  The trace anomaly of quantum fields in electromagnetic or gravitational
backgrounds implies the existence of massless scalar poles in physical
amplitudes involving the stress-energy tensor. Considering first the axial
anomaly and using QED as an example, we compute the full one-loop triangle
amplitude of the fermionic stress tensor with two current vertices, <T^{mn} J^a
J^b>, and exhibit the scalar pole in this amplitude associated with the trace
anomaly, in the limit of zero electron mass m -> 0. To emphasize the infrared
aspect of the anomaly, we use a dispersive approach and show that this
amplitude and the existence of the massless scalar pole is determined
completely by its ultraviolet finite terms, together with the requirements of
Poincare invariance of the vacuum, Bose symmetry under interchange of J^a and
J^b, and vector current and stress tensor conservation. We derive a sum rule
for the appropriate positive spectral function corresponding to the
discontinuity of the triangle amplitude, showing that it becomes proportional
to delta function of k^2, and therefore contains a massless scalar intermediate
state in the conformal limit of zero electron mass. The effective action
corresponding to the trace of the triangle amplitude can be expressed in local
form by the introduction of two scalar auxiliary fields which satisfy massless
wave equations. These massless scalar degrees of freedom couple to classical
sources, contribute to gravitational scattering processes, and can have long
range gravitational effects.
