Determination of coupling patterns by parallel searches for $\mu^-\to e^+$ and $\mu^-\to e^-$ in muonic atoms

We investigate a possibility that the $\mu^-\to e^+$ conversion is discovered prior to the $\mu^-\to e^-$ conversion, and its implications to the new physics search. We focus on the specific model including the mixing of the $SU(2)_L$ doublet- and singlet-type scalar leptoquarks, which induces not only the lepton flavor violation but also the lepton number violation. Such a structure is motivated by R-parity violating (RPV) supersymmetric models, where a sbottom mediates the conversion processes. We formulate the $\mu^-\to e^+$ rate in analogy with the muon capture in a muonic atom, and numerically evaluate it using several target nuclei. The lepton flavor universality test of pion decay directly limits the $\mu^-\to e^+$ rate, and the maximally allowed $\mu^-\to e^+$ branching ratio is $\sim 10^{-18}$ under the various bounds on RPV parameters. We show that either $\mu^-\to e^-$ or $\mu^-\to e^+$ signals can be discovered in near future experiments. This indicates that parallel searches for these conversions will give us significant information on the pattern of coupling constants.


I. INTRODUCTION
The standard model (SM), where all neutrinos are left-handed and massless, has the accidental global U (1) symmetries which ensure to conserve the lepton flavor numbers, L e , L µ , and L τ . Nonetheless, the lepton flavor violation (LFV) was established by the discovery of neutrino oscillation, which implies that the three global symmetries are broken and the SM should be extended to include LFV sources.
In lots of extended models, LFV sources cause not only the flavor violation among charged leptons (called CLFV) but also the lepton number violation (LNV). One may presume that the LNV processes are minor compared with CLFV, because, aside from the flavor number, the particle number must be violated. However, we know situations where it does not hold. A well-known example is the Majorana mass of the neutrinos; the branching ratio of an LNV process µ − → e + in nuclei could be much larger than that of LFV process µ → eγ due to the GIM suppression in the flavor changing neutral current [1][2][3]. Therefore, both the LFV and LNV processes should be investigated.
The muonic atom is a good probe to both the LFV and LNV; an LFV process µ − → e − conversion, µ − (Z, A) → e − (Z, A), and an LNV process µ − → e + conversion, µ − (Z, A) → e + (Z −2, A). See Ref. [4] for the recent review of the µ − → e + conversion. The experimental signals of these modes is single monoenergetic electron (positron), which is highly clean signal with little SM background. In near future experiments, the searches for these modes are planned by using a number of muonic atoms (COMET [5], Mu2e [6], and PRISM/PRIME [7]).
In this article, we investigate a possibility that the µ − → e + conversion could be discovered prior to the µ − → e − conversion. An interesting example to address the possibility is leptoquarks with the mixing of SU (2) doublet and singlet. The condition is satisfied by sbottoms in R-parity violating (RPV) supersymmetric (SUSY) model [8]. When the sbottomb has the RPV interactionb q and the mixing of SU (2) doubletb L and singletb R , the lepton number is not conserved and the µ − → e + conversion can be induced at tree level. We formulate the µ − → e + conversion rate for the sbottom mediation, and numerically evaluate it under the experimental bounds on RPV parameters. We see that importance to search for and analyze the non-standard reactions of muonic atoms without prejudice that the LFV reactions are always leading compared with the LNV ones.
The contents of this article are as follows: In Sec. II, we introduce leptoquarks inspired by sbottom in RPV SUSY and discuss current constraints on the coupling constants. We show the formula for the rate of the µ − → e − and µ − → e + conversions in a muonic atom in Sec. III. The results are shown in Sec. IV, and finally, the article is summarized in Sec. V.

