# Quantum Field Theory, Worldline Theory,
and Spin Magnitude Change in Orbital Evolution

Zvi Bern  Mani L. Bhaumik Institute for Theoretical Physics, University of
California at Los Angeles, Los Angeles, CA 90095, USA  Dimitrios Kosmopoulos
Département de Physique Théorique, Université de Genève, CH-1211 Geneva,
Switzerland  Andres Luna  Niels Bohr International Academy, Niels Bohr
Institute, University of Copenhagen, Blegdamsvej 17, DK-2100, Copenhagen Ø,
Denmark  Radu Roiban  Institute for Gravitation and the Cosmos, Pennsylvania
State University, University Park, PA 16802, USA  Trevor Scheopner  Mani L.
Bhaumik Institute for Theoretical Physics, University of California at Los
Angeles, Los Angeles, CA 90095, USA  Fei Teng  Institute for Gravitation and
the Cosmos, Pennsylvania State University, University Park, PA 16802, USA
Justin Vines  Mani L. Bhaumik Institute for Theoretical Physics, University of
California at Los Angeles, Los Angeles, CA 90095, USA

###### Abstract

A previous paper [1] identified a puzzle stemming from the amplitudes-based
approach to spinning bodies in general relativity: additional Wilson
coefficients appear compared to current worldline approaches to conservative
dynamics of generic astrophysical objects, including neutron stars. In this
paper we clarify the nature of analogous Wilson coefficients in the simpler
theory of electrodynamics. We analyze the original field-theory construction,
identifying definite-spin states some of which have negative norms, and
relating the additional Wilson coefficients in the classical theory to
transitions between different quantum spin states. We produce a new version of
the theory which also has additional Wilson coefficients, but no negative-norm
states. We match, through $\mathcal{O}(\alpha^{2})$ and $\mathcal{O}(S^{2})$,
the Compton amplitudes of these field theories with those of a modified
worldline theory with extra degrees of freedom introduced by releasing the
spin supplementary condition. We build an effective two-body Hamiltonian that
matches the impulse and spin kick of the modified field theory and of the
worldline theory, displaying additional Wilson coefficients compared to
standard worldline approaches. The results are then compactly expressed in
terms of an eikonal formula. Our key conclusion is that, contrary to standard
approaches, while the magnitude of the spin tensor is still conserved, the
magnitude of the spin vector can change under conserved Hamiltonian dynamics
and this change is governed by the additional Wilson coefficients. For
specific values of Wilson coefficients the results are equivalent to those
from a definite spin obeying the spin supplementary condition, but for generic
values they are physically inequivalent. These results warrant detailed
studies of the corresponding issues in general relativity.

###### Contents

  1. I Introduction
    1. I.1 General Overview
    2. I.2 Summary of Results
  2. II Field Theory
    1. II.1 Minimal Lagrangian in Electrodynamics
    2. II.2 Classical Asymptotic States and Coherent States
    3. II.3 The Transverse $(s,s)$ Representation
    4. II.4 The Nontransverse $(s,s)$ Representation
    5. II.5 Nonminimal Lagrangian
  3. III Scattering Amplitudes
    1. III.1 Three-Point Amplitudes
    2. III.2 Four-Point Compton Amplitudes
      1. III.2.1 The Linear-in-Spin Compton Amplitudes
      2. III.2.2 The Quadratic-in-Spin Compton Amplitudes
    3. III.3 Two-Body Amplitudes
      1. III.3.1 Tree Level
      2. III.3.2 One Loop
  4. IV Worldline Theories
    1. IV.1 Worldline Action with Dynamical Mass Function
    2. IV.2 Worldline Theory with SSC
      1. IV.2.1 Worldline Spin Degrees of Freedom
      2. IV.2.2 Equations of Motion with SSC
    3. IV.3 Worldline Theory with no SSC
      1. IV.3.1 Removing the SSC
      2. IV.3.2 Equations of Motion with no SSC
      3. IV.3.3 Worldline Compton Amplitude
      4. IV.3.4 Worldline Impulses
  5. V Effective Hamiltonian Including Lower-Spin States
    1. V.1 Hamiltonian 1: Solely Spinning Degrees of Freedom
    2. V.2 Hamiltonian 2: Inclusion of Boost Operator
    3. V.3 Amplitudes from the Effective Hamiltonian
    4. V.4 Hamiltonian Coefficients from Matching to Field Theory
    5. V.5 Observables from the Equations of Motion
    6. V.6 Observables from an Eikonal Formula
    7. V.7 Comparison to Observables from the Worldline Theory
    8. V.8 On the Reality of K
  6. VI Wilson coefficients and propagating degrees of freedom
    1. VI.1 Resolution of the Identity and Amplitudes with Lower-Spin States
    2. VI.2 Lower-Spin States and their Scaling in the Classical Limit
    3. VI.3 Lower-Spin States in the Compton Amplitude
  7. VII Discussion and Conclusion

##  I Introduction

###  I.1 General Overview

The landmark detection of gravitational waves by the LIGO/Virgo collaboration
[2, 3] opened a new era in astronomy, cosmology and perhaps even particle
physics. As gravitational-wave detectors become more sensitive [4, 5, 6], the
spin of objects such as black holes and neutron stars will play an
increasingly important role in identifying and interpreting signals. Spin also
leads to much richer three-dimensional dynamics because of the exchange of
angular momentum between bodies and their orbital motion. Its precise
definition leads to interesting and subtle theoretical questions, some of
which we address here.

The study of the dynamics of spinning objects in general relativity [7, 8, 9,
10] has a long history, in both the post-Newtonian (PN) framework [11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32,
33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51,
52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62], where observables are
simultaneously expanded in Newton’s constant $G$ and in the velocity $v$, and
the post-Minkowskian (PM) framework [63, 64, 65, 66, 67, 68, 69, 70, 71, 72,
73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 1, 83,
91, 92, 93, 94, 95, 96, 97, 98, 99, 100], where observables are expanded only
in Newton’s constant with exact velocity dependence. In both approaches the
interaction of spinning objects with the gravitational field is described in
terms of a set of higher-dimension operators whose Wilson coefficients encode
the detailed properties of the objects. For the interesting case of black
holes, the values of these coefficients at $\mathcal{O}(G)$ are known [72],
with proposals for the additional coefficients at $\mathcal{O}(G^{2})$
recently given based on a shift symmetry [89, 1, 95, 96] already present at
$\mathcal{O}(G)$.111Through $O(S^{4})$ Refs. [101, 102] find that the Compton
amplitude derived by solving the Teukolsky equation agrees with with these
previous results. However, the predictions based on shift symmetry at ${\cal
O}(S^{5})$ are in tension with results from the Teukolsky equation, though the
latter involve a subtle analytic continuation between the black-hole and
naked-singularity regimes. The electromagnetic case is similar in structure
[103, 104, 105, 106, 107, 108, 109] (see also Refs. [110, 111] for non-abelian
generalisations), with the post-Coulombian (PC) and post-Lorentzian (PL)
expansions being the respective analogs of the gravitational PN and PM
expansions.

A primary purpose of this paper is to explore puzzles identified in Ref. [1]
regarding the description of spinning bodies in general relativity. In that
paper, results for the conservative two-body scattering angle were obtained
through fifth power in the spin using a scattering-amplitudes-based method. A
rather striking outcome, which follows from the fact that the field-theory
Lagrangian is not directly expressed in terms of particles’ spin tensor, is
that the field-theory approach of Ref. [76] has a larger number of independent
Wilson coefficients for a given power of spin than standard (worldline)
methods. While at 1PM (tree) level the number of independent Wilson
coefficients is identical in the two approaches, matching of physical
observables starting at 2PM (one loop) and third power of the spin can only be
attained by setting some of the field-theory Wilson coefficients to definite
numerical values, so that they are no longer independent. This implies that
the field theory contains a larger number of physically-relevant independent
Wilson coefficients. For the special case of Kerr black holes it appears that
the additional Wilson coefficients present in the field theory are not needed
[72, 82, 83, 89, 1]. In electrodynamics we find a similar situation for the
root-Kerr solution [112], related to the Kerr solution via the double copy
[113, 114, 115, 116, 117, 118].

The connection between scattering amplitudes and effective two-body
interactions has been known for some time [119, 120, 121, 122, 123, 124, 125,
126, 127, 128, 129]. Recent years have seen the construction of new systematic
methods for extracting potentials and physical observables at high orders from
scattering amplitudes [130, 106, 131, 132, 133, 134, 135], which leverage
modern methods for calculating scattering amplitudes, including generalized
unitarity [136, 137, 138, 139, 140, 141], the double copy [113, 114, 115, 118]
and advanced integration techniques [142, 143, 144, 145, 145, 146, 147]. The
extraction of classical physics from quantum scattering is greatly simplified
by concepts from effective field theories (EFTs), systematized for the
gravitational-wave problem in Ref. [148] and applied to the PM framework in
Ref. [130]. By manifestly maintaining Lorentz invariance, the amplitudes
approach fits naturally in the PM or PL frameworks, and produced the first
conservative spinless two-body Hamiltonian at $\mathcal{O}(G^{3})$ and
$\mathcal{O}(G^{4})$ [131, 132, 149, 150] (see also Refs. [151, 152, 153, 154,
155, 156, 157, 97]). Such methods also led to new perspectives on the
gravitational interactions of spinning particles [158, 74, 73, 66, 75, 69, 70,
76, 71, 82, 1] and on tidal effects [159, 160, 161, 162, 163, 164, 165, 166].

Here we use both the amplitudes-based method and the more standard worldline
approach [148, 41, 167, 46, 168, 169] to study the interactions of spinning
particles. Since they describe the same physics, one may expect that there is
a (usually nontrivial) correspondence between the operators (as well as
between their Wilson coefficients) describing these interactions in the two
approaches. Each type of object, whether a Kerr black hole or neutron star, is
described by particular values for the Wilson coefficients, which are
determined by an appropriate matching calculation. In the worldline approach
one imposes a spin supplementary condition (SSC) [170] that identifies the
three physical spin degrees of freedom. This condition has been interpreted in
terms of a spin-gauge symmetry which formalizes the freedom to shift the
worldline in the ambient space [171, 46, 172] without changing the physics. An
important aspect of an SSC is that it reduces the number of possible
independent operators—and consequently the number of Wilson coefficients—by
equating operators whose difference is proportional to the SSC. Here we use
the dynamical mass function formalism of Ref. [168] to explore the
consequences of relaxing the SSC and to help interpret the additional degrees
of freedom.

An interesting subtlety in the amplitudes approach is whether the complete
description of a spinning compact body is provided by a single quantum spin
$s\gg 1$ or by a suitable combination of multiple quantum spins, with possible
transitions between them. For the sake of simplicity, the field theory of Ref.
[76]—meant to be valid only in the classical limit—is based on the matter
states forming an irreducible representation of the Lorentz group but a
reducible representation of the rotation group; some of its components have
negative norm. One might worry these negative-norm states might lead to some
difficulties in the classical limit [173]. In addition, projecting onto the
physical states of a quantum spin $s$ [173, 95] appears to effectively remove
the additional Wilson coefficients, leaving only those included in the
worldline framework, which we affirm here. Field-theory approaches [82, 83,
89, 96] based on the massive-spinor-helicity amplitudes [112] are a convenient
means for restricting the propagation to a single irreducible quantum spin.
Here we use physical state projectors [174, 175] for the same purpose.

The results of Ref. [1] raises several questions:

  1. 1.

What is a complete description of a spinning body in general relativity?

  2. 2.

Can one construct a worldline theory that matches field-theory descriptions
containing extra independent Wilson coefficients? If so, what extra degrees of
freedom are needed?

  3. 3.

The field-theory construction of Ref. [1] uses propagating reducible
representations of the rotation group (spin representations), some with
negative norm. In the context of this construction, what happens if only a
single quantum spin propagates?

  4. 4.

Can one build a field theory based on positive-norm irreducible
representations of the rotation group that also contain extra independent
Wilson coefficients?

  5. 5.

Should a classical spin be modeled as a definite-spin field or as a
superposition of fields with different spins? A related question on the latter
case is whether transitions between different spins are allowed that change
the magnitude of the spin vector even in the conservative sector.222With
dissipation and absorption included the spin magnitude is, of course, not
preserved (see e.g. Refs. [176, 177, 178] for recent discussions).

  6. 6.

Can one build an effective two-body Hamiltonian with extra degrees of freedom
whose physical observables match field-theory results containing extra Wilson
coefficients?

  7. 7.

What is the physical interpretation of the operators associated with
additional Wilson coefficients?

To address these questions we turn to electrodynamics, which has been useful
as a toy model for gravity [103, 104, 105, 106, 107, 108, 109]. While
electrodynamics cannot answer all questions about gravity, the overlap is more
than sufficient to make this a useful test case. In addition to the absence of
photon self-interactions, electrodynamics is particularly helpful for our
questions because the additional independent operators and their Wilson
coefficients affect observables already at the first order spin, rather than
at third order as for gravity, greatly simplifying the analysis.

We use various field theories, worldline theories and effective two-body
Hamiltonians, comparing and contrasting the results from each. In particular,
to help identify the origin of the extra Wilson coefficients we evaluate
Compton amplitudes and scattering angles for three related but distinct field
theories of electrodynamics coupled with higher-spin fields:

  1. FT1:

The setup from Refs. [76, 1], except for electrodynamics instead of general
relativity. The matter states of this theory form an irreducible
representation of the Lorentz group and a reducible representation of the
rotation group, thereby as a quantum theory it carries more degrees of freedom
than those of a fixed-spin particle, including negative-norm states. In this
theory we consider FT1s with classical asymptotic states having spin tensors
obeying the covariant spin supplementary condition (SSC),
$S_{\mu\nu}p^{\nu}=0$, and FT1g with classical asymptotic states having
unconstrained spin tensors. This is equivalent to relaxing the covariant SSC,
so that the resulting amplitudes explicitly contain factors of
$S_{\mu\nu}p^{\nu}$. When we do not need to distinguish between FT1s and FT1g,
we collectively refer to them as FT1. The results of FT1s are obtained from
those of FT1g simply by imposing the covariant SSC on the initial and final
spin tensors.

  2. FT2:

The higher-spin field is constrained to contain a single irreducible spin-$s$
representation of the rotation group [174]. The external massive states are
traceless and transverse due to the equation of motion. In contrast to FT1s
and FT1g, only positive-norm states propagate, and as we shall see, the
covariant SSC is automatically imposed on the spin tensors.

  3. FT3:

The same construction as for FT2 except that two positive-norm irreducible
representations of the rotation group, one with spin-$s$ and the other with
spin-$(s-1)$, are considered. While this field content allows us to reliably
capture effects linear in spin, it is sufficient to demonstrate that such
field theories support more Wilson coefficients than FT2. We include suitable
couplings between matter fields of different spin. Similarly to FT1, we
consider FT3s with asymptotic states having spin tensors obeying the covariant
SSC and FT3g with asymptotic states being a particular combination of the
asymptotic states of the two fields. When we do not need to distinguish
between FT3s and FT3g, we collectively refer to them as FT3.

The above field-theory constructions do not exhaust the ways to adjust the
spectrum of propagating states. For example, one can use the chiral
construction of Ref. [179], based on the representation $(2s,0)$ of the
Lorentz group leading to the same number $(2s+1)$ of propagating degrees as a
quantum spin-$s$ particle. We note that FT1 are not fully consistent as
quantum theories because of the appearance of propagating negative-norm
states. Because of this we use them only in the classical limit, as envisioned
in Ref. [76]. We moreover see that there is a close relation between them and
FT3, which is constructed using only positive-norm states.

To address the question of what kind of worldline theory has the same
observables as field theories with extra Wilson coefficients we consider two
worldline theories:

  1. WL1:

The standard worldline construction with the covariant SSC imposed. We use the
formalism of Ref. [168].

  2. WL2:

A modified worldline construction with no SSC imposed and consequently with
extra degrees of freedom. In the absence of an SSC we can include additional
operators and Wilson coefficients equivalent to the additional ones that can
be included in FT1g through the constructed orders.

Finally, we construct two two-body effective field-theory Hamiltonians by
matching the amplitudes of field theories with different number of internal
and asymptotic degrees of freedom. This allows us to directly construct
observables for these field theories and compare them with worldline theories:

  1. EFT1:

The two-body Hamiltonian of the type in Ref. [76] containing only the spin
vector S for each body. The parameters of this Hamiltonian can be adjusted to
match either FT2 or WL1. We may also match this Hamiltonian to FT1, FT3, and
WL2 when the additional Wilson coefficients are set to specific values.

  2. EFT2:

A two-body Hamiltonian containing both a spin vector S and a Lorentz boost
vector K, interpreted as a mass dipole and inducing an electric dipole. With
suitable parameters this Hamiltonian matches FT1g, FT3g and WL2.

###  I.2 Summary of Results

We compute and compare electrodynamics Compton amplitudes, impulses, spin
kicks and scattering angles in the theories outlined above. With $\alpha$
denoting the fine structure constant, the results of these computations
through ${\cal O}(\alpha^{2}S)$ for two-body observables and through ${\cal
O}(\alpha S^{2})$ for Compton amplitudes yield the following findings:

  1. 1.

In electrodynamics with the massive propagating degrees of freedom of a single
spin-$s$ particle realized as a symmetric traceless transverse $s$-index
tensor, as in FT2 and following Ref. [174], the number of Wilson coefficients
agrees with the standard worldline construction [46], in accord with Refs.
[173, 95].

  2. 2.

By including additional degrees of freedom either by relaxing the
transversality constraint of fields or/and by replacing the $s$-index
symmetric tensor by a more general $(l,r)$ representation of the Lorentz
group, as in FT1, additional Wilson coefficients can appear in the classical
limit. Thus, the additional Wilson coefficients reflect the additional degrees
of freedom present in nontransverse fields.

  3. 3.

We demonstrate that additional propagating positive-norm degrees of freedom in
the form of symmetric traceless transverse lower-rank tensor, as in FT3, also
lead to additional Wilson coefficients in the classical limit. Thus, the
additional Wilson coefficients are not tied specifically to nontransverse
fields, but are a manifestation of additional propagating degrees of freedom.

  4. 4.

By relaxing the SSC constraint on the worldline, the Compton amplitudes as
well as two-body physical observables such as the impulse and spin kick, match
the corresponding results of field theories FT1 and FT3.

  5. 5.

