Penguin Contributions Research Articles

Elimination of penguin contributions to CP asymmetries in B decays through isospin analysis

Isospin symmetry in B d 0→ π + π −, B d 0→ π 0 π 0, B +→ π + π 0 has been shown to remove the theoretical uncertainty due to penguin diagrams in the predictions for CP asymmetries in these decays. We apply this method in a critical analysis to two other examples with a larger number of isospin-related processes, B→ πK and B→ πϱ. It is shown that certain ambiguities in the determination of the weak phase α remain in these cases due to the larger number of processes. As a consequence of this and of the larger accomulated experimental error, the determination of α from these processes is expected to be less accurate than from B→ ππ.

Interference effects in Kη and Kη′ decay modes of heavy mesons. Clues to understanding weak transitions and CP violation

Interference between amplitudes producing η and η′ mesons via their strange and nonstrange components give relative suppression factors of an order of magnitude in certain diagrams and can be used to identify penguin contributions to B→Kη and B→Kη′ decays as well as annihilation diagram contributions to corresponding D decays. Similar suppression factors also arise in D 0→π 0η and D 0→π 0η′ decays and in charmonium decays to K +K −η and K +K −η′.

B-meson exclusive decays into baryons

A detailed description is presented of the method used and the results obtained for measuring various decay widths of the B-mesons into baryon-antibaryon pairs: (i) into two light baryons: B → p p, Δ p, … , (ii) into one light and one charmed baryon: B → p Σ c , p Λ c ,… , (iii) into two charmed baryons: B → Σ c , Ξ c , Λ c Ξ c ,… , (iv) the decays induced by the b → s and b → d penguin contributions: B → p Σ, p Λ, ΔΣ,… . As a by-product, we present the result for the B-meson decay constant, f B, obtained from the new QCD sum rule and the values of many other analogous constants for transitions of various currents into baryons.

The electromagnetic penguin contribution to ϵ′/ ϵ for large top quark mass

We investigate the electromagnetic penguin contribution to ϵ′/ ϵ when the top quark mass is large. We find it depends sensitively on the top quark mass.

The electromagnetic penguin contribution to ∈′/∈ for large top quark mass

Abstract We evaluate the electromagnetic penguin contribution to ∈′ under the assumption that the top quark mass can be large. We find a significant effect for top masses larger than the mass of the W. We also consider other operators which can affect the standard model prediction or χ′.

Review of weak matrix elements on the lattice

We review the status of attempts to calculate weak matrix elements on the lattice. Both Wilson and staggered fermion techniques are described, with special emphasis given to the inherent difficulties of each method and how these difficulties are being overcome. In particular we discuss the “mixing problem” ( i.e., how different lattice operators need to be combined to form the relevant continuum operators), and the complications introduced therein by chiral breaking (Wilson fermions) and flavor breaking (staggered fermions). We present and critically analyze results from various groups on K 0 − K 0 mixing (“B parameters”), K → ππ ΔI = 1 2 and 3 2 amplitudes, penguin and electromagnetic penguin contributions to ϵ 1, and heavy-meson mixing and pseudoscalar decay constants. For the electromagnetic penguins and for the heavy-meson parameters (for D mesons if not B mesons), the groups are in good agreement, and the systematic errors, though fairly large (≈20 to 30%), are reasonably well understood. In the other cases, the picture is much less clear. Still, the evidence seems to favor three qualitative statements: (1) there is a large ΔI = 1 2 enhancement in K → ππ decays which comes from “eye” graphs and is of the right order of magnitude, (2) “large” values (≈0.5 to 1.2) are preferred for B ll in the K system, and (3) the lattice K → ππ ΔI = 3 2 amplitude seems to be a factor of ≈2 larger than one would expect from experiment, at least when isospin-violating and electromagnetic effects are ignored.

Isospin breaking effects on the [formula omitted] amplitudes

Without including isospin breaking effects, the experimental data of K→2π yield A 0/ A 2 = (22.2±0.1) and δ 0− δ 2 = (56.5±3.0)°. The latter is substantially larger than the isospin phase shift difference determined from strong-interaction ππ scatterings; this seems to imply the importance of isospin violation. However, we show that effects of the π−η−η′ mixing and the electromagnetic penguin contributions do not modify δ 0− δ 2, but they account for about 17% of the observed K +→π +π 0 amplitude.

