Case studies of near-conformal $\beta$-functions

We present updated results for the non-perturbative $\beta$-function of SU(3) gauge theories with $N_f = 12$ or 10 massless flavors in the fundamental rep or $N_f = 2$ in the sextet rep, measured with staggered fermions. New data at finer lattice spacing and our previously introduced method, the infinitesimal $\beta$-function, strengthen the case that the $N_f = 12$ model has no infrared fixed point up to $g^2 = 7.2$. We show how underestimated cutoff dependence in one domain wall study for $N_f = 10$ has been corrected, which is now consistent with staggered results showing a monotonically increasing $\beta$-function. A consistent theme is that too small volumes can lead to apparent fixed points which vanish towards the continuum limit. We also apply the infinitesimal $\beta$-function method to the $N_f = 10$ model, finding consistent behavior with the finite-step $\beta$-function. Ongoing simulations and analysis for the sextet model confirm our previous results from weak to strong coupling with a non-zero $\beta$-function throughout, in quantitative difference to Wilson fermion simulations~\cite{Hasenfratz:2015ssa}.