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Higher-order diagrams required for radiative corrections to mixed electroweak and QCD processes at the LHC and anticipated future colliders will require numerically stable representations of the associated Feynman diagrams. The hypergeometric representation supplies an analytic framework that is useful for deriving such stable representations. We discuss the reduction of Feynman diagrams to master integrals, and compare integration-by-parts methods to differential reduction of hypergeometric functions. We describe the problem of constructing higher-order terms in the epsilon expansion, and characterize the functions generated in such expansions., Comment: 12 pages, Proceedings of DPF2011, Brown University, Providence, R.I., Aug. 9-13, 2011. Version 2 adds references, corrects the author list, and fixes several misprints. Version 3 has some relatively minor corrections

Stable reduction methods will be important in the evaluation of high-order perturbative diagrams appearing in QCD and mixed QCD-electroweak radiative corrections at the LHC. Differential reduction techniques are useful for relating hypergeometric functions with shifted values of the parameters. We present a proposition relating the number of master integrals in the expansion of a Feynman diagram to the number of derivatives in a differential reduction., Comment: 3 pages, presented at 35th International Conference of High Energy Physics - ICHEP2010, July 22-28, 2010, Paris, France

We will present some (formal) arguments that any Feynman diagram can be understood as a particular case of a Horn-type multivariable hypergeometric function. The advantages and disadvantages of this type of approach to the evaluation of Feynman diagrams is discussed., Comment: 7 pages, LaTeX, PoS style; based on a talk given by M.Kalmykov at ACAT 2008, Erice, Italy, Nov. 3-7, 2008; v2: few references added; v3: few references and some details added and conclusion is improved

We review the hypergeometric function approach to Feynman diagrams. Special consideration is given to the construction of the Laurent expansion. As an illustration, we describe a collection of physically important one-loop vertex diagrams for which this approach is useful., Comment: 15 pages, Latex, 1-eps figure; v2: few references added; v3: style corrections; v4: typo in Eq.(2) is corrected; based on a talk given by M.Kalmykov at 15th International Seminar on High-Energy Physics: Quarks-2008,Sergiev Posad, Russia, 23-29 May, 2008. The FORM code is available via the WWW at http://theor.jinr.ru/~kalmykov/hypergeom/hyper.html

Hypergeometric functions provide a useful representation of Feynman diagrams occuring in precision phenomenology. In dimension regularization, the epsilon-expansion of these functions about d=4 is required. We discuss the current status of differential reduction algorithms. As an illustration, we consider the construction of the all-order epsilon-expansion of the Appell hypergeometric function about integer values of the parameters and present an explicit evaluations of the first few terms., Comment: Poster presentation by S.A. Yost at the 34th International Conference on High Energy Physics, Philadelphia, August 2008. 3 pages, no figures, revtex4, slac_one.rtx. Version 4 corrects the arguments of one equation (between eqs. 5 and 6). Version 5 updates the template for ICHEP08

We continue our study of the construction of analytical coefficients of the epsilon-expansion of hypergeometric functions and their connection with Feynman diagrams. In this paper, we apply the approach of obtaining iteratated solutions to the differential equations associated with hypergeometric functions to prove the following result (Theorem 1): The epsilon-expansion of a generalized hypergeometric function with integer values of parameters is expressible in terms of generalized polylogarithms with coefficients that are ratios of polynomials. The method used in this proof provides an efficient algorithm for calculatiing of the higher-order coefficients of Laurent expansion., Comment: 12 pages, Latex + amsmath, JHEP3 class packages. This revision adds references 1 and 19. The FORM code is available via the WWW at http://theor.jinr.ru/~kalmykov/hypergeom/hyper.html