To match the worldline and field-theory amplitudes with additional asymptotic
degrees of freedom and Wilson coefficients, a two-body EFT Hamiltonian with
both spin and boost degrees of freedom are required.

  6. 6.

In the systems with additional degrees of freedom and additional Wilson
coefficients, the magnitudes of spin vectors are not preserved333We note that
the non-conservation of the magnitude of the intrinsic angular momentum of
subsystems of gravitationally-interacting conservative many-body systems has
been known for some time, see e.g. [180, 181, 182].  in the scattering process
while the magnitudes of spin tensors are preserved.

  7. 7.

For specific choices of Wilson coefficients, such as the root-Kerr solution
[112], the extra degrees of freedom decouple and the system can be described
by removing the boost degrees of freedom.

These results are rather striking. Dropping the SSC would seem to contradict
the standard interpretation of the worldline spin gauge symmetry, where local
shifts in the worldline are interpreted as a symmetry [171, 46, 172]. Here we
are reinterpreting this in terms of certain degrees of freedom of extended
nonrigid objects, in much the same way as the spin is interpreted as an
internal degree of freedom. As we discuss in Sect. IV, in the electromagnetic
case there is a natural explanation in terms of an induced electric dipole
moment correlated to the mass dipole.

This paper is organized as follows: In Sect. II we present the field-theory
constructions FT1. FT2 and FT3 for electrodynamics, giving a nonminimal
Lagrangian that contains additional Wilson coefficients compared to the
standard worldline approaches. We also describe the classical asymptotic
states in terms of coherent states and discuss the effect of using different
Lorentz representations. The purpose of the various field theories is to
identify the source of the extra Wilson coefficients. Sect. III then gives the
field-theory amplitudes associated with these theories, including the Compton
tree amplitudes needed to build the one-loop two-body amplitudes, which are
also presented. To interpret these results in the context of the more standard
worldline framework, in Sect. IV we construct the two worldline theories WL1
and WL2 and compare their Compton amplitudes with the field-theory ones. In
Sect. V we construct two-body EFT Hamiltonians so that the scattering
amplitudes of the corresponding EFTs match those of the various field
theories. One Hamiltonian contains only the usual spin operator and the other
also contains a boost operator. The impulse and spin kick derived from the
latter are the same as those following from the SSC-less worldline theory. A
remarkably compact form of physical observables is given in terms of an
eikonal formula. Sect. VI describes the link between extra Wilson coefficients
and the degrees of freedom that propagate in the field theory. In Sect. VII we
summarize our conclusions. Finally, an ancillary file is included containing
mathematica expressions for Hamiltonian coefficients and observables [183].

##  II Field Theory

In this section we construct the field theories FT1, FT2 and FT3 listed in
Sect. I that we use to track the source of additional degrees of freedom and
Wilson coefficients. We begin by discussing the covariantization of the free
matter Lagrangians, which we refer to as the “minimal” Lagrangians, first in
the framework of Refs. [76, 1] where the propagating states form a reducible
representation of the rotation group, and then in the framework of Ref. [174],
in which the only propagating states are only the $2s+1$ physical states of a
spin-$s$ field. After summarizing the coherent-state description of the
classical asymptotic states and the propagators, we then discuss nonminimal
interactions which are linear in the photon field strength and the
corresponding three-point amplitudes. The scaling of massive momenta $p$,
massless transferred momentum $q$, impact parameter $b$ and spins $S$ for
obtaining the classical limit are [76]

$\displaystyle p\rightarrow p,\qquad q\rightarrow\lambda q,\qquad
b\rightarrow\lambda^{-1}b,\qquad S\rightarrow\lambda^{-1}S\,,$ (1)

and the classical part of the $L$-loop two-body amplitude scales as
$\lambda^{-2+L}$ while Compton amplitudes scale as $\lambda^{0}$. The
connection of field theories FT1, FT2 and FT3 to worldline theories will be
discussed in Sect. IV.

###  II.1 Minimal Lagrangian in Electrodynamics

The extension of the construction of Refs. [76] to QED and thus the definition
of the covariantization of the free Lagrangians for FT1 is straightforward,
with the main difference from the gravitational case being that the fields
must be complex. The minimal coupling involves only the standard two-
derivative kinetic term444We are using the mostly minus signature. The
$(-1)^{s}$ factor makes the spin-$s$ component physical.

$\mathcal{L}_{\text{EM}}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\,,\hskip
42.67912pt\mathcal{L}_{\text{min}}=-(-1)^{s}\phi_{s}(D^{2}+m^{2})\bar{\phi}_{s}\,,$
(2)

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$, and the
covariant derivative is defined as

$D_{\mu}\phi_{s}=\partial_{\mu}\phi_{s}-iQA_{\mu}\phi_{s}\,,\hskip
42.67912ptD_{\mu}\bar{\phi}_{s}=\partial_{\mu}\bar{\phi}_{s}+iQA_{\mu}\bar{\phi}_{s}\,.$
(3)

Without the loss of generality, we take all the massive bodies as carrying the
same charge $Q$, and define the effective “fine structure constant”555Note
that this differs from the standard definition of the fine structure constant
in terms of the electron charge. To simplify subsequent formulae, we absorb in
$\alpha$ the charge of macroscopic bodies. as $\alpha=Q^{2}/(4\pi)$. The PL
framework expands observables in powers of $\alpha$ keeping the exact velocity
dependence. In $\mathcal{L}_{\text{min}}$, the fields $\phi_{s}$ and
$\bar{\phi}_{s}$ can be in generic representations of the Lorentz group as
long as their product is a Lorentz-singlet. The most general choice is that
both fields are in the $(l,r)$ representation, i.e. they are represented as

$\displaystyle\phi_{s}=\phi_{\alpha_{1}\alpha_{2}\ldots\alpha_{l}}^{\dot{\beta}_{1}\dot{\beta}_{2}\ldots\dot{\beta}_{r}}\,,\hskip
42.67912pt\bar{\phi}_{s}=\bar{\phi}^{\alpha_{1}\alpha_{2}\ldots\alpha_{l}}_{\dot{\beta}_{1}\dot{\beta}_{2}\ldots\dot{\beta}_{r}}\,,$
(4)

where $l+r=2s$ and $\phi_{s}$ and $\bar{\phi}_{s}$ are symmetric in the
$\alpha_{i}$ and $\dot{\beta}_{i}$ indices, which transform in the two-
dimensional representation of $SU(2)_{L}$ and $SU(2)_{R}$, respectively. The
covariantized free Lagrangian $\mathcal{L}_{\text{min}}$ in Eq. 2 treats
uniformly all the representations of the rotation group that are part of
$\phi_{s}$. Thus, the propagator derived from $\mathcal{L}_{\text{min}}$ is
proportional to the identity operator $1\kern-2.5pt\text{l}_{(l,r)}$ in the
$(l,r)$ representation. For $\phi_{s}$ in the $(s,s)$ representation, it is

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1\kern-2.5pt\text{l}_{(s,s)}\,.$ (5)

Consequently, there is no explicit dependence on the value of $s$ in the
amplitudes that follow from $\mathcal{L}_{\text{min}}$, making the large-spin
limit appropriate for classical physics convenient in this construction.

When evaluated on an $(s,s)$ representation, the Lagrangian (2) contains
propagating degrees of freedom beyond the $2s+1$ associated with a single
massive spin-$s$ particle and some of them have negative norm in Minkowskian
signature. While such a theory is not consistent as a quantum theory because
of difficulties with unitarity, we use this Langangian and its nonminimal
extension described below to find only classical observables, so that the
issue is not directly relevant. One may nevertheless worry that the negative
norm states might cause some inconsistency even in the classical limit, and
very likely they are the origin of the additional Wilson coefficients [173].
As we will see in Sect. VI, the key to the additional Wilson coefficients is
the presence of propagating degrees of freedom beyond those of a single
quantum spin-$s$ particle. This is independent of the sign of the norm of the
extra states. Moreover, there is a direct simple map which connects amplitudes
in this theory with amplitudes in a theory in which all states have positive
norm.

FT2 is designed to probe the relation between the extra Wilson coefficients
and the presence of states beyond those of a spin-$s$ representation of the
rotation group. To define it and to compare straightforwardly with the Singh-
Hagen Lagrangian [174] for a single spin-$s$ particle it is convenient to
choose $\phi_{s}$ in the $(s,s)$ representation, which is realized a symmetric
traceless rank-$s$ tensor,666Throughout the paper, the symmetrization is
defined as
$f_{(\mu_{1}\mu_{2}\ldots\mu_{s})}\equiv\frac{1}{s!}(f_{\mu_{1}\mu_{2}\ldots\mu_{s}}+\text{
permutations})$.

$\displaystyle\phi_{s}\equiv\phi_{\alpha_{1}\alpha_{2}\ldots\alpha_{s}}^{\dot{\beta}_{1}\dot{\beta}_{2}\ldots\dot{\beta}_{s}}\propto\phi^{(\mu_{1}\mu_{2}\ldots\mu_{s})}(\sigma_{\mu_{1}}){}_{(\alpha_{1}}{}^{(\dot{\beta}_{1}}\dots(\sigma_{\mu_{s}}){}_{\alpha_{s})}{}^{\dot{\beta}_{s})}\,,$
(6)

where as usual the parenthesis on the indices signify that they are
symmetrized. We primarily focus on this representation in subsequent sections,
especially when carrying out calculations at fixed values of the spin.

We ensure that only the $2s+1$ states of a spin-$s$ field are propagating by
imposing the requisite constraints with auxiliary fields, following the
strategy of Ref. [174]. The net effect of imposing transversality is that the
minimal Lagrangian $\mathcal{L}_{\text{min}}$ in Eq. (2) is modified to

$\displaystyle\mathcal{L}^{s}_{\text{min}}=-(-1)^{s}\Big{[}\phi_{s}(D^{2}+m^{2})\bar{\phi}_{s}+s(D\phi_{s})(D\bar{\phi}_{s})+\ldots\Big{]}\,,$
(7)

where $(D\phi_{s})\equiv D_{\mu}\phi^{\mu\mu_{2}\ldots\mu_{s}}$ and the
ellipsis stand for terms that remove unwanted states, as explained below.

The coupling $s(D\phi_{s})(D\bar{\phi}_{s})$ originates from integrating out
an auxiliary $\phi_{s-1}$ field that impose transversality via the equation of
motion. To see this, we add to free part of $\mathcal{L}_{\text{min}}$ the
term $a\phi_{s-1}\partial\phi_{s}$ as well as a standard quadratic term for
$\phi_{s-1}$, where $a$ is a normalization. The equations of motion are

$\displaystyle(\partial^{2}+m^{2})\phi_{s}^{\mu_{1}\mu_{2}\ldots\mu_{s}}=a\,\partial^{(\mu_{1}}\phi^{\mu_{2}\ldots\mu_{s})}_{s-1}\,,\qquad(b\,\partial^{2}+c\,m^{2})\phi_{s-1}=(\partial\phi_{s})\,,$
(8)

where we introduced two additional normalization constants $b$ and $c$. A
solution to the equation of motion is $\phi_{s-1}=\partial\phi_{s}=0$.
Requiring that this is the only solution gives $b=0$ and $a=scm^{2}$ such that

$\displaystyle(\partial^{2}+m^{2})\phi_{s}^{\mu_{1}\mu_{2}\ldots\mu_{s}}=s\,\partial^{(\mu_{1}}(\partial\phi_{s})^{\mu_{2}\ldots\mu_{s})}\,.$
(9)

Covariantization with respect to the photon gauge symmetry follows as usual,
by replacing the partial derivatives with the appropriate covariant
derivatives, leading to the $s(D\phi_{s})(D\bar{\phi}_{s})$ term in Eq. 7.

The process continues, as transversality of $\phi_{s}$ implies
$\partial\partial\phi_{s}=0$, which must also be imposed through an equation
of motion. More auxiliary fields are therefore needed, and this process can be
carried out recursively [174]. The resulting couplings involving traces,
multiple-divergences like $D_{\mu}D_{\nu}\phi^{\mu\nu\mu_{3}\ldots\mu_{s}}$,
and auxiliary fields with lower spins are collected in the ellipsis in Eq. 7.
Up to $s=3$, the Lagrangians generated by this procedure are

$\displaystyle\mathcal{L}_{s=1}$
$\displaystyle=\phi^{\mu_{1}}(D^{2}+m^{2})\bar{\phi}_{\mu_{1}}+(D_{\mu}\phi^{\mu})(D^{\nu}\bar{\phi}_{\nu})\,,$
(10a) $\displaystyle\mathcal{L}_{s=2}$
$\displaystyle=-\phi^{\mu_{1}\mu_{2}}(D^{2}+m^{2})\bar{\phi}_{\mu_{1}\mu_{2}}-2(D_{\mu}\phi^{\mu\mu_{2}})(D^{\nu}\bar{\phi}_{\nu\mu_{2}})+\phi_{\mu}{}^{\mu}(D^{2}+m^{2})\bar{\phi}{}^{\nu}{}_{\nu}$
$\displaystyle\quad-\phi_{\mu}{}^{\mu}D^{\rho}D^{\lambda}\bar{\phi}_{\rho\lambda}-\bar{\phi}{}^{\mu}{}_{\mu}D_{\rho}D_{\lambda}\phi^{\rho\lambda}\,,$
(10b) $\displaystyle\mathcal{L}_{s=3}$
$\displaystyle=\phi^{\mu_{1}\mu_{2}\mu_{3}}(D^{2}+m^{2})\bar{\phi}_{\mu_{1}\mu_{2}\mu_{3}}+3(D_{\mu}\phi^{\mu\mu_{2}\mu_{3}})(D^{\nu}\bar{\phi}_{\nu\mu_{2}\mu_{3}})$
$\displaystyle\quad-3\phi_{\mu}{}^{\mu\mu_{3}}(D^{2}+m^{2})\bar{\phi}{}^{\nu}{}_{\nu\mu_{3}}+3\phi_{\mu}{}^{\mu\mu_{3}}D^{\rho}D^{\lambda}\bar{\phi}_{\rho\lambda\mu_{3}}+3\bar{\phi}{}^{\mu}{}_{\mu\mu_{3}}D_{\rho}D_{\lambda}\phi^{\rho\lambda\mu_{3}}$
$\displaystyle\quad+\frac{3}{2}(D_{\mu}\phi^{\mu\rho}{}_{\rho})(D_{\nu}\bar{\phi}{}^{\nu\lambda}{}_{\lambda})+2\varphi(D^{2}+4m^{2})\bar{\varphi}+m(\varphi
D_{\mu}\bar{\phi}{}^{\mu\lambda}{}_{\lambda}+\bar{\varphi}D_{\mu}\phi^{\mu\lambda}{}_{\lambda})\,,$
(10c)

where $\varphi$ and $\bar{\varphi}$ in $\mathcal{L}_{s=3}$ are ghost-like
scalar auxiliary fields. The $\mathcal{L}_{s=1}$ and $\mathcal{L}_{s=2}$ here
are the Proca [184] and Fierz-Pauli Lagrangian [185], respectively, and
$\mathcal{L}_{s=3}$ was first obtained by Chang [186]. We note that the
construction in Ref. [174] uses only symmetric and traceless fields, and we
have absorbed certain auxiliary fields into the trace of $\phi_{s}$. We use
Eq. 7 — which is the arbitrary-spin generalization of Eq. (10) — as the
covariantization of the free Lagrangian of FT2.

FT3 is constructed to probe whether the extra Wilson coefficients in FT1 are
due to the unphysical nature of the extra states of this theory. Thus, we
define the covariantization of the free part of FT3 as being given, up to
nonminimal terms, by the sum of Lagrangians for physical transverse fields
with spins $s$, $s-1$, $\dots,0$. For simplicity, here we consider a
Lagrangian that involves only spin $s$ and $s-1$,

$\displaystyle\mathcal{L}^{s,s-1}_{\text{min}}=\mathcal{L}^{s}_{\text{min}}+\mathcal{L}^{s-1}_{\text{min}}$
$\displaystyle=-(-1)^{s}\Big{[}\phi_{s}(D^{2}+m^{2})\bar{\phi}_{s}+s(D\phi_{s})(D\bar{\phi}_{s})+\ldots\Big{]}$
(11)
$\displaystyle\quad-(-1)^{s-1}\Big{[}\phi_{s-1}(D^{2}+m^{2})\bar{\phi}_{s-1}+(s-1)(D\phi_{s-1})(D\bar{\phi}_{s-1})+\ldots\Big{]}\,.$

We show below that this Lagrangian is sufficient to describe classical physics
at $\mathcal{O}(S^{1})$ up to the one-loop order. We assume that $\phi_{s}$
and $\phi_{s-1}$ have the same minimal coupling to the photon. Somewhat
loosely, one may interpret this Lagrangian as being obtained from Eq. (2) upon
separating $\phi_{s}$ into fields obeying transversality constraints and
dropping the derivative factors that are responsible for the negative norms of
the $s-(2k+1)$ components.

The minimal Lagrangians of FT2 and FT3 make explicit reference to the value of
$s$, as can be seen in the explicit expressions in Eq. 10, and consequently
the propagators (and vertices) have the same property. The propagators for
massive $s=1$ and $s=2$ fields can be easily derived from the quadratic part
of $\mathcal{L}_{s=1}$ and $\mathcal{L}_{s=2}$. They are

$\mu$$\nu$
$\displaystyle=\frac{-i\mathcal{P}_{\mu,\nu}}{p^{2}-m^{2}}=\frac{-i\Theta_{\mu\nu}}{p^{2}-m^{2}}=\frac{-i}{p^{2}-m^{2}}\left(\eta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{m^{2}}\right)\,,$
(12) $\mu_{1}\mu_{2}$$\nu_{1}\nu_{2}$
$\displaystyle=\frac{i\mathcal{P}_{\mu_{1}\mu_{2},\nu_{1}\nu_{2}}}{p^{2}-m^{2}}=\frac{i}{p^{2}-m^{2}}\,\frac{1}{2}\left[\Theta_{\mu_{1}\nu_{1}}\Theta_{\mu_{2}\nu_{2}}+\Theta_{\mu_{1}\nu_{2}}\Theta_{\mu_{2}\nu_{1}}-\frac{2}{3}\Theta_{\mu_{1}\mu_{2}}\Theta_{\nu_{1}\nu_{2}}\right],$
(13)

where $\Theta_{\mu\nu}=\eta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{m^{2}}$. The
numerators are instances of the spin-$s$ state projector $\mathcal{P}$; its
general closed-form expression [174],

$\displaystyle\mathcal{P}_{\mu(s)}^{\nu(s)}=\sum_{j=0}^{\lfloor
s/2\rfloor}\frac{(-1)^{j}s!(2s-2j-1)!!}{2^{j}j!(s-2j)!(2s-1)!!}\Theta_{(\mu_{1}\mu_{2}}\Theta^{(\nu_{1}\nu_{2}}\ldots\Theta_{\mu_{2j-1}\mu_{2j}}\Theta^{\nu_{2j-1}\nu_{2j}}\Theta_{\mu_{2j+1}}^{\nu_{2j+1}}\ldots\Theta_{\mu_{s})}^{\nu_{s})}\,,$
(14)

is manifestly symmetric, transverse and traceless on-shell.

