Decay of a Bound Muon Into a Bound Electron

PHYSICAL REVIEW D 102, 073001 (2020)

Decay of a bound muon into a bound electron

M. Jamil Aslam ,1,2 Andrzej Czarnecki ,1 Guangpeng Zhang ,1 and Anna Morozova 1 1Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E1, Canada 2Department of Physics, Quaid-i-Azam University, Islamabad 44000, Pakistan

(Received 14 May 2020; accepted 23 September 2020; published 13 October 2020)

When a muon bound in an atom decays, there is a small probability that the daughter electron remains bound. That probability is evaluated. Surprisingly, a significant part of the rate is contributed by the negative energy component of the wave function, neglected in a previous study. A simple integral representation of the rate is presented. In the limit of close muon and electron masses, an analytic formula is derived.

DOI: 10.1103/PhysRevD.102.073001

I. INTRODUCTION nucleus (we neglect screening or Pauli blocking due to − other electrons). Electrostatic binding of a muon μ in an atom changes its − − The decay ðZμ Þ → ðZe Þνμν¯ was previously studied decay characteristics. Coulomb attraction decreases the e phase space available to the decay products but enhances in the very elegant and detailed paper [4]. We reevaluate it and find discrepancies with that pioneering study, particu- the daughter electron wave function. Muon motion smears larly for large values of Z. This is most likely explained by the energy spectrum of electrons. All these effects largely negative energy components of the Dirac wave functions, cancel in the lifetime of the muon [1] but they do slow neglected in [4] (as discussed in its Appendix A). Here down the decay by a factor that, for small atomic numbers we use exact Dirac wave functions in the Coulomb field Z, reads of a pointlike nucleus. Effects of extended nuclear charge distribution were found to be very small in [4] so we 2 − ðZαÞ neglect them. ΓððZμ Þ → eνμν¯ NÞ¼ 1 − Γðμ → eνμν¯ Þ; ð1Þ e 2 e Earlier studies of the differences between the decay of a free and of a bound muon include [5–8]. More recently, the spectrum of produced electrons was determined in [9–11]. and can be interpreted as the time dilation; the characteristic This paper is organized as follows. In Sec. II, velocity of the bound muon is Zα. momentum space wave functions are used to compute Another possible effect, of primary interest in this paper, − − the rate Γ½ðZμ Þ → ðZe Þνμν¯ , as in Ref. [4]. Significant is the decay into an electron that remains bound to the e differences are found so the result is checked with nucleus N. For the actual small ratio of electron to muon position space wave functions in Sec. III. That approach masses, m =mμ ≃ 1=207, that process is very rare, espe- e turns out to be much simpler; a one-dimensional integral cially for weak binding in atoms with moderate Z. representation is found, replacing the triple integral of We study it as a part of a program of characterizing Ref. [4]. In the limit of nearly equal masses, m → mμ,the bound muon decays, motivated by upcoming experiments e remaining integration is done analytically and a closed COMET [2] and Mu2e [3]. formula for the rate is obtained in Sec. IV.SectionV Throughout this paper we use c ¼ ℏ ¼ 1 and treat the presents conclusions and the Appendix summarizes the nucleus N as static, spin 0, and pointlike, neglecting effects formalism of Ref. [4]. of its recoil and finite size. We denote its number of protons In order to deal with binding effects, we describe the by Z. The notation ðZμ−Þ or ðZe−Þ denotes a muon or an initial and the final states by solutions of a stationary Dirac electron bound in the 1s state, forming a hydrogen-like equation. We treat the weak interaction that leads to the atom. We assume that no other particles are bound to the decay as a harmonic perturbation. This description is exact to all orders in Zα, including relativistic and thus positron and antimuon effects. We expect corrections to the result to Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. be suppressed by powers of the fine structure constant n Further distribution of this work must maintain attribution to OðαðZαÞ Þ, n>0, due to real and virtual radiative effects. the author(s) and the published article’s title, journal citation, Additional corrections due to the finite mass of the nucleus and DOI. Funded by SCOAP3. and its structure are also expected.

