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Renormalization-group equations of the LEFT at two loops: dimension-six baryon-number-violating operators

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Pith's one-line read This paper computes the two-loop renormalization-group equations for all baryon-number-violating dimension-six operators of the low-energy effective field theory (LEFT), in both the 't Hooft-Veltman and naive dimensional regularization…

desk verdict Solid two-loop RGE calculation for the ΔB≠0 LEFT sector, with new HV results and a real but narrow verification gap in the NDR QED sector pending the revised version of Ref. [62]. read the letter →

arxiv 2505.03871 v2 pith:OKG3UNXA submitted 2025-05-06 hep-ph hep-exhep-lat

Luca Naterop , Peter Stoffer This is my paper · ORCID
classification hep-phhep-exhep-lat
keywords two-loopanomalousdimensionslow-energyeffectivefieldtheorybaryon-numberviolationprotondecayrenormalization-groupequationsevanescentoperatorsnaivedimensionalregularizationtHooft-Veltmanscheme
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper completes the two-loop anomalous-dimension calculation for the baryon-number-violating sector of the low-energy effective field theory (LEFT) at dimension six. It provides renormalization-group equations for all sixteen operator classes that violate baryon number, so that their scale running can be resummed at next-to-leading-logarithmic accuracy. Results are given in two schemes: the algebraically consistent 't Hooft-Veltman scheme for $\gamma_5$, corrected by finite renormalizations to restore chiral symmetry and compensate evanescent operators, and naive dimensional regularization, which the authors show is consistent in this sector because no ill-defined $\gamma_5$-odd traces occur. If correct, these results enable a more precise EFT-based reanalysis of proton-decay constraints on physics beyond the Standard Model.

What carries the argument

The engine of the calculation is a renormalization-group master formula that combines two-loop counterterm poles with derivatives of one-loop counterterms and with finite renormalizations that compensate chiral-symmetry-breaking effects and evanescent-operator insertions. In the 't Hooft-Veltman scheme, the finite counterterms from the authors' earlier work restore chiral symmetry in the spurion sense; in the NDR scheme, the evanescent operators are defined through antisymmetrized products of Dirac matrices with $a_{ev}=-1/2$ and $e_{ev}=3/2$, so that the flavor symmetries of the Wilson coefficients are preserved under transposition of Dirac chains and the calculation becomes independent of the fermion-line reading direction.

What would settle it

Recompute the two-loop QCD contribution to a same-chirality operator such as $\dot L^{S,LL}_{udd}$ in the NDR scheme with a completely independent setup and check the coefficient of $g^4$ against eq. (A.18); a mismatch would signal a missing evanescent or finite counterterm. Alternatively, repeat the calculation with $a_{ev}=1$ and verify that the running coefficients shift exactly as predicted by eq. (6.8).

Watch

Extended reading notes

Core claim

The central discovery is a complete set of two-loop $\beta$ functions for the 16 baryon-number-violating dimension-six LEFT operators, for generic flavor numbers and charges. The authors derive explicit RGEs in both the chirally symmetric 't Hooft-Veltman scheme and the NDR scheme, with the evanescent scheme fixed by $a_{ev}=-1/2$ and $e_{ev}=3/2$. They show that the difference between the two schemes takes the compact form $[\dot L]_{\mathrm{NDR}}-[\dot L]_{\mathrm{HV}} = \left(8 e^4 b^e_{0,0}(q_p q_r+q_s q_t)-\frac{16}{3} g^4 b^g_{0,0}\right)L$, affecting only the $O(g^4)$ and $O(e^4)$ terms. They also show that the NDR scheme is consistent in this sector because the only fermionic traces at two loops are vacuum-polarization insertions on four-fermion operators, and that flavor symmetries restrict the RGEs to the same block-diagonal mixing structure as at one loop.

Load-bearing premise

The load-bearing premise is that the evanescent-operator sets used to define each scheme are complete enough, and in particular that in NDR no ill-defined $\gamma_5$-odd trace can appear at two loops.

Editorial extensions

If this is right

  • The running of all 16 $\Delta B\neq 0$ dimension-six operator classes is fixed at next-to-leading-logarithmic accuracy, so EFT analyses can now resum large logarithms between the electroweak scale and proton-decay scales.
  • The NDR results agree with the independent calculation in Ref. [62] after its revision, while the 't Hooft-Veltman results are new.
  • The scheme difference is summarized by a simple formula, so results can be translated between the two schemes through the described finite renormalizations.
  • The pure QCD contributions agree with existing two- and three-loop literature, providing a nontrivial cross-check of the computation.

