  We report the time-resolved spectral analysis of a bright near-infrared and
moderate X-ray flare of Sgr A*. We obtained light curves in the $M$-, $K$-, and
$H$-bands in the mid- and near-infrared and in the $2-8~\mathrm{keV}$ and
$2-70~\mathrm{keV}$ bands in the X-ray. The observed spectral slope in the
near-infrared band is $\nu L_\nu\propto \nu^{0.5\pm0.2}$; the spectral slope
observed in the X-ray band is $\nu L_\nu \propto \nu^{-0.7\pm0.5}$. We tested
synchrotron and synchrotron self-Compton (SSC) scenarios. The observed
near-infrared brightness and X-ray faintness, together with the observed
spectral slopes, pose challenges for all models explored. We rule out a
scenario in which the near-infrared emission is synchrotron emission and the
X-ray emission is SSC. A one-zone model in which both the near-infrared and
X-ray luminosity are produced by SSC and a model in which the luminosity stems
from a cooled synchrotron spectrum can explain the flare. In order to describe
the mean SED, both models require specific values of the maximum Lorentz factor
$\gamma_{max}$, which however differ by roughly two orders of magnitude: the
SSC model suggests that electrons are accelerated to $\gamma_{max}\sim 500$,
while cooled synchrotron model requires acceleration up to
$\gamma_{max}\sim5\times 10^{4}$. The SSC scenario requires electron densities
of $10^{10}~\mathrm{cm^{-3}}$ much larger than typical ambient densities in the
accretion flow, and thus require in an extraordinary accretion event. In
contrast, assuming a source size of $1R_s$, the cooled synchrotron scenario can
be realized with densities and magnetic fields comparable with the ambient
accretion flow. For both models, the temporal evolution is regulated through
the maximum acceleration factor $\gamma_{max}$, implying that sustained
particle acceleration is required to explain at least a part of the temporal
evolution of the flare.
