Cabibbo-Kobayashi-Maskawa Matrix Elements Research Articles

First determination of the Jarlskog invariant of CP violation from the moduli of the CKM matrix elements

We find that the precision and accuracy of current experimental data on the moduli of nine Cabibbo-Kobayashi-Maskawa (CKM) quark flavor mixing matrix elements allow us to numerically determine the correct size of the Jarlskog invariant of CP violation from four of them in eight different ways for the first time without making any special assumptions. This observation implies a remarkable self-consistency of the correlation between CP-conserving and CP-violating quantities of the CKM matrix as guaranteed by its unitarity.

Towards a discovery of BSM physics from the Cabibbo angle anomaly

New developments in both the theories and experiments related to the extraction of the top-row Cabibbo–Kobayashi–Maskawa matrix elements [Formula: see text] and [Formula: see text] led to a series of new anomalies, for instance, the apparent violation of the top-row unitarity relation. It is important to further reduce all the associated Standard Model theory uncertainties in order to better understand whether such observations point towards the possibility of physics beyond the Standard Model, or rather some unexpectedly large Standard Model effects. This requires improved studies of tree-level and higher-order Standard Model corrections that enter the beta decays of pions, neutron, nuclei and kaons. We will briefly review the recent progress along this direction and discuss possible improvements in the future.

Mapping out the space for new physics with leptonic and semileptonic B_{(c)} decays

Decays of B mesons with leptons in the final state offer an interesting laboratory to search for possible effects of physics from beyond the Standard Model. In view of puzzling patterns in experimental data, the violation of lepton flavour universality is an interesting option. We present a strategy, utilising ratios of leptonic and semileptonic B decays, where the elements |V_{ub}| and |V_{cb}| of the Cabibbo–Kobayashi–Maskawa (CKM) matrix cancel, to constrain the short-distance coefficients of (pseudo)-scalar, vector and tensor operator contributions. The individual branching ratios allow us then to extract also the CKM matrix elements, even in the presence of new-physics contributions. Bounds on unmeasured leptonic and semileptonic decays offer important additional constraints. In our comprehensive analysis, we give also predictions for decays which have not yet been measured in a variety of scenarios.

Measurement of the branching fraction of the {{B} ^0} {rightarrow }{{D} ^+_{s}} {{pi } ^-} decay

A branching fraction measurement of the {{B} ^0} {rightarrow }{{D} ^+_{s}} {{pi } ^-} decay is presented using proton–proton collision data collected with the LHCb experiment, corresponding to an integrated luminosity of 5.0,text {fb} ^{-1} . The branching fraction is found to be {mathcal {B}} ({{B} ^0} {rightarrow }{{D} ^+_{s}} {{pi } ^-} ) =(19.4 pm 1.8pm 1.3 pm 1.2)times 10^{-6}, where the first uncertainty is statistical, the second systematic and the third is due to the uncertainty on the {{B} ^0} {rightarrow }{{D} ^-} {{pi } ^+} , {{D} ^+_{s}} {rightarrow }{{K} ^+} {{K} ^-} {{pi } ^+} and {{D} ^-} {rightarrow }{{K} ^+} {{pi } ^-} {{pi } ^-} branching fractions. This is the most precise single measurement of this quantity to date. As this decay proceeds through a single amplitude involving a b{rightarrow }u charged-current transition, the result provides information on non-factorisable strong interaction effects and the magnitude of the Cabibbo–Kobayashi–Maskawa matrix element V_{ub}. Additionally, the collision energy dependence of the hadronisation-fraction ratio f_s/f_d is measured through {{overline{B}} {}^0_{s}} {rightarrow }{{D} ^+_{s}} {{pi } ^-} and {{B} ^0} {rightarrow }{{D} ^-} {{pi } ^+} decays.

Semileptonic decays of heavy mesons with artificial neural networks

Experimental checks of the second row unitarity of the Cabibbo-Kobayashi-Maskawa (CKM) matrix involve extractions of the matrix element $V_{cd}$, which may be obtained from semileptonic decay rates of $D$ to $\pi$. These decay rates are proportional to hadronic form factors which parameterize how the quark $c \to d$ transition is realized in $D \to \pi$ meson decays. The form factors can not yet be analytically computed over the whole range of available momentum transfer $q^2$, but can be parameterized with a varying degree of model dependency. We propose using artificial neural networks trained from experimental pseudo-data to predict the shape of these form factors with a prescribed uncertainty. We comment on the parameters of several commonly-used model parameterizations of semileptonic form factors. We extract shape parameters and use unitarity to bound the form factor at a given $q^2$, which then allows us to bound the CKM matrix element $|V_{cd}|$.

