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 $\beta$ functions differ to two loops by novel “anomalous” terms. At equilibrium we find a roughness $\zeta =0.208\mathrm{}29804\epsilon +0.006\mathrm{}858{\epsilon }^2$ (random bond), $\zeta =\epsilon /3$ (random field). At depinning we prove two-loop renormalizability and that random field attracts shorter range disorder. We find $\zeta =\frac{\epsilon }{3}(1+0.14331\epsilon )$, $\epsilon =4-d$, in violation of the conjecture $\zeta =\epsilon /3$, solving the discrepancy with simulations. For long range elasticity $\zeta =\frac{\epsilon }{3}(1+0.39735\epsilon )$, $\epsilon =2-d$, much closer to the experimental value ( $\approx 0.5$ both for liquid helium contact line depinning and slow crack fronts) than the standard prediction $1/3$.