A new proof of perturbative renormalizability and infrared finiteness for a scalar massless theory is obtained from a formulation of renormalized field theory based on the Wilson renormalization group. The loop expansion of the renormalized Green functions is deduced from the Polchinski equation of renormalization group. The resulting Feynman graphs are organized in such a way that the loop momenta are ordered. It is then possible to analyse their ultraviolet and infrared behaviours by iterative methods. The necessary subtractions and the corresponding counterterms are automatically generated in the process of fixing the physical conditions for the "relevant" vertices at the normalization point. The proof of perturbative renormalizability and infrared finiteness is simply based on dimensional arguments and does not require the usual analysis of topological properties of Feynman graphs.
We analyze a formulation of QED based on the Wilson renormalization group. Although the "effective Lagrangian" used at any given scale does not have simple gauge symmetry, we show that the resulting renormalized Green's functions correctly satisfies Ward identities to all orders in perturbation theory. The loop expansion is obtained by solving iteratively the Polchinski's renormalization group equation. We also give a new simple proof of perturbative renormalizability. The subtractions in the Feynman graphs and the corresponding counterterms are generated in the process of fixing the physical conditions.
By using the exact renormalization group formulation we prove perturbatively
the Slavnov-Taylor (ST) identities in SU(2) Yang-Mills theory. This results
from two properties: {\it locality}, i.e. the ST identities are valid if their
local part is valid; {\it solvability}, i.e. the local part of ST identities is
valid if the couplings of the effective action with non-negative dimensions are
properly chosen.Comment: 9 pages, LaTex, to be published in Phys. Lett.
We study the formulation of the Wilson renormalization group (RG) method for a non-Abelian gauge theory. We analyze the simple case of SU(2) and show that the local gauge symmetry can be implemented by suitable boundary conditions for the RG flow. Namely we require that the relevant couplings present in the physical effective action, i.e. the coefficients of the field monomials with dimension not larger than four, are fixed to satisfy the Slavnov-Taylor identities. The full action obtained from the RG equation should then satisfy the same identities. This procedure is similar to the one we used in QED. In this way we avoid the cospicuous fine tuning problem which arises if one gives instead the couplings of the bare Lagrangian. To show the practical character of this formulation we deduce the perturbative expansion for the vertex functions in terms of the physical coupling g at the subtraction point µ and perform one loop calculations. In particular we analyze to this order some ST identities and compute the nine bare couplings. We give a schematic proof of perturbative renormalizability. * Research supported in part by MURST, Italy
In the exact renormalization group (RG) flow in the infrared cutoff Λ one needs boundary conditions. In a previous paper on SU(2) Yang-Mills theory we proposed to use the nine physical relevant couplings of the effective action as boundary conditions at the physical point Λ = 0 (these couplings are defined at some non-vanishing subtraction point µ = 0). In this paper we show perturbatively that it is possible to appropriately fix these couplings in such a way that the full set of Slavnov-Taylor (ST) identities are satisfied. Three couplings are given by the vector and ghost wave function normalization and the three vector coupling at the subtraction point; three of the remaining six are vanishing (e.g. the vector mass) and the others are expressed by irrelevant vertices evaluated at the subtraction point. We follow the method used by Becchi to prove ST identities in the RG framework. There the boundary conditions are given at a non-physical point Λ = Λ ′ = 0, so that one avoids the need of a non-vanishing subtraction point.
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