Concurrent exploration of Axion-like particle interactions with gauge bosons at the LHC

Article Content

Abstract

Axion-like particles (ALPs) are pseudo Nambu-Goldstone bosons associated with spontaneously broken global symmetries incorporated in the Standard Model (SM) Lagrangian in many models beyond the SM. The existence of a light ALP is plausible due to the long-standing problems that the SM has not been able to address, such as the dark matter (DM) problem and the observed matter-antimatter asymmetry. There are many proposals in recent decades considering the ALP as a solution to some of these shortcomings. Motivated by such potential, we search for ALPs with a mass of 1 MeV at the LHC in a model-independent fashion. We explore two complementary production modes: ALP production in association with a pair of electroweak gauge bosons (ZZ or WW) and ALP production in association with a single gauge boson (W or Z) plus jets. For the

V V + a

final state, signal and dominant SM backgrounds are generated, and a realistic detector response simulation is performed. A multivariate analysis is employed to discriminate the

V V + a

signal from background processes, and the expected 95% confidence level (CL) exclusion limits in two-dimensional parameter spaces involving the ALP couplings are subsequently derived. The V+jets channel is interpreted using LHC measurements in regimes where the ALP escapes detection, appearing as missing energy. The two analyses are complementary: both the

V V + a

and the V+jets channels probe simultaneously the ALP couplings to gluons and to electroweak gauge bosons. While the

V V + a

channel offers clean multi-lepton final states and direct reconstruction of the ZZ or WW system, the V+jets channel benefits from larger production cross sections, enabling stronger constraints in certain regions of the parameter space. Comparing the limits obtained in this study with current and projected limits derived from LHC searches, it is seen that competitive sensitivities to the ALP couplings are achieved, and a significant region of previously unexplored parameter space becomes accessible.

1. Introduction

The Standard Model (SM) of particle physics is a successful theory with a large number of experimentally confirmed predictions. However, there are some problems that the SM is not able to address. For example, the dark matter (DM) problem, baryon asymmetry problem, strong CP problem, anomalous magnetic dipole moment of muon, etc. An extensive effort has been made to solve some of the SM shortcomings by developing models beyond the SM. One of the ideas that has attracted much attention is the existence of Axions which was originally proposed to address the strong CP problem [1], [2], [3], [4], [5], [6], [7], [8], [9]. Axions are pseudo Nambu-Goldstone bosons associated with a spontaneously broken global

U (1)

symmetry. The Axion mass is stringently related to the scale of the global symmetry breaking. Setting this relation aside, Axion-like particles (ALPs) with broad-ranging masses and couplings are introduced. ALPs exist in any model with a spontaneously broken global symmetry. They can be viable candidates for a DM particle [10], [11], [12], and can also explain the observed matter-antimatter asymmetry [13], [14]. In addition, ALPs provide the possibility to explain the anomalous magnetic dipole moment of muon [15], [16], the anomalous decays of the excited Beryllium

{}^{8}{Be}^{⁎}

[17], [18], and the excess of events observed in the rare K mesons searches performed by the KOTO experiment [19]. There are proposals that the neutrino mass problem may also be solved by ALPs through a coupling to neutrinos [20], [21], [22]. Being motivated by such possibilities, the ALP parameter space has been extensively probed to date by collider searches, low-energy experiments, cosmological observations, etc. [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42].

Light ALPs with masses below the threshold of the electron pair production (≈1 MeV) predominantly decay into a pair of photons. Other decay channels, e.g. decays into leptons and hadrons, become available for heavier ALPs. ALPs with large enough masses decay promptly after production and can be reconstructed by their decay products. Many studies have searched for such ALPs. The exotic Higgs decays

h \to a Z

and

h \to a a

followed by the ALP decays

a \to γ γ

and

a \to ℓ ℓ

have been studied at the LHC [29], [43], [44], [45]. The tri-γ production,

γ a \to 3 γ

, and the associated production of an ALP and a Higgs boson (or a top quark pair) have been studied in Refs. [23], [28], [29], [46]. The

Z \to γ γ

data from LEP has been analyzed providing significant sensitivities to the ALP mass range MeV to 90 GeV [46]. There are also a number of studies that utilizes the ultra-peripheral heavy-ion collisions at the LHC to search for ALPs with masses from 100 MeV to 100 GeV that decay into photons promptly [47].

The width of the ALP decay into a photon pair scales as the third power of the ALP mass,

m_{a}^{3}

. Moreover, the ALP coupling to photons is strongly constrained by observations [29]. Therefore, for light ALPs with masses ≲1 MeV, which predominantly decay into photons, the decay length becomes so large. This motivates helioscope experiments such as SUMICO [48], CAST [49] and IAXO [50] which search for solar ALPs. Proton beam-dump experiments have also been performed to probe long-lived ALPs with masses in the MeV-GeV range [51]. In addition, collider studies have been performed in search of long-lived ALPs. Mono-γ and mono-jet plus missing energy have been shown to be promising signals to probe ALP masses above the KeV scale [28]. Mono-W, mono-Z, Wγ, Zγ, WW and WWγ plus missing energy signals have been studied at the LHC to probe the ALP mass and couplings to electroweak gauge bosons [23], [24], [52], [53], [54], [55], [56], [57], [58]. Other experimental searches for ALPs have been performed by ATLAS and CMS, including in anomalous Higgs decays to photon pairs [59], light-by-light scattering in heavy-ion collisions [60], and nonresonant diboson production such as ZZ and ZH final states [61]. These studies place direct constraints on ALP couplings to photons, gluons, and electroweak gauge bosons across a broad mass range.

Although much effort has been devoted to probing light ALPs, a significant region of their parameter space remains unexplored. Such ALPs offer the potential to address persistent issues in the SM, such as the dark matter (DM) problem [10], [11], [12], the baryon asymmetry problem [13], [14], and the anomalous magnetic dipole moment of the muon [15], [16]. Motivated by these possibilities, we conduct a collider search for long-lived ALPs with a mass of 1 MeV in this work. These ALPs can be produced at colliders through their couplings to SM particles. In the first part of this work, we assume the collider to be the High Luminosity Large Hadron Collider (HL-LHC) [62] operating at a center-of-mass energy of 14 TeV. Given that the assumed ALP mass is much smaller than the characteristic energy of the interactions, the study’s results would remain largely unaffected by considering even lighter ALP masses. ALPs can be produced in association with single or pairs of electroweak gauge bosons, such as ZZ or WW, at the LHC. In this work, we focus on simultaneously probing ALP couplings with gluons and W bosons. Probing multiple couplings concurrently provides a more comprehensive understanding of the ALP parameter space and can reveal interactions that might be missed when considering individual couplings in isolation. This approach allows for more stringent and realistic constraints on ALP properties, thereby improving the sensitivity of our search and significantly expanding the explored parameter space. Ref. [63] presents concurrent bounds on ALP couplings to electroweak gauge bosons, derived from Vector Boson Scattering (VBS) processes at the LHC. The analysis focuses on non-resonant ALP-mediated VBS, where the ALP serves as an off-shell mediator. This process becomes relevant when the ALP is too light to be produced resonantly, exploiting the derivative nature of ALP interactions with the electroweak bosons of the SM. The study investigates the production of vector boson (VV) pairs with large invariant masses, accompanied by two jets. Operating within a gauge-invariant framework, the analysis extracts upper limits on ALP couplings to electroweak bosons through a reinterpretation of public CMS VBS results from Run 2.

The focus of this study is on the long-lived ALPs tend to be stable at the detector and manifest themselves as missing energy. Searching for the missing energy signature, we constrain the ALP couplings to the SM particles, namely couplings to W bosons and gluons. We provide several sets of limits obtained independently in the search for ZZa and WWa signals. The ZZa analysis uses both the fully leptonic and semi-leptonic final states, and in the aWW analysis, only the fully leptonic final state is considered. The analyses use a realistic detector response simulation and deploy a multivariate technique to discriminate the signal from background. We provide prospects for the HL-LHC with the integrated luminosity of 3

a b^{- 1}

as well as the expected limits with Run 2 LHC data which corresponds to an integrated luminosity of 138 fb⁻¹. Additionally, we probe the parameter space of the ALP using interpretations of the LHC experimental data from W+jets and Z+jets measurements. These production processes are sensitive to ALP couplings to W bosons, gluons, and fermions. As a result, the limits in this study are obtained assuming two non-zero ALP coupling constants at a time and are provided in two-dimensional planes. This approach enables us to provide concurrent limits on the ALP couplings, rather than on a single coupling. Comparing the limits obtained in this study with the current and projected limits inferred from the LHC searches, it will be shown that the present study improves the limits significantly and can be used to probe a vast region of unprobed parameter space.

