2. The Renormalizability Puzzle

In qft, Lagrangians are used to calculate quantities of empirical interest. The hitch is that, naively pursued, these calculations deliver infinities, rather than finite numbers to test against experiment. The conventional fix—renormalization—is to pursue more sophisticated calculations. Not every Lagrangian can be cured by the conventional fix: not every Lagrangian is renormalizable. Indeed, in some sense, most aren’t. The good news is that Standard Model Lagrangians are: they can be renormalized to yield finite predictions. The bad news is that the good news stuck us with the Renormalizability Puzzle, that “it seemed incredibly lucky that we could describe so much in particle physics using renormalizable QFTs” (Williams Reference Williams2023, 45).Footnote ³ This section offers a minimally technical introduction to the Renormalizability Puzzle.

A Lagrangian is a function of a collection of spacetime fields, where each field is map from points of a spacetime (which I’ll assume to be four-dimensional) to a value, and their spacetime derivatives. Throughout, I’ll adopt the drastically simplifying assumption that a Lagrangian involves only one field, a scalar-valued field $\phi $ , and its derivatives $\partial \phi $ .Footnote ⁴ Thus, a generic scalar field Lagrangian ${\cal L}$ is a sum of mononomials of $\phi $ and $\partial \phi $ , modified by coupling coefficients. A schematic example, shaped by some physical background knowledge and a few additional simplifying assumptions,Footnote ⁵ is

(1) $${\cal L} = {(\partial \phi )^2} + {m^2}{\phi ^2} + {g_1}{\phi ^4} + {g_2}{\phi ^6} + {g_3}{\phi ^6} + \cdots $$

The physical background knowledge is that

$${{\cal L}_0} = {(\partial \phi )^2} + {m^2}{\phi ^2}$$

is a cracker-jack Lagrangian, the free-field lagrangian, under complete analytic control. We can draw upon analogies with more familiar theories to recognize its kinetic term ${(\partial \phi )^2}$ as describing the kinetic energy of the free field, and its mass term ${m^2}{\phi ^2}$ as describing a potential energy associated with the mass $m$ of the field. And that’s not all! We can explicitly characterize solutions to the classical equations of motion ${{\cal L}_0}$ generates, and use the classical solution space to construct a Hilbert space undergirding a quantum field theory based on that Lagrangian—see Wald (Reference Wald1994) for more on the construction.

Alas, the qft thus obtained is deathly boring. Free fields don’t interact; they don’t scatter; they pass unaltered through our particle accelerators. For the sake of characterizing interesting physics, we need to add more terms to ${{\cal L}_0}$ , interaction terms third-order or higher in $\phi $ . A sum of ${{\cal L}_0}$ and a host of interaction terms, (1)’s generic Lagrangian ${\cal L}$ amends the free theory ${{\cal L}_0}$ by adding interactions ${\phi ^4}$ , ${\phi ^6}$ , ${\phi ^8}$ , … multiplied by coupling coefficients ${g_1}$ , ${g_2}$ , ${g_3}$ , … that indicate the strength of those interactions. Think of ${\cal L}$ as residing in a space ${\cal G}$ of Lagrangians; each Lagrangian in ${\cal G}$ is specified by a coupling vector listing the coefficientsFootnote ⁶ ${g_{\rm{k}}},{g_{\rm{m}}},{g_1},{g_2},{g_3}, \ldots $ modifying the monomials it depends on.

Now we’ve got an interacting qft that might describe interesting physics! Only there’s a hitch. Even the simplest interacting qft, one that adds just a ${\phi ^4}$ interaction to ${{\cal L}_0}$ ,

(2) $${{\cal L}_{{\phi ^4}}} = {{\cal L}_0} + {g_1}{\phi ^4},$$

eludes our analytical reach. We don’t know how to construct a Hilbert space from its classical solution space. So we pursue quantum-theoretic results by other means. We undertake to calculate quantities of empirical interest perturbatively. That is, we assume that the coupling ${g_1}$ is small, so that ${{\cal L}_{{\phi ^4}}}$ represents a small perturbation to ${{\cal L}_0}$ . And we approximate a quantity ${\cal M}$ of empirical interest—a scattering amplitude, say—by performing a perturbative expansion, in orders of ${g_1}$ , around the (usually very dull) precise value ${{\cal M}_0}$ the free Lagrangian predicts:

