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RI-(S)MOM to $\overline{\rm MS}$ conversion for $B_K$ at two-loop order

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper completes the two-loop RI-(S)MOM to $\overline{\mathrm{MS}}$ matching for the $\Delta S=2$ operator and reports $\hat{B}_K^{(f=3)}=0.7627(60)$, $\hat{B}_K^{(f=4)}=0.7759(84)$, with an updated Standard Model prediction…

desk verdict Genuinely new two-loop RI-(S)MOM to MS matching for B_K, with a solid internal validation, but the headline NNLO claim is diluted by mixed-order lattice inputs and an unverified external convention assumption. read the letter →

arxiv 2411.19861 v2 pith:L5S4M5DC submitted 2024-11-29 hep-ph hep-lat

Martin Gorbahn , Sebastian Jäger , Sandra Kvedaraitė This is my paper · ORCID
classification hep-phhep-lat
keywords kaonbagparameterRI-SMOMschemeRI-MOMMS-bartwo-loopmatchingfour-quarkoperatorepsilon_KlatticeQCD
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 matching between the RI-(S)MOM and $\overline{\mathrm{MS}}$ renormalization schemes for the $\Delta S=2$ four-quark operator that controls neutral-Kaon mixing, and uses that matching to convert all published 3- and 4-flavour lattice determinations of the Kaon bag parameter to a common scheme. The central quantitative claims are the conversion factors in Eq. (5.1) with the NNLO coefficients in Table 3, the world averages $\hat{B}_K^{(f=3)}=0.7627(60)$ and $\hat{B}_K^{(f=4)}=0.7759(84)$, and the updated value $|\epsilon_K|=2.171(65)(71)(153)\times 10^{-3}$. A sympathetic reader should care because $\hat{B}_K$ is the main non-perturbative input into indirect CP violation in the Kaon system; moving that input to NNLO and combining 3- and 4-flavour lattice results tightens the constraints on the CKM matrix and on new physics. The paper also notes a PDG rescaling factor of 1.28, reflecting a mild tension among lattice inputs after the NNLO conversion.

What carries the argument

The central object is the amputated four-point Green's function $\Lambda$ for the operator $Q$, with momentum-subtraction conditions defined by the projectors $P_{(\gamma_\mu)}$ and $P_{(q\!\!/)}$. The key mechanism is a projection-first calculation: the paper defines dimensionally unambiguous projectors $\Pi_\mu,\Pi_{11},\Pi_{12},\Pi_{22}$ that act before tensor reduction, avoiding $\gamma_5$ and Levi-Civita ambiguities, and it constructs evanescent operators that vanish under all projectors. This decomposes the two-loop amplitude into coefficients $A_i,\tilde{A}_i$ in front of tree-level matrix elements, from which the (S)MOM projections are recovered. The scheme conversion itself uses the relation $C_{B_K}^{S\to\overline{\mathrm{MS}}}=\lambda(\overline{\mathrm{MS}},l)\,[\sigma(\overline{\mathrm{MS}},s)]^2$, where $\lambda$ is the projected four-point amplitude and $\sigma$ is the scalar two-point amplitude.

What would settle it

Recompute the two-loop off-shell box master integrals with an independent method and re-extract the NNLO coefficients in Table 3; if any coefficient shifts outside its quoted uncertainty, or if the final $\hat{B}_K$ averages change when the evanescent-operator basis is changed, the claimed two-loop matching is falsified.

Watch

Extended reading notes

Core claim

The paper establishes that the two-loop RI-(S)MOM-to-$\overline{\mathrm{MS}}$ conversion factors for the $\Delta S=2$ bag parameter are now complete, with the explicit NNLO coefficients listed in Table 3 for all six momentum-subtraction variants, evaluated in Landau gauge at $N_c=3$. These factors take the form $C_{B_K}^{S\to\overline{\mathrm{MS}}}(\mu,\nu)=1+(\alpha_s/4\pi)C_{B_K}^{S,\mathrm{NLO}}+(\alpha_s/16\pi^2)[C_{B_K}^{S,\mathrm{NNLO}}+\ldots]$, and they are combined with the existing two-loop anomalous dimensions and charm-threshold matching to convert every lattice value of $B_K$ to the renormalization-group-invariant bag parameter. Combining all available 3- and 4-flavour lattice results, the paper obtains the averages $\hat{B}_K^{(f=3)}=0.7627(60)$ and $\hat{B}_K^{(f=4)}=0.7759(84)$, and from the 3-flavour average it updates the Standard Model prediction for indirect CP violation in the Kaon system to $|\epsilon_K|=2.171(65)(71)(153)\times 10^{-3}$.