2470-0010=2020=102(7)=073001(7) 073001-1 Published by the American Physical Society ASLAM, CZARNECKI, ZHANG, and MOROZOVA PHYS. REV. D 102, 073001 (2020) This theoretical framework has been used for other 1 0 where ϕ ¼ and ϕ− ¼ are two-component atomic processes such as electromagnetic decays of excited þ 0 1 atomic states and interactions of an atom with an external spinors describing spin projections 1=2 on the z axis. ϕ field [12–18]. We assume that the muon decays in the state þ.Wewill Our work is somewhat analogous to the first study use the simplified notation f; g ¼ f˜ðkÞ; g˜ðkÞ and the of the bound electron magnetic moment, by Gregory dimensionless variable q ¼ k , mαZ Breit [12],p whoffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi determined the gyromagnetic ratio sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 g ¼ 2½1 þ 2 1 − ðZαÞ =3. In his study, a bound electron 2γþ1Γ γ 1 π 1 γ ð þ Þ ð þ Þ 1 − −γ−1 interacts with an external field, just like in the decay f ¼ 3 2 Imð iqÞ ; ð4Þ qðmα Þ = Γð1 þ 2γÞ Z → 0 g Z process considered here. In the limit , the -factor sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tends to the free-electron value, g → 2. Breit’s result is 2γþ1ð1 − γÞΓðγÞ πð1 þ γÞ exact to all orders in Zα, just like we claim our result is. g ¼ α 2 α 3=2 Γ 1 2γ Breit’s 1928 calculation was analyzed from the Zq ðm ZÞ ð þ Þ point of view of a nonrelativistic effective theory [19]. ×Imf½1 − iqðγ þ 1Þð1 − iqÞ−γ−1g; ð5Þ That analysis reveals negative-energy contributions pffiffiffiffiffiffiffiffiffiffiffiffiffi [see Fig. 2(b) and the discussion following Eq. (12) in α α γ 1 − α2 α ≃ 1 137 where Z ¼ Z , ¼ Z,and = is the fine Ref. [19]]. We conclude that the negative-energy con- structure constant. Equations (4) and (5) are numerically tributions are correctly included in a treatment based on identical with Eqs. (A3) and (A4) in [4]; the functional the Dirac wave function. ’ form we present seems to lead to slightly faster numerical Our present study is conceptually parallel to Breit s. The integrations. We employ the basis [[22],Eq.(3.7)], only differences are that the initial and final states have 0 1 0 1 different masses; and, instead of an external magnetic field, 1 0 the electroweak field induces the transition. In both cases B C B C B C B 1 C B 0 C B C the full result is given by a double series, in powers of 1 k B C 2 k B C w ð Þ¼c kz ;wð Þ¼c k− ; ð6Þ Zα (describing the binding) and of α (describing self- B 0 C B 0 C @ k þm A @ k þm A interactions and, in case of the muon decay, real photon kþ kz 0 − radiation). The structure of this double series is shown, for k þm k0þm example, in Eq. (5) in Ref. [20]. We expect a similar series 0 1 0 1 kz k− to exist for the present problem of the bound muon decay. B k0þm C B k0þm C B C B C The soundness of the underlying theory has been tested by kþ kz 3 B 0 C 4 B − 0 C experiments providing the best determination of the elec- w ðkÞ¼cB k þm C;wðkÞ¼cB k þm C; ð7Þ B 1 C B 0 C tron mass [21]. @ A @ A 0 1 qffiffiffiffiffiffiffiffiffi II. MOMENTUM SPACE DERIVATION pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k0þm 0 2 k2 OF THE DECAY RATE with c ¼ 2m , k ¼ m þ ,andk ¼ kx iky.In A. Wave function and its normalization analogy with Eq. (A6) in [4], we expand the bound wave in momentum space function in this basis, rffiffiffiffiffiffiffi We consider the muon in the ground state of a 2m 1 2 Φ k A w k A−w k hydrogen-like ion and are interested in the final-state þð Þ¼ 2 0½ þ ð Þþ ð Þ electron also in the ground state. Both muon and electron k ⋆ 4 −k ⋆ 3 −k wave functions are 1s solutions of the Dirac equation and þ Bþw ð ÞþB−w ð Þ ð8Þ differ only by the mass, respectively mμ and m .Below 2 0 1 0 1 e 1 0 we present formulas for a generic mass m. The position sffiffiffiffiffiffiffiffiffiffiffiffiffiffi6 B C B C 6 B 0 C B 1 C space wave function ΦðxÞ will be presented below in k0 m6 B C B C þ 6 B C B C Eq. (18). Taking its Fourier transform (see Appendix 1 in ¼ 0 Aþ kz þ A− k− 2k 6 B 0 C B 0 C Ref. [4]) we obtain 4 @ k þm A @ k þm A kþ kz 0 − 0 Z 0 1k þm 0 1k3þm ˜ 3 −ik·x k− kz Φ ðkÞ¼ d xΦðxÞe ð2Þ − 0 − 0 B k þm C B k þm C7 B C B C7 kz kþ ⋆ B 0 C ⋆ B − 0 C7 k þm k m þ BþB C þ B−B þ C7: ð9Þ ˜ ϕ B 0 C B 1 C7 fðkÞ @ A @ A5 ¼ k ¼jkj; ð3Þ ˜ σ·k ϕ 1 0 gðkÞ k

