Holonomies of gauge fields in twistor space 6: incorporation of massive fermions

Following the previous paper arXiv:1205.4827, we formulate an S-matrix functional for massive fermion ultra-helicity-violating (UHV) amplitudes, i.e., scattering amplitudes of positive-helicity gluons and a pair of massive fermions. The S-matrix functional realizes a massive extension of the Cachazo-Svrcek-Witten (CSW) rules in a functional language. Mass-dimension analysis implies that interactions among gluons and massive fermions should be decomposed into three-point massive fermion subamplitudes. Namely, such interactions are represented by combinations of three-point UHV and next-to-UHV (NUHV) vertices. This feature is qualitatively different from the massive scalar amplitudes where the number of involving gluons can be arbitrary.Comment: 24 pages; v2. references added; v3. supplemental paragraph inserted below (3.42), typos corrected, published versio

A note on generalized hypergeometric functions, KZ solutions, and gluon amplitudes

Some aspects of Aomoto's generalized hypergeometric functions on Grassmannian spaces $Gr(k+1,n+1)$ are reviewed. Particularly, their integral representations in terms of twisted homology and cohomology are clarified with an example of the $Gr(2,4)$ case which corresponds to Gauss' hypergeometric functions. The cases of $Gr(2, n+1)$ in general lead to $(n+1)$-point solutions of the Knizhnik-Zamolodchikov (KZ) equation. We further analyze the Schechtman-Varchenko integral representations of the KZ solutions in relation to the $Gr(k+1, n+1)$ cases. We show that holonomy operators of the so-called KZ connections can be interpreted as hypergeometric-type integrals. This result leads to an improved description of a recently proposed holonomy formalism for gluon amplitudes. We also present a (co)homology interpretation of Grassmannian formulations for scattering amplitudes in ${\cal N} = 4$ super Yang-Mills theory.Comment: 51 pages; v2. reference added; v3. minor corrections, published versio