Measurement of CKM matrix elements in single top quark t-channel production in proton-proton collisions at [formula omitted

The first direct, model-independent measurement is presented of the modulus of the Cabibbo–Kobayashi–Maskawa (CKM) matrix elements |η|Vtb, |η|Vtd, and |η|Vts, in final states enriched in single top quark t-channel events. The analysis uses proton-proton collision data from the LHC, collected during 2016 by the CMS experiment, at a centre-of-mass energy of 13 TeV, corresponding to an integrated luminosity of 35.9 fb−1. Processes directly sensitive to these matrix elements are considered at both the production and decay vertices of the top quark. In the standard model hypothesis of CKM unitarity, a lower limit of |η|Vtb>0.970 is measured at the 95% confidence level. Several theories beyond the standard model are considered, and by releasing all constraints among the involved parameters, the values |η|Vtb=0.988±0.024, and |η|Vtd2+|η|Vts2=0.06±0.06, where the uncertainties include both statistical and systematic components, are measured.

Dark origin of the quark flavor hierarchy and mixing

We provide a dynamical mechanism for the generation of the Cabibbo-Kobayashi-Maskawa matrix in the context of a recently proposed model of flavor. The framework, based on the paradigm that Yukawa couplings are effective low energy couplings generated radiatively by dark sector interactions, is here extended to include a new scalar field which plays the role of a dark flavon. Being singlet under the Standard Model gauge group but charged under the $U(1)$ symmetry of the dark sector, this particle sources new two- and three-loop diagrams that result in off-diagonal Yukawa interactions, thereby providing a simple explanation for the observed structure of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. By using an effective parametrization of these new loop contributions, we show that the CKM matrix elements can be correctly reproduced for perturbative values of the involved couplings. New implications for dark matter phenomenology are briefly outlined.

Effective theory analysis for vector-like quark model

We study a model with a down-type SU(2) singlet vector-like quark (VLQ) as a minimal extension of the standard model (SM). In this model, flavor changing neutral currents (FCNCs) arise at tree level and the unitarity of the $3\times 3$ Cabibbo-Kobayashi-Maskawa (CKM) matrix does not hold. In this paper, we constrain the FCNC coupling from $b\rightarrow s$ transitions, especially $B_s\rightarrow \mu^+\mu^-$ and $\bar{B}\rightarrow X_s\gamma$ processes. In order to analyze these processes, we derive an effective Lagrangian which is valid below the electroweak symmetry breaking scale. For this purpose, we first integrate out the VLQ field and derive an effective theory by matching Wilson coefficients up to one-loop level. Using the effective theory, we construct the effective Lagrangian for $b\rightarrow s\gamma^{(*)}$. It includes the effects of the SM quarks and the violation of the CKM unitarity. We show the constraints on the magnitude of the FCNC coupling and its phase by taking account of the current experimental data on $\Delta M_{B_s}$, $\mathrm{Br}[B_s\rightarrow\mu^+\mu^-]$, $\mathrm{Br}[\bar{B}\rightarrow X_s\gamma]$ and CKM matrix elements as well as theoretical uncertainties. We find that the constraint from the $\mathrm{Br}[B_s\rightarrow\mu^+\mu^-]$ is more stringent than that from the $\mathrm{Br}[\bar{B}\rightarrow X_s\gamma$]. We also obtain the bound for the mass of the VLQ and the strength of the Yukawa couplings related to the FCNC coupling of $b\rightarrow s$ transition. Using the CKM elements which satisfy above constraints, we show how the unitarity is violated on the complex plane.