073001-2 DECAY OF A BOUND MUON INTO A BOUND ELECTRON PHYS. REV. D 102, 073001 (2020)

For example, for the spin projection þ1=2, sffiffiffiffiffiffiffiffiffiffiffiffiffiffi k0 þ m kg A ¼ f þ ; ð10Þ þ 2k0 k0 þ m

A− ¼ 0; ð11Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffi k0 þ m f g B⋆ ¼ k − þ ; ð12Þ þ 2k0 þ k0 þ m k sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ⋆ k þ m f g B− ¼ k − þ ; ð13Þ 2k0 z k0 þ m k FIG. 1. Rate of the bound-to-bound decay ðZμ−Þ → ðZe−Þþ ⋆ in agreement with (A7) in [4] except for B−,forwhichwe νν¯ normalized to the free muon decay rate Γ0, as a function of the find the opposite overall sign. We proceed to check the atomic number Z. The solid line shows our results found using normalization, complete wave functions. Red dots are the values obtained by neglecting negative energy components of wave functions. Errors Z 3 due to this approximation grow with the atomic number Z. d k 2 2 2 2 ðjA j þjA−j þjB j þjB−j Þ¼1: ð14Þ ð2πÞ3 þ þ proceed to check them in the position space. As a reward, We confirm that the B part of this integral is very small, we find that alternative method to be simpler. It will allow 0.16% even for Z ¼ 80, in agreement with a comment us to derive a closed formula for the rate in the limit of close below (A8) in [4]. Indeed, the B part of the normali- electron and muon masses. zation integral, interpreted as the probability of finding a O α5 → 0 μ − e − positron in the atom, is ð ZÞ when Z . For this III. BOUND TO BOUND DECAY IN reason, the negative energy components of the wave POSITION SPACE function were neglected in [4]. The positive and negative In this section, we calculate the bound state transition energy components can be separated by acting on the − − rate ðZμ Þ → ðZe Þνμν¯ using position space wave func- wave function with Casimir projectors, e tions for the decaying muon and the produced electron [24], m † † =k þ m γ0 PA ¼ 0 ðw1w1 þ w2w2Þ¼ 0 ; ð15Þ fðrÞ k 2k ΦðrÞ¼ψ 1 1 ðr; θ; ϕÞ¼pffiffiffiffiffiffi u ; ð18Þ n¼ ;j¼2; 4π =k − m P ¼ γ0 ; ð16Þ B 2k0 where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi such that P2 P and P P 1.Wefindthatthe 1 γ A;B ¼ A;B A þ B ¼ 3=2 þ γ−1 fðrÞ¼ð2mαZÞ ð2mrαZÞ exp ð−mrαZÞ; rate calculated with PA-projected wave functions is 2Γð1 þ 2γÞ substantially larger than when full wave functions are used. These results are compared in Fig. 1, where we plot ð19Þ the rate divided by and the mass m is either mμ for the muon or me for the 2 5 electron. In the Dirac representation GFmμ Γ0 ; ¼ 192π3 ð17Þ ϕ u ¼ =ρ ; ð20Þ the free muon decay rate at tree level, in the limit of a 0 pffiffi 2g2 massless electron; GF ¼ 8M2 is the Fermi constant [23]. 1−γ W with ρμ ¼ðρ0; ρÞ¼ð1;i rˆÞ. Since this approach differs The solid line in Fig. 1 shows the full wave function Zα from Ref. [4], we present it in some detail. result, and the dots show the result with projectors PA. For small and moderate Z,uptoZ ≃ 40, the results are close, and start to diverge quite strongly for larger nuclei. A. Factorizing neutrinos Since the B contribution to the normalization is small It is convenient to decompose the decay into two stages: even for large Z, these results are unexpected. We thus first the muon decays into the electron and a fictitious

