EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2019-212 LHCb-PAPER-2019-031 December 22, 2019

Search for $A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ decays

LHCb collaboration†††Authors are listed at the end of this Letter.

Searches are performed for both prompt-like and long-lived dark photons,
$A^{\prime}$, produced in proton-proton collisions at a center-of-mass energy
of 13$\mathrm{\,Te\kern-1.00006ptV}$. These searches look for
$A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ decays using a data sample corresponding
to an integrated luminosity of 5.5$\mbox{\,fb}^{-1}$ collected with the LHCb
detector. Neither search finds evidence for a signal, and 90% confidence-level
exclusion limits are placed on the $\gamma$–$A^{\prime}$ kinetic-mixing
strength. The prompt-like $A^{\prime}$ search explores the mass region from
near the dimuon threshold up to 70$\mathrm{\,Ge\kern-1.00006ptV}$, and places
the most stringent constraints to date on dark photons with
$214<m(A^{\prime})\lesssim 740\mathrm{\,Me\kern-1.00006ptV}$ and
$10.6<m(A^{\prime})\lesssim 30\mathrm{\,Ge\kern-1.00006ptV}$. The search for
long-lived $A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ decays places world-leading
constraints on low-mass dark photons with lifetimes
$\mathcal{O}(1)\mbox{\,ps}$.

Published as Physical Review Letters 124 (2020) 041801

© CERN on behalf of the LHCb collaboration, licence CC-BY-4.0.

Substantial effort has been dedicated recently [1, 2, 3] to searching for the
dark photon, $A^{\prime}$, a hypothetical massive vector boson that could
mediate the interactions of dark matter particles[4], similar to how the
ordinary photon, $\gamma$, mediates the electromagnetic (EM) interactions of
charged Standard Model (SM) particles. The dark photon does not couple
directly to SM particles; however, it can obtain a small coupling to the EM
current due to kinetic mixing between the SM hypercharge and $A^{\prime}$
field strength tensors [5, 6, 7, 8, 9, 10, 11, 12]. This coupling, which is
suppressed relative to that of the photon by a factor labeled $\varepsilon$,
would provide a portal through which dark photons can be produced in the
laboratory, and also via which they can decay into visible SM final states. If
the kinetic mixing arises due to processes described by one- or two-loop
diagrams containing high-mass particles, possibly even at the Planck scale,
then $10^{-12}\lesssim\varepsilon^{2}\lesssim 10^{-4}$ is expected [2].
Exploring this few-loop $\varepsilon$ region is one of the most important
near-term goals of dark-sector physics.

Dark photons will decay into visible SM particles if invisible dark-sector
decays are kinematically forbidden. Constraints have been placed on visible
$A^{\prime}$ decays by previous beam-dump [13, 14, 15, 16, 17, 18, 19, 20, 12,
21, 22, 23, 24, 25, 26, 27, 28], fixed-target [29, 30, 31, 32], collider [33,
34, 35, 36, 37, 38], and rare-meson-decay [39, 40, 41, 42, 43, 44, 45, 46, 47,
48] experiments. These experiments ruled out the few-loop region for dark-
photon masses $m(A^{\prime})\lesssim 10\mathrm{\,Me\kern-1.00006ptV}$ ($c=1$
throughout this Letter); however, most of the few-loop region at higher masses
remains unexplored. Constraints on invisible $A^{\prime}$ decays can be found
in Refs. [49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61]; only the
visible scenario is considered here.

Many ideas have been proposed to further explore the
$[m(A^{\prime}),\varepsilon^{2}]$ parameter space [62, 63, 64, 65, 66, 67, 68,
69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82]. The LHCb
collaboration previously performed a search based on the approach proposed in
Ref. [76] using data corresponding to 1.6$\mbox{\,fb}^{-1}$ collected in 2016
[83]. The constraints placed on prompt-like dark photons, where the dark-
photon lifetime is small compared to the detector resolution, were the most
stringent to date for $10.6<m(A^{\prime})<70\mathrm{\,Ge\kern-1.00006ptV}$ and
comparable to the best existing limits for
$m(A^{\prime})<0.5\mathrm{\,Ge\kern-1.00006ptV}$. The search for long-lived
dark photons was the first to achieve sensitivity using a displaced-vertex
signature, though only small regions of $[m(A^{\prime}),\varepsilon^{2}]$
parameter space were excluded.

This Letter presents searches for both prompt-like and long-lived dark photons
produced in proton-proton, $pp$, collisions at a center-of-mass energy of
13$\mathrm{\,Te\kern-1.00006ptV}$, looking for
$A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ decays using a data sample corresponding
to an integrated luminosity of 5.5$\mbox{\,fb}^{-1}$ collected with the LHCb
detector in 2016–2018. The strategies employed in these searches are the same
as in Ref. [83], though the three-fold increase in integrated luminosity,
improved trigger efficiency during 2017–2018 data taking, and improvements in
the analysis provide much better sensitivity to dark photons. The prompt-like
$A^{\prime}$ search is performed from near the dimuon threshold up to
70$\mathrm{\,Ge\kern-1.00006ptV}$, achieving a factor of 5 (2) better
sensitivity to $\varepsilon^{2}$ at low (high) masses than Ref. [83]. The
long-lived $A^{\prime}$ search is restricted to the mass range
$214<m(A^{\prime})<350\mathrm{\,Me\kern-1.00006ptV}$, where the data sample
potentially has sensitivity, and provides access to much larger regions of
$[m(A^{\prime}),\varepsilon^{2}]$ parameter space.

Both the production and decay kinematics of the
$A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ and $\gamma^{*}\\!\to\\!\mu^{+}\mu^{-}$
processes are identical, since dark photons produced in $pp$ collisions via
$\gamma$–$A^{\prime}$ mixing inherit the production mechanisms of off-shell
photons with $m(\gamma^{*})=m(A^{\prime})$. Furthermore, the expected
$A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ signal yield is related to the observed
prompt $\gamma^{*}\\!\to\\!\mu^{+}\mu^{-}$ yield in a small $\pm\Delta m$
window around $m(A^{\prime})$, $n_{\rm ob}^{\gamma^{*}}[m(A^{\prime})]$, by
[76]

$n_{\rm
ex}^{A^{\prime}}[m(A^{\prime}),\varepsilon^{2}]=\varepsilon^{2}\left[\frac{n_{\rm
ob}^{\gamma^{*}}[m(A^{\prime})]}{2\Delta
m}\right]\mathcal{F}[m(A^{\prime})]\,\epsilon_{{}^{\gamma^{*}}}^{{}_{A^{\prime}}}[m(A^{\prime}),\tau(A^{\prime})],$
(1)

where the dark-photon lifetime, $\tau(A^{\prime})$, is a known function of
$m(A^{\prime})$ and $\varepsilon^{2}$, $\mathcal{F}$ is a known
$m(A^{\prime})$-dependent function, and
$\epsilon_{{}^{\gamma^{*}}}^{{}_{A^{\prime}}}[m(A^{\prime}),\tau(A^{\prime})]$
is the $\tau(A^{\prime})$-dependent ratio of the
$A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ and $\gamma^{*}\\!\to\\!\mu^{+}\mu^{-}$
detection efficiencies. For prompt-like dark photons,
$A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ decays are experimentally
indistinguishable from prompt $\gamma^{*}\\!\to\\!\mu^{+}\mu^{-}$ decays,
resulting in
$\epsilon_{{}^{\gamma^{*}}}^{{}_{A^{\prime}}}[m(A^{\prime}),\tau(A^{\prime})]=1$.
This facilitates a fully data-driven search where most experimental systematic
effects cancel, since the observed $A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ yields,
$n_{\rm ob}^{A^{\prime}}[m(A^{\prime})]$, can be normalized to $n_{\rm
ex}^{A^{\prime}}[m(A^{\prime}),\varepsilon^{2}]$ to obtain constraints on
$\varepsilon^{2}$ without any knowledge of the detector efficiency or
luminosity. When $\tau(A^{\prime})$ is larger than the detector decay-time
resolution, $A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ decays can potentially be
reconstructed as displaced from the primary $pp$ vertex (PV) resulting in
$\epsilon_{{}^{\gamma^{*}}}^{{}_{A^{\prime}}}[m(A^{\prime}),\tau(A^{\prime})]\neq
1$; however, only the $\tau(A^{\prime})$ dependence of the detection
efficiency is required to use Eq. (1). Finally, Eq. (1) is altered for large
$m(A^{\prime})$ to account for additional kinetic mixing with the $Z$ boson
[84, 85].

The LHCb detector is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, described in detail in Refs. [86, 87]. The
prompt-like $A^{\prime}$ search is based on a data sample that employs a novel
data-storage strategy, made possible by advances in the LHCb data-taking
scheme introduced in 2015 [88, 89], where all online-reconstructed particles
are stored, but most lower-level information is discarded, greatly reducing
the event size. In contrast, the data sample used in the long-lived
$A^{\prime}$ search is derived from the standard LHCb data stream. Simulated
data samples, which are used to validate the analysis, are produced using the
software described in Refs. [90, *Sjostrand:2007gs, 92, 93,
*Agostinelli:2002hh].

