# Energetic long-lived particles in the CMS muon chambers

Andrea Mitridate  Walter Burke Institute for Theoretical Physics, California
Institute of Technology, Pasadena, CA 91125, USA  Deutsches Elektronen-
Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany  Michele Papucci
Walter Burke Institute for Theoretical Physics, California Institute of
Technology, Pasadena, CA 91125, USA  Christina W. Wang  California Institute
of Technology, Pasadena, CA 91125, USA  Cristián Peña  Fermi National
Accelerator Laboratory, Batavia, IL 60510, USA  Si Xie  California Institute
of Technology, Pasadena, CA 91125, USA  Fermi National Accelerator Laboratory,
Batavia, IL 60510, USA

###### Abstract

We present a recast in different benchmark models of the recent CMS search
that uses the endcap muon detector system to identify displaced showers
produced by decays of long-lived particles (LLPs). The exceptional shielding
provided by the steel between the stations of the muon system drastically
reduces the Standard Model background that limits other existing ATLAS and CMS
searches. At the same time, by using the muon system as a sampling
calorimeter, the search is sensitive to LLPs energies rather than masses. We
show that, thanks to these characteristics, this new search approach is
sensitive to LLPs masses even lighter than a GeV, and can be complementary to
proposed and existing dedicated LLP experiments.

††preprint: CALT-TH-2023-001††preprint: DESY-23-054

###### Contents

  1. I Introduction
  2. II Analysis and recast strategy
    1. II.1 CMS endcap muon system analysis
    2. II.2 Event generation
    3. II.3 Detector simulation with dedicated Delphes modules
    4. II.4 Analysis strategy
    5. II.5 Limit calculation and validation
  3. III Benchmark models
    1. III.1 Light scalar singlet
    2. III.2 Abelian hidden sector
    3. III.3 Inelastic Dark Matter
    4. III.4 Axion-like particles
    5. III.5 Hidden Valley
  4. IV Results
  5. V Discussion
  6. A ATLAS mono-jet limit for Axion-like Particles coupled to gluons

##  I Introduction

Despite its successes, the Standard Model (SM) of particle physics has several
shortcomings. Most notably, it fails to explain the nature of Dark Matter, the
baryon asymmetry in the Universe, and the origin of neutrino masses. Because
of this, several extensions of the SM have been proposed, a common byproduct
of which are long-lived particles (LLPs). This is especially true for _dark
sectors_ comprised of new neutral particles with masses below the electroweak
scale that interact with the SM through suppressed renormalizable couplings,
heavy mediators, or both. In recent years, several searches for neutral LLPs
have been conducted by ATLAS, CMS, LHCb, and other dedicated LLP experiments
and a vibrant search program is being developed [1, 2, 3, 4, 5, 6, 7, 8, 9,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27].
However, most ATLAS and CMS searches are known to face challenges for LLPs
with masses at or below a GeV. While some room for improvement may be present
in existing analyses [28, 29, 14, 30, 14, 31, 32, 33], looking for tracks from
displaced vertices will ultimately be limited by irreducible SM backgrounds.
This is especially true in the long lifetime regime, where the few tracks
produced by the decays of light LLPs are not enough to discriminate them
against SM LLP decays such as those of $K_{L}$.

However, recently the CMS collaboration published a search [34] for neutral
LLPs which uses the CMS endcap muon detectors (EMDs) as a sampling
calorimeter. Thanks to the unique design of the CMS EMDs (made of stations of
cathode strip chambers (CSCs) interleaved with steel return-yoke), LLPs decay
products can induce hadronic and electromagnetic showers that give rise to
high hit multiplicity in localized detector regions. Because of this, the LLPs
signature tracks the LLP energy instead of its mass. This, together with the
exceptional shielding provided by the CMS calorimeters and the steel in the
front layers of the EMD, allows this search to be sensitive to LLPs with
masses smaller than a GeV. Despite this search currently lacking a dedicated
trigger to maximize its potential, its reach is competitive with many proposed
dedicated LLP experiments [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47,
48, 49, 50, 51] due to its large geometric acceptance.

The original CMS paper presented the results on a benchmark model motivated by
the twin Higgs scenario, where the SM Higgs boson decays to a pair of neutral
long-lived scalars, each of which decays in turn to a pair of bottom quarks,
down quarks, or $\tau$ leptons. Masses of the scalar LLPs were probed down to
7 GeV. In this paper, we use the parameterized reconstruction and selection
efficiency functions provided in the HEPData entry [52] of the CMS paper to
recast the analysis in different benchmark models and explore its strengths
and weaknesses. The goal of this recast is to inform future iterations of the
CMS analysis and inform the choice of benchmarks for other proposed LLP
experiments away from already covered regions of the parameter space. Although
we expect the barrel muon detector to also produce the shower signature seen
in the CSCs, in this study, we will not consider the barrel region. The barrel
muon detector is made of drift tubes, which are a different detector
technology and system from the CSCs. Therefore, the response to LLPs will be
different, and we will wait for CMS to release efficiency functions for the
barrel muon detector for future studies.

The paper is structured as follows. In Section II, we briefly review the CMS
analysis, we outline the generation and simulation framework, as well as the
validation of the framework against the CMS result. In Section III, we discuss
all the benchmark models that will be considered. Finally, in Section IV, we
discuss the results of this paper.

##  II Analysis and recast strategy

In this section, we describe the details of the analysis. We start by
summarizing the CMS analysis [34] and then we discuss and validate the recast
procedure, including the event generation and detector simulation framework,
signal selections, signal and background yield estimate, and statistical
analysis to evaluate the upper limits.

###  II.1 CMS endcap muon system analysis

The missing transverse momentum, $p_{\mathrm{T}}^{\text{miss}}$, is calculated
by CMS by using only tracker and calorimeter information. Therefore, LLPs
decaying beyond the calorimeters can produce large
$p_{\mathrm{T}}^{\text{miss}}$ when they are produced in association with
other prompt, visible objects or recoiled against an initial state radiation
(ISR) jet. Due to this feature and the lack of a dedicated trigger, events are
triggered by requiring $p_{\mathrm{T}}^{\text{miss}}>120$ GeV. A further
requirement of $p_{\mathrm{T}}^{\text{miss}}>200$ GeV is then applied offline.
To further suppress the background from $W$ and top quark production, the
search also requires at least one jet from ISR with transverse momentum
$p_{\mathrm{T}}>50$ GeV and pseudorapidity $|\eta|<2.4$, and no leptons in the
event.

By clustering CSC hits using the DBSCAN algorithm [53], the search then
identifies displaced showers produced by LLPs decaying in the endcap muon
system. To suppress background from punch-through jets and muon
bremsstrahlung, clusters that are geometrically matched (i.e. within a cone of
radius $\Delta R=\sqrt{(\Delta\phi)^{2}+(\Delta\eta)^{2}}<0.4$) to jets
(muons) with $p_{T}>10\,(20)$ GeV are rejected. Furthermore, a number of
active vetoes are applied to veto clusters with hits or track segments in the
muon detector stations with the least amount of shielding. Finally, to
suppress the muon bremsstrahlung background originating from muons beyond the
detector acceptance, clusters with $|\eta|>2.0$ are vetoed. Clusters are
required to be consistent with an in-time interaction by restricting the
cluster time ($-5.0<t_{cluster}<12.5$ ns). To reject clusters composed of hits
from multiple bunch crossings, the root mean square spread of a cluster’s hit
times is required to be less than $20$ ns.

A cut-based ID that distinguishes signal from background clusters is defined
by using several features, including the cluster $\eta$ position and the
number of stations occupied by the cluster. The details required to simulate
the cut-based ID efficiency have been provided in the HEPData entry of the CMS
paper [52].

Finally, the number of hits in the cluster, $N_{\mathrm{hits}}$, and the
azimuthal angle between $p_{\mathrm{T}}^{\text{miss}}$ and the cluster
location, $\Delta\phi$, are used to make the final discrimination between
signal and background. The signal is required to have large
$N_{\mathrm{hits}}>130$ and small $\Delta\phi<0.75$. For the chosen signal the
bulk of the $p_{\mathrm{T}}^{\text{miss}}$ is produced by the LLP when they
are produced in association with other prompt, visible objects or recoiled
against an ISR jet, while for the backgrounds, $\Delta\phi$ is independent of
$N_{\mathrm{hits}}$. The independence of the two variables for the background
enables the use of the matrix (ABCD) method to predict the background yield.
Using the matrix method and assuming no signal contribution, 2.0 $\pm$ 1.0
background events were predicted in the signal-enriched region, and 3 events
were observed. No excess of events above the SM background was observed.

