Master Formulae for Delta F=2 NLO-QCD Factors in the Standard Model and Beyond

Andrzej J. Buras; Joerg Urban; Sebastian Jager

REVIEW

Master Formulae for Delta F=2 NLO-QCD Factors in the Standard Model and Beyond

Andrzej J. Buras, Sebastian Jager, Joerg Urban

High Energy Physics - Phenomenology · Submitted 26 Feb 2001 · 15 references

Pith's verdict: A solid standard-reference paper: complete analytic NLO ΔF=2 evolution matrices in NDR, with one flagged convention in the ρ-sector that deserves a caution but not alarm.

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CitationAndrzej J. Buras, Joerg Urban, Sebastian Jager. "Master Formulae for Delta F=2 NLO-QCD Factors in the Standard Model and Beyond." arXiv:hep-ph/0102316 (2001). https://pith.science/paper/VNBWGHR6

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Editor's note

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This is the paper people cite when they need NLO QCD running for ΔF=2 operators outside the VLL sector. It delivers what it promises: closed-form NLO evolution matrices valid for arbitrary μ (Section 3), the ρ-factors for running lattice matrix elements from μ_K up to 2 GeV (Section 4), NDR B-parameters converted from LRI lattice input (7.18), and master formulas for ΔM_K, ΔM_B and ε_K in terms of B_i and C_i(μ_t) (7.27–7.32). Prior work had LO analytic expressions and NLO evolution evaluated numerically at fixed scales; the arbitrary-μ closed forms and the ρ-sector are the genuinely new pieces, and they are what made this a working tool for constraining squark mass matrices and other new physics in mixing. The LO structure checks out by hand: 6/23 in (3.1), 3/23 and −24/23 in (3.3) and (3.6), and the 2/3 in (3.5) all follow from the one-loop anomalous dimensions with β_0 = 23/3. The paper is also careful about a real trap — stripping spurious O(α_s^2) terms when evolution matrices are multiplied across quark thresholds (Appendix B). That is exactly the kind of bookkeeping that quietly produces 5% discrepancies between groups, and they handle it explicitly. The soft spot is the log(η_3) term in (4.10). Ĵ is singular at f=3 in the LR sector, and the finite result comes from taking f→3 and classifying the log as NLO under the convention (B.6)–(B.7). It is flagged in the text and cited to [13], so it is supported rather than smuggled in. But the intermediate steps of the limit are not shown, so a reader cannot check whether all NLO pieces, including possible log^2 or non-commuting contributions, are captured. That is a real but narrow concern, and it lands on the ρ-sector and Table 3, not on the η(μ_L) master formula; the stress-test note is right to correct that framing. Two smaller deductions: (7.18) quotes NDR B-parameters without propagated errors even though the LRI inputs in (7.16) carry 5–15% uncertainties, and the one-loop scheme conversion carries unquantified higher-order error, which is standard practice but worth a sentence. This is for flavor practitioners who need the numbers without redoing the running, and for experts checking a calculation. It is a reference, not a read-through. Accept and send to referees; I would cite it.

Referee report

Significance

This is a genuinely useful reference paper: it converts a body of published two-loop anomalous-dimension work into explicit, checkable formulae that any practitioner can evaluate for arbitrary input parameters, and it makes the truncation prescription explicit rather than leaving it to individual implementations. The master formulae (7.27)–(7.32) with universal P^a_i are a clean separation of short-distance new physics (in C^a_i(μ_t)) from the QCD evolution, and the observation that P^LR_i and P^SLL_i exceed P^VLL_1 by two orders of magnitude in the K system is a substantive physics point that motivates the whole LR/SLL programme. The scheme-conversion relations (7.8)–(7.15) and the NDR values (7.18) are a further service to the lattice community. I checked the LO structure independently and it is internally consistent: the VLL exponent 6/23 in (3.1) equals γ^(0)/(2β_0) = 4/(46/3); the LR exponents 3/23 and −24/23 in (3.3) and (3.6) follow from γ^(0)_11 = 2 and γ^(0)_22 = −16; and the 2/3 prefactor in (3.5) follows correctly from the transposed off-diagonal entry γ^(0)_12 = 12 combined with the eigenvalue spacing 27/23.