The online event selection is performed by a trigger [95] consisting of a
hardware stage using information from the calorimeter and muon systems,
followed by a software stage that performs a full event reconstruction. At the
hardware stage, events are required to have a muon with momentum transverse to
the beam direction ${\mbox{$p_{\rm T}$}(\mu)\gtrsim
1.8\mathrm{\,Ge\kern-1.00006ptV}}$, or a dimuon pair with $\mbox{$p_{\rm
T}$}(\mu^{+})\mbox{$p_{\rm
T}$}(\mu^{-})\gtrsim(1.5\mathrm{\,Ge\kern-1.00006ptV})^{2}$. The long-lived
$A^{\prime}$ search also uses events selected at the hardware stage due to the
presence of a high-$p_{\rm T}$ hadron that is not associated to the
$A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ candidate. In the software stage, where
the $p_{\rm T}$ resolution is substantially improved cf. the hardware stage,
$A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ candidates are built from two oppositely
charged tracks that form a good-quality vertex and satisfy stringent muon-
identification criteria, though these criteria were loosened considerably in
the low-mass region during 2017–2018 data taking. Both searches require
$\mbox{$p_{\rm T}$}(A^{\prime})>1\mathrm{\,Ge\kern-1.00006ptV}$ and
$2<\eta(\mu)<4.5$. The prompt-like $A^{\prime}$ search uses muons that are
consistent with originating from the PV, with ${\mbox{$p_{\rm
T}$}(\mu)>1.0\mathrm{\,Ge\kern-1.00006ptV}}$ and momentum
${p(\mu)>20\mathrm{\,Ge\kern-1.00006ptV}}$ in 2016, and $\mbox{$p_{\rm
T}$}(\mu)>0.5\mathrm{\,Ge\kern-1.00006ptV}$,
${p(\mu)>10\mathrm{\,Ge\kern-1.00006ptV}}$, and ${\mbox{$p_{\rm
T}$}(\mu^{+})\mbox{$p_{\rm
T}$}(\mu^{-})>(1.0\mathrm{\,Ge\kern-1.00006ptV})^{2}}$ in 2017–2018. The long-
lived $A^{\prime}$ search uses muons that are inconsistent with originating
from any PV with $\mbox{$p_{\rm T}$}(\mu)>0.5\mathrm{\,Ge\kern-1.00006ptV}$
and ${p(\mu)>10\mathrm{\,Ge\kern-1.00006ptV}}$, and requires
$2<\eta(A^{\prime})<4.5$ and a decay topology consistent with a dark photon
originating from a PV.

The prompt-like $A^{\prime}$ sample is contaminated by prompt
${\gamma^{*}\\!\to\\!\mu^{+}\mu^{-}}$ production, various resonant decays to
$\mu^{+}\mu^{-}$, whose mass-peak regions are avoided in the search, and by
the following types of misreconstruction: ($hh$) two prompt hadrons
misidentified as muons; ($h\mu_{Q}$) a misidentified prompt hadron combined
with a muon produced in the decay of a heavy-flavor quark, $Q$, that is
misidentified as prompt; and ($\mu_{Q}\mu_{Q}$) two muons produced in
$Q$-hadron decays that are both misidentified as prompt. Contamination from a
prompt muon and a misidentified prompt hadron is negligible, though it is
accounted for automatically by the method used to determine the sum of the
$hh$ and $h\mu_{Q}$ contributions. The impact of the
$\gamma^{*}\\!\to\\!\mu^{+}\mu^{-}$ background is reduced, cf. Ref. [83], by
constraining the muons to originate from the PV when determining
$m(\mu^{+}\mu^{-})$. This improves the resolution,
$\sigma[m(\mu^{+}\mu^{-})]$, by about a factor of $2$ for small
$m(A^{\prime})$. The misreconstructed backgrounds are highly suppressed by the
stringent requirements applied in the trigger; however, substantial
contributions remain for $m(A^{\prime})\gtrsim
1.1\mathrm{\,Ge\kern-1.00006ptV}$. In this mass region, dark photons are
expected to be predominantly produced in Drell–Yan processes, from which they
would inherit the well-known signature of dimuon pairs that are largely
isolated. Therefore, the signal sensitivity is enhanced by applying the
anti-$k_{\rm T}$-based [96, 97, 98] isolation requirement described in Refs.
[83, 99] for $m(A^{\prime})>1.1\mathrm{\,Ge\kern-1.00006ptV}$.

The observed prompt-like $A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ yields, which are
determined from fits to the $m(\mu^{+}\mu^{-})$ spectrum, are normalized using
Eq. (1) to obtain constraints on $\varepsilon^{2}$. The $n_{\rm
ob}^{\gamma^{*}}[m(A^{\prime})]$ values in Eq. (1) are obtained from binned
extended maximum likelihood fits to the ${\rm
min}[\chi^{2}_{\mathrm{IP}}(\mu^{\pm})]$ distributions, where
$\chi^{2}_{\mathrm{IP}}(\mu)$ is defined as the difference in the vertex-fit
$\chi^{2}$ when the PV is reconstructed with and without the muon. The ${\rm
min}[\chi^{2}_{\mathrm{IP}}(\mu^{\pm})]$ distribution provides excellent
discrimination between prompt muons and the displaced muons that constitute
the $\mu_{Q}\mu_{Q}$ background. The $\chi^{2}_{\mathrm{IP}}(\mu)$ quantity
approximately follows a $\chi^{2}$ probability density function (PDF), with
two degrees of freedom, and therefore, the ${\rm
min}[\chi^{2}_{\mathrm{IP}}(\mu^{\pm})]$ distributions have minimal dependence
on mass for each source of dimuon candidates. The prompt-dimuon PDFs are taken
directly from data at $m({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}})$ and
$m(Z)$, where prompt resonances are dominant. Small corrections are applied to
obtain these PDFs at all other $m(A^{\prime})$, which are validated near
threshold, at $m(\phi)$, and at $m[\mathchar 28935\relax(1S)]$, where the data
predominantly consist of prompt dimuon pairs. Based on these validation
studies, a shape uncertainty of 2% is applied in each ${\rm
min}[\chi^{2}_{\mathrm{IP}}(\mu^{\pm})]$ bin. Same-sign $\mu^{\pm}\mu^{\pm}$
candidates provide estimates for the PDF and yield of the sum of the $hh$ and
$h\mu_{Q}$ contributions, where each involves misidentified prompt hadrons.
The $\mu^{\pm}\mu^{\pm}$ yields are corrected to account for the difference in
the production rates of $\pi^{+}\pi^{-}$ and $\pi^{\pm}\pi^{\pm}$, which are
determined precisely from data using dipion candidates weighted to account for
the kinematic dependence of the muon misidentification probability, since the
$hh$ background largely consists of $\pi^{+}\pi^{-}$ pairs where both pions
are misidentified. The uncertainty due to the finite size of the
$\mu^{\pm}\mu^{\pm}$ sample in each bin is included in the likelihood.
Simulated $Q$-hadron decays are used to obtain the $\mu_{Q}\mu_{Q}$ PDFs,
where the dominant uncertainties are from the relative importance of the
various $Q$-hadron decay contributions at each mass. Example ${\rm
min}[\chi^{2}_{\mathrm{IP}}(\mu^{\pm})]$ fits are provided in Ref. [99], while
the resulting prompt-like candidate categorization versus $m(\mu^{+}\mu^{-})$
is shown in Fig. 1. Finally, the $n_{\rm ob}^{\gamma^{*}}[m(A^{\prime})]$
yields are corrected for bin migration due to bremsstrahlung, which is
negligible except near the low-mass tails of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\mathchar
28935\relax(1S)$, and the small expected Bethe–Heitler contribution is
subtracted [76], resulting in the $n_{\rm
ex}^{A^{\prime}}[m(A^{\prime}),\varepsilon^{2}]$ values shown in Fig. S2 of
Ref. [99].

Figure 1:  Prompt-like mass spectrum, where the categorization of the data as
prompt $\mu^{+}\mu^{-}$, $\mu_{Q}\mu_{Q}$, and $hh+h\mu_{Q}$ is determined
using the ${\rm min}[\chi^{2}_{\mathrm{IP}}(\mu^{\pm})]$ fits described in the
text (examples of these fits are provided in the Supplemental Material). The
anti-$k_{\rm T}$-based isolation requirement is applied for
$m(A^{\prime})>1.1\mathrm{\,Ge\kern-1.00006ptV}$.