Beyond $s=2$ the off-diagonal nature of the quadratic terms in
$\mathcal{L}^{s}_{\text{min}}$ makes the construction of propagators more
involved. For example, the $\mathcal{L}^{s=3}_{\text{min}}$ Lagrangian
contains quadratic mixing between $\phi_{\mu_{1}\mu_{2}\mu_{3}}$ and the
auxiliary scalar $\varphi$; thus to derive the propagators it is necessary
diagonalize the quadratic terms, effectively summing over all possible
insertions of such two-point vertices. We represent the resummed propagators
by a cross in the middle,

$\mu_{1}\mu_{2}\mu_{3}$$\nu_{1}\nu_{2}\nu_{3}$$\times$
$\displaystyle=\frac{-i\,\mathcal{P}_{\mu_{1}\mu_{2}\mu_{3}}^{\nu_{1}\nu_{2}\nu_{3}}}{p^{2}-m^{2}}+\frac{i\,\mathcal{Q}_{\mu_{1}\mu_{2}\mu_{3}}^{\nu_{1}\nu_{2}\nu_{3}}}{40m^{6}}\,,$
(15a) $\mu_{1}\mu_{2}\mu_{3}$ $\times$
$\displaystyle=\frac{(p^{2}-m^{2})\eta_{(\mu_{1}\mu_{2}}p_{\mu_{3})}-4p_{\mu_{1}}p_{\mu_{2}}p_{\mu_{3}}}{40m^{5}}\,,$
(15b) $\times$ $\displaystyle=\frac{i(p^{2}+5m^{2})}{40m^{4}}\,.$ (15c)

Apart from the non-local term encoding the energy-momentum relation, the
propagator for the physical $s=3$ particle also has an additional local
contribution, with tensor structure

$\displaystyle\mathcal{Q}_{\mu_{1}\mu_{2}\mu_{3}}^{\nu_{1}\nu_{2}\nu_{3}}=\eta_{(\mu_{1}\mu_{2}}p_{\mu_{3})}\eta^{(\nu_{1}\nu_{2}}p^{\nu_{3})}(p^{2}-7m^{2})-4p_{\mu_{1}}p_{\mu_{2}}p_{\mu_{3}}\eta^{(\nu_{1}\nu_{2}}p^{\nu_{3})}-4\eta_{(\mu_{1}\mu_{2}}p_{\mu_{3})}p^{\nu_{1}}p^{\nu_{2}}p^{\nu_{3}}\,.$
(16)

Meanwhile, the contribution from the auxiliary field is completely local,
indicating that they carry no physical (asymptotic) degrees of freedom, as
expected.

###  II.2 Classical Asymptotic States and Coherent States

The standard description of the asymptotic states of (massive) spinning fields
is in terms of Lorentz tensors labeled by the (massive) little group.
Extending Ref. [76], we first consider the asymptotic state $\mathcal{E}$ and
its conjugate $\bar{\mathcal{E}}$ for general $(l,r)$ representations of the
Lorentz group with $l+r=2s$. Subsequently, we specialize to integer $s$ and
consider the $(s,s)$ representation, in which we identify general consequences
of transversality. In the classical limit, these states are chosen to minimize
the dispersion of the Lorentz generators, $\mathcal{E}\cdot
M^{\mu\nu}\cdot\bar{\mathcal{E}}$, where $M^{\mu\nu}$ satisfies the Lorentz
algebra

$\displaystyle[M^{\mu\nu},M^{\rho\lambda}]=-i(\eta^{\mu\rho}M^{\lambda\nu}+\eta^{\nu\rho}M^{\mu\lambda}-\eta^{\mu\lambda}M^{\rho\nu}-\eta^{\nu\lambda}M^{\mu\rho})\,.$
(17)

In the rest frame, the state $\mathcal{E}$ generalizes the spin coherent
states of $SU(2)$ implicit in the construction of Ref. [76] to those of
$SU(2)_{L}\times SU(2)_{R}$. We start by representing the states in terms of
spinors such that777We note that the $SU(2)$ indices are raised and lowered by
$\epsilon_{\alpha\beta}=\epsilon_{\dot{\alpha}\dot{\beta}}=\left(\begin{array}[]{cc}0&-1\\\
1&0\end{array}\right)\qquad\epsilon^{\alpha\beta}=\epsilon^{\dot{\alpha}\dot{\beta}}=\left(\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right).$

$\displaystyle\mathcal{E}(p)_{\alpha(l)\dot{\beta}(r)}$
$\displaystyle=\xi(p)_{\alpha_{1}}\dots\xi(p)_{\alpha_{l}}\chi(p)_{\dot{\beta}_{1}}\dots\chi(p)_{\dot{\beta}_{r}}\,,$
$\displaystyle\bar{\mathcal{E}}(p)^{\alpha(l)\dot{\beta}(r)}$
$\displaystyle=\tilde{\xi}(p)^{\alpha_{1}}\dots\tilde{\xi}(p)^{\alpha_{l}}\tilde{\chi}(p)^{\dot{\beta}_{1}}\dots\tilde{\chi}(p)^{\dot{\beta}_{r}}\,.$
(18)

Here, we choose $\mathcal{E}$ to be null for convenience. We note that the
final result does not rely on $\mathcal{E}$ being null. As with the spin
coherent states, the spinors $\xi(p)$, $\chi(p)$, $\tilde{\xi}(p)$ and
$\tilde{\chi}(p)$ are constructed by boosting their rest frame counterparts
$\xi_{0}$, $\chi_{0}$, ${\tilde{\xi}}_{0}$ and ${\tilde{\chi}}_{0}$,

$\displaystyle\xi(p)_{\alpha}$
$\displaystyle=\exp(i\eta{\hat{p}}^{k}{\hat{K}}{}^{k}_{L})_{\alpha}{}^{\beta}\xi_{0\beta}\,,$
$\displaystyle\quad\tilde{\xi}(p)_{\alpha}$
$\displaystyle=\exp(i\eta{\hat{p}}^{k}{\hat{K}}{}_{L}^{k})_{\alpha}{}^{\beta}\tilde{\xi}_{0\beta}\,,$
$\displaystyle\chi(p)^{\dot{\alpha}}$
$\displaystyle=\exp(i\eta{\hat{p}}^{k}{\hat{K}}{}^{k}_{R})^{\dot{\alpha}}{}_{\dot{\beta}}\chi_{0}^{\dot{\beta}}\,,$
$\displaystyle\quad\tilde{\chi}(p)^{\dot{\alpha}}$
$\displaystyle=\exp(i\eta{\hat{p}}^{k}{\hat{K}}{}_{R}^{k})^{\dot{\alpha}}{}_{\dot{\beta}}\tilde{\chi}_{0}^{\dot{\beta}}\,,$
(19)

where
$(\hat{K}^{k}_{L})_{\alpha}{}^{\beta}=(i/2)(\sigma^{k})_{\alpha}{}^{\beta}$
and
$(\hat{K}^{k}_{R})^{\dot{\alpha}}{}_{\dot{\beta}}=(-i/2)(\sigma^{k})^{\dot{\alpha}}{}_{\dot{\beta}}$
are the left/right-handed boost operators, $\eta$ is the rapidity and
${\hat{p}}^{k}$ are the components of the unit vector along the spatial part
of the momentum.

The rest frame coherent-state spinors are [187]

$\displaystyle\xi_{0\alpha}$
$\displaystyle=\exp(z_{L}\hat{N}^{L}_{+}-z_{L}^{*}\hat{N}^{L}_{-}){}_{\alpha}{}^{\beta}\xi^{+}_{0\beta}\,,\qquad$
$\displaystyle\tilde{\xi}_{0\alpha}$
$\displaystyle=\exp(z_{L}\hat{N}^{L}_{+}-z_{L}^{*}\hat{N}^{L}_{-})_{\alpha}{}^{\beta}\xi^{-}_{0\beta}\,,$
$\displaystyle\chi_{0}^{\dot{\alpha}}$
$\displaystyle=\exp(z_{R}\hat{N}^{R}_{+}-z_{R}^{*}\hat{N}{}^{R}_{-})^{\dot{\alpha}}{}_{\dot{\beta}}\chi^{+,\dot{\beta}}_{0}\,,\qquad$
$\displaystyle\tilde{\chi}_{0}^{\dot{\alpha}}$
$\displaystyle=\exp(z_{R}\hat{N}^{R}_{+}-z_{R}^{*}\hat{N}{}^{R}_{-})^{\dot{\alpha}}{}_{\dot{\beta}}\chi^{-,\dot{\beta}}_{0}\,,$
(20)

where $(\hat{N}^{L}_{\pm})_{\alpha}{}^{\beta}=(1/2)(\sigma^{1}\pm
i\sigma^{2})_{\alpha}{}^{\beta}$ and
$(\hat{N}^{R}_{\pm})^{\dot{\alpha}}{}_{\dot{\beta}}=(1/2)(\sigma^{1}\pm
i\sigma^{2})^{\dot{\alpha}}{}_{\dot{\beta}}$ are the generators of $SU(2)_{L}$
and $SU(2)_{R}$, $\xi_{0}^{\pm}$ and $\chi_{0}^{\pm}$ are the eigenvectors of
$\sigma^{3}$ with eigenvalues $\pm 1$, and

$z_{L,R}\equiv-(\theta_{L,R}/2)e^{-i\phi_{L,R}}\,,$ (21)

are coherent-state parameters. The rest frame spinors are normalized as
$\xi_{0}^{\alpha}\tilde{\xi}_{0\alpha}=\chi_{\dot{\alpha}}\tilde{\chi}^{\dot{\alpha}}=-1$,
such that $\mathcal{E}(p)\cdot\bar{\mathcal{E}}(p)=(-1)^{r}$. They are related
to unit vectors via

$\displaystyle
n_{L}^{i}=\xi_{0}^{\alpha}(\sigma^{i}){}_{\alpha}{}^{\beta}\tilde{\xi}_{0}{}_{\beta}\equiv\xi_{0}\sigma^{i}\tilde{\xi}_{0}\,,$
$\displaystyle\boldsymbol{n}_{L}=(\sin\theta_{L}\cos\phi_{L},\sin\theta_{L}\sin\phi_{L},\cos\theta_{L})\,,$
$\displaystyle
n_{R}^{i}=\chi_{0}{}_{\dot{\alpha}}(\sigma^{i}){}^{\dot{\alpha}}{}_{\dot{\beta}}\tilde{\chi}_{0}^{\dot{\beta}}\equiv\chi_{0}\sigma^{i}\tilde{\chi}_{0}\,,$
$\displaystyle\boldsymbol{n}_{R}=(\sin\theta_{R}\cos\phi_{R},\sin\theta_{R}\sin\phi_{R},\cos\theta_{R})\,.$
(22)

The rotation and boost generators in the $(l,r)$ representation is given by

$\displaystyle\hat{\boldsymbol{S}}$
$\displaystyle=\hat{\boldsymbol{S}}_{L}+\hat{\boldsymbol{S}}_{R}\,,\qquad\hat{\boldsymbol{K}}=\hat{\boldsymbol{K}}_{L}+\hat{\boldsymbol{K}}_{R}\,,$
$\displaystyle\hat{S}^{k}_{L}$
$\displaystyle=\frac{1}{2}\sum_{k=1}^{l}1\kern-2.5pt\text{l}\otimes\dots
1\kern-2.5pt\text{l}\otimes\sigma^{k}\otimes 1\kern-2.5pt\text{l}\dots\otimes
1\kern-2.5pt\text{l}\,,\qquad\hat{S}^{k}_{R}=\frac{1}{2}\sum_{k=1}^{r}1\kern-2.5pt\text{l}\otimes\dots
1\kern-2.5pt\text{l}\otimes\sigma^{k}\otimes 1\kern-2.5pt\text{l}\dots\otimes
1\kern-2.5pt\text{l}\,,$ (23) $\displaystyle\hat{K}^{k}_{L}$
$\displaystyle=\frac{i}{2}\sum_{k=1}^{l}1\kern-2.5pt\text{l}\otimes\dots
1\kern-2.5pt\text{l}\otimes\sigma^{k}\otimes 1\kern-2.5pt\text{l}\dots\otimes
1\kern-2.5pt\text{l}\,,\qquad\hat{K}^{k}_{R}=-\frac{i}{2}\sum_{k=1}^{r}1\kern-2.5pt\text{l}\otimes\dots
1\kern-2.5pt\text{l}\otimes\sigma^{k}\otimes 1\kern-2.5pt\text{l}\dots\otimes
1\kern-2.5pt\text{l}\,.$

With these definitions, the expectation values of the rotation and boost
generator under the rest frame spin coherent states are

$\displaystyle\mathcal{E}_{0}\cdot\hat{\boldsymbol{S}}\cdot\bar{\mathcal{E}}_{0}=\frac{1}{2}(l\,\boldsymbol{n}_{L}+r\,\boldsymbol{n}_{R})\equiv\boldsymbol{S}\,,\qquad\mathcal{E}_{0}\cdot\hat{\boldsymbol{K}}\cdot\bar{\mathcal{E}}_{0}=\frac{i}{2}(l\,\boldsymbol{n}_{L}-r\,\boldsymbol{n}_{R})\equiv
i\boldsymbol{K}\,.$ (24)

We identify the former with the classical rest-frame spin vector
$S^{\mu}_{0}=(0,\boldsymbol{S})$ and the latter with the boost vector
$K^{\mu}_{0}=(0,\boldsymbol{K})$. If $\mathcal{E}$ is not null, we view Eq. 24
as the definition for the classical spin and boost vector. The classical rest-
frame spin tensor given by

$\displaystyle\mathsf{S}^{\mu\nu}_{0}=\mathcal{E}_{0}\cdot
M^{\mu\nu}\cdot\bar{\mathcal{E}}_{0}=S^{\mu\nu}_{0}+iK^{\mu\nu}_{0}\,,$ (25)

does not obey the SSC, where

$\displaystyle
S^{\mu\nu}_{0}=\frac{1}{m}\varepsilon^{\mu\nu\rho\lambda}p_{0\rho}S_{\lambda}\,,\hskip
42.67912ptK^{\mu\nu}_{0}=\frac{1}{m}(p_{0}^{\mu}K_{0}^{\nu}-p_{0}^{\nu}K_{0}^{\mu})\,.$
(26)

It contains888Our convention for the Levi-Civita tensor is
$\epsilon_{0123}=+1$. an SSC-obeying component $S^{\mu\nu}_{0}$ and an SSC-
violating one $K^{\mu\nu}_{0}$, where $p_{0}^{\mu}=(m,0)$ is the rest-frame
momentum. An important feature for generic $(l,r)$ representations is that
$\boldsymbol{K}$ no longer vanishes identically, so that
${\mathsf{S}}^{\mu\nu}_{0}$ no longer satisfies the convariant SSC condition.
By suitably choosing $l$ and $r$ the norm $|\boldsymbol{K}|$ can be subleading
in the classical limit or commensurate with that of the spin vector. In this
way, the appearance of $\boldsymbol{K}$ in the classical limit appears
natural, simply by adjusting the Lorentz representation in the underlying
quantum system. For generic values, the classical limit is independent of the
details of the representation. However, for the special case of the
irreducible transverse $(s,s)$ representation then $\boldsymbol{K}$ vanishes,
as noted in Appendix C of Ref. [75].

The next step is to restore the momentum dependence of various quantities by
boosting the particle out of the rest frame. It is somewhat tedious but
straightforward to use Eq. 19 and the properties of the Pauli matrices to
boost products of polarization tensors and Lorentz generators for any $(l,r)$
representation, as well as Eq. 25 and its two components to arbitrary frames.
To leading order in the classical limit, we find

$\displaystyle\mathcal{E}_{1}\cdot\\{M^{\mu_{1}\nu_{1}},\dots,M^{\mu_{n}\nu_{n}}\\}\cdot\bar{\mathcal{E}}_{2}=\mathsf{S}(p_{1})^{\mu_{1}\nu_{1}}\dots\mathsf{S}(p_{n})^{\mu_{n}\nu_{n}}\mathcal{E}_{1}\cdot\bar{\mathcal{E}}_{2}+{\cal
O}(q^{1-n})\,,$ (27)

where $\mathcal{E}_{i}\equiv\mathcal{E}(p_{i})$, $q=p_{2}-p_{1}$ is the
momentum transfer and $\mathsf{S}(p_{i})^{\mu\nu}$ is the boost of Eq. 25 to
the frame moving with momentum $p_{i}$. The spin tensor scales as
$\mathsf{S}\sim q^{-1}$, and we neglect all the subleading
$\mathcal{O}(q^{1-n})$ terms. The symmetric product of Lorentz generators
$\\{M^{\mu_{1}\nu_{1}},\dots,M^{\mu_{n}\nu_{n}}\\}$ is defined as

$\displaystyle\\{M_{\mu_{1}\nu_{1}},M_{\mu_{2}\nu_{2}}\\}=\frac{1}{2}(M_{\mu_{1}\nu_{1}}M_{\mu_{2}\nu_{2}}+M_{\mu_{2}\nu_{2}}M_{\mu_{1}\nu_{1}})\,,$
$\displaystyle\\{M_{\mu_{1}\nu_{1}},M_{\mu_{2}\nu_{2}},\ldots,M_{\mu_{n}\nu_{n}}\\}=\frac{1}{n!}(M_{\mu_{1}\nu_{1}}M_{\mu_{2}\nu_{2}}\ldots
M_{\mu_{n}\nu_{n}}+\text{permutations})\,.$ (28)

They form a basis for arbitrary product of Lorentz generators under the
Lorentz algebra. The factorization (27) of the expectation value of the
product of Lorentz generators into the product of individual expectation
values is a reflection of the classical nature of the asymptotic states
$\mathcal{E}$ and $\bar{\mathcal{E}}$. In LABEL:{eq:generalS}, the product of
polarization tensors is given by

$\displaystyle(-1)^{r}\mathcal{E}_{1}\cdot\bar{\mathcal{E}}_{2}=\exp\left[-\frac{1}{m}\boldsymbol{q}\cdot\boldsymbol{K}\right]\exp\left[-i\frac{\epsilon_{rijk}u_{1}^{i}q^{j}S^{k}}{m(1+\sqrt{1+\boldsymbol{u}_{1}^{2}})}+{\cal
O}(q^{2})\right]+{\cal O}(q)\,,$ (29)

where $u^{k}_{i}=p^{k}_{i}/m$, generalizing the corresponding expression in
Ref. [76] to general $\boldsymbol{K}$. Eq. (29) captures the leading terms in
the classical limit and, apart from the sign on the left-hand side, it is
agnostic to the $(l,r)$ representation chosen for the fields.

