aainstitutetext: Department of Physics and Astronomy, Uppsala University, Box
516, 75120 Uppsala, Sweden,bbinstitutetext: Nordita, Stockholm University and
KTH Royal Institute of Technology,
Hannes Alfvéns väg 12, 10691 Stockholm, Sweden,ccinstitutetext: London
Institute for Mathematical Sciences, 21 Albemarle Street, London W1S 4BS,
UK,ddinstitutetext: School of Physical Science and Technology, ShanghaiTech
University,
393 Middle Huaxia Road, Shanghai 201210, China,eeinstitutetext: Centre for
Theoretical Physics, Department of Physics and Astronomy,
Queen Mary University of London, Mile End Road, London E1 4NS,
UK,ffinstitutetext: Service de Physique de l’Univers, Champs et Gravitation,
Université de Mons, 20 place du Parc, 7000 Mons, Belgium,gginstitutetext:
Lebedev Institute of Physics, Leninsky avenue 53, 119991 Moscow, Russia

# From higher-spin gauge interactions to Compton amplitudes for root-Kerr

Lucile Cangemi a Marco Chiodaroli a,b Henrik Johansson c,d Alexander Ochirov
a,e Paolo Pichini f,g and Evgeny Skvortsov<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We develop massive higher-spin theory as a framework for describing dynamics
of rotating compact objects, such as Kerr black holes. In this paper, we
explore gauge interactions up to quartic order and corresponding Compton
amplitudes of higher-spin massive objects coupled to electromagnetism and
Yang-Mills theory. Their classical counterparts are known as root-Kerr gauge-
theory solutions, whose amplitudes are closely related to those of Kerr black
holes. We use three distinct approaches: (i) massive higher-spin gauge
symmetry to introduce cubic interactions for all spins and the quartic
interactions up to spin 3, which is implemented both off shell and via Ward
identities; (ii) a chiral higher-spin approach to construct quartic
Lagrangians with correct degrees of freedom to all spins; (iii) on-shell
functional patterns before and after taking the classical limit to constrain
the Compton amplitudes. As final results, we arrive at simple local formulae
for the candidate root-Kerr Compton amplitudes both in the quantum regime and
classical limit, to all orders in spin. This is a precursor to the
gravitational Kerr case, which is presented in a follow-up paper.

††preprint: UUITP-35/23

##  1 Introduction

Higher-spin theory aims to describe interacting particles with spin quantum
numbers exceeding those of the fundamental force carriers — gauge bosons and
gravitons. At the fundamental level, the existence of such particles seems
crucial for theories of quantum gravity, such as string theory (for a review
see e.g. ref. Bekaert:2022poo ). More relevant to this work, theories with
massive higher-spin fields in Minkowski background are often understood as
effective field theories (EFTs) valid below certain energy scales
Ferrara:1992yc ; Cucchieri:1994tx ; Klishevich:1997pd ; Giannakis:1998wi ;
Buchbinder:1999be ; Buchbinder:2000ta ; Bekaert:2003uc ; Zinoviev:2001dt ;
Zinoviev:2006im ; Zinoviev:2008ck ; Francia:2007ee ; Metsaev:2007rn ;
Francia:2008ac ; Metsaev:2012uy ; Hassan:2012wr ; Cortese:2013lda ;
Bernard:2015uic ; Fukuma:2016rru ; Bonifacio:2018aon ; Afkhami-Jeddi:2018apj ;
Kaplan:2020ldi .

In recent years, the application of scattering-amplitude methods to
gravitational-wave physics (see e.g. the review Buonanno:2022pgc ) has
provided a new arena for higher-spin theory. Indeed, the long-range physics of
compact objects can be captured by spinning point particles. For example,
consider a three-point amplitude involving two particles of spin $s$ and mass
$m$, and a massless force carrier. Its dependence on the $2s{+}1$ physical
spin states of massive particle $\bm{1}$ may be encoded in a symmetrized set
$\\{a_{1},\dots,a_{2s}\\}$ of ${\rm SU}(2)$ indices, and likewise for particle
$\bm{2}$. A particularly interesting infinite family of such amplitudes,
proposed by Arkani-Hamed, Huang and Huang (AHH) Arkani-Hamed:2017jhn , may be
written in terms of massive Weyl spinors Conde:2016vxs ; Arkani-Hamed:2017jhn
(related to the momenta via
$|p^{a}\rangle_{\alpha}[p_{a}|_{\dot{\beta}}=p_{\mu}\sigma_{\alpha\dot{\beta}}^{\mu}$)
simply as111Pauli matrices appear as
$\sigma^{\mu}_{\alpha\dot{\beta}}=(1,\sigma^{1},\sigma^{2},\sigma^{3})=\bar{\sigma}_{\mu}^{\dot{\alpha}\beta}$.
Lorentz ${\rm SL}(2,\mathbb{C})$ indices and little-group ${\rm SU}(2)$
indices are raised and lowered via left-multiplication by Levi-Civita tensors,
normalized as $\epsilon^{12}=1=\epsilon_{21}$. The Lorentz invariants formed
out of Weyl spinors are $\langle
1_{a}2^{b}\rangle:=\epsilon_{\alpha\beta}\langle 1_{a}|^{\alpha}\langle
2^{b}|^{\beta}$ and
$[1_{a}2^{b}]:=\epsilon^{\dot{\alpha}\dot{\beta}}[1_{a}|_{\dot{\alpha}}[2^{b}|_{\dot{\beta}}$.

$\displaystyle{\cal A}_{\text{AHH}}(\bm{1}_{a_{1}\dots
a_{2s}},\bar{\bm{2}}^{b_{1}\dots b_{2s}}\\!,3^{+})$
$\displaystyle=\tfrac{1}{m^{2s}}\langle
1_{(a_{1}}2^{(b_{1}}\rangle\cdots\langle 1_{a_{2s})}2^{b_{2s})}\rangle{\cal
A}_{3}^{(0)}\,,$ (1a) $\displaystyle{\cal A}_{\text{AHH}}(\bm{1}_{a_{1}\dots
a_{2s}},\bar{\bm{2}}^{b_{1}\dots b_{2s}}\\!,3^{-})$
$\displaystyle=\tfrac{1}{m^{2s}}\,[1_{(a_{1}}2^{(b_{1}}]\cdots[1_{a_{2s})}2^{b_{2s})}]\,{\cal
A}_{3}^{(0)}\,.$ (1b)

Here ${\cal A}_{3}^{(0)}$ corresponds to a minimally coupled massive scalar,
e.g. ${\cal A}_{3}^{(0)}=2Q\,\varepsilon_{3}^{\pm}\\!\cdot p_{1}$ in the
electromagnetic (EM) case. For $s\leq 1$, eq. (1) matches all EM interactions
in the Standard Model Chung:2018kqs  and, upon switching the gauge group, also
the strong interaction Ochirov:2018uyq .

For ${\cal A}_{3}^{(0)}=-\sqrt{32\pi
G_{\text{Newton}}}(\varepsilon_{3}^{\pm}\\!\cdot p_{1})^{2}$, eq. (1) gives
equally simple amplitudes for massive spin-$s$ particles interacting
gravitationally. Remarkably, they were shown to reproduce the classical
behavior of rotating Kerr black holes (BHs) Guevara:2018wpp , which helped
solidify the link between quantum scattering amplitudes and classical dynamics
of compact objects Holstein:2008sw ; Holstein:2008sx ; Vaidya:2014kza ;
Guevara:2017csg ; Vines:2017hyw ; Kosower:2018adc ; Guevara:2018wpp ;
Chung:2018kqs ; Bautista:2019tdr ; Maybee:2019jus ; Guevara:2019fsj ;
Aoude:2020onz ; Aoude:2021oqj . In the context of the classical double copy
Monteiro:2014cda ; Arkani-Hamed:2019ymq ; Guevara:2020xjx ; Bern:2010ue , this
matching to the Kerr metric has a gauge-theory analogue, where the interaction
(1) implies the asymptotic EM field of a rotating charged disk (or ring). In
view of this link between the Kerr metric and the gauge-theory counterpart, we
refer to eq. (1) as the three-point $\sqrt{\text{Kerr}}$ amplitudes Arkani-
Hamed:2019ymq . Note that the $\sqrt{\text{Kerr}}$ solution is expected to
also describe the EM field sourced by a Kerr-Newman BH Chung:2019yfs . Here we
will, however, focus on a non-gravitational origin of $\sqrt{\text{Kerr}}$.

Besides such BH-like objects, there exist genuine microscopic higher-spin
particles that interact electromagnetically, such as atomic nuclei. For
instance, a ground state of ${}^{10}\text{Boron}$ has spin 3, and an
observationally stable isotope of ${}^{180}\text{Tantalum}$ has spin 9.
Interactions of a given nucleus state may be described by an EFT, whose
validity range depends on its lifetime. It may be too naive to expect nuclei
EFTs to possess any kind of simplicity. Nonetheless, the exploration of the
parameter space of such theories should perhaps start with those EFTs that do
exhibit simplicity, such as those related to $\sqrt{\text{Kerr}}$ amplitudes.

Despite the striking all-spin pattern of the three-point AHH amplitudes, the
related higher-point amplitudes and higher-order BH interactions are still
poorly understood. An important research direction is the study of Compton
amplitudes in the classical limit (see refs. Guevara:2017csg ; Guevara:2018wpp
; Chung:2018kqs ; Aoude:2020onz ; Haddad:2020tvs ; Chiodaroli:2021eug ;
Aoude:2022trd ; Bern:2022kto ; Aoude:2022thd ; Chen:2022clh ; Cangemi:2022abk
; Saketh:2022wap ; Bjerrum-Bohr:2023jau ; Haddad:2023ylx ; Brandhuber:2023hhy
; Alessio:2023kgf ; Aoude:2023vdk ; Bern:2023ity ; Bjerrum-Bohr:2023iey ;
Brandhuber:2023hhl ). These four-point tree amplitudes are building blocks for
calculating various observables related to gravitational radiation (see e.g.
refs. Maybee:2019jus ; Chung:2019duq ; Bern:2020buy ; Bern:2021xze ;
Kosmopoulos:2021zoq ; Chen:2021kxt ; Alessio:2022kwv ; Bautista:2023szu ;
DeAngelis:2023lvf ; Aoude:2023dui ). Unfortunately, a straightforward
extension of eq. (1) to four points Arkani-Hamed:2017jhn ; Johansson:2019dnu
is affected by local contact-term ambiguities Chung:2018kqs ;
Falkowski:2020aso ; Chiodaroli:2021eug ; Aoude:2022trd . This is often
highlighted as a problem of spurious poles in the opposite-helicity Compton
amplitudes. However, the problem is not the presence of the unphysical poles,
but that removing them can at best be done up to contact terms that cannot be
unambiguously determined Chung:2018kqs . It is thus important to identify
symmetry principles that may guide the identification of the correct contact
terms. In turn, this can potentially lead to new insights into BH physics.

In this paper, we combine several methods in higher-spin field theory in order
to construct and constrain the Compton amplitudes that we believe to closely
correspond to the gauge interactions of a $\sqrt{\text{Kerr}}$ object. Our
approach starts at the quantum level using off-shell actions either in terms
of chiral Ochirov:2022nqz  or non-chiral massive higher-spin fields
Fierz:1939ix ; Singh:1974qz ; Singh:1974rc . The latter are endowed with
higher-spin gauge symmetry Zinoviev:2001dt , which we employ as a natural
guiding principle for constraining the interactions and amplitude calculations
Cangemi:2022bew .

Quantum field theory of massive higher spins has been a challenging research
direction. This is in large part because the conventionally chosen traceless
symmetric tensor field $\Phi_{\mu_{1}\dots\mu_{s}}$ contains unphysical
longitudinal modes. Even for a free action Fierz:1939ix ; Singh:1974qz ;
Singh:1974rc , ensuring their non-propagation by imposing transversality
$\partial^{\lambda}\Phi_{\lambda\mu_{2}\ldots\mu_{s}}=0$ requires the
introduction of auxiliary fields. The multitude of compensating unphysical
degrees of freedom complicates the introduction of consistent interactions.
One significant improvement proposed by Zinoviev Zinoviev:2001dt ;
Zinoviev:2006im ; Zinoviev:2008ck ; Zinoviev:2009hu ; Zinoviev:2010cr ;
Buchbinder:2012iz  is to convert all second-class constraints into first-class
ones, corresponding to a new _massive higher-spin gauge symmetry_ à la
Stückelberg. In this formalism, any gauge-invariant interaction is consistent.
The price to pay is that the field content describing a single massive
spin-$s$ particle is now even larger, including $s{+}1$ double-traceless
symmetric tensors $\Phi_{\mu_{1}\dots\mu_{k}}$ of rank $k=0,\ldots,s$.
Complementary to this non-chiral formalism, we will also use a powerful
_chiral approach_ Ochirov:2022nqz  for constructing massive higher-spin
theories in $4d$. It relies on choosing a spinor field
$\Phi_{\alpha_{1}\ldots\alpha_{2s}}$ in the $(2s,0)$ representation of the
Lorentz group ${\rm SL}(2,\mathbb{C})$. This approach sidesteps the issue of
unphysical degrees of freedom entirely. Parity, however, is no longer
automatic and requires extra care.

Our work combines Zinoviev’s massive higher-spin symmetry and the chiral
approach with classical-limit considerations and heuristic assumptions, so as
to construct the Compton amplitudes, relevant for describing well-separated
non-decaying charged massive higher-spin fields. This paper gives an extended
exposition of the results in the letter Cangemi:2022bew , where higher-spin
gauge invariance was identified as a key property for elucidating BH dynamics.
We also go well beyond that work and complete the study of Compton amplitudes
in the $\sqrt{\text{Kerr}}$ theory. We give new perspectives on previous work,
such as the current constraint Ferrara:1992yc ; Porrati:1993in ;
Cucchieri:1994tx  proposed to single out BH amplitudes Chiodaroli:2021eug ,
and the multitude of subtleties of the classical limit. Here we only focus on
processes where the spin magnitude of the compact objects is conserved; spin-
magnitude-changing processes in the presence of gravity are more subtle
Aoude:2023fdm ; Bern:2023ity , but also an interesting avenue for further
research. Our closely related results, regarding gravitationally interacting
massive higher-spin theories and Compton amplitudes for Kerr BHs, will be
reported in ref. Upcoming2 .

The paper proceeds as follows. We warm up in Section 2 by discussing a
$\sqrt{\text{Kerr}}$ model involving a massive charged spin-1 field, which can
be derived as a limit of a spontaneously broken Yang-Mills theory. This is a
top-down example for the massive gauge symmetry originating from a fundamental
theory with massless gauge symmetry. Then we flip the perspective and show how
to obtain the same result in a bottom-up approach by imposing massive gauge
symmetry (off shell), or via the associated Ward identities (on shell).

In Section 3, we extend these bottom-up techniques to arbitrary spin. Note
that the massive higher-spin symmetry allows for general interactions, so
additional constraints are used in concert to restrict to the
$\sqrt{\text{Kerr}}$ amplitudes. These constraints are compatible with
imposing an improved high-energy behavior Chiodaroli:2021eug ; Cangemi:2022bew
via power counting and tree-level unitarity Ferrara:1992yc ; Porrati:1993in ,
similar to the behavior of fundamental theories. We give an explicit
construction of the spin-2 (see also ref. Zinoviev:2009hu ) and spin-3 cubic
Lagrangians that reproduce the $\sqrt{\text{Kerr}}$ amplitudes (1) using off-
shell massive gauge symmetry. We also present compelling evidence that the
$\sqrt{\text{Kerr}}$ three-point amplitudes are unique to any spin, once the
massive Ward identities and the additional constraints are employed. The on-
shell part of the cubic action is also worked out explicitly for all spins.
The on-shell method of Ward identities turns out to be more computationally
efficient than the Lagrangian approach at the quartic order, and we discuss
its implementation for spin-2 and spin-3 theories.

To find contact interactions to arbitrarily high spins, we turn to the chiral
approach of ref. Ochirov:2022nqz . In Section 4, we analyze possible parity-
preserving interactions that determine the non-minimal cubic
$\sqrt{\text{Kerr}}$ terms. A general-spin chiral Lagrangian is parameterized
up to the quartic level, which includes interactions linear in the field
strength $F_{\mu\nu}$, as well as generic $F_{\mu\nu}^{2}$ interactions that
are relevant for the opposite-helicity Compton amplitudes.

In Section 5, using plausible constraints and choices for the contact terms in
the $\sqrt{\text{Kerr}}$ theory, we present a manifestly local form for the
quantum Compton amplitudes for all spins and gauge groups. The amplitudes are
expressed in terms of complete homogeneous symmetric polynomials, which we
observe to be a defining feature of $\sqrt{\text{Kerr}}$ amplitudes. Contact
terms in the abelian sector are needed for consistency of the classical limit,
and we build them out of the symmetric polynomials guided by the constraints
inherited from massive gauge symmetry and other desirable properties.

The classical limit of the candidate Compton amplitude, studied in Section 6,
provides perhaps the strongest physical constraint on the quartic interactions
at high spin. The abelian amplitude develops a divergence in the large-spin
limit, unless contact terms are added in a specific way. We analyze two
variations of the classical limit for spin: the $s\to\infty$ limit of quantum
spin operators Guevara:2019fsj ; Cangemi:2022abk ; Cangemi:2022bew  and the
wavefunction scaling of coherent spin states Aoude:2021oqj . Given our choice
of constraints, we extract from the quantum amplitudes a unique classical
color-dressed Compton amplitude to all orders in spin. Possible modifications
of the constraints and the consequences for the classical amplitude are also
discussed.

Conclusions can be found in Section 7. The main text is supplemented with
appendices to keep track of the notation in Appendix A; to discuss the contact
terms within the chiral approach in Appendix B; and to collect some remarks on
factorization poles in Appendix C.

##  2 Spin-one warm-up

Before delving into the general case of higher-spin fields, let us set the
stage using the familiar spin-1 case. We will discuss three constructive
approaches for dealing with massive vector fields coupled to electromagnetism:
spontaneous symmetry breaking, massive gauge invariance order by order, and
Ward identities. The point of this section is to show the importance of a
gauge symmetry for constraining the EM interactions of massive fields.

###  2.1 Stückelberg, Proca and Higgs

A minimal model for our purposes involves EM interactions of a Proca field
$W_{\mu}$. In a bottom-up construction, one could start from the free
Stückelberg theory

${\cal
L}_{\text{St\"{u}ck.}}=-\frac{1}{2}|W_{\mu\nu}|^{2}+|mW_{\mu}-\partial_{\mu}\varphi|^{2},\qquad\quad
W_{\mu\nu}=\partial_{\mu}W_{\nu}-\partial_{\nu}W_{\mu}\,,$ (2)

where $\varphi$ is a complex version of Stückelberg’s auxiliary field
Stueckelberg:1938hvi . We use the mostly-minus metric convention
$\eta_{\mu\nu}=\text{diag}(1,-1,-1,-1)$ and work in four spacetime dimensions.
The action is invariant under the symmetry

$\delta W_{\mu}=\partial_{\mu}\xi\,,\qquad\quad\delta\varphi=m\xi\,,$ (3)

where $\xi$ is a complex gauge parameter. It can be used to remove the
auxiliary field $\varphi$ entirely, which gives the Proca Lagrangian ${\cal
L}_{\text{Proca}}:={\cal L}_{\text{St\"{u}ck.}}|_{\varphi=0}$. The equations
of motion that follow from ${\cal L}_{\text{Proca}}$ then generate the
standard transversality condition $\partial\cdot W=0$, which is needed to
reduce the degrees of freedom down to three (complex) physical polarizations.
This condition can also be obtained as the gauged-fixed version of the field
equation for $\varphi$, or as the Noether identities for the above gauge
symmetry.

The Lagrangian (2) is also invariant under a global U(1) symmetry, which we
can gauge and thus couple it to an EM field $A_{\mu}$. However, the standard
procedure of adding minimal interactions using covariant derivatives breaks
the massive gauge symmetry (3). In the next section, we will explain how to
introduce EM interactions order by order in the coupling while ensuring
compatibility with both the U(1) and massive gauge symmetries. In the spin-1
case, however, this can be done to all orders at once — using the knowledge
that such interactions should come from a spontaneously broken ${\rm SO}(3)$
gauge theory.

