Energy of knots and the infinitesimal cross ratio

This is a survey article on two topics. The Energy E of knots can be obtained by generalizing an electrostatic energy of charged knots in order to produce optimal knots. It turns out to be invariant under Moebius transformations. We show that it can be expressed in terms of the infinitesimal cross ratio, which is a conformal invariant of a pair of 1-jets, and give two kinds of interpretations of the real part of the infinitesimal cross ratio.Comment: This is the version published by Geometry & Topology Monographs on 19 March 200

On the cross section ratio sigma_n/sigma_p in eta photoproduction

The recently discovered enhancement of eta photoproduction on the quasi-free neutron at energies around sqrt(s)~1.67 GeV is addressed within a SU(3) coupled channel model. The quasi-free cross sections on proton and neutron, sigma_n and sigma_p, can be quantitatively explained. In this study, the main source for the peak in sigma_n/sigma_p is a coupled channel effect in S-wave that explains the dip-bump structure in gamma n --> eta n. In particular, the photon coupling to the intermediate meson-baryon states is important. The stability of the result is extensively tested and consistency with several pion- and photon-induced reactions is ensured.Comment: Version accepted by Phys. Lett.

Analytic expressions of amplitudes by the cross-ratio identity method

In order to obtain the analytic expression of an amplitude from a generic CHY-integrand, a new algorithm based on the so-called cross-ratio identities has been proposed recently. In this paper, we apply this new approach to a variety of theories including: non-linear sigma model, special Galileon theory, pure Yang-Mills theory, pure gravity, Born-Infeld theory, Dirac-Born-Infeld theory and its extension, Yang-Mills-scalar theory, Einstein-Maxwell theory as well as Einstein-Yang-Mills theory. CHY-integrands of these theories which contain higher-order poles can be calculated conveniently by using the cross-ratio identity method, and all results above have been verified numerically.Comment: 22 page

Infinitesimal Liouville currents, cross-ratios and intersection numbers

Many classical objects on a surface S can be interpreted as cross-ratio functions on the circle at infinity of the universal covering. This includes closed curves considered up to homotopy, metrics of negative curvature considered up to isotopy and, in the case of interest here, tangent vectors to the Teichm\"uller space of complex structures on S. When two cross-ratio functions are sufficiently regular, they have a geometric intersection number, which generalizes the intersection number of two closed curves. In the case of the cross-ratio functions associated to tangent vectors to the Teichm\"uller space, we show that two such cross-ratio functions have a well-defined geometric intersection number, and that this intersection number is equal to the Weil-Petersson scalar product of the corresponding vectors.Comment: 17 page

Measurement of the Lambda(b) cross section and the anti-Lambda(b) to Lambda(b) ratio with Lambda(b) to J/Psi Lambda decays in pp collisions at sqrt(s) = 7 TeV

The Lambda(b) differential production cross section and the cross section ratio anti-Lambda(b)/Lambda(b) are measured as functions of transverse momentum pt(Lambda(b)) and rapidity abs(y(Lambda(b))) in pp collisions at sqrt(s) = 7 TeV using data collected by the CMS experiment at the LHC. The measurements are based on Lambda(b) decays reconstructed in the exclusive final state J/Psi Lambda, with the subsequent decays J/Psi to an opposite-sign muon pair and Lambda to proton pion, using a data sample corresponding to an integrated luminosity of 1.9 inverse femtobarns. The product of the cross section times the branching ratio for Lambda(b) to J/Psi Lambda versus pt(Lambda(b)) falls faster than that of b mesons. The measured value of the cross section times the branching ratio for pt(Lambda(b)) > 10 GeV and abs(y(Lambda(b))) < 2.0 is 1.06 +/- 0.06 +/- 0.12 nb, and the integrated cross section ratio for anti-Lambda(b)/Lambda(b) is 1.02 +/- 0.07 +/- 0.09, where the uncertainties are statistical and systematic, respectively.Comment: Submitted to Physics Letters