# Probing the Interactions of Axion-Like Particles with Electroweak Bosons and
the Higgs Boson in the High Energy Regime at LHC

Tisa Biswas<EMAIL_ADDRESS>

Department of Physics, University of Calcutta, 92 Acharya Prafulla Chandra
Road,
Kolkata 700009, India

###### Abstract

We study the interactions of Axion-Like Particles (ALPs) with the Standard
Model particles, aiming to probe their phenomenology via non-resonant searches
at the LHC. These interactions are mediated by higher dimensional effective
operators within two possible frameworks of linearly and non-linearly realised
electroweak symmetry breaking. We consider the ALPs to be light enough to be
produced on-shell and exploit their derivative couplings with the SM Higgs
boson and the gauge bosons. We will use the high momentum transfer processes,
namely $hZ,Z\gamma,WW$ and $WW\gamma$ production from $pp$ collisions. We
derive upper limits on the gauge-invariant interactions of ALPs with the
electroweak bosons and/or Higgs boson that contribute to these processes, from
the re-interpretation of the latest Run 2 available LHC data. The constraints
we obtain are strong for ALP masses below 100 GeV. These allowed effective
interactions in the ALP parameter space yield better significance at HL-LHC
and thus, offer promising avenues for subsequent studies. Furthermore, we
augment our cut-based analysis with gradient-boosted decision trees, which
improve the statistical significance distinctly across these interaction
channels. We briefly compare the results with the complementary probe of these
couplings via direct production of ALPs in association with the Higgs boson or
a vector boson.

##  I Introduction

Originally motivated by the efforts to solve the strong CP problem
Peccei:1977hh ; Peccei:1977ur ; Weinberg:1977ma ; Hook:2014cda ;
Wilczek:1977pj , pseudo-Nambu-Goldstone bosons (pNGBs) generically arise in a
variety of new physics (NP) scenarios. Their implications are many, including
the dynamic generation of small neutrino masses (Majorons) Chikashige:1980ui ,
attempting to solve the flavor problem (Flavons) Froggatt:1978nt ,
contributing to composite Higgs models, and extra-dimensional theories
Izawa:2002qk . The pNGBs also play a role in addressing the long standing
anomaly of muon magnetic moment Chang:2000ii , the hierarchy problem
Graham:2015cka  and electroweak baryogenesis Jeong:2018jqe . In addition, they
can serve as potential dark matter candidates or provide a portal connecting
the Standard Model (SM) particles to dark sector Abbott:1982af ; Dine:1982ah ;
Preskill:1982cy . Typically, pNGBs exhibit symmetry under a continuous or, in
some cases, a discrete shift of the field. These pNGBs, which enjoy a variety
of origins, interactions, and masses, are often grouped together in a much
broader class of the Axion-Like Particles (ALP).

Owing to diverse origins, ALPs connect different sectors in high-energy
physics. Studies aiming to detect ALPs in a range of masses and interactions
have guided the current and future direction of experiments for beyond the
Standard Model (BSM) physics, as discussed in recent reviews Irastorza:2018dyq
; Graham:2015ouw ; Bruggisser:2023npd . ALPs can manifest through a variety of
traces in experiments running at different energy scales. At the LHC, ALP
interactions are probed through signatures including new resonances or missing
energy Georgi:1986df ; Mimasu:2014nea ; Brivio:2017ije ; Bauer:2017ris , or
interactions with top quarks Esser:2023fdo , and via Higgs decays
Bauer:2016zfj ; Biekotter:2022ovp . Meson decay experiments typically provide
the conventional constraints on ALP-QCD couplings. In the context of flavor
experiments, ALPs with masses below a few GeV can be resonantly produced via
meson Bjorkeroth:2018dzu ; Dobrich:2018jyi ; MartinCamalich:2020dfe ;
Bandyopadhyay:2021wbb  and lepton decays Bauer:2019gfk ; Calibbi:2020jvd  or
directly in $e^{+}e^{-}$ interactions Dolan:2017osp ; Acanfora:2023gzr .
Fixed-target settings Dobrich:2015jyk ; Dobrich:2019dxc  further enable the
search of sub-GeV ALPs. In cosmological and astrophysical probes, still
lighter ALPs manifest through observable phenomena Cadamuro:2011fd ;
Millea:2015qra ; Depta:2020wmr .

In this work, we aim to probe the effects of non-resonant ALP-mediated
production processes involving SM final states only, at the LHC. The ALP
serves as an off-shell propagator in these $s$-channel scattering processes.
We analyze the behavior of differential cross-sections at high energies for
scatterings that produce electroweak gauge bosons and the Higgs boson from
$pp$ collisions. As we will see, the enhanced high-energy sensitivity of the
LHC enables us to impose significant constraints on ALP-interactions, from
such processes as their rates grow with energy. This deviates from the SM
scenarios, which exhibit a decrease in the production rates with the
collisional center-of-mass energy($\sqrt{\hat{s}}$), as $1/\hat{s}$. The
explicit dependence of the derivative interactions of the ALPs with SM
particles lead to enhanced cross-sections at increasing energies, especially
when $\sqrt{\hat{s}}\gg m_{a}$, the mass of the ALP.

While presently, ALP interactions with massive gauge bosons are accessible
primarily through indirect channels, such as loop corrections to low-energy
processes Izaguirre:2016dfi ; Bauer:2017ris ; Alonso-Alvarez:2018irt ;
Ebadi:2019gij ; Gavela:2019wzg ; Ertas:2020xcc ; Gori:2020xvq ; Kelly:2020dda
; Galda:2021hbr ; Bonilla:2021ufe ; Bauer:2021mvw , or direct detection at
colliders, based on the ALP’s mass and decay width, these interactions might
be observed in associated production processes involving a gauge boson and an
ALP. The ALP may either go undetected Mimasu:2014nea ; Brivio:2017ije  or
participate in resonant decays Craig:2018kne ; Wang:2021uyb . In the context
of the kinematics discussed above in presence of the low-lying ALPs, the LHC
stands out as a key facility for exploring events that yield SM bosons in the
final state, including electroweak gauge bosons $(W,Z,\gamma)$, gluons and the
Higgs boson. The earlier investigations on processes where the ALP acts an
off-shell propagator included $ZZ~{}({\rm semileptonic~{}decay~{}channel}),WW$
and $Z\gamma$ in leptonic channel, $\gamma\gamma$ and di-jet production
Gavela:2019cmq ; Carra:2021ycg ; Bonilla:2022pxu . In this work, we will
consider that the ALP is too lighter to be produced resonantly and study the
gluon-initiated scatterings such as $pp(gg)\rightarrow Zh,Z\gamma,WW$ and
$WW\gamma$ that proceed through an intermediate ALP. The search strategy to
probe the ALP interactions with the SM particles entails identifying
deviations from the SM expectations within the high-energy tails of the
invariant mass distributions of the boosted final states.

Focusing on a particular ALP-mass range of below $100$ GeV, we highlight the
novel features of our study:

  * •

We carry out a comprehensive analysis of non-resonant ALP-mediated
$Zh,Z\gamma,WW$ and the $WW\gamma$ production processes at the LHC, aiming to
probe at high energy regime. In each of the $2\rightarrow 2$ processes,
atleast one of the heavy SM particles undergoes hadronic decay. We study these
processes for ALP interactions by contrasting the results with the latest
experimental data. By taking advantage of the derivative ALP couplings and the
boosted state of heavy bosons in the final state, we investigate for new
physics effects in the $Zh$ process spanning the final state invariant mass
spectrum from 320 GeV to 2.8 TeV, the $WW$ process from 900 GeV to 4 TeV, and
the $Z\gamma$ process between 800 GeV to 2.1 TeV. Previous studies, to our
knowledge, have focused on leptonic channels in lower energy ranges,
particularly for the $Z\gamma$ and $WW$ processesCarra:2021ycg . Our study
improves the ALP limits obtained in these production processes. Prospects at
HL-LHC are also discussed.

  * •

For the Higgs-strahlung and $Z\gamma$ production processes, where the Higgs
and Z boson, respectively, are boosted, result in a ‘fat-jet’ comprising two
b-tagged subjets. Similarly, in the semi-leptonic channel of the $WW$
production, one of the W bosons is manifested as a fat-jet. We explore final
state topologies featuring such fat-jets in combination with a hard photon or
an electroweak boson to search for signatures of new interactions differing
from the SM. These heavy particles, decaying hadronically, are abundantly
produced at the LHC but are challenging to detect through conventional
searches due to overwhelming QCD backgrounds. Tagging these jets and employing
jet-substructure techniques enhances the sensitivity of LHC to previously
unexplored kinematic regions for axion-like particle mediated processes.

  * •

This is the first detailed analysis of non-resonant ALP-mediated $WW\gamma$
production at the LHC. This $2\rightarrow 3$ process is dominantly produced
from gluon-fusion under the allowed constraints of ALP couplings from
available data. We explore how this process could probe $g_{a\gamma\gamma}$,
$g_{aZ\gamma},g_{aWW}$ couplings, either through $WWZ,WW\gamma$ triple gauge
coupling or via a four-point interaction of $a\gamma WW$.

  * •

We also briefly compare the limits on the ALP couplings obtained from the
aforementioned non-resonant production processes with those from direct probe
of “mono-$X$” signatures $(X=Z,W^{\pm},h)$ through the production of ALP in
association with a Higgs or a vector boson.

  * •

To enhance the distinction between signal and background in the processes
under study, we employ a multivariate analysis using the Boosted Decision Tree
(BDT) technique. This approach, transcending the conventional cut-based
method, exhibits a marked improvement in signal significance, as will be
explicitly demonstrated in the subsequent sections.

In this work, we adopt a model-independent effective field theory (EFT)
approach. Considering that the Higgs boson observed at LHC is still part of a
$SU(2)_{L}$ doublet, as in the SM, then any electroweak (EW) physics that
extends beyond the SM can be systematically examined using a linear EFT
expansion Buchmuller:1985jz ; Grzadkowski:2010es . The setup of a linear EFT
includes the SM and an ALP Georgi:1986df ; Choi:1986zw ; Brivio:2017ije , and
also contrasts with the framework of a chiral EFT when considering
interactions involving the ALP and the Higgs boson Brivio:2017ije ;
Bauer:2017ris . The current experimental results do not preclude the existence
of a Higgs component that deviates from this doublet structure, at least
within a 10% uncertainty margin ATLAS:2019slw , thus making the non-linear EFT
methodology equally pertinent for exploration Feruglio:1992wf ; Alonso:2012jc
; Azatov:2012bz ; Alonso:2012px ; Alonso:2012pz ; Buchalla:2013rka ;
Brivio:2013pma . We will mainly focus on the linear EFT framework in this
paper, while also consider the chiral EFT context to assess the ALP-Higgs
interactions.

In future LHC runs, the non-resonant ALP searches are set to become
increasingly competitive. This improvement is expected not just because of the
significant growth of available data on the high luminosity frontier but also
due to progress inspired by the SMEFT which promote a generalised, systematic
approach to probing new physics Cepeda:2019klc . While the SMEFT presumes that
new physics manifests through particles that are too heavy to be produced on-
shell Biswas:2021qaf ; Biswas:2022fsr , non-resonant ALP searches aim to look
for ALPs too light to undergo resonant decays. This distinct approach enables
non-resonant ALP searches to explore complementary areas of the parameter
space, depending on minimal assumptions about the ALP decay width.

The plan of the paper is the following. In section II, we describe the ALP
effective theory and set the framework for our analysis. This has been
followed by a discussion on the general features of non resonant ALP EW
processes considered in this study in section III. In section IV, we undertake
a detailed collider analysis studying the kinematical features of the signal
and background processes. We present the constraints derived on the parameters
of the ALP Lagrangian using measurements from the latest available Run 2 LHC
data. We discuss the validity range of our analysis. Thereafter, we define
some benchmark scenarios for ALP signals and discuss the projected
sensitivities to the effective couplings in the upcoming HL-LHC run. We also
discuss the constraints arising from direct probe of these couplings through
the production of ALP in association with a Higgs or a vector boson. In
section V, numerical results and their interpretations, along with detailed
discussions on cross-section parameter dependencies, are covered. In section
VI, the use of boosted decision trees to enhance the cut-based results is
explored. Finally, we draw our conclusions in section VIII.

##  II ALP Effective Lagrangian

We consider an ALP, denoted by $a$, which is a pseudo-scalar state. Its
interactions are constructed to respect the invariance under shifts
$a(x)\rightarrow a(x)+\alpha$, where $\alpha$ is a constant (reflecting to be
of the form $J^{\mu}\partial_{\mu}\alpha$, consistent with its Goldstone
nature). Within the EFT framework, we express all ALP interactions with
suppression factors which are inversely proportional to the characteristic
scale $f_{a}\gg m_{a}$ (mass of the ALP), that is unknown and naturally close
to the mass scale of the heavy sector the ALP originates from. Also, it is
implicitly assumed that $f_{a}\gg v$ where $v$ denotes the EW scale. We
require all ALP interactions to be invariant under the full SM gauge group.
For linear EWSB realization, the most general linear bosonic Lagrangian,
incorporating next-to-leading order (NLO) effects related to $a$, is given by

$\mathcal{L}_{\text{eff}}^{\text{linear}}=\mathcal{L}_{\text{LO}}+\,\delta\mathcal{L}_{\mathrm{eff}}^{\text{bosonic}}\,,$
(1)

where the leading order Lagrangian now comprises the SM Lagrangian along with
the ALP kinetic term,

$\mathcal{L}_{\text{LO}}=\mathcal{L}_{\text{SM}}\,+\frac{1}{2}(\partial_{\mu}a)(\partial^{\mu}a)\,,$
(2)

while the NLO bosonic corrections due to the ALP interactions with the SM
fields are included in the effective Lagrangian :

$\displaystyle\begin{split}\delta\mathcal{L}_{\mathrm{eff}}^{\text{bosonic}}&\supset
c_{\tilde{G}}\,\mathcal{O}_{\tilde{G}}+c_{\tilde{B}}\,\mathcal{O}_{\tilde{B}}+c_{\tilde{W}}\,\mathcal{O}_{\tilde{W}}\,+{c_{a\Phi}}\,\mathcal{O}_{a\Phi}\,,\end{split}$
(3)

Eq. (3) contains a complete and non-redundant set of dimension-5 bosonic
operators which are given by:

$\displaystyle\begin{split}\mathcal{O}_{\tilde{G}}\equiv-\dfrac{a}{f_{a}}G_{\mu\nu}\widetilde{G}^{\mu\nu}\,,\quad&\mathcal{O}_{\tilde{W}}\equiv-\dfrac{a}{f_{a}}W_{\mu\nu}^{a}\widetilde{W}^{\mu\nu}_{a}\,,\\\\[3.99994pt]
\mathcal{O}_{\tilde{B}}\equiv-\dfrac{a}{f_{a}}B_{\mu\nu}\widetilde{B}^{\mu\nu}\,,\quad&\mathcal{O}_{a\Phi}\equiv
i\dfrac{\partial^{\mu}a}{f_{a}}\,\Phi^{\dagger}\overleftrightarrow{D}_{\mu}\Phi\,.\end{split}$
(4)

Here, $G_{\mu\nu}$, $W_{\mu\nu}$ and $B_{\mu\nu}$ are the generic field
strength tensors corresponding to the SM gauge groups $SU(3)_{c}$,
$SU(2)_{L}$, and $U(1)_{Y}$ respectively. The dual field strength tensors
$\widetilde{X}^{\mu\nu}$ are defined by
$\tilde{X}^{\mu\nu}\equiv\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}X{\rho\sigma}$,
with $\varepsilon^{0123}=1$. The associated operator coefficients $c_{i}$ in
Eq. (3) are real constants. $\Phi$ is the SM Higgs doublet, with
$\Phi\overleftrightarrow{D}_{\mu}\Phi\equiv\Phi^{\dagger}\big{(}D_{\mu}\Phi\big{)}-\big{(}D_{\mu}\Phi\big{)}^{\dagger}\Phi{}$.
The first three operators in Eq. (3) induce ALP couplings to the gluon, the
photon and the Z and W bosons as given by :

$\displaystyle\delta\mathcal{L}_{\mathrm{eff}}$ $\displaystyle\supset$
$\displaystyle-\frac{g_{agg}}{4}a\,G_{\mu\nu}\widetilde{G}^{\mu\nu}-\frac{g_{a\gamma\gamma}}{4}a\,F_{\mu\nu}\widetilde{F}^{\mu\nu}-\frac{g_{aZ\gamma}}{4}a\,F_{\mu\nu}\widetilde{Z}^{\mu\nu}$
(5) $\displaystyle-$
$\displaystyle\frac{g_{aZZ}}{4}a\,Z_{\mu\nu}\widetilde{Z}^{\mu\nu}-\frac{g_{aWW}}{4}a\,W_{\mu\nu}\widetilde{W}^{\mu\nu}\,,$

The coupling strengths are defined as:

$\displaystyle g_{agg}$ $\displaystyle=\frac{4}{f_{a}}\,c_{\tilde{G}}\,,\quad
g_{a\gamma\gamma}=\frac{4}{f_{a}}\,\big{(}s_{w}^{2}\,c_{\tilde{W}}+c_{w}^{2}\,c_{\tilde{B}}\big{)}$
(6a) $\displaystyle g_{aWW}$
$\displaystyle=\frac{4}{f_{a}}\,c_{\tilde{W}}\,,\quad
g_{aZZ}=\frac{4}{f_{a}}\,({c_{w}^{2}}\,c_{\tilde{W}}+{s_{w}^{2}}\,c_{\tilde{B}})$
(6b) $\displaystyle g_{a\gamma Z}$
$\displaystyle=\frac{8}{f_{a}}\,s_{w}c_{w}(c_{\tilde{W}}-c_{\tilde{B}})\,,$
(6c)

with $s_{w}$ and $c_{w}$ denoting the sine and cosine of the Weinberg angle,
respectively.

