Probing the Gauge-boson Couplings of Axion-like Particle at the LHC and High-Luminosity LHC

In this work, we calculate the sensitivities on the gauge-boson couplings $g_{aZZ}$, $g_{aZ\gamma}$, and $g_{aWW}$ of an axion-like particle (ALP) that one can achieve at the LHC with $\sqrt{s}=14$ TeV and integrated luminosities of 300 fb$^{-1}$ (current run) and 3000 fb$^{-1}$ (High-Luminosity LHC). We focus on the associated production processes $pp\to Za \to (l^+l^-)(\gamma\gamma)$ and $pp\to W^\pm a \to (l^\pm \nu)(\gamma\gamma)$. We show that better sensitivities on these gauge couplings can be achieved at the LHC for $M_a = 1-100$ GeV, down to the level of $10^{-4}\,{\rm GeV}^{-1}$. In conclusion, this study emphasizes the significance of the investigated channels in constraining the ALP couplings at the LHC, offering valuable insights for future experiments dedicated to ALP detection.


I. INTRODUCTION
One of the long-standing problems in the standard model (SM) is the strong CP problem [1].It arises from the term θG µν Gµν in QCD, in which θ ∼ O (1).Such a term contributes substantially to the neutron electric-dipole moment (EDM) d n (θ) ≈ 2.4 × 10 −16 θ e • cm.
Nevertheless, non-observation of the neutron EDM gives an upper limit |d n | < 1.8 × 10 −26 e • cm [2], which requires θ < 10 −10 .Such a small coefficient in the Lagrangian is unnatural, which is coined as the strong CP problem.One of the best solutions is by introducing a global Peccei-Quinn (PQ) symmetry U (1) P Q symmetry [1], which was spontaneously broken by a dynamical axion field.The resulting pseudo-Nambu-Goldstone boson is known as the QCD axion [1,3,4].The neutron EDM constraint demands the breaking scale of the PQ symmetry to be very high with f a > 10 9 GeV, implying a tiny mass to the axion and very small couplings to the SM particles.
The axion can also serve as a dark matter candidate [5][6][7].If we do not require the pseudo-Nambu-Goldstone boson to be the solution of the strong CP problem, the mass of the axion is not restricted by the breaking scale f a .Such a hypothetical particle, called axion-like particle (ALP), is also a pseudoscalar boson.The ALP has a much wider range of mass and couplings that it can serve as the dark matter candidate.The axion mass and couplings to SM particles can extend over many orders of magnitude, which are constrained by astrophysical and cosmological observations, as well as collider experiments (for a comprehensive summary of constraints please see https://cajohare.github.io/AxionLimits/).
The ALP as a dark matter candidate is not the motivation of this work, unless the couplings of the ALP are extremely small such that the lifetime is longer than the age of the Universe.On the other hand, the ALP as a low-scale inflaton is another possibility that the ALP can decay [8].In some models [9][10][11][12][13][14], the axion or ALP can serve as the mediator to the dark-matter sector regardless of its mass or lifetime.In this work, we consider the potential sensitivities on the parameter space of the ALP model that one can achieve at the current LHC (L = 300 fb −1 ) and the future High-Luminosity LHC (L = 3000 fb −1 ).In this work, we focus on the gauge couplings g aZZ , g aW W , and g aZγ of the axion a.In principle, due to gauge invariance, it can also lead to the sensitivity on g aγγ .Instead, we obtain the sensitivities in a model-independent manner.
In this work, we focus on the associated production of the axion a with a Z or W boson, followed by the leptonic decay of the Z or W boson and the decay of a → γγ, i.e., pp → Za → (l + l − )(γγ) and pp → W ± a → (l ± ν)(γγ) , where l = e, µ.The associated production pp → Za can proceed via a Z or γ propagator, which can then probe g aZZ and g aZγ , respectively.On the other hand, the process pp → W ± a probes g aW W .We obtain the sensitivities on these gauge couplings for M a from 1 to 100 GeV.In the analysis, we found that when M a ≤ 25 GeV, the two photons from axion decay are quite close to each other, which form, what we called, a photon-jet.While for M a ≥ 25 GeV the two photons from axion decay are well separated.Therefore, we choose different selection procedures for low-mass M a and high-mass M a .More details are given in Sec.IV.
The organization is as follows.In the next section, we describe the relevant interactions of the ALP.In Sec.III, we summarize the existing constraints on g aZZ , g aZγ , and g aW W .In Sec.IV, we discuss in details the signal and background analysis.In Sec.V, we show our results.We conclude in Sec.VI.

