Characters and group invariant polynomials of (super)fields: road to “Lagrangian”

Abstract

AbstractThe dynamics of the subatomic fundamental particles, represented by quantum fields, and their interactions are determined uniquely by the assigned transformation properties, i.e., the quantum numbers associated with the underlying symmetry of the model under consideration. These fields constitute a finite number of group invariant operators which are assembled to build a polynomial, known as the Lagrangian of that particular model. The order of the polynomial is determined by the mass dimension. In this paper, we have introduced an automated $${\texttt {Mathematica}}^{\tiny \textregistered }$$ Mathematica ® package, GrIP, that computes the complete set of operators that form a basis at each such order for a model containing any number of fields transforming under connected compact groups. The spacetime symmetry is restricted to the Lorentz group. The first part of the paper is dedicated to formulating the algorithm of GrIP. In this context, the detailed and explicit construction of the characters of different representations corresponding to connected compact groups and respective Haar measures have been discussed in terms of the coordinates of their respective maximal torus. In the second part, we have documented the user manual of GrIP that captures the generic features of the main program and guides to prepare the input file. We have attached a sub-program CHaar to compute characters and Haar measures for $$SU(N), SO(2N), SO(2N+1), Sp(2N)$$ S U ( N ) , S O ( 2 N ) , S O ( 2 N + 1 ) , S p ( 2 N ) . This program works very efficiently to find out the higher mass (non-supersymmetric) and canonical (supersymmetric) dimensional operators relevant to the effective field theory (EFT). We have demonstrated the working principles with two examples: the standard model (SM) and the minimal supersymmetric standard model (MSSM). We have further highlighted important features of GrIP, e.g., identification of effective operators leading to specific rare processes linked with the violation of baryon and lepton numbers, using several beyond standard model (BSM) scenarios. We have also tabulated a complete set of dimension-6 operators for each such model. Some of the operators possess rich flavour structures which are discussed in detail. This work paves the way towards BSM-EFT.