{"id":"6760d895-a656-42e0-a15c-251be225f49f","arxiv_id":"2505.03871","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":1,"one_line_summary":"The two-loop anomalous dimensions for all dimension-six baryon-number-violating LEFT operators are derived in the 't Hooft-Veltman and naive dimensional regularization schemes.","lead":"This paper computes the two-loop renormalization-group equations for the baryon-number-violating dimension-six operators in the low-energy effective field theory below the electroweak scale. The results are the missing ingredient for next-to-leading-logarithmic analyses of proton-decay constraints on new physics.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Independent confirmation of the NDR results rests on a forthcoming revision of Ref. [62]; the public v1 disagrees in the pure-QCD terms, so the O(e^4) sector is not yet externally verified.","rationale":"The paper's central claim is well supported internally: two independent implementations, gauge-parameter independence, agreement with published pure-QCD results [71], and the nontrivial HV/NDR scheme-difference relation (6.2). I do not find an internal inconsistency that would overturn the results. The most load-bearing gap is the external confirmation of the NDR results: Ref. [62] v1 is the only near-complete independent calculation, but it disagrees in the pure-QCD contributions and some O(e^4) terms; the paper reports a private agreement and a forthcoming revision. This is not evidence of error, but it means the O(e^4) part of the central claim is not yet publicly verified. Footnote 3 also shows that evanescent-operator bookkeeping in the preceding Ref. [53] contained an error, which makes an external check of the QED sector substantive rather than ceremonial. The reader's weakest assumption about NDR trace consistency is reasonable and I agree it is not the main issue; the conditional status is appropriately driven by the pending confirmation. Thus I recommend no change to the CONDITIONAL verdict.","tokens_in":27045,"tokens_out":14689,"duration_ms":153866,"concrete_test":"When the updated version of Ref. [62] is released, compare its numerical two-loop coefficients for n_u=2, n_d=n_e=3 with Eqs. (A.34)-(A.35), including the aev-dependent terms of Eq. (6.8); full agreement for all 16 operator classes closes the verification gap. If the revision does not appear, recompute the O(e^4) coefficient for one representative class (e.g., LS,LL_udd, Eq. (A.18)) with an independent toolchain and compare.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is conditionally supported by an external confirmation that is not yet in the public record. Section 6.3 and footnote 6 state that arXiv:2501.08384v1 disagreed with the present results in the pure-QCD g^4 contributions and in several O(e^4) terms, and that the authors of Ref. [62] have privately agreed after revisiting their calculation. Until the updated version appears, the only published independent checks are the pure-QCD two-loop results of Gracey [71], which cover g^4 but not the O(e^4) and mixed g^2e^2 parts, and the internal HV/NDR scheme-difference relation (6.2). The O(e^4) two-loop anomalous dimensions, which are part of the central claim, therefore currently rest on the authors' own computation and on a private communication. This is a verification gap rather than a demonstrated error; however, footnote 3 shows that evanescent-operator bookkeeping in the preceding Ref. [53] contained an error, so an external check of the QED sector is substantive rather than ceremonial.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":27226,"tokens_out":4933,"duration_ms":48520,"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":[{"comment":"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.","section":"6.3, footnote 6, Conclusions"},{"comment":"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.","section":"2.3, Eq. (2.4)"},{"comment":"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.","section":"6.3, Ref. [70]"}],"minor_comments":[{"comment":"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.","section":"Section 3, footnote 3"},{"comment":"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.","section":"Section 2.3, Eq. (2.7)"},{"comment":"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.","section":"Appendix A, Eqs. (A.34)-(A.35)"},{"comment":"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.","section":"Section 6.3"}],"recommendation":"major_revision","confidential_remarks":"The main reservation is the status of the independent confirmation of the NDR results. The manuscript cites a private communication and a forthcoming revision of Ref. [62] as 'confirmed independently', which is not verifiable by the reader. Once the updated version of Ref. [62] appears, or the O(e^4) check is otherwise made public, the paper would, in my view, meet the bar for publication. The internal consistency checks (two implementations, gauge-parameter independence, chiral-symmetry restoration, and agreement with Gracey in pure QCD) are strong, and the issue is a verification gap rather than a demonstrated error."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper completes the two-loop RGEs for the dimension-six baryon-number-violating sector of the LEFT. What is genuinely new: the HV-scheme results, which are first, and the explicit scheme dependence and translation relations. The NDR results agree with the existing pure-QCD literature (Gracey, Nihei–Arafune) and the authors are transparent that full independent confirmation of the O(e^4) sector awaits the revision of Ref. [62].