Non-BPS Monopoles and Dyons via Resurgent Transseries

TL;DR

The paper addresses non-BPS monopoles and dyons in SU(2) Yang–Mills–Higgs theory by formulating their far-field behavior as resurgent transseries. Using Costin’s transseries framework, the authors express solutions as infinite sums of exponentially decaying terms, each dressed by factorially divergent fluctuations; at the BPS limit ($\beta\to0$) these fluctuations truncate. For the $\beta=1$ monopole, they derive a leading pair of fluctuations $(w_1,h_1)$ obeying linear ODEs, with higher orders generated recursively by inhomogeneous equations whose sources are products of lower-order terms; the second-order corrections can be written in closed form via integrals against the leading homogeneous solutions. When generalized to arbitrary $\beta$, the transseries acquires a double-exponential structure with exponents $e^{-r}$ and $e^{-\beta r}$, while for dyons an additional electric-sector sector enters and renormalizes the fluctuation system through Whittaker-type functions. The results provide a systematic analytic bridge between far-field resurgent expansions and near-field numerics, with potential extensions to other solitons and quantum corrections.

Abstract

Radially symmetric non-BPS 't Hooft-Polyakov monopoles and dyons are constructed as resurgent transseries: infinite sums of exponentially decaying terms, each multiplied by a factorially divergent fluctuation factor. All higher exponential terms are explicitly expressed in terms of the leading order solutions. In the BPS limit all fluctuation terms truncate.

Figures (4)

• Figure 1: Numerical solutions generated from the origin. We use a combination of explicit fourth-order Runge-Kutta and linearly implicit Euler methods due to the stiffness of the differential equations \ref{['eq:Weqn']}-\ref{['eq:Heqn']}. $W(r)$ is given by the blue curve, and $H(r)$ by the orange curve. The asymptotic value of 1 is denoted by a dashed black line. These solutions are shown in the region $r \in (10^{-6},7).$
• Figure 2: $W(r)$ solutions for $\sigma_h=1.905$, $\sigma_w=3.336$. The numerical solution from $r=0$ is given by the blue solid curve. First transseries order contributions are given by the yellow dashed curve, with second and third transseries order contributions given by the green dash-dotted and red dotted curves respectively. The asymptotic value of 1 is denoted by the thinly dotted black line. Note the increased agreement as $r \to 0$ with the inclusion of additional terms in the transseries.
• Figure 3: $H(r)$ solutions for $\sigma_h=1.905$, $\sigma_w=3.336$. The numerical solution from $r=0$ is given by the blue solid curve. First transseries order contributions are given by the yellow dashed curve, with second and third transseries order contributions given by the green dash-dotted and red dotted curves respectively. The asymptotic value of 1 is denoted by the thinly dotted black line. Note the increased agreement as $r \to 0$ with the inclusion of additional terms in the transseries.
• Figure 4: Integrand of the monopole mass (\ref{['eq:mass']}) for $\beta = 1$. This is an intermediate curve between the $\beta \to 0$ and $\beta \to \infty$ plots presented in Figure 2 of zachos.