From correlation functions to scattering amplitudes

Abstract: We study the correlators of half-BPS protected operators in N=4 super-Yang-Mills theory, in the limit where the positions of the adjacent operators become light-like separated. We compute the loop corrections by means of Lagrangian insertions. The divergences resulting from the light-cone limit are regularized by changing the dimension of the integration measure over the insertion points. Switching from coordinates to dual momenta, we show that the logarithm of the correlator is identical with twice the logarithm of the matching MHV gluon scattering amplitude. We present a number of examples of this new relation, at one and two loops.

Overview of the Paper: From Correlation Functions to Scattering Amplitudes

The research conducted by Burkhard Eden, Gregory P. Korchemsky, and Emery Sokatchev centers on the relationship between correlation functions of half-BPS protected operators in $\mathcal{N}=4$ super-Yang-Mills (SYM) theory and scattering amplitudes. The paper presents a novel perspective on these correlators by employing the light-cone limit, where the positions of adjacent operators become light-like separated.

Key Insights and Methodology

• Correlation Functions and Light-Cone Divergences: The paper examines the correlators of half-BPS operators in $\mathcal{N}=4$ SYM in the context of light-cone divergences. The researchers propose two methods to regulate these divergences: using the small distances as a cutoff or employing standard dimensional regularization by setting the distances to zero initially.
• Duality with MHV Scattering Amplitudes: A pivotal element of the study is the identification of a duality between correlation functions of gauge-invariant composite operators on the light cone and Maximally Helicity Violating (MHV) gluon scattering amplitudes. This duality is formulated using T-duality transformation and dimensional regularization techniques, revealing that the logarithm of the correlator coincides with twice the logarithm of the matching MHV gluon scattering amplitude.
• Dual Conformal Symmetry: The paper explores the dual conformal symmetry inherent in light-like Wilson loops, which extends to planar scattering amplitudes through an anomalous mechanism. The authors provide evidence for this symmetry through analytic expressions for loop momentum integrals in the correlators expressed in dual space.
• Lagrangian Insertions Method: The study utilizes the Lagrangian insertion approach for calculating loop corrections to the correlators. This method treats loop corrections as derivatives with respect to the coupling constant, facilitating calculations via Born-level correlators with extra insertions.

Implications and Future Perspectives

• Understanding of Dual Conformal Symmetry: The findings suggest a deeper understanding of dual conformal symmetry within $\mathcal{N}=4$ SYM theory, illustrating how the symmetry manifests in both amplitudes and correlators.
• Potential for New Duality Relations: The evidence for a duality between correlators and amplitudes may lead to further exploration of the relationship between different physical quantities in gauge theories, broadening the scope of duality in theoretical physics.
• Applications in Field Theory and String Theory: The results could have implications for the development of computational techniques in quantum field theory and string theory, particularly in exploring connections between AdS/CFT correspondence and gauge theories.

Conclusion

This paper provides insightful contributions to the field of theoretical physics by establishing connections between correlation functions and scattering amplitudes within the framework of $\mathcal{N}=4$ SYM theory. The robust mathematical framework utilized offers a profound understanding of the symmetries in play and suggests avenues for future research in exploring dualities and computational methods in higher loop corrections.