Electroweak Penguin Operators Research Articles

Singularity-free next-to-leading order ΔS = 1 renormalization group evolution and ϵ K ′ /ϵK in the Standard Model and beyond

The standard analytic solution of the renormalization group (RG) evolution for the $\Delta S = 1$ Wilson coefficients involves several singularities, which complicate analytic solutions. In this paper we derive a singularity-free solution of the next-to-leading order (NLO) RG equations, which greatly facilitates the calculation of $\epsilon_K^{\prime}$, the measure of direct $CP$ violation in $K\to \pi\pi$ decays. Using our new RG evolution and the latest lattice results for the hadronic matrix elements, we calculate the ratio $\epsilon_{K}^{\prime}/\epsilon_{K}$ (with $\epsilon_{K}$ quantifying indirect $CP$ violation) in the Standard Model (SM) at NLO to $\epsilon_{K}^{\prime}/\epsilon_{K} = (1.06 \pm 5.07) \times 10^{-4} $, which is $2.8\,\sigma$ below the experimental value. We also present the evolution matrix in the high-energy regime for calculations of new physics contributions and derive easy-to-use approximate formulae. We find that the RG amplification of new-physics contributions to Wilson coefficients of the electroweak penguin operators is further enhanced by the NLO corrections: If the new contribution is generated at the scale of 1-10 TeV, the RG evolution between the new-physics scale and the electroweak scale enhances these coefficients by 50-100 %. Our solution contains a term of order $\alpha_{EM}^2/\alpha_s^2$, which is numerically unimportant for the SM case but should be included in studies of high-scale new-physics.

New physics patterns in ε′/ε and ε K with implications for rare kaon decays and ∆M K

The SM prediction for the ratio $\varepsilon^\prime/\varepsilon$ appears to be significantly below the experimental data. Also $\varepsilon_K$ in the SM tends to be below the data. Any NP removing these anomalies will first of all have impact on flavour observables in the $K$ meson system, in particular on rare $K$ decays and $\Delta M_K$. Restricting the operators contributing to $\varepsilon^\prime/\varepsilon$ to the SM ones and to the corresponding primed operators, NP contributions are quite generally dominated either by QCD penguin (QCDP) operators $Q_6(Q_6^\prime)$ or electroweak penguin (EWP) operators $Q_8(Q_8^\prime)$ with rather different implications for other flavour observables. We discuss general models with tree-level $Z$ and $Z^\prime$ flavour violating exchanges and few specific models. We find that simultaneous enhancements of $\varepsilon^\prime/\varepsilon$, $\varepsilon_K$, $\mathcal{B}(K_L\to\pi^0\nu\bar\nu)$ and $\mathcal{B}(K^+\to\pi^+\nu\bar\nu)$ in $Z$ scenarios are only possible in the presence of both LH and RH flavour-violating couplings. In $Z^\prime$ scenarios this is not required but the size of NP effects and the correlation between $\mathcal{B}(K_L\to\pi^0\nu\bar\nu)$ and $\mathcal{B}(K^+\to\pi^+\nu\bar\nu)$ depends strongly on whether QCDP or EWP dominate NP contributions to $\varepsilon^\prime/\varepsilon$. In the QCDP case possible enhancements of both branching ratios are much larger than for EWP scenario and take place only on the branch parallel to the Grossman-Nir bound, which is in the case of EWP dominance only possible in the absence of NP in $\varepsilon_K$. QCDP and EWP scenarios of NP in $\varepsilon^\prime/\varepsilon$ can also be uniquely distinguished by the size and the sign of NP contribution to $\Delta M_K$, elevating the importance of the precise calculation of $\Delta M_K$ in the SM.

The Revival of Kaon Flavour Physics

After years of silence we should witness in the rest of this decade and in the next decade the revival of kaon flavour physics. This is not only because of the crucial measurements of the branching ratios for the rare decays $K^+\to\pi^+\nu\bar\nu$ and $K_{L}\to \pi^0\nu\bar\nu$ by NA62 and KOTO that being theoretically clean and very sensitive to new physics (NP) could hint for new phenomena even beyond the reach of the LHC without any significant theoretical uncertainties. Indeed simultaneously the advances in the calculations of perturbative and in particular non-perturbative QCD effects in $\varepsilon'/\varepsilon$, $\varepsilon_K$, $\Delta M_K$, $K_L\to\mu^+\mu^-$ and $K_L\to\pi^0\ell^+\ell^-$ will increase the role of these observables in searching for NP. In fact the hints for NP contributing to $\varepsilon'/\varepsilon$ have been already signalled last year through improved estimates of hadronic matrix elements of QCD and electroweak penguin operators $Q_6$ and $Q_8$ by lattice QCD and large $N$ dual QCD approach. This talk summarizes in addition to this new flavour anomaly the present highlights of this field including some results from concrete NP scenarios.

