# Scouting for dark showers at CMS and LHCb

Susan Born  Department of Physics, University of Illinois Urbana-Champaign,
Urbana, IL 61801  Rohith Karur  Theoretical Physics Group, Lawrence Berkeley
National Laboratory, Berkeley, CA 94720, USA  Berkeley Center for Theoretical
Physics, Department of Physics, University of California, Berkeley, CA 94720,
USA  Simon Knapen  Theoretical Physics Group, Lawrence Berkeley National
Laboratory, Berkeley, CA 94720, USA  Berkeley Center for Theoretical Physics,
Department of Physics, University of California, Berkeley, CA 94720, USA
Jessie Shelton  Department of Physics, University of Illinois Urbana-
Champaign, Urbana, IL 61801  Illinois Center for Advanced Studies of the
Universe, University of Illinois Urbana-Champaign, Urbana, IL 61801  Center
for Theoretical Physics, Department of Physics, Massachusetts Institute of
Technology, Cambridge, MA 02139

###### Abstract

We assess the capabilities of the CMS and LHCb searches for low-$p_{T}$
displaced dimuon pairs to discover hidden valley models. To do so, we develop
a new benchmark model featuring a light dark photon with dark flavor-violating
couplings, which realizes a range of dimuon vertex topologies. We show that
the data scouting techniques used in these searches provide unique sensitivity
and we make some additional suggestions to further extend the scope of future
experimental searches.

††preprint: MIT-CTP/5537

###### Contents

  1. I Introduction
  2. II Benchmark dark shower model
    1. II.1 Production
    2. II.2 Dark sector spectrum
    3. II.3 Vertex topologies
    4. II.4 Parameter choices and hadron multiplicities
  3. III Analysis
    1. III.1 CMS displaced dimuon search
    2. III.2 LHCb displaced dimuon search
    3. III.3 Simulation framework
  4. IV Results
    1. IV.1 Scenario A
    2. IV.2 Scenario B
    3. IV.3 Scenario C
  5. V Summary and recommendations
  6. A A two-flavor hidden valley
    1. A.1 Quark sector
    2. A.2 Meson sector
    3. A.3 Decay rates and branching ratios
    4. A.4 Pythia 8 implementation
  7. B Signal efficiency plots

##  I Introduction

In its second decade of operation the Large Hadron Collider (LHC) is now
transitioning to its high-intensity phase, which will deliver substantial new
sensitivity for precision Standard Model (SM) measurements as well as new
opportunities to search for very rare beyond-the-SM phenomena in the energy
range from the GeV scale up to a few hundreds of GeV. In relation to the
search for new physics, it has long been appreciated that trigger design can
and should play a dynamical role in developing analyses, as opposed to a
static obstruction when searching for soft signatures. All experiments
therefore strive to perform as much event reconstruction and analysis as
possible at the trigger level, as this allows for more informed decisions as
to which events to commit to tape.

Because more and more of the event reconstruction can be performed in real
time, ATLAS, CMS and LHCb have also been able to record partial or reduced
versions of the events reconstructed by their trigger-level algorithms. This
allows for a much higher output rate, and thus greatly reduced trigger
thresholds. This strategy is referred to as the Turbo Stream Aaij:2016rxn ;
Aaij:2019uij , Data Scouting CMS:2016ltu  or Trigger-Level Analysis
ATLAS:2018qto  by LHCb, CMS and ATLAS respectively. At LHCb in particular,
this online-analysis concept is a critical component of their ongoing and
future data taking strategies, with an $\mathcal{O}(1)$ fraction of all events
already being recorded through the Turbo Stream Aaij:2019uij . This has
enabled LHCb to measure (for example) certain exclusive charm decays and
allowed them to perform searches for new, low mass dimuon resonances
Aaij:2017rft ; Aaij:2019bvg ; Aaij:2020ikh . Fully online strategies have also
been deployed to great effect in searches at ATLAS and CMS, in particular for
low-mass dijet CMS:2016ltu ; ATLAS:2018qto  and (displaced) dimuon resonances
PhysRevLett.124.131802 ; CMS:2021sch ; CMS-PAS-EXO-21-005 . A scouting search
for low mass diphoton resonances would furthermore be very well motivated
Knapen:2021elo , though may be technically challenging. Broadly speaking, data
scouting strategies will become even more powerful tools during the HL-LHC
phase, provided that the existing capabilities can be maintained and,
optimally, expanded, in the context of the higher instantaneous luminosity
planned for future operation.

Data scouting is moreover of great interest from a theoretical point of view.
The emerging appreciation that new physics may be relatively light and coupled
only feebly to the SM (see e.g. the reviews Alexander:2016aln ;
Lanfranchi:2020crw ) places a renewed emphasis on searches for new phenomena
in relatively low-energy final states. Data scouting techniques are very well
suited to making these challenging, high-rate kinematic regimes accessible.
Moreover, searches relying on data scouting are _inherently inclusive_ ,
insofar as (e.g.) a dimuon scouting trigger will record information about all
dimuon pairs passing its selection criteria, independent of the properties of
the rest of the event. This property makes data scouting searches particularly
suitable to capture a wide range of unexpected phenomena. These two features
are particularly important in the current and upcoming phases of LHC
operations, as theory priors are nowadays much weaker than those during the
initial phase of the LHC.

Hidden valley (HV) models are a well-motivated and physically rich class of
dark sector models Strassler:2006im , whose defining feature is the existence
of interactions within their dark sector that generate a high-multiplicity
final state. This is typically due to the presence of a confining gauge group
in the dark sector that gives rise to dark showering and hadronization, in
broad analogy to the QCD sector in the Standard Model (SM). Hidden valleys
specifically have been developed as components of models seeking to address
the origin of dark matter Hur:2007uz ; Kribs:2009fy ; Beauchesne:2018myj ;
Francis:2018xjd ; Bernreuther:2019pfb , the matter-antimatter asymmetry
Bai:2013xga , and the stabilization of the electroweak scale Chacko:2005pe ;
Burdman:2006tz ; Craig:2015pha . More broadly, they represent a rather generic
example of particle physics beyond the SM that could have flown under the
radar of most existing searches at the LHC. Signatures of such hidden valleys
often feature non-isolated objects, high-multiplicity and/or soft final
states, and long-lived species, and thus frequently represent a challenging
target for existing analysis strategies at the LHC. These “dark shower”
signatures can give rise to spectacular and nonstandard final states, see e.g.
Strassler:2006im ; Strassler:2008bv ; Kang:2008ea ; Harnik:2008ax ;
Cohen:2015toa ; Schwaller:2015gea ; Knapen:2016hky , and are an emerging
frontier for upcoming runs at the LHC Albouy:2022cin .

A rigorous and comprehensive map of the theoretical possibilities within the
dark shower framework is not yet possible. However, some useful mileage can be
gained by systematically classifying and constraining the portals through
which dark hadrons may decay back to the SM Renner:2018fhh ; Li:2019ulz ;
Knapen:2021eip ; Bensalem:2021qtj . Notably, this strategy has demonstrated
that models featuring (sub-)GeV-scale dark mesons can only give rise to a high
multiplicity of visible final particles in a subset of possible portal
scenarios. In particular, soft muonic final states are well-motivated,
especially if the dark states are short- or moderately long-lived
($c\tau\lesssim 1$ cm). If the event is sufficiently energetic, these soft
muonic final states can be searched for inside (semi-visible) jets
Pierce:2017taw ; Cheng:2019yai ; Zhang:2021orr ; Cheng:2021kjg ;
Cazzaniga:2022hxl . On the other hand, scouting techniques are the natural
choice for scenarios where e.g. the $H_{T}$, dijet or missing transverse
energy (MET) triggers are not very effective. The sensitivity of LHCb in
particular to low-mass resonances produced in dark showers has been noted
earlier Pierce:2017taw ; Cheng:2019yai ; Aaij:2020ikh ; Cheng:2021kjg , as
well potential additional sensitivity at Belle II Bernreuther:2022jlj . In
this work we take a close look at the complementarity of the recent CMS low-
mass displaced dimuon search CMS:2021sch  with that at LHCb Aaij:2020ikh , and
show that both are powerful probes of hidden valley models in which the dark
particles have tracker-scale lifetimes. We also suggest a number of extensions
of these analyses, which could extend their discovery potential to hidden
valley models with different vertex topologies.

We carry out this analysis in the context of a benchmark model that has two
flavors of light dark quarks and an elementary dark photon. This benchmark
model has features that are reminiscent of both neutral naturalness models on
one hand Chacko:2005pe  and models of strongly-interacting massive particle
(SIMP) dark matter with a massive dark photon on the other Lee:2015gsa ;
Hochberg:2015vrg ; Berlin:2018tvf ; Katz:2020ywn . However, the primary aim of
the model developed here is to serve as a useful reference for comparing and
developing experimental sensitivities to dark shower events in the challenging
low-mass regime. Our two-flavor dark photon model is designed to realize a
variety of possible dimuon signatures with a relatively limited number of
model parameters. It is thus proposed somewhat in the spirit of a simplified
model approach to analysis design; however, we emphasize that it is a
theoretically self-consistent, UV-complete model (as far as hadronization
uncertainties allow).

The remainder of this paper is organized in two parts: the phenomenology and
experimental sensitivity estimates are discussed in the bulk of the paper,
while we reserve model-building aspects to the appendices. In Sec. II we
summarize the essential features of our benchmark model; theorists interested
in the model itself may find it beneficial to read the appendices here. We
describe our procedure for recasting the existing CMS and LHCb searches in
Sec. III and present results in Sec. IV. Our conclusions and recommendations
are presented in Sec. V. Appendix A contains further theoretical details of
the benchmark model and its Pythia implementation; some additional figures can
be found in Appendix B.

##  II Benchmark dark shower model

Here we develop a relatively simple benchmark model that can realize a range
of distinct dimuon signatures depending on the choices made for its mass
spectrum. In this section we provide an overview of the model’s collider
phenomenology, with further model details given in Appendix A.

###  II.1 Production

Throughout this study, we assume that the dark shower is initiated through an
exotic decay of the SM Higgs boson. This is theoretically the simplest
production mode, as it does not require the introduction of a new, heavy
mediator to initiate the dark shower. The SM Higgs ($h$) is moreover very
narrow, making it a sensitive probe of beyond the Standard Model physics in
general Curtin:2013fra ; Cepeda:2021rql  and hidden valley models in
particular Strassler:2006ri . Studying dark shower production in exotic Higgs
decays is especially well-motivated given the development of hidden valley
models that aim to address the hierarchy problem, such as neutral naturalness
models Chacko:2005pe ; Burdman:2006tz ; Craig:2015pha . From an experimental
point of view, Higgs production is also one of the more challenging
possibilities owing to the relatively low-$p_{T}$ final states that it
generates. For our present purposes, this makes exotic Higgs decays an
excellent benchmark scenario to illustrate the power of data scouting
techniques, as alternative trigger paths for these events would need to rely
on associated SM objects produced along with the Higgs in vector boson fusion
or associated $VH$ production, with accordingly smaller production cross
sections.

The Higgs boson initiates dark showers by decaying into pairs of dark quarks
through an interaction of the form $h\bar{q}_{D}q_{D}$. This interaction can
be readily constructed in the benchmark model described in detail in App. A.1;
in this benchmark, moreover, the size of the dark quark couplings to the SM
Higgs can be adjusted without significant impact on the dark hadron masses and
lifetimes. For our purposes, the branching ratio $\text{Br}(h\to\text{HV})$
into dark showers is the primary quantity of interest, and can be treated as a
free parameter. We will therefore present our results in terms of this
branching ratio.

