aainstitutetext: Theoretical Physics, Blackett Laboratory, Imperial College,
London, SW7 2AZ, UKbbinstitutetext: CERCA, Department of Physics, Case Western
Reserve University, 10900 Euclid Ave, Cleveland, OH 44106, USAccinstitutetext:
Perimeter Institute for Theoretical Physics, 31 Caroline St N, Waterloo,
Ontario, N2L 6B9, Canadaddinstitutetext: Institute for Theoretical Physics,
ETH Zürich, Wolfgang-Pauli-Strasse 27, 8093, Zürich,
Switzerlandeeinstitutetext: Institute for Theoretical Physics, Heidelberg
University, Philosophenweg 16, 69120 Heidelberg, Germany

# Positivity bounds in vector theories

Claudia de Rham d Laura Engelbrecht c,d,e Lavinia Heisenberg d Alice Lüscher
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Assuming unitarity, locality, causality, and Lorentz invariance of the,
otherwise unknown, UV completion, we derive a new set of constraints on the
effective field theory coefficients for the most general, ghost-free
Generalized Proca and Proca Nuevo massive vector models. For the Generalized
Proca model, we include new interactions that had not been previously
considered in the context of positivity bounds and find these additional terms
lead to a widened parameter space for the previously considered interactions.
Although, the Generalized Proca and Proca Nuevo models are inequivalent, we
find interesting analogues between the coefficients parameterizing the two
models and the roles they play in the positivity bounds.

##  1 Introduction

Effective field theories (EFTs), describe low energy (IR) physics, up to a
certain cutoff $\Lambda$ beyond which additional ingredients needs to be
considered. In constructing an EFT, every interaction comes with a
coefficient, which from the EFT point of view, can take any value, leading to
a huge parameter space. When considering experiments or observations where
searching an infinite parameter space may be costly or infeasible, narrowing
down the parameter space can be invaluable, allowing for more targets searches
or even allowing one to theoretically rule out certain theories as
incompatible with a standard UV completion. Even if the full (UV complete)
model at high energies is unknown, the parameter space of an EFT may be
restricted using constraints arising from hypotheses on the UV completion.

The physical requirements of unitarity, locality, causality, and Lorentz
invariance of the UV completion of an EFT translate into unitarity,
analyticity and crossing symmetry of scattering amplitudes. Put together,
these properties give rise to a dispersion relation, whose positivity
constrains the amplitudes. Schematically, analyticity allows one to write the
amplitude as a contour integral and to deform the contour to obtain an
integral of the imaginary part, which is positive by unitarity (optical
theorem) and crossing relations,

$\frac{1}{2}\mathcal{A}^{\prime\prime}(s)=\frac{1}{2\pi
i}\oint_{\mathcal{C}}ds^{\prime}\frac{\mathcal{A}(s^{\prime})}{(s^{\prime}-s)^{3}}=\frac{1}{\pi}\int_{\text{cuts}}d\mu\frac{\imaginary
A(\mu)}{(\mu-s)^{3}}>0\,.$ (1)

In practice, $\mathcal{A}^{\prime\prime}(s)$ is computed in a given EFT,
whereas the last inequality is not usually computed explicitly, but its
positivity is guaranteed by the assumptions on the UV completion and in turn
ensures the positivity of the $\mathcal{A}^{\prime\prime}(s)$. This procedure
provides positivity bounds which can be used to constrain the parameter space
of effective field theories.

It has long been known that analyticity and dispersion relations lead to
positivity constraints, but it is only quite recently that Adams_2006
exploited these constraints to bound EFT coefficients. Since their
foundational work, which applied to scalar theories in the forward limit, some
extensions have been worked out both away from the forward limit de_Rham_2017
and for spinning particles Bellazzini2017 ; de_Rham_2018 ; davighi2021natural
. These bounds have been applied to massive particles of various spin
de_Rham_2017gal ; Bellazzini_2019spin3 ; Bonifacio_2016 ; de_Rham_2019 , with
a particular attention on massive spin-2 fields Cheung2016 ; Bonifacio_2016 ;
de_Rham_2018improved ; Bellazzini_2018 ; de_Rham_2019 ; Alberte2020 ;
Alberte20202 ; Wang2021 . Some improved positivity bounds, which carve the
parameter space in a more systematical manner making use of non-linear bounds
and full crossing-symmetry, have also been developed Bellazzini_2021 ;
tolley2021new ; Caron-Huot2021 ; Arkani-Hamed:2020blm ; Chiang2021 ; Du_2021 .
Using these methods, the massless and massive Galileon have been found to
admit no standard UV completion Adams_2006 ; tolley2021new . Likewise, in
Bonifacio_2016 , it was shown that the simplest vector Galileon model also
cannot admit a standard UV completion. Nonetheless, as argued in Cheung2016 ;
de_Rham_2018improved ; de_Rham_2019 , it does not prevent massive gravity (or
GP, PN) to possibly admit one, even though the massive Galileon emerges as
their helicity-0 mode in the decoupling limit. Moreover, a violation of the
bounds is not dramatic in itself. It simply means that if a UV completion of
the model is to exist, it will not be a local, Lorentz invariant field theory,
but have some more exotic properties.

Due to their relevance in modern physics, some EFTs have been particularly
popular for the application of the bounds. Positivity bounds have been
extensively studied in the case of (massless) gravity, where both pure gravity
Bern_2021 ; Chiang2022 ; CaronHuot2022 ; Herrero_Valea_2021  and gravity in
the presence of other fields Alberte_2020spin2 ; Tokuda_2020 ;
Caron_Huot_2021GR ; Alberte2021 ; Alberte_202222 . Positivity bounds for
gravity require special care due to its massless nature and the presence of a
pole in the $t$-channel exchange, leading to additional complications.
Moreover, positivity bounds in gravity can be related to the Weak Gravity
Conjecture Alberte_2020spin2 ; Cheung_2014 ; Hamada_2019 ; Bellazzini_2019WGC
; Arkani_Hamed_2022 ; Henriksson2022 . The Standard Model EFT (SMEFT) has also
drawn a great deal of attention in the positivity bounds community Zhang_2019
; Zhou2019 ; Bellazzini_2018SM ; Remmen_2019 ; Zhang_2020 ; Zhou2021 ;
Fuks_2021 ; Remmen_2020 ; Bonnefoy_2021 ; Chala2021 . Constraining the SMEFT
parameter space makes both the experimental searches and the theoretical
interpretations of the data more efficient. If some violations of the bounds
were detected, it would indicate that some assumptions on the UV-complete
theory are erroneous. The application of the bounds in cosmology, as conducted
in Melville_2020 ; Noller2021 ; Kim_2019 ; Herrero_Valea_2019 ; Ye_2020 ;
melville2022positivity ; Creminelli:2022onn , can be combined with
observational data to strongly constrain the parameter space of dark energy
and modified gravity models. (See deRham:2022hpx  for a recent review of the
positivity bounds and their applications.)

The aim of this work is to provide new constraints on vector EFTs via the
positivity bounds technology. Positivity bounds on certain Proca interactions
have been considered in Bonifacio_2016 ; de_Rham_2019 , but these were not the
most general interactions allowed within these models. We consider the most
general possible parity even interaction terms allowed for Generalized Proca
theories which include terms with scalar-vector mixing in the decoupling
limit. Positivity bounds on Proca Nuevo models have never previously been
considered and are studied here for the first time.

We begin by presenting explicitly the two massive vector models of interest,
the Generalized Proca Heisenberg2014  and Proca-Nuevo derham2020  in section
2. Then, we review the spinning positivity bounds from de_Rham_2018  in
section 3. The reader familiar with these topics may skip these first two
sections. Finally, we present our results for the constraints on the two Proca
theories in section 4. We work in a $3+1$ dimensional flat space using the
mostly-plus signature $(-,+,+,+)$.

##  2 Vector Models

There were initial attempts to write down derivative self-interactions for a
massless abelian vector field, but it was concluded that the Maxwell term is
the only possible term compatible with gauge invariance and leading to second
order equation of motion (EOM) on flat space, hence a no go theorem for
massless vector Galileons nogo2014 111Even though a non-minimal coupling of
the gauge-invariant field strength to gravity exists, for most of the relevant
applications it is unstableBeltranJimenez:2013btb . On the other hand, it is
possible to construct derivative self-interactions with second order EOM if we
give up gauge symmetry and consider massive vector theories, also called
Generalized Proca theories. The Generalized Proca (GP) model Heisenberg2014 ;
Tasinato2013 ; Allys2016 ; Jimenez2016  is the most general abelian self-
interacting massive spin-1 theory222A non-abelian version of the Generalized
Proca has been investigated in BeltranJimenez:2016afo ; Allys_2016 . with
second order EOM, even if the interaction terms from the Lagrangian appear to
have higher order derivatives (see Heisenberg:2018vsk  for a review).
Moreover, it enjoys a Vainshtein screening mechanism, making it compatible
with solar system tests of gravity De_Felice_2016 . These models have found
various phenomenological applications in astrophysics Chagoya_2017 ; Kase_2018
; Kase_2020 ; Garcia_Saenz_2021 ; Brihaye_2022  notably for black holes
Chagoya_2016 ; Minamitsuji_2016 ; Cisterna_2016 ; Chagoya_2017 ;
Heisenberg_2017 ; Minamitsuji2017 ; Heisenberg_20172 ; Kase_20182 ; Kase_20183
; Rahman_2019 , and in cosmology Felice_2016 ; De_Felice_20162 ;
Heisenberg_2016 ; Nakamura_2017 ; Emami_2017 ; De_Felice_2017 ; Kase_20184 ;
Dom_nech_2018 ; Nakamura_2019 ; Oliveros_2019 ; Felice_2020 ; Heisenberg_2021
.

In order to avoid instabilities, a massive vector theory should propagate
three degrees of freedom. This is achieved by imposing a constraint, such as
the degeneracy of the Hessian matrix. In the Generalized Proca model, this
constraint manifests itself by the non-propagation of the temporal component,
and the absence of Ostrogradski instability is made apparent by the second
order EOM. However, there is no reason that the constraint may not be realized
in another way (in fact see Heisenberg:2016eld  and BeltranJimenez:2019wrd ).

Another kind of self-interacting massive vector theory has recently been
discovered derham2020 . In fact, a Proca theory different from GP emerges from
the decoupling limit of massive gravity on AdS space Laura2018 , and inspired
the construction of a new Proca theory on flat space, dubbed Proca-Nuevo (PN)
derham2020 . As intended, this model propagates three degrees of freedom, but
realizes the constraint non-linearly as opposed to GP, where the constraint is
realized order by order. Moreover, both the GP and PN theories lead to a time-
dependent vector condensate which could play the role of a dark energy fluid,
driving the accelerated expansion of the universe that we currently observe.
Additionally, these models have a technically natural vector mass and dark
energy scale Zosso2021 ; derham2021quantum . The application of an extended PN
model in cosmology have been successfully considered in derham2021cosmology ;
Pozsgay2022 .

###  2.1 Generalized Proca

The Generalized Proca interactions Heisenberg2014 , sometimes referred to as
vector Galileons, are massive vector self-interactions built out of two
requirements. One, that the equations of motion are second order. Two, that
the temporal component is not dynamical. In turn, these guarantee that only
three healthy degrees of freedom propagate.

A way to construct this theory is to write down all possible interactions at
each order in derivatives and tune the coefficients to have a degenerate
Hessian Heisenberg2014 . This procedure produces the following model for the
massive vector field $A_{\mu}$

$\mathcal{L}_{\text{GP}}=\sum_{n=2}^{6}\mathcal{L}_{n}\,,$ (2)

with

$\displaystyle\mathcal{L}_{2}$
$\displaystyle=f_{2}(A_{\mu},F_{\mu\nu},\tilde{F}_{\mu\nu})$ (3)
$\displaystyle\mathcal{L}_{3}$ $\displaystyle=f_{3}(A^{2})(\partial\cdot A)$
$\displaystyle\mathcal{L}_{4}$ $\displaystyle=f_{4}(A^{2})[(\partial\cdot
A)^{2}-\partial_{\mu}A_{\nu}\partial^{\nu}A^{\mu}]$
$\displaystyle\mathcal{L}_{5}$ $\displaystyle=f_{5}(A^{2})[(\partial\cdot
A)^{3}-3(\partial\cdot
A)\partial_{\mu}A_{\nu}\partial^{\nu}A^{\mu}+2\partial_{\mu}A_{\nu}\partial^{\nu}A^{\rho}\partial_{\rho}A^{\mu}]$
$\displaystyle+\tilde{f}_{5}(A^{2})\tilde{F}^{\mu\alpha}\tilde{F}^{\nu}_{\phantom{\nu}\alpha}\partial_{\mu}A_{\nu}$
$\displaystyle\mathcal{L}_{6}$
$\displaystyle=\tilde{f}_{6}(A^{2})\tilde{F}^{\mu\nu}\tilde{F}^{\alpha\beta}\partial_{\alpha}A_{\mu}\partial_{\beta}A_{\nu}\,,$

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$, and
$\tilde{F}^{\alpha\beta}=\frac{1}{2}\epsilon^{\alpha\beta\mu\nu}F_{\mu\nu}$ is
the dual field-tensor. The $f_{n}$’s are arbitrary functions of their
arguments. In particular, $f_{2}$ contains the kinematic and mass terms. This
theory is the most general one respecting the two requirements. Any other
possible interactions are related to these by total derivatives and disformal
transformations of the metric.

Moreover, expanding the $f_{n}(A^{2})$ in power series of $A^{2}$, the full
Lagrangian can be expressed perturbatively as

$\mathcal{L}_{\text{GP}}=\sum_{n=2}^{\infty}\frac{1}{\Lambda_{2}^{2(n-2)}}\mathcal{L}_{\text{GP}}^{(n)}\,,$
(4)

where each $\mathcal{L}^{(n)}$ contains the $n$-point interactions, and
$\Lambda_{p}=(m^{p-1}M_{\text{Pl}})^{1/p}$ is a dimensionful scale. In this
work, we are interested in computing tree-level 2-2 scattering amplitudes,
such that it is sufficient to consider only up to quartic interactions. The
first terms of the perturbative expansion can be expressed as

$\displaystyle\mathcal{L}_{\text{GP}}^{(2)}$
$\displaystyle=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-\frac{1}{2}m^{2}A^{2}$ (5)
$\displaystyle\mathcal{L}_{\text{GP}}^{(3)}$
$\displaystyle=a_{1}m^{2}A^{2}\partial_{\mu}A^{\mu}+a_{2}\tilde{F}^{\mu\alpha}\tilde{F}^{\nu}_{\
\alpha}\partial_{\mu}A_{\nu}$ $\displaystyle\mathcal{L}_{\text{GP}}^{(4)}$
$\displaystyle=b_{1}m^{4}A^{4}+b_{2}m^{2}A^{2}F_{\mu\nu}^{2}+b_{3}m^{2}A^{2}[(\partial\cdot
A)^{2}-\partial_{\mu}A_{\nu}\partial^{\nu}A^{\mu}]+b_{4}m^{2}A_{\mu}A^{\nu}F^{\alpha\mu}F_{\alpha\nu}$
$\displaystyle\,+b_{5}F^{\mu\nu}F^{\alpha\beta}F_{\mu\alpha}F_{\nu\beta}+b_{6}(F_{\mu\nu}^{2})^{2}+b_{7}\tilde{F}^{\alpha\beta}\tilde{F}^{\mu\nu}\partial_{\alpha}A_{\mu}\partial_{\beta}A_{\nu}\,,$

where the coefficients $a_{i}$ and $b_{i}$ are dimensionless coupling
constants. There exist different ways to define the theory perturbatively, but
they only differ by field redefinitions and total derivatives, and are
therefore equivalent. In any formulation, there are respectively 2 and 7
independent terms at cubic and quartic orders. In this work, we follow the
formulation of derham2020 , which offers a good comparison with the Proca-
Nuevo model.

