Dilaton Effective Field Theory across the Conformal Edge

TL;DR

This work tests whether dilaton EFT (dEFT) can diagnose whether near-edge gauge theories are confining or infrared conformal by fitting lattice data at finite fermion mass. The framework uses a dilaton field $\chi$ with a potential $V(\chi)=A\chi^4+B\chi^{\Delta}$ and conformal-coupled pNGBs, with relations such as $F_{\pi}^2 = P F_d^2$ and $M_{\pi}^2 = R F_d^{y}/F_{\pi}^2$ linking EFT parameters to lattice observables. Fitting to two lattice studies near the conformal edge yields contrasting conclusions: SU(3) with $N_f=8$ (fundamental) favors confinement outside the conformal window ($\Delta<4$, $A>0$, $B<0$), while SU(2) with $N_f=1$ adjoint favors infrared conformality ($\Delta>4$, $A>0$, $B<0$, $y\approx2$). The results demonstrate that dEFT can serve as a diagnostic tool across the edge of the conformal window, though the analysis is preliminary and limited by lattice artifacts and the leading-order EFT truncation; future high-precision lattice data could sharpen the distinctions and extend the framework to additional theories.

Abstract

Dilaton effective field theory (dEFT) can be employed to analyze lattice data in gauge theories that lie in close proximity of the lower edge of the conformal window. Under special conditions, we show that it can be used as a diagnostic tool to distinguish near-conformal, yet confining, theories from infrared conformal ones. We demonstrate this efficacy by analyzing two sets of lattice measurements taken from the literature. For the $SU(3)$ theory coupled to $N_f=8$ Dirac fermions transforming in the fundamental representation, our analysis favors confinement. For the $SU(2)$ theory with $N_f=1$ adjoint fermion, our fits favor infrared conformal behavior. We discuss future lattice measurements, and analysis refinements, that can further test this framework.

Figures (6)

• Figure 1: Contours of $\Delta\chi^2$ indicating confidence intervals (equivalent to $1 \sigma$, $2\sigma$, and $3 \sigma$, respectively), for $A$ and $\Delta$ in the $SU(3)$ gauge theory with $N_f = 8$ Dirac fermions in the fundamental representation. The central value denoted by the red dot is reported in Table \ref{['Tab:deft']}. For each point of the plot, we minimize the $\chi^2$ in respect to the other four fit parameters. The region $A > 0$ and $\Delta < 4$ is favored.
• Figure 2: Contours of $\Delta\chi^2$ indicating confidence intervals (equivalent to $1 \sigma$, $2\sigma$, and $3 \sigma$, respectively), for $B$ and $\Delta$, in the $SU(3)$ gauge theory with $N_f = 8$ Dirac fermions in the fundamental representation. The central value denoted by the red dot is reported in Table \ref{['Tab:deft']}. For each point of the plot, we minimize the $\chi^2$ in respect to the other four fit parameters. The region with $B < 0$ and $\Delta < 4$ is favored, consistent with being outside the conformal window.
• Figure 3: Plot of $a^3M^2_\pi F_\pi$ versus $a F_\pi$, in the $SU(3)$ theory with $N_f = 8$ fermions in the fundamental representation LatticeStrongDynamics:2023bqpSU(3)-data_release. The red curve is based on the central values of Table \ref{['Tab:deft']}, with the five data points denoted by the black discs. The blue shaded region represents the uncertainty in the fitting curve, equivalent to the $1\sigma$ confidence level. The red curve reaches the horizontal axis at a nonzero value of $a F_{\pi}$, corresponding to the limit $m \rightarrow 0$ in a confining gauge theory. For larger values of $F_{\pi} \propto F_d$, we have $M_{\pi}^2 F_{\pi} \propto V^{\prime}(F_d)$. The curve is extrapolated downward, where it becomes negative, by continuing Eq. (\ref{['eq2']}) to unphysical values of $F_{\pi}$, demonstrating that the region near the origin is unstable, as shown in the inset.
• Figure 4: Contours of $\Delta\chi^2$ indicating confidence intervals (equivalent to $1 \sigma$, $2\sigma$, and $3 \sigma$, respectively), for $A$ and $\Delta$ in the $SU(2)$ gauge theory with $N_f = 1$ Dirac fermion in the adjoint representation. The fit includes only data taken at the three lightest fermion mass points. The central value is reported in Table \ref{['Tab:adjdeft']}. For each point of the plot, we minimize the $\chi^2$ in respect to the other four fit parameters. The region with $\Delta>4$ and $A>0$ is favored, consistently with infrared conformal behavior.
• Figure 5: Contours of $\Delta\chi^2$ indicating confidence intervals (equivalent to $1 \sigma$, $2\sigma$, and $3 \sigma$, respectively), for $B$ and $\Delta$ in the $SU(2)$ gauge theory with $N_f = 1$ Dirac fermion in the adjoint representation. The fit includes only data taken at the three lightest fermion mass points. The central value is reported in Table \ref{['Tab:adjdeft']}. For each point of the plot, we minimize the $\chi^2$ in respect to the other four fit parameters. The region with $\Delta>4$ and $B<0$ is favored, consistently with infrared conformal behavior.