# Exploration of the Muon $g-2$ and Light Dark Matter explanations in NA64
with the CERN SPS high energy muon beam

Yu. M. Andreev  0000-0002-7397-9665 Authors affiliated with an institute
covered by a cooperation agreement with CERN  D. Banerjee  0000-0003-0531-1679
CERN, European Organization for Nuclear Research, CH-1211 Geneva, Switzerland
B. Banto Oberhauser  0009-0006-4795-1008 ETH Zürich, Institute for Particle
Physics and Astrophysics, CH-8093 Zürich, Switzerland  J. Bernhard
0000-0001-9256-971X CERN, European Organization for Nuclear Research, CH-1211
Geneva, Switzerland  P. Bisio  /0009-0006-8677-7495 INFN, Sezione di Genova,
16147 Genova, Italia  Università degli Studi di Genova, 16126 Genova, Italia
N. Charitonidis  0000-0001-9506-1022 CERN, European Organization for Nuclear
Research, CH-1211 Geneva, Switzerland  P. Crivelli  0000-0001-5430-9394
<EMAIL_ADDRESS>ETH Zürich, Institute for Particle Physics and
Astrophysics, CH-8093 Zürich, Switzerland  E. Depero  0000-0003-2239-1746 ETH
Zürich, Institute for Particle Physics and Astrophysics, CH-8093 Zürich,
Switzerland  A. V. Dermenev  0000-0001-5619-376X Authors affiliated with an
institute covered by a cooperation agreement with CERN  S. V. Donskov
0000-0002-3988-7687 Authors affiliated with an institute covered by a
cooperation agreement with CERN  R. R. Dusaev  0000-0002-6147-8038 Authors
affiliated with an institute covered by a cooperation agreement with CERN  T.
Enik  0000-0002-2761-9730 Authors affiliated with an international laboratory
covered by a cooperation agreement with CERN  V. N. Frolov  Authors affiliated
with an international laboratory covered by a cooperation agreement with CERN
R. B. Galleguillos Silva  Center for Theoretical and Experimental Particle
Physics, Facultad de Ciencias Exactas, Universidad Andres Bello, Fernandez
Concha 700, Santiago, Chile  Millennium Institute for Subatomic Physics at
High-Energy Frontier (SAPHIR), Fernandez Concha 700, Santiago, Chile  A.
Gardikiotis  0000-0002-4435-2695 Physics Department, University of Patras, 265
04 Patras, Greece  S. V. Gertsenberger  0009-0006-1640-9443 Authors affiliated
with an international laboratory covered by a cooperation agreement with CERN
S. Girod  CERN, European Organization for Nuclear Research, CH-1211 Geneva,
Switzerland  S. N. Gninenko  0000-0001-6495-7619 Authors affiliated with an
institute covered by a cooperation agreement with CERN  Center for Theoretical
and Experimental Particle Physics, Facultad de Ciencias Exactas, Universidad
Andres Bello, Fernandez Concha 700, Santiago, Chile  M. Hösgen  Universität
Bonn, Helmholtz-Institut für Strahlen-und Kernphysik, 53115 Bonn, Germany  V.
A. Kachanov  0000-0002-3062-010X Authors affiliated with an institute covered
by a cooperation agreement with CERN  Y. Kambar  0009-0000-9185-2353 Authors
affiliated with an international laboratory covered by a cooperation agreement
with CERN  A. E. Karneyeu  0000-0001-9983-1004 Authors affiliated with an
institute covered by a cooperation agreement with CERN  E. A. Kasianova
Authors affiliated with an international laboratory covered by a cooperation
agreement with CERN  G. Kekelidze  0000-0002-5393-9199 Authors affiliated with
an international laboratory covered by a cooperation agreement with CERN  B.
Ketzer  0000-0002-3493-3891 Universität Bonn, Helmholtz-Institut für Strahlen-
und Kernphysik, 53115 Bonn, Germany  D. V. Kirpichnikov  0000-0002-7177-077X
Authors affiliated with an institute covered by a cooperation agreement with
CERN  M. M. Kirsanov  0000-0002-8879-6538 Authors affiliated with an institute
covered by a cooperation agreement with CERN  V. N. Kolosov  Authors
affiliated with an institute covered by a cooperation agreement with CERN  V.
A. Kramarenko  0000-0002-8625-5586 Authors affiliated with an institute
covered by a cooperation agreement with CERN  Authors affiliated with an
international laboratory covered by a cooperation agreement with CERN  L. V.
Kravchuk  0000-0001-8631-4200 Authors affiliated with an institute covered by
a cooperation agreement with CERN  N. V. Krasnikov  0000-0002-8717-6492
Authors affiliated with an institute covered by a cooperation agreement with
CERN  Authors affiliated with an international laboratory covered by a
cooperation agreement with CERN  S. V. Kuleshov  0000-0002-3065-326X Center
for Theoretical and Experimental Particle Physics, Facultad de Ciencias
Exactas, Universidad Andres Bello, Fernandez Concha 700, Santiago, Chile
Millennium Institute for Subatomic Physics at High-Energy Frontier (SAPHIR),
Fernandez Concha 700, Santiago, Chile  V. E. Lyubovitskij  0000-0001-7467-572X
Authors affiliated with an institute covered by a cooperation agreement with
CERN  Universidad Técnica Federico Santa María and CCTVal, 2390123 Valparaíso,
Chile  Millennium Institute for Subatomic Physics at High-Energy Frontier
(SAPHIR), Fernandez Concha 700, Santiago, Chile  V. Lysan  0009-0004-1795-1651
Authors affiliated with an international laboratory covered by a cooperation
agreement with CERN  V. A. Matveev  0000-0002-2745-5908 Authors affiliated
with an international laboratory covered by a cooperation agreement with CERN
R. Mena Fredes  Millennium Institute for Subatomic Physics at High-Energy
Frontier (SAPHIR), Fernandez Concha 700, Santiago, Chile  Universidad Técnica
Federico Santa María and CCTVal, 2390123 Valparaíso, Chile  R. G. Mena Yanssen
Millennium Institute for Subatomic Physics at High-Energy Frontier (SAPHIR),
Fernandez Concha 700, Santiago, Chile  Universidad Técnica Federico Santa
María and CCTVal, 2390123 Valparaíso, Chile  L. Molina Bueno
0000-0001-9720-9764<EMAIL_ADDRESS>Instituto de Fisica Corpuscular
(CSIC/UV), Carrer del Catedratic Jose Beltran Martinez, 2, 46980 Paterna,
Valencia, Spain  M. Mongillo  0009-0000-7331-4076 ETH Zürich, Institute for
Particle Physics and Astrophysics, CH-8093 Zürich, Switzerland  D. V.
Peshekhonov  0009-0008-9018-5884 Authors affiliated with an international
laboratory covered by a cooperation agreement with CERN  V. A. Polyakov
0000-0001-5989-0990 Authors affiliated with an institute covered by a
cooperation agreement with CERN  B. Radics  0000-0002-8978-1725 York
University, Toronto, Canada  K. M. Salamatin  0000-0001-6287-8685 Authors
affiliated with an international laboratory covered by a cooperation agreement
with CERN  V. D. Samoylenko  Authors affiliated with an institute covered by a
cooperation agreement with CERN  D. A. Shchukin  0009-0007-5508-3615 Authors
affiliated with an institute covered by a cooperation agreement with CERN  O.
Soto  Departamento de Fisica, Facultad de Ciencias, Universidad de La Serena,
Avenida Cisternas 1200, La Serena, Chile  Millennium Institute for Subatomic
Physics at High-Energy Frontier (SAPHIR), Fernandez Concha 700, Santiago,
Chile  H. Sieber  0000-0003-1476-4258<EMAIL_ADDRESS>ETH Zürich,
Institute for Particle Physics and Astrophysics, CH-8093 Zürich, Switzerland
V. O. Tikhomirov  0000-0002-9634-0581 Authors affiliated with an institute
covered by a cooperation agreement with CERN  I. V. Tlisova
0000-0003-1552-2015 Authors affiliated with an institute covered by a
cooperation agreement with CERN  A. N. Toropin  0000-0002-2106-4041 Authors
affiliated with an institute covered by a cooperation agreement with CERN  M.
Tuzi  0009-0000-6276-1401 Instituto de Fisica Corpuscular (CSIC/UV), Carrer
del Catedratic Jose Beltran Martinez, 2, 46980 Paterna, Valencia, Spain  M. B.
Veit  Johannes Gutenberg Universitaet Mainz, Germany  P. V. Volkov
0000-0002-7668-3691 Authors affiliated with an international laboratory
covered by a cooperation agreement with CERN  V. Yu. Volkov
0009-0005-3500-5121 Authors affiliated with an institute covered by a
cooperation agreement with CERN  I. V. Voronchikhin  0000-0003-3037-636X
Authors affiliated with an institute covered by a cooperation agreement with
CERN  J. Zamora-Saá  0000-0002-5030-7516 Center for Theoretical and
Experimental Particle Physics, Facultad de Ciencias Exactas, Universidad
Andres Bello, Fernandez Concha 700, Santiago, Chile  Millennium Institute for
Subatomic Physics at High-Energy Frontier (SAPHIR), Fernandez Concha 700,
Santiago, Chile  A. S. Zhevlakov  0000-0002-7775-5917 Authors affiliated with
an international laboratory covered by a cooperation agreement with CERN

