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 Æ1=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 ,andkÆ ¼ kx Æ 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 Æ 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 Æ @ A @ A5 ¼ k ¼jkj; ð3Þ ˜ σ·k ϕ 1 0 gðkÞ k Æ 073001-2 DECAY OF A BOUND MUON INTO A BOUND ELECTRON PHYS.