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2.1 Perturbation theory and continuum limit

Perturbatively renormalizable field theories are a degenerate special case of the Wilson–Kadanoff framework. The main advantage of perturbation theory is that the UV cutoff Λ can be removed exactly and independently of the properties of the coupling flow. The existence of a Λ → ∞ limit with the required Properties (PTC1) can often be rigorously proven, in contrast to most nonperturbative techniques where this can only be established approximately by assembling evidence. With Criterion (PTC1) satisfied, the coupling flow then can be studied in a second step and used to probe whether or not the Criterion (PTC2) for the existence of a genuine continuum limit as anticipated in Section 1.3 is also satisfied. The main disadvantage of perturbation theory is that everything is initially defined as a formal power series in the loop counting parameter. Even if one trades the latter for a running coupling, the series in this coupling remains a formal one, typically non-convergent and not Borel-summable. It is generally believed, however, that provided Criterion (PTC2) is satisfied for a perturbative Gaussian fixed point, the series is asymptotic to the (usually unknown) exact result. In this case the perturbative analysis should indicate the existence of a genuine continuum limit based on an underlying Gaussian fixed point proper. Our main reason for going through this in some detail below is to point out that in a situation with several couplings the very same rationale applies if the perturbative fixed point is a non-Gaussian rather than a Gaussian one.

As mentioned, in perturbation theory one initially only aims at defining the expectations (A.1View Equation) as a formal power series in the loop counting parameter λ, where the sum of all ℓ-loop contributions to a quantity is assigned a factor λℓ. For the reasons explained in [106Jump To The Next Citation Point] the loop expansion does not necessarily coincide with an expansion in powers of Planck’s constant ℏ. For example when massless fields are involved, 1-loop diagrams can contribute to the classical limit 0 O (ℏ ). The loop counting parameter λ refers to a set of free fields of mass μ such that the formal expansion of the exponential in Equation (A.1View Equation) gives expectations whose computation can be reduced to the evaluation of Gaussians. We denote the (quadratic) action of this set of free fields by S∗,μ[χ]. The interaction is described by a set of monomials Pi [χ ], i ∈ Ep.c., which are “power counting renormalizable”. The latter means that their mass dimension − di is such that di ≥ 0. It is also assumed that the Pi are functionally independent, so that the corresponding couplings are essential. The so-called “bare” action functional then is ∑ S Λ = S∗,μ + i∈Ep.c.ui(Λ )Pi, where ui(Λ ) are the essential “bare” couplings (including masses) corresponding to the interaction monomials Pi[χ ]. Inessential parameters are generated by subjecting S Λ to a suitable class of field redefinitions. In more detail one writes

∑ ui(Λ ) = ui(μ )Vi,0(μ) + λℓVi,ℓ(u(μ),Λ, μ), ℓ≥1 ∑ (2.1 ) χΛ = χ μ + λ ℓΞℓ(χμ; u(μ),Λ, μ). ℓ≥1
Here ui(μ ) are the renormalized couplings which are Λ-independent and the Vi(u(μ),Λ, μ) are counterterms which diverge in the limit Λ → ∞. This divergence is enforced by very general properties of QFTs. Similarly the χμ are called renormalized fields and the Ξ ℓ(χ μ;u(μ),Λ, μ) are local functionals of the χ μ with coefficients depending on u(μ ),Λ, μ; the coefficients again diverge in the limit Λ → ∞. Often one aims at “multiplicative renormalizability”, which means the ansatz for the Ξ ℓ is taken to be linear in the fields Ξ ℓ(χ μ;u(μ),Λ, μ) = Z ℓ(u(μ ),Λ,μ)χ μ and Z ℓ is the ℓ-loop “wave function renormalization” constant. One should emphasize, however, that multiplicative renormalizability can often not be achieved, and even in field theories where it can be achieved, it evidently will work only with a particular choice of field coordinates (see [41Jump To The Next Citation Point] for a discussion).

