On character expansion and Gaussian regularization of Itzykson-Zuber measure

Character expansions are among the most important approaches to modern quantum field theory, which substitute integrals by combinations of peculiar special functions from the Schur-Macdonald family. These formulas allow various deformations, which are not transparent in integral formulation. We analyze from this point of view the Itzykson-Zuber integral over unitary matrices which is exactly solvable, but difficult to deform in $\beta$ and $(q,t)$ directions. Character expansion straightforwardly resolves this problem. However, taking averages with the so defined measure can look problematic, because integrals of individual expansion terms often diverge and well defined is only the sum of them. We explain a way to overcome this problem by Gaussian regularization, which can have a broad range of further applications.