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

Abstract

For commutative Euclidean time, we study the existence of field configurations that {\it a)} are formal power series expansions in $hθ^{\m\n}$, {\it b)} go to ordinary (anti-)instantons as $hθ^{\m\n}\to 0$, 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 $hθ^{\m\n}$. However, we obtain all the deformations --called first-order-in-$θ$-deformed instantons-- of the ordinary instanton that, at first order in $hθ^{\m\n}$, 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 $hθ^{\m\n}$. Other issues analyzed in this paper are the existence at higher orders in $hθ^{\m\n}$ 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 $hθ^{\m\n}$.