We continue the study of the construction of analytical coefficients of the epsilon-expansion of hypergeometric functions and their connection with Feynman diagrams. In this paper, we show the following results: Theorem A: The multiple (inverse) binomial sums of arbitrary weight and depth (see Eq. (1.1)) are expressible in terms of Remiddi-Vermaseren functions. Theorem B: The epsilon expansion of a hypergeometric function with one half-integer value of parameter (see Eq. (1.2)) is expressible in terms of the harmonic polylogarithms of Remiddi and Vermaseren with coefficients that are ratios of polynomials. Some extra materials are available via the www at this http://theor.jinr.ru/~kalmykov/hypergeom/hyper.html, Comment: 24 pages, latex with amsmath and JHEP3.cls; v2: some typos corrected and a few references added; v3: few references added

It is proved that the Laurent expansion of the following Gauss hypergeometric functions, 2F1(I1+a*epsilon, I2+b*ep; I3+c*epsilon;z), 2F1(I1+a*epsilon, I2+b*epsilon;I3+1/2+c*epsilon;z), 2F1(I1+1/2+a*epsilon, I2+b*epsilon; I3+c*epsilon;z), 2F1(I1+1/2+a*epsilon, I2+b*epsilon; I3+1/2+c*epsilon;z), 2F1(I1+1/2+a*epsilon,I2+1/2+b*epsilon; I3+1/2+c*epsilon;z), where I1,I2,I3 are an arbitrary integer nonnegative numbers, a,b,c are an arbitrary numbers and epsilon is an arbitrary small parameters, are expressible in terms of the harmonic polylogarithms of Remiddi and Vermaseren with polynomial coefficients. An efficient algorithm for the calculation of the higher-order coefficients of Laurent expansion is constructed. Some particular cases of Gauss hypergeometric functions are also discussed., Comment: 18 pages, JHEP3.cls The code (FORM) is available via the www http://theor.jinr.ru/~kalmykov/hypergeom/hyper.html v2: Appendix and a few references added

The Gauss hypergeometric functions 2F1 with arbitrary values of parameters are reduced to two functions with fixed values of parameters, which differ from the original ones by integers. It is shown that in the case of integer and/or half-integer values of parameters there are only three types of algebraically independent Gauss hypergeometric functions. The epsilon-expansion of functions of one of this type (type F in our classification) demands the introduction of new functions related to generalizations of elliptic functions. For the five other types of functions the higher-order epsilon-expansion up to functions of weight 4 are constructed. The result of the expansion is expressible in terms of Nielsen polylogarithms only. The reductions and epsilon-expansion of q-loop off-shell propagator diagrams with one massive line and q massless lines and q-loop bubble with two-massive lines and q-1 massless lines are considered. The code (Mathematica/FORM) is available via the www at this URL http://theor.jinr.ru/~kalmykov/hypergeom/hyper.html, Comment: 19 pages, LaTeX, 1-eps figure; v5: The code (Mathematica/FORM) is available via the www http://theor.jinr.ru/~kalmykov/hypergeom/hyper.html

For certain dimensionally-regulated massive two- and three-loop propagator-type diagrams the higher order epsilon-expansion is constructed., Comment: 17 pages, LaTeX, 1-eps figure; v3: journal version; new references added; some unessential typos corrected

Generalized log-sine functions appear in higher order epsilon-expansion of different Feynman diagrams. We present an algorithm for numerical evaluation of these functions of real argument. This algorithm is implemented as C++ library with arbitrary-precision arithmetics for integer 0 < k < 9 and j > 1. Some new relations and representations for the generalized log-sine functions are given., Comment: 16 pages,LaTeX; to be published in Comp.Phys.Comm. v.2: typos in Eq.(3.1) corrected; new section(3.3) added with detailed description of the installation instructions

Recent progress in analytical calculation of the multiple [inverse, binomial, harmonic] sums, related with epsilon-expansion of the hypergeometric function of one variable are discussed., Comment: 5 pages, to appear in the proceedings of 7th DESY Workshop on Elementary Particle Theory "Loops and Legs in Quantum Field Theory", April 25 -30, 2004, Zinnowitz (Usedom Island), Germany

