aainstitutetext: Department of Mathematical Sciences, University of Liverpool,
Liverpool L69 3BX, United Kingdombbinstitutetext: Università degli Studi di
Milano-Bicocca, Department of Physics “Giuseppe Occhialini”, & INFN, Sezione
di Milano-Bicocca, Piazza della Scienza 3, Milano 20126, Italy

# Systematic analysis of search strategies for $L_{\mu}-L_{\tau}$ gauge bosons
at Belle II

Connor Brown a, b , Juri Fiaschi a , Oliver Fischer a , Thomas Teubner
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Extensions of the Standard Model with masses at or below the GeV scale are
motivated by searches for dark matter and precision measurements in the quark
and lepton flavour sectors, including that of the muon anomalous magnetic
moment. An excellent experimental environment to test such light new physics
is given by the Belle II experiment, which foresees to take up to 50 ab-1 of
data. Here we consider a model with an additional gauged
U(1)${}_{L_{\mu}-L_{\tau}}$ symmetry that introduces a neutral gauge boson, a
Dark Photon, with possibly large couplings to muon- and tau-flavored leptons,
including neutrinos. Dark Photon mixing with the Standard Model photon is loop
induced, allowing it to couple to electrically charged fermions other than
muons and taus. We systematically investigate the possible search strategies
for Dark Photons with four fermion final states. We identified search channels
with muons as the most promising ones, and we analyse the kinematic
distributions to obtain cuts that optimise the sensitivity of Belle II
searches for the Dark Photon. Summarising the sensitivities from the most
promising search channels we provide a comprehensive overview of future
searches at Belle II.

###### Keywords:

New Light Particles, Specific BSM Phenomenology, Dark Matter at Colliders

††preprint: LTH-1372††arxiv: 2406.10204

###### Contents

  1. 1 Introduction
  2. 2 Dark Photon Model and Decays
  3. 3 Dark photon signatures at Belle II
  4. 4 Analysis setup
  5. 5 Dark photon search channels
    1. 5.1 Monophoton
    2. 5.2 Two muons plus missing energy
    3. 5.3 Four muons prompt
    4. 5.4 Displaced vertices
    5. 5.5 Signatures with tau leptons
  6. 6 Results
  7. 7 Conclusions

##  1 Introduction

The Standard Model (SM) of particle physics is a remarkably successful theory,
thoroughly tested by many experiments in the past decades. The theory appears
incomplete as it is unable to explain numerous phenomena - such as the
existence of dark matter, the baryon asymmetry of the universe, and neutrino
masses - while the increasing precision of modern measurements has revealed
significant tensions - as the recent measurements of $g-2$, the muon anomalous
magnetic moment, at Fermilab Muong-2:2021ojo ; Muong-2:2023cdq ;
Muong-2:2024hpx , or the measurement of the $W$-boson mass by the CDF
collaboration at the Tevatron CDF:2022hxs . New physics below the electro-weak
(EW) scale, coupling very weakly to the SM, could solve tensions between SM
predictions and experimental observations, including anomalies in the flavour
sector. Importantly, new light states, which are singlet under the SM gauge
group, can play the role of portals to hidden or dark sectors, hence
connecting ordinary and dark matter.

A minimal extension of the SM, connecting the visible to a dark sector, can be
constructed by adding an U(1)X Abelian gauge group which mixes with the SM
U(1)Y hypercharge field Bauer:2022nwt . The $L_{\mu}-L_{\tau}$ gauge symmetry
is a popular choice for the new Abelian group, as it naturally predicts a
small kinetic mixing with the SM neutral gauge bosons, while keeping the
theory renormalisable and anomaly free Essig:2009nc . Such lepton non-
universal interactions could also - at least partially - account for the
discrepancy between theory Aoyama:2020ynm ; Colangelo:2022jxc  and
measurements Muong-2:2021ojo ; Muong-2:2023cdq  of the muon anomaly.
Currently, the status of the $g-2$ discrepancy is unclear, due to significant
tensions between data used in Aoyama:2020ynm  as input in the data-driven
evaluation of the hadronic vacuum polarisation contributions to $g-2$ and the
recent two-pion data from CMD-3 CMD-3:2023alj ; CMD-3:2023rfe , and due to
tensions with the lattice predictions, in particular from the BMW
collaboration Borsanyi:2020mff . However, with a lot of efforts on new and
improved low-energy hadronic cross section measurements and on further lattice
determinations, the comparison between the $g-2$ measurement and its SM
prediction is expected to be consolidated and could still indicate a
significant discrepancy, or, in turn, strongly constrain physics beyond the
SM.

Neutral vector bosons with masses below the GeV scale, generally called Dark
Photons (DPs), can be probed looking at final states involving photons and
leptons Curtin:2014cca . Experiments at colliders provide the strongest
constraints for DP masses in the MeV to GeV range Ilten:2018crw . The LHC main
experiments have performed dedicated DP searches, see e.g. LHCb LHCb:2017trq ;
LHCb:2019vmc , ATLAS ATLAS:2019tkk ; ATLAS:2022xlo , CMS CMS:2023hwl ;
CMS:2020krr ; CMS:2019ajt ; CMS:2018yxg . However, due to the leptophilic
nature of the $L_{\mu}-L_{\tau}$ symmetry, lepton colliders present unique
opportunities to test this model. Because of the vanishing DP tree-level
couplings to coloured particles, purely leptonic final states with high
multiplicity are the most promising channels for investigating this model. In
order to probe very low couplings, high intensity beams are needed to produce
DPs at a detectable rate. High luminosity experiments like BaBar BaBar:2014zli
; BaBar:2016sci , KLOE KLOE-2:2014qxg , Belle Belle:2021feg  and Belle II
Ban:2020uii ; Belle-II:2023ydz  at the meson factories DA$\Phi$NE, PEP-II and
KEK, are able to collect very large data samples, hence providing the highest
sensitivity for $L_{\mu}-L_{\tau}$ DPs.

The DP associated with the U(1)${}_{L_{\mu}-L_{\tau}}$ symmetry group can also
act as a portal to a dark sector through its direct couplings to new invisible
light states, such as dark chiral fermions or sterile/Majorana right-handed
neutrinos, which can (at least partially) account for the observed dark matter
relic abundance. DP portals to dark sectors have been probed through
signatures with missing energy at lepton colliders (BaBar BaBar:2017tiz ) and
beam dump experiments (E137 Batell:2014mga , NA64 NA64:2024klw ), or through
their interactions with neutrinos (in indirect detection at IceCube
Kamada:2015era ; Hooper:2023fqn , in neutrino trident production
Altmannshofer:2014pba , at LSND deNiverville:2011it , CHARM-II CHARM-
II:1990dvf , CCFR CCFR:1991lpl , Borexino Gninenko:2020xys ; Bellini:2011rx ,
see also Amaral:2020tga ) and dark matter (XENON Essig:2012yx ; Essig:2017kqs
) in scintillator and direct detection experiments Figueroa:2024tmn . Dark
matter coupled to the DP carries complementary constraints Aboubrahim:2022qln
from astrophysics and cosmology Foldenauer:2024cdp ; Giovanetti:2021izc ;
Escudero:2019gzq ; Foldenauer:2019dai ; Kamada:2018zxi ; Bauer:2018onh ;
Feng:2016ijc  as well as indirect detection (FERMI-LAT Fermi-LAT:2015att )
surveys.

Light DPs weakly coupled to the SM naturally feature macroscopic decay
lengths, which lead to detectable displaced decay vertices. This is the case
if the DP has a narrow width, which still holds in models where the DP acts as
a portal to an extended dark sector, as long as the dark particles are heavy
enough and any additional decay channels for the DP are kinematically
forbidden. Signatures with displaced vertices represent a smoking gun for the
discovery of long-lived particles. A very active ongoing programme at the LHC
focuses on improving the sensitivity on long-lived particles LHCb:2020ysn ;
Lee:2018pag ; Alimena:2019zri ; Bernreuther:2020xus , also exploiting the CERN
Super Proton Synchrotron beam dump (NA64++ Gninenko:2014pea ; Gninenko:2018tlp
; Gninenko:2018num , NA48/2 NA482:2015wmo ), the muon beam (MUonE
Galon:2022xcl ), or novel dedicated forward detectors (FASER Feng:2017uoz ;
Feng:2018pew ; FASER:2018eoc , MATHUSLA Curtin:2017izq ; Curtin:2018mvb ,
CODEX-b Gligorov:2017nwh ).

Similarly, dedicated searches for displaced signatures at lepton colliders
Bossi:2013lxa  have been pursued by the KLOE KLOE-2:2018kqf , BaBar
BaBar:2015jvu ; BaBar:2016sci , and Belle II Belle-II:2010dht ; Duerr:2019dmv
; Duerr:2020muu ; Bernreuther:2022jlj ; Ferber:2022ewf ; Belle-II:2023ueh
experiments. In the future, the light long-lived DP scenario can be best
tested in the clean environment of $e^{+}e^{-}$ collisions with the Belle II
detector. Thanks to its large acceptance and with the prospect of high
luminosity in the forthcoming years, we can expect a statistically significant
sample to study long-lived DP signatures. The high energy asymmetric
$e^{+}e^{-}$ collisions generate a boost of the centre-of-mass frame,
enhancing the lifetime of the final state system and the probability to
observe events with a displaced decay vertex. The recast of the experimental
displaced vertices search in the DP framework therefore provides a powerful
probe to rule out yet unexplored regions in the parameter space for such BSM
scenarios.

In this paper we consider the sensitivity of the Belle II target luminosity on
the minimal BSM construction featuring a DP from an additional
U(1)${}_{L_{\mu}-L_{\tau}}$ gauge group, connected to the SM through kinetic
mixing generated at loop-level (for studies on less constrained model see e.g.
Bandyopadhyay:2022klg ; Bandyopadhyay:2023lvo ). In Sect. 2 we introduce the
DP model and its phenomenological key aspects, and discuss existing
constraints. In Sect. 3 we assess the potential of DP searches at Belle II,
with detailed analyses of the relevant signatures involving the new light
neutral boson, including displaced vertex searches for long-lived DPs. In
Sect. 6 we project the sensitivity of the ultimate Belle II target luminosity
on the parameter space of the model, and compare it with the existing
constraints. Finally, in Sect. 7 we draw our conclusions.

##  2 Dark Photon Model and Decays

We introduce a minimal BSM construction extending the SM gauge group with the
local abelian group U(1)${}_{L_{\mu}-L_{\tau}}$, with $L_{\alpha}$ being the
lepton number of flavour $\alpha$ Heeck:2011wj . The corresponding gauge boson
$\hat{X}^{\mu}$ interacts with the SM lepton current
$j_{\mu}^{L_{\mu}-L_{\tau}}$, i.e. with the SM muon and tau charged and
neutral leptons. The effective Lagrangian after EW symmetry breaking contains
the terms

${\cal L}\ \supset\
-\frac{1}{4}\hat{F}_{\mu\nu}\hat{F}^{\mu\nu}-\frac{1}{4}\hat{X}_{\mu\nu}\hat{X}^{\mu\nu}-\frac{\epsilon}{2}\hat{F}_{\mu\nu}\hat{X}^{\mu\nu}-\frac{1}{2}m_{X}\hat{X}^{\mu}\hat{X}_{\mu}\,,$
(1)

where $\hat{F}_{\mu\nu}$ and $\hat{X}_{\mu\nu}$ are the field strength tensors
corresponding to the SM photon field $\hat{A}_{\mu}$ and the dark photon
$\hat{X}_{\mu}$, respectively.

Figure 1: Loop induced mixing of the dark photon $\hat{X}$ with the SM photon
$\hat{A}$.

The U(1) kinetic mixing term in the Lagrangian effectively arises, at loop
level, from the interaction between the $\hat{X}$ boson and the
$j_{\mu}^{L_{\mu}-L_{\tau}}$ current, as illustrated in Fig. 1. While in the
general case the kinetic mixing parameter $\epsilon$ can be treated as a free
parameter of the Lagrangian, in this specific construction it is given by
Bauer:2018onh

$\epsilon=\frac{eg_{X}}{2\pi^{2}}f(m_{\mu},m_{\tau},q)\,,$ (2)

with the loop factor

$f(m_{\mu},m_{\tau},q)\ =\
\int\limits_{0}^{1}dx\,x(1-x)\ln\left.\frac{m_{\mu}^{2}-q^{2}x(1-x)}{m_{\tau}^{2}-q^{2}x(1-x)}\right.\,.$
(3)

The explicit mass $m_{X}$ of the DP may arise e.g. from higher scale
interactions with an extended Higgs sector. In the following we will assume
$m_{X}$ in the few MeV to few GeV range. We do not consider additional
explicit mass mixing with the much heavier $Z$-boson field, which is strongly
constrained by electroweak precision measurements Heeck:2011wj ; Babu:1997st .
Non-diagonal elements in the mass matrix inherited from the small kinetic
mixing $\epsilon$ can also be safely neglected. With this choice, the mass
eigenstates ($\hat{A},\hat{X}$) are diagonal in the interaction basis ($A,X$).

Theoretical consistency requires anomaly cancellation. Even after adding
right-handed neutrinos fields, which allow for the observed neutrino
oscillations, this minimal construction is automatically anomaly free. We give
Dirac masses of ${\cal O}(1)$ eV to muon and tau neutrinos, but do not
introduce Majorana masses, such that the mass eigenstates remain light.