###  II.2 Event generation

We generated signal events using MadGraph5 v2.9.3 [54], and performed the
parton shower and hadronization with Pythia v8.244 [55], while keeping the LLP
stable. Samples with different jet multiplicities were merged according to the
MLM algorithm [56, 57]. Generator-level cuts were applied to the events in
order to increase the statistics in the phase space regions selected by
$p_{\mathrm{T}}^{\text{miss}}$ cut in the CMS analysis. Additional details on
the samples, including the specific generator-level cuts, are given in Sec.
III for each of the benchmark models considered in this work.

To efficiently decay the LLP, we used the fact that the reconstruction
efficiency parameterization provided by the CMS search is spatially binned in
a small number of regions with simple shapes ( which are defined by the
intersection of ranges in the radial direction, $r$, longitudinal direction,
$z$, and pseudorapidity, $\eta$), for which the probability of decaying inside
a region can be computed analytically. For a region determined by the
conditions

$\eta_{0}\leq\eta\leq\eta_{1},\qquad r_{0}\leq r\leq r_{1},\qquad z_{0}\leq
z\leq z_{1},$ (1)

the probability, $P$, to decay inside the region for a particle traveling with
momentum $p^{\mu}$ and proper decay length $c\tau$ is

$P=e^{-y_{min}}-e^{-y_{max}}$ (2)

where we have defined

$\displaystyle y_{min}$ $\displaystyle=\frac{1}{\beta\gamma c\tau}{\rm
max}\Big{(}\min{\left(z_{0}\coth{\eta},z_{1}\coth{\eta}\right)},r_{0}\cosh{\eta}\Big{)},$
(3) $\displaystyle y_{max}$ $\displaystyle=\frac{1}{\beta\gamma c\tau}{\rm
min}\Big{(}\max{\left(z_{0}\coth{\eta},z_{1}\coth{\eta}\right)},r_{1}\cosh{\eta},\beta\gamma
c\,t_{cut}\left(\beta\gamma+\sqrt{1+(\beta\gamma)^{2}}\right)\Big{)},$ (4)

with $\eta$ being the pseudorapidity of the particle, and
$\beta\gamma=|\vec{p}|/m$. We have further introduced a timing requirement
such that $t_{decay}-d_{decay}/c<t_{cut}$ where $t_{decay}$ and $d_{decay}$
are the decay time and distance from the origin respectively.

The LLPs generated in the events were made to decay at fixed positions within
each region using Pythia, as the presence of the decay vertex in a given
region is the only geometrical information used by the detector simulation.
For a given proper decay length $c\tau$, the probability for the LLP to decay
in a given region was assigned to the event in the form of an event weight.
The decay probability for each LLP in an event is independent, so the event-
level probability is the product of the decay probability of the LLPs in an
event. Therefore, for each input event with an undecayed LLP, the decay
program generated as many decayed events as the number of regions intersected
by the LLP trajectory. We used the multi-weight capabilities of the HepMC
event format to perform a scan in $c\tau$ without having to reprocess the
events. In case there were multiple LLPs in the same event (as in the case of
Higgs decays to pairs of dark vectors or scalars), we also included decays
outside the signal regions. In the case of LLP decays in the inner detector,
where precise knowledge of the decay vertex position is used by the detector
simulation, we generated the decay vertex position according to the decay
probability distribution instead of keeping it fixed.

The events were subsequently passed to a simplified detector simulation based
on Delphes v3.4.2 [58] using the publicly available CMS configuration card for
the reconstruction of prompt objects supplemented by a dedicated module,
discussed in subsection II.3, simulating the LLP decay reconstruction and
selection using the information provided by the CMS search.

###  II.3 Detector simulation with dedicated Delphes modules

We based the fast simulation of the detector response to standard particle
flow (PF) candidates on the CMS detector response provided by the CMS
configuration card in the parametric Delphes framework. The detector
simulation of the hit clusters in the CSC endcap muon detector was based on a
dedicated Delphes module and class for the CSC cluster objects that we
developed [59] based on the parameterized detector response functions provided
in the HEPData entry [52]. Based on the recasting instructions provided in the
HEPData entry, the simulation of cluster-level selection efficiencies was
divided into three components.

The first component is cluster efficiency; which includes the cluster
reconstruction efficiency, muon veto, active veto, time spread, and
$N_{\mathrm{hits}}$ cut efficiency. This cluster efficiency is provided as a
function of the LLP electromagnetic and hadronic energy in two separate LLP
decay regions in the CSC detector. Building upon the existing Efficiency
module that is already used by all other PF candidates, we implemented a
dedicated CscClusterEfficiency module in Delphes which encodes this
parameterized function.

The second component is cluster identification efficiency. We implemented the
CscClusterID module following the code function provided by the CMS HEPData
entry.

The third and last component includes the model-dependent cluster time
requirement, jet veto, and the $\Delta\phi$ requirement. The values of the
three variables affected by these cuts were calculated using generator-level
information and saved as part of the CscCluster class. Specifically, the
cluster time was determined by calculating the LLP travel time from the
production to the decay vertex in the lab frame. The jet veto was implemented
by requiring no PF jets with $p_{T}>10$ GeV within a cone of $\Delta=0.4$
around the LLP. Finally, $\Delta\phi$ was computed as the azimuthal angle
difference between the LLP momentum and missing transverse momentum (MET)
using the result from the standard Delphes simulation modules. All these
requirements were then made at a later stage of the analysis workflow.

Finally, we modified the standard CMS configuration card from Delphes to
include the CSCClusterEfficiency and CSCClusterID module in the processing
sequence. The modules require only generator-level LLP information and can be
used for the recasting of the result for any other model. The implementation
can be found in [59].

###  II.4 Analysis strategy

In this section, we discuss the procedure used to recast the CMS Run 2 results
and to make projections for high-luminosity LHC (HL-LHC).

For the recasting of the Run 2 results, we used the exact same selection and
cuts used in the CMS paper and summarized in Sec. II.1. For all the standard
particle flow objects, we used the default CMS configuration card from
Delphes, which has been validated [58] to reproduce the object resolutions
from Run 2. The $p_{\mathrm{T}}^{\text{miss}}$calculation implemented in
Delphes is accurate when the LLP decays outside of the calorimeters that it’s
treated as invisible and when the LLP decays sufficiently close to the
interaction point that the energy of the decay particles are measured by the
calorimeters. In models where there is only one LLP in the event the LLP is
required to decay in the muon detector, so the LLPs are treated consistently
as invisible in both Delphes and CMS simulations. For models with two or more
LLPs per event, in the parameter space explored in this reinterpretation
study, this approximation for the $p_{\mathrm{T}}^{\text{miss}}$calculation
leads to a systematic error of 20% or less on the selection efficiency. For
the CSC cluster objects, we ran the CSCClusterEfficiency and CscClusterID
modules to select clusters that would pass the corresponding selections. We
then applied the $\Delta\phi<0.75$ selection, jet veto, and time cut for CSC
clusters. We used the number of signal events passing these signal selections
as our estimate of the signal yield. Finally, by using this estimate for the
signal yield together with the background yield ($2\pm 1$ background event)
and observed data (3 observed events) obtained in the CMS analysis, we derived
our constraints.

To further inform experimental studies and compare to other proposed LLP
experiments, we projected the sensitivity of this analysis to Phase 2
conditions. To simulate the effect on signal yield from the increased number
of pileup interactions during Phase 2, we modified the mean pileup number in
the CMS configuration card from 32 to 200. We observed that, due to the larger
number of jets from pileup interactions, the probability that a CSC cluster in
the signal region is accidentally matched to and vetoed by a pileup jet with
$p_{T}>10$ GeV is 20% higher. Signal region CSC clusters are concentrated in
the region with $|\eta|<$ 1.6, while most pileup jets are concentrated in the
high $\eta$ region, so the increase in the probability of accidental matching
is only 20%. Furthermore, the resolution of $p_{\mathrm{T}}^{\text{miss}}$is
degraded by the large number of pileup jets, such that the MET cut efficiency
increased by a factor of 2 due to fake MET. However, a more realistic
assumption would be that with the help of the new MIP Timing Layer (MTD) and
pileup removal algorithms, the MET resolution will be kept at the same level.
Similarly, we assumed that with the help of the MTD and new reconstruction
algorithms, the efficiency and resolution would be kept at the same level for
all PF candidates. Therefore, a simple projection for Phase 2 constraints can
be derived by scaling the signal (and background) yield by the increased
integrated luminosity, and applying an $80\%$ correction to the signal yield
per cluster due to larger number of pileup jets while assuming the same
efficiency and resolution for all PF candidates.