It is notable that while the

V V + a

channel directly probes interactions involving two electroweak gauge bosons and an ALP, the V+jets channel (with

V = Z, W^{\pm}

) benefits from higher production cross sections and more inclusive event selection criteria, enabling stronger constraints in certain regions of the parameter space. These two channels are therefore complementary in their sensitivity to different coupling structures and kinematic regimes. By concurrently investigating both channels within the same framework, we provide a more comprehensive and robust exploration of ALP-gauge boson interactions. This dual approach allows us to set simultaneous constraints on multiple ALP couplings and offers a clearer understanding of their interplay at the LHC.

This paper is structured as follows. In Section 2, the effective Lagrangian describing an ALP and its interactions with the SM particles is introduced. In section 3, the signal processes under consideration and the relevant SM background processes are discussed. In section 4, analysis details are discussed. In particular, section 4.1 describes the event generation method and tools, section 4.2 discusses the validity of the effective Lagrangian, section 4.3 discusses the object identification and event selection, and section 4.4 provides the details of the signal-background discrimination. The method used to calculate the limits and the obtained results are provided in section 5. Section 6 focuses on interpreting the CMS experiment measurements of the differential cross sections for the W+jets and Z+jets processes to investigate the couplings of the ALP with gluons and W bosons. Finally, the summary and conclusions of the paper are presented in Section 7.

2. ALPs effective Lagrangian

The QCD axion is the pseudo Nambu-Goldstone boson associated with the spontaneously broken anomalous global

U {(1)}_{P Q}

symmetry incorporated in the SM Lagrangian in the model proposed by Peccei and Quinn [1]. Providing the possibility of resolving the strong CP problem is the main motivation behind proposing the existence of the QCD axion. There is a strict relation between the mass and couplings of the QCD axion. ALPs are hypothetical particles resulting from eluding the relation connecting the mass and couplings of the QCD axion. Neglecting ALP couplings to fermions, the most general effective Lagrangian up to dimension-5 operators describing the interaction between the ALP and the SM fields is given by [23](1)

\begin{matrix} L_{eff} & = L_{SM} + \frac{1}{2} (\partial^{μ} a) (\partial_{μ} a) - \frac{1}{2} m_{a, 0}^{2} a^{2} \\ + c_{a Φ} \frac{i \partial^{μ} a}{f_{a}} (Φ^{†} {\overset{\leftrightarrow}{D}}_{μ} Φ) - c_{B B} \frac{a}{f_{a}} B_{μ ν} {\tilde{B}}^{μ ν} - c_{W W} \frac{a}{f_{a}} W_{μ ν}^{A} {\tilde{W}}^{μ ν, A} - c_{G G} \frac{a}{f_{a}} G_{μ ν}^{A} {\tilde{G}}^{μ ν, A}, \end{matrix}

where

G_{μ ν}^{A}

W_{μ ν}^{A}

and

B_{μ ν}

respectively represent the gauge field strength tensors corresponding to

S U {(3)}_{c}

S U {(2)}_{L}

and

U {(1)}_{Y}

symmetries, the dual field strength tensors

{\tilde{X}}^{μ ν}

are defined as

{\tilde{X}}^{μ ν} \equiv \frac{1}{2} ϵ^{μ ν ρ σ} X_{ρ σ}

with

ϵ^{μ ν ρ σ}

being the Levi-Civita symbol,

f_{a}

denotes the scale associated with the breakdown of the global

U (1)

symmetry, a and Φ are respectively the ALP field and the Higgs boson doublet, and

Φ^{†} {\overset{\leftrightarrow}{D}}_{μ} Φ \equiv Φ^{†} D_{μ} Φ - {(D_{μ} Φ)}^{†} Φ

. The ALP field undergoes the translation

a (x) \to a (x) + α

, with α being a constant, under a

U (1)

transformation (the symmetry corresponding to the a field translation is called the shift symmetry). As a result, a

U (1)

invariant effective Lagrangian should only contain the derivative of the ALP field. This is not the case here since the Lagrangian has a chiral anomaly induced by the

a G \tilde{G}

operator (the last operator in

L_{eff}

, Eq. (1)) which couples the ALP field to the gluon density

G \tilde{G}

. This operator induces a total divergence contribution to

δ L_{eff}

under the ALP field translation. This total divergence results in a non-zero change in the action due to the QCD vacuum structure and instanton effects. The shift symmetry is also broken (softly) by the ALP mass term present in

L_{eff}

. The ALP mass term makes ALP acquire more mass than is obtained from non-perturbative dynamics and adds a mass to the dynamically generated ALP mass,

m_{a}^{2} = m_{a, 0}^{2} + m_{a, dyn}^{2}

. This leads to a large parameter space and thus a rich phenomenology for ALPs. After electroweak symmetry breaking (EWSB), the operator

c_{a Φ} \frac{i \partial^{μ} a}{f_{a}} (Φ^{†} {\overset{\leftrightarrow}{D}}_{μ} Φ)

L_{eff}

gives rise to a term of the form

Z_{μ} \partial^{μ} a

, which complicates the direct interpretation of the effective Lagrangian. A field redefinition of the form(2)

Φ \to e^{i α_{Φ} a / f_{a}} Φ, ψ_{L} \to e^{i α_{ψ L} a / f_{a}} ψ_{L}, ψ_{R} \to e^{i α_{ψ R} a / f_{a}} ψ_{R},

where

ψ_{L} = {Q_{L}, L_{L}}

ψ_{R} = {u_{R}, d_{R}, e_{R}}

α_{Φ}

is a real constant and

α_{ψ L, R}

are

3 \times 3

hermitian matrices in flavor space, is thus applied to interpret the Lagrangian. The a–Z term can be redefined away in favor of the fermionic couplings of the ALP with the choice

α_{Φ} = c_{a Φ}

. The structure of the fermionic couplings induced in this way (Yukawa-like, vector-axial or a combination of them) can be controlled by tuning the parameters

α_{ψ L, R}

[23]. Applying the field redefinitions in Eq. (2) with the assumptions

α_{Φ} = c_{a Φ}

and

α_{ψ L, R} = 0

leads to(3)

\begin{matrix} L_{SM} \to L_{SM} & + c_{a Φ} [i ({\bar{Q}}_{L} Y_{U} \tilde{Φ} u_{R} - {\bar{Q}}_{L} Y_{D} Φ d_{R} - {\bar{L}}_{L} Y_{E} Φ e_{R}) \frac{a}{f_{a}} + h.c.] \\ - c_{a Φ} \frac{i \partial^{μ} a}{f_{a}} (Φ^{†} {\overset{\leftrightarrow}{D}}_{μ} Φ), \end{matrix}

where

\tilde{Φ} = i σ^{2} Φ^{⁎}

and

Y_{U}

Y_{D}

and

Y_{E}

are respectively

3 \times 3

matrices in flavor space containing Yukawa coupling constants for up-type quarks, down-type quarks and charged leptons. The Yukawa-like ALP-fermion operators in the first line of Eq. (3), which stem from the SM Yukawa Lagrangian, can be conveniently written as(4)

c_{a Φ} [i \frac{a}{f_{a}} \sum_{ψ = Q, L} ({\bar{ψ}}_{L} Y_{ψ} Φ σ_{3} ψ_{R}) + h.c.],

where

Q_{R} \equiv {u_{R}, d_{R}}

L_{R} \equiv {0, e_{R}}

σ_{3}

acts on weak isospin space, and the block matrices Φ and

Y_{ψ}

are given by(5)

Φ = diag (\tilde{Φ}, Φ), Y_{Q} \equiv diag (Y_{U}, Y_{D}), Y_{L} \equiv diag (0, Y_{E}) .