The normalizations in Equation (2.1View Equation) can be chosen such that ui(μ = Λ ) = ui(Λ) and χ μ=Λ = χ Λ, but one is really interested in the regime where μ ≪ Λ. Inserting these parameterizations into SΛ[χΛ ] gives an expression of the form

( ) ∑ ∑ S Λ[χΛ] = S∗,μ [χ μ] + λ ℓuα,ℓ(u(μ ),Λ, μ) P α[χμ], (2.2 ) α ℓ≥0
where the sum over α includes terms of the form appearing on the right-hand-side of Equation (A.8View Equation). Often the μ-dependence in the fields can be traded for one carried by (inessential) parameters zi(μ), i ∈ I. Then Equation (2.2View Equation) takes the form ∑ SΛ[χΛ ] = ′ uα′(g (μ ),z(μ),Λ,μ )Pα′[χ ] α, with some μ-independent fields, χ = χ μ0, say. The right-hand-side clearly resembles Equation (A.7View Equation) with the difference that modulo field redefinitions only power counting renormalizable interaction monomials occur.

So far the counterterms in Equation (2.1View Equation) have been left unspecified. The point of introducing them is of course as a means to absorb the cut-off dependence generated by the regularized functional integral in Equation (A.1View Equation). Specifically, one replaces the Boltzmann factor by its power series expansion in λ, i.e. ∑ ℓ exp {− SΛ[χΛ ]} = exp {− S∗,μ[χμ]}(1 + ℓ≥1 λ Qℓ[χμ]), and aims at an evaluation of multipoint functions ⟨χ Λ(x1)...χ Λ(xn)⟩SΛ as formal power series in λ. After inserting Equation (2.1View Equation) and the expansion of e−SΛ[χΛ] this reduces the problem to an evaluation of the free multipoint functions ⟨χμ(x1) ...χμ(xn)Ql[χμ ]⟩S ∗,μ computed with the quadratic action S∗,μ on the field space with cutoff Λ. The free multipoint functions will contain contributions which diverge in the limit Λ → ∞. On the other hand via the parameterization (2.1View Equation, 2.2View Equation) the coefficients carry an adjustable Λ dependence. In a renormalizable QFT the Λ dependence in the coefficients can be chosen such as to cancel (for μ ≪ Λ) that generated by the multipoint functions ⟨χ (x ) ...χ (x )Q [χ ]⟩ μ 1 μ n l μ S∗,μ. With this adjustment the limits

∑ ∑ λℓ lim ⟨χΛ(x1) ...χΛ(xn)⟩SΛ,ℓ =: λ ℓ⟨χ μ(x1)...χ μ(xn)⟩Sμ,ℓ, (2.3 ) ℓ≥0 Λ →∞ ℓ≥0
exist and define the renormalized multipoint functions. As indicated they can be interpreted as referring to the renormalized action lim Λ→ ∞ SΛ [χ Λ] = Sμ[χμ]. Equation (2.3View Equation) highlights the main advantage of renormalized perturbation theory: The existence of the infinite cutoff limit (2.3View Equation) is often a provable property of the system, while this is not the case for most nonperturbative techniques. In the terminology introduced in Section 1.3 the Criterion (PTC1) is then satisfied. In order for this to be indicative for the existence of a genuine continuum limit, however, the additional Condition (PTC2) must be satisfied, whose rationale we proceed to discuss now.

Since the renormalization scale μ is arbitrary, changing its value must not affect the values of observables. The impact of a change in μ can most readily be determined from Equation (2.1View Equation). The left-hand-sides are μ independent, so by differentiating these relations with respect to μ and extracting the coefficients in a power series in (say) Λ and/or log Λ consistency conditions arise for the derivatives d- μ dμui and d- μ dμ χμ. The ones obtained from the leading order are the most interesting relations. For the couplings one obtains a system of ordinary differential equations which define their renormalization flow under a change of μ. As usual it is convenient to work with dimensionless couplings gi := uiμ −di, where di is the mass dimension of ui. The flow equations then take the form

d μ dμgi = βi(g(μ)), (2.4 )
where the βi are the perturbative beta functions. The flow equations for the renormalized fields are familiar only in the case of multiplicatively renormalizable fields, where one can work with scale independent fields and have the scale dependence carried by the wave function renormalization constant. In general however the fields are scale dependent. For example this ensures that the renormalized action evaluated on the renormalized fields is scale independent: d μ dμSμ[χμ] = 0.

