# Exploring interference effects between two ALP effective operators at the
LHC

Kingman Cheung<EMAIL_ADDRESS>Department of Physics, National Tsing
Hua University, Hsinchu 30013, Taiwan  Center for Theory and Computation,
National Tsing Hua University, Hsinchu 30013, Taiwan  Division of Quantum
Phases and Devices, School of Physics, Konkuk University, Seoul 143-701,
Republic of Korea  Chih-Ting Lu<EMAIL_ADDRESS>Department of Physics and
Institute of Theoretical Physics, Nanjing Normal University, Nanjing, 210023,
P. R. China  C.J. Ouseph<EMAIL_ADDRESS>Department of Physics, National
Tsing Hua University, Hsinchu 30013, Taiwan  Center for Theory and
Computation, National Tsing Hua University, Hsinchu 30013, Taiwan  Priyanka
Sarmah<EMAIL_ADDRESS>Department of Physics, National Tsing
Hua University, Hsinchu 30013, Taiwan  Center for Theory and Computation,
National Tsing Hua University, Hsinchu 30013, Taiwan

###### Abstract

We observe that most studies of axion-like particle (ALP) production channels
at the Large Hadron Collider (LHC) focus on a single type of ALP operator for
each process in the effective field theory framework. In this work, we propose
an alternative approach that considers two or more types of relevant ALP
effective operators together in some specific ALP production channels and
study their interference effects. Using the $pp\rightarrow tja$ process with
$a\rightarrow\gamma\gamma$ as an example, we show that this approach allows us
to constrain the ALP interactions with both the $W$ boson and the top quark,
as well as their interference in a single process. For the final state with
two isolated photons and a top quark decaying semi-leptonically, we predict
that the future bounds on the ALP decay constant can reach around $f_{a}\sim
10\;(20)$ TeV for $25$ GeV $<M_{a}<100$ GeV at the LHC with 300 (3000) fb1
luminosity.

##  I Introduction

Axion-like particles (ALPs) are hypothetical particles that arise in several
extensions of the Standard Model (SM) of particle physics as pseudo-Goldstone
bosons with a shift symmetry Preskill  _et al._ (1983); Abbott and Sikivie
(1983); Dine and Fischler (1983); Bagger  _et al._ (1994); Svrcek and Witten
(2006); Arvanitaki  _et al._ (2010); Cicoli  _et al._ (2012); Visinelli and
Vagnozzi (2019); Chang  _et al._ (2000); Bastero-Gil  _et al._ (2021). They
are characterized by a low mass and a weak coupling to ordinary matter, which
makes them difficult to detect. However, their existence is predicted by
various theories that attempt to solve some of the outstanding problems in
physics, such as the strong CP problem Peccei and Quinn (1977a, b); Weinberg
(1978); Wilczek (1978); Kim (1979); Shifman  _et al._ (1980); Dine  _et al._
(1981), the nature of dark matter Duffy and van Bibber (2009); Chadha-Day  _et
al._ (2022) and experimental anomalies Chang  _et al._ (2021); Yuan  _et al._
(2022); Liu  _et al._ (2023); Cheung and Ouseph (2024).

In the effective field theory (EFT) framework Brivio  _et al._ (2017a),
possible interaction operators of ALP(s) and SM particles start at dimension-
five Georgi  _et al._ (1986) and continue to higher order ones Bauer  _et al._
(2019). We can explore the effect of each term separately or some of them
collectively. On the other hand, the ALP mass range in the general EFT extends
from almost massless to the electroweak scale or above. Therefore, there are
various search strategies for different ALP interactions and masses including
laboratory-based Sikivie (1983), beam-dump Riordan  _et al._ (1987); Döbrich
_et al._ (2019), Higgs factories Bauer  _et al._ (2017); Cheung and Ouseph
(2023), and collider Bauer  _et al._ (2017, 2019) experiments, as well as
cosmological and astrophysical observations Anastassopoulos  _et al._ (2017)
(for a recent summary please see Ref. O’Hare (2024)).

