Sunday, 25 May 2008

Producing Neutrinos

Neutrinos are produced only in the Weak interaction. This is because they possess no electric charge (so do not couple to the electromagnetic interaction) and no colour charge (so they don't couple to the strong interaction). They do possess a small amount of mass, so they should be affected by gravity, but we tend to ignore gravity in particle physics because it is much weaker than the other three interactions. There are three mechanisms for neutrino production, described in the Feynman diagrams below for electron-neutrinos (similar diagrams exist for the muon and tau for the W+ and W- versions, and the Z0 version can create all three neutrino flavours, by producing neutrino/anti-neutrino pairs).

Production of a charged electron and an anti-neutrino (to conserve lepton number):


Production of a charged positron (anti-electron) and a neutrino:



Production of neutral neutrino/anti-neutrino pair:



The W+, W- and Z0 are all boson (fields) of the Weak interaction. The Weak interaction has what is called a (V-A) structure, which describes the parity transformation of Weak currents.

The parity operation
\hat{P}
inverts space, so that (x, y, z) --> (-x, -y, -z). Vectors pick up a minus sign under parity, e.g. momentum p (and velocity, v):

\begin{align*}
\mathbf{p}=m\mathbf{v} \\
\mathbf{v}=\mathbf{\dot{r}}\\
\Rightarrow \hat{P}\mathbf{v} = -\mathbf{\dot{r}} = -\mathbf{v}\\
\Rightarrow \hat{P}\mathbf{p} = -m\mathbf{v} = -\mathbf{p}
\end{align*}

The V in (V-A) represents some component of the current which acts like a vector under parity. As for the A, this represents an axial vector (or pseudovector). An axial vector is one which retains its sign under parity, an example is the spin of a particle:

\begin{align*}
\mathbf{\sigma} = \mathbf{r}\times\mathbf{p} \\
\hat{P}\mathbf{\sigma} = -\mathbf{r}\times-\mathbf{p}\\
= \mathbf{r}\times\mathbf{p}\\
= \mathbf{\sigma}
\end{align*}


It is no coincidence that I chose to describe the vector and axial transformations under parity using momentum and spin as examples. Indeed, there is a quantity called Helicity, which is a projection of the spin of a particle along the momentum vector of the particle:

\hat{h} = \frac{\mathbf{\sigma}\cdot\mathbf{p}}{|\mathbf{p}|}

The helicity operator returns eigenvalues +1 (spin aligned with momentum) and -1 (spin anti-aligned with momentum). The +1 state is termed Right-Handed, and the -1 Left-Handed. Helicity commutes with the Hamiltonian, i.e.

\left[\hat{h},\hat{H}\right] = \hat{h}\hat{H}-\hat{H}\hat{h} = 0

which means that helicity states are not affected by motion of the particle (for an example of states which do not commute with the Hamiltonian, see my discussion on neutrino oscillations, particularly that weak flavour eigenstates are not (necessarily) the eigenstates of the Hamiltonian).

There is, however, a problem with helicity. It is possible to make a Lorentz boost (i.e. a relativistic transformation) into an inertial frame where the spin and momentum go from aligned to anti-aligned, or vice versa. This means that helicity is not Lorentz invariant. Lorentz invariance is rather useful when we're talking about relativistic particles, so we define the concept of chirality. A particle is chiral if you cannot superimpose it with its "mirror image". As an example, your left hand is a parity-transformed version of your right hand. Holding the two facing the same direction, it is impossible to superimpose the two and regain the same shape (you can do it by turning one hand around, which is analogous to the parity transformation).

Chirality and helicity are identical only for massless particles, since a massless particle must travel at the speed of light, and no matter how you Lorentz boost, you're not going to make it look any different (constancy of the speed of light in all inertial frames is one of the fundamental postulates of relativity). Chirality is Lorentz invariant, but it is not an eigenstate of the Hamiltonian (except for massless particles, in which case it is identical to Helicity). The chirality operator is
\gamma^5
.

Now, you may be wondering what all this has to do with neutrino interactions. It turns out that the V-A structure of the weak interaction can be represented in terms of gamma matrices. The electromagnetic interaction has a current which looks something like:

J^\mu_{em} = \bar{e}\gamma^\mu e

The Weak interaction, by contrast, has a current which looks something like:

J^\mu_{W} = \bar{e}\frac{1}{2}\gamma^\mu(1-\gamma^5)e

Here, the quantity
\gamma^\mu
transforms as a vector (V) under parity, and
\gamma^5
transforms as an axial vector (A) under parity. This provides the Weak interaction with the desired Vector - Axial (V-A) structure.

An important consequence of this comes from our definition of the chirality operator as
\gamma^5
. Since this operator exists within the Weak current, only left-handed particle states (or right-handed antiparticle states) couple to the Weak interaction, which means that neutrinos can only be produced as left-handed neutrinos (their spin is in the opposite direction to their momentum) or right-handed anti-neutrinos (their spin is in the same direction as their momentum).

Actually, it's a little more complicated than this; the neutral current (Z0) couples to right-handed charged leptons too, but doesn't couple to right-handed neutrinos. This is because the Z0 is actually a combination of two other neutral bosons and provides coupling to the electromagnetic interaction as well as weak interactions. This forms the basis of electroweak unification, and is (for now) beyond the scope of this blog.

If you'd like to know more about neutrinos, I can highly recommend the following:
Neutrino Physics, K. Zuber (Taylor & Francis, 2004) [Amazon]

For more about the V-A structure, interaction currents and electroweak unification, I refer you to the two excellent books by Aitchison and Hey on gauge theories:
Gauge Theories in Particle Physics (Vol. I), I. J. R. Aitchison and A. J. G. Hey (Taylor & Francis, 2003) [Amazon]
Gauge Theories in Particle Physics (Vol. II), I. J. R. Aitchison and A. J. G. Hey (Taylor & Francis, 2004) [Amazon]

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*}