Physics 401 - Spring 2015



The HET Group







George Siopsis - April 21, 2015


The weak force is responsible for $\beta$-decay, \[ n \to p + e + \bar\nu_e \] It looks like a point interaction (no potential involved) of strength \[ \frac{G_F}{\hbar c} \sim \frac{10^{-5}}{m_p^2} \] where $F$ is for Fermi and $m_p$ is the mass of the proton.

This is similar to the gravitational interactions, whose strength is given by Newton's constant, \[ \frac{G_N}{\hbar c} \sim \frac{1}{m_P^2} \ \ , \ \ \ \ m_P \approx 10^{16} m_p \] where $P$ is for Planck. Evidently, $G_F \gg G_N$, i.e., the weak force is much stronger than the gravitational force, which is a mystery of Nature. Thus, the weak force appears to be associated with a mass scale of about $100~m_p$. The corresponding Compton wavelength is \[ \lambda \sim \frac{\hbar}{100 m_p c} \sim 10^{-18} m \] It is possible for this to be the range of weak interactions, because it is so small that $\beta$-decay would still look like a point interaction. To probe such a short distance, one would need high energies. This short range was postulated theoretically without any experimental evidence. It was needed because the quantum theory of point interactions did not make sense at high energies (even though they could not be reached at the time). It implied the existence of a weak potential of the Yukawa type, \[ V \sim \frac{e^{-r/\lambda}}{r} \] and consequently the existence of a weak photon of mass $\sim 100 m_p$. Notice that as the mass goes to zero, $\lambda \to \infty$ and $V\sim \frac{1}{r}$, which is the Coulomb potential generated by massless photons in electromagnetism.

To have a massive photon for the weak interactions, we need to modify the Maxwell equations. The simplest way to do this is by adding to the charge and current densities, \[ \rho \to \rho - \frac{\epsilon_0}{\lambda^2} V \ \ , \ \ \ \ \vec J \to \vec J - \frac{1}{\mu_0 \lambda^2} \vec A \] If you follow the steps that led to the wave equation for the photon in electromagnetism, you can show that with weak interactions, the wave equation is \[ \frac{1}{c^2} \frac{\partial^2 \vec E}{\partial t^2} = \nabla^2 \vec E - \frac{1}{\lambda^2} \vec E \] which, as we have already seen, gives rise to a massive particle (weak photon).

Also, from Gauss's Law, we deduce for a static charge distribution \[ \nabla^2 V + \frac{1}{\lambda^2} V = - \frac{\rho}{\epsilon_0} \] whose solution for a point charge is \[ V = \frac{q}{4\pi\epsilon_0}\, \frac{e^{-r/\lambda}}{r} \] as desired.

I should caution you that in the above (and subsequent) discussion, I am omitting important details, such as the fact that the weak photon is electrically charged (it has to be, since in $\beta$-decay a neutron turns into a proton). However, these details are not needed to understand the fundamental ideas behind the construction of the theory of weak interactions.

By fitting with experimental data, the mass of the weak photon was predicted. It was subsequently (many years later, when high energies were attained that allowed us to probe distances $\sim 10^{-18} m$) observed at CERN in Geneva, Switzerland, and found to have precisely the mass that was predicted theoretically!

So did I successfully modify the Maxwell equations and applied them to weak interactions? Unfortunately, not quite. I added terms to the charge and current densities that were proportional to the potentials which are not invariant under gauge transformations. Thus the electric and magnetic fields no longer uniquely define the system. There are serious consequences to breaking gauge invariance: the quantum theory is badly defined. Instead of breaking the gauge symmetry so clumsily, I shall break it spontaneously. This is called the Higgs mechanism and implies the existence of a Higgs particle, which was detected, once again at CERN in Geneva, Switzerland (at the Large Hadron Collider (LHC)) in 2011.



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