On MHV Form Factors in Superspace for $\mathcal{N}=4$ SYM Theory

TL;DR

This paper develops a manifestly N=4 supersymmetric unitarity framework for form factors of operators in the chiral truncation of the stress-tensor multiplet, introducing the concept of a superstate–super form factor and establishing a link between zero-supermomentum form factors and the logarithmic derivative of the superamplitude with respect to the coupling. It provides explicit tree- and one-loop MHV results for n-point form factors, including a complete tree-level MHV form factor and its SUSY covariant expression, and demonstrates correctness of the soft limit relation to amplitudes. A reduction procedure for dual pseudoconformal scalar integrals is proposed to generate the basis of scalar integrals for form factors, enabling a two-loop three-point MHV ansatz built from a reduced integral basis. The work lays groundwork for higher-loop, higher-point analyses and enhances understanding of the interplay between operator insertions, form factors, and amplitudes in N=4 SYM, with potential implications for amplitude–Wilson loop dualities and correlation-function structures.

Abstract

In this paper we develop a supersymmetric version of unitarity cut method for form factors of operators from the chiral truncation of the the $\mathcal{N}=4$ stress-tensor current supermultiplet $T^{AB}$. The relation between the superform factor with supermomentum equals to zero and the logarithmic derivative of the superamplitude with respect to the coupling constant is discussed and verified at tree- and one-loop level for any MHV $n$-point ($n \geq 4$) superform factor involving operators from chiral truncation of the stress-tensor energy supermultiplet. The explicit $\mathcal{N}=4$ covariant expressions for n-point tree- and one-loop MHV form factors are obtained. As well, the ansatz for the two-loop three-point MHV superform factor is suggested in the planar limit, based on the reduction procedure for the scalar integrals suggested in our previous work. The different soft and collinear limits in the MHV sector at tree- and one-loop level are discussed.

Figures (6)

• Figure 1: Some coordinate superspace Feynman diagrams for the $F_3=\langle \phi_1^{AB} \phi_2^{AB} \phi_3^{AB}|\mathcal{O}^{(3)}_{AB}|0\rangle$ form factor in two loop order. The number of red lines is equal to $m_{in}$, the number of green lines equal to $m_{fin}$. The straight lines correspond to the chiral propagators $\langle \bar{\Phi}_I^a\Phi_J^b \rangle$, the wavy lines correspond to the vector propagator $\langle V^aV^b \rangle$ of $\mathcal{N}=1$ superfields BKV_FormFN=1. The lower bold line represents the insertion of the corresponding operator $\mathcal{O}^{(3)}_{AB}$.
• Figure 2: The basis of scalar integrals through $O(\epsilon)$ at one (A) and two (B,C) loops for the even part of the five-point MHV amplitude. Green ark corresponds to the presence of the numerator.
• Figure 3: Scalar integrals obtained by the reduction procedure. We conjecture that they form a basis of scalar integrals for the $n=3$-point MHV form factor at one (A) and two (B,C) loops.
• Figure 4: The configuration of external momenta for $\textbf{G}_5$ and $\textbf{G}_4$ scalar integrals.
• Figure 5: All two particle iterated cuts for the $n=3$ point MHV form factor at one loop. Dark grey vertex corresponds to the MHV tree form factor, light grey vertex corresponds to the MHV tree amplitude.