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Dispersive Approach to Power-Behaved Contributions in QCD Hard Processes

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Yu.L. Dokshitzer, G. Marchesini, B.R. Webber

High Energy Physics - Phenomenology · Submitted 18 Dec 1995 · 41 references

Pith's verdict: The paper that made power corrections bookkeeping rather than folklore: Eq. (3.5) and Table 1 are the durable deliverables, and the soft spots are the ones the authors themselves flag.

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arXiv IDhep-ph/9512336 (version 3)
Pith IDAMPNC5Y2, Pith's permanent record of this paper and its review
CitationB.R. Webber, G. Marchesini, Yu.L. Dokshitzer. "Dispersive Approach to Power-Behaved Contributions in QCD Hard Processes." arXiv:hep-ph/9512336 (1995). https://pith.science/paper/AMPNC5Y2
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The paper

The paper claims

A single infrared-finite effective QCD coupling may generate all power corrections to hard-process observables, with process-dependent powers and coefficients fixed by one-loop diagrams containing a gluon of small mass.

The reading

Hard-scattering predictions in QCD are made as series in the strong coupling, and the terms that fall off as powers of the hard scale — rather than logarithms — are the ones that limit precision and, at the same time, carry information about non-perturbative physics. This paper asks whether those power terms can be organized by a single object: an infrared-finite effective coupling $alpha_{eff}(mu^2)$ defined through a dispersion relation, so that the running coupling keeps a meaning all the way down to zero momentum. The authors show that for any process that is quark-dominated at the Born level, the observable can be written as an integral over $mu^2$ of $alpha_{eff}(mu^2)$ times the logarithmic derivative of a process-specific characteristic function, and that this characteristic function is nothing more than the one-loop diagram computed with a gluon of mass $mu$. The leading power correction is then read off from the leading non-analytic term in the small-$mu^2/Q^2$ expansion of that function, giving a power and a coefficient for each observable in terms of a few universal moment integrals of the low-scale coupling. If this holds, a table of measured power corrections across quite different processes becomes a set of simultaneous measurements of the same handful of numbers.

What carries the argument

The dispersive representation of the running coupling: $alpha_s(k^2) = -int_0^infty dmu^2, rho_s(mu^2)/(mu^2+k^2)$, with the spectral density rewritten through an effective coupling as $rho_s = dalpha_{eff}/dln mu^2$. This converts the awkward question of what the coupling means at low momentum into a convolution against a function that vanishes at $mu^2 = 0$, so the ill-defined $alpha_s(0)$ drops out. The second piece of machinery is the 'soft confinement' constraint: the non-perturbative shift $deltaalpha_{eff}$ must have vanishing moments $int (dmu^2/mu^2)(mu^2)^p deltaalpha_{eff} = 0$ for $p = 1, 2, dots, p_{max}$, otherwise the operator product expansion at high scales would be spoiled. The consequence is that only terms in $dot F$ that are non-analytic in $mu^2$ — non-integer powers, or logarithms — can generate power corrections.

Extended reading notes

Core claim

The paper's central result is Eq. (3.5): for a quark-dominated hard observable $F$, one has $F(Q^2,{x}) = int_0^infty (dmu^2/mu^2), alpha_{eff}(mu^2), dot F(Q^2,{x};mu^2)$ with $dot F = -partial F/partial ln mu^2$, where $F(mu^2)$ is the one-loop contribution computed with a gluon of mass $mu$. The power-behaved part follows from the small-$epsilon$ behaviour $dot F(epsilon) = epsilon^p [f(ln epsilon) + epsilon g(ln epsilon) + dots]$, $epsilon = mu^2/Q^2$: the leading non-analytic term fixes the power $p$ and a coefficient built from the universal moment integrals $A_{2p}$, $A'_{2p}$, $A''_{2p}$ of the coupling modification $deltaalpha_{eff}$. The authors carry this through for the $e^+e^-$ total cross section, the $tau$ hadronic width, DIS structure functions and sum rules, $e^+e^-$ fragmentation functions, event-shape means (thrust, Durham and JADE three-jet resolution) and the Drell-Yan $K$-factor, and collect the resulting powers and coefficients in Table 1. The pattern is that $e^+e^-$ total cross section starts at $Q^{-6}$, DIS and Drell-Yan at $Q^{-2}$ with logarithmic enhancement near phase-space edges, and thrust at $Q^{-1}$ from a square-root non-analyticity.

