Electron Beam Dump Constraints on Light Bosons with Lepton Flavor Violating Couplings

We study constraints on light and feebly interacting bosons with lepton flavor violation from electron beam dump experiments. Scalar, vector, and dipole interactions of the bosons are analyzed, respectively, and excluded regions from the searches for decays into electron-positron pairs are derived. It is found that parameter regions unconstrained by flavor violating decays of muon can be excluded using the results of the E137 experiment. We also discuss the impact of the search for flavor violating decays of the light bosons in electron beam dump experiments.


I. INTRODUCTION
For the last decade, the searches for new physics beyond the Standard Model (SM) have been primarily led by the LHC with particular attention to the regions of TeV-scale mass with O(1) coupling. Despite a great deal of experimental and theoretical efforts has been made in that direction, no signals have been found so far. Given the fact, recently, there is a growing interest in a neutral boson having sub-GeV mass and feeble interactions with the SM particles. In fact, there are several theoretical motivations to consider such a feeblyinteracting light boson; For instance, it plays a crucial role in the dark matter problem [1][2][3][4][5][6][7], the muon g − 2 anomaly [8][9][10], the Hubble tension [11][12][13][14], and it is also claimed that the existence of the light boson helps us understand the observed energy spectrum of high energy cosmic neutrino [15][16][17][18][19][20][21][22][23].
Feebly-interacting light bosons are expected to be long-lived and leave very displaced vertex signals at experiments. Among possible experiments, beam dump experiments are especially suitable for searching such bosons. The bosons can be created from incoming beam on a target, travel long distance, and decay to electron-positron pairs near the detector distant from the target. In the 1980s, several beam dump experiments were carried out to search for light neutral scalar bosons, such as axions, by using proton beam [24] and electron beam [25][26][27][28][29], and their experimental constraints were later translated to those on gauge bosons in Refs. [30] and [31,32], respectively. At present, the results of the beam dump experiments have been applied to various types of light boson models, and comprehensive analyses including constraints from collider experiments have also been done, e.g., see Refs. [33], [34,35], and [36] for the B − L and the L α − L β gauge boson (α, β = e, µ, τ ), the dark photon, and the dark higgs boson, respectively. Also, in Refs. [37][38][39][40], the prospect of a new beam dump experiment at the future ILC experiment is studied.
Thought, beam dump experiments are probably the best probe of searching for feeblyinteracting light bosons, the constraints are derived by assuming the simplest model setup, that is, only a single particle and a single coupling constant are introduced. In the literature, however, several extensions have been proposed recently.
In this paper, given the current situation, we attempt to enlarge the availability of beam dump experiments for the extended scenarios. We are especially interested in extensions including Charged Lepton Flavor Violation (CLFV) [41][42][43][44]. As is well known, CLFV is one of the most evident signals for new physics beyond the SM. Many experimental searches have been carried out, and in most cases muonic processes like µ → eee and µ → eγ place the tightest upper bounds on CLFV couplings. On the other hand, in the sub-GeV mass regions, electron beam dump experiments can possibly exclude parameter regions below the upper bounds from the muonic processes, and new bounds could be obtained. To illustrate  the exclusion regions of electron beam dump experiments, we consider three benchmark   scenarios for a new light boson, i.e., a scalar boson having Yukawa interactions, a gauge   boson having vectorial interactions, and a gauge boson having dipole interactions. For these, we introduce CLFV couplings in the electron (e) and muon (µ) sector as well as Charged Lepton Flavor Conserving (CLFC) couplings, and derive the constraints from the E137 experiment, which usually provides the strongest constraints. This paper is organized as follows. In Sec. II, we introduce three types of CLFV interactions analyzed in this work. In Sec. III, the production cross sections of light bosons through bremssthralung processes and the formula to calculate the number of signal events are given.
In Sec. IV, the constraints on the light bosons with CLFV coupling by the E137 experiment are derived, and the results are compared with the existing bounds from the muon CLFV decays. In Sec. V, the impact of searches for flavor violating decays are discussed. Section VI is devoted to summary.

