text:
\section{\label{sec:level1}Introduction}

Recently, there has been great interest in models of
beyond-the-Standard Model (BSM) physics involving new
low-mass matter and mediator particles.  A promising
method for testing these models is the use of beam
experiments, where the focus is typically on processes
with relatively low momentum transfer, in which the
low-mass mediators are produced at high longitudinal
boost (see, for example,~\cite{Batell:2017kty,Bauer:2018onh,Abdullah:2018ykz}).
Indeed, these types of processes are the typical
production mechanisms for mediators which arise in
the low-energy effective field theory.
The visible decays of long-lived mediators can then be
observed at displaced detectors (e.g., FASER~\cite{FASER:2019aik}), with the intervening
shielding serving to remove the background arising from the beam itself.

The Large Hadron Collider (LHC), however, excels at producing new heavy particles in high momentum transfer processes. Thus, current LHC searches for light mediators $a$ (e.g. axion-like particles) are carried out in the context of SM Higgs decays ($H \to aa(Za)$), $B$ meson decays ($B \to Xa$), or in proton-proton ($pp$) production mechanisms resulting in associated top quarks. One strategy often pursued in those analyses is a dilepton selection targeting the $a \to \ell\ell$ decay mode. However, the model parameter space (mass, lifetime, and coupling) that can be probed by those searches is often limited, especially for an MeV-scale mediator, where $a \to \ell\ell$ is not kinematically allowed. The sensitivity to new physics models with low mass mediators may be further suppressed by the low momentum attributed to the $a$ decay products, making it difficult to suppress SM backgrounds and experimentally trigger on interesting events.

In this paper, we devise a LHC search strategy for a light mediator particle produced in association with a TeV-scale top-partner particle. The heavy top-partner particle can be copiously produced at the LHC because it couples to the SM quarks and gluons, and its highly energetic decay products can be observed in association with a mediator particle containing substantial transverse momentum. If the mediator decays promptly into the SM particles, then the resulting SM particles will be detected in the central region of the detector and will be energetic enough to significantly reduce SM backgrounds. As a result of this search strategy, the LHC can expand the reach of heavy top-partners and may discover even MeV-scale mediators, which are otherwise difficult to probe at a hadron collider using traditional search strategies.

We illustrate the feasibility of probing such an MeV-scale mediator particle at the LHC using a benchmark model.
The benchmark model is a low-scale model of new physics (a new gauge group
$U(1)_{T3R}$~\cite{Pati:1974yy,Mohapatra:1974gc,Senjanovic:1975rk} with a sub-GeV dark
Higgs boson~\cite{Dutta:2019fxn,Dutta:2020enk,Dutta:2021afo}) along with a UV completion and the universal seesaw mechanism~\cite{Berezhiani:1983hm,Chang:1986bp,Davidson:1987mh,Rajpoot:1987fca,Babu:1988mw,Babu:1989rb}.
Although this model is well-motivated and realizable in nature, we use it as a benchmark scenario to define the
strategies for discovering the decays of light particles at the LHC.
Here, we introduce a gauge group $U(1)_{T3R}$ under
which only a set of right-handed SM fermions are charged.
Because this symmetry now protects the fermion masses, the dark Higgs boson
($\phi$), whose vacuum expectation value (vev) breaks $U(1)_{T3R}$, must couple to the SM fermions.
The particle excitation about this vev ($\phi'$) can be light, even
sub-GeV. Since both the SM Higgs boson ($V$) and the dark Higgs boson ($\phi'$) must couple to the SM fermions, the effective Yukawa coupling is actually a nonrenormalizable dimension-5 operator.  One can write a UV completion of this theory using the universal seesaw mechanism, in which one adds heavy vector-like fermions. A vector-like fermion can couple to the corresponding SM fermion via exchange of $H$ or $\phi'$. The effective Yukawa couplings then arise from integrating out the heavy fermions. We consider the case in which the top quark ($t$) couples to a light scalar $\phi'$ and a heavy vector-like quark ($T$). The new heavy quarks of this model can be produced at the LHC, and their decay products observed in association with an energetic $\phi'$ in the central regions of the CMS~\cite{Chatrchyan:2008aa} and ATLAS~\cite{Aad:2008zzm} detectors.
The decay $\phi' \rightarrow \gamma \gamma$ will then produce a pair of collimated photons.  The large SM background can thus be reduced by requiring the photons to be energetic, and by tagging $t$ and bottom quarks ($b$) which also necessarily appear.

The rest of this paper is structured as follows. In Section II, we describe the details of our new physics model. Section III provides an overview of current results from searches for vector-like fermions at the LHC. Section IV provides details of how the Monte Carlo samples were produced for this study. In Section V we discuss event selection criteria, and in Section VI, main results. We end with a short discussion in Section VII.

\section{$U(1)_{T3R}$ and the Universal Seesaw Mechanism}

We consider a scenario in which a set of right-handed SM fermions are charged under the gauge group $U(1)_{T3R}$, with up-type and down-type fermions having charge $\pm Q$.  Provided
that we couple an up-type quark, down-type quark, charged
lepton, and right-handed neutrino to $U(1)_{T3R}$, all
gauge anomalies will manifestly cancel.  For the purpose of
our analysis, we assume that right-handed
$t$-quarks couple to
$U(1)_{T3R}$.  The identity of the other fermions will not
affect our analysis; for simplicity, we will refer to the
down-type quark, charged lepton, and right-handed neutrino
as $b$, $\ell$, and $\nu_R$ respectively.

We assume that a dark Higgs field $\phi$ also has
charge $Q$ under $U(1)_{T3R}$ and is neutral under all
other gauge symmetries.  $U(1)_{T3R}$ is
spontaneously broken when this field gets a vev
$\langle \phi \rangle =V$.  We may write the excitation
about this vev as $\phi = V + (1/\sqrt{2}) \phi'$,
where $\phi'$ is a real scalar field (the imaginary
part of the excitation about the vev is the Goldstone mode,
and is absorbed into the longitudinal polarization of the
dark photon, which is the gauge boson of
$U(1)_{T3R}$).

Each charged fermion which couples to
$U(1)_{T3R}$ has a mass which is now protected by
both $U(1)_{T3R}$ and $SU(2)_L \times U(1)_Y$, and thus
is proportional to the Higgs vev ($v$) and the
dark Higgs vev ($V$)~\footnote{The mass terms for the
neutrino are more complicated, since the neutrino can
also have a Majorana mass term.  For our analysis, we
will not need to consider them further.}.  In the
effective field theory defined at the electroweak
scale, the charged
fermion masses thus arise from dimension 5 operators
given by
\begin{eqnarray}
{\cal L}_{\rm eff} &=& ... - \frac{\lambda_f}{\Lambda_f}
H^* \phi (\bar f P_R f ) + h.c.,
\end{eqnarray}
with
\begin{eqnarray}
m_f &=& \frac{\lambda_f \langle H \rangle
\langle \phi \rangle}{\Lambda_f} .
\end{eqnarray}
This non-renormalizable operator  can be
derived from a UV-complete renormalizable theory if
we introduce a new vector-like fermion $\tilde \chi_f$, which
is a singlet under $U(1)_{T3R}$ and has the
same charges under the SM gauge group
$SU(3)_C \times SU(2)_L \times U(1)_Y$ as the
right-handed fermion $f_R$.

In the UV-complete theory, we may write the operators
\begin{eqnarray}
{\cal L} &=& ... - \tilde m_{\chi_f} \bar{\tilde{\chi}}_f
\tilde \chi_f
- \lambda_{fL} H (\bar{\tilde{\chi}}_f P_L \tilde f_L)
\nonumber\\
&\,&
- \lambda_{fR} \phi^* (\bar{\tilde{\chi}}_f P_R \tilde f)
+h.c.,
\end{eqnarray}
where $\tilde f$ is a flavor eigenstate, and
$\tilde f_L$ is the $SU(2)_L$ doublet of which
$\tilde f$ is
a component.  Taking the vevs and couplings to be real, these  Lagrangian terms yield a mass
matrix for the flavor eigenstates $\tilde f$ and
$\tilde \chi_f$ given by
\begin{eqnarray}
M_f &=& \left(
        \begin{array}{cc}
          \tilde m_{\chi_f} & \lambda_{fL} \langle H \rangle \\
          \lambda_{fR} \langle \phi \rangle & 0 \\
        \end{array}
      \right) .
\end{eqnarray}
The lightest mass eigenvalue, $m_f$, is the mass of the
SM fermion, while $m_{\chi_f}$ is the heavier mass
eigenvalue.

We are interested in the case where $f=t$.
To be consistent with experimental literatures, we will
denote the vector-like heavy quark by $\chi_t = T$ (we replace the Lagrangian mass parameter $\tilde{m}_{\chi_t}$ with $\tilde{m}_{T}$).
We use the notation $m(T)$ for the mass of the heavy quark eigenstate, to
avoid confusion with the transverse mass kinematic variable, which we will denote
as usual by $m_T$.
Since
$T$ is charged under $SU(3)_C$, we will assume that
it is heavy, in order to avoid current LHC constraints.
If we assume $m_t^2 \ll m(T)^2$,
we then find
\begin{eqnarray}
m_t &\sim& \frac{\lambda_{tL} \langle H \rangle  \lambda_{tR} \langle \phi \rangle}{\sqrt{\tilde m_{T}^2 + |\lambda_{tL} \langle H \rangle|^2 +  |\lambda_{tR} \langle \phi \rangle|^2 } },
\nonumber\\
m(T) &\sim& \sqrt{\tilde m_{T}^2 + |\lambda_{tL} \langle H \rangle|^2 +  |\lambda_{tR} \langle \phi \rangle|^2 }.
\label{eqn:seesaw_mass}
\end{eqnarray}
Defining
\begin{eqnarray}
t_{L} &=& \cos \theta_L \tilde t_L + \sin \theta_L
\tilde T_L ,
\nonumber\\
t_{R} &=& \cos \theta_R \tilde t_R + \sin \theta_R
\tilde T_{R} ,
\nonumber\\
T_{L} &=& -\sin \theta_L \tilde t_L + \cos \theta_L \tilde T_{L} ,
\nonumber\\
T_{R} &=& -\sin \theta_R \tilde t_R + \cos \theta_R \tilde T_{R} ,
\end{eqnarray}
we find
\begin{eqnarray}
\tan \theta_L &= & \frac{1}{\tilde{m}_{T}} \left(
\frac{m_t^2}{\lambda_{tL}\langle H \rangle}
-\lambda_{tL}\langle H \rangle \right) ,
\nonumber\\
\tan \theta_R &= & \frac{1}{\tilde{m}_{T}} \left(
\frac{m_t^2}{\lambda_{tR}\langle \phi \rangle}
-\lambda_{tR}\langle \phi \rangle \right) .
\end{eqnarray}

The potential for $\phi$ and $H$ can be expressed as
\begin{eqnarray}
V_{\rm pot} &=& -\frac{\mu_\phi^2}{2} \phi \phi^* -\frac{\mu_H^2}{2} H H^*
+ \frac{\lambda}{4} (\phi \phi^*)^2
\nonumber\\
&\,& + \frac{\lambda_h}{4} (HH^*)^2
+ \frac{\lambda'}{4} (\phi \phi^*) (H H^*) ,
\label{eq:TreeLevelPotential}
\end{eqnarray}
where we will assume $\lambda', \lambda, \lambda_H >0$ and $\mu_{\phi,H}^2 >0$.  In this case,
we find
\begin{eqnarray}
\langle \phi \rangle  &=&
\sqrt{(\mu_\phi^2 - \lambda' \langle H \rangle^2/2)
/ \lambda} ,
\nonumber\\
\langle H \rangle  &=&
\sqrt{(\mu_H^2 - \lambda' \langle \phi \rangle^2/2) / \lambda_H} ,
\nonumber\\
\tilde m_{\phi'}^2 &=& 2 \lambda \langle \phi \rangle^2 ,
\nonumber\\
\tilde m_{H}^2 &=& 2 \lambda_h \langle H \rangle^2 .
\end{eqnarray}
Provided $\tilde m_{\phi'} < \langle \phi \rangle$, $\tilde m_{H} < \langle H \rangle$, we can
choose parameters $\mu_{\phi, H}$ and $\lambda, \lambda',
\lambda_H < 1$ consistent with any choice of
$\langle \phi \rangle$, $\langle H \rangle$,
$\tilde m_{\phi',H}^2$

From Eq.~\ref{eqn:seesaw_mass}, we must have $\langle \phi \rangle \gtrsim {\cal O}(\, {\rm TeV})$ if the Yukawa couplings are to be perturbative.
Since $m_{\phi'} \lesssim {\cal O}(\, {\rm GeV})$ in our study, the
quartic $\lambda$ is necessarily small.
Thus, one has to worry that the quartic coupling could
run negative at higher energies, resulting in
quantum corrections to the potential which could
destabilize the vacuum.  To avoid this possibility, we
assume that the particle content at masses above
$\langle \phi \rangle$ is sufficient to ensure that quartic
remain positive.  Since we take $\langle \phi \rangle
\gtrsim {\cal O}(\, {\rm TeV})$, there are few experimental
constraints on the particle content above this energy.

The mass matrix for the scalars can be written as
\begin{eqnarray}
M_s^2 &=& \left(
        \begin{array}{cc}
          \tilde m_{\phi}^2/2 & \lambda' \langle H \rangle \langle \phi \rangle / 2 \\
          \lambda' \langle H \rangle \langle \phi \rangle / 2 & \tilde m_{H}^2/2 \\
        \end{array}
      \right) .
\end{eqnarray}
Thus, there is a contribution to
$\phi'-H$ mixing which is
$\sim \lambda' \langle H \rangle \langle \phi \rangle
/\tilde m_H^2$.
In addition to the tree-level term,
one expects a one-loop contribution to $\lambda'$
to be generated by diagrams where a
fermion runs in the loop.  This contribution will be
$\sim (\lambda_{tL}^2 \lambda_{tR}^2 / 16\pi^2) $,
yielding a contribution to the mixing angle of magnitude
$\sim (\lambda_{tL} \lambda_{tR} /16\pi^2) (m_t m(T)/
\tilde m_H^2)$.
In order to
avoid constraints on anomalous Higgs decay~\cite{HiggsDecayBoundCMS1,HiggsDecayBoundATLAS1,HiggsDecayBoundCMS2}, it is necessary
for the mixing angle to be less then ${\cal O}(0.01)$, which
is realizable in this scenario. In keeping with our notation for $T$, we will henceforth use
$m(\phi')$ to denote
the mass of the light scalar eigenstate.

We will consider
the LHC processes in which a low-mass $\phi'$ is produced
in association with a heavy $T$.
The dark photon ($A'$) can also be produced at the LHC, but its
dominant visible decays will be to charged fermions,
through kinetic mixing.
We will focus on topologies involving $\phi'$,
leaving $A'$ production for future work.