Recent progress in the calculation of the two-loop on-shell mass counterterms within the electroweak Standard Model (SM) for the massive particles are discussed. We are in progress of developing a package for full two-loop SM calculations of 2 -> 2 fermion processes, with emphasis on the analytical approach where feasible. The complete two-loop on-shell renormalization is implemented. Substantial progress has been made in calculating the master integrals. We are able to compute in an efficient and stable manner up to a few thousands of diagrams of very complex mass structure., Comment: 4 pages, 1 style file. To appear in the proceedings of 9th International Workshop on Advanced Computing and Analysis Techniques in Physics Research (ACAT 03), Tsukuba, Japan, 1-5 Dec 2003

The O(\alpha \alpha_s) contribution to the relationship between the MS- and the pole-mass of the t-quark propagator within the Standard Model is reviewed. At the same order also the corrections to the top-Yukawa coupling is discussed. We furthermore present the exact analytic expression for the gaugeless limit., Comment: appolb.cls, 10 pages; contribution to the Proceedings of XXVII International Conference of Theoretical Physics, Ustron, Poland, September 15-21, 2003; v2: typos in Eq.(16) corrected

We have calculated the O(\alpha \alpha_s) contributions to the relationship between the MS-mass and the pole of the t-quark propagator in the Standard Model in the limit of a diagonal CKM matrix and for a massless b-quark. Analytical results for the so far unknown master-integrals appearing in the calculation are also given., Comment: 23 pages, LaTeX, 5-eps figures; In v.2 the typos in Eqs.(4.41), (5.54), (5.56), (5.57) and Fifure 4 corrected. New references are added

When calculating higher terms of the epsilon-expansion of massive Feynman diagrams, one needs to evaluate particular cases of multiple inverse binomial sums. These sums are related to the derivatives of certain hypergeometric functions with respect to their parameters. Exploring this connection and using it together with an approach based on generating functions, we analytically calculate a number of such infinite sums, for an arbitrary value of the argument which corresponds to an arbitrary value of the off-shell external momentum. In such a way, we find a number of new results for physically important Feynman diagrams. Considered examples include two-loop two- and three-point diagrams, as well as three-loop vacuum diagrams with two different masses. The results are presented in terms of generalized polylogarithmic functions. As a physical example, higher-order terms of the epsilon-expansion of the polarization function of the neutral gauge bosons are constructed., Comment: 54 pages, LaTeX, 2 eps figures; v4: some typos corrected; notations in Eqs.(C.7),(D.9), (E.3) improved

We have calculated the fermion contributions to the shift of the position of the poles of the massive gauge boson propagators at two-loop order in the Standard Model. Together with the bosonic contributions calculated previously the full two-loop corrections are available. This allows us to investigate the full correction in the relationship between MS and pole masses of the vector bosons Z and W. Two-loop renormalization and the corresponding renormalization group equations are discussed. Analytical results for the master-integrals appearing in the massless fermion contributions are given. A new approach of summing multiple binomial sums has been developed., Comment: 62 pages, LaTeX, 18 eps-figures, version to appear in Nucl.Phys.B

We discuss progress in SM two-loop calculations of the pole position of the massive gauge-boson propagators., Comment: 6 pages

Some problems related to the structure of higher terms of the epsilon-expansion of Feynman diagrams are discussed., Comment: 11 pages, LaTeX, contribution to the Proceedings of CPP2001 (Tokyo, Japan, November 2001)

The relationship between MS and pole masses of the vector bosons Z and W is calculated at the two-loop level in the Standard Model. We only consider the purely bosonic contributions which represents a gauge invariant subclass of diagrams. All calculations were performed in the linear $R_\xi$ gauge with three arbitrary gauge parameters utilizing the method of asymptotic expansions. The results are presented in analytic form as series in the small parameters $\sin^2\theta_W$ and mass ratio $m_Z^2/m_H^2$. As a byproduct we obtain the bosonic two-loop contributions to the renormalization of the weak mixing parameter $\sin^2\theta_W$ and of the Fermi constant. The running of Fermi constant will become important at high energy colliders., Comment: LaTeX, 36 p., 10 fig.; in v3 more technical details about renormalization procedure are added

