We present global fits of an effective field theory description of real, and
complex scalar dark matter candidates. We simultaneously take into account all
possible dimension 6 operators consisting of dark matter bilinears and gauge
invariant combinations of quark and gluon fields. We derive constraints on the
free model parameters for both the real (five parameters) and complex (seven)
scalar dark matter models obtained by combining Planck data on the cosmic
microwave background, direct detection limits from LUX, and indirect detection
limits from the Fermi Large Area Telescope. We find that for real scalars
indirect dark matter searches disfavour a dark matter particle mass below 100
GeV. For the complex scalar dark matter particle current data have a limited
impact due to the presence of operators that lead to p-wave annihilation, and
also do not contribute to the spin-independent scattering cross- section.
Although current data are not informative enough to strongly constrain the
theory parameter space, we demonstrate the power of our formalism to
reconstruct the theoretical parameters compatible with an actual dark matter
detection, by assuming that the excess of gamma rays observed by the Fermi
Large Area Telescope towards the Galactic centre is entirely due to dark matter
annihilations. Please note that the excess can very well be due to
astrophysical sources such as millisecond pulsars. We find that scalar dark
matter interacting via effective field theory operators can in principle
explain the Galactic centre excess, but that such interpretation is in strong
tension with the non-detection of gamma rays from dwarf galaxies in the real
scalar case. In the complex scalar case there is enough freedom to relieve the
tension.