aainstitutetext: Department of Physics and Astronomy, Uppsala University,
Box 516, 75120 Uppsala, Swedenbbinstitutetext: Nordita, Stockholm University
and KTH Royal Institute of Technology,
Hannes Alfvéns väg 12, 10691 Stockholm, Sweden

# Compton Black-Hole Scattering for $s\leq 5/2$

Marco Chiodaroli a,b Henrik Johansson a and Paolo Pichini
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Quantum scattering amplitudes for massive matter have received new attention
in connection to classical calculations relevant to gravitational-wave
physics. Amplitude methods and insights are now employed for precision
computations of observables needed for describing the gravitational dynamics
of bound massive objects such as black holes. An important direction is the
inclusion of spin effects needed to accurately describe rotating (Kerr) black
holes. Higher-spin amplitudes introduced by Arkani-Hamed, Huang and Huang at
three points have by now a firm connection to the effective description of
Kerr black-hole physics. The corresponding Compton higher-spin amplitudes
remain however an elusive open problem. Here we draw from results of the
higher-spin literature and show that physical insights can be used to uniquely
fix the Compton amplitudes up to spin $5/2$, by imposing a constraint on a
three-point higher-spin current that is a necessary condition for the
existence of an underlying unitary theory. We give the unique effective
Lagrangians up to spin $5/2$, and show that they reproduce the previously-
known amplitudes. For the multi-graviton amplitudes analogous to the Compton
amplitude, no further corrections to our Lagrangians are expected, and hence
such amplitudes are uniquely predicted. As an essential tool, we introduce a
modified version of the massive spinor-helicity formalism which allows us to
conveniently obtain higher-spin states, propagators and compact expressions
for the amplitudes.

††preprint:  UUITP-34/21
NORDITA 2021-013

##  1 Introduction

More than a century after its inception, General Relativity continues to reap
successes by accurately describing new physical phenomena, such as
gravitational waves from inspiraling black holes LIGOScientific:2016aoc  and
neutron stars LIGOScientific:2017vwq . However, despite having received
considerable attention, the classical scattering of massive spinning objects
in General Relativity is still poorly understood. The classical scattering
problem is closely related to bound-orbit dynamics, as both depend on the
calculation of a two-body effective potential, which may include radiative
effects. In the regime in which the compact objects are well separated, the
effective potential is most naturally presented in a post-Minkowskian or post-
Newtonian expansion.

The seemingly-unrelated problem of quantum scattering amplitudes, accompanied
by a classical limit procedure, has provided a versatile novel perspective to
tackle these challenges BjerrumBohr:2002kt ; Bjerrum-Bohr:2013bxa ;
Neill:2013wsa ; Akhoury:2013yua ; Vaidya:2014kza ; Luna:2016due ; Luna:2016idw
; Luna:2017dtq ; Cachazo:2017jef ; Damour:2017zjx ; Bjerrum-Bohr:2018xdl ;
Cheung:2018wkq ; Bern:2019nnu ; KoemansCollado:2019ggb ; Arkani-Hamed:2019ymq
; Cristofoli:2019neg ; Bern:2019crd ; Damgaard:2019lfh ; DiVecchia:2019kta ;
Damour:2019lcq ; Bjerrum-Bohr:2019kec ; Bern:2020gjj ; Chung:2020rrz ; Parra-
Martinez:2020dzs ; Aoude:2020mlg ; Bini:2020uiq ; DiVecchia:2020ymx ;
Damour:2020tta ; Bern:2020uwk ; Aoude:2020ygw ; AccettulliHuber:2020dal ;
Bini:2020rzn ; Bern:2021dqo ; Herrmann:2021lqe ; Kosmopoulos:2021zoq ;
Bjerrum-Bohr:2021vuf ; DiVecchia:2021bdo ; Herrmann:2021tct ;
Cristofoli:2021vyo ; Bautista:2021wfy ; Chen:2021huj .111See also refs. Iwa ;
Oka ; Donoghue:1993eb ; Donoghue:1994dn  for earlier related work. Amplitude
methods draw from ideas that originate from the effective-field-theory (EFT)
approach to quantum theories and are carried over to gravitational physics
Goldberger:2004jt ; Goldberger:2007hy ; Kol:2007bc ; Goldberger:2009qd ;
Foffa:2013qca ; Porto:2016pyg ; Foffa:2016rgu ; Levi:2018nxp . Like in other
analytical approaches, the goal is to describe the inspiral phase of a merging
binary system, thus taking full advantage of the separation of scales inherent
to the problem.

Powerful techniques developed for amplitude computations over the past three
decades are now being routinely employed for calculations in classical
gravity, including Britto-Cachazo-Feng-Witten (BCFW) on-shell recursion
Britto:2005fq , loop-level unitarity methods Bern:1994zx ; Bern:1994cg ;
Bern:1995db ; Bern:1997sc  and the double-copy construction Kawai:1985xq ;
Bern:2008qj ; Bern:2010ue  (see ref. Bern:2019prr  for a comprehensive
review). Most notably, scattering-amplitude methods have been instrumental in
the calculation of the two-body effective potential up to the third and fourth
post-Minkowskian orders $\mathcal{O}(G_{N}^{3})$ and $\mathcal{O}(G_{N}^{4})$
for non-spinning black-hole-like objects Bern:2019nnu ; Bern:2019crd ;
Bern:2020gjj ; Bern:2021dqo , where $G_{N}$ is Newton’s constant.222See also
current state-of-the-art post-Newtonian calculations of the conservative
effective potential up to fifth and sixth orders Bini:2019nra ;
Blumlein:2020znm ; Bini:2020wpo ; Bini:2020nsb . While the connection of new
techniques to more traditional approaches deserves an in-depth discussion,
this is beyond of the scope of this paper.  The two-body effective potential
can in turn be used as analytical input for the calculation of gravitational-
wave templates (see e.g. refs. Antonelli:2019ytb ; Buonanno:2014aza ).

While many of the current applications of amplitude techniques concern
scattering of objects without spin (e.g. Schwarzschild black holes), an
important research direction aims at quantifying the spin effects of rotating
compact objects (e.g. Kerr black holes). In the EFT framework, the study of
spin effects was initiated in ref. Porto:2005ac  and further developed over
the following years Porto:2006bt ; Porto:2008tb ; Porto:2008jj ; Levi:2008nh ;
Porto:2010tr ; Porto:2010zg ; Levi:2010zu ; Levi:2011eq ; Porto:2012as ;
Levi:2014gsa ; Levi:2014sba ; Levi:2015msa ; Levi:2015uxa ; Levi:2015ixa ;
Levi:2016ofk ; Maia:2017yok ; Maia:2017gxn ; Levi:2017oqx ; Levi:2017kzq ;
Levi:2018nxp ; Levi:2019kgk ; Kalin:2019rwq ; Kalin:2019inp ; Levi:2020kvb ;
Levi:2020uwu ; Levi:2020lfn ; Liu:2021zxr ; Cho:2021mqw ; Jakobsen:2021lvp .
More recently, quantum scattering amplitudes involving particles of arbitrary
spin were re-analyzed by Arkani-Hamed, Huang and Huang using the massive
spinor-helicity formalism Arkani-Hamed:2017jhn . Those results gave rise to an
emergent research direction that seeks to establish a link between classical
spin corrections to the two-body effective potential and quantum scattering
amplitudes for spinning matter Guevara:2017csg ; Vines:2017hyw ; Vines:2018gqi
; Guevara:2018wpp ; Chung:2018kqs ; Guevara:2019fsj ; Arkani-Hamed:2019ymq ;
Chung:2019duq ; Siemonsen:2019dsu ; Guevara:2020xjx ; Aoude:2020onz  (see also
related work Kosower:2018adc ; Maybee:2019jus ; Damgaard:2019lfh ;
Huang:2019cja ; Chung:2019yfs ; Haddad:2020tvs ; Emond:2020lwi ;
Brandhuber:2021kpo ). In particular, certain three-point amplitudes found in
ref. Arkani-Hamed:2017jhn  were later shown to reproduce the classical
scattering of spinning Kerr black holes via exponentiation of soft-graviton
exchanges Guevara:2018wpp , as well as through other direct matchings
Chung:2018kqs ; Arkani-Hamed:2019ymq ; Aoude:2020onz .

Quantum amplitudes for massive particles of spin $s$ can be used to access
spin corrections of power $(S^{\mu})^{2s}$ in the two-body effective
potential, where $S^{\mu}$ is the classical spin vector. Such calculations
will in general need input from loop-level Feynman diagrams where the
classical limit is taken such that the mass, momentum and spin of each
scattered object become large macroscopic quantities. Alternatively, by using
the factorization properties of loop diagrams and taking into account that the
large scale separation of the massive objects implies small graviton exchange
momenta, it is possible to show that all the necessary information for
computing the classical process is contained within tree-level scattering
between a single black hole (counted as two states in an out-out formalism)
and $n{-}2$ gravitons.

The natural higher-spin three-point amplitudes were sewn together via BCFW on-
shell recursion to produce simple candidates for the four-point Compton
amplitudes in refs. Arkani-Hamed:2017jhn ; Johansson:2019dnu . However, for
spin $s\geq 2$, spurious poles develop in the kinematic space of the Compton
amplitudes, indicating that certain contact terms are not correctly included
by the naive factorization.333BCFW recursion should, in principle, also
include residues at infinity of the complex plane, which would remove the
spurious poles. In general, these contributions cannot be a priori calculated.
Indeed, it is well-known that factorization properties alone do not completely
fix higher-point tree amplitudes, which is especially true for generic spin.
In this case, the ambiguity in the Compton amplitude is linked to the
proliferation of possible Lagrangian contact terms that respect all known
symmetries of the problem. Thus, it becomes important to find physical
principles to constrain amplitudes that go beyond three-point factorization.

A well-established approach for constraining amplitudes is to consider the
high-energy limit, as done for the three-point amplitudes of ref. Arkani-
Hamed:2017jhn , where they were deemed natural based on having a certain good
high-energy behavior. A further interpretation that can be inferred from ref.
Arkani-Hamed:2017jhn  is that these three-point amplitudes provide a “minimal”
description of the higher-spin states,444In this paper, we use the term
“minimal coupling” to describe interactions to matter that are induced by
covariantizing the derivatives of the free-theory Lagrangian, and this should
not be confused with the similar term used to describe the amplitudes in ref.
Arkani-Hamed:2017jhn , which require non-minimal Lagrangian terms for spin
large enough. and that the inclusion of other possible three-point
contributions would correspond to having additional physical quantities (e.g.
Wilson coefficients) that further characterize the massive spinning states.
While these are very reasonable properties that seem to resonate with what we
know about black holes, it is important to establish the precise constraints
that should be imposed on amplitudes at higher points. Furthermore, it would
be desirable to have a covariant Lagrangian understanding of the properties of
the amplitudes in ref. Arkani-Hamed:2017jhn  and, in general, determine what
makes them special.

The higher-spin literature has long studied similar theories from a different
angle Ferrara:1992yc ; Cucchieri:1994tx ; Klishevich:1997pd ; Giannakis:1998wi
; Buchbinder:2000ta ; Buchbinder:1999be ; Deser:2001dt ; Metsaev:2007rn ;
Francia:2007ee ; Porrati:2008ha ; Porrati:2008gv ; Francia:2008ac ;
Sagnotti:2010at ; Hassan:2012wr ; Buchbinder:2012iz ; Cortese:2013lda ;
Bernard:2015uic ; Rahman:2015pzl ; Fukuma:2016rru ; Bonifacio:2018aon ;
Afkhami-Jeddi:2018apj ; Kaplan:2020ldi  (for a review, see ref. Rahman:2013sta
). Theories with massive higher-spin fields and Minkowski background are
understood as effective theories that exist below a certain energy scale. This
occurs when the spin-$s$ fields are coupled to gravity ($s>2$), or to
electromagnetism ($s>1$), and thus the force carrier is not the field of
highest spin. The presence of an intrinsic cutoff in these theories is linked
to acausal behavior at high energy and, ultimately, leads to an obstruction to
having a local fundamental theory with a finite number of elementary higher-
spin fields. Cucchieri, Deser and Porrati showed in ref. Cucchieri:1994tx
that a theory with elementary higher-spin fields of mass $m$ violates tree-
level unitarity at energies well below the Planck scale. They attempted to
restore tree-level unitarity by adding non-minimal higher-derivative terms
$\sim\partial/m$ to the Lagrangian, and argued that a necessary condition for
the theory to be well defined up to the Planck scale is that the longitudinal
part of the higher-spin current $J^{\mu}$ vanishes in the $m\rightarrow 0$
limit,555This is equivalent to requiring higher-spin gauge invariance in the
massless limit. For spin $s\geq 5/2$ this formula will be restricted to the
traceless part, as we explain later.

$\partial_{\mu}J^{\mu}={\cal O}(m)\ .$ (1.1)

Since the higher-spin current can be used to compute scattering amplitudes
after dressing the free indices with on-shell polarizations, this condition
has a direct link to the problem of finding natural and well-behaved Compton
amplitudes.

In this paper, we argue that the problem of fixing contact-term ambiguities of
spinning black-hole Compton amplitudes is directly tied to established results
in the higher-spin literature. We show that, given some standard physical
assumptions on higher-derivative operators, the condition (1.1) uniquely fixes
three-point and Compton amplitudes for massive spin $s\leq 5/2$ fields in
General Relativity and, analogously, for massive charged spin $s\leq 3/2$
fields coupled to electromagnetism. The gravitational amplitudes that we find
should be useful for computing $(S^{\mu})^{\leq 5}$ corrections in the two-
body effective potential for binary Kerr black holes. We also give the
Lagrangians for the spin $s\leq 5/2$ fields coupled to General Relativity, and
argue that there exist no further terms that can be added to improve the tree-
level unitarity properties (as addressed by eq. (1.1)).

It should be emphasized that the kinematical regime in which classical black-
hole calculations are carried out (mass and black-hole momenta large in Planck
units) is quite different from the one originally envisioned in the higher-
spin literature, which has masses well-below the Planck scale. This seems to
require some mild reinterpretation of the precise connection between the tree-
level unitarity argument and the (quantum) effective theory perspective. In
order to practically calculate observables relevant to classical black-hole
physics, the Lagrangians given in this paper are ultimately meant to be used
in a classical limit where the momentum transfer (carried by gravitons) is
smaller than the inverse Schwarzschild radius, which is in turn is smaller
than the inverse Compton length of the black hole. This kinematical limit may
be taken after obtaining the loop integrand of the amplitude under
consideration, which can be done via Feynman diagrams or by unitarity-cut
methods. In Section 3.4, the classical limit for tree-level amplitudes is
briefly reviewed in our notation but, for a thorough exhibition of this
procedure at the multi-loop level, we refer the reader to the unitarity-cut
methods used in refs. Vaidya:2014kza ; Guevara:2017csg ; Bern:2019crd ;
Bern:2019nnu ; Bern:2020buy ; Kosmopoulos:2021zoq ; Herrmann:2021lqe ;
Herrmann:2021tct .

In this paper, we make heavy use of a novel notation obtained as an extended
version of the massive spinor-helicity formalism Arkani-Hamed:2017jhn ;
Craig:2011ws . In particular, the massive spinor-helicity formalism is
simplified by dressing the little-group indices with auxiliary variables,
which turn expressions involving spinors or polarizations into polynomials, or
in some situations, rational functions. Higher-spin states and amplitudes can
be easily constructed in this notation, and through the completeness relation
of such states one can infer important details of the off-shell propagators
for higher-spin fields. As a non-trivial application, we construct generating
functions for all higher-spin projectors, as well as all covariant three-point
amplitudes that match the known spinor-helicity expressions Arkani-
Hamed:2017jhn .

The paper is structured as follows: in Section 2, we review our notation and
introduce an extended version of the spinor-helicity formalism. In Section 3,
we review the massive higher-spin three-point amplitudes in both spinor-
helicity and covariant notation, and show how to use the current constraint
(1.1) to build Ansätze that uniquely reproduce these amplitudes up to spin
$5/2$. In Section 4, we review the Compton amplitudes and display the new
spurious-pole-free results. In Section 5, we consider Lagrangians that are
compatible with the current constraint (1.1), and show that they are unique
for the theories that we are considering.

##  2 Massive spinor-helicity for higher-spin states

In this section, we present an extension of the massive spinor-helicity
formalism from ref. Arkani-Hamed:2017jhn  (see also Craig:2011ws  for an early
version) which is designed to conveniently describe higher-spin states and
amplitudes. We recycle many of the conventions introduced in ref. Arkani-
Hamed:2017jhn , such as using bold spinors for massive states with suppressed
little-group indices. Hence, the translation should be straightforward.

###  2.1 Massive spinor parametrization

Consider a four-dimensional momentum $p^{\mu}$ that obeys the on-shell
condition $p^{2}=m^{2}$. Let us decompose it in terms of the null vectors
$k^{\mu},q^{\mu}$,

$p^{\mu}=k^{\mu}+\frac{m^{2}}{2p\cdot q}q^{\mu}\,,$ (2.2)

where we take $q^{\mu}$ to be an arbitrary reference vector, and $k^{\mu}$ is
then defined by the decomposition. Note that the decomposition implies the
identity $p\cdot q=k\cdot q$.

Using the fact that $k,q$ are null, we may now employ the massless spinor-
helicity formalism. First, we rewrite eq. (2.2) into bi-spinors by contracting
the momenta with $\sigma^{\mu}_{\alpha\dot{\alpha}}$ matrices,

$\sigma\cdot p=|k\rangle[k|+\frac{m^{2}}{2p\cdot
q}|q\rangle[q|\equiv|p^{a}\rangle[p_{a}|\,.$ (2.3)

We intentionally suppress the $(\alpha,\dot{\alpha})$ spinor indices of the
Lorentz group $SL(2,\mathbb{C})\sim SO(1,3)$. We then recognize that the two
terms can be reinterpreted as the contraction of two massive spinors that
carry $a,b,\ldots$ indices of the little group $SU(2)\sim SO(3)$. The massive
spinors can be identified as

$\displaystyle|p^{a}\rangle=\left(\begin{matrix}~{}|q\rangle\frac{m}{\left\langle
k\,q\right\rangle}\\\
|k\rangle~{}\end{matrix}\right)\,,~{}~{}~{}~{}~{}~{}~{}|p^{a}]=\left(\begin{matrix}|k]~{}\\\
~{}|q]\frac{m}{\left[k\,q\right]}\end{matrix}\right)\,.$ (2.4)

The mirrored spinors $\langle p^{a}|$ and $[p^{a}|$ are obtained, as implied
by the notation, by transposing the massless spinors:
$|k\rangle{\rightarrow}\langle k|$. Since the little group is $SU(2)$, we
lower and raise those indices using the rules
$|p_{a}\rangle=\epsilon_{ab}|p^{b}\rangle$ and
$|p^{a}\rangle=\epsilon^{ab}|p_{b}\rangle$; that is, we always multiply with
the Levi-Civita symbols from the left. The antisymmetric Levi-Civita symbols
are normalized as $\epsilon^{12}=\epsilon_{21}=1$. For real momentum $p$ with
$E>0$, $m^{2}>0$, the angle and square spinors are complex conjugates of each
other, up to a similarity transform. More specifically,
$(|p^{a}])^{*}=\Omega|p_{a}\rangle$, $(|p_{a}\rangle)^{*}=\Omega^{T}|p^{a}]$,
where $\Omega$ is a 2-by-2 unitary matrix. Because the massive on-shell
spinors are related by complex conjugation, one is justified to think of them
as the two chiral components of a massive Majorana spinor.

In general, we will consider amplitudes that depend on many momenta
$p_{i}^{\mu}$, with particle labels $i=1,\ldots,n$, which makes it convenient
to simplify the notation by only indicating the particle label inside the
spinor

$|i^{a}\rangle\equiv|p_{i}^{a}\rangle\,,~{}~{}~{}|i^{a}]\equiv|p_{i}^{a}]\,.$
(2.5)

For each particle $i$ we have the associated reference vector $q_{i}^{\mu}$
and mass $m_{i}$.

