Hidden analytic relations for two-loop Higgs amplitudes in QCD

Qingjun Jin,Gang Yang

Table 1. The universal maximally transcendental properties for Higgs amplitudes or form factors of length-2 and 3 operators with three partons, and minimal form factors with higher length operators are summarized. The color-singlet operators are classified according to their lengths and representative examples are provided. We also indicate the external on-shell partons.

Length-2 Length-3 Higher length Operators Examples $\mathrm{tr}({F}^{2})$ $\bar{\psi }\psi $ $\bar{\phi }\phi $ $\begin{array}{c}\mathrm{tr}({F}^{3}),\\ \mathrm{tr}({F}_{\mu }^{\,\nu }{D}_{\sigma }{F}_{\nu }^{\,\rho }{D}^{\sigma }{F}_{\rho }^{\,\mu })\end{array}$ $\begin{array}{c}{F}_{\mu \nu }{D}^{\mu }(\bar{\psi }{\gamma }^{\nu }\psi ),\\ {F}_{\mu \nu }(\bar{\psi }{\gamma }^{\mu \nu }\psi )\end{array}$ $\mathrm{tr}({F}^{L}),L\geqslant 4$ $\bar{\psi }({F}^{L})\psi ,L\geqslant 2$ External Partons $(g,g,g),(\bar{\psi },\psi ,g)$ $(\bar{\psi },\psi ,g)$ $(\bar{\phi },\phi ,g)$ $(g,g,g)$ $(\bar{\psi },g,\psi )$ $({g}_{1},\ldots ,{g}_{L})$ $(\bar{\psi },{g}_{1},\ldots ,{g}_{L},\psi )$ $\begin{array}{c}{\rm{Max.}}\,{\rm{Trans.}}\\ {\rm{Remainder}}\\ ({\rm{with}}\ {C}_{F}\to {C}_{A})\end{array}$ ${R}_{{\rm{L}}2;4}(u,v,w)$ ${R}_{\mathrm{len}-3;4}(u,v,w)$ ${\sum }_{i}{{ \mathcal R }}_{{\rm{density}};4}^{(2)}({u}_{i},{v}_{i},{w}_{i})$