Unraveling the Universe: Numerics in Lattice QCD

One of the most captivating questions of our time is: What is the world made of, and what holds it together?

Quantum Chromodynamics (QCD) endeavors to provide an answer to this profound question. First proposed in 1973, QCD constitutes the sector of the standard model responsible for describing the strong nuclear force, which binds quarks and gluons within hadrons. The theory confines the constituents of hadrons, which are never directly observed in experiments. 

Consequently, establishing connections between experimental phenomena and the intricacies of strongly coupled quantum field theory remains a formidable challenge.In particular, the nature of confinement and the physical signatures of confined gluons remain poorly understood. Over the preceding decades, QCD has developed into an interdisciplinary research area, with close connections between theoretical physics and numerical analysis. 
Certain aspects of QCD cannot be described from a perturbative point of view. Hence, QCD physics is simulated via a discretization in space-time on a four-dimensional hypercubic lattice. This scientific discipline is called lattice QCD. No way of computing the path integral of lattice QCD analytically is known. Even on a finite lattice, it amounts to solving a very high-dimensional integral. The integral receives large contributions only from minuscule fractions of the space of integration variables, which suggests one way to estimate the path integral is through importance sampling using Markov chain Monte Carlo (MCMC) methods. 

Mostly, the Hybrid Monte Carlo (HMC) method is used to produce a Markov chain of configurations, which forms the basis for the computation of the desired expectation values. The HMC algorithm consists of a molecular dynamics (MD) step and an acceptance step. In the MD step, Hamiltonian equations of motion have to be solved using numerical integration schemes. On the one hand, time-reversibility and volume preservation of the numerical integration scheme are essential to reach the equilibrium distribution of the Markov chain. On the other hand, gluons are described as elements of the special unitary group SU(3), a non-Abelian Lie group. Thus, the Hamiltonian equations of motion arise in the form of differential equations on SU(3) and the associated Lie algebra. Therefore, the numerical integration scheme also has to satisfy the closure property, ensuring that the computed configurations are elements in SU(3).

The geometric numerical integration in the MD step causes the main computational cost in the HMC algorithm. Thus, the derivation of less expensive numerical integrators is crucial to efficiently utilize the computing power of high-performance computers. 

Efficient geometric integration schemes for lattice QCD are obtained by exploiting the particular structure of the Hamiltonian system. 
Firstly, it is a separable Hamiltonian system, enabling the use of operator splitting techniques so that the flows of the subsystems can be computed exactly. Force-gradient integrators provide a tool to obtain more efficient splitting methods. They can be regarded as common splitting methods, applied to a modified potential, resulting in a more accurate approximation with respect to the original Hamiltonian system.
Secondly, the potential part can be split into different parts with respect to their dynamical behavior and evaluation cost. In particular, the overall force can be split into the gauge force and the fermion force. Whereas the gauge force is characterized by cheap evaluation cost, the fermion force is very expensive to evaluate since it demands the inverse of the discretized Dirac operator. At the same time, the fermion force does not contribute much to the dynamics of the solution. Consequently, the use of multiple time-stepping techniques results in a more efficient computational process. One approach to multiple time-stepping techniques is within the framework of splitting (and thus force-gradient) methods. In the physics literature, these are known as nested integration techniques

Adapted nested force-gradient integrators combine the ideas of force-gradient integrators and multiple time-stepping techniques. They have been successfully applied to the Schwinger model, a well-known benchmark problem that shares many features of QCD simulations. 

As part of the research unit FOR5269: Future methods for studying confined gluons in QCD financed by the German Research Foundation (DFG)Michael Günther (professor) and Kevin Schäfers (Phd student) from the University of Wuppertal work on a generalized framework for adapted nested force-gradient integrators, enabling the derivation of even more efficient geometric integrators by using appropriate tuning models. In order to make all algorithms available to the Lattice community, they will be implemented in the open-source software package openQCD.

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