Understand
We extend the position-space renormalization procedure, where renormalization factors are calculated from Green's functions in position space, by introducing a technique to take the average of Green's functions over spheres.
- In addition to reducing discretization errors, this technique enables the resulting position-space correlators to be evaluated at any physical distance, making them continuous functions similar to the $O(4)$-symmetric position-space Green's functions in the continuum theory but with a residual dependence on a regularization parameter, the lattice spacing $a$.
- We can then take the continuum limit of these renormalized quantities calculated at the same physical renormalization scale $|x|$ and investigate the resulting $|x|$-dependence to identify the appropriate renormalization window.
- As a numerical test of the spherical averaging technique, we determine the renormalized light and strange quark masses by renormalizing the scalar current.