nLab Tr(ϕ³)

 Idea

$Tr(\phi^3)$ is a variant of Yang-Mills theory consisting of spin-zero gluons. The name of the theory comes from the cubic self-interaction term $Tr(\phi^3)$ which appears in the Lagrangian of the theory:

$$\mathcal{L} \;=\; Tr(\partial \phi)^2 + m^2 Tr(\phi^2) + g Tr(\phi^3) \,.$$

The theory has been used by Nima Arkani-Hamed and collaborators to extend techniques of scattering amplitudes to non-supersymmetric Yang-Mills theory.

Related concepts

• S-matrix

• scattering amplitudes

• Yang-Mills theory

• amplituhedron

• associahedron

• cosmohedron

• surfaceology

References

• Freddy Cachazo, Song He, Ellis Ye Yuan, Scattering of Massless Particles: Scalars, Gluons and Gravitons, Journel of High Energy Physics 2014, 33 (2014). [doi:10.1007/JHEP07(2014)033, arXiv:1309.0885]

• Nima Arkani-Hamed, Hadleigh Frost, Giulio Salvatori, Pierre-Guy Plamondon, Hugh Thomas, All Loop Scattering As A Counting Problem [arXiv:2309.15913]

• Nima Arkani-Hamed, Hadleigh Frost, Giulio Salvatori, Pierre-Guy Plamondon, Hugh Thomas, All Loop Scattering For All Multiplicity [arXiv:2311.09284]

• Nima Arkani-Hamed, Qu Cao, Jin Dong, Carolina Figueiredo, Song He, Hidden zeros for particle/string amplitudes and the unity of colored scalars, pions and gluons [arXiv:2312.16282]

• Nima Arkani-Hamed, Qu Cao, Jin Dong, Carolina Figueiredo, Song He, Scalar-Scaffolded Gluons and the Combinatorial Origins of Yang-Mills Theory [arXiv:2401.00041]

• Nima Arkani-Hamed, Carolina Figueiredo, Hadleigh Frost, Giulio Salvatori, Tropical Amplitudes For Colored Lagrangians [arXiv:2402.06719]

• Nima Arkani-Hamed, Carolina Figueiredo, All-order splits and multi-soft limits for particle and string amplitudes [arXiv:2405.09608]

Discussion on the relation between the nonlinear $\sigma$-model and $Tr(\phi^3)$:

• Nima Arkani-Hamed, Qu Cao, Jin Dong, Carolina Figueiredo, Song He, NLSM $\subset$ Tr($\phi^3$) [arXiv:2401.05483]

• Nima Arkani-Hamed, Carolina Figueiredo, Circles and Triangles, the NLSM and Tr($\phi^3$) [arXiv:2403.04826]

Discussion on cosmohedra in $Tr(\phi^3)$:

• Nima Arkani-Hamed, Carolina Figueiredo, Francisco Vazão, Cosmohedra [arXiv:2412.19881]

• Ross Glew, Tomasz Lukowski, Amplitubes: Graph Cosmohedra [arXiv:2502.17564]