On the s→d self-energy transition ≈ αsGF and the relevance for the [formula omitted] and ϵ′/ ge puzzles

I have considered the s→d self-energy transition diagrams ≈ α s G F- or “self-penguins” - recently proposed by Shabalin as t he explanation of the ΔI= 1 2 rule for K→ππ decays. The effect of such a self-energy transition on physical amplitudes is considered in terms of a chiral quark loop model for K→π. A self-penguin contribution proportional to the dot product of the pion and kaon four-momenta is found. However, this contribution can at most account for 5% of the observed Δ I= 1 2 amplitude. The self-penguin contribution to the CP-violating quantity ϵ′/ ϵ calculated within the same framework could be as large as half of the standard penguin contribution.

On the contribution of electromagnetic penguins to ϵ′

I discuss the importance of the electromagnetic penguin (EMP) contribution to ϵ′. I confirm the corrections to earlier calculations found by Buras and Gerard (BG). Incorporating these corrections, I calculate the Wilson coefficients of the EMP operators using the full anomalous dimension matrices, and also using the large- N (number of colors) approach of BG. I disagree with BG on the coefficient of the EMP operator dominant at large N: my result is of opposite sign and smaller in magnitude. This means that for large N the EMP contribution increases ϵ′, but by less than 1%. I also find that this Wilson coefficient is poorly estimated in the large- N approach. I agree with the BG result for the coefficient of the EMP operator subdominant at large N. This coefficient is large, and is well estimated using the large- N anomalous dimension matrices. I emphasize that the crucial factor determining the EMP contribution to ϵ′ is the size of the hadronic matrix elements of the subdominant EMP operator. Though suppressed at large N, it increases ϵ′ by 5–20% if one assumes vacuum insertion approximation, and by up to 100% if one uses the results from a recent lattice calculation.

Isospin breaking contributions to ϵ′/ϵ

We reanalyze the π 0−η−η′ mixing and the QED Penguin operator contributions to ϵ′ ϵ , using the 1 N expansion to estimate the relevant hadronic matrix elements. We find that the existing analyses of the QED Penguin operators contain errors both in the Wilson coefficient functions and in the corresponding hadronic matrix elements. The true QED Penguin operator contributions to ϵ′/ϵ turn out to be substantially smaller than previously estimated, and consequently the dominant isospin breaking effect on ϵ′ ϵ comes from the π 0−η−η′ mixing. Taking all these effects into account we find that they reduce the standard model estimate of ϵ′ ϵ based on the pure QCD Penguin contribution by roughly 25%, which should be contrasted with a factor of two or three reduction found by Donoghue et al. We also emphasize that in addition to the value of the top quark mass, a better determination of the strange quark mass is required before a firm value for ϵ′/ϵ within the standard model can be obtained.

The K → ππ decays in the large- N limit quark evolution

We use the large- N limit to obtain a consistent picture of the K → ππ amplitudes, A(K → ππ). We emphasize that the full evaluation of A(K → ππ) should include two contributions: (i) the evolution from M W down to μ ≅ O(1 GeV) done in the quark picture and (ii) the evolution from μ ≅ O(1 GeV) down to low energy O( λ QCD, m π , m K) which must be done in the meson picture. Using chiral perturbation theory at the tree level, we confirm that the matrix elements of the dominant Q 6 penguin operator are enhanced by an order of magnitude over those of the current-current operators. We identify an additional (with respect to standard estimates) enhancement of the penguin contributions to the real parts of A(K → ππ) relevant for the ΔI = 1 2 rule, which comes from an imcomplete GIM cancellation above the charm quark mass. The corresponding contributions to the imaginary parts and to the ratio ε' ε are found to be slightly suppressed. Performing the short distance analysis to first nontrivial order in 1/ N, we demonstrate its validity and simplicity. We obtain encouraging results for A(K → ππ) amplitudes and point out that further improvements can be obtained through the inclusion of the low energy meson evolution which is presented in a separate publication. We find ε′ ε ≅ 5 · 10 −3 for m t = 40 GeV.

Nonleptonic K decays in broken SU(3)×SU(3)

Using the soft-pion theorem and chiral lagrangian technique, it is shown that terms first order in SU(3)×SU(3) breaking, obtained from the penguin graphs in the vacuum insertion approxiamtion, contribute to the K decay amplitudes only through their momentum-dependent part, which is O( m K 2 ʌ 2 ) relative to the leading terms. Neglecting this correction, it is found that only 25% of the ΔI= 1 2 amplitude can be accounted for by the short-distance and penguin contributions.