\subsection{Decay of $T$ and $\phi'$}
\label{sec:decay}

As a result of fermion
mixing, $T$ can decay to $b W$ or $tZ$, as well
$tH$ or $t\phi'$,
provided those states are kinematically accessible.
{Similarly, $T$ can also potentially decay to
$t A'$, if that state is kinematically accessible.
However, we leave the signals associated with this
state to future work.}
{The branching fraction to all of
these final states can be comparable.  The
branching fraction to electroweak gauge bosons depends on powers of small mixing angles, but this is compensated by enhancements to the
production of the longitudinal polarization.  This is
not surprising, as the coupling of $T$ to $H$ and to the longitudinal
polarizations of $W$ and $Z$ are related,
as a result of the Goldstone Equivalence Theorem.}

The dark Higgs $\phi'$ couples to
a right-handed up-type quark, down-type quark,
charged lepton, and right-handed neutrino. However, because
we are interested in the case where $m(\phi')$ is
small ($\lesssim {\cal O}(100\, {\rm MeV})$), we will assume that
no tree-level decays mediated by renormalizable
operators are kinematically allowed.  Instead
we will consider the decay process $\phi' \rightarrow
\gamma \gamma$, mediated by the higher-dimensional
operator $(1 / \Lambda_{\phi'}) \phi' F_{\mu \nu} F^{\mu \nu}$ in the effective field theory defined at low
energies.

The coefficient $\Lambda_{\phi'}^{-1}$ would receive
contributions from particles heavier than $\phi'$
running in loops,
including SM particles as well as
other potential new physics couplings to $U(1)_{T3R}$
and to $U(1)_{\rm em}$.  As such, $\Lambda_{\phi'}$ will
be model-dependent.
As a benchmark, we can consider the case in
which the higher-dimensional operator is generated by
integrating out a single heavy
fermion, coupled to $\phi'$
and $\gamma$, running in a one-loop diagram.  This diagram
will give a contribution to $\Gamma_{\phi'} / m(\phi')$
which is $\sim {\cal O}(10^{-7}) (m(\phi') / V)^2$ (see, for example, Ref.~\cite{Dutta:2019fxn}).

\section{Review of vector-like fermion searches}

Previous LHC searches for $T$ have been performed by ATLAS and CMS using $pp$ collisions at $\sqrt{s} = 8$ and $13$ TeV [10–24]. Those results focused on pair production via gluon mediated QCD processes from $q\bar{q}$ annihilation (Fig.~\ref{fig:sfig1}) or single $T$ production from electroweak processes containing associated quarks (Fig.~\ref{fig:sfig2}), followed by the $T$ decay to $bW$, $tZ$, or $tH$. The analyses which targeted the $bW$ decay mode utilized jet substructure observables in order to identify hadronic decays of boosted $W$ bosons, which allows for the reconstruction of the $T$ candidate. The analyses targeting the $tZ$ and $tH$ decay modes used a multiclassification boosted event shape algorithm to identify jets originating from boosted $t$, $Z$, and $H$. The $T$ branching fractions depend on the particular choice of model, and therefore the ATLAS and CMS searches are performed for various possible combinations of the $T$ branching fractions, resulting in 95\% confidence level upper limits on their masses ranging from 740 GeV to about 1.3 TeV.

In the case of $T$ pair production via QCD processes, the cross sections are known and only depend on the mass of the vector-like quark. Assuming a narrow $T$ decay width $(\Gamma/m(T) < 0.05$ or $0.1)$ and a $100$\% branching fraction to $bW$, $tZ$, or $tH$, those searches have placed stringent bounds on $m(T)$, excluding masses below about $1.2$ TeV at 95\% confidence level (CL). On the other hand, the excluded mass range is less stringent
for $\{\mathrm{Br}(T\to tZ),\mathrm{Br}(T\to bW),\mathrm{Br}(T\to tH)\} = \{0.2,0.6,0.2\}$, excluding masses below about $0.95$ TeV. However, if the $T \rightarrow \phi' t$ decay is allowed, those limits must be re-evaluated.

\begin{figure}[h]
\subfloat[QCD Production]{\label{fig:sfig1}%
  \includegraphics[width=0.5\linewidth]{qqbar_chi_t_pair_production.pdf}
}
\subfloat[Electroweak Production]{\label{fig:sfig2}%
  \includegraphics[width=0.5\linewidth]{qqbar_chi_t_EWK_production.pdf}
}
\caption{Feynman diagrams for $T$ pair production via gluon-mediated QCD (left) and $T$ production via electroweak processes (right). These processes have been studied in previous experimental analyses.}
\label{fig:production_diagrams}
\end{figure}

 \begin{figure}[h]
 \begin{center}
 \includegraphics[width=0.4\textwidth]{Dark_Photon_Chit_t_phi_to_photons_semileptonic_decay_Update.pdf}
 \end{center}
 \caption{``$T$-$t$ Fusion'' Feynman diagram.}
 \label{fig:FeynmanDiagram}
 \end{figure}

\section{Samples and simulation}

Given that this study is interested in scenarios where $m(\phi')$ is small, it is experimentally necessary to target an interaction topology which can provide ample boost to the $\phi'$ to aid in the reconstruction of its decay products. For this reason, the $T$-$t$ fusion interaction, as shown in Fig.~\ref{fig:FeynmanDiagram}, was studied due to its sufficiently large cross section over a wide range of $T$ masses, its unique topology that allows for sufficient background suppression, and the large boost given to the $\phi'$ and subsequent decay products via the more massive $T$ and $t$. In particular, we note that the $T$-$t$ fusion cross section is somewhat larger than the $T$ pair production cross section.  This is to be expected, as we are considering the parameter space for which $m(T) \gg m_t$.

The $T$-$t$ fusion mechanism occurs when the protons at the LHC approach the collision point, and the gluons that make up each proton split their energy into a $t\bar{t}$ pair and a $T\bar{T}$ pair. One top quark and one heavy vector-like fermion interact with each other to produce a $\phi'$, while the other $t$ and $T$ create visible decay products in the detector. We target events in which the top quark decays purely hadronically into a bottom quark and dijet pair ($t \to b W \to b q \bar{q}'$), the $T$ decays semileptonically into a $b$ quark, lepton, and neutrino, and the $\phi'$ produces two photons. Since the mass of the $t$-$T$ system which produces the $\phi'$ is much larger than $m(\phi')$ for the regions of parameter space studied in this paper, the large rest energy is converted into large momentum values for the $\phi'$.

As a benchmark scenario, we take the branching fraction $\mathrm{Br}(T \rightarrow bW)$ to
be $50\%$. Figure~\ref{fig:BranchingFraction} shows the calculated $T\to bW$ branching fractions as a function of $m(T)$ and $\lambda_{tL}$, assuming $m(\phi')$ = 100 MeV and $\lambda_{tR}=1$ (assuming a negligible branching
fraction to $tA'$, and assuming no mixing between the $b$-quark and new heavy fermions). We thus see that $\mathrm{Br}(bW) \sim 50\%$
is realizable with appropriate, physically motivated choices of the parameters. Table~\ref{tab:Width} lists the relevant $T$ decay widths for a benchmark scenario with $m(T) = 2$ TeV, $m(\phi') = 100$ MeV, and $\lambda_{tL,tR} = 1$. The $T \to bW$ decay width is relatively small compared to the mass of $T$ ($\frac{\Gamma_{T}}{m(T)} < 1$\%), and thus the ability to experimentally reconstruct the mass of $T$ is dominated by experimental/detector resolution.

{Note that the signal topology we consider can also be produced by events in which $T$ decays to
$tZ$, with $t$ decaying semileptonically and with
$Z$ decaying invisibly.  Since the relative branching
fraction between the $bW$ and $tZ$ channels is  model-dependent, we will preserve the generality of this study by considering a conservative
analysis in which signal events from the $tZ$ channel are
ignored.  Since the branching fraction for
$Z \rightarrow \nu \nu$ is small, this conservative
analysis is nevertheless a good approximation.}

 \begin{figure}[h]
 \begin{center}

  \includegraphics[width=0.5\textwidth]{U1T3R_bWBranchingRatio_vs_mT_vs_lambdaL.pdf}
 \end{center}
 \caption{$T\to bW$ branching fraction as a function of $m(T)$ and $\lambda_{tL}$, for the benchmark scenario with $m(\phi')$ = 100 MeV
 and $\lambda_{tR}=1$ (assuming no mixing between the $b$-quark and new heavy fermions).}
 \label{fig:BranchingFraction}
 \end{figure}

\begin{table}[]
\setlength{\tabcolsep}{7pt}
\caption{$T$ decay width for a benchmark scenario with $\lambda_{tL, tR} = 1$, $m(T) = 2$ TeV, and $m(\phi') = 100$ MeV. We assume that the $b$-quark has no mixing with new heavy fermions.\\}
\begin{tabular}{c|c}
\hline \hline
Process & Decay Width [GeV] \\
\hline
\multirow{4}{1pt}{}
$T \to bW$ & 1.225 \\
$T \to tH$ & 0.0874 \\
$T \to tZ$ & 1.191 \\
$T \to t\phi'$ & 0.0881 \\
\hline \hline
\end{tabular}
\label{tab:Width}
\end{table}

Given the large boost of the $\phi'$, the $\phi'\to\gamma\gamma$ decay results in two collimated photons which are almost always reconstructed as a single merged photon system, depending on the spatial resolution of the electromagnetic calorimeter (ECAL). For example, the CMS ECAL is a homogeneous and hermetic calorimeter containing lead tungstate ($\textrm{PbWO}_{4}$) scintillating crystals with front face cross sections of around 2.2 cm $\times$ 2.2 cm, resulting in a diphoton spatial resolution of $\Delta R_{\gamma\gamma} =\sqrt{(\Delta\phi_{\gamma\gamma})^2+(\Delta\eta_{\gamma\gamma})^2} \sim 0.04$, where $\Delta\phi_{\gamma\gamma}$ is the azimuthal angle between the two photons, and $\Delta\eta_{\gamma\gamma}$ is the pseudorapidity gap between the two photons.
As shown in Fig.~\ref{fig:photon_DeltaR}, for any $m(\phi') < 1$ GeV, the two photons produced from the $\phi'$ decay will almost always have $\Delta R<0.04$. These diphoton pairs will therefore be reconstructed as a single photon in the detector.
These results are consistent with those found in Ref.~\cite{Sheff:2020jyw}.
In order to emulate this effect, any diphoton pairs in both the signal and background samples with $\Delta R<0.04$ were clustered into a single photon object in the detector simulation software (described later).

 \begin{figure}[h]

 \begin{center}
 \includegraphics[width=0.5\textwidth]{photon_DeltaR.pdf}
 \end{center}
 \caption{The distribution of $\Delta R_{\gamma\gamma}$ for $m(\phi') = 1~\, {\rm MeV}$ (blue),
 $100~\, {\rm MeV}$ (red), $1~\, {\rm GeV}$ (green) and $10~\, {\rm GeV}$ (cyan), assuming $m(T) = 500~\, {\rm GeV}$.  The normalization of the $y$-axis  is arbitrary.  The vertical dashed orange line denotes $\Delta R_{\gamma\gamma} = 0.04$; for events to the left of this line, the two photons cannot be discriminated.}
 \label{fig:photon_DeltaR}
 \end{figure}

A similar effect is observed for the energetic $t$ produced through the $T$-$t$ fusion interaction, in which a portion of the $t$ decay products cannot be fully reconstructed independently of each other. This results in three different scenarios of $t$ reconstruction: a fully merged scenario, in which the decay products of the $W$ boson and the $b$ quark are so collimated that they are reconstructed as a single ``fat jet'' (henceforth referred to as a FatJet, FJ); a partially merged scenario, in which the decay products of the $W$ boson form a single FatJet but the $b$ quark can still be separately identified; and an un-merged scenario, in which all decay products can be independently identified. Each scenario has an associated tagging efficiency and misidentification rate, which depends on the boosted $t$/$W$ algorithm. Following Refs.~\cite{CMSboostedWTopTaggingPaper,ATLASboostedWTopTaggingPaper}, we consider two possible ``working points'' for the identification of the fully merged top decays: ($i$) a ``Loose'' working point with 85\% top tagging efficiency and 11-25\% misidentification rate, depending on the FatJet transverse momentum ($p_{T}^{FJ}$); and ($ii$) a ``Tight'' working point with 50\% top tagging efficiency and 4-10\% misidentification rate, depending on $p_{T}^{FJ}$. Similarly, for the partially merged scenario we consider a ``Loose'' working point with 50\% $W$ tagging efficiency and 1-2\% misidentification rate, and a ``Tight'' working point with 25\% $W$ tagging efficiency and 0.2-0.5\% misidentification rate. The choice of boosted $t$/$W$ working points is determined through an optimization process which maximizes discovery reach, as discussed in the next section. The ``Loose'' working  points for both the fully merged $t$ and partially merged $W$ were ultimately shown to provide the best sensitivity and therefore chosen for this study. For now we note that the contribution from SM backgrounds with a misidentified boosted $t$/$W$ is negligible, and thus our discovery projections are not sensitive to uncertainties related to the boosted $t$/$W$ misidentification rates.

Due to the above considerations, the ultimate final-state of interest will consist of: a single highly energetic photon, a (possibly boosted) top tagged system, a $b$ quark, a lepton, and large missing transverse momentum ($\vec{p}_{T}^{\textrm{~miss}}$). For the partially merged and un-merged scenarios, there will be two $b$ quarks present in the final state (one of which is part of the top tagged system). It is important to be able to differentiate between the $b$ quark associated with the hadronic decay of the top quark and the $b$ quark associated with the $T$ decay, for instance when attempting to reconstruct/measure the mass of the $T$ using its decay products. An identification scheme was therefore employed in which the $b$ quark located closest in $\Delta R$ to the dijet pair (from $W\to jj$) was labeled the ``hadronic decay'' $b$ quark ($b_{h}$), while the further $b$ quark would be referred to as the ``leptonic decay'' $b$ quark ($b_{\ell}$). This scheme mimics what is possible experimentally, where one cannot identify the exact decay origins of any particular particle. It was found that this scheme correctly labeled each $b$ quark with an accuracy of greater than 80\%, depending on $m(T)$.

Sample events for $pp$ collisions at $\sqrt{s}=13$ TeV for both signal and background were generated using MadGraph5\_aMC (v2.7.3)~\cite{MADGRAPH}. Hadronization was performed with PYTHIA (v8.2.05) \cite{Sjostrand:2014zea}. Detector effects were included through Delphes (v3.4.1) \cite{deFavereau:2013fsa}, using the CMS input card with 140 average pileup interactions. Signal samples were created for $m(T)$ ranging from 500 GeV to 2500 GeV. The large mass difference between $T$ and $\phi'$ resulted in kinematic distributions for $\phi'$ that were similar regardless of $m(\phi')$ mass (for masses up to 10 GeV).

