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Exploration of the Muon and Light Dark Matter explanations in NA64 with the CERN SPS high energy muon beam
在 CERN SPS 高能量μ子束下，探索 NA64 中的μ子 和轻暗物质解释

\author{ \作者{
Yu. M. Andreevब, D. Banerjee B. Banto Oberhauser J. Bernhard P. Bisio,, N. Charitonidis

R. B. Galleguillos Silva, A. Gardikiotis S. V. Gertsenberger S. Girod, S. N. Gninenko M. Hösgen,

D. V. Kirpichnikovఠ,, M. M. Kirsanove, V. N. Kolosov, V. A. Kramarenkoө,, L. V. Kravchuk

R. Mena Fredes, R. G. Mena Yanssen, L. Molina Bueno M. Mongilloo, D. V. Peshekhonov
V. A. Polyakov B. Radics K. M. Salamatin V. D. Samoylenko, D. A. Shchukin O. Soto,,

P. V. Volkov®, V. Yu. Volkov I. V. Voronchikhin J. Zamora-Saá and A. S. Zhevlakov
P. V. Volkov®, V. Yu. Volkov I. V. Voronchikhin J. Zamora-Saá 和 A. S. Zhevlakov
Authors affiliated with an institute covered by a cooperation agreement with CERN
作者隶属于与 CERN 签署合作协议的机构
CERN, European Organization for Nuclear Research, CH-1211 Geneva, Switzerland
CERN，瑞士日内瓦 CH-1211，欧洲核子研究组织
ETH Zürich, Institute for Particle Physics and Astrophysics, CH-8093 Zürich, Switzerland
ETH 苏黎世理工学院，粒子物理和天体物理研究所，瑞士苏黎世 8093
INFN, Sezione di Genova, 16147 Genova, Italia
INFN，热内瓦分部，意大利热内瓦 16147
Università degli Studi di Genova, 16126 Genova, Italia
热内瓦大学，意大利热内瓦 16126
Authors affiliated with an international laboratory covered by a cooperation agreement with CERN
与欧洲核子研究组织（CERN）合作协议覆盖的国际实验室有关的 作者
Center for Theoretical and Experimental Particle Physics, Facultad de Ciencias Exactas,
理论与实验粒子物理中心，智利圣地亚哥 Exactas 科学学院
Universidad Andres Bello, Fernandez Concha 700, Santiago, Chile
智利圣地亚哥 Fernandez Concha 700 号，安德烈斯贝略大学
Millennium Institute for Subatomic Physics at High-Energy Frontier (SAPHIR), Fernandez Concha 700, Santiago, Chile
智利圣地亚哥费尔南德斯孔查 700 号高能前沿亚原子物理千年研究所（SAPHIR）
Physics Department, University of Patras, 265 04 Patras, Greece
希腊帕特拉斯大学物理系，帕特拉斯 265 04
Universität Bonn, Helmholtz-Institut für Strahlen-und Kernphysik, 53115 Bonn, Germany
德国波恩大学，辐射与核物理赫尔姆霍兹研究所，波恩 53115
Universidad Técnica Federico Santa María and CCTVal, 2390123 Valparaíso, Chile
智利瓦尔帕莱索 2390123 年圣玛利亚联邦理工大学和 CCTVal
Instituto de Fisica Corpuscular (CSIC/UV), Carrer del Catedratic Jose Beltran Martinez, 2, 46980 Paterna, Valencia, Spain
西班牙瓦伦西亚 46980 年帕特尔纳，何塞·贝尔特兰·马丁内斯教授大街 2 号，CSIC/UV 粒子物理研究所
York University, Toronto, Canada
加拿大多伦多约克大学
Departamento de Fisica, Facultad de Ciencias,
物理系，理学院，
Universidad de La Serena, Avenida Cisternas 1200, La Serena, Chile
智利拉塞雷纳大学，Cisternas 大街 1200 号，拉塞雷纳
Johannes Gutenberg Universitaet Mainz, Germany
约翰内斯·古腾堡大学，德国
}
(Dated: January 4, 2024)
(日期：2024 年 1 月 4 日)

Abstract 摘要

We report on a search for a new 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 are directed to an active target. A signal event is a single scattered muon with momentum in the final state, accompanied by missing energy, i.e. no detectable activity in the downstream calorimeters. For a total statistic of muons on target, no event is observed in the expected signal region. This allows us to set new limits on part of the remaining parameter space which could provide an explanation for the muon 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.
