F. Ruiz

Gauge-fixing independence of IR divergences in non-commutative U(1), perturbative tachyonic instabilities and supersymmetry

Abstract It is argued that the quadratic and linear non-commutative IR divergences that occur in U (1) theory on non-commutative Minkowski space–time for small non-commutativity matrices θ μν are gauge-fixing independent. This implies in particular that the perturbative tachyonic instability produced by the quadratic divergences of this type in the vacuum polarization tensor is not a gauge-fixing artifact. Supersymmetry can be introduced to remove from the renormalized Green functions at one loop, not only the non-logarithmic non-commutative IR divergences, but also all terms proportional to θ μν p ν .

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.

Trouble with space-like noncommutative field theory

Abstract It is argued that the one-loop effective action for space-like noncommutative (i) λ φ 4 scalar field theory and (ii) U ( 1 ) gauge theory does not exist. This indicates that such theories are not renormalizable already at one-loop order and suggests supersymmetrization and reinvestigating other types of noncommutativity.

Position-dependent noncommutative products: Classical construction and field theory

Abstract We look in Euclidean R 4 for associative star products realizing the commutation relation [ x μ , x ν ] = i Θ μ ν ( x ) , where the noncommutativity parameters Θ μ ν depend on the position coordinates x. We do this by adopting Rieffels deformation theory (originally formulated for constant Θ and which includes the Moyal product as a particular case) and find that, for a topology R 2 × R 2 , there is only one class of such products which are associative. It corresponds to a noncommutativity matrix whose canonical form has components Θ 12 = − Θ 21 = 0 and Θ 34 = − Θ 43 = θ ( x 1 , x 2 ) , with θ ( x 1 , x 2 ) an arbitrary positive smooth bounded function. In Minkowski space–time, this describes a position-dependent space-like or magnetic noncommutativity. We show how to generalize our construction to n ⩾ 3 arbitrary dimensions and use it to find traveling noncommutative lumps generalizing noncommutative solitons discussed in the literature. Next we consider Euclidean λ ϕ 4 field theory on such a noncommutative background. Using a zeta-like regulator, the covariant perturbation method and working in configuration space, we explicitly compute the UV singularities. We find that, while the two-point UV divergences are nonlocal, the four-point UV divergences are local, in accordance with recent results for constant Θ.

UV/IR mixing and the Goldstone theorem in noncommutative field theory

Noncommutative IR singularities and UV/IR mixing in relation with the Goldstone theorem for complex scalar field theory are investigated. The classical model has two coupling constants, lambda(1) and lambda(2), associated to the two noncommutative extensions phi* star phi star phi* star phi and phi* star phi* star phi star phi of the interaction term \phi\(4) on commutative spacetime. It is shown that the symmetric phase is one loop renormalizable for all lambda(1) and lambda(2) compatible with perturbation theory, whereas the broken phase is proved to exist at one loop only if lambda(2) = 0, a condition required by the Ward identities for global U(I) invariance. Explicit expressions for the noncommutative IR singularities in the 1PI Green functions of both phases are given. They show that UV/IR duality does not hold for any of the phases and that the broken phase is free of quadratic noncommutative IR singularities. More remarkably, the pion selfenergy does not have noncommutative IR singularities at all, which proves essential to formulate the Goldstone theorem at one loop for all values of the spacetime noncommutativity parameter theta.

Seiberg-Witten maps for SO(1,3) gauge invariance and deformations of gravity

A family of diffeomorphism-invariant Seiberg-Witten deformations of gravity is constructed. In a first step Seiberg-Witten maps for an SO(1,3) gauge symmetry are obtained for constant deformation parameters. This includes maps for the vierbein, the spin connection, and the Einstein-Hilbert Lagrangian. In a second step the vierbein postulate is imposed in normal coordinates and the deformation parameters are identified with the components theta(mu nu)(x) of a covariantly constant bivector. This procedure gives for the classical action a power series in the bivector components which by construction is diffeomorphism invariant. Explicit contributions up to second order are obtained. For completeness a cosmological constant term is included in the analysis. Covariant constancy of theta(mu nu)(x), together with the field equations, imply that, up to second order, only four dimensional metrics which are direct sums of two two dimensional metrics are admissible, the two-dimensional curvatures being expressed in terms of theta(mu nu). These four-dimensional metrics can be viewed as a family of deformed emergent gravities.

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.

Noncommutative Einstein-Maxwell pp-waves

The field equations coupling a Seiberg-Witten electromagnetic field to noncommutative gravity, as described by a formal power series in the noncommutativity parameters theta(alpha beta), is investigated. A large family of solutions, up to order one in theta(alpha beta), describing Einstein-Maxwell null pp-waves is obtained. The order-one contributions can be viewed as providing noncommutative corrections to pp-waves. In our solutions, noncommutativity enters the spacetime metric through a conformal factor and is responsible for dilating/contracting the separation between points in the same null surface. The noncommutative corrections to the electromagnetic waves, while preserving the wave null character, include constant polarization, higher harmonic generation, and inhomogeneous susceptibility. As compared to pure noncommutative gravity, the novelty is that nonzero corrections to the metric already occur at order one in theta(alpha beta).

Exact solution for the Schwinger model at finite temperature

Abstract An exact solution for the Schwinger model at finite temperature is given. Path integral methods and thermo-field dynamics are used. The explicit dependence of the complete propagators with temperature is found.

Unitarity violation in non-abelian Pauli-Villars regularization

We regularize QCD using the combination of higher covariant derivatives and Pauli-Villars determinants proposed by Slavnov. It is known that for pure Yang-Mills theory the Pauli-Villars determinants generate unphysical logarithmic radiative corrections at one loop that modify the beta function. Here we prove that when the gauge fields are coupled to fermions so that one has QCD, these unphysical corrections translate into a violation of unitarity. We provide an understanding of this by showing that Slavnovs choice for the Pauli-Villars determinants introduces extra propagating degrees of freedom that are responsible for the unitarity breaking. This shows that Slavnovs regularization violates unitarity, hence that it should be rejected.