###### Abstract

We report on a search for a new $Z^{\prime}$ ($L_{\mu}-L_{\tau}$) vector boson
performed at the NA64 experiment employing a high energy muon beam and a
missing energy-momentum technique. Muons from the M2 beamline at the CERN
Super Proton Synchrotron with a momentum of 160 GeV/c are directed to an
active target. A signal event is a single scattered muon with momentum $<$ 80
GeV/c in the final state, accompanied by missing energy, i.e. no detectable
activity in the downstream calorimeters. For a total statistic of $(1.98\pm
0.02)\times 10^{10}$ muons on target, no event is observed in the expected
signal region. This allows us to set new limits on part of the remaining
$(m_{Z^{\prime}},\ g_{Z^{\prime}})$ parameter space which could provide an
explanation for the muon $(g-2)_{\mu}$ anomaly. Additionally, our study
excludes part of the parameter space suggested by the thermal Dark Matter
relic abundance. Our results pave the way to explore Dark Sectors and light
Dark Matter with muon beams in a unique and complementary way to other
experiments.

Dark Sectors (DS) are a promising paradigm to address open questions of the
Standard Model (SM) such as the Dark Matter (DM) origin [1]. In this
framework, one postulates a new sector of particles below the electroweak
scale that are not charged under the SM but could have a phenomenology of
their own [2, 3, 4, 5, 6, 7]. In addition to gravity, the interactions between
DS states and the SM could proceed through portal mediators [8, 9, 10, 11,
12]. If one assumes that DM is made by the lightest stable DS particles, the
resulting feeble interaction between the two sectors is compatible with
cosmological observations and, thus, would accommodate a solution to the DM
problem. DS models became an extremely fertile domain of exploration with many
different techniques tackling the very large parameter space of possible DM
candidates (see e.g. for recent reviews [13, 14, 15, 16]). Models with lepton
numbers $L_{\mu}-L_{\tau}$ gauging are very attractive to explain the origin
of DM and, at the same time, provide an explanation for the long-standing
$g-2$ muon anomaly [17]. The $Z^{\prime}$ vector boson originates from the
broken $U(1)_{L_{\mu}-L_{\tau}}$ symmetry and couples directly to the second
and third lepton generations, and their corresponding left-handed neutrinos
through the coupling $g_{Z^{\prime}}$ [18, 19, 20, 21, 22, 23]. The extension
of this model to interactions with DM candidates, being consistent in
predicting the observed DM relic density [24, 25, 26, 27] , is achieved by
adding to the Lagrangian a term of the type $\mathcal{L}\supseteq-
g_{\chi}Z_{\mu}^{\prime}J_{\chi}^{\mu}$ with the current $J_{\chi}^{\mu}$ and
the coupling $g_{\chi}$ of the $Z_{\mu}^{\prime}$ to the DM candidates. In the
case where $m_{Z^{\prime}}>m_{\chi}$ (away from the near on-shell resonant
enhancement $m_{Z^{\prime}}\simeq 2m_{\chi}$), the relic density is driven by
$\bar{\chi}\chi(\rightarrow Z^{(\ast)\prime}\rightarrow)\bar{f}f$,
$f=\mu,\tau,\nu$, with the relevant $s-$channel annihilation cross-section
scaling as [21, 26] $\langle\sigma
v\rangle\propto(g_{\chi}g_{Z^{\prime}})^{2}m_{\chi}^{2}/m_{Z^{\prime}}^{4}=ym_{\chi}^{-2}$.
Below the resonance, $m_{Z^{\prime}}<2m_{\chi}$, the $t-$channel annihilation
is $\bar{\chi}\chi\rightarrow Z^{\prime}Z^{\prime}$, with $\langle\sigma
v\rangle\propto g_{\chi}^{4}/m_{\chi}^{2}$.
Within this framework, the discrepancy between the experimental [28] and SM
predicted [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45,
46, 47] $(g-2)_{\mu}$ values can also be explained through loop corrections
[5, 48, 49, 50, 51, 52, 53]. The current bounds for $m_{Z^{\prime}}>2m_{\mu}$
arise from direct searches, sensitive to the kinematically allowed visible
decay channel $Z^{\prime}\rightarrow\mu^{+}\mu^{-}$ [54, 55, 56, 57]. Neutrino
scattering experiments [58, 59] and missing energy searches through
$Z^{\prime}\rightarrow\bar{\chi}\chi$ [60, 61] provide constraints for
$m_{Z^{\prime}}<2m_{\mu}$. The lower bound is set through the $Z^{\prime}$
contribution to the radiation density of the Universe through $\Delta
N_{\text{eff}}$, with its value being defined from both the CMB spectrum [27]
and the Big Bang nucleosynthesis (BBN) [62, 57, 63] to $m_{Z^{\prime}}>3-10$
MeV [64] and $g_{Z^{\prime}}\sim 10^{-4}-10^{-3}$.
In this Letter, we report on the first results of the NA64 experiment muon
program, dubbed NA64$\mu$, looking for Dark Sectors weakly coupled to muons.
The experimental set-up and working principle are schematically shown in Fig.
1.  Figure 1: Schematic illustration of the NA64$\mu$ set-up. a) The
spectrometer in the _upstream_ region is used for identifying incoming muons
with momentum $p_{\text{in}}\simeq 160$ GeV/c. b) The _downstream_ part
composed of calorimeters and a second spectrometer measures the momentum of
the scattered muons to search for the $Z^{\prime}$ vector boson production. c)
Sketch of the bremsstrahlung-like reaction $\mu N\rightarrow\mu
N(Z^{\prime}\rightarrow\text{invisible})$ of 160 GeV/c incident muons on the
ECAL target. If a $Z^{\prime}$ boson exists, it could be produced in the
bremsstrahlung-like reaction of a high energy muon scattering off atomic
nuclei in a target ($N$), followed by its prompt invisible decay $\mu
N\rightarrow\mu NZ^{\prime};Z^{\prime}\rightarrow\text{invisible}$, with only
$Z^{\prime}\rightarrow\bar{\nu}\nu$ in the _vanilla_ model, and additionally,
$Z^{\prime}\rightarrow\bar{\chi}\chi$ for DM candidates [65, 66, 67]. For a
value of $g_{\chi}=5\times 10^{-2}$ one can accommodate in the same parameter
space the muon g-2 and the DM relic prediction [68]. In the region of interest
(below $m_{\chi}<$1 GeV) $g_{\chi}\gg g_{Z^{\prime}}$, the branching ratio to
DS invisible final states can be assumed to be
$\text{Br}(Z^{\prime}\rightarrow\bar{\chi}\chi)\simeq 1$, while the ones in
visible states ($Z^{\prime}\rightarrow\mu^{+}\mu^{-}$) and neutrinos can be
neglected.
The search for signal events is based on a missing energy-momentum technique
