  The bubble size distribution below a breaking wave is of paramount interest
when quantifying mass exchanges between the atmosphere and oceans. Mass fluxes
at the interface are driven by bubbles that are small compared to the Hinze
scale $d_h$, the critical size below which bubbles are stable, even though
individually these are negligible in volume. Combining experimental and
numerical approaches, we report a power law scaling 3/2 of the diameter of
small bubble size distribution, for sufficiently large separation of scales
between the injection size and the Hinze scale. From an analysis of individual
bubble break-up events, we show that break-ups generating small bubbles are
driven by capillary effects, and that their break-up time scales with the
diameter to the 3/2, which physically explains the sub-Hinze scaling observed.
