Correlation functions of local composite operators from generalized unitarity

TL;DR

This paper extends generalized unitarity to the construction of momentum-space correlation functions of local gauge-invariant operators, using on-shell form factors and generalized form factors as the fundamental building blocks. By applying maximal-cut techniques and supersymmetric form factors, it reproduces known results for four-point BPS correlators and generates infinite families of $n$-point BPS and mixed BPS/non-BPS correlators at LO and NLO in ${\cal N}=4$ SYM, while clarifying regularization/renormalization and potential color/kinematics duality constraints. The authors also connect these correlators to energy/charge flow observables and to the gravitational effective action via stress-tensor form factors, offering a framework to explore strong-coupling extensions and possible position-space formulations. Overall, the work unifies amplitude-based methods with correlation-function computations, enabling systematic, on-shell construction of intricate operator correlators and shedding light on the interplay between conformal symmetry, integrability, and dualities in gauge theories.

Abstract

We describe the use of generalized unitarity for the construction of correlation functions of local gauge-invariant operators in general quantum field theories and illustrate this method with several calculations in N=4 super-Yang-Mills theory involving BPS and non-BPS operators. Form factors of gauge-invariant operators and their multi-operator generalization play an important role in our construction. We discuss various symmetries of the momentum space presentation of correlation functions, which is natural in this framework and give examples involving non-BPS and any number of BPS operators. We also discuss the calculation of correlators describing the energy flow in scattering processes as well as the construction of the effective action of a background gravitational field.

Figures (18)

• Figure 1: Generalized unitarity guarantees the presence of multi-loop amplitudes in the correlation function calculation. Additional contributions involves at least one form factor with more fields than the corresponding operator.
• Figure 2: The two generic terms in the MHV vertex expansion of the generalized super-form factor with two insertions of the chiral stress tensor. Grey blobs represent regular single-operator MHV super-form factors, the hashed blob represents an MHV generalized super-form factor and the white blob represents a regular MHV scattering super-amplitude. The momenta carried by the two operators are $q_{\tilde{1}}$ and $q_{\tilde{2}}$.
• Figure 3: Disconnected and connected quadruple cuts.
• Figure 4: Color-dressed cuts for connected part of the next-to-leading order 4-point correlator. There are 2 cuts of type $(a)$, 4 cuts of type $(b)$, 4 cuts of type $(c)$ and 2 cuts of type $(d)$. Other members of each set are related by relabeling. Each blob with an external line is a (generalized) color-dressed form factor; blobs with no external lines are color-dressed on-shell scattering amplitudes.
• Figure 5: 2-loop integrals: $\text{BTie}(4,1|2,3)$, $\text{DB}(4,1|2,3)$, $\text{TriP}(4,1|2,3)$ and $\text{TriB}(4|1|2,3)$.