Neutrino mediated muon--electron conversion in nuclei revisited

The non-photonic neutrino exchange mechanism of the lepton flavor violating muon-electron conversion in nuclei is revisited. First we determine the nucleon coupling constants for the neutrino exchange mechanism in a relativistic quark model taking into account quark confinement and chiral symmetry requirements. This includes a new, previously overlooked tree-level contribution from neutrino exchange between two quarks in the same nucleon. Then for the case of an additional sterile neutrino we reconsider the coherent mode of this process. The presence of a mixed sterile-active neutrino state heavier than the quark confinement scale Lambda_c (~1 GeV) may significantly improve the prospects for observation of this process in future experiments as compared to the conventional scenario with only light neutrinos. Turning the arguments around we derive new experimental constraints on |U_(mu h)| and |U_(e h)| mixing matrix elements from the non-observation of muon--electron conversion.


non-observation of muon-electron conversion.
The recent results from the Super-Kamiokande [1] and SNO [2] experiments on atmospheric and solar neutrinos give convincing evidence on neutrino oscillation and hence on neutrino masses and lepton flavor violation (LFV). As is known, experimental searches for rare processes offer complimentary information on the LFV. This can shed additional light on the physics underlying this phenomenon, discriminating various models beyond the standard model (SM) (for a review see [3]). The muon-electron [(µ − , e − )] conversion in nuclei [3][4][5][6][7][8], is one of the most prominent lepton flavor changing reaction. Experiments searching for this process in the coherent channel with a monoenergetic final state electron have reached an unprecedented level of sensitivity. Presently, the most stringent upper limits on the branching ratio R µe related to the (µ − , e − ) conversion has been set by the SINDRUM II collaboration [9]: R µe = Γ µe Γ µνµ < 6.1 × 10 −13 (target : 48 T i), 2.0 × 10 −11 (target : 79 Au), where Γ µe and Γ µνµ are the rates of the (µ − , e − ) conversion and ordinary muon capture, respectively. Future experiments will significantly improve these limits. There are proposals of the SINDRUM II collaboration to reduce the current limits on the ratio R µe for 48 T i and 197 Au down to 10 −14 and 6 × 10 −13 [9], respectively. A new Muon Electron COnversion (MECO) experiment on 27 Al is planned at BNL [10] with an expected sensitivity on the branching ratio of about 2 × 10 −17 . Another future project PRIME [11] for the (µ − , e − ) conversion on 48 T i is going to reach a sensitivity of 10 −18 . The realization of these projects would allow to set new stringent constraints on the LFV interactions relevant for the (µ − , e − ) conversion. This process can be triggered by the LFV interactions associated with the exchange of neutrinos and/or new heavy particles (neutralinos, charginos, leptoquarks etc.) predicted in models beyond the SM [3][4][5][6][7]. In general all the (µ − , e − ) conversion mechanisms can be separated into photonic and non-photonic ones. A mechanism is photonic if it involves a virtual photon line connecting the effective leptonic LFV current with the electromagnetic nuclear current, otherwise a mechanism is non-photonic. These classes of mechanisms differ significantly on their particle and nuclear physics sides and are usually studied independently.
In this letter we concentrate on the non-photonic neutrino exchange mechanism. We are studying a model with three left-handed, weak doublet neutrinos ν ′ Li = (ν ′ Le , ν ′ Lµ , ν ′ Lτ ) and a certain number n of the SM singlet, right-handed sterile neutrinos ν ′ Ri = (ν ′ R1 , ...ν ′ Rn ). Due to mixing they form n + 3 neutrino mass eigenstates N i with masses m i related to the weak  GeV was previously studied in connection with K ± and τ semileptonic decays [12].
The analysis of the nuclear (µ − , e − ) conversion starts with the elementary nucleon process In models with non-trivial neutrino mixing this process can be realized at the quark level according to the diagrams of Fig. 1. The diagrams of Fig. 1a,b are the well-known one-quark box diagrams [3] while that of Fig. 1c is the new tree-level two-quark diagram. These lowest order diagrams represent the complete set of the neutrino exchange diagrams on quark-level relevant for the above nucleon process.
In the process considered the typical momentum transfer Q 2 to the nucleon is small comparable to the scale set by the muon mass with Q 2 ∼ m 2 µ . Therefore, the quarks in the diagrams considered cannot be treated as free particles as would be the case in the asymptotic region Q 2 ≫ Λ 2 c . An appropriate treatment should deal with quarks as states which are confined in the nucleon. Due to the lack of a rigorous theory for confinement in QCD one has to engage phenomenological models. In this work we are using the perturbative chiral quark model (PCQM) [13,14] treating quarks as extended objects, the constituent quarks, which are confined in the nucleon. In this model each quark vertex acquires a form factor with the characteristic momentum scale Λ c ∼ 1 GeV related to the confinement length l c ∼ Λ −1 c . In the diagrams of Fig. 1 these form factors set the scale for the loop momentum q ν of the virtual neutrino. This is in contrast to the previous analysis [3] of the diagrams Fig. 1a,b associated with different (neutrino mass dependent) terms of the (µ − , e − ) conversion amplitude, where the q ν scale is set by the W-boson mass due to the presence of the corresponding propagators in these diagrams.
