Roland Seneor

Decay properties and Borel summability for the Schwinger functions in

For the truncated Schwinger functions of theP(Φ)2 field theories, we show strong decrease in the separation of points. This shows uniqueness of theories withP of degree four. We also extend the domain of analyticity in the coupling constant. For theories withP of degree four, the combination of these two results gives Borel summability.

P(\phi)_{2}

The Euclidean massive Gross-Neveu model in two dimensions is just renormalizable and asymptotically free. Thanks to the Pauli principle, bare perturbation theory with an ultra-violet cut-off (and the correct ansatz for the bare mass) is convergent in a disk, whose radius corresponds by asymptotic freedom to a small finite renormalized coupling constant. Therefore, the theory can be fully constructed in a perturbative way. It satisfies the O.S. axioms and is the Borel sum of the renormalized perturbation expansion of the model

theories

We construct the thermodynamic limit of the critical (massless) φ4 model in 4 dimensions with an ultraviolet cutoff by means of a “partly renormalized” phase space expansion. This expansion requires in a natural way the introduction of effective or “running” constants, and the infrared asymptotic freedom of the model, i.e. the decay of the running coupling constant, plays a crucial rôle. We prove also that the correlation functions of the model are the Borel sums of their perturbation expansion.

A renormalizable field theory: the massive Gross-Neveu model in two dimensions

We provide the basis for a rigorous construction of the Schwinger functions of the pure SU(2) Yang-Mills field theory in four dimensions (in the trivial topological sector) with a fixed infrared cutoff but no ultraviolet cutoff, in a regularized axial gauge. The construction exploits the positivity of the axial gauge at large field. For small fields, a different gauge, more suited to perturbative computations is used; this gauge and the corresponding propagator depends on large background fields of lower momenta. The crucial point is to control (in a non-perturbative way) the combined effect of the functional integrals over small field regions associated to a large background field and of the counterterms which restore the gauge invariance broken by the cutoff. We prove that this combined effect is stabilizing if we use cutoffs of a certain type in momentum space. We check the validity of the construction by showing that Slavnov identities (which express infinitesimal gauge invariance) do hold non-perturbatively.

Construction and Borel summability of infrared Φ 4 4 by a phase space expansion

LetG be a Euclidean Feynman graph containingL(G) lines. We prove that ifG has massive propagators and does not contain any divergent subgraphs its value is bounded byKL(G). We also prove the infrared analogue of this bound.

Construction of

We prove two bounds on the value of renormalized Euclidean Feynman graphs. One is a relatively crude but widely applicable bound; the other a finer bound applicable toφ44-like models.

{\rm YM}_4

Resume Nous calculons le comportement asymptotique a grandes distances des fonctions de correlation du modele (∇ Φ ) 3 4 .

with an infrared cutoff

A classification of “first order” deformations of Lie algebra representations by the use of a cohomology group is studied. A method is proposed for calculating this group for the case of algebras which are semi-direct products. The role of unitarity of the representations is exhibited. Applications are made for the Poincaré andE(3) algebras.

Bounds on Completely Convergent Euclidean Feynman Graphs

We bound the large order behavior of these pieces of the renormalized perturbation expansion for ρ44 which do not contain “renormalons” [1]. The bound we obtain has the form of the leading asymptotic behavior computed by the Lipatov method, with the exact value of the “Lipatov constant.” Therefore, this paper is a step towards the rigorous study of the large order behavior of Φ44 and towards a proof of existence of the first “renormalon” singularity which should prevent the theory from being Borel summable. Using the results of this paper and the techniques of [15], one can for instance improve [17] the estimate [18] on the finiteness of the radius of convergence of the Borel transform of renormalized Φ44 and obtain that this radius is at least the exact value conjectured in [1].

Bounds on renormalized Feynman graphs

Abstract We report on a rigorous construction of the Schwinger functions of the pure SU (2) Yang-Mills field theory in four dimensions (in the trivial topological sector) with a fixed infrared cutoff but no ultraviolet cutoff, in a regularized axial gauge. We briefly review the difficulties related to Gribov ambiguities and then we give an outline of our construction.