Renormalons

M. Beneke (CERN)

REVIEW

Renormalons

M. Beneke (CERN)

High Energy Physics - Phenomenology · Submitted 21 Jul 1998 · 164 references

Pith's verdict: The standard reference on renormalons, and it earns that status: the scaling-not-normalization thesis is right, and the paper is unusually honest about where the arguments stop being proofs.

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arXiv IDhep-ph/9807443 (version 2)

Pith IDL2QMX3P5, Pith's permanent record of this paper and its review

CitationM. Beneke (CERN). "Renormalons." arXiv:hep-ph/9807443 (1998). https://pith.science/paper/L2QMX3P5

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Editor's note

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This is the review you cite when you need the renormalon story in one place, and it holds up. The organizing thesis is stated cleanly at [abstract:s3] and developed in §4.2.1: IR renormalons fix the power of Lambda/Q in a power correction, and they tell you essentially nothing about its normalization. That split is the useful part, and Beneke keeps it disciplined throughout — the calculable scaling is separated from the incalculable residue at every step, including the explicit statement in §4.2.1(i) that the QCD cancellation between coefficient-function ambiguity and operator matrix element cannot be traced non-perturbatively. He says so rather than papering over it, which is more than most reviews of this subject do. The diagrammatic half (§3) is the best pedagogical treatment of the flavour expansion I know, and the O(N) sigma-model section (§4.1) is the part that actually earns the interpretation: the Borel representation falling out of the exact OPE at Eq. 4.31, with IR singularities cancelling between coefficient functions and condensates across orders in the power expansion, is the closest thing to a demonstration that the whole picture is coherent. The DIS twist-4 matching in §4.2.2 is a nice concrete check of the same mechanism in a real theory. On the stress-test note: it is right, and the reader's flagged weak point is not load-bearing. The MS-regularity conjecture in §3.4 is a genuine unproved assumption, but Eqs. 3.62–3.65 show the singularity location and exponent are scheme-invariant; only the normalization moves, and the paper already refuses to predict that. The real soft joint is the chain at Eq. 2.13, which the paper itself labels 'not rigorous' — and which the sigma-model section is there to shore up. Two things I would flag. First, §4.2.2 rests on unpublished notes with Braun and Magnea; the calculation is checkable in principle but it is not in the literature, and a reader who wants to use the twist-4 matching should know that. Second, §3.2.2 concedes there is no complete non-abelian diagrammatic analysis, and §3.3 concedes no general classification of IR renormalons exists; both are honest, but they mean the 'known from Feynman diagrams' framing of the abstract is more aspirational for IR than for UV. I read the theory half closely and the phenomenology half (Section 5 onward) only in outline, so weight my comments accordingly. For anyone working on power corrections, event shapes, heavy-quark masses, or the OPE, this is required reading and the citation of record. Send it out; it deserves referee time, though as a published review the useful referee work is mostly about whether the phenomenology survey is complete and current.

Referee report

Significance

The review's central thesis is stated at [abstract:s3] and developed in §4.2.1: IR renormalons in perturbative coefficient functions fix the scaling of power corrections in Lambda/Q, while the overall normalization is not calculable and must be fixed by data or by matching to operator matrix elements. The value of the manuscript lies in the discipline with which that separation is maintained. The calculable part is derived twice, diagrammatically in §3.2 and by renormalization-group methods in §3.2.3, and the two treatments are shown to agree. The incalculable part is named as such at every point where it enters. The O(N) sigma-model chapter (§4.1) is the strongest single element: it supplies a non-perturbative setting in which the Borel representation emerges from an exact OPE (Eq. 4.31) and the cancellation of IR singularities between coefficient functions and condensates is exhibited rather than assumed. The one-loop matching of the DIS twist-4 quadratic divergences onto the leading-twist distribution in §4.2.2 (Eqs. 4.53-4.57) is a concrete, checkable worked example of the operator interpretation. The review is also candid about its own limits: §2.1 calls the inference chain of

Major objections (1)

