aainstitutetext: Fields and Strings Laboratory, Institute of Physics
École Polytechnique Fédéral de Lausanne (EPFL)
Route de la Sorge, CH-1015 Lausanne, Switzerlandbbinstitutetext:  Philippe
Meyer Institute, Physics Department
École Normale Supérieure (ENS), Université PSL
24 rue Lhomond, F-75231 Paris, France

# Spinning S-matrix Bootstrap in 4d

Aditya Hebbar a,b Denis Karateev a and João Penedones

###### Abstract

We review unitarity and crossing constraints on scattering amplitudes for
particles with spin in four dimensional quantum field theories. As an
application we study two to two scattering of neutral spin 1/2 fermions in
detail. Assuming Mandelstam analyticity of its scattering amplitude, we use
the numerical S-matrix bootstrap method to estimate various non-perturbative
bounds on quartic and cubic (Yukawa) couplings.

††dedication: Dedicated to the memory of André Martin

##  1 Introduction

The space of Quantum Field Theories (QFT) is vast and uncharted. The numerical
S-matrix Bootstrap is a nonperturbative approach to explore this space
Paulos:2016fap ; Paulos:2016but ; Paulos:2017fhb ; Doroud:2018szp ; He:2018uxa
; Cordova:2018uop ; Guerrieri:2018uew ; Homrich:2019cbt ; EliasMiro:2019kyf ;
Cordova:2019lot ; Bercini:2019vme ; Karateev:2019ymz ; Karateev:2020axc ;
Correia:2020xtr ; Bose:2020shm . This recent approach is a practical
implementation of the decades old idea of S-matrix Theory (see for example the
book Eden:1966dnq ). So far this method has only been applied to scattering
amplitudes of scalar particles.111In fact, most recent papers studied two
dimensional QFT where there is no spin. In this work, we develop the formalism
to study 2 to 2 scattering amplitudes of particles with spin in four
dimensional QFTs.

The main idea of the numerical S-matrix Bootstrap Paulos:2016but ;
Paulos:2017fhb  is to write a generic analytic and Lorentz invariant ansatz
for the scattering amplitude and then impose the constraints from crossing
symmetry and unitarity. In the case of identical scalar particles of mass $m$,
the interacting part of the scattering amplitude denoted by $T(s,t,u)$, where
$s$, $t$ and $u$ are the Mandelstam variables, can be written as

$T(s,t,u)=\sum_{a}\alpha_{i}\;F_{i}(s,t,u)\,,$ (1)

where $\alpha_{i}$ are undetermined real coefficients and $F_{i}(s,t,u)$ are
functions with appropriate analyticity properties.222In practice, we assume
maximal analyticity of the scattering amplitude, _i.e._ the only non-
analyticities are the ones predicted by Landau diagrams.  Crossing symmetry

$T(s,t,u)=T(t,s,u)=T(u,t,s)\,,$ (2)

imposes linear constraints on the coefficients $\alpha_{i}$ in the ansatz (1).
Finally, the unitarity requirement can be cast as the following semidefinite
positive condition333This is equivalent to requiring non-negativity of the
determinant of the matrix which leads to the unitarity constraint in a more
recognizable form $|\mathcal{S}_{\ell}(s)|^{2}\leq 1$.

$\begin{pmatrix}1&\mathcal{S}^{*}_{\ell}(s)\\\
\mathcal{S}_{\ell}(s)&1\end{pmatrix}\succeq 0,$ (3)

where $\ell=0,2,4,\ldots$ and $s\geq 4m^{2}$. The partial amplitudes
$\mathcal{S}_{\ell}(s)$ are defined as the projection of the scattering
amplitude onto definite spin via the Legendre Polynomials
$P_{\ell}(\cos\theta)$ as

$\mathcal{S}_{\ell}(s)\equiv
1+i\frac{\sqrt{s-4m^{2}}}{32\pi\sqrt{s}}\times\int_{0}^{\pi}d\theta\sin\theta
P_{\ell}(\cos\theta)T(s,t(s,\cos\theta),u(s,\cos\theta)).$ (4)

The integration variable $\theta$ is the scattering angle and its precise
relation to the Mandelstam variables is given in (67). By keeping a finite
number of terms in the ansatz (1) one can numerically explore the space of
consistent scattering amplitudes, namely the ones obeying (2) and (3). In
particular, extremizing any physical parameter linearly related to the
scattering amplitude translates into a numerical (primal semidefinite)
optimization problem.

In the presence of spin, the expressions (2) - (4) immediately become more
complicated, see Martin:102663  for a review. The goal of section 2 is to
setup the formalism which allows to study crossing and unitarity for
amplitudes with generic masses and spins. This section is mostly a review of
known results in a concise form.

Compared to the scalar case a generic scattering amplitude describing the
process $12\rightarrow 34$ has the form

$T_{12\rightarrow
34}{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p_{1},p_{2},p_{3},p_{4}),$
(5)

where $p_{i}$ are four-momenta and $\lambda_{i}$ are helicities of scattering
particles respectively. Given the spins $j_{i}$ of four particles, helicities
take values from the range $\lambda_{i}=-j_{i},\ldots+j_{i}$ with step one.
There are thus

$N_{4}\equiv(2j_{1}+1)(2j_{2}+1)(2j_{3}+1)(2j_{4}+1)$ (6)

components describing the scattering process $12\rightarrow 34$.444When the
scattering process is parity and/or time-reversal invariant, when it contains
identical and/or massless particles, the counting of independent amplitudes
becomes much more complicated. We discuss it in section 2.4. We would then
like to define $N_{4}$ scattering amplitudes which depend only on the
Mandelstam variables. This can be done in two different ways: using the center
of mass frame or using tensor structures. We denote these two options
respectively by

$T_{12\rightarrow
34}{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u)\qquad\text{and}\qquad
T_{12\rightarrow 34}^{I}(s,t,u),$ (7)

where $I=1,\ldots N_{4}$. The two sets of amplitudes in (7) are related by a
linear transformation which depends on the Mandelstam variables only. This
relation is completely fixed once the basis of tensor structures is chosen.
Both options have their advantages and disadvantages. In the center of mass
frame one can derive crossing equations once and for all spins and masses as
summarized in section 2.3 (see appendix E for details).555Generically,
crossing equations relate physical amplitudes to the analytic continuation of
other amplitudes beyond their physical domain. There is no general proof
(especially for particles with spin) that this continuation exists and how big
is the domain where crossing equations hold. For a review of results for
scalar amplitudes, see Sommer:1970mr . In this work we do not address this
issue and simply assume that all the amplitudes under consideration are
maximally analytic.  However, the analyticity properties of the center of mass
amplitudes are subtle and one is forced to deal with the issue of kinematic
singularities discussed in section 2.7. Analyticity is more straightforward
when using tensor structures, however one needs to study crossing case by case
due to non-trivial linear relations between covariant tensor structures. As a
result we cannot give it a completely general treatment in section 2 and
instead employ it only in the particular example of identical neutral spin
$\frac{1}{2}$ particles in section 3.4. In addition, the construction of
general spin tensor structures is discussed to some extent in appendix H.

The key element for imposing unitarity constraints are the partial amplitudes
(4). In the case of generic spin, (4) remains valid for center of mass
amplitudes, if the Legendre polynomial is replaced by the small Wigner
d-matrix given in (16) in full generality. This is explained in detail in
section 2.5. Finally, the unitarity constraints for generic spin are given in
section 2.6.

In section 3, we specialize to the case of scattering of neutral spin
$\frac{1}{2}$ massive fermions, also known as Majorana fermions
Majorana:1937vz . This section should be seen as the simplest application of
the formalism given in section 2. In section 4 we write an ansatz for the
scattering of identical Majorana particles in a parity invariant QFT (assuming
this is the lightest particle). We then setup the optimization problem using
crossing and unitarity constraints derived in section 3. Finally, we present
our numerical results for the allowed values of the non-perturbative quartic
and cubic (Yukawa) couplings defined from the physical scattering amplitude in
section 3.3. The universal bounds for the quartic coupling, given in (251),
and for the cubic couplings, shown in figures 5 and 6, are our main numerical
results.

Our work opens the ground for many other interesting bootstrap studies in 4
dimensions. For example, photon-photon, pion-nucleon or spinning glueball
scattering. We conclude in section 5 with a discussion of these examples and
other open questions for the future. We also include several appendices that
fill in the details of the presentation in the main text.

##  2 Review: spinning $S$-matrix approach

In this paper we study quantum systems invariant under the restricted Poincaré
symmetry group. In addition, we consider special situations where parity
and/or time-reversal symmetries are also present. Scattering amplitudes are
important physical observables in these systems. They are constrained by many
non-trivial consistency conditions. The program of deriving and exploiting
such theoretical constraints is referred to as the S-matrix bootstrap.

The S-matrix bootstrap program was extensively developed half a century ago.
In this section we review its kinematic aspects (the ones fixed by
symmetries).666For other sources covering this topic see for example
Weinberg:1995mt . Our goal is to provide an updated and easy to use practical
summary of the basic ingredients.

###  2.1 States

We work in Lorentzian metric with the mostly plus signature

$\eta_{\mu\nu}=\\{-+++\\}.$ (8)

The unitary irreducible representations of the restricted Poincaré group were
classified by Wigner.777For a textbook discussion see for example
Weinberg:1995mt ; Tung:1985na . Here we will work with a particular unitary
representation which is positive energy time-like. The basis for such a
representation is formed by the states

$|c,\vec{p};{\ell},{\lambda};\gamma\rangle\,.$ (9)

where the 3-momentum $\vec{p}\in\mathbb{R}^{3}$ and the helicity
$\lambda=-\ell,-\ell+1,\ldots,+\ell$. The other labels are fixed within the
irreducible representation and serve to specify a given state. The label $c>0$
is a continuous real parameter related to the energy $p^{0}>0$ as

$c^{2}=-p_{\mu}p^{\mu}=(p^{0})^{2}-(\vec{p}\,)^{2}.$ (10)

It is defined as the eigenvalue of the first Casimir of the Poincaré group
$c^{2}=-P^{2}$, where $P^{\mu}$ are the generators of translations. Instead of
$c$ one can equivalently label the basis of states (9) using the energy
$p^{0}$. We will often use both labels throughout the text interchangeably.
The spin label $\ell=0,\,\tfrac{1}{2},\,1,\ldots$ is a non-negative integer or
half integer related to the eigenvalue of the second Casimir of the Poincaré
group $W^{2}=4c^{2}\,\ell(\ell+1)$, where $W^{\mu}$ is the Pauli–Lubanski
pseudovector.888In our convention
$W_{\mu}\equiv\epsilon_{\mu\nu\rho\sigma}M^{\nu\rho}P^{\sigma}$, where
$M^{\nu\rho}$ are the generators of the Lorentz group.  The helicity label
$\lambda$ is the projection of the spin vector along $\vec{p}$. Finally the
label $\gamma$ stands for any other additional (discrete or continuous) labels
which might be required to fully specify the state. We will see one particular
case of their importance in section 2.1.2.

We will often need to use spherical coordinates in which case the three
momentum is parametrized as

$\vec{p}=(\boldsymbol{p}\sin\theta\cos\phi,\;\boldsymbol{p}\sin\theta\sin\phi,\;\boldsymbol{p}\cos\theta),\quad\boldsymbol{p}\equiv|\vec{p}|,\quad
0\leq\phi<2\pi,\quad 0\leq\theta\leq\pi,$ (11)

The helicity basis states (9) with non-zero three momentum $\vec{p}$ are
defined as follows

$|c,\vec{p};{\ell},{\lambda};\gamma\rangle=e^{-i\phi J_{3}}e^{-i\theta
J_{2}}e^{+i\phi J_{3}}e^{-i\eta
K_{3}}|c,\vec{0};{\ell},{\lambda};\gamma\rangle,$ (12)

where $J_{a}$ and $K_{a}$ with $a=1,2,3$ are the generators of spatial
rotations and boosts respectively. The rapidity parameter $\eta$ is related to
the energy and momentum of the states as

$\cosh\eta=\frac{p^{0}}{c},\qquad\sinh\eta=\frac{\boldsymbol{p}}{c}.$ (13)

Lastly the helicity ${\lambda}$ in the zero three momentum state
$|c,\vec{0};j,{\lambda};\gamma\rangle$ is a projection of the spin ${\ell}$
onto the $z$ axis. In other words these states are eigenstates of the $J_{3}$
generators. Given a generic 3d rotation characterized by the three Euler
angles $(\alpha,\beta,\gamma)$ (where $\alpha$ and $\gamma$ parametrize
rotations around the $z$-axis and $\beta$ parametrizes rotations around the
$y$-axis) we have the following transformation property

$R(\alpha,\beta,\gamma)|c,\vec{0};{\ell},{\lambda};\gamma\rangle=\sum_{\lambda^{\prime}}\mathscr{D}^{(\ell)}_{\lambda^{\prime}\lambda}(\alpha,\beta,\gamma)|c,\vec{0};{\ell},{\lambda}^{\prime};\gamma\rangle,\quad
R(\vec{a})\equiv e^{-i\alpha J_{3}}e^{-i\beta J_{2}}e^{-i\gamma J_{3}}.$ (14)

The object $\mathscr{D}^{(\ell)}_{\lambda^{\prime}\lambda}$ is known as the
large Wigner D matrix and reads as

$\mathscr{D}^{(\ell)}_{\lambda^{\prime}\lambda}(\alpha,\beta,\gamma)=e^{-i\alpha\lambda^{\prime}}d^{(\ell)}_{\lambda^{\prime}\lambda}(\beta)e^{-i\gamma\lambda}.$
(15)

Here the object $d^{({\ell})}_{\lambda^{\prime}\lambda}$ is the small Wigner d
matrix, its explicit expression reads999Note that Mathematica implements the
small Wigner d matrices (16) with indices $\lambda$ and $\lambda^{\prime}$
flipped. In other words $d^{({\ell})}_{\lambda^{\prime}\lambda}(\beta)$ is
generated by the command
$\text{WignerD}[\\{{\ell},\lambda,\lambda^{\prime}\\},\beta]$.

$\displaystyle d^{({\ell})}_{\lambda^{\prime}\lambda}(\beta)$
$\displaystyle=\sqrt{({\ell}+\lambda)!({\ell}-\lambda)!({\ell}+\lambda^{\prime})!(\ell-\lambda^{\prime})!}$
$\displaystyle\times\sum_{\nu=0}^{2j}(-1)^{\nu}\frac{\left(\cos(\beta/2)\right)^{2j+\lambda-\lambda^{\prime}-2\nu}\left(-\sin(\beta/2)\right)^{\lambda^{\prime}-\lambda+2\nu}}{\nu!(\ell-\lambda^{\prime}-\nu)!({\ell}+\lambda-\nu)!(\nu+\lambda^{\prime}-\lambda)!}.$
(16)

In case both helicities $\lambda$ and $\lambda^{\prime}$ are equal to zero,
the small Wigner d matrix reduces to the usual Legendre polynomial

$d^{(\ell)}_{00}(\beta)=P_{\ell}(\cos\beta).$ (17)

The Wigner D matrices satisfy orthogonality. They also obey various useful
relations, see for example appendix A.2 in Martin:102663 .

Consider now a generic Poincaré transformation which consists of a translation
fixed by four Lie parameters $a^{\mu}$ and the Lorentz transformation
$\Lambda^{\mu}{}_{\nu}$ fixed by six Lie parameters $\rho_{[\mu\nu]}$. It
transforms the 4-momentum $p^{\mu}$ into
$p^{\prime\mu}=\Lambda^{\mu}{}_{\nu}p^{\nu}$. Under the Poincaré
transformation $(a,\rho)$ the states (9) transform according to

$U(a,\rho)|c,\vec{p};{\ell},{\lambda};\gamma\rangle=e^{ia\cdot
p^{\prime}}\times\sum_{\lambda^{\prime}}\mathscr{D}^{(\ell)}_{\lambda^{\prime}\lambda}(\alpha,\beta,\gamma)|c,\vec{p}^{\,\prime};{\ell},{\lambda}^{\prime};\gamma\rangle,$
(18)

where the three Wigner angles $(\alpha,\beta,\gamma)$ can be expressed in
terms of the six Lie parameters $\rho_{[\mu\nu]}$. We provide an example of a
Wigner angles computation in appendix A.4. For the special case of pure 3d
rotations, the Wigner D matrix reduces to a phase and therefore the helicity
of the state remains unchanged101010Notice that this is compatible with (14)
because the limit $\vec{p}\to 0$ of $|c,\vec{p};j,{\lambda};\gamma\rangle$ is
only equal to $|c,\vec{0};j,{\lambda};\gamma\rangle$ if $\vec{p}$ is parallel
to the $z$-axis.

$R(\vec{a})|c,\vec{p};{\ell},{\lambda};\gamma\rangle=e^{-i\lambda\,\xi(\vec{a},\phi,\theta)}|c,\vec{p}\,^{\prime};{\ell},{\lambda};\gamma\rangle,$
(19)

where the phase $\xi$ has a complicated dependence on its arguments.

##### Particles

So far we discussed states and their properties in a generic Poincaré
invariant quantum theory. The related notion of particle does not exist in
every QFT. The simplest context where particles are well defined is in free
QFTs. A free theory is described by a set of one particle states (describing
freely propagating particles with a given mass and spin). Taking tensor
products of these states we form a complete basis of states spanning the
Hilbert space of the theory. In interacting QFTs, particles can still be
defined in the asymptotic far past and future if interactions decay
sufficiently fast with distance. This is the case for theories with a mass gap
and massless theories with soft interactions like Goldstone bosons or photons
(in the absence of charged particles in the asymptotic states). In this paper,
we will only consider those theories where particles do exist. Then, one can
define two types of asymptotic states: the $in$ states (far past) and the
$out$ states (far future). Both the $in$ states and the $out$ states span the
full Hilbert space of the theory. A natural basis of $in/out$ states is given
by the tensor product of one particle $in/out$ states.

####  2.1.1 One particle states (1PS)

One particle states (1PS) are special cases of (9) where $c$ takes only
discrete values corresponding to the masses of stable particles in the theory.
For this reason, we will set $c=m$ from now on. If global symmetries are
present, the label $\gamma$ describes the charge or more generically the
representation under the global symmetry group. For simplicity, in this
section we ignore global symmetries, thus the 1PS will not carry any extra
labels $\gamma$.

We introduce a shorthand notation for one particle states

$|\kappa\rangle\equiv|m,\vec{p};j,\lambda\rangle.$ (20)

For indicating spin we use the label $j$ instead of ${\ell}$ here to ease the
visual distinction between the 1PS and generic irreps. We normalize 1PS as
follows

$\displaystyle\langle
m^{\prime},\vec{p}^{\,\prime};j^{\prime},\lambda^{\prime}|m,\vec{p};j,\lambda\rangle$
$\displaystyle=$
$\displaystyle(2\pi)^{3}\,2p^{0}\,\delta^{3}(\vec{p}^{\,\prime}-\vec{p})\,\delta_{m^{\prime}m}\delta_{j^{\prime}j}\delta_{\lambda^{\prime}\lambda}$
(21) $\displaystyle\equiv$ $\displaystyle\delta(\kappa^{\prime}-\kappa),$

where in accordance with (10) the energy of the states is

$p^{0}=\sqrt{m^{2}+p^{2}}.$ (22)

In the second line of (21) we have introduced the shorthand notation
$\delta(\kappa^{\prime}-\kappa)$ for the set of Kronecker and Dirac delta
functions. The transformation rule for a one particle state under a Poincare
transformation remains the same as in (18).

The case of massless particles should be treated separately since $c\neq 0$ in
(9). Skipping details, the following statement holds: massless particles can
be described by the states (20) with $m=0$ and the range of helicities
restricted to only two values $\lambda=-j$ and $\lambda=+j$.

