Neutrino Oscillations Theory (in vacuum and in matter)theorie.ikp.physik.tu-darmstadt.de/nhc/pages/lectures/rhiseminar05-06/wagner.pdf · Neutrino Oscillations - Theory (in vacuum

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)


Contents

Summary of the earlier talks

Theory of neutrinos with mass mν 6= 0

Oscillations in vacuum

Oscillations in matter

Results

Markus Wagner



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




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




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



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



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



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



Example

Example: 2 flavour oscillations

Mixing matrix:

U =

(cosθ sinθ−sinθ cosθ

)

Markus Wagner



Example


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


quantum mechanics


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



Results


Probability

P(νe → νe) = |〈ve |ve〉|2 = 1−(

sin2(2θ)sin2

(δM2

4

L

E

))

Markus Wagner



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



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





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



Example


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



Example


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



MSW

Level Crossing

Mixing angle

sin2(2θmat) =sin2(2θ)(

2√


)2

+ sin2(2θ)

Maximum at(

2√


)2

= 0

⇒ Ne = cos(2θ)δM2

2√

2GF E∼ ρ

L0,m = L0sin(2θm)sin(2θ)

Markus Wagner



MSW

Neutrino emmitted from the sun

νe = blue

νµ = red

ντ = yellow

Markus Wagner



MSW

Day-Night difference

Markus Wagner



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



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



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


Documents

Neutrino Oscillations Theory (in vacuum and in matter)theorie.ikp.physik.tu-darmstadt.de/nhc/pages/lectures/rhiseminar05-06/wagner.pdf · Neutrino Oscillations - Theory (in vacuum