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arxiv: 2501.08384 · v2 · submitted 2025-01-14 · ✦ hep-ph · hep-ex

Two-Loop Anomalous Dimensions in the LEFT: Dimension-Six Four-Fermion Operators in NDR

Jason Aebischer , Pol Morell , Marko Pesut , Javier Virto This is my paper

Pith reviewed 2026-05-23 04:57 UTC · model grok-4.3

classification ✦ hep-ph hep-ex
keywords four-fermion operatorsanomalous dimensionsLEFTNDR schemetwo-loop mixingdimension-six operatorsJMS basisQCD and QED corrections
0
0 comments X

The pith

The complete two-loop anomalous dimension matrix for all dimension-six four-fermion operators in the LEFT has been derived in the NDR scheme.

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

The paper computes the full set of two-loop anomalous dimensions that govern how four-fermion operators mix under renormalization in the Low Energy Effective Field Theory. These dimensions include corrections of order alpha_s squared, alpha_s times alpha, and alpha squared. A sympathetic reader cares because the running of Wilson coefficients from a high matching scale down to hadronic energies depends on this matrix, and two-loop accuracy improves precision in flavor and collider analyses. The calculation uses known ultraviolet poles from two-loop QCD diagrams and extends an existing technique for handling gamma5 traces under naive dimensional regularization. Results are given in the JMS operator basis and supplied for theories with five, four, or three active quark flavors, with the matrix implemented in the public code DsixTools.

Core claim

By combining known two-loop ultraviolet poles in QCD with an extended treatment of gamma5 traces in the NDR scheme, the authors obtain the complete two-loop anomalous dimension matrix for the full set of dimension-six four-fermion operators in the JMS basis of the LEFT, including all O(alpha_s^2), O(alpha_s alpha), and O(alpha^2) entries, and present the matrix explicitly for LEFT with five, four, and three active quark flavors.

What carries the argument

The two-loop Anomalous Dimension Matrix (ADM) for four-fermion operators in the JMS basis, obtained by feeding known QCD UV poles into an extended NDR gamma5-trace procedure.

If this is right

  • Wilson coefficients of four-fermion operators can now be evolved from a high scale to the electroweak or hadronic scale with full two-loop accuracy in both QCD and QED.
  • Global fits to flavor observables that rely on LEFT can incorporate the new mixing effects at order alpha_s^2 and mixed orders.
  • The results apply directly to LEFT realizations with five, four, or three active quark flavors without further calculation.
  • The matrix has been encoded in DsixTools, allowing immediate numerical use in matching and running routines.

Where Pith is reading between the lines

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

  • Higher-order matching conditions from specific ultraviolet models onto the LEFT can now be run down with reduced theoretical uncertainty from operator mixing.
  • Similar techniques may allow two-loop anomalous dimensions to be obtained for other classes of dimension-six operators once their one-loop mixing is known.
  • Discrepancies between LEFT predictions and data at the percent level could be re-examined with the new two-loop running to check whether they persist.

Load-bearing premise

The known two-loop UV pole results from QCD diagrams remain valid when inserted into the LEFT calculation and the extension of the gamma5 handling method correctly reproduces the NDR scheme.

What would settle it

An independent two-loop calculation of any single entry in the anomalous dimension matrix, performed in a different regularization scheme such as dimensional reduction, that yields a numerically different coefficient from the one reported here.

read the original abstract

We derive the complete set of two-loop anomalous dimensions describing the mixing of four-fermion operators in the Low Energy Effective Field Theory (LEFT). The calculation is performed in Naive Dimensional Regularization with anticommuting $\gamma_5$ (the NDR scheme), and the results are given in the "JMS basis" of dimension-six operators. The derivation relies on known results for UV poles in two-loop diagrams in QCD, which are then used to derive the two-loop Anomalous Dimension Matrix (ADM) for the full set of four-fermion operators including $O(\alpha_s^2)$, $O(\alpha_s\alpha)$ and $O(\alpha^2)$ corrections. The method employed is an extension of a common approach to deal with traces containing $\gamma_5$ in NDR. Our results have been implemented in the public code DsixTools. We also discuss and provide the results in the LEFT with 5, 4 and 3 active quark flavors.

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.

Referee Report

0 major / 2 minor

Summary. The manuscript derives the complete two-loop anomalous dimension matrix for the mixing of dimension-six four-fermion operators in the Low Energy Effective Field Theory (LEFT) in the Naive Dimensional Regularization (NDR) scheme with anticommuting γ5. Results are given in the JMS basis and cover the O(α_s²), O(α_s α), and O(α²) sectors by feeding known two-loop QCD UV poles into the LEFT operator basis; an extension of the standard NDR γ5 trace method is employed. The calculation is performed for 3, 4, and 5 active quark flavors, with all results implemented in the public DsixTools code.

Significance. If correct, the results supply a necessary ingredient for NNLO precision calculations in the LEFT, which is central to flavor-physics phenomenology and BSM searches. The public code release constitutes a clear strength, enabling direct reproducibility and independent checks of algebraic or scheme-dependent errors. The reuse of established UV-pole results and the explicit treatment of multiple flavor thresholds further increase the practical value of the work.

minor comments (2)
  1. [Introduction] The manuscript would benefit from an explicit statement, perhaps in the introduction or results section, of how the chosen NDR γ5 extension differs from the prescriptions used in the input two-loop QCD calculations.
  2. [Results] A short table or paragraph comparing the new two-loop entries with the known one-loop ADM (for at least one representative operator) would help readers assess the size of the corrections.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report and the recommendation of minor revision. We appreciate the recognition of the significance of our results for NNLO calculations in the LEFT and the value of the public DsixTools implementation.

Circularity Check

0 steps flagged

No significant circularity; derivation uses external QCD inputs

full rationale

The paper states that it derives the two-loop ADM for four-fermion operators by feeding known UV poles from two-loop QCD diagrams into the LEFT basis (abstract and method description), together with an extension of a standard NDR γ5 prescription. No equations or steps reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations whose validity depends on the present work. The public DsixTools implementation supplies an independent check. This is the normal non-circular case for a calculation paper that reuses established diagram results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The calculation rests on standard QFT regularization and previously published two-loop QCD pole results; no new free parameters or invented entities are introduced.

axioms (2)
  • standard math Standard QCD and QED Feynman rules together with NDR regularization using anticommuting γ5
    The calculation is performed in the NDR scheme as stated in the abstract.
  • domain assumption Known results for UV poles in two-loop diagrams in QCD
    The derivation explicitly relies on these known results to obtain the LEFT ADM.

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    Three-loop perturbative corrections to B and D meson lifetime ratios are calculated, producing values that agree with experiment when using HQET sum rules or lattice inputs.

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