aainstitutetext: Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607
Hamburg, Germanybbinstitutetext: Institut für Experimentalphysik, Universität
Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germanyccinstitutetext:
Institute for Theoretical Physics, Heidelberg University, 69120 Heidelberg,
Germanyddinstitutetext: Institute for Mathematics, Astrophysics and Particle
Physics, Radboud University, 6500 GL Nijmegen, The Netherlandseeinstitutetext:
Nikhef, Science Park 105, 1098 XG Amsterdam, The Netherlandsffinstitutetext:
Department of Physics and Astronomy, Uppsala University, 75120 Uppsala, Sweden

# Top Secrets: Long-Lived ALPs in Top Production

L. Rygaard  0000-0003-3192-1622 a J. Niedziela  0000-0002-9514-0799 c R.
Schäfer c,f S. Bruggisser a J. Alimena  0000-0001-6030-3191 c,d,e S. Westhoff
a,b F. Blekman  0000-0002-7366-7098<EMAIL_ADDRESS>
###### Abstract

We investigate the discovery potential for long-lived particles produced in
association with a top-antitop quark pair at the (High-Luminosity) LHC.
Compared to inclusive searches for a displaced vertex, top-associated signals
offer new trigger options and an extra handle to suppress background. We
design a search strategy for a displaced di-muon vertex in the tracking
detectors, in association with a reconstructed top-antitop pair. For axion-
like particles with masses above the di-muon threshold, we find that the
(High-Luminosity) LHC can probe effective top-quark couplings as small as
$|c_{tt}|/f_{a}=0.03~{}(0.002)/$TeV and proper decay lengths as long as
$20~{}(300)$ m, assuming a cross section of $1\,$fb, with data corresponding
to an integrated luminosity of 150 fb-1 (3 ab-1). Our predictions suggest that
searches for top-associated displaced di-muons will explore new terrain in the
current sensitivity gap between searches for prompt di-muons and missing
energy.

††arxiv: 2306.08686††preprint: DESY-23-088, Nikhef-2023-006

##  1 Introduction

Collider searches for long-lived particles (LLPs) Alimena:2019zri  explore the
fourth space-time dimension where discoveries might hide: time. Particles with
a long lifetime statistically decay at an observable distance from their
production point. If the decay is inside the detector, their (charged) decay
products can be reconstructed as a displaced vertex, a characteristic LLP
signal.

Searches for displaced vertices bridge the sensitivity gap in decay length
between searches for promptly decaying particles and searches with missing
energy. Displaced vertex searches can even lead in sensitivity, especially if
prompt and missing energy signals are plagued by a large Standard Model (SM)
background. Exploiting the full reach of displaced vertex searches is key to
maximizing the discovery potential for LLPs and to taking full advantage of
colliders like the LHC.

At the LHC, currently, most searches for displaced vertices are either
inclusive with respect to the remainder of the event CMS:2022qej ; CMS:2021sch
; CMS:2021tkn ; CMS:2020iwv ; ATLAS:2018rjc ; ATLAS:2019fwx ; ATLAS:2023oti ;
ATLAS:2020xyo ; ATLAS:2019tkk ; ATLAS:2019qrr ; LHCb:2017xxn  or target model-
specific signatures of LLPs produced in Higgs decays ATLAS:2018tup ;
ATLAS:2019jcm ; ATLAS:2022gbw ; ATLAS:2022zhj ; ATLAS:2021jig ; ATLAS:2018pvw
; CMS:2021yhb ; LHCb:2016buh . However, inclusive searches often face trigger
challenges Alimena:2021mdu , while model-specific searches are hard to
reinterpret in other scenarios. LLP searches in association with other
particles are a promising alternative, as demonstrated by recent pioneering
searches CMS:2023ovx ; CMS:2022fut ; ATLAS:2019kpx .

In this work, we explore the LHC’s discovery potential for displaced vertices
produced in association with prompt particles. We will call them _associated
displaced vertices_ in what follows. Such signatures offer the option to
trigger on the prompt part of the signal, which allows one to optimize the
selection criteria for the displaced vertex. Compared to model-specific
searches that rely on reconstructed event kinematics of the displaced vertex,
associated displaced vertex searches can be kept fairly general by only
applying loose kinematic selections on the prompt part of the signal. All that
is needed is an efficient LLP production channel with a fully reconstructable
final state. Examples are top-antitop production, the Drell-Yan process,
Vector Boson Fusion, or di-jet production, which probe LLPs with couplings to
top quarks, weak gauge bosons, and light quarks or gluons, respectively.

In this study, we focus on long-lived axion-like particles (ALPs, or “$a$”)
produced in association with a top-antitop pair, $pp\to t\bar{t}a$, and
decaying to a displaced di-muon, $a\to~{}\mu^{+}\mu^{-}$. In general, such a
signature is particularly sensitive to new light scalar or pseudo-scalar
particles, which in many scenarios have flavor-hierarchical couplings to
quarks Froggatt:1978nt ; Georgi:1986df ; Gherghetta:2000qt ; Agashe:2004rs .
In this paper, we therefore focus on ALPs with masses in the GeV range and
only top couplings $|c_{tt}|/f_{a}$ at high energies, see e.g. Ref.
Bauer:2021mvw ; Esser:2023fdo . The decays of these and similar light top-
philic particles are loop-suppressed, leading to a long decay length. They may
thus have escaped searches for prompt resonances in association with top
quarks ATLAS:2023ofo ; CMS:2019lwf ; CMS:2022arx  and still not be long-lived
enough to be caught in searches for tops and missing energy CMS:2015zwg ;
CMS:2017dcx ; CMS:2017qxu ; CMS:2019zzl ; CMS:2018ysw ; ATLAS:2018cjd ;
ATLAS:2014bba ; ATLAS:2018nda ; ATLAS:2017hoo ; ATLAS:2014dbf . For ALPs with
flavor-changing couplings, a complementary search for top-associated displaced
ALP decays in the calorimeter has been recently put forward in Ref.
Carmona:2022jid .

This article is structured as follows. In Sec. 2, we discuss the basic
characteristics of the signal and introduce the ALP benchmark model. In Sec.
3, we present a detailed analysis of the signal and background kinematics and
propose an efficient event selection. In Sec. 4, we predict the sensitivity to
top-associated displaced di-muon events at the LHC during Run 2 and the High-
Luminosity LHC (HL-LHC). We conclude in Sec. 5.

##  2 Long-lived particles in top-antitop production

New scalars or pseudo-scalars are predicted in many extensions of the SM.
Unlike new vector particles, scalars are generally expected to have flavor-
hierarchical couplings to quarks and leptons, with the strongest coupling to
top quarks. The reason for this assumption is that scalar couplings to
fermions break the flavor symmetry of the Lagrangian, just like the Higgs
Yukawa couplings. To preserve the observed mass hierarchies and flavor mixing
of the fermions, new scalar couplings should therefore not deviate much from
the Yukawa-like flavor pattern in the SM.

At the LHC, top-antitop production is a natural place to search for new
(pseudo-)scalar particles. If they couple mostly to top quarks, but are too
light to decay into a top-antitop pair, their decay is suppressed. This
suppression can be so strong that particles produced from top-antitop
production decay with a significant displacement from their production point.
In Fig. 1, we show a representative Feynman diagram of the LHC process

$\displaystyle pp\to t\bar{t}a,\,a\to\mu\bar{\mu}.$ (1)

A resonant long-lived ALP is produced in top-antitop events from proton-proton
collisions and decays into a displaced muon-antimuon pair. In general, other
decay channels like $e^{+}e^{-}$, $\tau^{+}\tau^{-}$, $\gamma\gamma$, or
$q\bar{q}$ are possible, depending on the mass and couplings of the LLP. In
this work, we focus on the $\mu^{+}\mu^{-}$ final state, which can be
reconstructed well in the LHC detectors.

Figure 1: Example Feynman diagram for the production and decay of a long-lived
ALP at the LHC via $pp\to t\bar{t}a,\,a\to\mu\bar{\mu}$. The dotted line
illustrates that ALP decays at a distance from its production point. The small
black circle indicates the effective ALP-top coupling, and the large white
circle indicates the loop-induced ALP coupling to muons.

###  2.1 Signal event rate

The expected number of top-associated LLPs “$a$” produced at the LHC with a
given integrated luminosity $\mathcal{L}$ is

$\displaystyle N=\mathcal{L}\,\sigma(pp\to t\bar{t}a),$ (2)

where $\sigma$ is the hadronic cross section at a fixed proton-proton
collision energy. The decay length of the LLP in the laboratory frame,
$d_{a}=\beta\gamma c\tau_{a}$, is determined by its lifetime $\tau_{a}$ and
Lorentz boost $\beta\gamma$. For an LLP with decay length $d_{a}$, produced in
the direction $\vec{r}_{a}$, the probability to decay within a radial interval
$\Delta r_{a}=r_{a}^{\rm out}-r_{a}^{\rm in}$ is

$\displaystyle P_{a}(d_{a};\vec{r}_{a})=\exp\left(-\frac{r_{a}^{\rm
in}}{d_{a}}\right)-\exp\left(-\frac{r_{a}^{\rm out}}{d_{a}}\right).$ (3)

In practice, $r_{a}^{\rm in}$ and $r_{a}^{\rm out}$ are the distances from the
production point to where the particle’s trajectory intersects with the
detector boundaries. The decay distribution of a sample of particles is thus
exponential, with the slope determined by the proper lifetime $\tau$ and the
average boost of the particles. On the other hand, displacement is a property
of each particle. It refers to where in the detector that particle decays. We
will refer to particles that decay after reaching at least 200 $\mu$m as
_displaced_ , and as _prompt_ otherwise.