The prompt-like $n_{\rm ob}^{A^{\prime}}[m(A^{\prime})]$ mass spectrum is
scanned in steps of $\sigma[m(\mu^{+}\mu^{-})]$/2 searching for
$A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ contributions [99], using the strategy
from Ref. [83]. At each mass, a binned extended maximum likelihood fit is
performed in a $\pm 12.5\,\sigma[m(\mu^{+}\mu^{-})]$ window around
$m(A^{\prime})$. The profile likelihood is used to determine the $p$-value and
the upper limit at 90% confidence level (CL) on $n_{\rm
ob}^{A^{\prime}}[m(A^{\prime})]$. The signal is well modeled by a Gaussian
distribution whose resolution is determined with 10% precision using a
combination of simulated $A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ decays and the
observed $p_{\rm T}$-dependent widths of the large resonance peaks in the
data. The mass-resolution uncertainty is included in the profile likelihood.
The method of Ref. [100] selects the background model from a large set of
potential components, which includes all Legendre modes up to tenth order and
dedicated terms for known resonances, by performing a data-driven process
whose uncertainty is included in the profile likelihood following Ref. [101].
No significant excess is found in the prompt-like $m(A^{\prime})$ spectrum,
after accounting for the trials factor due to the number of signal hypotheses.

Dark photons are excluded at 90% CL where the upper limit on $n_{\rm
ob}^{A^{\prime}}[m(A^{\prime})]$ is less than $n_{\rm
ex}^{A^{\prime}}[m(A^{\prime}),\varepsilon^{2}]$. Figure 2 shows that the
constraints placed on prompt-like dark photons are the most stringent for
$214<m(A^{\prime})\lesssim 740\mathrm{\,Me\kern-1.00006ptV}$ and
$10.6<m(A^{\prime})\lesssim 30\mathrm{\,Ge\kern-1.00006ptV}$. The low-mass
constraints are the strongest placed by a prompt-like $A^{\prime}$ search at
any $m(A^{\prime})$. These results are corrected for inefficiency and changes
in the mass resolution that arise due to $\tau(A^{\prime})$ no longer being
negligible at such small values of $\epsilon^{2}$. The high-mass constraints
are adjusted to account for additional kinetic mixing with the $Z$ boson [84,
85], which alters Eq. (1). Since the LHCb detector response is independent of
which $q\bar{q}\to A^{\prime}$ process produces the dark photon above
10$\mathrm{\,Ge\kern-1.00006ptV}$, it is straightforward to recast the results
in Fig. 2 for other models [102, 103].

Figure 2:  Regions of the $[m(A^{\prime}),\varepsilon^{2}]$ parameter space
excluded at 90% CL by the prompt-like $A^{\prime}$ search compared to the best
published [35, 38, 83] and preliminary [104] limits.

For the long-lived $A^{\prime}$ search, contamination from prompt particles is
negligible due to a stringent criterion applied in the trigger on ${\rm
min}[\chi^{2}_{\mathrm{IP}}(\mu^{\pm})]$ that requires muons be inconsistent
with originating from any PV. Therefore, the dominant background contributions
are: photons that convert into $\mu^{+}\mu^{-}$ in the silicon-strip vertex
detector that surrounds the $pp$ interaction region, known as the VELO [105];
$b$-hadron decay chains that produce two muons; and the low-mass tail from
${K^{0}_{\mathrm{\scriptscriptstyle S}}}\to\pi^{+}\pi^{-}$ decays, where both
pions are misidentified as muons (all other strange decays are negligible). A
$p$-value is assigned to the photon-conversion hypothesis for each long-lived
$A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ candidate using properties of the decay
vertex and muon tracks, along with a high-precision three-dimensional material
map produced from a data sample of secondary hadronic interactions [106]. A
$m(A^{\prime})$-dependent requirement is applied to these $p$-values that
results in conversions having negligible impact on the sensitivity, though
they are still accounted for to prevent pathologies when there are no other
background sources. The remaining backgrounds are highly suppressed by the
decay topology requirement applied in the trigger. Furthermore, since muons
produced in $b$-hadron decays are often accompanied by additional displaced
tracks, events are rejected if they are selected by the inclusive heavy-flavor
software trigger [107, 108] independently of the presence of the
$A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ candidate. In addition, boosted decision
tree classifiers are used to reject events containing tracks consistent with
originating from the same $b$-hadron decay as the signal muon candidates
[109].

The long-lived $A^{\prime}$ search is also normalized using Eq. (1); however,
$\epsilon_{{}^{\gamma^{*}}}^{{}_{A^{\prime}}}[m(A^{\prime}),\tau(A^{\prime})]$
is not unity, in part because the efficiency depends on the decay time, $t$.
The kinematics are identical for $A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ and
prompt $\gamma^{*}\\!\to\\!\mu^{+}\mu^{-}$ decays for
$m(A^{\prime})=m(\gamma^{*})$; therefore, the $t$ dependence of
$\epsilon_{{}^{\gamma^{*}}}^{{}_{A^{\prime}}}[m(A^{\prime}),\tau(A^{\prime})]$
is obtained by resampling prompt $\gamma^{*}\\!\to\\!\mu^{+}\mu^{-}$
candidates as long-lived $A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ decays, where all
$t$-dependent properties, e.g. ${\rm min}[\chi^{2}_{\mathrm{IP}}(\mu^{\pm})]$,
are recalculated based on the resampled decay-vertex locations (the impact of
background contamination in the prompt $\gamma^{*}\\!\to\\!\mu^{+}\mu^{-}$
sample is negligible). This approach is validated using simulation, where
prompt $A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ decays are used to predict the
properties of long-lived $A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ decays. The
relative uncertainty on
$\epsilon_{{}^{\gamma^{*}}}^{{}_{A^{\prime}}}[m(A^{\prime}),\tau(A^{\prime})]$
is estimated to be 5%, which arises largely due to limited knowledge of how
radiation damage affects the performance of the VELO as a function of the
distance from the $pp$ interaction region. The looser kinematic, muon-
identification, and hardware-trigger requirements applied to long-lived
$A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ candidates, cf. prompt-like candidates,
also increase the efficiency. This $t$-independent increase in efficiency is
determined using a control data sample of dimuon candidates consistent with
originating from the PV, but otherwise satisfying the long-lived criteria. The
$n_{\rm ex}^{A^{\prime}}[m(A^{\prime}),\varepsilon^{2}]$ values obtained using
these data-driven
$\epsilon_{{}^{\gamma^{*}}}^{{}_{A^{\prime}}}[m(A^{\prime}),\tau(A^{\prime})]$
values (discussed in more detail in Ref. [99]), along with the expected
prompt-like $A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ yields, are shown in Fig. 3.

Figure 3:  Expected reconstructed and selected long-lived
$A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ yield.

The long-lived $m(A^{\prime})$ spectrum is also scanned in discrete steps of
$\sigma[m(\mu^{+}\mu^{-})]$/2 looking for $A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$
contributions [99]; however, discrete steps in $\tau(A^{\prime})$ are also
considered here. Binned extended maximum likelihood fits are performed to the
three-dimensional feature space of $m(\mu^{+}\mu^{-})$, $t$, and the
consistency of the decay topology as quantified in the decay-fit
$\chi^{2}_{\rm DF}$, which has three degrees of freedom. The photon-conversion
contribution is derived in each $[m(\mu^{+}\mu^{-}),t,\chi^{2}_{\rm DF}]$ bin
from the number of dimuon candidates that are rejected by the conversion
criterion. Both the $b$-hadron and $K^{0}_{\mathrm{\scriptscriptstyle S}}$
contributions are modeled in each $[t,\chi^{2}_{\rm DF}]$ bin by second-order
polynomials of the energy released in the decay,
$\sqrt{m(\mu^{+}\mu^{-})^{2}-4m(\mu)^{2}}$. These contributions are validated
using the following large control data samples: candidates that fail the
$b$-hadron suppression requirements; and candidates that fail, but nearly
satisfy, the stringent muon-identification requirements. The profile
likelihood is used to obtain the $p$-values and confidence intervals on
$n_{\rm ob}^{A^{\prime}}[m(A^{\prime}),\tau(A^{\prime})]$. No significant
excess is observed in the long-lived $A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$
search (the three-dimensional data distribution and the background-only pull
distributions are provided in Ref. [99]).

Since the relationship between $\tau(A^{\prime})$ and $\varepsilon^{2}$ is
known at each mass [76], the upper limits on $n_{\rm
ob}^{A^{\prime}}[m(A^{\prime}),\tau(A^{\prime})]$ are easily translated into
limits on $n_{\rm ob}^{A^{\prime}}[m(A^{\prime}),\varepsilon^{2}]$. Regions of
the $[m(A^{\prime}),\varepsilon^{2}]$ parameter space where the upper limit on
$n_{\rm ob}^{A^{\prime}}[m(A^{\prime}),\varepsilon^{2}]$ is less than $n_{\rm
ex}^{A^{\prime}}[m(A^{\prime}),\varepsilon^{2}]$ are excluded at 90% CL.
Figure 4 shows that sizable regions of $[m(A^{\prime}),\varepsilon^{2}]$
parameter space are excluded, which are much larger than those excluded in
Ref. [83].

