aainstitutetext: Theoretical Physics Department, CERN, 1211 Geneva 23,
Switzerlandbbinstitutetext: Fields and Strings Laboratory, Institute of
Physics
École Polytechnique Fédéral de Lausanne (EPFL)
Route de la Sorge, CH-1015 Lausanne, Switzerlandccinstitutetext: CPHT, CNRS,
École Polytechnique
Institut Polytechnique de Paris
Route de Saclay, 91128 Palaiseau, Franceddinstitutetext:  Départment de
Physique Théorique, Université de Genève,
24 quai Ernest-Ansermet, 1211 Genève 4, Switzerland

# Bounds on photon scattering

Kelian Häring cb Aditya Hebbar d b Denis Karateev d b Marco Meineri b João
Penedones<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study 2-to-2 scattering amplitudes of massless spin one particles in $d=4$
space-time dimensions, like real world photons. We define a set of non-
perturbative observables (Wilson coefficients) which describe these amplitudes
at low energies. We use full non-linear unitarity to construct various novel
numerical bounds on these observables. For completeness, we also rederive some
bounds using positivity only. We discover and explain why some of these Wilson
coefficients cannot be bounded.

CERN-TH-2022-161

##  1 Introduction and summary

There has been a lot of new progress in the non-perturbative S-matrix
bootstrap, driven by the development of efficient numerical tools, see
Paulos:2016fap ; Paulos:2016but ; Paulos:2017fhb ; Doroud:2018szp ; He:2018uxa
; Cordova:2018uop ; Guerrieri:2018uew ; Paulos:2018fym ; Homrich:2019cbt ;
EliasMiro:2019kyf ; Cordova:2019lot ; Bercini:2019vme ; Gabai:2019ryw ;
Bose:2020shm ; Bose:2020cod ; Correia:2020xtr ; Kruczenski:2020ujw ;
Guerrieri:2020bto ; Hebbar:2020ukp ; Karateev:2019ymz ; Karateev:2020axc ;
Guerrieri:2020kcs ; Tourkine:2021fqh ; Guerrieri:2021ivu ; He:2021eqn ;
EliasMiro:2021nul ; Guerrieri:2021tak ; Chen:2021pgx ; Cordova:2022pbl ;
Albert:2022oes ; Sinha:2020win ; Chowdhury:2021ynh ; Karateev:2022jdb ;
Chen:2022nym .111For an overview of recent results and discussion of some
future directions, see Kruczenski:2022lot .

In this paper we focus on $4$ space-time dimensions. We consider a class of
theories which have a single photon-like particle, namely a massless particle
with spin one. We denote this particle by $\gamma$. The goal of this paper is
to study the 2-to-2 scattering amplitudes of the process

$\gamma(\lambda_{1})\gamma(\lambda_{2})\rightarrow\gamma(\lambda_{3})\gamma(\lambda_{4})\,,$
(1)

where $\lambda_{i}$ is the helicity of the in/out particles. We are agnostic
about the high energy behaviour of the theory apart from the requirement that
the 2-to-2 scattering amplitude must obey the usual S-matrix bootstrap
principles: Lorentz invariance, unitarity and analyticity.

Scattering amplitudes of particles with spin in $4d$ was reviewed
systematically in Hebbar:2020ukp .222This problem was studied in the 60s by
many authors Jacob:1959at ; Trueman:1964zzb ; Hara:1970gc ; Hara:1971kj , see
the older review Martin:102663  for a more comprehensive list. See also
Bellazzini:2016xrt ; deRham:2017zjm  for more recent discussions. We will use
their language in this work. Each photon in (1) has two helicities, as a
result there are 16 scalar amplitudes which fully describe the process (1).
For simplicity we assume parity invariance in this work.333Due to CPT symmetry
in the case of neutral identical particles (as in this paper) parity
invariance implies time-reversal invariance. Taking into account the fact that
the particles under consideration are also identical we are left with only 5
different amplitudes:

$++\to++,\qquad++\to--,\qquad+-\to+-,\qquad+-\to-+,\qquad++\to+-,$ (2)

where $\pm$ correspond to the helicities of the particles in (1). We denote
the corresponding amplitudes in the center of mass frame by

$\Phi_{1}(s,t,u),\qquad\Phi_{2}(s,t,u),\qquad\Phi_{3}(s,t,u),\qquad\Phi_{4}(s,t,u),\qquad\Phi_{5}(s,t,u),$
(3)

where $s$, $t$ and $u$ are the Mandelstam variables describing the scattering
process and obeying the standard relation

$s+t+u=0.$ (4)

Due to crossing equations only the $\Phi_{1}$, $\Phi_{2}$, and $\Phi_{5}$
amplitudes are independent, the rest can be related to these as

$\Phi_{3}(s,t,u)=\Phi_{1}(u,t,s),\qquad\Phi_{4}(s,t,u)=\Phi_{1}(t,s,u).$ (5)

Moreover, crossing also implies that $\Phi_{1}(s,t,u)$ is symmetric under
$t-u$ permutation and the amplitudes $\Phi_{2}(s,t,u)$ and $\Phi_{5}(s,t,u)$
are fully symmetric under permutations of their arguments. We give the precise
definition of the amplitudes (3) in appendix A. There we also derive their
crossing equations and unitarity constraints. For completeness we also explain
how to define the amplitudes via tensor structures in appendix B both in
vector and spinor formalisms.

##### Non-perturbative observables

At low energy, the amplitudes $\Phi_{1}$, $\Phi_{2}$ and $\Phi_{5}$ have the
following expansion:444Let us emphasize that even though we write
$\mathcal{O}(s^{n})$, the expansion is at small $s\sim t\sim u$ and $n$ is the
total power of $s,t,u$.

$\displaystyle\Phi_{1}(s,t,u)$
$\displaystyle=g_{2}s^{2}+g_{3}s^{3}+g_{4}s^{4}+g_{4}^{\prime}s^{2}tu+L_{1}(s|t,u)+\mathcal{O}(s^{5}),$
(6) $\displaystyle\Phi_{2}(s,t,u)$
$\displaystyle=f_{2}(s^{2}+t^{2}+u^{2})+f_{3}stu+f_{4}(s^{2}+t^{2}+u^{2})^{2}$
$\displaystyle+L_{2}(s,t,u)+\mathcal{O}(s^{5}),$
$\displaystyle\Phi_{5}(s,t,u)$
$\displaystyle=h_{3}\,stu+\mathcal{O}(s^{5})~{}.$

The two functions $L_{1}(s,t,u)$ and $L_{2}(s,t,u)$ are fixed by unitarity in
terms of the polynomial terms in (6), see appendix C:

$\displaystyle L_{1}(s|t,u)$ $\displaystyle\equiv
s^{2}\left(\beta_{1,1}s^{2}+\beta_{1,2}tu\right)\log\left(-s\sqrt{g_{2}}\right)+\beta_{1,3}s^{2}\left(t^{2}\log\left(-t\sqrt{g_{2}}\right)+u^{2}\log\left(-u\sqrt{g_{2}}\right)\right),$
$\displaystyle L_{2}(s,t,u)$
$\displaystyle\equiv\beta_{2}\left(s^{4}\log\left(-s\sqrt{g_{2}}\right)+t^{4}\log\left(-t\sqrt{g_{2}}\right)+u^{4}\log\left(-u\sqrt{g_{2}}\right)\right),$
(7)

where the coefficients $\beta$ read as

$\displaystyle\beta_{1,1}$
$\displaystyle=-\frac{14f_{2}^{2}+5g_{2}^{2}}{160\pi^{2}},\qquad$
$\displaystyle\beta_{1,2}=\frac{f_{2}^{2}}{240\pi^{2}}$ (8)
$\displaystyle\beta_{1,3}$ $\displaystyle=-\frac{g_{2}^{2}}{80\pi^{2}},\qquad$
$\displaystyle\beta_{2}=-\frac{5f_{2}g_{2}}{48\pi^{2}}.$

Equations (6) - (8) follow from a few simple assumptions, which are compactly
encoded by an effective field theory (EFT), to be discussed below.555The
amplitudes (6) can also be derived from softness, unitarity, crossing symmetry
and kinematical contraints. Softness is the assumption that the 2-to-2
amplitude scales like $(\textup{energy})^{4}$, and that the 2-to-$(n\geq 3)$
amplitude scales at least like $(\textup{energy})^{6}$ at low energy. Then
unitarity fixes the coefficient of the $\log$ terms as in (8), see appendix C.
Crossing symmetry restricts the polynomials of $s$, $t$ and $u$ that can
appear in (6). Finally, the kinematical contraints discussed in Hebbar:2020ukp
(see eq. (2.139)), imply that $\Phi_{3}=u^{2}(...)$ and $\Phi_{5}=stu(...)$.
For now, it suffices to know that the main ingredient is the absence of other
massless degrees of freedom beyond the spin one particle $\gamma$.

We refer to the real parameters $g_{n}$, $f_{n}$ and $h_{n}$ in (16) as non-
perturbative observables. Our notation is almost identical to the one of
Henriksson:2021ymi ; Henriksson:2022oeu 666The difference between our
observables (in black) and the ones of Henriksson:2021ymi ; Henriksson:2022oeu
(in blue) is the coefficients $g_{4}$ and $g_{4}^{\prime}$ which are related
to their
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}g_{4,1}}$
and
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}g_{4,2}}$
as follows  $\displaystyle
g_{4}={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}g_{4,1}+2g_{4,2}},\qquad
g_{4}^{\prime}={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}-2g_{4,2}}.$
apart from the fact that we take into account the logarithmic branch cuts
associated to intermediate massless particles. As it will be shown, the
parameter $g_{2}$ is always non-negative, thus the object $\sqrt{g_{2}}$
entering inside the log terms is unambiguous.

For complete clarity, let us emphasize that the observables $g_{n}$, $f_{n}$
and $h_{n}$ are well defined and measurable in terms of derivatives of the
non-perturbative scattering amplitude. At the $n\leq 3$ level we have

$g_{2}=\frac{1}{2}\partial_{s}^{2}\Phi_{1}(0,0,0),\qquad
g_{3}=\frac{1}{3!}\partial_{s}^{3}\Phi_{1}(0,0,0),$ (9)

together with

$2f_{2}=\frac{1}{2}\partial_{s}^{2}\Phi_{2}(0,0,0),\qquad-
f_{3}=\frac{1}{2}\partial_{s}^{2}\partial_{t}\Phi_{2}(0,0,0),\qquad-
h_{3}=\frac{1}{2}\partial_{s}^{2}\partial_{t}\Phi_{5}(0,0,0).$ (10)

At the $n=4$ level we have instead

$\displaystyle g_{4}$ $\displaystyle=\lim_{t\rightarrow 0}\lim_{s\rightarrow
0}\frac{1}{4!}\partial_{s}^{4}\left(\Phi_{1}(s,t,-s-t)-L_{1}(s|t,-s-t)\right),$
(11) $\displaystyle-g^{\prime}_{4}$ $\displaystyle=\lim_{t\rightarrow
0}\lim_{s\rightarrow
0}\frac{1}{3!}\partial_{s}^{3}\partial_{t}\left(\Phi_{1}(s,t,-s-t)-L_{1}(s|t,-s-t)\right),$
$\displaystyle 4f_{4}$ $\displaystyle=\lim_{t\rightarrow 0}\lim_{s\rightarrow
0}\frac{1}{4!}\partial_{s}^{4}\left(\Phi_{2}(s,t,-s-t)-L_{2}(s,t,-s-t)\right).$

We choose $g_{2}$ to define an energy scale. It is then convenient to define
the following dimensionless observables

$\bar{f}_{2}\equiv\frac{f_{2}}{g_{2}},\quad\bar{f}_{3}\equiv\frac{f_{3}}{g_{2}^{3/2}},\quad\bar{g}_{3}\equiv\frac{g_{3}}{g_{2}^{3/2}},\quad\bar{h}_{3}\equiv\frac{h_{3}}{g_{2}^{3/2}},\quad\overline{f}_{4}\equiv\frac{f_{4}}{g_{2}^{2}},\quad\overline{g}_{4}\equiv\frac{g_{4}}{g_{2}^{2}},\quad\overline{g}_{4}^{\prime}\equiv\frac{g_{4}^{\prime}}{g_{2}^{2}}.$
(12)

##### Effective field theory

From the QFT perspective there is only one consistent way to describe a
massless spin one particle, namely as a $U(1)$ gauge theory. We use this fact
to construct an effective field theory (EFT) Lagrangian density that can be
used to describe the process (1) at low energies. This is given by summing all
possible linearly independent Lorentz invariants built out of the
electromagnetic tensor $F_{\mu\nu}$ with some generic coefficients. The most
general form of such a Lagrangian density reads as

$\displaystyle\mathcal{L}_{\text{EFT}}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\mathcal{L}_{6}+\mathcal{L}_{8}+\mathcal{L}_{10}+\mathcal{L}_{12}+\ldots,$
(13)

where $\mathcal{L}_{n}$ denotes terms with mass dimension $n$. Explicitly they
read as777We do not include terms with more than four factors of $F_{\mu\nu}$
because these do not contribute to the process (1) to the order in $s,\,t,\,u$
we are considering.

$\displaystyle\mathcal{L}_{6}$ $\displaystyle=0,$
$\displaystyle\mathcal{L}_{8}$
$\displaystyle=c_{1}(F_{\mu\nu}F^{\nu\mu})(F_{\alpha\beta}F^{\beta\alpha})+c_{2}(F_{\mu\nu}F^{\nu\rho}F_{\rho\sigma}F^{\sigma\mu}),$
(14) $\displaystyle\mathcal{L}_{10}$
$\displaystyle=c_{3}(F_{\alpha\beta}\partial^{\beta}F_{\mu\nu}\partial^{\alpha}F^{\nu\rho}{F_{\rho}}^{\mu})+c_{4}(\partial_{\alpha}F_{\mu\nu}\partial^{\alpha}F^{\nu\mu})(F_{\rho\sigma}F^{\sigma\rho})+c_{5}(\partial_{\alpha}F_{\mu\nu}F^{\nu\rho}\partial^{\alpha}F_{\rho\sigma}F^{\sigma\mu}),$
$\displaystyle\mathcal{L}_{12}$
$\displaystyle=c_{6}(\partial_{\nu}F_{\alpha\rho}\partial^{\alpha}F^{\nu\sigma}\partial^{\delta}F^{\beta\rho}\partial_{\beta}F_{\delta\sigma})+c_{7}(F_{\rho\beta}\partial_{\alpha}F_{\sigma\gamma}\partial^{\beta}F^{\sigma\delta}\partial_{\delta}\partial^{\gamma}F^{\rho\alpha})+c_{8}(F_{\alpha\beta}F_{\sigma\gamma}\delta^{\alpha}\delta_{\rho}F^{\gamma\delta}\partial^{\beta}\partial_{\delta}F^{\rho\sigma}).$

The EFT description is valid up to some cut-off scale which we denote by $M$.
The real dimensionful coefficients $c_{i}$ are called Wilson coefficients.
They have the following mass dimensions

$[c_{1}]=[c_{2}]=-4,\qquad[c_{3}]=[c_{4}]=[c_{5}]=-6,\qquad[c_{6}]=[c_{7}]=[c_{8}]=-8.$
(15)

Some recent experimental bounds on some of these Wilson coefficients can be
found in Aaboud:2017bwk ; Ellis:2022uxv .

Using the EFT Lagrangian density we can compute scattering amplitudes of
massless particles $\Phi_{i}(s,t,u)$. We denote them by
$\Phi^{\text{EFT}}_{i}(s,t,u)$. The details of this computation, to 1-loop
order, are provided in appendix D. The amplitudes
$\Phi^{\text{EFT}}_{i}(s,t,u)$ are good approximations of the full non-
perturbative amplitudes $\Phi_{i}(s,t,u)$ in the regime $|s|\lesssim M^{2}$,
namely

$i=1,2,5:\qquad\Phi_{i}(s,t,u)\approx\Phi^{\text{EFT}}_{i}(s,t,u),\qquad|s|\lesssim
M^{2}.$ (16)

This is depicted in figure 1. The amplitudes $\Phi^{\text{EFT}}_{i}(s,t,u)$
will have precisely the same form as in (6) given the relations

$\displaystyle g_{2}=2(4c_{1}+3c_{2}),\qquad f_{2}=2(4c_{1}+c_{2}),$ (17)
$\displaystyle g_{3}=4c_{4},\qquad f_{3}=6(c_{3}+2c_{4}-c_{5}),\qquad
h_{3}=\frac{3}{2}c_{3},$ $\displaystyle
g_{4}=\frac{1}{4}(-3c_{6}+2c_{7}),\qquad
g_{4}^{\prime}=\frac{1}{2}(3c_{6}-2c_{7}-c_{8}),\qquad
f_{4}=-\frac{c_{6}}{8}.$

In the computation above we used dimensional regularization and we have chosen
our renormalization scale $\mu$ to be $\mu^{2}=\frac{1}{\sqrt{g_{2}}}$.

$s$$\bullet$$0$$\bullet$$-t$$M^{2}$ Figure 1: Analytic structure of the full
non-perturbative amplitudes $\Phi_{i}(s,t,u)$ in the complex plane $s$ at a
fixed value of $t<0$. There are two branch cuts along the real axis, namely,
for $s>0$ and for $u=-s-t>0$. The EFT amplitudes are good approximations of
the full non-perturbative amplitude inside the grey region corresponding to
$|s|\lesssim M^{2}$. The full non-perturbative amplitude may have other branch
points on the real axis. For example, in QED there are branch points at
$s=4m_{e}^{2}$ and $u=4m_{e}^{2}$ associated with the threshold for electron-
positron production.

##### Partial amplitudes and unitarity

Given the scattering amplitudes (3) one defines partial amplitudes as follows

${\Phi_{i}^{\ell}}(s)\equiv\frac{1}{32\pi}\int_{0}^{\pi}d\theta\sin\theta\,d^{\,\ell}_{\lambda_{12},\lambda_{34}}(\theta)\;\Phi_{i}\left(s,t(s,\theta),u(s,\theta)\right)$
(18)

where the scattering angle $\theta$ is related to the Mandelstam variables as

$t=-\frac{s}{2}(1-\cos\theta),\quad u=-\frac{s}{2}(1+\cos\theta).$ (19)

The small Wigner $d$-matrix is defined in (119). The differences of helicities
are defined as

$\lambda_{12}\equiv\lambda_{1}-\lambda_{2},\qquad\lambda_{34}\equiv\lambda_{3}-\lambda_{4}.$
(20)

The values of helicities for the amplitudes $\Phi_{i}(s,t,u)$ are given in
(2).

