# Unitarity constraints on ALP interactions

I. Brivio<EMAIL_ADDRESS>Institut für Theoretische Physik,
Universität Heidelberg.
Philosophenweg 16, 69120 Heidelberg (Germany)  O. J. P. Éboli<EMAIL_ADDRESS>Instituto de Física, Universidade de São Paulo, São Paulo – SP 05508-090,
Brazil  M. C. González-García  Institució Catalana de Recerca i Estudis
Avançats (ICREA),
Departament d’Estructura i Constituents de la Matèria, Universitat de
Barcelona,
647 Diagonal, E-08028 Barcelona, Spain  C.N. Yang Institute for Theoretical
Physics, SUNY at Stony Brook,
Stony Brook, NY 11794-3840, USA<EMAIL_ADDRESS>
###### Abstract

We derive partial-wave unitarity constraints on gauge-invariant interactions
of an Axion-Like Particle (ALP) up to dimension-6 from all allowed $2\to 2$
scattering processes in the limit of large center-of-mass energy. We find that
the strongest bounds stem from scattering amplitudes with one external ALP and
only apply to the coupling to a pair of $SU(2)_{L}$ gauge bosons. Couplings to
$U(1)_{Y}$ and $SU(3)_{C}$ gauge bosons and to fermions are more loosely
constrained.

††preprint: YITP-SB-2021-8

##  I Introduction

Axion-like Particles (ALPs) are generic pseudo-Goldstone bosons, that can
emerge from the spontaneous breaking of some global symmetry at energies well
above the electroweak (EW) scale $v$. While the main representative is the QCD
axion (either "invisible" Peccei and Quinn (1977a, b); Weinberg (1978);
Wilczek (1978); Kim (1979); Shifman et al. (1980); Dine et al. (1981);
Zhitnitsky (1980) or in modern setups where the tie between axion mass and
couplings is relaxed Rubakov (1997); Berezhiani et al. (2001); Hsu and Sannino
(2005); Hook (2015); Fukuda et al. (2015); Chiang et al. (2016); Gherghetta et
al. (2016); Dimopoulos et al. (2016); Kobakhidze (2016); Agrawal and Howe
(2018a, b); Gaillard et al. (2018); Buen-Abad and Fan (2019); Csáki et al.
(2020); Hook et al. (2020); Gherghetta and Nguyen (2020); Hook (2018); Di
Luzio et al. (2021a, b)), this class of particles encompasses a large number
of exotic states, that can emerge in composite Higgs models Gripaios et al.
(2009, 2017); Merlo et al. (2018); Chala et al. (2017); Brivio et al. (2019),
models with spontaneous breaking of lepton number ("Majorons") Chikashige et
al. (1981); Gelmini and Roncadelli (1981), dynamical flavor theories
("axiflavons") Davidson and Wali (1982); Wilczek (1982); Calibbi et al.
(2017); Ema et al. (2017), string theory Svrcek and Witten (2006); Arvanitaki
et al. (2010) and many other scenarios.

ALPs are usually studied within a model-independent Effective Field Theory
(EFT) framework Georgi et al. (1986); Choi et al. (1986). Their pseudo-
Goldstone nature justifies the assumption that ALPs are the only light remnant
of a much heavier new physics sector, whose interactions are suppressed by a
characteristic scale $f_{a}\gg v$. A priori, the allowed parameter space spans
several orders of magnitude both in the ALP mass $m_{a}$ and in the couplings
to Standard Model (SM) particles that, within the EFT approach, enter at
lowest order as dimension 5 operators.

The interest in the ALP Lagrangian as a self-consistent EFT has grown
recently, leading to several studies of its theoretical properties. For
instance, the Renormalization Group (RG) evolution and the matching to the ALP
EFT valid below the EW scale were derived in Choi et al. (2017); Chala et al.
(2021); Bauer et al. (2021). The interplay between dimension-5 ALP
interactions and dimension-6 operators in the Standard Model EFT (SMEFT) was
explored in Galda et al. (2021). The matching of the ALP EFT to concrete QCD
axion models was examined in Ref. Alonso-Álvarez et al. (2021), that pointed
out theoretical subtleties when applying the EFT approach to loop processes.

In this work we examine the validity range of the ALP EFT at high energies on
the basis of its perturbative partial-wave unitarity properties. It is well
known that classically non-renormalizable interactions give rise to rapid
growth of the scattering amplitudes with energy, which leads to partial-wave
unitarity violation at some large value of the center-of-mass energy
$\sqrt{S}$. Generically, partial-wave unitarity violation signals the
breakdown of the low-energy description and indicates that extra fundamental
degrees of freedom or the onset of the nonperturbative regime must be present
around or below the apparent unitarity violation scale in order to restore the
physical behavior of scattering amplitudes. Paradigmatic examples of the use
of unitarity relations to derive constraints on the validity of a theory
include the seminal work of Lee, Quigg and Thacker Lee et al. (1977a, b) that
imposed an upper bound on the Higgs mass by analyzing the unitarity of the
standard model and was used to build a case in favor of the construction of
the present generation of colliders. Another classical example are the bounds
on new fermions obtained by Chanowitz, Furman and Hinchliffe Chanowitz et al.
(1978). On a more formal front, unitarity arguments have also been employed,
for example, in connection with the requirement of gauge invariance Cornwall
et al. (1974).

In the last decades, partial-wave unitarity has been employed ubiquitously to
constrain effective interactions, in particular in the electroweak sector (see
for example Bilchak et al. (1988); Gounaris et al. (1994); Gounaris et al.
(1995a, b); Degrande (2013); Baur and Zeppenfeld (1988); Dahiya et al. (2016);
Ghosh et al. (2017)). Recently, Refs. Corbett et al. (2015, 2017); Almeida et
al. (2020) presented a general systematic study of unitarity bounds for the
case of effective interactions in the SMEFT and Higgs EFT (HEFT). Generically,
unitarity preservation imposes consistency conditions on the theory such that,
for the EFT to be valid up to a given $\sqrt{S}$, the effective couplings
(scale) need to be smaller (larger) than a certain threshold. Conversely, for
given values of the EFT coefficients and scale, unitarity imposes an upper
limit on the energy scales at which the EFT can be applied. In that respect
unitarity bounds are crucial for the interpretation of actual experiments,
which study tails of kinematical distributions, since one can infer unphysical
bounds that are too strong if these limits are not respected.

For the case of ALP EFT, the rapid growth of the scattering amplitudes with
energy, that leads to partial-wave unitarity violation, is particularly
enhanced. The reason for that is the pseudo-Goldstone nature of the ALPs which
requires all their interactions to be classically invariant under shifts
$a(x)\mapsto a(x)+\alpha$, _i.e._ to be of the form $J^{\mu}\partial_{\mu}a$.
As a consequence, an explicit momentum dependence is present in all ALP
couplings.

A partial analysis of unitarity constraints on ALP couplings was presented in
Refs. Marciano et al. (2016); Cornella et al. (2020). Here we adopt a more
systematic approach and derive maximal constraints on all ALP interactions of
dimension 5 and 6 from partial-wave unitarity, examining all allowed $2\to 2$
scattering processes in the limit of large center-of-mass energy. We adopt a
procedure analogous to the one employed in Corbett et al. (2015, 2017);
Almeida et al. (2020) for the case of effective interactions in the SMEFT and
HEFT.

The outline of this article is as follows. We present the relevant Lagrangian
employed in Sec. II and briefly discuss the number of relevant operators we
consider. The core of the results is contained in Sec. III where we derive
first the most general bounds for the ALP couplings to SM gauge bosons which
are obtained from the partial-wave analysis of the scattering of boson pairs
in Sec. III.1. Section III.2 contains our derivation of the most general
independent constraints on ALP couplings to SM fermions which are obtained
with the partial-wave analysis of scatterings involving fermion pairs. We
briefly discuss the results in Sec. IV. Explicit expressions of the helicity
amplitudes for all the relevant processes are presented in the Appendix.

##  II ALP effective Lagrangian

We consider the SM extended by the ALP effective Lagrangian Georgi et al.
(1986); Choi et al. (1986); Brivio et al. (2017)

$\displaystyle\mathcal{L}_{ALP}$
$\displaystyle=\frac{1}{2}\partial_{\mu}a\partial^{\mu}a-\frac{m_{a}^{2}}{2}a^{2}+C_{\tilde{B}}\,O_{\tilde{B}}+C_{\tilde{W}}\,O_{\tilde{W}}+C_{\tilde{G}}\,O_{\tilde{G}}+C_{a\Phi}\,O_{a\Phi}$
(1)
$\displaystyle+\big{[}C_{u\Phi}\,O_{u\Phi}+C_{d\Phi}\,O_{d\Phi}+C_{e\Phi}\,O_{e\Phi}+\rm
h.c.\big{]}$ $\displaystyle+C_{a\Phi}^{(2)}O_{a\Phi}^{(2)}\,,$

where the effective operators

$\displaystyle O_{\tilde{B}}$
$\displaystyle=\frac{a}{f_{a}}B_{\mu\nu}\tilde{B}^{\mu\nu}\,,$ $\displaystyle
O_{u\Phi}$
$\displaystyle=i\,\frac{a}{f_{a}}\;\bar{q}\,Y_{u}\tilde{\Phi}\,u\,,$ (2)
$\displaystyle O_{\tilde{W}}$
$\displaystyle=\frac{a}{f_{a}}W^{i}_{\mu\nu}\tilde{W}^{i\mu\nu}\,,$
$\displaystyle O_{d\Phi}$
$\displaystyle=i\,\frac{a}{f_{a}}\;\bar{q}\,Y_{d}\Phi\,d\,,$ (3)
$\displaystyle O_{\tilde{G}}$
$\displaystyle=\frac{a}{f_{a}}G^{a}_{\mu\nu}\tilde{G}^{a\mu\nu}\,,$
$\displaystyle O_{e\Phi}$
$\displaystyle=i\,\frac{a}{f_{a}}\;\bar{l}\,Y_{e}\Phi\,e\,,$ (4)
$\displaystyle O_{a\Phi}$
$\displaystyle=\,\frac{\partial_{\mu}a}{f_{a}}(i\Phi^{\dagger}(D_{\mu}\Phi)-i(D_{\mu}\Phi)^{\dagger}\Phi)$
$\displaystyle O_{a\Phi}^{(2)}$
$\displaystyle=\frac{\partial_{\mu}a\partial^{\mu}a}{f_{a}^{2}}(\Phi^{\dagger}\Phi)\,,$
(5)