我们报告了在 NA64 实验中进行的一项寻找新 矢量玻色子的研究，该实验采用高能量的μ子束和缺失能量-动量技术。来自 CERN 超级质子同步加速器 M2 束线的动量为 的μ子被定向到一个活动靶。信号事件是一个在最终态中动量为 的单个散射μ子，伴随着缺失能量，即在下游量热器中没有可检测的活动。对于总共 个靶上的μ子的统计数据，没有观察到预期信号区域中的任何事件。这使我们能够在剩余 参数空间的一部分上设定新的限制，这部分参数空间可能为μ子 异常提供解释。此外，我们的研究排除了热暗物质遗留丰度所建议的参数空间的一部分。我们的结果为探索暗扇区和轻暗物质提供了一条道路，通过μ子束以一种独特和互补的方式进行，与其他实验不同。

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 . In addition to gravity, the interactions between DS states and the SM could proceed through portal mediators [8-12]. 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 [13-16]). Models with lepton numbers gauging are very attractive to explain the origin of DM and, at the same time, provide an explanation for the long-standing muon anomaly [17]. The vector boson originates from the broken symmetry and couples directly to the second and third lepton generations, and their corresponding left-handed neutrinos through the coupling . The extension of this model to interactions with DM candidates, being consistent in predicting the observed DM relic density [24-27] , is achieved by adding to the Lagrangian a term of the type with the current and the coupling of the to the DM candidates. In the case
暗物质（DS）是解决标准模型（SM）中的一些未解问题的一个有前途的范式，比如暗物质（DM）的起源[1]。在这个框架中，人们假设在电弱尺度以下存在一个新的粒子部门，这些粒子不受 SM 的电荷影响，但可能有自己的现象学 。除了引力外，DS 状态与 SM 之间的相互作用可以通过门户中介体进行[8-12]。如果假设 DM 是由最轻的稳定 DS 粒子组成的，那么两个部门之间的微弱相互作用与宇宙学观测相一致，因此可以解决暗物质问题。DS 模型成为一个非常富有成果的探索领域，许多不同的技术处理可能的暗物质候选者的非常庞大的参数空间（例如，参见最近的评论[13-16]）。具有量子数 的模型非常有吸引力，可以解释暗物质的起源，并同时为长期存在的 μ子异常提供解释[17]。 矢量玻色子源自破缺的 对称性，并直接耦合到第二和第三代轻子，以及它们对应的左手中微子，通过耦合 。将该模型扩展到与 DM 候选相互作用，通过添加一个类似于 的项到拉格朗日量中，以及 的电流和 与 DM 候选的耦合，从而一致地预测观测到的 DM 遗留密度[24-27]。在这种情况下
where (away from the near on-shell resonant enhancement ), the relic density is driven by , with the relevant -channel annihilation cross-section scaling as . Below the resonance, , the -channel annihilation is , with .
当 (远离近似共振增强 )时，遗留密度由 驱动，相关的 -通道湮灭截面按比例缩放为 。在共振点以下， ， -通道湮灭是 ，具有 。
Within this framework, the discrepancy between the experimental and SM predicted values can also be explained through loop corrections [5, 4853]. The current bounds for arise from direct searches, sensitive to the kinematically allowed visible decay channel . Neutrino scattering ex- periments and missing energy searches through provide constraints for . The lower bound is set through the contribution to the radiation density of the Universe through , with its value being defined from both the CMB spectrum and the Big Bang nucleosynthesis (BBN) to and .
在这个框架内，实验 和 SM 预测的 值之间的差异也可以通过环修正来解释[5, 4853]。 的当前边界来自于直接搜索，对动力学允许的可见衰变通道 敏感。中微子散射实验 和通过 进行的失能量搜索为 提供了约束。下限是通过 对宇宙辐射密度的贡献设定的，其值是从 CMB 谱 和宇宙大爆炸核合成（BBN） 到 和 定义的。
In this Letter, we report on the first results of the NA64 experiment muon program, dubbed NA64 , looking for Dark Sectors weakly coupled to muons. The experimental set-up and working principle are schematically shown in Fig. 1.
