Perturbative (one- and two- loop) results for the beta function for three different gauge actions

In this file, we present the perturbative results for the beta function for three different gauge actions: Wilson Gauge Action, Symanzik Improved Action (tree-level) and Iwasaki Improved Action. The results correspond to the O(am) contributions and are evaluated up to two loops. The perturbative expressions and conventions follow the definitions presented in https://arxiv.org/abs/arXiv:2503.00463. The beta function describes the running of the bare coupling constant with the lattice spacing. It is defined as: β(g0) = − a dg0 / da, one can introduce a modified bare coupling through tilded{g0^2} = g0^2(1 + bg(g0^2) a mq). In perturbation theory the beta function admits an expansion in powers of the coupling constant: β(g0) = − b0 g0^3 − b1 g0^5 + b2 g0^7 + O(g0^9) Writing bg(g0) as an expansion in powers of g0: tilded{g0^2} = g0^2(1 + (bg1 g0^2 + bg2 g0^4 + ... ) a mq). where bg1 and bg2 correspond to the one–loop and two–loop coefficients, respectively. Variables / symbols used in the file: g: coupling constant Nc: number of colors Nf: number of flavors csw: clover parameter pisq16: 16 Pi^2 q2: q^2 q2Log2q2: q^2 (Log[q^2])^2