Optical activity of a neutrino sea in the Standard Model
We present the Standard Model calculation of the optical activity of a neutrino sea. The idea that intergalactic space is a birefrigent medium for light due to the presence of a neutrino sea has been contemplated for a long time. Some thirty years ago, Royer [1] computed in V − A theory an effect of order G F. Later, Stodolsky noted that due to a theorem of Gell-Mann [2], there can be no such effect with massless neutrinos and a point-like coupling, and his observation was recorded by one of the present authors in a review [3]. In the early 1980's data on propagation of radio waves through intergalactic Space put a stringent upper bound on possible optical activity of the neutrino sea [4] and [5] and this led to renewed estimates for the size of such effect on the assumption of a neutrino magnetic moment, which occur for a massive neutrino [5]. More recently [6] an evaluation was made for a off-mass shell photons within the Standard Model (SM). Here we note that there is a frequency dependent effect even for real photons and we also comment on the effect for virtual photons (both within the SM). Thus we shall consider three possible effects: a magnetic moment effect, an effect for virtual photons, an effect for real photons. 1 If the neutrino has a magnetic moment, this moment will interact with the magnetic field of an electromagnetic wave: L int = (eµ/2m)¯ ν(p 2)σ αβ q β ν(p 1)ǫ(q) , (1) In this equation ǫ(q) is the polarization vector of a photon of momenta q, while ν(p) is the neutrino bispinor, and µ is the neutrino magnetic moment in Bohr magnetons e/2m. With this interaction (1) the forward scattering amplitude of a photon of momentum k from a neutrino of momentum p is equal to T = 4i(eµ/m) 2 ǫ µναβ ǫ µ (k)ǫ * ν (k)p α k β. (2) This amplitude differs for left-handed and right-handed photons: T LL = −T RR = 4(eµ/m e) 2 (pk) , (3) and gives rise to a nonzero optical activity for a polarizes neutrino gas. In the Standard Model the neutrino has no intrinsic magnetic moment, but a magnetic moment can emerge as a one loop effect if the neutrino is massive(See figure 1).