form a complete basis of CP-even ALP interactions up to
$\mathcal{O}(f_{a}^{-3})$ terms. Here, $B_{\mu},W_{\mu}^{i}$ and $G_{\mu}^{a}$
are the gauge bosons of the $U(1)_{Y}\times SU(2)_{L}\times SU(3)_{c}$ SM
symmetry respectively, and the dual field strengths are defined by
$\tilde{X}_{\mu\nu}=\frac{1}{2}\varepsilon_{\mu\nu\rho\sigma}X^{\rho\sigma}$.
$\Phi$ denotes the $SU(2)_{L}$ Higgs doublet while
$\tilde{\Phi}=i\tau^{2}\Phi^{*}$ is its dual (being $\tau^{i}$ the Pauli
matrices). Upon EW symmetry breaking,
$\langle\Phi^{\dagger}\Phi\rangle=(v+H)^{2}/2$ with $H$ the physical Higgs
boson. The left (right)-handed fermion multiplets are denoted by $q,l$
($u,d,e$) and $Y_{u},Y_{d},Y_{e}$ are the $3\times 3$ Yukawa matrices. All
index contractions were left implicit and repeated indices are summed over
unless otherwise specified. A mass term $m_{a}$ for the ALP was introduced,
which is generically induced in the presence of soft breaking of shift-
invariance, such as non-perturbative instanton effects in the case of the QCD
axion Bardeen et al. (1978); Shifman et al. (1980); Di Vecchia and Veneziano
(1980); Grilli di Cortona et al. (2016).

We neglect CP violating effects such that all Wilson coefficients $C_{i}$ are
real scalar quantities. Although this is not manifest in Eqs. (2)–(5), all ALP
interactions are classically shift-invariant: the interactions to bosons can
be written as $\partial_{\mu}aJ^{\mu}_{X}$ by integration by parts, where
$J_{X}^{\mu}$ is the Chern-Simons current associated to the $X=\\{B,W,G\\}$
gauge boson.111In the $G$ case, only a discrete version of the shift-
invariance is preserved due to the presence of non-vanishing instanton
configurations. The operators with fermions were taken to follow the minimal-
flavor-violation ansatz Chivukula and Georgi (1987); Hall and Randall (1990);
D’Ambrosio et al. (2002), i.e. to respect a $U(3)^{5}$ global symmetry that is
only broken by insertions of the Yukawa couplings. With this flavor structure,
they could also be equivalently traded for a set of chirality-conserving ones
of the form
$(\partial_{\mu}a)(\bar{\psi}_{p}\gamma^{\mu}\psi_{r})\delta^{pr}$, with $p,r$
flavor indices Chala et al. (2021); Bonilla et al. (2021).

The operator $O_{a\Phi}$ is actually redundant Georgi et al. (1986); Brivio et
al. (2017); Bauer et al. (2021); Bonilla et al. (2021):

$O_{a\Phi}=O_{u\Phi}-O_{d\Phi}-O_{e\Phi}+\rm h.c.\,.$ (6)

Nevertheless, it is often retained because the set
$\\{O_{\tilde{B}},O_{\tilde{W}},O_{\tilde{G}},O_{a\Phi}\\}$ forms a complete
and non-redundant operator basis at dimension 5 in the bosonic sector that can
be of phenomenological interest.

The operator $O_{a\Phi}^{(2)}$ has been previously considered in Draper and
McKeen (2012); Bauer et al. (2017, 2019); Davoudiasl et al. (2021) and it is
the only shift-invariant operator222One more operator structure is present at
dimension 6, namely $(\partial_{\mu}\partial^{\mu}a)^{2}$. However, applying
the ALP equation of motion, this can be fully reabsorbed into a redefinition
of the ALP mass. We have checked the completeness of the dimension-6 set with
BasisGen Criado (2019). that can be constructed at dimension 6. SMEFT
operators of dimension 6 are neglected: we assume them to be suppressed by a
scale $\Lambda_{\rm SMEFT}\neq f_{a}$ and work consistently at order
$(f_{a}^{-2}\,\Lambda_{\rm SMEFT}^{0})$. Discussing the interplay of the two
expansions is beyond the scope of this work.333As will become clear from the
discussion in Sec. III, the bounds on $C_{\tilde{W}}$, $C_{f\Phi}$,
$C_{a\Phi}$ and $C_{a\Phi}^{(2)}$ are not expected to change significantly in
the presence of dimension 6 SMEFT operators, independently of the interplay
between the SMEFT and ALP expansions. This is because all these bounds are
dominated by scatterings with one or two external ALPs.

##  III Analysis of unitarity constraints

###  III.1 Helicity amplitudes for the scattering of pairs of bosons

Consider the two-to-two scattering of bosons $V_{i}$ with helicities
$\lambda_{i}$

${V_{1}}_{\lambda_{1}}{V_{2}}_{\lambda_{2}}\to{V_{3}}_{\lambda_{3}}{V_{4}}_{\lambda_{4}}\,,$
(7)

where we denote by $V$ either gauge bosons, Higgs or ALP. The corresponding
helicity amplitude can be expanded in partial waves in the center–of–mass
system as Jacob and Wick (1959)

$\displaystyle\mathcal{M}({V_{1}}_{\lambda_{1}}{V_{2}}_{\lambda_{2}}\to{V_{3}}_{\lambda_{3}}{V_{4}}_{\lambda_{4}})=16\pi\sum_{J}\left(2J+1\right)$
$\displaystyle~{}\sqrt{1+\delta_{{V_{1}}_{\lambda_{1}}}^{{V_{2}}_{\lambda_{2}}}}\sqrt{1+\delta_{{V_{3}}_{\lambda_{3}}}^{{V_{4}}_{\lambda_{4}}}}\,\times$
(8) $\displaystyle\times
d_{\lambda\mu}^{J}(\theta)~{}e^{iM\varphi}~{}T^{J}({V_{1}}_{\lambda_{1}}{V_{2}}_{\lambda_{2}}\to{V_{3}}_{\lambda_{3}}{V_{4}}_{\lambda_{4}})\;\;,$

where $\lambda=\lambda_{1}-\lambda_{2}$, $\mu=\lambda_{3}-\lambda_{4}$,
$M=\lambda_{1}-\lambda_{2}-\lambda_{3}+\lambda_{4}$, and $\theta$ ($\varphi$)
is the polar (azimuthal) scattering angle. $d$ is the usual Wigner rotation
matrix. This expression holds for gauge bosons with $\lambda=0,\pm 1$, and for
scalars (Higgs or ALP) with $\lambda\equiv 0$; the fermion case will be
addressed below. For further details and conventions see Ref. Corbett et al.
(2015).

In the limit $S\gg(M_{V_{1}}+M_{V_{2}})^{2}$, partial-wave unitarity for a
given _elastic_ channel requires that

$|T^{J}({V_{1}}_{\lambda_{1}}{V_{2}}_{\lambda_{2}}\to{V_{1}}_{\lambda_{1}}{V_{2}}_{\lambda_{2}})|\leq
1\,.$ (9)

The most stringent bounds are obtained by diagonalizing $T^{J}$ in the
particle and helicity space and then applying the condition in Eq. (9) to each
of the eigenvalues. This is the approach which we follow.

We start by calculating the scattering amplitudes for all possible
combinations of bosons and helicities generated by the SM extended with the
Lagrangian in Eq. (1) for a given total electric charge $Q=2,1,0$ and that
give non-vanishing projections on a given partial wave $J$ proportional to
some ALP coupling. Conservation of color implies that initial or final states
with color have to be considered independently of those in a color singlet
state. So one is led to consider separately the $T^{J}$ ($\overline{T}^{J}$)
amplitude matrices for processes with color singlet (octet) in the initial and
final states. One must also take into account that parity conservation at tree
level implies the relation

$T^{J}({V_{1}}_{\lambda_{1}}{V_{2}}_{\lambda_{2}}\to{V_{3}}_{\lambda_{3}}{V_{4}}_{\lambda_{4}})=(-1)^{\lambda_{1}-\lambda_{2}-\lambda_{3}+\lambda_{4}}T^{J}({V_{1}}_{-\lambda_{1}}{V_{2}}_{-\lambda_{2}}\to{V_{3}}_{-\lambda_{3}}{V_{4}}_{-\lambda_{4}})\;\;.$
(10)

and leads to a reduction of the number of independent helicity amplitudes.
Time-reversal invariance further reduces the number of helicity amplitudes
that need to be evaluated.

Altogether, the initial/final states contributing a priori to the $T^{J}$
matrices for each value of $Q$ and $J$ are:

$\footnotesize\begin{array}[]{c |ccccccccccccccccl}(Q,J)&{\rm
States}&&&&&&&&&&&&&&&&\rm{Total}\\\
\hline\cr\hline\cr(2,0)&W^{+}_{\pm}W^{+}_{\pm}&W^{+}_{0}W^{+}_{0}&&&&&&&&&&&&&&&3\\\\[2.84544pt]
\hline\cr(2,1)&W^{+}_{\pm}W^{+}_{0}&W^{+}_{0}W^{+}_{\pm}&&&&&&&&&&&&&&&4\\\\[2.84544pt]
\hline\cr(1,0)&W^{+}_{\pm}Z_{\pm}&W^{+}_{0}Z_{0}&W^{+}_{\pm}\gamma_{\pm}&W^{+}_{0}a&&&&&&&&&&&&&6\\\\[2.84544pt]
\hline\cr(1,1)&{W^{+}_{0}Z_{0}}&{W^{+}_{\pm}Z_{0}}&{W^{+}_{0}Z_{\pm}}&W^{+}_{\pm}Z_{\pm}&{W^{+}_{0}\gamma_{\pm}}&W^{+}_{\pm}\gamma_{\pm}&{W^{+}_{\pm}a}&{W^{+}_{0}a}&W^{+}_{\pm}H&&&&&&&&16\\\\[2.84544pt]
\hline\cr(0,0)&W^{+}_{\pm}W^{-}_{\pm}&W^{+}_{0}W^{-}_{0}&Z_{\pm}Z_{\pm}&Z_{0}Z_{0}&Z_{\pm}\gamma_{\pm}&\gamma_{\pm}\gamma_{\pm}&Z_{0}H&G_{\pm}G_{\pm}&Z_{0}a&Ha&aa&HH&&&&&17\\\\[2.84544pt]
\hline\cr(0,1)&{W^{+}_{0}W^{-}_{0}}&{W^{+}_{\pm}W^{-}_{0}}&{W^{+}_{0}W^{-}_{\pm}}&W^{+}_{\pm}W^{-}_{\pm}&Z_{\pm}Z_{0}&Z_{0}Z_{\pm}&Z_{\pm}\gamma_{\pm}&Z_{0}\gamma_{\pm}&Z_{0}H&Z_{\pm}H&\gamma_{\pm}H&{Z_{0}a}&{Z_{\pm}a}&{\gamma_{\pm}a}&Ha&&26\\\\[2.84544pt]
\end{array}$ (11)

and correspondingly the states contributing to the $\overline{T}^{J}$ matrices
are:

$\begin{array}[]{c |cccccc l}(Q,J)&{\rm States}&&&&&\rm{Total}\\\
\hline\cr\hline\cr(1,0)&{W^{+}_{\pm}G_{\pm}}&&&&&2\\\\[2.84544pt]
\hline\cr(1,1)&{W^{+}_{0}G_{\pm}}&{W^{+}_{\pm}G_{\pm}}&&&&4\\\\[2.84544pt]
\hline\cr(0,0)&{Z_{\pm}G_{\pm}}&{\gamma_{\pm}G_{\pm}}&&&&4\\\\[2.84544pt]
\hline\cr(0,1)&G_{\pm}G_{\pm}&{G_{\pm}a}&{Z_{\pm}G_{\pm}}&{Z_{0}G_{\pm}}&{\gamma_{\pm}G_{\pm}}&10\\\\[2.84544pt]
\end{array}$ (12)

where upper indices indicate charge and lower indices helicity. We also
display in Eqs. (11) and (12) the dimensionality of the particle and helicity
matrix for each independent $(Q\,,J)$ channel. In Eq. (11) the states $HH$ and
$W^{\pm}H$ are only present when the dimension 6 operator is considered.

We list in Tables 1– 4, that are shown in the Appendix, the expressions for
the most $S$-divergent part of the amplitudes for the channels which give the
dominant contribution to the $T^{J}$ and $\overline{T}^{J}$ matrices.

Bounds on individual operators. As seen in Tables 1– 3, for processes with
zero or two ALPs as external states, the most energy-divergent amplitudes
occur for scattering of transversely polarized gauge bosons, as expected.
These amplitudes are all proportional to the product of two axion couplings,
therefore, the two powers of their momentum involved in the coupling of ALP to
the gauge boson generate the leading $S/f_{a}^{2}$ dependence. A good fraction
of them contributes to $J=0$ matrices, which are, a priori, expected to lead
the strongest bounds. Furthermore, for amplitudes with gluon pairs, the
strongest bounds are obtained for the gluon pair in the singlet color state
$\frac{1}{N_{C}^{2}-1}\displaystyle\sum_{a=1}^{N_{C}^{2}-1}\ket{G^{a}G^{a}}$.

Altogether from the diagonalization of the $J=0$ matrices and assuming only
one non-zero coupling at a time we find that the largest eigenvalues
correspond to the $Q=0$, $T^{0}$ matrix and read

$\displaystyle\frac{1+\sqrt{97}}{16{\pi}}\,\frac{S}{f_{a}^{2}}\,C_{\tilde{W}}^{2}\,,$
$\displaystyle\frac{1+\sqrt{33}}{16{\pi}}\,\frac{S}{f_{a}^{2}}\,C_{\tilde{B}}^{2}\,,$
$\displaystyle\frac{4\,(N_{C}^{2}-1)}{{\pi}}\,\frac{S}{f_{a}^{2}}\,C_{\tilde{G}}^{2}\,$
$\displaystyle{\rm
and}\qquad\frac{1}{32\pi}\frac{S}{f_{a}^{2}}C_{a\Phi}^{(2)}\,,$ (13)

respectively. Applying the condition in Eq. (9) to each of these eigenvalues
we obtain the bounds

$\displaystyle|C_{\tilde{W}}|$ $\displaystyle\leq$
$\displaystyle{2.1}\,\frac{f_{a}}{\rm TeV}\,\left(\frac{\rm
TeV}{\sqrt{S}}\right)\,,$ (14) $\displaystyle|C_{\tilde{B}}|$
$\displaystyle\leq$ $\displaystyle{2.7}\,\frac{f_{a}}{\rm
TeV}\,\left(\frac{\rm TeV}{\sqrt{S}}\right)\,,$ (15)
$\displaystyle|C_{\tilde{G}}|$ $\displaystyle\leq$
$\displaystyle{0.31}\,\frac{f_{a}}{\rm TeV}\,\left(\frac{\rm
TeV}{\sqrt{S}}\right)\,,$ (16) $\displaystyle|C_{a\Phi}^{(2)}|$
$\displaystyle\leq$ $\displaystyle
101\,\frac{f_{a}^{2}}{\mathrm{TeV^{2}}}\,\left(\frac{\rm
TeV^{2}}{S}\right)\,.$ (17)

We observe that the constraint on the dimension 6 operator $O_{a\Phi}^{(2)}$
is dominated by scattering amplitudes with 2 ALP external states.

Unlike the amplitudes with even number of ALP in the external states, some
helicity amplitudes with only one ALP in either the initial or final state
have a leading behavior $S^{\frac{3}{2}}/(f_{a}\,M_{W}^{2})$, as seen in Table
4 444This is at variance with what is found for effective interactions in the
SMEFT Corbett et al. (2015, 2017); Almeida et al. (2020) for which all
operators of a given dimension lead to most divergent amplitudes with the same
power of $S$, that is, $S$ for dimension-six operators and $S^{2}$ for
dimension-8 operators.. These amplitudes involve two longitudinally polarized
gauge bosons, whose polarization vectors are proportional to $\sqrt{S}$, and
one transversely polarized gauge boson whose momentum contributes another
power of $\sqrt{S}$. This configuration can only be generated by a combination
of the SM vertices and those induced by $O_{\tilde{W}}$ and, consequently, the
amplitudes involve a single power of the $C_{\tilde{W}}$ coupling and of the
SM coupling $e$, and do not depend on any other Wilson coefficient. As seen
from the scattering angle dependence of the amplitudes in Table 4, they
contribute only to the $T^{J=1}$ matrix with either $Q=0$ or $Q=1$.
Diagonalizing these we find that the largest eigenvalue is

$\frac{\sqrt{1+c_{w}^{2}+2\,c_{w}^{4}}}{24\,\pi}\,\frac{S^{3/2}}{f_{a}\,M_{W}^{2}}\,e\,C_{\tilde{W}}\;,$
(18)

where $c_{w}$ is the cosine of the weak mixing angle. Therefore, the condition
in Eq. (9) implies the constraint on $C_{\tilde{W}}$

$|C_{\tilde{W}}|\leq 0.14\,\frac{f_{a}}{\rm TeV}\left(\frac{\rm
TeV}{\sqrt{S}}\right)^{3}\;.$ (19)

Comparing the bounds on $C_{\tilde{W}}$ from $J=0$-wave unitarity, Eq. (14),
and from $J=1$-wave unitarity, Eq. (19), we find that the constraint derived
from the $J=1$ amplitudes is the strongest for

$\sqrt{S}>260\;{\rm GeV}\;.$

Including multiple operators simultaneously. Fixing $C_{a\Phi}^{(2)}=0$ (or
equivalently barring cancellations between dimension 5 and 6 terms) and
allowing multiple dimension 5 operators to vary simultaneously does not alter
significantly the bounds reported above. For $C_{\tilde{W}}$ this is obvious,
because the leading constraint Eq. (19) is genuinely independent of the other
Wilson coefficients. For $C_{\tilde{B}}$ and $C_{\tilde{G}}$ this can be
understood considering that $C_{\tilde{G}}$ is dominantly constrained by
$G_{\pm}G_{\pm}\to G_{\pm}G_{\pm}$ scattering in the color singlet channel,
which is independent of $C_{\tilde{B}}$. We have also verified this explicitly
by diagonalizing the $Q=0$ $T^{J=0}$ matrix with $C_{\tilde{B}}$,
$C_{\tilde{G}}$ present at the same time. The diagonalization can still be
done analytically though the resulting expressions for the eigenvalues are not
particularly illuminating. Imposing the unitarity limits on those eigenvalues
yields the same bounds as in Eqs. (15) and (16).

Allowing all operators of dimension 5 and 6 to be present simultaneously (i.e.
allowing cancellations between both orders) we find that the largest
eigenvalues are

$\displaystyle\frac{5}{8{\pi}}\,\frac{S}{f_{a}^{2}}\,C_{\tilde{W}}^{2}\,,$
$\displaystyle\frac{1}{8{\pi}}\,\frac{S}{f_{a}^{2}}\,C_{\tilde{B}}^{2}\,,$
$\displaystyle\frac{4\,(N_{C}^{2}-1)}{{\pi}}\,\frac{S}{f_{a}^{2}}\,C_{\tilde{G}}^{2}\,$
$\displaystyle{\rm
and}\qquad\qquad\frac{1}{32\pi}\frac{S}{f_{a}^{2}}C_{a\Phi}^{(2)}\,,$ (20)

and correspondingly the unitarity limits on the Wilson coefficients are:

$\displaystyle|C_{\tilde{W}}|$ $\displaystyle\leq$
$\displaystyle{2.2}\,\frac{f_{a}}{\rm TeV}\,\left(\frac{\rm
TeV}{\sqrt{S}}\right)\,,$ (21) $\displaystyle|C_{\tilde{B}}|$
$\displaystyle\leq$ $\displaystyle{5.0}\,\frac{f_{a}}{\rm
TeV}\,\left(\frac{\rm TeV}{\sqrt{S}}\right)\,,$ (22)
$\displaystyle|C_{\tilde{G}}|$ $\displaystyle\leq$
$\displaystyle{0.31}\,\frac{f_{a}}{\rm TeV}\,\left(\frac{\rm
TeV}{\sqrt{S}}\right)\,,$ (23) $\displaystyle|C_{a\Phi}^{(2)}|$
$\displaystyle\leq$ $\displaystyle
101\,\frac{f_{a}^{2}}{\mathrm{TeV^{2}}}\,\left(\frac{\rm
TeV^{2}}{S}\right)\,.$ (24)

These results hold irrespective of whether $C_{\tilde{W}},\,C_{\tilde{B}}$ and
$C_{\tilde{G}}$ are included simultaneously or individually. It is also worth
noting that the bounds on $C_{\tilde{G}}$ and $C_{a\Phi}^{(2)}$ are unchanged
compared to the individual limits (16) and (17). Considering that
$C_{\tilde{W}}$ is always dominantly constrained by Eq. (19), we conclude that
only the unitarity constraints on $C_{\tilde{B}}$ depends significantly on
whether $C_{a\Phi}^{(2)}$ is included or not.