The operator

c_{a Φ} \frac{i \partial^{μ} a}{f_{a}} (Φ^{†} {\overset{\leftrightarrow}{D}}_{μ} Φ)

L_{eff}

introduces a derivative coupling between the ALP and the Higgs field. However, this term can be eliminated by a field redefinition of the Higgs doublet,

Φ \to e^{i c_{a Φ} a / f_{a}} Φ

, which generates an operator with opposite sign that cancels the original term, as shown in Eq. (3). Applying the field redefinitions, the effective Lagrangian up to dimension-5 operators becomes [23]:(6)

\begin{matrix} L_{eff} = & L_{SM} + \frac{1}{2} (\partial^{μ} a) (\partial_{μ} a) - \frac{1}{2} m_{a, 0}^{2} a^{2} \\ + c_{a Φ} [i \frac{a}{f_{a}} \sum_{ψ = Q, L} ({\bar{ψ}}_{L} Y_{ψ} Φ σ_{3} ψ_{R}) + h.c.] \\ - c_{B B} \frac{a}{f_{a}} B_{μ ν} {\tilde{B}}^{μ ν} - c_{W W} \frac{a}{f_{a}} W_{μ ν}^{A} {\tilde{W}}^{μ ν, A} - c_{G G} \frac{a}{f_{a}} G_{μ ν}^{A} {\tilde{G}}^{μ ν, A} . \end{matrix}

In this expression,

L_{SM}

is the SM Lagrangian, the second and third terms describe the canonical kinetic and mass terms for the ALP a, and the fourth term encodes ALP-fermion interactions via Yukawa couplings after electroweak symmetry breaking (EWSB). The remaining terms in

L_{eff}

correspond to anomalous ALP couplings to the SM gauge bosons: hypercharge (

B_{μ ν}

), weak (

W_{μ ν}^{A}

), and gluon (

G_{μ ν}^{A}

) field strengths. After EWSB, the neutral gauge bosons

B_{μ}

and

W_{μ}^{3}

mix to form the photon and Z boson. Consequently, the operators involving

B_{μ ν}

and

W_{μ ν}^{A}

translate into effective couplings between the ALP and the physical γ and Z bosons:(7)

L_{eff} ∋ - c_{γ γ} \frac{a}{f_{a}} F_{μ ν} {\tilde{F}}^{μ ν} - c_{γ Z} \frac{a}{f_{a}} F_{μ ν} {\tilde{Z}}^{μ ν} - c_{Z Z} \frac{a}{f_{a}} Z_{μ ν} {\tilde{Z}}^{μ ν},

where

F_{μ ν}

and

Z_{μ ν}

denote the electromagnetic and Z boson field strength tensors, respectively. The Wilson coefficients

c_{γ γ}

c_{γ Z}

, and

c_{Z Z}

are given by:(8)

c_{γ γ} = c_{θ}^{2} c_{B B} + s_{θ}^{2} c_{W W}, c_{γ Z} = s_{2 θ} (c_{W W} - c_{B B}), c_{Z Z} = s_{θ}^{2} c_{B B} + c_{θ}^{2} c_{W W},

with

s_{θ} = \sin θ_{W}

c_{θ} = \cos θ_{W}

, and

s_{2 θ} = 2 s_{θ} c_{θ}

, where

θ_{W}

is the weak mixing angle. These relations show how the original couplings

c_{B B}

and

c_{W W}

control the ALP interactions with photons and Z bosons after EWSB.

In general, when an ALP is produced, it may decay inside the detector (possibly reconstructed by its decay products) or leave the detector (manifesting itself as missing energy). The hadronic and leptonic decays of the ALP are not kinematically allowed here since the assumed ALP mass is well below the pion mass (≈135 MeV) and the on-shell electron pair production threshold (

2 m_{e} \approx 1.022

MeV). The decay into a pair of photons is, therefore, the dominant decay mode for the assumed ALP. At the leading order, the decay rate of an ALP into a pair of photons is given by(9)

Γ_{a \to γ γ} = \frac{m_{a}^{3}}{4 π} {(\frac{c_{γ γ}}{f_{a}})}^{2} .

As seen, the only Wilson coefficient contributing to this decay rate is

c_{γ γ}

. Other Wilson coefficients in the effective Lagrangian may also contribute to the decay of an ALP into a pair of photons if one takes into account beyond the tree-level effects. At the one-loop level, all the Wilson coefficients in the effective Lagrangian except for

c_{G G}

contribute to the ALP decay to photons via fermion loops and electroweak loops. The ALP-gluon coupling starts to contribute to the ALP decay to photons at two-loop level. To take into account loop-induced effects,

c_{γ γ}

in Eq. (9) can be replaced with the effective Wilson coefficient

c_{γ γ}^{eff}

which includes higher-order effects [29]. The effective coefficient

c_{γ γ}^{eff}

depends on the ALP mass. However, this dependence is so mild that the decay rate scales as

m_{a}^{3}

to an excellent approximation. The probability that an ALP decays inside the detector,

P_{a}^{\det}

, is given by(10)

P_{a}^{\det} = 1 - e^{- L_{\det} / L_{a}^{⊥} (θ)},

where

L_{\det}

represents the transverse distance of the calorimeter from the collision point, θ is the angle between the ALP’s velocity and the beam axis and

L_{a}^{⊥}

denotes the decay length of the ALP perpendicular to the beam axis given by(11)

L_{a}^{⊥} (θ) = γ β τ \sin θ = \frac{\sqrt{γ^{2} - 1}}{Γ_{a}} \sin θ \equiv L_{a} \sin θ,

where γ is the Lorentz boost factor, τ is the ALP proper lifetime,

Γ_{a}

is the ALP width, and β is the ALP speed. For small decay widths, the ALP escapes the detector before its decay leading to a large missing energy. The Lorentz boost factor, which depends on the experimental setup, also affects the detectability of ALPs. An ALP with mass 1 MeV can experience the visible decays

a \to γ γ

and

a \to γ ν \bar{ν}

. However, the decay mode

a \to γ ν \bar{ν}

, mediated by the a–Z–γ interaction, is negligible compared with the

a \to γ γ

mode. This is because the limit on

Γ (a \to γ ν \bar{ν})

, deduced from the experimental limit on the Wilson coefficient

c_{γ Z}

[23], is many orders of magnitude smaller than that of

Γ (a \to γ γ)

[29], [43]. The total decay width of such an ALP can be therefore approximated by the decay width of the ALP into a di-photon. Consequently, the ALP decay length in the laboratory frame can be estimated as(12)

L_{a} \approx \frac{| {\vec{p}}_{a} |}{m_{a}} \frac{1}{Γ_{a \to γ γ}} = \frac{4 π}{m_{a}^{4}} {(\frac{f_{a}}{c_{γ γ}})}^{2} | {\vec{p}}_{a} |,

where

{\vec{p}}_{a}

is the ALP three-momentum. Based on the current experimental limits, the parameter space region

c_{γ γ} / f_{a} > 1 \times 10^{- 9} Te V^{- 1}

has been experimentally excluded for an ALP of mass 1 MeV [29], [43]. Using Eq. (12) and the upper limit on

c_{γ γ} / f_{a}

, one can obtain the lower limit(13)

L_{a} ≳ (\frac{2.3 | {\vec{p}}_{a} |}{GeV}) 10^{21} m

on the ALP decay length. The ALP momentum

| {\vec{p}}_{a} |

depends on many factors, e.g. the collider center-of-mass energy, triggers used in the analysis, event selection criteria imposed to enrich the signal, etc. Based on the Monte Carlo simulations performed in this study (described in section 4.1), the magnitude of the ALP momentum is of

O (100)

GeV for the assumed experimental setup. As a result, one can find the lower limit

L_{a} ≳ 2.3 \times 10^{23} m

on the ALP decay length using Eq. (13). Using the obtained lower limit in Eq. (10), it can be seen that ALPs assumed in this work are likely to escape the detector and manifest themselves as missing energy. For ALPs lighter than 1 MeV, the decay length becomes larger leading to more tendency to be invisible at the detector. It can be concluded that a large fraction of ALPs produced at the collider are invisible leading to a large missing energy signature. A very small fraction of ALPs decay inside the detector. The effect of the decaying ALPs is taken into account in this study. It is worth noting that for light enough ALPs, the lifetime can become larger than the age of the universe thanks to the third power of the ALP mass in Eq. (9). Such ALPs can be viable candidates for DM.

