Counting operators in Effective Field Theories

When applying physical calculations to the real world, it is always necessary to make some approximations. In the context of field theories, in particular the quantum field theories used in particle physics, a systematic treatment of this problem is provided by the framework of Effective Field Theories. Working in a given energy range, it is enough to consider only a couple of particles, while the existence of the rest only contributes some corrections. This is reflected in the structure of Effective Field Theory Lagrangians given as perturbation series, which are in general composed of all possible operators consistent with locality, unitarity, and symmetry assumptions. Obviously, it is advantageous to work with a minimal set of operators covering all physical phenomena, however the construction of the operator basis is very hard in general. Simpler step is to at least count all independent operators with a given number of derivatives and particle fields. To this end, the generating function (termed the Hilbert series) is introduced, with coefficients precisely being the number of independent operators of a given type. One can consider only scalar operators, that is Lorentz and gauge invariant operators, moreover consistent with other additional symmetries. To count independent operators it is furthermore..