Lecture 15 Mixing of three neutrinos New ideas about neutrino

Lecture 15 Mixing of three neutrinos New ideas about neutrino sector

Neutrino oscillations: each neutrino is a superposition of 3 neutrino states with definite mass. Neutrinos are produced in weak process in flavor eigenstates – definite flavor As neutrino propagates in space, the quantum mechanical phase advances differently due to different masses. Neutrino produced as an electron neutrino can become after some distance a muon neutrino. Since mass differences are small, the effect becomes visible only at large distances.

∑ |να > =

Uα*i | νi >
i

neutrino with definite flavor: α = e, μ, τ

∑ |νi > =

Uαi |να >
α

neutrino with definite mass: i=1, 2, 3

Unitary transformation can be treated as a pro| duct of transformations between two flavors

General Derivation
mixing as function of time, distance and difference of masses

na = åUai ni
i

• Where na are the flavor eigenstates, ni are the

mass eigenstates with mass mi and Uai is the
neutrino analogue of the CKM matrix, i.e., the

mixing matrix.

å v(t) =

U e e n !" " -iEit i p×x

ai

i

i
• Gives the!eigenstates at a later time t and position x .

Intensity

• Let us assume the neutrino interacts weakly at time t, and we tag it as a flavor eigenstate vb . Then we have an intensity:

Iba =

å 2
vb v(t) =

2
UaiU b*ie-iEit

i

• We use the ultra-relativistic limit so that:
m2 Ei = p + i
2p

Mixing Equation

• So that:

å Iba =

U U e m2 2 -i i t * 2p ai bi

i

• Which will serve as the standard mixing equation.

Two Generation Mixing

• If only two generations (say, electron and

muon) participate, then:

æ cosq sinq ö

U = çè -sinq

cos

q

÷ ø

• Setting a =1 for the initial state, there are
two intensities, one for each value of b
related by:
I11 = 1- I21

Two Generation Mixing
• And
I21 = 4 cos2 q sin2 q sin2 éêë (m12 4-pm22 )t ùúû = sin2 2q sin2 éêë (m12 4-pm22 )t ùúû
• Which has three important limits:
• 1)
When we are close to the source (small t) , no oscillations are noticeable.

Other Cases

• 2)

m12 − m22 t ∼ 1 4p

• A pattern is noticeable as t varies, so the precise

calculation of

m2 - m2

1

2

is possible.

• 3)

The experiment will average over the

rapid oscillations, resulting in

We have oscillations, but cannot measure the mass difference.

Three Generation Mixing

• For three generation mixing, oscillations can be described in terms of four angles: one CPviolating phase and three differences of masses squared, only two of which are independent.

• Experimental evidence suggests that two of the mass eigenstates are more degenerate with each other than they are with the third:

Dm2 13

!

Dm2 23

"

Dm2 12

Three Generation Mixing

• This simplifies the mixing equation so that:

I

=4U

U*

2

sin

2

(

Dm2 t 23

)

ba

a3 b3

4p

• Which can be rewritten as: