 Monopole core and mass scales in an SU(5) transition

When a simply connected $SU(5)$ gauge group undergoes spontaneous symmetry breaking to the connected physical Standard Model subgroupthe vacuum manifold is $G/H$. The long exact sequence of homotopy groups of a fibration gives ${\pi }_2(G/H)\cong {\pi }_1(H)\cong \mathbb{Z}$. Thus the theory admits sectors with nonzero integer magnetic monopole topological charge; a charged configuration cannot relax continuously to the vacuum while its asymptotic field remains on the vacuum manifold. This classifies permitted stable defects. Their actual formation and abundance also depend on the transition and subsequent dynamics, as in the Kibble mechanism.^[ref]

In natural units, let $\eta$ be the magnitude setting the broken-symmetry Higgs field, $g$ its gauge coupling and $\lambda$ a characteristic dimensionless coefficient of the Higgs field potential. Up to normalization-dependent constants, the massive gauge bosons and radial Higgs field excitation have masses $g\eta$ and $\sqrt{\lambda }\eta$. Their inverse masses set the gauge and scalar core radii, respectively:Rescaling a static classical magnetic monopole profile by $g\eta$ gives its energy in the form $M_M=(4\pi \eta /g)f$ for a dimensionless profile factor $f$; its value depends on dimensionless ratios of scalar couplings to $g^2$, the Higgs field potential and the charge sector. Weak gauge coupling therefore makes these defects much heavier than the associated massive gauge bosons, rather than assigning them the elementary-particle scale $g\eta$.^[ref]

// Target: cosmology.bigb