1.3.6. Questions about Halo Problem¶

1.3.6.1. Differential Cross Section¶

../../_images/neutrino-scattering.png — Fig. 1.188 Scattering of neutrinos off electrons.¶

Giunti derived the general form of differential cross section for all possible neutrino scatterings off electrons, equation 5.19 in his book [Giunti2007]. We plugin differential forms of Mandelstam variables to get the differential cross section related to neutrino scattering angles,

\frac{d σ}{d (2 E_{ν} E_{ν}^{'} \cos θ_{ν})} = \frac{G_{F}}{π} (g_{1}^{2} + g_{2}^{2} (1 - \frac{E_{ν}^{'} \cos θ_{ν}}{m_{e}}) - g_{1} g_{2} \frac{E_{ν}^{'} \cos θ_{ν}}{E_{ν}}),

where $E_{ν} = p_{ν}$ and $E_{ν}^{'} = p_{ν}^{'}$ are the energies of neutrinos before and after scattering.

Applying conservation of four momentum, we have

p_{ν}^{'} = \frac{m_{e} (m_{e} + p_{ν})}{m_{e} + p_{ν} - p_{ν} \cos θ_{ν}} .

We are interested in the energy scale that $p_{ν} ≫ m_{e}$ . The energy of scattered neutrinos equations simplify to

p_{ν}^{'} = \frac{m_{e}}{1 - \cos θ_{ν}} .

Plugin this into the differential cross section and keep only 0 orders of $m_{e} / E_{ν}$ , we have

\frac{d σ}{d \cos θ_{ν}} = \frac{2 E_{ν} m_{e}}{(1 - \cos θ_{ν})^{2}} \frac{G_{F}}{π} (g_{1}^{2} + g_{2}^{2} \frac{1 - 2 \cos θ_{ν}}{1 - \cos θ_{ν}})

For neutral current, Giunti shows that the values of $g_{i}$ are [Giunti2007]

\begin{aligned} g_{1} & = g_{2} = - 0.27 \\ {\bar{g}}_{1} & = {\bar{g}}_{2} = 0.23, \end{aligned}

where bar indicates the values for antineutrinos.

Giunti’s formula

The differential cross section formula 5.29 in [Giunti2007] shows

\frac{d σ}{d \cos θ_{e}} = σ_{0} \frac{4 E_{ν}^{2} (m_{e} + E_{ν})^{2} \cos θ_{e}}{[(m_{e} + E_{ν})^{2} - E_{ν}^{2} \cos^{2} θ_{e}]^{2}} [g_{1}^{2} + g_{2}^{2} {(1 - \frac{2 m_{e} E_{ν} \cos^{2} θ_{e}}{(m_{e} + E_{ν})^{2} - E_{ν}^{2} \cos^{2} θ_{e}})}^{2} - g_{1} g_{2} \frac{2 m_{e}^{2} \cos^{2} θ_{e}}{(m_{e} + E_{ν})^{2} - E_{ν}^{2} \cos^{2} θ_{e}}] .

We are interested in supnernova neutrinos whose energy is usually larger than mass of electrons. We set $m_{e} / E_{ν} \to 0$ .

\frac{d σ}{d \cos θ_{e}} = σ_{0} \frac{4 E_{ν}^{2} \cos θ_{e}}{[1 - \cos^{2} θ_{e}]^{2}} .

We need a relation between $θ_{e}$ and $θ_{ν}$ . The way to derive it is to use conservation of four momentum.

\begin{aligned} E_{ν}^{'} + \sqrt{p_{e}^{' 2} - m_{e}^{2}} & = m_{e} + E_{ν} \\ p_{ν}^{'} \sin θ_{ν} + p_{e}^{'} \sin θ_{e} & = 0 \\ p_{ν}^{'} \cos θ_{ν} + p_{e}^{'} \cos θ_{e} & = p_{ν} . \end{aligned}

We imediately notice that for backward scattering, $θ_{e} = π + θ_{ν}$ .

Giunti2007(1,2,3): Guinti & Kim, Fundamentals of Neutrinos and Astrophysics

1.3.6.2. MISC¶

Reflection coefficients for neutrinos and anti-neutrinos are different.