arxiv: 2604.13464 · v1 · submitted 2026-04-15 · ✦ hep-th · hep-ph

Recognition: unknown

Four-loop Anomalous Dimensions of Scalar-QED Theory from Operator Product Expansion

Rijun Huang , Qingjun Jin , Yi Li

Authors on Pith no claims yet

Pith reviewed 2026-05-10 13:36 UTC · model grok-4.3

classification ✦ hep-th hep-ph

keywords operator product expansionanomalous dimensionsscalar QEDfour-loop perturbation theoryfixed-charge operatorsrenormalizationbeta functions

0 comments

The pith

The OPE algorithm computes the four-loop anomalous dimension of the fixed-charge operator in scalar QED.

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

This paper applies the operator product expansion to the renormalization of scalar quantum electrodynamics. It calculates the anomalous dimension of the operator formed by the scalar field raised to the power Q up to four loops in the modified minimal subtraction scheme. The computation extends the previously known three-loop result and simultaneously determines the beta functions along with the mass and field anomalous dimensions at four loops. The work also introduces an alternative method for constructing loop integrands using graph decomposition and skeleton expansions. This constitutes the first application of the OPE approach to a gauge theory beyond pure scalar models.

Core claim

Within the OPE framework, the anomalous dimension of the φ^Q operator is perturbatively computed to four-loop order in the modified minimal subtraction scheme, extending beyond the previously available three-loop result. The beta functions, as well as the mass and field anomalous dimensions, are also computed at this order. An alternative loop-integrand construction method is proposed, based on graph decomposition and skeleton expansion techniques, for deriving the integrands of one-particle-irreducible correlation functions.

What carries the argument

The Operator Product Expansion (OPE) algorithm applied to operator mixing and renormalization of the fixed-charge operator φ^Q in scalar QED.

Load-bearing premise

The OPE algorithm previously developed for pure scalar theories applies directly to scalar QED without new obstructions or extra counterterms at four-loop order.

What would settle it

An independent four-loop calculation of the anomalous dimension of φ^Q using conventional Feynman-diagram methods that produces a numerically different result would falsify the OPE-derived value.

read the original abstract

We apply the Operator Product Expansion (OPE) algorithm to the renormalization of scalar-QED theory, with a specific focus on the fixed-charge operator $\phi^Q$. Within the OPE framework, the anomalous dimension of the $\phi^Q$ operator is perturbatively computed to four-loop order in the modified minimal subtraction scheme, extending beyond the previously available three-loop result. The beta functions, as well as the mass and field anomalous dimensions, are also computed at this order. An alternative loop-integrand construction method is proposed, based on graph decomposition and skeleton expansion techniques, for deriving the integrands of one-Particle-Irreducible correlation functions. This work represents the first non-trivial validation of the OPE algorithm for higher-loop renormalization beyond pure scalar theories. The present successful computations further confirm the efficiency and versatility of the OPE algorithm in renormalization analysis.

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

Four-loop OPE results for the fixed-charge operator in scalar QED are a direct extension of prior scalar-only work, but the gauge-field extension rests on an untested carry-over assumption.

read the letter

The paper computes the four-loop anomalous dimension of the φ^Q operator in scalar QED, together with the beta function and the mass and field anomalous dimensions, all in the MSbar scheme. It also sketches an alternative integrand construction using graph decomposition and skeleton expansion. This is the first reported use of the OPE algorithm outside pure scalar theories, so the numerical results at four loops are new for this model. If the method transfers without hidden adjustments, the numbers should be usable for RG studies that need this precision. The alternative integrand method is a practical addition that could speed up similar calculations elsewhere. The main soft spot is the lack of visible cross-checks. The abstract claims success but does not show matching to known three-loop terms, independent diagram evaluations, or explicit tests for gauge invariance and operator mixing introduced by the photon. The stress-test concern about new counterterm structures or diagram classes is reasonable; without those details in the text, it is difficult to judge whether the four-loop coefficients are complete. The work is aimed at a narrow group of people already using OPE for multi-loop renormalization in scalar and gauge theories. Most readers outside that niche will not need the specific numbers. It is a concrete computation rather than a conceptual advance, but the claim is falsifiable and the method is laid out enough to be checked. I would send it to referees.

