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.     

The asymptotic behavior of massless Feynman integrals in the limit of several large scales

A general method is developed which permits the ealculation of the asymptotic behavior of Feynman integrals in the limit of several large seales, All those contributions which are power law suppressed are neglected. The parametric representation of Feynman integrals is employed, and is reviewed concisely. The main tool is the multiple Mellin transform of Feynman integrals with respect to their external momenta. A transformation of the Feynman parameters is represented which allows the systematic isolation of the leading poles in the multiple Mellin plane.     

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.     

Nonanalyticity of the perturbative expansion for super-renormalizable massless field theories

The complete infrared expansion of Feynman amplitudes is established at any dimension d. The so called infrared finite parts develop poles at rational d. We prove a conjecture by Parisi by constructing an infrared subtraction procedure which defines finite amplitudes in such dimensions. The corresponding counterterms are associated to nonlocal operators and are generated in a nonperturbative way for super-renormalizable theories. We determine at all orders the perturbative expansion which contains powers and logarithms of the coupling constant.     

State space formalism of Feynman diagram theory for acousto-optic interactions:Ⅱ. multifrequency case

A Feynman diagram theory for acousto-optic (AO) interactions is established, which provides a general method to calculate the scattering amplitudes and intensities for both single-frequency and multifrequcncy AO interactions. The method is based on counting the number of allowable Feynman diagrams. The following important assertion has been proved rigorously in this paper. The ratios of the numbers of Feynman diagrams allowable in various B ragg diffractions (isotropie, nondegeneratc birefringent, and degenerate birefringent) to that in Raman-Nath diffraction are independent of the number of different acoustic frequencies, being a function only of the order of the Feynman diagram and the diffraction order of the final state. General expressions for these ratios arc obtained. Based on this, complete perturbation solutions for the scattering amplitudes and intensities are obtained for any kind of AO interactions, any number of acoustic frcquencies, and any final state. This theory gives all results obtain     

Numerical evaluation of tensor Feynman integrals in Euclidean kinematics

For the investigation of higher order Feynman integrals, potentially with tensor structure, it is highly desirable to have numerical methods and automated tools for dedicated, but sufficiently ‘simple’ numerical approaches. We elaborate two algorithms for this purpose which may be applied in the Euclidean kinematical region and in d=4?2ε dimensions. One method uses Mellin–Barnes representations for the Feynman parameter representation of multi-loop Feynman integrals with arbitrary tensor rank. Our Mathematica package AMBRE has been extended for that purpose, and together with the packages MB (M. Czakon) or MBresolve (A.V. Smirnov and V.A. Smirnov) one may perform automatically a numerical evaluation of planar tensor Feynman integrals. Alternatively, one may apply sector decomposition to planar and non-planar multi-loop ε-expanded Feynman integrals with arbitrary tensor rank. We automatized the preparations of Feynman integrals for an immediate application of the package sector_decomposition (C. Bogner and S. Weinzierl) so that one has to give only a proper definition of propagators and numerators. The efficiency of the two implementations, based on Mellin–Barnes representations and sector decompositions, is compared. The computational packages are publicly available.     

The Polyakov path integral over bordered surfaces (the open string amplitudes)

We use the Feynman functional quantization scheme adapted to the gauge theories with reparametrization invariance to the functional covariant first quantization of the open bosonic BDHP string in a position representation. The consistent functional integral representation of the open string propagator is derived and evaluated. This result is used as a starting point for two kinds of constructions of the off-shell multiloop open string amplitudes. The general idea of the presented approach is to consider the off-shell amplitudes as functionals on the space of contours endowed with an intrinsic metric or on the space /₊.     

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.     

Bosonic colored group field theory

Bosonic colored group field theory is considered. Focusing first on dimension four, namely the colored Ooguri group field model, the main properties of Feynman graphs are studied. This leads to a theorem on optimal perturbative bounds of Feynman amplitudes in the “ultraspin” (large spin) limit. The results are generalized in any dimension. Finally, integrating out two colors we write a new representation, which could be useful for the constructive analysis of this type of models.     