II. THE MODEL
The axion, as a pseudo-Goldstone boson, has derivative couplings to fermions, as well as CP -odd couplings to the gauge field strengths.Before rotating the B and W i fields to the physical γ, Z, W ± , the interactions of the axion are given by following equations: [22-24] where where a represents the ALP field, f a is the ALP decay constant, A = 1, ....8 is the SU (3) color index and i = 1, 2, 3 is the SU (2) index.The term L f describes the fermionic couplings of the ALP.In principle, there can be non-trivial flavor structures in the ALP couplings such as g ij (∂µa) fa ( fi γ µ γ 5 f j ), where i, j denote the generation indices.Nevertheless, we set them all to zero in our study.The B, W 3 fields are rotated into γ, Z by where c w ,s w are cosine and sine of the Weinberg angle.The axion interactions with the fermions and the physical gauge bosons are given by The dimensionful couplings associated with ALP interactions from Eq. ( 7) are given by: Note that g aγγ , g aZZ , g aW W , and g aZγ are not independent if we assume the SU (2) symmetry relation as in Eq. (1).By choosing O(1) coefficients for C W W and C BB we can convert the existing constraints on g aγγ to the others.

III. EXISTING CONSTRAINTS ON ALPS
In this section, we discuss the existing constraints on the ALP-weak gauge boson couplings.Given that we focus on the mass range 1 GeV ≤ M a ≤ 100 GeV, we summarize the constraints for this ALP mass range.Various ALP-photon coupling limits have been established in various collider experiments [9,21,[25][26][27][28][29].These limits can be converted into g aZZ , g aZγ , and g aW W using Eqs.( 8)-( 11) by choosing O(1) coefficients for C W W and C BB .
The corresponding plots are presented in Fig. 7, labeled as "photons (various)".The label "photons (various)" includes the following: the bound established by the L3 collaboration, focusing on hadronic final states accompanied by a hard photon, was surpassed by exclusions from the LHC [25].The exclusion for the "Flavour" region was based on data from Babar and LHCb [30,31].At high axion masses near the TeV scale, the LHC bounds exceeded those from LEP due to enhanced axion production via gluon-gluon fusion.Strong limits are obtained from run 1 data [26,32].
There are other limits on the g aγγ in additional to those included in "photon (various)" in Fig. 7.Both CMS [33] and ATLAS [34], based on the γγ → a → γγ process in PbPb collisions, constrained g aγγ or 1/f a ≤ 0.1 − 1 TeV −1 for M a ∼ 10 − 100 GeV.On the other hand, the CMS and TOTEM [35], using diphoton production, excluded 1/f a > 0.03 − 1 TeV −1 for M a = 500 − 2000 GeV.Another search by ATLAS based on lightby-light scattering mediated by an ALP constrained 1/f a < 0.04 − 0.09 TeV −1 for M a = 150 − 1600 GeV [36].If we were to convert these constraints on g aγγ to g aZZ,aZγ based on gauge invariance, it would be similar to the shaded regions in Fig. 7. On the other hand, there are also constraints on the flavor-changing neutral current (FCNC) interactions of the ALP from the kaon and B-meson rare decays in invisible ALP and visible ALP channels [37,38].The invisible ALP channel can contribute to K → πν ν while the visible ALP channel can give rise to a displaced vertex in the decays such as These constraints are weaker than the shaded regions shown in Fig. 7.
A. Limits on g aZZ versus M a In Ref. [39], the process pp → triboson at the LHC was investigated to explore the limit on f a .By employing Eqs. ( 9) - (11), the constraint on g aZZ can be derived by selecting specific coupling coefficients C W W and C BB , labeled as "triboson (LHC)" in Fig. 7.The CMS Collaboration [40] used the ALP-mediated non-resonant ZZ pair production at the LHC and obtained a constraint on g aZZ of approximately 6.6 × 10 −4 GeV −1 .This set of constraints on g aZZ are shown in upper-left panel of Fig. 7.

