Determination of the bottom quark mass from the $\Upsilon(1S)$ system

We approximately compute the normalization constant of the first infrared renormalon of the pole mass (and the singlet static potential). Estimates of higher order terms in the perturbative relation between the pole mass and the $\MS$ mass (and in the relation between the singlet static potential and $\alpha_s$) are given. We define a matching scheme (the renormalon subtracted scheme) between QCD and any effective field theory with heavy quarks where, besides the usual perturbative matching, the first renormalon in the Borel plane of the pole mass is subtracted. A determination of the bottom $\MS$ quark mass from the $\Upsilon(1S)$ system is performed with this new scheme and the errors studied. Our result reads $m_{b,\MS}(m_{b,\MS})=4 210^{+90}_{-90}({\rm theory})^{-25}_{+25}(\alpha_s)$ MeV. Using the mass difference between the $B$ and $D$ meson, we also obtain a value for the charm quark mass: $m_{c,\MS}(m_{c,\MS})=1 210^{+70}_{-70}({\rm theory})^{+65}_{-65}(m_{b,\MS})^{-45}_{+45}(\lambda_1)$ MeV. We finally discuss upon eventual improvements of these determinations.