(3) $${\cal M} = {{\cal M}_0} + {g_1}{{\cal M}_1} + {({g_1})^2}{{\cal M}_2} + {({g_1})^3}{{\cal M}_3} + \cdots $$

The ${{\cal M}_n}$ in (3) are $n$ th-order perturbative corrections. Perturbation theory is tasked with directing us in their calculation. Assuming ${g_1}$ is small, we hope that truncating the perturbative expansion after a few orders yields an approximation to ${\cal M}$ accurate enough to test in our particle accelerators.

Path integrals are a standard method for extracting empirical content from qfts. To calculate a quantity ${\cal M}$ of empirical interest, one evaluates a path integral whose integrand depends on the qft Lagrangian.Footnote ⁷ Schematically,

(4)

The $D\left[ \phi \right]$ denotes a functional integration over field configurations $\phi $ whose energy and momentum depend on $\phi $ ’s frequency. The path integral defining ${\cal M}$ integrates over field configurations of all energies/momenta/frequencies, including arbitrarily high ones.

If ${\cal L}$ is a free Lagrangian, we can execute the path integral defining ${\cal M}$ . If ${\cal L}$ contains interaction terms, however, we can’t. Instead, we resort to the perturbative approximation (3). Feynman diagrams guide our calculation of perturbative corrections ${{\cal M}_1}$ , ${{\cal M}_2}$ , … Starting with the “loop-order” correction ${{\cal M}_2}$ , Feynman diagrams instruct us to calculate corrections by evaluating integrals that conspicuously and tragically diverge. Brought on by integrating over arbitrarily large momenta/frequencies, these catastrophes are known as ultraviolet divergences. Rather than the small adjustments to empirical quantities presupposed by our perturbative approximation, corrections plagued by ultraviolet divergence contribute infinite ones. Ultraviolet divergences render perturbative approximation meaningless.

Renormalization, the conventional way to address the catastrophic divergences, is a recipe for an order-by-order doctoring of a Lagrangian’s couplings that eventuates, at each order, in finite perturbative corrections. An essential ingredient in the recipe is a finite set of measurements that serve to constrain the doctoring at all orders. Other ingredients struck even the framers of perturbative renormalization approaches as suspiciously ad hoc, even desperate.Footnote ⁸ Notoriously, renormalization restores empirical sanity to perturbative approximations by subtracting one infinity from another to obtain a finite answer.

And yet, for some Lagrangians, renormalization techniques succeed, stunningly well. Call Lagrangians that surrender finite predictions when subject to renormalization techniques renomalizable. Dyson articulated a criterion for identifying such amiable Lagrangians. The criterion hinges on the notion of mass dimension. It’s standard operating procedure in high-energy physics to set Planck’s constant equal to the speed of light equal to the dimensionless number $1$ . This convention means we need only one physical dimension, which particle physicists declare to be $\left[ {{\rm{mass}}} \right]$ .Footnote ⁹ Scalar fields and their derivatives have mass dimension $1$ , as—unsurprisingly!—does the mass $m$ . Thus each term in ${{\cal L}_0}$ has mass dimension $4$ , as does the ${\phi ^4}$ interaction. Higher-order interactions ( ${\phi ^6}$ , ${\phi ^8}$ , …) have mass dimension $ \gt 4$ .

The notion of an action supplies a nomic scaffolding for path-integral approaches. (4) implicates ${\cal L}$ in empirical calculations by using it to define an action via $S = {\rm{exp}}\left( { - \smallint {\cal L}\;{d^4}x} \right)$ . The definition requires $\smallint {\cal L}\;{d^4}x$ to be dimensionless. Because ${d^4}x = {(1/\partial )^4}$ has mass dimension $ - 4$ , this obliges each term in ${\cal L}$ to have mass dimension $4$ . Thus, an interaction term with mass dimension $d$ must be multiplied by a coupling $g$ of mass dimension $4 - d$ .

