The Ba{\l}aban variational problem in the non-linear sigma model

The minimization of the action of a QFT with a constraint dictated by the block averaging procedure is an important part of Ba{\l}aban's approach to renormalization. It is particularly interesting for QFTs with non-trivial target spaces, such as gauge theories or non-linear sigma models on a lattice. We analyze this step for the $O(4)$ non-linear sigma model in two dimensions and demonstrate, in this case, how various ingredients of Ba{\l}aban's approach play together. First, using variational calculus on Lie groups, the equation for the critical point is derived. Then, this non-linear equation is solved by the Banach contraction mapping theorem. This step requires detailed control of lattice Green functions and their integral kernels via random walk expansions.