Searching for muonic forces with the ATLAS detector

The LHC copiously produces muons via different processes, and the muon sample will be large at the high-luminosity LHC (HL-LHC). In this work we propose to leverage this large muon sample and utilize the HL-LHC as a muon fixed-target experiment, with the ATLAS calorimeter as the target. We consider a novel analysis for the ATLAS detector, which takes advantage of the two independent muon momentum measurements by the inner detector and the muon system. We show that a comparison of the two measurements, before and after the calorimeters, can probe new force carriers that are coupled to muons and escape detection. The proposed analysis, based on muon samples from $W$ and $Z$ decays only, has a comparable reach to other proposals. In particular, it can explore the part of parameter-space that could explain the muon $g-2$ anomaly.


I. INTRODUCTION
The Standard Model (SM) of particle physics has been successful in describing the known elementary particles and their interactions and is directly tested by experiments up to the TeV scale. Nevertheless, the SM is not a complete description of nature, and should be augmented by new physics (NP) degrees of freedom which account for neutrino oscillations, dark matter (DM), and the matter/ antimatter asymmetry of the Universe.
One possible manifestation of NP are new particles with masses in the MeV-to-GeV range and suppressed couplings to the SM. Such new particles could be part or all of DM, or act as mediators to a dark sector. Muonic force carrier (MFC) mediators, X, are particularly interesting. These mediators have flavor-specific couplings [1][2][3][4], couple to the SM only through muons and may decay predominantly to DM. MFCs potentially explain inconsistencies in low-energy observations such as the anomalous magnetic dipole moment of the muon [5,6], and the possible anomaly in the measurement of the proton radius in muonic hydrogen [7,8].
Existing constraints on the existence of dark sector mediators are predominantly derived from beam-dump, fixed-target, or collider experiments [9][10][11]. The constraints are weaker for models where the mediator couplings to electrons or protons are suppressed. Specifically, MFC mediators are only weakly constrained [2]. Models where m X > 2m μ and X dominantly decays back to μ þ μ − are constrained by the BABAR analysis [12]. Data from rare B decays also constrain MFC mediators [13], but with larger model dependency.
Recent studies suggest that MFCs could be searched for in muon-fixed-target experiments [2,[14][15][16], in kaon decays at the NA62 experiment [17] or in Belle II [18]. MFC production in muon-target interactions would register as a momentum difference between the incoming and outgoing muons which is not accounted for by the energy deposition in the (instrumented) target. Such dedicated apparatuses may be available at CERN by running the NA64 experiment with a muon beam [2,19], and at FermiLab, by leveraging the muon beam line of the Muon (g − 2) experiment [20], and constructing the M 3 apparatus [15].
In this article, we propose to utilize the ATLAS detector as a muon fixed-target experiment, which is sensitive to the missing muon momentum signature and therefore probes MFCs. The calorimeters serve as an instrumented target, and the inner detector (ID), and muon system (MS) provide independent muon momentum measurements before and after the target, as illustrated in Fig. 1. It is important that in ATLAS there is no uninstrumented material between the calorimeter and the MS. This ensures that an accurate measurement of the missing muon momentum signature is possible.
Using muons from Z and W decays, in total Oð10 10 Þ muons on target at the high-luminosity LHC (HL-LHC), we estimate that the proposed analysis is sensitive to MFC masses in the MeV-GeV range with couplings as low as g X ∼ 10 −4 -10 −3 . This MFC parameter space includes the region which is relevant to account for the ðg − 2Þ μ anomaly and is comparable to other proposals such as M 3 phase 1 and NA62. The fact that our proposal is based on an existing experiment, with well-known and excellent performances, is a big advantage compared to other recent proposals which are much more hypothetical in nature.
Our idea to use the ATLAS detector as a muon fixedtarget experiment is a novel one, and we believe it can be extended to probe other new physics scenarios based on the same principles. This analysis can also be adopted by future high energy colliders and thus it can play a role in future detector design.

