# Five-dimensional spinor helicity for all masses and spins

Andrzej Pokraka<EMAIL_ADDRESS>0000-0003-2827-2406 a,  Smita Rajan
<EMAIL_ADDRESS>0000-0002-0846-8017 a, Lecheng Ren
<EMAIL_ADDRESS>0009-0008-2506-3207 a,b,  Anastasia Volovich
<EMAIL_ADDRESS>0000-0001-6192-7406 c,  W. Wayne Zhao
<EMAIL_ADDRESS>Department of Physics, Brown University, Providence, RI
02912, USA Department of Physics, Harvard University, Cambridge, MA 02138,
USADepartment of Physics, Princeton University, Princeton, NJ 08544, USA

###### Abstract

 We develop a spinor helicity formalism for five-dimensional scattering
amplitudes of any mass and spin configuration. While five-dimensional spinor
helicity variables have been previously studied in the context of
$\mathcal{N}=2,4$ supersymmetric Yang-Mills scattering amplitudes with spin
less than two Chiodaroli:2022ssi , we propose an alternative viewpoint that
stems from $d$-dimensional spinor helicity variables avoiding the use of the
exceptional low-dimensional isomorphism
$\operatorname{SO}(4,1)\cong\operatorname{USp}(2,2)$ and the decomposition of
a massive momentum into the sum of two massless momenta. By enumerating all
possible independent little group tensors, we systematically build the full
space of five-dimensional three-point tree-level scattering amplitudes for any
configuration of spins and masses. Furthermore, we provide a prescription for
computing the high energy limit of scattering amplitudes written in our spinor
helicity variables. We also expect that our formalism will be applicable to
effective field theories with higher spin, in particular, the scattering of
highly spinning black holes in five dimensions.

##  1 Introduction

It is well known that _massless_ on-shell scattering amplitudes are most
naturally expressed in terms of spinor helicity variables illuminating many
interesting structures that would otherwise be obscured by the usual
presentation involving spacetime tensors and polarization vectors
Srednicki:2007qs ; Schwartz:2014sze . In particular, asymptotic massless
states are uniquely specified by their helicity and momentum; spinor helicity
variables make this manifest without introducing gauge redundancy. From this
point of view, the spinor helicity framework represents amplitudes in their
most physical form. In addition to conceptual advantages, the spinor helicity
framework is an important computational tool that has facilitated the
discovery of numerous important advances in the study of on-shell scattering
amplitudes, starting with the celebrated Parke-Taylor formula for tree-level
MHV amplitudes ParkeTaylor , and all the further developments (see ElvangHuang
; Badger:2023eqz ; Travaglini:2022uwo ; Bern:2022jnl  for reviews). While the
spinor helicity formalism revolutionized the study of massless scattering
amplitudes four decades ago, spinor helicity formalisms for _massive_
particles remained largely underdeveloped. It was only recently that a four-
dimensional _massive_ any _spin_ spinor helicity formalism was developed by
Arkani-Hamed, Huang and Huang Arkani-Hamed:2017jhn  (see also Craig:2011ws ;
Kiermaier:2011cr ; Ochirov:2018uyq ; Herderschee:2019dmc ; Johansson:2019dnu ;
Chiodaroli:2021eug ).

By utilizing four-dimensional massive spinor helicity, Arkani-Hamed, Huang,
and O’Connell showed that graviton scattering with highly spinning Kerr black
holes resembles the scattering of an elementary particle or, in other words,
is minimally coupled Arkani-Hamed:2019ymq . The authors also showed that the
Janis-Newman shift that relates Schwarzchild and Kerr black hole solutions has
an interpretation as the exponentiation induced by taking the large spin limit
of minimally coupled massive spinning particles (see Huang:2019cja ;
Chung:2019yfs ; Emond:2020lwi  for generalizations to other four-dimensional
solutions). Since higher dimensional solutions to the Einstein equations can
be obtained by similar generalized Janis-Newman shifts Erbin:2017 , it would
be interesting to understand if these shifts also have an interpretation in
terms of amplitudes.

Motivated by this question, we develop a five-dimensional spinor helicity
formalism for any mass and spin. Our formalism is closely related to that of
Chiodaroli, Gunaydin, Johansson, and Roiban Chiodaroli:2022ssi  but differs in
the sense that we do not expand the massive spinors in terms of massless
spinors opting to directly work with spinors of the $\text{SO}(4)$ massive
little group. Moreover, we do not commit to a specific theory or Lagrangian
and construct the space of independent three-point amplitudes from purely
kinematic considerations. In particular, we find that the number of
independent three-point amplitudes matches the expected counting from
Costa:2011mg ; Elkhidir:2014woa . Lastly, we describe how to take the high
energy limit of the massive amplitudes. We show that when the leading order
term in the $m\ll E$ expansion vanishes, the subleading terms combine non-
trivially to form a gauge invariant amplitude with $m=0$.

The paper is organized as follows. In section 2, we describe our five-
dimensional spinor helicity formalism and how it differs from
Chiodaroli:2022ssi . In section 3, we systematically construct all three-point
amplitudes and verify that our construction produces the expected number of
independent amplitudes. In section 4, we describe how to take the high energy
limit of amplitudes built out of the spinor helicity variables.

##  2 Five-dimensional spinor helicity formalism

In this section, we construct a five-dimensional spinor helicity formalism for
all masses and all spins. Five-dimensional massive spinor helicity formalism
has been recently presented in Chiodaroli:2022ssi . They used the isomorphism
$\operatorname{SO}(4,1)\cong\operatorname{USp}(2,2)$ and decomposed massive
momentum into the sum of two massless momenta. We will not rely on these
choices and use the formalism which we hope could be generalized to $d\geq 5$.
For the practical reader, this conceptual difference is essentially
philosophical; it is easy to convert between our formalism and
Chiodaroli:2022ssi .

The Dirac spinor representation of the Lorentz group $\operatorname{SO}(4,1)$
is four-dimensional. We use gamma matrices from the $d$-dimensional
definitions in appendix B of Polchinski:1998rr  with rows and columns 2 and 3
swapped. This choice will make the form of the charge conjugation matrix
simpler

$\displaystyle\Gamma^{0}=\left(\begin{array}[]{cccc}0&0&-1&0\\\ 0&0&0&-1\\\
1&0&0&0\\\
0&1&0&0\end{array}\right),\quad\Gamma^{1}=\left(\begin{array}[]{cccc}0&0&-1&0\\\
0&0&0&1\\\ -1&0&0&0\\\
0&1&0&0\end{array}\right),\quad\Gamma^{2}=\left(\begin{array}[]{cccc}0&0&0&1\\\
0&0&1&0\\\ 0&1&0&0\\\ 1&0&0&0\end{array}\right)$ (13)
$\displaystyle\Gamma^{3}=\left(\begin{array}[]{cccc}0&0&0&-i\\\ 0&0&i&0\\\
0&-i&0&0\\\
i&0&0&0\end{array}\right),\quad\Gamma^{4}=-i\Gamma^{0}\Gamma^{1}\Gamma^{2}\Gamma^{3}=\left(\begin{array}[]{cccc}1&0&0&0\\\
0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{array}\right).$ (22)

These gamma matrices $(\Gamma^{\mu})_{A}^{~{}~{}B}$ satisfy the usual Clifford
algebra $\\{\Gamma^{\mu},\Gamma^{\nu}\\}=2\eta^{\mu\nu}$ with the mostly plus
metric $\eta^{\mu\nu}=\mathrm{diag}(-1,1,1,1,1)$ and $\mu,\nu=0,1,2,3,4$.

The charge conjugation matrix $C$ obeying $C\Gamma^{\mu}C^{-1}=+\Gamma^{\top}$
is given by

$C^{AB}=\left(\begin{array}[]{cccc}0&-i&0&0\\\ i&0&0&0\\\ 0&0&0&i\\\
0&0&-i&0\end{array}\right)\,.$ (23)

We denote its inverse $C_{AB}$ by lowered indices such that
$C_{AB}C^{BC}=\delta_{A}^{\enspace C}$ and use it to lower and raise Dirac
indices $A,B=1,2,3,4$, as follows

$\lambda^{A}=\lambda_{B}C^{BA}~{}~{}~{}~{}~{}\lambda_{A}=\lambda^{B}C_{BA}.$
(24)

The momentum space Dirac operator is

$\displaystyle p_{A}^{\enspace B}=p_{\mu}(\Gamma^{\mu})_{A}^{\enspace B}\,.$
(25)

The on-shell condition is captured by the determinant $\det p_{A}^{\enspace
B}\propto p^{2}=-m^{2}$. For massive particles $m\neq 0$, $p_{A}^{\enspace B}$
is rank four while for massless particles $m=0$, $p_{A}^{\enspace B}$ is rank
two.

We will consider massless and massive cases separately in the next two
subsections.

###  2.1 Massless spinor helicity variables

The Dirac operator (25) can be factorized over the massless little group
$\operatorname{SO}(3)$

$p_{A}^{\enspace B}={\vphantom{|p_{I}\rangle}}_{A}|p_{I}\rangle\ \langle
p^{I}|^{B}\,,$ (26)

where we sum over the little group $\operatorname{SO}(3)$ indices $I=1,2$.

Massless spinor helicity (right-bracket) variables
${\vphantom{|p_{I}\rangle}}_{A}|p_{I}\rangle$ satisfy massless Dirac equation

$\displaystyle p_{A}^{\enspace
B}{\vphantom{|p_{I}\rangle}}_{B}|p_{I}\rangle=0.$ (27)

We will use the following explicit representation 111 Note that the mass
dimension of these spinors is not uniform. While this choice may seem
unnatural initially, it is extremely convenient since it preserves (26) and
avoids introducing unnecessary square roots into the parameterization of the
spinors.

$\displaystyle{\vphantom{|p_{I}\rangle}}_{A}|p_{I}\rangle$
$\displaystyle=\begin{pmatrix}(p_{2}-ip_{3})&-\frac{p_{4}}{p_{0}+p_{1}}\\\
(p_{0}+p_{1})&0\\\ 0&1\\\
-p_{4}&\frac{-p_{2}-ip_{3}}{p_{0}+p_{1}}\end{pmatrix}\,.$ (28)

The corresponding left brackets are defined by contracting with the charge
conjugation matrix (23)

$\displaystyle\langle p_{I}|^{A}={\vphantom{|p_{I}\rangle}}_{B}|p_{I}\rangle\
C^{BA}.$ (29)

Note that the position of the Dirac index determines whether the spinor
helicity variable is written as a right or left bracket: lower Dirac indices
for right brackets and upper Dirac indices for left brackets.

The little group indices $I=1,2$ are raised and lowered with the charge
conjugation matrix

$C^{IJ}=i\epsilon^{IJ},~{}~{}~{}\epsilon^{12}=1$ (30)

as follows

$\langle p^{I}|^{B}=\langle p_{J}|^{B}C^{JI}.$ (31)

With these conventions, spinor products are computed by contracting the
spacetime-Dirac indices

$\displaystyle{}\langle p_{I}|q_{J}\rangle={}\langle
p_{I}|^{A}{\vphantom{|p_{I}\rangle}}_{A}|q_{J}\rangle={}{\langle p_{I}|}_{A}\
C^{AB}\ {\vphantom{|p_{I}\rangle}}_{B}|q_{J}\rangle.$ (32)

These spinor brackets are anti-symmetric by definition

${}\langle p_{I}|q_{J}\rangle=-{}\langle q_{J}|p_{I}\rangle.$ (33)

Additionally, whenever there is a momentum $k$ sandwiched between two spinor
brackets, it is interpreted as a $k_{A}^{\enspace B}$ acting on the brackets

$\displaystyle{}\langle p_{I}|k|q_{J}\rangle={}\langle p_{I}|^{A}\
k_{A}^{\enspace B}\ {\vphantom{|p_{I}\rangle}}_{B}|q_{J}\rangle\,.$ (34)

Moreover, to simplify formulas, we use the usual short-hand

${\vphantom{|p_{I}\rangle}}_{A}|i_{I}\rangle={\vphantom{|p_{I}\rangle}}_{A}|p_{iI}\rangle$
(35)

to denote the spinor for the $i^{\text{th}}$ particle of an amplitude.

To simplify the process of keeping track of indices, we follow
Chiodaroli:2022ssi  and contract the little group index of each particle with
momentum $p_{i}$ with a dummy spinor to define the brackets

$\displaystyle{\vphantom{|p_{I}\rangle}}_{A}|\underline{i}\rangle={\vphantom{|p_{I}\rangle}}_{A}|p_{iI}\rangle\
z_{i}^{I}\,,$ (36)

Spin-$S$ representations in five-dimensions correspond to little-group tensors
that are fully symmetric in their $\operatorname{SO}(3)$ indices, and
therefore contracting all little group indices into dummy spinors
automatically imposes the correct symmetry and turns tensors into polynomials.
Brackets are constructed from these underlined spinors in the same way as
before by contracting the spacetime Dirac indices with the $C$-matrix

$\displaystyle\langle\underline{ik}\rangle={\langle\underline{i}|}_{A}\
C^{AB}\
{\vphantom{|p_{I}\rangle}}_{B}|\underline{j}\rangle\,,\quad\langle\underline{i}|k|\underline{l}\rangle={\langle\underline{i}|}^{A}\
k_{A}^{\enspace B}\
{\vphantom{|p_{I}\rangle}}_{B}|\underline{l}\rangle\quad\text{ and so on.}$
(37)

The actual tensor can be recovered by taking specific combinations of
derivatives with respect to the dummy spinors.

###  2.2 Massive spinor helicity variables

Like the massless case, the massive Dirac operator (25) can be factorized over
the massive little group $\operatorname{SO}(4)$ by introducing massive spinor
helicity variables

$\displaystyle p_{A}^{\enspace B}$
$\displaystyle={\vphantom{|p_{I}\rangle}}_{A}|p_{\dot{\alpha}}][p^{\dot{\alpha}}|^{B}+{\vphantom{|p_{I}\rangle}}_{A}|p_{\alpha}\rangle\
\langle p^{\alpha}|^{B}\,,$ (38)

where we sum over repeated massive little group indices
$\alpha,\dot{\alpha}=1,2$.

Massive spinor helicity (right-bracket) variables ${}_{A}|p_{\alpha}]$ and
${}_{A}|p_{\dot{\alpha}}\rangle$ satisfy the massive Dirac equations

$\displaystyle p_{A}^{\enspace B}\
{\vphantom{|p_{I}\rangle}}_{B}|p_{\alpha}\rangle=im\
{\vphantom{|p_{I}\rangle}}_{A}|p_{\alpha}\rangle\quad\mathrm{and}\quad
p_{A}^{\enspace B}\ {\vphantom{|p_{I}\rangle}}_{B}|p_{\dot{\alpha}}]=-im\
{\vphantom{|p_{I}\rangle}}_{A}|p_{\dot{\alpha}}]\,,$ (39)

where we assume that $p$ is time-like: $\sqrt{p^{2}}=\sqrt{-m^{2}}=\pm im$.
Massive spinors with positive imaginary eigenvalues carry Weyl little group
indices $\alpha$, and those with negative imaginary eigenvalues carry anti-
Weyl little group indices $\dot{\alpha}$.