Total event yields are parameterized using $N=\sigma\times\mathcal{L}\times\epsilon$, where $N$ represents the total number of events, $\mathcal{L}$ is the integrated luminosity scenario being considered (for this study, 150 fb$^{-1}$ and 3000 fb$^{-1}$), and $\epsilon$ represents any efficiencies which might reduce the total event yield (reconstruction efficiencies, etc.). All production cross sections were
computed at tree-level.  As the $k$-factors associated with higher-order corrections to QCD production cross sections are typically greater than one, our estimates of the sensitivity are conservative.

Note that the sample of signal events
we generate include $T$-$t$ fusion processes (see Fig.~\ref{fig:FeynmanDiagram}), as well as $\bar T T$
pair-production, with $T$ decaying to $t\phi'$.  There
is little interference between these processes, which
can be distinguished because, in the case of $\bar T T$
pair-production, the invariant mass of the $t \phi'$
(or, equivalently $t\gamma \gamma$) system is $m(T)$.
The cross section for this latter process depends on
the $\mathrm{Br}(T \rightarrow t\phi')$, which depends on
details of the model which we do not fix.  For
simplicity and to be conservative with our discovery projections,, we set $\mathrm{Br}(T \rightarrow t\phi') = 1\%$, which
is reasonable and easily realizable in
the parameter space of interest (e.g., see Table I).  We find that these fraction of events arising from $\bar T T$ pair-production
is negligible.

The detection efficiency for $\phi' \rightarrow \gamma \gamma$ decays depends on the decay length. The distance the $\phi'$ particle travels before it decays varies depending on the decay width and boost of the $\phi'$. For scenarios with both sufficiently light and sufficiently boosted $\phi'$, a large portion of $\phi'$s begin to decay outside the detector, representing a change in the final state being reconstructed. This effect represents a lower bound of $\phi'$ mass that can be probed by the proposed diphoton final state.
To determine the range in $m(\phi')$ at which the long lifetime becomes important, we compute the $\phi'$ decay length perpendicular to the $pp$ beam axis, which has the form $L_{\phi',\perp} = \frac{\sqrt{\gamma_{\phi'}^{2}-1}}{\Gamma_{\phi'}}\sin\theta$. In this equation, $\theta$ is the scattering angle relative to the beam axis and $\gamma_{\phi'}$ is the relativistic boost factor. This quantity is calculated per simulated signal event by utilizing the $\phi'$ pseudorapidity distribution and the momentum of the particle in the laboratory frame. We conservatively require the $\phi'$ to decay before the CMS ECAL. Therefore, events with $L_{\phi',\perp}$ values greater than 1.29 m cannot be used. The $\phi'$ decay length efficiency was computed by simulating large statistics samples of events for given choices of $m(T)$, $m(\phi')$ and $\Gamma_{T}$, and
calculating the fraction of events in which the $\phi'$
decays within the detector. Figure~\ref{fig:Phi_Decay_Length_100Contour} shows these values as a function of both $m(T)$ and $m(\phi')$, before any selection cuts were applied, taking the $\phi'$
lifetime to scale as $t_{\phi'}\sim(10^{-14}\, {\rm s})(\mathrm{MeV}/m_{\phi'})^3$.
As this figure illustrates, the sensitivity will drop
rapidly with $m_{\phi'}$ once the $\phi'$ becomes light enough that an appreciable fraction of $\phi'$ decay outside the detector.

Table~\ref{tab:lifetime} shows the longest lifetime the $\phi'$ can have for which a 5$\sigma$ discovery is still possible. The exact event selection criteria will be defined in the next section. For now, we note that for longer lifetimes, too few $\phi'$s decay inside the detector to achieve such sensitivity.
We see that, in order for a significant number of
$\phi'$ to decay within the detector, one would need
$\Gamma_{\phi'}$ to be significantly larger than the
benchmark value considered in subsection~\ref{sec:decay}.  But if
one includes QCD-charge factors and the particle
multiplicities of a more complicated dark sector,
prompt decays are realizable.

\captionsetup[subtable]{position=top}
\begin{table}[]
\caption{Longest lifetime of $\phi'$ [s] for which 5$\sigma$ discovery can be achieved. ``N/A'' indicates no 5$\sigma$ discovery is possible (due to small cross sections).}
\subfloat[150 fb$^{-1}$ integrated luminosity.]{\label{tab:lifetime_150}%
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{|l|c|c|c|c|c|}
\cline{3-6}
\multicolumn{2}{c|}{} & \multicolumn{4}{c|}{$\bm{m}(\phi')$ $\textbf{[MeV]}$} \\
\cline{3-6}
\multicolumn{2}{c|}{} & 0.1 & 1 & 10 & 100 \\
\cline{1-6}
\multirow{7}{*}{\rotatebox[origin=c]{90}{$\bm{m(T)}$ $\mathbf{[GeV]}$}} & 500  & 2.6$\times10^{-15}$ & 2.6$\times10^{-14}$ & 2.6$\times10^{-13}$ & 2.6$\times10^{-12}$ \\
\cline{2-6}
& 750  & 2.4$\times10^{-15}$ & 2.4$\times10^{-14}$ & 2.4$\times10^{-13}$ & 2.4$\times10^{-12}$ \\
\cline{2-6}
& 1000 & 1.4$\times10^{-15}$ & 1.4$\times10^{-14}$ & 1.4$\times10^{-13}$ & 1.4$\times10^{-12}$ \\
\cline{2-6}
& 1250 & 5.6$\times10^{-16}$ & 5.6$\times10^{-15}$ & 5.6$\times10^{-14}$ & 5.6$\times10^{-13}$ \\
\cline{2-6}
& 1500 & N/A & N/A & N/A & N/A \\
\cline{2-6}
& 1750 & N/A & N/A & N/A & N/A \\
\cline{2-6}
& 2000 & N/A & N/A & N/A & N/A \\ \hline
\end{tabular}
}
\par
\subfloat[3000 fb$^{-1}$ integrated luminosity.]{\label{tab:lifetime_3000}%
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{|l|c|c|c|c|c|}
\cline{3-6}
\multicolumn{2}{c|}{} & \multicolumn{4}{c|}{$\bm{m}(\phi')$ $\mathbf{[MeV]}$} \\
\cline{3-6}
\multicolumn{2}{c|}{} & 0.1 & 1 & 10 & 100 \\
\cline{1-6}
\multirow{7}{*}{\rotatebox[origin=c]{90}{$\bm{m(T)}$ $\mathbf{[GeV]}$}} & 500 & 2.7$\times10^{-15}$ & 2.7$\times10^{-14}$ & 2.7$\times10^{-13}$ & 2.7$\times10^{-12}$ \\
\cline{2-6}
& 750  & 2.6$\times10^{-15}$ & 2.6$\times10^{-14}$ & 2.6$\times10^{-13}$ & 2.6$\times10^{-12}$ \\
\cline{2-6}
& 1000 & 2.3$\times10^{-15}$ & 2.3$\times10^{-14}$ & 2.3$\times10^{-13}$ & 2.3$\times10^{-12}$ \\
\cline{2-6}
& 1250 & 1.5$\times10^{-15}$ & 1.5$\times10^{-14}$ & 1.5$\times10^{-13}$ & 1.5$\times10^{-12}$ \\
\cline{2-6}
& 1500 & 9.0$\times10^{-16}$ & 9.0$\times10^{-15}$ & 9.0$\times10^{-14}$ & 9.0$\times10^{-13}$ \\
\cline{2-6}
& 1750 & 4.8$\times10^{-16}$ & 4.8$\times10^{-15}$ & 4.8$\times10^{-14}$ & 4.8$\times10^{-13}$ \\
\cline{2-6}
& 2000 & N/A & N/A & N/A & N/A \\ \hline
\end{tabular}
}
\label{tab:lifetime}
\end{table}
\captionsetup[subtable]{position=bottom}

For this study, various SM backgrounds are considered, including $W$+jets, $Z$+jets, $\gamma$+jets, $\gamma\gamma$+jets, QCD multijet, $t\bar{t}+\gamma$, $t\bar{t}+\gamma\gamma$, and $t\bar{t}+H$ ($H\rightarrow\gamma\gamma$) events. The production of $W$+jets, $Z$+jets, $\gamma$+jets, $\gamma\gamma$+jets, and QCD multijet events are found to be a negligible contribution to the proposed search region due to the low probability of having multiple light quark or gluon jets in each event to be misidentified as hadronic $t$/$W$ decays, photons, and/or leptons. Additionally, the majority of the QCD multijet processes contain no genuine missing momentum from neutrinos. The SM processes with higgs decays to photons do not contribute because the relatively low boost of the higgs boson results in two clean and well separated photons. It was found that $t\bar{t}+\gamma$ (with the $\gamma$ coming from initial state or final state radiation), as shown in Fig.~\ref{fig:BackgroundFeynmanDiagram}, represented the dominant irreducible background and $> 99$\% of the total background.

 \begin{figure}[h]
 \begin{center}
 \includegraphics[width=0.4\textwidth]{ttbar_plus_photon.pdf}
 \end{center}
 \caption{Typical $t\bar{t}+\gamma$ Feynman diagram, shown with final state radiation (FSR).}
 \label{fig:BackgroundFeynmanDiagram}
 \end{figure}

 \begin{figure}[h]
 \begin{center}
 \includegraphics[width=0.5\textwidth]{Phi_Decay_Length_100Contour.png}
 \end{center}
 \caption{$\phi'$ decay length efficiency, showing $90\%$, $50\%$, and $10\%$ efficiency levels, assuming $\Gamma_T$ and $\Gamma_{\phi'}$ as described in the text.
 The contours represent the fraction of events decaying inside the ECAL detector.}
 \label{fig:Phi_Decay_Length_100Contour}
 \end{figure}

 \section{Event selection criteria}
 In order to optimize the kinematic selections for maximum discovery potential, a cut-based selection procedure was performed, in which cuts were applied sequentially to optimize signal significance, defined as $N_S/\sqrt{N_S+N_B}$ where $N_S$ and $N_B$ are the number of events in signal and background respectively. We note this particular definition of signal significance is only used for the purpose of optimizing the selections. The final discovery reach is determined with a shape based analysis (described later) using the full reconstructed $T$ mass spectrum. The reconstructed $T$ mass is defined as the transverse mass $m_{T}(b_{\ell},\ell,\vec{p}_{T}^{\textrm{miss}}) = \sqrt{E_{T}(T)^{2} - p_{T}(T)^{2}}$, where $E_{T}(T)$ and $p_{T}(T)$ are the energy and momentum of the resulting Lorentz vector produced by adding together the 4-momentum vectors of the $b$ quark $b_{\ell}$, the lepton, and the missing transverse momentum. For the SM background, the bulk of the $m_{T}(b_{\ell},\ell,\vec{p}_{T}^{\textrm{miss}})$ distribution lies at relatively low reconstructed mass values and decreases exponentially as $m_{T}(b_{\ell},\ell,\vec{p}_{T}^{\textrm{miss}})$ increases. On the other hand, the $m_{T}(b_{\ell},\ell,\vec{p}_{T}^{\textrm{miss}})$ signal distribution appears as a localized bump centered at approximately $m(T)$ (see Results section). For the purpose of the optimization procedure, the values of $N_{S}$ and $N_{B}$ in the significance metric are derived by selecting events near the reconstructed signal $T$ mass peak, $m(T)-2\Gamma(T) < m_{T}(b_{\ell},\ell,\vec{p}_{T}^{\textrm{miss}}) < m(T)+4\Gamma(T)$, where $\Gamma(T)$ is the $T\to bW$ decay width. A benchmark scenario of $m(T) =500~\, {\rm GeV}$ and $m(\phi')=100~\, {\rm MeV}$ was chosen with which to perform the optimization, and the cut values derived therein were applied to all subsequent mass points. The order of the cut optimization was determined by which variables showed the greatest discriminating power between signal and background. Some examples of relevant kinematic distributions are shown in Figs.~\ref{fig:Photon_Pt},~\ref{fig:lepton_MET_delta_phi}, and~\ref{fig:lepton_pt} (all distributions have been normalized to unity).

 Figure~\ref{fig:Photon_Pt} shows the $p_T (\gamma)$ distribution for signal versus background, with the background shape being a falling exponential which is characteristic of initial or final state radiation, while the signal shape has a pronounced hump structure, which is characteristic of decays coming from a boosted object. The signal significance is optimized for $p_{T}(\gamma) > 160$ GeV. Figure~\ref{fig:lepton_MET_delta_phi} shows the $\Delta\phi (\ell,\vec{p}_{T}^{\textrm{~miss}})$ distribution. For signal, the lepton and neutrino which produces the $\vec{p}_{T}^{\textrm{~miss}}$ are the decay products of a very boosted $W$ system (which carries large momentum from the decay of the massive $T$). We therefore expect there to be a small (although non-zero) $\Delta\phi$ between these two very boosted objects. For background however, the semileptonically decaying $t$ carries comparatively less momentum, and therefore its decay products have a distribution with a wider spread of $\Delta\phi(\ell,\vec{p}_{T}^{\textrm{~miss}})$ values. We determined the optimal $\Delta\phi(\ell,\vec{p}_{T}^{\textrm{miss}})$ to be $|\Delta\phi(\ell,\vec{p}_{T}^{\textrm{miss}})| \leq 1$ radians. Figure~\ref{fig:lepton_pt} shows the $p_{T} (\ell)$ distributions  for signal versus background, where the signal lepton carries more momentum due to it being a final decay product of the massive $T$ and therefore peaks at a larger value than the background process. By a similar methodology, we determined the optimal threshold for the $p_{T}$ of the lepton to be $p_{T}(\ell) > 50$ GeV.

The identification of leptons and boosted top quarks plays an important role in the signal acceptance and reduction of SM backgrounds, and thus also the projected High-Luminosity LHC (HL-LHC) discovery reach. In the case of the HL-LHC, the reconstruction and identification of leptons and the hadronic decay products of the top quark may be non-trivial due to the presence of a large number of pileup interactions. The importance of pileup mitigation at CMS and ATLAS has been outlined in many papers, such as Ref.~\cite{CMS:2013ega}. While the expected performance of the upgraded ATLAS and CMS detectors for the HL-LHC is beyond the scope of this work, the studies presented in this paper do attempt to provide reasonable expectations by conservatively assuming some degradation in lepton and hadron identification efficiencies, based on Ref.~\cite{CMS:2013ega}, and considering the case of 140 average pileup interactions. For muons (electrons) with $|\eta|< 1.5$, the assumed identification efficiency is 90\% (85\%), with a 0.3\% (0.6\%) misidentification rate. The performance degrades linearly with $\eta$ for $1.5 < |\eta| < 2.5$, and we assume an identification efficiency of 65\% (60\%), and a 0.5\% (1.0\%) misidentification rate, at $|\eta| = 2.5$. Similarly, the charged hadron tracking efficiency, which contributes to the jet clustering algorithm and $\vec{p}_{T}^{\textrm{~miss}}$ calculation, is 97\% for $1.5 < |\eta| < 2.5$, and degrades to about 85\% at $|\eta| = 2.5$. These potential inefficiencies due to the presence of secondary $pp$ interactions contribute to how well the lepton and top kinematics can be reconstructed.

