## Higher-Order Scheme-Independent Series Expansions of $gamma_{barpsipsi,IR}$ and $beta’_{IR}$ in Conformal Field Theories

Preprint number: CP3-Origins-2017-10 DNRF90
Authors: Thomas A. Ryttov (CP3-Origins & DIAS) and R Shrock (C. N. Yang Institute for Theoretical Physics)

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, $gamma_{barpsipsi,IR}$, to
$O(Delta_f^4)$ and (ii) the derivative $beta’ = dbeta/dalpha$, 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 $G={rm SU}(N_c)$ and $R$ equal to the fundamental, adjoint,
and symmetric and antisymmetric rank-2 tensor representations. It is shown
that for all of these representations, $gamma_{barpsipsi,IR}$, 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
$gamma_{barpsipsi,IR}$ and $beta’_{IR}$ 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 $O(Delta_f^4)$.