By construction the perturbative beta functions have a fixed point at g∗i = 0, which is called the perturbative Gaussian fixed point. Nothing prevents them from having other fixed points, but the Gaussian one is built into the construction. This is because a free theory has vanishing beta functions and the couplings −di gi = uiμ have been introduced to parameterize the deviations from the free theory with action S ∗,μ. Not surprisingly the stability matrix Θij = ∂ βi∕∂gj|g∗=0 of the perturbative Gaussian fixed point just reproduces the information which has been put in. The eigenvalues come out to be − di modulo corrections in the loop coupling parameter, where − di are the mass dimensions of the corresponding interaction monomials. For the eigenvectors one finds a one-to-one correspondence to the unit vectors in the ‘coupling direction’ gi, again with power corrections in the loop counting parameter. One sees that the couplings ui not irrelevant with respect to the stability matrix Θ computed at the perturbative Gaussian fixed point are the ones with mass dimensions di ≥ 0, i.e. just the power counting renormalizable ones.

The attribute “perturbative Gaussian” indicates that whenever in a nonperturbative construction of the renormalization flow in the same ‘basis’ of interaction monomials g∗= 0 i is also a fixed point (called the Gaussian fixed point), the perturbatively defined expectations are believed to provide an asymptotic (nonconvergent) expansion to the expectations defined nonperturbatively based on the Gaussian fixed point, schematically

∑ ⟨𝒪 ⟩Gaussian FP ∼ λ ℓ⟨𝒪 ⟩ℓ. (2.5 ) ℓ≥0
Here ⟨𝒪 ⟩ℓ is the perturbatively computed ℓ-loop contribution after a so-called renormalization group improvement. Roughly speaking the latter amounts to the following procedure: One assigns to the loop counting parameter λ a numerical value (ultimately related to the value of Planck’s constant in the chosen units; see however [106]) and solves μ dgi-= βi(g(μ)) dμ as an ordinary differential equation. One of the functions obtained, say g1(μ ), is used to eliminate λ in favor of μ and an integration constant Λbeta (not to be confused with the cutoff, which is gone for good). So ∑ ¯ ℓ ⟨𝒪 ⟩L := ℓ≤L λ(μ,Λbeta) ⟨𝒪 ⟩ℓ,μ at this point carries a two-fold μ-dependence, the one which comes out of the renormalization procedure (2.3View Equation) and the one carried now by ¯λ(μ, Λbeta). For an observable quantity 𝒪 both dependencies cancel out, modulo terms of higher order, leaving behind a dependence on the integration constant Λbeta. We write ⟨𝒪 ⟩L(Λbeta) to indicate this dependence. One then uses the expectation of one, suitably chosen, observable 𝒪0 to match its value ⟨𝒪0 ⟩ (measured or otherwise known) with that of ⟨𝒪0⟩L(Λbeta) to a given small loop order L (typically not larger than 2). For a well chosen 𝒪0 this allows one to replace Λbeta by a physical mass scale m phys. Eliminating Λ beta in favor of m phys gives the perturbative predictions for all other observables. Apart from residual scheme dependencies (which are believed to be numerically small) this defines the right-hand-side of Equation (2.5View Equation) unambigously as a functional over the observables.

Nevertheless, except for some special cases, it is difficult to give a mathematically precise meaning to the ‘∼’ in Equation (2.5View Equation). Ideally one would be able to prove that perturbation theory is asymptotic to the (usually unknown) exact answer for the same quantity. For lattice theories on a finite lattice this is often possible; the problems start when taking the limit of infinite lattice size (see [159] for a discussion). In the continuum limit a proof that perturbation theory is asymptotic has been achieved in a number of low-dimensional quantum field theories: the superrenormalizable P2 (φ ) and 4 φ3 theories [6943] and the two-dimensional Gross–Neveu model, where the correlation functions are the Borel sum of their renormalized perturbation expansion [8789]. Strong evidence for the asymptotic correctness of perturbation theory has also been obtained in the O(3) nonlinear sigma-model via the form factor bootstrap [22]. In four or higher-dimensional theories unfortunately no such results are available. It is still believed that whenever the above g1 is asymptotically free in perturbation theory, that the corresponding series is asymptotic to the unknown exact answer. On the other hand, to the best of our knowledge, a serious attempt to establish the asymptotic nature of the expansion has never been made, nor are plausible strategies available. The pragmatic attitude usually adopted is to refrain from the attempt to theoretically understand the domain of applicability of perturbation theory. Instead one interprets the ‘∼’ in Equation (2.5View Equation) as an approximate numerical equality, to a suitable loop order L and in a benign scheme, as long as it works, and attributes larger discrepancies to the ‘onset of nonperturbative physics’. This is clearly unsatisfactory, but often the best one can do. Note also that some of the predictive power of the QFT considered is wasted by this procedure and that it amounts to a partial immunization of perturbative predictions against (experimental or theoretical) refutation.

