Long-range multi-scalar models at three loops

We compute the three-loop beta functions of long-range multi-scalar models with general quartic interactions. The long-range nature of the models is encoded in a kinetic term with a Laplacian to the power 0 < zeta < 1, rendering the computation of Feynman diagrams much harder than in the usual short-range case (zeta = 1). As a consequence, previous results stopped at two loops, while seven-loop results are available for short-range models. We push the renormalization group analysis to three loops, in an epsilon = 4 zeta - d expansion at fixed dimension d < 4, extensively using the Mellin-Barnes representation of Feynman amplitudes in the Schwinger parametrization. We then specialize the beta functions to various models with different symmetry groups: O(N), (Z(2))(N) x S-N and O(N) x O(M). For such models, we compute the fixed points and critical exponents.