Comment on “ Neutrinos with Magnetic Moment : Depolarization Rate in Plasma ”

In a recent contribution of Elmfors, Enqvist, Raffelt and Sigl, the rate of populating right–handed neutrinos at Big Bang Nucleosynthesis was calculated by using the real–time framework of Thermal Field Theory (amongst the different approaches presented there). In a calculation with only bare propagators, photon exchange diagrams give rise to logarithmically infrared divergent integrals, which were treated by the authors by ad hoc prescription for screening of the form ln(k max /k min). Here we describe how the ambiguities due to the choices of k max and k min disappear when a complete calculation to leading order in α is applied. Recently Elmfors, Enqvist, Raffelt and Sigl [1] have reexamined a cosmological limit on the magnetic moments of neutrinos by recalculating the depolarization rate for electron (anti)neutrinos through the scattering process ν eL + e ∓ −→ ν eR + e ∓. They used the real–time approach of Thermal Field Theory (TFT) [2] and calculated the rate of populating right–handed neutrinos through the imaginary part of the sterile–neutrino self–energy. However, to obtain a consistent TFT perturbative result at leading order in α one should amplify the theory by the resummation program for the soft regime (Pisarski [3], Braaten and Pisarski [4] and Frenkel and Taylor [5]), together with the practical method for computing the effects of screening in a hot gauge theory, as developed by Braaten and Yuan [6]. First, we point out several inconsistencies appearing in and from Eqs.(5.3) and (5.4) in the paper (hereafter, all of their notation is adopted). Since the expressions for polarization functions Π T,L as given by Eqs.(5.1) and (5.2) are valid only for soft momenta, k 0 , k < ∼ eT , all terms in the denominator of (3.9) are now of order e 2 T 2 , and all of them are to taken into account. In addition, the upper bound in the integral over k in Eqs.(5.3) and (5.4) should be restricted to be soft as well. This translates into inconsistent choices of k max and k min. The k max was chosen to be hard (of order T) and the infrared cutoff was provided by the plasmon mass eT /3. The screening mass for space–like longitudinal photons is provided by the Debye mass (k D) and not by the plasmon mass. Although these two quantities are not related, they become so for ultra–relativistic plasmas, k 2 D = 3M …