Chiral non-Abelian domain walls in the Ginzburg-Landau theory

TL;DR

The paper investigates chiral non-Abelian domain walls in a Ginzburg-Landau theory for dense QCD CFL matter by treating the GL couplings as free parameters and exploring an unconventional sector with asymmetric left/right condensates. In a strong-coupling sigma-model limit, the dynamics reduce to sine-Gordon-type equations, yielding a domain wall between $(voldsymbol{1}_3,0)$ and $(0,voldsymbol{1}_3)$ with a kink-free solution governed by a sine-Gordon profile; a nonexistence result for domain walls with a nonzero Josephson coupling is proven, while kink solitons and analytic solutions appear at a fine-tuned point. The authors complement the analytic results with full numerical solutions of the matrix equations of motion, finding good agreement with the sigma-model predictions for large $oldsymbol{ extlambda}_1$ in most cases, and identifying notable exceptions in the Josephson-kink regime. The work elucidates the structure of non-Abelian domain walls in CFL-like GL theories and suggests connections to non-Abelian vortices and D-brane solitons in QCD-like systems, offering a framework for exploring exotic vacua beyond perturbative QCD.

Abstract

In this paper, we study chiral non-Abelian domain walls in a phase of unconventional vacua of the Ginzburg-Landau model for dense QCD, by considering a wider range of parameters space not directly deduced from QCD. The phase is characterized by asymmetric vacuum-expectation values (VEVs), for example with the left scalar field, corresponding to the left quark-quark condensate, having a nonvanishing VEV and the right field having a vanishing one. The domain wall soliton interpolates between this vacuum and another where the left and right scalar fields switch roles. We study this formal possibility, but not any mechanism to generate these vacua non-perturbatively at finite density or finite temperature. Using a strong-coupling, or sigma-model limit, we are able to reduce the full dynamical complex matrix valued equations of motion to the sine-Gordon, a generalization of the sine-Gordon and a generalization of the double sine-Gordon equations. In this limit, we prove nonexistence of domain walls in one of the vacua studied here and we find full numerical computations to converge to the sigma-model limit for many cases, with some exceptions that we discuss.

Figures (9)

• Figure 1: DW interpolating the $(\Phi_{\rm L},\Phi_{\rm R})=(v\mathds{1}_3,0)$ and the $(\Phi_{\rm L},\Phi_{\rm R})=(0,v\mathds{1}_3)$ ground states. Increasing the DW profile angle $\theta$ further would interpolate to "parity siblings" of the first two ground states.
• Figure 2: The kink soliton interpolates between the vacuum at $\theta=0$ and the sign-flipped vacuum at $\theta=\pi$. It is fundamentally different from the DW solution that swaps the two vacua (up to an overall sign for negative $\gamma_1$), which would correspond to a solution that interpolates between the ground state at $\theta=0$ and the would-be ground state at $\theta=\frac{\pi}{2}$. Instead, the DW solution without the correct signs at $\frac{\pi}{2}$, i.e. a solution that goes to $\Phi_{\rm L}=-w_-\mathds{1}_3$, $\Phi_{\rm R}=-\mathop{\mathrm{sign}}\nolimits(\gamma_1)w_+\mathds{1}_3$ exists, but that is not a ground state. The ground states in the figure at $\theta=\frac{\pi}{2}$ and $\frac{3\pi}{2}$ are thus marked with $\times$, whereas the true ground states that are reachable by the kink are marked with gray-filled circles.
• Figure 3: The kink soliton interpolates between the vacuum at $\theta=0$ and the sign-flipped vacuum at $\theta=\pi$, at the fine-tuned point in the theory: The two nonvanishing VEVs of $|\Phi_{\rm L}|=|\Phi_{\rm R}|$ equal each other up to a sign in this fine-tuned ground state. Like the kink soliton of fig. \ref{['fig:Josephson_vacua']}, there is no solution interpolating between $\theta=0$ and $\theta=\frac{\pi}{2}$, but there is a solution going to $\Phi_{\rm L}=-\frac{v}{\sqrt{2}}\mathds{1}_3$, $\Phi_{\rm R}=-\mathop{\mathrm{sign}}\nolimits(\gamma_1)\frac{v}{\sqrt{2}}\mathds{1}_3$, which however is not a ground state. The would-be ground states in the figure at $\theta=\frac{\pi}{2}$ and $\frac{3\pi}{2}$ are thus marked with $\times$.
• Figure 4: DWs in the chirally symmetric but asymmetric ground states interpolating between $(\Phi_{\rm L},\Phi_{\rm R})=(v\mathds{1}_3,0)$ and $(\Phi_{\rm L},\Phi_{\rm R})=(0,v\mathds{1}_3)$ (see fig. \ref{['fig:DW_vacua']}), for a variety of couplings $\lambda_1=1,4,9,49,99$: (a) the diagonal part (all three elements are equal) of the scalar fields $\Phi_{{\rm L},{\rm R}}$, (b) the sigma-model constraint \ref{['eq:sigma_model_constraint']}. We compare the full computations (solid lines) with the sigma-model limit (dashed lines) and take $\lambda_4=\lambda_1+1$ so that $\lambda_4-\lambda_1=1>0$, but $\lambda_1$ is increased (the sigma-model limit corresponds to $\lambda_1\to\infty$). (c) Total energy (tension) of the DW with the solid line displaying the sigma-model limit result \ref{['eq:Esigmamodel_lambda']} and the points showing the energies of full computations. In this figure $m=\sqrt{2}$, $\lambda_{2,3}=0$, $\lambda_4=\lambda_1+1$, $\gamma_{1,2,3}=0$.
• Figure 5: DWs in the chirally broken and asymmetric ground states interpolating between $(\Phi_{\rm L},\Phi_{\rm R})=(v\mathds{1}_3,0)$ and $(\Phi_{\rm L},\Phi_{\rm R})=(0,v\mathds{1}_3)$, for a variety of couplings $\lambda_1=1,4,9,49,99$: (a) the diagonal part (all three elements are equal) of the scalar fields $\Phi_{{\rm L},{\rm R}}$, (b) the sigma-model constraint \ref{['eq:sigma_model_constraint']}, (c) the scalar fields with $\gamma_3=\tfrac{1}{4}$ minus the corresponding $\gamma_3=0$ solution ($\Phi_{{\rm L},{\rm R}}^{\gamma_3=1/4}-\Phi_{{\rm L},{\rm R}}^{\gamma_3=0}$), (d) the sigma model constraint \ref{['eq:sigma_model_constraint']} computed for the difference of fields displayed in panel (c). We compare the full computations (solid lines) with the sigma-model limit (dashed lines). (e) The sigma-model limit result $\theta$. (f) Total energy (tension) of the DW with the solid line displaying the sigma-model limit and the points showing the energies of full computations. In this figure $m=\sqrt{2}$, $\lambda_{2,3}=0$, $\lambda_4=\lambda_1+1$, $\gamma_{1,2}=0$ and $\gamma_3=\tfrac{1}{4}$.

Theorems & Definitions (3)

• Theorem 1
• Lemma 2
• Corollary 3