Figure 1: Schematic representation of particle content and decay chains in the
two-flavor hidden valley benchmark model developed in this work. The lowest-
lying dark hadrons are an approximate dark-isospin triplet
$(\pi_{1},\pi_{2},\pi_{3})$ and heavier pseudoscalar $\eta$. The $\cdots$
represent heavier mesons and baryons. The green arrow represents the dark
parton shower and hadronization steps, which result in multiple $\eta$s and
$\pi_{i}$s per event. Finally, the dark mesons couple to an elementary dark
photon $A^{\prime}$, which controls the (displaced) decays of the $\eta$
and/or some of the $\pi_{i}$ to final states with SM fermions, as detailed in
Sec. II.3

###  II.2 Dark sector spectrum

Once the Higgs has decayed to a pair of dark quarks, those quarks will undergo
showering and hadronization within the dark sector, which we model with the
hidden valley module Carloni:2010tw ; Carloni:2011kk  in Pythia 8
Sjostrand:2014zea . The hidden valley module recently received a major update
expanding its capabilities to describe dark meson multiplets with broken
flavor symmetries Albouy:2022cin , which was essential to implement the model
we develop in this work.

We fix the number of colors and flavors to $N_{c}=3$ and $N_{f}=2$
respectively, and allow for the running of the dark gauge coupling. The two-
flavor case is the most minimal setup that generates the range of displaced
vertex topologies that we study in Sec. II.3. We further add an additional,
massive $U(1)$ interaction to facilitate the various decay chains described
below. This $U(1)$ is spontaneously broken, and the resulting massive dark
photon kinetically mixes with the SM photon to supply a muon-philic decay
portal back to the SM. Fig. 1 shows a schematic representation of the spectrum
and the most important decay chains; the model is defined in full detail in
Appendix A.

Figure 2: Summary of the $\eta$ phase space and available decay chains. The
$\eta\to 3\pi$ decay (gray region) is always prompt, while the three-body
decay $\eta\to\pi_{2}\bar{f}f$ (red) tends to be displaced, with $f\bar{f}$
representing SM final states. The remaining modes can be either prompt or
displaced, depending on the choice of model parameters. If the $\eta$ is taken
to be long-lived in the blue region, the $\eta\to A^{\prime}\pi_{2}$ mode
always dominates. The (A), (B1), (B2) and (C) labels refer to the decay
topologies in Sec. II.3.

In a bit more detail, we assume that the confining dark gauge group
dynamically breaks the hidden sector $SU(2)\times SU(2)$ flavor symmetry down
to its diagonal subgroup, as is the case in the SM pion sector. This leads to
three pseudo-goldstone bosons, which we label $\pi_{1},\pi_{2}$ and $\pi_{3}$.
There is also a heavier $\eta$ meson, which corresponds to the would-be
goldstone boson of the anomalous axial $U(1)$. We further assume that the
couplings of $A^{\prime}$ to dark sector quarks break parity, charge
conjugation, and isospin. This isospin breaking will lead to small mass
splittings between the various $\pi_{i}$, which are not relevant for the
phenomenology we are interested in here. The charges of the $A^{\prime}$ are
however chosen such that the following interaction is non-vanishing:

$\mathcal{L}\supset g\sin\theta
A^{\prime\mu}\left(\pi_{2}\partial_{\mu}\eta-\eta\partial_{\mu}\pi_{2}\right),$
(1)

where $g$ is the $U(1)$ gauge coupling and $\sin\theta$ a mixing angle
parameterizing the isospin violation. (See Appendix A.2 for details). This
interaction enables the decay $\eta\to A^{\prime}\pi_{2}$. Both the $\pi_{3}$
and $\eta$ can also decay to a pair of $A^{\prime}$ through the dark sector
chiral anomaly, provided this channel is kinematically open. Meanwhile, the
$\pi_{1}$ and $\pi_{2}$ are stable on collider timescales and contribute MET
to the event.

Since the $\eta$ is heavier than the $\pi_{i}$, one expects hidden sector
hadronization to produce it less frequently. Hadronization, as always, is one
of the major sources of theoretical uncertainty in hidden valley models. In
particular, the relative probability of producing different particle species
in dark hadronization cannot be calculated from first principles, and thus the
choices made in the hadronization component of the Pythia HV module must also
be considered as part of the definition of any benchmark model; for an up-to-
date discussion of these choices see Albouy:2022cin . The Pythia cards for our
model points are generated with our public python script code ; more detail
about the Pythia implementation can be found in Appendix A.4.

The available decay channels for the $\eta$ are shown schematically in Fig. 2.
If $m_{\eta}>3m_{\pi}$, the prompt decay $\eta\to 3\pi$ will always dominate.
This case leads to a marginally enhanced multiplicity of $\pi$’s, where the
$\pi_{3}$ decays as $\pi_{3}\to A^{\prime}A^{\prime}\to\mathrm{SM}$. The
$\eta\to A^{\prime}A^{\prime}$ decay mode is only open for
$m_{\eta}>2m_{A^{\prime}}$, while $\eta\to A^{\prime}\pi_{2}$ may occur if
$m_{\eta}>m_{A^{\prime}}+m_{\pi_{2}}$. If both channels are available (blue
region in Fig. 2), the $\eta\to A^{\prime}\pi_{2}$ channel dominates,
especially if the $\eta$ is long-lived. Finally, if none of these conditions
are satisfied, the $\eta$ can decay through an off-shell $A^{\prime}$ into a
SM final state plus $\pi_{2}$. One expects this decay to occur with a proper
decay length of $\mathcal{O}(\mathrm{cm})$ or larger. We refer to Appendix A.3
for more details.

###  II.3 Vertex topologies

Our benchmark model can generate three possible vertex topologies, as
summarized in Fig. 3. Concretely, we consider the following scenarios:

  * •

Scenario A (resonant, pointing): This scenario is realized when the $\eta\to
3\pi$ channel is open (gray region in Fig. 2) such that the $\pi_{3}$ meson is
the only dark sector meson that decays directly to the SM. We assume that this
occurs promptly, through the $\pi_{3}\to A^{\prime}A^{\prime}$ channel, while
the $A^{\prime}$ decays to the SM by kinetically mixing with the SM photon. If
this mixing angle is small, $A^{\prime}$ can however be long-lived, and we
treat its lifetime ($c\tau_{A^{\prime}}$) as a free parameter. This scenario
produces resonant dimuon pairs, for which the vector sum of their momenta
points back to the beamline.

  * •

Scenario B (resonant, non-pointing): This case occurs when on-shell dark
photons are produced in the displaced decay of a long-lived parent particle.
The vector sum of the momentum of the resulting resonant dimuon pairs does not
point back to the beamline. We realize this in two different ways in our
benchmark model:

Scenario B1 (higher visible multiplicity): In this case we keep the $\eta\to
3\pi$ and $\pi_{3}\to A^{\prime}A^{\prime}$ decay channels open, but choose
the dark gauge coupling small enough that the $\pi_{3}\to
A^{\prime}A^{\prime}$ decay becomes displaced (see Appendix A.2). For a given
meson mass spectrum, this case has the same average multiplicity of displaced
dimuon pairs as Scenario A. When the dark photon lifetime is taken to be
prompt, the dimuon pairs are produced at the same vertex as the visible
particles that come from the decay of the other dark photon.

Scenario B2 (lower visible multiplicity): In this scenario, we instead take
the $\eta\to 3\pi$ and $\pi_{3}\to A^{\prime}A^{\prime}$ channels to be
kinematically closed, while $\eta\to\pi_{2}A^{\prime}$ is allowed (blue and
green regions in Fig. 2). We further assume that the $\eta$ is long-lived, but
that the $A^{\prime}$ decays promptly to the SM, again by mixing with the SM
photon. We treat the $\eta$ lifetime ($c\tau_{\eta}$) as a free parameter. In
this scenario, all three dark pions ($\pi_{1,2,3}$) now escape the detector,
so the dimuon pairs are the only visible particles associated with the
displaced vertex. In this case the average dimuon multiplicity depends on the
average $\eta$ multiplicity $\langle N_{\eta}\rangle$, and is substantially
smaller than in Scenario A.

  * •

Scenario C (non-resonant): This scenario holds when the $\eta$ can only decay
through the three-body process involving an off-shell dark photon
$\eta\to\pi_{2}f\bar{f}$ (red region in Fig. 2). The $\eta$ is typically long-
lived in this case (see Appendix A.3). This scenario produces non-resonant
dimuon pairs, with the same dependence on $\langle N_{\eta}\rangle$ as in
Scenario B2.111Displaced vertices with non-resonant muon pairs can also be
realized for $m_{\eta}>3m_{\pi}$ and $2m_{A^{\prime}}>m_{\pi_{3}}$, in which
case $\pi_{3}$ has three-body decays $\pi_{3}\to A^{\prime}f\bar{f}$; however
in this case the $\pi_{3}$ proper lifetime tends to be prohibitively long in
the kinematic regime relevant for scouting searches. In this scenario all dark
pions ($\pi_{1,2,3}$) are again detector-stable.

As we will see, the existing CMS and LHCb searches already have excellent
sensitivity for scenario A, but the CMS search is not currently targeting
scenario B. Neither search currently targets scenario C.

Figure 3: Schematic representation of the decay topologies considered in this
work, with the $f$ ($\bar{f}$) SM (anti-)fermions, e.g. $\mu^{-}$ ($\mu^{+}$).
The green lines indicate macroscopic decay lengths.

###  II.4 Parameter choices and hadron multiplicities

Our $N_{f}=2$ confining dark sector has a substantial number of parameters,
some of which are related to one another by chiral perturbation theory, as
worked out in Appendices A.1 and A.2. We choose the following parameters as
our maximal independent set: the number of dark sector colors ($N_{c}$), the
mass of $\pi_{2}$ ($m_{\pi_{2}}$), the mass of the $\eta$ ($m_{\eta}$), the
dark sector confinement scale ($\Lambda$), the scale of chiral symmetry
breaking ($f$), the degree of isospin breaking $(\sin\theta)$, the dark photon
mass ($m_{A^{\prime}}$), the dark $U(1)$ gauge coupling ($g$) and the mixing
parameter of the $A^{\prime}$ with the SM photon ($\epsilon$). The mass
spectrum and decay tables of the dark sector $\pi_{i}$, $\eta$ and
$A^{\prime}$ are fully specified in terms of these parameters, as provided in
the appendices and by our python tool code .

We follow Pythia’s definition of the confinement scale $\Lambda$, where it is
given by the dimensional transmutation parameter of the one-loop
$\beta$-function of the dark sector $SU(N_{c})$ coupling.222While this work
was in the final stages of completion, Pythia version 8.309 was released,
which now also includes the two- and three-loop $SU(N_{c})$ $\beta$-functions.
All our results were obtained with Pythia version 8.308. One may expect this
scale to roughly correspond to the scale at which chiral perturbation theory
ceases to be reliable, in other words

$4\pi f\approx\Lambda.$ (2)

For definitiveness, we assume this relation to hold exactly; changing this
assumption would lead to $\mathcal{O}(1)$ variations in the decay widths of
some of the dark mesons. In addition, our experience with the SM suggests that
the meson associated with the anomalous axial $U(1)$ flavor symmetry has a
mass comparable to the confinement scale. In our model, this role is played by
the $\eta$-meson and as such we expect

$m_{\eta}\approx\Lambda.$ (3)

This expectation famously does not hold for theories in the large-$N_{c}$
limit Witten:1980sp . We will take Eq. 3 to hold as an equality as well and
choose $N_{c}=3$. We further take $\sin\theta=0.1$, such that isospin breaking
is relatively small and

$m_{\pi_{1}}\approx m_{\pi_{2}}\approx m_{\pi_{3}}$ (4)

for all our benchmark points. Choosing a small value of the isospin-breaking
parameter $\sin\theta$ is also important for ensuring that the Pythia hidden
valley module provides a reasonable approximation to the hadronization
process, as we detail in Appendix A.4.