Indeed, consider an unbroken non-abelian theory with gauge field
$\vec{A}_{\mu}$ and Higgs field $\vec{\phi}$, both in the vector
representation of ${\rm SO}(3)$,

${\cal L}_{{\rm
SO}(3)}=-\frac{1}{4}\|\vec{F}_{\mu\nu}\|^{2}+\frac{1}{2}\|D_{\mu}\vec{\phi}\|^{2}+\frac{\mu^{2}\\!}{2}\|\vec{\phi}\|^{2}-\frac{\lambda}{4!}\|\vec{\phi}\|^{4}\,,\qquad
D_{\mu}\vec{\phi}=\partial_{\mu}\vec{\phi}-g\vec{A}_{\mu}\times\vec{\phi}\,.$
(4)

Here the non-abelian Lie algebra is encoded in the cross product
$(\vec{A}_{\mu}\times\vec{\phi})_{j}=iA_{\mu}^{k}t^{k}_{jl}\phi^{l}$, with
hermitian generators $t^{k}_{jl}=-i\epsilon_{jkl}$, and
$\vec{F}_{\mu\nu}=2\partial_{[\mu}\vec{A}_{\nu]}-g\vec{A}_{\mu}\times\vec{A}_{\nu}$.
The minima of the Higgs potential parameterize the sphere of size
$\|\vec{\phi}\|^{2}=v^{2}:=\frac{6\mu^{2}\\!}{\lambda}$. Therefore, we may
reparameterize the scalars in the standard way,

$\vec{\phi}=e^{\frac{i}{v}(\pi_{1}t^{1}+\pi_{2}t^{2})}\left(\begin{smallmatrix}0\\\
0\\\ v+h\end{smallmatrix}\right),$ (5)

using the Goldstone bosons $\pi_{i}$ and the massive Higgs boson $h$. Since we
are not interested in the Higgs boson, we may decouple it in the infinite-mass
limit, where $\lambda,\mu\to\infty$ but $v$ stays finite (see e.g. ref.
Schwartz:2014sze ). The resulting Lagrangian is

$\displaystyle{\cal L}_{\rm U(1)}=$
$\displaystyle-\frac{1}{4}(F_{\mu\nu})^{2}\\!-\frac{1}{2}|W_{\mu\nu}|^{2}\\!+|mW_{\mu}\\!-D_{\mu}\varphi|^{2}-iQ\overline{W}\\!_{\mu}F^{\mu\nu}W_{\nu}\\!-Q^{2}|\overline{W}_{[\mu}W_{\nu]}|^{2}$
(6) $\displaystyle-\frac{1}{4|\varphi|^{2}}\bigg{[}\bigg{(}{\rm
sinc}\Big{(}\\!\tfrac{\sqrt{2}Q|\varphi|}{m}\Big{)}[\overline{\varphi}D^{\mu}\varphi-\varphi\overline{D^{\mu}\varphi}]-m\cos\\!\Big{(}\\!\tfrac{\sqrt{2}Q|\varphi|}{m}\Big{)}[\overline{\varphi}W^{\mu}\\!-\overline{W}^{\mu}\\!\varphi]\\!\bigg{)}^{\\!2}$
$\displaystyle\qquad\qquad\qquad~{}\>\qquad-\Big{(}[\overline{\varphi}D^{\mu}\varphi-\varphi\overline{D^{\mu}\varphi}]-m[\overline{\varphi}W^{\mu}\\!-\overline{W}^{\mu}\\!\varphi]\Big{)}^{2}\bigg{]}\,,$

where $F_{\mu\nu}=2\partial_{[\mu}A_{\nu]}$ is the field strength of the
residual U(1) gauge symmetry (along the VEV direction, $A_{\mu}=A^{3}_{\mu}$).
The remaining two gauge bosons are given mass $m=gv$ and correspond to the
complex field $W_{\mu}=\tfrac{1}{\sqrt{2}}(A^{1}_{\mu}+iA^{2}_{\mu})$, with
field strength $W_{\mu\nu}=2D_{[\mu}W_{\nu]}$. The Goldstone bosons are
equivalent to the complex Stückelberg field considered previously,
$\varphi=\tfrac{1}{\sqrt{2}}(\pi_{1}+i\pi_{2})$. The covariant derivatives
$D_{\mu}=\partial_{\mu}-iQA_{\mu}$, with EM charge $Q=g$, now correspond to
the ${\rm U}(1)$ symmetry. The apparent non-locality in $|\varphi|^{2}$ is
canceled by series expanding in $Q$.

Clearly, the Lagrangian (6) is invariant under the ${\rm U}(1)$ gauge
transformations. However, it also inherits the following massive gauge
symmetry:

$\displaystyle\delta W_{\mu}$ $\displaystyle=D_{\mu}\xi=\partial_{\mu}\xi-
iQA_{\mu}\xi\,,$ (7) $\displaystyle\delta A_{\mu}$
$\displaystyle=iQ(\overline{W}\\!_{\mu}\xi-\bar{\xi}W_{\mu})\,,$
$\displaystyle\delta\varphi$
$\displaystyle=m\xi+\frac{m}{2\overline{\varphi}}\Big{(}1-\tfrac{\sqrt{2}Q|\varphi|/m}{\tan(\sqrt{2}Q|\varphi|/m)}\Big{)}(\bar{\xi}\varphi-\overline{\varphi}\xi)=m\xi+\frac{Q^{2}\varphi}{3m}(\bar{\xi}\varphi-\overline{\varphi}\xi)+{\cal
O}(Q^{4})\,.$

Note that the variation $\delta A_{\mu}$ is not coming from the EM gauge
transformation (since it is the massless spin-1 field of the system), which is
clear from the fact that the variation is strictly quadratic in the fields. In
contrast, both $\delta W_{\mu}$ and $\delta\varphi$ contain linear terms,
agreeing with the linearized massive gauge symmetry previously considered in
eq. (3).

We may use the above symmetry to set $\varphi=0$, which in the familiar
language of weak interactions corresponds to the unitary gauge, in which the
Goldstone bosons are “eaten” by the massive vector field, and only the terms
in the first line of the Lagrangian (6) survive. Note that in the context of
higher-spin generalizations, however, not all auxiliary fields can be gauged
away in this way Fierz:1939ix ; Singh:1974qz ; Singh:1974rc . It is also not
clear how to obtain a higher-spin analogue of the Brout-Englert-Higgs
mechanism. What is discussed in the following sections of paper can be viewed
as the first terms of a higher-spin generalization of the non-linear sigma
model (6), where the Higgs field has already been decoupled.

It is useful to briefly analyze the theory (6) from the point of view of its
high-energy behavior. For small coupling $Q\ll 1$, the tree-level amplitudes
do not grow too large given that the characteristic momenta is smaller than
the VEV scale, $p\leq v=\frac{m}{Q}$. Above this energy scale, tree-level
unitarity is violated and instead the amplitudes must be unitarized by loop-
level resummation. Because at these energy scales the original Lagrangian (4),
with $\mu\to 0$, is the better description, and since $\lambda\to\infty$ the
theory must be treated non-perturbatively. Alternatively, when we tune the
quartic coupling to the perturbative regime $\lambda<1$, then tree-level
unitarity is restored if we include the Higgs boson, which now can contribute
to tree amplitudes at energy scales of the Higgs mass
$m_{h}=\sqrt{2}\mu=\frac{m}{Q}\sqrt{\frac{\lambda}{3}}$.

Note that the high-energy limit is in the opposite direction of the classical
limit that we will consider in Section 6. Indeed, the classical limit is a
low-energy limit (large-distance limit), and as such it does not obviously
care about tree-level unitarity and completion by the Higgs boson.
Nevertheless, we find that the principles that govern the high-energy behavior
also appears to be good principles for constraining classical physics.

###  2.2 Spin-1 order by order

In the higher-spin case, we do not have access to a mechanism analogous to
spontaneous symmetry breaking. Nevertheless, the concept of massive higher-
spin gauge symmetry has proven to be very fruitful, even if it means working
out non-linear corrections order by order in the fields. In this section, we
will pretend to be unaware of the Higgs construction for massive spin-1 bosons
and will instead build their EM interactions and gauge symmetry anew order by
order. At the same time, we will use this example to introduce the main
features of our approach to higher spins used in Section 3.

The goal of this paper is to describe well-separated and non-decaying massive
spinning fields. Therefore, we may omit all self-interactions involving three
or more massive fields, physical or auxiliary, as well as interactions with a
single massive field (production or decay). In other words, the matter part of
any Lagrangian must be strictly quadratic in the massive fields. For instance,
the contact term $Q^{2}|\overline{W}_{[\mu}W_{\nu]}|^{2}$ and most of the
second and third lines of the spin-1 Lagrangian (6) are irrelevant for our
purposes. Moreover, since the infinitesimal gauge parameters $\xi$ shift the
auxiliary fields, they should be treated as massive fields, which forces other
massive fields to appear at most linearly in combination with $\xi$. For
example, the non-linear terms in the variation $\delta\varphi$ in eq. (7) are
irrelevant.

The general strategy is to start from a quadratic Lagrangian ${\cal L}_{2}$
and the linearized massive gauge variation $\delta_{0}$, both minimally
coupled/covariantized with respect to the EM gauge field. Then we
systematically add corrections involving more fields,

$\displaystyle{\cal L}$ $\displaystyle={\cal L}_{2}+{\cal L}_{3}+{\cal
L}_{4}+\ldots\,,$ (8) $\displaystyle\delta$
$\displaystyle=\delta_{0}+\delta_{1}+\delta_{2}+\ldots\,,$

where ${\cal L}_{n}$ and $\delta_{n-1}$ are covariant polynomials of degree
$n$ in the fields222When the homogeneity degree is mentioned, “fields” means
both fields and gauge parameters. Since the first step in introducing
interactions is to covariantize the derivatives in ${\cal L}_{2}$, it is
convenient to consider these $(D\Phi)^{2}$-terms still be a part of ${\cal
L}_{2}$, even though it is no longer of degree 2.  (massive or massless). The
gauge field $A_{\mu}$ only appears in the form of the field strength
$F_{\mu\nu}$ or inside the covariant derivative
$D_{\mu}=\partial_{\mu}-iQA_{\mu}$, thus ensuring U(1) gauge symmetry.
Importantly, we choose to regard $D_{\mu}$ as a derivative rather than a field
when counting the interaction degree.

For spin 1, we start from a minimally coupled Stückelberg action (2),

${\cal
L}_{2}=-\frac{1}{4}(F_{\mu\nu})^{2}-\frac{1}{2}|W_{\mu\nu}|^{2}+|mW_{\mu}-D_{\mu}\varphi|^{2}\,,$
(9)

where $W_{\mu\nu}=2D_{[\mu}W_{\nu]}$, and consider the linearized massive
gauge transformation (3), with covariantized derivatives

$\delta_{0}W_{\mu}=D_{\mu}\xi\,,\qquad\quad\delta_{0}\varphi=m\xi\,,\qquad\quad\delta_{0}A_{\mu}=0\,.$
(10)

However, once the minimal interactions are included, $Q\neq 0$, the massive
gauge symmetry $\delta_{0}$ is broken by the term $-|W_{\mu\nu}|^{2}/2$. To
restore it, we add a non-minimal deformation to both the Lagrangian and the
gauge transformations. This is done by suitable ansätze

${\cal
L}_{3}=-ic_{1}Q\overline{W}^{\mu}F_{\mu\nu}W^{\nu}\,,\qquad\delta_{1}W_{\mu}=0=\delta_{1}\varphi\,,\qquad\delta_{1}A_{\mu}=ic_{2}Q[\overline{W}\\!_{\mu}\xi-\bar{\xi}W_{\mu}]\,,$
(11)

where $c_{i}$ are parameters to be determined. The ansätze may appear as fine-
tuned, given the few non-vanishing terms. However, to obtain the non-minimal
interactions and transformations we rely on a set of guiding principles, which
we will consistently use for higher-spin cases later encountered. These
guiding principles include:

(SI)

Self-interactions of massive fields are suppressed (as motivated by the
classical limit).

(PS)

Parity symmetry is imposed on interactions. Specifically, the Levi-Civita
tensor $\epsilon_{\mu\nu\rho\sigma}$ or (anti-)self-dual field strengths do
not appear.

(PC)

Power counting of derivatives: at most $s+s^{\prime}-1$ derivatives in ${\cal
L}_{3}$, where $s$ and $s^{\prime}$ are the ranks of the two massive fields,
e.g. rank 1 for $W_{\mu}$ and rank 0 for $\varphi$.

(PC2)

Power counting: at most $s+s^{\prime}-1$ derivatives in $\delta_{1}$, where
$s$ and $s^{\prime}$ are the ranks of the massive fields and their gauge
parameters.333Note that gauge variation thus has at least one less derivative
than the corresponding vertex, which is required in order to preserve the
structure of the constraints that follow from the gauge symmetry. In fact, it
is not necessary to add this constraint if we first fix the on-shell part of
${\cal L}_{3}$, since this bound comes out of the general properties of the
formalism ($k$ derivatives in a vertex plus one derivative in $\delta_{0}$
minus two derivatives in ${\cal L}_{0}$ result in $k-1$ derivatives for
$\delta_{1}$, in general), see also below.

(MC)

Minimal-coupling interactions and linearized gauge transformations
$\delta_{0}$ are fixed. This follows from disallowing the kinetic terms ${\cal
L}_{2}$ to be modified. Combined with (SI) it implies that the non-minimal
interactions behave as ${\cal L}_{n}\propto(F_{\mu\nu})^{n-2}$, and for the
non-minimal gauge transformations, this implies that
$\delta_{n}\Phi\propto(F_{\mu\nu})^{n}$, where $\Phi$ is a massive field, e.g.
$\Phi\in\\{W_{\mu},\varphi\\}$.

From the power-counting constraint (PC), the unique non-minimal interaction
is given in eq. (11), where we count the derivatives appearing in
$F_{\mu\nu}$. Note that (SI) implis that the massive field variations,
$\delta W_{\mu}$ and $\delta\varphi$, should not contain multiple powers of
massive fields or gauge parameters, because they would mix with the massive
self-interactions that we have suppressed. Combined with assumptions (PC2)
and (MC), this implies $\delta_{1}W_{\mu}=0=\delta_{1}\varphi$. The variation
$\delta_{1}A_{\mu}$ is quadratic in the fields, including the gauge parameters
$\xi$, $\bar{\xi}$ that must appear linearly. Hence the power-counting
constraint (PC2), together with the Lorentz index and reality property of the
field $A_{\mu}$, implies the compact ansatz in eq. (11).

Finally, the free parameters $c_{1}$ and $c_{2}$ are uniquely fixed by
requiring massive gauge invariance to linear order in the coupling $Q$,

$(\delta_{0}+\delta_{1})({\cal L}_{2}+{\cal L}_{3})={\cal
O}(Q^{2})\qquad\Rightarrow\qquad c_{1}=c_{2}=1\,.$ (12)

This is of course consistent with the all-order results (6) and (7), which in
view of condition (SI) become

${\cal L}_{n\geq 4}=0\,,\qquad\quad\delta_{n\geq 2}=0\,.$ (13)

If we wanted to proceed systematically to ${\cal O}(Q^{2})$, we would need to
formulate appropriate ansätze for ${\cal L}_{4}$ and $\delta_{2}$. For the
spin-1 example we can be cavalier about the details, since the correct
solution is ${\cal
L}_{4}=\delta_{2}A_{\mu}=\delta_{2}W_{\mu}=\delta_{2}\varphi=0$, but for
higher-spin cases it is unclear what are the appropriate bounds we should
impose on the derivative power counts. We cannot rely on the higher-spin
literature for guidance, so we lack clear higher-order analogues of
assumptions (PC) and (PC2). Instead, we proceed by gradually increasing the
allowed derivative counts in ${\cal L}_{4}$ and $\delta_{2}$ until a solution
to the massive gauge transformation is obtained. However, we require that the
pieces of ${\cal L}_{4}$ that contribute to the Compton four-point amplitude
should not exceed the derivative count implied by the ${\cal L}_{3}$
interactions. Thus the higher-order interactions should not make the high-
energy behavior worse than implied by the cubic interactions.

Note that the cubic Lagrangian obeying eq. (12) yields the following three-
point amplitude ${\cal A}(\bm{1}^{s=1}\\!,\bar{\bm{2}}^{s=1}\\!,3)$, for two
massive spin-1 fields with momenta $p_{1}$ and $p_{2}$ and one photon with
momentum $p_{3}$,

${\cal
A}(\bm{1}^{s=1}\\!,\bar{\bm{2}}^{s=1}\\!,3)=-2Q(\bm{\mathbf{\varepsilon}}_{1}\\!\cdot\bm{\mathbf{\varepsilon}}_{2}\;\varepsilon_{3}\\!\cdot\\!p_{1}+\bm{\mathbf{\varepsilon}}_{2}\\!\cdot\varepsilon_{3}\;\bm{\mathbf{\varepsilon}}_{1}\\!\cdot\\!p_{2}+\varepsilon_{3}\\!\cdot\bm{\mathbf{\varepsilon}}_{1}\;\bm{\mathbf{\varepsilon}}_{2}\\!\cdot\\!p_{3})\,.$
(14)

It is instructive to plug in the on-shell spinor expressions for the massive
polarization vectors, parameterized in Appendix A among other definitions and
conventions, allowing us to write the physical amplitudes simply as

${\cal
A}(\bm{1}^{s=1}\\!,\bar{\bm{2}}^{s=1}\\!,3^{+})=2Q(\varepsilon_{3}^{+}\cdot
p_{1})\frac{\langle\bm{12}\rangle^{2}\\!}{m^{2}}\,,\qquad{\cal
A}(\bm{1}^{s=1}\\!,\bar{\bm{2}}^{s=1}\\!,3^{-})=2Q(\varepsilon_{3}^{-}\cdot
p_{1})\frac{[\bm{12}]^{2}\\!}{m^{2}}\,,$ (15)

where $\langle\bm{12}\rangle:=z_{1a}\langle 1^{a}2^{b}\rangle z_{2b}$,
$[\bm{12}]:=z_{1a}[1^{a}2^{b}]z_{2b}$, and for convenience we used
polarizations vectors with saturated SU(2) indices Chiodaroli:2021eug ,

${\bm{\varepsilon}}_{i,\mu}:=\varepsilon_{i,\mu}^{ab}z_{ia}z_{ib}=\frac{\langle
i^{a}|\sigma_{\mu}|i^{b}]}{\sqrt{2}m}z_{ia}z_{ib}\,.$ (16)

We observe that these spin-1 amplitudes belong to the $\sqrt{\text{Kerr}}$
family (1). Notably, the $\sqrt{\text{Kerr}}$ family does not accommodate
Chung:2018kqs  the amplitudes from the minimal EM coupling (9), which is not
entirely surprising: despite the name, the minimal EM coupling is not a
consistent interaction, as it violates the massive gauge symmetry or, in other
words, it sources unphysical degrees of freedom. As we will discuss in more
detail in the rest of this work, the same holds for arbitrary spin. Namely,
the minimal EM coupling for massive particles with spin $s\geq 1$ does not
give rise to the $\sqrt{\text{Kerr}}$ three-point amplitudes and it is found
to violate massive gauge invariance. Moreover, we checked up to $s=4$ that the
conditions (SI), (PS), (PC), (PC2) and (MC), imposed on an off-shell
cubic Lagrangian, yield the $\sqrt{\text{Kerr}}$ three-point amplitudes as
unique solution. For $s>4$ constructing the off-shell massive-gauge-invariant
Lagrangians is computationally challenging, so we will instead define a set of
on-shell massive Ward identities, introduced in the next section.

Note that, although it is a matter of straightforward computation at low
spins, the relationship between the Lagrangians and the on-shell amplitude
expressions (1) tends to be obscure. A way Chalmers:1997ui ; Chalmers:2001cy ;
Ochirov:2022nqz  to make it more transparent at the expense of explicit parity
is discussed in Section 4.1.

###  2.3 Massive Ward identities

As an alternative to the fully off-shell approach above, we can phrase massive
gauge invariance in terms of (generalized) Ward identities for momentum-space
correlation functions that are partially on shell. To write the sort of
identities most convenient for our purposes, we take the perspective that _the
purpose of Ward identities is to prevent pure-gauge fields to be sourced by
physical fields when solving the equations of motion._ This is of course true
for Ward identities in massless gauge theory, which set to zero any scattering
amplitude involving a single longitudinally polarized gauge boson. In the same
way, we may view the massive gauge symmetry (3) as an unphysical solution
$W_{\mu}=\partial_{\mu}\xi$, $\varphi=m\xi$ of the free theory (2), which
decouples from the remaining fields in any S-matrix element. Note that this
unphysical solution is not constrained by the usual mass-shell condition,
hence Ward identities typically involve one off-shell leg corresponding to the
pure-gauge field.