Among various kinds of ALP searches, the collider experiments are sensitive to
probe the GeV to TeV scale ALPs. Although there were already a number of
studies to explore the properties of ALPs at the Large Hadron Collider(LHC)
Jaeckel and Spannowsky (2016); Brivio  _et al._ (2017b); Bauer  _et al._
(2017); Ebadi  _et al._ (2019); d’Enterria (2021); Ren  _et al._ (2021);
Cheung  _et al._ (2024) and other future colliders Bauer  _et al._ (2019); Bao
_et al._ (2022); Zhang  _et al._ (2021); Liu and Yan (2023); Han  _et al._
(2022); Lu (2022); Lu  _et al._ (2023); Cheung  _et al._ (2024), we find that
most of the previous studies of ALP production channels at the LHC only focus
on a single ALP effective operator in each process. However, people often
overlook the potential for interference effects among different ALP operators.
An interesting example is the associated production of the Higgs boson with a
single top quark Barger  _et al._ (2010); Biswas  _et al._ (2013a, b); Farina
_et al._ (2013); Agrawal  _et al._ (2013); Ellis  _et al._ (2014); Englert and
Re (2014); Chang  _et al._ (2014), in which the $HWW$ coupling interferes non-
trivially with the top-Yukawa coupling and experimentally it can be
disentangled. With the same spirit, we propose an alternative approach that
takes into account two or more relevant ALP effective operators simultaneously
in a single production process and explore the interference effects. In
particular, we study the ALP-$W^{+}W^{-}$ and the ALP-$t\overline{t}$
couplings in the process $pp\to tja$, where $j$ denotes a forward hadronic
jet, at the LHC Ebadi  _et al._ (2019).

A similar process $pp\rightarrow j\gamma a$ was considered in Ref. Ebadi  _et
al._ (2019), in which the ALP-quark pair, ALP-gluon pair, ALP-$ZZ$, and
ALP-$Z\gamma$ are involved. In this work, we focus on the process
$pp\rightarrow tja$, in which only the ALP-$W^{+}W^{-}$ and
ALP-$t\overline{t}$ couplings are involved. Although this process is similar
to the associated production of the Higgs boson with a single top quark, the
relevant ALP coupling types are different from the Higgs boson ones.
Therefore, the process $pp\rightarrow tja$ can generate quite distinct
predictions. Our proposed approach allows us to explore novel ALP production
processes that involve multiple ALP operators and investigate their
interference effects.

The organization is as follows. In the next section, we describe the ALP
interactions relevant to this study in the EFT framework. In Sec. III, we
explore the interference effects between the ALP-$W^{+}W^{-}$ and
ALP-$t\overline{t}$ couplings. In Sec. IV, we describe the experimental setup
for discriminating the signal from the related SM backgrounds. We give the
numerical results and sensitivity reach of the ALP couplings in Sec. V.
Finally, we conclude in Sec. VI.

##  II Theoretical setup

In this study, the relevant ALP operators include the ALP-gauge boson pair and
the ALP-top quark pair couplings, which start at dimension-five. The $CP$-odd
couplings of the ALP to the electroweak gauge boson fields are given by

$\mathcal{L}_{\text{EW}}\supset-\frac{a}{f_{a}}\left(C_{WW}W_{\mu\nu}^{i}\tilde{W}^{i\mu\nu}+C_{BB}B_{\mu\nu}\tilde{B}^{\mu\nu}\right),$
(1)

where $i=1,2,3$ represents the $SU(2)$ index, and $\tilde{W}^{i\mu\nu}$ and
$\tilde{B}^{\mu\nu}$ are the dual field strength tensors. Here the ALP field
and its decay constant are represented by $a$ and $f_{a}$, respectively. After
transforming $W^{i}$ and $B$ to the physical fields $\gamma,Z,W^{\pm}$, the
interactions in Eq. (1) can be written as

$\mathcal{L}_{\text{EW}}\supset-\frac{1}{4}a\left(g_{a\gamma\gamma}F_{\mu\nu}\tilde{F}^{\mu\nu}+g_{a\gamma
Z}F_{\mu\nu}\tilde{Z}^{\mu\nu}+g_{aZZ}Z_{\mu\nu}\tilde{Z}^{\mu\nu}+g_{aWW}W_{\mu\nu}\tilde{W}^{\mu\nu}\right)\,,$
(2)

where $F_{\mu\nu}$, $W_{\mu\nu}$, and $Z_{\mu\nu}$ are the field strength
tensors of the photon, $W^{\pm}$, and $Z$ bosons, respectively. Thus, the
dimensionful couplings of the photon and the electroweak gauge bosons to the
ALP can be written in terms of $C_{WW}$ and $C_{BB}$ Brivio  _et al._ (2017b);
Georgi  _et al._ (1986); Ren  _et al._ (2021); Cheung and Ouseph (2023); Lu
_et al._ (2023),

$g_{a\gamma\gamma}=\frac{4}{f_{a}}(C_{BB}c_{w}^{2}+C_{WW}s_{w}^{2}),$ (3)
$g_{aWW}=\frac{4}{f_{a}}C_{WW},$ (4)
$g_{aZZ}=\frac{4}{f_{a}}(C_{BB}s_{w}^{2}+C_{WW}c_{w}^{2}),$ (5)
$g_{aZ\gamma}=\frac{8}{f_{a}}s_{w}c_{w}(C_{WW}-C_{BB})\,,$ (6)

where $c_{w}$ and $s_{w}$ are cosine and sine of the Weinberg angle that is
related to the rotation between the electroweak fields and the physical fields
as in

