The one-loop worldsheet S-matrix for the AdS_n × Sⁿ × T¹⁰⁻²ⁿ superstring

We compute the massive-sector worldsheet S-matrix for superstring theories in AdS_n × Sⁿ × T¹⁰⁻²ⁿ (with n = 2, 3, 5) in the near BMN expansion up to one-loop order in inverse string tension. We show that, after taking into account the wave function renormalization, the one-loop S-matrix is UV finite. In an appropriate regularization scheme the S-matrix is consistent with the underlying symmetries of the superstring theory, i.e. for the n=3,5casesitcoincideswiththeoneimpliedbythelight-conegaugesymmetrieswiththe dressing phases determined from the crossing equations. For the n = 2, 3 cases we observe that the massless modes decouple from the one-loop calculation of massive mode scattering, i.e. the 2n-dimensional supercoset sigma model and the full 10-dimensional superstring happen to have the same massive one-loop S-matrix.