Using partial amplitudes we can write full non-linear unitarity constraints in
simple positive semi-definite form as

$\begin{pmatrix}\mathbb{I}&\mathbb{S}^{\dagger}(s)\\\
\mathbb{S}(s)&\mathbb{I}\end{pmatrix}\succeq 0,$ (21)

where $\mathbb{I}$ is the (appropriate) identity matrix and

$\mathbb{S}(s)=\begin{cases}1+i(\Phi_{1}^{0}(s)+\Phi_{2}^{0}(s)),\qquad\ell=0,\\\
1+i(\Phi_{1}^{\ell}(s)-\Phi_{2}^{\ell}(s)),\qquad\ell\geq 0\text{ (even)},\\\
1+2i\Phi_{3}^{\ell}(s),\qquad\qquad\quad\;\;\;\ell\geq 3\text{ (odd)},\\\
\begin{pmatrix}1&0\\\
0&1\end{pmatrix}+i\begin{pmatrix}\Phi_{1}^{\ell}(s)+\Phi_{2}^{\ell}(s)&\qquad
2\Phi_{5}^{\ell}(s)\\\ 2\Phi_{5}^{\ell}(s)&\qquad
2\Phi_{3}^{\ell}(s)\end{pmatrix},\qquad\ell\geq 2\text{ (even)}.\end{cases}$
(22)

These conditions hold in the physical regime $s\geq 0$.

There is a simpler subset of the above constraints (called positivity) given
by

$\operatorname{Im}\mathbb{F}(s)\succeq 0,$ (23)

where $\mathbb{F}$ is defined via $\mathbb{S}$ as follows

$\mathbb{S}=\mathbb{I}+i\mathbb{F}\,.$ (24)

A subset (the linear part) of these positivity constraints read as

$\begin{split}&\ell\geq 0\text{
(even)}:\qquad\operatorname{Im}(\Phi_{1}^{\ell}(s)+\Phi^{\ell}_{2}(s))\geq
0,\\\ &\ell\geq 0\text{
(even)}:\qquad\operatorname{Im}(\Phi^{\ell}_{1}(s)-\Phi^{\ell}_{2}(s))\geq
0,\\\ &\ell\geq 2:\qquad\qquad\quad\operatorname{Im}\Phi^{\ell}_{3}(s)\geq
0.\end{split}$ (25)

##### Goal of the paper

In this paper we study bounds on the dimensionless observables (12) coming
from full non-linear unitarity (21).

##### Bounds from positivity

Using dispersion relations it is relatively easy to incorporate the positivity
constraints (25) and obtain analytic bounds on the observables. This is
addressed in section 2. Here we briefly summarize the results from that
section.

For the observables $g_{2}$ and $f_{2}$ we obtain the following rigorous
bounds

$-1\leq\frac{f_{2}}{g_{2}}\leq 1,\qquad g_{2}\geq 0.$ (26)

We could not derive any bound on $g_{3}$, $f_{3}$ and $h_{3}$ using dispersion
relations and positivity. There are simple dispersion relations for $g_{4}$
and $f_{4}$, however due to the presence of log terms in (6) we cannot derive
bounds on these observables. More precisely, we find

$\displaystyle\bar{g}_{4}\pm 2\bar{f}_{4}$ $\displaystyle\geq
0+\underbrace{\frac{42\bar{f}_{2}^{2}\pm
50\bar{f}_{2}+21}{480\pi^{2}}}_{>0}\log(\hat{s}\sqrt{g_{2}})+\mathcal{O}(\hat{s}\sqrt{g_{2}})\,,$
(27)

where $0<\hat{s}\lesssim M^{2}$. The first (zero) term in the right-hand side
of (27) was obtained in Henriksson:2021ymi ; Henriksson:2022oeu  neglecting
the branch cuts from photon loops. The second term in these expressions is a
novel result obtained by taking into account the $log$ terms. We would like to
consider $\hat{s}\rightarrow 0$ to drop the error term. However, in this
limit, the bound (27) is useless because the $log$ terms diverge.

More bounds similar to these can be derived—see for example Guerrieri:2020bto
; Bellazzini:2020cot ; Bellazzini:2021oaj  for bounds including IR _logs_. For
bounds on EFTs from positivity derived in various other contexts, see Caron-
Huot:2020cmc ; Caron-Huot:2021rmr ; Davighi:2021osh ; deRham:2021fpu ;
Henriksson:2021ymi ; Henriksson:2022oeu ; Caron-Huot:2022ugt ; Tolley:2020gtv
; deRham:2021bll ; deRham:2022hpx ; Chiang:2022ltp ; Caron-Huot:2022jli .

##### Bounds from full non-linear unitarity

We use the primal numerical approach of Paulos:2017fhb ; Homrich:2019cbt  to
bound the observables (12) using full non-linear unitarity (21). This is done
in section 3. We briefly summarize our results here.

First, we found numerically that the bound on $\bar{f}_{2}$ is identical to
(26) which was found by using positivity only. Second, we found that neither
upper nor lower bounds exist on the obervables $\bar{g}_{3}$, $\bar{f}_{3}$
and $\bar{h}_{3}$. Finally, we discovered that there exists a lower bound on
$\bar{g}_{4}$. The bound is presented in figures 2 \- 4. There the lower bound
on $\bar{g}_{4}$ is constructed as a function of $\bar{f}_{2}$, $\bar{f}_{4}$
and $\bar{g}_{4}^{\prime}$ respectively.

All our numerical data can be downloaded from
https://doi.org/10.5281/zenodo.7308006.

##### The absence of bounds

As stated in the previous paragraph, we often see that there is no bound on a
given observable. We explain this fact in section 4 by explicitly constructing
weakly coupled theories which satisfy all our assumptions and have unbounded
observables. For instance we show analytically that no bounds exist on
$\bar{g}_{3}$, $\bar{f}_{3}$ and $\bar{h}_{3}$.

Figure 2: Lower bound on the observable $\bar{g}_{4}$ as a function of
$\bar{f}_{2}$. The allowed region is shaded in light blue. The estimated error
is indicated by dark blue.

Figure 3: Lower bound on the $\bar{g}_{4}$ observable as a function
$\bar{f}_{4}$ for a given value of $\bar{f}_{2}$. The allowed region lies
above the colored lines. The dashed line described by
$\bar{g}_{4}=2|\bar{f}_{4}|$ is placed for reference.

Figure 4: Lower bound on the $\bar{g}_{4}$ observable as a function
$\bar{g}_{4}^{\prime}$ for a given value of $\bar{f}_{2}$. The allowed region
lies above the colored lines.

##  2 Bounds from positivity and constraints on EFTs

In this section we will write relations which allow to express the observables
(12) or (92) as integrals of the amplitudes $\Phi_{i}(s,t,u)$. These relations
are called dispersion relations. For simplicity we will work in the forward
limit $t=0$. Combining dispersion relations with positivity (25) allows us to
bound our observables. In what follows we will focus only on $g_{2}$, $f_{2}$,
$g_{4}$ and $f_{4}$.

##### Positivity constraints

Positivity constraints were given in (25) in terms of partial amplitudes.
These constraints can be equivalently translated into amplitudes in the
forward limit using the inverse of (18). We get then

$\operatorname{Im}(\Phi_{1}(s)\pm\Phi_{2}(s))\geq
0,\qquad\operatorname{Im}\Phi_{3}(s)\geq 0.$ (28)

We independently derive these constraints also in appendix A.4. Here and in
the rest of this section we use the following short-hand notation for the
forward amplitudes

$\Phi_{i}(s)\equiv\Phi_{i}(s,\,t=0,\,u=-s).$ (29)

##### Dispersion relations

Let us start by defining the following functions

$V_{n}^{\pm}(s)\equiv\frac{\Phi_{1}(s)\pm\Phi_{2}(s)+\Phi_{3}(s)}{s^{n+1}}$
(30)

in the $s$ complex plane. The functions $V_{n}^{\pm}(s)$ have an analytic
structure inherited from the functions $\Phi_{i}(s,t,u)$ as depicted in figure
1 and an additional pole at $s=0$. The functions $V_{n}^{\pm}(s)$ obey
$V_{n}^{\pm}(-s)=-V_{n}^{\pm}(s)$ for even $n$ due to $s-u$ crossing symmetry
in equation (5) and the discussion below it. Finally, the imaginary part of
the functions $V_{n}^{\pm}(s)$ is non-negative due to positivity constraints
(28) for $s>0$.

Integrating $V_{n}^{\pm}(s)$ over a closed contour as depicted in figure 5 we
get

$\oint dsV_{n}^{\pm}(s)=0$ (31)

since $V_{n}^{\pm}(s)$ are analytic inside this contour of integration. The
contour has several pieces: the small arc with radius $\hat{s}$ denoted by
$\gamma$, the big arc with infinitely large radius denoted by $\Gamma$ and two
horizontal stretches. Splitting the integral in (31) into these pieces we get

$\int_{\infty}^{\hat{s}}dsV_{n}^{\pm}(s)+\int_{-\hat{s}}^{-\infty}dsV_{n}^{\pm}(s)+\int_{\gamma}dsV_{n}^{\pm}(s)+\int_{\Gamma}dsV_{n}^{\pm}(s)=0.$
(32)

Assuming the analogue of the Martin-Froissart bound for spin one massless
particles888For a recent discussion of the Froissart bound in the case of
massless spin two particles see Haring:2022cyf . we get

$\lim_{|s|\rightarrow\infty}s\,V_{n}^{\pm}(s)=0\,,\quad\text{for }n\geq 2$
(33)

As a result the integral over the large arc $\Gamma$ vanishes

$\int_{\Gamma}dsV_{n}^{\pm}(s)=0.$ (34)

Using $s-u$ crossing symmetry we get999The first integral is evaluated
slightly above the right branch cute. By crossing symmetry, the left integral
is related to the integral slightly below the right branch cut. The sum of the
two terms gives a discontinuity which in turn is related to the imaginary part
of the amplitude.

$-\left(\int_{\infty}^{\hat{s}}dsV_{n}^{\pm}(s)+\int_{-\hat{s}}^{-\infty}dsV_{n}^{\pm}(s)\right)=2i\,\int_{\hat{s}}^{\infty}ds\operatorname{Im}V_{n}^{\pm}(s).$
(35)

In order to evaluate the third integral in (32) we use the representation (6)
of the amplitudes which is valid for $|s|\lesssim M^{2}$. Performing the
change of variables $s=\hat{s}\cos\phi$, where $\phi\in[0;\pi]$, we get

$\displaystyle\int_{\gamma}dsV_{2}^{\pm}(s)$ $\displaystyle=\pi i\,2(g_{2}\pm
f_{2})+\mathcal{O}(\hat{s}^{2}/M^{8}),$ (36)
$\displaystyle\int_{\gamma}dsV_{4}^{\pm}(s)$ $\displaystyle=\pi
i\,2\left(g_{4}\pm 2f_{4}+\left(\beta_{1,1}+\beta_{1,3}\
\pm\beta_{2}\right)\log\left(\hat{s}\sqrt{g_{2}}\right)\right)+\mathcal{O}(\hat{s}/M^{10})\,,$

where we estimated the error using the expected EFT scaling of Wilson
coefficients with the cutoff scale $M$.

Plugging equations (34) - (36) into (32) we finally obtain the following
dispersion relations

$g_{2}\pm
f_{2}+\mathcal{O}(\hat{s}^{2}/M^{8})=\frac{1}{\pi}\int_{\hat{s}}^{\infty}ds\frac{\operatorname{Im}[\Phi_{1}(s)\pm\Phi_{2}(s)+\Phi_{3}(s)]}{s^{3}},$
(37)

together with

$g_{4}\pm 2f_{4}+\left(\beta_{1,1}+\beta_{1,3}\
\pm\beta_{2}\right)\log\left(\hat{s}\sqrt{g_{2}}\right)+\mathcal{O}(\hat{s}/M^{10})=\\\
\frac{1}{\pi}\int_{\hat{s}}^{\infty}ds\frac{\operatorname{Im}[\Phi_{1}(s)\pm\Phi_{2}(s)+\Phi_{3}(s)]}{s^{5}}.$
(38)

$\gamma$$\Gamma$$s$$\hat{s}$$\bullet$ Figure 5: Integration contour of the
analytic functions $V_{n}^{\pm}$ defined in (30) in the $s$ complex plane. The
functions $V_{n}^{\pm}$ have the same analytic structures as in figure 1.
Compared to figure 1 they have however an additional pole at $s=0$ indicated
by a green dot.

##### Bounds on $g_{2}$ and $f_{2}$

Consider the dispersion relation (37) and take the limit $\hat{s}\rightarrow
0$. In this limit the combination $g_{2}\pm f_{2}$ is related to a certain
integral of the amplitudes. Due to positivity in the form (28) this integral
is non-negative and we obtain the following rigorous inequalities

$g_{2}\geq 0,\qquad-1\leq\frac{f_{2}}{g_{2}}\leq 1.$ (39)

##### Bounds on $g_{4}$ and $f_{4}$

Consider now the dispersion relation (38). Naively it is impossible to take
the limit $\hat{s}\rightarrow 0$ because there is a log term in the left-hand
side which diverges in this limit. Nevertheless, using positivity conditions
(28) we arrive at

$g_{4}\pm 2f_{4}+\underbrace{\left(\beta_{1,1}+\beta_{1,3}\
\pm\beta_{2}\right)}_{<0}\log\left(\hat{s}\sqrt{g_{2}}\right)+\mathcal{O}(\hat{s}/M^{10})\geq
0.$ (40)

This relation holds for any $\hat{s}$ in the range $\hat{s}\lesssim M^{2}$.
Unfortunately this relation is not very useful because either the error term
is large (for $\hat{s}$ large but still below the cutoff scale $M^{2}$) or the
$\log$ term is large (for $\hat{s}$ small). Dividing by $g_{2}^{2}$, we obtain
a dimensionless version of (40):

$\displaystyle\bar{g}_{4}\pm 2\bar{f}_{4}$ $\displaystyle\geq
0+\underbrace{\frac{42\bar{f}_{2}^{2}\pm
50\bar{f}_{2}+21}{480\pi^{2}}}_{>0}\log(\hat{s}\sqrt{g_{2}})+\mathcal{O}(\hat{s}\sqrt{g_{2}})\,,$
(41)

which we quoted in the introduction. Here we further assumed that $g_{2}\sim
M^{-4}$ to simplify the error term.

It is actually possible to take the limit $\hat{s}\rightarrow 0$ inside (38).
To this end, consider the following representation of the logarithm

$\log(x)=\int_{x}^{\infty}dy\frac{-1}{y(1+y)}+\mathcal{O}(x).$ (42)

It allows to bring the result (38) into the following form

$g_{4}\pm
2f_{4}=\frac{1}{\pi}\int_{\hat{s}}^{\infty}ds\left(\frac{\operatorname{Im}[\Phi_{1}(s)\pm\Phi_{2}(s)+\Phi_{3}(s)]}{s^{5}}+\frac{\pi(\beta_{1,1}+\beta_{1,3}\pm\beta_{2})}{s(1+s\sqrt{g_{2}})}\right)+\mathcal{O}(\hat{s}/M^{10}).$
(43)

The integrand here is well-defined at $\hat{s}\rightarrow 0$ since the
divergence of the first term governed by (6) cancels the divergence of the
second term. We can thus write an explicit integral form of $g_{4}\pm 2f_{4}$
as

$g_{4}\pm
2f_{4}=\frac{1}{\pi}\int_{0}^{\infty}ds\left(\frac{\operatorname{Im}[\Phi_{1}(s)\pm\Phi_{2}(s)+\Phi_{3}(s)]}{s^{5}}+\frac{\pi(\beta_{1,1}+\beta_{1,3}\pm\beta_{2})}{s(1+s\sqrt{g_{2}})}\right).$
(44)

Let us now notice that

$\beta_{1,1}+\beta_{1,3}\pm\beta_{2}<0.$ (45)

This means that the second term in the integrand in (44) is negative whereas
the first term in the integrand is non-negative due to positivity. As a result
the integrand in (44) does not have definite sign and no positivity bound can
be deduced on the simple combination $g_{4}\pm 2f_{4}$.

##  3 Bounds from full non-linear unitarity

In this section we present our numerical bounds. We start in subsection 3.1 by
explaining our numerical setup. We will then use it to bound the coefficient
$\bar{f}_{2}$ in subsection 3.2. We will explore bounds on $\bar{g}_{3}$,
$\bar{f}_{3}$ and $\bar{h}_{3}$ in subsection 3.3. We will study bounds on
$\bar{g}_{4}$, $\bar{g}_{4}^{\prime}$ and $\bar{f}_{4}$ in subsection 3.4. In
subsection 3.5 we show in all generality that parameters appearing linearly in
a scattering amplitude are subject at most to one-sided bounds in the non-
perturbative S-matrix bootstrap. Finally, in subsection 3.6 we test the low
spin dominance conjecture using our data.

We make our numerical data public. It can be downloaded from
https://doi.org/10.5281/zenodo.7308006.

###  3.1 Numerical setup

We consider the following non-perturbative ansatz for the three independent
amplitudes101010The prefactors in front of $\Phi_{1}\,,\Phi_{5}$ use the
variable $\chi$ defined below (48). They are included so that at low energy,
the prefactors become those of the spinor-helicity amplitudes (265).

$\displaystyle\Phi_{1}(s,t,u)$
$\displaystyle=\chi_{s}^{2}\sum_{a,b,c=0}^{N_{\text{max}}}\alpha^{1}_{abc}\rho_{s}^{a}\rho_{t}^{b}\rho_{u}^{c}+\mathbb{L}_{1}(s|t,u),$
(46) $\displaystyle\Phi_{2}(s,t,u)$
$\displaystyle=\sum_{a,b,c=0}^{N_{\text{max}}}\alpha^{2}_{abc}\rho_{s}^{a}\rho_{t}^{b}\rho_{u}^{c}+\mathbb{L}_{2}(s,t,u),$
$\displaystyle\Phi_{5}(s,t,u)$
$\displaystyle=(-\chi_{s}\chi_{t}\chi_{u})\sum_{a,b,c=0}^{N_{\text{max}}}\alpha^{5}_{abc}\rho_{s}^{a}\rho_{t}^{b}\rho_{u}^{c}.$

Here $\alpha^{i}_{abc}$ are real dimensionless parameters. Due to crossing
symmetry $\alpha^{1}_{abc}=\alpha^{1}_{acb}$ and $\alpha^{2}_{abc}$,
$\alpha^{5}_{abc}$ are fully symmetric in their indices. The $\rho$-variable
is defined as

$\rho_{s}\equiv\frac{\sqrt{-s_{0}}-\sqrt{-s}}{\sqrt{-s_{0}}+\sqrt{-s}}.$ (47)

where $s_{0}<0$ is a free real parameter. The $\chi$-variable is defined as

$\chi_{s}\equiv\frac{1}{4}(\rho_{s}-1)^{2}-\frac{1}{4}(\rho_{s}-1)^{3}=\frac{s}{s_{0}}-3\left(\frac{s}{s_{0}}\right)^{2}+\mathcal{O}(s^{5/2}).$
(48)

This variable was introduced in Guerrieri:2020bto , and it is built in such a
way that it becomes a constant at high energy. It is also convenient to define
the following object

$P\equiv\frac{1}{8}(1+\rho_{s})(1+\rho_{t})(1+\rho_{u}).$ (49)

At fixed scattering angle $\theta$ (defined using the Mandelstam variables in
(105)) and small value of $s$ it behaves as $1$ and at fixed angle and large
value of $s$ it decays as $\mathcal{O}(s^{-3/2})$. Finally the functions
$\mathbb{L}_{1}$ and $\mathbb{L}_{2}$ are defined as111111The prefactor $P$ is
introduced here in order to make the functions $\mathbb{L}_{1}(s|t,u)$ and
$\mathbb{L}_{2}(s,t,u)$ decay fast enough at large values of $s$. This allows
to use the large energy constraints derived in appendix H.2.