$\displaystyle W_{\mu}^{3}=c_{w}Z_{\mu}+s_{w}A_{\mu},\quad
B_{\mu}=-s_{w}Z_{\mu}+c_{w}A_{\mu}\,.$

On the other hand, the ALP-top quark pair interaction is given by Esser  _et
al._ (2023)

$\mathcal{L}_{at\overline{t}}=C_{a\phi}\frac{\partial_{\mu}{a}}{2f_{a}}(\bar{t}\gamma^{\mu}\gamma^{5}t).$
(7)

After applying the equation of motion, the above Lagrangian, Eq. (7), can be
written as

$\mathcal{L}_{at\overline{t}}=-iC_{a\phi}\frac{m_{t}a}{f_{a}}(\bar{t}\gamma^{5}t).$
(8)

where $m_{t}$ is the top quark mass. As we can see, the ALP-quark pair
coupling is proportional to the mass of the quark. Therefore, for the similar
size of $C_{a\phi}/f_{a}$, the $at\overline{t}$ coupling can provide stronger
interaction than other $aq\overline{q}$ couplings. Equipped ourselves with
these theoretical setups, we are now ready to discuss the interference effects
between the $aW^{+}W^{-}$ and the $at\overline{t}$ couplings in the process
$pp\to tja$.

##  III Production and Interference effects in $pp\to tja$

Figure 1:  Two key contributing Feynman diagrams for the process $pp\to tja$
at the LHC.  Figure 2: Production cross sections for the signal process $pp\to
j~{}t~{}a$ with $a\to\gamma\gamma$ and $t\to bW,~{}W\to l\nu_{l}$ at the LHC
($\sqrt{s}=14~{}\rm TeV$) for $M_{a}=25,50,100$ GeV. We fix the $aW^{+}W^{-}$
coupling by setting $C_{WW}=C_{BB}=1$ and $f_{a}=10$ TeV. The $at\overline{t}$
coupling $C_{a\phi}$ varies from $-10$ to $+10$.  Figure 3:  Two key
contributing Feynman diagrams for the process $pp\to tjba$ at the LHC.  Figure
4:  Two key contributing Feynman diagrams for the process $pp\to tWa$ at the
LHC.  Figure 5:  Two key contributing Feynman diagrams for the process $pp\to
tba$ at the LHC.

In this section, our focus is on investigating the interference effects
between the $aW^{+}W^{-}$ and the $at\overline{t}$ operators in the process
$pp\to tja$ at the LHC with $\sqrt{s}=14$ TeV. Two key contributing Feynman
diagrams for $pp\to tja$ are shown in Fig. 1. The ALP can bremsstrahlung off a
$W$ propagator and also off a top-quark leg 111Here we do not involve the
Feynman diagrams in which the ALP attached to the $b$ quark or light quarks
since their contributions are much smaller than that from the $at\overline{t}$
coupling with the same $C_{a\phi}/f_{a}$ value. Thus, these two sets of
diagrams can interfere.

We apply MadGraph5_aMC@NLO Alwall  _et al._ (2011) with the $5$-flavor scheme
($u,d,s,c,b$) to calculate the production cross sections for the process
$pp\to tja$ fixing the parameters: $C_{WW}=C_{BB}=1$ and $f_{a}=10$ TeV, and
vary the $at\overline{t}$ coupling labeled by $C_{a\phi}$ from $-10$ to $10$.
We show in Fig. 2 the production cross sections for $M_{a}=25,50,100$ GeV at
the LHC, including the branching ratios for $a\to\gamma\gamma$ and $t\to
bW,~{}W\to l\nu_{l}$ ($l=e,\mu$). The basic cuts for the final state particles
are shown in Appendix A.

It is not difficult to see the interference effects when we look at the cross-
section curves at both ends ($-10$ and $+10$), although the effects are
moderate at only about 10% difference. Moreover, we have observed that the
contribution from the $at\overline{t}$ interaction with $C_{a\phi}\sim 1$ is
smaller than that from the $aW^{+}W^{-}$ interaction with $C_{WW}=C_{BB}=1$ in
this process. Meanwhile, the constraints for the $aW^{+}W^{-}$ coupling are
much stronger than that for the $at\overline{t}$ coupling as shown in Refs.
Alonso-Álvarez  _et al._ (2019); d’Enterria (2021); Esser  _et al._ (2023).

Figure 6: Production cross sections for the signal processes $pp\to
t~{}j~{}a$, $pp\to t~{}j~{}b~{}a$, $pp\to t~{}W~{}a$, and $pp\to t~{}b~{}a$ at
the LHC ($\sqrt{s}=14~{}\mathrm{TeV}$) for $M_{a}=50~{}\mathrm{GeV}$. We fix
the ALP-gauge boson pair coupling by setting $C_{WW}=C_{BB}=1$ and
$f_{a}=10~{}\mathrm{TeV}$. The $at\overline{t}$ coupling $C_{a\phi}$ varies
from $-10$ to $+10$.