###  2.2 Bookkeeping of little-group indices, polarizations and projectors

It is convenient to define massive bosonic spinors that have no free little-
group indices,

$|\,{\bf i}\,\rangle\equiv|i^{a}\rangle z_{i,a}\,,~{}~{}~{}~{}~{}~{}|\,{\bf
i}\,]\equiv|i^{a}]z_{i,a}$ (2.6)

where $z_{i,a}=\epsilon_{ab}\,z_{i}^{b}$ are complex Grassmann-even auxiliary
variables that transform as spinors under the little group. Because we take
the $z$ variables to be complex, the two chiral spinors are no longer related
by complex conjugation, and thus the spinors (2.6) can be interpreted as the
chiral components of a massive Dirac spinor.

The spinor products are antisymmetric under label swaps,

$\langle{\bf 1}{\bf 2}\rangle=-\langle{\bf 2}{\bf
1}\rangle\,,~{}~{}~{}~{}[{\bf 1}{\bf 2}]=-[{\bf 2}{\bf 1}]\,,$ (2.7)

implying that spinor products with repeated indices vanish, e.g. $\langle{\bf
1}{\bf 1}\rangle=0$. Many other identities familiar from the massless spinor-
helicity formalism still hold, such as Schouten and Fierz identities:

$\langle{\bf 1}{\bf 2}\rangle\langle{\bf 3}{\bf 4}\rangle+\langle{\bf 2}{\bf
3}\rangle\langle{\bf 1}{\bf 4}\rangle+\langle{\bf 3}{\bf 1}\rangle\langle{\bf
2}{\bf 4}\rangle=0\,,~{}~{}~{}~{}~{}\langle{\bf 1}|\sigma^{\mu}|{\bf
2}]\langle{\bf 3}|\sigma_{\mu}|{\bf 4}]=2\langle{\bf 1}{\bf 3}\rangle[{\bf
4}{\bf 2}]\,.$ (2.8)

Because all indices are absent and the spinors are bosonic, we can now take
arbitrary powers of the spinors, e.g.

$\langle{\bf 1}{\bf 2}\rangle^{2s}=\text{degree-}4s\text{~{}polynomial
in~{}}(z_{1}^{a},z_{2}^{a})\,,$ (2.9)

which makes it possible to write down analytic functions with the spinors as
arguments. As a first example, consider the polarization vector for a massive
vector boson, which we define as

$\bm{\mathbf{\varepsilon}}_{i}^{\mu}=\frac{\langle{\bf i}|\sigma^{\mu}|{\bf
i}]}{\sqrt{2}m_{i}}=\frac{[{\bf i}|\bar{\sigma}^{\mu}|{\bf
i}\rangle}{\sqrt{2}m_{i}}=(z_{i}^{1})^{2}\varepsilon_{i,-}^{\mu}-\sqrt{2}z_{i}^{1}z_{i}^{2}\varepsilon_{i,L}^{\mu}-(z_{i}^{2})^{2}\varepsilon_{i,+}^{\mu}\,.$
(2.10)

Here $\varepsilon_{i,\pm}=\varepsilon_{i,\pm}(k_{i},q_{i})$ are standard
(massless) polarization vectors for the null momenta $k_{i}$, with $q_{i}$ as
the reference vector that appeared in eq. (2.2), and $\varepsilon_{i,L}$ is a
longitudinal polarization. Explicit expressions can be given, e.g.
$\sqrt{2}\varepsilon_{i,+}^{\mu}=\langle q_{i}|\sigma^{\mu}|k_{i}]/\langle
q_{i}k_{i}\rangle$ and $\varepsilon_{i,L}=k_{i}/m_{i}-m_{i}q_{i}/(2p_{i}\cdot
q_{i})$. Note that the massive polarization
$\bm{\mathbf{\varepsilon}}_{i}^{\mu}$ is still a null vector, since
$\bm{\mathbf{\varepsilon}}_{i}^{2}\propto\left\langle\bf i\,\bf
i\right\rangle\left[\bf i\,\bf i\right]$, and $\left\langle\bf i\,\bf
i\right\rangle=\left[\bf i\,\bf i\right]=0$. Also, since the $z$ variables are
complex, the polarization $\bm{\mathbf{\varepsilon}}_{i}^{\mu}$ naturally
describes a complex massive vector boson. In the massless limit, the
longitudinal polarization will behave as $\varepsilon_{L}^{\mu}\sim p^{\mu}/m$
and is thus singular, whereas the transverse polarizations $\varepsilon_{\pm}$
are well defined. It is interesting to note that the need of an arbitrary
reference vector $q$ to describe a massless polarization vector is easy to
understand from the ambiguity of the parametrization (2.2) we used for the
massive spinors.

To check the completeness relation for the polarization vectors, we need to
introduce polarizations that are complex conjugated,

$\bar{\bm{\mathbf{\varepsilon}}}_{i}^{\mu}=\frac{\big{(}[{\bf
i}|\sigma^{\mu}|{\bf
i}\rangle\big{)}^{*}}{\sqrt{2}m_{i}}\equiv-(\bar{z}_{i}^{1})^{2}\varepsilon_{i,-}^{\mu}+\sqrt{2}\bar{z}_{i}^{1}\bar{z}_{i}^{2}\varepsilon_{i,L}^{\mu}+(\bar{z}_{i}^{2})^{2}\varepsilon_{i,+}^{\mu}\,,$
(2.11)

where we have used $(z_{a})^{*}=\bar{z}^{a}$, $(z^{a})^{*}=-\bar{z}_{a}$ and
$(\varepsilon_{i,-}^{\mu})^{*}=\varepsilon_{i,+}^{\mu}$. We then get the
following non-zero Lorentz product

$\bm{\mathbf{\varepsilon}}_{i}\cdot\bar{\bm{\mathbf{\varepsilon}}}_{i}=-(z_{i}^{a}\bar{z}_{i,a})^{2}\,,$
(2.12)

where the sign can be traced to the mostly-minus metric $\eta^{\mu\nu}={\rm
diag}(1,-1,-1,-1)$.

If we contract the little-group indices using a derivative operator, we get
the completeness relation for the transverse part of the Lorentz group,

$-\frac{1}{4}\left(\frac{\partial^{2}}{\partial
z_{i,a}\partial\bar{z}_{i}^{a}}\right)^{2}\bm{\mathbf{\varepsilon}}_{i}^{\mu}\bar{\bm{\mathbf{\varepsilon}}}_{i}^{\nu}=\eta^{\mu\nu}-\frac{p^{\mu}_{i}p^{\nu}_{i}}{m^{2}_{i}}\,.$
(2.13)

The result should be familiar as the massive spin-$1$ projector, or as the
tensor structure of the massive spin-$1$ propagator. We will see that a direct
generalization of the derivative operator introduced here will be convenient
for computing state sums that contribute to amplitude factorization residues
or loop-level unitarity cuts, for any spin states.

###  2.3 Bosonic higher-spin states

Polarization tensors for bosonic spin-$s$ fields are simply products of $s$
polarization vectors

$\bm{\mathbf{\varepsilon}}_{i}^{\mu_{1}\mu_{2}\cdots\mu_{s}}\equiv\bm{\mathbf{\varepsilon}}_{i}^{\mu_{1}}\bm{\mathbf{\varepsilon}}_{i}^{\mu_{2}}\cdots\bm{\mathbf{\varepsilon}}_{i}^{\mu_{s}}=\text{degree-}2s\text{~{}polynomial
in~{}}z_{i}^{a}\,.$ (2.14)

Polarization tensors are automatically symmetric, traceless and transverse.
Transversality $p_{i,\mu_{1}}\varepsilon_{i}^{\mu_{1}\mu_{2}\cdots\mu_{s}}=0$
follows from the fact that $\langle i^{a}|p_{i}|i^{b}]\propto\epsilon^{ab}$,
which vanishes after contracting with the symmetric object
$z_{i}^{a}z_{i}^{b}$.

Contracting two CPT-conjugate polarizations gives the little-group
completeness relation

$\bm{\mathbf{\varepsilon}}_{i}^{\mu_{1}\mu_{2}\cdots\mu_{s}}\bar{\bm{\mathbf{\varepsilon}}}_{i,\mu_{1}\mu_{2}\cdots\mu_{s}}=(-1)^{s}(z_{i}^{a}\bar{z}_{i,a})^{2s}\,,$
(2.15)

where again the sign is needed due to our mostly-minus signature. For spin
$2$, we get the following completeness relation for the Lorentz structure:

$\frac{1}{(4!)^{2}}\left(\frac{\partial^{2}}{\partial
z_{i,a}\partial\bar{z}_{i}^{a}}\right)^{4}\bm{\mathbf{\varepsilon}}_{i}^{\mu\nu}\bar{\bm{\mathbf{\varepsilon}}}_{i}^{\rho\sigma}=\frac{1}{2}\Big{(}\tilde{\eta}^{\mu\rho}\tilde{\eta}^{\nu\sigma}+\tilde{\eta}^{\mu\sigma}\tilde{\eta}^{\nu\rho}-\frac{2}{3}\tilde{\eta}^{\mu\nu}\tilde{\eta}^{\sigma\rho}\Big{)}\,,$
(2.16)

where
$\tilde{\eta}^{\mu\nu}\equiv\eta^{\mu\nu}-\frac{p^{\mu}_{i}p^{\nu}_{i}}{m^{2}_{i}}$
is a shorthand notation for the spin-$1$ projector that appeared in eq.
(2.13). The above eq. (2.16) is the expected state projector for the five
physical degrees of freedom of a massive spin-$2$ field (e.g. massive
graviton).

For general bosonic spin $s$, we have the following state sum to evaluate:

$\frac{(-1)^{s}}{(2s)!^{\,2}}\\!\\!\left(\frac{\partial^{2}}{\partial
z_{i,a}\partial\bar{z}_{i}^{a}}\right)^{2s}\\!\\!\\!\\!\bm{\mathbf{\varepsilon}}_{i}^{\mu_{1}\mu_{2}\cdots\mu_{s}}\bar{\bm{\mathbf{\varepsilon}}}_{i}^{\nu_{1}\nu_{2}\cdots\nu_{s}}=\frac{1}{s!}(\tilde{\eta}^{\mu_{1}\nu_{1}}\tilde{\eta}^{\mu_{2}\nu_{2}}\\!{\ldots}\tilde{\eta}^{\mu_{s}\nu_{s}}+\text{perms})+\ldots\equiv
P_{(s)}^{\vec{\mu}\vec{\nu}}\,.$ (2.17)

Here $P_{(s)}^{\vec{\mu}\vec{\nu}}$ is a compact notation for the state
projector of an on-shell symmetric and traceless spin-$s$ state. (See, e.g.,
ref. Dobrev:1975ru  for early work on projectors.) Considered as a matrix, the
projector should satisfy

$\displaystyle P_{(s)}P_{(s)}=P_{(s)}\,,~{}~{}P_{(s)}^{T}=P_{(s)}\,,~{}~{}{\rm
tr}P_{(s)}=2s{+}1\,,~{}~{}p_{\mu_{i}}P_{(s)}=0\,,$ $\displaystyle
P_{(s)}\big{|}_{\mu_{i}\leftrightarrow\mu_{j}}=P_{(s)}\,,~{}~{}~{}{\eta_{\mu_{i}\mu_{j}}}P_{(s)}=0\,,~{}~{}{\eta_{\mu_{s}\nu_{s}}}P_{(s)}=\frac{2s{+}1}{2s{-}1}P_{(s{-}1)}\,,$
(2.18)

where ${\rm tr}P_{(s)}=2s{+}1$ is the number of degrees of freedom for a
massive spin-$s$ state. From these properties, we can find a recursive formula
that computes the projector:

$P_{(s)}^{\vec{\mu}\vec{\nu}}=\frac{1}{(s!)^{2}}\sum_{{\rm
perm}\,\vec{\mu}}\sum_{{\rm
perm}\,\vec{\nu}}\Big{(}\tilde{\eta}^{\mu_{s}\nu_{s}}P_{(s{-}1)}^{\vec{\mu}\vec{\nu}}-\frac{(s-1)^{2}}{(2s-1)(2s-3)}\tilde{\eta}^{\mu_{s}\mu_{s-1}}\tilde{\eta}^{\nu_{s}\nu_{s-1}}P_{(s{-}2)}^{\vec{\mu}\vec{\nu}}\Big{)}\,,$
(2.19)

where $P_{(0)}=1$. It is convenient to simplify the notation by contracting
the free Lorentz indices with auxiliary (symmetric) polarizations
$\epsilon^{\vec{\mu}}\equiv\prod_{i}\epsilon^{\mu_{i}}$ and
$\bar{\epsilon}^{\vec{\nu}}\equiv\prod_{i}\bar{\epsilon}^{\nu_{i}}$, where the
complex vector $\epsilon^{\mu}$ is a dummy variable. We then get the simpler
recursive formula for the scalar-valued projector
$P_{(s)}(\epsilon,\bar{\epsilon})=\epsilon_{\vec{\mu}}P_{(s)}^{\vec{\mu}\vec{\nu}}\bar{\epsilon}_{\vec{\nu}}\,$,

$P_{(s)}(\epsilon,\bar{\epsilon})=\epsilon.{\bar{\epsilon}}P_{(s{-}1)}(\epsilon,\bar{\epsilon})-\frac{(s-1)^{2}}{(2s-1)(2s-3)}\epsilon^{2}{\bar{\epsilon}}^{2}P_{(s{-}2)}(\epsilon,\bar{\epsilon})\,,$
(2.20)

where all dot products are taken using the spin-$1$ projector
$\tilde{\eta}^{\mu\nu}$; that is, the “lower dot product” is defined as
$\epsilon.{\bar{\epsilon}}\equiv\epsilon_{\mu}\tilde{\eta}^{\mu\nu}{\bar{\epsilon}}_{\nu}$,
with $\epsilon^{2}=\epsilon.\epsilon$ and
$\bar{\epsilon}^{2}=\bar{\epsilon}.\bar{\epsilon}$. This compact notation will
only be used in the current section. The recursive formula (2.20) can be
solved in terms of the hypergeometric function ${}_{2}F_{1}$,

$\displaystyle P_{(s)}(\epsilon,\bar{\epsilon})$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}(\epsilon.{\bar{\epsilon}})^{s-2j}\Big{(}{-}\frac{\epsilon^{2}{\bar{\epsilon}}^{2}}{2}\Big{)}^{j}\frac{s!(2s-2j-1)!!}{j!(s-2j)!(2s-1)!!}$
(2.21) $\displaystyle=$
$\displaystyle(\epsilon.{\bar{\epsilon}})^{s}{}_{2}F_{1}\Big{(}\frac{1-s}{2},-\frac{s}{2},\frac{1}{2}-s,\frac{\epsilon^{2}{\bar{\epsilon}}^{2}}{(\epsilon.{\bar{\epsilon}})^{2}}\Big{)}\
.$

Alternatively, the spin-$s$ projector can be written directly in terms of
Legendre polynomials $P_{n}(x)$,

$P_{(s)}(\epsilon,\bar{\epsilon})=\frac{\big{(}\epsilon^{2}{\bar{\epsilon}}^{2}\big{)}^{s/2}s!}{(2s-1)!!}P_{s}(x)\,,$
(2.22)

where
$x^{2}=\frac{(\epsilon.{\bar{\epsilon}})^{2}}{\epsilon^{2}{\bar{\epsilon}}^{2}}$.
Using Rodrigues’ formula for Legendre polynomials, we can simplify the state
projector so that it is manifestly local in the auxiliary vectors,

$P_{(s)}(\epsilon,\bar{\epsilon})=\frac{s!}{(2s)!}\Big{(}\frac{\partial}{\partial\epsilon.{\bar{\epsilon}}}\Big{)}^{s}\big{(}(\epsilon.{\bar{\epsilon}})^{2}-\epsilon^{2}{\bar{\epsilon}}^{2}\big{)}^{s}\,.$
(2.23)

Since the spin-$s$ projector is unique for on-shell states, we can use it to
construct a propagator for a spin-$s$ field, which takes the form

$\Delta_{(s)}(\epsilon,\bar{\epsilon})=i\frac{P_{(s)}(\epsilon,\bar{\epsilon})+\ldots}{p^{2}-m^{2}}\,,$
(2.24)

where the ellipsis denote possible terms of ${\cal O}(p^{2}{-}m^{2})$ that
vanish on shell. Such off-shell terms depend on the precise details of the
Lagrangian formulation used to describe the theory, and they can be altered by
field redefinitions.

###  2.4 Fermionic higher-spin states

Polarization spinors for fermionic spin-$(s+\frac{1}{2})$ fields are built
analogously to the bosonic ones. From products of the massive chiral spinors
and polarization vectors we get two chiral spin-$(s+\frac{1}{2})$
polarizations,

$\displaystyle|{\bf i}\rangle^{\mu_{1}\mu_{2}\cdots\mu_{s}}$
$\displaystyle\equiv$ $\displaystyle|{\bf
i}\rangle\bm{\mathbf{\varepsilon}}_{i}^{\mu_{1}}\bm{\mathbf{\varepsilon}}_{i}^{\mu_{2}}\cdots\bm{\mathbf{\varepsilon}}_{i}^{\mu_{s}}\,,$
$\displaystyle|{\bf i}]^{\mu_{1}\mu_{2}\cdots\mu_{s}}$ $\displaystyle\equiv$
$\displaystyle|{\bf
i}]\bm{\mathbf{\varepsilon}}_{i}^{\mu_{1}}\bm{\mathbf{\varepsilon}}_{i}^{\mu_{2}}\cdots\bm{\mathbf{\varepsilon}}_{i}^{\mu_{s}}\,,$
(2.25)

which are both degree $(2s+1)$ polynomials in $z_{i}^{a}$. The spinors are
automatically symmetric and traceless in the Lorentz indices, but also gamma-
traceless. They are gamma-traceless because $\sigma_{\mu_{1}}|{\bf
i}\rangle^{\mu_{1}\mu_{2}\cdots\mu_{s}}=\langle{\bf i}{\bf i}\rangle|{\bf
i}]^{\mu_{2}\cdots\mu_{s}}$ and $\langle{\bf i}{\bf i}\rangle=0$, with an
analogous relation for the other spinor.

The chiral spinors can be assembled into a spin-$(s+\frac{1}{2})$ Dirac
tensor-spinor as follows,

$v^{\mu_{1}\mu_{2}\cdots\mu_{s}}(p_{i})~{}\equiv~{}v(p_{i})\,\bm{\mathbf{\varepsilon}}_{i}^{\mu_{1}}\bm{\mathbf{\varepsilon}}_{i}^{\mu_{2}}\cdots\bm{\mathbf{\varepsilon}}_{i}^{\mu_{s}}~{}=~{}\left(\begin{matrix}|{\bf
i}\rangle^{\mu_{1}\mu_{2}\cdots\mu_{s}}\\\ -|{\bf
i}]^{\mu_{1}\mu_{2}\cdots\mu_{s}}\end{matrix}\right)\,,$ (2.26)

where $v$ is an on-shell Dirac spinor,

$v(p_{i})=\left(\begin{matrix}|i^{a}\rangle\\\
-|i^{a}]\end{matrix}\right)z_{i,a}\,,~{}~{}~{}~{}~{}~{}~{}(/\\!\\!\\!p_{i}+m_{i})v(p_{i})=0\,,$
(2.27)

where, as before, the auxiliary $z_{i,a}$ soak up the $SU(2)$ little-group
indices. The on-shell Dirac (tensor-)spinors are Grassmann even, and the
fermionic statistical factors have to be added by hand in any calculation,
which is the standard textbook treatment for fermionic polarizations.

Let us be precise and write out the Dirac conjugate in this notation,

$\bar{v}(p_{i})=\bar{z}_{i,a}\left(\begin{matrix}\,\langle
i^{a}|\,,&\,\,{}[i^{a}|\,\,\end{matrix}\right)\,.$ (2.28)

The inner product of two Dirac spinors of the same momentum is then

$\bar{v}(p_{i})v(p_{i})=2m_{i}\bar{z}^{a}_{i}z_{i,a}\,.$ (2.29)

The completeness relation is the familiar one,

$\frac{\partial^{2}}{\partial
z_{i,a}\partial\bar{z}_{i}^{a}}v(p_{i})\bar{v}(p_{i})=\left(\begin{matrix}\,-m_{i}\,&\,\sigma\cdot
p_{i}\,\\\ \,\bar{\sigma}\cdot
p_{i}\,&\,-m_{i}\,\end{matrix}\right)=/\\!\\!\\!p_{i}-m_{i}\,.$ (2.30)

The corresponding $u(p_{i})$ and $\bar{u}(p_{i})$ Dirac spinors are defined in
Appendix A.