which consists of the detection of a primary beam muon with a momentum of 160
GeV/c in the initial state, and a single muon scattered off an active target
with missing momentum $>$ 80 GeV/c in the final state, accompanied by missing
energy, i.e. no detectable electromagnetic or hadronic activity in the
downstream calorimeters.
The 160 GeV/c muons are delivered by the M2 beamline at the CERN Super Proton
Synchrotron (SPS)[69]. The beam optics comprises a series of quadrupoles
focusing the beam before the target with a divergency $\sigma_{x}\sim 0.9$ and
$\sigma_{y}\sim 1.9$ cm [70]. The incoming muon momentum is reconstructed
through a magnetic spectrometer (MS1) consisting of three 5 T$\cdot$m bending
magnets (BEND6), together with four micro-mesh gas detectors (Micromegas,
MM1-4), two straw tubes chambers (ST5,4) and six scintillator hodoscopes, the
beam momentum stations (BMS1-6). The obtained resolution is
$\sigma_{p_{\text{in}}}/p_{\text{in}}\simeq 3.8\%$. The target is an active
electromagnetic calorimeter (ECAL) composed of Shashlik-type modules made of a
lead-scintillator (Pb-Sc) resulting in 40 radiation lengths ($X_{0}$). The
ECAL is followed by a large $55\times 55$ cm2 high-efficiency veto counter
(VETO) and a 5 nuclear interaction lengths ($\lambda_{\text{int}}$) copper-Sc
(Cu-Sc) hadronic calorimeter (VHCAL) with a hole in its middle. The outgoing
muon momentum is reconstructed through a second magnetic spectrometer
consisting of a single 1.4 T$\cdot$m bending magnet (MS2) together with four
gaseous electron multiplier trackers (GEM1-4), two additional straw chambers
(ST2,1) and three 20$\times$8 cm2 Micromegas (MM5-7) yielding a resolution of
$\sigma_{p_{\text{out}}}/p_{\text{out}}\simeq 4.4\%$. To identify and remove
any residuals from interactions in the detectors upstream MS2 and ensure
maximal hermeticity, two large 120$\times$60 cm2, $\lambda_{\text{int}}\simeq\
30$ iron-Sc (Fe-Sc) HCAL modules (HCAL1,2) are placed at the end of the set-up
together with a 120$\times$60 cm2 UV straw, ST11. The trigger system is
defined by a veto counter with a hole (V1) and a set of Sc counters (S0-1)
before the target, together with two $20\times 20$ cm2 and $30\times 30$ cm2
Sc counters (S4 and Sμ) sandwiching the HCAL modules, shifted from the
undeflected beam axis (referred to as _zero-line_) to detect the scattered
muons.
The data were collected in two trigger configurations
(S${}_{0}\times$S${}_{1}\times\overline{\text{V}_{1}}\times$S${}_{4}\times$Sμ)
with different S4 and Sμ distances to the zero-line along the deflection axis
$\hat{x}$, namely S${}_{\mu}\hat{x}=-152$ mm and S${}_{\mu}\hat{x}=-117$ mm
with a similar S${}_{4}\hat{x}=-65$ mm. The corresponding measured rate is
$0.04\%$ and $0.07\%$ of the calibration trigger
(S${}_{0,1}\times\overline{\text{V}_{1}}$) coincidences at a beam intensity of
$2.8\times 10^{6}$ $\mu/$spill. In each configuration, we recorded
respectively $(11.7\pm 0.1)\times 10^{9}$ and $(8.1\pm 0.1)\times 10^{9}$
muons on target (MOT) yielding a total accumulated statistics of $(1.98\pm
0.02)\times 10^{10}$ MOT.
A detailed GEANT4-based [71, 72] Monte Carlo (MC) simulation is performed to
study the main background sources and the response of the detectors and the
muon propagation. In the latter case, the full beam optics developed by the
CERN BE-EA beam department is encompassed in the simulation framework using
separately both the TRANSPORT, HALO and TURTLE programs [73, 74, 75], as well
the GEANT4 compatible beam delivery simulation (BDSIM) program [76, 77, 78] to
simulate secondaries interactions in the beamline material. The signal
acceptance is carefully studied using the GEANT4 interface DMG4 package [79],
including light mediators production cross-sections computations through muon
bremsstrahlung [67]. The placements of $\text{S}_{4}$ and $\text{S}_{\mu}$ are
optimized to compensate for the low signal yield at high masses,
$\sigma_{Z^{\prime}}\sim g_{Z^{\prime}}^{2}\alpha Z^{2}/m_{Z^{\prime}}^{2}$,
with $\alpha$ the fine structure constant and $Z$ the atomic number of the
target, through angular acceptance being maximized for a scattered muon angle
$\psi_{\mu}^{\prime}\sim 10^{-2}$ rad after ECAL. In addition, the trigger
counters downstream of MS2 account for the expected 160 GeV/c mean deflected
position at the level of S4, estimated at $\langle\delta x\rangle\simeq-12.0$
mm from a detailed GenFit-based [80, 81] Runge-Kutta (RK) extrapolation
scheme.
The signal box, $p_{\text{out}}^{\text{cut}}\leq 80$ GeV/c and
$E_{\text{CAL}}^{\text{cut}}<12$ GeV, is optimized with signal simulations and
data to maximize the sensitivity. The cut on the total energy deposit in the
calorimeters, $E_{\text{CAL}}^{\text{cut}}$, is obtained from the sum of the
minimum ionizing particle (MIP) peaks of the related energy spectra.
To minimize the background, the following set of selection criteria is used.
(i) The incoming momentum should be in the momentum range $160\pm 20$ GeV/c.
(ii) A single track is reconstructed in each magnetic spectrometer (MS1 and
MS2) to ensure that a single muon traverses the full set-up. (iii) At most one
hit is reconstructed in MM5-7 and ST1 (no multiple hits) and the corresponding
extrapolated track to the HCAL face is compatible with a MIP energy deposit in
the expected cell. This cut verifies that no energetic enough secondaries from
interactions upstream MS2 arrive at the HCAL. (vi) The energy deposit in the
calorimeters and the veto should be compatible with a MIP. This cut enforces
the selection of events with no muon nuclear interactions in the calorimeters.
The aforementioned cut-flow is applied to events distributed in the outgoing
muon momentum and total energy deposit plane, $(p_{\text{out}},\
E_{\text{CAL}})$, as shown in Fig. 2.

Figure 2: Event distribution in the $(p_{\text{out}},\ E_{\text{CAL}})$ plane
before the MIP-compatible requirement selection criterion. The signal box is
defined as the shaded green rectangular area and the controlled region
labelled with $A$ through $D$ (see text).

