Decay of a bound muon into a bound electron

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.


Introduction
Electrostatic binding of a muon µ − in an atom changes its decay characteristics. Coulomb attraction decreases the phase space available to the decay products but enhances the daughter electron wave function. Muon motion smears the energy spectrum of electrons. All these effects largely cancel in the lifetime of the muon [1] but they do slow down the decay by a factor that, for small atomic numbers Z, reads and can be interpreted as the time dilation; the characteristic velocity of the bound muon is Zα. Another possible effect, of primary interest in this paper, is the decay into an electron that remains bound to the nucleus N . For the actual small ratio of electron to muon masses, me /mµ 1 /207, that process is very rare, especially for weak binding in atoms with moderate Z. We study it as a part of a program of characterizing bound muon decays, motivated by upcoming experiments COMET [2] and Mu2e [3]. Throughout this paper we use c = = 1 and treat the nucleus N as static, spin 0, and point-like, neglecting effects of its recoil and finite size. We denote its number of protons by Z. The notation (Zµ − ) or (Ze − ) denotes a muon or an electron bound in the 1s state, forming a hydrogen-like atom. We assume that no other particles are bound to the nucleus (we neglect screening or Pauli blocking due to other electrons). The decay (Zµ − ) → (Ze − ) ν µ ν e was previously studied in the very elegant and detailed paper [4]. We reevaluate it and find discrepancies with that pioneering study, particularly for large values of Z. This is most likely explained by negative energy components of the Dirac wave functions, neglected in [4] (as discussed in its Appendix A). Here we use exact Dirac wave functions in the Coulomb field of a point-like nucleus. Effects of extended nuclear charge distribution were found to be very small in [4] so we neglect them. Earlier studies of the differences between the decay of a free and of a bound muon include [5,6,7,8].
More recently, the spectrum of produced electrons was determined in [9,10,11]. This paper is organized as follows. In Section 2, momentum space wave functions are used to compute the rate Γ [(Zµ − ) → (Ze − ) ν µ ν e ], as in Ref. [4]. Significant differences are found so the result is checked with position space wave functions in Section 3. That approach turns out to be much simpler; a one-dimensional integral representation is found, replacing the triple integral of Ref. [4]. In the limit of nearly equal masses, m e → m µ , the remaining integration is done analytically and a closed formula for the rate is obtained in Section 4. Section 5 presents conclusions and Appendix A summarizes the formalism of Ref. [4].
2 Momentum space derivation of the decay rate

Wave function and its normalization in momentum space
We consider the muon in the ground state of a hydrogen-like ion and are interested in the final-state electron also in the ground state. Both muon and electron wave functions are 1s solutions of the Dirac equation and differ only by the mass, respectively m µ and m e . Below we present formulas for a generic mass m. The position space wave function Φ (x) will be presented below in eq. (18). Taking its Fourier transform we obtain where φ + = 1 0 and φ − = 0 1 are two-component spinors describing spin projections ± 1 /2 on the z axis. We assume that the muon decays in the state φ + . We will use the simplified notation f, g = f (k) , g (k) and the dimensionless variable q = k mα Z , where α Z = Zα, γ = 1 − α 2 Z , and α 1/137 is the fine structure constant. We employ the basis ( [12], eq. (3.7)), with c = k 0 +m 2m , k 0 = √ m 2 + k 2 , and k ± = k x ± ik y . In analogy with equation (A6) in [4], we expand the bound wave function in this basis For example, for the spin projection + 1 /2, in agreement with (A7) in [4] except for B − , for which we find the opposite overall sign. We proceed to check the normalization, We confirm that the B ± part of this integral is very small, 0.16% even for Z = 80, in agreement with a comment below (A8) in [4]. Indeed, the B ± part of the normalization integral, interpreted as the probability of finding a positron in the atom, is O (α 5 Z ) when Z → 0. For this reason, the negative energy components of the wave function were neglected in [4]. The positive and negative energy components can be separated by acting on the wave function with Casimir projectors, such that P 2 A,B = P A,B and P A + P B = 1. We find that the rate calculated with P A -projected wave functions is substantially larger than when full wave functions are used. These results are compared in Fig. 1, where we plot the rate divided by the free muon decay rate at tree level, in the limit of a massless electron; is the Fermi constant [13]. The solid line Fig. 1 shows the full wave function result, and the dots -the result with projectors P A . For small and moderate Z, up to Z 40, the results are close, and start to diverge quite strongly for larger nuclei.
Since the B ± contribution to the normalization is small even for large Z, these results are unexpected. We thus proceed to check them in the position space. As a reward, we find that alternative method to be simpler. It will allow us to derive a closed formula for the rate in the limit of close electron and muon masses.
3 Bound µ − to bound e − decay in position space In this section, we calculate the bound state transition rate (Zµ − ) → (Ze − ) ν µνe using position space wave functions for the decaying muon and the produced electron [14], where and the mass m is either m µ for the muon or m e for the electron. In the Dirac representation with ρ µ = (ρ 0 , ρ) = 1, i 1−γ Zαr . Since this approach differs from Ref. [4], we present it in some detail.