The inner product of two spin-$(s+\frac{1}{2})$ tensor-spinors of the same
momentum, but with conjugate auxiliary $z$’s, is given by

$\bar{v}^{\mu_{1}\mu_{2}\cdots\mu_{s}}(p_{i})v_{\mu_{1}\mu_{2}\cdots\mu_{s}}(p_{i})=2m_{i}\,(-1)^{s}(\bar{z}_{i}^{a}z_{i,a})^{2s+1}\,,$
(2.31)

where
$\bar{v}^{\mu_{1}\mu_{2}\ldots\mu_{s}}(p_{i})=\bar{\bm{\mathbf{\varepsilon}}}_{i}^{\mu_{1}}\bar{\bm{\mathbf{\varepsilon}}}_{i}^{\mu_{2}}\cdots\bar{\bm{\mathbf{\varepsilon}}}_{i}^{\mu_{s}}\bar{v}(p_{i})$
is the standard Dirac conjugate. Similarly, the completeness relation for
spin-$(s+\frac{1}{2})$ is

$\frac{(-1)^{s}}{(2s+1)!^{\,2}}\left(\frac{\partial^{2}}{\partial
z_{i,a}\partial\bar{z}_{i}^{a}}\right)^{2s+1}\\!\\!v^{\mu_{1}\mu_{2}\cdots\mu_{s}}(p_{i})\bar{v}^{\nu_{1}\nu_{2}\cdots\nu_{s}}(p_{i})\equiv
P_{(s+1/2)}^{\vec{\mu}\vec{\nu}}\,,$ (2.32)

where we have introduced the on-shell state projector. The state projector for
fermionic higher-spin states can be worked out. It satisfies the same
properties as the bosonic one (2.18), but, in addition, it is gamma-traceless:
$\gamma_{\mu_{i}}P_{(s+1/2)}^{\vec{\mu}\vec{\nu}}=0$.

By considering all the constraints on $P_{(s+1/2)}^{\vec{\mu}\vec{\nu}}$ one
can find a recursive formula that can be solved explicitly. The solution can
be written down using the same compact notation as in the bosonic case,
$P_{(s+1/2)}(\epsilon,\bar{\epsilon})=\epsilon_{\vec{\mu}}P_{(s+1/2)}^{\vec{\mu}\vec{\nu}}\bar{\epsilon}_{\vec{\nu}}$,

$P_{(s+1/2)}(\epsilon,\bar{\epsilon})=(/\\!\\!\\!p+m)\,A^{(s)}+(/\\!\\!\\!\epsilon+\frac{p\cdot\epsilon}{m})(/\\!\\!\\!p-m)(/\\!\\!\\!\bar{\epsilon}+\frac{p\cdot\bar{\epsilon}}{m})\,B^{(s)}\,,$
(2.33)

where all objects that contain gamma matrices are explicitly displayed, and
thus $A^{(s)},B^{(s)}$ are scalar-valued functions that depend on the dummy
vector $\epsilon^{\mu}$ and momentum $p^{\mu}$. They satisfy the relations

$\displaystyle A^{(s)}$ $\displaystyle=$
$\displaystyle\epsilon.\bar{\epsilon}\,A^{(s-1)}-\frac{s+1}{(2s-1)(2s+1)}\epsilon^{2}{\bar{\epsilon}}^{2}A^{(s-2)}\,,$
$\displaystyle B^{(s)}$ $\displaystyle=$
$\displaystyle\frac{A^{(s-1)}}{2s+1}\,,$ (2.34)

with $A^{(0)}=1$, $B^{(0)}=0$. The solution can again be given in terms of a
hypergeometric function,

$\displaystyle A^{(s)}$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}(\epsilon.{\bar{\epsilon}})^{s-2j}(-\frac{\epsilon^{2}{\bar{\epsilon}}^{2}}{2})^{j}\frac{s!(2s-2j+1)!!}{j!(s-2j)!(2s+1)!!}$
(2.35) $\displaystyle=$
$\displaystyle(\epsilon.{\bar{\epsilon}})^{s}{}_{2}F_{1}\Big{(}\frac{1-s}{2},-\frac{s}{2},-\frac{1}{2}-s,\frac{\epsilon^{2}{\bar{\epsilon}}^{2}}{(\epsilon.{\bar{\epsilon}})^{2}}\Big{)}\,.$

As before, the modified dot product in
$\epsilon.\bar{\epsilon},\epsilon^{2},\bar{\epsilon}^{2}$ include the spin-$1$
state projector $\tilde{\eta}^{\mu\nu}$, whereas the dot products
$p\cdot\epsilon$ and $p\cdot\bar{\epsilon}$ in eq. (2.33) are standard Lorentz
products.

The function $A^{(s)}$ can be related to the associated Legendre functions
$P_{s+1}^{1}(x)$, and there is a Rodrigues’ formula for it,

$A^{(s)}=\frac{s!}{(2s+2)!}\Big{(}\frac{\partial}{\partial\epsilon.{\bar{\epsilon}}}\Big{)}^{s+2}\big{(}(\epsilon.{\bar{\epsilon}})^{2}-\epsilon^{2}{\bar{\epsilon}}^{2}\big{)}^{s+1}\,,$
(2.36)

which makes manifest that it is a polynomial.

Again, the fermionic spin-$(s+1/2)$ projector is unique for on-shell states,
and it can be used to construct a propagator

$\Delta_{(s+1/2)}(\epsilon,\bar{\epsilon})=i\frac{P_{(s+1/2)}(\epsilon,\bar{\epsilon})+\ldots}{p^{2}-m^{2}}\,,$
(2.37)

where in general there are additional terms of ${\cal O}(p^{2}{-}m^{2})$ in
the numerator that contribute off shell. These terms depend on the details of
the Lagrangian formulation of the theory.

For the case of spin $1/2$ and spin $3/2$, additional terms are not expected,
and the propagators are

$\displaystyle\Delta_{(1/2)}(\epsilon,\bar{\epsilon})$ $\displaystyle=$
$\displaystyle i\frac{/\\!\\!\\!p+m}{p^{2}-m^{2}}\,,$
$\displaystyle\Delta_{(3/2)}(\epsilon,\bar{\epsilon})$ $\displaystyle=$
$\displaystyle
i\frac{(/\\!\\!\\!p+m)\,\epsilon.\bar{\epsilon}+\frac{1}{3}(/\\!\\!\\!\epsilon+\frac{p\cdot\epsilon}{m})(/\\!\\!\\!p-m)(/\\!\\!\\!\bar{\epsilon}+\frac{p\cdot\bar{\epsilon}}{m})}{p^{2}-m^{2}}\,,$
(2.38)

with
$\epsilon.\bar{\epsilon}=\epsilon^{\mu}(\eta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{m^{2}})\bar{\epsilon}^{\nu}$.
The propagators with free Lorentz indices can be obtained by taking an
appropriate number of derivatives $\frac{\partial}{\partial\epsilon^{\mu}}$
and $\frac{\partial}{\partial\bar{\epsilon}^{\nu}}$ that act on
$\Delta_{(s+1/2)}(\epsilon,\bar{\epsilon})$. This will automatically
symmetrize the Lorentz indices on each side of the propagator matrix.

##  3 Higher-spin three-point amplitudes

We now consider amplitudes for a pair of spin-$s$ particles using the massive
spinor-helicity formalism. To avoid displaying unimportant overall
normalization factors in the spinor-helicity formulae, we denote amplitudes
with either straight or calligraphic symbols. The calligraphic ones, ${\cal
A}(1,2,\ldots,n)$ for gauge theory and ${\cal M}(1,2,\ldots,n)$ for gravity,
are more suitable for covariant formulae that use polarization vectors. The
straight ones, $A(1,2,\ldots,n)$ and $M(1,2,\ldots,n)$, are more suitable for
spinor-helicity expressions. Their relative normalizations are

$\begin{split}{\cal A}(1,2,\ldots,n)&=(-1)^{\lceil
s\rceil}\Big{(}\sqrt{2}e\Big{)}^{n-2}A(1,2,\ldots,n)\,,\\\ {\cal
M}(1,2,\ldots,n)&=(-1)^{\lceil
s\rceil}\Big{(}\frac{\kappa}{2}\Big{)}^{n-2}M(1,2,\ldots,n)\,.\end{split}$
(3.39)

where $e$ is the gauge theory (electric) coupling, $\kappa$ is the
gravitational coupling, with $\kappa^{2}=32\pi G_{N}$. The ceiling function
${\lceil s\rceil}$ takes into account phases that depend on the spin of the
massive particle, which appear due to our mostly-minus metric signature
choice. Furthermore, sometimes it is convenient to set $e=1$ or $\kappa=1$, in
which case the two normalizations simply differ by powers of $\sqrt{2}$ and
signs.

###  3.1 Spinor-helicity three-point amplitudes

It was proposed by Arkani-Hamed, Huang and Huang Arkani-Hamed:2017jhn  that
the most natural three-point amplitudes between two massive higher-spin
particles and a gauge boson should be the following maximally-chiral objects:

$A(1\phi^{s},2\bar{\phi}^{s},3A^{+})=mx\frac{\langle{\bf 1}{\bf
2}\rangle^{2s}}{m^{2s}}\,,~{}~{}~{}~{}~{}~{}A(1\phi^{s},2\bar{\phi}^{s},3A^{-})=\frac{m}{x}\frac{[{\bf
1}{\bf 2}]^{2s}}{m^{2s}}\,,$ (3.40)

where $\phi^{s}$ is a (complex) spin-$s$ field (bosonic or fermionic) of mass
$m$, and $A^{\pm}$ is a massless photon with helicity $\pm 1$. The $x$
notation was introduced Arkani-Hamed:2017jhn  as a shorthand notation for the
dimensionless scalar amplitude, and in our normalization convention it is
given by

$A(1\phi,2\bar{\phi},3A^{\pm})=\frac{i}{\sqrt{2}}(p_{1}-p_{2})\cdot\varepsilon_{3}^{\pm}\equiv
mx^{\pm 1}\,.$ (3.41)

The amplitudes (3.40) were identified as being special due to their properties
in the high-energy limit, and because they agree with well-known gauge-theory
amplitudes for low spin $s=0,1/2,1$.

It is instructive to consider the high-energy limit in some detail. In this
limit, the massive spinors degenerate into massless spinors, up to corrections
${\cal O}(m)$. We can choose the momenta such that $\langle{\bf 1}{\bf
2}\rangle\sim p$, $[{\bf 1}{\bf 2}]\sim m$, and $x\sim m/p$, where $p\gg m$ is
some scale that characterizes the momenta. (Note that three-point kinematics
require complex momenta, thus the spinors are not related by complex
conjugation). The amplitudes (3.40) behave in this limit as,

$mx\frac{\langle{\bf 1}{\bf
2}\rangle^{2s}}{m^{2s}}\sim\frac{p^{2s-1}}{m^{2s-2}}\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{m}{x}\frac{[{\bf
1}{\bf 2}]^{2s}}{m^{2s}}\ \sim p\,,$ (3.42)

and we can compare this to generic amplitude terms

$A^{\rm
generic}_{s^{\prime},s^{\prime\prime}}(1\phi^{s},2\bar{\phi}^{s},3A^{\pm})\sim
mx^{\pm 1}\frac{\langle{\bf 1}{\bf
2}\rangle^{2s^{\prime}}}{m^{2s^{\prime}}}\frac{[{\bf 1}{\bf
2}]^{2s^{\prime\prime}}}{m^{2s^{\prime\prime}}}\ \sim\frac{p^{2s^{\prime}\mp
1}}{m^{2s^{\prime}-1\mp 1}}\,,$ (3.43)

where $s^{\prime},s^{\prime\prime}=0,1/2,1,3/2,\ldots$ are allowed powers and
$s=s^{\prime}+s^{\prime\prime}$. We see that, in this high-energy limit, the
most singular positive-helicity terms and least singular negative-helicity
terms coincide with the amplitudes from ref. Arkani-Hamed:2017jhn
.666Alternatively, one can consider the opposite limit $\langle{\bf 1}{\bf
2}\rangle\sim m$, $[{\bf 1}{\bf 2}]\sim p$ which is closer to the
considerations of ref. Arkani-Hamed:2017jhn . However, the two limits are
related by CPT conjugation.

It is interesting to note that some of the generic terms are both more
singular and more soft than the ones from ref. Arkani-Hamed:2017jhn  (in a
helicity averaged amplitude). For example, amplitude contributions $\sim
x^{-1}\langle{\bf 1}{\bf 2}\rangle^{2s-1}[{\bf 1}{\bf 2}]$ are more singular
than the terms $\sim x\langle{\bf 1}{\bf 2}\rangle^{2s}$, and the terms $\sim
x[{\bf 1}{\bf 2}]^{2s-1}\langle{\bf 1}{\bf 2}\rangle$ are softer than $\sim
x^{-1}[{\bf 1}{\bf 2}]^{2s}$ . For $s=1$, such amplitude contributions do
appear in a spin-$1$ Lagrangian with non-minimal coupling $\Delta{\cal
L}\sim\alpha\overline{W}^{\mu}W^{\nu}F_{\mu\nu}$, and they only cancel out for
$\alpha=1$, which corresponds to a spontaneously-broken Yang-Mills theory,
i.e. in a theory where the $W$-boson acquired its mass through the Higgs
mechanism. See Section 5.1.1 for further details on the spin-$1$ Lagrangian.

###  3.2 Constraining covariant three-point amplitudes and currents

In this paper, we argue that the three-point amplitudes (3.40) admit a
covariant description in terms of off-shell higher-spin currents that appear
to be associated to softly-broken higher-spin (gauge) symmetry. That is, we
can find an off-shell current $J^{\mu}$, associated to a massive higher-spin
field, that appropriately vanishes as $m\rightarrow 0$ when contracted with
one power of its momentum,

$p_{\mu}J^{\mu}\big{|}_{\rm traceless}={\cal O}(m)\,.$ (3.44)

We restrict the condition to hold for the traceless part (gamma-traceless part
for fermions) of the contracted current. Since the current constraint holds
off-shell, it is a powerful tool for constraining the higher-spin Lagrangian
that gives the three-point amplitudes (3.40). See Section 5 for Lagrangian
constructions based on this constraint, here we only focus on the current and
amplitudes. Naively, the property (3.44) is somewhat surprising, since the
higher-spin amplitudes (3.40) have arbitrarily high powers of the mass in the
denominator, and the massless limits are expected to be singular rather than
soft, at least beyond spin $2$.

We will not give the complete details of the off-shell currents here, since
they are quite lengthy objects and contain ambiguities due to their off-shell
nature. We will however show a few steps and then quote the final results for
the three-point case. In Section 5, more details are given using Lagrangians
up to spin $5/2$.

As a first step, we will find a covariant on-shell formula for the amplitudes
(3.40). This formula will reveal the number of derivatives that are present in
the three-point interactions, and also make manifest that all interactions are
parity even. We will quote the formula now without further delay, and then
prove it later. To give the covariant formula for all bosonic spin-$s$
amplitudes, it is easiest to construct a generating series,

$\sum_{s=0}^{\infty}A(1\phi^{s},2\bar{\phi}^{s},3A)=A_{\phi\phi
A}+\frac{A_{WWA}-(\bm{\mathbf{\varepsilon}}_{1}\cdot\bm{\mathbf{\varepsilon}}_{2})^{2}\,A_{\phi\phi
A}}{(1+\bm{\mathbf{\varepsilon}}_{1}\cdot\bm{\mathbf{\varepsilon}}_{2})^{2}+\frac{2}{m^{2}}\bm{\mathbf{\varepsilon}}_{1}\cdot
p_{2}\,\bm{\mathbf{\varepsilon}}_{2}\cdot p_{1}}\,,$ (3.45)

where the spin-$0$ and spin-$1$ amplitudes appear abbreviated on the right
hand side, their explicit covariant forms are

$A_{\phi\phi A}\equiv i\sqrt{2}\,\varepsilon_{3}\cdot
p_{1}\,,~{}~{}~{}A_{WWA}\equiv
i\sqrt{2}\,(\bm{\mathbf{\varepsilon}}_{1}\cdot\bm{\mathbf{\varepsilon}}_{2}\,\varepsilon_{3}\cdot
p_{2}+\bm{\mathbf{\varepsilon}}_{2}\cdot\varepsilon_{3}\,\bm{\mathbf{\varepsilon}}_{1}\cdot
p_{3}+\varepsilon_{3}\cdot\bm{\mathbf{\varepsilon}}_{1}\,\bm{\mathbf{\varepsilon}}_{2}\cdot
p_{1})\,.$ (3.46)

Note that the massive polarization vectors are polynomials of $z_{i}^{a}$ and
hence the generating function is a rational function in these variables.
Series expansion around $z_{i}^{a}=0$ will return the amplitudes for different
spins.

To prove that eq. (3.45) is the covariant formula for the resummed amplitudes
(3.40), we simply evaluate the polarization vectors in terms of the spinor-
helicity variables. The denominator evaluates to

$(1+\bm{\mathbf{\varepsilon}}_{1}\cdot\bm{\mathbf{\varepsilon}}_{2})^{2}+\frac{2}{m^{2}}\bm{\mathbf{\varepsilon}}_{1}\cdot
p_{2}\,\bm{\mathbf{\varepsilon}}_{2}\cdot p_{1}=\Big{(}1-\frac{\langle{\bf
1}{\bf 2}\rangle^{2}}{m^{2}}\Big{)}\Big{(}1-\frac{[{\bf 1}{\bf
2}]^{2}}{m^{2}}\Big{)}\,,$ (3.47)

and the numerator becomes
$A_{WWA}-(\bm{\mathbf{\varepsilon}}_{1}\cdot\bm{\mathbf{\varepsilon}}_{2})^{2}A_{\phi\phi
A}=x\langle{\bf 1}{\bf 2}\rangle^{2}(m^{2}-[{\bf 1}{\bf 2}]^{2})/m^{3}$ for
positive helicity $\varepsilon_{3}^{+}$ (the negative helicity case is
obtained by swapping square and angle spinors, and letting $x\rightarrow
1/x$). The factor with square brackets cancels out between the denominator and
numerator, and the generating function simplifies to

$\sum_{s=0}^{\infty}A(1\phi^{s},2\bar{\phi}^{s},3A^{+})=\frac{mx}{1-\frac{\langle{\bf
1}{\bf 2}\rangle^{2}}{m^{2}}}\,,$ (3.48)

which is indeed the geometric series that describes
$A(1\phi^{s},2\bar{\phi}^{s},3A^{+})$ in eq. (3.40). The negative helicity
amplitudes $A(1\phi^{s},2\bar{\phi}^{s},3A^{-})$ are obtained analogously.
This proves the assertion that eq. (3.45) gives the covariant bosonic
amplitudes.

Let us address the question of uniqueness of eq. (3.45). It is straightforward
to check at low spin that there exist no other covariant and parity-even
formulae with lower number of momentum powers (i.e. derivatives in the
Lagrangian). Also, one can confirm that it is not possible to write down Gram-
determinant expressions for the five independent vectors
$\\{\bm{\mathbf{\varepsilon}}_{1},\bm{\mathbf{\varepsilon}}_{2},\varepsilon_{3},p_{1},p_{2}\\}$,
since such terms would necessarily be quadratic in the massless polarization
vector $\varepsilon_{3}$ and hence describe interactions of a graviton rather
than a gauge boson. Finally, we note that, given that we assume parity-even
and gauge-invariant interactions, there are only four independent
dimensionless variables that can be used to construct the amplitude: two of
them involve $\varepsilon_{3}$, and can be chosen to be the amplitudes
$A_{\phi\phi A}/m$ and $A_{WWA}/m$, and the other two can be chosen as
$\bm{\mathbf{\varepsilon}}_{1}\cdot\bm{\mathbf{\varepsilon}}_{2}$ and
$\bm{\mathbf{\varepsilon}}_{1}\cdot p_{2}\,\bm{\mathbf{\varepsilon}}_{2}\cdot
p_{1}/m^{2}$. The spinor-helicity expressions appear to only involve three
dimensionless variables: $\langle{\bf 1}{\bf 2}\rangle/m$, $[{\bf 1}{\bf
2}]/m$, and $x$. However, because the amplitudes in eq. (3.40) give two
equations in these variables, there is a unique covariantization of the
spinor-helicity amplitudes.