Knowing the characteristic scale q 0 ∼ Λ c of the neutrino momentum q ν in the diagrams of Fig. 1, we consider the general structure of the (µ − , e − ) conversion amplitude A µe . In the (ν i , ν h ) neutrino scenario introduced above one can write Here G(q 2 /q 2 0 ) is a characteristic function suppressing the contribution for q 2 ≫ q 2 0 in the loop momentum integration. Then it follows that as a consequence of the unitarity of the mixing matrix with i U µi U * ei = 0. Previously, only the case m h ≪ q 0 of Eq. (4) was considered in the literature [3,5,15].
Because of the smallness of the ratio m 2 i /q 2 0 the neutrino exchange mechanism leads to rates for the (µ − , e − ) conversion which are out of reach for ongoing and near future experiments.
The situation changes if there exists a heavy neutrino state ν h with mass m h ≫ q 0 and with a non-vanishing admixture of active flavors ν µ,e . In this case the suppression factor associated with the small neutrino masses is replaced by the product of mixing matrix elements U µh U * eh as indicated in Eq. (4). Since the existing experimental constraints on U µh U * eh are not stringent [16] one may expect much larger rates for (µ − , e − ) conversion in the (ν l , ν h ) scenario than for the case without an intermediate mass ν h state.
Following the standard approach [7] we consider the effective nucleon Lagrangian written as Here m µ is the mass of the muon. The partial contributions of the diagrams Fig. 1a,b,c are contained in the coupling coefficients as In the present work we restrict ourselves to the dominant mode of (µ − , e − ) conversion, where the axial-vector current contribution [15] is neglected and, thus, only the vector form factors f p,n V are relevant for the subsequent analysis.
We evaluated the form factors f p,n V within the perturbative chiral quark model (PCQM), a relativistic quark model suggested in [13] and extended in [14] for the study of low-energy properties of baryons. The model operates with relativistic quark wave functions and takes into account quark confinement as well as chiral symmetry requirements. The PCQM was successfully applied to σ-term physics and to the electromagnetic properties of the nucleon [14]. In the present analysis we included the contributions from both the one-body (Fig.   1a, b) and the two-body (Fig. 1c) diagrams neglecting the external three-momenta of the leptons. For the one-body diagrams of Fig. 1a and Fig. 1b we restrict the expansion of the quark propagator to the ground state eigenmode: where E 0 and u 0 ( x) are the quark ground state energy and wave function; that is we restrict the intermediate baryon states to N and ∆ configurations. In Ref. [14] we showed that this approximation for the quark propagator works quite well in the phenomenology of low-energy nucleon physics.
With above approximations the partial contributions of the diagrams of Fig. 1a,b,c to the coupling constant of the vector current are: Here i and j are the quark indices, Φ 0 ( x) =ū 0 ( x)γ 0 u 0 ( x) and Φ( x) =ū 0 ( x) γu 0 ( x) are the time and spatial components of the quark vector current. The single components are projected onto the three-quark state building up the nucleon state |N >. As in Refs. [14] we use the variational Gaussian ansatz [17] for the quark ground state wave function given by: where χ s , χ f , χ c refer to the spin, flavor and color spinors. The constant N = [π 3/2 R 3 (1 + 3ρ 2 /2)] −1/2 is fixed by the normalization condition.
Our Gaussian ansatz contains two model parameters: the dimensional parameter R and the dimensionless parameter ρ. The parameter ρ can be related to the axial coupling constant g A calculated in zeroth-order (or 3q-core) approximation: where γ is a relativistic reduction factor The parameter R can be physically understood as the mean radius of the three-quark core and is related to the charge radius < r 2 E > P LO of the proton in the leading-order (or zerothorder) approximation as [14] < r 2 In our calculations we use the tree-level value g A =1.25 as obtained in Chiral Perturbation Theory [18] and the averaged value of R = 0.6 fm [14] corresponding to < r 2 E > P LO = 0.6 fm 2 . A straightforward analytical evaluation of the expressions in Eqs. (7)-(9) results in the following values for the partial isospin dependent vector coupling constants: which are just the sum of the partial contributions.
Starting from the effective Lagrangian of Eq. (5) the branching ratio of the coherent (µ − , e − ) conversion is derived as Here, E e (E e = m µ − ε b , ε b is the muon binding energy) and p e (p e = | p e |) are energy and momentum of the outgoing electron. F (Z, p e ) is the relativistic Coulomb factor [19] and the nuclear structure factor is defined as In our analysis we used values for the nuclear matrix elements M p,n as derived in Ref. [7]. Using the calculated vector coupling constants as an input, in Table I we   Significant sensitivity, down to 10 −4 − 10 −3 , will be hopefully achieved by the future MECO (target 27 Al) and PRIME (target 48 T i) experiments.
The mixing of massive neutrinos, like ν h with active ν e,µ,τ flavors was previously looked for in various experiments except for (µ − , e − ) conversion. An extensive list for the constraints on the |U eh | and |U µh | mixing matrix elements for various masses m h is given in Ref. [16]. In summary, we have studied the non-photonic neutrino exchange mechanism of coherent (µ − , e − ) conversion in nuclei in the presence of sterile neutrinos. We found a new tree level contribution to the (µ − , e − ) conversion (Fig. 1c) which is as important as the previously known box-type contributions (Fig. 1a,b). The nucleon form factors, parameterizing the effective nucleon Lagrangian, have been analyzed within the perturbative chiral quark model [14]. In this model the momentum scale of the virtual neutrino is set by the quark confinement with Λ c ∼ 1 GeV in the three diagrams. This significantly differs from the previous analysis [3] of the diagrams Fig. 1a The experimental values of the total muon capture rate Γ µνµ are from Ref. [20]. Nucleus