073001-3 ASLAM, CZARNECKI, ZHANG, and MOROZOVA PHYS. REV. D 102, 073001 (2020) spin-one boson A; next, boson A decays into the νν¯ pair. Γ½ðZμ−Þ → ðZe−ÞþA The kinematically allowed range of the mass mA of the 2 mμg 2 2 2 2 boson A, parametrized by a dimensionless variable z, ¼ kAðNa þ N þ Fa þ F Þ; ∈ 0 − 2π b b mA ¼ zmμ,isz ½ ;zmax ¼ðEμ EeÞ=mμ, where Eμ;e ffiffiffi p zmax 2 2 are the muon and electron energies; they should be replaced Na ¼ 2 ½4a ðC2 − S3Þþð1 þ a ÞS1; by muon and electron masses in the case of a free muon z pffiffiffi z kA 2 decay. The decay rate is an integral over , N ¼ 2 ð1 þ a ÞS1; b z 2 2 Z Fa ¼ 4a ðC2 − S3Þ − 2ð1 − a ÞS1; 256πΓ z Γ μ → νν¯ 0 max Γ μ− → − 3 4 − ð e Þ¼ 2 ½ e Az dz; ð21Þ Fb ¼ aðS2 C1Þ; ð25Þ g mμ 0

where the quantities Cn (and analogously Sn with cos → sin) are where g is the weak coupling constant. One advantage of Eq. (21) is that it holds both for a free and for a bound γ 1 1 þ γ 4δ þ2 Γð1 þ 2γ − nÞ muon decay. It is simpler to deal with a two-body decay C n ¼ 8 1 δ 2 nΓ 1 2γ μ → eA than with μ → eνν¯. Binding effects as well as ð þ Þ k ð þ Þ 2 n−1−γ radiative corrections (ignored in the present paper) affect × ð1 þ k Þ 2 cos ½ð1 þ 2γ − nÞ arctan k; ð26Þ only Γðμ → eAÞ. As an example of using (21) consider a free muon decay. 1 − δ δ kA Then zmax ¼ , ¼ me=mμ, and k ¼ : ð27Þ αZð1 þ δÞ

− − 2 The rate Γ½ðZμ Þ → ðZe Þþνν¯ can now be expressed as Γ μ → g 1 2 − 2 4 δ2 2 − 2δ2 δ4 qðzÞ ð eAÞ¼32π ð þ z z þ z þ Þ 2 ; a single integral over z, a variable equivalent to the invariant z γ 1 − δ mass of the neutrinos, from zero to zmax ¼ ð Þ, ð22Þ 1 Γ½ðZμ−Þ → ðZe−Þþνν¯ Γ0 where qðzÞ is the momentum of A; for a free muon decay, Z zmax λ1=2ð1;δ2;z2Þ 128 2 2 2 2 3 qðzÞ¼ 2 mμ with the Käll´en function λðx; y; zÞ¼ ¼ ðNa þ Nb þ Fa þ FbÞkAz dz: ð28Þ 0 x2 þ y2 þ z2 − 2ðxy þ yz þ zxÞ. Integration over z gives This is the main result of this paper. Note that the position space calculation results in a single integral representation Γ μ → νν¯ Γ 1 − 8δ2 − 24δ4 δ 8δ6 − δ8 ð e Þ¼ 0ð ln þ Þ; ð23Þ for the rate. This is much simpler than the result of Ref. [4], where two additional integrations remain, over the magni- tude and the polar angle of the argument of the momentum reproducing the correct electron mass dependence [25]. space wave function. Those integrations seem to be more involved than the corresponding radial and angular inte- B. Decay rate grations in the position space. In the following section we perform the remaining integration over z in the limit of The amplitude for the Zμ → Ze A transition is ð Þ ð Þþ close electron and muon masses.

Z g 3 ¯ λ ⋆ IV. LIMITING CASE OF SIMILAR ELECTRON M ¼ pffiffiffi d r exp ðiq · rÞΦμðrÞ=ϵ A LΦ ðrÞ; ð24Þ 2 e AND MUON MASSES We expect out results to agree with Ref. [4] for small Z, since the only conceptual difference between our approaches 1−γ5 where L ¼ 2 and λA labels the polarization state of A. involves negative energy components, and those are sup- r The triple integration is done analytically. Angular pressedbypowersofαZ. Here we demonstrate this agree- integrations lead to spherical Bessel functions, and the ment with a simple closed formula in the limiting case of r-integration results in a relatively compact formula.pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi After nearly equal masses. We write 2 − 2 squaring the amplitude, we find, using kA ¼ zmax z 1−γ − ϵ and a ¼ , mμ me ¼ mμ; ð29Þ αZ

073001-4 DECAY OF A BOUND MUON INTO A BOUND ELECTRON PHYS. REV. D 102, 073001 (2020) and consider a hypothetical situation where ϵ is small. In the The analogous limit of the free muon decay rate, Eq. (23), limit ϵ → 0, the decay rate computed in Ref. [4] [cf. Eq. (A7) 64 5 5 1þγþγ2 is 5 ϵ . Thus the binding effects given by γ 3 ¼ in the Appendix] is 1–3α2 O α4 Z þ ð ZÞ in the case of the decay into a bound electron are more pronounced than in the case of the decay Γ μ− → − ν ν¯ ½ðZ Þ ðZe Þ μ e into a free electron, given in Eq. (1). For a free electron, Z 2 2 m1−m2 G q the effect can be approximated by a single factor of q F q 2 4 ¼ dj j 3 Kðj jÞ γ ≃ 1 − α O α 0 12π Z þ ð ZÞ. 2 5 Numerical evaluation of Eq. (33) is given in the last G mμ −1 γ5ϵ5 F 0 − 0 2 0 0 column of Table I. For small Z,wehave L ≈ γ and ¼ 3 f½F1ð Þ F2ð Þ þ F1ð ÞF2ð Þg: ð30Þ h i 15π hence the corresponding results coincide in this case of equal muon and electron masses. This is not the case for In this approximation of almost equal muon and electron −1 large Z, where hL i is larger than γ. masses, the momentum transferred to the neutrinos q is approximately zero. Therefore, the form factors F1 and F2 V. CONCLUSION given in Eq. (30) are evaluated at jqj¼0 which gives We have calculated the decay rate of a bound muon to Z 3 0 d k 2k þ mμ 2 1 bound electron using Dirac wave functions for different 0 ψ k ψ k −1 F1ð Þ¼ 3 μð Þ eð Þ 0 ¼ þ hL i; Z ð2πÞ 3k 3 3 values of in two formalisms. Numerical results in momentum and in position space coincide, provided that Z 3 0 − μ 1 1 d k k m −1 complete wave functions (both positive and negative F2ð0Þ¼ ψ μðkÞψ ðkÞ ¼ − hL i; ð2πÞ3 e 3k0 3 3 energy components) are used. If the negative energy parts ð31Þ of the wave functions are neglected, as was done in Ref. [4], the results are significantly larger. For Z ¼ 80 the difference is about 38%. where the mean inverse Lorentz factor L−1 is h i This is surprising since the probability of finding Z 3 positrons in a hydrogen-like atom or ion is very small mμ −1 d k ψ k 2 even for Z ¼ 80. Our tentative interpretation is that the hL i¼ 3 j μð Þj 0 : ð2πÞ k1 probability of the decay into a bound electron is very suppressed and that this suppression is relatively less severe Hence, Eq. (30) becomes for the negative energy components. We note that the decay vertex couples positive and negative energy components Γ 64 1 þhL−1iþhL−1i2 without a suppression factor of α . Also, the decay rate ¼ ϵ5γ5 : ð32Þ Z Γ0 5 3 involves an interference of large A with small B wave function terms, whereas the normalization integral involves Γ μ− → − ν ν¯ ½ðZ Þ ðZe Þ μ e B Numerical results for the limiting case of the small only in second powers, thus greatly decreas- Γ0 ing their contribution [see Eq. (14)]. almost equal masses are shown in Table I. In order to check this unexpected result, we developed a q → 0 In the equal mass limit, using j jr , our momentum position space approach. It resulted in a simple one- space as well as position space treatments lead to the dimensional integral representation of the rate, Eq. (28). expression The remaining integral has been done in the limiting case of

