A Tower Weak Gravity Conjecture from Infrared Consistency

TL;DR

The paper demonstrates that infrared consistency conditions—causality and analyticity—in EFTs with gravity and multiple U(1) gauge fields impose nontrivial bounds on the charge-to-mass ratios of massive charged states. In 3D, these conditions yield a convex-hull type WGC and, when comparing different U(1) bases, enforce the presence of bifundamental states; in 4D, they reproduce analogous bounds and introduce UV-sensitive parameters that govern the strength of the constraints. Upon KK compactification, the constraints become stronger and force an infinite tower of states with bounded charge-to-mass ratios, i.e., a Tower WGC, with the tower required to include bifundamentals but not necessarily to occupy a full charge lattice. The findings suggest a version of the WGC that sits between the convex-hull and lattice formulations, dependent on UV data encoded in the higher-derivative EFT coefficients, and provide a framework for connecting infrared consistency to swampland criteria in a spectrum-wide manner.

Abstract

We analyze infrared consistency conditions of 3D and 4D effective field theories with massive scalars or fermions charged under multiple $U(1)$ gauge fields. At low energies, one can integrate out the massive particles and thus obtain a one-loop effective action for the gauge fields. In the regime where charge-independent contributions to higher-derivative terms in the action are sufficiently small, it is then possible to derive constraints on the charge-to-mass ratios of the massive particles from requiring that photons propagate causally and have an analytic S-matrix. We thus find that the theories need to contain bifundamentals and satisfy a version of the weak gravity conjecture known as the convex-hull condition. Demanding self-consistency of the constraints under Kaluza-Klein compactification, we furthermore show that, for scalars, they imply a stronger version of the weak gravity conjecture in which the charge-to-mass ratios of an infinite tower of particles are bounded from below. We find that the tower must again include bifundamentals but does not necessarily have to occupy a charge (sub-)lattice.

Figures (4)

• Figure 1: Typical diagrams for the effective $F^4$ operator after integrating out scalars/fermions. In the left, the scalar/fermion loop induces $F^4$ through four gauge couplings. In the middle, the loop induces an $RF^2$ term through two gauge couplings and one gravitational coupling, hence it is $\mathcal{O}(z^2)$. After using the tree-level equation of motion, $R\sim F^2$, it is converted to $F^4$. In the right, the loop induces $R^2$, which is converted to $F^4$ with an $\mathcal{O}(z^0)$ coefficient.
• Figure 2: The blue curve in the left figure is the integration contour for Eq. \ref{['int_IR']}, which captures the IR physics. On the other hand, the one in the right figure is for Eq. \ref{['int_UV']}, which carries the UV information. The integrand accommodates a pole at the origin and discontinuities on the real axis associated to on-shell intermediate states (depicted by red).
• Figure 3: Positivity constraints for 3D EFTs with two $U(1)$'s and particles with charge-to-mass vectors $\vec{z}_a$ (orange arrows). The first example does not satisfy the convex-hull condition (with the positivity bound indicated by the blue circle), and the second one does not have bifundamental particles for all basis choices of the $U(1)$ gauge fields. The third example is consistent with both positivity constraints.
• Figure 4: Starting with a tower of 4D particles, after compactification the charges form a 2D lattice. Consistency requires to sum over all modes such that $m_{nl}^2=m^2+\frac{n^2}{r^2}+l^2\mu^2 \le \Lambda^2$.