  While radiative corrections of infrared origin normally depress high energy
amplitudes (Sudakov form factors), we find that in some cases resummation of
leading effects produces exponentials with positive exponents, giving rise to
amplitudes that grow indefinitely with energy. The effect happens in broken
gauge theories like the electroweak sector of the Standard Model, and is
related to the existence of amplitudes that do not respect the gauge symmetry,
and that contrary to expectations do not vanish in the very high energy limit,
but rather become dominant. As a working example we consider a model with two
chiral abelian gauge groups $U'(1)\otimes U(1)$ with large mass splitting
$M_{Z'} \gg M_{Z} $, and we compute leading radiative corrections corrections
to the decay of the heavy extra ${Z'}$ boson into light fermions. For proper
fermionic charges, the chirality breaking magnetic dipole moment, although mass
suppressed, becomes the dominant contribution to the $Z'$ width at very high
energies.
