Plane-chain coupling in YBa$_{2}$Cu$_{3}$O$_{7}$ : impurity effect on the critical temperature

We have studied the effect of impurities on $T_{c}$ for a model of YBCO involving pairing both in the CuO$_{2}$ planes and in the CuO chains. In this model pairing in the planes is due to phonons, while Coulomb repulsion induces in the chains an order parameter with opposite sign. Due to the anticrossing produced by hybridization between planes and chains, the order parameter changes sign on a single sheet of the Fermi surface resulting in nodes in the gap. We find that $T_{c}$ is much less sensitive to impurities than in standard d-wave models. One reason is that impurities produce essentially plane-plane and chain-chain scattering, which does not affect the critical temperature. $T_{c }$ is reduced by plane-chain scattering, which is smaller by a factor of order hybridization coupling over Fermi energy. In the specific case which we have studied in details and which reduces to the two-band model, we have found a further reduction of the sensitivity of $T_{c}$ to impurities with a behaviour which can vary continuously from s-wave like to d-wave like depending on the parameters. We expect a similar reduction to occur in the general case.