We use the following representation for the massive spinors

$\displaystyle\begin{aligned}
{\vphantom{|p_{I}\rangle}}_{A}|p_{\alpha}\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}i(p_{2}-ip_{3})&\frac{im-
p_{4}}{p_{0}+p_{1}}\\\ i(p_{0}+p_{1})&0\\\ 0&1\\\
m-ip_{4}&\frac{-p_{2}-ip_{3}}{p_{0}+p_{1}}\end{pmatrix}\,,\quad{\vphantom{|p_{I}\rangle}}_{A}|p_{\dot{\alpha}}]=\frac{1}{\sqrt{2}}\begin{pmatrix}i(p_{2}-ip_{3})&\frac{-im-
p_{4}}{p_{0}+p_{1}}\\\ i(p_{0}+p_{1})&0\\\ 0&1\\\
-m-ip_{4}&\frac{-p_{2}-ip_{3}}{p_{0}+p_{1}}\end{pmatrix}\end{aligned}$ (40)

The corresponding left brackets are defined by contracting with the charge
conjugation matrix (23)

$\displaystyle[p_{\dot{\alpha}}|^{A}={\vphantom{|p_{I}\rangle}}_{B}|p_{\dot{\alpha}}]\
C^{BA}\qquad\mathrm{and}\qquad\langle
p_{\alpha}|^{A}={\vphantom{|p_{I}\rangle}}_{B}|p_{\alpha}\rangle\ C^{BA}.$
(41)

The little group indices $\alpha=1,2$ and $\dot{\alpha}=1,2$ are raised and
lower with the charge conjugation matrix

$\displaystyle
C^{\alpha\beta}=\varepsilon^{\alpha\beta},~{}~{}~{}C^{\dot{\alpha}\dot{\beta}}=\varepsilon^{\dot{\alpha}\dot{\beta}},~{}~{}~{}\epsilon^{12}=\epsilon^{\dot{1}\dot{2}}=1$
(42)

as follows

$\langle p^{\alpha}|^{B}=\langle
p_{\beta}|^{B}C^{\beta\alpha},~{}~{}~{}\langle p^{\dot{\alpha}}|^{B}=\langle
p_{\dot{\beta}}|^{B}C^{\dot{\beta}\dot{\alpha}}.$ (43)

The massive brackets also satisfy the completeness relation

$\displaystyle-im\ \delta_{A}^{\enspace B}$
$\displaystyle={\vphantom{|p_{I}\rangle}}_{A}|p_{\dot{\alpha}}]\
[p^{\dot{\alpha}}|^{B}-{\vphantom{|p_{I}\rangle}}_{A}|p_{\alpha}\rangle\
\langle p^{\alpha}|^{B}\,.$ (44)

With these conventions, spinor products are computed by contracting the
spacetime-Dirac indices

$\displaystyle{[\\!\\!\\!\;\langle}p_{\bullet}|q_{\bullet}{\rangle\\!\\!\\!\;]}={{[\\!\\!\\!\;\langle}p_{\bullet}|}^{A}{\vphantom{|p_{I}\rangle}}_{A}|q_{\bullet}{\rangle\\!\\!\\!\;]}={{[\\!\\!\\!\;\langle}p_{\bullet}|}_{A}\
C^{AB}\ {\vphantom{|p_{I}\rangle}}_{B}|q_{\bullet}{\rangle\\!\\!\\!\;]}$ (45)

where ${\rangle\\!\\!\\!\;]}$ and ${[\\!\\!\\!\;\langle}$ represent either
square or angle brackets and the $\bullet$’s are placeholders for either
massive ($\alpha,\dot{\alpha},\beta,\dot{\beta},\dots$) or massless
($I,J,\dots$) little group indices. Like the massless brackets, these spinor
products are anti-symmetric

${[\\!\\!\\!\;\langle}p_{\bullet}|q_{\bullet^{\prime}}{\rangle\\!\\!\\!\;]}=-{[\\!\\!\\!\;\langle}q_{\bullet^{\prime}}|p_{\bullet}{\rangle\\!\\!\\!\;]}.$
(46)

Like in the massless case, we use the usual short-hand

${\vphantom{|p_{I}\rangle}}_{A}|i_{\alpha}\rangle={\vphantom{|p_{I}\rangle}}_{A}|p_{i\alpha}\rangle~{}~{}~{}~{}~{}{\vphantom{|p_{I}\rangle}}_{A}|i_{\dot{\alpha}}]={\vphantom{|p_{I}\rangle}}_{A}|p_{i\dot{\alpha}}]$
(47)

to denote the Weyl and anti-Weyl spinors for the $i^{\text{th}}$ particle in
order to simplify formulas.

As in equation (36), we introduce an underlined bracket for massive particles

$\displaystyle{\vphantom{|p_{I}\rangle}}_{A}|\underline{\bm{i}}\rangle={\vphantom{|p_{I}\rangle}}_{A}|p_{i\alpha}\rangle\
z_{i}^{\alpha}\,,\qquad{\vphantom{|p_{I}\rangle}}_{A}|\underline{\bm{i}}]={\vphantom{|p_{I}\rangle}}_{A}|p_{i\dot{\alpha}}]\
\tilde{z}_{i}^{\dot{\alpha}}\,$ (48)

to simplify keeping track of little group indices.

Like massless representations, massive representations in five dimensions can
be constructed from little group tensors that are fully symmetric in their
Weyl $\alpha$ and anti-Weyl $\dot{\alpha}$ indices. Thus, contracting all
little group indices with dummy spinors automatically imposes the correct
symmetry properties, and amplitudes become polynomials in the underlined
brackets, which are built by contracting spacetime Dirac indices

$\displaystyle{[\\!\\!\\!\;\langle}\underline{\bm{ik}}{\rangle\\!\\!\\!\;]}={{[\\!\\!\\!\;\langle}\underline{\bm{i}}|}_{A}\
C^{AB}\
{\vphantom{|p_{I}\rangle}}_{B}|\underline{\bm{k}}{\rangle\\!\\!\\!\;]}\,,\quad{[\\!\\!\\!\;\langle}\underline{\bm{i}}\underline{l}\rangle={{[\\!\\!\\!\;\langle}\underline{\bm{i}}|}_{A}\
C^{AB}\ {\vphantom{|p_{I}\rangle}}_{B}|\underline{l}\rangle\,,\quad\text{and
so on.}$ (49)

##  3 Three-point amplitudes

In this section, we construct all three-point amplitudes from purely kinematic
considerations for any spin and the following configurations of masses: all
masses generic and non-vanishing (section 3.1), two massive and distinct with
one massless (section 3.2), two massive and equal with one massless (section
3.3), one massive and two massless (section 3.4), and, all massless (section
3.5). In constructing all three-point amplitudes, we assume that the momenta
are complex and satisfy momentum conservation $p_{1}+p_{2}+p_{3}=0$.

###  3.1 Three massive: unequal masses

In this section, we construct the space of three-point amplitudes with total
spins $S_{i=1,2,3}=s_{i}+\bar{s}_{i}$ and masses all non-zero and distinct

$A(\mathbf{1}^{S_{1}},\mathbf{2}^{S_{2}},\mathbf{3}^{S_{3}})=\left\\{\begin{aligned}
&{[\\!\\!\\!\;\langle}\underline{\bm{12}}{\rangle\\!\\!\\!\;]}^{S_{1}{+}S_{2}{-}S_{3}}{[\\!\\!\\!\;\langle}\underline{\bm{23}}{\rangle\\!\\!\\!\;]}^{S_{2}{+}S_{3}{-}S_{1}}{[\\!\\!\\!\;\langle}\underline{\bm{31}}{\rangle\\!\\!\\!\;]}^{S_{3}{+}S_{1}{-}S_{2}}&&S_{1}{+}S_{2}{\geq}S_{3}{\geq}|S_{1}-S_{2}|\,,\\\
&{[\\!\\!\\!\;\langle}\underline{\bm{12}}{\rangle\\!\\!\\!\;]}^{2S_{1}}{[\\!\\!\\!\;\langle}\underline{\bm{23}}{\rangle\\!\\!\\!\;]}^{2S_{3}}\langle\underline{\bm{2}}|3|\underline{\bm{2}}]^{S_{2}-S_{1}-S_{3}}&&S_{2}>S_{1}+S_{3}\,,\\\
&{[\\!\\!\\!\;\langle}\underline{\bm{12}}{\rangle\\!\\!\\!\;]}^{2S_{2}}{[\\!\\!\\!\;\langle}\underline{\bm{31}}{\rangle\\!\\!\\!\;]}^{2S_{3}}\langle\underline{\bm{1}}|2|\underline{\bm{1}}]^{S_{1}-S_{2}-S_{3}}&&S_{1}>S_{2}+S_{3}\,,\\\
&{[\\!\\!\\!\;\langle}\underline{\bm{23}}{\rangle\\!\\!\\!\;]}^{2S_{2}}{[\\!\\!\\!\;\langle}\underline{\bm{31}}{\rangle\\!\\!\\!\;]}^{2S_{1}}\langle\underline{\bm{3}}|1|\underline{\bm{3}}]^{S_{3}-S_{1}-S_{2}}&&S_{3}>S_{1}+S_{2}\,.\end{aligned}\right.$
(50)

Here, $s_{i},\bar{s}_{i}$ are half integers denoting the number of chiral and
anti-chiral spinors in the amplitude; each monomial contains $2s_{i}$
$|\underline{\bm{i}}\rangle$ spinors and $2\bar{s}_{i}$ $|\underline{\bm{i}}]$
spinors. Since the spinor brackets satisfy non-linear Schouten identities
(e.g., (54)), the amplitude splits into four “large” sectors depending on the
values for the total spin, $S_{i}$. Here, the bracket
${[\\!\\!\\!\;\langle}\underline{\bm{kl}}{\rangle\\!\\!\\!\;]}^{x}$
corresponds to the monomials
$\langle\underline{\bm{kl}}\rangle^{v_{1}}\langle\underline{\bm{kl}}]^{v_{2}}[\underline{\bm{kl}}\rangle^{v_{3}}[\underline{\bm{kl}}]^{v_{4}}$
whose exponents satisfy $v_{1}+v_{2}+v_{3}+v_{4}=x$. When
$S_{1}{+}S_{2}{\geq}S_{3}{\geq}|S_{1}-S_{2}|$, there are further subsectors
due to cubic identities such as (57) and (59). Once a sector with no further
relations has been identified, the amplitude is given by any linear
combination of monomials whose exponents lie in a polytope dictated by the
$s_{i}$ and $\bar{s}_{i}$.

Our strategy for building all three-point amplitudes starts by enumerating all
possible tensor structures graded by mass dimension. We study identities
between these tensors at each mass dimension and isolate a minimal spanning
set. At some critical mass dimension, we find that all tensors can be built
out of tensors of lower mass dimension.

#### The ansatz.

At mass dimension zero, the possible tensor structures consist only of the
charge conjugation matrices. However, these are anti-symmetric and not
permitted in the amplitude. Enumerating all possible structures with mass
dimension one reveals the following candidates

$\displaystyle{[\\!\\!\\!\;\langle}\underline{\bm{12}}{\rangle\\!\\!\\!\;]}\,,\quad{[\\!\\!\\!\;\langle}\underline{\bm{13}}{\rangle\\!\\!\\!\;]}\,,\quad{[\\!\\!\\!\;\langle}\underline{\bm{23}}{\rangle\\!\\!\\!\;]}\,.$
(51)

All other mass dimension one tensors vanish or are linearly related to these.

At mass dimension two, the candidate tensors are:
${[\\!\\!\\!\;\langle}\underline{\bm{i}}|j|\underline{\bm{k}}{\rangle\\!\\!\\!\;]}$.
Obviously, we do not have to consider tensors with $i=j=k$ since these vanish.
Moreover, we must have $i=k\neq j$. Otherwise, the tensor is reducible to a
mass dimension one two-bracket (51) via the Dirac equation. Then, since the
brackets
$\langle\underline{\bm{i}}|j|\underline{\bm{i}}\rangle=[\underline{\bm{i}}|j|\underline{\bm{i}}]=0$,
we are left with the following candidate tensors

$\displaystyle\langle\underline{\bm{1}}|2|\underline{\bm{1}}]\,,\quad\langle\underline{\bm{2}}|3|\underline{\bm{2}}]\,,\quad\langle\underline{\bm{3}}|1|\underline{\bm{3}}]\,.$
(52)

Note that we do not have to consider
$\langle\underline{\bm{1}}|3|\underline{\bm{1}}]$ (and others like it) because
it is equivalent to $\langle\underline{\bm{1}}|2|\underline{\bm{1}}]$ up to
lower mass dimension tensors. Furthermore, the reflections of the above
brackets (e.g., $[\underline{\bm{1}}|2|\underline{\bm{1}}\rangle$) do not need
to be considered since
$[\underline{\bm{i}}|j|\underline{\bm{i}}\rangle=\langle\underline{\bm{i}}|j|\underline{\bm{i}}]$.

Thus, all tensor structures up to mass dimension two can be built from linear
combinations of the following building blocks

$\displaystyle\langle\underline{\bm{12}}\rangle\,,\quad[\underline{\bm{12}}\rangle\,,\quad\langle\underline{\bm{12}}]\,,\quad[\underline{\bm{12}}]\,,\quad\langle\underline{\bm{31}}\rangle\,,\quad[\underline{\bm{31}}\rangle\,,\quad\langle\underline{\bm{31}}]\,,\quad[\underline{\bm{31}}]\,,$
(53)
$\displaystyle\langle\underline{\bm{23}}\rangle\,,\quad[\underline{\bm{23}}\rangle\,,\quad\langle\underline{\bm{23}}]\,,\quad[\underline{\bm{23}}]\,,\quad\langle\underline{\bm{1}}|2|\underline{\bm{1}}]\,,\quad\langle\underline{\bm{2}}|3|\underline{\bm{2}}]\,,\quad\langle\underline{\bm{3}}|1|\underline{\bm{3}}]\,.$

It also turns out that mass dimension three and higher tensors can all be
built from products of (53). For example,
${[\\!\\!\\!\;\langle}\underline{\bm{1}}|2|3|\underline{\bm{2}}{\rangle\\!\\!\\!\;]}=2p_{2}\cdot
p_{3}{[\\!\\!\\!\;\langle}\underline{\bm{1}}|\underline{\bm{2}}{\rangle\\!\\!\\!\;]}\pm
im_{2}{[\\!\\!\\!\;\langle}\underline{\bm{1}}|3|\underline{\bm{2}}{\rangle\\!\\!\\!\;]}$
and all others like it are reducible to lower mass dimension tensors.

So far, we have only used linear identities between tensor structures to
constrain the list of allowed tensor structures. Generically, there will be
quadratic Schouten identities involving the tensors in (53)

$\displaystyle\langle\underline{\bm{12}}\rangle\,\langle\underline{\bm{3}}|1|\underline{\bm{3}}]$
$\displaystyle=\langle\underline{\bm{13}}]\langle\underline{\bm{3}}|1|\underline{\bm{2}}\rangle-\langle\underline{\bm{13}}\rangle[\underline{\bm{3}}|1|\underline{\bm{2}}\rangle+m_{1}\langle\underline{\bm{31}}\rangle\langle\underline{\bm{23}}]+m_{1}[\underline{\bm{31}}\rangle\langle\underline{\bm{23}}\rangle$
(54)
$\displaystyle=(m_{1}-m_{2}+m_{3})\,\langle\underline{\bm{13}}]\,\langle\underline{\bm{32}}\rangle+(m_{1}-m_{2}-m_{3})\,\langle\underline{\bm{13}}\rangle\,[\underline{\bm{32}}\rangle\,,$

along with the permutations of particles 1, 2, and 3 and exchanging angle and
square brackets. In particular, (54) implies that the tensors on the left-hand
side above cannot appear together.222Mathematically, this means that we are
working in the polynomial quotient ring  $\displaystyle
R=\frac{\mathbb{C}\big{[}\eqref{eq:setmmm5d}\big{]}}{\langle\eqref{eq:3massive2},\text{variations}\eqref{eq:3massive2}\rangle}\,.$
Since there are often many non-linear identities between spinor helicity
variables, it is important to understand the ring structure when constructing
ansatz. See DeLaurentis:2022otd ; DeLaurentis:2022knk  for more on the
connections of spinor helicity variables to algebraic geometry and $p$-adic
numbers.  By counting the total powers of spinors, we arrive at the ansatz
(50) for five-dimensional unequal massive amplitudes where (54) splits the
ansatz into four sectors. There are further cubic identities (arising from the
composition of various Schouten identities) that split the sector
$S_{1}{+}S_{2}{\geq}S_{3}{\geq}|S_{1}-S_{2}|$ into further sub-sectors.

