Andrea Quadri

Algebraic properties of BRST coupled doublets

We characterize the dependence on doublets of the cohomology of an arbitrary nilpotent differential s (including BRST differentials and classical linearized Slavnov-Taylor (ST) operators) in terms of the cohomology of the doublets-independent component of s. All cohomologies are computed in the space of integrated local formal power series. We drop the usual assumption that the counting operator for the doublets commutes with s (decoupled doublets) and discuss the general case where the counting operator does not commute with s (coupled doublets). The techniques used are purely algebraic and do not rely on power-counting arguments. The main result is that the full cohomology that includes the doublets can be obtained directly from the cohomology of the doublets-independent component of s. This turns out to be a very useful property in many problems in Algebraic Renormalization.

Chern-Simons in the Seiberg-Witten map for non-commutative Abelian gauge theories in 4D

A cohomological BRST characterization of the Seiberg-Witten (SW) map is given. We prove that the coefficients of the SW map can be identified with elements of the cohomology of the BRST operator modulo a total derivative. As an example, it will be illustrated how the first coefficients of the SW map can be written in terms of the Chern-Simons three form. This suggests a deep topological and geometrical origin of the SW map. The existence of the map for both abelian and non-abelian case is discussed. By using a recursive argument and the associativity of the -product, we shall be able to prove that the Wess-Zumino consistency condition for non-commutative BRST transformations is fulfilled. The recipe of obtaining an explicit solution by use of the homotopy operator is briefly reviewed in the abelian case.

Weak Power-Counting Theorem for the Renormalization of the Nonlinear Sigma Model in Four Dimensions

The formulation of the non linear σ-model in terms of flat connection allows the construction of a perturbative solution of a local functional equation by means of cohomological techniques which are implemented in gauge theories. In this paper we discuss some properties of the solution at the one-loop level in D = 4. We prove the validity of a weak power-counting theorem in the following form: although the number of divergent amplitudes is infinite only a finite number of counterterms parameters have to be introduced in the effective action in order to make the theory finite at one loop, while respecting the functional equation (fully symmetric subtraction in the cohomological sense). The proof uses the linearized functional equation of which we provide the general solution in terms of local functionals. The counterterms are expressed in terms of linear combinations of these invariants and the coefficients are fixed by a finite number of divergent amplitudes. These latter amplitudes contain only insertions of the composite operators φ0 (the constraint of the non linear σ-model) and Fμ (the flat connection). The structure of the functional equation suggests a hierarchy of the Green functions. In particular once the amplitudes for the composite operators φ0 and Fμ are given all the others can be derived by functional derivatives. In this paper we show that at one loop the renormalization of the theory is achieved by the subtraction of divergences of the amplitudes at the top of the hierarchy. As an example we derive the counterterms for the four-point amplitudes.

THE SU(2) ⊗ U(1) ELECTROWEAK MODEL BASED ON THE NONLINEARLY REALIZED GAUGE GROUP

In the present paper, that is the second part devoted to the construction of an electroweak model based on a nonlinear realization of the gauge group SU(2) U(1), we study the tree-level vertex functional with all the sources necessary for the functional formulation of the relevant symmetries (Local Functional Equation, Slavnov‐Taylor identity, Landau Gauge Equation) and for the symmetric removal of the divergences. The Weak Power Counting criterion is proven in the presence of the novel sources. The local invariant solutions of the functional equations are constructed in order to represent the counterterms for the one-loop subtractions. The bleaching technique is fully extended to the fermion sector. The neutral sector of the vector mesons is analyzed in detail in order to identify the physical fields for the photon and the Z boson. The identities necessary for the decoupling of the unphysical modes are fully analyzed. These latter results are crucially bound to the Landau gauge used throughout the paper.

Massive Yang-Mills theory based on the nonlinearly realized gauge group

We propose a subtraction scheme for a massive Yang-Mills theory realized via a nonlinear representation of the gauge group [here

FURTHER COMMENTS ON THE SYMMETRIC SUBTRACTION OF THE NONLINEAR SIGMA MODEL

SU(2)

The Background Field Method as a Canonical Transformation

]. It is based on the subtraction of the poles in

Algebraic Aspects of the Background Field Method

D\ensuremath{-}4

Direct algebraic restoration of Slavnov-Taylor identities in the Abelian Higgs-Kibble model

of the amplitudes, in dimensional regularization, after a suitable normalization has been performed. Perturbation theory is in the number of loops, and the procedure is stable under iterative subtraction of the poles. The unphysical Goldstone bosons, the Faddeev-Popov ghosts, and the unphysical mode of the gauge field are expected to cancel out in the unitarity equation. The spontaneous symmetry breaking parameter is not a physical variable. We use the tools already tested in the nonlinear sigma model: hierarchy in the number of Goldstone boson legs and weak-power-counting property (finite number of independent divergent amplitudes at each order). It is intriguing that the model is naturally based on the symmetry

Renormalization of the non-linear sigma model in four dimensions. A two-loop example

SU(2{)}_{L}