3. ALP production in association with electroweak gauge bosons

At a proton-proton collider, an ALP can be produced in association with two electroweak gauge bosons,

p p \to a V V

, where VV can be ZZ or WW. Fig. 1 shows the representative Feynman diagrams contributing to these production processes at the leading order. BSM interactions are shown with the blue filled circles in the Feynman diagrams. These processes are governed by the effective Lagrangian, Eq. (6). The interactions

a - G - G

a - W - W

a - Z - Z

and

a - q - q

contribute to these processes, and thus these processes are sensitive to the Wilson coefficients

c_{G G}

c_{W W}

c_{B B}

and

c_{a Φ}

(

c_{Z Z}

is a function of

c_{W W}

and

c_{B B}

as seen in Eq. (8)). Assuming the HL-LHC operating at the center-of-mass energy of 14 TeV, we study these processes and set upper bounds on the involved Wilson coefficients for an ALP of mass 1 MeV.

The processes considered in this analysis provide an opportunity to simultaneously probe both the ALP–gluon and ALP–electroweak couplings through a distinctive BSM topology involving three effective vertices. For example, the ZZa signal process, in which the ALP interacts with gluons and electroweak gauge bosons, proceeds predominantly via gluon fusion, producing an ALP in association with two Z bosons. This results in characteristic kinematic features such as large invariant mass

m_{Z Z}

and central rapidities. For the ZZa final state, the squared matrix element for the dominant t-channel contribution scales as

| M |^{2} \propto \frac{c_{G G}^{4} c_{Z Z}^{2}}{f_{a}^{6}} \times {\hat{s}}^{2}

, where

\hat{s}

is the partonic center-of-mass energy. The corresponding cross section is enhanced by both the energy dependence of the squared matrix element and the steep rise of gluon PDFs at low Bjorken-x.

In contrast, there exists another process,

q \bar{q} \to Z Z a

, which results in a similar final state but proceeds through ALP radiation from a final-state Z boson via an effective

a Z \tilde{Z}

vertex, involving only a single non-zero ALP coupling. Its squared matrix element scales as

| M |^{2} \propto \frac{g^{4} c_{Z Z}^{2}}{f_{a}^{2}} \times \hat{s}

. However, this contribution is significantly suppressed in the high-

m_{Z Z}

region due to phase-space limitations and the rapidly falling quark PDFs at large momentum fractions. Moreover, the Z bosons and ALP produced in this process tend to be softer and more forward than those in the BSM signal, resulting in much lower acceptance under our selection criteria, which favor large

m_{Z Z}

and central leptons. Beyond these kinematic suppressions, the numerical impact of this process is negligible. For instance, for benchmark values

c_{G G} = c_{W W} = 0.5 {TeV}^{- 1}

, the cross section for the BSM-induced ZZa process is approximately

2.27 pb

, while the cross section for

q \bar{q} \to Z Z a

is only

0.0122 pb

(with

c_{W W} = 0.5 {TeV}^{- 1}

), corresponding to less than 1% of the signal rate. Thus, even if included, the

q \bar{q} \to Z Z a

process would have no meaningful impact on our signal region. Furthermore, the

q \bar{q} \to Z Z a

topology probes only a single coupling (e.g.,

c_{Z Z}

) and contributes at

O (c_{i}^{2})

, dominating in the low-coupling regime. Including such terms would bias the interpretation of the exclusion limits by reducing sensitivity to the interplay between multiple couplings and leading to exclusion contours dominated by a single direction in parameter space. This would undermine the central objective of simultaneously constraining multiple ALP interactions in a model-independent framework. By restricting our analysis to signal diagrams that involve three effective ALP vertices—and which do not interfere with SM amplitudes at leading order—we ensure a clean, interpretable extraction of two-dimensional exclusion limits in the

(c_{G G}, c_{W W})

(c_{G G}, c_{Z Z})

planes.

As shown in Fig. 2, the WWa and ZZa production processes exhibit greater sensitivity to the coupling

c_{G G}

compared to

c_{W W}

. Notably, both processes demonstrate minimal sensitivity to the coupling

c_{a Φ}

which arise from sub-processes of

q \bar{q}

in the initial state. For instance, in WWa production, when

c_{W W}

and

c_{a Φ}

are fixed at 1.0, the corresponding cross section is of the order of

10^{- 6}

. The main reason for such a small cross section is that the ALP coupling to quarks (fermions) is proportional to the Yukawa coupling constants (see Eq. (6)). The smallness of the Yukawa coupling constants corresponding to the up and down quarks significantly suppresses the cross section of sub-processes involving the ALP coupling to quarks. Hence, sub-processes involving the ALP coupling to gluons make the dominant contribution to the WWa and ZZa production with respect to the sub-processes with

q \bar{q}

in the initial state hence the presence of

c_{a Φ}

unlikely produces any discernible experimental effect, justifying its omission from further detailed exploration in this study.

The production cross section for the process

p p \to Z Z a

exhibits a dependence on the Wilson coefficient

c_{B B}

, as detailed in Eqs. (7) and (8). Fig. 3 illustrates the variation of the cross section

σ (p p \to Z Z a)

as a function of both

c_{B B}

and

c_{W W}

, under the assumption that the gluonic Wilson coefficient is fixed at

c_{G G} = 0.5 {TeV}^{- 1}

. A key observation is that the sensitivity to

c_{B B}

is notably weaker compared to

c_{W W}

. This can be attributed to the specific coupling structure inherent to the interaction terms, particularly the presence of a suppression factor proportional to

\sin^{2} θ_{W}

(denoted

s_{θ}^{2}

) in the

c_{B B}

contribution to

c_{Z Z}

. In contrast, the

c_{W W}

contribution involves a factor of

\cos^{2} θ_{W}

(denoted

c_{θ}^{2}

), which leads to a comparatively stronger enhancement for

c_{W W}

in processes involving neutral gauge bosons. As a result, the variation in the production rate of

p p \to Z Z a

with respect to

c_{B B}

remains minimal, and the overall cross section is both low and exhibits limited sensitivity to changes in

c_{B B}

. This reduced sensitivity suggests that

c_{B B}

does not significantly contribute to deviations in the cross section within the explored parameter space. Therefore, while the impact of

c_{B B}

on the cross section is presented and quantified, it remains subdominant compared to

c_{W W}

and no further detailed comparison is necessary for the purpose of setting the main exclusion limits.

A scan over the ALP mass shows that the production cross sections for both WWa and ZZa processes remain nearly independent of the ALP mass for values below a few GeV. This is because the ALP mass in this range is small compared with the typical energy scale of the process. It is also seen that the kinematics of the WWa and ZZa events is roughly independent of the ALP mass for such light ALPs. As a result, the constraints obtained in this study (provided in section 5) hold not only for an ALP of mass 1 MeV, but also for lighter ALPs to an excellent approximation.

The fully leptonic final state is considered for the WWa signal in this study. For the ZZa signal, the fully leptonic and semi-leptonic final states are studied. The dominant SM processes contributing to the background for the WWa production in the fully leptonic final state and the ZZa production in the semi-leptonic final state are assumed to be

t \bar{t}

, tW, WW, WZ, ZZ, ttZ, WWZ and ttH. The dominant SM backgrounds for the ZZa production in the fully leptonic final state are assumed to be ZZ, ttZ, WWZ, ttH,

W W W W

and

t t t t

. In these SM processes, neutrinos coming from jets or leptonic decays of W and Z bosons cause missing energies that mimic the signal missing energy signature.

4. Analysis

In this section, we discuss the details of the analysis. The analysis begins by the generation of the signal and background events. In the simulated events, objects such as jets and isolated leptons are identified. Then, events are selected and analyzed to separate the signal from background. A condition is also imposed to insure the validity of the effective Lagrangian.

4.1. Event generation

The effective Lagrangian, Eq. (6), has been implemented into FeynRules [64] and the generated Universal FeynRules Output (UFO) [65] model has been passed to MadGraph5_aMC@NLO [66] to compute the cross section and generate hard events for the signal and background processes. NNPDF23 [67] is used as the proton PDF and the center-of-mass energy of the proton beams is set to 14 TeV. Pythia 8.2.43 is used to perform parton showering, hadronization and decays of unstable particles. Showered events are then passed to Delphes 3.4.2 [68] for fast detector simulation, using the official CMS detector card delphes_card_CMS.tcl¹ provided in the Delphes distribution. While the present study relies on fast detector simulation using Delphes with the CMS detector card, a future extension of this work foresees the use of a full GEANT4-based simulation [69] to more accurately model jet reconstruction and low-energy particle effects, thereby validating the robustness of the present results.