[§4.1 and §4.2.1] The review's thesis, that IR renormalons are tied to non-perturbative power corrections, is supported non-perturbatively only by the O(N) sigma model of §4.1, and the review itself observes that this model is special in a way that does not carry over to QCD. After Eq. (4.31) the text notes that the leading factorizable 1/N contributions to the condensates are unambiguous, so that the power-like ambiguities of the perturbative expansions are parametrically smaller in 1/N than the actual condensates, and then states that 'a similar parametric suppression of perturbative ambiguities does not seem to take place in neither the large-Nc nor the large-Nf limit of QCD.' That is precisely the regime in which the phenomenological applications of Section 5 operate, where the renormalon ambiguity is used as a model for the magnitude of the power correction. The caveat is present and correctly stated, but it appears at the end of §4.1, whereas the numbered conclusions (i)-(vi) of §4.2.1 are written in a register that reads as established. I would ask the author to carry the caveat forward explicitly into §4.2.1, for instance as a sentence attached to conclusion (ii), so that a reader who consults §4.2.1 alone is not left with the impression that the sigma-model check transfers to QCD at large Nc or large Nf. This is a request for signposting, not for a new calculation.

Minor objections (10)

[§2.4, Fig. 3 caption] The caption is unreadable as supplied: 'Instan ton/-an ti/-instan ton singularities at t /= /4 /; /8 /; /: /: /:' and 'at t /= /BnZr m/=//0/, m /= /2 /; /3 /: /: /:'. The mathematical symbols have been lost, so the figure cannot be interpreted from its caption. The caption must be regenerated with proper symbols before publication.
[Throughout] The text contains numerous broken words and corrupted expressions: 'know n' in the abstract, 'div ergence' and 'renor malons' in the abstract, 'fo llows' in §2.2, 'fo ussed' and 'requiresd' in §3.2.1, 'onl y' and 'u niversal' in §3.2.1, 'the the' in §2.2, 'expect' for 'except' in §2.4, 'emphasis' for 'emphasized' in the opening of §4, and 'ˆg(µ)22' for 'ˆg(µ)^2/2' in Eq. (4.19). A clean copy-editing pass is needed.
[§2.4, footnote 14] The footnote reads 'Read: The process dependence factorizes and is calculable, see Section 3.2.' This is phrased as an author's marginal note rather than as prose and should be rewritten as a normal sentence, or folded into the body text.
[§3.4, Eq. (3.60)] The parameter S is introduced in Eq. (3.60) without a defining sentence, and it then carries the scheme-invariance discussion of Eqs. (3.62)-(3.65). Please state explicitly that S is the inverse of the singularity location in the Borel plane, so that the invariance of S in Eq. (3.63) can be read as the invariance of the power of Lambda/Q.
[§3.2.1, footnote 19] The statement 'After summation of all diagrams, the anomalous dimension of O6 is found to vanish' is given without a reference. A citation would let the reader check it against the mixing matrix quoted in Eq. (3.40).
[§3.3.1, Table 2] The footnote marker in the table is explained in the body text rather than in the caption, and the columns are not labelled by diagram. Please label the columns by the corresponding figure and move the explanation of the dagger into the caption.
[§2.3, Eq. (2.29) and §3.3.1] The gluon condensate is written once as <0|G^A_{mu nu} G^{A,mu nu}|0> and once as O = alpha_s G G. Please fix the normalization once, including the convention for the alpha_s factor, and use it consistently.
[§3.2.1, Eq. (3.12)] The operator O6 carries an explicit factor 1/g_s^2 while F^{mu nu} is described as the field strength of an external abelian gauge field. A sentence on the convention would help the reader reconcile Eq. (3.12) with the basis of Eq. (3.39), where the same factor appears.
[§2.2, Eq. (2.18)] The sign convention beta_0 < 0, which is essential for reading the Borel-plane figure and for the statement that IR renormalons lie on the positive real axis, is stated only in the caption of Fig. 3. It would be better placed at Eq. (2.18).
[§3.2.1] The phrase 'the two latter quantities' has no clear antecedent in the surrounding paragraph. Please name the quantities explicitly.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

The factorial divergence of perturbative QCD series is not noise: it encodes where non-perturbative power corrections must appear, and at what power of the hard scale.

The reading

Perturbative expansions in QCD do not converge. The coefficients grow like $K a^n n! n^b$, and the standard way to talk about such series is the Borel transform, whose singularities in the Borel plane are called renormalons. This review argues that those singularities are not a technical irritation but a physical signal: their positions are fixed by the first coefficient of the $\beta$ function, and the infrared ones sit at exactly the powers of $1/Q$ at which non-perturbative corrections to a hard process must appear. If that is right, one can read off the scaling of a power correction from a purely perturbative calculation, without any model of the non-perturbative dynamics. What the same calculation cannot deliver is the overall normalization of the correction, which must be measured or matched to an operator matrix element. The review then audits where this programme has worked for real processes and where renormalon-inspired models of power corrections have failed.

What carries the argument

The Borel transform $B[R](t) = sum_n r_n t^n / n!$ turns factorial growth into singularities, and the variable $u = -beta_0 t$ places infrared renormalons at positive integer or half-integer $u$ and ultraviolet ones at negative $u$. The calculational engine is the flavour ($1/N_f$) expansion, in which a chain of fermion bubbles becomes an analytically regularized gluon propagator of power $1+u$; the singularity structure of multi-chain diagrams then follows from a forest formula identical in shape to the standard ultraviolet renormalization recursion, with dimension-6 operators playing the role of counterterms. Renormalization-group analysis upgrades this to a statement valid beyond the flavour expansion: the coefficient functions of the compensating operators solve a linear equation whose kernel is the $\beta$ function and the anomalous-dimension matrix of dimension-6 operators, so everything except the integration constants is determined. For phenomenology the review prefers the dispersive or 'massive gluon' representation, in which the whole bubble chain is resummed into an effective coupling $Phi(xi)$ integrated against a characteristic function $T(xi)$; non-analytic terms in the small-$xi$ expansion of $T$ map one-to-one onto power corrections, so the correspondence can be read off without evaluating the Borel integral.

Extended reading notes

Core claim

The review's organizing thesis is that infrared renormalons in perturbative coefficient functions are the perturbative shadow of non-perturbative power corrections. Concretely: if a skeleton diagram behaves as $F(k^2) + O(k^a)$ at small loop momentum, then dressing it with fermion bubbles produces a Borel singularity at $u = a/2$ (with $u = -beta_0 t$) and a corresponding Borel-integral ambiguity scaling as $(Lambda/Q)^a$. The position of the singularity, and hence the power of the correction, is calculable; the residue is not, because it receives unsuppressed contributions from diagrams with arbitrarily many chains. The review supports this with a solvable case, the $O(N)$ non-linear $\sigma$ model in the $1/N$ expansion, where the cancellation between the renormalon ambiguity of the coefficient function and the power-divergent part of the operator matrix element can be followed exactly, and with a one-loop QCD check in which the quadratic divergence of twist-4 three-particle operators reproduces the leading infrared renormalon of the longitudinal deep-inelastic structure function. A parallel argument covers ultraviolet renormalons: they can be compensated by higher-dimension operator insertions, in direct analogy with Symanzik improvement of lattice actions, and in QCD they are sign-alternating and harmless.

Load-bearing premise

The quantitative reading of the Borel plane assumes that the subtraction scheme used to define the coupling is 'regular', meaning its subtraction constants do not themselves diverge factorially; the review states that this is reasonable and true to leading order in the flavour expansion, but that there is no proof for the minimal-subtraction scheme, since that scheme is defined only order by order.