The DP couples directly to muon and tau lepton flavours with coupling strength
$\alpha_{X}=g_{X}^{2}/(4\pi)$, and to all electrically charged particles with
effective coupling strength $\epsilon^{2}\alpha_{\rm em}$. It decays into all
fermion anti-fermion pairs with fermion masses below $m_{X}/2$, with a partial
decay width

$\Gamma_{i}\ =\ \frac{1}{3}m_{X}(\delta_{i,(\mu,\tau)}\alpha_{\rm
X}+\epsilon^{2}\alpha_{\rm
em}|q_{i}|N_{c})\left(1+\frac{2m_{i}^{2}}{m_{X}^{2}}\right)\sqrt{1-\frac{4m_{i}^{2}}{m_{X}^{2}}}\,\theta(m_{X}-2m_{i})\,,$
(4)

where $m_{i}$ and $q_{i}$ denote the fermion’s mass and charge respectively,
and $N_{c}$ is the number of colours (1 for leptons and 3 for quarks). The
$\delta_{i,(\mu,\tau)}$ indicates that the tree-level contribution is present
only for decays into muon and tau leptons. The DP decay into neutrino states
(both left- and right-handed), neglecting their mass, also occurs at tree-
level via its coupling to $\nu_{\mu}$ and $\nu_{\tau}$ flavours:

$\Gamma_{\nu_{i}}\ =\
\frac{1}{3}m_{X}\delta_{\nu_{i},(\nu_{\mu},\nu_{\tau})}\alpha_{\rm X}\,.$ (5)

The left panel of Fig. 2 shows the DP branching ratios as a function of its
mass, clearly demonstrating the strong suppression of decay channels that
proceed via kinetic mixing compared to those into fermions charged under
$L_{\mu}-L_{\tau}$. The DP hadronic branching ratio is obtained by
appropriately rescaling the experimental data for the ratio
$R=\frac{\sigma(e^{+}e^{-}\rightarrow\rm{hadr.})}{\sigma(e^{+}e^{-}\rightarrow\mu^{+}\mu^{-})}$
ParticleDataGroup:2022pth . Its contribution is small and irrelevant for our
analysis, as the DP width is dominated by decays into muon and tau flavoured
leptons above their mass threshold, and by decays into electrons below 500
MeV.

Figure 2: Branching ratio of DP as a function of its mass (left), and DP mean
lifetime and decay length in the DP rest frame multiplied by $g_{X}^{2}$ as a
function of its mass (right).

The mean lifetime $\tau_{X}$ and decay length $\lambda_{X}$ of the DP in its
rest frame are calculated as:

$\tau_{X}=\frac{\lambda_{X}}{c}=\frac{\hbar}{\Gamma_{X}},\qquad\text{with}\qquad\Gamma_{X}=\sum\limits_{s}\Gamma_{i}$
(6)

and are shown in the right panel of Fig. 2; in the plot we show the quantities
$g_{X}^{2}\tau_{X}$ and $g_{X}^{2}\lambda_{X}$ which are independent of
$g_{X}$.

##  3 Dark photon signatures at Belle II

Dark photons with masses between MeV and GeV are testable in low energy
$e^{+}e^{-}$ collisions. In the following, we focus on associated DP
production at the Belle II experiment via the processes $e^{+}e^{-}\to XY$,
where $Y=\gamma,f^{+}f^{-}$ and $f=e,\mu,\tau$ or hadrons. Belle II is a
general purpose detector at the $e^{-}e^{+}$ collider KEKB which runs at the
centre-of-mass energy $\sqrt{s}$ = 10.58 GeV with an anti-symmetric beam
energy setup (E($e^{-}$) = 7 GeV, E($e^{+}$) = 4 GeV). Operating mostly at the
$Y(4S)$ resonance, its primary focus is to study $B$ meson properties Belle-
II:2018jsg . The Belle II detector covers an asymmetric phase space, with the
polar angle spanning from $\theta_{l}=17^{\circ}$ to $\theta_{u}=150^{\circ}$,
with the positive $z$-axis in the direction of the $e^{-}$ beam. The Central
Drift Chamber (CDC) has an inner tracker radius $R_{T}$ = 0.168 m (with
$R_{T}^{2}=R_{x}^{2}+R_{y}^{2}$) and barrel length $R_{z}$ = 1.15 m. Its
current data set corresponds to an integrated luminosity of $L$ = 427 fb-1,
while the ultimate target luminosity is $L$ = 50 ab-1 Bertacchi:2023jzv .

In the following we discuss the relevant experimental signatures in the Belle
II environment which are sensitive to DPs within the model introduced in Sect.
2. The final states considered comprise combinations of photons, leptons with
prompt or displaced decay vertices, and of missing transverse energy/momentum
originating from invisible DPs decays. Because of the DP’s suppressed coupling
to electrons, cf. Eq. (2), and the large SM background, we do not consider
resonant processes with two body final states, $e^{+}e^{-}\to X\to
f^{+}f^{-}$. Nevertheless, DPs can be produced through their unsuppressed
couplings to muon and tau leptons, via the processes
$e^{+}e^{-}\to\ell^{+}\ell^{-}X$ with $\ell=\mu,\tau$, i.e. when radiated off
a heavy lepton final state. They can also be produced through kinetic mixing
in association with a SM photon, via the one-loop induced process
$e^{+}e^{-}\to\gamma X$, with amplitude proportional to $\epsilon^{2}$. The
corresponding Feynman diagrams are shown in Fig. 3.

Figure 3: Feynman diagrams depicting the considered DP production processes at
$e^{+}e^{-}$ colliders. Left: Radiation off a muon or tau lepton. Right: Mono
photon process.

Because of their very narrow width, we will exclusively consider DP on-shell
production and set the momentum transfer to $q^{2}=m_{X}^{2}$ when evaluating
$\epsilon$ in Eq. (2). For example, in the regime where $m_{X}\ll m_{\mu}$, we
obtain $\epsilon\simeq 0.015\,g_{X}$, whereas $m_{X}=10$ GeV yields
$\epsilon\simeq 6\times 10^{-4}g_{X}$. In Fig. 4 the cross sections, divided
by the coupling constant, $g_{X}^{2}$, for the three main production
mechanisms, are shown as function of the DP mass in the Belle II detector
acceptance region, including also a lower cut on the photon energy of 1 GeV.

Figure 4: Dark photon production cross sections at Belle II (with
$\sqrt{s}=10.58$ GeV), divided by the U(1)${}_{L_{\mu}-L_{\tau}}$ coupling
constant, $g_{X}^{2}$, for the production mechanisms in $e^{+}e^{-}$
collisions as described in the text. The Belle II geometric detector
acceptance has been imposed on the leptons and the photon, as well as a
minimum energy cut of 1 Gev for the photon.

The relevant processes for DP searches at Belle II are obtained combining the
decay channels of the DP $X$, depicted in the left panel of Fig. 2, with the
possible associated particles $Y$. Among the possible combinations, processes
where both $Y$ and the DP decay involve a combination of $\mu,\tau,\nu$
fermions are most relevant, due to their large production cross section and
${\cal O}(1)$ branching ratio. These signatures are expected to give the
largest sensitivity to DP searches and could become “discovery channels”. In
this group, we also include the so-called “monophoton” channel, where
$Y=\gamma$ and the DP decays invisibly, see e.g. Kaneta:2016uyt ;
Araki:2017wyg  for earlier works. This signature is somehow exceptional
because, as we will discuss below, excellent background suppression can be
achieved, and background free analyses are justified. For completeness we also
consider the associated production of a DP with $Y=\nu\bar{\nu}$. This process
occurs via weak interactions and is suppressed by a factor
$r_{W}=s^{2}G_{f}^{2}\simeq 10^{-6}$ compared to QED processes with Fermi’s
constant, $G_{f}=1.166\times 10^{-5}$ Gev2 ParticleDataGroup:2022pth . The
cross sections for processes where $Y\neq\mu,\tau,\nu$ are loop-suppressed by
a factor $r_{\epsilon}=\epsilon^{2}/g_{X}^{2}<10^{-4}$ compared to the
discovery channels, and are therefore expected to provide lesser sensitivity.
Final states stemming from DP decays into electrons or hadrons have branching
ratios loop-suppressed by the same factor $r_{\epsilon}$, therefore their
cross sections are much smaller compared to the discovery channels. The
discussed final states are summarised in Tab. 1, with their corresponding
suppression factors with respect to the SM QED background.

$Y$ $X\to$ | $\nu$ |  $e$/hadrons | $\mu/\tau$
---|---|---|---
$\nu$ | invisible | $r_{W}r_{\epsilon}$ | $r_{W}$
$e/\gamma/$hadrons | $r_{\epsilon}$ | $r_{\epsilon}^{2}$ | $r_{\epsilon}$
$\mu/\tau$ | 1 | $r_{\epsilon}$ | 1
Table 1: Four fermion final states from the process $e^{+}e^{-}\to XY$ as
discussed in the text. Entries denote suppression factors due to the loop-
induced coupling $\epsilon$, or weak interactions. Leading processes are
labelled “1”.

Existing Belle II searches for light DPs mainly target final states with
missing energy from the invisible decays of the DP. Analyses of the process
$e^{+}e^{-}\to\mu^{+}\mu^{-}X$ with $X\to$ invisibles have been performed
using a dataset corresponding to an integrated luminosity of 276 pb-1 Belle-
II:2019qfb  and, more recently, of 79.7 fb-1 Belle-II:2022yaw , achieving
exclusion limits between $g_{X}\simeq 3\times 10^{-3}$ for DPs with $m_{X}\ll
1$ GeV and $g_{X}\simeq 1$ for DPs as heavy as 8 Gev. For the ultimate Belle
II integrated luminosity goal of 50 ab-1, an exclusion limit on the DP
coupling down to $g_{X}\simeq{\cal O}(10^{-4})$ Belle-II:2019qfb  is projected
for this channel. Analyses targeting the specific $L_{\mu}-L_{\tau}$ gauge
symmetry, exploiting alternative signatures, such as the monophoton channel
with an assumed integrated luminosity of 20 fb-1 DePietro:2018sgg , and final
states with DPs decaying into muon or tau pairs with an assumed integrated
luminosity of 100 fb-1 Laurenza:2022rjm , have demonstrated a promising
sensitivity, comparable with the $2\mu+\not{E}$ final state.

Previous studies from the BaBar collaboration also reported searches for new
light physics in final states with photons and leptons. They probed a general
model with kinetic mixing between the SM photon and DPs, using the processes
$e^{+}e^{-}\to\gamma\ell^{+}\ell^{-}$ with $\ell=e,\,\mu$, based on their
complete dataset corresponding to a total luminosity of 514 fb-1. Upper limits
at 90% confidence level (CL) on $\epsilon$ were drawn between $10^{-4}$ and
$10^{-3}$ for DP masses between 0.02 GeV and 10.2 GeV BaBar:2014zli . In the
same mass range, a dedicated analysis on SM extensions with extra U(1)′
$L_{\mu}-L_{\tau}$ gauge symmetry, using the four muon final state
Godang:2016gna , yielded the current, most stringent upper bound,
$g_{X}=7\times 10^{-4}$.

##  4 Analysis setup

We have implemented the minimal DP model introduced in Sect. 2 into Feynrules
Christensen:2008py  and generated the model file for the Monte Carlo event
generator WHIZARD Kilian:2007gr ; Moretti:2001zz ; Christensen:2010wz . We
then used WHIZARD to generate samples containing between $10^{5}$ and $10^{7}$
events, and to calculate cross sections for leading order signal and
background processes. The complete information on the kinematics of each event
is recorded in the four-vectors of the final state particles which are then
used to compute the relevant observables for each of the analyses described in
the following.

The finite resolution of the Belle II detector for particles’ momenta and
energies is modelled by smearing the four-momenta following a normal
distribution with standard deviations of 0.5% for muons and electrons, and of
5% for photon energies Ferber:2022rsf . The energy resolution for charged
particles is taken to be 5 MeV. Particle identification is highly important in
order to study different event topologies at Belle II Kuhr:2018lps . We
consider the reconstruction efficiencies of photons as 100% for $E_{A}={\cal
O}(1)$ GeV, of muons $\sim 90\%$ and of electrons $\sim 95\%$ Belle:2021feg ;
Belle-II:2021rni . Contributions from mis-reconstructed particles are
neglected, since for muons they are as low as 0.9%, while the pion fake rate
is 1.4% Abashian:2002bd . Complicated backgrounds such as decay chains of $D$
mesons can also lead to the mis-identification of pions as muons or electrons,
with a likelihood as high as 10% for pion momenta $P_{\pi}\leq 3$ GeV. This
contribution is not included in our analyses, since we do not include QCD
processes which are suppressed in this model.

The SM QED processes representing the background for DP searches and their
fiducial cross sections are listed in Tab. 2.

| final state | $\sigma_{b}$
---|---|---
di-photon | $\gamma\gamma$ | 0.167 nb
radiative Bhabha scattering | $\gamma e^{+}e^{-}$ | 63 nb
radiative neutrinos | $\gamma\bar{\nu}\nu$ | 2.41 fb
di-tau | $\tau^{+}\tau^{-}$ | 0.8 nb
radiative di-muon | $\mu^{+}\mu^{-}\gamma$ | 0.32 nb
di-electron di-muon | $e^{+}e^{-}\mu^{+}\mu^{-}$ | 14.9 nb
four muon | $2\mu^{+}2\mu^{-}$ | 155 fb
di-tau di-muon | $\tau^{+}\tau^{-}\mu^{+}\mu^{-}$ | 130 fb
Table 2: SM QED background processes for DP searches. Cross sections are
computed for the fiducial volume of the Belle II detector, including the
angular acceptance cut on the visible particles in the final states, and a
minimum energy cut of 1 GeV for the photons.

Most search channels are affected by enormous SM background rates which
dominate the signatures for DP signals. In the following analyses we find
optimised kinematic cuts to enhance the signal-over-background ratio. Specific
observables are considered to discriminate the DP signals from the background.
These are used to extract the experimental sensitivity on the model parameters
at different luminosity stages. When background rates are much larger than the
signal, $N_{b}\gg N_{s}$, the optimised combination of kinematic cuts for each
phase space region is identified by minimising the critical number of signal
events for which the BSM hypothesis is excluded at a certain Confidence Level
(CL):

$N_{s}^{\rm crit}=\frac{n\,\left(\sum_{i}\sigma_{i}\epsilon_{i}\times
L\right)^{\frac{1}{2}}}{\epsilon_{s}}\,,$ (7)

where the sum is over all contributing background processes with their
respective cross sections $\sigma_{i}$ and selection cut efficiencies
$\epsilon_{i}$. We choose $n=2$ for a 95% one-sided CL, and $\epsilon_{s}$ is
the overall selection cut efficiency for the signal. When the number of
background events is less than ${\cal O}(100)$, we evaluate the critical
number of signal events assuming a Poisson distribution, and calculate
$N_{s}^{\rm crit}$ as

$\sum\limits_{k=0}^{N_{b}}\frac{\lambda^{k}e^{-\lambda}}{k!}\ =\ 1-CL\,,$ (8)

with $\lambda=N_{s}^{\rm crit}+N_{b}$, and CL = 95%. In the special case of a
completely reducible background, i.e. in background free analyses, we obtain
$N_{s}^{\rm crit}=2.99$. From $N_{s}^{\rm crit}$ we extract the critical
signal cross section by $\sigma_{s}^{\rm crit}=N_{s}^{\rm
crit}/(\epsilon_{s}L)$, from which in turn we obtain the critical value of
$g_{X}$ above which the signal is observable at the chosen confidence level.

