Energy-momentum tensors and z-diagrams

Abstract

The discussion of Z-diagrams is extended to equal-time commutators of the charges Q D α (0) = ∫ d 3 x x m T 0 m α ( x) with space and time components of currents J μ and their divergences. (We have denoted by T μν α any symmetric energy-momentum tensor within the Callan-Coleman-Jackiw class parametrized by α.) The requirement of absence of Z-diagrams in the infinite momentum limit for boson matrix elements of either of the equal time commutators [ Q D α (0), ∂ μ J μ (0)], [ T μ αμ (0)] or [ Q D α (0), J k (0) restricts the energy momentum tensor to the new and improved one. This tensor T μν satisfies the preceeding requirement and, as a consequence, Z-diagrams do not contribute in the infinite momentum limit to boson matrix elements of the commutators [ Q D(0), ∂ μ J μ (0)], [ T μ μ (0), Q(0)] and [ Q D(0), J k (0)] (with Q D the dilatation charge). For fermion matrix elements of [ Q D(0), ∂ μ J μ (0)] Z-diagrams do give a contribution in the infinite momentum limit. Furthermore, in this limit no Z-diagrams contribute to fermion matrix elements of [ Q D(0), J μ (0)] and to boson matrix elements of [ Q D α (0), J o (0)] (for any α). Moreover, assuming one particle saturation in the infinite momentum limit of both meson and fermion matrix elements of [ Q D(0), ∂ μ J μ (0)] and [ T μ μ (0), Q(0)] one consistently derives that ∂ μ J μ has dimension two if it has a dimension at all.