String Scattering Amplitudes in the Field Theory Limit

A research project on the low-energy degeneration of perturbative string amplitudes to quantum field theory amplitudes, with particular emphasis on ultraviolet structure, modular forms, and induced regulator behaviour.

This project studies how perturbative string scattering amplitudes reduce to quantum field theory amplitudes in the low-energy or field theory limit, and what this degeneration reveals about ultraviolet structure, regularisation, and the mathematical organisation of amplitudes.

A key feature of the research is the investigation into a new, top-down selection principle: ultraviolet (UV) finiteness in quantum gravity should constrain (and in favourable cases determine) the finite parts of low-energy quantum field theory (QFT) calculations. Recent developments in $\eta$ regularisation (missing reference) have already provided insights into what such a general class of regulators should look like from the bottom-up. This project aims to take steps forward by utilising string theory amplitudes, together with recent advances in tropical and positive geometry and analytic number theory, to derive from the top-down quantum gravity induced regulator classes for field theory amplitudes. In doing so, the research aims to provide a unifying picture and clarify when finite remainders are physically mandated, as questioned in related work.

  • Can top-down UV-finite structures constrain low-energy regularisation?

The project is motivated by the view that the field theory limit of string theory is not merely an approximation procedure, but a structured limit in which modular and geometric properties of the worldsheet may leave non-trivial traces in the effective low-energy description. Of particular interest is the use of tropical geometry, which exists at the intersection of algebraic geometry and combinatorics, to describe the degeneration of the string worldsheet and the decomposition of its modular data. This involves the use of tropical and metric graph methods to analyse how string amplitudes descend to Schwinger-parameter formulations of low-energy field theory, and the study of how induced regulator structure constrains the finite parts of low-energy amplitudes as directly descended from quantum gravity.

Additional research questions:

  • How do perturbative string amplitudes degenerate into field-theory amplitudes?
  • What geometric structure survives in the low-energy limit?
  • Can the field-theory limit induce preferred regulator classes?
  • What does the string/QFT interface reveal about UV completion?
  • How do tropical and metric-graph methods organise this limit?

As string theory is currently the most developed and well-defined theory of quantum gravity, with well-defined scattering amplitudes, a key aim of this project is to derive string theory induced classes of $\eta$ regulators (Smith, 2025) by modifying the integration measure on spaces of metric graphs, which is the space of Feynman graphs. The key hypothesis is that the stringy corrections encoded in the proposed measure admits a unique geometric interpretation not yet covered in the literature, but has well-motivated relation to positive geometry and recent work on the amplituhedron and related structures. Starting from one-loop examples, this should lead to a new geometric and combinatorial perspective on regularisation that can enhance computational techniques for string and field theory amplitudes, and inspire new UV complete toy models in the field theory limit of high-energy theory.

In parallel, generalising the $\eta$ framework to supersymmetric theories by extending the gauge consistency methodology to a supersymmetric setting and identifying preferred classes of supersymmetric $\eta$ regulators could also be interesting. Such regulators are expected to arise as field theory limits of modular invariant worldsheet theories, providing a unified view of supersymmetry, modularity, and regularisation.

<div class="row justify-content-sm-center">
  <div class="col-sm-8 mt-3 mt-md-0">
    {% include figure.liquid path="assets/img/6.jpg" title="example image" class="img-fluid rounded z-depth-1" %}
  </div>
  <div class="col-sm-4 mt-3 mt-md-0">
    {% include figure.liquid path="assets/img/11.jpg" title="example image" class="img-fluid rounded z-depth-1" %}
  </div>
</div>