Two-Loop QCD Anomalous Dimensions of Flavour-Changing Four-Quark Operators Within and Beyond the Standard Model

Andrzej J. Buras; Joerg Urban; Mikolaj Misiak

REVIEW

Two-Loop QCD Anomalous Dimensions of Flavour-Changing Four-Quark Operators Within and Beyond the Standard Model

Andrzej J. Buras, Mikolaj Misiak, Joerg Urban

High Energy Physics - Phenomenology · Submitted 18 May 2000 · 23 references

Pith's verdict: The SLL two-loop ADMs are the real deliverable and they hold up; the penguin block needed a 2024 fix, and the section 6 cross-check is less independent than it reads.

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arXiv IDhep-ph/0005183 (version 2)

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CitationAndrzej J. Buras, Joerg Urban, Mikolaj Misiak. "Two-Loop QCD Anomalous Dimensions of Flavour-Changing Four-Quark Operators Within and Beyond the Standard Model." arXiv:hep-ph/0005183 (2000). https://pith.science/paper/IA7Z3WEU

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Editor's note

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The thing to take away is eqs. (2.20) and (3.9): the two-loop NDR-MSbar SLL anomalous dimensions for Delta F=2 and Delta F=1. That was the last missing block in the SM four-quark ADM, and these have become the standard reference values. The VLL, VLR and SLR blocks are explicitly flagged as reproductions of prior work; the new content is SLL, the beyond-SM penguin mixing in (4.8)-(4.9), and the MS->RI transformation matrices in section 5. The paper is unusually explicit about which is which, which is not universal in this literature and is worth crediting. The scaffolding is also real: two independent diagrammatic methods (external momentum versus an IR mass regulator), gauge-parameter independence, and the section 6 re-derivation of the Delta F=2 ADM from the Delta F=1 one through the plus-minus basis. On the soft spots, in proportion. The stress-test is right that section 6 is a consistency check, not an independent computation: both routes share the appendix A/B evanescent conventions, and the paper says as much itself when it notes the appendix B/C operators combine into the appendix A ones. A correlated finite-subtraction error would pass. What section 6 does verify is the transformation machinery (W-hat, c-hat, eq. 6.10) and the Fierz-symmetry structure of the VLL/VLR/SLR blocks, which is worth something but is not a second calculation of the SLL entries. The genuinely external check is the LRI recovery in 7.3, and that runs through the one-loop Delta r-hat of section 5, so a one-loop error there could in principle mask a two-loop error in gamma^(1). Unlikely, but that is the logic. The 2024 correction of eq. (4.9) matters here: Appendix F states plainly that the transformation (F.14) was missed in the 2000 version and was caught by Morell and Virto. That is exactly a missed finite subtraction, in the one block with no independent cross-check. It does not touch the SLL numbers, but it is a real signal about the penguin sector, and the corrected version is the one to cite. On Ciuchini et al., the private communication is unverifiable from the published record, but the structural argument in 7.3 stands on its own: after c->d and ubar->sbar, Q-_i vanish identically in D dimensions, so they cannot mix into Q+_i. That is the stronger half of the section. This is for flavour practitioners doing NLO matching for K and B mixing, MSSM flavour constraints, and lattice RI matching. Not a general-interest paper. Send it to referees; the SLL results are load-bearing and cross-checked enough to trust.

Referee report

Significance

If the SLL entries hold, Eqs. (2.20) and (3.9) become the standard reference values for MSSM and generic beyond-SM analyses of neutral meson mixing, and the Delta r-hat matrices of Section 5 are directly usable by lattice groups matching NDR-MSbar Wilson coefficients to RI/MOM renormalized matrix elements. The paper earns credit for several things that are not universal in this literature: the calculation is performed by two independent infrared treatments (external momentum versus a common mass regulator), gauge-parameter independence is checked, the evanescent operators that define the scheme are written out explicitly in Appendices A-C so the scheme is reproducible rather than implicit, and the 2024 revision is transparent about a real error, confines it to Eq. (4.9), and supplies Appendix F deriving the corrected result. The explicit statement in Section 7.3 that the Ciuchini et al. NDR-MSbar discrepancy is a basis-definition issue rather than a physics disagreement is also a service to the community.

