Equivalence between small distance and high energy I see in a lot of particle physics literature statements along the lines of: 'This is valid for high energies (small distances)'. Exactly what do we mean by small distances in this case? From QFT, QM and other physics courses, the connection between small distance and high energy makes intuitive sense, but I am wondering if there is a concrete connection that authors have in mind?
 A: In HEP, where (in units of c=1) momenta are much larger than masses, so they are tantamount to energies, the fundamental scale relation of QM, $[x,p]=i\hbar$, comes to bear. The de Broglie relation ${\lambda\over 2\pi}= \hbar/p$ of the particles/waves involved relates distances to momenta/energies inversely.
Consequently, "small distance" is a synonym for "high energy" or momentum. So, analogously to diffractive crystallography, one probes ever smaller distance scales with ever higher enery/momentum scattering probes.
In HEP natural units, (ℏ=1) this is codified by the duality of the units for distance or energy,
such as the  radius of a proton being 4 GeV$^{-1}$ actually meaning
$$
4 \mathrm {GeV}^{-1} (\hbar c) = 8\cdot 10^{-16}{\mathrm m}~.
$$
A: Connections between energy and distance tend to involve the factor
$$
\hbar c= \rm 197\ eV\ nm=197\ MeV\ fm
$$
and some additional factors from doing algebra. The algebraic factors tend to be values like $\frac13$ or $\sqrt 6$, but it is unusual to do a page of first-principles algebra and come up a purely numerical factor of a thousand.
The most common example is a Yukawa-type force mediated by a massive particle, which has potential energy
$$
V=\alpha\hbar c\frac{e^{-r/r_0}}{r},
\qquad\text{where }r_0=\frac{\hbar c}{m c^2}
$$
Electromagnetism, where the photon is massless, is a Yukawa interaction in the limit $r_0\to\infty$, with $\alpha_\text{e.m.}≈10^{-2}$ the fine structure constant. In a meson-mediated model of nucleon-nucleon interactions, the longest-range component comes from the pion, whose mass $mc^2=140\rm\ MeV$ corresponds to a distance $r_0=1.4\ \rm fm$. For nucleons many femtometers apart, the pion-mediated attraction is exponentially suppressed. At shorter distances, you have to include heavier mesons in your calculations. At very high energies, you even have “forces” which mediated by the giga-eV weak bosons.