II. INTERACTION LAGRANGIAN
We start our discussion by introducing interaction Lagrangians analyzed in this work.
We consider three types of interactions, i.e., scalar-, vector-and dipole-type interactions, and briefly discuss the origin of these interactions in mind a two Higgs doublet model, a gauged L µ − L τ model [45][46][47] and a dark photon model, respectively. In the following, we restrict our analyses only to CLFC and CLFV in the eµ sector.

A. Scalar interactions in two Higgs doublet model
Firstly, we give the scalar-type interaction Lagrangian. Such an interaction can be obtained in models with an extra leptophilic Higgs doublet scalar. The new Higgs doublet scalar is assumed to couple to only leptons and to contribute to mass generation of the charged leptons. In this case, the mass matrix of the charged leptons consists of two parts originating from each Higgs doublet after developing vacuum expectation values (VEVs). In general, those two mass matrices are not necessarily diagonalized simultaneously, and the misalignment of the mass matrices generates CLFV interactions after the diagonalization.
To avoid large contributions from the CLFV interaction by the SM Higgs boson, we assume the VEV of the extra Higgs doublet field to be much smaller than the electroweak scale.
Then, the charged lepton masses are mostly determined by mass matrix with the VEV of the SM Higgs boson. In such a case, the CLFV interactions are given from the Yukawa terms of the extra scalar boson.
The relevant interaction Lagrangian is given by where φ is the extra scalar boson, and y and y eµ(µe) are the CLFC and CLFV coupling constants, respectively. Here we omit the interactions obtained from the mixing between φ and the SM Higgs boson, due to the assumption of a small VEV for φ.
From Eq. (1), the total decay width of φ is given by Here the partial decay widths into the charged leptons are given by [48,49] where S = y or y eµ(µe) for the CLFC-and CLFV-decays, respectively.  [47]. At loop level, CLFV vector interactions will be induced through the CLFV scalar loop for massive gauge bosons. In this case, it is unnecessary for the gauge symmetry to be flavor-dependent. In this work, we only study the CLFV vector interactions at the tree level. However, analyses for the loop-induced CLFV interactions are essentially the same and will be translated by replacing the couplings with loop-induced ones in our results.
For general discussions, we parametrize the CLFV coupling by θ which is the mixing angle between electron and muon. The mass and flavor eigenstates are connected by this mixing angle. Then, the Lagrangian of the vector interaction in mass eigenstate is given by L vector = g Z ρ (s 2 eγ ρ e + c 2 µγ ρ µ + sc µγ ρ e + sc eγ ρ µ) where Z is the gauge boson and g is the gauge coupling constant of the U(1) Lµ−Lτ gauge symmetry, and s = sin θ and c = cos θ, respectively. Here ν µ and ν τ are left-handed muon and tau neutrinos. It should be noted that, in general, there also can exist interactions through the kinetic mixing. Such interactions conserve lepton flavor and are independent of the mixing angle. Then, the flavor conserving productions and decays of Z are modified.
Although analyses of such a situation will lead more general constraints, the increase of the parameters will make the analyses complicated. For simplicity, we assume that contributions from the kinetic mixing can be negligible, and omit the kinetic mixing throughout this paper.
Given the Lagrangian in Eq. (4), the total decay width of Z is obtained as where the partial decay width into the neutrinos is given by in the limit of massless neutrinos 1 , while those into the charged leptons are where V = g s 2 (g c 2 ) or g sc for the decays into ee (µµ) or eμ andēµ, respectively, and λ(a, b) is the Kallen function defined as follows : where µ and µ are CLFC and CLFV dipole couplings, respectively, and A ρσ stands for the field strength of A . We assume that the dipole couplings are real. One may imagine that there should exist similar CLFV interactions in which external A is replaced by photon, which are strictly constrained by the MEG [50] and BaBar [51] experiments. However, it will be possible to suppress such dangerous electromagnetic dipole operators when the new gauge boson has an interaction vertex with neutral CP-even and odd scalars since the same vertex does not exist for the photon. One of such examples is the so-called dark photon model with dark Higgs particles [52].
Given the Lagrangian in Eq. (9), the total decay width of A is given by where D = µ or µ for CLFC or CLFV decays, respectively, and λ(a, b) is given in Eq. (8). interactions with electron and muon, they also can be produced through flavor violating bremsstrahlung processes shown in Fig. 1. The bremsstrahlung production process can be evaluated by using the cross section of electron-photon scattering, e − + γ → + X ( = e, µ), in the Weizsäcker-Williams approximation [53][54][55]. In this section, we show the differential cross sections of the scalar and vector boson production process for the interactions given in the previous section. Then, we give formulae to calculate the number of events in electron beam dump experiments.

