The Schr\"odinger functional running coupling with staggered fermions and its application to many flavor QCD

We discuss the Schr\"odinger functional in lattice QCD with staggered fermions and relate it, in the classical continuum limit, to the Schr\"odinger functional regularized with Wilson fermions. We compute the strong coupling constant defined via the Schr\"odinger functional with staggered fermions at one loop and show that it agrees with the continuum running coupling constant in the Schr\"odinger functional formalism. We compute this running coupling in the ``weak coupling phase'' of many flavor QCD numerically at several values of the bare coupling and for several system sizes from $L/a=4$ to 12. The results indicate that the $\beta$-function for 16 flavors has the opposite sign than for few flavor QCD, in agreement with a recent claim, and with the perturbative prediction.