Editorial extensions

If this is right

The leading power correction to a class of observables is fixed in its $Q$-dependence by the position of the nearest infrared renormalon, giving $1/Q$ corrections to event-shape variables, $1/m$ to the heavy-quark pole mass, and $1/Q^2$ to twist-4 structure functions without non-perturbative input.
For observables that share the same operator, the ratio of power corrections is computable in perturbation theory from the lowest-order coefficient functions, so only one overall number per operator has to be taken from data.
Ultraviolet renormalons can be absorbed into higher-dimension operators with calculable anomalous dimensions, so their factorial growth does not require new non-perturbative parameters, only a choice of renormalization scale.
The scheme dependence of the uncalculable normalization is one-loop and known explicitly, so a normalization fixed in one scheme can be transported to another once the relation between couplings is specified.
Where a hard process admits an operator product expansion, renormalons add nothing beyond what the condensate expansion already says; their value lies in processes such as fragmentation and event shapes that have no operator interpretation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the normalization is genuinely non-computable in QCD, the natural place to obtain it is the lattice: a numerical determination of the gluon condensate or of the relevant non-local soft matrix element would convert renormalon-inspired models from one-parameter fits into parameter-free predictions, and would test the whole chain rather than just its scaling.
The position prediction needs only the first beta-function coefficient, so the same bookkeeping should transfer to any asymptotically free theory with a known one-loop beta function, including lower-dimensional or condensed-matter critical systems; the cheap part of the prediction travels, the hard part (the residue) does not.
The pattern of successes and failures catalogued for event shapes suggests the modelling is most trustworthy when the leading infrared region is a single soft gluon coupling to eikonal lines, and least trustworthy when several partons go soft at once; that is a testable regularity, since one can sort observables by the number of lines in their soft subgraph and check whether the model's accuracy d
Because the review ties the infrared renormalon to the ultraviolet power divergence of the corresponding operator, a lattice calculation that subtracts power divergences non-perturbatively and one that does not should disagree by exactly the renormalon ambiguity; that disagreement is a measurable quantity rather than a bookkeeping artefact.

Original abstract

A certain pattern of divergence of perturbative expansions in quantum field theories, related to their small and large momentum behaviour, is known as renormalons. We review formal and phenomenological aspects of renormalon divergence. We first summarize what is known about ultraviolet and infrared renormalons from an analysis of Feynman diagrams. Because infrared renormalons probe large distances, they are closely connected with non-perturbative power corrections in asymptotically free theories such as QCD. We discuss this aspect of the renormalon phenomenon in various contexts, and in particular the successes and failures of renormalon-inspired models of power corrections to hard processes in QCD.

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What the abstract itself states, as probabilities: claims first 0.02 · states limitation 0.21 · human participants 0.02 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.20 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

A certain pattern of divergence of perturbative expansions in quantum field theories, related to their small and large momentum behaviour, is known as renormalons. We review formal and phenomenological aspects of renormalon divergence. We first summarize what is known about ultraviolet and infrared renormalons from an analysis of Feynman diagrams. Because infrared renormalons probe large distances, they are closely connected with non-perturbative power corrections in asymptotically free theories such as QCD. We discuss this aspect of the renormalon phenomenon in various contexts, and in particular the successes and failures of renormalon-inspired models of power corrections to hard processes in QCD.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	A certain pattern of divergence of perturbative expansions in quantum field theories, related to their small and large momentum behaviour, is known as renormalons.	background 1.00	0.04	0.05	asserted 1.00
s1	We review formal and phenomenological aspects of renormalon divergence.	contribution 0.98	0.64	0.05	asserted 0.51
s2	We first summarize what is known about ultraviolet and infrared renormalons from an analysis of Feynman diagrams.	method 0.46	0.17	0.04	asserted 0.93
s3	Because infrared renormalons probe large distances, they are closely connected with non-perturbative power corrections in asymptotically free theories such as QCD.	background 0.90	0.07	0.11	asserted 1.00
s4	We discuss this aspect of the renormalon phenomenon in various contexts, and in particular the successes and failures of renormalon-inspired models of power corrections to hard processes in QCD.	contribution 0.97	0.54	0.14	asserted 0.52

What kind of paper the classifier reports this is: review 1.00 · evidence from literature 0.82.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** The simplest set of ‘bubble’ diagrams for the Adler function consists of all diagrams with any number of fermion loops inserted into a single gluon line. Proceeding with the diagrams of

**Figure 2.** The integrand of (2.19) for n = 0 and n = 2 as function of ˆk 2 . The vertical scale is arbitrary. Note that UV and IR finiteness of the Adler function implies that F must have a powerlike approach to zero for both large and small ˆk 2 . The integrand of (2.19) is shown in