After electroweak symmetry breaking, the last operator in Eq. (3),
$\mathcal{O}_{a\Phi}$ results in a two-point function effect, equivalent to
$a$ serving as an extra contribution on the longitudinal component of the
electroweak gauge fields. To assess its effect on observables, one approach is
to substitute it with a fermionic vertex Georgi:1986df . This substitution can
involve a vertex that either preserves or alters chirality, or a combination
of both. For illustration, the Higgs field redefinition:

$\Phi\rightarrow e^{ic_{\Phi}\,a/f_{a}}\Phi$ (7)

when applied to the bosonic Lagrangian Eq. (1), leads to a modification
originating from the Higgs kinetic energy term in the SM. This modification
precisely negates $\mathcal{O}_{a\Phi}$ up to $\mathcal{O}(a/f_{a})$.
Meanwhile, the Yukawa terms in the SM induce a new coupling between Yukawa and
axion, allowing for a complete substitution of $\mathcal{O}_{a\Phi}$. The
overall effect is, the replacement in Eq. (3) by:

$\displaystyle
i\dfrac{a}{f_{a}}\Big{[}\overline{Q}Y_{u}\widetilde{\Phi}u_{R}-\overline{Q}Y_{d}\Phi
d_{R}-\overline{L}Y_{\ell}\Phi\ell_{R}\Big{]}+\text{h.c.},$ (8)

where $Y_{u,d,\ell}$ are the SM Yukawa matrices. In this work, we focus on
experimental signatures that involve ALPs and SM bosons ($W,\,Z,\,\gamma$ and
$h$). We do not consider the CP-violating terms and direct ALP-fermion
interactions (stemming from the $\mathcal{O}_{a\Phi}$ operator) since such
interactions are markedly suppressed at tree-level due to their
proportionality to the involved fermions’ Yukawa couplings 111For light
fermions as initial states, the effects of these contributions are effectively
negligible..

Within the framework of non-linear (chiral) electroweak theory, the
interactions of the ALP with SM fields at leading order are captured by the
following expression:

$\mathcal{L}=\mathcal{L}^{\text{HEFT}}_{\text{LO}}+\mathcal{L}^{\text{ALP}}_{\text{LO}}\,.$
(9)

Here, $\mathcal{L}^{\text{HEFT}}_{\text{LO}}$ denotes the chiral Lagrangian
within the Higgs Effective Field Theory (HEFT) Alonso:2012px ; Brivio:2016fzo
; Herrero:2020dtv ; Herrero:2021iqt  framework. In this model, the Higgs boson
is treated as a singlet field, while the Goldstone bosons $\pi^{a}$ are
encapsulated non-linearly through the matrix $U$ given by

$U(\pi^{a})\equiv\exp\left({i\pi^{j}\tau^{j}/v}\right)\,,$ (10)

with the Pauli matrices represented by $\tau^{j}$ where $j=1,2,3$. The $U$
matrix which transforms as a bi-fundamental under $SU(2)_{L}\times SU(2)_{R}$:

$U(\pi^{a})\ =\
\exp\left(\frac{i\pi^{j}\tau^{j}}{v}\right)\quad\xrightarrow[L\times R]{}\quad
L\>U(\pi^{a})\>R^{\dagger},$ (11)

The series expansion of $U$ is as follows:

$U(\pi^{a})={\mathbb{1}}_{2}+i\frac{\pi^{a}}{v}\tau^{a}-\frac{2G^{+}G^{-}+G^{0}G^{0}}{2v^{2}}{\mathbb{1}}_{2}+\dots\,,$
(12)

where $G^{\pm}$ and $G^{0}$ are defined as $G^{\pm}=(\pi^{2}\pm
i\pi^{1})/\sqrt{2}$ and $G^{0}=-\pi^{3}$, respectively. We work under this
framework to study novel ALP-Higgs interactions that can probe the unique
singlet nature of the Higgs boson as described by the HEFT Lagrangian. Now the
leading order Lagrangian for ALP interactions is expressed as:

$\mathcal{L}^{\text{ALP}}_{\text{LO}}=\frac{1}{2}\partial_{\mu}a\partial^{\mu}a-\frac{1}{2}m_{a}^{2}a^{2}+c_{2D}\mathcal{A}_{2D}(h),~{}~{}{\rm{with}}~{}\mathcal{A}_{2D}(h)=\left[iv^{2}{\rm
Tr}[\mathcal{T}\mathcal{V}_{\mu}]\partial^{\mu}\frac{a}{f_{a}}\mathcal{F}_{2D}(h)\right]\,,$
(13)

where the fields $\mathcal{V}_{\mu}(x)$ and $\mathcal{T}(x)$ are defined by
the relations :
$\mathcal{V}_{\mu}(x)\equiv\left(D_{\mu}U(x)\right)U(x)^{\dagger}$ and
$\mathcal{T}(x)\equiv U(x)\sigma_{3}U(x)^{\dagger}$. In this framework, as
stated, the Higgs boson is introduced as a gauge-singlet scalar field, with
its interactions incorporated by functions such as:

$\mathcal{F}_{2D}(h)=1+a_{2D}h/v+b_{2D}(h/v)^{2}+\mathcal{O}(h^{3}/v^{3}),$
(14)

where coefficients $a_{2D}$ and $b_{2D}$ are independent constants. The term
$\mathcal{A}_{2D}$ serves as the chiral analogue to the linear operator
$\mathcal{O}_{a\Phi}$, with a distinct feature: it facilitates not only ALP-
fermion interactions comparable to those in Eq. 8, but it also induces new
interactions at leading order between the ALP, electroweak gauge bosons, and
the Higgs, such as the trilinear $aZh$, $a\gamma h$ coupling. Exploring these
interaction phenomena yield an understanding of the process of electroweak
symmetry breaking and its interplay with axion-like states. Within the linear
paradigm, such interactions emerge at the next-to-next-to-leading order
(NNLO), corresponding to mass dimension seven, and thus, their effects are
expected to be relatively subdominant. Furthermore, within the chiral
framework, the operators $\mathcal{O}_{\tilde{G}}$, $\mathcal{O}_{\tilde{W}}$,
and $\mathcal{O}_{\tilde{B}}$ (in Eq. 4) also become relevant at NLO.

##  III ALP mediated processes

We focus exclusively on processes with off-shell production of ALP into SM
final states only. These processes are production of $Zh$, $Z\gamma$,
$W^{\pm}W^{\mp}$ and $W^{\pm}W^{\mp}\gamma$ from $pp$ collisions. They all
probe different operator combinations within the ALP EFT parameter space. To
facilitate our discussion, we present in Fig. 1, the Feynman diagrams which,
by virtue of higher dimensional operators, contribute to the aforementioned
processes. The blobs on the vertices of diagrams (a)-(i) stand for possible
inclusion of one of the higher dimensional operators listed in Eqs. (5) and
(13). ALP production in these processes is dominated by gluon-gluon fusion as
the $q\bar{q}$ induced process for these final states is proportional to the
quark masses from the operator $\mathcal{O}_{a\phi}$ (See Eq. (8)) and thus,
highly suppressed. These channels have been studied for resonant searches in
the differential measurements of the invariant mass of the final state by the
CMS and ATLAS collaborations. No excess of events have been found and we shall
reinterpret these measurements for the ALP mediated processes. We particularly
aim to probe the boosted regime with at least one of the weak bosons or the
Higgs boson decaying hadronically. This ensures we have a large fraction of
events and reduce uncertainties and yet maintain a balance with clean
environment, using the jet substructure techniques for tagging the heavy
bosons. Such boosted regimes with improved techniques are useful for
identifying lighter ALPs that get rejected by the selection criteria of the
cross-section measurements. The $WW\gamma$ channel final state is an exception
for which we will study the fully leptonic final state. All the four processes
receive contributions from s-channel mediated non-resonant ALP. The $WW\gamma$
process receives additional contribution from initial quark states. We
included these diagrams in our calculation for consistency. However, we have
checked that their contribution is significantly lower compared to those
initiated by gluons. We investigate for the first time into non-resonant
triboson production, mediated through ALP. It is known that the resonant
triboson production puts stringent constraints on ALP couplings for
$m_{a}>100$ GeV Craig:2018kne .

|  |  |
---|---|---|---
(a) | (b) | (c) | (d)
(a)

|  |  |  |
---|---|---|---|---
(e) | (f) | (g) | (h) | (i)
(b)

Figure 1: Representative Feynman diagrams depicting the production of (a)
$gg\rightarrow Zh$, (b) $gg\rightarrow Z\gamma$, (c) $gg\rightarrow
W^{+}W^{-}$ and (d)-(i) $gg(q\bar{q})\rightarrow W^{+}W^{-}\gamma$ mediated by
an off-shell ALP. Each of the diagram consistently involves a double insertion
of ALP operators.

As the ALP is always off-shell, its propagator acts as a suppression in the
hadronic scattering amplitudes. However, the explicit momentum dependence of
the ALP interactions under discussion, the ALP couplings lead to higher energy
growth with the invariant mass of the event final states than the
corresponding SM backgrounds.

All the diagrams in Fig. 1 must arise with double insertions of ALP operators.
This results for the amplitude to scale as $f_{a}^{-2}$, and cross sections in
the order $f_{a}^{-4}$. In all generality, the contributions from bosonic ALP
couplings in Eq. (5) interfere with the SM amplitudes. Thus, a generic cross-
section when expressed as polynomial functions of Wilson coefficients
$c_{i}\over f_{a}$, including both SM and EW ALP contributions, has the
structure

$\displaystyle\sigma_{\rm{ALP}}$
$\displaystyle=\sigma_{\rm{SM}}+\frac{1}{f_{a}^{2}}\,\sigma_{\rm
int}+\frac{1}{f_{a}^{4}}\,\sigma_{\rm NP}\,.$ (15)

When the ALP couplings are relatively small, their interference with the SM
background may become comparable with the pure ALP-signal, and thus must be
considered in evaluation of the process. The coupling value at which this
interference becomes significant varies based on the specific final state
being analyzed. In scenarios with electroweak diboson final states, the ALP
signal interferes with the SM processes occurring at one-loop. The nature of
interference could be constructive or destructive and it depends on the
relative sign of the couplings $g_{agg}$ and $g_{aV_{1}V_{2}}$ (new vertices
in the diagram, Fig 1). Currently, however, the magnitude of ALP couplings
accessible at LHC are loose and the interference effect is suppressed in the
total cross-section estimation. The quartic dependence from the pure ALP
interactions dominate and result in large-$\hat{s}$ enhancement in the cross-
section, $\sigma_{\rm{ALP}}\sim{\hat{s}}/{f_{a}^{4}}$. Such energy scaling is
valid only as long as the energies involved in the scattering remain below the
cutoff scale of EFT, $\sqrt{\hat{s}}<f_{a}$. On the other hand, the SM
backgrounds usually scale as $1/\hat{s}$ well above the resonance of the
s-channel. In hadronic collisions, the calculation of any cross-section
involves a convolution of this partonic cross-section with the parton
distribution functions (PDFs). These PDFs exhibit a declination with the
increase in energy. Taking this effect into account, the ALP mediated rates
show a slower decrement with the invariant mass compared to the SM background.
This allows to distinguish ALP-mediated processes from the SM background as
discussed in the following sections.

##  IV Collider Analysis

The effective Lagrangian has been implemented into FeynRules Alloul:2013bka
to generate the UFO model file Brivio:2017ije  for the event generator
Madgraph5_aMC@NLO Alwall:2014hca . MadGraph was employed for producing all
signal and background sample events. These events are generated at leading
order (LO) and subsequently processed by Pythia (v8) Sjostrand:2014zea  for
parton showering and hadronization. For event generation, NNPDFNLO parton
distribution functions pdf  are utilized, setting both factorization and
renormalization scales dynamically to half the sum of all final state
transverse energies in the scattering processes. The matching parameter, QCUT,
was specifically determined for the different processes as discussed in Ref.
MLMMerging . Detector effects are incorporated by passing the events through
Delphes-v3.4.1 deFavereau:2013fsa . Jets are reconstructed using
Fastjet-v3.3.2 Cacciari:2011ma . We impose a set of cuts at the generator
level on the final state particles, namely,

$\displaystyle p_{T_{(l,j)}}$ $\displaystyle>30~{}\rm{GeV}\,,$ $\displaystyle
p_{T_{\gamma}}$ $\displaystyle>20~{}\rm{GeV}\,,$ $\displaystyle\Delta
R_{(\gamma,j)}$ $\displaystyle>0.4\,,$ $\displaystyle\Delta R_{(j,j/ll/jl)}$
$\displaystyle>0.2\,,$ (16) $\displaystyle\eta_{j}$ $\displaystyle<5\,,$
$\displaystyle\eta_{(l,\gamma)}$ $\displaystyle<2.5\,,$ $\displaystyle m_{ll}$
$\displaystyle>50~{}\rm{GeV}\,$

all processes, except for the $WW\gamma$ channel where we require
$m_{ll^{\prime}}>10~{}\rm{GeV}$. The angular separation between two particles
is defined as $\Delta R=\sqrt{\Delta\eta^{2}+\Delta\phi^{2}}$, where $\eta$ is
the pseudorapidity and $\phi$ is its azimuthal angle of each particle. The ALP
signal events are generated fixing $m_{a}=1~{}\rm{MeV}$, treating the ALP as
effectively massless at LHC energies. The ALP width $\Gamma_{a}$ is assumed to
be considerably smaller than $m_{a}$. The specific choices of the ALP’s mass
and decay width have negligible impact in the non-resonant regime. We generate
signal samples with pure ALP-mediated production and the interference between
the ALP and the SM processes.

###  IV.1 13 TeV LHC probes

In this section, we will present the details of the simulations and derive
constraints on $g_{aV_{1}V_{2}}$ couplings via the non-resonant ALP-mediated
processes mentioned above, utilizing publicly available data from the ATLAS
and CMS collaborations at Run 2, 13 TeV LHC as listed in Table 1.

Channel | Distribution | # bins | Int. Lum. |  Data set
---|---|---|---|---
$Zh\rightarrow\ell^{+}\ell^{-}b\bar{b}$ |  $m_{VH}$ Fig. 4(d) | 16 | 139 fb-1 | ATLAS 13 TeV ATLAS:2022enb
$Z\gamma\rightarrow b\bar{b}+\gamma$ |  $m^{J\gamma}$, Fig. 5a | 18 | 36.1 fb-1 | ATLAS 13 TeV ATLAS:2018sxj
$WW\rightarrow\mu^{\pm}+J+\not{E}_{T}\;$ |  $m_{WW}$, Fig. 4 (middle, right) | 16 | 35.9 fb-1 | CMS 13 TeV CMS:2019ppl
$WW\rightarrow e^{\pm}+J+\not{E}_{T}\;$ |  $m_{WW}$, Fig. 4 (middle, left) | 16 | 35.9 fb-1 | CMS 13 TeV CMS:2019ppl
$WW\gamma\rightarrow e^{\pm}\mu^{\mp}+\gamma+\not{E}_{T}\;(0j)$ |  $m_{T}^{WW}$, Fig. 3 (left) | 12 | 138 fb-1 | CMS 13 TeV CMS:2023rcv
Table 1: Experimental data used in our study to constrain the ALP couplings.
The third column indicates the number of bins used in our analysis, always
starting from the highest.

####  IV.1.1 $pp\rightarrow Zh$

This process yields a powerful probe of the ALP-Higgs coupling through the
operator $\mathcal{A}_{2D}$ in Eq. 13. It may be expected among the leading
signals and a conclusive evidence if the underlying EWSB enjoys a non-linear
character. In fact, this operator also induces $ah\gamma$ interaction, and
thus, one can question for a process of $pp\rightarrow h\gamma$ signal
mediated by the ALP. However, the ALP forces the interaction to be derivative
and the photon being transverse, on-shell, leading to a vanishing cross-
section. There can be further probes of double Higgs production where this
operator contributes.

In order to probe the current-reach of the LHC in constraining this coupling
through $pp\rightarrow Zh$, we optimize a hadron-level analysis to obtain the
sensitivity to the BSM signal, which is well-pronounced in the high energy
bins. To achieve this, we consider the $Z(\ell^{+}\ell^{-})h$ production and
scrutinize the $h\rightarrow b\bar{b}$ decay channel. The dominant backgrounds
consist of $Zb\bar{b}$ and the irreducible SM production of $Zh$. Reducible
contributions arise from $Z+$ jets production ($c$-quarks included but not
explicitly tagged), where the light jets can be misidentified as $b$-jets, and
the fully leptonic decay for $t\bar{t}$. Rather than performing a resolved
analysis with two distinct b-tagged jets, our method focuses on a single fat-
jet with a cone-radius $R=1.0$. We apply the BDRS method Butterworth:2008iy
with some minor modifications to enhance sensitivity. This technique merges
jets using the CA algorithm, from a significantly large cone radius to
encapsulate all decay products of a resonance (like the Higgs boson). The
process involves breaking the primary jet $J$ into two subjets, $j_{1}$ and
$j_{2}$with $j_{1}$ being the heavier of the two. We impose a mass drop
condition, $m_{j_{1}}>m_{j_{2}}$. And a mass drop condition is applied such
that $m_{j_{1}}<\mu m_{J}$ where $\mu=0.66$ (with $m_{J}$ as the fat-jet’s
mass), alongside a symmetry criterion
$\frac{min(p_{T,j_{1}}^{2},p_{T,j_{2}}^{2})}{m_{J}^{2}}\Delta
R^{2}_{j_{1},j_{2}}>0.09$ between the subjets. If the condition fails, the
lighter subjet, $j_{2}$, is removed, and the process repeats with $j_{1}$.
This iteration continues until a final jet $J$ that satisfies the mass drop
condition is found. This selection is effective in filtering out QCD jets but
can still be impacted by the underlying events at the high energies and
luminosities of the LHC. To further eliminate rare QCD events and effects from
hard gluon emissions or underlying events, we refine the Higgs vicinity by
recombining the components of $j_{1}$ and $j_{2}$ using the CA algorithm with
a reduced radius $R_{\text{filt}}=\min(0.2,R_{b\bar{b}}/2)$222We find that
this choice of $R_{\text{filt}}$ is effective in background reduction.. We
keep only the three strongest filtered subjets for resonance reconstruction.
Overall, this approach effectively distinguishes boosted electroweak-scale
resonances from significant QCD backgrounds.