B. Limits on g aZγ versus M a
The constraint on g aZγ was initially established in Sec.6.1 of Ref. [22] by considering the uncertainty of the total Z boson width, Γ(Z → BSM) < ∼ 2MeV at 95% confidence level.The derived limit was |g aZγ | < 1.8 TeV −1 for ALP masses below the Z boson mass.Another limit was presented in Ref. [25] using the e + e − → Z → γ + hadron channel, with improvements observed in the region 10 GeV < ∼ M a < M Z .The constraint on g aZγ can also be implicated from the triboson limit in Ref. [39] by selecting ALP production constrained the ALP-weak boson coupling, as demonstrated in Ref. [41], further improved the limit on g aZγ , pushing it down to approximately 4×10 −4 GeV −1 .They are all summarized in the upper-right panel of Fig. 7.

C. Limits on g aW W versus M a
Reference [39] also provided the limit on g aW W with the triboson channel, estimating the constraint on g aW W to be approximately 10 −2 GeV −1 in the large mass region.A more refined result on g aW W was presented in Ref. [41] to about 5 × 10 −4 GeV −1 in the mass range shown.They are all summarized in the lower panel of Fig. 7.

IV. EXPERIMENT AND SIMULATION
To simulate the signal events, we utilize the UFO file 1 [22], as detailed in the effective Lagrangian presented in Eq. (7).The generation of parton-level signal and background events is carried out using MadGraph5aMC@NLO [42] at the leading order.To ensure the accuracy of the simulations, specific cuts are applied during the parton-level event generation, as outlined in the run card.datcuts (we used the default cuts outlined in the run card.dat).A total of 10 4 signal events are generated at the center-of-mass energy √ s = 14 TeV.As for the background events, we generate 10 5 or 10 6 for different background channels to maintain the accuracy.More details of background events are discussed in Sec.IV A 2 and Sec.
IV B 2. The subsequent steps involve parton showering using Pythia8 [43] and detection simulations conducted with Delphes3 [44], incorporating the delphes card ATLAS.tcl for accuracy and consistency.The basic parton-level selection cuts for photons, charged leptons, and jets are We focus on the ALP that exclusively couples to electroweak gauge bosons with the mass range spanning from 1 GeV to 100 GeV.One of the prominent production channels for ALPs at the LHC is the ALP-strahlung process, specifically, pp → Za where l = e, µ, as illustrated in the left of Fig. 1. 2 Note that for M a > M Z the ALP can decay into Zγ.However, for the choice of C W W and C BB (g aZγ and g aγγ are of similar size) the branching ratio into a → Zγ is of order 10 −3 such that the branching ratio of a → γγ is practically 1.Also, for the parameter space of g aγγ and M a considered in this work, the decay of the ALP is prompt.

Signal Events
The interaction of the ALP with gauge bosons are listed in Eq. ( 7).The parameters chosen for this simulation are set to specific benchmark values: C BB = 1, and C g = g af = 0.It is noteworthy that the deliberate assignment of different values to C W W and C BB allows for the presence of the coupling constant g aZγ .Because of very similar topology in the q q → Z * , γ * → Za → (l + l − )(γγ), the event kinematic distributions for other values of C W W and C BB would be very much the same, thus the signal selection efficiency is independent of the choice of C W W and C BB .
The mass range for the ALP spans from 1 GeV to 100 GeV.The decay width of the ALP is set to "auto" in MadGraph5aMC@NLO, indicating that the decay width of the ALP is calculated based on the Lagrangian in Eq. (7).Specifically, the leptonic decay modes of electrons and muons are employed for the Z boson.Note that although the hadronic and neutrino decay modes could give a higher signal event rates, the detection is far more challenging and suffered from more background.On the other hand, for the ALP, under our specified benchmark parameter space, the dominant decay mode is a pair of photons.The corresponding Feynman diagram illustrating the ALP-strahlung process with the decay of the Z boson and the ALP, pp → Za (Z → l + l − ), (a → γγ), is depicted in the left panel of Fig. 1.The production cross section σ including the branching ratios versus the ALP mass M a is shown in Fig. 2. In Fig. 3, we show the 2-D plot of the cross sections as a function of (C W W , C BB ) for f a = 1 TeV and M a = 10 GeV, 100 GeV.We mark the cross point of the figure, which are our benchmark values.The cross section simply scales as 1/f 2 a and would be of similar pattern for other values of M a .To delve into the physics behind this simulation, the photon propagator is specifically employed to scrutinize the coupling constant g aZγ , while the Z propagator is utilized to investigate the coupling constant g aZZ .This detailed analysis helps in understanding the behavior and interactions of the ALP with the electroweak gauge bosons in the given parameter space.
In the detection simulation, we opt for the delphes card ATLAS.tcl and employ the jet angle parameter R = 0.4 for clustering jets using FastJet [45] with the anti − k T algorithm [46].To enhance our analytical capabilities, we also calculate Nsubjettiness [47] in FastJet.