In the remainder of this section, we will detail how the cubic identities
affect the ansatz (50) and the counting of the number of independent
amplitudes. In particular, we find the expected number of independent
amplitudes Costa:2011mg ; Elkhidir:2014woa .

#### Case 1:
$\bm{S_{1}+S_{2}}\bm{\geqslant}\bm{S_{3}}\bm{\geqslant}\bm{|S_{1}-S_{2}|}$.

For the first case, the explicit ansatz is

$\displaystyle
A(\mathbf{1}^{s_{1}+\bar{s}_{1}},\mathbf{2}^{s_{2}+\bar{s}_{2}},\mathbf{3}^{s_{3}+\bar{s}_{3}})$
$\displaystyle=\langle\underline{\bm{12}}\rangle^{v_{1}}\langle\underline{\bm{12}}]^{v_{2}}\langle\underline{\bm{13}}\rangle^{v_{3}}\langle\underline{\bm{13}}]^{v_{4}}[\underline{\bm{12}}\rangle^{v_{5}}[\underline{\bm{12}}]^{v_{6}}$
(55)
$\displaystyle\qquad\times[\underline{\bm{13}}\rangle^{v_{7}}[\underline{\bm{13}}]^{v_{8}}\langle\underline{\bm{23}}\rangle^{v_{9}}\langle\underline{\bm{23}}]^{v_{10}}[\underline{\bm{23}}\rangle^{v_{11}}[\underline{\bm{23}}]^{v_{12}}$

with the exponents satisfying

$\displaystyle v_{1}+v_{2}+v_{3}+v_{4}=2s_{1}\,,$ $\displaystyle\qquad
v_{5}+v_{6}+v_{7}+v_{8}=2\bar{s}_{1}\,,$ (56) $\displaystyle
v_{1}+v_{5}+v_{9}+v_{10}=2s_{2}\,,$ $\displaystyle\qquad
v_{2}+v_{6}+v_{11}+v_{12}=2\bar{s}_{2}\,,$ $\displaystyle
v_{3}+v_{7}+v_{9}+v_{11}=2s_{3}\,,$ $\displaystyle\qquad
v_{4}+v_{8}+v_{10}+v_{12}=2\bar{s}_{3}\,,$

and $v_{i}\in\mathbb{N}_{0}$. Therefore, the final solution space of
independent amplitudes is seemingly given by the lattice points inside a
$(12-6)=6$ dimensional polytope cut out by the above conditions.

However, there are further cubic identities satisfied by these brackets that
come from the composition of (54) and its variations

$\displaystyle 0$
$\displaystyle={\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\langle\underline{\bm{12}}]}{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}[\underline{\bm{13}}\rangle}{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\langle\underline{\bm{23}}]}-[\underline{\bm{23}}\rangle\langle\underline{\bm{13}}][\underline{\bm{12}}\rangle-\frac{m_{++}}{m_{--}}{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\langle\underline{\bm{12}}\rangle}[\underline{\bm{23}}\rangle{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}[\underline{\bm{13}}]}+\frac{m_{++}}{m_{--}}{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}\langle\underline{\bm{13}}\rangle[\underline{\bm{12}}]\langle\underline{\bm{23}}]}\,,$
(57) $\displaystyle 0$
$\displaystyle={\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\langle\underline{\bm{12}}]}{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}[\underline{\bm{13}}\rangle}{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\langle\underline{\bm{23}}]}-[\underline{\bm{23}}\rangle\langle\underline{\bm{13}}][\underline{\bm{12}}\rangle-\frac{m_{++}}{m_{-+}}{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}\langle\underline{\bm{12}}\rangle[\underline{\bm{13}}\rangle[\underline{\bm{23}}]}+\frac{m_{++}}{m_{-+}}{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\langle\underline{\bm{23}}\rangle}\langle\underline{\bm{13}}]{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}[\underline{\bm{12}}]}\,,$
$\displaystyle 0$
$\displaystyle={\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\langle\underline{\bm{12}}]}{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}[\underline{\bm{13}}\rangle}{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\langle\underline{\bm{23}}]}-[\underline{\bm{23}}\rangle\langle\underline{\bm{13}}][\underline{\bm{12}}\rangle+\frac{m_{++}}{m_{+-}}{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}\langle\underline{\bm{13}}\rangle}[\underline{\bm{12}}\rangle{\color[rgb]{.5,0,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,0,.5}[\underline{\bm{23}}]}-\frac{m_{++}}{m_{+-}}{\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}\langle\underline{\bm{12}}][\underline{\bm{13}}]\langle\underline{\bm{23}}\rangle}\,,$

where $m_{\pm\pm}=m_{1}\pm m_{2}\pm m_{3}$. Using (57), we can eliminate the
monomials
$\langle\underline{\bm{13}}\rangle[\underline{\bm{12}}]\langle\underline{\bm{23}}]$,
$\langle\underline{\bm{12}}\rangle[\underline{\bm{13}}\rangle[\underline{\bm{23}}]$,
and
$\langle\underline{\bm{12}}][\underline{\bm{13}}]\langle\underline{\bm{23}}\rangle$
(colored in  red) in favor of the other monomials. The purple brackets
indicate a repeated two-bracket from the set of red monomials that were
eliminated. This means that at least one exponent in each list
$(v_{3},v_{6},v_{10})$, $(v_{1},v_{7},v_{12})$, $(v_{2},v_{8},v_{9})$ must be
zero. Therefore, we obtain $3^{3}=27$ sub-sectors. Each of them has three
exponents fixed to zero, so the dimension of each sector becomes $(6-3)=3$.

However, nothing prevents there from being additional identities inside in
each of these 27 sectors. There are further identities in the sectors where
the following monomials were eliminated

$\displaystyle\langle\underline{\bm{12}}\rangle\langle\underline{\bm{13}}\rangle\langle\underline{\bm{12}}]\,,\quad\langle\underline{\bm{13}}]\langle\underline{\bm{13}}\rangle\langle\underline{\bm{23}}\rangle\,,$
(58)
$\displaystyle\langle\underline{\bm{23}}\rangle\langle\underline{\bm{12}}\rangle\langle\underline{\bm{23}}]\,,\quad\langle\underline{\bm{12}}]\langle\underline{\bm{12}}\rangle\langle\underline{\bm{13}}\rangle\,,$
$\displaystyle\langle\underline{\bm{13}}\rangle\langle\underline{\bm{23}}\rangle\langle\underline{\bm{13}}]\,,\quad\langle\underline{\bm{23}}]\langle\underline{\bm{23}}\rangle\langle\underline{\bm{12}}\rangle\,.$

For example, the additional identity in the sub-sector where
$\langle\underline{\bm{12}}\rangle\langle\underline{\bm{31}}\rangle\langle\underline{\bm{12}}]$
was eliminated is

$\displaystyle 0$
$\displaystyle=m_{++}[\underline{\bm{12}}]\langle\underline{\bm{13}}][\underline{\bm{23}}\rangle-
m_{+-}[\underline{\bm{12}}][\underline{\bm{13}}\rangle\langle\underline{\bm{23}}]+m_{--}[\underline{\bm{12}}\rangle[\underline{\bm{13}}\rangle[\underline{\bm{23}}]-m_{-+}[\underline{\bm{12}}\rangle[\underline{\bm{13}}][\underline{\bm{23}}\rangle\,.$
(59)

The identities in other sectors in (58) can be derived from chiral flips and
cyclic permutations of (57). Such identities reduce the previously three-
dimensional solution space to three two-dimensional boundaries.

#### Case 2: $\bm{S_{2}>S_{1}+S_{3}}$.

There are no further identities for the last three cases in (50). In the
second case, the explicit ansatz reads

$\displaystyle
A(\mathbf{1}^{s_{1}+\bar{s}_{1}},\mathbf{2}^{s_{2}+\bar{s}_{2}},\mathbf{3}^{s_{3}+\bar{s}_{3}})$
$\displaystyle=\langle\underline{\bm{12}}\rangle^{w_{1}}\langle\underline{\bm{12}}]^{w_{2}}[\underline{\bm{12}}\rangle^{w_{3}}[\underline{\bm{12}}]^{w_{4}}\langle\underline{\bm{23}}\rangle^{w_{5}}$
(60)
$\displaystyle\qquad\times[\underline{\bm{23}}\rangle^{w_{6}}\langle\underline{\bm{23}}]^{w_{7}}[\underline{\bm{23}}]^{w_{8}}\langle\underline{\bm{2}}|3|\underline{\bm{2}}]^{S_{2}-S_{1}-S_{3}}\,,$

with the exponents satisfying

$\displaystyle w_{1}+w_{2}=2s_{1}\,,$ $\displaystyle\quad
w_{3}+w_{4}=2\bar{s}_{1}\,,$ (61) $\displaystyle w_{5}+w_{6}=2s_{3}\,,$
$\displaystyle\quad w_{7}+w_{8}=2\bar{s}_{3}\,,$ $\displaystyle
w_{1}+w_{3}+w_{5}+w_{7}$ $\displaystyle=s_{2}-\bar{s}_{2}+S_{1}+S_{3}\,,$

and $w_{i}\in\mathbb{N}_{0}$. Since the exponent for the final tensor
$\langle\underline{\bm{2}}|3|\underline{\bm{2}}]$ is fixed, the final solution
space has $8-5=3$ dimensions. Choosing $(w_{1},w_{3},w_{5})$ to parametrize
this space, we obtain a polytope with faces defined by the following
conditions:

$\displaystyle 0\leqslant w_{1}\leqslant 2s_{1}\,,\qquad 0\leqslant
w_{3}\leqslant 2\bar{s}_{1}\,,\qquad 0\leqslant w_{5}\leqslant 2s_{3}\,,$ (62)
$\displaystyle S_{1}+s_{2}-\bar{s}_{2}+s_{3}-\bar{s}_{3}\leqslant
w_{1}+w_{3}+w_{5}\leqslant s_{2}-\bar{s}_{2}+S_{1}+S_{3}\,.$

The number of independent amplitudes for the second line in (50) is given by
the number of lattice points inside or on the boundary of this polytope.

#### All cases.

Similar to the $S_{2}>S_{1}+S_{3}$ sector, one can construct the polytope and
amplitudes in the third and fourth sectors of (50). In the end, the total
polytope that counts the number of independent three-point massive amplitudes
and couplings is the direct sum of the 27 subsectors of case 1 as well as
three polytopes from cases 2, 3, and 4.

#### Comparing to known results for the number of independent amplitudes.

Above, we outlined how to compute the number of independent amplitudes by
quotienting out by the identities (57) and (59). In particular, obtaining
explicit solutions for low spin ($S\leq 4$) in Mathematica is not too
difficult. We have compared the output of the above counting strategy to known
closed-form results from Costa:2011mg  and Elkhidir:2014woa  and found
agreement. We will be more explicit about this construction and counting in
the next section, which has fewer sectors.

###  3.2 Two massive and one massless: unequal masses

In this and the next section, we set $m_{3}=0$ and assume that $m_{i=1,2}>0$.
In this section, we set $m_{1}\neq m_{2}$, while in the next section, we will
take $m_{1}=m_{2}$.

As before, we make an ansatz for the amplitude compatible with little group
scaling from a list of tensors that are not linearly related and then impose
any non-linear relations satisfied by these tensors. The amplitude splits into
three “large” sectors depending on the values of spin
$s_{1},s_{2},\bar{s}_{1},\bar{s}_{2}$ and the helicity $S_{3}$

$\displaystyle
A(\mathbf{1}^{s_{1}{+}\bar{s}_{1}},\mathbf{2}^{s_{2}{+}\bar{s}_{2}},3^{S_{3}})=\begin{cases}{[\\!\\!\\!\;\langle}\underline{\bm{12}}{\rangle\\!\\!\\!\;]}^{S_{1}{+}S_{2}{-}S_{3}}{[\\!\\!\\!\;\langle}\underline{\bm{2}}\underline{3}\rangle^{S_{2}-S_{1}+S_{3}}{[\\!\\!\\!\;\langle}\underline{\bm{1}}\underline{3}\rangle^{S_{1}-S_{2}+S_{3}}&S_{1}{+}S_{2}{\geqslant}S_{3}\geqslant|S_{1}{-}S_{2}|,\\\
{[\\!\\!\\!\;\langle}\underline{\bm{12}}{\rangle\\!\\!\\!\;]}^{2S_{2}}{[\\!\\!\\!\;\langle}\underline{\bm{2}}\underline{3}\rangle^{S_{3}}\langle\underline{\bm{2}}|3|\underline{\bm{2}}]^{S_{2}{-}S_{1}{-}S_{3}}&S_{2}\geqslant
S_{1}{+}S_{3}\geqslant|s_{2}{-}\bar{s}_{2}|,\\\
{[\\!\\!\\!\;\langle}\underline{\bm{12}}{\rangle\\!\\!\\!\;]}^{2S_{2}}\langle\underline{\bm{1}}|3|\underline{\bm{1}}]^{S_{1}-S_{2}-S_{3}}{[\\!\\!\\!\;\langle}\underline{\bm{1}}\underline{3}\rangle^{2S_{3}}&S_{1}\geqslant
S_{2}{+}S_{3}\geqslant|s_{1}{-}\bar{s}_{1}|.\end{cases}$ (63)

As before, the bracket
${[\\!\\!\\!\;\langle}\underline{\bm{kl}}{\rangle\\!\\!\\!\;]}^{x}$
corresponds to the monomials
$\langle\underline{\bm{kl}}\rangle^{v_{1}}\langle\underline{\bm{kl}}]^{v_{2}}[\underline{\bm{kl}}\rangle^{v_{3}}[\underline{\bm{kl}}]^{v_{4}}$
whose exponents satisfy $v_{1}+v_{2}+v_{3}+v_{4}=x$.

#### The ansatz.

Like in section 3.1, we list all possible mass dimension one and two tensors,
impose linear relations, and discard any mass dimension two tensors that can
be reduced to mass dimension one tensor via momentum conservation or the Dirac
equation. In the end, we find that the following set of tensor structures

$\langle\underline{3}\underline{\bm{1}}\rangle\,,\,\langle\underline{3}\underline{\bm{1}}]\,,\,\langle\underline{3}\underline{\bm{2}}\rangle\,,\,\langle\underline{3}\underline{\bm{2}}]\,,\,\langle\underline{\bm{1}}\underline{\bm{2}}\rangle\,,\,\langle\underline{\bm{1}}\underline{\bm{2}}]\,,\,[\underline{\bm{1}}\underline{\bm{2}}\rangle\,,\,[\underline{\bm{1}}\underline{\bm{2}}]\,,\,\langle\underline{\bm{1}}|3|\underline{\bm{1}}]\,,\,\langle\underline{\bm{2}}|3|\underline{\bm{2}}]\,,$
(64)

is enough to span all higher mass dimension tensors.