Major objections (2)

[§3.3 and Eq. (3.39), Table 3] Section 3.3 declines to present the analytic η(μ_K) factors and instead refers the reader to η̂(μ_K) = ρ̂(μ_K)η̂(μ_L) in (3.39), with the instruction that 'terms of O(α_s^2) should be removed'. Table 3 then reports numerical values for these factors. Because the product involves four distinct coupling ratios (η_3, η_4, η_5 and tilde-η_4) and both ρ̂^(0)η̂^(1) and ρ̂^(1)η̂^(0) contribute at the retained order, the truncation is not unambiguous from the text alone: the reader cannot tell whether, for example, the log(η_3) term of (4.10) is to be kept when it multiplies a leading-order factor, or whether factors of η_4 and η_5 appearing inside the NLO brackets are to be expanded. Since Table 3 is one of the paper's two headline deliverables and the analytic route to it is deliberately withheld, the paper should state the truncation rule explicitly (which combinations of α_s(μ_K), α_s(μ_c), α_s(μ_L), α_s(μ_b) and α_s(μ_t) count as O(α_s) and which as O(α_s^2)) so that Table 3 can be reproduced independently. A short worked example for one entry, e.g. [η(μ_K)]^VLL, would settle the point.
[§4, Eq. (4.10) and Eqs. (B.4)–(B.7)] The NLO coefficient [ρ^(1)_21(μ_K)]^LR in (4.10) contains a term −0.7407 log(η_3), which the authors correctly flag: the matrix Ĵ is singular in the LR sector at f = 3, so the standard form (B.4) does not apply, and the finite result is obtained by taking the limit f → 3 and classifying the logarithm as NLO under the convention log(η) = O(α_s^0) of (B.7). The limit result is attributed to ref. [13], and I do not ask for that published derivation to be reproduced here. The residual question is completeness rather than provenance: the paper does not indicate whether the f → 3 limit can also generate log^2(η_3) or other structures that the single displayed logarithm would miss, and the numerical size of the term (it enters [ρ_21]^LR, the largest ρ entry in the LR sector, and feeds Table 4 and hence the μ_K-to-μ_L conversion) makes the question worth answering. A concrete and inexpensive test would be to compare the analytic [ρ̂(μ_K)]^LR against a direct numerical integration of the f = 3 RGE for the LR sector over the same range, or to state explicitly that the limit produces no higher powers of log(η_3). Either would close the point without expanding the paper.

Minor objections (7)

[§2, Eqs. (2.3)–(2.4) versus §3] The hatted notation [η̂^(0)(μ)]_a introduced in (2.3) is dropped without comment in Section 3, where the same objects appear as [η^(0)(μ_b)]^VLL and [η^(0)ij(μ_b)]^LR. A sentence at the start of Section 3 stating that hats are suppressed from here on would prevent confusion with the unhatted η_B and η_2 of Section 6.3.
[Eqs. (3.19), (3.20) and (4.1)] The coupling ratios η_3, η_4, η_5, tilde-η_4 and η_6 (the last in Appendix C) are defined at four different places and two of them (η_4 and tilde-η_4) differ only by the scales entering the ratio. A single collected definition table, with the flavour label and the two scales for each, would materially reduce the chance of misreading the formulae in Sections 3 and 4.
[§7.1, Eqs. (7.16)–(7.18) and (7.35)–(7.37)] The lattice B-parameters of (7.16) are quoted with errors, but the NDR values (7.18) and the resulting P^a_i in (7.35)–(7.37) are given without propagated uncertainty. Since the LR and SLL scheme shifts are themselves 30% and 20%, a brief statement of the error on P^LR_i and P^SLL_i, or an explicit note that the errors are dominated by the scheme conversion and by quenching, would help users judge how seriously to take the two-orders-of-magnitude hierarchy quoted below (7.37).
[§7.1, Eq. (7.16)] The B_i values are taken from the lattice work of ref. [10]; the text should state that these are quenched results and that the associated systematic is not included in the numbers of (7.18) and (7.35)–(7.37). This is a caveat on the illustrative numbers, not on the master formulae, but it belongs in the text where the numbers are presented.
[Table 3 caption] The caption should state that the η(μ_K) entries are obtained from (3.39) using the ρ-factors of Table 4 and the η(μ_L) factors of Table 2, since Section 3.3 does not give the corresponding analytic expressions. As written, a reader consulting Table 3 alone has no pointer to its provenance.
[§7.2, Eq. (7.27)] The text notes that CKM factors are absorbed into C_i(μ_t), but for the K system it would be helpful to state explicitly that the overall factor is (V*_ts V_td)^2 in the SM, so that (7.27) can be compared directly with (1.1) and with the B-system analogue. The same remark applies to the B_s system, for which no explicit formula is written out.
[Figs. 1 and 2] The figures are described in Section 5 as showing [η_ij]^LR and [η_ij]^SLL versus μ, but the plotted ranges of μ and the values of the individual entries at the endpoints are not stated in the caption. Given that the tables give only three fixed scales, the captions should record the μ range and identify which curve is which entry.

