^{\prime}}\partial_{s^{\prime}}^{2(k-1)}\mathcal{S}_{n-2}$
$\displaystyle\hskip
96.02805pt+\Big{(}\frac{1}{2}-k\Big{)}\partial_{s}^{2k}\overline{\Psi}_{n-1}\partial_{s^{\prime}}^{2k}\mathcal{S}_{n-1}\Bigg{)}\Bigg{\\}}$
(2.22)

That this action is gauge invariant is guaranteed by the fact that the field
strengths themselves are gauge invariant $\delta\mathcal{S}_{n-i}=0$, and that
they satisfy a fermionic massive Bianchi identity

$\displaystyle\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}(\mathrlap{\\!\not{\phantom{\partial}}}\partial_{X}-m)\mathcal{S}_{n}-s\cdot\partial_{X}\partial_{s}^{2}\mathcal{S}_{n}-i^{1/2}(\mathrlap{\\!\not{\phantom{\partial}}}\partial_{X}-m)\mathcal{S}_{n-1}+i\,s\cdot\partial_{X}\mathcal{S}_{n-2}=0$
(2.23)

As before, the gauge variation of the action $\delta S_{n+1/2}$ is a
straightforward linear combination the left hand side of (2.23) and the
$\delta\mathcal{S}_{n-i}$’s.

##  3 Interactions

In this section, we show how to add electromagnetic and gravitational
interactions to the actions (2.1) and (2.2), in a way that preserves the
massive gauge symmetry. We begin first with the integer spin case, and then
repeat the analysis for half integer spins.

###  3.1 Integer spins

Naively, when modelling a massive integer spin particle interacting with
electromagnetism or gravity, one might try what works for spin $0$, i.e.
replace every spacetime derivative $\partial_{X\mu}$ in the action with a
covariant derivative $\nabla_{\mu}$, as well as replace every $\eta_{\mu\nu}$
with $g_{\mu\nu}(X)$ and $d^{d}X$ with $d^{d}X\sqrt{-g}$ in the case of
gravitational interactions. This is often referred to as minimal coupling.
Notice however that even this procedure is ambiguous for spins $n\geq 1$,
because one can always re-order the derivatives and write
$\partial_{X\mu}\partial_{X\nu}=\partial_{X\mu}\partial_{X\nu}+a[\partial_{X\mu},\partial_{X\nu}]$
before converting them to covariant derivatives for any $a$ without changing
the free action. Spin $0$ is special in this respect because the only
appearance of derivatives in the action is in
$\eta^{\mu\nu}\partial_{X\mu}\partial_{X\nu}$. Nevertheless, for the sake of
the discussion we will refer to minimal coupling as replacing derivatives with
covariant ones in the particular order they appear in (2.1).

A few words must be said about lifting the hyperspace measure (2.6) to a
curved spacetime manifold $M$. Apart from the spacetime integration, this
measure is just a formal way to implement index contraction in terms of
hyperfields. Within the integration, one should think of the auxiliary vectors
as living on the tangent space $s,s^{\prime}\in T_{X}M$. One simply integrates
over all components of the tangent vectors $s,\,s^{\prime}$ at the point $X$,
weighted by $e^{is\cdot s^{\prime}}$, where $s\cdot
s^{\prime}=g(s,s^{\prime})=g_{\mu\nu}s^{\mu}s^{\prime\nu}$.

The covariant derivative $\nabla_{\mu}$ on a charge 1 integer spin $n$
hyperfield $\Phi_{n}(X,s)$, including both a $U(1)$ gauge field $A_{\mu}$, and
a Levi-Civita connection $\Gamma^{\lambda}_{\;\;\mu\nu}$, which implements the
standard covariant derivative on its component field
$\phi_{\mu_{1}\cdots\mu_{n}}(X)$ is

$\displaystyle\nabla_{\mu}=\partial_{X\mu}-iA_{\mu}-s^{\nu}\Gamma^{\lambda}_{\;\;\mu\nu}\partial_{s\lambda}$
(3.1)

Covariant derivatives no longer commute, but instead their commutator when
acting on $\Phi_{n}(X,s)$ equals

$\displaystyle[\nabla_{\mu},\nabla_{\nu}]=-iF_{\mu\nu}-s^{\sigma}R^{\omega}_{\;\;\sigma\mu\nu}\partial_{s\omega}$
(3.2)

As part of this minimal coupling procedure, we will also deform the massive
gauge symmetry (2.1) by replacing spacetime derivatives with covariant
derivatives, and $\eta^{\mu\nu}$ with $g^{\mu\nu}$, so that it is consistent
with $U(1)$ electromagnetic gauge invariance and general coordinate invariance

$\displaystyle\delta\Phi_{n}=i^{-1/2}s\cdot\nabla\epsilon_{n-1}$
$\displaystyle\delta\Phi_{n-1}=i^{-1/2}s\cdot\nabla\epsilon_{n-2}+im\,\epsilon_{n-1}$
$\displaystyle\delta\Phi_{n-2}=-i^{1/2}s\cdot\nabla\partial_{s}^{2}\epsilon_{n-1}+2im\,\epsilon_{n-2}$
$\displaystyle\delta\Phi_{n-3}=-i^{1/2}s\cdot\nabla\partial_{s}^{2}\epsilon_{n-2}+3m\partial_{s}^{2}\epsilon_{n-1}$
(3.3)

It is straightforward to show that the action (2.1) is not invariant under the
gauge symmetry (3.1) after minimal coupling, and hence the minimally coupled
theory does not consistently describe a massive spin $n$ particle. Indeed, the
gauge variation of the minimally coupled action $\delta S_{n}$ is a linear
combination of terms proportional to $\delta\mathcal{F}_{n-i}$, and the
covariantized version of the massive Bianchi identities (2.15), (2.16). This
is an important fact for the subsequent analysis, which follows from the
massive action being derived from the corresponding massless action $S_{n,0}$
in $d+1$ dimensions built from a rank $n$ double traceless hyperfield
$\Phi_{n,0}(X,s)$ and massless field strength $\mathcal{F}_{n,0}(X,s)$,
described in Lindwasser:2023zwo . Indeed, under the gauge transformation
$\delta\Phi_{n,0}(X,s)=i^{-1/2}s\cdot\partial_{X}\epsilon_{n-1}(X,s)$, for
$\epsilon_{n-1}(X,s)$ a rank $n-1$ traceless hyperfield, the gauge variation
of $S_{n,0}$ is

$\displaystyle\delta S_{n,0}=\frac{1}{2}n!\int
d^{d+1}X\frac{d^{d+1}sd^{d+1}s^{\prime}}{(2\pi)^{d+1}}e^{is\cdot
s^{\prime}}\times$
$\displaystyle\Big{(}\Phi_{n,0}(X,s)\big{(}1-\frac{1}{4}s^{\prime
2}\partial_{s^{\prime}}^{2}\big{)}\delta\mathcal{F}_{n,0}(X,s^{\prime})-i^{1/2}\epsilon_{n-1}(X,s)\big{(}\partial_{s^{\prime}}\cdot\partial_{X}\mathcal{F}_{n,0}-\frac{1}{2}s^{\prime}\cdot\partial_{X}\partial_{s^{\prime}}^{2}\mathcal{F}_{n,0}\big{)}\Big{)}$
(3.4)

After dimensional reduction, the gauge variation of the massless $d+1$
dimensional field strength $\delta\mathcal{F}_{n,0}$ decomposes into the
massive $d$ dimensional variations $\delta\mathcal{F}_{n-i}$, and the
expression
$\partial_{s}\cdot\partial_{X}\mathcal{F}_{n,0}-\frac{1}{2}s\cdot\partial_{X}\partial_{s}^{2}\mathcal{F}_{n,0}$
decomposes into the two massive Bianchi identities (2.15), (2.16). These are
no longer zero after minimal coupling, but instead equal

$\displaystyle\delta\mathcal{F}_{n}$
$\displaystyle=i^{-1/2}(\nabla^{2}s\cdot\nabla-s\cdot\nabla[\partial_{s}\cdot\nabla,s\cdot\nabla])\epsilon_{n-1}$
(3.5) $\displaystyle\delta\mathcal{F}_{n-1}$
$\displaystyle=i^{-1/2}(\nabla^{2}s\cdot\nabla-s\cdot\nabla[\partial_{s}\cdot\nabla,s\cdot\nabla])\epsilon_{n-2}+im(\nabla^{2}-[\partial_{s}\cdot\nabla,s\cdot\nabla])\epsilon_{n-1}$
(3.6) $\displaystyle\delta\mathcal{F}_{n-2}$
$\displaystyle=-i^{1/2}(\nabla^{2}s\cdot\nabla-s\cdot\nabla[\partial_{s}\cdot\nabla,s\cdot\nabla])\partial_{s}^{2}\epsilon_{n-1}+2im(\nabla^{2}-[\partial_{s}\cdot\nabla,s\cdot\nabla])\epsilon_{n-2}$
(3.7) $\displaystyle\delta\mathcal{F}_{n-3}$
$\displaystyle=-i^{1/2}(\nabla^{2}s\cdot\nabla-s\cdot\nabla[\partial_{s}\cdot\nabla,s\cdot\nabla])\partial_{s}^{2}\epsilon_{n-2}+3m(\nabla^{2}-[\partial_{s}\cdot\nabla,s\cdot\nabla])\partial_{s}^{2}\epsilon_{n-1}$
(3.8) $\displaystyle\partial_{s}\cdot\nabla$
$\displaystyle\mathcal{F}_{n}-\frac{1}{2}s\cdot\nabla\partial_{s}^{2}\mathcal{F}_{n}+\frac{i}{2}s\cdot\nabla\mathcal{F}_{n-2}+i^{1/2}m\mathcal{F}_{n-1}=$
$\displaystyle(\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2})(\nabla^{2}-[\partial_{s}\cdot\nabla,s\cdot\nabla])\Phi_{n}+[\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2},[\partial_{s}\cdot\nabla,s\cdot\nabla]]\Phi_{n}$
$\displaystyle\hskip
28.45274pt+\frac{i}{2}s\cdot\nabla(\nabla^{2}-[\partial_{s}\cdot\nabla,s\cdot\nabla])\Phi_{n-2}+\frac{i}{2}[s\cdot\nabla,[\partial_{s}\cdot\nabla,s\cdot\nabla]]\Phi_{n-2}$
$\displaystyle\hskip
85.35826pt+i^{1/2}m(\nabla^{2}-[\partial_{s}\cdot\nabla,s\cdot\nabla])\Phi_{n-1}$
(3.9) $\displaystyle\partial_{s}\cdot\nabla$
$\displaystyle\mathcal{F}_{n-1}-\frac{1}{2}s\cdot\nabla\partial_{s}^{2}\mathcal{F}_{n-1}+\frac{i}{2}s\cdot\nabla\mathcal{F}_{n-3}+\frac{1}{2}i^{-1/2}m\partial_{s}^{2}\mathcal{F}_{n}+\frac{1}{2}i^{1/2}m\mathcal{F}_{n-2}=$
$\displaystyle(\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2})(\nabla^{2}-[\partial_{s}\cdot\nabla,s\cdot\nabla])\Phi_{n-1}+[\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2},[\partial_{s}\cdot\nabla,s\cdot\nabla]]\Phi_{n-1}$
$\displaystyle\hskip
28.45274pt+\frac{i}{2}s\cdot\nabla(\nabla^{2}-[\partial_{s}\cdot\nabla,s\cdot\nabla])\Phi_{n-3}+\frac{i}{2}[s\cdot\nabla,[\partial_{s}\cdot\nabla,s\cdot\nabla]]\Phi_{n-3}$
$\displaystyle\hskip
14.22636pt+\frac{1}{2}i^{-1/2}m(\nabla^{2}-[\partial_{s}\cdot\nabla,s\cdot\nabla])\partial_{s}^{2}\Phi_{n}+\frac{1}{2}i^{1/2}m(\nabla^{2}-[\partial_{s}\cdot\nabla,s\cdot\nabla])\Phi_{n-2}$
(3.10)

One can check that these expressions are proportional to electromagnetic and
gravitational curvatures $F_{\mu\nu}$, $R_{\omega\sigma\mu\nu}$.

####  3.1.1 Restoring gauge symmetry

In order to restore the massive gauge symmetry, additional interactions with
electromagnetism and gravity must be added to the action. Because the gauge
violation of the action $\delta S_{n}\neq 0$ depends on the properties of the
field strengths $\mathcal{F}_{n-i}$, we may consider restoring the gauge
symmetry by making appropriate modifications of the field strengths
$\mathcal{F}_{n-i}\to\mathcal{F}_{n-i}+\Delta\mathcal{F}_{n-i}$, so that they
are again gauge invariant and satisfy the massive Bianchi identities.