Load-bearing premise

The calculation assumes that the three-loop quark-field renormalization constants it imports from other papers use exactly the same conventions as its own four-quark operator scheme; if that assumption is wrong, all the NNLO conversion numbers move.

Editorial extensions

If this is right

  • The NNLO conversion factors complete the scheme-conversion chain for $\hat{B}_K$ at $O(\alpha_s^2)$, so any existing or future lattice result in any of the six RI-(S)MOM variants can be converted to $\overline{\mathrm{MS}}$ at the same perturbative accuracy.
  • The averages $\hat{B}_K^{(f=3)}=0.7627(60)$ and $\hat{B}_K^{(f=4)}=0.7759(84)$ combine all 3- and 4-flavour lattice inputs, using charm-threshold matching at $O(\alpha_s^2)$, and supersede previous flavour-separated averages with a smaller uncertainty.
  • The updated $|\epsilon_K|$ prediction shifts to $2.171(65)(71)(153)\times 10^{-3}$ and reduces the non-perturbative uncertainty from roughly 3.5% to about 3.28%, making long-distance contributions the dominant non-perturbative error.
  • Residual scale dependence is smallest for the SMOM scheme with slashed-momentum projectors at both 3 and 4 flavours, so future lattice determinations using that scheme will incur the smallest conversion uncertainty.
  • The same conversion factors also yield $\overline{\mathrm{MS}}$ and renormalization-group-invariant bag parameters for $D$-meson mixing from the existing four-flavour lattice input.

Reading between the lines

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

  • An extension of the paper's approach is that the same projection-first technique could produce two-loop RI-(S)MOM conversions for the other four-quark operators in $\Delta F=2$ effective Hamiltonians, where such NNLO conversions are not yet available.
  • If the imported bilinear conversion constants carry an undetected convention mismatch, the spread among the six schemes after conversion would reveal it: a consistent matching must make all schemes agree on $\hat{B}_K$, so a residual scheme-dependent spread beyond quoted errors would point to that input.
  • The quoted PDG rescaling factor of 1.28 is a testable diagnostic: if the mild lattice tension persists after future lattice inputs are added, the averaged error should be rescaled again, whereas a resolved tension would confirm that the NNLO conversion absorbed the previous scheme discrepancy.
  • The 4-flavour average $\hat{B}_K^{(f=4)}=0.7759(84)$ is directly usable for future phenomenology that treats the charm quark non-perturbatively, so improvements in long-distance lattice calculations should translate almost entirely into a smaller $\epsilon_K$ uncertainty.
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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. The paper computes the two-loop conversion factors between six RI-(S)MOM schemes and the MS-bar scheme for the Delta S = 2 four-quark operator that defines the kaon bag parameter. The four-quark projected amplitudes are computed from first principles in this paper, using a projector-based method that avoids gamma5 ambiguities, integration-by-parts reduction, and partly numerical evaluation of master integrals. The two-loop quark-field (bilinear) factors sigma(MS,s) are not computed here but are imported from Refs. [27-29]. The resulting conversion factors are then used to convert lattice bag parameters to the RGI quantity B_hat_K, to combine 3- and 4-flavour lattice results through a charm-threshold matching, and to produce updated world averages B_hat_K^{(f=3)} = 0.7627(60) and B_hat_K^{(f=4)} = 0.7759(84), together with an updated Standard Model prediction |epsilon_K| = 2.171(65)_pert.(71)_non-pert.(153)_param. x 10^{-3}.

Significance. If the results are correct, this is a valuable and timely piece of precision phenomenology. The two-loop RI-(S)MOM to MS-bar conversion for B_K is a natural completion of the NLO program, and the paper contains a number of concrete validation checks: the NLO results reproduce Refs. [17,19,20], the NNLO 1/epsilon poles cancel within the quoted numerical uncertainties and match the analytic pole structure of Ref. [6], and the residual scale dependence is mapped for all six schemes. The paper also makes a genuine effort to combine all available lattice inputs, including a 3/4-flavour matching at O(alpha_s^2). These strengths should be credited. The main reservation concerns not the four-quark calculation itself but the imported bilinear two-loop factors, which enter every NNLO conversion coefficient squared and whose convention compatibility with the MS-bar/NDR scheme used here is not demonstrated.