Region $A$ is inherent to events with MIP-compatible energy deposits in all of
the calorimeters, resulting in $p_{\text{in}}\simeq p_{\text{out}}\simeq 160$
GeV/c. By design, most of unscattered beam muons do not pass through the S4
and Sμ counters, however, the trigger condition can be fulfilled by
sufficiently energetic residual ionization $\mu N\rightarrow\mu N+\delta e$
originating from the downstream trackers MM5-7 or last HCAL2 layers. The
accumulation of events in region $C$ is associated with large energy
deposition of the full-momentum scattered muon in the HCAL, while region $B$
corresponds to a hard scattering/bremsstrahlung in the ECAL, with a soft
outgoing muon and full energy deposition in either the active target or HCAL.
The small number of events between $p_{\text{out}}\geq 50$ GeV/c and
$p_{\text{out}}\leq 100$ GeV/c associated with hard muon bremsstrahlung
events, $\mu N\rightarrow\mu N+\gamma$, with $\psi_{\mu}^{\prime}\ll 10^{-2}$
rad, is a result of the trigger optimization for signal events emitted at
larger angles. The events in the region $D$ are associated with muon nuclear
interactions in the ECAL, $\mu N\rightarrow\mu+X$, with $X$ containing any
combination of $\pi^{\prime}$s, $K$, $p$, $n$…, with low-energy charged
hadrons being deflected away in MS2, going out of the detector acceptance
(typically the HCAL modules).

Background source | Background, $n_{b}$
---|---
(I) Momentum mis-reconstruction | $0.05\pm 0.03$
(II) $K\rightarrow\mu+\nu$, … in-flight decays | $0.010\pm 0.001$
(III) Calorimeter non-hermeticity | $<0.01$
Total $n_{b}$ (conservatively) | $0.07\pm 0.03$

Table 1: Expected main background level within the signal box, together with
its statistical error, for the 2022 muon pilot run corresponding to $\sim
2\times 10^{10}$ MOT.

An exhaustive discussion of background sources is given in [82, 70]. The main
processes are summarised in Table 1, with the dominant background contribution
being associated with (I) momentum mis-reconstruction of the scattered muon in
MS2. An incoming muon with 160 GeV/c is reconstructed after the target with
momentum $\leq 80$ GeV/c, whereas it truly is 160 GeV/c. This background is
evaluated from data by selecting a sample of incoming muons within a $\sim
2\sigma_{p_{\text{in}}}$ window around its nominal momentum $\langle
p_{\text{in}}\rangle=160$ GeV/c and extrapolating the tails of the
corresponding downstream momentum distribution $p_{\text{out}}$ towards 80
GeV/c. The second most important background process is (II) kaons decays to
(semi-)leptonic final states with muons, $K\rightarrow\mu\nu,\ ...$, before
the ECAL target. Because of the level of hadron contamination in the M2
beamline, $P_{h}\simeq 5\times 10^{-5}$ [69], incoming kaons could be
reconstructed through MS1 with a momentum passing the selection criterion (i)
and subsequently decaying to muons with energy $\leq$ 80 GeV, with the
neutrino carrying away the remaining energy. This contribution is estimated
from MC with the hadron contamination being extracted from existing data [69].
Pion decays do not contribute to this background since due to kinematics the
muon momentum is always $\geq$ 80 GeV. Another background source is associated
with (III) non-hermeticity in the calorimeters due to muon nuclear
interactions in the target. As such, a leading hadron with energy $E_{h}\geq
80$ GeV could be produced and escape the ECAL with lesser energetic charged
secondaries and the scattered muon. Because of the non-zero charge of the
particles and the trigger acceptance, low-energy secondaries are deflected
away through MS2 resulting in missing energy events. This background is
extrapolated to the signal region from region D of Fig. 2. After applying all
selection criteria (i-iv) and summing up the processes contributing to the
background, the expected background level is found to be $0.07\pm 0.03$ for
the total statistics of $\sim 2\times 10^{10}$ MOT.
The upper limits on the coupling $g_{Z^{\prime}}$ as a function of its mass
$m_{Z^{\prime}}$ are estimated at 90% confidence level (CL) following the
modified frequentist approach. In particular, the RooFit/RooStats-based [83,
84, 85] profile likelihood ratio statistical test is used in the asymptotic
approximation [86]. The total number of signal events falling within the
signal box is given by the sum of the two trigger configurations $t$

$N_{Z^{\prime}}=\sum_{t=1,2}N_{Z^{\prime}}^{t}=\sum_{t=1,2}N_{\text{MOT}}^{t}\times\epsilon_{Z^{\prime}}^{t}\times
N_{Z^{\prime}}^{t}(m_{Z^{\prime}},g_{Z^{\prime}}),$ (1)

where $N_{\text{MOT}}^{t}$ is the number of MOT for trigger configuration $t$,
$N_{Z}^{t}$ the number of signals per MOT produced in the ECAL target,
depending on the mass/coupling parameters $m_{Z^{\prime}}$ and
$g_{Z^{\prime}}$, and $\epsilon_{Z^{\prime}}^{t}$ the trigger-dependent signal
efficiency.
The main systematic effects contributing to the signal yield defined in Eq.
(1) are studied in detail. The uncertainty on $N_{\text{MOT}}^{t}$ is
conservatively set to 1%. The systematics associated with the $Z^{\prime}$
production cross-section are extracted from the uncertainty introduced by the
Weiszäcker-Williams (WW) approximation and from QED corrections to the exact
tree-level (ETL) expression. In the former case, the relative error in
assessing the number of produced $Z^{\prime}$ ($N_{Z^{\prime}}^{t}$) is found
to be $2\%$ [66, 67]. In the latter case, both the running of $\alpha$ at the
upper bound $Q^{2}\simeq m_{Z^{\prime}}\sim\mathcal{O}(1)$ GeV and higher
order corrections from soft photon emissions are estimated to contribute
through respectively $\Delta
N_{Z^{\prime}}\sim\alpha^{2}g_{Z^{\prime}}^{2}Z^{2}$ and through the Sudakov
factor $\Delta N_{\text{soft}}\sim\exp(-\alpha/\pi)$ at the level of 2.4% and
1.4%. Uncertainties relative to the Pb purity of the ECAL target are addressed
at the level of 1%. The systematics on $\epsilon_{Z^{\prime}}^{t}$ are
evaluated by comparing the detector responses in MC and data around the MIP-
compatible peak, in particular in the ECAL and HCAL. Through spectra
integration and peak ratio, it is found that the related cumulative
uncertainty does not exceed 4%. Because of the strong dependence of the
efficiency $\epsilon_{Z^{\prime}}^{t}$ on the trigger configuration $t$, in
particular on the distance from the zero-line, additional uncertainties due to
S4 and Sμ misalignment are studied through the change in efficiency as a
response to small displacements of the Sc counters. Because of the
$m_{Z^{\prime}}$ mass-dependence of the trigger rate [70], the resulting
uncertainty reaches up to $\leq 5\%$. As such the total systematic in the
signal yield of Eq. (1) is $\leq 8\%$ . The acceptance loss due to accidentals
(pile-up events, $\sim 13\%$) entering the trigger time window is taken into
account in the final efficiency computations. The signal efficiency peaks at
its maximum of $\sim 12\%$ for the mass range $\mathcal{O}(100\ \text{MeV}-1\
\text{GeV})$.
After unblinding, no event compatible with $Z^{\prime}$ production is found in
the signal box. This allows us to set the 90% CL exclusion limits on
$g_{Z^{\prime}}$ which are plotted in Fig. 3 in the $(m_{Z^{\prime}},\
g_{Z^{\prime}})$ parameter space, together with the values of $\Delta a_{\mu}$
compatible with the muon $g-2$ anomaly, within $\pm 2\sigma$. The band is
computed using the latest results of the Muon $g-2$ collaboration for the
combined Runs 2 and 3 (2019-2020), $a_{\mu}(\text{Exp})=116\ 592\
059(22)\times 10^{-11}$ [28] and the SM prediction of $a_{\mu}(\text{SM})=116\
591\ 810(43)\times 10^{-11}$ from the Muon $g-2$ Theory Initiative (TI) [87,
29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47].
It is worth noticing that the latest results from the CMD-3 collaboration [88,
89] on the $\pi^{+}\pi^{-}$ disagree within the 2.5-5$\sigma$ level with the
TI value and recent lattice QCD computations from the BMW collaboration [90]
are in tension by 2.1$\sigma$. Our results, excluding masses
$m_{Z^{\prime}}\gtrsim 40$ MeV and coupling $g_{Z^{\prime}}\gtrsim 6\times
10^{-4}$, are the first search for a light $Z^{\prime}$ (_vanilla_
$L_{\mu}-L_{\tau}$ model) with a muon beam using the missing energy-momentum
technique (see Fig. 3).