However, this simple recasting strategy significantly underestimates the
potential sensitivity of a Phase 2 analysis. Realistically, given the larger
dataset, we would apply tighter cuts to achieve near zero background.
Therefore, we considered a second recasting strategy where we apply a tighter
$N_{\mathrm{hits}}$ selection, the main discriminator of the analysis, until
the expected background reaches zero. To estimate the signal and background
yield with a tighter $N_{\mathrm{hits}}$ cut, we used the $N_{\mathrm{hits}}$
distributions in the auxiliary materials from the CMS analysis. We fitted the
$N_{\mathrm{hits}}$ distribution for the background with an exponential
function to extrapolate the background yield at higher $N_{\mathrm{hits}}$
cuts. We found that a $N_{\mathrm{hits}}$ of 200 would suppress the background
yield to below 1 for the expected Phase 2 integrated luminosity. Similarly, we
found that increasing the cut from 130 to 210 would give a signal efficiency
of about $80\%$. Therefore, for this recast strategy, we scaled the signal
yield by an additional $0.8$ with respect to the simple recasting strategy
previously described and assumed a background yield of $0.2$. The majority of
the background comes from low $p_{T}$ kaons and pions from the recoil, pileup,
or underlying events that produce the clusters. The rate of low $p_{T}$
particles from pileup would increase linearly with the number of pileup, but
the background rate from the recoil in the main collision wouldn’t change with
higher pileup. Therefore, we produce a conservative estimate of the effect of
higher pileup by increasing the normalization of the $N_{\mathrm{hits}}$
distribution by the increase in number of pileup and recomputing the
$N_{\mathrm{hits}}$ threshold. The estimated effect on the signal yield is at
most 20%.

Finally, we considered a search strategy that would be enabled by a new
dedicated Level-1 and High Level Trigger targeting this signature starting
from the beginning of Run 3 [60]. The trigger selects for events with at least
one CSC cluster with a large number of hits and is already in operation and
validated. In this strategy, we remove the high $p_{\mathrm{T}}^{\text{miss}}$
selection (which is necessary during Run 2 to trigger the events) and require
at least two CSC clusters. In addition, we remove the requirement of at least
one jet with $p_{T}>50$ GeV and $\Delta\phi(\textrm{cluster, MET})<0.75$ that
has high signal efficiency only in the high MET phase space. For this
strategy, due to the double cluster requirement, we assumed that zero
background can be achieved.

For all the analyses discussed here, we assigned $20\%$ signal systematic
uncertainty which is of the same order of signal systematic for the CMS
result. There it is dominated by missing higher order QCD corrections, which
have a size of 21% for the gluon fusion production mode. The background
uncertainty is dominated by statistical uncertainty, and we assign no
additional background systematic uncertainty.

The result of our recast, for both Run 2 and the projection for Phase 2, are
shown for all the benchmark models in Section IV. Unless differently stated,
the first recasting strategy for Phase 2 is indicated with solid lines, and
the second (third) scenario with dot-dashed (dashed) lines.

Figure 1: Comparison of the 95% CL upper limits on the branching fraction
${\rm Br}(h\to SS)$ as functions of $c\tau$ derived with the standalone
workflow (dashed lines) and the CMS search (solid lines). In deriving these
limits we have considered a $15\;$\mathrm{G}\mathrm{e}\mathrm{V}$$ LLP
decaying into $d$-quark pairs (left), $b$-quark pairs (center), and $\tau$
pairs (right). The limits from this work are shown to agree with the CMS
search to within $30\%$.

###  II.5 Limit calculation and validation

Before moving to the discussion of the benchmark models, we present a
validation of our recast analysis. Specifically, we derived the 95% confidence
level (CL) limits on the branching fraction ${\rm Br}(h\to SS)$ for different
scenarios and compared them to the one derived in the CMS analysis. The
observed $95\%$ CL upper limits on the branching fraction ${\rm Br}(h\to SS)$
for $15\;$\mathrm{G}\mathrm{e}\mathrm{V}$$ LLP as functions of $c\tau$ for the
$S\to d\bar{d}$, $S\to b\bar{b}$, and $S\to\tau\bar{\tau}$ decay modes were
compared against the CMS results, as shown in Fig. 1. The limits evaluated
using the fast simulation from Delphes agree with the CMS result to within
$30\%$ for all lifetimes evaluated. Additionally, we also validate the
$\Delta\phi(\textrm{cluster, MET})$ distributions for
$15\;$\mathrm{G}\mathrm{e}\mathrm{V}$$ and 1 m proper decay length LLP
decaying to $d\bar{d}$ against the auxiliary materials provided by CMS, as
shown in Fig. 2. The signal yield when requiring $\Delta\phi(\textrm{cluster,
MET})<$1 agrees within 7%.

Figure 2: Comparison of the $\Delta\phi(\textrm{cluster, MET})$ distributions
derived with the standalone workflow (dashed line) and the CMS search (solid
line) for a $15\;$\mathrm{G}\mathrm{e}\mathrm{V}$$ and 1 m proper decay length
LLP decaying into $d$-quark pairs with ${\rm Br}(h\to SS)=0.01$ . The signal
yield when requiring $\Delta\phi(\textrm{cluster, MET})<$1 agrees within 7%.

##  III Benchmark models

In this section, we briefly describe the benchmark models considered in this
work. Each of these models has been chosen to showcase the strengths and
limitations of the current analysis in concrete examples exhibiting different
kinematics and signal topologies. Specifically, we want to investigate what
happens to the analysis reach when lower values of LLP masses are chosen, when
the LLP energy, $E_{LLP}$, is reduced, or when the LLP momentum is not
correlated in magnitude or direction with the missing transverse energy. We
also want to investigate what happens when there are multiple LLPs produced
roughly in the same direction, potentially leading to failed isolation cuts in
a non-trivial way.

Concretely, the models we consider are:

  * •

Exotic Higgs decays into dark photons or light scalars. These are the closest
models to the one considered in the original CMS analysis and are
characterized by a production rate decoupled from the exclusive decay channels
and LLP lifetime. Besides being commonly chosen benchmarks to compare the
performance of different experiments in LLP searches, these benchmarks will
allow us to probe the reach for LLP masses lighter than those presented in the
CMS analysis, for a fixed production rate and using more realistic decay
modes.

  * •

Axion-like particles (ALPs) coupled to SM gauge bosons. In this model, the
coupling to the SM is provided by a dimension five operator. A single
parameter (the ALP decay constant) controls the production rate and lifetime.
These models are characterized by a production cross section which is enhanced
for energetic LLPs, irrespective of the light LLP mass.

  * •

Inelastic Dark Matter (DM). In these models, the LLP is provided by an almost
degenerate partner of the DM, and the amount of energy carried out by the LLP
is controlled by the DM-LLP mass splitting and decoupled from the missing
transverse energy (MET). This allows us to probe the reach in the low
$E_{LLP}$ region while allowing the other selection requirements to be passed
without much of a penalty.

  * •

Confining Hidden Valley models where jets of LLPs are produced in perturbative
hidden showers, analogously to the case of QCD. This benchmark allows us to
study the impact of the jet veto in models where multiple LLPs are produced in
the same detector region.

In addition to the models considered in this paper, the CMS analysis has also
been shown to have non-trivial reach for Heavy Neutral Leptons (HNL) [61].

###  III.1 Light scalar singlet

The most minimal extension of the SM is provided by adding a real scalar
singlet ($S$) that mixes with the SM Higgs through renormalizable operators.
The Lagrangian for this model reads [62]:

$\mathcal{L}_{SH}=\mathcal{L}_{\rm SM}+\mathcal{L}_{\rm
DS}-\left(A_{HS}\,\hat{S}+\lambda_{HS}\,\hat{S}^{2}\right)\hat{H}^{\dagger}\hat{H}$
(5)

where $\mathcal{L}_{\rm SM}$ is the SM Lagrangian, $H$ is the complex Higgs
doublet, and the dark sector Lagrangian is given by

$\begin{split}\mathcal{L}_{\rm
DS}=\frac{1}{2}\partial_{\mu}\hat{S}\,\partial^{\mu}\hat{S}-\frac{\mu_{S}^{2}}{2}\hat{S}^{2}+...\end{split}$
(6)

where we have omitted possible self-interactions of the scalar singlet, which
we assume have been chosen in such a way that $S$ does not have a vacuum
expectation value. Here and in the following, we indicate with a hat the
original fields with non-canonical kinetic terms, before any field
redefinition is performed.