Factorizing neutrinos
It is convenient to decompose the decay into two stages: first the muon decays into the electron and a fictitious boson A; next, boson A decays into the νν pair. The kinematically allowed range of the mass m A of the boson A, parametrized by a dimensionless variable z, m A = zm µ , is z ∈ [0, z max = (E µ − E e ) /m µ ] where E µ,e are the muon and electron energies; they should be replaced by muon and electron masses in the case of a free muon decay. The decay rate is an integral over z, where g is the weak coupling constant. One advantage of eq. (21) is that it holds both for a free and for a bound muon decay. It is simpler to deal with a two-body decay µ → eA than with µ → eνν. Binding effects as well as radiative corrections (ignored in the present paper) affect only Γ (µ → eA).

Decay rate
The amplitude for the (Zµ) → (Ze) + A transition is where L = 1−γ 5 2 and λ A labels the polarization state of A. The triple r integration is done analytically. Angular integrations lead to spherical Bessel functions, and the r-integration results in a relatively compact formula. After squaring the amplitude, we find, using k A = z 2 max − z 2 and a = 1−γ α Z , where the quantities C n (and analogously S n with cos → sin) are The rate Γ [(Zµ − ) → (Ze − ) + νν] can now be expressed as a single integral over z, a variable equivalent to the invariant mass of the neutrinos, from zero to z max = γ (1 − δ), This is the main result of this paper. Note that the position space calculation results in a single integral representation for the rate. This is much simpler than the result of Ref. [4], where two additional integration remain, over the magnitude and the polar angle of the argument of the momentum space wave function. Those integrations seem to be more involved than the corresponding radial and angular integrations in the position space. In the following section we perform the remaining integration over z in the limit of close electron and muon masses.