Measurement of the inclusive electron spectrum from B meson decays and determination of |Vub|

Based on the full BaBar data sample of 466.5 million BBbar pairs, we present measurements of the electron spectrum from semileptonic B meson decays. We fit the inclusive electron spectrum to distinguish Cabibbo-Kobayashi-Maskawa (CKM) suppressed B -> Xu e nu decays from the CKM-favored B -> Xc e nu decays, and from various other backgrounds, and determine the total semileptonic branching fraction BR(B -> X e nu) = (10.34 +- 0.04_stat +- 0.26_syst)%, averaged over B+- and B0 mesons. We determine the spectrum and branching fraction for charmless B -> Xu e nu decays and extract the CKM element |Vub|, by relying on four different QCD calculations based on the heavy quark expansion. While experimentally, the electron momentum region above 2.1GeV/c is favored, because the background is relatively low, the uncertainties for the theoretical predictions are largest in the region near the kinematic endpoint. Detailed studies to assess the impact of these four predictions on the measurements of the electron spectrum, the branching fraction, and the extraction of the CKM matrix element |Vub| are presented, with the lower limit on the electron momentum varied from 0.8GeV/c to the kinematic endpoint. We determine |Vub| using each of these different calculations and find, |Vub| = (3.794 +- 0.107_exp +0.292-0.219_SF +0.078-0.068_theory)*10^{-3} (De Fazio and Neubert), (4.563 +- 0.126_exp +0.230-0.208_SF +0.162-0.163_theory)*10^{-3} (Bosh, Lange, Neubert, and Paz), (3.959 +- 0.104_exp +0.164-0.154_SF +0.042-0.079_theory)*10^{-3} (Gambino, Giordano, Ossola, and Uraltsev), (3.848 +- 0.108_exp +0.084-0.070_theory)*10^{-3} (dressed gluon exponentiation), where the stated uncertainties refer to the experimental uncertainties of the partial branching fraction measurement, the shape function parameters, and the theoretical calculations.

B(s)0-mixing matrix elements from lattice QCD for the Standard Model and beyond

We calculate---for the first time in three-flavor lattice QCD---the hadronic matrix elements of all five local operators that contribute to neutral $B^0$- and $B_s$-meson mixing in and beyond the Standard Model. We present a complete error budget for each matrix element and also provide the full set of correlations among the matrix elements. We also present the corresponding bag parameters and their correlations, as well as specific combinations of the mixing matrix elements that enter the expression for the neutral $B$-meson width difference. We obtain the most precise determination to date of the SU(3)-breaking ratio $\xi = 1.206(18)(6)$, where the second error stems from the omission of charm sea quarks, while the first encompasses all other uncertainties. The threefold reduction in total uncertainty, relative to the 2013 Flavor Lattice Averaging Group results, tightens the constraint from $B$ mixing on the Cabibbo-Kobayashi-Maskawa (CKM) unitarity triangle. Our calculation employs gauge-field ensembles generated by the MILC Collaboration with four lattice spacings and pion masses close to the physical value. We use the asqtad-improved staggered action for the light valence quarks, and the Fermilab method for the bottom quark. We use heavy-light meson chiral perturbation theory modified to include lattice-spacing effects to extrapolate the five matrix elements to the physical point. We combine our results with experimental measurements of the neutral $B$-meson oscillation frequencies to determine the CKM matrix elements $|V_{td}| = 8.00(34)(8) \times 10^{-3}$, $|V_{ts}| = 39.0(1.2)(0.4) \times 10^{-3}$, and $|V_{td}/V_{ts}| = 0.2052(31)(10)$, which differ from CKM-unitarity expectations by about 2$\sigma$. These results and others from flavor-changing-neutral currents point towards an emerging tension between weak processes that are mediated at the loop and tree levels.

Quark flavor mixings from hierarchical mass matrices

In this paper, we extend the Fritzsch ansatz of quark mass matrices while retaining their hierarchical structures and show that the main features of the Cabibbo-Kobayashi-Maskawa (CKM) matrix $V$, including $|V^{}_{us}| \simeq |V^{}_{cd}|$, $|V^{}_{cb}| \simeq |V^{}_{ts}|$ and $|V^{}_{ub}|/|V^{}_{cb}| < |V^{}_{td}|/|V^{}_{ts}|$, can be well understood. This agreement is observed especially when the mass matrices have non-vanishing $(1,3)$ and $(3,1)$ off-diagonal elements. The phenomenological consequences of these for the allowed texture content and gross structural features of `hierarchical' quark mass matrices are addressed from a model independent prospective under the assumption of factorizable phases in these. The approximate and analytical expressions of the CKM matrix elements are derived, and a detailed analysis reveals that such structures are in good agreement with the observed quark flavor mixing angles and the CP-violating phase at the $1\sigma$ level and call upon a further investigation of the realization of these structures from a top-down prospective.