###  II.3 The Transverse $(s,s)$ Representation

We now consider the special case of the $(s,s)$ representation, which
corresponds to symmetric-traceless fields. The coherent-state polarization
tensors have an equal number of dotted and undotted indices; in the rest
frame, they can be written as

$\displaystyle(\mathcal{E}_{0}^{(s)})_{\alpha(s)\dot{\beta}(s)}=(\mathcal{E}_{0}^{(s)})^{\mu_{1}\mu_{2}\ldots\mu_{s}}(\sigma_{\mu_{1}})_{\alpha_{1}\dot{\beta}_{1}}\ldots(\sigma_{\mu_{s}})_{\alpha_{s}\dot{\beta}_{s}}=\xi_{0\alpha_{1}}\ldots\xi_{0\alpha_{s}}\chi_{0\dot{\beta}_{1}}\ldots\chi_{0\dot{\beta}_{s}}\,.$
(30)

With this definition, we can explore the additional restrictions on the
coherent states required by the transversality of $\mathcal{E}_{0}^{(s)}$. It
suffices to analyze it in the rest frame, where it reads,999In this form
transversality can be imposed on the polarization tensor of a general $(l\neq
0,r\neq 0)$ state.

$\displaystyle
p_{0\mu}\mathcal{E}_{0}^{\mu\mu_{2}\ldots\mu_{s}}=0\qquad\Longleftrightarrow\qquad(p_{0\mu}\sigma^{\mu}){}^{\alpha}{}_{\dot{\beta}}(\mathcal{E}_{0})_{\alpha\alpha_{2}\dots\alpha_{s}}^{\dot{\beta}\dot{\beta}_{2}\dots\dot{\beta}_{s}}=0\,.$
(31)

Using the explicit form of the rest-frame momentum, $p_{0}=(m,0,0,0)$, and
that $(\sigma^{0}){}^{\alpha}{}_{\dot{\beta}}$ is numerically equal to the
$2\times 2$ Levi-Civita, it follows that

$\displaystyle
0=(p_{0\mu}\sigma^{\mu}){}^{\alpha}{}_{\dot{\beta}}(\mathcal{E}_{0})_{\alpha\alpha_{2}\dots\alpha_{s}}^{\dot{\beta}\dot{\beta}_{2}\dots\dot{\beta}_{s}}\propto\xi_{0\alpha}\epsilon^{\alpha}{}_{\dot{\alpha}}\chi_{0}^{\dot{\alpha}}\,.$
(32)

The solution, accounting for normalization, is

$\displaystyle\xi_{0\alpha}=\chi_{0}^{\dot{\alpha}}\quad\text{as column
vectors,}$ (33)

which in turn implies $z_{L}=z_{R}$ and hence the equality of the left-handed
and right-handed unit vectors $\boldsymbol{n}_{L}$ and $\boldsymbol{n}_{R}$ in
Sect. II.2. Together with Eq. 24 this implies that

$\displaystyle\boldsymbol{K}=0\quad\Longleftrightarrow\quad{\mathsf{S}}^{\mu\nu}_{0}=S^{\mu\nu}_{0}\,,$
(34)

for the transverse $(s,s)$ representation, and therefore, cf. Eq. 25,
$\mathcal{E}_{0}^{(s)}\cdot M^{\mu\nu}\cdot\bar{\mathcal{E}}_{0}^{(s)}$
becomes an SSC-satisfying spin tensor. On the other hand, if we do not impose
transversality, then the discussion for a generic $(l,r)$ representation also
applies to $(s,s)$, such that $\boldsymbol{K}$ does not vanish and hence that
covariant SSC is not obeyed. We thus see that the $(s,s)$ transverse
asymptotic states chosen in Ref. [76] can be replaced with more general
nontransverse ones. The polarization tensor for the transverse $(s,s)$
representation can be written as a direct product of transverse $s=1$ coherent
state vectors

$\displaystyle\mathcal{E}^{(s)}(p)^{\mu_{1}\mu_{2}\ldots\mu_{s}}=\varepsilon(p)^{\mu_{1}}\varepsilon(p)^{\mu_{2}}\ldots\varepsilon(p)^{\mu_{s}}\,,\qquad\varepsilon(p)^{\mu}(\sigma_{\mu})_{\alpha\dot{\beta}}=\xi_{\alpha}(p)\chi_{\dot{\beta}}(p)\,,$
(35)

where the spinors are boosted from the rest frame ones that satisfy the
condition (33), and we normalize the polarization vectors as
$\varepsilon\cdot\bar{\varepsilon}=-1$. For such external states, the
expectation value of Eq. 27 becomes

$\displaystyle
K^{\mu\nu}(p)=0\quad\Longleftrightarrow\quad\mathsf{S}^{\mu\nu}(p)=S^{\mu\nu}(p)\,,$
(36)

which is simply the counterpart of the rest frame relation (34). The product
(29) simplifies to

$\displaystyle(-1)^{s}\mathcal{E}^{(s)}_{1}\cdot\bar{\mathcal{E}}^{(s)}_{2}=(-\varepsilon_{1}\cdot\bar{\varepsilon}_{2})^{s}=\exp\left[-i\frac{\epsilon_{rsk}u_{1}^{r}q^{s}S^{k}}{m(1+\sqrt{1+\boldsymbol{u}_{1}^{2}})}+{\cal
O}(q^{2})\right]+{\cal O}(q)\,.$ (37)

The transverse $(s,s)$ representation is used in FT2 and FT3. Because the
Lagrangian depends explicitly on $s$, we need to use the explicit form of the
Lorentz generators,

$\displaystyle(M^{\mu\nu})_{\alpha(s)}{}^{\beta(s)}=-2i\delta_{(\alpha_{1}}^{[\mu}\eta^{\nu](\beta_{1}}\delta_{\alpha_{2}}^{\beta_{2}}\ldots\delta_{\alpha_{s})}^{\beta_{s})}\,.$
(38)

Consequently, the results are given in terms of various symmetric and
antisymmetric combinations of the polarization vector $\varepsilon$ and
momenta. To convert them into spin tensors, we need to compute the left-hand
side of Eq. 27 and identify the resulting structures with spin tensors.

We first consider the transverse $\mathcal{E}^{(s)}$. Starting with
$\mathcal{O}(S^{1})$, we have

$\displaystyle\mathcal{E}_{1}^{(s)}\cdot
M^{\mu\nu}\cdot\bar{\mathcal{E}}_{2}^{(s)}$
$\displaystyle=-2is(\varepsilon_{1}\cdot\bar{\varepsilon}_{2})^{s-1}\varepsilon_{1}^{[\mu}\bar{\varepsilon}_{2}^{\nu]}\,.$
(39)

According to Eq. 27, this combination should be identified with
$S(p_{1})^{\mu\nu}$, such that

$\displaystyle(\varepsilon_{1}\cdot\bar{\varepsilon}_{2})^{s-1}\varepsilon_{1}^{[\mu}\bar{\varepsilon}_{2}^{\nu]}=\frac{iS(p_{1})^{\mu\nu}}{2s}(\varepsilon_{1}\cdot\bar{\varepsilon}_{2})^{s}+\mathcal{O}(q^{0})\,.$
(40)

In amplitudes, we can use this relation to turn antisymmetric combination of
polarization vectors into spin tensors. The classical amplitude is obtained by
further taking the $s\rightarrow\infty$ limit. Similarly, at
$\mathcal{O}(S^{2})$, we can use the following identity,

$\displaystyle\mathcal{E}_{1}^{(s)}\cdot\\{M^{\mu\nu},M^{\rho\lambda}\\}\cdot\bar{\mathcal{E}}_{4}^{(s)}$
$\displaystyle=-4s(s-1)(\varepsilon_{1}\cdot\bar{\varepsilon}_{4})^{s-2}\varepsilon_{1}^{[\mu}\bar{\varepsilon}_{4}^{\nu]}\varepsilon_{1}^{[\rho}\bar{\varepsilon}_{4}^{\lambda]}$
(41)
$\displaystyle\quad-s(\varepsilon_{1}\cdot\bar{\varepsilon}_{4})^{s-1}\left(\eta^{\mu\lambda}\varepsilon_{1}^{(\nu}\bar{\varepsilon}_{4}^{\rho)}+\eta^{\nu\rho}\varepsilon_{1}^{(\mu}\bar{\varepsilon}_{4}^{\lambda)}-\eta^{\mu\rho}\varepsilon_{1}^{(\nu}\bar{\varepsilon}_{4}^{\lambda)}-\eta^{\nu\lambda}\varepsilon_{1}^{(\mu}\bar{\varepsilon}_{4}^{\rho)}\right).$

In the large $s$ limit, the second term is subleading, such that we have

$\displaystyle(\varepsilon_{1}\cdot\bar{\varepsilon}_{2})^{s-2}\varepsilon_{1}^{[\mu}\bar{\varepsilon}_{2}^{\nu]}\varepsilon_{1}^{[\rho}\bar{\varepsilon}_{2}^{\lambda]}\xrightarrow{\;\text{large
}s\;}-\frac{(\varepsilon_{1}\cdot\bar{\varepsilon}_{2})^{s}}{4s^{2}}S(p_{1})^{\mu\nu}S(p_{1})^{\rho\lambda}+\mathcal{O}(q^{-1})\,,$
(42)

which is equivalent to applying Eq. 40 twice. Contracting the Lorentz indices
on the spin tensors once leads to

$\displaystyle(\varepsilon_{1}\cdot\bar{\varepsilon}_{2})^{s-1}\varepsilon_{1}^{(\mu}\bar{\varepsilon}_{2}^{\nu)}\xrightarrow{\;\text{large
}s\;}-\frac{(\varepsilon_{1}\cdot\bar{\varepsilon}_{2})^{s}}{2s^{2}}S(p_{1})^{\mu\rho}S(p_{1})_{\rho}{}^{\nu}+\mathcal{O}(q^{-1})\,.$
(43)

Similar identities have been previously used in e.g. Refs. [188, 37]. They are
sufficient for amplitudes up to $\mathcal{O}(S^{2})$.

###  II.4 The Nontransverse $(s,s)$ Representation

For a nontransverse field in the $(s,s)$ representation we can use the general
results obtained for an $(l,r)$ representation. In particular, it has
$\boldsymbol{K}\neq 0$. It is instructive to identify the origin of
$\boldsymbol{K}$, and thus the structures governing its covariant version
$K^{\mu\nu}$, in terms of the lower-spin (longitudinal) components of
$\mathcal{E}_{\mu_{1}\ldots\mu_{s}}$. This will be important when discussing
FT3 which has physical lower-spin fields.

The coherent state in FT1 can be decomposed as

$\displaystyle\mathcal{E}_{\mu_{1}\ldots\mu_{s}}=\mathcal{E}^{(s)}_{\mu_{1}\ldots\mu_{s}}+\left(u\mathcal{E}^{(s-1)}\right)_{\mu_{1}\ldots\mu_{s}}+\left(u^{2}\mathcal{E}^{(s-2)}\right)_{\mu_{1}\ldots\mu_{s}}+\ldots\,,$
(44)

where the spin-$(s-k)$ component is represented by

$\displaystyle\left(u^{k}\mathcal{E}^{(s-k)}\right)_{\mu_{1}\ldots\mu_{s}}={s\choose
k}^{1/2}u_{(\mu_{1}}\ldots
u_{\mu_{k}}\varepsilon_{\mu_{k+1}}\ldots\varepsilon_{\mu_{s})}\,.$ (45)

The states with even $k$ have positive norm and those with odd $k$ have
negative norm. We may now compute products involving these polarization
tensors. We have

$\displaystyle\mathcal{E}^{(s)}_{1}\cdot\left(u_{2}\bar{\mathcal{E}}_{2}^{(s-1)}\right)$
$\displaystyle=-\frac{\sqrt{s}(\varepsilon_{1}\cdot{\bar{\varepsilon}}_{2})^{s-1}\varepsilon_{1}\cdot
q}{m}+\mathcal{O}(q^{2})\,.$ (46)

At $\mathcal{O}(S^{1})$, we plug Eq. 44 into Eq. 27 and find that

$\displaystyle\mathcal{E}^{(s)}_{1}\cdot
M^{\mu\nu}\cdot\left(u_{2}\bar{\mathcal{E}}^{(s-1)}_{2}\right)=i\sqrt{s}\left(\varepsilon_{1}\cdot\bar{\varepsilon}_{2}\right)^{s-1}\left(u_{2}^{\mu}\varepsilon_{1}^{\nu}-u_{2}^{\nu}\varepsilon_{1}^{\mu}\right)+\mathcal{O}(q)\,,$
(47) $\displaystyle\left(u_{1}\mathcal{E}^{(s-1)}_{1}\right)\cdot
M^{\mu\nu}\cdot\bar{\mathcal{E}}^{(s)}_{2}=-i\sqrt{s}(\varepsilon_{1}\cdot\bar{\varepsilon}_{2})^{s-1}\left(u_{1}^{\mu}\bar{\varepsilon}_{2}^{\nu}-u_{1}^{\nu}\bar{\varepsilon}_{2}^{\mu}\right)+\mathcal{O}(q)\,,$

while all the other $\mathcal{E}^{(s)}_{1}\cdot
M^{\mu\nu}\cdot\left(u^{k}_{2}\bar{\mathcal{E}}^{(s-k)}_{2}\right)$ vanish at
the classical order. We note that contractions like
$\left(u_{1}^{k}\mathcal{E}^{(s-k)}_{1}\right)\cdot
M^{\mu\nu}\cdot\left(u_{2}^{k}\bar{\mathcal{E}}^{(s-k)}_{2}\right)$ and
$\left(u_{1}^{k}\mathcal{E}^{(s-k)}_{1}\right)\cdot
M^{\mu\nu}\cdot\left(u_{2}^{k}\bar{\mathcal{E}}^{(s-k-1)}_{2}\right)$ give
identical result as Eqs. 39 and 47 in the limit $s\geqslant k$. They
contribute an overall factor that can be absorbed in the normalization of the
states. In this sense, we can identify the above combination with $K^{\mu\nu}$
or $K^{\mu}$. More precisely, we have

$\displaystyle-(\varepsilon_{1}\cdot\bar{\varepsilon}_{2})^{s-1}\Big{[}u_{1}^{\mu}\left(\varepsilon_{1}^{\nu}+\bar{\varepsilon}_{2}^{\nu}\right)-u_{1}^{\nu}\left(\varepsilon_{1}^{\mu}+\bar{\varepsilon}_{2}^{\mu}\right)\Big{]}$
$\displaystyle\xrightarrow{\;\text{large
}s\;}\frac{\mathcal{E}_{1}\cdot\bar{\mathcal{E}}_{2}}{\sqrt{s}}K(p_{1})^{\mu\nu}\,,$
$\displaystyle-(\varepsilon_{1}\cdot\bar{\varepsilon}_{2})^{s-1}\left(\varepsilon_{1}^{\mu}+\bar{\varepsilon}_{2}^{\mu}\right)$
$\displaystyle\xrightarrow{\;\text{large
}s\;}\frac{\mathcal{E}_{1}\cdot\bar{\mathcal{E}}_{2}}{\sqrt{s}}K(p_{1})^{\mu}\,,$
(48)

Using these two relations in the product
$\mathcal{E}_{1}\cdot\bar{\mathcal{E}}_{2}$, we find that

$\displaystyle\mathcal{E}_{1}\cdot\bar{\mathcal{E}}_{2}$
$\displaystyle=(\varepsilon_{1}\cdot\bar{\varepsilon}_{2})^{s}+\sqrt{s}(\varepsilon_{1}\cdot
u_{2}+\bar{\varepsilon}_{2}\cdot
u_{1})(\varepsilon_{1}\cdot\bar{\varepsilon}_{2})^{s-1}+\ldots$
$\displaystyle=(\varepsilon_{1}\cdot\bar{\varepsilon}_{2})^{s}+\frac{q\cdot
K}{m}\mathcal{E}_{1}\cdot\bar{\mathcal{E}}_{2}+\ldots$ (49)

where the first term comes from
$\mathcal{E}_{1}^{(s)}\cdot\left(u_{2}\bar{\mathcal{E}}_{2}^{(s-1)}\right)+\left(u_{1}\mathcal{E}_{1}^{(s-1)}\right)\cdot\bar{\mathcal{E}}_{2}$
and the $\ldots$ contains the contraction between $\mathcal{E}^{(s)}$ and the
states with spin less than $s-1$. Again, similar contractions between lower-
spin states contribute an overall factor that can be normalized away. We thus
get the relation between the transverse
$\mathcal{E}_{1}^{(s)}\cdot\bar{\mathcal{E}}_{2}^{(s)}=(\varepsilon_{1}\cdot\bar{\varepsilon}_{2})^{s}$
and the full result $\mathcal{E}_{1}\cdot\bar{\mathcal{E}}_{2}$ up to the
first order in $q$ and $K$,

$\displaystyle(\varepsilon_{1}\cdot\varepsilon_{2})^{s}=\left(1-\frac{q\cdot
K}{m}\right)(\mathcal{E}_{1}\cdot\bar{\mathcal{E}}_{2})+\mathcal{O}(q^{2},K^{2})\,,$
(50)

which is of course consistent with Eqs. 29 and 37. The relations (II.4) and
(50) are used to extract the $\mathcal{O}(K^{1})$ terms in FT3. More
generally, the contractions

$\displaystyle\mathcal{E}_{1}^{(s)}\cdot\left(u_{2}^{k}\bar{\mathcal{E}}_{2}^{(s-k)}\right)$
$\displaystyle={s\choose
k}^{1/2}(\varepsilon_{1}\cdot\bar{\varepsilon}_{2})^{s-k}\left(-\frac{q\cdot\varepsilon_{1}}{m}\right)^{k}\,,$
(51)
$\displaystyle\mathcal{E}_{1}^{(s)}\cdot\left\\{M^{\mu_{1}\nu_{1}},\ldots,M^{\mu_{k}\nu_{k}}\right\\}\cdot\left(u_{2}^{k}\bar{\mathcal{E}}_{2}^{(s-k)}\right)$
$\displaystyle={s\choose
k}^{1/2}(k!)(\varepsilon_{1}\cdot\bar{\varepsilon}_{2})^{s-k}\prod_{j=1}^{k}\left(2iu_{2}^{[\mu_{i}}\varepsilon_{1}^{\nu_{i}]}\right)$
$\displaystyle\rightarrow
s^{k/2}(\varepsilon_{1}\cdot\bar{\varepsilon}_{2})^{s-k}\prod_{j=1}^{k}\left(2iu_{2}^{[\mu_{i}}\varepsilon_{1}^{\nu_{i}]}\right),$
(52)

can be used to show that for the $s$ to $s-k$ amplitudes,

$\displaystyle(q\cdot K)^{k}\rightarrow
s^{k/2}(\varepsilon_{1}\cdot\bar{\varepsilon}_{2})^{s-k}\left(\frac{q\cdot\varepsilon_{1}}{m}\right)^{k}\,,$
(53) $\displaystyle\mathcal{E}_{1}^{(s)}\cdot\\{\underbrace{M,\ldots
M}_{m}\\}\cdot\left(u_{2}^{k}\bar{\mathcal{E}}_{2}^{(s-k)}\right)\sim
K^{k}S^{m-k}\,,$ (54)

which are necessary to identify the structures related to $K^{k}S^{m-k}$ in
the amplitudes. Recall that the notation
$u_{2}^{k}\bar{\mathcal{E}}_{2}^{(s-k)}$ includes a factor $s^{k/2}$, cf. Eq.
(45). We leave for future work the detailed study of structures of higher
orders in spin.