The [formula omitted] rule in the large- N limit

The large- N limit is used to obtain a consistent picture of the K → ππ matrix elements. It is confirmed that the dominant Q 6 penguin matrix elements are enhanced by an order of magnitude compared to those of the current-current operators and that this result is consistent with chiral perturbation theory. An additional enhancement of the penguin contributions is identified with respect to standard estimates which comes from an incomplete GIM cancellation above the charm quark mass. Performing the short-distance analysis to first nontrivial order in 1 N its validity and simplicity is demonstrated. Encouraging results are obtained for A(K → ππ) amplitudes and it is pointed out that further improvements can be obtained through the inclusion of the low-energy evolution which necessarily must be done in a meson rather than a quark picture.

Formula omitted] expansion for kaons

We calculate various parameters of the kaon system in the leading order in 1/ N expansion, ( N being the number of colors). We find that dropping the non-leading terms enhances the ratio A(K → ππ ( I = 0))/ A (K → ππ ( I = 2)) by an additional factor 1.5 which together with the usual short-distance enhancement and penguin contributions helps in explaining the ΔI = 1 2 rule. The B parameter is found to be 3 4 , the long-distance contribution to the K L − K S mass difference ΔM is positive and amounts to 30% of the measured value. The corresponding long-distance contribution to ε is at most 10%. The experimental data for ε and ΔM can easily be reproduced but ε′/ε is found to be O(10 −2) if the ΔI = 1 2 rule is to be respected. The K L → γγ amplitude is reproduced and gives rise to a small CP-violating effect.

On the short-distance double penguin contribution to the K 0- [formula omitted] mixing

We investigate the short-distance contribution of the double penguin box diagram to the K 0- K 0 mixing. Compared with previous work of other authors, the present paper takes into account (i) the momentum dependence of the box loop, (ii) both the non-local part of the penguin and the previously considered local part and (iii) the crossed diagrams, which make a class of double penguin diagrams complete. With such new ingredients, we arrive at the conclusion that the SD part of the double penguin is physically irrelevant. After eliminating this particular mechanism for the K 0- K 0 mixing, we point out other potentially relevant mechanisms.

Measurement of the weak neutral current via nuclear parity violation.

Present low-energy experiments which detect parity violation in nuclei are analyzed, and a consistent set of weak couplings is extracted. Using symmetries and the quark model, it is shown that a parity-violating weak-neutral-current contribution is required by the data, and we extract values ${\mathrm{sin}}^{2}$${\mathrm{\ensuremath{\theta}}}_{\mathrm{W}}$=0.0\ifmmode\pm\else\textpm\fi{}0.4 or \ensuremath{\rho}=1.1\ifmmode\pm\else\textpm\fi{}0.2, consistent with other determinations. The error quoted is statistical only, and does not include a possibly sizable theoretical uncertainty. On the other hand, if we assume the world-average values for \ensuremath{\rho} and ${\ensuremath{\theta}}_{W}$, we can obtain from the data (with minimal quark-model assumptions) the matrix element of an operator which is closely related to operators involved in the so-called ``penguin'' diagrams of \ensuremath{\Delta}S=1 hyperon decay. The matrix element so obtained is consistent with that calculated in a modified form of the vacuum-saturation technique, and has a magnitude such that the penguin contributions to hyperon decays are moderate but not dominant.

Bilinear terms in the penguin contribution to K → 2 π

We show that the chiral constraints on the penguin K- π matrix element are satisfied by a cancellation between the well known vacuum insertion estimate of Shifman, Vainshtein and Zakharov and currently omitted matrix elements of bilinear operators. We also show how the usual factorization prescription for K → 2 π does satisfy the chiral constraints.

Penguins and the parity-violating Ω − → Ξ −γ and Ξ − → Σ −γ modes

The Ω − → Ξ −γandΞ − → Σ −γ decays are analyzed in a non-relativistic quark model with negative parity baryons as intermediate states, the latter being uniquely specified by exact weak and e.m. “selection rules”. Ω − → Ξ −γ, for which an upper limit on the branching ratio (< 3.1 × 10 −3) was recently reported by Bourquin et al., is found to hold promise as the best choice for the estimation of the magnitude of the Wilson coefficients for the penguin contributions in the QCD corrected non-leptonic hamiltonian.

Penguins in ΔS = 1 nonleptonic weak decays

We discuss the use of the operator product expansion in computing nonleptonic weak decays. The estimation of “penguin” contributions is improved by a careful treatment of the u−c cancellation. Using the vacuum insertion technique and equations of motion to estimate the operator matrix element we find the penguin contribution is only 1 10 of the experimentally observed ΔI = 1 2 amplitude in kaon or hyperon decays.