Figure 3: NA64$\mu$ 90% CL exclusion limits on the coupling $g_{Z^{\prime}}$
as a function of the $Z^{\prime}$ mass, $m_{Z^{\prime}}$, for the vanilla
$L_{\mu}-L_{\tau}$ model. The $\pm 2\sigma$ band for the $Z^{\prime}$
contribution to the $(g-2)_{\mu}$ discrepancy is also shown. Existing
constraints from BaBar [91, 92] and from neutrino experiments such as BOREXINO
[57, 93, 94] and CCFR [95, 58] are plotted.

Figure 4 shows the obtained limits at 90% CL in the target parameter space
$(m_{\chi},\ y)$ with freeze-out parameter
$y=(g_{\chi}g_{Z^{\prime}})^{2}(m_{\chi}/m_{Z^{\prime}})^{4}$ for accelerator-
based experiments probing thermal DM for $m_{Z^{\prime}}=3m_{\chi}$, away from
the resonant enhancement $m_{Z^{\prime}}\simeq 2m_{\chi}$, and
$g_{\chi}=5\times 10^{-2}$. The thermal targets for favored $y$ values are
plotted for scalar, pseudo-Dirac, and Majorana DM candidate scenarios, and
obtained from the integration of the underlying Boltzmann equation [96]. The
results indicate that NA64$\mu$ excludes a portion of the $(m_{\chi},\ y)$
parameter space, below the current CCFR [95, 58] limits, constraining for a
choice of masses $m_{\chi}\lesssim 40$ MeV the dimensionless parameter to
$y\lesssim 6\times 10^{-12}$.

Figure 4: The 90% CL exclusion limits obtained by the NA64$\mu$ experiment in
the $(m_{\chi},\ y)$ parameters space for thermal Dark Matter charged under
$U(1)_{L_{\mu}-L_{\tau}}$ with $m_{Z^{\prime}}=3m_{\chi}$ and the coupling
$g_{\chi}=5\times 10^{-2}$ for $2\times 10^{10}$ MOT. The branching ratio to
invisible final states is assumed to be
$\text{Br}(Z^{\prime}\rightarrow\text{invisible})\simeq 1$ (see text for
details). Existing bounds obtained through the CCFR experiment [58, 95] are
shown for completeness. The thermal targets for the different scenarios are
taken from [96].

In summary, for a total statistics of $(1.98\pm 0.02)\times 10^{10}$ MOT, no
event falling within the expected signal region is observed. Therefore, 90% CL
upper limits are set in the $(m_{Z^{\prime}},\ g_{Z^{\prime}})$ parameter
space of the $L_{\mu}-L_{\tau}$ vanilla model, constraining viable mass values
for the explanation of the $(g-2)_{\mu}$ anomaly to $6-7\ \text{MeV}\lesssim
m_{Z^{\prime}}\lesssim 40$ MeV, with $g_{Z^{\prime}}\lesssim 6\times 10^{-4}$.
New constraints on light thermal DM for values $y\gtrsim 6\times 10^{-12}$ for
$m_{\chi}\gtrsim 40$ MeV are also obtained. With improvements in the
experimental set-up, such as an additional magnetic spectrometer to reduce by
more than an order of magnitude the background from momentum mis-
recontruction, and an increase in statistics, NA64$\mu$ is expected to fully
cover the $(g-2)_{\mu}$ compatible parameter space and to boost its coverage
in the search for thermal Dark Matter complementing the world wide effort for
DS searches [13, 14, 15, 16]. The use of a muon beam demonstrated in this work
opens a new window to explore other well-motivated New Physics scenarios such
as benchmark dark photon models in the mass region ($0.1-1$) GeV [97], scalar
portals [67], millicharged particles [98] or $\mu\rightarrow e$ or
$\mu\rightarrow\tau$ processes involving Lepton Flavour Conversion [99, 100,
101].
We gratefully acknowledge the support of the CERN management and staff, in
particular the help of the CERN BE-EA department. We are grateful to C.
Menezes Pires and R. Joosten for their support with the beam momentum
stations. We are also thankful for the contributions from HISKP, University of
Bonn (Germany), ETH Zurich, and SNSF Grants No. 186181, No. 186158, No.
197346, No. 216602, (Switzerland), ANID— Millennium Science Initiative
Program—ICN2019 044 (Chile), RyC-030551-I and PID2021-123955NA-100 funded by
MCIN/AEI/FEDER, UE (Spain).

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$\displaystyle\;\Pi_{\checkmark}-(\Pi_{1,{\checkmark}}-\Delta_{1}^{\mathsf{x}_{1}})\otimes\cdots\otimes(\Pi_{N,{\checkmark}}-\Delta_{N}^{\mathsf{x}_{N}})\;.$
(118)

From this follows the bound:

$\displaystyle\big{|}\big{|}{\Delta^{\vec{\mathsf{x}}}}\big{|}\big{|}\;$
$\displaystyle\leq\;1-(1-\epsilon_{1})\cdots(1-\epsilon_{N})\;=:\;\epsilon_{\mathrm{tot}}\;.$
(119)

which is the approximate strong fair sampling condition given in Eq. (63),
where the approximation error is $\epsilon_{\mathrm{tot}}$. Assuming that all
the $\epsilon_{k}$ are small we have:

$\displaystyle\epsilon_{\mathrm{tot}}\;$
$\displaystyle=\;\epsilon_{1}+\ldots+\epsilon_{N}\;+\;O\bigg{(}\Big{(}{\sum}_{k}\,\epsilon_{k}\Big{)}^{\\!2}\bigg{)}\;.$
(120)

This means that Proposition 4 can be directly applied to a multipartite
setting, and the error is upper bounded by $\epsilon_{\mathrm{tot}}$, which is
roughly equal to the sum of the deviations from exact fair sampling for each
one of the individual measurement devices.

###  F Approximate fair sampling for Bell operators

Consider $N$ spatially separated and non-communicating measurement devices
that are used to measure the expectation value of a Bell operator $B$. This
expectation value is entirely determined by the outcome probabilities of the
measurement devices and for linear Bell inequalities it can be written as:

$\displaystyle\langle\,B\,\rangle_{\Pr(\cdot|\cdot)}\;=\;\sum_{{\vec{\mathsf{a}}},\vec{\mathsf{x}}}c_{\vec{\mathsf{a}}}^{\vec{\mathsf{x}}}\;\Pr(\vec{\mathsf{a}}\,|\,\vec{\mathsf{x}})\;,$
(121)

for some coefficients $c_{\vec{\mathsf{a}}}^{\vec{\mathsf{x}}}\in\mathds{R}$.
Two sets of probability distributions are of interest in our case: the post-
selected probability distributions $\Pr_{\mathrm{p.s.}}$ obtained by the
device “in the lab”, where we consider a “good outcome” one where all the $N$
detectors click in a given experimental round; and the probability
distributions $\Pr_{\mathrm{ideal}}$ that would be obtained with lossless
devices. The devices satisfy an approximate fair sampling condition.

We want to bound the quantity

$\displaystyle\frac{1}{\beta_{\mathrm{max}}}\;\big{|}\,\langle\,B\,\rangle_{\mathrm{p.s.}}-\langle\,B\,\rangle_{\mathrm{ideal}}\,\big{|}$
(122)

where $\beta_{\mathrm{max}}$ is the algebraic bound for the Bell operator $B$,
and is introduced here to compensate for arbitrary rescaling of the
coefficients $c_{\vec{\mathsf{a}}}^{\vec{\mathsf{x}}}$. The algebraic bound is
defined as:

$\displaystyle\beta_{\mathrm{max}}\;:=$
$\displaystyle\;\max_{\\{\Pr(\cdot|\cdot)\\}}\left|\sum_{{\vec{\mathsf{a}}},\vec{\mathsf{x}}}c_{\vec{\mathsf{a}}}^{\vec{\mathsf{x}}}\;\Pr(\vec{\mathsf{a}}\,|\,\vec{\mathsf{x}})\right|$
(123)

where the maximum is over the set of probability distributions
$\Pr(\vec{\mathsf{a}}|\vec{\mathsf{x}})$. We also suppose that the total
variation distance between $\Pr_{\mathrm{p.s.}}(\cdot|\vec{\mathsf{x}})$ and
$\Pr_{\mathrm{ideal}}(\cdot|\vec{\mathsf{x}})$ is bounded by
$\epsilon_{\mathrm{tot}}$ for all collections of settings $\vec{\mathsf{x}}$:

$\displaystyle\left|\left|\;{\Pr}_{\mathrm{p.s.}}(\,\cdot\,|\,\vec{\mathsf{x}})-{\Pr}_{\mathrm{ideal}}(\,\cdot\,|\,\vec{\mathsf{x}})\;\right|\right|_{TV}\;\leq\;\epsilon_{\mathrm{tot}}\;.$
(124)

Then we have for all $\vec{\mathsf{x}}$:

$\displaystyle{\Pr}_{\mathrm{p.s.}}(\,\cdot\,|\,\vec{\mathsf{x}})-{\Pr}_{\mathrm{ideal}}(\,\cdot\,|\,\vec{\mathsf{x}})\;=\;\mu_{\vec{\mathsf{x}}}\;q_{1}(\,\cdot\,|\,\vec{\mathsf{x}})-\mu_{\vec{\mathsf{x}}}\;q_{2}(\,\cdot\,|\,\vec{\mathsf{x}})$
(125)

where $q_{1}(\cdot|\vec{\mathsf{x}})$ and $q_{2}(\cdot|\vec{\mathsf{x}})$ are
probability distributions having disjoint support (i.e.,
$q_{1}(\cdot|\vec{\mathsf{x}})=0$ if $q_{2}(\cdot|\vec{\mathsf{x}})\neq 0$ and
vice-versa) and $\mu_{\vec{\mathsf{x}}}\in[0,\epsilon_{\mathrm{tot}}]$ is a
re-scaling constant. We can then compute:

$\displaystyle\frac{1}{\beta_{\mathrm{max}}}\;\big{|}\,\langle\,B\,\rangle_{\mathrm{p.s.}}-\langle\,B\,\rangle_{\mathrm{ideal}}\,\big{|}\;$
$\displaystyle=\;\frac{1}{\beta_{\mathrm{max}}}\left|\sum_{{\vec{\mathsf{a}}},\vec{\mathsf{x}}}c_{\vec{\mathsf{a}}}^{\vec{\mathsf{x}}}\;\Big{[}{\Pr}_{\mathrm{p.s.}}(\vec{\mathsf{a}}\,|\,\vec{\mathsf{x}})-{\Pr}_{\mathrm{ideal}}(\vec{\mathsf{a}}\,|\,\vec{\mathsf{x}})\Big{]}\right|$
(126)
$\displaystyle=\;\frac{1}{\beta_{\mathrm{max}}}\left|\sum_{{\vec{\mathsf{a}}},\vec{\mathsf{x}}}c_{\vec{\mathsf{a}}}^{\vec{\mathsf{x}}}\,\mu_{\vec{\mathsf{x}}}\;\Big{[}q_{1}(\vec{\mathsf{a}}\,|\,\vec{\mathsf{x}})-q_{2}(\vec{\mathsf{a}}\,|\,\vec{\mathsf{x}})\Big{]}\right|$
(127)
$\displaystyle\leq\;\frac{1}{\beta_{\mathrm{max}}}\left[\;\epsilon_{\mathrm{tot}}\left|\sum_{{\vec{\mathsf{a}}},\vec{\mathsf{x}}}c_{\vec{\mathsf{a}}}^{\vec{\mathsf{x}}}\;q_{1}(\vec{\mathsf{a}}\,|\,\vec{\mathsf{x}})\right|+\epsilon_{\mathrm{tot}}\left|\sum_{{\vec{\mathsf{a}}},\vec{\mathsf{x}}}c_{\vec{\mathsf{a}}}^{\vec{\mathsf{x}}}\;q_{1}(\vec{\mathsf{a}}\,|\,\vec{\mathsf{x}})\right|\;\right]$
(128) $\displaystyle\leq\;2\,\epsilon_{\mathrm{tot}}\;.$ (129)

In line (128) we have used the triangle inequality and in line (129) the
definition of $\beta_{\mathrm{max}}$.

###  G Computation for approximate fair sampling with a polarization analyser

We want to show that the POVM $M_{\checkmark}^{\theta}$ defined in Eq. (68) in
Section 8 satisfies approximate fair sampling when employing the operator
$M_{Q,{\checkmark}}=\mathds{1}-R^{\hat{N}}$. We start computing
$\big{|}\big{|}\widetilde{M}_{\checkmark}^{\theta}\big{|}\big{|}$, where
$\widetilde{M}_{\checkmark}^{\theta}=(M_{Q,{\checkmark}})^{-1/2}\,M_{\checkmark}^{\theta}\,(M_{Q,{\checkmark}})^{-1/2}$.
As usual, the inverse square roots are defined on the support of the operator
$M_{Q,{\checkmark}}$, in this case on the subspace orthogonal to
$\left|{\mathrm{vac}}\right\rangle$. We use $[\hat{N},\hat{N}_{\theta}]=0$,
hence we have:

$\displaystyle\widetilde{M}_{\checkmark}^{\theta}\;$
$\displaystyle=\;(M_{Q,{\checkmark}})^{-1}\,M_{\checkmark}^{\theta}$
$\displaystyle=\;\big{(}\mathds{1}-R^{\hat{N}}\big{)}^{-1}\,\big{[}\mathds{1}-R^{\hat{N}}(1+\delta)^{\hat{N}_{\theta}}\big{]}\;.$
(130)

From $(1+\delta)^{\hat{N}_{\theta}}\succcurlyeq\mathds{1}$ we have
$[\mathds{1}-R^{\hat{N}}(1+\delta)^{\hat{N}_{\theta}}]\preccurlyeq(\mathds{1}-R^{\hat{N}})$
and thus $\widetilde{M}_{\checkmark}^{\theta}\preccurlyeq\mathds{1}$,
equivalently,
$\big{|}\big{|}\widetilde{M}_{\checkmark}^{\theta}\big{|}\big{|}\leq 1$. This
upper bound is saturated, e.g., by the state
$|{\theta^{\bot}}\rangle=b_{\theta^{\bot}}^{\dagger}\left|{\mathrm{vac}}\right\rangle$,
since
$\langle{\theta^{\bot}}|\widetilde{M}_{\checkmark}^{\theta}|{\theta^{\bot}}\rangle=(1-R)^{-1}(1-R)=1$,
and thue we conclude
$\big{|}\big{|}\widetilde{M}_{\checkmark}^{\theta}\big{|}\big{|}=1$.