Upper bounds on ε′/ε parameters B 6 (1/2) and B 8 (3/2) from large N QCD and other news

We demonstrate that in the large N approach developed by the authors in collaboration with Bardeen, the parameters B_6^{(1/2)} and B_8^{(3/2)} parametrizing the K\to\pi\pi matrix elements <Q_6>_0 and <Q_8>_2 of the dominant QCD and electroweak operators receive both negative O(1/N) corrections such that B_6^{(1/2)} < B_8^{(3/2)}<1 in agreement with the recent lattice results of the RBC-UKQCD collaboration. We also point out that the pattern of the size of the hadronic matrix elements of all QCD and electroweak penguin operators Q_i contributing to the K\to \pi \pi amplitudes A_0 and A_2, obtained by this lattice collaboration, provides further support to our large N approach. In particular, a very precise lattice result for the matrix element <Q_8>_0 implies for the corresponding parameter B_8^{(1/2)}=1.0\pm 0.2 to be compared with large N value B_8^{(1/2)}=1.1\pm 0.1. We discuss briefly the implications of these findings for the ratio epsilon'/epsilon. In fact, with the precise value for B_8^{(3/2)} from RBC-UKQCD collaboration, our upper bound on B_6^{(1/2)} implies epsilon'/epsilon in the SM roughly by a factor of two below its experimental value (16.6\pm 2.3)\times 10^{-4}. We also briefly comment on the parameter \hat B_K and the \Delta I=1/2$ rule.

ANALYSIS OF THE $B_q\to D_q(D_q^*)P$ AND $B_q\to D_q(D_q^*)V$ DECAYS WITHIN THE FACTORIZATION APPROACH IN QCD

Using the factorization approach and considering the contributions of the current–current, QCD penguin and electroweak penguin operators at the leading approximation, the decay amplitudes and decay widths of [Formula: see text] and [Formula: see text] transitions, where q = u, d, s and P and V are pseudoscalar and vector mesons, are calculated in terms of the transition form factors of the Bq→Dq and [Formula: see text]. Having computed those form factors in three-point QCD sum rules, the branching fraction for these decays are also evaluated. A comparison of our results with the predictions of the perturbative QCD as well as the existing experimental data is presented.

Improved method for CKM constraints in charmless three-bodyBandBsdecays

Recently Ciuchini, Pierini, and Silvestrini proposed a method for constraining CKM parameters in $B\ensuremath{\rightarrow}K\ensuremath{\pi}\ensuremath{\pi}$ and ${B}_{s}\ensuremath{\rightarrow}K\ensuremath{\pi}\ensuremath{\pi}$ through phase measurements of amplitudes involving $I=3/2$ ${K}^{*}\ensuremath{\pi}$ final states. We show that complementary information on CKM parameters may be obtained by studying the phases of $\ensuremath{\Delta}I=1$ $B\ensuremath{\rightarrow}({K}^{*}\ensuremath{\pi}{)}_{I=1/2}$, ${B}_{s}\ensuremath{\rightarrow}({K}^{*}\overline{K}{)}_{I=1}$ and ${B}_{s}\ensuremath{\rightarrow}({\overline{K}}^{*}K{)}_{I=1}$ amplitudes. Hadronic uncertainties in these constraints from electroweak penguin operators ${O}_{9}$ and ${O}_{10}$, studied using flavor SU(3), are shown to be very small in $B\ensuremath{\rightarrow}K\ensuremath{\pi}\ensuremath{\pi}$ and ${B}_{s}\ensuremath{\rightarrow}K\ensuremath{\pi}\ensuremath{\pi}$ and somewhat larger in ${B}_{s}\ensuremath{\rightarrow}K\overline{K}\ensuremath{\pi}$. The first processes imply a precise linear relation between $\overline{\ensuremath{\rho}}$ and $\overline{\ensuremath{\eta}}$, with a measurable slope and an intercept at $\overline{\ensuremath{\eta}}=0$ involving a theoretical error of 0.03. The decays ${B}_{s}\ensuremath{\rightarrow}K\ensuremath{\pi}\ensuremath{\pi}$ permit a measurement of $\ensuremath{\gamma}$ involving a theoretical error below a degree. We note that while time-dependence is required when studying ${B}^{0}$ decays at the $\ensuremath{\Upsilon}(4S)$, it may not be needed when studying ${B}_{s}$ decays at hadronic colliders.

