Consider the pseudoscalar Yukawa theory in Eq. (7.6.11) with a \(\phi^{4}\) interaction, \[\begin{equation*}\mathcal{L}=\frac{1}{2}\left(\partial_{\mu} \phi \partial^{\mu} \phi-m_{\phi}^{2} \phi^{2}\right)+\bar{\psi}(i ot \partial-m) \psi-\frac{\lambda}{4 !} \phi^{4}-i g \phi \bar{\psi} \gamma^{5} \psi . \tag{8.9.10}\end{equation*}\]

(a) Calculate the one-loop self-energy for the scalar and for the fermion and write down the \(\overline{\mathrm{MS}}\) counterterms.

(b) Calculate the \(\overline{\mathrm{MS}}\) one-loop \(\beta\)-functions for both couplings, \(\beta_{\lambda}(g, \lambda)\) and \(\beta_{g}(g, \lambda)\).