Two mass eigenstates of definite energy combine into the flavour states and

PHYSICS · Multi-part · Fermi University Physics · Particle Physics & Quantum Fields · Flavour, neutrinos and CP violation · Flavour, neutrinos and CP violation

Symmetry and Flavour, Ch Neutrino masses, mixing and oscillations

Problem

Two mass eigenstates ∣ν1⟩,∣ν2⟩|\nu_1\rangle,|\nu_2\rangle of definite energy combine into the flavour states ∣να⟩=cos⁡θ∣ν1⟩+sin⁡θ∣ν2⟩|\nu_\alpha\rangle=\cos\theta|\nu_1\rangle+\sin\theta|\nu_2\rangle and ∣νβ⟩=−sin⁡θ∣ν1⟩+cos⁡θ∣ν2⟩|\nu_\beta\rangle=-\sin\theta|\nu_1\rangle+\cos\theta|\nu_2\rangle, for a mixing angle θ\theta. Work in natural units (ℏ=c=1\hbar=c=1) throughout part ((a)) . (a) The text takes both mass states with the same momentum. Take them instead with the same energy EE, so that each travels as eipixe^{ip_ix} with pi=E2−mi2p_i=\sqrt{E^2-m_i^2} and m1≪m2≪Em_1\ll m_2\ll E. Starting from ∣να(x)⟩=cos⁡θ eip1x∣ν1⟩+sin⁡θ eip2x∣ν2⟩|\nu_\alpha(x)\rangle=\cos\theta\,e^{ip_1x}|\nu_1\rangle+\sin\theta\,e^{ip_2x}|\nu_2\rangle, derive P(να→νβ)P(\nu_\alpha\to\nu_\beta) after a distance LL and show it is again sin⁡22θsin⁡2(Δm2L/4E)\sin^22\theta\sin^2(\Delta m^2L/4E) with Δm2=m22−m12\Delta m^2=m_2^2-m_1^2. ((b)) A real beam has a spread of L/EL/E, so the phase ϕ=Δm2L/4E\phi=\Delta m^2L/4E is not fixed. Take ϕ\phi uniformly distributed between ϕ0−Δ\phi_0-\Delta and ϕ0+Δ\phi_0+\Delta. Show that the average of sin⁡2ϕ\sin^2\phi is 12[1−cos⁡2ϕ0 sin⁡2Δ2Δ]\tfrac12\left[1-\cos2\phi_0\,\dfrac{\sin2\Delta}{2\Delta}\right], and state what it becomes for Δ→0\Delta\to0 and for Δ≫1\Delta\gg1. ((c)) Take the limit of many oscillation lengths (L→∞L\to\infty) so that sin⁡2(⋅)\sin^2(\cdot) averages to 12\tfrac12, and hence show the electron-neutrino survival probability averages to 1−12sin⁡22θ1-\tfrac12\sin^22\theta. Evaluate this for sin⁡2θ12=0.308\sin^2\theta_{12}=0.308 (NuFIT 6.0, 2024) and compare with the low-energy solar νe\nu_e survival probability of about 0.550.55 measured by Borexino for pppp and 7^7Be neutrinos, where matter effects are negligible.

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