The signal selection criterion is as follows. We constructed fat-jets with a
minimum transverse momentum, $p_{T}>$ 100 GeV, and a rapidity cut of
$|y|<2.5$. Leptons ($e,\mu$) are isolated within a $R=0.3$ radius, with their
$p_{T}>25$ GeV and $|y|<2.5$. The events with exactly two isolated, opposite-
charge, same-flavor leptons, conforming to the $Z$-peak with invariant mass
between 80 and 100 GeV and a leptonic separation $\Delta R>$ 0.2 are selected.
For Higgs reconstruction, we required at least one fatjet with a minimum of
two $B$-meson tracks ($p_{T}>15$ GeV) and a fatjet $p_{T}>250$ GeV. After
mass-drop and filtering criteria, events needed exactly two $b$-tagged
subjets, well-separated from isolated leptons. The Higgs invariant mass was
set within 75 to 145 GeV. To minimize background, both reconstructed $Z$ and
Higgs were required to have $p_{T}>200$ GeV, and the $t\bar{t}$ background was
significantly reduced by setting a $\not{E}_{T}/\sqrt{H_{T}}<(1.15+(8\times
10^{-3}).m_{Zh})/1$ GeV limit, where all the events are required to have a
minimum invariant mass of Z and Higgs of 320 GeV. The ATLAS provides a
measurement of invariant mass of the $Zh$ system in the 2 leptons+2 b-jets
final state ATLAS:2022enb . The bins extend in varying steps from 320 GeV to
2.8 TeV. The corresponding signal and background distributions with the ATLAS
data are shown in Fig. 2 (a). The SM background and the experimental data have
been obtained from hepdata .

####  IV.1.2 $pp\rightarrow Z\gamma$

We then consider the signal produced by an off-shell ALP contribution to
$Z\gamma$ production and its impact on the measurement of $Z\gamma$ production
performed by ATLAS in Ref. ATLAS:2018sxj . This process receives contributions
from bosonic operator coefficients $c_{\tilde{B}}$ and $c_{\tilde{W}}$. These
coefficients also affect $a\gamma\gamma$ and $aZZ$ vertices. Hence, to fully
understand $aZ\gamma$ vertex modification, assumptions on $g_{a\gamma\gamma}$
and $g_{aZZ}$ are necessary as we elaborate later. In this study, we consider
the regime where both the Z boson and the photon are significantly boosted,
leading to the total hadronic decay products of Z being encompassed within a
large radius jet. Consequently, the final state features a fat-jet in recoil
against a hard photon. We employ jet substructure techniques to reconstruct
the Z jet from its invariant mass, with the fat-jet radius estimated by the
relation $R\sim\frac{2m_{h}}{p_{T_{h}}}$. The following SM processes can mimic
the $Z\gamma$ signal. Continuum $\gamma j$ process emerges as the most
dominant background. The $Z/W\gamma+$jets process, while having a similar
topology to the signal, is less prevalent due to its lower cross-section.
Production of $t\bar{t}\gamma$ with hadronic decays of the top quarks, and
single top productions like $tj\gamma,tb\gamma$ However, demanding a high
photon and Z tagging can suppress these backgrounds. The $pp\rightarrow
h(\rightarrow b\bar{b})\gamma$ associated production in the SM has a nominal
rate, either due to the very small couplings of Higgs with the initial state
quarks or because the process is predominantly a higher-order one-loop
contribution.

The ATLAS ATLAS:2018sxj  Collaboration has searched for a resonance decaying
into Z and photon. No significant excess over the SM expectation has been
reported. In the signal from $800~{}\rm GeV<m_{J\gamma}<2~{}\rm TeV$, the
ATLAS has collected 55 events with $\int\mathcal{L}dt=36.1~{}\rm fb^{-1}$.
With the SM background expectation, we compare Fig. 5 (a) of Ref.
ATLAS:2018sxj  as shown in Fig. 2 (b).

Cut flow | jets+ $\gamma$ | $W/Z+\gamma$ | $t\bar{t}\gamma+tj\gamma$ | $\rm Signal$
---|---|---|---|---
Atleast one fat-jet with two B-meson tracks with $p_{T}>200$ GeV | $0.034$ | $0.13$ | $0.79$ | $0.46$
Atleast 1 isolated photon and lepton veto | $0.79$ | $0.84$ | $0.83$ | $0.92$
Photon transverse momentum $>200$ GeV | $0.45$ | $0.55$ | $0.58$ | $0.96$
Atleast 1 fat-jet with two B-meson tracks ($p_{T}>250$ GeV) | $0.58$ | $0.74$ | $0.78$ | $0.94$
Two mass drop subjets and at least two filtered subjets | $0.32$ | $0.48$ | $0.71$ | $0.45$
Two b-tagged subjets | $0.17$ | $0.59$ | $0.18$ | $0.78$
$110.0<m_{H}<140.0$ GeV | $0.87$ | $0.53$ | $0.89$ | $0.49$
$\Delta R(\gamma,b_{i})>0.4,\not{E_{T}}<30\rm GeV,|\eta_{h}|<2.5$ | $0.38$ | $0.68$ | $0.38$ | $0.42$
$\Delta R(\gamma,h)>2.4$ | $0.97$ | $0.99$ | $0.98$ | $0.98$
Table 2: The cut-flow table for $Z+$photon production at $\sqrt{s}=13$ TeV.

####  IV.1.3 $pp\rightarrow WW$

The process $pp\rightarrow a^{*}\rightarrow WW$ depends on only one bosonic
operator $\mathcal{O}_{\tilde{W}}$. We consider final states where one $W$
decays leptonically $(e\nu$ or $\mu\nu)$ and the other $W$ decays
hadronically. The fully leptonic decay channel has been recently studied in
Ref. Carra:2021ycg . Although the hadronic decay channel of a gauge boson
suffers from the presence of background processes with significantly larger
cross-sections, it has a larger branching fraction than the leptonic decay
channel. The semileptonic final state, therefore, offers a good balance
between efficiency and purity. It also allows a full kinematic reconstruction
of the diboson system, using the W mass to constrain the combined four-
momentum of the lepton and neutrino. Since the effects of the ALPs are most
dramatic at high boson momenta, we consider highly Lorentz-boosted vector
bosons where the hadronization products of the two final state quarks overlap
in the detector to form a single, large-radius jet. Dominant background to
this signal come from SM processes: $l^{\pm}\nu+$0,1 jets (including
contributions from both on-shell and off-shell $W$ boson), $t\bar{t}$ (where
one of the top quark decays hadronically and the other semi-leptonically),
single top quark production ($t(\bar{t})\,j,\,\,tW$), $W^{+}W^{-}+$ jets
($W\rightarrow l\nu$, $W\rightarrow jj$), $t\bar{t}W+$ jets (when both top
quarks decay hadronically and $W\rightarrow l\nu$) and $WZ$ (with
$W\rightarrow l\nu$, $Z\rightarrow\nu\bar{\nu}/jj)$. To reject other
subdominant backgrounds from Drell–Yan and fully leptonic tt events, we reject
events that contain more than one lepton. Jets are clustered by the anti-kT
algorithm with radius parameter R=0.8 and required to have a hard $p_{T}>200$
GeV. The $\vec{p}_{T}^{miss}$ is required to be larger than 110 GeV to reject
QCD multijet background events.

The leptonic $W$ boson candidate is constructed from the lepton and the
$\vec{p}_{T}^{miss}$. The longitudinal momentum of the neutrino can be solved
for by appying the $W$ boson mass constraint, assuming that the neutrino is
the sole contributor to the ${p}_{T}^{miss}$. We follow the CMS analysis
method CMS:2019ppl . The transverse component of the neutrino momentum comes
directly from the $\vec{p}_{T}^{miss}$. Fixing the mass of the $W$ boson
candidate to its pole mass value, one can relate the four-momentum of the $W$
boson to those of the lepton and neutrino via a quadratic equation, which can
have two real or complex solutions. In the case of two real solutions, the
solution with the smaller absolute value is assigned as the neutrino
longitudinal momentum, whereas in the case of two complex solutions, the real
part common to both is instead assigned. The leptonic and hadronic boson
candidates are combined into a diboson system by adding their four-momenta.
Because signal events are expected to have a back-to-back topology in the
detector, we require events in the signal region to satisfy the following
requirements: $\Delta R(J,\text{lepton})>\pi/2$,
$\Delta\phi(J,\vec{p}_{T}^{miss})>2$, and $\Delta\phi(J,W_{l})>2$, where
$W_{l}$ denotes the reconstructed leptonic $W$ boson candidate. Additionally,
we require $m_{WW}>900~{}\rm{GeV}$ to isolate the signal events. The CMS
collaboration presents a measurement of the $m_{WW}$ distribution in the
lepton+jets+missing energy channel, employing a dataset of $137$ fb-1 from the
Run II LHC. This analysis spans $m_{WW}$ bins up to 4 TeV. The invariant mass
of the reconstructed diboson system, $m_{WV}$, is the chosen event variable
for the signal extraction. The comparison of the ALP signal with CMS data is
illustrated in Fig. 2 (c).

(a) (b)

(c) (d)

Figure 2: (a) The differential distribution of events at 13 TeV LHC and an
integrated luminosity of 139 fb-1 with respect to reconstructed $M_{Zh}$ for
the SM+ALP signal (red line) as well as the total SM background as given by
the ATLAS measurement in Ref ATLAS:2022enb . The signal corresponds to
coefficients $a_{2D}=0.1$ and $f_{a}=10$ TeV (b) Invariant mass distribution
of $J\gamma$ with 36.1 fb-1 data at 13 TeV run of LHC. The total SM prediction
(blue line) and the data (black dots with error bars) are taken from the
analysis by the ATLAS collaboration in Ref. ATLAS:2018sxj . The signal (red
line) correspond to coefficients $c_{\tilde{W}}=-c_{\tilde{B}}=1$ and
$f_{a}=5$ TeV, (c) $m_{WW}$ distribution in $1\ell+J+\not{E}_{T}$ channel,
incorporating data points and total SM background from Ref. CMS:2019ppl  by
the CMS measurement at 13 TeV and an integrated luminosity of 35.9 fb-1. The
solid red line represents the ALP signal for $c_{\tilde{W}}=1$ and $f_{a}=5$
TeV and (d) Comparison of ALP signal events ($c_{\tilde{W}}=-c_{\tilde{B}}=1$
and $f_{a}=5$ TeV) and the total SM expectation along with the CMS measurement
data points for the transverse mass distribution of $WW$ system from the
production of $WW\gamma$ in $2\ell+\not{E}_{T}+\gamma$ channel CMS:2023rcv  at
13 TeV and 138 fb-1 integrated luminosity.

####  IV.1.4 $pp\rightarrow WW\gamma$

We now consider the non-resonant ALP mediated production of triboson states of
$W^{+}W^{-}\gamma$ from $pp$ collisions and both the W bosons decaying
leptonically. We find that even for an elusive ALP mass window of
$m_{a}\subset[0.1,100]$ GeV, the process $W^{+}W^{-}\gamma$ deviates from the
SM case as it gets modified due to the presence of ALP-bosonic couplings
$\\{g_{aZ\gamma},g_{aWW},g_{aWW\gamma},g_{a\gamma\gamma}\\}$. We look into
final states with two different flavour, opposite sign (DFOS) leptons and one
photon along with $\not{E}_{T}$. Amongst the existing analyses for the same
final state carried by the experimental collaborations, the CMS analysis has
recently reported the first observation of SM $W^{+}W^{-}\gamma$ production in
the leptonically decay channel CMS:2023rcv  and hence, we have implemented the
same set of cuts as described below for our analysis to constrain the new
physics parameter space. The main SM backgrounds arise from
$WW\gamma,WZ\gamma,Z\gamma$ and $t\bar{t}\gamma$ and processes with non-prompt
leptons and photons. The final state events comprise of a photon having a
transverse momentum of $p_{T}^{\gamma}>20$ GeV and $|{\eta^{\gamma}}|<2.5$.
There should be exactly one pair of DFOS leptons requiring $|{\eta^{l}}|<2.5$
and $p_{T}^{l}>$ 20 $\rm{GeV}$. We also require
${p}_{T}^{\rm{miss}}>20~{}\rm{GeV}$. To minimise backgrounds from $WZ\gamma$
and relevant top quark processes, events are rejected that contain an
additional muon or electron with $p_{T}>10~{}\rm{GeV}$ or at least one
$b$-jet. The photon and the lepton must be well separated, such that $\Delta
R(l,\gamma)>0.5$. To further suppress background contributions, we impose
specific criteria: the dilepton invariant mass ($m_{ll}$) $>10$ GeV, the
dilepton transverse momentum ($p_{T}^{ll}$) $>15$ GeV, and the transverse
mass,
$m_{T}^{WW}=\smash[b]{\sqrt{2p_{T}^{ll}\vec{p}_{T}^{miss}[1-\cos\Delta\phi(p_{T}^{ll},\vec{p}_{T}^{miss})]}}>10~{}\rm{GeV}$.
The observed distributions of $m_{T}^{WW}$ in the bins of the invariant mass
of dilepton-photon system ($m_{ll\gamma}$) are compared with the ALP signal
(as shown in Fig. 2 (d) for one such benchmark case of ALP scenario) to derive
constraints on its couplings.

We take the experimental measurements in Table 1 as input and our theoretical
expectations for the observables in the ALP model. We fit the Wilson
coefficients described in Sec. II in the ALP EFT to quantify their effects in
the processes processes $Zh,Z\gamma,W^{+}W^{-}$, and $W^{\pm}W^{\mp}\gamma$.
We calculate uncorrelated $\chi^{2}$ functions of
$(m_{a},\bar{C}=c_{i}/f_{a})$ to differential cross section measurements for
these processes, which is defined as follows :

$\chi^{2}(\bar{C})=\sum_{i}^{\text{bins}}\frac{(\epsilon_{i\alpha}N(\bar{C})_{i\alpha}^{\text{theo}}-N_{i\alpha}^{\text{exp}})^{2}}{(\sigma^{\text{exp}}_{i\alpha})^{2}}\,,$
(17)

where $N_{i\alpha}^{exp}$ represents the observed event count,
$N_{i\alpha}(\bar{C})^{theo}$ estimates the expected event count, assumed to
be altered from the SM due to non-zero EFT couplings in the $i^{\text{th}}$
bin of dataset $\alpha$. We introduce an efficiency factor,
$\epsilon_{i\alpha}$, to account for an overall scaling of the simulation
data, where $\epsilon_{i\alpha}$ is calculated by the ratio of the
experimental SM differential cross section to our predicted differential
cross-section under SM conditions for each bin in dataset $\alpha$. The term
$\sigma_{i}^{exp}$ denotes the error, comprising both statistical and
systematic uncertainties, combined in quadrature.

$\sigma_{i}^{exp}=\sqrt{N_{i}^{exp}+\Delta_{sys}N_{i}^{exp}}$

We minimise the $\chi^{2}$ function to find the best fit value of
$c_{i}/f_{a}$ and identify the 95% confidence-level (C.L.) intervals by
requiring $\Delta\chi^{2}=\chi^{2}-\chi^{2}_{\rm min}\leq 3.84$.

The bounds extracted from these four process analyses constrain the products,
$g_{agg}g_{aV_{1}V_{2}}$ and $g_{agg}g_{aZh}$. For the $Zh$ process, we obtain
$a_{2D}<0.078$ TeV-1 at 95% C.L. The 95% C.L. limits on the coupling product
$g_{agg}g_{aWW}$ are determined to be : $g_{agg}g_{aWW}<0.59$ TeV-2 from WW
analysis and $g_{agg}g_{aWW}<0.27$ TeV-2 from $WW\gamma$ analysis. In
addition, the $WW\gamma$ process induces a four point interaction of
$aWW\gamma$ and the analysis puts a constraint on it of
$g_{agg}g_{aWW\gamma}<0.18$ TeV-2. The $Z\gamma$ process analysis yields a 95%
C.L. exclusion limit of $g_{agg}g_{aZ\gamma}<0.25$ TeV-2. These limits can be
interpreted as constraints on $g_{aVV}$, assuming a constant $g_{agg}$ value
of 1 TeV-1. A smaller $g_{agg}$ would result in more stringent limits on
$g_{aV_{1}V_{2}}$. It is noteworthy that these operator effect limits are more
significantly constrained by the higher energy data bins.

###  IV.2 Validity of EFT

In this work, we explore the non-resonant $s$-channel ALP signatures, which
have several interesting characteristics. When the momentum transfer through
the ALP propagator obeys $\sqrt{|p_{a}^{2}|}\gg m_{a},\Gamma_{a}$ where
$\Gamma_{a}$ is its decay width, the cross-section and differential
distribution of the ALP signal remain largely independent of $m_{a}$’s actual
value. This implies that our search technique retains its validity over a wide
range of ALP masses, particularly those significantly below the energy scale
of the experiment. For the LHC searches we investigated, this translates into
a consistent detection capability for ALP masses below 100 GeV.