Before ending this section, we would like to discuss some other associated ALP
production with a single top quark processes. The first one is $pp\to
t~{}j~{}b~{}a$ (Fig. 3), which can be regarded as a higher-order correction
from $pp\to t~{}j~{}a$ when the $b$-quark is not tagged in the final state. To
identify this process from $pp\to t~{}j~{}a$ and avoid the collinear
divergence, the following cuts are applied to the $b$ and $j$ in the final
state:

$P_{T_{b}}>25~{}\text{GeV},\quad|\eta_{b}|<2.5,\quad
P_{T_{j}}>10~{}\text{GeV},\quad|\eta_{j}|<5.$ (9)

The second and the third ones are $pp\to t~{}W~{}a$ (Fig. 4) and $pp\to
t~{}b~{}a$ (Fig. 5) processes, respectively. In order to fairly compare the
production cross-sections of these processes, we do not impose any cuts for
them here, except for $pp\to t~{}j~{}b~{}a$ with the cuts in Eq. (9) to avoid
the double-counting. The production cross-sections for the processes $pp\to
t~{}j~{}a$, $pp\to t~{}j~{}b~{}a$, $pp\to t~{}W~{}a$, and $pp\to t~{}b~{}a$ at
the LHC ($\sqrt{s}=14~{}\mathrm{TeV}$) for $M_{a}=50~{}\mathrm{GeV}$ are shown
in Fig. 6. We fix the ALP-gauge boson pair coupling by setting
$C_{WW}=C_{BB}=1$ and $f_{a}=10~{}\mathrm{TeV}$ and vary the $at\overline{t}$
coupling $C_{a\phi}$ from $-10$ to $+10$. Firstly, the shape of $pp\to
t~{}j~{}b~{}a$ is similar to $pp\to t~{}j~{}a$, but the cross-section is
smaller as we expect it to be a higher-order correction. Secondly, $pp\to
t~{}W~{}a$ is also a promising process which can show obvious interference
effects. However, its cross-section is less sensitive to the variation of
$C_{a\phi}$ than the one from $pp\to t~{}j~{}a$ and the decay modes of $W$
should be taken into account. Finally, $pp\to t~{}b~{}a$ displays sizable
interference effects as well, but its cross-section is much smaller than the
other three processes. Therefore, we will stick with the process $pp\to
t~{}j~{}a$ for the analysis in this study.

##  IV Experimental setup and Simulations

In this section, we describe the calculation and experimental setup for
discriminating the signal from various SM backgrounds. We show the event rates
for the center-of-mass energy $\sqrt{s}=14$ TeV and integrated luminosities of
$300~{}\rm fb^{-1}$ (current run) and $3000~{}\rm fb^{-1}$ (High-Luminosity
LHC)Apo (2015).

###  IV.1 Signal and relevant SM background processes

The Monte Carlo simulations of signal and relevant SM background events are
calculated utilizing MadGraph5_aMC@NLO. The UFO model file of the ALP EFT
framework (Eqs. (2) and (8)) is employed for the signal event simulation
Brivio  _et al._ (2017a)222This UFO model file is publicly accessible for
download at https://feynrules.irmp.ucl.ac.be/wiki/ALPsEFT. The setting in
run_card.dat file of MadGraph5_aMC@NLO can be found in Appendix A. In our
simulation, $10^{4}$ events are generated for the signal process and $10^{5}$
events for each SM background process. We perform the signal-to-background
analysis at parton level and use momentum smearing to approximate the detector
effects 333We adopt the energy smearing of the jets given by $\Delta
E/E=\delta_{1}/\sqrt{E}\oplus\delta_{2}$, where $\oplus$ means adding in
quadrature. Here we set $\delta_{1}=0.2$ and $\delta_{2}=0.01$ for hadronic
jets Aad  _et al._ (2021). . The output LHE files Alwall  _et al._ (2007) from
MadGraph5_aMC@NLO are passed to the Python-based tool Pylhe Heinrich  _et al._
for further analysis.

In order to investigate the final-state signature of two isolated photons, we
focus on the ALP within a mass range spanning from $25~{}\rm GeV$ to
$100~{}\rm GeV$444Two photons from the ALP decay will become too collimated to
pass the isolation criteria when $M_{a}\lesssim 20$ GeV. In this situation, a
photon-jet forms in the final state Ren  _et al._ (2021); Lu (2022); Cheung
_et al._ (2024) instead of two isolated photons.. In this simulation, some
specific benchmark values are assigned to the model parameters:
$f_{a}=10~{}\rm TeV$, $C_{WW}=C_{BB}=1$, and a scan of $C_{a\phi}$ from $-10$
to $10$. A nonzero $C_{a\phi}$ together with $C_{WW}=1$ initiates ALP
production from top bremsstralung, alongside with ALP production from
$W$-boson fusion. In contrast, the choice of $C_{a\phi}=0$ prohibits the top
bremsstralung into ALP. The variation $C_{a\phi}$ offers additional insights
into the interference effects among these ALP operators. We have already shown
the interference effects in Fig. 2 in the last section.