Apart from states created by the operators101010We denote by
$a^{\dagger}_{(s)}$ the creation operators of the field $\phi_{(s)}$
corresponding to the state labeled by the rest-frame polarization tensors in
Eqs. (II.2) and (19). $\mathcal{E}^{(s)}\cdot a^{\dagger}_{(s)}$ and
$\mathcal{E}^{(s-1)}\cdot a^{\dagger}_{(s-1)}$ of the fields with definite
spin, in FT3 we may also choose asymptotic states with indefinite spin, which
are a normalized linear combinations of these definite-spin states (and, in
general, also of lower-spin fields). In a quantum theory such a choice is
disfavored as it breaks the little-group symmetry. In the classical theory,
effectively with a single asymptotic state, this is not an issue. We therefore
also evaluate amplitudes in FT3 with asymptotic states

$\displaystyle|g\rangle=\frac{1}{\sqrt{2}}\left(\mathcal{E}^{(s)}\cdot
a^{\dagger}_{(s)}+\mathcal{E}^{(s-1)}\cdot
a^{\dagger}_{(s-1)}\right)|0\rangle\,.$ (55)

Similar states have also been considered in Refs. [79, 178]. We will refer to
these amplitudes as ${\cal A}^{\text{FT3g}}$; in terms of definite-spin states
they are

$\displaystyle{\cal A}^{\text{FT3g}}=\frac{1}{2}\left({\cal
A}^{\text{FT3s}}_{s\rightarrow s}+{\cal A}^{\text{FT3s}}_{s-1\rightarrow
s-1}+{\cal A}^{\text{FT3s}}_{s-1\rightarrow s}+{\cal
A}^{\text{FT3s}}_{s\rightarrow s-1}\right)\,.$ (56)

There is no simple polarization tensor that can be assigned to the state
$|g\rangle$; the closest analog of the sandwich of Lorentz generators and
polarization tensors is the expectation value of the (field) generator of
Lorentz transformations in the state $|g\rangle$. While does not have a simple
interpretation in terms of the $\bm{S}$ and $\bm{K}$ vectors, the interaction
(61) will supply the requisite factors of momenta for such an interpretation
to be possible, cf. Sect. II.4.

###  II.5 Nonminimal Lagrangian

We are primarily interested in amplitudes in the classical limit, where the
spin $s$ is taken to be large. We expect that the relevant interaction terms
do not depend on a particular representation of the spin, and thus are Lorentz
singlets constructed from covariant derivatives, photon field strengths,
$\phi_{s}$ and Lorentz generators (17) in the same representation as
$\phi_{s}$. Moreover, we consider for the time being only those interactions
that survive in the classical limit. The close relation in Eq. (27) between
Lorentz generators and the spin tensor and the scaling of momenta in the
classical limit imply that the number of derivatives on the photons must be
equal to the number of Lorentz generators. Under these guidelines, we can
write down the following nonminimal linear-in-$F_{\mu\nu}$ interactions up to
two powers of spins,

$\displaystyle(-1)^{s}\mathcal{L}_{\text{non-min}}$
$\displaystyle=QC_{1}F_{\mu\nu}\phi_{s}M^{\mu\nu}\bar{\phi}_{s}+\frac{QD_{1}}{m^{2}}F_{\mu\nu}(D_{\rho}\phi_{s}M^{\rho\mu}D^{\nu}\bar{\phi}_{s}+\text{c.c})$
(57)
$\displaystyle\quad-\frac{iQC_{2}}{2m^{2}}\partial_{(\mu}F_{\nu)\rho}(D^{\rho}\phi_{s}\mathbb{S}^{\mu}\mathbb{S}^{\nu}\bar{\phi}_{s}-\text{c.c})-\frac{iQD_{2}}{2m^{2}}\partial_{\mu}F_{\nu\rho}(D_{\alpha}\phi_{s}M^{\alpha\mu}M^{\nu\rho}\bar{\phi}_{s}-\text{c.c})\,,$

where for later convenience we choose111111Neutral particles can also have
nonminimal couplings analogous to those in Eq. (57). The corresponding
Lagrangian is obtained by the double-scaling limit $Q\rightarrow 0$,
$C_{i},D_{i}\rightarrow\infty$ with fixed products $QC_{i}$ and $QD_{i}$.  to
scale the Wilson coefficients by $Q$ so that at each order amplitudes display
overall powers of $\alpha$, and the Pauli-Lubanski spin operator
$\mathbb{S}^{\mu}$ is defined as

$\displaystyle\mathbb{S}^{\mu}\equiv\frac{-i}{2m}\,\varepsilon^{\mu\nu\rho\sigma}M_{\rho\sigma}D_{\nu}\,.$
(58)

We note that the $C_{i}$ operators are the electrodynamics analogs of
operators [41, 42, 46] of general relativity, and the $D_{i}$’s are the
electrodynamics analogs of the typical examples of “extra Wilson coefficients”
of Ref. [1]. From the effective-field-theory point of view, we can write down
another operator that contributes classically at the second order in spin,

$\displaystyle\mathcal{L}_{D_{2b}}=\frac{iQD_{2b}}{2m^{4}}\partial_{(\mu}F_{\nu)\rho}(D_{\lambda}\phi_{s}M^{\lambda\mu}M^{\nu}{}_{\sigma}D^{(\sigma}D^{\rho)}\bar{\phi}_{s}-\text{c.c})\,.$
(59)

While $C_{2}$ and $D_{2}$ give independent contribution to three-point
amplitudes at $\mathcal{O}(S^{2})$ and $\mathcal{O}(S^{1}K^{1})$, see Sect.
III.1, the above $D_{2b}$ operator gives independent contribution at
$\mathcal{O}(K^{2})$. Since the purpose of our current work is to understand
the existence of extra Wilson coefficients, for simplicity we will not
consider this operator further.

Field theory | Lagrangian |  Amplitude | External state
---|---|---|---
FT1 | $\mathcal{L}_{\text{EM}}+\mathcal{L}_{\text{min}}+\mathcal{L}_{\text{non-min}}$ | $\mathcal{A}^{\text{FT1s}}$ | spin-$s$
$\mathcal{A}^{\text{FT1g}}$ | generic
FT2 | $\mathcal{L}_{\text{EM}}+\mathcal{L}^{s}_{\text{min}}+\mathcal{L}_{\text{non-min}}$ | $\mathcal{A}^{\text{FT2}}$ | spin-$s$
FT3 | $\;\mathcal{L}_{\text{EM}}+\mathcal{L}^{s,s-1}_{\text{min}}+\mathcal{L}_{\text{non-min}}^{s,s-1}\;$ | $\mathcal{A}^{\text{FT3s}}$ |  spin-$s$
$\mathcal{A}^{\text{FT3g}}$ |  indefinite spin
Table 1: Field-theory amplitudes, corresponding Lagrangians and external
states. The Lagrangians are given in Eqs. (2), (7), (11), (57) and (61).

Table 1 collects the Lagrangians of the four effective field theories that are
our focus noting also the notation we use for their corresponding amplitudes.
FT1 are described by the same Lagrangian,
$\mathcal{L}_{\text{EM}}+\mathcal{L}_{\text{min}}+\mathcal{L}_{\text{non-
min}}$. To compute amplitudes in these two theories, we do not need to specify
a particular value for $s$. As discussed in Sect. II.3, the representations of
the rotation group with spin $s-2k$ that are part of the field $\phi_{s}$ have
positive norm and are therefore physical, while those with spin $s-(2k+1)$
have negative norm. In contrast, FT2 is described by the Lagrangian
$\mathcal{L}_{\text{EM}}+\mathcal{L}^{s}_{\text{min}}+\mathcal{L}_{\text{non-
min}}$, and contains only the physical spin-$s$ degrees of freedom. When
computing the amplitude $\mathcal{A}^{\text{FT1s}}$, we restrict the external
states to be the physical spin-$s$ states, which are transverse such that the
resultant spin tensors satisfy the covariant SSC according to Sect. II.3.
Meanwhile, we keep the external states generic in $\mathcal{A}^{\text{FT1g}}$.
As a result, the amplitude $\mathcal{A}^{\text{FT1g}}$ contains explicit SSC-
violating terms compared to $\mathcal{A}^{\text{FT1s}}$.

We show in Sect. III that despite having the same physical spin-$s$ external
states, $\mathcal{A}^{\text{FT1s}}$ and $\mathcal{A}^{\text{FT2}}$ are
different for four-point Compton scattering in the classical limit. In
particular, the Compton amplitudes from FT2 depend only on $C_{1}$ and $C_{2}$
while $\mathcal{A}^{\text{FT1s}}$ also depend on $D_{1}$ and $D_{2}$, similar
to the appearance of additional nontrivial Wilson coefficients in general
relativity [76]. The differences between these amplitudes vanish for

$\displaystyle C_{1}=C_{2}=1\qquad D_{1}=D_{2}=0\,,$ (60)

and reproduce the root-Kerr amplitudes of Ref. [112], so that the additional
$D_{i}$ operators do not contribute, in much the same way that additional
operators do not contribute to the Kerr black hole. The similarity of the
root-Kerr solution in electromagnetism and the Kerr solution in general
relativity follows from the double copy.

These results indicate that additional lower-spin degrees of freedom are the
origin of the extra Wilson coefficients. We consider an interpolation between
FT1 and FT2 in Sect. VI to understand the effect of the state projector. In
FT1 these degrees of freedom have negative norm; a natural question is whether
lower-spin states with positive norm have similar consequences. FT3 explores
this question. With some foresight which is justified in Sect. VI, we choose
the nonminimal interactions of $\phi_{s}$ and $\phi_{s-1}$ in Eq. 11, valid
through the quadratic order in spin, to be

$\displaystyle\mathcal{L}_{\text{non-min}}^{s,s-1}$
$\displaystyle=QC_{1}F_{\mu\nu}\phi_{s}M^{\mu\nu}\bar{\phi}_{s}-\frac{2iQ\widetilde{C}_{1}\sqrt{s}}{m}F_{\mu\nu}\Big{[}(\phi_{s})^{\mu}{}_{\alpha_{2}\ldots\alpha_{s}}D^{\nu}\bar{\phi}_{s-1}^{\alpha_{2}\ldots\alpha_{s}}-\text{c.c}\Big{]}$
(61)
$\displaystyle-\frac{iQC_{2}}{2m^{2}}\partial_{(\mu}F_{\nu)\rho}(D^{\rho}\phi_{s}\mathbb{S}^{\mu}\mathbb{S}^{\nu}\bar{\phi}_{s}-\text{c.c})-\frac{2iQ\widetilde{C}_{2}\sqrt{s}}{m}F_{\mu\nu}\Big{[}(\phi_{s})^{\mu}{}_{\alpha_{2}\ldots\alpha_{s}}D^{\alpha_{2}}\bar{\phi}_{s-1}^{\nu\alpha_{3}\ldots\alpha_{s}}-\text{c.c}\Big{]}\,,$

and the Lagrangian of FT3 is given by the third line of Table 1. We shall see
in Sect. III that the Wilson coefficients $\widetilde{C}_{1}$ and
$\widetilde{C}_{2}$ appear at $\mathcal{O}(K^{1})$ and
$\mathcal{O}(S^{1}K^{1})$ order of the Compton amplitudes respectively, and
that there exists an effective map between the $D_{i}$ and $\widetilde{C}_{i}$
coefficients. As discussed in Sect. II.3, we need to include couplings between
$\phi_{s}$ and $\phi_{s-2}$ to access the $\mathcal{O}(K^{2})$ interactions,
which we omit for simplicity.

Similar to gravity, operators describing tidal deformations under the
influence of external fields are necessary to describe the electromagnetic
interactions of generic spinning bodies. Simple counting of classical scaling
indicates that in QED they first appear $\mathcal{O}(S^{2})$. At this order in
spin three independent operators are

$\displaystyle(-1)^{s}\mathcal{L}_{F^{2}}$
$\displaystyle=\frac{Q^{2}E_{1}}{m^{2}}F_{\mu\nu}F_{\rho\sigma}\phi_{s}M^{\mu\nu}M^{\rho\sigma}\bar{\phi}_{s}+\frac{Q^{2}E_{2}}{m^{2}}F_{\mu\nu}F_{\rho}{}^{\mu}\phi_{s}M^{\nu\lambda}M_{\lambda}{}^{\rho}\bar{\phi}_{s}$
$\displaystyle\quad+\frac{Q^{2}E_{3}}{m^{4}}F_{\mu\nu}F_{\rho\sigma}D^{\mu}\phi_{s}M^{\nu\lambda}M_{\lambda}{}^{\rho}D^{\sigma}\bar{\phi}_{s}+{\cal
O}(M^{3})\,.$ (62)

Including them we find that all $E_{i}$ Wilson coefficients vanish for the
root-Kerr states in much the same way as the $D_{i}$ coefficients vanish for
these states in FT1.

##  III Scattering Amplitudes

In this section we first compute the 1PL (tree) Compton amplitudes of the
higher-spin effective Lagrangians introduced in the previous section and
summarized in Table 1. We then use them as the basic building blocks of the
$\mathcal{O}(\alpha^{2})$ two-body amplitudes through generalized
unitarity.121212For simplicity, we suppress a factor of $Q$ in the three-point
Compton amplitudes, and a factor of $Q^{2}$ in the four-point Compton
amplitudes, where $Q$ is the electric charge of the massive body. In addition,
classical Compton amplitudes are also observables that can be directly
compared with worldline computations along the lines of Ref. [189]. The
comparison will be given in Sect. IV.

###  III.1 Three-Point Amplitudes

We start with computing and comparing the three-point Compton amplitudes from
the theories in Table 1. Assuming that all the momenta are outgoing, the
Feynman rules for FT1 are given by

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$\displaystyle=(-1)^{s}\mathcal{E}_{1}\cdot\mathbb{M}_{3}(p_{1},p_{2},q_{3},\epsilon_{3})\cdot\bar{\mathcal{E}}_{2}\,,$
(63) $\displaystyle\mathbb{M}_{3}(p_{1},p_{2},q_{3},\epsilon_{3})$
$\displaystyle=2\epsilon_{3}\cdot
p_{1}\mathbbm{1}-2iC_{1}M_{\mu\nu}q_{3}^{\mu}\epsilon_{3}^{\nu}-\frac{2iD_{1}}{m^{2}}\epsilon_{3}\cdot
p_{1}M_{\mu\nu}p_{1}^{\mu}q_{3}^{\nu}$
$\displaystyle\quad+\frac{C_{2}}{m^{2}}\epsilon_{3}\cdot
p_{1}\\{M^{\mu\nu},M_{\mu}{}^{\rho}\\}q_{3}^{\nu}q_{3}^{\rho}-\frac{C_{2}}{m^{4}}\epsilon_{3}\cdot
p_{1}\\{M_{\mu\nu},M_{\rho\lambda}\\}p_{1}^{\mu}q_{3}^{\nu}p_{1}^{\rho}q_{3}^{\lambda}$
$\displaystyle\quad+\frac{2D_{2}}{m^{2}}\\{M_{\mu\nu},M_{\rho\lambda}\\}p_{1}^{\mu}q_{3}^{\nu}q_{3}^{\rho}\epsilon_{3}^{\lambda}\,.$

The symmetric product between Lorentz generators is defined in Sect. II.2.

