Renormalization of the commutative scalar theory with harmonic term to all orders

TL;DR

This paper proves the all-order renormalizability of a commutative scalar field theory with harmonic term, compares its beta function to the noncommutative case, and provides detailed one-loop calculations.

Contribution

It establishes the renormalizability of the commutative harmonic scalar theory to all orders and compares its beta function with the noncommutative Moyal space case.

Findings

The commutative harmonic scalar theory is renormalizable to all orders.

The one-loop beta function of the commutative theory is computed.

Comparison shows differences with the vanishing beta function in the noncommutative case.

Abstract

The noncommutative scalar theory with harmonic term (on the Moyal space) has a vanishing beta function. In this paper, we prove the renormalizability of the commutative scalar field theory with harmonic term to all orders by using multiscale analysis in the momentum space. Then, we consider and compute its one-loop beta function, as well as the one on the degenerate Moyal space. We can finally compare both to the vanishing beta function of the theory with harmonic term on the Moyal space.