In order to study the decoupling limit of the theory, we perform the
Stückelberg procedure, introducing the Stückelberg field $\phi$,

$A_{\mu}\to A_{\mu}+\frac{1}{m}\partial_{\mu}\phi\,,$ (6)

such that, in the decoupling limit, the 3 helicities ($\lambda=\pm 1,0$) of
the massive vector decomposes into 2 of a massless vector $A_{\mu}$
(helicity-1 modes) and 1 of a massless scalar $\phi$ (helicity-0 mode). Then,
taking the decoupling limit corresponds to send the mass to zero and the scale
to infinity while keeping the lowest interaction scale constant:

$m\to
0,\quad\Lambda_{2}\to\infty,\quad\text{while}\quad\Lambda_{3}\equiv(m\Lambda_{2}^{2})^{1/3}=\text{const}.$
(7)

In these limits, the perturbative Lagrangian reads

$\mathcal{L}_{\text{DL
GP}}=\sum_{n=2}^{\infty}\frac{1}{\Lambda_{3}^{3(n-2)}}\mathcal{L}_{\text{DL
GP}}^{(n)}\,,$ (8)

with

$\displaystyle\mathcal{L}_{\text{DL GP}}^{(2)}$
$\displaystyle=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-\frac{1}{2}(\partial\phi)^{2}$
(9) $\displaystyle\mathcal{L}_{\text{DL GP}}^{(3)}$
$\displaystyle=a_{1}(\partial\phi)^{2}\Box\phi+a_{2}\tilde{F}^{\mu\alpha}\tilde{F}^{\nu}_{\
\alpha}(\partial_{\mu}\partial_{\nu}\phi)$ $\displaystyle\mathcal{L}_{\text{DL
GP}}^{(4)}$
$\displaystyle=b_{3}(\partial\phi)^{2}[(\Box\phi)^{2}-(\partial_{\mu}\partial_{\nu}\phi)^{2}]+b_{7}\tilde{F}^{\alpha\beta}\tilde{F}^{\mu\nu}(\partial_{\alpha}\partial_{\mu}\phi)(\partial_{\beta}\partial_{\nu}\phi).$

$\mathcal{L}^{(2)}$ contains the decoupled kinetic terms of the massless
vector and scalar modes. The terms in $a_{1}$ and $b_{3}$ correspond
respectively to the cubic and quartic Galileon interactions, whereas those in
$a_{2}$ and $b_{7}$ mix the scalar and vector modes. In particular,
de_Rham_2019 ; Bonifacio_2016  use models that only gives Galileons in the
decoupling limit, i.e. with $a_{2}$, $b_{7}=0$ to derive some positivity
bounds for the Generalized Proca.

###  2.2 Proca-Nuevo

The Proca-Nuevo model derham2020  is built from the same determinant
formulation as dRGT massive gravity deRham:2010kj . There, the reference
metric is defined as
$f_{\mu\nu}=\partial_{\mu}\phi^{a}\partial_{\nu}\phi^{b}\eta_{ab}$, where $a$
is a Lorentz index, such that the quadruplet of Stückelberg fields $\phi^{a}$
is identified as a Lorentz vector, which can subsequently be decomposed to

$\phi^{\mu}=x^{\mu}+\frac{1}{\Lambda_{2}^{2}}A^{\mu}\,,$ (10)

for a vector field $A_{\mu}$. Then, in terms of this vector field,
$f_{\mu\nu}$ reads

$f_{\mu\nu}[A]=\eta_{\mu\nu}+2\frac{\partial_{(\mu}A_{\nu)}}{\Lambda_{2}^{2}}+\frac{\partial_{\mu}A_{\alpha}\partial_{\nu}A_{\beta}\eta^{\alpha\beta}}{\Lambda_{2}^{4}}\,.$
(11)

Next, we keep the same deformed determinant as in massive gravity, but with
$g_{\mu\nu}=\eta_{\mu\nu}$ the flat Minkowski metric and $f_{\mu\nu}$
expressed in terms of the vector field as defined in (11),

$\mathcal{K}^{\mu}_{\ \nu}=\left(\sqrt{\eta^{-1}f[A]}\right)^{\mu}_{\
\nu}-\delta^{\mu}_{\ \nu}\,.$ (12)

Finally, the full Lagrangian of this theory, defined by $\det(\delta^{\mu}_{\
\nu}+\mathcal{K}^{\mu}_{\ \nu})$, is given by

$\mathcal{L}_{\text{PN}}=\Lambda_{2}^{4}\sum_{n=0}^{4}\alpha_{n}(A^{2})\mathcal{L}_{n}[\mathcal{K}]\,,$
(13)

with $\mathcal{L}_{n}[\mathcal{K}]$, the elementary Lagrangians

$\displaystyle\mathcal{L}_{0}$ $\displaystyle=4!$ (14)
$\displaystyle\mathcal{L}_{1}$ $\displaystyle=3![\mathcal{K}]$
$\displaystyle\mathcal{L}_{2}$
$\displaystyle=2!([\mathcal{K}]^{2}-[\mathcal{K}^{2}])$
$\displaystyle\mathcal{L}_{3}$
$\displaystyle=[\mathcal{K}]^{3}-3[\mathcal{K}][\mathcal{K}^{2}]+2[\mathcal{K}^{3}]$
$\displaystyle\mathcal{L}_{4}$
$\displaystyle=[\mathcal{K}]^{4}-6[\mathcal{K}]^{2}[\mathcal{K}^{2}]+3[\mathcal{K}^{2}]^{2}+8[\mathcal{K}][\mathcal{K}^{3}]-6[\mathcal{K}^{4}]\,.$

Note that, despite its similar construction to massive gravity, by the
definition (11) of $f_{\mu\nu}$ in terms of a vector field, this model
contains no tensor degrees of freedom. It is a pure vector model, with an
infinite tower of self-interactions. These interactions lead to higher order
equation of motion, such that one may worry about the potential propagation of
an Ostrogradski ghost. However, the authors of derham2020  exhibited a null
eigenvector of the Hessian, implying the presence of a constraint that removes
the ghostly degree of freedom. Therefore, the above construction gives rise to
a self-interacting massive vector theory which propagates only three healthy
degrees of freedom. This theory is called Proca-Nuevo.

Additionally, the Proca-Nuevo theory is inequivalent to the Generalized Proca.
This can be proved explicitly by trying to match the scattering amplitudes of
both theories, as was done in derham2020 . This is not in contradiction with
the uniqueness of GP as the defining hypotheses differ. Indeed, GP is
characterized by having second order equations of motion, while PN violates
this condition. Moreover, the realisation of the constraint is imposed at each
order in GP, whereas all orders are needed for PN to realise it. As a
consequence of this last point, the null eigenvectors of the two theories are
fundamentally different, such that the constraint can not be simultaneously
realised derham2020 . Therefore, a model including both the GP and PN
interactions would suffer from ghost instabilities unless some non-trivial
restrictions are made (at least in the absence of gravity) derham2021cosmology
.

Furthermore, the PN Lagrangian can also be written perturbatively,

$\mathcal{L}_{\text{PN}}=\sum_{n=2}^{\infty}\frac{1}{\Lambda_{2}^{2(n-2)}}\mathcal{L}_{\text{PN}}^{(n)}\,,$
(15)

where the expressions for $\mathcal{L}^{(n)}$ are obtained by expanding the
arbitrary functions $\alpha_{n}$ in power of $A^{2}$

$\alpha_{n}(A^{2})=\bar{\alpha}_{n}+\frac{m^{2}}{\Lambda_{2}^{4}}\bar{\gamma}_{n}A^{2}+\frac{m^{4}}{\Lambda_{2}^{6}}\bar{\lambda}_{n}A^{4}+\dots$
(16)

The dimensions are contained in the scale $\Lambda_{2}$, while the
coefficients are dimensionless. In order to get the usual normalization for
the Maxwell and mass terms, we set
$\bar{\alpha}_{1}=-\frac{1}{3}(1-2\bar{\alpha}_{2})$ and
$\bar{\gamma}_{0}=-\frac{1}{48}$. We observe that the combination
$(1+4\bar{\alpha}_{2}-6\bar{\alpha}_{3})$ appears repeatedly in the results
and thus redefine $\bar{\alpha}_{2}$ accordingly. We also rescale the
coefficients $\bar{\gamma}_{1}$, $\bar{\gamma}_{2}$, and $\bar{\lambda}_{0}$
so that they later compare nicely to the GP parameters $a_{1}$, $b_{3}$, and
$b_{1}$ respectively. In sum, we define the following parameters

$\displaystyle\alpha_{2}^{\prime}=1+4\bar{\alpha}_{2}-6\bar{\alpha}_{3},\quad\alpha_{3}^{\prime}=3(\bar{\alpha}_{3}-4\bar{\alpha}_{4}),$
(17)
$\displaystyle\gamma_{1}^{\prime}=6\bar{\gamma}_{1},\quad\gamma_{2}^{\prime}=2\bar{\gamma}_{2},\quad\lambda_{0}^{\prime}=24\bar{\lambda}_{0}.$

Under this redefinition, the perturbative PN Lagrangian (15) reads

$\displaystyle\mathcal{L}_{\text{PN}}^{(2)}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-\frac{1}{2}m^{2}A^{2}$
(18)
$\displaystyle\mathcal{L}_{\text{PN}}^{(3)}=\gamma_{1}^{\prime}m^{2}A^{2}\partial_{\mu}A^{\mu}+\frac{1}{8}(\alpha_{2}^{\prime}-1)[F^{2}][\partial
A]+\frac{1}{4}(2-\alpha_{2}^{\prime})F^{2}_{\mu\nu}\partial^{\mu}A^{\nu}$
$\displaystyle\mathcal{L}_{\text{PN}}^{(4)}=\lambda_{0}^{\prime}m^{4}A^{4}+m^{2}A^{2}\Big{[}\gamma_{2}^{\prime}[\partial
A]^{2}-\frac{1}{2}\left(\frac{1}{2}\gamma_{1}^{\prime}+\gamma_{2}^{\prime}\right)\partial_{\mu}A_{\nu}\partial^{\nu}A^{\mu}+\frac{1}{2}\left(\frac{1}{2}\gamma_{1}^{\prime}-\gamma_{2}^{\prime}\right)\partial_{\mu}A_{\nu}\partial^{\mu}A^{\nu}\Big{]}$
$\displaystyle\qquad+\frac{1}{128}(-1+\alpha_{2}^{\prime}-2\alpha_{3}^{\prime})[F^{2}]^{2}+\frac{1}{64}(10-5\alpha_{2}^{\prime}-14\alpha_{3}^{\prime})F^{2}_{\mu\nu}F^{2\mu\nu}$
$\displaystyle\qquad+\frac{1}{8}\Big{[}\alpha_{3}^{\prime}[F^{2}]([\partial
A]^{2}-\partial_{\mu}A_{\nu}\partial^{\nu}A^{\mu})+\left(-2+\alpha_{2}^{\prime}+4\alpha_{3}^{\prime}\right)F^{2\mu\nu}\partial^{\alpha}A_{\mu}\partial_{\alpha}A_{\nu}$
$\displaystyle\qquad\qquad+\left(1-\alpha_{2}^{\prime}-4\alpha_{3}^{\prime}\right)F^{2\mu\nu}[\partial
A]\partial_{\mu}A_{\nu}+\left(-2+\alpha_{2}^{\prime}+2\alpha_{3}^{\prime}\right)F^{\mu\nu}F^{\alpha\beta}\partial_{\mu}A_{\alpha}\partial_{\nu}A_{\beta}\Big{]},$

where we used the notation $F^{2}_{\mu\nu}=F_{\mu}^{\ \alpha}F_{\nu\alpha}$
and $[F^{2}]=F^{\mu\nu}F_{\mu\nu}$. Following this reformulation, all of the
combinations of the parameters
$\\{\bar{\alpha}_{2},\bar{\alpha}_{3},\bar{\alpha}_{4}\\}$ in the Lagrangian
reduce to combinations of only
$\\{\alpha_{2}^{\prime},\alpha_{3}^{\prime}\\}$. We mentioned before that the
PN interactions have to be related in a specific way for the constraint to be
respected. This is well illustrated in the perturbative Lagrangian, where the
coefficients do not correspond to a specific interaction, but rather
intertwine them. This is in contrast with GP in (9) where each interaction has
its own coefficient.

Finally, taking the decoupling limit (7) of this theory gives

$\mathcal{L}_{\text{DL
PN}}=\sum_{n=2}^{\infty}\frac{1}{\Lambda_{3}^{3(n-2)}}\mathcal{L}_{\text{DL
PN}}^{(n)}\,,$ (19)

with

$\displaystyle\mathcal{L}_{\text{DL
PN}}^{(2)}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-\frac{1}{2}(\partial\phi)^{2}$
(20) $\displaystyle\mathcal{L}_{\text{DL
PN}}^{(3)}=\gamma_{1}^{\prime}(\partial\phi)^{2}\Box\phi+\frac{1}{8}(\alpha_{2}^{\prime}-1)[F^{2}]\Box\phi+\frac{1}{4}(2-\alpha_{2}^{\prime})F^{2}_{\mu\nu}(\partial^{\mu}\partial^{\nu}\phi)$
$\displaystyle\mathcal{L}_{\text{DL
PN}}^{(4)}=\gamma_{2}^{\prime}(\partial\phi)^{2}[(\Box\phi)^{2}-(\partial_{\mu}\partial_{\nu}\phi)^{2}]$
$\displaystyle\>+\frac{1}{8}\Big{[}\alpha_{3}^{\prime}[F^{2}][(\Box\phi)^{2}-(\partial_{\mu}\partial_{\nu}\phi)^{2}]+\left(-2+\alpha_{2}^{\prime}+4\alpha_{3}^{\prime}\right)F^{2\mu\nu}(\partial^{\alpha}\partial_{\mu}\phi)(\partial_{\alpha}\partial_{\nu}\phi)$
$\displaystyle\>+\left(1-\alpha_{2}^{\prime}-4\alpha_{3}^{\prime}\right)F^{2\mu\nu}\Box\phi(\partial_{\mu}\partial_{\nu}\phi)+\left(-2+\alpha_{2}^{\prime}+2\alpha_{3}^{\prime}\right)F^{\mu\nu}F^{\alpha\beta}(\partial_{\mu}\partial_{\alpha}\phi)(\partial_{\nu}\partial_{\beta}\phi)\Big{]}\,.$

The scalar modes are the $\bar{\gamma}_{1}$ and $\bar{\gamma}_{2}$ terms and
correspond to the cubic and quartic Galileon interactions respectively. The
other terms match the vector-scalar sector of the decoupling limit of massive
gravity derham2020 .

