Matrix Elements Of Four-fermion Operators Research Articles

Branching ratios and directCPasymmetries inD→PVdecays

We study the two-body hadronic $D\to PV$ decays, where $P$ ($V$) denotes a pseudoscalar (vector) meson, in the factorization-assisted topological-amplitude approach proposed in our previous work. This approach is based on the factorization of short-distance and long-distance dynamics into Wilson coefficients and hadronic matrix elements of four-fermion operators, respectively, with the latter being parametrized in terms of several nonperturbative quantities. We further take into account the $\rho$-$\omega$ mixing effect, which improves the global fit to the branching ratios involving the $\rho^0$ and $\omega$ mesons. Combining short-distance dynamics associated with penguin operators and the hadronic parameters determined from the global fit to branching ratios, we predict direct $CP$ asymmetries. In particular, the direct $CP$ asymmetries in the $D^0\to K^0\overline{K}^{*0},~\overline{K}^0K^{*0}$, $D^+\to\pi^+\rho^0$, and $D_s^+\to K^+\omega,~K^+\phi$ decays are found to be of ${\cal O}(10^{-3})$, which can be observed at the LHCb or future Belle II experiment. We also predict the $CP$ asymmetry observables of some neutral $D$ meson decays.

Branching ratios and directCPasymmetries inD→PPdecays

We propose a theoretical framework for analyzing two-body nonleptonic $D$ meson decays, based on the factorization of short-distance (long-distance) dynamics into Wilson coefficients (hadronic matrix elements of four-fermion operators). The parametrization of hadronic matrix elements in terms of several nonperturbative quantities is demonstrated for the $D\to PP$ decays, $P$ denoting a pseudoscalar meson. We consider the evolution of Wilson coefficients with energy release in individual decay modes, and the Glauber strong phase associated with the pion in nonfactorizable annihilation amplitudes, that is attributed to the unique role of the pion as a Nambu-Goldstone boson and a quark-anti-quark bound state simultaneously. The above inputs improve the global fit to the branching ratios involving the $\eta'$ meson, and resolves the long-standing puzzle from the $D^0\to\pi^+\pi^-$ and $D^0\to K^+K^-$ branching ratios, respectively. Combining short-distance dynamics associated with penguin operators and the hadronic parameters determined from the global fit to branching ratios, we predict direct CP asymmetries, to which the quark loops and the scalar penguin annihilation give dominant contributions. In particular, we predict $\Delta A_{\rm CP}\equiv A_{\rm CP}(K^+K^-)-A_{\rm CP}(\pi^+\pi^-)=-1.00\times 10^{-3}$, lower than the LHCb and CDF data.

Electromagnetic logarithms in [formula omitted

The B ¯ → X s ℓ + ℓ − decay rate is known at the next-to-next-to-leading order in QCD. It is proportional to α em ( μ ) 2 and has a ± 4 % scale uncertainty before including the O ( α em ln ( M W 2 / m b 2 ) ) electromagnetic corrections. We evaluate these corrections and confirm the earlier findings of Bobeth et al. Furthermore, we complete the calculation of logarithmically enhanced electromagnetic effects by including also QED corrections to the matrix elements of four-fermion operators. Such corrections contain a collinear logarithm ln ( m b 2 / m ℓ 2 ) that survives integration over the low dilepton invariant mass region 1 GeV 2 < m ℓ ℓ 2 < 6 GeV 2 and enhances the integrated decay rate in this domain. For the low- m ℓ ℓ 2 integrated branching ratio in the muonic case, we find B ( B → X s μ + μ − ) = ( 1.59 ± 0.11 ) × 10 −6 , where the error includes the parametric and perturbative uncertainties only. For B ( B → X s e + e − ) , in the current BaBar and Belle setups, the logarithm of the lepton mass gets replaced by angular cut parameters and the integrated branching ratio for the electrons is expected to be close to that for the muons.

