  We first analyse the integrable scattering theory describing the massless
excitations of $AdS_2 \times S^2 \times T^6$ superstrings in the relativistic
limit. The matrix part of the S-matrix is obtained in the BMN limit from the
conjectured exact expression, and compared with known S-matrices with N=1
supersymmetry. A dressing factor, yet unknown for the complete theory, is here
constructed based on relativistic crossing symmetry. We derive a Bethe-ansatz
condition by employing a transfer-matrix technique based on the so-called
free-fermion condition. This is known to overcome the problem of lack of a
reference state. We then generalise the method to the massless non-relativistic
case, and compare the resulting Bethe-ansatz condition with a simple massless
limit of the one conjectured by Sorokin, Tseytlin, Wulff and Zarembo.
