Gauge theories with overlap fermions in an arbitrary representation: Evaluation of the 3-loop β-function

This work presents the calculation of the relation between the bare coupling constant g₀ and the M Smacr -renormalized coupling g_{M Smacr} , g₀=Z_g(g₀,aμ)g_{M Smacr} , to 2 loops in perturbation theory, with fermions in an arbitrary representation of the gauge group SU(N). Our calculation is performed using overlap fermions and Wilson gluons, and the background field technique has been chosen for convenience. The corresponding results in the fundamental representation appear in our longer publication [M. Constantinou and H. Panagopoulos, Phys. Rev. D 76, 114504 (2007)PRVDAQ0556-282110.1103/PhysRevD.76.114504]. The 3-loop coefficient of the bare β-function, b₂^L, is extracted using the 2-loop expression for Z_g, and it is presented as a function of the overlap parameter ρ, the number of fermion flavors (N_f) and the number of colors (N). We also provide the expression for the ratio of energy scales Λ_L/Λ_{M Smacr} , in an arbitrary representation. A plot of Λ_L/Λ_{M Smacr} is given in the adjoint representation.