Renormalization of Pinned Elastic Systems: How Does It Work Beyond One Loop?

We study the field theories for pinned elastic systems at equilibrium and at depinning. Their β functions differ to two loops by novel “anomalous” terms. At equilibrium we find a roughness ζ = 0.20829804ε+0.006858ε² (random bond), ζ = ε/3 (random field). At depinning we prove two-loop renormalizability and that random field attracts shorter range disorder. We find ζ = ε/3(1+0.14331ε), ε = 4-d, in violation of the conjecture ζ = ε/3, solving the discrepancy with simulations. For long range elasticity ζ = ε/3(1+0.39735ε), ε = 2-d, much closer to the experimental value ( ≈0.5 both for liquid helium contact line depinning and slow crack fronts) than the standard prediction 1/3.