  The upcoming Laser Interferometer Space Antenna (LISA) is expected to detect
gravitational waves (GWs) from massive black hole binaries (MBHB). Finding the
electromagnetic (EM) counterparts for these GW events will be crucial for
understanding how and where MBHBs merge, measuring their redshifts,
constraining the Hubble constant and the graviton mass, and for other novel
science applications. However, due to poor GW sky localisation,
multi-wavelength, time-dependent electromagnetic (EM) models are needed to
identify the right host galaxy among many candidates. We studied merging MBHBs
embedded in a circumbinary disc using high-resolution two-dimensional
simulations, with a $\Gamma$-law equation of state, incorporating viscous
heating, shock heating, and radiative cooling. We simulate the binary from
large separation until after merger, allowing us to model the decoupling of the
binary from the circumbinary disc (CBD). We compute the EM signatures and
identify distinct features before, during, and after the merger. Our main
result is a multi-band EM signature: we find that the MBHB produces strong
thermal X-ray emission until 1-2 days prior to the merger. However, as the
binary decouples from the CBD, the X-ray-bright minidiscs rapidly shrink in
size, become disrupted, and the accretion rate drops precipitously. As a
result, the thermal X-ray luminosity drops by orders of magnitude, and the
source remains X-ray dark for several days after the merger, regardless of any
post-merger effects such as GW recoil or mass loss. Looking for the abrupt
spectral change where the thermal X-ray disappears is a tell-tale EM signature
of LISA mergers that does not require extensive pre-merger monitoring.