##  5 Dark photon search channels

In the following we describe optimised analyses for each relevant search
channel for DP detection at Belle II, i.e. for monophoton, two muon plus
missing energy, four muon prompt, and four muon displaced signatures. In each
channel, SM backgrounds partly arising because of the limited detector
acceptance, will be discussed. To extract sensitivity contours in the model
parameter space, the ultimate Belle II luminosity goal of 50 ab-1 is assumed,
and DP masses are scanned in the range from 1 MeV to 10 GeV.

###  5.1 Monophoton

In this section we consider the monophoton signature consisting of a single
observed photon together with missing transverse energy, i.e.
$e^{+}e^{-}\to\gamma+\not{E}$, see e.g. Ref. DePietro:2018sgg . The DP
contributes to this signature as $\not{E}$ when it decays into an invisible
final state or outside of the detector acceptance. The dominant contribution
to this signature stems from the decays into $\mu$ and $\tau$ neutrinos,
$X\to\nu_{\alpha}\bar{\nu}_{\alpha},$ with $\alpha=\mu,\tau$, which have a
large ${\cal O}(1)$ branching ratio, while the fraction of decays outside the
detector phase space contributes always below one percent to the total DP
signal and can thus be neglected.

The most relevant SM background processes contributing to the monophoton
signature are: di-photon processes ($e^{+}e^{-}\to\gamma\gamma$) where one
photon escapes detection; radiative Bhabha scattering ($e^{+}e^{-}\to\gamma
e^{+}e^{-}$) where the final state leptons are undetected; and processes with
a photon and one off-shell $Z$-boson decaying invisibly ($e^{+}e^{-}\to
Z\gamma$ with $Z\to\nu\bar{\nu}$). The inclusive cross sections of these SM
processes are given in Tab. 2, taking into account the geometric acceptance of
the detector and including a photon energy lower cut of 1 Gev to suppress
various backgrounds DePietro:2018sgg ; Belle-II:2023cal . The large di-photon
and radiative Bhabha production cross sections also lead to a large number of
monophoton events due to the gap in acceptance coverage between the forward
and backward hemispheres of the Belle II detector. Monophoton signatures from
these final states occur when one photon, or the lepton pair, is missed. Such
a scenario is depicted in Fig. 5, where one photon, $\gamma_{1}$, is in the
forward direction with an angle larger than $\theta_{l}=17^{\circ}$, while the
second photon, $\gamma_{2}$, escapes detection in the backward direction with
an angle larger than $\theta_{u}=150^{\circ}$.

Figure 5: Schematic picture in the lab frame of a monophoton background event
arising from the di-photon process $e^{+}e^{-}\to\gamma\gamma$ due to the
detector geometry.

The background contribution from the electro-weak process with a $Z$-boson
decaying into neutrinos was not considered in previous analyses
DePietro:2018sgg  because of its comparatively small cross section.
Nevertheless, we expect a significant contribution from this SM background
process because of the high target luminosity of the Belle II experiment, and
we therefore include this contribution in our analysis.

We simulate event samples at tree-level containing $10^{6}$ events for the DP
signal in the mass range 10 MeV$\leq m_{X}\leq 10$ GeV, $10^{6}$ events for
the SM di-photon and off-shell $Z$ into neutrinos processes, and $10^{5}$
events for the radiative Bhabha process. Di-photon and radiative Bhabha
samples are produced with no angular acceptance cuts on the final state
particles. We extract contributions to the monophoton SM background from these
two processes, selecting the events where respectively one of the photons or
the electron-positron pair are outside of the detector acceptance. To the di-
photon sample, we apply the following selection condition:

$(\theta_{l}<\theta_{\gamma_{i}}<\theta_{u})\quad\text{and}\quad(\theta_{\gamma_{j}}<\theta_{l}\text{
or }\theta_{\gamma_{j}}>\theta_{u})\,,\quad\text{with}\quad i\neq
j\quad\text{and}\quad i,j=1,2\,,$ (9)

where $\theta_{\gamma_{i}}$ is the angle of photon $\gamma_{i}$ with respect
to the positive beam axis, and $\theta_{l}=17^{\circ}$ and
$\theta_{u}=150^{\circ}$. The surviving monophoton events amount to just 1% of
the original di-photon sample. They populate a characteristic phase space
region, with photons being highly energetic $E_{\gamma}\approx$ 7 GeV and in a
narrow forward region $\theta_{l}<\theta_{\gamma}<\pi-\theta_{u}$ or photons
with low energy $E_{\gamma}\approx$ 4 GeV confined to a narrow backward region
$\pi-\theta_{l}<\theta_{\gamma}<\theta_{u}$. Similarly, from the radiative
Bhabha sample, we select monophoton events when electron and positron both
have angles bigger than $\theta_{u}$. This selection cut is very effective and
reduces the sample by a factor of $10^{-5}$, confining the photon to the same
phase space region as the di-photon sample. In Fig. 6 we show the distribution
in angle and energy for the three SM background processes, and for two
representative signals with $m_{X}=10$ MeV and $m_{X}=3.0$ GeV.

$m_{X}$ [MeV] |  $E_{\mu\mu}$ [GeV] |  $\theta_{X}$ [rad] | $\epsilon_{\gamma\nu\bar{\nu}}(\%)$ | $\epsilon_{X}(\%)$ |  $\sigma_{X}^{\rm crit}$ [fb]
---|---|---|---|---|---
1 | [4.10, 6.90] | [0.33, 2.58] | 0.0747 | 92.18 | 16.33
2 | [4.10, 6.90] | [0.33, 2.58] | 0.0747 | 92.24 | 16.33
3 | [4.10, 6.90] | [0.33, 2.58] | 0.0746 | 92.21 | 16.31
4 | [4.10, 6.90] | [0.33, 2.58] | 0.0747 | 92.16 | 16.33
5 | [4.10, 6.90] | [0.33, 2.58] | 0.0745 | 92.12 | 16.29
6 | [4.10, 6.90] | [0.33, 2.58] | 0.0747 | 92.22 | 16.33
7 | [4.10, 6.90] | [0.33, 2.58] | 0.0747 | 92.14 | 16.33
8 | [4.10, 6.90] | [0.33, 2.58] | 0.0747 | 92.20 | 16.33
9 | [4.10, 6.90] | [0.33, 2.58] | 0.0747 | 92.19 | 16.33
10 | [4.10, 6.90] | [0.33, 2.58] | 0.0747 | 92.19 | 16.33
20 | [4.10, 6.90] | [0.33, 2.58] | 0.0747 | 92.18 | 16.33
30 | [4.10, 6.90] | [0.33, 2.58] | 0.0747 | 92.19 | 16.33
40 | [4.10, 6.90] | [0.33, 2.58] | 0.0747 | 92.15 | 16.33
50 | [4.10, 6.90] | [0.33, 2.58] | 0.0747 | 92.19 | 16.33
60 | [4.10, 6.90] | [0.33, 2.58] | 0.0747 | 92.22 | 16.33
70 | [4.10, 6.90] | [0.33, 2.58] | 0.0747 | 92.16 | 16.33
80 | [4.10, 6.90] | [0.33, 2.58] | 0.0747 | 92.21 | 16.33
90 | [4.10, 6.90] | [0.33, 2.58] | 0.0748 | 92.16 | 16.35
100 | [4.10, 6.90] | [0.33, 2.58] | 0.0748 | 92.20 | 16.35
200 | [4.10, 6.90] | [0.33, 2.58] | 0.0753 | 92.20 | 16.42
300 | [4.09, 6.89] | [0.33, 2.58] | 0.0769 | 92.13 | 16.71
400 | [4.09, 6.89] | [0.33, 2.58] | 0.0781 | 92.18 | 16.93
500 | [4.09, 6.89] | [0.33, 2.58] | 0.0802 | 92.20 | 17.31
600 | [4.08, 6.88] | [0.33, 2.58] | 0.0822 | 92.19 | 17.65
700 | [4.08, 6.88] | [0.33, 2.58] | 0.0852 | 92.23 | 18.13
800 | [4.07, 6.87] | [0.33, 2.58] | 0.0886 | 92.21 | 18.40
900 | [4.07, 6.52] | [0.33, 2.58] | 0.0931 | 92.19 | 18.76
1000 | [4.06, 6.51] | [0.33, 2.58] | 0.0980 | 92.20 | 19.16
2000 | [3.95, 6.40] | [0.33, 2.58] | 0.2116 | 92.38 | 27.50
3000 | [3.77, 6.22] | [0.33, 2.58] | 0.4903 | 92.53 | 41.18
4000 | [3.51, 5.61] | [0.33, 2.58] | 0.9472 | 92.77 | 56.63
5000 | [3.18, 5.28] | [0.33, 2.58] | 1.7943 | 93.06 | 77.35
6000 | [2.78, 4.53] | [0.33, 2.58] | 3.0736 | 93.34 | 100.74
7000 | [2.30, 3.70] | [0.33, 2.58] | 5.4986 | 93.58 | 134.21
8000 | [1.76, 2.81] | [0.33, 2.58] | 10.2567 | 93.83 | 182.72
Table 3: Selection cuts in the monophoton channel for the considered DP masses
for the two observables: DP energy $E_{X}$ and angle w.r.t the beam axis
$\theta_{X}$. Column $\epsilon_{\gamma\nu\bar{\nu}}$ denotes the selection
efficiency for the SM background $\nu\bar{\nu}$ final state, which already
includes the efficiency of the $20^{\circ}<\theta_{\gamma}<148^{\circ}$ cut.
Column $\epsilon_{X}$ contains the selection efficiency on the signal, and the
last column, $\sigma_{X}^{\rm crit}$, shows the critical cross section above
which the signal is visible at 95% CL. Figure 6: Distribution in angle and
energy of the monophoton events from DP signals with $m_{X}=10$ MeV and
$m_{X}=3.0$ GeV (blue and purple respectively), from SM QED di-photon and
radiative Bhabha processes (green), and for the irreducible EW $Z$ to
neutrinos process (yellow). The displayed parameter space reflects the central
detector’s angular photon acceptance of 17∘.

In the DP signal samples, the photon’s angle and energy are tightly correlated
and all the signal events are bound to a narrow band in the
$\theta_{\gamma}$-$E_{\gamma}$ plane. The background events from di-photon and
radiative Bhabha samples populate a defined phase space region, and can be
entirely removed with a simple selection cut
$20^{\circ}<\theta_{\gamma}<148^{\circ}$, which has selection efficiencies on
the signal $\epsilon_{S}=0.92-0.94$, across the DP mass range, and on the
processes with a photon and one off-shell $Z$-boson
$\epsilon_{\gamma\nu\bar{\nu}}=0.93$. The off-shell $Z$-boson sample
constitutes an irreducible background for the DP signal in this channel, as
its topology is identical to the DP signal, despite the $Z$-boson being much
heavier than the DP in this model. The heavy $Z$-boson takes most of the
energy of the process and the photon is preferentially soft, and we observe no
clear correlation between its angle and energy.

Exploiting these kinematic features of signal and background, for each DP mass
choice we define an optimal search region which maximises our sensitivity to
the DP signal. For this, we consider bins of width $\Delta E$ = 0.35 Gev for
photon energies between 2 GeV and 7 GeV, compatible with the detector energy
resolution of 5% for $E\sim 7$ GeV Belle-II:2018jsg . For each energy bin we
identify an interval in the angle $\theta_{\gamma}$ which encloses all DP
signal events in the particular bin. In each phase space area we evaluate the
remaining number of background events from the SM process
$\gamma\bar{\nu}\nu$, finding a suppression of ${\cal O}(10^{-5})$ for all
$m_{X}$ considered. The sensitivity of Belle II to DP parameters is evaluated
according to Sect. 4, using $\epsilon_{\gamma}=0.99$ for the reconstruction
efficiency of a photon with energy at or above 2 GeV Henrikas:2604 . The
resulting sensitivity contour, for an integrated luminosity of 50 ab-1, is
displayed in Fig. 11 (dotted dark blue line) and discussed in Sect. 6.

###  5.2 Two muons plus missing energy

We now discuss the signature of two muons with missing transverse energy,
$e^{+}e^{-}\to\mu^{+}\mu^{-}+\not{E}$. The signal in this channel originates
from a DP, radiated from a muon and then decaying invisibly into muon or tau
neutrinos:
$e^{+}e^{-}\to\mu^{+}\mu^{-}+X\to\mu^{+}\mu^{-}\nu_{\mu,\tau}\bar{\nu}_{\mu,\tau}$.
The contribution from DPs decaying outside the detector acceptance is at most
at the sub-percent level for most of the considered parameter space, and will
therefore be neglected. Two SM background processes contribute to this
signature with sizeable event rates and constitute irreducible backgrounds for
the analysis. The first arises from di-tau production where both taus decay
into muons, $e^{+}e^{-}\to\tau^{+}\tau^{-}$ and
$\tau^{\pm}\to\mu^{\pm}\nu_{\mu}\bar{\nu}_{\tau}$, with cross section
$\sigma_{\tau\tau\to\mu\mu}=0.8~{}{\rm nb}\times{\rm
Br}(\tau\to\mu)^{2}\times\epsilon_{\mu\mu}$, where ${\rm
Br}(\tau\to\mu)=17.39\%$ is the branching ratio of the tau lepton into muons.
An efficiency of $\epsilon_{\mu\mu}=0.87$ for this background, due to the
geometric acceptance for the two muons, has been determined from simulation.
The second SM background is di-muon production with a hard photon radiated
outside the detector acceptance region,
$e^{+}e^{-}\to\mu^{+}\mu^{-}\gamma_{\rm inv}$, with the cross section
estimated to be $\sigma_{\mu\mu\gamma}=0.64$ nb (see. Tab. 2).

