Investigating the conformal behavior of SU(2) with one adjoint Dirac flavor

We present a major update on our investigations of SU(2) gauge theory with one Dirac flavor in the adjoint representation on the lattice. In particular, we consider larger volumes as well as four different values of the gauge coupling. We provide results for the spectrum including gluonic, fermionic, and hybrid observables; Polyakov loops; and the anomalous dimension of the fermionic condensate from the Dirac mode number. These data confirm that the theory is close to the lower boundary of the conformal window for adjoint fermions. Our investigations provide important insights regarding the realization of different infrared scenarios that have been conjectured for this theory.

Figure 1

The histograms of the average Polyakov loop for the ensembles DB1M10, DB2M7, DB3M8, and DB4M11 belonging to the set shown in the label of each subfigure, for the four space-time directions. Single peaks indicate an unbroken center symmetry.

Figure 2

Monte Carlo history of $Q$ (left panel) and resulting histogram (right panel) computed on 4000 configurations extracted using gradient flow at gradient flow time $\tau$ corresponding to $a\sqrt{8{\tau }_{\mathrm{flow}}}=L/2$ for the ensembles DB1M10, DB2M7, DB3M8, and DB4M11.

Figure 3

The topological susceptibility measured for the largest combination of each ($\beta$, $m$), in units of the string tension $\sqrt{\sigma }$. Included for comparison are data for pure SU(2) gauge theory and the SU(2) theory with two adjoint Dirac flavors, using data from Ref. [50].

Figure 4

The mass spectrum observed at the largest volume for each combination of ($\beta$, $m$), showing baryons, glueballs, the hybrid fermion $\breve{g}$, and the string tension $\sqrt{\sigma }$, as a function of the PCAC fermion mass, in units of the gradient flow scale $w_0$.

Figure 5

The decay constant of the $2^{+}$ baryon observed at the largest volume for each combination of ($\beta$, $m$), as a function of the PCAC fermion mass, in units of the gradient flow scale $w_0$.

Figure 6

The ratio of the mass of the $0^{++}$ glueball to that of the $2^{+}$ scalar baryon as a function of the PCAC fermion mass scaled by the gradient flow scale $w_0$.

Figure 7

Finite-size hyperscaling plots showing the mass of the $2^{+}$ baryon as a function of the product $L{m}_{\mathrm{PCAC}}^{1/(1+{\gamma }_{*})}$ for the stated values of ${\gamma }_{*}$. Data for different values of $\beta$ align at different values of ${\gamma }_{*}$, with the extrema being $\beta =2.05$ data aligning at ${\gamma }_{*}=0.943(71)$ and $\beta =2.2$ data aligning at ${\gamma }_{*}=0.717(43)$.

Figure 8

Results of the bootstrapped fits of the mode number for the ensembles DB1M8, DB1M10, DB2M7, DB3M8, and DB4M11 as a function of the lower end of the fit window. Colors represent the length of the window $\mathrm{\Delta }\mathrm{\Omega }={\mathrm{\Omega }}_{\mathrm{UE}}-{\mathrm{\Omega }}_{\mathrm{LE}}$, and opacity represents the proportion of fits for that window that converged.

Figure 9

The results of fitting the $2^{+}$ baryon mass (scaled by the gradient flow scale $w_0$) for the largest-volume ensemble for each combination of ($\beta$, $m$) with chiral perturbation theory.

Figure 10

The ratio $R=\frac{M_{2^{++}}}{M_{0^{++}}}$ measured at all volumes studied for each combination of ($\beta$, $m$). The shaded region indicates the range suggested by the model presented in Ref. [38], when the values for ${\gamma }_{*}$ in Table 1 are used. In the case of $\beta =2.05$ and 2.1, the upper bound on $R$ is infinite, as ${\gamma }_{*}$ is compatible with 1, at which point the model is singular.