It turns out that the generalization to fermionic gauge-theory amplitudes is
almost identical to the bosonic case. We only need to identify two low-spin
amplitudes, say, spin $1/2$ and spin $3/2$, and then the other fermionic
amplitudes are linear combinations of these. In fact, we observe that it is
exactly the same linear combinations that appeared in the bosonic generating
function (3.45). Hence, the fermionic amplitudes can be formally resummed as

$\sum_{s=0}^{\infty}A(1\phi^{s+1/2},2\bar{\phi}^{s+1/2},3A)=A_{\lambda\lambda
A}+\frac{A_{\psi\psi
A}-(\bm{\mathbf{\varepsilon}}_{1}\cdot\bm{\mathbf{\varepsilon}}_{2})^{2}\,A_{\lambda\lambda
A}}{(1+\bm{\mathbf{\varepsilon}}_{1}\cdot\bm{\mathbf{\varepsilon}}_{2})^{2}+\frac{2}{m^{2}}\bm{\mathbf{\varepsilon}}_{1}\cdot
p_{2}\,\bm{\mathbf{\varepsilon}}_{2}\cdot p_{1}}\,,$ (3.49)

where $A_{\lambda\lambda
A}=\frac{i}{\sqrt{2}}\bar{u}_{2}\not{\varepsilon}_{3}v_{1}$ is the standard
electron-photon three-point amplitude in quantum electrodynamics, and
$A_{\psi\psi A}$ is a well-known gravitino-photon amplitude that is unique in
supergravity with spontaneously-broken supersymmetry. The details of this
spin-$3/2$ amplitude are given in Section 5.1.2; for the current purpose we
only need to know that $A_{\psi\psi A}$ comes from parity-even interaction
terms that are linear in the momenta. That the covariant formula eq. (3.49) is
correct follows from analogous arguments to the ones used below eq. (3.47).
The uniqueness is less clear since Gram-determinant expressions can be
constructed using the vectors and higher-rank forms that come from the fermion
bilinears $\bar{u}_{2}\gamma^{\mu_{1}\ldots\mu_{n}}v_{1}$. Nevertheless, at
spin $1/2$ and spin $3/2$, the covariant amplitudes are unique as will be
discussed in Section 5.1.2.

Let us now switch the discussion to gravitational higher-spin amplitudes. In
ref. Arkani-Hamed:2017jhn  Arkani-Hamed, Huang and Huang gave the following
three-point amplitudes:

$\displaystyle
M(1\phi^{s},2\bar{\phi}^{s},3h^{+})=im^{2}x^{2}\frac{\langle\bm{1}\bm{2}\rangle^{2s}}{m^{2s}}\,,$
$\displaystyle
M(1\phi^{s},2\bar{\phi}^{s},3h^{-})=i\frac{m^{2}}{x^{2}}\frac{[\bm{1}\bm{2}]^{2s}}{m^{2s}}\,.$
(3.50)

It is clear from the spinor-helicity expressions that they are related to the
gauge-theory amplitudes by the double copy Kawai:1985xq ,

$M(1\phi^{s},2\bar{\phi}^{s},3h^{\pm})=iA(1\phi^{s_{\rm
L}},2\bar{\phi}^{s_{\rm L}},3A^{\pm})A(1\phi^{s_{\rm R}},2\bar{\phi}^{s_{\rm
R}},3A^{\pm})\,,$ (3.51)

where $s=s_{\rm L}+s_{\rm R}$, and equivalent formulae are obtained for any
$s_{\rm L},s_{\rm R}\geq 0$. Since this relation is insensitive to the
helicity of the massless state, it follows that the covariant gravitational
amplitudes are also given by a double copy of the covariant gauge-theory
amplitudes.

Using the double copy, it is a small step to show that a generating function
for the covariant gravitational amplitudes can be constructed out of the
gauge-theory generating functions. However, because the spin $s$ can be
decomposed into $s_{\rm L}+s_{\rm R}$ in multiple different ways, there is
some ambiguity on how to write it. We find that it is convenient to write a
combined generating function for bosonic and fermionic covariant gravitational
amplitudes, which has the form

$\sum_{2s=0}^{\infty}M(1\phi^{s},2\bar{\phi}^{s},3h)=M_{0\oplus
1/2}+iA_{WWA}\Big{(}A_{0\oplus
1/2}+\frac{A_{1\oplus\,3/2}-(\bm{\mathbf{\varepsilon}}_{1}\cdot\bm{\mathbf{\varepsilon}}_{2})^{2}\,A_{0\oplus
1/2}}{(1+\bm{\mathbf{\varepsilon}}_{1}\cdot\bm{\mathbf{\varepsilon}}_{2})^{2}+\frac{2}{m^{2}}\bm{\mathbf{\varepsilon}}_{1}\cdot
p_{2}\,\bm{\mathbf{\varepsilon}}_{2}\cdot p_{1}}\Big{)}\,,$ (3.52)

where the sum runs over both integer and half-integer spin $s$. Here we use
the shorthands $A_{0\oplus 1/2}=A_{\phi\phi A}+A_{\lambda\lambda A}$,
$A_{1\oplus\,3/2}=A_{WWA}+A_{\psi\psi A}$ and $M_{0\oplus 1/2}=iA_{\phi\phi
A}A_{0\oplus 1/2}$, which combine the independent low-spin amplitudes that
appeared in the bosonic (3.45) and fermionic (3.49) generating functions. The
reason for multiplying the gauge-theory generating function by $A_{WWA}$,
rather than by $A_{\phi\phi A}$, is that this way of writing it exposes the
correct maximal momentum power counting of the gravity amplitudes.

As an aside, it is interesting to note that we have already identified the
gauge-theory amplitudes $A_{WWA}$ and $A_{\psi\psi A}$, as well-known three-
point amplitudes in theories with spontaneously broken gauge symmetry ($s=1$)
and supersymmetry $(s=3/2)$, respectively. From this we should expect that
some of the low-spin gravitational amplitudes also have an interesting
physical interpretation. Indeed, it is known from previous work on the double
copy Chiodaroli:2017ehv ; Chiodaroli:2018dbu  that these massive spin-$1$ and
spin-$3/2$ gravitational amplitudes, $A_{\phi\phi A}\times A_{WWA}$ and
$A_{\lambda\lambda A}\times A_{WWA}$, are precisely those that appear in
R-symmetry gauged supergravity, and the massive spin-$2$ amplitude
$(A_{WWA})^{2}$ is precisely the one that appears in Kaluza-Klein gravity
Chiodaroli:2015rdg ; Momeni:2020vvr ; Johnson:2020pny ; Momeni:2020hmc ;
Hang:2021fmp ; Chi:2021mio . Indeed, in Section 5.2.4, we obtain a spin-$2$
Lagrangian from Kaluza-Klein compactification, and show that it is compatible
with the expected spin-$2$ current and Compton amplitude.

###  3.3 Off-shell current construction

Inspecting the covariant formulae (3.45), (3.49) and (3.52), we learn that the
gauge theory higher-spin three-point interactions are parity invariant, and
they have an upper bound for the number of derivatives. For integer $s\geq 1$,
the bosonic and fermionic covariant gauge-theory amplitudes have a maximal
power of momentum (derivatives) that scale as

$A(1\phi^{s},2\bar{\phi}^{s},3A)\sim
p^{2s-1}\,,~{}~{}~{}~{}A(1\phi^{s+1/2},2\bar{\phi}^{s+1/2},3A)\sim
p^{2s-1}\,.$ (3.53)

For integer $s\geq 2$, the bosonic and fermionic covariant gravity amplitudes
have a maximal power of momentum (derivatives) that scale as

$M(1\phi^{s},2\bar{\phi}^{s},3h)\sim
p^{2s-2}\,,~{}~{}~{}~{}M(1\phi^{s+1/2},2\bar{\phi}^{s+1/2},3h)\sim
p^{2s-2}\,.$ (3.54)

This matches the scaling that we found in eq. (3.42) for gauge-theory bosons.
For fermions, an extra power of momentum can be attributed to the wave
functions, but it is not counted here since we now focus on the momentum which
is explicit in the covariant formulae, as this will correspond to the
derivatives in a putative Lagrangian. The gravity momentum scaling is also
compatible with eq. (3.42) after accounting for the extra factor of $x\sim
m/p$ in the gravitational amplitudes (3.2). Note that the covariant amplitudes
have a slightly different scaling in the gauge-theory $s<1$ and gravity $s<2$
cases. In these cases, the massive particle is not the highest spin state, and
thus the gauge and gravity theories are expected to be completely standard,
that is, with minimally-coupled matter.

Let us now discuss off-shell versions of the covariant amplitudes. Consider
constructing an Ansatz for the off-shell correlation function between two
bosonic higher-spin fields $\phi^{s},\bar{\phi}^{s}$ and a gauge boson $A^{w}$
of spin $w$, that is we combine the discussion of gauge theory ($w=1$) and
gravity ($w=2$), so that $A^{w}=(A^{\mu},h^{\mu\nu})$. The discussion
simplifies if we saturate all the free Lorentz indices with dummy vectors
$\epsilon^{\mu}_{i}$, in analogy with the projector discussion in Section 2.
The off-shell correlation function for bosonic higher-spin fields then takes
the form

$\langle\phi^{s}(p_{1})\bar{\phi}^{s}(p_{2})A^{w}(p_{3})\rangle=m^{w}\,{\rm
Poly}(\epsilon_{i}\cdot\epsilon_{j},\,\epsilon_{i}\cdot p_{j}/m,\,p_{i}\cdot
p_{j}/m^{2})\,,$ (3.55)

where ${\rm Poly}(\ldots)$ is a polynomial Ansatz of the Lorentz invariants
that appear in the arguments. The range of leg labels is $i,j=1,2,3$; we
assume momentum conservation, $p_{1}+p_{2}+p_{3}=0$, but no other relations
exist among the arguments. All the arguments are dimensionless, and we have
factored out $w$ powers of the mass, so that the correlator is of the correct
dimension for a gauge/gravity theory. For fermionic higher-spin fields, the
correlation function is almost the same, except that the polynomial Ansatz
(3.55) also contains contractions between the dimensionless vectors
$p_{i}^{\mu}/m,\epsilon_{i}^{\mu}$ and the dimensionless fermion bilinears
$\bar{u}_{2}\gamma^{\mu_{1}\ldots\mu_{n}}v_{1}/m$, where $\bar{u}_{2},v_{1}$
are temporarily treated as unconstrained off-shell spinors. Note that every
term in the polynomial Ansatz must contain exactly one fermion bilinear, and
since the polarizations and momenta constitute five independent vectors, the
tensor rank of the fermion bilinear is at most five.

We will now give a list of constraints on the off-shell correlator that is
compatible with the covariant amplitudes (3.45), and hence compatible with the
Arkani-Hamed, Huang and Huang amplitudes Arkani-Hamed:2017jhn . In fact, the
constraints will uniquely pin down these amplitudes up to spin-$3/2$ gauge
theory, and spin-$5/2$ gravity. We constrain the polynomial (3.55) such that:

  * •

Each term contains $s$ powers of the higher-spin polarizations,
$(\epsilon_{1})^{s}$, $(\epsilon_{2})^{s}$, and $w$ powers of the gauge boson
$(\epsilon_{3})^{w}$;

  * •

Each term have at most $p^{2s-w}$ powers of momenta;

  * •

Gauge invariance is imposed for the gauge current $p_{3}\cdot J_{\rm
gauge}=0$;

  * •

The current constraint is imposed on the higher-spin currents; for leg 1 this
means $p_{1}\cdot J\big{|}_{\rm traceless}={\cal O}(m)$, and similarly for leg
2;

  * •

Standard kinetic-term behavior is imposed on interactions.

Let us clarify what this list of constraints means. Firstly, the gauge current
and higher-spin current are defined as

$\displaystyle J_{\rm gauge}^{\mu}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial\epsilon_{3}^{\mu}}\langle\phi^{s}(p_{1})\bar{\phi}^{s}(p_{2})A^{w}(p_{3})\rangle\Big{|}_{1\&2~{}\text{on
shell}}\,,$ $\displaystyle J^{\mu}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial\epsilon_{1}^{\mu}}\langle\phi^{s}(p_{1})\bar{\phi}^{s}(p_{2})A^{w}(p_{3})\rangle\Big{|}_{2\&3~{}\text{on
shell}}\,,$ (3.56)

and the higher-spin current for leg 2 is obtained by swapping
$1\leftrightarrow 2$. The currents are simply defined as the correlator where
a Lorentz index is exposed corresponding to an off-shell leg, and then the
remaining two legs are subject to on-shell conditions. The on-shell conditions
for the higher-spin legs $i=1,2$ are $\\{p_{i}^{2}=m^{2},\epsilon_{i}\cdot
p_{i}=0,\epsilon_{i}^{2}=0\\}$ and for the gauge-field leg 3 they are
$\\{p_{3}^{2}=0,\epsilon_{3}\cdot p_{3}=0,\epsilon_{3}^{2}=0\\}$. For
fermions, the on-shell conditions are supplemented by
$\\{(\not{p}_{1}+m)v_{1}=\bar{u}_{2}(\not{p}_{2}-m)=\not{\epsilon}_{1}v_{1}=\bar{u}_{2}\not{\epsilon}_{2}=0\\}$.
Note that tracelessness, $\epsilon_{i}^{2}=0$, and gamma-tracelessness,
$\not{\epsilon}_{1}v_{1}=\bar{u}_{2}\not{\epsilon}_{2}=0$, are not imposed for
the off-shell legs in the current. However, after contracting the higher-spin
current into its momentum, we remove the trace components from the current
constraint: $p_{1}\cdot J\big{|}_{\rm traceless}={\cal O}(m)$. Without
removing the trace components, the current constraint is too restrictive.

Finally, we need to impose the existence of a standard kinetic term. This may
sound strange since the correlator Ansatz (3.55) only contains the three-point
interactions. It is clear that some of the terms in the three-point correlator
will originate from the non-linear terms of the kinetic term, whereas other
terms originate from higher-derivative gauge-invariant cubic operators
(involving the gauge-boson field strength). To have a standard kinetic term,
we need to forbid interaction terms that originate from possible higher-
derivative kinetic terms. This can be done by demanding that all higher-
derivative terms obey gauge invariance for the full off-shell correlator
(hence they originate from the field strength, and not from the covariant
derivatives of the kinetic term). The standard kinetic term behavior is
imposed on the off-shell correlator by the constraint

$p_{3}^{\mu}\frac{\partial}{\partial\epsilon_{3}^{\mu}}\langle\phi^{s}(p_{1})\bar{\phi}^{s}(p_{2})A^{w}(p_{3})\rangle={\cal
O}(m^{0})\,.$ (3.57)

The right hand side of this equation involves no inverse powers of the mass,
since such terms come from higher-derivative operators, which should be fully
gauge invariant off-shell. A standard kinetic term would have no poles in $m$,
and the corresponding interaction terms need not be gauge invariant for an
off-shell correlator.

After having applied the above constraints on the correlator Ansatz (3.55),
all legs can be placed on-shell to obtain a covariant amplitude. Without any
input from Lagrangians or spinor-helicity formulae, this gives unique tree-
point amplitudes for spin-$1$ and spin-$3/2$ gauge theory, as well as spin-$2$
and spin-$5/2$ gravity, precisely agreeing with the covariant amplitudes
(3.49) and (3.52).777The Ansatz procedure also leads to unique amplitudes for
the low spin cases, $s<1$ gauge theory and $s<2$ gravity, but the derivative
counting needs to be adjusted upwards to be compatible with the kinetic term;
hence these are all minimal-coupling theories. For higher spin, beyond
spin-$3/2$ gauge theory and beyond spin-$5/2$ gravity, the procedure does not
result in unique amplitudes, but within the parameter space of the allowed
solution one can always find the amplitudes (3.49) and (3.52). We explicitly
constructed the correlator and currents up to spin 5 for both gravity and
gauge theory, and found them to be compatible with the covariant amplitudes.
Thus there is strong evidence that the spinor-helicity amplitudes of ref.
Arkani-Hamed:2017jhn  originate from higher-spin theories that obey the
current constraint.

Thus far, we have ignored the subtleties that come from four-dimensional
kinematics, so let us comment on this. For spin-$5/2$ gravity, one can find a
unique solution to the current constraint, but this requires that we use the
following Gram-determinant identity:

$G=\epsilon_{1\mu}f_{3\nu\rho}f_{2\sigma\kappa}\bar{u}_{2}\gamma^{[\mu\nu}v_{1}f^{\rho\sigma}_{3}\epsilon_{2}^{\kappa]}~{}\stackrel{{\scriptstyle
D=4}}{{=}}~{}0\,,$ (3.58)

where the five upper Lorentz indices are antisymmetrized, and
$f_{i}^{\mu\nu}=p_{i}^{\mu}\epsilon_{i}^{\nu}-p_{i}^{\nu}\epsilon_{i}^{\mu}$
is a field strength of particle $i$ (the corresponding linearized “Riemann
tensor” is the square $f_{i}^{\mu\nu}f_{i}^{\rho\sigma}$). The current
constraint then holds modulo this four-dimensional identity

$p_{1}\cdot J_{5/2}\Big{|}_{\rm traceless}~{}\sim~{}\frac{G}{m}+{\cal
O}(m)\,.$ (3.59)

Note that $G$ is one of the first Gram determinants that can appear for low-
spin correlators. In fact, for three-point correlators in a bosonic gauge
theory one cannot write down any Gram determinant compatible with our Ansatz.
Similarly, in a bosonic gravity theory one can write down only one,
$G(\epsilon_{1},\epsilon_{2},\epsilon_{3},p_{1},p_{2})$, which is quadratic in
all three polarizations. This object can first show up in a current constraint
for spin-$3$ particles; however, we find that the current constraint can be
solved without imposing the vanishing of the Gram determinant. For fermionic
gauge theory correlators, one could in principle write down
$G^{\prime}=f_{3\rho\sigma}f_{2\kappa\lambda}\epsilon_{1\nu}\bar{u}_{1}\gamma^{[\nu\rho\sigma\kappa}v_{2}\epsilon_{2}^{\lambda]}$,
which also corresponds to spin $5/2$. However, surprisingly, we find that we
do not need to make use of such identities. It is curious that it is only for
the fermionic gravity correlators that it seems necessary to use the Gram
determinants to satisfy the current constraint.

###  3.4 Classical limit

The gravitational amplitudes discussed in Section 3 have been shown to match
the classical energy-momentum tensor of a Kerr black hole Guevara:2018wpp ;
Arkani-Hamed:2019ymq . Here we briefly review that argument using our
notation, and we will also make an interesting comparison to the higher-spin
classical amplitudes recently computed in ref. Bern:2020buy .

The linearized energy-momentum tensor for a Kerr black hole can be given in
momentum space as Vines:2017hyw ; Guevara:2018wpp

$T^{\mu\nu}(-k)=2\pi\,\delta(p\cdot
k)\,p^{(\mu}{\exp(m^{-1}S*ik)^{\nu)}}_{\rho}\,p^{\rho}\,,$ (3.60)

where $p^{\mu}$ and $S^{\mu}$ are the black hole momentum and classical spin
vector (Pauli-Lubanski vector), respectively, and
${(S*k)^{\mu}}_{\nu}={\epsilon^{\mu}}_{\nu\rho\sigma}S^{\rho}k^{\sigma}$.

We can contract the tensor with an on-shell graviton polarization to obtain
what one may call a classical amplitude Guevara:2018wpp ,

$\varepsilon_{\mu\nu}(k)T^{\mu\nu}(-k)=(2\pi)^{2}\,\delta(p\cdot
k)\,\delta(k^{2})(\varepsilon\cdot
p)^{2}\exp\\!\Big{(}\\!-i\frac{k_{\mu}\varepsilon_{\nu}S^{\mu\nu}}{\varepsilon\cdot
p}\Big{)}\,,$ (3.61)

where $\varepsilon_{\mu\nu}(k)=\varepsilon_{\mu}(k)\varepsilon_{\nu}(k)$ and
$S^{\mu\nu}=m^{-1}\epsilon^{\mu\nu\rho\sigma}p_{\rho}S_{\sigma}$.

