Cherenkov radiation by neutrinos

We discuss the Cherenkov process ν → νγ in the presence of a homogeneous magnetic field. The neutrinos are taken to be massless with only standard-model couplings. The magnetic field fulfills the dual purpose of inducing an effective neutrino-photon vertex and of modifying the photon dispersion relation such that the Cherenkov condition ω < |k| is fulfilled. For a field strength Bcrit = m 2 e/e = 4.41×1013 Gauss and for E = 2me the Cherenkov rate is about 6× 10−11 s−1. In many astrophysical environments the absorption, emission, or scattering of neutrinos occurs in dense media or in the presence of strong magnetic fields [1]. Of particular conceptual interest are those reactions which have no counterpart in vacuum, notably the decay γ → ν̄ν and the Cherenkov process ν → νγ. These reactions do not occur in vacuum because they are kinematically forbidden and because neutrinos do not couple to photons. In the presence of a medium or B-field, neutrinos acquire an effective coupling to photons by virtue of intermediate charged particles. In addition, media or external fields modify the dispersion relations of all particles so that phase space is opened for neutrino-photon reactions of the type 1 → 2 + 3. If neutrinos are exactly massless as we will always assume, and if medium-induced modifications of their dispersion relation can be neglected, the Cherenkov decay ν → νγ is kinematically possible whenever the photon four momentum k = (ω,k) is space-like, i.e. k −ω2 > 0. Often the dispersion relation is expressed by |k| = nω in terms of the refractive index n. In this language the Cherenkov decay is kinematically possible whenever n > 1. Around pulsars field strengths around the critical value Bcrit = m 2 e/e = 4.41×1013 Gauss. The Cherenkov condition is satisfied for significant ranges of photon frequencies. In addition, the magnetic field itself causes an effective ν-γ-vertex by standard-model neutrino couplings to virtual electrons and positrons. Therefore, we study the Cherenkov effect entirely within the particle-physics standard model. This process has been calculated earlier in [2]. However, we do not agree with their results. Our work is closely related to a recent series of papers [3] who studied the neutrino radiative decay ν → ν ′γ in the presence of magnetic fields. Our work is also related to the process of photon splitting that may occur in magnetic fields as discussed, for example, in Refs. [4, 5]. Photons couple to neutrinos by the amplitudes shown in Figs. 1(a) and (b). We limit our discussion to field strengths not very much larger than Bcrit = m 2 e/e. Therefore, we keep only electron in the loop. Moreover, we are interested in neutrino energies very much smaller than the W and Z-boson masses, allowing us to use the limit of infinitely heavy gauge bosons and thus an effective four-fermion interaction (Fig. 1(c)). The matrix element has the form