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Unformatted text preview: 1 MUON DECAY PARAMETERS Revised August 2009 by W. Fetscher and H.-J. Gerber (ETH Zurich). Introduction: All measurements in direct muon decay, e + 2 neutrals, and its inverse, + e + 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 ( and e ), and hence the doublet assignments ( e e ) L and ( ) L , have been de- termined from experiments [2,3]. All considerations on muon decay are valid for the leptonic tau decays + + e with the replacements m m , 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 and + d with respect to the muon polarization vector P , and with its spin parallel to the arbitrary direction b , neglecting radiative corrections, is given by d 2 dx d cos = m 4 3 W 4 e G 2 F q x 2 x 2 ( F IS ( x ) P cos F AS ( x )) h 1 + b P e ( x, ) i . (1) Here, W e = max( E e ) = ( m 2 + m 2 e ) / 2 m is the maximum e energy, x = E e /W e is the reduced energy, x = m e /W e = 9 . 67 10 3 , and P = | P | is the degree of muon polarization. b 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, ), may be parametrized by the Michel parameter , by , by and [5,6], etc. These are bilinear combinations of the coupling constants g , 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 ( ) measured by a polarization insensitive detector, is given by d 2 dx d cos x 2 3(1 x ) + 2 3 (4 x 3) + 3 x (1 x ) /x P cos 1 x + 2 3 (4 x 3) . (2) Here, is the angle between the electron momentum and the muon spin, and x 2 E e /m . For the Standard Model coupling, we obtain = = 3 / 4, = 1, = 0 and the differential decay rate is d 2 dx d cos = G 2 F m 5 192 3 [3 2 x P cos (2 x 1)] x 2 . (3) The coecient in front of the square bracket is the total decay rate....
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This note was uploaded on 06/07/2011 for the course PHYS 4132 taught by Professor Kutter during the Spring '11 term at University of Florida.
- Spring '11