Muon-electron lepton-flavor violating transitions: shell-model calculations of transitions in 27Al

In this paper we present the results of large-scale shell-model calculations of muon-to-electron lepton-flavor violating transitions for the case of the target nucleus 27Al. We extend the previous shell-model calculations, done in the sd model space, by including also the p orbitals in order to see whether the negative-parity states produce any significant effect in the conversion rate. The analysis of the results shows the dominance of coherent transitions mediated by isovector operators and going by the ground state of the target, with practically null influence of excited positive- or negative-parity states.


I. INTRODUCTION
The non-zero mass of the neutrino allows for a variety of lepton-flavor violating processes, forbidden in the standard model of particle physics. One of these many processes is µ → e conversion [1][2][3], in which a bound 1S muon is captured by the nucleus and an electron is emitted with energy E e ≈ m µ . Other lepton-flavor violating processes include µ → eγ, µ → eēe, τ → µγ, and τ → eγ. In minimal extensions of the standard model, in which massive neutrinos are included, the lepton-flavor violating processes would be suppressed by the ratio of the masses of the neutrino and the weak boson, (m ν /m W ) 4 ∼ 10 −48 -10 −50 , and therefore be experimentally unobservable. The observation of such a conversion would therefore point to the existence of new massive particles, as was pointed out in [4].
In the present article we extend the shell-model valence space to include both the p and sd shells using a realistic effective interaction. In this way the potentially significant incoherent contributions with negative parity final states can be included. This also changes the occupation of the p-shell orbitals, which affects the important monopole part in the dominating coherent channel. The larger model space, however, increases the required computational burden significantly. For example the m-scheme dimensions for the 1/2 + states is 80 115 in the sd shell but 61 578 146 in the p − sd model space. Therefore, some limitations to the number of computed shell-model states are made. The coherent and incoherent contributions are calculated in order to find the ratio between the coherent and total µ − → e − conversion rates. This ratio is needed, since experimentally only the coherent channel is measurable due to the lack of background events such as muon decay in orbit and radiative muon capture followed by e + e − pair creation [13].
Since the formalism [14][15][16][17][18][19][20] is rather well known we shall restrict the presentation of it in Section II to the minimum needed to allow for a comprehensible reading of the manuscript. In the following sections we shall introduce the definitions of the participant nuclear matrix elements as well as the basic ingredients of the shell-model procedure.
The results of the calculations are presented and discussed in Section IV and our conclusions are drawn in Section V.

II. FORMALISM
Various extensions to the standard model of particle physics allows for several possible mechanisms for ν − → e − conversion, such as the exchange of virtual photons, a W boson, or a neutral Z boson [1][2][3], conversion by Higgsparticle exchange [15,16], by supersymmetric (SUSY) particles [21], or by R-parity violating mechanisms [22]. The operators related to the hadronic verticies of the relevant Feynman diagrams, needed for the calculation of the µ − → e − conversion mechanism mediated by interactions with nucleons, are of vector and axial-vector type. They can be written as where the summations run over all nucleons and the couplings (C vector (k), C axial−vector (k)) depend on the adopted mechanism for the µ − → e − conversion [3]. Their values are given in Table I. We use the same coupling constants as in the previous shell-model study [12]. Therefore, differences between the present and previous results are purely due to the differences in the nuclear-structure calculations, and not blurred by adjustments made to the couplings. The square of the matrix elements of the vector and axial-vector operators entering the vertices where the leptonic and nucleonic currents exchange bosons [23] are expressed in terms of the summation where the initial and final states belong to the target nucleus and m µ is the muon rest mass. The tensor operators which result from the plane-wave expansion of the factor e −iqr k are of the form [24] T λκγ,µ (qr In the particle representation they are written in the form where with i and f we denote the complete set of the corresponding quantum numbers needed to define a state in the single-particle basis, that is i(f ) = (n, l, j, m). In standard notation n is the number of nodes, l is the orbital angular momentum and (j, m) are the total angular momentum and its z projection, respectively, of the single particle orbit. In the previous equations the momentum transferred from the muon to the nucleons is defined by q f = m µ − E b − E excitation , that is by the difference between the muon mass (105.6 MeV), the binding energy of the muon in the 1S orbit (0.47 MeV) and the excitation energy of each of the final states of the nucleus which participate in the process, respectively. The ratio of the experimentally measurable coherent channel and the total conversion rate is where M coh is the matrix element for the coherent transition and M tot is the matrix element for the total conversion rate.

