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Two Notions of Naturalness

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My aim in this paper is twofold: (i) to distinguish two notions of naturalness employed in beyond the standard model (BSM) physics and (ii) to argue that recognizing this distinction has methodological consequences. One notion of naturalness is an “autonomy of scales” requirement: it prohibits sensitive dependence of an effective field theory’s low-energy observables on precise specification of the theory’s description of cutoff-scale physics. I will argue that considerations from the general structure of effective field theory provide justification for the role this notion of naturalness has played in BSM model construction. A second, distinct notion construes naturalness as a statistical principle requiring that the values of the parameters in an effective field theory be “likely” given some appropriately chosen measure on some appropriately circumscribed space of models. I argue that these two notions are historically and conceptually related but are motivated by distinct theoretical considerations and admit of distinct kinds of solution.

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Notes

  1. One sometimes calls this problem the “little Hierarchy problem,” reserving the title “Hierarchy problem” for the problem of the hierarchy between the EWSB scale and the Planck scale. Nothing hinges on this distinction in this paper.

  2. I neglect here the position that advocates simply staying the course, preserving the emphasis that has been placed on naturalness at the expense of focusing on more complicated natural extensions of the Standard Model.

  3. This position is also described in [7, 11, 64], among others.

  4. Like many such calculations in the 1930s, Weisskopf made use of the hole-theoretic formalism of Dirac; see [56, Chap. 2] for calculational methods in the 1930s.

  5. Weisskopf is here considering a theory to be inconsistent if the series expansion of the self-energy contribution W does not converge.

  6. This is not to suggest that Weisskopf’s treatment of the scalar theory is unique in this regard; as shown above, he makes a similar argument in the case of the purely fermionic theory.

  7. For example, see [9, 10, 50, 52, 62, 72].

  8. See [70, pp. 1825–6] for the assumptions about the behavior of the \(\beta \)-function underlying his analysis. He notes that “there is no way of knowing whether these assumptions are true for quantum electrodynamics or any other given field theory.”

  9. Of course, we now recognize this was a mistake. Quantum chromodynamics, the theory of the strong interaction, is asymptotically free, while quantum electrodynamics is not.

  10. In terminology that is now familiar, but which Wilson introduces immediately prior to this discussion, this is just to say that \(x_1\) is an ultraviolet stable fixed point.

  11. Wilson also cites [35] and [14] in support of the quotation below.

  12. Wilson has elsewhere defined the “renormalized charge” as the charge renormalized at \(\lambda = 0\).

  13. Indeed, ’t Hooft’s emphasis on the role of symmetry in his notion of naturalness has led Grinbaum to suggest that “based upon ’t Hooft’s definition, [naturalness] could have received a...conceptual foundation similar to that of symmetry” [41, p. 616]; see also [43] in which the “conventional” approach to naturalness is described akin to a symmetry principle. I do not think this kind of justification for naturalness is compelling, but these remarks illustrate the extent to which ’t Hooft’s notion of naturalness has become identified with his remarks about symmetries.

  14. Wells offers an interesting counterfactual history that argues strict insistence on Absolute Naturalness could have grounded a series of inferential steps leading from quantum electrodynamics to the Standard Model.

  15. This understanding of naturalness is also advocated in [11, 36,37,38].

  16. What follows is not primarily aimed at historically accurate exigesis: whether Susskind or ’t Hooft, or others actually were motivated by an autonomy of scales expectation is not of primary importance here. That said, I think there is fairly good evidence that they were, some of which will be briefly presented here.

  17. ’t Hooft assumes that all the gauge theories he is investigating in [60] have a UV cutoff, which he refers to as the “Naturalness Mass Breakdown Scale” and estimates to be at about 1 TeV.

  18. A referee suggests that ’t Hooft’s claim that the smallness of a parameter must be accounted for with a symmetry was motivated by reflection on the central methodological role that symmetries occupy in constructing quantum field theories. I think this would have been a strong argument for ’t Hooft to have given, but I am unable to find textual support for it in [60]. However, I think it is an interesting and plausible conjecture about ’t Hooft’s attitude toward quantum field theories at the time and I mention it here as an invitation for interested parties to take up the question.

  19. See [26] for a complementary discussion of the way that dimensional analysis informs intuitions about naturalness.

  20. The claim that all couplings of the Standard Model except the Higgs boson mass are natural is false if one is inclined to consider the cosmological constant as a Standard Model coupling.

  21. In [20, Chap. 8], for example, a pedagogical proof of the decoupling theorem is proven using a scalar field theory in which the couplings in the low-energy effective theory are quadratically sensitive to the high-energy physics that is integrated out.

  22. Recall Zee’s somewhat relaxed attitude on this score; see also [68, p. 103] for a similarly relaxed attitude.

  23. The reader is encouraged to see Grinbaum [41, Sect. 3] for a complementary discussion of the evolution of quantitative measures of naturalness and how the concept of naturalness itself underwent modifications through this process.

  24. In the original example, the low-energy observable is \(M_Z\), standing in for the scale of EWSB, and they consider variations of parameters \(\alpha _i\) related to the scale at which supersymmetry is broken in a given model. The prefactor \(\frac{\alpha _i}{{\mathcal {M}}}\) is included to remove an overall dependence on the scale of \(\alpha _i\) and \({\mathcal {M}}\).

  25. See, for example, [2,3,4, 8, 19, 23].

  26. A number of such choices have to be made in any attempt to construct a quantitative measure of naturalness, as is discussed in some detail in [34] and [21]. As I said above, I think there is a plausible argument to be made that this inevitable sense that one is making arbitrary choices stems from trying to impose unwarranted mathematical precision on an imprecise physical heuristic.

  27. Anderson recalls that their understanding of naturalness at the time was that “if you imagine that the fundamental (Lagrangian) parameters had some smooth probability distribution, an observable parameter would be unnatural if the measured value of that parameter was only within some characteristic range around the measured value for an unusually small part of the parameter space relative to other values” (personal communication). This has no essential connection to a notion of interscale sensitivity.

  28. For pedagogical discussion of the details of compactification mechanisms and the origin of the landscape, see [46] or [24].

  29. My thanks to a referee for urging me to address this.

  30. See also [55] for a discussion of this problem that is more directly focused on the string landscape.

  31. See also [1] for an early investigation of the properties an effective field theory should satisfy in order to have a UV completion.

  32. For what it is worth, the attitude expressed by [6] seems to me quite reasonable: “One might think that low-energy SUSY with \(m_S\sim \text {TeV}\) is preferred, since this does not entail a large fine-tuning to keep the Higgs light. However, this conclusion is unwarranted\(\ldots \) without a much better understanding of the structure of the landscape, we can’t decide whether low-energy SUSY breaking is preferred to SUSY broken at much higher energies.”

  33. My thanks to a referee for offering this objection.

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Acknowledgements

I would like to thank Tony Duncan, Michael Miller, and an insightful referee for this journal for helpful comments on an earlier draft, and Greg Anderson and Diego Castaño for helpful correspondence about the motivation for the notion of naturalness they introduced in [3]. I would also like to thank audiences at the Aachen workshop “Naturalness, Hierarchy, and Fine-tuning,” the University of Michigan workshop “Foundations of Modern Physics: the Standard Model after the Discovery of the Higgs Boson,” and at Balliol College, Oxford for their valuable feedback.

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    Porter Williams

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Williams, P. Two Notions of Naturalness. Found Phys 49, 1022–1050 (2019). https://doi.org/10.1007/s10701-018-0229-1

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