5.2. MSW Refraction, Resonance and Non-adiabacity¶

5.2.1. 2 Flavor Neutrino Oscillations and Matter Effect¶

Interaction with matter plays a big role in neutrino oscillation. As shown previously, the interaction only affects (anti) electron neutrinos. In other words, the interaction term in flavor basis is

where \(\Delta = \sqrt{2} G_F n\) and \(n\) is the number density of the electrons. However, to do calculations, since identity matrix doesn’t change the survival probability, we can always make the hamiltonian traceless, which becomes

5.2.2. Constant Electron Number Density¶

Suppose we have an environment with constant electron number density, the term \(- i \mathbf{U_m^{-1}} ( \partial_x \mathbf{U_m} )\) goes away. All we have is the diagonalized new Hamiltonian \(\mathbf{H_{md}}\) and the eigenvalues are easily obtained which are

The final result for these two eigenvalues are

Meanwhile the eigenstates are denoted as \(\ket{\nu_{c1}}\) and \(\ket{\nu_{c2}}\).

As for the survival probability for the initial condition that \(\Psi(x=0)=\ket{\nu_{c1}}\), the result has the same form as the vacuum case, which is

where

\(\theta_m = \theta(x)\) is the effective mixing angle which in fact doesn’t depend on \(x\) if the matter profile is constant.

5.2.3. Adiabatic Limit¶

In some astrophysical environments the electron number density changes very slowly which means the term \(\mathbf{U_m^{-1}} \partial_x \mathbf{U_m}\) is much smaller than \(\mathbf{H_{md}}\). By intuition we would expect that this term could be dropped to the lowest order.

The eigen energies are slowing changing with the position of neutrinos,

When the term \(\hat\Delta\) is very small \(1-2\hat\Delta\cos 2\theta_v\) will dominate and the whole term decreases. On the other hand as \(\hat\Delta\) becomes large, \(\hat\Delta^2\) will dominate and the whole term grows. Mathematically we could find the region when the part \(\sqrt{\hat\Delta^2 + 1 - 2 \hat\Delta \cos 2\theta_v}\) decreases and increases.

The survival probability for the light neutrinos would be

The survival probability for electron flavor neutrino is

if the neutrinos are produced in dense region and the detection happens in vacuum.

5.2.4. General Discussion of Matter Effect¶

This part is a very general discussion of the matter effect [Parke1986].

To work in flavor basis, we use the subscript \({}_{mf}\) to denote the flavor basis representation with mass effect. The equation of motion in flavor basis can be written down as

where

There are three stages for neutrinos to travel from the core of the sun to vacuum.

1. At the core, electron neutrinos are produced. The electron flavor state should be projected onto heavy and light instantaneous mass eigenstates. What fallows is the that the propagation is adiabatic until the transition happens. As we have seen in adiabatic situation, the states will stay in heavy and light states all along the evolution if the system starts from heavy or light state,

   \[\begin{split}\ket{\nu_{a1}(x)} &= \exp(-i \int_0^x \frac{\omega_m(x)}{2} dx ) \ket{\nu_L(x)} \\ \ket{\nu_{a2}(x)} &= \exp(i\int_0^x \frac{\omega_m(x)}{2} dx) \ket{\nu_H(x)},\end{split}\]

   where the heavy and light states are defined in the adiabatic situation previously. This is what happens before the passing through of the resonance.

2. At the resonance point, light instantaneous mass eigenstate has a probability to jump to the heavy state and vice versa. When it comes to the resonance point which is non-adiabatic propagation, the transition between the states \(\ket{\nu_L}\to a_L \ket{\nu_L(x)} + a_H \ket{\nu_H(x)} \ket{\nu_1(x)}\) and \(\ket{\nu_H}\to b_L \ket{\nu_L(x)} + b_H \ket{\nu_H(x)}\) will mix the heavy and light state up.

   \[\begin{split}\ket{\nu_1(x)} &= a_L \exp(-i \int_{x_r}^x \omega_m(x')/2 dx' ) \ket{\nu_L(x)} + a_H \exp(i\int_{x_r}^x \omega_m(x')/2 dx') \ket{\nu_H(x)} \\ \ket{\nu_2(x)} &= b_L \exp(-i \int_{x_r}^x \omega_m(x')/2 dx' ) \ket{\nu_L(x)} + b_H \exp(i\int_{x_r}^x \omega_m(x')/2 dx') \ket{\nu_H(x)},\end{split}\]

   where the relations between the constants are determined using the condition that \(\ket{\nu_1(x)}\) and \(\ket{\nu_2(x)}\) are orthonormal, which leads to the conclusion that

   \[\begin{split}b_L &= -a_H^* \\ b_H &= a_L^* \\ \lvert a_L \rvert^2 &= - \lvert a_H \rvert^2 .\end{split}\]