\n\nWhat the paper does well: the technical execution is careful. Two independent implementations, gauge-parameter independence, chiral-symmetry restoration in the HV scheme, and a clean derivation of the NDR consistency condition (no ill-defined γ5-odd traces at two loops) all support the central claim. The authors also correct an error in their own earlier scheme definition (footnote 3) without obscuring it. The results are presented for generic flavor numbers and charges, which makes them immediately usable. The simple relation between HV and NDR results, eq. (6.2), is a nice cross-check and a useful guide for scheme translation.\n\nThe soft spots are real but not damning. The main one is the verification gap: section 6.3 and footnote 6 say the authors of Ref. [62] privately agree after revisiting their calculation, but the public v1 disagrees in the QCD and several O(e^4) terms. Until the revised version appears, the O(e^4) two-loop anomalous dimensions rest on this private communication plus the internal HV/NDR consistency checks. That is a verification gap, not a demonstrated error. A second, minor caveat: the mixed-chirality operators cannot be compared directly to Ref. [70] because the basis transformation involves evanescent terms the authors did not compute. They say this clearly. The NDR consistency argument is reasonable; it would become fragile only if a future calculation in this sector produced a γ5-odd trace, which they argue does not happen here.\n\nWho should read this: anyone working on proton-decay EFT analyses at next-to-leading-log accuracy, and people doing two-loop γ5 scheme work. It is a specialist paper, not a paradigm shift, but it fills a necessary gap in the LEFT program.\n\nFor peer review: yes, I would send it out. The referee should check the internal consistency of Appendix A with eq. (6.2), verify a few of the NDR terms against Gracey where possible, and insist that the authors update the footnote about Ref. [62] once the revised version is public. The core computation looks solid.","headline":"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].","tokens_in":27799,"tokens_out":1440,"would_cite":true,"duration_ms":16387,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["two-loop anomalous dimensions","low-energy effective field theory","baryon-number violation","proton decay","renormalization-group equations","evanescent operators","naive dimensional regularization","t Hooft-Veltman scheme"],"falsifier":"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).","tokens_in":26826,"feed_emoji":"⚛️","tokens_out":5269,"duration_ms":50060,"temperature":0.7,"pith_summary":"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.","feed_headline":"All 16 baryon-violating operator classes now have two-loop running","feed_subtitle":"New results cover the dimension-six LEFT sectors in both gamma5 schemes, enabling next-to-leading-log resummation for proton-decay searches.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the one-loop RGEs, operator basis, and flavor-symmetry conventions that the two-loop results must extend and reproduce in the leading order.","marker":"[43]"},{"why":"Defines the 't Hooft-Veltman scheme with the finite renormalizations and evanescent operators that this paper uses and partially corrects in footnote 3.","marker":"[53]"},{"why":"Describes the semi-automated calculation method and the master formula at dimension five that are extended here to the baryon-number-violating dimension-six sector.","marker":"[60]"},{"why":"Independent NDR two-loop calculation of LEFT four-fermion operators; the present NDR results are compared with it and found to agree after revision.","marker":"[62]"},{"why":"Earlier two-loop QCD anomalous dimensions for same-chirality operators; the pure QCD results of the present paper agree with it.","marker":"[70]"},{"why":"Three-loop QCD results in the Larin scheme whose two-loop pieces serve as a cross-check of the NDR QCD contributions here.","marker":"[71]"},{"why":"Provides the evanescent-reduction rules and scheme parameters such as $a_{ev}$ that enter the NDR evanescent scheme chosen in this paper.","marker":"[44]"}],"fun_headline_variants":["Two-loop RGEs for all 16 baryon-violating operators","Two-loop anomalous dimensions for proton-decay operators","Complete two-loop running for baryon-violating operators","Two-loop RGEs make proton-decay searches NLL-precise"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Two-loop RGEs for all 16 baryon-violating operators","Two-loop anomalous dimensions for proton-decay operators","Complete two-loop running for baryon-violating operators","Two-loop RGEs make proton-decay searches NLL-precise"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000704,"raw_usage":{"total_tokens":3167,"prompt_tokens":928,"completion_tokens":2239,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":544,"completion_tokens_details":{"reasoning_tokens":2168}},"tokens_in":544,"tokens_out":2239,"duration_ms":16556,"temperature":1.0,"reasoning_tokens":2168,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:43:28.464742+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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).","supporting_citations":[],"review_version":1}