14729063
The Gauge anomaly and the Seiberg-Witten map
Abstract The consistent form of the gauge anomaly is worked out at first order in θ for the noncommutative three-point function of the ordinary gauge field of certain noncommutative chiral gauge theories defined by means of the Seiberg–Witten map. We obtain that for any compact simple Lie group the anomaly cancellation condition of this three-point function reads Tr T a T b T c =0, if one restricts the type of noncommutative counterterms that can be added to the classical action to restore the gauge symmetry to those which are renormalizable by power-counting. On the other hand, if the power-counting renormalizability paradigm is relinquished and one admits noncommutative counterterms (of the gauge fields, its derivatives and θ ) which are not power-counting renormalizable, then, the anomaly cancellation condition for the noncommutative three-point function of the ordinary gauge field becomes the ordinary one: Tr T a { T b , T c }=0.
118569185
Renormalisability of noncommutative GUT inspired field theories with anomaly safe groups
We consider noncommutative GUT inspired field theories formulated within the enveloping-algebra formalism for anomaly safe compact simple gauge groups. Our theories have only gauge fields and fermions, and we compute the UV divergent part of the one-loop background-field effective action involving two fermionic fields at first order in the noncommutativity parameter θ. We show that, if the second-degree Casimir has the same value for all the irreducible group representations furnished by the fermionic multiplets of the model, then, that UV divergent part can be renormalised by carrying out multiplicative renormalisations of the coupling constant, θ and the fields, along with the inclusion of θ-dependent counterterms which vanish upon imposing the equations of motion. These θ-dependent counterterms have no physical effect since they vanish on-shell. This result along with the vanishing of the UV divergent part of the fermionic four-point functions leads to the unexpected conclusion that the one-loop matter sector of the background-field effective action of these theories is one-loop multiplicatively renormalisable on-shell. We also show that the background-field effective action of the gauge sector of the theories considered here receives no θ-dependent UV divergent contributions at one-loop. We thus conclude that these theories are on-shell one-loop multiplicatively renormalisable at first order in θ.
18871973
Noncommutative SU(N) theories, the axial anomaly, Fujikawa's method and the Atiyah-Singer index
Abstract Fujikawa's method is employed to compute at first order in the noncommutative parameter the U ( 1 ) A anomaly for noncommutative SU ( N ) . We consider the most general Seiberg–Witten map which commutes with hermiticity and complex conjugation and a noncommutative matrix parameter, θ μ ν , which is of “magnetic” type. Our results for SU ( N ) can be readily generalized to cover the case of general nonsemisimple gauge groups when the symmetric Seiberg–Witten map is used. Connection with the Atiyah–Singer index theorem is also made.
16073719
Higher covariant derivative Pauli-Villars r egularization does not lead to a consistent QCD
Abstract We compute the beta function at one loop for Yang-Mills theory using as regulator the combination of higher covariant derivatives and Pauli-Villars determinants proposed by Faddeev and Slavnov. This regularization prescription has the appealing feature that it is manifestly gauge invariant and essentially four-dimensional. It happens however that the one-loop coefficient in the beta function that it yields is not − 11 3 , as it should be, but − 23 6 . The difference is due to unphysical logarithmic radiative corrections generated by the Pauli-Villars determinants on which the regularization method is based. This no-go result discards the prescription as a viable gauge invariant regularization, thus solving a long-standing open question in the literature. We also observe that the prescription can be modified so as to not generate unphysical logarithmic corrections, but at the expense of losing manifest gauge invariance.
17165052
Shift versus no-shift in local regularizations of Chern-Simons theory
Abstract We consider a family of local BRS-invariant higher covariant derivative regularizations of SU ( N ) Chern-Simons theory that do not shift the value of the Chern-Simons parameter k to k + sign ( k ) c v at one loop.
15974679
Anomaly cancellation and the gauge group of the Standard Model in NCG
Abstract It is well known that anomaly cancellation almost determines the hypercharges in the Standard Model. A related (and somewhat more stronger) phenomenon takes place in Connes' NCG framework: unimodularity (a technical condition on elements of the algebra) is strictly equivalent to anomaly cancellation; and this in turn reduces the symmetry group of the theory to the standard SU (3) × SU (2) × U (1).
119536533
Axial gauge Chern-Simons theory in the Leibbrandt-Mandelstam prescription for a certain class of UV regulators
Abstract We show that the complete perturbative effective action of an SU ( N ) Chern-Simons theory in an homogenous axial-type gauge ( nA a =0, n 2 >0, n 2 n 2 =0) with the generalized Leibbrandt-Mandelstam prescription is the tree-level action for a certain class of regulators. The Gauss linking number is expressed in terms of the axial propagators for the gauge fields.
120724746
DOES STOCHASTIC ANALYTIC REGULARIZATION PROVIDE QUANTUM FIELD THEORIES
We analyze whether the so-called method of stochastic analytic regularization is suitable as an intermediate step for constructing perturbative renormalized quantum field theories. We choose a λϕ3 in six dimensions to prove that this regularization method does not in general provide a quantum field theory. This result seems to apply to any field theory with a quadratically UV-divergent stochastic two-point function, for instance λϕ4 and gauge theories in four dimensions.
15154154
Chern-Simons theory as the large-mass limit of topologically massive Yang-Mills theory
We study quantum Chern-Simons theory as the large-mass limit of the limit D --> 3 of dimensionally regularized topologically massive Yang-Mills theory. This approach can also be interpreted as a BRS-invariant hybrid regularization of Chern-Simons theory, consisting of a higher-covariant derivative Yang-Mills term plus dimensional regularization. Working in the Landau gauge, we compute radiative corrections up to second order in perturbation theory and show that there is no two-loop correction to the one-loop shift k --> k + c(v), k being the bare Chern-Simons parameter. In passing, we also prove by explicit computation that topologically massive Yang Mills theory is UV finite.
121689569
Dimensional regularisation of Chern-Simons field theory
Abstract We discuss the dimensional regularisation program as applied to a pure Chern-Simons theory in Minkowski space. In order to make this regularisation program feasible, we propose adding a Yang-Mills term to the pure Chern-Simons action. It is argued that the pure Chern-Simons theory is recovered in a certain limit. Explicit computations are carried out at the one-loop level in the background field gauge.
120228894
Perturbative Chern-Simons theory in the light cone gauge: The One loop vacuum polarization tensor in a gauge invariant formalism
Abstract Perturbative Chern-Simons (CS) theory, with gauge group SU( N ), is studied in the physical light-cone gauge n σ A σ a = 0, n 2 = 0. Using a gauge-invariant regularization procedure, we compute the vacuum polarization tensor to one-loop order. The regularized theory is consistent with BRS invariance. It is shown that the vacuum polarization tensor is UV finite and transverse, but contains one non-local term which is strictly gauge-dependent. A finite, non-multiplicative (gauge-dependent) wave function renormalization enables us to express the one-loop effective action as Γ one-loop ( A μ (ren) , k ) = (( k + sign( k ) c v )/4 π ) S CS ( A μ (ren) , where S CS is the classical action and k is the bare Chern-Simons parameter.