2018-12-12 LeifLönnblad Neutrinooscillations

Neutrino oscillations

Leif Lönnblad

Institutionen för Astronomi och teoretisk fysik Lunds Universitet

2018-12-12

The massless meutrino

The neutrino is assumed to be massless in the SM.

We know that the mass is small from the Curie plot in β decays.

Remember weak decays

d Γµ= 2G²_Fm³_µ

(2π)⁵ π²d |~k |d |~q|

with |~k |, |~q| < m_µ/2.

(modified by masses)

Previous limits on the neutrino mass:

I m_νe <3 eV (Tritium beta decay)

I m_νµ <0.19 MeV (π⁺→ µ⁺ν_µ)

I mντ <18 MeV (τ → ντ+nπ)

I P m_ν . 0.23 eV (CMB)

Masive neutrino in the Lagragian

We could simply introduce a right-handed neutrino L = . . . + g_fν¯LφνR

and get a (Dirac) mass term m_Dν¯_Lν_R.

I Zero charge

I Weak singlet (T₃=0)

I Colour singlet

I . . . interacts only with gravity and with the higgs field

Masive neutrino in the Lagragian

We could simply introduce a right-handed neutrino L = . . . + g_fν¯LφνR

and get a (Dirac) mass term m_Dν¯_Lν_R.

I Zero charge

I Weak singlet (T₃=0)

I Colour singlet

I . . . interacts only with gravity and with the higgs field

Majorana mass term

What if the neutrino is its own anti-particle?

L = . . . + mMν¯Lν_L^c where ν_L^c is right-handed.

We would get fermion number violation and Neutrino-less double-beta decays

(n → pW⁻)

(n → pW⁻)

e⁻

e⁻

¯ νe

νe

?

(A, Z ) → (A, Z + 1) + e⁻+ ¯ν_e

→ (A, Z + 1) + e⁻+ ν_e

→ (A, Z + 2) + e⁻e⁻

Not seen: m_M<0.2 eV

Effects of massive neutrinos

Consider transversely polarized light (| ⊥i) in an optically active medium.

The medum has different refraction index (∆n) for right (|+i) and left (|−i) handed circular polarisations.

After a polarizer we have at t = 0

| ⊥_θi = (e^iθ|+i + e^−iθ|−i)/√ 2

Propagating (e^{ix ·p}) a distance L, The ± components will have travelled a time t±, with δt = ∆nL/c, so we will have a relative phase shift δθ = E ∆nL/c, changing the transverse polarization angle.

What if the weak eigenstates of the neutrinos are different from the mass eigenstates?

Denoting the mass eigenstates ν_i we get (c.f. d-type quarks)



 νe

νµ

ν_τ



=U



 ν1

ν₂ ν₃





U is the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix

U =





1 0 0

0 c23 s23

0 −s₂₃ c23



×





c13 0 s13e^−iδ

0 1 0

−s₁₃e^iδ 0 c₁₃



×





c12 s12 0

−s₁₂ c12 0

0 0 1





with s_ij =sin θ_ij and c_ij =cos θ_ij. (c.f. CKM)

Now consider a neutrino would be produced (eg. in π⁺ → µ⁺νµ) in a weak eigenstate at time 0:

νµ(0) = U_µ,1ν1(0) + U_µ,2ν2(0) + U_µ,3ν3(0)

but then we will have a propagation of the mass eigenstates, so that

νµ(t) = U_µ,1ν1(0)e^iE1t +U_µ,2ν2(0)e^iE2t +U_µ,3ν3(0)e^iE3t

Let’s simplify and only use two generations, which means that we can describe the mixing matrix with only one angle, α

 ν_e νµ



=

 cos α sin α

− sin α cos α

  ν₁ ν2



Now, if we start out with a pure νewe have νe(0) = ν₁(0) cos α + ν₂(0) sin α and

νµ(0) = −ν₁(0) sin α + ν₂(0) cos α = 0

but it is the mass eigenstates which propagates, so after some time t we will have

