  In four dimensional unitary scale invariant theories, arguments based on the
proof of the a-theorem suggest that the trace of the energy-momentum tensor T
vanishes when the momentum is light-like, p^2=0. We show that there exists a
local operator O such that the trace is given as T=\partial^2 O, which
establishes the equivalence of scale and conformal invariance. We define the
operator as O=\partial^{-2} T, and explain why this is a well-defined local
operator. Our argument is based on the assumptions that: (1) A kind of crossing
symmetry for vanishing matrix elements holds regardless of the existence of the
S-matrix. (2) Correlation functions in momentum space are analytic functions
other than singularities and branch cuts coming from on-shell processes. (3)
The Wightman axioms are sufficient criteria of the locality of an operator.
