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Higgs Mechanism with Type-II Nambu-Goldstone Bosons at Finite Chemical Potential

When the spontaneous symmetry breaking occurs for systems without Lorentz covariance, there arises possible mismatch, $N_{\rm NG} < N_{\rm BG}$, between numbers of Nambu-Goldstone (NG) bosons ($N_{\rm NG}$) and the numbers of broken generators ($N_{\rm BG}$. In such a situation, so-called type-II NG bosons emerge. We study how the gauge bosons acquire masses through the Higgs mechanism under this mismatch by employing gauge theories with complex scalar field at finite chemical potential and by enforcing "charge" neutrality. To separate the physical spectra from unphysical ones, the $R_ξ$ gauge is adopted. Not only massless NG bosons but also massive scalar bosons generated by the chemical potential are absorbed into spatial components of the gauge bosons. Although the chemical potential induces a non-trivial mixings among the scalar bosons and temporal components of the gauge bosons, it does not affect the structure of the physical spectra, so that the total number of physical modes is not modified even for $N_{\rm NG} < N_{\rm BG}$.