Figure 4:  Ratio of the observed upper limit on $n_{\rm
ob}^{A^{\prime}}[m(A^{\prime}),\varepsilon^{2}]$ at 90% CL to the expected
dark-photon yield, $n_{\rm ex}^{A^{\prime}}[m(A^{\prime}),\varepsilon^{2}]$,
where regions less than unity are excluded. The only constraints in this
region are from (hashed) the previous LHCb search [83].

In summary, searches are performed for prompt-like and long-lived dark photons
produced in $pp$ collisions at a center-of-mass energy of
13$\mathrm{\,Te\kern-1.00006ptV}$. Both searches look for
$A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ decays using a data sample corresponding
to an integrated luminosity of 5.5$\mbox{\,fb}^{-1}$ collected with the LHCb
detector during 2016–2018. No evidence for a signal is found in either search,
and 90% CL exclusion regions are set on the $\gamma$–$A^{\prime}$ kinetic-
mixing strength. The prompt-like $A^{\prime}$ search is performed from near
the dimuon threshold up to 70$\mathrm{\,Ge\kern-1.00006ptV}$, and produces the
most stringent constraints on dark photons with $214<m(A^{\prime})\lesssim
740\mathrm{\,Me\kern-1.00006ptV}$ and $10.6<m(A^{\prime})\lesssim
30\mathrm{\,Ge\kern-1.00006ptV}$. The long-lived $A^{\prime}$ search is
restricted to the mass range
$214<m(A^{\prime})<350\mathrm{\,Me\kern-1.00006ptV}$, where the data sample
potentially has sensitivity, and places world-leading constraints on low-mass
dark photons with lifetimes $\mathcal{O}(1)\mbox{\,ps}$. The three-fold
increase in integrated luminosity, improved trigger efficiency during
2017–2018 data taking, and improvements in the analysis result in the searches
presented in this Letter achieving much better sensitivity to dark photons
than the previous LHCb results [83]. The prompt-like $A^{\prime}$ search
achieves a factor of 5 (2) better sensitivity to $\varepsilon^{2}$ at low
(high) masses than Ref. [83], while the long-lived $A^{\prime}$ search
provides access to much larger regions of $[m(A^{\prime}),\varepsilon^{2}]$
parameter space.

These results demonstrate the excellent sensitivity of the LHCb experiment to
dark photons, even using a data sample collected with a hardware-trigger stage
that is highly inefficient for low-mass $A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$
decays. The removal of this hardware-trigger stage in Run 3, along with the
planned increase in luminosity, should increase the potential yield of
$A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ decays in the low-mass region by a factor
$\mathcal{O}(100)$ compared to the 2016–2018 data sample. Given that most of
the parameter space shown in Fig. 4 would have been accessible if the data
sample was only three times larger, these upgrades will greatly increase the
dark-photon discovery potential of the LHCb experiment.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST
and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN
(Italy); NWO (Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MSHE
(Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC
(United Kingdom); DOE NP and NSF (USA). We acknowledge the computing resources
that are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN
(Italy), SURF (Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and
Yandex LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-
GRID (Poland) and OSC (USA). We are indebted to the communities behind the
multiple open-source software packages on which we depend. Individual groups
or members have received support from AvH Foundation (Germany); EPLANET, Marie
Skłodowska-Curie Actions and ERC (European Union); ANR, Labex P2IO and OCEVU,
and Région Auvergne-Rhône-Alpes (France); Key Research Program of Frontier
Sciences of CAS, CAS PIFI, and the Thousand Talents Program (China); RFBR, RSF
and Yandex LLC (Russia); GVA, XuntaGal and GENCAT (Spain); the Royal Society
and the Leverhulme Trust (United Kingdom).

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Supplemental Material for LHCb-PAPER-2019-031

Search for $A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ decays

Additional details and figures for the prompt-like and long-lived searches are
presented in this Supplemental Material.

Isolation Criterion

For masses above the $\phi(1020)$ meson mass, dark photons are expected to be
predominantly produced in Drell–Yan processes in $pp$ collisions at the LHC. A
well-known signature of Drell–Yan production is dimuon pairs that are largely
isolated, and a high-mass dark photon would inherit this property. The signal
sensitivity for $m(A^{\prime})>1.1\mathrm{\,Ge\kern-1.00006ptV}$ is enhanced
by applying the jet-based isolation requirement used in Ref. [83]. Jet
reconstruction is performed by clustering charged and neutral particle-flow
candidates [98] using the anti-$k_{\rm T}$ clustering algorithm [96] with
$R=0.5$ as implemented in FastJet [97]. Muons with $\mbox{$p_{\rm
T}$}(\mu)/\mbox{$p_{\rm T}$}({\rm jet})<0.7$ are rejected, where the
contribution to $\mbox{$p_{\rm T}$}({\rm jet})$ from the other muon is
excluded if both muons are clustered in the same jet, as this is found to
provide nearly optimal sensitivity for all $m(A^{\prime})>m(\phi)$.

Prompt-Like Fits

The prompt-like fit strategy is the same as described in detail in the
Supplemental Material of Ref. [83]. This strategy was first introduced in Ref.
[100], where it is denoted by aic-o. The $m(\mu^{+}\mu^{-})$ spectrum is
scanned in steps of $\sigma[m(\mu^{+}\mu^{-})]$/2 searching for
${A^{\prime}\\!\to\\!\mu^{+}\mu^{-}}$ contributions. At each mass, a binned
extended maximum likelihood fit is performed, and the profile likelihood is
used to determine the $p$-value and the confidence interval on $n_{\rm
ob}^{A^{\prime}}[m(A^{\prime})]$. The prompt-like-search trials factor is
obtained using pseudoexperiments. As in Ref. [100], each fit is performed in a
$\pm 12.5\,\sigma[m(\mu^{+}\mu^{-})]$ window around the scan-mass value using
bins with widths of $\sigma[m(\mu^{+}\mu^{-})]/20$. Near threshold, the energy
released in the decay, $\sqrt{m(\mu^{+}\mu^{-})^{2}-4m(\mu)^{2}}$, is used
instead of the mass since it is easier to model. The confidence intervals are
defined using the bounded likelihood approach, which involves taking
$\Delta\log{\mathcal{L}}$ relative to zero signal, rather than the best-fit
value, if the best-fit signal value is negative. This approach enforces that
only physical (nonnegative) upper limits are placed on $n_{\rm
ob}^{A^{\prime}}[m(A^{\prime})]$, and prevents defining exclusion regions that
are much better than the experimental sensitivity in cases where a large
deficit in the background yield is observed.

The signal models are determined at each $m(A^{\prime})$ using a combination
of simulated $A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ decays and the widths of the
large resonance peaks that are clearly visible in the data. The background
models take as input a large set of potential background components, then the
data-driven model-selection process of Ref. [100] is performed, whose
uncertainty is included in the profile likelihood following Ref. [101].
Specifically, the method labeled aic-o in Ref. [100] is used, where the log-
likelihood of each background model is penalized for its complexity (number of
parameters). The confidence intervals are obtained from the profile
likelihoods, including the penalty terms, where the model index is treated as
a discrete nuisance parameter, as originally proposed in Ref. [101]. In this
analysis, the set of possible background components includes all Legendre
modes with $\ell\leq 10$ at every $m(A^{\prime})$. Additionally, dedicated
background components are included for sizable narrow SM resonance
contributions. The use of 11 Legendre modes adequately describes every double-
misidentified peaking background that contributes at a significant level, and
therefore, these do not require dedicated background components. In mass
regions where such complexity is not required, the data-driven model-selection
procedure reduces the complexity which increases the sensitivity to a
potential signal contribution. Therefore, the impact of the background-model
uncertainty on the size of the confidence intervals is mass dependent, though
on average it is about 30%. As in Ref. [100], all fit regions are transformed
onto the interval $[-1,1]$, where the scan $m(A^{\prime})$ value maps to zero.
After such a transformation, the signal model is (approximately) an even
function; therefore, odd Legendre modes are orthogonal to the signal
component, which means that the presence of odd modes has minimal impact on
the variance of $n_{\rm ob}^{A^{\prime}}[m(A^{\prime})]$. In the prompt-like
fits, all odd Legendre modes up to ninth order are included in every
background model, while only a subset of the even modes is selected for
inclusion in each fit.

Regions in the mass spectrum where large known resonance contributions are
observed are vetoed in the prompt-like $A^{\prime}$ search. Furthermore, the
regions near the $\eta^{\prime}$ meson and the excited $\mathchar 28935\relax$
states (beyond the $\mathchar 28935\relax(3S)$ meson) are treated specially.
For example, since it is not possible to distinguish between
$A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ and $\eta^{\prime}\\!\to\\!\mu^{+}\mu^{-}$
contributions at $m(\eta^{\prime})$, the $p$-values near this mass are
ignored. The small excess at $m(\eta^{\prime})$ is treated as signal when
setting the limits on $n_{\rm ob}^{A^{\prime}}[m(A^{\prime})]$, which is
conservative in that an $\eta^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ contribution
will weaken the constraints on $A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ decays. The
same strategy is used near the excited $\mathchar 28935\relax$ masses.