Background analysis
The final state is l + l − γγ and thus the predominant backgrounds are (i) pp → l + l − γγ (referred to as llγγBG), (ii) pp → l + l − γj (referred to as llγjBG) with the jet faking a photon with a fake rate f j→γ ≃ 5 × 10 −4 [48], and (iii) pp → l + l − j (referred to as lljBG).Note that the j here is the parton-level jet produced by MadGraph5aMC@NLO including gluon and light quarks, followed by showering.In the detector simulation level, the jets are clustered by FastJet with anti-k T algorithm.The first background is irreducible while in the second background the jet can fake a photon with a fake rate f j→γ ∼ 5 × 10 −4 [48].In the third background, the jet can fake a photon-jet for the case of light M a .The importance of incorporating the lljBG background lies in the potential for the photon pair from a lowmass ALP to exhibit behavior akin to a jet when the angular separation is sufficiently small.
A cutoff is imposed at the ALP mass M a = 25 GeV, below which the first and the second backgrounds have to be taken into account in the signal-background analysis.Conversely, if the ALP mass is heavier than 25 GeV cutoff, the first and the third backgrounds is considered in the analysis.
Initial computations involve the evaluation of signal s and background event b rates at √ s = 14 TeV, as described by the following equation: Here, σ s and σ b denote the cross-sections of signal and background, respectively, including decay ratios.The ratio N selected N sim represents the selection efficiency, and L signifies the integrated luminosity.Figure 2 illustrates the cross sections of the signal process along with the corresponding backgrounds.The simulation involves the generation of N sim = 10 4 signal events, N sim = 10 5 for llγγBG and llγjBG background events.For lljBG background events, to maintain the accuracy, we generate 10 6 events for this background channel.Additionally, two distinct integrated luminosities are considered, namely 300 fb −1 (current run) and 3000 fb −1 (high-luminosity run).GeV.

M a > 25 GeV
To minimize background event rates in the analysis, we examine the kinematic properties between the signal and backgrounds to establish a set of useful selection cuts.Within the ALP mass region M a > 25 GeV, we implement the following event selection cuts: • two photon selection The final state consists of two isolated photons and two charged leptons; therefore, it is imperative to apply the first two aforementioned cuts during the selection procedures.C BB = 1, and C g = g af = 0. "Before cuts" in the first row denotes the total number of events with only the parton-level cuts with the integrated luminosity of 300 fb −1 calculated by Eq. ( 12).
In llγjBG, we have applied the jet-fake rate from the left panel of Fig. 4 that a significant portion of the background (BG) arises from the Z boson.Additionally, the right panel of Fig. 4 shows the significance of applying p Tγγ > 80 GeV cut to mitigate the background events.The final cut is fixed by the ALP mass window.For the upper and lower bounds of the mass, we opt for a selection of 10% on each side.The cut flow table for the ALP mass M a = 100 GeV is presented in Table I.
In Table .I, the first two cuts are essential for achieving the detection of the photon and the charged lepton pairs.The selection cuts of M ll and p Tγγ retain over 77% of signal events and reduce background events to only 1.65%.Finally, the ALP mass-window cut further reduces the background by a factor of 7.