major comments (3)
  1. [Sec. 5.5, Table 8] The NNLO coefficients in Table 3 are built from C^{S->MS}_{BK} = lambda(MS,l) [sigma(MS,s)]^2, with sigma(MS,s) taken at NNLO from Refs. [27-29] after the substitutions CA=Nc, TF=1/2, w=1, r=1. The paper does not list the adapted sigma expressions, nor does it demonstrate that the RI' and RI/SMOM field-renormalization conventions used in [27-29] coincide with the projectors and the BG evanescent MS-bar scheme defined in Secs. 2.2 and 4.5. Since sigma enters squared, any convention mismatch or finite scheme piece in those imported results shifts every entry of Table 3, propagates through Eq. (5.2), and moves the final averages in Eqs. (5.11)-(5.12). This is a load-bearing input assumption, structurally separate from the two-loop four-quark calculation. A concrete fix is to display the explicit NNLO sigma(MS,s) expansions used in this paper and to verify them in the same Landau-gauge, NDR, BG-evanescent convention, for example by checking that the full conversion factor reproduces the pole structure of Ref. [6] without relying on cancellations between the four-quark and bilinear pieces.
  2. [Sec. 5.5, Table 8] Only four of the six lattice entries entering the global average are actually converted with the new two-loop factors. The SWME 15A and Laiho 11 results are in the MS-bar scheme and are converted with the external factor 1.369 taken from FLAG [13], as indicated by the asterisk in Table 8. The abstract states that the world averages combine the complete set of lattice results, but the role of these external conversions should be stated more prominently, and the numerical sensitivity of the final average to using NNLO versus NLO conversion for these two entries should be quantified. As written, the claim that the precision of B_hat_K is improved to NNLO is only partially supported for these two inputs.
  3. [Sec. 5.5, Eq. (5.8)] The correlation between the two RBC/UKQCD 24 SMOM projections is inferred from the chi^2 value of the two-projection average: the text states that chi^2 = 0.671 is used to estimate a 33.9% correlation. This is an indirect, model-dependent way to assign the covariance, and the same procedure is used for the f = 4 average with a correlation of 68.2%. Since the quoted uncertainties in Eqs. (5.11)-(5.12) depend directly on this correlation, the paper should report the range of central values and errors obtained as the correlation is varied over its allowed interval, or obtain the covariance from the lattice collaboration's published error budget.
minor comments (4)
  1. [Eq. (5.1)] The coefficients written as 3/9 f, 4/9 f, and 12/9 f should be reduced to 1/3 f, 4/9 f, and 4/3 f, and the definition of f should be restated in the equation caption to avoid ambiguity.
  2. [Abstract and Sec. 5.5] The abstract refers to a 'PDG rescaling factor of 1.28', but the text in Sec. 5.5 only introduces it as multiplication of the error by sqrt(chi^2/dof). The explicit connection between the two statements should be made in the text.
  3. [Table 5] The table caption uses the notation (delta_alpha_s/delta_nu) to report the ratio of uncertainties, but this notation is not defined in the text; a phrase such as 'ratio of the alpha_s uncertainty to the scale-variation uncertainty' would be clearer.
  4. [Sec. 5.1, Table 3] The NNLO coefficients are quoted with numerical errors from the integral evaluation, but the corresponding conversion factors in Table 4 are given without propagated uncertainties. Please state explicitly whether the errors in Table 4 include the Table 3 coefficient errors and the alpha_s error separately.

Circularity Check

0 steps flagged · score 0.0 of 10
full rationale

The central claim is the two-loop RI-(S)MOM-to-MS conversion for the four-quark operator, obtained from Eq. (2.17), C = lambda(MS,l) [sigma(MS,s)]^2. The four-quark projected amplitude lambda(MS,l) is computed in this paper from explicit two-loop diagrams, IBP reduction, master integrals, and MS counterterms, as described in Sections 3 and 4 and Appendices D and E. The bilinear factors sigma(MS,s) are taken from Refs. [17,27,28,29], which are independent published perturbative results for quark-field and quark-mass renormalization; they are not fitted to the lattice BK data and do not presuppose the four-quark conversion being derived. The lattice bag parameters enter as external measurements, not as fitted inputs, and the world averages in Eqs. (5.11) and (5.12) are ordinary combinations of those external values with the computed conversion factors. The RG evolution and charm-threshold matching elements taken from Ref. [7] are prior, documented perturbative calculations; even though one author overlaps with the present paper, the present two-loop conversion does not reduce to them, and they are not invoked as an unexamined uniqueness theorem or ansatz. The only substantive concern is convention compatibility of the imported bilinear results with the NDR/BG scheme, which is an input assumption that could affect correctness of the numerical coefficients but is not a circularity in the derivation. Accordingly, no circular step is present.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

