Right-handed Electrons in Radiative Muon Decay

Electrons emitted in the radiative decay mu^- ->e^- anti-nu_e nu_mu gamma have a significant probability of being right-handed, even in the limit m_e ->0. Such ``wrong-helicity'' electrons, arising from helicity-flip bremsstrahlung, contribute an amount alpha/(4 pi) Gamma_0 to the muon decay width (Gamma_0 = G_F^2 m^5_mu/ (192 pi^3)). We use the helicity-flip splitting function D_hf (z) of Falk and Sehgal (Phys. Lett. B 325, 509 (1994)) to obtain the spectrum of the right-handed electrons and the photons that accompany them. For a minimum photon energy E_gamma = 10 MeV (20 MeV), approximately 4% (7%) of electrons in radiative mu-decay are right-handed.

It is usually thought that in V-A theory, electrons emitted in muon decay are purely left-handed, in the limit m e → 0. This statement, however, is not true for electrons in the radiative decay µ − → e −ν e ν µ γ, where the photon is the result of inner bremsstrahlung. We show in this Letter that radiative muon decay contains a well-defined constituency of right-handed electrons, contributing an amount α 4π Γ 0 (Γ 0 ≡ G 2 F m 5 µ /(192π 3 )) to the decay width. We calculate the spectrum of these "wrong-helicity" electrons, and of the photons that accompany them. These spectra are compared with the unpolarized spectra, summed over electron helicities. This comparison provides a quantitative measure of the right-handed fraction and its distribution in phase space.
The appearance of "wrong-helicity" electrons in the decay µ − → e −ν e ν µ γ, even in the limit m e → 0, is a consequence of helicity-flip bremsstrahlung in quantum electrodynamics, a feature first noted by Lee and Nauenberg [1]. It was found that in the radiative scattering of electrons by a Coulomb field, the probability of helicity-flip (i.e. e L → e R or e R → e L ) did not vanish in the limit m e → 0. This unexpected result, in apparent contradiction to the naive expectation of helicity conservation in the m e → 0 limit, arises from the fact, that the helicity-flip cross section for bremsstrahlung at small angles has the form dσ dθ 2 ∼ me Ee 2 me Ee which, when integrated over angles, gives a finite non-zero answer in the limit m e → 0.
In Ref. [2], Falk and Sehgal examined the helicity structure of the bremsstrahlung process in an equivalent particle approach, and showed that helicity-flip radiation e − L → e − R + γ(z), in the limit m e → 0, can be described by a simple and universal splitting (or fragmentation) function where z = E γ /E e is the ratio of the photon energy to the energy of the radiating electron. This function is analogous to the familiar Weizsäcker-Williams function describing helicity-conserving (non-flip) bremsstrahlung Several applications of the helicity-flip function D hf (z) were considered in [2], including the process e − R + p → ν L + γ + X ("fake right-handed currents") and e − λ e + λ → ff γ (wrong-helicity e + e − annihilation). It was shown that the splitting function approach reproduced the results of the usual bremsstrahlung calculation in which the limit m e → 0 was taken at the end [3,4]. Subsequently, the equivalent-particle technique has been successfully applied to other helicity-flip processes such as π − → e − Lν γ and Z 0 → e − L e + L γ [5].
Recently, in an analysis of radiative corrections to the electron spectrum in muon decay, µ − → e −ν e ν µ , Fischer et al. [6] have noted that the radiative correction to the helicity of the electron, calculated in an early paper by Fischer and Scheck [7],can be reproduced in a simple way using the helicity-flip function D hf (z). This has motivated us to examine the helicity-dependence of the radiative decay µ − → e −ν e ν µ γ, to determine the incidence and spectrum of wrong-helicity (right-handed) electrons in this channel.