**Figure 3.** Singularities in the Borel plane of Π(Q2 ), the current-current correlation function in QCD. Shown are the singular points, but not the cuts attached to each of them. Recall that β0 < 0 according to (2.18). plane and hence governs the large-order behaviour of the series expansion of the Adler function. According to (2.4) the minimal term is of order Λ2/Q2 , using (2.27). A more precise analysis (Beneke & Zakharov 1992) shows that it is of order δDUV ∼ Q2Λ 2 µ4 × logarithms. (2.31) However, since UV renormalons produce sign-alternating factorial divergence in QCD, we do not take them…

**Figure 4.** Pair creation of quarks by an external current: (a) Leading order in the flavour expansion; (b) representatives at next-to-leading order. Chains are displayed as dashed lines. The chain propagator becomes particularly useful after applying Borel transformation (Beneke 1993a). Using the definition (2.5), one has B[αsDµν](u) = (−i) k 2

**Figure 5.** Leading order contribution in the flavour expansion to the two point function of vector currents jV (left) and its reduced diagrams with operator insertions (right). The momentum p is the loop momentum for the fermion loop. normalization of (3.11). In (3.12) F µν is the field strength of an external abelian gauge field that relates to the external vector current through ∂µF µν = j ν V . The factor 1/g2 s is convention. Note that there is only a limited number of 1PI one-loop graphs Γ, which can have a UV renormalon pole at u = −1. The condition is ω(Γ) ≥ −2. This generalizes to all…

**Figure 6.** A two-chain vertex integral and the contributions to its UV renormalon singularity. The straight arrows indicate the contractions that lead to insertions of dimension-6 operators. The arrows to the right indicate contractions that correspond to logarithmic operator mixing among the dimension-6 operators. F2 : 1 1 + u1 + u2 1 1 + u1 + u2 −→ 1 (1 + u) 2 . (3.15) The arrows indicate the resulting singularity of the Borel transform after integration over u1,2 according to (3.8). The second forest, containing the box subgraph, results in a double pole, to be compared with the single pole…

**Figure 7.** Some non-abelian vertex diagrams (a-e) at next-to-leading order in the flavour expansion. The long-dashed circle denotes a ghost loop. hierarchy of loop momenta. The contribution from the forest F2 is treated by this method in (Vainshtein & Zakharov 1994). 3.2.2 QCD There exists no complete analysis of non-abelian diagrams at the time of writing. The analysis of multi-chain diagrams in QED showed that higher order corrections in 1/Nf do not modify the location of UV renormalon singularities, but only their ‘strength’, specified by b in (3.1). Not so in QCD, where one expects that…

**Figure 8.** Infrared regions that give rise to infrared renormalons. (a) For a current-current twopoint function at euclidian momentum. The external currents are shown as dashed lines. (b) For an event-shape variable in e +e − annihilation near the two-jet limit. Wavy lines represent collections of soft lines. analytic regularization that the Borel transform has IR renormalon singularities at nonnegative integer u. 23 However, the structure of the singularity in terms of subgraphs is different from (3.10) as different notions of irreducibility apply to ultraviolet and infrared properties. The…

**Figure 9.** Two diagrams at higher order in the flavour expansion. the residue of the IR renormalon singularity receives contributions from arbitrarily complicated graphs in the shaded circle and remains uncalculable (Grunberg 1993; Beneke & Zakharov 1993) despite the simpler over-all diagram structure compared to the UV renormalon case. A graph-by-graph comparison of some contributions to the first IR and first UV renormalon is summarized in

**Figure 10.** Infrared regions that contribute to the first IR renormalon in the mass shift δm. Our discussion has focused on the current-current correlation functions. The generalization to other off-shell quantities is straightforward. 3.3.2 On-shell processes For on-shell, minkowskian processes the classification of IR sensitive regions of a Feynman integral is more complicated than for off-shell quantities. As is well known, in addition to soft, zero-momentum lines, collinear configurations of massless lines (‘jets’) have to be considered. Furthermore, on-shell propagators do not give power…

**Figure 11.** The three different types of cuts relevant for bubble graphs, here for e +e − → hadrons. The cuts may be weighted to give an event shape variable. rn are polynomials in Nf : rn = rn0 + rn1Nf + . . . + rnnN n f . (3.69) The set of fermion loop diagrams (‘bubble diagrams’) is gauge-invariant and gives the coefficient rnn with the largest power of Nf , the number of light flavours. In the following we do not consider the other terms in (3.69). In general, the first-order correction to R may be the sum of a one-loop virtual and a one-gluon real emission contribution. The fermion bubble…

