Degenerate kinks and kink-instantons in two-dimensional scalar field models with $\mathcal{N}=1$ and $\mathcal{N}=2$ supersymmetry

TL;DR

The paper analyzes degenerate kink configurations in two-dimensional scalar field theories with ${\mathcal N}=1$ and ${\mathcal N}=2$ supersymmetry. By examining the MSTB model and an ADS-motivated ${\mathcal N}=(2,2)$ Wess-Zumino model, it demonstrates that kink mixing via kink-instantons occurs in the purely bosonic sector but is suppressed in the supersymmetric theories due to localized fermionic zero modes. The work derives bosonic instanton actions, analyzes zero modes, and reveals how perturbative and nonperturbative effects—such as anomalies in the superpotential and irregularities of the non-holomorphic superpotential—shape the spectrum and stability of kinks and sphalerons. These results illuminate the intricate interplay between solitons, instantons, and supersymmetry in low-dimensional systems and establish a framework for instanton calculus in kink backgrounds with nontrivial topology. The findings have broader implications for nonperturbative phenomena in SUSY QFTs and may inform analogous structures in higher-dimensional theories and condensed-mmatter analogs.

Abstract

Models with classically degenerate vacua often support quasiclassical configurations of nontrivial topology. In (0+1)-dimensional quantum mechanics with a double-well potential, for example, instantons induce mixing between the two perturbative ground states in the purely bosonic case, while in the supersymmetric version, the tunneling amplitude is suppressed. In this work, we investigate (1+1)-dimensional models featuring classically Bogomol'nyi-Prasad-Sommerfield saturated kinks with degenerate masses and identical topology. Recent studies suggest that such kinks may undergo mixing mediated by scalar-field instantons. We analyze this phenomenon in a supersymmetric framework and demonstrate that, whereas mixing indeed occurs in the bosonic theory, the presence of fermionic zero modes in the supersymmetric case leads to the vanishing of the transition amplitude. To illustrate these results, we examine two examples featuring Wess-Zumino models with two and four supercharges. The latter example is motivated by the Affleck-Dine-Seiberg superpotential. We also present a number of developments of instanton calculus in the case of instantons in kink backgrounds.

Figures (11)

• Figure 1: (Super)Potentials for the MSTB model for $\lambda = m = 1$. (\ref{['fig:MSTB_potential_V']}) Potential \ref{['MSTB_potential']} at $\kappa=0.8$. (\ref{['fig:MSTB_potential_W1']}) and (\ref{['fig:MSTB_potential_W2']}) Superpotential \ref{['MSTB_superpotential']} at $\kappa = 0.3$ (the same plot from two different points of view). The values of $\kappa$ are chosen so that the characteristic features are most prominent visually. The thin gray contour lines mark the height of $U$ or $\mathcal{W}$. The profiles of the BPS degenerate kinks \ref{['MSTB_kinks']} are shown by the thick solid lines, magenta and green respectively. The sphaleron profile \ref{['MSTB_sphaleron']} is shown by the thick dashed red line. On (\ref{['fig:MSTB_potential_W1']}) and (\ref{['fig:MSTB_potential_W2']}), the BPS (and semi-BPS) profiles --- the thick lines --- can be seen as trajectories of viscous honey trickling down on the superpotential landscape (different trajectories correspond to different infinitesimal initial speeds).
• Figure 2: Field space spanned by $\phi_a$, $a=1,2$. Vacua \ref{['MSTB_vacua']} are shown by two cyan dots. Kinks \ref{['MSTB_kinks']} are shown by the blue (upper) and red (lower) solid lines respectively (see also Eq. \ref{['ellipse_trajectory']}). The sphaleron \ref{['MSTB_sphaleron']} is shown by green line going through the center.
• Figure 3: Eigenvalues of the mass matrix \ref{['mm_1']} as functions of bosonic fields $\phi_1,\phi_2$. For this plot, $\lambda = m = 1$ and $\kappa=0.9$. (\ref{['fig:MSTB_potential_V']}) First eigenvalue $\lambda_{\mathcal{M}}^{(1)}$. (\ref{['fig:MSTB_potential_W1']}) Second eigenvalue $\lambda_{\mathcal{M}}^{(2)}$. Labeling of the eigenvalues is the same as in Eq. \ref{['mass_matrix_singularity']}. Trajectories of the degenerate kinks \ref{['MSTB_kinks']} are shown by solid lines, magenta and green respectively. Trajectory of the sphaleron \ref{['MSTB_sphaleron']} is shown as a dashed red line. It is seen that the first eigenvalue $\lambda_{\mathcal{M}}^{(1)}$ is negative in both vacua, while the second eigenvalue $\lambda_{\mathcal{M}}^{(2)}$ changes sign along the soliton trajectories. Moreover, symmetry w.r.t. $\phi_2 \to -\phi_2$ is also evident.
• Figure 4: The first and the second critical values of the coupling, Eqs. \ref{['lambda_crit_1']} and \ref{['lambda_crit_2']}. The quantum correction \ref{['W_anom_MSTB']} does not spoil the theory as long as we stay below the solid line. Above this line, one of the vacua becomes lifted. Above the dashed line, both vacua are lifted.
• Figure 5: Quantum-corrected (super)potentials as functions of $\phi_1$ (slice $\phi_2=0$). (\ref{['fig:phi4_W_kappa0']}) shows the first derivative of the superpotential Eq. \ref{['W_quantum_corrected']} w.r.t. $\phi_1$, which is proportional to Eq. \ref{['quartic_vac_eq_1']}. (\ref{['fig:phi4_W_kappa10']}) shows the potential Eq. \ref{['V_quantum_corrected']}. For this plot, $\kappa = 0.4$. This figure shows the vicinity of one of the vacua, when the coupling is near the first critical value Eq. \ref{['lambda_crit_1']}. For the other vacuum and the coupling around Eq. \ref{['lambda_crit_2']}, the picture is similar.