The reward for this slog through the niceties of mass dimension is that Dyson’s renormalizability criterion is simply stated. A Lagrangian is renormalizable only if it contains no interactions of mass dimension $ \gt 4$ , and an interaction is renormalizable only if it has mass dimension $ \le 4$ . But the vast majority of interactions appearing in our generic Lagrangian (1) violate this criterion! Renormalizable Lagrangians are confined to a tiny corner of theory space ${\cal G}$ : the measly three-dimensional subspace ${{\cal G}^{{\rm{PR}}}}$ spanned by the handful of terms with mass dimension $ \le 4$ .

Hence the Renormalizability Puzzle. It somehow just so happens that the Lagrangians figuring in the Standard Model—not to mention other useful Lagrangians, such as the ${\phi ^4}$ Lagrangian—are renormalizable. They’re residents of that tiny region ${{\cal G}^{{\rm{PR}}}}$ . Particularly given how suspicious renormalization techniques seemed even to their framers, this seems absolutely extraordinary. Why should it be that the Lagrangians that govern real-world physics providentially belong to the small but exclusive subspace ${{\cal G}^{{\rm{PR}}}}$ ?

3. The RG Explanation

Viewed through the lens of RG approaches, the space ${\cal G}$ of Lagrangians is pierced by a flow of RG trajectories; the RG Explanation uses the structure of this flow to explain why so many useful Lagrangians are renormalizable. Here’s how.

Recall the schematic (4). Integrating over all energies, we can predict amplitudes for scattering experiments conducted at all energies. Only we haven’t got the wherewithal to conduct arbitrarily high-energy experiments. What we have the wherewithal to conduct are experiments at energies we can achieve in the particle accelerators we’ve managed to construct. Introduce an effective energy scale ${\rm{\Lambda }}$ that sets an upper bound on the energies at which we can test amplitudes extracted from our qft.

Now indulge in what I’ll call the autonomy-of-scales hope. This is the hope that there is some way to describe scale- ${\rm{\Lambda }}$ effective physics that prescinds from the details of ultra-high-energy physics—that to generate predictions empirically adequate at scale ${\rm{\Lambda }}$ , we need only survey scale- ${\rm{\Lambda }}$ fields: there is a Lagrangian ${\cal L}$ such that we can describe amplitudes for scattering experiments conducted at energies up to ${\rm{\Lambda }}$ , à la (5), by integrating ${\cal L}$ against a domain of field configurations that’s cut off at ${\rm{\Lambda }}$ . $\left( {{\cal L},{\rm{\Lambda }}} \right)$ is a scale- $\Lambda $ effective theory: its domain, both of definition and of application, is limited, but it succeeds in that domain.

Revised to implement the scale- ${\rm{\Lambda }}$ effective theory inspired by our autonomy-of-scales, (4) becomes

(5) $${\cal M} = \mathop \smallint \nolimits_{\phi \lt {\rm{\Lambda }}} D\left[ \phi \right]\left\{ { \ldots {\cal L} \ldots } \right\},$$

where the subscript $\phi \lt {\rm{\Lambda }}$ flags the cutoff we’ve imposed on the domain of integration. Evaluating (5) requires performing a path integral over field configurations of energies up to—but not beyond—the cutoff energy ${\rm{\Lambda }}$ . Issuing amplitudes for scattering experiments conducted at energies up to (but not above) ${\rm{\Lambda }}$ , (5) is a recipe for surveying fields at scale ${\rm{\Lambda }}$ in order to derive predictions concerning phenomena at that scale. Comparing (4) with (5) makes manifest that qft Lagrangians do not describe phenomena on their own, but only in conjunction with a range of field energies over which they’re integrated to yield amplitudes.

