17.1. Basis

Calculating neutrino oscillations requires a lot of rotations between different bases.

17.1.1. Rotation of States

There are three most used bases, vacuum basis which diagonalizes the vacuum Hamiltonian, flavor basis where the flavor states stay, and instataneous matter basis which diagonializes the Hamiltonian with constant matter interaction or the instataneous matter interaction.

  • The rotation from vacuum basis wavefunction Ψv to flavor basis Ψf is

    Ψf=Rv2f(θv)Ψv,

    where the rotation is

    Rv2f(θv)=(cosθvsinθvsinθvcosθv).
  • The rotation from matter basis wavefunction Ψm to flavor basis wavefunction Ψf is

    Ψf=Rm2f(θm)Ψm,

    where the roation is

    Rm2f(θm)=(cosθmsinθmsinθmcosθm).

    The matter mixing angle is determined by the matter potential λ=2GFne,

    tan2θm=sin2θvcos2θvλ/ω,

    where ω=δm2/2E is the vacuum frequency of neutrinos.

Rotate Twice

A rotation from vacuum basis to flavor basis then to matter basis is simply the addition of the two rotation, i.e.,

Rv2f2m(θv,θm)=(cos(θvθm)sin(θvθm)sin(θvθm)cos(θvθm)).

17.1.2. Rotation of Pauli Matrices

Rotate σ1 from vacuum basis to flavor basis,

Rv2f(θ)σ1Rv2f=sin2θσ3+cos2θσ1.

Rotate σ3 from vacuum basis to flavor basis,

Rv2f(θ)σ3Rv2f=(cosθsinθsinθcosθ)(1001)(cosθsinθsinθcosθ)=cos2θσ3sin2θσ1.

17.1.3. Rotation of Hamiltonian

Given the vacuum basis Hamiltonian Hv, we can rotation it to flavor basis by using the rotation Rv2f(θv)

Hf=Rv2f(θv)HvRv2f1(θv).

Similarly, the flavor basis Hamiltonian Hf can also be calculated from matter basis Hamiltonian Hm ,

Hf=Rm2f(θm)HmRv2f1(θm).

Vacuum Basis Hamiltonian

The Hamiltonian in this basis is composed of vacuum Hamiltonian which is diagonalized and the matter potential rotated from flavor basis,

Hv=ω2σ3+λ2(cosθsinθsinθcosθ)σ3(cosθsinθsinθcosθ)=ω2σ3+λ2cos2θvσ3+λ2sin2θvσ1.

Flavor Basis Hamiltonian

Hf=ω2cos2θσ3+ω2sin2θσ1+λ2σ3.

As a consistancy check, we now rotate Hamiltonian in vacuum basis to flavor basis.

Hf=(cosθsinθsinθcosθ)Hv(cosθsinθsinθcosθ)=(ω2+λ2cos2θ)(cosθsinθsinθcosθ)σ3(cosθsinθsinθcosθ)+λ2sin2θ(cosθsinθsinθcosθ)σ1(cosθsinθsinθcosθ)=(ω2+λ2cos2θ)(cos2θσ3sin2θσ1)+λ2sin2θ(sin2θσ3+cos2θσ1)=ω2cos2θσ3+ω2sin2θσ1+λ2(cos22θ+sin22θ)σ3=ω2cos2θσ3+ω2sin2θσ1+λ2σ3.

Numerical Calculation of The Rotations

To write clean code, it is better to define and test there rotations first.

17.1.4. Examples of Rotating Basis

17.1.4.1. Rotate From Vacuum to Another Basis

In vacuum basis, Hamiltonian with matter interaction is

Hv=ω2σ3+λ2cos2θvσ3+λ2sin2θvσ1,

where we have got a contribution of σ3 from matter interaction. By carefully defining a transformation that removes this contribution, we can define a new basis in which the wavefunction is Ψb, which is related to the vacuum basis wavefunction in the following way,

(ψv1ψv2)=(eiη(x)x00eiη(x)x)(ψb1ψb2),

where η(x) is a function of position. We can find the requirement of it by plugging the wavefunction into Schrodinger equation, which results in

η+xdηdx=λ2cos2θv.

Derivation of η

Plug the transformation into Schrodinger equation,

LHS=iddx[(eiη(x)x00eiη(x)x)(ψb1ψb2)]=i(iη(x)eiη(x)xixdη(x)dxeiη(x)x00iη(x)eiη(x)x+ixdη(x)dxeiη(x)x)(ψb1ψb2)+i(eiη(x)x00eiη(x)x)ddx(ψb1ψb2)

Multiplying on both sides of the equation the Hermitian conjugate of the transformation matrix

(eiη(x)x00eiη(x)x),

the two sides becomes

(eiη(x)x00eiη(x)x)LHS=i(iηxixdη(x)dx00iη(x)+ixdη(x)dx)(ψb1ψb2)+iddx(ψb1ψb2)(eiη(x)x00eiη(x)x)RHS=(ω2σ3+λ(x)2cos2θvσ3+λ(x)2sin2θv(0e2iη(x)xe2iη(x)x0))(ψb1ψb2).

We choose the condition that

η(x)+xdη(x)dx=λ(x)2cos2θv,

which removes the second term in the Hamiltonian in vacuum basis. Finally we have the equation of motion in this new basis

iddx(ψb1ψb2)=(ω2σ3+λ(x)2sin2θv(0e2iη(x)xe2iη(x)x0))(ψb1ψb2).

We could even remove all the σ3 terms using this method by choosing

η(x)+xdη(x)dx=ω2+λ(x)2cos2θv.

The general solution of η(x) is

η(x)=Constantx+1x1xcos2θv2λ(τ)dτ,

where the constant can always be set to 0, which tells us that

η(x)=1x1xcos2θv2λ(τ)dτ.

Constant Matter Density

As a check, for constant λ, we have

η(x)=cos2θv2xλ(x1).

In this new basis, the Hamiltonian becomes

Hb=ω2σ3+λ2sin2θv(0ei2η(x)xei2η(x)x0)=ω2σ3+λ2sin2θvcos(2η(x)x)σ1λ2sin2θvsin(2η(x)x)σ2.

17.1.4.2. Constant Matter Eigen Basis

For a Hamiltonian with matter interection,

Hv=ω2σ3+λ2cos2θvσ3+λ2sin2θvσ1,

where λ(x)=λ0+λ1(x). We rotate it into constant matter basis where the Hamiltonian is diagonalized with only λ0,

Hm=Rv2mHvRv2m,

where U rotates the state from vacuum basis to constant matter basis.

In general the Hamiltonian after the rotation is written in a form

Hm=H0+Rf2mλ1,f(x)Rf2m,

in which Rf2m is the rotation from flavor basis to constant matter basis and λ1,f is the perturbation of matter profile in flavor basis. We also have

H0=(ωm100ωm2)

and

λ1,f(x)=(δλ000).

Another Form of Matter Potential

For the perturbation we could also make it traceless without changing the probabilities.

Position Dependent Rotation

If the rotation is position dependent, i.e., the matter profile λ0 is not position independent, we have, in general,

Hm=H0iRf2mddxRf2m+Rf2mλ1(x)Rf2m.

In the case of discussion here, ddxRf2m=0.

17.1.5. Refs & Notes


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