在这封信中，我们报告了 NA64 实验μ子计划的第一批结果，代号为 NA64 ，寻找与μ子弱耦合的暗扇区。实验设置和工作原理如图 1 所示。
Figure 1. Schematic illustration of the NA64 set-up. a) The spectrometer in the upstream region is used for identifying incoming muons with momentum . b) The downstream part composed of calorimeters and a second spectrometer measures the momentum of the scattered muons to search for the vector boson production. c) Sketch of the bremsstrahlung-like reaction invisible) of incident muons on the ECAL target.
图 1. NA64 设置的示意图。a）上游区域的谱仪用于识别动量为 的入射μ子。b）由量热器和第二个谱仪组成的下游部分测量散射μ子的动量，以寻找 矢量玻色子产生。c）关于 入射μ子在 ECAL 靶上的类辐射反应 （不可见）的草图。
If a boson exists, it could be produced in the bremsstrahlung-like reaction of a high energy muon scattering off atomic nuclei in a target , followed by its prompt invisible decay invisible, with only in the vanilla model, and additionally, for DM candidates . For a value of one can accommodate in the same parameter space the muon and the DM relic prediction [68]. In the region of interest (below , the branching ratio to DS invisible final states can be assumed to be , while the ones in visible states and neutrinos can be neglected.
如果存在 玻色子，它可能是由高能量μ子在靶 上散射后类辐射反应产生的，随后是其即时不可见衰变 （不可见），在香草模型中仅有 ，另外， 用于 DM 候选 。对于 的一个值，可以在相同的参数空间中容纳μ子 和 DM 残留预测[68]。在感兴趣的区域（低于 ，DS 不可见最终态的分支比可以假定为 ，而可忽略可见状态 和中微子的状态。
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 in the initial state, and a single muon scat- tered off an active target with missing momentum in the final state, accompanied by missing energy, i.e. no detectable electromagnetic or hadronic activity in the downstream calorimeters.
信号事件的搜索基于一种缺失能量-动量技术，其中包括检测具有 动量的初级束缪子在初始状态下，以及一个散射到带有缺失动量 的活动靶上的单个缪子在最终状态下，伴随着缺失能量，即在下游量热器中没有可检测的电磁或强子活动。
The 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 and . The incoming muon momentum is reconstructed through a magnetic spectrometer (MS1) consisting of three bending magnets (BEND6), together with four micro-mesh gas detectors (Micromegas, ), two straw tubes chambers and six scintillator hodoscopes, the beam momentum stations . The obtained resolution is . The target is an active electromagnetic
缪子由 CERN 超级质子同步加速器（SPS）的 M2 束线提供[69]。束流光学包括一系列四极透镜，在靶前聚焦束流，发散度为 和 。入射缪子动量通过由三个 弯曲磁铁（BEND6）、四个微网气体探测器（Micromegas， ）、两个吸管室 和六个闪烁体闪烁体组成的磁谱仪（MS1）重建，束流动量站 。获得的分辨率为 。靶是一个活跃的电磁体。
calorimeter (ECAL) composed of Shashlik-type modules made of a lead-scintillator resulting in 40 radiation lengths . The ECAL is followed by a large high-efficiency veto counter (VETO) and a 5 nuclear interaction lengths 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 bending magnet (MS2) together with four gaseous electron multiplier trackers , two additional straw chambers and three Micromegas yielding a resolution of . To identify and remove any residuals from interactions in the detectors upstream MS2 and ensure maximal hermeticity, two large iron-Sc (Fe-Sc) HCAL modules are placed at the end of the set-up together with a UV straw, . The trigger system is defined by a veto counter with a hole and a set of Sc counters before the target, together with two and counters and sandwiching the HCAL modules, shifted from the undeflected beam axis (referred to as zero-line) to detect the scattered muons.
电量计（ECAL）由铅闪烁体模块组成，导致 40 个辐射长度。ECAL 后面是一个大型高效率否决计数器（VETO）和一个 5 个核相互作用长度的铜-闪烁体（Cu-Sc）强子量能器（VHCAL），中间有一个孔。通过第二个磁谱仪重建出射的μ子动量，该磁谱仪由一个单一的 1.4 弯曲磁铁（MS2）以及四个气体电子倍增器跟踪器、两个额外的吸管室和三个微网电离室组成，分辨率为。为了识别和清除上游 MS2 探测器中的任何残留相互作用，并确保最大的全封闭性，两个大型铁-闪烁体（Fe-Sc）HCAL 模块与一个 UV 吸管放置在设置的末端。触发系统由一个带孔的否决计数器和一组靶前的闪烁体计数器定义，以及两个和计数器夹在 HCAL 模块之间，从未偏转的光束轴线（称为零线）偏移，以侦测散射的μ子。
The data were collected in two trigger configurations with different and distances to the zero-line along the deflection axis , namely and with a similar . The corresponding measured rate is and of the calibration trigger coincidences at a beam intensity of spill. In each configuration, we recorded respectively and muons on target (MOT) yielding a total accumulated statistics of MOT.
数据是在两种触发配置 中收集的，它们沿偏转轴 的零线具有不同的 和 距离，即 和 ，具有类似的 。相应的测量速率是 和 ，在 泄漏的束强度下，校准触发 的巧合。在每种配置中，我们分别记录了 和 个靶上的μ子（MOT），产生了 个 MOT 的总累积统计数据。
A detailed GEANT4-based 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 [73-75], as well the GEANT4 compatible beam delivery simulation (BDSIM) program [76-78] 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 and are optimized to compensate for the low signal yield at high masses, , with the fine structure constant and the atomic number of the target, through angular acceptance being maximized for a scattered muon angle after ECAL. In addition, the trigger counters downstream of MS2 account for the expected mean deflected position at the level of , estimated at from a detailed GenFit-based Runge-Kutta extrapolation scheme.
使用基于 GEANT4 的详细 蒙特卡洛（MC）模拟来研究主要背景源以及探测器和μ子传播的响应。在后一种情况下，使用由 CERN BE-EA 束流部门开发的完整束流光学，分别使用 TRANSPORT、HALO 和 TURTLE 程序[73-75]，以及 GEANT4 兼容的束流传输模拟（BDSIM）程序[76-78]来模拟束线材料中的二次相互作用。使用 GEANT4 接口 DMG4 包[79]仔细研究信号接受度，包括通过μ子布伦斯特劳伦产生光介质产生截面计算[67]。通过优化 和 的位置来补偿高质量下低信号产量， ，其中 是精细结构常数， 是目标的原子序数，通过在 ECAL 后最大化散射μ子角度 的角度接受度。 此外，MS2 账户下游的触发器计数器考虑了期望的 平均偏转位置，估计在 级，从基于详细 GenFit 的 龙格-库塔 外推方案中估计为 。
The signal box, and , is optimized with signal simulations and data to maximize the sensitivity. The cut on the total energy deposit in the calorimeters, , is obtained from the sum of the minimum ionizing particle (MIP) peaks of the related energy spectra.
信号箱 和 经过信号模拟和数据优化，以最大化灵敏度。在量热器中的总能量沉积上的切割 是从相关能谱的最小电离粒子（MIP）峰的总和中获得的。
To minimize the background, the following set of selection criteria is used. (i) The incoming momentum should be in the momentum range 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 and (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, , as shown in Fig. 2.
为了最小化背景，使用以下一组选择标准。 (i) 入射动量应在动量范围 c 内。 (ii) 在每个磁谱仪 (MS1 和 MS2) 中重建单个径迹，以确保单个μ子穿过整个设置。 (iii) 在 和 中最多重建一个击中 (没有多重击中)，并且对应的外推径迹到 HCAL 面与预期单元中的 MIP 能量沉积相容。 此切割验证了来自 MS2 上游相互作用的足够高能次级粒子未到达 HCAL。 (vi) 在量热器和否决器中的能量沉积应与 MIP 相容。 此切割强制选择在量热器中没有μ子核相互作用的事件。 上述切割流应用于分布在出射μ子动量和总能量沉积平面 中的事件，如图 2 所示。
Figure 2. Event distribution in the ( ) 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 through (see text).
图 2. 在满足 MIP 相容要求选择标准之前的 ( ) 平面中的事件分布。 信号框定义为阴影绿色矩形区域，受控区域标有 到 (见文本)。
Region is inherent to events with MIP-compatible energy deposits in all of the calorimeters, resulting in c. By design, most of unscattered beam muons do not pass through the and counters, however, the trigger condition can be fulfilled by sufficiently energetic residual ionization originating from the downstream trackers or last layers. The accumulation of events in region is associated with large energy deposition of the fullmomentum scattered muon in the HCAL, while region corresponds to a hard scattering/bremsstrahlung in the
区域 是所有量热器中具有 MIP 兼容能量沉积的事件固有的，导致 c。按设计，大多数未散射的束流μ子不会穿过 和 计数器，然而，触发条件可以通过来自下游跟踪器 或最后 层的足够高能残留电离 来实现。在区域 中事件的积累与 HCAL 中全动量散射μ子的大能量沉积相关，而区域 对应于 ECAL 中的硬散射/辐射。
ECAL, with a soft outgoing muon and full energy deposition in either the active target or HCAL. The small number of events between and associated with hard muon bremsstrahlung events, , with , is a result of the trigger optimization for signal events emitted at larger angles. The events in the region are associated with muon nuclear interactions in the ECAL, , with containing any combination of s, , with low-energy charged hadrons being deflected away in MS2, going out of the detector acceptance (typically the HCAL modules).
ECAL 中，有一个软出射μ子和全部能量沉积在活动靶或 HCAL 中的硬散射事件之间的少量事件， ，带有 ，是对发射角较大的信号事件进行触发优化的结果。区域 中的事件与 ECAL 中的μ子核相互作用相关， ，带有 包含任何组合的 s， ，低能带电强子在 MS2 中被偏转出去，超出探测器接受范围（通常是 HCAL 模块）。
Background source 背景源 Background,
(I) Momentum mis-reconstruction
(I) 动量误重建
(II) in-flight decays
(II) 飞行衰变
(III) Calorimeter non-hermeticity
(III) 阳极非完全覆盖
Total (conservatively)
总共 （保守估计）
Table I. Expected main background level within the signal box, together with its statistical error, for the 2022 muon pilot run corresponding to MOT.
表 I. 信号框内预期的主要背景水平，以及对应于 MOT 的 2022 年μ子试运行的统计误差。
An exhaustive discussion of background sources is given in . The main processes are summarised in Table I, with the dominant background contribution being associated with (I) momentum mis-reconstruction of the scattered muon in MS2. An incoming muon with is reconstructed after the target with momentum , whereas it truly is c. This background is evaluated from data by selecting a sample of incoming muons within a window around its nominal momentum and extrapolating the tails of the corresponding downstream momentum distribution towards . The second most important background process is (II) kaons decays to (semi-)leptonic final states with muons, , before the ECAL target. Because of the level of hadron contamination in the M2 beamline, [69], incoming kaons could be reconstructed through MS1 with a momentum passing the selection criterion (i) and subsequently decaying to muons with energy , 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 . Another background source is associated with (III) nonhermeticity in the calorimeters due to muon nuclear interactions in the target. As such, a leading hadron with energy 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 miss- ing energy events. This background is extrapolated to the signal region from region 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 for the total statistics of MOT.
在 中提供了对背景来源的详尽讨论。主要过程总结在表 I 中，主要背景贡献与 MS2 中散射的μ子的动量误重建有关。一个具有 的入射μ子在靶后被重建为动量 ，而实际上它是 c。通过在其名义动量 周围选择一组入射μ子样本，并将相应下游动量分布 的尾部向 外推，从数据中评估了这种背景。第二重要的背景过程是（II）K 介子在 ECAL 靶前衰变为（半）轻子末态与μ子， 。由于 M2 束线中的强子污染水平， [69]，入射 K 介子可以通过 MS1 重建，其动量符合选择标准（i），随后衰变为能量 的μ子，中微子带走剩余能量。这一贡献是通过 MC 估计的，其中强子污染从现有数据中提取。 Pion 衰变不会对这种背景产生影响，因为由于动力学原因，μ子的动量始终为 。另一个背景源与（III）与靶中μ子核相互作用导致的量热器非完全性有关。因此，可能产生一个能量为 的主导强子，并逃离 ECAL，带有较少能量的带电次级粒子和散射的μ子。由于粒子的非零电荷和触发器接受度，低能次级粒子通过 MS2 被偏转，导致能量缺失事件。这种背景从图 2 的 区域外推到信号区域。在应用所有选择标准（i-iv）并总结导致背景的过程后，发现总统计量为 MOT 的预期背景水平为 。
The upper limits on the coupling as a function of its mass are estimated at confidence level (CL) following the modified frequentist approach. In particular, the RooFit/RooStats-based [83-85] 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
在修改后的频率主义方法下，根据修饰频率主义方法，以 置信水平（CL）估计耦合 的上限作为其质量 的函数。具体来说，采用基于 RooFit/RooStats 的[83-85]轮廓似然比统计检验在渐近近似[86]中使用。信号箱内的信号事件总数由两个触发器配置 的总和给出。
where is the number of MOT for trigger configuration the number of signals per MOT produced in the ECAL target, depending on the mass/coupling parameters and , and the trigger-dependent signal efficiency.
