1 Introduction

Observations in cosmology and astrophysics imply the existence of a dark sector potentially containing new particles that could weakly couple to Standard Model (SM) particles [1]. Neutrino oscillations coupled with nonzero neutrino masses provide experimental evidence of lepton flavor violation. Furthermore, the existing discrepancy between the measured [2] and expected [3, 4] value of the muon anomalous magnetic moment provides strong motivation for new physics searches with muons [5]. Inspired by these developments, a certain class of new theories proposes the search for charged lepton flavor violation (CLFV), which is heavily suppressed in the SM. In the coming decades, a new generation of experiments will be conducted to search for CLFV [6,7,8,9,10,11]. In parallel, next-generation long-baseline neutrino oscillation experiments [12, 13] will study neutrino oscillations while at the same time producing an intense muon beam dump with its beam line.

In this work, we study the sensitivity potential of muon-on-target experiments to new physics using a CLFV benchmark model introduced in a recent work [14], which uses a light, scalar boson associated with \(\mu -\tau \) conversion. While the previous work derived constraints from data in existing beam-dump experiments (LSND [15], NuTeV [16], CHARM [17]) and for the future SHiP experiment [18], here we focus on two further experiments: the coming neutrino experiment, DUNE [12] deploying a high-intensity beam from the Long-Baseline Neutrino Facility (LBNF) [19], and a fixed-target experiment, NA64\(\mu \) [20, 21], which searches for new physics with the high-energy muon beam from the CERN Super Proton Synchrotron (SPS) accelerator. Hence, we study new physics searches using two complementary muon beam setups. Even though we focus on one particular benchmark model scenario, there are also other possibilities to probe hidden sectors with muon beams and using similar techniques [22,23,24,25,26]. We also note that other groups have also performed studies of (charged) lepton flavor physics, including the role of mesons or bosons with different spin-parity assignments [27,28,29,30,31,32].

2 Flavor-changing scalar with long lifetime

The model proposed by [14] considers a new complex scalar field, \(\phi \), with a mass window \([m_{\tau } - m_{\mu }, m_{\tau } + m_{\mu }]\) that couples to \((\mu , \tau )\). With such a mass range, long lifetimes can be achieved with propagation distances on the order of tens of kilometers. The effective Lagrangian interaction terms describe the coupling of the new scalar field with leptons,

$$\begin{aligned} \mathcal {L_{I}} = \phi {\bar{\mu }}(g_{V} + g_{A}\gamma ^{5})l + \phi ^{*}{\bar{l}}(g_{V}^{*} - g_{A}^{*}\gamma ^{5})\mu \end{aligned}$$
(1)

with vector and axial vector coupling \(g_{V}\), \(g_{A}\), respectively. This model produces a benchmark parameter region explaining the muon \(g-2\) anomaly with a typical value of \(|g_{V}|\simeq 3\times 10^{-3}\), also used in this work. There are multiple production modes to produce \(\phi \) at beam dump experiments: direct electroweak process, heavy meson decay, and high-energy muons hitting a fixed target. In this work, we focus on the third case (the so-called \(\mu \)-on-target scenario). In this mode, muons pass through dense material and can produce the \(\phi \) boson via the exchange of a virtual photon with the nuclei of the target, \(\mu (p) N(P_i) \rightarrow \tau (p') \phi (k) N(P_f)\). Once created, the bosons produce a missing energy signature or propagate long distances and may be detected in a detector downstream from the target. When the incoming beam energy is much higher than the particle mass, the double-differential cross section of the \(2\rightarrow 3\) production process can be estimated using the equivalent photon approximation [33],

$$\begin{aligned}&\frac{\textrm{d}^{2}\sigma (p+P_{i}\rightarrow p' + k + P_{f})}{\textrm{d}E_{\phi }\textrm{d}\cos \theta _{\phi }} \nonumber \\&\quad = \frac{\alpha \chi }{\pi }\frac{E_{\mu }x\beta _{\phi }}{1-x}\frac{\textrm{d}\sigma (p+q\rightarrow p' + k)}{d(p\cdot k)}\biggr |_{t=t_{\textrm{min}}}, \end{aligned}$$
(2)

where \(E_{\mu }\) is the initial muon energy, \(E_\phi \) is the energy and \(\theta _{\phi }\) the scattering angle of \(\phi \) with respect to the initial muon in the lab frame, \(x = E_{\phi }/E_\mu \), \(q = P_i - P_f\), \(t = -q^2\) is the momentum transfer, \(\alpha = 1/137\) is the fine structure constant, \(\beta _{\phi }\) is the relativistic factor for \(\phi \), and \(\chi \) is the effective photon flux, defined as

