Interplay between the muon $g-2$ anomaly and the PTA nHZ gravitational waves from domain walls in next-to minimal supersymmetric standard model

With some explicitly $Z_3$ breaking terms in the NMSSM effective superpotential and scalar potential, domain walls (DWs) from spontaneously breaking of the discrete symmetry in approximate $Z_3$-invariant NMSSM can collapse and lead to observable stochastic gravitational wave (GW) background signals. In the presence of a hidden sector, such terms may originate from the geometric superconformal breaking with holomorphic quadratic correction to frame function when the global scale-invariant superpotential is naturally embedded into the canonical superconformal supergravity models. The smallness of such mass parameters in the NMSSM may be traced back to the original superconformal invariance. Naive estimations indicate that a SUSY explanation to muon $g-2$ anomaly can have tension with the constraints on SUSY by PTA data, because large SUSY contributions to $\mathrm{\Delta }{a}_{\mu }$ in general needs relatively light superpartners while present ${\mathrm{\Omega }}_{gw}^0$ can set the lower bounds for $m_{soft}$. We calculate numerically the signatures of GW produced from the collapse of DWs and find that the observed nHZ stochastic GW background by NANOGrav, etc., can indeed be explained with proper tiny values of $\chi m_{3\mathrm{/}2}\sim {10}^{-14}\mathrm{e}\mathrm{V}$ for $\chi S^2$ case (and $\chi m_{3\mathrm{/}2}\sim {10}^{-10}\mathrm{e}\mathrm{V}$ for $\chi H_u{H}_d$ case), respectively. Besides, there are still some parameter points, whose GW spectra intersect with the NANOGrav signal region, that can explain the muon $g-2$ anomaly to $1\sigma$ range.