$\displaystyle\mathbb{L}_{1}(s|t,u)$ $\displaystyle\equiv
s_{0}^{4}\,P\,\chi_{s}^{2}\left[\beta_{1,1}\chi_{s}^{2}\log\chi_{s}+\beta_{1,2}\chi_{t}\chi_{u}\log\chi_{s}+\beta_{1,3}(\chi_{t}^{2}\log\chi_{t}+\chi_{u}^{2}\log\chi_{u})\right],$
(50) $\displaystyle\mathbb{L}_{2}(s,t,u)$ $\displaystyle\equiv
s_{0}^{4}\,P\,\beta_{2}(\chi_{s}^{4}\log\chi_{s}+\chi_{t}^{4}\log\chi_{t}+\chi_{u}^{4}\log\chi_{u}).$

These functions have the following low energy expansion:

$\displaystyle\mathbb{L}_{1}(s|t,u)$
$\displaystyle=s^{2}\left[\beta_{1,1}s^{2}\log(s/s_{0})+\beta_{1,2}tu\log(s/s_{0})+\beta_{1,3}(t^{2}\log(t/s_{0})+u^{2}\log(u/s_{0}))\right]+\mathcal{O}(s^{5}),$
$\displaystyle\mathbb{L}_{2}(s,t,u)$
$\displaystyle=\beta_{2}(s^{4}\log(s/s_{0})+t^{4}\log(t/s_{0})+u^{4}\log(u/s_{0}))+\mathcal{O}(s^{5}).$

Comparing them with (7) we conclude that

$\displaystyle\mathbb{L}_{1}(s|t,u)$
$\displaystyle=L_{1}(s|t,u)-s^{2}\left[\beta_{1,1}s^{2}+\beta_{1,2}tu+\beta_{1,3}(t^{2}+u^{2})\right]\log(-s_{0}\sqrt{g_{2}})+\mathcal{O}(s^{5}),$
(51a) $\displaystyle\mathbb{L}_{2}(s,t,u)$
$\displaystyle=L_{2}(s,t,u)-\beta_{2}(s^{4}+t^{4}+u^{4})\log(-s_{0}\sqrt{g_{2}})+\mathcal{O}(s^{5}).$
(51b)

As was discussed in the introduction, scattering amplitudes of any massless
spin one particles in the vicinity of $s=0$ have the representation (6). We,
thus, need to expand (46) around $s=0$ taking into account (51) and match the
result with (6). This procedure will generate a set of linear constraints on
the parameters of the ansatz $\alpha$. Solving these constraints and plugging
the solution back into (46) we obtain the final form of the ansatz which
depends on the following parameters

$\\{s_{0};g_{2},g_{3},g_{4},g_{4}^{\prime},f_{2},f_{3},f_{4},h_{3},\,\text{rest
of }\alpha s\\}.$ (52)

All the coefficients in this list enter the ansatz linearly except for
$s_{0}$, $g_{2}$ and $f_{2}$. Squares of $g_{2}$ and $f_{2}$ multiply the log
terms. The former also enters inside the log terms.

The ansatz (46) has a finite number of terms controlled by the parameter
$N_{\text{max}}$. All the numerical results depend on this parameter. The true
bound is obtained by extrapolating the numerical results to
$N_{\text{max}}=\infty$.

In order to impose non-linear unitarity we first compute partial amplitudes by
plugging (46) into the definition (18). The integrals are evaluated
numerically in Mathematica. They will depend on the set of parameters (52).
Plugging them into (21) we obtain a set of unitarity constraints. We impose
these constraints for a finite number of spins $\ell=0,\ldots,L_{\text{max}}$.
All the numerical results also depend on the parameter $L_{\text{max}}$ and
thus require an extrapolation to $L_{\text{max}}=\infty$. We carefully discuss
the extrapolation procedure for both $N_{\text{max}}$ and $L_{\text{max}}$ in
subsection 3.4.1. In order to improve the convergence with $L_{\text{max}}$,
we add the unitarity constraints in the limit $L_{\text{max}}=\infty$. This is
discussed in appendix H.1. The convergence is also improved by adding the
amplitude positivity constraint in the forward limit (153). As
$L_{\text{max}}\to\infty$, these constraints are included in the full
unitarity constraints imposed numerically (21) via the positivity (25).
However, for any finite $L_{\text{max}}$, (153) contains more information.

The unitarity constraints (21) should be imposed for all $s\geq 0$. In
practice we pick a finite grid of $s$ values where we impose unitarity. The
points are chosen using the Chebyshev distribution in the $\rho_{s}$ variable
(see _e.g._ footnote 34 in Hebbar:2020ukp ). The number of points in this grid
is denoted by $N_{\text{grid}}$. In all the calculations we use
$N_{\text{grid}}=200$ for unitarity constraints with spins $\ell\leq 50$ and
$N_{\text{grid}}=50$ for unitarity constraints with spins $\ell>50$. We also
include analytic constraints at $s=\infty$, for details see appendix H.2. On
top of the unitarity constraints (21) we also impose the positivity
constraints (28) at $N_{\text{grid}}=200$ values of $s$.

The amplitudes in $d=4$ are dimensionless. We can thus rewrite the ansatz (46)
in terms of dimensionless quantities only. This is done by using $g_{2}$ as a
scale and defining

$\bar{s}\equiv\sqrt{g_{2}}\,s,\qquad\bar{t}\equiv\sqrt{g_{2}}\,t,\qquad\bar{u}\equiv\sqrt{g_{2}}\,u.$
(53)

Identical definitions are understood to hold for $s_{0}$. With these
definitions the explicit dependence on $g_{2}$ completely disappears in (46)
and the parameters we are left with are precisely the ones given in (12). In
other words the ansatz depends only on the following coefficients

$\\{\bar{s}_{0};\bar{g}_{3},\bar{g}_{4},\bar{g}_{4}^{\prime},\bar{f}_{2},\bar{f}_{3},\bar{f}_{4},\bar{h}_{3},\,\text{rest
of }\alpha s\\}$ (54)

and $g_{2}$ is not a free variable. We can chose the parameter $\bar{s}_{0}$
at our will. A particular choice does not play any role for significantly
large values of $N_{\text{max}}$. It can happen however that for some values
of these parameters the numerics converge better and it is desirable to search
for their optimal values. In practice, this search is unfeasible since we do
not want to numerically evaluate the partial amplitudes for different choices
of $\bar{s}_{0}$.

There is however a trick which allows to keep a tunable parameter in the
numerical setup. To achieve that, we keep the ansatz in the form where $g_{2}$
enters explicitly. We then set

$s_{0}=-1.$ (55)

This sets the scale of the problem which makes all the other variables and
parameters (like $s$ or $g_{2}$) effectively dimensionless. The parameter
$g_{2}$ is now free and can be set to any value. The bounds on (54) do not
depend on this choice in the limit $N_{\text{max}}=\infty$. However, for a
particular value of $g_{2}$ the numerics will converge faster. Faster
convergence means in practice that at a fixed $N_{\text{max}}$ there is some
value of $g_{2}$ which leads to a better result (_e.g._ if we look for a
minimal value of some parameter, better result means lower minimum of this
parameter). For each optimization problem, we always perform a scan in $g_{2}$
first. This strategy was already used in Guerrieri:2020bto  and
Guerrieri:2021ivu .121212We thank Andrea Guerrieri for emphasizing this idea.
The explicit details of this scan are explained below. Scanning over
$\bar{s}_{0}$ and scanning over $g_{2}$ given (55) is equivalent. This can be
seen by using the definition (53) and (55) which lead to

$\bar{s}_{0}=-\sqrt{g_{2}}.$ (56)

We solve the following optimization problem: find the values of parameters
(54) such that the unitarity conditions (21) are satisfied and one of the
parameters in (54) has a minimal or maximal value. We construct optimization
problems in Mathematica. For solving the optimization problems we use SDPB
Simmons-Duffin:2015qma ; Landry:2019qug . The summary of the parameters used
in our numerics can be found in table 1.

SDPB precision |  $832$ (binary)
---|---
dualityGapThreshold | $10^{-8}$
integral precision |  $35$ (decimal)
mathematica internal precision |  $200$ (decimal)
$N_{\text{grid}}$ | $\left\\{\begin{matrix}200&\ell\leq 50\\\ 50&\ell>50\end{matrix}\right.$
Table 1: Parameters in the numerical setup.

###  3.2 Bounds on $\bar{f}_{2}$

The very first optimization problem we would like to address is: what are the
maximal and minimal allowed values of the observable $\bar{f}_{2}$. For
solving this optimization problem, we set the $\beta$’s as free linear
parameters of our ansatz (50).131313One may think that including the
constraints (8) can lead to stronger bounds on $\bar{f}_{2}$. However, the
extremal amplitudes that give $\bar{f}_{2}\to\pm 1$ are very weakly coupled
and therefore the 1-loop $\log$ terms are irrelevant.  The numerical solution
of this optimization problem at a fixed value $N_{\text{max}}=20$ and several
values of $g_{2}$ is presented in table 2. Here we choose $L_{\text{max}}=50$.
The numerics converges extremely fast for this run and no extrapolation to
$N_{\text{max}}\rightarrow\infty$ and $L_{\text{max}}\rightarrow\infty$ is
needed. From table 2 we conclude that

$-1\lessapprox\bar{f}_{2}\lessapprox+1,$ (57)

which is in perfect agreement with the positivity bound (39). The solution is
very stable and depends weakly on $g_{2}$. Nevertheless, we see that when
$g_{2}\rightarrow 0$ we get slightly better results (smaller minimum and
greater maximum). This suggests that the extremal values of the bound
$\bar{f}_{2}=\pm 1$ are saturated by free theories, which have
$g_{2}=0$.141414Strictly speaking, the amplitude vanishes in free theory, and
there is no meaning for the observables (12). What we mean by “free theories”
in this section is, more precisely, the free limit of weakly coupled
amplitudes, for which ratios of Wilson coefficients can have well defined
limits.

Notice that the solution presented in table 2 is symmetric under the
$\bar{f}_{2}\leftrightarrow-\bar{f}_{2}$ exchange. This comes from the fact
that the unitarity constraints (21) are symmetric under
$\Phi_{2}\leftrightarrow-\Phi_{2}$ when $\Phi_{5}=0$. Investigating the
solution leading to table 2 we indeed see that $\Phi_{5}\approx 0$
numerically. For further discussions of the symmetries of the unitarity
constraints, see the last paragraph of appendix A.3.

The result obtained here can be seen as a non-trivial check of our numerical
setup. Interestingly, using full unitarity we do not get stronger bounds on
$\bar{f}_{2}$ than equation (39), which follows from positivity only. This is
easy to understand in light of the observation above: the bounds are saturated
by the free limit of weakly coupled theories, for which the constraints coming
from full non-linear unitarity are irrelevant. We review the free theories
saturating this bound in subsection 4.3.

$g_{2}$ | $\min(\bar{f}_{2})$ | $\max(\bar{f}_{2})$
---|---|---
$10^{-2}$ | $-0.998710$ | $0.998710$
$1$ | $-0.998619$ | $0.998619$
$10^{2}$ | $-0.991237$ | $0.991237$
Table 2: Minimal and maximal allowed values of the $\bar{f}_{2}$ observable
for different choices of $g_{2}$. These results are obtained with
$N_{\text{max}}=20$ and $L_{\text{max}}=50$.

###  3.3 Bounding the $(\bar{g}_{3},\bar{f}_{3},\bar{h}_{3})$ space

Let us now study the upper and lower bounds on the $\bar{g}_{3}$,
$\bar{f}_{3}$ and $\bar{h}_{3}$ observables.

Recall that our ansatz has log terms encoded into the objects
$\mathbb{L}_{1}(s|t,u)$ and $\mathbb{L}_{2}(s,t,u)$ defined in (50). The
coefficients $\beta$ in these expressions depend on $g_{2}$ and $f_{2}$
quadratically according to (8). Notice however that we can use our numerical
procedure to determine only observables entering linearly in our ansatz. The
terms with $g_{2}^{2}$ are not a problem since we always fix the value of
$g_{2}$. However, we need to take some care of terms containing $f_{2}^{2}$.
One option is to simply set $\mathbb{L}_{1}(s|t,u)=0$ and
$\mathbb{L}_{2}(s,t,u)=0$ in the ansatz since these terms only become
important for bounding $\bar{g}_{4}$, $\bar{g}_{4}^{\prime}$ and
$\bar{f}_{4}$. Another option is to also fix the value of $f_{2}$ or
equivalently the value of $\bar{f}_{2}$.151515Yet another option is to keep
$\beta$ as free linear parameters of the ansatz (50). We have realized both
options in practice, in this section we present only the results of the second
one. It is important to notice, though, that these options are inequivalent in
principle. We will comment on this fact in subsection 3.5.

Let us now look for the minimal and maximal values of the $\bar{g}_{3}$
observable at some fixed value of $\bar{f}_{2}$. Concretely, we will use two
values $\bar{f}_{2}=0$ and $\bar{f}_{2}=3/10$. We work at $N_{\text{max}}=20$
and $L_{\text{max}}=50$, and do not perform any extrapolation in these
parameters. The solution of this optimization problem is presented in table 3
for different values of $g_{2}$. From this table we see that the optimal value
of $g_{2}$ is at $g_{2}=0$ (when $g_{2}\rightarrow 0$ we get the lowest
minimum and the highest maximum). The results of table 3 suggest that the
optimal solution is a free theory (because $g_{2}=0$) with

$-\infty\leq\bar{g}_{3}\leq+\infty.$ (58)

Exactly the same conclusion holds for $\bar{f}_{3}$ and $\bar{h}_{3}$. We will
confirm this finding by analytically constructing free theories with
$g_{2}=0$, $\bar{g}_{3}=\pm\infty$, $\bar{f}_{3}=\pm\infty$ and
$\bar{h}_{3}=\pm\infty$ in subsection 4.4.

| $\bar{f}_{2}=0$ | $\bar{f}_{2}=3/10$
---|---|---
$g_{2}$ | $\min(\bar{g}_{3})$ | $\max(\bar{g}_{3})$ | $\min(\bar{g}_{3})$ | $\max(\bar{g}_{3})$
$10^{-2}$ | $-546.9$ | $470.5$ | $-390.2$ | $466.4$
$1$ | $-54.67$ | $47.00$ | $-39.00$ | $46.59$
$10^{2}$ | $-5.397$ | $4.628$ | $-3.840$ | $4.585$
Table 3: Minimal and maximal allowed values of $\bar{g}_{3}$ at a fixed value
of $\bar{f}_{2}$ for different choices of $g_{2}$. These results are obtained
with $N_{\text{max}}=20$ and $L_{\text{max}}=50$.

###  3.4 Bounding the $(\bar{g}_{4},\bar{g}^{\prime}_{4},\bar{f}_{4})$ space

Let us finally study bounds on the observables $\bar{g}_{4}$,
$\bar{g}_{4}^{\prime}$ and $\bar{f}_{4}$. As explained in the previous
subsection, all the bounds are obtained at some fixed values of $\bar{f}_{2}$.
In subsection 3.4.1 we present our numerical results for the lower bound on
$\bar{g}_{4}$ as a function of $\bar{f}_{2}$. In subsection 3.4.2 we present
our lower bound on $\bar{g}_{4}$ as a function of $\bar{f}_{2}$ and
$\bar{f}_{4}$. Finally, in subsection 3.4.3 we present our lower bound on
$\bar{g}_{4}$ as a function of $\bar{f}_{2}$ and $\bar{g}_{4}^{\prime}$. No
upper bound exists on $\bar{g}_{4}$ and $\bar{g}_{4}^{\prime}$, and from the
results below one can conclude that neither upper nor lower bounds exist on
$\bar{f}_{4}$, as in the case of the $\bar{g}_{3}$, $\bar{f}_{3}$ and
$\bar{h}_{3}$ observables studied in the previous subsection. As for the lower
bound on $\bar{g}_{4}^{\prime}$, the numerics in that region converge poorly,
so no conclusion can be drawn. We will partly explain the presence of
unbounded directions in subsection 4.5, by explicitly constructing free
theories with infinitely large values for the Wilson coefficients. In
particular, we will also show that $\bar{g}_{4}^{\prime}$ cannot be bounded
from below.

####  3.4.1 $\min{\bar{g}_{4}}$ vs. $\bar{f}_{2}$

We start by minimizing $\bar{g}_{4}$ at several fixed values of $\bar{f}_{2}$.
The results of the numerical optimization are given in table 4. This data is
computed using $N_{\text{max}}=20$ and $L_{\text{max}}=50$. It is clear from
the table that the best convergence is achieved at $g_{2}\sim 100$, since this
value of $g_{2}$ gives the lowest minimum. In the reminder of subsection 3.4
we will always use $g_{2}=100$.

| $\min(\bar{g}_{4})$
---|---
$g_{2}$ | $\bar{f}_{2}=0$ | $\bar{f}_{2}=3/10$ | $\bar{f}_{2}=6/10$ | $\bar{f}_{2}=9/10$
$10^{-2}$ | $1.068$ | $1.176$ | $1.839$ | $11.41$
$1$ | $0.02478$ | $0.03205$ | $0.05527$ | $0.2837$
$10$ | $0.01711$ | $0.02142$ | $0.03974$ | $0.1706$
$100$ | $0.01588$ | $0.01993$ | $0.03646$ | $0.1581$
$200$ | $0.01583$ | $0.01995$ | $0.03645$ | $0.1631$
$1000$ | $0.01589$ | $0.02025$ | $0.03947$ | $0.3270$
Table 4: Lower bound on $\bar{g}_{4}$ at different values of $g_{2}$ and
$\bar{f}_{2}$. The optimal lowest minimum is achieved at $g_{2}\sim 100$. The
results are obtained with $N_{\text{max}}=20$ and $L_{\text{max}}=50$.

In practice our bounds depend on the parameters $N_{\text{max}}$ and
$L_{\text{max}}$. The correct bound is obtained only in the limit
$N_{\text{max}}\to\infty$ and $L_{\text{max}}\to\infty$. Let us carefully
discuss how one can estimate the correct lower bound on $\bar{g}_{4}$ in this
limit.

Let us begin by using the strategy employed in Guerrieri:2021ivu . In figure 7
we study the dependence of the $\bar{g}_{4}$ bound on $L_{\text{max}}$ at
fixed values of $N_{\text{max}}$. For concreteness we take $\bar{f}_{2}=-3/11$
which is the value in QED at leading order in the coupling. The numerical data
is indicated by colored dots. In the left plot of figure 7 we see that the
bound gets stronger when we increase $L_{\text{max}}$. For $N_{\text{max}}=20$
the bound stabilizes around $L_{\text{max}}=30$ and diverges around
$L_{\text{max}}=100$. In the interval $L_{\text{max}}\in(30,100)$ the bound is
linear. We refer to this interval as the plateau. For larger values of
$N_{\text{max}}$ the divergence of the bounds begins at larger values of
$L_{\text{max}}$. This can be explained as follows - for a fixed finite
$N_{\text{max}}$, as we increase $L_{\text{max}}$, the number of unitarity
constraints that the ansatz has to satisfy increases. At some point, the
ansatz is not big enough to satisfy all of these constraints and therefore the
bound that we get diverges. But if we then increase $N_{\text{max}}$, the
ansatz is now bigger and therefore it can satisfy unitarity constraints for
larger values of $L_{\text{max}}$. Nevertheless, even with this bigger
$N_{\text{max}}$ and hence a bigger ansatz, the bound will still diverge at
some (larger) $L_{\text{max}}$ and we must disregard the data after this
point.

