Dynamical Symmetry Breaking in Supersymmetric SU(n_c) and USp(2n_c) Gauge Theories

TL;DR

This work analyzes how N=1 vacua in SU(n_c) and USp(2n_c) gauge theories dynamically break flavor symmetries, by perturbing exact N=2 theories with a small adjoint mass mu and quark masses. The authors connect semiclassical vacua, large-mu effective theories, and Seiberg–Witten curve singularities to construct a cohesive microscopic picture: confinement and flavor breaking arise via monopole condensation for r=1 and via magnetic-quark condensation for larger r, while certain vacua remain in a free magnetic or conformal phase for large flavor numbers. The results yield precise vacuum counting and symmetry-breaking patterns across regimes, including two distinct groups of vacua (finite-VEV and vanishing-VEV) at large mu, and a rich microscopic structure at small mu involving nonlocal conformal points. The work provides a unified framework tying together semiclassical analysis, exact curve techniques, and effective Lagrangian descriptions, with implications for confinement mechanisms and the realization of global symmetries in supersymmetric gauge theories.

Abstract

We find the phase and flavor symmetry breaking pattern of each N=1 supersymmetric vacuum of SU(n_c) and USp(2 n_c) gauge theories, constructed from the exactly solvable N=2 theories by perturbing them with small adjoint and generic bare hypermultiplet (quark) masses. In SU(n_c) theories with n_f \leq n_c the vacua are labelled by an integer r, in which the flavor U(n_f) symmetry is dynamically broken to U(r) \times U(n_f-r) in the limit of vanishing bare hyperquark masses. In the r=1 vacua the dynamical symmetry breaking is caused by the condensation of magnetic monopoles in the n_f representation. For general r, however, the monopoles in the {}_{n_f}C_r representation, whose condensation could explain the flavor symmetry breaking but would produce too-many Nambu--Goldstone multiplets, actually "break up" into "magnetic quarks": the latter with nonabelian interactions condense and induce confinement and dynamical symmetry breaking. In USp(2n_c) theories with n_f \leq n_c + 1, the flavor SO(2n_f) symmetry is dynamically broken to U(n_f), but with no description in terms of a weakly coupled local field theory. In both SU(n_c) and USp(2 n_c) theories, with larger numbers of quark flavors, besides the vacua with these properties, there exist also vacua in free magnetic phase, with unbroken global symmetry.

Figures (12)

• Figure 1: Five vacua of the $SU(3)$, $N=1$ theory with $n_{f}=1$ flavors, plotted as the projection $(\hbox{\rm Re} \, u, \hbox{\rm Im} \, u, \hbox{\rm Re} \, v)$ of the QMS. ($\Lambda_{1} = 2$, $m= 1/64$, $u\equiv {1\over 2} \hbox{\rm Tr} \langle \Phi^{2} \rangle$, $v\equiv {1\over 3} \hbox{\rm Tr} \langle \Phi^{3} \rangle$).
• Figure 2: In the left figure are the eight vacua of the $SU(3)$ theory with $n_{f}=2$, plotted as the projection $(\hbox{\rm Re} \, u, \hbox{\rm Im} \, u, \hbox{\rm Re} \, v)$ of the QMS. ($\Lambda_{2} = 2, \, m_{1}= 1/64$, $m_{2}= i/64$). The same in the right figure with equal masses $m_{1}=m_{2}=1/64$.
• Figure 3: Twelve vacua of the $SU(3)$ theory with $n_{f}=3$ in the projection $(\hbox{\rm Re} \, u, \hbox{\rm Im} \, u, \hbox{\rm Re} \, v)$. $\Lambda_{3} = 2, \, m_{1}= 1/64, \, m_{2}= i/64$, $m_{3} = - i/64$. The same projection in the right with equal masses: $m_i = 1/64$.
• Figure 4: The seventeen vacua of the $SU(3)$ theory with $n_{f}=4$ in the $(\hbox{\rm Re} \, u, \hbox{\rm Im} \, u, \hbox{\rm Re} \, v)$ projection. $\Lambda_{4} = 2, \, m_{1}= 1/64, \, m_{2}= - 1/64, \, m_{3}= i/64$, $m_{4}=-i/64$. On the right, the same plot in the equal masses case with $m_{i}= 1/64$.
• Figure 5: Five vacua of the $USp(4)$ with $n_{f}=1$ flavors, plotted in $(\hbox{\rm Re} u, \hbox{\rm Im} u, \hbox{\rm Re} v)$ projection. $\Lambda_{1} = 2^{1/5}$, $m = 1/64$, while $u=\phi_1^2 + \phi_2^2$ and $v=\phi_1^2 \phi_2^2$.