##  3 Positivity Bounds

In this section, we review how the physical assumptions of unitarity,
causality, and Lorentz invariance on the UV completion of an EFT gives rise to
bounds on the 2-2 scattering amplitudes. We consider four identical particles
of mass $m$ and integer spin $S$. The generalization to particles with
distinct mass or spin follows a similar derivation and can be found in
de_Rham_2018 , so does the fermionic case. The amplitudes are expressed in
terms of the Mandelstam variables $(s,t,u)$ Mandelstam1958 , where $s$ is the
center of mass energy, $t$ the momentum transfer, and $u=4m^{2}-s-t$ their
conjugate variable. In particular, $t$ is related to the scattering angle
$\theta$ as $\cos\theta=1+\frac{2t}{s-4m^{2}}$. Further definitions of the
kinematic variables are presented in Appendix A.

A difficulty that arises in deriving bounds for spinning particles comes from
the crossing relations which are trivial in the scalar case, but not in the
spinning case (except in the forward limit $t=0$). Therefore, to derive
spinning bounds beyond the forward limit, it has been suggested in
de_Rham_2018  to introduce a basis, known as transversity basis, which
diagonalizes the spinning crossing relations.

The singularities of the 2-2 amplitudes are well-known: there are simple poles
at the physical mass in the different exchange channels $s,t,u=m^{2}$ and a
branch cut starting at $s=4m^{2}$ corresponding to multi-particle production.
Crossing symmetry between the $s$ and $u$ channels implies that $A(s,t)$ has
two branch cuts on the real $s$-axis, from $s=-\infty$ to $-t$ and $s=4m^{2}$
to $\infty$, which are referred to as the left hand (LH) and right hand (RH)
cuts respectively. The amplitudes are usually assumed to be otherwise analytic
in the whole Mandelstam complex plane Mandelstam1958 . Moreover, the spinning
amplitudes have the same domain of analyticity as the scalar ones MartinSpin ;
Hara64 . Finally, the relation between causality and analyticity has been
established for a long time Bogoliubov:1959bfo ; Bremermann1958 . It is
reviewed in the Appendix A of de_Rham_2018 , and we shall not derive it again.

#### Tree-level

The bounds presented in this section are valid to all orders in loops.
However, in practice, it turns out to be convenient to work at tree-level. The
main difference is that, at tree-level, $\imaginary[A(s,t)]$ can only be non-
zero for $s\geq\Lambda^{2}$, where $\Lambda$ is the mass of the lightest state
outside of the EFT, i.e. the cutoff of the EFT de_Rham_2017 . In this case, we
can take the integration along the cuts to start at $\Lambda^{2}$. In the
following, we write the integrals on the branch cuts to run from $\mu_{b}$ to
$\infty$, where $\mu_{b}=4m^{2}$, but in general, one can take
$\mu_{b}=\Lambda^{2}$ at tree-level.

In this work, we compute the bounds at tree-level. This is justified if we
assume the presence of a weak coupling in the theory, corresponding to the
suppression factor of the loops. As tree and loop effects are mixed in the
bounds, it would make no sense to only consider tree-level amplitudes if they
did not dominate the loop contributions de_Rham_2017gal . We consider weak
coupling, but there are recent works exploring beyond it Bellazzini_2021 ;
Bellazzini2021 .

###  3.1 Transversity Formalism

Here we review the transversity basis introduced in de_Rham_2018 . The total
bosonic333There is an additional change of signs in the crossing relations for
fermions. amplitude for the process $AB\to CD$ associated with the $s$-channel
is related to the corresponding amplitude for $A\bar{D}\to C\bar{B}$ (where
the bar denotes anti-particles) associated with the $u$-channel by a
reordering of the particles de_Rham_2019 ,

$\mathcal{A}^{s}(p_{1},p_{2},p_{3},p_{4})=\mathcal{A}^{u}(p_{1},-p_{4},p_{3},-p_{2})\,.$
(21)

The right hand side (RHS) can be expressed in terms of the usual
$(p_{1},p_{2},p_{3},p_{4})$ configuration via a Lorentz transformation (hence
the need for the hypothesis that the UV-theory is Lorentz invariant). Such a
Lorentz transformation is trivial for scalar amplitudes,
$A^{s}(s,t)=A^{u}(u,t)$, but not for spinning particles.

Figure 1: The difference between the helicity and transversity formalism, as
introduced in de_Rham_2018 . The helicity is the spin projection along the
direction of motion, whereas the transversity is defined as its projection
along the direction transverse to the interaction plane.

Usually, amplitudes are expressed in helicity basis JACOB1959404 . However,
the crossing relations in this basis are not convenient to deal with, except
in the forward limit where positivity bounds for spinning particles have
therefore been derived and used in the helicity formalism Cheung2016 ;
Bellazzini2017 ; Bonifacio_2016 . Away from the forward limit, they are not
diagonal, and not sign-definite Trueman:1964zzb ; Hara:1970gc ; Hara71 , such
that it is difficult to conclude on the positivity along the LH cut in this
formalism, which is a crucial ingredient to derive the positivity bounds.
Therefore, to derive bounds beyond the forward limit, we use the transversity
formalism Kotanski66 , which have simplified crossing relations Kotanski70 .
Helicity polarizations, denoted by $\lambda$, are defined along the momentum
direction, while transversity polarizations, denoted by $\tau$, are defined
along the direction transverse to the interaction plane. This is depicted in
Fig. 1. Amplitudes in the transverity basis, denoted by $\mathcal{T}$, are
related to the ones in helicity basis, denoted by $\mathcal{H}$, by

$\mathcal{T}_{\tau_{1}\tau_{2}\tau_{3}\tau_{4}}=\sum_{\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}}u_{\lambda_{1}\tau_{1}}^{S}u_{\lambda_{2}\tau_{2}}^{S}u_{\lambda_{3}\tau_{3}}^{S*}u_{\lambda_{4}\tau_{4}}^{S*}\mathcal{H}_{\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}}\,,$
(22)

where $u_{\lambda\tau}^{S}=D^{S}_{\lambda\tau}(\pi/2,\pi/2,-\pi/2)$ with
$D^{S}$ a Wigner $D$-matrix wignergruppentheorie . These transversity
amplitudes follow nice crossing relations which further simplifiy for elastic
scattering or in the forward limit.

$\mathcal{T}^{s}_{\tau_{1}\tau_{2}\tau_{3}\tau_{4}}(s,t,u)=e^{i\pi\sum_{i}\tau_{i}}e^{-i\chi_{u}\sum_{i}\tau_{i}}\mathcal{T}^{u}_{-\tau_{1}-\tau_{2}-\tau_{3}-\tau_{4}}(u,t,s)\,,$
(23)

with

$e^{\pm i\chi_{u}}=\frac{-su\mp
2im\sqrt{stu}}{\sqrt{s(s-4m^{2})u(u-4m^{2})}}\,.$ (24)

#### Kinematical singularities

By its definition (24), the factor $e^{i\chi_{u}\sum_{i}\tau_{i}}$ in the
crossing relation (23) introduces additional singularities of order
$\sum_{i}\tau_{i}\leq 4S$. Namely, poles at $s,u=0$, $4m^{2}$ and a branch
point at $\sqrt{stu}=0$. These are kinematical singularities, and it is more
convenient to subtract them.

Let us consider these singularities separately. First, it has been shown in
osti_4534874  that helicity amplitudes are regular at $s=0$. Then, by (22), so
are the transversity amplitudes. Second, the pole at $s=4m^{2}$ can be removed
by multiplying by a factor $\sqrt{s(s-4m^{2})}^{\sum_{i}\tau_{i}}$. In
practice we use the maximal possible value of the exponent, that is $4S$, so
that it works for any configuration of polarizations. Finally, as
$\sqrt{stu}\sim\sin\theta$, we have that $\sqrt{stu}\to-\sqrt{stu}$ under
$\theta\to-\theta$, such that any even function of $\theta$ does not contain
the branch cut. Altogether, these considerations imply that the quantity

$\mathcal{T}_{\tau_{1}\tau_{2}\tau_{3}\tau_{4}}^{+}(s,\theta)=\left(s(s-4m^{2})\right)^{2S}\left(\mathcal{T}_{\tau_{1}\tau_{2}\tau_{3}\tau_{4}}(s,\theta)+\mathcal{T}_{\tau_{1}\tau_{2}\tau_{3}\tau_{4}}(s,-\theta)\right)$
(25)

is free of kinematical singularities. This regularized amplitude plays a
similar role to a scalar amplitude in the derivation of the bounds which
follows.

###  3.2 Spinning Bounds

#### Unitarity and analyticity

The requirement of unitarity imposes, via the optical theorem, that
$\text{Abs}_{s}\mathcal{T}^{+}_{\tau_{1}\tau_{2}}(s,0)$ together with its
$t$-derivatives is positive on the RH cut. Thanks to the crossing relations
(23) of the transversity amplitudes, this consideration can be extended to the
LH cut.444This extension is not trivial and can be found in de_Rham_2018 .
Note that
$\text{Abs}_{s}\mathcal{T}^{+}(s)=\frac{1}{2i}\text{Disc}\mathcal{T}^{+}(s)=\frac{1}{2i}\lim_{\epsilon\to
0}[\mathcal{T}^{+}(s+i\epsilon)-\mathcal{T}^{+}(s-i\epsilon)]$ denotes the
absorptive part of the amplitude, which is equal to the imaginary part if the
theory is time reversal invariant. Next, analyticity allows one to
analytically continue these properties away from $t=0$, such that

$\displaystyle\frac{\partial^{n}}{\partial
t^{n}}\text{Abs}_{s}\mathcal{T}^{+}_{\tau_{1}\tau_{2}}(s,t)>0,\quad\forall\,n\geq
0,\,s\geq 4m^{2},\,0\leq t<m^{2},$ (26)
$\displaystyle\frac{\partial^{n}}{\partial
t^{n}}\text{Abs}_{u}\mathcal{T}^{+}_{\tau_{1}\tau_{2}}(s,t)>0,\quad\forall\,n\geq
0,\,u\geq 4m^{2},\,0\leq t<m^{2}.$

Furthermore, unitarity and analyticity imply the presence of a Froissart bound
Froissart ; Martin1963 , which also exists for spinning particles Hara64  and
can be extended to non zero $t$ JinMartin . With $\epsilon(t)<1$ for $0\leq
t<m^{2}$, it reads

$\lim_{s\to\infty}|\mathcal{T}_{\tau_{1}\tau_{2}\tau_{3}\tau_{4}}(s,t)|<s^{1+\epsilon(t)}\,\implies\,\lim_{s\to\infty}|\mathcal{T}^{+}_{\tau_{1}\tau_{2}\tau_{3}\tau_{4}}(s,t)|<s^{N_{S}}.$
(27)

The implication for $\mathcal{T}^{+}(s,t)$ comes from its definition in (25)
where it has an additional $s^{4S}$ factor compared to $\mathcal{T}(s,t)$,
such that we define

$N_{S}=4S+2.$ (28)

Note that $\mathcal{T}^{+}$ could have been defined with a minimal factor of
$s^{\sum_{i}\tau_{i}}$, in which case $N_{S}=\sum_{i}\tau_{i}+2$ is
sufficient. However, this definition depending on the configuration of
polarizations is not really convenient, especially to study indefinite
transversities, so we shall use the definition (28) which works for any
polarization. It has been argued in de_Rham_2019  that not using the minimal
number of subtractions only leads to small differences and does not change the
qualitative form of the bounds.

#### Dispersion relation

We want to derive a dispersion relation for the regularized amplitude
$\mathcal{T}^{+}(s,t)$ using Cauchy’s theorem. First, we define the pole
subtracted amplitude

$\tilde{\mathcal{T}}^{+}_{\tau_{1}\tau_{2}}(s,t)=\mathcal{T}^{+}_{\tau_{1}\tau_{2}}(s,t)-\frac{\text{Res}\mathcal{T}^{+}_{\tau_{1}\tau_{2}}(s=m^{2},t)}{s-m^{2}}-\frac{\text{Res}\mathcal{T}^{+}_{\tau_{1}\tau_{2}}(s=3m^{2}-t,t)}{s+t-3m^{2}}\,,$
(29)

which is analytic in the whole $s$-complex plane (minus the branch cuts) and
can therefore be expressed via a contour integral, see Fig. 2,

$\tilde{\mathcal{T}}^{+}_{\tau_{1}\tau_{2}}(s,t)=\frac{1}{2\pi
i}\oint_{\mathcal{C}}ds^{\prime}\;\frac{\mathcal{T}^{+}_{\tau_{1}\tau_{2}}(s^{\prime},t)}{(s^{\prime}-s)}\,.$
(30)

Next, deforming the contour to $\mathcal{C}^{\prime}$, the amplitude is given
by the arcs at infinity and the contributions along the cuts. According to
(27), the arc contributions can be dropped by performing $N_{S}$ subtractions.
This introduces subtraction functions $a_{n}(t)$ and additional powers of $s$.
Then, the contributions along the cuts are given by the discontinuity of the
amplitude along the cuts; that is the absorptive part. All in all, we can
express the contour integral as

$\displaystyle\tilde{\mathcal{T}}^{+}_{\tau_{1}\tau_{2}}(s,t)=\sum_{n=0}^{N_{S}-1}a_{n}(t)s^{n}$
$\displaystyle+\frac{s^{N_{S}}}{\pi}\int_{\mu_{b}}^{\infty}d\mu\frac{{\text{Abs}}_{s}\mathcal{T}^{+}_{\tau_{1}\tau_{2}}(\mu,t)}{\mu^{N_{S}}(\mu-s)}$
(31)
$\displaystyle+\frac{u^{N_{S}}}{\pi}\int_{\mu_{b}}^{\infty}d\mu\frac{{\text{Abs}}_{u}\mathcal{T}^{+}_{\tau_{1}\tau_{2}}(4m^{2}-t-\mu,t)}{\mu^{N_{S}}(\mu-u)}\,.$

Figure 2: Contours in the complex $s$-plane. The initial contour $\mathcal{C}$
encloses the $s$\- and $u$-channel poles situated at $s=m^{2}$ and
$s=3m^{2}-t$ respectively. The branch cuts, from multi-particle production,
are along the real axis $s\geq 4m^{2}$, and by crossing symmetry, $s\leq-t$.
The deformed contour $\mathcal{C}^{\prime}$ goes around them and to
$|s|\to\infty$. The red and blue regions represent the regions of validity of
the EFT and UV-theory respectively.

