summary of the talks before Oscillation in vacuum Oscillation in matter Results References
Neutrino Oscillations-
Theory (in vacuum and in matter)
Markus Wagner
5.1.2006
Markus Wagner
Neutrino Oscillations - Theory (in vacuum and in matter)
summary of the talks before Oscillation in vacuum Oscillation in matter Results References
Contents
Summary of the earlier talks
Theory of neutrinos with mass mν 6= 0
Oscillations in vacuum
Oscillations in matter
Results
Markus Wagner
Neutrino Oscillations - Theory (in vacuum and in matter)
summary of the talks before Oscillation in vacuum Oscillation in matter Results References
Characteristics of neutrinos
More and more questions
So far: mν = 0, but what are theconsequences if neutrinos arenot massless?
Solar neutrino problem: can thistheory solve the problem?
Do massive neutrinos contribute significants to dark matter?
Markus Wagner
Neutrino Oscillations - Theory (in vacuum and in matter)
summary of the talks before Oscillation in vacuum Oscillation in matter Results References
Characteristics of neutrinos
mν = 0
3 flavours: ντ , νµ, νe
left-handed neutrinos (spin opposite to momentum)
right-handed antineutrinos (spin parallel to momentum)
vν = c
Markus Wagner
Neutrino Oscillations - Theory (in vacuum and in matter)
summary of the talks before Oscillation in vacuum Oscillation in matter Results References
Characteristics of neutrinos
mν 6= 0
3 flavours: ντ , νµ, νe
left-handed as well as right-handed neutrinos and antineutrinos
ρν ≈ 0.2ρdark?
vν . c
Markus Wagner
Neutrino Oscillations - Theory (in vacuum and in matter)
summary of the talks before Oscillation in vacuum Oscillation in matter Results References
Theory in vacuum
Theory in vacuum
α states of flavour: |να〉 with α = µ, τ, e
i states of mass: |νi 〉 with i = 1, 2, 3
states of mass 6= states of flavourSuperposition: |να〉 =
∑i Uiα|νi 〉
Markus Wagner
Neutrino Oscillations - Theory (in vacuum and in matter)
summary of the talks before Oscillation in vacuum Oscillation in matter Results References
Theory in vacuum
Theory in vacuum
|να〉 =∑
i Uiα|νi 〉 =⇒ |νi 〉 =∑
α(U†)iα|να〉
highly relativistic: E ≈ p, pi =√
E 2 −M2i ≈ E − M2
i
2E
|vi (t)〉 = e−iHt+ipix |vi (x = 0, t = 0〉 highlyrelativistic= e−iEt+ipix |vi 〉 =
e−iE(t−x)−iM2
i2E x |vi 〉 ∝ e−i
M2i
2E x |vi 〉projection: |vα(t, x = L)〉 = e−iHt+piL|vα〉 =
∑i e
−iHt+ipiL|vi 〉〈vi |vα〉
∝∑
α′∑
i e−i
M2i
2E L|vα′〉〈vα′ |vi 〉〈vi |vα〉 =∑α′ |vα′〉
(∑i e
−iM2
i2E LUα′iU
†iα
)
Markus Wagner
Neutrino Oscillations - Theory (in vacuum and in matter)
summary of the talks before Oscillation in vacuum Oscillation in matter Results References
Theory in vacuum
Theory in vacuum
Transition probability
P(α → α′) = |〈vα′ |vα〉|2 =
∣∣∣∣∑i e−i
M2i
2E LUα′iU†iα
∣∣∣∣2
Markus Wagner
Neutrino Oscillations - Theory (in vacuum and in matter)
summary of the talks before Oscillation in vacuum Oscillation in matter Results References
Example
Example: 2 flavour oscillations
Mixing matrix:
U =
(cosθ sinθ−sinθ cosθ
)
Markus Wagner
Neutrino Oscillations - Theory (in vacuum and in matter)
summary of the talks before Oscillation in vacuum Oscillation in matter Results References
Example
Example: 2 flavour oscillations
Hα = UH iU† with H i =M2
i
2E δij
e−iHαt−ipix |v(t = 0, x = 0)〉 = |v(t, x)〉, ct ≈ L, c = 1
P(νe → νµ) = |〈ve |vµ〉|2 = sin2(2θ)sin2(
δM2
4LE
)P(νe → νe) = 1− |〈ve |vµ〉|2 = 1− sin2(2θ)sin2
(δM2
4LE
)δM2 = M2
2 −M21
oscillation length: L0 = 4πEδM2
L � L0: no oscillation
L � L0: P(νe → νµ) = 0.5sin2(2θ), but why?
Markus Wagner
Neutrino Oscillations - Theory (in vacuum and in matter)
quantum mechanics
summary of the talks before Oscillation in vacuum Oscillation in matter Results References
EPR
Π+ decay
Π+ → µ + νµ
so far: only the νµ was accounted for
but the µ’s were not considered
we need more information about the process (where it happens)
one has to detect both (µ, νµ)one has to know the location of the decay
otherwise one has to average =⇒ no oscillation!