For a sample of $N$ produced LLP events, the fraction of LLPs that decay
within a detector volume $V$ is

$\displaystyle
N_{a}(V)=\mathcal{L}\,\frac{1}{N}\sum_{a=1}^{N}\sigma\left(pp\to
t\bar{t}\,a\,|d_{a};\vec{r}_{a}\right)P_{a}\left(d_{a};\vec{r}_{a}\right),$
(4)

where $\sigma\left(pp\to t\bar{t}\,a\,|d_{a};\vec{r}_{a}\right)$ is the cross
section for an LLP produced in a small interval around the direction
$\vec{r}_{a}$ with boost $\beta\gamma=d_{a}/c\tau_{a}$. The expected number of
$t\bar{t}(\mu\bar{\mu})$ signal events is finally given by

$\displaystyle N_{\rm
sig}=N_{a}(V)\,\mathcal{B}(a\to\mu\bar{\mu}),\qquad\mathcal{B}(a\to\mu\bar{\mu})=\frac{\Gamma(a\to\mu\bar{\mu})}{\Gamma_{a}},$
(5)

where $\Gamma_{a}=\tau_{a}^{-1}$ and $\Gamma(a\to\mu\bar{\mu})$ are the total
decay width and the partial decay width of the LLP into muons and
$\mathcal{B}(a\to\mu\bar{\mu})$ is the LLP’s branching ratio into muons.

In general, the signal rate $N_{\rm sig}$ depends on four fundamental
properties of the LLP: the mass $m_{a}$, the coupling to top quarks $c_{tt}$,
the decay width $\Gamma_{a}$, and the branching ratio
$\mathcal{B}(a\to\mu\bar{\mu})$. In specific models, some of these quantities
might be related.

###  2.2 Axion-like particles

ALPs are pseudo-scalars whose couplings preserve a shift symmetry $a\to a+c$.
At energies below a cutoff scale $\Lambda=4\pi f_{a}$, the phenomenology is
described by an effective Lagrangian Georgi:1986df

$\displaystyle\mathcal{L}_{\rm eff}(\mu)$
$\displaystyle=\frac{1}{2}(\partial_{\mu}a)(\partial^{\mu}a)-\frac{m_{a}^{2}}{2}a^{2}+\sum_{V}c_{VV}(\mu)\,\frac{a}{f_{a}}\frac{g^{2}}{(4\pi)^{2}}\,V_{\mu\nu}\widetilde{V}^{\mu\nu}$
(6)
$\displaystyle\quad+\sum_{F=Q,L}\frac{\partial^{\mu}a}{f_{a}}\,\bar{F}_{i}\,(C_{F})_{ij}\,\gamma_{\mu}\,F_{j}+\sum_{f=U,D,E}\frac{\partial^{\mu}a}{f_{a}}\,\bar{f}_{i}\,(C_{f})_{ij}\,\gamma_{\mu}\,f_{j}.$

In general, the scale $f_{a}$ and the mass of the ALP, $m_{a}$, are free
parameters.111In the special case of the QCD axion, $f_{a}$ corresponds to the
axion’s decay constant, and the axion mass scales as $m_{a}\propto 1/f_{a}$.
In our analysis, we set $f_{a}=1\,$TeV. The parameters $c_{VV}$ label ALP
couplings to gauge fields $V=G,W,B$ associated with the $SU(3)_{C}$,
$SU(2)_{L}$ and $U(1)_{Y}$ groups, with field-strength tensor $V_{\mu\nu}$ and
dual $\widetilde{V}_{\mu\nu}=\epsilon_{\mu\nu\rho\sigma}V^{\rho\sigma}$. The
ALP couplings $C_{F}$ and $C_{f}$ to left-handed and right-handed fermions,
respectively, are in general $3\times 3$ matrices in flavor space.222The
$SU(2)_{L}$ doublets are defined as $Q=(u_{L},d_{L})^{\top}$ and
$L=(\nu_{L},\ell_{L})^{\top}$; the $SU(2)_{L}$ singlets are $U=u_{R}$,
$D=d_{R}$, $E=\ell_{R}$. Among the flavor-diagonal couplings, only the axial-
vector combinations $(C_{F})_{ii}-(C_{f})_{ii}$ are physical; vector couplings
are incompatible with the shift symmetry. The derivative $\partial^{\mu}a$
implies that ALP couplings to on-shell fermions are proportional to the
fermion mass $m_{f}$ and therefore generally flavor-hierarchical.

We choose to work in a basis where the interaction and mass eigenstates of up-
type quarks coincide. In this basis, we define the flavor-diagonal ALP
couplings to up-type and down-type quarks as well as to charged leptons as

$\displaystyle c_{ii}$ $\displaystyle=(C_{u})_{ii}-(C_{Q})_{ii},\qquad\quad\
\,i=\\{u,c,t\\}$ (7) $\displaystyle c_{jj}$
$\displaystyle=(C_{d})_{jj}-(V^{\dagger}C_{Q}V)_{jj},\quad j=\\{d,s,b\\}$
$\displaystyle c_{kk}$ $\displaystyle=(C_{\ell})_{kk}-(C_{L})_{kk},\qquad\ \
\,k=\\{e,\mu,\tau\\},$

where $V$ is the CKM matrix. Flavor off-diagonal couplings will not play a
role in this work.

The top-quark coupling $c_{tt}=(C_{u})_{33}-(C_{Q})_{33}$ plays an important
role in the ALP effective theory at low energies. In the renormalization group
(RG) evolution of the effective couplings, $c_{tt}$ generates ALP couplings to
all other fermions through mixing and renormalization effects Choi:2017gpf ;
MartinCamalich:2020dfe ; Chala:2020wvs ; Bauer:2020jbp . For
$\Lambda=4\pi\,$TeV, the ALP couplings at the top mass scale $\mu=m_{t}$ are

$\displaystyle c_{uu,cc}(m_{t})$
$\displaystyle=c_{uu,cc}(\Lambda)-0.116\,c_{tt}(\Lambda)$ (8) $\displaystyle
c_{dd,ss}(m_{t})$ $\displaystyle=c_{dd,ss}(\Lambda)+0.116\,c_{tt}(\Lambda)$
$\displaystyle c_{bb}(m_{t})$
$\displaystyle=c_{bb}(\Lambda)+0.097\,c_{tt}(\Lambda)$ $\displaystyle
c_{\ell\ell}(m_{t})$
$\displaystyle=c_{\ell\ell}(\Lambda)+0.116\,c_{tt}(\Lambda),\quad\ell=\\{e,\mu,\tau\\}.$

To a good approximation, these relations also hold in the perturbative regime
below the weak scale. The ALP-top coupling can thus be probed in low-energy
observables through RG effects in effective ALP couplings to light fermions.
Moreover, the ALP-top coupling induces ALP couplings to photons and gluons
through top loops, which however play a minor role in this work. Notice that
also the top coupling itself receives RG corrections at energies below the
cutoff scale. For numerical predictions of the ALP couplings at different
energy scales, we use our public code Schafer_UFO  that implements the results
of Ref. Bauer:2020jbp .

A very predictive scenario is obtained by assuming that the top coupling is
the only coupling of the ALP at the cutoff scale, so that

$\displaystyle c_{tt}(\Lambda)\equiv c_{tt},\qquad c_{f\\!f}(\Lambda)=0\
(f\neq t),\qquad c_{VV}(\Lambda)=0.$ (9)

We will call this scenario the  _top scenario_ in what follows. Such a
scenario arises from ultraviolet completions of the ALP effective theory with
top-philic heavy particles that generate the effective coupling
$c_{tt}(\Lambda)$. In the top scenario, all ALP couplings are proportional to
$|c_{tt}|/f_{a}$. As a consequence, all partial decay widths of the ALP are
determined by $|c_{tt}|/f_{a}$. Moreover, the ALP decay rate is fully
correlated with the production from top quarks. The signal rate is thus
determined by only two parameters: the ALP mass $m_{a}$ and the effective top
coupling $|c_{tt}|/f_{a}$.

The total decay width of the ALP is in general given by 333Decays into
neutrinos are in principle possible, but extremely suppressed by the small
neutrino mass.

$\displaystyle\Gamma_{a}=\Gamma(a\to\gamma\gamma)+\sum_{\ell=e,\mu,\tau}\Gamma(a\to\ell\bar{\ell})\,\Theta(m_{a}-2m_{\ell})+\Gamma(a\to\text{had})\,\Theta(m_{a}-3m_{\pi}).$
(10)

The ALP decay into leptons and hadrons is open only above the kinematic
threshold, as indicated by the Heaviside function $\Theta$. The decay rate
into fermion pairs is

$\displaystyle\Gamma(a\to
f\\!\bar{f})=\frac{m_{a}}{8\pi}\frac{|c_{f\\!f}(m_{a})|^{2}m_{f}^{2}}{f_{a}^{2}}\left(1-\frac{4m_{f}^{2}}{m_{a}^{2}}\right)^{\frac{1}{2}}.$
(11)

In the top scenario, $c_{f\\!f}(m_{a})\approx c_{f\\!f}(\mu_{w})$ for ALP
masses below the weak scale $\mu_{w}$, and Eq. 8 applies to a good
approximation. For hadronic decays, Eq. 11 applies only in the regime of
perturbative QCD, _i.e._ , for $m_{a}\gtrsim 2\,$GeV. To make predictions at
lower scales, the ALP effective theory has to be matched onto the chiral
perturbation theory Georgi:1986df ; Bauer:2021wjo .

In Fig. 2 (left), we show the proper decay length of the ALP in the top
scenario (Eq. 9) as a function of its mass. Since all partial decay rates are
determined by the top coupling, the decay length scales as $c\tau_{a}\propto
f_{a}^{2}/c_{tt}^{2}$. The ALP branching ratio into any specific final state
is thus independent of $c_{tt}$. The top scenario (Eq. 9) predicts the longest
possible decay length of ALPs produced from top quarks. Any additional
coupling would increase the decay width, resulting in a shorter decay length.