The parameters $g$ and $\epsilon$ on the other hand are responsible for
setting the lifetimes of the $A^{\prime}$, $\pi_{3}$ and $\eta$. As such, all
our benchmark points in this paper only differ by different choices for the
masses $m_{\pi_{3}}$, $m_{\eta}$ and $m_{A^{\prime}}$, and the choices for the
lifetimes of the corresponding particles ($c\tau_{\pi_{3}}$, $c\tau_{\eta}$
and $c\tau_{A^{\prime}}$). Concretely, the individual scenarios are defined by
choosing

  * •

Scenario A: $c\tau_{\pi_{3}}=c\tau_{\eta}=0$, $m_{\eta}>3m_{\pi_{2}}$

  * •

Scenario B1: $c\tau_{A^{\prime}}=c\tau_{\eta}=0$, $m_{\eta}>3m_{\pi_{2}}$

  * •

Scenario B2: $c\tau_{A^{\prime}}=0$, $m_{\eta}>m_{\pi_{2}}+m_{A^{\prime}}$,
$\pi_{3}$ detector-stable

  * •

Scenario C: $c\tau_{A^{\prime}}=0$, $m_{\eta}<m_{\pi_{2}}+m_{A^{\prime}}$,
$\pi_{3}$ detector-stable

where “zero” $c\tau$ is meant as shorthand for “promptly decaying”, which is
consistent with experimental constraints on visible dark photons for
$m_{A^{\prime}}\gtrsim 10$ MeV Gori:2022vri . The remaining parameters are
specified in the relevant figures.

With these assumptions, we can investigate average particle multiplicities, as
generated by the Pythia 8 hidden valley module. We show the average meson
multiplicities in Fig. 4 as a function of pion mass for two example choices of
the mass ratio $m_{\eta}/m_{\pi_{2}}$. Naturally, the multiplicity of all
meson species rises as the meson mass and thus the confinement scale are
lowered. We furthermore see that the $\pi_{i}$ multiplicity is a factor of
several higher than the $\eta$ multiplicity. This is in part due to the lower
mass of the $\pi_{i}$, which means it is more likely to be produced in the
hadronization model. Moreover, the dark sector isospin triplet vector mesons,
the analogues of the $\rho$-mesons in the SM, are assumed to decay promptly to
the $\pi_{i}$, further enhancing their multiplicity.

At the level of the meson mass spectrum, the difference between scenarios (A,
B1) vs. scenarios (B2, C) is whether the $\eta\to 3\pi$ channel is
kinematically open. The former case comes with a slightly higher pion
multiplicity. In scenarios B2 and C, the $\eta\to A^{\prime}\pi_{2}$ decay
mode explicitly breaks isospin, and is the reason why the $\pi_{2}$
multiplicity is higher than the $\pi_{1,3}$ multiplicities in the right-hand
panel of Fig. 4.

Figure 4: Meson multiplicities for two representative choices of the
$m_{\eta}/m_{\pi_{2}}$ mass ratio. The left panel corresponds to scenarios A
and B1, while the right panel corresponds to scenarios B2 and C.

##  III Analysis

We consider two searches for low-mass displaced dimuon resonances from LHCb
Aaij:2020ikh  and CMS CMS:2021sch . Both analyses are powerful probes of the
dark shower signatures presented in the previous section. Here we briefly
describe both searches, as well as our procedures for event generation and
recasting.

###  III.1 CMS displaced dimuon search

The recent CMS search CMS:2021sch  records displaced low mass dimuon pairs
with $p_{T}$ thresholds as low as $\sim 3$ GeV, at a rate of 3 kHz. This large
rate is feasible because CMS only records a very small fraction of the total
event information, which includes the muon four-momenta, number of hits per
muon track, muon isolation, and track-quality information. The search used 101
$\text{fb}^{-1}$ of data at 13 TeV.

The most important selection criteria in the CMS search are listed in Tab. 1.
These include cuts on the transverse momentum ($p_{T}$) and pseudorapidity
($\eta$) of the muons, as well as a cut on the distance between the dimuon
vertex and the beamline ($L_{xy}$). In addition CMS also requires the vector
sum of the muon momentum vectors to point back to the beamline in order to
suppress backgrounds from fake vertices. Concretely, they impose a hard cut on
the azimuthal angle between the vector sum of the muon momenta and the vector
connecting the beamline to the displaced vertex. We will call this variable
$\Delta\phi(\sum\vec{p}_{T}(\mu),\vec{x})$, with $\vec{x}$ representing the
location of the displaced vertex in the detector frame. This pointing
requirement is automatically satisfied at truth-level for scenario A in Sec.
II.3, but not for scenarios B and C. We will speculate on loosening this
restriction in Sec. IV.2. CMS also imposes further selection criteria, such as
track and vertex quality requirements; we refer to CMS:2021sch  for more
details. In addition to this set of baseline cuts, CMS defines a set of
increasingly restrictive signal regions, the most relevant of which are listed
in Tab. 1. In particular, an event is put in the “isolated” category if the
scalar sum of the $p_{T}$ of all tracks within a cone of $\Delta R<0.3$ around
each muon does not exceed 20% of the $p_{T}$ of the muon.

CMS provides trigger and offline efficiencies for each signal region, as well
as the number of excluded events. We use this data in our re-interpretation,
as described in Sec. III.3. The collaboration evaluates signal efficiencies
for two well-motivated benchmark topologies: an exotic $B$-meson and exotic
Higgs decay, respectively $B\to X_{s}S$ and $h\to A^{\prime}A^{\prime}$, where
the scalar ($S$) and vector $(A^{\prime})$ are assumed to decay to
$\mu^{+}\mu^{-}$. We validated our reinterpretation procedure by simulating
both signal models and verifying that for each model the exclusions we
estimate by applying the provided efficiencies are in excellent agreement with
the experimental results. Since we expect the muon $p_{T}$ spectrum to be
rather soft for most low-mass hidden valley models, we consider $B$-meson
decay to be the more appropriate comparison model.

| fiducial cuts | signal regions
---|---|---
CMS |  $p_{T}(\mu)>3$ GeV | 0.0 cm $<L_{xy}$ (w/ and w/o isolation)
$|\eta(\mu)|<2.4$ | 0.2 cm $<L_{xy}$ (w/ and w/o isolation)
$L_{xy}<$ 11 cm | 1.0 cm $<L_{xy}$ (w/ and w/o isolation)
$\Delta\phi(\sum\vec{p}_{T}(\mu),\vec{x})<0.02$ | 2.4 cm $<L_{xy}$ (w/ and w/o isolation)
| 3.0 cm $<L_{xy}$ (w/ and w/o isolation)
| 7.0 cm $<L_{xy}$ (w/ and w/o isolation)
LHCb |  $p_{T}(\mu)>0.5$ GeV, $|\vec{p}(\mu)|>10$ GeV | 2 GeV $<p_{T}(A^{\prime})<3$ GeV (w/ and w/o pointing)
$2<\eta(\mu)<4.5$ | 3 GeV $<p_{T}(A^{\prime})<5$ GeV (w/ and w/o pointing)
$\sqrt{p_{T}(\mu^{+})p_{T}(\mu^{-})}>$ 1.5 GeV | 5 GeV $<p_{T}(A^{\prime})<10$ GeV (w/ and w/o pointing)
1.2 cm $<L_{xy}<$ 3 cm |
$\alpha(\mu^{+},\mu^{-})>$ 3 mrad |
Table 1: Most important fiducial cuts for the CMS CMS:2021sch  and LHCb
Aaij:2020ikh  searches, as well as most relevant signal regions for the
purpose of the study in this work. The CMS signal regions are non-exclusive,
while the LHCb signal regions are exclusive.

###  III.2 LHCb displaced dimuon search

The LHCb detector is specifically optimized to search for low mass resonances
and has consistently been on the forefront of the usage of online analysis
strategies. This has resulted in excellent sensitivity to light beyond-the-SM
particles, in particular if these particles have a substantial branching ratio
to muons LHCb:2015nkv ; Aaij:2016qsm ; Aaij:2019bvg ; Aaij:2020ikh . Here we
focus on LHCb’s recent inclusive search for displaced dimuon resonances
Aaij:2020ikh , which was carried out using 5.1 $\textrm{fb}^{-1}$ of data at
13 TeV. The most important fiducial cuts are listed in Tab. 1. The main
differences with respect to the CMS selection are: _i)_ a substantially lower
$p_{T}$ cut on the muons; _ii)_ the more forward acceptance of the LHCb
detector; _iii)_ the smaller $L_{xy}$ range; and _iv)_ a loose cut on the
opening angle between the two muons $\alpha(\mu^{+},\mu^{-})$.

###  III.3 Simulation framework

For our signal Monte Carlo samples, we generated Higgs bosons through the
gluon fusion channel using Pythia 8.308 Sjostrand:2014zea  and implemented the
decay mode to dark quarks. The dark quarks are subsequently showered and
hadronized with the Pythia 8 hidden valley module Carloni:2010tw ;
Carloni:2011kk . Further detail on the Pythia implementation of our benchmark
model is given in Appendix A.4. All signal samples were generated with
promptly-decaying particles and the lifetime dependence of the acceptance was
accounted for by the reweighting procedure described in Knapen:2022afb .
Concretely, for each event, we identify all vertices that pass the selection
cuts in Tab. 1, except for the $L_{xy}$ requirements. For the CMS analysis, we
moreover compute the isolation criteria on a vertex-by-vertex basis and
classify events accordingly. We work with the truth-level four-momenta of the
muons, effectively neglecting smearing due to the finite detector resolution.
Reconstruction and trigger efficiencies were however incorporated to the
extent possible, as described below.

To efficiently account for the cuts on $L_{xy}$, we assign an acceptance
weight $w_{\mathrm{acc}}(v)$ to each vertex $v$ that passed these cuts:

$w_{\mathrm{acc}}(v)=e^{-\frac{L_{xy}^{-}}{\beta\gamma\cosh(\eta)c\tau}}-e^{-\frac{L_{xy}^{+}}{\beta\gamma\cosh(\eta)c\tau}},$
(5)

which corresponds to the probability that the long-lived particle
corresponding to $v$ decays within two co-axial cylinders with inner and outer
radii $L_{xy}^{-}$ and $L_{xy}^{+}$. In other words, here the values
$[L_{xy}^{-},L_{xy}^{+}]$ represent the edges of the $L_{xy}$ bins used in the
analysis. Here $c\tau$, $\beta\gamma$, and $\eta$ are respectively the proper
lifetime, boost, and pseudorapidity of the long-lived particle. The full
weight of the vertex is obtained after multiplying $w_{\mathrm{acc}}(v)$ by
the probabilities that the vertex would pass the trigger and offline
selections, denoted by $\epsilon_{\mathrm{trig}}(v)$ and
$\epsilon_{\mathrm{off}}(v)$, or

$w(v)\equiv
w_{\mathrm{acc}}(v)\times\epsilon_{\mathrm{trig}}(v)\times\epsilon_{\mathrm{off}}(v).$
(6)

For the CMS analysis, $\epsilon_{\mathrm{trig}}(v)$ and
$\epsilon_{\mathrm{off}}(v)$ are provided in the supplementary material
attached to the analysis, for the signal regions listed in Tab. 1. For the
LHCb analysis we assume
$\epsilon_{\mathrm{trig}}(v)\approx\epsilon_{\mathrm{off}}(v)\approx 1$, in
line with the study in Ilten:2015hya .

