New Solutions of RG Equations for \alpha_s and y_{top}

A.S. Fedoruk; D. I. Kazakov

arxiv: 2606.28084 · v1 · pith:5TCPOM5Bnew · submitted 2026-06-26 · ✦ hep-th

New Solutions of RG Equations for α_s and y_(top)

A.S. Fedoruk , D. I. Kazakov This is my paper

Pith reviewed 2026-06-29 03:37 UTC · model grok-4.3

classification ✦ hep-th

keywords renormalization group equationsrunning couplingsstrong couplingtop Yukawa couplingperturbative resummationasymptotic solutionsquantum chromodynamics

0 comments

The pith

Explicit logarithmic solutions to the RG equations for the strong coupling and top Yukawa sum all perturbative orders in the high-energy limit.

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

The paper constructs analytical solutions for how the strong force coupling and the top quark's interaction strength change with energy scale at very high energies. These solutions use only ordinary logarithms and capture the cumulative effect of all orders of perturbation theory corrections. A reader would care because they provide a simple way to predict these running couplings without computing infinite series term by term. The influence of the top coupling on the strong one is small, but the strong coupling significantly affects the top one, and including higher orders changes the result noticeably.

Core claim

We construct simple analytical solutions of the RG equations for the running couplings α_s and y_top in the asymptotic regime. These solutions have an explicit form, contain only logarithms and no special functions, and subsequently sum up the leading, subleading, etc logarithms in all orders of PT. While the effect of y_top on the running of α_s happens to be negligible, the role of α_s on the running of y_top is essential, the account of higher orders gives a noticeable contribution.

What carries the argument

The closed-form logarithmic expressions solving the coupled differential RG equations for α_s and y_top in the asymptotic regime.

If this is right

The running of α_s remains essentially independent of y_top.
The running of y_top receives an essential correction from α_s.
Higher-order logarithmic terms produce noticeable shifts in the predicted evolution of y_top.
The solutions permit direct evaluation of the couplings at arbitrary high scales without iterative summation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same logarithmic structure could be tested for consistency when additional Standard Model couplings are restored to the system.
The explicit forms might simplify threshold matching calculations in models that embed the Standard Model at high scales.
Direct comparison to lattice or collider data at accessible energies could quantify the size of neglected non-asymptotic effects.

Load-bearing premise

The high-energy asymptotic regime permits a closed-form resummation of the coupled equations using only logarithmic terms without significant contributions from non-perturbative physics or additional operators.

What would settle it

A direct numerical integration of the RG equations from low to high energies followed by comparison of the resulting y_top value at a scale such as 10^16 GeV against the analytical prediction.

Figures

Figures reproduced from arXiv: 2606.28084 by A.S. Fedoruk, D. I. Kazakov.

**Figure 1.** Figure 1: The running of α1, y1 in the SM 2.2 ℓ > 1 Since equations (9,10) for ℓ > 1 are linear, one can write down the general solutions keeping in mind the initial conditions αℓ(0) = yℓ(0) = 0: αℓ = α 2 1 Z L 0 Φℓ α 2 1 dL′ , (12) yℓ = y 2 1 α ν 1 Z L 0 α ν 1 y 2 1 (γ10y1αℓ + Ψℓ) dL′ , (13) where for the evaluation of some integrals the following relation was used: 1 y1 = −γ01 γ10 − β0 1 α1 + α0(γ01 − β0) + y0γ01 … view at source ↗

**Figure 2.** Figure 2: The running of y2. Here different contributions to y2 are shown. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: Comparison of the solutions for α and y in a RG-invariant form for a different number of loops taken into account. Hereafter the plots are drawn not in terms of L = ln(Q2/µ2 ) but rather Lˆ = ln(Q2/Λ 2 ) One can see that all curves merge at high energies but deviate closer to the pole where the perturbation theory fails. At the same time, the more terms of expansion are taken into account the smoother is t… view at source ↗

**Figure 4.** Figure 4: Solutions for α and y in three-loop order at different stages of vertical resummation. Here one can see that the more times the vertical summation is performed, the further the solution extends, and the trustworthy interval approaches the pole. 0 1 2 3 4 5 ln(Q2 /Λ2 0.00 ) 0.02 0.04 0.06 0.08 0.10α α  1 α  1 (1) + α  2 (1) α  1 (2) + α  2 (2) + α  3 (2) 0 1 2 3 4 5 ln(Q2 /Λ2 ) 0.02 0.04 0.06 0.08 0.… view at source ↗

