Musings

Recall that the Weak Gravity Conjecture (WGC) puts a lower bound on the charge-to-mass ratio of the lightest particle charged under an (unbroken) $U(1)$ gauge group:

In a theory which violates this bound, you could form a blackhole (from nonsingular initial data), which would evaporate to form a stable remnant with $$\frac{q_{\text{BH}}}{M_{\text{BH}}} = \frac{1}{M_{\text{pl}}}$$ (the extremal Reissner-Nordstrom bound). For a variety of reasons (explained in the original WGC paper), this is anathema, and the spectrum of any low-energy effective field theory, containing such a $U(1)$, should also contain the requisite light charged states, which satisfy the bound and allow the blackhole to evaporate completely.

Cheung and Remmen point out something peculiar: namely that, while the numerator of (1) is multiplicatively renormalized, the denominator is — in some theories — additively renormalized, and UV-sensitive. A classic example is scalar QED: $$\mathcal{L} = -\frac{1}{4}F^2 + |D_\mu \phi|^2 - m^2 |\phi|^2 - \lambda |\phi|^4$$ The scalar mass receive quadratically-divergent corrections, and it is unnatural for the mass to be much less than the UV cutoff of the effective theory, $m^2 \ll \Lambda^2$. Another example (more of which, anon) is a charged fermion which receives its mass via the Higgs mechanism. The Higgs VEV is UV-sensitive, and it is unnatural for it to be much below the cutoff scale. So, again, it is technically natural for the numerator of (1) to be arbitrarily small, but naturalness suggests that the denominator cannot be, leading to a violation of the bound.

As an example, they consider a variant of the Standard Model with right-handed neutrinos, $\nu_R$, and an unbroken gauged $U(1)_{B-L}$. The neutrino masses, $m_\nu \sim y_\nu \langle H\rangle$, are entirely due to the Higgs mechanism $$\mathcal{L} = ... + y_\nu \nu_R H L + \text{h.c.}$$ and the lightest neutrino ($m_\nu\sim 0.1 \text{eV}$) must satisfy (1) for $U(1)_{B-L}$. This means that the $U(1)_{B-L}$ can be as small as $g\sim 10^{-29}$ — an absurdly small, but technically natural value. If it were this small, then – for fixed $y_\nu$ – the Higgs VEV could not be made much larger than $\langle H\rangle\sim 246 \text{GeV}$, without violating the bound (1).

In other words, whatever mysterious UV physics embeds this variant of the SM in quantum gravity, it must be such as to solve the Hierarchy Problem and guarantee a low electroweak scale.

That sounds crazy. But I think the actually crazy part is the innocuous-sounding assumption that we can take the $U(1)$ gauge coupling to be arbitrarily weak. String Theory puts real impediments in the way of achieving such a scenario.

In F-theory, for instance, 4D gauge theories are associated to 7-branes wrapped on divisors, $D \subset B$, where $B$ is a complex 3-fold (the base of an elliptically-fibered Calabi-Yau 4-fold). The 4D gauge coupling is $$g^2 \sim \frac{1}{\tilde{V}(D)}$$ where $\tilde{V}(D)$ is the volume of $D$ in 10D Planck units. On the other hand, the 4D Planck scale is $$M_{\text{pl}}^2 \sim M_{10}^2 \tilde{V}(B)$$ where $\tilde{V}(B)$ is the volume of $B$ in 10D Planck units and $M_{10}$ is the 10D Planck mass. We would like $\tilde{V}(B) \gg 1$, so that we can use 10D supegravity, algebraic geometry techniques etc., to study the F-theory compactification. But we don’t want $\tilde{V}(B)\ggg 1$; otherwise the cutoff (the Kaluza-Klein scale $\Lambda\sim M_{10}{\tilde{V}(B)}^{-1/6} \sim M_{\text{pl}} {\tilde{V}(B)}^{-2/3}$), beyond which the 4D effective field theory description breaks down, is too low.

But once we put an upper bound on $\tilde{V}(B)$, we’re no longer free to contemplate an arbitrarily large $\tilde{V}(D)$. So we can’t make the gauge coupling arbitrarily weak.

Similar considerations obtain in other string theory scenarios: arbitrarily weak gauge couplings imply a UV cutoff way below the 4D Planck scale. Indeed this was one of the points of the original Weak Gravity paper, as I discussed — at some length — back in the day. This “unexpectedly” lower cutoff scale always intervenes to prevent the would-be conflict in (1), between naturalness and the Weak Gravity Conjecture.