In addition to the $p_{T}(\gamma)$, $|\Delta\phi(\ell,\vec{p}_{T}^{\textrm{miss}})|$, and $p_{T}(\ell)$ requirements motivated by Figs.~7-9, events with a fully merged top quark system must have a reconstructed FatJet mass compatible with the top quark mass, requiring $m_\textrm{reco}(t) \in [120,220]$. Finally, we impose a modest missing momentum requirement of $p_{T}^{\textrm{miss}} > 20$ GeV. While the $m_\textrm{reco}(t)$ and $p_{T}^{\textrm{miss}}$ selections are highly efficient for signal events and $t\bar{t}+\gamma$ background events, they help to ensure the negligible contribution from other SM backgrounds with pairs of vector bosons, $Z/\gamma^{*}\to \ell\ell$ with associated jets, $W\to \ell\nu$ with associated jets, and QCD multijet production of light quarks. Table~\ref{tab:Selection_Cuts} shows the final derived selection cuts and values, with the listing order indicating the order in which the cuts were optimized.

 \begin{center}
 \begin{table}
 \caption{Selection cut values for optimized signal significance. A benchmark scenario of $m(T) = 500~\, {\rm GeV}$, $m(\phi') = 100~\, {\rm MeV}$ was considered.}
\bigskip
\begin{tabular}{c|c}
\hline \hline
Selection Parameter & Value \\
\hline
$p_{T}(\gamma)$ & $\geq 160$ GeV\\
$|\Delta\phi(\ell,\vec{p}_{T}^{\textrm{~miss}})|$ & $\leq$ 1 rads\\
$p_{T}(\ell)$& $\geq 50$ GeV\\
$\eta(\ell)$ & $\leq 2.5$\\
$p_{T}(b_{\ell})$ & $\geq 140$ GeV\\
$\eta(b_{\ell})$ & $\leq 2.4$\\
$|\vec{p}_{T}^{\textrm{~miss}}|$ & $\geq$ 20 GeV \\
$m_\textrm{reco}(t)$  & [120,220] GeV\\
$m_T(b_{\ell},\ell,\vec{p}_{T}^{\textrm{~miss}})$ lower bound & $m(T)-2*\Gamma(T)$\\
$m_T(b_{\ell},\ell,\vec{p}_{T}^{\textrm{~miss}})$ upper bound & $m(T)+4*\Gamma(T)$\\
 \hline \hline
\end{tabular}
 \label{tab:Selection_Cuts}
\end{table}
\end{center}

 \begin{figure}[h]
 \begin{center}
 \includegraphics[width=0.5\textwidth]{Photon_Pt.pdf}
 \end{center}
 \caption{$p_{T} (\gamma)$ distribution for signal (blue) and background (red) samples.
 Signal events were generated assuming  $m(T) = 500~\, {\rm GeV}$, $m(\phi') = 100~\, {\rm MeV}$.
 The $y$-axis normalization is arbitrary.
 The vertical black dashed line denotes the location of the
 selection cut.}
 \label{fig:Photon_Pt}
 \end{figure}

 \begin{figure}[h]
 \begin{center}
 \includegraphics[width=0.5\textwidth]{lepton_MET_delta_phi.pdf}
 \end{center}
 \caption{Distribution of the angle between
between $\ell$ and $\vec{p}_{T}^{\textrm{~miss}}$ ($\Delta\phi (\ell,\vec{p}_{T}^{\textrm{~miss}})$) for
signal (blue) and background (red) samples.
  Signal events were generated assuming  $m(T) = 500~\, {\rm GeV}$, $m(\phi') = 100~\, {\rm MeV}$.
 The $y$-axis normalization is arbitrary.
 The vertical black dashed lines denotes the location of the
 selection cut.}
 \label{fig:lepton_MET_delta_phi}
 \end{figure}

 \begin{figure}[h]
 \begin{center}
 \includegraphics[width=0.5\textwidth]{lepton_pt.pdf}
 \end{center}
 \caption{Distribution of $p_{T}(\ell)$ for signal (blue) and background (red) samples.
 Signal events were generated assuming  $m(T) = 500~\, {\rm GeV}$, $m(\phi') = 100~\, {\rm MeV}$.
 The $y$-axis normalization is arbitrary.
 The vertical black dashed line denotes the location of the
 selection cut.}
 \label{fig:lepton_pt}
 \end{figure}

  \begin{figure}[h]
 \begin{center}
 \includegraphics[width=0.5\textwidth]{SR_Plot.pdf}
 \end{center}
 \caption{Distribution of $m_{T}(b_{\ell},\ell,\vec{p}_{T}^{miss})$ for signal and background (red) after applying selection cuts on all parameters except $m_{T}(b_{\ell},\ell,\vec{p}_{T}^{miss})$, assuming $\mathcal{L} = 3000$ fb$^{-1}$. Two $m(T)$ mass points are shown: $m(T) = 1000$ GeV (blue) \& $m(T) = 1750$ GeV (green). In both signal benchmark scenarios, $m(\phi^{\prime})$ is 100 MeV.}
 \label{fig:SR_Plot}
 \end{figure}

 \section{Results}\label{sec:Results}

 Figure~\ref{fig:SR_Plot} shows the expected background and signal yields in bins of $m_{T}(b_{\ell},\ell,\vec{p}_{T}^{\textrm{miss}})$. A few signal benchmark points are considered and the yields are normalized to cross section times integrated luminosity of 3000 fb$^{-1}$. The predicted background yield in the $m_{T}(b_{\ell},\ell,\vec{p}_{T}^{\textrm{miss}})$ mass window of [930,1140] GeV is $N_{B} = 250.5 \pm 22.8$, while the signal yield for $m(T)=1000$ GeV within the same mass window is $N_{S} = 2933.3 \pm 52.3$. For the $m_{T}(b_{\ell},\ell,\vec{p}_{T}^{\textrm{miss}})$ mass window of [1627.5,1995] GeV, the predicted background yield is $N_{B} = 27.2 \pm 7.5$, while the signal yield for $m(T)=1750$ GeV is $N_{S} = 61.6 \pm 0.9$. The uncertainties on these numbers represent statistical uncertainties due to the number of raw simulated events passing the selection criteria.

 To assess the expected experimental sensitivity of this search at the LHC, we followed a profile binned likelihood test statistic approach, using the expected bin-by-bin yields in the reconstructed $m_{T}(\ell,b,\vec{p}_{T}^{\textrm{~miss}})$ distribution. The signal significance $z$ is determined using the probability of obtaining the same test statistic with the background-only hypothesis and the signal plus background hypothesis, defined as the local p-value. The value of $z$ corresponds to the point where the integral of a Gaussian distribution between $z$ and $\infty$ results in a value equal to the local p-value. The calculation of the signal significance is performed assuming two values for the total integrated luminosity at the LHC: ($i$) 150 fb$^{-1}$, which is approximately the amount of $pp$ data already collected by ATLAS and CMS; and ($ii$) 3000 fb$^{-1}$, expected by the end of the High Luminosity LHC era. The $m_{T}$-based signal significance calculations were performed using the ROOFit \cite{ROOTFit} toolkit developed by CERN.

Various sources of systematic uncertainties are considered in the signal significance calculation. These uncertainties, based on both experimental and theoretical constraints, are incorporated in the test statistic as nuisance parameters, assuming log-normal priors for normalization parameters, and Gaussian priors for uncertainties on the shape of the $m_{T}(\ell,b,\vec{p}_{T}^{\textrm{~miss}})$ distribution. Experimental systematic uncertainties on $\gamma$ identification, and on reconstruction and identification of $b$-jets and top-tagged jets were considered. For $\gamma$ identification, Refs.~\cite{Khachatryan:2016hje,Aad:2014cka} reports a systematic uncertainty of about 10\%. However, to account for possible effects from the merged photon system, we assume a conservative 15\% uncertainty which is uncorrelated between signal and background processes, and correlated across $m_{T}$ bins for each process. For experimental uncertainties related to the reconstruction and identification of boosted top quarks, a 20\% value was included (independent of $p_{T}$ and $\eta$ of the top-tagged system), following the results from Refs.~\cite{CMSboostedWTopTaggingPaper,ATLASboostedWTopTaggingPaper}. The uncertainty from $b$-jet identification is 10\%, while efficiencies for the electron and muon reconstruction, identification, and isolation requirements have an uncertainty of 2\%. The uncertainties on the reconstruction of $\vec{p}_{T}^{\textrm{~miss}}$ are due to the uncertainties on proper jet energy measurements. We assumed 2-5\% jet energy scale uncertainties, depending on $\eta$ and $p_{T}$, resulting in shape-based uncertainties on $m_{T}$ that range from 3\% to 6\%, depending on the $m_{T}$ bin.

 \begin{figure}[h]
 \begin{center}
 \includegraphics[width=0.5\textwidth]{U1T3R_Signal_Significance_1plot.pdf}
 \end{center}
 \caption{Signal significance as a function of $m(T)$, for scenarios with $m(\phi')=1$ MeV (magenta) and  $m(\phi')=100$ MeV (green) and for scenarios with $\mathcal{L}_{int}=150\;\mathrm{fb}^{-1}$ (solid lines) and $\mathcal{L}_{int}=3000\;\mathrm{fb}^{-1}$ (dashed lines).   The dashed vertical yellow line indicates the lower limit on $m(T)$ found in~\cite{Cacciapaglia:2019zmj},
 assuming $\mathrm{Br}(T \rightarrow tH) = 50\%$.}
 \label{fig:Signal_Significance}
 \end{figure}

 \begin{figure}[h]
 \begin{center}
 \includegraphics[width=0.5\textwidth]{SignalSig_BF_mT.png}
 \end{center}
 \caption{Signal significance as a function of $m(T)$ and Br($T\rightarrow Wb$), for $m(\phi')$ = 100 MeV. The heat map levels correspond to signal significance, while overlaid dashed lines correspond to the 95\% confidence level (yellow), 3$\sigma$ discovery potential (green), and 5$\sigma$ discovery potential (red), assuming $\mathcal{L}_{int}=3000$ fb$^{-1}$. }
 \label{fig:SignalSig_BF_mT}
 \end{figure}

In addition, theoretical uncertainties were included in order to account for the set of parton distribution functions (PDFs) used to produce the simulated  signal and background samples. The PDF uncertainty was calculated following the PDF4LHC prescription~\cite{Butterworth:2015oua}, and results in a 3-10\% systematic uncertainty, depending on the process. The effect of the chosen PDF set on the shape of the $m_{T}$ distribution is negligible. The absence of higher-order contributions to the signal cross sections affect the signal acceptance calculation. This effect is estimated by varying the renormalization and factorization scales by a factor of two with respect to the nominal value, and by considering the full change in the yields. They are found to be small (3.5\%).

Figure~\ref{fig:Signal_Significance} shows the results of the expected signal significance for different  $m(T)$ and $m(\phi')$ scenarios.
For the 150 fb$^{-1}$ scenario, it is feasible to exclude (at 95\% confidence level)  $ m(T) < 1.7$ $(1.1)$ TeV for $m(\phi') = 100$ $(1)$ MeV. In order to compare the expected reach of the proposed search with existing searches from CMS and ATLAS, Figure~\ref{fig:Signal_Significance} also shows the current observed upper limit on $m(T)$ from CMS/ATLAS under similar branching fraction considerations. The expected 95\% exclusion of $m(T) < 1.7$ TeV  from the proposed search is to be compared with the $m(T) < 875$ GeV exclusion from the current LHC searches~\cite{Cacciapaglia:2019zmj}. Similarly, for the 3000 fb$^{-1}$ scenario, the 5$\sigma$ ($3\sigma$) reach includes $m(T) < 1.8$ ($2$) TeV for $m(\phi') = 100$ MeV. For $m(\phi') \lesssim 1~\, {\rm MeV}$, a large fraction of the signal events are lost due to an average $\phi'$ decay length that exceeds 1.29 m, thus preventing a photon signature in the electromagnetic calorimeter. Therefore, for 3000 fb$^{-1}$ and $m(\phi') = 1~\, {\rm MeV}$, the 5$\sigma$ ($3\sigma$) reach degrades to $m(T) < 1.2$ ($1.3$) TeV. Figure~\ref{fig:SignalSig_BF_mT} further demonstrates how signal significance changes when one now varies the branching fraction of $\mathrm{Br}(T\to Wb)$ for various $m(T)$ (assuming $\mathcal{L}_{int}=3000$ fb$^{-1}$).

{Note that, in this scenario, the search for light scalars in association with
heavy quarks can provide better detection prospects than the search either light scalars or
heavy quarks individually.}

 \section{Discussion}

 We have demonstrated that the LHC can discover the visible
 decays of new sub-GeV particles.  The key to its
 discovery potential is the production of heavy QCD-coupled particles, which then decay to the
 light particles with large central boost.  The large
 boost allows the signal from visible decays to be seen
 above the lower-energy background.

 There are many laboratory experiments, either current
 or planned, which focus on the detection of new
 low-mass particles.  But the LHC can fill a niche
 which is not necessarily covered by other strategies.
 Most beam experiments focus on the production of
 light particles through processes with low momentum
 transfer, in which the produced new particles are
 largely co-linear with the beam.  These experiments
 must either deal with the large backgrounds which come
 with the beam itself, or are limited to the search
 for long-lived particles which decay at displaced
 detectors that are protected from beam backgrounds
 by shielding.

 The LHC search strategy described here can be used to
 discover the prompt decay of new light particles,
 because they are produced from processes with large
 momentum transfer, and thus are emitted in the
 central region, away from the beam.  We thus see that
 detection prospects for low-mass particles are
 enhanced when it is kinematically possible to access
 the heavy degrees of freedom which arise in the
 UV completion of the low-energy model.  Indeed, this
 scenario, in which the dominant coupling of the light
 scalar is to the top quark, is an example which would
 be difficult to directly probe at low energy beam
 experiments.

 We considered the specific example of a new gauge
 group $U(1)_{T3R}$, in which the dark Higgs is
 a light (sub-GeV) particle that couples to a heavy
 new vector-like quark.  This new fermion appears as
 the seesaw partner of the top quark, and appears as
 part of the UV completion of the low-energy model.  But
 this scenario was introduced as a toy-model, and
 the basic idea will generalize to other models.

 For example, if the scalar is slightly heavier than
 the range considered here, decays to $\mu^+ \mu^-$
 will be possible.  This channel should
 also provide
 good detection prospects.  It would be interesting to
 study these prospects in a future work.

\vskip .1in
{\bf Acknowledgments}
\vskip .1in
We are grateful to Noah Steinberg and James D.~Wells
for useful discussions.
The work of BD, SG and TK is supported in part by DOE grant DE-SC0010813. The work of SG is also supported in part by National Research Foundation of Korea (NRF), grant no. 6N021413.
The work of JK is supported in part by DOE grant DE-SC0010504. AG and DJ are supported in part by NSF Award PHY-1945366.

