Sidebands to mHz QPOs in 4U 1626$-$67 in the second spin-down state

We report results from an $AstroSat$ Target-of-Opportunity (ToO) observation of 4U 1626$-$67, performed on 2023 May 18, soon after the discovery of torque reversal to spin-down in the source. The X-ray emission exhibited significant dependence on both energy and torque state. This work highlights the comparison of timing features of 4U 1626$-$67 with a previous $AstroSat$ observation from 2018, when the neutron star was in the spin-up state. The power density spectrum (PDS) of the 2023 observation comprised a sharp peak corresponding to $\nu_{\rm NS}\sim$130 mHz X-ray pulsations along with a prominent quasi-periodic oscillation (QPO) feature at $\nu_{\rm QPO}\sim$46 mHz with $\sim$20\% rms amplitude, which was positively correlated with energy. We also report the detection of sidebands to QPO occurring at a beat frequency ($\nu_{\rm NS}-\nu_{\rm QPO}$) of $\sim$83 mHz with $\sim$8\% rms amplitude, having $>3\sigma$ detection significance. Additionally, we utilized $Nuclear ~Spectroscopic ~Telescope ~ARray$ ($NuSTAR$) observations from the same torque state (2023 May-July) to analogize the presence and energy dependence of sidebands. The source retains timing properties in this spin-down torque state, similar to those seen in the previous spin-down phase. In sharp contrast, PDS from the 2018 observation was dominated by red noise, an absence of QPOs and a broadening in the wings of the pulse frequency peak, indicating a coupling between periodic and low-frequency aperiodic variability. Furthermore, we detected the known cyclotron resonance scattering feature (CRSF) at 37 keV in the Large Area X-ray Proportional Counter (LAXPC) spectrum. We explore various mechanisms that could possibly explain the presence of QPOs exclusively during the spin-down state.