The electron spectrum in ordinary (non-radiative) muon decay µ − → e −ν e ν µ has, in Born appoximation, the well-known form where x = 2E e /m µ and θ e is the angle of the electron relative to the spin of the muon. We can obtain from this the spectrum of the radiative channel µ − → e −ν e ν µ γ, using the splitting functions D hf or D nf . In the specific case of right-handed electrons in the final state, the spectrum, in the limit m e → 0 (collinear bremsstrahlung), is given by (5) where the θ-function in the integrand has been inserted to allow for a minimum energy cut on the photon: The result of the integration is where Integrating over cos θ e and x e (0 ≤ x e ≤ 1 − x γ0 ), we obtain If no cut is imposed on the photon energy (i.e. x γ0 = 0) the spectrum of right-handed electrons given in Eq. (7) which coincides with the result obtained by Fischer and Scheck [7].
The helicity-flip fragmentation function also gives a simple way of calculating the spectrum of photons accompanying right-handed electrons in µ − → e −ν e ν µ γ. In the collinear limit (m e → 0), we have Integrating over all photon energies and over cos θ γ we get Γ rad e − R = α 4π Γ 0 , which is the same as Eq.(9) for x γ0 = 0.
The decay width into right-handed electrons, for a given minimum energy x γ0 (Eq.(9)), can be compared with the width summed over electron helicities. The helicity-summed photon spectrum in µ − → e −ν e ν µ γ was calculated by Kinoshita and Sirlin [8] and Eckstein and Pratt [9], and the integrated width, for x γ > x γ0 is [9] Γ rad This function is plotted in Fig.(1) and compared with the right-handed width Γ rad e − R (x γ0 ) calculated in Eq. (9). The right-handed fraction Γ rad Fig.(2), as a function of x γ0 . For a photon energy cut E γ > 10MeV (20 MeV ), this fraction is approximately 4% (7%). [It may be noted here that the branching ratio of µ − → e −ν e ν µ γ, summed over electron spins, with a photon energy cut E γ > 10 MeV , was measured in Ref. [10] to be (1.4 ± 0.4)%. The theoretical expression Eq.(12) yields for this quantity the value 1.3%].
A complete analysis of the channel µ − → e −ν e ν µ γ involves a study of the decay intensity in all kinematical variables. A variable of particular interest is the angle θ eγ between the electron and the photon. For helicity-flip radiation, the characteristic angular distribution is [2] dD hf (z, which is maximum at θ = 0 (forward direction). By contrast, the helicityconserving bremsstrahlung has the spectrum [2] dD nf (z, θ 2 ) dθ 2 ≈ α 2π which peaks at θ ≈ m 2 e /E 2 e . This suggests that in the decay µ − → e −ν e ν µ γ, the distribution in the angle θ eγ between the electron and photon could be a useful discriminant in separating the two electron helicities. A full analysis of the helicity-dependent decay spectrum in different kinematical variables will be reported elsewhere. Summary: (i) The decay µ − → e −ν e ν µ γ contains in the final state a constituency of right-handed electrons, which contribute an amount α 4π Γ 0 to the decay width, in the limit m e → 0. (ii) The spectrum of the right-handed electrons is given by Eq. (7), and reduces to Eq. (10) if no cut on photon energy is imposed. The latter differs in a characteristic way from the spectrum of left-handed electrons, which (on account of the soft 1/x γ nature of helicity-conserving bremsstrahlung) tends to follow the non-radiative pattern Eq.(4). Thus the energy spectra, integrated over angles are (dΓ/dx e ) R ∼ (1 − x e ) 2 (5 − 2x e ), (dΓ/dx e ) L ∼ x 2 e (3 − 2x e ), while the angular distribution, integrated over energies, is (dΓ/d cos θ e ) L,R ∼ (1 − 1 3 cos θ e ), the same for e − L and e − R . (iii) The photon spectrum associated with right-handed electrons is , and is hard compared to that accompanying left-handed electrons (dΓ/dx has been calculated as a function of the photon energy cut x γ0 , and amounts to 4% (7%) for E γ > 10 MeV (20 MeV ).
(v) The radiatively corrected decay width of the muon, usually written as can be regarded as a sum of two mutually exclusive helicity contributions [6] where Γ µ (e − L ) = Γ 0 1 + (vi) A full analysis of µ − → e −ν e ν µ γ, aimed at finding regions of phase space with enhanced concentration of right-handed electrons will be reported elsewhere.