We are ready to embark on an RG analysis of the space ${\cal G}$ of Lagrangians.Footnote ¹⁰ Our first move is to pick a very-high-energy ${{\rm{\Lambda }}_{{\rm{uv}}}}$ . Take a theory defined by a Lagrangian ${\cal L}$ and the UV cutoff energy ${{\rm{\Lambda }}_{{\rm{uv}}}}$ . We’re being studiously ecumenical about what form UV physics might take: even Lagrangians that aren’t renormalizable might play the role of ${\cal L}$ .

The theory $\left( {{\cal L},{{\rm{\Lambda }}_{{\rm{uv}}}}} \right)$ predicts values for low (scale- ${\rm{\Lambda }}$ ) energy amplitudes. So augment our autonomy-of-scales hope: posit a scale- ${\rm{\Lambda }}$ effective theory $\left( {{\cal L}{\rm{'}},{\rm{\Lambda }}} \right)$ that reproduces the low-energy amplitudes $\left( {{\cal L},{{\rm{\Lambda }}_{{\rm{uv}}}}} \right)$ predicts:

(6)

where ${ = _{ \lt {\rm{\Lambda }}}}$ denotes agreement about low-energy amplitudes.

RG equations identify theories $\left( {{\cal L},{{\rm{\Lambda }}_{{\rm{uv}}}}} \right)$ and $\left( {{\cal L}{\rm{'}},{\rm{\Lambda }}} \right)$ that satisfy the hopes just expressed. Given a high-energy (scale- ${{\rm{\Lambda }}_{{\rm{uv}}}}$ ) Lagrangian ${\cal L}$ , RG equations identify a low-energy (scale- ${\rm{\Lambda }}$ ) Lagrangian ${\cal L}{\rm{'}}$ that mimics ${\cal L}$ ’s low-energy empirical implications—where (6) explicates the mimicry relations. RG equations induce a map ${R_{{\rm{\Lambda }}{{\rm{\Lambda }}_{{\rm{uv}}}}}}$ on ${\cal G}$ :  ${\cal L}{\rm{'}} = {R_{{\rm{\Lambda }}{{\rm{\Lambda }}_{{\rm{uv}}}}}}{\cal L}$ . It’s important to keep the pair of energy scales indexing the RG map in view: Although the map acts on the space ${\cal G}$ of Lagrangians, it’s establishing an association (explicated by (6)) between theories $\left( {{\cal L},{{\rm{\Lambda }}_{{\rm{uv}}}}} \right)$ and $\left( {{\cal L}{\rm{'}},{\rm{\Lambda }}} \right)$ .

RG equations are devised by starting with the high-energy theory $\left( {{\cal L},{{\rm{\Lambda }}_{{\rm{uv}}}}} \right)$ and “integrating out” field configurations with energies between ${{\rm{\Lambda }}_{{\rm{uv}}}}$ and the new cutoff ${\rm{\Lambda }}$ —roughly speaking, blurring $\left( {{\cal L},{{\rm{\Lambda }}_{{\rm{uv}}}}} \right)$ ’s high-energy content by replacing individual high-energy field configurations with averages over the collection of high-energy fields—to identify the low-energy theory that reproduces $\left( {{\cal L},{{\rm{\Lambda }}_{{\rm{uv}}}}} \right)$ low-energy amplitudes.Footnote ¹¹ Because low-energy amplitudes are approximated perturbatively, so are RG equations: a first-order RG equation identifies low- and high-energy theories whose low-energy amplitudes, calculated to tree order, agree; a second- or loop-order RG equation identifies theories whose low-energy amplitudes, calculated to loop order, agree, and so on.

In the space ${\cal G}$ , a Lagrangian is coded by the coupling vector giving the coefficients of its kinetic, mass, and interaction terms. The central dogma of the RG approach is that “changing [the cutoff scale ${\rm{\Lambda }}$ ] and demanding the the physics be the same (since ${\rm{\Lambda }}$ is arbitrary) means the the couplings in the theory … must depend on ${\rm{\Lambda }}$ ” (Schwartz Reference Schwartz2014, 418). Poetically put, RG equations describe the running of the couplings. More prosaically, for each coupling $g$ , they determine a function $g\left( {\rm{\Lambda }} \right)$ that tracks how that coupling evolves as the cutoff scale is lowered from ${{\rm{\Lambda }}_{{\rm{uv}}}}$ to ${\rm{\Lambda }}$ .Footnote ¹²