For certain dimensionally-regulated one-, two- and three-loop diagrams, problems of constructing the epsilon-expansion and the analytic continuation of the results are studied. In some examples, an arbitrary term of the epsilon-expansion can be calculated. For more complicated cases, only a few higher terms in epsilon are obtained. Apart from the one-loop two- and three-point diagrams, the examples include two-loop (mainly on-shell) propagator-type diagrams and three-loop vacuum diagrams. As a by-product, some new relations involving Clausen function, generalized log-sine integrals and certain Euler--Zagier sums are established, and some useful results for the hypergeometric functions of argument 1/4 are presented., Comment: 49 pages, LaTeX, two figures (one in eps format), v3 is journal version; in v3 the typos in Egs.(2.17), (4.16), (B.12) corrected; in v4 the typos in Egs.(2.7),(4.10),(4.25),(B.11) corrected (see p.30 of present version)

Some problems related to construction of the epsilon-expansion of dimensionally regulated Feynman integrals are discussed. For certain classes of diagrams, an arbitrary term of the epsilon-expansion can be expressed in terms of log-sine integrals related to the polylogarithms. It is shown how the analytic continuation of these functions can be constructed in terms of the generalized Nielsen polylogarithms., Comment: 6 pages, requires espcrc2.sty, contribution to the Zeuthen Workshop "Loops and Legs in Gauge Theories" (Bastei, Germany, April 2000)

The $\epsilon$-expansion of several two-loop self-energy diagrams with different thresholds and one mass are calculated. On-shell results are reduced to multiple binomial sums which values are presented in analytical form., Comment: 10 pp LaTeX, misprints in app. A and minor misprints in the text corrected

Exploring the idea of Broadhurst on the ``sixth root of unity'' we present an ansatz for construction of a basis of transcendental numbers for the epsilon-expansion of single mass scale diagrams with two particle massive cut. As example, several new two- and three-loop master integrals are calculated., Comment: 13 pages, LaTeX, two eps-figures. in v.3 misprint in epsilon-part of Eq.(10) is corrected

A FORM based package (ON-SHELL2) for the calculation of two loop self-energy diagrams with one nonzero mass in internal lines and the external momentum on the same mass shell is elaborated. The algorithm, based on recurrence relations obtained from the integration-by-parts method, allows us to reduce diagrams with arbitrary indices (powers of scalar propagators) to a set of master integrals. The SHELL2 package is used for the calculation of special types of diagrams. Analytical results for master integrals are collected., Comment: 24 pages, elsart.sty; LaTeX, 4 eps-figures; in v4 the typos in Appendix A and C corrected

A set of recurrence relations for on-shell two-loop self-energy diagrams with one mass is presented, which allows to reduce the diagrams with arbitrary indices (powers of scalar propagators) to a set of the master integrals. The SHELL2 package is used for the calculation of special types of diagrams. A method of calculation of higher order \epsilon-expansion of master integrals is demonstrated., Comment: Talk given by J.Fleischer at 6th International Workshop on Software Engineering, Artificial Intelligence, Neural Nets, Genetic Algorithms, Symbolic Algebra, Automatic Calculation (AIHENP 99), Heraklion, Crete, Greece, 12-16 April, 1999; 8 pages, LaTeX, 3 eps-figures

Analytic results for the complete set of two-loop self-energy master integrals on shell with one mass are calculated., Comment: 14 pages, LaTeX, one eps-figure; in v5. misprints in the Eq.(2) and Table II corrected

In this paper we discuss the connection between the geometric and tetrad approaches in the quantum affine-metric gravity. The corresponding transition formulas are obtained at the one-loop level. As an example, the one-loop counterterms are calculated in the tetrad formalism in the theory with terms quadratic in the torsion field. This model possesses the extra local symmetries connected with transformation of the connection field. It is shown that the special gauge can be chosen so that the corresponding additional ghosts do not contribute to the one-loop divergent terms., Comment: LaTeX, 10 pages, to appear in Mod.Phys.Lett.A

In this paper, the dependence of the Einstein gravity with the cosmological constant as well as of this theory in the first-order formalism on the gauge and parametrization is been analyzed. The one-loop counterterms off the mass shell have been plainly calculated in arbitrary gauge and parametrization. The tensor package of analytic calculations, written in REDUCE, allowed all the calculations to be carried out. A method of renormalization group functions calculations off shell is discussed., Comment: LaTeX, 28 pages

