low-energy hadron spectrum or matrix elements to set the scale. 

In the lattice calculation, there are variety of choices for the quantity to be expanded in α$_{s}$ ( µ ). In order to achieve good enough accuracy, the perturbative expansion must be known to higher orders or evaluated at very high energies. The latter may be achieved by calculating the scaling towards the high energy regime non-perturbatively using the so- called step-scaling technique (see, e.g. [5, 33]). The former is numerically less intensive but requires analytic perturbative calculations beyond one-loop level. 

This work demonstrates that the vacuum polarization function can be used for the precise determination of α$_{s}$ . The important points are (i) the perturbative expansion can be done in the continuum theory and is known to O ( α 3 $_{s}$), and (ii) the non-perturbative lattice calculation with controlled systematic errors is possible. The discretization error was a concern as the large Q 2 points have to be calculated, but it turned out to be under control with currently available lattice setups by careful estimates of systematic effects. 

The use of the overlap fermion is certainly desirable as the massless limit of quarks is uniquely defined and the use of the continuum OPE is justified. With the lattice fermions that violates chiral symmetry, one expects dangerous terms such as ma − $^{3}$/Q$^{4}$, whose nu- merical impact has to be carefully studied. 

Extension of this work is straightforward. Since the largest uncertainty comes from the scale determination, a consistent determination of the lattice scale with various low- energy inputs is necessary in order to significantly improve the accuracy. This requires extensive simulations at larger volumes, smaller quark masses and smaller lattice spacings. The discretization effect in VPF considered in this work will also be significantly reduced by going to finer lattice that will become available within a few years. 

## Acknowledgments 

E.S. thanks Y. Kikukawa for correspondence and discussion. The work of HF was sup- ported by the Global COE program of Nagoya University called “QFPU” from JSPS and MEXT of Japan. Numerical calculations are performed on IBM System Blue Gene Solution and Hitachi SR11000 at High Energy Accelerator Research Organization (KEK) under a sup- port of its Large Scale Simulation Program (No. 07-16). This work is supported by the Grant- in-Aid of the Japanese Ministry of Education (No. 18740167, 19540286, 19740121, 20105005, 