We simulate $10^{6}$ events for the signal and each background process. The
following observables are constructed from the muons’ four-momenta: di-muon
energy $E_{\mu\mu}$, three momentum $P_{\mu\mu}$, and angle $\theta_{\mu\mu}$
with respect to the electron beam direction, and the di-muon opening angle
$\alpha_{\mu\mu}$. In addition we also construct the recoil mass of the di-
muons, which is defined as

$m_{\rm recoil}^{2}\ =\ \left(E_{e^{-}}-E_{\mu\mu}\right)^{2}-\Delta
E^{2}-P_{\mu\mu}^{2}+2\Delta E\,\cos(\theta_{\mu\mu})P_{\mu\mu}\,,$ (10)

where $E_{e^{-}}=11$ GeV is the electron beam energy, and $\Delta E=3$ GeV is
the difference between electron and positron beam energies. The recoil mass
peaks at $m_{X}^{2}$ for the DP signal, at zero for the radiative di-muon
background, and it is smeared from 0 to $s$ for the di-tau background due to
the energy carried away by the neutrinos from the tau decays. Optimal cuts in
the observables $\alpha_{\mu\mu}$, $\theta_{\mu\mu}$ and $P_{\mu\mu}$ were
found to improve the sensitivity only marginally, and will hence not be
discussed in the following. We study the distributions of the observables for
the signal with 10 MeV$\leq m_{X}\leq$ 7 Gev, and for the SM backgrounds.
Optimal cuts for each DP mass are found by minimising the number of background
events to obtain a visible signal at 95% CL according to Eq. (7).

Figure 7: Event rates as function of the recoil mass $m^{2}_{\textrm{recoil}}$
(left) and of the di-muon energy $E_{\mu\mu}$ (right), normalised as indicated
in the axes captions. The blue signal curves correspond to the choices $m_{X}$
= 10, 100, 1000 and 3000 MeV and $g_{X}=0.0007$. The green curves denote the
di-tau background and include the decay Br($\tau\to\mu$). The yellow curves
denote the radiative di-muon background for this signature.

We find that the two most relevant observables for signal-over-background
enhancement are the recoil mass $m_{\rm recoil}^{2}$ and $E_{\mu\mu}$, whose
distributions are shown in Fig. 7 for DP masses of
$m_{X}=10,\,100,\,1000,\,3000$ MeV and $g_{X}=0.0007$. Given the shape of the
recoil mass distribution, we find that appropriate cuts in this observable can
almost completely remove contributions from the di-muon background for
$m_{X}\geq 1$ GeV, in agreement with the results of Ref. DePietro:2018sgg .
For DPs with $m_{X}\geq 1$ GeV, the most effective way to disentangle the
signal from the background is through optimal cuts in the $E_{\mu\mu}$
observable. In Tab. 4 we list the considered DP masses, the selection cuts for
each search regions, their efficiencies for signal and the two SM backgrounds.
The critical signal cross sections in the last column are obtained assuming an
integrated luminosity of 50 ab-1, and they translate in the 95% CL exclusion
solid dark blue contour line in Fig. 11.

$m_{X}$ [MeV] |  $m_{\rm recoil}^{2}$ [GeV2] |  $E_{\mu\mu}$ [GeV] | $\epsilon_{\mu\mu\gamma}(\%)$ | $\epsilon_{\tau\tau}(\%)$ | $\epsilon_{X}(\%)$ |  $\sigma_{X}^{\rm crit}$ [fb]
---|---|---|---|---|---|---
1 | [-4.2, 3.1] | [7.2, 11] | 16.1 | 0.56 | 29.3 | 8.82
2 | [-4.4, 3.3] | [5.5, 11] | 54.1 | 0.69 | 99.6 | 4.75
3 | [-3.7, 3.2] | [5.5, 11] | 51.3 | 0.63 | 99.6 | 4.63
4 | [-4.2, 3.3] | [5.2, 11] | 49.7 | 0.67 | 99.7 | 4.55
5 | [-4.3, 3.6] | [5.2, 11] | 20.6 | 0.83 | 42.2 | 6.93
6 | [-4.0, 3.6] | [5.2, 11] | 20.6 | 0.80 | 43.9 | 6.66
7 | [-4.5, 3.3] | [5.2, 11] | 46.0 | 0.69 | 99.7 | 4.38
8 | [-4.2, 3.4] | [5.2, 11] | 20.6 | 0.72 | 46.7 | 6.26
9 | [-4.4, 3.7] | [5.2, 11] | 44.4 | 0.87 | 99.7 | 4.30
10 | [-4.4, 3.4] | [5.2, 11] | 43.7 | 0.73 | 99.7 | 4.27
20 | [-4.4, 3.8] | [4.8, 11] | 39.4 | 0.92 | 99.8 | 4.05
30 | [-4.3, 3.7] | [4.8, 11] | 36.8 | 0.84 | 99.8 | 3.92
40 | [-4.6, 3.8] | [4.8, 11] | 34.9 | 0.90 | 99.8 | 3.81
50 | [-4.2, 3.5] | [4.8, 11] | 33.6 | 0.77 | 99.8 | 3.74
60 | [-5.6, 3.8] | [4.8, 11] | 20.1 | 0.93 | 69.3 | 4.17
70 | [-5.2, 3.6] | [4.8, 11] | 20.0 | 0.83 | 70.8 | 4.07
80 | [-5.0,-0.1] | [7.5, 11] | 4.9 | 0.003 | 23.8 | 5.98
90 | [-4.8, 3.9] | [4.8, 11] | 19.7 | 0.96 | 73.6 | 3.88
100 | [-4.4,-0.1] | [7.5, 11] | 5.5 | 0.003 | 27.2 | 5.55
200 | [-4.3,-0.3] | [7.4, 11] | 3.1 | 0.001 | 18.9 | 6.00
300 | [0.08, 4.1] | [7.4, 11] | 5.1 | 0.85 | 39.6 | 3.68
400 | [0.05, 4.2] | [7.4, 11] | 5.7 | 0.89 | 48.8 | 3.14
500 | [0.38, 4.6] | [7.9, 10] | 1.3 | 0.91 | 23.5 | 3.18
600 | [0.26, 4.5] | [7.6, 10] | 2.4 | 0.98 | 43.1 | 2.35
700 | [0.43, 4.7] | [7.8, 10] | 1.3 | 0.98 | 37.3 | 1.96
800 | [0.10, 5.0] | [5.3, 10] | 6.0 | 1.45 | 75.5 | 2.10
900 | [0.10, 5.1] | [5.0,9.8] | 5.5 | 1.48 | 72.9 | 2.08
1000 | [0.25, 5.5] | [7.3, 10] | 2.5 | 1.38 | 65.1 | 1.59
2000 | [2.1, 8.3] | [6.7,9.0] | 0.1 | 2.65 | 71.6 | 0.349
3000 | [5.9, 13] | [3.8,8.0] | 0.0 | 5.81 | 99.8 | 0.307
4000 | [13, 20] | [3.5,7.0] | 0.0 | 7.33 | 99.7 | 0.345
5000 | [22, 28] | [3.2,6.0] | 0.0 | 7.85 | 99.7 | 0.357
6000 | [34, 39] | [2.8,5.0] | 0.0 | 7.61 | 99.6 | 0.352
7000 | [47, 52] | [2.3,3.9] | 0.0 | 6.29 | 92.2 | 0.346
8000 | [62, 67] | [1.8,2.9] | 0.0 | 4.51 | 86.7 | 0.312
9000 | [80, 83] | [1.9,1.9] | 0.0 | 0.05 | 21.6 | 0.135
Table 4: Selection cuts in the di-muon plus missing energy channel for the
considered DP masses for the two observables: recoil mass $m_{\rm recoil}^{2}$
and di-muon energy $E_{\mu\mu}$. Columns labelled $\epsilon_{\mu\mu\gamma}$
and $\epsilon_{\tau\tau}$ denote the selection efficiency for the SM
background processes with $\mu\mu\gamma$ and $\tau\tau$ final states
respectively. Column $\epsilon_{X}$ contains the selection efficiency on the
signal, and the last column, $\sigma_{X}^{\rm crit}$, shows the critical cross
section above which the signal is visible at 95% CL.

###  5.3 Four muons prompt

We study the sensitivity of the Belle II experiment on the DP signal in the
final state with two muon pairs,
$e^{+}e^{-}\rightarrow\mu^{+}\mu^{+}\mu^{-}\mu^{-}$, which we will refer to as
the $4\mu$ channel in the following. The DP signal in this channel stems from
a DP radiated from a muon and decaying into a muon pair, see the left panel of
Fig. 3. The main SM background is from the QED process with two photons
decaying into muon pairs and has a cross section of $\sigma_{4\mu}=155$ fb.
Here we neglect additional possible backgrounds with hadronic final states
where pions are misidentified as muons, which has a probability of 1.4% per
pion only.

We simulate 106 events within the geometric acceptance for the background, and
$10^{6}$ signal events in each of the suitably chosen bins in the mass range
220 MeV $\leq m_{X}\leq$ 8 GeV, see Tab. 5 for details. In the signal samples,
the DP propagator is taken to be on-shell as the DP is very narrow. The DP
kinematics is then reconstructed by identifying the muon pair with an
invariant mass $m_{X}\pm 5$ MeV, thus taking into account the finite detector
resolution. With this procedure, the signal selection efficiency is basically
100% in all bins. Due to the tight selection, the resulting background
suppression ranges from around 2% in bins with $m_{X}$ close to the di-muon
threshold, to $\sim$ 0.2% for $m_{X}\simeq\sqrt{s}$. This is visible in Fig. 8
where the number of background events within each search window is shown.

Figure 8: Distribution of 107 background events using the bin scheme as
defined in Table 5. The events in each bin represent the background for each
DP mass search region.

Figure 9: Energy and angular distributions of dark photon signals (blue) and SM background (yellow) in the 4$\mu$ final state at Belle II with integrated luminosity of 50 ab-1. Signal curves are shown for $m_{X}$ = 300, 1000 and 3000 MeV. $m_{X}$ [MeV] |  $m_{\mu\mu}$ [GeV] | $\epsilon_{m_{\mu\mu}}(\%)$ |  $E_{X}$ [GeV] |  $\theta_{X}$ [rad] | $\epsilon_{4\mu}(\%)$ | $\epsilon_{X}(\%)$ |  $\sigma_{X}^{\rm crit}$ [fb]
---|---|---|---|---|---|---|---
220 | [0.215, 0.225] | 1.57 | [0.22, 3.88] | [0.00007 , 2.94] | 65.09 | 90.35 | 0.0101
230 | [0.225, 0.235] | 2.06 | [0.23, 3.88] | [0.00003 , 3.12] | 64.28 | 90.18 | 0.0115
240 | [0.235, 0.245] | 2.28 | [0.24, 3.91] | [0.00007 , 3.13] | 64.07 | 90.24 | 0.0121
250 | [0.245, 0.255] | 2.37 | [0.25, 3.88] | [0.00021 , 3.13] | 63.06 | 89.87 | 0.0123
260 | [0.255, 0.265] | 2.40 | [0.26, 3.89] | [0.00039 , 3.13] | 62.81 | 89.77 | 0.0123
270 | [0.265, 0.275] | 2.41 | [0.27, 3.91] | [0.00032 , 3.12] | 62.43 | 89.71 | 0.0123
280 | [0.275, 0.285] | 2.37 | [0.28, 3.92] | [0.00065 , 3.13] | 62.05 | 89.63 | 0.0122
290 | [0.285, 0.295] | 2.33 | [0.29, 3.92] | [0.00027 , 2.98] | 61.66 | 89.49 | 0.0121
300 | [0.295, 0.305] | 2.28 | [0.30, 3.92] | [0.00074 , 2.97] | 61.09 | 89.31 | 0.0119
350 | [0.345, 0.355] | 2.00 | [0.35, 3.94] | [0.0019 , 3.13] | 58.92 | 88.73 | 0.0110
400 | [0.395, 0.405] | 1.73 | [0.40, 3.96] | [0.0019 , 3.14] | 57.24 | 88.21 | 0.0102
450 | [0.445, 0.455] | 1.53 | [0.45, 3.66] | [0.0027 , 2.98] | 50.81 | 84.17 | 0.0094
500 | [0.495, 0.505] | 1.37 | [0.50, 3.70] | [0.0031 , 2.98] | 49.13 | 83.79 | 0.0088
550 | [0.545, 0.555] | 1.25 | [0.55, 3.72] | [0.0082 , 2.98] | 47.39 | 83.24 | 0.0083
600 | [0.595, 0.605] | 1.14 | [0.60, 3.75] | [0.17 , 2.98] | 45.87 | 82.87 | 0.0079
650 | [0.645, 0.655] | 1.06 | [0.65, 3.78] | [0.32 , 3.14] | 44.99 | 82.55 | 0.0076
700 | [0.695, 0.705] | 0.99 | [0.70, 3.80] | [0.34 , 3.14] | 43.06 | 82.24 | 0.0072
750 | [0.745, 0.755] | 0.94 | [0.75, 3.83] | [0.35 , 2.98] | 41.86 | 81.69 | 0.0069
800 | [0.795, 0.805] | 0.89 | [0.80, 3.55] | [0.06 , 2.98] | 36.19 | 76.72 | 0.0067
850 | [0.845, 0.855] | 0.86 | [0.85, 3.58] | [0.47 , 3.00] | 35.18 | 76.26 | 0.0065
900 | [0.895, 0.905] | 0.82 | [0.90, 3.61] | [0.49 , 3.00] | 34.05 | 76.02 | 0.0063
1000 | [0.995, 1.005] | 0.78 | [1.00, 3.67] | [0.54 , 3.14] | 32.80 | 75.42 | 0.0060
2000 | [1.995, 2.005] | 0.57 | [2.00, 4.02] | [1.04 , 2.72] | 22.95 | 62.36 | 0.0052
3000 | [2.995, 3.005] | 0.46 | [3.00, 4.69] | [1.38 , 2.79] | 28.43 | 60.59 | 0.0054
4000 | [3.995, 4.005] | 0.35 | [4.00, 6.26] | [1.38 , 3.05] | 70.72 | 88.79 | 0.0051
5000 | [4.995, 5.005] | 0.28 | [5.00, 6.97] | [1.60 , 3.14] | 78.48 | 91.86 | 0.0046
6000 | [5.995, 6.005] | 0.23 | [6.00, 7.67] | [1.80 , 3.14] | 83.36 | 93.74 | 0.0042
7000 | [6.995, 7.005] | 0.20 | [7.00, 8.44] | [1.95 , 3.08] | 90.61 | 95.24 | 0.0040
8000 | [7.995, 8.005] | 0.19 | [8.00, 9.25] | [2.09 , 3.14] | 100.00 | 100.00 | 0.0039
9000 | [8.995, 9.005] | 0.19 | [9.08, 9.86] | [2.30 , 3.10] | 99.86 | 99.57 | 0.0040
Table 5: Selection cuts for the 4$\mu$ channel for the considered DP masses.
Column $m_{\mu\mu}$ contains the invariant mass selection cuts, and
$\epsilon_{m_{\mu\mu}}$ the corresponding fraction of background events in
that interval. Columns $E_{X}$ and $\theta_{X}$ give the selection cuts on the
two observables. Columns $\epsilon_{4\mu}$ and $\epsilon_{X}$ contain the
efficiency of the combined cuts on the background and signal respectively. The
last column, $\sigma_{X}^{\rm crit}$, shows the critical cross section above
which the signal is visible at 95% CL.

