Explore the vast range of titles available.

For the precision calculations in perturbative Quantum Chromodynamics (QCD) gigantic expressions (several GB in size) in terms of highly complicated divergent multi-loop Feynman integrals have to be calculated analytically to compact expressions in terms of special functions and constants. In this article we derive new symbolic tools to gain large-scale computer understanding in QCD. Here we exploit the fact that hypergeometric structures in single and multiscale Feynman integrals emerge in a wide class of topologies. Using integration-by-parts relations, associated master or scalar integrals have to be calculated. For this purpose it appears useful to devise an automated method which recognizes the respective (partial) differential equations related to the corresponding higher transcendental functions. We solve these equations through associated recursions of the expansion coefficient of the multivalued formal Taylor series. The expansion coefficients can be determined using either the package Sigma in the case of linear difference equations or by applying heuristic methods in the case of partial linear difference equations. In the present context a new type of sums occurs, the Hurwitz harmonic sums, and generalized versions of them. The code HypSeries transforming classes of differential equations into analytic series expansions is described. Also partial difference equations having rational solutions and rational function solutions of Pochhammer symbols are considered, for which the code solvePartialLDE is designed. Generalized hypergeometric functions, Appell-, Kampé de Fériet-, Horn-, Lauricella-Saran-, Srivasta-, and Exton–type functions are considered. We illustrate the algorithms by examples.

We present the scheme-invariant unpolarized and polarized flavor non-singlet evolution equation to N\\(^3\\)LO for the structure functions \\(F_2(x,Q^2)\\) and \\(g_1(x,Q^2)\\) including the charm- and bottom quark effects in the asymptotic representation. The corresponding evolution is based on the experimental measurement of the non-singlet structure functions at a starting scale \\(Q_0^2\\). In this way the evolution does only depend on the strong coupling constant \\(\\alpha_s(M_Z)\\) or the QCD scale \\(\\Lambda_{\\rm QCD}\\) and the charm and bottom quark masses \\(m_c\\) and \\(m_b\\) and provides one of the cleanest ways to measure the strong coupling constant in future high luminosity deep-inelastic scattering experiments. The yet unknown parts of the 4-loop anomalous dimensions introduce only a marginal error in this analysis.

We compute the logarithmic contributions to the polarized massive Wilson coefficients for deep-inelastic scattering in the asymptotic region \\(Q^2 \\gg m^2\\) to 3-loop order in the fixed-flavor number scheme and present the corresponding expressions for the polarized massive operator matrix elements needed in the variable flavor number scheme. The calculation is performed in the Larin scheme. For the massive operator matrix elements \\(A_{qq,Q}^{(3),\\rm PS}\\) and \\(A_{qg,Q}^{(3),\\rm S}\\) the complete results are presented. The expressions are given in Mellin-\\(N\\) space and in momentum fraction \\(z\\)-space.

Hypergeometric structures in single and multiscale Feynman integrals emerge in a wide class of topologies. Using integration-by-parts relations, associated master or scalar integrals have to be calculated. For this purpose it appears useful to devise an automated method which recognizes the respective (partial) differential equations related to the corresponding higher transcendental functions. We solve these equations through associated recursions of the expansion coefficient of the multivalued formal Taylor series. The expansion coefficients can be determined using either the package {\\tt Sigma} in the case of linear difference equations or by applying heuristic methods in the case of partial linear difference equations. In the present context a new type of sums occurs, the Hurwitz harmonic sums, and generalized versions of them. The code {\\tt HypSeries} transforming classes of differential equations into analytic series expansions is described. Also partial difference equations having rational solutions and rational function solutions of Pochhammer symbols are considered, for which the code {\\tt solvePartialLDE} is designed. Generalized hypergeometric functions, Appell-,~Kampé de Fériet-, Horn-, Lauricella-Saran-, Srivasta-, and Exton--type functions are considered. We illustrate the algorithms by examples.

Theoretical predictions for particle production cross sections and decays at colliders rely heavily on perturbative Quantum Chromodynamics (QCD) calculations, expressed as an expansion in powers of the strong coupling constant \\(\\alpha_s\\). The current \\(\\mathcal{O}(1\\%)\\) uncertainty of the QCD coupling evaluated at the reference Z boson mass, \\(\\alpha_s(m_Z) = 0.1179 \\pm 0.0009\\), is one of the limiting factors to more precisely describe multiple processes at current and future colliders. A reduction of this uncertainty is thus a prerequisite to perform precision tests of the Standard Model as well as searches for new physics. This report provides a comprehensive summary of the state-of-the-art, challenges, and prospects in the experimental and theoretical study of the strong coupling. The current \\(\\alpha_s(m_Z)\\) world average is derived from a combination of seven categories of observables: (i) lattice QCD, (ii) hadronic \\(\\tau\\) decays, (iii) deep-inelastic scattering and parton distribution functions fits, (iv) electroweak boson decays, hadronic final-states in (v) \\(e^+e^-\\), (vi) e-p, and (vii) p-p collisions, and (viii) quarkonia decays and masses. We review the current status of each of these seven \\(\\alpha_s(m_Z)\\) extraction methods, discuss novel \\(\\alpha_s\\) determinations, and examine the averaging method used to obtain the world-average value. Each of the methods discussed provides a ``wish list'' of experimental and theoretical developments required in order to achieve the goal of a per-mille precision on \\(\\alpha_s(m_Z)\\) within the next decade.

We present the two-mass QCD contributions to the polarized pure singlet operator matrix element at three loop order in \\(x\\)-space. These terms are relevant for calculating the polarized structure function \\(g_1(x,Q^2)\\) at \\(O(\\alpha_s^3)\\) as well as for the matching relations in the variable flavor number scheme and the polarized heavy quark distribution functions at the same order. The result for the operator matrix element is given in terms of generalized iterated integrals. These integrals depend on the mass ratio through the main argument, and the alphabet includes square--root valued letters.

A survey is given on the new 2- and 3-loop results for the heavy flavor contributions to deep-inelastic scattering in the unpolarized and the polarized case. We also discuss related new mathematical aspects applied in these calculations.

We compute the two-mass contributions to the polarized massive operator matrix element \\(A_{gg,Q}^{(3)}\\) at third order in the strong coupling constant \\(\\alpha_s\\) in Quantum Chromodynamics analytically. These corrections are important ingredients for the matching relations in the variable flavor number scheme and for the calculation of Wilson coefficients in deep--inelastic scattering in the asymptotic regime \\(Q^2 \\gg m_c^2, m_b^2\\). The analytic result is expressed in terms of nested harmonic, generalized harmonic, cyclotomic and binomial sums in \\(N\\)-space and by iterated integrals involving square-root valued arguments in \\(z\\) space, as functions of the mass ratio. Numerical results are presented. New two--scale iterative integrals are calculated.