Higher-Loop Structural Properties of the $\beta$ Function in Asymptotically Free Vectorial Gauge Theories

We investigate some higher-loop structural properties of the $\beta$ function in asymptotically free vectorial gauge theories. Our main focus is on theories with fermion contents that lead to an infrared (IR) zero in $\beta$. We present analytic and numerical calculations of the value of the gauge coupling where $\beta$ reaches a minimum, the value of $\beta$ at this minimum, and the slope of $\beta$ at the IR zero, at two-, three-, and four-loop order. The slope of $\beta$ at the IR zero is relevant for estimates of a dilaton mass in quasiconformal gauge theories. Some inequalities are derived concerning the dependence of the above quantities on loop order. A general inequality is derived concerning the dependence of the shift of the IR zero of $\beta$, from the $n$-loop to the $(n+1)$-loop order, on the sign of the $(n+1)$-loop coefficient in $\beta$. Some results are also given for gauge theories with ${\cal N}=1$ supersymmetry.