Lattice Analysis for the Energy Scale of QCD Phenomena

We formulate a new framework in lattice QCD to study the relevant energy scale of QCD phenomena. By considering the Fourier transformation of link variable, we can investigate the intrinsic energy scale of a physical quantity nonperturbatively. This framework is broadly available for all lattice QCD calculations. We apply this framework for the quark-antiquark potential and meson masses in quenched lattice QCD. The gluonic energy scale relevant for the confinement is found to be less than 1 GeV in the Landau or Coulomb gauge.

Figure 1

The two-dimensional schematic figure of the UV cut ${\Lambda }_{\mathrm{UV}}$ and the IR cut ${\Lambda }_{\mathrm{IR}}$ on momentum-space lattice. The momentum-space link variable ${\tilde{U}}_{\mu }(p)$ is replaced with the free one ${\tilde{U}}_{\mu }^{\mathrm{free}}(p)$ in the cut (shaded) regions. $a_p$ is the momentum-space lattice spacing.

Figure 2

The $Q\overline{Q}$ potential $V(R)$ with the IR cut ${\Lambda }_{\mathrm{IR}}$ against the interquark distance $R$. The unit is scaled with the lattice spacings $a\simeq 0.10\mathrm{fm}$ and $a_p\simeq 0.77\mathrm{GeV}$. The broken line is the original $Q\overline{Q}$ potential in lattice QCD.

Figure 3

The $Q\overline{Q}$ potential with the UV cut ${\Lambda }_{\mathrm{UV}}$. The notation is the same as Fig. 2.

Figure 4

The $Q\overline{Q}$ potential $V(R)$ with the IR cut ${\Lambda }_{\mathrm{IR}}$ by $\max (p_1,p_2,p_3,p_4)$ instead of $\sqrt{p^2}$ in Eq. (2). The notation is the same as Fig. 2.

Figure 5

Pion mass $m_{\pi }$ and $\rho$-meson mass $m_{\rho }$ with the IR cut ${\Lambda }_{\mathrm{IR}}$. The results of three hopping parameters $\kappa =0.1200$, 0.1300, and 0.1340 are shown. The unit is scaled with the lattice spacings $a\simeq 0.10\mathrm{fm}$ and $a_p\simeq 0.77\mathrm{GeV}$. The values at ${\Lambda }_{\mathrm{IR}}/a_p=0$ are calculated in original lattice QCD [9].