Truncating at dimension-5. Finally, it can be interesting to investigate
bounds on the dimension-5 interactions only. As we have seen above, the most
stringent bounds on $C_{\tilde{W}}$ originates from processes exhibiting just
one dimension-5 vertex, therefore, it is not modified when we truncate the EFT
expansion to $\mathcal{O}(f_{a}^{-1})$.

In order to obtain limits on $C_{\tilde{B}}$ and $C_{\tilde{G}}$ independently
of assumptions about $C_{a\Phi}^{(2)}$, we can restrict our analysis to a
subspace of initial states such that contributions of the dimension-6 operator
are negligible for all the scattering amplitudes retained. This is achieved by
eliminating “flavor” states in Eqs. (11) and (12) that lead to processes
containing two ALP external legs. We can re-derive the constraints on this
flavor subspace and we obtain that the largest eigenvalues come from the
$Q=0$, $T^{0}$ matrix and they coincide with those in Eq. (20), leading to the
bounds in Eqs. (21)–(23). The result is the same irrespective of whether
$C_{\tilde{B}}$ and $C_{\tilde{G}}$ are included simultaneously or
individually.

###  III.2 Helicity amplitudes involving fermions

ALP couplings to fermions can contribute to processes

${f_{1}}_{\sigma_{1}}{\bar{f_{2}}}_{\sigma_{2}}\to{V_{3}}_{\lambda_{3}}{V_{4}}_{\lambda_{4}}\,,$
(25)

which can also violate unitarity. In this case the partial-wave expansion is
given by

$\mathcal{M}({f_{1}}_{\sigma_{1}}{\bar{f_{2}}}_{\sigma_{2}}\to{V_{3}}_{\lambda_{3}}{V_{4}}_{\lambda_{4}})=16\pi\sum_{J}\left(2J+1\right)~{}\delta_{\sigma_{1},\sigma_{2}}d_{\sigma_{1}-\sigma_{2},\lambda_{3}-\lambda_{4}}^{J}(\theta)~{}T^{J}({f_{1}}_{\sigma_{1}}{\bar{f_{2}}}_{\sigma_{2}}\to{V_{3}}_{\lambda_{3}}{V_{4}}_{\lambda_{4}})\,.$
(26)

In principle,
${f_{1}}_{\sigma_{1}}{\bar{f_{2}}}_{\sigma_{2}}\to{V_{3}}_{\lambda_{3}}{V_{4}}_{\lambda_{4}}$
amplitudes of a given $J$ partial wave can be incorporated together with the
${V_{1}}_{\lambda_{1}}{V_{2}}_{\lambda_{2}}\to{V_{3}}_{\lambda_{3}}{V_{4}}_{\lambda_{4}}$
amplitudes in the corresponding $T^{J}$ matrix by extending the basis of
states to incorporate the relevant
${f_{1}}_{\sigma_{1}}{\bar{f_{2}}}_{\sigma_{2}}$ combinations contributing to
a given $Q$; see, for example, Ref. Chanowitz et al. (1978). However, we find
that the most energy divergent amplitudes for fermion-antifermion scattering
grow at most as $\sqrt{S}$ and, therefore, the contributions from the ALP-
fermion couplings enter with different power of $S$ with respect to the ALP-
gauge-boson couplings in the eigenvalues of this generalized $T^{J}$ matrices.
Thus, in order to derive independent unitarity constraints on the $C_{f\Phi}$
couplings we find it more convenient to follow the alternative procedure
presented in Ref. Baur and Zeppenfeld (1988) and relate the corresponding
${f_{1}}_{\sigma_{1}}{\bar{f_{2}}}_{\sigma_{2}}\to{V_{3}}_{\lambda_{3}}{V_{4}}_{\lambda_{4}}$
amplitude to that of the elastic process

${f_{1}}_{\sigma_{1}}{\bar{f_{2}}}_{\sigma_{2}}\to{f_{1}}_{\sigma_{1}}{\bar{f_{2}}}_{\sigma_{2}}\,.$
(27)

In this case the unitarity relation is

$\displaystyle{\rm
Im}[T^{J}({f_{1}}_{\sigma_{1}}{\bar{f_{2}}}_{\sigma_{2}}\to{f_{1}}_{\sigma_{1}}{\bar{f_{2}}}_{\sigma_{2}})]$
$\displaystyle=$
$\displaystyle\left|T^{J}({f_{1}}_{\sigma_{1}}{\bar{f_{2}}}_{\sigma_{2}}\to{f_{1}}_{\sigma_{1}}{\bar{f_{2}}}_{\sigma_{2}})\right|^{2}$
$\displaystyle+\sum_{{V_{3}}_{\lambda_{3}},{V_{4}}_{\lambda_{4}}}\left|T^{J}({f_{1}}_{\sigma_{1}}{\bar{f_{2}}}_{\sigma_{2}}\to{V_{3}}_{\lambda_{3}}{V_{4}}_{\lambda_{4}})\right|^{2}+\sum_{N}\left|T^{J}({f_{1}}_{\sigma_{1}}{\bar{f_{2}}}_{\sigma_{2}}\to
N)\right|^{2}\,,$

where we take the limit
$S\gg(M_{V_{3}}+M_{V_{4}})^{2}\;,\;(M_{f_{1}}+M_{f_{2}})^{2}$. $N$ represents
any state which ${f_{1}}_{\sigma_{1}}{\bar{f_{2}}}_{\sigma_{2}}$ can
annihilate into which does not consist of two bosons. Eq. (III.2) is a
quadratic equation for ${\rm
Im}[T^{J}({f_{1}}_{\sigma_{1}}{\bar{f_{2}}}_{\sigma_{2}}\to{f_{1}}_{\sigma_{1}}{\bar{f_{2}}}_{\sigma_{2}})]$
which only admits a solution if

$\sum_{{V_{3}}_{\lambda_{3}},{V_{4}}_{\lambda_{4}}}\left|T^{J}({f_{1}}_{\sigma_{1}}{\bar{f_{2}}}_{\sigma_{2}}\to{V_{3}}_{\lambda_{3}}{V_{4}}_{\lambda_{4}})\right|^{2}\leq\frac{1}{4}\;.$
(29)

The strongest bounds can be found by considering some optimized linear
combinations

$\ket{X}=\sum_{f_{1},\sigma_{1}}x_{f_{2},\sigma_{2}}\ket{{f_{1}}_{\sigma_{1}}{\bar{f_{2}}}_{\sigma_{2}}}$
(30)

with the normalization condition $\sum_{f\sigma}|x_{f\sigma}|^{2}=1$, for
which the amplitude $T^{J}(X\to{V_{3}}_{\lambda_{3}}{V_{4}}_{\lambda_{4}})$ is
the largest.

In this approach, processes of fermion scattering into one gauge boson and one
ALP provide independent constraints on the ALP-fermion coupling. As mentioned
above, the most divergent relevant helicity amplitudes grow as $\sqrt{S}$ and
are listed in Table 5. For couplings to leptons of a given generation the
strongest bounds are obtained with
$\ket{X}=\frac{1}{\sqrt{2}}\ket{e^{-}_{+}\,e^{+}_{+}\,+\,{e^{-}_{-}\,e^{+}_{-}}}$
(or equivalently with $\ket{X}=\ket{{\nu_{-}\,e^{+}_{-}}}$). For couplings to
quarks of a given generation, accounting for the $N_{C}=3$ color states, the
strongest bounds are obtained with
$\ket{X}=\dfrac{1}{\sqrt{2N_{c}}}\displaystyle\sum_{a=1}^{N_{c}}\ket{q^{a}_{+}\,\bar{q}^{a}_{+}\,+{\,q^{a}_{-}\,\bar{q}^{a}_{-}}}$,
or equivalently with
$\dfrac{1}{\sqrt{N_{c}}}\displaystyle\sum_{a=1}^{N_{c}}\ket{{u^{a}_{+}\,\bar{d}^{a}_{+}}}$
and
$\dfrac{1}{\sqrt{N_{c}}}\displaystyle\sum_{a=1}^{N_{c}}\ket{u^{a}_{-}\,\bar{d}^{a}_{-}}$
). Furthermore, within the assumed flavour symmetry of the axion coupling to
fermions, the strongest bounds correspond to the processes with fermions of
the third generation and they read

$\displaystyle\left|C_{a\Phi}-C_{e\Phi}\right|$
$\displaystyle\leq\frac{16\,\pi}{|Y_{\tau}|}\left(\frac{f_{a}}{\sqrt{S}}\right)$
$\displaystyle=\frac{50}{|Y_{\tau}|}\frac{f_{a}}{\rm TeV}\left(\frac{\rm
TeV}{\sqrt{S}}\right)\;,$ (31) $\displaystyle\left|C_{a\Phi}+C_{u\Phi}\right|$
$\displaystyle\leq\frac{16\,\pi}{\sqrt{N_{C}}\,|Y_{t}|}\left(\frac{f_{a}}{\sqrt{S}}\right)$
$\displaystyle=\frac{29}{|Y_{t}|}\frac{f_{a}}{\rm TeV}\left(\frac{\rm
TeV}{\sqrt{S}}\right)\;,$ (32) $\displaystyle\left|C_{a\Phi}-C_{d\Phi}\right|$
$\displaystyle\leq\frac{16\,\pi}{\sqrt{N_{C}}\,|Y_{b}|}\left(\frac{f_{a}}{\sqrt{S}}\right)$
$\displaystyle=\frac{29}{|Y_{b}|}\frac{f_{a}}{\rm TeV}\left(\frac{\rm
TeV}{\sqrt{S}}\right)\;.$ (33)

Because these bounds are inversely proportional to the Yukawa coupling of the
fermion and involve larger coefficients, we conclude that the unitarity
constraints on the ALP-fermion couplings are orders of magnitude weaker than
those on ALP-gauge-boson couplings even for the coupling to the up-quarks.
Moreover, the operator ${\cal O}_{a\Phi}$ should only be considered in a
scenario where the fermionic operators are absent. In this case, the most
stringent unitarity constraint on its coupling originates from Eq. (32).

