# I have some confusion over neutrino oscillations?

1. Mar 9, 2012

### Doofy

I'm trying to learn the basic theory of neutrino oscillations at a postgraduate level. I have a few things that are bothering me.

1) All of the papers & textbooks I have looked at start out by just assuming that each neutrino flavour eigenstate is a superposition of the mass eigenstates. However, I can't work out where this has come from - what led people to this conclusion?
Or was it just that someone happened to be playing around and said "let's just make this assumption, then we can predict oscillations should occur" - then the evidence came along to support it, so now all the stuff I've been reading starts out by just stating that this assumption is true?

2) I'm trying to understand are where the equation for the probability of oscillation between lepton flavours comes from, $P_\nu_\alpha_-_>_\nu_\beta = sin^2\theta sin^2(1.27\frac{\deltam_\alpha_\beta L}{E} )$

I am following this paper's treatment: http://arxiv.org/pdf/hep-ph/0409230.pdf
I understand all the way down until equation 2.7, but have been unable to find any paper that gets me from that equation to the familiar form of the oscillation probability equation I have written in this post above.

Can anyone help me out here?
Thanks.

2. Mar 9, 2012

### CompuChip

I've once written a report on neutrino oscillations for a cosmology course, if you remind me I can how much detail that went into.

3. Mar 9, 2012

Well, I haven't studied this formally, but as I have recently read a SM textbook that describes this and think I just about understand the equations, I will attempt to give a simpler example and, without full rigour, show the derivation. Hopefully the others here will pipe up if I've got any of this wrong!

Consider a neutrino that is created as an electron neutrino with energy E at t=0, z=0, and travels along the z axis. It propogates as a mass eigenstate $\nu$i with mass mi. Assuming it is relativistic, its momentum can be approximated as

$\large p_{i} = E\ (1 - \frac{m^{2}_{i}}{2E})$​

Assuming the neutrino propagates as a Dirac wavefunction with negative helicity, this wavefunction will be of the form

$\large \psi_{L} e^{-i[Et - p_{i}z]}\ \ =\ \ \psi_{L} e^{-i[Et - Ez + \frac{m^{2}_{i}}{2E}z]}$​

where the spinor $\psi_{L} = (0, 1)^{T}$. We will suppress the latter as it has no effect on this immediate calculation.

If I have a detector at z=D, the amplitude for the electron neutrino to propagate as a $\nu_{i}$ eigenstate and then be found as a $\nu_{\alpha} (\alpha = e/\nu/\tau)$ if detected in my detector is

$\large V^{*}_{\alpha i}\ e^{-i[Et - ED + \frac{m^{2}_{i}}{2E}D]}\ V_{ei}$​

where $V_{\alpha i}$ are the PMNS matrix coefficients. To simplify matters, we will approximate these using

$\large sin (\theta_{e2}) = s \approx .84,\ \ cos (\theta_{e2}) = c \approx .54,\ \ sin (\theta_{\mu3}) \approx cos (\theta_{\mu3}) \approx 1/\sqrt{2},\ \ sin (\theta_{e3}) = 0\ \$so$\large \ \ \delta =$ irrelevant​

$V_{e3} = 0$ in this approximation, so the original neutrino can only enter the $\nu_{1}$ and $\nu_{2}$ states, and the total amplitude for being in the $\nu_{\alpha}$ state if detected is thus

$\large V^{*}_{\alpha 1} e^{-i[Et - ED + \frac{m^{2}_{1}}{2E}D]} V_{e1}\ \ +\ \ V^{*}_{\alpha 2} e^{-i[Et - ED + \frac{m^{2}_{2}}{2E}D]} V_{e2}$​

and hence

$\large P (e → \alpha)\ \ =\ \ |(V^{*}_{\alpha 1} e^{-i[Et - ED + \frac{m^{2}_{1}}{2E}D]} V_{e1}\ \ +\ \ V^{*}_{\alpha 2} e^{-i[Et - ED + \frac{m^{2}_{2}}{2E}D]} V_{e2})|^{2}$​

$\large =\ \ (V^{*}_{\alpha 1} e^{-i[Et - ED + \frac{m^{2}_{1}}{2E}D]} V_{e1}\ \ +\ \ V^{*}_{\alpha 2} e^{-i[Et - ED + \frac{m^{2}_{2}}{2E}D]} V_{e2})\ (V_{\alpha 1} e^{\textbf{+}i[Et - ED + \frac{m^{2}_{1}}{2E}D]} V^{*}_{e1}\ \ +\ \ V_{\alpha 2} e^{\textbf{+}i[Et - ED + \frac{m^{2}_{2}}{2E}D]} V^{*}_{e2})$​

Each term in the first bracket has a phase factor of $e^{-i[Et - ED]}$ which is cancelled out by its conjugate on each term in the second bracket, so on multiplying out the brackets we get

$\large P (e → \alpha)\ \ =\ \ |V^{2}_{\alpha 1}| |V^{2}_{e 1}|\ \ +\ \ V^{*}_{\alpha 1}V_{\alpha 2}V_{e 1}V^{*}_{e 2}\ e^{+i[\frac{m^{2}_{2} - m^{2}_{1}}{2E}D]}\ \ +\ \ V_{\alpha 1}V^{*}_{\alpha 2}V^{*}_{e 1}V_{e 2}\ e^{-i[\frac{m^{2}_{2} - m^{2}_{1}}{2E}D]}\ \ +\ \ |V^{2}_{\alpha 2}| |V^{2}_{e 2}|$​

For today's simple example, our approximated PMNS matrix elements are all real, so we can drop the stars on the $V_{\alpha i}$ and we have

$\large P (e → \alpha)\ \ \approx\ \ V^{2}_{\alpha 1} V^{2}_{e 1}\ \ +\ \ V^{2}_{\alpha 2} V^{2}_{e 2}\ \ +\ \ 2 V_{\alpha 1}V_{\alpha 2}V_{e 1}V_{e 2}\ \ cos(\frac{\Delta m^{2}_{12} D}{2E})$​

Plugging in our approximate numbers, for $\alpha = e$ this is approximately

$\large P (e → e)\ \ \approx\ \ c^{4} + s^{4} + 2 c^{2} s^{2} cos(\frac{\Delta m^{2}_{12} D}{2E})\ \ \approx\ \ 0.5 + 0.1 + 0.4\ cos(\frac{\Delta m^{2}_{12} D}{2E})$​