For searches requiring just a single vertex, we can simply define the weight
of the whole event $w(e)$ as

$\displaystyle w(e)$ $\displaystyle\equiv
1-\prod_{v\in\\{\mathrm{vertices}\\}}(1-w(v))$ (7)
$\displaystyle\approx\sum_{v\in\\{\mathrm{vertices}\\}}w(v)$ (8)

where the approximation is justified whenever $w(v)\ll 1$ for all vertices.
For the CMS analysis, we will also consider a signal region where at least
_two_ vertices are reconstructed in the event. In this case, the event weight
is defined as

$\displaystyle w(e)\equiv$ $\displaystyle
1-\\!\\!\\!\prod_{v\in\\{\mathrm{vertices}\\}}\\!\\!\\!(1-w(v))$
$\displaystyle-\\!\\!\\!\sum_{v\in\\{\mathrm{vertices}\\}}\\!\\!\\!w(v)\\!\\!\\!\prod_{v^{\prime}\in\\{\mathrm{vertices}\\}\setminus\\{v\\}}\\!\\!\\!(1-w(v^{\prime})).$
(9)

This reweighting strategy allows us to efficiently compute limits for
arbitrary values of $c\tau$, without the need to regenerate the Monte Carlo
samples.

The procedure outlined above gives us the signal efficiency as a function of
the model parameters. We show the signal efficiency for select benchmark
points in Fig. 12 of Appendix B. We can directly compare this with the number
of excluded events in the case of CMS CMS:2021sch  or with the limit on the
fiducial cross section for LHCb Aaij:2020ikh  and extract a bound on the
branching ratio of the SM Higgs to Hidden Valley model under consideration for
each signal region. CMS and LHCb present their limits binned in terms of
$L_{xy}$ and $p_{T}$ respectively (see Tab. 1). In both cases we take the bin
with the strongest limit to represent our limit. CMS only reports results for
dimuon pairs whose reconstructed momentum satisfies the pointing cut in Tab.
1. For LHCb, we use the pointing selection for Scenario A and the non-pointing
selection for Scenarios B1 and B2.

##  IV Results

The relative advantage of a scouting trigger is best seen by looking at the
$p_{T}$ spectrum of the softer of the two muons in each vertex, as shown in
Fig. 5 for an example working point. While signal events may be muon-rich,
their muons are generally soft, and, depending on the parent particle
lifetime, very often displaced. The $p_{T}$ treshholds for displaced muon
triggers are prohibitively high for this signal, as Fig. 5 demonstrates, while
the scouting trigger can record events with trailing muon $p_{T}$ as low as
3.5 GeV. Thus the scouting trigger is uniquely suited to capture the regime
where signals yield soft, displaced muon pairs.

For muon pairs with a sufficiently small displacement to be picked up by
prompt dilepton triggers, a softer trailing muon $p_{T}$ cut of 10 GeV is more
reflective of ATLAS and CMS capabilities (e.g., CMS:2021pcy ; ATLAS:2023vxg ).
But even this $p_{T}$ threshold still yields an acceptance of only a few
percent for the model point in Fig. 5. For the low c$\tau$/prompt regime, the
scouting analysis in CMS-PAS-EXO-21-005  is therefore very pertinent. While
the scouting trigger has much better signal acceptance, it also collects more
background events than the traditional dimuon trigger, in particular for muon
pairs produced at low to no displacement. In some models, the signal may
contain additional handles such as jet substructure information or hadronic
displaced vertices that could help with background discrimination, but which
are not retained in the scouting stream. In such cases, a fully offline
analysis may prove to be more powerful. For the benchmark scenarios that we
consider in this paper, we do not expect such handles to add much
discriminating power in the regime where all dark decays are prompt. We thus
suspect that even in the limit of prompt decays the scouting analysis in CMS-
PAS-EXO-21-005  would outperform the standard dimuon trigger, although this
statement is model-dependent. We leave a quantitative study of scouting for
dark showers in the prompt regime to future work.

Figure 5: The $p_{T}$ spectra of the trailing muon in each vertex for an
example Scenario A model point, within pseudorapidity ranges covered by the
CMS tracker and LHCb VELO detectors (red and purple respectively). Dashed
lines indicate the $p_{T}$ threshold of the corresponding analyses, see Sec.
III.1 for details. The dot-dashed lines indicate the threshold for the prompt
and displaced dimuon triggers used in respectively CMS:2022qej  and
CMS:2021pcy ; ATLAS:2023vxg . The Pythia 8 card used to generate the events in
this figure is included in Appendix A.4.

###  IV.1 Scenario A

Figure 6:  Limits on the branching ratio for the SM Higgs decaying to the
hidden valley model described in Sec. II, with
$\Lambda=m_{\eta}=4m_{\pi_{3}}$, $\sin\theta=0.1$ and $g=0.05$. (See Appendix
A.2 for details.) The bounds are shown as a function of the lifetime of the
dark photon $c\tau_{A^{\prime}}$, from our recast of the CMS dimuon scouting
analysis CMS:2021sch  and the LHCb low mass dimuon analysis Aaij:2020ikh . The
dashed yellow line indicates the limit set by CMS:2021sch  when requiring two
dimuon vertices per event, under the assumption that this suffices to suppress
the background to negligible levels. (See text for details.)

We first discuss Scenario A, which produces resonant dimuon pairs whose
reconstructed momenta point back to the beamline. The bounds we obtain from
both LHCb and CMS for benchmark dark shower models produced in exotic Higgs
decays is shown in Fig. 6 for a few example mass points. The blue and red
shaded regions correspond to the bounds set by the CMS analysis with and
without imposing isolation, respectively. The isolation efficiency for these
signal benchmarks is roughly 40%, which reflects that in this scenario the
dark photons are produced in pairs. We see that imposing isolation is somewhat
beneficial for relatively low $m_{A^{\prime}}$, where the SM backgrounds are
substantial. For $1\;\mathrm{GeV}\gtrsim m_{A^{\prime}}\gtrsim 3$ GeV both
selections perform comparably. For $m_{A^{\prime}}\gtrsim 3$ GeV, however, the
background drops off sharply CMS:2021sch , and the more inclusive selection
sets the best limit. The best sensitivity of the LHCb analysis is at somewhat
lower $c\tau_{A^{\prime}}$, as the dark photons in the LHCb acceptance tend to
be more boosted than those in the CMS acceptance. Despite its much lower
luminosity, the LHCb analysis outperforms the CMS displaced analysis for
$m_{A^{\prime}}\gtrsim 1$ GeV and $c\tau_{A^{\prime}}\lesssim 1$ mm.333The CMS
prompt scouting search CMS-PAS-EXO-21-005  appeared when our manuscript was in
its final stage of completion. It likely has some sensitivity in the low
$c\tau$ regime; we defer a detailed analysis of its reach for dark shower
models to future work.

One of the key features of dark shower topologies is that they tend to produce
multiple long-lived particles per event. If the proper decay length of these
long-lived particles is comparable to or smaller than the size of the detector
and if the vertex reconstruction efficiency is sufficiently high, one expects
that requiring an additional displaced dimuon vertex would be a powerful
handle to reduce the backgrounds. To illustrate this, we derive a tentative
bound from the CMS analysis by assuming that _i)_ the vertex reconstruction
efficiencies $\epsilon_{\mathrm{trig}}(v)\times\epsilon_{\mathrm{off}}(v)$ are
independent between different vertices and that _ii)_ requiring an additional
vertex suffices to reduce the background to negligible levels for the whole
range of $m_{A^{\prime}}$. If both assumptions are satisfied, one obtains the
yellow curve in Fig. 6. We see that demanding an additional vertex may be a
very powerful handle at low to moderate $c\tau_{A^{\prime}}$. At higher values
of $c\tau_{A^{\prime}}$, a larger fraction of the long-lived $A^{\prime}$
escape the tracker before decaying, so that experimental sensitivity is
dominated by the single-dimuon vertex search.

Fig. 7 shows how the sensitivity of the CMS scouting search changes as we vary
both $m_{\pi_{3}}$ and $m_{A^{\prime}}$ while keeping $m_{\pi_{3}}/\Lambda$
fixed. Varying $m_{\pi_{3}}$ has two effects: a larger $m_{\pi_{3}}$ reduces
the available phase space for dark mesons, yielding a lower vertex
multiplicity. The larger mass however provides a somewhat higher boost to the
$A^{\prime}$ and therefore the muons, increasing the likelihood that said
vertices will be reconstructed. Varying $m_{A^{\prime}}$ primarily affects the
$A^{\prime}\to\mu\mu$ branching ratio, when it comes to signal acceptance;
however, $m_{A^{\prime}}$ also plays a primary role in determining the
background, which is larger at lower dimuon invariant masses. From Fig. 7 we
conclude that the dependence on $m_{\pi_{3}}$ is fairly mild, while the
results are more sensitive to $m_{A^{\prime}}$. In particular, we see a dip in
sensitivity when $m_{A^{\prime}}$ is close to the mass of the SM $\rho$ meson,
since resonant mixing with the SM $\rho$ enhances the
$A^{\prime}\to\pi^{+}\pi^{-}$ partial width and accordingly suppresses the
branching ratio of the dark photon into muons. In this figure we again see
that a double vertex analysis is particularly powerful at low
$c\tau_{A^{\prime}}$ and loses sensitivity faster at high
$c\tau_{A^{\prime}}$, as expected.

Figure 7: Limits on the Higgs branching ratio to the hidden valley model in
Scenario A from our recast of Ref. CMS:2021sch , with
$\Lambda=m_{\eta}=4m_{\pi_{3}}$, $\sin\theta=0.1$ and $g=0.05$. (See Appendix
A.2 for details.) The “single vertex” column indicates the single vertex
isolated or inclusive signal region, whichever gives the strongest limit for
the point in question. The “double vertex” limit makes important and
unvalidated assumptions about correlations in the vertex reconstruction
efficiencies and the SM background, as described in the text. It should
therefore be interpreted as a sensitivity estimate rather than as a robust
limit.

While we have studied a specific benchmark dark shower model, thanks to the
inclusive nature of both the LHCb and CMS searches, we can draw several broad
conclusions. In particular, we see that the scouting search provides the
leading sensitivity to low-mass dark shower signatures when the muons are
produced in the decay of a parent particle with lifetime $c\tau\gtrsim$ mm,
and for $c\tau\gtrsim$ cm the sensitivity is likely dominated by final states
with a single dimuon vertex. Since the scouting search depends only on the
kinematics and multiplicity of dimuon pairs, these properties will hold
broadly across a range of dark shower models that produce low-mass, muon-
philic final states, including those considered in Pierce:2017taw ;
Cheng:2019yai ; Cheng:2021kjg . The relative advantage of single- vs double-
vertex searches will depend weakly on the mass of the mediator initiating the
shower as well as the dark gauge coupling, both of which impact the
multiplicity of hidden hadrons. However, acceptance alone ensures that in the
longer-lifetime regime the single-vertex search will come to dominate the
sensitivity. We can thus conclude that scouting searches are a uniquely
sensitive tool for discovering low-mass hidden valley models, providing
leading sensitivity in the well-motivated but not universal cases where such
models give rise to displaced dimuon resonances that point back to the primary
vertex. A double-vertex search would moreover provide important additional
sensitivity at intermediate parent lifetimes.

