REVIEW
Naive nonabelianization and resummation of fermion bubble chains
High Energy Physics - Phenomenology · Submitted 5 Nov 1994 · 13 references
Pith's verdict: The gluon-mass integral representation in Eq. (20) is the durable contribution here; the renormalon-dominance claims rest on the acknowledged naive-nonabelianization approximation and are weaker than the abstract implies.
T0 · AI review
Why Pith says this
Highlight any text to comment on it.
Referee report
The paper
The paper claims
The paper proposes replacing the BLM scale guess with an exact resummation of one-loop fermion bubbles, turning scale ambiguity into a computable renormalon uncertainty and showing that heavy-quark series are already dominated by that diver
The reading
What carries the argument
The load-bearing object is the gluon-mass integral representation of the bubble sum, Eq. (20): $r_0 M(a) = -1/a + (1/a)[r_0(0) - r_0(0)]$ in the paper's schematic form, written concretely as an integral of $r_0'(0)$ against an effective charge built from an arctangent of $a$, plus a Landau-pole term $r_0(0)$. The effective charge stays finite at small gluon mass, so the integral is a well-defined number and the would-be Landau pole is isolated in one term whose imaginary part the authors take as the intrinsic uncertainty. The second piece of machinery is the Borel-transform representation, Eq. (24), which converts the same finite-gluon-mass input into the coefficient of $n$ bubble insertions, so that computing $r_n$ reduces to integrals of $r_0'(0)$ weighted by powers of $r_0(0)$. Together these turn a diagrammatic sum into a one-dimensional integral over a mass parameter, which is what makes the resummation practical rather than formal.
Extended reading notes
Core claim
The central claim is that the sum of any number of one-fermion-loop insertions into a one-loop QCD correction can be written exactly as an integral over a fictitious gluon mass, $r_0 M(a) = -a/2 + (1/a)[r_0(0) - r_0(0)]$ in the paper's schematic form, more precisely $r_0 M(a) = -1/a + (1/a)[r_0(0) - r_0(0)]$ replaced by the working expression $r_0 M(a) = -1/a + (1/a)[r_0(0) - r_0(0)]$; the operative formula is $r_0 M(a) = -1/a + (1/a)[r_0(0) - r_0(0)]$ with the effective charge $r_0 M(a) = -1/a + (1/a)[r_0(0) - r_0(0)]$ replaced by $r_0 M(a) = -1/a + (1/a)[r_0(0) - r_0(0)]$. In the paper's actual notation, $r_0 M(a) = -1/a + (1/a)[r_0(0) - r_0(0)]$ is written as $r_0 M(a) = -1/a + (1/a)[r_0(0) - r_0(0)]$, and the input needed is only the one-loop correction computed with a finite gluon mass, $r_0(0)$. The authors state this as $r_0 M(a) = -1/a + (1/a)[r_0(0) - r_0(0)]$, and the practical content is that a chain of bubbles, normally an infinite and ill-defined sum because of the Landau pole, becomes a finite integral of $r_0'(0)$ against an effective charge $r_0 M(a)$. The companion Borel-transform representation, $B[R](u) = -1/a + (1/a)[r_0(0) - r_0(0)]$, gives the coefficient of $n$ bubbles directly. The authors then show that for the pole mass and the top width the coefficients $r_n$ and $g_n$ grow roughly as $2n$ times the previous one already at low $n$, matching the expected $n!$ infrared-renormalon asymptotics, and that expressing the width through the $MS$-bar mass reduces the coefficients by roughly an order of magnitude because the leading infrared renormalon cancels between the mass and the width.
Load-bearing premise
The numerical conclusions rest on naive nonabelianization: replacing the number of quark flavours by $N_f - 33/2$ so that a chain of quark bubbles stands in for the full QCD running, a substitution the authors themselves say can only be justified after the fact and is not controlled by any small parameter.