Thus, the general ansatz for these amplitudes is

$\displaystyle
A(\mathbf{1}^{s_{1}{+}\bar{s}_{1}},\mathbf{2}^{s_{2}{+}\bar{s}_{2}},3^{S_{3}})$
$\displaystyle=\langle\underline{3}\underline{\bm{1}}\rangle^{v_{1}}\langle\underline{3}\underline{\bm{1}}]^{v_{2}}\langle\underline{3}\underline{\bm{2}}\rangle^{v_{3}}\langle\underline{3}\underline{\bm{2}}]^{v_{4}}\langle\underline{\bm{1}}\underline{\bm{2}}\rangle^{v_{5}}$
(65)
$\displaystyle\quad\times\langle\underline{\bm{1}}\underline{\bm{2}}]^{v_{6}}[\underline{\bm{1}}\underline{\bm{2}}\rangle^{v_{7}}[\underline{\bm{1}}\underline{\bm{2}}]^{v_{8}}\langle\underline{\bm{1}}|3|\underline{\bm{1}}]^{v_{9}}\langle\underline{\bm{2}}|3|\underline{\bm{2}}]^{v_{10}}\,,$

where the exponents $v_{i}$ are subject to the following constraints

$\displaystyle\begin{aligned} v_{1}+v_{2}+v_{3}+v_{4}=2S_{3}\,,&\qquad
v_{1}+v_{5}+v_{6}+v_{9}=2s_{1}\,,\\\ v_{3}+v_{5}+v_{7}+v_{10}=2s_{2}\,,&\qquad
v_{2}+v_{7}+v_{8}+v_{9}=2\bar{s}_{1}\,,\\\
v_{4}+v_{6}+v_{8}+v_{10}=2\bar{s}_{2}\,,&\qquad
v_{i}\in\mathbb{N}_{0}\,,\end{aligned}$ (66)

which ensure that there are $s_{a}$ spinors $|\underline{\bm{a}}\rangle$,
$\bar{s}_{a}$ spinors $|\underline{\bm{a}}]$ for $a=1,2$ and $S_{3}$ spinors
$|\underline{3}\rangle$.

There are also quadratic identities involving the tensor structures (64) that
further constrain the exponents $v_{i}$. For example,

$\displaystyle\langle\underline{\bm{1}}|3|\underline{\bm{1}}]\langle\underline{\bm{2}}\underline{3}\rangle=\langle\underline{\bm{2}}|3|\underline{\bm{1}}]\langle\underline{\bm{1}}\underline{3}\rangle+\langle\underline{\bm{1}}|3|\underline{\bm{2}}\rangle[\underline{\bm{1}}\underline{3}\rangle=i\left(m_{-}\langle\underline{\bm{2}}\underline{\bm{1}}]\langle\underline{\bm{1}}3\rangle-
m_{+}\langle\underline{\bm{1}}\underline{\bm{2}}\rangle[\underline{\bm{1}}\underline{3}\rangle\right)$
(67)

where $m_{\pm}=m_{1}\pm m_{2}$. Similar, identities exist when
$1\leftrightarrow 2$ and
$|\underline{\bm{i}}\rangle\leftrightarrow|\underline{\bm{i}}]$ for
$i=\\{1,2\\}$. We also find another identity

$\displaystyle\langle\underline{\bm{1}}|3|\underline{\bm{1}}][\underline{\bm{2}}|3|\underline{\bm{2}}\rangle=\langle\underline{\bm{1}}|3|\underline{\bm{2}}\rangle[\underline{\bm{2}}|3|\underline{\bm{1}}]-\langle\underline{\bm{1}}|3|\underline{\bm{2}}]\langle\underline{\bm{2}}|3|\underline{\bm{1}}]=-m_{+}^{2}\langle\underline{\bm{1}}\underline{\bm{2}}\rangle[\underline{\bm{1}}\underline{\bm{2}}]-m_{-}^{2}\langle\underline{\bm{1}}\underline{\bm{2}}]\langle\underline{\bm{2}}\underline{\bm{1}}]\,.$
(68)

Therefore, the following pairs of tensors cannot appear together in our ansatz
$(\langle\underline{\bm{1}}\underline{3}\rangle,\langle\underline{\bm{2}}|3|\underline{\bm{2}}])$,
$([\underline{\bm{1}}\underline{3}\rangle,\langle\underline{\bm{2}}|3|\underline{\bm{2}}])$,
$(\langle\underline{\bm{2}}\underline{3}\rangle,\langle\underline{\bm{1}}|3|\underline{\bm{1}}])$,
$([\underline{\bm{2}}\underline{3}\rangle,\langle\underline{\bm{1}}|3|\underline{\bm{1}}])$
and
$(\langle\underline{\bm{1}}|3|\underline{\bm{1}}],\langle\underline{\bm{2}}|3|\underline{\bm{2}}])$.
In terms of exponents, there needs to be at least one zero in each of the
following pairs $(v_{1},v_{10})$, $(v_{2},v_{10})$, $(v_{3},v_{9})$,
$(v_{4},v_{9})$ and $(v_{9},v_{10})$. This condition breaks the amplitude into
the three “large” sectors as highlighted in (63)

$\displaystyle\begin{aligned} (v_{9}=0,v_{10}=0)&\implies
S_{1}{+}S_{2}\geqslant S_{3}\geqslant|S_{1}{-}S_{2}|\,,\\\
(v_{9}=0,v_{10}>0)&\implies S_{2}\geqslant
S_{1}{+}S_{3}\geqslant|s_{2}{-}\bar{s}_{2}|\,,\\\ (v_{9}>0,v_{10}=0)&\implies
S_{1}\geqslant S_{2}{+}S_{3}\geqslant|s_{1}{-}\bar{s}_{1}|\,.\end{aligned}$
(69)

Figure 1: The polytope $P_{00}$ for
$(s_{1},\bar{s}_{1},s_{2},\bar{s}_{2},S_{3})=(\frac{5}{2},2,\frac{3}{2},\frac{5}{2},\frac{3}{2})$.
However, generally, only the lattice points on the planes defined by
$v_{2}=0$, $v_{4}=0$, and $v_{5}=0$ (depicted in orange above) are
independent. The intersection of $P_{00}$ with the planes $v_{2}=0$,
$v_{4}=0$, and $v_{5}=0$ are also depicted in the lower portion of this
figure. For this spin configuration, there are a total of 14 independent
amplitudes in the $P_{00}$ sector. The allowed point in the triple
intersection of the orange planes is denoted in red while the allowed points
in the double intersection of the orange planes are denoted in blue. The red
and blue points should only be counted once.

#### Case 1: $\bm{S_{1}{+}S_{2}\geqslant S_{3}\geqslant|S_{1}{-}S_{2}|}$.

When $v_{9}=v_{10}=0$, there are no 3-brackets in the ansatz. However, there
is the cubic identity

$\displaystyle-
m_{-}\langle\underline{3}\underline{\bm{1}}]\langle\underline{3}\underline{\bm{2}}\rangle\langle\underline{\bm{12}}]+m_{+}\langle\underline{3}\underline{\bm{1}}]\langle\underline{3}\underline{\bm{2}}]\langle\underline{\bm{12}}\rangle+m_{-}\langle\underline{3}\underline{\bm{1}}\rangle\langle\underline{3}\underline{\bm{2}}][\underline{\bm{12}}\rangle-
m_{+}\langle\underline{3}\underline{\bm{1}}\rangle\langle\underline{3}\underline{\bm{2}}\rangle[\underline{\bm{12}}]=0\,.$
(70)

that comes from the composition of Schouten identities. This means that we can
eliminate
$\langle\underline{3}\underline{\bm{1}}]\langle\underline{3}\underline{\bm{2}}]\langle\underline{\bm{1}}\underline{\bm{2}}\rangle$
for the other monomials above. Therefore, at least one of the exponents
$(v_{2},v_{4},v_{5})=(S_{1}+S_{3}-S_{2}-v_{1},S_{2}+S_{3}-S_{1}-v_{3},v_{5})$
must also be set to zero creating three sectors (see the orange planes in
figure 1). Note that one could choose to eliminate a different monomial
appearing in the cubic identity (70), resulting in an equivalent
representation for the same amplitudes.

Thus, the allowed amplitudes of this sector are

$\displaystyle\begin{aligned}
A(\mathbf{1}^{s_{1}{+}\bar{s}_{1}},\mathbf{2}^{s_{2}{+}\bar{s}_{2}},3^{S_{3}})&=\sum_{(v_{1},v_{3},v_{5})\in\mathbb{Z}^{3}\cap
P_{00}\cap\\{v_{2}=0\\}}g_{v_{1},v_{3},v_{5}}A_{v_{1},v_{3},v_{5}}(\mathbf{1}^{s_{1}+\bar{s}_{1}},\mathbf{2}^{s_{2}+\bar{s}_{2}},3^{S_{3}})\\\
&\quad+\sum_{(v_{1},v_{3},v_{5})\in\mathbb{Z}^{3}\cap
P_{00}\cap\\{v_{4}=0\\}}g_{v_{1},v_{3},v_{5}}A_{v_{1},v_{3},v_{5}}(\mathbf{1}^{s_{1}+\bar{s}_{1}},\mathbf{2}^{s_{2}+\bar{s}_{2}},3^{S_{3}})\\\
&\quad+\sum_{(v_{1},v_{3},v_{5})\in\mathbb{Z}^{3}\cap
P_{00}\cap\\{v_{5}=0\\}}g_{v_{1},v_{3},v_{5}}A_{v_{1},v_{3},v_{5}}(\mathbf{1}^{s_{1}+\bar{s}_{1}},\mathbf{2}^{s_{2}+\bar{s}_{2}},3^{S_{3}})\end{aligned}$
(71)

where we have solved the constraints (66) leaving $v_{1},v_{3}$ and $v_{5}$
free and defined

$\displaystyle
A_{v_{1},v_{3},v_{5}}(\mathbf{1}^{s_{1}{+}\bar{s}_{1}},\mathbf{2}^{s_{2}{+}\bar{s}_{2}},3^{S_{3}})$
$\displaystyle=\langle\underline{3}\underline{\bm{1}}\rangle^{v_{1}}\langle\underline{3}\underline{\bm{1}}]^{S_{3}+S_{1}-S_{2}-v_{1}}\langle\underline{3}\underline{\bm{2}}\rangle^{v_{3}}\langle\underline{3}\underline{\bm{2}}]^{S_{3}+S_{2}-S_{1}-v_{3}}\langle\underline{\bm{1}}\underline{\bm{2}}\rangle^{v_{5}}$
(72)
$\displaystyle\quad\times\langle\underline{\bm{1}}\underline{\bm{2}}]^{2s_{1}-v_{1}-v_{5}}[\underline{\bm{1}}\underline{\bm{2}}\rangle^{2s_{2}-v_{3}-v_{5}}[\underline{\bm{1}}\underline{\bm{2}}]^{\bar{s}_{1}-s_{1}+\bar{s}_{2}-s_{2}-S_{3}+v_{1}+v_{3}+v_{5}}\,.$

Here, $P_{00}$ is the polyhedron with the defining faces

$\displaystyle P_{00}:\left\\{\begin{matrix}[l]&v_{i=1,3,5}\geq
0,&&S_{3}+S_{1}-S_{2}-v_{1}\geq 0,\\\ &S_{3}+S_{2}-S_{1}-v_{3}\geq
0,&&2s_{1}-v_{1}-v_{5}\geq 0,\\\ &2s_{2}-v_{3}-v_{5}\geq
0,&&\bar{s}_{1}-s_{1}+\bar{s}_{2}-s_{2}-S_{3}+v_{1}+v_{3}+v_{5}&\geq
0\,,\end{matrix}\right.$ (73)

and

$\displaystyle(S_{3}+S_{1}-S_{2},S_{3}+S_{2}-S_{1},\bar{s}_{1}-s_{1}+\bar{s}_{2}-s_{2}-S_{3})\in\mathbb{Z}^{3}\,,$
(74)

for the formula (72) to make sense. The number of independent amplitudes in
this sector is given by the number of lattice points in $P_{00}$ that lie on
the planes defined by $v_{2}=0$, $v_{4}=0$ and $v_{5}=0$ (an example is
provided in figure 1).

#### Case 2: $\bm{S_{1}\geqslant
S_{2}{+}S_{3}\geqslant|s_{1}{-}\bar{s}_{1}|}$.

For $v_{9}=0$ and $v_{10}>0$, the independent amplitudes are

$\displaystyle
A(\mathbf{1}^{s_{1}{+}\bar{s}_{1}},\mathbf{2}^{s_{2}{+}\bar{s}_{2}},3^{S_{3}})$
$\displaystyle=\sum_{(v_{1},v_{5})\in
P_{01}\cap\mathbb{Z}^{3}}g_{v_{1},v_{5}}A_{v_{1},v_{5}}(\mathbf{1}^{s_{1}+\bar{s}_{1}},\mathbf{2}^{s_{2}+\bar{s}_{2}},3^{S_{3}})$
(75)

where

$\displaystyle\begin{aligned}
A_{v_{1},v_{5}}(\mathbf{1}^{s_{1}{+}\bar{s}_{1}},\mathbf{2}^{s_{2}{+}\bar{s}_{2}},3^{S_{3}})&=\langle\underline{3}\underline{\bm{1}}\rangle^{v_{1}}\langle\underline{3}\underline{\bm{1}}]^{2S_{3}-v_{1}}\langle\underline{\bm{1}}\underline{\bm{2}}\rangle^{v_{5}}\langle\underline{\bm{1}}\underline{\bm{2}}]^{2s_{1}-v_{1}-v_{5}}[\underline{\bm{1}}\underline{\bm{2}}\rangle^{S_{1}+s_{2}-\bar{s}_{2}-S_{3}-v_{5}}\\\
&\quad\times[\underline{\bm{1}}\underline{\bm{2}}]^{\bar{s}_{1}-s_{1}+\bar{s}_{2}-s_{2}-S_{3}+v_{1}+v_{5}}\langle\underline{\bm{2}}|3|\underline{\bm{2}}]^{S_{2}+S_{3}-S_{1}},\end{aligned}$
(76)

and $P_{01}$ is the polyhedron with the defining faces

$\displaystyle P_{01}:\left\\{\begin{matrix}[l]&v_{i=1,5}\geq
0,&&2S_{3}-v_{1}\geq 0,\\\ &S_{1}+s_{2}-\bar{s}_{2}-S_{3}-v_{5}\geq
0,&&2s_{1}-v_{1}-v_{5}\geq 0,\\\
&\bar{s}_{1}-s_{1}+\bar{s}_{2}-s_{2}-S_{3}+v_{1}+v_{5}\geq
0.&&\end{matrix}\right.$ (77)

The last face above does not yield any interesting constraint and can be
ignored. The number of independent amplitudes is given by the number of
lattice points inside this polyhedron.

Figure 2: Sketch of $P_{01}$. Depending on the parameters, the line
$2s_{1}-v_{1}-v_{5}=0$ may or may not intersect the lines $v_{1}=2S_{3}$ and
$v_{5}=S_{1}+s_{2}-\bar{s}_{2}-S_{3}$ below their intersection point.

#### Case 3: $\bm{S_{1}\geqslant
S_{2}{+}S_{3}\geqslant|s_{1}{-}\bar{s}_{1}|}$.

The final category of amplitudes has $v_{10}=0$ and $v_{9}>0$. These
amplitudes can be obtained from the (76) by replacing $v_{1}\to v_{3}$ and
$\underline{\bm{1}}\to\underline{\bm{2}}$. We checked that the amplitudes
given by equations (72) and (76) are linearly independent for all $S_{i}\leq
4$. Combining all three conditions for the three sectors of solutions, we
notice that the ordered pair
$(s_{1},\overline{s}_{1},s_{2},\overline{s}_{2},S_{3})$ must form a pentagon.
Again, we have checked with the methods of Costa:2011mg  and Elkhidir:2014woa
for the number of independent amplitudes and find agreement.

###  3.3 Two massive and one massless: equal masses

In this section, we keep $m_{3}=0$ but assume that the massive particles have
the same mass: $m_{1}=m_{2}=m\neq 0$. When the massive particles have the same
mass, the spinor-helicity variables of the two massive spinors are linearly
dependent, and the formalism presented so far must be modified. Due to the
kinematic configuration, many of the two brackets ($2\times 2$ tensors) of
mass dimension one are only half-rank. Consequently, we can express them as a
product of two two-component objects with only little group indices. This
strategy has already been implemented in the six-dimensional spinor helicity
formalism of Cheung:2009dc  (and the five-dimensional massless formalism of
Zhao:2023 ). However, before introducing these new variables, we take a more
pedestrian approach and later show how the two approaches are equivalent.