Figure 2: Proper decay length, $c\tau_{a}$, (left) and branching ratios
(right) of an ALP in the top scenario (Eq. 9) as a function of its mass,
$m_{a}$. For a fixed mass, all decay modes are fully determined by the top-
quark coupling $|c_{tt}(\Lambda)|/f_{a}$, defined at the cutoff scale
$\Lambda=4\pi\,$TeV. In the left panel, vertical lines for $c\tau_{a}$
indicate benchmark ALP masses used in this study. The dip in the decay length
around $m_{a}=m_{\pi}$ is due to ALP-pion mixing.

The branching ratios of ALP decays in the top scenario are shown in Fig. 2
(right), as a function of the ALP mass. For $m_{a}<2m_{\mu}$, the ALP can only
decay into photons and, if kinematically allowed, into electrons. For
$2m_{\mu}<m_{a}<1\,\text{GeV}$, the ALP width is dominated by the decay into
muons; decays into electrons and photons are suppressed by the small electron
mass and the electromagnetic coupling, respectively. For ALP masses around the
GeV scale, predictions are affected by large uncertainties in hadronic decays.
For the exact implementation of the ALP decay channels, we refer the reader to
App. A of Ref. Ferber:2022rsf .

Where perturbative predictions are possible, the ALP branching ratio into
muons in the top scenario is given by

$\displaystyle
2m_{\mu}<m_{a}<1\,\text{GeV}:\qquad\mathcal{B}(a\to\mu\bar{\mu})$
$\displaystyle\approx 1$ (12) $\displaystyle
m_{a}>2\,\text{GeV}:\qquad\mathcal{B}(a\to\mu\bar{\mu})$
$\displaystyle\approx\frac{\Gamma(a\to\mu\bar{\mu})}{\sum_{f}\Gamma(a\to
f\\!\bar{f})\,\Theta(m_{a}-2m_{f})}\,.$

Additional couplings beyond $c_{tt}$ can significantly change the branching
ratios.

For our analysis of displaced vertices, we focus on ALPs within the mass range
$2m_{\mu}<m_{a}<2m_{b}$. Below the di-muon threshold, the decay into muons is
kinematically forbidden; above the $b\bar{b}$ threshold, decays into bottom-
antibottom pairs result in essentially prompt decays. Within the mass range,
we define the following benchmarks:

$\displaystyle
m_{a}=\\{0.3,\,0.35,\,0.5,\,0.9,\,1.25,\,2,\,2.5,\,2.75,\,3,\,4,\,8,\,8.25,\,8.5,\,9,\,10\\}\,\text{GeV}.$
(13)

In the first two benchmarks, the ALP decays 100% of the time into muons. In
all other benchmarks, additional decay channels are kinematically allowed and
reduce the branching ratio to muons. The impact of the various decay channels
on the proper decay length is shown in Fig. 2. We will use these benchmarks to
study the dependence of the signal features on the ALP mass and lifetime.

When comparing these benchmarks in the top scenario with concrete UV
completions of the ALP effective theory, it is important to know for which
scenarios they are good proxies. Additional ALP couplings at the cutoff scale
affect the lifetime and branching ratios of the ALP. In Sec. 4.2, we therefore
interpret our results for ALPs with arbitrary lifetimes. Additional ALP
couplings can also lead to new production channels. In the quark sector, the
phenomenology is usually still dominated by the top coupling, because ALP
couplings to on-shell fermions are proportional to the fermion mass. Besides
resonant top production, sensitive observables are, for instance, flavor-
changing $B$ meson and kaon decays, which probe the top coupling in loops. In
the lepton sector, the mass proportionality can be outrun by precision probes
of the ALP-electron coupling. To obtain maximal information about the ALP
couplings in a specific UV completion, a global fit of the ALP effective
Lagrangian to data is the way to go Bruggisser:2023npd ; Biekotter:2023mpd .

##  3 Signal and background features

The goal of this analysis is to determine the LHC sensitivity to a long-lived
pseudo-scalar particle produced in association with a top-antitop pair, as in
Eq. 1. We perform a dedicated analysis of the signal and background kinematics
for the ALP benchmarks from Eq. 13 in the top scenario, Eq. 9. The results are
then generalized to ALPs with an arbitrary lifetime.

In the top scenario, the phenomenology of the ALP is determined by two
parameters, the mass $m_{a}$ and the top-quark coupling
$|c_{tt}(\Lambda)|/f_{a}$. Throughout the analysis, we set
$f_{a}~{}=~{}1\,$TeV. In our analysis of kinematics and decay distributions,
we fix $c_{tt}(\Lambda)~{}=~{}1$, such that the signal rate only depends on
the ALP mass.

The main SM background processes that can mimic a di-muon vertex in
association with a top-antitop pair are:

  * •

$pp\to t\bar{t}Z^{(\ast)},\ Z^{(\ast)}\to\mu\bar{\mu}$: top-antitop production
in association with a virtual or resonant $Z$ boson (or photon)
decaying/converting into a di-muon;

  * •

$pp\to t\bar{t}j$: top-antitop production in association with one or more
jets, where hadrons inside the jet decay into two opposite-sign muons.

The $t\bar{t}Z^{(\ast)}$ process is an important background for prompt or
nearly prompt di-muons. For top-associated displaced vertices,
$t\bar{t}Z^{(\ast)}$ can still be a background because the parton shower
produces mesons with a significant lifetime that decay into displaced di-
muons. The $t\bar{t}j$ background is relevant at larger displacements because
the muons are produced from decays of mesons inside the jet with a significant
boost. In both cases, di-muons can originate from the same particle (e.g. a
$J/\Psi$ meson), which we will refer to as "resonant". On the other hand, di-
muons can come from decays of two different mesons, called "non-resonant" in
what follows.

Muons from cosmic rays produce displaced muons in the LHC detectors, and
therefore in principle, cosmic muons that randomly cross and produce a
displaced vertex could be a background for this analysis. However, the rate of
this process is small and in practice, there are many ways for the experiments
to reject the vast majority of this background CMS:2022qej ; CMS:2021sch ;
ATLAS:2018rjc ; ATLAS:2019fwx . As a result, we do not consider background
from cosmic muons in this analysis.

We focus on the extraction of the displaced di-muon vertex and assume a
perfect reconstruction of the top-antitop pair (in all possible final states)
and a 100% effective top trigger. We also assume perfect identification of
muons coming from top quark decays, meaning not only muons originating from
the W boson decay, but also those originating from the hadronization of the b
quark. We estimate the impact of this assumption on the sensitivity to LLPs in
Sec. 4.

###  3.1 Event simulation

All signal and background processes are generated using MadGraph5_aMC@NLO v.
3.2.0 Alwall:2014  to simulate proton-proton collisions at $\sqrt{s}=13\,$TeV,
with NNPDF31_NLO_as_0118 parton distribution functions Ball_2017 . All
simulations of the hard scattering process are performed at leading order in
QCD. Hadronization and particle showering are simulated using PYTHIA8 v.8.306
PYTHIA8 , with the default Monash 2013 tune Skands_2014 . The top mass is set
to $m_{t}=172.0\,$GeV.

For the generation of the signal process $pp\to t\bar{t}a$, we use a UFO model
Schafer_UFO  that implements the effective Lagrangian (Eq. 6) in the top
scenario. We set the ALP-top coupling to $|c_{tt}(\Lambda)|/f_{a}=1$/TeV and
all other ALP couplings to zero. The displaced ALP decay is simulated by
PYTHIA8 based on the decay width, as described in Sec. 2.1 and 2.2. ALPs with
decays to hadrons or tau leptons can produce secondary muon pairs from meson
or tau decays. For the tau channel, this occurs in less than 3 percent of all
events. We ignore such secondary muons in our study; their impact on our
results is negligible.

For the background process $pp\to t\bar{t}Z^{(\ast)},\
Z^{(\ast)}\to\mu\bar{\mu}$, we require the muons to have a transverse momentum
$p_{T}^{\mu}>5\,$GeV and pseudo-rapidity $|\eta^{\mu}|<3.5$ at generator
level.

For $pp\to t\bar{t}j$, we first generate $pp\rightarrow t\bar{t}j$ in
MadGraph5, requiring that the jet has $p_{T}^{j}>20\,$GeV and $|\eta^{j}|<3$.
These criteria are chosen such that the particles can be well reconstructed at
the LHC detectors while maintaining optimal use of CPU resources by not
generating events that do not fall within the typical kinematic acceptance of
the LHC. Then we use PYTHIA8 to add additional ISR/FSR jets. However, the
cross section we obtain from MadGraph5 only includes the $pp\to t\bar{t}+1j$
process. To include more jets we use the measured cross sections for the
$t\bar{t}$ process with a different number of additional jets (0-4) from CMS
CMS:2017ttbar . Since the kinematic selections in this analysis somewhat
differ from ours and since CMS only included the semi-leptonic decay of the
top quark, we use the ratio of multi-jet versus one-jet emission from CMS to
re-scale our one-jet cross section obtained from MadGraph5 (MG). Finally, our
prediction of the $t\bar{t}+(1-4)j$ cross section is

$\displaystyle\sigma_{pp\to t\bar{t}+(1-4)j}=\sigma^{\mathrm{MG}}_{pp\to
t\bar{t}+1j}\cdot\frac{\sigma^{\mathrm{CMS}}_{pp\to
t\bar{t}+(1-4)j}}{\sigma^{\mathrm{CMS}}_{pp\to
t\bar{t}+1j}}=395~{}\mathrm{pb}\cdot\frac{132~{}\mathrm{pb}}{77~{}\mathrm{pb}}=667~{}\mathrm{pb}.$
(14)

###  3.2 Event selection

We apply selection criteria to the event samples in two stages: pre-selection
and signal selection. In the pre-selection, we select events that contain a
displaced di-muon; in the signal selection, we subsequently apply specific
requirements on the pre-selected di-muons to increase the signal sensitivity.
Although we apply selections on the truth level objects, the selection
requirements were tuned based on known detector resolutions, using the current
CMS detector as a reference. A summary of the selection criteria is given in
Table 1.

