  If the excess events from the CoGeNT experiment arise from elastic
scatterings of a light dark matter off the nuclei, crossing symmetry implies
non-vanishing annihilation cross-sections of the light dark matter into
hadronic final states inside the galactic halo, which we confront with the
anti-proton spectrum measured by the PAMELA collaboration. We consider two
types of effective interactions between the dark matter and the quarks: 1)
contact interactions from integrating out heavy particles and 2) long-range
interactions due to the electromagnetic properties of the dark matter. The lack
of excess in the anti-proton spectrum results in tensions for a scalar and, to
a less extent, a vector dark matter interacting with the quarks through the
Higgs portal.
