Brief Summary of MSW Resonance
The Hamiltonian for matter effect is
where
What does it mean?
Now think about the vacuum Hamiltonian in flavor basis,
We actually spot this similarity between matter Hamiltonian and vacuum Hamiltonian. Thus we define new oscillation frequency
so that the Hamiltonian has the form
What are the significances of
So
So the condition for MSW resonance also minimizes the oscillation angular frequency in matter.
This basis, which we call matter basis
An investigation into the effective mixing angles in matter
We also notice
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
equal mixing happens;
There are more about MSW effect, which is related to non-adiabatic conversion. Read the main text for more information.
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
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
Using Pauli matrices, I can decompose this to
Note
As a reminder,
Note
The red part is from the charged current Feynman diagram. We have a
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
Note
Eigenvalues of
and
As we have mentioned, this Hamiltonian is in flavour basis. When mixing angle
Interesting Limits
Before we really solve the equation of motion, some interesting limits can be shown here.
Interaction
Interaction
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
This new rotation in matrix form is
Diagonalize Hamiltonian
To diagonilize it, we need to multiply on both sides the rotation matrix and its inverse,
The second step is to set the off diagonal elements to zero. By solving the equaions we can find the
where
Set the off-diagonal elements to zero,
So the solutions are
Plug in
Define
We also have
which leads to the result of the diagonalized Hamiltonian
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
Note
As
With the newly defined heavy-light mass eigenstates, we can calculate the propagatioin of neutrinos,
where the
only returns
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
Combining the two terms on RHS,
where
The only part inside
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.
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.
In the small mixing angle limit,
which is very close to an identity matrix. This implies that electron neutrino is more like mass eigenstate
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.
Hamiltonian with matter effect is
and new basis is defined
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.
Applying the flavor isospin method (Neutrino Flavour Isospin), we can visualize the flavor conversions.
The equation of motion in flavor basis is
where
Writing down the dimensionless equation, we have
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
The numerical results can be obtains by plugging this density profile into the differential equation solver.
Since we can easily predict the survival probability using simple theory. Here are some comparision. The following figures are for matter profile
Vacuum part of the Hamiltonian is
The matter interaction in flavor basis is
Thus to work in vacuum mass eigenstates, we need a transformation
Then the Hamiltonian becomes
Trace of this Hamiltonian is
where
The traceless part of Hamiltonian becomes
Define the following quantities where only two of them are linearly independent
We define an energy scale related to the radius of the sun
The EoM can be written in a dimensionless manner,
where
The parameters for this calculation in units of
For these parameters there is only resonance for
A quick check over the different energy scales.
Vacuum energy scales in normal hierarchy
Matter related scale for density profile
Matter related scale for density profile
Applying a number density function
An interesting notion is the survival probability for the instantaneous eigenstates.
Lower matter density will have less suppression on vacuum oscillations.
Ohlsson, T., & Snellman, H. (1999). Three flavor neutrino oscillations in matter, 2768(2000), 25. doi:10.1063/1.533270
High matter density suppresses the vacuum oscillations which is clearly shown on a ternary diagram.
MSW effect is verified by serveral experiments, SNO, Borexino and many others.
Finally, one of the interesting things about MSW effect is that we could find a triangle where the survival probability is low.
Wolfenstein, L. (1978). Neutrino oscillations in matter. Physical Review D, 17(9), 2369–2374. doi:10.1103/PhysRevD.17.2369
Wolfenstein, L. (1979). Neutrino oscillations and stellar collapse. Physical Review D, 20(10), 2634–2635. doi:10.1103/PhysRevD.20.2634
Parke, S. J. (1986). Nonadiabatic Level Crossing in Resonant Neutrino Oscillations. Physical Review Letters, 57(10), 1275–1278. doi:10.1103/PhysRevLett.57.1275
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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