Major objections (2)

[Section 6, Eqs. (6.1)-(6.13)] The Delta F=2 ADM is re-derived from the Delta F=1 ADM as a consistency check, but the two routes are not independent with respect to the evanescent-operator scheme they are meant to validate. The transformation uses W-hat (6.2) and c-hat (6.12), and the text itself notes that the Appendix B/C evanescent operators 'can be linearly combined to the ones that are either symmetric or antisymmetric under d <-> c interchange' and reduce to those of Appendix A. A correlated finite-subtraction error in the Fierz-vanishing operators E_SLL1, E_SLL2 would therefore pass this check. The manuscript should say so explicitly and identify the LRI comparison of Section 7.3, via Eq. (7.5), as the genuinely external numerical check on the SLL entries. A concrete test that would strengthen the section: verify that the Delta F=2 result is unchanged when the evanescent basis of Appendix A is replaced by an equivalent basis differing by terms of O(eps) in the subtraction coefficients, and report the size of the residual.
[Appendix F and Eq. (4.9)] Appendix F documents that the transformation (F.14), a finite subtraction of exactly the class that can shift two-loop entries, was missed in the original version and that four entries of Eq. (4.9) were consequently wrong. The correction is confined to the penguin matrix, and the paper states that the corrected values now agree with Ref. [23]. Please state explicitly, in Appendix F or in Section 8, whether the same class of missed finite subtraction was searched for in the SLL entries of Eqs. (2.20) and (3.9), and if it cannot affect them, why not. Given that these two equations are the paper's headline results, an explicit statement that they were re-checked against this failure mode, rather than an implicit one, is warranted.

Minor objections (6)

[Footnote 3 and Eq. (4.9)] Footnote 3 refers to 'the four integers marked in red in Eq. (4.9)', but colour marking does not survive all renderings of the article. Please list the four corrected entries explicitly by row and column so that readers comparing against the v1 arXiv version can identify them without relying on typography.
[Section 7.3, Ref. [23]] Ref. [23] is cited as the independent confirmation of the corrected Eq. (4.9), but it is an arXiv preprint. Please state its status, or note that the agreement is with a preprint, so that the evidential weight of that confirmation is clear to the reader.
[Section 7.3] The resolution of the disagreement with Ref. [4] rests on a private communication about which operator basis their NDR-MSbar conditions were actually imposed in. This is a reasonable resolution, but the operational consequence for users of Ref. [4]'s published numbers should be stated in one sentence: which of their tabulated NDR-MSbar entries should be used with which basis, and which should be replaced by the values given here.
[Eqs. (2.20), (3.9), (4.9)] The main products of the paper are long matrices of rational coefficients that other groups will transcribe by hand. Please consider supplying them in a machine-readable form (supplementary file or plain-text table) alongside the typeset equations; this materially reduces the risk of transcription errors propagating into phenomenological and lattice analyses.
[Section 8, conclusions] The conclusions contain the run-together word 'correctlycalculated'. A careful proofread of the final version is advisable, since this is the version that will be cited.
[Section 4, Eqs. (4.2)-(4.7)] The argument that the basis (4.2) suffices even at two loops, with Q'11 - Q11 carried as an evanescent operator, is correct but compressed. A sentence pointing forward to Appendix F at the point where Q'11 is introduced would help the reader see that the subtlety is resolved rather than deferred.

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The paper

The paper claims

The paper computes the full two-loop QCD anomalous dimension matrix for every flavour-changing four-quark operator of dimension six, in one fixed scheme, and uses it to explain a published disagreement.

The reading

Weak decays of strange and bottom mesons are governed by four-quark operators whose coefficients are run down from the electroweak scale by QCD. To reach the precision now demanded by K and B mixing phenomenology, that running has to be done at two loops, which means knowing the two-loop anomalous dimension matrix (the matrix that tells how each operator mixes into the others under renormalization). This paper supplies that matrix in the NDR-$overline{MS}$ scheme for all dimension-six flavour-changing four-quark operators, not just the handful that appear in the Standard Model, so that supersymmetric and other extensions can be analysed at the same accuracy. The genuinely new entries are the two-loop results for the scalar/pseudoscalar sector (the operators built from $P_L$ rather than $gamma_mu P_L$) and the penguin-induced mixing of operators that exist only beyond the Standard Model. The paper also diagnoses why an earlier published two-loop result differed: the two calculations were implicitly using different definitions of the same operators in $D neq 4$ dimensions, a difference that is invisible in four dimensions but shifts the two-loop answer. A 2024 note appended to the manuscript records that four entries of the penguin matrix were wrong in the original version and were corrected after an external check.