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The paper

The paper claims

The paper supplies closed-form next-to-leading-order QCD evolution matrices for all $Delta F=2$ four-quark operators, so that one universal set of factors carries any model's short-distance Wilson coefficients at the top-mass scale down to,

The reading

The paper sets out to remove a bottleneck in predicting how neutral kaons and neutral $B$ mesons mix. Any model of new physics hands you, at a high scale $mu_t = O(m_t)$, a list of short-distance coefficients $C_i(mu_t)$ for the $Delta F=2$ four-quark operators; experiment, however, lives at scales of 1 to 5 GeV, and the bridge between the two is QCD running. Buras, Jäger and Urban write that bridge down in closed analytic form at next-to-leading order in the strong coupling, for all the chirality sectors of the operator basis, so the same factors serve the Standard Model and every extension of it. They do the same for the separate bridge between lattice matrix elements quoted at 2 GeV and matrix elements at scales below 1 GeV, where hadronic models naturally live. The payoff is a set of master formulae for $Delta M_K$, $Delta M_B$ and $epsilon_K$ in which all model dependence sits in the $C_i(mu_t)$ and all the QCD sits in a handful of universal coefficients.

What carries the argument

The load-bearing object is the evolution matrix written as $hat U(mu_1,mu_2) = (1 + frac{alpha_s(mu_1)}{4pi}hat J) hat U^{(0)} (1 - frac{alpha_s(mu_2)}{4pi}hat J)$ with the leading-order solution $hat U^{(0)} = (alpha_s(mu_2)/alpha_s(mu_1))^{hatgamma^{(0)T}/(2beta_0)}$, built from the one-loop anomalous dimension matrix and the two-loop matrix of the NDR calculation, plus the $hat J$ matrix constructed from the two-loop anomalous dimensions and the $\beta$ function. Because the operators do not mix across chirality sectors, the problem splits into a scalar VLL entry and $2times 2$ matrices for LR and SLL, and crossing quark thresholds is handled by a product of such factors in the $f=5$, $f=4$ and $f=3$ effective theories, with the convention that powers and logarithms of the coupling ratios count as $O(alpha_s^0)$ so that all $O(alpha_s^2)$ terms are consistently dropped. The $rho$ matrices are the same construction run in the opposite direction, from 2 GeV down to $mu_K$, and the scheme-conversion matrices $Delta hat r$ let the results be translated to other renormalization schemes.

Extended reading notes

Core claim

The central claim is that the next-to-leading-order evolution matrices $hat U(mu, mu_t)$ and $hat U(mu_K, mu_L)$ for the $Delta F=2$ dimension-six four-quark operators can be given in explicit closed form in the NDR scheme, for the VLL, LR and SLL sectors (VRR and SRR follow by chirality), with no numerical integration required. Sections 3 and 4 present these as algebraic expressions in the coupling ratios $eta_3$, $eta_4$, $eta_5$ and $tildeeta_4$, separating a leading-order power $eta^P$ from an $O(alpha_s)$ correction; Sections 5 and 6 give numerical tables, the scheme-conversion rules to other renormalization schemes, and the relation to the older $eta_B$ and $eta_2$ factors. The authors report that the running is mild in the vector-vector sector but large in the left-right and scalar sectors: at $mu_K = 1$ GeV the diagonal entry $[eta_{22}]^{LR}$ is enhanced by about a factor of 5.6, $[eta_{11}]^{SLL}$ by about 2.9, while $[eta_{22}]^{SLL}$ is suppressed to about 0.26, and the NLO corrections are typically 5 to 15 percent, reaching about 25 percent for $[eta_{21}]^{LR}$, $[eta_{22}]^{LR}$ and $[eta_{22}]^{SLL}$. Section 7 turns this into master formulae for $Delta M_K$, $Delta M_B$ and $epsilon_K$ in terms of non-perturbative $B_i$ parameters and the Wilson coefficients at $mu_t$, with universal coefficients $P^{VLL}_1 = 0.48$, $P^{LR}_1 = -36.1$, $P^{LR}_2 = 59.3$, $P^{SLL}_1 = -18.1$, $P^{SLL}_2 = -32.2$ in the kaon system.