\section{\label{sec:level1}Introduction}

Recently, there has been great interest in models of
beyond-the-Standard Model (BSM) physics involving new
low-mass matter and mediator particles.  A promising
method for testing these models is the use of beam
experiments, where the focus is typically on processes
with relatively low momentum transfer, in which the
low-mass mediators are produced at high longitudinal
boost (see, for example,~\cite{Batell:2017kty,Bauer:2018onh,Abdullah:2018ykz}).
Indeed, these types of processes are the typical
production mechanisms for mediators which arise in
the low-energy effective field theory.
The visible decays of long-lived mediators can then be
observed at displaced detectors (e.g., FASER~\cite{FASER:2019aik}), with the intervening
shielding serving to remove the background arising from the beam itself.

The Large Hadron Collider (LHC), however, excels at producing new heavy particles in high momentum transfer processes. Thus, current LHC searches for light mediators $a$ (e.g. axion-like particles) are carried out in the context of SM Higgs decays ($H \to aa(Za)$), $B$ meson decays ($B \to Xa$), or in proton-proton ($pp$) production mechanisms resulting in associated top quarks. One strategy often pursued in those analyses is a dilepton selection targeting the $a \to \ell\ell$ decay mode. However, the model parameter space (mass, lifetime, and coupling) that can be probed by those searches is often limited, especially for an MeV-scale mediator, where $a \to \ell\ell$ is not kinematically allowed. The sensitivity to new physics models with low mass mediators may be further suppressed by the low momentum attributed to the $a$ decay products, making it difficult to suppress SM backgrounds and experimentally trigger on interesting events.

In this paper, we devise a LHC search strategy for a light mediator particle produced in association with a TeV-scale top-partner particle. The heavy top-partner particle can be copiously produced at the LHC because it couples to the SM quarks and gluons, and its highly energetic decay products can be observed in association with a mediator particle containing substantial transverse momentum. If the mediator decays promptly into the SM particles, then the resulting SM particles will be detected in the central region of the detector and will be energetic enough to significantly reduce SM backgrounds. As a result of this search strategy, the LHC can expand the reach of heavy top-partners and may discover even MeV-scale mediators, which are otherwise difficult to probe at a hadron collider using traditional search strategies.

We illustrate the feasibility of probing such an MeV-scale mediator particle at the LHC using a benchmark model.
The benchmark model is a low-scale model of new physics (a new gauge group
$U(1)_{T3R}$~\cite{Pati:1974yy,Mohapatra:1974gc,Senjanovic:1975rk} with a sub-GeV dark
Higgs boson~\cite{Dutta:2019fxn,Dutta:2020enk,Dutta:2021afo}) along with a UV completion and the universal seesaw mechanism~\cite{Berezhiani:1983hm,Chang:1986bp,Davidson:1987mh,Rajpoot:1987fca,Babu:1988mw,Babu:1989rb}.
Although this model is well-motivated and realizable in nature, we use it as a benchmark scenario to define the
strategies for discovering the decays of light particles at the LHC.
Here, we introduce a gauge group $U(1)_{T3R}$ under
which only a set of right-handed SM fermions are charged.
Because this symmetry now protects the fermion masses, the dark Higgs boson
($\phi$), whose vacuum expectation value (vev) breaks $U(1)_{T3R}$, must couple to the SM fermions.
The particle excitation about this vev ($\phi'$) can be light, even
sub-GeV. Since both the SM Higgs boson ($V$) and the dark Higgs boson ($\phi'$) must couple to the SM fermions, the effective Yukawa coupling is actually a nonrenormalizable dimension-5 operator.  One can write a UV completion of this theory using the universal seesaw mechanism, in which one adds heavy vector-like fermions. A vector-like fermion can couple to the corresponding SM fermion via exchange of $H$ or $\phi'$. The effective Yukawa couplings then arise from integrating out the heavy fermions. We consider the case in which the top quark ($t$) couples to a light scalar $\phi'$ and a heavy vector-like quark ($T$). The new heavy quarks of this model can be produced at the LHC, and their decay products observed in association with an energetic $\phi'$ in the central regions of the CMS~\cite{Chatrchyan:2008aa} and ATLAS~\cite{Aad:2008zzm} detectors.
The decay $\phi' \rightarrow \gamma \gamma$ will then produce a pair of collimated photons.  The large SM background can thus be reduced by requiring the photons to be energetic, and by tagging $t$ and bottom quarks ($b$) which also necessarily appear.

The rest of this paper is structured as follows. In Section II, we describe the details of our new physics model. Section III provides an overview of current results from searches for vector-like fermions at the LHC. Section IV provides details of how the Monte Carlo samples were produced for this study. In Section V we discuss event selection criteria, and in Section VI, main results. We end with a short discussion in Section VII.

\section{$U(1)_{T3R}$ and the Universal Seesaw Mechanism}

We consider a scenario in which a set of right-handed SM fermions are charged under the gauge group $U(1)_{T3R}$, with up-type and down-type fermions having charge $\pm Q$.  Provided
that we couple an up-type quark, down-type quark, charged
lepton, and right-handed neutrino to $U(1)_{T3R}$, all
gauge anomalies will manifestly cancel.  For the purpose of
our analysis, we assume that right-handed
$t$-quarks couple to
$U(1)_{T3R}$.  The identity of the other fermions will not
affect our analysis; for simplicity, we will refer to the
down-type quark, charged lepton, and right-handed neutrino
as $b$, $\ell$, and $\nu_R$ respectively.

We assume that a dark Higgs field $\phi$ also has
charge $Q$ under $U(1)_{T3R}$ and is neutral under all
other gauge symmetries.  $U(1)_{T3R}$ is
spontaneously broken when this field gets a vev
$\langle \phi \rangle =V$.  We may write the excitation
about this vev as $\phi = V + (1/\sqrt{2}) \phi'$,
where $\phi'$ is a real scalar field (the imaginary
part of the excitation about the vev is the Goldstone mode,
and is absorbed into the longitudinal polarization of the
dark photon, which is the gauge boson of
$U(1)_{T3R}$).

Each charged fermion which couples to
$U(1)_{T3R}$ has a mass which is now protected by
both $U(1)_{T3R}$ and $SU(2)_L \times U(1)_Y$, and thus
is proportional to the Higgs vev ($v$) and the
dark Higgs vev ($V$)~\footnote{The mass terms for the
neutrino are more complicated, since the neutrino can
also have a Majorana mass term.  For our analysis, we
will not need to consider them further.}.  In the
effective field theory defined at the electroweak
scale, the charged
fermion masses thus arise from dimension 5 operators
given by
\begin{eqnarray}
{\cal L}_{\rm eff} &=& ... - \frac{\lambda_f}{\Lambda_f}
H^* \phi (\bar f P_R f ) + h.c.,
\end{eqnarray}
with
\begin{eqnarray}
m_f &=& \frac{\lambda_f \langle H \rangle
\langle \phi \rangle}{\Lambda_f} .
\end{eqnarray}
This non-renormalizable operator  can be
derived from a UV-complete renormalizable theory if
we introduce a new vector-like fermion $\tilde \chi_f$, which
is a singlet under $U(1)_{T3R}$ and has the
same charges under the SM gauge group
$SU(3)_C \times SU(2)_L \times U(1)_Y$ as the
right-handed fermion $f_R$.

In the UV-complete theory, we may write the operators
\begin{eqnarray}
{\cal L} &=& ... - \tilde m_{\chi_f} \bar{\tilde{\chi}}_f
\tilde \chi_f
- \lambda_{fL} H (\bar{\tilde{\chi}}_f P_L \tilde f_L)
\nonumber\\
&\,&
- \lambda_{fR} \phi^* (\bar{\tilde{\chi}}_f P_R \tilde f)
+h.c.,
\end{eqnarray}
where $\tilde f$ is a flavor eigenstate, and
$\tilde f_L$ is the $SU(2)_L$ doublet of which
$\tilde f$ is
a component.  Taking the vevs and couplings to be real, these  Lagrangian terms yield a mass
matrix for the flavor eigenstates $\tilde f$ and
$\tilde \chi_f$ given by
\begin{eqnarray}
M_f &=& \left(
        \begin{array}{cc}
          \tilde m_{\chi_f} & \lambda_{fL} \langle H \rangle \\
          \lambda_{fR} \langle \phi \rangle & 0 \\
        \end{array}
      \right) .
\end{eqnarray}
The lightest mass eigenvalue, $m_f$, is the mass of the
SM fermion, while $m_{\chi_f}$ is the heavier mass
eigenvalue.

We are interested in the case where $f=t$.
To be consistent with experimental literatures, we will
denote the vector-like heavy quark by $\chi_t = T$ (we replace the Lagrangian mass parameter $\tilde{m}_{\chi_t}$ with $\tilde{m}_{T}$).
We use the notation $m(T)$ for the mass of the heavy quark eigenstate, to
avoid confusion with the transverse mass kinematic variable, which we will denote
as usual by $m_T$.
Since
$T$ is charged under $SU(3)_C$, we will assume that
it is heavy, in order to avoid current LHC constraints.
If we assume $m_t^2 \ll m(T)^2$,
we then find
\begin{eqnarray}
m_t &\sim& \frac{\lambda_{tL} \langle H \rangle  \lambda_{tR} \langle \phi \rangle}{\sqrt{\tilde m_{T}^2 + |\lambda_{tL} \langle H \rangle|^2 +  |\lambda_{tR} \langle \phi \rangle|^2 } },
\nonumber\\
m(T) &\sim& \sqrt{\tilde m_{T}^2 + |\lambda_{tL} \langle H \rangle|^2 +  |\lambda_{tR} \langle \phi \rangle|^2 }.
\label{eqn:seesaw_mass}
\end{eqnarray}
Defining
\begin{eqnarray}
t_{L} &=& \cos \theta_L \tilde t_L + \sin \theta_L
\tilde T_L ,
\nonumber\\
t_{R} &=& \cos \theta_R \tilde t_R + \sin \theta_R
\tilde T_{R} ,
\nonumber\\
T_{L} &=& -\sin \theta_L \tilde t_L + \cos \theta_L \tilde T_{L} ,
\nonumber\\
T_{R} &=& -\sin \theta_R \tilde t_R + \cos \theta_R \tilde T_{R} ,
\end{eqnarray}
we find
\begin{eqnarray}
\tan \theta_L &= & \frac{1}{\tilde{m}_{T}} \left(
\frac{m_t^2}{\lambda_{tL}\langle H \rangle}
-\lambda_{tL}\langle H \rangle \right) ,
\nonumber\\
\tan \theta_R &= & \frac{1}{\tilde{m}_{T}} \left(
\frac{m_t^2}{\lambda_{tR}\langle \phi \rangle}
-\lambda_{tR}\langle \phi \rangle \right) .
\end{eqnarray}

The potential for $\phi$ and $H$ can be expressed as
\begin{eqnarray}
V_{\rm pot} &=& -\frac{\mu_\phi^2}{2} \phi \phi^* -\frac{\mu_H^2}{2} H H^*
+ \frac{\lambda}{4} (\phi \phi^*)^2
\nonumber\\
&\,& + \frac{\lambda_h}{4} (HH^*)^2
+ \frac{\lambda'}{4} (\phi \phi^*) (H H^*) ,
\label{eq:TreeLevelPotential}
\end{eqnarray}
where we will assume $\lambda', \lambda, \lambda_H >0$ and $\mu_{\phi,H}^2 >0$.  In this case,
we find
\begin{eqnarray}
\langle \phi \rangle  &=&
\sqrt{(\mu_\phi^2 - \lambda' \langle H \rangle^2/2)
/ \lambda} ,
\nonumber\\
\langle H \rangle  &=&
\sqrt{(\mu_H^2 - \lambda' \langle \phi \rangle^2/2) / \lambda_H} ,
\nonumber\\
\tilde m_{\phi'}^2 &=& 2 \lambda \langle \phi \rangle^2 ,
\nonumber\\
\tilde m_{H}^2 &=& 2 \lambda_h \langle H \rangle^2 .
\end{eqnarray}
Provided $\tilde m_{\phi'} < \langle \phi \rangle$, $\tilde m_{H} < \langle H \rangle$, we can
choose parameters $\mu_{\phi, H}$ and $\lambda, \lambda',
\lambda_H < 1$ consistent with any choice of
$\langle \phi \rangle$, $\langle H \rangle$,
$\tilde m_{\phi',H}^2$

From Eq.~\ref{eqn:seesaw_mass}, we must have $\langle \phi \rangle \gtrsim {\cal O}(\, {\rm TeV})$ if the Yukawa couplings are to be perturbative.
Since $m_{\phi'} \lesssim {\cal O}(\, {\rm GeV})$ in our study, the
quartic $\lambda$ is necessarily small.
Thus, one has to worry that the quartic coupling could
run negative at higher energies, resulting in
quantum corrections to the potential which could
destabilize the vacuum.  To avoid this possibility, we
assume that the particle content at masses above
$\langle \phi \rangle$ is sufficient to ensure that quartic
remain positive.  Since we take $\langle \phi \rangle
\gtrsim {\cal O}(\, {\rm TeV})$, there are few experimental
constraints on the particle content above this energy.

The mass matrix for the scalars can be written as
\begin{eqnarray}
M_s^2 &=& \left(
        \begin{array}{cc}
          \tilde m_{\phi}^2/2 & \lambda' \langle H \rangle \langle \phi \rangle / 2 \\
          \lambda' \langle H \rangle \langle \phi \rangle / 2 & \tilde m_{H}^2/2 \\
        \end{array}
      \right) .
\end{eqnarray}
Thus, there is a contribution to
$\phi'-H$ mixing which is
$\sim \lambda' \langle H \rangle \langle \phi \rangle
/\tilde m_H^2$.
In addition to the tree-level term,
one expects a one-loop contribution to $\lambda'$
to be generated by diagrams where a
fermion runs in the loop.  This contribution will be
$\sim (\lambda_{tL}^2 \lambda_{tR}^2 / 16\pi^2) $,
yielding a contribution to the mixing angle of magnitude
$\sim (\lambda_{tL} \lambda_{tR} /16\pi^2) (m_t m(T)/
\tilde m_H^2)$.
In order to
avoid constraints on anomalous Higgs decay~\cite{HiggsDecayBoundCMS1,HiggsDecayBoundATLAS1,HiggsDecayBoundCMS2}, it is necessary
for the mixing angle to be less then ${\cal O}(0.01)$, which
is realizable in this scenario. In keeping with our notation for $T$, we will henceforth use
$m(\phi')$ to denote
the mass of the light scalar eigenstate.

We will consider
the LHC processes in which a low-mass $\phi'$ is produced
in association with a heavy $T$.
The dark photon ($A'$) can also be produced at the LHC, but its
dominant visible decays will be to charged fermions,
through kinetic mixing.
We will focus on topologies involving $\phi'$,
leaving $A'$ production for future work.