Rephasing invariants of the Cabibbo-Kobayashi- Maskawa matrix

The paper is motivated by the importance of the rephasing invariance of the CKM (Cabibbo-Kobayashi-Maskawa) matrix observables. These observables appear in the discussion of the CP violation in the standard model (Jarlskog invariant) and also in the renormalization group equations for the quark Yukawa couplings. Our discussion is based on the general phase invariant monomials built out of the CKM matrix elements and their conjugates. We show that there exist 30 fundamental phase invariant monomials and 18 of them are a product of 4 CKM matrix elements and 12 are a product of 6 CKM matrix elements. In the main theorem we show that a general rephasing invariant monomial can be expressed as a product of at most five factors: four of them are fundamental phase invariant monomials and the fifth factor consists of powers of squares of absolute values of the CKM matrix elements. We also show that the imaginary part of any rephasing invariant monomial is proportional to the Jarlskog’s invariant J or is 0.

Determination of |V| from exclusive baryonic B decays

We use the exclusive baryonic $B$ decays to determine the Cabibbo-Kobayashi-Maskawa (CKM) matrix element $V_{ub}$. From the relation $|V_{ub}|^2/|V_{cb}|^2=({\cal B}_\pi/{\cal B}_D){\cal R}_{ff}$ based on $B^-\to p\bar p \pi^-$ and $\bar B^0\to p\bar p D^0$ decays, where $|V_{cb}|$ and ${\cal B}_\pi/{\cal B}_D\equiv{\cal B}(B^-\to p\bar p \pi^-)/{\cal B}(\bar B^0\to p\bar p D^0)$ are the data input parameters, while ${\cal R}_{ff}$ is the one fixed by the $B\to p\bar p$ transition matrix elements, we find $|V_{ub}|=(3.48^{+0.87}_{-0.63}\pm 0.40\pm 0.07)\times 10^{-3}$ with the errors corresponding to the uncertainties from ${\cal R}_{ff}$, ${\cal B}_\pi/{\cal B}_D$ and $|V_{cb}|$, respectively. Being independent of the previous results, our determination of $|V_{ub}|$ has the central value close to those from the exclusive $\bar B\to \pi\ell\bar \nu_\ell$ and $\Lambda_b\to p\mu^-\bar \nu_\mu$ decays, but overlaps the one from the inclusive $\bar B\to X_u \ell\bar \nu_\ell$ with the current uncertainties. The extraction of $|V_{ub}|$ in the baryonic $B$ decays is clearly very useful for the complete determination of the CKM matrix elements as well as the exploration of new physics.

Phenomenology of semileptonicB-meson decays with form factors from lattice QCD

We study the exclusive semileptonic $B$-meson decays $B\to K(\pi)\ell^+\ell^-$, $B\to K(\pi)\nu\bar\nu$, and $B\to\pi\tau\nu$, computing observables in the Standard model using the recent lattice-QCD results for the underlying form factors from the Fermilab Lattice and MILC Collaborations. These processes provide theoretically clean windows into physics beyond the Standard Model because the hadronic uncertainties are now under good control for suitably binned observables. For example, the resulting partially integrated branching fractions for $B\to\pi\mu^+\mu^-$ and $B\to K\mu^+\mu^-$ outside the charmonium resonance region are 1-2$\sigma$ higher than the LHCb Collaboration's recent measurements, where the theoretical and experimental errors are commensurate. The combined tension is 1.7$\sigma$. Combining the Standard-Model rates with LHCb's measurements yields values for the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements $|V_{td}|=7.45{(69)}\times10^{-3}$, $|V_{ts}|=35.7(1.5)\times10^{-3}$, and $|V_{td}/V_{ts}|=0.201{(20)}$, which are compatible with the values obtained from neutral $B_{(s)}$-meson oscillations and have competitive uncertainties. Alternatively, taking the CKM matrix elements from unitarity, we constrain new-physics contributions at the electroweak scale. The constraints on the Wilson coefficients ${\rm Re}(C_9)$ and ${\rm Re}(C_{10})$ from $B\to\pi\mu^+\mu^-$ and $B\to K\mu^+\mu^-$ are competitive with those from $B\to K^* \mu^+\mu^-$, and display a 2.0$\sigma$ tension with the Standard Model. Our predictions for $B\to K(\pi)\nu\bar\nu$ and $B\to\pi\tau\nu$ are close to the current experimental limits.

