  We reveal a new mechanism of conformal symmetry breaking at Born level. It
occurs in generalized form factors with several local operators and an on-shell
state of massless particles. The effect is due to hidden singularities on
collinear configurations of the momenta. This conformal anomaly is different
from the holomorphic anomaly of amplitudes. We present a number of examples in
four and six dimensions. We find an application of the new conformal anomaly to
finite loop momentum integrals with one or more massless legs. The collinear
region around a massless leg creates a contact anomaly, made visible by the
loop integration. The anomalous conformal Ward identity for an $\ell-$loop
integral is a 2nd-order differential equation whose right-hand side is an
$(\ell-1)-$loop integral. We show several examples, in particular the
four-dimensional scalar double box.