After electroweak symmetry breaking the Higgs scalar, $\hat{h}$, mixes with
the singlet $\hat{S}$. The resulting physical states, $h$ and $S$, obtained by
diagonalizing the mass matrix, are given by the linear combination

$\left(\begin{array}[]{c}h\\\
S\end{array}\right)=\left(\begin{array}[]{cc}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{array}\right)\left(\begin{array}[]{c}\hat{h}\\\
\hat{S}\end{array}\right)\,,$ (7)

where the mixing angle is controlled by the parameters $A_{HS}$, and
explicitly given by

$\tan\theta=\frac{x}{1+\sqrt{1+x^{2}}}\qquad\qquad
x=\frac{2vA_{HS}}{\mu_{H}^{2}-\mu_{S}^{2}-\lambda_{HS}\,v^{2}}\,,$ (8)

with $v$ being the Higgs vacuum expectation value (vev), and
$\mu_{H}^{2}=\lambda_{H}v^{2}$ with $\lambda_{H}$ the Higgs quartic coupling.
The mass eigenvalues can also be expressed in terms of the small parameter $x$
as

$m^{2}_{h,S}=\left(\frac{\mu_{H}^{2}+\mu_{S}^{2}+\lambda_{HS}\,v^{2}}{2}\right)\pm\left(\frac{\mu_{H}^{2}-\mu_{S}^{2}-\lambda_{HS}\,v^{2}}{2}\right)\sqrt{1+x^{2}}\,,$
(9)

which for $x\ll 1$ reduce to $m^{2}_{h}\simeq\mu_{H}^{2}$ and
$m^{2}_{s}\simeq\mu_{S}^{2}+\lambda_{HS}\,v^{2}$.

Due to the mixing in Eq. (7), $S$ inherits all the couplings of the SM Higgs,
modulo a suppression factor, $\sin\theta$, which is controlled by the
parameter $A_{HS}$. Therefore, the decay width of the singlet can be obtained
by rescaling the one of a SM Higgs of the same mass. Specifically, we follow
references [63, 64] to derive the singlet branching ratios used in this work.

In general, the production cross section is fixed by a combination of the
parameters $A_{HS}$ and $\lambda_{HS}$. The former controls the production via
the $b\to s$ penguin diagram (allowed for $m_{S}<m_{B}-m_{K}$) [65, 66, 67]
and $s\to d$ penguins for (allowed for $m_{S}<m_{K}-m_{\pi}$); while the
latter fixes the double $S$ production via the $b\to s$ penguin diagram with
an off-shell SM Higgs [68], or through direct Higgs decay. Indeed, in presence
of a non-vanishing $\lambda_{HS}$ (and for $2m_{S}<m_{h}$) the Higgs can decay
into a couple of $S$ with a width given by [62]

$\Gamma_{h\to
SS}=\frac{\lambda_{HS}^{2}v^{2}}{8\pi\,m_{h}}\sqrt{1-4{m_{S}^{2}/m_{h}^{2}}}\,.$
(10)

When $b\to s$ transitions dominate the production channel, decay and
production are controlled by the same parameter, $\theta$, and the model
parameter space is given by $\\{\sin\theta,\,m_{S}\\}$. However, the analysis
discussed in this work has no reach for the products of $b\to s$ transitions,
as the LLPs would be mostly produced inside (or near) $b$-jets and fail
isolation cuts. Therefore, we will concentrate on the limit where the
production is dominated by Higgs decays to two $S$, which is controlled by the
parameter $\lambda_{HS}$. Therefore, production and decay channels will be
decoupled and the model parameter space given by
$\\{\lambda_{HS},\,\sin\theta,\,m_{S}\\}$.

Concretely, we generated events for Higgs production from gluon fusion in
association with up to two jets and decayed the Higgs to two scalars. No
generator level cuts are imposed and the Higgs $p_{T}$ distribution is
reweighed to the NNLO prediction.

We conclude by noticing that, given
$m^{2}_{S}\simeq\mu_{S}^{2}+\lambda_{HS}\,v^{2}$, some level of fine-tuning is
required for $m^{2}_{S}<\lambda_{HS}\,v^{2}$. Measuring the degree of fine-
tuning in terms of the parameter $\Delta\equiv
m_{S}^{2}/(\lambda_{HS}\,v^{2})$, we can write the branching ratio for the
exotic decay $h\to SS$ as

${\rm Br}\left(h\to SS\right)\simeq\frac{\Gamma_{h\to SS}}{\Gamma^{\rm
SM}_{h}}\simeq 6\cdot 10^{-3}\bigg{(}\frac{m_{S}}{2.5\,{\rm
GeV}}\bigg{)}^{4}\bigg{(}\frac{0.1}{\Delta}\bigg{)}^{2}\,,$ (11)

where $\Gamma^{\rm SM}_{h}$ is the total SM Higgs width.

###  III.2 Abelian hidden sector

The next benchmark model that we consider consists in extending the SM by
adding a _dark_ $\rm{U}(1)$ gauge symmetry which is spontaneously broken by a
dark Higgs field, $S$. The dark ${\rm U}(1)$ is mediated by a _dark_ photon,
$X$, which kinetically mixes with the SM hypercharge as:

$\mathcal{L}_{SH}=\mathcal{L}_{\rm SM}+\mathcal{L}_{\rm
DS}-\lambda_{HS}\,\hat{S}^{2}\hat{H}^{\dagger}\hat{H}-\frac{\epsilon}{2\cos\theta_{W}}\hat{X}_{\mu\nu}\hat{B}^{\mu\nu}\,,$
(12)

where $\hat{B}_{\mu\nu}$ and $\hat{X}_{\mu\nu}$ are the field strengths of the
hypercharge and the new $\rm{U}(1)$ gauge group respectively. The dark sector
Lagrangian is

$\mathcal{L}_{\rm
DS}=-\frac{1}{4}\hat{X}_{\mu\nu}\hat{X}^{\mu\nu}+\mu_{S}^{2}\hat{S}^{2}-\lambda_{S}\hat{S}^{4}+|(\partial_{\mu}+ig_{D}\hat{X}_{\mu})\hat{S}|^{2}\,.$
(13)

As before, we indicate with a hat the original fields with non-canonical
kinetic terms, before any field redefinition is performed. The dark
$\rm{U}(1)$ is spontaneously broken by the vev of the dark Higgs, $\langle
S\rangle=v_{S}/\sqrt{2}$, which generate a mass for the dark photon
$m_{X,0}=g_{D}v_{S}$.

After electroweak symmetry breaking the kinetic mixing between the dark photon
and the hypercharge induces a coupling of the dark photon to the SM fermions
which, in the $m_{X}^{2}<<m_{Z}^{2}$ limits, reads

$\mathcal{L}_{Xf\bar{f}}=\epsilon e\,Q_{f}X_{\mu}\bar{f}\gamma^{\mu}f\,,$ (14)

where $Q_{f}$ is the fermion electric charge. This coupling, controlled by the
small parameter $\epsilon$, provides the decay channel in visible states for
the dark photon. Specifically, we compute the dark photon branching ratios by
using the package provided in [69].

The diagonalization of the scalar sector proceeds similarly to what was
discussed in the previous section, with the only difference that now we are
interested in the regime where $m_{S}\gg m_{h}$, so that the dark Higgs
decouples from the phenomenology of the model. Given the non vanishing
coupling between $S$ and $X$, the mixing between the SM and dark Higgs
generates a non-zero $hXX$ coupling which gives rise to the exotic Higgs decay
$h\to XX$, with a width given by

$\Gamma(h\to
XX)=\frac{\lambda_{HS}^{2}}{32\pi}\frac{m_{h}v^{2}}{m_{S}^{2}}\sqrt{1-\frac{4m_{X}^{2}}{m_{h}^{2}}}\frac{(m_{h}^{2}+2m_{X}^{2})^{2}-8(m_{h}^{2}-m_{X}^{2})m_{X}^{2}}{m_{h}^{4}}\,.$
(15)

In the limit of small $\epsilon$ (which will be the relevant limit for our
analysis), this dominates over Drell-Yan and $h\to ZX$ production and becomes
the dominant dark photon production channel. In this limit the decay channel,
controlled by $\epsilon$, and the production channel, controlled by
$\lambda_{HS}$, are decoupled; and the model parameter space is given by
$\\{\epsilon,\lambda_{HS},m_{X}\\}$. The event generation for this benchmark
was performed similarly to the light scalar singlet case.