The signal samples were generated at leading order using the ALP linear UFO model,² which implements the effective interactions of the ALP with SM gauge bosons in the linear EFT framework. All SM background processes were also generated at leading order. The generation of WWa and ZZa signal samples has been performed assuming two non-zero Wilson coefficients,

c_{W W}

and

c_{G G}

, at a time. The WWa and ZZa samples are both used to derive exclusion limits in the

(c_{W W}, c_{G G})

plane. The ALP mass,

m_{a}

, is set to 1 MeV for all samples. As discussed in section 3, a very small fraction of ALPs produced in the collider decay inside the detector. The effect of the decaying ALPs is taken into account in an event-by-event manner with the use of the decay probability,

P_{a}^{\det}

, defined in Eq. (10).

4.2. Validity of the ALP effective Lagrangian

To ensure the validity of the effective Lagrangian, the mass scale of new physics, $f_{a}$ , should be significantly larger than the typical energy scale, $\sqrt{\hat{s}}$ , of the process under study. The strict requirement for the validity of the effective description is, therefore, to satisfy the condition $\sqrt{\hat{s}} < f_{a}$ . However, $\sqrt{\hat{s}}$ cannot be experimentally measured due to the presence of invisible ALPs in the final state. One may use the correlation between $\sqrt{\hat{s}}$ and the missing transverse energy, obtained using the Monte Carlo simulated events, to naively ensure validity of the EFT by requiring

, where

is the highest missing transverse energy data bin in the analysis. In this work, the strict EFT validity condition $\sqrt{\hat{s}} < f_{a}$ is imposed by discarding events that do not satisfy the condition

4.3. Object identification and event selection

Generated WWa and ZZa samples are independently analyzed to constrain BSM couplings. Jet reconstruction is performed using the anti-

k_{t}

algorithm [70] implemented in FastJet 3.3.2 [71], with a radius parameter

R = 0.4

that defines the jet cone size in the rapidity-azimuthal angle (y–ϕ) plane, such that

Δ R = \sqrt{{(Δ y)}^{2} + {(Δ ϕ)}^{2}}

. The transverse momentum and pseudorapidity of reconstructed jets are required to satisfy the conditions

p_{T} > 30

GeV and

| η | < 2.5

. Isolated electrons and muons are identified using the relative isolation variable,

I_{r e l}

. This variable is defined as

I_{r e l} = \sum p_{T}^{i} / p_{T}^{P}

, where P is the candidate particle for which

I_{r e l}

is calculated, and i is the summation index running over all particles (excluding the particle P) within a cone of radius

R = 0.5

centered on the candidate particle, defined in the pseudorapidity-azimuthal angle plane as

Δ R = \sqrt{{(Δ η)}^{2} + {(Δ ϕ)}^{2}}

. All particles with a transverse momentum greater than 0.5 GeV are taken into account in the summation. An electron (muon) is identified as an isolated lepton if

I_{r e l} < 0.12

(0.25). Isolated electrons and muons should satisfy the conditions

p_{T} > 20

GeV and

| η | < 2.4

To analyze the WWa signal in the fully leptonic final state, events are required to have exactly two isolated oppositely-charged leptons (electron or muon) and no jet. The flavors of the leptons can be the same or different. For the ZZa analysis, both the fully leptonic and semi-leptonic final states are studied. Two signal regions corresponding to these final states are defined as follows. SR1 is defined by the requirement that events should have exactly two pairs of isolated oppositely-charged same-flavor leptons (electron or muon) and no jet. This includes events with the final states $e e μ μ$ , $e e e e$ and $μ μ μ μ$ . SR2 is defined by requiring exactly one pair of isolated oppositely-charged same-flavor leptons (ee or μμ) and at least two jets. The missing transverse energy,

, which shows the energy imbalance in the transverse plane is required to be larger than 30 GeV for all events for the ZZa SR2 and aWW analyses. In the analysis, jets from all parton flavors are considered, with no flavor-based selection, and jets found within a cone of $Δ R < 0.4$ around any selected lepton are removed to avoid overlap. In SR1, lepton pairs are required to have invariant masses consistent with the Z boson mass window. Fake leptons and jet misidentification effects are not included in the fast detector simulation via Delphes and are therefore beyond the scope of this study.

Applying the above-mentioned cuts, event selection efficiencies presented in Table 1 are obtained for the signal and background samples. As seen, the SM production of

W W W W

, ZZ and

t t t t

have the highest selection efficiencies in the SR1 signal region of the ZZa analysis. For the SR2 signal region, the processes ttZ, WWZ and

t \bar{t}

have the highest efficiencies. Finally, the processes tW,

t \bar{t}

and WW are the most important backgrounds in the aWW analysis.

Table 1. Event selection efficiencies after applying all selection cuts obtained for the signal (ZZa, WWa) and different background processes. Two efficiencies are presented for each signal process. For the signal processes the couplings values are c_WW = 0.1, c_GG = 0.1.

ZZa (SR1)	ZZ	ttZ	WWZ	ttH	WWWW	tttt
0.071	0.0024	0.00043	0.00047	1.5e-6	0.0035	0.0013

ZZa (SR2)		tW	WW	WZ	ZZ	ttZ	WWZ	ttH
0.396	0.034	0.024	0.0022	0.013	0.015	0.082	0.043	0.0028

WWa		tW	WW	WZ	ZZ	ttZ	WWZ	ttH
0.478	0.199	0.235	0.128	0.00091	0.0017	0.096	0.113	0.013

4.4. Signal-background discrimination

Signal events with non-zero $c_{W W}$ and $c_{G G}$ are analyzed independently for the ZZa (SR1 and SR2) and WWa processes resulting in exclusion limits in the $c_{W W}$ – $c_{G G}$ plane, respectively. Different variables are defined and used to discriminate between the selected events in the signal and background samples. To obtain the best signal-background discrimination, a multivariate technique using the TMVA (Toolkit for Multivariate Data Analysis) package [72], [73], [74] is deployed. The discrimination power of all the multivariate classification algorithms available in the TMVA package is compared using the receiver operating characteristic (ROC) curve to find the algorithm with the highest discrimination power. The Boosted Decision Trees (BDT) algorithm [75] was found to be the most powerful algorithm for both the ZZa and WWa analyses and is thus used in this study. The distributions obtained for the discriminating variables are passed to the BDT algorithm, and the BDT performs the training process considering all background processes according to their respective weights. The TMVA overtraining check is performed to ensure overtraining does not occur. Kolmogorov-Smirnov (K-S) test is also performed to ensure the consistency of the BDT responses obtained for the training and test samples. The discriminating variables defined to be used for the SR1 signal region in the ZZa analysis are:

•

Missing transverse energy,

.
•
Invariant mass of the four reconstructed charged leptons, $m_{Z Z}$ .
•
Magnitude of the sum of transverse momentum vectors of the four reconstructed charged leptons, $p_{T}^{Z Z}$ .
•
Azimuthal separation between the two reconstructed Z bosons, $Δ Φ_{Z_{1} Z_{2}}$ . Each Z boson is reconstructed using a pair of oppositely-charged same-flavor charged leptons. If two lepton pairs of the same flavor exist, i.e. $e e e e$ and $μ μ μ μ$ , the pair of oppositely-charged leptons with minimum $| m_{ℓ ℓ} - m_{Z} |$ (with $m_{Z} \approx 91$ being the Z boson mass) is used to reconstruct the first Z boson (Z1), and the remaining pair is used to reconstruct the second Z boson (Z2).

Distributions obtained from the selected events for the above variables are shown in Fig. 4. According to the BDT output,

, $p_{T}^{Z Z}$ and $m_{Z Z}$ are the best variables in terms of the discrimination power for the ZZa SR1 analysis. The discriminating variables listed below are used for the WWa and ZZa SR2 analyses:

•

Missing transverse energy,

.
•
Invariant mass of the two reconstructed charged leptons, $m_{ℓ ℓ}$ .
•
Magnitude of the sum of transverse momentum vectors of the two reconstructed charged leptons, $p_{T}^{ℓ ℓ}$ .
•
Spatial separation, $Δ R = \sqrt{Δ η^{2} + Δ ϕ^{2}}$ , with η and ϕ being the pseudorapidity and azimuth angle, between the two reconstructed charged leptons, $Δ R_{ℓ_{1} ℓ_{2}}$ .
•
Number of reconstructed jets, $N_{j e t}$ .