Fig.3 (left panel) verifies the off-shell approximation for the processes. It
shows the ALP signal cross section at $\sqrt{s}=13\rm{TeV}$, applying the cuts
defined in Eq. (16), plotted against $m_{a}$ for fixed values of $c_{2D}$,
$c_{\tilde{W}}$, $c_{\tilde{B}}$, and $f_{a}$. Here, $\Gamma_{a}$ is
implicitly determined at each data point, dependent on $m_{a}$ and the ALP
couplings, following the relation $\Gamma_{a}\propto
m_{a}^{3}(c_{i}/f_{a})^{2}$. The extended lines running almost parallel with
the mass of the ALP in Fig.3 (left panel) confirm that our simulations are
relevant even for extremely small $m_{a}$ values. We perform the analyses on
the assumption that the ALP contributes only off-shell in all the processes we
considered, setting the ALP mass and decay width in our simulations at
$m_{a}=1~{}{\rm{MeV}},\Gamma_{a}=0$.

As the mass $m_{a}$ increases, the cross sections for processes like
$Zh,Z\gamma,W^{\pm}W^{\mp}$, and $W^{\pm}W^{\mp}\gamma$ shows a resonance
effect when the propagator becomes predominantly influenced by the ALP mass.
This is particularly noticeable for all the processes. The chosen point values
for $c_{i},f_{a}$ facilitate ALP exchange in the $s$-channel
$Z\gamma,W^{\pm}W^{\mp},W^{\pm}W^{\mp}\gamma$ at masses above 100 GeV, close
to 150 GeV, and around 250 GeV, respectively. The slight shifts in the
$WW\gamma$ and $Z\gamma$ processes can be attributed to the photon $p_{T}$
preselection cut. We evaluated the $Z\gamma$ channel at a point
$(c_{\tilde{W}}=1,c_{\tilde{B}}=-0.305)$ to ensure a “photophobic” interaction
(where $g_{a\gamma\gamma}=0$) and to explore the resonant case induced by
$g_{aZ\gamma}$ coupling. In the $Zh$ process, the resonance effect is apparent
near 300 GeV. These observations serve as a validation that our results hold
for ALP masses up to approximately $100~{}\rm{GeV}$. At this mass, the cross
sections for all four mentioned processes deviate by less than 5% from their
asymptotic values when $m_{a}$ approaches zero.

Furthermore, an important distinction of the non-resonant process is its lack
of dependence on specific assumptions about extra couplings not directly
contributing to the process and any other model specific parameters. This is
in contrast to on-shell analyses, which are usually limited to particular mass
and width ranges and where the impact of extra ALP couplings becomes evident
in their partial decay widths. Conventionally, studies on ALP limits from
resonant processes have focused on a single independent $g_{aV_{1}V_{2}}$
coupling, as outlined in Eq.(5)Mimasu:2014nea ; Jaeckel:2015jla ;
Knapen:2016moh ; Dobrich:2015jyk ; Izaguirre:2016dfi . However, recent studies
have started to explore scenarios incorporating two or three independent
couplings simultaneously Mariotti:2017vtv ; Alonso-Alvarez:2018irt ;
Gavela:2019wzg . This approach increases the model-independence of non-
resonant searches, making them more effective in detecting new physics
phenomena.

Estimating the validity of the EFT expansion is crucial for collider bounds,
especially given the broad range of energies encountered in hadron collider
experiments. We now consider the range of Wilson coefficients constrained and
check whether they allow for a valid EFT interpretation of cross-sections.
Theoretically, the $g_{aV_{1}V_{2}}$ couplings depend only on the ratio
$c_{i}/f_{a}$ (as detailed in Eqs.(6a)-(6c)). However, the value of $f_{a}$ is
important in assessing the validity of EFT, which in turn restricts the energy
range feasible for LHC searches, such as energy bins where
$\sqrt{\hat{s}}<f_{a}$. If the underlying BSM theory operates in a weak
coupling regime, leading to the operators in Eq.(4) at one-loop, the
coefficients might be attenuated by an additional $16\pi^{2}$ factor. This
would considerably restrict the valid energy bins for LHC searches.

For illustration, in the $Z\gamma$ production process where the Z decays into
two bottom quarks and is detected as a fat-jet, the energy scale of the
collision is determined by the invariant mass of the jet-photon system
($m_{J\gamma}$). The $Z\gamma$ EFT expansion validity can be maintained by
ensuring $m_{J\gamma}$ stays below the cutoff scale ($f_{a}$). However,
precisely defining the EFT cutoff scale in a model-independent manner is
difficult without taking into account the specific details of the underlying
UV-complete theory. To have an idea of the cutoff scale, we adopt a
methodology based on Ref. Busoni:2013lha ; Bhattacharya:2015vja . If
$m_{J\gamma}$ is consistently smaller than $f_{a}$ in most collisions, the
ratio $R_{Minv}$, defined subsequently, would approach 1.

$R_{Minv}\equiv\frac{\int^{m_{J\gamma}<{m^{max}_{J\gamma}}}\frac{d\sigma}{dm_{J\gamma}}\;dm_{J\gamma}}{\int\frac{d\sigma}{dm_{J\gamma}}\;dm_{J\gamma}}$
(18)

In Fig.3 (right panel), we see this effect in the process involving non-zero
EFT coupling at $\sqrt{s}=$14 TeV. As for the $Z\gamma$ process, ratio of
$R_{Minv}$ close to 1 suggest that the energy exchange in the process remains
considerably below ${m^{max}{J\gamma}}$, the maximum value allowed for
$m{J\gamma}$. Identifying such peak $m_{J\gamma}$ values provides a practical
reference for the EFT cutoff scale, $f_{a}$. Additionally, we examine similar
variations for the $Zh$ and $WW$ processes, using their respective invariant
mass measurements. In cases involving both the Higgs chiral operator and
linear bosonic operators, the ratio $R$ is approaching unity when
$m^{max}_{J\gamma}>$ 2.0TeV.

Figure 3: Left: Total cross sections at $\sqrt{s}=13~{}\rm{TeV}$ for the ALP
contributions to the different scattering processes as a function of the ALP
mass. The value of $f_{a}$ is taken to be $4$ TeV. The ‘$Zh^{\prime}$ curve is
evaluated at $c_{2D}=1~{}\rm{TeV^{-1}}$ and ‘$WW$’ curve at
$c_{\tilde{W}}=1~{}\rm{TeV^{-1}}$. For the ‘$Z\gamma$’ and ‘$WW\gamma$’ cases,
they are evaluated at $c_{\tilde{W}}=-c_{\tilde{B}}=1~{}\rm{TeV^{-1}}$ point.
At each point in the plot, the ALP decay width was re-estimated as a function
of $m_{a}$ and the Wilson coefficients. Right:  The ratio $R_{Minv}$ variation
as a function of maximum invariant mass of the final state system in $Zh$
(red), $Z\gamma$ (yellow) and $WW$ (blue) production processes.

###  IV.3 Collider Analysis with HL-LHC probes

We will discuss the results of our cut-based analysis for a few benchmark
points (BPs) to accentuate the distinguishability of the ALP signal from the
backgrounds. The BPs are so chosen such that they obey the experimental
constraints obtained from the 13 TeV data. The selected benchmark points are
listed in Table 3. As some of the operator coefficients probe more than one
process at a time, we choose these points to highlight specific regions of
parameter space so that they probe one coupling at at time for a specific
process as detailed below.

Signal | Coupling parameter | Process
---|---|---
BP1 |  $a_{2D}=0.2$ TeV-1, $f_{a}=5$ TeV | $pp\rightarrow Zh$
BP2 |  $c_{\tilde{W}}=0.5$ TeV-1, $c_{\tilde{B}}=-0.5$ TeV${}^{-1},f_{a}=5$ TeV | $pp\rightarrow Z\gamma$
BP3 |  $c_{\tilde{W}}=0.5$ TeV-1, $c_{\tilde{B}}=-1.639$ TeV${}^{-1},f_{a}=5$ TeV | $pp\rightarrow Z\gamma$
BP4 |  $c_{\tilde{W}}=0.5$ TeV-1, $f_{a}=5$ TeV | $pp\rightarrow W^{\pm}W^{\mp}$
BP5 |  $c_{\tilde{W}}=0.5$ TeV-1, $c_{\tilde{B}}=0.5$ TeV${}^{-1},f_{a}=5$ TeV | $pp\rightarrow W^{\pm}W^{\mp}\gamma$
BP6 |  $c_{\tilde{W}}=0.5$ TeV-1, $c_{\tilde{B}}=-0.152$ TeV${}^{-1},f_{a}=5$ TeV | $pp\rightarrow W^{\pm}W^{\mp}\gamma$
Table 3: Summary of selected benchmark points for the study

For the $Zh$ production, we have only operator that contributes at LO and we
choose the corresponding operator coefficient value for $a_{2D}=0.5$ TeV-1.
The $Z\gamma$ production receives contribution from $g_{aZ\gamma}$ coupling
which in turn receives contribution from $c_{\tilde{W}}$ and $c_{\tilde{B}}$.
We choose BP2 such that $c_{\tilde{W}}=-c_{\tilde{B}}$ and BP3 such that
$c_{\tilde{W}}=-c_{\tilde{B}}t_{\theta}^{2}~{}i.e.,~{}g_{aZZ}=0$. The
$W^{\pm}W^{\mp}$ production receives bosonic contribution from $g_{aWW}$
coupling only and thus, depends on $c_{\tilde{W}}$. $W^{\pm}W^{\mp}\gamma$
production receives contributions from $g_{a\gamma\gamma},g_{aZ\gamma}$ and
$a_{aWW},g_{aWW\gamma}$ couplings. BP5 corresponds to $g_{aZ\gamma}=0$ while
BP6 corresponds to
$c_{\tilde{W}}=-c_{\tilde{B}}/t_{\theta}^{2},~{}i.e.,~{}g_{a\gamma\gamma}=0$.
Couplings $a_{aWW},g_{aWW\gamma}$ are proportional to $c_{\tilde{W}}$ only.
Equipped with these benchmark points, we now discuss some kinematic
differences between the ALP signal and the SM backgrounds for each of the
above mentioned processes.

We first consider the Higgs-strahlung process, which has a radius R=1 fat-jet
and two leptons in the final state. Fig. 4 (a) shows the mass of the leading
fat-jet for the signal BP1 and the dominant backgrounds. It is evident from
these distributions that the peak around $115-140~{}\rm GeV$ reflect the Higgs
peak for the signal process whereas for most of the backgrounds, the peaks are
below 50 GeV reflecting that the fat-jet mimicing from either single prong
hard QCD jet or a peak around 90 GeV reflects $Z$ boson or peak about 165-185
GeV from a top. The variable $p_{T_{J}}$ (Fig. 4(b)) is quite efficient in
distinguishing the new interactions from most of the SM backgrounds. The
availability of larger parton center-of-mass energy in these derivative
interactions pushes the transverse momentum of fat-jet ($p_{T_{J}}$) to higher
values. We thus, put slightly tighter constraints on these variables compared
to the 13 TeV analysis, namely, $p_{T_{J}}>250$ GeV and $115<m_{{J}}<140$ GeV.
We also select events satisfying $m_{Zh}>500$ GeV.

(a) (b)

Figure 4: Normalized distributions of (a) the mass $m_{J}$ and (b) the
transverse momentum $p_{T_{J}}$ of the jet, both for the $pp\rightarrow hZ$
ALP mediated signal and SM backgrounds. For the ALP mediated signal, we have
chosen BP1 with $a_{2D}=0.2$ and $f_{a}=5$ TeV (blue).

For $Z\gamma$, the photon $p_{\textrm{T}}$ is a strong discriminator. The
distributions for the ALP signals for BP2 and BP3 and various SM backgrounds
are shown in Fig. 5 (a). The photons in the signal events exhibit a hard
$p_{T}$. The $E_{\gamma}$ distributions extend to 1 TeV. In the signal
process, despite being an s-channel process, we see significant enhancement
from the SM backgrounds at high energy tails of the distribution due to the
contribution of type of dimension-5 operators. In presence of the effective
operators, the cross-section grows faster at higher energies compared to the
SM backgrounds whose effect diminish with increasing energy.

(a) (b)

Figure 5: Normalized distributions of (a) the transverse energy $p_{T_{J}}$ of
the photon and (b) N-subjettiness of the jet, both for the $pp\rightarrow
Z\gamma$ ALP mediated signal and SM backgrounds. For the ALP mediated signal,
we have chosen BP2 with with $c_{\tilde{W}}=1,c_{\tilde{B}}=-1,f_{a}=5$ TeV
(blue) and BP3 with with $c_{\tilde{W}}=0.5,c_{\tilde{B}}=-1.639,f_{a}=5$ TeV
(red).

The fat-jet resulting from the $Z\rightarrow b\bar{b}$ decay can potentially
retain details about its two-pronged structure. This characteristic is
captured by the jet-shape variable known as N-subjettiness Thaler:2010tr ;
Thaler:2011gf , which is defined as follows:

$\displaystyle\tau_{N}^{(\beta)}=\frac{1}{\mathcal{N}_{0}}\sum\limits_{i}p_{i,T}\,\min\left\\{\Delta
R_{i1}^{\beta},\Delta R_{i2}^{\beta},\cdots,\Delta R_{iN}^{\beta}\right\\}$
(19)

where, $N$ refers to the number of subjet axes taken within the fat-jet. The
index $i$ runs over the individual jet constituents, and $p_{i,T}$ represents
their transverse momenta. $\Delta
R_{ij}=\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}$ measures the separation in
the $\eta-\phi$ plane between a possible subjet $j$ candidate and a
constituent particle $i$. The normalization factor, $\mathcal{N}{0}$, is
computed as $\sum{i}p_{i,T}R_{0}$, where $R_{0}$ denotes the fat-jet radius.
Essentially, the ratio $\tau_{N}/\tau_{N-1}$ serves to differentiate between
jets that likely contain $N$ internal energy clusters versus those with $N-1$
clusters. Specifically in our analysis, the jet emanating from the Z boson is
observed to exhibit smaller values for $\tau_{21}$ in comparison to typical
QCD jets, a pattern evident in Fig 5(b). Thus, a cut of $\tau_{21}<0.45$ can
reduce a significant amount of background while retaining most of the signal
events.

We analyze the $pp\rightarrow W^{\pm}W^{\mp}$ process, in which one W boson
undergoes a leptonic decay and the other a hadronic decay. A key variable in
this context is the effective mass, $m_{eff}$, which is pivotal in searches
for Beyond the Standard Model (BSM) physics. It is computed as follows:

$m_{eff}=\sum_{i}|\vec{p}{T{i}}|+\not{E}_{T}$ (20)

In this equation, $i$ encompasses all entities in the event, including
reconstructed jets. This comprehensive variable, which doesn’t rely on
specific event topology, proves to be highly useful, especially given that
signal events typically have a significantly higher parton-level center-of-
mass energy compared to most Standard Model (SM) background processes. Fig. 6
(a) showcases the distribution of this effective mass for the process under
study. It is evident that for most SM backgrounds (with the exception of
$t\bar{t}$+jets), the distributions tend to peak at lower values than in the
ALP scenario. It’s important to note that these are normalized distributions,
providing qualitative insights into potential additional cuts on these
variables, rather than quantitative ones.

In Fig. 6 (b), we display the $\Delta\phi(jet,\not{E}_{T})$ distribution for
both the signal and background processes. The distribution peaks around
$\sim\pi$ for the signal, as well as for the SM $WW$ and $W+$jets backgrounds.

We now consider the process $pp\rightarrow W^{\pm}W^{\mp}$ where only one W
boson decays leptonically while the other decays hadronically. Here, we
consider the $m_{e}ff$ or the effective mass variable which is an important
variable for BSM searches. It defined as follows :

$m_{eff}=\sum_{i}|\vec{p}_{T_{i}}|+\not{E}_{T}$ (21)

Here, $i$ encompasses all entities in the event, including reconstructed jets
and $p_{T}$ refers their transverse momenta and $\not{E}_{T}$ is the total
transverse missing energy in the event. This global variable, which does not
rely on specific event topology, proves to be highly useful, especially given
that signal events typically receive a high parton-level center-of-mass energy
compared to most SM background processes. In Fig. 6 (a), we present the
effective mass of the process distribution for this channel for BP4. It is
evident that for most SM backgrounds, the distributions tend to peak at lower
values than in the ALP scenario. It is to be noted that these are normalized
distributions, providing qualitative insights into potential additional cuts
on these variables, rather than quantitative ones.

In Fig. 6 (b), we display the $\Delta\phi(jet,\not{E}_{T})$ distribution for
both the signal and background processes. The distribution peaks around
$\sim\pi$ for the signal, as well as for the SM $WW$ and $W+$jets backgrounds.

|
---|---
(a) | (b)
(a)  Figure 6: Normalized distributions of (a) the effective mass $m_{eff}$
variable and (b) $\Delta\phi(jet,\not{E}_{T})$, both for the $pp\rightarrow
WW$ ALP mediated signal and SM backgrounds. For the ALP mediated signal, we
have chosen BP4 with $c_{\tilde{W}}=0.5,f_{a}=5$ TeV (blue).

Next, we examine the $pp\rightarrow W^{\pm}W^{\mp}\gamma$ process in a fully
leptonic channel, characterized by two different-flavor opposite-sign (DFOS)
leptons, a photon, and missing energy in the final state. In Fig. 7 (a), the
invariant mass distribution of the dilepton-photon system is depicted. In case
of the signal, the leptons originating from the W bosons are boosted due to
the influence of the ALP coupling. As a result, the distribution for the
signal shows a prominent extension towards higher values of the system
invariant mass, in contrast to the SM background processes. We also show the
distribution of $\not{E}_{T}$ in Fig. 7 (b) for both the pure SM backgrounds
and the ALP signal events. It is evident that for all the benchmark points
considered, the event distribution in the presence of an ALP is shifted
towards increased missing transverse energy, distinguishing it from the
typical SM scenarios. Thus, these variables play a significant role in
isolating the ALP interaction effects in the events.

