Saturday, 24 May 2008

The Quantum Mechanics of Neutrino Oscillations

Neutrinos are neutral (i.e. carry no electric charge) leptons with a very small (but non-zero!) mass. They are produced only in Weak interactions (those involving W+, W- or Z0 bosons) and have incredibly low interaction cross-sections. A neutrino produced in the core of the Sun, for example, passes almost straight out without interacting. By contrast, photons produced in the core of the Sun scatter many times and take upwards of
10^5
years to reach the surface. Neutrinos therefore provide an almost real-time picture of the fusion processes occurring in the core of the Sun (since neutrinos are produced in a number of the processes involved in the fusion cycles within the Solar core).

An interesting phenomenon in neutrino physics is neutrino flavour oscillation. As you may know, charged leptons exist in three flavours: electron, muon and tau... denoted
(e, \mu, \tau)
. For each lepton flavour, there is an associated neutrino:
(\nu_e, \nu_\mu, \nu_\tau)
. What is interesting here is that the weak flavour states (which determine the physically observed neutrino states) are not eigenstates of the Hamiltonian.

In slightly less technical terms, this means that the mass states of neutrinos are not the same as their flavour states, and that it is the mass states which are responsible for propagation (since the Hamiltonian describes the motion of the system). It is the flavour states (here, I'm talking about quantum mechanical eigenstates) which are responsible for neutrino interactions, but it is the mass eigenstates which are responsible for neutrino propagation.

Each flavour state corresponds to a linear combination of mass states, so a neutrino produced in the Sun as an electron-neutrino has components of all three mass states. As it propagates, the mass states (which differ in mass [and therefore momentum] by a very small amount) begin to lead (or lag) each other, introducing a phase difference. At some later point, if the neutrino wavepacket wants to interact, it must condense down to a flavour eigenstate. The flavour state it chooses is the one which most closely matches the relative phase of the mass states at that moment in time.

It is possible to derive the quantum mechanics of neutrino oscillations by treating the neutrino as a wavepacket and performing a full quantum mechanical calculation of the time evolution of such a wavepacket, and this approach can be seen in the paper On the quantum mechanics of neutrino oscillation, B. Kayser, Phys. Rev. D, 24(1), 1981.

Here, I will take a less fundamental approach, but one that is hopefully simpler to understand without detailed quantum theory. I will also work with just two neutrino flavours, and two neutrino masses, rather than treating all three. To do this with three neutrinos is almost identical, but the maths gets longer!

Firstly, we define the neutrino flavour states as a linear combination of mass states. This can be done for two neutrino flavour states (and two neutrino mass states) using a standard 2D rotation matrix:

\left( \begin{array}{c} |\nu_\alpha\rangle \\ |\nu_\beta\rangle \end{array}\right) = \left( \begin{array}{cc} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{array} \right) \left( \begin{array}{c} |\nu_1\rangle \\ |\nu_2\rangle \end{array}\right)


In order to consider the propagation of a neutrino, we need to write the mass states (responsible for propagation) as a linear combination of the flavour states (responsible for generation of the neutrino in the first place). We can do this almost directly from the above equation, giving:

\left( \begin{array}{c} |\nu_1(0,0)\rangle \\ |\nu_2(0,0)\rangle \end{array}\right) = \left( \begin{array}{cc} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{array} \right) \left( \begin{array}{c} |\nu_\alpha(0,0)\rangle \\ |\nu_\beta(0,0)\rangle \end{array}\right)

Where
|\nu_\alpha(x,t)\rangle
is a flavour state at position x and time t, and
|\nu_1(x,t)\rangle
is a mass state at position x and time t.

Now, propagation can be modeled as a plane-wave, such that:

|\nu_k(x,t)\rangle = e^{i\phi_k}|\nu_k(0,0)\rangle

(Where k denotes one of the two mass states). We can once again use matrix form to express the propagation of the two mass states:

\left( \begin{array}{c} |\nu_1(x,t)\rangle \\ |\nu_2(x,t)\rangle \end{array}\right) = \left( \begin{array}{cc} e^{i\phi_1} & 0 \\ 0 & e^{i\phi_2} \end{array} \right) \left( \begin{array}{c} |\nu_1(0,0)\rangle \\ |\nu_2(0,0)\rangle \end{array}\right)


Finally, we can use the first matrix we wrote down to write the final flavour eigenstates in terms of the final mass eigenstates. Combining it all:

\left( \begin{array}{c} |\nu_\alpha(x,t)\rangle \\ |\nu_\beta(x,t)\rangle \end{array} \right) = \left( \begin{array}{cc} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{array} \right) \left( \begin{array}{cc} e^{i\phi_1} & 0 \\ 0 & e^{i\phi_2} \end{array} \right) \left( \begin{array}{cc} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{array}\right) \left( \begin{array}{c} |\nu_\alpha(0,0)\rangle \\ |\nu_\beta(0,0)\rangle \end{array}\right)


