Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Backreacted Coulomb energy in the Skyrme model

Sven Bjarke Gudnason, James Martin Speight

TL;DR

This work extends the Skyrme model by including a Maxwell term with a source term that enforces a Gell-Mumann-Nishijima–consistent coupling, and studies the backreaction of the electric field on static Skyrmions in the isospin-0 sector for $B=4,8,12,16,40$. The authors formulate the static energy in Skyrme units with two parameters $m$ and $\\kappa$, derive Gauss's law $\\Delta V = B^0$, and provide a gradient formula that includes a nonlocal Coulomb contribution. Through a calibration to Carbon-12, they obtain $m \\approx 0.650$ and $\\kappa \\approx 0.737$, achieving ground-state masses within $\\sim 1.86\%$ of data and Coulomb energies within 3–22% of phenomenological fits, with radii accurate to ~15%. The results show Coulomb backreaction is more significant for large Skyrmions, affecting dynamics more than ground states, and hint at rich structure for large nuclei when anomalies and gauged terms are further considered.

Abstract

The Skyrme model is extended with the Maxwell action and a source term for the gauge field. We consider the specialized case of vanishing isospin states, such that only an electric potential is turned on and study the backreaction onto the Skyrme fields. In particular, we study Skyrmions with baryon numbers B=4,8,12,16 and 40. We find, in agreement with physical expectations, that the Coulomb backreaction is most pronounced for large Skyrmions and find furthermore that the dynamics of the theory is more sensitive to the backreaction than the ground states (global minimizers of the energy). Calibrating the model to Carbon-12, we find excellent agreement of the masses of the studied Skyrmions - within 1.86% of experimental data. The Coulomb energies are slightly larger than phenomenological fits suggest, but only by about 3-22%, whereas the radii are within 15% errors, with the largest errors on the smallest baryon number (B=4) and the smallest errors on the large baryon numbers.

Backreacted Coulomb energy in the Skyrme model

TL;DR

This work extends the Skyrme model by including a Maxwell term with a source term that enforces a Gell-Mumann-Nishijima–consistent coupling, and studies the backreaction of the electric field on static Skyrmions in the isospin-0 sector for . The authors formulate the static energy in Skyrme units with two parameters and , derive Gauss's law , and provide a gradient formula that includes a nonlocal Coulomb contribution. Through a calibration to Carbon-12, they obtain and , achieving ground-state masses within of data and Coulomb energies within 3–22% of phenomenological fits, with radii accurate to ~15%. The results show Coulomb backreaction is more significant for large Skyrmions, affecting dynamics more than ground states, and hint at rich structure for large nuclei when anomalies and gauged terms are further considered.

Abstract

The Skyrme model is extended with the Maxwell action and a source term for the gauge field. We consider the specialized case of vanishing isospin states, such that only an electric potential is turned on and study the backreaction onto the Skyrme fields. In particular, we study Skyrmions with baryon numbers B=4,8,12,16 and 40. We find, in agreement with physical expectations, that the Coulomb backreaction is most pronounced for large Skyrmions and find furthermore that the dynamics of the theory is more sensitive to the backreaction than the ground states (global minimizers of the energy). Calibrating the model to Carbon-12, we find excellent agreement of the masses of the studied Skyrmions - within 1.86% of experimental data. The Coulomb energies are slightly larger than phenomenological fits suggest, but only by about 3-22%, whereas the radii are within 15% errors, with the largest errors on the smallest baryon number (B=4) and the smallest errors on the large baryon numbers.

Paper Structure

This paper contains 13 sections, 1 theorem, 74 equations, 50 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The Skyrme model with Coulomb energy
  3. The field theory
  4. Geometric approach to the variational problem
  5. Derrick scaling
  6. Topological energy bound
  7. Numerical algorithm
  8. Calibration
  9. Skyrmion solutions
  10. The Coulomb energy and the effect of its backreaction
  11. Conclusion and discussion
  12. Proof of Lemma \ref{['loclem']}
  13. The remaining B=40 solutions

Key Result

Lemma 1

Let $C>0$ and $\rho:\mathbb{R}^3\rightarrow\mathbb{R}$ be any smooth function such that $|\rho(x)|\leq Ce^{-|x|/C}$ for all $x$. Let $V:\mathbb{R}^3\rightarrow\mathbb{R}$ be the electrostatic potential induced by $\rho$. Then there exists $K>0$ such that, for all $x\in\mathbb{R}^3$,

Figures (50)

  • Figure 1: Dependence of the (a) radius, (b) total energy and (c) Coulomb energies on $\kappa$ in Skyrme units for the $B=4_a$ cube. The four curves correspond to $m=0.5, 1, 1.5, 2$, respectively.
  • Figure 2: Dependence of the (a) radius, (b) total energy and (c) Coulomb energies on $\kappa$ in Skyrme units for the $B=12_a$ chain. The four curves correspond to $m=0.5, 1, 1.5, 2$, respectively.
  • Figure 3: $B=12$ Skyrmion solutions in order of increasing mass (energy), for $\kappa=0$ and pion mass $m=1$. These figures are taken from ref. Gudnason:2022jkn
  • Figure 4: $B=12$ Skyrmion solutions as functions of the pion mass $m$ for $\kappa=0$. Since the energy grows drastically with $m$, we display the energies divided by their topological energy bound, see eq. \ref{['eq:Ebound']}. When a curve stops, the solution ceases to exist, but when a curve drops to another existing curve, the solution decays or transforms itself to that solution.
  • Figure 5: Stable $B=12$ Skyrmion solutions (global energy minimizers) for $\kappa=0$ and pion mass in the interval $m\in[0,1]$.
  • ...and 45 more figures

Theorems & Definitions (1)

  • Lemma 1