Long-Lived Fits

The long-lived fit strategy is also the same as described in the Supplemental
Material to Ref. [83]. The signal yields are determined from binned extended
maximum likelihood fits performed on all long-lived
$A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ candidates using the three-dimensional
feature space of $m(\mu^{+}\mu^{-})$, $t$, and $\chi^{2}_{\rm DF}$. As in the
prompt-like $A^{\prime}$ search, a scan is performed in discrete steps of
$\sigma[m(\mu^{+}\mu^{-})]$/2; however, in this case, discrete steps in
$\tau(A^{\prime})$ are also considered. The profile likelihood is again used
to obtain the $p$-values and the confidence intervals on $n_{\rm
ob}^{A^{\prime}}[m(A^{\prime}),\tau(A^{\prime})]$. The binning scheme involves
four bins in $\chi^{2}_{\rm DF}$: [0,2], [2,4], [4,6], and [6,8]. Eight bins
in $t$ are used: [0.2,0.6], [0.6,1.1], [1.1,1.6], [1.6,2.2], [2.2,3.0], [3,5],
[5,10], and $>\\!10\mbox{\,ps}$. The binning scheme used for
$m(\mu^{+}\mu^{-})$ depends on the scan $m(A^{\prime})$ value, and is chosen
such that the vast majority of the signal falls into a single bin. Signal
decays mostly have small $\chi^{2}_{\rm DF}$ values, with about 50% (80%) of
$A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ decays satisfying $\chi^{2}_{\rm
DF}<2\,(4)$. Background from $b$-hadron decays populates the small $t$ region
and is roughly uniformly distributed in $\chi^{2}_{\rm DF}$, whereas
background from $K^{0}_{\mathrm{\scriptscriptstyle S}}$ decays is signal-like
in $\chi^{2}_{\rm DF}$ and roughly uniformly distributed in $t$. Figure S5
shows the three-dimensional distribution of all long-lived
$A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ candidates.

The expected contribution in each bin from photon conversions is derived from
the number of candidates rejected by the conversion criterion. Two large
control data samples are used to develop and validate the modeling of the
$b$-hadron and $K^{0}_{\mathrm{\scriptscriptstyle S}}$ contributions. Both
contributions are well modeled by second-order polynomials in
$\sqrt{m(\mu^{+}\mu^{-})^{2}-4m(\mu)^{2}}$. While no evidence for $t$ or
$\chi^{2}_{\rm DF}$ dependence is observed for these parameters in either the
$b$-hadron or $K^{0}_{\mathrm{\scriptscriptstyle S}}$ control sample, all
parameters are allowed to vary independently in each $[t,\chi^{2}_{\rm DF}]$
region in the fits used in the long-lived $A^{\prime}$ search.

Figure S6 shows the long-lived $A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ candidates,
along with the pull values obtained from fits performed to the data where no
signal contributions are included. These distributions are consistent with
those observed in pseudoexperiments where no signal component is generated.
Specifically, the data are consistent with being predominantly due to
$b$-hadron decays at small $t$, and due to $K^{0}_{\mathrm{\scriptscriptstyle
S}}$ decays for large $t$ and $m(\mu^{+}\mu^{-})\gtrsim
280\mathrm{\,Me\kern-1.00006ptV}$. The remaining few candidates at large $t$
and small $m(\mu^{+}\mu^{-})$ are likely photon conversions. N.b., the same
dominant backgrounds are expected in the upgraded LHCb detector in each region
of mass and decay time.

Long-Lived Efficiency Determination

The value of
$\epsilon_{{}^{\gamma^{*}}}^{{}_{A^{\prime}}}[m(A^{\prime}),\tau(A^{\prime})]$
is not unity in the long-lived $A^{\prime}$ search, in part because the
efficiency depends on the decay time, $t$. The kinematics are identical for
$A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ and prompt
$\gamma^{*}\\!\to\\!\mu^{+}\mu^{-}$ decays for $m(A^{\prime})=m(\gamma^{*})$;
therefore, the $t$ dependence of
$\epsilon_{{}^{\gamma^{*}}}^{{}_{A^{\prime}}}[m(A^{\prime}),\tau(A^{\prime})]$
is obtained by resampling prompt $\gamma^{*}\\!\to\\!\mu^{+}\mu^{-}$
candidates as long-lived $A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ decays, where all
$t$-dependent properties, e.g. ${\rm min}[\chi^{2}_{\mathrm{IP}}(\mu^{\pm})]$,
are recalculated based on the resampled decay-vertex locations. The impact of
background contamination in the prompt $\gamma^{*}\\!\to\\!\mu^{+}\mu^{-}$
sample is shown to be negligible by recomputing
$\epsilon_{{}^{\gamma^{*}}}^{{}_{A^{\prime}}}[m(A^{\prime}),\tau(A^{\prime})]$
using same-sign $\mu^{\pm}\mu^{\pm}$ candidates, which are purely background,
instead of prompt $\gamma^{*}\\!\to\\!\mu^{+}\mu^{-}$ candidates in this
procedure. The high-precision three-dimensional material map produced from a
data sample of secondary hadronic interactions in Ref. [106], which forms the
basis of the photon-conversion criterion applied in the selection, is used
here to determine where each muon would hit active sensors, and thus, have
recorded hits in the VELO. The resolution on the vertex location and other
$t$-dependent properties varies strongly with the location of the first VELO
hit on each muon track, though this dependence is largely geometric, making
rescaling the resolution of prompt tracks straightforward. Finally, the values
used for the decay-fit $\chi^{2}_{\rm DF}$ are resampled from the observed
distribution for $K^{0}_{\mathrm{\scriptscriptstyle S}}$ decays in data, which
is similar to that expected for an ideal $\chi^{2}$ distribution with three
degrees of freedom, to ensure that non-Gaussian effects are properly
considered.

This approach is validated using simulation, where prompt
$A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ decays are used to predict the properties
of long-lived $A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ decays. Figure S1 shows one
such example using $A^{\prime}$ decays generated with zero lifetime to predict
the radial flight distance of a sample produced with
$\tau(A^{\prime})=2\mbox{\,ps}$. Here, the full long-lived selection is
applied except for the photon-conversion criterion, which is validated
separately as discussed below. Since this validation study involves only
simulated decays, the material map produced in Ref. [106] is replaced by the
geometry used in the simulation. The integrated long-lived
$A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ yield predictions agree to $\approx 2\%$
with the actual values in the $[m(A^{\prime}),\tau(A^{\prime})]$ region
explored in the long-lived $A^{\prime}$ search. Both the short-distance turn-
on (driven by the ${\rm min}[\chi^{2}_{\mathrm{IP}}(\mu^{\pm})]$ criterion)
and long-distance turn-off (driven by the minimum number of VELO hits required
to form a track) curves are well described. The dominant uncertainty here
arises due to limited knowledge of how radiation damage has affected the
performance of the VELO as a function of the distance from the $pp$
interaction region. This uncertainty is estimated to be 5% by rerunning the
resampling method many times under different radiation-damage hypotheses.

To validate the efficiency of the photon-conversion criterion, the predicted
efficiency for a dark photon with the same mass and lifetime as the
$K^{0}_{\mathrm{\scriptscriptstyle S}}$ meson is compared to the efficiency
observed in a control data sample of $K^{0}_{\mathrm{\scriptscriptstyle S}}$
decays. The predicted and observed efficiencies agree to 1%. Additionally, in
Ref. [106] the expected performance of the photon-conversion criterion was
shown to agree with the performance observed in an independent control data
sample of photon conversions out to the $\mathcal{O}(10^{-4})$ level. The
expected performance here was obtained in a similar manner to how the dark
photon efficiencies are obtained, except starting from a pure sample of
conversions instead of prompt $\gamma^{*}\\!\to\\!\mu^{+}\mu^{-}$ candidates.

The looser kinematic, muon-identification, and hardware-trigger requirements
applied to long-lived $A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ candidates, cf.
prompt-like candidates, also increase the efficiency. This $t$-independent
increase in efficiency is determined using a control data sample of dimuon
candidates consistent with originating from the PV, but otherwise satisfying
the long-lived criteria. Specifically, ${\rm
min}[\chi^{2}_{\mathrm{IP}}(\mu^{\pm})]$ fits like those used to determine
$n_{\rm ob}^{\gamma^{*}}[m(A^{\prime})]$ in the prompt-like $A^{\prime}$
search are run on this control sample and on the subset of its candidates that
satisfy the more stringent criteria applied in the prompt-like $A^{\prime}$
search. The decrease in the $\gamma^{*}\\!\to\\!\mu^{+}\mu^{-}$ yield observed
when applying the more stringent criteria at each mass is used to determine
the $t$-independent increase in efficiency.

Figure S1:  Validation of the data-driven approach used to predict the long-
lived $A^{\prime}$ properties from prompt $\gamma^{*}\\!\to\\!\mu^{+}\mu^{-}$
candidates in data is shown using two simulated
$A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ samples, each generated with the same
production kinematics, for (black points) $\tau(A^{\prime})=2\mbox{\,ps}$
reconstructed and satisfying the long-lived selection criteria except the
photon-conversion criterion, and (red line) the prediction for
$\tau(A^{\prime})=2\mbox{\,ps}$ based only on the prompt
$A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ sample. N.b., there is no free
normalization factor, the red yield is predicted entirely from the prompt
sample.

