allow an extra phenomenological parameter ?

Do neutrino oscillations allow an extra phenomenological parameter? Abstract The quantity ξ introduced recently in the phenomenological description of neu-trino oscillations is in fact not a free parameter, but a fixed number. The literature on phenomenology of neutrino oscillations is vast (see, e.g., [1]-[6] and references therein). In a recent paper [7] Giunti and Kim in the case of two-flavour mixing have introduced a new phenomenological parameter ξ. According to [7], ξ = 0 corresponds to the so-called equal momentum assumption [1, 2], while ξ = 1 corresponds to equal energy assumption [5, 6]. Authors of [7] emphasize that ξ disappears from final expressions for the neutrino oscillation probability. The aim of this note is to indicate that parameter ξ is fixed by energy-momentum conservation in the process which is responsible for neutrino emission, as explicitly assumed in ref. [7]. Following ref. [7] we will consider the decay π → µν in the framework of two-flavour toy model. The parameter ξ is defined in [7] for the pion rest-frame by considering the auxiliary case of absolutely massless neutrinos and denoting the energy of such neutrinos as E, ξ = 1 / 2 (1 + m 2 µ /m 2 π) , (1) where m µ and m π are the masses of the muon and the pion. Then for massive (but light!) neutrinos they get: E 1,2 = E + (1 − ξ)m 2 1,2 /2E , (2) p 1,2 = E − ξm 2 1,2 /2E. are the energies, momenta and masses of the neutrinos, respectively. Wherefrom the above statement about ξ = 0, 1 follows: E 1 = E 2 for ξ = 1 and p 1 = p 2 for ξ = 0. (4) Thus, the equal energy and equal momentum assumptions in the form ∆E ≡ E 1 − E 2 = 0 and ∆p ≡ p 1 − p 2 = 0, respectively, are treated by authors of ref. [7] as particular cases of the general kinematical relations (2) and (3): ∆E = (1 − ξ)∆m 2 /2E = 0 for ξ = 1 , (5)