Multiplying out the matrices, we end up with (using
c = \cos\theta, s = \sin\theta
):

\left( \begin{array}{c} |\nu_\alpha(x,t)\rangle \\ |\nu_\beta(x,t)\rangle \end{array} \right) = \left( \begin{array}{cc} c^2 e^{i\phi_1} + s^2 e^{i\phi_2} & -cs e^{i\phi_1} + cs e^{i\phi_2} \\ -cs e^{i\phi_1} + cs e^{i\phi_2} & s^2 e^{i\phi_1} + c^2 e^{i\phi_2} \end{array} \right) \left( \begin{array}{c} |\nu_\alpha(0,0)\rangle \\ |\nu_\beta(0,0)\rangle \end{array}\right)


We can now compute the matrix elements
\langle\nu_\alpha(x,t)|\nu_\alpha(0,0)\rangle>
(survival) or
\langle\nu_\beta(x,t)|\nu_\alpha(0,0)\rangle
(disappearance). The matrix elements squared correspond to the survival probability (e.g. electron-neutrinos remaining as electron-neutrinos) and the disappearance probability (e.g. electron-neutrino to muon-neutrino oscillation). For the example of survival:

\langle\nu_\alpha(x,t)| = \left(c^2 e^{-i\phi_1}+s^2 e^{-i\phi_2}\right)\langle\nu_\alpha(0,0)| + \left(-cs e^{-i\phi_1}+cs e^{-i\phi_2}\right)\langle\nu_\beta(0,0)|


The survival amplitude is:

\begin{align*}
\langle\nu_\alpha(x,t)|\nu_\alpha(0,0)\rangle = \left(c^2 e^{-i\phi_1}+s^2 e^{-i\phi_2}\right)\langle\nu_\alpha(0,0)|\nu_\alpha(0,0)\rangle \\
+ \left(-cs e^{-i\phi_1}+cs e^{-i\phi_2}\right)\langle\nu_\beta(0,0)|\nu_\alpha(0,0)\rangle
\end{align*}


For a source of pure "alpha type" neutrinos,

\begin{align*}
\langle\nu_\alpha(0,0)|\nu_\alpha(0,0)\rangle = 1 \\
\langle\nu_\beta(0,0)|\nu_\alpha(0,0)\rangle = 0
\end{align*}


Giving:

\langle\nu_\alpha(x,t)|\nu_\alpha(0,0)\rangle = \left(c^2 e^{-i\phi_1}+s^2 e^{-i\phi_2}\right)


\begin{align*}
\left| \langle\nu_\alpha(x,t)|\nu_\alpha(0,0)\rangle \right|^2 = \left(c^2 e^{-i\phi_1}+s^2 e^{-i\phi_2}\right)\left(c^2 e^{+i\phi_1}+s^2 e^{+i\phi_2}\right) \\
= c^4 + s^4 + c^2s^2\left[e^{i(\phi_2-\phi_1)} + e^{-i(\phi_2-\phi_1)}\right]\\
= (c^2+s^2)^2 - 2c^2s^2\left[1-\cos(\phi_2-\phi_1)\right] \\
= 1 - \sin^2(2\theta)\sin^2\left(\frac{\phi_2-\phi_1}{2}\right)
\end{align*}

The form of the phases is:

\phi_i(x,t) = E_i t - p_i x

After some rearrangement, using relativistic approximations and introducing a baseline length L from generation to detection, the survival probability can be written as:

\begin{align*}
P(\nu_\alpha \rightarrow \nu_\alpha) = 1 - \sin^2(2\theta)\sin^2\left(\frac{\Delta m^2 L}{4E}\right) \\
= 1 - \sin^2(2\theta)\sin^2\left(1.27\frac{\Delta m^2 L}{E}\right)
\end{align*}

Where the final form takes energy E in GeV, mass difference
\Delta m^2 ~\mathrm{in~eV^2}
and L in km. This final form can be used to tune a neutrino oscillation experiment for a maximum in the oscillation probability (which is 1 - survival probability):

\begin{align*}
P(\nu_\alpha \rightarrow \nu_\beta) = 1 - P(\nu_\alpha \rightarrow \nu_\alpha) \\
= \sin^2(2\theta)\sin^2\left(1.27\frac{\Delta m^2 L}{E}\right)
\end{align*}

1 comment:

  1. Woo! Sciencing at it's finest! Shame it was 30 minutes for 11 marks. Godsdammits.

    ReplyDelete