For a flexible implementation of such Ward identities, we use _off-shell
vertices_ $V(\dots)$, which we regard as pre-contracted with the corresponding
planewave polarization-tensor _placeholders_

$\epsilon_{i}^{\mu_{1}\dots\mu_{s_{i}}}=\epsilon_{i}^{\mu_{1}}\\!\cdots\epsilon_{i}^{\mu_{s_{i}}}\qquad\text{for
each particle }i.$ (17)

For $s>1$, this automatically implies full symmetrization over the Lorentz
indices. On the mass shell ($p_{i}^{2}=m_{i}^{2}$, all momenta are understood
as incoming), the placeholders are simply replaced by the physical
polarization vectors:

$\displaystyle\epsilon_{i,\mu}~{}\to~{}\varepsilon_{i,\mu}^{h}(p,q)\hskip
28.45274pt\text{particle }i\text{ is massless, with helicity }h=\pm,$ (18)
$\displaystyle\epsilon_{i,\mu}~{}\to~{}\varepsilon_{i,\mu}^{ab}(p)\hskip
38.41139pt\text{particle }i\text{ is massive, with ${\rm SU}(2)$ spin indices
}a,b=1,2.$

Note that only $\varepsilon_{i}$ are assumed to obey the usual properties
expected from polarization vectors, namely $\varepsilon_{i}\cdot
p_{i}=\varepsilon_{i}^{2}=0$. For example, $V(W_{1}\overline{W}\\!_{2}A_{3})$
is a scalar function of $\epsilon_{i,\mu}$ and momenta $p_{i}$, $i=1,2,3$,
containing all off-shell information about the EM interaction vertex of the
spin-1 field, while the corresponding amplitude is obtained by putting every
leg on shell, which we write as

${\cal
A}(\bm{1}^{s=1}\\!,\bar{\bm{2}}^{s=1}\\!,3)\,:=\,V(W_{1}\overline{W}\\!_{2}A_{3})\big{|}_{(1,2,3)}\,.$
(19)

In this notation, a spin-1 Ward identity due to the Stückelberg symmetry (3)
looks like

$mV(\varphi_{1}\overline{W}\\!_{2}A_{3})-ip_{1}^{\mu}\,\frac{\partial~{}}{\partial\epsilon_{1}^{\mu}}V(W_{1}\overline{W}\\!_{2}A_{3})\bigg{|}_{(2,3)}\\!=0\,,$
(20)

where we used $\frac{\partial}{\partial x^{\mu}}\leftrightarrow-ip_{\mu}$
assuming incoming momenta. Throughout this work we use the subscript notation
$(i,j,\dots)$ to indicate that the on-shell conditions
$p_{i}^{2}-m_{i}^{2}=\varepsilon_{i}\cdot p_{i}=\varepsilon_{i}^{2}=0$ are
applied to particles $\\{i,j,\dots\\}$, in this case particles 2 and 3, for
all terms displayed. Note that requiring that the expression vanishes in the
fully off-shell sense would be a different and overly strong claim, since it
is violated even by the Lagrangian (6). Indeed, the Ward identity (20) follows
from

$\left(\delta_{0}{\cal L}_{3}+\delta_{1}{\cal L}_{2}\right)\big{|}_{\text{on-
shell}}=\delta_{0}{\cal L}_{3}\big{|}_{\text{on-shell}}={\cal O}(Q^{2})\,,$
(21)

which is weaker than the full off-shell gauge invariance of the action at this
order, as expressed by eq. (12). Hereabove, $\delta_{1}{\cal
L}_{2}=\delta_{1}(\delta S_{2}/\delta\Phi)$ is a multiple of the free
equations of motion to this order444It is worth recalling that ${\cal L}_{2}$
is chosen to contain the minimal interactions introduced via $\partial\to D$.
Nevertheless, we can neglect the cubic part of ${\cal L}_{2}$ in eq. (21),
since its variation is multiplied by $\delta_{1}$, which is bilinear in the
fields.  in $Q$. Therefore, the second term vanishes when the corresponding
legs are put on-shell. On the other hand, once we found an ${\cal L}_{3}$ that
is gauge invariant on shell, one can take its (non-unique) off-shell extension
and read off $\delta_{1}$, cf. footnote 3. Indeed, $\delta_{0}{\cal
L}_{3}\big{|}_{\text{on-shell}}=0$ means that $\delta_{0}{\cal L}_{3}$ is a
multiple of the free equations of motion and, hence, can be compensated by a
suitable $\delta_{1}$.

In order to constructively use eq. (20), we first come up with an ansatz for
the vertices, schematically,

$\displaystyle V(W_{1}\overline{W}\\!_{2}A_{3})$ $\displaystyle\sim
m\,\epsilon_{1}\,\epsilon_{2}\,\epsilon_{3}\,\Big{(}\frac{p}{m}\Big{)}\quad\to\quad
6\text{ terms}\,,$ (22a) $\displaystyle
V(\varphi_{1}\overline{W}\\!_{2}A_{3})$ $\displaystyle\sim
m\,\epsilon_{2}\,\epsilon_{3}\,\big{(}p^{0}\big{)}\qquad\;\,\to\quad 1\text{
term}\,,$ (22b)

where we specify the power counting in various ingredients but omit momentum
labels, index contractions and free coefficients. We have already used the
derivative-counting assumption (PC) and parity (absence of the Levi-Civita
tensor). Furthermore imposing the reality condition
$V(W_{1}\overline{W}\\!_{2}A_{3})=-V(W_{2}\overline{W}\\!_{1}A_{3})$, as per
assumption (PS), determines three free coefficients. Imposing the Ward
identity (20) reduces the remaining four coefficients to one. This last free
parameter is fixed to the coupling $Q$ by identifying the minimal-coupling
(MC) terms that originate from an appropriate kinetic term
$|D_{[\mu}W_{\nu]}|^{2}$. This gives the final result

$\displaystyle V(W_{1}\overline{W}\\!_{2}A_{3})$
$\displaystyle=-Q\big{[}\epsilon_{1}\\!\cdot\epsilon_{2}\;\epsilon_{3}\\!\cdot\\!(p_{1}-p_{2})+\epsilon_{2}\\!\cdot\epsilon_{3}\;\epsilon_{1}\\!\cdot\\!(p_{2}-p_{3})+\epsilon_{3}\\!\cdot\epsilon_{1}\;\epsilon_{2}\\!\cdot\\!(p_{3}-p_{1})\big{]}\,,$
(23a) $\displaystyle V(\varphi_{1}\overline{W}\\!_{2}A_{3})$
$\displaystyle=iQm\,\epsilon_{2}\cdot\epsilon_{3}\,.$ (23b)

The kinematic form of the spin-1 vertex is identical to the 3-gluon Feynman
rule, which is no coincidence in view of the discussed construction of the
Lagrangian (6) from non-abelian gauge theory, which results in the same
trivalent vertices as above. The on-shell amplitude ${\cal
A}(\bm{1}^{s=1}\\!,\bm{2}^{s=1}\\!,3)$ can be computed from the vertex
$V(W_{1}\overline{W}\\!_{2}A_{3})$ and it agrees with eq. (14), thus
reproducing the $\sqrt{\text{Kerr}}$ result.

An important advantage of eq. (20) is that we do not need to construct an
ansatz for the non-minimal gauge transformations $\delta_{1}$, since it only
depends on the free-theory transformations $\delta_{0}|_{Q=0}$, as discussed
in eq. (21). This significantly reduces the number of free parameters and
hence allows us to compute amplitudes with higher multiplicity and higher
values of the spin $s$. For the purpose of computing the Compton diagram one
does not even have to solve Ward identities for all the vertices: one leg has
to be the physical spin-$s$ field, i.e. the vertices with two Stückelberg
fields and one gauge field $A$ can be omitted. However, such reduced Ward
identities are in general less constraining than imposing massive gauge
invariance on the full off-shell Lagrangian. Although eq. (20) is enough to
fix the three-point amplitude uniquely, for $s>1$ the reduced Ward identities
alone are not sufficient, and they will need to be supplemented by additional
on-shell constraints. We will discuss this in detail in the next section.

##  3 Higher-spin gauge theory with massive gauge symmetry

Having already outlined the problem of unphysical degrees of freedom, in this
section we discuss Zinoviev’s solution to it Zinoviev:2001dt  based on massive
gauge symmetry, which intertwines the primary higher-spin field with its
auxiliary-field descendants, as well as with the gauge field with which it
interacts.

###  3.1 General Lagrangian, gauge fixing and Feynman propagator

We are now ready to review how to construct general integer-spin theories
using massive gauge symmetry. In addition to the physical symmetric spin-$s$
field $\Phi_{\mu_{1}\dots\mu_{s}}$, which we will sometimes shorten to
$\Phi^{s}$, Zinoviev Zinoviev:2001dt  introduced auxiliary fields
$\Phi^{s-1},\Phi^{s-2},\dots,\Phi^{1},\Phi^{0}$, where the last one is a
scalar field. These fields are assumed to be complex and double-traceless
$\Phi^{\lambda\nu}{}_{\lambda\nu\mu_{5}\dots\mu_{k}}=0$, such that even the
physical, maximal-rank field $\Phi^{s}$ contains more off-shell degrees of
freedom than the field chosen in earlier literature Fierz:1939ix ;
Singh:1974qz ; Singh:1974rc .

The fields $\\{\Phi^{k}\\}_{k=0}^{s}$ are subject to massive gauge
transformations

$\delta_{0}\Phi_{\mu_{1}\dots\mu_{k}}=\partial_{(\mu_{1}}\xi_{\mu_{2}\dots\mu_{k})}+m\alpha_{k}\xi_{\mu_{1}\dots\mu_{k}}+m\beta_{k}\eta_{(\mu_{1}\mu_{2}}\xi_{\mu_{3}\dots\mu_{k})}\,,$
(24)

where the complex gauge parameters are traceless,
$\xi^{\lambda}{}_{\lambda\mu_{3}\dots\mu_{k}}=0$, and the numerical
coefficients are

$\alpha_{k}=\sqrt{\frac{(s-k)(s+k+d-3)}{(k+1)(2k+d-2)}}\,,\qquad\quad\beta_{k}=\frac{k\alpha_{k-1}}{2k+d-6}\,,\qquad\quad
d=4\,.$ (25)

Note that $\alpha_{s}=0$, implying that the physical field $\Phi^{s}$ cannot
be set to zero by a gauge choice, and the relevant parameters are
$\\{\xi^{k}\\}_{k=0}^{s-1}$. Of course, for $s=1$ we simply recover
Stückelberg’s massive gauge transformations (3).

In ref. Cangemi:2022bew , we found that the free Lagrangian for a complex
massive spin-$s$ field that is invariant under the massive gauge
transformations (24) can be written as

${\cal L}_{\text{free}}={\cal L}_{\rm F}-{\cal L}_{\text{gf}},$ (26a) with
${\cal
L}_{\text{gf}}=-\sum_{k=0}^{s-1}(-1)^{k}(k+1)\overline{G_{\mu(k)}}G^{\mu(k)}\,,$
(26b)

where ${\cal L}_{\text{F}}={\cal L}_{\text{free}}+{\cal L}_{\text{gf}}$ is the
Lagrangian in Feynman gauge, obtained by adding the gauge-fixing term ${\cal
L}_{\text{gf}}$ to the gauge-invariant Lagrangian ${\cal L}_{\text{free}}$.
The Feynman-gauge Lagrangian ${\cal L}_{\text{F}}$ is completely diagonal,

${\cal
L}_{\text{F}}=-\sum_{k=0}^{s}(-1)^{k+1}\bigg{[}\bar{\Phi}_{\mu(k)}(\Box+m^{2})\Phi^{\mu(k)}-\frac{k(k-1)}{4}\bar{\tilde{\Phi}}_{\mu(k-2)}(\Box+m^{2})\tilde{\Phi}^{\mu(k-2)}\bigg{]}\,,$
(27)

with $\tilde{\Phi}_{\mu(k-2)}:=\Phi^{\lambda}{}_{\lambda\mu_{3}\dots\mu_{k}}$
denoting single traces.

The gauge-fixing functions $G^{\mu(k)}=G^{(\mu_{1}\dots\mu_{k})}$ contain the
off-diagonal contributions and are given by

$G^{\mu(k)}\\!:=\partial_{\lambda}\Phi^{\lambda\mu(k)}-\frac{k}{2}\partial^{\mu}\tilde{\Phi}^{\mu(k-1)}+m\alpha_{k}\Phi^{\mu(k)}-m\alpha_{k+1}\frac{k{+}1}{2}\tilde{\Phi}^{\mu(k)}-m\alpha_{k}\frac{k{-}1}{4}\eta^{\mu\mu}\tilde{\Phi}^{\mu(k-2)}\,,$
(28)

where we use the convention that repeated all-upper (or all-lower) indices
$\eta^{\mu\mu}\Phi^{\mu(k)}$, $\partial^{\mu}\Phi^{\mu(k)}$ are a compact
notation for symmetrization of indistinguishable indices. Note that traces of
rank-0,1 fields are defined to be zero,
$\tilde{\Phi}^{\mu(-2)}=\tilde{\Phi}^{\mu(-1)}:=0$.

The gauge-fixing functions have a simple gauge transformation following from
eq. (24),

$\delta_{0}G^{\mu(k)}=\frac{1}{k+1}(\Box+m^{2})\xi^{\mu(k)}\,.$ (29)

The appearance of the Klein-Gordon operator $(\Box+m^{2})$ implies that gauge-
fixing constraints, such as the equivalent of Lorenz gauge $G^{\mu(k)}=0$, are
invariant under gauge transformations with parameters $\xi^{\mu(k)}$ that
satisfy the mass-shell constraint $p^{2}=m^{2}$. Hence, a residual gauge
freedom remains present.

The diagonal form of the gauge-fixed action (27) is particularly advantageous
when extending our analysis to four-point amplitudes, where the propagator for
each field is required. The Feynman-gauge propagator $\Delta^{(s)}$ for a
spin-$s$ field is diagonal and has trivial momentum dependence. It is unique,
and we find that in $d=4$ dimensions it is given by the simple generating
function Cangemi:2022bew

$\Delta(\epsilon,\bar{\epsilon})=\sum_{s=0}^{\infty}(\epsilon)^{s}{\cdot}\Delta^{\\!(s)}{\cdot}(\bar{\epsilon})^{s}=\frac{i}{p^{2}{-}m^{2}{+}i0}\,\frac{1-\frac{1}{4}\epsilon^{2}\bar{\epsilon}^{2}}{1+\epsilon\cdot\bar{\epsilon}+\frac{1}{4}\epsilon^{2}\bar{\epsilon}^{2}}\,,$
(30)

where, as before, $\epsilon_{\mu},\bar{\epsilon}_{\mu}$ are auxiliary vectors
regarded as independent, and $\epsilon\cdot\bar{\epsilon}$, $\epsilon^{2}$,
$\bar{\epsilon}^{2}$ are standard Lorentz contractions. Remarkably, this
propagator already includes the appropriate projectors that enforce the
double-tracelessness of the rank-$s$ fields.555While the Feynman-gauge kinetic
term does not a have unique inverse, the additional requirement that the
propagator acts as a double-traceless projector gives the unique formula.
Note that the Feynman propagator does not discern between physical and
auxiliary fields, but treats them all on the same footing, which is also clear
from the Feynman-gauge-fixed Lagrangian (27).

In order to introduce interactions in the spin-$s$ theory (26) one can first
attempt to minimally couple it to electromagnetism. As before, we covariantize
the free Lagrangian, and we obtain the minimally coupled interacting spin-$s$
theory,

${\cal L}_{2}={\cal L}_{\text{free}}\Big{|}_{\partial_{\mu}\to
D_{\mu}=\partial_{\mu}-iQA_{\mu}}\,,$ (31)

where the subscript on ${\cal L}_{2}$ is to emphasize that this comes from the
quadratic terms.666We emphasize that, unlike ${\cal L}_{2}$, the gauge-fixing
term is kept quadratic. Alternatively, one can also covariantize the gauge-
fixing term with respect to massless gauge transformations, but this approach
will not be pursued here.  However, this step alone will not give rise to a
consistent theory. The minimal interactions spoil the counting of the degrees
of freedom, since we have not ensured that the massive gauge symmetry is
preserved.

As was the case in Section 2.2, covariantizing ${\cal L}_{2}$ and $\delta_{0}$
is not sufficient, and we must add non-minimal interactions777Note that
neither the “minimal EM” nor non-minimal terms are consistent interactions
when considered independently of each other. Only very specific linear
combinations of such terms form consistent interactions. Those can be found be
requiring the longitudinal modes to decouple, which is equivalent to imposing
the massive higher-spin gauge symmetry. , ${\cal L}_{3}+...$, and gauge
transformations, $\delta_{1}+...$. In Section 2.2, we discussed the systematic
approach for fixing the theory at the Lagrangian level, as shown in eq. (8).
This involves constructing the action and gauge transformations order by
order, fully off-shell. For instance, at cubic order, we need to consider two
types of contributions to the off-shell cubic vertex,

$V(\Phi^{s}_{1}\Phi^{s^{\prime}}_{2}A_{3})=V_{{\cal
L}_{2}}(\Phi^{s}_{1}\Phi^{s^{\prime}}_{2}A_{3})+V_{{\cal
L}_{3}}(\Phi^{s}_{1}\Phi^{s^{\prime}}_{2}A_{3})\,,$ (32)

where $V_{{\cal L}_{2}}(\Phi^{s}_{1}\Phi^{s^{\prime}}_{2}A_{3})$ is extracted
from the minimally-coupled Lagrangian ${\cal L}_{2}$ in eq. (31), and
$V_{{\cal L}_{3}}(\Phi^{s}_{1}\Phi^{s^{\prime}}_{2}A_{3})$ from the non-
minimal cubic interactions ${\cal L}_{3}$ linear in the field strength of the
massless boson $F_{\mu\nu}$. The ${\cal L}_{2}$ term can be written out
explicitly,

$\displaystyle V_{{\cal L}_{2}}(\Phi^{s}_{1}\bar{\Phi}^{s^{\prime}}_{2}A_{3})$
$\displaystyle=2\delta^{ss^{\prime}}(-1)^{s}(ip_{2}\cdot\epsilon_{3})(\epsilon_{1}\cdot\epsilon_{2})^{s-2}\bigg{[}(\epsilon_{1}\cdot\epsilon_{2})^{2}-\frac{s(s-1)}{4}\epsilon_{1}^{2}\epsilon_{2}^{2}\bigg{]}$
$\displaystyle+V_{{\cal
L}_{\text{gf}}}(\Phi^{s}_{1}\bar{\Phi}^{s^{\prime}}_{2}A_{3})$ (33)

where the first line comes from ${\cal L}_{\rm F}$ and the second line comes
from the $\overline{G_{\mu(k)}}G^{\mu(k)}$ term. Note that if the gauge-fixing
function is covariantized it alter the second line; however, we choose to work
with a strictly quadratic gauge-fixing function, which does not alter the
interaction terms.

The $V_{{\cal L}_{3}}$ interactions are constructed as an ansatz, together
with the non-minimal gauge transformations $\delta_{1}$, and the free
coefficients are fixed by requiring off-shell massive gauge invariance. We
will do this explicitly for spin-2 and spin-3 fields in Section 3.3, and we
will see that it is a significant computational challenge even at low spins.

On the other hand, in Section 2.3 we introduced an approach influenced by on-
shell techniques, working with identities at the level of the vertices. This
approach only requires the free-field massive gauge transformations, which
lets us bypass ansatzing the non-linear gauge transformations and thus
provides a significant computational advantage. Moreover, there is an
immediate connection to scattering amplitudes, which is an advantage
considering our goal of comparing to the $\sqrt{\text{Kerr}}$ three-point
amplitudes and extending them to four points. We will use this method to
compute arbitrary-spin three-point amplitudes in Section 3.3, and four-point
Compton amplitudes for spin-2 and spin-3 particles in Section 3.4.

Note, however, that solving the Ward identities, which are partially on-shell,
and finding a fully off-shell gauge-invariant Lagragian are two equivalent
approaches. Indeed, consider the cubic vertex in eq. (32), the Ward identities
$\delta_{0}V|_{\text{on-shell}}=0$ guarantee that $\delta_{0}V$ is a multiple
of the free equations of motion and, hence, $\delta_{1}$ can be read off once
the variation is kept off-shell, see eq. (21). Different ways of extending
$V_{{\cal L}_{3}}$ off-shell will lead to different $\delta_{1}$, which are
related via field redefinitions. Therefore, the ambiguity in the off-shell
extension does not carry any information about the physics. Let us note,
however, that, in practice, we solve the reduced Ward identities where one of
the fields is the physical spin-$s$ field and additional constraints are
required, which are discussed below.

