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If a global symmetry gets both spontaneously and explicitly broken, the explicit symmetry breaking pattern is crucial for understanding the formation of topological defects.

For example, in the axion solution to the strong CP problem, the explicit breaking of the Peccei-Quinn (PQ) symmetry due to non-perturbative effects, $U(1)_{\rm PQ}\rightarrow Z_N$, leads to the formation of domain walls. Independently, the spontaneous breaking of $U(1)_{\rm PQ}$ by the vacuum expectation value of the axion field yields cosmic strings.

In QCD, the flavor $SU(3)\times SU(3)$ symmetry gets explicitly broken down to $U(1)\times U(1)\times U(1)$ by the non-zero quark masses $m_u\neq m_d\neq m_s$. Independently, the quark condensate spontaneously breaks $SU(3)\times SU(3)\rightarrow SU(3)$, leading to the formation of skyrmions.

Now my question is: why doesn't the explicit symmetry breaking in QCD lead to the formation of global monopoles? Is that because the $U(1)\times U(1)\times U(1)$ is further broken by weak effects?

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