The signal final state consists of the decay of the ALP and the top quark, as
well as a hadronic jet. The dominant decay mode of the ALP is into a pair of
isolated photons for the ALP mass range from $25-100$ GeV for the setting
$C_{WW}=C_{BB}=1$. We choose the semi-leptonic decay of the top quark: $t\to
Wb,~{}\;W\to l\nu_{l}$ in this study. For such a final state, we consider two
main SM backgrounds: (i) $p~{}p\to t~{}j~{}\gamma~{}\gamma$ (labeled as BG1)
and (ii) $p~{}p\to W~{}j~{}j~{}\gamma~{}\gamma$ (labeled as BG2). Note that
BG1 emerges as the predominant background in comparison to BG2. In order to
estimate the sensitivity reach of the ALP coupling, we evaluate the total
number of signal and background events at the LHC with a center-of-mass energy
of $\sqrt{s}=14~{}\rm TeV$. The total number of events is defined as:

$N_{s},N_{b}=\sigma_{s,b}\times\frac{N_{\text{selected}}}{N_{\text{sim}}}\times\mathcal{L}\times\eta_{b-\text{tag}}\;,$
(10)

where $\sigma_{b}$ and $\sigma_{s}$ denote the cross-sections of background
and signal events, respectively. The ratio
$\frac{N_{\text{selected}}}{N_{\text{sim}}}$ represents the selection rate,
and $\mathcal{L}$ is the integrated luminosity. Here, $\eta_{b-\text{tag}}$
represents the $b$-quark tagging efficiency. For the signal ($p~{}p\to
j~{}t~{}a$) and BG1 ($p~{}p\to t~{}j~{}\gamma~{}\gamma$), the $b$-quark
tagging efficiency $\eta_{b-\text{tag}}$ is set to $0.75$, while the $b$-quark
mistag probability for BG2 ($p~{}p\to W~{}j~{}j~{}\gamma~{}\gamma$) is set at
$0.01$ Aaboud  _et al._ (2018).

###  IV.2 Event Selections

In an effort to reduce these two main SM background events, we scrutinize the
kinematic characteristics between the signal and background events, aiming to
determine a suitable threshold. As discussed in the preceding section, we
explore ALP masses ranging from $M_{a}=25~{}\rm GeV$ to $M_{a}=100~{}\rm GeV$.
To illustrate interference effects, we select three benchmark values for the
ALP mass, namely $M_{a}=10,25,~{}\text{and}~{}100~{}\rm GeV$, and keeping
$C_{W}=C_{B}=1$ fixed. Additionally, we examine the cases with
$C_{a\phi}=-10,-5,0,5,10$. By observing variations in the number of events (as
already depicted in Fig. 2, where the event rate quantifies the interference
effect) for couplings with the same magnitude but opposite signs, we can
readily discern the interference effect. Finally, the results for the
integrated luminosities 300 fb-1 (current run) and 3000 fb-1 (High-Luminosity
LHC) will be shown in the next section.

Figure 7: Kinematical distributions for the signal with $M_{a}=50$ GeV and two
main SM backgrounds BG1 and BG2. Here $f_{a}=10$ TeV and $C_{WW}=C_{BB}=1$ are
fixed. The “leading” in $P^{\rm leading}_{T_{\gamma}}$ refers to the photon
with the highest transverse momentum.

Various kinematical distributions for the signal case of $M_{a}=50$ GeV along
with two main SM backgrounds are shown in Fig. 7. The signal cases for
$M_{a}=25$ and $100$ GeV show similar behavior. It is easy to see that the
leading photon (with the highest $p_{T}$) has higher $p_{T}$ and more central
rapidity than those two main SM backgrounds. We thus impose the cuts
$P_{T_{\gamma}}^{\rm leading}>60$ GeV and $|\eta_{\gamma}^{\rm leading}|<1.5$
to them. The signal also demonstrates the forward-jet behavior as shown by the
$|\eta_{j}|$ distribution. We therefore require $2.5<|\eta_{j}|<4.7$ for the
highest $p_{T}$ hadronic jet which is not identified as the $b$-jet. Then we
require $bl$ coming from the top-quark decay by $M_{bj}<200$ GeV and
$|\eta_{b}|<1.5$. Finally, we impose the invariant-mass window cut on the
diphoton from the ALP decay: $|M_{\gamma\gamma}-M_{a}|<5$ GeV.