Referee Report

2 major / 2 minor

Summary. The manuscript applies the Operator Product Expansion (OPE) algorithm to scalar-QED, computing the four-loop anomalous dimension of the fixed-charge operator φ^Q in the modified minimal subtraction scheme (extending the prior three-loop result). It also reports four-loop results for the beta functions and the mass and field anomalous dimensions. An alternative integrand construction method based on graph decomposition and skeleton expansion is proposed for one-particle-irreducible correlation functions. The work is presented as the first non-trivial validation of the OPE algorithm beyond pure scalar theories.

Significance. If the four-loop results hold, the paper supplies new perturbative data for scalar-QED and demonstrates that the OPE renormalization procedure can be extended to an abelian gauge theory. The proposed alternative integrand method is a concrete technical contribution that could improve efficiency in higher-loop calculations. These elements would strengthen the case for using OPE-based techniques in models with gauge fields.

major comments (2)

The central claim that the OPE algorithm extends without new obstructions to scalar-QED at four loops (including photon propagators and charged-scalar interactions) is load-bearing for both the anomalous-dimension results and the 'first non-trivial validation' statement. The manuscript does not appear to provide an explicit check that the counterterm basis or integrand construction remains unmodified, nor does it quantify potential gauge-fixing artifacts or operator-mixing structures absent in pure-scalar cases.
No cross-verification of the four-loop results against independent methods, known lower-order expansions, or numerical estimates is visible. Without such checks (e.g., reproduction of the three-loop anomalous dimension or comparison with existing literature values), the reliability of the new four-loop numbers cannot be assessed.

minor comments (2)

Notation for the fixed-charge operator φ^Q and the precise definition of the modified minimal subtraction scheme should be stated explicitly at first use to avoid ambiguity for readers unfamiliar with the OPE literature.
The description of the alternative loop-integrand method would benefit from a short flowchart or pseudocode illustrating the graph-decomposition steps.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive comments on our manuscript. We address each major point below and will incorporate revisions to clarify the validation aspects of the OPE application in scalar-QED.

read point-by-point responses

Referee: The central claim that the OPE algorithm extends without new obstructions to scalar-QED at four loops (including photon propagators and charged-scalar interactions) is load-bearing for both the anomalous-dimension results and the 'first non-trivial validation' statement. The manuscript does not appear to provide an explicit check that the counterterm basis or integrand construction remains unmodified, nor does it quantify potential gauge-fixing artifacts or operator-mixing structures absent in pure-scalar cases.

Authors: We agree that an explicit discussion would strengthen the manuscript. The successful four-loop computation itself demonstrates that the OPE procedure carries over without new obstructions, as the counterterm basis for the relevant operators (including those involving the photon field) is identical to the pure-scalar case up to the gauge-fixing terms already accounted for in the Lagrangian. In the revised version we will add a dedicated paragraph in Section 2 detailing the counterterm structures, confirming that the graph-decomposition method for integrand construction requires no modification, and briefly addressing gauge-fixing artifacts by noting that the Landau-gauge choice eliminates longitudinal contributions at the orders considered. Operator mixing is suppressed by U(1) charge conservation for the fixed-charge operator φ^Q, with no additional mixing structures appearing at four loops. revision: yes
Referee: No cross-verification of the four-loop results against independent methods, known lower-order expansions, or numerical estimates is visible. Without such checks (e.g., reproduction of the three-loop anomalous dimension or comparison with existing literature values), the reliability of the new four-loop numbers cannot be assessed.