The renormalization group method in $R^2$-gravity without matter fields is discussed. A criterion for the existence of the renormalization constant for the metric has been found, two-loop higher order poles have been calculated, a relation which allows us to find the one-loop renormalization constant of the Newtonian constant has been suggested., Comment: LaTeX, 8 pages

The charge $F_1(0)$ and the magnetic $F_2(0)$ form factors of heavy charged leptons have been shown in the framework of the perturbation theory to have imaginary part. The imaginary parts of the form factors for muon and tau lepton have been calculated at the two-loop level in the Standard Model. The effects where these imaginary parts could manifest themselves are discussed., Comment: New references added, the last section modified; final version to appear in Phys. Lett. B; LaTeX, 10 pages, 5 eps-figures included

The method of the large mass expansion (LME) has the technical advantage that two-loop integrals occur only as bubbles with large masses. In many cases only one large mass occurs. In such cases these integrals are expressible in terms of $\Gamma$-functions, i.e. they can be handled completely analytically avoiding even recursions and therefore this approach may find a wide field of application. We consider it necessary to investigate the precision of this method and test it for several two-loop vertex functions ocurring in the $Z \to b\bar{b}$ decay by comparing it with the small momentum expansion. It turns out that in general high order approximants have to be taken into account for a sufficient accuracy., Comment: LaTeX, 8 pages including 5 eps figures, uses epsf.sty

A method to calculate two-loop self-energy diagrams of the Standard Model is demonstrated. A direct physical application is the calculation of the two-loop electroweak contribution to the anomalous magnetic moment of the muon ${{1/2}(g-2)}_{\mu}$. Presently, we confine ourselves to a ``toy'' model with only $\mu$, $\gamma$ and a heavy neutral scalar particle (Higgs). The algorithm is implemented as a FORM-based program package. For generating and automatically evaluating any number of two-loop self-energy diagrams, a special C-program has been written. This program creates the initial FORM-expression for every diagram generated by QGRAF, executes the corresponding subroutines and sums up the final results., Comment: LaTeX, 20 pages, 7 eps-figures included; extended version of talk given at AIHEN96, Lausanne, 1-6 Sept. 1996; detailed description of C-program is given; accepted for publication in Comp.Phys.Comm

A new way how to calculate the off-shell renormalization functions within the $R^2$-gravity has been proposed. The one-loop renormalization group equations in the approach suggested have been constructed. The behaviour of effective potential for an massless scalar field interacting with the quantum gravitational field has been analyzed in this approach., Comment: LaTeX, 12 pages; accepted for publication in Phys. Lett. B

Pole masses of quarks in the quantum chromodynamics are calculated to the two-loop order in the framework of the regularization by dimensional reduction. For the diagram with a light quark loop, the non-Euclidean asymptotic expansion is constructed with the external momentum on the mass shell of a heavy quark., Comment: the version accepted for publication in Nucl.Phys.B. The new references are added, new Appendix are presented, little misprints fixed in eqs. (2), (14), (17). LaTex, 23 pages, including 1 eps-figure

A method to calculate two-loop self-energy diagrams of the Standard Model is demonstrated. A direct physical application is the calculation of the two-loop electroweak contribution to the anomalous magnetic moment of the muon ${1/2}(g-2)_{\mu}$. Presently we confine ourselves to a ``toy'' model with only $\mu$, $\gamma$ and a scalar particle (Higgs). The algorithm is implemented as a package of computer programs in FORM. For generating and automatically evaluating any number of two-loop self-energy diagrams, a special C-program has been written. This program creates the initial FORM-expression for every diagram generated by QGRAF, executes the corresponding subroutines and sums up the final results., Comment: LaTeX, 6 pades, including 1 eps-figure; Contribution to Proceeding of 5th Workshop on Software Engineering and Artificial Intelligence for High Energy And Nuclear Physics (AIHEN96), Lausanne, 1-6 Sept. 1996