Finally we state the transformation properties of 1PS under parity and time-
reversal. For further details see appendix B. One has

$\displaystyle\mathcal{P}|m,\vec{p};j,\lambda\rangle$
$\displaystyle=\eta(-1)^{j-\lambda}\exp(2i\phi\lambda)|m,-\vec{p};j,-\lambda\rangle,$
(23) $\displaystyle\mathcal{T}|m,\vec{p};j,\lambda\rangle$
$\displaystyle=\varepsilon(-1)^{2j}\exp(-2i\phi\lambda)|m,-\vec{p};j,\lambda\rangle,$
(24)

where $\phi$ is the spherical angle defined in (11), see also (34), $\eta$ and
$\epsilon$ are pure phases. Their values are yet other quantities
characterizing the state. Let us briefly discuss the phase $\eta$ called
intrinsic parity. One can argue as in section 3.3 of Weinberg:1995mt , that
one can always define ${\mathcal{P}}$ in such a way that
${\mathcal{P}}^{2}=+1$ or ${\mathcal{P}}^{2}=-1$. As a result one has

$\eta=\pm 1\quad\text{or}\quad\eta=\pm i.$ (25)

The imaginary values of intrinsic parities are only possible for fermions,
however no such fermions have been discovered so far in the nature.

####  2.1.2 Two particle states (2PS)

We define the two particle states (2PS) by taking the ordered tensor product
of two 1PS

$|\kappa_{1},\kappa_{2}\rangle\equiv|m_{1},\vec{p}_{1};j_{1},\lambda_{1}\rangle\otimes|m_{2},\vec{p}_{2};j_{2},\lambda_{2}\rangle.$
(26)

The normalization of the 2PS defined above follows from that of the 1PS:

$\displaystyle\langle\kappa_{1},\kappa_{2}|\kappa_{3},\kappa_{4}\rangle$
$\displaystyle=\delta(\kappa_{1}-\kappa_{3})\delta(\kappa_{2}-\kappa_{4}).$
(27)

Two particle states do not form an irreducible representation of the
restricted Poincaré group. However, they can be decomposed into a direct sum
of states (9) transforming in irreducible representations. This is done by
injecting the completeness relation into 2PS

$|\kappa_{1},\kappa_{2}\rangle=\int\frac{d^{4}p}{(2\pi)^{4}}\,\theta(p^{0})\theta(-p^{2})\sum_{\gamma}\sum_{\ell,\lambda}|c,\vec{p};{\ell},{\lambda};\gamma\rangle\,\langle
c,\vec{p};{\ell},{\lambda};\gamma|\kappa_{1},\kappa_{2}\rangle,$ (28)

where we normalize the states (9) as follows111111The Kronecker deltas follow
from the fact that the states here are simultaneous eigenstates of the two
Casimirs (285) and the helicity operator (305).

$\langle
c^{\prime},\vec{p}^{\,\prime};{\ell}^{\prime},{\lambda}^{\prime};\gamma^{\prime}|c,\vec{p};{\ell},{\lambda};\gamma\rangle=(2\pi)^{4}\delta^{4}({p^{\prime}}^{\mu}-p^{\mu})\,\delta_{\ell^{\prime}\ell}\,\delta_{\lambda^{\prime}\lambda}\,\delta_{\gamma^{\prime}\gamma},$
(29)

where $p^{\mu}=(p^{0},\,\vec{p})$ is the 4-momentum and the symbolic
expression $\delta_{\gamma^{\prime}\gamma}$ will be properly specified when
the additional labels $\gamma$ and $\gamma^{\prime}$ are defined. We use the
normalization (29) for all irreducible Poincaré representations, with the
exception of 1PS which are the only states where the label $c$ takes
particular discrete values. Looking at the right-hand side of (28), we see
that the Clebsch-Gordon coefficients of this decomposition (due to translation
invariance) obey

$\langle
c,\vec{p};{\ell},{\lambda};\gamma|\kappa_{1},\kappa_{2}\rangle\propto(2\pi)^{4}\delta^{4}(p^{\mu}-p_{1}^{\mu}-p_{2}^{\mu}).$
(30)

This delta function completely removes the integration over $p$ in (28). The
label $\gamma$ is the multiplicity label. In the case of 2PS decomposition the
multiplicity label $\gamma$ consists of only discrete parameters121212This is
no longer the case for the decompositions of three or more particle states. In
these cases the multiplicity label $\gamma$ would also contain continuous
parameters associated with the relative momenta of the component particles.
and reads as

$\gamma=(m_{1},j_{1},\lambda_{1};\,m_{2},j_{2},\lambda_{2}).$ (31)

Thus the label $\gamma$ keeps track of which particles and what helicities
were used to make the two particle state. In what follows we will almost
always drop the explicit mass and spin labels in order to simplify the
formulas. However, when dealing with particles of different masses and spins,
the mass and spin labels are important. Finally we choose to normalize the
states appearing in (28) according to (29) with

$\delta_{\gamma^{\prime}\gamma}=\delta_{m_{1}^{\prime}m_{1}}\delta_{m_{2}^{\prime}m_{2}}\delta_{j_{1}^{\prime}j_{1}}\delta_{j_{2}^{\prime}j_{2}}\times\delta_{\lambda_{1}^{\prime}\lambda_{1}}\delta_{\lambda_{2}^{\prime}\lambda_{2}}.$
(32)

##### COM two particle states

We can always use Lorentz invariance to go to the frame where the total
momentum of the two particles is 0. This frame is called the centre of mass
(COM) frame. Therefore, we do not need to know the most general decomposition
(28), and instead it is enough to focus on the special case of 2PS in the
center of mass (COM) frame, namely the states (26) obeying the constraint
$\vec{p}_{1}=-\vec{p}_{2}$. We give a special label to such two particles
states

$\displaystyle|(\boldsymbol{p},\theta,\phi);\lambda_{1},\lambda_{2}\rangle\equiv|m_{1},+\vec{p};j_{1},\lambda_{1}\rangle\otimes|m_{2},-\vec{p};j_{2},\lambda_{2}\rangle,$
(33)

where the three-momenta have the following spherical coordinates131313 Notice
that given the vector $+\vec{p}$ in spherical coordinates, the vector opposite
to it is defined as $-\vec{p}=(\boldsymbol{p},\,\pi-\theta,\,\pi+\phi)$ for
$\phi\in[0,\pi]$ and $-\vec{p}=(\boldsymbol{p},\,\pi-\theta,\,-\pi+\phi)$ for
$\phi\in(\pi,2\pi]$. In other words in order to describe the vector $-\vec{p}$
in spherical coordinates one needs two different descriptions, one for
$\phi\in[0,\pi]$ and one for $\phi\in(\pi,2\pi]$. We do not indicate it in the
main text, since all the consequent formulas remain uniform in the whole range
$\phi\in[0,2\pi]$. The reason for that is the choice of the helicity basis
(12) and the fact that $R(\phi,\theta,-\phi)$ is $2\pi$ periodic, see footnote
42. Notice also that there is a special case when $\theta=\pi$. For this
particular point, we choose the spherical angles of the first state to be
$+\vec{p}=(\boldsymbol{p},\,\theta=\pi,\,\phi=\pi)$, whereas the spherical
angles of the second state are
$-\vec{p}=(\boldsymbol{p},\,\theta=0,\,\phi=0)$.

$+\vec{p}=(\boldsymbol{p},\,\theta,\,\phi),\quad-\vec{p}=(\boldsymbol{p},\,\pi-\theta,\,\pi+\phi),\quad\theta\in[0,\pi],\quad\phi\in[0,2\pi].$
(34)

The states (33) are normalized according to (27). By performing a change of
variables we can rewrite the normalization in terms of spherical coordinates
as

$\langle(\boldsymbol{p},\theta,\phi);\lambda_{1},\lambda_{2}|(\boldsymbol{p}^{\prime},0,0);\lambda_{1}^{\prime},\lambda_{2}^{\prime}\rangle=(2\pi)^{4}\delta^{4}(0)\times\frac{16\pi^{2}\sqrt{s}}{\sqrt{\boldsymbol{p}\boldsymbol{p}^{\prime}}}\frac{\delta(\theta)\delta(\phi)}{\sin\theta}\times\\\
\delta_{m_{1}^{\prime}m_{1}}\delta_{m_{2}^{\prime}m_{2}}\delta_{j_{1}^{\prime}j_{1}}\delta_{j_{2}^{\prime}j_{2}}\times\delta_{\lambda_{1}\lambda_{1}^{\prime}}\delta_{\lambda_{2}\lambda_{2}^{\prime}},$
(35)

where $s=(p_{1}^{0}+p_{2}^{0})^{2}$ is the square of the COM energy. In
appendix A.2.4, we compute the Clebsch-Gordon coefficients (30) for the COM
states (33). Here we present only the result which reads as

$|(\boldsymbol{p},\theta,\phi);\lambda_{1},\lambda_{2}\rangle=\sum_{\ell,\lambda}C_{\ell}(\boldsymbol{p})\,e^{i\phi(\lambda_{1}+\lambda_{2}-\lambda)}d^{(\ell)}_{\lambda\lambda_{12}}(\theta)|c,0;{\ell},{\lambda};\gamma\rangle,$
(36)

where we have

$\lambda_{12}\equiv\lambda_{1}-\lambda_{2},\quad
c=\sqrt{s}=\sqrt{m_{1}^{2}+\boldsymbol{p}^{2}}+\sqrt{m_{2}^{2}+\boldsymbol{p}^{2}},$
(37)

with the multiplicity labels $\gamma$ given in (31) and
$d^{(\ell)}_{\lambda\lambda_{12}}(\theta)$ given by (16). Using the
orthogonality of the small Wigner $d$ matrix and the exponential function, we
can invert equation (36) as follows

$|c,0;{\ell},{\lambda};\gamma\rangle=\frac{2\ell+1}{4\pi
C_{\ell}(\boldsymbol{p})}\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta\sin\theta\,e^{-i\phi(\lambda_{1}+\lambda_{2}-\lambda)}d^{(\ell)}_{\lambda\lambda_{12}}(\theta)|(\boldsymbol{p},\theta,\phi);\lambda_{1},\lambda_{2}\rangle.$
(38)

The coefficient $C_{\ell}$ is completely fixed by the consistency requirement
that the left-hand side of (36) satisfies the normalization condition (27) and
the state in the right-hand side of (36) satisfies the normalization condition
(29). For non-identical particles it reads as

$\displaystyle
C_{\ell}(\boldsymbol{p})^{2}=4\pi\,(2\ell+1)\times\frac{c}{\boldsymbol{p}}.$
(39)

##### Identical particles

The discussion presented above should be slightly modified when the two
particle state is composed of identical particles. In the latter case it must
satisfy

$|\kappa_{1},\kappa_{2}\rangle_{id}=(-1)^{2j}|\kappa_{2},\kappa_{1}\rangle_{id}.$
(40)

We have added the subscript $id$ to explicitly indicate that the state
describes a system of two identical particles. In order to incorporate (40),
we have (instead of simply taking an ordered product) to take either
symmetrized (in case of bosons) or anti-symmetrized (in case of fermions)
tensor product. Such a state will thus have the following form

$|\kappa_{1},\kappa_{2}\rangle_{id}\equiv\frac{1}{\sqrt{2}}\left(|m,\vec{p}_{1};j,\lambda_{1}\rangle\otimes|m,\vec{p}_{2};j,\lambda_{2}\rangle+(-1)^{2j}|m,\vec{p}_{2};j,\lambda_{2}\rangle\otimes|m,\vec{p}_{1};j,\lambda_{1}\rangle\right).$
(41)

The normalization of the state (41) follows from (21)

${}_{id}\langle\kappa_{1},\kappa_{2}|\kappa_{3},\kappa_{4}\rangle_{id}=\delta(\kappa_{1}-\kappa_{3})\delta(\kappa_{2}-\kappa_{4})+(-1)^{2j}\delta(\kappa_{1}-\kappa_{4})\delta(\kappa_{2}-\kappa_{3}).$
(42)

As before we need to define the identical 2PS in the center of momentum.
Adapting (33) to the case of identical particles we get

$|(\boldsymbol{p},\theta,\phi);\lambda_{1},\lambda_{2}\rangle_{\text{id}}\equiv\\\
\frac{1}{\sqrt{2}}\left(|m,+\vec{p};j,\lambda_{1}\rangle\otimes|m,-\vec{p};j,\lambda_{2}\rangle+(-1)^{2j}|m,-\vec{p};j,\lambda_{2}\rangle\otimes|m,\vec{p};j,\lambda_{1}\rangle\right).$
(43)

The normalization of the identical COM states (43) is fixed by (42). It is
still given by (29) but with

$\delta_{\gamma^{\prime}\gamma}=\frac{1}{2}\left(\delta_{\lambda_{1}\lambda_{1}^{\prime}}\delta_{\lambda_{2}\lambda_{2}^{\prime}}+(-1)^{\ell+\lambda_{1}-\lambda_{2}}\delta_{\lambda_{1}\lambda_{2}^{\prime}}\delta_{\lambda_{2}\lambda_{1}^{\prime}}\right).$
(44)

We would now like to decompose the identical two particle state (43) into
irreducible representations

$|c,0,{\ell},{\lambda};\lambda_{1},\lambda_{2}\rangle_{id}\equiv\frac{1}{2}\left(|c,0,{\ell},{\lambda};\lambda_{1},\lambda_{2}\rangle+(-1)^{\ell+\lambda_{1}-\lambda_{2}}|c,0,{\ell},{\lambda};\lambda_{2},\lambda_{1}\rangle\right).$
(45)

The decomposition of identical 2PS is obtained straightforwardly by applying
(36) to both terms in the right-hand side of (43) which leads to

$|(\boldsymbol{p},\theta,\phi);\lambda_{1},\lambda_{2}\rangle_{\text{id}}=\sqrt{2}\sum_{\ell,\lambda}C_{\ell}(\boldsymbol{p})e^{i(\lambda_{1}+\lambda_{2}-\lambda)\phi}d^{(\ell)}_{\lambda\lambda_{12}}(\theta)|c,0;{\ell},{\lambda};\lambda_{1},\lambda_{2}\rangle_{id},$
(46)

where the coefficient $C_{\ell}$ is given by (39). For the detailed derivation
of these equations see appendix C.

###  2.2 S-matrix elements

Given a generic state in the reference frame at $t=0$, an observer in another
reference frame in the far past ($t=-\infty$) or far future ($t=+\infty$) will
see the same state as a (linear combination of) tensor product of one particle
states which we refer to as in or out asymptotic states respectively.
Asymptotic states have a complicated time evolution. One can however establish
a formal one to one map between these states and those of some free theory
(which evolve trivially with time) by means of a pair of unitary operators
$\Omega_{-}$ and $\Omega_{+}$ called the Møller operators. See section 2.1 in
Karateev:2019ymz  for a recent discussion. In their notation

$|\kappa\rangle_{in}=\Omega_{-}|\kappa\rangle_{free},\quad|\kappa\rangle_{out}=\Omega_{+}|\kappa\rangle_{free}.$
(47)

Let us now discuss inner products between asymptotic states. Since the Møller
operators are isometric,141414Recall that an isometric operator $O$ on a
Hilbert space preserves distances. This implies that $O^{\dagger}O=1$, however
the operator need not be surjective and $OO^{\dagger}\neq 1$ generically. A
surjective isometric operator is unitary. the inner products of only $in$
states or only $out$ states are fixed by the normalization conditions (29) and
(21). This means that the only non-trivial matrix elements must include both
$in$ and $out$ states.

##### Four-particle amplitudes

We start with the most important object for our work

${}_{out}\langle\kappa_{3},\kappa_{4}|\kappa_{1},\kappa_{2}\rangle_{in}={}_{free}\langle\kappa_{3},\kappa_{4}|S|\kappa_{1},\kappa_{2}\rangle_{free},$
(48)

where the scattering operator $S$ is defined via (47) as

$S\equiv\Omega_{+}^{\dagger}\Omega_{-}.$ (49)

Møller operators are isometric which implies the unitarity of the scattering
operator151515This is a non-trivial statement, see taylor2012scattering  for
details.

$S^{\dagger}S=1.$ (50)

Poincaré invariance implies that161616See the discussion around (18).

$U(a,\rho)\,S\,U^{-1}(a,\rho)=S,$ (51)

where $U$ represents a generic Poincaré transformation in the Hilbert space.
Finally it is convenient to split the scattering operator into the trivial
part (identity operator) and the interacting part as

$S=1+iT.$ (52)

If $T=0$ we simply recover the free theory. The matrix element (48) describes
scattering of two particles. Factoring out the overall delta function due to
translation invariance we can define the two to two scattering amplitude as

$(2\pi)^{4}\delta^{(4)}(p_{1}^{\mu}+p_{2}^{\mu}-p_{3}^{\mu}-p_{4}^{\mu})\times
S_{12\rightarrow
34}{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p_{1},p_{2},p_{3},p_{4})\equiv{}_{free}\langle\kappa_{3},\kappa_{4}|S|\kappa_{1},\kappa_{2}\rangle_{free}.$
(53)

Equivalently we define the interacting part of the two to two scattering
amplitude

$(2\pi)^{4}\delta^{(4)}(p_{1}^{\mu}+p_{2}^{\mu}-p_{3}^{\mu}-p_{4}^{\mu})\times
T_{12\rightarrow
34}{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p_{1},p_{2},p_{3},p_{4})\equiv{}_{free}\langle\kappa_{3},\kappa_{4}|T|\kappa_{1},\kappa_{2}\rangle_{free}.$
(54)

Since the (interacting) scattering amplitude is defined via the one particle
states, all the 4-momenta are on-shell

$p_{i}^{2}=-m_{i}^{2}.$ (55)

We study these amplitudes and their properties in depth in section 2.3. We
sometimes drop the subscript $12\rightarrow 34$ when it is clear from the
context which scattering process we describe. However, it is necessary to keep
this subscript when relating amplitudes describing different processes. Using
(52) we can relate the scattering amplitude (53) with its interacting part
(54) as

$S_{12\rightarrow
34}{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p_{1},p_{2},p_{3},p_{4})=\\\
\left[\frac{{}_{free}\langle\kappa_{3},\kappa_{4}|\kappa_{1},\kappa_{2}\rangle_{free}}{(2\pi)^{4}\delta^{(4)}(p_{1}^{\mu}+p_{2}^{\mu}-p_{3}^{\mu}-p_{4}^{\mu})}\right]+iT_{12\rightarrow
34}{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p_{1},p_{2},p_{3},p_{4}),$
(56)

where the first term $[\ldots]$ is a schematic expression which can be
evaluated straightforwardly.171717 For example in the COM frame defined in
(65) using the normalization condition in spherical coordinates (35) it is
straightforward to write
$\left[\frac{{}_{free}\langle\kappa_{3},\kappa_{4}|\kappa_{1},\kappa_{2}\rangle_{free}}{(2\pi)^{4}\delta^{(4)}(p_{1}^{\mu}+p_{2}^{\mu}-p_{3}^{\mu}-p_{4}^{\mu})}\right]_{com}=\frac{8\pi\sqrt{s}}{\sqrt{\boldsymbol{p}\boldsymbol{p}^{\prime}}}\times\frac{\delta(\theta)\delta(\phi)}{\sin\theta}\delta_{m_{1}m_{3}}\delta_{m_{2}m_{4}}\delta_{j_{1}j_{3}}\delta_{j_{2}j_{4}}\times\delta_{\lambda_{1}\lambda_{3}}\delta_{\lambda_{2}\lambda_{4}}.$
Similarly in case of identical particles with mass $m$ and spin $j$ using
(468) we have
$\left[\frac{{}_{free}\langle\kappa_{3},\kappa_{4}|\kappa_{1},\kappa_{2}\rangle_{free}}{(2\pi)^{4}\delta^{(4)}(p_{1}^{\mu}+p_{2}^{\mu}-p_{3}^{\mu}-p_{4}^{\mu})}\right]_{com}=\frac{32\pi^{2}\sqrt{s}}{\sqrt{s-4m^{2}}}\times\left(\frac{\delta(\theta)\delta(\phi)}{\sin\theta}\delta_{\lambda_{1}\lambda_{3}}\delta_{\lambda_{2}\lambda_{4}}+(-1)^{2j}\frac{\delta(\pi-\theta)\delta(\phi+\pi)}{\sin(\pi-\theta)}\delta_{\lambda_{1}\lambda_{4}}\delta_{\lambda_{2}\lambda_{3}}\right).$
(57) This piece is not a function but a distribution. Due to the relation
(56), in practice we will never need to discuss the full amplitude $S$ and
instead we will only need the interacting $T$ amplitude.