Additional Figures

This section provides additional figures from the analysis, and more detailed
comparisons of the new constraints presented in the Letter with previous
results.

Figure S2:  Expected reconstructed and selected prompt-like
$A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ yield divided by $\varepsilon^{2}$, where
the displayed uncertainties include the systematic contributions. The gray
boxes cover the regions with large SM resonance contributions, where no search
for dark photons is performed. The anti-$k_{\rm T}$-based isolation
requirement is applied for $m(A^{\prime})>1.1\mathrm{\,Ge\kern-1.00006ptV}$.

Figure S3:  Example ${\rm min}[\chi^{2}_{\mathrm{IP}}(\mu^{\pm})]^{1/2}$
distributions with fit results overlaid for prompt-like candidates near (left)
$m(A^{\prime})=0.5$, (middle) 5, and (right)
50$\mathrm{\,Ge\kern-1.00006ptV}$. The square root of ${\rm
min}[\chi^{2}_{\mathrm{IP}}(\mu^{\pm})]$ is used in the fits to increase the
bin occupancies at large ${\rm min}[\chi^{2}_{\mathrm{IP}}(\mu^{\pm})]$
values. The background categories, as described in the Letter, are as follows:
($hh$) two prompt hadrons misidentified as muons; ($h\mu_{Q}$) a misidentified
prompt hadron combined with a muon produced in the decay of a heavy-flavor
quark, $Q$, that is misidentified as prompt; and ($\mu_{Q}\mu_{Q}$) two muons
produced in $Q$-hadron decays that are both misidentified as prompt.  Figure
S4:  Signed local significances at each scan mass from (black points) all fits
and (red) the expected distribution; if the best-fit signal-yield estimator is
negative, the signed significance is negative and vice versa. The error bars
account for the correlation between nearest-neighbor fit results, which often
produces outliers in pairs due to the $\sigma[m(\mu^{+}\mu^{-})]/2$ step size.
Figure S5:  Three-dimensional distribution of $\chi^{2}_{\rm DF}$ versus $t$
versus $m(\mu^{+}\mu^{-})$, which is fit to determine the long-lived signal
yields. The data are consistent with being predominantly due to $b$-hadron
decays at small $t$, and due to $K^{0}_{\mathrm{\scriptscriptstyle S}}$ decays
for large $t$ and $m(\mu^{+}\mu^{-})\gtrsim 280\mathrm{\,Me\kern-1.00006ptV}$.
The remaining few candidates at large $t$ and small $m(\mu^{+}\mu^{-})$ are
likely photon conversions.

Figure S6:  Long-lived $A^{\prime}\\!\to\\!\mu^{+}\mu^{-}$ candidates (black
points) showing $t$ versus $m(\mu^{+}\mu^{-})$ in bins of $\chi^{2}_{\rm DF}$,
compared to the pulls from binned fits performed without a signal contribution
(color axis). Positive pulls denote an excess of data candidates.  Figure S7:
Comparison of the results presented in this Letter to existing constraints
from previous experiments in the few-loop $\varepsilon$ region (see Ref. [102]
for details about previous experiments).  Figure S8:  Comparison of the
results presented in this Letter to existing constraints from previous
experiments in the few-loop $\varepsilon$ region (see Ref. [102] for details
about previous experiments), restricted to the mass region motivated by self-
interacting dark matter [4].

