Beyond Coleman's Instantons

Abstract

In the absence of gravity, Coleman's theorem states that the $O(4)$-symmetric instanton solution, which is regular at the origin and exponentially decays at infinity, gives the lowest action. Perturbatively, this implies that any small deformation from $O(4)$-symmetry gives a larger action. In this letter we investigate the possibility of extending this theorem to the situation where the $O(4)$-symmetric instanton is singular, provided that the action is finite. In particular, we show a general form of the potential around the origin, which realizes a singular instanton with finite action. We then discuss a concrete example in which this situation is realized, and analyze non-trivial anisotropic deformations around the solution perturbatively. Intriguingly, in contrast to the case of Coleman's instantons, we find that there exists a deformed solution that has the same action as the one for the $O(4)$-symmetric solution up to the second order in perturbation. Our result implies that there exist non-$O(4)$-symmetric solutions with finite action beyond Coleman's instantons, and gives rise to the possibility of the existence of a non-$O(4)$-symmetric instanton with a lower action.

Figures (3)

• Figure 1: $V(\Phi)/(\alpha\Phi_\star)^4$ in (\ref{['eq:piecewise_potential']}) as a function of $\Phi$ with $m_1 = 0.98\Phi_\star$, $m_2 = 7.11 \Phi_\star$ and $\Phi_{\rm M} = -2.35\Phi_\star$. These values of parameters are chosen such that the matching conditions for Eqs. (\ref{['eq:bg_exp']})--(\ref{['eq:sol_phi_0_V2']}) are satisfied. The solid red line represents the exponential potential, while the blue dashed and the green dot-dashed lines refer to the potentials $V_1$ and $V_2$ in Eq. (\ref{['eq:piecewise_potential']}) respectively. The two black points in the plot refer to the two matching locations at $\Phi = 0$ and at $\Phi = \Phi_2 = -2.34\Phi_\star$. The false vacuum for this particular choice of parameters is located at $\Phi_{\rm M} = -2.35\Phi_\star$.
• Figure 2: Parameter space of $m_1/\Phi_\star$ and $m_2/\Phi_\star$. The condition (\ref{['eq:con_bg']}) is satisfied in the yellow region. The green line refers to the condition $\bar{\Phi}(\tilde{\rho}_2) = \Phi_2$, the red curve corresponds to the condition $c_8 = 0$. We choose $\tilde{\rho}_2 = 5$ and $\alpha = 0.5$. The black dot is $\{0.98,7.11\}$, for which the fluctuation is regular everywhere (their corresponding $\Phi_2$ and $\Phi_{\rm M}$ are respectively given by $\Phi_2 = -2.34\Phi_\star, \Phi_{\rm M} = -2.35\Phi_\star$). The small figure explicitly shows the intersection point.
• Figure 3: The solutions for $\bar{\Phi}$ (solid line) and $f_2(\tilde{\rho})$ (dashed line). The red, blue and green colors refer to the solutions in the exponential potential, $V_1$ and $V_2$ regimes, respectively. We set $\alpha = 0.5$, $m_1 = 0.98\Phi_\star$ and $m_2 = 7.11\Phi_\star$, so that $\Phi_2 = -2.34\Phi_\star$ and $\Phi_{\rm M} = -2.35\Phi_\star$. The two matching points (black dots) are located at $\tilde{\rho} = 1$ and $\tilde{\rho}_2 = 5$.