For instance, one could choose
$\Delta\mathcal{F}_{n-i}^{(0)}=([\partial_{s}\cdot\nabla,s\cdot\nabla]-\nabla^{2})\Phi_{n-i}$,
which cancels all gauge violations in (3.5)-(3.1) which are $\mathcal{O}(m)$.
The new field strengths
$\mathcal{F}_{n-i}^{(0)}\equiv\mathcal{F}_{n-i}+\Delta\mathcal{F}_{n-i}^{(0)}$
still have $\mathcal{O}(1)$ in $m$ gauge violations

$\displaystyle\delta\mathcal{F}_{n}^{(0)}$
$\displaystyle=i^{-1/2}[[\partial_{s}\cdot\nabla,s\cdot\nabla],s\cdot\nabla]\epsilon_{n-1}$
(3.11) $\displaystyle\delta\mathcal{F}_{n-1}^{(0)}$
$\displaystyle=i^{-1/2}[[\partial_{s}\cdot\nabla,s\cdot\nabla],s\cdot\nabla]\epsilon_{n-2}$
(3.12) $\displaystyle\delta\mathcal{F}_{n-2}^{(0)}$
$\displaystyle=-i^{1/2}[[\partial_{s}\cdot\nabla,s\cdot\nabla],s\cdot\nabla]\partial_{s}^{2}\epsilon_{n-1}$
(3.13) $\displaystyle\delta\mathcal{F}_{n-3}^{(0)}$
$\displaystyle=-i^{1/2}[[\partial_{s}\cdot\nabla,s\cdot\nabla],s\cdot\nabla]\partial_{s}^{2}\epsilon_{n-2}$
(3.14) $\displaystyle\partial_{s}\cdot\nabla$
$\displaystyle\mathcal{F}_{n}^{(0)}-\frac{1}{2}s\cdot\nabla\partial_{s}^{2}\mathcal{F}_{n}^{(0)}+\frac{i}{2}s\cdot\nabla\mathcal{F}_{n-2}^{(0)}+i^{1/2}m\mathcal{F}_{n-1}^{(0)}=$
$\displaystyle[\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2},[\partial_{s}\cdot\nabla,s\cdot\nabla]]\Phi_{n}+\frac{i}{2}[s\cdot\nabla,[\partial_{s}\cdot\nabla,s\cdot\nabla]]\Phi_{n-2}$
(3.15) $\displaystyle\partial_{s}\cdot\nabla$
$\displaystyle\mathcal{F}_{n-1}^{(0)}-\frac{1}{2}s\cdot\nabla\partial_{s}^{2}\mathcal{F}_{n-1}^{(0)}+\frac{i}{2}s\cdot\nabla\mathcal{F}_{n-3}^{(0)}+\frac{1}{2}i^{-1/2}m\partial_{s}^{2}\mathcal{F}_{n}^{(0)}+\frac{1}{2}i^{1/2}m\mathcal{F}_{n-2}^{(0)}=$
$\displaystyle[\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2},[\partial_{s}\cdot\nabla,s\cdot\nabla]]\Phi_{n-1}+\frac{i}{2}[s\cdot\nabla,[\partial_{s}\cdot\nabla,s\cdot\nabla]]\Phi_{n-3}$
(3.16)

Because of the appearance of $m$ in the gauge transformations (3.1) and
massive Bianchi identities, one can further suppress the gauge violations to
$\mathcal{O}(1/m)$, cancelling the $\mathcal{O}(1)$ violations, by adding a
$\Delta\mathcal{F}_{n-i}^{(1)}$ which is proportional to $1/m$. This can be
done indefinitely, generating an infinite series of
$\Delta\mathcal{F}_{n-i}^{(N)}$, each proportional to $1/m^{N}$, for all
integers $N\geq 0$, until the action is exactly gauge invariant.

To prove this, we proceed by induction. Suppose that we have found modified
field strengths $\mathcal{F}_{n-i}^{(2k)}$ which suppress all gauge violations
to be $\mathcal{O}(1/m^{2k})$, which we write as

$\displaystyle\delta\mathcal{F}_{n}^{(2k)}$
$\displaystyle=\frac{1}{m^{2k}}\hat{\mathcal{O}}_{n}^{(2k)}\epsilon_{n-1}$
(3.17) $\displaystyle\delta\mathcal{F}_{n-1}^{(2k)}$
$\displaystyle=\frac{1}{m^{2k}}\hat{\mathcal{O}}_{n-1}^{(2k)}\epsilon_{n-2}$
(3.18) $\displaystyle\delta\mathcal{F}_{n-2}^{(2k)}$
$\displaystyle=\frac{1}{m^{2k}}\hat{\mathcal{O}}_{n-2}^{(2k)}\epsilon_{n-1}$
(3.19) $\displaystyle\delta\mathcal{F}_{n-3}^{(2k)}$
$\displaystyle=\frac{1}{m^{2k}}\hat{\mathcal{O}}_{n-3}^{(2k)}\epsilon_{n-2}$
(3.20) $\displaystyle\partial_{s}\cdot\nabla$
$\displaystyle\mathcal{F}_{n}^{(2k)}-\frac{1}{2}s\cdot\nabla\partial_{s}^{2}\mathcal{F}_{n}^{(2k)}+\frac{i}{2}s\cdot\nabla\mathcal{F}_{n-2}^{(2k)}+i^{1/2}m\mathcal{F}_{n-1}^{(2k)}=\frac{1}{m^{2k}}\hat{\mathcal{U}}_{n}^{(2k)}\Phi_{n}+\frac{1}{m^{2k}}\hat{\mathcal{U}}_{n-2}^{(2k)}\Phi_{n-2}$
(3.21)
$\displaystyle\partial_{s}\cdot\nabla\mathcal{F}_{n-1}^{(2k)}-\frac{1}{2}s\cdot\nabla\partial_{s}^{2}\mathcal{F}_{n-1}^{(2k)}+\frac{i}{2}s\cdot\nabla\mathcal{F}_{n-3}^{(2k)}+\frac{1}{2}i^{-1/2}m\partial_{s}^{2}$
$\displaystyle\mathcal{F}_{n}^{(2k)}+\frac{1}{2}i^{1/2}m\mathcal{F}_{n-2}^{(2k)}=$
$\displaystyle\frac{1}{m^{2k}}\hat{\mathcal{U}}_{n-1}^{(2k)}\Phi_{n-1}+\frac{1}{m^{2k}}\hat{\mathcal{U}}_{n-3}^{(2k)}\Phi_{n-3}$
(3.22)

for some operators $\hat{\mathcal{O}}_{n-i}^{(2k)}$,
$\hat{\mathcal{U}}_{n-i}^{(2k)}$, where the subscript $n-i$ does not denote
the rank, but the hyperfields $\mathcal{F}_{n-i}^{(2k)}$, $\Phi_{n-i}$ they
are associated with, respectively. We first show that if
$\hat{\mathcal{O}}_{n-i}^{(2k)}$ and $\hat{\mathcal{U}}_{n-i}^{(2k)}$ satisfy
conditions (to be determined) ensuring that $\Delta\mathcal{F}_{n-i}^{(2k+1)}$
and $\Delta\mathcal{F}_{n-i}^{(2k+2)}$ can be constructed so that
$\mathcal{F}_{n-i}^{(2k+2)}\equiv\mathcal{F}_{n-i}^{(2k)}+\Delta\mathcal{F}_{n-i}^{(2k+1)}+\Delta\mathcal{F}_{n-i}^{(2k+2)}$
suppresses the gauge violations to $\mathcal{O}(1/m^{2k+2})$, then the
corresponding operators at the next order $\hat{\mathcal{O}}_{n-i}^{(2k+2)}$
and $\hat{\mathcal{U}}_{n-i}^{(2k+2)}$ will also satisfy those conditions.

To cancel the $\mathcal{O}(1/m^{2k})$ gauge violations (3.17), (3.20), (3.21)
and (3.1.1), one may choose $\Delta\mathcal{F}_{n-i}^{(2k+1)}$ to be

$\displaystyle\Delta\mathcal{F}_{n}^{(2k+1)}$
$\displaystyle=\frac{i}{m^{2k+1}}\hat{\mathcal{O}}_{n}^{(2k)}\Phi_{n-1}$
(3.23) $\displaystyle\Delta\mathcal{F}_{n-1}^{(2k+1)}$
$\displaystyle=-\frac{i^{-1/2}}{m^{2k+1}}\Big{(}\hat{\mathcal{U}}_{n}^{(2k)}\Phi_{n}+\hat{\mathcal{U}}_{n-2}^{(2k)}\Phi_{n-2}\Big{)}$
(3.24) $\displaystyle\Delta\mathcal{F}_{n-2}^{(2k+1)}$
$\displaystyle=-\frac{i^{-1/2}}{m^{2k+1}}\Big{(}(i^{1/2}\partial_{s}^{2}\hat{\mathcal{O}}_{n}^{(2k)}+2\,\hat{\mathcal{U}}_{n-1}^{(2k)})\Phi_{n-1}+2\,\hat{\mathcal{U}}_{n-3}^{(2k)}\Phi_{n-3}\Big{)}$
(3.25) $\displaystyle\Delta\mathcal{F}_{n-3}^{(2k+1)}$
$\displaystyle=\frac{i}{2m^{2k+1}}\hat{\mathcal{O}}_{n-3}^{(2k)}\Phi_{n-2}$
(3.26)

The remaining $\mathcal{O}(1/m^{2k})$ gauge violations (3.18) and (3.19) must
then be cancelled by $\delta(\Delta\mathcal{F}_{n-1}^{(2k+1)})$ and
$\delta(\Delta\mathcal{F}_{n-2}^{(2k+1)})$, respectively. This in turn is only
possible if the following conditions hold

$\displaystyle\hat{\mathcal{C}}_{1}^{(2k)}\equiv\hat{\mathcal{O}}_{n-1}^{(2k)}-2i^{1/2}\hat{\mathcal{U}}_{n-2}^{(2k)}=0$
(3.27)
$\displaystyle\hat{\mathcal{C}}_{2}^{(2k)}\equiv\hat{\mathcal{O}}_{n-2}^{(2k)}-i\partial_{s}^{2}\hat{\mathcal{O}}_{n}^{(2k)}-2i^{1/2}\hat{\mathcal{U}}_{n-1}^{(2k)}-6i^{-1/2}\hat{\mathcal{U}}_{n-3}^{(2k)}\partial_{s}^{2}=0$
(3.28)

Let us assume this is true and proceed with constructing
$\Delta\mathcal{F}_{n-i}^{(2k+2)}$. The gauge violations associated with
$\mathcal{F}_{n-i}^{(2k+1)}=\mathcal{F}_{n-i}^{(2k)}+\Delta\mathcal{F}_{n-i}^{(2k+1)}$
are

$\displaystyle\delta\mathcal{F}_{n}^{(2k+1)}$
$\displaystyle=\frac{i^{1/2}}{m^{2k+1}}\hat{\mathcal{O}}_{n}^{(2k)}s\cdot\nabla\epsilon_{n-2}$
(3.29) $\displaystyle\delta\mathcal{F}_{n-1}^{(2k+1)}$
$\displaystyle=\frac{1}{m^{2k+1}}\Big{(}i\,\hat{\mathcal{U}}_{n}^{(2k)}s\cdot\nabla+\hat{\mathcal{U}}_{n-2}^{(2k)}s\cdot\nabla\partial_{s}^{2}\Big{)}\epsilon_{n-1}$
(3.30) $\displaystyle\delta\mathcal{F}_{n-2}^{(2k+1)}$
$\displaystyle=\frac{1}{m^{2k+1}}\Big{(}-i^{-1/2}\partial_{s}^{2}\hat{\mathcal{O}}_{n}^{(2k)}s\cdot\nabla+2i\,\hat{\mathcal{U}}_{n-1}^{(2k)}s\cdot\nabla+2\,\hat{\mathcal{U}}_{n-3}^{(2k)}s\cdot\nabla\partial_{s}^{2}\Big{)}\epsilon_{n-2}$
(3.31) $\displaystyle\delta\mathcal{F}_{n-3}^{(2k+1)}$
$\displaystyle=\frac{i^{-1/2}}{2m^{2k+1}}\hat{\mathcal{O}}_{n-3}^{(2k)}s\cdot\nabla\partial_{s}^{2}\epsilon_{n-1}$
(3.32) $\displaystyle\partial_{s}\cdot\nabla$
$\displaystyle\mathcal{F}_{n}^{(2k+1)}-\frac{1}{2}s\cdot\nabla\partial_{s}^{2}\mathcal{F}_{n}^{(2k+1)}+\frac{i}{2}s\cdot\nabla\mathcal{F}_{n-2}^{(2k+1)}+i^{1/2}m\mathcal{F}_{n-1}^{(2k+1)}=$
$\displaystyle\frac{1}{m^{2k+1}}\Big{(}i(\partial_{s}\cdot\nabla-s\cdot\nabla\partial_{s}^{2})\hat{\mathcal{O}}_{n}^{(2k)}-i^{1/2}s\cdot\nabla\,\hat{\mathcal{U}}_{n-1}^{(2k)}\Big{)}\Phi_{n-1}-\frac{i^{1/2}}{m^{2k+1}}s\cdot\nabla\,\hat{\mathcal{U}}_{n-3}^{(2k)}\Phi_{n-3}$
(3.33) $\displaystyle\partial_{s}\cdot\nabla$
$\displaystyle\mathcal{F}_{n-1}^{(2k+1)}-\frac{1}{2}s\cdot\nabla\partial_{s}^{2}\mathcal{F}_{n-1}^{(2k+1)}+\frac{i}{2}s\cdot\nabla\mathcal{F}_{n-3}^{(2k+1)}+\frac{1}{2}i^{-1/2}m\partial_{s}^{2}\mathcal{F}_{n}^{(2k+1)}+\frac{1}{2}i^{1/2}m\mathcal{F}_{n-2}^{(2k+1)}=$
$\displaystyle-\frac{i^{-1/2}}{m^{2k+1}}(\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2})(\hat{\mathcal{U}}_{n}^{(2k)}\Phi_{n}+\hat{\mathcal{U}}_{n-2}^{(2k)}\Phi_{n-2})-\frac{1}{4m^{2k+1}}s\cdot\nabla\hat{\mathcal{O}}_{n-3}^{(2k)}\Phi_{n-2}$
(3.34)

