Near-SUSY to Non-SUSY Crossover

TL;DR

This work argues that softly broken SUSY QCD (ASQCD) with anomaly mediation at $m \\ll \\Lambda$ is continuously connected to the non-SUSY regime at $m \\gg \\Lambda$, analogous to a BCS--BEC crossover. By deriving the chiral Lagrangian and WZW term across ADS, QM, s-confinement, and conformal regimes, and computing quark and gluon condensates, the authors demonstrate consistent large-$N_c$ scaling and a matching of topological susceptibility with $m_{\eta'}$ and $m_\pi$. They provide explicit scaling predictions for $f_\pi^2$ and $\langle GG\rangle$ as functions of $N_c$ and $N_f$, including regimes where $N_f/N_c = O(1)$, and confirm a Witten-Veneziano-type relation within ASQCD. Mass-spectrum analyses in representative cases reveal phase-specific spectra while supporting the proposed crossover, with a universal scalar-pseudoscalar sum rule linking $m_\sigma^2$ and $m_\pi^2$. Overall, the paper furnishes a coherent, testable framework for relating near-SUSY and non-SUSY dynamics in SU$(N_c)$ gauge theories across a wide range of flavors and colors. If borne out, these results offer a deeper understanding of nonperturbative QCD-like dynamics and potential insights into real-world QCD from SUSY-perturbed theories.

Abstract

Gauge theories can be solved exactly slightly away from the supersymmetric (SUSY) limit softly broken by anomaly mediation when the size of SUSY breaking is much smaller than the dynamical scale ($m \ll Λ$). We show empirical evidence that the near-SUSY limit is continuously connected to the non-SUSY limit ($m \gg Λ$) in $\mathrm{SU}(N_c)$ gauge theories with $N_f$ quarks in the fundamental representation. The evidence includes the behavior of quark bi-linear condensate and gluon condensates, light hadron spectra, and consistency with the large $N_c$ limit. In addition, we present new predictions when $N_f/N_c \gtrsim O(1)$.

Figures (15)

• Figure 1: The illustration of BCS--BEC crossover, by changing the attractive interaction $g$. Only the left-most point $g=0$ does not break the $\mathrm{U}(1)$ symmetry, while from the BCS state with small $g$ to the BEC with infinite $g$ is a crossover and the universality class of broken $\mathrm{U}(1)$ remains unchanged. We expect a similar crossover for the deformation of near-SUSY SQCD ($m\ll \Lambda$) into non-SUSY QCD ($m\rightarrow \infty$), while SQCD itself ($m=0$) belongs to a different universality class.
• Figure 2: The dependence of $\log_{N_c}(f_\pi^2)$ on $N_f/N_c$ is shown, in the limit $N_c\to\infty$ with $N_f/N_c$ held fixed, for $N_f<3N_c/2$. We use \ref{['eq:ADSVEV', 'eq:FreeMagVEV']}. For $N_f<N_c$ the expected scaling behavior $f_\pi^2\propto N_c$ as predicted by 't Hooft diagram approach is observed. In contrast, for $N_f>N_c$ the scaling is suppressed. Although this plot is constructed in the $N_c\to\infty$ limit, our results can also be used to predict the dependence on $N_f/N_c$ at finite $N_c$.
• Figure 3: The $N_f/N_c$ dependence of $\log_{N_c}(f_\pi^2)$ is shown in the $N_c\to\infty$ limit with $N_f/N_c$ held fixed, near the lower edge of the Conformal Window. We use \ref{['eq:FreeMagVEV', 'eq:LCWVEV']}. The point at exactly $N_f=3N_c/2$ is omitted, reflecting the fact that AMSB cannot be applied to obtain the non-SUSY limit due to the classical conformal invariance there. An interpolation between the chiral symmetry breaking results for $N_f<3N_c/2$ and $N_f>3N_c/2$ is expected.
• Figure 4: The $N_f/N_c$ dependence of $\log_{N_c}(f_\pi^2)$ is shown in the $N_c\to\infty$ limit with $N_f/N_c$ held fixed for the full range $0<N_f/N_c < 3$. We use \ref{['eq:ADSVEV', 'eq:FreeMagVEV', 'eq:LCWVEV', 'eq:UCWVEV']}. In the conformal window, results from perturbing around the electric and magnetic Banks-Zaks fixed points are shown in red and orange, respectively, and a smooth interpolation between them is shown in dashed gray. Note that $f_\pi^2 \to 0$ as $N_f\to 3N_c$ which is because the AMSB becomes ineffective when the SUSY limit is classically conformal.
• Figure 5: The scaling of the gluon condensate in the $N_c\to\infty$ limit with $N_f/N_c$ held fixed is shown. We have used \ref{['eq:ADSgluoncondense', 'eq:Freemaggluoncondense']} The expected $N_c^2$ scaling is observed when $N_f < N_c$, whereas the power law decreases when $N_f>N_c$. The region $0<N_f/N_c<1.5$ is shown here.