The Balaban 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 Bałaban’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 Bałaban’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.