$$\begin{aligned} \chi = \int _{t_{\textrm{min}}}^{t_{\textrm{max}}} \textrm{d}t\frac{t - t_{\textrm{min}}}{t^2} F^2(t), \end{aligned}$$
(3)

where \(F(t) = Z^2/(1+t/d)^2\) is the form factor, with \(d = 0.164\) GeV\(^2 A^{-2/3}\), and the limits are given in the appendix of [14]. We calculate the analytical expression for \(\chi \) using Mathematica [34] and obtain

$$\begin{aligned} \chi = Z^2 \left[ \frac{t_{\textrm{min}}}{t} + \frac{d+t_{\textrm{min}}}{d+t} + \frac{(d+2t_{\textrm{min}})}{d}\frac{\ln {t}}{\ln {(d + t)}} \right] _{t_{\textrm{min}}}^{t_{\textrm{max}}}.\nonumber \\ \end{aligned}$$
(4)

In Eq. (2), \(\sigma (p+q\rightarrow p' + k)\) is the cross section for the \(2\rightarrow 2\) scattering process, \(\mu (p)\gamma (q)\rightarrow \tau (p')\phi (k)\),

$$\begin{aligned} \frac{\textrm{d}\sigma (p+q\rightarrow p'+k)}{d(p\cdot k)} = \frac{|\mathcal {{\bar{A}}}_{2\rightarrow 2}|^2}{8\pi s^2}, \end{aligned}$$
(5)

where \(|\mathcal {{\bar{A}}}|^2\) is the amplitude squared, which we calculate using the FeynCalc tools [35] for Mathematica. We obtain the following expression for the \((\mu , \tau )\) case

$$\begin{aligned}&|\mathcal {{\bar{A}}}_{2\rightarrow 2}|^2 = -\frac{e^{2} m_\mu m_\tau (g_{A} g_{A}^* - g_{V} g_{V}^*)}{(m_\mu ^2 - s)^2 (m_\tau ^2 - u)^2} \nonumber \\&\quad \times \left[ m_\mu ^4 (m_\phi ^2 + u)+ 2 m_\mu ^3 (m_\tau ^3- m_\tau u) \right. \nonumber \\&\quad + m_\mu ^2 \left( m_\tau ^4 - 2 m_\phi ^2 s - 2m_\tau ^2 u + u(u-2s) \right) \nonumber \\&\quad + \left. 2 m_\mu m_\tau s(u - m_\tau ^2 ) + s \left( m_\phi ^2 s + m_\tau ^4 - 2 m_\tau ^2 u + u(s+u) \right) \right] , \end{aligned}$$
(6)

where \(e = \sqrt{4\pi \alpha }\), \(m_{\phi }\), \(m_{\tau }\), and \(m_\mu \) are the masses of the boson \(\phi \), \(\tau \), and \(\mu \), respectively, and s, u are the Mandelstam variables, which can be evaluated in the laboratory frame,

$$\begin{aligned} s&= (p+q)^2 \simeq m_\mu ^2 - \frac{u - m_{\tau }^2}{1 - x} \nonumber \\ u&= (p-k)^2 \simeq -E_\mu x \theta _\phi ^2 - \frac{1-x}{x} m_{\phi }^2 + (1-x)m_\mu ^2. \end{aligned}$$
(7)

To calculate the lifetime of the \(\phi \) boson, \(\tau _{\phi }\), we use Equations 3.2\(-\)3.5 in [14] adapted to the \((\mu , \tau )\) case. We calculate the cross section and decay width using the GNU Scientific Library [36]. The production cross section for the \(\phi \) boson as a function of the incoming lepton energy is shown in Fig. 1. Assuming \(m_{\phi } \simeq m_{\tau }\), the threshold for the production is given by \(E_{\mu } > [(2m_{\tau } + m_{N})^{2} - m_{\mu }^{2} - m_{N}^{2}]/2m_{N} \simeq 3.8\) GeV for Pb, above which the cross section steeply rises. In the following, we use the corresponding muon beam flux of each experiment (NA64\(\mu \) and DUNE) to evaluate the expressions for the \(\phi \) production via the variables defined in Eq. (7).