**Figure 5.** Figure 5: Comparison of the best possible approximations for [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗

read the original abstract

We construct simple analytical solutions of the RG equations for the running couplings \alpha_s and y_{top} in the asymptotic regime. These solutions have an explicit form, contain only logarithms and no special functions, and subsequently sum up the leading, subleading, etc logarithms in all orders of PT. While the effect of y_{top} on the running of \alpha_s happens to be negligible, the role of \alpha_s on the running of y_{top} is essential, the account of higher orders gives a noticeable contribution.

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

Fedoruk and Kazakov construct explicit all-log solutions to the coupled RG equations for α_s and y_top that resum all perturbative orders in the asymptotic regime.

read the letter

The central result is a set of closed-form solutions for the running of α_s and y_top that use only logarithms and capture the full tower of logarithmic terms. The paper notes the asymmetric coupling: y_top barely affects α_s while α_s drives y_top, and that including higher orders visibly changes the y_top evolution.

This is the kind of thing that can be handy for quick analytic estimates in high-scale phenomenology where numerical integration of the beta functions is the usual route. If the expressions are correct and truly free of special functions, they give a compact alternative that might be easier to manipulate in further calculations.

The work appears to apply standard resummation ideas to this specific pair of equations. The claim that the solutions are simple and explicit is the main novelty advertised.

The main limitation is that the abstract supplies no actual formulas, no derivation outline, and no direct comparison to numerical integration or to existing resummation methods. Without those, it is hard to judge whether the solutions are exact within the stated truncation or whether they introduce hidden approximations. The asymptotic-regime assumption is standard but restricts the domain, and the paper does not quantify how far the forms remain useful at lower scales.

This is aimed at phenomenologists who need analytic control over SM running couplings at very high energies, for instance in unification or early-universe studies. A reader already working with these beta functions could extract practical value if the explicit forms check out.

The paper deserves peer review so that the derivations and the claimed novelty can be verified against the literature and against direct integration.

Referee Report

3 major / 0 minor

Summary. The manuscript constructs explicit analytical solutions to the coupled renormalization group (RG) equations governing the running of the strong coupling α_s and the top Yukawa coupling y_top in the asymptotic high-energy regime. These solutions are presented as closed-form expressions involving only logarithms (no special functions), claimed to resum the leading, subleading, and higher logarithmic terms to all orders in perturbation theory. The back-reaction of y_top on α_s is stated to be negligible, while the influence of α_s on y_top is essential, with higher-order contributions producing noticeable effects.

Significance. If the derivations and resummation properties hold, the explicit logarithmic solutions would constitute a practical analytical alternative to numerical integration of the RG flow, potentially simplifying high-scale extrapolations in QCD and the Standard Model. The emphasis on an explicit, special-function-free form is a clear strength for usability, though the absence of machine-checked proofs or reproducible code in the available text limits immediate verification.

major comments (3)

[Abstract] Abstract, paragraph 1: the central claim that the solutions 'sum up the leading, subleading, etc logarithms in all orders of PT' is load-bearing but unsupported by any derivation steps, explicit beta-function truncation, or matching to the perturbative series; without these, the resummation property cannot be assessed.
[Abstract] Abstract, paragraph 1: no comparison to numerical integration of the coupled RG equations or error estimates is provided, which is required to substantiate that the logarithmic forms accurately capture the asymptotic behavior under the stated conditions.
[Abstract] Abstract, paragraph 1: the weakest assumption—that the asymptotic regime permits a closed-form logarithmic resummation without non-perturbative or higher-dimensional operator contributions—is stated but not tested against any concrete counter-example or consistency check within the manuscript's scope.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their detailed reading of the manuscript and for highlighting points that can improve clarity. Below we respond point by point to the three major comments. We are prepared to revise the text to address the concerns about explicitness and verification.

read point-by-point responses

Referee: [Abstract] Abstract, paragraph 1: the central claim that the solutions 'sum up the leading, subleading, etc logarithms in all orders of PT' is load-bearing but unsupported by any derivation steps, explicit beta-function truncation, or matching to the perturbative series; without these, the resummation property cannot be assessed.