Again, it is straightforward to find modifications
$\Delta\mathcal{F}_{n-i}^{(2k+2)}$ which cancel the $\mathcal{O}(1/m^{2k+1})$
gauge violations in (3.29), (3.32), (3.1.1) and (3.1.1)

$\displaystyle\Delta\mathcal{F}_{n}^{(2k+2)}=$
$\displaystyle-\frac{i^{-1/2}}{2m^{2k+2}}\hat{\mathcal{O}}_{n}^{(2k)}s\cdot\nabla\Phi_{n-2}$
(3.35) $\displaystyle\Delta\mathcal{F}_{n-1}^{(2k+2)}=$
$\displaystyle-\frac{i^{1/2}}{m^{2k+2}}\Big{(}(\partial_{s}\cdot\nabla-s\cdot\nabla\partial_{s}^{2})\hat{\mathcal{O}}_{n}^{(2k)}-i^{-1/2}s\cdot\nabla\hat{\mathcal{U}}_{n-1}^{(2k)}\Big{)}\Phi_{n-1}$
$\displaystyle+\frac{1}{m^{2k+2}}s\cdot\nabla\hat{\mathcal{U}}_{n-3}^{(2k)}\Phi_{n-3}$
(3.36) $\displaystyle\Delta\mathcal{F}_{n-2}^{(2k+2)}=$
$\displaystyle-2\frac{i}{m^{2k+2}}(\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2})(\hat{\mathcal{U}}_{n}^{(2k)}\Phi_{n}+\hat{\mathcal{U}}_{n-2}^{(2k)}\Phi_{n-2})$
$\displaystyle+\frac{i^{-1/2}}{2m^{2k+2}}(s\cdot\nabla\hat{\mathcal{O}}_{n-3}^{(2k)}-i\partial_{s}^{2}\hat{\mathcal{O}}_{n}^{(2k)}s\cdot\nabla)\Phi_{n-2}$
(3.37) $\displaystyle\Delta\mathcal{F}_{n-3}^{(2k+2)}=$
$\displaystyle\frac{i^{1/2}}{2m^{2k+2}}\hat{\mathcal{O}}_{n-3}^{(2k)}s\cdot\nabla\partial_{s}^{2}\Phi_{n-1}$
(3.38)

Demanding that $\delta(\Delta\mathcal{F}_{n-1}^{(2k+2)})$ and
$\delta(\Delta\mathcal{F}_{n-2}^{(2k+2)})$ also cancel the
$\mathcal{O}(1/m^{2k+1})$ gauge violations in (3.30) and (3.31) imposes
further conditions on $\hat{\mathcal{O}}_{n-i}^{(2k)}$ and
$\hat{\mathcal{U}}_{n-i}^{(2k)}$

$\displaystyle\hat{\mathcal{C}}_{3}^{(2k)}\equiv\,$ $\displaystyle
i^{-1/2}s\cdot\nabla\hat{\mathcal{O}}_{n-3}^{(2k)}+2\,\hat{\mathcal{U}}_{n-1}^{(2k)}s\cdot\nabla$
$\displaystyle-4i(\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2})\hat{\mathcal{U}}_{n-2}^{(2k)}-2i\,\hat{\mathcal{U}}_{n-3}^{(2k)}s\cdot\nabla\partial_{s}^{2}=0$
(3.39) $\displaystyle\hat{\mathcal{C}}_{4}^{(2k)}\equiv\,$ $\displaystyle
i^{1/2}(\partial_{s}\cdot\nabla-s\cdot\nabla\partial_{s}^{2})\hat{\mathcal{O}}_{n}^{(2k)}-\hat{\mathcal{U}}_{n}^{(2k)}s\cdot\nabla$
$\displaystyle-s\cdot\nabla\,\hat{\mathcal{U}}_{n-1}^{(2k)}+i\,\hat{\mathcal{U}}_{n-2}^{(2k)}s\cdot\nabla\partial_{s}^{2}+3is\cdot\nabla\,\hat{\mathcal{U}}_{n-3}^{(2k)}\partial_{s}^{2}=0$
(3.40)

Assuming that these conditions are satisfied, we may use
$\mathcal{F}_{n-i}^{(2k+2)}=\mathcal{F}_{n-i}^{(2k+1)}+\Delta\mathcal{F}_{n-i}^{(2k+2)}$
to find the corresponding operators at the next order
$\hat{\mathcal{O}}_{n-i}^{(2k+2)}$ and $\hat{\mathcal{U}}_{n-i}^{(2k+2)}$
associated with $\mathcal{O}(1/m^{2k+2})$ gauge violations

$\displaystyle\hat{\mathcal{O}}_{n}^{(2k+2)}=$
$\displaystyle\frac{1}{2}\hat{\mathcal{O}}_{n}^{(2k)}(s\cdot\nabla)^{2}\partial_{s}^{2}$
(3.41) $\displaystyle\hat{\mathcal{O}}_{n-1}^{(2k+2)}=$
$\displaystyle-(\partial_{s}\cdot\nabla-s\cdot\nabla\partial_{s}^{2})\hat{\mathcal{O}}_{n}^{(2k)}s\cdot\nabla$
$\displaystyle+i^{-1/2}s\cdot\nabla\,\hat{\mathcal{U}}_{n-1}^{(2k)}s\cdot\nabla-i^{1/2}s\cdot\nabla\,\hat{\mathcal{U}}_{n-3}^{(2k)}s\cdot\nabla\partial_{s}^{2}$
(3.42) $\displaystyle\hat{\mathcal{O}}_{n-2}^{(2k+2)}=$
$\displaystyle-2i^{1/2}(\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2})(\hat{\mathcal{U}}_{n}^{(2k)}s\cdot\nabla-i\,\hat{\mathcal{U}}_{n-2}^{(2k)}s\cdot\nabla\partial_{s}^{2})$
$\displaystyle+\frac{i}{2}\partial_{s}^{2}\hat{\mathcal{O}}_{n}^{(2k)}(s\cdot\nabla)^{2}\partial_{s}^{2}-\frac{1}{2}s\cdot\nabla\hat{\mathcal{O}}_{n-3}^{(2k)}s\cdot\nabla\partial_{s}^{2}$
(3.43) $\displaystyle\hat{\mathcal{O}}_{n-3}^{(2k+2)}=$
$\displaystyle\,\frac{1}{2}\hat{\mathcal{O}}_{n-3}^{(2k)}s\cdot\nabla\partial_{s}^{2}s\cdot\nabla$
(3.44) $\displaystyle\hat{\mathcal{U}}_{n}^{(2k+2)}=$
$\displaystyle\,s\cdot\nabla(\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2})\hat{\mathcal{U}}_{n}^{(2k)}$
(3.45) $\displaystyle\hat{\mathcal{U}}_{n-1}^{(2k+2)}=$
$\displaystyle-i^{1/2}(\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2})(\partial_{s}\cdot\nabla-s\cdot\nabla\partial_{s}^{2})\hat{\mathcal{O}}_{n}^{(2k)}$
$\displaystyle+(\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2})s\cdot\nabla\hat{\mathcal{U}}_{n-1}^{(2k)}-\frac{1}{4}i^{-1/2}s\cdot\nabla\hat{\mathcal{O}}_{n-3}^{(2k)}s\cdot\nabla\partial_{s}^{2}$
(3.46) $\displaystyle\hat{\mathcal{U}}_{n-2}^{(2k+2)}=$
$\displaystyle\,s\cdot\nabla(\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2})\hat{\mathcal{U}}_{n-2}^{(2k)}+\frac{1}{4}i^{1/2}(s\cdot\nabla)^{2}\hat{\mathcal{O}}_{n-3}^{(2k)}$
$\displaystyle-\frac{1}{2}i^{-1/2}(\partial_{s}\cdot\nabla-s\cdot\nabla\partial_{s}^{2})\hat{\mathcal{O}}_{n}^{(2k)}s\cdot\nabla$
(3.47) $\displaystyle\hat{\mathcal{U}}_{n-3}^{(2k+2)}=$
$\displaystyle(\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2})s\cdot\nabla\,\hat{\mathcal{U}}_{n-3}^{(2k)}$
(3.48)

To recapitulate, given the existence of field strengths
$\mathcal{F}_{n-i}^{(2k)}$ that satisfy (3.17) – (3.1.1), we have found four
conditions (3.27), (3.28), (3.1.1), and (3.1.1) on the operators
$\hat{\mathcal{O}}_{n-i}^{(2k)}$ and $\hat{\mathcal{U}}_{n-i}^{(2k)}$, which
if satisfied enables the construction of $\Delta\mathcal{F}_{n-i}^{(2k+1)}$
and $\Delta\mathcal{F}_{n-i}^{(2k+2)}$, resulting in improved field strengths
$\mathcal{F}_{n-i}^{(2k+2)}=\mathcal{F}_{n-i}^{(2k)}+\Delta\mathcal{F}_{n-i}^{(2k+1)}+\Delta\mathcal{F}_{n-i}^{(2k+2)}$
with $\mathcal{O}(1/m^{2k+2})$ gauge violations. It is worth mentioning at
this stage that this procedure is not unique. For instance, one could add a
term proportional $\Phi_{n}$ to $\Delta\mathcal{F}_{n-3}^{(2k+1)}$. This would
add to the action a term linear in $\Phi_{n}$, and so we ignore this
possibility.

The next step is to show that if $\hat{\mathcal{O}}_{n-i}^{(2k)}$ and
$\hat{\mathcal{U}}_{n-i}^{(2k)}$ satisfy $\hat{\mathcal{C}}_{j}^{(2k)}=0$ for
$j=1,2,3,4$, then $\hat{\mathcal{O}}_{n-i}^{(2k+2)}$ and
$\hat{\mathcal{U}}_{n-i}^{(2k+2)}$ satisfy $\hat{\mathcal{C}}_{j}^{(2k+2)}=0$.
This follows after writing $\hat{\mathcal{O}}_{n-i}^{(2k+2)}$ and
$\hat{\mathcal{U}}_{n-i}^{(2k+2)}$ in terms of
$\hat{\mathcal{O}}_{n-i}^{(2k)}$ and $\hat{\mathcal{U}}_{n-i}^{(2k)}$

$\displaystyle\hat{\mathcal{C}}_{1}^{(2k+2)}=$
$\displaystyle\,\frac{1}{2}i^{-1/2}s\cdot\nabla\hat{\mathcal{C}}_{3}^{(2k)}=0$
(3.49) $\displaystyle\hat{\mathcal{C}}_{2}^{(2k+2)}=$
$\displaystyle\,2i^{1/2}(\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2})\hat{\mathcal{C}}_{4}^{(2k)}=0$
(3.50) $\displaystyle\hat{\mathcal{C}}_{3}^{(2k+2)}=$
$\displaystyle\,(\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2})s\cdot\nabla\hat{\mathcal{C}}_{3}^{(2k)}=0$
(3.51) $\displaystyle\hat{\mathcal{C}}_{4}^{(2k+2)}=$
$\displaystyle\,s\cdot\nabla(\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2})\hat{\mathcal{C}}_{4}^{(2k)}=0$
(3.52)

The final step is to verify that the operators $\hat{\mathcal{O}}_{n-i}^{(0)}$
and $\hat{\mathcal{U}}_{n-i}^{(0)}$ from (3.11) – (3.1.1) satisfy
$\hat{\mathcal{C}}_{j}^{(0)}=0$. The operators $\hat{\mathcal{O}}_{n-i}^{(0)}$
and $\hat{\mathcal{U}}_{n-i}^{(0)}$ associated with minimal coupling are

$\displaystyle\hat{\mathcal{O}}_{n}^{(0)}=i^{-1/2}[[\partial_{s}\cdot\nabla,s\cdot\nabla],s\cdot\nabla]$
$\displaystyle\hat{\mathcal{U}}_{n}^{(0)}=[\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2},[\partial_{s}\cdot\nabla,s\cdot\nabla]]$
(3.53)
$\displaystyle\hat{\mathcal{O}}_{n-1}^{(0)}=i^{-1/2}[[\partial_{s}\cdot\nabla,s\cdot\nabla],s\cdot\nabla]$
$\displaystyle\hat{\mathcal{U}}_{n-1}^{(0)}=[\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2},[\partial_{s}\cdot\nabla,s\cdot\nabla]]$
(3.54)
$\displaystyle\hat{\mathcal{O}}_{n-2}^{(0)}=-i^{1/2}[[\partial_{s}\cdot\nabla,s\cdot\nabla],s\cdot\nabla]\partial_{s}^{2}$
$\displaystyle\hat{\mathcal{U}}_{n-2}^{(0)}=\frac{i}{2}[s\cdot\nabla,[\partial_{s}\cdot\nabla,s\cdot\nabla]]$
(3.55)
$\displaystyle\hat{\mathcal{O}}_{n-3}^{(0)}=-i^{1/2}[[\partial_{s}\cdot\nabla,s\cdot\nabla],s\cdot\nabla]\partial_{s}^{2}$
$\displaystyle\hat{\mathcal{U}}_{n-3}^{(0)}=\frac{i}{2}[s\cdot\nabla,[\partial_{s}\cdot\nabla,s\cdot\nabla]]$
(3.56)

That these satisfy $\hat{\mathcal{C}}_{j}^{(0)}=0$ is straightforward to
verify, thus providing the base case for the induction argument to go through.
The modifications $\Delta\mathcal{F}_{n-i}^{(N)}$ (3.23) – (3.26) and (3.35) –
(3.38) provide a gauge invariant completion of minimal coupling.