B+ and B0 Direct CP Asymmetries difference in a sequential Fourth Generation scenario

Direct CP violation in B0 → K+π− decay has emerged at −10% level, but the asymmetry in B+ → K+π0 mode is consistent with zero. This difference points towards possible New Physics in the electroweak penguin operator. We point out that a sequential fourth generation, with sizable V*t′s Vt′b and near maximal phase, could be a natural cause. We compare t′ effects on direct CP violation in B+ → K+π0 with b → sl+ l− and Bs mixing. Such large effects in b ↔ s transitions would affect s ↔ d transitions, as kaon constraints would demand Vt′d ≠ 0. Using Γ(Z → b b̄) to bound |Vt′b|, we infer sizable |Vt′s| ≲ |Vt′b| ≲ |Vus|. Imposing ϵK, K+ → π+ v v̄ and ϵ′/ϵ constraints, we find V*t′d Vt′s ∼ few ×104 with large phase, enhancing KL → π0 vv̄ to 5 × 10−10 or even higher. Interestingly, ΔmBd and sin 2ΦBd are not much affected, as |V*t′d Vt′b| << |V*td Vtb| ∼ 0.01.

K0−K¯0mixing beyond the standard model andCP-violating electroweak penguins in quenched QCD with exact chiral symmetry

We present results for the $\ensuremath{\Delta}S=2$ matrix elements which are required to study neutral kaon mixing in the standard model (SM) and beyond . We also provide leading chiral order results for the matrix elements of the electroweak penguin operators which give the dominant $\ensuremath{\Delta}I=3/2$ contribution to direct $CP$ violation in $K\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$ decays. Our calculations were performed with Neuberger fermions on two sets of quenched Wilson gauge configurations at inverse lattice spacings of approximately 2.2 GeV and 1.5 GeV. All renormalizations were implemented nonperturbatively in the regularization-independent/momentum (RI/MOM) scheme, where we accounted for subleading operator product expansion corrections and discretization errors. We find ratios of non-SM to SM matrix elements which are roughly twice as large as in the only other dedicated lattice study of these amplitudes. On the other hand, our results for the electroweak penguin matrix elements are in good agreement with two recent domain-wall fermion calculations. As a by-product of our study, we determine the strange quark mass. Our main results are summarized and discussed in Sec. VII. Within our statistics, we find no evidence for scaling violations.

Branching ratio andCPAsymmetry ofB→ρη(′)decays in the perturbative QCD approach

In this paper,we calculate the branching ratios and CP-violating asymmetries for $B^0 \to \rho^0 \eta^{(\prime)}$ and $B^+\to \rho^+ \etap$ decays in the perturbative QCD factorization approach. In this approach, we not only calculate the usual factorizable contributions, but also evaluate the non-factorizable and annihilation type contributions. Besides the current-current operators, the contributions from the QCD and electroweak penguin operators are also taken into account. The theoretical predictions for the branching ratios are $Br(B^+ \to \rho^+ \eta^{(\prime)}) \approx 9 \times 10^{-6}$ and $Br(B^0 \to \rho^0 \eta^{(\prime)}) \approx 5 \times 10^{-8}$, which agree well with the measured values and currently available experimental upper limits. We also predict large CP-violating asymmetries in these decays: $A_{CP}^{dir}(\rho^\pm \eta)\approx -13 %$, $A_{CP}^{dir}(\rho^\pm \eta^\prime)\approx -18 %$, $A_{CP}^{dir}(\rho^0 \eta)\approx -41%$, $A_{CP}^{dir}(\rho^0 \eta^\prime)\approx -27%$, $A_{CP}^{mix}(\rho^0 \eta)\approx +25%$, and $A_{CP}^{mix}(\rho^0 \eta^\prime) \approx +11%$, which can be tested by the current or future B factory experiments.

