Summer semester 2016
1
Excercises zu Neutrino Physics
1. Let pi (i = 1, 2) be 4-momentum vectors with p2i = m2i 6= 0. Prove that then the
following relation holds: p1 · p2 = m1 m2 ⇔ m1 p2 = m2 p1 .
1 point
2. Let ` be a charged particle. Demonstrate that in vacuum the process ` → ` + γ is
kinematically impossible.
1 point
3. What is the minimal energy of a muon (in MeV) so that it produces Cherenkov
light in water?
1 point
4. Compute the energy Ee of the positron in inverse β-decay as a function of the
neutrino energy Eν and the angle θ between the direction of the neutrino and the
positron.
2 points
Hint: In order to facilitate writing, use the notation ∆0 ≡ (m2n − m2p − m2e )/(2mp ),
x ≡ Eν /mp .
5. In the Weyl basis the Dirac gamma matrices are
µ
γ =
0 σµ
σ̄ µ 0
!
with σ µ = (1, ~σ ),
σ̄ µ = (1, −~σ ).
Show that in this basis the solutions of (p/ − m)u = 0 and (p/ + m)v = 0 with the
4-momentum vector p (p2 = m2 ) are given by
!
√
p·σξ
√
p · σ̄ ξ
u(p, ξ) =
and v(p, ξ) =
!
√
p·ση
√
,
− p · σ̄ η
respectively, with arbitrary ξ, η ∈ C2 .
1 point
6. Compute ūu and v̄v for the normalization kξk = kηk = 1.
1 point
7. Prove the Fierz identity
(γ µ γL )ab (γµ γL )cd = − (γµ γL )cb (γ µ γL )ad ,
where γL = 21 (1 − γ5 ) and the indices a, b, c, d are Dirac indices. Conclude furthermore from this Fierz identity that
ψ̄1 γ µ γL ψ2 ψ̄3 γµ γL ψ4 = ψ̄3 γµ γL ψ2 ψ̄1 γ µ γL ψ4
holds for Dirac fields ψj (j = 1, 2, 3, 4).
2 points
A
µ
µν
µ
Hint: Use the fact that the matrices Γ = 1, γ , σ (µ < ν), γ γ5 , γ5 form a basis
in the space of complex 4 × 4 matrices. Therefore, one can make the ansatz
(γ µ γL )ab (γµ γL )cd =
X
A
(cA )cb ΓA
ad
and determine the coefficients (cA )cb by multiplying the ansatz with (ΓB )da and
summing over a, d.
Summer semester 2016
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8. Consider the current j µ (x) = : ψ̄(x)γ µ γL ψ(x) :, where ψ is a free Dirac field. Compute the expectation value hf, 1|j µ (x)|f, 1i in the one-particle state
|f, 1i =
Z
d3 p f (~p ) b† (p, s)|0i with
Z
d3 p |f (~p )|2 = 1.
Now assume that the spin state is independent of p~ and corresponds to ξs ∈ C2 .
Then show that in the non-relativistic limit, where f = 0 for |~p | > p0 and p0 m,
this expectation value approaches
1
hf, 1|j (x)|f, 1i −→ |fˆ(x)|2
2
NR
µ
with
1
−~n
!
1 Z 3 i~p·~x
d p e f (~p ) and ~n = ξs†~σ ξs .
(2π)3/2
fˆ(~x ) =
2 points
Note: The Dirac field is given by
ψ(x) =
d3 p
XZ
q
(2π)3 2E
s
u(p, s)e−ip·x b(p, s) + v(p, s)eip·x d† (p, s) .
Therefore, the creation and annihilation operators feature the anticommutation relations {b(p, s), b† (p0 , s0 )} = δss0 δ (3) (~p − p~ 0 ), etc.
†
9. Show that ~n = ξ †~σ ξ is a unit vector if kξk = 1 and that ξ⊥
~σ ξ⊥ = −~n for ξ⊥ ⊥ ξ.
1 point
10. Let S be a Lorentz transformation on Dirac spinors, C the charge conjugation matrix
and σµν = 2i [γµ , γν ]. Show that
T
Cσµν
C −1 = −σµν
and S T C −1 S = C −1 .
1 point
11. The charge conjugation operation on spin 1/2 fields is defined as ψ c = Cγ0T ψ ∗ . Show
that, if γL ψ = ψ, then γR ψ c = ψ c and vice versa.
1 point
12. Show that ψ cc = ψ.
1 point
13. Show that ψ T C −1 = −ψ c and use this relation to demonstrate the identity
χTL C −1 χL + H.c. = −χ̄χ
where χ = χL + (χL )c .
14. Show that from (p/ − m)u = 0 it follows that (p/ + m)uc = 0.
1 point
1 point
Summer semester 2016
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15. Show that the Euler–Lagrange equation to the action
S=
Z
1 d4 x L with L = iχ̄L γ µ ∂µ χL + m χTL C −1 χL + H.c.
2
is the Dirac equation for the Majorana field χ = χL + (χL )c .
2 points
16. Show that every 2 × 2 unitary matrix U can be written as
iα̂
U =e
!
cos θ sin θ
− sin θ cos θ
eiβ̂
with diagonal matrices of phase factors eiα̂ and eiβ̂ .
1 point
17. Consider neutrino oscillations with three flavours. Let δE be the uncertainty in
the measurement of the neutrino energy, Ē the mean neutrino energy and Losc the
oscillation length with respect to the smallest mass-squared difference. Argue that
for L Losc Ē/δE all oscillating terms in the oscillation probability Pνα →νβ are
averaged out and the effective oscillation probability is given by
P̄να →νβ =
3
X
|Uαj |2 |Uβj |2 .
j=1
1 point
18. Assume that, apart from external phase factors, the lepton mixing matrix is given
by
 2

√
√1
0
6
3


√1
√1
√1  .
−
−
U =

6
3
2 
− √16 √13 √12
Assume further that, in a very distant galaxy with an active galactic nucleus, neutrino fluxes are produced with ratios (νe ) : (νµ ) : (ντ ) = 1 : 2 : 0. Compute with
these assumptions the neutrino flux ratios measured on earth.
2 points
19. In the computation of (0νββ)-decay via Majorana neutrinos one is lead to the
T
effective neutrino propagator M(x1 − x2 ) = h0|T νeL (x1 )νeL
(x2 )|0i. Show that,
P
because νeL = j Uej νjL and the mass eigenfields νj are Majorana fields, the effective
propagator is given by
M(x1 − x2 ) = −i
X
(Uej )2 mj
j
Z
d4 p e−ip·(x1 −x2 ) 1 − γ5
C.
(2π)4 p2 − m2j
2
2 points
20. A real symmetric 2 × 2 matrix M = (mij ) is diagonalized by an orthogonal matrix
R=
cos θ sin θ
− sin θ cos θ
!
.
Summer semester 2016
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Show that the angle θ is determined by
tan(2θ) =
2m12
.
m22 − m11
1 point
21. Consider the solar survival probability Pνe →νe and allow for a non-zero probability
Pc for level crossing |ψm1 i → |ψm2 i. Show that in this case the adiabatic formula
gets modified in the following way:
Pνe →νe =
1
1
1
(1 + cos 2θm (x0 ) cos 2θ) → +
− Pc cos 2θm (x0 ) cos 2θ.
2
2
2
2 points
22. Let B be an arbitrary complex m × n matrix such that all eigenvalues of B † B are
smaller than one. Show that in this case
!
√
1m − BB † √ B
W =
−B †
1n − B † B
is an (m + n) × (m + n) unitary matrix. In W , 1m and 1n are m × m and n × n
unit matrices, respectively.
2 points