Fig. 1
figure 1

Production cross section of the \(\phi \) boson, with mass \(m_{\phi } = m_{\tau }\) and coupling constant \(|g_{V}|=3\times 10^{-3}\), as a function of the incoming muon energy

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3 \(\mu \)-on-target experiments

We estimate the projected sensitivity of an experiment by finding the pair of parameter values \((g_{V}, m_\phi )\), where \(N_\phi \) appropriate signal events are produced either directly in the target (NA64\(\mu \)) or in the detector (DUNE), as explained later, after a given exposure. In this work we exploit the \(\phi \) boson production process using different and complementary \(\mu \)-on-target experiments: NA64\(\mu \) with the active dump technique (using missing energy as a signal) compared to proton beam-dump experiments such as the DUNE neutrino experiment using the LBNF proton beam (direct detection through interaction as a signal). Although the underlying production mechanism is the same, the two techniques differ in the flux of the muon beam, \(\Phi _{\mu }(E)\), and the target thickness and materials.

In a general case, \(\phi \) bosons are generated by the \(\mu \)-on-target process, and after production, a fraction of them decay inside a detector volume and can be detected. The number of such signal events is

$$\begin{aligned} N_{\phi } = \int \textrm{d}E_{\phi } \Phi _{\phi }(E_\phi ) \times \frac{l_\textrm{det}}{\gamma \beta c \tau _{\phi }}, \end{aligned}$$
(8)

where \(\gamma \) is the relativistic Lorentz factor, \(l_{\textrm{det}}\) is the length within which bosons decay, \(l_\textrm{det}/\gamma \beta c \tau _{\phi }\) is the fraction of bosons decaying in flight to produce a signal in the detector, and \(\Phi _{\phi }(E_\phi )\) is the flux of \(\phi \) bosons at the detector, estimated as

$$\begin{aligned} \Phi _{\phi }(E_{\phi })&= \int \textrm{d}E\Phi _{\mu }(E) \int _{E_{\textrm{min}}}^{E} \textrm{d}E_{l}\frac{n_A}{-\textrm{d}E/\textrm{d}l} \nonumber \\&\quad \times \int _{0}^{\theta _{\textrm{det}}}\textrm{d}\theta _{\phi }\sin \theta _{\phi }\frac{\textrm{d}^{2}\sigma (E_l, E_\phi )}{\textrm{d}E_{\phi }\textrm{d}\cos \theta _{\phi }}. \end{aligned}$$
(9)

Here, \(\Phi _{\mu }(E)\) is the flux of the muon beam as a function of energy, \(n_{A}\) is the number of target atoms per volume, \(E_{l}\) is the muon energy after traveling a length l in the target and losing energy according to the stopping power \(-\textrm{d}E/\textrm{d}l\), \(E_\textrm{min}\) is the energy of the muon at the end of the target, and \(\theta _{\textrm{det}}\) is the angular acceptance of the detector.

In the following subsections, we separately describe the two experimental scenarios and the assumptions in the derived sensitivity limits.

3.1 NA64\(\mu \) scenario

NA64\(\mu \) is a fixed-target experiment at CERN looking for new particles of dark matter and portal interactions produced in electromagnetic showers and coupled to muons. The experiment uses the secondary 160-GeV muons from the interactions of 400-GeV protons from the CERN SPS with a target. A set of beam scintillators and veto counters, low-material-budget trackers, and dipole magnets allow us to precisely constrain the momentum of the incoming 160-GeV muons impinging on an active target. The main detector, where \(\phi \) may be produced, consists of an electromagnetic calorimeter with 40 \(X_0\) radiation length. Downstream, the detector is further equipped with veto counters and a \(\sim \)30-interaction-length hadronic calorimeter. New particles could be produced by the muon beam scattering in the target and later decay to visible SM particles that could be seen by their signatures in a downstream detector. The current work is based on the detection of missing energy and momentum carried away by the produced hypothetical, long-lived \(\phi \) boson, leaving a scattered muon as experimental signature (with the momentum of the scattered muon in the range of 10–80 GeV/c). The sensitivity in the search for the \(\phi \) boson is higher with respect to the beam-dump approach due to the lower power in the coupling strength without a decay vertex. Thus, in NA64\(\mu \), only the number of events at the production target needs to be estimated. Therefore, the number of events is given by \(N_{\phi } = \int \textrm{d}E_{\phi } \Phi _{\phi }(E_\phi )\). Furthermore, the production target thickness is small, and the muon energy loss can be neglected. As a result, we use the following expression to estimate the \(\phi \) boson flux

$$\begin{aligned} \Phi _{\phi }(E_{\phi })&= l_\textrm{target} n_{A}\int dE\Phi _{\mu }(E) \nonumber \\&\quad \times \int _{0}^{\theta _{\textrm{det}}}\textrm{d}\theta _{\phi }\sin \theta _{\phi }\frac{\textrm{d}^{2}\sigma (E, E_\phi )}{\textrm{d}E_{\phi }\textrm{d}\cos \theta _{\phi }}, \end{aligned}$$
(10)

where \(l_\textrm{target}\) is the thickness of the target.

