We have the following Lagrangian density:

$$ \mathcal{L} = \bar{\psi}_i i \gamma^\mu \partial_\mu \psi_i + \frac{g^2}{2} \left( \bar{\psi}_i \psi_i \right)^2 $$

which corresponds to the two-dimensional massless Gross-Neveu theory.

We also consider the following gamma matricies:

$$ \gamma^0 = \sigma^2 = \left( \begin{array}{cc} 0 & -i \\ i & 0 \end{array} \right) , \hspace{0,3cm} \gamma^1 = i \sigma^1 = \left( \begin{array}{cc} 0 & i \\ i & 0 \end{array} \right) , \hspace{0,3cm} \gamma^5 = \sigma^3 = \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right) $$

where $\sigma^i$ are the Pauli matricies. After showing that it is invariant over the transformation $\psi_i \rightarrow \gamma_5 \psi_i$, how can we prove that this theory is renormalizable?

  • 1
    $\begingroup$ Hint: Start by showing that $g$ is dimensionless. $\endgroup$
    – Qmechanic
    May 27 at 22:04
  • $\begingroup$ @Qmechanic How do we do that? Could you provide an answer $\endgroup$ May 28 at 11:41
  • 1
    $\begingroup$ Look at the kinetic term first. What is the dimension of ψ? Its square? Its fourth power? Surely you've covered power counting in your course. $\endgroup$ May 28 at 21:39
  • $\begingroup$ @CosmasZachos Nevermind, I arrived at $[g]=0$, but that condition although is suficient it is not enough to prove renormalizability. What do I do next? $\endgroup$ May 30 at 10:41
  • $\begingroup$ Explain. Sufficient normally means enough… $\endgroup$ May 30 at 10:51


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