II. BENCHMARK MODEL
We introduce a benchmark SUSY model wherein the reaction rates of µ − → e + conversion and µ − → e − conversion are comparable to each other.
The gauge invariant superpotential contains the RPV terms [9][10][11], and one of them could be a source of LFV, Here D i is a SU (2) L singlet superfield, and L i and Q i are SU (2) L doublet superfields. Indices i, j, and k represent the generations. The interaction terms related with LFV and LNV processes are whered j is the SUSY partner of down-type quark d j . We assume the simple situation that only the lighter sbottom contributes to low-energy observables, which is motivated by that, in many SUSY scenarios, it is lighter than the first and second generation squarks [12]. Thus, j (k) in d jL ( d * kR ) must be 3. The left-and right-handed sbottom (b L andb R ) are mixed each other after the SU (2) L symmetry breaking, and it could be large as m 2 Here A b is so-called the trilinear scalar coupling, µ is the higgsino mass parameter, and tan β is the ratio of Higgs field vevs. The mixing is parametrized through the diagonalization of sbottom mass as where we set m 1 ≤ m 2 and take the mixing angle θb as Thus the RPV interaction Lagrangian in terms of mass eigenstates is where we defineλ i31 = λ i31 cos θb andλ i13 = λ i13 sin θb.
The lepton flavors are no longer defined as conserved quantities with the interactions in Eq. (1). Then, the µ − → e − conversion in nuclei is induced by the exchange ofb L as shown in Fig. 1.
When theb L -b R mixing exists in addition to the RPV interactions, the lepton number conservation is violated: if the mixing is absent, the lepton number −1 (+1) can be assigned tob L (b R ). The µ − → e + conversion in nuclei arises via the LFV vertex and theb L -b R mixing (Fig. 2). It is important to emphasize that, when either λ 213 or λ 113 is zero, the µ − → e − conversion rate goes to zero, but the µ − → e + conversion could be observable.
The experimental bounds on the RPV parameters are set by independent measurements. We summarize the bounds in the rest of this section. The measurements of atomic parity violation (APV) and parity violating electron scattering (PVES) test the parity violating interaction, and set the bound on λ 131 [13]. The parity violating interaction is parametrized as −(G F / √ 2)C 1iē γ µ γ 5 eq i γ µ q i , where G F = 1.166 × 10 −5 GeV −2 is the Fermi coupling constant. The sbottom interferes with the photon and Z boson in APV and PVES, and the effective coupling is 1 is obtained by including the APV results in the global fit incorporating the Qweak collaboration result and PVES database, C 1d = 0.3389 ± 0.0025 (1σ) [15]. With sin 2 θ w = 0.2382 1 We neglect the QED corrections to the C 1d because it is small, [14], and the resultant effect on the λ 131 bound is negligible.
at the experimental scale, the bound is which depends on the assumption of the stop mass mt L . If the stop is sufficiently heavy, substantially there is no constraint on the coupling. In the analysis of this article, we will set mt L = 1 TeV to have a bound, λ 131 < 0.69.
The sbottom exchange subprocess via λ 231 interferes with the SM neutrino deep inelastic scattering (DIS) ν µ d R → ν µ d R [13]. Taking into account the interference, the coupling for the neutral current connecting ν µ and d R is . The precision measurement of the neutrino DIS provides g d R = −0.027 +0.077 −0.048 [16], which excludes nonzero λ 231 at the 1σ level. The bound at the 2σ level is The direct search sets the limits on sbottom mass and RPV couplings. The decay width of RPV channelb 1 → e lL u L is Here λ(x, y, z) = x 2 + y 2 + z 2 − 2xy − 2yz − 2zx. The decay width of R-parity conserving where Y L and Y R are the hypercharge for left-and right-handed bottom, mχ0 is the neutralino mass, and m b is the bottom mass. Setting the mass scales by maximally small ones m 1 = 200 GeV and mχ 0 = 160 GeV [17], the direct search [18,19] is transferred to the bound on RPV coupling as

D. Lepton flavor universality of pion decays
The RPV interactions, Eq. (1), could violate the lepton flavor universality of pion decays (Fig. 3). The RPV contributions δΓ e and δΓ µ are related to the ratio of decay rates as where β = Γ SM e /Γ SM µ , e = δΓ e /Γ SM e , and µ = δΓ µ /Γ SM µ . Here we assumed e , µ 1. After the straightforward calculation, we obtain Here V ud is the u-d component of the CKM matrix. The first term in the parenthesis comes from the diagrams (a) and (b) in Fig. 3 for e , and from the diagrams (e) and (f) for µ . These initial states form a scalar state with u L and d R , and their contributions are much bigger than other diagram's ones by m 2 π /(m u + m d ) 2 , which is so-called chiral enhancement effect [20]. In the parameter region we are interested in, the first terms dominate e and µ . Besides, the direct search limit (9) is more stringent than the limits from the second and third terms of Eqs. (11) and (12). Then the second and third terms are irrelevant in our analysis. The experimental constraint is given by R π,exp e/µ = 1.2327(23) × 10 −4 according to Ref. [21]. With the SM prediction R π,SM e/µ = 1.2352 × 10 −4 [22,23], we set the constraint as where we allow for a discrepancy of 2σ. The RPV interactions also affect the decay π 0 → e + e − . Since the RPV interactions lead to a (pseudo-)vector state for the initial state, this decay mode does not receive the chiral enhancement. It means that, as long as |λ | 2 /m 2 1 G F , the RPV effects do not appear on this mode. It is because even the Z 0 exchange channel is negligible compared with leading channel, i.e., the electromagnetic loop one [24].