###  3.2 Massive Ward identities again

The discussion of the Ward identities in Section 2.3 can easily be extended to
any spin $s$. At the cubic order we have the vertices (32) that encodes
interactions between two fields from the massive spin-$s$ multiplets and a
gauge field, $V(\Phi^{k}\\!,\overline{\Phi}^{n}\\!,A)$, where $k,n=1,...,s$
run over the auxiliary fields $k,n=1,...,s{-}1$ and the physical field
$k,n=s$. In practice, one factorizes fields via the unconstrained
polarizations $\epsilon^{\mu}_{i}$ introduced in eq. (17). As a result, in
momentum space $V(\Phi^{k}\overline{\Phi}^{n}A)$ becomes a scalar function of
$\epsilon_{i,\mu}$ and $p_{i}$, $i=1,2,3$, which we denote
$V(\Phi^{k}_{1}\overline{\Phi}^{n}_{2}A_{3})$ with the subscript referring to
the $i$-index of $\epsilon_{i,\mu}$.888Note that it is not necessary to impose
double-tracelessness of the fields $\Phi_{i}^{\mu(k)}$, since either the on-
shell conditions or the Feynman propagator (30) will automatically enforce it.
We can and will use the same polarization vectors $\epsilon_{i}$ to factorize
the gauge parameters $\xi^{\mu(k)}$, $k=0,...,s{-}1$ since $\epsilon_{i,\mu}$
is merely a book-keeping device. The tracelessness of the gauge parameters is
ensured by further imposing $\epsilon_{i}^{2}=0$.

We then consider the linearized gauge transformation $\delta_{0}$ in eq. (24),
and exhibit all fields that pick up a change proportional to the gauge
parameter $\xi_{i}^{\mu(k)}$. Recalling that the same polarization vectors can
be used for fields and gauge parameters, we arrive at the following
implementation of the gauge transformations

$\displaystyle\delta_{0}\Phi^{\mu(k)}_{1}$
$\displaystyle=m\alpha_{k}\xi^{\mu(k)}_{1}$ $\displaystyle\to$ $\displaystyle
m\alpha_{k}$ $\displaystyle
V(\Phi^{\mu(k)}_{1}\\!\cdots)\Big{|}_{\epsilon_{1}^{2}=0}\,,$ (34)
$\displaystyle\delta_{0}\Phi^{\mu(k+1)}_{1}$
$\displaystyle=\partial^{\mu}\xi^{\mu(k)}_{1}$ $\displaystyle\to$
$\displaystyle\frac{1}{k+1}\frac{\partial~{}}{\partial
x_{\nu}}\frac{\partial~{}}{\partial\epsilon_{1}^{\nu}}$ $\displaystyle
V(\Phi^{\mu(k+1)}_{1}\\!\cdots)\Big{|}_{\epsilon_{1}^{2}=0}\,,$
$\displaystyle\delta_{0}\Phi^{\mu(k+2)}_{1}$
$\displaystyle=m\beta_{k+2}\eta^{\mu\mu}\xi^{\mu(k)}_{1}$ $\displaystyle\to$
$\displaystyle\frac{m\beta_{k+2}}{(k+2)(k+1)}\frac{\partial~{}}{\partial\epsilon_{1}}\cdot\frac{\partial~{}}{\partial\epsilon_{1}}$
$\displaystyle V(\Phi^{\mu(k+2)}_{1}\\!\cdots)\Big{|}_{\epsilon_{1}^{2}=0}\,.$

Here, we displayed the operators that realize the gauge transformations due to
$\xi^{\mu(k)}$ for the $3$-point case. Generalization to $n$-point case is
straightforward.

Now we consider the (reduced) massive Ward identity for a process involving
three plane waves, and apply the gauge transformation $\delta_{0}$ that is due
to $\xi^{\mu(k)}$ to the first leg. First, using the relations (34), and
plugging in $\beta_{k+2}=\frac{1}{2}a_{k+1}(k{+}2)/(k{+}1)$, we construct an
off-shell version of the gauge-transformed three-point vertex,

$V(\xi_{1}^{k}\bar{\Phi}_{2}^{s}A_{3}):=m\alpha_{k}V(\Phi_{1}^{k}\bar{\Phi}_{2}^{s}A_{3})-\frac{ip_{1}}{k\\!+\\!1}\cdot\frac{\partial~{}}{\partial\epsilon_{1}}V(\Phi_{1}^{k+1}\bar{\Phi}_{2}^{s}A_{3})+\frac{m\alpha_{k+1}}{2(k{+}1)^{2}}\bigg{(}\\!\frac{\partial~{}}{\partial\epsilon_{1}}\\!\bigg{)}^{\\!\\!2}V(\Phi_{1}^{k+2}\bar{\Phi}_{2}^{s}A_{3}),$
(35)

where the notation reminds us that we are effectively scattering the gauge
parameter $\xi_{1}^{\mu(k)}$. Then, the massive Ward identity at three points
is given by setting the above vertex to zero after imposing on-shell
conditions on legs $2,3$ and tracelessness on leg $1$.

Since we plan to work with the reduced Ward identities, we now state the
additional constraints that we will impose on the three point interactions for
the spin-$s$ theory:

(WI)

Massive Ward identities due to Zinoviev’s gauge symmetry (24):

$V(\xi_{1}^{k}\bar{\Phi}_{2}^{s}A_{3})\big{|}_{(2,3),\epsilon_{1}^{2}=0}=0\,.$
(36)

(CC)

Current constraint following from high-energy unitarity, as discussed in ref.
Chiodaroli:2021eug :

$p_{1}\cdot\frac{\partial~{}}{\partial\epsilon_{1}}V(\Phi_{1}^{s}\bar{\Phi}_{2}^{s}A_{3})\bigg{|}_{(2,3),\epsilon_{1}^{2}=0}={\cal
O}(m)\,.$ (37)

(ND)

Near-diagonal interactions of two massive fields. The cubic vertices differ at
most by one unit:999In practice, it is applied to $n=s$ since we consider only
the reduced Ward identities where the physical spin-$s$ field is present.

$V(\Phi_{1}^{n-k}\bar{\Phi}_{2}^{n}A_{3}^{h})\big{|}_{k>h}=0\,.$ (38)

In addition, the previously introduced constraints (SI), (PS), (PC), (MC)
should still be imposed on the spin-$s$ theory.

Let us discuss the new constraints. As we will see in Section 3.3.3, they are
needed, in combination with the reduced massive Ward identities, to fix the
three-point $\sqrt{\text{Kerr}}$ amplitudes uniquely.101010Note that leading-
Regge open-string higher-spin amplitudes violate the constraints that we are
imposing. Specifically, it is the combination of (PC)+(CC) that are too
constraining for string amplitudes above $s>2$. In particular, string
amplitudes involve non-minimal interactions.  However, in the off-shell
Lagrangian approach, the $\sqrt{\text{Kerr}}$ amplitudes follow uniquely from
massive gauge invariance without the need to impose additional constraints, as
we checked up to $s=4$. Therefore, it is possible that (CC) and (ND) encode
information about massive gauge symmetry, which is somehow missed by the
reduced on-shell Ward identities, since they do not probe all three-point
vertices but only those with one leg being the physical spin-$s$ field. To
confirm this, we would need to push the off-shell computation to higher values
of spin, $s>4$.

The (ND) constraint is obeyed by the minimally coupled theories that follow
from the free Lagrangians (26), so we propose that it should be extended to
the non-minimal interactions. The current constraint (CC) has been argued
Ferrara:1992yc ; Porrati:1993in ; Cucchieri:1994tx  to give rise to scattering
amplitudes with an improved high-energy limit. For instance, in the spin-1
theory discussed in Section 2, the massive propagator in unitary gauge (UG)
becomes

$\Delta_{\mu\nu}^{\rm
UG}=-i\frac{\eta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{m^{2}}}{p^{2}-m^{2}+i0}\,.$
(39)

In the high-energy limit $p/m\to\infty$, it is clear that the second term in
the propagator numerator give rise to a divergence. The implication for a
physical process, such as the Compton amplitude ${\cal
A}(\bm{1}^{s=1}\\!,\bar{\bm{2}}^{s=1}\\!,3^{-}\\!,4^{+})$, is that such
divergences appear in the massive-exchange diagrams and are proportional to
the contraction of the exchanged momentum $p_{\mu}$ and the three-point
vertices, namely
$\frac{p}{m}\cdot\frac{\partial~{}}{\partial\epsilon_{p}}V(\Phi_{p}^{s}\bar{\Phi}_{i}^{s}A_{j})\big{|}_{(i,j)}$.
The current constraint (37) cures this type of divergences by removing the
factor of the mass $m$ from the denominator.

On the other hand, in Feynman gauge the propagators (30) do not contain
momentum-dependent terms. In that case, the divergences appear in the vertices
of the theory. For instance, we can modify the $s=1$ Lagrangian in Section 2.2
by adding the term

$iQ\tilde{c}_{1}F^{\mu\nu}\left(\overline{W}_{\mu}-\frac{1}{m}\overline{D_{\mu}\varphi}\right)\left(W_{\nu}-\frac{1}{m}D_{\nu}\varphi\right)\,,$
(40)

where $\tilde{c}_{1}$ is a free coefficient. The resulting Lagrangian is still
gauge-invariant at cubic order, since the structure in the brackets is
invariant under the free gauge transformations (10). However, it violates the
current constraint, namely,

$p_{1}\cdot\frac{\partial~{}}{\partial\epsilon_{1}}V(W_{1}\overline{W}_{2}A_{3})\Big{|}_{(2,3)}=Q\tilde{c}_{1}(2\bm{\mathbf{\varepsilon}}_{2}\\!\cdot\\!p_{1}\,\varepsilon_{3}\\!\cdot\\!p_{1}-p_{1}^{2}\,\bm{\mathbf{\varepsilon}}_{2}\\!\cdot\\!\varepsilon_{3})+{\cal
O}(m)\,.$ (41)

In Feynman gauge, the terms in eq. (40) proportional to $1/m$ give rise to
high-energy divergences in scattering amplitudes, unless $\tilde{c}_{1}=0$ as
required by the current constraint.

Vertex (40) is also an illustration of the fact that the massive higher-spin
gauge symmetry needs to be supplemented with appropriate constraints on the
number of derivatives in the interactions. Indeed, one can take any function
of $W_{\mu}$, $\overline{W}_{\mu}$ and replace them by the expressions in the
brackets of eq. (40), the resulting vertex being gauge-invariant. However,
such a cavalier approach induces higher derivatives for the auxiliary fields.
For example, eq. (40) is a three-derivative vertex, while the minimal
interaction is a two-derivative one. The current constraint is a way to
improve the derivative counting at least at the cubic order. Black holes
should correspond to the best possible scenario, where the number of
derivatives is kept the lowest at all orders.

###  3.3 Massive gauge symmetry at cubic order

####  3.3.1 Example: spin-2 $\sqrt{\rm Kerr}$ theory

In this section, we apply the methods outlined in Section 2.2 to the case of a
massive spin-2 field coupled to a photon. We work in $d=4$ dimensions and
start with the free massive spin-$2$ Lagrangian ${\cal L}_{2}$ and then add
non-minimal terms ${\cal L}_{3}$. For convenience, and following ref.
Zinoviev:2009hu , we decompose the spin-$s$ Lagrangians as ${\cal
L}^{(s)}_{n,k}$, where $n$ is the number of fields (the gauge field in
$D_{\mu}$ is not counted) and $k$ is the number of derivatives (counting both
$D_{\mu}$ and the derivative in $F_{\mu\nu}$).

The minimally coupled spin-2 Lagrangian is then

${\cal L}_{2}^{(2)}={\cal L}_{2,2}^{(2)}+{\cal L}_{2,1}^{(2)}+{\cal
L}_{2,0}^{(2)}\,,$ (42)

where

$\displaystyle{\cal L}_{2,2}^{(2)}$
$\displaystyle=\overline{D_{\alpha}\Phi_{\mu\nu}}D^{\alpha}\Phi^{\mu\nu}-2\overline{D\cdot\Phi_{\mu}}D\cdot\Phi^{\mu}+\overline{D\cdot\Phi_{\nu}}D^{\nu}\tilde{\Phi}+\overline{D_{\nu}\tilde{\Phi}}D\cdot\Phi^{\nu}-\overline{D_{\mu}\tilde{\Phi}}D^{\mu}\tilde{\Phi}+\overline{D_{\alpha}\varphi}D^{\alpha}\varphi$
$\displaystyle\quad-\overline{D_{\mu}B_{\nu}}D^{\mu}B^{\nu}+\overline{D\cdot
B}\,D\cdot B\,,$ $\displaystyle{\cal L}_{2,1}^{(2)}$
$\displaystyle=\sqrt{3}m(\overline{D\cdot B}\varphi+\bar{\varphi}D\cdot
B)-\sqrt{2}m(\overline{D\cdot\Phi_{\nu}}B^{\nu}+\overline{B}_{\nu}D\cdot\Phi^{\nu})-\sqrt{2}m(\overline{\tilde{\Phi}}D\cdot
B+\overline{D\cdot B}\tilde{\Phi})\,,$ $\displaystyle{\cal L}_{2,0}^{(2)}$
$\displaystyle=-m^{2}(\overline{\Phi}_{\mu\nu}\Phi^{\mu\nu}-\overline{\tilde{\Phi}}\tilde{\Phi})-\sqrt{\frac{3}{2}}m^{2}(\overline{\tilde{\Phi}}\varphi+\bar{\varphi}\tilde{\Phi})+2m^{2}\bar{\varphi}\varphi\,.$
(43)

Here we recall the shorthand notation
$D\cdot\Phi^{\mu_{1}\dots\mu_{k}}:=D_{\rho}\Phi^{\rho\mu_{1}\dots\mu_{k}}$ and
$\tilde{\Phi}^{\mu_{1}\dots\mu_{k}}:=\eta_{\rho\sigma}\Phi^{\rho\sigma\mu_{1}\dots\mu_{k}}$,
and for clarity we renamed the Stückelberg fields to $B_{\mu}$ and $\varphi$.

The minimally coupled massive gauge transformations $\delta_{0}$ are given by

$\displaystyle\delta_{0}\Phi_{\mu\nu}$
$\displaystyle=D_{(\mu}\xi_{\nu)}+\frac{m}{\sqrt{2}}\eta_{\mu\nu}\zeta\,,$
(44) $\displaystyle\delta_{0}B_{\mu}$
$\displaystyle=D_{\mu}\zeta+\frac{m}{\sqrt{2}}\xi_{\mu}\,,$
$\displaystyle\delta_{0}\varphi$ $\displaystyle=\sqrt{3}m\zeta\,.$

As before, for $Q\neq 0$ this gauge symmetry is broken and we need to add non-
minimal deformations to restore it. Assuming (SI), (MC), (PC) and (PC2)
the allowed ansatz for the Lagrangian contains 37 free parameters, whereas the
ansatz for the gauge transformations contains 42 free parameters. Imposing
massive gauge invariance leaves 12 free parameters in the Lagrangian and 18 in
the gauge transformations. The leftover free parameters arise due to field-
redefinition redundancies and higher-point effects, since none of them
contributes to the on-shell three-point amplitude ${\cal
A}(\Phi_{1}\bar{\Phi}_{2}A_{3}^{-})$, which is uniquely predicted and it
matches the known $\sqrt{\text{Kerr}}$ answer,

$\displaystyle{\cal A}(\bm{1}^{s=2}\\!,\bar{\bm{2}}^{s=2}\\!,3^{+})$
$\displaystyle=\frac{4Q}{m^{2}}{\bm{\mathbf{\varepsilon}}_{1}}\\!\cdot\\!{p_{2}}{\bm{\mathbf{\varepsilon}}_{2}}\\!\cdot\\!{p_{1}}({\bm{\mathbf{\varepsilon}}_{1}}\\!\cdot\\!{\varepsilon_{3}}\,{\bm{\mathbf{\varepsilon}}_{2}}\\!\cdot\\!{p_{3}}+{\bm{\mathbf{\varepsilon}}_{2}}\\!\cdot\\!{\varepsilon_{3}}\,\bm{\mathbf{\varepsilon}}_{1}\\!\cdot\\!p_{2}+{\bm{\mathbf{\varepsilon}}_{1}}\\!\cdot\\!{\bm{\mathbf{\varepsilon}}_{2}}\,{\varepsilon_{3}}\\!\cdot\\!{p_{1}})$
(45)
$\displaystyle\quad+Q{\bm{\mathbf{\varepsilon}}_{1}}\\!\cdot\\!{\bm{\mathbf{\varepsilon}}_{2}}(4{\bm{\mathbf{\varepsilon}}_{1}}\\!\cdot\\!{\varepsilon_{3}}\,{\bm{\mathbf{\varepsilon}}_{2}}\\!\cdot\\!{p_{3}}+4{\bm{\mathbf{\varepsilon}}_{2}}\\!\cdot\\!{\varepsilon_{3}}\,{\bm{\mathbf{\varepsilon}}_{1}}\\!\cdot\\!{p_{2}}+2\bm{\mathbf{\varepsilon}}_{1}\\!\cdot\\!\bm{\mathbf{\varepsilon}}_{2}\,{\varepsilon_{3}}\\!\cdot\\!{p_{1}})=2Q\,\varepsilon_{3}^{+}\\!\cdot\\!p_{1}\frac{\langle\bm{12}\rangle^{4}\\!}{m^{4}}.$

One explicit solution for the full cubic Lagrangian ${\cal L}_{3}$ is

${\cal L}_{3}^{(2)}={\cal L}_{3,3}^{(2)}+{\cal L}_{3,2}^{(2)}+{\cal
L}_{3,1}^{(2)}\,,$ (46)

where

$\displaystyle{\cal L}_{3,3}^{(2)}$
$\displaystyle=\frac{2iQ}{m^{2}}F^{\mu\nu}\Big{(}\overline{D_{\mu}\Phi_{\alpha\beta}}D^{\alpha}\Phi{}^{\beta}{}_{\nu}\,-\overline{D_{\alpha}\Phi_{\beta\nu}}D_{\mu}\Phi^{\alpha\beta}+\overline{D_{\alpha}\Phi_{\beta\mu}}D^{\alpha}\Phi{}^{\beta}{}_{\nu}\,-2\overline{D_{\alpha}\Phi_{\beta\mu}}D^{\beta}\Phi{}^{\alpha}{}_{\nu}\,$
$\displaystyle\quad-\overline{D_{\mu}\Phi_{\alpha\beta}}D_{\nu}\Phi^{\alpha\beta}+\overline{D_{\mu}\Phi_{\nu\alpha}}D\cdot\Phi^{\alpha}-\overline{D\cdot\Phi_{\alpha}}D_{\mu}\Phi{}_{\nu}{}^{\alpha}\,+\overline{D\cdot\Phi_{\mu}}D\cdot\Phi_{\nu}$
$\displaystyle\quad-\overline{D\cdot\Phi_{\mu}}D_{\nu}\tilde{\Phi}+\overline{D_{\nu}\tilde{\Phi}}D\cdot\Phi_{\mu}-\overline{D_{\mu}\Phi_{\nu\alpha}}D^{\alpha}\tilde{\Phi}+\overline{D_{\alpha}\tilde{\Phi}}D_{\mu}\Phi{}_{\nu}{}^{\alpha}\,+\overline{D_{\mu}\Phi}D_{\nu}\tilde{\Phi}\Big{)}\,,$
(47a) $\displaystyle{\cal L}_{3,2}^{(2)}$
$\displaystyle=\frac{i\sqrt{2}Q}{m}F^{\mu\nu}\left(\overline{D\cdot\Phi_{\mu}}B_{\nu}-\bar{B}_{\nu}D\cdot\Phi_{\mu}-\overline{D_{\mu}\tilde{\Phi}}B_{\nu}+\bar{B}_{\nu}D_{\mu}\tilde{\Phi}\right)$
(47b) $\displaystyle{\cal L}_{3,1}^{(2)}$
$\displaystyle=iQF^{\mu\nu}\left(2\bar{\Phi}_{\mu\alpha}\Phi{}_{\nu}{}^{\alpha}\,+\bar{B}_{\mu}B_{\nu}\right)\,,$
(47c)

where we recall that $\tilde{\Phi}:=\Phi^{\mu}_{\mu}$ and
$F_{\mu\nu}:=2\partial_{[\mu}A_{\nu]}$.

