We show that the scattering amplitude of four open string scalars or tachyons on the world-volume of a D$_p$-brane in the bosonic string theory can be written in a universal form. The difference between this amplitude and the corresponding amplitude in the superstring theory is in an extra tachyonic pole. We show that in an $\\\\alpha\\\'$ expansion and for slowly varying fields, the amplitude is consistent with the tachyonic DBI action in which the even part of the tachyon potential is $V(T)=e^{-(\\\\sqrt{\\\\pi} T/\\\\alpha)^2}$ with $\\\\alpha=1$ for bosonic theory and $\\\\alpha=\\\\sqrt{2}$ for superstring theory.

Keywords