We summarize the above cuts for the signal and background event selections:

  * •

$P_{T_{\gamma}}^{\rm leading}>60~{}\rm GeV$,

  * •

$|\eta^{\rm leading}_{\gamma}|<1.5$,

  * •

$2.5<|\eta_{j}|<4.7$,

  * •

$M_{bl}<200~{}\rm GeV$,

  * •

$|\eta_{b}|<1.5$,

  * •

$|M_{\gamma\gamma}-M_{a}|<5$ GeV.

The cut-flow tables for $M_{a}=25,50$, and 100 GeV are given in Tables 1, 2 3,
respectively. The ALP invariant-mass window cut is the strongest one to reduce
events from both BG1 and BG2 but keep the signal events.

Cut | BG1 | BG2 | Signal
---|---|---|---
|  |  | $C_{a\phi}=-10$ | $C_{a\phi}=-5$ | $C_{a\phi}=0$ | $C_{a\phi}=5$ | $C_{a\phi}=10$
NT | 2318.39 | 307.86 | 59.17 | 24.68 | 7.12 | 27.53 | 63.69
$P^{\rm leading}_{T_{\gamma}}>60~{}\rm GeV$ | 1181.01 | 170.03 | 41.28 | 17.96 | 6.04 | 20.45 | 45.44
$|\eta^{\rm leading}_{\gamma}|<1.5$ | 853.03 | 124.99 | 32.76 | 13.97 | 4.15 | 15.66 | 35.88
$2.5<|\eta_{j}|<4.7$ | 384.44 | 8.92 | 19.38 | 7.78 | 1.39 | 8.26 | 20.54
$M_{bl}<200~{}\rm GeV$ | 384.44 | 7.03 | 19.38 | 7.78 | 1.39 | 8.26 | 20.54
$|\eta_{b}|<1.5$ | 266.20 | 4.68 | 14.85 | 5.93 | 1.00 | 6.28 | 15.69
$20~{}\rm GeV<M_{\gamma\gamma}<30~{}\rm GeV$ | 4.08 | 0.06 | 14.85 | 5.93 | 1.00 | 6.28 | 15.69
Table 1: Cutflow table for the SM backgrounds (BG1: $p~{}p\to t~{}j~{}\gamma~{}\gamma$ and BG2: $p~{}p\to W~{}j~{}j~{}\gamma~{}\gamma$), and the signal: $p~{}p\to j~{}t~{}a$ with various $C_{a\phi}$ couplings. Here we set $M_{a}=25~{}\text{GeV},~{}f_{a}=10~{}\text{TeV},~{}C_{WW}=C_{BB}=1$, and $C_{a\phi}=-10,-5,0,5,10$ GeV. NT in the first row denotes the total number of events with basic cuts shown in Appendix A before further event selections. The number of events are calculated by Eq. (10) and integrated luminosity is set to $\mathcal{L}=3000~{}\rm fb^{-1}$. It is noteworthy that the number of events with luminosity $\mathcal{L}=3000~{}\rm fb^{-1}$ is ten times that of the one corresponding to $\mathcal{L}=300~{}\rm fb^{-1}$.  Cut | BG1 | BG2 | Signal
---|---|---|---
|  |  | $C_{a\phi}=-10$ | $C_{a\phi}=-5$ | $C_{a\phi}=0$ | $C_{a\phi}=5$ | $C_{a\phi}=10$
NT | 2318.39 | 307.86 | 120.56 | 43.26 | 15.73 | 49.69 | 131.68
$P^{\rm leading}_{T_{\gamma}}>60~{}\rm GeV$ | 1181.01 | 170.03 | 99.22 | 36.80 | 14.83 | 42.53 | 109.11
$|\eta^{\rm leading}_{\gamma}|<1.5$ | 853.03 | 124.99 | 81.24 | 29.35 | 10.90 | 33.61 | 88.86
$2.5<|\eta_{j}|<4.7$ | 384.44 | 8.92 | 47.97 | 15.80 | 3.45 | 17.05 | 51.62
$M_{bl}<200~{}\rm GeV$ | 384.44 | 7.03 | 47.97 | 15.80 | 3.45 | 17.05 | 51.62
$|\eta_{b}|<1.5$ | 266.20 | 4.68 | 37.94 | 12.38 | 2.56 | 13.47 | 40.46
$45~{}\rm GeV<M_{\gamma\gamma}<55~{}\rm GeV$ | 8.95 | 0.15 | 37.94 | 12.38 | 2.56 | 13.47 | 40.46
Table 2: Cutflow table for the SM backgrounds (BG1: $p~{}p\to t~{}j~{}\gamma~{}\gamma$ and BG2: $p~{}p\to W~{}j~{}j~{}\gamma~{}\gamma$) and the signal: $p~{}p\to j~{}t~{}a$ with different $C_{a\phi}$ couplings. $M_{a}=50~{}GeV,~{}f_{a}=10~{}TeV,~{}C_{WW}=C_{BB}=1$. The number of events are calculated by Eq. (10) and luminosity is set to $\mathcal{L}=3000~{}\rm fb^{-1}$.  Cut | BG1 | BG2 | Signal
---|---|---|---
|  |  | $C_{a\phi}=-10$ | $C_{a\phi}=-5$ | $C_{a\phi}=0$ | $C_{a\phi}=5$ | $C_{a\phi}=10$
NT | 2318.39 | 307.86 | 124.01 | 44.55 | 18.81 | 50.40 | 136.64
$P^{\rm leading}_{T_{\gamma}}>60~{}\rm GeV$ | 1181.01 | 170.03 | 115.96 | 42.07 | 18.39 | 47.90 | 128.89
$|\eta^{\rm leading}_{\gamma}|<1.5$ | 853.03 | 124.99 | 97.00 | 34.28 | 14.13 | 38.80 | 106.93
$2.5<|\eta_{j}|<4.7$ | 384.44 | 8.92 | 58.41 | 17.88 | 4.16 | 19.29 | 62.29
$M_{bl}<200~{}\rm GeV$ | 384.44 | 7.03 | 58.41 | 17.88 | 4.16 | 19.29 | 62.29
$|\eta_{b}|<1.5$ | 266.20 | 4.68 | 46.02 | 14.12 | 3.19 | 15.37 | 49.38
$95~{}\rm GeV<M_{\gamma\gamma}<105~{}\rm GeV$ | 19.40 | 0.31 | 45.76 | 14.07 | 3.18 | 15.28 | 49.23
Table 3: Cutflow table for the SM background (BG1: $p~{}p\to
t~{}j~{}\gamma~{}\gamma$ and BG2: $p~{}p\to W~{}j~{}j~{}\gamma~{}\gamma$) and
the signal: $p~{}p\to j~{}t~{}a$ with different $C_{a\phi}$ couplings.
$M_{a}=100~{}GeV,~{}f_{a}=10~{}TeV,~{}C_{WW}=C_{BB}=1$. The number of events
are calculated by Eq. (10) and luminosity is set to $\mathcal{L}=3000~{}\rm
fb^{-1}$.

