A Closer Look at Constrained Instantons

Abstract

Instantons play a crucial role in understanding non-perturbative dynamics in quantum field theories, including those with spontaneously broken gauge symmetries. In the broken phase, finite-size instanton-like configurations are no longer exact stationary points of the Euclidean action, in contrast to the symmetric phase. Non-perturbative effects in this setting are therefore typically studied within the constrained instanton framework. However, a previous study pointed out a possible difficulty in constructing consistent constrained instanton solutions based on conventional gauge-invariant constraints. In this work, we revisit the asymptotic structure of constrained instantons and re-examine the claimed difficulty. By carefully tracking the behavior of the solutions near the spatial origin and at infinity, we show that the required boundary conditions can be satisfied without encountering the inconsistency. We explicitly construct consistent constrained instantons in both massive $φ^4$ theory and Yang--Mills theory with spontaneous symmetry breaking, and we support our analytic matching procedure with numerical solutions. Our results establish that conventional gauge-invariant constraints can be consistently employed in semiclassical computations when asymptotic expansions are treated properly.

Figures (9)

• Figure 1: (Left) Particle-mechanics interpretation of the radial equation of motion in the inverted potential $-V(\phi)$. With initial conditions $\phi(0)$ arbitrary and $\phi'(0)=0$, the trajectory monotonically approaches $\phi\to 0$ as $r\to\infty$ in the massless theory, corresponding to the instanton solution. (Right) Adding a mass term eliminates the finite-action instanton solution. For any arbitrary initial value $\phi(0)$ with $\phi'(0)=0$, the motion results in damped oscillations.
• Figure 2: (Left) Effect of the positive $\phi^6$ constraint term in the particle-mechanics picture for $m^2=0$: the solution overshoots, and the finite-action instanton solution disappears. (Right) Adding both a mass term and a positive $\phi^6$ term restores a finite-action constrained instanton solution.
• Figure 3: Schematic illustration of the matching procedure. We extend the inner solution to $r\ll m^{-1}$ and the outer solution to $r\gg \rho$. The inner and outer solutions are matched in the overlap region $\rho\ll r\ll m^{-1}$.
• Figure 4: Illustration of the $(n,k)$ matching conditions between the coefficients $f_n^{(k)}$ and $g_{n+k-2}^{(k)}$. The coefficient $f_n^{(k)}$ vanishes for $k<2-n$. The dashed lines indicate the orders appearing in the outer expansion in Eq. \ref{['eq: outer solution perturbed']}, while the horizontal axis represents the order $n$ in the inner expansion in Eq. \ref{['eq: phi perturbative series']}. The $(0,2)$ condition corresponds to the LO matching condition, while $(0,4)$, $(2,0)$, and $(2,2)$ correspond to the NLO matching conditions.
• Figure 5: Dimensionless Lagrange multiplier $m^2\sigma$ as a function of the instanton size $\rho m$. The numerical result is compared with the analytic NLO prediction in Eq. \ref{['eq:sigma2']}, showing agreement for sufficiently small $\rho m$.