  Context: HESS J1731-347 has been identified as one of the few TeV-bright
shell-type supernova remnants (SNRs). These remnants are dominated by
nonthermal emission, and the nature of TeV emission has been continuously
debated for nearly a decade.
  Aims: We carry out the detailed modeling of the radio to gamma-ray spectrum
of HESS J1731-347 to constrain the magnetic field and energetic particles
sources, which we compare with those of the other TeV-bright shell-type SNRs
explored before.
  Methods: Four years of data from Fermi Large Area Telescope (LAT)
observations for regions around this remnant are analyzed, leading to no
detection correlated with the source discovered in the TeV band. The Markov
Chain Monte Carlo method is used to constrain parameters of one-zone models for
the overall emission spectrum.
  Results: Based on the 99.9% upper limits of fluxes in the GeV range, one-zone
hadronic models with an energetic proton spectral slope greater than 1.8 can be
ruled out, which favors a leptonic origin for the gamma-ray emission, making
this remnant a sibling of the brightest TeV SNR RX J1713.7-3946, the Vela
Junior SNR RX J0852.0-4622, and RCW 86. The best-fit leptonic model has an
electron spectral slope of 1.8 and a magnetic field of about 30 muG, which is
at least a factor of 2 higher than those of RX J1713.7-3946 and RX
J0852.0-4622, posing a challenge to the distance estimate and/or the energy
equipartition between energetic electrons and the magnetic field of this
source. A measurement of the shock speed will address this challenge and has
implications on the magnetic field evolution and electron acceleration driven
by shocks of SNRs.
