Question of a Possible Infrared Zero in the Beta Function of the Finite-$N$ Gross-Neveu Model

We investigate whether the beta function of the finite-$N$ Gross-Neveu model, as calculated up to the four-loop level, exhibits evidence for an infrared zero. As part of our analysis, we calculate and analyze Padé approximants to this beta function and evaluate effects of scheme dependence. From our study, we find that in the range of coupling where the perturbative calculation of the four-loop beta function is reliable, it does not exhibit robust evidence for an infrared zero.

Figure 1

Plot of the $n$-loop $\beta$ function ${\beta }_{a,n\ell }$ of the Gross-Neveu model as a function of $a$ for $N=3$ and (i) $n=2$ (red), (ii) $n=3$ (green), and (iii) $n=4$ (blue). At $a=0.5$, going from bottom to top, the curves are ${\beta }_{4\ell }$, ${\beta }_{3\ell }$, and ${\beta }_{2\ell }$.

Figure 2

Plot of the $n$-loop $\beta$ function ${\beta }_{a,n\ell }$ of the Gross-Neveu model as a function of $a$ for $N=10$ and (i) $n=2$ (red), (ii) $n=3$ (green), and (iii) $n=4$ (blue). At $a=0.2$, going from bottom to top, the curves are ${\beta }_{4\ell }$, ${\beta }_{2\ell }$, and ${\beta }_{3\ell }$.