Pierre Grangé

Taylor–Lagrange renormalization and gauge theories in four dimensions

The treatment of gauge theories within the recently proposed Taylor–Lagrange renormalization scheme (TLRS) is examined in detail. The conservation of gauge symmetry is demonstrated directly at the physical dimension D = 4f or specific examples of fermion and boson self energies and vertices of QED and QCD. Comparisons with dimensional regularization and Bogoliubov– Parasiuk–Hepp–Zimmermann (BPHZ) substractions, improved by algebraic regularization based on the quantum action principle, exhibit clearly the important mathematical properties of the TLRS leading to conservation of this fundamental gauge symmetry.

Taylor-Lagrange renormalization scheme: Application to light-front dynamics

The recently proposed renormalization scheme based on the definition of field operators as operator valued distributions acting on specific test functions is shown to be very convenient in explicit calculations of physical observables within the framework of light-front dynamics. We first recall the main properties of this procedure based on identities relating the test functions to their Taylor remainder of any order expressed in terms of Lagranges formulas, hence the name given to this scheme. We thus show how it naturally applies to the calculation of state vectors of physical systems in the covariant formulation of light-front dynamics. As an example, we consider the case of the Yukawa model in the simple two-body Fock state truncation.

Continuum version ofφ1+14theory in light-front quantization

A genuine continuum treatment of the massive \phi^4_{1+1}-theory in light-cone quantization is proposed. Fields are treated as operator valued distributions thereby leading to a mathematically well defined handling of ultraviolet and light cone induced infrared divergences and of their renormalization. Although non-perturbative the continuum light cone approach is no more complex than usual perturbation theory in lowest order. Relative to discretized light cone quantization, the critical coupling increases by 30% to a value r = 1.5. Conventional perturbation theory at the corresponding order yields r_1=1, whereas the RG improved fourth order result is r_4 = 1.8 +-0.05.

Light cone quantization and 1N expansion

Abstract For the φ4 scalar field theory with O(N) symmetry we exploit the possibility of a 1 N expansion to solve the equations of motion for the field operators in the framework of light cone quantization in the symmetric phase. Due to the triviality of the light cone vacuum the field operators can be explicitly constructed order by order in 1 N . This is at variance with the conventional treatment where the appearance of non-trivial vacuum states renders the corresponding expansion practically very difficult beyond the 1 N correction. We argue that the proposed scheme will also be very useful for the treatment of the broken phase.

UV AND IR behaviour for QFT and LCQFT with fields as operator valued distributions: Epstein and Glaser revisited

Following Epstein-Glasers work we show how a QFT formulation based on operator valued distributions (OPVD) with adequate test functions treats original singularities of propagators on the diagonal in a mathematically rigourous way. Thereby UV and/or IR divergences are avoided at any stage, only a finite renormalization finally occurs at a point related to the arbitrary scale present in the test functions. Some well known UV cases are exemplified. The power of the IR treatment is shown for the free massive scalar field theory developed in the (conventionally hopeless) mass perturbation expansion. It is argued that the approach should prove most useful for non-pertubative methods where the usual determination of counterterms is elusive.

Field dynamics on the light cone: Compact versus continuum quantization

Compact canonical quantization on the light cone (DLCQ) is examined in the limit of infinite periodicity lenth L. Pauli Jordan commutators are found to approach continuum expressions with marginal non causal terms of order

Broken phase of the O(N) φ4) model in light cone quantization and 1N expansion

L^{-3/4}

Taylor-Lagrange renormalization scheme, Pauli-Villars subtraction, and light-front dynamics

traced back to the handling of IR divergence through the elimination of zero modes. In contrast direct quantization in the continuum (CLCQ) in terms of field operators valued distributions is shown to provide the standard causal result while at the same time ensuring consistent IR and UV renormalization.

Critical properties of

Abstract The solution of the O(N) φ4 scalar field theory in the broken phase is given in the framework of light cone quantization and a 1 N expansion. It involves the successive building of operator solutions to the equation of motion and constraints including operator zero modes of the fields which are the LC counterpart to the equal time non trivial vacuum effects. The renormalization of the procedure is accomplished up to 2nd order in the 1 N expansion for the equal of motion and constraints. In addition the renormalization of the divergent contributions of the two-point and four-point functions is performed in a covariant way. The presence of zero modes leads to genuine non perturbative renormalization features.

\Phi^4_{1+1}

We show how the recently proposed Taylor-Lagrange renormalization scheme can lead to extensions of singular distributions which are reminiscent of the Pauli-Villars subtraction. However, at variance with the Pauli-Villars regularization scheme, no infinite mass limit is performed in this scheme. As an illustration of this mechanism, we consider the calculation of the self-energy in second order perturbation theory in the Yukawa model, within the covariant formulation of light-front dynamics. We show, in particular, how rotational invariance is preserved in this scheme.