Pre-selection |
---|---
Muon kinematics |  $p_{T}^{\mu}>5\,$GeV,$\ |\eta^{\mu}|<2.5$
Muon displacement |  $l_{xy}>200\,\mu$m
At least one opposite-sign di-muon
Signal selection |
Muon kinematics |  $p_{T}^{\mu}>10\,$GeV
Di-muon mass | $m_{\mu\bar{\mu}}\neq m_{J/\Psi},m_{\Psi(2S)}$
Di-muon vertex | $R_{lxy}<0.05$

Table 1: Selection criteria applied on the signal and background events.

####  3.2.1 Pre-selection

In the pre-selection, we request two opposite-sign muons, each passing basic
kinematic selections $|\eta^{\mu}|<2.5$ and $p_{T}^{\mu}>5\,$GeV. We only
consider muons that do not originate from $W$ bosons from top quark decays.
Due to the re-balancing of momentum during hadronization in PYTHIA, muons
coming from ALP decays may experience a slight modification of their four-
momenta and be included multiple times in the event with different status
codes. The modification in the four-momenta results in a di-muon mass spectrum
with a non-negligible width, as will be shown below. Furthermore, to avoid
non-physical distortions of the ALP resonance caused by this effect, for the
signal we select the muon four-momentum before these modifications occur. For
the background, we work with final-state muons.

The main criterion to identify a displaced muon is the transverse displacement
$l_{xy}$, defined in terms of the position coordinates $x$ and $y$ of the muon
production point as

$\displaystyle l_{xy}=\sqrt{x^{2}+y^{2}}.$ (15)

In our analysis, we select displaced muons with $l_{xy}>200\,\mu$m. This value
is roughly based on the transverse impact parameter resolution at CMS (20–75
$\mu$m for tracks with $1\,\text{GeV}<p_{T}<10\,$GeV CMS:2014pgm ). We choose
the conservative value of 200 $\mu$m.

This selection criterion significantly reduces the prompt $t\bar{t}Z$
background, with events passing mainly due to meson decays within jets,
similarly to the $t\bar{t}j$ background. Given the small cross section of
$t\bar{t}Z$, the expected number of passing events is negligible.
Consequently, we do not consider this process any further in our analysis.

The event passes the pre-selection if there is at least one opposite-sign di-
muon remaining after the above-mentioned criteria.

####  3.2.2 Signal selection

In the second stage, we apply additional selection criteria that enhance the
signal sensitivity. These criteria are motivated by the kinematic
distributions of the signal and background processes. In what follows, we
discuss the event kinematics in detail. The event rates displayed in the
figures correspond to an integrated luminosity of 150 fb-1.

Figure 3 shows the transverse momentum distribution of the muon with the
lowest $p_{T}^{\mu}$ in each event. The signal event rate depends little on
the muon transverse momentum, with a slight decline for light ALPs at low
$p_{T}^{\mu}$. The $t\bar{t}j$ background rate decreases at high
$p_{T}^{\mu}$. To increase the signal sensitivity, we therefore tighten the
selection of the muon transverse momentum to $p_{T}^{\mu}>10\,$GeV.

Figure 3: The transverse momentum distribution of the muon with the lowest
$p_{T}^{\mu}$ in each event, for various ALP signal mass benchmarks (colored
lines) and background from $t\bar{t}j$ events (gray regions). The distribution
is shown after the full event selection (upper right corner), excluding the
$p_{T}^{\mu}$ criterion. The last bin includes the overflow at higher
$p_{T}^{\mu}$.

The di-muon invariant mass is shown in Fig. 4. The signal peaks around the ALP
mass, as expected. The observable width for light ALPs is due to the four-
momenta modifications in PYTHIA described in Sec. 3.2.1. The decrease in the
number of signal events for higher ALP masses is mostly due to the lower
branching ratio $\mathcal{B}(a\to\mu\bar{\mu})$. The peaks in the background
are identified as the resonances of the $J/\Psi$ and $\Psi$(2S) mesons. We
reject events with di-muon invariant masses within 5% above and below the
tabulated PDG masses of these mesons Workman:2022ynf .

Figure 4: The invariant mass distribution of the pre-selected di-muon in each
event, for various ALP signal mass benchmarks (colored lines) and background
from $t\bar{t}j$ events (gray regions). The distribution is shown after the
full event selection (upper right corner), excluding the $m_{\mu\mu}$
criterion.

In experiments, the displaced vertex is reconstructed from the decay products.
However, in this phenomenological study, we determine if two displaced muons
originate from the same vertex by calculating the proximity of their
production points. As a selection criterion, we use the difference in
transverse displacement between the two muons. To reduce the dependency on
detector resolution assumptions, we define the ratio

$\displaystyle
R_{lxy}=\frac{\sqrt{(x_{\mu}-x_{\bar{\mu}})^{2}+(y_{\mu}-y_{\bar{\mu}})^{2}}}{\sqrt{(|x_{\mu}|+|x_{\bar{\mu}}|)^{2}+(|y_{\mu}|+|y_{\bar{\mu}}|)^{2}}}$
(16)

for the transverse coordinates of the muon, $x_{\mu}$ and $y_{\mu}$, and
antimuon, $x_{\bar{\mu}}$ and $y_{\bar{\mu}}$. To ensure a smooth behavior for
$x_{\mu}=-x_{\bar{\mu}},\,y_{\mu}=-y_{\bar{\mu}}$, we use the absolute values
in the denominator of $R_{lxy}$.

The ratio of the transverse displacement between the muons, $R_{lxy}$, is
displayed in Fig. 5. As can be seen, the signal and resonant $t\bar{t}j$
background accumulate in the first bin, while the non-resonant background is
nearly flat in $R_{lxy}$. The ratio $R_{lxy}$ is therefore an efficient
selection variable to reduce the number of events with muons originating from
different production points. To account for detector resolution effects
affecting the reconstruction of the di-muons, we pick events for which the di-
muon with the smallest value of $R_{lxy}$ passes a relatively loose selection
criterion of $R_{lxy}<0.05$.

Figure 5: The ratio of the transverse displacement, $R_{lxy}$, between the
muons of the pre-selected di-muons, for various ALP signal mass benchmarks
(colored lines) and background from $t\bar{t}j$ events (gray regions). The
distribution is shown after the full event selection (upper right corner),
excluding the $R_{lxy}$ criterion.

Table 2 shows the cumulative efficiency after each applied criterion in the
event selection, in order from top to bottom, and the number of events passing
the event selection, for four benchmark signals. Similarly, the cumulative
background efficiency after each selection stage and the number of passing
events are given in Table 3. The few remaining $t\bar{t}Z^{(\ast)}$ events
after the selection mainly include muons from meson decays. The event
selection effectively reduces the overall number of background events, while
keeping the number of signal events stable.

Signal efficiency |  $m_{a}=0.35$ GeV |  $m_{a}=0.9$ GeV |  $m_{a}=2$ GeV |  $m_{a}=8$ GeV
---|---|---|---|---
Pre-selection | (8.92 $\pm$ 0.01)$\times$$10^{-1}$ | (7.94 $\pm$ 0.01)$\times$$10^{-1}$ | (6.40 $\pm$ 0.01)$\times$$10^{-1}$ | (7.25 $\pm$ 0.03)$\times$$10^{-2}$
$p_{T}^{\mu}>10$ GeV | (7.99 $\pm$ 0.01)$\times$$10^{-1}$ | (6.79 $\pm$ 0.01)$\times$$10^{-1}$ | (5.58 $\pm$ 0.01)$\times$$10^{-1}$ | (6.87 $\pm$ 0.03)$\times$$10^{-2}$
$m_{\mu\bar{\mu}}\neq m_{J/\Psi},m_{\Psi(2S)}$ | (7.99 $\pm$ 0.01)$\times$$10^{-1}$ | (6.79 $\pm$ 0.01)$\times$$10^{-1}$ | (5.58 $\pm$ 0.01)$\times$$10^{-1}$ | (6.86 $\pm$ 0.03)$\times$$10^{-2}$
$R_{lxy}<0.05$ | (7.99 $\pm$ 0.01)$\times$$10^{-1}$ | (6.79 $\pm$ 0.01)$\times$$10^{-1}$ | (5.58 $\pm$ 0.01)$\times$$10^{-1}$ | (6.86 $\pm$ 0.03)$\times$$10^{-2}$
Events passing pre-selection | 19793 $\pm$ 21 | 17697 $\pm$ 20 | 2516 $\pm$ 3 | 1.66 $\pm$ 0.01
Events passing signal selection | 17740 $\pm$ 20 | 15116 $\pm$ 18 | 2193 $\pm$ 3 | 1.57 $\pm$ 0.01

Table 2: The cumulative selection efficiency and the number of events after
the selection for four ALP signal benchmark masses, for an integrated
luminosity $\mathcal{L}=150\,$fb-1 and $\sqrt{s}=13\,$TeV. The listed
uncertainties are statistical.

