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 scheme-independent 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 $N_f \to \infty$ 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 $1/N_f$.