The L-R Correlator and its Chiral Condensates in the MHA and MHA+V' Approximations to Large- N c QCD

In this talk, we describe part of a recent work [1] on the correlation function of a V − A current with a V + A current in the framework of QCD in the limit of a large number of colours N c . The discussion takes place within two successive approximations of this theory, called MHA and MHA+V'. Results concerning the evaluation of chiral condensates of dimension six and eight, as well as matrix elements of the Q 7 and Q 8 electroweak penguin operators, are given.

Branching ratio and CP asymmetry of [formula omitted] decays in the perturbative QCD approach

We calculate the branching ratios and CP-violating asymmetries for B 0 → π 0 η ( ′ ) and B + → π + η ( ′ ) decays in the perturbative QCD (pQCD) factorization approach here. We not only calculate the usual factorizable contributions, but also evaluate the non-factorizable and annihilation type contributions. Besides the current–current operators, the contributions from the QCD and electroweak penguin operators are also taken into account. The pQCD results for the CP-averaged branching ratios are Br ( B + → π + η ) ≈ 4.1 × 10 −6 , Br ( B + → π + η ′ ) ≈ 2.4 × 10 −6 , and Br ( B 0 → π 0 η ( ′ ) ) ≈ 0.2 × 10 −6 , which agree very well with the measured values or currently available experimental upper limits. We also predict large CP-violating asymmetries in these decays: A CP dir ( π ± η ( ′ ) ) ∼ A CP dir ( π 0 η ( ′ ) ) ∼ − 0.35 , and A CP mix ( π 0 η ( ′ ) ) ∼ 0.67 , but with large errors. The pQCD prediction for A CP dir ( π ± η ) ( A CP dir ( π ± η ′ ) ) has the same (opposite) sign with the primary measured values. Further improvements in both theory and experiments are needed to clarify this discrepancy.

ΔS = 0 effective weak chiral Lagrangianfrom the instanton vacuum

We investigate the ΔS = 0 effective chiral Lagrangian from the instanton vacuum. Based on the ΔS = 0 effective weak Hamiltonian from the operator product expansion and renormalization group equations, we derive the strangeness-conserving effective weak chiral Lagrangian from the instanton vacuum to order ${{\mathcal{O}}}(p^2)$ and the next-to-leading order in the 1/N c expansion at the quark level. We find that the quark condensate and a dynamical term which arise from the QCD and electroweak penguin operators appear in the next-to-leading order in the 1/N c expansion for the ΔS = 0 effective weak chiral Lagrangian, while they are in the leading order terms in the ΔS = 1 case. Three different types of form factors are employed and we find that the dependence on the different choices of the form factor is rather insensitive. The low-energy constants of the Gasser-Leutwyler type are determined and discussed in the chiral limit.

Difference inB+andB0DirectCPAsymmetry as an Effect of a Fourth Generation

Direct CP violation in B0 --> K+ pi- decay has emerged at the -10% level, but the asymmetry in B+ --> K+ pi0 mode is consistent with zero. This difference points towards possible new physics in the electroweak penguin operator. We point out that a sequential fourth generation, with sizable V(t' s)* V(t' b) and near maximal phase, could be a natural cause. We use the perturbative QCD factorization approach for B --> Kpi amplitudes. While the B0 --> K+ pi- mode is insensitive to t', we critically compare t' effects on direct CP violation in B+ --> K+ pi0 with b --> sl+ l- and B(s) mixing. If the K+ pi0 - K+ pi- asymmetry difference persists, we predict sin2phi(Bs) to be negative.

An exploratory lattice study of [formula omitted][formula omitted] decays at next-to-leading order in the chiral expansion

We present the first direct evaluation of Δ I = 3 / 2 K → π π matrix elements with the aim of determining all the low-energy constants at NLO in the chiral expansion. Our numerical investigation demonstrates that it is indeed possible to determine the K → π π matrix elements directly for the masses and momenta used in the simulation with good precision. In this range however, we find that the matrix elements do not satisfy the predictions of NLO chiral perturbation theory. For the chiral extrapolation we therefore use a hybrid procedure which combines the observed polynomial behavior in masses and momenta of our lattice results, with NLO chiral perturbation theory at lower masses. In this way we find stable results for the quenched matrix elements of the electroweak penguin operators ( 〈 π π | O 8 | K 0 〉 I = 2 = ( 0.68 ± 0.09 ) GeV 3 and 〈 π π | O 7 | K 0 〉 I = 2 = ( 0.12 ± 0.02 ) GeV 3 in the NDR- MS ¯ scheme at the scale 2 GeV), but not for the matrix elements of O 4 (for which there are too many low-energy constants at NLO for a reliable extrapolation). For all three operators we find that the effect of including the NLO corrections is significant (typically about 30%). We present a detailed discussion of the status of the prospects for the reduction of the systematic uncertainties.