Figure 1: Summary of unitarity bounds derived in this work. The shaded regions
indicate allowed values of $f_{a}/C$ ($f_{a}/\sqrt{C}$ in the case of
$C_{a\Phi}^{(2)}$) and $\sqrt{S}$ for each bosonic interaction. For
$C_{\tilde{W}}$ we plot the most stringent bound between Eq. (14) and (19).
The solid red line indicates the bound on $C_{\tilde{B}}$ in Eq.(22), derived
allowing $C_{a\Phi}^{(2)}$ to vary simultaneously. The individual bound on
$C_{\tilde{B}}$ given in Eq. (15) is drawn as a dotted line in the same color.
The bounds on fermionic operators are always subdominant and are only marked
as solid lines. The bound on $C_{a\Phi}$, that should only considered in a
setup where the fermionic operators are absent, coincides with that on
$C_{u\Phi}$. Finally, the grey dashed line marks the diagonal for reference.

##  IV Conclusions

We have derived maximal constraints on the effective interactions of Axion-
Like-Particles from partial-wave unitarity in $2\to 2$ scattering processes.
Our results are summarized in Fig. 1. They hold in the kinematic regime where
$\sqrt{S}\gg v$ and the ALP mass was also implicitly taken to be
$m_{a}\ll\sqrt{S}$. Furthermore the consistency of the ALP EFT expansion
requires $\sqrt{S}\ll f_{a}$.

We find that, for fixed $C/f_{a}$, the most stringent unitarity bound is
imposed on $C_{\tilde{W}}$ in $VV\to Va$ scattering processes, while the
weakest limits are on ALP-fermion interactions. The constraints exhibit only a
limited dependence on whether the effective operators are taken individually
or allowed to vary simultaneously, signaling that each of them is dominantly
constrained in a class of scattering amplitudes that is nearly orthogonal to
the others.

The constraints we have derived can be particularly relevant for ALP searches
at colliders Jaeckel et al. (2013); Mimasu and Sanz (2015); Jaeckel and
Spannowsky (2016); Brivio et al. (2017); Bauer et al. (2017, 2019); Frugiuele
et al. (2018); Craig et al. (2018); Ebadi et al. (2019); Yue et al. (2019);
Gavela et al. (2020); İnan and Kisselev (2020); Haghighat et al. (2020);
Goncalves and Sauter (2020); Flórez et al. (2021); Bonilla et al.  where,
depending on their masses, ALP particles could be observed in $a\to XX$
decays, with $X$ a generic SM state, in $Xa$ associated production (with the
ALP either going undetected or decaying to photons or light fermions) or in
non-resonant $2\to 2$ scattering processes, where the ALP appears as an off-
shell internal line Gavela et al. (2020); Flórez et al. (2021); Bonilla et al.
.

In this respect, let us stress that our results should not be interpreted as
strict unitarity constraints on any specific process used in the ALP searches,
in the sense that it might be difficult to directly identify the kinematic
information available with the subprocess center-of-mass energy of an
individual $2\to 2$ scattering. Notwithstanding, unitarity bounds must be
satisfied in the event generation and, consequently, can affect the shapes of
expected distributions used in the searches.

For example, recently, the ATLAS collaboration has searched for axions in
events with an energetic jet Aad et al. (2021a) or a photon Aad et al. (2021b)
and missing transverse momentum. The monojet analysis Aad et al. (2021a)
constrains the axion coupling to gluons to satisfy $C_{\tilde{G}}/f_{a}<0.008$
TeV-1 at 95% CL. Using Eq. (23), we find that for the largest allowed coupling
in this search unitarity is preserved up to center mass-of-mass energy of 39
TeV, clearly beyond the LHC reach. On the other hand, the mono-photon analysis
limits the $C_{\tilde{W}}$ coupling to satisfy $C_{\tilde{W}}/f_{a}<0.12$
TeV-1 at 95% CL. From Eq. (19) we read that for $C_{\tilde{W}}/f_{a}$ at the
95% CL boundary, unitarity is violated in subprocesses with center-of-mass
energy greater than 1.04 TeV. We conclude that the tail of the expected
missing $E_{T}$ distribution should be analyzed cautiously and the unitarity
constraints could have an impact in the derivation of the mono-photon search
bound.

The unitarity bounds derived in this work would be also relevant in the event
that an ALP signal will be detected in the future (independently of the energy
regime at which the experimental search is conducted), leading to a defined
measurement of one or more ALP couplings. In this case, unitarity bounds would
provide an upper limit to the mass scale of the new physics sector the ALP
originates from and motivate further searches in this energy region.

## Acknowledgments

O.J.P.E. is supported in part by Conselho Nacional de Desenvolvimento
Centífico e Tecnológico (CNPq) and by Fundação de Amparo à Pesquisa do Estado
de São Paulo (FAPESP) grant 2019/04837-9. M.C.G-G is supported by spanish
grant PID2019-105614GB-C21, by USA-NSF grant PHY-1915093, and by AGAUR
(Generalitat de Catalunya) grant 2017-SGR-929. The authors acknowledge the
support of European ITN grant H2020-MSCA-ITN-2019//860881-HIDDeN.

## Helicity amplitudes at leading order in $S$

We present here the list of unitarity violating amplitudes for all the
$2\rightarrow 2$ scattering processes considered in the evaluation of the
unitarity constraints.

$\begin{array}[]{|c|cccc|c|}\hline\cr&\lambda_{1}&\lambda_{2}&\lambda_{3}&\lambda_{4}&{\cal
M}\left(\times\frac{S}{f_{a}^{2}}\right)\\\ \hline\cr W^{+}W^{+}\rightarrow
W^{+}W^{+}&1&-1&1&1&2\,C_{\tilde{W}}^{2}\\\ &-1&1&\mp 1&\pm
1&X_{\pm}\,C_{\tilde{W}}^{2}\\\ \hline\cr W^{+}Z\rightarrow W^{+}Z&-1&\mp
1&1&\pm 1&\pm 2\,C_{\tilde{W}}\;{\rm
X_{-}}\left(\,s_{w}^{2}\,C_{\tilde{B}}\,+\,c_{w}^{2}\,C_{\tilde{W}}\right)\\\
\hline\cr W^{+}Z\rightarrow W^{+}\gamma&-1&\mp 1&1&\pm 1&\mp
2\,c_{w}\,C_{\tilde{W}}\,s_{w}\;{\rm
X_{-}}(\,C_{\tilde{B}}-\,C_{\tilde{W}})\\\ \hline\cr W^{+}\gamma\rightarrow
W^{+}\gamma&-1&\mp 1&1&\pm 1&\pm 2\,C_{\tilde{W}}\;{\rm
X_{-}}\left(\,c_{w}^{2}\,C_{\tilde{B}}\,+\,s_{w}^{2}\,C_{\tilde{W}}\right)\\\
\hline\cr W^{+}W^{-}\rightarrow W^{+}W^{-}&-1&-1&-1&-1&4\,C_{\tilde{W}}^{2}\\\
&-1&\mp 1&1&\pm 1&-2\,C_{\tilde{W}}^{2}\;{\rm X_{\pm}}\\\ \hline\cr
W^{+}W^{-}\rightarrow ZZ&-1&-1&\mp 1&\mp 1&\pm
2\sqrt{2}\,C_{\tilde{W}}\left(\,s_{w}^{2}\,C_{\tilde{B}}+\,c_{w}^{2}\,C_{\tilde{W}}\right)\\\
\hline\cr W^{+}W^{-}\rightarrow Z\gamma&-1&-1&\pm 1&\pm 1&\pm
4\,c_{w}\,C_{\tilde{W}}\,s_{w}(\,C_{\tilde{B}}-\,C_{\tilde{W}})\\\ \hline\cr
W^{+}W^{-}\rightarrow\gamma\gamma&-1&-1&\mp 1&\mp 1&\pm
2\sqrt{2}\,C_{\tilde{W}}\left(\,c_{w}^{2}\,C_{\tilde{B}}\,+\,s_{w}^{2}\,C_{\tilde{W}}\right)\\\
\hline\cr ZZ\rightarrow
ZZ&-1&-1&-1&-1&2\left(\,c_{w}^{2}\,C_{\tilde{W}}-\,s_{w}^{2}\,C_{\tilde{B}}\right)^{2}\\\
&-1&1&\mp 1&\pm
1&-X_{\pm}\,\left(\,c_{w}^{2}\,C_{\tilde{W}}-\,s_{w}^{2}\,C_{\tilde{B}}\right)^{2}\\\
\hline\cr ZZ\rightarrow
Z\gamma&-1&-1&-1&-1&2\sqrt{2}\,c_{w}\,s_{w}(\,C_{\tilde{B}}-\,C_{\tilde{W}})\
\left(\,s_{w}^{2}\,C_{\tilde{B}}-\,c_{w}^{2}\,C_{\tilde{W}}\right)\\\
&-1&-1&\mp 1&\pm
1&-\sqrt{2}\,X_{\pm}\,c_{w}\,s_{w}(\,C_{\tilde{B}}-\,C_{\tilde{W}})\
\left(\,s_{w}^{2}\,C_{\tilde{B}}-\,c_{w}^{2}\,C_{\tilde{W}}\right)\\\
\hline\cr
ZZ\rightarrow\gamma\gamma&-1&-1&-1&-1&-2\left(\,(C_{\tilde{B}}^{2}+C_{\tilde{W}}^{2})\,c_{w}^{2}\,s_{w}^{2}-\,C_{\tilde{B}}\,C_{\tilde{W}}\
\left(\,c_{w}^{4}+\,s_{w}^{4}\right)\right)\\\
&-1&-1&1&1&-2\,C_{\tilde{B}}\,C_{\tilde{W}}\\\ &-1&1&\mp 1&\pm
1&X_{\pm}\,(C_{\tilde{B}}-C_{\tilde{W}})^{2}\,c_{w}^{2}\,s_{w}^{2}\\\
\hline\cr Z\gamma\rightarrow
Z\gamma&-1&-1&-1&-1&4\,c_{w}^{2}\,s_{w}^{2}(\,C_{\tilde{B}}-\,C_{\tilde{W}})^{2}\\\
&-1&-1&1&1&-2\,C_{\tilde{B}}\,C_{\tilde{W}}\;{\rm X_{-}}\\\ &-1&1&\mp 1&\pm
1&-2\,X_{\pm}\,(C_{\tilde{B}}-C_{\tilde{W}})^{2}\,c_{w}^{2}\,s_{w}^{2}\\\
\hline\cr\gamma\gamma\rightarrow\gamma\gamma&-1&-1&-1&-1&2\left(\,c_{w}^{2}\,C_{\tilde{B}}\,+\,s_{w}^{2}\,C_{\tilde{W}}\right)^{2}\\\
&-1&1&\mp 1&\pm
1&-X_{\pm}\,\left(\,c_{w}^{2}\,C_{\tilde{B}}\,+\,s_{w}^{2}\,C_{\tilde{W}}\right)^{2}\\\
\hline\cr W^{+}W^{-}\rightarrow G^{a}G^{b}&-1&-1&\mp 1&\mp 1&\pm
8\sqrt{2}\,C_{\tilde{G}}\,C_{\tilde{W}}\ \,\delta_{ab}\\\ \hline\cr
ZZ\rightarrow G^{a}G^{b}&-1&-1&\mp 1&\mp 1&\pm
8\,C_{\tilde{G}}\left(\,s_{w}^{2}\,C_{\tilde{B}}+\,c_{w}^{2}\,C_{\tilde{W}}\right)\,\delta_{ab}\\\
\hline\cr Z\gamma\rightarrow G^{a}G^{b}&-1&-1&\mp 1&\mp 1&\mp
8\sqrt{2}\,C_{\tilde{G}}\,c_{w}\,s_{w}(\,C_{\tilde{B}}-\,C_{\tilde{W}})\
\,\delta_{ab}\\\ \hline\cr\gamma\gamma\rightarrow G^{a}G^{b}&-1&-1&\mp 1&\mp
1&\pm
8\,C_{\tilde{G}}\left(\,s_{w}^{2}\,C_{\tilde{B}}\,+\,c_{w}^{2}\,C_{\tilde{W}}\right)\,\delta_{ab}\\\
\hline\cr G^{a}G^{b}\rightarrow
G^{c}G^{d}&-1&-1&-1&-1&32\,C_{\tilde{G}}^{2}\,\delta_{ab}\,\delta_{cd}\\\
&-1&-1&1&1&16\,C_{\tilde{G}}^{2}(\;{\rm
X_{-}}\,\delta_{ac}\,\delta_{bd}+\;{\rm
X_{+}}\,\delta_{ad}\,\delta_{bc}-2\,\delta_{ab}\,\delta_{cd})\\\
&-1&1&-1&1&-4\,X_{+}\,C_{\tilde{G}}^{2}\,\delta_{ac}\,\delta_{bd}\\\
&-1&1&1&-1&-4\,X_{-}\,C_{\tilde{G}}^{2}\,\delta_{ad}\,\delta_{bc}\\\
\hline\cr\end{array}$