To achieve the best sensitivity for DP signals in this channel, we consider
four observables for the DP candidates: energy $E_{X}$, momentum, angle
$\theta_{X}$ with respect to the beam axis, and opening angle of the two
additional muons. The distributions for $E_{X}$ and $\theta_{X}$ are shown in
Fig. 9 for the background and for signals from DP masses $m_{X}$ = 300, 1000
and 3000 MeV. For each DP mass used in the scan, optimal cuts on the four
kinematic observables are determined to maximise the 95% CL signal exclusion,
following Eq. (7). We find that cuts on $E_{X}$ and $\theta_{X}$ lead to the
best signal over background enhancement. A list of the optimised cuts for some
representative DP masses is given in Tab. 5, together with the corresponding
signal and background efficiencies due to the optimised cuts but after the
initial invariant mass selection.111Note that because of the Gaussian smearing
$\theta_{X}$ can assume values smaller than zero. The last column gives the
critical signal cross sections for an integrated luminosity of 50 ab-1, from
which we obtain the 95% CL exclusion dashed-dotted dark blue contour line in
Fig. 11, following the procedure described in Sect. 4.

###  5.4 Displaced vertices

Figure 10: Schematic description of the process
$e^{+}e^{-}\to\mu^{+}\mu^{-}X$, where $X$ propagates a macroscopic distance
and decays into $\ell^{+}\ell^{-}$. The primary vertex is denoted by ‘PV’ and
is identified with the intersection of the two muon tracks (red arrows), the
secondary vertex is denoted with ‘SV’ and identified with the intersection of
the two lepton tracks (blue arrows). Uncertainties in the PV and SV positions
are indicated by the yellow error bars. The electron and positron beams are
denoted by black arrows, the detector acceptance is indicated by the purple
lines, where the grey regions are invisible.

We consider a special signature in the 4$\ell$ final state which features two
interaction vertices separated by a measurable distance. The position of the
decay vertices is reconstructed from the information collected in the Vertex
detector (VXD), which is the closest sub-detector to the interaction point
Belle-IISVD:2017cpj , and whose main purpose is to study the decay of tau
leptons and $B$ and $D$ mesons. The design and the performance of the VXD
Belle-IISVD:2022upf ; Belle-II:2020mvd  allow a displacement resolution in the
transverse plane as precise as a few micrometers Wieduwilt:2021phh . A
displaced signature is observed when the distance between the two vertices
exceeds the detector resolution. This is schematically depicted in Fig. 10,
where the Primary Vertex (PV) is on the nominal beam axis while the Secondary
Vertex (SV) is displaced.222The position of the vertices along the beam
direction is reconstructed less precise than that in the transverse plane. For
our analysis, we will therefore only make use of the transverse displacement.
We have checked that including information in the longitudinal displacement,
even under optimistic assumptions, adds only marginally to the derived
sensitivities. In our analysis for displaced vertices, we consider spatial
detector resolutions of 5 and 10 $\mu$m, which is somewhat optimistic compared
to the recent study Maiti:2023axl , but potentially achievable in the future
as dedicated studies on the VXD performances are ongoing Belle-IISVD:2024gjd .

In the $L_{\mu}-L_{\tau}$ model considered here, a light DP with mass $\sim$
200 MeV and gauge coupling $g_{X}\sim 10^{-3}$ has a decay length in its rest
frame $\lambda_{X}\sim 0.02~{}\mu$m (see Fig. 2). The additional boost into
the Belle II laboratory frame can enhance the DP decay length up to a factor
10. Therefore, with the assumed vertex resolution and the high integrated
luminosity collected at Belle II, a statistically significant number of DP
decays, which populate the tail of the decay distributions, will appear as
events with a displaced vertex. In principle, DP decays into any charged
lepton contribute, but decays into electrons are strongly suppressed by the
factor $\epsilon^{2}$ (cf. Eq. (2)) compared to muons, whereas decays into
taus require $m_{X}>2m_{\tau}\simeq 3.5$ GeV, where the DP lifetime is small.
We will therefore focus on displaced signatures with four muons.

SM background contributions with displaced final states include intermediate
particles with a finite life-time such as $B$ mesons and $\tau$ leptons.
However, the Belle II detector is designed to perform precision measurements
on the former, which have a very different topology compared to the DP signals
considered, since on average $B$ decays yield many tracks and include several
mesons. Since the likelihood to misidentify several pions (or kaons) as muons
is tiny, we assume that this background is sufficiently suppressed by the
requirement of a four muon final state. The SM process
$e^{+}e^{-}\to\tau^{+}\tau^{-}\mu^{+}\mu^{-}$ with
$\tau^{\pm}\to\mu^{\pm}\bar{\nu}_{\mu}\nu_{\tau}$ has a cross section of about
4 fb and can feature displaced vertices because of the finite decay length of
the relativistic $\tau$ lepton $\lambda_{\tau}\simeq 10^{-4}$ m. This
background can be vetoed by requiring the total deposited energy to be within
3% of $\sqrt{s}$, reducing the predicted number of such events to 0.1 for 50
ab-1. The only relevant background for our analysis originates from the SM QED
process with four muons in the final state where, because of the finite
detector resolution, a virtual photon decay may appear with a visible
displacement.

We generate $10^{7}$ background events, and $10^{6}$ signal events for the DP
masses in the restricted range $m_{X}$ = [220, 400] MeV, as detailed in Tab.
6. Because heavy DPs have an average shorter decay length, in our model we
lose sensitivity at higher masses, where the decay length is too short for the
determination of the displaced vertex. The background analysis in the
displaced signature case closely follows the procedure defined for the prompt
channel, starting with the reconstruction of the muon pair with an invariant
mass $m_{X}\pm 5$ MeV. The fraction of background events within each search
region is identical to the prompt analysis, and their distribution is shown in
Fig. 8.

$m_{X}$ [MeV] |  $m_{\mu\mu}$ [GeV] | $\epsilon_{m_{\mu\mu}}(\%)$ | BFD [$\mu$m] | $\epsilon_{X}(\%)$ | Max $N_{\rm sig}$
---|---|---|---|---|---
220 | [0.215, 0.225] | 1.6 | 15.18 | 22.70 | 4.78
230 | [0.225, 0.235] | 2.1 | 13.86 | 23.07 | 5.91
240 | [0.235, 0.245] | 2.3 | 13.99 | 21.02 | 5.89
250 | [0.245, 0.255] | 2.4 | 14.96 | 16.70 | 4.89
260 | [0.255, 0.265] | 2.4 | 13.82 | 17.42 | 5.18
270 | [0.265, 0.275] | 2.4 | 13.72 | 16.27 | 4.87
280 | [0.275, 0.285] | 2.4 | 13.91 | 14.53 | 4.36
290 | [0.285, 0.295] | 2.3 | 14.20 | 32.26 | 3.81
300 | [0.295, 0.305] | 2.3 | 13.53 | 12.84 | 3.76
310 | [0.305, 0.315] | 2.2 | 12.17 | 14.97 | 4.36
320 | [0.315, 0.325] | 2.2 | 13.78 | 28.18 | 3.21
Table 6: Selection cuts for the 4$\mu$ displaced channel for the considered DP
masses. Column $m_{\mu\mu}$ contains the invariant mass selection cuts, and
$\epsilon_{m_{\mu\mu}}$ the corresponding fraction of background events in
that interval. Column ‘BFD’ shows the background free distance above which no
displaced background events are observed for a detector resolution of 5
$\mu$m. Column $\epsilon_{X}$ lists the efficiencies of the cuts on the
signal, and the last column, Max $N_{\rm sig}$, shows the number of surviving
displaced signal events at the BFD.

In the simulation of background and signal samples, random displacements are
assigned to each muon track origin, accounting for both the finite detector
resolution and finite life-time of the DP. The effect of the finite detector
resolution is simulated by drawing all displacements from a normal
distribution with standard deviation equal to the considered detector
resolution. For this, the track origins projected on the transverse $x$-$y$
plane are obtained by multiplying the random Gaussian shifts by
$p_{i}/|\mathbf{p}|$, with $i=x,y$.333We chose to rescale the displacement
according to the total momentum, rather than the transverse momentum, in order
to account for the boost component along the beam axis. This factor has been
chosen to model the information on the position of the origin of the muons,
which is derived from the particles’ reconstructed tracks and momenta
according to their energy deposition in the ECAL layers. Effectively, with
this procedure, the origin of each muon track lays in the direction of the
particle momentum. In the signal samples, an additional displacement is
assigned to the muon pair which originates from the DP. To obtain the correct
boost, the displacements are calculated on an event-by-event basis, following
the DP exponential decay law, with laboratory frame decay length
$\lambda_{X}^{\textrm{lab}}$. The displacement is then scaled according to the
particle momenta as before.

In order to assess the experimental sensitivity in each search region we
proceed as follows. In the first step, we reject all events in both background
and signal samples where the relative distance between the (already paired)
muon and antimuon in either the PV or SV is larger than the experimental
resolution. This selection removes events with wrongly paired muon-antimuon
which effectively do not originate from the same vertex. Secondly, we define
the position of the PV and SV as the middle point between the paired muon and
antimuon, then calculate the overall displacement as the relative distance
between PV and SV. We identify the displacement threshold above which the
number of background events drops below one, i.e. where the background is
completely reducible and a background free analysis applies.

By construction, the determination of the end point of the distribution of
background events suffers from large statistical fluctuations. In order to
reduce these fluctuations we repeat the assignment of random displacements to
the background samples $10^{4}$ times, and then consider the average
distribution of events as function of their displacement. The end point of the
distribution is obtained adopting a regression fitting method inspired by
machine learning: we evaluate the number of surviving events above fixed
multiples of the chosen resolution444Since the simulated muon origins in the
background sample were drawn from a Gaussian distribution which depends solely
on the detector resolution, the end point of the distribution can be given in
units of this resolution parameter., and fit the logarithm of this
distribution using a quadratic polynomial. The parameters of the quadratic
polynomial are obtained applying a linear regression algorithm, trained using
a gradient descent minimiser with squared error cost function, and the
optimised parameters are then used to estimate the distribution end point.
This procedure is repeated for each search region, to obtain the background
free thresholds for each simulated DP mass, and gives more accurate results
than a direct exponential fit to the background distribution. The background
free thresholds are reported in units of the detector resolution in column
four of Tab. 6. It turns out that for the displayed mass range, they are
almost constant. Only at higher masses, not displayed due to a loss of signal
sensitivity, do the thresholds drop to smaller values as expected.

The fractions of signal events with displacement above the background free
thresholds are given in the last column of Tab. 6. They are used to obtain the
sensitivity contour as a function of mass $m_{X}$ and the gauge coupling
$g_{X}$, following the statistical analysis described in Sect. 4, and
specifically applying Eq. 8. The resulting 95% CL exclusion contour, for an
assumed vertex resolution of 5 $\mu$m and an integrated luminosity of 50 ab-1,
is displayed in Fig. 11 by the dashed dark blue line and will be discussed in
section 6.

###  5.5 Signatures with tau leptons

Because of the underlying $L_{\mu}-L_{\tau}$ symmetry, processes involving the
production of DPs in association with tau leptons, as in
$e^{+}e^{-}\to\tau^{+}\tau^{-}X$, have a cross section proportional to
$g_{X}^{2}$, similar in magnitude to those discussed above involving muons
(see Fig. 4 and Tab. 1). Analogous search channels for DP detection can be
defined replacing muons with taus in the final state: di-tau plus missing
energy, di-taus plus two muons, four taus prompt, and two taus plus two
displaced muons. Tau leptons are not observable directly, but are detected
reconstructing their decay products. Their decays necessarily include light
tau neutrinos which carry away a part of the kinematic information in form of
missing energy or momentum. Tau observables are therefore more ‘smeared out’
compared to the analogous muon channels and have smaller selection and
identification efficiencies, which eventually leads to an overall lower
sensitivity. Dedicated searches for light resonances decaying into tau pairs
are being performed at Belle II Belle-II:2023ydz , which provide the existing
most stringent limits in this signature. Those constraints are however
superseded by analogous analyses on signatures with muon final states.

For these reasons, in this study we do not further investigate the sensitivity
of the Belle II detector to channels with tau leptons, but leave this for
future work. Indeed, if a positive signal would be discovered in any channel,
signatures involving tau leptons could be decisive to fingerprint the
underlying model.

