When CoGeNT met PAMELA
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.