Lepton flavour violating μ-e conversion rate for various nuclei

Ryuichiro Kitano†‡§, Masafumi Koike†and Yasuhiro Okada†‡

† Theory Group, KEK, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan

‡ Department of Particle and Nuclear Physics, The Graduate University for Advanced Studies, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan

Abstract. The coherent μ-e conversion rate for general LFV interactions for various nuclei is calculated. The branching ratio increases for the light nuclei such as Z ≲ 30, are largest for Z = 30 – 60, and decreases for heavy nuclei with Z ≳ 60. Although this tendency of Z dependence is the same for different types of coupling constants, there are significant differences in Z dependence of branching ratios. The atomic number dependence of the conversion ratio calculated here is useful to distinguish theoretical models with lepton flavor violation.

1. Introduction

The search for the lepton flavour violation (LFV) is one of the most interesting subjects in particle physics. The μ-e conversion in muonic atoms, or neutrinoless muon capture, is an expected signal of the LFV to be observed. The μ-e conversion rate has been calculated by several authors [1, 2], and its experimental upper bound is given to be 6.1 × 10^{−13} [3]. Further experiments are ongoing and planned to improve the bound down to 10^{−17} − 10^{−18} [4, 5]. The PRISM project [5] in the JAERI-KEK joint project is of particular interest among them since it is planned to make use of a muon storage ring, the muon beam from which is free from contamination of electrons and pions. It will then be possible to carry out the experiments with various target atom, and thus to optimize the target atom for the search. We thereby evaluate the μ-e conversion rate for nuclei of a wide range of atomic numbers. The atomic number dependence of the conversion ratio calculated here turns out to be useful to distinguish theoretical models with LFV. More details are found in the reference [6].

2. Formula of μ-e conversion rate

It is necessary to specify the interaction Lagrangian and the initial and final states in order to calculate the μ-e conversion amplitudes and rates. The LFV interaction Lagrangian is taken to be most general form which contributes to the μ-e conversion in nuclei [7]. It consists of photonic interaction terms and four-Fermi interaction terms, and the magnitude of coupling constant in each term depends on the interaction of the new physics in which the lepton flavour conservation is violated. The initial state in the μ-e conversion process is the 1s state of the muonic atom, and the final electron

§ Present address: Institute for Advanced Study, Princeton, NJ 08540, USA

state is the eigenstate with an energy of m_{μ} − ε_{b}, where ε_{b} is the binding energy of the 1s state of muonic atom. The final state of the nucleus is taken to be same as the initial state, which is referred to as the coherent process; the fraction of this process is generally larger than incoherent one approximately by a factor of the mass number of the target nuclei.

Then the formula of the μ-e conversion rate can be written in terms of the five overlap integrals D, S^{(p,n)} and V^{(p,n)} as

ω_{conv} = 2G_{F}^{2} |A_{R}^{∗}D + g̃_{LS}^{(p)}S^{(p)} + g̃_{LS}^{(n)}S^{(n)} + g̃_{LV}^{(p)}V^{(p)} + g̃_{LV}^{(n)}V^{(n)}|^{2}
+ 2G_{F}^{2} |A_{L}^{∗}D + g̃_{RS}^{(p)}S^{(p)} + g̃_{RS}^{(n)}S^{(n)} + g̃_{RV}^{(p)}V^{(p)} + g̃_{RV}^{(n)}V^{(n)}|^{2} . (1)

Here G_{F} is the Fermi constant, and A_{R,L}^{∗} and g̃_{LS,LV,RS,RV}^{(p,n)} are the coupling constants relevant to the LFV interaction. The overlap integrals are defined by

D = \frac{4}{\sqrt{2}}m_{μ} ∫_{0}^{∞} drr^{2}[−E(r)](g_{e}^{−}f_{μ}^{−} + f_{e}^{−}g_{μ}^{−}),  (2)
S^{(p)}, V^{(p)} = \frac{1}{2\sqrt{2}} ∫_{0}^{∞} drr^{2}Zρ^{(p)}(g_{e}^{−}g_{μ}^{−} ∓ f_{e}^{−}f_{μ}^{−}),  (3)
S^{(n)}, V^{(n)} = \frac{1}{2\sqrt{2}} ∫_{0}^{∞} drr^{2}(A − Z)ρ^{(n)}(g_{e}^{−}g_{μ}^{−} ∓ f_{e}^{−}f_{μ}^{−}),  (4)

where m_{μ} is the muon mass, E(r) is the electric field in nuclei, ρ^{(p,n)}(r) is the normalized density distribution of the proton and neutron in the nuclei, and g_{μ,e}^{−}(r) and f_{μ,e}^{−}(r) are the radial part of the wave function of the muon and the electron. The minus and plus sign in (3) and (4) corresponds to S^{(p,n)} and V^{(p,n)}, respectively.