We therefore restrict our attention to the plateau161616 We find the plateau
using the discrete derivative $\texttt{dDer}(1/L_{\text{max}})$. Then, we
select points such that $\texttt{dDer}(1/L_{\text{max}})\leq
X\cdot\min(\texttt{dDer})$, where $X$ is a factor that can be chosen. In
practice we chose $X=3$. and use linear extrapolation in $1/L_{\text{max}}$ to
obtain the lower bound of $\bar{g}_{4}$ at $L_{\text{max}}\rightarrow\infty$
for fixed values of $N_{\text{max}}$. The extrapolations are indicated by
solid lines in figure 7. The right plot in figure 7 is a zoomed in version of
the left plot. The circles around the numerical data there indicate the points
which were included in the plateau.

Once we obtain the extrapolated bounds at $L_{\text{max}}\rightarrow\infty$ we
also do a linear extrapolation in $1/N_{\text{max}}$. The result is depicted
in figure 7 in black. Black dots indicate extrapolated
$L_{\text{max}}\rightarrow\infty$ values. Black lines indicate linear
extrapolation in $1/N_{\text{max}}$. From previous experience we expect the
numerical data to be linear in $1/N_{\text{max}}$ starting from
$N_{\text{max}}\sim 20$. This is compatible with the results depicted in
figure 7.

Figure 6: Lower bound on $\bar{g}_{4}$ at fixed $\bar{f}_{2}=-3/11$ (QED
point). Colored dots indicate the numerical data. Solid lines indicate linear
extrapolations in $1/L_{\text{max}}$. The right plot is a zoomed version of
the left one. The circles around the numerical data indicate where the data is
linear in $1/L_{\text{max}}$. The linear regions are called the plateaus. The
linear extrapolation is performed only within each plateau.

Figure 7: Lower bound on $\bar{g}_{4}$ at fixed $\bar{f}_{2}=-3/11$ (QED
point). Colored dots denote the numerical data obtained at various fixed
values of $L_{\text{max}}$. Black dots are obtained instead by performing
$L_{\text{max}}\rightarrow\infty$ extrapolation in figure 7. Dashed lines
indicate linear extrapolations in $1/N_{\text{max}}$ to
$N_{\text{max}}\rightarrow\infty$. Figure 8: Lower bound on $\bar{g}_{4}$ as a
function of $\bar{f}_{2}$. Colored dots represent the numericsl data obtained
with $L_{\text{max}}$ defined in (60) and various values of $N_{\text{max}}$.
Black bars represent the extrapolation of the numerical results to
$N_{\text{max}}\rightarrow\infty$ and indicate the estimated errors.

Performing the above extrapolation in $L_{\text{max}}$ and $N_{\text{max}}$ is
computationally very expensive since it requires obtaining a lot of numerical
data. Let us explore a cheaper alternative. One could compute bounds at
various values of $N_{\text{max}}$ with

$L_{\text{max}}=aN_{\text{max}}-b,$ (59)

with $a$ and $b$ some integer numbers. Once the data with this
$L_{\text{max}}$ is obtained, we can perform a linear extrapolation in
$1/N_{\text{max}}$ to $N_{\text{max}}\rightarrow\infty$. In figure 7 we make
several choices of $a$ and $b$. All the resulting extrapolations lie very
close to the one obtained by using the previous approach. Notice, however,
that the choice of $a$ and $b$ cannot be completely arbitrary, the values of
$L_{\text{max}}$ should always be inside the plateau. In figure 7 we also
indicate the extrapolation using the points on the plateau right before the
divergence. We denote this choice by “End of plateaus”. In the rest of this
section, we shall use

$L_{\text{max}}=5N_{\text{max}}-50\,,$ (60)

to perform extrapolations to $N_{\text{max}}\rightarrow\infty$. From now on,
we assume that (60) is a good choice for other values of $\bar{f}_{2}$ and all
the other bounds computed below.

We are finally in a position to present our lower bound on $\bar{g}_{4}$ as a
function of $\bar{f}_{2}$. It is given in figure 8. The numerical data for
different $N_{\text{max}}$ is depicted by colored dots. The extrapolated
values are given by black bars which also reflect a rough error of the
extrapolation.171717To estimate the error on the extrapolated bound, we
perform a linear interpolation using $N_{\text{max}}\geq N_{*}$ where $18\leq
N_{*}\leq 24$ and compute the maximal difference between interpolated values.
Therefore, this error indicates if the linear interpolation is a reasonable
extrapolation of the bound to $N_{\text{max}}\to\infty$.  Figure 2 presented
in the introduction is obtained from 8 by using only the extrapolated results
and connecting the points.

The lower bound on $\bar{g}_{4}$ is symmetric under
$\bar{f}_{2}\to-\bar{f}_{2}$ as a consequence of the
$\Phi_{2}\leftrightarrow-\Phi_{2}$ symmetry in the unitarity constraints (21)
when $\Phi_{5}=0$. Indeed, we observe that the optimal solution leads to
$\Phi_{5}\approx 0$ and thus the numerical solution is
$\Phi_{2}\leftrightarrow-\Phi_{2}$ symmetric. The same remains true in
subsections 3.4.2 and 3.4.3. The absolute minimum of the bound in figure 2 is
achieved at $f_{2}=0$ and is given by

$\bar{g}_{4}\geq 0.0138\pm 0.0003.$ (61)

For larger values of $\bar{f}_{2}$ the errors quickly increase. In order to
understand this, recall from subsection 3.2 that the boundary values
$\bar{f}_{2}=\pm 1$ are saturated by free theories with $g_{2}=0$. Moreover,
we shall prove in subsection 4.3 that free theories are in fact the only
theories with $\bar{f}_{2}=\pm 1$. In order to efficiently obtain bounds in
this region we would need to keep $g_{2}\approx 0$, while in the present
section we fix $g_{2}=100$. This leads to bad convergence when we approach
$\bar{f}_{2}=\pm 1$.

####  3.4.2 $\min{\bar{g}_{4}}$ vs. $\bar{f}_{2}$ and $\bar{f}_{4}$

The lower bound on $\bar{g}_{4}$ as a function of $\bar{f}_{4}$ at two fixed
values $\bar{f}_{2}=0$ and $\bar{f}_{2}=0.3$ are shown in figures 10 and 10.
The colored lines indicate the numerical data for various values of
$N_{\text{max}}$. The gray line indicates the extrapolated bound. The width of
this line represents the extrapolation error.

As we can see from figures 10 and 10, the extrapolated bound has exactly the
same behavior as the finite $N_{\text{max}}$ one and lies very close to it. In
what follows we will focus on $N_{\text{max}}=20$ and $L_{\text{max}}=50$ in
order to reduce the cost of our numerical computation. This allows us to scan
over more values of $\bar{f}_{2}$. In figures 11 and 12 we present the lower
bound on $\bar{g}_{4}$ at $\bar{f}_{2}=n/10,\ n=0,1,...9$. Using the
$\Phi_{2}\leftrightarrow-\Phi_{2}$ symmetry discussed above we automatically
obtain the bound also for $n=-9,...-1$.

It is interesting to notice that all our bounds in figures 10 \- 10 satisfy
the naive condition

$\bar{g}_{4}\pm 2\bar{f}_{4}\geq 0$ (62)

which is obtained from (27) by dropping the log terms. We indicate this
condition by a gray dashed line.

Figure 9: Lower bound on $\bar{g}_{4}$ as a function of $\bar{f}_{4}$ at the
fixed value $\bar{f}_{2}=0$. Colored lines represent the numerical data
obtained with $L_{\text{max}}$ defined in (60) and various $N_{\text{max}}$.
The gray line represents the extrapolated bound to
$N_{\text{max}}\rightarrow\infty$. The width of the gray line represents the
estimated error. The extrapolated bound has the form
$\bar{g_{4}}\geq(0.0129\pm 0.0001)+(1.999\pm 0.001)|\bar{f}_{4}|$. The dashed
line is drawn for reference.

Figure 10: Lower bound on $\bar{g}_{4}$ as a function of $\bar{f}_{4}$ at the
fixed value $\bar{f}_{2}=0.3$. The extrapolated bound to the right of the
minimum has the form $\bar{g}_{4}\geq(0.0055\pm 0.0001)+(2.000\pm
0.001)\bar{f}_{4}$. The extrapolated bound to the left of the minimum has the
form $\bar{g}_{4}\geq(0.0240\pm 0.0001)+(-1.997\pm 0.002)\bar{f}_{4}$ .
Figure 11: Lower bound on $\bar{g}_{4}$ as a function of $\bar{f}_{4}$ at
fixed values of $\bar{f}_{2}$. This bound is constructed at
$N_{\text{max}}=20$ and $L_{\text{max}}=50$. Figure 12: Lower bound on
$\bar{g}_{4}$ as a function of $\bar{f}_{4}$ and $\bar{f}_{2}$. The
transparent gray V-shape represents $\bar{g}_{4}=2\bar{f}_{4}$, it is reported
for reference. The color gradient indicates places with the same values of
$\bar{f}_{2}$ and is equivalent to the one used in figure 11. The solid lines
indicate places with the same values of $\bar{g}_{4}$. The bound is
constructed at $N_{\text{max}}=20$ and $L_{\text{max}}=50$.

####  3.4.3 $\min{\bar{g}_{4}}$ vs. $\bar{f}_{2}$ and $\bar{g}_{4}^{\prime}$

The lower bound on $\bar{g}_{4}$ as a function of $\bar{g}_{4}^{\prime}$ at
two fixed values $\bar{f}_{2}=0$ and $\bar{f}_{2}=0.3$ are shown in figures 14
and 14. The colored lines indicate the numerical data for various values of
$N_{\text{max}}$. The gray line indicates the extrapolated bound. The width of
this line represents the extrapolation error. On these plots, we observed that
for negative values of $\bar{g}_{4}^{\prime}$ the extrapolation error grows
very quickly. As a result, the extrapolation cannot be trusted in this region.
A quick investigation shows that in this case there is no plateau in
$L_{\text{max}}$ analogous to figure 7. Further work is needed in order to
understand the issue.

Fixing $N_{\text{max}}=20$ and $L_{\text{max}}=50$ allows us to scan over more
values of $\bar{f}_{2}$. As in the previous subsection, exactly the same
bounds hold when $\bar{f}_{2}\to-\bar{f}_{2}$. We present our results in
figures 16 and 16. There, we focus on the region where one could have
performed a controlled extrapolation.

Figure 13: Lower bound on $\bar{g}_{4}$ as a function of
$\bar{g}^{\prime}_{4}$ at the fixed value $\bar{f}_{2}=0$. Colored lines (and
dots in the zoomed part) represent the numerical data obtained with
$L_{\text{max}}$ defined in (60) and various $N_{\text{max}}$. The gray line
(and black bars in the zoomed part) represent the extrapolated bound to
$N_{\text{max}}\rightarrow\infty$. The width of the gray line represents the
estimated error. The extrapolation to the left of the minimum has an enormous
estimated error and, thus, cannot be trusted.

Figure 14: Lower bound on $\bar{g}_{4}$ as a function of
$\bar{g}^{\prime}_{4}$ at the fixed value $\bar{f}_{2}=0.3$. The gray line
represents the extrapolation to $N_{\text{max}}\rightarrow\infty$.

Figure 15: Lower bound on $\bar{g}_{4}$ as a function of
$\bar{g}_{4}^{\prime}$ at fixed values of $\bar{f}_{2}$. This bound is
constructed at $N_{\text{max}}=20$ and $L_{\text{max}}=50$.

Figure 16: Lower bound on $\bar{g}_{4}$ as a function of
$\bar{g}_{4}^{\prime}$ and $\bar{f}_{2}$. The color gradient indicates places
with the same values of $\bar{f}_{2}$ and is equivalent to the one used in
figure 16. The solid lines indicate places with the same values of
$\bar{g}_{4}$. The bound is constructed at $N_{\text{max}}=20$ and
$L_{\text{max}}=50$.

###  3.5 Linear constraints and the numerical bounds

Let us now make a general remark on a limitation specific to convex
optimization, when it comes to bounding observables like the ones in (12). For
simplicity, consider the simplified setting of a single amplitude
$\mathcal{T}(s)$, function of a single Mandelstam invariant—for instance, a
scalar 2-to-2 amplitude in two dimensions. Given an amplitude which satisfies
unitarity and crossing, it is easy to see that the one parameter family

$\mathcal{T}_{\lambda}(s)=\lambda\mathcal{T}(s)~{},\qquad\lambda\in[0,1]~{},$
(63)

does as well. This simple fact implies that _no two-sided bound is possible
for dimensionless ratios of coefficients of different mass dimension, if they
appear linearly in the amplitude._

Concretely, suppose that the amplitude has a low energy expansion of the form

$\mathcal{T}(s)=g_{2}s^{2}+g_{3}s^{3}+\dots~{}.$ (64)

Then, equation (63) generates a one-parameter family of allowed values for the
dimensionless ratio:

$\bar{g}_{3}(\lambda)=\frac{g_{3}(\lambda)}{{[g_{2}(\lambda)]}^{3/2}}=\frac{1}{\lambda^{1/2}}\frac{g_{3}}{g_{2}^{3/2}}~{}.$
(65)

If $g_{3}$ is positive (negative) in the original solution to unitarity and
crossing, $\bar{g}_{3}$ is unbounded from above (below).

It is worth emphasizing that the argument above is not sufficient to forbid
two-sided bounds in physical EFTs: elastic unitarity imposes that Wilson
coefficients appear nonlinearly in the low energy expansion—see (6) - (8).
Hence, if particle production happens at some higher order in $s$, the
reasoning must be modified. As mentioned in subsection 3.3, we did add
nonlinear terms in the ansatz, in particular involving $g_{2}$, with the
exception of the runs to bound $\bar{f}_{2}$. However, as a matter of fact, in
our numerical runs we only found two-sided bounds precisely for $\bar{f}_{2}$.
The latter is a ratio of coefficients with the same mass dimension, and does
not scale with the parameter $\lambda$.

This raises the question whether the absence of two-sided bounds is indeed a
hallmark of Wilson coefficients of any EFT. In a certain region of parameter
space, the answer is affirmative, as we shall see in detail in the next
section. Indeed, weakly coupled theories precisely come equipped with a small
parameter $\lambda$, and the low energy Wilson coefficients are linear in it
at leading order—see table 9. Elastic unitarity is perturbatively satified
order by order in $\lambda$, hence one can consider these theories—if they can
be UV completed—as a non-linear completion of the example in (63). We will
indeed use weakly coupled theories to explain some features of the plots
presented in this section.

###  3.6 Numerical tests of low spin dominance

In this subsection, we test low spin dominance which was conjectured in
Bern:2021ppb .181818See also Chowdhury:2021ynh ; Ghosh:2022net  where low spin
dominance was shown to follow from locality in tree level EFTs. Consider the
observables

$\displaystyle\langle\rho_{\ell}^{+}\rangle_{k}$ $\displaystyle\equiv
16(\ell+1)\int_{0}^{\infty}\frac{\text{Im}\left(\Phi_{1}^{\ell}(s)+\Phi_{2}^{\ell}(s)\right)}{s^{k+1}}ds\,,$
(66) $\displaystyle\langle\rho_{\ell}^{-}\rangle_{k}$ $\displaystyle\equiv
16(\ell+1)\int_{0}^{\infty}\frac{\text{Im}\left(\Phi_{1}^{\ell}(s)-\Phi_{2}^{\ell}(s)\right)}{s^{k+1}}ds\,,$
$\displaystyle\langle\rho_{\ell}^{3}\rangle_{k}$ $\displaystyle\equiv
16(\ell+1)\int_{0}^{\infty}\frac{\text{Im}\left(\Phi_{3}^{\ell}(s)\right)}{s^{k+1}}ds\,.$

In our case, only the $k=2$ integrals converge because for higher values of
$k$, the logarithms in (6) spoil convergence. Note that in terms of these
positive observables, the sum rules (37) can be written (after taking the
limit $\hat{s}\rightarrow$ 0 there) as

$\displaystyle g_{2}+f_{2}$
$\displaystyle=\sum_{l=0,2\ldots}\langle\rho_{\ell}^{+}\rangle_{2}+\sum_{l=2,3\ldots}\langle\rho_{\ell}^{3}\rangle_{2}\,,$
(67) $\displaystyle g_{2}-f_{2}$
$\displaystyle=\sum_{l=0,2\ldots}\langle\rho_{\ell}^{-}\rangle_{2}+\sum_{l=2,3\ldots}\langle\rho_{\ell}^{3}\rangle_{2}\,.$

The weak version of low spin dominance states that in each channel, the
contribution of the lowest spin dominates over all the higher spins i.e.

$\frac{\langle\rho_{0}^{+}\rangle_{2}}{\langle\rho_{\ell>0}^{+}\rangle_{2}}\geq
1,\quad\frac{\langle\rho_{0}^{-}\rangle_{2}}{\langle\rho_{\ell>0}^{-}\rangle_{2}}\geq
1,\quad\frac{\langle\rho_{0}^{3}\rangle_{2}}{\langle\rho_{\ell>0}^{3}\rangle_{2}}\geq
1\,.$ (68)

The strong version of low spin dominance instead states that in fact the
contribution from the lowest spin is much larger than the other spins

$\frac{\langle\rho_{0}^{+}\rangle_{2}}{\langle\rho_{\ell>0}^{+}\rangle_{2}}\geq\alpha,\quad\frac{\langle\rho_{0}^{-}\rangle_{2}}{\langle\rho_{\ell>0}^{-}\rangle_{2}}\geq\alpha,\quad\frac{\langle\rho_{0}^{3}\rangle_{2}}{\langle\rho_{\ell>0}^{3}\rangle_{2}}\geq\alpha\,,$
(69)

with $\alpha\sim 100$ in Bern:2021ppb . In our work, since we construct
scattering amplitudes, we test this conjecture and find that the weak version
of low spin dominance seems to always be true, however the strong version need
not be. More precisely we find amplitudes where $\alpha<10$. We illustrate
this by considering two points from figure 10 in tables 5 and 6. Note that in
this subsection we analyzed the data at $N_{\text{max}}=26$ and
$L_{\text{max}}=80$ and checked that the behavior is stable as we change
$L_{\text{max}}$ and $N_{\text{max}}$. In table 5 which is the absolute
minimum value of $\bar{g}_{4}$, we see that191919Note that when
$\bar{f}_{2}=0$ and $\bar{f}_{4}=0$, the amplitude $\Phi_{2}=0$ and therefore
$\Phi^{-}_{\ell}=\Phi^{+}_{\ell}$.

