# A Strong Scalar Weak Gravity Conjecture and Some Implications

## TL;DR

This paper introduces a new scalar Weak Gravity Conjecture that constrains scalar field interactions, leading to implications for inflation, compactification, and string theory models, emphasizing the necessity of certain bounds for consistency with quantum gravity.

## Contribution

It proposes a novel differential constraint for scalar fields in quantum gravity, connecting extremal states, extra dimensions, and swampland conjectures, with broad implications for theoretical physics.

## Key findings

- Scalar self-interactions must be stronger than gravity.
- Constraints on axion decay constants and inflation potentials.
- Implications for neutrino masses and string compactifications.

## Abstract

We propose a new version of the scalar Weak Gravity Conjecture (WGC) which would apply to any scalar field coupled to quantum gravity. For a single scalar it is given by the differential constraint $V''\leq (2V'''^2/V''-V'''')M_{\text{p}}^2$. It corresponds to the statement that self-interactions of a scalar must be stronger than gravity for any value of the scalar field. We find that the solutions which saturate the bound correspond to towers of extremal states with mass $m^2(\phi)=m_0^2/((R/m)^2+1/(nR)^2)$, with $R^2=e^\phi$, consistent with the emergence of an extra dimension at large or small $R$ and the existence of extended objects (strings). These states act as WGC states for the scalar $\phi$. It is also consistent with the distance swampland conjecture with a built-in duality symmetry. From this constraint one can derive several swampland conjectures. In particular one finds that an axion potential is only consistent if $f\leq M_{\text{p}}$. The conjecture has far reaching consequences and applies to several interesting physical systems: i) Among chaotic inflation potentials only those asymptotically linear may survive. ii) If applied to the radion of the circle compactification of the Standard Model to 3D, the constraint implies that the 4D cosmological constant scale must be larger than the mass of the lightest neutrino, which must be Dirac and in normal hierarchy. iii) It also constraints simplest moduli fixing string models. The simplest KKLT model is compatible with the constraints but the latter may be relevant for some choices of parameters.