The Cutkosky Rule of Three Dimensional Noncommutative Field Theory in Lie Algebraic Noncommutative Spacetime

We have investigated the unitarity of three dimensional noncommutative scalar field theory in Lie algebraic noncommutative spacetime [xi, xj] = 2iκeijkxk, (i, j, k = 0, 1, 2). This noncommutative field theory possesses an SL(2, R)/Z2 group momentum space, which leads to a Hopf algebraic translational symmetry. We have checked the Cutkosky rule of the one‐loop self‐energy diagrams in the noncommutative φ3 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 have found that the Cutkosky rule is satisfied if the mass of the scalar field is less than 1/2κ, which however leads to be violations of the Cutkosky rule for smaller masses in more complicated diagrams.