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Caravel: A C++ Framework for the Computation of Multi-Loop Amplitudes with Numerical Unitarity

S. Abreu, J. Dormans, F. Febres Cordero, H. Ita, M. Kraus, B. Page, E. Pascual, M. S. Ruf, V. Sotnikov

TL;DR

Caravel introduces the first public C++17 framework for computing multi-loop scattering amplitudes via numerical unitarity, combining D-dimensional integrand decomposition, finite-field arithmetic, and functional reconstruction. The paper details a modular software architecture with core arithmetic, tree-level currents, integrand-space construction, and master-integral libraries, plus example programs spanning tree, one-loop, two-loop, and finite-remainder calculations in QCD and Einstein gravity. It also showcases analytic reconstruction of amplitudes from numerical data, using MPI for parallelization and finite-field rational-function reconstruction to obtain analytic results. While not yet fully automated for arbitrary multi-loop multi-leg processes, Caravel provides a flexible platform to extend master integrals and process-specific inputs, with practical impact for precision predictions at the LHC and gravity scattering studies.

Abstract

We present the first public version of Caravel, a C++17 framework for the computation of multi-loop scattering amplitudes in quantum field theory, based on the numerical unitarity method. Caravel is composed of modules for the $D$-dimensional decomposition of integrands of scattering amplitudes into master and surface terms, the computation of tree-level amplitudes in floating point or finite-field arithmetic, the numerical computation of one- and two-loop amplitudes in QCD and Einstein gravity, and functional reconstruction tools. We provide programs that showcase Caravel's main functionalities and allow to compute selected one- and two-loop amplitudes.

Caravel: A C++ Framework for the Computation of Multi-Loop Amplitudes with Numerical Unitarity

TL;DR

Caravel introduces the first public C++17 framework for computing multi-loop scattering amplitudes via numerical unitarity, combining D-dimensional integrand decomposition, finite-field arithmetic, and functional reconstruction. The paper details a modular software architecture with core arithmetic, tree-level currents, integrand-space construction, and master-integral libraries, plus example programs spanning tree, one-loop, two-loop, and finite-remainder calculations in QCD and Einstein gravity. It also showcases analytic reconstruction of amplitudes from numerical data, using MPI for parallelization and finite-field rational-function reconstruction to obtain analytic results. While not yet fully automated for arbitrary multi-loop multi-leg processes, Caravel provides a flexible platform to extend master integrals and process-specific inputs, with practical impact for precision predictions at the LHC and gravity scattering studies.

Abstract

We present the first public version of Caravel, a C++17 framework for the computation of multi-loop scattering amplitudes in quantum field theory, based on the numerical unitarity method. Caravel is composed of modules for the -dimensional decomposition of integrands of scattering amplitudes into master and surface terms, the computation of tree-level amplitudes in floating point or finite-field arithmetic, the numerical computation of one- and two-loop amplitudes in QCD and Einstein gravity, and functional reconstruction tools. We provide programs that showcase Caravel's main functionalities and allow to compute selected one- and two-loop amplitudes.

Paper Structure

This paper contains 32 sections, 29 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Computational methodology
  3. Integrand parametrization
  4. Integrand factorization and cut equations
  5. Special functions
  6. Analytic structure in the dimensional regulators
  7. Finite fields and functional reconstruction
  8. Internal modules
  9. Installation and setup
  10. Example programs
  11. Helicity amplitudes
  12. Specifying program input
  13. Specifying particles
  14. Specifying amplitudes
  15. Specifying kinematics
  16. ...and 17 more sections

Figures (2)

  • Figure 1: Illustration of the interplay of the different modules in Caravel in generic calculations of scattering amplitudes. A black arrow indicates a dependence, a blue arrow means input for a module, and a red arrow the capacity of a module to deliver a given component of the calculation. Modules surrounded by a red box rely on external input to operate. All modules depend on the Core module.
  • Figure 2: Specifying momentum routing for the double-box tensor integral.