Gauge invariance and finite width effects in radiative two-pionτlepton decay

Abstract

The contribution of the ρ ± vector meson to the τ → ππνγ decay is considered as a potential source for the determination of the magnetic dipole moment of this light vector meson. In order to keep gauge-invariance of the whole decay amplitude, a procedure similar to the fermion loop-scheme for charged gauge bosons is implemented to incorporate the finite width effects of the ρ ± vector meson. The absorptive pieces of the one-loop corrections to the propagators and electromagnetic vertices of the ρ ± meson and W ± gauge boson have identical forms in the limit of massless particles in the loops, suggesting this to be a universal feature of spin-one unstable particles. Modeldependent contributions to the τ → ππνγ decay are suppressed by fixing the two-pion invariant mass distribution at the rho meson mass value. The resulting photon energy and angular distribution is relatively sensitive to the effects of the ρ magnetic dipole moment.

Absorptive one-loop corrections and the complex-mass prescription for the Δ resonance propagator

“…By considering the absorptive part of the one-loop self-energy corrections to the propagator, in this paper we show how the CMS prescription advocated in our previous papers can be derived in the limit of massless particles appearing in loop corrections (similar results were obtained in the case of spin-1 resonances in [6,8]). The effects on the cross section of elastic π + p scattering due to the absorptive corrections beyond the ones considered in the CMS expression of the propagator turn out to be negligibly small.…”

“…In the following we will consider only the absorptive (imaginary) parts of the self-energy correction, i.e. we will assume [6,8] that the parameter m represents the 'renormalized' mass of (we place quotation marks as a reminder that the Lagrangian is not renormalizable; only the absorptive corrections are finite in this case). If we denote by p the 4-momentum of and compute the one-loop absorptive corrections by applying the cutting rules to figure 2, we obtain…”

“…The naive introduction of the finite width effects in the resonance propagator can spoil the gauge invariance of the amplitude. As has been extensively discussed in the literature in the case of unstable gauge bosons [5,6] and of hadronic resonances [7,8], there are several procedures to render the amplitude gauge invariant in the presence of finite width effects of unstable particles. One of these prescriptions is the so-called complex-mass scheme (CMS), which consists in replacing m 2 → m 2 −im (or equivalently, m → m − i /2) in all the Feynman rules where the mass of the unstable particle appears.…”

“…In previous studies of the elastic and radiative π + p scattering [10], pion photoproduction off nucleons [11] and of the one-pion production in νN reactions [12] in the resonance region, we have assumed the CMS prescription [7] to introduce the finite width of the resonance in the tree-level expression of the amplitudes. It is worth noting that the finite width effects can be introduced also by considering the absorptive one-loop corrections in the propagator and electromagnetic vertex of the resonance (see for example [6,8]). This gives rise to gauge-invariant amplitudes as these corrected Green functions satisfy the electromagnetic Ward identity.…”

Closing the gap on RD* by including longitudinal effects

“…The complete 4-body amplitude, on the other hand, is given by where D µν is the D * propagator which, upon considering the absorptive correction (dominated by the Dπ mode as discussed before), can be set in terms of the transverse and longitudinal part as follows [15][16][17]:…”