The HESS instrument has observed a diffuse flux of TeV gamma -rays from a large solid angle around the Galactic center (GC). This emission is correlated with the distribution of gas in the region suggesting that the gamma -rays originate in collisions between cosmic ray hadrons (CRHs) and ambient matter. Of particular interest, HESS has detected gamma -rays from the Sagittarius (Sgr) B Molecular Cloud Complex. Prompted by the suggestion of a hadronic origin for the gamma rays, we have examined archival 330 and 74 MHz Very Large Array radio data and 843 MHz Sydney University Molonglo Sky Survey data covering Sgr B, looking for synchrotron emission from secondary electrons and positrons (expected to be created in the same interactions that supply the observed gamma rays). Intriguingly, we have uncovered non-thermal emission, but at a level exceeding expectation. Adding to the overall picture, recent observations by the Atacama Pathfinder Experiment telescope show that the cosmic ray ionization rate is ten times greater in the Sgr B2 region of Sgr B than the local value. Lastly, Sgr B2 is also a very bright X-ray source. We examine scenarios for the spectra of CRHs and/or primary electrons that would reconcile all these different data. We determine that (i) a hard ( E-2.2), high-energy (> TeV) population CRHs is unavoidably required by the HESS gamma -ray data and (ii) the remaining broad-band, non-thermal phenomenology is explained either by a rather steep ( E-2.9) spectrum of primary electrons or a ( E-2.7) population of CRHs. No single, power-law population of either leptons or hadrons can explain the totality of broadband, non-thermal Sgr B phenomenology.