(a) (b)

Figure 7: Normalized distributions of (a) the invariant mass of the dilepton
and photon system, $m_{ll\gamma}$ and (b) transverse missing energy
$\not{E}_{T}$, both for the $pp\rightarrow WW\gamma$ ALP mediated signal and
SM backgrounds in a fully leptonic decay channel. For the ALP mediated signal,
we have chosen BP5 with with $c_{\tilde{W}}=0.5,c_{\tilde{B}}=-0.5,f_{a}=5$
TeV (blue) and BP6 with with $c_{\tilde{W}}=0.5,c_{\tilde{B}}=-0.152,f_{a}=5$
TeV (red).

It is relevant to mention here that the kinematic distributions for BP5 and
BP6 look quite similar. As the mass of the ALPs are same and it indicates that
the process receives dominant contribution from $g_{aWW}$ as we re-iterate
that the benchmark points have been chosen such that BP5 leads to
$g_{aZ\gamma}=0$ and BP6 leads to $g_{a\gamma\gamma}=0$.

We now explore the relationship between two variables in the $WW\gamma$ final
state: the invariant mass of the dilepton-photon system, $m_{ll\gamma}$, and
the $\Delta R$ separation between the two leptons. Fig. 8 highlights how the
populated regions in the phase-space shift with the inclusion of new physics
effects from higher dimensional operators. The following observations emerge
from this figure:

  * •

In the background scenarios, such as $WZ\gamma$ and events with non-prompt
electrons or jets, the $m_{ll\gamma}$ distribution typically decreases
smoothly and rapidly. However, in scenarios involving new physics, this
distribution fall more slowly. We observe that background dileptons are more
likely to appear in the same hemisphere, contrasting with the signal events.
In the case of the ALP signal, there is a notable increase in event density as
$\Delta R_{ll}$ approaches $\pi$, indicating that the leptons from the W
bosons are more likely to be back-to-back (indicated by red color for a higher
number of events). Implementing a cut on the invariant mass of the jet and
photon at 1 TeV would distinctly highlight these new phase space regions.
Additionally, an angular separation cut of $\Delta R_{ll}\geq 2.5$ could
effectively filter out a significant portion of the background events, which
tend to cluster at lower angular separations.

  * •

This implies suggests that ALP interactions which result in both the dilepton
and the photon gaining higher energy, also results in the angular separation
between the leptons tending to be larger compared to that in the SM
background. This correlation is especially evident in Fig.8(a), where the most
populated event regions are around $\Delta R_{ll}\sim\pi$, especially in high
$m_{ll\gamma}$ regions (around 200 GeV).

(a) (b)

Figure 8: Two dimensional histograms showing the correlation between invariant
mass of the dilepton and photon system, $m_{ll\gamma}$ and the separation
between the two leptons, $\Delta R_{ll}$. The $z$-axis indicates the
normalized frequency of events, in arbitrary units. (a) represents BP6 ALP
scenario with $c_{\tilde{W}}=0.5,c_{\tilde{B}}=-0.152,f_{a}=5$ TeV and (b)
represents the SM backgrounds, comprising, $SM~{}WW\gamma$, $WZ\gamma$,
$V\gamma$, $t\bar{t}\gamma$ and backgrounds from non-prompt leptons and non-
prompt photon.

We assess the sensitivity reach for various benchmark points at the 14 TeV
LHC. To quantify the signal significance, we use the following definition:

${\mathcal{S}}=\sqrt{2[(S+B)ln(1+S/B)-S]}$ (22)

Here $S$ and $B$ are the numbers of observed signal and background events,
respectively, corresponding to the residual signal and background cross
sections after applying the selection criteria that isolates the signal events
from the backgrounds.

The calculated signal significance for each benchmark point across the four
different processes is presented in Table 4. This includes for different
possible choices of integrated luminosities, specifically at
${\mathcal{L}}=300,1000$, and $3000$ fb-1.

| Statistical significance | Required Luminosity
---|---|---
|  (${\mathcal{S}}$) |  (in $fb^{-1}$)
Signal |  ${\mathcal{L}}=300$ fb-1 |  ${\mathcal{L}}=1000$ fb-1 |  ${\mathcal{L}}=3000$ fb-1 | ${\mathcal{S}}=3{\sigma}$ | ${\mathcal{S}}=5{\sigma}$
BP1 | 1.781 | 3.253 | 5.634 | 860.69 | 2363.02
BP2 | 2.053 | 3.748 | 6.493 | 640.44 | 1778.75
BP3 | 2.443 | 4.469 | 7.726 | 452.02 | 1256.90
BP4 | 1.812 | 3.319 | 5.748 | 817.01 | 2269.48
BP5 | 1.754 | 3.203 | 5.548 | 877.10 | 2436.39
BP6 | 1.911 | 3.489 | 6.044 | 739.21 | 2053.36
Table 4: Signal statistical significance at various benchmark points for the
distinct four processes of our study at the 14 TeV LHC. The significance
levels evaluated for integrated luminosities of ${\mathcal{L}}=300,1000,$ and
$3000$ fb-1. We also estimate the integrated luminosity required to attain a
$3\sigma$ and $5\sigma$ excess for each benchmark point at the LHC running at
$\sqrt{s}=14$ TeV.

We can see from Table 4 that BP1 for the ALP mediated Higgs-strahlung signal
will have substantial significance at $3000$ fb-1 luminosity. The major reason
is the large production cross-section of the ALP signal. Detecting signatures
of the $aZh$ interaction in this process, a phenomenon not expected in linear
expansions up to NNLO, would essentially serve as the smoking gun evidence for
non-linearity. The $Z\gamma$ process via BP2 and BP3 shows to have the most
prominent separation between the signal and the background. The $WW\gamma$ BP6
is slightly more sensitive in probing $g_{aWW}$ coupling.

###  IV.4 Direct probes of ALP coupling

In this subsection, we compare the constraints derived from direct ALP
searches with those obtained from non-resonant ALP-mediated processes. We
particularly focus on the ‘ALP-strahlung’ process, which involves the
production of ALPs in association with Higgs or vector bosons, and study the
constraints it puts on ALP-Higgs and ALP-vector boson interactions. We assume
the ALP to be stable within the collider, meaning that it has a sufficiently
long lifetime to leave the detector without decaying. This assumption depends
on the decay modes available to the ALP, which in turn depend on its mass and
couplings. For an ALP with a mass around 1 MeV, decaying to fermions or
heavier particles is not kinematically possible. The possible decay channels
include $a\rightarrow\nu\bar{\nu}\nu\bar{\nu}$ (indistinguishable from a pure
missing energy signature), $a\rightarrow\gamma\gamma$, and
$a\rightarrow\gamma\nu\bar{\nu}$. Both of the latter decays would typically
allow the ALP to traverse distances much greater than the detector’s
dimensions before decaying. When the ALP mass exceeds 1 MeV, new decay
channels to fermions become feasible once the ALP mass becomes greater than
twice the mass of the fermions in the final state. For example, $m_{a}\geq
3m_{\pi}$ ($\sim 0.5$ GeV) would enable hadronic decay channels. However, this
introduces a dependence on complex model-specific factors, which we do not
delve into in this study. Direct ALP probes involve additional model-based
assumptions, limiting the generality of fit results. For our simulations in
the MG5aMC_@NLO framework, we assume an ALP mass of 1 MeV, consistent with our
non-resonant analysis, and treat the ALPs as stable within the collider for
the purposes of detector simulation.

##### ATLAS measurement of Higgs boson production in association with missing
energy and h decays to b-quarks:

We will study the ALP signal $pp\rightarrow h(\rightarrow b\bar{b})+a$ and
reinterpret the ATLAS search for dark matter search in association with a SM
Higgs boson ATLAS:2021shl  in the context of the ALP signal. ATLAS has
recently provided measurements of the missing transverse energy
($E_{T}^{miss}$) distribution in events with a large-radius jet with two
b-tags and missing energy, using the Run-II data from the LHC at $\sqrt{s}=13$
TeV with an integrated luminosity ($\mathcal{L}$) of 139 fb-1, along with an
estimation of the SM background. The analysis is confined to a fiducial
region, which is closely replicated by the phase space cuts outlined in Table
5.

Data are categorized into 5 bins, with the bin widths increasing with higher
values of $E_{T}^{miss}$. The boundaries of these bins are set at (150, 200,
350, 500, 750) GeV. For the ALP signal simulation, we model the process

$pp\rightarrow h+a,\left(h\rightarrow b\bar{b}\right),$ (23)

with the reconstructed Higgs jet having a radius parameter of 1. The
comparative results between the ALP signal and the ATLAS data are depicted in
Fig. 9 (a), showcasing a slight increase in energy across the $E_{T}^{miss}$
bins. By studying the indirect probing in the $m_{Zh}$ bins, we have an
enhanced sensitivity to the ALP-Higgs coupling. We perform a $\chi^{2}$ fit to
establish a limit of :

$\left|\frac{f_{a}}{a_{2D}}\right|>1.75\,{\rm TeV}\qquad{\rm
at}\,\,95\%\,~{}{\rm C.L.}$ (24)

(a) Mono-Higgs | (b) Mono-Z | (c) Mono-W
---|---|---
Observable | Selection | Observable | Selection | Observable | Selection
$p_{T}$ leading jet |  $>100$ GeV | $N_{l}$ | $2$ |  $p_{T}$ of leading lepton |  $>55$ GeV ($\mu^{-}$),$>60$ GeV ($e^{-}$)
$E_{T}^{miss}$ |  $>150$ GeV |  $p_{T}$ of lepton |  $>20$ GeV |  $|\eta|$ of leading lepton |  $<2.5$ ($\mu^{-}$),$<2.47$ ($e^{-}$)
Lepton & extended $\tau$-lepton vetos |  | Reconstructed $Z$ boson mass |  $|m_{ll}-m_{Z}|<15$ GeV |  | excluding $1.37<|\eta|<1.52$
$\Delta\phi$(jet,$E_{T}^{miss}$) |  $>20$ rad | $p_{T}^{ll}$ |  $>60$ GeV | Exactly 1 lepton |
$E_{T}^{miss}$ |  $>500$ GeV | $p_{T}^{miss}$ |  $>100$ GeV | Jet veto |
large-R jet | $\geq 1$ | b-jet & extended $\tau$-lepton vetos |  |  |
b-tagged subjets within the R-jet | $>2$ | $\Delta\phi({\vec{p}_{T}^{j},\vec{p}_{T}^{miss}})$ | $>0.5$ | $E_{T}^{miss}$ |  $>55$ GeV ($\mu^{-}$),$>65$ GeV ($e^{-}$)
Reconstructed Higgs jet mass |  $\in[50,270]$ GeV | $\Delta\phi({\vec{p}_{T}^{ll},\vec{p}_{T}^{miss}})$ | $>2.6$ | $m_{T}$ |  $>110$ GeV ($\mu^{-}$),$>130$ GeV ($e^{-}$)
|  | $|p_{T}^{miss}-p_{T}^{ll}|/p_{T}^{ll}$ | $<0.4$ |  |
|  | $\Delta R_{ll}$ | $<1.8$ |  |

Table 5: Phase space cuts defining the selection criteria for signal region in
(a) ATLAS search for $h(\rightarrow b\bar{b})$ \+ MET ATLAS:2021shl , (b) CMS
search for $Z(\rightarrow l^{+}l^{-})$ \+ MET CMS:2017nxf  and (c) ATLAS
search for $W(\rightarrow l\nu_{l})$ \+ MET ATLAS:2019lsy .

(a) (b)
(c)

Figure 9: (a) The differential distribution of $E_{T}^{\rm miss}$ for
$h(\rightarrow b\bar{b})+E_{T}^{\rm miss}$ signal and background for
$\sqrt{s}=13$ TeV and $139~{}{\rm{fb}}^{-1}$ of integrated luminosity,
following the selection cuts from Table 5 (a). The total SM $E_{T}^{\rm miss}$
background (blue) distribution is obtained from ATLAS:2021shl , and the signal
$pp\rightarrow ah$ ($h\rightarrow b\bar{b}$) $E_{T}^{\rm miss}$ distribution
corresponds to coefficients $a_{2D}=1$ and $f_{a}=4$ TeV along with SM
contribution (red). (b) Distribution of $p_{T}^{\rm miss}$ for $a\,Z$
($Z\rightarrow\ell^{+}\ell^{-}$) production, with coefficients
$c_{\tilde{W}}=1,c_{\tilde{B}}=-1,f_{a}=5$ TeV (red) and the experimental data
and SM backgrounds from the CMS analysis CMS:2017nxf  at 13 TeV and
$35.9\,\mathrm{fb}^{-1}$. (c) Distribution of transverse mass $m_{T}$ for
$a\,W^{\pm}$ ($W^{\pm}\rightarrow\ell^{\pm}\nu_{\ell}$) production in the
$\mu+\not{E}_{T}$ final state, obtained with $c_{\tilde{W}}=1,f_{a}=5$ TeV
(red), compared with experimental data and SM backgrounds from the ATLAS
analysis ATLAS:2019lsy  at 13 TeV, and $139\,\mathrm{fb}^{-1}$ integrated
luminosity.

##### CMS search for new physics events with Z production with large missing
energy:

We consider now ALP production in association with a $Z$ boson, in hadronic
collisions. We will study the impact of the ALP signal on the CMS measurement
of $Z+\not{E}_{T}$ search CMS:2017nxf  with $\sqrt{s}=13$ TeV and integrated
luminosity $35.9\,\mathrm{fb}^{-1}$. This time, we will be considering a
measurement in the leptonic channel to assess the sensitivity to the effective
ALP interaction.

We will use the $\not{E}_{T}$ distribution as a key kinematic discriminator
between signal and background. Data within the fiducial region, as opposed to
the full phase space, will be used to refine the search. The selection cuts
from the second and third column of Table 5 will be employed. The comparison
of signal and background $\not{E}_{T}$ distributions for $\ell=\mu$ can be
seen in Fig. 9 (b), with the maximum $\not{E}_{T}$ bin being set at 600 GeV.
To ensure the EFT applicability, we remove events in each bin where
$\sqrt{\hat{s}}$ exceeds $2\not{E}_{T}^{\text{max}}$.

The ALP-photon-$Z$ and ALP-$Z$-$Z$ couplings could potentially lead to a
mono-$Z$ final state. This indicates contributions from both Wilson
coefficients $c_{\tilde{W}}$ and $c_{\tilde{B}}$ to this process. We establish
constraints on $c_{\tilde{W}}$, assuming
$c_{\tilde{B}}=-t_{\theta}^{2}c_{\tilde{W}}$. Similar to previous analyses, a
$\chi^{2}$ fit, as outlined in Eq.(17), will be used to derive constraints on
$c_{\tilde{W}}$, giving :

$\left|\frac{f_{a}}{c_{\tilde{W}}}\right|>7.75,{\rm TeV}\qquad{\rm
at},,95\%~{}{\rm C.L.}$ (25)

The mono-$Z$ search proves to be a robust tool in restricting
$c_{{\tilde{W}}}/f_{a}$. Notably, in the higher-energy regime of $p_{T}^{\rm
miss}>250$ GeV, the ALP contribution becomes considerable, especially in the
tail of the $p_{T}^{\rm miss}$ distribution, which is where the most
significant constraints originate. Nonetheless, this results in a constraint
on $c_{\tilde{W}}$ that is less stringent than what is derived from the CMS
$m_{WW}$ and $WW\gamma$ distributions.

##### ATLAS measurement of charged lepton with missing energy:

Let us now concentrate on the ALP production in association with a $W$ boson.
We reinterpret the ATLAS search for $W^{\prime}$ decaying to $l+\not{E}_{T}$
final states with $139$ fb-1 integrated luminosity ATLAS:2019lsy . We employ
the transverse mass distribution of the leptonically decaying W for our
analysis, as depicted in Fig.9(c). To study the influence of the ALP signal on
the $m_{T}$ distribution, we apply the selection criteria outlined in the
final two columns of Table5. The figure also includes the $m_{T}$ spectrum of
the SM background. The ALP coupling involved in this signal is
$c_{\tilde{W}}$, with the high-$m_{T}$ bins playing a significant role in
shaping the constraints on $c_{\tilde{W}}$.

Background data for the electron and muon samples are taken from
Ref.ATLAS:2019lsy , and the depicted bins correspond to those with available
experimental background information, following $m_{T}<m_{T}^{\text{max}}=2.6$
TeV for electrons and $m_{T}<m_{T}^{\text{max}}=3$ TeV for muons. From this
analysis, we derive a constraint of:

$\left|\frac{f_{a}}{c_{\tilde{W}}}\right|,>,9.11,{\rm TeV}\qquad{\rm
at},,95\%~{}{\rm C.L.}$ (26)

from the ATLAS $m_{T}$ data. Consequently, the mono-$W$ analysis yields
stronger constraints than those obtained from the non-resonant $pp\rightarrow
WW$ process.

It is important to note that most direct bounds usually depend on specific
model assumptions, which often involve setting all other coefficients to zero,
unlike indirect bounds. As such, the indirect limits presented in this study
act as a good complementary probe, proving useful even in instances where
direct probes might initially seem more stringent.

##  V Projected Sensitivities on ALP EFT couplings

##### Sensitivity from Zh:

The results presented in Table 4 provide the sensitivities for different
benchmark points. This section outlines the sensitivity projections within the
parameter space of ALP couplings using the relevant ALP-mediated non-resonant
production processes.