###  III.3 Inelastic Dark Matter

Inelastic Dark Matter (iDM) models are characterized by a DM candidate that
couples with the SM only through interactions with a nearly degenerate state.
A simple realization of this scenario can be obtained by adding to the model
discussed in the previous section a Dirac pair of Weyl fermions, $\eta$ and
$\xi$, that couple to the dark photon, $X$, with opposite charges. As before,
the Higgs provides a source of U(1) breaking, generating a mass for the dark
photon and a Majorana mass, $\delta$, for the two Weyl fermions. A Dirac mass,
$m_{D}$, involving the two Weyl fermions is also allowed, so that at energies
below the dark ${\rm U}(1)$ breaking scale, the mass terms for the dark
fermions are

$\displaystyle\mathcal{L}\supset-
m_{D}\,\eta\xi-\frac{\delta}{2}(\eta^{2}+\xi^{2})\,+{\rm h.c.}.$ (16)

For a technically natural small Majorana mass, these mass terms can be
perturbatively diagonalized to give the physical states

$\displaystyle\chi_{1}\simeq\frac{i}{\sqrt{2}}(\eta-\xi)\qquad\qquad\chi_{2}\simeq\frac{1}{\sqrt{2}}(\eta+\xi)\,,$
(17)

which have nearly degenerate masses $m_{1,2}\simeq m_{D}\pm\delta$. These mass
eigenstates couple off-diagonally to the dark photon, i.e.

$\displaystyle\mathcal{L}\supset
ie_{D}\hat{X}_{\mu}\,\bar{\chi}_{1}\gamma^{\mu}\chi_{2}+\mathcal{O}\left(\frac{\delta}{m_{D}}\right)\,,$
(18)

where we have written $\chi_{1,2}$ as Majorana spinors using four-component
notation. Therefore, if $m_{X}>m_{1}+m_{2}$ and
$\alpha_{D}\gg\epsilon\alpha_{\rm em}$, once produced dark photons decay into
$\chi_{1}\chi_{2}$ pairs with a rate given by
$\Gamma_{X\to\chi_{1}\chi_{2}}\simeq\alpha_{D}m_{X}$, and provide the dominant
production channel for $\chi_{1}\chi_{2}$ pairs at LHC. For the values of
$\epsilon$ we are interested in this analysis, the dominant production channel
for dark photons is provided by Drell-Yan processes and scales as
$\epsilon^{2}$.

The lightest state, $\chi_{1}$, is stable and once produced leaves the
detector as missing energy; $\chi_{2}$ can decay into $\chi_{1}$ plus a pair
of SM particles through an off-shell dark photon, possibly leaving a
detectable signature. The rate for decays with leptonic final states is given
by [70]:

$\displaystyle\Gamma_{\chi_{2}\to\chi_{1}l\bar{l}}=\epsilon^{2}\alpha_{\rm
em}\alpha_{D}\int_{4m_{l}^{2}}^{(m_{1}\Delta)^{2}}ds\frac{|\vec{p}_{1}|(m_{1}^{2}\Delta^{2}-s)(2s+m_{1}^{2}(2+\Delta)^{2})(s+2m_{l}^{2})(s-4m_{l}^{2})^{1/2}}{6\pi
m_{2}^{2}s^{3/2}(s-m_{X}^{2})^{2}}$ (19)

where $s$ is the invariant mass of the lepton pair, $\vec{p}_{1}$ is the
momentum of $\chi_{1}$ in the rest frame of $\chi_{2}$, and we have introduced
the dimensionless parameter $\Delta\equiv(m_{2}-m_{1})/m_{1}$. The rate for
decays involving hadronic final states can be derived by setting
$m_{l}=m_{\mu}$ and multiplying the integrand of Eq. (19) by the
experimentally measured quantity $R(s)\equiv\sigma(e^{+}e^{-}\to{\rm
hadrons})/\sigma(e^{+}e^{-}\to\mu^{+}\mu^{-})$.

For this benchmark, events were generated using a MadGraph5 $Z^{\prime}$ model
for $X$ via production in association with up to three jets. A generator level
cut $p_{T}>100$ GeV was applied on the $X$, as the truth-level
$p_{\mathrm{T}}^{\text{miss}}$ is given by $p_{T}$ of the $Z^{\prime}$ (its
decay products are one DM particle and an LLP decaying in the muon chambers).

###  III.4 Axion-like particles

Axion-like particles (ALPs) extend the axion scenario to include any
pseudoscalar particle that couples to the SM through dimension five operators.
The naturally suppressed couplings make them a natural candidate for LLP
searches. The general Lagrangian for these kinds of models is given by

$\displaystyle{\cal L}=$ $\displaystyle{\cal
L}_{SM}+\frac{1}{2}\left(\partial_{\mu}a\right)^{2}-\frac{1}{2}m_{a}^{2}a^{2}+\frac{c_{q}^{ij}}{2f}(\partial_{\mu}a)\bar{q}_{i}\gamma^{\mu}\gamma^{5}q_{j}+\frac{c_{\ell}^{ij}}{2f}(\partial_{\mu}a)\bar{\ell}_{i}\gamma^{\mu}\gamma^{5}\ell_{j}$
$\displaystyle+\frac{a}{4\pi
f}\left(\alpha_{s}c_{GG}\,G_{\mu\nu}^{a}\,\widetilde{G}^{a,\mu\nu}+\alpha_{2}c_{WW}\,W_{\mu\nu}^{a}\widetilde{W}^{a,\mu\nu}+\alpha_{1}c_{BB}\,B_{\mu\nu}\widetilde{B}^{\mu\nu}\right)+\cdots$
(20)

where $\widetilde{G}_{\mu\nu}=1/2\,\epsilon_{\mu\nu\rho\sigma}G^{\rho\sigma}$
where $G^{\rho\sigma}$ is the gluon field strength, and similarly for
$\widetilde{W}$ and $\widetilde{B}$. In the broken, phase the couplings to $W$
and $B$ bosons induce couplings to photons and $Z$-bosons which are given by:

$\displaystyle c_{ZZ}=c_{WW}+c_{BB}\qquad c_{\gamma
Z}=c_{w}^{2}c_{WW}-s_{w}^{2}c_{BB}\qquad
c_{\gamma\gamma}=c_{w}^{4}c_{WW}+s_{w}^{4}c_{BB}\,.$ (21)

In this work, we will focus on benchmark models in which the ALP couples only
to gauge bosons $(c_{q}^{ij}=c_{\ell}^{ij}=0)$. Since the focus is on the
production of energetic, isolated LLPs, this choice is sufficient to capture
most of the dominant production modes at the LHC. Specifically, we will
consider the three following scenarios: ALP coupled to $W$ ($c_{WW}\neq
0,\,c_{GG}=c_{BB}=0$), photophilic ALP ($c_{\gamma\gamma}\neq 0,\,c_{\gamma
Z}=c_{GG}=0$), and ALP coupled to gluons ($c_{GG}\neq 0,\,c_{BB}=c_{WW}=0$).
The latter is a well-studied benchmark model in the context of light LLP
searches, yielding the highest production rate at the LHC. The photophilic
model chosen here is one of the (infinite) possible choices of UV-completion
at LHC energies of the well-studied low-energy benchmark of “ALP coupled to
photons”. The conservative choice $c_{\gamma Z}=0$ is to focus on the
parameter region where the existing LEP bounds are the weakest. Finally, the
$c_{W}\neq 0$ benchmark provides a better UV-motivated benchmark than the
photophilic choice, where associated ALP production with all the gauge bosons
is allowed.

For the ALP coupled to gluons, we generated events where the LLP is produced
in association with up to 3 jets, and imposed a $p_{T}>100$ GeV and a
$|\eta|<3$ generator-level cuts on the transverse momentum and pseudorapidity
of the ALP. The MadGraph5 model used here has been described in [71], and we
have only adapted the normalization of the couplings to the one used above. We
did not include ALP production in the shower (i.e. where the ALP is produced
at intermediate scales between the hard process collision and the QCD
confinement scale) which was first estimated in [45], as there are not yet
reliable event generators that can be used to keep track of the angular
separation between the ALP and QCD jets (necessary for the jet veto
requirements of the analysis). Therefore, for this benchmark, our limits
should be considered conservative estimates for the reach of this CMS
analysis, as they miss an important production channel. Production from meson
mixing and meson decay was also neglected because it yields softer and non-
isolated ALPs, for which this analysis has no sensitivity. For the lifetime
and exclusive decay branching ratios of this benchmark, we used the estimates
of [72].

For the case of the other two ALP benchmarks, we considered ALP production in
association with either a $W$, a $Z$, or a photon and up to 2 extra jets. We
kept the same generator-level pseudorapidity cut but lowered the $p_{T}$ cut
to $50$ GeV as some of the missing transverse energy can be produced by the
decay products of the $W$ and $Z$ bosons. In these two benchmarks, the ALP
decays predominantly into two photons.

