Noncommutative SU Research Articles

UV/IR mixing in noncommutative SU(N) Yang\u2013Mills theory

We show that there are one-loop IR singularities arising from UV/IR mixing in noncommutative SU(N) Yang–Mills theory defined by means of the theta -exact Seiberg–Witten map. This is in spite of the fact that there are no ordinary U(1) gauge fields in the theory and this is at variance with the noncommutative U(N) case, where the two-point part of the effective action involving the ordinary SU(N) fields do not suffer from those one-loop IR singularities.

Weyl equation and (non)-commutative SU(n + 1) BPS monopoles

We apply the ADHMN construction to obtain the SU(n + 1) (for generic values of n) spherically symmetric BPS monopoles with minimal symmetry breaking. In particular, the problem simplifies by solving the Weyl equation, leading to a set of coupled equations, whose solutions are expressed in terms of the Whittaker functions. Next, this construction is generalized for non-commutative SU(n + 1) BPS monopoles, where the corresponding solutions are given in terms of the Heun B functions.

Noncommutative 𝒩 = 1 super Yang-Mills, the Seiberg-Witten map and UV divergences

Classically, the dual under the Seiberg-Witten map of noncommutative U(N), {\cal N}=1 super Yang-Mills theory is a field theory with ordinary gauge symmetry whose fields carry, however, a \theta-deformed nonlinear realisation of the {\cal N}=1 supersymmetry algebra in four dimensions. For the latter theory we work out at one-loop and first order in the noncommutative parameter matrix \theta^{\mu\nu} the UV divergent part of its effective action in the background-field gauge, and, for N>=2, we show that for finite values of N the gauge sector fails to be renormalisable; however, in the large N limit the full theory is renormalisable, in keeping with the expectations raised by the quantum behaviour of the theory's noncommutative classical dual. We also obtain --for N>=3, the case with N=2 being trivial-- the UV divergent part of the effective action of the SU(N) noncommutative theory in the enveloping-algebra formalism that is obtained from the previous ordinary U(N) theory by removing the U(1) degrees of freedom. This noncommutative SU(N) theory is also renormalisable.

Noncommutative QCD, first-order-in-θ-deformed instantons and 't Hooft vertices

For commutative Euclidean time, we study the existence of field configurations that {\it a)} are formal power series expansions in $h\theta^{\m\n}$, {\it b)} go to ordinary (anti-)instantons as $h\theta^{\m\n}\to 0$, and {\it c)} render stationary the classical action of Euclidean noncommutative SU(3) Yang-Mills theory. We show that the noncommutative (anti-)self-duality equations have no solutions of this type at any order in $h\theta^{\m\n}$. However, we obtain all the deformations --called first-order-in-$\theta$-deformed instantons-- of the ordinary instanton that, at first order in $h\theta^{\m\n}$, satisfy the equations of motion of Euclidean noncommutative SU(3) Yang-Mills theory. We analyze the quantum effects that these field configurations give rise to in noncommutative SU(3) with one, two and three nearly massless flavours and compute the corresponding 't Hooft vertices, also, at first order in $h\theta^{\m\n}$. Other issues analyzed in this paper are the existence at higher orders in $h\theta^{\m\n}$ of topologically nontrivial solutions of the type mentioned above and the classification of the classical vacua of noncommutative SU(N) Yang-Mills theory that are power series in $h\theta^{\m\n}$.

Renormalizability of noncommutativeSU(N) gauge theory

We analyze the renormalizability properties of pure gauge noncommutative SU(N) theory in the $\theta$-expanded approach. We find that the theory is one-loop renormalizable to first order in $\theta$.

U(1)Aanomaly in noncommutativeSU(N)theories

We work out the one-loop $U(1)_A$ anomaly for noncommutative SU(N) gauge theories up to second order in the noncommutative parameter $\theta^{\mu\nu}$. We set $\theta^{0i}=0$ and conclude that there is no breaking of the classical $U(1)_A$ symmetry of the theory coming from the contributions that are either linear or quadratic in $\theta^{\mu\nu}$. Of course, the ordinary anomalous contributions will be still with us. We also show that the one-loop conservation of the nonsinglet currents holds at least up to second order in $\theta^{\mu\nu}$. We adapt our results to noncommutative gauge theories with SO(N) and U(1) gauge groups.