There has been extensive study of simulating and evaluating the production of dark matter particles with a muon beam at NA64 [26, 37, 38]. We use the same method in this work to estimate the sensitivity reach of the experiment. We assume a muon beam with a mean of 160 GeV energy and a width of 4.3 GeV hitting a lead target with total data of \(\sim 3\times 10^{13}\) muons-on-target (MOT).

3.2 DUNE/LBNF scenario

The Deep Underground Neutrino Experiment is a next-generation, wide-energy-beam long-baseline neutrino experiment at Fermilab. It will use an intense (anti)neutrino beam that passes through a near detector at Fermilab and a far detector 1300 km away in South Dakota. The neutrino beam line of DUNE is a result of exhaustive design and optimization work [19]. The beam is produced by a 60–120-GeV proton beam hitting a graphite target, after which the produced pions and kaons decay to leptons and neutrinos in a \(\sim 220\)-m-long decay pipe. At the end of the pipe, a dedicated \(\sim 30\)-m-long stainless-steel structure acts as a beam dump to stop all muons 300 m upstream from the near detector. We use the simulation of the neutrino beam production to trace particles along the beam axis. The muon flux used in the calculation is estimated from a dedicated tracking plane, which is located at the end of the decay pipe in the simulation. An example of the obtained muon energy spectrum is illustrated in Fig. 2. The peak of the spectrum is at \(E_{\mu } \simeq 2.5\) GeV; however, the long high-energy tail is responsible for the majority of \(\phi \) boson production due to the rapid increase in the cross section with the incoming lepton energy (see Fig. 1). At the lower-energy region, the production rate is much smaller. From the neutrino flux simulation we estimate an integrated muon flux of \(\Phi _{\mu } \simeq 5\times 10^{19}\) muons for \(1.1\times 10^{21}\) protons on target (POT), corresponding to 1 year of data-taking.

In DUNE, a signal could be detected in the near detector from the decay of the \(\phi \) boson that was produced by the muons hitting the stainless-steel dump. We consider the decay channel \(\phi \rightarrow \mu ^{+}\mu ^{-}\nu _{\mu }\nu _{\tau }\) with a branching ratio of \(17\%\) [14], motivated by the dimuon results from NuTeV [16]. Possible backgrounds leading to a dimuon signature include deep inelastic scattering (DIS) and resonance production of mesons in charged-current (CC) muon–neutrino (\(\nu _\mu \)) interactions with a target nucleus. The mesons could decay in semi-leptonic mode, producing an extra muon. In order to analyze such potential background processes, we performed simulations with the GENIE [39] Monte Carlo (MC) event generator that provides a comprehensive neutrino interaction modeling in the \(E_{\nu } \sim 100\) MeV − few-hundred GeV neutrino energy region, including quasi-elastic, resonance, and DIS processes. Unlike the \(\phi \) boson decay, events with DIS or resonant meson production are accompanied by additional activity in the final state. Similarly to previous findings for NuTeV, after discriminating for low-multiplicity events with two muons, we found that these backgrounds could be completely suppressed in an MC simulation of 400 million events of neutrino interactions on an argon target (corresponding to \(\sim 5\) years of operation of DUNE at the nominal intensity). However, further studies are planned with a full detector simulation to gain a detailed understanding of the possible bounds on the background rejection.

Fig. 2
figure 2

Energy spectrum of muons at the end of the decay pipe from the full DUNE neutrino beam line simulation [19], see explanation in the text

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4 Results

We illustrate the sensitivity potential for the benchmark CLFV scenario with the complementary muon beams at NA64\(\mu \) and DUNE in Fig 3. For both experiments, the double-differential cross section in Eqs. (9) or (10) is evaluated given the energy spectra of each experiment, \(\Phi _{\mu }(E)\), and the kinematic limits on the final-state \(\phi \)-boson fractional energy, x, which is constrained by the masses of the boson \(\phi \) and the \(\tau \).