##  V Numerical results

Figure 8: Exclusion regions at 95% confidence level (C.L.) for the ALP cutoff
scale $f_{a}$ derived from the process $pp\to j~{}t~{}a$ followed by
$a\to\gamma\gamma$ and $t\to bl\nu_{l}$ at the LHC with $\sqrt{s}=14$ TeV.
This analysis is conducted with different choices of $C_{a\phi}=0,\,10$ and
under two sets of integrated luminosities, $\mathcal{L}=300~{}\rm fb^{-1}$ and
$3000~{}\rm fb^{-1}$. Solid lines in the plot represent sensitivity curves
with a $10\%$ systematic uncertainty, while the dashed lines depict the curves
without incorporating systematic uncertainty. The gray areas represent the
existing limits from LEP Acciarri  _et al._ (1995); Abbiendi  _et al._ (2003),
CDF Aaltonen  _et al._ (2014), and LHC Clark (2011); Masciovecchio (2016).

After imposing the event selections provided in the last section, the number
of signal events are comparable to, if not larger than, the background events.
It is thus meaningful to calculate the significance of the signal and set
limits on the cutoff scale $f_{a}$. The relation between the ALP signal
events, $N_{s}$, and the ALP cutoff scale, $f_{a}$, is given as follows:

$N_{s}\propto f_{a}^{-2}\;.$ (11)

Therefore, we can rescale the factor $f_{a}$ to match the expected signal
events $N_{s}$. The significance of the signal is given by Cheung and Ouseph
(2023)

$Z=\sqrt{2\left[(N_{s}+N_{b})\ln\left(\frac{(N_{s}+N_{b})(N_{b}+\sigma_{B}^{2})}{{N_{b}}^{2}+(N_{s}+N_{b})\sigma_{B}^{2}}\right)-\frac{N_{b}^{2}}{\sigma_{B}^{2}}\ln\left(1+\frac{\sigma_{B}^{2}N_{s}}{N_{b}(N_{b}+\sigma_{B}^{2})}\right)\right]}\;,$
(12)

where $N_{s}$, $N_{b}$ are the number of signal and background events, and
$\sigma_{B}$ is the systematic uncertainty in background estimation, which is
taken to be zero and $0.1N_{b}$ in the presentation. The $95\%$ confidence
level (C.L.) sensitivity curves for the ALP cutoff scale $f_{a}$ to the ALP
mass $M_{a}$ are obtained by requiring the significance $Z>2$.