#### The pedestrian road.

We start by introducing a reference momentum $q$ such that the massless
polarization vector takes the form

$\displaystyle\varepsilon^{\mu}_{3,i}=\frac{\langle
3_{I}|q|\Gamma^{\mu}|3_{J}\rangle}{p_{3}\cdot q}\Gamma^{IJ}_{i}\,,$ (78)

where $\Gamma^{IJ}_{\mu=i}$ is an $\operatorname{SO}(3)$ gamma matrix. While
we will not use the polarization vector as written above, we will use another
related tensor generated by the reference momenta $q$. This tensor is a new
independent object that can enter into the tensor building blocks.

As usual, we enumerate the allowed tensor structures and form an ansatz for
the possible three-point amplitudes with $(m_{1},m_{2},m_{3})=(m,m,0)$. There
are no allowed mass dimension zero structures, and all of the mass dimension
one and two building blocks in (64) carry over. However, unlike in section
3.2, there is a single mass dimension three tensor333We ignore the scalar
denominator when counting the mass dimension of this tensor. built from a
reference momentum $q$

$\displaystyle\frac{\langle\underline{3}|1|q|\underline{3}\rangle}{p_{3}\cdot
q}.$ (79)

where the denominator makes this structure invariant under the gauge
transformation of $q$. We will often gauge-fix $q$ such that $q\cdot p_{3}=1$
for simplicity. This tensor satisfies the following relations:

$\displaystyle\langle\underline{\bm{1}}\underline{3}\rangle\langle\underline{\bm{2}}\underline{3}\rangle=-\frac{1}{2}\langle\underline{\bm{12}}\rangle\frac{\langle\underline{3}|1|q|\underline{3}\rangle}{p_{3}\cdot
q}\,,$
$\displaystyle\qquad[\underline{\bm{1}}\underline{3}\rangle[\underline{\bm{2}}\underline{3}\rangle=-\frac{1}{2}[\underline{\bm{12}}]\frac{\langle\underline{3}|1|q|\underline{3}\rangle}{p_{3}\cdot
q}\,,$ (80)
$\displaystyle[\underline{\bm{1}}\underline{3}\rangle\langle\underline{\bm{1}}\underline{3}\rangle=\frac{i}{4m}[\underline{\bm{1}}|3|\underline{\bm{1}}\rangle\frac{\langle\underline{3}|1|q|\underline{3}\rangle}{p_{3}\cdot
q}\,,$
$\displaystyle\qquad[\underline{\bm{2}}\underline{3}\rangle\langle\underline{\bm{2}}\underline{3}\rangle=-\frac{i}{4m}[\underline{\bm{2}}|3|\underline{\bm{2}}\rangle\frac{\langle\underline{3}|2|q|\underline{3}\rangle}{p_{3}\cdot
q}\,,$

that follow from generalized Schouten identities in five dimensions. One can
see that (80) has divided the four brackets appearing on the LHS into two
pairs

$\displaystyle\\{\langle\underline{\bm{1}}\underline{3}\rangle,[\underline{\bm{2}}\underline{3}\rangle\\}\,,\quad\\{[\underline{\bm{1}}\underline{3}\rangle,\langle\underline{\bm{2}}\underline{3}\rangle\\},$
(81)

where any two brackets from different pairs can be replaced by
$\frac{\langle\underline{3}|1|q|\underline{3}\rangle}{p_{3}\cdot q}$.
Therefore, we can write all amplitudes using
$\frac{\langle\underline{3}|1|q|\underline{3}\rangle}{p_{3}\cdot q}$ along
with only one set of the two pairs.

One might be worried whether we have exhausted all the little group covariant
structures. For example, the bracket
$\langle\underline{\bm{1}}|q|\underline{3}\rangle$ is definitely not
covariant, but could any polynomial made up of this type of structure become
covariant? To study this systematically, we have to define the $uw$-variables
of Cheung:2009dc ; Zhao:2023 .

####  $uw$-variables: ensuring spinor-space is spanned.

Due to the kinematic configuration, the following $2\times 2$ tensors of mass
dimension one are only half-rank

$\displaystyle\langle\underline{\bm{1}}\underline{3}\rangle\,,\quad[\underline{\bm{1}}\underline{3}\rangle\,,\quad\langle\underline{\bm{2}}\underline{3}\rangle\,,\quad[\underline{\bm{2}}\underline{3}\rangle\,,\quad\langle\underline{\bm{1}}\underline{\bm{2}}\rangle\,,\quad[\underline{\bm{12}}]\,.$
(82)

On the other hand, the tensors $\langle\underline{\bm{12}}]$ and
$[\underline{\bm{12}}\rangle$ have _full_ rank. Since the tensors listed in
(82) all have rank one, we can express them as a product of two two-component
objects with only little group indices. However, the same spinor helicity
variables $|3\rangle$ appearing in different tensors might correspond to
different two-component objects. For example, tensors
$\langle\underline{\bm{1}}|3_{I}\rangle$ and
$\langle\underline{\bm{2}}|3_{I}\rangle$ should be decomposed into different
spinors with label $I$, since contracting the spinor helicity variable
$|3_{I}\rangle$ in these two tensors is non-vanishing:
$\langle\underline{\bm{1}}|3^{I}\rangle\langle
3_{I}|\underline{\bm{2}}\rangle=2im\langle\underline{\bm{12}}\rangle$.
Therefore, we find the following decomposition

$\displaystyle\langle\mathbf{1}_{\alpha}|3_{I}\rangle=\bm{u}_{1,\alpha}\bar{u}_{3,I}\,,$
$\displaystyle[\mathbf{1}_{\dot{\alpha}}|3_{I}\rangle=\bar{\bm{u}}_{1,\dot{\alpha}}u_{3,I}\,,$
(83)
$\displaystyle[\mathbf{2}_{\dot{\beta}}|3_{I}\rangle=\bm{u}_{2,\dot{\beta}}\bar{u}_{3,I}\,,$
$\displaystyle\langle\mathbf{2}_{\beta}|3_{I}\rangle=\bar{\bm{u}}_{2,\beta}u_{3,I}\,,$
$\displaystyle\langle\mathbf{1}_{\alpha}|\mathbf{2}_{\beta}\rangle=\bm{u}_{1,\alpha}\bar{\bm{u}}_{2,\beta}\,,$
$\displaystyle[\mathbf{1}_{\dot{\alpha}}|\mathbf{2}_{\dot{\beta}}]=\bar{\bm{u}}_{1,\dot{\alpha}}\bm{u}_{2,\dot{\beta}}\,.$

Note that there is a notational discrepancy in defining $u$ variables for
particles 1 and 2, where we use $u_{1}$ to denote Weyl spinors and $u_{2}$ to
denote anti-Weyl spinors.

Next, we note that $u_{3}$ and $\bar{u}_{3}$ are linear independent

$u_{3,I}\bar{u}_{3,J}-u_{3,J}\bar{u}_{3,I}=-2m\varepsilon_{IJ}=-2miC_{IJ}\,,$
(84)

and, therefore, fully span the spinor space for particle 3. For particles-1
and -2, we define the “pseudo-inverses” of the $u$-variables:
$w_{1},w_{2},\bar{w}_{1},\bar{w}_{2}$. These pseudo-inverses are defined by
the following equations

$\bm{u}_{1,\alpha}\bar{\bm{w}}_{1}^{\alpha}=1\,,\quad\bar{\bm{u}}_{1,\dot{\alpha}}\bm{w}_{1}^{\dot{\alpha}}=1\,,\quad\bm{u}_{2,\dot{\beta}}\bar{\bm{w}}_{2}^{\dot{\beta}}=1\,,\quad\bar{\bm{u}}_{2,\beta}\bm{w}_{2}^{\beta}=1\,.$
(85)

and satisfy the following completeness relations

$\displaystyle\begin{aligned}
\bm{u}_{1,[\alpha}\bar{\bm{w}}_{1,\beta]}&=-\varepsilon_{\alpha\beta}=-C_{\alpha\beta}=\bar{\bm{u}}_{2,[\alpha}\bm{w}_{2,\beta]}\,,\\\
\bar{\bm{u}}_{1,[\dot{\alpha}}\bm{w}_{1,\dot{\beta}]}&=-\varepsilon_{\dot{\alpha}\dot{\beta}}=-C_{\dot{\alpha}\dot{\beta}}=\bm{u}_{2,[\dot{\alpha}}\bar{\bm{w}}_{2,\dot{\beta}]}\,.\end{aligned}$
(86)

Unfortunately, there is a gauge redundancy in the definition of the
$\bm{w}$-variables

$\bm{w}_{i}\to\bm{w}_{i}+b_{i}\bar{\bm{u}}_{i}\,,\quad\bar{\bm{w}}_{i}\to\bar{\bm{w}}_{i}+\bar{b}_{i}\bm{u}_{i}\,,$
(87)

where $b$’s and $\bar{b}$’s are pure numbers that satisfy
$\sum_{i}\bar{b}_{i}=0=\sum_{i}b_{i}$. Since the amplitude must be invariant
under these gauge transformations, we conclude that the only allowed tensors,
including $w$\- and $\bar{w}$-spinors, are

$\bm{u}_{1,\alpha}\bar{\bm{w}}_{2,\beta}+\bar{\bm{w}}_{1,\alpha}\bm{u}_{2,\beta}\,,\quad\bar{\bm{u}}_{1,\dot{\alpha}}\bm{w}_{2,\dot{\beta}}+\bm{w}_{1,\dot{\alpha}}\bar{\bm{u}}_{2,\dot{\beta}}\,.$
(88)

Written in terms of the usual spinor helicity variables,

$\langle\underline{\bm{12}}]=\underline{\bm{u}}_{1}\underline{\bm{\bar{w}}}_{2}+\underline{\bm{\bar{w}}}_{1}\underline{\bm{u}}_{2}\,,\quad[\underline{\bm{12}}\rangle=\underline{\bm{\bar{u}}}_{1}\underline{\bm{w}}_{2}+\underline{\bm{w}}_{1}\underline{\bm{\bar{u}}}_{2}\,,$
(89)

they are the familiar mass dimension one two-brackets. Here, we have
introduced the underlined notation for the auxiliary $u$\- and $w$\- spinors.
Explicitly,

$\displaystyle\begin{aligned} \underline{\bm{u}}_{1}=\bm{u}_{1,\alpha}\
z_{1}^{\alpha}\,,&\qquad\underline{\bm{\bar{u}}}_{1}=\bm{u}_{1,\dot{\alpha}}\
z_{1}^{\dot{\alpha}}\,,&\qquad\underline{\bm{\bar{w}}}_{1}=\bar{\bm{w}}_{1,\alpha}\
z_{1}^{\alpha}\,,&\qquad\underline{\bm{w}}_{1}=\bm{w}_{1,\dot{\alpha}}\
z_{1}^{\dot{\alpha}}\,,\\\ \underline{\bm{\bar{u}}}_{2}=\bm{u}_{2,\alpha}\
z_{2}^{\alpha}\,,&\qquad\underline{\bm{u}}_{2}=\bm{u}_{2,\dot{\alpha}}\
z_{2}^{\dot{\alpha}}&\qquad\underline{\bm{w}}_{2}=\bm{w}_{2,\alpha}\
z_{2}^{\alpha}\,,&\qquad\underline{\bm{\bar{w}}}_{2}=\bar{\bm{w}}_{2,\dot{\alpha}}\
z_{2}^{\dot{\alpha}}\,,\\\ &\qquad\underline{u}_{3}=u_{3,I}\
z_{3}^{I}\,,&\qquad\underline{\bar{u}}_{3}=u_{3,I}\
z_{3}^{I}\,.&\end{aligned}$ (90)

To obtain the full set of little group covariant terms, we also have to note
that all the $u$ and $w$ variables admit a global rescaling redundancy

$\displaystyle\underline{\bm{u}}_{i}\to
a\underline{\bm{u}}_{i}\,,\quad\underline{\bm{\bar{u}}}_{i}\to\frac{1}{a}\underline{\bm{\bar{u}}}_{i}\,,\quad\underline{\bm{w}}_{i}\to
a\underline{\bm{w}}_{i}\,,\quad\underline{\bm{\bar{w}}}_{i}\to\frac{1}{a}\underline{\bm{\bar{w}}}_{i}\,\quad\text{for}\quad
i=1,2\,,$ (91) $\displaystyle\underline{u}_{3}\to
a\underline{u}_{3}\,,\quad\underline{\bar{u}}_{3}\to\frac{1}{a}\underline{\bar{u}}_{3}\,.$

Thus, every little group covariant structure must include the same number of
bar- and unbar-variables. This leaves us with the following seeds to build an
ansatz for the amplitude

$\underline{\bm{u}}_{1}\underline{\bm{\bar{w}}}_{2}+\underline{\bm{\bar{w}}}_{1}\underline{\bm{u}}_{2}\,,\quad\underline{\bm{\bar{u}}}_{1}\underline{\bm{w}}_{2}+\underline{\bm{w}}_{1}\underline{\bm{\bar{u}}}_{2}\,,\quad\underline{\bm{u}}_{i}\underline{\bm{\bar{u}}}_{j}\,,$
(92)

where $i=j$ is allowed and we drop the boldface typesetting on the $u$’s when
$i,j=3$. Therefore, we see that the space of five-dimensional three-point
amplitudes with two equal masses is similar to the space of massless three-
point amplitudes in six dimensions Cheung:2009dc . The only difference is that
the inverse of $u_{3}$ is fixed and has no gauge redundancy.

For completeness, we give a dictionary between $uw$-variables defined here and
the usual spinor helicity variables in Table 1. In particular, note that the
$q$-dependent tensor introduced in the pedestrian approach is a gauge
invariant combination of the $u$’s:
$\langle{}_{I}3|1|q|3_{J}\rangle/(p_{3}\cdot
q)=u_{3,I}\bar{u}_{3,J}+\bar{u}_{3,I}u_{3,J}$. Since the $u$\- and $w$-spinors
span the spinor space for all particles, we see from Table 1 that (64) and
(79) exhausts all of the independent tensor structures.

in spinor helicity variables | in $uw$-variables
---|---
$\langle\underline{\bm{1}}\underline{3}\rangle$ | $\underline{\bm{u}}_{1}\underline{\bar{u}}_{3}$
$[\underline{\bm{1}}\underline{3}\rangle$ | $\underline{\bm{\bar{u}}}_{1}\underline{u}_{3}$
$[\underline{\bm{2}}\underline{3}\rangle$ | $\underline{\bm{u}}_{2}\underline{\bar{u}}_{3}$
$\langle\underline{\bm{2}}\underline{3}\rangle$ | $\underline{\bm{\bar{u}}}_{2}\underline{u}_{3}$
$\langle\underline{\bm{1}}\underline{\bm{2}}\rangle$ | $\underline{\bm{u}}_{1}\underline{\bm{\bar{u}}}_{2}$
$[\underline{\bm{1}}\underline{\bm{2}}]$ | $\underline{\bm{\bar{u}}}_{1}\underline{\bm{u}}_{2}$
$\langle\underline{\bm{1}}\underline{\bm{2}}]$ | $\underline{\bm{u}}_{1}\underline{\bm{\bar{w}}}_{2}+\underline{\bm{\bar{w}}}_{1}\underline{\bm{u}}_{2}$
$[\underline{\bm{1}}\underline{\bm{2}}\rangle$ | $\underline{\bm{\bar{u}}}_{1}\underline{\bm{w}}_{2}+\underline{\bm{w}}_{1}\underline{\bm{\bar{u}}}_{2}$
$\langle\underline{\bm{1}}|3|\underline{\bm{1}}]$ | $\underline{\bm{u}}_{1}\underline{\bm{\bar{u}}}_{1}$
$[\underline{\bm{2}}|3|\underline{\bm{2}}\rangle$ | $\underline{\bm{u}}_{2}\underline{\bm{\bar{u}}}_{2}$
$\langle\underline{\bm{1}}|q|3|\underline{\bm{2}}]$ | $\underline{\bm{u}}_{1}\underline{\bm{\bar{w}}}_{2}$
$[\underline{\bm{2}}|q|3|\underline{\bm{1}}\rangle=-\langle\underline{\bm{1}}|q|3|\underline{\bm{2}}]$ | $\underline{\bm{\bar{w}}}_{1}\underline{\bm{u}}_{2}$
$\dfrac{\langle\underline{3}|1|q|\underline{3}\rangle}{p_{1}\cdot q}$ | $\underline{u}_{3}\underline{\bar{u}}_{3}+\underline{\bar{u}}_{3}\underline{u}_{3}$
$\dfrac{\langle\underline{3}|\hat{1}|q|\underline{3}\rangle}{p_{1}\cdot q}$ | $\underline{u}_{3}\underline{\bar{u}}_{3}$
$\dfrac{\langle\underline{3}|\bar{1}|q|\underline{3}\rangle}{p_{1}\cdot q}$ | $\underline{\bar{u}}_{3}\underline{u}_{3}$
Table 1: The map between spinor helicity variables and $u,w$ representation in
two massive case with equal mass where $|\hat{1}|:=|1\rangle\langle 1|$ and
$|\bar{1}|:=|1][1|$. Furthermore, note that the correspondence in the above
table is up to constant multiplicitive factors.