Background efficiency | $t\bar{t}j$ | $t\bar{t}Z^{(\ast)}$
---|---|---
Pre-selection | (2.55 $\pm$ 0.05)$\times$$10^{-4}$ | (1.89 $\pm$ 0.04)$\times$$10^{-4}$
$p_{T}^{\mu}>10$ GeV | (7.4 $\pm$ 0.2)$\times$$10^{-5}$ | (9.4 $\pm$ 0.3)$\times$$10^{-5}$
$m_{\mu\bar{\mu}}\neq m_{J/\Psi},m_{\Psi(2S)}$ | (6.8 $\pm$ 0.2)$\times$$10^{-5}$ | (5.8 $\pm$ 0.2)$\times$$10^{-5}$
$R_{lxy}<0.05$ | (7.1 $\pm$ 0.8)$\times$$10^{-6}$ | (4.9 $\pm$ 0.7)$\times$$10^{-6}$
Events passing pre-selection | 25917 $\pm$ 458 | 0.59 $\pm$ 0.01
Events passing signal selection | 721 $\pm$ 76 | 0.015 $\pm$ 0.002

Table 3: The cumulative selection efficiency and the number of events after
the selection for the background, for an integrated luminosity
$\mathcal{L}=150\,$fb-1 and $\sqrt{s}=13\,$TeV. The listed uncertainties are
statistical.

To analyze the signal sensitivity in terms of the decay length of the ALPs, we
separate the event samples into several bins of the transverse displacement
$l_{xy}$. For this analysis, we concentrate on the tracker region and define
displacement bins up to $l_{xy}=1.3\,$m, with the bins for the pixel tracker
defined as in Ref. CMS:2021sch . The signal and background $l_{xy}$
distributions of the least displaced muon for events passing all selections
are shown in Fig. 6.

Figure 6: The transverse displacement $l_{xy}$ of the least displaced muon
after event selection, for various ALP signal benchmarks (colored lines) and
background from $t\bar{t}j$ events (gray regions). Displaced decays within the
pixel tracker (in 6 bins) and the outer tracker are shown, with the binning
used in the analysis.

In the last step, we generalize the top scenario and treat the lifetime of the
ALP as an independent parameter. To this end, we explicitly enforce
c$\tau_{a}$ for several values in the range $10^{-4}\,$ – $10^{0}\,$m. The
corresponding $l_{xy}$ distributions for $m_{a}=0.35$ GeV are shown in Fig. 7.
In this case, the distributions are binned in four large detector regions,
approximately based on the CMS detector geometry CMS:2008xjf . In the top
scenario, the ALPs decay predominantly within the tracker region (see Fig. 6),
and the background is contained within the pixel tracker. In an experiment, we
would expect some small amount of instrumental background also in the outer
tracker, calorimeters, and the muon system. However, we verified that a few
background events at higher $l_{xy}$ do not significantly affect the
sensitivity of this study. For ALPs and, more generally, for top-associated
LLPs with longer lifetimes, searches for displaced decays in the calorimeters
and in the muon system can further increase the sensitivity.

Figure 7: The transverse displacement $l_{xy}$ of the least displaced muon
after the event selection, for the $t\bar{t}j$ background (gray regions) and
ALP signal benchmarks with fixed $m_{a}=0.35\,$GeV and varying proper lifetime
$c\tau$ (colored lines). Events are normalized to unity.

##  4 Predictions for LHC Run 2 and HL-LHC

After the selection procedure outlined in Sec. 3.2, we predict the expected
sensitivity of the LHC and HL-LHC to top-associated displaced vertices. In
Sec. 4.1, we discuss our results for ALPs in the top scenario, which is
described by two parameters, $m_{a}$ and $|c_{tt}|/f_{a}$. In Sec. 4.2, we
generalize these results to pseudo-scalars with an arbitrary production rate
and lifetime. We compare the predicted sensitivity with existing searches and
indirect probes for similar scenarios in Sec. 4.3.

To calculate the expected limits, we use the combine tool CMS-NOTE-2011-005 ,
which implements the frequentist AsymptoticLimits method Cowan_2011 . Limits
were calculated for ALP mass points (including all 15 benchmarks used in this
analysis) ranging from $0.3$ to 10 GeV. Between mass points, we use a linear
interpolation.

The uncertainty in the luminosity measurement in CMS is around $2-3\%$
CMS:LUM-21-001 . However, in order to accommodate other uncertainties
affecting both signal and background in the same way (such as displaced vertex
reconstruction resolution, or di-muon mass resolution), we assume a
conservative overall systematic uncertainty of 10% in the signal efficiency
and 30% in the background efficiency when computing expected limits.

Note that all of the results are presented assuming $\sqrt{s}=13\,$TeV,
although a center-of-mass energy of 14 TeV is expected at the HL-LHC. Thus,
the results presented here for the HL-LHC are conservative, considering the
increase in the cross section expected at higher $\sqrt{s}$.

###  4.1 ALP benchmark scenario

To calculate the expected upper limits on the cross section $\sigma(pp\to
t\bar{t}a)\times\mathcal{B}(a\to\mu\bar{\mu})$ and on the ALP-top coupling
$|c_{tt}|/f_{a}$, we use the events that pass the event selection shown in
Tables 2 and 3. We categorize these events based on the displacement of the
reconstructed vertex. These categories include six bins for the pixel tracker
and one bin for the outer tracker (see Fig. 6).

The results are shown in Fig. 8, where we display the expected 95% confidence
level (CL) upper limits on $\sigma(pp\to
t\bar{t}a)\times\mathcal{B}(a\to\mu\bar{\mu})$ (left panel) and on
$|c_{tt}|/f_{a}$ (right panel), as a function of the ALP mass.

Figure 8: Expected 95% CL upper limits on the cross section times branching
ratio (left) and the ALP-top coupling $|c_{tt}(\Lambda)|/f_{a}$ (right) in the
top scenario, as a function of the ALP mass $m_{a}$. The signal rate and
lifetime of the ALP are set by the two model parameters $m_{a}$ and
$c_{tt}(\Lambda)/f_{a}$, with $f_{a}=1\,$TeV and $\Lambda=4\pi f_{a}$.
Predictions are shown for an integrated luminosity of 150 fb-1 (3 ab-1),
corresponding to the LHC Run 2 (HL-LHC). ALP masses around the $J/\Psi$ and
$\Psi(2S)$ resonances are removed by the event selection, as indicated by the
interruptions of the curves. For ALP masses above $8.25\,$GeV, the expected
event rate with $150\,$fb-1 is close to zero and these points are therefore
excluded.

The predicted limits correspond to an integrated luminosity of $150\,$fb-1
collected at the LHC during Run 2. To obtain projections for the HL-LHC, we
simply upscale the data luminosity to $3\,$ab-1. The sensitivity is highest
for small ALP masses and drops quickly at larger masses. This tendency can be
understood intuitively: light ALPs have longer lifetimes and statistically
decay in regions further away from the production point, which are less
affected by SM background (see Fig. 6). Furthermore, for light ALPs the
branching ratio into muons is larger than for heavier ALPs, where other decay
channels are kinematically accessible. This results in much lower di-muon
event rates for heavy ALPs (see Table 2). The sensitivity to heavy ALPs could
be enhanced by up to 2 orders of magnitude in certain cases by searching for
other final states, for instance for di-taus or multi-hadron final states. The
exact gain from including non-muon final states would require a dedicated
study. A dedicated search for promptly decaying pseudo-scalars produced in
association with a top-antitop pair (see Sec. 4.3) could also be sensitive to
ALPs with masses above the GeV scale, since most of them decay very close to
the production point.

As indicated in Sec. 3, throughout our analysis, we have assumed that we can
identify muons coming from the top quark decay with an efficiency of
$\epsilon_{t\bar{t}}=100\%$. We have also assumed that top quarks can be fully
reconstructed regardless of their decay chain. In reality, both signal and
background will be affected by the top reconstruction efficiency. We expect
that a limited efficiency will scale down the number of signal and background
events in a similar way, because they both contain $t\bar{t}$, and low-mass
ALPs do not affect their kinematics. This will reduce the overall sensitivity
in Fig. 8 and shift the limits to a higher effective cross section
$\sigma\times\mathcal{B}/\epsilon_{t\bar{t}}$. In a similar way, the choice of
the top decays in an analysis will scale the reach proportionally to the top
branching ratio. For instance, if the overall top-pair reconstruction
efficiency was at the level of 10%, the limits on $\sigma\times\mathcal{B}$ in
Fig. 8 would shift upwards by about an order of magnitude.

###  4.2 General pseudo-scalars with top couplings

Using the predicted event rates for arbitrary lifetimes of the ALP, see Fig.
7, we interpret our analysis for general pseudo-scalar particles with
couplings to top quarks.444The shift symmetry $a\to a+c$ of the ALP Lagrangian
(6) does not affect the event kinematics in $pp\to t\bar{t}a$ production. An
ALP with generic couplings and lifetime has therefore the same phenomenology
as any pseudo-scalar that does not feature a shift symmetry. Unlike in the top
scenario, the cross section and lifetime of the pseudo-scalar are not
correlated.

In Figure 9, we show the expected 95% CL upper limits on the proper decay
length of the pseudo-scalar, $c\tau_{a}$, as a function of its mass, $m_{a}$,
for different assumptions on $\sigma(pp\to
t\bar{t}a)\times\mathcal{B}(a\to\mu\bar{\mu})$. With data corresponding to
$150\,$fb-1 of integrated luminosity and cross sections around $1\,$fb,
pseudo-scalars with proper lifetimes up to around 1–$10\,$m can be probed in
the considered mass range. For longer lifetimes, a large fraction of particles
decays outside of the detector and escapes the analysis acceptance. At the HL-
LHC with an integrated luminosity of $3\,$ab-1, the reach can be extended by
more than an order of magnitude in the proper lifetime. For very short
lifetimes of $c\tau_{a}\lesssim 100\,\mu$m, the signal is difficult to
distinguish from the meson-induced background. As a consequence, at lifetimes
below those displayed in Fig. 9, the signal sensitivity drops.