##### Partial amplitudes

The second matrix element we need is between the $in$ and $out$ states in the
irreducible representation

${}_{out}\langle
c^{\prime},\vec{p}\,^{\prime};{\ell}^{\prime},{\lambda}^{\prime};\gamma^{\prime}|c,\vec{p};{\ell},{\lambda};\gamma\rangle_{in}={}_{free}\langle
c^{\prime},\vec{p}\,^{\prime};{\ell}^{\prime},{\lambda}^{\prime};\gamma^{\prime}|S|c,\vec{p};{\ell},{\lambda};\gamma\rangle_{free}.$
(58)

Again factoring out the overall delta function due to translational invariance
we can define the partial amplitude with a definite spin $\ell$ as

$(2\pi)^{4}\delta^{(4)}(p^{\mu}-{p^{\prime}}^{\mu})\delta_{\ell\ell^{\prime}}\delta_{\lambda\lambda^{\prime}}\times
S_{\ell}{}_{\gamma}^{\gamma^{\prime}}(c)={}_{free}\langle
c^{\prime},\vec{p}\,^{\prime};{\ell}^{\prime},{\lambda}^{\prime};\gamma^{\prime}|S|c,\vec{p};{\ell},{\lambda};\gamma\rangle_{free}.$
(59)

Equivalently we can define the interacting part of the partial amplitude as

$(2\pi)^{4}\delta^{(4)}(p^{\mu}-{p^{\prime}}^{\mu})\delta_{\ell\ell^{\prime}}\delta_{\lambda\lambda^{\prime}}\times
T_{\ell}{}_{\gamma}^{\gamma^{\prime}}(c)={}_{free}\langle
c^{\prime},\vec{p}\,^{\prime};{\ell}^{\prime},{\lambda}^{\prime};\gamma^{\prime}|T|c,\vec{p};{\ell},{\lambda};\gamma\rangle_{free}.$
(60)

We prove that the left-hand side of (59) and (60) must contain
$\delta_{\ell\ell^{\prime}}\delta_{\lambda\lambda^{\prime}}$ factor in
appendix D.1. In a generic situation due to (29) the two partial amplitudes
are simply related as

$S_{\ell}{}_{\gamma}^{\gamma^{\prime}}(c)=\delta_{\gamma^{\prime}\gamma}+iT_{\ell}{}_{\gamma}^{\gamma^{\prime}}(c).$
(61)

In practice we will only need to consider partial amplitudes where the irreps
come from the decomposition of COM two particle states (36). In that case the
additional labels $\gamma$ are multiplicities given by (31). The Kronecker
delta for distinct particles is given by (32) and for identical particles by
(44). We examine partial amplitudes and their properties in detail in section
2.5. In addition, we also derive the relation between partial and scattering
amplitudes.

###  2.3 Scattering amplitudes and crossing

In this section we carefully study various aspects of the scattering
amplitudes (53) and its interacting part (54).

We start with the transformation property under the Poincaré group. It
directly follows from the transformation property of each state given by (18).
In the most generic case it reads as

$\displaystyle\leavevmode\resizebox{422.77661pt}{}{$\displaystyle{T_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p_{1},p_{2},p_{3},p_{4})=\sum_{\lambda_{i}^{\prime}}\mathscr{D}^{(j_{1})}_{\lambda^{\prime}_{1}\lambda_{1}}(\vec{\omega}_{1})\mathscr{D}^{(j_{2})}_{\lambda^{\prime}_{2}\lambda_{2}}(\vec{\omega}_{2})\mathscr{D}^{*(j_{3})}_{\lambda^{\prime}_{3}\lambda_{3}}(\vec{\omega}_{3})\mathscr{D}^{*(j_{4})}_{\lambda^{\prime}_{4}\lambda_{4}}(\vec{\omega}_{4})T_{\lambda_{1}^{\prime},\lambda_{2}^{\prime}}^{\lambda_{3}^{\prime},\lambda_{4}^{\prime}}(p_{1}^{\prime},p_{2}^{\prime},p_{3}^{\prime},p_{4}^{\prime})}$},$
(62)

where $\vec{\omega}_{i}\equiv(\alpha_{i},\beta_{i},\gamma_{i})$ are the Wigner
angles for each one particle state defining the (interacting) scattering
amplitude.

Let us now introduce the Mandelstam variables which are invariant quantities
under Lorentz transformations

$s\equiv-(p_{1}+p_{2})^{2},\quad t\equiv-(p_{1}-p_{3})^{2},\quad
u\equiv-(p_{1}-p_{4})^{2},\quad s+t+u=\sum_{i=1}^{4}m_{i}^{2}.$ (63)

Using these variables one can split the scattering amplitude (54) into parts
invariant under Lorentz transformation and parts transforming non-trivially,

$T_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p_{1},p_{2},p_{3},p_{4})=\sum_{I=1}^{(2j_{1}+1)\ldots(2j_{4}+1)}T_{I}(s,t,u)\times\mathbb{T}_{I}{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p_{1},p_{2},p_{3},p_{4}).$
(64)

We refer to the quantities $T_{I}(s,t,u)$ as the scalar components of the
scattering amplitudes and $\mathbb{T}_{I}$ are called tensor structures. The
latter ensures the correct transformation property of the amplitude as
dictated by (62). We abuse notation and call both the full amplitude and its
scalar components by the same symbol $T$. It should be clear which is which by
the presence of indices and arguments. We construct tensor structures
explicitly for a particular example in section 3.4.1. For a general approach
see appendix H.

Instead of defining tensor structures and scalar components of the amplitude
as in (64) one can evaluate the full amplitude in a particular frame. The
standard choice for this frame is the center of mass (COM) defined as

$\displaystyle p_{1}^{\text{com}}=(E_{1},0,0,+\boldsymbol{p}),$ (65)
$\displaystyle p_{2}^{\text{com}}=(E_{2},0,0,-\boldsymbol{p}),$ $\displaystyle
p_{3}^{\text{com}}=(E_{3},+\boldsymbol{p}^{\prime}\sin\theta,0,+\boldsymbol{p}^{\prime}\cos\theta),$
$\displaystyle
p_{4}^{\text{com}}=(E_{4},-\boldsymbol{p}^{\prime}\sin\theta,0,-\boldsymbol{p}^{\prime}\cos\theta).$

Here the angle $\theta\in[0,\pi]$. All the parameters in (65) can be expressed
in terms of the Mandelstam variables. In the simplest case where all four
particles have the same mass $m$ we have

$E_{i}=\frac{\sqrt{s}}{2},\quad\boldsymbol{p}=\boldsymbol{p}^{\prime}=\sqrt{\frac{s}{4}-m^{2}},\quad\sin\theta=\frac{2\sqrt{tu}}{s-4m^{2}},\quad\cos\theta=\frac{t-u}{s-4m^{2}}.$
(66)

Instead of using $(s,t,u)$ variables to characterize the scattering process
one can also use $(s,\theta)$ by using the relations

$t=-\frac{s-4m^{2}}{2}(1-\cos\theta),\quad
u=-\frac{s-4m^{2}}{2}(1+\cos\theta).$ (67)

From these expressions it is clear that the physical range of the Mandelstam
variables is

$s\geq 4m^{2},\quad t\in[4m^{2}-s,\,0],\quad u\in[4m^{2}-s,\,0].$ (68)

For the definition of the center of mass frame for the most generic case see
appendix D.3.

Using the COM frame we can define the scalar components of the interacting
scattering amplitude in either of the two equivalent ways181818In order to see
the equivalence of two definitions simply compare (54) evaluated in the COM
frame (65) with (33).

$\displaystyle T_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u)$
$\displaystyle\equiv
T_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p^{\text{com}}_{1},p^{\text{com}}_{2},p^{\text{com}}_{3},p^{\text{com}}_{4}),$
(69) $\displaystyle
T_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u)\times(2\pi)^{4}\delta^{(4)}(0)$
$\displaystyle\equiv{}_{free}\langle(\boldsymbol{p}^{\prime},\theta,0);\lambda_{3},\lambda_{4}|T|(\boldsymbol{p},0,0);\lambda_{1},\lambda_{2}\rangle_{free},$

As in (64) we abuse notation and call the full (interacting) amplitude in a
generic frame and its particular form in the COM frame by the same symbol. The
difference should always be clear from the arguments. Given the interacting
scattering amplitude in the COM frame one can unambiguously obtain the
interacting scattering amplitude in a generic frame by using (62). As an
example, let us apply a rotation by an angle $\phi$ around the z axis to (69).
One gets the following relation

$e^{i(\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4})\phi}T_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u)\times(2\pi)^{4}\delta^{(4)}(0)=\\\
{}_{free}\langle(\boldsymbol{p}^{\prime},\theta,\phi);\lambda_{3},\lambda_{4}|T|(\boldsymbol{p},0,0);\lambda_{1},\lambda_{2}\rangle_{free}$
(70)

This result will be useful when we compute the partial wave decomposition of
the scattering amplitude in section 2.5.

We note that the scalar components of the scattering amplitude defined in (64)
and the COM frame amplitude defined in (69) are simply related by a linear
transformation which depends only on $s$, $t$ and $u$ variables. This relation
can be found by evaluating (64) in the center of mass frame and comparing with
(69). We will see an explicit example of this in section 3.4.1.

#### Parity and time-reversal

Let us now discuss additional constraints which appear if the system is parity
or time-reversal invariant. In terms of the $S$ operator (and hence also the
$T$ operator), the following must hold

${\mathcal{P}}S{\mathcal{P}}^{\dagger}=S,\quad{\mathcal{T}}S{\mathcal{T}}^{\dagger}=S^{\dagger}.$
(71)

At the level of COM amplitudes these translate into the following conditions

$\displaystyle T_{12\rightarrow
34}{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u)$
$\displaystyle=\eta_{1}\eta_{2}\eta_{3}^{*}\eta_{4}^{*}(-1)^{j_{1}+j_{2}+j_{3}+j_{4}}(-1)^{\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4}}T_{12\rightarrow
34}{}_{-\lambda_{1},-\lambda_{2}}^{-\lambda_{3},-\lambda_{4}}(s,t,u),$ (72)
$\displaystyle T_{12\rightarrow
34}{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u)$
$\displaystyle=\varepsilon_{1}^{*}\varepsilon_{2}^{*}\varepsilon_{3}\varepsilon_{4}(-1)^{\lambda_{1}-\lambda_{2}-\lambda_{3}+\lambda_{4}}T_{34\rightarrow
12}{}_{\lambda_{3},\lambda_{4}}^{\lambda_{1},\lambda_{2}}(s,t,u).$ (73)

We derive them in appendix B.

#### Crossing

Our goal now is to formulate crossing equations. The case of particles with
generic spin was first addressed in Trueman:1964zza . It was further discussed
in Martin:102663 ; Hara:1970gc ; Hara:1971kj . For a recent discussion see
also deRham:2017zjm . All the results presented below are carefully derived in
appendix E.

Consider the scattering process of four particles. We denote it schematically
by

$12\rightarrow 34.$ (74)

Each particle is characterized by its mass, spin, helicity and 3-momentum, for
instance $1=(m_{1},j_{1},\lambda_{1};\vec{p}_{1})$. There exist five other
related process

$\displaystyle\bar{4}2\rightarrow
3\bar{1},\qquad\qquad\bar{3}2\rightarrow\bar{1}4,$ (75) $\displaystyle
1\bar{3}\rightarrow\bar{2}4,\qquad\qquad 1\bar{4}\rightarrow 3\bar{2}.$

together with $\bar{3}\bar{4}\rightarrow\bar{1}\bar{2}$. Here if particle $i$
has a charge (or more generally transforms in some representation of a global
group) then particle $\bar{i}$ has the opposite charge (transforms in the
conjugate representation). In other words $\bar{i}$ is the antiparticle of
particle $i$. The scattering process (74) is described by the following
interacting part of the amplitude

$T_{12\rightarrow
34}{\;}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p_{1},p_{2},p_{3},p_{4}),$
(76)

whereas the scattering processes in (75) are described by

$\displaystyle T_{\bar{4}2\rightarrow
3\bar{1}}{\;}_{\lambda_{4},\lambda_{2}}^{\lambda_{3},\lambda_{1}}(p_{4},p_{2},p_{3},p_{1}),\qquad
T_{\bar{3}2\rightarrow\bar{1}4}{\;}_{\lambda_{3},\lambda_{2}}^{\lambda_{1},\lambda_{4}}(p_{3},p_{2},p_{1},p_{4}),$
(77) $\displaystyle
T_{1\bar{3}\rightarrow\bar{2}4}{\;}_{\lambda_{1},\lambda_{3}}^{\lambda_{2},\lambda_{4}}(p_{1},p_{3},p_{2},p_{4}),\qquad
T_{1\bar{4}\rightarrow
3\bar{2}}{\;}_{\lambda_{1},\lambda_{4}}^{\lambda_{3},\lambda_{2}}(p_{1},p_{4},p_{3},p_{2}).$

Here all 4-momenta have positive energies $p_{i}^{0}>0$ and are on-shell (55).

Under the assumption that the amplitudes in (76) and (77) can be analytically
continued in $p_{i}$ and defined in some common domain of $p_{i}$ values, one
can write a set of crossing equations

$\displaystyle T_{12\rightarrow
34}{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p_{1},p_{2},p_{3},p_{4})$
$\displaystyle=\epsilon_{14}T_{\bar{4}2\rightarrow
3\bar{1}}{}_{-\lambda_{4},+\lambda_{2}}^{+\lambda_{3},-\lambda_{1}}(-p_{4},p_{2},p_{3},-p_{1}),$
(78) $\displaystyle T_{12\rightarrow
34}{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p_{1},p_{2},p_{3},p_{4})$
$\displaystyle=\epsilon_{23}T_{1\bar{3}\rightarrow\bar{2}4}{}_{+\lambda_{1},-\lambda_{3}}^{-\lambda_{2},+\lambda_{4}}(p_{1},-p_{3},-p_{2},p_{4}).$
(79)

which are referred to as the $s-t$ crossing equations and

$\displaystyle T_{12\rightarrow
34}{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p_{1},p_{2},p_{3},p_{4})$
$\displaystyle=\epsilon_{13}T_{\bar{3}2\rightarrow\bar{1}4}{}_{-\lambda_{3},+\lambda_{2}}^{-\lambda_{1},+\lambda_{4}}(-p_{3},p_{2},-p_{1},p_{4}),$
(80) $\displaystyle T_{12\rightarrow
34}{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p_{1},p_{2},p_{3},p_{4})$
$\displaystyle=\epsilon_{24}T_{1\bar{4}\rightarrow
3\bar{2}}{}_{+\lambda_{1},-\lambda_{4}}^{+\lambda_{3},-\lambda_{2}}(p_{1},-p_{4},p_{3},-p_{2}),$
(81)

which are referred to as the $s-u$ crossing equations. Here $\epsilon_{14}$,
$\epsilon_{23}$, $\epsilon_{13}$ and $\epsilon_{24}$ are some phases which can
be fixed in principle using the LSZ procedure. Finally the amplitude for the
process $\bar{3}\bar{4}\rightarrow\bar{1}\bar{2}$ can be related to that of
the $12\rightarrow 34$ process by using (80) and (81) one after the other.
There are two distinct ways for the analytic continuation of amplitudes
discussed in section E.1. In writing (78) - (81) we have made a particular
choice, more precisely the one given by (531).

Let us now focus on the 23- and 24-crossing equations given by (79) and (81)
respectively and evaluate them in the standard COM (65). We can then use the
definition of the center of mass amplitude (69) in the left-hand side of (79)
and (81) but not in the right-hand side. The right-hand side is not in the
center of mass frame of particles 13 and 14 respectively. In order to bring
them to this frame we need to perform a Lorentz transformation. The
23-crossing equation then reads

$\displaystyle T_{12\rightarrow
34}{\;}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u)$
$\displaystyle=\epsilon_{23}^{\prime}\,\sum_{\lambda_{i}^{\prime}}e^{i\pi(\lambda_{1}^{\prime}+\lambda_{4}^{\prime})}$
$\displaystyle\times
d^{(j_{1})}_{\lambda^{\prime}_{1}\lambda_{1}}(\alpha_{1})d^{(j_{2})}_{\lambda^{\prime}_{2}\lambda_{2}}(\alpha_{2})d^{(j_{3})}_{\lambda^{\prime}_{3}\lambda_{3}}(\alpha_{3})d^{(j_{4})}_{\lambda^{\prime}_{4}\lambda_{4}}(\alpha_{4})T_{1\bar{3}\rightarrow\bar{2}4}{\;}_{\lambda_{1}^{\prime},\lambda_{3}^{\prime}}^{\lambda_{2}^{\prime},\lambda_{4}^{\prime}}(t,s,u),$
(82)

where the angles $\alpha_{i}$ in the equal mass case are given by

$\displaystyle+\cos\alpha_{1}=-\cos\alpha_{2}=-\cos\alpha_{3}=+\cos\alpha_{4}$
$\displaystyle=+\frac{st}{\sqrt{s(s-4m^{2})}\sqrt{t(t-4m^{2})}},$ (83)
$\displaystyle+\sin\alpha_{1}=-\sin\alpha_{2}=+\sin\alpha_{3}=-\sin\alpha_{4}$
$\displaystyle=-\frac{2m\,\sqrt{stu}}{\sqrt{s(s-4m^{2})}\sqrt{t(t-4m^{2})}}.$

For the most general case see (597) and (598). Similarly the $24$-crossing
reads

$\displaystyle T_{12\rightarrow
34}{\;}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u)$
$\displaystyle=\epsilon_{24}^{\prime}\,\sum_{\lambda_{i}^{\prime}}e^{i\pi(\lambda_{1}^{\prime}+\lambda_{3}^{\prime})}$
$\displaystyle\times
d^{(j_{1})}_{\lambda^{\prime}_{1}\lambda_{1}}(\beta_{1})d^{(j_{2})}_{\lambda^{\prime}_{2}\lambda_{2}}(\beta_{2})d^{(j_{3})}_{\lambda^{\prime}_{3}\lambda_{3}}(\beta_{3})d^{(j_{4})}_{\lambda^{\prime}_{4}\lambda_{4}}(\beta_{4})T_{1\bar{4}\rightarrow
3\bar{2}}{\;}_{\lambda_{1}^{\prime},\lambda_{4}^{\prime}}^{\lambda_{3}^{\prime},\lambda_{2}^{\prime}}(u,t,s),$
(84)

where the angles $\beta_{i}$ in the equal mass case are given by

$\displaystyle+\cos\beta_{1}=-\cos\beta_{2}=+\cos\beta_{3}=-\cos\beta_{4}$
$\displaystyle=+\frac{su}{\sqrt{s(s-4m^{2})}\sqrt{u(u-4m^{2})}},$ (85)
$\displaystyle+\sin\beta_{1}=-\sin\beta_{2}=-\sin\beta_{3}=+\sin\beta_{4}$
$\displaystyle=+\frac{2m\,\sqrt{stu}}{\sqrt{s(s-4m^{2})}\sqrt{u(u-4m^{2})}}.$

For the most general case see (612) and (613). Notice that small Wigner
d-matrices are $4\pi$ periodic. As a result the angles $\alpha_{i}$ and
$\beta_{i}$ are not completely fixed by the equations (83) and (85) since
sines and cosines are $2\pi$ periodic. Different choices of $\alpha_{i}$ and
$\beta_{i}$ in the crossing equations satisfying (83) and (85) lead to
different additional phases which we combine with $\epsilon_{23}$ and
$\epsilon_{24}$ introduced in (79) and (81) and denote by
$\epsilon_{23}^{\prime}$ and $\epsilon_{24}^{\prime}$. For each scattering
process these phases can be computed at the leading order in perturbation
theory and since the phases are purely kinematic in nature, the result can
then be used non-perturbatively. We also notice that the form of crossing
equations in a general frame, (79) and (81), depends on the choice of the
analytic continuation, however both analytic continuations lead to the same
expressions in the center of mass frame (82) and (84).

