[Paper Review] One-loop Double Copy Relation in String Theory

Stephan Stieberger|arXiv (Cornell University)|Oct 11, 2023

Computational Physics and Python Applications4 citations

TL;DR

This paper establishes a one-loop double copy relation in string theory, generalizing the tree-level Kawai-Lewellen-Tye (KLT) relations to loop amplitudes. It shows that one-loop closed string amplitudes can be expressed as a sum over products of one-loop open string amplitudes, with a structure analogous to the double copy in quantum field theory, incorporating loop momentum integrals and modular invariance via theta functions and KLT kernels.

ABSTRACT

We discuss relations between closed and open string amplitudes at one-loop. While at tree-level these relations are known as Kawai-Lewellen-Tye (KLT) and/or double copy relations, here we investigate how such relations are manifested at one-loop. While there exist examples of one-loop closed string amplitudes that can strikingly be written as sum over squares of one-loop open string amplitudes, generically the one-loop closed string amplitudes assume a form reminiscent from the one-loop doubly copy structure of gravitational amplitudes involving a loop momentum. This double copy structure represents the one-loop generalization of the KLT relations.

Motivation & Objective

• To extend the Kawai-Lewellen-Tye (KLT) relations from tree-level to one-loop amplitudes in string theory.
• To establish a one-loop generalization of the double copy structure, analogous to field-theory loop-level double copy constructions.
• To demonstrate how closed string one-loop amplitudes can be expressed as sums over squares of open string one-loop amplitudes, incorporating loop momentum and modular invariance.
• To derive a manifestly modular-invariant expression for one-loop amplitudes using analytic continuation of vertex operator positions and torus integrals.
• To connect the string-level double copy to field-theory limits, recovering known color-kinematics duality at loop level.

Proposed method

• Derives one-loop closed string amplitudes as integrals over the torus moduli space, using the complex structure of the worldsheet and modular properties.
• Performs analytic continuation of closed string vertex coordinates into real boundary coordinates, linking them to open string amplitudes on the boundary.
• Expresses the one-loop amplitude as a sum over products of one-loop open string integrals, with a kernel structure generalizing the tree-level KLT kernel.
• Incorporates loop momentum integrals via a Fourier expansion of the worldsheet Green's function, leading to a representation involving sums over momenta and winding modes.
• Uses theta functions and Dedekind eta functions to express modular invariants, ensuring modular covariance of the final amplitude.
• Applies level-matching conditions and delta-function constraints to enforce consistency with the physical spectrum, leading to a field-theory limit.

Experimental results

Research questions

• RQ1 How can the KLT relations be generalized from tree-level to one-loop amplitudes in string theory?
• RQ2 What is the structure of one-loop closed string amplitudes in terms of open string amplitudes and loop momentum integrals?
• RQ3 How does the double copy structure manifest at the one-loop level in string theory, particularly in relation to modular invariance?
• RQ4 Can the one-loop closed string amplitude be written as a sum over products of one-loop open string amplitudes, and what is the role of the KLT kernel in this context?
• RQ5 What is the field-theory limit of the one-loop double copy, and how does it recover color-kinematics duality?

Key findings

• The one-loop closed string amplitude for two external states is expressed as a sum over products of one-loop open string amplitudes, with a structure that generalizes the tree-level KLT relations.
• The amplitude takes the form of a loop integral over momentum space, with a kernel involving theta functions and modular forms, ensuring modular invariance.
• The final expression for the amplitude includes delta-function constraints from level-matching, leading to a field-theory limit that matches known results.
• The derivation confirms that the one-loop closed string amplitude can be written as a sum over squares of one-loop open string amplitudes, with the KLT kernel extended to loop-level via loop momentum integration.
• The result is consistent with the field-theory limit, recovering the double copy structure in quantum gravity and confirming the presence of color-kinematics duality at one loop.
• The method applies to both massless and massive states, and to both supersymmetric and non-supersymmetric cases, showing broad generality.