ParFORM: recent development

We report on the status of our project of parallelization of the symbolic manipulation program FORM. We have now parallel versions of FORM running on Cluster- or SMP-architectures. These versions can be used to run arbitrary FORM programs in parallel.Comment: 5 pages, 6 Encapsulated postscript figures, LaTeX2e, uses espcrc2.sty (included). Talk given at ACAT0

A practical criterion of irreducibility of multi--loop Feynman integrals

A practical criterion for the irreducibility (with respect to integration by part identities) of a particular Feynman integral to a given set of integrals is presented. The irreducibility is shown to be related to the existence of stable (with zero gradient) points of a specially constructed polynomial

Adler Function, Sum Rules and Crewther Relation of Order O(alpha_s^4): the Singlet Case

The analytic result for the singlet part of the Adler function of the vector current in a general gauge theory is presented in five-loop approximation. Comparing this result with the corresponding singlet part of the Gross-Llewellyn Smith sum rule [1], we successfully demonstrate the validity of the generalized Crewther relation for the singlet part. This provides a non-trivial test of both our calculations and the generalized Crewther relation. Combining the result with the already available non-singlet part of the Adler function [2,3] we arrive at the complete ${\cal O}(\alpha_s^4)$ expression for the Adler function and, as a direct consequence, at the complete ${\cal O}(\alpha_s^4)$ correction to the $e^+ e^-$ annihilation into hadrons in a general gauge theory.Comment: 4 pages, 1 figure. Final published versio

Solving Recurrence Relations for Multi-Loop Feynman Integrals

We study the problem of solving integration-by-parts recurrence relations for a given class of Feynman integrals which is characterized by an arbitrary polynomial in the numerator and arbitrary integer powers of propagators, {\it i.e.}, the problem of expressing any Feynman integral from this class as a linear combination of master integrals. We show how the parametric representation invented by Baikov can be used to characterize the master integrals and to construct an algorithm for evaluating the corresponding coefficient functions. To illustrate this procedure we use simple one-loop examples as well as the class of diagrams appearing in the calculation of the two-loop heavy quark potential.Comment: 24 pages, 5 ps figures, references added, minor modifications, published versio

Four Loop Massless Propagators: an Algebraic Evaluation of All Master Integrals

The old "glue--and--cut" symmetry of massless propagators, first established in [1], leads --- after reduction to master integrals is performed --- to a host of non-trivial relations between the latter. The relations constrain the master integrals so tightly that they all can be analytically expressed in terms of only few, essentially trivial, watermelon-like integrals. As a consequence we arrive at explicit analytical results for all master integrals appearing in the process of reduction of massless propagators at three and four loops. The transcendental structure of the results suggests a clean explanation of the well-known mystery of the absence of even zetas (zeta_{2n}) in the Adler function and other similar functions essentially reducible to the massless propagators. Once a reduction of massless propagators at five loops is available, our approach should be also applicable for explicit performing the corresponding five-loop master integrals.Comment: 34 pages, few typos have been fixed, references and acknowledgements have been updated. Results for master integrals (together with some auxiliary information) are now available in http://www-ttp.physik.uni-karlsruhe.de/Progdata/ttp10/ttp10-18

Adler Function, DIS sum rules and Crewther Relations

The current status of the Adler function and two closely related Deep Inelastic Scattering (DIS) sum rules, namely, the Bjorken sum rule for polarized DIS and the Gross-Llewellyn Smith sum rule are briefly reviewed. A new result is presented: an analytical calculation of the coefficient function of the latter sum rule in a generic gauge theory in order O(alpha_s^4). It is demonstrated that the corresponding Crewther relation allows to fix two of three colour structures in the O(alpha_s^4) contribution to the singlet part of the Adler function.Comment: Talk presented at 10-th DESY Workshop on Elementary Particle Theory: Loops and Legs in Quantum Field Theory, W\"orlitz, Germany, 25-30 April 201

Equivalence of Recurrence Relations for Feynman Integrals with the Same Total Number of External and Loop Momenta

We show that the problem of solving recurrence relations for L-loop (R+1)-point Feynman integrals within the method of integration by parts is equivalent to the corresponding problem for (L+R)-loop vacuum or (L+R-1)-loop propagator-type integrals. Using this property we solve recurrence relations for two-loop massless vertex diagrams, with arbitrary numerators and integer powers of propagators in the case when two legs are on the light cone, by reducing the problem to the well-known solution of the corresponding recurrence relations for massless three-loop propagator diagrams with specific boundary conditions.Comment: 8 pp., LaTeX with axodraw.st

The criterion of irreducibility of multi-loop Feynman integrals

The integration by parts recurrence relations allow to reduce some Feynman integrals to more simple ones (with some lines missing). Nevertheless the possibility of such reduction for the given particular integral was unclear. The recently proposed technique for studying the recurrence relations as by-product provides with simple criterion of the irreducibility.Comment: LaTeX, 6 pages, no figures, the complete paper, including figures, is also available via anonymous ftp at ftp://ttpux2.physik.uni-karlsruhe.de/ttp99/ttp99-52/ or via www at http://www-ttp.physik.uni-karlsruhe.de/Preprints

Optimal renormalization and the extraction of strange quark mass from semi-leptonic τ\tau-decay

We employ optimal renormalization group analysis to semi-leptonic $\tau$-decay polarization functions and extract the strange quark mass from their moments measured by the ALEPH and OPAL collaborations. The optimal renormalization group makes use of the renormalization group equation of a given perturbation series which then leads to closed form sum of all the renormalization group-accessible logarithms which have reduced scale dependence. Using the latest theoretical inputs we find $m_s(2{\rm GeV}) = 106.70 \pm 9.36~{\rm MeV}$ and $m_s(2{\rm GeV}) = 74.47 \pm 7.77~{\rm MeV}$ for ALEPH and OPAL data respectively.Comment: 3 pages, Contribution to the proceedings of the XXII DAE-BRNS High Energy Physics Symposium, University of Delhi, Dec. 12-16, 201

R(s) and hadronic tau-Decays in Order alpha_s^4: technical aspects

We report on some technical aspects of our calculation of alpha_s^4 corrections to R(s) and the semi-leptonic tau decay width [1-3]. We discuss the inner structure of the result as well as the issue of its correctness. We demonstrate recently appeared independent evidence positively testing one of two components of the full result