$\bar{f}_{2}=0\,,\bar{f}_{4}=0\,:\quad\frac{\langle\rho_{0}^{+}\rangle_{2}}{\langle\rho_{\ell>0}^{+}\rangle_{2}}=5.7,\quad\quad\frac{\langle\rho_{0}^{3}\rangle_{2}}{\langle\rho_{\ell>0}^{3}\rangle_{2}}=6.3\,,$
(70)

which indicates some low spin dominance but not the strong form. We contrast
this in table 6 with the point $\bar{f}_{2}=0$ and $\bar{f}_{4}=0.05$ which
lies to right hand side of the minimum where the bound between $\bar{f}_{4}$
and $\bar{g}_{4}$ appears to be linear. Here we observe much stronger low spin
dominance, albeit only in the $\rho_{\ell}^{+}$ channel:

$\bar{f}_{2}=0\,,\bar{f}_{4}=0.05\,:\quad\frac{\langle\rho_{0}^{+}\rangle_{2}}{\langle\rho_{\ell>0}^{+}\rangle_{2}}=21.9,\quad\frac{\langle\rho_{0}^{-}\rangle_{2}}{\langle\rho_{\ell>0}^{-}\rangle_{2}}=3.3,\quad\frac{\langle\rho_{0}^{3}\rangle_{2}}{\langle\rho_{\ell>0}^{3}\rangle_{2}}=8.8\,.$
(71)

|  $\bar{f}_{2}=0$, $\bar{f}_{4}=0$, $\bar{g}_{4}\approx 0.015$ |
---|---|---
| $\ell=0$ | $\ell=1$ | $\ell=2$ | $\ell=3$ | $\ell=4$ | $\ell=20$ | $\sum_{\ell}$
$\langle\rho_{\ell}^{+}\rangle_{2}$ | $27.9$ |  | $4.90$ |  | $1.62$ | $0.03$ | $36.1$
$\langle\rho_{\ell}^{-}\rangle_{2}$ | $27.9$ |  | $4.90$ |  | $1.62$ | $0.03$ | $36.1$
$\langle\rho_{\ell}^{3}\rangle_{2}$ |  |  | $50.4$ | $8.04$ | $2.86$ | $0.01$ | $63.9$
Table 5: Sum rules for the minimum point on the boundary of allowed region in figure 10. Note that at this point, the amplitude $\Phi_{2}=0$ and therefore $\Phi^{+}=\Phi^{-}$. The last column is the sum of partial wave contributions up to spin $\ell=20$ and we see that (67) is saturated already, keeping in mind that $g_{2}=100$ and $f_{2}=0$. | $\bar{f}_{2}=0$, $\bar{f}_{4}=0.05$, $\bar{g}_{4}\approx 0.11$ |
---|---|---
| $\ell=0$ | $\ell=1$ | $\ell=2$ | $\ell=3$ | $\ell=4$ | $\ell=20$ | $\sum_{\ell}$
$\langle\rho_{\ell}^{+}\rangle_{2}$ | $40.6$ |  | $1.85$ |  | $0.46$ | $0.01$ | $43.4$
$\langle\rho_{\ell}^{-}\rangle_{2}$ | $25.3$ |  | $7.69$ |  | $5.12$ | $0.02$ | $43.4$
$\langle\rho_{\ell}^{3}\rangle_{2}$ |  |  | $47.8$ | $5.42$ | $1.51$ | $0.02$ | $56.6$
Table 6: Sum rules for a point on the boundary of the allowed region in figure
10. The last column is the sum of partial wave contributions up to spin
$\ell=20$ and we once again see that (67) is saturated already, since
$g_{2}=100$ and $f_{2}=0$.

We also display two points from figure 10 in tables 7 and 8 where we observe
similar phenomena. In particular notice in table 8 that at the point
$\bar{f}_{2}=0.3$ and $\bar{f}_{4}=0.05$ which lies in the linear region, the
$\rho^{+}_{\ell}$ channel has strong low spin dominance while the
$\rho^{-}_{\ell}$ channel is opposite.

$\displaystyle\bar{f}_{2}=0.3\,,\bar{f}_{4}=0.0045\,:\quad\frac{\langle\rho_{0}^{+}\rangle_{2}}{\langle\rho_{\ell>0}^{+}\rangle_{2}}$
$\displaystyle=4.7,\quad\frac{\langle\rho_{0}^{-}\rangle_{2}}{\langle\rho_{\ell>0}^{-}\rangle_{2}}=12.9,\quad\frac{\langle\rho_{0}^{3}\rangle_{2}}{\langle\rho_{\ell>0}^{3}\rangle_{2}}=8.7\,,$
(72)
$\displaystyle\bar{f}_{2}=0.3\,,\bar{f}_{4}=0.05\,:\quad\frac{\langle\rho_{0}^{+}\rangle_{2}}{\langle\rho_{\ell>0}^{+}\rangle_{2}}$
$\displaystyle=53.1,\quad\frac{\langle\rho_{0}^{-}\rangle_{2}}{\langle\rho_{\ell>0}^{-}\rangle_{2}}=2.2,\quad\frac{\langle\rho_{0}^{3}\rangle_{2}}{\langle\rho_{\ell>0}^{3}\rangle_{2}}=9.4\,.$

We also remark that if we instead consider points with negative values of
$\bar{f}_{4}$, say $\bar{f}_{4}=-0.05$, we find that it is now the
$\rho^{-}_{\ell}$ channel which has strong low spin dominance while the
$\rho^{+}_{\ell}$ channel has weaker low spin dominance.

|  $\bar{f}_{2}=0.3$, $\bar{f}_{4}=0.0045$, $\bar{g}_{4}\approx 0.02$ |
---|---|---
| $\ell=0$ | $\ell=1$ | $\ell=2$ | $\ell=3$ | $\ell=4$ | $\ell=20$ | $\sum_{\ell}$
$\langle\rho_{\ell}^{+}\rangle_{2}$ | $50.5$ |  | $10.8$ |  | $5.05$ | $0.03$ | $70.7$
$\langle\rho_{\ell}^{-}\rangle_{2}$ | $9.33$ |  | $0.72$ |  | $0.27$ | $0.02$ | $10.7$
$\langle\rho_{\ell}^{3}\rangle_{2}$ |  |  | $49.8$ | $5.72$ | $2.01$ | $0.02$ | $59.3$
Table 7: Sum rules for the minimum point on the boundary of figure 10. The last column is the sum of partial wave contributions up to spin $\ell=20$ and we see that (67) is saturated already, keeping in mind that $g_{2}=100$ and $f_{2}=30$. |  $\bar{f}_{2}=0.3$, $\bar{f}_{4}=0.05$, $\bar{g}_{4}\approx 0.11$ |
---|---|---
| $\ell=0$ | $\ell=1$ | $\ell=2$ | $\ell=3$ | $\ell=4$ | $\ell=20$ | $\sum_{\ell}$
$\langle\rho_{\ell}^{+}\rangle_{2}$ | $86.5$ |  | $1.63$ |  | $0.37$ | $0.01$ | $88.9$
$\langle\rho_{\ell}^{-}\rangle_{2}$ | $14.6$ |  | $6.66$ |  | $4.06$ | $0.02$ | $28.8$
$\langle\rho_{\ell}^{3}\rangle_{2}$ |  |  | $34.9$ | $3.70$ | $1.00$ | $0.02$ | $41.1$
Table 8: Sum rules for a point on the boundary of the allowed region in figure
10. The last column is the sum of partial wave contributions up to spin
$\ell=20$ and we once again see that (67) is saturated already, keeping in
mind that $g_{2}=100$ and $f_{2}=30$.

##  4 Allowed amplitudes from perturbation theory

In the first part of this work, we saw how to explore the space of Wilson
coefficients using the numerical S-matrix bootstrap. This section is dedicated
to answering the following question: how much of this space can be understood
analytically? Our main tool will be a set of effective field theories which
are under perturbative control, and which will be used to populate the space
of Wilson coefficients. Most of the models we consider were shown in
Henriksson:2021ymi  to be compatible with analytic positivity bounds, however
we also discuss two cases which are in tension with the same bounds. These
theories are obtained by weakly coupling the photon to a heavy particle at
tree level (Yukawa-like theories) or at one loop (QED-like theories). The
resulting theories have the following feature: they all have a dimensionless
parameter $\lambda$, such that at the point $\lambda=0$ the heavy particle
decouples and the photons are free. These theories are not necessarily UV
complete on their own when $\lambda\neq 0$, but for some of them one can
exhibit explicit UV completions.202020More precisely, what one can concoct is
a renormalizable action with marginal couplings. This still leaves open the
possibility of a Landau pole, see appendix F. Furthermore, they all obey the
classical Regge growth conjecture Chowdhury:2019kaq , which states that the
Regge intercept should be bounded by two for tree-level scattering amplitudes.
The non-perturbative version of the bound is not known in the case of massless
spin one particles, but the same statement was recently proven for graviton
scattering Haring:2022cyf , and we shall assume that it holds for photons as
well. Even when we do not exhibit an explicit UV completion, we take the Regge
boundedness as a reason to trust in its existence.

The rest of the section is organized as follows. In subsection 4.1, we gather
the Wilson coefficients of interest, while a detailed analysis of the theories
that produce them is relegated to appendix E. The following subsection is
dedicated to explaining which linear combinations of the basis of amplitudes
are compatible with unitarity. In the three subsections 4.3, 4.4, 4.5, we rule
in portions of the space of physical observables defined in (12).

###  4.1 Wilson coefficients of models with a small parameter

As mentioned above, the class of theories we consider are obtained by
integrating out a particle at tree level or at one-loop. At tree level, we
consider resonances of spin bounded by two. The criterion for this choice is
the requirement of Regge boundedness discussed above. In appendix E we give
more details on the construction of these amplitudes, and in subsections 4.3,
4.4 we will further comment on the case of a spin two resonance, which leads
to some subtleties. The second class of theories comprises the coupling of a
photon to an electrically charged particle: the particle is only created in
pairs, and therefore contributes to the photon EFT at one loop. We collect the
Wilson coefficients in table 9, where $\lambda$ is a dimensionless parameter.
In all cases, the coefficients are only correct to leading order in $\lambda$,
which should therefore be taken to be small. In the following, we shall often
denote the couplings by the dimension of the corresponding operators in the
Lagrangian (13): $(g_{2},f_{2})$ are the dimension 8 Wilson coefficients,
$(g_{3},f_{3},h_{3})$ are the dimension 10 and $(g_{4},g_{4}^{\prime},f_{4})$
are the dimension 12 coefficients.

$\displaystyle g_{2}$ $\displaystyle f_{2}$ $\displaystyle g_{3}$
$\displaystyle f_{3}$ $\displaystyle h_{3}$ $\displaystyle g_{4}$
$\displaystyle g^{\prime}_{4}$ $\displaystyle f_{4}$ Spin $0$ scalar
$\displaystyle\frac{\lambda^{2}}{m^{4}}$
$\displaystyle\frac{\lambda^{2}}{m^{4}}$
$\displaystyle\frac{\lambda^{2}}{m^{6}}$
$\displaystyle\frac{3\lambda^{2}}{m^{6}}$ $\displaystyle 0$
$\displaystyle\frac{\lambda^{2}}{m^{8}}$ $\displaystyle 0$
$\displaystyle\frac{\lambda^{2}}{2m^{8}}$ axion
$\displaystyle\frac{\lambda^{2}}{m^{4}}$
$\displaystyle-\frac{\lambda^{2}}{m^{4}}$
$\displaystyle\frac{\lambda^{2}}{m^{6}}$
$\displaystyle-\frac{3\lambda^{2}}{m^{6}}$ $\displaystyle 0$
$\displaystyle\frac{\lambda^{2}}{m^{8}}$ $\displaystyle 0$
$\displaystyle-\frac{\lambda^{2}}{2m^{8}}$ Spin $2$ parity even I
$\displaystyle\frac{\lambda^{2}}{m^{4}}$
$\displaystyle\frac{\lambda^{2}}{m^{4}}$
$\displaystyle\frac{\lambda^{2}}{m^{6}}$ $\displaystyle 0$ $\displaystyle 0$
$\displaystyle\frac{\lambda^{2}}{m^{8}}$
$\displaystyle-\frac{6\lambda^{2}}{m^{8}}$
$\displaystyle\frac{\lambda^{2}}{2m^{8}}$ parity even II
$\displaystyle\frac{\lambda^{2}}{m^{4}}$ $\displaystyle 0$
$\displaystyle-\frac{\lambda^{2}}{m^{6}}$ $\displaystyle 0$ $\displaystyle 0$
$\displaystyle\frac{\lambda^{2}}{m^{8}}$
$\displaystyle-\frac{2\lambda^{2}}{m^{8}}$ $\displaystyle 0$ parity odd
$\displaystyle\frac{\lambda^{2}}{m^{4}}$
$\displaystyle-\frac{\lambda^{2}}{m^{4}}$
$\displaystyle\frac{\lambda^{2}}{m^{6}}$ $\displaystyle 0$ $\displaystyle 0$
$\displaystyle\frac{\lambda^{2}}{m^{8}}$
$\displaystyle-\frac{6\lambda^{2}}{m^{8}}$
$\displaystyle-\frac{\lambda^{2}}{2m^{8}}$ One loop scalar QED
$\displaystyle\frac{2\lambda^{2}}{45m^{4}}$
$\displaystyle\frac{\lambda^{2}}{30m^{4}}$
$\displaystyle\frac{\lambda^{2}}{210m^{6}}$
$\displaystyle\frac{\lambda^{2}}{63m^{6}}$
$\displaystyle\frac{\lambda^{2}}{630m^{6}}$
$\displaystyle\frac{11\lambda^{2}}{9450m^{8}}$
$\displaystyle-\frac{\lambda^{2}}{3780m^{8}}$
$\displaystyle\frac{\lambda^{2}}{1890m^{8}}$ spinor QED
$\displaystyle\frac{11\lambda^{2}}{45m^{4}}$
$\displaystyle-\frac{\lambda^{2}}{15m^{4}}$
$\displaystyle\frac{4\lambda^{2}}{315m^{6}}$
$\displaystyle-\frac{2\lambda^{2}}{63m^{6}}$
$\displaystyle-\frac{\lambda^{2}}{315m^{6}}$
$\displaystyle\frac{13\lambda^{2}}{2700m^{8}}$
$\displaystyle-\frac{\lambda^{2}}{378m^{8}}$
$\displaystyle-\frac{\lambda^{2}}{945m^{8}}$ vector QED
$\displaystyle\frac{14\lambda^{2}}{5m^{4}}$
$\displaystyle\frac{\lambda^{2}}{10m^{4}}$
$\displaystyle-\frac{47\lambda^{2}}{630m^{6}}$
$\displaystyle\frac{\lambda^{2}}{21m^{6}}$
$\displaystyle\frac{\lambda^{2}}{210m^{6}}$
$\displaystyle\frac{131\lambda^{2}}{3150m^{8}}$
$\displaystyle-\frac{23\lambda^{2}}{420m^{8}}$
$\displaystyle\frac{\lambda^{2}}{630m^{8}}$

Table 9: Wilson coefficients of photons EFTs obtained by integrating out
particles at tree level or one loop.

###  4.2 The rules of the game

If the amplitudes in table 9 are to be treated as UV complete, their convex
hull should lie within the numerical bounds found in section 3. However,
violations of unitarity only disappear in the $\lambda\to 0$ limit, in which
case all the dimensionless observables defined in (12) are pushed to infinity,
with the exception of $\bar{f_{2}}$. For instance, the non-vanishing
observables associated with the first row of table 9 are

$\bar{f}_{2}=1~{},\quad\bar{g}_{3}=\frac{1}{\lambda}~{},\quad\bar{f}_{3}=\frac{3}{\lambda}~{},\quad\bar{g}_{4}=\frac{1}{\lambda^{2}}~{},\quad\bar{f}_{4}=\frac{1}{2\lambda^{2}}~{}.$
(73)

We conclude that the amplitudes considered in this section mark _unbounded
directions_ in the space of photon EFTs at dimension larger than 8.

In the rest of this section, we shall find such directions. A simplifying
feature of the problem is that the convex hull of these amplitudes degenerates
into a cone in the weak coupling limit. Indeed, let us define the vectors
$v_{i}(\lambda_{i})$, where the index $i$ labels the rows of table 9 and whose
components are the Wilson coefficients:

$v=(g_{2},f_{2}\,|\,g_{3},f_{3},h_{3}\,|\,g_{4},g^{\prime}_{4},f_{4})=(v^{(2)}\,|\,v^{(3)}\,|\,v^{(4)})~{}.$
(74)

The vectors spanning subspaces at each mass dimension are distinguished by the
upper index. The following obvious equality,

$\sum_{i}\alpha_{i}v_{i}(\lambda_{i})=\sum_{i}\beta_{i}v_{i}(\lambda^{\prime}_{i})~{},\qquad\beta_{i}=\frac{\alpha_{i}}{(\kappa_{i})^{2}}~{},\quad\lambda^{\prime}_{i}=\lambda_{i}\kappa_{i}~{},\quad\kappa_{i}>0~{},$
(75)

makes the condition $\sum_{i}\alpha_{i}=1$ irrelevant. In the $\lambda_{i}\to
0$ limit, all linear combinations of the amplitudes with coefficients
$\beta_{i}>0$ are allowed. In other words, perturbative amplitudes are only
constrained by positivity, rather than by full nonlinear unitarity.

A second simplification comes from the fact that we are allowed to take
_independent_ linear combinations at dimension 8 and 10, or at dimension 8 and
12. This follows from the freedom of choosing the mass of the exchanged
particle, which introduces new dimensionless parameters $m_{i}/m_{j}$. For
instance, the following change of variables,

$x=\frac{\lambda^{2}}{m^{4}}~{},\quad y=\frac{\lambda^{2}}{m^{6}}~{},$ (76)

allows to write the Wilson coefficients at dimension 8 and 10 as linear
functions of different parameters. Referring to the definition (74),

$v^{(2)}=x\,\hat{v}^{(2)}~{},\quad v^{(3)}=y\,\hat{v}^{(3)}~{},\quad
v^{(4)}=\frac{y^{2}}{x}\hat{v}^{(4)}~{},$ (77)

with hatted vectors being purely numerical. Then we can extend the observation
(75) to

$\sum_{i}\alpha_{i}v_{i}(x_{i},y_{i})=\sum_{i}\left(\beta_{i}^{(2)}v_{i}^{(2)}(x^{\prime}_{i})\,|\,\beta_{i}^{(3)}v_{i}^{(3)}(y^{\prime}_{i})\,|\,\beta^{(4)}_{i}v_{i}^{(4)}(y^{\prime
2}_{i}/x^{\prime}_{i})\right)~{},\\\
\beta_{i}^{(2)}=\frac{\alpha_{i}}{\kappa^{(2)}_{i}}~{},\
\beta_{i}^{(3)}=\frac{\alpha_{i}}{\kappa^{(3)}_{i}}~{},\
\beta_{i}^{(4)}=\alpha_{i}\frac{\kappa^{(2)}_{i}}{\big{(}\kappa^{(3)}_{i}\big{)}^{2}}~{},\qquad
x^{\prime}_{i}=x_{i}\kappa^{(2)}_{i}~{},\
y^{\prime}_{i}=y_{i}\kappa^{(3)}_{i}~{},\quad\kappa_{i}^{(2,3)}>0~{}.$ (78)

This means that, for any choice of positive
$\beta_{i}^{(2)},\,\beta_{i}^{(3)}$, one can find an amplitude in the convex
hull of the perturbative theories defined by table 9 as $\lambda\to
0.$212121Notice that instead
$\beta_{i}^{(4)}=\big{(}\beta_{i}^{(3)}\big{)}^{2}/\beta_{i}^{(2)}.$ An
analogous statement holds for the subspace of dimension 8 and dimension 12
coefficients.