Distributions obtained for these variables in the WWa analysis (for example) are shown in Fig. 5.

As discussed in section 3, the dominant fraction of ALPs produced at the collider decay outside the detector. The resulting missing energy signature plays an important role in separating the signal from background in both the WWa and ZZa analyses. The only source of missing energy in the SM background is the missing neutrinos which result in missing energies typically lower than that of the signal (see Figs. 4a, 5a). The magnitude of the sum of transverse momentum vectors of the reconstructed leptons, i.e. $p_{T}^{Z Z}$ and $p_{T}^{ℓ ℓ}$ , are also powerful variables as the final state ALP in the ZZa and WWa processes prevents the produced W and Z bosons from being emitted in a back-to-back configuration (unlike the case of ZZ and WW production) resulting in relatively large momenta sum values. Furthermore, the dominance of the s-channel diagrams contribution to the signal cross section results in W and Z bosons mostly emitted near the transverse plane leading to high transverse momenta sums. As a result, the ZZ and WW production processes, which are respectively the dominant backgrounds for the ZZa and WWa signals, are significantly suppressed by the momenta sum variables (see Figs. 4b, 5b). As seen in Fig. 5e, the number of jets, $N_{j e t}$ , is particularly useful for suppressing backgrounds with high jet multiplicities like ttZ, WWZ, ttH and tt. According to the BDT output, $m_{ℓ ℓ}$ , $Δ R_{ℓ_{1} ℓ_{2}}$ and

(

, $N_{j e t}$ and $p_{T}^{ℓ ℓ}$ ) are the most powerful variables for the ZZa SR2 (WWa) analysis to discriminate the signal from background.

Using the trained model, BDT responses for the signal and background events are obtained. Fig. 6 shows the obtained results for the ZZa SR1, ZZa SR2 and WWa analyses. As seen, the BDT performs well in signal-background discrimination. A comparison shows that the best signal-background separation is achieved for the ZZa SR1 analysis, and the worst one is obtained for the WWa analysis.

5. Expected limits and comparison with existing bounds

Using the obtained BDT response distributions, 95% CL exclusion limits in the

c_{W W} - c_{G G}

plane are derived. A threshold on the BDT output is applied to define the signal region. The expected number of signal and background events passing this cut—computed using event weights based on cross section, luminosity, and number of generated events—are used to construct Poisson likelihood functions under the background-only and signal-plus-background hypotheses. The CL_s method [76], [77], implemented in the RooStats package [78], is used to derive the exclusion limits. Specifically, the log-likelihood ratio test statistic

Q = 2 \ln (L_{signal+bkg} / L_{bkg})

is used to compute p-values for both hypotheses, and the exclusion is set at 95% confidence level by requiring

{CL}_{s} = \frac{P_{signal+bkg} (Q > Q_{0})}{1 - P_{bkg} (Q < Q_{0})} \leq 0.05,

where

Q_{0}

is the observed (or expected) value of the test statistic. Limits in the ZZa (SR1 and SR2) and WWa analyses are computed independently. To account for systematic uncertainties, an overall uncertainty of 10% is applied to both signal and background yields, reflecting detector effects such as energy scale, resolution, pile-up, and reconstruction efficiency. Fig. 7 shows the expected 95% CL limits in the

c_{W W} - c_{G G}

plane for the ZZa SR1, ZZa SR2, and WWa analyses, assuming integrated luminosities of 138 fb⁻¹ and 3 ab⁻¹. We have verified that removing the 10% systematic uncertainty leads to a

5 - 10 %

improvement in the exclusion limits across all channels, confirming the robustness of the results.

The limits have been derived under the assumption that the kinematics of the signal events remain independent of the values of the non-zero Wilson coefficients. A comparative analysis reveals that the ZZa SR1 search provides slightly stronger constraints than both the ZZa SR2 and WWa analyses. This can be attributed to more effective background suppression and greater discriminating power of the BDT classifier in the fully leptonic ZZa final state. Although the cross section for the ZZa process is larger than for WWa, the key factor driving the difference in exclusion power lies in the distribution of the missing transverse energy (

E_{T}^{miss}

), which serves as a crucial input to the BDT. In the ZZa SR1 channel, the dominant SM ZZ background leads to very low MET, whereas in the WWa channel, the main backgrounds such as WW and

t \bar{t}

produce significant MET from neutrinos, making separation from signal more challenging. The ZZa SR2 and WWa analyses yield nearly comparable limits in the

c_{W W} - c_{G G}

plane.

To assess the significance of our results, we compare them quantitatively with existing experimental and theoretical bounds on the ALP couplings available in the literature. The loop-induced corrections to electroweak precision observables, encapsulated by the well-known oblique parameters S, T, and U, were utilized in Ref. [29] to simultaneously probe the Wilson coefficients

c_{W W} - c_{B B}

and

c_{γ γ} - c_{γ Z}

. Notably, it was found that the coefficient

c_{B B} / f_{a}

remains substantially unconstrained, whereas

c_{W W} / f_{a}

is confined to a much narrower parameter space. At the 99% CL, an upper bound of 8 TeV⁻¹ was placed on

c_{W W} / f_{a}

. The limits on the Wilson coefficients in the plane of

c_{W W} - c_{B B}

, obtained from the combination of VBS

q_{1} q_{2} \to q_{1}^{'} q_{2}^{'} V_{1} V_{2}

processes, where

V_{1, 2} = W, Z, γ

, are presented in Ref. [63]. Assuming

M_{V_{1} V_{2}} < 4

TeV, values of

c_{W W} / f_{a}

and

c_{B B} / f_{a}

above 0.75 TeV⁻¹ and 1.59 TeV⁻¹, respectively, are excluded. In Ref. [35], the analysis was extended by imposing simultaneous two-dimensional constraints on

c_{W W} / f_{a}

and

c_{a Φ} / f_{a}

using the measured cross sections of the

t \bar{t}

and tW processes at the LHC. Specifically,

c_{a Φ} / f_{a}

is constrained to values around 40 TeV⁻¹, while

c_{G G} / f_{a}

is limited to less than 4 TeV⁻¹. One-dimensional limits on

| c_{W W} / f_{a} |

for an integrated luminosity of 3

a b^{- 1}

, based on LHC searches for mono-W, mono-Z, Wγ, Zγ, WW, and WWγ final states with missing energy, are available in Refs. [23], [24], [52], [53], [54], [55], [56], [57]. The ATLAS search for mono-W events in the leptonic final state, with an integrated luminosity of 139

f b^{- 1}

, places an upper bound of

| c_{W W} / f_{a} | < 0.11

TeV⁻¹ at 95% CL [24], [55]. Similarly, the CMS search for mono-Z events using 35.9

f b^{- 1}

of data provides a slightly weaker constraint of

| c_{W W} / f_{a} | < 0.129

TeV⁻¹ at 95% CL [24], [58]. Additionally, a 95% CL limit of

| c_{W W} / f_{a} | < 0.44

TeV⁻¹ can be inferred from the uncertainty on the Z boson width [79], [23].

Constraints on the ALP coupling to gluons,

c_{G G} / f_{a}

, have been derived from various experimental measurements. The measurement of

K^{+} \to π^{+} ν_{ℓ} {\bar{ν}}_{ℓ}

provides a tighter limit of

| c_{G G} / f_{a} | < 0.00769

TeV⁻¹, while the process

B \to K^{⁎} ν_{ℓ} {\bar{ν}}_{ℓ}

yields a constraint of

| c_{G G} / f_{a} | < 2.1

TeV⁻¹ [80], [81]. Mono-jet analyses at the 8 TeV LHC, using 19.6 fb⁻¹ of data, result in a 95% CL upper limit of

| c_{G G} / f_{a} | < 0.025

TeV⁻¹ [28].