So how do the couplings run? A celebrated analysis of RG flows in the space of scalar field theories due to Polchinski (Reference Polchinski1984) supports a striking answer. Fix a UV cutoff scale ${{\rm{\Lambda }}_{{\rm{uv}}}}$ . Pick a Lagrangian ${\cal L}$ and start the RG flow. At first the news is bad. “Although the [cutoff scale ${{\rm{\Lambda }}_{{\rm{uv}}}}$ ] Lagrangian might start with a simple form, at lower scales it becomes complicated” (1984, 277), as interactions of all sorts, not just renormalizable ones, scramble to encapsulate the empirical implications of high-energy modes integrated out of $\left( {{\cal L},{{\rm{\Lambda }}_{{\rm{uv}}}}} \right)$ . But good news is coming: “At scales far below ${{\rm{\Lambda }}_{{\rm{uv}}}}$ … a great simplification will occur…. no matter what initial Lagrangian we start with (within limits), the Lagrangian will be strongly attracted toward a three-dimensional submanifold in the infinite-dimensional space of possible Lagrangians” (1984, 277). For ${\rm{\Lambda }} \ll {{\rm{\Lambda }}_{{\rm{uv}}}}$ , ${\cal L}{\rm{'}} = {R_{{\rm{\Lambda }}{{\rm{\Lambda }}_{{\rm{uv}}}}}}{\cal L}$ will be well-approximated by a Lagrangian featuring just three terms.Footnote ¹³ Additional terms in the high-energy Lagrangian ${\cal L}$ are suppressed—multiplied by factors of ${\rm{\Lambda }}/{{\rm{\Lambda }}_{{\rm{uv}}}}$ —by the RG flow. Although they won’t be strictly speaking absent from ${\cal L}{\rm{'}}$ , these additional terms have so little empirical impact that we might as well write them in invisible ink. As far as effective physics within experimental tolerances is concerned, it’ll be described by a Lagrangian in a three-dimensional subspace of ${\cal G}$ .

What’s more, according to Polchinski’s analysis, the exponentially suppressed terms are exactly the interactions declared non-renormalizable by Dyson’s criterion. In other words, to all intents and purposes, Lagrangians participating in effective physics reside in ${{\cal G}^{{\rm{PR}}}}$ . “In light of the above RG analysis,” Williams remarks, “it no longer seems like incredible luck that so many useful QFT [Lagrangians] are renormalizable” (Reference Williams2023, 54).

And that’s the RG Explanation. Useful physics is low-energy-effective physics, and the RG flow suppresses the contributions of non-renormalizable interactions to low-energy-effective Lagrangians. Polchinski tells us that this happy outcome “follows in a very general way when dimensional analysis is applied to the RG equation for an effective Lagrangian” (1984, 274). Where ${g_i}$ is a coupling vector specifying a scale- ${{\rm{\Lambda }}_{{\rm{uv}}}}$ Lagrangian, RG analysis yields an account ${g_i}\left( {\rm{\Lambda }} \right)$ of how each coupling runs as high-energy modes are integrated out of the high-energy theory. We expect that couplings are going to mix under the action of the RG—that after an RG transformation, the coupling for each interaction will typically be a function of the complete set of pre-transformation couplings. If the couplings $\left\{ {g_j\left( {\rm{\Lambda }} \right)} \right\}$ have different mass dimensions, what sense can we make of the mass dimension of their combination? An easy way to avert this awkwardness is to have the RG act on dimensionless couplings. “It is,” after all, as Weinberg remarks, “convenient to work with dimensionless parameters” (Reference Weinberg1995, 525).