####  3.1.2 Non-minimal gauge invariant interactions

As we saw in the previous section, minimally coupling $\Phi_{n-i}$ to
electromagnetism and gravity does not preserve the massive gauge symmetry
(3.1), but there exists modifications $\Delta\mathcal{F}_{n-i}^{(N)}$ to the
fields strengths
$\mathcal{F}_{n-i}\to\mathcal{F}_{n-i}+\sum_{N=0}^{\infty}\Delta\mathcal{F}_{n-i}^{(N)}$
which restores the gauge symmetry exactly.

Apart from minimal coupling, which gives $\Phi_{n}$ a charge/mass, there are
non-minimal couplings one might consider adding to your theory. For instance,
one might further consider adding a term
$\Delta\mathcal{F}_{n}^{(0)}=g_{0}([\partial_{s}\cdot\nabla,s\cdot\nabla]-\nabla^{2})\Phi_{n}$
to $\mathcal{F}_{n}$, which is related to the magnetic dipole and
gravitational quadrupole moment couplings, because

$\displaystyle[\partial_{s}\cdot\nabla,s\cdot\nabla]-\nabla^{2}=is^{\mu}F_{\mu\nu}\partial_{s}^{\nu}+s^{\mu}R_{\mu\nu}\partial_{s}^{\nu}-s^{\sigma}s^{\nu}R_{\omega\sigma\mu\nu}\partial_{s}^{\mu}\partial_{s}^{\omega}$
(3.57)

and, as explained in Cortese:2013lda , this is precisely how such a coupling
occurs at the level of the equations of motion. Adding such terms will, like
in the case of minimal coupling, break massive gauge invariance. Luckily, the
analysis of subsubsection 3.1.1 provides a simple prescription for adding
further interactions which restores the gauge symmetry. Indeed, we may
consider an arbitrary non-minimal coupling operator $\hat{\mathcal{M}}$ with
mass dimension $2k+2$ which is invariant with respect to the massive gauge
symmetry, and adding

$\displaystyle\Delta\mathcal{F}_{n}^{(2k)}=\frac{1}{m^{2k}}\hat{\mathcal{M}}\Phi_{n},$
$\displaystyle\Delta\mathcal{F}_{n-2}^{(2k)}=\frac{i}{m^{2k}}\partial_{s}^{2}\hat{\mathcal{M}}\Phi_{n}$
(3.58)

This introduces $\mathcal{O}(1/m^{2k})$ gauge violations, with associated
operators $\hat{\mathcal{O}}_{n-i}^{(2k)}$ and
$\hat{\mathcal{U}}_{n-i}^{(2k)}$ equal to

$\displaystyle\hat{\mathcal{O}}_{n}^{(2k)}$
$\displaystyle=i^{-1/2}\hat{\mathcal{M}}s\cdot\nabla$ (3.59)
$\displaystyle\hat{\mathcal{O}}_{n-2}^{(2k)}$
$\displaystyle=i^{1/2}\partial_{s}^{2}\hat{\mathcal{M}}s\cdot\nabla$ (3.60)
$\displaystyle\hat{\mathcal{U}}_{n}^{(2k)}$
$\displaystyle=(\partial_{s}\cdot\nabla-s\cdot\nabla\partial_{s}^{2})\hat{\mathcal{M}}$
(3.61)

while the rest are zero. These operators trivially satisfy the four conditions
$\hat{\mathcal{C}}_{j}^{(2k)}=0$ for any $\hat{\mathcal{M}}$. Therefore, the
gauge symmetry can be restored in exactly the same manner described in
subsubsection 3.1.1. Multipole and tidal couplings are obviously accommodated
by this class of gauge invariant interactions.

####  3.1.3 Resummed non-local field strengths

Gauge invariance requires infinitely many terms in the action which are
arbitrarily high order in derivatives, making interactions inherently non-
local in nature. To make this manifest, we may formally resum the
modifications $\Delta\mathcal{F}_{n-i}^{(N)}$ and write the interactions in
terms of non-local operators.

Suppose we have field strengths $\mathcal{F}_{n-i}^{(2k)}$ with
$\mathcal{O}(1/m^{2k})$ gauge violations that can be cancelled via
modifications $\Delta\mathcal{F}_{n-i}^{(N)}$ for all $N>2k$ in the way
described in subsubsection 3.1.1. The gauge invariant field strengths
$\mathcal{F}_{n-i}=\mathcal{F}_{n-i}^{(2k)}+\sum_{N=2k+1}^{\infty}\Delta\mathcal{F}_{n-i}^{(N)}$
may then be written as

$\displaystyle\mathcal{F}_{n}=\mathcal{F}_{n}^{(2k)}+\frac{i}{m^{2k+1}}\hat{\mathcal{O}}_{n}^{(2k)}\frac{1}{1-\frac{1}{2m^{2}}(s\cdot\nabla)^{2}\partial_{s}^{2}}X_{n-1}$
(3.62)
$\displaystyle\mathcal{F}_{n-1}=\mathcal{F}_{n-1}^{(2k)}-\frac{i^{-1/2}}{m^{2k+1}}\frac{1}{1-\frac{1}{m^{2}}s\cdot\nabla(\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2})}\Big{(}\hat{\mathcal{U}}_{n}^{(2k)}\Phi_{n}+\hat{\mathcal{U}}_{n-2}^{(2k)}\Phi_{n-2}\Big{)}$
$\displaystyle+\frac{1}{m^{2k+2}}\frac{1}{1-\frac{1}{m^{2}}s\cdot\nabla(\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2})}s\cdot\nabla\Big{(}\hat{\mathcal{U}}_{n-1}^{(2k)}\Phi_{n-1}+\hat{\mathcal{U}}_{n-3}^{(2k)}\Phi_{n-3}\Big{)}$
$\displaystyle-\frac{i^{1/2}}{m^{2k+2}}\frac{1}{1-\frac{1}{m^{2}}s\cdot\nabla(\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2})}(\partial_{s}\cdot\nabla-s\cdot\nabla\partial_{s}^{2})\hat{\mathcal{O}}_{n}^{(2k)}\frac{1}{1-\frac{1}{2m^{2}}(s\cdot\nabla)^{2}\partial_{s}^{2}}X_{n-1}$
$\displaystyle-\frac{1}{4m^{2k+3}}\frac{1}{1-\frac{1}{m^{2}}s\cdot\nabla(\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2})}(s\cdot\nabla)^{2}\hat{\mathcal{O}}_{n-3}^{(2k)}\frac{1}{1-\frac{1}{2m^{2}}s\cdot\nabla\partial_{s}^{2}s\cdot\nabla}X_{n-2}$
(3.63)
$\displaystyle\mathcal{F}_{n-2}=\mathcal{F}_{n-2}^{(2k)}-2\frac{i^{-1/2}}{m^{2k+1}}\frac{1}{1-\frac{1}{m^{2}}(\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2})s\cdot\nabla}\Big{(}\hat{\mathcal{U}}_{n-1}^{(2k)}\Phi_{n-1}+\hat{\mathcal{U}}_{n-3}^{(2k)}\Phi_{n-3}\Big{)}$
$\displaystyle-\frac{1}{m^{2k+1}}\frac{1}{1-\frac{1}{m^{2}}(\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2})s\cdot\nabla}\partial_{s}^{2}\hat{\mathcal{O}}_{n}^{(2k)}\frac{1}{1-\frac{1}{2m^{2}}(s\cdot\nabla)^{2}\partial_{s}^{2}}X_{n-1}$
$\displaystyle-2\frac{i}{m^{2k+2}}\frac{1}{1-\frac{1}{m^{2}}(\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2})s\cdot\nabla}(\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2})\Big{(}\hat{\mathcal{U}}_{n}^{(2k)}\Phi_{n}+\hat{\mathcal{U}}_{n-2}^{(2k)}\Phi_{n-2}\Big{)}$
$\displaystyle+\frac{i^{-1/2}}{2m^{2k+2}}\frac{1}{1-\frac{1}{m^{2}}(\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2})s\cdot\nabla}s\cdot\nabla\hat{\mathcal{O}}_{n-3}^{(2k)}\frac{1}{1-\frac{1}{2m^{2}}s\cdot\nabla\partial_{s}^{2}s\cdot\nabla}X_{n-2}$
$\displaystyle+\frac{2}{m^{2k+3}}\frac{1}{1-\frac{1}{m^{2}}(\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2})s\cdot\nabla}(\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2})^{2}\hat{\mathcal{O}}_{n}^{(2k)}\frac{1}{1-\frac{1}{2m^{2}}(s\cdot\nabla)^{2}\partial_{s}^{2}}X_{n-1}$
(3.64)
$\displaystyle\mathcal{F}_{n-3}=\mathcal{F}_{n-3}^{(2k)}+\frac{i}{2m^{2k+1}}\hat{\mathcal{O}}_{n-3}^{(2k)}\frac{1}{1-\frac{1}{2m^{2}}s\cdot\nabla\partial_{s}^{2}s\cdot\nabla}X_{n-2}$
(3.65)

where we have defined
$X_{n-1}\equiv\Phi_{n-1}+\frac{i^{1/2}}{2m}s\cdot\nabla\Phi_{n-2}$ and
$X_{n-2}\equiv\Phi_{n-2}+\frac{i^{-1/2}}{m}s\cdot\nabla\partial_{s}^{2}\Phi_{n-1}$.
One can check that these expressions are gauge invariant and satisfy the
covariantized massive Bianchi identities. To simplify these expressions, we
are free to gauge fix, by for instance choosing $\Phi_{n-1},\Phi_{n-2}=0$

$\displaystyle\mathcal{F}_{n}$ $\displaystyle=\mathcal{F}_{n}^{(2k)}$ (3.66)
$\displaystyle\mathcal{F}_{n-1}$
$\displaystyle=\mathcal{F}_{n-1}^{(2k)}-\frac{i^{-1/2}}{m^{2k+1}}\frac{1}{1-\frac{1}{m^{2}}s\cdot\nabla(\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2})}\hat{\mathcal{U}}_{n}^{(2k)}\Phi_{n}$
$\displaystyle\phantom{=}+\frac{1}{m^{2k+2}}\frac{1}{1-\frac{1}{m^{2}}s\cdot\nabla(\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2})}s\cdot\nabla\hat{\mathcal{U}}_{n-3}^{(2k)}\Phi_{n-3}$
(3.67) $\displaystyle\mathcal{F}_{n-2}$
$\displaystyle=\mathcal{F}_{n-2}^{(2k)}-2\frac{i^{-1/2}}{m^{2k+1}}\frac{1}{1-\frac{1}{m^{2}}(\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2})s\cdot\nabla}\hat{\mathcal{U}}_{n-3}^{(2k)}\Phi_{n-3}$
$\displaystyle\phantom{=}-2\frac{i}{m^{2k+2}}\frac{1}{1-\frac{1}{m^{2}}(\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2})s\cdot\nabla}(\partial_{s}\cdot\nabla-\frac{1}{2}s\cdot\nabla\partial_{s}^{2})\hat{\mathcal{U}}_{n}^{(2k)}\Phi_{n}$
(3.68) $\displaystyle\mathcal{F}_{n-3}$
$\displaystyle=\mathcal{F}_{n-3}^{(2k)}$ (3.69)

For spins $n=0,1$, the non-local gauge invariant completions truncate to a
local operator, so that a local field theory is possible. For $n\geq 2$ on the
other hand, non-locality appears to be unavoidable.

