5.1. MSW Effect

Brief Summary of MSW Resonance

The Hamiltonian for matter effect is

H(x)=(λ(x)2ωv2cos2θv)σ3+ωv2sin2θvσ1,

where λ(x)=2GFne(x) is the matter potential and ωv=Δm22E is the vacuum oscillation angular frequency. MSW resonance happens when the diagonal terms disappear, i.e.,

λ=ωvcos2θv.

What does it mean?

Now think about the vacuum Hamiltonian in flavor basis,

Hvacuum=ωvcos2θv2σ3+ωvsin2θv2σ1.

We actually spot this similarity between matter Hamiltonian and vacuum Hamiltonian. Thus we define new oscillation frequency ωm and mixing angle θm,

ωm=ωv(λωvcos2θv)2+sin22θv,tan2θm=sin2θvcos2θvλ/ωv,

so that the Hamiltonian has the form

H(x)=ωmcos2θm2σ3+ωmsin2θm2σ1.

What are the significances of θm and ωm? We can look at them in constant matter profile, which means they are constant. In such a case we can find a basis in which the Hamiltonian is diagonalized.

Hdiagonalized=ωm2σ3.

So ωm is the oscillation frequency in matter. However, a caveat is that Hamiltonian with matter interaction is not always diagonalizable. Noneless, we know for constant matter profile and adiabatic approximation, diagonalizing the Hamiltonian is doable.

So the condition for MSW resonance also minimizes the oscillation angular frequency in matter.

This basis, which we call matter basis {|νL,|νH}, is related to flavor basis through this rotation,

(|νe|νx)=(cosθmsinθmsinθmcosθm)(|νL|νH).

An investigation into the effective mixing angles in matter θm shows that resonance condition also maximizes the sin2θm. The reason is that λ(x)ωvcos2θv=0 will make tan2θm infinity. Then we have 2θm=π2 which means sin2θm=1.

We also notice θm=π4 will also lead to equal mixing, i.e., electron flavor state consists of equal fraction of light state and heavy state.

In summary, the MSW resonance happens when

  • diagonal elements of Hamiltonian in flavor basis vanish;

  • energy split between heavy and light states is minimized;

  • flavor oscillation (angular) frequency is minimized or oscillation length is maximized;

  • mixing angle in matter θm=π4;

  • equal mixing happens;

  • sin2θm is maximized which means the oscillation amplitude happens.

There are more about MSW effect, which is related to non-adiabatic conversion. Read the main text for more information.

5.1.1. An Introduction to MSW Effect

First of all, MSW means Mikheyev–Smirnov–Wolfenstein.

The symmetry breaking between neutral current and charged current will change the evolution and makes the states more electron neutrino.

This is the reason of MSW effect.

In other words, the first requirement of MSW effect is that the electrons interacts with neutrinos and makes it in a specific state that is heavy if the electron density is strong enough. Meanwhile, if the mixing angle is not that large, a level crossing could happen making the state a light state as the density becomes vacuum. The other requirement, which is obvious, is that the density change should be adiabatic, the meaning of which is the density profile of matter gently reduces to vacuum, leaving enough reaction time for the neutrinos.

The Hamiltonian for neutinos with neutrino-matter interaction (in flavour basis) is

H=Δm24E(cos2θsin2θsin2θcos2θ)+Δ2σ3+ΔI,

where the last term (green part) can be neglected because this term will only shift all the eigenvalues with the same amount without changing the eigenvectors.

Define a quantities like ω=Δm22E for neutrinos ( ω¯=Δm22E for antineutrinos) and Δ=2GFn(x) (which might be denoted by ν=2GFnν in other lituratures).

Using Pauli matrices, I can decompose this to

H=ω2(cos2θσ3+sin2θσ1)+Δ2σ3+ΔI

Note

As a reminder, Δ=2GFn(x).

Note

The red part is from the charged current Feynman diagram. We have a σ3 matrix instead of an matrix like

(1000)

because we rewrite this matrix with Pauli matrices and identy. Then the identities are neglected.

This can be done properly because Pauli matrice and Identy matrix form a complete basis.

In a more compact form, this Hamiltonian is

H=Δm24E(cos2θσ3+sin2θσ1)+Δ2σ3=(Δ2Δm24Ecos2θ)σ3+Δm24Esin2θσ1

Note

Eigenvalues of σ3 are 1 and -1 with corresponding eigenvectors

(10)

and

(01).

