High-energy behavior of scattering due to massless (neutrino) exchange

We devise a method of calculating the high-energy behavior of some processes where massless particles are being exchanged and the normal theorems such as the Froissart bound cannot be directly applied. The processes are ν-ν and ν-ν̅ scattering, with a long-range potential due to the exchange of neutrinos. It is shown that as the energy E tends to infinity, (i) the ν-ν and ν-ν̅ total cross sections behave as constants, (ii) the ν-ν and ν-ν̅ cross sections approach each other (a Pomeranchuk theorem), and (iii) the long-range two-neutrino exchange force is of the form E/r5. The calculation is made with the assumption that the amplitudes obey the Mandelstam representation, but we do not restrict ourselves to any specific order in the weak coupling constant G, or to any specific form of the Hamiltonian. The high-energy behavior is deduced by constructing a consistency condition to it. Using analyticity and unitarity, the potential due to the massless neutrino exchange is shown to depend on the high-energy behavior of the ̅-̅ and ̅-̅processes. Since, in turn, the high-energy cross sections of these processes are themselves dependent on the potential, a consistency requirement is available on the former, leading to the results mentioned above