Large N_c QCD and Weak Matrix Elements

por Eduardo de Rafael (CPT Marseille)

miércoles 10 mar. 1999 16:30 → 17:30 Europe/Madrid

Sala Seminarios (Dpto. de Fisica Teorica (UVEG))

The first part of my talk will be an overview of the progress made and the problems which remain in deriving an effective Lagrangian which describes the non--leptonic weak interactions of the Nambu--Goldstone degrees of freedom ($K$, $\pi$ and $\eta$) of the spontaneous $SU(3)_{L}\times SU(3)_{R}$ symmetry breaking in the Standard Model. I shall show, with examples, how the coupling constants of the $\vert\Delta S\vert=1$ and $\vert\Delta S\vert=2$ effective low energy Lagrangian are given by {\it integrals} of Green's functions of QCD currents and density currents, while those of the strong sector (QCD only) are more simply related to the coefficients of the Taylor expansions of QCD Green's functions. The study of these issues within the framework of the $1/N_c$ expansion, and in the {\it lowest meson dominance} approximation described in ref.~\cite{PPdeR98}, seems a very promising path. The second part of my talk will be dedicated to a review of the properties of the $\Pi_{LR}(Q^2)$ correlation function in the large--$N_c$ limit as recently discussed in ref.~\cite{KdeR97}. This is the correlation function which governs the {\it electroweak} $\pi^{+}-\pi^{0}$ mass difference. Following the discussion of ref.~\cite{KPdeR98}, I shall show how the calculation of this observable, which requires non--trivial contributions from next--to--leading terms in the $1/N_c$ expansion, provides an excellent theoretical laboratory for studying issues of long-- and short-- distance matching in calculations of weak matrix elements of four--quark operators. The third part of my talk will be dedicated to recent work reported in ref~\cite{KPdeR98d}, where it is shown that the $K\rightarrow\pi\pi$ matrix elements of the four--quark operator $Q_7$, generated by the electroweak penguin--like diagrams of the Standard Model, can be calculated to first non--trivial order in the chiral expansion and in the $1/N_c$ expansion. I shall compare the results to recent numerical evaluations from lattice--QCD. \begin{thebibliography}{12} \bibitem{PPdeR98} S.~Peris, M.~Perrottet and E.~de Rafael, JHEP {\bf 05} (1998) 011. \bibitem{KdeR97} M.~Knecht and E.~de Rafael, Phys. Lett. {\bf B424} (1998) 335. \bibitem{KPdeR98} M.~Knecht, S.~Peris and E.~de Rafael, Phys. Lett.{\bf B443} (1998) 255. \bibitem{KPdeR98d} M.~Knecht, S.~Peris and E.~de Rafael, hep-ph/9812471. \end{thebibliography}