Optimization of perturbation series in QCD for physical quantities using the renormalization group: necessary conditions and partial results

D. Kotlorz; S. V. Mikhailov

arxiv: 2606.31592 · v1 · pith:55NGKUMHnew · submitted 2026-06-30 · ✦ hep-ph

Optimization of perturbation series in QCD for physical quantities using the renormalization group: necessary conditions and partial results

D. Kotlorz , S. V. Mikhailov This is my paper

Pith reviewed 2026-07-01 04:49 UTC · model grok-4.3

classification ✦ hep-ph

keywords QCD perturbation theoryrenormalization groupBjorken sum ruleAdler functionperturbative series optimizationphysical quantities

0 comments

The pith

Numerical optimization of perturbative QCD series segments is explored using the renormalization group for the Bjorken sum rule and Adler function.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates methods to numerically optimize segments of perturbative series in QCD by using the renormalization group. It applies these methods to the coefficient function of the Bjorken polarized sum rule and the Adler function. Various techniques proposed in the literature are used, and the consequences of such optimization are discussed. A sympathetic reader would care because these approaches aim to improve the reliability of perturbative predictions for physical quantities in strong interactions.

Core claim

We explore approaches to numerically optimize a segment of the perturbative series for physical quantities using the QCD renormalization group. We apply these methods to the perturbative series for the coefficient function C_Bjps of the Bjorken polarized sum rule and the Adler function D_A. Using various techniques proposed in the literature, we discuss the consequences of optimization.

What carries the argument

The QCD renormalization group applied to numerically optimize segments of perturbative series expansions for physical quantities.

Load-bearing premise

That numerical optimization of a perturbative segment via renormalization-group techniques yields meaningfully improved or more reliable physical predictions for the chosen quantities without independent validation against non-perturbative results.

What would settle it

A comparison of optimized perturbative predictions for the Bjorken sum rule coefficient or Adler function against exact non-perturbative calculations or high-precision data that shows no improvement or reduced accuracy.

read the original abstract

We explore approaches to numerically optimize a segment of the perturbative series for physical quantities using the QCD renormalization group. We apply these methods to the perturbative series for the coefficient function $C_{Bjps}$ of the Bjorken polarized sum rule and the Adler function $D_A$. Using various techniques proposed in the literature, we discuss the consequences of ``optimization.''

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper applies known optimization techniques from the literature to the Bjorken coefficient and Adler function series, then discusses the numerical outcomes without claiming improved physical predictions.

read the letter

The paper takes optimization approaches already proposed elsewhere and applies them numerically to the perturbative series for C_Bjps in the Bjorken polarized sum rule and for the Adler function D_A. It uses the QCD renormalization group to adjust a segment of each series and then examines what changes. The authors are explicit that the goal is exploration and discussion of consequences rather than a demonstration of better accuracy.

What the work does is run several existing techniques on two standard quantities that appear often in phenomenology. This produces concrete series after optimization and lets the reader see how the coefficients shift under different choices. For readers already working on perturbative QCD series handling, that kind of side-by-side comparison can be useful as a reference point.

The main limitation is the absence of any external check. There is no comparison to non-perturbative lattice results, no test against known exact limits, and no demonstration that the optimized series reduce the theoretical uncertainty in a measurable way. Because the methods involve additional scale or parameter choices, it remains open whether the procedure adds information or simply reparameterizes the truncation. The abstract itself frames the contribution as discussion rather than a new derivation or first-principles result, so the incremental nature is consistent with what is shown.

This is a paper for a narrow group already following work on QCD perturbative optimization. A broader hep-ph reader will not find new observables, new methods, or resolved tensions with data. It is formally grounded in the sense that it cites prior techniques and applies them consistently, but the evidential weight is light.

I would send it to peer review for a specialized journal that publishes short technical notes on series methods, mainly so that the community can see the numerical outcomes and judge their utility directly. It does not rise to the level of a high-priority refereeing exercise.

Referee Report

0 major / 2 minor

Summary. The paper explores numerical approaches to optimize segments of perturbative series for physical quantities in QCD by employing the renormalization group. It applies these methods to the coefficient function C_Bjps of the Bjorken polarized sum rule and the Adler function D_A, and discusses the consequences of optimization using various techniques proposed in the literature.