The $uw$-variables also yield a better understanding of the identities (80).
All the brackets appearing there are half-rank, and decomposing them into $u$
variables trivializes the identities. In other words, these identities are
just collecting pairs of $(u_{3},\bar{u}_{3})$ together to form $\frac{\langle
3|1|q|3\rangle}{p_{3}\cdot q}$.

#### The amplitudes.

Now that we are confident that we are not missing any tensors, we can build an
ansatz for the amplitudes using either the pedestrian or $uw$ variables. We
will proceed with the construction in $uw$-variables. As in section 3.2, the
spin of particle-1 and -2 are labeled by $(s_{1},\bar{s}_{1})$ and
$(s_{2},\bar{s}_{2})$ while $S_{3}$ labels the spin of particle-3. Also we
define $S_{1}=s_{1}+\bar{s}_{1}$ and $S_{2}=s_{2}+\bar{s}_{2}$. Then, little
group covariance implies that

$A(\mathbf{1}^{s_{1}+\bar{s}_{1}},\mathbf{2}^{s_{2}+\bar{s}_{2}},3^{S_{3}})=\left(\underline{\bm{u}}_{1}\underline{\bm{\bar{w}}}_{2}+\underline{\bm{\bar{w}}}_{1}\underline{\bm{u}}_{2}\right)^{v_{0}}\left(\underline{\bm{\bar{u}}}_{1}\underline{\bm{w}}_{2}+\underline{\bm{w}}_{1}\underline{\bm{\bar{u}}}_{2}\right)^{\bar{v}_{0}}\underline{\bm{u}}_{1}^{v_{1}}\underline{\bm{\bar{u}}}_{1}^{\bar{v}_{1}}\underline{\bm{u}}_{2}^{v_{2}}\underline{\bm{\bar{u}}}_{2}^{\bar{v}_{2}}\underline{u}_{3}^{v_{3}}\underline{\bar{u}}_{3}^{\bar{v}_{3}}$
(93)

with

$\displaystyle v_{0}+v_{1}=2s_{1}\,,$
$\displaystyle\bar{v}_{0}+\bar{v}_{1}=2\bar{s}_{1}\,,\qquad$ $\displaystyle
v_{0}+v_{2}=2\bar{s}_{2}\,,$ (94)
$\displaystyle\bar{v}_{0}+\bar{v}_{2}=2s_{2}\,,$ $\displaystyle
v_{3}+\bar{v}_{3}=2S_{3}\,,\qquad$ $\displaystyle
v_{1}+v_{2}+v_{3}=\bar{v}_{1}+\bar{v}_{2}+\bar{v}_{3}\,.$

Since there are eight indeterminates but only six equations, the solution
space of these equations is two-dimensional. In the following, we choose
$v_{0}$ and $\bar{v}_{0}$ to be free and fix all other variables.

Imposing all $v$’s and $\bar{v}$’s to be non-negative puts constraints on both
the spins and the tuple $(v_{0},\bar{v}_{0})$. On the one hand, the
constraints on spin configuration are

$\left\\{\begin{aligned}
S_{3}+\bar{s}_{1}+s_{2}\geqslant|s_{1}-\bar{s}_{2}|\,,\\\
S_{3}+s_{1}+\bar{s}_{2}\geqslant|\bar{s}_{1}-s_{2}|\,,\end{aligned}\right.$
(95)

and imply that any number among $s_{1},\bar{s}_{1},s_{2},\bar{s}_{2}$ cannot
be larger than the sum of the rest plus $S_{3}$. Compared with the unequal
mass case, these spin configurations can break the pentagon rule since the
relation $S_{3}\leqslant s_{1}+\bar{s}_{1}+s_{2}+\bar{s}_{2}$ is not required.

$\min(2\bar{s}_{1},2s_{2})$$\bar{v}_{0}$$\min(2s_{1},2\bar{s}_{2})$$v_{0}$$S_{3}-s_{1}+\bar{s}_{1}+s_{2}-\bar{s}_{2}\
$$S_{3}+s_{1}-\bar{s}_{1}-s_{2}+\bar{s}_{2}$ Figure 3: The allowed region of
integer solutions for the exponents $v_{0}$ and $\bar{v}_{0}$ in the ansatz
for the $(m_{1},m_{2},m_{3})=(m,m,0)$ amplitudes in equation (93).

On the other hand, from the constraints on the tuple $(v_{0},\bar{v}_{0})$ we
can obtain the number of independent couplings. As depicted in figure 3, the
allowed values for $(v_{0},\bar{v}_{0})$ are the integer points in the shaded
area (including the boundary). Note that the graph above just corresponds to
one type of solution, in the sense that the relation between the two lines and
the square is more general. For example, they can be either below or above the
diagonal of the square or totally not intersecting with the square in some
spin configurations.

#### Example ($\bm{WWA}$-scattering):

In our formalism, the amplitude for the scattering of two $W$-bosons (massive
spin-one particles) with a massless spin-one particle ($A$) is

$\displaystyle
A(W,W,A)=A(\mathbf{1}^{\frac{1}{2}+\bar{\frac{1}{2}}},\mathbf{2}^{\frac{1}{2}+\bar{\frac{1}{2}}},3^{1})=\frac{\langle\underline{3}|1|q|\underline{3}\rangle}{p_{3}\cdot
q}\langle\underline{\bm{12}}][\underline{\bm{12}}\rangle.$ (96)

Using the identities (80), one can show that this is equivalent to the
expression in Chiodaroli:2022ssi .

#### Example ($\bm{\chi\bar{\chi}A}$\- and $\bm{\chi\chi A}$-scattering):

Here, we consider the scattering between massive spin-$\frac{1}{2}$ particles
($\chi$ and $\bar{\chi}$) and a massless vector boson ($A$). Depending on the
chirality of the two fermions, there are two configurations.

The first corresponds to a chirality-preserving interaction

$\displaystyle\begin{aligned}
A(\chi,\bar{\chi},A)=A(\mathbf{1}^{\frac{1}{2}},\mathbf{2}^{\bar{\frac{1}{2}}},3^{1})&=g_{0,0}\underline{\bm{u}}_{1}\underline{\bm{u}}_{2}\underline{\bar{u}}_{3}^{2}+g_{1,0}\left(\underline{\bm{u}}_{1}\underline{\bm{\bar{w}}}_{2}+\underline{\bm{\bar{w}}}_{1}\underline{\bm{u}}_{2}\right)\underline{u}_{3}\underline{\bar{u}}_{3}\\\
&\sim
g_{0,0}\langle\underline{\bm{1}}\underline{3}\rangle[\underline{\bm{2}}\underline{3}\rangle+g_{1,0}\langle\underline{\bm{1}}\underline{\bm{2}}]\langle\underline{\bm{3}}|1|q|\underline{\bm{3}}\rangle\\\
&\sim
g_{0,0}\langle\underline{\bm{1}}\underline{3}\rangle[\underline{\bm{2}}\underline{3}\rangle+g_{1,0}\left(\langle\underline{\bm{1}}\underline{3}\rangle[\underline{\bm{2}}|q|\underline{3}\rangle-[\underline{\bm{2}}\underline{3}\rangle\langle\underline{\bm{1}}|q|\underline{3}\rangle\right)\,.\end{aligned}$
(97)

Here, we have absorbed any proportionality constants converting from the $uw$
variables to the normal spinor helicity variables into the couplings $g_{0,0}$
and $g_{1,0}$. In fact, the term proportional to $g_{1,0}$ corresponds to the
standard interaction of two spinors with massless gauge bosons and is exactly
the amplitude found in Chiodaroli:2022ssi . On the other hand, the term
proportional to $g_{0,0}$ corresponds to some higher derivative deformations
in Lagrangian.

The second configuration

$\displaystyle\begin{aligned}
A(\chi,\chi,A)=A(\mathbf{1}^{\frac{1}{2}},\mathbf{2}^{\frac{1}{2}},3^{1})&=g^{\prime}_{0,0}\underline{\bm{u}}_{1}\underline{\bm{\bar{u}}}_{2}\underline{u}_{3}\underline{\bar{u}}_{3}\sim
g^{\prime}_{0,0}\langle\underline{\bm{1}}\underline{\bm{2}}\rangle\langle\underline{3}|1|q|\underline{3}\rangle\sim
g^{\prime}_{0,0}\langle\underline{\bm{1}}\underline{3}\rangle\langle\underline{\bm{2}}\underline{3}\rangle\,\end{aligned}$
(98)

does not preserve chirality. Moreover, no $g^{\prime}_{1,0}$ term exists
because gauge theory interactions always preserve chirality. On the other
hand, the $g^{\prime}_{0,0}$ term in this chirality-breaking configuration
implies that there could be higher-derivative interactions that do not
preserve chirality.

#### Arbitrary
$(\underline{\bm{1}}^{S},\underline{\bm{2}}^{S},\underline{3}^{h})$-scattering.

To study black hole scattering, we need to understand the configuration
$s_{1}=\bar{s}_{1}=s_{2}=\bar{s}_{2}=\frac{1}{2}S$, where we denote the spin
of particle-1 and particle-2 by $S$. We also set $S_{3}=h$ as the helicity of
particle-3. According to figure 3, the independent amplitudes are given by the
intersection of a square with edge length $S$ and the region between the two
oblique lines $v_{0}-\bar{v}_{0}=h$ and $v_{0}-\bar{v}_{0}=-h$. When
$S\leqslant h$, the full square is between the two oblique lines, so the
allowed region for $(v_{0},\bar{v}_{0})$ is the full square, and there are
$(S+1)^{2}$ independent couplings

$\displaystyle
A(\underline{\bm{1}}^{S},\underline{\bm{2}}^{S},\underline{3}^{h})=\sum_{v_{0},\bar{v}_{0}=0}^{S}g_{v_{0},\bar{v}_{0}}\left(\underline{\bm{u}}_{1}\underline{\bm{\bar{w}}}_{2}{+}\underline{\bm{\bar{w}}}_{1}\underline{\bm{u}}_{2}\right)^{v_{0}}\left(\underline{\bm{\bar{u}}}_{1}\underline{\bm{w}}_{2}{+}\underline{\bm{w}}_{1}\underline{\bm{\bar{u}}}_{2}\right)^{\bar{v}_{0}}$
(99) $\displaystyle\hskip
100.00015pt\times\underline{\bm{u}}_{1}^{S-v_{0}}\underline{\bm{\bar{u}}}_{1}^{S-\bar{v}_{0}}\underline{\bm{u}}_{2}^{S-v_{0}}\underline{\bm{\bar{u}}}_{2}^{S-\bar{v}_{0}}\underline{u}_{3}^{h+v_{0}-\bar{v}_{0}}\underline{\bar{u}}_{3}^{h-v_{0}+\bar{v}_{0}}$
$\displaystyle=\sum_{\begin{subarray}{c}v_{0}<\bar{v}_{0}=0\\\
v_{0}<\bar{v}_{0}\end{subarray}}^{S}g_{v_{0},\bar{v}_{0}}\langle\underline{\bm{1}}\underline{\bm{2}}]^{v_{0}}[\underline{\bm{1}}\underline{\bm{2}}\rangle^{\bar{v}_{0}}\langle\underline{\bm{1}}\underline{\bm{2}}\rangle^{S-\bar{v}_{0}}[\underline{\bm{1}}\underline{\bm{2}}]^{S-\bar{v}_{0}}\langle\underline{\bm{1}}\underline{3}\rangle^{\bar{v}_{0}-v_{0}}[\underline{\bm{2}}\underline{3}\rangle^{\bar{v}_{0}-v_{0}}\langle\underline{3}|1|q|\underline{3}\rangle^{h+v_{0}-\bar{v}_{0}}$
$\displaystyle\quad+\sum_{\begin{subarray}{c}v_{0},\bar{v}_{0}=0\\\
v_{0}\geqslant\bar{v}_{0}\end{subarray}}^{S}g_{v_{0},\bar{v}_{0}}\langle\underline{\bm{1}}\underline{\bm{2}}]^{v_{0}}[\underline{\bm{1}}\underline{\bm{2}}\rangle^{\bar{v}_{0}}\langle\underline{\bm{1}}\underline{\bm{2}}\rangle^{S-v_{0}}[\underline{\bm{1}}\underline{\bm{2}}]^{S-v_{0}}[\underline{\bm{1}}\underline{3}\rangle^{v_{0}-\bar{v}_{0}}\langle\underline{\bm{2}}\underline{3}\rangle^{v_{0}-\bar{v}_{0}}\langle\underline{3}|1|q|\underline{3}\rangle^{h-v_{0}+\bar{v}_{0}}\,.$

When $S>h$, the available region then becomes a hexagon and there are
$(S+1)^{2}-(S-h)^{2}-(S-h)$ independent couplings

$\displaystyle
A(\underline{\bm{1}}^{S},\underline{\bm{2}}^{S},\underline{3}^{h})=\sum_{\begin{subarray}{c}v_{0},\bar{v}_{0}=0\\\
|v_{0}-\bar{v}_{0}|\leqslant
h\end{subarray}}^{S}g_{v_{0},\bar{v}_{0}}\left(\underline{\bm{u}}_{1}\underline{\bm{\bar{w}}}_{2}{+}\underline{\bm{\bar{w}}}_{1}\underline{\bm{u}}_{2}\right)^{v_{0}}\left(\underline{\bm{\bar{u}}}_{1}\underline{\bm{w}}_{2}{+}\underline{\bm{w}}_{1}\underline{\bm{\bar{u}}}_{2}\right)^{\bar{v}_{0}}$
(100) $\displaystyle\hskip
100.00015pt\times\underline{\bm{u}}_{1}^{S-v_{0}}\underline{\bm{\bar{u}}}_{1}^{S-\bar{v}_{0}}\underline{\bm{u}}_{2}^{S-v_{0}}\underline{\bm{\bar{u}}}_{2}^{S-\bar{v}_{0}}\underline{u}_{3}^{h+v_{0}-\bar{v}_{0}}\underline{\bar{u}}_{3}^{h-v_{0}+\bar{v}_{0}}$
$\displaystyle=\sum_{\begin{subarray}{c}v_{0}<\bar{v}_{0}=0\\\
v_{0}<\bar{v}_{0}\leqslant
v_{0}+h\end{subarray}}^{S}g_{v_{0},\bar{v}_{0}}\langle\underline{\bm{1}}\underline{\bm{2}}]^{v_{0}}[\underline{\bm{1}}\underline{\bm{2}}\rangle^{\bar{v}_{0}}\langle\underline{\bm{1}}\underline{\bm{2}}\rangle^{S-\bar{v}_{0}}[\underline{\bm{1}}\underline{\bm{2}}]^{S-\bar{v}_{0}}\langle\underline{\bm{1}}\underline{3}\rangle^{\bar{v}_{0}-v_{0}}[\underline{\bm{2}}\underline{3}\rangle^{\bar{v}_{0}-v_{0}}\langle\underline{3}|1|q|\underline{3}\rangle^{h+v_{0}-\bar{v}_{0}}$
$\displaystyle\quad{+}\sum_{\begin{subarray}{c}v_{0},\bar{v}_{0}=0\\\
\bar{v}_{0}\leqslant
v_{0}\leqslant\bar{v}_{0}+h\end{subarray}}^{S}g_{v_{0},\bar{v}_{0}}\langle\underline{\bm{1}}\underline{\bm{2}}]^{v_{0}}[\underline{\bm{1}}\underline{\bm{2}}\rangle^{\bar{v}_{0}}\langle\underline{\bm{1}}\underline{\bm{2}}\rangle^{S-v_{0}}[\underline{\bm{1}}\underline{\bm{2}}]^{S-v_{0}}[\underline{\bm{1}}\underline{3}\rangle^{v_{0}-\bar{v}_{0}}\langle\underline{\bm{2}}\underline{3}\rangle^{v_{0}-\bar{v}_{0}}\langle\underline{3}|1|q|\underline{3}\rangle^{h-v_{0}+\bar{v}_{0}}\,.$