Load-bearing premise

The left-right $rho$ coefficient in eq. (4.10) contains a term proportional to $log(eta_3)$ that arises only by taking the limit $f to 3$ of the general evolution formula, because the matrix $hat J$ is singular there; the authors classify that limit term as a next-to-leading contribution under their convention, and if that bookkeeping is not the right way to organize the expansion the LR entry of the kaon master formula is mis-assigned.

Editorial extensions

If this is right

Once a model's Wilson coefficients $C_i(mu_t)$ are computed in the NDR scheme, $Delta M_K$, $Delta M_B$ and $epsilon_K$ follow directly from the universal $P^a_i$ coefficients, with no further QCD work by the user.
In the kaon system the left-right and scalar coefficients are two orders of magnitude larger than the vector-vector one ($P^{LR}_1 = -36.1$, $P^{LR}_2 = 59.3$, $P^{SLL}_1 = -18.1$, $P^{SLL}_2 = -32.2$ against $P^{VLL}_1 = 0.48$), so even small new-physics contributions to those operators can dominate the mixing amplitude.
Lattice matrix elements quoted at 2 GeV and hadronic-model matrix elements computed below 1 GeV can be placed on a common scale and compared directly.
The familiar Standard Model factors $eta_B$ and $eta_2$ are not the same objects as $[eta(mu_b)]^{VLL}$ and $[eta(mu_K)]^{VLL}$; the paper supplies the explicit conversion between them.
The paper also provides the evolution matrix for running upward from $mu_t$ to a higher new-physics scale $mu_s$ in a six-flavour effective theory, so that models with very heavy particles can be matched in stages.

Reading between the lines

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

Because the $P^a_i$ are universal numbers, they can be reused as a fixed lookup table by any future model builder; the practical bottleneck then shifts entirely to computing $C_i(mu_t)$ and to the lattice $B$ parameters in the LR and SLL sectors, which Section 7 shows are the least controlled, with scheme shifts of roughly 30 and 20 percent respectively.
The size of the LR and SLL enhancements suggests that meson-mixing observables are far more sensitive to chirality-flipped operators than to the standard vector-vector one, which is an argument that a future discrepancy in $epsilon_K$ or $Delta M_K$ would more plausibly point to scalar or left-right new physics than to a shift in the standard operator.
The treatment of the $f=3$ limit in the LR sector implies that 'next-to-leading order' in a multi-threshold evolution is partly a bookkeeping convention; two groups using different conventions could legitimately differ at the few-percent level the introduction warns about, without either being wrong.
A useful cross-check that the paper does not perform would be to evolve a set of lattice matrix elements from 2 GeV down to 1 GeV and back with the $rho$ matrices and verify closure at the stated accuracy; a failure of closure would localize the problem to a specific sector.

Original abstract

We present analytic formulae for the QCD renormalization group factors relating the Wilson coefficients C_i(mu_t) and C_i(mu), with mu_t = O(m_t) and mu < mu_t, of the Delta F=2 dimension six four-quark operators Q_i in the Standard Model and in all of its extensions. Analogous analytic formulae for the QCD factors relating the matrix elements <Q_i(2 GeV)> and <Q_i(mu_K)> with mu_K < 2GeV are also presented. The formulae are given in the NDR scheme. The strongest renormalization-group effects are found for the operators with the Dirac structures (1 - gamma_5)*(1 + gamma_5) and (1 - gamma_5)*(1 - gamma_5). We calculate the matrix elements <\bar K^0|Q_i|K^0> in the NDR scheme using the lattice results in the LRI scheme. We give expressions for the mass differences Delta M_K and Delta M_B and the CP-violating parameter epsilon_K in terms of the non-perturbative parameters B_i and the Wilson coefficients C_i(mu_t). The latter summarize the dependence on new physics contributions.