The non-minimal gauge transformations $\delta_{1}$ are

$\begin{split}\delta_{1}\Phi_{\mu\nu}=&\,{-}\frac{iQ}{m^{2}}\Big{\\{}(F{}_{\mu}{}^{\alpha}\,D_{[\alpha}\xi_{\nu]}+F{}_{\nu}{}^{\alpha}\,D_{[\alpha}\xi_{\mu]})+\frac{1}{2}\eta_{\mu\nu}F^{\alpha\beta}D_{[\alpha}\xi_{\beta]}\Big{\\}},\\\
\delta_{1}A_{\mu}=&\,{-}\frac{iQ}{m^{2}}\Big{\\{}2\overline{D_{\alpha}\Phi_{\beta\mu}}D^{[\alpha}\xi^{\beta]}-2\overline{D_{[\alpha}\xi_{\beta]}}D^{\alpha}\Phi{}^{\beta}{}_{\mu}\,+\sqrt{2}m(\overline{B}^{\alpha}D_{[\mu}\xi_{\alpha]}-\overline{D_{[\mu}\xi_{\alpha]}}B^{\alpha})\\\
&\qquad\quad+m^{2}(\bar{\Phi}_{\mu\alpha}\xi^{\alpha}-\bar{\xi}^{\alpha}\Phi_{\mu\alpha})-m^{2}(\bar{B}_{\mu}\zeta-\bar{\zeta}B_{\mu})\Big{\\}}\,.\end{split}$
(48)

The next step is to study the spin-2 theory in the language of massive Ward
identities, as outlined in Section 3. From the free-theory part of eq. (44),
we compute the gauge-transformed vertices

$\displaystyle\\!\\!V(\xi_{1}\bar{\Phi}_{2}A_{3})$
$\displaystyle=-\frac{ip_{1}}{2}\\!\cdot\\!\frac{\partial}{\partial\epsilon_{1}}V(\Phi_{1},\bar{\Phi}_{2},A_{3})+\frac{m}{\sqrt{2}}V(B_{1},\bar{\Phi}_{2},A_{3})\,,$
(49a) $\displaystyle\\!\\!V(\zeta_{1}\bar{\Phi}_{2}A_{3})$
$\displaystyle=\frac{m}{2\sqrt{2}}\frac{\partial}{\partial\epsilon_{1}}\\!\cdot\\!\frac{\partial}{\partial\epsilon_{1}}V(\Phi_{1}\bar{\Phi}_{2}A_{3})-ip_{1}\\!\cdot\frac{\partial}{\partial\epsilon_{1}}\\!V(B_{1}\bar{\Phi}_{2}A_{3})+\sqrt{3}mV(\varphi_{1}\bar{\Phi}_{2}A_{3})\,,$
(49b)

and impose the massive Ward identities

$V(\xi_{1}\overline{\Phi}_{2}A_{3})\big{|}_{(2,3)}=V(\zeta_{1}\overline{\Phi}_{2}A_{3})\big{|}_{(2,3)}=0$
(50)

together with the assumptions (SI), (MC), (PC) and (PC2). The solution
yields a three-point amplitude ${\cal
A}(\bm{1}^{s=2}\\!,\bar{\bm{2}}^{s=2}\\!,3)$ with a single free parameter $c$:

$\displaystyle{\cal A}(\bm{1}^{s=2}\\!,\bar{\bm{2}}^{s=2}\\!,3)=$
$\displaystyle\,Q\bigg{(}\frac{c}{m^{2}}\bm{\mathbf{\varepsilon}}_{1}\\!\cdot\\!p_{2}\,\bm{\mathbf{\varepsilon}}_{2}\\!\cdot\\!p_{1}+\frac{4\\!+\\!c}{2}\bm{\mathbf{\varepsilon}}_{1}\\!\cdot\\!\bm{\mathbf{\varepsilon}}_{2}\\!\bigg{)}\\!\big{(}\bm{\mathbf{\varepsilon}}_{1}\\!\cdot\\!\bm{\mathbf{\varepsilon}}_{2}\,\varepsilon_{3}\\!\cdot\\!p_{1}+\bm{\mathbf{\varepsilon}}_{2}\\!\cdot\\!\varepsilon_{3}\,\bm{\mathbf{\varepsilon}}_{1}\\!\cdot\\!p_{2}+\varepsilon_{3}\\!\cdot\\!\bm{\mathbf{\varepsilon}}_{1}\,\bm{\mathbf{\varepsilon}}_{2}\\!\cdot\\!p_{3}\big{)}$
$\displaystyle-\frac{Q\alpha}{2}(\bm{\mathbf{\varepsilon}}_{1}\\!\cdot\\!\bm{\mathbf{\varepsilon}}_{2})^{2}\,\varepsilon_{3}\\!\cdot\\!p_{1}\,.$
(51)

If we additionally impose the constraints (CC) and (ND), we find $c=4$,
matching eq. (45).

In summary, the combination of massive Ward identities, the current constraint
and the (ND) assumption encodes the same information on the cubic theory as
the off-shell Lagrangian gauge invariance, but the latter is considerably
simpler to implement and easier to push to higher orders.

####  3.3.2 Example: spin-3 $\sqrt{\rm Kerr}$ theory

We can repeat the above analysis for a massive spin-$3$ field coupled to
electromagnetism. We begin with the minimally coupled Lagrangian

${\cal L}_{2}^{(3)}={\cal L}_{2,2}^{(3)}+{\cal L}_{2,1}^{(3)}+{\cal
L}_{2,0}^{(3)}\,,$ (52)

where

$\displaystyle{\cal L}_{2,2}^{(3)}=$
$\displaystyle-\overline{D_{\mu}\Phi_{\nu\rho\sigma}}D^{\mu}\Phi^{\nu\rho\sigma}+3\overline{D\\!\cdot\\!\Phi_{\mu\nu}}D\\!\cdot\\!\Phi^{\mu\nu}+\frac{3}{2}\overline{D\\!\cdot\\!\tilde{\Phi}}D\\!\cdot\\!\tilde{\Phi}+3\overline{D_{\mu}\tilde{\Phi}_{\nu}}D^{\mu}\tilde{\Phi}^{\nu}$
(53a)
$\displaystyle-3\overline{D_{\mu}\tilde{\Phi}_{\nu}}D\\!\cdot\\!\Phi^{\mu\nu}-3\overline{D\\!\cdot\\!\Phi^{\mu\nu}}D_{\mu}\tilde{\Phi}_{\nu}+\overline{D_{\mu}H_{\nu\rho}}D^{\mu}H^{\nu\rho}-2\overline{D\\!\cdot\\!H_{\mu}}D\\!\cdot\\!H^{\mu}$
$\displaystyle+\overline{D_{\mu}\tilde{H}}D\\!\cdot\\!H^{\mu}+\overline{D\\!\cdot\\!H^{\mu}}D_{\mu}\tilde{H}-\overline{D_{\mu}\tilde{H}}D^{\mu}\tilde{H}-\overline{D_{\mu}B_{\nu}}D^{\mu}B^{\nu}+\overline{D\\!\cdot\\!B}D\\!\cdot\\!B+\overline{D_{\mu}\varphi}D^{\mu}\varphi\,,$
$\displaystyle{\cal L}_{2,1}^{(3)}=$
$\displaystyle\,2\sqrt{3}m\overline{\tilde{\Phi}}_{\mu}D\\!\cdot\\!H^{\mu}+2\sqrt{3}m\overline{D\\!\cdot\\!H^{\mu}}\tilde{\Phi}_{\mu}+\sqrt{3}m\overline{D\\!\cdot\\!\Phi_{\mu\nu}}H^{\mu\nu}+\sqrt{3}m\overline{H^{\mu\nu}}D\\!\cdot\\!\Phi_{\mu\nu}$
$\displaystyle+\frac{\sqrt{3}}{2}m\overline{D\\!\cdot\\!\tilde{\Phi}}\tilde{H}+\frac{\sqrt{3}}{2}m\overline{\tilde{H}}D\\!\cdot\\!\tilde{\Phi}-\sqrt{5}m\overline{\tilde{H}}D\\!\cdot\\!B-\sqrt{5}m\overline{D\\!\cdot\\!B}\tilde{H}-\sqrt{5}m\overline{D\\!\cdot\\!H_{\mu}}B^{\mu}$
$\displaystyle-\sqrt{5}m\bar{B}^{\mu}D\\!\cdot\\!H_{\mu}+\sqrt{6}m\overline{D\\!\cdot\\!B}\varphi+\sqrt{6}m\bar{\varphi}D\\!\cdot\\!B\,,$
(53b) $\displaystyle{\cal L}_{2,0}^{(3)}=$
$\displaystyle\,m^{2}\bar{\Phi}_{\mu\nu\rho}\Phi^{\mu\nu\rho}-3m^{2}\overline{\tilde{\Phi}}_{\mu}\tilde{\Phi}^{\mu}+\frac{3}{2}m^{2}\overline{\tilde{H}}\tilde{H}-\frac{3}{2}m^{2}\bar{B}_{\mu}B^{\mu}+5m^{2}\bar{\varphi}\varphi+\frac{\sqrt{15}}{2}m^{2}\overline{\tilde{\Phi}}_{\mu}B^{\mu}$
$\displaystyle+\frac{\sqrt{15}}{2}m^{2}\bar{B}^{\mu}\tilde{\Phi}_{\mu}-\sqrt{\frac{15}{2}}m^{2}\overline{\tilde{H}}\varphi-\sqrt{\frac{15}{2}}m^{2}\bar{\varphi}\tilde{H}\,.$
(53c)

As before, we use the shorthands
$D\cdot\Phi^{\mu_{1}\dots\mu_{k}}:=D_{\rho}\Phi^{\rho\mu_{1}\dots\mu_{k}}$ and
$\tilde{\Phi}^{\mu_{1}\dots\mu_{k}}:=\eta_{\rho\sigma}\Phi^{\rho\sigma\mu_{1}\dots\mu_{k}}$.
The minimally coupled gauge transformations are

$\displaystyle\delta\Phi_{\lambda\mu\nu}$
$\displaystyle=\partial_{(\lambda}\xi_{\mu\nu)}+\frac{\sqrt{3}}{4}m\eta_{(\lambda\mu}\xi_{\nu)}\,,$
(54) $\displaystyle\delta H_{\mu\nu}$
$\displaystyle=\partial_{(\mu}\xi_{\nu)}+\frac{m}{\sqrt{3}}\xi_{\mu\nu}+\frac{\sqrt{5}}{2}m\eta_{\mu\nu}\xi\,,$
$\displaystyle\delta B_{\mu}$
$\displaystyle=\partial_{\mu}\xi+\frac{\sqrt{5}}{2}m\xi_{\mu}\,,$
$\displaystyle\delta\varphi$ $\displaystyle=\sqrt{6}m\xi\,,$

in terms of the gauge parameters $\xi_{\mu\nu},\xi_{\mu}$ and $\xi$, where
$\xi^{\mu}_{\mu}=0$.

In order to restore the gauge symmetry for $Q\neq 0$, we construct an ansatz
for the non-minimal cubic Lagrangian ${\cal L}_{3}^{(3)}$ and the non-minimal
gauge transformations $\delta_{1}$. Assuming (SI), (MC), (PC) and (PC2),
the allowed ansatz contains 432 free parameters in the Lagrangian and 436 in
the gauge transformations. Imposing massive gauge invariance leaves 141 free
parameters in the Lagrangian and 260 in the gauge transformations. As before,
the leftover free parameters are due to field redefinitions and higher-point
effects, since they do not contribute to the on-shell three-point amplitude
given by

$\displaystyle{\cal A}(\bm{1}^{s=3}\\!,\bar{\bm{2}}^{s=3}\\!,3^{+})=$
$\displaystyle-\frac{8Q}{m^{4}}(\bm{\mathbf{\varepsilon}}_{1}\\!\cdot\\!p_{2})^{2}(\bm{\mathbf{\varepsilon}}_{2}\\!\cdot\\!p_{1})^{2}(\bm{\mathbf{\varepsilon}}_{1}\\!\cdot\\!{\varepsilon_{3}}\,{\bm{\mathbf{\varepsilon}}_{2}}\\!\cdot\\!{p_{3}}+{\bm{\mathbf{\varepsilon}}_{2}}\\!\cdot\\!{\varepsilon_{3}}\,\bm{\mathbf{\varepsilon}}_{1}\\!\cdot\\!p_{2}+{\bm{\mathbf{\varepsilon}}_{1}}\\!\cdot\\!{\bm{\mathbf{\varepsilon}}_{2}}\,{\varepsilon_{3}}\\!\cdot\\!{p_{1}})$
$\displaystyle-\frac{4Q}{m^{2}}{\bm{\mathbf{\varepsilon}}_{1}}\\!\cdot\\!{\bm{\mathbf{\varepsilon}}_{2}}\,{\bm{\mathbf{\varepsilon}}_{1}}\\!\cdot\\!{p_{2}}\,{\bm{\mathbf{\varepsilon}}_{2}}\\!\cdot\\!{p_{1}}\left(4{\bm{\mathbf{\varepsilon}}_{1}}\\!\cdot\\!{\varepsilon_{3}}\,{\bm{\mathbf{\varepsilon}}_{2}}\\!\cdot\\!{p_{3}}+4{\bm{\mathbf{\varepsilon}}_{2}}\\!\cdot\\!{\varepsilon_{3}}\,{\bm{\mathbf{\varepsilon}}_{1}}\\!\cdot\\!{p_{2}}+3{\bm{\mathbf{\varepsilon}}_{1}}\\!\cdot\\!{\bm{\mathbf{\varepsilon}}_{2}}\,{\varepsilon_{3}}\\!\cdot\\!{p_{1}}\right)$
$\displaystyle-2Q({\bm{\mathbf{\varepsilon}}_{1}}\\!\cdot\\!{\bm{\mathbf{\varepsilon}}_{2}})^{2}\left(3{\bm{\mathbf{\varepsilon}}_{1}}\\!\cdot\\!{\varepsilon_{3}}\,{\bm{\mathbf{\varepsilon}}_{2}}\\!\cdot\\!{p_{3}}+3{\bm{\mathbf{\varepsilon}}_{2}}\\!\cdot\\!{\varepsilon_{3}}\,{\bm{\mathbf{\varepsilon}}_{1}}\\!\cdot\\!{p_{2}}+{\bm{\mathbf{\varepsilon}}_{1}}\\!\cdot\\!{\bm{\mathbf{\varepsilon}}_{2}}\,{\varepsilon_{3}}\\!\cdot\\!{p_{1}}\right)$
$\displaystyle=$
$\displaystyle\,\,2Q\,\varepsilon_{3}^{+}\\!\cdot\\!p_{1}\frac{\langle\bm{12}\rangle^{6}\\!}{m^{6}}\,.$
(55)

This matches the known $\sqrt{\text{Kerr}}$ amplitude (1).

The explicit solutions for the non-minimal Lagrangian and gauge
transformations are rather cumbersome, so we omit them in this work. Instead,
we can reproduce the above results in a simpler fashion using massive Ward
identities. From eq. (54) we read off the gauge-transformed vertices

$\displaystyle V(\xi_{1}^{\mu\nu}\bar{\Phi}_{2}A_{3})$
$\displaystyle=-\frac{ip_{1}}{3}\\!\cdot\\!\frac{\partial}{\partial\epsilon_{1}}V(\Phi_{1},\bar{\Phi}_{2},A_{3})+\frac{m}{\sqrt{3}}V(H_{1},\bar{\Phi}_{2},A_{3})\,,$
(56) $\displaystyle V(\xi_{1}^{\mu}\bar{\Phi}_{2}A_{3})$
$\displaystyle=\frac{m}{8\sqrt{3}}\frac{\partial}{\partial\epsilon_{1}}\\!\cdot\\!\frac{\partial}{\partial\epsilon_{1}}V(\Phi_{1}\bar{\Phi}_{2}A_{3})-\frac{ip_{1}}{2}\\!\cdot\frac{\partial}{\partial\epsilon_{1}}\\!V(H_{1}\bar{\Phi}_{2}A_{3})+\frac{\sqrt{5}m}{2}V(B_{1}\bar{\Phi}_{2}A_{3})\,,$
$\displaystyle V(\xi_{1}\bar{\Phi}_{2}A_{3})$
$\displaystyle=\frac{\sqrt{5}m}{4}\frac{\partial}{\partial\epsilon_{1}}\\!\cdot\\!\frac{\partial}{\partial\epsilon_{1}}V(H_{1}\bar{\Phi}_{2}A_{3})-ip_{1}\\!\cdot\frac{\partial}{\partial\epsilon_{1}}\\!V(B_{1}\bar{\Phi}_{2}A_{3})+\sqrt{6}mV(\varphi_{1}\bar{\Phi}_{2}A_{3})\,,$

and impose the (reduced) massive Ward identities

$V(\xi_{1}^{\mu\nu}\bar{\Phi}_{2}A_{3})\big{|}_{(2,3)}=V(\xi_{1}^{\mu}\bar{\Phi}_{2}A_{3})\big{|}_{(2,3)}=V(\xi_{1}\bar{\Phi}_{2}A_{3})\big{|}_{(2,3)}=0\,,$
(57)

together with the assumptions (SI), (MC), (PC), (PC2). This leaves two
free parameters in the on-shell amplitude. Additionally imposing (CC) and
(ND) fixes the leftover freedom and reproduces the amplitude (3.3.2).

####  3.3.3 Uniqueness of spin-$s$ $\sqrt{\rm Kerr}$

In the arbitrary-spin case it is hard to obtain expressions for the non-
minimal off-shell Lagrangian and gauge transformations. The main issue is that
such objects are not uniquely defined, because they depend on field
redefinitions. One option is to study fixed values of spin and choose field
redefinitions that manifest some nice properties of the Lagrangians, in the
hope that such special choices can be extrapolated to arbitrary spin. In this
work, we instead focus on reduced Ward identities, which produced the same on-
shell results as the Lagrangian approach in the cases discussed above, but are
much simpler to implement for arbitrary spin. We write down ansätze for the
vertices $V(\Phi_{1}^{k}\Phi_{2}^{s}A_{3}^{h})$, use them to compute the gauge
variations $V(\xi_{1}^{k}\Phi_{2}^{s}A_{3}^{h})$ as discussed in Section 3.1,
and impose combinations of the conditions (MC), (PC), (WI), (CC), (ND).

Imposing (MC)+(PC)+(WI) in spin-$s$ $\sqrt{\rm Kerr}$ theory and extracting
the on-shell three-point amplitude, we find

${\cal A}(\bm{1}^{s}\\!,\bar{\bm{2}}^{s}\\!,3^{+})={\cal
A}_{3}^{(0)}\frac{\langle\bm{12}\rangle^{2s}}{m^{2s}}\left\\{1+\sum_{k=1}^{s-1}c_{k}\left(\frac{[\bm{12}]^{k}}{\langle\bm{12}\rangle^{k}}-1\right)\right\\},$
(58)

where $c_{k}$ are free parameters and ${\cal
A}_{3}^{(0)}=2Q(\varepsilon_{3}\cdot p_{1})$ is the scalar amplitude. Imposing
(ND) does not modify eq. (58), whereas (CC) requires $\sum_{k}c_{k}=0$.
However, (CC)+(ND) fix $c_{k}=0$ and yield eq. (3.3.2).

The calculations outlined above for the $\sqrt{\text{Kerr}}$ theories were
performed explicitly up to $s\leq 10$. However, we do not expect any new
feature beyond this spin and believe the above results to be valid for any
spin. Note that the off-shell analysis done in the previous sections lands us
unambiguously on the $\sqrt{\text{Kerr}}$ amplitude, and it only required the
conditions (SI), (PS), (PC), (PC2) and (MC). We believe that a complete
set of Ward identities is equivalent to the off-shell constraints, and hence
it will only require the conditions (MC)+(PC)+(WI) to derive the
$\sqrt{\text{Kerr}}$ amplitudes. By contrast, the reduced Ward identities
(with one leg always having spin $s$) leave us with free parameters, which
are, nevertheless, fixed with the help of the constraints (CC) and (ND).
Therefore, provided we follow the off-shell approach or impose all Ward
identities, the latter constraints are redundant.

####  3.3.4 On-shell cubic $\sqrt{\rm Kerr}$ Lagrangians

As discussed, it is difficult to obtain arbitrary-spin expressions for off-
shell Lagrangians, due to their redundant nature. However, it is much easier
to write down the part of the arbitrary-spin Lagrangian that contributes to
the on-shell three-point amplitude. In this subsection, we derive expressions
for arbitrary-spin on-shell Lagrangians that reproduce the
$\sqrt{\text{Kerr}}$ amplitudes (1), in terms of a convenient choice of
variables.

It is convenient to re-express the spin-$s$ field as a tensor product of $s$
spin-1 fields in the (generalized) double-copy sense:

$\Phi^{s}=\Phi^{\mu_{1}\mu_{2}\ldots\mu_{s}}=W^{(\mu_{1}}\otimes
W^{\mu_{2}}\otimes\cdots\otimes W^{\mu_{s})}\,,$ (59)

where the tensor product is understood to hold in momentum space, or
equivalently as a convolution in position space. We expand the $\sqrt{\rm
Kerr}$ gauge theory Lagrangian in terms of operators contributing at different
orders in the coupling $g$

${\cal L}_{\sqrt{\rm Kerr}}^{(s)}=-\frac{1}{4}(F_{\mu\nu})^{2}+{\cal
L}_{2}^{(s)}+ig{\cal L}_{3}^{(s)}+g^{2}{\cal L}_{4}^{(s)}+\ldots\,.$ (60)

Up to cubic order we may assume, without loss of generality, that the non-
trivial interactions are abelian and hence $g=Q$. Moreover, we will only
analyze the Lagrangian terms that contribute on shell at three points.