In the classical limit, the massive spinning particles are described by the
spin coherent states (30). We first consider FT1g with generic coherent states
that do not satisfy the transversality. The expectation values of Lorentz
generators are given by Eq. 27, which lead to the classical spin tensor
$\mathsf{S}_{\mu\nu}$ that do not satisfy the covariant SSC. The three-point
amplitude is

$\displaystyle\mathcal{A}_{3}^{\text{FT1g}}$
$\displaystyle=(-1)^{s}\mathcal{E}_{1}\cdot\bar{\mathcal{E}}_{2}\left[2\epsilon_{3}\cdot
p_{1}-2iC_{1}\mathsf{S}_{\mu\nu}q_{3}^{\mu}\epsilon_{3}^{\nu}+\frac{C_{2}}{m^{2}}\epsilon_{3}\cdot
p_{1}\mathsf{S}_{\mu\nu}q_{3}^{\nu}\mathsf{S}^{\mu}{}_{\lambda}q_{3}^{\lambda}\right.$
$\displaystyle\qquad-\left.\mathsf{S}_{\mu\nu}p_{1}^{\mu}q_{3}^{\nu}\left(\frac{2iD_{1}}{m^{2}}\epsilon_{3}\cdot
p_{1}+\frac{C_{2}}{m^{4}}\epsilon_{3}\cdot
p_{1}\mathsf{S}_{\lambda\sigma}p_{1}^{\lambda}q_{3}^{\sigma}-\frac{2D_{2}}{m^{2}}\mathsf{S}_{\lambda\sigma}q_{3}^{\lambda}\epsilon_{3}^{\sigma}\right)\right]$
$\displaystyle=2(-1)^{s}\mathcal{E}_{1}\cdot\bar{\mathcal{E}}_{2}\left[\epsilon_{3}\cdot
p_{1}-iC_{1}S_{\mu\nu}q_{3}^{\mu}\epsilon_{3}^{\nu}-(C_{1}-D_{1})\epsilon_{3}\cdot
p_{1}\frac{q\cdot K}{m}\vphantom{\left(\frac{q\cdot K}{m}\right)^{2}}\right.$
$\displaystyle\qquad+\left.\frac{C_{2}}{2m^{2}}\epsilon_{3}\cdot
p_{1}S_{\mu\nu}q_{3}^{\nu}S^{\mu}{}_{\lambda}q_{3}^{\lambda}+iD_{2}S_{\mu\nu}q_{3}^{\mu}\epsilon_{3}^{\nu}\frac{q\cdot
K}{m}+D_{2}\epsilon_{3}\cdot p_{1}\left(\frac{q\cdot K}{m}\right)^{2}\right],$
(64)

where in the second equal sign we have used Eq. 25 to expose the SSC
preserving $S$-part and the SSC violating $K$-part in $\mathsf{S}^{\mu\nu}$.
As expected, the extra Wilson coefficients $D_{i}$ appear with the SSC-
violating terms. If we further restrict the external states to be transverse,
the $K$-part becomes subleading in the classical limit and thus drops out.
This leads to the three-point amplitudes $\mathcal{A}_{3}^{\text{FT1s}}$ and
$\mathcal{A}_{3}^{\text{FT2}}$

$\displaystyle\mathcal{A}_{3}^{\text{FT1s}}=\mathcal{A}_{3}^{\text{FT2}}$
$\displaystyle=2(-\varepsilon_{1}\cdot\bar{\varepsilon}_{2})^{s}\left[\epsilon_{3}\cdot
p_{1}-iC_{1}S_{\mu\nu}q_{3}^{\mu}\epsilon_{3}^{\nu}+\frac{C_{2}}{2m^{2}}\epsilon_{3}\cdot
p_{1}S_{\mu\nu}q_{3}^{\nu}S^{\mu}{}_{\lambda}q_{3}^{\lambda}\right].$ (65)

This amplitude only depends on the $C_{i}$ Wilson coefficients. The fact that
$\mathcal{A}_{3}^{\text{FT1s}}$ does not contain any additional Wilson
coefficients is analogous to the three-point gravity amplitude of Ref. [76],
which did not contain any additional Wilson coefficients either, connected to
restricting the external states to traceless and transverse spin-$s$ ones.

For FT3, we can similarly restrict the external states to be spin-$s$. The
resulting amplitude $\mathcal{A}_{3}^{\text{FT3s}}$ is the same as Eq. 65,
i.e.

$\displaystyle\mathcal{A}_{3}^{\text{FT1s}}=\mathcal{A}_{3}^{\text{FT2}}=\mathcal{A}_{3}^{\text{FT3s}}\,.$
(66)

We may also choose the indefinite-spin states (55); the corresponding
amplitude $\mathcal{A}_{3}^{\text{FT3g}}$ receives contributions from both the
spin-$s$ and spin-$(s-1)$ external states,

$\displaystyle\mathcal{A}_{3}^{\text{FT3g}}$
$\displaystyle=\mathcal{A}_{3}^{\text{FT3s}}+\frac{2i\sqrt{s}}{m}(-\varepsilon_{1}\cdot\bar{\varepsilon}_{2})^{s-1}(\varepsilon_{1}\cdot
q_{3}+\bar{\varepsilon}_{2}\cdot
q_{3})\left[\widetilde{C}_{1}(\epsilon_{3}\cdot
p_{1})+\widetilde{C}_{2}(\mathcal{E}_{1}\cdot
f_{3}\cdot\bar{\mathcal{E}}_{4})\right]$
$\displaystyle=2(-1)^{s}\mathcal{E}_{1}\cdot\bar{\mathcal{E}}_{2}\left[\epsilon_{3}\cdot
p_{1}-iC_{1}S_{\mu\nu}q_{3}^{\mu}\epsilon_{3}^{\nu}+\frac{C_{2}}{2m^{2}}\epsilon_{3}\cdot
p_{1}S_{\mu\nu}q_{3}^{\nu}S^{\mu}{}_{\lambda}q_{3}^{\lambda}\right.$
$\displaystyle\qquad\qquad\qquad\qquad+\left.(i\widetilde{C}_{1}-1)\epsilon_{3}\cdot
p_{1}\frac{q\cdot
K}{m}+(iC_{1}-\widetilde{C}_{2})S_{\mu\nu}q_{3}^{\mu}\epsilon_{3}^{\nu}\frac{q\cdot
K}{m}\right],$ (67)

where we have used Sects. II.4 and 50 to obtain the final expression in terms
of the boost vector $K$. The appearance of the Wilson coefficients
$\widetilde{C}_{1}$ and $\widetilde{C}_{2}$ associated with SSC violation. We
find that, up to $\mathcal{O}(K^{2})$ terms, the additional Wilson
coefficients in FT1g and FT3g are related as

$\displaystyle\mathcal{A}_{3}^{\text{FT3g}}=\mathcal{A}_{3}^{\text{FT1g}}\qquad\text{for}\quad
i\widetilde{C}_{1}=1-C_{1}+D_{1}\;\text{ and
}\;i\widetilde{C}_{2}=D_{2}-C_{1}\,.$ (68)

The extra factor of $i$ in this map reflects the unphysical nature of the
spin-$(s-1)$ states in FT1 vs. their physical nature in FT3. While this map is
not an equivalence of Lagrangians, it provides a simple relation between the
amplitudes of FT1 and FT3. A similar map also exists at $\mathcal{O}(K^{2})$
if we include such interactions in both FT1 and FT3.

###  III.2 Four-Point Compton Amplitudes

$p_{1}$$p_{4}$$q_{2},\epsilon_{2}$$q_{3},\epsilon_{3}$$p_{1}$$p_{4}$$q_{3},\epsilon_{3}$$q_{2},\epsilon_{2}$$p_{1}$$p_{4}$$q_{2},\epsilon_{2}$$q_{3},\epsilon_{3}$

.

Figure 1: The three Feynman diagrams describing lowest-order Compton
scattering.

At four points, the Compton amplitudes are given by the three Feynman diagrams
in Fig. 1, with the relevant propagators and three- and four-point vertices
derived from our field theories. While the Lagrangian of FT1 is independent of
$s$ and therefore general properties properties of Lorentz generators and
coherent states are sufficient for amplitude calculations, the explicit
dependence on $s$ of FT2 and FT3 Lagrangians requires that for them we choose
a particular representation. As noted in Sect. II.3, we choose the $(s,s)$
representation, for which the coherent states are given by Eq. 35 and the
Lorentz generators are listed in Eq. 38. Specifically for four-point Compton
amplitudes, the Feynman rules for FT2 and FT3 simplify considerably because
every vertex has at least one on-shell massive particle represented by the
symmetric, traceless and transverse polarization tensor. Thus, when deriving
the three- and four-point vertex rules we can ignore all the interactions
covered by the ellipsis in Eq. 7 because they only include traces or/and
longitudinal modes of the external on-shell particle. For the same reason, we
can also ignore all the (resummed) propagators that involve lower-spin
auxiliary fields.

The spin-independent part of the Compton amplitude is common to FT1 though
FT3,

$\displaystyle\mathcal{A}_{4,\text{
cl.}}^{\text{FT1,2,3}}\Big{|}_{S^{0}}=(-1)^{s}\mathcal{E}_{1}\cdot\bar{\mathcal{E}}_{4}\frac{2(p_{1}\cdot
f_{2}\cdot f_{3}\cdot p_{1})}{(p_{1}\cdot q_{2})^{2}}\,,$ (69)

where
$f_{i}^{\mu\nu}\equiv\varepsilon_{i}^{\mu}q_{i}^{\nu}-\varepsilon_{i}^{\nu}q_{i}^{\mu}$.
For the $(s,s)$ representation, we do not need to distinguish
$\mathcal{E}_{1}\cdot\bar{\mathcal{E}}_{4}$ and
$(\varepsilon_{1}\cdot\bar{\varepsilon}_{4})^{s}$ as their difference is
higher order in $S$ and $K$. Up to the overall factor
$\mathcal{E}_{1}\cdot\bar{\mathcal{E}}_{4}$, Eq. (69) reproduces the classical
limit of the scalar QED Compton amplitude given in, for example, Eq. (2.8) of
Ref. [108].131313In Ref. [108] higher order in $q_{i}$ terms are also included
since they are needed when feeding the Compton amplitudes into unitarity cuts
for building higher PL two-body amplitudes.

####  III.2.1 The Linear-in-Spin Compton Amplitudes

Consider now spin-dependent parts of amplitudes of the four field theories.
For FT1, here and after we choose $\phi_{s}$ to be in the $(s,s)$
representation to streamline the comparison with FT2 and FT3. We first
consider $\mathcal{A}^{\text{FT1s}}$ in which the external states are
transverse. Evaluating the linear-in-spin part of the three Feynman diagrams
in Fig. 1 with the propagators and vertices following from the Lagrangian of
FT1 leads to

$\displaystyle\mathcal{A}_{4,\text{
cl.}}^{\text{FT1s}}\Big{|}_{S^{1}}=(-\varepsilon_{1}\cdot\bar{\varepsilon}_{4})^{s}S(p_{1})_{\mu\nu}$
$\displaystyle\left[\frac{iC_{1}}{(p_{1}\cdot
q_{2})^{2}}(f_{2}^{\mu\nu}q_{2\rho}f_{3}^{\rho\lambda}+f_{3}^{\mu\nu}q_{3\rho}f_{2}^{\rho\lambda})p_{1\lambda}+\frac{2iC_{1}^{2}}{p_{1}\cdot
q_{2}}f_{2}^{\nu\rho}f_{3\rho}{}^{\mu}\right.$
$\displaystyle\;+\left.\frac{2iD_{1}(2C_{1}-D_{1}-2)}{(p_{1}\cdot
q_{2})m^{2}}p_{1\rho}f_{2}^{\rho\mu}f_{3}^{\nu\lambda}p_{1\lambda}\right].$
(70)

The amplitude $\mathcal{A}_{\text{4, cl}}^{\text{FT1}}\Big{|}_{S^{1}}$ depends
on both the $C_{1}$ and $D_{1}$ Wilson coefficients.141414The gravitational
analog of this amplitude is of ${\cal O}(S^{2})$. We note that $D_{1}$ appears
only together with the combination
$p_{1\rho}f_{2}^{\rho\mu}f_{3}^{\nu\lambda}p_{1\lambda}$. Repeating the
calculation while relaxing the transversality on external states leads to the
Compton amplitude,

$\displaystyle\mathcal{A}_{4,\text{
cl.}}^{\text{FT1g}}\Big{|}_{S^{1}}=(-1)^{s}\mathcal{E}_{1}\cdot\bar{\mathcal{E}}_{4}\left\\{\mathsf{S}(p_{1})_{\mu\nu}\left[\frac{iC_{1}}{(p_{1}\cdot
q_{2})^{2}}(f_{2}^{\mu\nu}q_{2\rho}f_{3}^{\rho\lambda}+f_{3}^{\mu\nu}q_{3\rho}f_{2}^{\rho\lambda})p_{1\lambda}+\frac{2iC_{1}^{2}}{p_{1}\cdot
q_{2}}f_{2}^{\nu\rho}f_{3\rho}{}^{\mu}\right.\right.$
$\displaystyle\quad+\left.\frac{2iD_{1}(2C_{1}-D_{1}-2)}{(p_{1}\cdot
q_{2})m^{2}}p_{1\rho}f_{2}^{\rho\mu}f_{3}^{\nu\lambda}p_{1\lambda}\right]+\mathsf{S}(p_{1})_{\mu\nu}p_{1}^{\nu}\left[\frac{2iD_{1}(C_{1}+1)}{(p_{1}\cdot
q_{2})m^{2}}p_{1\rho}(f_{3}^{\rho\lambda}f_{2\lambda}{}^{\mu}-f_{2}^{\rho\lambda}f_{3\lambda}{}^{\mu})\right.$
$\displaystyle\quad+\left.\left.\frac{2iD_{1}}{(p_{1}\cdot
q_{2})^{2}m^{2}}p_{1\rho}p_{1\lambda}(f_{3}^{\rho\mu}q_{3\sigma}f_{2}^{\sigma\lambda}+f_{2}^{\rho\mu}q_{2\sigma}f_{3}^{\sigma\lambda})\right]\right\\}.$
(71)

We note that the first term is formally identical to Sect. III.2.1 for FT1
except for the replacement $S\rightarrow\mathsf{S}$, while the second term is
proportional to the SSC condition $\mathsf{S}_{\mu\nu}p_{1}^{\nu}$.

Proceeding to FT2 of a single transverse spin-$s$ field, we extract the
classical $\mathcal{O}(S^{1})$ Compton amplitude from explicit calculations
for $s=1,2,3$ using the Lagrangian given in the third row of Table 1. Unlike
FT1, the amplitudes are now given in terms of explicit polarization vectors
instead of spin tensors such that we need to convert the former into the
latter. Since the classical amplitudes in terms of spin tensors scale as
$\mathcal{O}(q^{0})$ and the spin tensors scale as $\mathcal{O}(q^{-1})$, in a
fixed-spin calculation, the classical part of the $\mathcal{O}(S^{1})$
amplitude is among the $\mathcal{O}(q)$ terms of the full quantum amplitude
[76]. At this order, the massive polarization vectors appear in two
structures, $(\varepsilon_{1}\cdot\bar{\varepsilon}_{4})^{s}$ and
$(\varepsilon_{1}\cdot\bar{\varepsilon}_{4})^{s-1}\varepsilon_{1}^{[\mu}\bar{\varepsilon}_{4}^{\nu]}$.
The terms proportional to $(\varepsilon_{1}\cdot\bar{\varepsilon}_{4})^{s}$
belong to the quantum spinless amplitude, which can be ignored here. We then
use the relation (40) to convert the second structure to spin tensors. The
final classical amplitude is obtained by extrapolating the finite-spin results
to generic $s$ and taking the $s\rightarrow\infty$ of that expression. At
$\mathcal{O}(S^{1})$, the amplitude after the replacement (40) is in fact
independent of $s$, as we have explicitly checked for $s\leqslant 3$. After
identifying the classical part, the final answer for the classical Compton
amplitude is

$\displaystyle\mathcal{A}_{\text{4,
cl}}^{\text{FT2}}\Big{|}_{S^{1}}=(-\varepsilon_{1}\cdot\bar{\varepsilon}_{4})^{s}S(p_{1})_{\mu\nu}$
$\displaystyle\left[\frac{iC_{1}}{(p_{1}\cdot
q_{2})^{2}}(f_{2}^{\mu\nu}q_{2\rho}f_{3}^{\rho\lambda}+f_{3}^{\mu\nu}q_{3\rho}f_{2}^{\rho\lambda})p_{1\lambda}+\frac{2iC_{1}^{2}}{p_{1}\cdot
q_{2}}f_{2}^{\nu\rho}f_{3\rho}{}^{\mu}\right.$
$\displaystyle\;+\left.\frac{2i(C_{1}-1)^{2}}{(p_{1}\cdot
q_{2})m^{2}}p_{1\rho}f_{2}^{\rho\mu}f_{3}^{\nu\lambda}p_{1\lambda}\right].$
(72)

Notably, this amplitude is independent of $D_{1}$, and it can be obtained from
Sect. III.2.1 by setting $D_{1}$ to a special value,

$\mathcal{A}_{\text{4, cl}}^{\text{FT1s}}\Big{|}_{S^{1}}=\mathcal{A}_{\text{4,
cl}}^{\text{FT2}}\Big{|}_{S^{1}}\quad\text{for}\quad D_{1}=C_{1}-1\,.$ (73)

In other words, for this special value of $D_{1}$, item FT1 effectively
propagates only the spin-$s$ states. Moreover, the special value $C_{1}=1$ and
$D_{1}=0$ reproduces the root-Kerr Compton amplitudes [112]. The appearance of
additional Wilson coefficients in $\mathcal{A}^{\text{FT1s}}$ compared with
$\mathcal{A}^{\text{FT2}}$ can be attributed to the additional propagating
degrees of freedom.151515In Ref. [173], a similar computation was carried out
for $s=1$ and observed a similar effect. Their amplitudes are equivalent to
our $D_{1}=0$ case.