##  6 Results

In this section we present our projections for the Belle II sensitivity on
$L_{\mu}-L_{\tau}$ DPs in the search channels discussed above, assuming an
integrated luminosity of 50 ab-1. The exclusion limits derived through the
analyses described above are displayed in Fig. 11 in the two dimensional
parameter space of the model, scanning over the mass $m_{X}$ and coupling
$g_{X}$ of the DP. For comparison, we report the most recent existing
constraints from the collider experiments Belle II, BaBar, KLOE, CMS and LHCb,
from the CCFR neutrino experiment, and indirect constraints from Fermilab
measurements on the muon $g-2$. Experimental analyses on general models, with
an additional neutral vector boson connected to the SM through kinetic mixing
$\epsilon$ are recast into our minimal $L_{\mu}-L_{\tau}$ framework using Eq.
2, i.e. dividing $\epsilon$ by the loop factor in Eq. 2 to obtain the gauge
coupling $g_{X}$, while rescaling the excluded cross sections by the
appropriate branching ratios.

Low mass region: Very light DPs, with mass below the di-muon threshold, can be
probed only indirectly. In this mass region, the dominant exclusion limits are
indeed given by the NA64 experiment, by the CCFR neutrino trident experiment,
and by measurements of the muon $g-2$ anomaly. The most stringent constraints
from NA64 arise from a recent analysis of approximately $2\times 10^{10}$
muons generated by the CERN Super Proton Synchrotron and scattered off fixed-
target atomic nuclei. In this analysis, a DP is radiated in a bremsstrahlung-
like process from the muon after the scattering, and is searched for in events
with a final state comprising a muon accompanied by missing energy which is
attributed to the DP decaying into either neutrino or dark matter pairs
NA64:2024klw  (solid yellow line). The CCFR neutrino trident experiment
studies the rare process $\nu N\to\nu\mu^{+}\mu^{-}$, with the
$\mu^{+}\mu^{-}$ pair produced from muon-neutrino scattering off the Coulomb
field of a nucleus. It is therefore particularly sensitive to additional
contributions from neutral currents coupling to the muon flavour. At present,
measurements on SM neutrino cross sections from CCFR Altmannshofer:2014pba
exclude gauge couplings above $g_{X}\simeq 10^{-3}$ for DP masses up to the
di-muon threshold, and above $g_{X}\simeq 10^{-2}$ for DP masses of the order
of 10 Gev (light grey shaded area of Fig. 11). The purple band indicates the
parameter space which would explain, with 90% CL, the muon $g-2$ discrepancy
arising at the $4.2\sigma$ level from the comparison of the Run-1 measurements
at FNAL Muong-2:2021ojo  with the SM prediction from the Muon $g-2$ Theory
Initiative Aoyama:2020ynm ; Aoyama:2012wk ; Aoyama:2019ryr ; Czarnecki:2002nt
; Gnendiger:2013pva ; Davier:2017zfy ; Keshavarzi:2018mgv ; Colangelo:2018mtw
; Hoferichter:2019mqg ; Davier:2019can ; Keshavarzi:2019abf ; Kurz:2014wya ;
Melnikov:2003xd ; Masjuan:2017tvw ; Colangelo:2017fiz ; Hoferichter:2018kwz ;
Gerardin:2019vio ; Bijnens:2019ghy ; Colangelo:2019uex ; Blum:2019ugy ;
Colangelo:2014qya . As mentioned in the introduction, the size and
significance of the discrepancy is currently not fully confirmed and the band
should be taken as an indication only. The muon $g-2$ constraints are mostly
relevant in the low mass region, as is clear from Fig. 11. Constraints from
collider experiments in this low mass region are only possible considering
final states with missing energy or momentum. The exclusion limit from the
Belle II analysis on the 2$\mu$ plus missing energy signature with an
integrated luminosity dataset of 79.7 fb-1 Belle-II:2022yaw  is represented by
the solid light pink curve. For light DPs with masses below 1 Gev, it excludes
gauge couplings above $g_{X}\simeq 3\cdot 10^{-3}$, while for heavier DPs the
sensitivity reduces exponentially to $g_{X}\simeq$ 1 for masses of 8 Gev. Our
optimised analysis on the 2$\mu$ plus missing energy channel (solid dark blue
line), shows comparable sensitivity with respect to the current Belle II
analysis, and it could test couplings above $g_{X}\simeq 3\cdot 10^{-3}$ for
DP masses up to 2 Gev, and above $g_{X}\simeq 10^{-1}$ for DP masses up to 8
Gev.

Two muon final states: Analyses on final states with only two muons are, in
general, the least constraining. Searches for DPs at KLOE are sensitive to
masses between 520 MeV to 990 MeV, and include both decay modes into muon and
pion pairs KLOE-2:2018kqf . In the context of the $L_{\mu}-L_{\tau}$ model,
only the former is relevant, and the contour obtained with an integrated
luminosity of 1.93 fb-1, represented by the solid purple line, excludes gauge
couplings up to $g_{X}\simeq 5\cdot 10^{-2}$. The dashed brown curve is
derived from the CMS analysis on direct production of a light resonance
decaying into a muon pair, with integrated luminosity of 96.6 fb-1 CMS:2023hwl
. It uses a trigger selection and muon identification optimized for the low-
mass region considered, and it is sensitive to DP masses between 1.1 Gev and
7.9 Gev, excluding the region between 2.6 and 4.2 Gev, i.e. around the
$J/\Psi$, $\Psi$(2S), and $\Upsilon$(1S) resonances. It excludes gauge
couplings as small as $g_{X}\simeq 2\cdot 10^{-2}$ for very light DPs, with
exponentially decreasing sensitivity to $g_{X}\simeq 1$ for heavier DPs. The
LHCb analysis on light resonances decaying into a muon pair is obtained with a
dataset corresponding to an integrated luminosity of 5.5 fb-1 LHCb:2019vmc .
Both prompt and displaced signatures are considered, and the exclusion limits
are represented by the dashed black and solid black lines respectively. The
LHCb prompt analysis shows a similar sensitivity to the analogous CMS one, but
also covering the region below 1 Gev, down to the di-muon threshold, where it
excludes gauge couplings $g_{X}\simeq 8\cdot 10^{-3}$. In the mass region
between the di-muon threshold and 300 MeV, the analysis with displaced
signatures gives the best sensitivity, excluding gauge couplings down to
$g_{X}\simeq 10^{-3}$. In comparison with these experimental results, we
observe that our analysis on displaced DP decays (dashed dark blue curve)
shows a shape similar to the analogous LHCb result, and a significantly
improved sensitivity to gauge couplings below $g_{X}\simeq 10^{-4}$.

Four muon final states: Analyses on final states with 4 muons provide strong
constraints in the $L_{\mu}-L_{\tau}$ model parameter space, especially for DP
heavier than 1 Gev. The BaBar exclusion curve (solid orange line) has been
obtained from the analysis of the 4$\mu$ prompt signature, considering the
dataset corresponding to an integrated luminosity of 514 fb-1, taken at the
$\Upsilon$(4S) resonance and in the neighborhood of the $\Upsilon$(3S) and
$\Upsilon$(2S) peaks BaBar:2016sci . It covers the mass region above the di-
muon threshold, excluding gauge couplings below $g_{X}\simeq 10^{-3}$ up to
masses of 2 Gev, while for heavier DPs the sensitivity reduces exponentially
to $g_{X}\simeq 1$ for masses of 8 Gev. An analogous analysis from Belle II
with a dataset corresponding to an integrated luminosity of 643 fb-1
Belle:2021feg  shows a similar sensitivity (solid red line). Belle II performs
slightly worse than BaBar for DP lighter than 2 Gev, while marginally
improving BaBar exclusions for DPs heavier than 2 Gev. Very recently, an
analysis of 178 fb-1 in the four muon final state has been released Belle-
II:2024wtd  by Belle II. The results (solid dark green line) show a
sensitivity similar to BaBar and previous Belle II analyses, and further
improving upon CCFR. The combination of all three results sets stringent
exclusion limits for DP masses above 3 Gev. The CMS analysis of the four muons
prompt final state with an integrated luminosity of 77.3 fb-1 CMS:2018yxg
covers only DPs heavier than 5 Gev (solid brown line), and provides the
strongest exclusion down to $g_{X}\simeq 5\cdot 10^{-3}$. Our results on the
four muons prompt signature (dot-dashed dark blue line) show a significant
improvement with respect to BaBar and previous Belle II analyses, increasing
the gauge coupling exclusion by roughly a factor 2. For DPs as heavy as 5-6
Gev, our analysis is sensitive to gauge couplings down to $g_{X}\simeq 5\cdot
10^{-3}$, and comparable with the CMS high mass analysis.

A special mention is deserved by our study of the monophoton channel, whose
sensitivity curve is given by the dotted dark blue line. The interpretation of
this signature within the $L_{\mu}-L_{\tau}$ framework shows a remarkable
potential for constraining specifically the minimal construction we have
considered. The underlying process is indeed very sensitive to the kinetic
mixing $\epsilon$, which in our case is not a free parameter, but instead
directly connected to the gauge coupling $g_{X}$ through the kinetic mixing
loop factor, $f(m_{\mu},m_{\tau},q)$. This analysis predicts the strongest
sensitivity for DP masses below the di-muon threshold, excluding down to
$g_{X}\simeq 4\cdot 10^{-4}$, which is comparable or better than the strongest
indirect bound from the muon $g-2$. For heavier DPs, the sensitivity of the
monophoton analysis closely follows the one of the four muons prompt analysis.

Figure 11: Summary of existing constraints and projected 95% CL sensitivities
for Belle II with integrated luminosity of 50 ab-1. Constraints across the
whole energy range are provided by the NA64 experiment with muon beam
NA64:2024klw  (solid yellow line), CCFR (solid light grey line) probing the
neutrino trident production Altmannshofer:2014pba  and by Belle II, probing
the 2$\mu+\not{E}$ process (solid light pink line) Belle-II:2022yaw .
Constraints that probe generic new light vector boson models are given by KLOE
(solid purple line) KLOE-2:2018kqf  and CMS (dashed brown line) CMS:2023hwl ,
both probing the 2$\mu+\not{E}$ final state, and by LHCb (dashed black and
solid black lines respectively) LHCb:2019vmc  with a search in the prompt and
displaced 4$\mu$ channels; with values of $\epsilon$ being recast into the
$U(1)_{L_{\mu}-L_{\tau}}$ model considered here. Other relevant constraints,
in addition to CCFR, are given by Babar BaBar:2016sci , Belle II Belle:2021feg
; Belle-II:2024wtd  and CMS CMS:2018yxg , which look for resonances in the
prompt 4$\mu$ channel. The purple band indicates the parameter range which
would solve, at 90% confidence level, the $4.2\sigma$ muon $g-2$ discrepancy,
see the text for further details.

##  7 Conclusions

In this work we have explored the sensitivity of the Belle II experiment on a
minimal dark photon (DP) model with an underlying $L_{\mu}-L_{\tau}$ extra
U(1) gauge group. Assuming the 50 ab-1 target luminosity of Belle II, we
projected the sensitivity contours of multiple signatures on the model’s
parameter space which, in its minimal configuration, reduces to the DP’s mass-
coupling plane $m_{X}-g_{X}$. We considered the most promising signatures
providing the best sensitivity at lepton colliders. They include final states
where the DP appears as missing energy, such as the monophoton and two muons
plus missing energy channels, and also multi-lepton final states where a pair
of muons originates from the DP decay. The latter can arise as a prompt
signature, i.e. as a resonant excess in a narrow invariant mass bin, or as a
displaced signature, where the long-lived DP decays at a measurable distance
with respect to the principal interaction point. For the analysis of this
particular signature, we included the simulation of displaced decay events in
both signal and background samples. In all the simulated events we considered
the detector’s fiducial acceptance and reconstruction efficiencies for each
particle, and for the VXD a resolution of 5 $\mu$m is assumed. The sensitivity
projections are compared to the most recent results from direct searches of
DPs at lepton colliders from the BaBar, Belle and KLOE experiments, at the LHC
from the CMS and LHCb experiments, and to indirect probes using CCFR neutrino
trident production data and the muon anomalous magnetic moment $g-2$.

We find that with its ultimate 50 ab-1 target dataset, the Belle-II
sensitivity on light DPs will improve on all existing exclusion limits. For
very light DPs, i.e. below the two muon decay threshold, we observe the
strongest sensitivity in the monophoton channel, which will overlap or extend
the sensitivity band of the muon $g-2$ indirect probe. When the DP is heavy
enough to decay into a pair of muons, multi-leptonic final states will provide
the most sensitive signatures, and only for heavy DPs ($m_{X}>8$ Gev) the LHC
sensitivity becomes dominant.

The displaced decay signature in particular reveals a significant potential
for the searches of narrow, long-lived DPs. Despite its limited sensitivity
region, it allows to probe parameter regions not accessible through other
signatures. The shape of the sensitivity contour is strongly dependent on the
choice of the minimal VXD spacial resolution. In the minimal
$L_{\mu}-L_{\tau}$ model, the typical DP decay displacements are indeed
comparable with the VXD resolution. Assuming 5 $\mu$m, we showed that the
displaced analysis can test DP couplings $g_{X}\sim 10^{-4}$ and below,
improving current bounds by one order of magnitude. While this assumption may
seem optimistic, future developments in the muon track reconstruction may
allow this level of precision or even better. In fact, even small further
improvements of the VXD resolution would lead to critical reductions of the SM
fake displaced background rates. This in turn would significantly increase the
sensitivity of long-lived particles searches and eventually turn this
signature into a very promising channel for the discovery of such BSM signals.

While $L_{\mu}-L_{\tau}$ DPs would also decay into tau pairs with a
significant rate, we did not focus on signatures involving tau leptons,
because the experimental challenges connected to their reconstruction
generally decrease the overall sensitivity in comparison with purely muonic
final states. Yet, we stress that a combined fit of different search channels
can help to identify the specific characteristics of the new physics model,
such as an additional BSM symmetry group. For instance, the equivalence of DP
decay rates into muons and taus would provide strong indications of an
underlying $L_{\mu}-L_{\tau}$ symmetry. Similarly, assessing the BSM
contribution to the monophoton channel gives a direct handle on the kinetic
mixing parameter $\epsilon$. This parameter is very sensitive to the presence
of additional hidden states, and can help to distinguish between BSM models
with extra gauged $U(1)_{X}$ symmetries and generic models featuring
additional vector boson resonances.