ν_µ(t) = −ν₁(0)e^−iE1tsin α + ν₂(0)e^−iE2tcos α and there is a probability that the νehas turned into a νµ

|hν_e(0)|ν_µ(t)i|² = sin²α cos²α  

e^−iE1t − e^−iE2t  

= sin²2α sin²(t(E₂− E1)/2) where

E₂− E₁= E₂²− E₁²

E₁+E₂ ≈ m₂²− m²₁

2E = ∆m²

2E ≈ 2m∆m

2E = ∆m

γ where the latter can be seen as a boosted version of the fact that also a neutrino at rest has no definite weak eigenstates.

Including also the third family we can write the probability for oscillation from family α to β

P_α→β = δ_αβ − 4X

i>j

<(U_αi^?U_βiU_αjU_βj^?)sin²

"

1.27∆m²_ij eV²

L km

GeV E

#

+ 2X

i>j

=(U_αi^?U_βiU_αjU_βj^?)sin²

"

2.54∆m²_ij eV²

L km

GeV E

#

Neutrinos also interacts (very weakly) with matter

I e⁻ν →e⁻ν (Z⁰exchange) same for all species.

I e⁻ν_e → e⁻ν_e(W exchange) special for ν_e.

The latter will give an effective mass, different from the one in vacuum

id dt

 νe

ν_µ



=U

 E₁ 0 0 E₂

 U^†

 νe

ν_µ

 +

 √

2G_FN_e 0

0 0

  νe

ν_µ



where N_e is the density of electrons in the medium.

which will give us

P(νe→ ν_µ) = sin²2θ

W² sin² 1.27 · W∆m²_ij eV²

L km

GeV E

!

with

W²=sin²2θ +

√

2G_FN_e 2E

∆m²− cos 2θ

2

Measurements of neutrino oscillations

Sources of neutrinos:

I The sun: Abundant source of νe, but low energies. The main process p + p → d + e⁺+ ν_e+ γgives too low energies to detect. But⁷Be + p →⁸Be + e⁺+ νe+ γ works.

I Cosmic rays: Gives high energies, approximately 1:2 ratio of ν_eand ν_µand corresponding anti particles.

I Accelerators: Generate eg. a beam of π⁺which decays to µ⁺νµ, with controllable energy.

I Reactors: Radioactive materials, mainly ν_ewith different energies.

Detecting neutrinos (far away)

I Homesteke Mine (USA) (Ray Davies NP2002):

νe+³⁷Cl →³⁷Ar + e⁻,³⁷Ar is radioactive so that we can count them.

I Sudbury Neutrino Observatory (Canada) (Arthur McDonald NP2015): Heavy water using the inverse pp cycle. But also netral current interactions with d , and elastic scattering on electrons. In all cases look for tiny flashes of light.

I Super Kamiokande (Japan) (Takaaki Kajita NP2015):

Huge underwater cave, with walls covered by photo multipliers, filled with normal water.

What are we looking for?

I Appearance of ν_µwhere we only expect νe I Disappearance of ν_e(e.g. from the sun)

I Disappearance of ν_µ(from long baseline beams) Different sources, different energies, different lengths to detector, different detectors.

Current status

I sin²(2θ₁₃) =0.093 ± 0.008

I sin²(2θ₁₂) =0.846 ± 0.021 (a.k.a θ_sol)

I sin²(2θ₂₃) >0.92 (90%) (a.k.a θ_atm)

I ∆m²₁₂= (7.53 ± 0.18) · 10⁻⁵eV²

I  ∆m²₃₁

≈ ∆m²₃₂

(2.44 ± 0.05) · 10⁻³eV² We only measure differences: m₁<m₂ m₃or m₃ m₁<m₂?

What about δ

So far there has been no measurement of the (CP-violating) phase factor δ

For Majorana masses there are also two additional phases α₁ and α₂.

Several experiments are planned:

DUNE (US), HyperKamiokande (JP), and . . .

ESSnuSB (SE)

(European Spallation Source Neutrino Super Beam)