**Figure 12.** (a, b) σ-self-energy diagrams at order 1/N. (c, d) Connected contributions to the α propagator at order 1/N. (e) Non-factorizable contribution to the vacuum expectation value of O2 ∝ α. (f) Non-factorizable contribution to the vacuum expectation value of O4 ∝ α 2 . The solid lines represent the σ propagator, the wavy lines the leading order α propagator (4.8). In the following we drop the disconnected term proportional to hαi 2 . At leading order in 1/N, the expansion 1 4π Dα(p) p 2 = ˆg(p) + m2 p 2 2ˆg(p) − 2ˆg(p) 2 + m4 p 4 −2ˆg(p) − gˆ(p) 2 + 4ˆg(p) 3 + . . . (4.13)…

**Figure 13.** (a) Diagrams that contribute to the twist-2 coefficient function in the operator product expansion of the hadronic tensor. (Wave-function renormalization on the external quark legs is not shown.) The wavy lines denote the external current with momentum q. When the gluons are dressed with fermion loops, these diagrams contribute at leading order in the flavour expansion. (b) Diagrams that give contributions to the matrix elements of twist-4 operators to leading order in the flavour expansion. The third diagram is scaleless and vanishes. (David 1984; Martinelli & Sachrajda 1996).…

**Figure 14.** δ (pt) as a function of αs(mτ ) (MS scheme) for various truncations/partial resummations of the perturbative expansion: (i) Large-β0 resummation according to (5.25). (ii) Fixed-order perturbation theory up to (including) α 3 s . (iii) Resummation of running coupling effects from the contour integral only (see text for discussion). (iv) Fixed-order perturbation theory up to α 2 s . The shaded bar gives the experimental measurement with experimental errors only. The figure is an update from (Ball et al. 1995a). How reliable is the large-β0 approximation for the unknown higher order

**Figure 15.** Relative twist-4 contribution D2(x) (called C(x) here) defined by (5.35) to the structure function F2 in the ‘renormalon model’ compared with the data analysis of (Virchaux & Milsztajn 1992). Plot taken from (Dokshitzer et al. 1996)

**Figure 16.** Relative twist-4 contribution D2(x) (called Cp,d(x) here) defined by (5.35) to the proton (deuteron) structure function F2 in the ‘renormalon model’ (dashed line) compared with proton (filled circles) and deuteron (empty circles) data (Virchaux & Milsztajn 1992). Plot taken from (Maul et al. 1997). The solid curve shows the literal estimate of the renormalon ambiguity. 99

**Figure 17.** Twist-4 correction to xF3 as extracted from the (revised) CCFR data. The three plots show the effect of including leading order (LO), next-to-leading order (NLO) and nextto-next-to-leading order (NNLO) QCD corrections in the twist-2 term. The data points are quoted from the analysis of (Kataev et al. 1997). Overlaid is the shape obtained from the ‘renormalon model’ for the 1/Q2 power correction. 101

**Figure 18.** Primary (left) and secondary (right) quark fragmentation diagrams (in cut diagram representation) in the large-β0 approximation or the approximation of single gluon emission. Note that the figure to the right appears to have two chains of fermion loops, but should nonetheless be interpreted as a single chain diagram. violations in Q. In addition, there exists power-like scaling violations (‘power corrections’) due to multi-parton correlations (Balitsky & Braun 1991). However, contrary to DIS, the moments of these multi-parton correlations are not related to matrix elements of…

**Figure 19.** Shape of Λ2/Q2 power corrections to the longitudinal fragmentation cross section as a function of x. Primary quark fragmentation (dashed line), secondary quark fragmentation (dotted line) and their sum (solid line). in (Dasgupta & Webber 1997; Beneke et al. 1997b) for longitudinal and transverse components separately. In the first paper a simplified prescription was adopted in which all contributions were calculated with a finite gluon mass. In the second paper the diagrams of

