Past exam of the particle physics course of the University of Oxford / 2021 / 3 / d

For a positive electron neutrino matter potential and the normal ordering implicit in positive $K$, choose $0\le {\theta }_m\le \pi /2$ by continuous diagonalization of the matter Hamiltonian. With $D=[(\cos 2\theta -K\rho {)}^2+{\sin }^{2}2\theta {]}^{1/2}$, the quadrant is fixed byAt high density $\cos 2{\theta }_m\to -1$, soTaking a principal-value arctangent alone and concluding ${\theta }_m\to 0$ would select the wrong branch. The production state is the second matter propagation eigenstate, rather than literally the vacuum mass eigenstate while still inside the dense medium.

Assume the solar density decreases slowly enough for adiabatic matter-enhanced neutrino conversion, and neglect subsequent Earth matter effects. The state follows ${\nu }_{2m}$ continuously into the vacuum mass eigenstate ${\nu }_2$. Since $⟨{\nu }_e|{\nu }_2⟩=\sin \theta$,This is substantially below the phase-averaged vacuum result $0.75$. There is no interference between two populated vacuum mass eigenstates in this ideal adiabatic limit. More generally, after phase averaging, $P_{ee}=\frac{1}{2}+\frac{1}{2}\cos 2\theta \cos 2{\theta }_m$ at production if the evolution is adiabatic. In the high-density limit, a nonadiabatic level-jump probability $P_c$ instead gives $P_{ee}=(1-P_c){\sin }^2\theta +P_c{\cos }^2\theta$. Thus the supplied core-density condition alone does not guarantee the numerical Earth result: the usual solar adiabatic assumption is also needed. These are the stipulated two-flavor parameters, not a precision prediction for the measured three-flavor solar spectrum.