On the uncertain nature of the core of $\boldsymbol{\alpha}$ Cen A

High-quality astrometric, spectroscopic, interferometric and, importantly, asteroseismic observations are available for $\alpha$ Cen A, which is the closest binary star system to earth. Taking all these constraints into account, we study the internal structure of the star by means of theoretical modelling. Using the Aarhus STellar Evolution Code (ASTEC) and the tools of Computational Bayesian Statistics, in particular a Markov chain Monte Carlo algorithm, we perform statistical inferences for the physical characteristics of the star. We find that $\alpha$ Cen A has a probability of approximately 40% of having a convective core. This probability drops to few percents if one considers reduced rates for the $^{14}$N(p,$\gamma$)$^{15}$O reaction. These convective cores have fractional radii less than 8% when overshoot is neglected. Including overshooting also leads to the possibility of a convective core mostly sustained by the ppII chain energy output. We finally show that roughly 30% of the stellar models describing $\alpha$ Cen A are in the subgiant regime.