The power of power corrections

Perturbation theory is a widely used approach to obtain approximate solutions to problems in physics. However, in quantum field theory (QFT), perturbative expansions often exhibit factorially growing coefficients, leading to divergent asymptotic series that limit predictive power. Resurgence, a mathematical framework that has recently gained attention in the physics community, provides a systematic way to extract physical information from such divergent series. In paricular, resurgence reveals the connection between non-perturbative and perturbative phenomena.
In this thesis, we study this connection through so-called large order relations. We show that with an appropriate resummation prescription, large order relations are often exact: they can be used to exactly compute perturbative coefficients - not just their large order growth.
Then we consider the role of resurgence in QFT, focusing on its application to renormalons, which are non-perturbative power corrections that appear in QFTs. In particular, we compute non-perturbative contributions to the Adler function - the derivative of the vacuum polarization function in gauge theory - using resurgence methods. We study how the presence of non-perturbative sectors influences the perturbative expansions in neighbouring sectors.
Beyond resurgence, we also investigate another type of power correction using factorization. We investigate the factorization properties of the massive fermion form factor in QED, to next-to-leading power in the fermion mass, and up to two-loop order. For this purpose we define new jet functions that have multiple connections to the hard part as operator matrix elements, and compute them to second order in the coupling. We test our factorization formula using these new jet functions in a region-based analysis and find that factorization indeed holds.