The dispersion relation (31) allows us to derive positivity constraints on the
derivatives of $\tilde{\mathcal{T}}^{+}_{\tau_{1}\tau_{2}}(s,t)$, using (26).
Indeed, defining

$f_{\tau_{1}\tau_{2}}(v,t)=\frac{1}{N_{S}!}\frac{\partial^{N_{S}}}{\partial
s^{N_{S}}}\tilde{\mathcal{T}}^{+}_{\tau_{1}\tau_{2}}(s,t)\Big{|}_{s=v+2m^{2}-t/2}\,,$
(32)

where the $N_{S}$ derivatives get rid of the subtraction functions, the
following quantities are positive

$\displaystyle\partial_{v}^{2N}f_{\tau_{1}\tau_{2}}(v,t)>0\quad\forall\,N\geq
0,$ (33)
$\displaystyle\partial_{t}f_{\tau_{1}\tau_{2}}(v,t)+\frac{N_{S}+1}{2\mathcal{M}^{2}}f_{\tau_{1}\tau_{2}}(v,t)>0\,,$

where at tree level,

$|v|<\Lambda^{2},\;0\leq t<m^{2},\;\mathcal{M}^{2}=\Lambda^{2}\,.$ (34)

In particular, the tree-level bounds can be used to obtain constraints on the
EFT cutoff $\Lambda$, as performed in e.g. de_Rham_2017 . The details of the
derivation of (33) can be found in Appendix B.

#### Indefinite transversity bounds

We can also consider initial and final particles that are not polarized in a
definite transversity direction, but rather in some superpositions of
transversity polarizations. The elastic scattering amplitudes for these
indefinite transversity states are computed as

$\mathcal{T}_{\alpha\beta}(v,t)=\sum_{{\tau_{1}\tau_{2}\tau_{3}\tau_{4}}}\alpha_{\tau_{1}}\beta_{\tau_{2}}\alpha^{*}_{\tau_{3}}\beta^{*}_{\tau_{4}}\mathcal{T}_{\tau_{1}\tau_{2}\tau_{3}\tau_{4}}(v,t)\,,$
(35)

where $\alpha$ and $\beta$ are generic unit vectors. For given indefinite
polarizations, their components are the projection of the polarization along
the definite transversity polarization vectors. Next, we define the indefinite
polarization generalization of $f_{\tau_{1}\tau_{2}}$ as

$f_{\alpha\beta}(v,t)=\sum_{{\tau_{1}\tau_{2}\tau_{3}\tau_{4}}}\alpha_{\tau_{1}}\beta_{\tau_{2}}\alpha^{*}_{\tau_{3}}\beta^{*}_{\tau_{4}}f_{\tau_{1}\tau_{2}\tau_{3}\tau_{4}}(v,t)\,,$
(36)

where $f_{\tau_{1}\tau_{2}\tau_{3}\tau_{4}}$ is the inelastic generalization
of $f_{\tau_{1}\tau_{2}}$, i.e. it is computed by (32) for
$\tilde{\mathcal{T}}_{\tau_{1}\tau_{2}\tau_{3}\tau_{4}}$. The arguments we
used to prove the positivity of $f_{\tau_{1}\tau_{2}}$, and its
$v$-derivatives, are still valid for $v=0$ and in the forward limit $t=0$ (see
the Appendix A of de_Rham_2019 )555It is an interesting question whether the
condition $v=0$ may be relaxed.

$\partial_{v}^{2N}f_{\alpha\beta}(0,0)>0\quad\forall\quad N\geq 0\,.$ (37)

These bounds hold for indefinite transversities. Therefore, they also hold for
definite and indefinite helicities. Finally, note that it is sufficient to
study $f_{\alpha\beta}$ as it contains all the definite polarization
quantities: $f_{\tau_{1}\tau_{2}}$ corresponds to the
$\mathcal{O}(\alpha_{\tau_{1}}^{2}\beta_{\tau_{2}}^{2})$ terms.

##  4 Bounds on Proca Theories

In the following, we obtain some linear $f$-quantities for the vector theories
we study. We can express them as

$\displaystyle f_{\tau_{1}\tau_{2}}(v,t)$
$\displaystyle=f_{\tau_{1}\tau_{2}}(0,0)+\partial_{t}f_{\tau_{1}\tau_{2}}\cdot
t\equiv\frac{1}{\Lambda_{2}^{4}}\left[\mu_{\tau_{1}\tau_{2}}+\lambda_{\tau_{1}\tau_{2}}\frac{t}{m^{2}}\right],$
(38)

with the $\mu$ and $\lambda$ being some combinations of the EFT parameters.
There are four independent quantities for the definite elastic polarizations,
denoted

$f_{SS}\equiv f_{00}\,,\quad f_{SV}\equiv f_{0\pm 1}\,,\quad f_{V+}\equiv
f_{\pm 1\pm 1}\,,\quad f_{V-}\equiv f_{\pm 1\mp 1}\,.$ (39)

We can also write $f_{\alpha\beta}(0,0)$ as

$\displaystyle f_{\alpha\beta}(0,0)=\frac{1}{\Lambda_{2}^{4}}\Big{[}$
$\displaystyle\tilde{\mu}_{1}|\alpha_{+}|^{2}|\beta_{+}|^{2}$ (40)
$\displaystyle+$
$\displaystyle\tilde{\mu}_{2}\big{[}|\alpha_{+}|^{2}(1-|\beta_{+}|^{2})+|\beta_{+}|^{2}(1-|\alpha_{+}|^{2})\big{]}$
$\displaystyle+$
$\displaystyle\tilde{\mu}_{3}\big{[}|\alpha_{-}|^{2}|\beta_{0}|^{2}+|\alpha_{0}|^{2}|\beta_{-}|^{2}\big{]}$
$\displaystyle+$
$\displaystyle\tilde{\mu}_{4}\big{[}|\alpha_{-}|^{2}|\beta_{-}|^{2}+|\alpha_{0}|^{2}|\beta_{0}|^{2}\big{]}$
$\displaystyle+$ $\displaystyle
2(\tilde{\mu}_{3}-\tilde{\mu}_{4})\real(\alpha_{-}\alpha_{0}^{*})\real(\beta_{-}\beta_{0}^{*})$
$\displaystyle+$
$\displaystyle\tilde{\mu}_{5}\big{[}\real(\alpha_{-}\alpha_{+}^{*})\real(\beta_{-}\beta_{+}^{*})-\real(\alpha_{0}\alpha_{+}^{*})\real(\beta_{0}\beta_{+}^{*})\big{]}\Big{]}>0\,.$

Then, the bounds from the previous section translate into a set of 10
independent bounds:

$\displaystyle\tilde{\mu}_{1}>0\,,\quad\tilde{\mu}_{2}>0\,,\quad\tilde{\mu}_{3}>0\,,$
(41) $\displaystyle\mu_{SS}>0\,,\quad\mu_{V+}>0\,,\quad\mu_{V-}>0\,,$
$\displaystyle\lambda_{SS}\geq 0\,,\quad\lambda_{SV}\geq
0\,,\quad\lambda_{V+}\geq 0\,,\quad\lambda_{V-}\geq 0\,.$

The formal derivation of this statement is presented in Appendix C. The
$\mu$’s are bounds in the forward limit; those with a ∼ come from indefinite
polarizations. The $\lambda$ bounds correspond to the $t$-derivative bounds,
which are available thanks to the analysis beyond the forward limit. We shall
now compute these coefficients explicitly for the two types of Proca theories.

###  4.1 Generalized Proca

For convenience, we recall the perturbative interacting Lagrangian (5) for
Generalized Proca

$\displaystyle\mathcal{L}_{\text{GP}}^{(3)}$
$\displaystyle=a_{1}m^{2}A^{2}\partial_{\mu}A^{\mu}+a_{2}\tilde{F}^{\mu\alpha}\tilde{F}^{\nu}_{\
\alpha}\partial_{\mu}A_{\nu}$ (42)
$\displaystyle\mathcal{L}_{\text{GP}}^{(4)}$
$\displaystyle=b_{1}m^{4}A^{4}+b_{2}m^{2}A^{2}F_{\mu\nu}^{2}+b_{3}m^{2}A^{2}[(\partial\cdot
A)^{2}-\partial_{\mu}A_{\nu}\partial^{\nu}A^{\mu}]+b_{4}m^{2}A_{\mu}A^{\nu}F^{\alpha\mu}F_{\alpha\nu}$
$\displaystyle\,+b_{5}F^{\mu\nu}F^{\alpha\beta}F_{\mu\alpha}F_{\nu\beta}+b_{6}(F_{\mu\nu}^{2})^{2}+b_{7}\tilde{F}^{\alpha\beta}\tilde{F}^{\mu\nu}\partial_{\alpha}A_{\mu}\partial_{\beta}A_{\nu}\,.$

Computing the amplitudes from this Lagrangian, we obtain
$f_{\tau_{1}\tau_{2}}$ with the following coefficients, which must be
positive,

$\displaystyle\mu_{SS}=$ $\displaystyle
8[2b_{5}+4b_{6}+a_{2}(a_{1}+\frac{1}{4}a_{2})]>0,\qquad\mu_{SV}=b_{4}+4b_{5}-\frac{1}{2}a_{2}^{2}>0,$
(43) $\displaystyle\mu_{V+}=$ $\displaystyle
2[2b_{5}+4b_{6}+a_{2}(a_{1}+\frac{1}{4}a_{2})+a_{1}^{2}+b_{1}-2b_{2}+b_{3}]>0,$
$\displaystyle\mu_{V-}=$ $\displaystyle
2[2b_{5}+4b_{6}+a_{2}(a_{1}-\frac{1}{4}a_{2})-3a_{1}^{2}+b_{1}+2b_{2}-b_{3}+b_{4}]>0,$

and

$\displaystyle\lambda_{SS}=$ $\displaystyle\frac{3}{2}a_{2}^{2}\geq
0,\qquad\lambda_{SV}=\frac{1}{4}[3a_{2}^{2}-4b_{7}]\geq 0,$ (44)
$\displaystyle\lambda_{V+}=$
$\displaystyle\frac{3}{2}[b_{3}+a_{1}^{2}+a_{2}(a_{1}+\frac{1}{4}a_{2})]\geq
0,$ $\displaystyle\lambda_{V-}=$
$\displaystyle\frac{3}{2}[b_{3}+a_{1}^{2}-a_{2}(a_{1}+\frac{1}{12}a_{2})]\geq
0.$

The indefinite polarizations in the forward limit $f_{\alpha\beta}(0,0)$ are
of the form given by (40) with

$\displaystyle\tilde{\mu}_{1}$
$\displaystyle=8[b_{1}-a_{1}^{2}]>0,\qquad\tilde{\mu}_{2}=2b_{4}-a_{2}^{2}>0,$
(45) $\displaystyle\tilde{\mu}_{3}$
$\displaystyle=8b_{5}>0,\qquad\tilde{\mu}_{4}=8[2b_{5}+4b_{6}+a_{2}(a_{1}+\frac{1}{4}a_{2})]>0,$
$\displaystyle\tilde{\mu}_{5}$
$\displaystyle=-4[4b_{2}+b_{4}-2b_{3}-\frac{1}{2}a_{2}^{2}-4a_{1}^{2}]\,.$

Note that, as mentioned in (101), the constraint on $\tilde{\mu}_{5}$ is
encoded in $\mu_{V\pm}$. Additionally, $\mu_{SS}=\tilde{\mu}_{4}$ and
$\mu_{SV}\propto\tilde{\mu}_{2}+\tilde{\mu}_{3}$, so some of the previous
bounds are redundant.

To summarize, the constraints on the EFT coefficients from the bounds (41) for
Generalized Proca are given by

$\displaystyle b_{1}>a_{1}^{2},\quad 2b_{4}>a_{2}^{2},\quad b_{5}>0,$ (46)
$\displaystyle
b_{3}+a_{1}^{2}\geq\text{Max}\left[-a_{2}(a_{1}+\frac{1}{4}a_{2}),a_{2}(a_{1}+\frac{1}{12}a_{2})\right],\quad
4b_{7}\leq 3a_{2}^{2},$ $\displaystyle
2b_{5}+4b_{6}+a_{2}(a_{1}+\frac{1}{4}a_{2})>0,$ $\displaystyle
2b_{5}+4b_{6}+a_{2}(a_{1}+\frac{1}{4}a_{2})+a_{1}^{2}+b_{1}-2b_{2}+b_{3}>0,$
$\displaystyle
2b_{5}+4b_{6}+a_{2}(a_{1}-\frac{1}{4}a_{2})-3a_{1}^{2}+b_{1}+2b_{2}-b_{3}+b_{4}>0,$

where the first line shows the $\tilde{\mu}$ bounds, the second line the
$\lambda$ bounds (without the trivial $\lambda_{SS}$), and the rest are the
$\mu$ bounds. These results are in agreement with those of de_Rham_2019  (and
Bonifacio_2016 666Note that the $S$ and $V$ denote helicity polarizations in
their results, rather than transversity polarizations. in the forward limit)
up to some parameter redefinitions. We also have additional constraints
arising from the interactions parameterized by $a_{2}$ and $b_{7}$, which were
not previously considered.

Finally, although only $f_{\alpha\beta}(0,0)$ is relevant for the bounds we
consider, the full $f_{\alpha\beta}(v,t)$ contains a linear $t$ and $v$
dependence that is worth commenting on. In particular, the $v$ contribution
gather all of the $s^{3}$ terms in the amplitudes (every polarization is
included in $f_{\alpha\beta}$) and therefore indicates the scale at which
unitarity is perturbatively broken. For the Generalized Proca this
contributaion is given by

$\displaystyle
f_{\alpha\beta}^{v}=\frac{14}{\Lambda_{2}^{4}m^{2}}a_{2}\Big{[}$
$\displaystyle
a_{2}\imaginary(\alpha_{-}\alpha_{0}^{*})\imaginary(\beta_{0}\beta_{-}^{*})$
(47)
$\displaystyle+(2a_{1}+a_{2})[\imaginary(\alpha_{+}\alpha_{0}^{*})\imaginary(\beta_{0}\beta_{+}^{*})-\imaginary(\alpha_{-}\alpha_{+}^{*})\imaginary(\beta_{+}\beta_{-}^{*})]\Big{]}.$

Then, perturbative unitarity breaks at
$s^{3}\sim\Lambda_{2}^{4}m^{2}\sim\Lambda_{3}^{6}$, which confirms the
existence of non-trivial operators at the scale $\Lambda_{3}$. What is
interesting here is the overall $a_{2}$ coefficient which means that, for any
polarizations, the unitarity breaking term is parametrized by $a_{2}$.
Therefore, setting $a_{2}=0$ raises the cutoff of the model.