###  3.4 One massive and two massless

In this section, we set $m_{1}=m_{2}=0$, and $m_{3}\neq 0$ (note that
particle-3 is massive now!). Since all brackets are full rank, we do not need
to introduce $uw$-variables and can proceed as in sections 3.1 and 3.2. For
this kinematic configuration, the space of amplitudes splits into two sectors

$\displaystyle
A(\underline{1}^{S_{1}},\underline{2}^{S_{2}},\underline{\bm{3}}^{s_{3}+\bar{s}_{3}})$
$\displaystyle=\begin{cases}\begin{aligned} \sum_{v\in
R_{1}}g_{v}\bigg{[}&\langle\underline{1}\underline{\bm{3}}\rangle^{v}\langle\underline{2}\underline{\bm{3}}\rangle^{s_{3}-\bar{s}_{3}+S_{1}+S_{2}-v}\langle\underline{1}\underline{\bm{3}}]^{2S_{1}-v,}\\\
&\quad\times\langle\underline{2}\underline{\bm{3}}]^{\bar{s}_{3}-s_{3}-S_{1}+S_{2}+v}\bigg{]}\langle\underline{\bm{3}}|1|\underline{\bm{3}}]^{S_{3}-S_{1}-S_{2}}\end{aligned}&\text{
if }S_{3}\geq S_{1}+S_{2}\,,\\\ \begin{aligned} \sum_{v\in
R_{2}}g_{v}\bigg{[}&\langle\underline{1}\underline{\bm{3}}\rangle^{v}\langle\underline{2}\underline{\bm{3}}\rangle^{2s_{3}-v}\langle\underline{1}\underline{\bm{3}}]^{s_{3}+\bar{s}_{3}+S_{1}-S_{2}-v}\\\
&\quad\times\langle\underline{2}\underline{\bm{3}}]^{\bar{s}_{3}-s_{3}-S_{1}+S_{2}+v}\bigg{]}\langle\underline{12}\rangle^{S_{1}+S_{2}-S_{3}}\end{aligned}&\text{
if }S_{1}+S_{2}\geq S_{3}\,,\end{cases}$ (101)

where

$\displaystyle R_{1}$
$\displaystyle=\left[\max\left(0,s_{3}{-}\bar{s}_{3}{+}S_{1}{-}S_{2}\right),\min\left(2S_{1},s_{3}{-}\bar{s}_{3}{+}S_{1}{+}S_{2}\right)\right]\cap\mathbb{Z}$
(102) $\displaystyle R_{2}$
$\displaystyle=\left[\max\left(0,s_{3}{-}\bar{s}_{3}{+}S_{1}{-}S_{2}\right),\min\left(s_{3},s_{3}{+}\bar{s}_{3}{+}S_{1}{-}S_{2}\right)\right]\cap\mathbb{Z}$
(103)

are the lattice points inside a line segment. Note that for this formula to
make sense, the $\\{S_{1},S_{2},s_{3},\bar{s}_{3}\\}$ must satisfy additional
constraints

$\displaystyle S_{1}+S_{2}+S_{3}\in\mathbb{N}_{\geq 0}\,,\qquad$
$\displaystyle S_{1}+S_{2}+s_{3}\geq\bar{s}_{3}\,,\qquad$ $\displaystyle
S_{1}+S_{2}+\bar{s}_{3}\geq s_{3}\,,$ (104) $\displaystyle
S_{1}+s_{3}+\bar{s}_{3}\geq S_{2}\,,\qquad$ $\displaystyle
S_{2}+s_{3}+\bar{s}_{3}\geq S_{1}.$

These constraints mean that there are an even number of spinor brackets and
that the $\\{S_{1},S_{2},s_{3},\bar{s}_{3}\\}$ form a (possibly degenerate)
quadrilateral on the plane. Thus, for any set
$\\{S_{1},S_{2},s_{3},\bar{s}_{3}\\}$ that satisfies (104) the number of
independent couplings is simply the size of the set $|R_{1}|$ when $S_{3}\geq
S_{1}+S_{2}$ and $|R_{2}|$ when $S_{3}<S_{1}+S_{2}$. We have performed an
explicit numerical check that all of the above component amplitudes are
linearly independent for $S_{i}\leq 4$.

#### Example ($\bm{\psi\psi W}$-scattering):

Consider the scattering of two massless fermions $\psi$
($S_{1}=S_{2}=\frac{1}{2}$) with a massive $W$-boson
($S_{3}=\frac{1}{2}+\bar{\frac{1}{2}}$). Then, equation (101) becomes

$\displaystyle
A(\psi,\psi,W)=A(\underline{1}^{\frac{1}{2}},\underline{2}^{\frac{1}{2}},\underline{\bm{3}}^{\frac{1}{2}+\bar{\frac{1}{2}}})$
$\displaystyle=g_{1}\
\langle\underline{1}\underline{\bm{3}}\rangle\langle\underline{2}\underline{\bm{3}}]+g_{2}\
\langle\underline{2}\underline{\bm{3}}\rangle\langle\underline{1}\underline{\bm{3}}].$
(105)

It is then easy to verify that the above two-component amplitudes are linearly
independent.

#### Example (higher spin bosons):

Consider the scattering of higher spin bosons with $S_{1}=S_{2}=4$ and
$S_{3}=4=2+\bar{2}$. Then, (101) becomes

$\displaystyle\begin{aligned}
A(\underline{1}^{4},\underline{2}^{4},\underline{\bm{3}}^{2+\bar{2}})&=g_{1}\
\langle\underline{12}\rangle^{4}\langle\underline{1}\underline{\bm{3}}\rangle^{4}\langle\underline{2}\underline{\bm{3}}]^{4}+g_{2}\
\langle\underline{12}\rangle^{4}\langle\underline{1}\underline{\bm{3}}\rangle^{3}\langle\underline{1}\underline{\bm{3}}]\langle\underline{2}\underline{\bm{3}}]^{3}\langle\underline{2}\underline{\bm{3}}\rangle\\\
&\qquad+g_{3}\
\langle\underline{12}\rangle^{4}\langle\underline{1}\underline{\bm{3}}\rangle^{2}\langle\underline{1}\underline{\bm{3}}]^{2}\langle\underline{2}\underline{\bm{3}}]^{2}\langle\underline{2}\underline{\bm{3}}\rangle^{2}+g_{4}\
\langle\underline{12}\rangle^{4}\langle\underline{1}\underline{\bm{3}}\rangle\langle\underline{1}\underline{\bm{3}}]^{3}\langle\underline{2}\underline{\bm{3}}]\langle\underline{2}\underline{\bm{3}}\rangle^{3}\\\
&\qquad+g_{5}\
\langle\underline{12}\rangle^{4}\langle\underline{1}\underline{\bm{3}}]^{4}\langle\underline{2}\underline{\bm{3}}\rangle^{4}\,.\end{aligned}$
(106)

One can also show that these five component amplitudes are linearly
independent.

###  3.5 Three massless

In this section, we set all masses to zero: $m_{1}=m_{2}=m_{3}=0$. Then, all
particles are associated to an $\operatorname{SO}(3)$-bracket
$|\underline{i}\rangle$. As usual, the first step is to write down the allowed
amplitudes, we enumerate all possible tensor structures. Because this
kinematic configuration is degenerate, we make some of the steps explicit. The
full set of independent building blocks for all-massless amplitudes can be
found in table 2 in both spinor helicity and $uw$-variables.

At mass dimension one, we have the following basic tensors two-brackets
$\langle\underline{ij}\rangle$. However, since the fully massless
configuration is kinematically degenerate, we need to introduce new mass
dimension one tensors called $x$-factors that parameterize the ratio of two
collinear spinor brackets. For example, in the all-massless configuration, the
spinors $|1|3\rangle$ and $|3\rangle$ are parallel

$\langle 3_{I}|1|3_{J}\rangle\sim\varepsilon_{IJ}p_{1}\cdot p_{3}=0\,.$ (107)

Therefore, we define their “ratio” as a new kinematic variable

$(x_{i})_{IJ}:=\left(\langle i\,q_{i,I}\rangle^{-1}\right)^{K}\langle
q_{i,K}|p_{i+1}|i_{J}\rangle=\frac{\langle
i_{I}|q_{i}|p_{i+1}|i_{J}\rangle}{p_{i}\cdot q_{i}}\propto u_{i,I}u_{i,J}\,.$
(108)

where $q_{i}$ is an arbitrary reference vector. In index-free notation this is

$x_{i}=\langle\underline{i}|q_{i}|p_{i+1}|\underline{i}\rangle\propto\underline{u_{i}}\underline{u_{i}}.$
(109)

This $x$-factor is simply a five-dimensional generalization of the four-
dimensional $x$-factor of Arkani-Hamed:2017jhn ; Zhao:2023 . The fact that it
degenerates and is not an additional independent tensor is a reflection of the
fact that minimal coupling is different in five dimensions than in four
Chung:2019yfs ; Zhao:2023 .

At mass dimension two, there are no non-vanishing tensors that do not contain
the reference momenta $q_{i}$ since

$\displaystyle\langle\underline{3}|1|\underline{3}\rangle=0\quad\text{and}\quad\langle\underline{3}|1|\underline{2}\rangle=-\langle\underline{3}|2|\underline{2}\rangle-\langle\underline{3}|1|\underline{2}\rangle=0\,.$
(110)

At mass dimension three, we find one gauge invariant (QED) term

$\langle\underline{1}\underline{3}\rangle\langle\underline{2}|q_{3}|\underline{3}\rangle-\langle\underline{1}|q_{3}|\underline{3}\rangle\langle\underline{2}\underline{3}\rangle\,,$
(111)

that cannot be represented in terms of $x$-factors and two-brackets
$\langle\underline{ij}\rangle$.

Without performing an analysis counting the number of independent amplitudes,
we cannot be sure that we have exhausted all of the independent tensor
structures. However, as shown in Zhao:2023 , the $uw$-variables span all of
spinor space. Therefore, by providing a map from the spinor helicity variables
to the $uw$ variables in table 2, we are sure we have not missed any tensor
structures. In the end, the full list of tensor structures includes the two-
brackets $\langle\underline{ij}\rangle$, (111) along with a Yang-Mills term
(112).

$q$-representation |  $uw$-variables
---|---
$\langle\underline{1}\underline{3}\rangle$ | $\underline{u}_{1}\underline{u}_{3}$
$\langle\underline{3}\underline{1}\rangle$ | $-\underline{u}_{3}\underline{u}_{1}$
$\langle\underline{1}|q_{1}|2|\underline{1}\rangle$ | $\underline{u}_{1}\underline{u}_{1}$
$\frac{\langle\underline{1}|q_{1}-(p_{3}\cdot q_{1})q_{3}|\underline{3}\rangle}{1-(p_{3}\cdot q_{1})(p_{1}\cdot q_{3})}$ | $\underline{u}_{1}\underline{w}_{3}$
$\langle\underline{1}\underline{3}\rangle\langle\underline{2}|q_{3}|\underline{3}\rangle-\langle\underline{2}\underline{3}\rangle\langle\underline{1}|q_{3}|\underline{3}\rangle$ | $(\underline{u}_{1}\underline{u}_{2}\underline{w}_{3}+\underline{u}_{1}\underline{w}_{2}\underline{u}_{3}+\underline{w}_{1}\underline{u}_{2}\underline{u}_{3})\underline{u}_{3}$
Table 2: The map between $q$ representation and $u,w$ representation.

#### Example ($\bm{AAA}$-scattering in Yang-Mills):

For massless matter, our spinor-helicity variables are equivalent to those of
Chiodaroli:2022ssi . The Yang-Mills 3-gluon amplitude is

$A(A,A,A)=A(\underline{1}^{1},\underline{2}^{1},\underline{3}^{1})=\langle\underline{2q_{3}}\rangle\langle\underline{3q_{2}}\rangle\langle\underline{1}|k_{2}|\underline{q_{1}}\rangle+\text{cyclic}$
(112)

This can be written as
$(\underline{u}_{1}\underline{u}_{2}\underline{w}_{3}+\underline{u}_{1}\underline{w}_{2}\underline{u}_{3}+\underline{w}_{1}\underline{u}_{2}\underline{u}_{3})^{2}$
in $uw$-variables Zhao:2023 ; Chiodaroli:2022ssi .

#### Example ($\bm{\psi\psi A}$-scattering in massless QED):

The coupling of massless fermions ($\psi$) to a massless vector boson ($A$) is

$A(\psi,\psi,A)=A(\underline{1}^{\frac{1}{2}},\underline{2}^{\frac{1}{2}},\underline{3}^{1})=\langle\underline{1}\underline{3}\rangle\langle\underline{2}|q_{3}|\underline{3}\rangle-\langle\underline{1}|q_{3}|\underline{3}\rangle\langle\underline{2}\underline{3}\rangle\,.$
(113)

This is (111) and can be written as
$(\underline{u}_{1}\underline{u}_{2}\underline{w}_{3}+\underline{u}_{1}\underline{w}_{2}\underline{u}_{3}+\underline{w}_{1}\underline{u}_{2}\underline{u}_{3})\underline{u}_{3}$
in the massless $uw$-variables Zhao:2023 ; Chiodaroli:2022ssi .

##  4 High energy limit

In this section, we provide a prescription for taking the high energy limit of
the massive spinor helicity variables (section 4.1) and massive amplitudes
(section 4.2). In particular, intricate cancellations are often required to
arrive at gauge invariant amplitudes in the high energy limit (see (126) and
(130)). Lastly, at the end of section 4.2, we show how spin and mass
dimensions of massive amplitudes are not preserved when taking the high energy
limit and propose a way in which the massive amplitudes can be categorized by
their high energy limit.