###  III.5 Hidden Valley

Confining Hidden Valleys (HV) [73], with a perturbative evolution below the
scale mediating the interactions producing hidden sector particles, are a
generic hidden sector extension of the SM on which we can have some
theoretical control based on our knowledge of QCD-like theories. In general,
one expects jets of hidden sector partons to hadronize in HV particles, some
of which may decay back into SM final states, potentially as LLPs. Still,
large freedom exists in defining a specific model. From the field content of
the hidden sector and its symmetries, to the portal interactions mediating
both the production of HV states and their decay back to the SM [74]. Many
studies of search strategies at the LHC have been performed for different
incarnations of this paradigm [21].

In the context of this reinterpretation study, we choose one particular
realization as an example model generating the LLP-jet signature, aiming at
maximizing the multiplicity of LLP produced in a jet, while keeping a high
level of simplicity of reinterpretation. Therefore the example chosen is by no
means generic _per se_ , although the lessons learned about the CMS analysis
are. Specifically, we used the Hidden Valley module [75] implemented in Pythia
to generate events and choose a perturbative hidden sector with an $SU(N_{c})$
asymptotically free gauge group with $N_{f}$ hidden quark flavors, fixing
$N_{c}=3$ and $N_{f}=1$. The choice of $N_{f}=1$ is to guarantee the absence
of stable hidden mesons, therefore reducing the amount of collider stable
particles produced and maximizing the number of LLPs in a hidden jet. This has
a drawback, namely the lack of knowledge of the mass spectrum of such a theory
as it lacks chiral symmetry breaking which is an important handle used in
lattice simulation. In particular, the mass ratio between the first
(pseudo-)scalar, $\eta_{V}$, and vector, $\omega_{V}$, resonances are poorly
known, but expected to be ${\cal O}(1)$ [76, 77, 78]. Again, motivated by
maximizing LLP multiplicity, we choose
$m_{\omega_{V}}=2.5m_{\eta_{V}}=\Lambda_{HV}$ and assumed that the lowest
scalar state (which Pythia will not use in the hadronization of the HV
partons) is also able to decay to pairs of $\eta_{V}$. In this way, vector
resonances can promptly decay to pseudoscalar mesons $\eta_{V}$, which will be
the LLPs. For portals, we decide to decouple production and HV meson decays so
that we can study the effects of varying the LLP lifetime on the limits for a
fixed production rate. Specifically, we will produce hidden quarks in Higgs
decays and will decay back the hidden spin-0 mesons $100\%$ into pair of
photons. The latter choice is purely driven by the fact that the CMS analysis
is not too sensitive to the relative amount of hadronic vs electromagnetic
energy in LLP decays. At the same time, existing limits on light LLPs decaying
to pair of photons are quite weak, so we can focus on reinterpreting this
analysis without worrying about recasting other existing searches111The case
of a recent CMS search for trackless jets [79] provides likely the strongest
constraint for low values of $c\tau$ where a significant fraction of LLPs
decays in the inner detector. However, that analysis explicitly vetoes
signatures compatible with loose photons and photon conversion. The efficiency
for one or more light LLPs decaying into pairs of photons being identified as
loose photons is hard to recast, therefore we do not consider such analysis
when presenting our limits.. From a model building point of view, these
portals can be easily generated by introducing a heavy scalar and pseudoscalar
states $S$ and $A$, having Yukawa interactions with the HV vector-like quark
$q_{V}$. The scalar $S$ can then interact with the SM Higgs via a $|H|^{2}S$
cubic interaction, generating a Yukawa coupling between $q_{V}$ and the SM
Higgs and a $q_{V}$ mass after electroweak symmetry breaking. At the same
time, the pseudoscalar $A$ can have a coupling to the SM photons $AF\tilde{F}$
which in turn will induce a small decay width for $\eta_{V}$ via $\eta_{V}-A$
mixing.

##  IV Results

In this section, we present the results for the benchmark models discussed in
the previous section. We present both the current constraints, derived from
data collected from 2016 to 2018, corresponding to an integrated luminosity of
137 ${\mathrm{fb}}^{-1}$ and the projected constraints for Phase 2. The
different projections for Phase 2 are derived by using the three different
search strategies discussed in Section II.4. Specifically, solid lines
correspond to the search with the same selections as the CMS paper and a
background rescaled according to the higher luminosity, dot-dashed lines
correspond to the search with a higher $N_{\mathrm{hits}}$ cut and zero
background, and the dashed lines to the search with a dedicated trigger (that
no longer require the MET and isolation cuts, but the presence of two separate
LLP decays in the muon chambers) and zero background (see Section II.4 for a
more detailed discussion of the search strategies). Other existing and
projected limits shown in the following plots are all taken from the
literature, as referenced in the figure captions. The only exception is a
limit originating from an ATLAS mono-jet search for the case of the gluon-
coupled ALP, Fig. 9, whose mass dependence was derived in this work as
described in Appendix A.

In Fig. 3 we show the reach for the light scalar model (discussed in Section
III.1) with $\lambda=1.6\times 10^{-3}$. This choice of $\lambda$ corresponds
to an exotic Higgs branching fraction of ${\rm Br}(h\to SS)=0.01$, which is
roughly the future reach for the Higgs branching into invisible final states.
The present constraints are shown in the left panel, where we see that for low
masses the analysis probes a previously unconstrained region of the parameter
space, while at higher masses the constraints are similar to the ones of the
ATLAS search for displaced vertices in the muon chambers (indicated as ATLAS
mu-ROI in Fig. 3), whose reach was presented for $m_{S}>5$ GeV. In the right
panel, we show the projections for Phase 2 and compare them with the projected
constraints from other future experiments and upgrades. We can see that,
thanks to the different distance from the interaction point (IP), the
projected results are complementary to dedicated LLPs experiments such as
CODEX-b, FASER2, and MATHUSLA; all of which are positioned further away from
the IP. To give an idea of how the constraints depend on the value of ${\rm
Br}(h\to SS)$, in Fig. 4 we show the same constraints of Fig. 3 but for
different values of ${\rm Br}(h\to SS)$. We see that the current search start
to lose sensitivity for ${\rm Br}(h\to SS)\lesssim 3\times 10^{-3}$, while for
the future Phase 2 search we start to lose sensitivity for ${\rm Br}(h\to
SS)\lesssim 3\times 10^{-4}$. In all the plots we present also the values of
the LLP mass (function of ${\rm Br}(h\to SS)$) below which tuning of more than
10% is present. Alternatively, in Fig. 5 we show the limits for a different
slicing of the parameter space of this model, where the tree-level mass for
$S$ is absent and the LLP mass is fully controlled by $\lambda_{HS}$ and
therefore by ${\rm Br}(h\to SS)$. In this case, there is no tuning even for
lower masses, but the production rate varies with $m_{S}$ and searches for
$H\to{\rm inv.}$ set an upper bound on $m_{S}$. Finally, to compare with the
results of the CMS analysis, in Fig. 6 we report the present and future limits
on ${\rm Br}(h\to SS)$ as a function of the scalar lifetime.

In Fig. 7 we report the constraints for the Abelian hidden sector discussed in
Section III.2. As before, the value of the exotic Higgs branching ratio is
fixed to ${\rm Br}(h\to A^{\prime}A^{\prime})=0.01$. We see that our current
constraints (left panel) cover a mostly unconstrained region of the parameter
space, except for the overlap with the ATLAS mu-ROI search at high masses. As
for the scalar model, our projected constraints (right panel) well complement
dedicated LLPs searches thanks to the different baselines. To investigate
which is the lowest value of ${\rm Br}(h\to A^{\prime}A^{\prime})=0.01$ that
we can probe, in Fig. 8 we show present and future constraints for different
values of the exotic Higgs branching. For the current search we see that we
start to lose sensitivity for ${\rm Br}(h\to A^{\prime}A^{\prime})=3\times
10^{-3}$, while for the Phase 2 the constraints start to disappear for ${\rm
Br}(h\to A^{\prime}A^{\prime})=3\times 10^{-4}$. This is consistent with what
was found for the singlet scalar model and shows the relative insensitivity of
the analysis to the specific exclusive decay modes. The only significant
differences are around resonance mixing with hadronic resonances, which differ
between the scalar and vector LLPs, and affect the LLP lifetime; and in the
region between $200\,{\rm MeV}\lesssim m\lesssim 300\,{\rm MeV}$ where the
$2\mu$ final state, to which this analysis is not sensitive to, contributes to
${\cal O}(50\%)$ of the dark photon branching ratios.

The constraints for the three ALP models that we consider are shown in Fig. 9
\- 10. For both the gluon (Fig. 9) and electroweak (Fig. 10 and Fig. 11)
coupled scenarios, we find that the reinterpretation of the CMS analysis
covers new territory beyond previous monojet [80] and fixed target [81]
searches while being complementary to dedicated LLP experiments. Moreover, one
can expect the projections shown here to be underestimated, as dedicated
searches using the fact the ALP is produced in association with a photon or a
vector boson may allow us to relax some of the cuts and access softer LLPs
that are produced with higher rates, pushing the estimated limits towards
higher ALP masses and decay constants.