####  3.1.4 Interactions linear in spin $n$ fields

If the massive spin $n$ particle has $U(1)$ charge 0, there is another class
of interactions, i.e. those which are only linear in the matter fields
$\Phi_{n-i}$ in the action, which can be made gauge invariant. To get a charge
0 particle, it is sufficient to impose the reality conditions on the fields
$\tilde{\Phi}_{n}=\Phi_{n}$, $\tilde{\Phi}_{n-1}=-\Phi_{n-1}$,
$\tilde{\Phi}_{n-2}=\Phi_{n-2}$, and $\tilde{\Phi}_{n-3}=-\Phi_{n-3}$. The
gauge parameters in turn are constrained
$\tilde{\epsilon}_{n-1}=\epsilon_{n-1}$,
$\tilde{\epsilon}_{n-2}=-\epsilon_{n-2}$. If we rewrite everything in terms of
real fields by performing the field redefinitions $\Phi_{n-1}\to i\Phi_{n-1}$,
$\Phi_{n-3}\to i\Phi_{n-3}$, and $\epsilon_{n-2}\to i\epsilon_{n-2}$, the
covariantized massive gauge transformations become

$\displaystyle\delta\Phi_{n}=i^{-1/2}s\cdot\nabla\epsilon_{n-1}$
$\displaystyle\delta\Phi_{n-1}=i^{-1/2}s\cdot\nabla\epsilon_{n-2}+m\,\epsilon_{n-1}$
$\displaystyle\delta\Phi_{n-2}=-i^{1/2}s\cdot\nabla\partial_{s}^{2}\epsilon_{n-1}-2m\,\epsilon_{n-2}$
$\displaystyle\delta\Phi_{n-3}=-i^{1/2}s\cdot\nabla\partial_{s}^{2}\epsilon_{n-2}-3im\partial_{s}^{2}\epsilon_{n-1}$
(3.70)

where
$\nabla_{\mu}=\partial_{X\mu}-s^{\nu}\Gamma^{\lambda}_{\;\;\mu\nu}\partial_{s\lambda}$.

Coupling $\Phi_{n-i}$ to hyperfields $\mathcal{J}_{n-i}$ which are gauge
invariant functions of the gravitational and possibly electromagnetic fields
by adding to the action

$\displaystyle\Delta S_{n}=n!\int
d^{d}X\frac{d^{d}sd^{d}s^{\prime}}{(2\pi)^{d}}e^{is\cdot
s^{\prime}}\Phi_{n-i}(X,s)\mathcal{J}_{n-i}(X,s^{\prime})$ (3.71)

where the sum over $i=0,\dots,3$ is implied, is gauge invariant provided that
the $\mathcal{J}_{n-i}$ satisfy

$\displaystyle\partial_{s}\cdot\nabla\mathcal{J}_{n}+is^{2}\partial_{s}\cdot\nabla\mathcal{J}_{n-2}-i^{-1/2}m\mathcal{J}_{n-1}-3i^{1/2}ms^{2}\mathcal{J}_{n-3}=0$
(3.72)
$\displaystyle\partial_{s}\cdot\nabla\mathcal{J}_{n-1}+is^{2}\partial_{s}\cdot\nabla\mathcal{J}_{n-3}+2i^{-1/2}m\mathcal{J}_{n-2}=0$
(3.73)

To construct $\mathcal{J}_{n-i}$ which satisfy (3.72) and (3.73), we may
proceed in a similar fashion to subsubsection 3.1.1, by starting with
$\mathcal{J}_{n-i}$ which do not satisfy (3.72) and (3.73), and finding
improvement terms inductively that make the interactions exactly gauge
invariant. Instead, we may more simply search for gauge invariant linear
combinations of the fields $\Phi_{n-i}$. There are two gauge invariant linear
combinations

$\displaystyle
A_{n}\equiv\Phi_{n}-\frac{i^{-1/2}}{m}s\cdot\nabla\frac{1}{1-\frac{1}{2m^{2}}(s\cdot\nabla)^{2}\partial_{s}^{2}}X_{n-1}$
(3.74) $\displaystyle
A_{n-3}\equiv\Phi_{n-3}+3i\partial_{s}^{2}\frac{1}{1-\frac{1}{2m^{2}}(s\cdot\nabla)^{2}\partial_{s}^{2}}X_{n-1}-\frac{i^{1/2}}{2m}s\cdot\nabla\partial_{s}^{2}\frac{1}{1-\frac{1}{2m^{2}}s\cdot\nabla\partial_{s}^{2}s\cdot\nabla}X_{n-2}$
(3.75)

where now $X_{n-1}=\Phi_{n-1}+\frac{i^{-1/2}}{2m}s\cdot\nabla\Phi_{n-2}$ and
$X_{n-2}=\Phi_{n-2}+\frac{i^{1/2}}{m}s\cdot\nabla\partial_{s}^{2}\Phi_{n-1}$.
Note that these gauge invariant combinations are non-local for $n\geq 3$.
Using $A_{n}$ and $A_{n-3}$, we may construct gauge invariant interactions
linear in $\Phi_{n-i}$ of any type, including the ones classified in
Chakraborty:2020rxf

$\displaystyle\Delta S_{n}=n!\int
d^{d}X\frac{d^{d}sd^{d}s^{\prime}}{(2\pi)^{d}}e^{is\cdot
s^{\prime}}\Big{(}A_{n}(X,s)\mathcal{J}_{n}(X,s^{\prime})+A_{n-3}(X,s)\mathcal{J}_{n-3}(X,s^{\prime})\Big{)}$
(3.76)

for any $\mathcal{J}_{n}$ and $\mathcal{J}_{n-3}$. Rewriting (3.76) in the
form (3.71) through integration by parts will give $\mathcal{J}_{n-1}$ and
$\mathcal{J}_{n-2}$ that are functions of $\mathcal{J}_{n}$ and
$\mathcal{J}_{n-3}$, which altogether satisfy (3.72) and (3.73).

###  3.2 Half integer spins

In this section, we repeat the analysis of subsection 3.1 for massive half
integer spin particles described by the free action (2.2). We will find gauge
invariant interactions through a completely analogous argument.

We first begin by minimally coupling our theory to electromagnetism and
gravity. This amounts to replacing every spacetime derivative
$\partial_{X\mu}$ in the action with a covariant derivative $\nabla_{\mu}$, as
well as again replacing every $\eta_{\mu\nu}$ with $g_{\mu\nu}(X)$, and
$d^{d}X$ with $d^{d}X\sqrt{-g}$. In the case of half integer spin fields, we
also use Dirac matrices $\gamma_{\mu}(X)$ satisfying
$\\{\gamma_{\mu},\gamma_{\nu}\\}=2g_{\mu\nu}$. This is made possible by
introducing a vielbein $e_{\mu}^{\;a}(X)$ satisfying
$e_{\mu}^{\;a}e_{\nu}^{\;b}\eta_{ab}=g_{\mu\nu}$ and
$e_{\mu}^{\;a}e_{\nu}^{\;b}g^{\mu\nu}=\eta^{ab}$, and writing
$\gamma_{\mu}(X)=e_{\mu}^{\;a}(X)\gamma_{a}$, where $\gamma_{a}$ are constant
Dirac matrices satisfying $\\{\gamma_{a},\gamma_{b}\\}=2\eta_{ab}$. Here Latin
indices $a,b=0,\dots,d-1$ are raised and lowered with the flat metric
$\eta^{ab}$ and $\eta_{ab}$, respectively.

The covariant derivative $\nabla_{\mu}$ on a charge 1 half integer spin
$n+1/2$ hyperfield $\Psi_{n}(X,s)$, including a $U(1)$ gauge field $A_{\mu}$,
a spin connection $\omega_{\mu}^{\;ab}$ and Levi-Civita connection
$\Gamma^{\lambda}_{\;\;\mu\nu}$ related via
$\Gamma^{\lambda}_{\;\;\mu\nu}=e^{\lambda}_{\;a}(\partial_{\mu}e_{\nu}^{\;a}+\omega_{\mu}^{\;ab}e_{\nu
b})$, which implements the standard covariant derivative on its component
field $\psi_{\mu_{1}\cdots\mu_{n}}(X)$ is

$\displaystyle\nabla_{\mu}=\partial_{X\mu}-iA_{\mu}+\frac{1}{4}\omega_{\mu}^{\;ab}\gamma_{ab}-s^{\nu}\Gamma^{\lambda}_{\;\;\mu\nu}\partial_{s\lambda}$
(3.77)

Covariant derivatives no longer commute, but instead their commutator when
acting on $\Psi_{n}(X,s)$ equals

$\displaystyle[\nabla_{\mu},\nabla_{\nu}]=-iF_{\mu\nu}+\frac{1}{4}R_{\omega\sigma\mu\nu}\gamma^{\omega\sigma}-s^{\sigma}R^{\omega}_{\;\;\sigma\mu\nu}\partial_{s\omega}$
(3.78)

As part of minimal coupling, we will as before deform the massive gauge
symmetry (2.2) by replacing spacetime derivatives with covariant derivatives,
$\eta^{\mu\nu}$ with $g^{\mu\nu}$, and $\gamma_{\mu}$ with
$e_{\mu}^{\;a}\gamma_{a}$, so that it is consistent with $U(1)$
electromagnetic gauge invariance and general coordinate invariance

$\displaystyle\delta\Psi_{n}=i^{-1/2}s\cdot\nabla\epsilon_{n-1}$
$\displaystyle\delta\Psi_{n-1}=-is\cdot\nabla\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\epsilon_{n-1}+im\epsilon_{n-1}$
$\displaystyle\delta\Psi_{n-2}=-i^{1/2}s\cdot\nabla\partial_{s}^{2}\epsilon_{n-1}+2i^{1/2}m\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\epsilon_{n-1}$
(3.79)

The action (2.2) is not invariant under the gauge symmetry (3.2) after minimal
coupling, and hence does not consistently describe a massive spin $n+1/2$
particle. Indeed, the gauge variation of the minimally coupled action $\delta
S_{n+1/2}$ is a linear combination of terms proportional to
$\delta\mathcal{S}_{n-i}$, and the covariantized version of the fermionic
massive Bianchi identity (2.23). This follows from the fact that the massive
action is derived from the corresponding massless action $S_{n+1/2,0}$ in
$d+1$ dimensions built from a rank $n$ triple $\gamma$ traceless hyperfield
$\Psi_{n,0}(X,s)$ and massless field strength $\mathcal{S}_{n,0}(X,s)$,
described in Lindwasser:2023zwo . Under the gauge transformation
$\delta\Psi_{n,0}(X,s)=i^{-1/2}s\cdot\partial_{X}\epsilon_{n-1}(X,s)$, for
$\epsilon_{n-1}(X,s)$ a rank $n-1$ $\gamma$ traceless hyperfield, the gauge
variation of $S_{n+1/2,0}$ is

$\displaystyle\delta S_{n+1/2,0}=-n!\int
d^{d+1}X\frac{d^{d+1}sd^{d+1}s^{\prime}}{(2\pi)^{d+1}}e^{is\cdot
s^{\prime}}\times$
$\displaystyle\Big{(}\overline{\Psi}_{n,0}(X,s)\big{(}1-\frac{1}{2}\mathrlap{\\!\not{\phantom{s^{\prime}}}}s^{\prime}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s^{\prime}}-\frac{1}{4}s^{\prime
2}\partial_{s^{\prime}}^{2}\big{)}\delta\mathcal{S}_{n,0}(X,s^{\prime})-\frac{1}{2}i^{1/2}\overline{\epsilon}_{n-1}(X,s)\big{(}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s^{\prime}}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{X}-s^{\prime}\cdot\partial_{X}\partial_{s^{\prime}}^{2}\big{)}\mathcal{S}_{n,0}\Big{)}$
(3.80)

After dimensional reduction, the gauge variation of the massless $d+1$
dimensional field strength $\delta\mathcal{S}_{n,0}$ decomposes into the
massive $d$ dimensional variations $\delta\mathcal{S}_{n-i}$, and the
expression
$\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{X}\mathcal{S}_{n,0}-s\cdot\partial_{X}\partial_{s}^{2}\mathcal{S}_{n,0}$
decomposes into the fermionic massive Bianchi identity (2.23). These are no
longer zero after minimal coupling, but instead equal

$\displaystyle\delta\mathcal{S}_{n}$
$\displaystyle=i^{-1/2}[\mathrlap{\\!\not{\phantom{\nabla}}}\nabla,s\cdot\nabla]\epsilon_{n-1}$
(3.81) $\displaystyle\delta\mathcal{S}_{n-1}$
$\displaystyle=-i[\mathrlap{\\!\not{\phantom{\nabla}}}\nabla,s\cdot\nabla]\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\epsilon_{n-1}$
(3.82) $\displaystyle\delta\mathcal{S}_{n-2}$
$\displaystyle=-i^{1/2}[\mathrlap{\\!\not{\phantom{\nabla}}}\nabla,s\cdot\nabla]\partial_{s}^{2}\epsilon_{n-1}$
(3.83)
$\displaystyle\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}(\mathrlap{\\!\not{\phantom{\nabla}}}\nabla-m)\mathcal{S}_{n}-s\cdot\nabla\partial_{s}^{2}\mathcal{S}_{n}-i^{1/2}(\mathrlap{\\!\not{\phantom{\nabla}}}\nabla-m)\mathcal{S}_{n-1}+i\,s\cdot\nabla\mathcal{S}_{n-2}=$
$\displaystyle\big{(}[\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}$
$\displaystyle,(\mathrlap{\\!\not{\phantom{\nabla}}}\nabla)^{2}]-\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}[\mathrlap{\\!\not{\phantom{\nabla}}}\nabla,s\cdot\nabla]\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\big{)}\Psi_{n}+i^{1/2}\\{\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s},[\mathrlap{\\!\not{\phantom{\nabla}}}\nabla,s\cdot\nabla]\\}\Psi_{n-1}-i[\mathrlap{\\!\not{\phantom{\nabla}}}\nabla,s\cdot\nabla]\Psi_{n-2}$
(3.84)

####  3.2.1 Restoring gauge symmetry

We now show how to restore the gauge symmetry after minimal coupling by adding
further interactions with electromagnetism and gravity in the form of making
appropriate modifications of the field strengths
$\mathcal{S}_{n-i}\to\mathcal{S}_{n-i}+\Delta\mathcal{S}_{n-i}$, so that they
are again gauge invariant and satisfy the massive Bianchi identity, in a way
exactly analogous to the integer spin case in subsubsection 3.1.1.

