Planar Dominance in Non-commutative Field Theories at Infinite External Momentum

TL;DR

This paper investigates whether planar diagrams dominate in non-commutative field theories when external momentum becomes infinite, extending known results from the infinite non-commutativity parameter case to finite values.

Contribution

It demonstrates through explicit two-loop calculations that planar dominance also occurs at finite non-commutativity when external momentum tends to infinity.

Findings

Nonplanar diagrams vanish after renormalization at high external momentum.

Planar dominance extends beyond the infinite non-commutativity limit.

Confirmed in six-dimensional  theory through explicit two-loop analysis.

Abstract

In perturbative expansion of field theories on a non-commutative geometry, it is known that planar diagrams dominate when the non-commutativity parameter $\theta$ goes to infinity. We discuss whether the ``planar dominance'' occurs also in the case where $\theta$ is finite, but the external momentum goes to infinity instead. While this holds trivially at the one-loop level, it is not obvious at the two-loop level in particular in the presence of UV divergences. We perform explicit two-loop calculations in the six-dimensional $\phi^3$ theory, and confirm that nonplanar diagrams after renormalization do vanish in the above limit.