  Core excitons in solids have garnered increasing interest, yet their behavior
and decay mechanisms are not fully understood. Here, we use attosecond extreme
ultraviolet (XUV) transient absorption spectroscopy, performed with a broadband
25-45 eV sub-fs XUV pump pulse and a 500-1000 nm sub 5 fs near-infrared (NIR)
supercontinuum probe pulse to monitor the excitation, dynamics, and decay of
core excitons in CaF$_2$ at the Ca$^{2+}$ M$_{2,3}$ edge. The XUV pulses are
used to excite core excitons in CaF$_2$ based around the Ca$^{2+}$ and the
polarization of the medium is subsequently perturbed by the time-delayed NIR
pulses to measure the spectral changes and decays. A number of features are
identified in the transient absorption spectrum, which suggest transfer between
excitonic states, Stark shifts, and the emergence of light-induced states. We
find that various core excitons identified exhibit coherence lifetimes spanning
3-7 fs. Furthermore, a NIR-intensity-dependent analysis finds a negative
correlation with the coherence lifetime of various identified excitonic
features, supporting a phonon-mediated mechanism as responsible for the core
exciton decoherence. We present a computational band structure projection
analysis strategy to estimate the orbital structure of the core excitons and
determine which core excitonic transitions should be allowed by selection rules
with the probe beam. This strategy is found to successfully describe the
observed spectroscopic data. The outlined joint spectroscopic and computational
investigation of core excitons is a powerful technique that explains the
complex behavior of core excitons in solid-state materials.