Figure 9: The expected sensitivity for a pseudo-scalar with proper lifetime
$c\tau_{a}$ and mass $m_{a}$, for fixed values of the signal cross section
$\sigma(pp\to t\bar{t}a)\times\mathcal{B}(a\to\mu\bar{\mu})$. The shaded areas
can be excluded at 95% CL. The left (right) vertical axis corresponds to an
integrated luminosity of 150 fb-1 (3 ab-1).

###  4.3 Comparison with existing searches

In searches for long-lived particles, the sensitivity depends on both the
signal cross section and the decay length. A comparison with searches with
prompt or invisible final states is therefore not straightforward, but
possible within certain limitations. Here we attempt to compare our
predictions for top-associated displaced di-muons with (proposed) searches for
similar ALP scenarios at the LHC and other experiments. Most of these searches
are complementary in mass and/or lifetime to this study.

##### Promptly decaying pseudo-scalars in top-antitop production

To obtain an optimal sensitivity across a large range of lifetimes, LHC
searches for displaced vertices should be combined with searches for promptly
decaying resonances and searches with missing energy. The ATLAS and CMS
collaborations have performed searches for prompt di-muons from pseudo-scalars
produced via $pp\to t\bar{t}a,\,a\to\mu\bar{\mu}$ ATLAS:2023ofo ; CMS:2019lwf
; CMS:2022arx . The ATLAS analysis ATLAS:2023ofo  selects prompt di-muons with
invariant masses $15\,$GeV$\,<m_{\mu\mu}<72\,$GeV from final states with
multi-leptons and jets. For pseudo-scalars with masses in this range, the
search is sensitive to cross sections $\sigma(pp\to
t\bar{t}a)\times\mathcal{B}(a\to\mu\bar{\mu})$ of a few femtobarns with
$139\,$fb-1 of data. CMS achieves a similar sensitivity within the same mass
range CMS:2022arx . The reach is comparable with our predictions for displaced
di-muons from pseudo-scalars with $2m_{\mu}<m_{a}<10\,$GeV and proper decay
lengths $1\,\text{m}<c\tau_{a}<10\,$m in $150\,$fb-1 of data (see Fig. 9). For
pseudo-scalars with decay lengths $c\tau\lesssim 1\,$m, a larger sensitivity
in cross section can be reached with displaced-vertex searches, provided that
the displacement is large enough to be resolved.

##### Invisible pseudo-scalars in top-pair production

Searches for top quarks and missing energy in $t\bar{t}$ production
Haisch:2016gry  and single-top production Pinna:2017tay ; Plehn:2017bys  have
been proposed to probe (pseudo-)scalars with top couplings. The corresponding
searches by ATLAS and CMS CMS:2015zwg ; CMS:2017dcx ; CMS:2017qxu ;
CMS:2019zzl ; CMS:2018ysw ; ATLAS:2018cjd ; ATLAS:2014bba ; ATLAS:2018nda ;
ATLAS:2017hoo ; ATLAS:2014dbf  target heavy particles well above $10\,$GeV. In
Ref. Esser:2023fdo , the authors have re-interpreted an ATLAS search for top-
pairs and missing energy ATLAS:2021hza  for detector-stable ALPs in a broader
mass range (see also Ref. Ebadi:2019gij ). They find an upper bound on the
ALP-top coupling, $|c_{tt}|/f_{a}<1.8/$TeV at $95\%\,$CL, provided that the
ALP escapes detection. In the top scenario, this condition applies for ALPs
with masses well below the di-muon threshold. Missing energy and displaced
vertex searches therefore probe complementary mass regimes of ALPs with top
couplings. Alternatively, the ALP-top coupling can be probed indirectly
through NLO effects in $t\bar{t}$ production Galda:2021hbr ; Esser:2023fdo .
The sensitivity is comparable to top-associated missing energy searches.

##### Displaced ALPs from top decays

In Ref. Carmona:2022jid , the authors have proposed a search for displaced
ALPs produced from top decays via flavor-changing top-charm and top-up
couplings at the LHC. The analysis focuses on the process $pp\to
t\bar{t},\,t\to aq$, with subsequent hadronic ALP decays. The signature
consists of a trackless jet that could be detected in the hadronic calorimeter
or the muon spectrometer. Similar to our strategy, the authors suggest using
the top quark to trigger on these types of events. For $t\bar{t}$ events, this
analysis is sensitive to top quark branching ratios $\mathcal{B}(t\to
aq)\lesssim 10^{-4}$ for hadronically decaying ALPs with masses above
$1\,$GeV. A recast of a CMS search for flavor-changing top-quark couplings in
single top events yields a sensitivity to $\mathcal{B}(t\to aq)\approx
10^{-3}$. The sensitivity to flavor-changing ALP couplings is comparable to
our prediction for flavor-diagonal ALP couplings. In this way, both analyses
complement each other in probing the parameter space of the ALP effective
Lagrangian.

##### Inclusive di-muon searches

An alternative to top-associated di-muons is inclusive searches for
(displaced) di-muon resonances. Several displaced di-muon searches have probed
resonances with masses well above $10\,$GeV CMS:2022qej ; ATLAS:2018rjc . To
access di-muons with smaller invariant masses, a dedicated search for
displaced di-muon vertices using a high-rate data stream (“scouting”, “turbo”,
or “trigger-level” analysis) has been developed CMS:2021sch . On the other
hand, such scouting searches store reduced detector information, thus
deteriorating the tracking performance and limiting the range of observable
displacements. Moreover, di-muon searches based on scouting data do not allow
one to reconstruct additional top quarks. Therefore, we expect that inclusive
di-muon searches are less sensitive to top-philic ALPs or other light
particles, due to a larger amount of background. Notice, however, that a
direct comparison of inclusive and top-associated di-muon searches is largely
model-dependent because the relative event rates depend strongly on the
relevant couplings of the particles.

##### ALP signatures with virtual top quarks

Indirect bounds on the ALP-top coupling can be obtained by exploring loop-
induced ALP couplings to other particles through virtual tops Bauer:2020jbp ;
Bonilla:2021ufe . As mentioned in Sec. 2.2, an ALP-top coupling at the cutoff
scale $\Lambda=4\pi f_{a}$ of the ALP Effective Field Theory induces ALP
couplings to all other SM particles at lower energies. At the LHC, effective
ALP couplings to photons, weak gauge bosons, and the Higgs can be probed in,
for instance, di-boson production and $h\to Za$ decays Bauer:2017ris ;
Gavela:2019cmq ; CMS:2016369 . In the top scenario, the derived indirect
bounds on the ALP-top coupling range around $|c_{tt}|/f_{a}\lesssim
10-100/$TeV at $95\%\,$CL Esser:2023fdo . With the current experimental
status, the sensitivity to the ALP coupling to virtual tops is weaker than in
searches with resonant tops.

A very sensitive probe of ALPs below the GeV scale is meson decays like $B\to
Ka$, induced through top-quark loops. For flavor-universal couplings
$c_{f\\!f}/f_{a}$, the sensitivity of ALP decays into muons is stronger than
in top-associated ALP production Ferber:2022rsf . In the top scenario,
however, the decay length of the ALP is comparably larger, so that searches
for displaced di-muons from meson decays lose sensitivity. Proposed searches
for $B\to Ka$ decays with missing energy at Belle II Ferber:2022rsf  offer an
interesting alternative to probe the top scenario. A dedicated comparison of
flavor and high-energy observables in this scenario would give valuable
information on future prospects.

As described in this section, there exist a large variety of searches for
ALPs. However, all of them probe different regions of phase space (including
lower or higher masses), prompt decays, or do not require top-antitop pair in
an event, which results in overwhelming backgrounds. Therefore, we believe
that the analysis strategy proposed in this work is complementary to existing
studies and would nicely fit in the landscape of searches for ALPs.

##  5 Summary and outlook

This work aimed to establish the discovery potential of top-associated
displaced vertices at the LHC. To this end, we have performed an analysis of
ALPs produced in association with a top-antitop pair and decaying into a
displaced di-muon. In a benchmark study, the so-called top scenario, we have
assumed that the ALP couples only to top quarks at high energies, so that the
effective top coupling $c_{tt}/f_{a}$ and the mass $m_{a}$ of the ALP are the
only model parameters.

Based on simulation, we have designed a dedicated selection procedure to
discriminate between the top-associated di-muon signal and the dominant
background from $t\bar{t}Z^{(\ast)}$ and $t\bar{t}j$ production. For a fixed
ALP coupling $c_{tt}/f_{a}=1$/TeV, we have quantified step by step how the
event selection efficiently reduces the background while preserving most of
the signal. A crucial discriminator is the displacement of the reconstructed
di-muon vertex, which suppresses much of the background from meson decays with
observable displacements.

Assuming 100% effective identification of muons from top quarks and 100%
trigger efficiency for top quarks, as well as including all top quark decay
modes, we derive the expected reach of the LHC for top-philic ALPs in Fig. 8
and for generalized long-lived pseudo-scalars in Fig. 9. In the top scenario,
the (HL-)LHC is sensitive to ALP-top couplings as small as
$c_{tt}/f_{a}\approx 0.03(0.002/$TeV) for ALP masses $2m_{\mu}<m_{a}\lesssim
2\,$GeV with $150\,$fb-1 ($3\,$ab-1). More generally, top-philic pseudo-
scalars with proper lifetimes up to $20\,$m ($300\,$m) and cross sections
around $1\,$fb can be probed at the (HL-)LHC.