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, 5,732 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.12 · compares with prior results 0.20 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We present analytic formulae for the QCD renormalization group factors relating the Wilson coefficients C_i(mu_t) and C_i(mu), with mu_t = O(m_t) and mu < mu_t, of the Delta F=2 dimension six four-quark operators Q_i in the Standard Model and in all of its extensions. Analogous analytic formulae for the QCD factors relating the matrix elements <Q_i(2 GeV)> and <Q_i(mu_K)> with mu_K < 2GeV are also presented. The formulae are given in the NDR scheme. The strongest renormalization-group effects are found for the operators with the Dirac structures (1 - gamma_5)*(1 + gamma_5) and (1 - gamma_5)*(1 - gamma_5). We calculate the matrix elements <\bar K^0|Q_i|K^0> in the NDR scheme using the lattice results in the LRI scheme. We give expressions for the mass differences Delta M_K and Delta M_B and the CP-violating parameter epsilon_K in terms of the non-perturbative parameters B_i and the Wilson coefficients C_i(mu_t). The latter summarize the dependence on new physics contributions.

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 present analytic formulae for the QCD renormalization group factors relating the Wilson coefficients C_i(mu_t) and C_i(mu), with mu_t = O(m_t) and mu < mu_t, of the Delta F=2 dimension six four-quark operators Q_i in the Standard Model and in all of its extensions.	contribution 1.00	0.95	0.62	asserted 1.00
s1	Analogous analytic formulae for the QCD factors relating the matrix elements <Q_i(2 GeV)> and <Q_i(mu_K)> with mu_K < 2GeV are also presented.	contribution 0.96	0.92	0.10	asserted 1.00
s2	The formulae are given in the NDR scheme.	method 0.89	0.65	0.05	asserted 1.00
s3	The strongest renormalization-group effects are found for the operators with the Dirac structures (1 - gamma_5)(1 + gamma_5) and (1 - gamma_5)(1 - gamma_5).	result 0.98	0.87	0.13	asserted 1.00
s4	We calculate the matrix elements <\bar K^0\|Q_i\|K^0> in the NDR scheme using the lattice results in the LRI scheme.	method 0.86	0.93	0.05	asserted 1.00
s5	We give expressions for the mass differences Delta M_K and Delta M_B and the CP-violating parameter epsilon_K in terms of the non-perturbative parameters B_i and the Wilson coefficients C_i(mu_t).	contribution 0.65	0.89	0.22	asserted 1.00
s6	The latter summarize the dependence on new physics contributions.	interpretation 0.96	0.55	0.10	asserted 0.99

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

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

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.** The [ηij ]LR factors as functions of µ for α (5) s (MZ) = 0.118. [22℄SLL [21℄SLL [12℄SLL [11℄SLL [GeV℄ K m L 3 mb 10 30 MW mt 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 -0.5

**Figure 2.** The [ηij ] SLL factors as functions of µ for α (5) s (MZ) = 0.118 17

Figures reproduced from arXiv:hep-ph/0102316 by the authors.

References

Works cited (15)

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A.J. Buras, hep-ph/9806471, in Probing the Standard Model of Particle Interactions , eds. R. Gupta, A. Morel, E. de Rafael and F. David (Elsevier Science B.V., A msterdam, 1998), page 281; A.J. Buras, hep-ph/0101336, Erice Lectures, (2000)

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hep-ph 2013-10 conditional novelty 6.0 of 10

Four-quark scalar operators involving third-generation quarks can generate the C9 shift needed to explain the LHCb P'5 anomaly without violating Bs mixing or b to s gamma constraints.
Quark flavour mixing with right-handed currents: an effective theory approach
hep-ph 2010-07 accept novelty 6.0 of 10

Fitting a right-handed quark mixing matrix to charged-current data resolves the |V_ub| tension but forces S_psiK = sin 2 beta = 0.77 +/- 0.05, roughly 2 sigma above the measured 0.672 +/- 0.023.
Higgs-mediated FCNCs: Natural Flavour Conservation vs. Minimal Flavour Violation
hep-ph 2010-05 accept novelty 6.0 of 10

Minimal Flavour Violation, unlike Natural Flavour Conservation, is stable under quantum corrections in two-Higgs-doublet models, and with flavour-blind CP phases it accommodates a large Bs mixing phase while softening...
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...
Prospects for $K_{L}\to\pi^{0}\nu\bar{\nu}$, $K_S\to\mu^+\mu^-$, $K_L\to\pi^0 \ell^+\ell^-$ and $\varepsilon^{\prime}/\varepsilon$ after the new $K^{+}\to\pi^{+}\nu\bar{\nu}$ result from NA62
hep-ph 2026-07 conditional novelty 5.0 of 10

