How perturbative are heavy sea quarks?

Effects of heavy sea quarks on the low energy physics are described by an effective theory where the expansion parameter is the inverse quark mass, 1/$M$. At leading order in 1/$M$ (and neglecting light quark masses) the dependence of any low energy quantity on $M$ is given in terms of the ratio of $\Lambda$ parameters of the effective and the fundamental theory. We define a function describing the scaling with the mass $M$. We find that its perturbative expansion is very reliable for the bottom quark and also seems to work very well at the charm quark mass. The same is then true for the ratios of $\Lambda^{(4)}/\Lambda^{(5)}$ and $\Lambda^{(3)}/\Lambda^{(4)}$, which play a major r\^ole in connecting lattice determinations of $\alpha^{(3)}_{MSbar}$ from the three-flavor theory with $\alpha^{(5)}_{MSbar}(M_Z)$. Also the charm quark content of the nucleon, relevant for dark matter searches, can be computed accurately from perturbation theory. We investigate a very closely related model, namely QCD with $N_f=2$ heavy quarks. Our non-perturbative information is derived from simulations on the lattice, with masses up to the charm quark mass and lattice spacings down to about 0.023 fm followed by a continuum extrapolation. The non-perturbative mass dependence agrees within rather small errors with the perturbative prediction at masses around the charm quark mass. Surprisingly, from studying solely the massive theory we can make a prediction for the ratio $Q^{1/\sqrt{t_0}}_{0,2}=[\Lambda \sqrt{t_0(0)}]_{N_f=2}/[\Lambda \sqrt{t_0}]_{N_f=0}$, which refers to the chiral limit in $N_f=2$. Here $t_0$ is the Gradient Flow scale of [1]. The uncertainty for $Q$ is estimated to be 2.5%. For the phenomenologically interesting $\Lambda^{(3)}/\Lambda^{(4)}$, we conclude that perturbation theory introduces errors which are at most at the 1.5% level, far smaller than other current uncertainties.