In Fig. 10 (a), we present the variation of significance of ALP mediated $hZ$
signal with operator coefficient $\frac{{a}_{2D}}{f_{a}}$ for an integrated
luminosity of $139$ fb-1 at $\sqrt{s}=13$ TeV (red curve) and $3000$ fb-1 at
$\sqrt{s}=14$ TeV (yellow curve). Signal stands over the background a
$3\sigma$ level for $\frac{{a}_{2D}}{f_{a}}\simeq$ 0.095 TeV-1 (13 TeV) and
$\frac{{a}_{2D}}{f_{a}}\simeq$ 0.058 TeV-1 (14 TeV).

Figure 10: Left: Significance, $\mathcal{S}$ (calculated using total cross-
sections) as function of $\frac{a_{2D}}{f_{a}}$ (TeV)-1 at
$\sqrt{s}=13~{}$TeV, $139~{}$fb-1 (red) and $\sqrt{s}=14~{}$TeV, $3000~{}$fb-1
(yellow) for $pp\rightarrow Zh$. Right: Sensitivity contours at $2\sigma$
(red), $3\sigma$ (yellow) and $5\sigma$ (green) levels for the ALP mediated
$pp\rightarrow Zh$ signal at $\sqrt{s}=13$ TeV LHC and for an integrated
luminosity of $3000$ fb-1, in $a_{2D}$-$f_{a}$ plane. The green region
depicting $f_{a}<\sqrt{s}^{\mathrm{min}}$, is excluded by the criterion of EFT
validity. The blue region is excluded by the limits from
$\operatorname{Br}(h\rightarrow{\rm BSM})$ CMS:2022dwd . The dash-dotted lines
represent constant values of $f_{a}/a_{2D}$.

Figure 10 (b) shows sensitivity levels at $2\sigma$ (red), $3\sigma$ (yellow),
and $5\sigma$ (green) for the $pp\rightarrow Zh$ signal at the HL-LHC with
$\sqrt{s}=14~{}\rm{TeV}$ and for an integrated luminosity of
$3000~{}\rm{fb^{-1}}$ in the parameter space of $f_{a}$-${a}_{2D}$. The green
region represents $f_{a}\leq\sqrt{\hat{s}^{\rm min}}$ and is thus, excluded
since all signal events will break the validity criterion of EFT.

The $5\sigma$ sensitivity level is achieved for $f_{a}/a_{2D}\simeq 15{\,\rm
TeV}$ for an integrated luminosity of $3000{\,\rm fb^{-1}}$ of data. Thus, the
region is amenable to detection at the HL-LHC.

The dash-dotted reference lines correspond to constant values of
$f_{a}/{a}_{2D}$. In the region with higher values of $f_{a}$, we find that
the sensitivity curves run almost parallel to the lines of constant
$f_{a}/{a}_{2D}$ indicating a stable detection range for $f_{a}/{a}{2D}$ here,
despite a loose constraint on $a_{2D}$.

When $f_{a}$ decreases below $1,\rm TeV$, the sensitivity curves fall slowly
compared to the reference lines. This indicates that the analysis in this
lower $f_{a}$ region is limited to smaller $f_{a}/{a}{2D}$ ratios compared to
the higher $f_{a}$ regions. This change in sensitivity is attributed to the
reason that as $f_{a}$ decreases, more and more events from the higher energy
bins are excluded to ensure the applicability of the EFT in the region. This
leads to loss in discerning power of the signal.

The interaction between ALP and the Higgs boson also induces non-standard
decays of Higgs. These decay modes of the Higgs boson will put constraints on
ALP interactions through the unobserved Higgs branching fraction
(h$\rightarrow$ BSM). Considering that these exotic decays are the only LO
modifications to Higgs properties, the global signal strength measurements can
be used to constrain $a_{2D}/f_{a}$. This is because the invisible Higgs
branching fraction will be proportional to $1-\rm BR(h\rightarrow SM)$. The
latest combined CMS global signal strength measurement restricts $\rm
BR(h\rightarrow BSM)$ to less than 0.11 CMS:2022dwd ; ATLAS:2022vkf . Assuming
$\Gamma_{BSM}\simeq\Gamma_{h\rightarrow aZ}$, we obtain a limit of
$\Gamma_{h\rightarrow aZ}<0.5$ MeV at 95% C.L. This limit translates into a
constraint of $f_{a}/{\tilde{a}}_{2D}\geq 5.95$ TeV for $m_{a}\leq 34$ GeV.
However, this expected sensitivity is less stringent than the current limit
derived from the $pp\rightarrow a^{*}\rightarrow Zh$ process, as depicted in
the blue region of Fig. 10 (b).

##### Sensitivity from $Z\gamma,WW$ and $WW\gamma$ :

Fig. 11 presents the upper limits on the coefficients
$\frac{c_{\tilde{W}}}{f_{a}}$ and $\frac{c_{\tilde{B}}}{f_{a}}$ (in TeV-1),
derived from $Z\gamma,WW$, and $WW\gamma$ analyses. These limits were
calculated for each individual experimental channel and based on the
differential measurements of relevant energy-dependent variables (refer to
Sec. IV.1). The $Z\gamma$ process which gets modified by both $c_{\tilde{W}}$
and $c_{\tilde{B}}$, constrains the difference
$|c_{\tilde{B}}-c_{\tilde{W}}|<$ 0.074, as derived from the 13 TeV
$m_{Z\gamma}$ differential measurement. The $WW\gamma$ analysis imposes a
stricter limit on $c_{\tilde{W}}$. Combining the results from $WW$ and
$WW\gamma$, along with other disboson channels like $ZZ$, $W\gamma$, and
triboson channels such as $ZZ\gamma$, could potentially yield enhanced results
and is a prospect for global analysis in future work. The expected limit for
$WW\gamma$, based on $m_{WW}$, is $|c_{\tilde{W}}|<0.147$, which is twice as
stringent as that from the $WW$ analysis based on $m_{T}^{WW}$. The $WW$
process is not affected by $c_{\tilde{B}}$, whereas $WW\gamma$ has a slight
dependence on it, as seen in Fig. 11. The non-linear framework of ALP EFT can
induce other operators that could modify the interactions of the charged weak
bosons with the ALP. Exploring the $WW\gamma$ process further could help
disentangle more than one directions in the ALP parameter space, an endeavor
to be taken up in the follow-up. When all constraints are considered together,
only a narrow overlapping region near zero remains viable, with
$|c_{\tilde{W}}|<0.06$ and $|c_{\tilde{B}}|<0.072$. The limits from $Z\gamma$
measurement provide the most stringent constraints along the $c_{\tilde{B}}$
axis. These constraints can also be interpreted in terms of effective
couplings like $g_{a\gamma\gamma},g_{aZ\gamma}$, and $g_{aZZ}$ (using Eq.
(6)), which are depicted in the dashed, dotted, dot-dashed lines in the Fig.
11.

The upper limit on the width of $Z$ boson to exotic channels is
$\Gamma(Z\rightarrow\text{BSM})\lesssim 2{\rm MeV}$ at a 95% C.L.
Workman:2022ynf . This puts a conservative limit on the $Z\rightarrow a\gamma$
decay. This contribution $\Gamma(Z\rightarrow a\gamma)$ is given by:

$\Gamma(Z\rightarrow
a\gamma)=\frac{M_{Z}^{3}}{384\pi},g_{aZ\gamma}^{2}\left(1-\frac{m_{a}^{2}}{M_{Z}^{2}}\right)^{3},$
(27)

Using the $Z$ boson width data, the coefficient $g_{aZ\gamma}$ can be
constrained, which is largely independent of $m_{a}$ for values of $m_{a}$
less than 1 GeV:

$|g_{aZ\gamma}|<1.8~{}{\rm TeV^{-1}}\quad(95\%~{}{\rm C.L.})$ (28)

Constraints from LEP experiments on the $Z\rightarrow 3\gamma$ decay process
Jaeckel:2015jla  cosntrains a combination of $g_{a\gamma\gamma},g_{aZ\gamma}$.
However, based on the limits of $g_{a\gamma\gamma}$, the resulting bound on
$g_{aZ\gamma}$ turns out to be less stringent than the one derived in Eq.
(28).

Figure 11: Left: 95% CL region in $c_{\tilde{W}}$-$c_{\tilde{B}}$ plane from
13 TeV analysis of differential distribution measurements for $Z\gamma$ (dark
pink), $WW$ (blue) and $WW\gamma$ (yellow) productions. Right: Projected
$2\sigma$ limit (darker-shaded) and $5\sigma$ discovery level (light-shaded)
regions in $c_{\tilde{W}}$-$c_{\tilde{B}}$ plane for the respective ALP
mediated signals at $\sqrt{s}=14~{}\rm{TeV}$ and integrated luminosities of
$3000$. The thin dashed, dotted and dot-dashed lines represent the directions
of vanishing couplings for $g_{a\gamma\gamma}$, $g_{aZ\gamma}$ and $g_{aZZ}$
respectively. The vertical axis at $c_{\tilde{W}}=0$ respresents $g_{aWW}=0$.

We also examined the sensitivity of these ALP mediated processes at the HL-
LHC. Fig. 11 (b) shows the projected regions at$2\sigma$ (darker shaded
region) and $5\sigma$ (lighter shaded region) significance levels on the
$\frac{c_{\tilde{W}}}{f_{a}}-\frac{c_{\tilde{W}}}{f_{a}}$ plane for a 14 TeV
LHC and $3000,\rm{fb^{-1}}$ of data. At $2\sigma$ level, a more stringent
region for each channel is seen, with the $Z\gamma$ channel exhibiting the
most significant individual improvement ($|c_{\tilde{B}}-c_{\tilde{W}}|=0.05$
TeV-1). The combined constraints are mostly influenced by the $WW\gamma$ and
$Z\gamma$ channels.

The constraints on $c_{\tilde{B}}$ could improve by about a factor of 2
compared to the current limits, and those on $c_{\tilde{W}}$ could improve by
approximately a factor of 1.2. Additionally, Fig. 11 (b) highlights the
expected discovery threshold (lighter shaded region) at 14 TeV, where the SM
point would be excluded by 5 standard deviations if the measurements align
with the predicted ALP signal. This region is within the exclusion limits of
the current 13 TeV LHC data, and suggests that the absence of results from the
current LHC data does not necessarily rule out the possibility of a discovery
at the HL-LHC.

##  VI Multivariate analysis

After performing a cut-based analysis for each of the signals in the four
distinct non-resonant processes at the LHC in Sec. IV, we now delve into
investiagting for potential improvement in the analysis with some advanced
techniques like Gradient Boosted Decision Trees Coadou:2022nsh . The
usefulness of these methods have been extensively studied in recent studies
Baldi:2014kfa ; Oyulmaz:2019jqr , particularly in the Higgs sector
Bakhet:2015uca ; Lasocha:2020ctd  and have demonstrated better efficacy in
differentiating between signal and background characteristics compared to
traditional rectangular cut-based approaches. Their application in ALP
scenario searches at colliders is yet to be thoroughly explored. In our study,
we assess the possibility for maximizing signal significance in the specific
signal processes under our consideration. To achieve this, we utilized the
AdaBoost classifier from the scikit-learn library in Python.

At first, we discuss the details of our analysis for the Higgs-strahlung
process, considering the BP1 benchmark scenario for the ALP mediated signal.
We take into account for all relevant SM backgrounds in the process $-$ here,
$Z$+jets which includes $Z$+$b\bar{b}$ is the most dominant background for
$Zh$ production. To optimize the classifier’s performance in identifying the
signal region, we impose slightly looser cuts compared to the cut-based
analysis, thereby ensuring better training. The selection criteria we employed
are as follows:
$70~{}{\rm{GeV}}<m_{ll}<110~{}{\rm{GeV}},~{}p_{T_{ll}}>160~{}{\rm{GeV}},~{}\Delta
R_{ll}>0.2,~{}p_{T_{J}}>60~{}{\rm{GeV}},~{}95~{}{\rm{GeV}}<m_{J}<155~{}{\rm{GeV}},~{}\Delta
R_{b_{i},b_{j}}>0.4$ and $\not{E}_{T}<80~{}{\rm{GeV}}$. After these
preselections, we trained the classifier on the signal and background samples
with the following set of variables:

  * •

Transverse momenta ($p_{T}$) of the two isolated leptons

  * •

Resconstructed Z boson and its $p_{T}$

  * •

$\Delta R$ separation between the two b-tagged subjets ($\Delta
R_{b_{i},b_{j}}$), subjet $i$ and lepton $j$ ($\Delta R_{b_{i},l_{j}}$), and
two leptons ($\Delta R_{l_{i},l_{j}}$)

  * •

Scattering angle of reconstructed $Z$ boson.

  * •

N-subjettiness of the fat-jet ($\tau_{21}$)

  * •

$\Delta\phi$ separation between the fat-jet and the reconstructed Z boson

  * •

Mass of the reconstructed Higgs jet and its $p_{T}$

For the gradient boosted Decision Tree method of separation, we have taken
$\sim 1000$ estimators and maximum depth 4 with learning rate 0.1. We have
used 75% of the total dataset for training purpose and 25% for validation.

After implementing the BDT algorithm, we obtain the distribution of the
response of the BDT classifier for the signal and total background events for
Higgs-strahlung process as shown in Fig. 12. We can see a clear distinction
between the signal and the background distributions. We have checked that in
this process, $p_{T}$ distribution of the leading lepton plays the role of the
most important input variable and separation between the two b-tagged subjets
and $p_{T}$ distribution of the reconstructed Higgs jet are the second and
third best discriminator in this case, respectively. Thus, stronger transverse
momenta of the leading fatjet and the lepton are favourable to retain the
correct classification of these variables. We have plotted the Receiver
Operating Characteristic (ROC) curve (that estimates the degree of rejecting
the backgrounds with respect to the signal) for the benchmark signal process
BP1 in Fig. 12 (right). One of the possible demerits of these techniques is
over-training of the data sample. In case of over-training the training sample
gives extremely good accuracy but the test sample fails to achieve that. We
have explicitly checked that with our choice of parameters the algorithm does
not over-train. The ROC curve remains almost same for training and testing
sample. The area under the ROC curve is 0.90 for BP1. At the $\sqrt{s}=14$ TeV
LHC with 3000 fb-1 of integrated luminosity ($\mathcal{L}$), we expect to
observe 835 signal events and 9843 background events for an optimal cut of
0.1982 on our BDT output. Adding a 5% (20%) systematic uncertainty translates
to a significance (using the formula in Eq. (22)) of 8.106 (7.595). We present
our results for $\sqrt{s}=14$ TeV to make it easier to translate to the case
of Run-3 ($\sqrt{s}=13.6$ TeV) and HL-LHC ($\sqrt{s}=14$ TeV) as the cross-
sections are not expected to change much.

Figure 12: $pp\rightarrow Zh$. Left: The normalized BDT score distributions
for the signal and the background. Significance as a function of the BDT cut
value for BP1 at $\sqrt{s}=14~{}$TeV,
$\mathcal{L}_{int}=3000\,\mathrm{fb}^{-1}$. Right: ROC curve for BP1.

For the $Z\gamma$ process, we study the BP2 and BP3 ALP mediated signals. For
the MVA, we have adopted cuts that are slightly less stringent than those used
in the cut-based approach (detailed in Table 2). Along with the preliminary
selection cuts, we have applied a requirement for the leading fat-jet and the
photon to have a minimum transverse momentum of 175 GeV. We have set a minimum
threshold for the reconstructed fat-jet mass at 60 GeV. These criteria
effectively minimize the dominant background while retaining most of the
signal events. This approach is important because the MVA tends to be less
effective with only baseline cuts, given the small signal size relative to the
large background. It is also worth noting that the more stringent cuts from
Table 2 do not necessarily lead to better results in the MVA context.
Therefore, the cuts chosen for MVA are carefully calibrated to be neither too
strict nor too relaxed compared to the cut-based analysis.

The BDT classifier is configured with the following hyperparameters:
‘n_estimators’:800, ‘learning_rate’:0.1, ‘trees’:10, ‘max_depth’:4

For the training, we have selected a range of observables that are highly
effective at distinguishing between the signal and background. These
observables are chosen as input variables for the BDT to optimize its
discerning potency.

$\begin{split}p_{T}^{\gamma},~{}m_{J},~{}\tau_{21},~{}p_{T}^{J},~{}cos\theta_{\gamma}^{*},~{}\Delta
R_{J\gamma},~{}E^{\gamma},~{}\eta^{J},~{}p_{T}^{b_{i}},~{}\Delta
R_{b_{i}b_{j}}\end{split}$ (29)

where the symbols have their usual meaning. $p_{T}^{b_{i}}$ denotes the
transverse momentum of $i^{\rm th}$ b-tagged sub-jet and cos
$\theta_{\gamma}^{*}$ is the scattering angle of photon in the $Z\gamma$ rest
frame. Among these variables used, the four most important variables to
distinguish the ALP signal from the backgrounds are :
$m_{J},~{}\tau_{21},~{}p_{T}^{J},~{}E^{\gamma}$.

The classifier, after being trained with these kinematic variables, is used to
discriminate the signal benchmark from each background class by computing the
significance of observing the signal over the background events. We find that
the signal significance over background for the benchmark scenarios BP2 (shown
in Fig. 13) and BP3 are $15.06~{}(14.57)$ and $19.63~{}(18.06)$, respectively,
assuming zero $(20\%)$ systematic uncertainty at 14 TeV HL-LHC. To achieve the
discovery level significance of signal at $5\sigma$, less than $1000$
$fb^{-1}$ luminosity of data is required as shown in Table 6.