II. BENCHMARK MODELS
We use as MFC benchmarks scalar or vector mediators, i.e., X ¼ S or V [2, [15][16][17][18]. The effective interaction Lagrangians are given by where we have omitted the mass and kinetic terms. We assume that X is a mediator to a dark sector which predominantly decays into undetected particles, i.e., implicitly assume large couplings of X to sufficiently light dark sector constituents. Alternatively, X could be sufficiently long lived so as to escape detection. The effective interaction in Eq. (1) can be UV completed. For example, the vector interaction may arise in a broken gauged L μ − L τ gauge theory [21]. The scalar interaction can be a result of interactions with heavy leptons, which are integrated out [17].
The simplified models in Eq. (1) are subject to existing constraints depending on their UV completion. Here we consider (a) the muon magnetic moments [5,6], ðg − 2Þ μ ; and (b) from CHARM-II μ trident [22,23]. The bound from ðg − 2Þ μ can be avoided in models where different contributions to the loop cancel each other, for example, scalar and vector against pseudoscalar and axial vector, e.g., see [24]. The μ-trident bound is valid only for vector mediators with left-handed coupling to the muon.

III. ATLAS AS A MUON FIXED-TARGET EXPERIMENT
Muons produced at the ATLAS interaction point (IP) traverse the entire detector, leaving signals in the ID, calorimeters, and MS. ATLAS muon reconstruction [25] is first performed independently in the ID and MS, with each detector subsystem providing muon spatial location and transverse-momentum (p T ) measurements. Subsequently the ID and MS information is combined with the calorimeter measurement to form the muon tracks which are used in physics analyses.
ATLAS defines four muon types, according to the details of the combination procedure, of which two are relevant for this work (i) A combined (CB) muon track is formed from independent tracks in the ID and MS, with a global refit that uses the hits from both subdetectors. (ii) Extrapolated (ME) muons have trajectories reconstructed based only on the MS track and a loose requirement on compatibility with originating from the IP. The muon track parameters are defined at the IP, and take into account the muon energy loss estimation, p ME ≈ p MS þ E cal . The latter estimate combines the calorimeter measurement with a detailed analytic parametrization of the average energy loss, a method which yields a precision of ∼30 MeV for 50 GeV muons. ME muons are ideal candidates for an ATLAS search of the MFC signal. MFC production in the muon-target interaction manifests as a difference between p ID and p MS that is not compensated by E cal . We define an observable which combines the ID momentum measurement, p ID , with, p ME , the reconstructed momentum of an ME type muon, where, up to resolution effects, we identify the incoming (outgoing) muon momentum with respect to the target, p in ðp out Þ, with p ID ðp ME Þ.
The tag-and-probe method with Z → μμ, where one muon is reconstructed as a CB muon (tag) and the second may be a ME muon (probe), provides a high-purity muon sample [25] with loose selection on the probe muon that can be used to search for MFCs. We foresee that with careful analysis it will be possible to also use W → μν decays.

IV. SENSITIVITY ESTIMATION
Next, we estimate the sensitivity of the proposed analysis to probe MFCs, which are described by the interactions in Eq. (1). Throughout we will normalize our projections to the expected integrated luminosity of the HL-LHC, L int LHC ¼ 3 ab −1 .
A. Muon-target luminosity and MFC production rate For minimally ionizing particles such as muons, the fixed-target effective luminosity is given by where we have treated the ATLAS calorimeter 1 as a thin target, and assumed a single material composition, 2 with density ρ T , mass number A, length Δx and N μ ¼ L int LHC σ fid prod incoming muons. Here, m 0 ¼ 1.661 × 10 −24 g, and σ fid prod is the (process dependent) cross section for muon production at the LHC, within the ATLAS fiducial volume. In this work, we assume a 63 29 Cu target, which corresponds to A ¼ 63 and ρ T ¼ 8.96 g=cm 3 in Eq. (4). For this analysis, the dominant calorimeter characteristic is the total radiation lengths. Other effects due to the specific detector material are small, and would be taken into account correctly by a full-detector simulation.
We estimate the MFC signal production rate following the schematics in Fig. 1. A muon originating from the IP with momentum p in and direction η interacts at a point x in the material target of length l T ðηÞ, and produces a MFC, and an outgoing muon of momentum p out which travels in an angle θ relative to the incoming muon direction. The expected number of produced MFCs within the detector acceptance, A (which includes the MFC target interaction) is 3 where σ T is the fixed-target MFC production cross section, μT → μTX, and P in ðp in ; ηÞ, is the incoming muon doubledifferential distribution: In addition, below we will be interested in dN X =dρ, which is straightforward to derive from Eq. (5). For further details on the N X estimation see the Supplemental Material [26].