Authors: We acknowledge the value of explicit cross-checks. As part of the computation pipeline, the three-loop anomalous dimension of φ^Q was reproduced exactly and matches the known result in the literature; the same holds for the three-loop beta function and field anomalous dimension. In the revised manuscript we will include a dedicated subsection (or appendix) presenting these lower-order reproductions side-by-side with the literature values, together with a brief consistency check against the two-loop results for the mass anomalous dimension. This will make the validation of the four-loop numbers transparent without altering the central results. revision: yes

Circularity Check

0 steps flagged

No circularity: direct perturbative extension of OPE algorithm

full rationale

The paper applies the established OPE renormalization algorithm to scalar-QED, computing four-loop anomalous dimensions of φ^Q, beta functions, and field/mass anomalous dimensions in the MS-bar scheme. It proposes an alternative integrand construction via graph decomposition and skeleton expansion. No derivation step reduces by construction to its inputs: there are no self-definitional relations, fitted parameters renamed as predictions, load-bearing self-citations that render the central result tautological, or ansatzes smuggled via prior work. The extension beyond pure scalars is framed as a non-trivial validation relying on independent diagrammatic evaluation rather than circular logic. The derivation chain remains self-contained against external perturbative benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard perturbative expansion in quantum field theory and the validity of the OPE algorithm in the MS-bar scheme. No new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)

domain assumption Validity of the Operator Product Expansion for perturbative renormalization in scalar-QED
The entire computation applies the OPE algorithm to obtain anomalous dimensions at four loops.

pith-pipeline@v0.9.0 · 5441 in / 1199 out tokens · 40045 ms · 2026-05-10T13:36:17.249241+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

98 extracted references · 66 canonical work pages · 1 internal anchor

[1]

P. A. M. Dirac,The quantum theory of the emission and absorption of radiation,Proceedings of the Royal Society A114(1927), no. 767 243–265

1927
[2]

Zinn-Justin,Quantum Field Theory and Critical Phenomena: Fifth Edition

J. Zinn-Justin,Quantum Field Theory and Critical Phenomena: Fifth Edition. Oxford University Press, 04, 2021

2021
[3]

Shankar,Quantum Field Theory and Condensed Matter

R. Shankar,Quantum Field Theory and Condensed Matter. Cambridge University Press, 8, 2017

2017
[4]

P. W. Anderson,Plasmons, Gauge Invariance, and Mass,Phys. Rev.130(1963) 439–442

1963
[5]

Englert and R

F. Englert and R. Brout,Broken Symmetry and the Mass of Gauge Vector Mesons,Phys. Rev. Lett.13(1964) 321–323

1964
[6]

P. W. Higgs,Broken Symmetries and the Masses of Gauge Bosons,Phys. Rev. Lett.13(1964) 508–509

1964
[7]

G. S. Guralnik, C. R. Hagen, and T. W. B. Kibble,Global Conservation Laws and Massless Particles,Phys. Rev. Lett.13(1964) 585–587

1964
[8]

V. L. Ginzburg and L. D. Landau,On the Theory of superconductivity,Zh. Eksp. Teor. Fiz.20(1950) 1064–1082

1950
[9]

Hohenberg and A

P. Hohenberg and A. Krekhov,An introduction to the ginzburg–landau theory of phase transitions and nonequilibrium patterns,Physics Reports572(Apr., 2015) 1–42

2015
[10]

B. i. Halperin, T. C. Lubensky, and S.-k. Ma,First order phase transitions in superconductors and smectic A liquid crystals,Phys. Rev. Lett.32(1974) 292–295

1974
[11]

I. F. Herbut and Z. Tesanovic,Critical fluctuations in superconductors and the magnetic field penetration depth,Phys. Rev. Lett.76(1996) 4588, [cond-mat/9605185]

work page arXiv 1996
[12]

Wen and Y.-S

X.-G. Wen and Y.-S. Wu,Transitions between the quantum hall states and insulators induced by periodic potentials,Phys. Rev. Lett.70(Mar, 1993) 1501–1504

1993
[13]

M. C. Cross and P. C. Hohenberg,Pattern formation outside of equilibrium,Rev. Mod. Phys.65(Jul, 1993) 851–1112. – 21 –

1993
[14]

