### Gravity is Weak

I wasn’t going to post anything about the recent paper by Arkani-Hamed *et al*, figuring that Luboš is perfectly capable of explaining himself. But, after discussions with various people, it became clear that a few comments are in order.

They argue for two propositions which “must” be true in any theory of quantum gravity, but which are not obvious, at all, from the point of view of low-energy effective field theory. They both concern theories in which there is an unbroken $U(1)$ gauge symmetry in the low-energy theory.

First, they argue that there must exist charged particle(s) in the theory, whose charge-to-mass ratio exceeds^{1} the extremal bound on the charge-to-mass ratio of blackholes. *A-priori*, you could imagine that one could twiddle the masses and charges of fundamental particles arbitrarily. However, as they make clear, in the absence of particles which exceed the bound, charged blackholes cannot radiate away all their charge, and one is left with a large (possibly infinite) number of charged remnants.

This is a perfectly solid result, and one which can be understood quite clearly, once one takes account of blackholes and their evaporation. From it, they abstract away the slogan, “Gravity is the weakest force.” Which leads them to their second conjecture.

The strength of the effective gravitational force grows like a power-law in the UV. The strength of gauge-interactions vary only logarithmically (growing in the UV for abelian gauge theories and falling in the UV for asymptotically-free nonabelian gauge theories). If we go to high enough energies, the gravitational force, therefore, comes to dominate, or would do so if the theory were not cut off.

If we ignore the slow logarithmic running of the gauge coupling, demanding that gauge-interactions dominate over gravitational ones puts a cutoff on effective field theory, not at $M_{\text{pl}}$, but at a lower scale, $g M_{\text{pl}}$.

Now, it’s certainly true in all known string theories, 4D effective field theory breaks down below (often, well below) the 4D Planck scale. Indeed, as Arkani-Hamed knows well, the scale at which 4D effective field theory breaks down *could* be as low as several TeV. I firmly believe (along with the authors) that this is a general principle. But, to put a precise upper bound on the cutoff, at which 4D effective field theory must break down, does require taking account of the running of the gauge coupling.

To sharpen the conjecture, the authors assert that “$g$” in the above formula is the low-energy value of the gauge coupling below the mass of the lightest charged particle. This is not directly related to the “high-energy” value of the coupling (close to the cutoff scale). The rate at which the coupling runs depends on massive charged species at intermediate scales. Not just the magnitude, but even the *sign* of the $\beta$-function could change (if the abelian gauge theory is un-higgsed into a nonabelian one). So the scale at which gravity and gauge interactions become comparable in strength cannot be determined from low-energy data alone; it could be higher or lower than the “naïve” estimate of $g_{\text{IR}}M_{\pl}$.

Indeed, in many string backgrounds, $SU(2)\times U(1)$ is unbroken, and the quarks and leptons are massless. (In the same approximation, supersymmetry is, frequently, also unbroken.) In that case, the $U(1)$ gauge coupling is driven all the way to zero in the IR^{2}. But that does *not* mean that effective field theory has zero range of validity.

One can imagine a self-consistent bound of the form $\Lambda = g(\Lambda) M_{\text{pl}}$. That’s the form of the bound that they actually check in examples. (It’s $g(\Lambda)$ that is directly related to $g_{\text{st}}$, not $g_{\text{IR}}$.) In that form, as they verify, the bound holds^{3}. But it’s not a form that depends solely on low-energy data.

^{1} In the BPS case, saturates the bound.

^{2} The fact that a massless electron causes the gauge coupling to flow to zero in the IR does not contradict the previous argument about the charge-to-mass ratio. Even though both $m$ and $(\log(m))^{-1/2}$ vanish as $m\to 0$, the latter vanishes *more slowly*, so we preserve the fact that we have particles whose charge-to-mass ratio exceeds the extremal bound.

^{3} After some back-and-forth over email, Nima Arkani-Hamed agrees that this, rather than $\Lambda = g_{\text{IR}}M_{\text{pl}}$, is the bound. My argument is that it directly expresses the idea that $\Lambda$ is the scale at which gravity and gauge interactions become comparable in strength. Nima had a more sophisticated argument, involving the evaporation of magnetically-charged blackholes.