are no additional WGC H scalars present in the spectrum to verify a WGC. Rather the states playing that role will be towers of extremal collective objects, as described in section (\[clave\]).

Eq.(\[ours\]) looks like a condition which imposes that the strength of a scalar interaction must always be stronger than gravity. The presence of the quartic term is crucial to obtain the required consistency for the axion potential and is also justified a posteriori by the results in section (\[clave\]). In fact the factors and signs of the terms in the left are crucial in order to obtain the nice results in that section. We come short of having a Feynman graph explanation for the above differential constraint. One can motivate this expression by considering the short distance behaviour of the potential between two scalars, see fig.(\[higgsfinal\]). At short distances the first term comes from the exchange of the scalar $\phi$, which has the same attractive behaviour than the Newtonian term, $V\simeq -1/r$. The second term includes a direct quartic piece, which is repulsive and proportional to a Dirac delta, hence an UV contribution. On the other hand, in the IR regime, due to the trilinear coupling being super-renormalizable, 