The minimal Lagrangian, obtained from the covariantization of eq. (26), is

${\cal
L}_{2}^{(s)}=(-1)^{s}\Big{(}\overline{D_{\mu_{0}}\Phi_{\mu_{1}\ldots\mu_{s}}}D^{\mu_{0}}\Phi^{\mu_{1}\ldots\mu_{s}}-m^{2}\bar{\Phi}_{\mu_{1}\ldots\mu_{s}}\Phi^{\mu_{1}\ldots\mu_{s}}+\ldots\Big{)}\,,$
(61)

where the ellipsis correspond to terms that vanish on shell. Recall that the
covariant derivative is
$D_{\mu_{0}}\Phi_{\mu_{1}\ldots\mu_{s}}=\partial_{\mu_{0}}\Phi_{\mu_{1}\ldots\mu_{s}}-iQA_{\mu_{0}}\Phi_{\mu_{1}\ldots\mu_{s}}$
thus the minimal-coupling cubic interaction terms coming from ${\cal
L}_{2}^{(s)}$ are

$(-1)^{s}iQ\partial_{\mu_{0}}\Phi_{\mu_{1}\ldots\mu_{s}}A^{\mu_{0}}\bar{\Phi}^{\mu_{1}\ldots\mu_{s}}\\!-(\Phi\leftrightarrow\bar{\Phi})~{}\to~{}2Q(\varepsilon_{3}\cdot
p_{1})(-{\bm{\varepsilon}}_{1}\cdot{\bm{\varepsilon}}_{2})^{s}\,.$ (62)

In terms of spinor-helicity variables, the minimal-coupling gauge-theory
amplitude for a positive-helicity photon is then

${\cal
A}_{\text{min}}(\bm{1}^{s}\\!,\bar{\bm{2}}^{s}\\!,3^{+})=2Q(\varepsilon_{3}^{+}\\!\cdot
p_{1})\frac{\langle\bm{12}\rangle^{s}[\bm{12}]^{s}}{m^{2s}}\,.$ (63)

and the negative-helicity case is simply obtained by
$\varepsilon_{3}^{+}\to\varepsilon_{3}^{-}$. There is no additional
information in the negative-helicity amplitude since we will only use parity-
invariant operators, hence we can focus on the positive-helicity case.

It is now an easy task to compute the difference between the above amplitude
and the $\sqrt{\text{Kerr}}$ amplitude (1) and reverse-engineer the cubic
operators that correspond to the mismatch. We decorate the on-shell part of
the cubic Lagrangians with a tilde to emphasize that they need to be
supplemented by off-shell-completion terms to restore Zinoviev’s massive gauge
invariance,

${\cal L}_{3}^{(s)}=\tilde{\cal L}_{3}^{(s)}+\text{(gauge-invariance
completion terms)}\,.$ (64)

At spin $s=1$, one can identify two independent candidate operators that
contribute on shell,

$\displaystyle{\cal O}_{1}$
$\displaystyle=-\overline{W}_{\mu}F^{\mu\nu}W_{\nu}~{}~{}~{}~{}~{}~{}~{}~{}~{}\to~{}~{}~{}~{}~{}\,\frac{2Q}{m^{2}}(\varepsilon_{3}^{+}\\!\cdot
p_{1})\langle\bm{12}\rangle(\langle\bm{12}\rangle-[\bm{12}])\,,$ (65)
$\displaystyle{\cal O}_{2}$
$\displaystyle=-\frac{1}{m^{2}}\overline{W}_{\rho\mu}F^{\mu\nu}W_{\nu}{}^{\rho}~{}~{}\to~{}~{}-\frac{2Q}{m^{2}}(\varepsilon_{3}^{+}\\!\cdot
p_{1})[\bm{12}](\langle\bm{12}\rangle-[\bm{12}])\,,$

where $F_{\mu\nu}=2\partial_{[\mu}A_{\nu]}$ and
$W_{\mu\nu}=2\partial_{[\mu}W_{\nu]}$ are (linearized) field strengths for the
massive and massless fields (here $\Phi^{\mu}=W^{\mu}$). Using the spinor-
helicity representations on the right-hand side, it is clear that $\tilde{\cal
L}_{3}^{(1)}={\cal O}_{1}$ reproduces the $\sqrt{\rm Kerr}$ amplitude.

For $s>1$ we can construct the $\tilde{\cal L}_{3}^{(s)}$ operators
recursively, by tensoring the ${\cal O}_{1}$ and ${\cal O}_{2}$ operators with
spin-1 building blocks that are insensitive to the photon helicity. For three-
point on-shell kinematics there are only two such independent building blocks,

$\displaystyle u$
$\displaystyle=-\frac{1}{m^{2}}\overline{W}_{\mu\nu}W^{\mu\nu}\to\frac{1}{m^{2}}\big{(}\langle\bm{12}\rangle^{2}+[\bm{12}]^{2}\big{)}\,,$
(66) $\displaystyle v$
$\displaystyle=-\overline{W}_{\mu}W^{\mu}\to\frac{\langle\bm{12}\rangle[\bm{12}]}{m^{2}}\,.$

Using eq. (59) to decompose $\Phi_{\mu_{1}\ldots\mu_{s}}$ into spin-1 fields,
we can now write down $\tilde{\cal L}_{3}^{(s)}$ to any spin. Through spin 3,
the non-minimal interaction terms are

$\displaystyle\tilde{\cal L}_{3}^{(0)}$
$\displaystyle=0\,,\qquad~{}~{}\>\\!\quad\tilde{\cal L}_{3}^{(2)}=(u+v){\cal
O}_{1}+v{\cal O}_{2}\,,$ (67) $\displaystyle\tilde{\cal L}_{3}^{(1)}$
$\displaystyle={\cal O}_{1}\,,\qquad\quad\tilde{\cal
L}_{3}^{(3)}=(u+v)\big{(}u{\cal O}_{1}+v{\cal O}_{2}\big{)}\,.$

When combined with the the minimally-coupled interactions (62), these terms
reproduce the $\sqrt{\text{Kerr}}$ amplitudes up to spin 3.

For spin 4 and higher, we find that the non-minimal $\sqrt{\text{Kerr}}$
interaction terms satisfy the following recurrence formula,

$\displaystyle\tilde{\cal L}^{(s)}_{3}=(u^{2}-2v^{2})\tilde{\cal
L}^{(s-2)}_{3}-v^{4}\tilde{\cal L}^{(s-4)}_{3}+(2v+u)v^{s-2}\big{(}{\cal
O}_{1}+{\cal O}_{2}\big{)}\,.$ (68)

For example, for spin $s=4,5,6$ we have the non-minimal interactions

$\displaystyle\tilde{\cal L}_{3}^{(4)}$ $\displaystyle=u(u^{2}+uv-v^{2}){\cal
O}_{1}+uv(u+v){\cal O}_{2}\,,$ (69a) $\displaystyle\tilde{\cal L}_{3}^{(5)}$
$\displaystyle=(u^{2}-v^{2})(u^{2}+uv-v^{2}){\cal
O}_{1}+uv(u^{2}+uv-v^{2}){\cal O}_{2}\,,$ (69b) $\displaystyle\tilde{\cal
L}_{3}^{(6)}$ $\displaystyle=(u^{2}-v^{2})(u^{3}+u^{2}v-2uv^{2}-v^{3}){\cal
O}_{1}+v(u^{2}-v^{2})(u^{2}+uv-v^{2}){\cal O}_{2}\,.$ (69c)

As an alternative to the recursive formula, we can also explore a generating-
function description of the cubic on-shell Lagrangian $\tilde{\cal L}_{3}$.
Indeed, it is straightforward to obtain the generating function by summing eq.
(68) over all spins, giving an equation for the sum with solution

$\sum_{s=0}^{\infty}\tilde{\cal L}^{(s)}_{3}=\frac{{\cal O}_{1}+v{\cal
O}_{2}}{(1-v)(1-u+v^{2})}\,$ (70)

For simplicity, we have suppressed the auxiliary generating variable, since
this leads to no ambiguity.

In terms of standard Lagrangian notation, using the fields
$\Phi^{\mu_{1}\cdots\mu_{s}}$ and covariantizing the derivatives, the non-
minimal cubic interactions up to spin-3 take the form

$\displaystyle\tilde{\cal L}^{(1)}_{3}=$
$\displaystyle-\bar{\Phi}_{\mu}F^{\mu\nu}\Phi_{\nu}\,,$ (71a)
$\displaystyle\tilde{\cal L}^{(2)}_{3}=$
$\displaystyle\;\bar{\Phi}_{\mu\nu}F^{\nu}{}_{\rho}\Phi^{\rho\mu}+\frac{4}{m^{2}}\overline{D_{[\mu}\Phi_{\nu]\rho}}F^{\rho\sigma}D^{[\mu}\Phi^{\nu]}_{\
\
\sigma}+\frac{4}{m^{2}}\overline{D_{[\mu}\Phi_{\nu]\rho}}F^{\nu}{}_{\sigma}D^{[\sigma}\Phi^{\mu]\rho}\,,$
(71b) $\displaystyle\tilde{\cal L}^{(3)}_{3}=$
$\displaystyle-\frac{4}{m^{2}}\overline{D_{[\mu}\Phi_{\nu]\rho\lambda}}F^{\rho}{}_{\sigma}D^{[\mu}\Phi^{\nu]\sigma\lambda}-\frac{4}{m^{2}}\overline{D_{[\mu}\Phi_{\nu]\rho\lambda}}F^{\nu}{}_{\sigma}D^{[\sigma}\Phi^{\mu]\rho\lambda}$
(71c)
$\displaystyle\hbox{}-\frac{2^{4}}{m^{4}}\overline{D^{[\nu}D_{[\mu}\Phi_{\rho]}{}^{\sigma]\lambda}}F_{\lambda}{}^{\kappa}{D_{[\nu}D^{[\mu}\Phi^{\rho]}{}_{\sigma]\kappa}}-\frac{2^{4}}{m^{4}}\overline{D^{[\sigma}D_{[\mu}\Phi_{\rho]}{}^{\nu]\lambda}}F_{\nu}{}^{\kappa}{D_{[\kappa}D^{[\mu}\Phi^{\rho]}{}_{\sigma]\lambda}}\,.$

Note that the above Lagrangians have derivatives appearing only as curls of
the spin-$s$ field, which is the general structure of interactions of massless
fields, see e.g. ref. Grigoriev:2020lzu .

As mentioned above, eq. (71) contains the operators that contribute to the on-
shell three-point amplitude. In order to have the full cubic Lagrangian ${\cal
L}^{(s)}_{3}$, we would need to add additional terms that vanish on shell, yet
they are needed to preserve massive gauge invariance. For the highest
derivative terms in ${\cal L}^{(s)}_{3}$ we can find the completions, since
they correspond to having gauge invariance in the high-energy limit. The
following leading-derivative terms are compatible with the above on-shell
analysis and high-energy gauge invariance:

$\displaystyle{\cal L}^{(s)}_{3}\\!=(-1)^{s}\frac{2^{2s-2}}{m^{2s-2}}$
$\displaystyle\bigg{(}\,\overline{\Phi^{[\mu_{1};\nu_{1}][\mu_{2};\nu_{2}]\cdots[\mu_{s-1};\nu_{s-1}]\rho}}F_{\rho}{}^{\sigma}\Phi_{[\mu_{1};\nu_{1}][\mu_{2};\nu_{2}]\cdots[\mu_{s-1};\nu_{s-1}]\sigma}$
$\displaystyle+\overline{\Phi^{[\mu_{1};\sigma][\mu_{2};\nu_{2}]\cdots[\mu_{s-1};\nu_{s-1}]\rho}}F_{\mu_{1}}{}^{\nu_{1}}\Phi_{[\sigma;\nu_{1}][\mu_{2};\nu_{2}]\cdots[\mu_{s-1};\nu_{s-1}]\rho}$
(72)
$\displaystyle+\frac{1}{2}\overline{D_{\rho}\Phi^{[\mu_{1};\nu_{1}][\mu_{2};\nu_{2}]\cdots[\mu_{s-1};\nu_{s-1}]\rho}}F_{\mu_{1}}{}^{\sigma}\Phi_{[\mu_{2};\nu_{2}]\cdots[\mu_{s-1};\nu_{s-1}]\sigma\nu_{1}}$
$\displaystyle+\frac{1}{2}\overline{\Phi^{[\mu_{2};\nu_{2}]\cdots[\mu_{s-1};\nu_{s-1}]\rho\nu_{1}}\\!}F_{\rho}{}^{\mu_{1}}D^{\sigma}\Phi_{[\mu_{1};\nu_{1}][\mu_{2};\nu_{2}]\cdots[\mu_{s-1};\nu_{s-1}]\sigma}\bigg{)}\\!+{\cal
O}\Big{(}\frac{1}{m^{2s-4}}\\!\Big{)},$

where the $[\mu;\nu]$ notation is shorthand for curls, e.g. with one curl we
have
$\Phi^{\mu_{1}\mu_{2}\ldots[\mu_{i};\nu_{i}]\ldots\mu_{s}}:=D^{[\nu_{i}}\Phi^{\mu_{1}\mu_{2}\ldots\mu_{i}]\ldots\mu_{s}}$,
and similarly for taking curls repeatedly. The Stückelberg fields all enter
${\cal L}^{(s)}_{3}$ as lower derivative terms, thus they are hidden in the
${\cal O}({m^{-2s+4}})$ corrections.

###  3.4 Massive gauge symmetry at quartic order

Finding consistent quartic interactions for massive higher-spin fields is a
notoriously hard problem. In the Stückelberg formalism discussed in this work,
it requires solving gauge invariance to quadratic order in the coupling $Q$,

$(\delta_{0}+\delta_{1}+\delta_{2})({\cal L}_{2}+{\cal L}_{3}+{\cal
L}_{4})={\cal O}(Q^{3})\,.$ (73)

At linear order, the above equation reduces to eq. (12) and the implementation
of gauge invariance has already been discussed. The new information comes at
${\cal O}(Q^{2})$, where the non-minimal quartic interactions in ${\cal
L}_{4}$ starts contributing. At this order, we need to compute the variation
of the quartic vertices under the free-theory gauge transformations, namely
$\delta_{0}{\cal L}_{4}$ and also $\delta_{0}({\cal L}_{2}+{\cal L}_{3})$ from
expanding the covariant derivatives. We also need to vary the free Lagrangian
${\cal L}_{2}|_{Q=0}$ under ${\cal O}(Q^{2})$ corrections to the gauge
transformations, $\delta_{2}$, which are quadratic in the fields and linear in
the gauge parameters. Both ${\cal L}_{4}$ and $\delta_{2}$ require writing
down an ansatz and hence introducing new free parameters.

The number of derivatives required in the ansatz, and hence the number of free
parameters, increase rapidly with spin. To see this, we consider the
contribution of the three-point vertices to the four-point Compton amplitude,
given by

$\bigg{\\{}\Delta_{P}\Big{(}\frac{\partial}{\partial\epsilon_{P}},\frac{\partial}{\partial\epsilon_{{-}P}}\Big{)}\sum_{l=0}^{s}V(\Phi^{s}_{1}\bar{\Phi}^{l}_{P}A_{4})V(\Phi^{l}_{{-}P}\bar{\Phi}^{s}_{2}A_{3})~{}+~{}(3\leftrightarrow
4)\bigg{\\}}\bigg{|}_{\epsilon_{P},\epsilon_{{-}P}\to 0}\,,$ (74)

where the Feynman propagator-generating function
$\Delta_{P}(\epsilon,\bar{\epsilon})$ (30) has become an operator through the
substitution of its arguments by derivatives, which act on the polarization
vectors of the internal line. The lowest-derivative solution for the three-
point vertices contains $2s{-}1$ derivatives, as discussed in the previous
sections. Since eq. (74) contains two such vertices and a propagator, and the
latter subtracts two derivatives, we expect at least $4s{-}4$ derivatives in
the Compton amplitude and in the four-point contact terms in ${\cal L}_{4}$.
This will produce up to $4s{-}3$ derivatives in the term $\delta_{0}{\cal
L}_{4}$. We expect the same power of derivatives in the term $\delta_{2}{\cal
L}_{2}$, and since ${\cal L}_{2}$ has two derivatives, $\delta_{2}$ should
have $4s{-}5$.

Moreover, we need to consider the variation of the cubic vertices under the
first non-linear corrections to the gauge transformations, for instance in the
term $\delta_{1}{\cal L}_{3}$. Both $\delta_{1}$ and ${\cal L}_{3}$ are
partially fixed by the higher-spin generalization of eq. (12), but in general
there are leftover free parameters. Provided we fix the cubic amplitude, all
the leftover parameters are associated with field redefinitions and with
redefinitions of gauge parameters, i.e. they represent no physical information
and can be set at will, for instance, to zero. Nevertheless, eq. (73) at
${\cal O}(Q^{2})$ is a system of coupled equations containing many tensor
structures, and it is very hard to solve in general.

Alternatively, we can resort to the same Ward identities that provided
simplification in the cubic case. This requires computing a four-point Compton
amplitude using the three-point solutions to the cubic Ward identities and the
Feynman-gauge propagators defined in eq. (30). More precisely, we need to
compute the abelian amputated correlation function111111We refer to eq. (3.4)
as an amputated correlation function, since it can be obtained from an
ordinary correlation function by stripping the propagators off the external
legs.

$\displaystyle\langle\Phi^{k}_{1}\bar{\Phi}^{s}_{2}A_{3}A_{4}\rangle_{\rm
U(1)}\\!:=\,$
$\displaystyle\bigg{\\{}\Delta_{P}\Big{(}\frac{\partial}{\partial\epsilon_{P}},\frac{\partial}{\partial\epsilon_{{-}P}}\Big{)}\sum_{l=0}^{s}V(\Phi^{k}_{1}\bar{\Phi}^{l}_{P}A_{4})V(\Phi^{l}_{{-}P}\bar{\Phi}^{s}_{2}A_{3})+(3\leftrightarrow
4)\bigg{\\}}\Big{|}_{\epsilon_{P},\epsilon_{{-}P}\to 0}$
$\displaystyle\hbox{}+V_{{\cal
L}_{2}}(\Phi^{k}_{1}\bar{\Phi}^{s}_{2}A_{3}A_{4})+V_{{\cal
L}_{3}}(\Phi^{k}_{1}\bar{\Phi}^{s}_{2}A_{3}A_{4})+V_{{\cal
L}_{4}}(\Phi^{k}_{1}\bar{\Phi}^{s}_{2}A_{3}A_{4})\,.$ (75)

The terms $V_{{\cal L}_{2}}$ and $V_{{\cal L}_{3}}$ pick up the quartic terms
from ${\cal L}_{2}\sim(D\Phi)^{2}$ and ${\cal L}_{3}$ respectively, which are
due to $A_{\mu}$ inside the covariant derivatives $D_{\mu}$. In addition, we
need the four-point contact terms of form

$V_{{\cal L}_{4}}(\Phi^{k}_{1}\bar{\Phi}^{s}_{2}A_{3}A_{4})={\rm
Ansatz}^{(k,s)}(p_{i},\epsilon_{i})\,,$ (76)

where the ansatz for the contact terms must involve two field strengths ${\cal
L}_{4}\propto F_{\mu\nu}F_{\rho\sigma}$.

Let us also define a gauge-transformed amputated correlator for each gauge
parameter $\xi^{k}$, in the same fashion as in eq. (35),

$\langle\xi_{1}^{k}\bar{\Phi}_{2}^{s}A_{3}A_{4}\rangle\\!:=m\alpha_{k}\langle\Phi_{1}^{k}\bar{\Phi}_{2}^{s}A_{3}A_{4}\rangle-\frac{ip_{1}}{k\\!+\\!1}\cdot\frac{\partial~{}}{\partial\epsilon_{1}}\langle\Phi_{1}^{k+1}\bar{\Phi}_{2}^{s}A_{3}A_{4}\rangle+\frac{m}{2}\beta_{k+2}\bigg{(}\\!\frac{\partial~{}}{\partial\epsilon_{1}}\\!\bigg{)}^{\\!2}\\!\langle\Phi_{1}^{k+2}\bar{\Phi}_{2}^{s}A_{3}A_{4}\rangle.$
(77)

The four-point Ward identities then become

$\langle\xi^{k}_{1}\bar{\Phi}^{s}_{2}A_{3}A_{4}\rangle\big{|}_{(2,3,4),\epsilon_{1}^{2}=0}~{}=~{}0\,,$
(78)

where legs 2, 3 and 4 are on-shell states, meaning that they obey
$p_{i}^{2}-m_{i}^{2}=\varepsilon_{i}^{2}=\varepsilon_{i}\cdot p_{i}=0$, and
the leg 1 is traceless $\epsilon_{1}^{2}=0$. Likewise, the spin-$s$ amplitude
can be obtained from the amputated abelian correlator, where all legs are
taken on shell:

$A_{\rm
U(1)}(\bm{1}^{s}\\!,\bar{\bm{2}}^{s}\\!,3,4):=-\frac{1}{2}\langle\Phi^{s}\bar{\Phi}^{s}A_{3}A_{4}\rangle_{\rm
U(1)}\Big{|}_{(1,2,3,4),\;Q\to 1}\,,$ (79)

where have removed the coupling constant and tweaked the overall normalization
to be compatible with later formulae of color-stripped Compton amplitudes. In
general, $A(...)$ denote amplitudes without couplings and color factors, as
opposed to full amplitudes ${\cal A}(...)$.