No parameters are fitted to the target data in the perturbative part: the two ledger entries are a correlation estimate (33.9% for the RBC/UKQCD 24 projections, inferred from chi^2) and a hand-chosen scale-variation range (2 to 6 GeV) that sets the quoted higher-order uncertainty. No new physical entities are postulated; the GJK evanescent operators (Eqs. 3.17-3.24) and Fierz-evanescent combinations (Appendix A) are renormalization-scheme conventions with vanishing physical content. The central result rests on six axioms, of which the most fragile is the convention-compatibility of the imported three-loop bilinear results from [27-29].

free parameters (2)
  • Correlation between the two RBC/UKQCD 24 SMOM projections = 33.9%
    Inferred in Section 5.5 from chi^2 = 0.671 following the Schmelling procedure, rather than from a full correlated error budget; it sets the error of the RBC/UKQCD 24 sub-average in Eq. (5.8).
  • MS scale-variation range for higher-order uncertainty = nu in [2, 6] GeV
    Hand-chosen range in Section 5.3 to estimate missing N3LO corrections; it directly sets the scale-variation errors quoted in Table 5 and Figures 9-10.
assumptions (6)
  • domain assumption NDR with anti-commuting gamma5 and the GJK evanescent basis is a valid scheme; the listed evanescent structures are complete at two loops.
    Sections 2.2 and 3.4. The difference to the BG basis used for the Wilson coefficients is removed explicitly in Eq. (4.6), but the completeness of the twelve evanescent structures (Eqs. 3.17-3.24) is assumed.
  • domain assumption The two-loop MS renormalization constants extracted from the anomalous dimensions of [6,7] are correct.
    Eq. (E.5) uses Eq. (5.2) of Buras-Weisz [6], and Appendix C derives the ADM finiteness conditions; these external ADMs are load-bearing for the pole cancellation of the two-loop amplitude.
  • domain assumption The NNLO bilinear conversion factors of [27,28,29] are correct and mutually consistent when adapted with CA=Nc, TF=1/2, w=1, r=1.
    Section 5.1. These three-loop results supply sigma(MS,s) at NNLO and enter every conversion factor through Eqs. (2.17)-(2.18); this is the weakest assumption in the paper.
  • domain assumption The RG evolution and charm-threshold matching of Brod-Gorbahn [7] are correct.
    Eqs. (5.2)-(5.7) are taken from [7]; they perform the MS-to-RGI conversion and the 3-to-4 flavor matching, and every world average in Section 5.5 depends on them.
  • domain assumption The lattice bag parameters and their stated errors are reliable inputs.
    Section 5.5. The world average treats the five lattice results as faithful measurements with the published uncertainties and neglects correlations among different collaborations.
  • standard math The RI-(S)MOM to MS conversion factors are gauge-independent, so evaluating at Landau gauge is sufficient.
    Section 5.1 states all results are at Landau gauge with Nc = 3; gauge independence of physical scheme-conversion factors is a standard theorem but is not demonstrated in the paper.

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Pith. "Pith review of RI-(S)MOM to $\overline{\rm MS}$ conversion for $B_K$ at two-loop order." pith.science (2026). https://pith.science/paper/L5S4M5DC

@misc{pith2026241119861,
  author       = {Pith},
  title        = {Pith review of: RI-(S)MOM to $\overline\rm MS$ conversion for $B_K$ at two-loop order},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L5S4M5DC}},
  note         = {Machine review of arXiv:2411.19861}
}
abstract

The Kaon bag parameter $ {\hat{B}}_K $ plays a critical role in constraining the parameters of the CKM matrix and in probing physics beyond the Standard Model. In this work, we improve the precision of $ \hat{B}_K $ to next-to-next-to-leading order (NNLO) and provide world averages for both $3$- and $4$-flavour theories. In the course of this, as our main technical development, we carry out the two-loop matching between the RI-(S)MOM and $\overline{\mathrm{MS}}$ schemes. Our world averages combine all available lattice data, including conversion between the 3- and 4-flavour theories as appropriate. We obtain the result $\hat B_{K}^{(f=3)} = 0.7627(60)$, which comprises the complete set of $3$- and $4$-flavour lattice results and can be used directly in phenomenological applications. The error is dominated by lattice uncertainties and missing higher-order corrections (residual scale dependence). Our averages include a PDG rescaling factor of 1.28 reflecting a mild tension among the lattice inputs after inclusion of NNLO corrections in the scheme conversion and matching across flavour thresholds. Our averages imply an updated value $|\epsilon_K|=2.171(65)_\text{pert.}(71)_\text{non-pert.}(153)_\text{param.} \times 10^{-3}$. We briefly discuss applications of our results to $D$-meson mixing.

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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Reviewed August 12, 2026 · model on record in the stance chip above.