Thursday, July 18, 2013

New miraculous ways how F-theory achieves gauge coupling unification





What's going on? The gauge couplings \(g_1,g_2,g_3\) of the \(U(1),SU(2),SU(3)\) factors of the Standard Model group aren't quite dimensionless constants. Due to the quantum loop effects linked to the renormalization group, they slowly – essentially logarithmically – evolve with the characteristic energy scale:\[

\frac{1}{g_{1,2,3}^2 (E)} = \frac{1}{g_{1,2,3}^2 (E_{EW})} + B_{1,2,3} \ln\zav{\frac{E}{E_{EW}}}.

\] So the three inverse squared couplings are linear functions of \(\ln(E)\), the logarithm of the typical energy scale of the processes. We measure their values at the electroweak scale and extrapolate them to the high energies via the calculable slopes \(B_{1,2,3}\).

We effectively draw three lines in the energy-coupling plane. Each generic pair of lines intersects somewhere but three random lines usually don't intersect at a single point. Instead, they produce three pairwise intersections.

And that's indeed the case of the Standard Model. The three pairwise intersections are sort of close to each other, near a high-energy slightly sub-Planckian scale, but they're distinguishably separated.

However, when you include the superpartners of all Standard Model particles and the extra Higgs fields needed to produce the Minimal Supersymmetric Standard Model, the quantum loops involving the new particles in the Feynman diagrams modify the slopes and the result is that the three lines pretty much exactly intersect at a high value of \(E\) which we may define to be the GUT scale.

(The hypercharge \(U(1)\) coupling constant has to be rescaled by the right factor of \(\sqrt{3/5}\) which follows from the actual way how this \(U(1)\) is embedded into GUT groups such as \(SU(5)\) or larger ones.)

This coincidence had an a priori chance smaller than 1-in-100 to yield a happy end. Some people exaggerate how powerful this numerical coincidence is (note that only one real number has to be adjusted or to naturally have the right value for the third line to hit the intersection of the first two, to present it in one particular way) but it is fair that this coincidence is as strong as a 2-sigma excess, perhaps a larger one, that seems to support the idea of the gauge coupling unification with the minimal MSSM spectrum near the \(\TeV\) scale.

When you add (or remove) generic particles from the MSSM, chances are that the unification goes out of the window.

However, the authors of the new paper I just linked to show that F-theory is capable of adding new exotic fields. Some of them are living in the bulk, i.e. arise from the 78-dimensional representation, some of them are living on the matter curves i.e. arise from the 27-dimensional representation. But their contributions to the mismatch sometimes cancel so that the gauge coupling unification is restored. The unification wouldn't work if you only added the bulk exotics or if you only added the matter curve exotics but if you add both, it just works. As far as I understand, this restoration of the gauge coupling unification occurs despite the fact that the added fields don't form complete representations of the GUT group – which is a simple way how to keep the unification undisturbed.

Because of the minimality, the MSSM is often viewed as the canonical choice – although some people have already switched to different favored models such as the NMSSM (MSSM with an extra chiral superfield \(S\) which is just the higgsino parameter \(\mu\) promoted to a field). But the results of this paper show – using my words – that if you love the MSSM because of the happy coincidence with the gauge coupling unification, you should love several classes of the F-theory models as well. In fact, when symmetries are broken by the flux which is the "flagship" mechanism in the F-theory model building, the unification is restored rather generically.

And those models do predict new exotics at the \(\TeV\) scale so if they're true, things get extremely interesting at the LHC approximately in 2015. I should mention that these E6SSM models tend to complete the leptons and quarks to a complete 27-dimensional representation of \(E_6\) which is really wonderful.

We may see. Stay tuned.

A few words about another paper by Don Marolf and Joe Polchinski,
Gauge/Gravity Duality and the Black Hole Interior.
It's disturbing because I feel that I have heard these arguments several times and they're still equally wrong. They claim that the existence of firewalls may be proven even without a near-maximal entanglement. Joe has written me the argument almost a year ago but it is based on a flawed assumption that the microstates with excited and unexcited infalling modes are equally represented. In reality, among microstates of a black hole with the mass \(M\pm \delta M\) (and when talking about genericity of microstates, it's very important to clearly declare which set of microstates or which density matrix we are talking about!), the microstates which predict \(N_a=0\) for generic low-energy freely infalling modes are exponentially more represented than those for which \(N_a\gt 0\) – because the cleaned, empty black hole is the result of stabilization and thermalization that was gradually increasing the entropy of the former star towards the maximum. So there's no contradiction between getting \(N_a=0^+\) on one hand and preserving the (approximate) thermality of the \(b\)-modes on the other hand.

They also try to challenge the ER-EPR correspondence by the comment that EPR-entangled states with generic phases don't admit a smooth ER-bridge visualization. Well, if you add generic phases, it's like adding generic excitations (local, low-energy, but also stringy or high-energy) into the Einstein-Rosen bridge, as Maldacena and Susskind also rather explicitly said, or at least I did. Of course that the clean, simple wormhole-like geometries describing entangled states are sort of special and rare, and so are the corresponding EPR-entangled states. But the rest may be obtained by "excitations" of these special states much like a generic Fock state vector may be obtained by adding with creation operators on the vacuum. I don't understand what's Marolf's and Polchinski's problem with that. This observation doesn't seem to weaken the claim that a loophole in the firewall "proof" has been demonostrated.

One more comment. The ER-EPR correspondence is a rather explicit, cute new way to think about certain microstates but it is clear that this loophole existed even without this particular geometrization of the entangled states. So I think it's obvious that the Maldacena-Susskind, while very exciting, is in no way "necessary" to see that the AMPS(S) papers are invalid.


  1. Yep since I like the MSSM from my demystified book, I think I like this F-theory gauge coupling unification too ... as far as I understand the explanations in this TRF article (F-theory always picks me, I dont know why :-D) ... :-P :-)

    What do the new potentially observable particle look like? They puzzle me a bit.

  2. Dear Dilaton, the picture is a few years old. ;-)

    It's a good question. They look exotic because they are exotic. ;-) But they're not *too* different. Just some charges are a bit unusual for the quarks - different multiples of e/3 - and some of them may behave as leptoquarks i.e. carry both baryon and lepton number at the same moment. The particles in the adjoint 78 are generalizations of the W bosons of various types, some of them are colored etc.