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On integrands and loop momentum in string and field theory

Piotr Tourkine

TL;DR

The paper addresses how to define and utilize a global loop-momentum integrand in string theory and its field-theory limit, motivated by loop monodromies and the color/kinematics duality. It develops a precise loop-momentum construction based on a canonical dissection of the worldsheet, derives the corresponding field-theory (tropical) limit with uniform graph-by-graph labeling, and applies this to two- and three-loop monodromy relations that organize numerators into BCJ triplets with shifted momenta. These results strengthen the link between worldsheet loop dynamics and spacetime perturbative amplitudes, offering insights relevant for BCJ/KLT frameworks and ambitwistor/string approaches. The work also discusses broader implications, including potential connections to twisted strings and modular invariance, and outlines open questions for further study.

Abstract

The notion of a unique integrand does not a priori makes sense in field theory: different Feynman diagrams have different loop momenta and there should be no reason to compare them. In string theory, however, a global integrand is natural and allows, for instance, to make explicit the separation between left and right-moving degrees of freedom. However, the significance of this integrand had not really been investigated so far. It is even more important in view of the recently discovered loop monodromies that are related to the duality between color and kinematics in gauge and gravity loop amplitudes. This paper intends to start filling this gap, by presenting a careful definition of the loop momentum in string theory, and describing precisely the resulting global integrand obtained in the field theory limit. We will then apply this technology to write down some monodromy relations at two and three loops, and make contact with the color/kinematics duality.

On integrands and loop momentum in string and field theory

TL;DR

The paper addresses how to define and utilize a global loop-momentum integrand in string theory and its field-theory limit, motivated by loop monodromies and the color/kinematics duality. It develops a precise loop-momentum construction based on a canonical dissection of the worldsheet, derives the corresponding field-theory (tropical) limit with uniform graph-by-graph labeling, and applies this to two- and three-loop monodromy relations that organize numerators into BCJ triplets with shifted momenta. These results strengthen the link between worldsheet loop dynamics and spacetime perturbative amplitudes, offering insights relevant for BCJ/KLT frameworks and ambitwistor/string approaches. The work also discusses broader implications, including potential connections to twisted strings and modular invariance, and outlines open questions for further study.

Abstract

The notion of a unique integrand does not a priori makes sense in field theory: different Feynman diagrams have different loop momenta and there should be no reason to compare them. In string theory, however, a global integrand is natural and allows, for instance, to make explicit the separation between left and right-moving degrees of freedom. However, the significance of this integrand had not really been investigated so far. It is even more important in view of the recently discovered loop monodromies that are related to the duality between color and kinematics in gauge and gravity loop amplitudes. This paper intends to start filling this gap, by presenting a careful definition of the loop momentum in string theory, and describing precisely the resulting global integrand obtained in the field theory limit. We will then apply this technology to write down some monodromy relations at two and three loops, and make contact with the color/kinematics duality.

Paper Structure

This paper contains 18 sections, 66 equations, 8 figures.

Table of Contents

  1. Introduction
  2. String theory
  3. Classical solution.
  4. Field theory integrand.
  5. Single pinching of a Riemann surface.
  6. Graph labeling in the field theory limit
  7. Closed string.
  8. Open string.
  9. Non-planar graphs
  10. Application to the monodromy relations in open string and gauge theory at loop-level.
  11. Two loops
  12. A relation at three loops
  13. Discussion
  14. An integrand in field theory.
  15. Kawai-Lewellen-Tye
  16. ...and 3 more sections

Figures (8)

  • Figure 1: Canonical dissection along the homology cycles.
  • Figure 2: From the picture we see that $C \cup (a_2)^{-1} \cup b_1 \cup (b_1)^{-1} \cup c_1 \cup c_2 = \mathrm{id}$, hence $C=a_2 \cup (c_1)^{-1}\cup (c_2)^{-1}$ and the momentum flowing through $C$ is given by $\frac{1}{2\pi\alpha'}\oint_C (-\partial +\bar{\partial})X^\mu = \ell_2^\mu-k_1^\mu-k_2^\mu$.
  • Figure 3: Illustration of the plumbing fixture construction.
  • Figure 4: Illustration of the plumbing fixture construction.
  • Figure 5: Handle-representation drawing of the pinching that we studied in this section.
  • ...and 3 more figures