M a < 25 GeV
In the low ALP mass region, where M a ≤ 25 GeV, we implement the following cuts: • At least one jet • min( E had E EM ) < 0.02 As mentioned in Sec.IV A 2, the two photons decaying from a lower mass ALP form a photon-jet.Therefore, the first two selections aim at identifying the candidates for this type of ALP jets.Considering potential noise in hadron colliders, we accept multi-jet events.The candidate ALP jet should predominantly consists of photons, and thus the majority of its energy should be detected in the electromagnetic (EM) calorimeter.However, since there may be multiple jets in the event, a quantity, E had E EM , is defined for each jet.We choose to select jets with the lowest E had E EM that is less than 0.02.The third and fourth selection cuts target at the Z boson.
The quantity τ N represents the Nsubjettiness [47,49], which characterizes the substructure of a jet, defined by where where p T,k is the transverse momentum of the k th constituent particle, ∆η and ∆ϕ are the pseudorapidity and azimuthal angle between j th candidate subjet and k th constituent particle, respectively, and R 0 is the characteristic jet radius, which we set to be R 0 = 0.4 in FastJet.One identifies N candidate subjets, after which k iterates over all the other constituent particles in the jet to calculate this quantity.A jet with τ N = 0 indicates that there are only N or fewer constituent subjets in the jet, while a jet with τ N > 0 indicates the presence of additional radiation outside the N candidate subjets.The ratio τ N τ N −1 is a useful parameter to determine if a jet is composed of N substructures.The ideal ALP jet, consisting of two photons without any other initial radiation, will result in τ 2 being zero but with a non-zero τ 1 .The final selection involves the jet mass.We focus on the ALP jet with the lowest E EM E had as selected in the second cut.Instead of employing a symmetric mass window, the upper bound used is twice of the lower bound.The asymmetric mass window is motivated by the acceptance of minimal initial radiation, which may contribute additional mass to the jet.The selection of M jet for various ALP mass window is listed in Table II.
Taking M a = 10 GeV as an example, Table III  and (pp → llj) with M a = 10 GeV, with f a = 1 TeV, C W W = 2, C BB = 1, and C g = g af = 0.
"Before cuts" in the first row denotes the total number of events with only parton-level cuts with the integrated luminosity of 300 fb −1 calculated by Eq. (12).
jets in favor of ALP-induced jets.Additionally, the Nsubjettiness ratio cut effectively filters out jets resulting from the decay of light mesons.As a result, after applying these cuts, no background events pass the ALP mass selection.and C g = g af = 0, and the ALP decay width is set to "auto" in the parm card.dat of MadGraph5aMC@NLO, and its mass range is from M a = 1 − 100GeV.The production cross section of the signal including the branching ratios is showed in Fig. 5.

Background analysis
The final state consists of a charged lepton and missing energy from W decay and a pair of photons from the ALP decay.We consider two major backgrounds: (i) pp → l ± ν l γγ (referred to as lν l γγBG), (ii) pp → l ± ν l γj (referred to as lν l γjBG) with the jet faking a photon with a fake rate f j→γ ≃ 5 × 10 −4 , and (iii) pp → l ± ν l j (referred to as lν l jBG ).Similar to the Za process, we use a cutoff of 25 GeV, such that different background consideration is applied to the case M a < 25 GeV and the case M a > 25 GeV.

M a > 25 GeV
To differentiate between the signal and background events, we apply the following cuts for the ALP mass range of 25 GeV < M a ≤ 100 GeV.
• two photon selection • one lepton selection The first two cuts in Table IV identify one charged lepton and two photons in the final state.Furthermore, as illustrated in Fig. 6, p Tγγ is a useful kinematical variable to suppress the background.We choose p Tγγ > 50 GeV .Since there is missing energy in the final state due to the leptonic decay of the W boson, we utilize the transverse mass M T = (E T,l + E T,mis ) 2 − (⃗ p T,l + ⃗ p T,mis ) 2 to align with the Jacobian peak of the W boson decay.We choose M T > 58 GeV to differentiate from the background (as seen in Fig 6).
The mass window of the final selection cut depends on the ALP mass.Given that the average peak of M γγ typically aligns with M a and the peak width diminishes as the ALP mass decreases, we opt for ±10% of M a as the upper and lower limits for the mass window.
M a < 25 GeV In the low ALP mass region, where M a ≤ 25 GeV, we implement the following cuts: • At least one jet with M a = 100 GeV, with couplings f a = 1 TeV, C W W = 2, C BB = 1, and C g = g af = 0. "Before cuts" in the first row denotes the total number of events with only the parton-level cuts computed using Eq. ( 12), with the signal and background cross sections given in Fig. 5  The decay of low-mass ALPs into two photons results in the formation of photon-jets.
Thus, we have changed the selection criteria from two photons to selecting at least one jet.
To ensure that the selection includes jets composed of two photons, we select the jet with the smallest E had E EM value, which must be under 0.02.The third and fourth selection cuts are similar to the case of M a > 25 GeV to identify the decay of the W boson.The fifth selection cuts on the ratio of Nsubjettiness effectively minimizes the impact of the jet background.M a = 10 GeV, featuring couplings f a = 1 TeV, C W W = 2, C BB = 1, and C g = g af = 0. "Before cuts" in the first row denotes the total number of events with only the parton-level cuts computed using Eq. ( 12), with the signal and background cross sections given in Fig. 5 and the luminosity set at L = 300 fb −1 .
We employ an asymmetric mass window for the M jet cut, the same as the case of Za.The cut flow table for M a = 10 GeV is presented in Table V.

