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O. V. Tarasov

Publications and source records attributed to O. V. Tarasov.

At least 37 records · Page 2Linked to original sources
arXiv2002

Factorizing one-loop contributions to two-loop Bhabha scattering and automatization of Feynman diagram calculations

In higher order calculations a number of new technical problems arise: one needs diagrams in arbitrary dimension in order to obtain their needed $ε$-expansion, zero Gram determinants appear, renormalization produces diagrams with `dots' on the lines, i.e. higher order powers of scalar propagators. All these problems cannot be accessed by the `standard' Passarino-Veltman approach: there is not available what is needed for higher loops. We demonstrate our method of how to solve these problems.

hep-ph↗
arXiv2001

Application and explicit solution of recurrence relations with respect to space-time dimension

A short review of the method for the tensor reduction of Feynman integrals based on recurrence relations with respect to space-time dimension d- is given. A solution of the difference equation with respect to d for the n - point one-loop integrals with arbitrary momenta and masses is presented. The result is written as multiple hypergeometric series depending on ratios of Gram determinants. For the 3-point function a new expression in terms of the Appell hypergeometric function F_1 is presented.

hep-ph↗
arXiv1999

Algebraic reduction of one-loop Feynman graph amplitudes

An algorithm for the reduction of one-loop n-point tensor integrals to basic integrals is proposed. We transform tensor integrals to scalar integrals with shifted dimension and reduce these by recurrence relations to integrals in generic dimension. Also the integration-by-parts method is used to reduce indices (powers of scalar propagators) of the scalar diagrams. The obtained recurrence relations for one-loop integrals are explicitly evaluated for 5- and 6-point functions. In the latter case the corresponding Gram determinant vanishes identically for d=4, which greatly simplifies the application of the recurrence relations.

hep-ph↗
arXiv1999

Two-loop QCD corrections of the massive fermion propagator

The off-shell two-loop correction to the massive quark propagator in an arbitrary covariant gauge is calculated and results for the bare and renormalized propagator are presented. The calculations were performed by means of a set of new generalized recurrence relations proposed recently by one of the authors. From the position of the pole of the renormalized propagator we obtain the relationship between the pole mass and the \bar{MS} mass. This relation confirms the known result by Gray et al.. The bare amplitudes are given for an arbitrary gauge group and for arbitrary space-time dimensions.

hep-ph↗
arXiv1998

Reduction of Feynman graph amplitudes to a minimal set of basic integrals

An algorithm for the reduction of massive Feynman integrals with any number of loops and external momenta to a minimal set of basic integrals is proposed. The method is based on the new algorithm for evaluating tensor integrals, representation of generalized recurrence relations for a given kind of integrals as a linear system of PDEs and the reduction of this system to a standard form using algorithms proposed by G. Reid. Basic integrals reveal as parametric derivatives of the system in the standard form and the number of basic integrals in the minimal set is determined by the dimension of the solution space of the system of PDEs.

hep-ph↗
arXiv1998

Exact mass dependent two--loop $\barα_s(Q^2)$ in the background MOM renormalization scheme

A two-loop calculation of the renormalization group $β$--function in a momentum subtraction scheme with massive quarks is presented using the background field formalism. The results have been obtained by using a set of new generalized recurrence relations proposed recently by one of the authors (O.V.T.). The behavior of the mass dependent effective coupling constant is investigated in detail. Compact analytic results are presented.

hep-ph↗
arXiv1998

Two-loop three-gluon vertex in zero-momentum limit

The two-loop three-gluon vertex is calculated in an arbitrary covariant gauge, in the limit when one of the external momenta vanishes. The differential Ward-Slavnov-Taylor (WST) identity related to this limit is discussed, and the relevant results for the ghost-gluon vertex and two-point functions are obtained. Together with the differential WST identity, they provide another independent way for calculating the three-gluon vertex. The renormalization of the results obtained is also presented.

hep-ph↗
arXiv1997

Generalized Recurrence Relations for Two-loop Propagator Integrals with Arbitrary Masses

An algorithm for calculating two-loop propagator type Feynman diagrams with arbitrary masses and external momentum is proposed. Recurrence relations allowing to express any scalar integral in terms of basic integrals are given. A minimal set consisting of 15 essentially two-loop and 15 products of one-loop basic integrals is found. Tensor integrals and integrals with irreducible numerators are represented as a combination of scalar ones with a higher space-time dimension which are reduced to the basic set by using the generalized recurrence relations proposed in Ref.[1] (Phys.Rev.D54 (1996) 6479).

hep-ph↗
arXiv1996

Three-gluon vertex in arbitrary gauge and dimension

One-loop off-shell contributions to the three-gluon vertex are calculated, in arbitrary covariant gauge and in arbitrary space-time dimension, including quark-loop contributions (with massless quarks). It is shown how one can get the results for all on-shell limits of interest directly from the general off-shell expression. The corresponding general expressions for the one-loop ghost-gluon vertex are also obtained. They allow for a check of consistency with the Ward--Slavnov--Taylor identity.