3. Numerical results

Figure 1(a) shows the five overlap integrals as functions of the atomic number Z. Here the proton density is taken from the reference [8], and the neutron density is assumed to be equal to the proton density. This is not necessarily a good approximation to calculate S^{(n)} and V^{(n)} for heavy nuclei, but the result turns out to be consistent within 10% to the result obtained from the experimental neutron distribution data [6]. We see that the overlap integrals increase as functions of Z for light nuclei up to Z ∼ 30, and saturate or decrease for heavy nuclei. Although the tendency is the same, each overlap integral has different Z dependence, especially for heavy nuclei.

The conversion rate ω_{conv} and the branching ratio B_{μN→eN} ≡ ω_{conv}/ω_{capt}, where ω_{capt} is the rate of the ordinary muon capture, can be straightforwardly calculated through equation (1) once a theoretical model with LFV is specified. We consider the following three cases: (i) Only the photonic dipole LFV operator is relevant; (ii) Only the scalar four-Fermi LFV operator is relevant; (iii) Only the vector four-Fermi LFV coupling constants g̃_{LV}^{(p)} and/or g̃_{RV}^{(p)} are relevant.Figure 1(b) presents the μ-e conversion branching ratio normalized by the value for aluminum (Z = 13). We can see that, for all three types, the branching ratio increases with Z for Z ≲ 30, is largest for 30 ≲ Z ≲ 60, and decreases for Z ≳ 60. It is also seen that the conversion ratios are very different in heavy nuclei depending on the type of interaction. From this property we may be able to distinguish models beyond the Standard Model through several experiments with different targets.

Figure 1. (a): The overlap integrals are plotted as functions of the atomic number Z. Neutron density distributions are assumed to be equal to charge density distributions. (b): The μ-e conversion ratios for typical theoretical models are plotted as functions of the atomic number Z. The solid, the long dashed, and the dashed lines represent the cases where the photonic dipole, scalar, and vector operator dominates, respectively. The proton and neutron distribution are taken to be the same, and the conversion ratios are normalized by the conversion ratio in aluminum nuclei (Z = 13).

Acknowledgments

The authors are grateful to J. Hisano for useful discussions. This work is supported by the JSPS Research Fellowships for Young Scientists (R. K. and M. K.). The work of Y. O. is supported in part by a Grant-in-Aid of the Ministry of Education, Culture, Sports, Science and Technology, Government of Japan (No. 13640309), priority area “Supersymmetry and Unified Theory of Elementary Particles” (No. 707).

References

[1] Weinberg S and Feinberg G 1959 Phys. Rev. Lett 3 111; Marciano W J and Sanda A I 1977 Phys. Rev. Lett. 38 1512; Shanker O 1979 Phys. Rev. D 20 1608; Czarnecki A et al. Preprint hepph/9801218; Czarnecki A et al. 1998 Proc. of Workshop on Physics at the 1st Muon Collider and at the Front End of the Muon Collider, AIP Conf. Proc. No. 435 edited by S H Geer and R Raja (New York: American Institute of Physics) p 409

[2] Chiang H C et al. 1993 Nucl. Phys. A 559 526; Kosmas T S et al. 1994 Nucl. Phys. A 570 637; Kosmas T S et al. 1996 Phys. Rept. 264 251; Kosmas T S et al. 1998 Phys. Atom. Nucl. 61 1161; Kosmas T S et al. 2000 Nucl. Phys. A 665 183; Siiskonen T et al. 2000 Phys. Rev. C 62 035502; Kosmas T S 2001 Preprint nucl-th/0108045

[3] Wintz P 1998 Proc. 1st Int. Symp. on Lepton and Baryon Number Violation edited by H V Klapdor-Kleingrothaus and I V Krivosheina (Bristol/Philadelphia: Institute of Physics) p 534

[4] Bachman M et al. (MECO Collaboration) 1997 Experimental proposal E940 to Brookhaven National Laboratory Alternating Gradient Synchrotron

[5] Kuno Y 2000 Proc. of The 2nd Int. Workshop on Neutrino Oscillations and their Origin edited by Y Suzuki et al. (Singapore: World Scientific) p 253

[6] Kitano R et al. 2002 Phys. Rev. D 66 096022

[7] For review, see Kuno Y and Okada Y 2001 Rev. Mod. Phys. 73 151

[8] De Jager C W et al. 1987 Atom. Data Nucl. Data Tabl. 36 495; Fricke G et al. 1995 Atom. Data Nucl. Data Tabl. 60 177