Astroparticle Physics

6.6 Problems 1373. Radiation exposure due to solar neutrinos.a) Use (6.9) to work out the number of interactions ofsolar neutrinos in the human body (tissue densityϱ ≈ 1gcm −3 ).b) Neutrinos interact in the human body byν e + N → e − + N ′ ,where the radiation damage is caused by the electrons.Estimate the annual dose for a human underthe assumption that on average 50% of the neutrinoenergy is transferred to the electron.c) The equivalent dose is defined asH = (E/m) w R (6.87)(m is the mass of the human body, w R the radiationweighting factor (= 1 for electrons), [H ]=1Sv =1w R Jkg −1 ), and E the energy deposit in the humanbody). Work out the annual equivalent dose dueto solar neutrinos and compare it with the normalnatural dose of H 0 ≈ 2mSv/a.4. Neutrino oscillations. 15In the most simple case neutrino oscillations can be describedin the following way (as usual ¯h and c will be setto unity): in this scenario the lepton flavour eigenstatesare superpositions|ν e 〉=cos θ|ν 1 〉+sin θ|ν 2 〉 ,|ν µ 〉=−sin θ|ν 1 〉+cos θ|ν 2 〉of mass eigenstates |ν 1 〉 and |ν 2 〉. All these states areconsidered as wave packets with well-defined momentum.In an interaction, e.g., a ν e is assumed to be generatedwith momentum p, which then propagates as freeparticle, |ν e ; t〉 = e −iHt |ν e 〉. For the mass √eigenstatesone has e −iHt |ν i 〉=e −iE ν t i |ν i 〉 with E νi = p 2 + m 2 i ,i = 1, 2. The probability to find a muon neutrino after atime t isP νe →ν µ(t) =|〈ν µ |ν e ; t〉| 2 .a) Work out P νe →ν µ(t). After a time t the particle isat x ≈ vt = pt/E νi . Show that – under the assumptionof small neutrino masses – the oscillationprobability as given in (6.22) can be derived.15 This problem is difficult and its solution is mathematically demanding.