Then we can calculate

$\displaystyle\left|\left|\,\Pi_{\checkmark}-\frac{\widetilde{M}_{\checkmark}^{\theta}}{\big{|}\big{|}{\widetilde{M}_{\checkmark}^{\theta}}\big{|}\big{|}}\,\right|\right|\;$
$\displaystyle=\;\left|\left|\,\Pi_{\checkmark}-\big{(}\mathds{1}-R^{\hat{N}}\big{)}^{-1}\,\big{[}\mathds{1}-R^{\hat{N}}(1+\delta)^{\hat{N}_{\theta}}\big{]}\,\right|\right|$
(131)
$\displaystyle=\;\left|\left|\,\big{(}\mathds{1}-R^{\hat{N}}\big{)}^{-1}\,\big{[}\mathds{1}-R^{\hat{N}}-\mathds{1}+R^{\hat{N}}(1+\delta)^{\hat{N}_{\theta}}\big{]}\,\right|\right|$
(132)
$\displaystyle=\;\left|\left|\,\big{(}\mathds{1}-R^{\hat{N}}\big{)}^{-1}\,\big{[}R^{\hat{N}}(1+\delta)^{\hat{N}_{\theta}}-R^{\hat{N}}\big{]}\,\right|\right|$
(133) $\displaystyle=\;\max_{n\geq 1}\
\frac{R_{1}^{n}-R_{2}^{n}}{1-R_{2}^{n}}\;.$ (134)

In line (132) we have used
$\Pi_{\checkmark}=\big{(}\mathds{1}-R^{\hat{N}}\big{)}^{-1}\,\big{(}\mathds{1}-R^{\hat{N}}\big{)}$.
To obtain line (134) we have used the fact that the operator in line (133)
commutes with $\hat{N}$, thus it can be written as a direct sum of operators
having a definite number of photons; within an $n$-photon subspace, the
operator $(\mathds{1}-R^{\hat{N}})^{-1}$ takes value
$1/(1-R)^{n}=1/(1-R_{2})^{n}$, while the operator
$[R^{\hat{N}}(1+\delta)^{\hat{N}_{\theta}}-R^{\hat{N}}]$ attains the maximum
value $[R\,(1+\delta)]^{n}-R^{n}=R_{1}^{n}-R_{2}^{n}$ when all $n$ photons are
polarized in the $\theta$ direction. Maximizing over all $n$-photon subspaces
(with $n\neq 0$) one obtains the value of the operator norm. But the maximum
is always attained for $n=1$, since:

$\displaystyle\frac{R_{1}^{n}-R_{2}^{n}}{1-R_{2}^{n}}\;=\;\frac{R_{1}-R_{2}}{1-R_{2}}\;\frac{R_{1}^{n-1}+R_{1}^{n-2}R_{2}+~{}\cdots~{}+R_{2}^{n-1}}{\hfill
1~{}+\phantom{R_{1}^{n-2}}R_{2}+~{}\cdots~{}+R_{2}^{n-1}}\;\leq\;\frac{R_{1}-R_{2}}{1-R_{2}}\;.$
(135)

This finally results in:

$\displaystyle\left|\left|\,\Pi_{\checkmark}-\frac{\widetilde{M}_{\checkmark}^{\theta}}{\big{|}\big{|}{\widetilde{M}_{\checkmark}^{\theta}}\big{|}\big{|}}\,\right|\right|\;$
$\displaystyle=\;\frac{R_{1}-R_{2}}{1-R_{2}}\;=\;\frac{R\,\delta}{1-R}\;=\;\frac{1-\eta}{\eta}\,\delta\;.$
(136)

###  H Proof of Proposition 5

Consider a state $\hat{\rho}\in S(\mathcal{H}\oplus\mathcal{H}^{\bot})$ as in
Eq. (79): $\Pi\,\hat{\rho}\,\Pi=(1-\epsilon^{\prime})\rho$,
$\Pi^{\bot}\hat{\rho}\,\Pi^{\bot}=\epsilon^{\prime}\rho^{\bot}$, and
$\Pi\,\hat{\rho}\,\Pi^{\bot}=c$, where $\Pi,\Pi^{\bot}$ are the projectors on
the well-behaved subspace $\mathcal{H}$ and on the orthogonal subspace,
respectively. Preliminary, we establish the following Lemma.

######  Lemma 1.

The coherence term satisfies
$\left|\left|\,c\,\right|\right|_{\mathrm{Tr}}\leq\sqrt{\epsilon^{\prime}(1-\epsilon^{\prime})}$.

###### Proof.

We write $\hat{\rho}$ as a convex mixture of pure states,
$\hat{\rho}=\sum_{i}p_{i}|{\hat{\psi}_{i}}\rangle\\!\langle{\hat{\psi}_{i}}|$,
and define

$\displaystyle\sqrt{1-\epsilon_{i}}\left|{\psi_{i}}\right\rangle\;:=\;\Pi\,|{\hat{\psi}_{i}}\rangle\qquad\sqrt{\epsilon_{i}}\left|{\psi_{i}^{\bot}}\right\rangle\;:=\;\Pi^{\bot}|{\hat{\psi}_{i}}\rangle\;.$
(137)

We then have $\sum_{i}p_{i}\epsilon_{i}=\epsilon^{\prime}$,
$\sum_{i}p_{i}(1-\epsilon_{i})=1-\epsilon^{\prime}$ and
$c=\sum_{i}p_{i}\sqrt{\epsilon_{i}(1-\epsilon_{i})}|{\psi_{i}}\rangle\\!\langle{\psi_{i}^{\bot}}|$.
Using the triangle and Cauchy-Schwarz inequalities, we obtain the required
result:

$\displaystyle\left|\left|\,c\,\right|\right|_{\mathrm{Tr}}\;$
$\displaystyle\leq\;\sum_{i}p_{i}\sqrt{\epsilon_{i}(1-\epsilon_{i})}\;\left|\left|\,|{\psi_{i}}\rangle\\!\langle{\psi_{i}^{\bot}}|\,\right|\right|_{\mathrm{Tr}}$
(138)
$\displaystyle=\;\sum_{i}\sqrt{p_{i}\phantom{|\\!}\epsilon_{i}}\sqrt{p_{i}\,(1-\epsilon_{i})}$
(139) $\displaystyle\leq\;\sqrt{\sum_{i}p_{i}\,\epsilon_{i}}\
\sqrt{\sum_{i}p_{i}\,(1-\epsilon_{i})}$ (140)
$\displaystyle=\;\sqrt{\epsilon^{\prime}}\sqrt{1-\epsilon^{\prime}}\;.$ (141)