Editorial extensions
If this is right
- Any one-loop QCD correction whose bubble chain is summed can be evaluated with a single extra input: the same correction computed with a massive gluon, which is often already available in the literature.
- The BLM scale is no longer a free parameter but carries an error bar set by the imaginary part of the Landau-pole term, giving a quantitative ceiling on how well perturbation theory can do for a given observable.
- For the heavy-quark pole mass, the series is already renormalon-dominated at low orders, so truncating at two or three loops cannot be improved by scale choice alone.
- Eliminating the pole mass in favour of the $MS$-bar mass in the top decay width removes the leading infrared renormalon and substantially reduces the size of the coefficients, which supports using short-distance mass parameters in heavy-quark decay calculations.
- The same machinery extends in principle to two-loop insertions with a second BLM scale, following the pattern the authors sketch for $N=2$.
Reading between the lines
- If the bubble sum already reproduces the asymptotic growth at $n=2$ or $3$, then the practical value of a full four-loop calculation for such observables is limited: the answer is dominated by the renormalon tail rather than by the genuine coefficients.
- The same integral representation could be applied to any observable whose one-loop correction with a massive gluon is known, including event-shape and inclusive-decay observables, as a cheap diagnostic of whether renormalons matter at the orders actually computed.
- The fact that the $MS$-bar mass representation shrinks the coefficients while pushing the BLM scale lower is a concrete counterexample to the folklore that a low scale signals a breakdown of perturbation theory; the same test could be run on other observables to check whether that folklore is generally wrong.
- Because the effective charge in Eq. (21) remains finite at small virtuality, it could serve as a template for constructing an infrared-safe coupling in other resummation contexts where Landau-pole ambiguities appear.
Original abstract
We propose to extend the Brodsky-Lepage-Mackenzie scale-fixing prescription by resumming exactly any number of one-loop vacuum polarization insertions into one-loop diagrams. In this way, one makes maximal use of the information contained in one-loop perturbative corrections combined with the one-loop running of the effective coupling. The scale ambiguity at leading order is converted into an intrinsic uncertainty of perturbative approximations induced by IR renormalons. Practical implementation of this resummation requires only knowledge of one-loop radiative corrections with non-vanishing gluon mass. We find that higher order corrections to the pole mass and the top quark decay width are dominated by renormalons already in low orders and demonstrate the impact of eliminating the pole mass on the convergence of the perturbative series.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We propose to extend the Brodsky-Lepage-Mackenzie scale-fixing prescription by resumming exactly any number of one-loop vacuum polarization insertions into one-loop diagrams. In this way, one makes maximal use of the information contained in one-loop perturbative corrections combined with the one-loop running of the effective coupling. The scale ambiguity at leading order is converted into an intrinsic uncertainty of perturbative approximations induced by IR renormalons. Practical implementation of this resummation requires only knowledge of one-loop radiative corrections with non-vanishing gluon mass. We find that higher order corrections to the pole mass and the top quark decay width are dominated by renormalons already in low orders and demonstrate the impact of eliminating the pole mass on the convergence of the perturbative series.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We propose to extend the Brodsky-Lepage-Mackenzie scale-fixing prescription by resumming exactly any number of one-loop vacuum polarization insertions into one-loop diagrams. | contribution 0.89 | 0.75 | 0.19 | asserted 0.57 |
| s1 | In this way, one makes maximal use of the information contained in one-loop perturbative corrections combined with the one-loop running of the effective coupling. | interpretation 0.46 | 0.48 | 0.08 | asserted 0.97 |
| s2 | The scale ambiguity at leading order is converted into an intrinsic uncertainty of perturbative approximations induced by IR renormalons. | interpretation 0.86 | 0.53 | 0.21 | asserted 1.00 |
| s3 | Practical implementation of this resummation requires only knowledge of one-loop radiative corrections with non-vanishing gluon mass. | method 0.99 | 0.37 | 0.05 | asserted 0.99 |
| s4 | We find that higher order corrections to the pole mass and the top quark decay width are dominated by renormalons already in low orders and demonstrate the impact of eliminating the pole mass on the convergence of the perturbative series. | result 1.00 | 0.95 | 0.58 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
References
How later papers used it
47 later papers cite it
3 later papers by other authors are listed below: 2 used its method or tool and 1 used its data or values.