What carries the argument

The load-bearing device is the systematic treatment of evanescent operators. Because the Dirac algebra in $D neq 4$ dimensions does not preserve the four-dimensional Fierz identities, every Fierz rearrangement used to identify or reduce operators must be accompanied by an explicit evanescent operator that carries the difference, and that operator must be included as a counterterm. The paper writes these down for the $Delta F = 2$ case (appendix A) and the $Delta F = 1$ case (appendices B and C), and then uses the standard relation $hat{gamma}^{(1)} = 4hat{a}_{12} - 2hat{b}hat{c}$ to convert one- and two-loop counterterms into the anomalous dimension matrix, where $hat{c}$ is read off from the finite parts of one-loop matrix elements of the evanescent operators. The second device is the transformation law for the matrix under a change of scheme induced by adding evanescent operators to the physical ones, $hat{gamma}^{(1)} to hat{R}{hat{gamma}^{(1)} + [Deltahat{r}, hat{gamma}^{(0)}] + 2beta_0 Deltahat{r}}hat{R}^{-1}$, with $Deltahat{r} = -hat{W}hat{c}$. Setting $Deltahat{r} = 0$ is exactly the step that reproduces the earlier, discrepant result, which is how the paper localizes the source of the disagreement.

Extended reading notes

Core claim

The central claim is that the two-loop anomalous dimensions of the scalar/pseudoscalar ($SLL$) four-quark operators, given in eq. (2.20) for $Delta F = 2$ and eq. (3.9) for $Delta F = 1$, are correctly calculated here for the first time in the NDR-$overline{MS}$ scheme, together with the penguin mixing matrix of eqs. (4.8)-(4.9) for operators that appear only in extensions of the Standard Model. Everything else in the matrix is either reproduced from, or straightforwardly extracted from, earlier Standard Model calculations. The paper then shows that these new $Delta F = 2$ results can be re-derived from the $Delta F = 1$ results by passing to a plus/minus basis and setting the flavours equal, provided additional evanescent operators (operators that vanish in four dimensions by Fierz identities) are introduced along the way. That re-derivation is the consistency check, and it is also the tool used to explain the disagreement with the earlier literature: the earlier NDR-$overline{MS}$ numbers correspond to a different operator basis, one in which the Fierz rearrangement has been applied inside $D$ dimensions where it is not valid. The paper states that the earlier Regularization-Independent (LRI) results are reproduced exactly by transforming its own numbers with the conversion matrices listed at the end of section 5.

Load-bearing premise

The scalar/pseudoscalar results are only as good as the specific choice of Fierz-vanishing evanescent operators used to define the scheme, and the explanation of the earlier disagreement additionally rests on the other authors' account of which operator basis their code actually used, which an outside reader cannot check from the published text alone.

Editorial extensions

If this is right

Analyses of $K^0$-$bar{K}^0$ and $B^0$-$bar{B}^0$ mixing in supersymmetric and other beyond-Standard-Model frameworks can now be run at two-loop QCD accuracy for the full set of dimension-six four-quark operators, rather than only for the Standard Model ones.
The one-loop conversion matrices collected in section 5 let lattice groups translate non-perturbatively renormalized matrix elements obtained in an RI/MOM-type scheme into the $overline{MS}$ scheme for every operator in the basis, including the scalar/pseudoscalar ones.
The apparent discrepancy between the two published two-loop calculations is resolved without either side having made an arithmetic error: it is a basis-and-scheme mismatch traceable to the treatment of Fierz-vanishing evanescent operators.
The observation that the Fierz relations between the VLL, VLR and SLR sectors survive at two loops for current-current diagrams follows from the vanishing of the corresponding conversion matrix, which the paper traces to the vanishing of one-loop matrix elements of specific evanescent operators.

