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β-Functions for abelian and nonabelian gauge theories are studied in the regime where the large N flavor expansion is applicable. The first nontrivial order in the 1/N expansion is known for any value of Nα, and there are also various indications as to the nature of higher order effects. The singularity structure as a function of Nα has implications for the existence of nontrivial fixed points.
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Bob Holdom (2010) studied this question.
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