Numerology from Vafa and Visser

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In summary, θ is packaged with α in the three sum rules, which implies that θ-angle is 0 or π for QCD and QED.f
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Theta-problem and the String Swampland
Sergio Cecotti, Cumrun Vafa
(Submitted on 10 Aug 2018)
In the context of N=2 supergravity without vector multiplets coupled to hypermultiplets, the coupling constant of graviphoton τ is apriori a free parameter. Stringy realization of this and using a mathematical conjecture leads to the statement that j(τ)∈R so that the θ-angle is 0 or π. We conjecture that for any consistent realization of N=2 supergravity theories coupled only to hypermultiplets this is the case and the rest belong to the swampland. This leads to the speculation that the θ-angle for QCD or QED may also be fixed to 0 for quantum gravitational consistency.
The Pauli sum rules imply BSM physics
Matt Visser (Victoria University of Wellington)
(Submitted on 14 Aug 2018)
Some 67 years ago (1951) Wolfgang Pauli mooted the three sum rules:
$$\sum_n (−1)^{2S_n} g_n=0; \sum_n (−1)^{2S_n} g_n m_n^2 = 0; \sum_n (−1)^{2S_n} g_n m_n^4 = 0.$$ These three sum rules are intimately related to both the Lorentz invariance and the finiteness of the zero-point stress-energy tensor. Further afield, these three constraints are also intimately related to the existence of finite QFTs ultimately based on fermi--bose cancellations. (Supersymmetry is neither necessary nor sufficient for the existence of these finite QFTs; though softly but explicitly broken supersymmetry can be used as a book-keeping device to keep the calculations manageable.) In the current article I shall instead take these three Pauli sum rules as given, assume their exact non-perturbative validity, contrast them with the observed standard model particle physics spectrum, and use them to extract as much model-independent information as possible regarding beyond standard model (BSM) physics.
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Great papers (more or less unrelated as far as I can tell). Also, I don't think that calling these papers mere "numerology" is really fair.

The First Paper

I missed the first one, which is really intriguing.

I have long classified the Strong CP problem as a "why" problem of within the Standard Model physics that really isn't a "problem" much like naturalness of baryon asymmetry in the universe is a "why" problem but not an inherent problem unless you get into the business of telling the universe what its laws should be, which isn't a scientist's job.

The other notable reason to think that θ=0 for QCD and QED is that both have zero mass vector bosons. Zero mass bosons don't experience the passage of time per Special Relativity, in their own reference frame. So, it makes sense that a CP violation which is inherently arrow of time dependent shouldn't arise in those theories. In contrast, the weak force, which exhibits CP violation has a massive vector boson which does experience the flow time, so it makes sense that it can experience CP violation which is arrow of time dependent.

The paper's justification for quantum gravity consistency, or the simple fact that the strong CP problem isn't really a problem, or the massless boson justification as a resolution of the strong CP problem all eliminate the need for axions (so would a up quark with zero mass, which has been pretty much experimentally ruled out).

If we need no axion, then the theoretical motivation for the existence of an axion-dark matter candidates is greatly undermined, and axions start looking more like nothing more than very low mass sterile neutrinos with a lot of the justification for their particular alleged properties undermined.

The Second Paper

I did flag the second one when I saw it.

I very much like the statement that "Supersymmetry is neither necessary nor sufficient for the existence of these finite QFTs; though softly but explicitly broken supersymmetry can be used as a book-keeping device to keep the calculations manageable.", which I have long been a voice in the wilderness professing. My personal suspicion is that the mass constants and CKM matrix in the SM have values that precisely meet this condition making them much less free to have any possible value in relation to each other than they seem.

I also suspect that "the finiteness of the zero-point stress-energy tensor" as a limiting condition is the flaw and that this assumption is simply one more artifact of GR being formulated as a classical rather than a quantum physics theory. Thus, the bold conclusion of this paper that BSM physics is needed turns out to really merely add to the existing knowledge on other grounds that quantum gravity is necessary for the SM and a successor to GR to be consistent.