T. W. B. Kibble,Topology of Cosmic Domains and Strings,J. Phys. A9(1976) 1387–1398

1976
[15]

Hindmarsh, S

M. Hindmarsh, S. Stuckey, and N. Bevis,Abelian Higgs Cosmic Strings: Small Scale Structure and Loops, Phys. Rev. D79(2009) 123504, [arXiv:0812.1929]

work page arXiv 2009
[16]

Dufaux, D

J.-F. Dufaux, D. G. Figueroa, and J. Garcia-Bellido,Gravitational Waves from Abelian Gauge Fields and Cosmic Strings at Preheating,Phys. Rev. D82(2010) 083518, [arXiv:1006.0217]

work page arXiv 2010
[17]

Hindmarsh, J

M. Hindmarsh, J. Lizarraga, J. Urrestilla, D. Daverio, and M. Kunz,Scaling from gauge and scalar radiation in Abelian Higgs string networks,Phys. Rev. D96(2017), no. 2 023525, [arXiv:1703.06696]

work page arXiv 2017
[18]

Hindmarsh, J

M. Hindmarsh, J. Lizarraga, A. Urio, and J. Urrestilla,Loop decay in Abelian-Higgs string networks,Phys. Rev. D104(2021), no. 4 043519, [arXiv:2103.16248]

work page arXiv 2021
[19]

I. K. Affleck, O. Alvarez, and N. S. Manton,Pair Production at Strong Coupling in Weak External Fields, Nucl. Phys. B197(1982) 509–519

1982
[20]

G. V. Dunne and C. Schubert,Worldline instantons and pair production in inhomogeneous fields,Phys. Rev. D72(2005) 105004, [hep-th/0507174]

work page arXiv 2005
[21]

K. G. Wilson and M. E. Fisher,Critical exponents in 3.99 dimensions,Phys. Rev. Lett.28(1972) 240–243

1972
[22]

Pelissetto and E

A. Pelissetto and E. Vicari,Critical phenomena and renormalization-group theory,Phys. Rept.368(2002), no. 6 549–727

2002
[23]

Moshe and J

M. Moshe and J. Zinn-Justin,Quantum field theory in the large N limit: A Review,Phys. Rept.385(2003) 69–228, [hep-th/0306133]

work page arXiv 2003
[24]

A. A. Vladimirov and D. V. Shirkov,THE RENORMALIZATION GROUP AND ULTRAVIOLET ASYMPTOTICS,Sov. Phys. Usp.22(1979) 860–878

1979
[25]

R. M. J. van Damme,Most General Two Loop Counterterm for Fermion Free Gauge Theories With Scalar Fields,Phys. Lett. B110(1982) 239–241

1982
[26]

Kolnberger and R

S. Kolnberger and R. Folk,Critical fluctuations in superconductors,Phys. Rev. B41(Mar, 1990) 4083–4088

1990
[27]

Ihrig, N

B. Ihrig, N. Zerf, P. Marquard, I. F. Herbut, and M. M. Scherer,Abelian Higgs model at four loops, fixed-point collision and deconfined criticality,Phys. Rev. B100(2019), no. 13 134507, [arXiv:1907.08140]

work page arXiv 2019
[28]

D. V. Batkovich, K. G. Chetyrkin, and M. V. Kompaniets,Six loop analytical calculation of the field anomalous dimension and the critical exponentηinO(n)-symmetricφ 4 model,Nucl. Phys. B906(2016) 147–167, [arXiv:1601.01960]

work page arXiv 2016
[29]

Kompaniets and E

M. Kompaniets and E. Panzer,Renormalization group functions ofϕ 4 theory in the MS-scheme to six loops, PoSLL2016(2016) 038, [arXiv:1606.09210]

work page arXiv 2016
[30]

M. V. Kompaniets and E. Panzer,Minimally subtracted six loop renormalization ofO(n)-symmetricϕ 4 theory and critical exponents,Phys. Rev. D96(2017), no. 3 036016, [arXiv:1705.06483]

work page arXiv 2017
[31]

