Long-distance interactions ofD-brane bound states and longitudinal five-brane inM(atrix) theory

We discuss long-distance, low-velocity interaction potentials for processes involving a longitudinal $M$5-brane (corresponding in type IIA theory language to the 1/4 supersymmetric bound state of a 4-brane and 0-brane). We consider the following scattering configurations: (a) a $D=11\mathrm{}$ graviton off longitudinal $M5\mathrm{}$-brane, or, equivalently, 0-branes off marginal $4\Vert 0$ bound state; an (b) $M2\mathrm{}$-brane off longitudinal $M5\mathrm{}$-brane, or, equivalently, a nonmarginal $2+0$ bound state off marginal $4\Vert 0$ bound state; (c) two parallel longitudinal $M5\mathrm{}$-branes, or, equivalently, two $4\Vert 0$ marginal bound states. We demonstrate the equivalence between the classical string theory (supergravity) and $M$(atrix) model (one-loop super Yang-Mills) results for the leading terms in the interaction potentials. The supergravity results are obtained using a generalization of a classical probe method which allows one to treat bound states of $D$-branes as probes by introducing nonzero world-volume gauge field backgrounds.