QCD Penguin Amplitude Research Articles

Two-loop non-leptonic penguin amplitude in QCD factorization

We complete the calculation of the QCD penguin amplitude at next-to-next- to-leading order in the QCD factorization approach to non-leptonic B-meson decays. This provides the last missing piece in the computation of the QCD correction to direct CP asymmetries at leading power in the heavy-quark expansion.

Two-loop current–current operator contribution to the non-leptonic QCD penguin amplitude

The computation of direct CP asymmetries in charmless B decays at next-to-next-to-leading order (NNLO) in QCD is of interest to ascertain the short-distance contribution. Here we compute the two-loop penguin contractions of the current–current operators Q1,2 and provide a first estimate of NNLO CP asymmetries in penguin-dominated b→s transitions.

UpdatingB→PP,VPdecays in the framework of flavor symmetry

Current data of charmless B meson decays to two pseudoscalar mesons (PP) and one vector and one pseudoscalar mesons (VP) are analyzed within the framework of flavor SU(3) symmetry, a working principle that we have tested by allowing symmetry breaking factors in the decay amplitudes and found to be a good approximate symmetry. In the PP sector, the color-suppressed tree amplitude is found to be larger than previously known and has a strong phase of ~ -70^o relative to the color-favored tree amplitude. We have extracted for the first time the W-exchange and penguin-annihilation amplitudes. The former has a size of about the QCD-penguin amplitude and a phase opposite to that of the color-favored tree amplitude, while the latter is suppressed in magnitude but gives the dominant contribution to the $B_s^0 \to \pi^+ \pi^-$ and $\pi^0 \pi^0$ decays. In the VP sector, one striking feature is that the color-suppressed tree amplitude with the spectator quark ending up in the vector meson has a large size and a strong phase of ~ -90^o relative to the color-favored tree amplitudes. The associated electroweak penguin amplitude also has a similar strong phase and a magnitude comparable to the corresponding QCD penguin amplitude. This leads to a large branching fraction of order 10^{-6} for $B_s^0 \to \phi \pi^0$. In contrast, the color-suppressed tree, QCD penguin, and electroweak penguin amplitudes with the spectator quark ending up in the pseudoscalar meson have magnitudes more consistent with naive expectations. Besides, current data are not sufficiently precise for us to fix the W-exchange amplitudes. For both the PP and VP sectors, predictions of all the decay modes are made based upon our preferred fit results and compared with data and those made by perturbative approaches. We have identified a few observables to be determined experimentally in order to discriminate among theory calculations.

SU(3) symmetry breaking andCPviolation inD→PPdecays

Evidence of CP violation in the charm sector has been observed recently by the LHCb and CDF Collaborations. Adopting the topological diagram approach, we study flavor SU(3) symmetry breaking effects in the weak decay tree amplitudes of singly Cabibbo-suppressed $D\to PP$ decays. The symmetry breaking in the color-allowed and color-suppressed amplitudes is estimated with the help of the factorization ansatz, while that in the $W$-exchange amplitude is done by fitting to related branching fraction data. We find that the $W$-exchange amplitudes stay in the second quadrant relative to the color-allowed tree amplitude, albeit there are two possibilities for one type of $W$-exchange amplitude. The weak decay penguin amplitudes, on the other hand, are evaluated within the framework of QCD factorization. Using the input of topological tree amplitudes extracted from the Cabibbo-favored decay modes and the perturbative results for QCD penguin amplitudes, we make predictions for the branching fractions and CP asymmetries of singly Cabibbo-suppressed modes. The predictions of branching fractions are generally improved from those in the SU(3) limit. We conclude that the direct CP asymmetry difference between $D^0 \to K^+ K^-$ and $D^0 \to \pi^+ \pi^-$ is about $-(0.139\pm 0.004)%$ and $-(0.151\pm 0.004)%$ for the two solutions of $W$-exchange amplitudes, respectively. We also find that the CP asymmetry of $D^0\to K^0\ov K^0$ dominated by the interference between $W$-exchange amplitudes ranges from $-0.62\times 10^{-3}$ to $-1.82\times 10^{-3}$. We study phenomenological implications of two new physics scenarios for explaining the observed CP asymmetry in the charm sector, one with large penguin amplitudes and the other with a large chromomagnetic dipole operator. We find that the two scenarios can be discriminated by the measurements of CP asymmetries of a set of decay modes.

