When do neutrinos cease to oscillate ?

In order to investigate when neutrinos cease to oscillate in the framework of quantum field theory, we have reexamined the wave packet treatment of neutrino oscillations by taking different sizes of the wave packets of the particles involved in the production and detection processes. The treatment is shown to be considerably simplified by using the Grimus–Stockinger theorem which enables us to carry out the integration over the momentum of the propagating neutrino. Our new results confirm the recent observation by Kiers, Nussinov and Weiss that a precise measurement of the energies of the particles involved in the detection process would increase the coherence length. We also present a precise definition of the coherence length beyond which neutrinos cease to oscillate. PACS number(s): 13.15.+g, 14.60.Pq Typeset using REVTEX 1 A rigorous treatment of neutrino oscillations requires the study of the processes in which neutrinos are produced and detected [1–9]. Following the first attempts [1–3] to develop a proper use of the wave packet formalism, we have carried out detailed calculations in both quantum mechanics [4] and quantum field theory [5], with a quantitative derivation of the coherence length for neutrino oscillations. Due to the complexity of calculations, however, in [5] it was assumed that the sizes of the wave packets of the initial and final particles are all the same. Recently, Grimus and Stockinger [8] proved an elegant and very useful theorem that allows to simplify the wave packet treatment of neutrino oscillations. Taking advantage of this theorem, we have re-derived the neutrino oscillation formula in the general case in which all the particles involved in the production and detection processes have different wave packet sizes. Our new result confirms in the framework of a quantum field theoretical approach an interesting observation presented in [7] that an accurate measurement of the energies of the particles involved in the detection process leads to an increase of the coherence length for neutrino oscillations. Let us consider the flavor-changing process PI → PF + lα + να ↓ (να→νβ) νβ +DI → DF + l−β , (1) where PI and PF (DI and DF ) are the initial and final production (detection) particles. The process (1) takes place through the intermediate propagation of a neutrino, which oscillates from flavor α to flavor β (here α, β = e, μ, τ). In the process (1), the production and detection interactions are localized at the coordinates ( ~ XP , TP ) and ( ~ XD, TD), respectively. The form of the wave functions of the initial and final particles involved in the process (1) is determined by how the initial particles are prepared and how the final particles are detected. In the following, we will assume, for simplicity, Gaussian wave functions, whose wave packet forms in momentum and coordinate space are given in [5]. The wave packets in momentum space are assumed to be sharply peaked around their average momenta, which are denoted by 〈~ pk〉, where k = PI , PF , α,DI , DF , β. The corresponding average energies 〈Ek〉 are given by 〈Ek〉 = √ 〈~ pk〉2 +mk, where mk is the mass of the k particle. In order to make a realistic calculation, we will consider a different spatial width σxk for the wave packet of each particle involved in the process (1). Let us emphasize that the localization of the particles does not require necessarily the action of a man-made apparatus, but can be determined by the environment in which the process (1) takes place. Following the method presented in [5], the amplitude of the process (1) can be written as (see Eq.(8) of [5]) Aαβ(~ L, T ) ∝ ∑ a U αa Uβa ∫ dq (2π)4 UD q/ q2 −ma + iǫ VP exp [