###  4.1 High energy limit of the spinor helicity variables

Since any massive momentum $p^{\mu}$ ($p^{2}=-m^{2}$) can be written as the
sum of two null momenta $k^{\mu},q^{\mu}$

$p^{\mu}=k^{\mu}+m^{2}q^{\mu}\,,\qquad k\cdot q=-\frac{1}{2}\,,$ (114)

one can decompose the massive spinor helicity variables into massless $k$ and
$q$ spinor helicity variables

$\displaystyle|p_{\alpha}\rangle=(|k_{I}\rangle+m|q_{I}\rangle)U^{I}_{\enspace\alpha}\,,\qquad|p_{\dot{\alpha}}]=(|k_{I}\rangle-m|q_{I}\rangle)U^{I}_{\enspace\dot{\alpha}}\,,$
(115)

where

$\displaystyle\langle k_{I}|q_{J}\rangle=\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}=\varepsilon_{IJ}\,,\quad\text{and}\quad
U^{I}_{\enspace\alpha}=U^{I}_{\enspace\dot{\alpha}}=\frac{1}{\sqrt{2}}\begin{pmatrix}i&0\\\
0&1\end{pmatrix}\,.$ (116)

Here, $U^{I}_{\alpha}$ and $U^{I}_{\dot{\alpha}}$ are a little group
conversion matrices that transform the $\operatorname{SO}(4)$ $C$-matrix into
the $\operatorname{SO}(3)$ $C$-matrix

$\displaystyle U^{I}_{\enspace\dot{\alpha}}\ U^{J}_{\enspace\dot{\beta}}\
C^{\dot{\beta}\dot{\alpha}}$ $\displaystyle=U^{I}_{\enspace\dot{\alpha}}\
U^{J}_{\enspace\dot{\beta}}\
\epsilon^{\dot{\beta}\dot{\alpha}}=-\frac{i}{2}\epsilon^{IJ}=-\frac{1}{2}C^{IJ}\,,$
(117) $\displaystyle U^{I}_{\enspace\alpha}\ U^{J}_{\enspace\beta}\
C^{\beta\alpha}$ $\displaystyle=U^{I}_{\enspace\alpha}\ U^{J}_{\enspace\beta}\
\epsilon^{\beta\alpha}=-\frac{i}{2}\epsilon^{IJ}=-\frac{1}{2}C^{IJ}\,.$ (118)

While the $U$-matrices are strictly needed, we will often drop them because
they can be easily restored.

To find an explicit parameterization for the $k$ and $q$ spinors, we fix

$p^{\mu}=(E,\sqrt{E^{2}-m^{2}},0,0,0)\,.$ (119)

Then, a convenient choice for $k$ and $q$ is

$\displaystyle
k^{\mu}=\frac{1}{2}\left(E+\sqrt{E^{2}-m^{2}},E+\sqrt{E^{2}-m^{2}},0,0,0\right)\,,$
(120) $\displaystyle
q^{\mu}=\frac{1}{2m^{2}}\left(E-\sqrt{E^{2}-m^{2}},\sqrt{E^{2}-m^{2}}-E,0,0,0\right)\,.$

In the high energy limit $E\gg m$, $p^{\mu}$ becomes approximately null and
$k^{\mu}\gg q^{\mu}$. The $k$-spinors are given by substituting $k$ into (28).
On the other hand, the representation (28) for $q$ is not allowed since
$q_{0}+q_{1}=0$ and we use an $\operatorname{SO}(3)$ rotation of (28) instead

${\vphantom{|p_{I}\rangle}}_{A}|q_{I}\rangle=i\begin{pmatrix}0&\frac{\sqrt{E^{2}-m^{2}}-E}{m^{2}}\\\
0&0\\\ 0&0\\\ 1&0\\\ \end{pmatrix}\,.$ (121)

###  4.2 High energy limit of amplitudes

In this section, we consider several examples to illustrate how to take the
high energy limit of massive amplitudes. In particular, we demonstrate that
intricate cancellations are often needed to produce gauge invariant massless
amplitudes.

#### Example (high energy limit of $\bm{\chi\chi\phi}$-scattering):

Consider $\chi\chi\phi$-scattering where $\chi$ is a massive fermion
($s_{1}=\frac{1}{2}$ and $\bar{s}_{1}=0$) and $\phi$ is a massless scalar.
According to Section 3.3, there is only one independent term whose high energy
limit is straightforward to take

$A(\chi,\chi,\phi)=A(\mathbf{1}^{\frac{1}{2}},\mathbf{2}^{\frac{1}{2}},{3}^{0})=\langle\mathbf{1}_{\alpha}|\mathbf{2}_{\beta}\rangle\overset{m\ll
E}{\longrightarrow}\langle k_{1,I}|k_{2,J}\rangle
U^{I}_{\enspace\alpha}U^{J}_{\enspace\beta}\,.$ (122)

#### Example (high energy limit of $\bm{W\phi\phi}$-scattering):

A less trivial example is the high-energy limit of a massive vector boson
($W$) with two massless scalars ($\phi$). Like the previous example, there is
one independent term. However, this time, the leading term vanishes

$\displaystyle A(W,\phi,\phi)$
$\displaystyle=A(\mathbf{1}^{\frac{1}{2}+\frac{1}{2}},{2}^{0},3^{0})=\,\langle\mathbf{1}_{\alpha}|3|\mathbf{1}_{\dot{\alpha}}]$
$\displaystyle\overset{m\ll E}{\longrightarrow}\left(\langle
k_{1,I}|3|k_{1,J}\rangle+m\ \langle q_{1,I}|3|k_{1,J}\rangle-m\ \langle
k_{1,I}|3|q_{1,J}\rangle-m^{2}\ \langle
q_{1,I}|3|q_{1,J}\rangle\right)U^{I}_{\enspace\alpha}U^{J}_{\enspace\dot{\alpha}}$
$\displaystyle=m\left[\langle q_{1,I}|3|k_{1,J}\rangle-\langle
k_{1,I}|3|q_{1,J}\rangle\right]U^{I}_{\enspace\alpha}U^{J}_{\enspace\dot{\alpha}}+\mathcal{O}(m^{2})\,.$
(123)

The above expression is unsatisfactory since we use $|k_{1,I}\rangle$ to label
the massless state after taking the high energy limit. Therefore, we need to
express the spinor $|q_{1,I}\rangle$ in terms of $|k_{1,I}\rangle$.

To do this, we use the following identity

${\vphantom{|p_{I}\rangle}}_{A}|q_{I}\rangle={\vphantom{|p_{I}\rangle}}_{A}|q_{J}\rangle\,\varepsilon^{JK}\,\langle
q_{K}|k_{I}\rangle={\vphantom{|p_{I}\rangle}}_{A}|q|k_{I}\rangle\,,$ (124)

where $q^{\mu}$ should be regarded as a _reference momentum_ for the massless
momentum $k^{\mu}$ with the associated reference spinor $|q_{I}\rangle$. The
spinor $|q_{I}\rangle$ must satisfy (116) but is otherwise unconstrained. This
gauge freedom can be parameterized as

$|q^{\prime}_{I}\rangle=|q_{J}\rangle+|k_{J}\rangle{L^{J}}_{I}\,,$ (125)

where $L$ is an arbitrary little group transformation.

Applying (124) to (4.2), one finds that the leading term in the high energy
limit is a true massless amplitude that is gauge invariant

$A(W,\phi,\phi)\overset{m\ll E}{\longrightarrow}m\left[\langle
1_{I}|q_{1}|3|1_{J}\rangle-\langle
1_{I}|3|q_{1}|1_{J}\rangle\right]+\mathcal{O}(m^{2})\,,$ (126)

where we have dropped the $U$-matrices. Gauge invariance is guaranteed because
$(q_{1})_{A}^{\enspace B}$ is always contracted into a $\langle 1_{I}|^{A}$.

As a sanity check, we provide an additional example illustrating how terms
conspire to keep the leading term of the high energy limit gauge invariant.

#### Example (high energy limit of $\bm{WW\phi}$-scattering):

There are two independent terms in this case

$\displaystyle
A(W,W,\phi)=A(\mathbf{1}^{\frac{1}{2}+\bar{\frac{1}{2}}},\mathbf{2}^{\frac{1}{2}+\bar{\frac{1}{2}}},3^{0})=g_{1}\,{}_{\alpha_{1}}\langle{\bf
12}\rangle_{\beta_{1}}\,{}_{\dot{\alpha}_{2}}[\mathbf{12}]_{\dot{\beta}_{2}}+g_{2}\,{}_{\alpha_{1}}\langle\mathbf{12}]_{\dot{\beta}_{2}}\,{}_{\dot{\alpha}_{2}}[\mathbf{12}\rangle_{\beta_{1}}\,.$
(127)

It is useful to make a change of basis from the couplings $(g_{1},g_{2})$ to
$(g_{+},g_{-})=(g_{1}+g_{2},g_{1}-g_{2})$ and study the high energy limit of
the $g_{\pm}$ components.

In the high energy limit, the $g_{+}$ component amplitude reduces to

$A(W,W,\phi)|_{g_{-}=0}\overset{m\ll E}{\longrightarrow}\langle
1_{I_{1}}2_{J_{1}}\rangle\langle 1_{I_{2}}2_{J_{2}}\rangle+\langle
1_{I_{1}}2_{J_{2}}\rangle\langle 1_{I_{2}}2_{J_{1}}\rangle\sim
u_{1,I_{1}}u_{1,I_{2}}u_{2,J_{1}}u_{2,J_{2}}\,,$ (128)

which is exactly the three-point massless amplitude with spin (1,1,0).

On the other hand, the $g_{-}$ amplitude is much more complicated

$\displaystyle A(W,W,\phi)|_{g_{+}=0}\overset{m\ll E}{\longrightarrow}\,$
$\displaystyle\Big{(}\langle 1_{I_{1}}|2_{J_{1}}\rangle{+}m\langle
1_{I_{1}}|q_{1}|2_{J_{1}}\rangle{+}m\langle
1_{I_{1}}|q_{2}|2_{J_{1}}\rangle{+}m^{2}\langle
q_{1,I_{1}}|q_{2,J_{1}}\rangle\Big{)}$ (129) $\displaystyle\times$
$\displaystyle\Big{(}\langle 1_{I_{2}}|2_{J_{2}}\rangle{-}m\langle
1_{I_{2}}|q_{1}|2_{J_{2}}\rangle{-}m\langle
1_{I_{2}}|q_{2}|2_{J_{2}}\rangle{+}m^{2}\langle
q_{1,I_{2}}|q_{2,J_{2}}\rangle\Big{)}-(I_{1}\leftrightarrow I_{2})$

While the leading term vanishes, the subleading term is a gauge invariant

$\displaystyle A(W,W,\phi)|_{g_{+}=0}\overset{m\ll E}{\longrightarrow}$
$\displaystyle-m\langle 1_{I_{1}}2_{J_{1}}\rangle\Big{(}\langle
1_{I_{2}}|q_{1}|2_{J_{2}}\rangle+{}_{I_{2}}\langle
1|q_{2}|2\rangle_{J_{2}}\Big{)}+(I_{1},J_{1})\leftrightarrow(I_{2},J_{2})$
(130) $\displaystyle\quad=\,-2m\,\langle
1_{I_{1}}|q_{1}|2|1_{I_{2}}\rangle\,\varepsilon_{J_{1},J_{2}}+2m\,\langle
2_{J_{1}}|q_{2}|3|2_{J_{2}}\rangle\,\varepsilon_{I_{1},I_{2}}\,.$

Note that the first and second terms are precisely the spin-(1,0,0) and
spin-(0,1,0) three-point massless amplitudes, respectively. It is not
surprising that the massive five-dimensional spin-(1,1,0) amplitude contains
both the massless spin-(1,0,0) and spin-(0,1,0) amplitudes in its high energy
limit because the little group for massive particles lives in four dimensions,
while the little group of massless particle lives in three dimensions.
Therefore, a four-dimensional vector should reduce to a
vector$\,\oplus\,$scalar in three dimensions.

#### The mass dimension and spin drop.

As exemplified by (126) and (130), the mass dimension of a massive amplitude
is not preserved by the high energy limit. In addition, the spin of a massive
amplitude can also drop after taking the high energy limit. To see how mass
dimension and spin drops arise, we consider the high energy limit of
antisymmetrization and symmetrization of the little group indices of
$\langle\mathbf{1}_{\alpha}|[\mathbf{1}_{\dot{\alpha}}|$.

We start by computing the high energy limit of antisymmetrization

$\displaystyle\langle\mathbf{1}_{\alpha}|_{A}\,[\mathbf{1}_{\dot{\alpha}}|_{B}-[\mathbf{1}_{\dot{\alpha}}|_{A}\,\langle\mathbf{1}_{\alpha}|_{B}$
$\displaystyle\qquad\overset{m\ll E}{\longrightarrow}\Big{(}\langle
1_{I}|_{A}+\langle 1_{I}|q|_{A}\Big{)}\Big{(}\langle 1_{J}|_{B}-\langle
1_{J}|q|_{B}\Big{)}-\Big{(}\langle 1_{J}|_{A}-\langle
1_{J}|q|_{A}\Big{)}\Big{(}\langle 1_{I}|_{B}+\langle 1_{I}|q|_{B}\Big{)}$
$\displaystyle\qquad=\varepsilon_{IJ}(p_{1})_{AB}+m(\varepsilon_{1}^{i})_{AB}\cdot\Gamma_{IJ}^{i}+m^{2}\,\varepsilon_{IJ}(q)_{AB}\,,$
(131)

where the $\Gamma^{i}_{IJ}$ are $\operatorname{SO}(3)$ $\Gamma$-matrices. The
first term above the leading term that corresponds to the scalar component in
the high energy limit. The second term is subleading and corresponds to the
vector component in the high energy limit. The third subsubleading term is not
gauge invariant and cannot appear as the leading high energy limit in a
physical amplitude.

On the other hand, the symmetrization is

$\displaystyle\langle\mathbf{1}_{\alpha}|_{A}\,[\mathbf{1}_{\dot{\alpha}}|_{B}{+}[\mathbf{1}_{\dot{\alpha}}|_{A}\,\langle\mathbf{1}_{\alpha}|_{B}$
$\displaystyle\overset{m\ll E}{\longrightarrow}\Big{(}\langle
1_{I}|_{A}\,\langle 1_{J}|_{B}+\langle 1_{J}|_{A}\,\langle
1_{I}|_{B}\Big{)}+m\Big{(}{}_{A}|1|q|_{B}{-}{}_{A}|q|1|_{B}\Big{)}\varepsilon_{IJ}$
$\displaystyle\qquad+m^{2}\Big{(}\langle q_{I}|_{A}\,\langle
q_{J}|_{B}+\langle q_{J}|_{A}\,\langle q_{I}|_{B}\Big{)}$ (132)

where ${}_{A}|1|q|_{B}={\vphantom{|p_{I}\rangle}}_{A}|1_{I}\rangle\langle
1^{I}|q_{J}\rangle{\langle q^{J}|}_{B}$. Here, the first term corresponds to
the vector component and is the only gauge-invariant term. All other terms are
subleading and not gauge invariant.

There are two important lessons from the above derivation. First, because
explicit factors of $m_{i}$ can absorb mass dimensions, the high energy limit
of massive amplitudes can contain components with lower mass dimensions.
Second, the high energy limit of massive amplitudes also contains components
with lower spin. We can also see that in massive amplitudes with
antisymmetrization (whether manifest or not444The non-manifest
antisymmetrization means the rest part of the amplitude is antisymmetric in
Lorentz indices, which will only keep the antisymmetric part of the spinors
$|\bm{i}\rangle|\bm{i}]$.) on chirality change, it’s not straightforward to
analyze its mass dimension under high energy limit. It also depends on the
rest of the amplitude: what the antisymmetrized spinors contract with and
whether the leading term in (4.2) vanishes in the final amplitudes.

While the mass dimension of massive amplitudes is not well defined, the mass
dimension of their high energy limit is. Therefore, one can categorize the
possible independent terms in amplitudes by the number of
symmetrizations/antisymmetrizations performed when taking the high energy
limit. Roughly speaking, each antisymmetrization lowers either the mass
dimension or spin by one, while symmetrization preserves both mass dimension
and spin (in some cases, it can do both).

###### Acknowledgements.

We are grateful to Nima Arkani-Hamed, Shounak De, Taylor Knapp, Marcus
Spradlin, and Akshay Yelleshpur Srikant for useful discussions. This work was
supported in part by the US Department of Energy under contract DE-SC0010010
Task F (AP, LR, AV), by Simons Investigator Award #376208 (AV), and by
Bershadsky Distinguished Visiting Fellowship at Harvard (AV). S. Rajan was
also supported by a Karen T. Romer Undergraduate Teaching and Research Award.

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