Renormalization flow for unrooted forests on a triangular lattice

We compute in small temperature expansion the two-loop renormalization constants and the three-loop coefficient of the β-function, that is the first non-universal term, for the σ-model with O(N) invariance on the triangular lattice at N=−1. The partition function of the corresponding Grassmann theory is, for negative temperature, the generating function of unrooted forests on such a lattice, where the temperature acts as a chemical potential for the number of trees in the forest. To evaluate Feynman diagrams we extend the coordinate space method to the triangular lattice.