The Glashow resonance
The neutrino telescopes IceCube, Baikal-GVD, KM3NeT and P-ONE aim to identify astrophysical sources of ultra-high-energy neutrinos.
(i) Without neglecting \(q^2\) compared with \(m_W^2\) in the \(W\)-boson propagator, compute the cross sections of the deep inelastic reactions \(\nu_\ell + N\to \ell^-+X\) and \(\bar{\nu}_\ell + N\to \ell^++X\) (\(X\) stands for hadrons). Plot them as functions of the neutrino energy. Use, for example, the lhapdf library for the quark distributions.
(ii) Estimate the absorption length of neutrinos and antineutrinos. Find the energy above which the Earth is no longer transparent to them.
(iii) Compute the cross section of the reaction \(\bar{\nu}_e+e^-\to W^-\) (the Glashow resonance). Why is it called so? Compare this cross section with the cross sections from (i). Is there a similar resonance for \(\nu_e\)? Why?
(iv) Compute the cross sections of the reactions \(\bar{\nu}_e+e^-\to \bar{\nu}_\ell+\ell^-\) and \(\bar{\nu}_e+e^-\to X\) in the region of the Glashow resonance. Compare them with the cross sections from (i) on one plot.
(v) What can the observation of the Glashow resonance tell us? Give a short review of the experimental status.
(vi) Estimate the volume of a water telescope that would detect one such event per year.