It is interesting to consider the case when all four particles are massless.
Assuming the physical domain (68) of the Mandelstam variables for the process
$12\rightarrow 34$ in the limit $m\rightarrow 0$ the expressions (83) and (85)
lead to

$\displaystyle\alpha_{1}=\pi,\quad\alpha_{2}=0,\quad\alpha_{3}=0,\quad\alpha_{4}=\pi,$
(86)
$\displaystyle\beta_{1}=\pi,\quad\beta_{2}=0,\quad\beta_{3}=\pi,\quad\beta_{4}=0.$
(87)

Using the following properties of the small Wigner d-matrices

$d^{(j)}_{\lambda^{\prime}\lambda}(0)=\delta_{\lambda^{\prime},\lambda},\qquad
d^{(j)}_{\lambda^{\prime}\lambda}(\pi)=(-1)^{j-\lambda}\delta_{\lambda^{\prime},\,-\lambda}$
(88)

we get the following crossing equations

$\displaystyle T_{12\rightarrow
34}{\;}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u)$
$\displaystyle=\epsilon_{23}^{\prime\prime}T_{1\bar{3}\rightarrow\bar{2}4}{\;}_{-\lambda_{1},+\lambda_{3}}^{+\lambda_{2},-\lambda_{4}}(t,s,u),$
(89) $\displaystyle T_{12\rightarrow
34}{\;}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u)$
$\displaystyle=\epsilon_{24}^{\prime\prime}T_{1\bar{4}\rightarrow
3\bar{2}}{\;}_{-\lambda_{1},+\lambda_{4}}^{-\lambda_{3},+\lambda_{2}}(u,t,s),$
(90)

where $\epsilon_{23}^{\prime\prime}$ and $\epsilon_{24}^{\prime\prime}$ are
some new phases.

#### Neutral identical particles

In some practical applications one is required to study scattering processes
of identical neutral particles with mass $m$ and spin $j$. We discuss this
case in great detail in appendix C. Here we state only the most important
results.

We define the scattering amplitude of identical neutral particles in a generic
and in the center of mass frames as

$\displaystyle
T_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p_{1},p_{2},p_{3},p_{4})$
$\displaystyle\times(2\pi)^{4}\delta^{(4)}(p_{1}^{\mu}+p_{2}^{\mu}-p_{3}^{\mu}-p_{4}^{\mu})\equiv{}_{id\;free}\langle\kappa_{3},\kappa_{4}|T|\kappa_{1},\kappa_{2}\rangle_{id\;free},$
(91) $\displaystyle
T_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u)$
$\displaystyle\times(2\pi)^{4}\delta^{(4)}(0)\equiv{}_{id\;free}\langle(\boldsymbol{p}^{\prime},\theta,0);\lambda_{3},\lambda_{4}|T|(\boldsymbol{p},0,0);\lambda_{1},\lambda_{2}\rangle_{id\;free}.$

In comparison with (53) we drop the subscript $12\rightarrow 34$ since it does
not carry any useful information anymore. The two particle states formed from
identical particles were defined in (41). According to (40) they are
(anti)symmetric under the exchange of particles 12 and 34. Due to this
condition the following equations hold

$\displaystyle T_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u)$
$\displaystyle=(-1)^{-\lambda_{1}+\lambda_{2}-\lambda_{3}+\lambda_{4}}T_{\lambda_{2},\lambda_{1}}^{\lambda_{3},\lambda_{4}}(s,u,t),$
(92) $\displaystyle
T_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u)$
$\displaystyle=(-1)^{+\lambda_{1}-\lambda_{2}-\lambda_{3}+\lambda_{4}}T_{\lambda_{1},\lambda_{2}}^{\lambda_{4},\lambda_{3}}(s,u,t).$
(93)

See appendix C.3 for details. Since the variables $t$ and $u$ are flipped in
the left- and right-hand side we refer to them as the t-u crossing equations.
Applying them twice we get the following kinematic constraint

$T_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u)=T_{+\lambda_{2},+\lambda_{1}}^{+\lambda_{4},+\lambda_{3}}(s,t,u).$
(94)

In the case of distinct particles the crossing equations (82) and (84)
establish relations between different amplitudes. When particles are identical
there is only a single amplitude (91) and the crossing equations become
constraints on this single amplitude.

One can also combine together (80), (81) and (78), (79) to obtain

$\displaystyle
T_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p_{1},p_{2},p_{3},p_{4})$
$\displaystyle=\,T_{-\lambda_{3},-\lambda_{4}}^{-\lambda_{1},-\lambda_{2}}(-p_{3},-p_{4},-p_{1},-p_{2}),$
(95) $\displaystyle
T_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p_{1},p_{2},p_{3},p_{4})$
$\displaystyle=\,T_{-\lambda_{4},-\lambda_{3}}^{-\lambda_{2},-\lambda_{1}}(-p_{4},-p_{3},-p_{2},-p_{1}).$
(96)

Using the techniques of appendix E.4 one can bring both sides of these two
equations to the center of mass frame and show that

$\displaystyle T_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u)$
$\displaystyle=\,T_{-\lambda_{3},-\lambda_{4}}^{-\lambda_{1},-\lambda_{2}}(s,t,u),$
(97) $\displaystyle
T_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u)$
$\displaystyle=\,T_{-\lambda_{4},-\lambda_{3}}^{-\lambda_{2},-\lambda_{1}}(s,t,u).$
(98)

We refer to the conditions (94), (97) and (98) as the kinematic constraints
associated to the simultaneous permutation of particles (12)(34), (13)(24) and
(14)(23) respectively.

As presented, the transition from (95), (96) to (97), (98) is very difficult.
Strictly speaking the equations (97) and (98) might have some overall helicity
independent phase which we have little control over. There is a much simpler
way of deriving (97) however. As discussed in appendix B.3 one can use the CPT
theorem to obtain (97). The constraint (98) follows from combining (97) with
(94). This discussion indicates (but not proves) that CPT transformation is
equivalent to using crossing twice.

###  2.4 Counting scattering amplitudes

It is useful to count kinematically independent amplitudes in various cases.
Looking at the definition of the center of mass amplitude (69) it is obvious
that the number of all possible amplitudes $N_{4}$ for four different massive
particles is

$N_{4}=(2j_{1}+1)(2j_{2}+1)(2j_{3}+1)(2j_{4}+1),$ (99)

since there are no restrictions on the helicity values. Using (72) we can
further split the amplitudes into parity even and parity odd ones. This is
done by taking appropriate linear combinations of

$T{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}\qquad\text{and}\qquad
T{}_{-\lambda_{1},-\lambda_{2}}^{-\lambda_{3},-\lambda_{4}}.$ (100)

Since the scattering amplitude must always contain an even number of fermions
and due to (25) and the comment below, the product of intrinsic parities

$\eta_{1}\eta_{2}\eta_{3}^{*}\eta_{4}^{*}$ (101)

entering (72) is either $+1$ or $-1$191919Parity invariance implies that
$\eta_{1}\eta_{2}=\pm\eta_{3}\eta_{4}$.. Let us assume for concreteness that
the product in (101) is $+1$. (In the other case when (101) is $-1$ the role
of parity even and odd amplitudes constructed from (100) simply flips.) Having
constructed the appropriate linear combination from (100) the counting follows
straightforwardly. In the case when there are two or four fermions we have the
same number of parity even $N_{4}^{+}$ and parity odd amplitudes $N_{4}^{-}$
which read

$N_{4}^{\pm}=\frac{1}{2}(2j_{1}+1)(2j_{2}+1)(2j_{3}+1)(2j_{4}+1).$ (102)

In case all particles are bosons we get

$N_{4}^{\pm}=\frac{1}{2}\left((2j_{1}+1)(2j_{2}+1)(2j_{3}+1)(2j_{4}+1)\pm
1\right).$ (103)

The difference between (102) and (102) arises due to the fact that when all
particles are bosons there is always a parity even amplitude with all the zero
helicities $T{}_{00}^{00}$. The latter is no longer true in the presence of
fermions. Clearly, the following is obeyed

$N_{4}=N_{4}^{+}+N_{4}^{-}.$ (104)

If we impose parity as symmetry of our system only parity even amplitudes will
survive.

It is more difficult to perform general counting when particles are identical
since we need to take into account the relations (94) - (98). However it easy
to do for any particular case of interest by forming a linear system of all
the constraints (due presence of parity, identical and massless particles),
solving it and counting the number of independent amplitudes. For example in
the case of identical massive Majorana particles and identical massive spin
one particles we have

identical Majorana fermions: $\displaystyle N_{4}^{+}=5,\quad\;\;N_{4}^{-}=2,$
(105) identical spin one bosons: $\displaystyle N_{4}^{+}=17,\quad
N_{4}^{-}=10.$ (106)

Time-reversal does not further reduce these numbers. This can be intuitively
understood by noticing that P implies T invariance for neutral particles due
to the CPT symmetry.

In case a particle with spin $j$ is massless its helicity can only take two
values $+j$ and $-j$. As a result if all four particles in the scattering
process are massless and carry a non-zero spin we always have, independently
of the precise values of spin,

$\text{four different massless particles:}\quad N_{4}=2^{4}=16.$ (107)

In the case when all particles are identical, massless and carry a non-zero
spin we have

$\text{identical massless particles:}\quad N_{4}^{+}=5,\quad N_{4}^{-}=2.$
(108)

It was proposed in section 6 of Costa:2011mg  that the number of scattering
amplitudes in $d$ dimensions should be equal to the number of tensor
structures of four-point functions in $d-1$ conformal field theories, where
massless particles correspond to conserved operators. This correspondence got
an explanation in Kravchuk:2016qvl  where it was noted that the conformal
frame analysis of four point function is equivalent group theoretically to the
center of mass analysis of scattering amplitudes. When parity is involved or
particles are identical or massless the matching of CFT and amplitude counting
is more difficult to confirm. Here we explicitly verify this correspondence on
some particular examples. For instance (102) and (103) are in a perfect
agreement with the formulas (4.47) and (4.49) in Kravchuk:2016qvl , results
(105) and (106) match (2.40) and (4.58) in Kravchuk:2016qvl , finally the very
special case (108) matches (3.24) in Kravchuk:2016qvl .202020The number of
parity even conserved tensor structures in conformal field theories was first
computed in Dymarsky:2013wla , see table 1. In $d=3$ it is 5 as expected.

###  2.5 Partial amplitudes

As we explained in section 2.1.2, the two particle states are in a reducible
representation of the Poincaré group and can be expressed as a direct sum of
Poincaré irreps according to (28) or (36) (in the special case of COM states).
This leads to a decomposition (often referred to as the partial wave
decomposition) of scattering amplitudes into partial amplitudes.

We start from the definition of the center of mass amplitude (70) and
decompose the two particle states there according to (36). As a result we get

$e^{i(\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4})\phi}T_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u)\times(2\pi)^{4}\delta^{(4)}(0)=\sum_{\ell,\ell^{\prime},\lambda,\lambda^{\prime}}C_{\ell^{\prime}}(\boldsymbol{p}^{\prime})C_{\ell}(\boldsymbol{p})\\\
e^{-i(\lambda_{3}+\lambda_{4}-\lambda)\phi}d^{(\ell^{\prime})}_{\lambda^{\prime}\lambda_{34}}(\theta)\,d^{(\ell)}_{\lambda\lambda_{12}}(0)\langle
c,0,{\ell}^{\prime},{\lambda}^{\prime};\lambda_{3},\lambda_{4}|T|c,0,{\ell},{\lambda};\lambda_{1},\lambda_{2}\rangle,$
(109)

where we have defined

$\lambda_{12}\equiv\lambda_{1}-\lambda_{2},\quad\lambda_{34}\equiv\lambda_{3}-\lambda_{4}.$
(110)

The coefficient $C_{\ell}(\boldsymbol{p})$ was computed in (39). Using it we
can write

$C_{\ell}(\boldsymbol{p}^{\prime})C_{\ell}(\boldsymbol{p})=4\pi(2\ell+1)\frac{\sqrt{s}}{\sqrt{\boldsymbol{p}\boldsymbol{p}^{\prime}}}.$
(111)

Due to the standard property of the small Wigner d-matrix

$d^{(\ell)}_{\lambda\lambda_{12}}(0)=\delta_{\lambda\lambda_{12}}$ (112)

the dependence on the azimuthal angle $\phi$ in (109) cancels out on both
sides and we obtain

$T_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u)\times(2\pi)^{4}\delta^{(4)}(0)=\sum_{\ell,\ell^{\prime},\lambda^{\prime}}C_{\ell^{\prime}}(\boldsymbol{p}^{\prime})C_{\ell}(\boldsymbol{p})d^{(\ell^{\prime})}_{\lambda^{\prime}\lambda_{34}}(\theta)\langle
c,0,{\ell}^{\prime},{\lambda}^{\prime};\lambda_{3},\lambda_{4}|T|c,0,{\ell},\lambda_{12};\lambda_{1},\lambda_{2}\rangle.$

Using the definition of the partial amplitudes (60) the above can be written
in its final form

$\displaystyle T_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u)$
$\displaystyle=\sum_{\ell}C_{\ell}(\boldsymbol{p}^{\prime})C_{\ell}(\boldsymbol{p})d^{(\ell)}_{\lambda_{12}\lambda_{34}}(\theta)T_{\ell}{}^{\lambda_{3},\lambda_{4}}_{\lambda_{1},\lambda_{2}}(s).$
(113)

By using orthogonality of the small Wigner d matrix the decomposition (113)
can be inverted and leads to

$\displaystyle
T_{\ell}{}^{\lambda_{3},\lambda_{4}}_{\lambda_{1},\lambda_{2}}(s)$
$\displaystyle=\frac{2\ell+1}{2C_{\ell}(\boldsymbol{p}^{\prime})C_{\ell}(\boldsymbol{p})}\times\int_{0}^{\pi}d\theta\sin\theta
d^{(\ell)}_{\lambda_{12}\lambda_{34}}(\theta)T_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u).$
(114)

Note that $t$ and $u$ are functions of $s$ and $\theta$. In the equal mass
case one has for instance (67).

Analogously to (114) one can also write the decomposition of the full
amplitude

$\displaystyle
S_{\ell}{}^{\lambda_{3},\lambda_{4}}_{\lambda_{1},\lambda_{2}}(s)$
$\displaystyle=\frac{2\ell+1}{2C_{\ell}(\boldsymbol{p}^{\prime})C_{\ell}(\boldsymbol{p})}\times\int_{0}^{2\pi}\frac{d\phi}{2\pi}\int_{0}^{\pi}d\theta\sin\theta
d^{(\ell)}_{\lambda_{12}\lambda_{34}}(\theta)S_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u).$
(115)

We need to introduce the integration over the azimuthal angle $\phi$ because
the disconnected part of the scattering amplitude depends on it, see (56) and
footnote 17. A simple relation between (114) and (115) partial amplitudes
follows from (61) and (32). It reads212121Here we have used a simple fact that
$\delta_{\lambda_{i}\lambda_{k}}\delta_{\lambda_{j}\lambda_{l}}\delta_{\lambda_{ij}\lambda_{kl}}=\delta_{\lambda_{i}\lambda_{k}}\delta_{\lambda_{j}\lambda_{l}}.$

$S_{\ell}{}^{\lambda_{3},\lambda_{4}}_{\lambda_{1},\lambda_{2}}(s)=\delta_{m_{1}m_{3}}\delta_{m_{2}m_{4}}\delta_{j_{1}j_{3}}\delta_{j_{2}j_{4}}\delta_{\lambda_{1}\lambda_{3}}\delta_{\lambda_{2}\lambda_{4}}+iT_{\ell}{}^{\lambda_{3},\lambda_{4}}_{\lambda_{1},\lambda_{2}}(s).$
(116)

As a consistency check one can obtain this relation in a different way. One
can plug (56) evaluated in the COM frame together with the very first equation
in the footnote 17 into (115). The delta functions cancel all the integrals in
(115) and we simply arrive at (116).

##### Identical particles

In case either the incoming particles or the outgoing particles are identical
we also have relations between the partial amplitudes due to (45). If the
incoming particles 1 and 2 are identical we get

$S_{\ell}{}^{\lambda_{3},\lambda_{4}}_{\lambda_{1},\lambda_{2}}(s)=(-1)^{\lambda_{2}-\lambda_{1}-\ell}S_{\ell}{}^{\lambda_{3},\lambda_{4}}_{\lambda_{2},\lambda_{1}}(s).$
(117)

Similarly if the outgoing particles 3 and 4 are identical we get

$S_{\ell}{}^{\lambda_{3},\lambda_{4}}_{\lambda_{1},\lambda_{2}}(s)=(-1)^{\lambda_{4}-\lambda_{3}-\ell}S_{\ell}{}^{\lambda_{4},\lambda_{3}}_{\lambda_{1},\lambda_{2}}(s).$
(118)

For the case of all four particles being identical, we get
$\sqrt{\boldsymbol{p}\boldsymbol{p}^{\prime}}=\sqrt{s/4-m^{2}}$ due to (66),
and thus

$\displaystyle
S_{\ell}{}^{\lambda_{3},\lambda_{4}}_{\lambda_{1},\lambda_{2}}(s)$
$\displaystyle=\frac{\sqrt{s-4m^{2}}}{32\pi\sqrt{s}}\times\int_{0}^{2\pi}\frac{d\phi}{2\pi}\int_{0}^{\pi}d\theta\sin\theta
d^{(\ell)}_{\lambda_{12}\lambda_{34}}(\theta)S_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u),$
(119) $\displaystyle
T_{\ell}{}^{\lambda_{3},\lambda_{4}}_{\lambda_{1},\lambda_{2}}(s)$
$\displaystyle=\frac{\sqrt{s-4m^{2}}}{32\pi\sqrt{s}}\times\int_{0}^{\pi}d\theta\sin\theta
d^{(\ell)}_{\lambda_{12}\lambda_{34}}(\theta)T_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u).$
(120)

Notice that we have used here (46) which contains an additional $\sqrt{2}$
factor compared to a non-identical particle case. Analogously to (116) there
is a simple relation between the $S$ and $T$ partial amplitudes that follows
from (61) and (44). It reads222222See footnote 21.

$S_{\ell}{}^{\lambda_{3},\lambda_{4}}_{\lambda_{1},\lambda_{2}}(s)=\frac{1}{2}\left(\delta_{\lambda_{1}\lambda_{3}}\delta_{\lambda_{2}\lambda_{4}}+(-1)^{\ell-\lambda_{34}}\delta_{\lambda_{1}\lambda_{4}}\delta_{\lambda_{2}\lambda_{3}}\right)+iT_{\ell}{}^{\lambda_{3},\lambda_{4}}_{\lambda_{1},\lambda_{2}}(s).$
(121)

This result can also be obtained by plugging (56) evaluated in the COM frame
together with the second equation in the footnote 17 into (119). We also
notice that in the case of identical scalar particles we recover the standard
result, see for example equation (10) in Paulos:2017fhb .

##### Parity and time reversal

As usual parity and time reversal invariance lead to additional constraints.
The parity constraint follows directly from (420) and reads

$S_{\ell}{}^{\lambda_{3},\lambda_{4}}_{\lambda_{1},\lambda_{2}}(s)=\eta_{1}\eta_{2}\eta_{3}^{*}\eta_{4}^{*}(-1)^{j_{1}-j_{2}+j_{3}-j_{4}-2\ell}S_{\ell}{}^{-\lambda_{3},-\lambda_{4}}_{-\lambda_{1},-\lambda_{2}}(s).$
(122)

Similarly the time reversal constraint follows from (445) and reads

$S_{12\to
34\;\ell}{}^{\lambda_{3},\lambda_{4}}_{\lambda_{1},\lambda_{2}}(s)=\varepsilon_{1}^{*}\varepsilon_{2}^{*}\varepsilon_{3}\varepsilon_{4}S_{34\to
12\;\ell}{}^{\lambda_{1},\lambda_{2}}_{\lambda_{3},\lambda_{4}}(s).$ (123)

Note that we reintroduced the subscripts $12\rightarrow 34$ and $34\rightarrow
12$ since Time Reversal is a relation between these two distinct processes.