One can compare the above classical amplitude to the quantum three-point
amplitude obtained by Arkani-Hamed, Huang and Huang. Let us focus on a
positive-helicity graviton with momentum $k$ and polarization
$\varepsilon_{\mu\nu}(k)$ that scatters against two bosonic spin-$s$ particles
with momenta $p_{1}$ and $p_{2}$. We express a
$\langle\bm{1}\bm{2}\rangle^{2}$ factor in terms of massive polarizations to
get an expression reminiscent of an exponential,

$(-1)^{s}i\frac{m^{2}x^{2}}{2}\frac{\langle\bm{1}\bm{2}\rangle^{2s}}{m^{2s}}=-i(\varepsilon\cdot
p_{1})^{2}\left(\bm{\mathbf{\varepsilon}}_{2}\cdot\Big{(}1+\frac{1}{m}k\cdot\hat{S}+\frac{1}{m^{2}}(k\cdot\hat{S})^{2}\Big{)}\cdot\bm{\mathbf{\varepsilon}}_{1}\right)^{s}\,,$
(3.62)

where
$\hat{S}^{\mu}=(2m)^{-1}\epsilon^{\mu\nu\rho\sigma}p_{1\nu}M_{\rho\sigma}$ is
a spin operator, with
$(M^{\mu\nu})_{\alpha\beta}=2i\delta^{[\mu}_{\alpha}\delta^{\nu]}_{\beta}$
being the spin-$1$ Lorentz generator. The positive-helicity amplitude is
similar, and it can be obtained by the replacement
$\hat{S}\rightarrow-\hat{S}$ in the above expression.

We will now take the classical limit, which we can do in two steps. First take
the momentum of the graviton $k$ to be small compared to momenta of the
higher-spin particles. Using momentum conservation and imposing that
polarizations are null and transverse, one can relate $\varepsilon_{2}$ and
$\bar{\varepsilon}_{1}$ through a finite power series in $k$,888We thank
Lucile Cangemi for highlighting the importance of this relation.

$\bm{\mathbf{\varepsilon}}_{2}^{\mu}=-\bar{\bm{\mathbf{\varepsilon}}}_{1}^{\mu}+p_{1}^{\mu}\,\frac{\bar{\bm{\mathbf{\varepsilon}}}_{1}\cdot
k}{m^{2}}+k^{\mu}\,\frac{\bar{\bm{\mathbf{\varepsilon}}}_{1}\cdot
k}{2m^{2}}\,.$ (3.63)

Next, we identify the classical spin vector with the spin-operator expectation
value as
$-\bar{\bm{\mathbf{\varepsilon}}}_{1}\cdot\hat{S}^{\mu}\cdot\bm{\mathbf{\varepsilon}}_{1}\rightarrow
S^{\mu}/s$. We also assume that the polarizations are normalized such that
$\bar{\bm{\mathbf{\varepsilon}}}_{1}\cdot\bm{\mathbf{\varepsilon}}_{1}=-1$
(which imposes $(\bar{z}^{a}z_{a})^{2}=1$). The second step is to take the
large-spin limit. Combining both steps, we get the result

$~{}~{}\stackrel{{\scriptstyle k\ll
p}}{{\longrightarrow}}~{}~{}(\varepsilon\cdot
p)^{2}\Big{(}1+\frac{1}{ms}k\cdot S+{\cal
O}(S^{2}/s^{2})\Big{)}^{s}~{}~{}\stackrel{{\scriptstyle
s\rightarrow\infty}}{{\longrightarrow}}~{}~{}(\varepsilon\cdot
p)^{2}\exp\Big{(}\frac{1}{m}k\cdot S\Big{)}\,.$ (3.64)

The suppressed quadratic term corresponds to a quantum contribution and
vanishes in the large-$s$ limit. Finally we have to make use of the on-shell
relation $ik_{\mu}\varepsilon_{\nu}^{\pm}S^{\mu\nu}=\mp k\cdot
S\,\varepsilon^{\pm}{\cdot\,}p_{1}/m$ and then the match to eq. (3.61) works
up to the overall normalization.

It is instructive to compare to the results from ref. Bern:2020buy , where a
higher-spin Lagrangian formalism was used to obtain amplitudes that reproduces
the linearized energy-momentum tensor (3.60) in the classical limit. The
three-point amplitude, with the same momentum and polarization labeling as
above, can be written as Bern:2020buy

${\cal M}(1\phi^{s},2\bar{\phi}^{s},3h)=-i(\varepsilon\cdot
p_{1})^{2}\bm{\mathbf{\varepsilon}}_{2}^{s}\cdot\left(\exp\Big{(}\frac{1}{m}k\cdot\mathbb{S}\Big{)}-\frac{i}{m^{2}}p_{1}\cdot\mathbb{M}\cdot
k\right)\cdot\bm{\mathbf{\varepsilon}}_{1}^{s}\,,$ (3.65)

where $\mathbb{M}$ and $\mathbb{S}$ are the spin-$s$ representations of the
operators that we called $M$ and $\hat{S}$ above. Even before taking the
classical limit, we can note that the exponential term can be expanded out as

$\displaystyle\bm{\mathbf{\varepsilon}}_{2}^{s}\cdot\exp\left(\frac{1}{m}k\cdot\mathbb{S}\right)\cdot\bm{\mathbf{\varepsilon}}_{1}^{s}$
$\displaystyle=$
$\displaystyle\left(\bm{\mathbf{\varepsilon}}_{2}\cdot\exp\Big{(}\frac{1}{m}k\cdot\hat{S}\Big{)}\cdot\bm{\mathbf{\varepsilon}}_{1}\right)^{s}$
$\displaystyle=$
$\displaystyle\left(\bm{\mathbf{\varepsilon}}_{2}\cdot\Big{(}1+\frac{1}{m}k\cdot\hat{S}+\frac{1}{2m^{2}}(k\cdot\hat{S})^{2}\Big{)}\cdot\bm{\mathbf{\varepsilon}}_{1}\right)^{s}\,,$

where we used that the exponent is a nilpotent matrix $(k\cdot\hat{S})^{3}=0$.
This expression looks very similar to the quantum amplitude (3.62), except
that the term quadratic in $\hat{S}$ has a numerical coefficient smaller by a
factor of $1/2$. Since we already mentioned that this term vanishes in the
classical limit, it can be thought of as a quantum contribution. The same is
true of the second term in eq. (3.65) that contains $p_{1}\cdot\mathbb{M}\cdot
k$.

Let us check if the two contributions responsible for the quantum mismatch
between eq. (3.62) and eq. (3.65) are perhaps related. We find the relation

$\bm{\mathbf{\varepsilon}}_{2}^{s}\cdot\Big{(}\frac{i}{m^{2}}p_{1}\cdot\mathbb{M}\cdot
k\Big{)}\cdot\bm{\mathbf{\varepsilon}}_{1}^{s}=s(\bm{\mathbf{\varepsilon}}_{1}\cdot\bm{\mathbf{\varepsilon}}_{2})^{s-1}\bm{\mathbf{\varepsilon}}_{2}\cdot\Big{(}\frac{1}{m^{2}}(k\cdot\hat{S})^{2}\Big{)}\cdot\bm{\mathbf{\varepsilon}}_{1}\,,$
(3.67)

and for $s=1$ the two expressions indeed conspire in eq. (3.65) with numerical
coefficients $1/2-1$. However, this still does not add up to the unit
coefficient of this term in eq. (3.62), which through $s\leq 5/2$ should give
the unique theories that satisfy tree-level unitarity. That said, the terms
proportional to $p_{1}\cdot\mathbb{M}\cdot k$ or to
$\bm{\mathbf{\varepsilon}}_{2}\cdot(k\cdot\hat{S})^{2}\cdot\bm{\mathbf{\varepsilon}}_{1}$
are subleading in the classical limit and thus the quantum difference is
irrelevant for the purpose of describing astrophysical black holes. In
conclusion, this analysis confirms that eq. (3.65) and eq. (3.62) are
classically equivalent and match the Kerr black-hole dynamics.

##  4 Spinor-helicity Compton amplitudes for $s\leq 5/2$

In ref. Arkani-Hamed:2017jhn , three-point higher-spin amplitudes, which we
discussed in Section 3, were used together with BCFW recursion Britto:2004ap ;
Britto:2005fq  to construct candidates for the Compton amplitudes with
opposite-helicity photons/gravitons. In a later reference the equal-helicity
Compton amplitudes were obtained in the same way Johansson:2019dnu . Let us
start by inspecting the photon amplitudes

$\displaystyle A(1\phi^{s},2\bar{\phi}^{s},3A^{+},4A^{+})$
$\displaystyle=i\frac{\langle\bm{1}\bm{2}\rangle^{2s}[34]^{2}}{m^{2s-2}t_{13}t_{14}}\,,$
(4.68a) $\displaystyle A(1\phi^{s},2\bar{\phi}^{s},3A^{-},4A^{+})$
$\displaystyle=-i\frac{[4|p_{1}|3\rangle^{2-2s}([4\bm{1}]\langle
3\bm{2}\rangle+[4\bm{2}]\langle 3\bm{1}\rangle)^{2s}}{t_{13}t_{14}}\,,$
(4.68b)

where $s_{12}=(p_{1}+p_{2})^{2}$ and $t_{ij}=(p_{i}+p_{j})^{2}-m^{2}$. As was
discussed in ref. Arkani-Hamed:2017jhn , the opposite-helicity amplitude is
well behaved for $s\leq 1$, and starting at $s=3/2$ it develops a spurious
pole corresponding to the factor $[4|p_{1}|3\rangle^{2-2s}$. This pole is
unphysical, and must be canceled by adding a contact interaction to the
Compton amplitude, such that it has a compensating spurious pole. Exactly how
to do this in a unique way has not yet been firmly established. In contrast,
we see that the equal-helicity Compton amplitude does not have a spurious pole
for any spin. And this suggests that it should not be corrected by contact
terms, although a priori it cannot be ruled out that it receives corrections
that are manifestly free of momentum poles.

Next, let us quote the corresponding candidate Compton amplitudes for gravity,
which can be obtained via BCFW recursion in the same way Arkani-Hamed:2017jhn
; Johansson:2019dnu ,

$\displaystyle M(1\phi^{s},2\bar{\phi}^{s},3h^{+},4h^{+})$
$\displaystyle=i\frac{\langle\bm{1}\bm{2}\rangle^{2s}[34]^{4}}{m^{2s-4}s_{12}t_{13}t_{14}}\,,$
(4.69a) $\displaystyle M(1\phi^{s},2\bar{\phi}^{s},3h^{-},4h^{+})$
$\displaystyle=i\frac{[4|p_{1}|3\rangle^{4-2s}([4\bm{1}]\langle
3\bm{2}\rangle+[4\bm{2}]\langle 3\bm{1}\rangle)^{2s}}{s_{12}t_{13}t_{14}}\,.$
(4.69b)

Again, the opposite-helicity amplitudes develop a spurious pole
$[4|p_{1}|3\rangle^{4-2s}$ for spin $s>2$. The naive application of BCFW
recursion does not yield a physical answer, and an unknown contact term needs
to be added to cancel the spurious pole. The results are perhaps not
surprising, since naive application of BCFW recursion assumes a good high-
energy behavior with an absent residue at infinity. However, higher-spin
theories are effective theories, and one may expect that their high-energy
behavior requires a careful analysis. A possible choice for the residue at
infinity was discussed in ref. Arkani-Hamed:2017jhn , but it is not unique and
thus further physical criteria are needed for pinning down these
contributions, and by extension, finding the Compton amplitudes that describe
a Kerr black hole.

In Section 5, we will show the explicit details of using the $P\cdot J={\cal
O}(m)$ condition to uniquely obtain the three-point and Compton amplitudes up
to spin-$3/2$ gauge theory and spin-$5/2$ gravity. These exactly agree with
the spurious-pole-free amplitudes of refs. Arkani-Hamed:2017jhn ;
Johansson:2019dnu . In addition, we obtain unique contact term corrections to
the spin-$3/2$ gauge theory and spin-$5/2$ gravity Compton amplitudes. As a
spoiler of that discussion, here we will briefly quote the new spurious-pole-
free amplitudes, and in Section 5.3 we will revisit the details of the
correction terms.

First, let us pick a shorthand notation for some recurring spinor
combinations,

$\displaystyle N_{2}=[4\bm{1}]\langle 3\bm{2}\rangle+[4\bm{2}]\langle
3\bm{1}\rangle\,,$ $\displaystyle N^{\prime}_{2}=[4\bm{1}]\langle
3\bm{2}\rangle-[4\bm{2}]\langle 3\bm{1}\rangle\,,$ (4.70) $\displaystyle
N_{4}=[\bm{1}4][\bm{2}4]\langle\bm{1}3\rangle\langle\bm{2}3\rangle\,,$

where the subscripts are indicating the dimension counting. In the opposite-
helicity case, we obtain the following Compton amplitude for spin-$3/2$ matter
coupled to photons:

$A(1\phi^{3/2}\\!,2\bar{\phi}^{3/2}\\!,3A^{-}\\!,4A^{+})=\frac{iN_{2}}{m^{4}t_{13}t_{14}}(m^{3}N_{2}(\langle\bm{2}\bm{1}\rangle+[\bm{2}\bm{1}])+m^{2}[4|p_{1}|3\rangle\langle\bm{2}\bm{1}\rangle[\bm{2}\bm{1}]+[3|p_{1}|4\rangle
N_{4})\,.$ (4.71)

The corresponding equal-helicity Compton amplitude for spin-$3/2$ matter
receives no corrections, and hence eq. (4.68a) remains valid.

In the case of spin-$5/2$ matter coupled to gravitons, we obtain a corrected
opposite-helicity Compton amplitude given by

$M(1\phi^{5/2},2\bar{\phi}^{5/2},3h^{-},4h^{+})=iN_{2}\frac{m^{4}Q_{4}+m^{3}Q_{3}+m^{2}Q_{2}+Q_{0}}{m^{6}s_{12}t_{13}t_{14}}\,,$
(4.72)

where we have decomposed the contributions according to powers of the mass;
these are

$\displaystyle Q_{0}=-s_{12}N_{4}^{2}[3|p_{1}|4\rangle\,,$ $\displaystyle
Q_{3}=2N_{2}(\langle\bm{1}\bm{2}\rangle+[\bm{1}\bm{2}])\big{(}[4|p_{1}|3\rangle^{2}(\langle\bm{1}\bm{2}\rangle-[\bm{1}\bm{2}])^{2}+s_{12}N_{4}\big{)}\,,$
(4.73) $\displaystyle
Q_{4}=-2N_{2}^{2}[4|p_{1}|3\rangle(\langle\bm{1}\bm{2}\rangle-[\bm{1}\bm{2}])^{2}\,,$

and

$Q_{2}=\frac{1}{2}[4|p_{1}|3\rangle\left\\{2[4|p_{1}|3\rangle^{2}\langle\bm{1}\bm{2}\rangle[\bm{2}\bm{1}]\left(\langle\bm{1}\bm{2}\rangle^{2}+[\bm{1}\bm{2}]^{2}+3\langle\bm{1}\bm{2}\rangle[\bm{2}\bm{1}]\right)\right.\\\
\left.\hbox{}+[4|p_{1}|3\rangle(\langle\bm{1}\bm{2}\rangle^{2}+[\bm{1}\bm{2}]^{2})\big{(}[\bm{2}|p_{3,4}|\bm{1}\rangle(N^{\prime}_{2}+N_{2})+[\bm{1}|p_{3,4}|\bm{2}\rangle(N^{\prime}_{2}-N_{2})\big{)}\right.\\\
\left.\hbox{}+2N_{4}s_{12}\left(3\langle\bm{1}\bm{2}\rangle^{2}+3[\bm{1}\bm{2}]^{2}+4\langle\bm{1}\bm{2}\rangle[\bm{2}\bm{1}]\right)\right\\}\,,$
(4.74)

where $p_{3,4}\equiv p_{3}-p_{4}$. Again, the equal-helicity Compton amplitude
for spin-$5/2$ matter receives no corrections, and hence eq. (4.69a) remains
valid. Note that, although the opposite-helicity Compton amplitude is much
more complicated than the equal-helicity one, they are both given by the same
covariant expression obtained via the Lagrangian that we give in Section
5.2.5.

Before moving on to derive these expressions, let us comment about the
possibility of including additional contact terms. We first note that the
gauge theory amplitude in eq. (4.71) is dimensionless, and thus any additional
terms must be dimensionless and free of any kinematic poles. This means that
all the spinors need to appear in the numerator; for example, the schematic
minimal expression $|{\bf 1}\rangle^{3}|{\bf
2}\rangle^{3}|3\rangle^{2}|4]^{2}/m^{5}$ has the right little-group weights
and dimension to be a contact term corresponding to an operator quadratic in
field strength $\sim(F_{\mu\nu})^{2}$. However, it clearly has a different
$1/m$ behavior than the terms already present in eq. (4.71), hence it seems
implausible that such contact terms would conspire to further improve the
high-energy behavior or the tree-level unitarity properties of the Compton
amplitude.

The same analysis can be done for the gravitational amplitude (4.72), which is
of dimension two. A schematic minimal expression that has the correct little-
group weights and dimension is $|{\bf 1}\rangle^{5}|{\bf
2}\rangle^{5}|3\rangle^{4}|4]^{4}/m^{7}$, which corresponds to an operator
quadratic in the Riemann tensor $\sim(R_{\mu\nu\rho\sigma})^{2}$. However,
again it has a higher power of $1/m$ than the terms already present in eq.
(4.72) and thus there is no reason to think that the addition of such terms
can improve the high-energy behavior or the tree-level unitarity properties of
the Compton amplitude. For the theories we consider in the next section, we
will see that operators that are quadratic in the field strength or Riemann
tensor do not appear in Lagrangians that have a minimal number of derivatives
and satisfy the current constraint.

##  5 Lagrangians and Compton amplitudes from $P\cdot J={\cal O}(m)$

It is well known from the higher-spin literature that the minimal-coupling
prescription, i.e. only replacing ordinary derivatives with covariant ones, is
not the most natural choice for higher-spin interactions in gauge theory and
gravity. For instance, ref. Ferrara:1992yc  shows the necessity of non-minimal
terms to avoid tree-level unitarity violations, and how this imposes the
familiar value of $g=2$ for the gyromagnetic ratio.

In this section we will discuss the natural Lagrangians for spinning massive
matter, and how they are uniquely fixed by the current constraint $P\cdot
J={\cal O}(m)$ and counting of derivatives. We start with matter with low spin
and work our way up in spin until it exceeds the spin of the gauge bosons
(photon/graviton). While these Lagrangians are far from new, the purpose of
this exercise is to connect the Lagrangians to the three-point and four-point
amplitudes in Sections 3 and 4, which have been argued to describe the
dynamics of Kerr black holes, as well as “root-Kerr” gauge theory solutions
Monteiro:2014cda ; Arkani-Hamed:2019ymq .

Effective Lagrangians appropriate for describing black holes and other massive
objects will in general contain additional higher-derivative terms that behave
as $(\partial/m)^{w}$ for some positive power $w$. The scattering processes
that the effective Lagrangians are geared towards describing include the
Compton amplitude, but also any $n$-point process involving two massive
higher-spin states and $n{-}2$ gravitons (or gluons). This implies that we are
interested in higher-derivative operators that involve two higher-spin fields
and $n{-}2$ factors of field strengths $R_{\mu\nu\rho\sigma}$ (or
$F_{\mu\nu}$) for the massless gauge bosons. Thus, there is a minimum number
of derivatives that an $n$-point amplitude must contain in order for such an
operator to be present. In gravity it is $(\partial/m)^{2n-4}$ and in gauge
theory it is $(\partial/m)^{n-2}$ derivatives. Whether or not one should add
such $n$-point operators to the Lagrangian depends on the derivative counting
that is already present in the amplitude from previously-added operators, as
well as whether there are problems with the minimal-coupling Lagrangian that
the new operators will cure. In the cases that we consider, the tree-level
unitarity violation will be ameliorated by the addition of three-point
operators, which then dictate the derivative counting such that it forbids
possible $n{>}3$ higher-derivative operators. The details will be shown below.
Of course, one could still consider including such operators, but they are not
expected to help with the tree-level unitarity considerations, and instead
they could potentially re-introduce the tree-level unitarity problem. Thus, in
this section we will make the minimalistic choice to not consider operators
that are beyond linear in the field strength, up to spin-$5/2$ gravity and
spin-$3/2$ gauge theory.