Significance. If the optimization procedures satisfy the necessary conditions identified and yield consistent results independent of arbitrary choices, the work could provide a useful framework for handling truncated perturbative series in QCD observables. The emphasis on necessary conditions rather than empirical claims of improvement is a strength, as is the focus on two specific quantities with known perturbative expansions.

minor comments (2)

The abstract and title refer to 'necessary conditions and partial results,' but the manuscript should explicitly state in the introduction or conclusions which conditions are derived versus assumed, to clarify the scope.
Notation for the optimized series and the specific RG equations used should be introduced with a dedicated section or appendix for reproducibility, especially since multiple techniques from the literature are compared.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation for minor revision. No specific major comments are listed in the report.

Circularity Check

0 steps flagged

No significant circularity; exploratory discussion of RG-based optimization methods

full rationale

The paper states its goal as exploring numerical optimization approaches for perturbative series segments via the QCD renormalization group, applying literature techniques to C_Bjps and D_A, and discussing consequences of optimization. No load-bearing derivation chain is presented that reduces a claimed prediction or result to a fitted parameter, self-definition, or self-citation chain. The argument remains methodological and self-contained, with no equations or uniqueness theorems invoked that collapse to the paper's own inputs by construction. External benchmarks or independent validation are not asserted as outcomes.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no free parameters, axioms, or invented entities can be extracted from the provided text.

pith-pipeline@v0.9.1-grok · 5581 in / 937 out tokens · 35972 ms · 2026-07-01T04:49:13.912846+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

32 extracted references · 24 canonical work pages · 17 internal anchors

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[1] [1]

Kotlorz and S

D. Kotlorz and S. V. Mikhailov, Phys. Rev. D 100, 056007 (2019), 1810.02973

work page arXiv 2019

[2] [2]

P. A. Baikov, K. G. Chetyrkin, and J. H. Kuhn, Phys. Rev. Le tt. 101, 012002 (2008), 0801.1821

work page internal anchor Pith review Pith/arXiv arXiv 2008

[3] [3]

P. A. Baikov, K. G. Chetyrkin, and J. H. Kuhn, Phys. Rev. Le tt. 104, 132004 (2010), 1001.3606

work page internal anchor Pith review Pith/arXiv arXiv 2010

[4] [4]

P. A. Baikov, K. G. Chetyrkin, J. H. Kuhn, and J. Rittinger , JHEP 07, 017 (2012), 1206.1284

work page internal anchor Pith review Pith/arXiv arXiv 2012

[5] [5]

S. J. Brodsky, G. P. Lepage, and P. B. Mackenzie, Phys. Rev . D28, 228 (1983)

1983

[6] [6]

S. V. Mikhailov, JHEP 06, 009 (2007), hep-ph/0411397

work page internal anchor Pith review Pith/arXiv arXiv 2007

[7] [7]

A. L. Kataev and S. V. Mikhailov, Phys. Rev. D91, 014007 (2015), 1408.0122. – 19 –

work page internal anchor Pith review Pith/arXiv arXiv 2015

[8] [8]

P. A. Baikov and S. V. Mikhailov, JHEP 09, 185 (2022), 2206.14063

work page arXiv 2022

[9] [9]

S. V. Mikhailov, JHEP 10, 166 (2024), 2406.15014

work page arXiv 2024

[10] [10]

A Systematic All-Orders Method to Eliminate Renormalization-Scale and Scheme Ambiguities in PQCD

M. Mojaza, S. J. Brodsky, and X.-G. Wu, Phys. Rev. Lett. 110, 192001 (2013), 1212.0049

work page internal anchor Pith review Pith/arXiv arXiv 2013

[11] [11]

Di Giustino, S

L. Di Giustino, S. J. Brodsky, P. G. Ratcliﬀe, X.-G. Wu, a nd S.-Q. Wang, Prog. Part. Nucl. Phys. 135, 104092 (2024), 2307.03951

work page arXiv 2024

[12] [12]