Reading between the lines

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

The 2024 correction to four entries of the penguin matrix, flagged in the manuscript itself, suggests that the penguin sector is the numerically fragile part of any such calculation; an independent third-party recomputation of eq. (4.9) would be worth more than a recomputation of the current-current entries.
Because the Fierz symmetries of eq. (3.10) hold for current-current diagrams but are explicitly broken by penguins, the breaking looks structural rather than accidental, tied to whether the relevant conversion matrix vanishes; that would make it a useful diagnostic for future higher-order calculations.
If the corrected penguin entries feed into squark-mass-insertion bounds from $K$-$bar{K}$ mixing, the shifts are likely small but systematically directional, and would be worth propagating through existing beyond-Standard-Model fits rather than treated as a negligible bookkeeping change.
The same evanescent-operator bookkeeping will be needed for any future three-loop extension, since the Fierz problem does not disappear at higher order and the conversion matrices given here would become the input to that calculation.

Original abstract

We calculate the two-loop QCD anomalous dimension matrix (ADM) gamma^(1)_NDR in the NDR-MSbar scheme for all the flavour-changing four-quark dimension-six operators that are relevant in both the Standard Model and its extensions. Both current-current and penguin diagrams are included. Some of our NDR-MSbar results for Delta F=1 operators overlap with the previous calculations, but several others have never been published before. In the case of Delta F=2 operators, our results are compatible with the ones obtained by Ciuchini et al. in the Regularization-Independent renormalization scheme, but differ from their NDR-MSbar results. In order to explain the difference, we calculate the ADM of Delta F=2 operators again, extracting it from the ADM of Delta F=1 operators.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,332 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.13 · states limitation 0.38 · human participants 0.01 · controlled comparison 0.08 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.98 · mentions code or data availability 0.02.

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arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We calculate the two-loop QCD anomalous dimension matrix (ADM) gamma^(1)_NDR in the NDR-MSbar scheme for all the flavour-changing four-quark dimension-six operators that are relevant in both the Standard Model and its extensions. Both current-current and penguin diagrams are included. Some of our NDR-MSbar results for Delta F=1 operators overlap with the previous calculations, but several others have never been published before. In the case of Delta F=2 operators, our results are compatible with the ones obtained by Ciuchini et al. in the Regularization-Independent renormalization scheme, but differ from their NDR-MSbar results. In order to explain the difference, we calculate the ADM of Delta F=2 operators again, extracting it from the ADM of Delta F=1 operators.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We calculate the two-loop QCD anomalous dimension matrix (ADM) gamma^(1)_NDR in the NDR-MSbar scheme for all the flavour-changing four-quark dimension-six operators that are relevant in both the Standard Model and its extensions.	contribution 1.00	0.97	0.40	asserted 1.00
s1	Both current-current and penguin diagrams are included.	method 0.85	0.79	0.03	asserted 1.00
s2	Some of our NDR-MSbar results for Delta F=1 operators overlap with the previous calculations, but several others have never been published before.	gap 0.50	0.89	0.10	asserted 0.99
s3	In the case of Delta F=2 operators, our results are compatible with the ones obtained by Ciuchini et al. in the Regularization-Independent renormalization scheme, but differ from their NDR-MSbar results.	result 0.97	0.84	0.15	asserted 0.92
s4	In order to explain the difference, we calculate the ADM of Delta F=2 operators again, extracting it from the ADM of Delta F=1 operators.	method 0.88	0.97	0.09	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.87.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
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relations to other papers

levels run

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** One-loop diagrams consistently at the tree level, at one loop and (later) at two loops does not alter the final results for the renormalization constants. All the one- and two-loop diagrams considered in the present article are calculated using two different methods. In both of them, a covariant gauge-fixing term Lgf = − 1 2λ (∂ µG a µ )(∂ νG a ν ) (2.17) is used, and the physical masses are set to zero. In the first method, the external quarks are assumed to have momentum ±p. In the second method, the external momenta are set to zero, but a common mass parameter is introduced in…

Figures reproduced from arXiv:hep-ph/0005183 by the authors.