Reading between the lines

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

  • If these beta functions hold, existing one-loop proton-decay bounds in an EFT framework could shift at next-to-leading-logarithmic order, potentially by amounts comparable to current experimental sensitivity; a numerical implementation would quantify the shift.
  • The compact scheme-difference formula suggests that two-loop HV-versus-NDR differences for baryon-number-violating operators are chirality-blind and proportional to beta-function coefficients; it would be worth testing whether a similarly simple relation holds in the dimension-five or SMEFT sectors.
  • The method extends naturally to two-loop baryon-number-violating RGEs in SMEFT, which the authors note is work in progress; using the same $a_{ev}=-1/2$ evanescent scheme there should make one-loop matching to the LEFT consistent at the next-to-leading-log level.
  • The predicted $a_{ev}$ dependence in eq. (6.8) could be tested directly by repeating the calculation with the 'Greek projection' value $a_{ev}=1$ and checking that the running coefficients shift exactly as stated.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This paper computes the two-loop renormalization-group equations (RGEs) for the dimension-six baryon-number-violating operators of the low-energy effective field theory below the electroweak scale (LEFT). The calculation is performed in two schemes: the 't Hooft–Veltman (HV) scheme with finite counterterms that restore chiral symmetry and compensate evanescent operators, and naive dimensional regularization (NDR) with a specific choice of evanescent scheme (aev = -1/2, eev = 3/2). The authors provide explicit results for all 16 operator classes for an arbitrary number of flavors in Appendix A, and they derive a compact expression for the scheme difference between HV and NDR in Eq. (6.2). They compare their NDR results with existing literature, finding agreement with Gracey's pure-QCD results in Ref. [71] and partial agreement with Ref. [62], the latter pending an updated version.

Significance. If correct, this is a substantial technical advance: it completes the two-loop anomalous dimensions for the entire ΔB≠0 sector of the LEFT, enabling next-to-leading-logarithmic analyses of proton-decay and related baryon-number-violating observables in an EFT framework. The paper's strengths include two independent computational implementations (FORM and Symbolica), gauge-parameter independence checks, restoration of chiral symmetry in the HV scheme, and a transparent correction of an error in the authors' previous scheme definition (footnote 3). The derivation is first-principles within a defined scheme and involves no fitted parameters. The main caveat is that the independent confirmation of the NDR results in the QED sector is not yet in the public record, as discussed below.

major comments (3)
  1. [6.3, footnote 6, Conclusions] The Conclusions state that 'the NDR results are confirmed independently [62]', but the public version of Ref. [62] (arXiv:2501.08384v1) disagrees with the present results in the pure-QCD g^4 contribution and in several O(e^4) terms; the asserted agreement rests on a private communication about an unpublished revision. The O(e^4) two-loop anomalous dimensions are a central part of the claim in App. A.2 and are not covered by the independent check of Ref. [71], which is limited to pure QCD. The manuscript should either cite the updated version of Ref. [62] once it is available, provide the confirming calculation in a form the reader can verify, or explicitly qualify the status of the NDR results as not yet independently confirmed in the QED sector.
  2. [2.3, Eq. (2.4)] The consistency of the NDR scheme is asserted on the premise that 'the only fermionic traces come from vacuum-polarization corrections' and that no ill-defined γ5-odd traces appear at two loops. This premise is load-bearing for all NDR results in App. A.2, but the paper does not demonstrate it systematically. Please provide an explicit argument, for instance an enumeration of the two-loop diagram topologies showing that every closed fermion loop contains an even number of γ5 insertions, so that the reader can verify the absence of γ5-odd traces and hence the algebraic consistency of the NDR scheme in this sector.
  3. [6.3, Ref. [70]] For mixed-chirality operators, the comparison to Ref. [70] is not possible because the basis transformation involves Fierz-evanescent operators whose insertions were not computed. This means that the mixed-chirality NDR results are only checked against the pure-QCD results of Ref. [71] and the internal scheme-difference relation (6.2); the O(e^4) and mixed g^2e^2 contributions for these classes remain without independent public confirmation beyond the private communication with the authors of Ref. [62]. This is part of the verification gap identified above and should be stated clearly in the comparison subsection.
minor comments (4)
  1. [Section 3, footnote 3] The correction to Ref. [53] regarding the Fierz symmetry of the finite counterterms to LS,LL/RR ddd is only stated in a footnote. Since it changes the scheme definition used in the earlier paper, it would be helpful to summarize this correction in the main text, perhaps in a dedicated paragraph, so that readers relying on Ref. [53] are alerted.
  2. [Section 2.3, Eq. (2.7)] The symbol ε in Eq. (2.7) denotes the dimensional regulator, while the same symbol is used in the operator definitions for the Levi-Civita tensor (e.g., εαβγ in Table 1). This double use is potentially confusing; consider using a different symbol, such as (D-4)/2, in the Dirac-reduction rules.
  3. [Appendix A, Eqs. (A.34)-(A.35)] The numerical results for the benchmark flavor number nu=2, nd=ne=3 are useful, but a brief indication of how the values were obtained from the general formulas (for example, the values of be0,0 and bg0,0 used) would make the comparison to Ref. [62] easier to reproduce.
  4. [Section 6.3] The discussion of the parity-related discrepancies found in Ref. [62] (e.g., LS,LL udd versus LS,RR udd) is informative. A small table listing which operator classes agreed and which disagreed would improve the transparency of the comparison.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the two-loop RGEs are computed from first principles, with prior self-citations providing scheme definitions rather than the target results.