###  2.6 Unitarity

Unitarity of a quantum theory implies that the norm of any state must be non-
negative, _i.e._ $\forall$ $|\psi\rangle$ one has $\langle\psi|\psi\rangle\geq
0$. Now suppose we are given some set of $N$ states $|\psi_{a}\rangle$ with
$a=1,\ldots,N$. Unitarity then requires that the $N\times N$ hermitian matrix
with components $\langle\psi_{a}|\psi_{b}\rangle$ is positive semi-definite,
namely $\left(\langle\psi_{a}|\psi_{b}\rangle\right)\succeq 0$. This
formulation allows us to impose unitarity constraints on the partial
amplitudes straightforwardly.

We start by considering the incoming two particles state (formed from
particles 1 and 2) together with the outgoing two particle state (formed from
particles 3 and 4). We decompose each of these states into irreducible
representations according to (28). We then have the following
$N=N_{\text{in}}+N_{\text{out}}$ independent states transforming in the spin
$\ell$ irreducible representation of the Poincaré group

$\displaystyle|1\rangle_{\text{in}}\equiv|c,\vec{p},{\ell},{\lambda};j_{1},j_{2}\rangle_{\text{in}},\quad$
$\displaystyle|1\rangle_{\text{out}}\equiv|c,\vec{p},{\ell},{\lambda};j_{3},j_{4}\rangle_{\text{out}},$
$\displaystyle|2\rangle_{\text{in}}\equiv|c,\vec{p},{\ell},{\lambda};j_{1},j_{2}-1\rangle_{\text{in}},\quad$
$\displaystyle|2\rangle_{\text{out}}\equiv|c,\vec{p},{\ell},{\lambda};j_{3},j_{4}-1\rangle_{\text{out}},$
$\displaystyle\qquad\qquad\qquad\vdots$
$\displaystyle\qquad\qquad\qquad\vdots$ $\displaystyle|N_{\rm
in}\rangle_{\text{in}}\equiv|c,\vec{p},{\ell},{\lambda};-j_{1},-j_{2}\rangle_{\text{in}},\quad$
$\displaystyle|N_{\rm
out}\rangle_{\text{out}}\equiv|c,\vec{p},{\ell},{\lambda};-j_{3},-j_{4}\rangle_{\text{out}},$
(124)

where the number of incoming and outgoing irreducible states is

$N_{\text{in}}\equiv(2j_{1}+1)(2j_{2}+1),\qquad
N_{\text{out}}\equiv(2j_{3}+1)(2j_{4}+1).$ (125)

Thus for each spin $\ell$ we can construct the following hermitian $N\times N$
matrix

$\mathcal{H}_{\ell}(s)\times(2\pi)^{4}\delta^{4}(p^{\prime}-p)\delta_{\ell^{\prime}\ell}\delta_{\lambda^{\prime}\lambda}=\begin{pmatrix}{}_{\text{in}}\langle
a^{\prime}|b\rangle_{\text{in}}&{}_{\text{in}}\langle
a^{\prime}|b\rangle_{\text{out}}\\\ {}_{\text{out}}\langle
a^{\prime}|b\rangle_{\text{in}}&{}_{\text{out}}\langle
a^{\prime}|b\rangle_{\text{out}},\end{pmatrix}$ (126)

where $s=c^{2}$ is the square of the COM frame total energy and the primes
indicate that the conjugated states to (2.6) have all the labels $c$, $p$,
$\ell$ and $\lambda$ primed. In defining the matrix $\mathcal{H}_{\ell}(s)$ we
have also explicitly factored out the overall delta function appearing due to
translation invariance. According to the discussion above, unitarity then
implies that

$\mathcal{H}_{\ell}(s)\succeq 0,\qquad\forall\ell\quad\text{ and}\quad
s\geq\max(m_{1}+m_{2},m_{3}+m_{4})^{2}.$ (127)

Let us now discuss the components of the matrix (126). Since the Møller
operators introduced in section 2.2 to define the incoming and outgoing states
are isometric, the elements ${}_{\text{in}}\langle
a^{\prime}|b\rangle_{\text{in}}$ and ${}_{\text{out}}\langle
a^{\prime}|b\rangle_{\text{out}}$ are simply fixed by the normalization
condition (29)

$\displaystyle{}_{\text{in}}\langle
c_{1}^{\prime},\vec{p}^{\,\prime};{\ell}^{\prime},{\lambda}^{\prime};\lambda_{1}^{\prime},\lambda_{2}^{\prime}$
$\displaystyle|c_{1},\vec{p};{\ell},{\lambda};\lambda_{1},\lambda_{2}\rangle_{\text{in}}$
$\displaystyle=(2\pi)^{4}\delta^{(4)}({p^{\prime}}^{\mu}-p^{\mu})\,\delta_{\ell^{\prime}\ell}\,\delta_{\lambda^{\prime}\lambda}\,\delta_{\lambda_{1}^{\prime}\lambda_{1}}\,\delta_{\lambda_{2}^{\prime}\lambda_{2}},$
(128) $\displaystyle{}_{\text{out}}\langle
c_{1}^{\prime},\vec{p}^{\,\prime};{\ell}^{\prime},{\lambda}^{\prime};\lambda_{3}^{\prime},\lambda_{4}^{\prime}$
$\displaystyle|c_{1},\vec{p};{\ell},{\lambda};\lambda_{3},\lambda_{4}\rangle_{\text{out}}$
$\displaystyle=(2\pi)^{4}\delta^{(4)}({p^{\prime}}^{\mu}-p^{\mu})\,\delta_{\ell^{\prime}\ell}\,\delta_{\lambda^{\prime}\lambda}\,\delta_{\lambda_{3}^{\prime}\lambda_{3}}\,\delta_{\lambda_{4}^{\prime}\lambda_{4}}.$
(129)

On the other hand, the matrix elements ${}_{\text{out}}\langle
a^{\prime}|b\rangle_{\text{in}}$ and ${}_{\text{in}}\langle
a^{\prime}|b\rangle_{\text{out}}$ form partial amplitudes according to (59)

$\displaystyle{}_{\text{out}}\langle
c_{1}^{\prime},\vec{p}\,^{\prime};{\ell}^{\prime},{\lambda}^{\prime};\lambda_{3},\lambda_{4}$
$\displaystyle|c_{1},\vec{p};{\ell},{\lambda};\lambda_{1},\lambda_{2}\rangle_{\text{in}}$
$\displaystyle=(2\pi)^{4}\delta^{(4)}(p^{\mu}-{p^{\prime}}^{\mu})\delta_{\ell\ell^{\prime}}\delta_{\lambda\lambda^{\prime}}\,S_{\ell}{}^{\lambda_{3},\lambda_{4}}_{\lambda_{1},\lambda_{2}}(s),$
(130) $\displaystyle{}_{\text{in}}\langle
c_{1}^{\prime},\vec{p}^{\prime};{\ell}^{\prime},{\lambda}^{\prime};\lambda_{1},\lambda_{2}$
$\displaystyle|c_{1},\vec{p};{\ell},{\lambda};\lambda_{3},\lambda_{4}\rangle_{\text{out}}$
$\displaystyle=(2\pi)^{4}\delta^{(4)}(p^{\mu}-{p^{\prime}}^{\mu})\delta_{\ell\ell^{\prime}}\delta_{\lambda\lambda^{\prime}}\,S_{\ell}^{*}{}^{\lambda_{3},\lambda_{4}}_{\lambda_{1},\lambda_{2}}(s).$
(131)

Plugging these to (126) we can schematically write the unitarity condition
(127) as

$\begin{pmatrix}\delta_{ab}&S_{\ell}^{\dagger}{}_{ab}\\\
S_{\ell}{}_{ab}&\delta_{ab}\end{pmatrix}\succeq 0.$ (132)

##### Identical particles

In case either the incoming particles 1 and 2 or outgoing particles 3 and 4
are identical there exist relations (117) and (118) between the partial
amplitudes. This introduces redundancies into the condition (132). In order to
remove these redundancies we restrict our attention to states with
$\lambda_{1}\geq\lambda_{2}$ (in case of identical incoming particles) and
$\lambda_{3}\geq\lambda_{4}$ (in case of identical outgoing particles).

##### Parity invariance

In the presence of parity invariance various partial amplitudes entering (132)
are related according to (122). As a consequence the condition (132) again
becomes redundant. One can reformulate the condition (132) in an equivalent
but less redundant way by considering parity eigenstates. One then repeats the
procedure above taking into account the fact that Parity invariance forbids
transitions between parity even and parity odd states. As a result we get two
separate positivity conditions for parity even and parity odd states

$\mathcal{H}_{\ell}^{+}(s)\succeq 0,\qquad\mathcal{H}_{\ell}^{-}(s)\succeq 0.$
(133)

We will see an explicit example of this in section 3.2.

##### Time reversal invariance

Time reversal invariance relates the scattering amplitudes for the process
$12\rightarrow 34$ to the scattering amplitudes for the process $34\rightarrow
12$. Therefore in general time reversal does not have any implications for the
matrix $\mathcal{H}_{\ell}$ since all its elements are scattering amplitudes
for the $12\rightarrow 34$. However in the special case of elastic scattering
i.e $12\rightarrow 12$, time reversal invariance (123) implies that the sub-
matrix $S_{ij}$ is symmetric232323Note that
$|\varepsilon_{1}\varepsilon_{2}|^{2}=1$

$S_{\ell}{}^{\lambda_{3},\lambda_{4}}_{\lambda_{1},\lambda_{2}}(s)=S_{\ell}{}^{\lambda_{1},\lambda_{2}}_{\lambda_{3},\lambda_{4}}(s)$
(134)

###  2.7 Kinematic non-analyticities and constraints

This section is devoted to the study of the behaviour of COM interacting
scattering amplitudes defined in (69) at some very particular values of the
Mandelstam variables $s$, $t$ and $u$. For simplicity we focus on the case of
identical particles with mass $m$ and spin $j$.

Using (66) the center of mass frame (65) can be written in the following way

$\displaystyle
p_{1}^{\text{com}}=\left(\frac{\sqrt{s}}{2},0,0,+\sqrt{\frac{s}{4}-m^{2}}\right),$
(135) $\displaystyle
p_{2}^{\text{com}}=\left(\frac{\sqrt{s}}{2},0,0,-\sqrt{\frac{s}{4}-m^{2}}\right),$
$\displaystyle
p_{3}^{\text{com}}=\left(\frac{\sqrt{s}}{2},+\frac{\sqrt{tu}}{\sqrt{s-4m^{2}}},0,+\frac{t-u}{2\sqrt{s-4m^{2}}}\right),$
$\displaystyle
p_{4}^{\text{com}}=\left(\frac{\sqrt{s}}{2},-\frac{\sqrt{tu}}{\sqrt{s-4m^{2}}},0,-\frac{t-u}{2\sqrt{s-4m^{2}}}\right).$

The center of mass amplitude (69) is strictly defined in the physical domain
of the Mandelstam variables (68). If one attempts however to analytically
continue the COM frame amplitudes to arbitrary complex values of $s$, $t$ and
$u$, as can be already expected from (135), one will encounter non-
analyticities (poles and branch points) at

$s=4m^{2},\quad s=0,\quad t=0,\quad u=0.$ (136)

Some of these non-analyticities have a purely kinematic nature and have
nothing to do with the dynamics of the theory. Our goal here is to isolate
them. In what follows we will formulate the problem of kinematic non-
analyticities precisely and then discuss each of the special points (136) in
detail. (A concrete example will be presented in section 3.1.) For more
details on the subject see chapter 7.3 in Martin:102663  and references
therein.

Recall the definition of helicity states (12), two particle center of mass
states (33) and center of mass amplitudes (69). Using them we can write
explicitly the 1PS describing the center of mass scattering process as

$\displaystyle|m,+\vec{p}_{z};j,\lambda_{1}\rangle$ $\displaystyle\equiv
e^{-i\eta K_{3}}|m,\vec{0};j,\lambda_{1}\rangle,$ (137)
$\displaystyle|m,-\vec{p}_{z};j,\lambda_{2}\rangle$ $\displaystyle\equiv
e^{-i\pi J_{3}}e^{-i(\pi-0)J_{2}}e^{+i\pi J_{3}}e^{-i\eta
K_{3}}|m,\vec{0};j,\lambda_{2}\rangle,$
$\displaystyle|m,+\vec{p}_{\theta};j,\lambda_{3}\rangle$ $\displaystyle\equiv
e^{-i\theta J_{2}}e^{-i\eta K_{3}}|m,\vec{0};j,\lambda_{3}\rangle,$
$\displaystyle|m,-\vec{p}_{\theta};j,\lambda_{4}\rangle$ $\displaystyle\equiv
e^{-i\pi J_{3}}e^{-i(\pi-\theta)J_{2}}e^{+i\pi J_{3}}e^{-i\eta
K_{3}}|m,\vec{0};j,\lambda_{4}\rangle,$

where $\vec{p}_{z}$ is the 3-momentum in the positive $z$-direction,
$\vec{p}_{\theta}$ is the 3-momentum in the x-z plane, $J_{2}$ is the
generator of rotations around the $y$-axis, $K_{3}$ is the boost in the
$z$-direction. In the center of mass frame due to (66) the angle $\theta$ and
the rapidity $\eta$ defined in (13) can be related to the Mandelstam variables
as follows

$\cos\theta=\frac{t-u}{s-4m^{2}},\quad\sin\theta=\frac{2\sqrt{tu}}{s-4m^{2}},\quad\sinh\eta=\frac{\sqrt{s-4m^{2}}}{2m},\quad\cosh\eta=\frac{\sqrt{s}}{2m}.$
(138)

The non-analyticities at $s=4m^{2}$, $s=0$, $t=0$ and $u=0$ of these
expressions enter the center of mass amplitude (69) via (137).

The phenomena of kinematic non-analyticities is closely related to the
phenomena of kinematic constraints. When defining the COM scattering
amplitudes (69) we have used up all of the Lorentz symmetry to bring the
scattering particles to the x-z plane. However at the special points (136) we
get an enhancement of symmetry. For instance at $s=4m^{2}$ the system is
$SO(3)$ symmetric, at $s=0$ the system is $SO(1,1)$ symmetric and at $t=0$ and
$u=0$ the system is $SO(2)$ symmetric. This is straightforward to see from
(135).242424Notice that for the physical range of Mandelstam values
$t=O(s-4m^{2})$ and $u=O(s-4m^{2})$ which means that $\cos\theta$ and
$\sin\theta$ are finite. This is not the case anymore once we promote $s$ and
$t$ to independent complex variables during the analytic continuation process.
As a consequence of the enhanced symmetry the amplitudes rearrange themselves
into irreducible representations of the enhanced symmetry. Only the amplitudes
transforming in the trivial representations are allowed to be present, while
the rest must vanish. The latter requirement leads to kinematic
constraints.252525Kinematic constraints have recently appeared in a similar
context in conformal field theories when studying four point correlation
functions of local primary operators. They received a proper group theoretic
treatment in appendix A of Kravchuk:2016qvl  (see also appendix D of
Karateev:2019pvw ).

##### Special point: $\boldsymbol{s=4m^{2}}$

In order to isolate the singular behavior of the COM amplitudes at $s=4m^{2}$
we perform a simple rewriting of the states (137) as follows262626The relation
$e^{-i\chi J_{2}}e^{i\pi J_{3}}=e^{i\pi J_{3}}e^{+i\chi J_{2}}$ is used for
the states 2 and 4.

$\displaystyle|m,+\vec{p}_{z};j,\lambda_{1}\rangle$
$\displaystyle=e^{-i\frac{\theta}{2}J_{2}}X_{+}e^{+i\frac{\theta}{2}J_{2}}|m,\vec{0};j,\lambda_{1}\rangle,$
(139) $\displaystyle|m,-\vec{p}_{z};j,\lambda_{2}\rangle$
$\displaystyle=e^{-i\frac{\theta}{2}J_{2}}e^{+i\pi
J_{2}}X_{+}e^{+i\frac{\theta}{2}J_{2}}|m,\vec{0};j,\lambda_{2}\rangle,$
$\displaystyle|m,+\vec{p}_{\theta};j,\lambda_{3}\rangle$
$\displaystyle=e^{-i\frac{\theta}{2}J_{2}}X_{-}e^{-i\frac{\theta}{2}J_{2}}|m,\vec{0};j,\lambda_{3}\rangle,$
$\displaystyle|m,-\vec{p}_{\theta};j,\lambda_{4}\rangle$
$\displaystyle=e^{-i\frac{\theta}{2}J_{2}}e^{+i\pi
J_{2}}X_{-}e^{-i\frac{\theta}{2}J_{2}}|m,\vec{0};j,\lambda_{4}\rangle,$

where the operators $X_{\pm}$ are defined as

$X_{\pm}\equiv e^{\pm i\frac{\theta}{2}J_{2}}e^{-i\eta K_{3}}e^{\mp
i\frac{\theta}{2}J_{2}}=e^{-i\left(K_{3}\cos\frac{\theta}{2}\mp
K_{1}\sin\frac{\theta}{2}\right)\eta}.$ (140)

In (140) we have used the commutation properties of the Lorentz generators
(290). Writing

$\cos\frac{\theta}{2}=\sqrt{\frac{-u}{s-4m^{2}}},\qquad\sin\frac{\theta}{2}=\sqrt{\frac{-t}{s-4m^{2}}},$
(141)

we notice that the operators $X_{\pm}$ are completely regular at $s=4m^{2}$
since

$\eta\cos\frac{\theta}{2}=\frac{\sqrt{-u}}{2m}+O(s-4m^{2}),\qquad\eta\sin\frac{\theta}{2}=\frac{\sqrt{-t}}{2m}+O(s-4m^{2}).$
(142)

Plugging the states (139) into the definition of the COM amplitudes (69) and
using the fact that the scattering operator is invariant under the Poincaré
transformations272727The same property holds obviously true for the $T$
operators because of (52). The invariance we use here is
$e^{i\frac{\theta}{2}J_{2}}Te^{-i\frac{\theta}{2}J_{2}}=T.$ (51) and the
transformation property (14) we can write

$T_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u)=\sum_{\lambda^{\prime}}d^{(j)}_{\lambda^{\prime}_{1}\lambda_{1}}\left(-\frac{\theta}{2}\right)d^{(j)}_{\lambda^{\prime}_{2}\lambda_{2}}\left(-\frac{\theta}{2}\right)d^{(j)}_{\lambda^{\prime}_{3}\lambda_{3}}\left(\frac{\theta}{2}\right)d^{(j)}_{\lambda^{\prime}_{4}\lambda_{4}}\left(\frac{\theta}{2}\right)A_{\lambda^{\prime}_{1},\lambda^{\prime}_{2}}^{\lambda^{\prime}_{3},\lambda^{\prime}_{4}}(s,t,u),$
(143)

where the new scattering amplitude $A$ is defined as

$A_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u)\equiv\left(\langle\vec{0},\lambda_{3}|X_{-}^{\dagger}\otimes\langle\vec{0},\lambda_{4}|X_{-}^{\dagger}e^{-i\pi
J_{2}}\right)T\left(X_{+}|\vec{0},\lambda_{1}\rangle\otimes e^{+i\pi
J_{2}}X_{+}|\vec{0},\lambda_{2}\rangle\right).$ (144)

From this explicit expression we see that at $s=4m^{2}$ the amplitude $A$ is
completely regular due to (142), more precisely

$A_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u)=O\left((s-4m^{2})^{0}\right)\,.$
(145)

As a result the non-analytic behavior (141) at $s=4m^{2}$ enters the amplitude
$T$ only through the Wigner d-matrices in (143). Now in order to extract the
precise behaviour of the poles in the COM amplitude in practice we simply need
to expand (143) around $s=4m^{2}$ to the leading order using the explicit
expression of the Wigner d matrices (16), taking into account (141) and the
fact that the functions $A$ are regular at $s=4m^{2}$.

