Lorentz Invariance of Neutrino Oscillations

It is shown that, in spite of the appearances, the standard expression for the oscillation probability of ultrarelativistic neutrinos is Lorentz invariant. Flavor is the quantum number that distinguishes the different types of quarks and leptons. It is a Lorentz-invariant quantity. For example, an electron is seen as an electron by any observer, never as a muon. Therefore, the probability of flavor neutrino oscillations must be Lorentz invariant (as already remarked in Ref. [1]). However, it may appear that the standard expression for the probability of να → νβ transitions in vacuum, Pνα→νβ(E, L) = ∑ k |Uαk||Uβk| + 2Re ∑ k>j U∗ αkUβkUαjU ∗ βj e −i∆φkj(E,L) , (1) with the phase differences ∆φkj(E, L) = ∆mkj L 2 E , (2) is not Lorentz invariant. In Eq. (1) U is the mixing matrix that connects the flavor neutrino fields να (α = e, μ, τ) with the massive neutrino fields νk (k = 1, 2, 3) , L is the distance between the neutrino source and the neutrino detector, and E is the neutrino energy (see Refs. [2, 3]). In Eq. (2) ∆mkj ≡ mk −mj is the difference between the squared-masses of νk and νj. Let us assume that Eq. (1) is valid in the inertial system O with time axis t and the x axis in the direction of neutrino propagation. Consider another inertial system O′ with axes x′, t′ moving with respect to O with velocity v in the x direction. The Lorentz transformations of space and time intervals are ∆x′ = γ (∆x− v ∆t) , (3a) ∆t′ = γ (−v ∆x + ∆t) , (3b) In the simplest case of two-neutrino mixing with να = cosθν1 +sinθν2 and νβ = − sin θν1 +cosθν2, the transition probability in Eq. (1) reduces to the well known expression Pνα→νβ (E, L) = sin 2 2θ sin ( ∆m2L 4E ) . If the number of massive neutrinos is N > 3, the indices k, j run from 1 to N and α, β = e, μ, τ, s1, . . . , sN−3, with the indices s1, . . . , sN−3 denoting N − 3 sterile neutrino fields (see Refs. [2, 3]). We use natural units, with c = 1.