References

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Lepton Flavor Violation in Exclusive $b\to s$ Decays
hep-ph 2016-02 accept novelty 6.0 of 10

The paper derives complete angular distributions for lepton-flavor-violating b→sℓ1ℓ2 decays and bounds their branching fractions between 10^-8 and 10^-13 in Higgs-coupling and Z′ scenarios.
Matching of gauge invariant dimension 6 operators for $b\to s$ and $b\to c$ transitions
hep-ph 2015-12 accept novelty 6.0 of 10

The paper supplies the matching dictionary from SM gauge-invariant dimension-six operators to the b to s and b to c B physics Hamiltonian, covering tree level and leading one-loop contributions.
Addressing the LHC flavour anomalies with horizontal gauge symmetries
hep-ph 2015-03 conditional novelty 6.0 of 10

A single U(1)' horizontal gauge symmetry with charges (L_mu - L_tau) - a(B1 + B2 - 2B3) can simultaneously account for the B to K* mu mu, R(K), and h to mu tau anomalies, predicting tau to 3mu at a rate within reach o...
Strong P invariance, neutron EDM and minimal Left-Right parity at LHC
hep-ph 2014-07 conditional novelty 6.0 of 10

Strict parity in the minimal Left-Right model, combined with the neutron EDM and kaon CP violation, requires the right-handed W boson to be heavier than 20 TeV unless an axion or explicit parity breaking is added.
Renormalization Group Evolution of the Standard Model Dimension Six Operators III: Gauge Coupling Dependence and Phenomenology
hep-ph 2013-12 accept novelty 6.0 of 10

The authors compute the gauge-coupling terms of the one-loop anomalous dimension matrix for the 59 dimension-six SMEFT operators, completing the full 2499 x 2499 matrix.
Constraints on CP-violating Higgs couplings to the third generation
hep-ph 2013-10 accept novelty 6.0 of 10

Existing electric dipole moment data bound the CP-violating Higgs-top coupling to about 0.01 and the bottom and tau couplings to order one, with future experiments projected to reach 10^-4 for the top.
Low Energy Probes of PeV Scale Sfermions
hep-ph 2013-08 accept novelty 6.0 of 10

In mini-split supersymmetry, kaon mixing already probes PeV-scale squarks, and planned neutron-EDM, muon-to-electron, and charm-mixing measurements will reach 100 to 1000 TeV sfermion masses.
B-physics from Nf=2 tmQCD: the Standard Model and beyond
hep-lat 2013-08 accept novelty 6.0 of 10

Two-flavour lattice QCD gives m_b = 4.29(12) GeV, f_Bs = 228(8) MeV, f_B = 189(8) MeV, and the first unquenched determination of the full basis of B-mixing bag parameters, tightening new-physics bounds from B mixing.
Kaon Mixing Beyond the SM from Nf=2 tmQCD and model independent constraints from the UTA
hep-lat 2012-07 accept novelty 6.0 of 10

Lattice QCD with two dynamical quark flavours, extrapolated to the continuum, gives 3 to 7 percent determinations of all five Delta S = 2 kaon bag parameters, which push model-independent new-physics scale bounds as h...
The Impact of Flavour Changing Neutral Gauge Bosons on B->X_s gamma
hep-ph 2011-05 accept novelty 6.0 of 10

Two new contributions to B to X_s gamma from flavour-changing neutral gauge bosons are computed, including the first calculation of QCD mixing of neutral current-current operators into the dipole operator, yielding mo...
Continuum Limit of $B_K$ from 2+1 Flavor Domain Wall QCD
hep-lat 2010-12 accept novelty 6.0 of 10

Lattice QCD with 2+1 flavors of domain wall fermions at two lattice spacings yields the continuum-limit kaon bag parameter B_K-hat = 0.749(27), a 3.6 percent determination, using new non-exceptional RI-SMOM renormaliz...
Delta F=2 Observables and Fine-Tuning in a Warped Extra Dimension with Custodial Protection
hep-ph 2008-09 conditional novelty 6.0 of 10

In warped extra dimensions with custodial protection, kaon mixing forces a Kaluza-Klein scale near 20 TeV generically, but tuned corners of parameter space allow 3 TeV, where heavy Z_H and Z' exchanges rival Kaluza-Kl...
Exploring $Z/\gamma$-mediated heavy FCNCs at the FCC-ee
hep-ph 2026-08 conditional novelty 5.0 of 10