Table 1:  Leading contributions to the helicity amplitudes for the channels
involving SM bosons and even number of gluons in the initial or final state
and with projections in $J=0$. They contribute to the $T^{J}$ matrices with
$Q=2,1,0$ and $J=0,1$ In the expressions ${\rm X_{+}}=(1+\cos x)$ and ${\rm
X_{-}}=(1-\cos x)$ where $x$ is the polar angle.

$\begin{array}[]{|c|cccc|c|}\hline\cr&\lambda_{1}&\lambda_{2}&\lambda_{3}&\lambda_{4}&{\cal
M}\left(\times\frac{S}{f_{a}^{2}}\right)\\\ \hline\cr W^{+}a\rightarrow
W^{+}a&-1&0&-1&0&-2\,C_{\tilde{W}}^{2}\;{\rm X_{-}}\\\
&-1&0&1&0&4\,C_{\tilde{W}}^{2}\;{\rm X_{-}}\\\
&0&0&0&0&\frac{1}{2}C_{a\Phi}^{(2)}\;{\rm X_{-}}\\\ \hline\cr
W^{+}W^{-}\rightarrow aa&-1&-1&0&0&4\sqrt{2}\,C_{\tilde{W}}^{2}\\\
&-1&1&0&0&-2\sqrt{2}\,C_{\tilde{W}}^{2}\\\
&0&0&0&0&\frac{1}{\sqrt{2}}\,C_{a\Phi}^{(2)}\\\ \hline\cr ZZ\rightarrow
aa&-1&-1&0&0&4\left(\,s_{w}^{2}\,C_{\tilde{B}}^{2}\,+\,c_{w}^{2}\,C_{\tilde{W}}^{2}\right)\\\
&-1&1&0&0&-2\left(\,s_{w}^{2}\,C_{\tilde{B}}^{2}\,+\,c_{w}^{2}\,C_{\tilde{W}}^{2}\right)\\\
&0&0&0&0&C_{a\Phi}^{(2)}\\\ \hline\cr Z\gamma\rightarrow
aa&-1&-1&0&0&-4\sqrt{2}\,c_{w}\,s_{w}\left(C_{\tilde{B}}^{2}-\,C_{\tilde{W}}^{2}\right)\\\
&-1&1&0&0&2\sqrt{2}\,c_{w}\,s_{w}\left(C_{\tilde{B}}^{2}-\,C_{\tilde{W}}^{2}\right)\\\
\hline\cr Za\rightarrow Za&-1&0&-1&0&-2\;{\rm
X_{-}}\left(\,s_{w}^{2}\,C_{\tilde{B}}^{2}\,+\,c_{w}^{2}\,C_{\tilde{W}}^{2}\right)\\\
&-1&0&1&0&4\;{\rm
X_{-}}\left(\,s_{w}^{2}\,C_{\tilde{B}}^{2}\,+\,c_{w}^{2}\,C_{\tilde{W}}^{2}\right)\\\
&0&0&0&0&\frac{1}{2}C_{a\Phi}^{(2)}\;{\rm X_{-}}\\\ \hline\cr
Za\rightarrow\gamma a&-1&0&-1&0&2\,c_{w}\,s_{w}\;{\rm
X_{-}}\left(C_{\tilde{B}}^{2}-\,C_{\tilde{W}}^{2}\right)\\\
&-1&0&1&0&-4\,c_{w}\,s_{w}\;{\rm
X_{-}}\left(C_{\tilde{B}}^{2}-\,C_{\tilde{W}}^{2}\right)\\\
\hline\cr\gamma\gamma\rightarrow
aa&-1&-1&0&0&4\left(\,c_{w}^{2}\,C_{\tilde{B}}^{2}+\,s_{w}^{2}\,C_{\tilde{W}}^{2}\right)\\\
&-1&1&0&0&-2\left(\,c_{w}^{2}\,C_{\tilde{B}}^{2}+\,s_{w}^{2}\,C_{\tilde{W}}^{2}\right)\\\
\hline\cr\gamma a\rightarrow\gamma a&-1&0&-1&0&-2{\rm
X_{-}}\left(\,c_{w}^{2}\,C_{\tilde{B}}^{2}+\,s_{w}^{2}\,C_{\tilde{W}}^{2}\right)\\\
&-1&0&1&0&4{\rm
X_{-}}\left(\,c_{w}^{2}\,C_{\tilde{B}}^{2}+\,s_{w}^{2}\,C_{\tilde{W}}^{2}\right)\\\
\hline\cr G^{a}G^{b}\rightarrow
aa&-1&-1&0&0&16\,C_{\tilde{G}}^{2}\,\delta_{ab}\\\
&-1&11&0&0&-8\,C_{\tilde{G}}^{2}\,\delta_{ab}\\\ \hline\cr Ha\rightarrow
Ha&0&0&0&0&\frac{1}{2}C_{a\Phi}^{(2)}\;{\rm X_{-}}\\\ \hline\cr HH\rightarrow
aa&0&0&0&0&\frac{1}{2}C_{a\Phi}^{(2)}\\\ \hline\cr\end{array}$

Table 2:  Leading contributions to the helicity amplitudes for the channels
involving SM bosons and two ALPs and even number of gluons in the initial or
final state. They contribute to the $T^{J}$ matrices with $Q=1,0$ and $J=0$ or
$J=1$. In the expressions ${\rm X_{+}}=(1+\cos x)$ and ${\rm X_{-}}=(1-\cos
x)$ where $x$ is the polar angle.

$\begin{array}[]{|c|cccc|c|}\hline\cr&\lambda_{1}&\lambda_{2}&\lambda_{3}&\lambda_{4}&{\cal
M}\left(\times\frac{S}{f_{a}^{2}}\right)\\\ \hline\cr W^{+}G^{a}\rightarrow
W^{+}G^{b}&-1&\mp 1&1&\pm 1&\pm 8\,C_{\tilde{G}}\,C_{\tilde{W}}\;{\rm
X_{-}}\,\delta_{ab}\\\ \hline\cr ZG^{a}\rightarrow ZG^{b}&-1&\mp 1&1&\pm 1&\pm
8\,C_{\tilde{G}}\;{\rm
X_{-}}\left(\,s_{w}^{2}\,C_{\tilde{B}}+\,c_{w}^{2}\,C_{\tilde{W}}\right)\,\delta_{ab}\\\
\hline\cr ZG^{a}\rightarrow\gamma G^{b}&-1&\mp 1&1&\pm 1&\mp
8\,C_{\tilde{G}}\,c_{w}\,s_{w}\;{\rm
X_{-}}(\,C_{\tilde{B}}-\,C_{\tilde{W}})\,\delta_{ab}\\\ \hline\cr\gamma
G^{a}\rightarrow\gamma G^{b}&-1&\mp 1&1&\pm 1&\pm 8\,C_{\tilde{G}}\;{\rm
X_{-}}\left(\,c_{w}^{2}\,C_{\tilde{B}}\,+\,s_{w}^{2}\,C_{\tilde{W}}\right)\,\delta_{ab}\\\
\hline\cr G^{a}a\rightarrow G^{b}a&-1&0&-1&0&-8\,C_{\tilde{G}}^{2}\;{\rm
X_{-}}\,\delta_{ab}\\\ &-1&0&1&0&16\,C_{\tilde{G}}^{2}\;{\rm
X_{-}}\,\delta_{ab}\\\ \hline\cr\end{array}$

Table 3:  Leading contributions to the helicity amplitudes for the channels
with one gluon in the initial and final state. They contribute
$\overline{T}^{J}$ with $Q=1,0$ and $J=0,1$. ${\rm X_{+}}=(1+\cos x)$ and
${\rm X_{-}}=(1-\cos x)$ where $x$ is the polar angle.

