The Vector and Scalar Form Factors of the Pion to Two Loops

We calculate the vector and scalar form factors of the pion to two loops in Chiral Perturbation Theory. We estimate the unknown O(p^6) constants using resonance exchange. We make a careful comparison to the available data and determine two O(p^4) constants rather precisely, and two O(p^6) constants less precisely. We also use Chiral Perturbation Theory to two loops to extract in a model--independent manner the charge radius of the pion from the available data, and obtain \rpiV=0.437\pm 0.016 fm^2.Comment: Latex2e, JHEP.cls, 30 pages 5 figure

Renormalization of chiral perturbation theory to order p^6

The renormalization of chiral perturbation theory is carried out to next-to-next-to-leading order in the meson sector. We calculate the divergent part of the generating functional of Green functions of quark currents to O(p^6) for chiral SU(n), involving one- and two-loop diagrams. The renormalization group equations for the renormalized low-energy constants of O(p^6) are derived. We compare our results with previous two-loop calculations in chiral perturbation theory.Comment: 43 pages, LaTeX2e, 1 figur

The quark condensate from K_{e_4} decays

We show that, independently of the size of the quark condensate, chiral symmetry correlates the two S-wave \pi\pi scattering lengths. In view of this constraint, the new precision data on K_{e_4} decay allow a remarkably accurate determination of these quantities. The result confirms the hypothesis that the quark condensate is the leading order parameter.Comment: 4 pages (LaTex), 3 figure

Scalar form factors of light mesons

The scalar radius of the pion plays an important role in CHPT, because it is related to one of the basic effective coupling constants, viz. the one which controls the quark mass dependence of F_pi at one loop. In a recent paper, Yndurain derives a {\it robust lower bound} for this radius, which disagrees with earlier determinations. We show that such a bound does not exist: the "derivation" relies on an incorrect claim. Moreover, we discuss the physics of the form factors associated with the operators \ubar u, \dbar d and \sbar s and show that their structure in the vicinity of the K \Kbar threshold is quite different. Finally, we draw attention to the fact that the new data on the slope of the scalar K_l3 form factor confirm a recent, remarkably sharp theoretical prediction.Comment: 13 pages, 2 figures. v2: minor changes - version to appear on Phys. Lett. B. v3: Published version. Values of the measured K-pi radii correcte

Effective Lagrangian for Two-photon and Two-gluon Decays of PP-wave Heavy Quarkonium χc0,2\chi_{c0,2} and χb0,2\chi_{b0,2} states

In the traditional non-relativistic bound state calculation, the two-photon decay amplitudes of the $P$-wave $\chi_{c0,2}$ and $\chi_{b0,2}$ states depend on the derivative of the wave function at the origin which can only be obtained from potential models. However by neglecting the relative quark momenta, the decay amplitude can be written as the matrix element of a local heavy quark field operator which could be obtained from other processes or computed with QCD sum rules technique or lattice simulation. Following the same line as in recent work for the two-photon decays of the $S$-wave $\eta_{c}$ and $\eta_{b}$ quarkonia, we show that the effective Lagrangian for the two-photon decays of the $P$-wave $\chi_{c0,2}$ and $\chi_{b0,2}$ is given by the heavy quark energy-momentum tensor local operator or its trace, the $\bar{Q}Q$ scalar density and that the expression for $\chi_{c0}$ two-photon and two-gluon decay rate is given by the $f_{\chi_{c0}}$ decay constant and is similar to that of $\eta_{c}$ which is given by $f_{\eta_{c}}$. From the existing QCD sum rules value for $f_{\chi_{c0}}$, we get $5\rm keV$ for the $\chi_{c0}$ two-photon width, somewhat larger than measurement, but possibly with large uncertainties.Comment: v3, LaTeX, 5 pages, 1 figure, minor typos corrected, to appear in Physical Review

Strong Interactions at Low Energy

The lectures review some of the basic concepts relevant for an understanding of the low energy properties of the strong interactions: chiral symmetry, spontaneous symmetry breakdown, Goldstone bosons, quark condensate. The effective field theory used to analyze the low energy structure is briefly sketched. As an illustration, I discuss the implications of the recent data on the decay $K\to \pi\pi e\nu$ for the magnitude of the quark condensate.Comment: Lectures given at the school of physics "Understanding the structure of hadrons", Prague, July 2001, 20 p

Procedural \u3cem\u3eJus Cogens\u3c/em\u3e

Jus cogens are a species of supernorm in international law. They are universally binding and trump all contrary rules—such as treaties and customary international law. They are typically framed in terms of substantive prohibitions: no genocide, no slavery, no crimes against humanity, etc. This Article seeks to identify a procedural jus cogens; namely, the right to due process of law made up of notice, a hearing, and an impartial and independent decisionmaker. To do so, it draws from what are called “general principles of international law”; that is, principles common to legal systems around the world, which make up a source of international law. It argues that a comparative approach to these principles can reveal an empirically supported, objective underlying natural law right. In particular, by looking to rights states deem most important, hierarchically superior, and foundational to their legal systems as contained in their constitutions, this approach solves major seemingly intractable jurisprudential and practical dilemmas for the international law of jus cogens by providing an alternative to horizontal, consent-based positivistic law of treaties and custom.To make its argument it examines the 193 member states of the United Nations, plus Kosovo, the Republic of China (Taiwan), and the Vatican City (Holy See). Diligent research has revealed that virtually all states in the world secure the most basic requirements of due process: notice, a hearing, and an impartial and independent decisionmaker. More specifically, 189 states provide notice to the accused, 196 states provide for the right to a hearing, and 196 states provide for an impartial and independent decisionmaker. Moreover, the vast majority of these protections are constitutional. The right to notice is protected in 179 constitutions, the right to a hearing is protected in 193 constitutions, and the right to an impartial and independent decisionmaker is protected in 193 constitutions. This analysis easily satisfies the recent International Law Commission criteria that for a norm to qualify as jus cogens it must be accepted by “a very large majority of states . . . across regions, legal systems and cultures.”Discovering a procedural jus cogens would be revolutionary in some respects. A procedural jus cogens norm would expand the concept of jus cogens because such a norm would qualitatively differ from a substantive one since it is not merely a negative obligation on a state but imposes a positive duty to provide a right. Further, the Article’s argument holds powerful implications not just for international law but for domestic U.S. law as well. The Supreme Court long ago held that international law is part of our law, including the law of jus cogens, and mechanisms exist to enforce that law in U.S. courts

Supersymmetric contributions to direct CP violation in K -> pi pi gamma decays

We analyze the supersymmetric contributions to direct-CP-violating observables in $K \to \pi\pi\gamma$ decays induced by gluino-mediated magnetic-penguin operators. We find that \epsp_{+-\gamma} and the differential width asymmetry of $K^\pm \to \pi^\pm \pi^0 \gamma$ decays could be substantially enhanced with respect to their Standard Model values, especially in the scenario where \epsp/\eps is dominated by supersymmetric contributions. These observables could therefore provide a useful tool to search for New Physics effects in $|\Delta S|=1$ transitions, complementary to \epsp/\eps and rare decays.Comment: 14 pages, Latex. Numerical error in Eq.(36) corrected, conclusions unchanged; version to appear in Phys. Lett.