#### Some examples

  * •

First, in addition to raising the cutoff scale, setting $a_{2}=0$ greatly
simplifies the expression of the bounds. It (along with the $b_{7}$
interaction term) generates scalar-vector mixing term in the decoupling limit
(9), and, as seen from (47). The bounds (46) for $a_{2}=0$ are given by

$\displaystyle b_{1}>a_{1}^{2},\quad b_{4}>0,\quad b_{5}>0,\quad b_{3}\geq-
a_{1}^{2},\quad b_{7}\leq 0,\quad 2b_{5}+4b_{6}>0,$ (48) $\displaystyle
b_{1}-2b_{2}+b_{3}+a_{1}^{2}+2b_{5}+4b_{6}>0,\quad
b_{1}+2b_{2}-b_{3}+b_{4}-3a_{1}^{2}+2b_{5}+4b_{6}>0.$

Hence, having the dimension-6 operator represented by $a_{2}$ weakens the
positivity constraints. This seems to be aligned with the findings in
Zhang:2018shp .

  * •

Additionally, $b_{7}$ only appears in the beyond forward limit bound
$\lambda_{SV}\geq 0$. Note here the importance of the beyond forward limit
analysis. Without it $b_{7}$ would remain unconstrained. Therefore, if we
consider the case with $b_{7}=0$ (i.e. canceling the other scalar-vector
mixing term than the previous example), we get the bounds (46) but without the
constraint $4b_{7}\leq 3a_{2}^{2}$ which trivializes. Similarly, setting both
$a_{2}$ and $b_{7}$ to 0 gives again (48), but without the constraint
$b_{7}\leq 0$. In other words, the presence or absence of $b_{7}$ does not
influence the constraints on the other parameters.

  * •

Then, studying the simplest vector Galileon model, keeping only the
interactions in $a_{1}$ and $b_{3}$ leads to the inconsistent bounds

$3a_{1}^{2}<-b_{3}\leq a_{1}^{2}.$ (49)

As already noted in de_Rham_2019 , such a theory admits no standard UV
completion. In this work, they indicated that other interactions should be
included in order to obtain a window of parameters satisfying the positivity
bounds. Here, our analysis on indefinite polarizations provides a more precise
result on which interactions to include. The bounds for any model containing
the cubic vector Galileon can be satisfied by adding the purely quartic
interaction parametrized by $b_{1}$, which must be positive due to the
condition $b_{1}>a_{1}^{2}$, coming from $\tilde{\mu}_{1}$. This minimal model
is given by the Lagrangian

$\displaystyle\mathcal{L}=$
$\displaystyle-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-\frac{1}{2}m^{2}A^{2}+\frac{1}{\Lambda_{2}^{2}}a_{1}m^{2}A^{2}\partial_{\mu}A^{\mu}$
(50) $\displaystyle+\frac{1}{\Lambda_{2}^{4}}b_{3}m^{2}A^{2}[(\partial\cdot
A)^{2}-\partial_{\mu}A_{\nu}\partial^{\nu}A^{\mu}]+\frac{1}{\Lambda_{2}^{4}}b_{1}m^{4}A^{4}\,,$

with the bounds reducing to

$\displaystyle 3a_{1}^{2}-b_{1}<-b_{3}\leq a_{1}^{2}.$ (51)

The corresponding window of parameter is pictured on figure 3. Note that the
bounds on $\tilde{\mu}_{1}$ and $\mu_{V+}$ ($b_{1}>a_{1}^{2}$ and
$b_{1}+b_{3}+a_{1}^{2}>0$) are also non-trivial, but are automatically
fulfilled when (51) is respected.

Figure 3: Parameter space of the simplest vector Galileon. The region excluded
by $\lambda_{V\pm}$ is indicated in blue. The bound on $\mu_{V-}$ is pictured
for different values of the pure quartic coupling $b_{1}$. The allowed region
increases with $b_{1}$ and would vanish for $b_{1}=0$.

  * •

If we only consider a model containing the interaction terms leading to
scalar-vector mixing in the decoupling limit, with coefficients $a_{2}$ and
$b_{7}$, then, the bounds cannot be satisfied. The positivity bound,
$\tilde{\mu}_{2}>0$ requires the $b_{4}$ interaction to also be included, with
the constraints

$2b_{4}>a_{2}^{2},\quad 4b_{7}\leq 3a_{2}^{2}.$ (52)

With only these interactions $0\leq\lambda_{V-}=-a_{2}^{2}/8$ is violated.
Hence, we need to also add either $a_{1}$ (and therefore $b_{1}$) or $b_{3}$,
i.e. again the vector Galileon interactions. The minimal such model would
therefore contain $\\{a_{2},b_{3},b_{4},b_{7}\\}$ and be constrained by the
bounds

$\frac{a_{2}^{2}}{12}\leq b_{3}<b_{4}-\frac{a_{2}^{2}}{4}.$ (53)

  * •

Finally, by the previous considerations, the simplest model including both the
vector Galileon $\\{a_{1},b_{3}\\}$ and mixing interactions
$\\{a_{2},b_{7}\\}$ should also contain the quartic interactions with
coefficients $b_{1}$ (to satisfy $\tilde{\mu}_{1}>0$) and $b_{4}$ (to satisfy
$\tilde{\mu}_{2}>0$). The bounds for such a model correspond to setting
$b_{2},b_{5},b_{6}=0$ in (46). Note that we could also consider only the
mixing term $a_{2}$, setting $b_{7}=0$, which would give the same bounds.

  * •

Similarly, the simplest vector Galileon model with $\\{a_{1},b_{3}\\}$ and
only the mixing term $b_{7}$ should contain $b_{1}$ as well, and the
constraints would be given by (51) plus $b_{7}\leq 0$.

###  4.2 Proca-Nuevo

First, we recall the perturbative Proca-Nuevo Lagrangian, whose coefficients
we want to constrain.

$\displaystyle\mathcal{L}_{\text{PN}}^{(3)}=\gamma_{1}^{\prime}m^{2}A^{2}\partial_{\mu}A^{\mu}+\frac{1}{8}(\alpha_{2}^{\prime}-1)[F^{2}][\partial
A]+\frac{1}{4}(2-\alpha_{2}^{\prime})F^{2}_{\mu\nu}\partial^{\mu}A^{\nu}$ (54)
$\displaystyle\mathcal{L}_{\text{PN}}^{(4)}=\lambda_{0}^{\prime}m^{4}A^{4}+m^{2}A^{2}\Big{[}\gamma_{2}^{\prime}[\partial
A]^{2}-\frac{1}{2}\left(\frac{1}{2}\gamma_{1}^{\prime}+\gamma_{2}^{\prime}\right)\partial_{\mu}A_{\nu}\partial^{\nu}A^{\mu}+\frac{1}{2}\left(\frac{1}{2}\gamma_{1}^{\prime}-\gamma_{2}^{\prime}\right)\partial_{\mu}A_{\nu}\partial^{\mu}A^{\nu}\Big{]}$
$\displaystyle\qquad+\frac{1}{128}(-1+\alpha_{2}^{\prime}-2\alpha_{3}^{\prime})[F^{2}]^{2}+\frac{1}{64}(10-5\alpha_{2}^{\prime}-14\alpha_{3}^{\prime})F^{2}_{\mu\nu}F^{2\mu\nu}$
$\displaystyle\qquad+\frac{1}{8}\Big{[}\alpha_{3}^{\prime}[F^{2}]([\partial
A]^{2}-\partial_{\mu}A_{\nu}\partial^{\nu}A^{\mu})+\left(-2+\alpha_{2}^{\prime}+4\alpha_{3}^{\prime}\right)F^{2\mu\nu}\partial^{\alpha}A_{\mu}\partial_{\alpha}A_{\nu}$
$\displaystyle\qquad\qquad+\left(1-\alpha_{2}^{\prime}-4\alpha_{3}^{\prime}\right)F^{2\mu\nu}[\partial
A]\partial_{\mu}A_{\nu}+\left(-2+\alpha_{2}^{\prime}+2\alpha_{3}^{\prime}\right)F^{\mu\nu}F^{\alpha\beta}\partial_{\mu}A_{\alpha}\partial_{\nu}A_{\beta}\Big{]}.$

In contrast with the GP model, the PN parameters do not correspond to a
specific interaction (except for $\lambda_{0}^{\prime}$), but rather appear in
various combinations to parametrize several interactions. It is due to the
special tuning required to have a ghost-free theory despite higher order EOM.
However, note that $\lambda_{0}^{\prime}$ parametrizes a purely quartic
interaction that vanishes in the decoupling limit, $\gamma_{1}^{\prime}$ and
$\gamma_{2}^{\prime}$ give rise to the cubic and quartic Galileons
interactions respectively in the decoupling limit, and the interactions
entering with $\alpha$’s give rise to scalar-vector mixing terms in the
decoupling limit.

Then, the $f_{\tau_{1}\tau_{2}}(v,t)$ is again of the form (38) with
coefficients that must satisfy the positivity bounds. At constant order the
bounds are given by

$\displaystyle\mu_{SS}=\frac{1}{8}[2+\alpha_{2}^{\prime}(\alpha_{2}^{\prime}-4(1+4\gamma_{1}^{\prime}))]>0,\quad\mu_{SV}=\frac{1}{32}[8-2\alpha_{2}^{\prime}-\alpha_{2}^{\prime
2}+4\alpha_{3}^{\prime}]>0,$ (55)
$\displaystyle\mu_{V+}=\frac{1}{32}[10+\alpha_{2}^{\prime}(\alpha_{2}^{\prime}-4(1+4\gamma_{1}^{\prime}))-16\gamma_{1}^{\prime}(1-4\gamma_{1}^{\prime})+96\gamma_{2}^{\prime}+64\lambda_{0}^{\prime}]>0,$
$\displaystyle\mu_{V-}=\frac{1}{32}[2-\alpha_{2}^{\prime}(\alpha_{2}^{\prime}+4(1+4\gamma_{1}^{\prime}))-48\gamma_{1}^{\prime}(1+4\gamma_{1}^{\prime})-96\gamma_{2}^{\prime}+64\lambda_{0}^{\prime}]>0,$

and at linear $t$ order they are given by

$\displaystyle\lambda_{SS}=\frac{3}{32}\alpha_{2}^{\prime 2}\geq
0,\quad\lambda_{SV}=\frac{1}{64}[3\alpha_{2}^{\prime
2}-8\alpha_{2}^{\prime}+16\alpha_{3}^{\prime}+4]\geq 0,$ (56)
$\displaystyle\lambda_{V+}=\frac{3}{32}[16\gamma_{2}^{\prime}+16\gamma_{1}^{\prime
2}-\alpha_{2}^{\prime}(1+4\gamma_{1}^{\prime}-\frac{1}{4}\alpha_{2}^{\prime})+2]\geq
0,$
$\displaystyle\lambda_{V-}=\frac{3}{32}[16\gamma_{2}^{\prime}+16\gamma_{1}^{\prime
2}+\alpha_{2}^{\prime}(1+4\gamma_{1}^{\prime}-\frac{1}{12}\alpha_{2}^{\prime})-\frac{2}{3}]\geq
0.$

Additionally, the indefinite polarizations quantity $f_{\alpha\beta}(0,0)$ is
given by (40) with

$\displaystyle\tilde{\mu}_{1}=4[2\lambda_{0}^{\prime}-\gamma_{1}^{\prime}(1+2\gamma_{1}^{\prime})]>0,\quad\tilde{\mu}_{2}=\frac{1}{16}[4-\alpha_{2}^{\prime
2}]>0,$ (57)
$\displaystyle\tilde{\mu}_{3}=\frac{1}{8}[2-\alpha_{2}^{\prime}+2\alpha_{3}^{\prime}]>0,\quad\tilde{\mu}_{4}=\frac{1}{8}[2+\alpha_{2}^{\prime}(\alpha_{2}^{\prime}-4(1+4\gamma_{1}^{\prime}))]>0,$
$\displaystyle\tilde{\mu}_{5}=\frac{1}{8}[\alpha_{2}^{\prime
2}+16\gamma_{1}^{\prime}(1+8\gamma_{1}^{\prime})+96\gamma_{2}^{\prime}+4].$

Before analyzing the bounds, note that the contribution to $f_{\alpha\beta}$
linear in $v$, which corresponds to the $s^{3}$ terms in all of the
amplitudes, is given by

$\displaystyle
f_{\alpha\beta}^{v}=\frac{7}{8\Lambda_{2}^{4}m^{2}}\alpha_{2}^{\prime}\Big{[}$
$\displaystyle\alpha_{2}^{\prime}\imaginary(\alpha_{-}\alpha_{0}^{*})\imaginary(\beta_{0}\beta_{-}^{*})$
(58)
$\displaystyle+(\alpha_{2}^{\prime}-8\gamma_{1}^{\prime}-2)[\imaginary(\alpha_{+}\alpha_{0}^{*})\imaginary(\beta_{0}\beta_{+}^{*})-\imaginary(\alpha_{-}\alpha_{+}^{*})\imaginary(\beta_{+}\beta_{-}^{*})]\Big{]}.$

It has exactly the same structure as the GP contribution (47). Again, this
means that perturbative unitarity breaks at scale $\Lambda_{3}$. The overall
coefficient of these terms is now $\alpha_{2}^{\prime}$. Therefore, the tuning
$\alpha_{2}^{\prime}=0$ plays a special role as it raises the cutoff. Unlike
for GP, this tuning does not cancel a particular interaction term in the
Lagrangian, but rather relates the interactions in a specific way.

The positivity constraints (55)$-$(57) are highly redundant. They reduce to
only 5 independent bounds, which are given by

$\displaystyle\tilde{\mu}_{2}>0:$ $\displaystyle-2<\alpha_{2}^{\prime}<2\,,$
(59) $\displaystyle\lambda_{SV}\geq 0:$
$\displaystyle\alpha_{3}^{\prime}\geq\frac{1}{16}\left(-3\alpha_{2}^{\prime
2}+8\alpha_{2}^{\prime}-4\right)\,,$ $\displaystyle\mu_{SS}>0:$
$\displaystyle\alpha_{2}^{\prime}\gamma_{1}^{\prime}<\frac{1}{16}\left(\alpha_{2}^{\prime
2}-4\alpha_{2}^{\prime}+2\right)\,,$ $\displaystyle\lambda_{V-}\geq 0:$
$\displaystyle\gamma_{2}^{\prime}\geq-\gamma_{1}^{\prime
2}+\frac{1}{192}\alpha_{2}^{\prime}(\alpha_{2}^{\prime}-48\gamma_{1}^{\prime}-12)+\frac{1}{24}\,,$
$\displaystyle\mu_{V-}>0:$
$\displaystyle\lambda_{0}^{\prime}-\frac{3}{2}\gamma_{2}^{\prime}>3\gamma_{1}^{\prime
2}+\frac{3}{4}\gamma_{1}^{\prime}+\frac{1}{64}\alpha_{2}^{\prime}(\alpha_{2}^{\prime}+16\gamma_{1}^{\prime}+4)-\frac{1}{32}\,.$

Both the analysis of indefinite polarizations, via $\tilde{\mu}_{2}$, and
beyond forward limit, via $\lambda_{SV}$ and $\lambda_{V-}$, play an important
role in constraining the model. In the following, we comment on the
constraints (59) separately and show them graphically on figures
4$-$LABEL:fig.grid, where the region excluded by each bound is depicted in its
own color and the allowed region remains white.