We investigate the consistency of the background-field formalism when applying various regularizations and renormalization schemes. By an example of a two-dimensional $\sigma$ model it is demonstrated that the background-field method gives incorrect results when the regularization (and/or renormalization) is noninvariant. In particular, it is found that the cut-off regularization and the differential renormalization belong to this class and are incompatible with the background-field method in theories with nonlinear symmetries., Comment: Talk given at 9th International Workshop on High Energy Physics and Quantum Field Theory (NPI MSU95), Zvenigorod, Russia, 19-26 Sep 1995, 7 pages, LaTeX

We discuss the role of additional local symmetries related to the transformations of connection fields in the affine-metric theory of gravity. The corresponding BRST transformations connected with all symmetries (general coordinate, local Lorentz and extra) are constructed. It is shown, that extra symmetries give the additional contribution to effective action which is proportional to the corresponding Nielsen-Kallosh ghost one. Some arguments are given, that there is no anomaly associated with extra local symmetries., Comment: 14 pages in LATEX (The version of paper accepted for publication in Class. Quant. Grav.)

In the context of the standard SUSY GUT scenario, we present a detailed analysis of the softly broken finite supersymmetric grand unified theory. The model, albeit non-minimal, remains very rigid due to the requirement of finiteness. It is based on the $SU(5)$ gauge group and is UV finite to all orders of perturbation theory. It contains three generations of the matter fields together with four pairs of Higgses. The requirement of UV finiteness fixes all the Yukawa couplings at the GUT scale. Imposing the condition of universality on the soft couplings at the Planck scale and then extending the condition of finiteness to them, one gets a completely finite unified theory above $M_{GUT}.$ This makes the fine-tuning procedure more meaningful and leads to the usual Minimal Supersymmetric Standard Model below $M_{GUT}$. All the masses of the ordinary particles including Higgses are obtained due to the Higgs mechanism at the electro-weak scale. The hierarchy of quark and lepton masses is related to that of v.e.v.'s of the Higgs fields and is governed by the Higgs mixing matrix in the generation space. Superpartners develop their masses according to the RG equations starting from the soft terms at the Planck scale. The suggestion of complete finiteness and maximal simplicity of the unified theory leads to the connection between the initial values of soft SUSY breaking parameters, namely $m_0^2=1/3m_{1/2}^2,\ A_t=A_b=A_\tau=-m_{1/2}, \ B=-m_{1/2}$, so that the number of free parameters is less than that of the MSSM., Comment: LaTex, 22 pages, no figures; some refs are added, minor changes, version to appear in Nucl. Phys. B

The ambiguity in the calculations of one-loop counterterms by the background field method in nonrenormalizable theories of gravity is discussed. Some examples of such ambiguous calculations are given. The non-equivalence of the first and second order formalism in the quantum gravity is shown., Comment: 21 pages in LATEX

The parametrization and gauge dependencies of the one-loop counterterms on the mass-shell in the Einstein gravity are investigated. The physical meaning of the loop calculation results on the mass shell and the parametrization dependence of the renormgroup functions in the nonrenormalizable theories are discussed., Comment: 14 pages in LATEX (Some references added)

We investigate the role of the torsion field at the quantum level in the affine-metric theory of gravity. One-loop counterterms are calculated in the theory with terms quadratic in the torsion field., Comment: 13 pages in LATEX. (The title and references are changed. The discussion of the BRST-symmetry added.)

We investigate the consistency of the background-field formalism when applying various regularizations and renormalization schemes. By an example of a two-dimensional $\sigma$ model it is demonstrated that the background-field method gives incorrect results when the regularization (and/or renormalization) is noninvariant. In particular, it is found that the cut-off regularization and the differential renormalization belong to this class and are incompatible with the background-field method in theories with nonlinear symmetries., Comment: 12 pages, LATEX

We investigate the influence of the projective invariance on the renormalization properties of the theory. One-loop counterterms are calculated in the most general case of interaction of gravity with scalar field., Comment: 10 pages, LATEX