A simple consequence of this observation is that _the asymptotic allowed
values of the dimension 10 or of the dimension 12 coefficients do not depend
on $\bar{f_{2}}$._

More generally, we are lead to a simple recipe to find which directions are
necessarily unbounded in the space of couplings of each mass dimension:

_Consider the vectors of Wilson coefficients in table 9, restricted to a given
mass dimension larger than 8. Set $\lambda=m=1$. Compute the one-sided cone
generated by positive linear combinations of the numerical vectors thus
obtained. The intersection of this cone with the sphere at infinity marks the
unbounded directions._

The basic technology to solve this kind of problems is reviewed in Arkani-
Hamed:2020blm . We shall only report the results in the rest of the section.

###  4.3 The $\bar{f}_{2}$ space

The spin 0 exchanges in table 9 are sufficient to cover the space of couplings
at dimension 8, which is parametrized by $|\bar{f_{2}}|\leq 1$. Indeed, the
scalar amplitude has $\bar{f_{2}}=1$ and the axion amplitude has
$\bar{f_{2}}=-1$. Their convex hull generates the whole allowed space.

The other amplitudes in table 9 all lie inside this interval. In the case of
the spin 2 resonance, this fact deserves a comment. The exchange of a
resonance determines the amplitude uniquely only at the location of the pole.
In particular, the amplitudes constructed in appendix E are ambiguous up to
the addition of a polynomial in the Mandelstam invariants, of the kind showed
in (6). Such polynomials correspond to contact interactions for the photons,
as explained in the introduction. A polynomial of degree two can be tuned so
that the Regge limit of the spin two resonance obeys (33), and, as a
consequence, the dispersion relation (37). This implies $|\bar{f}_{2}|\leq 1$.
Let us emphasize that (33) is strictly weaker than the Regge bound mentioned
in the previous subsection, since it only holds at $t=0$, for a specific
linear combination of amplitudes. In fact, it is not possible to obtain a
Regge intercept strictly less than 2, for all values of $t$, by adding a
finite degree polynomial.

Let us finally comment on the constraints imposed by dispersion relations and
crossing onto the amplitudes living at the boundary of the allowed region. We
start from the boundary $\bar{f_{2}}=1$. Equations (37) and (28) imply

$\operatorname{Im}\left(\Phi_{1}(s)-\Phi_{2}(s)\right)=0~{},\qquad\operatorname{Im}\Phi_{3}(s)=0~{},$
(79)

where we recall that the notation means the amplitudes are evaluated at $t=0$.
Let us now decompose (79) in partial waves, using (121). The small Wigner
$d-$matrix are non negative at $\theta=0$, as it is easy to verify from the
definition (119). Together with the positivity constraint (25), this means
that (79) is valid at the level of partial waves:

$\operatorname{Im}\left(\Phi_{1}^{\ell}(s)-\Phi_{2}^{\ell}(s)\right)=0~{},\qquad\operatorname{Im}\Phi_{3}^{\ell}(s)=0~{}.$
(80)

Furthermore, the unitarity constraints (136) - (139) force these combinations
of partial waves to vanish:

$\left(\Phi_{1}^{\ell}(s)-\Phi_{2}^{\ell}(s)\right)=0~{},\qquad\Phi_{3}^{\ell}(s)=0~{}.$
(81)

This is easily seen for the $\Phi_{1}^{\ell}(s)-\Phi_{2}^{\ell}(s)$
combination, which obeys the usual spinless unitarity constraint
$|1+i\mathcal{T}|<1$.222222Recall that $t-u$ symmetry of $\Phi_{1}$ and
$\Phi_{2}$ and the property
$d^{\ell}_{0,0}(\pi-\theta)=(-1)^{\ell}d^{\ell}_{0,0}(\theta)$ make
$\Phi_{1}^{\ell}(s)-\Phi_{2}^{\ell}(s)$ vanish for $\ell$ odd. With some more
work, one can extract from (139) the following necessary conditions valid for
even spin:

$\left|1+i(\Phi_{1}^{\ell}(s)+\Phi_{2}^{\ell}(s))\right|<1~{},\qquad\left|1+2i\Phi_{3}^{\ell}(s)\right|<1~{},\qquad\left|\Phi_{5}^{\ell}(s)\right|<1~{}.$
(82)

The second of these conditions, together with (137), implies the third
equality in (81).232323Recall that $\Phi_{3}^{\ell}=0$ for $\ell=0,\,1$.

If we now plug (81) back into (139), we discover that also

$\Phi_{5}^{\ell}(s)=0~{}.$ (83)

All in all, we found that the following amplitudes vanish if $\bar{f_{2}}=1$:

$\Phi_{1}(s,t,u)-\Phi_{2}(s,t,u)=0~{},\qquad\Phi_{3}(s,t,u)=0~{},\qquad\Phi_{5}(s,t,u)=0~{}.$
(84)

Using crossing—equation (5)—we conclude that in fact all the amplitudes must
vanish. A similar reasoning also leads to the same conclusion for the other
boundary of the space allowed by unitarity, $\bar{f_{2}}=-1$.

Notice that this conclusion does not contradict the statement that scalar and
pseudoscalar resonances saturate these bounds. Indeed, as emphasized, the
theories considered in this section are only compatible with unitarity in the
limit $\lambda\to 0$, _i.e._ precisely when they become free. In other words,
in any interacting UV completion of the EFT obtained integrating out a scalar
or an axion, the value of $\bar{f_{2}}$ is corrected at higher orders in
$\lambda$.

###  4.4 The $(\bar{g}_{3},\bar{f}_{3},\bar{h}_{3})$ space

In subsection 3.3, we saw that the numerics indicates the absence of bounds
for the dimension 10 observables. Here, we show that the full space is in fact
covered by (linear combinations of) perturbative completions of the photon
effective lagrangian. Following the recipe given in subsection 4.2, we
consider the cone generated by the following amplitudes: scalar, axion, scalar
QED, spinor QED and vector QED. One can easily see that the cone coincides
with the whole three-dimensional space spanned by $g_{3}$, $f_{3}$ and
$h_{3}$, thus proving that the sphere at infinity in the normalized space
$(\bar{g}_{3},\bar{f}_{3},\bar{h}_{3})$ is populated by weakly coupled
theories.

Notice that, since the unitarity constraints are convex, no lower bound on
$\bar{g}_{3}$, $\bar{f}_{3}$, $\bar{h}_{3}$ is possible either. Let us
emphasize that it was not necessary to include spin two resonances in order to
achieve this result. In other words, bounds on dimension 10 Wilson
coefficients are impossible already within amplitudes with Regge intercept
smaller than two.

###  4.5 The $(\bar{g}_{4},\bar{g}^{\prime}_{4},\bar{f}_{4})$ space

The space of dimension 12 Wilson coefficients is the first one which is not
fully covered by (linear combinations of) weakly coupled amplitudes. This is
in accordance with the results of section 3, where we presented evidence of
non-trivial bounds in the $(\bar{g}_{4},\bar{g}^{\prime}_{4},\bar{f}_{4})$
space.

The boundaries of the cone generated by the perturbative amplitudes depend on
which portion of table 9 is included in the analysis. Let us begin by
including all the rows of the table. These vectors generate a cone bounded by
the following four faces:

$g_{4}-2f_{4}\geq 0~{},\qquad g_{4}+2f_{4}\geq 0~{},\qquad g_{4}^{\prime}\leq
0~{},\qquad g_{4}^{\prime}+6g_{4}\geq 0~{}.$ (85)

This implies in particular $g_{4}\geq 0$. If instead we restrict to theories
whose Regge intercept is strictly less than two, _i.e._ we exclude the spin
two resonances, we get the following inequalities:

$-10f_{4}+5g_{4}+2g_{4}^{\prime}\geq
0~{},\qquad-157f_{4}+85g_{4}+60g_{4}^{\prime}\geq 0~{},\qquad
g_{4}^{\prime}\leq 0~{},\qquad 230f_{4}+115g_{4}+94g_{4}^{\prime}\geq 0~{}.$
(86)

Again, only the half-space $g_{4}\geq 0$ is populated. The two cones are shown
in figure 17.

Figure 17: A portion of the cone generated by positive linear combinations of
the perturbative amplitudes of table 9, restricted to the space of dimension
12 Wilson coefficients. The smaller cone (yellow) only includes amplitudes
which grow slower than $s^{2}$ in the Regge limit. The wider cone (purple)
includes all the rows of table 9.

It is interesting to compare these findings with the amplitudes constructed
numerically in section 3, and with the analytic results obtained in Arkani-
Hamed:2020blm ; Henriksson:2021ymi ; Henriksson:2022oeu . The numerical
results are showcased in figures 10 \- 12 and 14 \- 16, for the
$(\bar{g}_{4},\bar{f}_{4})$ and for the $(\bar{g}_{4},\bar{g}^{\prime}_{4})$
planes, respectively. In particular, it is clear that the
$(\bar{g}_{4},\bar{f}_{4})$ space is bounded at infinity by the two directions
$\bar{g}_{4}/\bar{f}_{4}=\pm 2$, for all values of $\bar{f}_{2}$. This matches
both the wider and the narrower cones, equations (85) and (86). The two
boundaries are generated by the scalar and axion amplitudes respectively (and
also by the parity even I and by the parity odd amplitudes) in table 9.

The status of the $(\bar{g}_{4},\bar{g}^{\prime}_{4})$ plane appears less
clear. The bounds in figures 14 \- 16 are well converged only when
$\bar{g}^{\prime}_{4}\gtrsim 0$, while all the perturbative amplitudes we
consider have non-positive values for the same Wilson coefficient. The two
plots are therefore hardly comparable. It would be interesting to find a
perturbative amplitude with positive $g_{4}^{\prime}$ that could explain our
numerical results. For reference, the sections of the two cones in (85) and
(86) are bounded, respectively, as

$-6\leq\frac{g_{4}^{\prime}}{g_{4}}\leq 0~{},\qquad g_{4}\geq 0~{},$ (87)

and

$-\frac{345}{262}\leq\frac{g_{4}^{\prime}}{g_{4}}\leq 0~{},\qquad g_{4}\geq
0~{}.$ (88)

The lower bound in the first equation is reached by the parity even I and
parity odd spin 2 resonances. If those are excluded, smallest ratio belongs to
vector QED. One can also combine the numerical results to the weakly coupled
theories to obtain a larger convex hull. In particular, one can consider
figure 14, and draw a line with slope $-1/6$ or $-262/345$ (depending on the
reader’s opinion about the spin 2 resonances) which intersects the
$\bar{g}_{4}^{\prime}=0$ axis in correspondence of the numerical lower bound.
The region to the right of this line and above the bound is populated by
linear combinations of numerical and perturbative amplitudes.

Finally, in the $(\bar{g}_{4}^{\prime},\bar{f}_{4})$ plane, the perturbative
amplitudes cover the semi-circle at infinity with $\bar{g}_{4}^{\prime}<0$.
Therefore, their convex hull covers the whole half-plane
$(\bar{g}_{4}^{\prime}<0,\bar{f}_{4})$. Again, one can combine this knowledge
with the numerical results. From figures 14 and 14 it is clear that there is
no upper bound on $\bar{g}^{\prime}_{4}$. Consider then an allowed point with
$\bar{g}_{4}^{\prime}=x>0$ in the $(\bar{g}_{4}^{\prime},\bar{f}_{4})$ plane.
The convex hull of this point and the half plane $\bar{g}_{4}^{\prime}<0$ is
the region to the left of the line $\bar{g}_{4}^{\prime}=x$. Now, since $x$
can go to infinity, we conclude that there are no bounds in the
$(\bar{g}_{4}^{\prime},\bar{f}_{4})$ plane.

In Henriksson:2021ymi ; Henriksson:2022oeu , the space of Wilson coefficients
was bounded analytically, by means of linear positivity—see section 2—and tree
level crossing, _i.e._ null constraints. Depending on how many null
constraints were imposed, the following bounds were found:

$\displaystyle-\frac{18}{7}\leq\frac{\bar{g}_{4}^{\prime}}{\bar{g}_{4}}\leq\frac{370}{29}~{},\quad
g_{4}\geq 0~{},$ $\displaystyle\textup{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Henriksson:2021ymi}{\@@citephrase{(}}{\@@citephrase{)}}}}~{},$
(89)
$\displaystyle-\frac{12}{5}\leq\frac{\bar{g}_{4}^{\prime}}{\bar{g}_{4}}\leq
0~{},\quad g_{4}\geq 0~{},$
$\displaystyle\textup{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Henriksson:2022oeu}{\@@citephrase{(}}{\@@citephrase{)}}}}~{}.$
(90)

On the other hand, the paper Arkani-Hamed:2020blm  found the following bounds:

$-6\leq\frac{\bar{g}_{4}^{\prime}}{\bar{g}_{4}}\leq\frac{30}{7}~{},\quad
g_{4}\geq 0~{}.$ (91)

Figure 18: Comparison between (87), (88)—purple and orange solid lines
respectively—and the analytic bounds imposed by linear positivity and tree
level crossing. The allowed region (a) was derived in Arkani-Hamed:2020blm ,
(b) in Henriksson:2021ymi  and updated by the same authors in
Henriksson:2022oeu  (c). Notice that part of the purple region lies outside
the bounds which rely on null constraints. Figure 19: Comparison between the
numerical bounds obtained in section 3 and the analytic bounds (a) Arkani-
Hamed:2020blm , (b) Henriksson:2021ymi  and (c) Henriksson:2022oeu . The
allowed regions lie in the $\bar{g}_{4}>0$ half-plane, and are delimited by
the dashed lines (notice that the blue and green left edges almost coincide in
the picture).

The bounds (90) - (91) are compared to the weakly coupled theories (87) and
(88) in figure 18, and to the numerical results in figure 19. It is
interesting to notice that the cone generated by our perturbative amplitudes
extends beyond the bounds found in Henriksson:2021ymi ; Henriksson:2022oeu .
This is not a contradiction, since the use of null constraints requires the
Regge limit to be strictly softer than $s^{2}$ beyond the forward limit.
Indeed, excluding the spin 2 exchanges we get a convex hull squarely contained
within their bounds. However, Henriksson:2021ymi  did consider the exchange of
a massive graviton, finding it compatible with the analytic bounds. As we
further discuss in appendix E, the amplitude proposed in Henriksson:2021ymi
coincides with our parity even II up to operators of dimension 8 in the EFT.
One can easily check that the Wilson coefficients of dimension 10 and 12 in
table 9 coincide with the ones reported in table 2 of Henriksson:2021ymi . At
the level of our analysis, which is based on the classical Regge bound, there
is no reason to prefer this combination to the other two spin 2 resonances: it
would be interesting to understand if, in some way, the other amplitudes are
exonerated from the null constraints of Henriksson:2021ymi ;
Henriksson:2022oeu , while this one must obey them. In appendix E, we offer a
few additional comments on the status of the parity even I and parity odd
EFTs.

##  5 Discussion

Photons are the particles of light. To the best of our knowledge, they are
massless spin one particles. Assuming four dimensional Lorentz invariance,
quantum mechanics and the absence of other massless particles, the low energy
dynamics of photons must be described by the EFT Lagrangian (13). Of course,
we expect photons to also interact with gravitons, but this is a excindingly
small effect at reasonable energies.242424For example, the exclusive cross
section of two photons into two photons from 1-loop effects in QED is of order
$\sigma_{QED}\sim\alpha^{4}s^{3}/m_{e}^{8}$ and from tree-level graviton
exchange is of order $\sigma_{grav}\sim G_{N}^{2}s\sim s/m_{P}^{4}$.
Therefore, the ratio $\sigma_{grav}/\sigma_{QED}\sim
m_{e}^{8}/(m_{P}^{4}\alpha^{4}s^{2})\sim
10^{-58}\left(1\,eV/\sqrt{s}\right)^{4}$ is tiny for visible (or higher
frequency) photons.  The Wilson coefficients $c_{i}$ in the EFT Lagrangian
(13) parametrize our ignorance about the high energy behavior of the theory.
Of course, in the real world we expect these to be dominated by QED effects.
Nevertheless, in this paper we asked ourselves what values can these numbers
take compatible with the S-matrix bootstrap principles of Lorentz invariance,
unitarity and analyticity. Using a numerical algorithm, we estimated several
non-perturbative bounds as described in section 3. QED seems to live well
inside the allowed region. Some bounds are saturated by weakly coupled
amplitudes as we discuss in section 4 but others are not. For example, the
amplitude with minimal $\bar{g}_{4}$ is strongly coupled. Is there a physical
theory that realizes such a small value of $\bar{g}_{4}$? This is an open
question for the future.

Let us comment on the bounds derived in Henriksson:2021ymi  using positivity.
In that work, the authors assumed weak coupling below the scale $M$ and
neglected the branch cuts of the amplitude from photon loops. More precisely,
they assumed the analytic structure depicted in figure 20 and defined the
scale $M$ from the position of the branch point at $s=M^{2}$. This assumption
makes it possible to derive bounds on new dimensionless quantities like

$\frac{g_{3}M^{2}}{g_{2}},\qquad\frac{g_{4}M^{4}}{g_{2}},\qquad\frac{g_{4}^{\prime}M^{4}}{g_{2}},\qquad\ldots$
(92)

It would be interesting to derive similar bounds without completely ignoring
the $\log$ terms in (6). This requires a different definition of the scale
$M$. For example, we may impose that the discontinuity of the amplitude is
bounded by (6) up to $s=M^{2}$ and let it free for $s>M^{2}$ (still imposing
unitarity). Such scenario can be easily studied with our primal numerical
methods (see Miro:2022cbk  for a concrete implementation of a similar idea).
We leave this exploration for the future.

$s$$M^{2}$$\bullet$$-M^{2}-t$$\bullet$ Figure 20: Analytic structure of the
amplitudes $\Phi_{i}(s,t,u)$ assumed in the positivity study
Henriksson:2021ymi . The branch cuts associated to photon loops are neglected
due to the assumption of weak coupling. The beginning of the cut at $s=M^{2}$
defines an energy scale $M$ that is interpreted as the UV cutoff of the photon
EFT.

The primal numerical S-matrix bootstrap becomes very expensive in the presence
of massless particles. This has been noticed before in Guerrieri:2020bto ;
Guerrieri:2021ivu  but it was more extreme in this work. For this reason, we
are in great need of more efficient numerical methods. Hopefully, the dual
methods of Guerrieri:2021tak ; He:2021eqn  can be generalized to massless
particles.

It would be very interesting to consider the joint system of scattering
amplitudes between photons and charged particles. However, this idea collides
with the well known problem of IR divergences in 4 space-time dimensions. The
same applies to scattering amplitudes involving photons and gravitons. These
are also not well defined non-perturbatively in 4 dimensions. Hopefully,
better (inclusive?) observables can be constructed with properties amenable to
a bootstrap approach. In the meantime, one can extend the present S-matrix
bootstrap study in two main directions. Firstly, we can go to higher
dimensions to avoid the IR divergences. It would be very interesting to study
photon-graviton scattering in higher dimensions in the context of the weak
gravity conjecture Henriksson:2022oeu . Secondly, we can consider photons as
probes of any QFT with a continuous global symmetry. In particular, we can
study scattering of pions and photons in QCD in order to probe the chiral
anomaly, analogously to the use of dilatons to probe the $a$-anomaly
Karateev:2022jdb . Furthermore, this system can be further simplified by
working in the planar limit of QCD as in Albert:2022oes . In this case, non-
linear unitarity is reduced to positivity.