3. After the resonance point, the heavy and light states will continue on their adiabatic propagation.

Electron neutrinos are produced in a dense region as \(\ket{\nu_e}\), which are partially transformed to other the other neutrinos due to matter and the resonance then it propagates as if it satisfies the adiabatic condition again. The initial state in terms of light and heavy state is

The final state right before the resonance is

After the resonance the state is described by the general jumping

in which the \(x_{r-}\) is actually \(x_r\) thus

To calculate the survival probability it is easier to use flavor basis, hence we have another form of \(\ket{\Psi_m(x)}\) which is

Since \(\cos\theta_m\), \(\sin\theta_m\) and \(\omega_m\) are real while \(a_L\) and \(a_H\) are complex, survival amplitude of electron neutrinos is given by

where the coefficients are

The detection is in a region where matter density is very small, thus we use \(x\to\infty\) which means the effective mixing angle becomes vacuum mixing angle. The probability is the square of the amplitude,

where \(\phi\) is defined as

Note that for any complex number \((a+ib)e^{i\phi} \equiv \rho e^{i\psi}\),

which means that the previous result can be simplified to

with the definition that \(\psi_{LH}(x)\) is the argument of \(A_L^*(x)A_H(x)\).

However the coefficients \(a_L\) and \(a_H\) are still not known yet. The trick is to average over the detection and production position. The average over \(x\) removes the \(\cos\) term due to the dependent of \(x\) for \(\phi\) and averages \(\cos^2\theta_m(x)\) to \(\frac{1}{2}\), which results in

Applying the condition that \(\lvert a_L \rvert^2 + \lvert a_H \rvert^2 = 1\), the probability becomes

where \(\psi_{LH}\) is the argument of \(a_H a_L\) and \(\phi\) is \(\int_{x_0}^{x_r} \frac{\omega_m(x')}{2}dx'\) .

The average over production removes the last part.

Notice that in fact the detection happens in vacuum, which means \(\theta_m(x)=\theta_v\).

This means that the adiabatic result is of the form

Define a transition probability at resonance

which can be determined by the Landau-Zener transition analytically (first order) to the first order.

Parke1986

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

5.2.5. Landau-Zener Transition of Neutrinos¶

As discussed in the previous subsection, a transition probability between the two states \(\ket{\nu_L(x)}\) and \(\ket{\nu_H(x)}\) would change the final survival probabilty. Thus calculating this transition probability will be done in this subsection.

Recall that the effective potential is

where

\(\sin\theta(x)\) and \(\cos\theta(x)\) can be found by solving the equations. Plug in the results and applying the trick that

we have

Since we are dealing with resonance which is located at \(\hat\Delta =\cos 2\theta_v\), the quantities can be expanded around \(\hat\Delta - \cos 2\theta = 0\).

To keep only first order of in the effective potential, we have to expand around \(\hat\Delta = \cos 2\theta_v\)

The equation of motion up to first order of \(\hat\Delta\) becomes

We have already solved

where the eigenstates are \(\ket{\nu_L}\) and \(\ket{\nu_H}\) with eigenvalues \(\omega_{m1}\) and \(\omega_{m2}\) respectively.

To save typing we define

so that the effective potential reduces to a simple form

The general solution to the equation we need to solve can be written as

where

Hamiltonian applied to this state results in

Plug the state \(\ket{\Psi_m}\) into the Schrodinger equation, we have

in which \(omega_m\) is

The boundary condition for such a problem in general is

It should be made clear that the problem we will be discussing is the transition from one state \(\ket{\nu_L(x)}\) to another \(\ket{\nu_H(x)}\) in first order approximation. That means we will confine this system so that the initial condition is \(\ket{\Psi_m(-\infty)} = \ket{\nu_L}\). In terms of \(C_L\) and \(C_H\),

The first order differential equations of \(C_L(x)\) and \(C_H(x)\) can be combined and produce a second order differential equation.

If we use the approximation that \(\frac{d \hat\Delta }{dx}\) is a constant, where in fact we are assuming that \(n(x)\) is linearly depending on \(x\) which means \(\hat\Delta\) is a linear function of \(x\). Thus \(v\propto\frac{d\hat\Delta}{dx}\) is a constant. The equation simplifies to

where \(v=-\frac{\cot\theta_v}{4} \frac{d\hat\Delta}{dx}\) is constant.

In the paper by Zener, [Zener1932] we need to do substitution of the function \(C_L\) so that the equation reduces to Weber equation.

The eigenvalues are not varying very fast and satisfies the condition that

where \(\alpha = \delta \omega_m'(x_r)\) is a constant and comes from the first order of the expression.