A Z' model with both left- and right-handed sbar-d couplings can enhance K_L→π0ννbar by an order of magnitude while keeping K+→π+ννbar and ε_K SM-like.
Addressing $R_{D^{(*)}}$, $R_{K^{(*)}}$, muon $g-2$ and ANITA anomalies in a minimal $R$-parity violating supersymmetric framework
hep-ph 2020-02 conditional novelty 5.0 of 10

A single R-parity-violating supersymmetric parameter point can accommodate the R_D(*), R_K(*), muon g-2 and ANITA anomalies at 3 sigma, with third-generation superpartners in the 1-10 TeV range.
Complete Vector-like Fourth Family and new $\mathrm{U(1)^\prime}$ for Muon Anomalies
hep-ph 2019-06 conditional novelty 5.0 of 10

A single model with a new U(1)-prime gauge boson and a complete vector-like fourth family fits both the muon g-2 and b to s ll anomalies while remaining consistent with other precision observables.
Low-energy signatures of the $\mathrm{PS}^3$ model: from $B$-physics anomalies to LFV
hep-ph 2018-05 conditional novelty 5.0 of 10

Fitting the three-site Pati-Salam model to B-decay lepton-universality anomalies yields correlated predictions for tau to mu gamma, tau to 3 mu, B to K tau mu and Bs to tau mu at rates near current bounds.
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.
Long Lived Light Scalars as Probe of Low Scale Seesaw Models
hep-ph 2017-03 conditional novelty 5.0 of 10

In TeV-scale left-right seesaw models the B-L-breaking scalar can be a GeV-scale particle that lives long enough to produce displaced collimated diphoton vertices at the LHC.
New Physics Patterns in $\varepsilon^\prime/\varepsilon$ and $\varepsilon_K$ with Implications for Rare Kaon Decays and $\Delta M_K$
hep-ph 2015-12 conditional novelty 5.0 of 10

If new Z or Z-prime bosons explain the kaon CP anomalies, the sign of their contribution to the K-Kbar mass difference separates QCD-penguin from electroweak-penguin solutions.
Future sensitivity to new physics in B_d, B_s and K mixings
hep-ph 2013-09 accept novelty 5.0 of 10

Projected LHCb and Belle II data will tighten bounds on new physics in neutral meson mixing about fivefold, probing new-physics scales up to roughly 20 TeV for CKM-like tree-level couplings.
The Anatomy of Z' and Z with Flavour Changing Neutral Currents in the Flavour Precision Era
hep-ph 2012-11 conditional novelty 5.0 of 10

For a tree-level flavour-changing Z-prime, fixing quark couplings from meson-mixing data yields parameter-free predictions for rare K and B decays, and the resulting correlations can separate left-handed, right-handed...
The Anatomy of Quark Flavour Observables in 331 Models in the Flavour Precision Era
hep-ph 2012-11 conditional novelty 5.0 of 10

In the 331 model with a heavy Z-prime, four parameter regions survive current flavour data, and the rare decays B_d to mu mu plus the CP asymmetry in B_s to mu mu can in future identify which one is correct.
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.
Charged-Higgs phenomenology in the Aligned two-Higgs-doublet model
hep-ph 2010-06 accept novelty 5.0 of 10

Low-energy flavour data bound the three complex Yukawa alignment parameters of the aligned two-Higgs-doublet model, giving |ς_l|/M_H± ≤ 0.40 GeV^-1 and |ς_u|/M_H± < 0.011 GeV^-1 at 95% CL.
CP violation in B_s mixing from heavy Higgs exchange
hep-ph 2010-05 conditional novelty 5.0 of 10

Tree-level exchange of a heavy neutral Higgs in an uplifted two-Higgs-doublet model can produce the CP-violating B_s mixing asymmetry reported by D0, but the reported central value also requires new physics in other channels.
Precision calculations in the MSSM Higgs-boson sector with FeynHiggs 2.14
hep-ph 2018-11 conditional novelty 4.0 of 10

Version 2.14 of FeynHiggs adds a DR-renormalized stop sector, non-degenerate matching thresholds, and a linear interpolation over complex phases, improving MSSM Higgs-mass predictions across low and high SUSY scales.
Flavor constraints on the Two Higgs Doublet Models of $Z_2$ symmetric and aligned types
hep-ph 2015-11 conditional novelty 4.0 of 10

Updated flavor fits exclude a charged Higgs below roughly 490 GeV in the type II and Y two-Higgs-doublet models, and the aligned variant accommodates the muon g-2 and R(D) excesses only in one narrow parameter corner.
Towards the Identification of New Physics through Quark Flavour Violating Processes
hep-ph 2013-06 accept novelty 4.0 of 10

The paper proposes a twelve-step, correlation-based program and DNA charts for identifying new physics from quark-flavor observables, illustrated with CMFV, MFV, two-Higgs-doublet, and tree-level Z-prime and scalar scenarios.
New Constraints on SUSY Flavour Mixing in Light of Recent Measurements at the Tevatron
hep-ph 2006-04 accept novelty 4.0 of 10

New Tevatron bounds on B_s mixing strongly constrain right-handed squark flavor-changing parameters in the MSSM, especially at large tan beta.