###  5.1 Gauge-theory Lagrangians

As a warm-up, we here consider massive spinning fields in a gauge theory, and
we use the current constraint to find the allowed theories with the lowest
number of derivatives. In particular, in Section 3, we showed that the current
constraint can be solved at cubic order with at most $\partial^{2s-1}$
derivatives for a spin-$s$ boson. Assuming that additional quartic
interactions introduce no worse momentum scaling in the four-point amplitude,
this implies an expectation of
$\partial^{2s-1}\frac{1}{\Box}\partial^{2s-1}=\partial^{4s-4}$ derivatives for
the quartic terms. Here we assumed that the propagator schematically behave as
$\frac{1}{\Box}$ (in the high-energy limit), which can only be true if the
current constraint is obeyed, as we will elaborate on shortly. Similarly, for
a spin-$s$ fermion, the current constraint can be solved at cubic order with
at most $\partial^{2s-2}$ derivatives, which implies the expectation of
$\partial^{2s-2}\frac{1}{\partial}\partial^{2s-2}=\partial^{4s-5}$ derivatives
at quartic order.

More generally, we can do the same derivative counting in terms of schematic
new $n$-point operators, under the assumption that their existence is tied to
the non-minimal interactions at three points. For multiplicity $n\geq 2$ and
bosonic spin $s\geq 1$, the derivative counting of Section 3 permits operators

$\Delta{\cal
L}_{n}^{(s)}~{}\sim~{}D^{2(n-2)(s-2)+2}\,\bar{\phi}^{s}\Big{(}\frac{F_{\mu\nu}}{m^{2s-2}}\Big{)}^{n-2}\phi^{s}\,,$
(5.75)

or with fewer derivatives. Here $D$ is the covariant derivative, $\phi^{s}$ is
the higher-spin field, and $F_{\mu\nu}$ is the photon field strength. For
fermions, the same schematic formula is obtained except that the operators
have exactly one derivative less. We see that the number of covariant
derivatives becomes negative for $s\leq 3/2$ and $n\geq 4$, and thus such
operators are not considered.

####  5.1.1 Spin $1$ – the $W$ boson

Let us begin from the simplest case where a current for the matter field can
be first defined, that of a massive charged spin-$1$ field
$W^{\mu}\equiv\phi^{1}$, coupled to the photon $A^{\mu}$. From the general
considerations in Section 3, we expect that the current constraint can be
satisfied by cubic interactions with at most one derivative, and quartic
interactions with zero derivatives. The Lagrangian Ansatz is then

${\cal
L}=2D_{[\mu}\overline{W}_{\nu]}D^{[\mu}W^{\nu]}-m^{2}\overline{W}_{\mu}W^{\mu}+ie\alpha\overline{W}^{\mu}W^{\nu}F_{\mu\nu}\,,$
(5.76)

with covariant derivative $D_{\mu}=\partial_{\mu}-ieA_{\mu}$ and field
strength $F_{\mu\nu}=2\partial_{[\mu}A_{\nu]}$. The free parameter $\alpha$
multiplies the non-minimal interaction, and the choice $\alpha=0$ gives the
minimally-coupled theory. Since quartic operators must come with derivatives
in the form of $F^{2}$ terms, such operators are not allowed by our derivative
counting. For convenience, we work with $e=1$ in the rest of this section.

Let us compute the Compton amplitude, as well as the three-point current. To
do so, we will need the massive spin-$1$ propagator,

$\Delta^{\rho\sigma}_{(1)}(p)=\frac{i}{p^{2}-m^{2}}\Big{(}\eta^{\rho\sigma}-\frac{p^{\rho}p^{\sigma}}{m^{2}}\Big{)}.$
(5.77)

Following ref. Ferrara:1992yc , one notes that this propagator may introduce a
mass singularity in the Compton amplitude. This can be phrased as tree-level
unitarity problem: for momentum $p\gg m$, the amplitude may grow larger than
unity. The problem can be avoided by constraining the off-shell current that
is contracted into the propagator.

Let $J^{\mu}(-P,p_{2},p_{3})$ be the three-point current of the massive
spin-$1$ field, $\mu$ its free Lorentz index and $P=p_{2}+p_{3}$ its off-shell
momentum. The momenta $p_{2}$, $p_{3}$ correspond to an on-shell spin-$1$
boson and an on-shell photon, respectively. Suppose that the current satisfies

$P_{\mu}J^{\mu}(-P,p_{2},p_{3})=m\,X(-P,p_{2},p_{3})\,,$ (5.78)

with $X(-P,p_{2},p_{3})$ bounded as $m\to 0$. With these conditions, the
potentially dangerous mass singularity in the propagator factor $P^{\mu}/m$ is
neutralized when contracted into the current as one builds up the Feynman
diagrams. Hence, the current constraint is a necessary condition for avoiding
the tree-level unitarity violation.

Explicit computation using the Lagrangian (5.76) yields

$P_{\mu}J^{\mu}(-P,p_{2},p_{3})=im^{2}\bm{\mathbf{\varepsilon}}_{2}\cdot\varepsilon_{3}+i(1-\alpha)(\varepsilon_{3}\cdot
P\,\bm{\mathbf{\varepsilon}}_{2}\cdot
p_{3}-\bm{\mathbf{\varepsilon}}_{2}\cdot\varepsilon_{3}\,p_{3}\cdot P)\,,$
(5.79)

which confirms that tree-level unitarity requires $\alpha=1$. Not
surprisingly, this is precisely the value that is required for $W$-bosons in a
spontaneously-broken Yang-Mills theory, such as the Standard Model.

We may confirm that $\alpha=1$ reproduces the expected amplitudes. The three-
point amplitude, with $P\to-p_{1}$ on shell, is given by

${\cal
A}(1W,2\overline{W},3A)=-2i\bm{\mathbf{\varepsilon}}_{1}\cdot\bm{\mathbf{\varepsilon}}_{2}\,\varepsilon_{3}\cdot
p_{1}-i(1+\alpha)\bm{\mathbf{\varepsilon}}_{2\mu}f_{3}^{\mu\nu}\bm{\mathbf{\varepsilon}}_{1\nu}\,,$
(5.80)

where
$f_{3}^{\mu\nu}=p_{3}^{\mu}\varepsilon_{3}^{\nu}-p_{3}^{\nu}\varepsilon_{3}^{\mu}$.
With photon helicity $\varepsilon_{3}=\varepsilon_{3}^{-}$, we have

$A(1W,2\overline{W},3A^{-})=-\frac{(1+\alpha)x^{-1}}{2m}[\bm{1}\bm{2}]^{2}-\frac{(1-\alpha)x^{-1}}{2m}[\bm{1}\bm{2}]\langle\bm{1}\bm{2}\rangle\,,$
(5.81)

which reduces to eq. (3.40) for $\alpha=1$. Furthermore, at four points, one
finds that $\alpha=1$ reproduces exactly the spin-$1$ Compton amplitude in eq.
(4.68). Of course, we could have directly matched the amplitudes to eq. (3.40)
or eq. (4.68), which would have given the same $\alpha=1$ solution. However,
as we will demonstrate with more examples, the current constraint pins down
both the Lagrangians and the amplitudes with very little input.

Before moving on to higher spin, let us discuss some salient points. In the
$m\to 0$ limit, the current constraint (5.78) becomes the Ward identity
corresponding to gauge invariance of a massless spin-$1$ particle. It is not
surprising that the resulting Lagrangian (5.76) is built out of the standard
interaction terms for $W$-bosons, which in the massless limit reduces to non-
abelian Yang-Mills theory. Here we are not interested in the $W^{4}$ contact
terms and the Higgs field that are also present in such a theory because, with
an eye towards the application of classical scattering with very massive
compact objects (black holes in gravity), such terms would correspond to
quantum corrections that are irrelevant. Thus, we are only interested in
Lagrangians that are quadratic in the massive spinning field.

####  5.1.2 Spin $3/2$ – charged gravitino

We now discuss the Lagrangian for photons coupled to a massive charged
spin-$3/2$ field $\psi_{\mu}$, which for convenience we often call
“gravitino”. We follow the approach of ref. Deser:2000dz  and consider the
most general Lagrangian linear in the field strength, which precisely matches
our expectation on the maximum number of allowed derivatives, as discussed in
the beginning of this section. Although the considerations in Section 3 ruled
out parity-odd terms, we include them here for illustrative purposes. The
Lagrangian Ansatz consists of the Rarita-Schwinger Lagrangian Rarita:1941mf
plus non-minimal terms,

${\cal
L}=\bar{\psi}^{\mu}\gamma_{\mu\nu\rho}\Big{(}iD^{\nu}-\frac{1}{2}m\gamma^{\nu}\Big{)}\psi^{\rho}-\frac{ie}{m}\Big{(}l_{1}\bar{\psi}_{\mu}F^{\mu\nu}\psi_{\nu}+l_{2}\bar{\psi}_{\mu}F_{\rho\sigma}\gamma^{\rho}\gamma^{\sigma}\psi^{\mu}+l_{3}F^{\mu\nu}(\bar{\psi}_{\mu}\gamma_{\nu}\gamma\cdot\psi\\\
+\bar{\psi}\cdot\gamma\gamma_{\mu}\psi_{\nu})+l_{4}\bar{\psi}\cdot\gamma
F_{\rho\sigma}\gamma^{\rho}\gamma^{\sigma}\gamma\cdot\psi+il_{5}F^{\mu\nu}(\bar{\psi}_{\mu}\gamma_{\nu}\gamma\cdot\psi-\bar{\psi}\cdot\gamma\gamma_{\mu}\psi_{\nu})\Big{)}\,,$
(5.82)

with the same covariant derivative and field strength as defined in the
previous subsection. The gravitino is described by a vector-spinor field, with
the free equation of motion $\gamma\cdot\psi=(i\not{\partial}-m)\psi_{\mu}=0$.

It is known that the parameter values
$l_{1}=-2,l_{2}=1/2,l_{3}=1,l_{4}=-1/2,l_{5}=0$ uniquely match a truncation of
gauged ${\cal N}=2$ supergravity Freedman:1976aw . As a spoiler, this choice
precisely gives the three-point amplitude (3.40), as well as the Compton
amplitudes (4.68) and (4.71). Furthermore, these values are uniquely fixed by
the property $P\cdot J={\cal O}(m)$. With this said, let us clean up the
Lagrangian corresponding to this parameter choice,

${\cal
L}=\bar{\psi}^{\mu}\gamma_{\mu\nu\rho}\Big{(}iD^{\nu}-\frac{1}{2}m\gamma^{\nu}\Big{)}\psi^{\rho}+\frac{ie}{m}\bar{\psi}_{\mu}{\cal
F}^{\mu\nu}\psi_{\nu}\,,$ (5.83)

where ${\cal F}^{\mu\nu}\equiv
F^{\mu\nu}-i/2\,\gamma^{5}\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$ is a
gamma-matrix-augmented field strength, which is parity even.

Let us confirm the result. Consider the three-point amplitude in the
Lagrangian Ansatz, and note that $l_{3}$, $l_{4}$ and $l_{5}$ do not
contribute due to the gamma-tracelessness of on-shell gravitinos. Assuming
$\varepsilon_{3}=\varepsilon_{3}^{-}$, we have

${\cal
A}(1\phi^{3/2},2\bar{\phi}^{3/2},3A^{-})=i\bar{u}_{2\rho}\not{\varepsilon}_{3}v_{1}^{\rho}+i\frac{l_{1}}{m}\bar{u}_{2\mu}f_{3}^{\mu\nu}v_{1\nu}+i\frac{l_{2}}{m}\bar{u}_{2\mu}f_{3\rho\sigma}\gamma^{\rho}\gamma^{\sigma}v_{1}^{\mu}\,,$
(5.84)

which leads to the spinor-helicity expression

$A(1\phi^{3/2},2\bar{\phi}^{3/2},3A^{-})=-\frac{x^{-1}}{2m^{2}}[\bm{1}\bm{2}]^{3}\left(\\!-2\frac{\langle\bm{1}\bm{2}\rangle}{[\bm{1}\bm{2}]}+l_{1}\left(1\\!-\\!\frac{\langle\bm{1}\bm{2}\rangle^{2}}{[\bm{1}\bm{2}]^{2}}\right)+4l_{2}\left(\frac{\langle\bm{1}\bm{2}\rangle}{[\bm{1}\bm{2}]}\\!-\\!\frac{\langle\bm{1}\bm{2}\rangle^{2}}{[\bm{1}\bm{2}]^{2}}\right)\\!\right)\\!.$
(5.85)

One sees that $l_{1}=-2$ and $l_{2}=1/2$ reproduce eq. (3.40).

Next, consider the calculation of the Compton amplitude. We need the massive
gravitino propagator that we derived in eq. (2.4); here it is in full detail:

$\Delta^{\rho\sigma}_{(3/2)}(p)=\frac{i}{p^{2}-m^{2}}\left(\Big{(}\eta^{\rho\sigma}-\frac{p^{\rho}p^{\sigma}}{m^{2}}\Big{)}\left(\not{p}+m\right)+\frac{1}{3}\Big{(}\frac{p^{\rho}}{m}+\gamma^{\rho}\Big{)}\left(\not{p}-m\right)\Big{(}\frac{p^{\sigma}}{m}+\gamma^{\sigma}\Big{)}\right).$
(5.86)

Similar to the spin-$1$ case, the above propagator introduces mass divergences
in the Compton amplitude. Note that, with the field content considered, we
will not be able to find a special theory that has a finite massless limit. If
this were possible, it would result in a massless spin-$3/2$ field coupled to
the photon, and this is forbidden by various no-go results Johnson:1960vt ;
Velo:1969bt , including the Weinberg-Witten theorem Weinberg:1980kq . The only
known completion of a massless spin-$3/2$ theory is within the context of
supergravity, but that would require that we add a graviton as well as other
fields to the Lagrangian Das:1976ct ; Deser:1977uq ; Freedman:1976aw . For a
review on these topics, see ref. Rahman:2013sta .

Nonetheless, there is a notion of unitarity violation in the Lagrangian (5.82)
that we can cure. A detailed argument can be found in ref. Cucchieri:1994tx ,
here we give a brief summary. Imposing the constraint $P\cdot J={\cal O}(m)$
ensures that the mass divergences due to the longitudinal part of the current
drop out. However, the addition of non-minimal terms in the Lagrangian will in
general introduce explicit poles $1/m$, as can be seen in eq. (5.82), and
hence give new mass divergences in the transverse part of the current
$J_{\bot}$, which automatically obeys $P\cdot J_{\bot}=0$. One can argue that
this is a separate problem that needs to be solved via other methods, such as
introducing new fields, including possibly an infinite tower of higher-spin
fields Cucchieri:1994tx . Such an approach, however, cannot solve the
unitarity violation in the longitudinal current, and hence the current
constraint is yet again to be interpreted as a necessary condition for tree-
level unitarity.

Now we follow the same procedure as in the spin-$1$ case. Let us define the
three-point current $J^{\mu}(-P,p_{2},p_{3})$, where $P$ is off shell, and
$p_{2}$ and $p_{3}$ are on-shell. From eq. (5.82) we compute the current

$J^{\mu}(-P,p_{2},p_{3})v_{P\mu}=\\\
i\bar{u}_{2\mu}\gamma^{\mu\nu\rho}\varepsilon_{3\nu}v_{P\rho}+\frac{il_{1}}{m}f_{3}^{\mu\nu}\bar{u}_{2\mu}v_{P\nu}+\frac{2il_{2}}{m}\bar{u}_{2\mu}\not{p}_{3}\not{\varepsilon}_{3}v_{P}^{\mu}+\frac{il_{3}-l_{5}}{m}f_{3}^{\mu\nu}\bar{u}_{2\mu}\gamma_{\nu}\gamma\cdot
v_{P}\,,$ (5.87)

where $\bar{u}_{2\mu}$ is an on-shell vector-spinor, and $v_{P\mu}$ is an
auxiliary vector-spinor for the off-shell leg. To check the current
constraint, we replace $v_{P\mu}\rightarrow P_{\mu}v_{P}$, where $v_{P}$ is an
off-shell spinor, giving

$J^{\mu}(-P,p_{2},p_{3})P_{\mu}v_{P}=im\varepsilon_{3}\cdot\bar{u}_{2}v_{P}-i(1-l_{3}-il_{5})f_{3}^{\mu\nu}\bar{u}_{2\mu}\gamma_{\nu}v_{P}\\\
+\frac{i}{m}(l_{1}+2l_{3}+2il_{5})f_{3}^{\mu\nu}p_{2\mu}\bar{u}_{2\nu}v_{P}+\frac{i}{m}(2l_{2}-l_{3}-il_{5})k\cdot\bar{u}_{2}\not{\varepsilon_{3}}\not{p}_{3}v_{P}\,.$
(5.88)

To have $P\cdot J={\cal O}(m)$, we need all terms except the first one to
vanish. Since the $l_{i}$ are real parameters, this implies
$l_{1}=-2,l_{2}=1/2,l_{3}=1,l_{4}=-1/2,l_{5}=0$, as promised.

We proceed to compute the Compton amplitude for the truncated supergravity
theory. The ingredients required are the three-point vertex (5.87) and the
propagator (2.4). The final result is best presented in terms of spinor-
helicity notation, and it exactly gives the two formulae already quoted in
eqs. (4.68a) and (4.71) for the equal- and opposite-helicity cases,
respectively. This confirms that the equal-helicity amplitude is not modified
by the contact term that cures the spurious-pole problem of the opposite-
helicity Compton amplitude. Also, by the analysis of Lagrangian properties
(current constraint and derivative counting) we can rule out the presence of
$F^{2}$ operators in a well-behaved and hairless spin-$3/2$ gauge theory. By
“hairless” we mean that there are no physical properties (e.g. Wilson
coefficients) that characterize the spin-$3/2$ particle other than its mass,
spin and charge.

###  5.2 Gravity Lagrangians

We now proceed to analyze gravitational Lagrangians and amplitudes for
spinning matter. We find that imposing $P\cdot J={\cal O}(m)$ on the currents
uniquely fixes the Lagrangians up to spin 5/2, which in turn gives the tree-
point amplitudes in eq. (3.2), as well as the unique Compton amplitudes that
we presented in Section 4.

In analogy with the gauge-theory discussion around eq. (5.75), we will work
with gravitational theories that can satisfy the current constraint with the
fewest number of derivatives in the non-minimal interactions. Given that we
already know the three-point derivative counting from the discussion in
Section 3, we can infer the multiplicity-$n$ counting by assuming that they
give rise to no worse behavior. For spin-$s$ bosons we get that the derivative
counting is compatible with $n$-point operators of the schematic form

$\Delta{\cal
L}_{n}^{(s)}~{}\sim~{}\nabla^{2(n-2)(s-3)+2}\,\bar{\phi}^{s}\Big{(}\frac{R_{\mu\nu\rho\sigma}}{m^{2s-4}}\Big{)}^{n-2}\phi^{s}\,,$
(5.89)

where $\nabla$ is the covariant derivative, $\phi^{s}$ is the higher-spin
field, and $R_{\mu\nu\rho\sigma}$ is the Riemann tensor. For fermions, the
same schematic formula is obtained except that the operators have exactly one
derivative less. The number of derivatives becomes negative for $s\leq 5/2$
and $n\geq 4$, thus they are not considered. For further details on the non-
minimal terms, see ref. Chung:2019duq  where an on-shell action was given for
arbitrary spin up to $n=3$, which matches with the three-point amplitudes of
ref. Arkani-Hamed:2017jhn .