FCC-ee sensitivity projections are presented for six Z/gamma-mediated third-generation flavor-violating channels with dipole and Higgs-current SMEFT operators, together with a comparison to low-energy flavor bounds.
Flavor Constraints in a Generational Three Higgs Doublet Model
hep-ph 2025-02 conditional novelty 5.0 of 10

A generational three-Higgs-doublet model is constructed, and flavor constraints show that its additional Higgs bosons can be as light as about 1.5 TeV if the only flavor violation is the minimal amount needed to repro...
Unified Framework for $B$-Anomalies, Muon $g-2$, and Neutrino Masses
hep-ph 2020-09 conditional novelty 5.0 of 10

Two TeV-scale scalar leptoquarks plus a quadruplet Higgs can simultaneously fit the R_D(*), R_K(*), and muon g-2 anomalies and generate neutrino masses at one loop.
Explaining $b\to s\ell^+\ell^-$ and the Cabibbo Angle Anomaly with a Vector Triplet
hep-ph 2020-05 conditional novelty 5.0 of 10

A single SU(2)_L vector triplet with order-one couplings at 10 TeV can fit both the b to s lepton-pair anomalies and the Cabibbo Angle Anomaly, improving the global fit over the Standard Model.
The CKM parameters in the SMEFT
hep-ph 2018-12 conditional novelty 5.0 of 10

The paper defines tilde Wolfenstein parameters that absorb dimension-six SMEFT contributions to four chosen flavour observables, yielding numerical values with correlations and a consistent recipe for interpreting oth...
$B$ Meson Anomalies in a Pati-Salam Model within the Randall-Sundrum Background
hep-ph 2018-01 conditional novelty 5.0 of 10

A Pati-Salam model in a Randall-Sundrum background gives a 3 TeV vector leptoquark that fits the b to s mu mu anomalies, but D-Dbar mixing and tau to 3 mu cap the b to c tau nu effect below the measured R(D) and R(D*).
One constraint to kill them all?
hep-ph 2017-12 conditional novelty 5.0 of 10

Updated lattice inputs raise the Standard Model B_s mixing prediction to 20.0 ps^-1 against a measured 17.76 ps^-1, tightening Z-prime and leptoquark mass bounds by roughly a factor of five.
Flavoured $B-L$ Local Symmetry and Anomalous Rare $B$ Decays
hep-ph 2017-05 conditional novelty 5.0 of 10

A flavoured B-L gauge boson coupling only to the third generation can explain the LHCb lepton-universality anomaly while remaining perturbative to the Planck scale and consistent with leptogenesis.
Anomaly-free local horizontal symmetry and anomaly-full rare B-decays
hep-ph 2017-04 conditional novelty 5.0 of 10

A new gauge symmetry from a Pati-Salam unification framework can accommodate the reported lepton universality violation in rare B decays while staying within reach of LHC searches.
\Delta S=2 and \Delta C=2 bag parameters in the SM and beyond from Nf=2+1+1 twisted-mass LQCD
hep-lat 2015-05 accept novelty 5.0 of 10

Using Nf=2+1+1 twisted-mass lattice QCD, the ETM Collaboration determines the kaon bag parameter B_K^RGI = 0.717(24) and the full set of BSM bag parameters for K and D mixing.
Lepton-flavour violating $B$ decays in generic $Z^\prime$ models
hep-ph 2015-04 conditional novelty 5.0 of 10

In generic Z-prime models, existing B_s mixing, tau to 3 mu, tau to mu nu nu and B to K nu nu data bound lepton-flavour-violating B decay branching ratios at roughly 10^-6 for tau-mu final states and 10^-7 for mu-e, c...
D-meson lifetimes within the heavy quark expansion
hep-ph 2013-05 conditional novelty 5.0 of 10

Including NLO QCD and 1/m_c^4 corrections, the heavy quark expansion reproduces the D+/D0 and Ds+/D0 lifetime ratios within large hadronic uncertainties, with no sign of breakdown at the charm scale.
Flavor-phenomenology of two-Higgs-doublet models with generic Yukawa structure
hep-ph 2013-03 conditional novelty 5.0 of 10

