Harmony of Super Form Factors

TL;DR

The paper extends on-shell amplitude techniques to form factors of half-BPS operators in N=4 SYM, developing tree-level and supersymmetric form factor methods including MHV rules, BCFW-type recursion, and unitarity. It introduces a closed-form solution for split-helicity form factors via zig-zag diagrams, and uses harmonic and non-chiral superspace to derive Ward identities and compact expressions for the chiral and complete stress-tensor multiplet form factors, including the surprisingly simple maximally non-MHV cases. Supersymmetric MHV rules, recursion, and unitarity are generalized to form factors, with a detailed discussion of their large-z behavior and seeds. Dual MHV rules are extended to form factors in dual momentum space, revealing a periodic open polygon picture and enabling tree- and loop-level computations through dual diagrams. Collectively, the work bridges on-shell amplitude techniques with off-shell observables, offering practical computational tools and deeper structural insights into form factors and their supersymmetric structure.

Abstract

In this paper we continue our systematic study of form factors of half-BPS operators in N=4 super Yang-Mills. In particular, we extend various techniques known for amplitudes to the case of form factors, including MHV rules, recursion relations, unitarity and dual MHV rules. As an application, we present the solution of the recursion relation for split-helicity form factors. We then consider form factors of the stress-tensor multiplet operator and of its chiral truncation, and write down supersymmetric Ward identities using chiral as well as non-chiral superspace formalisms. This allows us to obtain compact formulae for families of form factors, such as the maximally non-MHV case. Finally we generalise dual MHV rules in dual momentum space to form factors.

Figures (18)

• Figure 1: The four MHV diagrams contributing to the NMHV form factor \ref{['1stffnmhv']}.
• Figure 2: The two recursive diagrams contributing to the NMHV form factor \ref{['1stffnmhv']}.
• Figure 3: The iterative structure of split-helicity form factors illustrated by a square lattice. The three coloured paths ending on the MHV line are in one-to-one correspondence with terms that appear in the iterated recursion of $F_{2,2}$. Similarly there will be three paths (terms) that end on the $\overline{\rm MHV}$ line.
• Figure 4: Correspondence of lattice paths and MHV zig-zags for NMHV form factors.
• Figure 5: (a) MHV diagram for a tree-level NMHV form factor. (b) MHV diagram for a one-loop MHV form factor.