Load-bearing premise

The load-bearing premise is that the QCD running coupling can be written in the Abelian-style dispersive form and continued to zero momentum as a universal effective coupling, and that its non-perturbative modification can be chosen with vanishing low moments; the paper argues that gauge-dependent pieces stay finite as the gluon momentum goes to zero but asserts this for the non-Abelian case rather than proving it.

Editorial extensions

If this is right

  • Power corrections to quite different observables become correlated: a measurement of the linear $1/Q$ term in thrust fixes the same parameter $A_1$ that appears in other event-shape means, and a measurement of the $1/Q^2$ higher-twist coefficient in $F_2$ fixes $A'_2$, which then predicts the corresponding coefficients in the Gross-Llewellyn-Smith sum rule, the fragmentation function and the Drell
  • The Adler sum rule receives neither perturbative nor power corrections in this framework, because an exact identity makes the $N=1$ moment of the characteristic function vanish for every $mu^2$.
  • Event-shape variables can be engineered to have no leading $1/Q$ term: the Durham three-jet resolution has a $ln Q/Q^2$ correction rather than $1/Q$, which the authors suggest is why it has needed smaller non-perturbative corrections than JADE-style definitions.
  • Power corrections grow sharply at phase-space edges — as $N/Q^2$ for DIS coefficient functions, $N^2/Q^2$ for the Drell-Yan $K$-factor, $1/((1-T)Q^2)$ for the differential thrust distribution — so the power term becomes order one where the final-state invariant mass is squeezed to about $Lambda^2$.
  • The scheme choice for the coupling is fixed by requiring that the two-loop splitting function contain no soft singularity, which relates the physical scheme to $overline{MS}$ through a constant $K$ in the scale $Lambda$.

Reading between the lines

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

  • The framework turns the infrared renormalon ambiguity from an obstruction into a bookkeeping device: the ambiguous $Q^{-2p}$ terms are precisely the slots where the non-perturbative moments enter, so the ambiguity and the power correction are two descriptions of the same missing physics.
  • Because the coefficients are computed from one-loop diagrams with a massive gluon, essentially any existing one-loop Monte Carlo or analytic one-loop result can be re-read as a prediction for the shape of the power correction, which makes the programme cheap to extend to observables the authors did not tabulate.
  • The moment-vanishing constraint suggests a sharp experimental test: if a single set of moments fits DIS higher twist and thrust simultaneously, that is evidence for a universal low-scale coupling; if the fitted moments are mutually inconsistent at the predicted coefficients, the universality premise fails rather than the individual calculations.
  • The same characteristic-function technology should apply to hadron-collider observables with quark-dominated Born terms, such as Drell-Yan $p_T$ spectra and vector-boson plus jet rates, where the small-$mu^2$ behaviour would similarly be governed by soft-gluon phase space.
Original abstract

We consider power-behaved contributions to hard processes in QCD arising from non-perturbative effects at low scales which can be described by introducing the notion of an infrared-finite effective coupling. Our method is based on a dispersive treatment which embodies running coupling effects in all orders. The resulting power behaviour is consistent with expectations based on the operator product expansion, but our approach is more widely applicable. The dispersively-generated power contributions to different observables are given by (log-)moment integrals of a universal low-scale effective coupling, with process-dependent powers and coefficients. We analyse a wide variety of quark-dominated processes and observables, and show how the power contributions are specified in lowest order by the behaviour of one-loop Feynman diagrams containing a gluon of small virtual mass. We discuss both collinear safe observables (such as the e+e- total cross section and \tau hadronic width, DIS sum rules, e+e- event shape variables and the Drell-Yan K-factor) and collinear divergent quantities (such as DIS structure functions, e+e- fragmentation functions and the Drell-Yan cross section).

