Robert Mann - An Introduction to Particle Physics and the Standard Model · Problem 8

Testing QED — Robert Mann

Physics — Particle Physics · scalar-spinor-coupled-theory-gauge-covariance

Problem

Consider a theory of one complex scalar particle with wavefunction

\varphi

and two spin-

\frac{1}{2}

particles

\psi

and

\chi

, each of which couples to the photon via the equations

\begin{aligned} (i\gamma^\mu D_\mu - m_\psi)\psi &= g\varphi\chi \\ (i\gamma^\mu D_\mu - m_\chi)\psi &= g\varphi^*\psi \\ D^\mu D_\mu\varphi + m^2_\varphi\varphi &= g\bar{\psi}\chi \end{aligned}

where

m^2_\varphi

m_\psi

, and

m_\chi

are the respective masses of the

\varphi

\psi

and

\chi

particles. (a) Find the most general local phase transformation of

\varphi

\psi

and

\chi

that leaves this system gauge-covariant. (b) What is the relationship between the charges of

\varphi

\psi

and

\chi

? (c) In Maxwell's equations

\partial^\mu F_{\mu\nu} = \mathcal{J}_\nu

what is the current

\mathcal{J}_\nu

for this theory?