Astroparticle Physics

6.2 Neutrino Astronomy 91For an assumed two-neutrino mixing of ν e and ν µ theweak eigenstates could be related to the mass eigenstates bythe following two equations:mixing angleν e = ν 1 cos θ + ν 2 sin θ,ν µ =−ν 1 sin θ + ν 2 cos θ.(6.19)The mixing angle θ determines the degree of mixing.This assumption requires that the neutrinos have non-zeromass and, in addition, m 1 ̸= m 2 must hold.In the framework of this oscillation model the probabilitythat an electron neutrino stays an electron neutrino, canbe calculated to be (see also Problem 4 in this section):(P νe →ν e(x) = 1 − sin 2 2θ sin 2 π x ), (6.20)L νwhere x is the distance from the source to the detector andL ν the oscillation lengthL ν =2.48 E ν [MeV](m 2 1 − m2 2 ) [eV2 /c 4 ] m . (6.21)The expression m 2 1 −m2 2 is usually abbreviated as δm2 . Equations(6.20) and (6.21) can be combined to give(P νe →ν e(x) = 1 − sin 2 2θ sin 2 1.27 δm 2 x )(6.22)E νwhere δm 2 is measured in eV 2 , x in km, and E ν in GeV.The idea of a two-neutrino mixing is graphically presentedin Fig. 6.13.For the general case of mixing of all three neutrinoflavours one obtains as generalization of (6.19)⎛⎝ ν ⎞ ⎛eν µ⎠ = U N⎝ ν ⎞1ν 2⎠ , (6.23)ν τ ν 3where U N is the (3 × 3) neutrino mixing matrix.The deficit of muon neutrinos can now be explained bythe assumption that some of the muon neutrinos transformthemselves during propagation from the point of productionto the detector into a different neutrino flavour, e.g., into tauneutrinos. The sketch shown in Fig. 6.13 demonstrated thatforanassumedmixingangleof45 ◦ all neutrinos of a certaintype have transformed themselves into a different neutrinoflavour after propagating half the oscillation length.Fig. 6.13Oscillation model for ν e –ν µmixing for different mixing anglesmixing matrixdeficit of muon neutrinos