Adler function and Bjorken polarized sum rule: Perturbation expansions in powers of $SU(N_c)$ conformal anomaly and studies of the conformal symmetry limit

G. Cvetič and A. L. Kataev, Phys. Rev. D94, 014006 (2016), 1604.00509

work page internal anchor Pith review Pith/arXiv arXiv 2016

[13] [13]

A. L. Kataev and V. S. Molokoedov, Phys. Part. Nucl. 54, 931 (2023), 2211.10242

work page arXiv 2023

[14] [14]

D. J. Broadhurst and A. L. Kataev, Phys. Lett. B 315, 179 (1993), hep-ph/9308274

work page internal anchor Pith review Pith/arXiv arXiv 1993

[15] [15]

Grunberg and A

G. Grunberg and A. L. Kataev, Phys. Lett. B 279, 352 (1992)

1992

[16] [16]

A. Deur, V. Burkert, J.-P. Chen, and W. Korsch, Phys. Let t. B650, 244 (2007), hep-ph/0509113

work page internal anchor Pith review Pith/arXiv arXiv 2007

[17] [17]

A. Deur, S. J. Brodsky, and G. F. de Teramond, Phys. Lett. B 757, 275 (2016), 1601.06568

work page internal anchor Pith review Pith/arXiv arXiv 2016

[18] [18]

Setting the Renormalization Scale in pQCD: Comparisons of the Principle of Maximum Conformality with the Sequential Extended Brodsky-Lepage-Mackenzie Approach

H.-H. Ma, X.-G. Wu, Y. Ma, S. J. Brodsky, and M. Mojaza, Ph ys. Rev. D91, 094028 (2015), 1504.01260

work page internal anchor Pith review Pith/arXiv arXiv 2015

[19] [19]

X.-G. Wu, S. J. Brodsky, and M. Mojaza, Prog. Part. Nucl. Phys. 72, 44 (2013), 1302.0599

work page internal anchor Pith review Pith/arXiv arXiv 2013

[20] [20]

I. O. Goriachuk, A. L. Kataev, and V. S. Molokoedov, JHEP 05, 028 (2022), 2111.12060

work page arXiv 2022

[21] [21]

D. J. Broadhurst and A. G. Grozin, Phys. Rev. D 52, 4082 (1995), hep-ph/9410240

work page internal anchor Pith review Pith/arXiv arXiv 1995

[22] [22]

A. P. Bakulev, S. V. Mikhailov, and N. G. Stefanis, JHEP 06, 085 (2010), 1004.4125

work page internal anchor Pith review Pith/arXiv arXiv 2010

[23] [23]

K. G. Chetyrkin, Nucl. Phys. B 985, 115988 (2022), 2206.12948

work page arXiv 2022

[24] [24]

D. J. Gross and F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973)

1973

[25] [25]

H. D. Politzer, Phys. Rev. Lett. 30, 1346 (1973)

1973

[26] [26]

W. E. Caswell, Phys. Rev. Lett. 33, 244 (1974)

1974

[27] [27]

D. R. T. Jones, Nucl. Phys. B75, 531 (1974)

1974

[28] [28]

O. V. Tarasov, A. A. Vladimirov, and A. Yu. Zharkov, Phys . Lett. 93B, 429 (1980)

1980

[29] [29]

S. A. Larin and J. A. M. Vermaseren, Phys. Lett. B303, 334 (1993), hep-ph/9302208

work page internal anchor Pith review Pith/arXiv arXiv 1993

[30] [30]

The four-loop beta-function in Quantum Chromodynamics

T. van Ritbergen, J. A. M. Vermaseren, and S. A. Larin, Ph ys. Lett. B400, 379 (1997), hep-ph/9701390

work page internal anchor Pith review Pith/arXiv arXiv 1997

[31] [31]

Tanabashi et al

M. Tanabashi et al. (Particle Data Group), Phys. Rev. D98, 030001 (2018)

2018

[32] [32]

K. G. Chetyrkin, J. H. Kuhn, and C. Sturm, Nucl. Phys. B 744, 121 (2006), hep-ph/0512060. – 20 –

work page internal anchor Pith review Pith/arXiv arXiv 2006