No room for minimal monopole dark matter

The magnetic monopole of a dark sector has been advocated as an appealing dark matter candidate. We revisit the computation of the monopole abundance <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mi mathvariant="normal">Ω</a:mi> <a:mi>M</a:mi> </a:msub> </a:math> , generated by a thermal phase transition in the minimal ’t Hooft-Polyakov model. We explore the three regimes where the phase transition is second order, weakly first order, or supercooled, identifying the parameter space regions where <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:msub> <d:mi mathvariant="normal">Ω</d:mi> <d:mi>M</d:mi> </d:msub> </d:math> can match the observed dark matter abundance. However, the dark sector necessarily contains a stable electrically charged particle, namely a massive vector boson, with a calculable abundance <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:msub> <g:mi mathvariant="normal">Ω</g:mi> <g:msup> <g:mi>W</g:mi> <g:mo>′</g:mo> </g:msup> </g:msub> </g:math> . We show that, under minimal assumptions, <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"> <j:msub> <j:mi mathvariant="normal">Ω</j:mi> <j:msup> <j:mi>W</j:mi> <j:mo>′</j:mo> </j:msup> </j:msub> </j:math> is always far larger than <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:msub> <m:mi mathvariant="normal">Ω</m:mi> <m:mi>M</m:mi> </m:msub> </m:math> : dark monopoles cannot constitute a sizeable fraction of dark matter.