Weak gravity conjecture and effective field theory

The weak gravity conjecture (WGC) is a proposed constraint on theories with gauge fields and gravity, requiring the existence of light charged particles and/or imposing an upper bound on the field theory cutoff $\mathrm{\Lambda }$. If taken as a consistency requirement for effective field theories (EFTs), it rules out possibilities for model building including some models of inflation. I demonstrate simple models which satisfy all forms of the WGC, but which through Higgsing of the original gauge fields produce low-energy EFTs with gauge forces that badly violate the WGC. These models illustrate specific loopholes in arguments that motivate the WGC from a bottom-up perspective; for example the arguments based on magnetic monopoles are evaded when the magnetic confinement that occurs in a Higgs phase is accounted for. This indicates that the WGC should not be taken as a veto on EFTs, even if it turns out to be a robust property of UV quantum gravity theories. However, if the latter is true, then parametric violation of the WGC at low energy comes at the cost of nonminimal field content in the UV. I propose that only a very weak constraint is applicable to EFTs, $\mathrm{\Lambda }\lesssim (\log \frac{1}{g}{)}^{-1/2}{M}_{\mathrm{pl}}$, where $g$ is the gauge coupling, motivated by entropy bounds. Remarkably, EFTs produced by Higgsing a theory that satisfies the WGC can saturate but not violate this bound.

Figure 1

Spectrum of charged particles (blue lines) and gauge fields (red lines) for the model of Sec. 3, along with the mass scales implied by the WGC (black lines). This spectrum satisfies the various WGCs for the $A$ and $B$ fields when the scalar of charge ($Z$, $1$) does not have a vacuum expectation value (VEV). However, when the ($Z$, $1$) field acquires a VEV, then most forms of the WGC (see Table 1) are violated for the remaining massless gauge field, which has coupling $g_{\mathrm{eff}}=g/Z$.

Figure 2

Magnetic monopoles of the $A$, $B$ gauge fields in the two phases of the model. Left: In the Coulomb phase, the magnetic WGC requires that magnetic monopoles of both the $A$ and $B$ fields exist, with magnetic charge $2\pi /g$ and size of order the cutoff length ${\mathrm{\Lambda }}^{-1}$, where $\mathrm{\Lambda }\lesssim g{M}_{\mathrm{pl}}$. Right: In the Higgs phase where a scalar of charge ($Z$, $1$) gets a VEV, magnetic flux of the massive gauge boson $A+B/Z$ is confined, and only net $B-A/Z$ flux can escape to infinity. A monopole of $B-A/Z$ charge can be formed by joining $Z$ monopoles of $B$ and one antimonopole of $A$, with the $A+B/Z$ field confined to flux tubes.

Figure 3

A sketch of a possible inflaton potential generated within the biaxion models discussed here and in [33], for an EFT cutoff that is only slightly above the compactification scale. In this case the potential may receive a suppressed “higher harmonic” contribution with a period $1/Z$ times the total field range, leading to observable oscillations in the primordial power spectrum. (The amplitude of this contribution is exaggerated in this figure; current data constrain it to be smaller than pictured.)