$\begin{array}[]{|c|cccc|c|}\hline\cr&\lambda_{1}&\lambda_{2}&\lambda_{3}&\lambda_{4}&{\cal
M}\left(\times e\,\frac{S^{3/2}}{f_{a}M_{W}^{2}}\right)\\\ \hline\cr
W^{+}W^{-}\rightarrow
Za&-1&0&0&0&\sqrt{2}\,\frac{c_{w}^{2}}{s_{w}}\,C_{\tilde{W}}\,{\rm Y}\\\
&0&-1&0&0&-\sqrt{2}\,\frac{c_{w}^{2}}{s_{w}}\,C_{\tilde{W}}\,{\rm Y}\\\
&0&0&-1&0&\sqrt{2}\,\frac{c_{w}}{s_{w}}\,C_{\tilde{W}}\,{\rm Y}\\\ \hline\cr
W^{+}W^{-}\rightarrow\gamma a&0&0&-1&0&\sqrt{2}\,C_{\tilde{W}}\,{\rm Y}\\\
\hline\cr W^{+}Z\rightarrow
W^{+}a&-1&0&0&0&-\sqrt{2}\,\frac{c_{w}^{2}}{s_{w}}\,C_{\tilde{W}}\,{\rm Y}\\\
&0&-1&0&0&\sqrt{2}\,\frac{c_{w}^{2}}{s_{w}}\,C_{\tilde{W}}\,{\rm Y}\\\
&0&0&-1&0&-\sqrt{2}\,\frac{c_{w}^{2}}{s_{w}}\,C_{\tilde{W}}\,{\rm Y}\\\
\hline\cr W^{+}\gamma\rightarrow W^{+}a&0&-1&0&0&\sqrt{2}\,C_{\tilde{W}}\,{\rm
Y}\\\ \hline\cr\end{array}$

Table 4:  Leading contributions to the helicity amplitudes for the channels
involving SM bosons and one ALP. They contribute to the $T^{J}$ matrices with
$Q=1,0$ and $J=1$ In these amplitudes ${\rm Y}=\sin x$ where $x$ is the polar
angle.

$\begin{array}[]{|c|cccc|c|}\hline\cr&\sigma_{1}&\sigma_{2}&\lambda_{3}&\lambda_{4}&{\cal
M}\left(i\frac{\sqrt{S}}{f_{a}}\right)\\\ \hline\cr
e_{r}^{-}\,e_{s}^{+}\rightarrow
Z\,a&-&-&0&0&\frac{1}{\sqrt{2}}\left(C_{a\Phi}-C_{e\Phi}\right)\,(Y^{*}_{e})_{rs}\\\
&+&+&0&0&\frac{1}{\sqrt{2}}\left(C_{a\Phi}-C_{e\Phi}\right)\,(Y_{e})_{sr}\\\
\hline\cr\nu_{r}\,e^{+}_{s}\rightarrow
W^{+}\,a&-&-&0&0&\left(C_{a\Phi}-C_{e\Phi}\right)\,(Y^{*}_{e})_{rs}\\\
\hline\cr u_{r}^{a}\,\bar{u}_{s}^{b}\rightarrow
Z\,a&-&-&0&0&\frac{1}{\sqrt{2}}\left(C_{a\Phi}+C_{u\Phi}\right)\,(Y^{*}_{u})_{rs}\,\delta_{ab}\\\
&+&+&0&0&\frac{1}{\sqrt{2}}\left(C_{a\Phi}+C_{u\Phi}\right)\,(Y_{u})_{sr}\,\delta_{ab}\\\
\hline\cr d_{r}^{a}\,\bar{d}_{s}^{b}\rightarrow
Z\,a&-&-&0&0&\frac{1}{\sqrt{2}}\left(C_{a\Phi}-C_{d\Phi}\right)\,(Y^{*}_{d})_{rs}\,\delta_{ab}\\\
&+&+&0&0&\frac{1}{\sqrt{2}}\left(C_{a\Phi}-C_{d\Phi}\right)\,(Y_{d})_{sr}\,\delta_{ab}\\\
\hline\cr u_{r}^{a}\,\bar{d}_{s}^{b}\rightarrow
W^{+}\,a&-&-&0&0&\left(C_{a\Phi}-C_{d\Phi}\right)\,(Y^{*}_{d})_{rs}\,\delta_{ab}\\\
&+&+&0&0&\left(C_{a\Phi}+C_{u\Phi}\right)\,(Y_{u})_{sr}\,\delta_{ab}\\\
\hline\cr\end{array}$

Table 5:  Leading contributions to the helicity amplitudes for the channels
with fermion scattering. $a,b$ denote color indices and $r,s$ denote flavour
indices.

