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A scalar weak gravity conjecture seems powerful

Stringy quantum gravity may be predicting an \(r=0.07\) BICEP triumph

Many topics in theoretical physics seem frustratingly understudied to me but one of those that are doing great is the Weak Gravity Conjecture (WGC) which is approaching 500 followups at the rate of almost a dozen per month. WGC hasn't ever been among the most exciting ideas in theoretical physics for me – which is why the activity hasn't been enough to compensate my frustration about the other, silenced topics – but maybe the newest paper has changed this situation, at least a little bit.

Nightingales of Madrid by Waldemar Matuška. Lidl CZ goes through the Spanish week now.

Eduardo Gonzalo and Luis E. Ibáñez (Zeman should negotiate with the Spanish king and conclude that our ň and their ñ may be considered the same letter! Well, the name should also be spelled Ibáněz then but I don't want to fix too many small mistakes made by our Spanish friends) just released:

A Strong Scalar Weak Gravity Conjecture and Some Implications
and it seems like a strong cup of tea to me, indeed. The normal WGC notices that the electron-electron electric force is some \(10^{44}\) times stronger than their attractive gravity and figures out that this is a general feature of all consistent quantum gravity (string/M/F-theory) vacua. This fact may be justified by tons of stringy examples, by the consistency arguments dealing with the stability of near-extremal black holes, by the ban on "almost global symmetries" in gravity which you get by adjusting the gauge coupling to too small values, and other arguments.

Other authors have linked the inequality to the Cosmic Censorship Conjecture by Penrose (they're almost the same thing in some contexts), to other swampland-type inequalities by Vafa, and other interesting ideas. However, for a single chosen Universe, the statement seems very weak: a couple of inequalities. The gravitational constant is smaller than the constant for this electric-like force, another electric-like force, and that's it.

Yes, this Spanish variation seems to be stronger. First, we want to talk about scalar interactions mediated by scalars instead of gauge fields. At some level, this generalization must work. A scalar may be obtained by taking a gauge field component \(A_5\) and compactifying the fifth dimension. If the force mediated by the gauge field was strong, so should be one mediated by the scalar.

To make the story short, they decide that the scalar self-interactions must be stronger than gravity as well and decide that an inequality for the scalar potential should hold everywhere, at every damn point of the configuration space\[

2(V''')^2 - V'''' \cdot V'' - \frac{(V'')^2}{M_P^2} \geq 0.

\] It's some inequality for the 2nd, 3rd, 4th derivatives of the potential. The self-interaction's being strong says that the third derivative should mostly dominate, in some quantitative sense. That's a bit puzzling for the purely quartic interactions. For \(A\phi^2+B\phi^4\), the inequality seems violated for \(\phi=0\) because there's a minus sign in front of the fourth derivative term and the "purely second" derivative term, too (the third derivative term vanishes in the middle). Do we really believe that this first textbook example of a QFT is prohibited? Does quantum gravity predict that the Higgs mechanism is unavoidable? And if it does, couldn't this line of reasoning solve even the hierarchy problem in a new way?

OK, they decide this is their favorite inequality in two steps: the fourth-derivative term is added a bit later, for some consistency with axions.

The very fact that they have this local inequality is quite stunning. In old-fashioned effective field theories, you could think that you may invent almost any potential \(V(\phi)\) and there were no conditions. But now, calculate the left hand side of the inequality above. You get some function and of course it's plausible that it's positive in some intervals and negative in others. It's unlikely that you avoid negative values of the left hand side everywhere. But if it's negative anywhere, this whole potential is banned by the new Spanish Inquisition, I mean the new Spanish condition! Clearly, a large majority of the "truly man-made" potentials are just eliminated.

Now, the authors try to find a potential that saturates their inequality. It has two parameters and is the imaginary part of the dilogarithm. It's pretty funny how complicated functions can be obtained just by trying to saturate such a seemingly elementary condition – gravitation is weaker than self-interactions of the scalars – that is turned into equations in the most consistent imaginable way.

The potentials they're led to interpolate between asympotically linear and perhaps asymptotically exponentially dropping potentials. They also derive some swampland conjectures and find a link to the distance swampland conjecture, another somewhat well-known example of Vafa's swampland program.