Limiting Case of Similar Electron and Muon Masses
We expect out relusts to agree with Ref. [4] for small Z, since the only conceptual difference between our approaches involves negative energy components, and those are suppressed by powers of α Z . Here we demonstrate this agreement with a simple closed formula in the limiting case of nearly equal masses. We write Z Eq. (39) Eq. (32) Eq. (33) 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 Table 1: Numerical values of Γ [(Zµ − ) → (Ze − ) ν µνe ] /Γ 0 for = 0.01 for a small Z = 10 and a large Z = 80: using the formalism of Ref. [4], Eq. (39) (second column), its limit for m e m µ , Eq. (32) (third column), and the m e m µ limit of our approach, Eq. (33) (forth column). As expected, the agreement is better for small Z (first line). and consider a hypothetical situation where is small. In the limit → 0, the decay rate computed in Ref. [4] (cf. Eq. (40) in the Appendix) is In this approximation of equal masses of muon and electron, the momentum transferred to the neutrinos q is approximately zero. Therefore, the form factors F 1 and F 2 given in Eq. (30) are evaluated at |q| = 0 which gives where the mean inverse Lorentz factor L −1 is Hence, Eq. (30) becomes Γ Γ 0 = 64 The numerical results for the limiting case of almost equal masses are stated in the Table 1. In equal mass limit, using |q| r → 0, our momentum space as well as position space treatments lead to the expression The analogous limit of the free muon decay rate, eq. (23), is 64 5 5 . Thus the binding effects given by γ 5 1+γ+γ 2 in case of the decay into a bound electron are more pronounced than in the case of the decay into a free electron, given in eq. (1). For a free electron, the effect can be approximated by a single factor of γ 1 − α 2 Z + O (α 4 Z ). Numerical evaluation of Eq. (33) is given in the last column of Table 1. For small Z, we have L −1 ≈ γ and hence the corresponding results coincide in this case of equal muon and electron masses. This is not the case for large Z, where L −1 is larger then γ.

Conclusion
We have calculated the decay rate of a bound muon to bound electron using Dirac wave functions for different values of Z in two formalisms. Numerical results in momentum and in position space coincide, provided that complete wave functions (both positive and negative energy components) are used. If the negative energy parts of the wave functions are neglected, as was done in Ref. [4], the results are significantly larger. For Z = 80 the difference is about 38%. This is surprising since the probability of finding positrons in a hydrogen-like atom or ion is very small even for Z = 80. Our tentative interpretation is that the probability of the decay into a bound electron is very suppressed and that this suppression is relatively less severe for the negative energy components. We note that the decay vertex couples positive and negative energy components without a suppression factor of α Z . Also, the decay rate involves an interference of large A ± with small B ± wave function terms, whereas the normalization integral involves the small B ± only in second powers, thus greatly decreasing their contribution (see eq. (14)). In order to check this unexpected result, we developed a position space approach. It resulted in a simple one-dimensional integral representation of the rate, eq. (28). The remaining integral has been done in the limiting case of close electron and muon masses, eq. (33), giving a closed formula valid for all α Z . In closing, we quote from Sidney Coleman's field theory lectures [16]: Dirac's theory gives excellent results to order ( v /c) 2 for the hydrogen atom, even without considering pair production and multiparticle intermediate states. This is a fluke. A Bound Muon Decay in the Formalism of Ref. [4] In this Appendix, we summarize the formalism of Ref. [4] for the transition B 1 → B 2 + X, where B 1 and B 2 are bound states. The invariant amplitude of (Zµ − ) → (Ze − ) ν µνe decay is where the subscripts B 1 and B 2 represent the (Zµ − ) and (Ze − ) states, respectively. The masses of the bound states are, with M denoting the nucleus mass (note that in our approach the nucleus is treated as infinitely heavy and M does not appear), where E b,µ(e) are the binding energies. In Eq. (34), the neutrino part is given by and the charged current part is Here, k 1 , k 2 , p νe and p νµ are the 4-momenta of the muon, electron, ν e and ν µ , respectively, and the subscripts r and s are spin indices (r, s = ±1/2). The corresponding decay rate for (Zµ − ) → (Ze − ) ν µνe can be calculated as After integration over the phase space and neglecting terms suppressed by 1/M , the decay rate is where and q 2 = q 2 0 − q 2 = (m 1 − m 2 ) 2 − q 2 . The form factors F i are defined as with h 1 = k 0 1 + m µ k 0 2 + m e + q 0 (1 − C) k 0 1 + m µ − C k 0 2 + m e + (B − C) q 2 0 + A, h 2 = (C − B) q 2 + 2A, h 4 = C k 0 2 + m e − (B − C) q 0 m µ and the expressions of A, B and C are given in Eq. (35) of [4]. We mention that in the expressions for h 1 and h 2 in Eq. (42), the sign of A is different than in [4] (cf. Eq. (34) there).