  Turbulence consists of interacting flow structures covering a wide range of
length and time scales. A long-standing question looms over pair diffusion of
particles in close proximity i.e. particle pair diffusion at small separations:
what range of turbulence length scales governs pair diffusion? Here, we attempt
to answer this question by addressing pair diffusion by both fine scales and
larger scale coherent structures in which the fine scales are embedded - we
unavoidably encounter a combination of both local and non-local interactions
associated with the small and large length scales. The local structures possess
length scales of the same order of magnitude as the pair separation $l$, and
they induce strong relative motion between the particle pair. However, the
non-local structures, possessing length scales much larger than $l$, also
induce (via Biot-Savart) significant relative motion, an effect ignored in
prior studies (based on Richardson-Obukhov R-O theory). This fundamentally
changes the interpretation of the pair diffusion process, giving the pair
diffusivity $K$ scaling as $K\sim l^{1.556}$ -- agreeing within $1\%$ of data.
The `R-O constant' $g_l$ is shown to be not a constant, although widely assumed
to be a constant. However, new constants (representing pair diffusivity $G_K$
and pair separation $G_l$) are identified, which we show to asymptote to
$G_K\approx 0.73$ and $G_l\sim 0.01$ at high Reynolds numbers. As an
application, we show that the radius of a cloud of droplets in a spray is
smaller by an order of magnitude as compared to R-O theory.
