Abstract
Naturalness, as a guiding principle for effective field theories (EFTs), requires that there be no sensitive correlations between phenomena at low- and high-energy scales. This essay considers four reasons to adopt this principle: (i) natural EFTs exhibit modest empirical success; (ii) unnatural EFTs are improbable; (iii) naturalness underwrites what Williams (Stud Hist Philos Mod Phys 51:82, 2015) calls a “central dogma” of EFTs; namely, that phenomena at widely separated scales should decouple; and (iv) naturalness underwrites a non-trivial notion of emergence. I argue that the first three are not compelling reasons, whereas the fourth is.
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Notes
In this essay by an EFT I mean an effective quantum field theory. Some authors use the term effective theory in a broader sense (e.g., [1]).
Schwarz [5, p. 418] points out that the Wilsonian approach to EFTs and renormalization has its origins in applications to condensed matter systems with explicit characteristic scales (e.g., atomic lattice spacing). One consequence of this, according to Schwarz [5, p. 411], is that "much of our intuition for fine-tuning and naturalness comes from condensed matter physics".
This suggests that a Wilsonian effective action is only well-defined in "top-down" cases in which a high-energy theory exists. To the contrary, a Wilsonian effective action can also be constructed via a "bottom-up" process in which one writes down an expansion of the form of (2), including in it all local operators consistent with what one takes to be low-energy symmetries, and then suppresses these terms by powers of an appropriate physical cutoff Λ. Weinberg [6, p. 329] has argued that the result of using such an action to calculate S-matrix elements "… will simply be the most general possible S-matrix consistent with analyticity, perturbative unitarity, cluster decomposition, and the assumed symmetry principles".
In a weakly interacting theory, the 0th order term S0 in the expansion (2) dominates the other terms, since it contains no couplings which are assumed to be very small. S0 only contains factors of ϕL and its derivatives, and it is dimensionless; and this suffices to determine the dimension of ϕL. See Footnote 7 below for a concrete example.
At this point, one thing that should be made clear is that, while a physical cutoff Λ suggests naturalness, it does not entail it. In other words, even though naturalness is part of the internal logic of a Wilsonian EFT, this does not mean that Wilsonian EFTs are necessarily characterized by it.
The first term is dimensionless. The d4x part of it has dimension − 4 and the \( \partial_{\mu }^{2} \) part has dimension 2. If the dimension of ΦL is δ, then 0 = − 4 + 2 + 2δ, hence δ = 1.
These are discussed in Williams [2] who argues that they all have the sensitivity prohibition in common, and it is the latter, as opposed to the former, that should be identified with naturalness. Briefly, according to Williams, formulation (a) risks viewing violations of naturalness as dependent on the choice of regularization scheme one adopts. Formulation (b) is sufficient but not necessary for a sensitivity prohibition: there are explanations of unnatural parameters that do not involve a violation of a symmetry condition (e.g., the technicolor proposed solution to the Hierarchy problem). And finally, formulation (c) is also sufficient but not necessary for a sensitivity prohibition: there are fine-tuning problems that have nothing to do with undue sensitivity between widely separated scales (e.g., the flatness and horizon problems in cosmology, and appeals to the low entropy state of the early universe). Moreover, to the extent that the fine-tuning of formulation (c) "… always concerns relevant operators not protected by any symmetry" [2, p. 88], examples of violations of naturalness that are not associated with relevant operators fall outside its purview (e.g., the Strong CP problem).
This is ultimately a result of the fact that fermions are typically represented mathematically by spinor (as opposed to scalar or tensor) fields, and the former have a "built-in" chiral symmetry. The symmetry formulation of naturalness was introduced by 't Hooft [10].
Admittedly, naturalness as a constraint on effective theories in general has had much more empirical success than its restricted application to effective quantum field theories (see, e.g., [1]). (Thanks to Sebastian Rivet for making this point.).
Details of these examples may be found in Giudice [9, pp. 168–169].
See, e.g., Dine [8, p. 48]. This upper limit is based on experimental limits on the electric dipole moment of the neutron, which can be derived as a function of θ.
The CP-violating term takes the form (θ/16π2)Gμν\( \tilde{G}_{\mu v} Z \), where Gμν is the QCD field strength and \( \tilde{G}_{\mu v} \) is its dual [8]. In the Wilsonian approach, Gμν\( \tilde{G}_{\mu v} Z \) is not a relevant operator, and this might be puzzling if one associated the failure of naturalness only with relevant operators. This puzzle can be resolved by noting that the association of the failure of naturalness with relevant operators is made in the context of a weakly-interacting perturbative analysis of the low-energy effective action, as described in Sect. 2. The θ-term in QCD represents a non-perturbative, non-local (in fact topological) contribution to the low-energy effective action that falls outside the analytical framework of Wilsonian EFTs; yet, arguably, it is unnatural in the sense adopted in this essay; i.e., it represents a sensitive correlation between low- and high-energy scales. (In this case, it is in general a correlation between low-energy observables and the global topology of the system. Evidently, we should take it seriously to the extent that we think, for instance, instanton solutions to the QCD equations of motion should be taken seriously.) (Thanks to an anonymous referee for raising these concerns.).