E. Neutrinoless double beta decay
We estimate the bound on RPV parameters along with the neutrinoless double beta decay (0ν2β) in analogy with that assuming the Majorana neutrinos. Extracting the LNV source part in each amplitude (Fig. 4), we find the relation λ 131λ 113 where m ee is the effective Majorana mass of electron neutrino and q is the momentum of internal neutrino. In our analysis we set q = 100 MeV, which is evaluated by the typical distance between nucleons in a nucleus. Applying the bound m ee 0.1 eV [21], above relation (14) leads to the bound on RPV parameters as

III. NEW PHYSICS SEARCHES USING MUONIC ATOMS
The muonic atom sheds light on not only the LFV but also the LNV through µ − → e − conversion and µ − → e + conversion.
The µ − → e − conversion in nuclei occurs with the combination ofλ 213 andλ 113 (Fig. 1). Applying the formula for µ − → e − conversion rate [25], the branching ratio in our scenario is obtained by The dimensionless overlap integral V (p,n) and the muonic-atom lifetimeτ µ are listed in Table I. The most stringent bound, B (µ − → e − ; Au) < 7 × 10 −13 [26], gives the limit by λ 213λ 113 < 1.6 × 10 −7 for m 1 = 200 GeV.  [25] and the lifetime of a muonic atom [27]. The combination of the LFV RPV couplings and theb L -b R mixing gives rise to the µ − → e + conversion in nuclei (Fig. 2). The reaction rate of µ − → e + conversion faces the nuclear transition matrix. For the Majorana-neutrino case, it is evaluated by the nuclear proton-neutron renormalized quasi-particle random phase approximation [28][29][30] and the shell model calculation [31]. The short-range effective operators inducing the µ − → e + conversion were discussed in Refs. [32][33][34]. The conversion rate for other types of operators have not been qualitatively investigated, also for the operator in this work. We therefore estimate the conversion rate in analogy with the muon capture µ − p → ν µ n in muonic atoms.
We adopt the phenomenological parametrization of capture rate for a nucleus of an atomic number Z and of a mass number A [27,35], Here Z eff is the effective atomic number for muonic atoms [36]. The Z eff dependence stems from the effective number of protons in a nucleus (Z eff ) and probability of a muon being at the nuclear center (Z 3 eff ); the latter can also be understood by the expression of the muon wave function, |ψ µ (0)| 2 = (m µ Z eff α) 3 /π. The parameter X 1 corresponds to the capture rate for muonic hydrogen, and X 2 parametrizes the Pauli blocking effect. The experimental data fit the parameters by X 1 = 170 s −1 and X 2 = 3.125 [27].
The µ − → e + conversion rate is inferred in an analogy of the muon capture rate as where N presents the initial nucleus, and (i, j) = (1, 3) , (3, 1). The factor 1/q 2 expresses the correlation function of active neutrino of momentum q. Since the process associates with the internal conversion 2p → 2n, it is expected that the rate is proportional to Z 2 eff . Note that the energy scale factor Q µ − →e + contains the nuclear transition strength in addition to the phase space volume, and its power is determined by the dimensional analysis. The branching ratio is given by B (µ − → e + ; N ) =τ µ Γ (µ − → e + ; N ). We use X 2 = 3.125 as the muon capture, and we take q = 100 MeV, which corresponds to the Fermi momentum of nucleon in the nucleus. The energy scale factor is set by Q µ − →e + = m µ .