As we have mentioned, this Hamiltonian is in flavour basis. When mixing angle θ0, the eigenvectors are almost eigenvectors of σ3 which are electron neutrinos and x type neutrinos.

Interesting Limits

Before we really solve the equation of motion, some interesting limits can be shown here.

Interaction Δ is much larger than cacuum mixing terms. In this case, the Hamiltonian becomes diagonalized and the neutrinos will stay on it’s flavour eigenstates in the propagation.

Interaction Δ is much smaller than vacuum mixing terms. The propagation reduces to vacuum case.

To see this effect quantitively, we need to diagonalize this Hamiltonian (Can we actually diagonalize the equation of motion? NO!). Equivalently, we can rewrite it in the basis of mass eigenstates {|νL(x),|νH(x)},

|νL(x)=cosθ(x)|νesinθ(x)|νμ|νH(x)=sinθ(x)|νecosθ(x)|νμ.

This new rotation in matrix form is

(|νL(x)|νH(x))=(cosθ(x)sinθ(x)sinθ(x)cosθ(x))(|νe|νx)=Ux1(|νe|νx)

Diagonalize Hamiltonian

To diagonilize it, we need to multiply on both sides the rotation matrix and its inverse,

Hxd=Ux1HUx.

The second step is to set the off diagonal elements to zero. By solving the equaions we can find the sin2θ(x) and cos2θ(x).

Hxd=Ux1(A1σ1+A3σ3)Ux=(A3cos2θ(x)A1sin2θ(x)A3sin2θ(x)+A1cos2θ(x)A3sin2θ(x)+A1cos2θ(x)A3cos2θ(x)+A1sin2θ(x)),

where

A3=Δ2δ2m4Ecos2θA1=δ2m4Esin2θ.

Set the off-diagonal elements to zero,

A3sin2θ(x)+A1cos2θ(x)=0

So the solutions are

sin2θ(x)=A1A12+A32cos2θ(x)=A3A12+A32.

Plug in A1 and A3

sin2θ(x)=sin2θv(Δω)2+12Δωcos2θvcos2θ(x)=cos2θvΔω(Δω)2+12Δωcos2θv.

Define Δ^=Δω with ω=Δm22E, which represents the matter interaction strength compared to the vacuum oscillation.

sin2θ(x)=sin2θvΔ^2+12Δ^cos2θvcos2θ(x)=cos2θvΔ^Δ^2+12Δ^cos2θv.

We also have

A3cos2θ(x)A1sin2θ(x)=ω2Δ^2+12Δ^cos2θv,

which leads to the result of the diagonalized Hamiltonian

Hxd=ω2Δ^2+12Δ^cos2θv(1001).

This diagonalizes the Hamiltonian LOCALLY. It’s not possible to diagonalize the Hamiltonian globally if the electron number density is not a constant.

The point is, for equation of motion, we have a differentiation with respect to position x! So even we diagonalize the Hamiltonian, the equation of motion won’t be diagonalized. An extra matrix will occur on the LHS and de-diagonalize the Hamiltonian on RHS.

Note

As Δ, A3 and sin2θ(x) vanishes. Thus the neutrino will stay on flavour eigenstates.

With the newly defined heavy-light mass eigenstates, we can calculate the propagatioin of neutrinos,

it|ψx(t)=ExtraMatrixFromLHSHxd|ψx(t),

where the ExtraMatrixFromLHS comes from the fact that changing from flavor basis Ψ(x) to heavy-light basis Ψm(x) using Um,

ix(UmΨm(x))=H(UmΨm(x))

only returns

ixΨm(x)=HmdΨm(x)iUm1(xUm)Ψm(x).

We imediately know the propagation is on the heavy-light mass eigenstates under adiabatic condition WITHOUT solving the equation. The eigenvalue of these states are A32+A12 and A32+A12. The absolute value of these solutions grow as Δ becomes large.

Combining the two terms on RHS,

ixΨm(x)=HmΨm(x),

where

Hm=HmdiUm1(xUm).

The only part inside Um(x) that is space dependent is the number density of the electrons n(x). Thus we know immediately that the Hamiltonian is diagonalized if the number density is constant.

