The $\beta$ -function in duality-covariant non-commutative $\phi^4$ -theory

Abstract

We compute the one-loop $\beta$ -functions describing the renormalisation of the coupling constant $\lambda$ and the frequency parameter $\Omega$ for the real four-dimensional duality-covariant non-commutative $\phi^4$ -model, which is renormalisable to all orders. The contribution from the one-loop four-point function is reduced by the one-loop wavefunction renormalisation, but the $\beta_\lambda$ -function remains non-negative. Both $\beta_\lambda$ and $\beta_\Omega$ vanish at the one-loop level for the duality-invariant model characterised by $\Omega = 1$ . Moreover, $\beta_\Omega$ also vanishes in the limit $\Omega\to 0$ , which defines the standard non-commutative $\phi^4$ -quantum field theory. Thus, the limit $\Omega\to 0$ exists at least at the one-loop level.