####  5.2.1 Spin $1/2$ – standard fermion

We start by considering the simplest example of a spin-$1/2$ massive field
$\lambda\equiv\phi^{s=1/2}$ coupled to gravity. While this case is somewhat
trivial, it gives a warm-up that highlights how fermions couple to gravity.
Since the spin-$1/2$ current will not have any free Lorentz indices, the
current constraint does not apply. However, it then automatically follows that
the minimally-coupled theory is free of tree-level unitarity problems. For
convenience, we work with $\kappa=1$ in this section.

The minimally-coupled Lagrangian is given by the fermion kinetic and mass
terms, ${\cal L}=\sqrt{-g}\,\bar{\lambda}(i\not{\nabla}-m)\lambda$, which we
expand out for convenience,

$\sqrt{-g}\,\bar{\lambda}(i\not{\nabla}-m)\lambda=\bar{\lambda}(i\not{\partial}-m)\lambda-\frac{i}{2}\bar{\lambda}\gamma^{a}h_{a}^{\mu}\partial_{\mu}\lambda+\frac{i}{8}\bar{\lambda}\gamma^{\mu}\omega^{ab}_{\mu}[\gamma_{a},\gamma_{b}]\lambda+{\cal
O}(h^{2},\eta^{\mu\nu}h_{\mu\nu}).$ (5.90)

We here distinguish between Greek indices $\\{\mu,\nu,\dots\\}$ used for
curved coordinate indices, and Latin indices $\\{a,b,\dots\\}$ used for flat
indices, such as in the tetrad $e_{a}^{\mu}$. However, when there is no
potential for confusion, such as in amplitude and current expressions, we
resort to using Greek indices for flat space contractions.

The covariant derivative is written explicitly as

$\nabla_{\mu}\lambda=\partial_{\mu}\lambda+{1\over
8}\omega^{ab}_{\mu}[\gamma_{a},\gamma_{b}]\lambda\,,$ (5.91)

in terms of the spin connection $({\omega^{ab}})_{\mu}\equiv
e^{a}_{\nu}\nabla_{\mu}e^{b\nu}=(1/2)(\eta^{bc}\Gamma^{a}_{c\mu}-\eta^{ac}\Gamma^{b}_{c\mu})+{\cal
O}(h^{2})$. Gamma matrices with curved indices are defined in terms of the
flat-space gamma matrices and the tetrad as
$\gamma^{\mu}=\gamma^{a}e_{a}^{\mu}$. In turn, the tetrads are expanded in
terms of the fluctuations of the metric $h_{\mu\nu}$ as
$e_{a}^{\mu}=\delta_{a}^{\mu}-(1/2)h_{a}^{\mu}+{\cal O}(h^{2})$. Hence, there
will be additional contributions to the vertices coming from the expansion of
the tetrad fields which dress the flat-space gamma matrices. This corresponds
to the second term in eq. (5.90).

Now we wish to compute the three-point amplitude. The connection one-form term
does not contribute if the graviton is on shell. This is because the symmetry
of the Christoffel symbol and the Clifford algebra relations can be used to
rewrite it in terms of $\eta^{\mu\nu}\Gamma^{\lambda}_{\mu\nu}$ and
$\Gamma^{\mu}_{\mu\nu}$, both vanishing on-shell. The second term instead
gives

${\cal M}(1\phi^{1/2},2\bar{\phi}^{1/2},3h)=\frac{i}{2}\varepsilon_{3}\cdot
p_{1}\,\bar{u}_{2}\not{\varepsilon}_{3}v_{1}\,.$ (5.92)

Assuming the helicity choice $\varepsilon_{3}=\varepsilon_{3}^{-}$, we have

${\cal
M}(1\phi^{1/2},2\bar{\phi}^{1/2},3h^{-})=-i\frac{mx^{-2}}{2}[\bm{1}\bm{2}]\,,$
(5.93)

in agreement with eq. (3.2).

The Compton amplitude can be computed by expanding the kinetic term to second
order in $h^{\mu\nu}$, and also considering the cubic graviton self-
interactions coming from the Einstein-Hilbert action. We will not give the
details of the calculation here, as it has little bearing on the higher-spin
current argument. We can instead rely on BCFW recursion or double copy to
argue that the Compton amplitude must agree with eq. (4.69). For minimally
coupled matter of spin $s\leq 2$, BCFW recursion is expected to give correct
Compton amplitudes with no contributions from infinity of the complex plane
(see e.g. refs. Arkani-Hamed:2008bsc ; Cohen:2010mi ; Arkani-Hamed:2017jhn ;
Johansson:2019dnu ). The double copy is also expected to hold for Compton
amplitudes of minimally coupled massive matter with spin $s\leq 2$ (see e.g.
refs. Chiodaroli:2015rdg ; Chiodaroli:2017ehv ; Chiodaroli:2018dbu ;
Ochirov:2018uyq ; Johansson:2019dnu ; Bautista:2019evw ; Edison:2020ehu ).

####  5.2.2 Spin $1$ – Proca boson

Like the previous subsection, the spin-$1$ case is somewhat trivial, but for
the sake of completeness we discuss it briefly. Consider a massive spin-$1$
field $W_{\mu}\equiv\Phi^{s=1}$ coupled to gravity. The Lagrangian is obtained
by covariantizing the Proca Lagrangian, which is essentially the quadratic
piece of the Lagrangian (5.76),

${\cal
L}=\sqrt{-g}\,\big{(}2\nabla_{[\mu}\overline{W}_{\nu]}\nabla^{[\mu}W^{\nu]}-m^{2}\overline{W}_{\mu}W^{\mu}\big{)}\
.$ (5.94)

Here the covariant derivative is given by

$\nabla_{\mu}W_{\rho}=\partial_{\mu}W_{\rho}-\Gamma^{\lambda}_{\rho\mu}W_{\lambda}\
.$ (5.95)

The current constraint is automatically satisfied in this case. This is not
surprising given that the Proca field can be thought of as a $W$ boson in a
spontaneously-broken U(1) gauge theory in the Stückelberg formulation.

Explicit computation using the Lagrangian (5.94) yields the amplitude

${\cal
M}(1W,2\overline{W},3h)=i\bm{\mathbf{\varepsilon}}_{1}\cdot\bm{\mathbf{\varepsilon}}_{2}\
p_{1}\cdot\varepsilon_{3}\,\varepsilon_{3}\cdot
p_{2}-ip_{1}\cdot\bm{\mathbf{\varepsilon}}_{2}\
\bm{\mathbf{\varepsilon}}_{1}\cdot\varepsilon_{3}\,\varepsilon_{3}\cdot
p_{2}-ip_{2}\cdot\bm{\mathbf{\varepsilon}}_{1}\
p_{1}\cdot\varepsilon_{3}\,\varepsilon_{3}\cdot\bm{\mathbf{\varepsilon}}_{2}\,.$
(5.96)

If we take $\varepsilon_{3}=\varepsilon_{3}^{-}$, we obtain the amplitude

${\cal M}(1W,2\overline{W},3h^{-})=-{i\over 2}x^{-2}[\bm{1}\bm{2}]^{2}\ ,$
(5.97)

in agreement with eq. (3.2). We will not work out the Compton amplitude, since
it was already checked in ref. Johansson:2019dnu  using Feynman rules that it
agrees with the amplitudes (4.69). That reference also showed that the double
copy provides a streamlined method for computing multi-graviton scattering
with massive spin-1 vectors, by means of computing the much simpler quark-
gluon diagrams of QCD Johansson:2015oia ; Johansson:2019dnu .

####  5.2.3 Spin $3/2$ – massive gravitino

We consider the minimally-coupled theory of a massive spin-$3/2$ field
$\psi_{\mu}\equiv\phi^{s=3/2}$ coupled to gravity. The Lagrangian can be read
off from the minimal part of eq. (5.82), and this is simply the
covariantization of the Rarita-Schwinger Lagrangian,

${\cal
L}=\sqrt{-g}\bar{\psi}_{\mu}\gamma^{\mu\rho\sigma}\Big{(}i\nabla_{\rho}-\frac{1}{2}m\gamma_{\rho}\Big{)}\psi_{\sigma}\equiv{\cal
L}_{free}+{\cal L}_{e}+{\cal L}_{\omega}+{\cal
O}(h^{2},\eta^{\mu\nu}h_{\mu\nu})\,,$ (5.98)

where

$\displaystyle{\cal L}_{e}$
$\displaystyle=-\frac{1}{2}\bar{\psi}_{\mu}({h^{\mu}}_{a}\gamma^{a\rho\sigma}+{h^{\rho}}_{a}\gamma^{\mu
a\sigma}+{h^{\sigma}}_{a}\gamma^{\mu\rho
a})i\partial_{\rho}\psi_{\sigma}-\frac{m}{2}\bar{\psi}_{\mu}({h^{\mu}}_{a}\gamma^{a\rho}+{h^{\rho}}_{a}\gamma^{\mu
a})\psi_{\rho}\,,$ $\displaystyle{\cal L}_{\omega}$
$\displaystyle=\frac{i}{8}\bar{\psi}_{\mu}\gamma^{\mu\rho\sigma}\omega^{ab}_{\rho}[\gamma_{a},\gamma_{b}]\psi_{\sigma}=\frac{i}{8}\bar{\psi}_{\mu}\gamma^{\mu\rho\sigma}\Gamma^{a}_{b\rho}[\gamma_{a},\gamma^{b}]\psi_{\sigma}\,.$
(5.99)

Note that, at ${\cal O}(h)$, there is no difference between the Latin and
Greek indices in ${\cal L}_{e}$ and ${\cal L}_{\omega}$, since
${e^{\mu}}_{a}={\delta^{\mu}}_{a}$ at zeroth order. Also note that the above
Lagrangian, in the massless limit, becomes the gravitino Lagrangian in
supergravity Freedman:1976xh ; Deser:1976eh . The covariant derivative here is
given by

$\nabla_{\mu}\psi_{\rho}=\partial_{\mu}\psi_{\rho}+{1\over
8}\omega^{ab}_{\mu}[\gamma_{a},\gamma_{b}]\psi_{\rho}-\Gamma^{\lambda}_{\rho\mu}\psi_{\lambda}\
,$ (5.100)

but the last term drops out due to the antisymmetry of
$\gamma^{\mu\rho\sigma}$.

We wish to compute ${\cal M}(1\phi^{3/2},2\bar{\phi}^{3/2},3h)$, but, having
the Compton amplitude in mind, we can start by keeping the gravitinos off-
shell and derive the three-point Feynman rules in momentum space. The three-
point current contribution from ${\cal L}_{e}$ is given by

$-\frac{i}{2}\\{\varepsilon_{3}\cdot\bar{u}_{2}\gamma(\varepsilon_{3},P,v_{P})+\varepsilon_{3}\cdot
P\gamma(\bar{u}_{2},\varepsilon_{3},v_{P})+\gamma(\bar{u}_{2},P,\varepsilon_{3})\varepsilon_{3}\cdot
v_{P}\\\
+m\varepsilon_{3}\cdot\bar{u}_{2}\gamma(\varepsilon_{3},v_{P})+m\gamma(\bar{u}_{2},\varepsilon_{3})\varepsilon_{3}\cdot
v_{P}\\},$ (5.101)

which for on-shell gravitinos reduces to

$\frac{i}{2}\varepsilon_{3}\cdot
p_{1}\bar{u}_{2\rho}\not{\varepsilon}_{3}v_{1}^{\rho}+\frac{i}{2}p_{3}\cdot\bar{u}_{2}\not{\varepsilon}_{3}\varepsilon_{3}\cdot
v_{1}\,.$ (5.102)

Similarly we can find the contribution from ${\cal L}_{\omega}$,

$\frac{i}{8}\bar{u}_{2\mu}(\gamma^{\mu}\varepsilon_{3}^{\nu}-\gamma^{\nu}\varepsilon_{3}^{\mu})f_{3ab}\gamma^{a}\gamma^{b}v_{P\nu}\,,$
(5.103)

which on-shell reduces to

$-\frac{i}{2}\varepsilon_{3}\cdot\bar{u}_{2}\not{\varepsilon}_{3}p_{3}\cdot
v_{1}\,.$ (5.104)

The graviton is on-shell in all the above expressions. Summing the
contributions in eq. (5.102) and eq. (5.104) yields

${\cal M}(1\phi^{3/2},2\bar{\phi}^{3/2},3h)=\frac{i}{2}\varepsilon_{3}\cdot
p_{1}\bar{u}_{2\rho}\not{\varepsilon_{3}}v_{1}^{\rho}+\frac{i}{2}f_{3\rho\sigma}\bar{u}_{2}^{\rho}\not{\varepsilon}_{3}v_{1}^{\sigma}\,.$
(5.105)

Assuming $\varepsilon_{3}=\varepsilon_{3}^{-}$, we have

$\frac{i}{2}\varepsilon_{3}\cdot
p_{1}\bar{u}_{2\rho}\not{\varepsilon_{3}}v_{1}^{\rho}+\frac{i}{2}f_{3\rho\sigma}\bar{u}_{2}^{\rho}\not{\varepsilon}_{3}v_{1}^{\sigma}=i\frac{x^{-2}}{2m}[\bm{1}\bm{2}]^{3}\,,$
(5.106)

matching eq. (3.2). Now we wish to check the property $P\cdot J={\cal O}(m)$.
We consider the current $J_{\mu}(-P,p_{2},p_{3})$ with two massive gravitinos
of momenta $-P$ and $p_{2}$ and one graviton of momentum $p_{3}$. From eq.
(5.101) and eq. (5.103) we have:

$J^{\mu}(-P,p_{2},p_{3})v_{P\mu}=\frac{i}{2}\Big{(}f_{3\mu\nu}\bar{u}_{2}^{\mu}\gamma^{\nu}\varepsilon_{3}\cdot
v_{P}-\varepsilon_{3}\cdot
p_{2}\bar{u}_{2\rho}\not{\varepsilon}_{3}v_{P}^{\rho}+\varepsilon_{3}\cdot\bar{u}_{2}\not{\varepsilon}_{3}P\cdot
v_{P}\\\
-\varepsilon_{3}\cdot\bar{u}_{2}\not{\varepsilon}_{3}\not{p_{3}}\not{v}_{P}-\frac{1}{2}\varepsilon_{3}\cdot\bar{u}_{2}\gamma^{\nu}\not{p}_{3}\not{\varepsilon}_{3}v_{P\nu}\Big{)}\,,$
(5.107)

where $v_{P}^{\nu}$ is a reference vector-spinor, and hence:

$J^{\mu}(-P,p_{2},p_{3})P_{\mu}v_{P}=\frac{im}{4}\varepsilon_{3}\cdot\bar{u}_{2}(\not{P}+m)\not{\varepsilon}_{3}v_{P}\,,$
(5.108)

where $v_{P}$ is a reference spinor.

The remaining task is to compute the Compton amplitude, and show that it
equals eq. (4.69). However, this is a standard text-book calculation,
equivalent to that of considering a massive gravitino amplitude in a
spontaneously broken supergravity, so we will not do it explicitly. We use an
indirect argument instead. Due to the property (5.108), the gravitino
propagator is not responsible for any $1/m$ poles in the amplitude. Any such
poles must come from the longitudinal part of the polarization vectors, at
most one $1/m$ factor for each external gravitino. Thus the Compton amplitude
obtained from the Lagrangian (5.98) at worst scale as $1/m^{2}$. Any local
ambiguity of the Compton amplitude must correspond to a gauge-invariant
operator, which on shell is a polynomial in the spinors. For example, consider
a schematic local expression $|{\bf 1}\rangle^{3}|{\bf
2}\rangle^{3}|3\rangle^{4}|4]^{4}/m^{5}$, which has the correct dimension at
the expense of too many powers of $1/m$. Thus eq. (4.69) is the only possible
answer compatible with the Lagrangian (5.98). As can be seen, the Compton
amplitudes (4.69) have in fact no $1/m$ poles, because in the $m\rightarrow 0$
limit there is a hidden symmetry that protects it, namely, supersymmetry.

####  5.2.4 Spin $2$ – Kaluza-Klein graviton

In this subsection, we discuss a convenient way to obtain the Lagrangian for a
massive spin-2 field $H_{\mu\nu}\equiv\phi^{s=2}$. The starting point is the
Einstein-Hilbert action in one dimension higher,

$S_{EH}=-2\\!\int\\!d^{5}x\,\sqrt{-g}\,R\,.$ (5.109)

The action is expanded in terms of the graviton field as
$g_{\hat{\mu}\hat{\nu}}=\eta_{\hat{\mu}\hat{\nu}}+h_{\hat{\mu}\hat{\nu}}$.
Upon dimensional reduction to four dimensions, the five-dimensional graviton
yields a rank-two symmetric tensor, a vector and a scalar,

$h_{\hat{\mu}\hat{\nu}}=\big{\\{}h_{\mu\nu}\ ,\ a_{\mu}\equiv h_{\mu 5}\ ,\
\phi\equiv h_{55}\big{\\}}\ .$ (5.110)

Masses are associated to the five-dimensional components of momenta. Massive
and massless modes can be separated as

$h_{\hat{\mu}\hat{\nu}}=h^{0}_{\mu\nu}+\sum_{m}H^{(m)}_{\mu\nu}\,.$ (5.111)

The summation runs over the whole Kaluza-Klein tower of states and involves an
infinite number of massive modes. Having Compton amplitudes in mind, we only
keep terms in the Lagrangian that have two massive fields with equal mass (up
to a sign).999Note that this is not a consistent truncation in the
conventional sense as general amplitudes from the truncated Lagrangian cannot
be obtained from the full Kaluza-Klein theory by simply inserting projectors
on the external states. However, for the subsector that contains the Compton
amplitude, as well as the multi-graviton generalizations of the Compton
amplitude, the truncation can be done consistently. We now indicate the
massive spin-$2$ fields as ${\overline{H}}_{\mu\nu}$,$H_{\mu\nu}$ and massive
vectors and scalars as $A_{\mu},\bar{A}_{\mu}$ and $\Phi,\overline{\Phi}$.

Using the equivalent of the $R_{\xi}$ unitarity gauge, we introduce a gauge-
fixing term of the form

${\cal
L}_{\text{gf}}=-{2\over\xi}\Big{|}\partial_{\nu}H^{\nu}_{\mu}-\partial_{\mu}H^{\nu}_{\nu}-im\xi
A_{\mu}-{1\over
2}\partial_{\mu}\Phi\Big{|}^{2}-{2\over\xi}\Big{|}\partial_{\mu}A^{\mu}+{3\over
2}imH^{\mu}_{\mu}-{i\over 2}m\xi\Phi\Big{|}^{2}\,.$ (5.112)

Additionally, the spin-2 field is redefined as

$H_{\mu\nu}\rightarrow H_{\mu\nu}+{1\over 2}\eta_{\mu\nu}\Phi\,.$ (5.113)

With these, the kinetic term becomes diagonal in the different fields. In the
$\xi\rightarrow\infty$ limit, vector and scalar become infinitely massive and
can be integrated out (in fact, these fields contain no physical states, they
are pure gauge). By carrying out an explicit calculation, we find that the
Compton amplitude from Kaluza-Klein gravity is reproduced by the simple
Lagrangian

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\sqrt{-g}\,\big{(}\nabla_{\mu}{\overline{H}}_{\nu\rho}\nabla^{\mu}H^{\nu\rho}-2\nabla_{\nu}{\overline{H}}_{\mu}^{\nu}\nabla^{\rho}H^{\mu}_{\rho}-{\overline{H}}^{\rho}_{\rho}\nabla_{\mu}\nabla_{\nu}H^{\mu\nu}-H^{\rho}_{\rho}\nabla_{\mu}\nabla_{\nu}{\overline{H}}^{\mu\nu}$
(5.114)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}-\nabla_{\mu}{\overline{H}}^{\nu}_{\nu}\nabla^{\mu}H_{\rho}^{\rho}-m^{2}{\overline{H}}_{\mu\nu}H^{\mu\nu}+m^{2}{\overline{H}}^{\mu}_{\mu}H^{\nu}_{\nu}-2R^{\mu\nu\rho\sigma}{\overline{H}}_{\mu\rho}H_{\nu\sigma}\big{)}\,.$

This is a complex version of the Fierz-Pauli Lagrangian Fierz:1939ix  with the
addition of a non-minimal coupling proportional to the Riemann tensor. It is
instructive to consider the case in which the non-minimal term has a free
coefficient,

${\cal L}_{\text{non-
min}}=-2\alpha\sqrt{-g}\,R^{\mu\nu\rho\sigma}{\overline{H}}_{\mu\rho}H_{\nu\sigma}\,.$
(5.115)

A direct calculation of the amplitude at three points leads to

${\cal M}(1H,2{\overline{H}},3h^{-})=i{x^{-2}\over 2m^{2}}[{\bf
12}]^{4}\Bigg{\\{}1+(\alpha-1)\Big{(}1-{\langle{\bf 12}\rangle\over[{\bf
12}]}\Big{)}^{2}\Bigg{\\}}\,.$ (5.116)

This reproduces the three-point amplitude from Arkani-Hamed, Huang and Huang
provided that $\alpha=1$ Arkani-Hamed:2017jhn .

