Ideal walking dynamics via a gauged NJL model

According to the ideal walking technicolor paradigm, large mass anomalous dimensions arise in gauged Nambu–Jona-Lasinio (NJL) models when the four-fermion coupling is sufficiently strong to induce spontaneous symmetry breaking in an otherwise conformal gauge theory. We therefore study the $SU(2)$ gauged NJL model with two adjoint fermions using lattice simulations. The model is in an infrared conformal phase at small NJL coupling while it displays a chirally broken phase at large NJL couplings. In the infrared conformal phase, we find that the mass anomalous dimension varies with the NJL coupling, reaching ${\gamma }_m\sim 1$ close to the chiral symmetry breaking transition, de facto making the present model the first explicit realization of the ideal walking scenario.

Figure 1

A sketch of the phase diagram at zero quark mass. Phases 1.a and 1.b are the physical infrared conformal and chirally broken regions. The solid line shows the large N ladder approximation result for the critical line separating the phases (the $\chi \mathrm{SB}$ line), and circles denote its measured locations. In region 2 at $\beta >{\beta }_{\max }$, the Polyakov loop grows with $\beta$, indicating significant finite size effects. At $L=16$, we find ${\beta }_{\max }\gtrsim 3$. In phase 3, there is a first order bulk transition instead of a critical line, and the quark mass cannot be taken to zero. The squares denote the measured boundaries of this phase.

Figure 2

The plaquette expectation value as a function of the bare mass at several values of $g$ and $\beta =2.25$ (left) and $\beta =1.7$ (middle) and the expectation value $⟨{\pi }_3⟩$ at $\beta =1.7$ (right). At the larger $\beta$, we observe only a crossover, and a critical line can be found. At $\beta =1.7$ and $g<0.3$, we see a first order transition into the bulk phase. In this case, the transition happens at a positive quark mass, preventing studies at the critical line. At larger $g=0.3$, we find no first order transition, and there is a critical line signaled by a nonzero expectation value $⟨{\pi }_3⟩$.

Figure 3

The Polyakov loop expectation value as a function of the gauge coupling $\beta$ at $L=8$, $g=0.1$, and $m_0=0$.

Figure 4

Left: The ${\overline{m}}_{ND}(t)$ as a function of $t$. The $t$ dependence of the condensate provides a measure of discretization effects. Right: The condensate ${\overline{m}}_{ND}$ as a function of bare mass with $\beta =2.25$ and $L=16$.

Figure 5

The expectation value $⟨{\pi }_3⟩$ (left) and the susceptibility ${\chi }_{\pi }$ (right) with $\beta =2.25$ and $g=0.3$.

Figure 6

The condensate ${\overline{m}}_{ND}$ (left) and the nondiagonal pseudoscalar ($NDP$) and vector ($NDV$) meson masses at $\beta =2.25$ (left) and $\beta =3$ (right).

Figure 7

A hyperscaling fit to the nondiagonal meson masses at $g=0.05$, 0.1, 0.2, and 0.25 and $\beta =2.25$. The data points with filled in symbols are included in the fit. The points corresponding to the vector meson have been shifted to the right by $\mathrm{\Delta }x=2$.