The Decay Lifetime of Polarized Fermions in Flight

Based on the parity violation in Standard model, we study the dependence of lifetime on the helicity of an initial-state fermion in weak interactions. It is pointed out that if the initial fermions in the decays are longitudinally polarized, then the decay lifetime of left-handed polarized fermions is different from that of right-handed polarized fermions in flight with a same velocity in a same inertial system. PACS numbers: 11.30.Er; 11.30.Rd;12.15.-y; 13.30.-a; 13.35.Bv The phenomena of parity violation are experimentally exhibited in two aspects. (1) The angular distribution of the particles emitted from decay processes is asymmetric. The experiment has established that the emission of beta particles is more favored in the direction opposite to that of the Co nuclei spin.[1] In muon decays, this asymmetry has also been observed.[2] The parity violation in the Λ-decay is manifested as an up-down asymmetry of the decay pion (or proton) relative to the production plane. It is worthy of special mention that the asymmetry of the angular distribution is dependent on the spin orientation of the initial decay particle. If the average spin component of the initial particle is equal to zero, the angular distribution is symmetric. (2) All of fermions emitted from decay processes are longitudinally polarized. It is well known that the neutrinos are left-handed (LH) polarized while the antineutrinos are right-handed (RH) polarized. In beta or muon decays it is found that the electrons are always LH polarized while the positrons are RH polarized.[3] Based on the two experimental facts above, naturally, it is thought that not only the final-state fermions in the decays, but also the initial-state fermion should be relevant to its longitudinal polarization. So we further could consider that the lifetime of fermions in the LH helicity state and the that of fermions in the RH helicity state should be different as well. However, The experimental and theoretical study on the polarization or helicity of the initial fermions in decays has not yet been discussed fully in the literature. In this paper we aim at exploring this point in some detail. Present address: Residence 10-2-7, Shaanxi Normal University, Xi’an 710062, China. E-mail address: [email protected] E-mail address: gj [email protected] I The charged weak currents in the SM In order to describe the parity violation in weak interactions, all of fundamental fermions are divided into two classes, LH chirality state and RH chirality state, in the standard model (SM). They are defined as ψ L = 1 2 (1 + γ5)ψ, ψR = 1 2 (1− γ5)ψ, (1) respectively. And we have ψ = ψ L + ψ R . (2) The LH chirality state is different from the RH chirality state. The former is the SU(2)-doublet state whereas the latter the SU(2)-singlet state, hence they have different gauge transformations. Especially, RH chirality state has zero weak isospin and is only present in neutral weak currents. Therefore, there exist the only LH chirality states in charged weak currents. The interaction Lagrangians for charged weak lepton current and charged weak quark current read, respectively Ll W = 1 √ 2 g2 eLγμW + μ νL + 1 √ 2 g2 νLγμW − μ eL, (3) LqW = 1 √ 2 g 2 d L γμW + μ uL + 1 √ 2 g 2 u L γμW − μ dL, (4) where g2 is the coupling constant corresponding to SU(2). Obviously, all of fermions are in the LH chirality states and all of antifermions while in the RH chirality states in decay processes. For example, the weak interaction in muon decay is successfully described by four-fermion interaction Hamiltonian. We denote the matrix element by M ∼ ∑ g εμ 〈eε|Γγ|(νe)n〉〈(νμ)m|Γγ|μμ〉, (5) where γ = S, V, T indicates a scalar, vector or tensor interaction; and ε, μ = R,L indicate a rightor left-handed chirality of the electron or muon. The chiralities n and m of the νe and νμ are then determined by the values of γ, ε and μ. All the coupling constants have been obtained entirely from experiments without any model assumption.[4] The experiments on muon decay show g RL , g RR , g LR to be zero, and at least one of the two coupling, g LL or g LL , to be nonzero. The experiments on inverse muon decay provides a lower limit for pure V −A interaction with |gV LL | > 0.960. Thus the measurements give a strong support to the standard model which sets g LL = 1 and all the others being zero, and then indicate that the charged weak current is dominated by a coupling to left-handed chirality fermions. Therefore, the negative muon decay can be written as μ− L −→ e− L + νeR + νμL . (6) And the matrix element (5) has the form M = g2 4m2W (νμLγμ μL)(eLγμ νeL). (7) Similarly, the neutron decay can be written as n L −→ p L + e L + ν R , (8) and the corresponding matrix element is M = g2 4m2W (p L γμ nL)(eLγμ νeL). (9) II Helicity and chirality The Dirac equation has the form in Pauli metric (γμ∂μ +mc )ψ = 0, (10) where m is the rest mass and ψ is four-component spinor. The projection of spin vector ~σ along the direction of fermion momentum is known as the helicity or polarization: h = ~σ · ~p | ~ p | , (11) where h is a constant of the motion with eigenvalues ±1. Taking the simplest case of ~ p : pz = p, when h = +1 and h = −1, we have the RH helicity state ψ Rh and the LH helicity state ψ Lh , respectively