1.1.2. MSW Effect Revisted

Pauli Matrices and Rotations

Given a rotation

U=(cosθsinθsinθcosθ),

its effect on Pauli matrices are

Uσ3U=cos2θσ3+sin2θσ1Uσ1U=sin2θσ3+cos2θσ1.

1.1.2.1. Flavor Basis

Vacuum Oscillations

Vacuum oscillations is already a Rabi oscillation at resonance with oscillation width ωvsin2θv.

Neutrino oscillation in matter has a Hamiltonian in flavor basis

H(f)=(12ωvcos2θv+12λ(x))σ3+12ωvsin2θvσ1.

The Schroding equation is

ixΨ(f)=H(f)Ψ(f).

To make connections to Rabi oscillations, we would like to remove the changing σ3 terms, using a transformation

T=(eiη(x)00eiη(x)),

which transform the flavor basis to another basis

(ψeψx)=(eiη(x)00eiη(x))(ψaψb).

The Schrodinger equation can be written into this new basis

ix(TΨ(r))=H(f)TΨ(r),

which is simplified to

ixΨ(r)=H(r)Ψ(r),

where

H(r)=12ωvcos2θvσ3+12ωvsin2θv(0e2iη(x)e2iη(x)0),

in which we remove the varying component of σ3 elements using

ddxη(x)=λ(x)2.

The final Hamiltonian would have some form

H(r)=12ωvcos2θvσ3+12ωvsin2θv(0ei0xλ(τ)dτ+2iη(0)ei0xλ(τ)dτ2iη(0)0),

where η(0) is chosen to conter the constant terms from the integral.

For arbitary matter profile, we could first apply Fourier expand the profile into trig function then use Jacobi-Anger expansion so that the system becomes a lot of Rabi oscillations.

Any transformations or expansions that decompose exp(i0xλ(τ)dτ) into many summations of exp(iax+b) would be enough for an Rabi oscillation interpretation.

Let’s discuss the constant matter profile, λ(x)=λ0. Thus we have

η(x)=12λ0x.

The Hamiltonian becomes

H(r)=12ωvcos2θvσ3+12ωvsin2θv(0eiλ0xeiλ0x0),

which is exactly a Rabi oscillation. The resonance condition is

λ0=ωvcos2θv.

1.1.2.2. Instanteneous Matter Basis

Neutrino oscillation in matter has a Hamiltonian in flavor basis

H(f)=(12ωvcos2θv+12λ(x))σ3+12ωvsin2θvσ1.

The Schroding equation is

ixΨ(f)=H(f)Ψ(f),

which can be transformed to instantaneous matter basis by applying a rotation U,

ix(UΨ(m))=H(f)UΨ(m),

where

U=(cosθmsinθmsinθmcosθm).

With a little algebra, we can write the system into

ixΨ(m)=H(m)Ψ(m)
H(m)=UH(f)UiUxU.

By setting the off-diagonal elements of the first term UH(f)U to zero, we can derive the relation

tan2θm=sin2θvcos2θvλ/ωv.

Furthermore, we derive the term

iUxU=θ˙mσ2.

We can calculate θ˙m by taking the derivative of tan2θm,

ddxtan2θm=2cos22θmθ˙m,

so that

θ˙m=12cos2(2θm)ddxtan2θm=12(cos2θvλ/ωv)2(λ/ωv)2+12λcos2θv/ωvddxsin2θvcos2θvλ/ωv=12(cos2θvλ/ωv)2(λ/ωv)2+12λcos2θv/ωvsin2θv(cos2θvλ/ωv)21ω)vddxλ(x)=12sin2θm1ωmddxλ(x).

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