Tracking the Long-Term GW Phase Evolution for HM Cancri-like Binaries with LISA

From prolonged X-ray and optical data of the ultra-compact binary HM Cancri, two groups recently measured the second derivative of its orbital frequency. The space gravitational wave (GW) detector LISA will detect $\sim10^4$ Galactic binaries and their second frequency derivatives will be interesting observational targets for LISA. Here, we forecast the GW signal analysis for HM Cancri, as an ideal reference system for these numerous binaries. We find that, in its nominal operation period $T\sim4$yr, LISA is unlikely to realize a sufficient measurement precision for the reported second frequency derivative of this binary. However, because of a strong dependence on the time baseline, the precision will be drastically improved by extending the operation period of LISA or combining it with other missions (e.g., Taiji and TianQin) in a sequential order.