Grinbaum [15] provides a detailed history of such measures.
Craig [17, p. 7] also reports that sensitivity measures can make intuitively incorrect judgments, labelling theories in which small energy scales are set by dynamical processes as unnatural.
Hossenfelder [12, p. 10] notes that Anderson and Castano [19, p. 302] are explicit in their admission of "an element of arbitrariness to the construction" of their probabilistic measure. One should also point out that Anderson and Castano's objective is not a justification of naturalness; rather, it is an attempt to construct a quantifiable measure of it. They are not concerned with answering the question "Why be natural?"; rather, they are concerned with the question "Given we should be natural, how can we quantitatively distinguish the natural theories from the unnatural theories?".
Given n mutually exclusive and exhaustive outcomes A1,…, An, and the conditional probability P(Ai|B) of outcome Ai given background evidence B, (finite) additivity requires \( \sum\nolimits_{\text{i = 1}}^{n} {P(A_{\text{i}} |B) = 1} \). Norton interprets this as meaning that B can favor one outcome or set of outcomes only if it disfavors others.
Franklin [20, p. 23] similarly claims that the "effectiveness" of EFTs is not due to naturalness, but rather to an invariance of the low-energy dynamics with respect to changes in the state at high energies, consistent with the high-energy dynamics. According to Franklin, this type of "autonomy from microstates" is due to (effective) renormalizability in the context of EFTs. This assessment conforms to the discussion in the text above, given that "autonomy from microstates" can be identified with what I subsequently refer to as "heuristic decoupling".
Franklin [20, p. 15] similarly stresses the significance of unnatural but (effectively) renormalizable EFTs.
Assumedly, the term "continuum EFT" refers to the absence of a physical cutoff in a mass-independent scheme.
These observables can take the form of scattering amplitudes, for instance.
The point here is that the Wilsonian cutoff Λ and the continuum mass scale μ = M are quantitatively distinct, as Schwarz [5, p. 444] notes: "The Wilsonian cutoff Λ should always be much larger than all relevant physical scales. This is in contrast to the μ in the continuum picture, which should be equal to a relevant physical scale."
Certainly, one can adopt a similar agnostic attitude towards Wilsonian EFTs. The point of the above discussion is that this attitude does not seem in keeping with the way high-energy effects are encoded in effective couplings under a physical interpretation of the Wilsonian cutoff Λ.
There are examples of EFTs for which this claim is problematic. For instance, in a "top-down" EFT in which there is a clear distinction in the high-energy theory between light fields and heavy fields, and the high-energy degrees of freedom (consisting of the heavy fields and the high-energy dynamics of the light fields) are represented in the effective Lagrangian as corrections to the low-energy dynamics of the light fields, the sense in which the low-energy dynamics is independent of the high-energy dynamics is fairly weak. Dynamical independence seems better motivated by examples of "top-down" EFTs for which there is no initial clear distinction between "heavy" and "light" degrees of freedom, and examples of "bottom-up" EFTs for which the high-energy degrees of freedom remain unknown.
Franklin [20, p. 22] also suggests that EFTs can exhibit emergence: "… EFTs are emergent if they are novel and autonomousms with respect to higher-energy theories", where "… novelty implies that new explanations are available which are not expressible in terms of the variables of the higher-energy theory." For Franklin, autonomyms (i.e., autonomy with respect to microstates) is underwritten by a renormalization scheme in which high-energy degrees of freedom are encoded in low-energy effective couplings. This is what I have identified above as heuristic decoupling. Thus, insofar as Franklin takes this to underwrite Crowther's Dependence criterion for emergence, we are in agreement. On the other hand, Franklin's notion of novelty as underwritten by a particular type of explanatory power seems to make emergence an epistemic notion. My preference is for an ontic notion of emergence that characterizes physical systems, as opposed to our knowledge of physical systems. Again, novelty, for me, involves the sort of robust dynamical independence that underwrites naturalness.
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Bain, J. Why be Natural?. Found Phys 49, 898–914 (2019). https://doi.org/10.1007/s10701-019-00249-z
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DOI: https://doi.org/10.1007/s10701-019-00249-z