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We study the possibility of an ultraviolet (UV) zero in the $n$-loop betafunction 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 differentpowers of $N_f$ in the $n$-loop term in the beta function is calculated. Ageneral scheme-independent criterion is given for determining whether or notthe $n$-loop beta function has a UV zero for large $N_f$. We compare theresults with exact integral representations of the leading terms in the betafunctions 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 analyticand numerical results are presented for certain coefficients, denoted$b_{n,n-1}$, that control the large-$N_f$ behavior at $n$-loop order in thebeta function. We also investigate various test functions incorporating apower-law and essential UV zero in the beta function and determine theirmanifestations in series expansions in powers of coupling and in powers of$1/N_f$.

Study of Possible Ultraviolet Zero of the Beta Function in Gauge Theories with Many Fermions