Analytical methods in Quantum Field Theories: from loop integrals to defect correlators

TL;DR

This work develops a unified framework to study the analytic structure of quantum field theory observables at weak coupling by combining a parametric-space IBP approach with polynomial-ideal techniques, and applies it to defect CFTs arising from a Maldacena-Wilson line in ${\mathcal N}=4$ SYM. It shows that bulk-defect and multipoint correlators up to next-to-next-to-leading order can be constrained or determined by symmetry (including superconformal Ward identities) and non-perturbative inputs such as localization, yielding results that are rational or governed by Goncharov polylogarithms in favorable kinematics. The thesis identifies classes of integrals that encode all perturbative information for these correlators, clarifies the role of boundary terms in parameter-space IBP, and demonstrates how ideals and Gröbner-basis techniques can substantially reduce the computational complexity of reduction and differential-equation constructions. It also exposes intriguing cancellations of transcendental terms at weak coupling, ties perturbative results to topological sectors, and advances a perturbative bootstrap program for multipoint defect correlators, including the critical six-point train-track integral. Overall, the work deepens our understanding of the function spaces that describe correlators in symmetric QFTs and provides practical, algebraically informed methods for master-integral reductions and defect-CFT bootstrap analyses with potential broad applicability to GKZ systems and beyond.

Abstract

This thesis expands the available techniques at weak coupling by investigating the linear space of Feynman integrals and the role that (super)symmetry plays in reducing the number of integrals necessary to calculate correlators in the presence of a one-dimensional extended operator - the line defect. In the first part, linear relations among Feynman parametrized integrals are derived from their properties as projective forms; these relations are then tested on one- and multi-loop examples, and their connection to the algebra of polynomial ideals is uncovered. In the second part, made of two chapters, the defect CFT formed by the N = 4 super Yang-Mills theory in the presence of a Maldacena-Wilson line is studied through bulk-defect-defect and multipoint correlation functions up to next-to-next-to-leading order in the perturbative expansion at weak coupling. The investigations into this defect CFT lead to the identification of classes of integrals containing all the perturbative information necessary to compute the correlators, which turn out to be either rational functions or Goncharov polylogarithms.

Figures (20)

• Figure 1: A visual representation of the holographic principle. We thank Julien Barrat for the permission to use his drawing. The original can be found in Ref. Barrat:2024nod.
• Figure 2: $3\times3$ block-spin transformations for the $2d$ Ising model at the critical point (blue), above it (red), and below it (green). The images were generated by our version of the code by Douglas Stanford at http://large.stanford.edu/courses/2008/ph372/stanford1/c/. On the left we reproduce the section of the original lattice; on the right, the block-spin transformed one. The value of the reduced coupling is reported below each image.
• Figure 3: Four-point function of local operators inserted on the Wilson line from a Witten diagram on the AdS2 worldsheet. Credits to Giulia Peveri from the image taken with permission from Peveri:2023qip.
• Figure 4: a) One-loop box diagram b) Two-loop sunrise diagram
• Figure 5: Example of two domains in $\mathbb{R}^3$ that are equivalent under projective transformations.

Theorems & Definitions (10)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1.1.1
• proof
• Theorem 1.1.2: Cheng-Wu theorem
• proof
• Definition 4
• Definition 5
• Theorem 2.1.1: Coleman-Mandula theorem