IV. RESULTS
Numerical analysis is shown in two cases; One is of negligible µ − → e − conversion rate and the other is more general ones. We adopt m 1 = 200 GeV. Free parameters are the four RPV couplings,λ 213 ,λ 131 ,λ 113 , andλ 231 . We evaluate the maximal B (µ − → e + ; N ). Decomposing B (µ − → e + ; N ) into the target dependent partB (Table II) and uncertain parts (q and Q µ − →e + ), it is rewritten as We the S/N ratio [38]. The shaded area shows the excluded parameter region.λ 131 is unbound unless the stop mass is given. Here we take mt L = 1 TeV. Then the direct search (9) and the measurement of APV-PVES (5) draw the boundaries. The bound onλ 131 gets looser for the heavier stop mass, and then the LFU test in pion decays makes the boundary. It is testable in near future experiments, and could shed light on the LFV and LNV sources. It is important to emphasize that the LFU in pion decays tightly correlated with the µ − → e + conversion (compare Figs. 2 and 3) in this scenario. When the violation of LFU is observed in pion decays, searches for the µ − → e + conversion would provide complementary information for new physics.  Table II for the pattern II, the maximal B (µ − → e + ; N ) is obtained by O(10 −21 ). This implies that the discovery of µ − → e + conversion at COMET, Mu2e, and PRISM experiments rules out the pattern II. Figure 5 (b) is the same as Fig. 5 (a) but for theλ 113 -λ 231 plane. The direct search (9) and the LFU test in pion decays (13) draw the boundaries to the excluded region.

B. General analysis including all four couplings
The bounds on RPV couplings from the searches for µ − → e − conversion and 0ν2β are also comprehended, in addition to the bounds discussed in Sec. IV A. The bounds derived from relevant observables are summarized in Table III.  Figure 6 shows the excluded region (shaded area) for each combination of RPV couplings. In each panel, the other RPV couplings are set to be zero. It has been already investigated for theλ 131 -λ 213 andλ 113 -λ 231 planes, wherein both the µ − → e − conversion and 0ν2β are turned off. The 0ν2β search draws the outline of excluded region in theλ 113 -λ 131 plane ( Fig. III (a)). The µ − → e − conversion search draws the outline of excluded region in thẽ λ 113 -λ 213 plane ( Fig. III (b)). These processes are therefore important ingredients for the analysis in the space of λ 113 ,λ 131 ,λ 213 ,λ 231 . The bounds and observables in Fig. 6 have actually more complicated correlations with each other. Figure 7 shows an example result. In the parameter space wherein  free from all experimental bounds except for the µ − → e − conversion, first, we lead the maximally allowed B (µ − → e + ; Ca). With these arrangements, the maximally allowed B (µ − → e − ; Al) is evaluated. For the region ofλ 113 4 × 10 −9 , as is close to the setting in Sec. IV A 1, B (µ − → e − ; Al) does not reach the PRISM/PRIME sensitivity. In this region, the maximized combinationλ 131λ 213 leads to the large B (µ − → e + ; Ca). For the region of 4 × 10 −9 λ  measurements of these conversions shed light on not only the LFV source but also the origin of LNV in new physics scenarios.

V. SUMMARY
We have investigated the possibility that the LNV process µ − → e + conversion is observed prior to the LFV process µ − → e − conversion. For a reference scenario of our interest, we have focused on RPV SUSY models wherein the SU (2) L doublet and singlet sbottom (b L andb R ) mixes each other.
When the conservation of lepton flavors is violated by the RPV interactions, they give rise to the µ − → e − conversion. Theb L -b R mixing flips the lepton number on the internal sbottom line, and hence the lepton number is no longer conserved. The µ − → e + conversion arises via the LFV vertex and theb L -b R mixing. It is important to emphasize that, when either λ 213 or λ 113 is zero, the µ − → e − conversion rate goes to zero, but the µ − → e + conversion still could be observable.
We have evaluated the rate of µ − → e + mediated by the sbottom in analogy of the muon capture process in muonic atom. Then we have investigated how could the µ − → e + rate be large under the experimental bounds on RPV parameters. Bounds come from the µ − → e − conversion search, the measurement of LFU in pion decays, the direct sbottom search at the LHC, and so on. Especially, we have found that the LFU in pion decays provides the direct constraints for the µ − → e + rate because they are connected through the same combinations of the couplings. The largest B (µ − → e + ; Ca) is achieved in the parameter region of small B (µ − → e − ; Al). In some parameter regions, both B (µ − → e + ; Ca) and B (µ − → e − ; Al) are experimentally reachable at next-generation experiments. Complementary measurements of these conversions shed light on not only the LFV source and also the origin of LNV in new physics scenarios. It is important to search for and analyze the non-standard reactions of muonic atoms without prejudice that the LFV reactions always are leading compared with the LNV ones.