Markus Wagner
Neutrino Oscillations - Theory (in vacuum and in matter)
summary of the talks before Oscillation in vacuum Oscillation in matter Results References
Results
Example: 2 flavour oscillations
Probability
P(νe → νe) = |〈ve |ve〉|2 = 1−(
sin2(2θ)sin2
(δM2
4
L
E
))
Markus Wagner
Neutrino Oscillations - Theory (in vacuum and in matter)
summary of the talks before Oscillation in vacuum Oscillation in matter Results References
3 flavour oscillations
3 flavour oscillations
U1 =
1 0 00 cosθ23 sinθ23
0 −sinθ23 cosθ23
U2 =
cosθ13 0 sinθ13e−iδ
0 1 0−sinθ13e
iδ 0 cosθ13
U3 =
cosθ12 sinθ12 0−sinθ12 cosθ12 0
0 0 1
U = U1U2U3
The phase δ includes the CP-effects!
Markus Wagner
Neutrino Oscillations - Theory (in vacuum and in matter)
summary of the talks before Oscillation in vacuum Oscillation in matter Results References
Theory of neutrinos in matter
Theory for neutrinos in matter
What is changing?
only νe can interact with electrons in non-neutral current process
neutral currents yield an overall phase factor
Markus Wagner
Neutrino Oscillations - Theory (in vacuum and in matter)
summary of the talks before Oscillation in vacuum Oscillation in matter Results References
Theory of neutrinos in matter
Theory of neutrinos in matter
MSW-effect: Mikheyev, Smirnov and Wolfenstein
Hαvac = UH iU† with H i =
M2i
2E δij
Hαmat =
√2GFNeδijδj1e
e−i(Hαvac+Hα
mat)t−ipix |v(t = 0, x = 0)〉 = |v(t, x)〉,ct ≈ L, c = 1
new basis: vi,mat
Markus Wagner
Neutrino Oscillations - Theory (in vacuum and in matter)
summary of the talks before Oscillation in vacuum Oscillation in matter Results References
Example
Example: 2 flavour oscillations
New mixing angle
sin2(2θmat) =sin2(2θ)(
2√
2GF NeEδM2 − cos(2θ)
)2
+ sin2(2θ)
New mass eigenvalues
δM2mat = δM2
√√√√(2√
2GFNeE
δM2− cos(2θ)
)2
+ sin2(2θ)
Markus Wagner
Neutrino Oscillations - Theory (in vacuum and in matter)
summary of the talks before Oscillation in vacuum Oscillation in matter Results References
Example
Example: 2 flavour oscillations
New probability
P(νe → νµ) = |〈ve |vµ〉|2 = sin2(2θmat)sin2
(δM2
mat
4
L
E
)
P(νe → νe) = 1− |〈ve |vµ〉|2 = 1− sin2(2θmat)sin2
(δM2
mat
4
L
E
)
Markus Wagner
Neutrino Oscillations - Theory (in vacuum and in matter)
summary of the talks before Oscillation in vacuum Oscillation in matter Results References
MSW
Level Crossing
Mixing angle
sin2(2θmat) =sin2(2θ)(
2√
2GF NeEδM2 − cos(2θ)
)2
+ sin2(2θ)
Maximum at(
2√
2GF NeEδM2 − cos(2θ)
)2
= 0
⇒ Ne = cos(2θ)δM2
2√
2GF E∼ ρ
L0,m = L0sin(2θm)sin(2θ)
Markus Wagner
Neutrino Oscillations - Theory (in vacuum and in matter)
summary of the talks before Oscillation in vacuum Oscillation in matter Results References
MSW
Neutrino emmitted from the sun
νe = blue
νµ = red
ντ = yellow
Markus Wagner
Neutrino Oscillations - Theory (in vacuum and in matter)
summary of the talks before Oscillation in vacuum Oscillation in matter Results References
MSW
Day-Night difference
Markus Wagner
Neutrino Oscillations - Theory (in vacuum and in matter)
summary of the talks before Oscillation in vacuum Oscillation in matter Results References
Mass and angles
Constraints on parameters
LSND: Liquid Scintillator NeutrinoDetector at Los Alamos
LSND: conflict with other experiments!
SK atmos: atmospheric neutrinos
solar: solar neutrinos
Markus Wagner
Neutrino Oscillations - Theory (in vacuum and in matter)
summary of the talks before Oscillation in vacuum Oscillation in matter Results References
Mass and angles
Resulting Parameters
|δM231| = 2, 0+0,8
−0,6 × 10−3eV 2, (sign of δM231 undetermined)
δM221 = 7, 1+2,1
−1,1 × 10−5eV 2
sin2(2θ23) = 1, 00+0.00−0,13, [θ23 = (45± 11)◦]
sin2(2θ12) = 0, 82+0.10−0,10, [θ12 = (33± 4)◦]
sin2(2θ13) = 0, 17, [θ13 < 12◦]
δ undetermined
Markus Wagner
Neutrino Oscillations - Theory (in vacuum and in matter)
summary of the talks before Oscillation in vacuum Oscillation in matter Results References
References
V. Barger, D. Marfatia, K. Whisnant - High Energy Physics -Phenomenology, abstract hep-ph/0308123
Chris Waltham - Teaching Neutrino oscillations
Haxton, Holstein - Neutrino physics
Bahcall, Kraslev, Smirnov - Where do we stand with solar neutrinooscillations? - Phys. Rev. D Vol. 58, 096016
Smirnov - Neutrino masses and mixing - hep-ph/0512303
Markus Wagner
Neutrino Oscillations - Theory (in vacuum and in matter)