Higher-Order Scheme-Independent Series Expansions of ${\gamma }_{\bar{\psi }\psi ,IR}$ and ${\beta }_{IR}^{\prime }$ in Conformal Field Theories

Abstract

We study a vectorial asymptotically free gauge theory, with gauge group $G$ and $N_f$ massless fermions in a representation $R$ of this group, that exhibits an infrared (IR) zero in its beta function, $\beta$, at the coupling $\alpha ={\alpha }_{IR}$ in the non-Abelian Coulomb phase. For general $G$ and $R$, we calculate the scheme-independent series expansions of (i) the anomalous dimension of the fermion bilinear, ${\gamma }_{\bar{\psi }\psi ,IR}$, to $O({\Delta }_f^4)$ and (ii) the derivative ${\beta }^{\prime }=d\beta /d\alpha$, to $O({\Delta }_f^5)$, both evaluated at ${\alpha }_{IR}$, where ${\Delta }_f$ is an $N_f$-dependent expansion variable. These are the highest orders to which these expansions have been calculated. We apply these general results to theories with $G=\mathrm{SU}(N_c)$ and $R$ equal to the fundamental, adjoint, and symmetric and antisymmetric rank-2 tensor representations. It is shown that for all of these representations, ${\gamma }_{\bar{\psi }\psi ,IR}$, calculated to the order ${\Delta }_f^p$, with $1\le p\le 4$, increases monotonically with decreasing $N_f$ and, for fixed $N_f$, is a monotonically increasing function of $p$. Comparisons of our scheme-independent calculations of ${\gamma }_{\bar{\psi }\psi ,IR}$ and ${\beta }_{IR}^{\prime }$ are made with our earlier higher $n$-loop values of these quantities, and with lattice measurements. For $R=F$, we present results for the limit $N_c\to \infty$ and $N_f\to \infty$ with $N_f/N_c$ fixed. We also present expansions for ${\alpha }_{IR}$ calculated to $O({\Delta }_f^4)$.

40 pages, latex, 12 figures

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