To facilitate this convenience, take a coupling $g$ that modifies a Lagrangian term whose mass dimension $d \ne 4$ . For reasons rehearsed in section 2, $g$ must have mass dimension $4 - d$ . To transmute $g$ to a dimensionless coupling $\tilde g$ , multiply it by the appropriate power of the cutoff ${\rm{\Lambda }}$ (which, recall, has mass dimension $1$ ) collaborating with the Lagrangian to define the effective theory. Thus,

(7) $$\tilde g: = {({\rm{\Lambda }})^{d - 4}}g;$$

$\tilde g$ is a duly dimensionless coupling whose running will be amenable to RG analysis. Because the expedient ${({\rm{\Lambda }})^{d - 4}}$ plays a major role in what follows, I’ll give it a name: the dimension-cancelling pre-factor.

For a brutalist approximation to how $\tilde g$ runs under the RG, suppose that the “physical” coupling $g$ itself carries no scale dependence. Then only the dimension-cancelling prefactor varies as the cutoff scale changes—assuming, that is, that the ${\rm{\Lambda }}$ in the dimension-cancelling prefactor slides along with the scale at which integrals like (5) are cut off. (This is an assumption we’ll revisit.) It follows that

(Naive) $${{\partial \tilde g} \over {\partial \Lambda }} = g{{\partial \left( {{\Lambda ^{d - 4}}} \right)} \over {\partial \Lambda }}.$$

Ignoring the fundamental moral of RG approaches, that physical couplings change as scales change, (Naive) gives a first-order approximation to the RG equations.Footnote ¹⁴ Higher-order approximations to the RG equations, approximations sensitive to the details of how qft Lagrangians underwrite predictions at higher orders of perturbation theory, could inflict corrections on (Naive).

But let this not distract us from the story (Naive) has to tell! According to (Naive). if $\tilde g\left( {{{\rm{\Lambda }}_{{\rm{uv}}}}} \right) = {({{\rm{\Lambda }}_{{\rm{uv}}}})^{d - 4}}g$ is a coupling’s value in a UV cutoff Lagrangian ${\cal L}$ , and we lower the cutoff from ${{\rm{\Lambda }}_{{\rm{uv}}}}$ to ${\rm{\Lambda }}$ , then the ${\rm{\Lambda }}$ scale coupling $\tilde g\left( {\rm{\Lambda }} \right)$ becomes ${({\rm{\Lambda }})^{d - 4}}g$ . But then

(8) $$\tilde g\left( \Lambda \right) = {\left({\Lambda \over {{\Lambda _{{\rm{uv}}}}}}\right)^{d - 4}}\tilde g\left( {{\Lambda _{{\rm{uv}}}}} \right);$$

$d$ is the dimension of the interaction $\tilde g$ modifies. Dyson’s criterion tells us that if $d \gt 4$ , that interaction is non-renormalizable. (8) tells us that if $d \gt 4$ , the RG flow suppresses—multiplies by positive powers of ${\rm{\Lambda }}/{{\rm{\Lambda }}_{{\rm{uv}}}}$ , where ${{\rm{\Lambda }}_{{\rm{uv}}}}$ is a lot bigger than ${\rm{\Lambda }}$ —that interaction’s coupling. So (8) is the happy result we’re after. (8) attests that under the RG flow toward the infrared, couplings of non-renormalizable interactions are dramatically suppressed.

They’re dramatically suppressed, at least, according to the (Naive) approximation to the RG equations. It could be that higher-order approximations complicate this pleasing upshot, by revealing $\tilde g$ —and by extension, the non-renormalizable interaction it modifies—to matter more to low-energy physics than (8) allows. This menace, that the higher-order corrections “overwhelm” (Polchinski, Reference Polchinski1984, 273) (Naive)’s account of the RG flow, is ruled out by Polchinski’s analysis. Approximating RG equations to all orders, he shows that (Naive) gets the basic story basically right. The brutalist approximation is a good guide to how couplings run under the RG flow. And that approximation tells us what we want to hear: couplings of non-renormalizable interactions are dramatically suppressed.

4. Some dimensions of the problem

This criminally brief concluding section asks: How happy should we be with the RG Explanation? What’s doing almost all of the work in the version of the explanation just sketched is the “convenient” dimension-cancelling pre-factor (7) introduced to ensure that all the couplings, whose running we treat via RG methods, are dimensionless. The pre-factor condemns non-renormalizable interactions to suppression under the RG flow. And that condemnation resolves the Puzzle. But can the dimension-cancelling prefactor really bear the explanatory weight the RG Explanation places on it?