Study of dynamics ofD0→K−e+νeandD0→π−e+νedecays

In an analysis of a 2.92 fb(-1) data sample taken at 3.773 GeV with the BESIII detector operated at the BEPCII collider, we measure the absolute decay branching fractions B(D-0 -> K(-)e(+)nu(e)) = (3.505 +/- 0.014 +/- 0.033)% and B(D-0 -> pi(-)e(+)nu(e)) = (0.295 +/- 0.004 +/- 0.003)%. From a study of the differential decay rates we obtain the products of hadronic form factor and the magnitude of the Cabibbo-Kobayashi-Maskawa (CKM) matrix element f(+)(K)(0)vertical bar V-cs vertical bar = 0.7172 +/- 0.0025 +/- 0.0035 and f(+)(pi)(0)vertical bar V-cd vertical bar = 0.1435 +/- 0.0018 +/- 0.0009. Combining these products with the values of vertical bar V-cs(d)vertical bar from the SM constraint fit, we extract the hadronic form factors f(+)(K)(0) = 0.7368 +/- 0.0026 +/- 0.0036 and f(+)(pi)(0) = 0.6372 +/- 0.0080 +/- 0.0044, and their ratio f(+)(pi)(0)/f(+)(K)(0) = 0.8649 +/- 0.0112 +/- 0.0073. These form factors and their ratio are used to test unquenched lattice QCD calculations of the form factors and a light cone sum rule (LCSR) calculation of their ratio. The measured value of f(+)(K(pi))(0)vertical bar V-cs(d)vertical bar and the lattice QCD value for f(+)(K(pi))(0) are used to extract values of the CKM matrix elements of vertical bar V-cs vertical bar = 0.9601 +/- 0.0033 +/- 0.0047 +/- 0.0239 and vertical bar V-cd vertical bar = 0.2155 +/- 0.0027 +/- 0.0014 +/- 0.0094, where the third errors are due to the uncertainties in lattice QCD calculations of the form factors. Using the LCSR value for f(+)(pi)(0)/f(+)(K)(0), we determine the ratio vertical bar V-cd vertical bar/vertical bar V-cs vertical bar = 0.238 +/- 0.004 +/- 0.002 +/- 0.011, where the third error is from the uncertainty in the LCSR normalization. In addition, we measure form factor parameters for three different theoretical models that describe the weak hadronic charged currents for these two semileptonic decays. All of these measurements are the most precise to date.

Moments in inclusive semileptonic B meson decays at the Belle experiment

Since my return to Austria in the year 2003, I have measured observables in inclusive B meson decays at the Belle experiment and worked together with theorists on the interpretation of these measurements in terms of the Cabibbo–Kobayashi–Maskawa matrix element |Vcb|. And in fact, only this memorial book project made me fully aware of Kolya Uraltsev's ground breaking theoretical contributions to this field. He was not a theorist who talked a lot to an experimentalist like me, and maybe this is not a bad thing for good science. I certainly remember his enthusiasm from conferences, e.g., when I was powerless to keep his presentation to the scheduled time as a session chair at the CKM2005 workshop in San Diego. Still I feel there is some amount of irony in the fact, that I know so little about a person whose work has been so decisive for my career in high energy physics. To commemorate Kolya Uraltsev's pioneering work on inclusive semileptonic B meson decays B → Xcℓν and on the Heavy Quark Expansion (HQE), which has already been paid tribute to in other articles in this volume, I will review the measurement of the electron energy and the hadronic mass moments in B → Xcℓν decays performed at the Belle experiment. These measurements allow to both test his theoretical calculations and to extract |Vcb| and non-perturbative quantities, such as the b-quark mass, from his formulae.

Determination of $$f_+^K(0)$$ f + K ( 0 ) and extraction of $$|V_{cs}|$$ | V c s | from semileptonic $$D$$ D decays