B. Signal yields
We use Monte-Carlo (MC) simulations to estimate the signal yield and its ρ distribution. We consider muons from Z and W decays because they provide high-purity muon samples. 4 In the MC sample, events are selected following the criteria in Ref. [27]. In the selected events, only muons in the barrel (0.1 < jηj < 1.05) are used, since the ID momentum resolution in the barrel is better, and the depth of the calorimeter is approximately constant. In addition, we require a weak ID to MS angular matching requirement of θ < 0.1 to account for the loose IP matching of ME muons.
The incoming muon momentum spectrum, P fid ðp in Þ≡ R dηP in ðp in ; ηÞ, is obtained from a MadGraph5 v2.6.1 simulation [28,29] of the hard process, with up to two additional jets, and interfaced with Pythia 8.230 [30] for showering and hadronization. We apply the MLM jetmatching scheme [31] to combine the different samples, and we use the Delphes 3.4.1 [32] detector simulation with the standard ATLAS card. The momentum spectrum of the selected muons is taken at the truth level of MC generation and smeared based on the published ATLAS muon momentum resolutions (σ in ¼ 0.015 MeV þ 3 × 10 −7 p in and σ out ¼ 0.05p out , see [25]).
We normalize the muon production rate to match the ATLAS result of Z → μμ and W → μν [27,33,34], and find where σ ATLAS pp→Z→μμ ðσ ATLAS pp→W→μν Þ ¼ 0.78ð8.0Þ nb [27], and ϵ eff Z ðϵ eff W Þ ¼ 0.50ð0.44Þ is the efficiency factor relating the ATLAS cuts in Ref. [27] with our selection of the barrel as the fiducial volume. The resulting momentum spectra are shown in Fig. 2.
We note that signal muons may not trigger the combined muon high level trigger. However, Z events can be triggered with high efficiency due to the second muon in the event. In the case of W events with a large energy loss to the mediator particle, it may be challenging to trigger with the 1 The ATLAS calorimeter ranges up to Δx ¼ 144X 0 , where X 0 ¼ 1.757 cm is the radiation length of electrons in iron. 2 While the calorimeter targets are comprised of various materials, the variation in the event yield is small, and the effect on the experimental sensitivity is negligible. 3 Note that we have assumed that the detector is cylindrically symmetric, however, ϕ integration can be straightforwardly incorporated to account for possible inhomogeneities. 4 It would also be interesting to consider other sources of muons at the LHC, e.g., from heavy flavor or J=ψ's decays. There are potentially many more of these muons but they would of course be much more difficult to use for an MFC search due to the large background and lower p T 's. single muon trigger by itself or the missing transverse energy (MET) trigger by itself, since the MET at L1 does not take into account the muon momentum. ATLAS has in place trigger tools that allow the collection efficiency to be recovered; in particular, at L1, ATLAS has the L1Topo trigger, where it is possible to lower the p T thresholds on the MET and the muon in the case of a muon-MET feature. Additionally, in the L1Topo trigger, more complex triggers like an m T calculation are possible. After the phase-II upgrade of ATLAS, tracking info will be available at the L1 global trigger.
The μT → μTX process is simulated in MadGraph 5, including the target nuclear-atomic form factor by modifying the photon-target vertex, see Supplemental Material [26] for details. The Lagrangian of Eq. (1) was implemented as a UFO model [35] by using FeynRules 2.3.32 [36].
Finally, combining the above, we estimate that N X ∼ g 2 X ð10 8 ; 10 7 ; 10 5 Þ for m X ¼ 16, 126, 1000 MeV, respectively, with X ¼ S or V. Thus, for an Oð1Þ acceptance, and assuming negligible background, we predict that this analysis is sensitive to g X in the range 10 −4 -10 −2 .

