Neutrino Trapping and Neutrino Mass Bounds

It has been shown recently that the exchange of virtual νν̄ pairs leads to an unphysically large energy-density in neutron stars and white dwarfs, unless neutrinos have a minimum mass,mν > ∼ 0.4 eV/c 2 . Here we consider the possibility that the presence of trapped low-energy neutrinos can suppress the exchange of virtual νν̄ pairs, thereby avoiding a large energy-density even for massless neutrinos. We show that a) there can be subvolumes in a neutron star or white dwarf where neutrino-trapping does not take place, and which can thus have an unphysically large energy density, and b) even in those volumes where trapping does occur, the resulting suppression can be too small to alter the conclusion that neutrinos must have a minimum mass. It is well known that the exchange of massless neutrino-antineutrino (νν̄) pairs gives rise to a long-range interaction among electrons, protons, and neutrinos [1-7]. The 2-body potential between electrons, V (2) ee (r), is given by [1,4,7] V (2) ee (r) = GF (2 sin 2 θW + 1 2 ) 2 4π3r5 = 3× 10eV (r/1m)5 . (1) Here r = |~r1 − ~r2| is the separation of the electrons, GF is the Fermi constant, and θW is the weak mixing angle. As can be seen from Eq.(1), the 2-body potential arising from neutrino-exchange is extremely weak, but its effects may nonetheless be detectable in equivalence principle experiments [6]. 1 Neutrino-exchange can also lead to many-body interactions, and these have been studied by Feynman [2] and Hartle [3]. Among other observations, Feynman noted that higher order (in GF ) many-body interactions could be important because they depend on higher powers of the masses of the interacting objects. It has been shown more recently [7] that the selfenergy of a compact object such as a white dwarf or neutron star arising from many-body neutrino-exchange interactions can become unphysically large, when calculated in the standard model. In the absence of some suppression mechanism, one is led to the conclusion that all neutrinos must have a minimum mass, mν > ∼ 0.4 eV/c. (2) In order to understand more clearly what any suppression mechanism would have to achieve, we briefly review the argument leading to the bound in Eq.(2). The self-energy of a spherical neutron star arising from the interaction of 4 particles, for example, must be of order (1/R)(GF /R ). Since the number of 4-particle interactions that can arise amongN particles (where N = O(10)) is given by the binomial coefficient