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]

## No comments:

## Post a Comment