Field-theoretical approach to multicritical behavior near free surfaces

The critical behavior of a semi-infinite $n$-vector model in $4-\epsilon$ dimensions near the special transition is studied. The renormalization of the relevant surface operators is discussed and explicitly carried out in two-loop order. It is found that the crossover exponent ${\phi }_s$ related to the square of the order parameter in the surface does not satisfy the relation ${\phi }_s=1-\nu$ due to Bray and Moore.