Our results suggest that searches for top-associated displaced di-muons bridge
the current sensitivity gap in the lifetime between searches with prompt di-
muons (short lifetimes) and those with missing energy (detector-stable
particles). In many scenarios, the lifetime of a particle is anti-correlated
with its mass. This suggests that top-associated displaced di-muons will lead
the sensitivity to top-philic ALPs within the range $2m_{\mu}<m_{a}\lesssim
2\,$GeV at the LHC. From an experimental perspective, displaced vertex
searches in association with reconstructed top quarks offer new options for
triggering. This opens new opportunities to extend the discovery potential of
the LHC to feebly interacting particles well below the weak scale.

###### Acknowledgements.

JA, FB, JN, and LR acknowledge support from DESY (Hamburg, Germany), a member
of the Helmholtz Association HGF, and support by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy – EXC 2121 "Quantum Universe" – 390833306. RS acknowledges
support of the DFG through the research training group Particle Physics Beyond
the Standard Model (GRK 1940). The research of SB and SW has been supported by
the DFG under grant no. 396021762–TRR 257.

## References

  * (1) J. Alimena et al., Searching for long-lived particles beyond the Standard Model at the Large Hadron Collider, J. Phys. G 47 (2020), no. 9 090501, [arXiv:1903.04497].
  * (2) CMS Collaboration, Search for long-lived particles decaying to a pair of muons in proton-proton collisions at $\sqrt{s}$ = 13 TeV, arXiv:2205.08582. Accepted by JHEP.
  * (3) CMS Collaboration, A. Tumasyan et al., Search for long-lived particles decaying into muon pairs in proton-proton collisions at $\sqrt{s}$ = 13 TeV collected with a dedicated high-rate data stream, JHEP 04 (2022) 062, [arXiv:2112.13769].
  * (4) CMS Collaboration, A. M. Sirunyan et al., Search for long-lived particles decaying to jets with displaced vertices in proton-proton collisions at $\sqrt{s}=$ 13 TeV, Phys. Rev. D 104 (2021), no. 5 052011, [arXiv:2104.13474].
  * (5) CMS Collaboration, A. M. Sirunyan et al., Search for long-lived particles using displaced jets in proton-proton collisions at $\sqrt{s}=$ 13 TeV, Phys. Rev. D 104 (2021), no. 1 012015, [arXiv:2012.01581].
  * (6) ATLAS Collaboration, M. Aaboud et al., Search for long-lived particles in final states with displaced dimuon vertices in $pp$ collisions at $\sqrt{s}=$ 13 TeV with the ATLAS detector, Phys. Rev. D 99 (2019), no. 1 012001, [arXiv:1808.03057].
  * (7) ATLAS Collaboration, G. Aad et al., Search for displaced vertices of oppositely charged leptons from decays of long-lived particles in $pp$ collisions at $\sqrt{s}$ =13 TeV with the ATLAS detector, Phys. Lett. B 801 (2020) 135114, [arXiv:1907.10037].
  * (8) ATLAS Collaboration, Search for long-lived, massive particles in events with displaced vertices and multiple jets in $pp$ collisions at $\sqrt{s}=13$ TeV with the ATLAS detector, arXiv:2301.13866.
  * (9) ATLAS Collaboration, G. Aad et al., Search for long-lived, massive particles in events with a displaced vertex and a muon with large impact parameter in $pp$ collisions at $\sqrt{s}=13$ TeV with the ATLAS detector, Phys. Rev. D 102 (2020), no. 3 032006, [arXiv:2003.11956].
  * (10) ATLAS Collaboration, G. Aad et al., Search for light long-lived neutral particles produced in $pp$ collisions at $\sqrt{s}=$ 13 TeV and decaying into collimated leptons or light hadrons with the ATLAS detector, Eur. Phys. J. C 80 (2020), no. 5 450, [arXiv:1909.01246].
  * (11) ATLAS Collaboration, M. Aaboud et al., Search for long-lived neutral particles in $pp$ collisions at $\sqrt{s}$ = 13 TeV that decay into displaced hadronic jets in the ATLAS calorimeter, Eur. Phys. J. C 79 (2019), no. 6 481, [arXiv:1902.03094].
  * (12) LHCb Collaboration, R. Aaij et al., Updated search for long-lived particles decaying to jet pairs, Eur. Phys. J. C 77 (2017), no. 12 812, [arXiv:1705.07332].
  * (13) ATLAS Collaboration, M. Aaboud et al., Search for long-lived particles produced in $pp$ collisions at $\sqrt{s}=13$ TeV that decay into displaced hadronic jets in the ATLAS muon spectrometer, Phys. Rev. D 99 (2019), no. 5 052005, [arXiv:1811.07370].
  * (14) ATLAS Collaboration, G. Aad et al., Search for long-lived neutral particles produced in $pp$ collisions at $\sqrt{s}=13$ TeV decaying into displaced hadronic jets in the ATLAS inner detector and muon spectrometer, Phys. Rev. D 101 (2020), no. 5 052013, [arXiv:1911.12575].
  * (15) ATLAS Collaboration, G. Aad et al., Search for events with a pair of displaced vertices from long-lived neutral particles decaying into hadronic jets in the ATLAS muon spectrometer in pp collisions at $\sqrt{s}$=13 TeV, Phys. Rev. D 106 (2022), no. 3 032005, [arXiv:2203.00587].
  * (16) ATLAS Collaboration, G. Aad et al., Search for neutral long-lived particles in $pp$ collisions at $\sqrt{s}$ = 13 TeV that decay into displaced hadronic jets in the ATLAS calorimeter, JHEP 06 (2022) 005, [arXiv:2203.01009].
  * (17) ATLAS Collaboration, G. Aad et al., Search for exotic decays of the Higgs boson into long-lived particles in $pp$ collisions at $\sqrt{s}$ = 13 TeV using displaced vertices in the ATLAS inner detector, JHEP 11 (2021) 229, [arXiv:2107.06092].
  * (18) ATLAS Collaboration, M. Aaboud et al., Search for the Higgs boson produced in association with a vector boson and decaying into two spin-zero particles in the $H\rightarrow aa\rightarrow 4b$ channel in $pp$ collisions at $\sqrt{s}=13$ TeV with the ATLAS detector, JHEP 10 (2018) 031, [arXiv:1806.07355].
  * (19) CMS Collaboration, A. Tumasyan et al., Search for long-lived particles produced in association with a Z boson in proton-proton collisions at $\sqrt{s}$ = 13 TeV, JHEP 03 (2022) 160, [arXiv:2110.13218].
  * (20) LHCb Collaboration, R. Aaij et al., Search for Higgs-like bosons decaying into long-lived exotic particles, Eur. Phys. J. C 76 (2016), no. 12 664, [arXiv:1609.03124].
  * (21) D. Acosta et al., Review of opportunities for new long-lived particle triggers in Run 3 of the Large Hadron Collider, arXiv:2110.14675.
  * (22) CMS Collaboration, Search for long-lived heavy neutral leptons with lepton flavour conserving or violating decays to a jet and an electron, muon, or tau lepton, Tech. Rep. CMS-PAS-EXO-21-013, 2023.  http://cds.cern.ch/record/2852843.
  * (23) CMS Collaboration, A. Tumasyan et al., Search for long-lived heavy neutral leptons with displaced vertices in proton-proton collisions at $\sqrt{\mathrm{s}}$ =13 TeV, JHEP 07 (2022) 081, [arXiv:2201.05578].
  * (24) ATLAS Collaboration, G. Aad et al., Search for heavy neutral leptons in decays of $W$ bosons produced in 13 TeV $pp$ collisions using prompt and displaced signatures with the ATLAS detector, JHEP 10 (2019) 265, [arXiv:1905.09787].
  * (25) C. D. Froggatt and H. B. Nielsen, Hierarchy of Quark Masses, Cabibbo Angles and CP Violation, Nucl. Phys. B 147 (1979) 277–298. http://cds.cern.ch/record/133050.
  * (26) H. Georgi, D. B. Kaplan, and L. Randall, Manifesting the Invisible Axion at Low-energies, Phys. Lett. B 169 (1986) 73–78. https://doi.org/10.1016/0370-2693(86)90688-X.
  * (27) T. Gherghetta and A. Pomarol, Bulk fields and supersymmetry in a slice of AdS, Nucl. Phys. B 586 (2000) 141–162, [hep-ph/0003129].
  * (28) K. Agashe, R. Contino, and A. Pomarol, The Minimal composite Higgs model, Nucl. Phys. B 719 (2005) 165–187, [hep-ph/0412089].
  * (29) M. Bauer, M. Neubert, S. Renner, M. Schnubel, and A. Thamm, Flavor probes of axion-like particles, JHEP 09 (2022) 056, [arXiv:2110.10698].
  * (30) F. Esser, M. Madigan, V. Sanz, and M. Ubiali, On the coupling of axion-like particles to the top quark, arXiv:2303.17634.
  * (31) ATLAS Collaboration, G. Aad et al., Search for a new pseudoscalar decaying into a pair of muons in events with a top-quark pair at $\sqrt{s}=13$~TeV with the ATLAS detector, arXiv:2304.14247.
  * (32) CMS Collaboration, A. M. Sirunyan et al., Search for physics beyond the standard model in multilepton final states in proton-proton collisions at $\sqrt{s}=$ 13 TeV, JHEP 03 (2020) 051, [arXiv:1911.04968].