The same Lagrangian, up to total-derivative terms, can be obtained by
calculating the longitudinal part of the off-shell current

$P^{\nu}J_{\mu\nu}(-P,p_{2},p_{3})=i{(\alpha-1)}\big{(}P^{2}\,{\bm{\mathbf{\varepsilon}}}_{2}\cdot\varepsilon_{3}+2P\cdot{\bm{\mathbf{\varepsilon}}}_{2}\,P\cdot\varepsilon_{3}\big{)}\big{(}p_{3}\cdot{\bm{\mathbf{\varepsilon}}}_{2}\
\varepsilon_{3\mu}-p_{3\mu}\,{\bm{\mathbf{\varepsilon}}}_{2}\cdot\varepsilon_{3}\big{)}+{\cal
O}(m^{2}).$ (5.117)

Imposing that the current vanishes as $m^{2}$ in the $m\rightarrow 0$ limit
fixes again $\alpha=1$. Starting with a more general Ansatz in which all terms
in the Lagrangian (5.114) have free coefficients, this condition fixes
completely the theory up to terms proportional to the mass.

The Compton amplitude can be calculated using the Feynman rules coming from
the Lagrangian (5.114). But it is easier to use the Kaluza-Klein trick
directly on a massless graviton amplitude. Starting from a covariant four-
graviton amplitude, two of the external momenta are chosen as
$p^{\hat{\mu}}_{1}=(p^{\mu}_{1},m)$, $p^{\hat{\mu}}_{2}=(p^{\mu}_{2},-m)$, and
the remaining momenta $p_{3},p_{4}$, as well as all the polarizations, are
chosen to only have four-dimensional components. After plugging in these
momenta and polarizations into the amplitude, and converting to spinor-
helicity notation, the answer agrees with eq. (4.69).

####  5.2.5 Spin 5/2

The Lagrangian for a spin-$5/2$ field minimally coupled to gravity can be
written in terms of a symmetric tensor-spinor
$\psi_{\mu\nu}\equiv\phi^{s=5/2}$ and an auxiliary spinor $\lambda$,

$e^{-1}{\cal L}_{\rm
min}=\bar{\psi}_{\mu\nu}(i\not{\nabla}-m)\psi^{\mu\nu}+2\bar{\psi}_{\mu\nu}\gamma^{\nu}(i\not{\nabla}+m)\gamma^{\rho}\psi_{\rho}^{\mu}-\frac{1}{2}\bar{\psi}_{\mu}^{\mu}(i\not{\nabla}-m)\psi_{\rho}^{\rho}\\\
~{}~{}~{}~{}~{}~{}-(2\bar{\psi}^{\rho\mu}i\nabla_{\rho}\gamma^{\nu}\psi_{\mu\nu}+2\bar{\psi}_{\mu\nu}\gamma^{\nu}i\nabla_{\rho}\psi^{\rho\mu})+(\bar{\psi}_{\mu}^{\mu}i\nabla_{\rho}\gamma_{\sigma}\psi^{\rho\sigma}+\bar{\psi}^{\rho\sigma}\gamma_{\sigma}i\nabla_{\rho}\psi_{\mu}^{\mu})\\\
+m(\bar{\psi}_{\mu}^{\mu}\lambda+\bar{\lambda}\psi_{\mu}^{\mu})-\frac{12}{5}\bar{\lambda}(i\not{\nabla}+3m)\lambda\,,$
(5.118)

where $e=\sqrt{-g}$ and the covariant derivatives are

$\displaystyle\nabla_{\mu}\psi_{\rho\sigma}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\psi_{\rho\sigma}+{1\over
8}\omega^{ab}_{\mu}[\gamma_{a},\gamma_{b}]\psi_{\rho\sigma}-\Gamma^{\lambda}_{\rho\mu}\psi_{\lambda\sigma}-\Gamma^{\lambda}_{\sigma\mu}\psi_{\rho\lambda}\
,$ (5.119) $\displaystyle\nabla_{\mu}\lambda$ $\displaystyle=$
$\displaystyle\partial_{\mu}\lambda+{1\over
8}\omega^{ab}_{\mu}[\gamma_{a},\gamma_{b}]\lambda\,.$ (5.120)

This Lagrangian can be derived from the one by Singh and Hagen Singh:1974rc
via a field redefinition. The free equations of motion are given by
$(i\not{\partial}-m)\psi_{\mu\nu}=\gamma^{\mu}\psi_{\mu\nu}=\partial^{\mu}\psi_{\mu\nu}=\lambda=0$.
In addition, by taking $m\to 0$ in the free Lagrangian one recovers a gauge
symmetry $\delta\psi_{\mu\nu}=\partial_{(\mu}\xi_{\nu)}$. Note that this is a
restricted gauge symmetry, since the spinor-vector gauge variable $\xi_{\mu}$
must satisfy the conditions $\gamma\cdot\xi=0$. This implies a modification of
the $P\cdot J={\cal O}(m)$ condition in the gravitational current, as we will
see shortly.

The propagator can be written as

$\epsilon_{\mu}\epsilon_{\nu}\Delta_{(5/2)}^{\mu\nu,\rho\sigma}\bar{\epsilon}_{\rho}\bar{\epsilon}_{\sigma}=\frac{i}{p^{2}-m^{2}}\left(P_{(5/2)}+(p^{2}-m^{2})\Delta^{(2)}\right)\,,$
(5.121)

where for convenience we have contracted the Lorentz indices with auxiliary
polarizations, following the discussion in Section 2. The first term of the
propagator corresponds to the on-shell spin-$5/2$ projector that we computed
in that section; we spell it out here in full detail

$P_{(5/2)}\\!=\\!(\not{p}+m)\Big{(}\\!P_{(1)}(\epsilon,\\!\bar{\epsilon})^{2}-\frac{1}{5}P_{(1)}(\epsilon,\\!\epsilon)P_{(1)}(\bar{\epsilon},\\!\bar{\epsilon})\\!\Big{)}\\!+\frac{2}{5}P_{(1)}(\epsilon,\\!\bar{\epsilon})\Big{(}\\!\not{\epsilon}\\!+\\!\frac{\epsilon\\!\cdot\\!p}{m}\Big{)}\\!(\not{p}-m)\\!\Big{(}\\!/\\!\\!\\!\bar{\epsilon}\\!+\\!\frac{\bar{\epsilon}\\!\cdot\\!p}{m}\Big{)},$
(5.122)

where
$P_{(1)}(\epsilon,\epsilon^{\prime})\equiv\epsilon\cdot\epsilon^{\prime}-(\epsilon\cdot
p)(\epsilon^{\prime}\cdot p)/m^{2}$ is the spin-$1$ projector. As expected,
the propagator becomes proportional to the spin-$5/2$ projector for
$p^{2}=m^{2}$. The second term depends on the details of the Lagrangian
(5.118), and in our case it is given by

$\displaystyle\Delta^{(2)}$ $\displaystyle=$
$\displaystyle\frac{\epsilon^{2}\bar{\epsilon}^{2}(\not{p}+13m)}{1280m^{2}}+\frac{\epsilon\cdot
p\,\bar{\epsilon}\cdot
p\,\not{\epsilon}(\not{p}+5m)/\\!\\!\\!\bar{\epsilon}}{80m^{4}}-\frac{\epsilon\cdot
p\,\bar{\epsilon}^{2}{\not{\epsilon}}(\not{p}+9m)}{160m^{3}}+\frac{(\bar{\epsilon}\cdot
p)^{2}\epsilon^{2}}{40m^{3}}$ (5.123)
$\displaystyle\hbox{}-\frac{(\bar{\epsilon}\cdot p)^{2}\,\epsilon\cdot
p{\not{\epsilon}}}{10m^{4}}+\big{\\{}\bar{\epsilon}\leftrightarrow\epsilon,\gamma^{\mu_{1}}{\cdots}\gamma^{\mu_{n}}\leftrightarrow\gamma^{\mu_{n}}{\cdots}\gamma^{\mu_{1}}\big{\\}}\,.$

The $\Delta^{(2)}$ term has important implications on the current constraint:
contrary to all the previous cases, not all mass divergences in the propagator
come with a factor of $p^{\mu}$ contracting into the vertices. Hence, we might
expect a modification of the $P\cdot J={\cal O}(m)$ condition. The correct
prescription turns out to be $P_{\mu}J^{\mu\nu}v_{P\nu}={\cal O}(m)$, where
$v_{P\nu}$ is a reference vector-spinor satisfying $\gamma\cdot v_{P}=0$,
recovering the Ward identity for restricted gauge invariance in the massless
limit Cucchieri:1994tx .

Therefore, we wish to add a non-minimal term to the Lagrangian (5.118), such
that we get $P_{\mu}J^{\mu\nu}v_{P\nu}={\cal O}(m)$ with $\gamma\cdot
v_{P}=0$. As we will see, this requires Cucchieri:1994tx

${\cal L}_{\text{non-min}}=-\frac{1}{m}\sqrt{-g}\,\bar{\psi}_{\mu\rho}{\cal
R}^{\mu\nu\rho\sigma}\psi_{\nu\sigma}\,,$ (5.124)

where ${\cal
R}^{\mu\nu\rho\sigma}=R^{\mu\nu\rho\sigma}-(i/2)\gamma^{5}\epsilon^{\rho\sigma\alpha\beta}{R^{\mu\nu}}_{\alpha\beta}$
is a gamma-matrix-augmented Riemann tensor, and ${R^{\mu}}_{\nu\rho\sigma}$ is
the standard Riemann tensor. The full Lagrangian is thus given by ${\cal
L}={\cal L}_{\text{min}}+{\cal L}_{\text{non-min}}$. Interestingly, the non-
minimal term is analogous to the one that appeared in gauge theory, eq.
(5.83), which is consistent with the observed double-copy structure of the
three-point amplitudes.

To check that the above is the theory we want, we start by computing the
three-point amplitude. Note that only the first term in eq. (5.118)
contributes, as well as the non-minimal term, since the others are zero on-
shell. The result is

${\cal M}(1\phi^{5/2},2\bar{\phi}^{5/2},3h)=\frac{i}{2}\varepsilon_{3}\cdot
p_{1}\,\bar{u}_{2\mu\kappa}\not{\varepsilon}_{3}v_{1}^{\mu\kappa}+if_{3}^{\kappa\lambda}\bar{u}_{2\mu\kappa}\not{\varepsilon}_{3}{v_{1}^{\mu}}_{\lambda}-\frac{i}{2m}f_{3}^{\kappa\lambda}\bar{u}_{2\mu\kappa}\mathfrak{F}_{3}^{\mu\nu}v_{1\nu\lambda}\,,$
(5.125)

where $\mathfrak{F}^{\mu\nu}_{j}\equiv
f^{\mu\nu}_{j}-(i/2)\gamma^{5}\epsilon^{\mu\nu\rho\sigma}f_{j\rho\sigma}$.

Using the on-shell identity
$\bar{u}_{2\mu}\mathfrak{F}_{3}^{\mu\nu}v_{1\nu}\,\bm{\mathbf{\varepsilon}}_{1}{\cdot}\bm{\mathbf{\varepsilon}}_{2}\,\varepsilon_{3}{\cdot}p_{1}{=}-m\,\bar{u}_{2\rho}\not{\varepsilon}_{3}v_{1}^{\rho}\,\bm{\mathbf{\varepsilon}}_{2\mu}f_{3}^{\mu\nu}\bm{\mathbf{\varepsilon}}_{1\nu}$,
this can be written as

$\displaystyle{\cal M}(1\phi^{5/2},2\bar{\phi}^{5/2},3h)$ $\displaystyle=$
$\displaystyle\frac{i}{2}(\bar{u}_{2\rho}\not{\varepsilon}_{3}v_{1}^{\rho}-\frac{1}{m}\bar{u}_{2\mu}\mathfrak{F}_{3}^{\mu\nu}v_{1\nu})(\bm{\mathbf{\varepsilon}}_{1}\cdot\bm{\mathbf{\varepsilon}}_{2}\,\varepsilon_{3}\cdot
p_{1}+\bm{\mathbf{\varepsilon}}_{2\mu}f_{3}^{\mu\nu}\bm{\mathbf{\varepsilon}}_{1\nu})$
(5.126) $\displaystyle=$ $\displaystyle-{i\over
2}A(1\phi^{3/2},2\bar{\phi}^{3/2},3A)A(1\phi^{1},2\bar{\phi}^{1},3A)\,.$

The right-hand side is a product of the amplitudes of eq. (3.40) for spins 1
and $3/2$, hence by double-copy the ${\cal
M}(1\phi^{5/2},2\bar{\phi}^{5/2},3h)$ amplitude will match eq. (3.2).

As we did in the spin-2 case, we can also consider a one-parameter family of
Lagrangians where the non-minimal term is given by

${\cal L}_{\text{non-
min}}=-\frac{\alpha}{m}\sqrt{-g}\,\bar{\psi}_{\mu\rho}{\cal
R}^{\mu\nu\rho\sigma}\psi_{\nu\sigma}\,.$ (5.127)

In this case, the three-point amplitude is

${\cal M}(1\phi^{5/2},2\bar{\phi}^{5/2},3h)=-{i\over 2}{x^{-2}\over
m^{3}}[{\bf 12}]^{5}\left\\{1+(\alpha-1)\Big{(}1-{\langle{\bf
12}\rangle\over[{\bf 12}]}\Big{)}^{2}\right\\}\,.$ (5.128)

Then we proceed to checking the $P_{\mu}J^{\mu\nu}v_{P\nu}={\cal O}(m)$
property. The current is easiest written with indices contracted with an off-
shell spinor-tensor $v_{P\mu\nu}$,

$J^{\mu\nu}(-P,p_{2},p_{3})v_{P\mu\nu}=-\frac{i}{2}\varepsilon_{3}\cdot
p_{2}\,\bar{u}_{2}^{\mu\nu}\not{\varepsilon}_{3}v_{P\mu\nu}-\frac{i}{2}p_{3\mu}\varepsilon_{3\nu}\bar{u}_{2}^{\mu\nu}\not{\varepsilon}_{3}v_{P\rho}^{\rho}+i\varepsilon_{3}\cdot\bar{u}_{2}^{\mu}\not{\varepsilon}_{3}P\cdot
v_{P\mu}\\\ \hskip 54.06006pt+2i\varepsilon_{3}\cdot
p_{2}\,\varepsilon_{3}\cdot\bar{u}_{2}^{\mu}\gamma\cdot
v_{P\mu}-i\varepsilon_{3\mu}\varepsilon_{3\nu}\bar{u}_{2}^{\mu\nu}\gamma_{\sigma}v_{P}^{\rho\sigma}p_{2\rho}-i\varepsilon_{3}\cdot\bar{u}_{2}^{\mu}\not{p}_{3}\varepsilon_{3}\cdot
v_{P\mu}\\\ \hskip
31.2982pt+\frac{i}{2}\varepsilon_{3\mu}\varepsilon_{3\nu}\bar{u}_{2}^{\mu\nu}\not{p}_{3}v_{P\rho}^{\rho}-im\varepsilon_{3}\cdot\bar{u}_{2}^{\mu}\not{\varepsilon}_{3}\gamma\cdot
v_{P\mu}-i\varepsilon_{3}\cdot\bar{u}_{2}^{\mu}\not{\varepsilon}_{3}\not{P}\gamma\cdot
v_{P\mu}\\\ \hskip
73.97733pt-\frac{i}{2}\varepsilon_{3}\cdot\bar{u}_{2}^{\mu}\not{p}_{3}\not{\varepsilon}_{3}\gamma\cdot
v_{P\mu}+if_{3}^{\rho\sigma}\bar{u}_{2\mu\rho}\not{\varepsilon}_{3}v_{P\sigma}^{\mu}-\frac{i}{2m}f_{3}^{\mu\nu}\bar{u}_{2\mu\kappa}\mathfrak{F}_{3}^{\kappa\lambda}v_{P\nu\lambda}\,,$
(5.129)

with $P=p_{2}+p_{3}$. We should now check the behavior of the current
contracted into its momentum, $P_{\mu}J^{\mu\nu}v_{P\nu}$. Let us quote an
identity that we need to use,

$\frac{1}{m}f_{3}^{\kappa\lambda}P_{\lambda}\bar{u}_{2\mu\kappa}\mathfrak{F}_{3}^{\mu\nu}=f_{3}^{\kappa\lambda}f_{3}^{\mu\nu}\bar{u}_{2\mu\kappa}\gamma_{\lambda}\
,$ (5.130)

which is valid for on-shell $p_{2}$ and $p_{3}$. This identity follows from
the Gram-determinant expression given in eq. (3.58), and hence it is an
identity specific to four dimensions. We get that the contracted current is

$J^{\mu\nu}(-P,p_{2},p_{3})P_{\mu}v_{P\nu}=-\frac{im}{2}\varepsilon_{3\mu}\varepsilon_{3\nu}\bar{u}_{2}^{\mu\nu}p_{2}\cdot
v_{P}-\frac{im}{4}\varepsilon_{3}\cdot\bar{u}_{2}^{\mu}\not{\varepsilon}_{3}(\not{p}_{3}{-}\,2m)v_{P\mu}\,,$
(5.131)

provided that the residual off-shell vector-spinor is gamma-tracless,
$\gamma\cdot v_{P}=0$. Therefore the theory satisfies the required current
constraint.

All that is left to do is to compute the Compton amplitude. We start from the
massive channel diagrams. The first one is the exchange of the physical
spin-$5/2$ field shown in Figure 1, and it can be computed via the Feynman
rules in eq. (5.121) and eq. (5.129). There are also diagrams involving the
auxiliary field $\lambda$, shown in Figure 2. Here we need the auxiliary field
propagator $\Delta_{\rm aux}$, the three-point current $J_{\rm aux}$ between
the fields $\\{h,\psi,\lambda\\}$, and the two-point vertex $V^{(2)}_{\rm
aux}$ between $\psi$ and $\lambda$.101010The auxiliary-field propagator is
simply the inverse of the term quadratic in $\lambda$ in the Lagrangian. This
is different from eq. (5.121), which is obtained by resumming all auxiliary-
field insertions. These are given below:

Figure 1:  Feynman diagram with an exchange of the massive spin-$5/2$ field.

$\begin{split}&\Delta_{\rm
aux}(p)=-\frac{5i}{12}\frac{\not{p}-3m}{p^{2}-9m^{2}}\,,\\\ &J_{\rm
aux}(-P,p_{2},p_{3})=-im\varepsilon_{3\mu}\varepsilon_{3\nu}\bar{u}_{2}^{\mu\nu}v^{(\lambda)}_{P}\,,\\\
&V^{(2)}_{\rm
aux}(p)=im(\bar{u}_{p})_{\rho}^{\rho}v^{(\lambda)}_{p}\,.\end{split}$ (5.132)

Note that the propagator $\Delta_{\rm aux}$ introduces spurious poles;
however, these cancel out in the sum of all diagrams.

Next, we compute the graviton-exchange diagram. This is simpler, since the
$\psi$ field is on-shell and the $\psi$ propagator does not appear. Aside from
the usual graviton Feynman rules, we need the three-point current $J_{\rm
grav}$ between the fields $\\{\bar{\psi},\psi,h\\}$, the latter being off-
shell with momentum $q^{\mu}=-p_{1}^{\mu}-p_{2}^{\mu}$,

$J_{\rm