Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Resurgent structure of the 't Hooft-Polyakov monopole

Michal Malinský

Abstract

In this letter we present a comprehensive analysis of the differential equations governing the spatial profile of the 't Hooft-Polyakov monopole from the viewpoint of resurgence theory. We note that the special shape of the gauge-component asymptotics, together with the simplicity of its Borel transform and the Green's kernels in the Volterra equations, provides a complete control over the proliferation of the Borel-plane singularities to all orders and for any $λ/e^2>0$, along with full information about the relevant logarithmic discontinuities. Hence, the resurgent structure of the non-BPS solutions turns out to be unexpectedly simple, and its resonance at the origin of the Borel plane even admits an analytic calculation of the normalization coefficients of their asymptotic forms to an arbitrary level of precision.

Resurgent structure of the 't Hooft-Polyakov monopole

Abstract

In this letter we present a comprehensive analysis of the differential equations governing the spatial profile of the 't Hooft-Polyakov monopole from the viewpoint of resurgence theory. We note that the special shape of the gauge-component asymptotics, together with the simplicity of its Borel transform and the Green's kernels in the Volterra equations, provides a complete control over the proliferation of the Borel-plane singularities to all orders and for any , along with full information about the relevant logarithmic discontinuities. Hence, the resurgent structure of the non-BPS solutions turns out to be unexpectedly simple, and its resonance at the origin of the Borel plane even admits an analytic calculation of the normalization coefficients of their asymptotic forms to an arbitrary level of precision.
Paper Structure (12 sections, 27 equations, 1 figure)

This paper contains 12 sections, 27 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The MNBPS monopole ($\beta\to\infty$)
  3. Proliferation of (leading) singularities
  4. The general case with $\beta >0$
  5. The BPS recess
  6. Few practical remarks
  7. Conclusions
  8. Universal $\beta>0$ gauge asymptotics
  9. The Borel-plane structure of $y_a(x)$
  10. The MNBPS $y(x)$ trans-series at $x\to\infty$
  11. The MNBPS $f(x)$ trans-series at $x=0$
  12. Borel-plane power-log towers at $t=0$

Figures (1)

  • Figure 1: Ratios of the theoretical asymptotic gauge profiles of Eq. (\ref{['eq:yasymptotics']}) to the corresponding true numerical solutions $y_\beta(x)$ for $\beta\to\infty$ (the lower curve) and $\beta=1$ (the upper one). Note that both $y_{\infty}$ and $y_{1}$ assymptotically behave like $\sqrt{x}K_\nu(x)$, albeit with different normalization constants $A_{\infty}$ and $A_{1}$.