Lecture 15 Mixing of three neutrinos New ideas about neutrino

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Lecture 15 Mixing of three neutrinos New ideas about neutrino

Transcript Of 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 >

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

∑ |νi > =

Uαi |να >

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

• 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



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


• 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) =

UaiU b*ie-iEit


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

Mixing Equation

• So that:

å Iba =

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


• 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



÷ 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



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:








Dm2 t 23



a3 b3


• Which can be rewritten as: