The emission processes active in the highly relativistic jets of gamma-ray bursts (GRBs) remain unknown. In this paper we propose a new measure to describe spectra: the width of the $EF_E$ spectrum, a quantity dependent only on finding a good fit to the data. We apply this to the full sample of GRBs observed by Fermi/GBM and CGRO/BATSE. The results from the two instruments are fully consistent. We find that the median widths of spectra from long and short GRBs are significantly different (chance probability $<10^{-6}$). The width does not correlate with either duration or hardness, and this is thus a new, independent distinction between the two classes. Comparing the measured spectra with widths of spectra from fundamental emission processes -- synchrotron and blackbody radiation -- the results indicate that a large fraction of GRB spectra are too narrow to be explained by synchrotron radiation from a distribution of electron energies: for example, 78% of long GRBs and 85% of short GRBs are incompatible with the minimum width of standard slow cooling synchrotron emission from a Maxwellian distribution of electrons, with fast cooling spectra predicting even wider spectra. Photospheric emission can explain the spectra if mechanisms are invoked to give a spectrum much broader than a blackbody.
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