  While observations have suggested that power-law electron energy spectra are
a common outcome of strong energy release during magnetic reconnection, e.g.,
in solar flares, kinetic simulations have not been able to provide definite
evidence of power-laws in energy spectra of non-relativistic reconnection. By
means of 3D large-scale fully kinetic simulations, we study the formation of
power-law electron energy spectra in non-relativistic low-$\beta$ reconnection.
We find that both the global spectrum integrated over the entire domain and
local spectra within individual regions of the reconnection layer have
power-law tails with a spectral index $p \sim 4$ in the 3D simulation, which
persist throughout the non-linear reconnection phase until saturation. In
contrast, the spectrum in the 2D simulation rapidly evolves and quickly becomes
soft. We show that 3D effects such as self-generated turbulence and chaotic
magnetic field lines enable the transport of high-energy electrons across the
reconnection layer and allow them to access several main acceleration regions.
This leads to a sustained and nearly constant acceleration rate for electrons
at different energies. We construct a model that explains the observed
power-law spectral index in terms of the dynamical balance between particle
acceleration and escape from main acceleration regions, which are defined based
upon a threshold for the curvature drift acceleration term. This result could
be important for explaining the formation of power-law energy spectrum in solar
flares.
