  We demonstrate laser wakefield acceleration of quasi-monoenergetic electron
bunches up to 15 MeV at 1 kHz repetition rate with 2.5 pC charge per bunch and
a core with < 7 mrad beam divergence. Acceleration is driven by 5 fs, < 2.7 mJ
laser pulses incident on a thin, near-critical density hydrogen gas jet. Low
beam divergence is attributed to reduced sensitivity to laser carrier envelope
phase slip, achieved in two ways using laser polarization and gas jet control:
(1) electron injection into the wake on the gas jet's plasma density downramp,
and (2) use of circularly polarized drive pulses. Under conditions of mild
wavebreaking in the downramp, electron beam profiles have a 2D Lorentzian shape
consistent with a kappa electron energy distribution. Such distributions had
previously been observed only in space or dusty plasmas. We attribute this
shape to the strongly correlated collisionless bunch confined by the quadratic
wakefield bubble potential, where transverse velocity space diffusion is
imparted to the bunch by the red-shifted laser field in the bubble.
