The shortest distance Norm on the forward mass-shell of mass $m$ is $$d(k,k')=m\ln\left[\frac{k\cdot k'}{m^2}+\sqrt{\frac{(k\cdot k')^2}{m^4}-1}\right],$$ which extends to a projective Norm for 4–vectors within the forward light–cone, $$d_0(k,k')=\ln\left[\frac{k\cdot k'}{mm'}+\sqrt{\frac{(k\cdot k')^2}{m^2{m'}^2}-1}\right],$$ where $k\cdot k=m^2,k'\cdot k'={m'}^2$.

I'd like to know of previous uses in Physics of either of these Norms and a citable reference in either the Physics or Mathematics literature.