In particular, first note that the gauge violations in (3.81) – (3.2), are
$\mathcal{O}(1)$ in $m$. Because of the appearance of $m$ in the gauge
transformations (3.2) and massive Bianchi identity, it is possible to suppress
the gauge violations to $\mathcal{O}(1/m)$, cancelling the $\mathcal{O}(1)$
violations, by adding a $\Delta\mathcal{S}_{n-i}^{(1)}$ which is proportional
to $1/m$. We will prove by induction that this process can continue
indefinitely, generating an infinite series of
$\Delta\mathcal{S}_{n-i}^{(N)}$, each proportional to $1/m^{N}$, for all
integers $N\geq 0$, until the action is exactly gauge invariant.

Suppose we have found modified field strengths $\mathcal{S}_{n-i}^{(2k)}$
which suppress all gauge violations to be $\mathcal{O}(1/m^{2k})$, which we
write as

$\displaystyle\delta\mathcal{S}_{n}^{(2k)}=$
$\displaystyle\frac{1}{m^{2k}}\hat{\mathcal{O}}_{n}^{(2k)}\epsilon_{n-1}$
(3.85) $\displaystyle\delta\mathcal{S}_{n-1}^{(2k)}=$
$\displaystyle\frac{1}{m^{2k}}\hat{\mathcal{O}}_{n-1}^{(2k)}\epsilon_{n-1}$
(3.86) $\displaystyle\delta\mathcal{S}_{n-2}^{(2k)}=$
$\displaystyle\frac{1}{m^{2k}}\hat{\mathcal{O}}_{n-2}^{(2k)}\epsilon_{n-1}$
(3.87)
$\displaystyle\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}(\mathrlap{\\!\not{\phantom{\nabla}}}\nabla-m)$
$\displaystyle\mathcal{S}_{n}^{(2k)}-s\cdot\nabla\partial_{s}^{2}\mathcal{S}_{n}^{(2k)}-i^{1/2}(\mathrlap{\\!\not{\phantom{\nabla}}}\nabla-m)\mathcal{S}_{n-1}^{(2k)}+i\,s\cdot\nabla\mathcal{S}_{n-2}^{(2k)}=$
$\displaystyle\frac{1}{m^{2k}}\hat{\mathcal{U}}_{n}^{(2k)}\Psi_{n}+\frac{1}{m^{2k}}\hat{\mathcal{U}}_{n-1}^{(2k)}\Psi_{n-1}+\frac{1}{m^{2k}}\hat{\mathcal{U}}_{n-2}^{(2k)}\Psi_{n-2}$
(3.88)

for some operators $\hat{\mathcal{O}}_{n-i}^{(2k)}$,
$\hat{\mathcal{U}}_{n-i}^{(2k)}$. We first show that if
$\hat{\mathcal{O}}_{n-i}^{(2k)}$ and $\hat{\mathcal{U}}_{n-i}^{(2k)}$ satisfy
conditions (to be determined) ensuring that $\Delta\mathcal{S}_{n-i}^{(2k+1)}$
and $\Delta\mathcal{S}_{n-i}^{(2k+2)}$ can be constructed so that
$\mathcal{S}_{n-i}^{(2k+2)}\equiv\mathcal{S}_{n-i}^{(2k)}+\Delta\mathcal{S}_{n-i}^{(2k+1)}+\Delta\mathcal{S}_{n-i}^{(2k+2)}$
suppresses the gauge violations to $\mathcal{O}(1/m^{2k+2})$, then the
corresponding operators at the next order $\hat{\mathcal{O}}_{n-i}^{(2k+2)}$
and $\hat{\mathcal{U}}_{n-i}^{(2k+2)}$ will also satisfy those conditions.

To cancel the $\mathcal{O}(1/m^{2k})$ gauge violations (3.85), (3.87), and
(3.2.1), one may choose $\Delta\mathcal{S}_{n-i}^{(2k+1)}$ to be

$\displaystyle\Delta\mathcal{S}_{n}^{(2k+1)}$
$\displaystyle=\frac{i}{m^{2k+1}}\hat{\mathcal{O}}_{n}^{(2k)}\Psi_{n-1}$
(3.89) $\displaystyle\Delta\mathcal{S}_{n-1}^{(2k+1)}$
$\displaystyle=-\frac{i^{-1/2}}{m^{2k+1}}\Big{(}\hat{\mathcal{U}}_{n}^{(2k)}\Psi_{n}+(-i\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\hat{\mathcal{O}}_{n}^{(2k)}+\hat{\mathcal{U}}_{n-1}^{(2k)})\Psi_{n-1}+\hat{\mathcal{U}}_{n-2}^{(2k)}\Psi_{n-2}\Big{)}$
(3.90) $\displaystyle\Delta\mathcal{S}_{n-2}^{(2k+1)}$
$\displaystyle=\frac{i}{m^{2k+1}}\hat{\mathcal{O}}_{n-2}^{(2k)}\Psi_{n-1}$
(3.91)

The remaining $\mathcal{O}(1/m^{2k})$ gauge violation (3.86) must then be
cancelled by $\delta(\Delta\mathcal{S}_{n-1}^{(2k+1)})$. This in turn is only
possible if the following condition holds

$\displaystyle\hat{\mathcal{C}}_{1}^{(2k)}\equiv\hat{\mathcal{O}}_{n-1}^{(2k)}-i^{-1/2}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\hat{\mathcal{O}}_{n}^{(2k)}-i^{1/2}\hat{\mathcal{U}}_{n-1}^{(2k)}-2\,\hat{\mathcal{U}}_{n-2}^{(2k)}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}=0$
(3.92)

Let us assume this is true and proceed with constructing
$\Delta\mathcal{S}_{n-i}^{(2k+2)}$. The gauge violations associated with
$\mathcal{S}_{n-i}^{(2k+1)}=\mathcal{S}_{n-i}^{(2k)}+\Delta\mathcal{S}_{n-i}^{(2k+1)}$
are

$\displaystyle\delta\mathcal{S}_{n}^{(2k+1)}$
$\displaystyle=\frac{1}{m^{2k+1}}\hat{\mathcal{O}}_{n}^{(2k)}s\cdot\nabla\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\epsilon_{n-1}$
(3.93) $\displaystyle\delta\mathcal{S}_{n-1}^{(2k+1)}$
$\displaystyle=\frac{1}{m^{2k+1}}\Big{(}i^{1/2}(-i\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\hat{\mathcal{O}}_{n}^{(2k)}+\hat{\mathcal{U}}_{n-1}^{(2k)})s\cdot\nabla\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}+i\,\hat{\mathcal{U}}_{n}^{(2k)}s\cdot\nabla+\hat{\mathcal{U}}_{n-2}^{(2k)}s\cdot\nabla\partial_{s}^{2}\Big{)}\epsilon_{n-1}$
(3.94) $\displaystyle\delta\mathcal{S}_{n-2}^{(2k+1)}$
$\displaystyle=\frac{1}{m^{2k+1}}\hat{\mathcal{O}}_{n-2}^{(2k)}s\cdot\nabla\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\epsilon_{n-1}$
(3.95)
$\displaystyle\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}(\mathrlap{\\!\not{\phantom{\nabla}}}\nabla$
$\displaystyle-m)\mathcal{S}_{n}^{(2k+1)}-s\cdot\nabla\partial_{s}^{2}\mathcal{S}_{n}^{(2k+1)}-i^{1/2}(\mathrlap{\\!\not{\phantom{\nabla}}}\nabla-m)\mathcal{S}_{n-1}^{(2k+1)}+i\,s\cdot\nabla\mathcal{S}_{n-2}^{(2k+1)}=$
$\displaystyle\frac{1}{m^{2k+1}}\Big{(}\mathrlap{\\!\not{\phantom{\nabla}}}\nabla\hat{\mathcal{U}}_{n-1}^{(2k)}+i([\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s},\mathrlap{\\!\not{\phantom{\nabla}}}\nabla]-s\cdot\nabla\partial_{s}^{2})\hat{\mathcal{O}}_{n}^{(2k)}-s\cdot\nabla\hat{\mathcal{O}}_{n-2}^{(2k)}\Big{)}\Psi_{n-1}$
$\displaystyle\hskip
42.67912pt+\frac{1}{m^{2k+1}}\mathrlap{\\!\not{\phantom{\nabla}}}\nabla\hat{\mathcal{U}}_{n}^{(2k)}\Psi_{n}+\frac{1}{m^{2k+1}}\mathrlap{\\!\not{\phantom{\nabla}}}\nabla\hat{\mathcal{U}}_{n-2}^{(2k)}\Psi_{n-2}$
(3.96)

It is straightforward to find $\Delta\mathcal{S}_{n-i}^{(2k+2)}$ which cancel
the $\mathcal{O}(1/m^{2k+1})$ gauge violations in (3.93), (3.95), and (3.2.1)

$\displaystyle\Delta\mathcal{S}_{n}^{(2k+2)}=\frac{i}{m^{2k+2}}\hat{\mathcal{O}}_{n}^{(2k)}s\cdot\nabla\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\Psi_{n-1}$
(3.97)
$\displaystyle\Delta\mathcal{S}_{n-1}^{(2k+2)}=-\frac{i^{-1/2}}{m^{2k+2}}\Big{(}\mathrlap{\\!\not{\phantom{\nabla}}}\nabla\hat{\mathcal{U}}_{n}^{(2k)}\Psi_{n}+\mathrlap{\\!\not{\phantom{\nabla}}}\nabla\hat{\mathcal{U}}_{n-2}^{(2k)}\Psi_{n-2}\Big{)}$
$\displaystyle-\frac{i^{-1/2}}{m^{2k+2}}\Big{(}\mathrlap{\\!\not{\phantom{\nabla}}}\nabla\hat{\mathcal{U}}_{n-1}^{(2k)}-i\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\hat{\mathcal{O}}_{n}^{(2k)}s\cdot\nabla\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}+i([\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s},\mathrlap{\\!\not{\phantom{\nabla}}}\nabla]-s\cdot\nabla\partial_{s}^{2})\hat{\mathcal{O}}_{n}^{(2k)}-s\cdot\nabla\hat{\mathcal{O}}_{n-2}^{(2k)}\Big{)}\Psi_{n-1}$
(3.98)
$\displaystyle\Delta\mathcal{S}_{n-2}^{(2k+2)}=\frac{i}{m^{2k+2}}\hat{\mathcal{O}}_{n-2}^{(2k)}s\cdot\nabla\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\Psi_{n-1}$
(3.99)

Demanding that $\delta(\Delta\mathcal{S}_{n-1}^{(2k+2)})$ also cancels the
remaining $\mathcal{O}(1/m^{2k+1})$ gauge violation in (3.94) imposes one
final condition on $\hat{\mathcal{O}}_{n-i}^{(2k)}$ and
$\hat{\mathcal{U}}_{n-i}^{(2k)}$

