Critical behavior of semi-infinite random systems at the special surface transition

We use a three-dimensional massive field theory up to the two-loop approximation to study the critical behavior of semi-infinite quenched random Ising-like systems at the special surface transition. Besides, we extend up to the next-to leading order, the previous first-order results of the $\sqrt{\epsilon }$ expansion obtained by Ohno and Okabe [Phys. Rev. B $46,$ 5917 (1992)]. The numerical estimates for surface critical exponents in both cases are computed by means of the Padé analysis. Moreover, in the case of the massive field theory we perform Padé-Borel resummation of the resulting two-loop series expansions for surface critical exponents. The most reliable estimates for critical exponents of semi-infinite systems with quenched bulk randomness at the special surface transition, which we can obtain in the frames of the present approximation scheme, are ${\eta }_{\Vert }=-0.238,$ ${\Delta }_1=1.098,$ ${\eta }_{\perp }=-0.104,$ ${\beta }_1=0.258,$ ${\gamma }_{11}=0.839,$ ${\gamma }_1=1.426,$ ${\delta }_1=6.521,$ and ${\delta }_{11}=\mathrm{4.249.}\mathrm{}$ These values are different from critical exponents for pure semi-infinite Ising-like systems and show that in a system with quenched bulk randomness the plane boundary is characterized by a new set of critical exponents at the special surface transition.