Is Adabatic Condition Valid Here?

Haxton’s paper.

Before going into the system, here is a discussion of adiabatic in thermodynamics.

From the two solutions we know there is a gap between the two trajectories. We draw a figure with electron number density as the horizontal axis and energy as the vertical axis.

../_images/msw.png

Fig. 5.1 Neutrino physics by Wick C. Haxton and Barry R. Holstein.

5.1.2. Solar Neutrinos and MSW Effect

The MSW effect itself can be made clear using the example of neutrino oscillations in our sun. In fact it is responsible for the missing solar neutrino problem.

Small Mixing Angle Limit

Just for fun.

Take two flavour mixing as an example.

(νeνx)=(cosθsinθsinθcosθ)(ν1ν2)

In the small mixing angle limit,

(νeνx)(1θθ1)(ν1ν2)

which is very close to an identity matrix. This implies that electron neutrino is more like mass eigenstate ν1. By ν1 we mean the state with energy Δm24E in vacuum.

We need this intuitive picture to understand MSW effect. Electron neutrinos are almost identical to the low mass neutrino mass eigenstate. However, as we will see, due to the matter interaction, the electron flavour neutrino is corresponding to the HEAVY mass eigenstate. This is the key idea in physics of MSW effect.

../_images/clorine-detector-solar-neutrinos.jpg

Fig. 5.2 Solar neutrino problem: chlorine detector (Homestake experiment) results and theory predictions. SNU: 1 event for 1036 target atoms per second. Source: Kenneth R. Lang (2010)

../_images/msw-and-density.png

Fig. 5.3 MSW effect of solar neutrinos. This figure is adapted from Smirnov 2003.

Hamiltonian with matter effect is

H=λ(x)ωvcos2θv2σ3+ωvsin2θv2σ1

and new basis is defined

(|νe|νμ)=(cosθmsinθmsinθmcosθm)(|νL|νH.)

Now we have two states in this matter basis, the heavy state and the light state. When we talk about adabatic propagation, we mean the system doesn’t jump between these heave and light states.

In the figure, we have dense matter on the left while the matter desnity approaching vacuum on the right. Upper bar means the probability of finding the system to be in heavy state and the lower bar means in light state. As the matter profile doesn’t change too fast, the system undergoes adiabatic propagation and the length of the bars doesn’t change. For example, if the system starts with completely heavy state , it will always remain on heavy state.

Since almost all neutrinos produced in the sun are electron neutrinos, and electron flavor neutrinos experience a big potential, electron flavor almost means heavy state. So we have the system starts with a state that is mostly heavy state and it remains this way. However, during the propagation, heavy state is going to have less electron flavor until some point, we have equal mixing which is MSW resonance. As it approaches vacuum, we have only about 1/3 of the probability to find the neutrinos to be on electron flavor state.

If we discuss more about this phenomenon, we have situations such as not so large density.

../_images/msw_and_density1.png

Fig. 5.4 MSW conversion for different matter profiles. Smirnov 2003.

5.1.3. Visualizing the Solar Neutrino Flavor Oscillations

Applying the flavor isospin method (Neutrino Flavour Isospin), we can visualize the flavor conversions.

../_images/matter-effect-large-density.png ../_images/matter-effect-adiabatic.png ../_images/matter-effect-adiabatic-3.png

5.1.4. Numerical Results

5.1.4.1. 2 Flavor Oscillation

The equation of motion in flavor basis is

ixΨmf(x)=HmfΨmf(x)

where

Hmf=(Δ2ω2cos2θv)σ3+ω2sin2θvσ1.

Writing down the dimensionless equation, we have

ix^Ψmf=RSω2((Δ^cos2θv)σ3+sin2θvσ1)Ψmf.

As for the data of the sun I use a simple exponential distribution. The data is also from the paper by Bahcall. The model using just exponential is not accurate however it is enough to make the point in MSW resonance. So I choose a solar model in which the core density is n(0)=1013GeV3. The distribution is

n=10134.3x^GeV3.

The numerical results can be obtains by plugging this density profile into the differential equation solver.

../_images/numericalMSW-model-3.png

Fig. 5.5 Numerical results for electron flavor neutrino probability and the other flavor neutrino probability when the electron density profile is 10144.3x^GeV3.

