*See also Jacques Distler's blog post for a more technical review of the paper.*

The Weak Gravity Conjecture (WGC) by Arkani-Hamed, Vafa, Nicolis, and your humble correspondent is arguably the most well-known example of Vafa's "swampland" restrictions – conditions obeyed in string theory and/or/i.e. consistent theories of quantum gravity that have no reason to be obeyed in general effective field theories. WGC and other swampland restrictions are testimonies of the bonus predictive power offered by string theory. (Ironically enough, WGC actually has more citations than the original swampland paper.)

WGC effectively claims that gravity has to be the weakest force (yes, just like it is the weakest one in the Universe around us). More specifically, for every other force, e.g. a \(U(1)\) gauge force, there has to exist a sufficiently light particle species for which the gravitational force between it and its identical partner is weaker than the gauge force. In normal variables, it means that a particle lighter than \(g_{U(1)}m_{\rm Pl}\) has to exist.

So very weakly coupled gauge theories may be "allowed" but they're disfavored in the sense that they inevitably have consequences for low-energy physics. The weaker gauge coupling you consider, the lower is the mass scale at which you may see additional consequences of the weak coupling.

I am still not sure what is the most universal form of the principle that is valid in a theory with many non-gravitational forces, magnetic monopoles, dyons, and so on; instead, I am nearly completely convinced that we haven't quite solved this problem. But I still think that at least at some level, the inequality has to be right.

The conjecture may be supported by diverse pieces of evidence. First, quantum gravity should forbid "global symmetries" and a gauge symmetry with a very weak \(g\) seems to be close and converge to a global symmetry for \(g\to 0\). That's a marginal offense and quantum gravity marginally fights against this offense by restricting the interval of energy scales where the original theory without new states may be valid.

Second, the "very light charged particles" whose existence WGC demands boast a charge-to-mass ratio that obeys the same inequality as that of the superextremal (forbidden) black holes. That's OK because the particles are formally "supertiny" black holes for which the extremality bound is inapplicable (the quantum corrections to these micro black holes are significant). In fact, the bound has to be "reverted" for ordinary large extremal black holes to be able to "shed" their charge and decay (Hawking evaporate). If they couldn't decay, there would be a de facto unlimited number of stable black hole microstates with an arbitrarily large electric charge. That would be like having an unlimited assortment of black hole remnants and such a situation would imply the same inconsistencies as black hole remnants in general. (Particularly dumb people such as Carlo Rovelli have failed to notice that there are any inconsistencies with remnants – for several decades. So you may see another paper with this breathtakingly stinky shit under a new name.)

One may also argue that a minimally charged magnetic monopole shouldn't be a black hole yet.

Finally, the inequality seems to be nontrivially obeyed by a number of classes of string-theoretical compactifications. The new light states predicted by WGC may be strings, monopoles, and other objects – or the inequality arises from the inequality guaranteeing the validity of an effective theory. See the paper for details.

The WGC inequality is natural and "holds" in the real world but it is a close relative of other inequalities that may look "harmful" to some eyes. For example, an inequality analogous to WGC seems to imply that inflation in string theory cannot be a slow-roll inflation: the inequality demanded by a generalized WGC is exactly the opposite one than the slow-roll condition. WGC is actually not the only reason to think that at the end, string theory could prefer Alan Guth's old (tunneling-based) inflation models relatively to Andrei Linde's new (slow-roll) inflation.

Finally, there is a new paper today about the "naturalness vs WGC":

Naturalness and the Weak Gravity Conjecture(Caltech Junior Professor) Clifford Cheung and (Hertz Foundation Fellow i.e. top student) Grant N. Remmen argue that the WGC conjecture reparameterized in a certain way seems to be at odds with the inequalities considered to define "naturalness". If true, you could say that at least at some heuristic level, string theory has always been predicting naturalness to break down.

They finally extend their observations to rather general theories with scalars and gauge fields and their final example – an extended electroweak theory – shows that naturalness may be rather sharply forbidden by WGC.

The reason for the stigmatization of naturalness is always hiding in the charged fundamental scalars. Scalar quantum electrodynamics is their simplest toy model to clarify the observation. The authors have to carefully decide what is the scale at which WGC demands \(m\lt q\) – it's the scale of the mass of the particle \(m\) itself. They analyze the RG running of the mass and conclude that WGC has particularly new implications if \(e^2\ll \lambda\) i.e. the quartic coupling is much larger than the squared gauge coupling. If that's so, WGC requires a mandatory fine-tuning \(b\to 0\).

I am of course hugely sympathetic to their proposal, or at least its general philosophy, because I've been saying that similar modifications of the common wisdom could arise in QG/ST for quite some time. For example, in a recent article on JFK conspiracy theorists and naturalness, I argued that (see e.g. the last paragraph) an effective field theory is only good at describing the effective phenomena, and it shouldn't be trusted when it comes to global questions such as "where it must break down" and "what values of the parameters are more natural than others".

Answers to such questions – about parameters etc. – may only come from a complete theory. The effective field theory may "favor" certain values of coefficients but quantum gravity may add a "pressure" for very different inequalities and correlations, so the ultimate patterns of the coupling constants we find in Nature may naturally be very different from those that would be considered "natural" by the effective field theory itself.

So far – or at least before this new paper – I would think that people were very naive about "what is actually natural" in Nature when they were completely overlooking all arguments that go beyond effective field theory (in particular, all issues of quantum gravity). We should get more mature about these matters and I hope that the new paper will help physicists in this process.

I tend to agree with (but I am not quite sure about) some of the criticism by Jacques. For example, they consider a tiny coupling of \(10^{-29}\) or so – which the present authors envision as a reason for the Higgs not to be too much heavier than \(246\GeV\) as they link the Higgs mass to the neutrino mass – also implies a tiny cutoff relatively to the Planck scale. Doesn't WGC imply just new states, while the cutoff may be much higher?

If the present authors suggest that some string vacua are ruled out, then I disagree with them, however. I think that in consistent stringy vacua, WGC always holds automatically. If that's really the essence of Jacques' critique, I am surely with him. Within the strict stringy landscape, WGC is vacuous, I believe.

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