First, we note that the particular case $\alpha_{2}^{\prime}=0$ greatly
simplifies the bounds. Even if this is not necessarily manifest from the
Lagrangians, $\alpha_{2}^{\prime}$ appears to play a similar role to $a_{2}$
from the GP model, both in the bounds and in the breaking of perturbative
unitarity. Setting $\alpha_{2}^{\prime}=0$, the bounds on $\tilde{\mu}_{2}$
and $\lambda_{SV}$ are trivially respected, and the other ones simplify to

$\displaystyle\lambda_{SV}\geq 0:$
$\displaystyle\alpha_{3}^{\prime}\geq-\frac{1}{4}\,,$ (60)
$\displaystyle\lambda_{V-}\geq 0:$
$\displaystyle\gamma_{2}^{\prime}\geq-\gamma_{1}^{\prime 2}+\frac{1}{24}\,,$
$\displaystyle\mu_{V-}>0:$
$\displaystyle\lambda_{0}^{\prime}-\frac{3}{2}\gamma_{2}^{\prime}>3\gamma_{1}^{\prime
2}+\frac{3}{4}\gamma_{1}-\frac{1}{32}\,.$

Figure 4: Region of the $\\{\alpha_{2}^{\prime},\alpha_{3}^{\prime}\\}$
parameter space allowed by the positivity bounds.
Figure 5: Region of the $\\{\alpha_{2}^{\prime},\gamma_{1}^{\prime}\\}$
parameter space allowed by $\tilde{\mu}_{2}$ and $\mu_{SS}$.

Now, back to the general $\alpha_{2}^{\prime}$ case, the first bound of (59),
$\tilde{\mu}_{2}>0$ restricts one of the parameter, $\alpha_{2}^{\prime}$, in
a $\mathcal{O}(1)$ interval, independently of the rest of the model. In
particular, the coefficients of the interactions $[F^{2}][\partial A]$ and
$F^{2}_{\mu\nu}\partial^{\mu}A^{\nu}$ in $\mathcal{L}_{\text{PN}}^{(3)}$ are
constrained to lie in the intervals $(-3/8,1/8)$ and $(0,1)$ respectively.

Then, $\alpha_{3}^{\prime}$ only appears in the second bound,
$\lambda_{SV}>0$, together with $\alpha_{2}^{\prime}$. The allowed region for
$\\{\alpha_{2}^{\prime},\alpha_{3}^{\prime}\\}$ is therefore entirely
determined by the first two bounds and is shown in figure 4. Although neither
of these coefficients corresponds to a specific interaction term by itself,
together they parametrize the last three lines of
$\mathcal{L}_{\text{PN}}^{(4)}$, as can be seen in (54).

Next, $\mu_{SS}>0$ relates $\gamma_{1}^{\prime}$ to $\alpha_{2}^{\prime}$ as
shown in figure 5. This region corresponds to

$\displaystyle\begin{cases}\gamma_{1}^{\prime}>F(\alpha_{2}^{\prime})\quad&-2<\alpha_{2}^{\prime}<0,\\\
\gamma_{1}^{\prime}\in\mathbb{R}&\alpha_{2}^{\prime}=0,\\\
\gamma_{1}^{\prime}<F(\alpha_{2}^{\prime})\quad&0<\alpha_{2}^{\prime}<2,\end{cases}\qquad
F(\alpha_{2}^{\prime})=\frac{1}{16}\frac{\alpha_{2}^{\prime
2}-4\alpha_{2}^{\prime}+2}{\alpha_{2}^{\prime}}\,.$ (61)

However, the coefficient $\gamma_{1}^{\prime}$ also appears in the remaining
bounds. Adding these constraints in figure 5 would reduce the allowed region.

Finally, $\gamma_{2}^{\prime}$ and $\lambda_{0}^{\prime}$ are successively
introduced in $\lambda_{V-}$ and $\mu_{V-}$. These constraints are considered
together with those on $\alpha_{2}^{\prime}$ and $\gamma_{1}$. These bounds
are shown both in the $\\{\gamma_{1}^{\prime},\gamma_{2}^{\prime}\\}$ and
$\\{\gamma_{1}^{\prime},\lambda_{0}^{\prime}\\}$ planes for various values of
the other parameters in the top and bottom lines of figure LABEL:fig.grid
respectively. These allow for a visualization of the influence of different
parameters on the allowed regions of the parameter space, shown in white. In
particular, we pick $\alpha_{2}^{\prime}$ to be successively negative, zero,
and positive. Regarding the top line, the islands are reproduced in figure 6
for various values of $\lambda_{0}^{\prime}$. We observe that the islands
shrink for decreasing values of $\lambda_{0}^{\prime}$, with no allowed region
for $\lambda_{0}^{\prime}<-0.0624$. This bound on $\lambda_{0}^{\prime}$,
which parametrizes the quartic interaction $A^{4}$ is similar to the bound on
$b_{1}$ in GP, which also parametrizes the $A^{4}$ interaction. However, in
GP, $b_{1}$ must be positive (or non-negative?), while $\lambda_{0}^{\prime}$
is still allowed to be very slightly negative. From the bottom line of figure
LABEL:fig.grid, we can see that the allowed region of the
$\\{\gamma_{1}^{\prime},\lambda_{0}^{\prime}\\}$ parameter space is the
parabola described by $\mu_{V-}$, which gets shifted upwards for increasing
$\gamma_{2}^{\prime}$. Also, a band around $\gamma_{1}^{\prime}=0$ is
forbidden by $\lambda_{V-}$ for negative $\gamma_{2}^{\prime}$. In particular,
the region $\lambda_{0}^{\prime}<-0.0624$ is always forbidden by one of the
bounds, in accordance with our previous remarks.

Figure 6: Allowed islands of the
$\\{\gamma_{1}^{\prime},\gamma_{2}^{\prime}\\}$ parameter space for various
values of $\lambda_{0}^{\prime}$ and $\alpha_{2}^{\prime}$. The regions shrink
for decreasing $\lambda_{0}^{\prime}$, to eventually become a point at
$\lambda_{0}^{\prime}=\lambda^{\prime}_{0\text{ Min}}$ and vanish for smaller
values. For $\alpha_{2}^{\prime}=0,-1,1$, this happens at
$\lambda^{\prime}_{0\text{ Min}}\approx-0.062,-0.042,-0.042$ at the points
$(\gamma_{1}^{\prime},\gamma_{2}^{\prime})\approx(-0.25,-0.02),(-0.29,-0.04),(-0.29,-0.01)$
respectively.

Figure 7: Boundaries of the $\lambda_{V-}$ and $\mu_{V-}$ bounds for the
allowed interval $-2<\alpha_{2}^{\prime}<2$. The curves stay close to one
another for the different values of $\alpha_{2}^{\prime}$, indicating that
$\alpha_{2}^{\prime}$ has only little influence on these bounds. In the right
plot, we set $\gamma^{\prime}_{2}=0$, as other values of $\gamma_{2}$ would
only shift the plot.

Figure 8: Regions excluded by the
$\mathord{\mathop{\mu}\limits^{\scriptscriptstyle(\sim)}}$ bounds in the
forward limit (blue) and the $\lambda$ bounds beyond (orange) in the
$\\{\alpha_{2}^{\prime},\alpha_{3}^{\prime}\\}$ and
$\\{\gamma_{1}^{\prime},\gamma_{2}^{\prime}\\}$ planes (for an arbitrary value
of $\alpha_{2}^{\prime}$ and $\lambda_{0}^{\prime}$). The final white regions
respectively correspond to the ones on figure 4 and figure LABEL:fig.grid up
left.

The value of $\alpha_{2}^{\prime}$ primarily influences the green region in
figure LABEL:fig.grid and the blue and red regions are fairly stable as
$\alpha_{2}^{\prime}$ changes. This can be seen explicitly by inspecting the
last two bounds for various values of $\alpha_{2}^{\prime}$ in its allowed
interval, see figure 7. Therefore, the $\lambda_{V-}$ and $\mu_{V-}$ bounds,
whose expressions may seem rather complicated at first glance on (59), are
well approximated by their simplified expressions at $\alpha_{2}^{\prime}=0$
in (60). Accordingly, the main role of $\alpha_{2}^{\prime}$ in the three last
bounds of (59) is roughly to pick the sign of $\gamma_{1}^{\prime}$ via (61).
This can be seen in figure 6 where only the left/right part of the full island
is allowed for positive/negative $\alpha_{2}^{\prime}$, but the rest of the
shape remains quite stable.

As a last comment, we investigate the importance of the beyond forward limit
analysis. It was observed in de_Rham_2019  that the bounds beyond the forward
limit had only little influence on the allowed island for dRGT massive
gravity. We want to check whether this is also the case for PN. Looking at
figure 8, we see that going away from forward limit is truly relevant for PN.
In particular, the island for $\\{\gamma_{1}^{\prime},\gamma_{2}^{\prime}\\}$
is reduced by more than a half. However, we should note that we have not used
the most stringent bounds in the forward limit, and by minimizing
$f_{\alpha\beta}$ numerically, it is possible that the blue region could be
closer to the orange region.

###  4.3 Comparison between GP and PN

Even though there exists no tuning of the coefficients that makes GP and PN
equivalent, we find some of the coefficients play similar roles in both
theories. This identification is rendered more straightforward by our
redefinition (17) of the PN coefficients. In particular, from our previous
results, there is an obvious analogy between $a_{2}$ and
$\alpha_{2}^{\prime}$, although they play a quite different at the Lagrangian
level, as they both control the scale at which perturbative unitarity breaks
in their respective theories. Also, the pure quartic interaction $A^{4}$ is
parametrized by $b_{1}$ and $\lambda_{0}^{\prime}$ in each model. Finally, by
looking at the decoupling limits (9) and (20), we see that $a_{1}$ and $b_{3}$
are analogous to $\gamma_{1}^{\prime}$ and $\gamma_{2}^{\prime}$, in the sense
that they correspond to the cubic and quartic Galileon interactions
respectively. However, they are not truly equivalent parameters, as
$\gamma_{1}^{\prime}$ and $\gamma_{2}^{\prime}$ appear also as the
coefficients of additional interactions away from the decoupling limit. In
sum, this analogy of coefficients is given by

$\displaystyle a_{1}\,\longleftrightarrow\,\gamma_{1}^{\prime}\,,\quad
a_{2}\,\longleftrightarrow\,\alpha_{2}^{\prime}\,,\quad
b_{1}\,\longleftrightarrow\,\lambda_{0}^{\prime}\,,\quad
b_{3}\,\longleftrightarrow\,\gamma_{2}^{\prime}\,.$ (62)

The results for the ten bounds (41) for GP and PN are reported in table 1.
Looking at these results, while keeping this analogy in mind, we observe a
similarity between the bounds.

| Generalized Proca | Proca-Nuevo
---|---|---
$\tilde{\mu}_{1}>0$ | $b_{1}-a_{1}^{2}>0$ | $2\lambda_{0}^{\prime}-\gamma_{1}^{\prime}(1+2\gamma_{1}^{\prime})>0$
$\tilde{\mu}_{2}>0$ | $2b_{4}-a_{2}^{2}>0$ | $4-\alpha_{2}^{\prime 2}>0$
$\tilde{\mu}_{3}>0$ | $b_{5}>0$ | $2-\alpha_{2}^{\prime}+2\alpha_{3}^{\prime}>0$
$\mu_{SS}>0$ | $2b_{5}+4b_{6}+a_{2}(a_{1}+\frac{1}{4}a_{2})>0$ | $2+\alpha_{2}^{\prime}(\alpha_{2}^{\prime}-4(1+4\gamma_{1}^{\prime}))>0$
$\mu_{V+}>0$ | $2b_{5}+4b_{6}+a_{2}(a_{1}+\frac{1}{4}a_{2})$ | $10+\alpha_{2}^{\prime}(\alpha_{2}^{\prime}-4(1+4\gamma_{1}^{\prime}))$
| $\quad+a_{1}^{2}+b_{1}-2b_{2}+b_{3}>0$ | $\quad-16\gamma_{1}^{\prime}(1-4\gamma_{1}^{\prime})+96\gamma_{2}^{\prime}+64\lambda_{0}^{\prime}>0$
$\mu_{V-}>0$ | $2b_{5}+4b_{6}+a_{2}(a_{1}-\frac{1}{4}a_{2})$ | $2-\alpha_{2}^{\prime}(\alpha_{2}^{\prime}+4(1+4\gamma_{1}^{\prime}))$
| $\quad-3a_{1}^{2}+b_{1}+2b_{2}-b_{3}+b_{4}>0$ | $\quad-48\gamma_{1}^{\prime}(1+4\gamma_{1}^{\prime})-96\gamma_{2}^{\prime}+64\lambda_{0}^{\prime}>0$
$\lambda_{SS}\geq 0$ | $a_{2}^{2}\geq 0$ | $\alpha_{2}^{\prime 2}\geq 0$
$\lambda_{SV}\geq 0$ | $a_{2}^{2}-4b_{7}\geq 0$ | $3\alpha_{2}^{\prime 2}-8\alpha_{2}^{\prime}+16\alpha_{3}^{\prime}+4\geq 0$
$\lambda_{V+}\geq 0$ | $b_{3}+a_{1}^{2}+a_{2}(a_{1}+\frac{1}{4}a_{2})\geq 0$ | $16\gamma_{2}^{\prime}+16\gamma_{1}^{\prime 2}-\alpha_{2}^{\prime}(1+4\gamma_{1}^{\prime}-\frac{1}{4}\alpha_{2}^{\prime})+2\geq 0$
$\lambda_{V-}\geq 0$ | $b_{3}+a_{1}^{2}-a_{2}(a_{1}+\frac{1}{12}a_{2})\geq 0$ | $16\gamma_{2}^{\prime}+16\gamma_{1}^{\prime 2}+\alpha_{2}^{\prime}(1+4\gamma_{1}^{\prime}-\frac{1}{12}\alpha_{2}^{\prime})-\frac{2}{3}\geq 0$
Table 1: Summary of the positivity bounds (43)$-$(45) for GP and (55)$-$(57)
for PN. The positive overall factors are omitted. The coefficients of the GP
and PN models are defined in (5) and (18) respectively.