A. Differential cross section of bremsstrahlung process
With the improved Weizsäcker-Williams approximation, the differential cross section of bremsstrahlung process of the light boson production by a target with atomic number Z, e − + Z → − + Z + X ( = e, µ, X = φ, Z , A ), is calculated by that of scattering one, e − + γ → − + X, as follows : where α is the electromagnetic fine structure constant, E 0 is the energy of the injected electron beam, β X = 1 − m 2 X /E 2 e is the kinematical factor, and x = E X /E e with E e(X) being the energy of incident electron (produced light boson). The effective photon flux is denoted as ξ [31], and the definition of ξ is given in Appendix A. The differential cross section of the scattering process with respect to x is given by where g X and m X are the coupling constant and mass of the light boson X, respectively. In the square bracket,Ũ n (n = 1-4) are functions of the maximal angle θ max determined by the angular acceptance of the detector and η given by where m is the mass of lepton . The definition and approximate form ofŨ n are given in Appendix B. The functions f 1 (x), f 2 (x) and f 3 (x) depend on the final state lepton and the types of the light boson X. For convenience, we define and Note that = e corresponds to the case of the CLFC interactions while = µ to the CLFV interactions. Thus, the difference between the CLFC and CLFV interactions arises in the third term of η as well as r and g X .
For the scalar interaction given in Eq. (1), X = φ and g φ = 1. The functions f 1,2,3 (x) in Eq. (13) are where S 1 = |y eµ | 2 + |y µe | 2 and S 2 = Re(y eµ y µe ) for the CLFV process ( = µ), while S 1 = 2|y e | 2 and S 2 = Re(y 2 e ) for the CLFC ( = e), respectively. For the vector interaction in Eq. (4), X = Z and g Z = g sc (g s 2 ) for the CLFV (CLFC) process. The functions in Eq. (13) are In the dipole case, X = A and g A = µ m A (µ e m A ) for the CLFV (CFLC) process, and function of the X boson mass. In the left panels, E e is fixed to be 20 GeV, and x is taken to be 1.0 (red), 0.9 (green), and 0.5 (blue), respectively. In the right panels, E e is varied as 20 (red), 10 (green), and 3 (blue) GeV, respectively, and x is fixed to be 1. Solid and dashed curves correspond to the CLFC and CLFV interactions, respectively.
In figure 2, the coupling constants are taken to be y e = y eµ = y µe = 10 −6 as an illustrating example. In the left panel, the CLFV differential cross sections are much smaller than the CLFC ones with the same x. This is becauseŨ n are decreasing functions of η , and η µ is larger than η e . The CLFC differential cross sections increase by order of magnitudes as x approaches unity because the second term in Eq. (13) is much larger than other terms.
These with x < 1 also rapidly decrease as m φ becomes large since η e scales as m 2 φ /E 2 e . On the other hand, the CLFV differential cross sections are rather constant because η µ scales as m 2 µ /E 2 e for m φ < m µ . For m φ larger than m µ , r µ 1 and η µ η e , therefore the CLFC and CLFV cross sections asymptotically approach to each other. In the right panel, the CLFC differential cross sections with x = 1 are similar to different E e while the CLFV ones change order of magnitudes. For x = 1, Eq. (13) for the CLFC process is approximated as where we have usedŨ e 2 1/2η e = E 2 e /2m 2 e . The effective photon flux is almost constant for m φ < 1 GeV. Thus, the electron energy dependence only comes from β φ which is almost unity unless close to the threshold. In the CLFV process, on the other hand, the electron energy dependence remains inŨ e 2 /E 2 e . From both panels, it can be understood that the main contributions to the signal event come from the CLFC process with 0.9 < ∼ x < ∼ 1.   case. Figure 4 also shows the same plots for the dipole interaction. The dipole moments are fixed to µ e = µ = 10 −6 GeV −1 . In this case, the differential cross sections are proportional to m 2 A . These increase as m A increases, then goes to zero as the mass reaches to the kinematical threshold. From Figs. 3 and 4, the dominant contributions to the signal events also come from the CLFC differential cross sections with 0.9 < ∼ x < ∼ 1 for the vector and dipole interaction case, respectively.