When using Ward identities, only the free-theory gauge transformations
$\delta_{0}\big{|}_{Q=0}$ contribute, so we do not have to worry about non-
linear corrections $\delta_{1}$ and $\delta_{2}$. As discussed, the number of
terms in the ansatz for $\delta_{2}$ increase rapidly with increasing spins,
so this provides a major simplification. In general, eq. (78) is also a
quadratic equations, since the three-point free parameters in
$V(\Phi_{i}^{s^{\prime}}\bar{\Phi}_{j}^{s}A_{k})$ get squared in eq. (3.4). In
practice, however, once we fix the on-shell three-point amplitude as described
in previous sections, the free parameters left in
$V(\Phi_{i}^{s^{\prime}}\bar{\Phi}_{j}^{s}A_{k})$ are four-point contact
terms, since they vanish in the factorization limit. Therefore, we can choose
to reabsorb them in the ansatz for
$V(\Phi^{k}_{1}\bar{\Phi}^{s}_{2}A_{3}A_{4})$ and thus linearize the system of
equations (78).

We can also study the non-abelian case. We work at the level of color-ordered
amplitudes, hence the cubic vertices are the same as the ones discussed in
previous sections, once the abelian charge $Q$ is replaced by the non-abelian
gauge coupling $Q\rightarrow g/\sqrt{2}$. Then we compute the non-abelian
color-ordered amputated correlation function

$\displaystyle\langle\Phi^{k}_{1}\bar{\Phi}^{s}_{2}A_{3}A_{4}\rangle_{\text{ord}}:=$
$\displaystyle\bigg{\\{}\Delta_{P}\bigg{(}\frac{\partial}{\partial\epsilon_{P}},\frac{\partial}{\partial\epsilon_{{-}P}}\bigg{)}\sum_{l=0}^{s}V(\Phi^{k}_{1}\bar{\Phi}^{l}_{P}A_{4})V(\Phi^{l}_{{-}P}\bar{\Phi}^{s}_{2}A_{3})\bigg{\\}}\bigg{|}_{\epsilon_{P},\epsilon_{{-}P}\to
0}$ (80)
$\displaystyle+\frac{1}{s_{12}}\frac{\partial}{\partial\epsilon_{q}}{\cdot}\frac{\partial}{\partial\epsilon_{-q}}V(\Phi^{k}_{1}\bar{\Phi}^{s}_{2}A_{q})V_{g}(A_{-q}A_{3}A_{4})$
$\displaystyle\hbox{}+\widetilde{V}_{{\cal
L}_{2}}(\Phi^{k}_{1}\bar{\Phi}^{s}_{2}A_{3}A_{4})+\widetilde{V}_{{\cal
L}_{3}}(\Phi^{k}_{1}\bar{\Phi}^{s}_{2}A_{3}A_{4})+\widetilde{V}_{{\cal
L}_{4}}(\Phi^{k}_{1}\bar{\Phi}^{s}_{2}A_{3}A_{4})\,.$

Note that in this case we only need one massive-exchange diagram, given by the
first line, since the other one in eq. (3.4) contributes to a different color
ordering. Instead, we need to add the gluon-exchange diagram, given by the
second line, where $V_{g}(A_{i}A_{j}A_{k})$ is the color-ordered three-gluon
vertex and the derivative operator
$(\partial/\partial{\epsilon_{q}})\cdot(\partial/\partial{\epsilon_{-q}})$
contracts the Lorentz indices corresponding to $A_{q}$ and $A_{-q}$
polarizations.

As before, $\widetilde{V}_{{\cal L}_{2}}$ and $\widetilde{V}_{{\cal L}_{2}}$
are the quartic terms coming from ${\cal L}_{2}$ and ${\cal L}_{3}$. Then we
need the four-point contact terms involving two field strengths,

$\widetilde{V}_{{\cal L}_{4}}(\Phi^{k}_{1}\bar{\Phi}^{s}_{2}A_{3}A_{4})={\rm
Ansatz}^{(k,s)}(p_{i},\epsilon_{i})\,.$ (81)

Note that in the abelian case (76) the vertex $V_{{\cal
L}_{4}}(\Phi^{k}_{1}\bar{\Phi}^{s}_{2}A_{3}A_{4})$ must be symmetric under
$1\leftrightarrow 2$ exchange and $3\leftrightarrow 4$ exchange separately. In
the non-abelian case, instead, the vertex is only symmetric under the
combination of both exchanges, therefore the ansatz will contain additional
free parameters.

In the rest of this section, we will apply the Ward identities outlined above
to the cases of spin-2 and spin-3 particles and will derive explicit abelian
and non-abelian Compton amplitudes. The latter are color-decomposed as
follows:

$\displaystyle{\cal
A}(\bm{1}^{s}\\!,\bar{\bm{2}}^{s}\\!,3,4):=2g^{2}\Big{[}T^{c_{3}}T^{c_{4}}A(\bm{1}^{s}\\!,\bar{\bm{2}}^{s}\\!,3,4)+T^{c_{4}}T^{c_{3}}A(\bm{1}^{s}\\!,\bar{\bm{2}}^{s}\\!,4,3)\Big{]}=\langle\Phi^{s}\bar{\Phi}^{s}A_{3}A_{4}\rangle\big{|}_{(1,2,3,4)}\,,$
$\displaystyle\langle\Phi^{s}\bar{\Phi}^{s}A_{3}A_{4}\rangle:=2T^{c_{3}}T^{c_{4}}\langle\Phi^{s}\bar{\Phi}^{s}A_{3}A_{4}\rangle_{\rm
ord}+2T^{c_{4}}T^{c_{3}}\langle\Phi^{s}\bar{\Phi}^{s}A_{4}A_{3}\rangle_{\rm
ord}\,.$ (82)

Namely, when extracting the ordered amplitudes from the four-point color-
dressed amplitude, we choose to strip $2g^{2}$ along with the color factor. We
can also relate the amputated correlators of the abelian and ordered type,
using $g^{2}=2Q^{2}$,

$\langle\Phi^{s}\bar{\Phi}^{s}A_{3}A_{4}\rangle_{\rm
U(1)}=\langle\Phi^{s}\bar{\Phi}^{s}A_{3}A_{4}\rangle_{\rm
ord}+\langle\Phi^{s}\bar{\Phi}^{s}A_{4}A_{3}\rangle_{\rm ord}\,,$ (83)

which is an echo of the usual Kleiss-Kuijf relation for color-ordered partial
amplitudes.

####  3.4.1 Quartic spin-2 interactions

We can consider the opposite-helicity amplitude
$A(\bm{1}^{s=2}\\!,\bar{\bm{2}}^{s=2}\\!,3^{-}\\!,4^{+})$ between two massive
spin-2 fields and two massless spin-1 bosons. We first construct an ansatz for
the off-shell quartic vertices with the following schematic derivative counts:

$\displaystyle V(\Phi^{2}_{1}\bar{\Phi}^{2}_{2}A_{3}A_{4})$
$\displaystyle\sim\partial^{4}\,,$ (84) $\displaystyle
V(\Phi^{1}_{1}\bar{\Phi}^{2}_{2}A_{3}A_{4})$
$\displaystyle\sim\partial^{5}\,,$ $\displaystyle
V(\Phi^{0}_{1}\bar{\Phi}^{2}_{2}A_{3}A_{4})$ $\displaystyle=0\,,$

where the last vertex is set to zero from the start, as it is consistent with
our guiding principles.

In the abelian case, imposing Ward identities yields the amplitude
Cangemi:2022bew

$\displaystyle A_{\rm
U(1)}(\bm{1}^{s=2}\\!,\bar{\bm{2}}^{s=2}\\!,3^{-}\\!,4^{+})=\frac{\langle
3|1|4]^{2}(U+V)^{4}}{m^{8}t_{13}t_{14}}-\frac{\langle\bm{1}3\rangle\langle
3|1|4][4\bm{2}]}{m^{8}t_{13}}4U(U^{2}+V^{2})$ (85)
$\displaystyle+\,\frac{\langle\bm{1}3\rangle\langle
3\bm{2}\rangle[\bm{1}4][4\bm{2}]}{m^{8}}(3U^{2}+V^{2})$
$\displaystyle+C^{(2)}_{-+}\,,$

where we use the variables
$U=\frac{1}{2}(\langle\bm{1}|4|\bm{2}]-\langle\bm{2}|4|\bm{1}])-m[\bm{12}]$,
$V=\frac{1}{2}(\langle\bm{1}|4|\bm{2}]+\langle\bm{2}|4|\bm{1}])$ introduced in
ref. Cangemi:2022bew , and the contact term is

$\displaystyle C^{(2)}_{-+}=\frac{c_{1}}{m^{6}}$
$\displaystyle(\langle\bm{12}\rangle+[\bm{12}])^{2}\langle\bm{1}3\rangle\langle
3\bm{2}\rangle[\bm{1}4][4\bm{2}]+\frac{c_{2}}{m^{6}}(\langle\bm{12}\rangle-[\bm{12}])^{2}\langle\bm{1}3\rangle\langle
3\bm{2}\rangle[\bm{1}4][4\bm{2}]$ $\displaystyle+\frac{c_{3}}{m^{8}}$
$\displaystyle\Big{(}\langle 3|1|4]\langle\bm{12}\rangle\big{(}\langle
3|1|4][\bm{12}]\langle\bm{12}\rangle^{2}+(\langle\bm{12}\rangle+[\bm{12}])(\langle\bm{1}|3|\bm{2}]\langle\bm{2}3\rangle[\bm{1}4]+\langle\bm{2}|3|\bm{1}]\langle\bm{1}3\rangle[4\bm{2}])\big{)}$
$\displaystyle~{}\;+s_{12}\langle\bm{12}\rangle[\bm{12}]\langle\bm{1}3\rangle\langle
3\bm{2}\rangle[\bm{1}4][4\bm{2}]\Big{)}\,.$ (86)

The three coefficients $c_{n}$ are free parameters not fixed by the Ward
identities, nor by off-shell gauge invariance. In the non-abelian case, were
we use the same power counting for the derivative in the vertices
$\tilde{V}(\Phi^{k}\bar{\Phi}^{2}A_{3}A_{4})$ as above, and after imposing the
Ward identities we obtain the color-ordered amplitude

$\displaystyle
A(\bm{1}^{s=2}\\!,\bar{\bm{2}}^{s=2}\\!,3^{-}\\!,4^{+})=\frac{\langle
3|1|4]^{2}(U+V)^{4}}{m^{8}s_{12}t_{14}}-\frac{\langle\bm{1}3\rangle\langle
3|1|4][4\bm{2}]}{m^{8}s_{12}}4U(U^{2}+V^{2})$ (87)
$\displaystyle\qquad+\frac{\langle\bm{1}3\rangle\langle
3\bm{2}\rangle[\bm{1}4][4\bm{2}]}{m^{4s}s_{12}}\big{[}t_{13}(3U^{2}+V^{2}+W_{-}^{2}-W_{+}^{2})-4m^{2}V(U+W_{+})\big{]}+\tilde{C}^{(2)}_{-+}\,,$

where we use the shorthand
$W_{\pm}=\frac{m}{2}(\langle\bm{12}\rangle\pm[\bm{12}])$. The contact term now
contains one additional free parameter,

$\tilde{C}^{(2)}_{-+}=C^{(2)}_{-+}+\frac{c_{4}}{m^{6}}\langle\bm{12}\rangle[\bm{12}]\big{(}\langle\bm{2}3\rangle^{2}[\bm{1}4]^{2}-\langle\bm{1}3\rangle^{2}[4\bm{2}]^{2}\big{)}\,,$
(88)

which we cannot fix in the absence of further constraints.

We can also consider the consequences for other helicity sectors, since our
procedure gives amplitudes expressed in terms of covariant polarization
vectors. Specializing to the positive same-helicity amplitude gives

$A^{\rm
abelian}(\bm{1}^{s=2}\\!,\bar{\bm{2}}^{s=2}\\!,3^{+}\\!,4^{+})=\frac{\langle\bm{12}\rangle^{4}[34]^{2}}{m^{2}t_{13}t_{14}}+C^{(2)}_{++}\,,$
(89)

where the contact term contains an additional five free parameters,

$\displaystyle C^{(2)}_{++}=$
$\displaystyle\,\frac{[34]^{2}}{m^{6}}(c_{5}\langle\bm{12}\rangle^{3}[\bm{12}]+c_{6}\langle\bm{12}\rangle^{2}[\bm{12}]^{2}+c_{7}\langle\bm{12}\rangle[\bm{12}]^{3})$
(90)
$\displaystyle+\frac{\langle\bm{12}\rangle}{m^{6}}(c_{8}[\bm{12}]+c_{9}\langle\bm{12}\rangle)([\bm{1}3]^{2}[\bm{2}4]^{2}+[\bm{1}4]^{2}[\bm{2}3]^{2})\,.$

Finally, the non-abelian color-ordered amplitude, in the positive same-
helicity sector, is given by

$A(\bm{1}^{s=2}\\!,\bar{\bm{2}}^{s=2}\\!,3^{+}\\!,4^{+})=\frac{\langle\bm{12}\rangle^{4}[34]^{2}}{m^{2}s_{12}t_{14}}+\tilde{C}^{(2)}_{++}\,,$
(91)

and there are no new free parameters compared to the abelian case, i.e.
$\tilde{C}^{(2)}_{++}=C^{(2)}_{++}$.

In later sections of this paper additional structures and patterns will be
used, such that we can fix these free parameters. In particular, we will
choose to use $c_{n>2}=0$ and $c_{1}=c_{2}=-1$.

####  3.4.2 Quartic spin-3 interactions

We can also consider the amplitude
$A(\bm{1}^{s=3}\\!,\bar{\bm{2}}^{s=3}\\!,3,4)$ between two massive spin-3
fields and two photons or gluons. The Lagrangian can be written in terms of a
rank-3 field $\Phi^{3}_{\mu\nu\rho}$ and three lower-rank Stückelberg fields
$\\{\Phi^{2}_{\mu\nu},\Phi^{1}_{\mu},\Phi^{0}\\}$. In the abelian case we
consider an ansatz for the off-shell quartic vertices with the following
derivative count,

$\displaystyle V(\Phi^{3}_{1}\bar{\Phi}^{3}_{2}A_{3}A_{4})$
$\displaystyle\sim\partial^{8}\,,$ (92) $\displaystyle
V(\Phi^{2}_{1}\bar{\Phi}^{3}_{2}A_{3}A_{4})$
$\displaystyle\sim\partial^{9}\,,$ $\displaystyle
V(\Phi^{1}_{1}\bar{\Phi}^{3}_{2}A_{3}A_{4})$ $\displaystyle=0\,,$
$\displaystyle V(\Phi^{0}_{1}\bar{\Phi}^{3}_{2}A_{3}A_{4})$
$\displaystyle=0\,.$

We will not show the details of this calculation, as it is much more involved
compared the lower-spin cases that we already elaborated on.

Solving the massive Ward identities yields 53 free parameters in the on-shell
amplitude $A(\bm{1}^{s=3}\\!,\bar{\bm{2}}^{s=3}\\!,3,4)$, with 21 free
parameters that only appear in the opposite-helicity amplitude
$A(\bm{1}^{s=3}\\!,\bar{\bm{2}}^{s=3}\\!,3^{-}\\!,4^{+})$ and 32 free
parameters that only appear in the same-helicity amplitude
$A(\bm{1}^{s=3}\\!,\bar{\bm{2}}^{s=3}\\!,3^{+}\\!,4^{+})$. We omit the
explicit expressions since they are rather cumbersome, and we will study
Compton amplitudes for spins $s\geq 3$ in more detail in the next section.

##  4 Chiral higher-spin gauge theory

In the previous section, we have seen how to simplify the study of massive-
gauge-invariant Lagrangians through the use of on-shell Ward identities. We
have applied these identities at three points to re-derive the
$\sqrt{\text{Kerr}}$ amplitudes from first principles, and we have extended
them to four points to compute new results for the Compton amplitudes. Ward
identities require constructing ansätze for all vertices
$V(\Phi^{k}_{1}\bar{\Phi}_{2}^{s}\dots)$ and fields $\Phi^{k}_{1}$ with $0\leq
k\leq s$. This quickly becomes computationally intense, and we were able to
find explicit results at four points only up to spin $s=3$.

In this section, we make use of a complementary new approach to massive
higher-spin theory Ochirov:2022nqz , which in four dimensions allows us to
forego auxiliary fields altogether. The approach amounts to trading the
SO(1,3) symmetric tensors $\Phi_{\mu_{1}\dots\mu_{s}}$ for ${\rm
SL}(2,\mathbb{C})$ chiral symmetric tensors
$\Phi_{\alpha_{1}\dots\alpha_{2s}}$. Recall that SO(1,3) symmetric traceless
tensors correspond to non-chiral tensors
$\Phi_{\alpha_{1}\dots\alpha_{s}\dot{\beta}_{1}\dots\dot{\beta}_{s}}:=\Phi_{\mu_{1}\dots\mu_{s}}\sigma_{\alpha_{1}\dot{\beta}_{1}}^{\mu_{1}}\\!\cdots\sigma_{\alpha_{s}\dot{\beta}_{s}}^{\mu_{s}}$
in ${\rm SL}(2,\mathbb{C})$ notation, hence switching to
$\Phi_{\alpha_{1}\alpha_{2}\dots\alpha_{2s}}$ as the fundamental field is a
radical reformation, that nevertheless encodes the same on-shell degrees of
freedom. In principle, any rank-$2s$ ${\rm SL}(2,\mathbb{C})$ tensor field is
capable of describing the degrees of freedom of a spin-$s$ particle. However,
in general, the off-shell field contains a lot of redundant components upon
reduction to the Wigner little group ${\rm SU}(2)$, and eliminating them
requires additional constraints. Preserving such constraints when interactions
are turned on has been the main difficulty that we were dealing with in the
previous sections. The advantage of the chiral formulation is that
$\Phi_{\alpha_{1}\dots\alpha_{2s}}$ contains exactly $2s+1$ components, which
is the number of physical degrees of freedom of the massive spin-$s$ field.
Therefore, no further constraints are needed. The price to pay is that the
parity properties of the interactions become highly non-trivial, and it is
best to treat the theory perturbatively as an expansion around the chiral
sector. Even the familiar spin-1 $W$-boson Lagrangian (6) develops spurious
poles in the mass and has an infinite geometric series expansion in the
fields, as we will see below.

###  4.1 Spin-1 in chiral formulation

As an illuminating example of the chiral framework, let us again consider the
spin-1 case, for which the transition from the non-chiral formulation is well
established Chalmers:2001cy . For simplicity, we use a real uncharged Proca
field, to be generalized shortly. Consider the standard Proca Lagrangian and
add a topological term (proportional to the derivative
$\partial_{\mu}[\epsilon^{\mu\nu\rho\sigma}W_{\nu}\partial_{\rho}W_{\sigma}]$),

$-\frac{1}{4}(W_{\mu\nu})^{2}+\frac{m^{2}\\!}{2}(W_{\mu})^{2}-\frac{i}{8}\epsilon_{\mu\nu\rho\sigma}W^{\mu\nu}W^{\rho\sigma}=-\frac{1}{4}W^{\alpha\beta}W_{\alpha\beta}+\frac{m^{2}\\!}{2}(W_{\mu})^{2}\,.$
(93)

The mass-independent kinetic term now only depends on the anti-self-dual field
strength in the ${\rm SL}(2,\mathbb{C})$ notation
$W_{\alpha}{}^{\beta}:=(\sigma^{\mu\nu})_{\alpha}{}^{\beta}\partial_{\mu}W_{\nu}$,
where $\sigma^{\mu\nu}=\sigma^{[\mu}\bar{\sigma}^{\nu]}$. We then “integrate
in” a new chiral symmetric spinor field $\Phi_{\alpha\beta}$ by effectively
adding the quadratic form

$\frac{1}{4}\big{(}\sqrt{2}m\Phi^{\alpha\beta}-W^{\alpha\beta}\big{)}\big{(}\sqrt{2}m\Phi_{\alpha\beta}-W_{\alpha\beta}\big{)}$
(94)

to the action (93). This corresponds to multiplying the path integral by an
irrelevant Gaussian integral. The new Lagrangian can be integrated by parts
and rewritten as

$\frac{m^{2}\\!}{2}\Phi^{\alpha\beta}\Phi_{\alpha\beta}+\frac{m^{2}\\!}{2}(W_{\mu})^{2}-\frac{m}{\sqrt{2}}W_{\mu}\operatorname*{tr}(\sigma^{\mu\nu}\partial_{\nu}\Phi)\,,$
(95)

where the trace is over the ${\rm SL}(2,\mathbb{C})$ indices. We can now
integrate out the original vector field $W_{\mu}$, which gives the new kinetic
term
$-\frac{1}{4}\operatorname*{tr}(\sigma^{\mu\nu}\partial_{\nu}\Phi)\operatorname*{tr}(\sigma_{\mu\rho}\partial^{\rho}\Phi)=-\frac{1}{2}\partial^{\mu}\Phi^{\alpha\beta}\partial_{\mu}\Phi_{\alpha\beta}$.
We thus obtain a chiral spin-1 Lagrangian121212The overall sign of the
Lagrangian (93) comes from that of eq. (93), which includes the same overall
$(-1)^{\lfloor s\rfloor}$ oscillation that can be observed e.g. in the non-
chiral action (27) and which comes from the mostly-minus metric convention. In
the chiral formulation, we choose to drop it starting from eq. (97).

