  Promising searches for new physics beyond the current Standard Model (SM) of
particle physics are feasible through isotope-shift spectroscopy, which is
sensitive to a hypothetical fifth force between the neutrons of the nucleus and
the electrons of the shell. Such an interaction would be mediated by a new
particle which could in principle be associated with dark matter. In so-called
King plots, the mass-scaled frequency shifts of two optical transitions are
plotted against each other for a series of isotopes. Subtle deviations from the
expected linearity could reveal such a fifth force. Here, we study
experimentally and theoretically six transitions in highly charged ions of Ca,
an element with five stable isotopes of zero nuclear spin. Some of the
transitions are suitable for upcoming high-precision coherent laser
spectroscopy and optical clocks. Our results provide a sufficient number of
clock transitions for -- in combination with those of singly charged Ca$^+$ --
application of the generalized King plot method. This will allow future
high-precision measurements to remove higher-order SM-related nonlinearities
and open a new door to yet more sensitive searches for unknown forces and
particles.