Scanning a type-III two-Higgs-doublet model against tree-level and loop flavor observables leaves only the up-type couplings epsilon^u_31 and epsilon^u_32 large, and those two suffice to fit the B to tau nu and B to D...
SU(3)-Flavor Anatomy of Non-Leptonic Charm Decays
hep-ph 2012-11 conditional novelty 5.0 of 10

SU(3)-flavor fits to charm two-body decay data accommodate roughly 30% breaking but require strongly enhanced penguin amplitudes to explain the observed CP asymmetries.
New Physics Models of Direct CP Violation in Charm Decays
hep-ph 2012-02 accept novelty 5.0 of 10

A systematic survey finds that only a handful of minimal new-physics models, chiefly those with chirally enhanced chromomagnetic penguins, can explain the LHCb charm CP asymmetry without violating D-meson mixing and c...
Delta F = 2 observables and B -> X_q gamma decays in the Left-Right Model: Higgs particles striking back
hep-ph 2011-11 conditional novelty 5.0 of 10

In the left-right model, tree-level heavy neutral Higgs exchange dominates Delta F = 2 observables even for Higgs masses near 20 TeV, and viable parameter regions exist for W_R masses of 2-3 TeV.
Anatomy and Phenomenology of FCNC and CPV Effects in SUSY Theories
hep-ph 2009-09 accept novelty 5.0 of 10

Correlation patterns among B, K, D and lepton observables separate supersymmetric flavour models with right-handed currents from those with left-handed currents, and both from Little-Higgs and Randall-Sundrum alternatives.

How later papers used it

109 later papers cite it

Evidence: substantial. 90 of the citing papers share no author with it, and 36 of those carry a confident label from Pith's citation classifier.

Pith read 21 of the 296 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

4 later papers by other authors are listed below: 4 used its method or tool. 1 more paper by the same authors is listed: 1 used its method or tool.

Uses its method or tool 5 of 32 papers with this label · 1 by the same authors

Used the rules for the scheme of this paper, as collected in an appendix of another work.

Toolkit for staggered $\Delta S=2$ matrix elements (2014) · 1.00 · paraphrased: the licence does not allow quoting
Extended the continuum basis to d=4-2ϵ dimensions using the same definition of evanescent operators as in this paper.

Kaon BSM B-parameters using improved staggered fermions from $N_f=2+1$ unquenched QCD (2015) · 0.96 · paraphrased: the licence does not allow quoting
Used perturbation theory to convert the operators into the NDR schemes with the treatment of evanescent operators as in this paper.

Continuum Limit of $B_K$ from 2+1 Flavor Domain Wall QCD (2010) · 1.00 · paraphrased: the licence does not allow quoting
Used this paper's one-loop and two-loop RGEs for the Wilson coefficients in question.

SMEFT interpretation of $\Delta$F = 2 transitions (2022) · 1.00 · paraphrased: the licence does not allow quoting

By the same authors

Treated evanescent operators appearing in the one-loop matching in the same manner as done in this paper.

SMEFT ATLAS of $\Delta F=2$ Transitions (2020) (same authors) · 0.85 · paraphrased: the licence does not allow quoting

How the literature uses it

109 arXiv papers cite this in 299 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 196 sentences without a confident label

Contests it 1 paper

However, in the Bern basis, the one-loop corrections to the current-current diagrams generate Dirac structures that cannot be reduced to physical and evanescent operators with the operators given in Ref. [ref] .

Towards a Global Analysis of the $b\to c\bar{u} q$ Puzzle · Evanescent operators · contest 0.73

Builds on it 4 papers

The purpose of this section is twofold: First, we present the transformation of the results of Ref. [ref] to the $(Q_1^SLL,Q_1^SLL)$ basis and the scheme corresponding to the evanescent operators of Eq. ([ref]), for which lattice groups quote their results.

The supersymmetric Higgs sector and B-Bbar mixing for large tan beta · Renormalization group and bag parameters · 0.82
In our paper [ref] we have therefore generalized our analysis of two-loop anomalous dimensions to the full set of $ΔF=1$ four-quark operators.

