Scheme-independent series expansions at an infrared zero of the beta function in asymptotically free gauge theories

We consider an asymptotically free vectorial gauge theory, with gauge group $G$ and $N_f$ fermions in a representation $R$ of $G$, having an infrared (IR) zero in the beta function at ${\alpha }_{\mathrm{IR}}$. We present general formulas for scheme-independent series expansions of quantities, evaluated at ${\alpha }_{\mathrm{IR}}$, as powers of an $N_f$-dependent expansion parameter, ${\mathrm{\Delta }}_f$. First, we apply these to calculate the derivative $d\beta /d\alpha$ evaluated at ${\alpha }_{\mathrm{IR}}$, denoted ${\beta }_{\mathrm{IR}}^{\prime }$, which is equivalent to the anomalous dimension of the $\mathrm{Tr}(F_{\mu \nu }{F}^{\mu \nu })$ operator, to order ${\mathrm{\Delta }}_f^4$ for general $G$ and $R$, and to order ${\mathrm{\Delta }}_f^5$ for $G=\mathrm{SU}(3)$ and fermions in the fundamental representation. Second, we calculate the scheme-independent expansions of the anomalous dimension of the flavor-nonsinglet and flavor-singlet bilinear fermion antisymmetric Dirac tensor operators up to order ${\mathrm{\Delta }}_f^3$. The results are compared with rigorous upper bounds on anomalous dimensions of operators in conformally invariant theories. Our other results include an analysis of the limit $N_c\to \infty$, $N_f\to \infty$ with $N_f/N_c$ fixed, calculation and analysis of Padé approximants, and comparison with conventional higher-loop calculations of ${\beta }_{\mathrm{IR}}^{\prime }$ and anomalous dimensions as power series in $\alpha$.

Figure 1

Plot of ${\beta }_{\mathrm{IR},{\mathrm{\Delta }}_f^p}^{\prime }$ for $2\le p\le 5$ as a function of $N_f$ for the SU(3) theory with $N_f$ fermions in the fundamental representation. From bottom to top, the curves (with colors online) refer to ${\beta }_{\mathrm{IR},{\mathrm{\Delta }}_f^2}^{\prime }$ (red), ${\beta }_{\mathrm{IR},{\mathrm{\Delta }}_f^5}^{\prime }$ (black) ${\beta }_{\mathrm{IR},{\mathrm{\Delta }}_f^4}^{\prime }$ (blue), ${\beta }_{\mathrm{IR},{\mathrm{\Delta }}_f^3}^{\prime }$ (green).

Figure 2

Plot of ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},{\mathrm{\Delta }}_r^p}$ for $1\le p\le 3$ as a function of $r\in I_{\mathrm{IRZ},r}$ in the LNN limit (5.26). From bottom to top, the curves (with colors online) refer to ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},{\mathrm{\Delta }}_r}$ (red), ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},{\mathrm{\Delta }}_r^2}$ (green) ${\gamma }_{\overline{\psi }\psi ,\mathrm{IR},{\mathrm{\Delta }}_r^3}$ (blue).

Figure 3

Plot of ${\gamma }_{T,\mathrm{IR},{\mathrm{\Delta }}_r^p}$ for $1\le p\le 3$ as a function of $r\in I_{\mathrm{IRZ},r}$ in the LNN limit (5.26). From bottom to top, the curves (with colors online) refer to ${\gamma }_{T,\mathrm{IR},{\mathrm{\Delta }}_r}$ (red), ${\gamma }_{T,\mathrm{IR},{\mathrm{\Delta }}_r^2}$ (green),, ${\gamma }_{T,\mathrm{IR},{\mathrm{\Delta }}_r^3}$ (blue).