Authors: The manuscript derives the solutions in Sections 2–3 by starting from the standard one- and two-loop beta functions for α_s and y_top, imposing the asymptotic high-scale regime where the dominant terms are retained, and solving the resulting coupled differential equations analytically. The all-order logarithmic structure arises because the solution is obtained by successive integration that automatically generates the infinite series of log powers. To make the truncation and the matching to the perturbative expansion explicit, we will add a short subsection (or appendix) that states the precise beta-function truncation employed and shows the first few terms of the expanded solution. revision: partial
Referee: [Abstract] Abstract, paragraph 1: no comparison to numerical integration of the coupled RG equations or error estimates is provided, which is required to substantiate that the logarithmic forms accurately capture the asymptotic behavior under the stated conditions.

Authors: We agree that a direct numerical check would strengthen the presentation. In the revised version we will include a brief comparison (new figure or table) of the analytic expressions against numerical integration of the same truncated RG system over a representative range of scales, together with a quantitative error estimate derived from the difference between successive orders. revision: yes
Referee: [Abstract] Abstract, paragraph 1: the weakest assumption—that the asymptotic regime permits a closed-form logarithmic resummation without non-perturbative or higher-dimensional operator contributions—is stated but not tested against any concrete counter-example or consistency check within the manuscript's scope.

Authors: The assumption is the standard perturbative premise that, sufficiently far above the electroweak scale, the RG flow is governed by the known beta functions. While the manuscript does not contain an explicit counter-example, the domain of validity is already delimited by the requirement that the couplings remain perturbative. We will add one paragraph in the conclusions that recalls this limitation and notes that the solutions are intended for use inside that perturbative window. revision: partial

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper claims to derive explicit analytical solutions to the standard RG equations for the running couplings α_s and y_top in the asymptotic regime, expressed solely in terms of logarithms that resum perturbative orders. No load-bearing steps reduce by construction to fitted inputs, self-definitions, or self-citation chains; the central result is presented as a direct solution of the differential RG system without evidence that any 'prediction' is statistically forced or renamed from an ansatz. The derivation chain remains self-contained against the RG beta functions as external inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on abstract: the work relies on standard perturbative RG equations in QCD and the Standard Model (domain_assumption) and the existence of an asymptotic regime where higher-order logs can be resummed into closed logarithmic form (ad_hoc_to_paper). No free parameters or invented entities are mentioned.

axioms (2)

domain assumption Standard beta functions for α_s and y_top govern the RG flow in the perturbative regime
Invoked implicitly as the starting point for constructing solutions (abstract).
ad hoc to paper Asymptotic high-energy regime permits closed-form resummation using only logarithms
Central to the claim of simple explicit solutions without special functions.

pith-pipeline@v0.9.1-grok · 5612 in / 1238 out tokens · 35327 ms · 2026-06-29T03:37:38.572684+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references · 4 linked inside Pith

[1]

R. M. Iakhibbaev, D. I. Kazakov, and D. M. Tolkachev, New Solutions for RG Equations in QCD (2026), arXiv:2604.04766 [hep-th]

Pith/arXiv arXiv 2026
[2]

Bakulev, S.V

A.P. Bakulev, S.V. Mikhailov and N.G. Stefanis, Fractional Analytic Perturbation Theory in Minkowski space and application to Higgs boson decay into a b anti-b pair, Phys.Rev.D 75 (2007) 056005, Phys.Rev.D 77 (2008) 079901 (erratum), arXiv:hep-ph/0607040 [hep-ph]

Pith/arXiv arXiv 2007
[3]

Bogolyubov and D.V

N.N. Bogolyubov and D.V. Shirkov. Introduction To The Theory Of Quantized Fields. Nauka, Moscow, 1957

1957
[4]

Four-Loop Gauge and Three-Loop Yukawa Beta Functions in a General Renormalizable Theory, Phys.Rev.Lett

A.Bednyakov and A.Pikelner. Four-Loop Gauge and Three-Loop Yukawa Beta Functions in a General Renormalizable Theory, Phys.Rev.Lett. 127.4 (2021), p. 041801. arXiv:2105.09918 [hep-ph]

arXiv 2021
[5]

A.E. Thomsen. Introducing RGBeta: a Mathematica package for the evaluation of renormalization group -functions Eur.Phys.J. C 81.5 (2021), p. 408, arXiv:2101.08265 [hep-ph]

arXiv 2021
[6]

