Approximations for strongly-coupled supersymmetric quantum mechanics

Abstract

We advocate a set of approximations for studying the finite temperature behavior of strongly-coupled theories in 0+1 dimensions. The approximation consists of expanding about a Gaussian action, with the width of the Gaussian determined by a set of gap equations. The approximation can be applied to supersymmetric systems, provided that the gap equations are formulated in superspace. It can be applied to large-N theories, by keeping just the planar contribution to the gap equations. We analyze several models of scalar supersymmetric quantum mechanics, and show that the Gaussian approximation correctly distinguishes between a moduli space, mass gap, and supersymmetry breaking at strong coupling. Then we apply the approximation to a bosonic large-N gauge theory, and argue that a Gross-Witten transition separates the weak-coupling and strong-coupling regimes. A similar transition should occur in a generic large-N gauge theory, in particular in 0-brane quantum mechanics.

Figures (7)

• Figure 1: The gap equation for the $\phi$ propagator. Heavy lines are the dressed propagator $\sigma^2$ and thin lines are the bare propagator $g^2$.
• Figure 2: One-loop gap equations. Heavy lines are dressed propagators and thin lines are bare propagators.
• Figure 3: Schwinger-Dyson equations for a theory with 3-point and 4-point couplings but no tadpoles. The solid blobs are dressed propagators; the empty circles are 1PI vertices. All external lines are amputated.
• Figure 4: $\beta F$ vs. $\beta$ for the $\Phi_1 \Phi_2 \Phi_3$ model, as given by the sum of the first three terms of the reorganized perturbation series. Numerical calculations were performed at the indicated points.
• Figure 5: Plots of $(-)$ individual terms in the free energy vs. $\beta$ in the low temperature regime. The scale is $\log$--$\log$, and we shifted $\beta F_0$ by a constant to display its power-law behavior. The top curve is $-<S - S_0>_0$, the middle curve is $+{1 \over 2}<(S - S_0)^2>_{{\hbox{C}},0}$, and the bottom curve is $-(\beta F_0 + 1.04)$.