Day Night Effect of Neutrinos

[thinkLamp]

Homework Statement

I'm trying to solve problem 3 from http://www.ictp-saifr.org/wp-content/uploads/2018/07/hw_ICTP-SAIFR-2.pdf The problem is as follows:
Assume that solar neutrinos arrive at the surface of the Earth in the $\left| \nu_2 \right>$ state (a mass eigenstate). Assume this neutrino propagates a distance L through the Earth before reaching the detector. Assume that the the electron number density in the neutrino's path is constant which means the oscillation frequency is
$$
\Delta_M = \sqrt{ \Delta^2 \sin^2{2 \theta} + (\Delta \cos{2 \theta} - \sqrt{2} G_F N_e )^2 }
$$

Compute the probability that this neutrino is detected as an electron-type neutrino.

Homework Equations

$$
\Delta_M = \sqrt{ \Delta^2 \sin^2{2 \theta} + (\Delta \cos{2 \theta} - \sqrt{2} G_F N_e )^2 }
$$

$$
\left| \nu_2 \right> = \cos{\theta} \left| \nu_e \right> - \sin{\theta} \left| \nu_{\mu} \right>
$$

The Attempt at a Solution

I started by writing

$$
\left| \nu_2 \right> = \cos{\theta} \left| \nu_e \right> - \sin{\theta} \left| \nu_{\mu} \right>
$$

Then, because it's the flavor states that's going to be oscillating, I wrote

$$
\left| \nu_e (t) \right> = \left| \nu_e \right> e^{-i \Delta_M t} = \left| \nu_e \right> e^{-i \Delta_M L}
$$
But then, that would mean, the probability that the $\left| \nu_2 \right>$ neutrino is detected as a $\left| \nu_e \right>$ neutrino is just
$$
\left| \left< \nu_2 (t) | \nu_e \right> \right|^2 = \cos^2{\theta} \left| e^{-i \Delta_M L} \right|^2 = \cos^2{\theta}
$$
But, this does not depend on $L$ even though in the next sub-problem, I need to use a numeric value of $L$ to compute the probability so it seems wrong.

Where does this dependency on $L$ come from?

 

[Orodruin]

Then, because it's the flavor states that's going to be oscillating, I wrote

$$
\left| \nu_e (t) \right> = \left| \nu_e \right> e^{-i \Delta_M t} = \left| \nu_e \right> e^{-i \Delta_M L}
$$

This is not correct. The electron neutrino is not an eigenstate of propagation in matter so you cannot just multiply it by a phase.

You need to use the full evolution of the neutrino state to the detector.

Also note that the frequency at the relevant energies is very rapid in $L/E$. For all practical purposes this means that you can average over the oscillation phase and your final result will not depend on $L$.

 

[thinkLamp]

Thanks for the help!

So I should be evolving my mass eigenstate instead. That means,
$$
\left| \nu_2, t \right> = \left| \nu_2, 0 \right> e^{-iE_2t}
= e^{-iE_2t} (\sin{\theta} \left| \nu_e \right> + \cos{\theta} \left| \nu_{\mu} \right> )
$$
So then when,
$$P_{2 \to e} = \left| \left< \nu_2, t | \nu_e \right> \right|^2 = \sin^2{\theta}$$

=But, then our time propagation didn't change the outcome at all. It was as if the neutrino didn't travel through the Earth at all. What am I missing?

 

[Orodruin]

So I should be evolving my mass eigenstate instead.

No, the mass eigenstates are only propagation eigenstates in vacuum. In the case of propagation in the Earth, the propagation eigenstates are the eigenstates that diagonalise the full Hamiltonian, which includes both the vacuum Hamiltonian as well as the one arising from coherent forward scattering in matter.

 

[thinkLamp]

Ah. How do I find by propagation eigenstates then? I'm assuming it'd be just the eigenstates of my Earth-propagation Hamiltonian? So I need to diagonalize my full Hamiltonian. But I'm not sure how to relate that to $\nu_e$ or $\nu_1$. As in, I took my Hamiltonian (which for this case is

where $A = \sqrt{2} G_F N_e$. So I found the eigenvalues and eigenvectors of this Hamiltonian but I'm not sure how to tie that back to the mass / flavor eigenstates.

 

[Orodruin]

What you have written down is the Hamiltonian in flavor basis. In the mass basis it would look different.

 

[thinkLamp]

Okay, does that mean the eigenstates of this Hamiltonian (which is in flavor basis) would relate to the flavor eigenstates? Or do I have to convert this in the mass basis, find its eigenstates, relate that to mass eigenstates? If my first statement is true, how does the eigenstates of my Hamiltonian relate to the flavor eigenstates?

 

[Orodruin]

If you write the Hamiltonian in flavor basis, then the flavor eigenstates are represented by $\begin{pmatrix}1 \\ 0\end{pmatrix}$ and $\begin{pmatrix}0 \\ 1\end{pmatrix}$.

 

[thinkLamp]

Right. But you said the Hamiltonian I wrote is in flavor basis so wouldn't that imply that its eigenstates are flavor states? Then, it'd be the flavor states that would propagate but that's incorrect like you pointed out in post #2.

 

[Orodruin]

Right. But you said the Hamiltonian I wrote is in flavor basis so wouldn't that imply that its eigenstates are flavor states?

