  Young, rapidly-rotating M dwarfs exhibit prominent starspots, which create
quasiperiodic signals in their photometric and Doppler spectroscopic
measurements. The periodic Doppler signals can mimic radial velocity (RV)
changes expected from orbiting exoplanets. Exoplanets can be distinguished from
activity-induced false positives by the chromaticity and long-term incoherence
of starspot signals, but these qualities are poorly constrained for
fully-convective M stars. Coherent photometric starspot signals on M dwarfs may
persist for hundreds of rotations, and the wavelength dependence of starspot RV
signals may not be consistent between stars due to differences in their
magnetic fields and active regions. We obtained precise multi-wavelength RVs of
four rapidly-rotating M dwarfs (AD Leo, G 227-22, GJ 1245B, GJ 3959) using the
near-infrared (NIR) Habitable-zone Planet Finder, and the optical Keck/HIRES
spectrometer. Our RVs are complemented by photometry from Kepler, TESS, and the
Las Cumbres Observatory (LCO) network of telescopes. We found that all four
stars exhibit large spot-induced Doppler signals at their rotation periods, and
investigated the longevity and optical-to-NIR chromaticity for these signals.
The phase curves remain coherent much longer than is typical for Sunlike stars.
Their chromaticity varies, and one star (GJ 3959) exhibits optical and NIR RV
modulation consistent in both phase and amplitude. In general, though, we find
that the NIR amplitudes are lower than their optical counterparts. We conclude
that starspot modulation for rapidly-rotating M stars frequently remains
coherent for hundreds of stellar rotations, and gives rise to Doppler signals
that, due to this coherence, may be mistaken for exoplanets.