Matrix elements of four-fermion operators with quenched Wilson fermions

We present results for the matrix elements of a variety of four-fermion operators calculated using quenched Wilson fermions. Our simulations are done on 170 lattices of size ${32}^{3}\ifmmode\times\else\texttimes\fi{}64$ at $\ensuremath{\beta}=6.0$. We find ${B}_{K}=0.74\ifmmode\pm\else\textpm\fi{}0.04\ifmmode\pm\else\textpm\fi{}0.05$, ${B}_{D}=0.78\ifmmode\pm\else\textpm\fi{}0.01$, ${B}_{7}^{3/2}=0.58\ifmmode\pm\else\textpm\fi{}{0.02}_{\ensuremath{-}0.03}^{+0.07}$, ${B}_{8}^{3/2}=0.81\ifmmode\pm\else\textpm\fi{}{0.03}_{\ensuremath{-}0.02}^{+0.03}$, with all results being in the NDR scheme at $\ensuremath{\mu}=2$ GeV. We also calculate the $B$ parameter for the operator ${\mathcal{Q}}_{s},$ which is needed in the study of the difference of $B$-meson lifetimes. Our best estimate is ${B}_{S}(\mathrm{NDR},\ensuremath{\mu}=1/a=2.33 \mathrm{GeV})$ $=0.80\ifmmode\pm\else\textpm\fi{}0.01$. This is given at the lattice scale since the required two-loop anomalous dimension matrix is not known. In all these estimates, the first error is statistical, while the second is due to the use of truncated perturbation theory to match continuum and lattice operators. Errors due to quenching and lattice discretization are not included. We also present new results for the perturbative matching coefficients, extending the calculation to all Lorentz scalar four-fermion operators, and using NDR as the continuum scheme.

Rigorous QCD analysis of inclusive annihilation and production of heavy quarkonium.

A rigorous QCD analysis of the inclusive annihilation decay rates of heavy quarkonium states is presented. The effective-field-theory framework of nonrelativistic QCD is used to separate the short-distance scale of annihilation, which is set by the heavy quark mass M, from the longer-distance scales associated with quarkonium structure. The annihilation decay rates are expressed in terms of nonperturbative matrix elements of four-fermion operators in nonrelativistic QCD, with coefficients that can be computed using perturbation theory in the coupling constant ${\mathrm{\ensuremath{\alpha}}}_{\mathit{s}}$(M). The matrix elements are organized into a hierarchy according to their scaling with v, the typical velocity of the heavy quark. An analogous factorization formalism is developed for the production cross sections of heavy quarkonium in processes involving momentum transfers of order M or larger. The factorization formulas are applied to the annihilation decay rates and production cross sections of S-wave states at next-to-leading order in ${\mathit{v}}^{2}$ and P-wave states at leading order in ${\mathit{v}}^{2}$.

Weak annihilation and the end-point spectrum in semileptonic B decays

A general relationship is formulated between the contributions of ‘Weak Annihilation’ (WA) to inclusive semileptonic decays of heavy flavour hadrons and the matrix elements of four-fermion operators. We argue that non-perturbative contributions from low-energy hadronic final states provide the dominant impact of WA on the semileptonic b → u width of B − mesons. In that case WA affects the lepton spectrum mainly in its end-point region - i.e. beyond the kinematical boundary for b → c transitions - and we expect the latter to look quite different in B − than in B d decays. At present we are unable to make a truly quantitative prediction; yet a detailed experimental comparison of the B − and the B d lepton spectra will enable us to determine the size of the relevant matrix elements and to check directly to what extent the factorization approximation works here. Using universal analyticity properties of the decay amplitude we analyse WA in the perturbative regime and rederive our earlier results about the absence of a power enhancement due to gluon emission in a more general way. We also address the problem of the invariant mass of hadrons in the final state in semileptonic B decays and identify non-trivial cancellations in 〈 M hadr 2〉.

Weak interactions on the lattice

We show that lattice QCD can be used to evaluate the matrix elements of four-fermion operators which are relevant for weak decays. A firrst comparison between the results obtained on the lattice and other determinations are also presented.