Weak decays of magnetized charged pions in the symmetric gauge

Abstract

We consider the decay π − → lν l (l = e − , µ − ) in the presence of an arbitrary large uniform magnetic field, using the symmetric gauge. The consequences of the axial symmetry of the problem and the issue of angular momentum conservation are discussed in detail. In particular, we analyze the projection of both the canonical and the mechanical total angular momenta along the direction of the magnetic field. It is found that while the former is conserved in the symmetric gauge, the latter is not conserved in both the symmetric and Landau gauges. We derive an expression for the integrated π − → lν l width that coincides exactly with the one found using the Landau gauge in Ref. [20], providing an explicit test of the gauge independence of that result. Such an expression implies that for nonzero magnetic fields the decay width does not vanish in the limit in which the outgoing charged leptons are massless, i.e. it does not exhibit the helicity suppression found in the case of no external field.

Magnetic field driven enhancement of the weak decay width of charged pions

“…The same expression was found in ref. [23] using the symmetric gauge, explicitly showing the gauge independence of the result.…”

“…Moreover, the outcomes obtained for B = 0 from angular momentum conservation do not apply for nonzero B. The analysis of the decay in terms of angular momenta of the initial and final states is not straightforward, since for nonzero B canonical angular momenta of charged particles turn out to be gauge dependent quantities, and total mechanical angular momentum is in general not conserved [23,[25][26][27]. A brief discussion on how this can be reconciled with the rotational invariance of the system is included in appendix A.…”

“…To have a better understanding of the situation, it is interesting to consider the case in which the magnitude of the magnetic field is large enough so that the outgoing charged lepton l − can only be in the LLL, n = 0 (the validity of this assumption will be discussed below). Considering the explicit form of the corresponding spinors [22,23], it is not hard to show (see appendix B) that in the m l → 0 limit one has…”

“…In this case, however, the transverse piece of the helicity operator provides in general a nonvanishing contribution. On one hand, there is no restriction for antineutrino helicity eigenstates to be in general a combination of the two available possible polarization states, Σ z (ν l ) = ±1 [22,23]. On the other hand, the antineutrino transverse momentum k ⊥ is in general nonvanishing, since, due to zero-point motion, the wavefunctions of charged particles in the LLL involve a superposition of various transverse momenta.…”

“…A model independent expression for the width has been obtained in refs. [22,23], taking the decaying pion to be in the LLL, with p z = 0. The main steps leading to this expression are summarized in the following subsections.…”