The expression (143) together with (144) can also be used to address kinematic
constraints. Expanding (143) to the next to leading order one finds that some
linear combinations should vanish as $O\big{(}(s-4m^{2})^{1}\big{)}$ instead
of $O\big{(}(s-4m^{2})^{0}\big{)}$ or $O\big{(}(s-4m^{2})^{-1}\big{)}$. For
simple examples as in section 3.1 such linear combinations can be found
manually. For more complicated cases one can invoke group theoretic arguments
similar to ones in Kravchuk:2016qvl .

##### Special points: $\boldsymbol{t=0}$ and $\boldsymbol{u=0}$

Kinematic branch points $\sqrt{t}$ and $\sqrt{u}$ enter the center of mass
scattering amplitudes via $\sin\theta$ given (138). Their presence can be
deduced by looking at the partial wave decomposition (113) and noticing that
the Wigner d matrix there can be written in the form

$d^{(\ell)}_{\lambda^{\prime}\lambda}(\theta)=\left(\cos\frac{\theta}{2}\right)^{|\lambda+\lambda^{\prime}|}\left(\sin\frac{\theta}{2}\right)^{|\lambda-\lambda^{\prime}|}P^{\ell}_{\lambda^{\prime}\lambda}(\cos\theta),$
(146)

where $P^{\ell}_{\lambda^{\prime}\lambda}$ is a polynomial whose precise
definition is irrelevant here but can be deduced from (4.1.19) and (4.1.23) in
Edmonds:1955fi . The important point is that the polynomial
$P^{\ell}_{\lambda^{\prime}\lambda}$ depends only on $\cos\theta$ and
therefore does not have any branch points as can be seen from (138). Using the
above, (141) and (113) we conclude that

$T_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u)\sim
d^{(\ell)}_{\lambda_{12}\lambda_{34}}(\theta)\sim\left(\sqrt{-u}\right)^{|\lambda_{12}+\lambda_{34}|}\left(\sqrt{-t}\right)^{|\lambda_{12}-\lambda_{34}|}.$
(147)

The $SO(2)$ enhancement of symmetry at $t=0$ ($\theta=0$) and $u=0$
($\theta=\pi$), see (135), leads to kinematic constraints at these two points.
This is the simplest case among all the special points and can be easily
addressed in full generality. In order to deduce the implications of this
$SO(2)$ invariance we inject the identity in the form $1=e^{-i\gamma
J_{3}}e^{+i\gamma J_{3}}$ to the left and to the right of the $T$ operator in
(69), where $\gamma$ is some angle. Using the invariance of the $T$ operator
(51) and the fact that the 1PS states along the $z$-direction are the
eigenstates of $J_{3}$, we arrive at

$\displaystyle T{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t=0)$
$\displaystyle=e^{i\gamma(\lambda_{1}-\lambda_{2}-\lambda_{3}+\lambda_{4})}T{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t=0),$
(148) $\displaystyle
T{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,u=0)$
$\displaystyle=e^{i\gamma(\lambda_{1}-\lambda_{2}+\lambda_{3}-\lambda_{4})}T{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,u=0).$
(149)

These constraints should be satisfied for any value of the angle $\gamma$.
Thus, the amplitudes must vanish unless
$\lambda_{1}-\lambda_{2}-\lambda_{3}+\lambda_{4}=0$ in the first case and
$\lambda_{1}-\lambda_{2}+\lambda_{3}-\lambda_{4}=0$ in the second case. This
is just conservation of angular momentum along the $z$–axis.

##### Special point: $\boldsymbol{s=0}$

Finally let us address the most complicated $s=0$ case. In the vicinity of
$s=0$, the rapidity parameter $\eta$ can be written as

$\eta=\frac{i\pi}{2}-\frac{i\sqrt{s}}{2m}+O(s^{3/2}).$ (150)

Let us now define the new rapidity $\xi$ as

$\xi\equiv\eta-\frac{i\pi}{2}=-\frac{i\sqrt{s}}{2m}+O(s^{3/2}).$ (151)

By using (295) - (297) and the following property of the small Wigner d-matrix

$d^{(j)}_{\lambda^{\prime}\lambda}(-\pi)=(-1)^{j+\lambda}\delta_{\lambda^{\prime},-\lambda},$
(152)

we can rewrite the states (137) in the following way

$\displaystyle|m,+\vec{p}_{z};j,\lambda_{1}\rangle$
$\displaystyle=e^{+\frac{\pi}{2}K_{3}}\left(e^{-i\xi
K_{3}}|m,\vec{0};j,\lambda_{1}\rangle\right),$ (153)
$\displaystyle|m,-\vec{p}_{z};j,\lambda_{2}\rangle$
$\displaystyle=(-1)^{j+\lambda_{2}}e^{-\frac{\pi}{2}K_{3}}\left(e^{+i\xi
K_{3}}|m,\vec{0};j,-\lambda_{2}\rangle\right),$
$\displaystyle|m,+\vec{p}_{\theta};j,\lambda_{3}\rangle$
$\displaystyle=e^{-i\theta J_{2}}e^{+\frac{\pi}{2}K_{3}}\left(e^{-i\xi
K_{3}}|m,\vec{0};j,\lambda_{3}\rangle\right),$
$\displaystyle|m,-\vec{p}_{\theta};j,\lambda_{4}\rangle$
$\displaystyle=(-1)^{j+\lambda_{4}}e^{-i\theta
J_{2}}e^{-\frac{\pi}{2}K_{3}}\left(e^{+i\xi
K_{3}}|m,\vec{0};j,-\lambda_{4}\rangle\right).$

These can be further rewritten as282828The steps involved here are as follows.
First, we inject the identity operators $\mathbb{I}=e^{\pm
i\frac{\pi}{2}J_{2}}e^{\mp i\frac{\pi}{2}J_{2}}$ to the left and right of the
$e^{\pm i\xi K_{3}}$ operator and use the following relations  $\displaystyle
e^{-iJ_{2}\frac{\pi}{2}}e^{\pm i\xi K_{3}}e^{+iJ_{2}\frac{\pi}{2}}=e^{\pm i\xi
K_{1}},\qquad e^{+iJ_{2}\frac{\pi}{2}}e^{\pm i\xi
K_{3}}e^{-iJ_{2}\frac{\pi}{2}}=e^{\mp i\xi K_{1}}.$ Second, we use the
following relations  $\displaystyle
e^{\pm\frac{\pi}{2}K_{3}}e^{+iJ_{2}\frac{\pi}{2}}=e^{+iJ_{2}\frac{\pi}{2}}e^{\pm\frac{\pi}{2}K_{1}},\qquad
e^{\pm\frac{\pi}{2}K_{3}}e^{-iJ_{2}\frac{\pi}{2}}=e^{-iJ_{2}\frac{\pi}{2}}e^{\mp\frac{\pi}{2}K_{1}}$
to bring all the exponents containing $J_{2}$ to the left. Finally, we use
$e^{-i\theta J_{2}}e^{\frac{\pi}{2}K_{1}}=e^{\frac{\pi}{2}K_{1}}e^{-\theta
K_{3}}.$

$\displaystyle|m,+\vec{p}_{z};j,\lambda_{1}\rangle$
$\displaystyle=e^{+\frac{\pi}{2}K_{1}}e^{+\frac{\pi}{2}K_{3}}e^{-i\xi
K_{1}}\left(e^{-i\frac{\pi}{2}J_{2}}|m,\vec{0};j,\lambda_{1}\rangle\right),$
(154) $\displaystyle|m,-\vec{p}_{z};j,\lambda_{2}\rangle$
$\displaystyle=e^{+\frac{\pi}{2}K_{1}}e^{-\frac{\pi}{2}K_{3}}e^{-i\xi
K_{1}}\left((-1)^{j+\lambda_{2}}e^{+i\frac{\pi}{2}J_{2}}|m,\vec{0};j,-\lambda_{2}\rangle\right),$
$\displaystyle|m,+\vec{p}_{\theta};j,\lambda_{3}\rangle$
$\displaystyle=e^{+\frac{\pi}{2}K_{1}}e^{-(\theta-\frac{\pi}{2})K_{3}}e^{-i\xi
K_{1}}\left(e^{-i\frac{\pi}{2}J_{2}}|m,\vec{0};j,\lambda_{3}\rangle\right),$
$\displaystyle|m,-\vec{p}_{\theta};j,\lambda_{4}\rangle$
$\displaystyle=e^{+\frac{\pi}{2}K_{1}}e^{-(\theta+\frac{\pi}{2})K_{3}}e^{-i\xi
K_{1}}\left((-1)^{j+\lambda_{4}}e^{+i\frac{\pi}{2}J_{2}}|m,\vec{0};j,-\lambda_{4}\rangle\right).$

We use now invariance of the scattering operator under boosts and the action
of rotations on the center of mass states (14), which for the second and
fourth particles become

$\displaystyle(-1)^{j+\lambda}e^{+i\frac{\pi}{2}J_{2}}|m,\vec{0};j,-\lambda\rangle$
$\displaystyle=(-1)^{j+\lambda}\sum_{\lambda^{\prime}}d^{j}_{\lambda^{\prime},-\lambda}(-\frac{\pi}{2})|m,\vec{0};j,\lambda^{\prime}\rangle$
(155)
$\displaystyle=(-1)^{2j}\sum_{\lambda^{\prime}}d^{j}_{\lambda^{\prime},\lambda}(+\frac{\pi}{2})|m,\vec{0};j,\lambda^{\prime}\rangle,$

to obtain the final expression

$T_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u)=\sum_{\lambda^{\prime}}d^{(j)}_{\lambda_{1}^{\prime}\lambda_{1}}\left(\frac{\pi}{2}\right)d^{(j)}_{\lambda_{2}^{\prime}\lambda_{2}}\left(\frac{\pi}{2}\right)d^{(j)}_{\lambda_{3}^{\prime}\lambda_{3}}\left(\frac{\pi}{2}\right)d^{(j)}_{\lambda_{4}^{\prime}\lambda_{4}}\left(\frac{\pi}{2}\right)B_{\lambda^{\prime}_{1},\lambda^{\prime}_{2}}^{\lambda^{\prime}_{3},\lambda^{\prime}_{4}}(s,t,u),$
(156)

where the amplitude $B$ is defined as

$B_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u)\equiv\left(\langle
m,\vec{0};j,\lambda_{3}|e^{+i\xi K_{1}}e^{+\frac{\pi}{2}K_{3}}\otimes\langle
m,\vec{0};j,\lambda_{4}|e^{+i\xi
K_{1}}e^{-\frac{\pi}{2}K_{3}}\right)e^{-\theta K_{3}}T\\\
\left(e^{+\frac{\pi}{2}K_{3}}e^{-i\xi
K_{1}}|m,\vec{0};j,\lambda_{1}\rangle\otimes e^{-\frac{\pi}{2}K_{3}}e^{-i\xi
K_{1}}|m,\vec{0};j,\lambda_{2}\rangle\right).$ (157)

Let us inspect the structure of this amplitude. We expand it around $\xi=0$ or
equivalently around $s=0$ according to (151). Schematically speaking, each
term in this expansion will contain $(\xi K_{1})^{n}$ with some non-negative
integer $n$. We then notice that $(\xi K_{1})^{n}$ are the only operators
which change helicities of particles.292929One can define the following
operators  $K_{\pm}\equiv K_{1}\pm iK_{2}\quad\Rightarrow\quad
K_{1}=\frac{1}{2}\,(K_{+}+K_{-}).$ According to (290) these operators rise and
lower helicities of the center of mass states as
$J_{3}K_{\pm}|m,\vec{0};j,\lambda\rangle=(\lambda\pm
1)K_{\pm}|m,\vec{0};j,\lambda\rangle.$ Now, the only non-zero terms will be
the ones with equal total helicity of the states to the left and to the right
of the scattering operator $T$. Given a set of helicities $\lambda_{i}$, the
leading term in the $\xi=0$ expansion will contain $(\xi K_{1})^{n}$ with
$n=|\lambda_{1}+\lambda_{2}-\lambda_{3}-\lambda_{4}|$. Using (151) we conclude
that

$B_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u)=\left(\sqrt{s}\right)^{|\lambda_{1}+\lambda_{2}-\lambda_{3}-\lambda_{4}|}\times
O(s^{0}).$ (158)

From (156) and (158) it follows that the COM amplitudes $T$ get a $\sqrt{s}$
branch point only for odd values of
$|\lambda_{1}+\lambda_{2}-\lambda_{3}-\lambda_{4}|$. The relations (156) and
(158) can be also used to address kinematic singularities. Expanding (156)
around $s=0$ at the leading order one can find linear combinations of the
amplitudes which behave as $O(s^{1})$ instead of $O(s^{0})$.

##  3 Application: identical Majorana fermions

We will now use the machinery set up in the previous chapter to study the two
to two scattering of identical neutral303030By neutral we mean particles not
carrying any $U(1)$ charge and in general not transforming in any non-trivial
representation of the global group. In common words it means that the particle
is its own antiparticle. spin $\frac{1}{2}$ fermions also known as Majorana
fermions. We will require invariance under parity. As a result we need to
specify the intrinsic parity $\eta$ defined in (23). In the two to two
scattering of identical particles we are sensitive only to the value of
$\eta^{2}$. According to (25) there are two possibilities

$\eta^{2}=-1\quad\text{or}\quad\eta^{2}=+1.$ (159)

For concreteness we assume the former in this section. In the latter situation
everything in this section still remains valid except that the meaning of
parity even and odd states in section 3.2 is flipped and the role of scalar
and pseudoscalar particles is exchanged in section 3.3. Helicity of a spin
$\frac{1}{2}$ particle takes only two values: $+\frac{1}{2}$ and
$-\frac{1}{2}$. Thus a priori, we have $2^{4}=16$ helicity amplitudes. However
due to the fact that the particles are all identical these amplitudes are
related according to (94) - (98). As a result we can write the following 9
relations

$\displaystyle T_{--}^{--}=T_{++}^{++},\qquad T_{--}^{+-}=T_{--}^{-+},\qquad
T_{-+}^{--}=T_{++}^{+-},\qquad T_{-+}^{-+}=T_{+-}^{+-},\qquad
T_{-+}^{+-}=T_{+-}^{-+},$ $\displaystyle T_{-+}^{++}=T_{--}^{-+},\qquad
T_{+-}^{--}=T_{++}^{+-},\qquad T_{+-}^{++}=T_{--}^{-+},\qquad
T_{++}^{-+}=T_{++}^{+-},$ (160)

where $+$ and $-$ stand for $+\frac{1}{2}$ and $-\frac{1}{2}$ helicities
respectively. Hence out of the 16 amplitudes we are left with 7 independent
ones. Requiring parity invariance and noticing that due to (159) the product
of intrinsic parities $\eta_{1}\eta_{2}\eta_{3}^{*}\eta_{4}^{*}=+1$, due to
(72) we get in addition the following 2 constraints

$T_{--}^{-+}=-T_{++}^{+-},\qquad T_{--}^{++}=+T_{++}^{--}.$ (161)

As a result out of the 16 helicity amplitudes we are left with only 5
independent ones, in agreement with (105), which we denote as

$\displaystyle\Phi_{1}(s,t,u)\equiv T_{++}^{++}(s,t,u),$ (162)
$\displaystyle\Phi_{2}(s,t,u)\equiv T_{++}^{--}(s,t,u),$
$\displaystyle\Phi_{3}(s,t,u)\equiv T_{+-}^{+-}(s,t,u),$
$\displaystyle\Phi_{4}(s,t,u)\equiv T_{+-}^{-+}(s,t,u),$
$\displaystyle\Phi_{5}(s,t,u)\equiv T_{++}^{+-}(s,t,u).$

It is interesting to note that the scattering of identical neutral fermions
preserving parity is automatically time-reversal invariant, this can be
intuitively understood from the CPT symmetry since charge conjugation is
trivial for neutral particles, see appendix B.3.

As discussed in section 2.3, in the case of scattering of uncharged identical
particles, the crossing equations (82) and (84) form highly non-trivial
constraints on the scattering amplitudes. For instance, in the case of
identical Majorana particles these crossing equations give rise to two sets of
$16$ linear equations. Taking into account the relations (160) and (161) we
simply obtain two sets of 5 linear equations on the independent amplitudes
(162). They read as

$\displaystyle\Phi_{I}(s,t,u)$
$\displaystyle=\sum_{J=1}^{5}C_{st}^{IJ}(s,t,u)\Phi_{J}(t,s,u),$ (163)
$\displaystyle\Phi_{I}(s,t,u)$
$\displaystyle=\sum_{J=1}^{5}C_{su}^{IJ}(s,t,u)\Phi_{J}(u,t,s),$ (164)

where the $s-t$ crossing matrix $C_{st}$ is given by

$C_{st}=-\frac{\epsilon_{23}^{\prime}}{2}\begin{pmatrix}-\sin^{2}\alpha&\sin^{2}\alpha&-\sin^{2}\alpha&1+\cos^{2}\alpha&4\cos\alpha\sin\alpha\\\
\sin^{2}\alpha&1+\cos^{2}\alpha&\sin^{2}\alpha&\sin^{2}\alpha&-4\cos\alpha\sin\alpha\\\
-\sin^{2}\alpha&\sin^{2}\alpha&1+\cos^{2}\alpha&-\sin^{2}\alpha&4\cos\alpha\sin\alpha\\\
1+\cos^{2}\alpha&\sin^{2}\alpha&-\sin^{2}\alpha&-\sin^{2}\alpha&4\cos\alpha\sin\alpha\\\
\cos\alpha\sin\alpha\ \ \ &-\cos\alpha\sin\alpha\ \ \ &\cos\alpha\sin\alpha\ \
&\cos\alpha\sin\alpha\ \ \ &2(\sin^{2}\alpha-\cos^{2}\alpha)\end{pmatrix},$
(165)

and the $s-u$ crossing matrix $C_{su}$ is given by

$C_{su}=-\frac{\epsilon_{24}^{\prime}}{2}\begin{pmatrix}-\sin^{2}\beta&\sin^{2}\beta&1+\cos^{2}\beta&-\sin^{2}\beta&4\cos\beta\sin\beta\\\
\sin^{2}\beta&1+\cos^{2}\beta&\sin^{2}\beta&\sin^{2}\beta&-4\cos\beta\sin\beta\\\
1+\cos^{2}\beta&\sin^{2}\beta&-\sin^{2}\beta&-\sin^{2}\beta&4\cos\beta\sin\beta\\\
-\sin^{2}\beta&\sin^{2}\beta&-\sin^{2}\beta&1+\cos^{2}\beta&4\cos\beta\sin\beta\\\
\cos\beta\sin\beta\ \ &-\cos\beta\sin\beta\ \ \ &\cos\beta\sin\beta\ \ \
&\cos\beta\sin\beta\ \ \ &2(\sin^{2}\beta-\cos^{2}\beta)\end{pmatrix}.$ (166)

The angles $\alpha$ and $\beta$ are defined as follows. Looking at the
expressions (83) and (85) we can make the following choice of Wigner angles

$\alpha_{1}=\alpha,\quad\alpha_{2}=\pi+\alpha,\quad\alpha_{3}=\pi-\alpha,\quad\alpha_{4}=-\alpha.$
(167)
$\beta_{1}=\beta,\quad\beta_{2}=\pi+\beta,\quad\beta_{3}=-\beta,\quad\beta_{4}=\pi-\beta,$
(168)

where the angles $\alpha$ and $\beta$ obey

$\cos\alpha=\frac{st}{\sqrt{s(s-4m^{2})}\sqrt{t(t-4m^{2})}},\quad\sin\alpha=-\frac{2m\,\sqrt{stu}}{\sqrt{s(s-4m^{2})}\sqrt{t(t-4m^{2})}},$
(169)
$\cos\beta=\frac{su}{\sqrt{s(s-4m^{2})}\sqrt{u(u-4m^{2})}},\quad\sin=+\frac{2m\,\sqrt{stu}}{\sqrt{s(s-4m^{2})}\sqrt{u(u-4m^{2})}}.$
(170)

The correct choice of the phases at (165) and (166) will be explained at the
end of section 3.1. Here we simply state the correct result, which is

$\epsilon_{23}^{\prime}=\epsilon_{24}^{\prime}=-1.$ (171)

There are two non-trivial consistency checks our matrices (165) and (166)
pass. First, these matrices are involutory, namely they satisfy the following
conditions313131More accurately one should write
$C_{st}(s,t,u)C_{st}(t,s,u)=1,\qquad C_{su}(s,t,u)C_{su}(u,t,s)=1.$ However,
these conditions reduce to (172) by noticing that the matrices $C_{st}(s,t,u)$
and $C_{st}(u,t,s)$ are symmetric in the exchange of $s\leftrightarrow t$ and
$s\leftrightarrow u$ respectively. This follows from the fact that the
expressions for the angles $\alpha$ and $\beta$ given by (169) and (170) obey
the symmetry $s\leftrightarrow t$ and $s\leftrightarrow u$ respectively.