In Fig. 8, we show $95\%$ C.L. exclusion region for the ALP cutoff scale,
$f_{a}$. This exclusion region is obtained through the process $pp\to
j~{}t~{}a$ with $a\to\gamma\gamma$ and $t\to bW,~{}W\to l\nu_{l}$ at the LHC
with $\sqrt{s}=14$ TeV. In the figure, we fix $C_{WW}=C_{BB}=1$ and set
$C_{a\phi}=0$ and $C_{a\phi}=10$ as two benchmark points. Two different sets
of integrated luminosities, $\mathcal{L}=300~{}\rm fb^{-1}$ and $3000~{}\rm
fb^{-1}$ are plotted. In the plot, solid lines denote sensitivity curves
accounting for a $10\%$ systematic uncertainty, while dashed lines represent
curves without incorporating systematic uncertainty. We can find that the
systematic uncertainty only slightly changes the predictions of future bounds
in this study. It indicates that our results are robust against possible
uncertainties. The gray shaded areas correspond to the existing limits from
LEP Acciarri  _et al._ (1995); Abbiendi  _et al._ (2003), CDF Aaltonen  _et
al._ (2014), and LHC Clark (2011); Masciovecchio (2016). The lines in Fig. 8
unmistakably demonstrate that sensitivity curves with non-zero values of
$C_{a\phi}$ can yield better limits on $f_{a}$ than those with $C_{a\phi}=0$.
It can also be observed in Fig. 2 in which the minimal production cross
section for the signal process appears at $C_{a\phi}=0$ and it enhances as the
$|C_{a\phi}|$ increases. Specifically, the sensitivity reach corresponding to
$C_{a\phi}=10$ and $\mathcal{L}=300\;(3000)~{}\rm fb^{-1}$ (depicted by the
red (green) line) exhibits enhanced sensitivity compared to current
constraints for $M_{a}>45$ GeV (the whole ALP mass range under consideration).
Even in the absence of the $C_{a\phi}$ coupling ($C_{a\phi}=0$), the HL-LHC
($\mathcal{L}=3000~{}\rm fb^{-1}$) can impose the sensitivity on $f_{a}$
(indicated by the blue curve) for $M_{a}>75~{}\rm GeV$.

##  VI Conclusions

In summary, we have presented a novel approach that considers simultaneous
presence of two or more ALP interaction operators in a single process at the
LHC. In particular, we demonstrated the interference effects of the ALP-gauge
boson pair and the ALP-top quark pair couplings in the process $pp\to tja$ as
shown in Fig. 2. Through a detailed analysis of $pp\rightarrow tja$, followed
by $a\rightarrow\gamma\gamma$ and semi-leptonic decay of the top quark, as a
case study, we demonstrated the efficacy of this approach in constraining ALP
interactions as well as their interference within a single process.

Our findings indicate that the sensitivity of the ALP cutoff scale $f_{a}$
could potentially reach down to the values around $1/f_{a}\sim 5\times
10^{-2}\;{\rm TeV}^{-1}$ for the ALP masses ranging from $25$ GeV to $100$ GeV
at the HL-LHC as shown in Fig. 8. It indicates that some uncovered parameter
space can be further explored from the process $pp\rightarrow tja$ in the near
future. Furthermore, once the absolute size of the ALP-gauge boson pair and
the ALP-top quark pair couplings can be pinned down by other ALP production
channels, this process can provide extra information about the relative sign
(or phase) between two coefficients of ALP interaction operators. In our case,
we observe a positive interference between the processes in fig. 1 which is
visible from the fact that the cross-section increases with $C_{a\phi}$.

##  Appendix A Settings of the run_card.dat file of MadGraph5_aMC@NLO

The basic cuts implemented in the run_card.dat file of MadGraph5_aMC@NLO
during the simulation are outlined below:
25.0 = ptj ! minimum pt for the jets
20.0 = pta ! minimum pt for the photons
25.0 = ptl ! minimum pt for the charged leptons
25.0 = ptb ! minimum pt for the b-quark
4.7 = etaj ! max rap for the jets
2.5 = etaa ! max rap for the photons
2.5 = etal ! max rap for the charged leptons
2.5 = etab ! max rap for the b-quark
0.4 = drjj ! min distance between jets
0.3 = draa ! min distance between gammas (photon)
0.4 = draj ! min distance between gamma (photon) and jet
0.4 = drjl ! min distance between jet and lepton
0.4 = dral ! min distance between gamma (photon) and lepton
0.4 = drbl ! min distance between b-quark and lepton
0.4 = drbj ! min distance between b-quark and jet
0.4 = drab ! min distance between gamma (photon) and b-quark

## Acknowledgment

The work of K.Cheung, P.Sarmah, and C.J. Ouseph is supported by the Taiwan
NSTC with grant no. MoST-110-2112-M-007-017-MY3. The work of C.-T. Lu is
supported by the National Natural Science Foundation of China (NNSFC) under
grant No. 12335005.

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