\subsection{Decay of $T$ and $\phi'$}
\label{sec:decay}

As a result of fermion
mixing, $T$ can decay to $b W$ or $tZ$, as well
$tH$ or $t\phi'$,
provided those states are kinematically accessible.
{Similarly, $T$ can also potentially decay to
$t A'$, if that state is kinematically accessible.
However, we leave the signals associated with this
state to future work.}
{The branching fraction to all of
these final states can be comparable.  The
branching fraction to electroweak gauge bosons depends on powers of small mixing angles, but this is compensated by enhancements to the
production of the longitudinal polarization.  This is
not surprising, as the coupling of $T$ to $H$ and to the longitudinal
polarizations of $W$ and $Z$ are related,
as a result of the Goldstone Equivalence Theorem.}

The dark Higgs $\phi'$ couples to
a right-handed up-type quark, down-type quark,
charged lepton, and right-handed neutrino. However, because
we are interested in the case where $m(\phi')$ is
small ($\lesssim {\cal O}(100\, {\rm MeV})$), we will assume that
no tree-level decays mediated by renormalizable
operators are kinematically allowed.  Instead
we will consider the decay process $\phi' \rightarrow
\gamma \gamma$, mediated by the higher-dimensional
operator $(1 / \Lambda_{\phi'}) \phi' F_{\mu \nu} F^{\mu \nu}$ in the effective field theory defined at low
energies.

The coefficient $\Lambda_{\phi'}^{-1}$ would receive
contributions from particles heavier than $\phi'$
running in loops,
including SM particles as well as
other potential new physics couplings to $U(1)_{T3R}$
and to $U(1)_{\rm em}$.  As such, $\Lambda_{\phi'}$ will
be model-dependent.
As a benchmark, we can consider the case in
which the higher-dimensional operator is generated by
integrating out a single heavy
fermion, coupled to $\phi'$
and $\gamma$, running in a one-loop diagram.  This diagram
will give a contribution to $\Gamma_{\phi'} / m(\phi')$
which is $\sim {\cal O}(10^{-7}) (m(\phi') / V)^2$ (see, for example, Ref.~\cite{Dutta:2019fxn}).

\section{Review of vector-like fermion searches}

Previous LHC searches for $T$ have been performed by ATLAS and CMS using $pp$ collisions at $\sqrt{s} = 8$ and $13$ TeV [10–24]. Those results focused on pair production via gluon mediated QCD processes from $q\bar{q}$ annihilation (Fig.~\ref{fig:sfig1}) or single $T$ production from electroweak processes containing associated quarks (Fig.~\ref{fig:sfig2}), followed by the $T$ decay to $bW$, $tZ$, or $tH$. The analyses which targeted the $bW$ decay mode utilized jet substructure observables in order to identify hadronic decays of boosted $W$ bosons, which allows for the reconstruction of the $T$ candidate. The analyses targeting the $tZ$ and $tH$ decay modes used a multiclassification boosted event shape algorithm to identify jets originating from boosted $t$, $Z$, and $H$. The $T$ branching fractions depend on the particular choice of model, and therefore the ATLAS and CMS searches are performed for various possible combinations of the $T$ branching fractions, resulting in 95\% confidence level upper limits on their masses ranging from 740 GeV to about 1.3 TeV.

In the case of $T$ pair production via QCD processes, the cross sections are known and only depend on the mass of the vector-like quark. Assuming a narrow $T$ decay width $(\Gamma/m(T) < 0.05$ or $0.1)$ and a $100$\% branching fraction to $bW$, $tZ$, or $tH$, those searches have placed stringent bounds on $m(T)$, excluding masses below about $1.2$ TeV at 95\% confidence level (CL). On the other hand, the excluded mass range is less stringent
for $\{\mathrm{Br}(T\to tZ),\mathrm{Br}(T\to bW),\mathrm{Br}(T\to tH)\} = \{0.2,0.6,0.2\}$, excluding masses below about $0.95$ TeV. However, if the $T \rightarrow \phi' t$ decay is allowed, those limits must be re-evaluated.

\begin{figure}[h]
\subfloat[QCD Production]{\label{fig:sfig1}%
  \includegraphics[width=0.5\linewidth]{qqbar_chi_t_pair_production.pdf}
}
\subfloat[Electroweak Production]{\label{fig:sfig2}%
  \includegraphics[width=0.5\linewidth]{qqbar_chi_t_EWK_production.pdf}
}
\caption{Feynman diagrams for $T$ pair production via gluon-mediated QCD (left) and $T$ production via electroweak processes (right). These processes have been studied in previous experimental analyses.}
\label{fig:production_diagrams}
\end{figure}

 \begin{figure}[h]
 \begin{center}
 \includegraphics[width=0.4\textwidth]{Dark_Photon_Chit_t_phi_to_photons_semileptonic_decay_Update.pdf}
 \end{center}
 \caption{``$T$-$t$ Fusion'' Feynman diagram.}
 \label{fig:FeynmanDiagram}
 \end{figure}

\section{Samples and simulation}

Given that this study is interested in scenarios where $m(\phi')$ is small, it is experimentally necessary to target an interaction topology which can provide ample boost to the $\phi'$ to aid in the reconstruction of its decay products. For this reason, the $T$-$t$ fusion interaction, as shown in Fig.~\ref{fig:FeynmanDiagram}, was studied due to its sufficiently large cross section over a wide range of $T$ masses, its unique topology that allows for sufficient background suppression, and the large boost given to the $\phi'$ and subsequent decay products via the more massive $T$ and $t$. In particular, we note that the $T$-$t$ fusion cross section is somewhat larger than the $T$ pair production cross section.  This is to be expected, as we are considering the parameter space for which $m(T) \gg m_t$.

The $T$-$t$ fusion mechanism occurs when the protons at the LHC approach the collision point, and the gluons that make up each proton split their energy into a $t\bar{t}$ pair and a $T\bar{T}$ pair. One top quark and one heavy vector-like fermion interact with each other to produce a $\phi'$, while the other $t$ and $T$ create visible decay products in the detector. We target events in which the top quark decays purely hadronically into a bottom quark and dijet pair ($t \to b W \to b q \bar{q}'$), the $T$ decays semileptonically into a $b$ quark, lepton, and neutrino, and the $\phi'$ produces two photons. Since the mass of the $t$-$T$ system which produces the $\phi'$ is much larger than $m(\phi')$ for the regions of parameter space studied in this paper, the large rest energy is converted into large momentum values for the $\phi'$.

As a benchmark scenario, we take the branching fraction $\mathrm{Br}(T \rightarrow bW)$ to
be $50\%$. Figure~\ref{fig:BranchingFraction} shows the calculated $T\to bW$ branching fractions as a function of $m(T)$ and $\lambda_{tL}$, assuming $m(\phi')$ = 100 MeV and $\lambda_{tR}=1$ (assuming a negligible branching
fraction to $tA'$, and assuming no mixing between the $b$-quark and new heavy fermions). We thus see that $\mathrm{Br}(bW) \sim 50\%$
is realizable with appropriate, physically motivated choices of the parameters. Table~\ref{tab:Width} lists the relevant $T$ decay widths for a benchmark scenario with $m(T) = 2$ TeV, $m(\phi') = 100$ MeV, and $\lambda_{tL,tR} = 1$. The $T \to bW$ decay width is relatively small compared to the mass of $T$ ($\frac{\Gamma_{T}}{m(T)} < 1$\%), and thus the ability to experimentally reconstruct the mass of $T$ is dominated by experimental/detector resolution.

{Note that the signal topology we consider can also be produced by events in which $T$ decays to
$tZ$, with $t$ decaying semileptonically and with
$Z$ decaying invisibly.  Since the relative branching
fraction between the $bW$ and $tZ$ channels is  model-dependent, we will preserve the generality of this study by considering a conservative
analysis in which signal events from the $tZ$ channel are
ignored.  Since the branching fraction for
$Z \rightarrow \nu \nu$ is small, this conservative
analysis is nevertheless a good approximation.}

 \begin{figure}[h]
 \begin{center}

  \includegraphics[width=0.5\textwidth]{U1T3R_bWBranchingRatio_vs_mT_vs_lambdaL.pdf}
 \end{center}
 \caption{$T\to bW$ branching fraction as a function of $m(T)$ and $\lambda_{tL}$, for the benchmark scenario with $m(\phi')$ = 100 MeV
 and $\lambda_{tR}=1$ (assuming no mixing between the $b$-quark and new heavy fermions).}
 \label{fig:BranchingFraction}
 \end{figure}

\begin{table}[]
\setlength{\tabcolsep}{7pt}
\caption{$T$ decay width for a benchmark scenario with $\lambda_{tL, tR} = 1$, $m(T) = 2$ TeV, and $m(\phi') = 100$ MeV. We assume that the $b$-quark has no mixing with new heavy fermions.\\}
\begin{tabular}{c|c}
\hline \hline
Process & Decay Width [GeV] \\
\hline
\multirow{4}{1pt}{}
$T \to bW$ & 1.225 \\
$T \to tH$ & 0.0874 \\
$T \to tZ$ & 1.191 \\
$T \to t\phi'$ & 0.0881 \\
\hline \hline
\end{tabular}
\label{tab:Width}
\end{table}

Given the large boost of the $\phi'$, the $\phi'\to\gamma\gamma$ decay results in two collimated photons which are almost always reconstructed as a single merged photon system, depending on the spatial resolution of the electromagnetic calorimeter (ECAL). For example, the CMS ECAL is a homogeneous and hermetic calorimeter containing lead tungstate ($\textrm{PbWO}_{4}$) scintillating crystals with front face cross sections of around 2.2 cm $\times$ 2.2 cm, resulting in a diphoton spatial resolution of $\Delta R_{\gamma\gamma} =\sqrt{(\Delta\phi_{\gamma\gamma})^2+(\Delta\eta_{\gamma\gamma})^2} \sim 0.04$, where $\Delta\phi_{\gamma\gamma}$ is the azimuthal angle between the two photons, and $\Delta\eta_{\gamma\gamma}$ is the pseudorapidity gap between the two photons.
As shown in Fig.~\ref{fig:photon_DeltaR}, for any $m(\phi') < 1$ GeV, the two photons produced from the $\phi'$ decay will almost always have $\Delta R<0.04$. These diphoton pairs will therefore be reconstructed as a single photon in the detector.
These results are consistent with those found in Ref.~\cite{Sheff:2020jyw}.
In order to emulate this effect, any diphoton pairs in both the signal and background samples with $\Delta R<0.04$ were clustered into a single photon object in the detector simulation software (described later).

 \begin{figure}[h]

 \begin{center}
 \includegraphics[width=0.5\textwidth]{photon_DeltaR.pdf}
 \end{center}
 \caption{The distribution of $\Delta R_{\gamma\gamma}$ for $m(\phi') = 1~\, {\rm MeV}$ (blue),
 $100~\, {\rm MeV}$ (red), $1~\, {\rm GeV}$ (green) and $10~\, {\rm GeV}$ (cyan), assuming $m(T) = 500~\, {\rm GeV}$.  The normalization of the $y$-axis  is arbitrary.  The vertical dashed orange line denotes $\Delta R_{\gamma\gamma} = 0.04$; for events to the left of this line, the two photons cannot be discriminated.}
 \label{fig:photon_DeltaR}
 \end{figure}

A similar effect is observed for the energetic $t$ produced through the $T$-$t$ fusion interaction, in which a portion of the $t$ decay products cannot be fully reconstructed independently of each other. This results in three different scenarios of $t$ reconstruction: a fully merged scenario, in which the decay products of the $W$ boson and the $b$ quark are so collimated that they are reconstructed as a single ``fat jet'' (henceforth referred to as a FatJet, FJ); a partially merged scenario, in which the decay products of the $W$ boson form a single FatJet but the $b$ quark can still be separately identified; and an un-merged scenario, in which all decay products can be independently identified. Each scenario has an associated tagging efficiency and misidentification rate, which depends on the boosted $t$/$W$ algorithm. Following Refs.~\cite{CMSboostedWTopTaggingPaper,ATLASboostedWTopTaggingPaper}, we consider two possible ``working points'' for the identification of the fully merged top decays: ($i$) a ``Loose'' working point with 85\% top tagging efficiency and 11-25\% misidentification rate, depending on the FatJet transverse momentum ($p_{T}^{FJ}$); and ($ii$) a ``Tight'' working point with 50\% top tagging efficiency and 4-10\% misidentification rate, depending on $p_{T}^{FJ}$. Similarly, for the partially merged scenario we consider a ``Loose'' working point with 50\% $W$ tagging efficiency and 1-2\% misidentification rate, and a ``Tight'' working point with 25\% $W$ tagging efficiency and 0.2-0.5\% misidentification rate. The choice of boosted $t$/$W$ working points is determined through an optimization process which maximizes discovery reach, as discussed in the next section. The ``Loose'' working  points for both the fully merged $t$ and partially merged $W$ were ultimately shown to provide the best sensitivity and therefore chosen for this study. For now we note that the contribution from SM backgrounds with a misidentified boosted $t$/$W$ is negligible, and thus our discovery projections are not sensitive to uncertainties related to the boosted $t$/$W$ misidentification rates.

Due to the above considerations, the ultimate final-state of interest will consist of: a single highly energetic photon, a (possibly boosted) top tagged system, a $b$ quark, a lepton, and large missing transverse momentum ($\vec{p}_{T}^{\textrm{~miss}}$). For the partially merged and un-merged scenarios, there will be two $b$ quarks present in the final state (one of which is part of the top tagged system). It is important to be able to differentiate between the $b$ quark associated with the hadronic decay of the top quark and the $b$ quark associated with the $T$ decay, for instance when attempting to reconstruct/measure the mass of the $T$ using its decay products. An identification scheme was therefore employed in which the $b$ quark located closest in $\Delta R$ to the dijet pair (from $W\to jj$) was labeled the ``hadronic decay'' $b$ quark ($b_{h}$), while the further $b$ quark would be referred to as the ``leptonic decay'' $b$ quark ($b_{\ell}$). This scheme mimics what is possible experimentally, where one cannot identify the exact decay origins of any particular particle. It was found that this scheme correctly labeled each $b$ quark with an accuracy of greater than 80\%, depending on $m(T)$.

Sample events for $pp$ collisions at $\sqrt{s}=13$ TeV for both signal and background were generated using MadGraph5\_aMC (v2.7.3)~\cite{MADGRAPH}. Hadronization was performed with PYTHIA (v8.2.05) \cite{Sjostrand:2014zea}. Detector effects were included through Delphes (v3.4.1) \cite{deFavereau:2013fsa}, using the CMS input card with 140 average pileup interactions. Signal samples were created for $m(T)$ ranging from 500 GeV to 2500 GeV. The large mass difference between $T$ and $\phi'$ resulted in kinematic distributions for $\phi'$ that were similar regardless of $m(\phi')$ mass (for masses up to 10 GeV).

