A theorem on asymptotic expansion of Feynman amplitudes

For any Feynman amplitude, where any subset of invariants and/or squared masses is scaled by a real parameter going to zero or infinity, the existence of an expansion in powers of and ln is proved, and a method is given for determining such an expansion. This is shown quite generally in euclidean metric, whatever the external momenta (exceptional or not) and the internal masses (vanishing or not) may be, and for some simple cases in minkowskian metric, assuming only finiteness of the — eventually renormalized — amplitude before scaling. The method uses what is called Multiple Mellin representation, the validity of which is related to a generalized power-counting theorem.On leave of absence from University of Bahia (Brazil). Fellow of CAPES, Brazil     

Asymptotic expansion of Feynman amplitudes

Employing the technique of Mellin transforms to scalar convergent Feynman amplitude in the Schwinger integral representation, we determine its asymptotic expansion for large Euclidean momenta. The determination of the coefficients of the expansion is effected via the use of generalized Taylor operators.     

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.     

Feynman rules for probability amplitudes

Starting with very simple assumptions, Feynman rules for the quantum mechanical amplitudes and the associated probabilities are derived. These rules emerge as the only consistent rules for manipulating complex amplitudes assigned to processes. The probability of a process to which an amplitudex has been assigned is determined asp(x)=|x| , 0<2. If virtual processes are allowed,=2.     

Feynman amplitudes with confinement included

Amplitudes for any multipoint Feynman diagram are written taking into account vacuum background confining field. Higher order gluon exchanges are treated within background perturbation theory. For amplitudes with hadrons in initial or final states vertices are shown to be expressed by the corresponding wave function with the renormalized z factors. Examples of two-point functions, three-point functions (form factors), and decay amplitudes are explicitly considered. The text was submitted by the author in English.     

Covariant string theory Feynman amplitudes

The covariant string amplitudes on a strip with the associated Feynman rules are considered. The contribution of the Faddeev-Popov ghost is evaluated. The overlap vertex (coming from a δ-function interaction) is constructed. The corresponding scattering diagrams are shown to give the standard Veneziano model answer. The ghosts provide the necessary contribution to the measure.     

P-adic Feynman and string amplitudes

We derive an explicit representation forp-adic Feynman and Koba-Nielsen amplitudes and we briefly outline the connection between the scalar models ofp-adic quantum field theory and Dyson's hierarchical models.This work was supported in part by the French Government     

Renormalization in the complete Mellin representation of Feynman amplitudes

The Feynman amplitudes are renormalized in the formalism of the CM representation. This Mellin-Barnes type integral representation, previously introduced for the study of asymptotic behaviours, is shown to have the following interesting property: in contrast with the usual subtraction procedures, the renormalization leaves the CM integrand unchanged, and only results into translations of the integration path. The explicit CM representation of the renormalized amplitudes is given. In addition, the dimensional regularization and the extension to spinor amplitudes are sketched.     

A note on covariant Feynman rules for closed strings

It is established that covariant Feynman rules for closed string fields must decompose the moduli space of a four-punctured sphere with each diagram giving two copies of a fundamental region of the group of anharmonic ratios. This may help determining if such decomposition is possible without an elementary four closed-string interaction.     

Renormalization of Feynman amplitudes and parametric integral representation

A new substraction formula is presented to renormalize Feynman amplitudes written in Schwinger's integral representation. The substractions are generated by an operator acting on the integrand, which only depends on the total number of internal lines but is completely independent of the structure of the graph. This formulation is also valid for non-renormalizable theories and is shown to reduce to Zimmermann'sR-operation for scalar theories. It satisfies in any case Bogoliubov's recursive formula and yields an explicit tool for actual computations of renormalized Feynman amplitudes with a minimal number of substractions.     

Infrared and ultraviolet dimensional meromorphy of Feynman amplitudes

By the concurrent use of dimensional and analytic regularizations with the complete Mellin (CM) representation, we find in a direct way the ultraviolet and infrared poles in space-time dimension, for any Feynman amplitude with an arbitrary subset of vanishing masses.     

Dual Feynman rules—Topological asymptotic freedom

Feynman-graph rules are formulated for the strong—interaction components of the topological expansion—defined as those graphs all of whose vertices are zero—entropy connected parts. These rules imply a “topological asymptotic freedom” and admit a corresponding perturbative evaluation where the zeroth order exhibits topological supersymmetry.     

Quark-gluon amplitudes in the dual expansion

The dual representation, which gives a simple analytical form for purely gluonic amplitudes, is extended to amplitudes which include a quark-antiquark pair. To minimize the calculations, supersymmetry is used to relate the purely gluonic amplitudes to those including a gluino pair from which the quark-antiquark amplitudes are easily deduced. We explicitly give simple analytical forms for the full amplitudes for those multi-parton processes which involve a quark-antiquark pair plus two, three and four gluons.     

Simple method to make asymptotic series of Feynman diagrams converge

We show that, for two nontrivial lambda phi(4) problems (the anharmonic oscillator and the Landau-Ginzburg hierarchical model), improved perturbative series can be obtained by cutting off the large field contributions. The modified series converge to values exponentially close to the exact ones. For lambda larger than some critical value, the method outperforms Padé's approximants and Borel summations. The method can also be used for series which are not Borel summable such as the double-well potential series. We show that semiclassical methods can be used to calculate the modified Feynman rules, estimate the error, and optimize the field cutoff.     

Concentratable functionals and asymptotic relations for scattering amplitudes

Very wide classes of concentratable functionals are considered, i.e. classes where the notion of concentrating, which satisfies a certain six axioms, can be introduced, and where it is possible to formulate the microcausality (or the local-commutativity) principle as the requirement of concentration of functionals in the corresponding space-time region. Every such class of functionals is essentially wider than the class of tempered distributions and is defined on some analytic functions only. It is proved that the asymptotic smoothed partially-microcausal amplitude of an arbitrary reaction, under corresponding conditions imposed only in the physical region of changing arguments, satisfies all the asymptotic relations.     

A note on the asymptotic master equations for systems with bound states

It is shown that, within the convolutionless formalism of Fuliski, the asymptotic form (in time) of the Liouville equation does not change with the existence of bound states in the energy spectrum of the system under consideration.     

Classification and asymptotic scaling of the light-cone wave-function amplitudes of hadrons

We classify the hadron light-cone wave-function amplitudes in terms of parton helicity, orbital angular momentum, and quark-flavor and color symmetries. We show in detail how this is done for the pion, meson, nucleon, and delta resonance up to and including three partons. For the pion and nucleon, we also consider four-parton amplitudes. Using the scaling law derived previously, we show how these amplitudes scale in the limit that all parton transverse momenta become large.Received: 16 October 2003, Published online: 29 January 2004     

A note on the summation of the virial expansion in two dimensions

The infinite subset of Cayley trees in the cluster expansion is considered and the expressions for two-dimensional pressure and adsorption isotherm are obtained.     

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.Supported by Bundesministerium für Forschung und Technologie