Sector decomposition

Sector decomposition. It is a mathematical technique primarily used in high-energy physics to numerically evaluate multi-loop Feynman integrals that are infrared divergent. The method systematically disentangles overlapping singularities in the parameter space of integral representations, transforming them into sums of integrals where singularities are factorized and can be extracted using dimensional regularization. This process is crucial for making higher-order perturbative calculations in the Standard Model feasible, enabling precise theoretical predictions for collider experiments at facilities like the Large Hadron Collider.

Introduction

The development of sector decomposition was driven by the need to compute complex scattering amplitudes beyond leading order in quantum chromodynamics and electroweak theory. Traditional methods, such as analytic continuation, often struggled with integrals featuring intricate, overlapping ultraviolet divergences and soft divergences. Pioneering work by Thomas Binoth and Gudrun Heinrich established the modern algorithmic approach, building on earlier ideas in mathematical physics. The technique's power lies in its ability to handle integrals arising from multi-leg processes, which are essential for accurate simulations at the Tevatron and the LHC. Its implementation has become a cornerstone in automated computation tools like FIESTA and SecDec.

Mathematical formulation

The core procedure operates on integrals represented in Feynman parametric form. Beginning with an integral over parameters $x\_i$, the first step involves decomposing the integration domain, typically the unit hypercube, into sectors where a specific ordering of variables is enforced. This is achieved through iterative splits, often guided by strategies like the Hepp sector method or the Speer sector formalism. Within each sector, a sequence of variable transformations is applied to factorize the integrand, isolating singularities as monomials in the epsilon expansion parameter of dimensional regularization. The final result is a Laurent series in $\backslash epsilon$, where coefficients are finite integrals suitable for numerical integration by libraries such as CUBA.

Applications in particle physics

Sector decomposition is indispensable for calculating next-to-next-to-leading order corrections to cross sections and differential distributions. It has been applied to critical processes like Higgs boson production via gluon fusion, top quark pair production, and vector boson scattering. These calculations, vital for interpreting data from the ATLAS experiment and the CMS experiment, test the Standard Model and constrain beyond the Standard Model physics. The method also facilitates computations in effective field theories like SCET, and has been used in precision electroweak studies at LEP and the SLAC.

Algorithmic implementation

Several public codes automate the sector decomposition workflow. The SecDec program, developed by Sophia Borowka and collaborators, and FIESTA, created by A.V. Smirnov, are widely used in the community. These programs integrate with Feynman diagram generators like FeynArts and QGRAF, performing the algebraic decomposition, singularity subtraction, and numerical integration. Key algorithmic challenges include optimizing the sector selection to minimize their number and managing the algebraic complexity of high-multiplicity processes. Recent advances leverage parallel computing on high-performance computing clusters and GPU acceleration to tackle integrals with many propagators.

Generalizations and related techniques

The formalism has been extended beyond multi-loop integrals. A notable generalization is amplitude decomposition, applied directly to Feynman integrals before parametric integration. Connections exist with techniques in algebraic geometry, such as resolution of singularities and the study of Newton polytopes. In mathematics, related concepts appear in the study of iterated integrals and periods. Other computational methods in perturbative QCD include integration by parts identities, solved using algorithms like Laporta's algorithm, and differential equations techniques advanced by Johannes Henn. Sector decomposition also finds parallels in statistical physics for evaluating certain critical exponents.

Category:Mathematical physics Category:Perturbation theory Category:Computational physics