The WGC-like thinking has been used to argue that string/M-theory prohibits "inflation with large excursions of the scalar field". The "large excursion" is basically prohibited in analogy with the "tiny gauge coupling", it's still morally the same inequality. And it's a "weak" inequality in the sense that there's one inequality per Universe.

But these Spaniards have a finer resolution and stronger claims – they study the inequalities locally on the configuration space. And in the case of inflation, they actually weaken some statements and say that large excursions of the inflaton are actually allowed if the potential is approximately linear. As you know, I do believe that inflation is probably necessary and almost established in our Universe. But the swampland reasoning has led Vafa and others to almost abandon inflation (and try to replace it with quintessence or something) because the swampland reasoning seemed to prohibit larger-than-Planck-distance excursions of the inflaton. Others were proposing monodromy inflation etc.

But these authors have a new loophole: asymptotically linear potentials are OK and allow the inflaton to go far and produce 50-60 \(e\)-foldings. If they were really relevant as potentials of the inflaton, you would have a very predictive theory. In particular, the tensor-to-scalar ratio should be \(r=0.07\) which is still barely allowed but could be discovered soon (or not). Do you remember the fights between BICEP2 and Planck? Planck has pushed BICEP2 to switch to publishing papers saying "we don't see anything" but I still see the primordial gravitational waves in their picture and \(r=0.07\) could explain why I do. According to some interpretations, Planck+BICEP2 still hint at \(r=0.06\pm 0.04\), totally consistent with the linear potential. BICEP3 and BICEP Array have been taking data in the recent year or two. Do they still see something? Perhaps I should ask: Do they see the tensor modes again? Hasn't the Brian guy who did it for the Nobel Prize given up? Are there others working on it?

These new authors also claim that a near-saturation of their inequality naturally produces the spectrum of strings on a circle, with momenta and windings related by T-duality. In the process, they deal with the function \(m^2\sim V''\) and substitute integers to some exponentially reparameterized formulae... Well, I don't really understand this argument, it looks like black magic. Why do they suddenly assume that some of the parameters are integers and these integers label independent states? But maybe even this makes some sense to those who analyze the meaning of the mathematical operations carefully.

We often hear about predictivity. The swampland program and the WGC undoubtedly produce some predictions (like "gravity is weak") – it's a reason I was naturally attracted to these things because by my nature, I usually and slightly prefer to disprove and debunk possibilities than to invent new ones – but these predictions have looked rather isolated and weak, a few inequalities or qualitative statements per Universe. But when studied more carefully, there may be tons of new consequences like inequalities that hold locally in the configuration space. Functions that nearly or completely saturate these conditions are obviously attractive choices of potentials (I finally avoided the adjective "natural" not to confuse it with more technical versions of "naturalness").

And these functions may have the ability to turn stringy inflation into a truly predictive theory because they would imply the \(r=0.07\) tensor modes. Maybe WGC is pretty exciting, after all. (Just to be sure, it's been known for a long time that the linear potentials produce this tensor-to-scalar ratio.)

If it is truly exciting, I am still comparing it to the uncertainty principle. Imagine that you have some inequalities that look like the uncertainty principle for various pairs of variables. Some of these inequalities might be a bit wrong, a bit too weak etc. But you also want to consolidate them (into the general inequality for any two observables) and derive something really sharp and deep, e.g. that the observables have nonzero commutators. (This is not how it happened historically, Heisenberg had the commutators first, in 1925, and the inequality was derived in 1927.)

Maybe we're in a similar situation. They're asking the reader whether the WGC is a property of the black holes only or quantum gravity. I surely think it's both and the latter is more general. Black holes are just important in quantum gravity – as some extreme and/or generic localized objects (which produce the whole seemingly empty interior and paradoxes associated with it). But at the end, I do think that the WGC or its descendants should be equivalent even to holography and other things that are not "just" about the black holes.

Quantum gravity is not quite the same as an effective field theory. And the difference between the two may be very analogous to the difference between classical and quantum physics. The WGC and its gradually thickening variations could be the first glimpses of a new understanding of quantum gravity – first glimpses that might hypothetically make the full discovery and understanding unavoidable.

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