###  IV.2 Scenario B

Both CMS and LHCb have selection criteria that require the vector sum of the
three-momenta of the muons in a given vertex to point back to the beam line,
as this is a good handle to reduce background from fake vertices and material
interactions. However, this pointing cut has poor efficiency on signals where
a long-lived state decays to a dimuon pair plus one or more other states.
Scenario B presents two such example cases, where we assume that the
$A^{\prime}$ decays promptly, but that the dark meson $\pi_{3}$ (scenario B1)
or $\eta$ (scenario B2) has a macroscopic lifetime.

We can characterize the degree of pointing through the variable
$\Delta\phi(\sum\vec{p}_{T}(\mu),\vec{x})$, which at truth-level corresponds
to the the azimuthal angle $\Delta\phi$ between the three-momenta of the
$A^{\prime}$ and its parent meson, as shown in Fig. 8 for two example points
in scenario B2. Though the distribution of
$\Delta\phi(\sum\vec{p}_{T}(\mu),\vec{x}))$ is still peaked at small values,
it is rather broad and the CMS analysis cut (dot-dashed line) is only
$\mathcal{O}(10\%)$ efficient. LHCb on the other hand also reports results for
an inclusive selection that does not require pointing. This selection comes
roughly with an order of magnitude more background than the selection
requiring pointing, which is acceptable if a large enough increase in signal
efficiency can be achieved.

We show the resulting bounds on both Scenarios B1 and B2 in Fig. 9 for two
example mass points. As before, blue and red shaded regions correspond to the
bounds set by the CMS analysis with and without imposing isolation,
respectively. The isolation efficiency for Scenario B1 is comparable to that
for Scenario A, roughly 40%, while for Scenario B2 it is nearly 100%.

Since the multiplicity and kinematics of dark photons produced in Scenarios A
and B1 are very similar, the differences between the bounds on these two
scenarios are mainly attributable to the dimuon pair not pointing to beamline.
Concretely, the LHCb limit on Scenario B1 using the inclusive selection is
generally weaker than the Scenario A limit, which uses the pointing selection
at the same dark photon mass, owing to larger backgrounds. The relative
stringency of the limits based on these two selections varies bin-by-bin,
reflecting background fluctuations; for the specific mass point shown in Fig.
9, the Scenario B1 and Scenario A limits are comparable. Meanwhile for the CMS
scouting search, the loss of acceptance from the cut on
$\Delta\phi(\sum\vec{p}_{T}(\mu),\vec{x})$ weakens the Scenario B1 limit by
roughly a factor of five compared to Scenario A.

For Scenario B2, the number of dimuon pairs per event is substantially
smaller, and accordingly the sensitivity of the existing analyses is further
reduced. In the bulk of parameter space for this scenario, the existing CMS
and LHCb searches do not have better sensitivity than that offered by indirect
constraints from global fits to Higgs properties, which currently constrain
the exotic Higgs branching fraction to be less than 15% ATLAS:2021vrm . It
would be very interesting for the CMS collaboration to attempt a search with
relaxed pointing criteria, as the enhanced signal efficiency may be sufficient
to offset the higher backgrounds, particularly for isolated dimuon pairs.

Figure 8:  Truth-level distribution of
$\Delta\phi(\sum\vec{p}_{T}(\mu),\vec{x})$ for two example benchmark points in
scenario B2. The dashed line indicates the cut imposed in the CMS scouting
analysis CMS:2021sch . The efficiency of this cut for the red (blue) benchmark
is 15% (10%).  Figure 9:  Limits on the branching ratio for the SM Higgs
decaying to the hidden valley models described in Sec. II, with
$\Lambda=m_{\eta}=4m_{\pi_{3}}$, $\sin\theta=0.1$ and $g=0.05$. (See Appendix
A.2 for details.) The bounds are shown as a function of the lifetime of the
dark photon $c\tau_{A^{\prime}}$, from our recast of the CMS dimuon scouting
analysis CMS:2021sch  and the LHCb low mass dimuon analysis Aaij:2020ikh .

###  IV.3 Scenario C

Finally, in scenario C, long-lived $\eta$ mesons produce a displaced dimuon
pair that is non-resonant and hence necessarily non-pointing. The kinematics
of the decay produce an endpoint in the dimuon mass at
$m_{\mu\mu,\mathrm{max}}=m_{\eta}-m_{\pi_{2}}$, as shown in Fig. 10. Because
of the off-shell dark photon and the three-body final state, the minimum
possible $\eta$ lifetime for this case is a strong function of the mass
splitting between the $\eta$ and the $\pi_{2}$. For splittings of the order
$m_{\eta}-m_{\pi_{2}}\sim$ few GeV, the minimum possible proper decay length
is $\mathcal{O}$(cm), where the exact value depends on the remaining model
parameters (see Appendix A.3). Very narrow splittings
($m_{\eta}-m_{\pi_{2}}\ll$ GeV) are therefore likely to have intractably low
acceptance.

As for scenario B, we are unable to reliably model the backgrounds for CMS in
the absence of the pointing cut and therefore do not attempt to make a
sensitivity projection. Of course, it is clear this is a much more challenging
signal than Scenario B, as the distribution of the signal events over a range
of dimuon masses both dilutes the overall statistical significance of the
signal as well as complicates the data-driven background estimation process.
An interesting open question at this juncture is to compare the sensitivities
of analyses based on scouting triggers to those using alternative trigger
strategies, which may have access to a smaller number of signal events, but
may contain more information that can be used to control
backgrounds.444Alternatively, one may attempt to expand the amount of useful
information recording in the scouting analysis, e.g. by making use of machine
learning-driven data compression methods Collins:2022qpr . The parked data set
CMS-DP-2019-043  is a particularly intriguing possibility in this context. For
the more traditional triggers, the relative sensitivities will necessarily
depend in detail on the production mechanism.

As a concrete example, our benchmark scenario of production in exotic Higgs
decays offers the chance to trigger on Higgs production in association with a
leptonically-decaying $W$ or $Z$ boson using prompt lepton triggers.555The
associated $ttH$ production cross-section, which is $0.5$ pb at $\sqrt{s}=13$
TeV LHCHiggsCrossSectionWorkingGroup:2016ypw , can provide a sizeable
contribution to Higgs production in association with one or more prompt
leptons. This production channel is often a challenge for SM analyses owing to
the combinatoric challenges of its final states, but it can be important for
recording beyond-the-SM Higgs decays to exotic final states. While the Higgs
production cross-section in semi-leptonic processes is two orders of magnitude
smaller than the inclusive Higgs production cross-section relevant for
scouting, events that arrive on a standard lepton trigger will contain
information about the whole event and will be subject to different
backgrounds. Without being able to reliably model the backgrounds for non-
pointing dimuon pairs, we cannot determine which analysis strategy offers the
best sensitivity.

We can, however, conclude that the Higgs production channel considered here is
perhaps the most optimistic choice when it comes to alternative trigger
pathways for low-mass dark showers. Other natural choices for light
($\lesssim$ few hundred GeV) mediators are _i)_ the $Z$ boson Cheng:2019yai ;
Cheng:2021kjg  and _ii)_ a new SM-singlet, whether vector or scalar (e.g.
Pierce:2017taw ); one could straightforwardly extend our benchmark model to
cover either case by adding additional heavy states, without substantially
changing the dark meson phenomenology. For a $Z$ boson mediator, associated
$WZ$ production is more suppressed compared to inclusive $Z$ production than
$VH$ is relative to inclusive $H$ production; meanwhile for a BSM mediator
such as a $Z^{\prime}$ or a new scalar, the best additional handle that one
could generically expect is energetic initial state radiation (ISR). In this
case, the presence of a moderately-hard ISR jet could enable a MET or a
standard dimuon trigger strategy, depending on the specifics of the model, at
the cost of substantially (and perhaps prohibitively) reducing the available
signal cross-section. An example of this strategy was very recently deployed
in the context of a search for inelastic dark matter CMS-PAS-EXO-20-010 ,
which relied on a MET trigger and additionally required a soft displaced
dimuon pair in the final state. As this search indicates, the thresholds
necessary for a MET trigger strategy are not small; Ref. CMS-PAS-EXO-20-010
placed an offline analysis cut of $E\\!\\!\\!/_{T}>200$ GeV.

Figure 10: Truth-level invariant mass of dimuon pair in scenario C, for two
benchmark model points with cm-scale proper decays lengths.

##  V Summary and recommendations

The search for hidden valleys at the LHC is a challenging and multi-faceted
program, thanks in part to the fantastic diversity of possible signatures.
Fortunately, search strategies for models with a dark confinement scale at the
few GeV scale admit a certain amount of streamlining: many of the portal
operators that could govern decays of GeV-scale dark hadrons to SM final
states predict lifetimes that are too long to leave much visible energy within
the main detectors Knapen:2021eip . For such low-mass dark shower events, the
operators that are easily compatible with high-multiplicity visible final
states tend to be muonphilic. Thus low-$p_{T}$ (displaced) dimuon pairs stand
out as one of the most promising signatures of low-mass hidden valley
theories.

Online analysis techniques, including CMS’ data scouting stream and the LHCb
low mass dimuon searches, are a powerful tools for probing the soft dimuons
characteristic of low-mass dark shower events. We establish the sensitivity of
CMS’ scouting search for displaced dimuon resonances CMS:2021sch  to dark
showers produced in exotic decays of the SM Higgs boson, and compare its reach
to that of an inclusive dimuon resonance search at LHCb Aaij:2020ikh . We
demonstrate the significant advancement in reach realized in the scouting
search in the context of a benchmark hidden valley model; however, the
inclusive nature of the experimental searches lets us make several
observations that hold across a broader class of low-mass dark shower
signatures. Concretely, CMS performs best for medium to long proper lifetimes
due to its larger integrated luminosity, while LHCb currently sets the
strongest bounds at short lifetimes. A reinterpretation of the prompt scouting
analysis by CMS CMS-PAS-EXO-21-005  in terms of hidden valley models would be
well-motivated. Though we cannot make sharp statements without more
information on the background, we expect that double vertex analysis selection
by CMS and potentially by LHCb could further improve the bounds.

Our simple and flexible benchmark model features two flavors of light dark
quarks and an elementary dark photon, and is capable of realizing several
different dimuon vertex topologies with relatively few parameters. The
simplest scenario occurs when dimuon resonances are produced from the decay of
a promptly-produced particle, and thus the reconstructed dimuon momentum
points back to the beamline. Our model also realizes scenarios where the
dimuon pair does not point back to the beamline, as well as a non-resonant
displaced dimuon signature from a three-body decay.

For the non-pointing case, the LHCb search currently has a suitable selection,
which has only marginal sensitivity with the current dataset. It would
therefore be important for both LHCb and CMS to (continue to) include a non-
pointing signal region in future searches with more integrated luminosity.
Finally, the non-resonant, non-pointing case is also well-motivated and always
implies a macroscopic lifetime. We recommend that it too be included as a
possibility in future searches. It remains an interesting open question
whether scouting searches or searches for (e.g.) MET in addition to non-
pointing dimuon vertices in events that arrive on higher-threshold trigger
streams can provide the best probe of these non-resonant scenarios. However
here scouting has the particular advantage that searches are independent of
assumptions about the production mode for these showers; alternate trigger
strategies depend more strongly on the model-dependent features of the
mediator responsible for initiating the shower.