V. NUMERICAL RESULTS
In this section, we are going to derive the sensitivity reach on the ALP-gauge couplings g aZZ , g aZγ , and g aW W using the processes pp → Za → (l + l − )(γγ) and pp → W a → (lν)(γγ) at the 14 TeV LHC with integrated luminosities of 300 fb −1 and 3000 fb −1 .In the last section, we have illustrated the signal events rates for a choice of f a = 1 TeV, C W W = 2, and C BB = 1 (C g = C af = 0).We use a simple scaling to estimate the sensitivity reach.
The correlation between the number of events and the ALP-gauge couplings is expressed through Eqs. ( 7) - (11): The 95% confidence level (C.L.) sensitivity for the ALP-gauge couplings can be determined by requiring the significance Z > 2, defined by [50,51]: Bottom Left: g aW W resulting from pp → W a(W → l ν l )(a → γγ).The other existing limits are described in Sec.III.
where s and b represent the numbers of signal and background events, respectively.Additionally, σ b denotes the systematic uncertainty associated with the SM background estimation.
We consider two scenarios for σ b = 0% and 10% of background events.For the process in which all background events are excluded, as illustrated in Table VI and VII of appendix A, the 95% C.L. is estimated by requiring 3 signal events.and g aZγ obtained from the pp → Za process, while the bottom panel represents the limit on g aW W derived from the pp → W a process.The shaded regions are those excluded by the current constraints.Our results are denoted by the blue curve for the 300 fb −1 integrated luminosity and by the orange curve for the 3000 fb −1 .
In the high-mass region, 25 GeV < M a < 100 GeV, our sensitivity curves demonstrate an improvement of approximately one order of magnitude compared to the current limits.
Despite that the signal process has a larger cross section than the background processes, the enhancement is not only attributed to this factor, but also the selection criteria, which also play a pivotal role.We have focused on the diphoton decay mode of the ALP, and notably, there are no SM particles decaying into diphotons within the mass range of 25 GeV to 100 GeV.
In the low-mass region, 1 GeV < M a < 25 GeV, even with an additional background process with substantial cross-sections, our results consistently exhibit an improvement of one to two orders of magnitude compared to the current limits, except for the case of M a = 1 GeV.In handling the features of jet substructure, we have adopted a more stringent approach, thus resulting in the exclusion of jet backgrounds.The sudden degradation in the limit at M a = 1 GeV can be attributed to the diphoton decay mode of the neutral pion, where the reconstructed mass aligns with our selection range.
For completeness, we show in Fig. 8 GeV −1 for a luminosity of L = 3000 fb −1 .This is attributed to the scarcity of background events in the region 10 GeV < M a < 25 GeV due to stringent selection criteria.Nevertheless, the signal cross sections, as depicted in Figs. 2 and 5, diminish with an increase in the ALP mass.