The analysis methods presented in this work are very general and can be
applied to various BSM scenarios across different collider experiments. The
generic procedure, to determine for each signature the optimal kinematic cuts
maximising the experimental sensitivity to a new physics signal, can be
applied to different detector configurations and BSM models. In addition, our
analysis of displaced signatures, which includes the simulation of both signal
and background contributions, can be easily adapted to specific experimental
environments and BSM constructions predicting long-lived particles. To fully
assess the ultimate experimental sensitivity, more accurate theoretical
predictions including higher order corrections and extra radiation are
required, together with detailed detector simulations. While this level of
analysis is beyond the scope of our current study, we plan to address some of
these aspects in future work.

## Acknowledgements

We thank Prof. Gianluca Inguglia for useful discussions on the Belle II
detector. CB is supported by an STFC PGR studentship. JF and TT were supported
by the STFC Consolidated Grant ST/T000988/1 and TT currently by ST/X000699/1.
JF acknowledges financial support from ICSC – Centro Nazionale di Ricerca in
High Performance Computing, Big Data and Quantum Computing, funded by the
European Union – NextGenerationEU.

## References

  * (1) Muon g-2 collaboration, _Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm_ , _Phys. Rev. Lett._ 126 (2021) 141801 [2104.03281].
  * (2) Muon g-2 collaboration, _Measurement of the Positive Muon Anomalous Magnetic Moment to 0.20 ppm_ , 2308.06230.
  * (3) Muon g-2 collaboration, _Detailed Report on the Measurement of the Positive Muon Anomalous Magnetic Moment to 0.20 ppm_ , 2402.15410.
  * (4) CDF collaboration, _High-precision measurement of the $W$ boson mass with the CDF II detector_, _Science_ 376 (2022) 170.
  * (5) M. Bauer and P. Foldenauer, _Consistent Theory of Kinetic Mixing and the Higgs Low-Energy Theorem_ , _Phys. Rev. Lett._ 129 (2022) 171801 [2207.00023].
  * (6) R. Essig, P. Schuster and N. Toro, _Probing Dark Forces and Light Hidden Sectors at Low-Energy e+e- Colliders_ , _Phys. Rev. D_ 80 (2009) 015003 [0903.3941].
  * (7) T. Aoyama et al., _The anomalous magnetic moment of the muon in the Standard Model_ , _Phys. Rept._ 887 (2020) 1 [2006.04822].
  * (8) G. Colangelo et al., _Prospects for precise predictions of $a_{\mu}$ in the Standard Model_, 2203.15810.
  * (9) CMD-3 collaboration, _Measurement of the $e^{+}e^{-}\to\pi^{+}\pi^{-}$ cross section from threshold to 1.2 GeV with the CMD-3 detector_, 2302.08834.
  * (10) CMD-3 collaboration, _Measurement of the pion formfactor with CMD-3 detector and its implication to the hadronic contribution to muon (g-2)_ , 2309.12910.
  * (11) S. Borsanyi et al., _Leading hadronic contribution to the muon magnetic moment from lattice QCD_ , _Nature_ 593 (2021) 51 [2002.12347].
  * (12) D. Curtin, R. Essig, S. Gori and J. Shelton, _Illuminating Dark Photons with High-Energy Colliders_ , _JHEP_ 02 (2015) 157 [1412.0018].
  * (13) P. Ilten, Y. Soreq, M. Williams and W. Xue, _Serendipity in dark photon searches_ , _JHEP_ 06 (2018) 004 [1801.04847].
  * (14) LHCb collaboration, _Search for Dark Photons Produced in 13 TeV $pp$ Collisions_, _Phys. Rev. Lett._ 120 (2018) 061801 [1710.02867].
  * (15) LHCb collaboration, _Search for $A^{\prime}\to\mu^{+}\mu^{-}$ Decays_, _Phys. Rev. Lett._ 124 (2020) 041801 [1910.06926].
  * (16) ATLAS collaboration, _Search for light long-lived neutral particles produced in $pp$ collisions at $\sqrt{s}=$ 13 TeV and decaying into collimated leptons or light hadrons with the ATLAS detector_, _Eur. Phys. J. C_ 80 (2020) 450 [1909.01246].
  * (17) ATLAS collaboration, _Search for dark photons from Higgs boson decays via $ZH$ production with a photon plus missing transverse momentum signature from $pp$ collisions at $\sqrt{s}$ = 13 TeV with the ATLAS detector_, _JHEP_ 07 (2023) 133 [2212.09649].
  * (18) CMS collaboration, _Search for direct production of GeV-scale resonances decaying to a pair of muons in proton-proton collisions at $\sqrt{s}$ = 13 TeV_, _JHEP_ 12 (2023) 070 [2309.16003].
  * (19) CMS collaboration, _Search for dark photons in Higgs boson production via vector boson fusion in proton-proton collisions at $\sqrt{s}$ = 13 TeV_, _JHEP_ 03 (2021) 011 [2009.14009].
  * (20) CMS collaboration, _Search for dark photons in decays of Higgs bosons produced in association with Z bosons in proton-proton collisions at $\sqrt{s}$ = 13 TeV_, _JHEP_ 10 (2019) 139 [1908.02699].
  * (21) CMS collaboration, _Search for an $L_{\mu}-L_{\tau}$ gauge boson using Z$\to 4\mu$ events in proton-proton collisions at $\sqrt{s}=$ 13 TeV_, _Phys. Lett. B_ 792 (2019) 345 [1808.03684].
  * (22) BaBar collaboration, _Search for a Dark Photon in $e^{+}e^{-}$ Collisions at BaBar_, _Phys. Rev. Lett._ 113 (2014) 201801 [1406.2980].
  * (23) BaBar collaboration, _Search for a muonic dark force at BABAR_ , _Phys. Rev. D_ 94 (2016) 011102 [1606.03501].
  * (24) KLOE-2 collaboration, _Search for light vector boson production in $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}\gamma$ interactions with the KLOE experiment_, _Phys. Lett. B_ 736 (2014) 459 [1404.7772].
  * (25) Belle collaboration, _Search for Z’→ $\mu$+$\mu$\- in the L$\mu$-L$\tau$ gauge-symmetric model at Belle_, _Phys. Rev. D_ 106 (2022) 012003 [2109.08596].
  * (26) K. Ban, Y. Jho, Y. Kwon, S.C. Park, S. Park and P.-Y. Tseng, _Search for new light vector boson using $J/\Psi$ at BESIII and Belle II_, _JHEP_ 04 (2021) 091 [2012.04190].
  * (27) Belle-II collaboration, _Search for a $\tau^{+}\tau^{-}$ resonance in $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}\tau^{+}\tau^{-}$ events with the Belle II experiment_, _Phys. Rev. Lett._ 131 (2023) 121802 [2306.12294].
  * (28) BaBar collaboration, _Search for Invisible Decays of a Dark Photon Produced in ${e}^{+}{e}^{-}$ Collisions at BaBar_, _Phys. Rev. Lett._ 119 (2017) 131804 [1702.03327].
  * (29) B. Batell, R. Essig and Z. Surujon, _Strong Constraints on Sub-GeV Dark Sectors from SLAC Beam Dump E137_ , _Phys. Rev. Lett._ 113 (2014) 171802 [1406.2698].
  * (30) NA64 collaboration, _First Results in the Search for Dark Sectors at NA64 with the CERN SPS High Energy Muon Beam_ , _Phys. Rev. Lett._ 132 (2024) 211803 [2401.01708].
  * (31) A. Kamada and H.-B. Yu, _Coherent Propagation of PeV Neutrinos and the Dip in the Neutrino Spectrum at IceCube_ , _Phys. Rev. D_ 92 (2015) 113004 [1504.00711].
  * (32) D. Hooper, J. Iguaz Juan and P.D. Serpico, _Signals of a new gauge boson from IceCube and the muon g-2_ , _Phys. Rev. D_ 108 (2023) 023007 [2302.03571].
  * (33) W. Altmannshofer, S. Gori, M. Pospelov and I. Yavin, _Neutrino Trident Production: A Powerful Probe of New Physics with Neutrino Beams_ , _Phys. Rev. Lett._ 113 (2014) 091801 [1406.2332].
  * (34) P. deNiverville, M. Pospelov and A. Ritz, _Observing a light dark matter beam with neutrino experiments_ , _Phys. Rev. D_ 84 (2011) 075020 [1107.4580].
  * (35) CHARM-II collaboration, _First observation of neutrino trident production_ , _Phys. Lett. B_ 245 (1990) 271.
  * (36) CCFR collaboration, _Neutrino Tridents and W Z Interference_ , _Phys. Rev. Lett._ 66 (1991) 3117.
  * (37) S. Gninenko and D. Gorbunov, _Refining constraints from Borexino measurements on a light Z’-boson coupled to L $\mu$-L$\tau$ current_, _Phys. Lett. B_ 823 (2021) 136739 [2007.16098].
  * (38) G. Bellini et al., _Precision measurement of the 7Be solar neutrino interaction rate in Borexino_ , _Phys. Rev. Lett._ 107 (2011) 141302 [1104.1816].
  * (39) D.W.P.d. Amaral, D.G. Cerdeno, P. Foldenauer and E. Reid, _Solar neutrino probes of the muon anomalous magnetic moment in the gauged $\mathrm{U}{(1)}_{L_{\mu}-{L}_{\tau}}$_, _JHEP_ 12 (2020) 155 [2006.11225].
  * (40) R. Essig, A. Manalaysay, J. Mardon, P. Sorensen and T. Volansky, _First Direct Detection Limits on sub-GeV Dark Matter from XENON10_ , _Phys. Rev. Lett._ 109 (2012) 021301 [1206.2644].
  * (41) R. Essig, T. Volansky and T.-T. Yu, _New Constraints and Prospects for sub-GeV Dark Matter Scattering off Electrons in Xenon_ , _Phys. Rev. D_ 96 (2017) 043017 [1703.00910].
  * (42) P. Figueroa, G. Herrera and F. Ochoa, _Direct detection of light dark matter charged under a $L_{\mu}-L_{\tau}$ symmetry_, 2404.03090.
  * (43) A. Aboubrahim, M.M. Altakach, M. Klasen, P. Nath and Z.-Y. Wang, _Combined constraints on dark photons and discovery prospects at the LHC and the Forward Physics Facility_ , 2212.01268.
  * (44) P. Foldenauer and J. Hoefken Zink, _How to rule out $(g-2)_{\mu}$ in $U(1)_{L_{\mu}-L_{\tau}}$ with White Dwarf Cooling_, 2405.00094.
  * (45) C. Giovanetti, M. Lisanti, H. Liu and J.T. Ruderman, _Joint Cosmic Microwave Background and Big Bang Nucleosynthesis Constraints on Light Dark Sectors with Dark Radiation_ , _Phys. Rev. Lett._ 129 (2022) 021302 [2109.03246].
  * (46) M. Escudero, D. Hooper, G. Krnjaic and M. Pierre, _Cosmology with A Very Light L μ $-$ Lτ Gauge Boson_, _JHEP_ 03 (2019) 071 [1901.02010].
  * (47) P. Foldenauer, _Dark Sectors from the Hidden Photon Perspective_ , _PoS_ LHCP2019 (2019) 179 [1907.10630].
  * (48) A. Kamada, K. Kaneta, K. Yanagi and H.-B. Yu, _Self-interacting dark matter and muon $g-2$ in a gauged U$(1)_{L_{\mu}-L_{\tau}}$ model_, _JHEP_ 06 (2018) 117 [1805.00651].
  * (49) M. Bauer, P. Foldenauer and J. Jaeckel, _Hunting All the Hidden Photons_ , _JHEP_ 07 (2018) 094 [1803.05466].
  * (50) J.L. Feng, J. Smolinsky and P. Tanedo, _Detecting dark matter through dark photons from the Sun: Charged particle signatures_ , _Phys. Rev. D_ 93 (2016) 115036 [1602.01465].
  * (51) Fermi-LAT collaboration, _Searching for Dark Matter Annihilation from Milky Way Dwarf Spheroidal Galaxies with Six Years of Fermi Large Area Telescope Data_ , _Phys. Rev. Lett._ 115 (2015) 231301 [1503.02641].
  * (52) LHCb collaboration, _Searches for low-mass dimuon resonances_ , _JHEP_ 10 (2020) 156 [2007.03923].
  * (53) L. Lee, C. Ohm, A. Soffer and T.-T. Yu, _Collider Searches for Long-Lived Particles Beyond the Standard Model_ , _Prog. Part. Nucl. Phys._ 106 (2019) 210 [1810.12602].
  * (54) J. Alimena et al., _Searching for long-lived particles beyond the Standard Model at the Large Hadron Collider_ , _J. Phys. G_ 47 (2020) 090501 [1903.04497].
  * (55) E. Bernreuther, J.C. Mejia, F. Kahlhoefer, M. Krämer and P. Tunney, _On the challenges of searching for GeV-scale long-lived particles at the LHC_ , _JHEP_ 04 (2021) 210 [2011.06604].
  * (56) S.N. Gninenko, N.V. Krasnikov and V.A. Matveev, _Muon g-2 and searches for a new leptophobic sub-GeV dark boson in a missing-energy experiment at CERN_ , _Phys. Rev. D_ 91 (2015) 095015 [1412.1400].
  * (57) S.N. Gninenko and N.V. Krasnikov, _Probing the muon $g_{\mu}$ \- 2 anomaly, $L_{\mu}-L_{\tau}$ gauge boson and Dark Matter in dark photon experiments_, _Phys. Lett. B_ 783 (2018) 24 [1801.10448].
  * (58) S. Gninenko, S. Kovalenko, S. Kuleshov, V.E. Lyubovitskij and A.S. Zhevlakov, _Deep inelastic $e-\tau$ and $\mu-\tau$ conversion in the NA64 experiment at the CERN SPS_, _Phys. Rev. D_ 98 (2018) 015007 [1804.05550].