CPviolation and kaon-pion interactions inB→Kπ+π−decays

We study $CP$ violation and the contribution of the strong kaon-pion interactions in the three-body $B\ensuremath{\rightarrow}K{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ decays. We extend our recent work on the effect of the two-pion $S$- and $P$-wave interactions to that of the corresponding kaon-pion ones. The weak amplitudes have a first term derived in QCD factorization and a second one as a phenomenological contribution added to the QCD penguin amplitudes. The effective QCD coefficients include the leading order contributions plus next-to-leading order vertex and penguins corrections. The matrix elements of the transition to the vacuum of the kaon-pion pairs, appearing naturally in the factorization formulation, are described by the strange $K\ensuremath{\pi}$ scalar ($S$-wave) and vector ($P$-wave) form factors. These are determined from Muskhelishvili-Omn\`es coupled channel equations using experimental kaon-pion $T$-matrix elements, together with chiral symmetry and asymptotic QCD constraints. From the scalar form factor study, the modulus of the ${K}_{0}^{*}(1430)$ decay constant is found to be $(32\ifmmode\pm\else\textpm\fi{}5)\text{ }\text{ }\mathrm{MeV}$. The additional phenomenological amplitudes are fitted to reproduce the $K\ensuremath{\pi}$ effective mass and helicity angle distributions, the $B\ensuremath{\rightarrow}{K}^{*}(892)\ensuremath{\pi}$ branching ratios and the $CP$ asymmetries of the recent data from Belle and BABAR collaborations. We use also the new measurement by the BABAR group of the phase difference between the ${B}^{0}$ and ${\overline{B}}^{0}$ decay amplitudes to ${K}^{*}(892)\ensuremath{\pi}$. Our predicted ${B}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}{K}_{0}^{*}(1430){\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}$, ${K}_{0}^{*}(1430)\ensuremath{\rightarrow}{K}^{\ifmmode\pm\else\textpm\fi{}}{\ensuremath{\pi}}^{\ensuremath{\mp}}$ branching fraction, equal to $(11.6\ifmmode\pm\else\textpm\fi{}0.6)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$, is smaller than the result of the analyzes of both collaborations. For the neutral ${B}^{0}$ decays, the predicted value is $(11.1\ifmmode\pm\else\textpm\fi{}0.5)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$. In order to reduce the large systematic uncertainties in the experimental determination of the $B\ensuremath{\rightarrow}{K}_{0}^{*}(1430)\ensuremath{\pi}$ branching fractions, a new parametrization is proposed. It is based on the $K\ensuremath{\pi}$ scalar form factor, well constrained by theory and experiments other than those of $B$ decays.

Spectator scattering at NLO in non-leptonic B decays: Leading penguin amplitudes

We complete the computation of the 1-loop ( α s 2 ) corrections to hard spectator scattering in non-leptonic B decays at leading power in Λ / m b by evaluating the penguin amplitudes. This extends the knowledge of these next-to-next-to-leading-order contributions in the QCD factorization formula for B decays to a much wider class of final states, including all pseudoscalar–pseudoscalar, pseudoscalar–vector, and longitudinally polarized vector–vector final states, except final states with η or η ′ mesons. The new 1-loop correction is significant for the colour-suppressed amplitudes, but turns out to be strongly suppressed for the leading QCD penguin amplitude α 4 p . We provide numerical values of the phenomenological P / T and C / T amplitude ratios for the ππ, πρ and ρρ final states, and discuss corrections to several relations between electroweak penguin and tree amplitudes.

Polarization in [formula omitted] decays

Factorizable amplitudes in B decays to light vector meson pairs give a longitudinal polarization satisfying 1−fL=O(1/mb2). This remains formally true when non-factorizable graphs are included in QCD factorization, and is numerically realized in B→ρρ. In ΔS=1 decays a QCD penguin annihilation graph can effectively contribute at leading power to the transverse and longitudinal amplitudes. The observed longitudinal polarization, fL(B→ϕK*)≈50%, can therefore be accounted for in the SM. The ratio of transverse rates Γ⊥/Γ∥ provides a sensitive test for new right-handed currents. The transverseb→sg dipole operator amplitudes are highly suppressed. CP violation measurements can therefore discriminate between new contributions to the dipole and four quark operators. SU(3)F violation in QCD penguin amplitudes can easily be O(1), in general, due to annihilation. Implications for B→ρK* polarization and New Physics searches are pointed out.

