Dimensional transmutation in QED? I am familiar with the QCD dimensional transmutation (QCD is dimensionless theory but via renormalisation we get dimensional parameter; we have to choose some referent scale...), but I'm not sure if there exists dimensional transmutation for QED? If not, why not?
 A: Pure QED, unlike Pure Yang Mills ('pure' in the sense that there is only an $F^2$ term in the lagrangian, and it doesn't couple to matter) is a free theory. That means that it's boring, there's no need for renormalization or perturbation theory or anything. So the coupling constant (in this case the wave function renormalization of the photon) doesn't run with energy scale, and QED retains the conformal properties of the classical theory. Pure Yang Mills, on the other hand, is not free because it is non-Abelian, and the self interactions of the gauge field present in ${\rm tr}F^2$ do induce a running of the Yang Mills coupling constant, breaking conformal invariance (which basically is dimensional transmutation). 
In normal QED with a photon + massive electrons, conformal invariance isn't even present to begin with.
You could ask about QED with a photon coupled to a massless electron. Classically that has conformal invariance. Since the one loop beta function for QED doesn't depend on hte mass of the electron ($\beta(\alpha)=2\alpha^3/3\pi$ using the conventions on wikipedia), $\alpha$ will still run even when we take the mass of the electron to zero and thus conformal invariance is broken / dimensional transmutation occurs. Generically you should expect conformal invariance in interacting theories to get broken by quantum corrections. The known 4d exceptions are very special theories with a lot of symmetry (e.g. $N=4$ super Yang Mills). Critical string theory is another example that of a very special theory that maintains conformal invariance after quantum corrections, where 'nearby' non-critical string theories don't have it.
