$27 \otimes 27$

Abstract

We study the decomposition into $SU(N)$ irreducible representations (irreps) of the tensor product $27 \otimes 27$, where $27$ is the highest-dimensional $SU(N)$ irrep present in a two-gluon system, and explicitly construct all Hermitian projectors on these irreps, as well as transition operators between them. This yields an explicit basis of the complete $27\otimes 27$ color space (defined as the space of $27\otimes 27\to 27 \otimes 27$ color maps) in terms of orthogonal multiplets. This study shows that even complex color structures can be addressed, with the help of the birdtrack pictorial technique, using only elementary tools. In particular, we highlight the usefulness of the quadratic Casimir operator, whose eigenspaces allow efficient filtering of all projectors and transition operators, and of the permutation operators that further improve this filtering. The product $27\otimes 27$ also has an interesting feature: three equivalent irreps $27$ appear in the decomposition, two of which are symmetric and can therefore be distinguished neither by the quadratic Casimir operator nor by their symmetry under permutation. In this case, it is convenient to use Clebsch-Gordan coefficients to derive the two associated, symmetric projectors. The latter are not uniquely determined (only their sum is), and we give the set of all solutions. Finally, we explicitly derive the soft anomalous dimension matrix associated with $27\otimes 27\to 27\otimes 27$, whose block-diagonal main structure is easy to understand, but whose detailed spectrum properties remain intriguing. The approach presented for $27 \otimes 27 \to 27 \otimes 27$ could in principle be applied to any product of $SU(N)$ irreps, and eventually automated.