An analysis of the time-dependent CP asymmetryin $B \to \pi \pi$ decays in the standard model

Measurements of the time-dependent CP-asymmetry in the decay \(B^0_d (t)\to \pi^ + \pi^-\) and its charge conjugate by the BELLE and BABAR collaborations currently yield \(C_{\pi \pi}^{ + -} = -0.46 \pm 0.13\) and \(S_{\pi \pi}^{ + -} = -0.74 \pm 0.16\), characterizing the direct and mixing-induced CP-asymmetries, respectively. We study the implication of these measurements on the CKM phenomenology taking into account the available information in the quark mixing sector. Our analysis leads to the results that the ratio |Pc/Tc| involving the QCD-penguin and tree amplitudes and the related strong phase difference \(\delta_c = \delta_c^P - \delta_c^T\) in the \(B^0_d/\bar{B}_d^0 \to \pi^ + \pi^-\) decays are quite substantial. Using the isospin symmetry to constrain |Pc/Tc| and \(\cos (2\theta)\), where \(2 \theta\) parameterizes the penguin-induced contribution, we present a fit of the current data including the measurements of \(S_{\pi \pi}^{ + -}\) and \(C_{\pi \pi}^{ + -}\). Our best-fits yield \(\alpha = 92^\circ\), \(\beta = 24^\circ\), \(\gamma = 64^\circ\), \(\vert P_c/T_{\mathrm c}\vert = 0.77\), and \(\delta_c = -43^\circ\). At 68% C.L., the ranges are \(81^\circ \leq \alpha \leq 103^\circ\), \(21.9^\circ \leq \beta \leq 25.5^\circ\), \(54^\circ \leq \gamma \leq 75^\circ\), \(0.43 \leq \vert P_c/T_{\mathrm c}\vert \leq 1.35\) and \(-64^\circ \leq \delta_c \leq -29^\circ\). Currently en vogue dynamical approaches to estimate the hadronic matrix elements in \(B \to \pi \pi\) decays do not provide a good fit of the current data.

Two-body charmlessBdecays involvingηandη′

We discuss the implications of recent experimental data for B decays into two pseudoscalar mesons, with an emphasis on those with $\ensuremath{\eta}$ and ${\ensuremath{\eta}}^{\ensuremath{'}}$ in the final states. Applying a U-spin argument, we show that tree and penguin amplitudes, both in ${B}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}\ensuremath{\eta}$ and in ${B}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\eta}}^{\ensuremath{'}},$ are of comparable magnitudes. Nontrivial relative weak and strong phases between the tree-level amplitudes and penguin-loop amplitudes in the ${B}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\eta}$ modes are extracted. We predict the possible values for the averaged branching ratio and CP asymmetry of the ${B}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}{\ensuremath{\eta}}^{\ensuremath{'}}$ modes. We test the assumption of a singlet-penguin amplitude with the same weak and strong phases as the QCD penguin amplitude in explaining the large branching ratios of ${\ensuremath{\eta}}^{\ensuremath{'}}K$ modes, and show that it is consistent with current branching ratio and CP asymmetry data of the ${B}^{+}\ensuremath{\rightarrow}({\ensuremath{\pi}}^{0},\ensuremath{\eta},{\ensuremath{\eta}}^{\ensuremath{'}}{)K}^{+}$ modes. We also show that the strong phases of the singlet-penguin and tree-level amplitudes can be extracted with further input of electroweak penguin contributions and a sufficiently well-known branching ratio of the $\ensuremath{\eta}{K}^{+}$ mode. Using $\mathrm{SU}(3)$ flavor symmetry, we also estimate required data samples to detect modes that have not yet been seen.

CLEO measurement ofB→π+π−and determination of weak phaseα

The CLEO collaboration has recently reported a (first) measurement of $BR (B \to \pi^+ \pi^-) = (4.7^{+1.8}_{-1.5} \pm 0.6) \times 10^{-6}$. We study, using recent results on QCD improved factorization, the implications of this measurement for the determination of the CKM phase $\alpha$ and also for the rate for $B \to \pi^0 \pi^0$. If the $B \to \pi^-$ form factor is large $(\stackrel{>}{\sim} 0.3)$, then we find that the CLEO measurement favors small $|V_{ub}/V_{cb}|$ so that the expected error (due to neglecting the QCD penguin amplitude) in the measurement of $\alpha$ using {\em only} the time-dependent analysis of the decay $B \to \pi^+ \pi^-$ is large $\sim 15^{\circ}$. However, if $|V_{ub}/V_{cb}|$ is known, then it is possible to determine the correct value of $\sin 2 \alpha$.

Penguin zoology in B → ππ and the extraction of the CKM angle α

We reanalyze the decay B d → π + π − without assuming dominance of QCD penguins with internal top-quark exchanges. In that case the weak phase of the b → d QCD penguin amplitude is not related in a simple way to the CKM angle β. Nevertheless it is still possible to extract the CKM angle α in a transparent and almost model-independent way from the time-dependent CP-violating asymmetry of the transition B d → π + π −. Aside from that CP asymmetry this approach needs as an input only amplitudes of decays with branching ratios of order 10 −5 and will thus be well within reach at future B-factories.