Climbing NLO and NNLO Summits of Weak Decays: 1988-2023 · $ΔF=1$ Non-leptonic Penguin Operators · 0.83
In the CBSM scenario, with our choice of operator basis and renormalization, including in particular anticommuting $γ_5$, the vectors $y⃗$, $z⃗$ and $C⃗_Ap$ read [equation] extending the SM case [ref] .

Charming New $B$-Physics · Renormalization-group evolution · 1.00

Uses its method or tool 34 papers

Using two independent private implementations of the algorithmic procedure to extract UV poles as detailed in [ref] and the classification of relevant current-current Feynman diagrams in [ref] , all one-loop and two-loop divergences are computed for the general operator in <ref>.

Universal Two-Loop Current--Current Renormalization of Four-Fermion Operators · Methodology · 1.00
The back-rotation [ref] is taken into account automatically at $$ when using wilson. * In the next step the complete LO [ref] and NLO QCD [ref] running of the LEFT Wilson coefficients down to lower scales $$ is taken into account following sec:LEFT-master. * Finally, at the low scale $$ the LEFT Wilson coefficients are combined with the hadronic matrix elements given in tab:DF2-me-input to calculate $[M_12^ij ]_BSM$.

SMEFT ATLAS of $\Delta F=2$ Transitions · Derivation of the SMEFT Master Formula · 0.95
The Wilson coefficients are evolved to the low scale $μ_low$ by solving their renormalization group equations [ref] .

A Supersymmetric Flavor Clockwork · Constraints from Meson Mixing · 0.98

Uses its data or values 9 papers

For $γ^1$ in $MS$ we used Ref. [ref] and for RI-MOM we used Ref. [ref] (that paper does not give the $11$ element; for this we used the value in Ref. [ref] ).

Kaon mixing beyond the standard model with physical masses · NLO scaling · 0.98
The corresponding LO and NLO ADMs in QCD are given by the $=2$ result in Eq. (2.21) of Ref. [ref] [equation]

General non-leptonic $\Delta F=1$ WET at the NLO in QCD · Anomalous dimensions · 0.93
In performing the NLO QCD evolutions we have used the $P_a^ij()$ from [ref] , see [ref], which are based on the two-loop anomalous dimensions of operators calculated in [ref] .

SMEFT ATLAS of $\Delta F=2$ Transitions · The Issue of Evanescent Operators · 0.83

Compares with it 10 papers

Our one-loop results for the physical sector agree with the results in the literature [ref] , after accounting for the different normalization of the operators and the different conventions in the covariant derivative.

Electric dipole moment constraints on CP-violating heavy-quark Yukawas at next-to-leading order · One-loop anomalous dimensions · 1.00
The result agrees with (2.15) and (2.22) of Reference [ref] (recall again that we are using $D=4+2ϵ$ and the authors of [ref] are using $D=4-2ϵ$). * The total logarithmic term is [equation] which agrees with the known anomalous dimension.

Continuum Limit of $B_K$ from 2+1 Flavor Domain Wall QCD · The Conversion Factor · 1.00
On the other hand, the corresponding evanescent operators in the BMU basis [ref] are defined with the explicit choice $a_ev = b_ev = c_ev = 1$, which coincides with the so-called Greek projection method [ref] .

General non-leptonic $\Delta F=1$ WET at the NLO in QCD · Basis change JMS$→$BMU · 0.81

Cites it as background 18 papers

These anomalous dimensions were presented in complete form for the first time in the highly relevant paper by Buras, Misiak and Urban (BMU) in the year 2000 [ref] .

On the two-loop penguin contributions to the Anomalous Dimensions of four-quark operators · Summary · 0.83
The BMU basis [ref] contains several sub-sectors of operators, depending on their quark-flavour content and Lorentz structures.

General non-leptonic $\Delta F=1$ WET at the NLO in QCD · BMU basis for $= 5$ · 0.84
* The one-loop ADMs relevant for the RG in WET [ref] and SMEFT [ref] are also known. * The two-loop QCD ADMs relevant for RG evolutions for both $ΔF=1$ and $ΔF=2$ transitions in WET are also known [ref] .

NLO QCD Renormalization Group Evolution for Non-Leptonic $\Delta F=2$ Transitions in the SMEFT · Introduction · 0.90

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