We now turn to the inelastic DM model results. The reinterpretation of this
model is fairly sensitive to the LLP energy, $E_{LLP}$, via the mass
splitting, $\Delta$. Unfortunately, the efficiency tables provided by the CMS
Collaboration in HepData are not granular enough at low deposited energies
$(E_{em},E_{had})$ to resolve the turn-on shoulder of the 2D efficiency
surface (the first bin is between 0 and 25 GeV). Therefore, our ability to
reliably recast this model is hampered by the lack of knowledge about the
minimal energy threshold for which the LLP visible decay products can produce
${\cal O}(20-30)$ charged particles emerging from a steel layer into the muon
stations. To estimate this energy threshold, we impose an additional cut
$E_{LLP}>5\,{\rm GeV}$ (which is approximately the energy needed for an
electron to produce ${\cal O}(20)$ charged particles at the shower development
maximum). The constraints for this choice of cut and using the model
parameters $\Delta=0.005$, $\alpha_{D}=0.1$, and $m_{A^{\prime}}=3m_{1}$, are
reported in Fig. 12. We see that the analysis covers previously unconstrained
regions of the parameter space near the $Z$-resonance at
$m_{A^{\prime}}=3m_{1}=m_{Z}$. We have also derived the constraints for a
model with $\Delta=0.01$ but decided not to show them since they are weaker
than already existing limits, as larger mass splittings produce lifetimes too
short to reach the CMS muon chambers. To further estimate the sensitivity of
these results to the lower cut on the LLP energy, we show in Fig. 13 the
effect of varying it between 0 and 10 GeV.

Finally, in Fig. 14 we report the limits on the exotic Higgs branching ratio
${\rm Br}(h\to Q\bar{Q})$ for the hidden valley model discussed in Section
III.5. We specifically choose a value for the HV confining scale
$\Lambda_{HV}=20\,{\rm GeV}$, which correspond to a pseudoscalar mass
$m_{\eta_{V}}=8\,{\rm GeV}$. Since in this model LLPs are produced within
dark-showers in LLP jets, we expect the jet veto to reduce the sensitivity of
the analysis. To quantify this effect, in the lower panels of Fig. 14 we show
the ratio of the signal efficiency of the CMS analysis divided by the signal
efficiency of the same analysis without the jet veto. As expected, this ratio
rapidly approaches zero for small LLP lifetimes, when it is more likely for
multiple LLPs to decay within the inner detector regions and the calorimeters
in front of the cluster in the muon chambers selected as a signal by the
analysis. Conversely, in the long lifetime area, the higher LLP multiplicity
renders the limit more stringent than the case of Higgs decay to pairs of
LLPs. Lowering the hidden confinement scale will increase the meson
multiplicity inside hidden jets and therefore amplify this behavior.

##  V Discussion

We have explored some of the strengths and limitations of a recent search for
LLPs using the muon chambers at CMS. The reinterpretation was made possible by
the additional information provided by the collaboration in HEPData, which was
embedded into Delphes modules.

We have shown that this analysis proves very effective at constraining light
LLPs, $m_{LLP}<{\cal O}({\rm GeV})$, as long as they can be produced
energetically in LHC hadronic collisions and have $c\tau\lesssim{\cal O}({\rm
m})$. In fact, we found that the current version of such a search strategy not
only provides a counterexample to the lore that LLP searches at ATLAS and CMS
are limited at low masses by irreducible SM backgrounds, but it is already
able to cover previously unconstrained parameter space in many models, see
Figs. 3, 7, 9, 11, 12, 14, competing with and complementing the reach of
dedicated LLP detectors.

Still, various avenues for improvement exist. As mentioned in Sec. IV,
producing signal categories with lower MET requirements but in association
with another object such as a photon, lepton(s) or b-jet, may improve limits
on specific models such as ALPs and HNLs. This will greatly increase the
coverage of the search for many other models. This is especially true given
the particular simplicity and reliability of the recasting provided by the
publicly released information in HepData [52]. In this respect, we encourage
the CMS Collaboration to provide more finely spaced efficiency maps at low
$(E_{had},\,E_{em})$ to fully capture the turn-on shoulders, which is required
in models where LLPs are producing less visible energy such as in the
inelastic Dark Matter benchmark shown here.

Perhaps the most important avenue of improvement may be the investigation of
how much the cluster isolation requirement can be relaxed. Many models, and
production modes within a model, produce LLPs inside ($b$-)jets. Examples
include the case of a light scalar model, where $S$ can be produced
efficiently in $b$ decays and would yield muon chamber clusters not isolated
from a $b$-jet; the case of ALPs produced in $b$-flavored hadron decays or in
hadronic showers via $\pi^{0}$\- $\eta^{(\prime)}$ mixing; or the case of
emerging jets [82] where showering within QCD and a Hidden Valley happens
concurrently. Extending this kind of searches into the non-isolated regime
will inevitably require some characterization and understanding of the origin
of SM backgrounds mimicking clusters in the CSC. This effort has also
implications and synergies beyond CMS itself. In fact, the amount of
(instrumented) shielding provided by the calorimeters and the steel layers in
the muon chambers is about $20-27$ nuclear interaction lengths, not far from
the required shielding of other proposals for dedicated LLP experiments, such
as e.g. the $30\lambda$ of active shield estimated to be necessary for CODEX-b
[40]. Therefore, any characterization of SM backgrounds for CMS LLP searches
would also benefit and inform the ongoing shielding design and simulation of
other experimental proposals such as CODEX-b.

Many benchmarks chosen here correspond to some of those selected to compare
present and future efforts in the LLP search program such as within the CERN
Physics Beyond Collider (PBC) [6]. Given the relevance of this novel type of
CMS search on the LHC reach for LLPs, we encourage the Collaboration to
produce official limits that can be included in the PBC comparison plots. The
capabilities of this kind of CMS search in probing light LLPs greatly extend
what was considered possible for general-purpose existing LHC experiments to
achieve and will undoubtedly complement and inform the broader future search
program for LLPs beyond the Standard Model.

###### Acknowledgements.

We would like to thank the CMS Collaboration and especially the CMS Exotica
group, where part of this work was presented in 2021, for their positive
feedback. CW and SX are partially supported by the U.S. Department of Energy,
Office of Science, Office of High Energy Physics, under Award Number DE-
SC0011925. This work has been discussed in the relevant Snowmass community
study subgroups in 2022 and is part of CW’s PhD thesis along with additional
reinterpretation studies based solely on CMS published results and Delphes
public codes [34, 52, 59]. CW, CP, and SX are grateful to the organizers and
participants of the “New ideas in detecting long-lived particles at the LHC”
workshop at LBNL in the Summer of 2018 where experimentalists and theorists
gathered to generate new ideas on triggers and analysis strategies for long-
lived particles searches at the LHC [83] as well as the Fermilab LPC LLP
group. AM was supported by the U.S. Department of Energy, Office of Science,
Office of High Energy Physics, under Award No. DE-SC0021431, the Quantum
Information Science Enabled Discovery (QuantISED) for High Energy Physics
(KA2401032), and the Deutsche Forschungsgemeinschaft under Germany’s
Excellence Strategy - EXC 2121 Quantum Universe - 390833306. MP was supported
by the U.S. Department of Energy, Office of Science, Office of High Energy
Physics, under Award Number DE-SC0011632 and by the Walter Burke Institute for
Theoretical Physics. MP would like to thank the Aspen Center for Physics where
part of this work was performed.

