Relic neutrinos
According to the modern cosmological model of the birth and evolution of the Universe, we are surrounded by the so-called relic neutrinos. Their temperature today is \(T_\nu\approx 1.95\) K (\(kT_\nu\approx 1.7\cdot 10^{-4}\) eV), which makes them very hard to observe. Nevertheless, let us think of something. A Nobel Prize is guaranteed!
(i) Compute the cross section of the reaction \(\nu_e +n\to p+e^-\) at the energy of a relic neutrino. Compute the mass of free neutrons needed to detect 100 such events per year. Do we have a suitable source of free neutrons of the required mass?
If not, suggest an atomic nucleus that could be used instead of free neutrons.
(ii) Consider another project: a particle accelerator. If an electron or a muon is accelerated in it to a high enough energy, the lepton can disappear after interacting with a relic antineutrino, producing a pion: \(\bar{\nu}_\ell +\ell^-\to \pi^-\).
a) Compute the lepton energy needed for this reaction to occur.
b) Compute the corresponding cross section.
c) Estimate the luminosity of the accelerator needed to detect a statistically significant number of pions.
(iii) Is there a hidden Glashow resonance in the reaction \(\bar{\nu}_\ell +\ell^-\to \pi^-\)? (Discuss this idea with the colleague working on that project.) What could follow from it?
(iv) Propose your own idea for detecting relic neutrinos and support it with calculations.