# Scattering Amplitudes I: KLT Relations and the Open–Closed Link

## Overview

RU is a certificate engine for universality and obstructions in constrained nonnegative propagation. Host structures supply finite inputs. Outputs are audited certificates or finite obstruction witnesses.

## RU fit

- **Fit level:** Low

## RU instantiation hooks

- Amplitude relations are structurally important but not a primary RU bridge lane because RU is a positivity-driven propagation engine.
- Retained as context; treat as out-of-scope for direct certificate instantiation unless a positive propagation reduction is explicitly built.

## Purpose

This document explains a very direct and widely reused export from string theory into mainstream quantum field theory:

- **KLT relations** connect closed-string and open-string tree amplitudes.
- In the low-energy limit, these relations imply deep structural links between **gravity** and **gauge theory** scattering.

This is a strong example of “universal host suitability for RU instantiation” because the resulting field-theory tools are used far beyond string theory itself.

## The core statement

Kawai, Lewellen, and Tye showed that any closed-string tree amplitude can be expressed as a sum over products of open-string tree amplitudes with a specific momentum-dependent kernel.

In slogans that are accurate at tree level:

- **closed string ≈ (open string) × (open string)**

This is not poetic. It is a concrete formula.

## Why the relation exists (conceptual picture)

A worldsheet view helps:

- Open-string amplitudes are built from holomorphic data with boundary conditions.
- Closed-string amplitudes factorize into left-moving and right-moving sectors.
- The KLT kernel encodes how the integration regions on the sphere relate to those on the disk.

You do not need the full derivation to use the result. What matters for application is that KLT provides a bridge from gauge-like building blocks to gravity-like outputs.

## The low-energy field-theory limit

When you take α′ → 0 (string length going to zero), the string amplitudes reduce to field-theory amplitudes:

- Open-string → Yang–Mills
- Closed-string → gravity

KLT then implies that tree-level gravity amplitudes can be written as combinations of products of Yang–Mills amplitudes.

This is one of the origins of the modern “gauge–gravity relations” program in amplitudes.

## Why this is reducible into RU certificates in practice

The host suitability for RU instantiation is concrete:

- Gravity amplitudes are often more complex than gauge-theory amplitudes.
- KLT turns gravity computations into combinations of gauge computations, which can be easier due to color ordering and known recursion structures.

Even when one does not care about strings, one cares about this relation because it:

- Organizes gravity calculations
- Reveals hidden cancellations
- Suggests deeper dualities exploited at loop level by modern methods

## What KLT is and is not

KLT is:

- A precise tree-level relation between amplitudes.

KLT is not:

- A claim that gravity equals gauge theory in all regimes.
- A loop-level statement by itself (though it inspires loop-level structures, especially when combined with unitarity methods).

## How this connects to the broader “universal host suitability for RU instantiation” arc

In the “host suitability for RU instantiation” story, KLT is a textbook example of a transferable method:

- Start in string theory where the amplitude structure is geometrically constrained.
- Take a limit that yields field-theory scattering.
- Export the resulting identities as tools in pure QFT and gravity calculations.

This is why string theory remains relevant even to researchers who are not doing string compactifications.

## References to start with

- Kawai, Lewellen, Tye, “A Relation Between Tree Amplitudes of Closed and Open Strings,” *Nuclear Physics* B269 (1986) 1–23.