Classical Upper Bounds for Grand-Unified Monopole Masses

In addition to being bounded from below by the Prasad-Sommerfiled limit, monopole masses are also bounded from above by the limit in which the scalar field variables are frozen to their vacuum values. This upper bound is close to the lower one: The single-, double-, and triple-strength SU(5) monopoles are found to have their masses bound in $m<~M<~m\times 1.7867$, $2m<~M<~2m\times 2.0741$, and $3m<~M<~3m\times 2.3155$, respectively, where $m=\frac{3M_x}{8\alpha }$.