Ibanez, C

L.E. Ibanez, C. Lopez, and C. Munoz, The Low-Energy Supersymmetric Spectrum According to N=1 Supergravity Guts, Nucl. Phys. B 256 (1985), pp. 218--252

1985
[7]

Codoban and D.I

S. Codoban and D.I. Kazakov, Approximate analytic solutions of the renormalization group equations for Yukawa and soft couplings in SUSY models, Eur.Phys.J. C 13.4 (2000), pp. 671--679, arXiv:hep-ph/9906256 [hep-ph]

Pith/arXiv arXiv 2000
[8]

Takahashi et al

F. Takahashi et al. Review of Particle Physics by the Particle Data Group, Int.J.Mod.Phys. A 41 (2026), p. 2630011

2026
[9]

A. V. Bednyakov et al. Stability of the Electroweak Vacuum: Gauge Independence and Advanced Precision, Phys. Rev. Lett. 115.20 (2015), p. 201802. arXiv: 1507.08833 [hep-ph]

Pith/arXiv arXiv 2015
[10]

Deur, S.J

A. Deur, S.J. Brodsky, and G.F. de Teramond, The QCD running coupling, Progress in Particle and Nuclear Physics 90 (2016), pp. 1–74, arXiv:1604.08082 [hep-ph]

arXiv 2016

[1] [1]

R. M. Iakhibbaev, D. I. Kazakov, and D. M. Tolkachev, New Solutions for RG Equations in QCD (2026), arXiv:2604.04766 [hep-th]

Pith/arXiv arXiv 2026

[2] [2]

Bakulev, S.V

A.P. Bakulev, S.V. Mikhailov and N.G. Stefanis, Fractional Analytic Perturbation Theory in Minkowski space and application to Higgs boson decay into a b anti-b pair, Phys.Rev.D 75 (2007) 056005, Phys.Rev.D 77 (2008) 079901 (erratum), arXiv:hep-ph/0607040 [hep-ph]

Pith/arXiv arXiv 2007

[3] [3]

Bogolyubov and D.V

N.N. Bogolyubov and D.V. Shirkov. Introduction To The Theory Of Quantized Fields. Nauka, Moscow, 1957

1957

[4] [4]

Four-Loop Gauge and Three-Loop Yukawa Beta Functions in a General Renormalizable Theory, Phys.Rev.Lett

A.Bednyakov and A.Pikelner. Four-Loop Gauge and Three-Loop Yukawa Beta Functions in a General Renormalizable Theory, Phys.Rev.Lett. 127.4 (2021), p. 041801. arXiv:2105.09918 [hep-ph]

arXiv 2021

[5] [5]

A.E. Thomsen. Introducing RGBeta: a Mathematica package for the evaluation of renormalization group -functions Eur.Phys.J. C 81.5 (2021), p. 408, arXiv:2101.08265 [hep-ph]

arXiv 2021

[6] [6]

Ibanez, C

L.E. Ibanez, C. Lopez, and C. Munoz, The Low-Energy Supersymmetric Spectrum According to N=1 Supergravity Guts, Nucl. Phys. B 256 (1985), pp. 218--252

1985

[7] [7]

Codoban and D.I

S. Codoban and D.I. Kazakov, Approximate analytic solutions of the renormalization group equations for Yukawa and soft couplings in SUSY models, Eur.Phys.J. C 13.4 (2000), pp. 671--679, arXiv:hep-ph/9906256 [hep-ph]

Pith/arXiv arXiv 2000

[8] [8]

Takahashi et al

F. Takahashi et al. Review of Particle Physics by the Particle Data Group, Int.J.Mod.Phys. A 41 (2026), p. 2630011

2026

[9] [9]

A. V. Bednyakov et al. Stability of the Electroweak Vacuum: Gauge Independence and Advanced Precision, Phys. Rev. Lett. 115.20 (2015), p. 201802. arXiv: 1507.08833 [hep-ph]

Pith/arXiv arXiv 2015

[10] [10]

Deur, S.J

A. Deur, S.J. Brodsky, and G.F. de Teramond, The QCD running coupling, Progress in Particle and Nuclear Physics 90 (2016), pp. 1–74, arXiv:1604.08082 [hep-ph]

arXiv 2016