Stable characteristic classes of smooth manifold bundles

Characteristic classes of oriented vector bundles can be identified with cohomology classes of the disjoint union of classifying spaces BSO_n of special orthogonal groups SO_n with n=0,1,... A characteristic class is stable if it extends to a cohomology class of a homotopy colimit BSO of classifying spaces BSO_n. Similarly, characteristic classes of smooth oriented manifold bundles with fibers given by oriented closed smooth manifolds of a fixed dimension d\ge 0 can be identified with cohomology classes of the disjoint union of classifying spaces BDiff M of orientation preserving diffeomorphism groups of oriented closed manifolds of dimension d. A characteristic class is stable if it extends to a cohomology class of a homotopy colimit of spaces BDiff M. We show that each rational stable characteristic class of oriented manifold bundles of even dimension d is tautological, e.g., if d=2, then each rational stable characteristic class is a polynomial in terms of Miller-Morita-Mumford classes.Comment: 52 page

Impact of QED radiative corrections on Parton Distribution Functions

The level of precision achieved by the experimental measurements at the LHC requires the inclusion of higher order electroweak effects to the processes of $pp$ scattering. In particular the photon-induced process $\gamma\gamma \to \ell^+\ell^-$ make a significant contribution ($\sim 10 \%$) to the dilepton invariant mass distribution. To evaluate the cross-section of this process one need to know the parton distribution function (PDF) of the photon in the proton $\gamma (x,\mu^2)$. The aim of the current study is to investigate the impact of QED corrections on PDFs and describe the implementation of QED-modified evolution equations into beta release of new version of {\tt QCDNUM} program. The {\tt APPLGRID} interface to {\tt SANC} Monte Carlo generator for fast evaluation of photon-induced cross-section is also outlined. The results were cross-checked with {\tt partonevolution} program, {\tt MRST2004QED} PDF set and {\tt APFEL} program. The described developments are planned to include into {\tt HERAFitter} package and can be used to determine the photon PDF using new data from the LHC experiments

Bases in the solution space of the Mellin system

Local holomorphic solutions z=z(a) to a univariate sparse polynomial equation p(z) =0, in terms of its vector of complex coefficients a, are classically known to satisfy holonomic systems of linear partial differential equations with polynomial coefficients. In this paper we investigate one of such systems of differential equations which was introduced by Mellin. We compute the holonomic rank of the Mellin system as well as the dimension of the space of its algebraic solutions. Moreover, we construct explicit bases of solutions in terms of the roots of p and their logarithms. We show that the monodromy of the Mellin system is always reducible and give some factorization results in the univariate case