Neutrino Physics 2010: Assignment 3

Neutrino Physics 2010: Assignment 3
(Given 14/04/2010, To be submitted 03/05/2010)
1. Consider a short baseline experiment, where only one ∆m 2 is relevant
(single mass-squared dominance approximation). Take the standard
parameterization of U P MNS , assume no CP violation.
(a) Calculate the survival probability P µµ ≡ P νµ→ν µ
. Comparing it
with the standard two-neutrino oscillation form, determine the
“effective disappearance mixing angle” θµµ.
eff
(b) Calculate the conversion probability P µe ≡ P νµ→νe . Comparing
it with the standard two-neutrino oscillation form, determine the
“effective appearance mixing angle” θµe.
eff
(c) Show the equi-probability contours of P νµ→νe
in the plane of
log(∆m 2 /eV 2 ) vs. sin 2 2θ eff , with P µe = 0, 0.2, 0.5.
Use L = 10 MeV, E = 1 km. Indicate the effects of decoherence
at high ∆m 2 .
2. We want to calculate how ∆ and θ change in the presence of matter
with constant potential V c , using the perturbation theory technique.
This is a warm-up problem with two flavours.
(a) Calculate the effective Hamiltonian in the flavour basis,
H f = UH vac U † + V , where
H vac =
(
0 0
0 2∆
)
, V =
and U is the neutrino mixing matrix.
(
Vc 0
0 0
(b) Separate H f as H 0 + H 1 , where H 0 = H f (V c = 0).
(c) Find eigenvalues ɛ (0)
j and corresponding eigenvectors v (0)
j of H 0 .
(d) Using perturbation theory techniques, calculate the first order corrections
to the eigenvalues and eigenvectors of H 0 , i.e. calculate
ɛ (1)
j and v (1)
j .
(e) Find the net eigenvalues ɛ j = ɛ (0)
j + ɛ (1)
j , hence determine ∆ m .
(f) Find the net eigenvectors v j = v (0)
j + v (1)
j . Normalize v j (only keep
terms to first order in the small parameter V c ) and determine the
new mixing matrix U m . Hence find the mixing angle θ m .
(g) Compare the expressions for ∆ m and θ m with the known exact
expressions.
)
1

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