Mixing of fermions and spectral representation of propagator

Abstract

We develop the spectral representation of propagator for $n$ mixing fermion fields in the case of $\mathsf{P}$-parity violation. The approach based on the eigenvalue problem for inverse matrix propagator makes possible to build the system of orthogonal projectors and to represent the matrix propagator as a sum of poles with positive and negative energies. The procedure of multiplicative renormalization in terms of spectral representation is investigated and the renormalization matrices are obtained in a closed form without the use of perturbation theory. Since in theory with $\mathsf{P}$-parity violation the standard spin projectors do not commute with the dressed propagator, they should be modified. The developed approach allows us to build the modified (dressed) spin projectors for a single fermion and for a system of fermions.Comment: 22 pages, this is refreshed and extended versio

LSZ-reduction, resonances and non-diagonal propagators: Gauge fields

“…The structure (97) of the pole of the mixed φh propagator and the lack of poles of the (vanishing identically in the Landau gauge) mixed scalar-vector propagator (cf. (18)) are correctly reproduced by φ φ φ j unph of (135), if…”

“…In what follows, it will be assumed that an IR regulator has been introduced, if necessary, so that the limits (23) and (26) are finite. 18 As before, we have to assume that each generalized eigenvector of M 2 V (m 2 V (λ) ) associated with the eigenvalue m 2 V (λ) is an ordinary eigenvector.…”

“…In the context of field mixing we should mention also the paper[17] in which the possibility of imposing on-shell renormalization conditions in systems with mixed scalar, vector and fermionic fields was studied. Mixing of fermions treated in this approach was re-examined in[18] with the aid of special parametrization of the propagator.…”

Neutrino propagation in media and axis of complete polarization

“…Such form of propagator gives the simplest and most convenient algebraic construction and means in fact a complete diagonalization. A spectral representation was discussed earlier for dressed fermion propagator in theory with parity violation [36] and for dressed matrix propagator with mixing of few fermionic fields [37]. Note that the problem of the neutrino propagator in the presence of matter, including account for possible effect of matter motion, was also discussed in [38] (see also [39]).…”

“…The appeared here generalized spin progectors [37] commute with propagator and in theories with parity violation take the form Σ ± (s) = 1 2 (1 ± γ 5ŝn ), n µ = p µ /W, W = p 2 .…”