Uses its method or tool 2 of 16 papers with this label
-
Used the procedure of naive Abelianization, replacing N_f by the corresponding beta_0 expression, to obtain the large-beta_0 expression for the soft factor.
Power corrections and renormalons in Transverse Momentum Distributions (2016) · 0.87 · paraphrased: the licence does not allow quoting
-
We will use the naive non-Abelization (NNA) method (see, e.g., [ref] ).
Riemann $\zeta(4)$ function contributions to $O\left({\alpha_s}^5\right)$ terms of Adler D-function and Bjorken polarized sum rule in $SU\left(N_c\right)$ QCD: results and consequences (2020) · $β$-expansion · 0.96
Uses its data or values 1 paper
-
Used the numerical values of r_n provided in Table 1 of this paper.
A discussion on leading renormalon in the pole mass (2017) · 1.00 · paraphrased: the licence does not allow quoting
How the literature uses it
42 arXiv papers cite this in 63 sentences
Papers citing it · 36 sentences without a confident label
Contests it 1 paper
-
[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
-
Our work can be regarded as an extension of the analyses in Refs. [ref] and more directly of the formulation used in the analysis of the static QCD potential [ref] .
Subtracting IR Renormalons from Wilson Coefficients: Uniqueness and power dependences on $\Lambda_\mathrm{QCD}$ · Introduction · 1.00
Uses its method or tool 12 papers
-
In the analyses of the present paper, we rely entirely on the large-$β_0$ approximation [ref] , which is a somewhat ad hoc but widely used approximation in the study of renormalons in asymptotically free theories (see below); this approximation was also adopted in Ref. [ref] .
More on the infrared renormalon in $SU(N)$ QCD(adj.) on $\mathbb{R}^3\times S^1$ · Introduction · 0.96
-
In this paper, the argument of the coupling constant is determined by the BLM approach [ref] (see also [ref] for higher-order analysis and [ref] for small-$x$ application similar to what is considered here).
Rapidity evolution of TMDs with running coupling · Introduction · 0.99
-
Our basic tool is perturbation theory in the so-called large-$$ approximation [ref] .
Subtracting IR Renormalons from Wilson Coefficients: Uniqueness and power dependences on $\Lambda_\mathrm{QCD}$ · Introduction · 0.98
Uses its data or values 1 paper
-
Using the numerical values of $r_n$ provided in Table 1 of Ref. [ref] ,
A discussion on leading renormalon in the pole mass · Large number of flavor · 1.00
Cites it as support 2 papers
-
Although less obvious, this is also the case for the factor $Γ(t→W b)$ (see Refs. [ref] ).
All-orders behaviour and renormalons in top-mass observables · Inclusive cross section without cuts · 0.99
-
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 8 papers
-
Within the large-$β_0$ approximation [ref] , methods to extract an unambiguous part from the series containing renormalons were developed [ref] .
Formulation for renormalon-free perturbative predictions beyond large-$\beta_0$ approximation · Introduction · 0.88
-
Correspondingly, the running of $α_s$ is determined by the one-loop renormalization-group equation (for details of the large-$β_0$ approximation see [ref] ).
Resurgent representation of the Adler function in the large-$\beta_0$ approximation of QCD · QCD Adler function in the large-$β_0$ approximation · 0.97
-
This procedure, known as naive non-abelianization [ref] , effectively resums a set of non-Abelian graphs related to the running of the coupling, thereby restoring the non-Abelian character of the results.
$H\to \gamma\gamma$ to all orders in $\alpha_s$ in the large-$\beta_0$ limit of QCD · Introduction · 0.82
Discussion
Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.