hep-ph↗
arXiv1996

A new approach to the momentum expansion of multiloop Feynman diagrams

We present a new method for the momentum expansion of Feynman integrals with arbitrary masses and any number of loops and external momenta. By using the parametric representation we derive a generating function for the coefficients of the small momentum expansion of an arbitrary diagram. The method is applicable for the expansion w.r.t. all or a subset of external momenta. The coefficients of the expansion are obtained by applying a differential operator to a given integral with shifted value of the space-time dimension $d$ and the expansion momenta set equal to zero. Integrals with changed $d$ are evaluated by using the generalized recurrence relations proposed in \cite{OVT1}. We show how the method works for one- and two-loop integrals. It is also illustrated that our method is simpler and more efficient than others.

hep-ph↗
arXiv1996

Connection between Feynman integrals having different values of the space-time dimension

A systematic algorithm for obtaining recurrence relations for dimensionally regularized Feynman integrals w.r.t. the space-time dimension $d$ is proposed. The relation between $d$ and $d-2$ dimensional integrals is given in terms of a differential operator for which an explicit formula can be obtained for each Feynman diagram. We show how the method works for one-, two- and three-loop integrals. The new recurrence relations w.r.t. $d$ are complementary to the recurrence relations which derive from the method of integration by parts. We find that the problem of the irreducible numerators in Feynman integrals can be naturally solved in the framework of the proposed generalized recurrence relations.

hep-th↗
arXiv1996

One-loop results for three-gluon vertex in arbitrary gauge and dimension

We review the calculation of one-loop contributions to the three-gluon vertex, for arbitrary (off-shell) external momenta, in arbitrary covariant gauge and in arbitrary space-time dimension. We discuss how one can get the results for all on-shell limits of interest directly from the general off-shell expression.

hep-ph↗
arXiv1996

Calculation of Feynman diagrams with zero mass threshold from their small momentum expansion

A method of calculating Feynman diagrams from their small momentum expansion [1] is extended to diagrams with zero mass thresholds. We start from the asymptotic expansion in large masses [2] (applied to the case when all $M_i^2$ are large compared to all momenta squared). Using dimensional regularization, a finite result is obtained in terms of powers of logarithms (describing the zero-threshold singularity) times power series in the momentum squared. Surprisingly, these latter ones represent functions, which not only have the expected physical `second threshold' but have a branchcut singularity as well below threshold at a mirror position. These can be understood as pseudothresholds corresponding to solutions of the Landau equations. In the spacelike region the imaginary parts from the various contributions cancel. For the two-loop examples with one mass $M$, in the timelike region for $q^2 \approx M^2$ we obtain approximations of high precision. This will be of relevance in particular for the calculation of the decay $Z \to b\bar{b}$ in the $m_b=0$ approximation.

hep-ph↗
arXiv1995

ALGORITHMIC CALCULATION OF TWO-LOOP FEYNMAN DIAGRAMS

In a recent paper \cite{ft} a new powerful method to calculate Feynman diagrams was proposed. It consists in setting up a Taylor series expansion in the external momenta squared. The Taylor coefficients are obtained from the original diagram by differentiation and putting the external momenta equal to zero. It was demonstrated that by a certain conformal mapping and subsequent resummation by means of Padé approximants it is possible to obtain high precision numerical values of the Feynman integrals in the whole cut plane. The real problem in this approach is the calculation of the Taylor coefficients for the arbitrary mass case. Since their analytic evaluation by means of CA packages uses enormous CPU and yields very lengthy expressions, we develop an algorithm with the aim to set up a FORTRAN package for their numerical evaluation. This development is guided by the possibilities offered by the formulae manipulating language FORM \cite{FORM}.

hep-ph↗
arXiv1994

Application of conformal mapping and Padé approximants $(ωP's)$ to the calculation of various two-loop Feynman diagrams

Feynman diagrams are calculated by means of their Taylor series expansion in terms of external momenta squared. It is demonstrated in various examples that by the application of conformal mapping and Padé approximants, it is possible to obtain high precision results in the spacelike as well as in the timelike region on the cut. Examples are given for two- and three-point functions, but in principle the method is applicable also to four-point functions.

hep-ph↗
arXiv1994

Two-Loop Gluon-Condensate Contributions To Heavy-Quark Current Correlators: Exact Results And Approximations

The coefficient functions of the gluon condensate $ $, in the correlators of heavy-quark vector, axial, scalar and pseudoscalar currents, are obtained analytically, to two loops, for all values of $z=q^2/4m^2$. In the limiting cases $z\to0$, $z\to1$, and $z\to-\infty$, comparisons are made with previous partial results. Approximation methods, based on these limiting cases, are critically assessed, with a view to three-loop work. High accuracy is achieved using a few moments as input. A {\em single} moment, combined with only the {\em leading} threshold and asymptotic behaviours, gives the two-loop corrections to better than 1% in the next 10 moments. A two-loop fit to vector data yields $<\frac{α_{\rm s}}πG^2>\approx0.021$ GeV$^4$.

hep-ph↗
arXiv1994

Calculation of Feynman Diagrams from their Small Momentum Expansion

A new powerful method to calculate Feynman diagrams is proposed. It consists in setting up a Taylor series expansion in the external momenta squared (in general multivariable). The Taylor coefficients are obtained from the original diagram by differentiation and putting the external momenta equal to zero, which means a great simplification. It is demonstrated that it is possible to obtain by analytic continuation of the original series high precision numerical values of the Feynman integrals in the whole cut plane. For this purpose conformal mapping and subsequent resummation by means of Padé approximants or Levin transformation are applied.

hep-ph↗