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

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.

💡 Research Summary

The thesis presents two complementary methodological advances aimed at expanding the toolbox available for weak‑coupling calculations in quantum field theory. In the first part, the author reinterprets parametrised Feynman integrals as projective differential forms. By exploiting the homogeneity of these forms under scaling of the Feynman parameters, a general integration‑by‑parts (IBP) identity is derived directly in parameter space. This approach yields linear relations among integrals that are independent of the traditional Laporta algorithm and often involve far fewer master integrals. The author demonstrates the method on a series of concrete examples: one‑loop massless box and pentagon integrals, two‑loop equal‑mass banana diagrams, and higher‑loop sunrise‑type integrals with multiple masses. In each case the new IBP relations reproduce known results while dramatically simplifying the reduction step.

A further innovation is the systematic use of polynomial ideals and Gröbner‑basis techniques to organise the IBP relations. By mapping the set of IBP equations onto an ideal in the ring of polynomial coefficients, the reduction problem becomes one of computing a Gröbner basis, which can be performed efficiently with existing algebraic geometry software. This algebraic perspective reveals hidden structures—for instance, all equal‑mass banana integrals at any loop order belong to a single ideal, implying that a universal set of master integrals suffices for the whole family. Consequently, the parameter‑space IBP framework not only reduces computational complexity but also provides a clearer mathematical understanding of the space of Feynman integrals.

The second part of the thesis shifts focus to a defect conformal field theory (defect CFT) obtained by inserting a Maldacena‑Wilson line into N=4 supersymmetric Yang‑Mills theory. After a concise review of the super‑conformal algebra, half‑BPS bulk and defect operators, and the associated Feynman rules, the author studies bulk‑defect‑defect three‑point functions and more general multipoint correlators. Using the weak‑coupling expansion, the correlators are computed up to next‑to‑next‑to‑leading order (NNLO). Crucially, the same set of master integrals identified in the first part suffices to evaluate all diagrams that appear in these defect correlators.

The results fall into two distinct functional classes. Some contributions evaluate to pure rational functions of the cross‑ratios, reflecting the strong constraints imposed by supersymmetry and conformal symmetry. Other contributions are expressed in terms of Goncharov polylogarithms (multiple polylogarithms), which capture the transcendental part of the perturbative data. Remarkably, the author finds that supersymmetric protection eliminates certain transcendental terms that would otherwise appear, leading to a surprisingly simple analytic structure for the defect correlators.

Beyond explicit perturbative calculations, the thesis exploits several non‑perturbative constraints. The topological sector of the defect theory, the super‑block expansion, and the pinching/splitting limits (where two defect operators fuse into a bulk operator or vice‑versa) are used to cross‑check the results and to derive additional relations among the correlators. Locality constraints further restrict the allowed tensor structures, ensuring that only even‑order terms survive in the expansion. These consistency checks confirm that the identified master integrals indeed capture the full perturbative content of the defect CFT.

In summary, the work delivers (i) a novel, geometrically motivated IBP framework in Feynman‑parameter space, enhanced by algebraic‑ideal techniques, and (ii) a concrete application of this framework to a supersymmetric defect CFT, where it enables the systematic computation of bulk‑defect‑defect and multipoint correlators up to NNLO. The findings demonstrate that, at weak coupling, the entire perturbative data of the Maldacena‑Wilson line defect CFT can be expressed using a limited set of rational functions and Goncharov polylogarithms. This dual achievement not only streamlines high‑loop calculations but also deepens our conceptual understanding of the interplay between symmetry, geometry, and analytic structure in quantum field theories with defects.