Electroweak sphaleron revisited. II. Study of decay dynamics

We present a comprehensive analysis of electroweak sphaleron decay dynamics, employing both analytical techniques and high-resolution numerical simulations. Using a spherically symmetric ansatz, we reformulate the system as a ( <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mn>1</a:mn> <a:mo>+</a:mo> <a:mn>1</a:mn> </a:mrow> </a:math> )-dimensional problem and analyze its stability properties with current Standard Model parameters ( <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msub> <c:mi>m</c:mi> <c:mi>H</c:mi> </c:msub> <c:mo>=</c:mo> <c:mn>125.1</c:mn> <c:mtext> </c:mtext> <c:mtext> </c:mtext> <c:mi>GeV</c:mi> </c:math> , <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msub> <e:mi>m</e:mi> <e:mi>W</e:mi> </e:msub> <e:mo>=</e:mo> <e:mn>80.4</e:mn> <e:mtext> </e:mtext> <e:mtext> </e:mtext> <e:mi>GeV</e:mi> </e:math> ). We identify precisely one unstable mode with eigenvalue <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:msubsup> <g:mi>ω</g:mi> <g:mo>−</g:mo> <g:mn>2</g:mn> </g:msubsup> <g:mo>≃</g:mo> <g:mo>−</g:mo> <g:mn>2.7</g:mn> <g:msubsup> <g:mi>m</g:mi> <g:mi>W</g:mi> <g:mn>2</g:mn> </g:msubsup> </g:math> and numerically evolve the full nonlinear field equations under various initial conditions. Through spectral decomposition, we quantify the particle production resulting from the sphaleron decay. Our results demonstrate that the decay process is dominated by transverse gauge bosons, which constitute approximately 80% of the total energy and multiplicity, while Higgs bosons account for only 7%–8%. On average, the sphaleron decays into <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mrow> <i:mn>49</i:mn> <i:mi>W</i:mi> </i:mrow> </i:math> bosons and 4 Higgs bosons. The particle spectra consistently peak at momenta <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi>k</k:mi> <k:mo>∼</k:mo> <k:mn>1</k:mn> <k:mi>–</k:mi> <k:mn>1.5</k:mn> <k:msub> <k:mi>m</k:mi> <k:mi>W</k:mi> </k:msub> </k:math> , reflecting the characteristic size of the sphaleron. Remarkably, these properties remain robust across different decay scenarios, suggesting that the fundamental structure of the sphaleron, rather than specific triggering mechanisms, determines the decay outcomes. These findings provide distinctive experimental signatures of nonperturbative topological transitions in the electroweak theory, with significant implications for baryon number violation in the early universe and potentially for high-energy collider physics.