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X-ray and radio observations of the AMXP MAXI J1957+032 covering the 2022-2025 outbursts

We presented a comprehensive multi-epoch timing and multiwavelength analysis of the accreting millisecond X-ray pulsar MAXI J1957+032, covering two major outbursts in 2022 and 2025. By reanalyzing the 2022 outburst data from the Neutron Star Interior Composition Explorer (NICER), we found the spin frequency and orbital parameters from the observations in 0.3-5 keV. For the 2025 outburst, we reported the detection of pulsations with the Einstein Probe (EP). Based on the $\sim$3-year baseline between these two outbursts, we measured a significant long-term spin-down rate of $\dotν= (-5.73 \pm 0.28) \times 10^{-14}~{\rm Hz~s^{-1}}$. Assuming that the quiescent spin-down is driven by magnetic dipole radiation, we inferred a spin-down luminosity of $L \approx 1.1 \times 10^{36}~{\rm erg~s^{-1}}$ and a surface dipolar magnetic field of $B \approx (7.3 - 10.4) \times 10^8$ G. Furthermore, we conducted a deep radio pulsation search with the Five-hundred-meter Aperture Spherical radio Telescope (FAST) during the X-ray quiescent state in 2024, resulting in a non-detection with a 7$σ$ flux density upper limit of 12.3 $μ$Jy. This corresponds to a radio efficiency upper limit of $ξ< 2.8 \times 10^{-10}$, which is significantly lower than that of typical millisecond pulsars with a similar spin-down power. This profound radio pulsation faintness can be explained by two primary scenarios: either a geometric effect, wherein the pulsar&#39;s radio beam is directed away from our line of sight, or a physical suppression of the emission mechanism, potentially caused by a persistent low-level accretion flow during the X-ray quiescent state.