../_images/numericalMSW-model-2flavor-minus13-1.png

Fig. 5.6 Number density profile n(x^)=10134.3x^GeV3.

Since we can easily predict the survival probability using simple theory. Here are some comparision. The following figures are for matter profile 10134.3x^GeV3 and vacuum oscillation frequency ω=1020GeV,1019GeV,1018GeV,1017GeV respectively. As we can see that in this two flavor special case, the problem doesn’t dependent on energy and mass difference independently but depends only on ω=Δm22E. If we choose Δm2=7.5×105eV2, the four figures are corresponding to energies 7.5MeV,0.75MeV,7.5×102MeV,7.5×103MeV.

../_images/compThNu13-1.png

Fig. 5.7 The grey line is theoretical survival probability at x^=1. In this calculation the vacuum oscillation frequency is set to ω=1020GeV.

../_images/compThNu13-2.png

Fig. 5.8 The grey line is theoretical survival probability at x^=1. In this calculation the vacuum oscillation frequency is set to ω=1019GeV.

../_images/compThNu13-3.png

Fig. 5.9 The grey line is theoretical survival probability at x^=1. In this calculation the vacuum oscillation frequency is set to ω=1018GeV.

../_images/compThNu13-4.png

Fig. 5.10 The grey line is theoretical survival probability at x^=1. In this calculation the vacuum oscillation frequency is set to ω=1017GeV.

5.1.4.2. Three flavor Oscillations

Vacuum part of the Hamiltonian is

Hmvv=12E(m12000m22000m32).

The matter interaction in flavor basis is

Vmf=2GFndiag1,0,0.

Thus to work in vacuum mass eigenstates, we need a transformation

Vmv=U1VmfU.

Then the Hamiltonian becomes

Hm=(m122E+ΔUe12ΔUe1Ue2ΔUe1Ue3ΔUe2Ue1m222E+ΔUe22ΔUe2Ue3ΔUe3Ue1ΔUe3Ue2m322E+ΔUe32)

Trace of this Hamiltonian is Tr(Hm)=m12+m22+m322E+Δ. To find the traceless part, we can use the relation [ohlsson2000]

M=Mtraceless+1NTr(M)I,

where N is the rank.

The traceless part of Hamiltonian becomes

Hm=(ΔUe1213Δ+13(m12m22+m12m322E)ΔUe1Ue2ΔUe1Ue3ΔUe2Ue1ΔUe2213Δ+13m22m12+m22m322EΔUe2Ue3ΔUe1Ue3ΔUe2Ue3ΔUe3213Δ+13m32m12+m32m222E).

Define the following quantities where only two of them are linearly independent

Δm122=m22m12Δm232=m32m22Δm132=m32m12.

We define an energy scale related to the radius of the sun

ϵS=1RS.

The EoM can be written in a dimensionless manner,

ix^Ψm=(Δ~Ue1213Δ~+13(Δm122+Δm1322EϵS)Δ~Ue1Ue2Δ~Ue1Ue3Δ~Ue2Ue1Δ~Ue2213Δ~+13Δm122+Δm2322EϵSΔ~Ue2Ue3Δ~Ue1Ue3Δ~Ue2Ue3Δ~Ue3213Δ~+13Δm132+Δm2322EϵS)Ψm,

where Δ~=Δ/ϵS.

The parameters for this calculation in units of GeVsomepower are

n(x)=10124.3xϵS=1024Δ~(x)=2GFn(x)/ϵSΔm122=7.6×105×1018Δm132=2.3×103×1018Δm232=Δm132Δm122E=103

For these parameters there is only resonance for Δm132+Δm232.

A quick check over the different energy scales.

  1. Vacuum energy scales in normal hierarchy

    ω12=Δm1222E=3.8×1020GeVω13=Δm1322E=1.7×1018GeVω23=Δm2322Eω13
  2. Matter related scale for density profile 10144.3x^

    Δ1=1.6×10194.3x^[1.6×1023.3,1.6×1019]
  3. Matter related scale for density profile 10134.3x^

    Δ1=1.6×10184.3x^[1.6×1022.3,1.6×1018]
../_images/numericalMSW3Flavor-normalization.png

Fig. 5.11 Normalization factor as a function of distance.