其中 是触发器配置 的 MOT 数量， 是在 ECAL 靶中产生的每个 MOT 的信号数，取决于质量/耦合参数 和 ， 是依赖于触发器的信号效率。
The main systematic effects contributing to the signal yield defined in Eq. (1) are studied in detail. The uncertainty on is conservatively set to . The systematics associated with the production crosssection 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 is found to be . In the latter case, both the running of at the upper bound and higher order corrections from soft photon emissions are estimated to contribute through respectively and through the Sudakov factor at the level of and . Uncertainties relative to the purity of the ECAL target are addressed at the level of . The systematics on are evaluated by comparing the detector responses in MC and data around the MIPcompatible 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 . Because of the strong dependence of the efficiency on the trigger configuration , in particular on the distance from the zero-line, additional uncertainties due to and misalignment are studied through the change in efficiency as a response to small displacements of the Sc counters. Because of the mass-dependence of the trigger rate [70], the resulting uncertainty reaches up to . As such the total systematic in the signal yield of Eq. (1) is . The acceptance loss due to accidentals (pile-up events, ) entering the trigger time window is taken into account in the final efficiency computations. The signal efficiency peaks at its maximum of for
在方程（1）中定义的信号产额中，主要的系统效应被详细研究。 的不确定性被保守地设定为 。与 产生截面相关的系统误差是从 Weiszäcker-Williams（WW）近似引入的不确定性和 QED 修正到精确树级（ETL）表达式中提取的。在前一种情况下，评估产生的 数量的相对误差被发现为 。在后一种情况下， 在上限 处的变化以及来自软光子发射的高阶修正分别通过 和 Sudakov 因子 在 和 水平上估计为贡献。与 ECAL 靶的 纯度相关的不确定性在 水平上得到解决。通过比较 MC 和数据中 MIP 兼容峰附近的探测器响应，特别是在 ECAL 和 HCAL 中，评估了 的系统误差。通过谱积分和峰比，发现相关的累积不确定性不超过 。 由于效率 对触发配置 的强依赖性，特别是对距离零线的距离，由于 和 的不对齐而产生的额外不确定性是通过对 Sc 计数器的小位移响应效率变化进行研究的。由于触发速率[70]的 质量依赖性，导致的不确定性高达 。因此，方程（1）中信号产量的总系统误差为 。由于意外事件（堆积事件， ）进入触发时间窗口导致的接受损失在最终效率计算中得到考虑。信号效率在其最大值 处达到峰值
the mass range .
质量范围 。
After unblinding, no event compatible with production is found in the signal box. This allows us to set the CL exclusion limits on which are plotted in Fig. 3 in the parameter space, together with the values of compatible with the muon anomaly, within . The band is computed using the latest results of the Muon collaboration for the combined Runs 2 and 3 (2019-2020), and the SM prediction of from the Muon Theory Initiative (TI) [29-47, 87]. It is worth noticing that the latest results from the CMD-3 collaboration on the disagree within the level with the TI value and recent lattice QCD computations from the BMW collaboration [90] are in tension by . Our results, excluding masses and coupling , are the first search for a light (vanilla model) with a muon beam using the missing energy-momentum technique (see Fig. 3).
解密后，在信号框中找不到与 产生相容的事件。这使我们能够在 上设置 CL 排除限制，这些限制在 参数空间的图 3 中绘制出来，与与μ子 异常相容的 值一起，在 内。该带是使用 Muon 合作伙伴关于联合 Runs 2 和 3（2019-2020 年）的最新结果， 以及 Muon 理论倡议（TI）[29-47, 87]的 SM 预测计算得出的。值得注意的是，来自 CMD-3 合作伙伴关于 的最新结果与 TI 值在 水平上不一致，而来自 BMW 合作伙伴关于最近格点 QCD 计算的结果则存在 的紧张关系。我们的结果，排除质量 和耦合 ，是使用缺失能量-动量技术（见图 3）对轻 （香草 模型）进行的第一次寻找，使用μ子束。
Figure 3. NA64 CL exclusion limits on the coupling as a function of the mass, , for the vanilla model. The band for the contribution to the discrepancy is also shown. Existing constraints from BaBar and from neutrino experiments such as BOREXINO and CCFR are plotted.
