Worlds Next Door: A Candidate Giant Planet Imaged in the Habitable Zone of α Cen A.1 II. Binary Star Modeling, Source Detection, and Sensitivity Analysis

The \emph{James Webb} Space Telescope (JWST) observed our closest solar twin, $\alpha$~Centauri~A (\acenA), with the Mid-InfraRed Instrument (MIRI) in the F1550C (15.5 $\mu$m) coronagraphic imaging mode at three distinct epochs between August 2024 and April 2025. For the first time with JWST, we demonstrate the application of reference star differential imaging to simultaneously subtract the coronagraphic image of a primary star (\acenA) and the point spread function (PSF) of its binary companion (\acenB) to conduct a deep search for exoplanets and exozodiacal dust emission (exozodi). We achieve a typical 5$\sigma$ point source contrast sensitivity between $\sim$$10^{-5}$--$10^{-4}$ at separations $\gtrsim$~1\arcsec\ and an signal-to-noise ratio (S/N)~$>$~5 exozodi sensitivity of $\sim$2$\times$ the Solar System's zodiacal cloud around \acenA. The latter is an extraordinary limit, representing the deepest sensitivity to exozodi achieved for any stellar system to date. Additionally, post-processing with the principal component analysis-based Karhunen–Loève Image-Processing algorithm reveals a point source, called \objS, in August 2024, detected at S/N~$=$~4--6, a projected separation of $\approx$ 1\farcs5 (2 au), and with a F1550C flux density (contrast) of $\approx$ 3.5 mJy ($\approx 5.5 \times 10^{-5}$). Various tests conducted with the available data show that \objS is unlikely to be a detector artifact or PSF subtraction artifact and confirm that it is neither a background nor a foreground object. \objS is not re-detected in two follow-up observations (February and April 2025) and, if real, has moved to a region of poor sensitivity due to orbital motion. We perform PSF injection-recovery tests and provide 2D sensitivity maps for each epoch to enable orbital completeness calculations. Additional observations, with JWST or upcoming facilities, are necessary to re-detect candidate \objS and confirm its nature as a planet orbiting our nearest solar-type neighbor, \acenA. More broadly, this program highlights the challenges associated with confirming short-period ($\sim$few years) planet candidates identified without prior orbital constraints in direct imaging searches. This paper is second in a series of two papers: Paper I (Beichman \& Sanghi et al. 2025, submitted) presents the observation strategy and discusses the astrophysical case (physical and orbital properties) for \objS as a planet candidate.