$\big{(}C_{st}(s,t,u)\big{)}^{2}=1,\qquad\big{(}C_{su}(s,t,u)\big{)}^{2}=1.$
(172)

Second, we can obtain the crossing matrix appearing in the $t-u$ crossing
equations as

$C_{tu}(s,t,u)=C_{st}(s,t,u)C_{su}(t,s,u)C_{st}(u,s,t).$ (173)

In our case it reads as

$C_{tu}(s,t,u)=\begin{pmatrix}1&0&0&0&0\\\ 0&1&0&0&0\\\ 0&0&0&1&0\\\
0&0&1&0&0\\\ 0&0&0&0&-1\end{pmatrix}.$ (174)

This is in perfect agreement with the result (93).

###  3.1 Improved amplitudes

It is important to study the analytic structure (presence of poles and branch
cuts) of helicty amplitudes when all the Mandelstam variables $s$, $t$ and $u$
are promoted to the full complex plane. As explained in section 2.7 in the
case of scattering of spinning particles such amplitudes develop non-analytic
behaviour purely due to kinematic reasons. In this section we show how to
isolate such kinematic features in the case of Majorana fermions and define
improved amplitudes which do not have them.

Due to (143) all the amplitudes have a pole at $s=4m^{2}$. Expanding (143)
around this point we get

$\displaystyle\Phi_{1}$
$\displaystyle=\frac{a_{1}}{s-4m^{2}}+b_{1}+O(s-4m^{2}),$ (175)
$\displaystyle\Phi_{2}$
$\displaystyle=\frac{a_{2}}{s-4m^{2}}+b_{2}+O(s-4m^{2}),$
$\displaystyle\Phi_{3}$
$\displaystyle=\frac{a_{3}}{s-4m^{2}}+b_{3}+O(s-4m^{2}),$
$\displaystyle\Phi_{4}$
$\displaystyle=\frac{a_{4}}{s-4m^{2}}+b_{4}+O(s-4m^{2}),$
$\displaystyle\Phi_{5}$
$\displaystyle=\frac{a_{5}}{s-4m^{2}}+b_{5}+O(s-4m^{2}),$

where $a_{i}$ and $b_{i}$ are some factors which are regular at $s=4m^{2}$. We
do not write them explicitly, their form can be obtained straightforwardly
using computer algebra.323232 Notice that identical particles and parity imply
constraints on the amplitudes $T$ according to (160) and (161). In order to
proceed with the expansion one needs to deduce the analogues of these
expressions on the regular $A(s,t,u)$ amplitudes entering (143) by solving an
appropriate system of linear equations. We only notice that

$a_{1}=-a_{2}=a_{3}=a_{4}=-ia_{5}.$ (176)

Using (175) we can verify the following kinematic relation at the singular
point333333We expand around $s=4m^{2}$ keeping $t$ independent. Then in the
right-hand side of (177) the leading and the next to leading order terms
appear to be proportional to the following expression
$t-u-2i\sqrt{tu}=2t+(s-4m^{2})-2i\sqrt{t(4m^{2}-s-t)}=0+O(s-4m^{2}).$ In the
last equality we have used the domain where $t<0$ and $s$ has a small positive
imaginary part.

$\Phi_{1}-\Phi_{2}+\Phi_{3}+\Phi_{4}+4i\Phi_{5}=0+O(s-4m^{2}).$ (177)

Now due to (156) the amplitude $\Phi_{5}$ also develops a branch point at
$s=0$ as

$\Phi_{5}\sim\sqrt{s},$ (178)

whereas all the other amplitudes are regular at $s=0$. Expanding (156) to the
leading order we can verify the following kinematical constraint

$\Phi_{1}+\Phi_{2}-\Phi_{3}-\Phi_{4}=0+O(s).$ (179)

Finally we consider the behavior of the amplitudes at $t=0$ and $u=0$ points.
Due to (147) the amplitudes $\Phi_{1}$, $\Phi_{2}$, $\Phi_{3}$ and $\Phi_{4}$
are all analytic at these points. In contrast the amplitude $\Phi_{5}$
develops a branch point both at $t=0$ and $u=0$ as

$\Phi_{5}\sim\sqrt{tu}.$ (180)

In addition due to (148) and (149) we have the following constraints

$\displaystyle\Phi_{4}=0+O(t),\,\quad\Phi_{5}=0+O(t),$ (181)
$\displaystyle\Phi_{3}=0+O(u),\quad\Phi_{5}=0+O(u).$

Now that we know precisely the non-analytic behaviour of the amplitudes, we
can define new improved amplitudes which are free of the kinematic pole at
$s=4m^{2}$ and kinematic branch points $\sqrt{s}$, $\sqrt{t}$ and $\sqrt{u}$.
We denote such improved amplitudes by $H_{I}(s,t,u)$. The old amplitudes and
the new improved amplitudes can be related as

$\Phi_{I}(s,t,u)=\sum_{J=1}^{5}M_{IJ}^{-1}(s,t,u)H_{J}(s,t,u),$ (182)

where $M(s,t,u)$ is some matrix to be determined. It is constructed by
requiring that

$\Phi_{I}\sim\frac{1}{s-4m^{2}},\quad\Phi_{5}\sim\sqrt{stu}$ (183)

and that the relations (175) along with (176), (177), (179) and (181) are
fulfilled. These requirements do not fix the matrix $M_{IJ}(s,t,u)$
completely. One possible choice is

$M_{IJ}(s,t,u)=\left(\begin{array}[]{ccccc}\frac{4}{s-4m^{2}}&\frac{-4}{s-4m^{2}}&\frac{2\,(1-t/u)}{s-4m^{2}}&\frac{2(1-u/t)}{s-4m^{2}}&\frac{s+4m^{2}}{s-4m^{2}}\times\frac{2(t-u)}{m\sqrt{stu}}\\\
0&0&\frac{2}{u}&-\frac{2}{t}&-\frac{8m}{\sqrt{stu}}\\\
0&0&\frac{2}{u}&-\frac{2}{t}&-\frac{2s}{m\sqrt{stu}}\\\
0&0&\frac{2}{u}&\frac{2}{t}&0\\\
-\frac{4}{s}&-\frac{4}{s}&\frac{2}{u}+\frac{4}{s}&\frac{2}{t}+\frac{4}{s}&\frac{2(t-u)}{m\sqrt{stu}}\\\
\end{array}\right).$ (184)

We motivate this choice in section 3.4.1.

Having established the relation (182) we can write the crossing equations
(163) and (164) directly in terms of the improved amplitudes as

$\displaystyle H_{I}(s,t,u)$
$\displaystyle=\sum_{J=1}^{5}\widetilde{C}_{st}^{IJ}(s,t,u)H_{J}(t,s,u),$
(185) $\displaystyle H_{I}(s,t,u)$
$\displaystyle=\sum_{J=1}^{5}\widetilde{C}_{su}^{IJ}(s,t,u)H_{J}(u,t,s),$
(186)

where the crossing matrices $\widetilde{C}_{st}$ and $\widetilde{C}_{su}$ read
as

$\displaystyle\widetilde{C}_{st}$ $\displaystyle\equiv
M(s,t,u)C_{st}(s,t,u)M^{-1}(t,s,u),$ (187) $\displaystyle\widetilde{C}_{su}$
$\displaystyle\equiv M(s,t,u)C_{su}(s,t,u)M^{-1}(u,t,s).$ (188)

Plugging here the explicit expressions (165), (166) and (184) we get

$\widetilde{C}_{st}=\left(\begin{array}[]{ccccc}-\frac{1}{4}&-1&\frac{3}{2}&1&-\frac{1}{4}\\\
-\frac{1}{4}&\frac{1}{2}&0&\frac{1}{2}&\frac{1}{4}\\\
\frac{1}{4}&0&\frac{1}{2}&0&\frac{1}{4}\\\
\frac{1}{4}&\frac{1}{2}&0&\frac{1}{2}&-\frac{1}{4}\\\
-\frac{1}{4}&1&\frac{3}{2}&-1&-\frac{1}{4}\\\
\end{array}\right),\qquad\widetilde{C}_{su}=\left(\begin{array}[]{ccccc}-\frac{1}{4}&1&-\frac{3}{2}&1&-\frac{1}{4}\\\
\frac{1}{4}&\frac{1}{2}&0&-\frac{1}{2}&-\frac{1}{4}\\\
-\frac{1}{4}&0&\frac{1}{2}&0&-\frac{1}{4}\\\
\frac{1}{4}&-\frac{1}{2}&0&\frac{1}{2}&-\frac{1}{4}\\\
-\frac{1}{4}&-1&-\frac{3}{2}&-1&-\frac{1}{4}\\\ \end{array}\right).$ (189)

It is remarkable that both matrices turn out to be purely numerical! Just like
the original matrices $C_{st}$ and $C_{su}$, the matrices $\widetilde{C}_{st}$
and $\widetilde{C}_{su}$ are also involutory, _i.e_ $\widetilde{C}_{st}^{2}=1$
and $\widetilde{C}_{su}^{2}=1$. This follows from the definitions (187), (188)
and the condition (172). Note that similar to (173) we can compute the $tu$
crossing matrix
$\widetilde{C}_{tu}=\widetilde{C}_{st}\widetilde{C}_{su}\widetilde{C}_{st}$,
it reads as

$\widetilde{C}_{tu}=\left(\begin{array}[]{ccccc}1&0&0&0&0\\\ 0&-1&0&0&0\\\
0&0&-1&0&0\\\ 0&0&0&1&0\\\ 0&0&0&0&1\\\ \end{array}\right).$ (190)

It says that the improved amplitudes defined via (182) are all eigenfunctions
of $tu$ crossing.

The overall sign of the crossing matrices (189) depends on the choice of
phases in (171). The choice made in (171) is the only correct one. In order to
see that we can take the Fermi lagrangian and compute the scattering of
Majorana fermions to the leading order. We do it in appendix F.1, the final
result for the improved amplitudes is given in (621). It automatically
satisfies the crossing equations (185) and (186). Any other choice of phases
(171) leads to crossing equations which are inconsistent with the perturbative
computation of appendix F.1. This phase choice is independent of any
particular model and holds non-perturbatively.

###  3.2 Unitarity

The general strategy for imposing unitarity constraints on scattering
amplitudes was provided in section 2.6. Here we apply that strategy to the
case of Majorana fermions. According to section 2.6 one needs to consider all
possible states transforming in the irreducible representations which appear
in the decomposition of the two particle state formed from two (identical)
Majorana particles. These are

$|c,\vec{p},{\ell},{\lambda};\lambda_{1},\lambda_{2}\rangle_{id},$ (191)

where $\lambda_{i}=\pm\frac{1}{2}$. Since the particles are identical we can
further restrict our attention to the states with
$\lambda_{1}\geq\lambda_{2}$. As a result we are left with only three states
of the form

$|c,\vec{p},{\ell},{\lambda};+,+\rangle_{id},\qquad|c,\vec{p},{\ell},{\lambda};-,-\rangle_{id},\qquad|c,\vec{p},{\ell},{\lambda};+,-\rangle_{id}.$
(192)

We further notice that due to (45), the first two states in (192) exist only
for even spins $\ell$, whereas the last state in (192) exists for both even
and odd spins $\ell$. Using (420) we can form the following three parity
eigenstates out of the states (192)

$\displaystyle|1\rangle\equiv\frac{1}{\sqrt{2}}\left(|c,\vec{p},{\ell},{\lambda};+,+\rangle_{id}+|c,\vec{p},{\ell},{\lambda};-,-\rangle_{id}\right),\quad\ell\geq
0\quad({\ell}\text{ even}),$ (193)
$\displaystyle|2\rangle\equiv\frac{1}{\sqrt{2}}\left(|c,\vec{p},{\ell},{\lambda};+,+\rangle_{id}-|c,\vec{p},{\ell},{\lambda};-,-\rangle_{id}\right),\quad\ell\geq
0\quad({\ell}\text{ even}),$ (194)
$\displaystyle|3\rangle\equiv\sqrt{2}\;|c,\vec{p},{\ell},{\lambda};+,-\rangle_{id},\quad\ell\geq
1.$ (195)

The state $|1\rangle$ is parity odd while the states $|2\rangle$ and
$|3\rangle$ are parity even. The states (193) - (195) can either be $in$ or
$out$ asymptotic states. We now form all possible inner products between such
states taking into account that parity eigenstates do not mix since we assumed
parity invariance. The states (193) lead to

$\ell\geq
0\;\;\text{(even)}:\qquad\mathcal{H}^{-}_{\ell}(s)\times\delta_{\ell\ell^{\prime}}\delta_{\lambda\lambda^{\prime}}(2\pi)^{4}\delta^{(4)}(p-p^{\prime})\equiv\begin{pmatrix}{}_{in}\langle
1^{\prime}|1\rangle_{in}&{}_{in}\langle 1^{\prime}|1\rangle_{out}\\\
{}_{out}\langle 1^{\prime}|1\rangle_{in}&{}_{out}\langle
1^{\prime}|1\rangle_{out}\end{pmatrix},$ (196)

where the primed states have the labels $c^{\prime}$, $\vec{p}\,^{\prime}$,
$\ell^{\prime}$ and $\lambda^{\prime}$. Analogously the states (194) for
$\ell=0$ and the states (195) for odd $\ell\geq 1$ lead to

$\displaystyle\ell=0:\qquad\mathcal{H}^{+}_{\ell}(s)$
$\displaystyle\times\delta_{\ell\ell^{\prime}}\delta_{\lambda\lambda^{\prime}}(2\pi)^{4}\delta^{(4)}(p-p^{\prime})\equiv\begin{pmatrix}{}_{in}\langle
2^{\prime}|2\rangle_{in}&{}_{in}\langle 2^{\prime}|2\rangle_{out}\\\
{}_{out}\langle 2^{\prime}|2\rangle_{in}&{}_{out}\langle
2^{\prime}|2\rangle_{out}\end{pmatrix},$ (197) $\displaystyle\ell\geq
1\;\;\text{(odd)}:\qquad\mathcal{H}^{+}_{\ell}(s)$
$\displaystyle\times\delta_{\ell\ell^{\prime}}\delta_{\lambda\lambda^{\prime}}(2\pi)^{4}\delta^{(4)}(p-p^{\prime})\equiv\begin{pmatrix}{}_{in}\langle
3^{\prime}|3\rangle_{in}&{}_{in}\langle 3^{\prime}|3\rangle_{out}\\\
{}_{out}\langle 3^{\prime}|3\rangle_{in}&{}_{out}\langle
3^{\prime}|3\rangle_{out}\end{pmatrix}.$ (198)

Finally for even $\ell\geq 2$, the states (194) and (195) can mix. They lead
to

$H^{+}_{\ell\geq
2}(s)\times\delta_{\ell\ell^{\prime}}\delta_{\lambda\lambda^{\prime}}(2\pi)^{4}\delta^{(4)}(p-p^{\prime})\equiv\begin{pmatrix}{}_{in}\langle
2^{\prime}|2\rangle_{in}&{}_{in}\langle
2^{\prime}|3\rangle_{in}&{}_{in}\langle
2^{\prime}|2\rangle_{out}&{}_{in}\langle 2^{\prime}|3\rangle_{out}\\\
{}_{in}\langle 3^{\prime}|2\rangle_{in}&{}_{in}\langle
3^{\prime}|3\rangle_{in}&{}_{in}\langle
3^{\prime}|2\rangle_{out}&{}_{in}\langle 3^{\prime}|3\rangle_{out}\\\
{}_{out}\langle 2^{\prime}|2\rangle_{in}&{}_{out}\langle
2^{\prime}|3\rangle_{in}&{}_{out}\langle
2^{\prime}|2\rangle_{out}&{}_{out}\langle 2^{\prime}|3\rangle_{out}\\\
{}_{out}\langle 3^{\prime}|2\rangle_{in}&{}_{out}\langle
3^{\prime}|3\rangle_{in}&{}_{out}\langle
3^{\prime}|2\rangle_{out}&{}_{out}\langle
3^{\prime}|3\rangle_{out}\end{pmatrix}.$ (199)

Let us write explicitly the components of these matrices. The inner product of
only $in$ or $out$ states are fixed by our normalization conventions, which
read

${}_{in}\langle a^{\prime}|b\rangle_{in}={}_{out}\langle
a^{\prime}|b\rangle_{out}=\delta_{ab}\times\delta_{\ell\ell^{\prime}}\delta_{\lambda\lambda^{\prime}}(2\pi)^{4}\delta^{(4)}(p-p^{\prime}).$
(200)

The inner products between $in$ and $out$ states lead to partial amplitudes,
we have

$\displaystyle{}_{out}\langle 1^{\prime}|1\rangle_{in}$
$\displaystyle=\delta_{\ell\ell^{\prime}}\delta_{\lambda\lambda^{\prime}}(2\pi)^{4}\delta^{(4)}(p-p^{\prime})\times\left(1+i\,\left(T_{\ell}{}_{++}^{++}(s)+T_{\ell}{}_{++}^{--}(s)\right)\right),$
(201) $\displaystyle{}_{out}\langle 2^{\prime}|2\rangle_{in}$
$\displaystyle=\delta_{\ell\ell^{\prime}}\delta_{\lambda\lambda^{\prime}}(2\pi)^{4}\delta^{(4)}(p-p^{\prime})\times\left(1+i\,\left(T_{\ell}{}_{++}^{++}(s)-T_{\ell}{}_{++}^{--}(s)\right)\right),$
$\displaystyle{}_{out}\langle 3^{\prime}|3\rangle_{in}$
$\displaystyle=\delta_{\ell\ell^{\prime}}\delta_{\lambda\lambda^{\prime}}(2\pi)^{4}\delta^{(4)}(p-p^{\prime})\times\left(1+2i\,T_{\ell}{}_{+-}^{+-}(s)\right),$
$\displaystyle{}_{out}\langle 3^{\prime}|2\rangle_{in}$
$\displaystyle={}_{out}\langle
2^{\prime}|3\rangle_{in}=\delta_{\ell\ell^{\prime}}\delta_{\lambda\lambda^{\prime}}(2\pi)^{4}\delta^{(4)}(p-p^{\prime})\times\left(2i\,T_{\ell}{}_{++}^{+-}(s)\right).$

The partial amplitudes entering these expressions are related to scattering
amplitudes via (120). We write here that expression again for the reader’s
convenience

$T_{\ell}{}^{\lambda_{3},\lambda_{4}}_{\lambda_{1},\lambda_{2}}(s)=\frac{\sqrt{s-4m^{2}}}{32\pi\sqrt{s}}\times\int_{0}^{\pi}d\theta\sin\theta
d^{(\ell)}_{\lambda_{12}\lambda_{34}}(\theta)T_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u),\quad\lambda_{ij}\equiv\lambda_{i}-\lambda_{j}.$
(202)

In (201) we used relations (160), (161) and properties of the Wigner d
matrices. As discussed in section 2.6 unitarity requires the matrices
$\mathcal{H}^{-}_{\ell}(s)$ and $\mathcal{H}^{+}_{\ell}(s)$ to be positive
semi-definite for all $s\geq 4m^{2}$ and $\ell$. In what follows we will write
these conditions in the final form.