Figure 3: Constraints on light scalars produced in Higgs decays for ${\rm
Br}(h\to SS)=0.01$. Left: Comparison of our current reach (red region) with
existing limits from LHCb (orange) [84], LSND (azure) [85], reinterpretation
[63] of the CHARM experiment (blue) [81], CMS “HT +2DV” search (green) [64,
10], and reinterpretation of ATLAS mu-ROI (purple) [64, 86]. Right:
Projections of our constraints for a luminosity of $3\,{\rm ab}^{-1}$ (red
region). The three red contours (solid, dashed, and dot-dashed) correspond to
the three search strategies discussed in the main text (rescaled CMS analysis,
dedicated trigger, and higher $N_{\mathrm{hits}}$). We compare our results
with current constraints (gray shaded region) and projections for MATHUSLA
[87], CODEX-b [45], FASER2 [88], and LHCb $300{\rm fb}^{-1}$ [89]. The
constraints for the projections are shown between the dashed lines with the
corresponding colors. The vertical orange line indicates the scalar mass below
which the model needs to be fine-tuned (see discussion around Eq. (11)).
Figure 4: Our limits for the light scalar model for different values of ${\rm
Br}(h\to SS)$. In the left panel we show the current reach, while in the right
panel we present the $3\;{\rm ab}^{-1}$ projections assuming that the same
selections of the original CMS analysis are used. As in the previous plot, the
vertical lines indicate the scalar mass below which the model needs to be
fine-tuned (see discussion around Eq. (11)). Figure 5: Constraints on the
singlet scalar model in absence of a tree-level mass for $S$ ($\mu_{S}=0$).
The solid red line shows the current constraints, while the other contours
(dashed, dot-dashed, and dotted) show $3\,{\rm ab}^{-1}$ projections derived
by using the three recast strategies discussed in Section II.4 (rescaled CMS
analysis, dedicated trigger, and higher $N_{\mathrm{hits}}$). The other
existing constraints appearing on the plot are the same of Figure 3. Figure 6:
Upper limits on the branching fraction ${\rm Br}(h\to SS)$ as functions of
$c\tau$. In the left panel, we report the current constraints set by our
analysis. In the other panels, we show the projected constraints for a
luminosity of $3\;{\rm ab}^{-1}$ derived from the three different search
strategies discussed in the main text. Figure 7: Constraints on dark-photons
produced in Higgs decays for ${\rm Br}(h\to SS)=0.01$. Left: Comparison of our
current reach (red region) with existing limits from BaBar (blue) [90], KLOE
(azure) [91], LHCb (purple) [92], NA48 (brown) [93], reinterpretation of ATLAS
$\mu$-ROI (yellow) [86], ATLAS search for displace dark-photon jets (yellow)
[94], and beam dump experiments (orange, gray, green, pink) [81, 95, 96, 97].
Most of the experimental constraints appearing in this plot have been
digitized with the help of darkcast [69]. Right: Projections of our
constraints for a luminosity of $3\,{\rm ab}^{-1}$ (red region). The three red
contours (solid, dashed, and dot-dashed) correspond to the three search
strategies discussed in the main text (rescaled CMS analysis, dedicated
trigger, and higher $N_{\mathrm{hits}}$). We compare our results with current
constraints (gray shaded region) and projections for MATHUSLA (orange) [87],
SHiP (azure) [41], DarkQuest (purple) [13, 98], NA62 in dump mode (green) [13,
99], LHCb upgrade (brown) [13, 100], and Belle II (blue) [101]. Figure 8: Our
limits for the dark photon model for different values of ${\rm Br}(h\to SS)$.
In the left panel we show the current reach, while in the right panel we
present the projections for $3\;{\rm ab}^{-1}$ assuming the same selections of
the original CMS analysis are used. Figure 9: Constraints on ALPs coupled to
gluons. Left: Comparison of our current reach (red region) with existing
limits from CHARM (orange) [45, 81], our reinterpretation of ATLAS (green)
[80], LEP [72], and flavor probes (purple) [102, 103, 104, 105, 106]. Right:
Projections of our constraints for a luminosity of $3\;{\rm ab}^{-1}$ (red
region). The solid and dot-dashed red contours correspond to the projections
derived by using the same selections of the original CMS analysis, and the one
derived by using a higher $N_{\mathrm{hits}}$ cut and assuming zero
background. We compare our results with current constraints (gray shaded
region) and projections for FASER (purple) [39], CODEX-b (orange) [45], and
Belle II (green) [104, 102] Figure 10: Constraints on ALPs coupled to $W$
bosons. Left: Comparison of our current reach (red region) with existing
constraints from star cooling constraints (green) [107], beam dump experiments
(yellow) [107], $Z$ invisible branching ratio (orange) [108, 109], limits on
$e^{+}e^{-}\to\gamma\gamma$ from LEP data (violet) [108, 110, 111, 112],
PrimeEX (purple) [113, 114], and Belle II (blue) [115]. Right: Projections of
our constraints for a luminosity of $3{\rm ab}^{-1}$ (red region). The solid
and dot-dashed red contours correspond to the projections derived by using the
same selections of the original CMS analysis, and the one derived by using a
higher $N_{\mathrm{hits}}$ cut and assuming zero background. We compare our
results with current constraints (gray shaded region) and projections for SHiP
(orange) [6, 41], PrimEX (purple) [113, 114], GlueX (violet) [113, 116], , and
Belle II (blue) [6, 117]. Figure 11: Constraints on ALPs coupled to
electroweak gauge bosons, and with $c_{\gamma Z}=0$. Left: Comparison of our
current reach (red region) with existing constraints from star cooling
constraints (green) [107], electron [108, 118, 96] and proton [6, 119, 81]
beam dump experiments (pink and brown), limits from mono-photon searches at
LEP (orange) [108, 120], NA64 (green) [97], PrimEX (purple) [114, 113], and
Belle II (blue) [115]. Right: Projections of our constraints for a luminosity
of $3{\rm ab}^{-1}$ (red region). The solid and dot-dashed red contours
correspond to the projections derived by using the same selections of the
original CMS analysis, and the one derived by using a higher
$N_{\mathrm{hits}}$ cut and assuming zero background. We compare our results
with current constraints (gray shaded region) and projections for FASER
(brown) [39], SHiP (orange) [41], PrimEX (purple) [114, 113], GlueX (green)
[113, 116], and Belle II (green) [117]. Figure 12: Constraints on inelastic DM
models, assuming a normalized mass splitting of $\Delta=0.05$, a dark coupling
$\alpha_{D}=0.1$, and mediator mass given by $m_{A^{\prime}}=3m_{1}$. Left:
Comparison of our current reach(red region) with existing constraints from
BaBar (green) [121, 122] and LEP (blue) [121, 123, 100, 124]. Right:
Projections of our constraints for a luminosity of $3{\rm ab}^{-1}$ (red
region). The solid and dot-dashed red contours correspond to the projections
derived by assuming the same selections of the original CMS analysis, and the
one derived by using a higher $N_{\mathrm{hits}}$ cut and assuming zero
background. We compare our results with current constraints (gray shaded
region) and projections for BelleII (pink) [117], SeaQuest (gray) [98], FASER
(blue) [38], MATHUSLA (orange) [35], CODEX-b (brown) [40], LHC (yellow and
purple) [125, 126, 121], and LHCb (green) [121, 127]. Figure 13: Constraints
on the inelastic DM model for different choices of the lower cut on the LLP
energy. The other constraints appearing in the plot are the same reported in
Figure 12. The projections in the right panel are for a luminosity of $3{\rm
ab}^{-1}$ luminosity, and assuming a tighter $N_{\rm hits}$ cut and zero
background. Figure 14: Current (upper left panel) and projected (upper right
panel) constraints on the Higgs exotic decay into dark quarks of a confining
hidden valley model. For the projections we report (solid, dashed, and dot-
dashed lines) the results obtained by using the three search strategies
discussed in the main text (rescaled CMS analysis, dedicated trigger, and
higher $N_{\mathrm{hits}}$).

##  Appendix A ATLAS mono-jet limit for Axion-like Particles coupled to gluons

Here we summarize the procedure used to reinterpret the ATLAS monojet limit on
ALPs coupled to gluons [80]. The ATLAS collaboration already provides a lower
limit on the ALP decay constant at a fixed ALP mass $m_{a}=1\,{\rm MeV}$ in
this particular model and claims that such limit should hold for ALP masses up
to approximately $1\,{\rm GeV}$. This claim is motivated from ALP literature
prior to the improved estimates on ALP lifetimes and branching ratios provided
in Ref. [72] and it is modified in the region $0.1-1\,{\rm GeV}$ due to the
non-trivial behavior from ALP mixing with the neutral pseudoscalar mesons. To
estimate the limit curve in this region we use our ALP+jet simulation to
extract the 2D LLP energy and pseudorapidity distributions, convolve that with
the lifetime model of [72], and require that the ALP does not decay in the
ATLAS detector volume, for a fixed value of $m_{a}$ and $f$. We then rescale
the ATLAS limit for the ratio of the two efficiencies described above computed
at $m_{a}=1\,{\rm MeV}$ and at a different mass point. This produces a
function of $(m_{a}/1\,{\rm MeV},f/f_{\rm limit,1{\rm MeV}})$. We then invert
this function to solve for the limit on $f$ as function of $m_{a}$ as shown in
Fig. 9. As expected the limit is fairly flat at low ALP masses but gets cut
off earlier than 1 GeV due to the ALP lifetime significantly changing after
the $m_{\eta}$ threshold. The steepness of the turn-off renders this limit
curve a little sensitive to the specific geometric dimensions considered for
the ATLAS detector.

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