Astrophysical observations impose stringent constraints on ALPs through their couplings to photons, electrons, and nucleons. For instance, stellar cooling processes set a lower bound on

f_{a}

that is several orders of magnitude stricter than those explored in this work. While these constraints are not directly on

c_{G G, W W} / f_{a}

, they inform the allowed parameter space for ALP models in a broader context [82], [83]. Regarding the mass range

m_{a} = 1

MeV, supernova constraints exclude couplings in the range

10^{- 8}

10^{- 5}

TeV⁻¹ [84]. These bounds, however, are not directly comparable to our results, as they do not address the simultaneous behavior of

c_{W W}

and

c_{G G}

6. Probe of the ALP couplings using the LHC measurements of W+jets and Z+jets

In this section, we analyze the sensitivity of the production of an ALP in association with a vector boson (V) and jets (V+jets+ALP), where

V = Z, W^{\pm}

. The leading-order Feynman diagrams for these processes are illustrated in Fig. 8. Similar to the previous analysis, we consider scenarios where the ALP does not decay within the detector, manifesting as missing transverse energy. Consequently, the final states resemble those of the SM processes

W^{\pm}

+jets and Z+jets, although with modified kinematics due to the presence of the ALP.

These processes are particularly sensitive to the couplings of the ALP with gluons (

c_{G G}

) and W bosons (

c_{W W}

). In V+jets+ALP production, the initial state primarily involves quark-antiquark annihilation and gluon-quark interactions. The parton distribution function (PDF) for gluons is significantly large, particularly at low momentum fractions (x), leading to substantial contributions from gluon-induced processes, even though quark-antiquark annihilation has remarkable contribution for the V+jets+ALP production.

These processes are crucial as they provide a probe of both

c_{G G}

and

c_{W W}

couplings. The transverse momentum (

p_{T}

) of the V boson is a key observable, offering insights into the initial state kinematics and the dynamics of the associated jet production. A higher

p_{T}

of the V boson typically corresponds to more energetic jets, reflecting the underlying parton-level interactions that characterize the final state. By analyzing the

p_{T}

spectrum of the V boson or the leading jet, one can explore the ALP interactions with gluons and vector bosons. In the subsequent subsections, we utilize the

p_{T}

spectra of the Z boson and leading jet, derived from CMS experiment measurements in Refs. [85], [86], to probe these interactions. The fiducial object definitions and event selection criteria were implemented manually based on the detailed descriptions provided in Refs. [85], [86], and validated by comparing the resulting kinematic distributions with the corresponding unfolded CMS measurements.

6.1. Z+jets+ALP process

In Ref. [85], a measurement of the differential cross section for Z+jets production at a center-of-mass energy of

\sqrt{s} = 13

TeV is presented, focusing on the dimuon final state

(Z \to μ^{+} μ^{-})

. The data used in this analysis were collected with the CMS detector in 2016, corresponding to an integrated luminosity of 35.9 fb⁻¹. The study investigates regions of phase space characterized by the production of Z bosons at large transverse momentum (

p_{T, Z}

) in association with at least one high

p_{T}

jet.

In the analysis, the Z boson is identified via its decay into a pair of muons (

μ^{+} μ^{-}

). Events for the Z+jets analysis are selected using a high-level trigger that requires a loosely isolated muon with a minimum

p_{T}

threshold of 24 GeV. Muon candidates are required to lie within the fiducial region

| η_{μ} | < 2.4

and to be separated from any jets in the event by a distance

Δ R (μ, jets) > 0.5

, where ΔR is defined as

Δ R = \sqrt{Δ η^{2} + Δ ϕ^{2}}

. Two oppositely charged muons are selected, and the invariant mass of the dimuon pair,

m_{μ^{+} μ^{-}}

, is required to be consistent with the Z boson mass, within the range

71 < m_{μ^{+} μ^{-}} < 111

GeV. Following the baseline selection, each muon is required to have

p_{T} > 30

GeV, and the dimuon system is required to have

p_{T} > 200

GeV with a rapidity

| y_{μ^{+} μ^{-}} | < 1.4

. The measured distributions are then unfolded from the detector level to the particle level. Jets are required to have

p_{T, j} > 40

GeV and

| η_{j} | < 2.4

, with at least one jet in the event required to have

p_{T} > 40

GeV. The differential cross section is evaluated as a function of the Z boson’s transverse momentum,

p_{T, Z}

. The experimental measurements of the differential

p_{T, Z}

distribution are used as input, along with theoretical predictions from the ALP model. A subsequent fit is performed to extract constraints on the ALP couplings

c_{G G}

and

c_{W W}

ALP signal events are generated using MadGraph5_aMC@NLO, with two of the couplings

c_{G G}

c_{W W}

set to a non-zero value at a time. These events are then processed through Pythia 8 for parton showering, hadronization, and the decay of stable particles. After applying the selection criteria, the normalized distribution of

p_{T, Z}

is depicted in the left panel of Fig. 9. The distributions for ALP signal events, with

c_{G G} / f_{a}

and

c_{W W} / f_{a}

couplings each set to 0.1 TeV⁻¹, are also shown, assuming an ALP mass of

m_{a} = 1

MeV.

To determine the upper limits on the ALP couplings, we calculate the uncorrelated

χ^{2}

functions of the ALP couplings and mass using the differential cross-section measurements of

p_{T, Z}

. The

χ^{2}

function is defined as:(14)

χ^{2} (\frac{c_{W W}}{f_{a}}, \frac{c_{G G}}{f_{a}}, m_{a}) = \sum_{i \in bins} \frac{{[Data (i) - Bkg (i) - {Pred}_{ALP} (i; \frac{c_{W W}}{f_{a}}, \frac{c_{G G}}{f_{a}}, m_{a})]}^{2}}{δ_{i}^{2}},

where

Data (i)

represents the measured differential distribution in the ith bin,

{Pred}_{ALP} (i)

and

Bkg (i)

represent the differential distributions of the ALP signal and the SM background, respectively, in the same bin. The term

δ_{i}

accounts for both systematic and statistical uncertainties. These uncertainties, along with the measured data distribution and the SM prediction, are derived from the CMS measurements of the differential cross-section for Z+jets production at the LHC, performed at a center-of-mass energy of 13 TeV with an integrated luminosity of 35.9 fb⁻¹ [85].

The observed upper limits on the couplings

c_{G G} / f_{a}

and

c_{W W} / f_{a}

at a 95% confidence level (CL) for an ALP mass

m_{a} = 1

MeV are shown in Fig. 10. This figure also presents the projected constraints on

c_{G G} / f_{a}

and

c_{W W} / f_{a}

at 95% CL for the High-Luminosity LHC (HL-LHC), assuming the systematic uncertainties remain consistent with those of the current CMS measurements.

6.2. $W^{\pm} +$ jets+ALP process

In this subsection, we utilize the differential cross sections for W boson production in association with jets, as reported in Ref. [86], to explore the parameter space of the ALP model. These measurements were based on an integrated luminosity of 2.2 fb⁻¹ of data collected with the CMS detector at the LHC, during proton-proton collisions at a center-of-mass energy of 13 TeV. The analysis in Ref. [86] measured the differential cross sections for W bosons decaying via the muon channel, as functions of jet transverse momentum and the absolute value of rapidity for the leading jets, as well as the scalar sum of the transverse momenta of all jets in each event (

H_{T}

). Additionally, the cross sections were measured as a function of the azimuthal separation between the muon direction and the leading jet.

The

W^{\pm}

+jets+ALP events were generated using MadGraph5_aMC@NLO, followed by simulation and tuning similar to those detailed in Ref. [86]. We consider events where the ALP does not decay within the detector, leading to a final state that mimics the SM

W^{\pm}

+jets process but with altered kinematics due to the presence of the ALP. The selection criteria applied are as follows: muons are required to have a transverse momentum

p_{T} > 25

GeV and be within the pseudorapidity range

| η_{μ} | < 2.4

. Jets must satisfy the selection criteria with

p_{T} > 30

GeV and

| y_{jets} | < 2.4

. Additionally, events are required to be in the transverse mass peak region for W bosons, defined by

m_{T}^{W}

, where

m_{T}^{W} = \sqrt{2 | {\vec{p}}_{T}^{μ} | | {\vec{p}}_{T}^{miss} | (1 - \cos (Δ ϕ))}

. Here, Δϕ is the azimuthal angle difference between the muon momentum direction and the missing transverse momentum

{\vec{p}}_{T}^{miss}

. As in the previous analysis, a fit is performed on the differential distributions to determine the sensitivity to the

c_{G G}

and

c_{W W}

couplings. Among the various differential distributions analyzed, the leading jet transverse momentum distribution provided the highest expected sensitivity. The differential cross-section measurement for the leading jet transverse momentum, along with the SM prediction, is shown in the right panel of Fig. 9. The leading-order prediction for an ALP with

m_{a} = 1

MeV is also presented.

