Towards a consistent noncommutative supersymmetric Yang-Mills theory: Superfield covariant analysis

Commutative four dimensional supersymmetric Yang-Mills (SYM) theory is known to be renormalizable for $\mathcal{N}=1,2$, and finite for $\mathcal{N}=4$. However, in the noncommutative version of the model the UV/IR mechanism gives rise to infrared divergences which may spoil the perturbative expansion. In this work we pursue the study of the consistency of the $\mathcal{N}=1,2,4$ noncommutative supersymmetric Yang-Mills theory with gauge group $\mathrm{U}(N)$ (NCSYM). We employ the covariant superfield framework to compute the one-loop corrections to the two- and three-point functions of the gauge superfield $V$. It is found that the cancellation of the harmful UV/IR infrared divergences only takes place in the fundamental representation of the gauge group. We argue that this is in agreement with the low energy limit of the open superstring in the presence of an external magnetic field. As expected, the planar sector of the two-point function of the $V$ superfield exhibits UV divergences. They are found to cancel, in the Feynman gauge, for the maximally extended $\mathcal{N}=4$ supersymmetric theory. This gives support to the belief that the $\mathcal{N}=4$ NCSYM theory is UV finite.

Figure 1

Free propagators.

Figure 2

Elementary vertices.

Figure 3

Graphs involving a quartic $V$ vertex.

Figure 4

Contributions involving two trilinear $V$ vertices.

Figure 5

Momentum flow for the $V$-loop.

Figure 6

Ghost tadpole contributions.

Figure 7

Diagrams involving a ghost loop.

Figure 8

Matter contributions.

Figure 9

Vector loop contributions to ${\Gamma }_{VVV}^{(1)}$.

Figure 10

Antisymmetric property of the triple vertex ${\overline{D}}^{2}D{V}^a{V}^{b}D{V}^c$.

Figure 11

Ghost loops contributions to ${\Gamma }_{VVV}^{(1)}$.

Figure 12

Matter loops contributions to ${\Gamma }_{VVV}^{(1)}$.