LHCb collaboration

R. Aaij31, C. Abellán Beteta49, T. Ackernley59, B. Adeva45, M. Adinolfi53, H.
Afsharnia9, C.A. Aidala79, S. Aiola25, Z. Ajaltouni9, S. Akar64, P.
Albicocco22, J. Albrecht14, F. Alessio47, M. Alexander58, A. Alfonso Albero44,
G. Alkhazov37, P. Alvarez Cartelle60, A.A. Alves Jr45, S. Amato2, Y. Amhis11,
L. An21, L. Anderlini21, G. Andreassi48, M. Andreotti20, F. Archilli16, J.
Arnau Romeu10, A. Artamonov43, M. Artuso67, K. Arzymatov41, E. Aslanides10, M.
Atzeni49, B. Audurier26, S. Bachmann16, J.J. Back55, S. Baker60, V.
Balagura11,b, W. Baldini20,47, A. Baranov41, R.J. Barlow61, S. Barsuk11, W.
Barter60, M. Bartolini23,47,h, F. Baryshnikov76, G. Bassi28, V. Batozskaya35,
B. Batsukh67, A. Battig14, V. Battista48, A. Bay48, M. Becker14, F.
Bedeschi28, I. Bediaga1, A. Beiter67, L.J. Bel31, V. Belavin41, S. Belin26, N.
Beliy5, V. Bellee48, K. Belous43, I. Belyaev38, G. Bencivenni22, E. Ben-
Haim12, S. Benson31, S. Beranek13, A. Berezhnoy39, R. Bernet49, D.
Berninghoff16, H.C. Bernstein67, E. Bertholet12, A. Bertolin27, C.
Betancourt49, F. Betti19,e, M.O. Bettler54, Ia. Bezshyiko49, S. Bhasin53, J.
Bhom33, M.S. Bieker14, S. Bifani52, P. Billoir12, A. Bizzeti21,u, M. Bjørn62,
M.P. Blago47, T. Blake55, F. Blanc48, S. Blusk67, D. Bobulska58, V. Bocci30,
O. Boente Garcia45, T. Boettcher63, A. Boldyrev77, A. Bondar42,x, N. Bondar37,
S. Borghi61,47, M. Borisyak41, M. Borsato16, J.T. Borsuk33, T.J.V. Bowcock59,
C. Bozzi20, S. Braun16, A. Brea Rodriguez45, M. Brodski47, J. Brodzicka33, A.
Brossa Gonzalo55, D. Brundu26, E. Buchanan53, A. Buonaura49, C. Burr47, A.
Bursche26, J.S. Butter31, J. Buytaert47, W. Byczynski47, S. Cadeddu26, H.
Cai71, R. Calabrese20,g, L. Calero Diaz22, S. Cali22, R. Calladine52, M.
Calvi24,i, M. Calvo Gomez44,m, A. Camboni44, P. Campana22, D.H. Campora
Perez47, L. Capriotti19,e, A. Carbone19,e, G. Carboni29, R. Cardinale23,h, A.
Cardini26, P. Carniti24,i, K. Carvalho Akiba31, A. Casais Vidal45, G. Casse59,
M. Cattaneo47, G. Cavallero47, R. Cenci28,p, J. Cerasoli10, M.G. Chapman53, M.
Charles12,47, Ph. Charpentier47, G. Chatzikonstantinidis52, M. Chefdeville8,
V. Chekalina41, C. Chen3, S. Chen26, A. Chernov33, S.-G. Chitic47, V.
Chobanova45, M. Chrzaszcz47, A. Chubykin37, P. Ciambrone22, M.F. Cicala55, X.
Cid Vidal45, G. Ciezarek47, F. Cindolo19, P.E.L. Clarke57, M. Clemencic47,
H.V. Cliff54, J. Closier47, J.L. Cobbledick61, V. Coco47, J.A.B. Coelho11, J.
Cogan10, E. Cogneras9, L. Cojocariu36, P. Collins47, T. Colombo47, A. Comerma-
Montells16, A. Contu26, N. Cooke52, G. Coombs58, S. Coquereau44, G. Corti47,
C.M. Costa Sobral55, B. Couturier47, D.C. Craik63, J. Crkovska66, A.
Crocombe55, M. Cruz Torres1, R. Currie57, C.L. Da Silva66, E. Dall’Occo31, J.
Dalseno45,53, C. D’Ambrosio47, A. Danilina38, P. d’Argent16, A. Davis61, O. De
Aguiar Francisco47, K. De Bruyn47, S. De Capua61, M. De Cian48, J.M. De
Miranda1, L. De Paula2, M. De Serio18,d, P. De Simone22, J.A. de Vries31, C.T.
Dean66, W. Dean79, D. Decamp8, L. Del Buono12, B. Delaney54, H.-P.
Dembinski15, M. Demmer14, A. Dendek34, V. Denysenko49, D. Derkach77, O.
Deschamps9, F. Desse11, F. Dettori26, B. Dey7, A. Di Canto47, P. Di Nezza22,
S. Didenko76, H. Dijkstra47, F. Dordei26, M. Dorigo28,y, A.C. dos Reis1, L.
Douglas58, A. Dovbnya50, K. Dreimanis59, M.W. Dudek33, L. Dufour47, G.
Dujany12, P. Durante47, J.M. Durham66, D. Dutta61, R. Dzhelyadin43,†, M.
Dziewiecki16, A. Dziurda33, A. Dzyuba37, S. Easo56, U. Egede60, V.
Egorychev38, S. Eidelman42,x, S. Eisenhardt57, R. Ekelhof14, S. Ek-In48, L.
Eklund58, S. Ely67, A. Ene36, S. Escher13, S. Esen31, T. Evans47, A.
Falabella19, J. Fan3, N. Farley52, S. Farry59, D. Fazzini11, M. Féo47, P.
Fernandez Declara47, A. Fernandez Prieto45, F. Ferrari19,e, L. Ferreira
Lopes48, F. Ferreira Rodrigues2, S. Ferreres Sole31, M. Ferrillo49, M. Ferro-
Luzzi47, S. Filippov40, R.A. Fini18, M. Fiorini20,g, M. Firlej34, K.M.
Fischer62, C. Fitzpatrick47, T. Fiutowski34, F. Fleuret11,b, M. Fontana47, F.
Fontanelli23,h, R. Forty47, V. Franco Lima59, M. Franco Sevilla65, M. Frank47,
C. Frei47, D.A. Friday58, J. Fu25,q, M. Fuehring14, W. Funk47, E. Gabriel57,
A. Gallas Torreira45, D. Galli19,e, S. Gallorini27, S. Gambetta57, Y. Gan3, M.
Gandelman2, P. Gandini25, Y. Gao4, L.M. Garcia Martin46, J. García Pardiñas49,
B. Garcia Plana45, F.A. Garcia Rosales11, J. Garra Tico54, L. Garrido44, D.
Gascon44, C. Gaspar47, D. Gerick16, E. Gersabeck61, M. Gersabeck61, T.
Gershon55, D. Gerstel10, Ph. Ghez8, V. Gibson54, A. Gioventù45, O.G. Girard48,
P. Gironella Gironell44, L. Giubega36, C. Giugliano20, K. Gizdov57, V.V.
Gligorov12, C. Göbel69, D. Golubkov38, A. Golutvin60,76, A. Gomes1,a, P.
Gorbounov38,6, I.V. Gorelov39, C. Gotti24,i, E. Govorkova31, J.P. Grabowski16,
R. Graciani Diaz44, T. Grammatico12, L.A. Granado Cardoso47, E. Graugés44, E.
Graverini48, G. Graziani21, A. Grecu36, R. Greim31, P. Griffith20, L.
Grillo61, L. Gruber47, B.R. Gruberg Cazon62, C. Gu3, E. Gushchin40, A. Guth13,
Yu. Guz43,47, T. Gys47, T. Hadavizadeh62, G. Haefeli48, C. Haen47, S.C.
Haines54, P.M. Hamilton65, Q. Han7, X. Han16, T.H. Hancock62, S. Hansmann-
Menzemer16, N. Harnew62, T. Harrison59, R. Hart31, C. Hasse47, M. Hatch47, J.
He5, M. Hecker60, K. Heijhoff31, K. Heinicke14, A. Heister14, A.M.
Hennequin47, K. Hennessy59, L. Henry46, J. Heuel13, A. Hicheur68, R. Hidalgo
Charman61, D. Hill62, M. Hilton61, P.H. Hopchev48, J. Hu16, W. Hu7, W. Huang5,
W. Hulsbergen31, T. Humair60, R.J. Hunter55, M. Hushchyn77, D. Hutchcroft59,
D. Hynds31, P. Ibis14, M. Idzik34, P. Ilten52, A. Inglessi37, A. Inyakin43, K.
Ivshin37, R. Jacobsson47, S. Jakobsen47, J. Jalocha62, E. Jans31, B.K.
Jashal46, A. Jawahery65, V. Jevtic14, F. Jiang3, M. John62, D. Johnson47, C.R.
Jones54, B. Jost47, N. Jurik62, S. Kandybei50, M. Karacson47, J.M. Kariuki53,
N. Kazeev77, M. Kecke16, F. Keizer54, M. Kelsey67, M. Kenzie54, T. Ketel32, B.
Khanji47, A. Kharisova78, K.E. Kim67, T. Kirn13, V.S. Kirsebom48, S. Klaver22,
K. Klimaszewski35, S. Koliiev51, A. Kondybayeva76, A. Konoplyannikov38, P.
Kopciewicz34, R. Kopecna16, P. Koppenburg31, I. Kostiuk31,51, O. Kot51, S.
Kotriakhova37, L. Kravchuk40, R.D. Krawczyk47, M. Kreps55, F. Kress60, S.
Kretzschmar13, P. Krokovny42,x, W. Krupa34, W. Krzemien35, W. Kucewicz33,l, M.
Kucharczyk33, V. Kudryavtsev42,x, H.S. Kuindersma31, G.J. Kunde66, A.K.
Kuonen48, T. Kvaratskheliya38, D. Lacarrere47, G. Lafferty61, A. Lai26, D.
Lancierini49, J.J. Lane61, G. Lanfranchi22, C. Langenbruch13, T. Latham55, F.
Lazzari28,v, C. Lazzeroni52, R. Le Gac10, R. Lefèvre9, A. Leflat39, F.
Lemaitre47, O. Leroy10, T. Lesiak33, B. Leverington16, H. Li70, X. Li66, Y.
Li6, Z. Li67, X. Liang67, R. Lindner47, F. Lionetto49, V. Lisovskyi11, G.
Liu70, X. Liu3, D. Loh55, A. Loi26, J. Lomba Castro45, I. Longstaff58, J.H.
Lopes2, G. Loustau49, G.H. Lovell54, Y. Lu6, D. Lucchesi27,o, M. Lucio
Martinez31, Y. Luo3, A. Lupato27, E. Luppi20,g, O. Lupton55, A. Lusiani28, X.
Lyu5, S. Maccolini19,e, F. Machefert11, F. Maciuc36, V. Macko48, P.
Mackowiak14, S. Maddrell-Mander53, L.R. Madhan Mohan53, O. Maev37,47, A.
Maevskiy77, K. Maguire61, D. Maisuzenko37, M.W. Majewski34, S. Malde62, B.
Malecki47, A. Malinin75, T. Maltsev42,x, H. Malygina16, G. Manca26,f, G.
Mancinelli10, R. Manera Escalero44, D. Manuzzi19,e, D. Marangotto25,q, J.
Maratas9,w, J.F. Marchand8, U. Marconi19, S. Mariani21, C. Marin Benito11, M.
Marinangeli48, P. Marino48, J. Marks16, P.J. Marshall59, G. Martellotti30, L.
Martinazzoli47, M. Martinelli24, D. Martinez Santos45, F. Martinez Vidal46, A.
Massafferri1, M. Materok13, R. Matev47, A. Mathad49, Z. Mathe47, V.
Matiunin38, C. Matteuzzi24, K.R. Mattioli79, A. Mauri49, E. Maurice11,b, M.
McCann60,47, L. Mcconnell17, A. McNab61, R. McNulty17, J.V. Mead59, B.
Meadows64, C. Meaux10, G. Meier14, N. Meinert73, D. Melnychuk35, S.
Meloni24,i, M. Merk31, A. Merli25, M. Mikhasenko47, D.A. Milanes72, E.
Millard55, M.-N. Minard8, O. Mineev38, L. Minzoni20,g, S.E. Mitchell57, B.
Mitreska61, D.S. Mitzel47, A. Mödden14, A. Mogini12, R.D. Moise60, T.
Mombächer14, I.A. Monroy72, S. Monteil9, M. Morandin27, G. Morello22, M.J.
Morello28,t, J. Moron34, A.B. Morris10, A.G. Morris55, R. Mountain67, H. Mu3,
F. Muheim57, M. Mukherjee7, M. Mulder31, D. Müller47, K. Müller49, V.
Müller14, C.H. Murphy62, D. Murray61, P. Muzzetto26, P. Naik53, T. Nakada48,
R. Nandakumar56, A. Nandi62, T. Nanut48, I. Nasteva2, M. Needham57, N.
Neri25,q, S. Neubert16, N. Neufeld47, R. Newcombe60, T.D. Nguyen48, C. Nguyen-
Mau48,n, E.M. Niel11, S. Nieswand13, N. Nikitin39, N.S. Nolte47, C. Nunez79,
A. Oblakowska-Mucha34, V. Obraztsov43, S. Ogilvy58, D.P. O’Hanlon19, R.
Oldeman26,f, C.J.G. Onderwater74, J. D. Osborn79, A. Ossowska33, J.M. Otalora
Goicochea2, T. Ovsiannikova38, P. Owen49, A. Oyanguren46, P.R. Pais48, T.
Pajero28,t, A. Palano18, M. Palutan22, G. Panshin78, A. Papanestis56, M.
Pappagallo57, L.L. Pappalardo20,g, C. Pappenheimer64, W. Parker65, C.
Parkes61,47, G. Passaleva21,47, A. Pastore18, M. Patel60, C. Patrignani19,e,
A. Pearce47, A. Pellegrino31, M. Pepe Altarelli47, S. Perazzini19, D.