Online analyses such as scouting therefore offer an unmatched discovery tool
for low-mass dark showers, which can otherwise easily evade traditional
detection strategies. Of course, in the event of a compelling excess,
recording more features of the events would be a top priority. The relatively
high-multiplicity final states that can be realized in our benchmark models
can also provide other handles to further identify and characterize the
signature, in particular displaced hadronic and/or electronic vertices and a
moderate amount of missing energy. Displaced hadronic final states in
particular are a locus of discussion for future trigger capabilities at LHCb
as well as ATLAS and CMS Alimena:2021mdu , and could offer further windows
onto this challenging class of signatures.

###### Acknowledgements.

We gratefully acknowledge useful conversations with Hsin-Chia Cheng, Matthew
Citron, Zeynep Demiragli, Lingfeng Li, Zoltan Ligeti, Steven Lowette, Mario
Masciovecchio, Titus Mombächer, Dean Robinson, Ennio Salvioni, Christiane
Scherb, Matthew Strassler, Indara Suarez, Mahiko Suzuki, Xabier Cid Vidal, and
Michael Williams. We thank Matthew Citron, Lingfeng Li, and Ennio Salvioni for
comments on the manuscript. SK would also like to thank Diego Redigolo,
Alberto Mariotti and Sam Junius for collaboration on related work. The work of
SB was supported by a grant from the United States-Israel Binational Science
Foundation (BSF). The work of SK was supported by the U.S. Department of
Energy, Office of Science under contract DE-AC02-05CH11231. Part of this work
was performed at the Aspen Center for Physics, which is supported by National
Science Foundation grant PHY-1607611. JS thanks the MIT Center for Theoretical
Physics and Laboratory of Nuclear Science for their generous hospitality
during the completion of this work.

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##  Appendix A A two-flavor hidden valley

Here we construct the benchmark two-flavor model with a light dark photon used
in this work. We first discuss the UV Lagrangian in terms of elementary dark
sector quarks and dark gluons, then consider the resulting meson spectrum and
use chiral perturbation theory to calculate the relevant branching ratios and
decay widths. We end with a discussion of the Pythia implementation.

###  A.1 Quark sector

We consider two dark sector quarks ($q_{i}$) and their corresponding anti-
quarks ($\bar{q}_{i}$), which are respectively in the fundamental and the
anti-fundamental representation of the dark sector $SU(N_{c})$ strong force.
We furthermore introduce a single, weakly coupled $U(1)$ gauge field
$A^{\prime}$. We will be interested in the case when this gauge field has a
mass. For scenarios A and B1, where the anomalous decay $\pi_{3}\to
A^{\prime}A^{\prime}$ is responsible for the visible signal, one can take this
mass to come from a Stückelberg mechanism, which would be compatible with
extending the model into a “neutral naturalness” solution to the hierarchy
problem. In order to realize the distinct vertex topologies in scenarios B2
and C, we additionally include a dark Higgs scalar $\phi$, whose vacuum
expectation value will contribute to the mass of $A^{\prime}$ as well as those
of the dark quarks. The charge assignments are as follows:

$\begin{array}[]{c|ccccc}&q_{1}&q_{2}&\bar{q}_{1}&\bar{q}_{2}&\phi\\\\[2.0pt]
\hline\cr\\\\[-8.0pt]
SU(N_{c})&\square&\square&\overline{\square}&\overline{\square}&1\\\\[2.0pt]
U(1)&1&-1&0&0&1\end{array}$ (10)

which, at the renormalizeable level, allow for the following set of
interactions

$\displaystyle\mathcal{L}\supset$ $\displaystyle
i\sum_{i}q^{\dagger}_{i}\not{D}q_{i}+i\sum_{i}\bar{q}^{\dagger}_{i}\not{D}\bar{q}_{i}+|D^{\mu}\phi|^{2}$
(11) $\displaystyle-\left(\\!\begin{array}[]{c}q_{1}\\\
q_{2}\end{array}\\!\right)^{T}\left(\\!\begin{array}[]{cc}y_{11}\phi^{\dagger}&y_{12}\phi^{\dagger}\\\
y_{21}\phi&y_{22}\phi\\\
\end{array}\\!\right)\left(\\!\begin{array}[]{c}\bar{q}_{1}\\\
\bar{q}_{2}\end{array}\\!\right)+H.c.$ (18) $\displaystyle-\epsilon
eA^{\prime}_{\mu}J_{EM}^{\mu}$ (19)

with $D_{\mu}\equiv\partial_{\mu}+igQA^{\prime}_{\mu}+ig_{s}T^{a}G^{a}_{\mu}$.
Here $g$ and $g_{s}$ are respectively the $U(1)$ and $SU(N_{c})$ gauge
couplings, with $Q$ and $T^{a}$ the $U(1)$ charge and the $SU(N_{c})$
generators respectively. The $G^{a}_{\mu}$ are the $SU(N_{c})$ gluons. For the
mass range of interest here, the dark photon can to good approximation be
taken to have a small coupling $\epsilon e$ to the SM electromagnetic current
$J_{EM}^{\mu}$, through the mixing of the SM photon with the $A^{\prime}$. We
further assume that the scalar field develops a vacuum expectation value $v$,
such that we can decompose it as

$\phi=\frac{1}{\sqrt{2}}(v+\varphi)e^{i\tilde{a}/v}.$ (20)

The vacuum expectation value contributes both to the mass of the $A^{\prime}$
and the masses of the quarks through the interactions in Eq. 11 and Eq. 18.
Concretely, expanding the kinetic term for $\phi$ we find

$|D^{\mu}\phi|^{2}\supset\frac{1}{2}g^{2}v^{2}A^{\prime}_{\mu}A^{\prime\mu}+gvA^{\prime}_{\mu}\partial^{\mu}\tilde{a}\left(1+\frac{\varphi}{v}\right)^{2}$
(21)

A Higgs portal coupling between SM and dark Higgses is allowed by all
symmetries and provides a natural UV completion of the coupling
$hq_{i}\bar{q}_{i}$ responsible for initiating the dark showers, as we now
sketch. Introducing a Higgs portal mixing between dark and SM Higgs of the
form $\mathcal{L}_{int}=\kappa|\phi|^{2}|H|^{2}$ gives rise to a dark Higgs-SM
Higgs mixing angle that in the limit $\kappa\ll 1$ can be expressed in terms
of the dark Higgs vev $v$ and mass $m_{\varphi}$ as

$\theta\approx\frac{\kappa vv_{h}}{m_{\varphi}^{2}-m_{h}^{2}},$ (22)

where $v_{h}$ is the SM Higgs vev. As a result of this mixing, the SM Higgs
$h$ picks up a coupling to dark quarks given by

$\mathcal{L}_{int}=\frac{y_{ij}}{\sqrt{2}}\theta h\bar{q}_{Di}q_{Dj}.$ (23)

We will assume for simplicity that the radial mode $\varphi$ decouples from
the subsequent phenomenology, and neglect it hereafter.666Adding the Higgs
portal coupling $\kappa$ also shifts the dark quark masses by a quantity of
order $\kappa v_{h}^{2}/m_{\varphi}^{2}$. For our purposes we can absorb this
shift into a redefinition of the quark mass parameters. In the absence of
chiral symmetry breaking, we could simply remove the Goldstone boson
$\tilde{a}$ by working in the unitary gauge. The mass of $A^{\prime}$ however
receives a contribution both from $v$ and from the confining $SU(N_{c})$
dynamics, which means that a linear combination of the elementary Goldstone
mode $\tilde{a}$ with one of the meson modes will furnish the longitudinal
component of the $A^{\prime}$ in the unitary gauge. This is best treated in
chiral perturbation theory, with the meson degrees of freedom; we describe it
in the next section.

The quark sector in Eq. 18 a priori leaves us with a lot of freedom and is
therefore rather unwieldy. Rather than mapping out the phenomenology of the
fully general case, we are interested in picking an example that is simple to
parametrize and that generates the signatures we are interested in. For this
reason we will unapologetically assume the following relations between the
Yukawa couplings in Eq. 18:

$\displaystyle y_{11}=y_{22}\quad\mathrm{and}\quad y_{12}=y_{21}.$ (24)

To track the gauge invariance in the low-energy chiral Lagrangian, it will be
useful to retain the explicit dependence on the goldstone mode $\tilde{a}$
coming from the fundamental Higgs boson. Thus we write the mass matrix as

$M=\Phi M_{0},$ (25)

where $M_{0}$ is the mass matrix with $\tilde{a}=0$, and
$\Phi\equiv\mathrm{diag}(e^{-i\tilde{a}/v},e^{i\tilde{a}/v})$ in the gauge
basis. If we move to the basis that diagonalizes $M_{0}$, the full mass matrix
$M$ then reads

$\mathcal{L}\supset-\left(\\!\\!\begin{array}[]{c}\bar{q}_{1}\\\
\bar{q}_{2}\end{array}\\!\\!\right)^{T}\left(\\!\\!\begin{array}[]{cc}m_{1}\cos\left(\frac{\tilde{a}}{v}\right)&-i\,m_{2}\sin\left(\frac{\tilde{a}}{v}\right)\\\
-i\,m_{1}\sin\left(\frac{\tilde{a}}{v}\right)&m_{2}\cos\left(\frac{\tilde{a}}{v}\right)\\\
\end{array}\\!\\!\right)\left(\\!\\!\begin{array}[]{c}q_{1}\\\
q_{2}\end{array}\\!\\!\right)+\mathrm{H.c.}$ (26)

with $m_{1}\equiv\frac{v}{\sqrt{2}}(y_{11}-y_{12})$ and
$m_{2}\equiv\frac{v}{\sqrt{2}}(y_{12}+y_{11})$. We explicitly retain the phase
$\tilde{a}/v$, as it will play a role in the chiral perturbation theory
calculations in the next section.

At this point it is convenient to switch to Dirac notation, which will make
the approximate $SU(2)\times SU(2)$ flavor symmetry manifest and facilitate
the matching onto chiral perturbation theory. The model is then specified by
(with $\tilde{a}=0)$

$\mathcal{L}\supset
i\sum_{i=1,2}\overline{Q}_{i}\not{D}Q_{i}-m_{i}\overline{Q}_{i}Q_{i}$ (27)

with the Dirac fermions $Q_{i}=(q_{i}\;\bar{q}_{i}^{\dagger})^{T}$. In this
basis, the interactions of the $A^{\prime}$ are now manifestly chiral, and the
covariant derivative is defined as

$D_{\mu}\equiv\partial_{\mu}+igQ_{L}P_{L}A^{\prime}_{\mu}+igQ_{R}P_{R}A^{\prime}_{\mu}+ig_{s}T^{a}G^{a}_{\mu}$
(28)

with the $P_{L,R}$ the chiral projection operators. The charge matrices
$Q_{L,R}$ are obtained by rotating the charges in Eq. LABEL:eq:chargetable to
the basis that diagonalizes $M_{0}$:

$Q_{L}=\left(\\!\\!\begin{array}[]{cc}&1\\\ 1&\\\
\end{array}\\!\\!\right)\quad\mathrm{and}\quad Q_{R}=0.$ (29)

In general, the charge matrices in the $M_{0}$ basis can be non-sparse and
contain non-integer numbers, as does the CKM matrix in the SM. In other words,
the simple form of Eq. 29 is non-generic and a consequence of our choices in
Eq. 24. This choice is intended to streamline the analysis in the next
section, without qualitatively changing the phenomenology. Note that the
$A^{\prime}$ interaction breaks both $C$ and $P$ symmetry, which is essential
to realize the $\eta\to A^{\prime}\pi_{2}$ decay channel. It preserves $CP$,
however, and we will assume that the $\theta$-angle associated with the
$SU(N_{c})$ dynamics is small enough to not meaningfully affect the
phenomenology, as is the case in the SM.

