Neutrino Physics 2010: Assignment 3
Neutrino Physics 2010: Assignment 3
Neutrino Physics 2010: Assignment 3
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
3. We want to calculate the conversion probability P νe→ν µ<br />
in three flavours<br />
in matter with constant potential V c .<br />
(a) Calculate the effective Hamiltonian in the flavour basis,<br />
H f = UH vac U † + V , where<br />
H vac = 2∆ 31<br />
⎛<br />
⎜<br />
⎝<br />
0 0 0<br />
0 α 0<br />
0 0 1<br />
⎞<br />
⎟<br />
⎠ ,<br />
V = 2∆ 31<br />
⎛<br />
⎜<br />
⎝<br />
 0 0<br />
0 0 0<br />
0 0 0<br />
Here  ≡ V c/(2∆ 31 ), α ≡ ∆ 21 /∆ 31 , and U is the neutrino mixing<br />
matrix.<br />
(b) Separate H f as H 0 + H 1 , where H 0 = H f (θ 13 = 0, α = 0).<br />
(c) Find eigenvalues ɛ (0)<br />
j and corresponding eigenvectors v (0)<br />
j of H 0 .<br />
(d) Using perturbation theory techniques, calculate the first order corrections<br />
to the eigenvalues and eigenvectors of H 0 , i.e. calculate<br />
ɛ (1)<br />
j and v (1)<br />
j .<br />
(e) Determine the net eigenvalues ɛ j = ɛ (0)<br />
j + ɛ (1)<br />
j and the net eigenvectors<br />
v j = v (0)<br />
j + v (1)<br />
j .<br />
(f) Normalize v j (only keep terms to first order in the small parameters<br />
θ 13 and α) and determine the new mixing matrix U m .<br />
(g) Using the net conversion probability<br />
calculate P eµ .<br />
∑<br />
P αβ =<br />
(U m ) ∗ βj(U m ) αj e −iɛ jL<br />
∣<br />
∣<br />
j<br />
4. Let there be a sterile neutrino, which is heavier than all the active<br />
neutrinos, with ∆m 2 ≈ 1 eV 2 . The net 4 × 4 neutrino mixing matrix<br />
is given by<br />
⎛<br />
⎞<br />
U e1 U e2 U e3 U e4<br />
U<br />
U = µ1 U µ2 U µ3 U µ4<br />
⎜<br />
⎟<br />
⎝ U τ1 U τ2 U τ3 U τ4 ⎠<br />
U s1 U s2 U s3 U s4<br />
Calculate<br />
(a) P µµ relevant for atmospheric neutrinos. Simplify to the extent<br />
possible by neglecting / averaging out appropriate terms.<br />
(b) P µe relevant for the LSND experiment. Simplify to the extent<br />
possible by neglecting / averaging out appropriate terms.<br />
(c) P ee relevant for the KamLAND experiment. Simplify to the extent<br />
possible by neglecting / averaging out appropriate terms.<br />
2<br />
,<br />
⎞<br />
⎟<br />
⎠<br />
2