Scheme-Independent Calculations of Physical Quantities in an $\mathcal{N}=1$ Supersymmetric Gauge Theory

We consider an asymptotically free, vectorial, $\mathcal{N}=1$ supersymmetric gauge theory with gauge group $G$ and $N_f$ pairs of chiral superfields in the respective representations $\mathcal{R}$ and $\bar{\mathcal{R}}$ of $G$, having an infrared fixed point (IRFP) of the renormalization group at ${\alpha }_{IR}$. We present exact results for the anomalous dimensions of various (gauge-invariant) composite chiral superfields ${\gamma }_{{\mathrm{\Phi }}_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{d}}}$ at the IRFP and prove that these increase monotonically with decreasing $N_f$ in the non-Abelian Coulomb phase of the theory and that scheme-independent expansions for these anomalous dimensions as powers of an $N_f$-dependent variable, ${\mathrm{\Delta }}_f$, exhibit monotonic and rapid convergence to the exact $\gamma_{{\Phi}{\rm prod}}$ throughout this phase. We also present a scheme-independent calculation of the derivative of the beta function, $d\beta/d\alpha |{\alpha=\alpha_{IR}}$,denoted$\beta’_{IR}$,upto$O(\Delta_f^3)$ for general $G$ and $\mathcal{R}$, and, for the case $G=\mathrm{S}\mathrm{U}(N_c)$, $\mathcal{R}=F$, we give an analysis of the properties of ${\beta }_{IR}^{\mathrm{\prime }}$ calculated to $O({\mathrm{\Delta }}_f^4)$.