How later papers used it

88 later papers cite it

Evidence: substantial. 71 of the citing papers share no author with it, and 34 of those carry a confident label from Pith's citation classifier.

Pith read 25 of the 164 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

6 later papers by other authors are listed below: 5 used its method or tool and 1 used its data or values. 3 more papers by the same authors are listed: 1 built on it, 1 used its method or tool and 1 used its data or values.

Builds on it 1 of 2 papers with this label · 1 by the same authors

By the same authors

Presented an update of the master formula for Delta F=2 processes in LEFT valid in any BSM scenario first presented in this paper.

SMEFT ATLAS of $\Delta F=2$ Transitions (2020) (same authors) · 0.86 · paraphrased: the licence does not allow quoting

Uses its method or tool 6 of 31 papers with this label · 1 by the same authors

Used Eqs. (7.24)-(7.27) of this paper for the Delta B=2 and Delta S=2 amplitudes to compute the observables in terms of Wilson coefficients of dimension-six operators.

Exploring the flavour structure of the high-scale MSSM (2019) · 1.00 · paraphrased: the licence does not allow quoting
Performed the running for O_1^SRR using the results of this paper.

Charged-Higgs phenomenology in the Aligned two-Higgs-doublet model (2010) · 0.90 · paraphrased: the licence does not allow quoting
Followed this paper to perform renormalization group evolution of the NP operators O_1^SLL, O_1^SRR and O_2^LR.

Flavor Violating Higgs Couplings in Minimal Flavor Violation (2018) · 1.00 · paraphrased: the licence does not allow quoting
Used the notation of Eq. (2.1) in this paper and expressed the Wilson coefficient of the operator Q_1 s^LR by performing a Fierz transformation to the operator Q_5 in the SUSY basis.

Revisiting R-parity violating interactions as an explanation of the B-physics anomalies (2018) · 0.88 · paraphrased: the licence does not allow quoting
Gave results in the continuum MS-NDR scheme at scale m_b using both the BBGLN and BJU choices of evanescent operators.

Neutral $B$ mixing from 2+1 flavor lattice-QCD: the Standard Model and beyond (2011) · 0.92 · paraphrased: the licence does not allow quoting

By the same authors

Obtained the values of the Wilson coefficients at scales O(mu_B) by the standard techniques of this paper.

EDMs vs. CPV in B_{s,d} mixing in two Higgs doublet models with MFV (2010) (same authors) · 0.91 · paraphrased: the licence does not allow quoting

Uses its data or values 2 of 19 papers with this label · 1 by the same authors

Employed the formulas provided by this paper, using eta_LL=0.65, eta_LR=0.99 and eta_LR'=-5.08 at the scale mu=1 GeV.

$\Delta S=2$ nonleptonic hyperon decays as probes of new physics (2023) · 0.83 · paraphrased: the licence does not allow quoting

By the same authors

Used by default the B_i^X values at the scale mu=2 GeV given in this paper using the NDR renormalization scheme.

SUSY_FLAVOR: a computational tool for FCNC and CP-violating processes in the MSSM (2010) (same authors) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

85 arXiv papers cite this in 159 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 · 91 sentences without a confident label

Builds on it 2 papers

In sec:LEFT-master we present an update of the master formula for $=2$ processes ($$, $$ and $$) in LEFT valid in any BSM scenario first presented in [ref] .