No. The Hamiltonian is the Hamiltonian whatever basis you write it in. If the flavor states were the eigenstates of the Hamiltonian and the Hamiltonian was written in flavor basis, then it would be diagonal.

 

[thinkLamp]

That makes sense.

So then, how do I go about using this Hamiltonian to propagate my neutrinos? If I diagonalize my Hamiltonian, that would give me my propagation eigenstate, which I can then propagate over distance L and then relate that state back to my flavor eigenstate to find the overlap with my electron neutrino. But, I get incredibly complicated eigenvectors so it feels wrong. Second, even if I find my propagation eigenvectors, how does that relate to the flavor eigenstates? Because at the end I need to calculate the overlap with an electron neutrino.

 

[Orodruin]

The initial state is a mass eigenstate so you need to find an expression for that and then propagate it using the Hamiltonian.

But, I get incredibly complicated eigenvectors so it feels wrong.

You can introduce a matter mixing angle $\theta_M$ that corresponds to the rotation between the flavor states and the matter eigenstates. This angle will have a fairly complicated expression, but doing so will make everything else relatively simple.

 

[thinkLamp]

The mater mixing angle is given as,
$$
\Delta_M \sin{2 \theta_M} = \Delta \sin{2 \theta}
$$
and
$$
\Delta_M \cos{2 \theta_M} = \Delta \cos{2 \theta} - \sqrt{2} G_F N_e
$$
So then, I can write, for example,
$$
\left| \nu_e \right> = \cos{\theta_M} \left| \nu_1 \right> + \sin{\theta_M} \left| \nu_2 \right>
$$

But how does this related to the propagation hamiltonian?

 

[Orodruin]

So then, I can write, for example,
$$
\left| \nu_e \right> = \cos{\theta_M} \left| \nu_1 \right> + \sin{\theta_M} \left| \nu_2 \right>
$$

No. You need to separate the mass eigenstates $|\nu_1\rangle$ and $|\nu_2\rangle$, which are the eigenstates of the vacuum Hamiltonian, from the matter eigenstates $|\nu_{1M}\rangle$ and $|\nu_{2M}\rangle$. These are different bases. However, if you can express the flavor eigenstates in both bases, then you should also be able to write the mass eigenstates (in particular your in state $|\nu_2\rangle$ in terms of the matter eigenstates.

The entire point of the matter eigenstates is that they diagonalise the Hamiltonian in matter so that the propagation is just given by a phase.

 

[thinkLamp]

Ah.
Okay, so
$$
\left| \nu_2 \right> = \cos{\theta} \left| \nu_e \right> - \sin{\theta} \left| \nu_{\mu} \right>
$$
$$
\left| \nu_e \right> = \cos{\theta_M} \left| \nu_1M \right> + \sin{\theta_M} \left| \nu_2 M\right> \\
\left| \nu_{\mu} \right> = - \sin{\theta_M} \left| \nu_1M \right> + \cos{\theta_M} \left| \nu_2 M\right>
$$
So,
$$
\left| \nu_2 \right> = \cos{\theta} \left( \cos{\theta_M} \left| \nu_1 M \right> + \sin{\theta_M} \left| \nu_2 M \right>\right) - \sin{\theta} \left( -\sin{\theta_M} \left| \nu_1 M \right> + \cos{\theta_M} \left| \nu_2 M \right>\right) \\
= \left( \cos{\theta} \cos{\theta_M} + \sin{\theta} \sin{\theta_M} \right) \left| \nu_1 M \right> + \left( \cos{\theta} \sin{\theta_M} - \sin{\theta} \cos{\theta_M} \right) \left| \nu_2 M \right>
$$
Then, we can evolve the 1M and 2M states with our propagation Hamiltonian,
$$
= \left( \cos{\theta} \cos{\theta_M} + \sin{\theta} \sin{\theta_M} \right) e^{-i E_1 t} \left| \nu_1 M \right> + \left( \cos{\theta} \sin{\theta_M} - \sin{\theta} \cos{\theta_M} \right) e^{-i E_2 t} \left| \nu_2 M \right>
$$
Then, using
$$
\left| \nu_e \right> = \cos{\theta_M} \left| \nu_1M \right> + \sin{\theta_M} \left| \nu_2 M\right>
$$
$$
\left< \nu_e | \nu_2, t \right> = \left( \cos{\theta} \cos^2{\theta_M} + \sin{\theta} \sin{\theta_M} \cos{\theta_M} \right) e^{-iE_1t} + \left( \cos{\theta} \sin^2{\theta_M} - \sin{\theta} \cos{\theta_M} \sin{\theta_M} \right) e^{-iE_2t}
$$

Is this the right idea?

 

[Orodruin]

I think you would benefit a lot by applying some trigonometric relations that you should probably be aware of. Otherwise yes.

 

[Orodruin]

Also, your expression for the second mass eigenstate in terms of the flavor states is not correct.

 

[thinkLamp]

Ah, it should be:
$$
\left| \nu_2 \right> = \sin{\theta} \left| \nu_e \right> + \cos{\theta} \left| \nu_{\mu} \right>
$$

Thank you so much for your help! You've been very kind and patient.

 

[Orodruin]

Well, this is a subject I have close at heart. I did my MSc thesis on three-flavour effects in the day-night effect.

 

[thinkLamp]

Oh wow! Nice!