− − 2 close electron and muon masses, Eq. (33), giving a closed Γ½ðZμ Þ → ðZe Þνμν¯ 64 1 þ γ þ γ α e ¼ ϵ5γ5 : ð33Þ formula valid for all Z. Γ0 5 3 In closing, we quote from Sidney Coleman’s field theory lectures [26]: Dirac’s theory gives excellent results to order 2 Γ μ− → − ν ν¯ Γ ðv=cÞ for the hydrogen atom, even without considering TABLE I. Numerical values of ½ðZ Þ ðZe Þ μ e= 0 in pair production and multi-particle intermediate states. This an artificial situation with the electron only slightly lighter that is a fluke. the muon, ϵ ¼ 1 − me=mμ ¼ 0.01, for a small Z ¼ 10 and a large Z ¼ 80: using the formalism of Ref. [4], Eq. (A6) (second ACKNOWLEDGMENTS column), its limit for me ≃ mμ, Eq. (32) (third column), and the ≃ me mμ limit of our approach, Eq. (33) (fourth column). As A. C. thanks Andrey Volotka for conversations that expected, the agreement is better for small Z (first line). stimulated this project. M. J. A. and A. C. gratefully Z Eq. (A6) Eq. (32) Eq. (33) acknowledge the hospitality of the Banff International Research Station (BIRS) where parts of this work were 10 1.25 × 10−9 1.26 × 10−9 1.26 × 10−9 80 3.85 × 10−10 3.83 × 10−10 3.72 × 10−10 done. This research was supported by the Natural Sciences and Engineering Research Council of Canada.

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APPENDIX: BOUND MUON DECAY IN THE and the charged current part is FORMALISM OF REF. [4] Z 3 ¯ ;k −q μ;k μ d k1 usðe 1 Þ μ urð 1Þ In this Appendix, we summarize the formalism of S ψ μ k1 ψ k1 −q pffiffiffiffiffiffiffi γ L pffiffiffiffiffiffiffi : sr ¼ 2π 3 ð Þ eð Þ 0 0 Ref. [4] for the transition B1 → B2 þ X, where B1 and ð Þ 2k2 2k1 B2 are bound states. − − ðA4Þ The invariant amplitude of ðZμ Þ → ðZe Þνμν¯e decay is

4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Here, k1, k2, pν and pν are the 4-momenta of the muon, M GffiffiffiF 4 μ e μ B1→B2 ¼ p mB1 mB2 NμSsr; ðA1Þ ν ν 2 electron, e and μ, respectively, and the subscripts r and s are spin indices ðr; s ¼1=2Þ. − − − ¯ where the subscripts B1 and B2 represent the ðZμ Þ and The corresponding decay rate for ðZμ Þ → ðZe Þνμνe ðZe−Þ states, respectively. The masses of the bound states can be calculated as are, with M denoting the nucleus mass (note that in our approach the nucleus is treated as infinitely heavy and M 1 2 dΓ ¼ dΦjM → j : ðA5Þ does not appear), 2 B1 B2 mB1

m ¼ M þ m1;m1 ¼ mμ − E μ; B1 b; After integration over the phase space and neglecting terms − 1 mB2 ¼ M þ m2;m2 ¼ me Eb;e; ðA2Þ suppressed by =M, the decay rate is [4]

2 Z where E μ are the binding energies. In Eq. (A1), the m1−m2 b; ðeÞ Γ GF q q2 q ¼ 3 dj j Kðj jÞ; ðA6Þ neutrino part is given by 12π 0

Nμ ¼ u¯ðpν ÞγμLυðpν Þ; ðA3Þ μ e where

2 q 2 2 − 2 2 2 q 4 − 2 − 2 2 2 Kðj jÞ ¼ ½q þ ðm1 m2Þ ðF1 þ F2Þþ 2 ½ ðm1 m2Þ q ðF3 þ F4Þ mμ 2 − − 6 2 q m1 m2 − − q F1F2 þ 2 F3F4 þ ðF1 F2ÞðF3 F4Þ ; ðA7Þ mμ mμ

2 2 2 2 2 and q ¼ q0 − q ¼ðm1 − m2Þ − q . The form factors Fi are defined as

Z 3 d k1 h 2 ψ k ψ k − q qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii Fiðq Þ¼ 3 μð 1Þ eð 1 Þ ; ðA8Þ ð2πÞ 0 0 0 0 4k1k2ðk1 þ mμÞðk2 þ meÞ with

0 0 0 0 0 2 h1 ¼ðk1 þ mμÞðk2 þ meÞþq ½ð1 − CÞðk1 þ mμÞ − Cðk2 þ meÞ þ ðB − CÞq0 þ A; 2 h2 ¼ðC − BÞq þ 2A; 0 h3 ¼½ð1 − CÞðk1 þ mμÞþðB − CÞq0mμ; 0 h4 ¼½Cðk2 þ meÞ − ðB − CÞq0mμ ðA9Þ and the expressions of A, B and C are given in Eq. (35) of [4]. We mention that in the expressions for h1 and h2 in Eq. (A9), the sign of A is different than in [4] [cf. Eq. (34) there].

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