The Number of Master Integrals is Finite

For a fixed Feynman graph one can consider Feynman integrals with all possible powers of propagators and try to reduce them, by linear relations, to a finite subset of integrals, the so-called master integrals. Up to now, there are numerous examples of reduction procedures resulting in a finite number of master integrals for various families of Feynman integrals. However, up to now it was just an empirical fact that the reduction procedure results in a finite number of irreducible integrals. It this paper we prove that the number of master integrals is always finite.     

First results for the Coulomb gauge integrals using NDIM

The Coulomb gauge has at least two advantages over other gauge choices in that bound states between quarks and studies of confinement are easier to understand in this gauge. However, perturbative calculations, namely Feynman loop integrations, are not well defined (there are the so-called energy integrals) even within the context of dimensional regularization. Leibbrandt and Williams proposed a possible cure to such a problem by splitting the space-time dimension into , i.e., introducing a specific parameter to regulate the energy integrals. The aim of our work is to apply the negative dimensional integration method (NDIM) to the Coulomb gauge integrals using the recipe of split-dimension parameters and present complete results – finite and divergent parts – to the one- and two-loop level for arbitrary exponents of the propagators and dimension. Received: 12 May 2000 / Revised version: 28 August 2000 / Published online: 15 March 2001     

Renormalization of QED with planar binary trees

The Dyson relations between renormalized and bare photon and electron propagators and are expanded over planar binary trees. This yields explicit recursive relations for the terms of the expansions. When all the trees corresponding to a given power of the electron charge are summed, recursive relations are obtained for the finite coefficients of the renormalized photon and electron propagators. These relations significantly decrease the number of integrals to carry out, as compared to the standard Feynman diagram technique. In the case of massless quantum electrodynamics (QED), the relation between renormalized and bare coefficients of the perturbative expansion is given in terms of a Hopf algebra structure. Received: 23 March 2000 / Revised version: 24 November 2000 / Published online: 6 April 2001     

Double-real corrections at $${{\mathcal {O}}(\alpha \alpha _s)}\,$$to single gauge boson production

We consider the \({{\mathcal {O}}(\alpha \alpha _s)}\,\)corrections to single on-shell gauge boson production at hadron colliders. We concentrate on the contribution of all the subprocesses where the gauge boson is accompanied by the emission of two additional real partons and we evaluate the corresponding total cross sections. The latter are divergent quantities, because of soft and collinear emissions, and are expressed as Laurent series in the dimensional regularization parameter. The total cross sections are evaluated by means of reverse unitarity, i.e. expressing the phase-space integrals in terms of two-loop forward box integrals with cuts on the final-state particles. The results are reduced to a combination of master integrals, which eventually are evaluated in terms of generalized polylogarithms. The presence of internal massive lines in the Feynman diagrams, due to the exchange of electroweak gauge bosons, causes the appearance of 14 master integrals which were not previously known in the literature and have been evaluated via differential equations.     

Two-loop relevant parts ofD-dimensional massive scalar one-loop integrals

The calculation of two-loop Feynman integrals within the dimensional regularization scheme requires the knowledge of scalar one-loop integrals up to the linear term inD-4. We give the corresponding explicit expressions in terms of polylogarithms for the general one-, two-and three-point function and for a special case of the fourpoint function needed for vertex corrections. Our results are valid in all kinematical regions for real masses and momenta.     

Calculation of the quark and gluon form factors to three loops in QCD

We describe the calculation of the three-loop QCD corrections to quark and gluon form factors. The relevant three-loop Feynman diagrams are evaluated and the resulting three-loop Feynman integrals are reduced to a small set of known master integrals by using integration-by-parts relations. Our calculation confirms the recent results by Baikov et al. for the three-loop form factors. In addition, we derive the subleading \( \mathcal{O}\left( \varepsilon \right) \) terms for the fermion-loop type contributions to the three-loop form factors which are required for the extraction of the fermionic contributions to the four-loop quark and gluon collinear anomalous dimensions. The finite parts of the form factors are used to determine the hard matching coefficients for the Drell-Yan process and inclusive Higgs-production in soft-collinear effective theory.     

Differential reduction of generalized hypergeometric functions from Feynman diagrams: One-variable case

The differential-reduction algorithm, which allows one to express generalized hypergeometric functions with parameters of arbitrary values in terms of the same functions with parameters whose values differ from the original ones by integers, is discussed in the context of evaluating Feynman diagrams. Where this is possible, we compare our results with those obtained using standard techniques. It is shown that the criterion of reducibility of multiloop Feynman integrals can be reformulated in terms of the criterion of reducibility of hypergeometric functions. The relation between the numbers of master integrals obtained by differential reduction and integration by parts is discussed.     