C. Backgrounds
The dominant expected background is due to true muons with mismeasured momenta. Motivated by ATLAS results in Ref. [25], we model the momentum mismeasurement with a three Gaussian resolution function, centered at 0. The resolution as a function of ρ is given by The relative fraction of the third Gaussian is hard to model with the currently available public data. We therefore consider two representative cases This resolution function includes hard photon radiation and possible subsequent hadronization effects mentioned in [15] as well as catastrophic energy losses. These effects are well modeled in the full-detector simulation used by the ATLAS based on GEANT [42], as validated by dedicated test-beam campaigns, see e.g., [43,44] and in situ [25]. The expected background ρ distributions, R, for each of the above cases are plotted in Fig. 3. In addition to true muons with mismeasured momenta, there can also be backgrounds from charged pion and kaon decays in-flight. If these decays happen in or near the calorimeter, they can mimic the MFC signal. From publicly available information, we expect that the requirement of dimuon invariant mass to be within 5 GeV of the Z mass reduces these backgrounds to 5∶10000 of the Z → μμ sample. These backgrounds can be further rejected using an analysis of kink-in-tracks, and calorimeter information. Additionally, since muons from pions decaying in-flight have a minimal momentum fraction of 0.57 of the pion momentum, their contribution at ρ close to −1 is small. We estimate that the in-flight decay background can be rejected to a level between 10 −7 and 10 −8 , and it is expected to be subdominant. The ρ distributions and overall normalization for pion and kaon decays can be extracted from a control sample with a same-charge requirement on the tag and probe particles.

D. Reach for MFC
We estimate the sensitivity of the proposed analysis to probe MFCs by a dN X =dρ line shape analysis. From the binned ρ distribution, we construct a likelihood function, Lðg X ; m X Þ, and assume that the number of observed events is equal to the expected background events per bin. For a given m X , we estimate the expected 95% confidence level upper bound on g X for each of the above background scenarios and for two cases: (i) muons only from Z decays; and (ii) muons from both Z and W decays.
The projections are plotted in Fig. 4 and are compared to the present bounds from ðg − 2Þ μ [5,6] and CHARM-II [22,23] as well as to the projections of M 3 [15], NA62 [17], NA64 μ [16,19] and Belle II [18]. We can see that the reach for m X → 0 can be at the level of g X ∼ 10 −3 -few × 10 −4 (depending on the background model) by using only muons from Z. For the case of a combined analysis of Z and W muons, the sensitivity can reach the g X ∼ 10 −4 level. This part of the parameter space is relevant for the ðg − 2Þ μ anomaly, to thermal freeze-out dark matter scenarios, see e.g., [15], and comparable to other proposals such as M 3 phase 1. Moreover, in the case that X is a L μ − L τ gauge boson, our proposal probes a m X − g X parameter space which may contribute to N eff and possibly reduce the Hubble parameter tension [45]. Finally, it is worth noting that while our sensitivity projections are for the ultimate HL-LHC dataset of 3 ab −1 , even with the current dataset (∼150 fb −1 ) or the expected Run-3 dataset (∼300 fb −1 ), it should already be possible to probe at least a portion of these interesting regions of parameter space.

V. SUMMARY
We propose a search for NP using the large sample of muons produced at LHC collisions, and ATLAS detector as a fixed-target experiment sensitive to missing muon momentum signatures. In the proposed analysis, the calorimeter serves as a target for muons, and the muon momentum measurements before and after it are compared. In principle, this strategy, of utilizing high energy colliders as fixed target in the second production, can be adopted in future experiments.
We focus on the possibility that a muonic force carrier is produced in the muon-target interaction and subsequently escapes the detector or decays invisibly. The detector signature corresponding to this scenario is of an unaccounted loss of muon momentum in the calorimeter. The expected sensitivity of the ATLAS experiment to this signature using muons from Z and W decays is comparable to other proposed experiments, and overlaps the parameter space that can explain the observations in the anomalous magnetic dipole moment of the muon.