We consider zero-temperature transitions from conformal to non-conformal
phases in quantum theories. We argue that there are three generic mechanisms
for the loss of conformality in any number of dimensions: (i) fixed point goes
to zero coupling, (ii) fixed point runs off to infinite coupling, or (iii) an
IR fixed point annihilates with a UV fixed point and they both disappear into
the complex plane. We give both relativistic and non-relativistic examples of
the last case in various dimensions and show that the critical behavior of the
mass gap behaves similarly to the correlation length in the finite temperature
Berezinskii-Kosterlitz-Thouless (BKT) phase transition in two dimensions, xi ~
exp(c/|T-T_c|^{1/2}). We speculate that the chiral phase transition in QCD at
large number of fermion flavors belongs to this universality class, and attempt
to identify the UV fixed point that annihilates with the Banks-Zaks fixed point
at the lower end of the conformal window.Comment: 30 pages, 6 figures; v2: typos fixed, references adde
