The matter-modified mixing angle for a two-flavour neutrino system in matter of…

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

The matter-modified mixing angle for a two-flavour neutrino system in matter of electron density

n_e

satisfies

\tan2\theta_m=\frac{\sin2\theta}{\cos2\theta-2\sqrt2G_Fn_eE/\Delta m^2},

with

V\equiv\sqrt2G_Fn_e=7.6\times10^{-14}\,\mathrm{eV}\times Y_e\rho[\mathrm{g\,cm^{-3}}]

. ((a)) Show that the resonance condition (

\theta_m=45^\circ

) gives a resonance energy

E_{\rm res}=\Delta m^2\cos2\theta/(2V)

. ((b)) Apply this to the

1

3

sector,

\Delta m^2_{31}=2.513\times10^{-3}\,\mathrm{eV^2}

and

\sin^2\theta_{13}=0.0222

(NuFIT 6.0, 2024), using the solar core value

V=7.6\times10^{-12}\,\mathrm{eV}

(for

Y_e\rho\approx100\,\mathrm{g\,cm^{-3}}

). Find

E_{\rm res}

\mathrm{MeV}

. ((c)) The

^8

B solar neutrino spectrum ends near

16\,\mathrm{MeV}

, and the rare hep neutrinos, the highest-energy solar component, reach about

18.8\,\mathrm{MeV}

. Explain what your result in (b) implies for a

1

3

resonance inside the Sun, and hence why the energy-dependent MSW transition (resonance near

1.9\,\mathrm{MeV}

) is governed by the

1

2

sector, with

\theta_{13}

entering only as a small correction.

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