Schnetz,Numbers and Functions in Quantum Field Theory,Phys

O. Schnetz,Numbers and Functions in Quantum Field Theory,Phys. Rev. D97(2018), no. 8 085018, [arXiv:1606.08598]

work page arXiv 2018
[32]

Schnetz,Graphical functions with spin,JHEP06(2025) 053 [2504.05850]

O. Schnetz,Graphical functions with spin,JHEP06(2025) 053, [arXiv:2504.05850]. – 22 –

work page arXiv 2025
[33]

J. A. Gracey,Four loop renormalization ofϕ 3 theory in six dimensions,Phys. Rev. D92(2015), no. 2 025012, [arXiv:1506.03357]

work page arXiv 2015
[34]

Kompaniets and A

M. Kompaniets and A. Pikelner,Critical exponents from five-loop scalar theory renormalization near six-dimensions,Phys. Lett. B817(2021) 136331, [arXiv:2101.10018]

work page arXiv 2021
[35]

Borinsky, J

M. Borinsky, J. A. Gracey, M. V. Kompaniets, and O. Schnetz,Five-loop renormalization ofϕ3 theory with applications to the Lee-Yang edge singularity and percolation theory,Phys. Rev. D103(2021), no. 11 116024, [arXiv:2103.16224]

work page arXiv 2021
[36]

Schnetz,ϕ 3 theory at six loops,Phys

O. Schnetz,ϕ3 theory at six loops,Phys. Rev. D112(2025), no. 1 016028, [arXiv:2505.15485]

work page arXiv 2025
[37]

De Prato, A

M. De Prato, A. Pelissetto, and E. Vicari,Third harmonic exponent in three-dimensional N vector models, Phys. Rev. B68(2003) 092403, [cond-mat/0302145]

work page arXiv 2003
[38]

Calabrese and P

P. Calabrese and P. Parruccini,Harmonic crossover exponents in O(n) models with the pseudo-epsilon expansion approach,Phys. Rev. B71(2005) 064416, [cond-mat/0411027]

work page arXiv 2005
[39]

Arias-Tamargo, D

G. Arias-Tamargo, D. Rodriguez-Gomez, and J. G. Russo,The large charge limit of scalar field theories and the Wilson-Fisher fixed point atϵ= 0,JHEP10(2019) 201, [arXiv:1908.11347]

work page arXiv 2019
[40]

Badel, G

G. Badel, G. Cuomo, A. Monin, and R. Rattazzi,The Epsilon Expansion Meets Semiclassics,JHEP11(2019) 110, [arXiv:1909.01269]

work page arXiv 2019
[41]

Antipin, J

O. Antipin, J. Bersini, F. Sannino, Z.-W. Wang, and C. Zhang,Charging theO(N)model,Phys. Rev. D102 (2020), no. 4 045011, [arXiv:2003.13121]

work page arXiv 2020
[42]

On the large charge sector in the critical O(N) model at large N,

S. Giombi and J. Hyman,On the large charge sector in the critical O(N) model at large N,JHEP09(2021) 184, [arXiv:2011.11622]

work page arXiv 2021
[43]

Arias-Tamargo, D

G. Arias-Tamargo, D. Rodriguez-Gomez, and J. G. Russo,On the UV completion of theO(N)model in6−ϵ dimensions: a stable large-charge sector,JHEP09(2020) 064, [arXiv:2003.13772]

work page arXiv 2020
[44]

Antipin, J

O. Antipin, J. Bersini, F. Sannino, Z.-W. Wang, and C. Zhang,More on the cubic versus quartic interaction equivalence in theO(N)model,Phys. Rev. D104(2021) 085002, [arXiv:2107.02528]

work page arXiv 2021
[45]

Long range, large charge, large N,

S. Giombi, E. Helfenberger, and H. Khanchandani,Long range, large charge, large N,JHEP01(2023) 166, [arXiv:2205.00500]

work page arXiv 2023
[46]

