  We investigate disorder relevance for the pinning of a renewal whose
inter-arrival law has tail exponent $\alpha>0$ when the law of the random
environment is in the domain of attraction of a stable law with parameter
$\gamma \in (1,2)$. We prove that in this case, the effect of disorder is not
decided by the sign of the specific heat exponent as predicted by Harris
criterion but that a new criterion emerges to decide disorder relevance. More
precisely we show that when $\alpha>1-\gamma^{-1}$ there is a shift of the
critical point at every temperature whereas when $\alpha< 1-\gamma^{-1}$, at
high temperature the quenched and annealed critical point coincide, and the
critical exponents are identical.
