The U(1)_L_μ-L_τ breaking phase transition, muon g-2, dark matter, collider and gravitational wave

Combining the dark matter and muon g-2 anomaly, we study the U(1)_L_μ-L_τ breaking phase transition, gravitational wave spectra, and the direct detection at the LHC in an extra U(1)_L_μ-L_τ gauge symmetry extension of the standard model. The new fields includes vector-like leptons (E_1,  E_2,  N), U(1)_L_μ-L_τ breaking scalar S and gauge boson Z', as well as the dark matter candidate X_I and its heavy partner X_R. A joint explanation of the dark matter relic density and muon g-2 anomaly excludes the region where both min(m_E_1,m_E_2,m_N,m_X_R) and min(m_Z',m_S) are much larger than m_X_I. In the parameter space accommodating the DM relic density and muon g-2 anomaly, the model can achieve a first order U(1)_L_μ-L_τ breaking phase transition, whose strength is sensitive to the parameters of Higgs potential. The corresponding gravitational wave spectra can reach the sensitivity of U-DECIGO. In addition, the direct searches at the LHC impose stringent bound on the mass spectra of the vector-like leptons and dark matter.