Total event yields are parameterized using $N=\sigma\times\mathcal{L}\times\epsilon$, where $N$ represents the total number of events, $\mathcal{L}$ is the integrated luminosity scenario being considered (for this study, 150 fb$^{-1}$ and 3000 fb$^{-1}$), and $\epsilon$ represents any efficiencies which might reduce the total event yield (reconstruction efficiencies, etc.). All production cross sections were
computed at tree-level.  As the $k$-factors associated with higher-order corrections to QCD production cross sections are typically greater than one, our estimates of the sensitivity are conservative.

Note that the sample of signal events
we generate include $T$-$t$ fusion processes (see Fig.~\ref{fig:FeynmanDiagram}), as well as $\bar T T$
pair-production, with $T$ decaying to $t\phi'$.  There
is little interference between these processes, which
can be distinguished because, in the case of $\bar T T$
pair-production, the invariant mass of the $t \phi'$
(or, equivalently $t\gamma \gamma$) system is $m(T)$.
The cross section for this latter process depends on
the $\mathrm{Br}(T \rightarrow t\phi')$, which depends on
details of the model which we do not fix.  For
simplicity and to be conservative with our discovery projections,, we set $\mathrm{Br}(T \rightarrow t\phi') = 1\%$, which
is reasonable and easily realizable in
the parameter space of interest (e.g., see Table I).  We find that these fraction of events arising from $\bar T T$ pair-production
is negligible.

The detection efficiency for $\phi' \rightarrow \gamma \gamma$ decays depends on the decay length. The distance the $\phi'$ particle travels before it decays varies depending on the decay width and boost of the $\phi'$. For scenarios with both sufficiently light and sufficiently boosted $\phi'$, a large portion of $\phi'$s begin to decay outside the detector, representing a change in the final state being reconstructed. This effect represents a lower bound of $\phi'$ mass that can be probed by the proposed diphoton final state.
To determine the range in $m(\phi')$ at which the long lifetime becomes important, we compute the $\phi'$ decay length perpendicular to the $pp$ beam axis, which has the form $L_{\phi',\perp} = \frac{\sqrt{\gamma_{\phi'}^{2}-1}}{\Gamma_{\phi'}}\sin\theta$. In this equation, $\theta$ is the scattering angle relative to the beam axis and $\gamma_{\phi'}$ is the relativistic boost factor. This quantity is calculated per simulated signal event by utilizing the $\phi'$ pseudorapidity distribution and the momentum of the particle in the laboratory frame. We conservatively require the $\phi'$ to decay before the CMS ECAL. Therefore, events with $L_{\phi',\perp}$ values greater than 1.29 m cannot be used. The $\phi'$ decay length efficiency was computed by simulating large statistics samples of events for given choices of $m(T)$, $m(\phi')$ and $\Gamma_{T}$, and
calculating the fraction of events in which the $\phi'$
decays within the detector. Figure~\ref{fig:Phi_Decay_Length_100Contour} shows these values as a function of both $m(T)$ and $m(\phi')$, before any selection cuts were applied, taking the $\phi'$
lifetime to scale as $t_{\phi'}\sim(10^{-14}\, {\rm s})(\mathrm{MeV}/m_{\phi'})^3$.
As this figure illustrates, the sensitivity will drop
rapidly with $m_{\phi'}$ once the $\phi'$ becomes light enough that an appreciable fraction of $\phi'$ decay outside the detector.

Table~\ref{tab:lifetime} shows the longest lifetime the $\phi'$ can have for which a 5$\sigma$ discovery is still possible. The exact event selection criteria will be defined in the next section. For now, we note that for longer lifetimes, too few $\phi'$s decay inside the detector to achieve such sensitivity.
We see that, in order for a significant number of
$\phi'$ to decay within the detector, one would need
$\Gamma_{\phi'}$ to be significantly larger than the
benchmark value considered in subsection~\ref{sec:decay}.  But if
one includes QCD-charge factors and the particle
multiplicities of a more complicated dark sector,
prompt decays are realizable.

\captionsetup[subtable]{position=top}
\begin{table}[]
\caption{Longest lifetime of $\phi'$ [s] for which 5$\sigma$ discovery can be achieved. ``N/A'' indicates no 5$\sigma$ discovery is possible (due to small cross sections).}
\subfloat[150 fb$^{-1}$ integrated luminosity.]{\label{tab:lifetime_150}%
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{|l|c|c|c|c|c|}
\cline{3-6}
\multicolumn{2}{c|}{} & \multicolumn{4}{c|}{$\bm{m}(\phi')$ $\textbf{[MeV]}$} \\
\cline{3-6}
\multicolumn{2}{c|}{} & 0.1 & 1 & 10 & 100 \\
\cline{1-6}
\multirow{7}{*}{\rotatebox[origin=c]{90}{$\bm{m(T)}$ $\mathbf{[GeV]}$}} & 500  & 2.6$\times10^{-15}$ & 2.6$\times10^{-14}$ & 2.6$\times10^{-13}$ & 2.6$\times10^{-12}$ \\
\cline{2-6}
& 750  & 2.4$\times10^{-15}$ & 2.4$\times10^{-14}$ & 2.4$\times10^{-13}$ & 2.4$\times10^{-12}$ \\
\cline{2-6}
& 1000 & 1.4$\times10^{-15}$ & 1.4$\times10^{-14}$ & 1.4$\times10^{-13}$ & 1.4$\times10^{-12}$ \\
\cline{2-6}
& 1250 & 5.6$\times10^{-16}$ & 5.6$\times10^{-15}$ & 5.6$\times10^{-14}$ & 5.6$\times10^{-13}$ \\
\cline{2-6}
& 1500 & N/A & N/A & N/A & N/A \\
\cline{2-6}
& 1750 & N/A & N/A & N/A & N/A \\
\cline{2-6}
& 2000 & N/A & N/A & N/A & N/A \\ \hline
\end{tabular}
}
\par
\subfloat[3000 fb$^{-1}$ integrated luminosity.]{\label{tab:lifetime_3000}%
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{|l|c|c|c|c|c|}
\cline{3-6}
\multicolumn{2}{c|}{} & \multicolumn{4}{c|}{$\bm{m}(\phi')$ $\mathbf{[MeV]}$} \\
\cline{3-6}
\multicolumn{2}{c|}{} & 0.1 & 1 & 10 & 100 \\
\cline{1-6}
\multirow{7}{*}{\rotatebox[origin=c]{90}{$\bm{m(T)}$ $\mathbf{[GeV]}$}} & 500 & 2.7$\times10^{-15}$ & 2.7$\times10^{-14}$ & 2.7$\times10^{-13}$ & 2.7$\times10^{-12}$ \\
\cline{2-6}
& 750  & 2.6$\times10^{-15}$ & 2.6$\times10^{-14}$ & 2.6$\times10^{-13}$ & 2.6$\times10^{-12}$ \\
\cline{2-6}
& 1000 & 2.3$\times10^{-15}$ & 2.3$\times10^{-14}$ & 2.3$\times10^{-13}$ & 2.3$\times10^{-12}$ \\
\cline{2-6}
& 1250 & 1.5$\times10^{-15}$ & 1.5$\times10^{-14}$ & 1.5$\times10^{-13}$ & 1.5$\times10^{-12}$ \\
\cline{2-6}
& 1500 & 9.0$\times10^{-16}$ & 9.0$\times10^{-15}$ & 9.0$\times10^{-14}$ & 9.0$\times10^{-13}$ \\
\cline{2-6}
& 1750 & 4.8$\times10^{-16}$ & 4.8$\times10^{-15}$ & 4.8$\times10^{-14}$ & 4.8$\times10^{-13}$ \\
\cline{2-6}
& 2000 & N/A & N/A & N/A & N/A \\ \hline
\end{tabular}
}
\label{tab:lifetime}
\end{table}
\captionsetup[subtable]{position=bottom}

For this study, various SM backgrounds are considered, including $W$+jets, $Z$+jets, $\gamma$+jets, $\gamma\gamma$+jets, QCD multijet, $t\bar{t}+\gamma$, $t\bar{t}+\gamma\gamma$, and $t\bar{t}+H$ ($H\rightarrow\gamma\gamma$) events. The production of $W$+jets, $Z$+jets, $\gamma$+jets, $\gamma\gamma$+jets, and QCD multijet events are found to be a negligible contribution to the proposed search region due to the low probability of having multiple light quark or gluon jets in each event to be misidentified as hadronic $t$/$W$ decays, photons, and/or leptons. Additionally, the majority of the QCD multijet processes contain no genuine missing momentum from neutrinos. The SM processes with higgs decays to photons do not contribute because the relatively low boost of the higgs boson results in two clean and well separated photons. It was found that $t\bar{t}+\gamma$ (with the $\gamma$ coming from initial state or final state radiation), as shown in Fig.~\ref{fig:BackgroundFeynmanDiagram}, represented the dominant irreducible background and $> 99$\% of the total background.

 \begin{figure}[h]
 \begin{center}
 \includegraphics[width=0.4\textwidth]{ttbar_plus_photon.pdf}
 \end{center}
 \caption{Typical $t\bar{t}+\gamma$ Feynman diagram, shown with final state radiation (FSR).}
 \label{fig:BackgroundFeynmanDiagram}
 \end{figure}

 \begin{figure}[h]
 \begin{center}
 \includegraphics[width=0.5\textwidth]{Phi_Decay_Length_100Contour.png}
 \end{center}
 \caption{$\phi'$ decay length efficiency, showing $90\%$, $50\%$, and $10\%$ efficiency levels, assuming $\Gamma_T$ and $\Gamma_{\phi'}$ as described in the text.
 The contours represent the fraction of events decaying inside the ECAL detector.}
 \label{fig:Phi_Decay_Length_100Contour}
 \end{figure}

 \section{Event selection criteria}
 In order to optimize the kinematic selections for maximum discovery potential, a cut-based selection procedure was performed, in which cuts were applied sequentially to optimize signal significance, defined as $N_S/\sqrt{N_S+N_B}$ where $N_S$ and $N_B$ are the number of events in signal and background respectively. We note this particular definition of signal significance is only used for the purpose of optimizing the selections. The final discovery reach is determined with a shape based analysis (described later) using the full reconstructed $T$ mass spectrum. The reconstructed $T$ mass is defined as the transverse mass $m_{T}(b_{\ell},\ell,\vec{p}_{T}^{\textrm{miss}}) = \sqrt{E_{T}(T)^{2} - p_{T}(T)^{2}}$, where $E_{T}(T)$ and $p_{T}(T)$ are the energy and momentum of the resulting Lorentz vector produced by adding together the 4-momentum vectors of the $b$ quark $b_{\ell}$, the lepton, and the missing transverse momentum. For the SM background, the bulk of the $m_{T}(b_{\ell},\ell,\vec{p}_{T}^{\textrm{miss}})$ distribution lies at relatively low reconstructed mass values and decreases exponentially as $m_{T}(b_{\ell},\ell,\vec{p}_{T}^{\textrm{miss}})$ increases. On the other hand, the $m_{T}(b_{\ell},\ell,\vec{p}_{T}^{\textrm{miss}})$ signal distribution appears as a localized bump centered at approximately $m(T)$ (see Results section). For the purpose of the optimization procedure, the values of $N_{S}$ and $N_{B}$ in the significance metric are derived by selecting events near the reconstructed signal $T$ mass peak, $m(T)-2\Gamma(T) < m_{T}(b_{\ell},\ell,\vec{p}_{T}^{\textrm{miss}}) < m(T)+4\Gamma(T)$, where $\Gamma(T)$ is the $T\to bW$ decay width. A benchmark scenario of $m(T) =500~\, {\rm GeV}$ and $m(\phi')=100~\, {\rm MeV}$ was chosen with which to perform the optimization, and the cut values derived therein were applied to all subsequent mass points. The order of the cut optimization was determined by which variables showed the greatest discriminating power between signal and background. Some examples of relevant kinematic distributions are shown in Figs.~\ref{fig:Photon_Pt},~\ref{fig:lepton_MET_delta_phi}, and~\ref{fig:lepton_pt} (all distributions have been normalized to unity).

 Figure~\ref{fig:Photon_Pt} shows the $p_T (\gamma)$ distribution for signal versus background, with the background shape being a falling exponential which is characteristic of initial or final state radiation, while the signal shape has a pronounced hump structure, which is characteristic of decays coming from a boosted object. The signal significance is optimized for $p_{T}(\gamma) > 160$ GeV. Figure~\ref{fig:lepton_MET_delta_phi} shows the $\Delta\phi (\ell,\vec{p}_{T}^{\textrm{~miss}})$ distribution. For signal, the lepton and neutrino which produces the $\vec{p}_{T}^{\textrm{~miss}}$ are the decay products of a very boosted $W$ system (which carries large momentum from the decay of the massive $T$). We therefore expect there to be a small (although non-zero) $\Delta\phi$ between these two very boosted objects. For background however, the semileptonically decaying $t$ carries comparatively less momentum, and therefore its decay products have a distribution with a wider spread of $\Delta\phi(\ell,\vec{p}_{T}^{\textrm{~miss}})$ values. We determined the optimal $\Delta\phi(\ell,\vec{p}_{T}^{\textrm{miss}})$ to be $|\Delta\phi(\ell,\vec{p}_{T}^{\textrm{miss}})| \leq 1$ radians. Figure~\ref{fig:lepton_pt} shows the $p_{T} (\ell)$ distributions  for signal versus background, where the signal lepton carries more momentum due to it being a final decay product of the massive $T$ and therefore peaks at a larger value than the background process. By a similar methodology, we determined the optimal threshold for the $p_{T}$ of the lepton to be $p_{T}(\ell) > 50$ GeV.

The identification of leptons and boosted top quarks plays an important role in the signal acceptance and reduction of SM backgrounds, and thus also the projected High-Luminosity LHC (HL-LHC) discovery reach. In the case of the HL-LHC, the reconstruction and identification of leptons and the hadronic decay products of the top quark may be non-trivial due to the presence of a large number of pileup interactions. The importance of pileup mitigation at CMS and ATLAS has been outlined in many papers, such as Ref.~\cite{CMS:2013ega}. While the expected performance of the upgraded ATLAS and CMS detectors for the HL-LHC is beyond the scope of this work, the studies presented in this paper do attempt to provide reasonable expectations by conservatively assuming some degradation in lepton and hadron identification efficiencies, based on Ref.~\cite{CMS:2013ega}, and considering the case of 140 average pileup interactions. For muons (electrons) with $|\eta|< 1.5$, the assumed identification efficiency is 90\% (85\%), with a 0.3\% (0.6\%) misidentification rate. The performance degrades linearly with $\eta$ for $1.5 < |\eta| < 2.5$, and we assume an identification efficiency of 65\% (60\%), and a 0.5\% (1.0\%) misidentification rate, at $|\eta| = 2.5$. Similarly, the charged hadron tracking efficiency, which contributes to the jet clustering algorithm and $\vec{p}_{T}^{\textrm{~miss}}$ calculation, is 97\% for $1.5 < |\eta| < 2.5$, and degrades to about 85\% at $|\eta| = 2.5$. These potential inefficiencies due to the presence of secondary $pp$ interactions contribute to how well the lepton and top kinematics can be reconstructed.