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.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,110 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.15 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.80 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

We consider power-behaved contributions to hard processes in QCD arising from non-perturbative effects at low scales which can be described by introducing the notion of an infrared-finite effective coupling. Our method is based on a dispersive treatment which embodies running coupling effects in all orders. The resulting power behaviour is consistent with expectations based on the operator product expansion, but our approach is more widely applicable. The dispersively-generated power contributions to different observables are given by (log-)moment integrals of a universal low-scale effective coupling, with process-dependent powers and coefficients. We analyse a wide variety of quark-dominated processes and observables, and show how the power contributions are specified in lowest order by the behaviour of one-loop Feynman diagrams containing a gluon of small virtual mass. We discuss both collinear safe observables (such as the e+e- total cross section and \tau hadronic width, DIS sum rules, e+e- event shape variables and the Drell-Yan K-factor) and collinear divergent quantities (such as DIS structure functions, e+e- fragmentation functions and the Drell-Yan cross section).

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We consider power-behaved contributions to hard processes in QCD arising from non-perturbative effects at low scales which can be described by introducing the notion of an infrared-finite effective coupling. contribution 0.54 0.61 0.16 asserted 0.74
s1 Our method is based on a dispersive treatment which embodies running coupling effects in all orders. method 0.99 0.83 0.09 asserted 1.00
s2 The resulting power behaviour is consistent with expectations based on the operator product expansion, but our approach is more widely applicable. interpretation 0.78 0.66 0.24 hedged possibility 0.65
s3 The dispersively-generated power contributions to different observables are given by (log-)moment integrals of a universal low-scale effective coupling, with process-dependent powers and coefficients. result 0.80 0.88 0.41 asserted 1.00
s4 We analyse a wide variety of quark-dominated processes and observables, and show how the power contributions are specified in lowest order by the behaviour of one-loop Feynman diagrams containing a gluon of small virtual mass. contribution 0.72 0.94 0.15 asserted 1.00
s5 We discuss both collinear safe observables (such as the e+e- total cross section and \tau hadronic width, DIS sum rules, e+e- event shape variables and the Drell-Yan K-factor) and collinear divergent quantities (such as DIS structure functions, e+e- fragmentation functions and the Drell-Yan cross section). contribution 0.97 0.73 0.07 asserted 0.91

What kind of paper the classifier reports this is: method or tool 0.72 · evidence from proof or derivation 0.98.

What was examined, and what was not

sources read
  • arXiv listing abstract v3
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

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. Characteristic function for e +e − total cross section, R
Figure 2. The behaviour for small and large ǫ is as follows: For ǫ ≪ 1 R = 3 2 (1 − ǫ 2 ) − 2ǫ 3 3  ln ǫ − 11 6  + O  ǫ 4 ln ǫ  . (4.10a) For ǫ → ∞ R = 2 3ǫ  ln ǫ + 11 6  + O
Figure 3. Derivative of characteristic function for
Figure 4. Derivative of characteristic function for DIS,
Figure 5. Second derivative of characteristic function for
Figure 6. Coefficient of higher-twist contribution to DIS.
Figure 7. Coefficient of power correction to e +e − fragmentation function. the light quark fragmentation function at Q = 22 GeV given in Ref. [37]. The coefficient function C˜ is larger than that for DIS, but qualitatively similar in form. The values of Q2 in e +e − annihilation being much larger, the predicted power correction is probably too small to be detectable. In Ref. [37], the non-perturbative contribution to the fragmentation function was parametrized as a small shift in the value of x, x → x + h0
Figure 8. Derivative of characteristic function for mean va
Figure 9. Derivative of characteristic function for mean va
Figure 10. Derivative of characteristic function for momen

Figures reproduced from arXiv:hep-ph/9512336 by the authors.

References
How later papers used it

75 later papers cite it

Evidence: substantial. 73 of the citing papers share no author with it, and 15 of those carry a confident label from Pith's citation classifier.

Pith read 17 of the 121 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

Builds on it 1 paper
Uses its method or tool 4 of 7 papers with this label
How the literature uses it

69 arXiv papers cite this in 111 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 · 81 sentences without a confident label

Builds on it 1 paper
  • We extend the parametrization of the time-like sector, Ref. [ref] , to the space like sector, which leads to α→α_eff(Q^2) = 4π/β_0 ł{ [ L_-^-1; for Q^2 0 ,; 1/2 - π^-1 atn( L_+/π) ] .̊

    Towards an understanding of the RHIC single electron data · Running coupling constant · 0.99

Uses its method or tool 7 papers
Compares with it 3 papers
Cites it as support 3 papers
Cites it as background 12 papers

Discussion

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