SMEFT ATLAS of $\Delta F=2$ Transitions · Introduction · 0.86
We quote all three matrices here, so that the formulae of Ref. [ref] can be easily extended to the $(Q_1^SLL,Q_1^SLL)$ basis: [equation]

The supersymmetric Higgs sector and B-Bbar mixing for large tan beta · NLO scheme transformation formulae · 0.88

Uses its method or tool 31 papers

To compute explicitly the $ΔO_i^NP(x)$ we use the expressions of the observables in terms Wilson Coefficients of dimension-six operators according to the following references: * $ΔB = 2$ and $ΔS=2$ amplitudes: Eqs. (7.24)–(7.27) in Ref. [ref] ; * $ΔC = 2$ amplitude: Eq. (3) in Ref. [ref] ; * $ϵ_K'/ϵ_K$: Eq. (82) in Ref. [ref] , * $ℬ (B→X_sγ)$: Eq. (33) in Ref. [ref] ; * $ℬ (K^+ →π^+ ν̅ ν)$: Eq. (24) in Ref. [ref] [We have contacted the authors and confirmed that the parameter $κ$ should be rescaled by a factor of $(2^3/2 πsin^2θ_W v/α)^2$.].

Exploring the flavour structure of the high-scale MSSM · Analysis strategy · 1.00
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
[ref] to perform renormalization group evolution of the NP operators $𝒪_1^SLL$, $𝒪_1^SRR$ and $𝒪_2^LR$.

Flavor Violating Higgs Couplings in Minimal Flavor Violation · Neutral $B$ and $K$ meson mixing · 1.00

Uses its data or values 19 papers

Thus, with $ΔM_d^exp=(0.507±0.005) ps^-1$ [ref] and $ΔM_d^SM=(0.560^+0.067_-0.076) ps^-1$ [ref] , using the approximation $|1+δ_d^|≃1+Re δ_d$, we can extract the $1 σ$ range [equation] Similarly, $ΔM_s^exp=(17.77±0.12) ps^-1$ [ref] and $ΔM_s^SM=(17.6^+1.7_-1.8) ps^-1$ [ref] translate into [equation] To proceed, in addition to $m_X^=214.3$ MeV, we use $m_b^=4.4$ GeV, $P_1^VLL=0.84$, $P_1^SLL=-1.47$, $P_1^LR=-1.62$, $P_2^LR=2.46$ [ref] , CKM parameters from Ref. [ref] , $f_B_d=190$ MeV, $f_B_s=228$ MeV, $η_B^=0.551$, $B_B_d=1.17$, and $B_B_s=1.23$ [ref] , as well as meson masses from Ref. [ref] .

Rare B Decays with a HyperCP Particle of Spin One · Constraints from $B_q$-$B̅_q$ mixing · 0.99
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
The bag parameters at the $μ_b = m_b$ scale are given as [ref] [equation] and the QCD corrections are $η_11 (μ_b)=1.654$ and $η_21 (μ_b)=-0.007$ [ref] .

Flavor constraints on the Two Higgs Doublet Models of $Z_2$ symmetric and aligned types · Input and experimental data · 1.00

Compares with it 1 paper

The result is [equation] The number in p2lrnum is significantly larger than $P_2^LR=2.46$ quoted in Ref. [ref] , because our value for $m_b$ is smaller and the lattice bag factors are larger than one.

The supersymmetric Higgs sector and B-Bbar mixing for large tan beta · Hadronic matrix elements and heavy-quark relations · 0.83

Cites it as background 5 papers

The strength of $B_s(d)-B̅_s(d)$ mixing is described by the mass difference $ΔM_s(d)$, defined by [ref] [equation] where $M_12^q$ denotes the off-diagonal element in the neutral $B_q$-meson mass matrix, and the effective weak Hamiltonian can be written in a general form as [ref] [equation] Within the $ℤ_3$-invariant NMSSM with NMFV, the following eight operators, as defined in Ref. [ref] , are all found to be relevant: [equation] where $α$ and $β$ are the colour indices, $σ_μν=1/2[γ_μ,γ_ν]$, and $P_L,R=(1∓γ_5)/2$.

$B_{s(d)}-\bar{B}_{s(d)}$ Mixing and $B_s\to\mu^+\mu^-$ Decay in the NMSSM with the Flavour Expansion Theorem · $B_s(d)-B̅_s(d)$ mixing · 0.83
There are in principle eight dimension-six four-quark operators that can contribute to down-type $ΔF=2$ processes [ref] .

Solving the flavour problem with hierarchical fermion wave functions · $ΔF=2$ operators · 0.92
The detailed formulae for $K_0-K̅_0$ mixing can be found in reference [ref] .

Flavour-alignment in an $S_3$-symmetric Higgs sector and its RG-behaviour · $K_0-K̅_0$ mixing · 0.91

Discussion

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