Noncommutative [formula omitted] theories, the axial anomaly, Fujikawa's method and the Atiyah–Singer index

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, θμν, 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.

On divergent 3-vertices in noncommutative SU(2) gauge theory

We analyse divergences in 2-point and 3-point functions for noncommutative θ-expanded SU(2)-gauge theory with massless fermions. We refine our previous argument on renormalizability of 3-vertices by explicit SW construction. We show that, after the field redefinition and renormalization of couplings, divergent terms remain.

Non-renormalizability of the noncommutative SU(2) gauge theory

We analyze the divergent part of the one-loop effective action for the noncommutative SU(2) gauge theory coupled to the fermions in the fundamental representation. We show that the divergencies in the 2-point and the 3-point functions in the $\theta$-linear order can be renormalized, while the divergence in the 4-point fermionic function cannot.

Non-commutative standard model: model building

A noncommutative version of the usual electro-weak theory is constructed. We discuss how to overcome the two major problems: 1) although we can have noncommutative U(n) (which we denote by $U_{\star}(n)$) gauge theory we cannot have noncommutative SU(n) and 2) the charges in noncommutative QED are quantized to just $0, \pm 1$. We show how the problem with charge quantization, as well as with the gauge group, can be resolved by taking $U_{\star}(3)\times U_{\star}(2)\times U_{\star}(1)$ gauge group and reducing the extra U(1) factors in an appropriate way. Then we proceed with building the noncommutative version of the standard model by specifying the proper representations for the entire particle content of the theory, the gauge bosons, the fermions and Higgs. We also present the full action for the noncommutative Standard Model (NCSM). In addition, among several peculiar features of our model, we address the {\it inherent} CP violation and new neutrino interactions.

Noncommutative quantum mechanics from noncommutative quantum field theory.

We derive noncommutative multiparticle quantum mechanics from noncommutative quantum field theory in the nonrelativistic limit. Particles of opposite charges are found to have opposite noncommutativity. As a result, there is no noncommutative correction to the hydrogen atom spectrum at the tree level. We also comment on the obstacles to take noncommutative phenomenology seriously and propose a way to construct noncommutative SU(5) grand unified theory.

Flavor Symmetry on Non-Commutative Compact Space and SU(6)×SU(2)_R Model

Flavor Symmetry on Non-Commutative Compact Space and SU(6)×SU(2)_R Model

Noncommutative SU(N) and gauge invariant baryon operator

We propose a constraint on the noncommutative gauge theory with U(N) gauge group which gives rise to a noncommutative version of the SU(N) gauge group. The baryon operator is also constructed.

Flavor Symmetry on Non-Commutative Compact Space and SU(6) x SU(2)R Model

In the four-dimensional effective theory from string compactification discrete flavor symmetries arise from symmetric structure of the compactified space and generally contain both the $R$ symmetry and non-$R$ symmetry. We point out that a new type of non-Abelian flavor symmetry can also appear if the compact space is non- commutative. Introducing the dihedral group $D_4$ as such a new type of flavor symmetry together with the $R$ symmetry and non-$R$ symmetry in $SU(6) \times SU(2)_R$ model, we explain not only fermion mass hierarchies but also hierarchical energy scales including the breaking scale of the GUT-type gauge symmetry, intermediate Majorana masses of R-handed neutrinos and the scale of $\mu$-term.

Bounding noncommutative QCD

Jurco, Moller, Schraml, Schupp, and Wess have shown how to construct noncommutative SU(N) gauge theories from a consistency relation. Within this framework, we present the Feynman rules for noncommutative QCD and compute explicitly the most dangerous Lorentz-violating operator generated through radiative corrections. We find that interesting effects appear at the one-loop level, in contrast to conventional noncommutative U(N) gauge theories, leading to a stringent bound. Our results are consistent with others appearing recently in the literature that suggest collider limits are not competitive with low-energy tests of Lorentz violation for bounding the scale of spacetime noncommutativity.