图 3. NA64 CL 排除耦合 的限制，作为 质量的函数， ，用于香草 模型。还显示了 带对 贡献到 差异的影响。来自 BaBar 和来自 BOREXINO 以及 CCFR 等中微子实验的现有约束被绘制出来。
Figure 4 shows the obtained limits at in the target parameter space with freeze-out parameter for accelerator-based experiments probing thermal DM for , away from the resonant enhancement , and . The thermal targets for favored 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 excludes a portion of the parameter space, below the current CCFR lim- its, constraining for a choice of masses the dimensionless parameter to .
图 4 显示了在 处获得的限制，位于目标参数空间 中，具有冻结参数 ，用于加速器实验探测 的热暗物质，远离共振增强 ，以及 。为标量、伪-狄拉克和马约拉纳暗物质候选方案绘制了受欢迎 值的热目标，并从基础玻尔兹曼方程的积分中获得[96]。结果表明，NA64 排除了 参数空间的一部分，低于当前 CCFR 限制，对于一组质量 的选择，将无量纲参数限制为 。
Figure 4. The 90% CL exclusion limits obtained by the NA64 experiment in the parameters space for thermal Dark Matter charged under with and the coupling for MOT. The branching ratio to invisible final states is assumed to be invisible (see text for details). Existing bounds obtained through the CCFR experiment are shown for completeness. The thermal targets for the different scenarios are taken from .
图 4. NA64 实验在热暗物质在 下带电的 参数空间中获得的 90% CL 排除限制，其中 和 的耦合为 MOT。假设到不可见末态的分支比为 不可见 （详见文本）。为了完整起见，通过 CCFR 实验获得的现有限制也显示出来。不同情景的热靶材取自 。
In summary, for a total statistics of MOT, no event falling within the expected signal region is observed. Therefore, CL upper limits are set in the parameter space of the vanilla model, constraining viable mass values for the explanation of the anomaly to , with . New constraints on light thermal DM for values for 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 is expected to fully cover the compatible parameter space and to boost its coverage in the search for thermal Dark Matter complementing the world wide effort for DS searches [13-16]. 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 , scalar portals [67], millicharged particles [98] or or processes involving Lepton Flavour Conversion [99-101].
总的来说，对于 MOT 的总统计数据，没有观察到落入预期信号区域的任何事件。因此，在 基础模型的 参数空间中设定了 CL 上限，限制了用于解释 异常的可行质量值为 ，具有 。还获得了关于轻热暗物质的新约束，其数值为 ，适用于 。随着实验设置的改进，例如增加一个额外的磁谱仪以将动量误重建的背景减少一个数量级以上，并增加统计数据，预计 NA64 将完全覆盖 兼容的参数空间，并增强其在寻找热暗物质方面的覆盖范围，从而补充全球 DS 搜索的努力[13-16]。本研究中展示的μ子束的使用为探索其他充分动机的新物理场景打开了一扇新窗户，例如在质量区域 的基准暗光子模型，标量门户[67]，微电荷粒子[98]或涉及轻子味转换的 或 过程[99-101]。
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
我们衷心感谢 CERN 管理和员工的支持，特别感谢 CERN BE-EA 部门的帮助。我们感谢 C. Menezes Pires 和 R. Joosten 在束流动量站方面的支持。
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).
我们还要感谢 HISKP，波恩大学（德国），苏黎世联邦理工学院，以及瑞士国家科学基金会资助的项目编号 186181，编号 186158，编号 197346，编号 216602，智利 ANID - 千禧科学倡议计划-ICN2019 044，西班牙 MCIN/AEI/FEDER，欧盟资助的 RyC-030551-I 和 PID2021-123955NA-100。
† e-mail:laura.molina.bueno@cern.ch
† 电子邮件：laura.molina.bueno@cern.ch
‡ e-mail:henri.hugo.sieber@cern.ch
‡ 电子邮件：henri.hugo.sieber@cern.ch
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