Study of possible ultraviolet zero of the beta function in gauge theories with many fermions

We study the possibility of an ultraviolet (UV) zero in the n-loop beta function of U(1) and non-Abelian gauge theories with N-f fermions for large N-f. The effect of scheme transformations on the coefficients of different powers of N-f in the n-loop term in the beta function is calculated. A general criterion is given for determining whether or not the n-loop beta function has a UV zero for large N-f. We compare the results with exact integral representations of the leading terms in the beta functions for the respective Abelian and non-Abelian theories in the limit N-f -> infinity limit with N-f alpha finite. As part of this study, new analytic and numerical results are presented for certain coefficients, denoted b(n,n-1), that control the large-N-f behavior at n-loop order in the beta function. We also investigate various test functions incorporating a power-law and essential UV zero in the beta function and determine their manifestations in series expansions in powers of coupling and in powers of 1/N-f.