**Figure 20.** Energy dependence of h1−Ti (upper) and the heavy mass hM2 H/Q2 i (lower) plotted as function of 1/Q. Data compilation from (Movilla Fern´andez et al. 1998a), see references there. Dotted line: second order perturbation theory with scale µ = Q. Solid line: second order perturbation theory with power correction added according to (5.59) and with µ = Q, µI = 2 GeV. For ¯α0(2 GeV) the fit values 0.543 for thrust and 0.457 for the heavy jet mass from (Movilla Fern´andez et al. 1998a) are taken. The dashed line shows second order perturbation theory at the very low scale 0.07Q with no…

**Figure 21.** Energy dependence of h1−Tci, h1−Tzi/2, the current jet broadening hBci and the current jet hemisphere invariant mass hρci in DIS compared to NLO perturbation theory with and without 1/Q power correction. Figure taken from (H1 collaboration 1997). Event shape variables in DIS are usually defined in the Breit frame, where the gauge boson momentum that induces the hard scattering process is purely space-like: q = (0, 0, 0, Q). In leading order the target remnant moves into the direction opposite to q and the struck parton moves into the direction of the virtual gauge boson. This…

**Figure 22.** One-loop corrections to the heavy quark potential. there is a close relation between the potential and the pole mass, as far as their leading IR renormalon divergence is concerned. The static potential in coordinate space, V (~r ), is defined through a Wilson loop WC(~r, T) of spatial extension ~r and temporal extension T with T → ∞. In this limit WC(~r, T) ∼ exp(−iTV (~r )). The potential in momentum space, V˜ (q), is the Fourier transform of V (~r ). We can compute the potential directly in momentum space from the on-shell quark-anti-quark scattering amplitude (divided by i) at…

**Figure 23.** (a) The subtracted plaquette expectation value as a function of loop order compared to the scaling of a 1/Q2 and 1/Q4 term. (b) Comparison of the all-order subtracted plaquette MC data with the scaling of a 1/Q2 and 1/Q4 term. Figure taken from (Burgio et al. 1998). where αeff(k) is supposed to be a physical definition of the coupling. The integral receives a contribution of order Λ4 /Q4 from k ∼ Λ and of order αeff(Q) from k ∼ Q. But if the effective coupling has a term of order Λ2/k2 in its own short-distance expansion, then this gives rise to a power correction of order Λ2/Q2

Figures reproduced from arXiv:hep-ph/9807443 by the authors.

References

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hep-ph 2017-04 accept novelty 5.0 of 10

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hep-ph 2017-01 conditional novelty 5.0 of 10

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See all 75 Pith citations

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219 later papers cite it

Evidence: substantial. 216 of the citing papers share no author with it, and 22 of those carry a confident label from Pith's citation classifier.

Pith read 16 of the 430 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

1 later paper by other authors is listed below: 1 used its method or tool.

Uses its method or tool 1 of 8 papers with this label

Estimated the large order behavior using the Naive non-abelianization, with delta_n approximately zero for n at least 4, to estimate d_n to all orders from the fermion bubble graphs.

The QCD Adler function and the muon $g-2$ anomaly from renormalons (2021) · 0.96 · paraphrased: the licence does not allow quoting

How the literature uses it

210 arXiv papers cite this in 414 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 243 sentences without a confident label

Contests it 3 papers

This is the first point where our construction is in sharp disagreement with the widely accepted perspective [ref] which asserts that the leading non-BPS semi-classic configuration in the $SU(N)$ theory must have an action $2S_I$

TQFT at work for IR-renormalons, resurgence and Lefschetz decomposition · Introduction · contest 0.79
(The heavy quark whose mass is considered here is decoupled.) The constants that appear in ([ref]) are given by [ref] $b=-β_1/(2β_0^2)$ and [equation] Note that we have corrected some misprints in the expression for $b$ and $s_2$ given in [ref] (eqs.

On the ultimate uncertainty of the top quark pole mass · Summary of formulae · contest 0.86
[ref] , where the scalar correlation function has been calculated in the large-$N_f$ approximation [ref] , or relatedly the large-$β_0$ approximation [ref] (for a review see [ref] ), to all orders in the strong coupling.[For historical reasons, we shall speak about the “large-$β_0$” approximation, although in the notation employed in this work, the leading coefficient of the $β$-function is termed $β_1$.] However, as will be discussed in more detail below, the large-$β_0$ approximation does not provide a satisfactory representation of the scalar correlator in full QCD.

