We discuss the transition radiation process $\nu \to \nu \gamma$ at an interface of two media. The medium fulfills the dual purpose of inducing an effective neutrino-photon vertex and of modifying the photon dispersion relation. The transition radiation occurs when at least one of those quantities have different values in different media. We present a result for the probability of the transition radiation which is both accurate and analytic. For $E_\nu =1$MeV neutrino crossing polyethylene-vacuum interface the transition radiation probability is about $10^{-39}$ and the energy intensity (deposition) is about $10^{-34}$eV. At the surface of the neutron stars the transition radiation probability may be $\sim 10^{-20}$. Our result on three orders of magnitude is larger than the results of previous calculations.}
In many astrophysical environments the absorption, emission, or scattering of neutrinos occurs in media, in the presence of magnetic fields 1 or at the interface of two media.
In the presence of a media, neutrinos acquire an effective coupling to photons by virtue of intermediate charged particles. The violation of the translational invariance at the direction from one media into another leads to the non conservation of the momentum at the same direction so that transition radiation becomes kinematically allowed.
The theory of the transition radiation by charged particle has been developed in 2 3 . It those articles authors used classical theory of electrodynamics. In 4 the quantum field theory was used for describing the phenomenon. The neutrinos have very tiny masses. Therefore one has to use the quantum field theory approach in order to study transition radiation by neutrinos.
The presence of a magnetic field induces an effective ν-γ-coupling. The Cherenkov decay in a magnetic field was calculated in 5 .
At the interface of two media with different refractive indices the transition radiation ν → νγ was studied in 6 with an assumption of existence of large (neutrino) magnetic dipole moment. We presently extend previous studies of the transition radiation to neutrinos with only standard-model couplings. The media changes the photon dispersion relation. In addition, the media causes an effective ν-γ-vertex by standard-model neutrino couplings to the background electrons. We neglect neutrino masses and medium-induced modifications of their dispersion relation due to their negligible role. Therefore, we study the transition radiation entirely within the particle-physics standard model.
A detailed literature search reveals that neutrino transition radiation has been studied earlier in 7 . They used vacuum induced ν-γ vertex (“neutrino toroid dipole moment”) for the ν → νγ matrix element. We do not agree with their treatment of the process. The media itself induces ν-γ vertex. The vacuum induced vertex can be treated as a radiation correction to the medium induced one. We found that the result of 7 for the transition radiation rate is more than three orders of magnitude, ( 8α π ) 2 , smaller than our result.
Let us consider a neutrino crossing the interface of two media with refraction indices n 1 and n 2 (see Fig. 1). In terms of the matrix element M the transition radiation probability of the process ν → νγ is
Here, p = (E, p), p ′ = (E ′ , p ′ ), and k = (ω, k) are the four momenta of the incoming neutrino, outgoing neutrino, and photon, respectively and β z = p z /E. The sum is over photon polarizations.
We shall neglect the neutrino masses and the deformation of its dispersion relations due to the forward scattering. Thus we assume that the neutrino dispersion relation is precisely light-like so that p 2 = 0 and E = |p|.
The formation zone length of the medium is
The integral over z in eq. ( 1) oscillates beyond the length of the formation zone. Therefore the contributions to the process from the depths over the formation zone length may be neglected.
The z momentum (p z -p ′ z -k z ) transfers to the media from the neutrino. Since photons propagation in the media suffers from the attenuation(absorption) the formation zone length must be limited by the attenuation length of the photons in the media when the later is shorter than the formation zone length.
After integration of (1) over p ′ and z we find
where β ′ z = p ′ z /E ′ , θ is the angle between the emitted photon and incoming neutrino. M (1,2) are matrix elements of the ν → νγ in each media. k As it will be shown below main contribution to the process comes from large formation zone lengths and ,thus, small angle θ. Therefore the rate of the process does not depend on the angle between the momenta of the incoming neutrino and the boundary surface of two media (if that angle is not close to zero). The integration over ϕ drops out and we may replace dϕ → 2π. k
here we have used n (1,2) = |k| (1,2) /ω. If the medium is isotropic and homogeneous the polarization tensor, π µν , is uniquely characterized by a pair of two polarization functions which are often chosen to be the longitudinal and transverse polarization functions. They can be projected from the full polarization matrix. In this paper we are interested in transverse photons, since they may propagate in the vacuum as well. The transverse polarization function is
The dispersion relation for the photon in the media is the location of its pole in the effective propagator (which is gauge independent)
3
In a media, photons couple to neutrinos via interactions to electrons by the amplitudes shown in Fig 2 . One may take into account similar graphs with nuclei as well, but their contribution are usually negligible. When photon energy is below weak scale (E ≪ M W ) one may use four-fermion interactions and the matrix element for the ν-γ vertex can be written in the form
here
π µν t i
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