terms of the effective AdS scale $\ell_{\rm eff} = \sqrt{- 3/\Lambda}$: $$\label{eq:lrunning}
{1\over\ell^2_{\rm eff}} = {1\over\ell^2_A} \left[ 1 - {4\pi G\over 3\ell^2_A} (K_0\ell^4_A)\ln {x^2_{\rm phys}\over\ell^2_0}\right] ~.$$ The combination $(K_0\ell^4_A)$ is a pure number that we have computed above for some specific fields. The most important part of this expression is the absence of power law corrections that would enter through the UV cutoff $\epsilon$. This would signal dependence on unknown UV physics. Instead we have nontrivial logarithmic quantum corrections that are computable within the low energy theory. [^5]

A good way to construct AdS supergravity is to gauge supergravity in flat space. This procedure identifies the gauge coupling constant as [@Freedman:1976aw; @deWit:1981yv] $$e^2 = {4\pi G\over\ell^2_A}~.$$ This coupling constant is small $e^2\ll 1$ when the AdS radius is much larger than Planck scale as we have implicitly presumed. Resumming the (possibly large) logarithms we can recast (\[eq:lrunning\]) as $$e^2 = {e^2_0\over 1 + {1\over 3}e^2_0 (K_0\ell^4_A)\ln {x^2_{\rm phys}\over\ell^2_0}} ~.$$ Comparing with standard formulae we can write an effective $\beta$-function for these theories $$\beta = - {b\over 16\pi^2} e^3~.$$ where $$b = - {1\over 3} (16\pi^2K_0\ell^4_A)~.$$ The $\beta$-function determines the running of a dimensionless version of the cosmological constant through the 