  * (33) CMS Collaboration, Search for dilepton resonances from decays of (pseudo)scalar bosons produced in association with a massive vector boson or top quark anti-top quark pair at $\sqrt{s}=13~{}\mathrm{TeV}$, . https://cds.cern.ch/record/2815307.
  * (34) CMS Collaboration, V. Khachatryan et al., Search for the production of dark matter in association with top-quark pairs in the single-lepton final state in proton-proton collisions at sqrt(s) = 8 TeV, JHEP 06 (2015) 121, [arXiv:1504.03198].
  * (35) CMS Collaboration, A. M. Sirunyan et al., Search for dark matter produced in association with heavy-flavor quark pairs in proton-proton collisions at $\sqrt{s}=13$ TeV, Eur. Phys. J. C 77 (2017), no. 12 845, [arXiv:1706.02581].
  * (36) CMS Collaboration, A. M. Sirunyan et al., Search for supersymmetry in proton-proton collisions at 13 TeV using identified top quarks, Phys. Rev. D 97 (2018), no. 1 012007, [arXiv:1710.11188].
  * (37) CMS Collaboration, A. M. Sirunyan et al., Search for dark matter produced in association with a single top quark or a top quark pair in proton-proton collisions at $\sqrt{s}=13$ TeV, JHEP 03 (2019) 141, [arXiv:1901.01553].
  * (38) CMS Collaboration, A. M. Sirunyan et al., Search for dark matter particles produced in association with a top quark pair at $\sqrt{s}=$ 13 TeV, Phys. Rev. Lett. 122 (2019), no. 1 011803, [arXiv:1807.06522].
  * (39) ATLAS Collaboration, M. Aaboud et al., Search for large missing transverse momentum in association with one top-quark in proton-proton collisions at $\sqrt{s}$ = 13 TeV with the ATLAS detector, JHEP 05 (2019) 041, [arXiv:1812.09743].
  * (40) ATLAS Collaboration, G. Aad et al., Search for invisible particles produced in association with single-top-quarks in proton-proton collisions at $\sqrt{s}$ = 8 TeV with the ATLAS detector, Eur. Phys. J. C 75 (2015), no. 2 79, [arXiv:1410.5404].
  * (41) ATLAS Collaboration, M. Aaboud et al., Search for dark matter in events with a hadronically decaying vector boson and missing transverse momentum in $pp$ collisions at $\sqrt{s}=13$ TeV with the ATLAS detector, JHEP 10 (2018) 180, [arXiv:1807.11471].
  * (42) ATLAS Collaboration, M. Aaboud et al., Search for dark matter produced in association with bottom or top quarks in $\sqrt{s}=13$ TeV pp collisions with the ATLAS detector, Eur. Phys. J. C 78 (2018), no. 1 18, [arXiv:1710.11412].
  * (43) ATLAS Collaboration, G. Aad et al., Search for dark matter in events with heavy quarks and missing transverse momentum in $pp$ collisions with the ATLAS detector, Eur. Phys. J. C 75 (2015), no. 2 92, [arXiv:1410.4031].
  * (44) A. Carmona, F. Elahi, C. Scherb, and P. Schwaller, The ALPs from the top: searching for long lived axion-like particles from exotic top decays, JHEP 07 (2022) 122, [arXiv:2202.09371].
  * (45) K. Choi, S. H. Im, C. B. Park, and S. Yun, Minimal Flavor Violation with Axion-like Particles, JHEP 11 (2017) 070, [arXiv:1708.00021].
  * (46) J. Martin Camalich, M. Pospelov, P. N. H. Vuong, R. Ziegler, and J. Zupan, Quark Flavor Phenomenology of the QCD Axion, Phys. Rev. D 102 (2020), no. 1 015023, [arXiv:2002.04623].
  * (47) M. Chala, G. Guedes, M. Ramos, and J. Santiago, Running in the ALPs, Eur. Phys. J. C 81 (2021), no. 2 181, [arXiv:2012.09017].
  * (48) M. Bauer, M. Neubert, S. Renner, M. Schnubel, and A. Thamm, The Low-Energy Effective Theory of Axions and ALPs, JHEP 04 (2021) 063, [arXiv:2012.12272].
  * (49) R. Schäfer, “ttalps.” https://gitlab.com/p8413/ttalps, 2022.
  * (50) M. Bauer, M. Neubert, S. Renner, M. Schnubel, and A. Thamm, Consistent Treatment of Axions in the Weak Chiral Lagrangian, Phys. Rev. Lett. 127 (2021), no. 8 081803, [arXiv:2102.13112].
  * (51) T. Ferber, A. Filimonova, R. Schäfer, and S. Westhoff, Displaced or invisible? ALPs from B decays at Belle II, JHEP 04 (2023) 131, [arXiv:2201.06580].
  * (52) S. Bruggisser, L. Grabitz, and S. Westhoff, Global Analysis of the ALP Effective Theory, arXiv:2308.11703.
  * (53) A. Biekötter, J. Fuentes-Martín, A. M. Galda, and M. Neubert, A global analysis of axion-like particle interactions using SMEFT fits, arXiv:2307.10372.
  * (54) J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H.-S. Shao, T. Stelzer, P. Torrielli, and M. Zaro, The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations, JHEP 2014 (2014), no. 7 [arXiv:1405.0301].
  * (55) R. D. Ball, V. Bertone, S. Carrazza, L. D. Debbio, S. Forte, P. Groth-Merrild, A. Guffanti, N. P. Hartland, Z. Kassabov, J. I. Latorre, E. R. Nocera, J. Rojo, L. Rottoli, E. Slade, and M. Ubiali, Parton distributions from high-precision collider data, Eur. Phys. J. C 77 (2017), no. 10 [arXiv:1706.00428].
  * (56) T. Sjöstrand, S. Ask, J. R. Christiansen, R. Corke, N. Desai, P. Ilten, S. Mrenna, S. Prestel, C. O. Rasmussen, and P. Z. Skands, An introduction to PYTHIA 8.2, Comput. Phys. Commun. 191 (2015) 159, [arXiv:1410.3012].
  * (57) P. Skands, S. Carrazza, and J. Rojo, Tuning PYTHIA 8.1: the monash 2013 tune, Eur. Phys. J. C 74 (2014), no. 8 [arXiv:1404.5630].
  * (58) CMS Collaboration, V. Khachatryan et al., Measurement of differential cross sections for top quark pair production using the lepton+jets final state in proton-proton collisions at 13 TeV, Phys. Rev. D 95 (2017), no. 9 092001, [arXiv:1610.04191].
  * (59) CMS Collaboration, S. Chatrchyan et al., Description and performance of track and primary-vertex reconstruction with the CMS tracker, JINST 9 (2014), no. 10 P10009, [arXiv:1405.6569].
  * (60) Particle Data Group Collaboration, R. L. Workman et al., Review of Particle Physics, PTEP 2022 (2022) 083C01. https://doi.org/10.1093/ptep/ptac097.
  * (61) CMS Collaboration, S. Chatrchyan et al., The CMS experiment at the CERN LHC, JINST 3 (2008) S08004. https://doi.org/10.1088/1748-0221/3/08/S08004.
  * (62) The ATLAS and CMS Collaborations and The LHC Higgs Combination Group Collaboration, Procedure for the LHC Higgs boson search combination in Summer 2011, tech. rep., CERN, Geneva, 2011.  https://cds.cern.ch/record/1379837.
  * (63) G. Cowan, K. Cranmer, E. Gross, and O. Vitells, Asymptotic formulae for likelihood-based tests of new physics, Eur. Phys. J. C 71 (2011) 1554, [arXiv:1007.1727]. [Erratum: Eur.Phys.J.C 73, 2501 (2013)].
  * (64) CMS Collaboration, CMS Collaboration, Luminosity determination using z boson production at the CMS experiment, CMS Physics Analysis Summary CMS-PAS-LUM-21-001, 2023.  https://cds.cern.ch/record/2852905.
  * (65) U. Haisch, P. Pani, and G. Polesello, Determining the CP nature of spin-0 mediators in associated production of dark matter and $t\overline{t}$ pairs, JHEP 02 (2017) 131, [arXiv:1611.09841].
  * (66) D. Pinna, A. Zucchetta, M. R. Buckley, and F. Canelli, Single top quarks and dark matter, Phys. Rev. D 96 (2017), no. 3 035031, [arXiv:1701.05195].
  * (67) T. Plehn, J. Thompson, and S. Westhoff, Dark Matter from Electroweak Single Top Production, Phys. Rev. D 98 (2018), no. 1 015012, [arXiv:1712.08065].
  * (68) ATLAS Collaboration, G. Aad et al., Search for new phenomena in events with two opposite-charge leptons, jets and missing transverse momentum in pp collisions at $\sqrt{\mathrm{s}}$ = 13 TeV with the ATLAS detector, JHEP 04 (2021) 165, [arXiv:2102.01444].
  * (69) J. Ebadi, S. Khatibi, and M. Mohammadi Najafabadi, New probes for axionlike particles at hadron colliders, Phys. Rev. D 100 (2019), no. 1 015016, [arXiv:1901.03061].
  * (70) A. M. Galda, M. Neubert, and S. Renner, ALP — SMEFT interference, JHEP 06 (2021) 135, [arXiv:2105.01078].
  * (71) J. Bonilla, I. Brivio, M. B. Gavela, and V. Sanz, One-loop corrections to ALP couplings, JHEP 11 (2021) 168, [arXiv:2107.11392].
  * (72) M. Bauer, M. Neubert, and A. Thamm, Collider Probes of Axion-Like Particles, JHEP 12 (2017) 044, [arXiv:1708.00443].
  * (73) M. B. Gavela, J. M. No, V. Sanz, and J. F. de Trocóniz, Nonresonant Searches for Axionlike Particles at the LHC, Phys. Rev. Lett. 124 (2020), no. 5 051802, [arXiv:1905.12953].
  * (74) CMS Collaboration, V. Khachatryan et al., Search for neutral resonances decaying into a Z boson and a pair of b jets or $\tau$ leptons, Phys. Lett. B 759 (2016) 369–394, [arXiv:1603.02991].