So far the discussion was independent of the nature of the running of ¯λ(μ,Λbeta) (which was traded for g1). The chances that the vague approximate relation ‘∼’ in Equation (2.5View Equation) can be promoted to the status of an asymptotic expansion are of course way better if ¯λ (μ,Λbeta) is driven towards ¯λ = 0 by the perturbative flow. Only then is it reasonable to expect that an asymptotic relation of the form (2.5View Equation) holds, linking the perturbative Gaussian fixed point to a genuine Gaussian fixed point defined by nonperturbative means. The perturbatively and the nonperturbatively defined coupling g1 can then be identified asymptotically and lie in the unstable manifold of the fixed point g1 = 0. On the other hand the existence of a Gaussian fixed point with a nontrivial unstable manifold is thought to entail the existence of a genuine continuum limit in the sense discussed before. In summary, if g1 is traded for a running ¯ λ(μ,Λbeta), a perturbative criterion for the existence of a genuine continuum limit is that the perturbative flow of g1 is regular with lim μ→ ∞ g1(μ) = 0. Since the beta functions of the other couplings are formal power series in λ without constant coefficients, the other couplings will vanish likewise as g → 0 1, and one recovers the local quadratic action S∗,μ[χ ] at the fixed point. The upshot is that the coupling with respect to which the perturbative expansion is performed should be asymptotically free in perturbation theory in order to render the existence of a nonperturbative continuum limit plausible.

The reason for going through this discussion is to highlight that is applies just as well to a perturbative non-Gaussian fixed point. This sounds like a contradiction in terms, but it is not. Suppose that in a situation with several couplings g1,...,gn the perturbative beta functions (which are formal power series in λ without constant coefficients) admit a nontrivial zero, ∗ ∗ g1(λ),...,gn (λ ). Suppose in addition that all the couplings lie in the unstable manifold of that zero, i.e. the flows gi(μ) are regular and lim μ→ ∞ gi(μ ) = g ∗i. We shall call a coupling with this property asymptotically safe, so that the additional assumption is that all couplings are asymptotically safe. As before one must assign λ a numerical value in order to define the flow. Since the series in λ anyhow has zero radius of convergence, the ‘smallness’ of λ is not off-hand a measure for the reliability of the perturbative result (the latter intuition in fact precisely presupposes Equation (2.5View Equation)). Any one of the deviations δg = g − g∗ i i i, which is of order λ at some μ can be used as well to parameterize the original loop expansion. By a relabeling or reparameterization of the couplings we may assume that this is the case for δg1. The original loop expansion can then be rearranged to read ∑ ℓ≥0(δg1)ℓ⟨𝒪 ⟩ℓ. However, if there is an underlying nonperturbative structure at all, it is reasonable to assume that it refers to a non-Gaussian fixed point,

∑ ℓ ⟨𝒪 ⟩non− Gaussian FP ∼ (δg1) ⟨𝒪 ⟩ℓ. (2.6 ) ℓ≥0
The rationale for Equation (2.6View Equation) is exactly the same as for Equation (2.5View Equation). What matters is not the value of the couplings at a perturbative fixed point, but their flow pattern. For a nontrivial fixed point the couplings ∗ gi in the above basis of interaction monomials are nonzero, but any one of the deviations ∗ δgi = gi − gi can be made arbitrarily small as μ → ∞. The relation ‘∼’ in Equation (2.6View Equation) then again plausibly amounts to an asymptotic expansion for the unknown exact answer, where the latter this time is based on a non-Gaussian fixed point.

Summarizing: In perturbation theory the removal of the cutoff can be done independently of the properties of the coupling flow, while in a non-perturbative setting both aspects are linked. Only if the coupling flow computed from the perturbative beta functions meets certain conditions is it reasonable to assume that there exists an underlying non-perturbative framework to whose results the perturbative series is asymptotic. Specifically we formulate the following criterion:

Criterion (Continuum limit via perturbation theory):
(PTC1) Existence of a formal continuum limit, i.e. removal of the UV cutoff is possible and the renormalized physical quantities are independent of the scheme and of the choice of interpolating fields, all in the sense of formal power series in the loop counting parameter.

(PTC2)  The perturbative beta functions have a Gaussian or a non-Gaussian fixed point and the dimension of its unstable manifold (as computed from the perturbative beta functions) equals the number of independent essential couplings. Equivalently, all essential couplings are asymptotically safe in perturbation theory.

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