Different accretion heating of the neutron star crust during multiple outbursts in MAXI J0556-332

The transient neutron star (NS) low-mass X-ray binary MAXI J0556$-$332 provides a rare opportunity to study NS crust heating and subsequent cooling for multiple outbursts of the same source. We examine {\it MAXI}, {\it Swift}, {\it Chandra}, and {\it XMM-Newton} data of MAXI J0556$-$332 obtained during and after three accretion outbursts of different durations and brightness. We report on new data obtained after outburst III. The source has been tracked up to $\sim$1800 d after the end of outburst I. Outburst I heated the crust strongly, but no significant reheating was observed during outburst II. Cooling from $\sim$333 eV to $\sim$146 eV was observed during the first $\sim$1200 d. Outburst III reheated the crust up to $\sim$167 eV, after which the crust cooled again to $\sim$131 eV in $\sim$350 d. We model the thermal evolution of the crust and find that this source required a different strength and depth of shallow heating during each of the three outbursts. The shallow heating released during outburst I was $\sim$17 MeV nucleon$^{-1}$ and outburst III required $\sim$0.3 MeV nucleon$^{-1}$. These cooling observations could not be explained without shallow heating. The shallow heating for outburst II was not well constrained and could vary from $\sim$0--2.2 MeV nucleon$^{-1}$, i.e., this outburst could in principle be explained without invoking shallow heating. We discuss the nature of the shallow heating and why it may occur at different strengths and depths during different outbursts.