## Acknowledgements

We thank Kausik Ghosh, Andrea Guerrieri, Sébastien Reymond, Lorenzo Ricci,
Francesco Riva, Biswajit Sahoo, Balt van Rees, Pedro Vieira and Alexander
Zhiboedov for very useful conversations. We also thank Andrea Guerrieri for
reading the draft and giving valuable comments.

DK is supported by the SNSF Ambizione grant PZ00P2_193411. The work of KH and
JP is supported by the Simons Foundation grant 488649 (Simons Collaboration on
the Nonperturbative Bootstrap) and by the Swiss National Science Foundation
through the project 200020_197160 and through the National Centre of
Competence in Research SwissMAP. MM is supported by the SNSF Ambizione grant
PZ00P2_193472. AH is supported by the Simons Foundation grant 488659 (Simons
Collaboration on the Nonperturbative Bootstrap). AH thanks the participants of
the ‘S-matrix Bootstrap IV’ workshop in Crete, Greece and the ‘Non-
perturbative Methods in Quantum Field Theory’ workshop at CERN, Switzerland,
where this work was presented, for interesting comments.

##  Appendix A Amplitudes, crossing and unitarity

In this appendix we define scattering amplitudes of spin one massless
particles. Focusing on the center of mass frame we will derive crossing
equations and unitarity conditions they must obey. For completeness, in
appendix B we will study these amplitudes in a generic frame and derive
crossing equations also there. We will find that the results of the two
appendices are in perfect agreement.

In this paper we work in the mostly plus metric

$\eta^{\mu\nu}=\\{-,+,\ldots,+,+\\}\,.$ (93)

Throughout our work we always focus on the case of $d=4$ space-time dimensions
except for appendices B.3.1 and B.3.3 where we stay in general number of
space-time dimensions $d$.

###  A.1 Amplitudes for spin one massless particles

We start by defining scattering amplitudes in a general frame. We then focus
on the center of mass frame.

##### Scattering amplitudes in a general frame

Consider the 2-to-2 (in-out) scattering process of massless spin one
particles, schematically $12\rightarrow 34$. It is described by the scattering
amplitude
$\mathcal{S}{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p_{1},p_{2},p_{3},p_{4})$
defined via the following matrix element

$(2\pi)^{4}\delta^{(4)}(p_{1}^{\mu}+p_{2}^{\mu}-p_{3}^{\mu}-p_{4}^{\mu})\times\mathcal{S}{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p_{1},p_{2},p_{3},p_{4})\equiv{}_{id}\langle\kappa_{3},\kappa_{4}|S|\kappa_{1},\kappa_{2}\rangle_{id}.$
(94)

Here $S$ is the scattering operator. The particles with 4-momenta $p_{1}$ and
$p_{2}$ are incoming, and the particles with momenta $p_{3}$ and $p_{4}$ are
outgoing. We adopt the convention that helicities of incoming particles are
always placed downstairs, instead helicities of outgoing particles are always
placed upstairs. The two-particle state describing the system of two identical
massless particles is defined as

$|\kappa_{1},\kappa_{2}\rangle_{id}\equiv\frac{1}{\sqrt{2}}\big{(}|\vec{p}_{1};\lambda_{1}\rangle\otimes|\vec{p}_{2};\lambda_{2}\rangle+|\vec{p}_{2};\lambda_{2}\rangle\otimes|\vec{p}_{1};\lambda_{1}\rangle\big{)},$
(95)

where the symbol $\otimes$ stands for the ordered tensor product. By
construction it obeys the condition
$|\kappa_{1},\kappa_{2}\rangle_{id}=|\kappa_{2},\kappa_{1}\rangle_{id}$. The
$\sqrt{2}$ factor in the definition is part of our conventions. The one-
particle states entering (95) are denoted by

$|\vec{p};\lambda\rangle\equiv|m=0,\vec{p};j=1,\lambda\rangle.$ (96)

The right-hand side of (96) is standard notation for one-particle states,
where $m$ is the mass of the particle, $j$ is its spin, $\vec{p}$ is the
spatial momentum and $\lambda$ is the helicity. For massless spin $j=1$
particles helicity can only take two values $\lambda=\pm 1$. The normalization
of one-particle states is given by

$\langle\vec{p}_{1};\lambda_{1}|\vec{p}_{2};\lambda_{2}\rangle=2\boldsymbol{p}_{1}\delta_{\lambda_{1}}^{\lambda_{2}}\times(2\pi)^{3}\delta^{(3)}(\vec{p}_{1}-\vec{p}_{2})\,,$
(97)

where we defined $\boldsymbol{p}_{i}=|\vec{p}_{i}|$.

Let us now define the interacting part of the scattering operator $T$ as
follows

$S=1+iT.$ (98)

This leads to the definition of the interacting scattering amplitude
$\mathcal{T}{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p_{1},p_{2},p_{3},p_{4})$,
namely

$(2\pi)^{4}\delta^{(4)}(p_{1}^{\mu}+p_{2}^{\mu}-p_{3}^{\mu}-p_{4}^{\mu})\times\mathcal{T}{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p_{1},p_{2},p_{3},p_{4})\equiv{}_{id}\langle\kappa_{3},\kappa_{4}|T|\kappa_{1},\kappa_{2}\rangle_{id},$
(99)

Using (98) we can write the relation between the scattering amplitude and its
interacting part. It reads

$\mathcal{S}{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p_{1},p_{2},p_{3},p_{4})=\frac{{}_{id}\langle\kappa_{3},\kappa_{4}|\kappa_{1},\kappa_{2}\rangle_{id}}{(2\pi)^{4}\delta^{(4)}(p_{1}^{\mu}+p_{2}^{\mu}-p_{3}^{\mu}-p_{4}^{\mu})}+i\mathcal{T}{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p_{1},p_{2},p_{3},p_{4}).$
(100)

The first term in the right-hand side of (100) is a formal expression. It can
be straightforwardly evaluated for example in the center of mass frame in
spherical coordinates (e.g. see footnote 15 in Hebbar:2020ukp ). The
normalization of two-particle states follows from (95) and (97). It reads

${}_{id}\langle\kappa_{3},\kappa_{4}|\kappa_{1},\kappa_{2}\rangle_{id}=4\boldsymbol{p}_{1}\boldsymbol{p}_{2}(2\pi)^{6}\left(\delta^{(3)}(\vec{p_{1}}-\vec{p_{3}})\delta^{(3)}(\vec{p_{2}}-\vec{p_{4}})\delta_{\lambda_{1}}^{\lambda_{3}}\delta_{\lambda_{2}}^{\lambda_{4}}+(3\leftrightarrow
4)\right).$ (101)

Using the 4-momenta $p_{i}^{\mu}$ one can form three scalar quantities called
the Mandelstam variables. For the in-out amplitudes their standard form reads
as

$s\equiv-(p_{1}+p_{2})^{2},\quad t\equiv-(p_{1}-p_{3})^{2},\quad
u\equiv-(p_{1}-p_{4})^{2},\quad s+t+u=0.$ (102)

Their physical range is

$s\geq 0,\qquad t\in[-s,\,0],\qquad u\in[-s,\,0].$ (103)

##### Scattering amplitudes in the center of mass frame

The center of mass (COM) frame is defined by the following configuration of
the 4-momenta

$\displaystyle p_{1}^{\text{com}}\equiv\frac{\sqrt{s}}{2}\times(1,0,0,+1),$
(104) $\displaystyle
p_{2}^{\text{com}}\equiv\frac{\sqrt{s}}{2}\times(1,0,0,-1),$ $\displaystyle
p_{3}^{\text{com}}\equiv\frac{\sqrt{s}}{2}\times(1,+\sin\theta,0,+\cos\theta),$
$\displaystyle
p_{4}^{\text{com}}\equiv\frac{\sqrt{s}}{2}\times(1,-\sin\theta,0,-\cos\theta),$

where $\theta\in[0,\pi]$ is the scattering angle. Plugging these into the
definition of the Mandelstam variables (102) we find that

$\sin\theta=\frac{2\sqrt{tu}}{s},\quad\cos\theta=\frac{t-u}{s}\quad\Leftrightarrow\quad
t=-\frac{s}{2}(1-\cos\theta),\quad u=-\frac{s}{2}(1+\cos\theta).$ (105)

The center of mass amplitudes are defined as

$\mathcal{T}{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u)\equiv\mathcal{T}{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p_{1}^{\text{com}},p_{2}^{\text{com}},p_{3}^{\text{com}},p_{4}^{\text{com}}).$
(106)

Due to the presence of identical particles and parity symmetry, there are only
5 distinct center of mass amplitudes, our choice here is

$\displaystyle\Phi_{1}(s,t,u)$
$\displaystyle\equiv\mathcal{T}^{++}_{++}(s,t,u),$ (107)
$\displaystyle\Phi_{2}(s,t,u)$
$\displaystyle\equiv\mathcal{T}^{--}_{++}(s,t,u),$
$\displaystyle\Phi_{3}(s,t,u)$
$\displaystyle\equiv\mathcal{T}^{+-}_{+-}(s,t,u),$
$\displaystyle\Phi_{4}(s,t,u)$
$\displaystyle\equiv\mathcal{T}^{-+}_{+-}(s,t,u),$
$\displaystyle\Phi_{5}(s,t,u)$
$\displaystyle\equiv\mathcal{T}^{+-}_{++}(s,t,u).$

The rest of the center of mass amplitudes are related to the above 5 ones via
the 11 relations. Due to the presence of identical particles we get the
following 9 constraints

$\displaystyle\mathcal{T}_{--}^{--}$
$\displaystyle=\Phi_{1},\quad\;\;\;\mathcal{T}_{--}^{+-}=\mathcal{T}_{--}^{-+},\quad\mathcal{T}_{-+}^{--}=\Phi_{5},\quad\;\;\;\mathcal{T}_{-+}^{-+}=\Phi_{3},\quad\mathcal{T}_{-+}^{+-}=\Phi_{4},$
(108) $\displaystyle\mathcal{T}_{-+}^{++}$
$\displaystyle=\mathcal{T}_{--}^{-+},\quad\mathcal{T}_{+-}^{--}=\Phi_{5},\quad\;\;\;\mathcal{T}_{+-}^{++}=\mathcal{T}_{--}^{-+},\quad\mathcal{T}_{++}^{-+}=\Phi_{5}.$

Due to the requirement of parity invariance we get in addition another two
relations which read

$\mathcal{T}_{--}^{-+}=\Phi_{5},\qquad\mathcal{T}_{--}^{++}=\Phi_{2}.$ (109)

Both (108) and (109) follow straightforwardly from equations (2.64), (2.86),
(2.89) and (2.90) in Hebbar:2020ukp .

###  A.2 Crossing equations in the center of mass frame

The $s-t$ and $s-u$ crossing equations for the center of mass amplitudes in
the case of massless $j=1$ particles can be obtained up to an overall phase
using the arguments of Trueman and Wick Trueman:1964zzb . They were derived in
detail for example in Hebbar:2020ukp , see equations (2.81) and (2.82).
Specializing to the case of photon scattering, their result reads

$\displaystyle\mathcal{T}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u)$
$\displaystyle=\chi_{st}\,\mathcal{T}_{-\lambda_{1},+\lambda_{3}}^{+\lambda_{2},-\lambda_{4}}(t,s,u),$
(110)
$\displaystyle\mathcal{T}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(s,t,u)$
$\displaystyle=\chi_{su}\,\mathcal{T}_{-\lambda_{1},+\lambda_{4}}^{-\lambda_{3},+\lambda_{2}}(u,t,s),$

where the overall phases $\chi_{st}$ and $\chi_{su}$ remain undetermined.
Using (107), (108) and (109) we can rewrite the crossing equations (110) in
terms of the 5 center of mass amplitudes $\Phi_{i}(s,t,u)$ only. They read

$\Phi_{I}(s,t,u)=\chi_{st}\sum_{J=1}^{5}C^{st}_{IJ}\Phi_{J}(t,s,u),\qquad\Phi_{I}(s,t,u)=\chi_{su}\sum_{J=1}^{5}C^{su}_{IJ}\Phi_{J}(u,t,s),$
(111)

where the crossing matrices read

$C^{st}\equiv\begin{pmatrix}0&0&0&1&0\\\ 0&1&0&0&0\\\ 0&0&1&0&0\\\
1&0&0&0&0\\\ 0&0&0&0&1\end{pmatrix},\qquad
C^{su}\equiv\begin{pmatrix}0&0&1&0&0\\\ 0&1&0&0&0\\\ 1&0&0&0&0\\\ 0&0&0&1&0\\\
0&0&0&0&1\end{pmatrix}.$ (112)

Both matrices have the following eigenvalues $\\{-1,1,1,1,1\\}$. They cannot
however be simultaneously diagonalized.

The easiest way to fix the unknown phases $\chi_{st}$ and $\chi_{su}$ in (111)
is to plug the explicit expressions of $\Phi_{i}(s,t,u)$ given in (6) at the
lowest order in $s$ (i.e. using only $s^{2}$ terms) into the crossing
equations (111). One then immediately concludes that

$\chi_{st}=\chi_{su}=+1.$ (113)

As an alternative approach, in appendix G, we will derive the crossing
equation (110) explicitly using LSZ. There, we also show that the phases are
given by (113).

###  A.3 Unitarity

We now discuss the constraints on the amplitudes due to unitarity. This
subsection is an application of the general construction presented in
Hebbar:2020ukp , which the reader is referred to for more details. We begin by
defining a short-hand notation for the two-particle state (95) evaluated in
the center of mass frame, namely

$|(\boldsymbol{p},\theta,\phi);\lambda_{1},\lambda_{2}\rangle_{id}\equiv\frac{1}{\sqrt{2}}(|\vec{p};\lambda_{1}\rangle\otimes|-\vec{p};\lambda_{2}\rangle+|-\vec{p};\lambda_{2}\rangle\otimes|\vec{p};\lambda_{1}\rangle).$
(114)

Here $(\theta,\phi)$ are the angular coordinates of $\vec{p}$ and
$\boldsymbol{p}\equiv|\vec{p}\,|$. The state (114) transforms in the reducible
representation of the Poincaré group. Let us now define a two particle state
which transforms in the irreducible representation of the Poincaré group
instead. It reads

$|c,\vec{0},\ell,\lambda;\lambda_{1},\lambda_{2}\rangle_{id}\equiv\\\
\frac{2\ell+1}{4\pi\sqrt{2}C_{\ell}}\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta\sin\theta
e^{-i\phi(\lambda_{1}+\lambda_{2}-\lambda)}d^{\,\ell}_{\lambda,\lambda_{12}}(\theta)|(\boldsymbol{p},\theta,\phi);\lambda_{1},\lambda_{2}\rangle_{id},$
(115)

where $\lambda_{12}\equiv\lambda_{1}-\lambda_{2}$, $\ell=0,1,2,\ldots$ and
$\lambda$ are the total spin and helicity, $c\equiv 2\boldsymbol{p}=\sqrt{s}$
is the center of mass energy and

$C_{\ell}\equiv\sqrt{8\pi(2\ell+1)}\,.$ (116)

Since the states (115) transform in irreducible representations of the
Poincaré group we conclude that their inner product with the scattering
operator $T$ have the following most general form

${}_{id}\langle
c^{\prime},\vec{p}\,^{\prime},\ell^{\prime},\lambda^{\prime};\lambda_{3},\lambda_{4}|T|c,\vec{p},\ell,\lambda;\lambda_{1},\lambda_{2}\rangle_{id}=(2\pi)^{4}\delta^{4}(p^{\mu}-p^{\prime\mu})\delta_{\ell\ell^{\prime}}\delta_{\lambda\lambda^{\prime}}{\mathcal{T}_{\ell}}^{\,\lambda_{3},\lambda_{4}}_{\,\lambda_{1},\lambda_{2}}(s).$
(117)

Here the functions
${\mathcal{T}_{\ell}}^{\,\lambda_{3},\lambda_{4}}_{\,\lambda_{1},\lambda_{2}}(s)$
are called (interacting part of) partial amplitudes. They are related to the
interacting part of scattering amplitudes via the following integral transform

${\mathcal{T}_{\ell}}^{\,\lambda_{3},\lambda_{4}}_{\,\lambda_{1},\lambda_{2}}(s)=\frac{1}{32\pi}\int_{0}^{\pi}d\theta\sin\theta\,d^{\,\ell}_{\lambda_{12},\lambda_{34}}(\theta)\mathcal{T}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}\Big{(}s,t(s,\theta),u(s,\theta)\Big{)},$
(118)

with $t(s,\theta)$ and $u(s,\theta)$ given in (105). Here
$d^{\,\ell}_{\lambda_{12},\lambda_{34}}(\theta)$ stand for small Wigner
d-matrices, they are defined by

$\displaystyle d^{\ell}_{\lambda\lambda^{\prime}}(\beta)$
$\displaystyle=\sqrt{(\ell+\lambda)!(\ell-\lambda)!(\ell+\lambda^{\prime})!(\ell-\lambda^{\prime})!}$
$\displaystyle\qquad\times\sum_{\nu=0}^{2j}(-1)^{3\nu+\lambda-\lambda^{\prime}}\frac{(\cos(\beta/2))^{2\ell+\lambda^{\prime}-\lambda-2\nu}(\sin(\beta/2))^{\lambda-\lambda^{\prime}+2\nu}}{\nu!(\ell-\lambda-\nu)!(\ell+\lambda^{\prime}-\nu)!(\nu+\lambda-\lambda^{\prime})!}$
(119)
$\displaystyle=\left(\cos\left(\frac{\beta}{2}\right)\right)^{\lambda^{\prime}+\lambda}\left(\sin\left(\frac{\beta}{2}\right)\right)^{\lambda^{\prime}-\lambda}\sqrt{\frac{(l-\lambda)!(\lambda^{\prime}+l)!}{(\lambda+l)!(l-\lambda^{\prime})!}}$
$\displaystyle\qquad\times\frac{\,{}_{2}F_{1}\left(\lambda^{\prime}-\ell,\ell+\lambda^{\prime}+1;-\lambda+\lambda^{\prime}+1;\sin^{2}\left(\frac{\beta}{2}\right)\right)}{\Gamma(-\lambda+\lambda^{\prime}+1)}\,,$
(120)

where for completeness, we wrote two equivalent definitions.252525Note that
Mathematica implements the small Wigner d-matrices with a different sign
convention
$\texttt{WignerD}[\\{j,\lambda,\lambda^{\prime}\\},\beta]=(-1)^{\lambda-\lambda^{\prime}}d^{j}_{\lambda\lambda^{\prime}}(\beta)$.
Using properties of the Wigner $d$-matrices, see for example appendix A.1 of
Hebbar:2020ukp , the integral transform (118) can be inverted and we obtain
the usual partial wave expansion

$\mathcal{T}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}\Big{(}s,t(s,\theta),u(s,\theta)\Big{)}=\sum_{\ell}16\pi(2\ell+1)\,{T_{\ell}}^{\,\lambda_{3},\lambda_{4}}_{\,\lambda_{1},\lambda_{2}}(s)\,d^{\,\ell}_{\lambda_{12},\lambda_{34}}(\theta).$
(121)

In the case of two identical spin one massless particles there are three
possible two particle Poincaré irreps. We list them using the notation $+$ for
helicity $+1$ and $-$ for helicity $-1$262626Note that
$|c,\vec{p},\ell,\lambda;-,+\rangle_{id}=(-1)^{\ell}|c,\vec{p},\ell,\lambda;+,-\rangle_{id}$
by Bose symmetry.