B. Number of signal events
The number of the signal events can be calculated by the following formula [31], where N e is the number of electrons in the injected beam, N avo 6 × 10 23 mol −1 the Avogadro's number, X 0 the radiation length of the target, A the target atomic mass in g/mol, and T sh ≡ ρ sh L sh /X 0 with ρ sh being the density of the shield. The length of shield and decay region are denoted as L sh and L dec , respectively. The decay length in laboratory frame and branching ratio of X are denoted by L X and Br(X → e + e − ), respectively. The energy distribution of electrons after passing through t radiation lengths in the beam dump is denoted by I e and given by [56] I e (E 0 , E e , t) = 1 E 0 where b = 4/3 . This energy distribution function sharply peaks around t 0 for E e = E 0 , and the t integration can diverge. To avoid such divergence, we split the t integration into two parts by a cut t cut , following Ref. [57].

IV. ELECTRON BEAM DUMP CONSTRAINTS
In this section, we show the constraints from electron beam dump experiments, E137 Using the continuous-slowing-down-approximation range [58], the range for the muon to be stopped is estimated at ∼ 50m at most for the E137 experiment. The range is 1/3 times shorter than the shield length. Therefore, the produced muon would be stopped in the shield.   Table II, and the partial widths of these CLFV decays are given in Appendix C. Note that constraints from µ → eγ are weaker than those from µ → eee since the decay width of µ → eγ is suppressed by the electromagnetic coupling and a loop factor in comparison with   that of µ → eee. As can be seen from the figure, most of the parameter regions are excluded by these constraints. Nevertheless, we find that our 95% C.L. limit further excludes the parameter regions unconstrained by µ → eee and µ → eφ. The exclusion region shifts to lighter m φ as y eµ /y e increases. As we explained in Sec. III, the differential cross sections are mainly determined by the CLFC ones for y eµ < 10. Therefore, in this case, the effects of the flavor violating couplings appear only in the total decay width. Once the threshold of φ → eµ opens, the decay length and branching ratio of φ → ee become smaller. Then, the expected number of the signal events reduces, resulting in the exclusion regions shown in Figure 5. For y eµ = 100, contributions from the CLFV differential cross sections are not negligible, making the exclusion regions wider especially in the small y e regions. Such a large y eµ , however, makes the decay length and the branching ratio so small that no signal events are expected above the threshold of µ → eφ.  The red and blue curves represent the same constraints as in Fig. 5. In contrast to the case of the scalar interaction, the exclusion regions become wider as the CLFV mixing becomes large. This is because a nonzero θ induces interactions between Z and electrons 3 , increasing the production cross sections and the branching ratio of Z → ee. For θ > 0.5, the constraint from E137 experiment excludes the parameter regions for m Z > m µ + m e , which are not constrained by µ → eee and µ → eZ . When θ = π/2, the interaction to muon vanishes and the exclusion region coincides with that for the minimal U(1) Le−Lτ model [33]. The CLFV dipole is taken to µ /µ e = 0 (solid), 1 (dashed), 10 (dotted) and 100 (dotdashed), respectively. In this case, the constraint from the E137 experiment also excludes the parameter region below the µ → eee limit. The behavior of the exclusion region is similar to the scalar interaction case.

