Distributional limits of renormalized Feynman integrals with zero-mass denominators

It is shown that the 0 limits of renormalized Feynman integrals exist and define Lorentz invariant tempered distributions in the external momenta. The proof applies to the case where some or all particle masses vanish.Research supported in part by the National Science Foundation Grant MPS-74-21778.Research supported in part by the National Science Foundation Grant GP-43758.     

Convergence theorems for renormalized Feynman integrals with zero-mass propagators

A general momentum-space subtraction procedure is proposed for the removal of both ultraviolet and infrared divergences of Feynman integrals. Convergence theorems are proved which allow one to define time-ordered Green functions, as tempered distributions, for a wide class of theories with zero-mass propagators.Research supported in part by National Science Foundation Grant MPS-74-21778.     

Open quantum systems and Feynman integrals: Some problems

A few open problems of mathematical physics are presented. They concern open quantum systems and Feynman path integrals; some of them are technical, while others are of conceptual importance.Dedicated to the 30th anniversary of the Joint Institute for Nuclear Research.     

Efficient numerical evaluation of Feynman integrals

Feynman loop integrals are a key ingredient for the calculation of higher order radiation effects, and are responsible for reliable and accurate theoretical prediction. We improve the efficiency of numerical integration in sector decomposition by implementing a quasi-Monte Carlo method associated with the CUDA/GPU technique. For demonstration we present the results of several Feynman integrals up to two loops in both Euclidean and physical kinematic regions in comparison with those obtained from FIESTA3. It is shown that both planar and non-planar two-loop master integrals in the physical kinematic region can be evaluated in less than half a minute with O(10~(-3))accuracy, which makes the direct numerical approach viable for precise investigation of higher order effects in multiloop processes, e.g. the next-to-leading order QCD effect in Higgs pair production via gluon fusion with a finite top quark mass.     

A recursive reduction of tensor Feynman integrals

We perform a new, recursive reduction of one-loop n-point rank R   tensor Feynman integrals [in short: (n,R)$(n,R)$-integrals] for n?6$n?6$ with R?n$R?n$ by representing (n,R)$(n,R)$-integrals in terms of (n,R−1)$(n,R-1)$- and (n−1,R−1)$(n-1,R-1)$-integrals. We use the known representation of tensor integrals in terms of scalar integrals in higher dimension, which are then reduced by recurrence relations to integrals in generic dimension. With a systematic application of metric tensor representations in terms of chords, and by decomposing and recombining these representations, a recursive reduction for the tensors is found. The procedure represents a compact, sequential algorithm for numerical evaluations of tensor Feynman integrals appearing in next-to-leading order contributions to massless and massive three- and four-particle production at LHC and ILC, as well as at meson factories.     

Symbology of Feynman integrals from twistor geometries

We study the symbology of planar Feynman integrals in dimensional regularization by considering geometric configurations in momentum twistor space corresponding to their leading singularities(LS). Cutting propagators in momentum twistor space amounts to intersecting lines associated with loop and external dual momenta, including the special line associated with the point at infinity, which breaks dual conformal symmetry. We show that cross-ratios of intersection points on these lines, especially...     

The monodromy rings of one loop Feynman integrals

The monodromy rings of Feynman integrals for one loop graphs with an arbitrary number of lines are determined.Research sponsored by the National Science Foundation, Grant No. GP-16147.     

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.     

A study of Feynman integrals by micro-differential equations

We use the holonomic character of Feynman integrals to describe their singularity structure explicitly in some simple cases. The results in §1 show that under moderate conditions Feynman amplitudes can be locally expressed essentially in terms of Legendre functions near the points where two positive- Landau-Nakanishi surfaces meet. Related topics such as hierarchical principle in perturbation theory are also discussed in terms of holonomic systems involved. In §4 we use the concrete expressions for Feynman amplitudes obtained in §1 to discuss the validity of Sato's conjecture.Supported in part by NSF MCS 75-2333Supported in part by NSF GP 36269     

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.     

Renormalization of lattice Feynman integrals with massless propagators

A renormalization procedure is proposed which applies to lattice Feynman integrals containing zero-mass propagators and is analogous to the BPHZL renormalization procedure for continuum Feynman integrals. The renormalized diagrams are infrared convergent for non-exceptional external momenta, if the vertices of the theory satisfy a general infrared constraint. Under the same conditions as in the massive case [4], the continuum limit of the renormalized theory exists and is independent of the details of the lattice action.     

Some fractional-calculus results for the \bar H-function associated with a class of Feynman integrals

In many recent works, several authors demonstrated the usefulness of fractional-calculus operators in many different directions. The main object of this paper is to present, in a unified manner, a number of key results for the general -function, which is associated with a certain class of Feynman integrals, involving the Riemann-Liouville, the Weyl, and other fractional-calculus operators such as those based on the Cauchy-Goursat Integral Formula. Various paricular cases and consequences of our main fractional-calculus results are also considered. Dedicated to the memory of Professor Boris Moiseevich Levitan (1914–2004)     

Some properties of the renormalized harmonic phonon approximation at high temperatures

Using the Einstein and the high temperature approximation the renormalized harmonic phonon approximation yields at constant pressure a first order phase transition similar to the van der Waals phase transition. Calculation of the compressibility by different methods leads to an inconsistency of about 55%.     

Holonomic character and local monodromy structure of Feynman integrals

We prove that the micro-local holonomic structure controls the local monodromy structure of functions involved. This result plays an essential role in investigating hierarchical principle in perturbation theory.Supported by National Science FoundationSupported by Miller Institute for Basic Research in Science     

Definition of renormalized mass in classical gravidynamics

InEinstein's theory of gravitation, energy and momentum of sufficientlyregular systems can be defined either asspace integrals of suitable densities, or alternatively assurface integrals of certain superpotentials. Forsingular fields, these definitions are in general not equivalent. This is shown explicitely for an electrically charged point particle (Reissner-Nordström metric). The first method produces, similar to classical electrodynamics, adivergent selfenergy, whereas the second method gives afinite result, thus leading automatically to arenormalized mass. Perhaps quantized gravity might cause similar improvements ofquantum field theory.     

Asymptotic expansions of Feynman integrals of exponentials with polynomial exponent

In the paper, an asymptotic expansion of path integrals of functionals having exponential form with polynomials in the exponent is constructed. The definition of the path integral in the sense of analytic continuation is considered.