Not yet reviewed by Pith; the record is open.

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.