I also suspect that Pauli's formulation of those equations could be flawed. Mathematical physics wasn't nearly as rigorous in his day as it is now and he could easily have missed some critical but subtle point in formulating them.
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Certainly there's much more than just numerology in these papers. But they actually do both engage in fine-structure constant numerology, that's what initially caught my eye.

In Cecotti and Vafa, θ is packaged with α in a single complex coupling τ - this is standard in theories with electric-magnetic duality and helps keep track of how the couplings change under duality - and at the end they speculate that the same number-theoretic constraints which set θ to 0 or π in string theory (in the special case of a rigid Calabi-Yau), should also constrain α to be one of a set of special values. α is one over the imaginary part of τ, so it's a little reminiscent of the non-trivial zeros of the Riemann zeta function, which (according to the million-dollar hypothesis) all share the same real part, 1/2, and differ only in the imaginary part.

As for Visser, in section 4 he is trying to estimate the scale at which BSM physics enters. Since the cosmological constant is so close to zero, the difference between the SM scale and the BSM scale can only be of order one (since the log of the ratio is almost zero, eqn 4.4) - and in the next section he will conclude for a different reason that some part of BSM physics must couple to the Higgs. Then he ends section 4 by observing that the small deviation from zero represented by dark energy, is around the nonperturbative electromagnetic scale exp(-1/α) in magnitude.

Visser is a little vague, but Abel and Stewart eqn 2.58 suggests how this could work in string theory - a supertrace sum rule, with a number-theoretic interpretation (modular invariance), in which a contribution to the cosmological constant depends on gauge and yukawa couplings.
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α is one over the imaginary part of τ, so it's a little reminiscent of the non-trivial zeros of the Riemann zeta function
Atiyah's attempt to get the fine-structure constant from number theory led me to revisit this paper by Cecotti and Vafa. Vafa gave a talk about it just this week (starts at 32:50). I am coming around to the view, by the way, that Vafa's "swampland conjectures" are another leap forward in the understanding of string theory, comparable to Polchinski's D-branes. In this case, it's all about understanding how the stringy version of quantum gravity is a heavy constraint on the kinds of field theory that can be realized as the low-energy limit of a string theory vacuum.

Cecotti and Vafa find a concrete example in the form of N=2 supergravity with hypermultiplets but no vector multiplets. This is a kind of field theory, and they then ask, what are the possible couplings of the graviphoton, the spin-1 superpartner of the graviton (spin 1 being two susy transformations away from spin 2). Field-theoretically, the coupling can be any complex number, but for about 50 examples that they have calculated from string theory, the real part of the coupling, which gives you the theta angle for the graviphoton, is always either 0 or π. They don't have an explanation but speculate it has something to do with CP symmetry.

Fine; but what caught my attention was the claim that the graviphoton fine-structure constant can also be calculated, and will be an element of a special number field. They say that the complex coupling will be "the vev of a field corresponding to the smallest eigenvalue of the Laplacian acting on (2,1) forms on the CY 3-fold". That is an exact statement, but it's still beyond my power to compute such a quantity. Fortunately, in his talk (at 43 minutes), Vafa tells us what the complex coupling is for a particular Calabi-Yau: τ = 1/2 + i (√3)/2. If the normalizations are the same as in the paper, where τ = θ/2π + 4πi/(e2), then for that Calabi-Yau, θ = π and αgraviphoton = e2/4π = 2/√3.

How close is this to the real world? Cecotti and Vafa remark that the "extreme infrared" limit of such theories - in which one only considers massless particles - is the same: gravitons and photons. So this suggests a research direction for physics numerologists working in string theory: start with this extreme infrared limit, and bear in mind how it works for the N=2 case, but instead slowly add standard-model degrees of freedom, while trying to preserve calculability, number-theoretic structure, and relevant physical principles.
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