Moreover, reestablishing the overall coefficients that were omitted in table 1
for readability, there exists a tuning that makes the bounds of both theories
perfectly equivalent, which is given by

$\begin{aligned}
&a_{1}=\pm(\gamma_{1}^{\prime}+\frac{1}{4}-\frac{1}{3}\alpha_{2}^{\prime-1})\\\
&a_{2}=\pm\frac{1}{4}\alpha_{2}^{\prime}\\\
&b_{1}=\lambda_{0}^{\prime}+\frac{1}{16}-\frac{1}{6}\alpha_{2}^{\prime-1}(1+4\gamma_{1}^{\prime})+\frac{1}{9}\alpha_{2}^{\prime-2}\\\
&b_{3}=\gamma_{2}^{\prime}-\frac{1}{2}\gamma_{1}^{\prime}-\frac{1}{48}+\frac{1}{6}\alpha_{2}^{\prime-1}(1+4\gamma_{1}^{\prime})-\frac{1}{9}\alpha_{2}^{\prime-2}\\\
&b_{7}=-\frac{1}{16}(1-2\alpha_{2}^{\prime}+4\alpha_{3}^{\prime})\end{aligned}\qquad\begin{aligned}
&b_{2}=-\frac{1}{4}\gamma_{2}^{\prime}+\frac{1}{8}\gamma_{1}^{\prime}-\frac{1}{96}\\\
&\qquad-\frac{1}{12}\alpha_{2}^{\prime-1}(1+4\gamma_{1}^{\prime})+\frac{1}{8}\alpha_{2}^{\prime-2}\\\
&b_{4}=\frac{1}{8}\\\
&b_{5}=\frac{1}{64}(2-\alpha_{2}^{\prime}+2\alpha_{3}^{\prime})\\\
&b_{6}=-\frac{1}{384}(11-3\alpha_{2}^{\prime}+6\alpha_{3}^{\prime})\,.\end{aligned}$
(63)

The first terms correspond to the instinctive matching of (62), but there are
additional corrections depending mainly on $\alpha_{2}^{\prime}$. We could
imagine that the tuning takes a simpler form if we consider the bounds with
$\alpha_{2}^{\prime}=0$. However, quite surprisingly, there is no such tuning
in this case. The existence of this matching does not mean that the theories
are equivalent as these bounds only contain the $s^{2}$ contributions to only
the elastic amplitudes. Inserting this solution in the full amplitudes does
not make them match. For instance, considering the same amplitude in helicity
basis as derham2020 ,

$\mathcal{A}_{++--}^{\text{GP}}(s,t)-\mathcal{A}_{++--}^{\text{PN}}(s,t)=\frac{1}{96\Lambda_{2}^{4}}\,s\left(3\cos(2\theta)\left(s-4m^{2}\right)+44m^{2}-3s\right).$
(64)

Therefore, there is no profound meaning to the solution of (63), it is simply
a tuning that makes the bounds equivalent.

Another consequence of the similarity of the bounds is that the study of the
simplest Galileon from GP with parameters $\\{a_{1},b_{3},b_{1}\\}$ relates
very well with the
$\\{\gamma_{1}^{\prime},\gamma_{2}^{\prime},\lambda_{0}^{\prime}\\}$ sector of
PN for $a_{2}=\alpha_{2}^{\prime}=0$. Indeed, keeping in mind the analogy (62)
and reporting these bounds from (51) and (60) for GP and PN respectively

$\displaystyle\lambda_{V-}\geq 0$ $\displaystyle\,:\,b_{3}\geq-
a_{1}^{2},\quad\qquad\gamma_{2}^{\prime}\geq-\gamma_{1}^{\prime
2}+\frac{1}{24}$ (65) $\displaystyle\mu_{V-}>0$
$\displaystyle\,:\,b_{1}-b_{3}>3a_{1}^{2},\quad\,\lambda_{0}^{\prime}-\frac{3}{2}\gamma_{2}^{\prime}>3\gamma_{1}^{\prime
2}+\frac{3}{4}\gamma_{1}^{\prime}-\frac{1}{32},$

we observe a similar structure. It can also be seen by comparing figure 3 with
the left of figure 6. Note that, even in this case, PN still has a full
additional sector parametrized by $\alpha_{3}^{\prime}$ that can not be
removed.

Finally, we found in both models that the quartic interaction $A^{4}$ is
bounded from below, needing to be positive for GP ($b_{1}>0$) and for PN,
$\lambda_{0}^{\prime}>-0.06$. Possibly, some more sringent bounds on PN would
constrain $\lambda_{0}^{\prime}$ to be strictly positive as well.

##  5 Conclusion

In this work, we have reviewed how EFT amplitudes for spinning particles are
constrained by assuming a unitary, causal, Lorentz invariant UV completion and
how this motivates the use of the transversity formalism due to its crossing
symmetry properties to derive positivity bounds away from the forward limit.
The application of these bounds to the Generalized Proca and Proca-Nuevo has
allowed us to strongly restrict the parameter space of these models. We have
first studied them separately, by exhibiting a set of ten inequalities each of
their coefficients had to satisfy. This set further reduced to five
independent bounds for PN, confining its parameters in islands, which we
displayed in several figures. In particular, this work furnishes the first
positivity bounds analysis of the PN model. Finally, we have highlighted an
analogy between certain coefficients between the two theories, which lead to
similar structures in the bounds for both theories. Overall, we have found
that PN is more constrained by the bounds than GP. This makes sense as the PN
coefficients are associated with multiple interactions, due to the non-linear
realization of the Hessian constraint, and therefore appear repetitively in
the bounds.

From a technical point of view, there would be various ways to tighten the
bounds we derived in this work. First, we could use some improved bounds from
davighi2021natural ; tolley2021new ; Arkani-Hamed:2020blm . However, not all
of their methods already apply to spinning particles or away from the forward
limit. Second, the minimization of the indefinite polarization bound (40)
could be made more precise, either using some numerical methods Cheung2016  or
via a more systematic analytical minimization Zhou2019 ; Zhou2021 . Finally,
we could also imagine including loop effects Bellazzini_2021 ; Arkani-
Hamed:2020blm ; Bellazzini2021 . This analysis would be facilitate by the
works of Zosso2021 ; derham2021quantum  where the one-loop corrections of GP
and PN have been worked out.

In practice, the bounds we derived could be used to restrict the parameter
space of dark energy models, in complement to observational data. This
analysis could also be combined with other constraints on the parameters
arising for example from the presence of a Vainshtein mechanism on spherically
symmetric background, as studied in De_Felice_2016  for the Generalized Proca,
or else from imposing subluminal propagation of gravitational waves
Melville_2020 ; Noller2021 . Similarly, these bounds could be joined with
restrictions coming from stability analysis of quantum corrections or
Swampland conjectures.

Finally, it may be interesting to further study the perturbative PN model (18)
under the redefinition of parameters we have proposed in (17). In particular,
it eliminates a free parameter. Moreover, the tuning $\alpha_{2}^{\prime}=0$,
which does not obviously play a particular role at the Lagrangian level, seems
to give an interesting realisation of the model, as we have notably shown in
(58) that it raises the cutoff.

Even though GP and PN are fundamentally different theories, if one restricts
their parameter space adequately they might share equivalent positivity bounds
and suffer from similar quantum corrections, which might then indicate that
they descent from the same more fundamental UV-complete theory. As was shown
in Zosso2021 ; derham2021quantum , both have the same high energy behaviour
when quantum corrections are included. Loop induced counter terms have the
exact same structure and scaling. In this work we have further shown, that
despite of the fact that PN is more constrained by the positivity bounds than
GP, similar structures in the bounds for both theories arise and this might
signal that their spin-1 field originates from the same UV theory and the
general properties of this fundamental theory give rise to the same
structures.

##  Appendix A Set-up

We consider 2-2 amplitudes of four identical particles of mass $m$ and spin
$S$ with momenta $p_{1}$, $p_{2}$, $p_{3}$ and $p_{4}$. Working in the center
of mass frame, considering the scattering in the $xz$-plane, and treating the
particles as all incoming by flipping the sign of the outgoing particles’
momenta, the momenta are given by

$p_{i}=(-1)^{\eta}(E,p\cos\theta_{i},0,p\sin\theta_{i})\,,$ (66)

with $\eta=0$ for the incoming momenta $p_{1}$ and $p_{2}$, $\eta=1$ for the
outgoing momenta $p_{3}$ and $p_{4}$, and

$\theta_{1}=0,\quad\theta_{2}=\pi\quad\theta_{3}=\theta,\quad\theta_{4}=\pi+\theta.$
(67)

We work with the Mandelstam variables
$s=-(p_{1}+p_{2})^{2},t=-(p_{1}+p_{3})^{2}$, and $u=-(p_{1}+p_{4})^{2}$, with
$s+t+u=m^{2}$. The parameters of (66) can be expressed in terms of the
Mandelstam variables as

$E=\frac{\sqrt{s}}{2},\quad
p=\frac{1}{2}\sqrt{s-4m^{2}},\quad\cos\theta=1+\frac{2t}{s-4m^{2}},\quad\sin\theta=2\frac{\sqrt{tu}}{(s-4m^{2})},$
(68)

such that the physical region corresponds to $s>4m^{2}$. Any two independent
variables are sufficient to express the theory. Examples of such sets are
$(p,\theta)$ or else $(s,t)$. Also, note that the forward limit $t=0$
corresponds to $\cos\theta=1$, or equivalently, $\theta=0$, hence the name
forward.

To compute amplitudes in transversity basis, we use the polarization vectors
from de_Rham_2018 , given by

$\displaystyle\epsilon^{\mu}_{\tau=\pm
1}(p_{i})=\frac{i}{\sqrt{2}m}(p,E\sin\theta_{i}\pm
im\cos\theta_{i},0,E\cos\theta_{i}\mp im\sin\theta_{i}),$ (69)
$\displaystyle\epsilon^{\mu}_{\tau=0}(p_{i})=(0,0,1,0).$

##  Appendix B Bounds Derivation

First, we define a quantity that gets rid off the subtraction functions by
taking $N_{S}$ derivatives on $\tilde{\mathcal{T}}^{+}$, defined in (25), as
follows

$f_{\tau_{1}\tau_{2}}(s,t)=\frac{1}{N_{S}!}\frac{\partial^{N_{S}}}{\partial
s^{N_{S}}}\tilde{\mathcal{T}}^{+}_{\tau_{1}\tau_{2}}(s,t)\,.$ (70)

Using the dispersion relation (31), or equivalently taking the derivatives on
the contour integral (30) and deforming the contour afterwards, we can write

$f_{\tau_{1}\tau_{2}}(s,t)=\frac{1}{\pi}\int_{\mu_{b}}^{\infty}d\mu\frac{{\text{Abs}}_{s}\mathcal{T}^{+}_{\tau_{1}\tau_{2}}(\mu,t)}{(\mu-s)^{N_{S}+1}}+\frac{1}{\pi}\int_{\mu_{b}}^{\infty}d\mu\frac{{\text{Abs}}_{u}\mathcal{T}^{+}_{\tau_{1}\tau_{2}}(4m^{2}-t-\mu,t)}{(\mu-u)^{N_{S}+1}}\,.$
(71)

We know from (26) that the numerators are positive for $0\leq t<m^{2}$. The
denominators are also positive for $s$ within the branch cuts, such that

$f_{\tau_{1}\tau_{2}}(s,t)>0\quad\text{for
}\;4m^{2}-t-\mu_{b}<s<\mu_{b},\;0\leq t<m^{2}.$ (72)

Now, let us consider this quantity in terms of the crossing symmetric variable
$v=s+t/2-2m^{2}$,

$f_{\tau_{1}\tau_{2}}(v,t)=f_{\tau_{1}\tau_{2}}(s=v+2m^{2}-t/2,t)=\frac{1}{N_{S}!}\frac{\partial^{N_{S}}}{\partial
s^{N_{S}}}\tilde{\mathcal{T}}^{+}_{\tau_{1}\tau_{2}}(s,t)\Big{|}_{s=v+2m^{2}-t/2}\,.$
(73)

Then,

$f_{\tau_{1}\tau_{2}}(v,t)=\frac{1}{\pi}\int_{\mu_{b}}^{\infty}d\mu\frac{{\text{Abs}}_{s}\mathcal{T}^{+}_{\tau_{1}\tau_{2}}(\mu,t)}{(\mu-2m^{2}+t/2-v)^{N_{S}+1}}+\frac{{\text{Abs}}_{u}\mathcal{T}^{+}_{\tau_{1}\tau_{2}}(4m^{2}-t-\mu,t)}{(\mu-2m^{2}+t/2+v)^{N_{S}+1}}\,,$
(74)

and we still have

$f_{\tau_{1}\tau_{2}}(v,t)>0\quad\text{for}\;|v|<\mu_{b}-2m^{2}+t/2,\,0\leq
t<m^{2}.$ (75)

Furthermore, the $v$-derivatives are given by

$\displaystyle\partial_{v}^{N}f_{\tau_{1}\tau_{2}}(v,t)=\frac{(N_{S}+N)!}{N!}\frac{1}{\pi}\int_{\mu_{b}}^{\infty}d\mu$
$\displaystyle\frac{{\text{Abs}}_{s}\mathcal{T}^{+}_{\tau_{1}\tau_{2}}(\mu,t)}{(\mu-2m^{2}+t/2-v)^{N_{S}+1+N}}$
(76) $\displaystyle+(-1)^{N}$
$\displaystyle\frac{{\text{Abs}}_{u}\mathcal{T}^{+}_{\tau_{1}\tau_{2}}(4m^{2}-t-\mu,t)}{(\mu-2m^{2}+t/2+v)^{N_{S}+1+N}}\,,$

such that every even $v$-derivative is positive

$\partial_{v}^{2N}f_{\tau_{1}\tau_{2}}(v,t)>0\quad\forall\,N\geq
0,\quad|v|<\mu_{b}-2m^{2}+t/2,\,0\leq t<m^{2}.$ (77)

Next, looking at the first $t$-derivative,

$\displaystyle\partial_{t}f_{\tau_{1}\tau_{2}}(v,t)=$
$\displaystyle\frac{1}{\pi}\int_{\mu_{b}}^{\infty}d\mu\frac{\partial_{t}{\text{Abs}}_{s}\mathcal{T}^{+}_{\tau_{1}\tau_{2}}(\mu,t)}{(\mu-2m^{2}+t/2-v)^{N_{S}+1}}+\frac{\partial_{t}{\text{Abs}}_{u}\mathcal{T}^{+}_{\tau_{1}\tau_{2}}(4m^{2}-t-\mu,t)}{(\mu-2m^{2}+t/2+v)^{N_{S}+1}}$
(78) $\displaystyle-\,\frac{N_{S}+1}{2}$
$\displaystyle\frac{1}{\pi}\int_{\mu_{b}}^{\infty}d\mu\frac{{\text{Abs}}_{s}\mathcal{T}^{+}_{\tau_{1}\tau_{2}}(\mu,t)}{(\mu-2m^{2}+t/2-v)^{N_{S}+2}}+\frac{{\text{Abs}}_{u}\mathcal{T}^{+}_{\tau_{1}\tau_{2}}(4m^{2}-t-\mu,t)}{(\mu-2m^{2}+t/2+v)^{N_{S}+2}}\,,$

we know by (26) that the first integral of the RHS is positive for $0\leq
t<m^{2}$. The second line looks like $f_{\tau_{1}\tau_{2}}(v,t)$, but with an
additional power of the denominator. Noting the integral inequality for any
positive definite function $\rho(\mu)$

$\int_{\mu_{b}}^{\infty}d\mu\frac{\rho(\mu)}{(\mu-2m^{2}+t/2\pm
v)^{N_{S}+2}}<\frac{1}{\mathcal{M}^{2}}\int_{\mu_{b}}^{\infty}d\mu\frac{\rho(\mu)}{(\mu-2m^{2}+t/2\pm
v)^{N_{S}+1}}\,,$ (79)

for

$\mathcal{M}^{2}={\rm Min}_{\mu\geq\mu_{b}}(\mu-2m^{2}+t/2\pm
v)=\mu_{b}-2m^{2}+t/2\pm v\,,$ (80)

we conclude that

$\partial_{t}f_{\tau_{1}\tau_{2}}(v,t)+\frac{N_{S}+1}{2\mathcal{M}^{2}}f_{\tau_{1}\tau_{2}}(v,t)>0\,.$
(81)

Similar arguments can be used to obtain bounds on higher derivatives in $t$.