The upper limits on the couplings

c_{G G} / f_{a}

and

c_{W W} / f_{a}

at 95% CL for

m_{a} = 1

MeV, obtained from the CMS measurements of the W+jets process, are presented in Fig. 10. Additionally, the projected limits for the High-Luminosity LHC (HL-LHC) are shown, assuming that the systematic uncertainties remain consistent with those in the current CMS measurements from Ref. [86].

To thoroughly understand the results presented in Fig. 10, it is crucial to delve into the comparative sensitivity of the Z+jets+ALP and

W^{\pm}

+jets+ALP channels to the ALP couplings,

c_{G G}

and

c_{W W}

. The results clearly indicate that both channels exhibit greater sensitivity to the

c_{G G}

coupling compared to

c_{W W}

. This enhanced sensitivity arises from the stronger dependence of the cross sections on

c_{G G}

. In the ALP model, the coupling

c_{G G}

governs interactions between the ALP and gluons, which are abundant in the initial state of proton-proton collisions. Consequently, processes involving

c_{G G}

tend to have larger cross sections, leading to a more pronounced impact on the differential distributions of observables like the transverse momentum of the V boson (where

V = Z, W^{\pm}

) and the leading jet. This results in a more significant deviation from the SM expectation when

c_{G G}

is non-zero, thereby enhancing the sensitivity of the analysis to this coupling.

When comparing the two channels under the same integrated luminosity of 3 ab⁻¹, the limits on the couplings are found to be relatively similar; however, the Z+jets+ALP channel consistently provides slightly better constraints than the

W^{\pm}

+jets+ALP channel. This difference in sensitivity can be attributed to the characteristics of the fiducial region in which the fit is performed. In the fiducial region for the Z+jets+ALP process, there is a more substantial contribution of the ALP signal relative to the SM background, compared to the

W^{\pm}

+jets+ALP process. This means that the presence of the ALP has a more pronounced effect on the kinematic distributions in the Z+jets+ALP channel, allowing for tighter constraints on the ALP couplings.

In summary, while both channels offer valuable insights into the ALP couplings, the Z+jets+ALP channel stands out as slightly more sensitive. This enhanced sensitivity is mostly due to more significant contribution of the ALP signal in the fiducial region where the analysis is conducted.

7. Summary and conclusions

Axion-like particles appear in theories with a spontaneously broken global

U (1)

symmetry. ALPs provide the possibility to address some of the long-lasting SM problems, e.g. the dark matter problem, baryon asymmetry problem, neutrino mass problem. Motivated by such a potential, we examined the capability of the 14 TeV HL-LHC to explore the parameter space of an ALP with a mass of 1 MeV. Such an ALP predominantly decays into a pair of photons. However, due to the small mass and the strong experimental limits on the ALP-photon coupling, the ALP decay width into photons is so small that it is likely to be stable at the detector manifesting itself as missing energy. Hence, the ALP production process features a relatively large missing energy. The assumed production processes are the production of an ALP in association with a pair of electroweak gauge bosons, i.e.

p p \to Z Z a

and

p p \to W W a

. These processes are sensitive to the Wilson coefficients

c_{W W}

and

c_{G G}

, and are used to set expected 95% CL exclusion limits in the

c_{W W} - c_{G G}

plane. The integrated luminosities assumed to calculate the limits are 138 fb⁻¹ corresponding to the data collected at Run 2 of the LHC and 3

{ab}^{- 1}

, which is the ultimate integrated luminosity planned for the HL-LHC. The signal regions SR1 and SR2 which respectively correspond to the fully leptonic (

Z Z \to 4 ℓ

) and semi-leptonic (

Z Z \to 2 ℓ 2 j

) final states are considered in the ZZa analysis. The WWa search is focused on the fully leptonic (

W W \to 2 ℓ 2 ν

) final state. Signal and dominant SM backgrounds events are generated with a realistic simulation of the detector response. With the help of a proper set of discriminating variables, the ALP production processes are separated from the SM background. To achieve the best signal-background discrimination, a multivariate technique is deployed. We presented the expected two-dimensional exclusion limits obtained in these analyses. Comparing the expected limits obtained in this study with the limits derived from the LHC searches, namely the searches for the mono-W and mono-Z plus missing energy signals, shows that a significant region of the parameter space which has been unprobed becomes accessible with the help of the present study. The presented results correspond to the ALP mass of 1 MeV. However, it can be seen that for lower (sub-MeV) ALP masses, the limits would not change significantly. It can be concluded that the present analysis can serve as a tool for searching for the associated production of an ALP and a pair of electroweak gauge bosons at the HL-LHC, and can be used to probe the ALP couplings to W bosons, gluons to an unprecedented extent in certain regions of the parameter space.

In the final part of this paper, we have thoroughly investigated the sensitivity of ALP production in association with a massive vector boson and jets, V+jets+ALP, where

V = Z, W^{\pm}

. We focused on scenarios where the ALP does not decay within the detector, resulting in missing transverse energy, leading to final states that mimic the SM processes

W^{\pm}

+jets and Z+jets, but with distinct kinematic signatures due to the ALP presence. We utilized differential cross-section measurements from the CMS experiment for

W^{\pm}

+jets and Z+jets processes to constrain the ALP parameter space, considering both

c_{W W}

and

c_{G G}

couplings simultaneously. Our analysis demonstrates that fitting these differential distributions enables probing of ALP couplings down to

10^{- 3}

TeV⁻¹ at the High-Luminosity LHC (HL-LHC). The Z+jets+ALP channel exhibits superior sensitivity with respect to

W^{\pm}

+jets, primarily due to the more contribution of the ALP signal in the fiducial region of the analysis. In addition to the relatively larger ALP signal contribution in the fiducial region, the Z+jets+ALP channel benefits from several experimental advantages that contribute to its superior sensitivity. The CMS Z+jets measurement used in this analysis is based on 35.9 fb⁻¹ of data, compared to only 2.2 fb⁻¹ for the W+jets measurement. Moreover, being a more recent result, the Z+jets analysis benefits from improved detector understanding and reduced systematic uncertainties. The key observable in the Z+jets channel, the transverse momentum of the Z boson, is reconstructed from muon kinematics, which inherently have smaller experimental uncertainties than jet-based or MET-related observables. These factors collectively enhance the precision and exclusion power of the Z+jets+ALP analysis.

The two channels studied-

V V + a

and V+jets-offer complementary sensitivity to ALP couplings, with the V+jets analysis yielding stronger constraints due to its enhanced kinematic reach and larger production rates. Importantly, both analyses constrain the ALP parameter space simultaneously in the

(c_{W W}, c_{G G})

plane, and their combination could further tighten these bounds. The consistent interpretation across both channels highlights the robustness of the limits obtained and underscores the value of multi-channel approaches in probing ALP interactions at the LHC.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors would like to thank Gh. Haghighat for his valuable assistance in the preparation of portions of the text, as well as for the insightful discussions and constructive comments.

Data availability

Data will be made available on request.

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¹: https://github.com/delphes/delphes/blob/master/cards/delphes_card_CMS.tcl.

²: https://feynrules.irmp.ucl.ac.be/attachment/wiki/ALPsEFT/ALP_linear_UFO.tar.gz.

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Article Content

Abstract

1. Introduction

2. ALPs effective Lagrangian

3. ALP production in association with electroweak gauge bosons

4. Analysis

4.1. Event generation

4.2. Validity of the ALP effective Lagrangian

4.3. Object identification and event selection

4.4. Signal-background discrimination

5. Expected limits and comparison with existing bounds

6. Probe of the ALP couplings using the LHC measurements of W+jets and Z+jets

6.1. Z+jets+ALP process

6.2. $W^{\pm} +$ jets+ALP process

7. Summary and conclusions

Declaration of Competing Interest

Acknowledgements

Data availability

References

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