Geometric approach to asymptotic expansion of Feynman integrals

We present an algorithm that reveals relevant contributions in non-threshold-type asymptotic expansion of Feynman integrals about a small parameter. It is shown that the problem reduces to finding a convex hull of a set of points in a multidimensional vector space.Comment: 6 pages, 2 figure

Problems of the Strategy of Regions

Problems that arise in the application of general prescriptions of the so-called strategy of regions for asymptotic expansions of Feynman integrals in various limits of momenta and masses are discussed with the help of characteristic examples of two-loop diagrams. The strategy is also reformulated in the language of alpha parameters.Comment: 12 pages, LaTeX with axodraw.st

On the Resolution of Singularities of Multiple Mellin-Barnes Integrals

One of the two existing strategies of resolving singularities of multifold Mellin-Barnes integrals in the dimensional regularization parameter, or a parameter of the analytic regularization, is formulated in a modified form. The corresponding algorithm is implemented as a Mathematica code MBresolve.mComment: LaTeX, 10 page

Decoupling of heavy quarks in HQET

Decoupling of c-quark loops in b-quark HQET is considered. The decoupling coefficients for the HQET heavy-quark field and the heavy-light quark current are calculated with the three-loop accuracy. The last result can be used to improve the accuracy of extracting f_B from HQET lattice simulations (without c-quark loops). The decoupling coefficient for the flavour-nonsinglet QCD current with n antisymmetrized gamma-matrices is also obtained at three loops; the result for the tensor current (n=2) is new.Comment: JHEP3 documentclass; the results in a computer-readable form can be found at http://www-ttp.physik.uni-karlsruhe.de/Progdata/ttp06/ttp06-25/ V2: a few typos corrected, a few minor text improvements, a few references added; V3: several typos in formulas fixe

An Algorithm to Construct Groebner Bases for Solving Integration by Parts Relations

This paper is a detailed description of an algorithm based on a generalized Buchberger algorithm for constructing Groebner-type bases associated with polynomials of shift operators. The algorithm is used for calculating Feynman integrals and has proven itself efficient in several complicated cases.Comment: LaTeX, 9 page

How to choose master integrals

The standard procedure when evaluating integrals of a given family of Feynman integrals, corresponding to some Feynman graph, is to construct an algorithm which provides the possibility to write any particular integral as a linear combination of so-called master integrals. To do this, public (AIR, FIRE, REDUZE, LiteRed, KIRA) and private codes based on solving integration by parts relations are used. However, the choice of the master integrals provided by these codes is not always optimal. We present an algorithm to improve a given basis of the master integrals, as well as its computer implementation; see also a competitive varian

Analytical evaluation of certain on-shell two-loop three-point diagrams

An analytical approach is applied to the calculation of some dimensionally-regulated two-loop vertex diagrams with essential on-shell singularities. Such diagrams are important for the evaluation of QED corrections to the muon decay, QCD corrections to top quark decays t->W^{+}b, t->H^{+}b, etc.Comment: 2 pages, LaTeX, contribution to proceedings of ACAT2002 (Moscow, June 2002

Analytical Result for Dimensionally Regularized Massless Master Double Box with One Leg off Shell

The dimensionally regularized massless double box Feynman diagram with powers of propagators equal to one, one leg off the mass shell, i.e. with non-zero q^2=p_1^2, and three legs on shell, p_i^2=0, i=2,3,4, is analytically calculated for general values of q^2 and the Mandelstam variables s and t. An explicit result is expressed through (generalized) polylogarithms, up to the fourth order, dependent on rational combinations of q^2,s and t, and a one-dimensional integral with a simple integrand consisting of logarithms and dilogarithms.Comment: 10 pages, LaTeX with axodraw.sty, one reference is correcte

Analytical Results for Dimensionally Regularized Massless On-shell Double Boxes with Arbitrary Indices and Numerators

We present an algorithm for the analytical evaluation of dimensionally regularized massless on-shell double box Feynman diagrams with arbitrary polynomials in numerators and general integer powers of propagators. Recurrence relations following from integration by parts are solved explicitly and any given double box diagram is expressed as a linear combination of two master double boxes and a family of simpler diagrams. The first master double box corresponds to all powers of the propagators equal to one and no numerators, and the second master double box differs from the first one by the second power of the middle propagator. By use of differential relations, the second master double box is expressed through the first one up to a similar linear combination of simpler double boxes so that the analytical evaluation of the first master double box provides explicit analytical results, in terms of polylogarithms \Li{a}{-t/s}, up to $a=4$, and generalized polylogarithms $S_{a,b}(-t/s)$, with $a=1,2$ and $b=2$, dependent on the Mandelstam variables $s$ and $t$, for an arbitrary diagram under consideration.Comment: LaTeX, 16 pages; misprints in ff. (8), (24), (30) corrected; some explanations adde

Some recent results on evaluating Feynman integrals

Some recent results on evaluating Feynman integrals are reviewed. The status of the method based on Mellin-Barnes representation as a powerful tool to evaluate individual Feynman integrals is characterized. A new method based on Groebner bases to solve integration by parts relations in an automatic way is described.Comment: 5 pages, LaTeX, Conference Proceedings Radcor 200