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     

On renormalization and complex space-time dimensions

The method of using the dimension of space-time as a complex parameter introduced recently to regularize Feynman amplitudes is extended to an arbitrary Feynman graph. The method has promise of being particularly well-suited to gauge theories. It is shown how the renormalized amplitude, together with the Lagrangian counter-terms, may be extracted directly, following the method of analytic renormalization.     

Feynman Diagram Theory of Degenerate Birefringent Bragg Diffraction

A Feynman diagram theory for acousto-optic (AO) interaction is established, which supplies a general method to calculate the scattering amplitudes and intensities for both single frequency and multifrequency AO interactions. The method L based on counting the number of allowable Feynman diagrams. Based on this, a complete solution for the scattering amplitudes and intensities an be obtained for any kind of AO interactions, any number of acoustic frequencies, and any final states. A geometrical representation for this Feynman diagram theory of AO interaction is set up too. The theory is ale0 verified by comparing with the experiments.     

Group field theory with noncommutative metric variables

We introduce a dual formulation of group field theories as a type of noncommutative field theories, making their simplicial geometry manifest. For Ooguri-type models, the Feynman amplitudes are simplicial path integrals for BF theories. We give a new definition of the Barrett-Crane model for gravity by imposing the simplicity constraints directly at the level of the group field theory action.     

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.     

Local existence of the Borel transform in euclidean massless Φ 4 4

We extend the methods of [1] to prove large order estimates on the renormalized Feynman amplitudes of massless ₄ ⁴ euclidean field theory, at non-exceptional momenta. The Borel transform of the perturbative series is analytic in a disk centered at the origin of the complex plane. This result is a step towards the rigorous investigation of the infra-red singularities in the Borel plane, for theories containing massless particles, like the gauge theories.     

Integral representation for the dimensionally renormalized Feynman amplitude

A compact convergent integral representation for dimensionally renormalized Feynman amplitudes is explicitly constructed. The subtracted integrand is expressed as a distribution in the Schwinger -parametric space, and is obtained by applying upon the bare integrand a new subtraction operatorR' which respects Zimmermann's forest structure.     

Near mass shell singularities in quantum electrodynamics

We discuss the near mass shell infrared behavior of QED by performing an explicit sum over all Feynman diagrams in the eikonal approximation. We review the infrared singularities of exclusive amplitudes in particular limits ((a) small photon mass or dimension ≠ 4, (b) equal off shell p_i², (c) large momentum transfers) as special cases of a general parametric formula. In the parametric representation the infrared singularities always exponentiate. This allows us to derive simple differential equations for Laplace transforms of the scattering amplitudes. Similar differential equations have been conjectured to hold in QCD and we summarize the present evidence regarding this assumption.     

State space formalism of Feynman diagram theory for acousto-optic interactions:I.single-frequency case

A Feynman diagram theory for acousto—optic(AO)interactions is es-tablished,which provides a general method to calculate the scatteringamplitudes and intensities for both single-frequency and multifrequency AOinteractions.The method is based on counting the number of allowableFeynman diagrams.Some important assertions have been proved rigorously inthis paper.     

Automated one-loop calculations with GoSam

We present the program package GoSam which is designed for the automated calculation of one-loop amplitudes for multi-particle processes in renormalisable quantum field theories. The amplitudes, which are generated in terms of Feynman diagrams, can be reduced using either D-dimensional integrand-level decomposition or tensor reduction. GoSam can be used to calculate one-loop QCD and/or electroweak corrections to Standard Model processes and offers the flexibility to link model files for theories Beyond the Standard Model. A standard interface to programs calculating real radiation is also implemented. We demonstrate the flexibility of the program by presenting examples of processes with up to six external legs attached to the?loop.