../_images/numericalMSW3Flavor-probabilities.png

Fig. 5.12 Probability for each flavor of neutrinos.

Applying a number density function n(x)=10134.3x to the system, the small scale oscillations are revived,

../_images/numericalMSW3Flavor-2-norm.png

Fig. 5.13 Normalization of the states for numerical 3 flavor oscillation in the sun with density profile 10134.3x.

../_images/numericalMSW3Flavor-2-probability.png

Fig. 5.14 Numerical results for 3 flavor oscillation in the sun with density profile 10134.3x.

../_images/numericalMSW3Flavor-minus13-Inst-Eigen-Energies.png

Fig. 5.15 Eigenenergies for density profile 10134.3x.

../_images/numericalMSW3Flavor-vac-eigen-prob.png

Fig. 5.16 Survival probabilities for different vacuum mass eigenstates for 3 flavor oscillation in the sun with density profile 10134.3x.

An interesting notion is the survival probability for the instantaneous eigenstates.

../_images/instanEigenstetes-minus13-Grid.png

Fig. 5.17 Probability for the instantaneous eigenstates for matter profile 10134.3x.

Lower matter density will have less suppression on vacuum oscillations.

../_images/numericalMSW3Flavor-minus14matter.png

Fig. 5.18 Numerical results for 3 flavor oscillation in the sun with density profile 10144.3x.

ohlsson2000

Ohlsson, T., & Snellman, H. (1999). Three flavor neutrino oscillations in matter, 2768(2000), 25. doi:10.1063/1.533270

5.1.4.3. Ternary Diagram

High matter density suppresses the vacuum oscillations which is clearly shown on a ternary diagram.

../_images/mass-1.png

Fig. 5.19 Ternary diagram for MSW effect shows that the vacuum oscillations in this case is suppressed. Comparing this with the survival probability, the survival probability for electron neutrino drops to a value and the rapid oscillations start to show up. The drop is the movement in the ternary diagram from the right-bottom cornor to the tau neutrino axis. These rapid oscillations corresponds to the T-shaped tip at the other end of the line.

../_images/mass-1-scatter.png

Fig. 5.20 Ternary diagram for MSW effect with homogeneously descretized position x with matter profile 10134.3xGeV3. We can see clearly that the system stays on the two ends to the line for a longer time that in the middle where the transition happens very quickly. This can also be seen in the survival probability plot.

../_images/matter-inst-eigen-e-1.png

Fig. 5.21 Ternary diagram for instantaneous eigenstates with matter profile 10134.3xGeV3. The system starts from almost the second instantaneous state then moves along the state that ν1=0.

../_images/matter-inverted-1.png

Fig. 5.22 Ternary diagram for MSW effect with inverted hierarchy Δm23=m32m22<0. The shape changes a lot since the frequencies changes a lot.

5.1.5. Experiments

MSW effect is verified by serveral experiments, SNO, Borexino and many others.

../_images/msw-experiments.png

Fig. 5.23 Comparison of MSW prediction and experiments. The uncertainties of MSW prediction line are on on mixing angles. Figure from Haxton et al 2013.

5.1.6. MSW Triangle

Finally, one of the interesting things about MSW effect is that we could find a triangle where the survival probability is low.

../_images/msw-triangle.png

Fig. 5.24 Code and illustration for the calculation can be downloaded on github https://github.com/NeuPhysics/codebase/blob/master/ipynb/matter/msw_triangle.ipynb .

5.1.7. Refs and Notes

  1. Wolfenstein, L. (1978). Neutrino oscillations in matter. Physical Review D, 17(9), 2369–2374. doi:10.1103/PhysRevD.17.2369

  2. Wolfenstein, L. (1979). Neutrino oscillations and stellar collapse. Physical Review D, 20(10), 2634–2635. doi:10.1103/PhysRevD.20.2634

  3. Parke, S. J. (1986). Nonadiabatic Level Crossing in Resonant Neutrino Oscillations. Physical Review Letters, 57(10), 1275–1278. doi:10.1103/PhysRevLett.57.1275

  4. Bethe, H. A. (1986). Possible Explanation of the Solar-Neutrino Puzzle. Physical Review Letters, 56(12), 1305–1308. doi:10.1103/PhysRevLett.56.1305


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