Finally, FT3 amplitudes also receive contributions from lower-spin states. We
first restrict the lower spins to only appear in the intermediate states.
Repeating the same steps as for FT2 we find that spin-$(s-1)$ intermediate
states contribute as

$\displaystyle\mathcal{A}_{\text{4, cl}}^{\text{FT3s}}\Big{|}_{S^{1}}$
$\displaystyle=\mathcal{A}_{\text{4,
cl}}^{\text{FT3}}\Big{|}_{S^{1}}+(-\varepsilon_{1}\cdot\bar{\varepsilon}_{4})^{s}S(p_{1})_{\mu\nu}\left[\frac{2i\widetilde{C}_{1}^{2}}{(p_{1}\cdot
q_{2})m^{2}}p_{1\rho}f_{2}^{\rho\mu}f_{3}^{\nu\lambda}p_{1\lambda}\right].$
(74)

We note that the $\mathcal{O}(S^{1})$ amplitude does not change if we include
intermediate states with spin less than $s-1$. Comparing with Sect. III.2.1,
we find that the two amplitudes are formally related by the same map as Eq.
68,

$\displaystyle\mathcal{A}_{4,\text{
cl.}}^{\text{FT3s}}\Big{|}_{S^{1}}=\mathcal{A}_{4,\text{
cl.}}^{\text{FT1s}}\Big{|}_{S^{1}}\quad\text{for}\quad
i\widetilde{C}_{1}=1-C_{1}+D_{1}\,.$ (75)

Furthermore, this map persists even for amplitudes with external lower-spin
states. To see this, we first rewrite Sect. III.2.1 using Eq. 25,

$\displaystyle\mathcal{A}_{4,\text{ cl.}}^{\text{FT1g}}\Big{|}_{S^{1}}$
$\displaystyle=(-1)^{s}\mathcal{E}_{1}\cdot\bar{\mathcal{E}}_{4}\left\\{S(p_{1})_{\mu\nu}\left[\frac{iC_{1}}{(p_{1}\cdot
q_{2})^{2}}(f_{2}^{\mu\nu}q_{2\rho}f_{3}^{\rho\lambda}+f_{3}^{\mu\nu}q_{3\rho}f_{2}^{\rho\lambda})p_{1\lambda}+\frac{2iC_{1}^{2}}{p_{1}\cdot
q_{2}}f_{2}^{\nu\rho}f_{3\rho}{}^{\mu}\right.\right.$
$\displaystyle+\left.\frac{2iD_{1}(2C_{1}-D_{1}-2)}{(p_{1}\cdot
q_{2})m^{2}}p_{1\rho}f_{2}^{\rho\mu}f_{3}^{\nu\lambda}p_{1\lambda}\right]+\frac{2K(p_{1})_{\mu}p_{1\nu}}{m}\left[\frac{D_{1}-C_{1}}{(p_{1}\cdot
q_{2})^{2}}(q_{2}^{\mu}+q_{3}^{\mu})f_{2}^{\nu\rho}f_{3\rho\lambda}p_{1}^{\lambda}\right.$
$\displaystyle-\left.\left.\frac{C_{1}(1-C_{1}+D_{1})}{p_{1}\cdot
q_{2}}(f_{2}^{\nu\rho}f_{3\rho}{}^{\mu}-f_{3}^{\nu\rho}f_{2\rho}{}^{\mu})\right]\right\\}.$
(76)

We then find $\mathcal{A}_{4}^{\text{FT3g}}$, with external states in Eq.
(55), from the Lagrangian of FT3. The momentum dependence of vertices is
essential to express the contributions with spin-$(s-1)$ external states in
terms of $K^{\mu}$, using Sects. II.4 and 50. The result is

$\displaystyle\mathcal{A}_{4}^{\text{FT3g}}\Big{|}_{S^{1}}$
$\displaystyle=(-1)^{s}\mathcal{E}_{1}\cdot\bar{\mathcal{E}}_{4}\left\\{\left[\vphantom{\frac{iC_{1}\widetilde{C}_{1}}{p_{1}\cdot
q_{2}}}S(p_{1})_{\mu\nu}\frac{iC_{1}}{(p_{1}\cdot
q_{2})^{2}}(f_{2}^{\mu\nu}q_{2\rho}f_{3}^{\rho\lambda}+f_{3}^{\mu\nu}q_{3\rho}f_{2}^{\rho\lambda})p_{1\lambda}+\frac{2iC_{1}^{2}}{p_{1}\cdot
q_{2}}f_{2}^{\nu\rho}f_{3\rho}{}^{\mu}\right.\right.$
$\displaystyle+\left.\frac{2i[(C_{1}-1)^{2}+\widetilde{C}_{1}^{2}]}{(p_{1}\cdot
q_{2})m^{2}}p_{1\rho}f_{2}^{\rho\mu}f_{3}^{\nu\lambda}p_{1\lambda}\right]+\frac{2K(p_{1})_{\mu}p_{1\nu}}{m}\left[\frac{i\widetilde{C}_{1}-1}{(p_{1}\cdot
q_{2})^{2}}(q_{2}^{\mu}+q_{3}^{\mu})f_{2}^{\nu\rho}f_{3\rho\lambda}p_{1}^{\lambda}\right.$
$\displaystyle-\left.\left.\frac{iC_{1}\widetilde{C}_{1}}{p_{1}\cdot
q_{2}}(f_{2}^{\nu\rho}f_{3\rho}{}^{\mu}-f_{3}^{\nu\rho}f_{2\rho}{}^{\mu})\right]\right\\}.$
(77)

Now comparing Sects. III.2.1 and III.2.1, we find that

$\displaystyle\mathcal{A}_{4,\text{
cl.}}^{\text{FT3g}}\Big{|}_{S^{1}}=\mathcal{A}_{4,\text{
cl.}}^{\text{FT1g}}\Big{|}_{S^{1}}\qquad\text{for}\quad
i\widetilde{C}_{1}=1-C_{1}+D_{1}\,.$ (78)

The robustness of this map demonstrates that the terms tagged by the extra
Wilson coefficients present in the amplitudes (and observables) of FT1 and FT3
carry new physical information compared to FT2.

####  III.2.2 The Quadratic-in-Spin Compton Amplitudes

Feynman-diagram calculations using the propagators and vertices of FT1 as well
as properties of transverse coherent states show that, at $\mathcal{O}(S^{2})$
the Compton amplitude depends on two distinct contractions of spin tensors:

$\displaystyle\mathcal{A}_{\text{4, cl.}}^{\text{FT1s}}\Big{|}_{S^{2}}$
$\displaystyle=(-\varepsilon_{1}\cdot\bar{\varepsilon}_{4})^{s}\big{(}S(p_{1})_{\mu\nu}S(p_{1})_{\lambda\sigma}\mathcal{X}^{\mu\nu\lambda\sigma}+S(p_{1})_{\mu\lambda}S(p_{1})^{\lambda}{}_{\nu}\mathcal{X}^{\mu\nu}\big{)}\,.$
(79)

Their kinematic coefficients are given by

$\displaystyle\mathcal{X}^{\mu\nu\lambda\sigma}$
$\displaystyle=\frac{C_{1}^{2}(q_{2}\cdot q_{3})}{2(p_{1}\cdot
q_{2})^{2}}f_{2}^{\mu\nu}f_{3}^{\lambda\sigma}+\frac{C_{1}D_{1}+D_{2}(C_{1}-D_{1}-1)}{(p_{1}\cdot
q_{2})m^{2}}p_{1\rho}(f_{3}^{\rho\mu}q_{2}^{\nu}f_{2}^{\lambda\sigma}-f_{2}^{\rho\mu}q_{3}^{\nu}f_{3}^{\lambda\sigma})\,,$
(80) $\displaystyle\mathcal{X}^{\mu\nu}$
$\displaystyle=\frac{C_{2}}{m^{2}}\left[\frac{p_{1\rho}p_{1\alpha}(f_{2}^{\rho\mu}q_{2}^{\nu}f_{3}^{\alpha\beta}q_{2\beta}+f_{3}^{\rho\mu}q_{3}^{\nu}f_{2}^{\alpha\beta}q_{3\beta})}{(p_{1}\cdot
q_{2})^{2}}+\frac{p_{1\rho}(f_{2}^{\rho\sigma}f_{3\sigma}{}^{\mu}_{\vphantom{2}}q_{3}^{\nu}-f_{3}^{\rho\sigma}f_{2\sigma}{}^{\mu}_{\vphantom{2}}q_{2}^{\nu})}{(p_{1}\cdot
q_{2})}\right.$ (81)
$\displaystyle\qquad\quad+\left.\frac{2C_{1}\,p_{1\rho}(f_{3}^{\rho\mu}f_{2}^{\nu\sigma}q_{3\sigma}-f_{2}^{\rho\mu}f_{3}^{\nu\sigma}q_{2\sigma})}{(p_{1}\cdot
q_{2})}+\frac{2(C_{1}-1)p_{1\rho}f_{2}^{\rho\mu}f_{3}^{\nu\sigma}p_{1\sigma}}{m^{2}}+2C_{1}f_{2\rho}{}^{\mu}_{\vphantom{2}}f_{3}^{\nu\rho}\right].$

The dependence on Wilson coefficients indicates that both terms originate from
both $\mathcal{L}_{\text{min}}$ and $\mathcal{L}_{\text{non-min}}$, and the
Lorentz algebra was used to reduce a product of three Lorentz generators to a
sum of irreducible (symmetric) products. We also note that the $D_{1}$ and
$D_{2}$ dependence only appear in $\mathcal{X}^{\mu\nu\lambda\sigma}$.

Choosing general asymptotic states instead of transverse ones leads to the
amplitude $\mathcal{A}^{\text{FT1g}}$. Apart from the replacement
$S\rightarrow\mathsf{S}$ in $\mathcal{A}_{\text{4,
cl}}^{\text{FT1s}}\Big{|}_{S^{2}}$, the amplitude contains terms proportional
to the covariant SSC conditions:

$\displaystyle\mathcal{A}_{4,\text{ cl.}}^{\text{FT1g}}\Big{|}_{S^{2}}$
$\displaystyle=(-1)^{s}\mathcal{E}_{1}\cdot\bar{\mathcal{E}}_{4}\Big{[}\mathsf{S}(p_{1})_{\mu\nu}\mathsf{S}(p_{1})_{\lambda\sigma}\mathcal{X}^{\mu\nu\lambda\sigma}+\mathsf{S}(p_{1})_{\mu\lambda}\mathsf{S}(p_{1})^{\lambda}{}_{\nu}\mathcal{X}^{\mu\nu}$
(82)
$\displaystyle\quad+\mathsf{S}(p_{1})_{\mu\nu}p_{1}^{\nu}\mathsf{S}(p_{1})_{\lambda\sigma}\mathcal{Y}^{\mu\lambda\sigma}+\mathsf{S}(p_{1})_{\mu\nu}p_{1}^{\nu}\mathsf{S}(p_{1})_{\lambda\sigma}p_{1}^{\sigma}\mathcal{Y}^{\mu\lambda}+\mathsf{S}(p_{1})_{\mu\lambda}\mathsf{S}(p_{1})^{\lambda}{}_{\nu}p_{1}^{\nu}\mathcal{Y}^{\mu}\Big{]},$

where the additional kinematic coefficients are given by

$\displaystyle\mathcal{Y}^{\mu\lambda\sigma}$
$\displaystyle=\frac{C_{1}(D_{1}-D_{2})}{(p_{1}\cdot
q_{2})m^{2}}(q_{2\rho}f_{3}^{\rho\mu}f_{2}^{\lambda\sigma}-q_{3\rho}f_{2}^{\rho\mu}f_{3}^{\lambda\sigma})+\frac{C_{1}D_{1}(q_{2}\cdot
q_{3})}{(p_{1}\cdot
q_{2})^{2}m^{2}}p_{1\rho}(f_{3}^{\rho\mu}f_{2}^{\lambda\sigma}-f_{2}^{\rho\mu}f_{3}^{\lambda\sigma})$
$\displaystyle\quad+\frac{2C_{1}D_{2}}{(p_{1}\cdot
q_{2})m^{2}}(q_{2}+q_{3})^{\mu}f_{2}^{\sigma\alpha}f_{3\alpha}{}^{\lambda}+\frac{2D_{1}^{2}}{(p_{1}\cdot
q_{2})m^{4}}p_{1\rho}p_{1\alpha}(f_{2}^{\rho\mu}f_{3}^{\alpha\lambda}q_{2}^{\sigma}-f_{3}^{\rho\mu}f_{2}^{\alpha\lambda}q_{3}^{\sigma})$
$\displaystyle\quad+\frac{2D_{2}}{(p_{1}\cdot
q_{2})^{2}m^{2}}p_{1\alpha}(q_{2}^{\lambda}f_{2}^{\sigma\mu}f_{3}^{\alpha\beta}q_{2\beta}+q_{3}^{\lambda}f_{3}^{\sigma\mu}f_{2}^{\alpha\beta}q_{3\beta})$
$\displaystyle\quad+\frac{2(C_{2}(C_{1}-D_{1}-1)-D_{1}D_{2})}{(p_{1}\cdot
q_{2})m^{4}}(q_{2}+q_{3})^{\mu}p_{1\rho}f_{3}^{\rho\lambda}f_{2}^{\sigma\alpha}p_{1\alpha}$
$\displaystyle\quad+\frac{C_{2}(C_{1}-D_{1}-1)}{m^{4}}(f_{2}^{\mu\lambda}f_{3}^{\sigma\rho}+f_{3}^{\mu\lambda}f_{2}^{\sigma\rho})p_{1\rho}+\frac{D_{1}D_{2}}{m^{4}}p_{1\rho}(f_{3}^{\rho\mu}f_{2}^{\lambda\sigma}+f_{2}^{\rho\mu}f_{3}^{\lambda\sigma})\,,$
(83) $\displaystyle\mathcal{Y}^{\mu\lambda}$
$\displaystyle=\frac{2D_{1}^{2}}{(p_{1}\cdot
q_{2})m^{4}}p_{1\rho}(f_{2}^{\rho\mu}q_{2\alpha}f_{3}^{\alpha\lambda}-f_{3}^{\rho\mu}q_{3\alpha}f_{2}^{\alpha\lambda})+\frac{2D_{1}^{2}(q_{2}\cdot
q_{3})}{(p_{1}\cdot
q_{2})^{2}m^{4}}p_{1\rho}f_{2}^{\rho\mu}p_{1\alpha}f_{3}^{\alpha\lambda}$
$\displaystyle\quad+\frac{C_{2}}{(p_{1}\cdot
q_{2})^{2}m^{4}}p_{1\rho}(q_{2}^{\mu}f_{2}^{\rho\lambda}p_{1\alpha}f_{3}^{\alpha\beta}q_{2\beta}+q_{3}^{\mu}f_{3}^{\rho\lambda}p_{1\alpha}f_{2}^{\alpha\beta}q_{3\beta})$
$\displaystyle\quad+\frac{C_{2}-2D_{1}D_{2}}{(p_{1}\cdot
q_{2})m^{4}}p_{1\alpha}(q_{3}^{\mu}f_{2}^{\alpha\beta}f_{3\beta}{}^{\lambda}-q_{2}^{\mu}f_{3}^{\alpha\beta}f_{2\beta}{}^{\lambda})+\frac{2C_{1}C_{2}}{(p_{1}\cdot
q_{2})m^{4}}p_{1\alpha}(q_{2}^{\mu}f_{2}^{\alpha\beta}f_{3\beta}{}^{\lambda}-q_{3}^{\mu}f_{3}^{\alpha\beta}f_{2\beta}{}^{\lambda})$
$\displaystyle\quad-\frac{2C_{1}C_{2}}{m^{4}}f_{2}^{\mu\alpha}f_{3\alpha}{}^{\lambda}-\frac{2C_{2}D_{1}}{m^{6}}p_{1\rho}p_{1\alpha}(f_{2}^{\rho\mu}f_{3}^{\alpha\lambda}+\eta^{\mu\lambda}f_{2}^{\rho\beta}f_{3\beta}{}^{\alpha})\,,$
(84) $\displaystyle\mathcal{Y}^{\mu}$
$\displaystyle=\frac{2C_{2}D_{1}}{(p_{1}\cdot
q_{2})m^{4}}p_{1\rho}p_{1\alpha}(f_{2}^{\rho\mu}f_{3}^{\alpha\beta}q_{2\beta}-f_{3}^{\rho\mu}f_{2}^{\alpha\beta}q_{3\beta})$
$\displaystyle\quad-\frac{C_{2}(C_{1}+D_{1}-1)}{m^{4}}p_{1\rho}(f_{3}^{\rho\alpha}f_{2\alpha}{}^{\mu}+f_{2}^{\rho\alpha}f_{3\alpha}{}^{\mu})\,.$
(85)

We proceed next to the $\mathcal{O}(S^{2})$ tree-level Compton amplitude of
FT2. Repeating at this order the classical scaling argument we described at
${\cal O}(S^{1})$ shows that, in a fixed-spin calculation, the classical tree-
level Compton amplitude is contained in the $\mathcal{O}(q^{2})$ terms of the
quantum tree-level Compton amplitude. Thus, we extract these terms from
explicit $s=1,2,3$ calculations, extrapolate them to large spin and keep only
the leading term. The massive polarization vectors now appear in four
structures,

$\displaystyle\hskip-5.69046pt(\varepsilon_{1}\cdot\bar{\varepsilon}_{4})^{s}\,,\hskip
25.6073pt(\varepsilon_{1}\cdot\bar{\varepsilon}_{4})^{s-1}\varepsilon_{1}^{[\mu}\bar{\varepsilon}_{4}^{\nu]}\,,\hskip
25.6073pt(\varepsilon_{1}\cdot\bar{\varepsilon}_{4})^{s-1}\varepsilon_{1}^{(\mu}\bar{\varepsilon}_{4}^{\nu)}\,,\hskip
25.6073pt(\varepsilon_{1}\cdot\bar{\varepsilon}_{4})^{s-2}\varepsilon_{1}^{[\mu}\bar{\varepsilon}_{4}^{\nu]}\varepsilon_{1}^{[\rho}\bar{\varepsilon}_{4}^{\lambda]}\,,\hskip
5.69046pt$ (86)

where the first two correspond to the quantum spinless and
$\mathcal{O}(S^{1})$ amplitudes that can be ignored here. We use the
replacement (42) and (43) for the last two structures. It turns that the
dependence on $s$ is simple so that we can extrapolate it to obtain the
general $s$ dependence and take $s\rightarrow\infty$ limit. The kinematic
coefficient of $S^{\mu\rho}S_{\rho}{}^{\nu}$ can be accessed by any
$s\geqslant 1$; in the large $s$ limit, it exactly reproduces the
$\mathcal{X}^{\mu\nu}$ shown in Eq. 80. The structure
$S^{\mu\nu}S^{\rho\lambda}$ appears for $s\geqslant 2$. A careful analysis
with $s=2$ and $3$ gives identical results, so that we postulate that the
coefficient of $S^{\mu\nu}S^{\rho\lambda}$ is independent of $s$. Thus we find
that the tree-level Compton amplitude of FT2 is

$\displaystyle\mathcal{A}_{\text{4, cl.}}^{\text{FT2}}\Big{|}_{S^{2}}$
$\displaystyle=(-\varepsilon_{1}\cdot\bar{\varepsilon}_{4})^{s}\left(S_{\mu\nu}S_{\lambda\sigma}\widetilde{\mathcal{X}}^{\mu\nu\lambda\sigma}+S_{\mu\lambda}S^{\lambda}{}_{\nu}\mathcal{X}^{\mu\nu}\right)\,,$
(87) $\displaystyle\widetilde{\mathcal{X}}^{\mu\nu\lambda\sigma}$
$\displaystyle=\frac{C_{1}^{2}(q_{2}\cdot q_{3})}{2(p_{1}\cdot
q_{2})^{2}}f_{2}^{\mu\nu}f_{3}^{\lambda\sigma}+\frac{C_{1}(C_{1}-1)}{(p_{1}\cdot
q_{2})m^{2}}p_{1\rho}(f_{3}^{\rho\mu}q_{2}^{\nu}f_{2}^{\lambda\sigma}-f_{2}^{\rho\mu}q_{3}^{\nu}f_{3}^{\lambda\sigma})\,,$

where $\mathcal{X}^{\mu\nu}$ is defined in Eq. (81). These coefficients depend
only on $C_{1}$ and $C_{2}$ and are independent of $D_{1}$ and $D_{2}$. We
again observe the same pattern as in the linear-in-spin case,

$\displaystyle\mathcal{A}_{4,\text{
cl.}}^{\text{FT1s}}\Big{|}_{S^{2}}=\mathcal{A}_{\text{4,
cl.}}^{\text{FT2}}\Big{|}_{S^{2}}\qquad\text{for}\quad D_{1}=C_{1}-1\,.$ (88)

We note that the special value of $D_{1}$ also removes the dependence on
$D_{2}$.

Similar to $\mathcal{O}(S^{1})$, the $\mathcal{O}(S^{2})$ Compton amplitudes
of FT3 receive contributions from lower-spin intermediate states. Keeping the
external states transverse, we get

$\displaystyle\mathcal{A}_{\text{4, cl}}^{\text{FT3s}}\Big{|}_{S^{2}}$
$\displaystyle=\mathcal{A}_{\text{4,
cl}}^{\text{FT2}}\Big{|}_{S^{2}}+(-\varepsilon_{1}\cdot\bar{\varepsilon}_{4})^{s}S(p_{1})_{\mu\nu}S(p_{1})_{\lambda\sigma}\left[\frac{\widetilde{C}_{1}\widetilde{C}_{2}}{(p_{1}\cdot
q_{2})m^{2}}p_{1\rho}(f_{3}^{\rho\mu}q_{2}^{\nu}f_{2}^{\lambda\sigma}-f_{2}^{\rho\mu}q_{3}^{\nu}f_{3}^{\lambda\sigma})\right].$
(89)

Just like the previous cases, the same formal relations hold between the
additional Wilson coefficients in FT1 and FT3. Indeed, comparing Eq. 89 and
Eq. 79, it is easy to see that

$\displaystyle\mathcal{A}_{\text{4,
cl}}^{\text{FT1s}}\Big{|}_{S^{2}}=\mathcal{A}_{\text{4,
cl}}^{\text{FT3s}}\Big{|}_{S^{2}}\qquad\text{for}\quad
i\widetilde{C}_{1}=1-C_{1}+D_{1}\;\text{ and
}\;i\widetilde{C}_{2}=D_{2}-C_{1}\,,$ (90)

which is identical to Eq. 68. This demonstrates that the relation between
extra Wilson coefficients and extra propagating degrees of freedom holds also
at $\mathcal{O}(S^{2})$. A comparison between $\mathcal{A}_{4,\text{
cl.}}^{\text{FT1g}}$ and $\mathcal{A}_{4,\text{ cl.}}^{\text{FT3g}}$ at
$\mathcal{O}(S^{1}K^{1})$ and $\mathcal{O}(K^{2})$ requires that we include in
the Lagrangian of FT3 a spin-$(s-2)$ field $\phi_{s-2}$, and additional
operators that contribute independently at $\mathcal{O}(K^{2})$, for example
Eq. 59 for FT1. This is because the effect of $\mathcal{O}(K^{2})$ operators
show up at $\mathcal{O}(S^{1}K^{1})$ in the four-point Compton amplitudes due
to the commutator $[K^{2},K]\sim SK$. Finally, we note that the spin-
transition Compton amplitude $\mathcal{A}_{4}^{s\rightarrow s-1}$ under a
fixed-spin calculation may superficially contain super-classical contribution
that does not cancel between the two matter channels. Consistency of the
theory requires however that in the large-spin limit this term is subleading.
We will assume that this cancellation holds as $s\rightarrow\infty$. It is
nontrivial to carry out explicit calculations to demonstrate this, but would
be worth investigating.

###  III.3 Two-Body Amplitudes

In previous subsections, we have explored and understood the effect of various
types of interactions between higher-spin fields and photons on Compton
amplitudes, and the number of Wilson coefficients necessary to describe such
interactions. We found that, under suitable conditions like allowing spin
magnitude change, this number is indeed larger than that required to describe
the interactions of SSC preserving spins. The rationale of this exercise is to
eventually understand their effects on two-body observables, such as the
momentum impulse and the spin kick. It was originally suggested in the context
of gravity that a larger number of Wilson coefficients may be required to
describle more general interactions [1]. We therefore proceed to expose the
photon-mediated two-body amplitudes and, in later sections, the observables
that follow from them as well as their comparison with a wordline perspective.
We use the generalized unitarity method [136, 137, 139] to construct the
relevant integrands, while taking advantage of the simplifications introduced
in Ref. [190]. To reduce the encountered loop integrals to known ones we make
use of integration by parts [142, 146] as implemented in FIRE [191, 192].

We use the momentum and mass variables

$\displaystyle\bar{m}_{1}=m_{1}^{2}-q^{2}/4\,,\hskip
45.52458pt\bar{m}_{2}^{2}=m_{2}^{2}-q^{2}/4\,,\hskip
45.52458pty=\frac{\bar{p}_{1}\cdot\bar{p}_{2}}{\bar{m}_{1}\bar{m}_{2}}\,,$
$\displaystyle\bar{p}_{1}=p_{1}+q/2=-p_{4}-q/2\,,\hskip
45.52458pt\bar{p}_{2}=p_{2}-q/2=-p_{3}+q/2\,,$ (91)

which are originally used for the expansion in the soft region of
gravitational amplitudes in [147]. We primarily focus on FT1 because this is
what we compare with a worldline theory. Unitarity guarantees that the two-
body amplitudes of FT2 can be obtained from those of FT1 by setting
$D_{1}=C_{1}-1$ and imposing the covariant SSC, while the two-body amplitudes
of FT3 can be obtained using the map between the extra Wilson coefficients
proposed in the previous subsection.

####  III.3.1 Tree Level

The structure of the two-body amplitude at tree-level and in the classical
limit is

$\displaystyle
i\mathcal{M}_{4,\text{cl.}}^{\text{tree}}=(4\pi\alpha)(\mathcal{E}_{1}\cdot\bar{\mathcal{E}}_{4})(\mathcal{E}_{2}\cdot\bar{\mathcal{E}}_{3})\left(\frac{d_{\leavevmode\hbox
to6.09pt{\vbox to6.09pt{\pgfpicture\makeatletter\hbox{\hskip
3.04544pt\lower-3.04544pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{-2.84544pt}{-2.84544pt}\pgfsys@lineto{2.84544pt}{-2.84544pt}\pgfsys@moveto{-2.84544pt}{2.84544pt}\pgfsys@lineto{2.84544pt}{2.84544pt}\pgfsys@moveto{0.0pt}{-2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}{q^{2}}\right),$
(92)

where $q$ is the photon momentum, and the numerator $d_{\leavevmode\hbox
to6.09pt{\vbox to6.09pt{\pgfpicture\makeatletter\hbox{\hskip
3.04544pt\lower-3.04544pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{-2.84544pt}{-2.84544pt}\pgfsys@lineto{2.84544pt}{-2.84544pt}\pgfsys@moveto{-2.84544pt}{2.84544pt}\pgfsys@lineto{2.84544pt}{2.84544pt}\pgfsys@moveto{0.0pt}{-2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$ is a function of
momenta and spin tensors which scales as $d_{\leavevmode\hbox to6.09pt{\vbox
to6.09pt{\pgfpicture\makeatletter\hbox{\hskip 3.04544pt\lower-3.04544pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{-2.84544pt}{-2.84544pt}\pgfsys@lineto{2.84544pt}{-2.84544pt}\pgfsys@moveto{-2.84544pt}{2.84544pt}\pgfsys@lineto{2.84544pt}{2.84544pt}\pgfsys@moveto{0.0pt}{-2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\sim q^{0}$ in the
classical limit. Our results for FT1g through the quadratic order in spin are
as follows:

$\displaystyle d_{\leavevmode\hbox to6.09pt{\vbox
to6.09pt{\pgfpicture\makeatletter\hbox{\hskip 3.04544pt\lower-3.04544pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{-2.84544pt}{-2.84544pt}\pgfsys@lineto{2.84544pt}{-2.84544pt}\pgfsys@moveto{-2.84544pt}{2.84544pt}\pgfsys@lineto{2.84544pt}{2.84544pt}\pgfsys@moveto{0.0pt}{-2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\Big{|}_{\text{spinless}}$
$\displaystyle=4iy\bar{m}_{1}\bar{m}_{2}\,,$ $\displaystyle
d_{\leavevmode\hbox to6.09pt{\vbox
to6.09pt{\pgfpicture\makeatletter\hbox{\hskip 3.04544pt\lower-3.04544pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{-2.84544pt}{-2.84544pt}\pgfsys@lineto{2.84544pt}{-2.84544pt}\pgfsys@moveto{-2.84544pt}{2.84544pt}\pgfsys@lineto{2.84544pt}{2.84544pt}\pgfsys@moveto{0.0pt}{-2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\Big{|}_{S_{1}^{1}S_{2}^{0}}^{\text{FT1g}}$
$\displaystyle=-4\bar{m}_{2}\mathsf{S}_{1}{}_{\mu\nu}\left(C_{1(1)}\bar{u}_{2}^{\mu}q^{\nu}-D_{1(1)}y\bar{u}_{1}^{\mu}q^{\nu}\right),$
$\displaystyle d_{\leavevmode\hbox to6.09pt{\vbox
to6.09pt{\pgfpicture\makeatletter\hbox{\hskip 3.04544pt\lower-3.04544pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{-2.84544pt}{-2.84544pt}\pgfsys@lineto{2.84544pt}{-2.84544pt}\pgfsys@moveto{-2.84544pt}{2.84544pt}\pgfsys@lineto{2.84544pt}{2.84544pt}\pgfsys@moveto{0.0pt}{-2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\Big{|}_{S_{1}^{1}S_{2}^{1}}^{\text{FT1g}}$
$\displaystyle=4iC_{1(1)}C_{1(2)}\mathsf{S}_{1}^{\mu\nu}q_{\nu}\mathsf{S}_{2\mu\rho}q^{\rho}$
(93)
$\displaystyle\quad-4i\mathsf{S}_{1}^{\mu\nu}q_{\nu}\mathsf{S}_{2}^{\rho\lambda}q_{\lambda}\left(C_{1(1)}D_{1(2)}\bar{u}_{2\mu}\bar{u}_{2\rho}+C_{1(2)}D_{1(1)}\bar{u}_{1\mu}\bar{u}_{1\rho}-D_{1(1)}D_{1(2)}y\bar{u}_{1\mu}\bar{u}_{2\rho}\right),$
$\displaystyle d_{\leavevmode\hbox to6.09pt{\vbox
to6.09pt{\pgfpicture\makeatletter\hbox{\hskip 3.04544pt\lower-3.04544pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{-2.84544pt}{-2.84544pt}\pgfsys@lineto{2.84544pt}{-2.84544pt}\pgfsys@moveto{-2.84544pt}{2.84544pt}\pgfsys@lineto{2.84544pt}{2.84544pt}\pgfsys@moveto{0.0pt}{-2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\Big{|}_{S_{1}^{2}S_{2}^{0}}^{\text{FT1g}}$
$\displaystyle=\frac{2i\bar{m}_{2}}{\bar{m}_{1}}\left[yC_{2(1)}\mathsf{S}_{1}^{\mu\nu}q_{\nu}\mathsf{S}_{1\mu\rho}q^{\rho}-yC_{2(1)}(\mathsf{S}_{1}^{\mu\nu}\bar{u}_{1\mu}q_{\nu})^{2}-2D_{2(1)}\mathsf{S}_{1}^{\mu\nu}\bar{u}_{1\mu}q_{\nu}\mathsf{S}_{1}^{\rho\lambda}\bar{u}_{2\rho}q_{\lambda}\right],$

where the spinless case agrees with Ref. [188, 106, 108]. The notation
$C_{i(j)}$ and $D_{i(j)}$ refers to the $C_{i}$ and $D_{i}$ coefficients
associated with body $j$. If the external states are transverse,
$(\mathcal{E}_{1}\cdot\bar{\mathcal{E}}_{4})(\mathcal{E}_{2}\cdot\bar{\mathcal{E}}_{3})=(\varepsilon_{1}\cdot\bar{\varepsilon}_{4})^{s}(\varepsilon_{2}\cdot\bar{\varepsilon}_{3})^{s}$,
then the spin tensor obeys the covariant SSC, such that,

$\displaystyle d_{\leavevmode\hbox to6.09pt{\vbox
to6.09pt{\pgfpicture\makeatletter\hbox{\hskip 3.04544pt\lower-3.04544pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{-2.84544pt}{-2.84544pt}\pgfsys@lineto{2.84544pt}{-2.84544pt}\pgfsys@moveto{-2.84544pt}{2.84544pt}\pgfsys@lineto{2.84544pt}{2.84544pt}\pgfsys@moveto{0.0pt}{-2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\Big{|}_{S_{1}^{1}S_{2}^{0}}^{\text{FT1s}}=d_{\leavevmode\hbox
to6.09pt{\vbox to6.09pt{\pgfpicture\makeatletter\hbox{\hskip
3.04544pt\lower-3.04544pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{-2.84544pt}{-2.84544pt}\pgfsys@lineto{2.84544pt}{-2.84544pt}\pgfsys@moveto{-2.84544pt}{2.84544pt}\pgfsys@lineto{2.84544pt}{2.84544pt}\pgfsys@moveto{0.0pt}{-2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\Big{|}_{S_{1}^{1}S_{2}^{0}}^{\text{FT2}}=d_{\leavevmode\hbox
to6.09pt{\vbox to6.09pt{\pgfpicture\makeatletter\hbox{\hskip
3.04544pt\lower-3.04544pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{-2.84544pt}{-2.84544pt}\pgfsys@lineto{2.84544pt}{-2.84544pt}\pgfsys@moveto{-2.84544pt}{2.84544pt}\pgfsys@lineto{2.84544pt}{2.84544pt}\pgfsys@moveto{0.0pt}{-2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\Big{|}_{S_{1}^{1}S_{2}^{0}}^{\text{FT3s}}=-4C_{1(1)}\bar{m}_{2}S_{1}^{\mu\nu}\bar{u}_{2\mu}q^{\nu}\,,$
$\displaystyle d_{\leavevmode\hbox to6.09pt{\vbox
to6.09pt{\pgfpicture\makeatletter\hbox{\hskip 3.04544pt\lower-3.04544pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{-2.84544pt}{-2.84544pt}\pgfsys@lineto{2.84544pt}{-2.84544pt}\pgfsys@moveto{-2.84544pt}{2.84544pt}\pgfsys@lineto{2.84544pt}{2.84544pt}\pgfsys@moveto{0.0pt}{-2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\Big{|}_{S_{1}^{1}S_{2}^{1}}^{\text{FT1s}}=d_{\leavevmode\hbox
to6.09pt{\vbox to6.09pt{\pgfpicture\makeatletter\hbox{\hskip
3.04544pt\lower-3.04544pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{-2.84544pt}{-2.84544pt}\pgfsys@lineto{2.84544pt}{-2.84544pt}\pgfsys@moveto{-2.84544pt}{2.84544pt}\pgfsys@lineto{2.84544pt}{2.84544pt}\pgfsys@moveto{0.0pt}{-2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\Big{|}_{S_{1}^{1}S_{2}^{1}}^{\text{FT2}}=d_{\leavevmode\hbox
to6.09pt{\vbox to6.09pt{\pgfpicture\makeatletter\hbox{\hskip
3.04544pt\lower-3.04544pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{-2.84544pt}{-2.84544pt}\pgfsys@lineto{2.84544pt}{-2.84544pt}\pgfsys@moveto{-2.84544pt}{2.84544pt}\pgfsys@lineto{2.84544pt}{2.84544pt}\pgfsys@moveto{0.0pt}{-2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\Big{|}_{S_{1}^{1}S_{2}^{1}}^{\text{FT3s}}=4iC_{1(1)}C_{1(2)}S_{1}^{\mu\nu}q_{\nu}S_{2\mu\rho}q^{\rho}\,,$
(94) $\displaystyle d_{\leavevmode\hbox to6.09pt{\vbox
to6.09pt{\pgfpicture\makeatletter\hbox{\hskip 3.04544pt\lower-3.04544pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{-2.84544pt}{-2.84544pt}\pgfsys@lineto{2.84544pt}{-2.84544pt}\pgfsys@moveto{-2.84544pt}{2.84544pt}\pgfsys@lineto{2.84544pt}{2.84544pt}\pgfsys@moveto{0.0pt}{-2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\Big{|}_{S_{1}^{2}S_{2}^{0}}^{\text{FT1s}}=d_{\leavevmode\hbox
to6.09pt{\vbox to6.09pt{\pgfpicture\makeatletter\hbox{\hskip
3.04544pt\lower-3.04544pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{} {}{}{}{{}}{} {}{}{}{{}}{}
{}{}{}\pgfsys@moveto{-2.84544pt}{-2.84544pt}\pgfsys@lineto{2.84544pt}{-2.84544pt}\pgfsys@moveto{-2.84544pt}{2.84544pt}\pgfsys@lineto{2.84544pt}{2.84544pt}\pgfsys@moveto{0.0pt}{-2.84544pt}\pgfsys@lineto{0.0pt}{2.84544pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\Big{|}_{S_{1}^{2}S_{2}^{0}}^{\text{FT2}}=d_{\leavevmode\hbox
to6.09pt{\vbox to6.09pt{\pgfpicture\makeatletter\hbox{\hskip
3.04544pt\lower-3.04544pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{