By globally analyzing all existing measured branching fractions and partial rates in different four momentum transfer-squared $$q^2$$ bins of $$D\rightarrow Ke^+\nu _e$$ decays, we obtain the product of the form factor and magnitude of Cabibbo–Kobayashi–Maskawa matrix element $$V_{cs}$$ to be $$f_+^K(0)|V_{cs}|=0.717\pm 0.004$$ . With this product, we determine the $$D\rightarrow K$$ semileptonic form factor $$f_+^K(0)=0.737\pm 0.004\pm 0.000$$ in conjunction with the value of $$|V_{cs}|$$ determined from the standard model global fit. Alternately, with the product together with the input of the form factor $$f_+^K(0)$$ calculated in lattice quantum chromodynamics (LQCD) recently, we extract $$|V_{cs}|^{D\rightarrow Ke^+\nu _e}=0.962\pm 0.005\pm 0.014$$ , where the error is still dominated by the uncertainty of the form factor calculated in LQCD. Combining the $$|V_{cs}|^{D_s^+\rightarrow \ell ^+\nu _\ell }=1.012\pm 0.015\pm 0.009$$ extracted from all existing measurements of $$D^+_s\rightarrow \ell ^+\nu _\ell $$ decays and $$|V_{cs}|^{D\rightarrow Ke^+\nu _e}=0.962\pm 0.005\pm 0.014$$ together, we find the most precisely determined $$|V_{cs}|$$ to be $$|V_{cs}|=0.983\pm 0.011$$ , which improves the accuracy of the PDG’2014 value $$|V_{cs}|^\mathrm{PDG'2014}=0.986\pm 0.016$$ by $$45\,\%$$ .

Form factors for B meson weak decays in QCD light cone sum rules with a chiral current correlator

We report some applications of QCD light cone sum rules (LCSR) to \(B\) meson weak decays. Special emphasis is on estimates of the form factors for \(B\) decays into a pseudoscalar (\(P\))/vector (\(V\)) meson, with a certain chiral current correlator. The main new ingredient, as compared with the case of the standard correlators, is that in the operator product expansion calculations, the contributions due to the twist-3 distribution amplitudes of the related light mesons, which are less known and would bring a larger uncertainty to the calculations with the standard correlators, cancel out fully in the \(B\rightarrow P\) case and do out partially in the \(B\rightarrow V\) one. An important observation, which is similar to that in soft collinear effective theory, is made in twist-3 approximation: whereas only one independent form factor is needed for parameterizing the hadronic matrix elements for a \(B\rightarrow P\) transition induced by all the relevant heavy-light quark currents, there exist two independent form factors under the condition of neglecting the terms suppressed by a factor of \(m_V^2\), for the \(B\rightarrow V\) transition. Therefore, the improved LCSR approach could be of stronger predictive power for the weak form factors. Also, this approach is employed to understand the \(B\rightarrow D\) transitions by introducing a leading twist-2 DA for an energetic \(D\) meson, combined with some of other QCD-based approaches. A detailed QCD next-to-leading order calculation of the \(B\rightarrow \pi \) form factors is presented for an illustrative purpose, and the sum rule results are used to extract the Cabibbo–Kobayashi–Maskawa matrix element \(|V_{ub}|\) from the latest BaBar data.

Measurement of the ratio [formula omitted] in pp collisions at [formula omitted

The ratio of the top-quark branching fractions R=B(t→Wb)/B(t→Wq), where the denominator includes the sum over all down-type quarks (q=b,s,d), is measured in the tt¯ dilepton final state with proton–proton collision data at s=8 TeV from an integrated luminosity of 19.7 fb−1, collected with the CMS detector. In order to quantify the purity of the signal sample, the cross section is measured by fitting the observed jet multiplicity, thereby constraining the signal and background contributions. By counting the number of b jets per event, an unconstrained value of R=1.014±0.003(stat.)±0.032(syst.) is measured, in a good agreement with current precision measurements in electroweak and flavour sectors. A lower limit R>0.955 at the 95% confidence level is obtained after requiring R≤1, and a lower limit on the Cabibbo–Kobayashi–Maskawa matrix element |Vtb|>0.975 is set at 95% confidence level. The result is combined with a previous CMS measurement of the t-channel single-top-quark cross section to determine the top-quark total decay width, Γt=1.36±0.02(stat.)−0.11+0.14(syst.) GeV.

Wolfenstein parametrization at higher order: Seeming discrepancies and their resolution

The precise determination of the Cabibbo–Kobayashi–Maskawa (CKM) matrix elements is extremely important towards the understanding of CP violation. We explicitly study seeming discrepancies between the CKM matrix elements at the higher order of the expansion parameter λ in different Wolfenstein parametrizations derived from different exact parametrizations. A systematic way of resolving the seeming discrepancies is proposed. We find that most of the discrepancies can be naturally resolved by a proper redefinition of the numerically small (of order λ) parameters. Our approach is further applied to the cases for the Wolfenstein-like parametrizations, such as the Qin–Ma parametrization.