In addition to the $p_{T}(\gamma)$, $|\Delta\phi(\ell,\vec{p}_{T}^{\textrm{miss}})|$, and $p_{T}(\ell)$ requirements motivated by Figs.~7-9, events with a fully merged top quark system must have a reconstructed FatJet mass compatible with the top quark mass, requiring $m_\textrm{reco}(t) \in [120,220]$. Finally, we impose a modest missing momentum requirement of $p_{T}^{\textrm{miss}} > 20$ GeV. While the $m_\textrm{reco}(t)$ and $p_{T}^{\textrm{miss}}$ selections are highly efficient for signal events and $t\bar{t}+\gamma$ background events, they help to ensure the negligible contribution from other SM backgrounds with pairs of vector bosons, $Z/\gamma^{*}\to \ell\ell$ with associated jets, $W\to \ell\nu$ with associated jets, and QCD multijet production of light quarks. Table~\ref{tab:Selection_Cuts} shows the final derived selection cuts and values, with the listing order indicating the order in which the cuts were optimized.

 \begin{center}
 \begin{table}
 \caption{Selection cut values for optimized signal significance. A benchmark scenario of $m(T) = 500~\, {\rm GeV}$, $m(\phi') = 100~\, {\rm MeV}$ was considered.}
\bigskip
\begin{tabular}{c|c}
\hline \hline
Selection Parameter & Value \\
\hline
$p_{T}(\gamma)$ & $\geq 160$ GeV\\
$|\Delta\phi(\ell,\vec{p}_{T}^{\textrm{~miss}})|$ & $\leq$ 1 rads\\
$p_{T}(\ell)$& $\geq 50$ GeV\\
$\eta(\ell)$ & $\leq 2.5$\\
$p_{T}(b_{\ell})$ & $\geq 140$ GeV\\
$\eta(b_{\ell})$ & $\leq 2.4$\\
$|\vec{p}_{T}^{\textrm{~miss}}|$ & $\geq$ 20 GeV \\
$m_\textrm{reco}(t)$  & [120,220] GeV\\
$m_T(b_{\ell},\ell,\vec{p}_{T}^{\textrm{~miss}})$ lower bound & $m(T)-2*\Gamma(T)$\\
$m_T(b_{\ell},\ell,\vec{p}_{T}^{\textrm{~miss}})$ upper bound & $m(T)+4*\Gamma(T)$\\
 \hline \hline
\end{tabular}
 \label{tab:Selection_Cuts}
\end{table}
\end{center}

 \begin{figure}[h]
 \begin{center}
 \includegraphics[width=0.5\textwidth]{Photon_Pt.pdf}
 \end{center}
 \caption{$p_{T} (\gamma)$ distribution for signal (blue) and background (red) samples.
 Signal events were generated assuming  $m(T) = 500~\, {\rm GeV}$, $m(\phi') = 100~\, {\rm MeV}$.
 The $y$-axis normalization is arbitrary.
 The vertical black dashed line denotes the location of the
 selection cut.}
 \label{fig:Photon_Pt}
 \end{figure}

 \begin{figure}[h]
 \begin{center}
 \includegraphics[width=0.5\textwidth]{lepton_MET_delta_phi.pdf}
 \end{center}
 \caption{Distribution of the angle between
between $\ell$ and $\vec{p}_{T}^{\textrm{~miss}}$ ($\Delta\phi (\ell,\vec{p}_{T}^{\textrm{~miss}})$) for
signal (blue) and background (red) samples.
  Signal events were generated assuming  $m(T) = 500~\, {\rm GeV}$, $m(\phi') = 100~\, {\rm MeV}$.
 The $y$-axis normalization is arbitrary.
 The vertical black dashed lines denotes the location of the
 selection cut.}
 \label{fig:lepton_MET_delta_phi}
 \end{figure}

 \begin{figure}[h]
 \begin{center}
 \includegraphics[width=0.5\textwidth]{lepton_pt.pdf}
 \end{center}
 \caption{Distribution of $p_{T}(\ell)$ for signal (blue) and background (red) samples.
 Signal events were generated assuming  $m(T) = 500~\, {\rm GeV}$, $m(\phi') = 100~\, {\rm MeV}$.
 The $y$-axis normalization is arbitrary.
 The vertical black dashed line denotes the location of the
 selection cut.}
 \label{fig:lepton_pt}
 \end{figure}

  \begin{figure}[h]
 \begin{center}
 \includegraphics[width=0.5\textwidth]{SR_Plot.pdf}
 \end{center}
 \caption{Distribution of $m_{T}(b_{\ell},\ell,\vec{p}_{T}^{miss})$ for signal and background (red) after applying selection cuts on all parameters except $m_{T}(b_{\ell},\ell,\vec{p}_{T}^{miss})$, assuming $\mathcal{L} = 3000$ fb$^{-1}$. Two $m(T)$ mass points are shown: $m(T) = 1000$ GeV (blue) \& $m(T) = 1750$ GeV (green). In both signal benchmark scenarios, $m(\phi^{\prime})$ is 100 MeV.}
 \label{fig:SR_Plot}
 \end{figure}

 \section{Results}\label{sec:Results}

 Figure~\ref{fig:SR_Plot} shows the expected background and signal yields in bins of $m_{T}(b_{\ell},\ell,\vec{p}_{T}^{\textrm{miss}})$. A few signal benchmark points are considered and the yields are normalized to cross section times integrated luminosity of 3000 fb$^{-1}$. The predicted background yield in the $m_{T}(b_{\ell},\ell,\vec{p}_{T}^{\textrm{miss}})$ mass window of [930,1140] GeV is $N_{B} = 250.5 \pm 22.8$, while the signal yield for $m(T)=1000$ GeV within the same mass window is $N_{S} = 2933.3 \pm 52.3$. For the $m_{T}(b_{\ell},\ell,\vec{p}_{T}^{\textrm{miss}})$ mass window of [1627.5,1995] GeV, the predicted background yield is $N_{B} = 27.2 \pm 7.5$, while the signal yield for $m(T)=1750$ GeV is $N_{S} = 61.6 \pm 0.9$. The uncertainties on these numbers represent statistical uncertainties due to the number of raw simulated events passing the selection criteria.

 To assess the expected experimental sensitivity of this search at the LHC, we followed a profile binned likelihood test statistic approach, using the expected bin-by-bin yields in the reconstructed $m_{T}(\ell,b,\vec{p}_{T}^{\textrm{~miss}})$ distribution. The signal significance $z$ is determined using the probability of obtaining the same test statistic with the background-only hypothesis and the signal plus background hypothesis, defined as the local p-value. The value of $z$ corresponds to the point where the integral of a Gaussian distribution between $z$ and $\infty$ results in a value equal to the local p-value. The calculation of the signal significance is performed assuming two values for the total integrated luminosity at the LHC: ($i$) 150 fb$^{-1}$, which is approximately the amount of $pp$ data already collected by ATLAS and CMS; and ($ii$) 3000 fb$^{-1}$, expected by the end of the High Luminosity LHC era. The $m_{T}$-based signal significance calculations were performed using the ROOFit \cite{ROOTFit} toolkit developed by CERN.

Various sources of systematic uncertainties are considered in the signal significance calculation. These uncertainties, based on both experimental and theoretical constraints, are incorporated in the test statistic as nuisance parameters, assuming log-normal priors for normalization parameters, and Gaussian priors for uncertainties on the shape of the $m_{T}(\ell,b,\vec{p}_{T}^{\textrm{~miss}})$ distribution. Experimental systematic uncertainties on $\gamma$ identification, and on reconstruction and identification of $b$-jets and top-tagged jets were considered. For $\gamma$ identification, Refs.~\cite{Khachatryan:2016hje,Aad:2014cka} reports a systematic uncertainty of about 10\%. However, to account for possible effects from the merged photon system, we assume a conservative 15\% uncertainty which is uncorrelated between signal and background processes, and correlated across $m_{T}$ bins for each process. For experimental uncertainties related to the reconstruction and identification of boosted top quarks, a 20\% value was included (independent of $p_{T}$ and $\eta$ of the top-tagged system), following the results from Refs.~\cite{CMSboostedWTopTaggingPaper,ATLASboostedWTopTaggingPaper}. The uncertainty from $b$-jet identification is 10\%, while efficiencies for the electron and muon reconstruction, identification, and isolation requirements have an uncertainty of 2\%. The uncertainties on the reconstruction of $\vec{p}_{T}^{\textrm{~miss}}$ are due to the uncertainties on proper jet energy measurements. We assumed 2-5\% jet energy scale uncertainties, depending on $\eta$ and $p_{T}$, resulting in shape-based uncertainties on $m_{T}$ that range from 3\% to 6\%, depending on the $m_{T}$ bin.

 \begin{figure}[h]
 \begin{center}
 \includegraphics[width=0.5\textwidth]{U1T3R_Signal_Significance_1plot.pdf}
 \end{center}
 \caption{Signal significance as a function of $m(T)$, for scenarios with $m(\phi')=1$ MeV (magenta) and  $m(\phi')=100$ MeV (green) and for scenarios with $\mathcal{L}_{int}=150\;\mathrm{fb}^{-1}$ (solid lines) and $\mathcal{L}_{int}=3000\;\mathrm{fb}^{-1}$ (dashed lines).   The dashed vertical yellow line indicates the lower limit on $m(T)$ found in~\cite{Cacciapaglia:2019zmj},
 assuming $\mathrm{Br}(T \rightarrow tH) = 50\%$.}
 \label{fig:Signal_Significance}
 \end{figure}

 \begin{figure}[h]
 \begin{center}
 \includegraphics[width=0.5\textwidth]{SignalSig_BF_mT.png}
 \end{center}
 \caption{Signal significance as a function of $m(T)$ and Br($T\rightarrow Wb$), for $m(\phi')$ = 100 MeV. The heat map levels correspond to signal significance, while overlaid dashed lines correspond to the 95\% confidence level (yellow), 3$\sigma$ discovery potential (green), and 5$\sigma$ discovery potential (red), assuming $\mathcal{L}_{int}=3000$ fb$^{-1}$. }
 \label{fig:SignalSig_BF_mT}
 \end{figure}

In addition, theoretical uncertainties were included in order to account for the set of parton distribution functions (PDFs) used to produce the simulated  signal and background samples. The PDF uncertainty was calculated following the PDF4LHC prescription~\cite{Butterworth:2015oua}, and results in a 3-10\% systematic uncertainty, depending on the process. The effect of the chosen PDF set on the shape of the $m_{T}$ distribution is negligible. The absence of higher-order contributions to the signal cross sections affect the signal acceptance calculation. This effect is estimated by varying the renormalization and factorization scales by a factor of two with respect to the nominal value, and by considering the full change in the yields. They are found to be small (3.5\%).

Figure~\ref{fig:Signal_Significance} shows the results of the expected signal significance for different  $m(T)$ and $m(\phi')$ scenarios.
For the 150 fb$^{-1}$ scenario, it is feasible to exclude (at 95\% confidence level)  $ m(T) < 1.7$ $(1.1)$ TeV for $m(\phi') = 100$ $(1)$ MeV. In order to compare the expected reach of the proposed search with existing searches from CMS and ATLAS, Figure~\ref{fig:Signal_Significance} also shows the current observed upper limit on $m(T)$ from CMS/ATLAS under similar branching fraction considerations. The expected 95\% exclusion of $m(T) < 1.7$ TeV  from the proposed search is to be compared with the $m(T) < 875$ GeV exclusion from the current LHC searches~\cite{Cacciapaglia:2019zmj}. Similarly, for the 3000 fb$^{-1}$ scenario, the 5$\sigma$ ($3\sigma$) reach includes $m(T) < 1.8$ ($2$) TeV for $m(\phi') = 100$ MeV. For $m(\phi') \lesssim 1~\, {\rm MeV}$, a large fraction of the signal events are lost due to an average $\phi'$ decay length that exceeds 1.29 m, thus preventing a photon signature in the electromagnetic calorimeter. Therefore, for 3000 fb$^{-1}$ and $m(\phi') = 1~\, {\rm MeV}$, the 5$\sigma$ ($3\sigma$) reach degrades to $m(T) < 1.2$ ($1.3$) TeV. Figure~\ref{fig:SignalSig_BF_mT} further demonstrates how signal significance changes when one now varies the branching fraction of $\mathrm{Br}(T\to Wb)$ for various $m(T)$ (assuming $\mathcal{L}_{int}=3000$ fb$^{-1}$).

{Note that, in this scenario, the search for light scalars in association with
heavy quarks can provide better detection prospects than the search either light scalars or
heavy quarks individually.}

 \section{Discussion}

 We have demonstrated that the LHC can discover the visible
 decays of new sub-GeV particles.  The key to its
 discovery potential is the production of heavy QCD-coupled particles, which then decay to the
 light particles with large central boost.  The large
 boost allows the signal from visible decays to be seen
 above the lower-energy background.

 There are many laboratory experiments, either current
 or planned, which focus on the detection of new
 low-mass particles.  But the LHC can fill a niche
 which is not necessarily covered by other strategies.
 Most beam experiments focus on the production of
 light particles through processes with low momentum
 transfer, in which the produced new particles are
 largely co-linear with the beam.  These experiments
 must either deal with the large backgrounds which come
 with the beam itself, or are limited to the search
 for long-lived particles which decay at displaced
 detectors that are protected from beam backgrounds
 by shielding.

 The LHC search strategy described here can be used to
 discover the prompt decay of new light particles,
 because they are produced from processes with large
 momentum transfer, and thus are emitted in the
 central region, away from the beam.  We thus see that
 detection prospects for low-mass particles are
 enhanced when it is kinematically possible to access
 the heavy degrees of freedom which arise in the
 UV completion of the low-energy model.  Indeed, this
 scenario, in which the dominant coupling of the light
 scalar is to the top quark, is an example which would
 be difficult to directly probe at low energy beam
 experiments.

 We considered the specific example of a new gauge
 group $U(1)_{T3R}$, in which the dark Higgs is
 a light (sub-GeV) particle that couples to a heavy
 new vector-like quark.  This new fermion appears as
 the seesaw partner of the top quark, and appears as
 part of the UV completion of the low-energy model.  But
 this scenario was introduced as a toy-model, and
 the basic idea will generalize to other models.

 For example, if the scalar is slightly heavier than
 the range considered here, decays to $\mu^+ \mu^-$
 will be possible.  This channel should
 also provide
 good detection prospects.  It would be interesting to
 study these prospects in a future work.

\vskip .1in
{\bf Acknowledgments}
\vskip .1in
We are grateful to Noah Steinberg and James D.~Wells
for useful discussions.
The work of BD, SG and TK is supported in part by DOE grant DE-SC0010813. The work of SG is also supported in part by National Research Foundation of Korea (NRF), grant no. 6N021413.
The work of JK is supported in part by DOE grant DE-SC0010504. AG and DJ are supported in part by NSF Award PHY-1945366.

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