Domain Walls and the $CP$ Anomaly in Softly Broken Supersymmetric QCD

TL;DR

The paper investigates CP violation and domain-wall dynamics in softly broken SQCD with $N_f<N$, highlighting how the light $\eta'$ field enriches wall trajectories and how an 't Hooft CP anomaly governs the existence of nontrivial wall excitations. By analyzing the ADS-based low-energy potential, moduli-space structure, and small SUSY-breaking corrections, it shows that wall tensions and stabilities map directly to anomaly data via the gcd$(N,N_f)$ structure, with special treatment of the $N_f=N-1$ case where approximately BPS walls emerge. The work connects holographic-like geometric constraints in the vacua to physical wall excitations and provides a phase diagram for wall stability under soft breaking. It also extends the discussion to adjoint QCD with fundamentals (and axions), illustrating how similar wall phenomena and discrete gauge symmetries shape nonperturbative dynamics across related theories.

Abstract

In ordinary QCD with light, degenerate, fundamental flavors, $CP$ symmetry is spontaneously broken at $θ=π$, and domain wall solutions connecting the vacua can be constructed in chiral perturbation theory. In some cases the breaking of $CP$ saturates an 't Hooft anomaly, and anomaly inflow requires nontrivial massless excitations on the domain walls. Analogously, $CP$ can be spontaneously broken in supersymmetric QCD with light flavors and small soft breaking parameters. We study $CP$ breaking and domain walls in softly broken SQCD with $N_f<N$ flavors. Relative to ordinary QCD, the supersymmetric case contains an extra light field, the $η^\prime$, which has interesting effects on the structure of the walls. Vanishing of the $CP$ anomaly is associated with the existence of multiple domain wall trajectories through field space, including walls which support no nontrivial massless excitations. In cases with an anomaly such walls are forbidden, and their absence in the relevant SQCD theories can be seen directly from the geometry of the low energy field space. In the case $N_f=N-1$, multiple approximately-BPS walls connect the vacua. Corrections to their tensions can be computed at leading order in the soft breaking parameters, producing a phase diagram for the stable wall trajectory. We also comment on domain walls in the similar case of QCD with an adjoint and fundamental flavors, and on the impact of adding an axion in this theory.

Figures (9)

• Figure 1: Examples of branches and vacua in SQCD with $N_f<N$ flavors. Left: $N=3,N_f=2,\theta=0$. Right: $N=6,N_f=2,\theta=0$. Lines denote the subspace of the low-energy meson theory parametrized by $\eta^\prime\equiv(\arg\det Q\bar{Q})^{1/N_f}$. For fixed $\eta^\prime$, there are $N-N_f$ branches of the gaugino bilinear ($\arg W_{ADS}$.) Wrapping around $\Delta\eta^\prime=2\pi$ smoothly connects branch $k$ to branch $k-N_f \mod (N-N_f)$, leading to $\gcd(N_f,N-N_f)$ sets of branches that are disconnected in the $\eta^\prime$ direction. $N$ vacua are denoted by purple dots and are located along the line $\arg W=\eta^\prime$. The distance between neighboring vacua in the $\eta^{\prime}$ direction is $2\pi(N-N_f)/N$.
• Figure 2: Cartoon of domain wall trajectories between nearest neighbor vacua in $N_f=2,N=4$. Arrows denote different possible wall trajectories in mixed $\eta^{\prime}$-pion directions (solid corresponds to the $\eta^{\prime}$ direction; dashed corresponds to motion "out of the page" in the $\pi_{-1}$ direction.) The $CP$ conjugate vacua cannot be connected by motion purely in the $\eta^{\prime}$ direction.
• Figure 3: Phase and modulus profiles of an $SU(N_f)_V$-breaking BPS domain wall in $N=4, N_f=2$, corresponding to the green path in Fig. \ref{['fig:Nf2N4cartoon']}. In this case, since $\gcd(N,N_f)>1$, only $SU(N_f)_V$-breaking walls are present between the $CP$ vacua. For definiteness we have set $\Lambda/m=4$. In the left-hand panel, the orange (blue) curve corresponds to the $\pi_{-1}$ ($\eta^{\prime}$). In the right-hand panel, the orange (blue) curve corresponds to $\rho_1$ ($\rho_2$).
• Figure 4: Cartoon of the domain wall trajectory between $CP$ vacua for $N_f=3,N=5$. Arrows denote a wall trajectory in the $\eta^{\prime}$ direction, passing through an intermediate vacuum at $\eta^{\prime}=\pi$. (Mixed $\pi_{-1}-\eta^{\prime}$ trajectories not shown.)
• Figure 5: Phase and modulus profile of a domain wall in $N=5, N_f=3$, corresponding to the green trajectory in Fig. \ref{['fig:Nf3N5cartoon']}. In this case, since $\gcd(N_f,N-N_f)=1$, an $SU(N_f)_V$-preserving wall is present between the $CP$ vacua. However, it is not BPS, and exists only for finite SUSY-breaking. We have taken $\Lambda/m=4$ and $B=-1/50$.