full rationale

The paper's two-loop anomalous dimensions are obtained by direct diagrammatic computation: QGRAF diagram generation, FORM/Symbolica implementations of the local R-operation, generic gauge parameters whose dependence cancels, and two independent code chains. The only quantities taken from prior work are the one-loop RGEs (Ref. [43], independently confirmed in Ref. [53]), the definition of the HV-scheme evanescent basis and finite renormalizations (Refs. [53,60]), and the standard master formula (5.1). These are scheme definitions and calculational tools, not the target two-loop coefficients; using one's own previously established scheme is legitimate reuse, not circularity. The NDR results are benchmarked against external calculations: pure-QCD two-loop results in Ref. [71] agree, and same-chirality QCD agrees with Ref. [70]; the relation (6.2) between schemes is a derived consistency relation, and the aev dependence in Eq. (6.8) follows from explicit finite-shift manipulations (6.5)-(6.7), not from fitting. Footnote 3 corrects an error in the authors' earlier evanescent bookkeeping, and Sect. 6.3 candidly reports that Ref. [62] v1 disagreed until a private recalculation; this is a verification gap, not circular reasoning. No fitted parameter is renamed as a prediction, and no central claim reduces by definition to an input.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The calculation rests on defining renormalization schemes (HV and NDR) with specific evanescent operators, on a master formula for two-loop RGEs from the authors' prior work, and on the completeness of the operator basis. No parameters are fitted to data; the only scheme parameter entering the final NDR results is aev.

free parameters (1)
  • aev = -1/2
    Evanescent-scheme parameter chosen by hand to preserve flavor symmetries under transposition of Dirac bilinears (Sect. 2.3). The dependence on generic aev is reinstated in eq. (6.8) and in eqs. (A.34)-(A.35). It is a scheme definition, not a fit to data.
assumptions (5)
  • standard math Dimensional regularization with the 't Hooft-Veltman prescription for gamma5 (eq. (2.3)) is algebraically consistent to all loop orders.
    Invoked as background; the paper references Refs. [65,66,67,68,69].
  • domain assumption In the NDR scheme, no ill-defined gamma5-odd traces occur at two loops in the B-violating sector (Sect. 2.3).
    Load-bearing for the NDR results; argued from the diagram structure, not proven in this paper.
  • domain assumption The master formula (5.1) correctly converts one- and two-loop counterterms, including finite renormalizations, into two-loop RGEs.
    Taken from the authors' previous paper Ref. [53]; stated without re-derivation.
  • domain assumption The operator and evanescent-operator bases of Refs. [39,53] and Tabs. 1-4 are complete at NLL for the B-violating sector.
    The paper fixes an error in the earlier evanescent scheme in footnote 3, indicating sensitivity to this assumption.
  • domain assumption The flavor symmetries of the Wilson coefficients (eqs. (3.2)-(3.3)) can be imposed without loss, with any violation shifted into evanescent operators via (3.5)-(3.7).
    Scheme choice that restricts the flavor structure of the RGEs.

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Cite this review

Pith. "Pith review of Renormalization-group equations of the LEFT at two loops: dimension-six baryon-number-violating operators." pith.science (2026). https://pith.science/paper/OKG3UNXA

@misc{pith2026250503871,
  author       = {Pith},
  title        = {Pith review of: Renormalization-group equations of the LEFT at two loops: dimension-six baryon-number-violating operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OKG3UNXA}},
  note         = {Machine review of arXiv:2505.03871}
}
abstract

We present the second part of a systematic calculation of the two-loop anomalous dimensions for the low-energy effective field theory below the electroweak scale (LEFT): the baryon-number-violating sector at dimension six in the power counting. We obtain the results in two different schemes: in the algebraically consistent 't Hooft-Veltman scheme for $\gamma_5$, corrected for evanescent as well as chiral-symmetry-breaking effects through finite renormalizations; and in naive dimensional regularization, which in the considered sector of the theory does not lead to any ill-defined $\gamma_5$-odd traces. Our results are of interest for a reanalysis of the constraints on physics beyond the Standard Model from proton-decay searches within an EFT framework at next-to-leading-logarithmic accuracy.

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