Loops in AdS from Conformal Field Theory (1612.03891v3)

Abstract: We propose and demonstrate a new use for conformal field theory (CFT) crossing equations in the context of AdS/CFT: the computation of loop amplitudes in AdS, dual to non-planar correlators in holographic CFTs. Loops in AdS are largely unexplored, mostly due to technical difficulties in direct calculations. We revisit this problem, and the dual $1/N$ expansion of CFTs, in two independent ways. The first is to show how to explicitly solve the crossing equations to the first subleading order in $1/N^2$, given a leading order solution. This is done as a systematic expansion in inverse powers of the spin, to all orders. These expansions can be resummed, leading to the CFT data for finite values of the spin. Our second approach involves Mellin space. We show how the polar part of the four-point, loop-level Mellin amplitudes can be fully reconstructed from the leading-order data. The anomalous dimensions computed with both methods agree. In the case of $\phi^4$ theory in AdS, our crossing solution reproduces a previous computation of the one-loop bubble diagram. We can go further, deriving part of the four-point function in $\phi^3+\phi^4$ theory in AdS which had never been computed. In the process, we show how to analytically derive anomalous dimensions from Mellin amplitudes with an infinite series of poles, and discuss applications to more complicated cases such as the ${\cal N}=4$ super-Yang-Mills theory.

• The paper proposes solving CFT crossing equations at first subleading order in 1/N², enabling a resummable expansion for finite spin.
• It reconstructs the polar parts of loop-level Mellin amplitudes, reproducing known φ⁴ one-loop bubble results and extending to φ³+φ⁴ computations in AdS.
• Both methods yield consistent anomalous dimensions, enhancing the toolkit for addressing non-planar corrections in holographic quantum field theories.

Overview of the Paper: Loops in AdS from Conformal Field Theory

This paper presents a novel approach to computing loop amplitudes in anti-de Sitter (AdS) space using conformal field theory (CFT) crossing equations, specifically within the context of the AdS/CFT correspondence. The authors aim to bridge the gap in understanding loop-level dynamics in AdS, which has been a challenging area due to the technical complexities involved in direct calculations. They propose a systematic technique to address this, offering two methods: solving the crossing equations at the first subleading order in $1/N^2$ and utilizing Mellin space to reconstruct the polar parts of four-point loop-level Mellin amplitudes.

Key Contributions

1. Solving Crossing Equations: The paper provides a detailed method to solve the crossing equations for CFT at first subleading order, given a leading order solution. The approach involves expanding in inverse powers of the spin, which is then resummable, leading to a solution that is valid for finite spin.
2. Mellin Space Reconstruction: The authors demonstrate the use of Mellin space to reconstruct loop-level amplitudes. This reconstruction is achieved through the determination of the polar part of the Mellin amplitudes, which can be inferred from leading-order data.
3. Agreement in Methods: Both methods yield the same results for anomalous dimensions, validating the internal consistency and robustness of the approaches. For instance, in the $\phi^4$ theory in AdS, the paper successfully reproduces known results for the one-loop bubble diagram, and further extends this to compute parts of the four-point function in $\phi^3+\phi^4$ theory in AdS.
4. Analytic Techniques: The paper details how to analytically extract anomalous dimensions from Mellin amplitudes with an infinite series of poles. This is particularly noteworthy as it highlights the potential to compute OPE data from complex Mellin amplitude structures.

Theoretical Implications

The methodologies advanced in this paper deepen the understanding of holographic CFT dynamics beyond the planar limit. By effectively mapping higher-order CFT corrections to AdS loop diagrams, the research opens pathways to explore the nature of AdS loop and scattering amplitudes. This is pivotal in comprehending the structure of amplitudes in curved spaces and aligning them with known flat space properties, potentially uncovering new organizational principles.

Practical Implications

Practically, the paper provides tools to compute non-trivial CFT data, which can be applied to more complex gauge theories and gravity duals. The ability to translate intricate solutions from crossing equations to physical quantities like loop amplitudes enhances the toolkit available for researchers in holography, promising insights into strongly coupled system behaviors in quantum field theories.

Future Prospects

The techniques and findings in this work could spearhead further developments in the representation theory of the conformal bootstrap and its applications in both theoretical studies and numerical simulations. By elucidating the structure of loop-level corrections, the paper sets the stage for advancements in understanding non-planar corrections in holographic theories, possibly paving the way for new discoveries in quantum gravity and high-energy physics.

In conclusion, this research represents a significant stride in holographic studies, providing practical solutions and theoretical insights that could extend into various applications within the field of high-energy theoretical physics. The combination of CFT methodologies with AdS insights showcases a powerful synergy in tackling long-standing computational challenges.