Perturbative Champions: Cohen and Hansen take it to next-to-next-to-next-to-next-to-next-to-next-to-next-to-next-to leading order.

In 1999, in a preprint on arXiv.org, Thomas D. Cohen and James M. Hansen, physicists at the University of Maryland, claimed the following:

If one insists on an accuracy of ∼ 20%, one estimates contributions at their nominal order and Λ is taken to be 300 MeV, then one has to work to order (Q/Λ)$^7$ , this corresponds to next-to-next-to-next-to-next-to-next-to-next-to-next-to-next-to leading order.

This octet is, to my knowledge, the largest string of next-to's ever to appear in scientific literature. As far as I can tell, the runner up is a recent paper by Eltern et al. in Physical Review C, with a paltry five next-to's, although this may be the champion of peer-reviewed literature. Below this, four next-to's is fairly common; it has its own notation, N4LO. I found a reference to N6LO in the literature, but it becomes hard to google these.

What does this actually mean, and why does it sound so silly? A powerful tool in physics is perturbation theory, where we approximate the solution to a problem with a power series, and then compute first the simplest terms, then terms of increasing complexity until we have a solution that is close enough to the exact solution that it's useful to us. There will be the zeroth-order solution that tells us the order of magnitude, the first-order solution with its basic dependences on system parameters, then second and higher order solutions for non-linear effects. In many cases, certain powers will be zero, so the "leading order" term might be the first non-zero term besides zeroth-order, although not necessarily first order. For example, the leading order term of something that is perturbed about equilibrium is second-order (which is one of the reasons why treating things as harmonic oscillators is so useful). Feynman diagrams are essentially a way to express perturbation theory in a graphic form, first drawing interactions with no loops, then one loop, then two loops, etc.

Anyway, it may have been the case that all of the even and odd powers of the perturbation series in Cohen and Hansen's paper were zero, they couldn't just say "eighth order," but thought they had to expand all those next-to's. Each term makes the approximation more accurate, and for their 20% desired accuracy, they needed to compute all these Feynman diagrams really far past leading order, but unfortunately, as they claim, "it is implausible that such calculations will ever prove tractable."

Another, similar situation involves describing sites on a lattice. For example, if you're sitting at a site on a square lattice (with lattice constant x), you have four "nearest neighbors" at distance x from you, four "next-nearest neighbors" across the diagonals at distance $\sqrt{2}$x, "next-next-nearest neighbors at distance 2x, etc. This paper uses the term "next next next next nearest neighbors" which is designates 4NN.

There was perhaps a better way to visualize this.

These record nextings are fairly insignificant but perhaps mildly interesting.