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Light fermions in color: why the quark mass is not the Planck mass
by Gustavo Pazzini de Brito, Astrid Eichhorn, Shouryya Ray
Submission summary
| Authors (as registered SciPost users): | Shouryya Ray |
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| Preprint Link: | https://arxiv.org/abs/2311.16066v1 (pdf) |
| Date submitted: | 2023-12-08 18:44 |
| Submitted by: | Ray, Shouryya |
| Submitted to: | SciPost Physics |
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| Academic field: | Physics |
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| Approach: | Theoretical |
Abstract
We investigate whether quantum gravity fluctuations can break chiral symmetry for fermions that are charged under a $U(1)$ and an $SU(N_\text{c})$ gauge symmetry and thus closely resemble Standard-Model fermions. Unbroken chiral symmetry in the quantum-gravity regime is a necessary prerequisite to recover the Standard Model from a joint gravity-matter theory; if chiral symmetry is broken by quantum gravity, fermions cannot generically be much lighter than the Planck mass and the theory is ruled out. To answer this, we work in a Fierz-complete basis of four-fermion interactions and explore whether they are driven to criticality. We discover that the interplay of quantum gravity with the non-Abelian gauge theory results in chiral symmetry breaking, because gravitational and gauge field fluctuations act together to produce bound states. Chiral symmetry breaking is triggered by four-fermion channels that first appear when non-Abelian charges are introduced and that become critical if the non-Abelian symmetry is gauged. Extrapolating our result to the Standard Model fermions, we conjecture that the non-Abelian gauge coupling, Abelian gauge coupling and Newton coupling are all bounded from above, if Standard Model fermions are to remain much lighter than the Planck mass. In contrast, fermions that are charged under a global non-Abelian group can remain light for arbitrarily large values of the Newton coupling. We find that different chiral symmetries are emergent at low energies, depending on the strength of the gravitational coupling. This is an example of fixed points with different degrees of enhanced global symmetry trading stability in fixed-point collisions driven by gravitational fluctuations.
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The theme of this paper, most broadly put, is: Can the hypothesis of asymptotic safe gravity be consistent with the experimental fact that the electroweak symmetry breaking scale is much lower than the Planck scale? Or more specifically, as being asked in this paper, can asymptotically safe gravity lead to a condensate of SM fermions that breaks EW symmetry at the Planck scale? If it can, what is the range of couplings in which such catastrophe is avoided?
The question is important in the framework of asymptotically safe gravity. The concern comes directly from the very assumption of asymptotically safe gravity, namely, the existence of an interacting UV fixed point in the space of couplings including gravitational couplings. Since gravitational interactions are irrelevant interactions around the trivial fixed point, we must accept that at the nontrivial fixed point the theory has to be strongly coupled. At least gravity for sure gets strong at the Planck scale simply due to its canonical scaling dimension. One might say that is only a weak coupling statement and who knows what the actual scaling dimension is at strong coupling, but then he/she has already bought the proposition that it is strongly coupled. Once gravity is strong, some other couplings may also get strong, and it would be problematic if such strong interactions lead to a condensate of SM fermions that breaks EW symmetry at the Planck scale (in a somewhat analogous way as the formation of q-qbar condensate by QCD that would break EW symmetry even in the absence of the Higgs VEV, although this one is due to a relevant interaction getting strong in the IR as opposed to an irrelevant interaction going strong in the UV). Therefore, the problem studied in this paper is a must-ask question in the framework of asymptotically safe gravity simply because of the very nature of this hypothesis. Toward this goal the paper aims to study the question in a somewhat simpler theory than the SM, where they look at vectorlike fermions charged under a gauged U(1) or SU(N) and study the fate of the chiral symmetry of the fermions in the UV.
What I am really having trouble with is that their analysis assumes that the anomalous dimensions are small and the scaling dimensions take near-canonical values. I understand that some approximation/truncation scheme is practically needed to study a strongly coupled problem, but the one approximation we don't want to make is that the couplings are weak! The beta functions presented in the paper have the form of "canonical scaling + 1-loop corrections", but the perturbative corrections cannot offset canonically irrelevant dimensions by O(1) to make them marginal, unless the couplings are large. But once the couplings are so large that 1-loop anomalous dimensions can balance against canonical (i.e., tree-level) scalings, the 1-loop calculations do not tell us anything quantitatively meaningful as 2-loop and higher-loop corrections should be just as important. The detailed numerical coefficients in the beta functions and the detailed numerical results as presented in this paper are not justified in the sense that they give the false impression that the results are quantitatively under control, whereas every term in their beta functions is at best only a very rough estimate (at the level of Naive Dimensional Analysis) modulo an unknown factor and even an unknown sign. The issue is somewhat analogous to keeping more significant digits in experimental measurements than justified and drawing conclusions based on those extra digits.
Therefore, I find the claims made by the authors in the paper too aggressive as I don't see a calculational framework by which such quantitative statements can be made.