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This PhD thesis is devoted to show that differential renormalization is asimple and useful renormalization method that we can use when dealing withgauge theories. In this work, it is shown how the one-loop results ofConstraint Differential Renormalization can be fruitfully applied whenrenormalizing two-loop amplitudes. As an example of this procedure, thetwo-loop beta functions of various relevant gauge theories are obtained (QED,SuperQED, Yang-Mills and N=1 SuperYM) without making use of the Wardidentities, which are mandatory in the standard differential renormalizationprocedure. Also, we study how to deal with expressions that have IR and UVdivergences, showing that both are renormalized with independent scales. Thisresult allow us to give new insight on the origin of the higher ordercorrections to the beta function of the N=1 SuperYM theory, as in this case wehave both divergences due to the form of the gauge propagator. This is anadvantage over usual dimensional regularization methods, where both divergencesbecome mixed.

The beta function of gauge theories at two loops in differential renormalization