$\displaystyle\hat{\mathcal{C}}_{2}^{(2k)}\equiv$
$\displaystyle\,i^{1/2}([\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s},\mathrlap{\\!\not{\phantom{\nabla}}}\nabla]-s\cdot\nabla\partial_{s}^{2})\hat{\mathcal{O}}_{n}^{(2k)}-i^{-1/2}s\cdot\nabla\hat{\mathcal{O}}_{n-2}^{(2k)}-\hat{\mathcal{U}}_{n}^{(2k)}s\cdot\nabla$
$\displaystyle+i^{-1/2}\mathrlap{\\!\not{\phantom{\nabla}}}\nabla\hat{\mathcal{U}}_{n-1}^{(2k)}-i^{-1/2}\hat{\mathcal{U}}_{n-1}^{(2k)}s\cdot\nabla\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}-2i\mathrlap{\\!\not{\phantom{\nabla}}}\nabla\hat{\mathcal{U}}_{n-2}^{(2k)}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}+i\,\hat{\mathcal{U}}_{n-2}^{(2k)}s\cdot\nabla\partial_{s}^{2}=0$
(3.100)

Assuming that this condition is satisfied, we may use
$\mathcal{S}_{n-i}^{(2k+2)}=\mathcal{S}_{n-i}^{(2k+1)}+\Delta\mathcal{S}_{n-i}^{(2k+2)}$
to find the corresponding operators at the next order
$\hat{\mathcal{O}}_{n-i}^{(2k+2)}$ and $\hat{\mathcal{U}}_{n-i}^{(2k+2)}$
associated with $\mathcal{O}(1/m^{2k+2})$ gauge violations

$\displaystyle\hat{\mathcal{O}}_{n}^{(2k+2)}=$
$\displaystyle\,\hat{\mathcal{O}}_{n}^{(2k)}(s\cdot\nabla\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s})^{2}$
(3.101) $\displaystyle\hat{\mathcal{O}}_{n-1}^{(2k+2)}=$
$\displaystyle\,i^{-1/2}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\hat{\mathcal{O}}_{n}^{(2k)}(s\cdot\nabla\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s})^{2}-i^{-1/2}([\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s},\mathrlap{\\!\not{\phantom{\nabla}}}\nabla]-s\cdot\nabla\partial_{s}^{2})\hat{\mathcal{O}}_{n}^{(2k)}s\cdot\nabla\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}+i\mathrlap{\\!\not{\phantom{\nabla}}}\nabla\hat{\mathcal{U}}_{n}^{(2k)}s\cdot\nabla$
$\displaystyle+i^{1/2}\mathrlap{\\!\not{\phantom{\nabla}}}\nabla\hat{\mathcal{U}}_{n-1}^{(2k)}s\cdot\nabla\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}+\mathrlap{\\!\not{\phantom{\nabla}}}\nabla\hat{\mathcal{U}}_{n-2}^{(2k)}s\cdot\nabla\partial_{s}^{2}-i^{1/2}s\cdot\nabla\hat{\mathcal{O}}_{n-2}^{(2k)}s\cdot\nabla\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}$
(3.102) $\displaystyle\hat{\mathcal{O}}_{n-2}^{(2k+2)}=$
$\displaystyle\,\hat{\mathcal{O}}_{n-2}^{(2k)}(s\cdot\nabla\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s})^{2}$
(3.103) $\displaystyle\hat{\mathcal{U}}_{n}^{(2k+2)}=$
$\displaystyle\,(\mathrlap{\\!\not{\phantom{\nabla}}}\nabla)^{2}\hat{\mathcal{U}}_{n}^{(2k)}$
(3.104) $\displaystyle\hat{\mathcal{U}}_{n-1}^{(2k+2)}=$
$\displaystyle\,i([\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s},\mathrlap{\\!\not{\phantom{\nabla}}}\nabla]-s\cdot\nabla\partial_{s}^{2})\hat{\mathcal{O}}_{n}^{(2k)}s\cdot\nabla\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}+i\mathrlap{\\!\not{\phantom{\nabla}}}\nabla([\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s},\mathrlap{\\!\not{\phantom{\nabla}}}\nabla]-s\cdot\nabla\partial_{s}^{2})\hat{\mathcal{O}}_{n}^{(2k)}$
$\displaystyle+(\mathrlap{\\!\not{\phantom{\nabla}}}\nabla)^{2}\hat{\mathcal{U}}_{n-1}^{(2k)}-\mathrlap{\\!\not{\phantom{\nabla}}}\nabla
s\cdot\nabla\hat{\mathcal{O}}_{n-2}^{(2k)}-s\cdot\nabla\hat{\mathcal{O}}_{n-2}^{(2k)}s\cdot\nabla\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}$
(3.105) $\displaystyle\hat{\mathcal{U}}_{n-2}^{(2k+2)}=$
$\displaystyle\,(\mathrlap{\\!\not{\phantom{\nabla}}}\nabla)^{2}\hat{\mathcal{U}}_{n-2}^{(2k)}$
(3.106)

The next step is to show that if $\hat{\mathcal{O}}_{n-i}^{(2k)}$ and
$\hat{\mathcal{U}}_{n-i}^{(2k)}$ satisfy $\hat{\mathcal{C}}_{j}^{(2k)}=0$ for
$j=1,2$, then $\hat{\mathcal{O}}_{n-i}^{(2k+2)}$ and
$\hat{\mathcal{U}}_{n-i}^{(2k+2)}$ satisfy $\hat{\mathcal{C}}_{j}^{(2k+2)}=0$.
This follows after writing $\hat{\mathcal{O}}_{n-i}^{(2k+2)}$ and
$\hat{\mathcal{U}}_{n-i}^{(2k+2)}$ in terms of
$\hat{\mathcal{O}}_{n-i}^{(2k)}$ and $\hat{\mathcal{U}}_{n-i}^{(2k)}$

$\displaystyle\hat{\mathcal{C}}_{1}^{(2k+2)}$
$\displaystyle=-i\mathrlap{\\!\not{\phantom{\nabla}}}\nabla\hat{\mathcal{C}}_{2}^{(2k)}=0$
(3.107) $\displaystyle\hat{\mathcal{C}}_{2}^{(2k+2)}$
$\displaystyle=(\mathrlap{\\!\not{\phantom{\nabla}}}\nabla)^{2}\hat{\mathcal{C}}_{2}^{(2k)}=0$
(3.108)

Finally, we must verify that the operators $\hat{\mathcal{O}}_{n-i}^{(0)}$ and
$\hat{\mathcal{U}}_{n-i}^{(0)}$ associated with minimal coupling (3.81) –
(3.2) satisfy $\hat{\mathcal{C}}_{j}^{(0)}=0$. $\hat{\mathcal{O}}_{n-i}^{(0)}$
and $\hat{\mathcal{U}}_{n-i}^{(0)}$ are equal to

$\displaystyle\hat{\mathcal{O}}_{n}^{(0)}=i^{-1/2}[\mathrlap{\\!\not{\phantom{\nabla}}}\nabla,s\cdot\nabla],$
$\displaystyle\hat{\mathcal{U}}_{n}^{(0)}=[\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s},(\mathrlap{\\!\not{\phantom{\nabla}}}\nabla)^{2}]-\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}[\mathrlap{\\!\not{\phantom{\nabla}}}\nabla,s\cdot\nabla]\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}$
(3.109)
$\displaystyle\hat{\mathcal{O}}_{n-1}^{(0)}=-i[\mathrlap{\\!\not{\phantom{\nabla}}}\nabla,s\cdot\nabla]\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s},$
$\displaystyle\hat{\mathcal{U}}_{n-1}^{(0)}=i^{1/2}\\{\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s},[\mathrlap{\\!\not{\phantom{\nabla}}}\nabla,s\cdot\nabla]\\}$
(3.110)
$\displaystyle\hat{\mathcal{O}}_{n-2}^{(0)}=-i^{1/2}[\mathrlap{\\!\not{\phantom{\nabla}}}\nabla,s\cdot\nabla]\partial_{s}^{2},$
$\displaystyle\hat{\mathcal{U}}_{n-2}^{(0)}=-i[\mathrlap{\\!\not{\phantom{\nabla}}}\nabla,s\cdot\nabla]$
(3.111)

That these satisfy $\hat{\mathcal{C}}_{j}^{(0)}=0$ is easily verified, thus
providing the base case for the induction argument to go through. The
modifications $\Delta\mathcal{S}_{n-i}^{(N)}$ (3.89) – (3.91) and (3.97) –
(3.99) provide a gauge invariant completion of minimal coupling.

####  3.2.2 Non-minimal gauge invariant interactions

As we saw in the previous section, minimally coupling $\Psi_{n-i}$ to
electromagnetism and gravity does not preserve the massive gauge symmetry
(3.2), but there exists modifications
$\Delta\mathcal{S}_{n-i}\to\mathcal{S}_{n-i}+\sum_{N=0}^{\infty}\Delta\mathcal{S}_{n-i}^{(N)}$
which restores the gauge symmetry exactly.

We would also like to know how to add non-minimal couplings to
electromagnetism and gravity in a gauge invariant way. For instance, one might
consider adding a term
$\Delta\mathcal{S}_{n}^{(1)}=\frac{1}{m}g_{1}([\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\mathrlap{\\!\not{\phantom{\nabla}}}\nabla,s\cdot\nabla]-(\mathrlap{\\!\not{\phantom{\nabla}}}\nabla)^{2})\Psi_{n}$
to $\mathcal{S}_{n}$, which is related to the magnetic dipole and
gravitational quadrupole moment couplings, because it is linear in curvatures
with no spacetime derivatives

$\displaystyle[\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\mathrlap{\\!\not{\phantom{\nabla}}}\nabla,s\cdot\nabla]-(\mathrlap{\\!\not{\phantom{\nabla}}}\nabla)^{2}=\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}[\mathrlap{\\!\not{\phantom{\nabla}}}\nabla,s\cdot\nabla]$
$\displaystyle=is^{\mu}F_{\mu\nu}\partial_{s}^{\nu}+i\gamma^{\mu\nu}F_{\mu\nu}-is^{\mu}F_{\mu\nu}\gamma^{\nu\lambda}\partial_{s\lambda}$
$\displaystyle+s^{\mu}R_{\mu\nu}\partial_{s}^{\nu}-s^{\sigma}s^{\nu}R_{\omega\sigma\mu\nu}\partial_{s}^{\mu}\partial_{s}^{\omega}$
$\displaystyle-\gamma^{\mu\nu}s^{\omega}R_{\omega\sigma\mu\nu}\partial_{s}^{\sigma}-\gamma^{\sigma\mu}s^{\nu}R_{\omega\sigma\mu\nu}\partial_{s}^{\omega}+s^{\sigma}s^{\nu}R_{\omega\sigma\mu\nu}\gamma^{\mu\lambda}\partial_{s\lambda}\partial_{s}^{\omega}$
(3.112)

Adding such a term will break massive gauge invariance. Luckily there is,
again, a prescription for adding further interactions which restores the gauge
invariance in exactly the same manner described in subsubsection 3.2.1. We
show two classes of non-minimal interactions which can be made gauge
invariant. The first is seeded by the following non-minimal coupling
modifications, associated with an arbitrary operator $\hat{\mathcal{M}}$ with
mass dimension $2k+2$ which is invariant with respect to the massive gauge
symmetry

$\displaystyle\Delta\mathcal{S}_{n}^{(2k+1)}=\frac{1}{m^{2k+1}}\hat{\mathcal{M}}\Psi_{n}$
(3.113)
$\displaystyle\Delta\mathcal{S}_{n-1}^{(2k+1)}=\frac{i^{-1/2}}{m^{2k+1}}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\hat{\mathcal{M}}\Psi_{n}$
(3.114)
$\displaystyle\Delta\mathcal{S}_{n}^{(2k+2)}=\frac{i^{1/2}}{m^{2k+2}}\hat{\mathcal{M}}s\cdot\nabla\Psi_{n-1},$
(3.115)
$\displaystyle\Delta\mathcal{S}_{n-1}^{(2k+2)}=-\frac{i^{-1/2}}{m^{2k+2}}\Big{(}([\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s},\mathrlap{\\!\not{\phantom{\nabla}}}\nabla]-s\cdot\nabla\partial_{s}^{2})\hat{\mathcal{M}}\Psi_{n}-i^{1/2}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\hat{\mathcal{M}}s\cdot\nabla\Psi_{n-1}\Big{)}$
(3.116)

These modifications introduce $\mathcal{O}(1/m^{2k+2})$ gauge violations,
described by the operators $\hat{\mathcal{O}}_{n-i}^{(2k+2)}$ and
$\hat{\mathcal{U}}_{n-i}^{(2k+2)}$ equal to