Define a new variable \(W\) which is determined by

Then we get a simple equation about \(W\),

which can be reduced to the standard form of Weber equation with the new parameters which are are found by using a single assumption that \(z=g(x- x_r + \frac{2\sin\theta_v}{\alpha})\),

where \(g^2\equiv -i\alpha'\) ( \(g=(1-i)\sqrt{\left\vert \alpha' \right\vert } /\sqrt{2}=\sqrt{\left\vert \alpha' \right\vert }e^{-i\pi/4}\) ) and \(\alpha' = -\alpha\). The equation we need to solve becomes

Parabolic Cylinder Function

The Weber equation has two independent solutions \(D_\nu(z)\) and \(D_{-\nu-1}(iz)\). They are also called Parabolic Cylinder Function on wolfram mathworld.

Since \(D_{\nu}(z)\) blows up for the line on complex plane \(z\propto e^{-\pi i/4}\), the solution that works is \(D_{-\nu-1}(iz)\). Then the solution to \(U_L\) is

\[U_L(x) = u_{+} D_{-iv^2/\alpha -1} (\frac{1-i}{\sqrt{2}} x) ,\]

or

\[U_L(x) = u_- D_{-iv^2/\alpha -1} ( - \frac{1-i}{\sqrt{2}} x) .\]

The asymptotic expression for \(D_{-\nu-1}\) on the line of \(e^{-i\pi/4}\) and \(e^{-3i\pi/4}\) at infinite contour radius on complex plane are

\[\begin{split}D_{-\nu-1}(i x e^{-3i\pi/4}) &\to e^{i(\nu+1)\pi/4} e^{i x^2/4} x^{-\nu-1} \\ D_{-\nu-1}(i x e^{-i\pi/4}) &\to e^{-i(\nu+1)\pi/4} e^{-i x^2/4} x^{-\nu-1}.\end{split}\]

So the real part of these asymptotic expressions are

\[\begin{split}e^{i\nu\pi/4} x^{-\nu-1} &= e^{-v^2\pi/4\alpha'} x^{-\nu-1} \\ e^{-i\nu\pi/4} x^{-\nu-1} &=e^{v^2\pi/4\alpha'} x^{-\nu-1}\end{split}\]

Apply the boundary condition we have the results of the coefficients.

\[\lvert u_+ \rvert = \lvert u_- \rvert = e^{-\gamma \pi/4}\sqrt{\gamma} ,\]

where \(\gamma = \frac{v^2}{\lvert \alpha \rvert}\).

What we need to find out is the state at \(x\to \infty\), which depends on the asymptotic values of \(D_{-\nu-1}\),

\[C_L(x) &\to \sqrt{\gamma} e^{-\gamma \pi/4} \left( e^{3\pi (\nu+1)i/4} e^{-ix'^2/4} x'^{-\nu-1} + \frac{\sqrt{2\pi}}{ \Gamma (\nu+1) } e^{i \pi\nu/4} e^{i x'^2 /4} x'^\nu \right) ,\]

or

\[C_L & \to \sqrt{\gamma} e^{-\gamma \pi/4} \left( e^{-3\pi (\nu+1)i/4} e^{ix'^2/4} x'^{-\nu-1} + \frac{\sqrt{2\pi}}{ \Gamma (\nu+1) } e^{i \pi\nu/4} e^{ - i x'^2 /4} x'^\nu \right).\]

The transition rate is determined by \(\lvert C_L \rvert^2\)

\[\lvert C_L(\infty) \rvert^2 = \gamma e^{-\pi\gamma} \frac{2\pi}{\Gamma(i\gamma +1) \Gamma(-i\gamma +1)} = 2e^{-\pi\gamma}\sinh \pi\gamma = 1-e^{-2\pi\gamma}.\]

Now we understand the transition probability is given by

\[P_{tran} = e^{-2\pi\gamma}.\]

Suppose we have the initial condition as \(\ket{\Psi_m(x=-\infty)} = \ket{\nu_L}\), the system can jump to \(\ket{\nu_H}\) since the state at arbitrary position \(x\) is a mixing of the two states. The probability of jumping is given by [SJParke1986] [Petcov1987]

The survival probability can be calculated by applying this transition probability to the result we had previously.

To be clear, if electron neutrinos are produced inside core of our sun, it will be almost the heavy state. Since the interaction with matter is very strong, it transfers to \(\ket{\nu_L}\) with probability \(P(x\to \infty, \ket{\nu_L}\to\ket{\nu_H})\) due to the gradient of the matter profile which works as the perturbation. Thus the final state will be a mixing of \(\ket{\nu_L}\) and \(\ket{\nu_H}\).

Zener1932

Zener, C. (1932). Non-Adiabatic Crossing of Energy Levels. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 137(833), 696–702. doi:10.1098/rspa.1932.0165

SJParke1986

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

Petcov1987

Petcov, S. T. (1987). On the non-adiabatic neutrino oscillations in matter. Physics Letters B, 191(3), 299–303. doi:10.1016/0370-2693(87)90259-0