∎

Now, we prove the following inequality:

$\displaystyle\sum_{\mathsf{a}\in\mathsf{A}}\left|\,\mathrm{Tr}(\hat{M}_{\mathsf{a}}^{\mathsf{x}}\hat{\rho})-\mathrm{Tr}(M_{\mathsf{a}}^{\mathsf{x}}\rho)\,\right|\;$
$\displaystyle\leq\;2\left|\left|c\right|\right|_{\mathrm{Tr}}\,+\,2\epsilon^{\prime}\;.$
(142)

From $\rho=\Pi\rho\Pi$ and
$M_{\mathsf{a}}^{\mathsf{x}}=\Pi\hat{M}_{\mathsf{a}}^{\mathsf{x}}\Pi$ it
follows that
$\mathrm{Tr}(M_{\mathsf{a}}^{\mathsf{x}}\rho)=\mathrm{Tr}(\hat{M}_{\mathsf{a}}^{\mathsf{x}}\rho)$.
Thus, we have:

$\displaystyle\sum_{\mathsf{a}\in\mathsf{A}}\left|\,\mathrm{Tr}(\hat{M}_{\mathsf{a}}^{\mathsf{x}}\hat{\rho})-\mathrm{Tr}(M_{\mathsf{a}}^{\mathsf{x}}\rho)\,\right|\;$
$\displaystyle=\;\sum_{\mathsf{a}\in\mathsf{A}}\left|\,\mathrm{Tr}\big{[}\,\hat{M}_{\mathsf{a}}^{\mathsf{x}}(\hat{\rho}-\rho)\,\big{]}\,\right|$
(143)
$\displaystyle=\;\sum_{\mathsf{a}\in\mathsf{A}^{+}}\mathrm{Tr}\big{[}\,\hat{M}_{\mathsf{a}}^{\mathsf{x}}(\hat{\rho}-\rho)\,\big{]}+\sum_{\mathsf{a}\in\mathsf{A}^{-}}\mathrm{Tr}\big{[}\,\hat{M}_{\mathsf{a}}^{\mathsf{x}}(\rho-\hat{\rho})\,\big{]}$
(144)
$\displaystyle=\;\mathrm{Tr}\big{[}\,(\hat{M}_{+}^{\mathsf{x}}-M_{-}^{\mathsf{x}})(\hat{\rho}-\rho)\,\big{]}$
(145)
$\displaystyle\leq\;\left|\left|\,\hat{\rho}-\rho\,\right|\right|_{\mathrm{Tr}}$
(146)

In line (144) we split the sum over subsets $\mathsf{A}^{+}$ and
$\mathsf{A}^{-}$ where
$\mathrm{Tr}[\hat{M}_{\mathsf{a}}^{\mathsf{x}}(\hat{\rho}-\rho)]$ is,
respectively, positive and negative, and in line (145) we defined
$\hat{M}_{\pm}^{\mathsf{x}}:=\sum_{\mathsf{a}\in\mathsf{A}^{\pm}}\hat{M}_{\mathsf{a}}^{\mathsf{x}}$.
We then used the operational characterization of the trace norm,
$\left|\left|\sigma\right|\right|_{\mathrm{Tr}}=\max_{X}\mathrm{Tr}(X\sigma)$,
where the maximum is taken over hermitian matrices $X$ with
$-\mathds{1}\preccurlyeq X\preccurlyeq\mathds{1}$. We then have:

$\displaystyle\left|\left|\,\hat{\rho}-\rho\,\right|\right|_{\mathrm{Tr}}\;$
$\displaystyle=\;\left|\left|\,\epsilon^{\prime}\rho^{\bot}-\epsilon^{\prime}\rho+c+c^{\dagger}\,\right|\right|_{\mathrm{Tr}}$
(147)
$\displaystyle\leq\;2\left|\left|\,c\,\right|\right|_{\mathrm{Tr}}\,+\,\epsilon^{\prime}\left|\left|\,\rho^{\bot}-\rho\,\right|\right|_{\mathrm{Tr}}$
(148)
$\displaystyle\leq\;2\left|\left|\,c\,\right|\right|_{\mathrm{Tr}}\,+\,2\,\epsilon^{\prime}\;.$
(149)

Next, using the shorthand
$f_{\mathsf{a}}=\mathrm{Tr}(\hat{M}_{\mathsf{a}}^{\mathsf{x}}\hat{\rho})$,
$g_{\mathsf{a}}=\mathrm{Tr}(M_{\mathsf{a}}^{\mathsf{x}}\rho)$,
$f=\sum_{\mathsf{a}}f_{\mathsf{a}}$, and $g=\sum_{\mathsf{a}}g_{\mathsf{a}}$,
we have:

$\displaystyle\left|\,f-g\,\right|\;\leq\;\sum_{\mathsf{a}\in\mathsf{A}}\left|\,f_{\mathsf{a}}-g_{\mathsf{a}}\,\right|\;\leq\;2\left|\left|\,c\,\right|\right|_{\mathrm{Tr}}+2\,\epsilon^{\prime}$
(150)

Finally, we can prove inequality (80):

$\displaystyle\sum_{\mathsf{a}\in\mathsf{A}}\frac{1}{2}\left|\,\frac{\mathrm{Tr}(\hat{M}_{\mathsf{a}}^{\mathsf{x}}\hat{\rho})}{\mathrm{Tr}(\hat{M}_{\checkmark}^{\mathsf{x}}\hat{\rho})}-\frac{\mathrm{Tr}(M_{\mathsf{a}}^{\mathsf{x}}\rho)}{\mathrm{Tr}(M_{\checkmark}^{\mathsf{x}}\rho)}\,\right|\;$
$\displaystyle=\;\sum_{\mathsf{a}\in\mathsf{A}}\frac{1}{2}\left|\,\frac{f_{\mathsf{a}}}{f}-\frac{g_{\mathsf{a}}}{g}\,\right|$
(151)
$\displaystyle=\;\sum_{\mathsf{a}\in\mathsf{A}}\frac{1}{2f}\left|\,f_{\mathsf{a}}-g_{\mathsf{a}}+g_{\mathsf{a}}-g_{\mathsf{a}}\frac{f}{g}\,\right|$
(152)
$\displaystyle\leq\;\sum_{\mathsf{a}\in\mathsf{A}}\frac{1}{2f}\left|\,f_{\mathsf{a}}-g_{\mathsf{a}}\,\right|+\sum_{\mathsf{a}\in\mathsf{A}}\frac{g_{\mathsf{a}}}{2f}\left|\,1-\frac{f}{g}\,\right|$
(153)
$\displaystyle\leq\;\sum_{\mathsf{a}\in\mathsf{A}}\frac{1}{2f}\left|\,f_{\mathsf{a}}-g_{\mathsf{a}}\,\right|+\frac{1}{2f}\left|\,g-f\,\right|$
(154)
$\displaystyle\leq\;\frac{1}{f}\left(2\left|\left|\,c\,\right|\right|_{\mathrm{Tr}}+2\,\epsilon^{\prime}\right)\;.$
(155)

By symmetry under exchange of $f_{\mathsf{a}}$ and $g_{\mathsf{a}}$, we also
have
$\frac{1}{g}(2\left|\left|\,c\,\right|\right|_{\mathrm{Tr}}+2\,\epsilon^{\prime})$
as an upper bound. Recognizing that
$f=\mathrm{Tr}(\hat{M}_{\checkmark}^{\mathsf{x}}\hat{\rho})$ and
$g=\mathrm{Tr}(M_{\checkmark}^{\mathsf{x}}\rho)$, the proof is concluded.