$S$ Parameter and Pseudo Nambu-Goldstone Boson Mass from Lattice QCD

We present a lattice calculation of $L_{10}$, one of the low-energy constants in chiral perturbation theory, and the charged-neutral pion squared-mass splitting, using dynamical overlap fermion. The exact chiral symmetry of the overlap fermion allows us to reliably extract these quantities from the difference of the vacuum polarization functions for vector and axial-vector currents. In the context of the technicolor models, these two quantities are read as the $S$ parameter and the pseudo Nambu-Goldstone boson mass, respectively, and play an important role in discriminating the models from others. This calculation can serve as a feasibility study of the lattice techniques for more general technicolor gauge theories.

Figure 1

${\widehat{q}}^2$ dependence of ${\Delta }_J$ ($J=V$ or $A$) (top) and their difference (bottom). The result for ${\widehat{m}}_q=0.015$ is shown.

Figure 2

${\widehat{q}}^2{\Pi }_{V-A}^{(1)}{|}_{{\widehat{q}}^2=0.038}$ as a function of quark mass. The fit results with (solid) and without (dashed) finite volume correction are shown.

Figure 3

The fit results with (9) (dashed curves) and (10) (solid curves). The results in the chiral limit are also shown. The statistical errors shown are smaller than the symbol size.