###  A.2 Meson sector

We assume that gauge coupling and masses in Sec. A.1 are a small perturbation
on the strong dynamics in the dark sector. In this case, the lightest dark
sector mesons are well described as the three pseudo-goldstone bosons
associated by the breaking of the $U(2)\times U(2)\to U(2)$ symmetry by the
dynamics of the confining gauge group. The mesons ($\tilde{\pi}_{i}$)
therefore make up the adjoint representation of the unbroken $U(2)$, such that
we can define the following matrix of meson fields

$\bm{\pi}\equiv\sum_{i=0}^{3}\lambda_{i}\tilde{\pi}_{i}$ (30)

with the $\lambda_{i}$ the Pauli matrices for $i=1,2,3$ and
$\lambda_{0}=\mathbb{1}_{2\times 2}$. In other words, the $\tilde{\pi}_{i}$
refer to the fields in the symmetry eigenbasis, as defined by Eq. 30. We will
reserve the $\pi_{i}$ notation for the fields in the mass eigenbasis, as
introduced below. We further define the matrix of fields

$\ \Sigma\equiv\frac{1}{2}fe^{i\bm{\pi}/f}.$ (31)

This field transforms as $\Sigma\to L^{\dagger}\Sigma R$ under the $U(2)\times
U(2)$ flavor symmetry of the UV theory and as $\Sigma\to V^{\dagger}\Sigma V$
under the unbroken $U(2)$. The meson effective theory is then described by

$\mathcal{L}=\mathcal{L}_{\mathrm{kin}}+\mathcal{L}_{\mathrm{mass}}+\mathcal{L}_{\mathrm{an}}$
(32)

with

$\displaystyle\mathcal{L}_{\mathrm{kin}}$
$\displaystyle=\mathrm{Tr}\left[D^{\mu}\Sigma^{\dagger}D_{\mu}\Sigma\right]$
(33) $\displaystyle\mathcal{L}_{\mathrm{mass}}$ $\displaystyle=\ c_{m}\Lambda
f\mathrm{Tr}\left[M\Sigma\right]+\mathrm{H.c.}$ (34)
$\displaystyle\mathcal{L}_{\mathrm{an}}$
$\displaystyle=\frac{m_{0}^{2}}{8}\mathrm{Tr}\left[\ln\Sigma-\ln\Sigma^{\dagger}\right]=-\frac{1}{2}m_{0}^{2}\tilde{\pi}_{0}^{2}$
(35)

with

$D_{\mu}\Sigma\equiv\partial_{\mu}\Sigma+igA^{\prime}_{\mu}(Q_{L}\Sigma-\Sigma
Q_{R}).$ (36)

Here $M$ is the quark mass matrix defined in Eqs. 25-26. The parameter
$\Lambda$ in Eq. 34 represents the dark sector’s confinement scale, while
$c_{m}$ is an $\mathcal{O}(1)$, dimensionless matching coefficient. We set
$c_{m}=1$ going forward.777We will always present our results in terms of the
physical meson masses; as such the value of $c_{m}$ will only enter when
converting the $m_{\pi_{i}}$ to the quark masses. The value of $c_{m}$
therefore does not affect the phenomenology, except when we verify that the
Yukawa couplings in Eq. 18 are perturbative for the meson spectra of our
choice. Eq. 35 reflects the fact the isospin singlet $\pi_{0}$ is not a true
Goldstone boson and receives a mass correction due to instanton effects
Witten:1980sp , as does the $\eta^{\prime}$ in the SM. The coefficient
$m_{0}^{2}$ scales as $1/N_{c}$ in the large $N_{c}$ limit. We will assume
that $m_{0}\gg m_{1,2}$ in our analysis. Here we have kept only the leading
terms in the momentum and $1/N_{c}$ expansions, an approximation that
reproduces the meson spectrum in the SM to within $\sim 20\%$ accuracy (see
e.g. Degrande:2009ps ).

We can obtain the mass matrix for the mesons by expanding Eq. 34 and Eq. 35 to
leading non-trivial order in $1/f$ and Eq. 26 to second order in $1/v$. This
results in

$\displaystyle\mathcal{L}\supset$
$\displaystyle-\frac{1}{2}\bar{m}\Lambda\left(\\!\\!\begin{array}[]{c}\tilde{a}\\\
\tilde{\pi}_{1}\\\
\end{array}\\!\\!\right)^{T}\left(\begin{array}[]{cc}\frac{f^{2}}{v^{2}}&\frac{f}{v}\\\
\frac{f}{v}&1\end{array}\\!\\!\right)\left(\\!\\!\begin{array}[]{c}\tilde{a}\\\
\tilde{\pi}_{1}\end{array}\right)-\frac{1}{2}\bar{m}\Lambda\tilde{\pi}_{2}^{2}$
(43)
$\displaystyle-\frac{1}{2}\left(\\!\\!\begin{array}[]{c}\tilde{\pi}_{0}\\\
\tilde{\pi}_{3}\end{array}\\!\\!\right)^{T}\left(\begin{array}[]{cc}m_{0}^{2}+\bar{m}\Lambda&\delta
m\Lambda\\\ \delta
m\Lambda&\bar{m}\Lambda\end{array}\\!\\!\right)\left(\\!\\!\begin{array}[]{c}\tilde{\pi}_{0}\\\
\tilde{\pi}_{3}\end{array}\right)$ (50)

with $\bar{m}\equiv m_{1}+m_{2}$ and $\delta m\equiv m_{2}-m_{1}$. The
isospin-conserving limit is retrieved by sending $\delta m\to 0$ and
$v\to\infty$, in which case we find a light, degenerate isospin triplet,
analogous to the SM pions, and a heavy isospin singlet, analogous to the SM
$\eta$. We will however work with broken isospin, i.e, with $\delta m\neq 0$,
such that the $\tilde{\pi}_{0}$ and $\tilde{\pi}_{3}$ eigenstates mix. We can
define the fields in the mass basis as

$\displaystyle a$
$\displaystyle\equiv\cos\phi\,\tilde{a}+\sin\phi\,\tilde{\pi}_{1}$ (51)
$\displaystyle\pi_{1}$
$\displaystyle\equiv\sin\phi\,\tilde{a}-\cos\phi\,\tilde{\pi}_{1}$ (52)
$\displaystyle\pi_{2}$ $\displaystyle\equiv\tilde{\pi}_{2}$ (53)
$\displaystyle\pi_{3}$
$\displaystyle\equiv\cos\theta\,\tilde{\pi}_{3}+\sin\theta\,\tilde{\pi}_{0}$
(54) $\displaystyle\eta$
$\displaystyle\equiv\sin\theta\,\tilde{\pi}_{3}-\cos\theta\,\tilde{\pi}_{0}$
(55)

with the mixing angles specified by

$\displaystyle\tan\phi$ $\displaystyle=\frac{f}{v}$ (56)
$\displaystyle\tan\theta$ $\displaystyle=\frac{\sqrt{m_{0}^{4}+4\delta
m^{2}\Lambda^{2}}-m_{0}^{2}}{2\delta m\Lambda}$ (57)
$\displaystyle\approx\frac{\Lambda\delta
m}{m_{0}^{2}}+\mathcal{O}\left(\frac{\delta
m^{3}\Lambda^{3}}{m_{0}^{6}}\right)$ (58)

The masses are

$\displaystyle m_{a}^{2}$ $\displaystyle=0$ (59) $\displaystyle
m_{\pi_{1}}^{2}$
$\displaystyle=\bar{m}\Lambda\left(1+\frac{f^{2}}{v^{2}}\right)$ (60)
$\displaystyle m_{\pi_{2}}^{2}$ $\displaystyle=\bar{m}\Lambda$ (61)
$\displaystyle m_{\pi_{3}}^{2}$
$\displaystyle=\bar{m}\Lambda\left(1-\tan\theta\frac{\delta
m}{\bar{m}}\right)$ (62) $\displaystyle m_{\eta}^{2}$
$\displaystyle=m_{0}^{2}+\bar{m}\Lambda\left(1+\tan\theta\frac{\delta
m}{\bar{m}}\right)$ (63)

The massless linear combination $a$ will furnish the longitudinal component of
the $A^{\prime}$ in the unitary gauge, as we show below. The isospin-
preserving limit corresponds to $\sin\theta\to 0$ and $v\to\infty$.

Turning now to the kinetic term in Eq. 33, we can add the kinetic term for the
Higgs field in Eq. 21, expand in $1/f$ and subsequently move to the mass
eigenbasis. Aside from canonical kinetic terms for the $a$, $\pi_{i}$ and
$\eta$ fields, this yields the following terms

$\displaystyle\mathcal{L}\supset$
$\displaystyle\frac{1}{2}m_{A^{\prime}}^{2}A^{\prime\mu}A^{\prime}_{\mu}+m_{A^{\prime}}A^{\prime\mu}\partial_{\mu}a$
(64) $\displaystyle+g\sin\theta
A^{\prime\mu}\left(\pi_{2}\partial_{\mu}\eta-\eta\partial_{\mu}\pi_{2}\right)$
(65) $\displaystyle+g\cos\theta
A^{\prime\mu}\left(\pi_{3}\partial_{\mu}\pi_{2}-\pi_{2}\partial_{\mu}\pi_{3}\right)+\cdots$
(66)

with $m_{A^{\prime}}\equiv g\sqrt{v^{2}+f^{2}}$. The $\cdots$ represent higher
order terms in the $1/v$ and $1/f$ expansions. We see that the $A^{\prime}$
mass term receives a contribution from the chiral condensate, as expected. The
massless linear combination $a$ moreover indeed corresponds to the mode that
is eaten by the $A^{\prime}$ in the unitary gauge.

As long as isospin is broken ($\sin\theta\neq 0$), the $\eta\to
A^{\prime}\pi_{2}$ decay mode is available. This decay is both $C$ and $P$
violating, similar to the $\eta^{\prime}\to\rho\pi_{0}$ mode in the SM. Unlike
the SM, the UV completion we chose in Appendix A.1 maximally breaks both $C$
and $P$, such that the $\eta\to A^{\prime}\pi_{2}$ decay in our dark sector is
allowed. The decay will always proceed to the longitudinal component of the
$A^{\prime}$, since the decays to the transverse polarizations are
incompatible with angular momentum conservation.

###  A.3 Decay rates and branching ratios

We consider three decay modes for the dark sector mesons $\pi_{3}$ and $\eta$
that can give rise to displaced muon pairs:

  1. i)

$\pi_{3},\eta\to A^{\prime}A^{\prime}$ through the $A^{\prime}$ chiral
anomaly.

  2. ii)

$\eta\to\pi_{2}A^{\prime}$ through Eq. 65.

  3. iii)

$\eta\to\pi_{2}A^{\prime\ast}\to\pi_{2}f\bar{f}$ through Eq. 65 as a three-
body decay, with the $A^{\prime}$ off-shell.

For simplicity, we will always choose our mass benchmark points such that
other pion decay modes such as $\pi_{2}\to\pi_{3}A^{\prime}$ are kinematically
closed. The other dark mesons $\pi_{1,2}$ are detector-stable and contribute
to missing energy.