VI. CONCLUSIONS
In this study, we have investigated the sensitivity potential of the current run and the future High-Luminosity run of the LHC (L = 300 fb −1 and 3000 fb −1 ) at the center-of-mass energy of √ s = 14 TeV, focusing on probing the dimensionful coupling constants g aZZ , g aZγ , and g aW W associated with the axion-like particle.This exploration was conducted through the processes pp → Za(Z → l + l − )(a → γγ) and pp → W a(W ± → l ± ν l )(a → γγ).Our results demonstrated that these channels provide the most stringent bounds for the couplings g aZZ , g aZγ , and g aW W .
To maintain generality, we adopted the values C W W = 2 and C BB = 1, ensuring that the couplings g aZZ , g aZγ , and g aW W are interrelated as shown in Eqs. ( 8)-( 11) with all of them being non-zero.The analysis can be readily extended to explore independent coupling strengths.
In conclusion, our study has culminated in the presentation of a summary plot (Fig. 7) illustrating the sensitivity of g aZZ , g aZγ , and g aW W achievable at the LHC, which are compared with existing constraints.Our estimations of the bounds on the axion-like particle gauge boson couplings versus the ALP mass from M a = 1 GeV to 100 GeV, and provide valuable insights for future experiments dedicated to the detection of ALPs.

ACKNOWLEDGMENT
The work was supported in part by NSTC under the grant number MOST-110-2112-M-007-017-MY3.
Appendix A: Event Rates In this appendix, we list the total number of signal and background events for various ALP masses from M a = 1 GeV to 100 GeV that before and after all the cuts mentioned in Sec.IV.We show in Total number of signal events for pp → Za, followed by Z → l + l − , a → γγ and background events of llγγBG and lljBG for the mass range M a = 1 − 100 GeV.The number of events are calculated by Eq. ( 12), where the cross sections of signal and backgrounds are shown in Fig. 2 and the integrated luminosity is set at L = 300 fb −1 .
M a (GeV) Total number of signal events for pp → W ± a, followed by W ± → l ± ν l , a → γγ and background events of lν l γγBG and lν l jBG for the mass range M a = 1 − 100 GeV.The number of events are calculated by Eq. ( 12), where the cross sections of signal and backgrounds are shown in Fig. 5 and the integrated luminosity is set at L = 300 fb −1 .

FIG. 2 .
FIG. 2. Production cross section for pp → Za (Z → l + l − )(a → γγ) with l = e, µ versus M a , including the branching ratios at √ s = 14 TeV.Here we also show the backgrounds llγγBG and lljBG.

Furthermore, the third
selection cut aims to identify an outgoing Z boson with a mass of M Z = 91.1876GeV.The distributions of the invariant mass M ll of charged lepton pair and the transverse momentum p Tγγ of the photon pair are depicted in Fig. 4. It is apparent Cut flow for the signal (pp → Za) with Z → l + l − and a → γγ and the background process (pp → l + l − γ γ) and (pp → l + l − γj) with M a = 100 GeV, with f a = 1 TeV, C W W = 2,

B
. pp → W a with W → lν l and a → γγ Another useful production channel is W a production followed by the leptonic decay of the W boson and a → γγ: pp → W ± a (W ± → l ± ν l ) , (a → γγ), as illustrated in the right panel of Figure.1 Similar to the case of pp → Za, we set the parameters f a = 1 TeV, C W W = 2, C BB = 1,

Figure 7
Figure 7 displays our final result for the limits on the ALP-gauge couplings alongside with several existing constraints.The upper-left and upper-right panels depict the limits on g aZZ the sensitivity region of 95% C.L. in the plane of (C W W , C BB ) with f a = 1 TeV for M a = 10 GeV (left panel) and M a = 100 GeV (right panel).The choice of M a = 10, 100 GeV corresponds to the two ranges of M a in our analysis with the corresponding cross-section shown in Fig. 3 .Since we have employed distinct signal selection criteria to address both heavier and lighter ALPs, a gap arises around the cutoff, M a = 25 GeV.The best sensitivity is observed around M a = 10 GeV, where g aZZ ≈ 2 × 10 −5 GeV −1 , g aZγ ≈ 8 × 10 −6 GeV −1 , and

TABLE II .
presents the cut flow.One of the key cuts in TableIIIis the second one, which significantly reduces the contribution from hadronization M jet mass window for the low ALP mass region

TABLE III
. Cut flow for the signal process (pp → Za) and background processes (pp → llγγ)

TABLE IV .
Cut flow for the signal pp → W ± a and the background (pp → lν l γγ) and (pp → lν l γj)

TABLE V .
Cut flow for the signal pp → W ± a and backgrounds pp → lν l γγ and pp → lν l j with

TABLE VI .
Table VI for Za channel and in Table VII for W a channel.