Scalar correlator, Higgs decay into quarks, and scheme variations of the QCD coupling · Introduction · contest 0.75

Builds on it 1 paper

[ref] , but closely follows the material already presented there and in part in the review [ref] .

Higher-order behaviour of two-point current correlators · General renormalon-pole structure · 0.90

Uses its method or tool 8 papers

We estimate the large order behavior in Eq. [ref] using the “Naive non-abelianization”, namely the $δ_n∼0$ for $n≥4$, such that we are able to estimate $d_n$ to all orders from the fermion bubble graphs [ref] .

The QCD Adler function and the muon $g-2$ anomaly from renormalons · Introduction · 0.96
Using standard RG arguments (see e.g. [ref] ), and evaluating the non-perturbative correction at $μ=Q$, we find

Renormalons in integrable field theories · Large order behavior from renormalons · 0.81
Indeed, in such cases the appearance of linear power corrections can be investigated by computing $𝒪(α_s)$ corrections to the process (and observable) under consideration that originate from virtual exchanges and real emissions of massive gluons, in the limit of a small gluon mass $λ$, see Ref. [ref] .[Since by assumption the underlying process does not contain gluons, this does not lead to any issue with gauge invariance.] The presence of terms that are linear in $λ$ implies that a particular observable receives leading power corrections that are of the type $𝒪(Λ_QCD/Q)$, while their absence can be interpreted as an indication that non-perturbative corrections are further suppressed.

On linear power corrections in certain collider observables · Introduction · 0.84

Uses its data or values 2 papers

The Adler function is known up to $n=3$ [ref] or up to $𝒪(α_s^4)$, from which the coefficients $d_n$ and $δ_n$ can be taken from Ref. [ref] .

The QCD Adler function and the muon $g-2$ anomaly from renormalons · Introduction · 0.83
where $ν=2β_1/β_0^2$, and the following large-$β_0$ result is used [ref] : C_0=3/4C_F e^10/3 .

Flavor dependence of normalization constant for an infrared renormalon · Introduction · 0.99

Compares with it 6 papers

This result is identical to the leading renormalon in the pole mass (see for instance Refs. [ref] ).

A discussion on leading renormalon in the pole mass · Leading renormalon from recurrence relation · 0.83
The recent estimate of the ambiguity in Ref. [ref] following the prescription of [ref] is about $60%$ smaller and coincides with the size of the minimal correction term $Δ(n_min)$ and also the scale variation at $(α_s^min)$ for $R=163$ GeV.

On the Light Massive Flavor Dependence of the Top Quark Mass · Pole Mass Ambiguity · 0.95
The coefficients $g_ℓ$ agree with the coefficients $s_ℓ$ given in Refs. [ref] .

The MSR Mass and the ${\cal O}(\Lambda_{\rm QCD})$ Renormalon Sum Rule · QCD beta-Function and Coefficients · 0.94

Cites it as support 2 papers

This procedure is scheme-independent and leads to results free from divergent renormalon terms [ref] .

Elimination of QCD Renormalization Scale and Scheme Ambiguities · Introduction · 0.97
The divergent pattern can be inferred alternatively from particular classes of Feynman diagrams, which indicate a factorial growth of the expansion coefficients in both QED [ref] , [ref] and QCD [ref] , [ref] , [ref] .

Conformal mappings in perturbative QCD · Introduction · 0.96

Cites it as background 105 papers

These singularities are the renormalons of perturbation theory [ref] .

Renormalons in integrated spectral function moments and $\alpha_s$ extractions · Introduction · 0.99
[ref] , based on renormalon calculus [ref] .

A note on renormalon models for the determination of alpha_s(M_tau) · Introduction · 0.86
Analytical models based on renormalons [ref] have in the past been met with great success in the description of LEP and HERA event-shape variables [ref] , but have not really been utilised outside that context.

Hadronisation corrections for jets in the k_t algorithm · Introduction · 0.97

Discussion

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