Antipin, J

O. Antipin, J. Bersini, and P. Panopoulos,Yukawa interactions at large charge,JHEP10(2022) 183, [arXiv:2208.05839]

work page arXiv 2022
[47]

Antipin, A

O. Antipin, A. Bednyakov, J. Bersini, P. Panopoulos, and A. Pikelner,Gauge Invariance at Large Charge, Phys. Rev. Lett.130(2023), no. 2 021602, [arXiv:2210.10685]

work page arXiv 2023
[48]

Antipin, J

O. Antipin, J. Bersini, P. Panopoulos, F. Sannino, and Z.-W. Wang,Infinite order results for charged sectors of the Standard Model,JHEP02(2024) 168, [arXiv:2312.12963]

work page arXiv 2024
[49]

Jack and D

I. Jack and D. R. T. Jones,Anomalous dimensions at large charge in d=4 O(N) theory,Phys. Rev. D103 (2021), no. 8 085013, [arXiv:2101.09820]

work page arXiv 2021
[50]

Jin and Y

Q. Jin and Y. Li,Five-loop anomalous dimensions ofϕ Q operators in a scalar theory with O(N) symmetry, JHEP10(2022) 084, [arXiv:2205.02535]. – 23 –

work page arXiv 2022
[51]

Bednyakov and A

A. Bednyakov and A. Pikelner,Six-loop anomalous dimension of theϕQ operator in the O(N) symmetric model,Phys. Rev. D106(2022), no. 7 076015, [arXiv:2208.04612]

work page arXiv 2022
[52]

Jack and D

I. Jack and D. R. T. Jones,Scaling dimensions at large charge for cubicϕ3 theory in six dimensions,Phys. Rev. D105(2022), no. 4 045021, [arXiv:2112.01196]

work page arXiv 2022
[53]

Huang, Q

R. Huang, Q. Jin, and Y. Li,From operator product expansion to anomalous dimensions,JHEP06(2025) 135, [arXiv:2410.03283]

work page arXiv 2025
[54]

Huang, Q

R. Huang, Q. Jin, and Y. Li,Five-loop anomalous dimensions of cubic scalar theory from operator product expansion,JHEP02(2026) 158, [arXiv:2508.13620]

work page arXiv 2026
[55]

Schnetz,Graphical functions and single-valued multiple polylogarithms,Commun

O. Schnetz,Graphical functions and single-valued multiple polylogarithms,Commun. Num. Theor. Phys.08 (2014) 589–675, [arXiv:1302.6445]

work page arXiv 2014
[56]

M. Golz, E. Panzer, and O. Schnetz,Graphical functions in parametric space,Lett. Math. Phys.107(2017), no. 6 1177–1192, [arXiv:1509.07296]

work page arXiv 2017
[57]

Borinsky and O

M. Borinsky and O. Schnetz,Graphical functions in even dimensions,Commun. Num. Theor. Phys.16 (2022), no. 3 515–614, [arXiv:2105.05015]

work page arXiv 2022
[58]

Schnetz and S

O. Schnetz and S. Theil,Notes on graphical functions with numerator structure,PoSLL2024(2024) 026, [arXiv:2407.17133]

work page arXiv 2024
[59]

Schnetz,HyperlogProceduresver.0.8, 2025

O. Schnetz,HyperlogProceduresver.0.8, 2025. Maple package available on the homepage of the author, https://www.math.fau.de/person/oliver-schnetz

2025
[60]

Huang, Q

R. Huang, Q. Jin, and Y. Li,On the seven-loop renormalization of Gross-Neveu model,JHEP06(2025) 134, [arXiv:2504.00713]

work page arXiv 2025
[61]

F. V. Tkachov,A theorem on analytical calculability of 4-loop renormalization group functions,Phys. Lett. B 100(1981) 65–68

1981
[62]

K. G. Chetyrkin and F. V. Tkachov,Integration by parts: The algorithm to calculateβ-functions in 4 loops, Nucl. Phys. B192(1981) 159–204

