Non-Abelian Confinement via Abelian Flux Tubes in Softly Broken N=2 SUSY QCD

TL;DR

The paper investigates confinement in softly broken $\mathcal{N}=2$ SUSY QCD with gauge group $SU(N_c)$ and $N_f$ flavors, concentrating on $SU(3)$ and weak coupling vacua. It derives the vacuum structure from a deformed superpotential and analyzes the low-energy Abelian theory, revealing two $U(1)$ factors and two light photon-axion pairs, with BPS flux tubes extending the Abrikosov-Nielsen-Olesen construction to a two-abelian setting. Flux tubes in ${\tt r}=1$ vacua are standard ANO strings, while ${\tt r}=2$ vacua support $(n,k)$-strings with two gauge fields; their tensions and stability are computed, forming a lattice tied to the $SU(3)$ root system. In theories with $N_f=4,5$ and equal quark masses, $SU(2)$ is non-perturbatively restored, which removes the ${\bf e}_0$-string and merges the ${\bf e}_1$ and ${\bf e}_2$ strings, yielding a meson spectrum with a single non-Abelian confinement trajectory, thereby linking Abelian flux tubes to non-Abelian confinement. The results illuminate how non-Abelian confinement can emerge from Abelian flux tubes and clarify the role of Higgs branches, monopole condensation, and flux-tube lattices across different vacua.

Abstract

We study confinement in softly broken N=2 SUSY QCD with gauge group SU(N_c) and N_f hypermultiplets of fundamental matter (quarks) when the Coulomb branch is lifted by small mass of adjoint matter. Concentrating mostly on the theory with SU(3) gauge group we discuss the N=1 vacua which arise in the weak coupling at large values of quark masses and study flux tubes and monopole confinement in these vacua. In particular we find the BPS strings in SU(3) gauge theory formed by two interacting U(1) gauge fields and two scalar fields generalizing ordinary Abrikosov-Nielsen-Olesen vortices. Then we focus on the SU(3) gauge theories with N_f=4 and N_f=5 flavors with equal masses. In these theories there are N=1 vacua with restored SU(2) gauge subgroup in quantum theory since SU(2) subsectors are not asymptotically free. We show that although the confinement in these theories is due to Abelian flux tubes the multiplicity of meson spectrum is the same as expected in a theory with non-Abelian confinement.

Figures (5)

• Figure 1: Root and weight vectors in the Cartan plane for $SU(3)$ group. We have depicted explicitly the root vectors ${\hbox{\boldmath $\alpha$}}_1$ and ${\hbox{\boldmath $\alpha$}}_2$ (the simple roots), the highest root ${\hbox{\boldmath $\alpha$}}_{12}={\hbox{\boldmath $\alpha$}}_1+{\hbox{\boldmath $\alpha$}}_2$ and the weights of the fundamental representation ${\hbox{\boldmath $\mu$}}_1\equiv{\hbox{\boldmath $\mu$}}_u$, ${\hbox{\boldmath $\mu$}}_2\equiv{\hbox{\boldmath $\mu$}}_d$ and ${\hbox{\boldmath $\mu$}}_3\equiv{\hbox{\boldmath $\mu$}}_s$, corresponding to $u$, $d$ and $s$ quarks respectively. We have also depicted the "normalized" roots (of unit length) ${\bf e}_1 = {\hbox{\boldmath $\alpha$}}_{12}/\sqrt{2}$ and ${\bf e}_2 = {\hbox{\boldmath $\alpha$}}_{2}/\sqrt{2}$ as well as normalized weights ${\bf u}$, ${\bf d}$ and ${\bf s}$ of the length $1/\sqrt{3}$.
• Figure 2: Strings and different phases in ${\tt r}=1$ vacuum. Black squares correspond to the monopole and quark states in confinement phase while white squares -- to the states in Higgs phase. Vectors ${\bf q}_u$ and ${\bf q}_2 = {\bf e}_2/2$ label the (charges of) magnetic and electric strings correspondingly.
• Figure 3: Lattice of string solutions in ${\tt r}=2$ vacuum. We have specified explicitly $(1,0)={\bf e}_1$, $(0,1)={\bf e}_2$, $(1,-1)={\bf e}_0$, $(1,1)={\bf e}_1+{\bf e}_2$ and $(2,-1)\equiv 2{\bf u}$ strings.
• Figure 4: Lattice of string solutions for $0<\omega<1$. The non-BPS strings are drawn by white dots while the BPS strings are depicted by black dots. All BPS strings are marginally stable bound states of "fundamental" ${\bf e}_1$- and ${\bf e}_2$-strings, corresponding to two "closest" roots of $SU(3)$ algebra. The non-BPS strings can be "crossed" by straight line $n+\omega k =0$ or ${\bf e}(\omega)$ when one moves parameter $\omega$ in the region $0<\omega<\infty$.
• Figure 5: Different values of the parameter $\omega$. Values $0<\omega<1$ are restricted by straight lines $\omega=0$ along the vector ${\bf e}_2$ and $\omega=1$ along ${\bf e}_0$, values $\omega>1$ are between the lines $\omega=1$ along ${\bf e}_0$ and $\omega=\infty$, which is along the vector ${\bf e}_1$.