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

Understand

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.