1981
[63]

High-precision calculation of multi-loop Feynman integrals by difference equations

S. Laporta,High-precision calculation of multiloop Feynman integrals by difference equations,Int. J. Mod. Phys. A15(2000) 5087–5159, [hep-ph/0102033]

work page Pith review arXiv 2000
[64]

A. V. Smirnov,Algorithm FIRE – Feynman Integral REduction,JHEP10(2008) 107, [arXiv:0807.3243]

work page arXiv 2008
[65]

Kira - A Feynman Integral Reduction Program

P. Maierh¨ ofer, J. Usovitsch, and P. Uwer,Kira—A Feynman integral reduction program,Comput. Phys. Commun.230(2018) 99–112, [arXiv:1705.05610]

work page Pith review arXiv 2018
[66]

Z. Wu, J. Boehm, R. Ma, H. Xu, and Y. Zhang,NeatIBP 1.0, a package generating small-size integration-by-parts relations for Feynman integrals,Comput. Phys. Commun.295(2024) 108999, [arXiv:2305.08783]

work page arXiv 2024
[67]

X. Guan, X. Liu, Y.-Q. Ma, and W.-H. Wu,Blade: A package for block-triangular form improved Feynman integrals decomposition,Comput. Phys. Commun.310(2025) 109538, [arXiv:2405.14621]

work page arXiv 2025
[68]

A. A. Vladimirov,Method for Computing Renormalization Group Functions in Dimensional Renormalization Scheme,Theor. Math. Phys.43(1980) 417. – 24 –

1980
[69]

K. G. Chetyrkin, A. L. Kataev, and F. V. Tkachov,New Approach to Evaluation of Multiloop Feynman Integrals: The Gegenbauer Polynomial x Space Technique,Nucl. Phys. B174(1980) 345–377

1980
[70]

W. E. Caswell and A. D. Kennedy,A Simple Approach to Renormalization Theory,Phys. Rev. D25(1982) 392

1982
[71]

K. G. Chetyrkin and F. V. Tkachov,Infrared R Operation and Ultraviolet Counterterms in the MS Scheme, Phys. Lett. B114(1982) 340–344

1982
[72]

Chetyrkin and V

K. Chetyrkin and V. Smirnov,R*-operation corrected,Physics Letters B144(1984), no. 5 419–424

1984
[73]

Larin and P

S. Larin and P. van Nieuwenhuizen,The Infrared R* operation,hep-th/0212315

work page arXiv
[74]

Two-loop mixing of dimension-five flavor-changing operators

M. Misiak and M. Munz,Two loop mixing of dimension five flavor changing operators,Phys. Lett. B344 (1995) 308–318, [hep-ph/9409454]

work page Pith review arXiv 1995
[75]

K. G. Chetyrkin, M. Misiak, and M. Munz,Beta functions and anomalous dimensions up to three loops,Nucl. Phys. B518(1998) 473–494, [hep-ph/9711266]

work page arXiv 1998
[76]

J. C. Collins,Renormalization: An Introduction to Renormalization, the Renormalization Group and the Operator-Product Expansion. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 1984

1984
[77]

B. Eden, P. Heslop, G. P. Korchemsky, V. A. Smirnov, and E. Sokatchev,Five-loop Konishi in N=4 SYM, Nucl. Phys. B862(2012) 123–166, [arXiv:1202.5733]

work page arXiv 2012
[78]

Proch´ azka and A

V. Proch´ azka and A. S¨ oderberg,Composite operators near the boundary,JHEP03(2020) 114, [arXiv:1912.07505]

work page arXiv 2020
[79]

’t Hooft,Dimensional regularization and the renormalization group,Nucl

G. ’t Hooft,Dimensional regularization and the renormalization group,Nucl. Phys. B61(1973) 455–468

1973
[80]

Mariño and R

M. Marino and R. Miravitllas,Trans-series from condensates,arXiv:2402.19356

work page arXiv

Showing first 80 references.