Breakdown of the Perturbative Renormalization Group at Certain Quantum Critical Points

We show that a controlled loop expansion for critical exponents can break down if multiple time scales are present at a quantum critical point. We present a mechanism that leads to a divergence of coefficients in a controlled $\epsilon$-expansion. This can invalidate results obtained from a finite-order perturbative renormalization group treatment. The mechanism is explained in terms of dangerous irrelevant variables. As a physically relevant example we discuss the quantum ferromagnetic transition in disordered metals.