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Unformatted text preview: – 1– MUON DECAY PARAMETERS Revised August 2009 by W. Fetscher and H.-J. Gerber (ETH Z¨urich). Introduction: All measurements in direct muon decay, μ − → e − + 2 neutrals, and its inverse, ν μ + e − → μ − + neutral, are successfully described by the “ V- A interaction,” which is a par- ticular case of a local, derivative-free, lepton-number-conserving, four-fermion interaction . As shown below, within this framework, the Standard Model assumptions, such as the V- A form and the nature of the neutrals ( ν μ and ¯ ν e ), and hence the doublet assignments ( ν e e − ) L and ( ν μ μ − ) L , have been de- termined from experiments [2,3]. All considerations on muon decay are valid for the leptonic tau decays τ → + ν τ + ¯ ν e with the replacements m μ → m τ , m e → m . Parameters: The differential decay probability to obtain an e ± with (reduced) energy between x and x + dx , emitted in the direction b x 3 at an angle between ϑ and ϑ + dϑ with respect to the muon polarization vector P μ , and with its spin parallel to the arbitrary direction b ζ , neglecting radiative corrections, is given by d 2 Γ dx d cos ϑ = m μ 4 π 3 W 4 eμ G 2 F q x 2 − x 2 × ( F IS ( x ) ± P μ cos ϑ F AS ( x )) × h 1 + b ζ · P e ( x, ϑ ) i . (1) Here, W eμ = max( E e ) = ( m 2 μ + m 2 e ) / 2 m μ is the maximum e ± energy, x = E e /W eμ is the reduced energy, x = m e /W eμ = 9 . 67 × 10 − 3 , and P μ = | P μ | is the degree of muon polarization. b ζ is the direction in which a perfect polarization-sensitive electron detector is most sensitive. The isotropic part of the spectrum, F IS ( x ), the anisotropic part F AS ( x ), and the electron polarization, P e ( x, ϑ ), may be parametrized by the Michel parameter ρ , by η , by ξ and δ [5,6], etc. These are bilinear combinations of the coupling constants g γ εμ , which occur in the matrix element (given below). If the masses of the neutrinos as well as x 2 are neglected, the energy and angular distribution of the electron in the rest CITATION: K. Nakamura et al. (Particle Data Group), JPG 37 , 075021 (2010) (URL: http://pdg.lbl.gov) July 30, 2010 14:34 – 2– frame of a muon ( μ ± ) measured by a polarization insensitive detector, is given by d 2 Γ dx d cos ϑ ∼ x 2 · ½ 3(1 − x ) + 2 ρ 3 (4 x − 3) + 3 η x (1 − x ) /x ± P μ · ξ · cos ϑ · 1 − x + 2 δ 3 (4 x − 3) ¸¾ . (2) Here, ϑ is the angle between the electron momentum and the muon spin, and x ≡ 2 E e /m μ . For the Standard Model coupling, we obtain ρ = ξδ = 3 / 4, ξ = 1, η = 0 and the differential decay rate is d 2 Γ dx d cos ϑ = G 2 F m 5 μ 192 π 3 [3 − 2 x ± P μ cos ϑ (2 x − 1)] x 2 . (3) The coeﬃcient in front of the square bracket is the total decay rate....
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- Spring '11
- Electron, muon decay, W. Fetscher