  * (59) NA48/2 collaboration, _Search for the dark photon in $\pi^{0}$ decays_, _Phys. Lett. B_ 746 (2015) 178 [1504.00607].
  * (60) I. Galon, D. Shih and I.R. Wang, _Dark Photons and Displaced Vertices at the MUonE Experiment_ , 2202.08843.
  * (61) J.L. Feng, I. Galon, F. Kling and S. Trojanowski, _ForwArd Search ExpeRiment at the LHC_ , _Phys. Rev. D_ 97 (2018) 035001 [1708.09389].
  * (62) J.L. Feng, I. Galon, F. Kling and S. Trojanowski, _Axionlike particles at FASER: The LHC as a photon beam dump_ , _Phys. Rev. D_ 98 (2018) 055021 [1806.02348].
  * (63) FASER collaboration, _FASER’s physics reach for long-lived particles_ , _Phys. Rev. D_ 99 (2019) 095011 [1811.12522].
  * (64) D. Curtin and M.E. Peskin, _Analysis of Long Lived Particle Decays with the MATHUSLA Detector_ , _Phys. Rev. D_ 97 (2018) 015006 [1705.06327].
  * (65) D. Curtin et al., _Long-Lived Particles at the Energy Frontier: The MATHUSLA Physics Case_ , _Rept. Prog. Phys._ 82 (2019) 116201 [1806.07396].
  * (66) V.V. Gligorov, S. Knapen, M. Papucci and D.J. Robinson, _Searching for Long-lived Particles: A Compact Detector for Exotics at LHCb_ , _Phys. Rev. D_ 97 (2018) 015023 [1708.09395].
  * (67) F. Bossi, _Dark Photon Searches Using Displaced Vertices at Low Energy $e^{+}e^{-}$ Colliders_, _Adv. High Energy Phys._ 2014 (2014) 891820 [1310.8181].
  * (68) KLOE-2 collaboration, _Combined limit on the production of a light gauge boson decaying into $\mu^{+}\mu^{-}$ and $\pi^{+}\pi^{-}$_, _Phys. Lett. B_ 784 (2018) 336 [1807.02691].
  * (69) BaBar collaboration, _Search for Long-Lived Particles in $e^{+}e^{-}$ Collisions_, _Phys. Rev. Lett._ 114 (2015) 171801 [1502.02580].
  * (70) Belle-II collaboration, _Belle II Technical Design Report_ , 1011.0352.
  * (71) M. Duerr, T. Ferber, C. Hearty, F. Kahlhoefer, K. Schmidt-Hoberg and P. Tunney, _Invisible and displaced dark matter signatures at Belle II_ , _JHEP_ 02 (2020) 039 [1911.03176].
  * (72) M. Duerr, T. Ferber, C. Garcia-Cely, C. Hearty and K. Schmidt-Hoberg, _Long-lived Dark Higgs and Inelastic Dark Matter at Belle II_ , _JHEP_ 04 (2021) 146 [2012.08595].
  * (73) E. Bernreuther, K. Böse, T. Ferber, C. Hearty, F. Kahlhoefer, A. Morandini et al., _Forecasting dark showers at Belle II_ , _JHEP_ 12 (2022) 005 [2203.08824].
  * (74) T. Ferber, C. Garcia-Cely and K. Schmidt-Hoberg, _BelleII sensitivity to long–lived dark photons_ , _Phys. Lett. B_ 833 (2022) 137373 [2202.03452].
  * (75) Belle-II collaboration, _Search for a long-lived spin-0 mediator in $b\to s$ transitions at the Belle II experiment_, 2306.02830.
  * (76) T. Bandyopadhyay, S. Chakraborty and S. Trifinopoulos, _Displaced searches for light vector bosons at Belle II_ , _JHEP_ 05 (2022) 141 [2203.03280].
  * (77) T. Bandyopadhyay, _Dark Photons from displaced vertices_ , 2311.16997.
  * (78) J. Heeck and W. Rodejohann, _Gauged $L_{\mu}-L_{\tau}$ Symmetry at the Electroweak Scale_, _Phys. Rev. D_ 84 (2011) 075007 [1107.5238].
  * (79) K.S. Babu, C.F. Kolda and J. March-Russell, _Implications of generalized Z - Z-prime mixing_ , _Phys. Rev. D_ 57 (1998) 6788 [hep-ph/9710441].
  * (80) Particle Data Group collaboration, _Review of Particle Physics_ , _PTEP_ 2022 (2022) 083C01.
  * (81) Belle-II collaboration, _The Belle II Physics Book_ , _PTEP_ 2019 (2019) 123C01 [1808.10567].
  * (82) Belle-II collaboration, _Recent results from Belle II_ , _Nucl. Part. Phys. Proc._ 324-329 (2023) 107.
  * (83) Y. Kaneta and T. Shimomura, _On the possibility of a search for the $L_{\mu}-L_{\tau}$ gauge boson at Belle-II and neutrino beam experiments_, _PTEP_ 2017 (2017) 053B04 [1701.00156].
  * (84) T. Araki, S. Hoshino, T. Ota, J. Sato and T. Shimomura, _Detecting the $L_{\mu}-L_{\tau}$ gauge boson at Belle II_, _Phys. Rev. D_ 95 (2017) 055006 [1702.01497].
  * (85) Belle-II collaboration, _Search for an Invisibly Decaying $Z^{\prime}$ Boson at Belle II in $e^{+}e^{-}\to\mu^{+}\mu^{-}(e^{\pm}\mu^{\mp})$ Plus Missing Energy Final States_, _Phys. Rev. Lett._ 124 (2020) 141801 [1912.11276].
  * (86) Belle-II collaboration, _Search for an invisible $Z^{\prime}$ in a final state with two muons and missing energy at Belle II_, 2212.03066.
  * (87) Belle-II collaboration, _First data at Belle II and Dark Sector physics_ , _PoS_ BEAUTY2018 (2018) 034 [1808.00776].
  * (88) Belle II collaboration, _Dark-sector physics at Belle II_ , _PoS_ PANIC2021 (2022) 047.
  * (89) BABAR collaboration, _Search for Muonic Dark Forces at BABAR_ , _EPJ Web Conf._ 141 (2017) 02005 [1611.07934].
  * (90) N.D. Christensen and C. Duhr, _FeynRules - Feynman rules made easy_ , _Comput. Phys. Commun._ 180 (2009) 1614 [0806.4194].
  * (91) W. Kilian, T. Ohl and J. Reuter, _WHIZARD: Simulating Multi-Particle Processes at LHC and ILC_ , _Eur. Phys. J. C_ 71 (2011) 1742 [0708.4233].
  * (92) M. Moretti, T. Ohl and J. Reuter, _O’Mega: An Optimizing matrix element generator_ , hep-ph/0102195.
  * (93) N.D. Christensen, C. Duhr, B. Fuks, J. Reuter and C. Speckner, _Introducing an interface between WHIZARD and FeynRules_ , _Eur. Phys. J. C_ 72 (2012) 1990 [1010.3251].
  * (94) T. Ferber, A. Filimonova, R. Schäfer and S. Westhoff, _Displaced or invisible? ALPs from $B$ decays at Belle II_, 2201.06580.
  * (95) Belle-II Framework Software Group collaboration, _The Belle II Core Software_ , _Comput. Softw. Big Sci._ 3 (2019) 1 [1809.04299].
  * (96) Belle-II collaboration, _Measurements of branching fractions and CP-violating charge asymmetries in multibody charmless $B$ decays reconstructed in 2019-2020 Belle II data_, 2109.10807.
  * (97) A. Abashian et al., _Muon identification in the Belle experiment at KEKB_ , _Nucl. Instrum. Meth. A_ 491 (2002) 69.
  * (98) Belle-II collaboration, _Photon Reconstruction in the Belle II Calorimeter Using Graph Neural Networks_ , _Comput. Softw. Big Sci._ 7 (2023) 13 [2306.04179].
  * (99) H. Svidras, M. Röhrken, S. Wehle and K. Tackmann, _Measurement of the data to mc ratio of photon reconstruction efficiency of the belle ii calorimeter using radiative muon pair events_ , .
  * (100) Belle-II SVD collaboration, _The Belle II silicon vertex detector assembly and mechanics_ , _Nucl. Instrum. Meth. A_ 845 (2017) 38.
  * (101) Belle-II SVD collaboration, _The design, construction, operation and performance of the Belle II silicon vertex detector_ , _JINST_ 17 (2022) P11042 [2201.09824].
  * (102) Belle-II collaboration, _Performance of the Belle II Silicon Vertex Detector_ , _PoS_ Vertex2019 (2020) 014.
  * (103) P. Wieduwilt et al., _Performance of production modules of the Belle II pixel detector in a high-energy particle beam_ , _Nucl. Instrum. Meth. A_ 991 (2021) 164978 [2101.10107].
  * (104) R.K. Maiti, _Search for muonic dark force and performance studies of lepton identification at Belle II_ , Ph.D. thesis, Institut of High Energy Physics, Vienna, Tech. U., 2023.
  * (105) Belle-II SVD collaboration, _The Silicon Vertex Detector of the Belle II experiment_ , _Nucl. Instrum. Meth. A_ 1061 (2024) 169131.
  * (106) T. Aoyama, M. Hayakawa, T. Kinoshita and M. Nio, _Complete Tenth-Order QED Contribution to the Muon $g-2$_, _Phys. Rev. Lett._ 109 (2012) 111808 [1205.5370].
  * (107) T. Aoyama, T. Kinoshita and M. Nio, _Theory of the Anomalous Magnetic Moment of the Electron_ , _Atoms_ 7 (2019) 28.
  * (108) A. Czarnecki, W.J. Marciano and A. Vainshtein, _Refinements in electroweak contributions to the muon anomalous magnetic moment_ , _Phys. Rev._ D67 (2003) 073006 [hep-ph/0212229].
  * (109) C. Gnendiger, D. Stöckinger and H. Stöckinger-Kim, _The electroweak contributions to $(g-2)_{\mu}$ after the Higgs boson mass measurement_, _Phys. Rev._ D88 (2013) 053005 [1306.5546].
  * (110) M. Davier, A. Hoecker, B. Malaescu and Z. Zhang, _Reevaluation of the hadronic vacuum polarisation contributions to the Standard Model predictions of the muon $g-2$ and ${\alpha(m_{Z}^{2})}$ using newest hadronic cross-section data_, _Eur. Phys. J._ C77 (2017) 827 [1706.09436].
  * (111) A. Keshavarzi, D. Nomura and T. Teubner, _Muon $g-2$ and $\alpha(M_{Z}^{2})$: a new data-based analysis_, _Phys. Rev._ D97 (2018) 114025 [1802.02995].
  * (112) G. Colangelo, M. Hoferichter and P. Stoffer, _Two-pion contribution to hadronic vacuum polarization_ , _JHEP_ 02 (2019) 006 [1810.00007].
  * (113) M. Hoferichter, B.-L. Hoid and B. Kubis, _Three-pion contribution to hadronic vacuum polarization_ , _JHEP_ 08 (2019) 137 [1907.01556].
  * (114) M. Davier, A. Hoecker, B. Malaescu and Z. Zhang, _A new evaluation of the hadronic vacuum polarisation contributions to the muon anomalous magnetic moment and to $\mathbf{\boldsymbol{\alpha}(m_{Z}^{2})}$_, _Eur. Phys. J._ C80 (2020) 241 [1908.00921].
  * (115) A. Keshavarzi, D. Nomura and T. Teubner, _$g-2$ of charged leptons, $\alpha(M^{2}_{Z})$ , and the hyperfine splitting of muonium_, _Phys. Rev. D_ 101 (2020) 014029 [1911.00367].
  * (116) A. Kurz, T. Liu, P. Marquard and M. Steinhauser, _Hadronic contribution to the muon anomalous magnetic moment to next-to-next-to-leading order_ , _Phys. Lett._ B734 (2014) 144 [1403.6400].
  * (117) K. Melnikov and A. Vainshtein, _Hadronic light-by-light scattering contribution to the muon anomalous magnetic moment revisited_ , _Phys. Rev._ D70 (2004) 113006 [hep-ph/0312226].
  * (118) P. Masjuan and P. Sánchez-Puertas, _Pseudoscalar-pole contribution to the $(g_{\mu}-2)$: a rational approach_, _Phys. Rev._ D95 (2017) 054026 [1701.05829].
  * (119) G. Colangelo, M. Hoferichter, M. Procura and P. Stoffer, _Dispersion relation for hadronic light-by-light scattering: two-pion contributions_ , _JHEP_ 04 (2017) 161 [1702.07347].
  * (120) M. Hoferichter, B.-L. Hoid, B. Kubis, S. Leupold and S.P. Schneider, _Dispersion relation for hadronic light-by-light scattering: pion pole_ , _JHEP_ 10 (2018) 141 [1808.04823].
  * (121) A. Gérardin, H.B. Meyer and A. Nyffeler, _Lattice calculation of the pion transition form factor with $N_{f}=2+1$ Wilson quarks_, _Phys. Rev._ D100 (2019) 034520 [1903.09471].
  * (122) J. Bijnens, N. Hermansson-Truedsson and A. Rodríguez-Sánchez, _Short-distance constraints for the HLbL contribution to the muon anomalous magnetic moment_ , _Phys. Lett._ B798 (2019) 134994 [1908.03331].
  * (123) G. Colangelo, F. Hagelstein, M. Hoferichter, L. Laub and P. Stoffer, _Longitudinal short-distance constraints for the hadronic light-by-light contribution to $(g-2)_{\mu}$ with large-$N_{c}$ Regge models_, _JHEP_ 03 (2020) 101 [1910.13432].
  * (124) T. Blum, N. Christ, M. Hayakawa, T. Izubuchi, L. Jin, C. Jung et al., _The hadronic light-by-light scattering contribution to the muon anomalous magnetic moment from lattice QCD_ , _Phys. Rev. Lett._ 124 (2020) 132002 [1911.08123].
  * (125) G. Colangelo, M. Hoferichter, A. Nyffeler, M. Passera and P. Stoffer, _Remarks on higher-order hadronic corrections to the muon $g-2$_, _Phys. Lett._ B735 (2014) 90 [1403.7512].
  * (126) Belle-II collaboration, _Search for a $\mu$+$\mu$\- resonance in four-muon final states at Belle II_, _Phys. Rev. D_ 109 (2024) 112015 [2403.02841].