V. FLAVOR VIOLATING DECAY SIGNAL
In the previous section, we have only considered the CLFC decay of X → ee as the signal of the vector and scalar boson, based on the setup and analyses of the E137 experiment.
Under the presence of the CLFV interactions, the light bosons can decay into eμ andēµ above its kinematical threshold. This decay mode is a smoking gun signature of CLFV interactions in the dark sector. Searches for these decays will bring further information on the CLFV couplings.
To illustrate the impact of the CLFV decay searches, we have demonstrated analyses for the scalar CLFV decays with the setup of the E137 experiment in Table. I. Figure 8 shows presumed excluded regions with 95% C.L. for φ → ee, eµ, µµ. The solid, dashed, dotted and dot-dashed curves correspond to y eµ /y e = 0, 1, 5 and 100, respectively. Compared with This is because, for m φ ≥ m e + m µ (2m µ ), the decay branch of φ → eµ (µµ) opens, and the decay length of the light boson shortens. The smaller coupling is, therefore, preferred to reach the detector of the E137 experiment. Since these regions cannot be excluded only by analyzing the ee decay search, the CLFV couplings can be further constrained in these regions. Thus, it is important to search for not only the CLFC decays but also the CLFV decays as signals in future beam dump experiments. The FASER experiment [63][64][65], which will start next year, will be able to perform the searches for the dark photon decay into eµ pairs. Theoretical analyses will be shown in [66].

VI. SUMMARY
We have studied the constraints from the electron beam dump experiment taking into account the charged lepton flavor violating interactions. The scalar, vector, and dipole type interactions are considered and the excluded regions have been derived for the search for X → ee. We have found that the parameter regions unconstrained by µ → eee and µ → eX can be excluded for three interactions. The exclusion regions depend on the CLFV and CLFC couplings as well as the boson mass, and new bounds can be derived when the ratios of the CLFV coupling to the CLFC one are < ∼ 100. For the illustrative purpose, we have also derived the presumed excluded region for the search of φ → ee, eµ, µµ decays in the same setup with the E137 experiment. It has been found that the excluded regions can be extended and the CLFV couplings can be further constrained in those regions. Such searches and analyses for the CLFV decays will be important to new physics searches in future experiments. The elastic electric form factor is given by where a = 111Z −1/3 /m e and d = 0.164 GeV 2 A −2/3 . The inelastic electric form factor is given by where a = 773Z −2/3 /m e , m p is the proton mass and µ p = 2.79.
The electric form factor is dominated by the elastic one in our study, thus it scales by Z 2 . See [31] for more detail.
where θ max is the maximal angle determined by angular acceptance of the detector, and η (x) is given in Eq. (14). For the small angle θ max ,Ũ n can be approximated by the following formulae :  where s 12 = (p − k 3 ) 2 and s 23 = (p − k 1 ) 2 . The integral ranges of s 12 and s 23 are given by The amplitude is written as where P X s 23(12) = s 23(12) − m 2 X −1 , X = φ, Z , A , and C = y e y eµ , g s 3 c and µ e µ for the scalar, the vector and the dipole interaction, respectively. For the scalar interaction, M 2 3 = 2s 12 s 23 + 4m β (m α + m β )(s 12 + s 23 ) − 2m β (m α + m β )(5m 5 β + 2m α m β + m 2 α ) , for the vector interaction, and for the dipole interaction,