Remarks on the hadronic matrix elements relevant to the supersymmetricK0−K0¯mixing amplitude

We compute the 1-loop chiral corrections to the bag parameters which are needed for the discussion of the SUSY K-Kbar mixing problem in both finite and infinite volume. We then show how the bag parameters can be combined among themselves and with some auxiliary quantities and thus sensibly reduce the systematic errors due to chiral extrapolations as well as those due to finite volume artefacts present in the results obtained from lattice QCD. We also show that in some cases these advantages remain as such even after including the 2-loop chiral corrections. Similar discussion is also made for the K --> pi electro-weak penguin operators.

Chiral Condensates, Q7and Q8Matrix Elements and Large-NcQCD

The correlation function of a V−A current with a V+A current is discussed within the framework of QCD in the limit of a large number of colours Nc. Applications to the evaluation of chiral condensates of dimension six and higher, as well as to the matrix elements of the Q7 and Q8 electroweak penguin operators are discussed. A critical comparison with previous determinations of the same parameters has also been made.

Branching ratio andCPasymmetry ofBs→π+π−decays in the perturbative QCD approach

In this paper, we calculate the decay rate and CP asymmetry of the $B_s \to \pi^+\pi^-$ decay in perturbative QCD approach with Sudakov resummation. Since none of the quarks in final states is the same as those of the initial $B_s$ meson, this decay can occur only via annihilation diagrams in the standard model. Besides the current-current operators, the contributions from the QCD and electroweak penguin operators are also taken into account. We find that (a) the branching ratio is about $4 \times 10^{-7}$; (b) the penguin diagrams dominate the total contribution; and (c) the direct CP asymmetry is small in size: no more than $3% $; but the mixing-induced CP asymmetry can be as large as ten percent testable in the near future LHC-b experiments.

/ at the NLO: 10 Years Later

During the last four years several parameters relevant for the analysis of the CP-violating ratio ?'/? improved and/or changed significantly. In particular, the experimental value of ?'/? and the strange quark mass decreased, the uncertainty in the CKM factor has been reduced, and for a value of the hadronic matrix element of the dominant electroweak penguin operator Q8, some consensus has been reached among several theory groups. In view of this situation, ten years after the first analyses of ?'/? at the next-to-leading order, we reconsider the analysis of ?'/? within the SM and investigate what can be said about the hadronic Q6 matrix element of the dominant QCD penguin operator on the basis of the present experimental value for ?'/? and todays values of all other parameters. Employing a conservative range for the reduced electroweak penguin matrix element R8 = 1.0?0.2 from lattice QCD, and present values for all other input parameters, on the basis of the current world average for ?'/?, we obtain the reduced hadronic matrix element of the dominant QCD penguin operator R6 = 1.23?0.16 implying Q60NDR(mc) ? ??0.8?Q82NDR(mc). We compare these results with those obtained in large-Nc approaches in which generally R6 ? R8 and Q60NDR(mc) is chirally suppressed relatively to Q82NDR(mc). We present the correlation between R6 and R8 that is implied by the data on ?'/? provided new physics contributions to ?'/? can be neglected.

QCD condensates for the light quarkV−Acorrelator

We use the procedure of pinched-weight Finite Energy Sum Rules (pFESR) to determine the OPE coefficients a_6, ...,a_16 of the flavor ud V-A correlator in terms of existing hadronic tau decay data. We show by appropriate weight choices that the error on the dominant d=6 contribution, which is known to be related to the K -> Pi Pi matrix elements of the electroweak penguin operator in the chiral limit, may be reduced to below the ~15% level. The values we obtain for the OPE coefficients with d>8 are shown to naturally account for the discrepancies between our results for the d=6 and d=8 terms and those of previous analyses, which were obtained neglecting d>8 contributions.

THE CHIRAL LIMIT K → ππ MATRIX ELEMENTS OF THE ELECTROWEAK PENGUIN OPERATORS Q7,8

The SU(3) chiral limit K → ππ matrix elements of the electroweak penguin operators, Q7,8, are determined using hadronic τ decay data, and dispersive and finite energy sum rules.