Our bounds are valid, in general, for the ranges of parameters

$|v|<2m^{2}+t/2,\;0\leq t<m^{2},\;\mathcal{M}^{2}=2m^{2}+t/2\pm v\,,$ (82)

and at tree level, with $\Lambda$ the EFT cutoff, for the ranges

$|v|<\Lambda^{2},\;0\leq t<m^{2},\;\mathcal{M}^{2}=\Lambda^{2}-2m^{2}+t/2\pm
v\approx\Lambda^{2}\,.$ (83)

##  Appendix C Linear Vector Bounds

In this part, we express a set of 10 bounds out of the analysis of section 3.
First, we recall the definition of the main quantities we use here,

$f_{\tau_{1}\tau_{2}}(v,t)=\frac{1}{N_{S}!}\frac{\partial^{N_{S}}}{\partial
s^{N_{S}}}\tilde{\mathcal{T}}^{+}_{\tau_{1}\tau_{2}}(s=v+2m^{2}-t/2,t)\,,$
(84)
$f_{\alpha\beta}(v,t)=\sum_{{\tau_{1}\tau_{2}\tau_{3}\tau_{4}}}\alpha_{\tau_{1}}\beta_{\tau_{2}}\alpha^{*}_{\tau_{3}}\beta^{*}_{\tau_{4}}f_{\tau_{1}\tau_{2}\tau_{3}\tau_{4}}(v,t)\,,$
(85)

where $\mathcal{T}^{+}_{\tau_{1}\tau_{2}\tau_{3}\tau_{4}}$ is the combination
of spinning amplitudes in transversity basis defined in (25). This work
focuses on spin-1 theories such that

$N_{S}=4S+2=6.$ (86)

The bounds can be summarized as follows777Some higher $t$-derivative bounds
could also be considered, but are not relevant here..

$\displaystyle\partial_{v}^{2N}f_{\tau_{1}\tau_{2}}(v,t)>0\quad\forall\quad
N\geq 0\,,$ (87)
$\displaystyle\partial_{t}f_{\tau_{1}\tau_{2}}(v,t)+\frac{N_{S}+1}{2\mathcal{M}^{2}}f_{\tau_{1}\tau_{2}}(v,t)>0\,,$
(88) $\displaystyle\partial_{v}^{2N}f_{\alpha\beta}(0,0)>0\quad\forall\quad
N\geq 0\,,$ (89)

where, at tree level,

$|v|<\Lambda^{2},\;0\leq t<m^{2},\;\mathcal{M}^{2}=\Lambda^{2}.$ (90)

Furthermore, the quantities we are going to find for the GP and PN models are
linear in $t$ and $v$. They may be written in the schematic form

$\displaystyle f_{\tau_{1}\tau_{2}}(v,t)$
$\displaystyle=f_{\tau_{1}\tau_{2}}(0,0)+\partial_{t}f_{\tau_{1}\tau_{2}}\cdot
t$ (91)
$\displaystyle\equiv\frac{1}{\Lambda_{2}^{4}}\left[\mu_{\tau_{1}\tau_{2}}+\lambda_{\tau_{1}\tau_{2}}\frac{t}{m^{2}}\right],$

and

$f_{\alpha\beta}(v,t)=f_{\alpha\beta}(0,0)+\partial_{t}f_{\alpha\beta}\cdot
t+\partial_{v}f_{\alpha\beta}\cdot v\,,$ (92)

where $f(0,0)$, $\partial_{t}f$, and $\partial_{v}f$ are linear combinations
of the EFT parameters with no kinematics dependence left, where we introduce
$\mu$ and $\lambda$ for the bounds that follow. These are statements coming
from our explicit results rather than from any theoretical inputs888The term
in $v$ may seem surprising. Indeed, if there is still an $s$ (hence $v$)
dependence after taking $N_{S}$ derivatives, it means that the amplitude was
not respecting the $s^{2}$ growth of the Froissart bound. That is alright, as
we are working with an EFT, and only indicates that unitarity may be
perturbatively broken. Our amplitudes in the definite transversity
polarization basis do not have such a dependence left..

Then, the only non-trivial bounds are the $N=0$ $v$-derivative bounds in (87)
and (89), and the first $t$-derivative bound (88). Moreover, the second term
of the latter is suppressed by a factor of $m^{2}/\Lambda^{2}$, but is assured
to be strictly positive by (87). Therefore, neglecting this term, we obtain a
non-strict inequality for $\partial_{t}f_{\tau_{1}\tau_{2}}$, later denoted by
$\lambda_{\tau_{1}\tau_{2}}$. Explicitly, the remaining bounds are

$f_{\tau_{1}\tau_{2}}(v,t)=f_{\tau_{1}\tau_{2}}(0,0)+\partial_{t}f_{\tau_{1}\tau_{2}}\cdot
t>0\,,\quad 0\leq t<m^{2},$ (93)

and

$\displaystyle\partial_{t}f_{\tau_{1}\tau_{2}}(v,t)\geq 0\,,$ (94)
$\displaystyle f_{\alpha\beta}(0,0)>0\,.$

Note that (93) is included in these two last bounds (as the definite
polarizations are just particular cases of indefinite ones, and $t\geq 0$).
Hence, we can focus on the bounds in (94), which have no dependence in $t$ or
$v$.

There are four independent quantities for the definite elastic polarizations

$\displaystyle f_{SS}\equiv f_{00}\,,$ (95) $\displaystyle f_{SV}\equiv
f_{0+1}=f_{0-1}=f_{+10}=f_{-10}\,,$ $\displaystyle f_{V+}\equiv
f_{+1+1}=f_{-1-1}\,,$ $\displaystyle f_{V-}\equiv f_{+1-1}=f_{-1+1}\,,$

where we use $S$ and $V$ to denote 0- and 1-transversity modes respectively.
This follows the convention used by Cheung2016 999Note that they were working
in helicity basis, such that $S$ and $V$ respectively denote 0- and 1-helicity
modes in this paper.. In the following, $\mu_{SS}=f_{SS}(0,0)$,
$\lambda_{SS}=\partial_{t}f_{SS}$ and similarly for the other polarizations.
Thus, the $t$-derivative bound correspond to four distinct bounds

$\lambda_{SS}\geq 0\,,\quad\lambda_{SV}\geq 0\,,\quad\lambda_{V+}\geq
0\,,\quad\lambda_{V-}\geq 0\,.$ (96)

To study the indefinite polarization bounds we introduce the quantities
$\alpha_{\pm 1,0}$, $\beta_{\pm 1,0}$, which designate the projections along
the definite polarization vectors in the transversity basis. Also,

$\alpha_{\pm}\equiv\frac{1}{\sqrt{2}}(\alpha_{-1}\pm\alpha_{+1})\,,\quad\beta_{\pm}\equiv\frac{1}{\sqrt{2}}(\beta_{-1}\pm\beta_{+1})\,,$
(97)

where we have the normalization condition

$|\alpha|^{2}=|\alpha_{-}|^{2}+|\alpha_{0}|^{2}+|\alpha_{+}|^{2}=1\,,$ (98)

and similarly for $\beta$. Then, we explicitly find that the indefinite
polarization $f_{\alpha\beta}(0,0)$ can be written in the following form for
both GP and PN

$\displaystyle f_{\alpha\beta}(0,0)=\frac{1}{\Lambda_{2}^{4}}\Big{[}$
$\displaystyle\tilde{\mu}_{1}|\alpha_{+}|^{2}|\beta_{+}|^{2}$ (99)
$\displaystyle+$
$\displaystyle\tilde{\mu}_{2}\big{[}|\alpha_{+}|^{2}(1-|\beta_{+}|^{2})+|\beta_{+}|^{2}(1-|\alpha_{+}|^{2})\big{]}$
$\displaystyle+$
$\displaystyle\tilde{\mu}_{3}\big{[}|\alpha_{-}|^{2}|\beta_{0}|^{2}+|\alpha_{0}|^{2}|\beta_{-}|^{2}\big{]}$
$\displaystyle+$
$\displaystyle\tilde{\mu}_{4}\big{[}|\alpha_{-}|^{2}|\beta_{-}|^{2}+|\alpha_{0}|^{2}|\beta_{0}|^{2}\big{]}$
$\displaystyle+$ $\displaystyle
2(\tilde{\mu}_{3}-\tilde{\mu}_{4})\real(\alpha_{-}\alpha_{0}^{*})\real(\beta_{-}\beta_{0}^{*})$
$\displaystyle+$
$\displaystyle\tilde{\mu}_{5}\big{[}\real(\alpha_{-}\alpha_{+}^{*})\real(\beta_{-}\beta_{+}^{*})-\real(\alpha_{0}\alpha_{+}^{*})\real(\beta_{0}\beta_{+}^{*})\big{]}\Big{]}>0\,,$

where the $\tilde{\mu}$’s are combinations of the EFT coefficients101010This
expression for $f_{\alpha\beta}$ is equivalent to (4.4) of de_Rham_2019 ,
where we redefined their coefficients in the following way
$\tilde{\mu}_{1}=\mu_{1}+2\mu_{2}$, $\tilde{\mu}_{2}=\mu_{2}$,
$\tilde{\mu}_{3}=\mu_{4}$, $\tilde{\mu}_{4}=\mu_{5}$ and
$\tilde{\mu}_{5}=\mu_{3}$.. The quantity $f_{\alpha\beta}$ has to be positive
for any polarization. In particular, we focus on some specific ones to deduce
some bounds. First, picking $|\alpha_{+}|^{2}=|\beta_{+}|^{2}=1$ implies, by
the normalization condition, that only the first line is non-zero, such that
(99) reduces to $\tilde{\mu_{1}}>0$. Similarly, picking $|\alpha_{+}|^{2}=1$,
$|\beta_{+}|^{2}=0$ implies that $\tilde{\mu_{2}}>0$,
$|\alpha_{-}|^{2}=|\beta_{0}|^{2}=1$ (or $|\alpha_{0}|^{2}=|\beta_{-}|^{2}=1$)
implies that $\tilde{\mu_{3}}>0$, and $|\alpha_{-}|^{2}=|\beta_{-}|^{2}=1$ (or
$|\alpha_{0}|^{2}=|\beta_{0}|^{2}=1$) implies that $\tilde{\mu_{4}}>0$. In
summary, we obtain the following bounds

$\tilde{\mu}_{1}>0\,,\quad\tilde{\mu}_{2}>0\,,\quad\tilde{\mu}_{3}>0\,,\quad\tilde{\mu}_{4}>0\,.$
(100)

A condition on $\tilde{\mu}_{5}$ is less straightforward to obtain. In order
to obtain the most stringent bound one could minimize $f_{\alpha\beta}$, e.g.
numerically Cheung2016 ; Wang2021 . Whereas for spin-2 theories only a
numerical approach is available, for spin-1 theories an analytic minimazation
is conceivable. It has been worked out for the gauge-bosons in the Standard
Model EFT in Zhou2019 ; Zhou2021 . It would be interesting to see how to apply
their analytical procedure in our models.

Here, we want to keep a simple analytical approach and aim to obtain only
sufficient conditions on $\tilde{\mu}_{5}$111111 A similar philosophy is
adopted in de_Rham_2019 , where they derive a slightly stronger sufficient
condition than ours in their Appendix C. We rather use ours, which is given by
the bound on definite polarizations, as it has a more straightforward physical
meaning.. We establish these conditions by looking at the bounds for the
definite polarizations $f_{\tau_{1}\tau_{2}}(0,0)>0$. Indeed, by considering
the $\mathcal{O}(\alpha^{2}_{\tau_{1}}\beta^{2}_{\tau_{2}})$ in
$f_{\alpha\beta}$, we see that

$\displaystyle\mu_{SS}=\tilde{\mu}_{4}>0\,,$ (101)
$\displaystyle\mu_{SV}=\frac{1}{2}[\tilde{\mu}_{2}+\tilde{\mu}_{3}]>0\,,$
$\displaystyle\mu_{V+}=\frac{1}{4}[\tilde{\mu}_{1}+2\tilde{\mu}_{2}+\tilde{\mu}_{4}+\tilde{\mu}_{5}]>0\,,$
$\displaystyle\mu_{V-}=\frac{1}{4}[\tilde{\mu}_{1}+2\tilde{\mu}_{2}+\tilde{\mu}_{4}-\tilde{\mu}_{5}]>0\,.$

Then, the bounds we are going to use for $\tilde{\mu}_{5}$ are the following

$|\tilde{\mu}_{5}|<\tilde{\mu}_{1}+2\tilde{\mu}_{2}+\tilde{\mu}_{4}\quad\Longleftrightarrow\quad\mu_{V\pm}>0\,.$
(102)

As already mentioned, stricter bounds could be found, in particular using
numerical methods. Furthermore, the bounds (100) derived from the indefinite
polarizations are respected by the definite bounds (as expected) on $\mu_{SS}$
and $\mu_{SV}$, but also provide additional information that would not be
available from the definite polarization analysis alone. Namely, the bound on
$\tilde{\mu}_{1}$, and the positivity of $\tilde{\mu}_{2}$ and
$\tilde{\mu}_{3}$ not only as a sum, but also separately.

In sum, we have derived a total of 10 bounds:

$\displaystyle\tilde{\mu}_{1}>0\,,\quad\tilde{\mu}_{2}>0\,,\quad\tilde{\mu}_{3}>0\,,$
(103) $\displaystyle\mu_{SS}>0\,,\quad\mu_{V+}>0\,,\quad\mu_{V-}>0\,,$
$\displaystyle\lambda_{SS}\geq 0\,,\quad\lambda_{SV}\geq
0\,,\quad\lambda_{V+}\geq 0\,,\quad\lambda_{V-}\geq 0\,,$

where the $\mu$’s are bounds in the the forward limit, those with
$\tilde{\mu}$’s come from indefinite polarizations, and the $\lambda$’s
correspond to the $t$-derivative bounds, which are available due to the
analysis beyond the forward limit.

###### Acknowledgements.

LH is supported by funding from the European Research Council (ERC) under the
European Unions Horizon 2020 research and innovation programme grant agreement
No 801781 and by the Swiss National Science Foundation grant 179740.

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