# Higher-Order Scheme-Independent Calculations of Physical Quantities in the Conformal Phase of a Gauge Theory

Thomas A. Ryttov, Robert Shrock

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We consider an asymptotically free vectorial SU($N_c$) gauge theory with $N_f$ massless fermions in a representation $R$, having an infrared fixed point (IRFP) of the renormalization group at $\alpha_{IR}$ in the conformal non-Abelian Coulomb phase. The cases with $R$ equal to the fundamental, adjoint, and symmetric rank-2 tensor representation are considered. We present scheme-independent calculations of the anomalous dimension $\gamma_{\bar\psi\psi,IR}$ to $O(\Delta_f^4)$ and $\beta'_{IR}$ to $O(\Delta_f^5)$ at this IRFP, where $\Delta_f$ is an $N_f$-dependent expansion parameter. Comparisons are made with conventional $n$-loop calculations and lattice measurements. As a test of the accuracy of the $\Delta_f$ expansion, we calculate $\gamma_{\bar\psi\psi,IR}$ to $O(\Delta_f^3)$ in ${\cal N}=1$ SU($N_c$) supersymmetric quantum chromodynamics and find complete agreement, to this order, with the exactly known expression. The $\Delta_f$ expansion also avoids a problem in which an IRFP may not be manifest as an IR zero of a higher $n$-loop beta function.