Figure 13: $pp\rightarrow Z\gamma$. Left: The normalized BDT score
distributions for the signal and the background. Significance as a function of
the BDT cut value for BP2 at $\sqrt{s}=14~{}$TeV,
$\mathcal{L}_{int}=3000\,\mathrm{fb}^{-1}$. Right: ROC curve for BP2.

For our MVA of the semileptonic channel of the WW production, we consider BP4
ALP mediated signal category and all the relevant background processes which
the mimic $1J+1\ell+\not{E}_{T}$ final state. In order to be quantitative, we
have applied some weak kinematical cuts than discussed in Sec. IV.1.3, eg.
$\not{E}_{T}>80$ GeV, $m_{W}^{lep}>65$ GeV, $p_{T,W^{lep}}>120$ GeV and
$p_{T,W^{had}}>100$ GeV on signal and background events in addition to the
pre-selection criteria mentioned in Sec. IV. The different backgrounds are
mixed according to their proper weights to obtain the kinematical
distributions for the combined background class. Upon inspecting various
kinematic distributions, we choose the following 11 variables for our
multivariate analysis:

$\tau_{21},\not{E}_{T},~{}m_{T}^{W},~{}m_{eff},~{}\Delta
R_{jj},~{}m_{J},~{}p_{T}^{J},~{}p_{T}^{l},~{}\Delta\phi(J,p_{T}^{\rm{miss}}),~{}\Delta
R(J,l),~{}m_{W}^{\rm{lep}}$

where $\Delta\phi(J,p_{T}^{\rm{miss}})$ implies the azimuthal angle separation
between the directions of leading fat-jet and missing $p_{T}$. The final
number of signal and background events along with the significance are listed
in Table 6. The four best discriminatory variables are
$\tau_{21},\;~{}m_{T}^{W},\;m_{eff}$ and $\Delta R_{jj}$.

Finally, with a judicious cut on the BDTD observable, we find $2749$ signal
and $49134$ background events, yielding a significance of $12.289$ upon
neglecting systematic uncertainties and a significance of $11.23$ taking into
account 20% systematic uncertainties. In Fig. 14, the ROC curve for the
benchmark BP4 is shown and an area of $\sim 89\%$(BP4) is obtained under ROC
curve.

Figure 14: $pp\rightarrow WW$ in semileptonic channel. Left: The normalized
BDT score distributions for the signal and the background. Significance as a
function of the BDT cut value for BP4 at $\sqrt{s}=14~{}$TeV,
$\mathcal{L}_{int}=3000\,\mathrm{fb}^{-1}$. Right: ROC curve for BP4.

Before concluding this subsection, we make an attempt to decipher the
potential of the leptonic final state for the $WW\gamma$ channel. We study the
benchmark scenarios BP5 and BP6 separately for the signal. We consider the
same set of cuts as for this channel before performing the multivariate
analysis as the cuts are neither too strong nor too relaxed. For this case, we
find the following variables to have the best discriminatory properties.

$m_{T}^{WW},\not{E}_{T},~{}m_{ll},~{}\Delta
R_{ll},~{}p_{T}^{l_{1}},~{}p_{T}^{l_{2}},~{}p_{T}^{\gamma},~{}\Delta\phi_{ll},~{}p_{T,ll},~{}E_{\gamma},~{}\Delta\eta_{ll},~{}m_{ll\gamma},~{}\eta^{\gamma}$

where $p_{T,ll}$ and $\Delta\eta_{ll}$ refer to the $p_{T}$ of the dilepton
system and the rapidity separation between the leptons respectively. The best
four variables are $\Delta R_{ll},~{}m_{T}^{WW},~{}\Delta\eta_{ll}$ and
$p_{T}^{\gamma}$.

Hence, in an analogous way to the WW case, we train the BDTD with the signal
and the background samples, albeit with proper weight factors for the
backgrounds. We find a similar significance of $6.78$ and $6.85$ for the
benchmark scenarios BP5 and BP6 respectively, assuming zero systematic
uncertainties. The results are summarised in Table 6. Assuming $20\%$
systematic uncertainties, we obtain a significance of $6.19$ and $6.29$ for
BP5 and BP6 respectively. The response of the classifier and the ROC curve for
BP6 are shown in Fig. 15.

Figure 15: $pp\rightarrow WW\gamma$ in fully leptonic channel. Left: The
normalized BDT score distributions for the signal and the background.
Significance as a function of the BDT cut value for BP5 at
$\sqrt{s}=14~{}$TeV, $\mathcal{L}_{int}=3000\,\mathrm{fb}^{-1}$. Right: ROC
curve for BP5.

The signal significance computed for all the benchmark points with Adaptiove
BDT algorithm is presented in Table 6. One can compare these results with the
ones quoted in Table 4. It is clear that in all cases there is significant
improvement from rectangular cut-based analysis. We particularly point out the
BP2 and BP3 in case of $Z\gamma$ production. In these two cases we observe
striking improvement from the cut-based results. The BDT algorithm finds the
best possible combination of feature variables to separate the signal and
background by choosing the best possible set of cuts on the most relevant
observables. We remark here that the data sample used by us for training
purpose may in principle be subjected to some pre-assigned additional cuts,
such as demanding specific invariant masses for opposite-sign dileptons in
$WW\gamma$ and using variables that directly are proportional to the scale of
the experiment, for instance, the invariant mass of the final state system in
$2\rightarrow 2$ scattering processes is one of the most important
distinguishing feature between signal and background. However, to minimize the
bias we do not use it as an input variable to the BDT. Thus, the analysis
always has the scope of improvement, by choosing a better set of variables and
cuts. However, the variables that we have used are good discriminators as
demonstrated in the following.

BPs | $\mathcal{N}_{S}^{bc}$ | $\mathcal{N}_{\textrm{Bkg}}$ | $\mathrm{BDT}_{opt}$ | $\mathcal{N}_{S}(\epsilon_{S})$ | $\mathcal{N}_{B}(\epsilon_{B})$ | $\mathcal{S}$ | $\mathcal{L}^{req}_{(5\sigma)}(fb^{-1})$
---|---|---|---|---|---|---|---
BP1 | 2104 | 283066 | 0.1982 | 835(0.397) | 9843(0.03477) | 8.301 | 1058.8
BP2 | 4237 | 464601 | 0.2399 | 1035(0.2443) | 3927(0.008452) | 15.06 | 274.94
BP3 | 5295 | 464601 | 0.2399 | 1293(0.2443) | $3927(0.008452)$ | 19.633 | 352.33
BP4 | 8237 | 3035249 | 0.1727 | 7657(0.3338) | 49134(0.01619) | 12.289 | 487.64
BP5 | 1642 | 214365 | 0.063 | 1517(0.9243) | 49579(0.2313) | 6.778 | 1615.8
BP6 | 1669 | 214365 | 0.065 | 1543(0.9244) | 49579(0.2313) | 6.849 | 1561.8
Table 6: Evaluation of signal and background events at 14 TeV LHC for an
integrated luminosity of 3000 fb-1. The table includes the number of signal
($\mathcal{N}_{S}^{bc}$) and background ($\mathcal{N}_{\text{Bkg}}$) events
before and after applying the optimal BDT cut ($\mathrm{BDT}_{\text{opt}}$),
along with the signal ($\epsilon_{S}$) and background ($\epsilon_{B}$)
acceptance efficiencies at the $\mathrm{BDT}_{\text{opt}}$ cut value are
given. The statistical significance ($\mathcal{S}$) for each benchmark point
and the required luminosity for a $5\sigma$ discovery
($\mathcal{L}^{req}_{(5\sigma)}$) are also presented.

##  VII ALP couplings and masses

Fig. 16 illustrates the constraints obtained in our study, plotting them
against the EW $g_{aVV}$ couplings as defined in Eqs.6b and 6c, and the ALP
mass $m_{a}$. We compare the constraints on $g_{aWW}$ and $g_{aZ\gamma}$ with
those from various other experiments (See Ref. Alonso-Alvarez:2018irt ). It is
important to note that most measurements often rely on multiple ALP couplings.
To depict these constraints on a two-dimensional plane of ($m_{a},g_{aVV}$),
it is necessary to employ a specific projection method or theoretical
assumptions, which can differ widely among various constraints. These
variations are important for an accurate comparison. The constraints, derived
from the permissible region in the
$\frac{c_{\tilde{W}}}{f_{a}}-\frac{c_{\tilde{B}}}{f_{a}}$ plane, inherently
incorporate gauge invariance relations. We assume that $g_{agg}$ is
significantly greater than either $g_{aWW}$ or $g_{aZ\gamma}$, thereby
minimizing the impact of the gluon coupling, as discussed in Ref.Alonso-
Alvarez:2018irt . The constraints depicted in brown-hatched region in Fig. 16
are obtained from the non-resonant $gg\rightarrow a^{*}\rightarrow V_{1}V_{2}$
processes. For visualization purposes, these are normalized to
$g_{agg}=1\rm{TeV^{-1}}$.

We will discuss constraints that incorporate more complex assumptions about
the ALP parameter space. The majority of these constraints, particularly those
relating to the ALP’s interactions with massive gauge bosons, assume that the
ALP is stable and focus on the mass range below 1 GeV. In such scenarios,
constraints are derived from mono-$W$ and mono-$Z$ observations at the LHC,
and for $g_{a\gamma Z}$, from the lack of observation of exotic
$Z\rightarrow\gamma+\text{inv.}$ decays at LEP Craig:2018kne  and LHC
ATLAS:2020uiq . It is to be noted that, resonant triboson constraints on
$g_{aWW}$ and $g_{a\gamma Z}$ are based on a photophobic ALP model
Craig:2018kne  and they provide dominant bounds for ALP masses above 100 GeV.

All these searches for a stable ALP (including mono-$W$, mono-$Z$,
$Z\rightarrow\gamma+{\rm inv.}$) implicitly assume a sufficiently small ALP
decay width, which, in the relevant mass range, implies certain assumptions
about its coupling to photons, electrons, and muons. If we move away from the
stable ALP assumption, a more conservative constraint arises from the total
$Z$ decay width measurements at LEP, applicable up to $m_{a}\lesssim
m_{Z}$Brivio:2017ije ; Craig:2018kne . The LEP constraints are predicated on
negligible decay rates into leptons. It is also important to note that this
bound cuts off at $m_{a}\leq 3m_{\pi}\simeq 0.5\rm{GeV}$, as beyond this
point, hadronic decay channels for the ALP become kinematically feasible. This
leads to potential $Z\rightarrow\gamma+\text{hadrons}$ decays L3:1992kcg ,
introducing additional dependence on $c_{\widetilde{G}}$ that would
necessitate a detailed analysis Alonso-Alvarez:2018irt , possibly weakening
the LEP constraints.

The coupling of ALPs to W bosons uniquely contributes to rare meson decays at
the 1-loop level, enabling a stringent limits for lighter ALPs
Izaguirre:2016dfi ; BNL-E949:2009dza . These decays fill a significant gap in
the $g_{aWW}$ coupling parameter space, particularly in the mass range of 0.4
GeV to 100 GeV. This mass range was previously difficult to analyse, as
earlier studies focusing exclusively on EW ALP couplings become less relevant
for masses above $m_{a}>3m_{\pi}$, due to the opening of the hadronic decay
channel. In the case of $g_{aZ\gamma}$, much of the mass range addressed by
this analysis was already covered by LEP studies. However, our analysis
expands the detection scope to lower couplings by nearly an order of
magnitude.

While the resonant triboson production yields stringent constraints in the
mass range above 100 GeV, the non-resonant $WW\gamma$ process provides
constraints valid over a mass window from 1 MeV to 100 GeV.

The constraints labeled “Photons” derive from a combination of beam dump
experiments, observations from supernova SN1987a, and LHC studies. For ALP
masses below the GeV scale, beam dump searches (in violet region)
Riordan:1987aw ; Bjorken:1988as ; Blumlein:2013cua —as compiled in
Ref.Dobrich:2015jyk —and energy-loss considerations related to supernova
SN1987a Payez:2014xsa ; Jaeckel:2017tud  set limits on $g_{aZ\gamma}$. These
parameters are primarily constrained by the absence of additional cooling and
a lack of photon bursts from decaying axions. The radiative corrections that
link axion-boson couplings to axion-photon couplings, once translated into
axion-photon coupling terms, help establish bounds on $g_{aWW}$ and
$g_{aZ\gamma}$, assuming minimal dependence on $f_{a}$ Alonso-Alvarez:2018irt
.

The use of MVA techniques and improved search strategies are also likely to
significantly enhance these constraints.

Figure 16: Summary of current constraints as a function of the ALP mass and
couplings $g_{aZ\gamma}$ (left) and $g_{aWW}$ (right). Limits derived in this
work are labeled “Non-resonant LHC” and shown in hatched region (red). These
constraints are normalised with $g_{agg}=1~{}\rm{TeV}^{-1}$. Existing
constraints from relevant astrophysical and terrestrial probes are shown :
Left: Supernova SN1987a (violet), LHC constraints from mono-Z (green) and
resonant triboson production (violet) and LEP constraints from Z width
(orange). Right : Supernova SN1987a (violet), constraints from rare meson
decays (green), LHC constraints from mono-W (orange) and resonant triboson
production (orange) . See the main text for more details.

Summarizing, the primary advantage of non-resonant searches lies in their
ability to directly probe ALP interactions with EW bosons at the tree level,
across a broad range of ALP masses, with minimal dependence on specific model
assumptions. This work included processes initiated by gluons, which are
influenced by the value of $g_{agg}$. For this analysis, we set $g_{agg}$ at 1
TeV-1. In cases where $g_{agg}$ falls below a certain level, non-resonant
constraints from EW processes, such as from vector boson scattering processes,
could become more prominent, depending on the specific EW coupling being
probed.

##  VIII Summary and Conclusions

Exploring the phenomenology of new, light, propagating particles such as the
axion-like particles are pivotal to the beyond the SM endeavours as
investigated at, for eg. the LHC. The LHC allows for a plethora of processes
that are sensitive to the ALPs, and in recent times, it has expanded the range
of probing the ALP interactions particularly with electroweak bosons and the
top quark. Similarly the Higgs particle, whose global understanding still
remains elusive, presents a vital area for potential discoveries in new
physics. As experiments at the LHC gain in their sensitivity to rare
phenomena, they may unveil evidence of new physics linked to the Higgs. Our
study focuses on the interactions of ALP with the SM Higgs boson and the gauge
bosons through non-resonant searches at the LHC. In particular, we have
studied the potential impact of ALP couplings in the effective theory
framework, on the production of $Zh,Z\gamma,WW$ and $WW\gamma$ processes at
the LHC. Here, the ALP serves as an off-shell mediator in these scattering
processes. The key strategy utilises the presence of explicit dependence of
derivative interaction of the ALP with the SM bosons. As a consequence, there
is a high energy growth of these scattering processes which deviates
significantly from the SM. This has been exhibited in the regime where
$\sqrt{\hat{s}}\gg v$ and the ALP mass respects $m_{a}\ll\sqrt{\hat{s}}$.
Additionally, we ensure the consistency of the ALP EFT expansion with
$\sqrt{\hat{s}}\ll f_{a}$. With reinterpretation of the public data from the
ATLAS and CMS collaborations at 13 TeV for the measurement of the
aforementioned SM processes, we obtained constraints on ALP couplings to SM
gauge bosons in the set $\\{g_{aZh},g_{aZ\gamma},g_{aWW},g_{aWW\gamma}\\}$. We
underline the importance of using information from differential distributions
in the high energy tails of the final system mass spectrum. The limits we get
are rigorous across a broad mass window of ALP from 1 MeV to 100 GeV, assuming
an ALP-gluonic coupling exists. For the $Zh$ and $Z\gamma$ production
processes, depending on the value of the scale $f_{a}$ and $g_{agg}=1$ TeV-1,
upper limits on the ALP coupling to $Zh$ and $Z\gamma$ of $a_{2D}=0.078$ TeV-1
and $|c_{\tilde{B}}-c_{\tilde{W}}|=0.073$ TeV-1 have been extracted. We also
carried out the analyses for $WW$ and $WW\gamma$ processes which provide a
handle to probe the coupling $c_{\tilde{W}}$. We find that these processes
impose a constraint of $c_{\tilde{W}}<$ 0.068 TeV-1 and 0.147 TeV-1,
respectively.

We have chosen a few representative benchmark points which give distinct
signatures from the SM backgrounds in the boosted regime. We examined the
potential of HL-LHC in probing these ALP interactions via non-resonant
searches and projections for integrated luminosities up to 3000 fb-1 at 14 TeV
LHC are presented. The upcoming HL-LHC program will allow for the examination
of ALPs through their relevant electroweak boson couplings. Detection of
statistically significant $Zh$ signal events mediated by the ALP at LHC would
essentially indicate an evidence of non-linear EWSB.

We also concluded that if the ALP is collider stable and escapes detection,
the $W+$ MET (mono-W) signature is more sensitive than the off-shell mediated
processes involving the ALP-W interactions while processes such as mono-Higgs
and mono-Z are less sensitive in the direct probes than the corresponding non-
resonant ALP signal analysis carried out. Nevertheless, a comprehensive global
analysis of both the direct and indirect ALP searches would yield more
information on the constraints of the various ALP operators, both in the
linear and non-linear frameworks, with emphasis on effects responsible for
electroweak symmetry breaking.

To explore for potential improvement of the sensitivity for the non-resonant
signals at the LHC, we employed a multivariate analysis. This method differs
from the rectangular cut-based analysis by considering all the input kinematic
variables at one go and providing an optimal separation between the signal and
background. We utilized a boosted decision tree network and trained it with a
variety of kinematic variables specific to each of the relevant process to
enhance the signal distinction. The results show a clear improvement in the
LHC sensitivity to detect new interactions using this method, especially for
the benchmark points we considered. We find that with 1000 $\rm fb^{-1}$ of
data, non-linear operators could be detected with a confidence level above
5$\sigma$. Identifying linear operators requires less data to achieve the same
level of confidence.

