QED factorization of two-body non-leptonic and semi-leptonic B to charm decays

The QCD×QED factorization is studied for two-body non-leptonic and semi-leptonic B decays with heavy-light final states. These non-leptonic decays, like $$ {\overline{B}}_{(s)}^0\to {D}_{(s)}^{+}{\pi}^{-} $$ B ¯ s 0 → D s + π − and $$ {\overline{B}}_d^0\to {D}^{+}{K}^{-} $$ B ¯ d 0 → D + K − , are among the theoretically cleanest non-leptonic decays as penguin loops do not contribute and colour-suppressed tree amplitudes are suppressed in the heavy-quark limit or even completely absent. Advancing the theoretical calculations of such decays requires therefore also a careful analysis of QED effects. Including QED effects does not alter the general structure of factorization which is analogous for both semi-leptonic and non-leptonic decays. For the latter, we express our result as a correction of the tree amplitude coefficient a1. At the amplitude level, we find QED effects at the sub-percent level, which is of the same order as the QCD uncertainty. We discuss the phenomenological implications of adding QED effects in light of discrepancies observed between theory and experimental data, for ratios of non-leptonic over semi-leptonic decay rates. At the level of the rate, ultrasoft photon effects can produce a correction up to a few percent, requiring a careful treatment of such effects in the experimental analyses.

Evaluating EYM amplitudes in four dimensions by refined graphic expansion

The recursive expansion of tree level multitrace Einstein-Yang-Mills (EYM) amplitudes induces a refined graphic expansion, by which any tree-level EYM amplitude can be expressed as a summation over all possible refined graphs. Each graph contributes a unique coefficient as well as a proper combination of color-ordered Yang-Mills (YM) amplitudes. This expansion allows one to evaluate EYM amplitudes through YM amplitudes, the latter have much simpler structures in four dimensions than the former. In this paper, we classify the refined graphs for the expansion of EYM amplitudes into N k MHV sectors. Amplitudes in four dimensions, which involve k + 2 negative-helicity particles, at most get non-vanishing contribution from graphs in N k′ (k′ ≤ k) MHV sectors. By the help of this classification, we evaluate the non-vanishing amplitudes with two negative-helicity particles in four dimensions. We establish a correspondence between the refined graphs for single-trace amplitudes with $$ \left({g}_i^{-},{g}_j^{-}\right) $$ g i − g j − or $$ \left({h}_i^{-},{g}_j^{-}\right) $$ h i − g j − configuration and the spanning forests of the known Hodges determinant form. Inspired by this correspondence, we further propose a symmetric formula of double-trace amplitudes with $$ \left({g}_i^{-},{g}_j^{-}\right) $$ g i − g j − configuration. By analyzing the cancellation between refined graphs in four dimensions, we prove that any other tree amplitude with two negative-helicity particles has to vanish.

REMARK ON PION SCATTERING LENGTHS

First it is shown that the tree amplitude for pion–pion scattering in the minimal linear sigma model has an exact expression which is proportional to a geometric series in the quantity [Formula: see text], where mB is the sigma mass which appears in the Lagrangian and is the only a priori unknown parameter in the model. This induces an infinite series for every predicted scattering length in which each term corresponds to a given order in the chiral perturbation theory counting. It is noted that, perhaps surprisingly, the pattern, though not the exact values, of chiral perturbation theory predictions for both the isotopic spin 0 and isotopic spin 2 s-wave pion–pion scattering lengths to orders p2, p4 and p6 seems to agree with this induced pattern. The values of the p8 terms are also given for comparison with a possible future chiral perturbation theory calculation. Further aspects of this approach and future directions are briefly discussed.

IS THERE ANY PUZZLE OF NEW PHYSICS IN B → Kπ DECAYS?

Due to re-parametrization invariance of decay amplitudes, any single new physics (NP) amplitude arising through either the electro-weak penguin (EWP) or the color-suppressed tree amplitude can be embedded simultaneously into both the color-suppressed tree and the EWP contribution in B → Kπ decays. We present a systematic method to extract each standard model (SM)-like hadronic parameter as well as new physics parameters in analytic way, so that one can pinpoint them once experimental data are given. Using the currently available experimental data for B → Kπ modes, we find two possible analytic results: one showing the large SM-like color-suppressed tree contribution and the other showing the large SM-like EWP contribution. The magnitude of the NP amplitude and its weak phase are quite large. For instance, we find |PNP/P′| = 0.39 ± 0.13, φNP = 92° ± 15° and δNP = 7° ± 26°, which are the ratio of the NP-to-SM contribution, the weak and the strong phase of the NP amplitude, respectively. We also investigate the dependence of the NP contribution on the weak phase γ and the mixing induced CP asymmetry of B0 → KSπ0, respectively

WEAK PHASES FROM AMPLITUDE PARAMETRIZATION

We determine the weak phases and the amplitudes by proposing an parametrization to each topologies of amplitudes in power of Wolfenstein parameter λ ~ 0.22. Comparing this parametrization with the experimental data, we can obtain the phase ϕ3 from the B → Kπ data up to theoretical uncertainty of O(λ3) ~ 5%. We find that these solutions indicate a large color-suppressed tree amplitude. The extraction of the phase ϕ2 and ϕ3 are consistent with the global unitarity triangle fit.

A new property of superstring tree amplitude

The N-point correlator of the superstring is decomposed into N – 3 four-point correlators in calculating the N-point tree amplitude.

A NOTE ON STRING AMPLITUDES AT HIGH ENERGY LIMIT

We discuss string amplitudes, in particular tree amplitude of bosonic open strings, using an approximated vertex of the light-cone field theory from the point of view of high energy.