Here, in rapid-fire, are a few reasons to think the answer isn’t obvious.

Start with a difference between the instances of dimensional analysis that lead to (i) the verdict that interactions of mass dimension $d$ must be accompanied in a Lagrangian by couplings of mass dimension $\left( {4 - d} \right)$ , and (ii) the dimension-cancelling pre-factor in (7). For the first verdict, a nomic commodity—the action $S: = \smallint {d^4}x{\cal L}$ —guides the conclusion reached by dimensional analysis. That analysis reveals how couplings must behave in order for the action, and the swathe of physical theorizing based on the action, to make sense. For verdict (ii), by contrast, something arguably less nomic guides dimensional analysis—the maneuver of using dimension-cancelling pre-factors to facilitate easy management of functions describing the running of the couplings. Whereas in case (i) dimensional analysis reveals how couplings must behave for the sake of nomic compliance, in case (ii) dimensional analysis reveals how dimension-cancelling pre-factors must behave for the sake of … convenience. It’s not obvious that we should trust conclusions reached through dimensional analysis guided by considerations of convenience as much as we trust those guided by nomic considerations.

But grant, for the sake of argument, that functions describing how couplings run should act on dimensionless couplings.Footnote ¹⁵ Why use powers of a sliding cutoff ${\rm{\Lambda }}$ as the dimension-cancelling pre-factor? The result (Naive) anchoring the RG explanation follows only if the dimension-cancelling pre-factor varies with scale—indeed, [NAIVE] is an account of how the dimension-cancelling pre-factor varies with scale! But a fixed cutoff— ${{\rm{\Lambda }}_{{\rm{uv}}}}$ , for instance—would serve just as well in a pre-factor that cancels $g\left( {\rm{\Lambda }} \right)$ ’s dimension as ${\rm{\Lambda }}$ does. It’s the mass dimension of the pre-factor, not its value, that effects the cancellation. The problem for the RG Explanation is that if we use a fixed value of the cutoff in the dimension-cancelling pre-factor, dimensionless couplings inherit no cutoff dependence from that pre-factor, eradicating the exponential suppression that’s the crux of the explanation.

Of course, if we use a fixed value of the cutoff in dimensional-cancelling pre-factors, it would be nice to have a reason for the value we choose. There’s one choice for which a reason comes ready-made with RG approaches. That choice is ${{\rm{\Lambda }}_{{\rm{uv}}}}$ . Standard accounts of RG approaches cast ${{\rm{\Lambda }}_{{\rm{uv}}}}$ as the energy at which Lagrangian field theories break down. The only “natural” energy in sight, ${{\rm{\Lambda }}_{{\rm{uv}}}}$ is like (or may even be) the Planck length, which physicists are accustomed to wield in the cause of dimensional analyses in other contexts.Footnote ¹⁶ Given that we can define dimensionless couplings using a fixed cutoff, and that it’s hardly anathema to RG approaches to identify ${{\rm{\Lambda }}_{{\rm{uv}}}}$ as the fixed cutoff, we needn’t accept the result (Naive) that secures the RG explanation.

There is an even shorter route to the suppression of non-renormalizable interactions than the one section 3 charted. Forget about dimensionless couplings. Adopt another perspective on (6). We’ve been using (6) to answer the question: Which low-energy theory $\left( {{\cal L}{\rm{'}},{\rm{\Lambda }}} \right)$ mimics the low-energy amplitudes defined by a high-energy theory $\left( {{\cal L},{{\rm{\Lambda }}_{{\rm{uv}}}}} \right)$ ? This is a question about theories with different (but overlapping!) domains. But let’s ask a different question, one about theories with the same domain: How do ${\cal L}$ and ${\cal L}{\rm{'}}$ , both understood as governing physics up to the UV cutoff ${\Lambda _{uv}}$ , compare? In condensed matter physics, asking the counterpart of this question has huge payoffs: if the answer is “they’re the same!” then you’ve found a fixed point of the RG flow and are well on your way to calculating critical exponents and explaining universality.