In the case of the NA64\(\mu \) experiment, a total integrated muon flux of \(\sim 3\times 10^{13}\) MOT is achievable [20]. The time needed to accumulate the assumed total MOT is estimated to be \(\sim 100\) days. This conservative estimation is based on the CERN SPS delivering on average 3500 spills per day and \(2\times 10^{8}\) muons per spill. Benefiting from the unique combination of a 160-GeV muon beam with missing-energy and momentum search, NA64\(\mu \) would be able to perform a competitive search for such a CLFV signal. Furthermore, the sensitivity reach critically depends on the length of the active target. We find that a 1-m-long target would already be able to explore a large part of the parameter space, \(g_{V} \ge 6\times 10^{-3}\). However, the experiment is highly modular, and a possible optimization of the setup could further enhance its potential: a feasible option is to increase the target length to 5 m which would make it possible to completely cover the \((g_{\mu }-2)\) preferred region and to probe a variety of other new physics scenarios involving muons. We also find that, already with a \(10^{12}\) MOT and a \(\sim 3\)-m-long target, the \(g_{V} \ge \times 10^{-2}\) parameter region can be covered. A detailed Monte Carlo simulation of an optimized setup will follow as a next step.

For the DUNE experiment, the \(\phi \)-boson production is driven by the high-end tail of the muon flux. Compared to NA64\(\mu \), the lower muon energies at DUNE and thus lower production cross section are compensated by the more intense muon flux. Assuming 10–20 years of operation at nominal intensity, DUNE would be able to explore a significant part of the parameter space, reaching into the \(g_{V} \simeq 10^{-2}\) region and potentially improving the constraints from NuTeV. However, an optimization of the neutrino beam line could further enhance the contribution of high-energy muons in the flux and subsequently approach the \(g_{V} \le 10^{-2}\) benchmark region. This scenario is partially motivated by a recent work exploring an alternative beam-dump operation mode to probe new physics with DUNE [40].

For comparison, we also show the projected sensitivity for the same \(\mu \)-on-target mode calculated by [14] for CHARM, NuTeV, and SHiP. Here, we do not show the constraints or projected sensitivity limits derived for the direct electroweak and heavy meson decay process, which was included in the previous work [14], since we only focus on the \(\mu \)-on-target scenario. In the case of SHiP, the assumed total data corresponds to \(2\times 10^{20}\) protons on target (POT). The worse sensitivity in DUNE relative to SHiP stems from the lower muon energies in the flux [41]. It is also noted that there are differences in the beam intensity between NA64\(\mu \) and SHiP. We assume \(10^{12}\) POT per spill intensity at the CERN SPS in the case of the muon beam line used by NA64\(\mu \), while for SHiP the proton beam intensity is generally expected to be an order of magnitude higher.

We note that a similar setup of NA64\(\mu \) is capable of searching for other new scalar particle candidates using the same muon beam [26]. In addition, the proposed Muon Missing Momentum (\(M^{3}\)) experiment at Fermilab [42] plans to probe new physics with a dedicated muon beam. Finally, a number of experiments also have the potential to search for hidden-sector scalar particles, such as SHADOWS [43], HIKE [44], and ATLAS [45].

Fig. 3
figure 3

Constraints from CHARM, NuTeV [16, 17], and projected sensitivity curves for SHiP [14, 18] (dash-dotted, magenta line), DUNE (solid brown and blue lines for 10 and 20 years of operation, respectively), and NA64\(\mu \) (dashed brown and black lines) in the \(\mu \)-on-target production mode (except for CHARM). In the case of NA64\(\mu \), the optimized setup with 1-m-long (brown dashed line) and 5-m-long (black dashed line) targets are also shown separately. The light blue region shows the benchmark parameter range explaining the muon \(g_{\mu }-2\) anomaly

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5 Conclusions

In summary, we present the sensitivity potential of two \(\mu \)-on-target experiments, NA64\(\mu \) and DUNE, as complementary modes of searching for new physics with muon beams. We find that both NA64\(\mu \) and DUNE have the potential to cover a significant portion of the benchmark model parameter space, (\(m_{\phi }, g_{V}\)). NA64\(\mu \) with an optimized setup could probe the coupling parameter down to \(g_{V}\simeq 3\times 10^{-3}\), completely covering the muon \(g_{\mu } - 2\) preferred region and thus providing a similar projected reach as SHiP. DUNE will also be able to cover unexplored parts of the parameter space, potentially improving on the obtained constraints from NuTeV. An optimization of the neutrino beam line, increasing the contribution from the high-energy tail, could allow us to further enhance the sensitivity of DUNE.

Although we use a given CLFV model as a benchmark in this work, we note that similar techniques can be used to study the sensitivity potential of experiments with muon beams to other physics scenarios. Beyond the \(g_{\mu }-2\) discrepancy and neutrino flavor oscillations, searches for dark sector particles are also motivated by the matter–antimatter asymmetry, the known dark matter abundance from astrophysical and cosmological observations, or by theoretical motivations strongly suggesting the existence of additional gauge groups weakly coupling to SM fields [1, 46].