The Cutkosky rule of three dimensional noncommutative field theory in Lie algebraic noncommutative spacetime

TL;DR

This paper examines the unitarity of a three-dimensional noncommutative scalar field theory with Lie algebraic spacetime structure, confirming the Cutkosky rule holds under certain mass conditions when incorporating braiding effects.

Contribution

It demonstrates the validity of the Cutkosky rule in a specific noncommutative field theory with Hopf algebraic symmetry, highlighting the importance of braiding at the quantum level.

Findings

Cutkosky rule is satisfied for masses less than 1/(√2)κ

Inclusion of braiding is essential for Hopf algebraic symmetry at quantum level

Unitarity is preserved under specified mass conditions

Abstract

We investigate the unitarity of three dimensional noncommutative scalar field theory in the Lie algebraic noncommutative spacetime [x^i,x^j]=2i kappa epsilon^{ijk}x_k. This noncommutative field theory possesses a SL(2,R)/Z_2 group momentum space, which leads to a Hopf algebraic translational symmetry. We check the Cutkosky rule of the one-loop self-energy diagrams in the noncommutative phi^3 theory when we include a braiding, which is necessary for the noncommutative field theory to possess the Hopf algebraic translational symmetry at quantum level. Then, we find that the Cutkosky rule is satisfied if the mass is less than 1/(2^(1/2)kappa).