The non-resonant searches offer a complementary probe for very light ALP
masses. The main advantage lies in the independence of specific assumptions on
the ALP characteristics. Exploring additional processes like di-higgs
production, $WWZ,ZZ\gamma$ processes, and other multi-particle productions
could further refine our understanding of the ALP parameter space,
particularly when considering the potential interactions with direct ALP
production. With the LHC entering a new phase with higher energy and
luminosity, it becomes increasingly important to focus on these rare
processes, which offer significant sensitivity to phenomena beyond the
standard paradigm.

## Acknowledgements

The author would like to thank the organisers of 2023 “ICTP Summer School on
Particle Physics” where this work was initiated and acknowledges the helpful
lectures on the relevant topics. TB is thankful to Prof. Anindya Datta for the
useful discussions and acknowledges helpful exchanges with Ilaria Brivio
during the course of the work. The work of TB is funded by the Council of
Scientific and Industrial Research, Government of India through Senior
Research Fellowship [File No. 09/028(1107)/2019-EMR-I].

## References

  * (1) R. D. Peccei and H. R. Quinn, CP Conservation in the Presence of Instantons, Phys. Rev. Lett. 38, 1440 (1977).
  * (2) R. D. Peccei and H. R. Quinn, Constraints Imposed by CP Conservation in the Presence of Instantons, Phys. Rev. D 16, 1791 (1977).
  * (3) S. Weinberg, A New Light Boson?, Phys. Rev. Lett. 40, 223 (1978).
  * (4) A. Hook,Anomalous solutions to the strong CP problem, Phys. Rev. Lett. 114, no. 14, 141801 (2015); [arXiv:1411.3325 [hep-ph]].
  * (5) F. Wilczek, Problem of Strong $P$ and $T$ Invariance in the Presence of Instantons, Phys. Rev. Lett. 40, 279 (1978).
  * (6) Y. Chikashige, R. N. Mohapatra and R. D. Peccei, Are There Real Goldstone Bosons Associated with Broken Lepton Number?, Phys. Lett. B 98 (1981), 265-268.
  * (7) C. D. Froggatt and H. B. Nielsen, Hierarchy of Quark Masses, Cabibbo Angles and CP Violation, Nucl. Phys. B 147 (1979), 277-298.
  * (8) K. I. Izawa, T. Watari and T. Yanagida, Higher dimensional QCD without the strong CP problem, Phys. Lett. B 534 (2002), 93-96.; [arXiv:hep-ph/0202171 [hep-ph]].
  * (9) D. Chang, W. F. Chang, C. H. Chou and W. Y. Keung, Large two loop contributions to g-2 from a generic pseudoscalar boson, Phys. Rev. D 63 (2001), 091301.; [arXiv:hep-ph/0009292 [hep-ph]].
  * (10) P. W. Graham, D. E. Kaplan and S. Rajendran, Cosmological Relaxation of the Electroweak Scale, Phys. Rev. Lett. 115 (2015) no.22, 221801.; [arXiv:1504.07551 [hep-ph]].
  * (11) K. S. Jeong, T. H. Jung and C. S. Shin, Adiabatic electroweak baryogenesis driven by an axionlike particle, Phys. Rev. D 101 (2020) no.3, 035009.; [arXiv:1811.03294 [hep-ph]].
  * (12) L. F. Abbott and P. Sikivie, A Cosmological Bound on the Invisible Axion, Phys. Lett. B 120 (1983), 133-136.
  * (13) M. Dine and W. Fischler, The Not So Harmless Axion, Phys. Lett. B 120 (1983), 137-141.
  * (14) J. Preskill, M. B. Wise and F. Wilczek, Cosmology of the Invisible Axion, Phys. Lett. B 120 (1983).
  * (15) P. W. Graham, I. G. Irastorza, S. K. Lamoreaux, A. Lindner and K. A. van Bibber, Experimental Searches for the Axion and Axion-Like Particles, Ann. Rev. Nucl. Part. Sci. 65 (2015), 485-514; [arXiv:1602.00039 [hep-ex]].
  * (16) S. Bruggisser, L. Grabitz and S. Westhoff, Global Analysis of the ALP Effective Theory; [arXiv:2308.11703 [hep-ph]].
  * (17) I. G. Irastorza and J. Redondo, New experimental approaches in the search for axion-like particles, Prog. Part. Nucl. Phys. 102 (2018); [arXiv:1801.08127 [hep-ph]].
  * (18) H. Georgi, D. B. Kaplan and L. Randall, Manifesting the Invisible Axion at Low-energies, Phys. Lett. B 169 (1986), 73-78.
  * (19) K. Mimasu and V. Sanz, ALPs at Colliders, JHEP 06 (2015), 173; [arXiv:1409.4792 [hep-ph]].
  * (20) I. Brivio, M. B. Gavela, L. Merlo, K. Mimasu, J. M. No, R. del Rey and V. Sanz, ALPs Effective Field Theory and Collider Signatures, Eur. Phys. J. C 77 (2017) no.8, 572; [arXiv:1701.05379 [hep-ph]].
  * (21) M. Bauer, M. Neubert and A. Thamm, Collider Probes of Axion-Like Particles, JHEP 1712, 044 (2017); [arXiv:1708.00443 [hep-ph]].
  * (22) F. Esser, M. Madigan, V. Sanz and M. Ubiali, On the coupling of axion-like particles to the top quark, JHEP 09 (2023), 063; [arXiv:2303.17634 [hep-ph]].
  * (23) M. Bauer, M. Neubert and A. Thamm, Analyzing the CP Nature of a New Scalar Particle via S$\rightarrow$Zh Decay, Phys. Rev. Lett. 117 (2016), 181801; [arXiv:1610.00009 [hep-ph]].
  * (24) A. Biekötter, M. Chala and M. Spannowsky, New Higgs decays to axion-like particles, Phys. Lett. B 834 (2022), 137465; [arXiv:2203.14984 [hep-ph]].
  * (25) F. Björkeroth, E. J. Chun and S. F. King, Flavourful Axion Phenomenology, JHEP 08 (2018), 117; [arXiv:1806.00660 [hep-ph]].
  * (26) B. Döbrich, F. Ertas, F. Kahlhoefer and T. Spadaro, Model-independent bounds on light pseudoscalars from rare B-meson decays, Phys. Lett. B 790 (2019), 537-544; [arXiv:1810.11336 [hep-ph]].
  * (27) J. Martin Camalich, M. Pospelov, P. N. H. Vuong, R. Ziegler and J. Zupan, Quark Flavor Phenomenology of the QCD Axion, Phys. Rev. D 102 (2020) no.1, 015023; [arXiv:2002.04623 [hep-ph]].
  * (28) T. Bandyopadhyay, S. Ghosh and T. S. Roy, ALP-Pions generalized, Phys. Rev. D 105 (2022) no.11, 115039; [arXiv:2112.13147 [hep-ph]].
  * (29) M. Bauer, M. Neubert, S. Renner, M. Schnubel and A. Thamm, Axionlike Particles, Lepton-Flavor Violation, and a New Explanation of $a_{\mu}$ and $a_{e}$, Phys. Rev. Lett. 124 (2020) no.21, 211803; [arXiv:1908.00008 [hep-ph]].
  * (30) L. Calibbi, D. Redigolo, R. Ziegler and J. Zupan, Looking forward to lepton-flavor-violating ALPs, JHEP 09 (2021), 173; [arXiv:2006.04795 [hep-ph]].
  * (31) M. J. Dolan, T. Ferber, C. Hearty, F. Kahlhoefer and K. Schmidt-Hoberg, Revised constraints and Belle II sensitivity for visible and invisible axion-like particles, JHEP 03 (2021), 190; [arXiv:1709.00009 [hep-ph]].
  * (32) F. Acanfora, R. Franceschini, A. Mastroddi and D. Redigolo, Fusing photons into nothing, a new search for invisible ALPs and Dark Matter at Belle II,[arXiv:2307.06369 [hep-ph]].
  * (33) B. Döbrich, J. Jaeckel, F. Kahlhoefer, A. Ringwald and K. Schmidt-Hoberg, ALPtraum: ALP production in proton beam dump experiments, JHEP 02 (2016), 018; [arXiv:1512.03069 [hep-ph]].
  * (34) B. Döbrich, J. Jaeckel and T. Spadaro, Light in the beam dump - ALP production from decay photons in proton beam-dumps, JHEP 05 (2019), 213; [arXiv:1904.02091 [hep-ph]]. [erratum: JHEP 10 (2020), 046]
  * (35) D. Cadamuro and J. Redondo, Cosmological bounds on pseudo Nambu-Goldstone bosons, JCAP 02 (2012), 032; [arXiv:1110.2895 [hep-ph]].
  * (36) M. Millea, L. Knox and B. Fields, New Bounds for Axions and Axion-Like Particles with keV-GeV Masses, Phys. Rev. D 92 (2015) no.2, 023010; [arXiv:1501.04097 [astro-ph.CO]].
  * (37) P. F. Depta, M. Hufnagel and K. Schmidt-Hoberg, Robust cosmological constraints on axion-like particles, JCAP 05 (2020), 009; [arXiv:2002.08370 [hep-ph]].
  * (38) E. Izaguirre, T. Lin and B. Shuve, Searching for Axionlike Particles in Flavor-Changing Neutral Current Processes, Phys. Rev. Lett. 118 (2017) no.11, 111802; [arXiv:1611.09355 [hep-ph]].
  * (39) G. Alonso-Alvarez, M. B. Gavela and P. Quilez, Axion couplings to electroweak gauge bosons, Eur. Phys. J. C 79 (2019) no.3, 223; arXiv:1811.05466 [hep-ph].
  * (40) M. B. Gavela, R. Houtz, P. Quilez, R. Del Rey and O. Sumensari, Flavor-changing constraints on electroweak ALP couplings, Eur. Phys. J. C 79 (2019) no.5, 369; arXiv:1901.02031 [hep-ph].
  * (41) J. Ebadi, S. Khatibi and M. Mohammadi Najafabadi, New probes for axionlike particles at hadron colliders, Phys. Rev. D 100 (2019) no.1, 015016; arXiv:1901.03061 [hep-ph].
  * (42) F. Ertas and F. Kahlhoefer, On the interplay between astrophysical and laboratory probes of MeV-scale axion-like particles, JHEP 07 (2020), 050; arXiv:2004.01193 [hep-ph].
  * (43) S. Gori, G. Perez and K. Tobioka, KOTO vs. NA62 Dark Scalar Searches, JHEP 08 (2020), 110; arXiv:2005.05170 [hep-ph].
  * (44) K. J. Kelly, S. Kumar and Z. Liu, Heavy axion opportunities at the DUNE near detector, Phys. Rev. D 103 (2021) no.9, 095002; arXiv:2011.05995 [hep-ph].
  * (45) A. M. Galda, M. Neubert and S. Renner, ALP — SMEFT interference, JHEP 06 (2021), 135; arXiv:2105.01078 [hep-ph].
  * (46) J. Bonilla, I. Brivio, M. B. Gavela and V. Sanz, One-loop corrections to ALP couplings, JHEP 11 (2021), 168; arXiv:2107.11392 [hep-ph].
  * (47) M. Bauer, M. Neubert, S. Renner, M. Schnubel and A. Thamm, Flavor probes of axion-like particles, JHEP 09 (2022), 056; arXiv:2110.10698 [hep-ph].
  * (48) D. Wang, L. Wu, J. M. Yang and M. Zhang, Photon-jet events as a probe of axionlike particles at the LHC, Phys. Rev. D 104 (2021) no.9, 095016; arXiv:2102.01532 [hep-ph].
  * (49) N. Craig, A. Hook and S. Kasko, The Photophobic ALP, JHEP 09 (2018), 028; [arXiv:1805.06538 [hep-ph]].
  * (50) M. B. Gavela, J. M. No, V. Sanz and J. F. de Trocóniz, Nonresonant Searches for Axionlike Particles at the LHC, Phys. Rev. Lett. 124 (2020) no.5, 051802; [arXiv:1905.12953 [hep-ph]].
  * (51) S. Carra, V. Goumarre, R. Gupta, S. Heim, B. Heinemann, J. Kuechler, F. Meloni, P. Quilez and Y. C. Yap, Constraining off-shell production of axionlike particles with Z$\gamma$ and WW differential cross-section measurementsn, Phys. Rev. D 104 (2021) no.9, 092005); [arXiv:2106.10085 [hep-ex]].
  * (52) J. Bonilla, I. Brivio, J. Machado-Rodríguez and J. F. de Trocóniz, Nonresonant searches for axion-like particles in vector boson scattering processes at the LHC, JHEP 06 (2022), 113; [arXiv:2202.03450 [hep-ph]].
  * (53) W. Buchmuller and D. Wyler, Effective Lagrangian Analysis of New Interactions and Flavor Conservation, Nucl. Phys. B 268 (1986) 621-653.
  * (54) . Grzadkowski, M. Iskrzynski, M. Misiak and J. Rosiek, Dimension-Six Terms in the Standard Model Lagrangian, JHEP 1010 (2010) 085; [arXiv:1008.4884 [hep-ph]].
  * (55) K. Choi, K. Kang and J. E. Kim, Effects of $\eta^{\prime}$ in Low-energy Axion Physics, Phys. Lett. B 181 (1986) 145.
  * (56) The ATLAS collaboration [ATLAS Collaboration],Combined measurements of Higgs boson production and decay using up to $80$ fb-1 of proton–proton collision data at $\sqrt{s}=$ 13 TeV collected with the ATLAS experiment; ATLAS-CONF-2019-005
  * (57) R. Alonso, M. B. Gavela, L. Merlo, S. Rigolin and J. Yepes, The Effective Chiral Lagrangian for a Light Dynamical ”Higgs Particle”, Phys. Lett. B 722 (2013) 330; Erratum: [Phys. Lett. B 726 (2013) 926][arXiv:1212.3305 [hep-ph]].
  * (58) F. Feruglio, The Chiral approach to the electroweak interactions, Int. J. Mod. Phys. A 8 (1993) 4937; [hep-ph/9301281].
  * (59) R. Alonso, M. B. Gavela, L. Merlo, S. Rigolin and J. Yepes, Minimal Flavour Violation with Strong Higgs Dynamics, JHEP 1206 (2012) 076; [arXiv:1201.1511 [hep-ph]].
  * (60) A. Azatov, R. Contino and J. Galloway, Model-Independent Bounds on a Light Higgs, JHEP 1204 (2012) 127; Erratum: [JHEP 1304 (2013) 140] [arXiv:1202.3415 [hep-ph]].
  * (61) R. Alonso, M. B. Gavela, L. Merlo, S. Rigolin and J. Yepes, Flavor with a light dynamical ”Higgs particle”, Phys. Rev. D 87 (2013) no.5, 055019; [arXiv:1212.3307 [hep-ph]].
  * (62) G. Buchalla, O. CatÃ and C. Krause, Complete Electroweak Chiral Lagrangian with a Light Higgs at NLO, Nucl. Phys. B 880 (2014) 552; Erratum: [Nucl. Phys. B 913 (2016) 475][arXiv:1307.5017 [hep-ph]].
  * (63) I. Brivio, T. Corbett, O. J. P. Éboli, M. B. Gavela, J. Gonzalez-Fraile, M. C. Gonzalez-Garcia, L. Merlo and S. Rigolin, Disentangling a dynamical Higgs, JHEP 1403 (2014) 024; [arXiv:1311.1823 [hep-ph]].
  * (64) M. Cepeda, S. Gori, P. Ilten, M. Kado, F. Riva, R. Abdul Khalek, A. Aboubrahim, J. Alimena, S. Alioli and A. Alves, et al. Report from Working Group 2: Higgs Physics at the HL-LHC and HE-LHC, CERN Yellow Rep. Monogr. 7 (2019), 221-584;  [arXiv:1902.00134 [hep-ph]].
  * (65) T. Biswas, A. Datta and B. Mukhopadhyaya, Following the trail of new physics via the vector boson fusion Higgs boson signal at the Large Hadron Collider, Phys. Rev. D 105 (2022) no.5, 055028; [arXiv:2107.05503 [hep-ph]].
  * (66) T. Biswas and A. Datta, Exploring Higgs-photon production at the LHC, JHEP 05 (2023), 104; [arXiv:2208.08432 [hep-ph]].
  * (67) I. Brivio, J. Gonzalez-Fraile, M. C. Gonzalez-Garcia and L. Merlo, The complete HEFT Lagrangian after the LHC Run I, Eur. Phys. J. C 76 (2016) no.7, 416; [arXiv:1604.06801 [hep-ph]].
  * (68) M. Herrero and R. A. Morales, Anatomy of Higgs boson decays into $\gamma\gamma$ and $\gamma Z$ within the electroweak chiral Lagrangian in the $R_{\xi}$ gauges, Phys. Rev. D 102 (2020) no.7, 075040; [arXiv:2005.03537 [hep-ph]].
  * (69) M. J. Herrero and R. A. Morales, One-loop renormalization of vector boson scattering with the electroweak chiral Lagrangian in covariant gauges, Phys. Rev. D 104 (2021) no.7, 075013; [arXiv:2107.07890 [hep-ph]].
  * (70) A. Alloul, N. D. Christensen, C. Degrande, C. Duhr and B. Fuks, _FeynRules 2.0 - A complete toolbox for tree-level phenomenology_ , _Comput. Phys. Commun._ 185, 2250 (2014), arXiv:1310.1921 [hep-ph].