III. SHELL-MODEL BASIS AND INTERACTIONS
The wave functions and one-body transition densities needed for transitions between states in 27 Al were computed assuming a 4 He core and using the 0p, 0d, and 1s orbitals as the valence space for both protons and neutrons. The Hamiltonian adopted to perform the calculations, labelled psdmod, was taken from Ref. [25]. It is a modified version of the interaction presented by Utsuno and Chiba in [26], which itself is a modification of the interaction PSDWBT of Warburton and Brown [27]. In the parametrization given by the psdmod Hamiltonian, the p − sd shell gap has been increased by 1 MeV from the original interaction of Utsuno and Chiba. The calculations were performed using the shell-model code NuShellX@MSU [28] on a computer cluster using four 12-core Intel Xeon E5-2680 v3 @ 2.50GHz CPUs and took approximately 48 hours. For each spin and parity ten states were calculated.
Including excitations over the p − sd shell gap allows the description of the negative parity states, which can not be described using only the sd shell. Therefore, the potentially significant spin-dipole contributions are included in the present study.

IV. RESULTS
The calculated spectrum of 27 Al, with displayed states up to 6 MeV, is shown in Fig. 1, where the theoretical results are compared with the available experimental data. As seen from the figure, the energy gap between the 5/2 + ground state and the lowest excited states, which is of the order of 1 MeV is reasonably well reproduced, although the splitting between the first 3/2 + state and the first 1/2 + state is larger in the calculations than in the data. For the rest of the spectrum both the sequence of the spin and parity values and the corresponding energies are reasonably well reproduced by the calculations. As can be seen from Fig. 1, in addition to the good reproduction of the positive-parity states, the shell model is able to reproduce the energies of the lowest 1/2 − and 3/2 − states at 4055 keV and 5156 keV reasonably well, with calculated values of the energies at 3764 keV and 5137 keV, respectively. Not seen in the figure, the energy of the first 5/2 − state at 5438 keV is also well reproduced by the shell model, placing it at 5.433 keV. This is strong evidence that the size of the p − sd shell gap is correct in the adopted Hamiltonian.
The shell-model occupancies for the single-particle states in the active orbitals are given in Table II. The 0s shell is taken to be fully occupied. In the next table III we show the results of the present calculations for the nuclear form factors F Z (q 2 ) and F N (q 2 ), calculated using the shell-model occupancies of Table II. The form factors are given by where p and n denote proton and neutron states, while the coherent form factors S N = N F N and S Z = ZF Z , where N = 14 and Z = 13 are the neutron and proton numbers in the case of 27 Al. With respect to the previous shell-model results (see [12]), the present value of the coherent matrix element squared is smaller but still some 10 percent larger than the experimental value. In Table IV we are listing the calculated values for the matrix elements of Eq. (3), for each of the processes which may contribute to the muon-to-electron conversion. From the results given in the table it is seen that the calculated values are practically exhausted by the ground-state contributions to each process, and that the spin-independent vector channel is the dominant transition for all the processes considered. The analysis of the contributions to these matrix elements shows that they are given in almost equal parts by proton and neutron configurations, and that the main contribution which yields a value practically equal to the final one is given by the ground state of the target nucleus.

V. CONCLUSIONS
The measurement of muon-to-electron conversion process in nuclei is a convenient tool to establish limits on the lepton-flavor violation. The nucleus 27 Al is one of the nuclei of choice for performing the experiments, and due to it we have calculated the nuclear matrix elements for the operators entering the muon-to-electron conversion mediated by W-exchange, SUSY particles, Z exchange and photon exchange between nucleus and leptons with production of neutrinos. From our results, based on large-scale shell-model calculations of the wave functions of the participant nuclear states in Al, we confirm the dominance of spin-independent, coherent ground-state transitions, with practically no contributions from the excited positive-parity or negative-parity states.