The $\eta\to 3\pi$ decay is isospin-violating, but otherwise proceeds through
the dark sector’s strong interaction. Provided it is kinematically allowed and
$\sin\theta$ is not tiny, we therefore assume that this decay happens
promptly. The leading terms in the chiral Lagrangian, Eqs. 32–35, predict
equal branching ratios to the final states $\pi_{1}\pi_{1}\pi_{3}$ and
$\pi_{2}\pi_{2}\pi_{3}$, up to phase space corrections due to the small mass
difference between the pions. However, these branching ratios will be
corrected by the higher-order operator
$\mathrm{Tr}\left(D_{\mu}\Sigma^{\dagger}D^{\mu}\Sigma(M\Sigma+\mathrm{h.c.})\right)$,
which contributes at the same order in isospin-breaking and derivatives, and
whose coefficient is a priori undetermined. For definiteness and simplicity,
when the $\eta\to 3\pi$ mode is kinematically open, we will assume

$\text{Br}[\eta\to\pi_{1}\pi_{1}\pi_{3}]=\text{Br}[\eta\to\pi_{2}\pi_{2}\pi_{3}]=\frac{1}{2},$
(67)

again up to phase space corrections.

The $\pi_{3},\eta\to A^{\prime}A^{\prime}$ decays are mediated by the
anomalous current

$\displaystyle\partial_{\mu}J^{0\mu}_{5}$
$\displaystyle=-\frac{\alpha^{\prime}}{8\pi}N_{c}\mathrm{Tr}\left[\lambda^{0}Q_{L}^{2}\right]F^{\prime\mu\nu}\tilde{F}^{\prime}_{\mu\nu}$
(68)
$\displaystyle=-\frac{\alpha^{\prime}}{4\pi}N_{c}F^{\prime\mu\nu}\tilde{F}^{\prime}_{\mu\nu}$
(69)

where in the first line the trace runs only over flavor indices, and
$\alpha^{\prime}\equiv g^{2}/4\pi$. We can identify the matrix elements by

$\displaystyle\langle 0|J^{0\mu}_{5}|\pi_{3}\rangle$
$\displaystyle=-i\sin\theta f_{\pi}p^{\mu}e^{-i\mathbf{x}\cdot\mathbf{p}}$
(70) $\displaystyle\langle 0|J^{0\mu}_{5}|\eta\rangle$
$\displaystyle=-i\cos\theta f_{\eta}p^{\mu}e^{-i\mathbf{x}\cdot\mathbf{p}}$
(71)

with $f_{\pi}$ and $f_{\eta}$ the $\pi_{3}$ and $\eta$ decay constants.
Consequently, we find

$\displaystyle\Gamma_{\pi_{3}\to A^{\prime}A^{\prime}}$
$\displaystyle=\frac{\alpha^{\prime
2}}{64\pi^{3}}N_{c}^{2}\sin^{2}\theta\frac{m_{\pi_{3}}^{3}}{f^{2}}$ (72)
$\displaystyle\Gamma_{\eta\to A^{\prime}A^{\prime}}$
$\displaystyle=\frac{\alpha^{\prime
2}}{64\pi^{3}}N_{c}^{2}\cos^{2}\theta\frac{m_{\eta}^{3}}{f^{2}}.$ (73)

where for definiteness we set $f_{\pi}\approx f_{\eta}\approx f$. For the
$\pi_{3}$ meson, this is the only decay mode, which proceeds with a proper
lifetime of

$\displaystyle c\tau\sim$ $\displaystyle\;1.7\times
10^{-7}\;\mathrm{cm}\times\left(\frac{0.01}{\alpha^{\prime}}\right)^{2}\times\left(\frac{0.1}{\sin\theta}\right)^{2}$
$\displaystyle\times\left(\frac{1\,\mathrm{GeV}}{m_{\pi_{3}}}\right)^{3}\times\left(\frac{f}{0.2\,\mathrm{GeV}}\right)^{2}.$
(74)

In other words, by dialing the gauge coupling $\alpha^{\prime}$, one can
choose the $\pi_{3}$ to be either prompt or very long-lived on collider-
relevant timescales.

The $\eta$ can a priori also decay through $\eta\to\pi_{2}A^{\prime}$,
provided that $m_{\pi_{2}}+m_{A^{\prime}}<m_{\eta}$. The partial width of this
channel is

$\Gamma_{\eta\to\pi_{2}A^{\prime}}=\frac{\alpha^{\prime}\sin^{2}\theta\,m_{\eta}^{3}}{4m_{A^{\prime}}^{2}}\left((1-x_{\pi}^{2}-x_{A}^{2})^{2}-4x_{\pi}^{2}x_{A}^{2}\right)^{3/2}$
(75)

with $x_{\pi}\equiv m_{\pi_{2}}/m_{\eta}$ and $x_{A}\equiv
m_{A^{\prime}}/m_{\eta}$. The decay width in Eq. 75 appears to diverge in the
$m_{A^{\prime}}\to 0$ limit, which is expected from the goldstone equivalence
theorem. Provided that there is no extreme phase space suppression
($x_{\pi}\ll 1$ and $x_{A}\ll 1$) and that the $\eta\to A^{\prime}A^{\prime}$
channel is either subleading or kinematically closed, the $\eta$ lifetime is

$\displaystyle c\tau\sim$ $\displaystyle\;3\times
10^{-11}\;\mathrm{cm}\times\left(\frac{0.01}{\alpha^{\prime}}\right)\times\left(\frac{0.1}{\sin\theta}\right)^{2}$
$\displaystyle\times\left(\frac{2\,\mathrm{GeV}}{m_{\eta}}\right)^{3}\times\left(\frac{m_{A^{\prime}}}{0.5\,\mathrm{GeV}}\right)^{2}.$
(76)

Also here we see that the decay can be made either prompt or displaced,
depending on the values adopted for $\alpha^{\prime}$ and $\sin\theta$.

Figure 11: Proper decay length and dimuon branching ratio for the three body
decay in Eq. 79, for a few benchmark model points as indicated in the figure.
For $3m_{\pi_{2}}<m_{\eta}$, the $\eta\to 3\pi$ decays will always dominate.

Finally, if the $\eta\to\pi_{2}A^{\prime}$ decay is kinematically closed, the
three-body decay $\eta\to\pi_{2}f\bar{f}$ will dominate. The partial width to
muons for this process is

$\displaystyle\frac{d\Gamma_{\eta\to\pi_{2}\mu\mu}}{dx}=$
$\displaystyle\frac{\epsilon^{2}\alpha^{\prime}\alpha\sin^{2}\theta}{12\pi}m_{\eta}\frac{x+2x_{\mu}^{2}}{(x-x_{A}^{2})^{2}}\sqrt{x-4x_{\mu}^{2}}$
$\displaystyle\times\left(\frac{(x-1)^{2}-2(x+1)x_{\pi}^{2}+x_{\pi}^{4}}{x}\right)^{3/2}$
(77)

with $\alpha$ the SM fine structure constant, $x_{\mu}\equiv m_{\mu}/m_{\eta}$
and $x=q^{2}/m^{2}_{\eta}$, where $q^{2}$ is the invariant mass formed by the
muon pair. The partial width and total width are obtained by evaluating

$\displaystyle\Gamma_{\eta\to\pi_{2}\mu\mu}$
$\displaystyle=\int_{4x_{\mu}^{2}}^{(1-x_{\pi})^{2}}\\!\\!\\!dx\frac{d\Gamma_{\eta\to\pi_{2}\mu\mu}}{dx},$
(78) $\displaystyle\Gamma_{\eta\to\pi_{2}f\bar{f}}$
$\displaystyle=\int_{4x_{\mu}^{2}}^{(1-x_{\pi})^{2}}\\!\\!\\!dx\frac{d\Gamma_{\eta\to\pi_{2}\mu\mu}}{dx}R(xm^{2}_{\eta})$
(79)

with $R(q^{2})$ the R-ratio, as extracted from experiment
ParticleDataGroup:2022pth . Fig. 11 shows the proper decay length and dimuon
branching ratio for a set of benchmark points, which were chosen to roughly
minimize $c\tau$ without resorting to unphysical choices for underlying
couplings. The photon-dark photon mixing parameter $\epsilon$ was chosen to
satisfy the existing constraints from direct searches for dark photons
Gori:2022vri . We see that this decay tends to be somewhat displaced, with
$c\tau\gtrsim 1$ cm. The branching ratio to muons hovers around 25% and is
relatively insensitive to $m_{\eta}$ and $m_{\pi_{2}}$. For completeness, we
include all underlying parameters for an example point in Tab. 2.

meson sector | $m_{\pi_{1}}$ | 0.403 GeV
---|---|---
$m_{\pi_{2}}$ | 0.4 GeV
$m_{\pi_{3}}$ | 0.396 GeV
$m_{\eta}$ | 1 GeV
$m_{A^{\prime}}$ | 1 GeV
$\sin\theta$ | 0.1
$f$ | 0.080 GeV
$g$ | 1
$\epsilon$ | $5\times 10^{-4}$
quark sector | $m_{1}$ | 0.072 GeV
$m_{2}$ | 0.088 GeV
$\Lambda$ | 1 GeV
$m_{0}$ | 1 GeV
$v$ | 0.997 GeV
$y_{11}$ | 0.012
$y_{12}$ | 0.11
| $c\tau$ | 2.8 cm
| $\mathrm{Br}[\eta\to\pi_{2}\mu^{+}\mu^{-}]$ | 0.37
Table 2: Example model point for which a relatively short-lived
$\eta\to\pi_{2}\mu^{+}\mu^{-}$ decay is realized.

###  A.4 Pythia 8 implementation

For Monte Carlo implementation of our model, we use the Pythia 8 hidden valley
module Carloni:2010tw ; Carloni:2011kk . Our model relies on a recent update
Albouy:2022cin , specifically the new `separateFlav = on` flag, which allows
us to set the masses of the pions individually. While our work was in the
final stages of preparation, Pythia version 8.309 was released, which contains
further updates to the hidden valley module. We did not benchmark versions
8.308 and 8.309 against each other, but we did update our python tool to make
it compatible with version 8.309 by adding the new required flag
`HiddenValley:setLambda = on`. For version 8.308 this line should be commented
out in the Pythia configuration cards. In the remainder of this appendix we
specify our full settings and mention some approximations and caveats.

Firstly, we note that the dark $\pi^{\pm}$ with PDG codes $\pm$`4900211` are
each other’s anti-particles and therefore degenerate in mass. In our case, the
mass eigenstates are the $\pi_{1}$ and $\pi_{2}$, which need not be
degenerate, as the dark $U(1)$ is broken spontaneously. We therefore always
work in the regime where $m_{\pi_{1}}\approx m_{\pi_{2}}$ and simply identify
those states with the $\pm$`4900211` states in the Pythia module.

There is substantial uncertainty in the hadronization probabilities of the
dark sector vector mesons, as well as its $\eta$-meson. For the vector mesons,
we choose `HiddenValley:probVector=0.75`, such that the hadronization
probability matches the naive expectation from the counting of the number of
degrees of freedom. For the $\eta$ meson, we fix `HiddenValley:probKeepEta1 =
1.0`, which corresponds to assuming that the suppression from taking
$m_{\eta}>m_{\pi}$ in the Lund string model Andersson:1983ia ;
Andersson:1983jt  is sufficient to model the $\eta$ hadronization probability.

We set the masses of the vector mesons and the $\eta$ meson to be equal to the
confinement scale $\Lambda$ and choose the constituent quark masses (`4900101`
and `4900102`) to be $\Lambda+m_{q}$, in line with the recommendations in
Albouy:2022cin . Self-consistent Pythia cards for this model can be generated
with our publicly-available python code code .

##  Appendix B Signal efficiency plots

We present signal efficiencies for the CMS scouting analysis as a function of
the lifetime of the long-lived particle, for four of the example benchmark
points discussed in the main text.

Figure 12: Signal efficiencies for the CMS analysis, for select benchmark
points. Solid (dashed) lines are signal are without (with) imposing the
isolation condition.