To compare ${\cal L}$ and ${\cal L}{\rm{'}}$ as Lagrangians for the same, high-energy domain, we need to rescale. ${\cal L}{\rm{'}}$ as it appears in (6) governs physics only up to a scale ${\rm{\Lambda }}$ . To broaden its coverage so that its domain coincides with the domain of ${\cal L}$ , multiply all the couplings that appear on the right-hand side of (6) with mass (= energy) dimension $d$ by a scale factor ${({{\rm{\Lambda }}_{{\rm{uv}}}}/{\rm{\Lambda }})^d}$ . After this rescaling, ${\cal L}{{\rm{'}}_{{\rm{rescaled}}}}$ has the same domain of validity as ${\cal L}$ . It’s ${\cal L}{{\rm{'}}_{{\rm{rescaled}}}}$ we should consider alongside ${\cal L}$ if our aim is to compare Lagrangians with a common domain. But soft! If ${\cal L}$ includes an interaction of mass dimension $d \gt 4$ —a non-renormalizable interaction—that interaction will be modified by a coupling of negative mass dimension. This coupling will appear in ${\cal L}{{\rm{'}}_{{\rm{rescaled}}}}$ multiplied by a positive power of ${\rm{\Lambda }}/{{\rm{\Lambda }}_{{\rm{uv}}}}$ . This is the exponential suppression key to the RG explanation—here exhibited as a consequence of the rescaling step of an RG transformation.Footnote ¹⁷

But why, in the context of thinking, fueled by the autonomy-of-scales hopes, about effective Lagrangians, should we take the rescaling step? The question of which low-energy theory $\left( {{\cal L}{\rm{'}},{\rm{\Lambda }}} \right)$ mimics the low-energy amplitudes defined by a high-energy theory $\left( {{\cal L},{{\rm{\Lambda }}_{{\rm{uv}}}}} \right)$ is the right one to ask in this context. The adjacent question of what high-energy theory would ${\cal L}{\rm{'}}$ be if ${\cal L}{\rm{'}}$ were a high-energy theory is not. It’s ${\cal L}{\rm{'}}$ as participant in the effective theory $\left( {{\cal L}{\rm{'}},{\rm{\Lambda }}} \right)$ we care about. And how we’ve expressed that care makes rescaling ${\cal L}{\rm{'}}$ to extend its domain up to the ultra-high-energy cutoff ${{\rm{\Lambda }}_{{\rm{uv}}}}$ weird. If physics breaks down at ${{\rm{\Lambda }}_{{\rm{uv}}}}$ , what (after rescaling) are we to make of the high-energy modes we (before rescaling) integrated out of ${\cal L}$ to obtain ${\cal L}{\rm{'}}$ ? This is hardly to deny what the previous paragraph acknowledged: that there are plenty of contexts where it’s illuminating to pursue the question about how ${\cal L}$ and ${\cal L}{\rm{'}}$ , considered as theories with the same domain, compare. It is, however, to suggest that the present context might not be one of them. And if it isn’t one of them, the RG Explanation that emerges from the pursuit rings hollow.

The foregoing are reasons, internal to the RG Explanation, to question its bona fides. I’ll close with an external reason. It’s that the RG Explanation explains too much. The RG explanation primes us to expect effective theories to be renormalizable. But paradigm examples of effective theories—the $4$ -Fermi theory of beta decay (which implements the electroweak theory at low energies), the chiral Lagrangian (which implements QCD at low energies)—aren’t renormalizable. This is adamantly not to deny that such theories play important roles, rewarding of methodological and conceptual analysis, in working physics, or that a broad effective theory philosophy helps to make sense of those roles. It is rather to observe that non-renormalizable effective theories sit awkwardly alongside a very specific line of thought about effective theories: the line of thought traced by the RG Explanation. As sketchy as the reservations I’ve raised about that explanation are, perhaps this awkwardness is a reason to take them seriously.