Feynman Integrals and Motives of Configuration Spaces

We formulate the problem of renormalization of Feynman integrals and its relation to periods of motives in configuration space instead of momentum space. The algebro-geometric setting is provided by the wonderful compactifications [`(Conf)]_G(X){\overline{Conf}_\Gamma(X)} of arrangements of subvarieties associated to the subgraphs of a Feynman graph Γ, with X a (quasi)projective variety. The motive and the class in the Grothendieck ring are computed explicitly for these wonderful compactifications, in terms of the motive of X and the combinatorics of the Feynman graph, using recent results of Li Li. The pullback to the wonderful compactification of the form defined by the unrenormalized Feynman amplitude has singularities along a hypersurface, whose real locus is contained in the exceptional divisors of the iterated blowup that gives the wonderful compactification. A regularization of the Feynman integrals can be obtained by modifying the cycle of integration, by replacing the divergent locus with a Leray coboundary. The ambiguities are then defined by Poincaré residues. While these residues give periods associated to the cohomology of the exceptional divisors and their intersections, the regularized integrals give rise to periods of the hypersurface complement in the wonderful compactification.     

Asymptotic expansions in limits of large momenta and masses

Asymptotic expansions of renormalized Feynman amplitudes in limits of large momenta and/or masses are proved. The corresponding asymptotic operator expansions for theS-matrix, composite operators and their time-ordered products are presented. Coefficient functions of these expansions are homogeneous within a regularization of dimensional or analytic type. Furthermore, they are explicitly expressed in terms of renormalized Feynman amplitudes (at the diagrammatic level) and certain Green functions (at the operator level).     

Feynman integral relations from parametric annihilators

We study shift relations between Feynman integrals via the Mellin transform through parametric annihilation operators. These contain the momentum space integration by parts relations, which are well known in the physics literature. Applying a result of Loeser and Sabbah, we conclude that the number of master integrals is computed by the Euler characteristic of the Lee–Pomeransky polynomial. We illustrate techniques to compute this Euler characteristic in various examples and compare it with numbers of master integrals obtained in previous works.

     

Implicit regularization beyond one-loop order: gauge field theories

We extend a constrained version of implicit regularization (CIR) beyond one-loop order for gauge field theories. In this framework, the ultraviolet content of the model is displayed in terms of momentum loop integrals order by order in perturbation theory for any Feynman diagram, while the Ward–Slavnov–Taylor identities are controlled by finite surface terms. To illustrate, we apply CIR to massless abelian gauge field theories (scalar and spinorial QED) to two-loop order and calculate the two-loop beta-function of spinorial QED. PACS  11.10.Gh; 11.15.Bt; 11.15.-q     

Convergence of Feynman integrals in Coulomb gauge QCD

At 2-loop order, Feynman integrals in the Coulomb gauge are divergent over the internal energy variables. Nevertheless, it is known how to calculate the effective action, provided that the external gluon fields are all transverse. We show that, for the two-gluon Greens function as an example, the method can be extended to include longitudinal external fields. The longitudinal Greens functions appear in the BRST identities. As an intermediate step, we use a flow gauge, which interpolates between the Feynman and Coulomb gauges.     

Hamiltonian Feynman-Kac and Feynman Formulae for Dynamics of Particles with Position-Dependent Mass

A Feynman formula is a representation of the semigroup, generated by an initial-boundary value problem for some evolutionary equation, by a limit of integrals over Cartesian powers of some space E, the integrands being some elementary functions. The multiple integrals in Feynman formulae approximate integrals with respect to some measures or pseudomeasures on sets of functions which take values in E and are defined on a real interval. Hence Feynman formulae can be used both to calculate explicitly solutions for such problems, to get some representations for these solutions by integrals over functions taking values in E (such representations are called Feynman-Kac formulae), to get approximations for transition probability of some diffusion processes and transition amplitudes for quantum dynamics and to get computer simulations for some stochastic and quantum dynamics. The Feynman formula is called a Hamiltonian Feynman formula if the space, Cartesian products of which are used, is the phase space of a classical Hamiltonian system; the corresponding Feynman-Kac formula is called a Hamiltonian Feynman-Kac formula. In the latter formula one integrates over functions taking values in the same phase space. In a similar way one can define Lagrangian Feynman formulae and Lagrangian Feynman-Kac formulae substituting the phase space by the configuration space.     

Functional equations for Feynman integrals

New types of equations for Feynman integrals are found. It is shown that the latter satisfy functional equations that relate integrals with different kinematics. A regular method for obtaining such relations is proposed. A derivation of the functional equations for one-loop two-, three-, and four-point functions with arbitrary masses and external momenta is given. It is demonstrated that the functional equations can be used to analytically continue Feynman integrals to various kinematical domains.     

Cut-off versus dimensional regularization in the light-cone gauge

The conventional cut-off method is applied to massless light-cone gauge Feynman integrals. Despite the presence of non-local terms in the unintegrated expression for the Yang-Mills self-energy, the cut-off procedure yields the same ultra-violet behaviour as the lengthier technique of dimensional regularization.     

有限温度下胶子场三圈真空图中的红外发散

用热传播子的实时形式对有限温度下胶子场的三圈真空图进行了详细的计算，在维数正规化方案下把真空图中的红外发散全部孤立出来了.     

Infrared Divergences of Three-Loop Vacuum Graghs of Gluon Field at Finite Temperature

The kinds of infrared divergent integrals in three-loop vacuum graghs of gluon field at finite temperature are pointed out and their regularization is discussed. All of the infrared divergences in three-loop vacuum graghs of gluon field are isolated.