$-\frac{1}{2}\partial^{\mu}\Phi^{\alpha\beta}\partial_{\mu}\Phi_{\alpha\beta}+\frac{m^{2}\\!}{2}\Phi^{\alpha\beta}\Phi_{\alpha\beta}\,=:\,{-}\frac{1}{2}\langle\partial^{\mu}\Phi|\partial_{\mu}\Phi\rangle+\frac{m^{2}\\!}{2}\langle\Phi|\Phi\rangle\,.$
(96)

where in the last step we converted to bra–ket notation,
$|\Phi\rangle:=\Phi_{\alpha\beta}$, instead of writing out the spinor indices.
This is convenient when the number of spinorial indices becomes high.

The chiral gauge-interacting Lagrangian for massive spin-1 was first
introduced in ref. Chalmers:2001cy  for the purpose of describing electroweak
vector bosons. Adapting it to a general non-abelian gauge group, with
$|\Phi\rangle$ transforming in a matter representation, we find the chiral
version of the action (6) to be given in the bra–ket notation by

$\displaystyle{\cal L}^{(1)}$
$\displaystyle=\langle\Phi|\bigg{\\{}|\overset{\leftarrow}{D}|\overset{\rightarrow}{D}|\otimes\frac{1}{1-\tfrac{ig}{m^{2}}|{F_{\;\\!\\!-}}|}\bigg{\\}}|\Phi\rangle-m^{2}\langle\Phi|\Phi\rangle+{\cal
O}(\Phi^{4})$ (97a) $\displaystyle=\langle
D_{\mu}\Phi|D^{\mu}\Phi\rangle-m^{2}\langle\Phi|\Phi\rangle+ig\langle\Phi|{F_{\;\\!\\!-}}|\Phi\rangle+\frac{ig}{m^{2}}\langle\Phi|\Big{\\{}|\overset{\leftarrow}{D}|\overset{\rightarrow}{D}|\otimes|{F_{\;\\!\\!-}}|\Big{\\}}|\Phi\rangle$
(97b)
$\displaystyle\qquad\qquad\qquad\qquad\qquad~{}\;\quad-\frac{g^{2}}{m^{4}}\langle\Phi|\Big{\\{}|\overset{\leftarrow}{D}|\overset{\rightarrow}{D}|\otimes|{F_{\;\\!\\!-}}|{F_{\;\\!\\!-}}|\Big{\\}}|\Phi\rangle+{\cal
O}(F^{3})+{\cal O}(\Phi^{4}).$

The arrows over the covariant derivatives

$\begin{aligned}
\overset{\leftarrow}{D}_{\alpha\dot{\beta}}&=\overset{\leftarrow}{\partial}_{\\!\alpha\dot{\beta}}+igA_{\alpha\dot{\beta}}\,,\\\
\overset{\rightarrow}{D}{}^{\dot{\alpha}\beta}&=\partial^{\dot{\alpha}\beta}-igA^{\dot{\alpha}\beta},\end{aligned}\qquad
A_{\mu}=A_{\mu}^{c}T^{c},\qquad
T^{c\dagger}=T^{c},\qquad\operatorname*{tr}T^{c}T^{c^{\prime}}=\frac{1}{2}\delta^{cc^{\prime}},$
(98)

indicate on which matter field they act, and this notation implicitly assumes
that the covariant derivatives do not act on the field strength factors
$|F_{-}|$ of the massless gauge boson. The minus label emphasizes that this is
the anti-self-dual part of the field strength,
$|F_{-}|:=F_{-\,\alpha}{}^{\beta}:=\frac{1}{2}(\sigma^{\mu}\bar{\sigma}^{\nu})_{\alpha}{}^{\beta}F_{\mu\nu}$,
where the (anti-)self-dual parts satisfy
$\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}F_{\pm}^{\rho\sigma}=\pm
iF^{\pm}_{\mu\nu}$. These labels are also correlated with the gluon helicities
that are allowed to appear in the tree-level amplitude. Hence, at leading
order interaction terms with $k$ factors of $|F^{-}|$ only contribute to
amplitudes that have $k$ negative-helicity gluons.

The gauge-group indices are subsumed by the indexless bra–ket notation, and we
can assume without loss of generality that kets $|\Phi\rangle$ belong to the
fundamental representation and bras $\langle\Phi|$ to the anti-fundamental
representation of the gauge group. (The Lagrangian is also valid for general
complex or real representations.) Although the indexless notation is
convenient for dealing with ${\rm SL}(2,\mathbb{C})$ indices, it admittedly
obscures the multiplication order of the non-abelian gauge-group generators.
To be more precise, let us write the generic term in the expansion of eq.
(97a) such that the non-abelian order is displayed, at the expense of having
explicit spinorial indices:

$\\!\\!\langle\Phi|\Big{\\{}|\overset{\leftarrow}{D}|\overset{\rightarrow}{D}|\otimes\underbrace{|{F_{\;\\!\\!-}}|\cdots|{F_{\;\\!\\!-}}|}_{n}\Big{\\}}|\Phi\rangle:=(D_{\alpha\dot{\varepsilon}}\Phi^{\alpha\beta})^{\star}({F_{\;\\!\\!-}}\;\\!\\!\\!\\!^{c_{1}})_{\beta}{}^{\zeta_{2}}\cdots({F_{\;\\!\\!-}}\;\\!\\!\\!\\!^{c_{n}})_{\zeta_{n}}\\!{}^{\delta}T^{c_{1}}\\!\cdots
T^{c_{n}}(D^{\dot{\varepsilon}\gamma}\Phi_{\gamma\delta})\,,\\!$ (99)

Here and below, the star is used to flip $g\to-g$ and conjugate the gauge-
group indices — but without conjugating the spacetime indices (which would
otherwise imply an unwanted chirality switch). For instance,

$(D_{\gamma\dot{\delta}}\Phi^{\alpha\beta})^{\star}_{\bar{\jmath}}=\big{(}\partial_{\gamma\dot{\delta}}\Phi^{\alpha\beta}\\!-igA_{\gamma\dot{\delta}}^{c}T^{c}\Phi^{\alpha\beta}\big{)}_{\bar{\jmath}}^{\star}:=\partial_{\gamma\dot{\delta}}(\Phi^{\star})^{\alpha\beta}_{\bar{\jmath}}\\!+ig(\Phi^{\star})^{\alpha\beta}_{\bar{\imath}}T^{c}_{i\bar{\jmath}}A_{\gamma\dot{\delta}}^{c}\,.$
(100)

Note that the action (97) obeys the following reality condition:

$\int\\!d^{4}x\,{\cal L}^{\star}=\int\\!d^{4}x\,{\cal L},$ (101)

which we will always demand of a chiral Lagrangian, and this will imply that
the theory can be equivalently described in a conventional (i.e. non-chiral)
formulation using a real-valued Lagrangian.131313Strictly speaking, the
equivalence between the chiral and the usual tensorial approaches has been
shown only for the lower spins and minimal interactions Chalmers:1997ui ;
Chalmers:2001cy . However, given that we do not see any obstruction in getting
any consistent cubic and quartic amplitudes within the chiral approach, it
seems plausible that it covers all possible interactions at all orders.  In
particular, this condition allows for a consistent projection of the theory to
the case of real-valued generators,141414In our current conventions, which are
tailored to hermitian generators, the real representations of ${\rm
SU}(N_{\text{c}})$ or ${\rm SO}(N_{\text{c}})$ actually have imaginary-valued
generators.  such as the adjoint representation of ${\rm SU}(N_{\text{c}})$ or
the fundamental representation of ${\rm SO}(N_{\text{c}})$, considered in ref.
Ochirov:2022nqz .

###  4.2 Cubic $\sqrt{\text{Kerr}}$ Lagrangian for arbitrary spin

After having introduced the chiral framework in the familiar spin-1 context,
it is now straightforward to generalize it to the spin-$s$ case. In
particular, we combine the kinetic terms and minimal cubic interactions of
ref. Ochirov:2022nqz , and then add appropriate cubic non-minimal interactions
to restore parity invariance of three-point amplitudes. The resulting cubic
Lagrangians are simple, and we give the new result without further ado.

The family of chiral spin-$s$ Lagrangians that reproduce the three-point
$\sqrt{\text{Kerr}}$ gauge-theory amplitudes (1) can be written compactly as

${\cal L}^{(s)}=\langle
D_{\mu}\Phi|D^{\mu}\Phi\rangle-m^{2}\langle\Phi|\Phi\rangle+\sum_{k=0}^{2s-1}\frac{ig}{m^{2k}}\langle\Phi|\Big{\\{}|\overset{\leftarrow}{D}|\overset{\rightarrow}{D}|^{\odot
k}\\!\otimes|{F_{\;\\!\\!-}}|\Big{\\}}|\Phi\rangle+{\cal O}(F^{2})\,.$ (102)

We have used the $\odot$ symbol to denote symmetrized tensor product
Guevara:2017csg . Here we additionally define it to symmetrize the internal
spinor indices. In the higher-spin multi-index notation, in which all indices
denoted by the same letter are understood as symmetrized among themselves, we
may write more explicitly

$\displaystyle\langle\Phi|\Phi\rangle$
$\displaystyle\>\\!\\!:=\big{(}\Phi^{\alpha(2s)}\big{)}^{\\!\star}\Phi_{\alpha(2s)}\,,$
(103) $\displaystyle\langle D_{\mu}\Phi|D^{\mu}\Phi\rangle$
$\displaystyle\>\\!\\!:=\big{(}D_{\mu}\Phi^{\alpha(2s)}\big{)}^{\\!\star}D^{\mu}\Phi_{\alpha(2s)}\,,$
$\displaystyle\langle\Phi|\big{\\{}|\overset{\leftarrow}{D}|\overset{\rightarrow}{D}|^{\odot
k}\\!\otimes\\!|{F_{\;\\!\\!-}}|\big{\\}}|\Phi\rangle$
$\displaystyle\>\\!\\!:=\big{(}D_{\alpha(k)\dot{\gamma}(k)}\Phi^{\alpha(k+1)\gamma(2s-k-1)}\big{)}^{\\!\star}({F_{\;\\!\\!-}}\;\\!\\!\\!\\!^{c})_{\alpha}{}^{\beta}T^{c}D^{\dot{\gamma}(k)\beta(k)}\Phi_{\beta(k+1)\gamma(2s-k-1)},$

where
$D_{\alpha(k)\dot{\beta}(k)}\\!:=D_{(\alpha_{1}(\dot{\beta}_{1}}\\!\cdots
D_{\alpha_{k})\dot{\beta}_{k})}$ and
$D^{\dot{\alpha}(k)\beta(k)}\\!:=D^{(\dot{\alpha}_{1}(\beta_{1}}\cdots
D^{\dot{\alpha}_{k})\beta_{k})}$.

As explained in ref. Ochirov:2022nqz , the on-shell wavefunctions for the
massive particles, described by the chiral fields, are simply
$|\Phi\rangle\to|{\bm{p}}\rangle^{2s}\\!/m^{s}$. To guide the reader, it may
be helpful to evaluate some of the Lagrangian terms by plugging in on-shell
wavefunctions, and obtain explicit matrix elements. For example, consider on-
shell three-point kinematics applied to the first two terms of eq. (102),

$\langle
D_{\mu}\Phi|D^{\mu}\Phi\rangle\Big{|}_{(2,3^{\pm}\\!,1)}\\!=ig\big{(}\langle
A_{\mu}\Phi|\partial^{\mu}\Phi\rangle-\langle\partial_{\mu}\Phi|A^{\mu}\Phi\rangle\big{)}\Big{|}_{(2,3^{\pm}\\!,1)}\\!=2gT^{c_{3}}(p_{1}\cdot\,\varepsilon_{3}^{\pm})\frac{\langle\bm{21}\rangle^{2s}\\!}{m^{2s}}\,.$
(104)

Here we generalize the subscript notation $(i_{1},\dots,i_{n})$ introduced in
eq. (20), to indicate that we extract the $n$-particle component of the
corresponding Lagrangian operators, and we impose the usual on-shell
conditions. Eq. (104) shows that the chiral “minimal-coupling” terms reproduce
the $\sqrt{\text{Kerr}}$ amplitudes (1) for a positive-helicity
Ochirov:2022nqz , but not for a negative-helicity gluon. Since the two
$\sqrt{\text{Kerr}}$ amplitudes are related by a parity transformation, the
minimal-coupling chiral Lagrangian explicitly breaks parity. However, let us
show now that the non-minimal terms restore it.

Let us evaluate the three-point matrix element of the non-minimal coupling
terms. They are non-vanishing only for the minus-helicity gluon and evaluate
to

$\frac{1}{m^{2k}}\langle\Phi|\big{\\{}|\overset{\leftarrow}{D}|\overset{\rightarrow}{D}|^{\odot
k}\\!\otimes\\!|{F_{\;\\!\\!-}}|\big{\\}}|\Phi\rangle\Big{|}_{(2,3^{-}\\!,1)}\\!=-i\sqrt{2}T^{c_{3}}\langle\bm{2}3\rangle\langle
3\bm{1}\rangle[\bm{21}]^{k}\frac{\langle\bm{21}\rangle^{2s-k-1}}{m^{2s}}\,.$
(105)

We can now sum the kinematic factors over all powers $k$:

$\sum_{k=0}^{2s-1}\langle\bm{2}3\rangle\langle
3\bm{1}\rangle[\bm{21}]^{k}\langle\bm{21}\rangle^{2s-k-1}=\langle\bm{2}3\rangle\langle
3\bm{1}\rangle\bigg{(}\frac{[\bm{21}]^{2s}\\!-\langle\bm{21}\rangle^{2s}}{[\bm{21}]-\langle\bm{21}\rangle}\bigg{)}=\sqrt{2}(p_{1}\\!\cdot\varepsilon_{3}^{-})\big{(}[\bm{21}]^{2s}\\!-\langle\bm{21}\rangle^{2s}\big{)},$
(106)

where we used the identity $\langle\bm{2}3\rangle\langle
3\bm{1}\rangle=\sqrt{2}(p_{1}\\!\cdot\varepsilon_{3}^{-})\big{(}[\bm{21}]-\langle\bm{21}\rangle\big{)}$
to make locality manifest. When the non-minimal matrix element is combined
appropriately with the negative-helicity amplitude (104), the
$\langle\bm{21}\rangle^{2s}$ factor is canceled out, and instead replaced by
$[\bm{21}]^{2s}$. We get the chiral-Lagrangian amplitudes as

${\cal
A}(\bm{1}^{s}\\!,\bar{\bm{2}}^{s}\\!,3^{+})=2gT^{c_{3}}(p_{1}\\!\cdot\varepsilon_{3}^{+})\frac{\langle\bm{21}\rangle^{2s}\\!}{m^{2s}}\,,\qquad\quad{\cal
A}(\bm{1}^{s}\\!,\bar{\bm{2}}^{s}\\!,3^{-})=2gT^{c_{3}}(p_{1}\\!\cdot\varepsilon_{3}^{-})\frac{[\bm{21}]^{2s}\\!}{m^{2s}}\,.$
(107)

In summary, we have shown that the action (102) restores parity at the three-
point level and yields the known $\sqrt{\text{Kerr}}$ amplitudes Arkani-
Hamed:2019ymq .

Although we refer to eq. (102) as the cubic Lagrangian, it also automatically
incorporates higher-point vertices. For instance, the non-minimal cubic terms
can be shown to give the following contact-term contributions to the opposite-
helicity Compton amplitude:

$\displaystyle\\!\\!\frac{ig}{m^{2k}}\langle\Phi|\big{\\{}|\overset{\leftarrow}{D}|\overset{\rightarrow}{D}|^{\odot
k}\\!\otimes\\!|{F_{\;\\!\\!-}}|\big{\\}}|\Phi\rangle\Big{|}_{(2,3^{-}\\!,4^{+}\\!,1)}\\!=\frac{g^{2}}{2m^{2s}}\langle\bm{21}\rangle^{2s-k-1}[\bm{21}]^{k}\langle\bm{2}|[\varepsilon^{-}_{3},\varepsilon^{+}_{4}]|\bm{1}\rangle[T^{c_{3}},$
$\displaystyle T^{c_{4}}]$
$\displaystyle+\,\frac{\sqrt{2}g^{2}}{m^{2s+k}}\langle\bm{21}\rangle^{2s-k-1}\langle\bm{2}3\rangle\langle
3\bm{1}\rangle\bigg{\\{}\frac{[\bm{2}|\varepsilon_{4}^{+}|\bm{1}\rangle}{[\bm{2}|4|\bm{1}\rangle}\big{(}[\bm{2}|1\\!+\\!4|\bm{1}\rangle^{k}{-}m^{k}[\bm{21}]^{k}\big{)}T^{c_{3}}$
$\displaystyle T^{c_{4}}$ (108)
$\displaystyle-\frac{\langle\bm{2}|\varepsilon_{4}^{+}|\bm{1}]}{\langle\bm{2}|4|\bm{1}]}\big{(}\langle\bm{2}|1\\!+\\!3|\bm{1}]^{k}{-}m^{k}[\bm{21}]^{k}\big{)}T^{c_{4}}$
$\displaystyle T^{c_{3}}\\!\bigg{\\}}\,.$

The first line comes from the non-linearity of the self-dual field strength,
and the other two lines are more complicated as they come from the gauge-field
dependence of the powers of covariant derivatives. Note that the expression is
local, as the spurious denominators can be canceled against the numerators
after expanding out the powers in $k$. Furthermore, note that this contact
term does not respect massless gauge invariance by itself, as it needs to be
combined with the corresponding cubic exchange diagrams in a gauge-invariant
amplitude. The gauge-invariant contact terms will be discussed in the next
subsection.

Eq. (4.2) illustrates a general property that the (anti-)self-duality of the
explicit field strength in the Lagrangian gives rise to a helicity-selection
rule, but only at the lowest multiplicity for which it starts to contribute to
an amplitude, whereas at higher multiplicities it includes both helicities.
Namely, the operator on the left-hand side contains only a negative-helicity
gluon at three points, but both helicities at four points and higher. However,
the leading-order helicity separation of the terms involving $|F_{\pm}|$ is
very helpful, since the strategy is to add such terms only when they are
needed. For instance, to restore parity in amplitudes with $k$ negative-
helicity gluons one should consider operators involving $k$ field strengths
$|F_{-}|$.

###  4.3 Parity of Compton amplitudes

Before we proceed to discussing quartic interactions in more detail, let us
recall that, by construction Ochirov:2022nqz , the first two “minimal-
coupling” terms of the Lagrangian (102) imply

${\cal
A}(\bm{1}^{s}\\!,\bar{\bm{2}}^{s}\\!,3^{+}\\!,4^{+})=-2g^{2}\frac{\langle\bm{21}\rangle^{2s}[34]^{2}}{m^{2s-2}s_{12}}\bigg{[}\frac{T^{c_{3}}T^{c_{4}}\\!}{t_{14}}+\frac{T^{c_{4}}T^{c_{3}}\\!}{t_{13}}\bigg{]}\,,$
(109a) where the Mandelstam variables are $s_{12}=(p_{1}+p_{2})^{2}$ and
$t_{ij}=2p_{i}\cdot p_{j}$. This same-helicity amplitude is known to follow
from its factorization limits Arkani-Hamed:2017jhn ; Johansson:2019dnu ;
Lazopoulos:2021mna  onto the three-point amplitudes (107), as well as to have
a well-behaved classical limit Aoude:2020onz , given below in eq. (186). From
minimality considerations, it is expected to correspond to a
$\sqrt{\text{Kerr}}$ object, so we do not wish to modify it in any way. In
particular, we have checked that the tower of non-minimal cubic terms in eq.
(102) does not contribute to this amplitude, which is not trivial in the non-