In (162) we denoted the five independent amplitudes by $\Phi_{I}(s,t,u)$. In
accordance we define the five partial amplitudes as

$\displaystyle\Phi^{\ell}_{1}(s)\equiv T_{\ell}{}_{++}^{++}(s),$ (203)
$\displaystyle\Phi^{\ell}_{2}(s)\equiv T_{\ell}{}_{++}^{--}(s),$
$\displaystyle\Phi^{\ell}_{3}(s)\equiv T_{\ell}{}_{+-}^{+-}(s),$
$\displaystyle\Phi^{\ell}_{4}(s)\equiv T_{\ell}{}_{+-}^{-+}(s),$
$\displaystyle\Phi^{\ell}_{5}(s)\equiv T_{\ell}{}_{++}^{+-}(s).$

Plugging (200) and (201) into (196) - (199) we can write the semi-definite
positivity conditions on the matrices $\mathcal{H}^{-}_{\ell}(s)$ and
$\mathcal{H}^{+}_{\ell}(s)$ as

$\displaystyle\ell\geq 0\;\;\text{(even)}:\qquad$
$\displaystyle\begin{pmatrix}1&1\\\
1&1\end{pmatrix}+i\begin{pmatrix}0&-\Phi_{1}^{\ell*}(s)-\Phi_{2}^{\ell*}(s)\\\
\Phi_{1}^{\ell}(s)+\Phi_{2}^{\ell}(s)&0\end{pmatrix}\succeq 0,$ (204)
$\displaystyle\ell=0:\qquad$ $\displaystyle\begin{pmatrix}1&1\\\
1&1\end{pmatrix}+i\begin{pmatrix}0&-\Phi_{1}^{\ell*}(s)+\Phi_{2}^{\ell*}(s)\\\
\Phi_{1}^{\ell}(s)-\Phi_{2}^{\ell}(s)&0\end{pmatrix}\succeq 0,$ (205)
$\displaystyle\ell\geq 1\;\;\text{(odd)}:\qquad$
$\displaystyle\begin{pmatrix}1&1\\\
1&1\end{pmatrix}+2i\begin{pmatrix}0&-\Phi_{3}^{\ell*}(s)\\\
\Phi_{3}^{\ell}(s)&0\end{pmatrix}\succeq 0.$ (206)

Finally, the matrix (199) leads to the following condition

$\ell\geq 2\;\;\text{(even)}:\qquad\begin{pmatrix}\mathbb{I}_{2\times
2}&\mathbb{S}^{\ell\dagger}_{2\times 2}(s)\\\ \mathbb{S}^{\ell}_{2\times
2}(s)&\mathbb{I}_{2\times 2}\end{pmatrix}\succeq 0,$ (207)

where we have defined

$\mathbb{I}_{2\times 2}\equiv\begin{pmatrix}1&0\\\
0&1\end{pmatrix},\qquad\mathbb{S}^{\ell}_{2\times
2}(s)\equiv\begin{pmatrix}1&0\\\
0&1\end{pmatrix}+i\begin{pmatrix}\Phi_{1}^{\ell}(s)-\Phi_{2}^{\ell}(s)&\qquad
2\Phi_{5}^{\ell}(s)\\\ 2\Phi_{5}^{\ell}(s)&\qquad
2\Phi_{3}^{\ell}(s)\end{pmatrix}.$ (208)

It is interesting to note that the equations above do not contain the partial
amplitude $\Phi_{4}^{\ell}(s)$ at all. This is because due to the $t-u$
crossing equations, see (174), one has

$\Phi_{4}(s,t,u)=\Phi_{3}(s,u,t)$ (209)

Using this inside (202) we get the following relation among the partial
amplitudes343434In order to show this, we use (67) and change the integration
variable in (202) as $\theta\rightarrow\pi-\theta$. Using the properties of
the small Wigner d matrix we get then
$T_{\ell}{}^{-,+}_{+,-}(s)=(-1)^{\ell+1}T_{\ell}{}^{+,-}_{+,-}(s)$.

$\Phi_{4}^{\ell}(s)=(-1)^{\ell+1}\Phi_{3}^{\ell}(s).$ (210)

###  3.3 Non-perturbative couplings

We can use the Majorana fermion scattering amplitude to define several non-
perturbative coupling constants. These are useful parameters to describe the
allowed space of QFTs.

##### Quartic coupling

We begin by considering the value of the amplitude at the crossing symmetric
point

$s=t=u=\frac{4m^{2}}{3}.$ (211)

At this point, the improved amplitudes $H_{I}$ must be invariant under both
the s-t (185) and the s-u (186) crossing equations

$\displaystyle
H_{I}\left(\frac{4m^{2}}{3},\frac{4m^{2}}{3},\frac{4m^{2}}{3}\right)=\sum_{J=1}^{5}\tilde{C}^{IJ}_{st}H_{J}\left(\frac{4m^{2}}{3},\frac{4m^{2}}{3},\frac{4m^{2}}{3}\right),$
(212) $\displaystyle
H_{I}\left(\frac{4m^{2}}{3},\frac{4m^{2}}{3},\frac{4m^{2}}{3}\right)=\sum_{J=1}^{5}\tilde{C}^{IJ}_{su}H_{J}\left(\frac{4m^{2}}{3},\frac{4m^{2}}{3},\frac{4m^{2}}{3}\right).$

The most general solution reads

$\vec{H}(4m^{2}/3,4m^{2}/3,4m^{2}/3)=\frac{\lambda}{m^{2}}\times\begin{pmatrix}1\\\
0\\\ 0\\\ 1\\\ -1\end{pmatrix},$ (213)

where $\vec{H}$ represents the five amplitudes collectively and $\lambda$ is
some parameter. We refer to $\lambda$ as the non-perturbative quartic
coupling. By comparing (213) with the perturbative result (621) we see that
$\lambda$ can be identified with the coupling in front of the
$(\overline{\Psi}\Psi)(\overline{\Psi}\Psi)$ interaction term in the Fermi
theory.

##### Cubic (Yukawa) couplings

Suppose now that our theory is described not only by the Majorana asymptotic
state but also by a scalar (spin zero) asymptotic state with mass $M$. We
restrict our attention to the values of $M$ in the range $(m,2m)$, where $m$
is the mass of the Majorana asymptotic state. This ensures that the Majorana
fermion is the lightest particle in the theory and that the scalar boson is
stable.

From general principles we expect such a scalar asymptotic state to manifest
itself as a simple pole in the improved scattering amplitudes of Majorana
fermions. In full generality one can then write

$\vec{H}_{bound}(s,t,u)=\begin{pmatrix}\frac{g^{2}_{1}}{s-M^{2}}\\\
\frac{g^{2}_{2}}{s-M^{2}}\\\ \frac{g^{2}_{3}}{s-M^{2}}\\\
\frac{g^{2}_{4}}{s-M^{2}}\\\
\frac{g^{2}_{5}}{s-M^{2}}\end{pmatrix}+\begin{pmatrix}\frac{g^{\prime
2}_{1}}{t-M^{2}}\\\ \frac{g^{\prime 2}_{2}}{t-M^{2}}\\\ \frac{g^{\prime
2}_{3}}{t-M^{2}}\\\ \frac{g^{\prime 2}_{4}}{t-M^{2}}\\\ \frac{g^{\prime
2}_{5}}{t-M^{2}}\end{pmatrix}+\begin{pmatrix}\frac{g^{\prime\prime
2}_{1}}{u-M^{2}}\\\ \frac{g^{\prime\prime 2}_{2}}{u-M^{2}}\\\
\frac{g^{\prime\prime 2}_{3}}{u-M^{2}}\\\ \frac{g^{\prime\prime
2}_{4}}{u-M^{2}}\\\ \frac{g^{\prime\prime 2}_{5}}{u-M^{2}}\end{pmatrix}+{\rm
regular},$ (214)

where $g_{I}$, $g_{I}^{\prime}$ and $g_{I}^{\prime\prime}$ are 15 arbitrary
parameters. In (214) we have written only the poles and omitted all the
regular terms at $s,t,u=M^{2}$. Due to the crossing equations (185) and (186),
the values of $g_{I}^{\prime}$ and $g_{I}^{\prime\prime}$ get fixed in terms
of $g_{I}$. As a result we are left with 5 undetermined parameters $g_{I}$. We
further require that the $s=M^{2}$ poles contribute only to the zero spin
partial amplitudes. This enforces the fact that the particle generating the
poles is a scalar. This leads to the following three additional constraints

$g_{2}=g_{3}=g_{4}=0.$ (215)

Thus we are left with only two parameters $g_{1}$ and $g_{5}$.

According to the discussion of section 3.2, more precisely due to the formulas
(204) and (205) one can take combinations of components of partial amplitudes
to form parity odd (-) and parity even (+) partial amplitudes which read as

$\Phi_{-}^{\ell}(s)\equiv\Phi_{1}^{\ell}+\Phi_{2}^{\ell},\qquad\Phi_{+}^{\ell}(s)\equiv\Phi_{1}^{\ell}-\Phi_{2}^{\ell}.$
(216)

The $s=M^{2}$ poles with the parameter $g_{1}$ contribute only to the parity
even partial amplitude $\Phi_{+}^{\ell=0}$. We thus interpret $g_{1}$ as the
non-perturbative coupling describing the interaction between two Majorana
particles and a scalar parity even particle. The corresponding pole structure
of the amplitude reads as

$\vec{H}_{scalar}(s,t,u)=\frac{1}{2}\,g^{2}\times\vec{P}_{scalar}(s,t,u)+{\rm
regular},$ (217)

where $g\equiv g_{1}$ and we have defined

$\vec{P}_{scalar}(s,t,u)\equiv\begin{pmatrix}-\frac{4}{s-M^{2}}+\frac{1}{t-M^{2}}+\frac{1}{u-M^{2}}\\\
\frac{1}{t-M^{2}}-\frac{1}{u-M^{2}}\\\ -\frac{1}{t-M^{2}}+\frac{1}{u-M^{2}}\\\
-\frac{1}{t-M^{2}}-\frac{1}{u-M^{2}}\\\
\frac{1}{t-M^{2}}+\frac{1}{u-M^{2}}\end{pmatrix}$ (218)

The $s=M^{2}$ poles with the parameter $g_{5}$ instead contribute only to the
parity odd partial amplitude $\Phi_{-}^{\ell=0}$. Thus, the second parameter
$g_{5}$ describes non-perturbatively the interaction between two Majorana
particles and a scalar parity odd (pseudoscalar) particle. The corresponding
pole structure of the amplitude reads as

$\vec{H}_{pseudoscalar}(s,t,u)=\frac{1}{2}\,\tilde{g}^{2}\times\vec{P}_{pseudoscalar}(s,t,u)+{\rm
regular},$ (219)

where $\tilde{g}\equiv g_{5}$ and we have defined

$\vec{P}_{pseudoscalar}(s,t,u)\equiv\begin{pmatrix}\frac{1}{t-M^{2}}+\frac{1}{u-M^{2}}\\\
-\frac{1}{t-M^{2}}+\frac{1}{u-M^{2}}\\\
-\frac{1}{t-M^{2}}+\frac{1}{u-M^{2}}\\\ \frac{1}{t-M^{2}}+\frac{1}{u-M^{2}}\\\
-\frac{4}{s-M^{2}}+\frac{1}{t-M^{2}}+\frac{1}{u-M^{2}}\end{pmatrix}.$ (220)

The cubic couplings $g\equiv g_{1}$ and $\tilde{g}\equiv g_{5}$ can also be
called the non-perturbative Yukawa coupling constants. We also remark that the
masses $M$ in (218) and (220) do not have to be the same.

In the discussion above we fixed the overall sign in (217) and (219) so that
the residue at $s=M^{2}$ has the appropriate sign in the unitarity equations
(204) and (205). Alternatively one can compare (217) and (219) to the
perturbative results (625) and (629). This comparison not only fixes the
correct signs but also shows that $g$ and $\tilde{g}$ here can be identified
with the couplings appearing in front of the $\varphi\overline{\Psi}\Psi$ and
$\tilde{\varphi}\overline{\Psi}\gamma^{5}\Psi$ Yukawa interactions
respectively in (622) and (626).

###  3.4 An alternative approach to crossing equations

We have so far carefully discussed the construction of crossing equations in
the COM frame and explicitly showed it in the case of Majorana particle
scattering. There is an alternative way of approaching this topic, namely
using the fully covariant language based on constructing tensor structures.
This relies on the most general representation of a scattering amplitude given
in (64).

In what follows we will construct tensor structures in the particular example
of Majorana particle scattering and re-derive the crossing equations. We will
describe a general procedure of constructing tensor structures for particles
with spin in appendix H.

####  3.4.1 Tensor structures

As we know from the COM analysis, there are 5 independent amplitudes in the
case of identical Majorana particles. As a result there will be 5 linearly
independent tensor structures. A particular choice of these tensor structures
was made in Gribov:1963gx . It reads

$\mathbb{T}_{I}{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p_{1},p_{2},p_{3},p_{4})=-\frac{1}{2}\,[\bar{u}_{\lambda_{4}}(p_{4})\mathcal{\mathcal{O}}_{I}v_{\lambda_{3}}(p_{3})]\cdot[\bar{v}_{\lambda_{2}}(p_{2})\mathcal{\mathcal{O}}_{I}u_{\lambda_{1}}(p_{1})],$
(221)

where the five $4\times 4$ matrices $\mathcal{O}_{I}$ are given by

$\mathcal{O}_{1}\equiv 1_{4\times
4},\quad\mathcal{O}_{2}\equiv\gamma^{\mu},\quad\mathcal{O}_{3}\equiv\sqrt{2}\,\sigma^{\mu\nu},\quad\mathcal{O}_{4}\equiv
i\,\gamma^{5}\gamma^{\mu},\quad\mathcal{O}_{5}\equiv\gamma^{5}.$ (222)

The symbol “$\cdot$” in (221) means contraction of all the Lorentz indices
among $\mathcal{O}_{I}$ matrices. Notice also that there is no summation over
the repeated index $I$ in (221). For the readers convenience we also write
(221) explicitly

$\displaystyle\mathbb{T}_{1}{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p_{1},p_{2},p_{3},p_{4})$
$\displaystyle=-\frac{1}{2}[\bar{u}_{\lambda_{4}}(p_{4})v_{\lambda_{3}}(p_{3})][\bar{v}_{\lambda_{2}}(p_{2})u_{\lambda_{1}}(p_{1})],$
(223)
$\displaystyle\mathbb{T}_{2}{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p_{1},p_{2},p_{3},p_{4})$
$\displaystyle=-\frac{1}{2}[\bar{u}_{\lambda_{4}}(p_{4})\gamma_{\mu}v_{\lambda_{3}}(p_{3})][\bar{v}_{\lambda_{2}}(p_{2})\gamma^{\mu}u_{\lambda_{1}}(p_{1})],$
$\displaystyle\mathbb{T}_{3}{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p_{1},p_{2},p_{3},p_{4})$
$\displaystyle=\;-\;\,[\bar{u}_{\lambda_{4}}(p_{4})\sigma_{\mu\nu}v_{\lambda_{3}}(p_{3})][\bar{v}_{\lambda_{2}}(p_{2})\sigma^{\mu\nu}u_{\lambda_{1}}(p_{1})],$
$\displaystyle\mathbb{T}_{4}{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p_{1},p_{2},p_{3},p_{4})$
$\displaystyle=+\frac{1}{2}[\bar{u}_{\lambda_{4}}(p_{4})\gamma_{5}\gamma_{\mu}v_{\lambda_{3}}(p_{3})][\bar{v}_{\lambda_{2}}(p_{2})\gamma_{5}\gamma^{\mu}u_{\lambda_{1}}(p_{1})],$
$\displaystyle\mathbb{T}_{5}{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p_{1},p_{2},p_{3},p_{4})$
$\displaystyle=-\frac{1}{2}[\bar{u}_{\lambda_{4}}(p_{4})\gamma_{5}v_{\lambda_{3}}(p_{3})][\bar{v}_{\lambda_{2}}(p_{2})\gamma_{5}u_{\lambda_{1}}(p_{1})].$

The objects $u_{\lambda}(p)$ and $v_{\lambda}(p)$ are the usual 4-spinor
solutions to the Dirac equation and

$\bar{u}_{\lambda}(p)\equiv
u_{\lambda}^{\dagger}(p)\gamma^{0},\quad\bar{v}_{\lambda}(p)\equiv
v_{\lambda}^{\dagger}(p)\gamma^{0}.$ (224)

We use the Weyl (also known as chiral) basis for the Dirac $\gamma^{\mu}$
matrices given in (351) and the helicity basis of states. With these
conventions the spinor solutions to the Dirac equation read

$\displaystyle u_{\frac{1}{2}}(p)$
$\displaystyle=\frac{1}{\sqrt{2}}\,\begin{pmatrix}\;\;\,\sqrt{p^{0}-\boldsymbol{p}}\,\cos\frac{\theta}{2}&\\\
\;\;\,\sqrt{p^{0}-\boldsymbol{p}}\,\sin\frac{\theta}{2}&e^{+i\phi}\\\
\;\;\,\sqrt{p^{0}+\boldsymbol{p}}\,\cos\frac{\theta}{2}&\\\
\;\;\,\sqrt{p^{0}+\boldsymbol{p}}\,\sin\frac{\theta}{2}&e^{+i\phi}\end{pmatrix},\;u_{-\frac{1}{2}}(p)=\frac{1}{\sqrt{2}}\,\begin{pmatrix}-\sqrt{p^{0}+\boldsymbol{p}}\,\sin\frac{\theta}{2}&e^{-i\phi}\\\
\,\;\;\sqrt{p^{0}+\boldsymbol{p}}\,\cos\frac{\theta}{2}&\\\
-\sqrt{p^{0}-\boldsymbol{p}}\,\sin\frac{\theta}{2}&e^{-i\phi}\\\
\,\;\;\sqrt{p^{0}-\boldsymbol{p}}\,\cos\frac{\theta}{2}&\end{pmatrix},$ (225)
$\displaystyle v_{\frac{1}{2}}(p)$
$\displaystyle=\frac{1}{\sqrt{2}}\,\begin{pmatrix}-\sqrt{p^{0}+\boldsymbol{p}}\,\sin\frac{\theta}{2}&e^{-i\phi}\\\
\;\;\,\sqrt{p^{0}+\boldsymbol{p}}\,\cos\frac{\theta}{2}&\\\
\;\;\,\sqrt{p^{0}-\boldsymbol{p}}\,\sin\frac{\theta}{2}&e^{-i\phi}\\\
-\sqrt{p^{0}-\boldsymbol{p}}\,\cos\frac{\theta}{2}&\end{pmatrix},\;v_{-\frac{1}{2}}(p)=\frac{1}{\sqrt{2}}\,\begin{pmatrix}-\sqrt{p^{0}-\boldsymbol{p}}\,\cos\frac{\theta}{2}&\\\
-\sqrt{p^{0}-\boldsymbol{p}}\,\sin\frac{\theta}{2}&e^{+i\phi}\\\
\;\;\,\sqrt{p^{0}+\boldsymbol{p}}\,\cos\frac{\theta}{2}&\\\
\;\;\,\sqrt{p^{0}+\boldsymbol{p}}\,\sin\frac{\theta}{2}&e^{+i\phi}\end{pmatrix}.$

We notice that the objects $u_{\lambda}(p)$ and $\bar{v}_{\lambda}(p)$
transform in the spin-1/2 representation, namely they get rotated by
$D^{(1/2)}_{\lambda^{\prime}\lambda}$, whereas the objects $v_{\lambda}(p)$
and $\bar{u}_{\lambda}(p)$ transform in the conjugate spin-1/2 representation,
namely they get rotated by $D^{*(1/2)}_{\lambda^{\prime}\lambda}$. The tensor
structures (223) are constructed in such a way that all the Lorentz indices