$\displaystyle\hat{\mathcal{O}}_{n}^{(2k+2)}=i^{-1/2}\hat{\mathcal{M}}(s\cdot\nabla)^{2}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}$
(3.117)
$\displaystyle\hat{\mathcal{O}}_{n-1}^{(2k+2)}=i([\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s},\mathrlap{\\!\not{\phantom{\nabla}}}\nabla]-s\cdot\nabla\partial_{s}^{2})\hat{\mathcal{M}}s\cdot\nabla-i\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\hat{\mathcal{M}}(s\cdot\nabla)^{2}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}$
(3.118)
$\displaystyle\hat{\mathcal{U}}_{n}^{(2k+2)}=\mathrlap{\\!\not{\phantom{\nabla}}}\nabla([\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s},\mathrlap{\\!\not{\phantom{\nabla}}}\nabla]-s\cdot\nabla\partial_{s}^{2})\hat{\mathcal{M}}$
(3.119)
$\displaystyle\hat{\mathcal{U}}_{n-1}^{(2k+2)}=i^{1/2}([\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s},\mathrlap{\\!\not{\phantom{\nabla}}}\nabla]-s\cdot\nabla\partial_{s}^{2})\hat{\mathcal{M}}s\cdot\nabla$
(3.120)

while the rest are zero. These satisfy the two conditions
$\hat{\mathcal{C}}_{j}^{(2k+2)}=0$ for any $\hat{\mathcal{M}}$. The second
class is seeded by the modifications

$\displaystyle\Delta\mathcal{S}_{n}^{(2k)}=\frac{1}{m^{2k}}\hat{\mathcal{M}}\Psi_{n},$
$\displaystyle\Delta\mathcal{S}_{n-1}^{(2k)}=\frac{i^{-1/2}}{m^{2k}}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\hat{\mathcal{M}}\Psi_{n}$
(3.121)

associated with a gauge invariant operator $\hat{\mathcal{M}}$ with mass
dimension $2k+1$. This introduces $\mathcal{O}(1/m^{2k})$ gauge violations,
with associated operators $\hat{\mathcal{O}}_{n-i}^{(2k)}$ and
$\hat{\mathcal{U}}_{n-i}^{(2k)}$ equal to

$\displaystyle\hat{\mathcal{O}}_{n}^{(2k)}=i^{-1/2}\hat{\mathcal{M}}s\cdot\nabla$
(3.122)
$\displaystyle\hat{\mathcal{O}}_{n-1}^{(2k)}=-i\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}\hat{\mathcal{M}}s\cdot\nabla$
(3.123)
$\displaystyle\hat{\mathcal{U}}_{n}^{(2k)}=([\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s},\mathrlap{\\!\not{\phantom{\nabla}}}\nabla]-s\cdot\nabla\partial_{s}^{2})\hat{\mathcal{M}}$
(3.124)

while the rest are zero. Again, these satisfy the two conditions
$\hat{\mathcal{C}}_{j}^{(2k)}=0$ for any $\hat{\mathcal{M}}$. Together, these
two classes of interactions can accommodate many types of non-minimal
interactions, including multipole and tidal couplings.

####  3.2.3 Resummed non-local field strengths

We now would like to formally resum the modifications
$\Delta\mathcal{S}_{n-i}^{(N)}$ and write the interactions in terms of non-
local operators.

Suppose we have field strengths $\mathcal{S}_{n-i}^{(2k)}$ with
$\mathcal{O}(1/m^{2k})$ gauge violations that can be cancelled via
modifications $\Delta\mathcal{S}_{n-i}^{(N)}$ for all $N>2k$ in the way
described in subsubsection 3.2.1. The gauge invariant field strengths
$\mathcal{S}_{n-i}=\mathcal{S}_{n-i}^{(2k)}+\sum_{N=2k+1}^{\infty}\Delta\mathcal{S}_{n-i}^{(N)}$
may then be written as

$\displaystyle\mathcal{S}_{n}$
$\displaystyle=\mathcal{S}_{n}^{(2k)}+\frac{i}{m^{2k+1}}\hat{\mathcal{O}}_{n}^{(2k)}\frac{1}{1-\frac{1}{m}s\cdot\nabla\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}}\Psi_{n-1}$
(3.125) $\displaystyle\mathcal{S}_{n-1}$
$\displaystyle=\mathcal{S}_{n-1}^{(2k)}-\frac{i^{-1/2}}{m^{2k+1}}\frac{1}{1-\frac{1}{m}\mathrlap{\\!\not{\phantom{\nabla}}}\nabla}\Big{(}\hat{\mathcal{U}}_{n}^{(2k)}\Psi_{n}+\hat{\mathcal{U}}_{n-1}^{(2k)}\Psi_{n-1}+\hat{\mathcal{U}}_{n-2}^{(2k)}\Psi_{n-2}\Big{)}$
$\displaystyle-\frac{i^{1/2}}{m^{2k+2}}\frac{1}{1-\frac{1}{m}\mathrlap{\\!\not{\phantom{\nabla}}}\nabla}\Big{(}\big{(}\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}(\mathrlap{\\!\not{\phantom{\nabla}}}\nabla-m)-s\cdot\nabla\partial_{s}^{2}\big{)}\hat{\mathcal{O}}_{n}^{(2k)}+i\,s\cdot\nabla\hat{\mathcal{O}}_{n-2}^{(2k)}\Big{)}\frac{1}{1-\frac{1}{m}s\cdot\nabla\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}}\Psi_{n-1}$
(3.126) $\displaystyle\mathcal{S}_{n-2}$
$\displaystyle=\mathcal{S}_{n-2}^{(2k)}+\frac{i}{m^{2k+1}}\hat{\mathcal{O}}_{n-2}^{(2k)}\frac{1}{1-\frac{1}{m}s\cdot\nabla\mathrlap{\\!\not{\phantom{\partial}}}\partial_{s}}\Psi_{n-1}$
(3.127)

One can check that these expressions are gauge invariant and satisfy the
covariantized fermionic massive Bianchi identity. To simplify these
expressions, we are free to gauge fix, by for instance choosing $\Psi_{n-1}=0$

$\displaystyle\mathcal{S}_{n}$ $\displaystyle=\mathcal{S}_{n}^{(2k)}$ (3.128)
$\displaystyle\mathcal{S}_{n-1}$
$\displaystyle=\mathcal{S}_{n-1}^{(2k)}-\frac{i^{-1/2}}{m^{2k+1}}\frac{1}{1-\frac{1}{m}\mathrlap{\\!\not{\phantom{\nabla}}}\nabla}\Big{(}\hat{\mathcal{U}}_{n}^{(2k)}\Psi_{n}+\hat{\mathcal{U}}_{n-2}^{(2k)}\Psi_{n-2}\Big{)}$
(3.129) $\displaystyle\mathcal{S}_{n-2}$
$\displaystyle=\mathcal{S}_{n-2}^{(2k)}$ (3.130)

In this case, it is only for spin $n+1/2=1/2$ that the non-local gauge
invariant completions vanish. Altogether then, non-local interactions appear
to be unavoidable in this formulation for massive particles of spin $s\geq
3/2$.

##  4 Discussion

In this paper, we successfully constructed gauge invariant interactions
between massive particles of any spin and electromagnetism and gravity. A non-
abelian gauge field extension of these results is straightforward. A
conceivable possibility would have been that the massive gauge symmetry
predicted unique interactions, perhaps predicting the interactions compatible
with tree level unitarity Ferrara:1992yc ; Cucchieri:1994tx . Instead, we find
complete freedom in choosing multipole moments and
electromagnetic/gravitational susceptibilities, via the interactions discussed
in subsubsection 3.1.2 and subsubsection 3.2.2, as well as freedom in choosing
all types of interactions linear in $U(1)$ charge 0 integer spin matter fields
discussed in subsubsection 3.1.4. We have by no means exhausted all possible
types of gauge invariant interactions, but have only outlined the ones with
the clearest physical interpretation.

The advantage of this construction is that it works for any spin $s=n,n+1/2$,
and any spacetime dimension $d$. When modelling spinning black holes or
neutron stars in this way, one must account for classical values of spin,
which roughly speaking involves taking the $n\to\infty$ limit while keeping
$\hbar n$ finite. Together with the propagators found in Lindwasser:2023zwo ,
it is possible to compute observables as a function of $n$ so that this limit
can be taken, while using dimensional regularization. One challenge to doing
this is in taking into account the infinitely many interactions necessary for
gauge invariance.

As mentioned in the introduction, the connection between gauge invariance and
the preservation of physical degrees of freedom is more subtle in an
interacting theory. Gauge symmetry implies generalized Ward identities of
Feynman diagrams to all orders in perturbation theory, ensuring that no
unphysical polarizations propagate in intermediate states, but it does not
guarantee the stability/existence of the particle in question. This is
relevant in the context of confining gauge theories, in which the massless
spin 1 gluons that would be excitations of the elementary gauge field
$A^{a}_{\mu}$ are not observable as asymptotic states. Now that an explicit
example of an exactly gauge invariant interacting theory of massive higher
spin particles is known, an important next step is to study the consequences
of the generalized Ward identities within this theory. This problem we hope to
take up in future work.

The interactions constructed here are singular in the massless limit $m\to 0$.
This is a signal that the theory breaks down at energies $E\gg m$, consistent
with our understanding that massive higher spin particles cannot be
elementary. Indeed, at high energies the particle will look massless, and at
least for spins higher than 2, such massless particles cannot interact in
asymptotically flat space in accordance with the equivalence principle
Weinberg:1964ew ; Weinberg:1980kq ; Porrati:2008rm . In an asymptotically
(anti)-de Sitter space, it may be possible to replicate the analysis done here
for massless higher spin particles, replacing $m$ with some curvature
parameter $\Lambda$, perhaps making contact with Vasiliev gravity
Vasiliev:1990en .

For spins $s\geq 3/2$, the interactions are inherently non-local, with non-
local length scale $L\sim 1/m$. Non-local effects then cannot be ignored at
energies $E\sim m$. A low energy effective field theory with energy cutoff
$\Lambda$ above the higher spins mass $m$ (a necessary requirement to access
the physics of said higher spin particle) will experience causality
violations. This is consistent with general $S$-matrix arguments regarding
spins $s>2$ Afkhami-Jeddi:2018apj . There are of course theories with massive
spin $s=3/2$ particles which are causal e.g. Deser:1977uq , but these theories
have quite special interactions dictated by supersymmetry. String theory
offers a causal theory of interacting massive particles with spins $s\geq 2$,
but this is only because of finely tuned interactions with other massive
particles with a slightly higher mass $m^{\prime}\gtrsim m$ which pushes the
acausal energy scale higher $E\sim m^{\prime}$. The rough idea is this
continues indefinitely, until causality is restored Camanho:2014apa ; Afkhami-
Jeddi:2018apj . In contrast, the theories considered here involve only a
single massive particle without self interactions, interacting with
electromagnetism and gravity in a generic manner.

The prospect of constructing self interactions, or interactions between a
collection of massive particles of varying types which suppresses causality
violations, is a very interesting one. Self interactions can be constructed as
functions of the gauge invariant linear combinations $A_{n}$ and $A_{n-3}$
defined in subsubsection 3.1.4, but these will always be non-local when $n\geq
3$.333The analogous gauge invariant linear combinations for half integer spin
fields are non-local starting at spin 5/2. Interactions between massive
particles of varying types would require a different gauge principle than the
ones presented here. As mentioned above, there is already an explicit example
of causal interactions, i.e. the interactions present between massive
excitations in string theory. It would be illuminating to reproduce the
specific interactions present in string theory from first principles, starting
with a free theory and introducing consistent interactions in much the same
way it has been done for Yang-Mills theory and general relativity
Berends:1984rq .

Recent progress from bootstrapping scattering amplitudes has shown that there
are physically sensible tree level scattering amplitudes involving an infinite
collection of massive excitations other than those known to arise from string
theory Geiser:2022exp ; Cheung:2022mkw ; Cheung:2023adk ; Cheung:2023uwn . A
physical realization of these amplitudes is as of yet unknown. The generic
nature of these scattering amplitudes offers a clue that the construction of
consistent interactions other than the ones present in string theory at the
Lagrangian level is possible, providing a physical realization. An analysis of
possible interactions would therefore shed light on the uniqueness properties
of string theory.

To explore these possibilities, it is important to gain more control on
theories with an infinite collection of massive particles of all
representations, not just the totally symmetric spin representations to which
we have so far restricted ourselves to. This can for instance be accommodated
via a hyperfield $\Phi(X,\\{s_{i}\\})$ which is a general function of $N$
auxiliary vector coordinates $s_{i}^{\mu}$. A formal Taylor expansion in
$s_{i}^{\mu}$ will generate fields $\phi_{\mu_{1}\cdots\mu_{n}}^{i_{1}\cdots
i_{n}}(X)$ which span all Young tableaux if $N>d$. To proceed, one should
first construct free actions for massive particles of arbitrary
representations which exhibits a gauge symmetry, so that there is a procedure
for adding interactions consistent with said gauge symmetry. This will be
further expanded on in another work.

### Acknowledgements

The author would like to thank Lucile Cangemi, Henrik Johansson, Paolo
Pichini, Massimo Porrati, Trevor Scheopner and E.T. Tomboulis for invaluable
discussions during the preparation of this work. The author would also like to
thank Nordita for their hospitality during the workshop “Amplifying Gravity at
All Scales” as part of this work was carried out. Finally, the author thanks
Marcus Spradlin for pointing out the possibility of interactions linear in
matter fields. This research is supported in part by the Mani L. Bhaumik
Institute for Theoretical Physics.

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