$|c,\vec{p},\ell,\lambda;+,+\rangle_{id},\quad|c,\vec{p},\ell,\lambda;-,-\rangle_{id},\quad|c,\vec{p},\ell,\lambda;+,-\rangle_{id}$
(122)

Bose symmetry of identical particles also implies the selection rule that the
first two states in the list above only exist for even $\ell$. The third state
exists for all spin $\ell\geq 2$.272727The total spin $\ell$ must always be
greater than the difference in helicity $\lambda_{1}-\lambda_{2}$ of the two
particles. Under parity transformation, the three states transform as follows

$\displaystyle\mathcal{P}|c,\vec{0},\ell,\lambda;+,+\rangle_{id}$
$\displaystyle=(-1)^{\ell}|c,\vec{0},\ell,\lambda;-,-\rangle_{id},$ (123)
$\displaystyle\mathcal{P}|c,\vec{0},\ell,\lambda;-,-\rangle_{id}$
$\displaystyle=(-1)^{\ell}|c,\vec{0},\ell,\lambda;+,+\rangle_{id},$
$\displaystyle\mathcal{P}|c,\vec{0},\ell,\lambda;+,-\rangle_{id}$
$\displaystyle=|c,\vec{0},\ell,\lambda;+,-\rangle_{id}.$

Since we consider parity invariant theories, it is convenient to define new
linear combinations which are parity eigenstates:

$\text{\bf parity even:}\quad\begin{aligned}
|1\rangle&\equiv\frac{1}{\sqrt{2}}\Big{(}|c,\vec{p},\ell,\lambda;+,+\rangle_{id}+|c,\vec{p},\ell,\lambda;-,-\rangle_{id}\Big{)},\quad\ell\geq
0\;(\text{even}),\\\
|2\rangle&\equiv\sqrt{2}\,|c,\vec{p},\ell,\lambda;+,-\rangle_{id},\quad\ell\geq
2,\end{aligned}$ (124) $\text{\bf parity
odd:}\quad|3\rangle\equiv\frac{1}{\sqrt{2}}\Big{(}|c,\vec{p},\ell,\lambda;+,+\rangle_{id}-|c,\vec{p},\ell,\lambda;-,-\rangle_{id}\Big{)},\quad\ell\geq
0\;(\text{even}).\\\ $ (125)

In a unitary quantum theory, the norm of any state in the theory must be non-
negative. Consider the following set of six states:

$|1\rangle_{in},\qquad|2\rangle_{in},\qquad|3\rangle_{in},\qquad|1\rangle_{out},\qquad|2\rangle_{out},\qquad|3\rangle_{out}.$
(126)

Any linear combination of these states must have non-negative norm. This
statement is equivalent to the statement that the $6\times 6$ Hermitian matrix
formed by the inner products between the six states is positive semi-definite.
Factoring out the overall delta functions we write

$\begin{pmatrix}{}_{in}\langle a^{\prime}|b\rangle_{in}&{}_{in}\langle
a^{\prime}|b\rangle_{out}\\\ {}_{out}\langle
a^{\prime}|b\rangle_{in}&{}_{out}\langle
a^{\prime}|b\rangle_{out}\end{pmatrix}=\mathcal{H}_{\ell}(s)\times(2\pi)^{4}\delta^{4}(p^{\mu}-p^{\prime\mu})\delta_{\ell\ell^{\prime}}\delta_{\lambda\lambda^{\prime}}\,.$
(127)

Unitarity as stated above then implies that

$\mathcal{H}_{\ell}(s)\succeq 0\,,\quad\forall\ \ell\quad\text{and}\quad s\geq
0\,.$ (128)

The inner products between two incoming states or two outgoing states are
fixed by the normalization of these states, namely282828The pre-factors in
(124) and (125) ensure that all three states have the same normalization

${}_{in}\langle a^{\prime}|b\rangle_{in}={}_{out}\langle
a^{\prime}|b\rangle_{out}=\delta_{a^{\prime}b}\times\delta_{\ell\ell^{\prime}}\delta_{\lambda\lambda^{\prime}}(2\pi)^{4}\delta^{4}(p^{\mu}-p^{\prime\mu})\,.$
(129)

The inner products between incoming and outgoing states are, by definition,
the matrix elements of the scattering operator $S=1+iT$ and therefore due to
(117) we have

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

Because of the invariance under parity there is no scattering between states
with different parity eigenvalues. Hence the inner products ${}_{out}\langle
3^{\prime}|1\rangle_{in}$ and ${}_{out}\langle 3^{\prime}|2\rangle_{in}$ are
all zero and the positive semi-definite condition (128) simplifies into
smaller matrices. Taking into account the parity selection rules, we arrive at
two separate sectors, namely the parity even and parity odd sectors.

##### Parity even sector

We begin by considering parity even eigenstates (124). For even spin $\ell\geq
2$, we have

$\displaystyle\mathcal{H}^{+}_{\ell}(s)\times\delta_{\ell\ell^{\prime}}\delta_{\lambda\lambda^{\prime}}(2\pi)^{4}\delta^{(4)}(p-p^{\prime})$
$\displaystyle\equiv\begin{pmatrix}{}_{in}\langle
1^{\prime}|1\rangle_{in}&{}_{in}\langle
1^{\prime}|2\rangle_{in}&{}_{in}\langle
1^{\prime}|1\rangle_{out}&{}_{in}\langle 1^{\prime}|2\rangle_{out}\\\
{}_{in}\langle 2^{\prime}|1\rangle_{in}&{}_{in}\langle
2^{\prime}|2\rangle_{in}&{}_{in}\langle
2^{\prime}|1\rangle_{out}&{}_{in}\langle 2^{\prime}|2\rangle_{out}\\\
{}_{out}\langle 1^{\prime}|1\rangle_{in}&{}_{out}\langle
1^{\prime}|2\rangle_{in}&{}_{out}\langle
1^{\prime}|1\rangle_{out}&{}_{out}\langle 1^{\prime}|2\rangle_{out}\\\
{}_{out}\langle 2^{\prime}|1\rangle_{in}&{}_{out}\langle
2^{\prime}|2\rangle_{in}&{}_{out}\langle
2^{\prime}|1\rangle_{out}&{}_{out}\langle
2^{\prime}|2\rangle_{out}\end{pmatrix}$ (131)

The case of spin $\ell=0$ is special because the state $|2\rangle$ does not
exist and therefore we get a smaller matrix:

$\mathcal{H}^{+}_{0}(s)\times\delta_{\lambda\lambda^{\prime}}(2\pi)^{4}\delta^{(4)}(p-p^{\prime})\equiv\begin{pmatrix}{}_{in}\langle
1^{\prime}|1\rangle_{in}&{}_{in}\langle 1^{\prime}|1\rangle_{out}\\\
{}_{out}\langle 1^{\prime}|1\rangle_{in}&{}_{out}\langle
1^{\prime}|1\rangle_{out}\end{pmatrix},$ (132)

We now consider odd $\ell\geq 3$, in which case the only state that exists is
state $|2\rangle$ and therefore we have

$\displaystyle\mathcal{H}^{+}_{\ell}(s)\times\delta_{\ell\ell^{\prime}}\delta_{\lambda\lambda^{\prime}}(2\pi)^{4}\delta^{(4)}(p-p^{\prime})$
$\displaystyle\equiv\begin{pmatrix}{}_{in}\langle
2^{\prime}|2\rangle_{in}&{}_{in}\langle 2^{\prime}|2\rangle_{out}\\\
{}_{out}\langle 2^{\prime}|2\rangle_{in}&{}_{out}\langle
2^{\prime}|2\rangle_{out}\end{pmatrix}$ (133)

##### Parity odd sector

We now turn to the parity odd eigenstate (125) which exists for even spin
$\ell\geq 0$. We have

$\mathcal{H}^{-}_{\ell}(s)\times\delta_{\ell\ell^{\prime}}\delta_{\lambda\lambda^{\prime}}(2\pi)^{4}\delta^{(4)}(p-p^{\prime})\equiv\begin{pmatrix}{}_{in}\langle
3^{\prime}|3\rangle_{in}&{}_{in}\langle 3^{\prime}|3\rangle_{out}\\\
{}_{out}\langle 3^{\prime}|3\rangle_{in}&{}_{out}\langle
3^{\prime}|3\rangle_{out}\end{pmatrix}.$ (134)

##### Final summary

We can now plug equations (129), (130) into (131) - (134) to obtain the final
form of the unitarity constraints. Below we will use the following notation
for the partial amplitudes in the center of mass frame

$\displaystyle\Phi^{\ell}_{1}(s)\equiv\mathcal{T}_{\ell}{}_{++}^{++}(s),$
(135)
$\displaystyle\Phi^{\ell}_{2}(s)\equiv\mathcal{T}_{\ell}{}_{++}^{--}(s),$
$\displaystyle\Phi^{\ell}_{3}(s)\equiv\mathcal{T}_{\ell}{}_{+-}^{+-}(s),$
$\displaystyle\Phi^{\ell}_{4}(s)\equiv\mathcal{T}_{\ell}{}_{+-}^{-+}(s),$
$\displaystyle\Phi^{\ell}_{5}(s)\equiv\mathcal{T}_{\ell}{}_{++}^{+-}(s).$

These are consistent with the definitions given in (107). We finally get

$\displaystyle\ell\geq 0\text{ (even)}:\quad$
$\displaystyle\begin{pmatrix}1&1\\\
1&1\end{pmatrix}+i\begin{pmatrix}0&-\Phi_{1}^{\ell*}(s)+\Phi_{2}^{\ell*}(s)\\\
\Phi_{1}^{\ell}(s)-\Phi_{2}^{\ell}(s)&0\end{pmatrix}\succeq 0\,,$ (136)
$\displaystyle\ell\geq 3\text{ (odd)}:\quad$
$\displaystyle\begin{pmatrix}1&1\\\
1&1\end{pmatrix}+2i\begin{pmatrix}0&-\Phi_{3}^{\ell*}(s)\\\
\Phi_{3}^{\ell}(s)&0\end{pmatrix}\succeq 0\,,$ (137)
$\displaystyle\ell=0:\quad$ $\displaystyle\begin{pmatrix}1&1\\\
1&1\end{pmatrix}+i\begin{pmatrix}0&-\Phi_{1}^{0*}(s)-\Phi_{2}^{0*}(s)\\\
\Phi_{1}^{0}(s)+\Phi_{2}^{0}(s)&0\end{pmatrix}\succeq 0\,,$ (138)
$\displaystyle\ell\geq 2\text{ (even)}:\quad$
$\displaystyle\begin{pmatrix}\mathbb{I}_{2\times
2}&\mathbb{S}^{\ell\dagger}_{2\times 2}(s)\\\ \mathbb{S}^{\ell}_{2\times
2}(s)&\mathbb{I}_{2\times 2}\end{pmatrix}\succeq 0,$ (139)

where in (139) we have defined

$\mathbb{I}_{2\times 2}\equiv\begin{pmatrix}1&0\\\
0&1\end{pmatrix},\qquad\mathbb{S}^{\ell}_{2\times
2}(s)\equiv\begin{pmatrix}1&0\\\
0&1\end{pmatrix}+i\begin{pmatrix}\Phi_{1}^{\ell}(s)+\Phi_{2}^{\ell}(s)&\qquad
2\Phi_{5}^{\ell}(s)\\\ 2\Phi_{5}^{\ell}(s)&\qquad
2\Phi_{3}^{\ell}(s)\end{pmatrix}\,.$ (140)

The unitarity constraints presented in (21) and (22) are a compact rewriting
of the above conditions.

We conclude this subsection by commenting on a curious symmetry of the above
unitarity constraints. The partial amplitude $\Phi_{5}^{\ell}$ appears only in
the $4$ by $4$ matrix (139) via (140). It is straightforward to see that
semidefinite positivity of (139) is invariant under
$\Phi_{5}\leftrightarrow-\Phi_{5}$. If in some particular theory $\Phi_{5}=0$
(and as a consequence $\Phi_{5}^{\ell}=0$), the matrix (139) can be brought
into a block diagonal form and as a result the semidefinite positivity then
simply reduces to a $2$ by $2$ condition

$\ell\geq 2\text{ (even)}:\quad\begin{pmatrix}1&1\\\
1&1\end{pmatrix}+i\begin{pmatrix}0&+\Phi_{1}^{\ell*}(s)+\Phi_{2}^{\ell*}(s)\\\
\Phi_{1}^{\ell}(s)+\Phi_{2}^{\ell}(s)&0\end{pmatrix}\succeq 0,$ (141)

and a $2$ by $2$ condition on $\Phi_{3}^{\ell}$, same as (137) but now also
for even spin

$\ell\geq 2\text{ (even)}:\quad\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\,.\\\ $ (142)

The unitarity conditions (136) - (138) together with (141) and (142) are now
$\Phi_{2}\leftrightarrow-\Phi_{2}$ symmetric.

###  A.4 Forward limit

Consider the scattering operator $S$. It is unitary, namely

$S^{\dagger}S=1.$ (143)

Splitting $S$ into its trivial and interacting part according to (98), the
above constraint can be rewritten in the following form

$T-T^{\dagger}=i\,T^{\dagger}T.$ (144)

Taking the expectation value in some state $|\text{state}\rangle$ we get

$\operatorname{Im}\langle\text{state}|T|\text{state}\rangle=\frac{1}{2}\langle\text{state}|TT^{\dagger}|\text{state}\rangle\geq
0.$ (145)

The last inequality holds because any norm in a unitary theory should be non-
negative. The result (145) is known as the optical theorem.

We have defined two-particle states of identical spin one massless particles
in (95). Let us consider the situation when these states are in the center of
mass (COM) frame given by (104). In what follows we will use the following
short-hand notation

$|\lambda_{1},\lambda_{2}\rangle\equiv|\kappa_{1},\kappa_{2}\rangle_{id}^{\text{COM}},\qquad\langle\lambda_{3},\lambda_{4}|\equiv{}_{id}^{\text{COM}}\langle\kappa_{3},\kappa_{4}|,$
(146)

where $\kappa_{i}$ are the particles participating in the scattering process
$12\rightarrow 34$ with the 4-momenta $p_{i}^{\mu}$. We would also like to
define the following state

$|\text{state}\rangle=\alpha_{1}|+,+\rangle+\alpha_{2}|-,-\rangle+\alpha_{3}|+,-\rangle+\alpha_{4}|-,+\rangle,$
(147)

with an analogous definition for $\langle\text{state}|$. Here
$\vec{\alpha}\equiv\\{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\\}$ is a
vector of complex numbers. Plugging (147) into (145), using the definitions
(107) and evaluating the expression in the forward limit given by $\theta=0$
(or equivalently $t=0$) we obtain the following constraint292929In writing
this equation we have dropped the overall delta-function
$(2\pi)^{4}\delta^{(4)}(0)$.

$\displaystyle\vec{\alpha}^{\dagger}\operatorname{Im}\begin{pmatrix}\Phi_{1}(s)&\Phi_{2}(s)&\Phi_{5}(s)&\Phi_{5}(s)\\\
\Phi_{2}(s)&\Phi_{1}(s)&\Phi_{5}(s)&\Phi_{5}(s)\\\
\Phi_{5}(s)&\Phi_{5}(s)&\Phi_{3}(s)&\Phi_{4}(s)\\\
\Phi_{5}(s)&\Phi_{5}(s)&\Phi_{4}(s)&\Phi_{3}(s)\end{pmatrix}\vec{\alpha}\geq
0.$ (148)

Here we have also defined the short-hand notation of the amplitudes in the
forward limit

$\Phi_{i}(s)\equiv\Phi_{i}(s,\,t=0,\,u=-s).$ (149)

In the forward limit we have a special situation because

$\Phi_{4}(s)=\Phi_{5}(s)=0.$ (150)

This follows from the representation of amplitudes in terms of partial
amplitudes (121) and the fact that

$d^{\ell}_{2,-2}(\theta=0)=d^{\ell}_{2,0}(\theta=0)=0.$ (151)

Using (150) the constraint (148) can be rewritten in its final form

$\displaystyle\operatorname{Im}\begin{pmatrix}\Phi_{1}(s)&\Phi_{2}(s)&0&0\\\
\Phi_{2}(s)&\Phi_{1}(s)&0&0\\\ 0&0&\Phi_{3}(s)&0\\\
0&0&0&\Phi_{3}(s)\end{pmatrix}\succeq 0.$ (152)

Due to the Sylvester’s criterion the positive semi-definite condition (152) is
equivalent to

$\left\\{\begin{split}&\operatorname{Im}(\Phi_{1}(s)+\Phi_{2}(s))\geq 0\\\
&\operatorname{Im}(\Phi_{1}(s)-\Phi_{2}(s))\geq 0\\\
&\operatorname{Im}\Phi_{3}(s)\geq 0\end{split}\right.\qquad\qquad\forall s\geq
0\ .$ (153)

The result (153) can be alternatively obtained from the unitarity constraints
(136) - (139) and (121) by taking the forward limit $\theta=0$.

##  Appendix B Tensor structures

Consider the in-out interacting amplitude
$\mathcal{T}{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p_{1},p_{2},p_{3},p_{4})$
describing the scattering process $12\rightarrow 34$. Recall that
$\lambda_{1}$ and $\lambda_{2}$ are helicities of the incoming particles with
4-momenta $p_{1}$ and $p_{2}$, and $\lambda_{3}$ and $\lambda_{4}$ are
helicities of the outgoing particles with 4-momenta $p_{3}$ and $p_{4}$. It is
convenient to factorize the interacting scattering amplitude of spinning
particles in the following way

$\mathcal{T}{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p_{1},p_{2},p_{3},p_{4})=\sum_{I=1}^{N}\mathcal{A}_{I}(s,t,u)\mathbf{T}_{I}{}_{\lambda_{1},\lambda_{2}}^{\lambda_{3},\lambda_{4}}(p_{1},p_{2},p_{3},p_{4}).$
(154)

Here the objects $\mathbf{T}_{I}$ take care of the correct Lorentz
transformation properties. They are called tensor structures. Their form is
completely fixed by kinematics. There are $N$ linearly independent tensor
structures. The objects $\mathcal{A}_{I}(s,t,u)$ are referred to as the
components of the interacting scattering amplitude and are invariant under
Lorentz transformations. As a result they depend only on the Mandelstam
variables defined in (102). Contrary to tensor structures, the components of
interacting amplitudes $\mathcal{A}_{I}(s,t,u)$ carry dynamical information of
a particular theory.

The goal of this appendix is to explicitly construct a linearly independent
basis of tensor structures $\mathbf{T}_{I}$ in the case of identical spin one
massless particles. We will do it using two different formalisms: the vector
formalism presented in appendix B.3 and the spinor formalism presented in
appendix B.4. While the former works for any space-time dimension, the latter