Uncovering the Combinatorial Origins of Yang-Mills Theory?

[mitchell porter]

TL;DR Summary

Perturbative Yang-Mills from combinatorics, by Nima Arkani-Hamed, Song He, and collaborators

For many years now, the theorist Nima Arkani-Hamed has lent his prestige and energy to a research program that aims to transform our understanding of quantum field theory, by using symmetries in the sums of Feynman diagrams to uncover perspectives on the theory not based in ordinary space-time. The best-known buzzword is "amplituhedron"...

Here is his latest paper:

"Scalar-Scaffolded Gluons and the Combinatorial Origins of Yang-Mills Theory" (Arkani-Hamed, Cao, Dong, Figueiredo, He)

In this paper they claim, among other things, to have brought this reformulation of QFT, definitively into contact with the theories we use to describe

major aspects of the world we see outside our windows ... we have freed ourselves from the shackles of supersymmetry and integrability and “dlog forms” (these being features of the amplituhedron - M.P.), along the way discovering a startling unity between non-supersymmetric gluons, pions, and the simplest theory of colored scalars.

Specifically, they say they have a way to get the Feynman scattering diagrams for gluon-like particles - the quanta of Yang-Mills theories.

From what I understand, they do this by adding to the pure Yang-Mills theory (which would only contain gluons), colored scalar particles. Recall that gluons have spin-1, they carry color charges, and in the real world these govern their interactions with quarks, which have spin-1/2 and also have color charges. These colored scalars are spin-0 particles with color charges (resembling the pions of the real world edit: cancelled this since pions don't have color charge).

They then recover the Feynman diagrams for n gluons, by considering Feynman diagrams for 2n colored scalars, and specifically the Feynman diagrams where pairs of colored scalars fuse into gluons (thus 2n scalars give rise to n gluons). They have an amplituhedron-like recipe for scattering of 2n scalars in terms of a 2n-gon (a 2n-sided polygon), and then they have a way of restricting it to an internal n-gon which represents the interaction of the n gluons (see figure 12, page 26). Mission accomplished.

They call the scalars, scaffolding for the gluons (a scaffold in construction is a temporary platform from which you build something more lasting). They say

We remark that the “scaffolding” method is a very general one, and can be used to describe particles of any mass and spin, since we can always imagine producing the spinning particles with a pair of scalars. We can for instance also “scaffold” gravitons, only the scaffolded avatar of the gauge-invariance/multilinearity conditions will change. Similarly we can scaffold massive string states, with some extra care to be taken to disentangle mass degeneracy in the string spectrum.

There is also a formulation of the scaffolding in which the colored scalars are gluons in formal extra dimensions.

There's a lot here, as one might imagine:

* The alternative to space-time is kinematic space, the combined space of possible momenta for the external particles. (This is a counterpart of holography for flat space-time; there should be connections with "celestial holography" here.)

* Amplitudes are obtained by integrating dy/(y^2) differential forms over combinatorial objects like "fat graphs" and polygons with internal "chords". (The amplituhedron involved logarithmic dy/y differential forms.)

* A "Mirzakhani trick" deriving from the work of the Fields medalist, is used to render tractable the nonplanar Feynman diagrams (the amplituhedron was restricted to planar Feynman diagrams, scattering processes which when drawn in two dimensions, don't contain any crossed lines.)

* These amplitudes are naturally stringy, but are turned into field theory amplitudes by a "tropicalization" of the geometry, tropical algebra being a simplification of ordinary arithmetic which is being used in many places (e.g. neural networks) to obtain a kind of discretized geometry.

Two of the authors are from Princeton, USA, three are from Beijing, China. The paper is a descendant of one that I noted here six years ago, the common authors being Arkani-Hamed and Song He, who has a few lectures on Youtube (albeit not as many as Nima).