Beta functions and anomalous dimensions up to three loops

Abstract

We derive an algorithm for automatic calculation of perturbative $β$-functions and anomalous dimensions in any local quantum field theory with canonical kinetic terms. The infrared rearrangement is performed by introducing a common mass parameter in all the propagator denominators. We provide a set of explicit formulae for all the necessary scalar integrals up to three loops.

Figures (4)

• Figure 1: Graphical representation of the integrals given in eqns. (\ref{['int1']})--(\ref{['int3']})
• Figure 2: Graphical representation of the integral $I^{(3)}_{111111}$. It is equivalent to the last diagram in fig. \ref{['vac.diag']} with $n_1 = ... = n_6 = 1$.
• Figure 3: Graphical representation of the integral $J^{(2)}_{11111}(q^2,m^2)$.
• Figure 4: Graphical representation of the three-loop integral with one nonpositive index