Mohamad-Ziad Charif - Antares

The eigenstate mixing can be written as following:|ν i 〉 = ∑ U in |ν n 〉 (2.22)n=1,2,3|ν n 〉 = ∑ Uin|ν ∗ i 〉 (2.23)i=1,2,3The parameters i and U in equations 2.22 and 2.23 refers to leptonic flavors e µand τ, and U being the leptonic mixing matrix[82].⎛U = ⎝c 13 c 12 c 13 s 12 s 13 e −iδ ⎞−s 12 c 23 − c 12 s 23 s 13 e iδ c 12 c 23 − s 12 s 23 s 13 e iδ s 23 c 13⎠The parameters in the U matrix are : c i j = cos(θ i j ), s i j = sin(θ i j ) where θ i j beingthe mixing angle between two leptonic flavor, and finally δ being the CP violatingphase.The Hamiltonian of neutrino propagation in vacuum contains solely the oscillationmatrix and can be written as followingH = 12E ·U · diag(0,∆m2 12,∆m 2 13) ·U † (2.24)The ∆m 2 terms in equation 2.24 are the neutrino squared mass differences. Thetemporal state of the neutrino can be written as |ν i (t)〉 = e −iHt |ν i (0)〉. While theneutrino oscillation does not depend whether it is in a vacuum or not, but whenpassing through matter ν e /barν e interacts with electrons found orbiting the atomsand altering the Hamiltonian in equation 2.24 and causing additional oscillations,this is called Mikheyev-Smirnov-Wolfenstein (MSW) effect[83, 84]. The newHamiltonian becomes:H = 12E ·U · diag(0,∆m2 12,∆m 2 13) ·U † ± diag( √ 2G F N e ,0,0) (2.25)The G F parameters in the new Hamiltonian in equation 2.25 is the Fermi couplingconstant and N e being the electron density.The value of the parameters θ i j , δ and ∆m 2 are obtained via global fits[85] fromthe different neutrino experiments. These values are :• θ 12 = 33.2 ◦ ± 4.9 ◦• θ 13 < 12.5 ◦• θ 23 = 45.0 ◦ ± 10.6 ◦ 30