large internal dimensions. The presence of the large internal dimensions is reflected in the fact that massive Kaluza-Klein modes run in loops of the four dimensional theory. This gives rise to a power law running of the couplings. In  it was suggested that this effect may be responsible for the power law running of couplings in the IIB background discussed in that study. In our case, there is no need for effects due to large internal dimensions and the power law running of the coupling is simply explained by instanton effects in the four dimensional field theory.

Before leaving this section we would like to make some comments on the supergravity interpretation of the leading $|a|^{-4}$ behaviour of the four derivative terms. The operator on the worldvolume which couples to the dilaton is given by $${\cal O}_{\phi}=-{1\over 4}F^{\mu\nu}F_{\mu\nu}.$$

By expanding the Born-Infeld action one finds a four derivative term in the effective potential that has the form

$${\cal O}_{\phi}{\cal O}_{\phi}\Big[{2N\pi\over (a\bar{a})^{2}}+
{\alpha Nm_{1}m_{2}m_{3}m_{4}\tau_{2cl}^{2}\over 4(a\bar{a})^{2}}
\Big({q\over a^{4}}+{\bar{q}\over\bar{a}^{4}}\Big)\Big].$$

The leading term of $|a|^{-4}$ comes from the static massless propagator in the six dimensional transverse space. This term is due to exchange of a dilaton and appears because the supergravity modes which couple 