Pereima38, P. Perret9, L. Pescatore48, K. Petridis53, A. Petrolini23,h, A.
Petrov75, S. Petrucci57, M. Petruzzo25,q, B. Pietrzyk8, G. Pietrzyk48, M.
Pikies33, M. Pili62, D. Pinci30, J. Pinzino47, F. Pisani47, A. Piucci16, V.
Placinta36, S. Playfer57, J. Plews52, M. Plo Casasus45, F. Polci12, M. Poli
Lener22, M. Poliakova67, A. Poluektov10, N. Polukhina76,c, I. Polyakov67, E.
Polycarpo2, G.J. Pomery53, S. Ponce47, A. Popov43, D. Popov52, S.
Poslavskii43, K. Prasanth33, L. Promberger47, C. Prouve45, V. Pugatch51, A.
Puig Navarro49, H. Pullen62, G. Punzi28,p, W. Qian5, J. Qin5, R. Quagliani12,
B. Quintana9, N.V. Raab17, R.I. Rabadan Trejo10, B. Rachwal34, J.H.
Rademacker53, M. Rama28, M. Ramos Pernas45, M.S. Rangel2, F. Ratnikov41,77, G.
Raven32, M. Ravonel Salzgeber47, M. Reboud8, F. Redi48, S. Reichert14, F.
Reiss12, C. Remon Alepuz46, Z. Ren3, V. Renaudin62, S. Ricciardi56, S.
Richards53, K. Rinnert59, P. Robbe11, A. Robert12, A.B. Rodrigues48, E.
Rodrigues64, J.A. Rodriguez Lopez72, M. Roehrken47, S. Roiser47, A.
Rollings62, V. Romanovskiy43, M. Romero Lamas45, A. Romero Vidal45, J.D.
Roth79, M. Rotondo22, M.S. Rudolph67, T. Ruf47, J. Ruiz Vidal46, J. Ryzka34,
J.J. Saborido Silva45, N. Sagidova37, B. Saitta26,f, C. Sanchez Gras31, C.
Sanchez Mayordomo46, B. Sanmartin Sedes45, R. Santacesaria30, C. Santamarina
Rios45, M. Santimaria22, E. Santovetti29,j, G. Sarpis61, A. Sarti30, C.
Satriano30,s, A. Satta29, M. Saur5, D. Savrina38,39, L.G. Scantlebury Smead62,
S. Schael13, M. Schellenberg14, M. Schiller58, H. Schindler47, M.
Schmelling15, T. Schmelzer14, B. Schmidt47, O. Schneider48, A. Schopper47,
H.F. Schreiner64, M. Schubiger31, S. Schulte48, M.H. Schune11, R. Schwemmer47,
B. Sciascia22, A. Sciubba30,k, S. Sellam68, A. Semennikov38, A. Sergi52,47, N.
Serra49, J. Serrano10, L. Sestini27, A. Seuthe14, P. Seyfert47, D.M.
Shangase79, M. Shapkin43, T. Shears59, L. Shekhtman42,x, V. Shevchenko75,76,
E. Shmanin76, J.D. Shupperd67, B.G. Siddi20, R. Silva Coutinho49, L. Silva de
Oliveira2, G. Simi27,o, S. Simone18,d, I. Skiba20, N. Skidmore16, T.
Skwarnicki67, M.W. Slater52, J.G. Smeaton54, A. Smetkina38, E. Smith13, I.T.
Smith57, M. Smith60, A. Snoch31, M. Soares19, L. Soares Lavra1, M.D.
Sokoloff64, F.J.P. Soler58, B. Souza De Paula2, B. Spaan14, E. Spadaro
Norella25,q, P. Spradlin58, F. Stagni47, M. Stahl64, S. Stahl47, P. Stefko48,
S. Stefkova60, O. Steinkamp49, S. Stemmle16, O. Stenyakin43, M. Stepanova37,
H. Stevens14, S. Stone67, S. Stracka28, M.E. Stramaglia48, M. Straticiuc36, S.
Strokov78, J. Sun3, L. Sun71, Y. Sun65, P. Svihra61, K. Swientek34, A.
Szabelski35, T. Szumlak34, M. Szymanski5, S. Taneja61, Z. Tang3, T. Tekampe14,
G. Tellarini20, F. Teubert47, E. Thomas47, K.A. Thomson59, M.J. Tilley60, V.
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A. Trisovic54, A. Tsaregorodtsev10, G. Tuci28,47,p, A. Tully48, N. Tuning31,
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Zucchelli19,e.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4School of Physics State Key Laboratory of Nuclear Physics and Technology,
Peking University, Beijing, China
5University of Chinese Academy of Sciences, Beijing, China
6Institute Of High Energy Physics (IHEP), Beijing, China
7Institute of Particle Physics, Central China Normal University, Wuhan, Hubei,
China
8Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy,
France
9Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France
10Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
11LAL, Univ. Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, Orsay, France
12LPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3,
Paris, France
13I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
14Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
15Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
16Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
17School of Physics, University College Dublin, Dublin, Ireland
18INFN Sezione di Bari, Bari, Italy
19INFN Sezione di Bologna, Bologna, Italy
20INFN Sezione di Ferrara, Ferrara, Italy
21INFN Sezione di Firenze, Firenze, Italy
22INFN Laboratori Nazionali di Frascati, Frascati, Italy
23INFN Sezione di Genova, Genova, Italy
24INFN Sezione di Milano-Bicocca, Milano, Italy
25INFN Sezione di Milano, Milano, Italy
26INFN Sezione di Cagliari, Monserrato, Italy
27INFN Sezione di Padova, Padova, Italy
28INFN Sezione di Pisa, Pisa, Italy
29INFN Sezione di Roma Tor Vergata, Roma, Italy
30INFN Sezione di Roma La Sapienza, Roma, Italy
31Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
32Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, Netherlands
33Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
34AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
35National Center for Nuclear Research (NCBJ), Warsaw, Poland
36Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
37Petersburg Nuclear Physics Institute NRC Kurchatov Institute (PNPI NRC KI),
Gatchina, Russia
38Institute of Theoretical and Experimental Physics NRC Kurchatov Institute
(ITEP NRC KI), Moscow, Russia, Moscow, Russia
39Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
40Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS),
Moscow, Russia
41Yandex School of Data Analysis, Moscow, Russia
42Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
43Institute for High Energy Physics NRC Kurchatov Institute (IHEP NRC KI),
Protvino, Russia, Protvino, Russia
44ICCUB, Universitat de Barcelona, Barcelona, Spain
45Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
46Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain
47European Organization for Nuclear Research (CERN), Geneva, Switzerland
48Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
49Physik-Institut, Universität Zürich, Zürich, Switzerland
50NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
51Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
52University of Birmingham, Birmingham, United Kingdom
53H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
54Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
55Department of Physics, University of Warwick, Coventry, United Kingdom
56STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
57School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
58School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
59Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
60Imperial College London, London, United Kingdom
61Department of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
62Department of Physics, University of Oxford, Oxford, United Kingdom
63Massachusetts Institute of Technology, Cambridge, MA, United States
64University of Cincinnati, Cincinnati, OH, United States
65University of Maryland, College Park, MD, United States
66Los Alamos National Laboratory (LANL), Los Alamos, United States
67Syracuse University, Syracuse, NY, United States
68Laboratory of Mathematical and Subatomic Physics , Constantine, Algeria,
associated to 2
69Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
70South China Normal University, Guangzhou, China, associated to 3
71School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
72Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 12
73Institut für Physik, Universität Rostock, Rostock, Germany, associated to 16
74Van Swinderen Institute, University of Groningen, Groningen, Netherlands,
associated to 31
75National Research Centre Kurchatov Institute, Moscow, Russia, associated to
38
76National University of Science and Technology “MISIS”, Moscow, Russia,
associated to 38
77National Research University Higher School of Economics, Moscow, Russia,
associated to 41
78National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
38
79University of Michigan, Ann Arbor, United States, associated to 67

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bLaboratoire Leprince-Ringuet, Palaiseau, France
cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
dUniversità di Bari, Bari, Italy
eUniversità di Bologna, Bologna, Italy
fUniversità di Cagliari, Cagliari, Italy
gUniversità di Ferrara, Ferrara, Italy
hUniversità di Genova, Genova, Italy
iUniversità di Milano Bicocca, Milano, Italy
jUniversità di Roma Tor Vergata, Roma, Italy
kUniversità di Roma La Sapienza, Roma, Italy
lAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
mLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
nHanoi University of Science, Hanoi, Vietnam
oUniversità di Padova, Padova, Italy
pUniversità di Pisa, Pisa, Italy
qUniversità degli Studi di Milano, Milano, Italy
rUniversità di Urbino, Urbino, Italy
sUniversità della Basilicata, Potenza, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità di Modena e Reggio Emilia, Modena, Italy
vUniversità di Siena, Siena, Italy
wMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines
xNovosibirsk State University, Novosibirsk, Russia
ySezione INFN di Trieste, Trieste, Italy
zSchool of Physics and Information Technology, Shaanxi Normal University
(SNNU), Xi’an, China
aaPhysics and Micro Electronic College, Hunan University, Changsha City, China
†Deceased