text. We start with the Hamiltonian $$\begin{aligned}
\mathcal{H}_0(k)&=& \frac{k^2}{2M_0} \Gamma^0+\epsilon_a(k) \Gamma^a  + \frac{d_4(k) \Gamma^4 + d_5(k) \Gamma^5}{2M_c} \ , \nonumber\end{aligned}$$ where we defined $\Gamma^0$ as the $4\times 4$ unit matrix to emphasize the matrix structure. The corresponding effective Lagrange density is $$\begin{aligned}
\mathcal{L}_{\rm eff} = \psi^{\dagger} (-i \omega_n  \Gamma^0
+\mathcal{H}_0(k) - \Sigma_{f}(k,i \omega_n))\psi \nonumber .\end{aligned}$$ Here we have included the one-loop self energy needed for the $\varepsilon$ expansion. It is given by $$\begin{aligned}
  && \Sigma_{\psi}(k, i \omega_n) = \\ 
  && -g^2 \int_q T \sum_{\Omega_n} G_f(k+q, i \omega_n+i \Omega_n) G_{\varphi} (q,i \Omega_n) \ ,\nonumber \end{aligned}$$ where $G_f^{-1}(k,i \omega) = -i \omega \Gamma^0 +\mathcal{H}_0(k)$, $G_{\varphi}^{-1}(q,i \Omega) = c_0 q^2$ are used. Note that the bosonic propagator does not have frequency dependence since it represents the instantaneous Coulomb interaction. By integrating out high momentum degrees of freedom, we investigate the renormalization of fermion parameters.

Frequency and momentum dependences of the self energy determine anomalous dimensions of the fermion field and mass terms, respectively. At one-loop level, the fermion self energy does not have frequency dependence due to the instantaneous Coulomb interaction, so no anomalous dimension of the fermion field appears. Moreover, the frequency-independence makes one mass coupling ($1/M_0$) unchanged at one loop level 