## References

  * Peccei and Quinn (1977a) R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. 38, 1440 (1977a).
  * Peccei and Quinn (1977b) R. D. Peccei and H. R. Quinn, Phys. Rev. D 16, 1791 (1977b).
  * Weinberg (1978) S. Weinberg, Phys. Rev. Lett. 40, 223 (1978).
  * Wilczek (1978) F. Wilczek, Phys. Rev. Lett. 40, 279 (1978).
  * Kim (1979) J. E. Kim, Phys. Rev. Lett. 43, 103 (1979).
  * Shifman et al. (1980) M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Nucl. Phys. B 166, 493 (1980).
  * Dine et al. (1981) M. Dine, W. Fischler, and M. Srednicki, Phys. Lett. B 104, 199 (1981).
  * Zhitnitsky (1980) A. R. Zhitnitsky, Sov. J. Nucl. Phys. 31, 260 (1980).
  * Rubakov (1997) V. A. Rubakov, JETP Lett. 65, 621 (1997), eprint hep-ph/9703409.
  * Berezhiani et al. (2001) Z. Berezhiani, L. Gianfagna, and M. Giannotti, Phys. Lett. B 500, 286 (2001), eprint hep-ph/0009290.
  * Hsu and Sannino (2005) S. D. H. Hsu and F. Sannino, Phys. Lett. B 605, 369 (2005), eprint hep-ph/0408319.
  * Hook (2015) A. Hook, Phys. Rev. Lett. 114, 141801 (2015), eprint 1411.3325.
  * Fukuda et al. (2015) H. Fukuda, K. Harigaya, M. Ibe, and T. T. Yanagida, Phys. Rev. D 92, 015021 (2015), eprint 1504.06084.
  * Chiang et al. (2016) C.-W. Chiang, H. Fukuda, M. Ibe, and T. T. Yanagida, Phys. Rev. D 93, 095016 (2016), eprint 1602.07909.
  * Gherghetta et al. (2016) T. Gherghetta, N. Nagata, and M. Shifman, Phys. Rev. D 93, 115010 (2016), eprint 1604.01127.
  * Dimopoulos et al. (2016) S. Dimopoulos, A. Hook, J. Huang, and G. Marques-Tavares, JHEP 11, 052 (2016), eprint 1606.03097.
  * Kobakhidze (2016) A. Kobakhidze (2016), eprint 1607.06552.
  * Agrawal and Howe (2018a) P. Agrawal and K. Howe, JHEP 12, 029 (2018a), eprint 1710.04213.
  * Agrawal and Howe (2018b) P. Agrawal and K. Howe, JHEP 12, 035 (2018b), eprint 1712.05803.
  * Gaillard et al. (2018) M. K. Gaillard, M. B. Gavela, R. Houtz, P. Quilez, and R. Del Rey, Eur. Phys. J. C 78, 972 (2018), eprint 1805.06465.
  * Buen-Abad and Fan (2019) M. A. Buen-Abad and J. Fan, JHEP 12, 161 (2019), eprint 1911.05737.
  * Csáki et al. (2020) C. Csáki, M. Ruhdorfer, and Y. Shirman, JHEP 04, 031 (2020), eprint 1912.02197.
  * Hook et al. (2020) A. Hook, S. Kumar, Z. Liu, and R. Sundrum, Phys. Rev. Lett. 124, 221801 (2020), eprint 1911.12364.
  * Gherghetta and Nguyen (2020) T. Gherghetta and M. D. Nguyen, JHEP 12, 094 (2020), eprint 2007.10875.
  * Hook (2018) A. Hook, Phys. Rev. Lett. 120, 261802 (2018), eprint 1802.10093.
  * Di Luzio et al. (2021a) L. Di Luzio, B. Gavela, P. Quilez, and A. Ringwald, JHEP 05, 184 (2021a), eprint 2102.00012.
  * Di Luzio et al. (2021b) L. Di Luzio, B. Gavela, P. Quilez, and A. Ringwald (2021b), eprint 2102.01082.
  * Gripaios et al. (2009) B. Gripaios, A. Pomarol, F. Riva, and J. Serra, JHEP 04, 070 (2009), eprint 0902.1483.
  * Gripaios et al. (2017) B. Gripaios, M. Nardecchia, and T. You, Eur. Phys. J. C 77, 28 (2017), eprint 1605.09647.
  * Merlo et al. (2018) L. Merlo, F. Pobbe, and S. Rigolin, Eur. Phys. J. C 78, 415 (2018), [Erratum: Eur.Phys.J.C 79, 963 (2019)], eprint 1710.10500.
  * Chala et al. (2017) M. Chala, G. Durieux, C. Grojean, L. de Lima, and O. Matsedonskyi, JHEP 06, 088 (2017), eprint 1703.10624.
  * Brivio et al. (2019) I. Brivio, M. B. Gavela, S. Pascoli, R. del Rey, and S. Saa, Chin. J. Phys. 61, 55 (2019), eprint 1710.07715.
  * Chikashige et al. (1981) Y. Chikashige, R. N. Mohapatra, and R. D. Peccei, Phys. Lett. B 98, 265 (1981).
  * Gelmini and Roncadelli (1981) G. B. Gelmini and M. Roncadelli, Phys. Lett. B 99, 411 (1981).
  * Davidson and Wali (1982) A. Davidson and K. C. Wali, Phys. Rev. Lett. 48, 11 (1982).
  * Wilczek (1982) F. Wilczek, Phys. Rev. Lett. 49, 1549 (1982).
  * Calibbi et al. (2017) L. Calibbi, F. Goertz, D. Redigolo, R. Ziegler, and J. Zupan, Phys. Rev. D 95, 095009 (2017), eprint 1612.08040.
  * Ema et al. (2017) Y. Ema, K. Hamaguchi, T. Moroi, and K. Nakayama, JHEP 01, 096 (2017), eprint 1612.05492.
  * Svrcek and Witten (2006) P. Svrcek and E. Witten, JHEP 06, 051 (2006), eprint hep-th/0605206.
  * Arvanitaki et al. (2010) A. Arvanitaki, S. Dimopoulos, S. Dubovsky, N. Kaloper, and J. March-Russell, Phys. Rev. D 81, 123530 (2010), eprint 0905.4720.
  * Georgi et al. (1986) H. Georgi, D. B. Kaplan, and L. Randall, Phys. Lett. B 169, 73 (1986).
  * Choi et al. (1986) K. Choi, K. Kang, and J. E. Kim, Phys. Lett. B 181, 145 (1986).
  * Choi et al. (2017) K. Choi, S. H. Im, C. B. Park, and S. Yun, JHEP 11, 070 (2017), eprint 1708.00021.
  * Chala et al. (2021) M. Chala, G. Guedes, M. Ramos, and J. Santiago, Eur. Phys. J. C 81, 181 (2021), eprint 2012.09017.
  * Bauer et al. (2021) M. Bauer, M. Neubert, S. Renner, M. Schnubel, and A. Thamm, JHEP 04, 063 (2021), eprint 2012.12272.
  * Galda et al. (2021) A. M. Galda, M. Neubert, and S. Renner (2021), eprint 2105.01078.
  * Alonso-Álvarez et al. (2021) G. Alonso-Álvarez, F. Ertas, J. Jaeckel, F. Kahlhoefer, and L. J. Thormaehlen (2021), eprint 2101.03173.
  * Lee et al. (1977a) B. W. Lee, C. Quigg, and H. B. Thacker, Phys. Rev. Lett. 38, 883 (1977a).
  * Lee et al. (1977b) B. W. Lee, C. Quigg, and H. B. Thacker, Phys. Rev. D 16, 1519 (1977b).
  * Chanowitz et al. (1978) M. S. Chanowitz, M. A. Furman, and I. Hinchliffe, Phys. Lett. B 78, 285 (1978).
  * Cornwall et al. (1974) J. M. Cornwall, D. N. Levin, and G. Tiktopoulos, Phys. Rev. D 10, 1145 (1974), [Erratum: Phys.Rev.D 11, 972 (1975)].
  * Bilchak et al. (1988) C. Bilchak, M. Kuroda, and D. Schildknecht, Nucl. Phys. B299, 7 (1988).
  * Gounaris et al. (1994) G. J. Gounaris, J. Layssac, and F. M. Renard, Phys. Lett. B332, 146 (1994), eprint hep-ph/9311370.
  * Gounaris et al. (1995a) G. J. Gounaris, J. Layssac, J. E. Paschalis, and F. M. Renard, Z. Phys. C66, 619 (1995a), eprint hep-ph/9409260.
  * Gounaris et al. (1995b) G. J. Gounaris, F. M. Renard, and G. Tsirigoti, Phys. Lett. B350, 212 (1995b), eprint hep-ph/9502376.
  * Degrande (2013) C. Degrande, EPJ Web Conf. 49, 14009 (2013), eprint 1302.1112.
  * Baur and Zeppenfeld (1988) U. Baur and D. Zeppenfeld, Phys. Lett. B 201, 383 (1988).
  * Dahiya et al. (2016) M. Dahiya, S. Dutta, and R. Islam, Phys. Rev. D93, 055013 (2016), eprint 1311.4523.
  * Ghosh et al. (2017) S. Ghosh, R. Islam, and A. Kundu (2017), eprint 1704.01867.
  * Corbett et al. (2015) T. Corbett, O. J. P. Éboli, and M. C. Gonzalez-Garcia, Phys. Rev. D 91, 035014 (2015), eprint 1411.5026.
  * Corbett et al. (2017) T. Corbett, O. J. P. Éboli, and M. C. Gonzalez-Garcia, Phys. Rev. D 96, 035006 (2017), eprint 1705.09294.
  * Almeida et al. (2020) E. d. S. Almeida, O. J. P. Éboli, and M. C. Gonzalez–Garcia, Phys. Rev. D 101, 113003 (2020), eprint 2004.05174.
  * Marciano et al. (2016) W. J. Marciano, A. Masiero, P. Paradisi, and M. Passera, Phys. Rev. D 94, 115033 (2016), eprint 1607.01022.
  * Cornella et al. (2020) C. Cornella, P. Paradisi, and O. Sumensari, JHEP 01, 158 (2020), eprint 1911.06279.
  * Brivio et al. (2017) I. Brivio, M. B. Gavela, L. Merlo, K. Mimasu, J. M. No, R. del Rey, and V. Sanz, Eur. Phys. J. C 77, 572 (2017), eprint 1701.05379.
  * Bardeen et al. (1978) W. A. Bardeen, S. H. H. Tye, and J. A. M. Vermaseren, Phys. Lett. B 76, 580 (1978).
  * Di Vecchia and Veneziano (1980) P. Di Vecchia and G. Veneziano, Nucl. Phys. B 171, 253 (1980).
  * Grilli di Cortona et al. (2016) G. Grilli di Cortona, E. Hardy, J. Pardo Vega, and G. Villadoro, JHEP 01, 034 (2016), eprint 1511.02867.
  * Chivukula and Georgi (1987) R. S. Chivukula and H. Georgi, Phys. Lett. B188, 99 (1987).
  * Hall and Randall (1990) L. J. Hall and L. Randall, Phys. Rev. Lett. 65, 2939 (1990).
  * D’Ambrosio et al. (2002) G. D’Ambrosio, G. F. Giudice, G. Isidori, and A. Strumia, Nucl. Phys. B645, 155 (2002), eprint hep-ph/0207036.
  * Bonilla et al. (2021) J. Bonilla, I. Brivio, M. B. Gavela, and V. Sanz (2021), eprint 2107.11392.
  * Draper and McKeen (2012) P. Draper and D. McKeen, Phys. Rev. D 85, 115023 (2012), eprint 1204.1061.
  * Bauer et al. (2017) M. Bauer, M. Neubert, and A. Thamm, JHEP 12, 044 (2017), eprint 1708.00443.
  * Bauer et al. (2019) M. Bauer, M. Heiles, M. Neubert, and A. Thamm, Eur. Phys. J. C 79, 74 (2019), eprint 1808.10323.
  * Davoudiasl et al. (2021) H. Davoudiasl, R. Marcarelli, N. Miesch, and E. T. Neil (2021), eprint 2105.05866.
  * Criado (2019) J. C. Criado, Eur. Phys. J. C 79, 256 (2019), eprint 1901.03501.
  * Jacob and Wick (1959) M. Jacob and G. C. Wick, Annals Phys. 7, 404 (1959).
  * Jaeckel et al. (2013) J. Jaeckel, M. Jankowiak, and M. Spannowsky, Phys. Dark Univ. 2, 111 (2013), eprint 1212.3620.
  * Mimasu and Sanz (2015) K. Mimasu and V. Sanz, JHEP 06, 173 (2015), eprint 1409.4792.
  * Jaeckel and Spannowsky (2016) J. Jaeckel and M. Spannowsky, Phys. Lett. B 753, 482 (2016), eprint 1509.00476.
  * Frugiuele et al. (2018) C. Frugiuele, E. Fuchs, G. Perez, and M. Schlaffer, JHEP 10, 151 (2018), eprint 1807.10842.
  * Craig et al. (2018) N. Craig, A. Hook, and S. Kasko, JHEP 09, 028 (2018), eprint 1805.06538.
  * Ebadi et al. (2019) J. Ebadi, S. Khatibi, and M. Mohammadi Najafabadi, Phys. Rev. D 100, 015016 (2019), eprint 1901.03061.
  * Yue et al. (2019) C.-X. Yue, M.-Z. Liu, and Y.-C. Guo, Phys. Rev. D 100, 015020 (2019), eprint 1904.10657.
  * Gavela et al. (2020) M. B. Gavela, J. M. No, V. Sanz, and J. F. de Trocóniz, Phys. Rev. Lett. 124, 051802 (2020), eprint 1905.12953.
  * İnan and Kisselev (2020) S. C. İnan and A. V. Kisselev, JHEP 06, 183 (2020), eprint 2003.01978.
  * Haghighat et al. (2020) G. Haghighat, D. Haji Raissi, and M. Mohammadi Najafabadi, Phys. Rev. D 102, 115010 (2020), eprint 2006.05302.
  * Goncalves and Sauter (2020) V. P. Goncalves and W. K. Sauter, Phys. Lett. B 811, 135981 (2020), eprint 2006.16716.
  * Flórez et al. (2021) A. Flórez, A. Gurrola, W. Johns, P. Sheldon, E. Sheridan, K. Sinha, and B. Soubasis, Phys. Rev. D 103, 095001 (2021), eprint 2101.11119.
  * (91) J. Bonilla, I. Brivio, J. Machado, and J. F. de Trocóniz, in preparation.
  * Aad et al. (2021a) G. Aad et al. (ATLAS) (2021a), eprint 2102.10874.
  * Aad et al. (2021b) G. Aad et al. (ATLAS), JHEP 02, 226 (2021b), eprint 2011.05259.