Lattice QCD

Lattice QCD is a well-established non-perturbative approach to solving the quantum chromodynamics (QCD) theory of quarks and gluons. It is a lattice gauge theory formulated on a grid or lattice of points in space and time. When the size of the lattice is taken infinitely large and its sites infinitesimally close to each other, the continuum QCD is recovered. Lattice QCD was developed in the 1970s by Nobel laureate Kenneth Wilson. It was developed within a short interval of time after the theory of quantum chromodynamics had been discovered.

Analytic or perturbative solutions in low-energy QCD are hard or impossible to obtain due to the highly nonlinear nature of the strong force and the large coupling constant at low energies. This formulation of QCD in discrete rather than continuous spacetime naturally introduces a momentum cut-off at the order 1/a, where a is the lattice spacing, which regularizes the theory. As a result, lattice QCD is mathematically well-defined. Most importantly, lattice QCD provides a framework for investigation of non-perturbative phenomena such as confinement and quark–gluon plasma formation.

In lattice QCD, fields representing quarks are defined at lattice sites (which leads to fermion doubling), while the gluon fields are defined on the links connecting neighboring sites. This approximation approaches continuum QCD as the spacing between lattice sites is reduced to zero. Because the computational cost of numerical simulations increases as the lattice spacing decreases, results must be extrapolated to a = 0 (the continuum limit) by repeated calculations at different lattice spacings a.

Numerical lattice QCD calculations using Monte Carlo methods can be extremely computationally intensive, requiring the use of the largest available supercomputers. To reduce the computational burden, the so-called quenched approximation can be used, in which the quark fields are treated as non-dynamic "frozen" variables. While this was common in early lattice QCD calculations, "dynamical" fermions are now standard.

At present, lattice QCD is primarily applicable at low baryon densities where the numerical sign problem does not interfere with calculations. Monte Carlo methods are free from the sign problem when applied to the case of QCD with gauge group SU(2) (QC<sub>2</sub>D).

Lattice QCD has already successfully agreed with many experiments. For example, the mass of the proton has been determined theoretically with an error of less than 2 percent. Lattice QCD predicts that the transition from confined quarks to quark–gluon plasma occurs around a temperature of (), within the range of experimental measurements.

Lattice QCD has also been used as a benchmark for high-performance computing, an approach originally developed in the context of the IBM Blue Gene supercomputer.

Techniques

Monte-Carlo simulations

After Wick rotation, the path integral for the partition function of QCD takes the form

<math display="block"> Z = \int \mathcal{D} U \, e^{-S[U]}

= \int \prod_{x, \mu} dU_\mu(x) \, e^{-S[U]} </math>

where the gauge links <math>U_\mu(x) \in \mathrm{SU}(3)</math> range over all the sites <math>x</math> and space-time directions <math>\mu</math> in a 4-dimensional space-time lattice, <math>S[U]</math> denotes the (Euclidean) action and <math>dU_\mu(x)</math> denotes the Haar measure on <math>\mathrm{SU}(3)</math>. Physical information is obtained by computing observables

<math display="block"> \left\langle \mathcal{O} \right\rangle

= \frac{1}{Z} \int \mathcal{D} U \, \mathcal{O}(U)

e^{-S[U]} </math>

For cases where evaluating observables pertubatively is difficult or impossible, a Monte Carlo approach can be used, computing an observable <math> \mathcal{O} </math> as

<math display="block"> \left\langle \mathcal{O} \right\rangle

\approx \frac{1}{N} \sum_{i=1}^{N} \mathcal{O}(U_i)

</math>

where <math>U_1, \dots, U_{N}</math> are i.i.d random variables distributed according to the Boltzman distribution <math> U_i \sim e^{-S[U_i]}/Z </math>. For practical calculations, the samples <math>\{U_i\}</math> are typically obtained using Markov chain Monte Carlo methods, in particular Hybrid Monte Carlo, which was invented for this purpose.

Fermions on the lattice

Lattice QCD is a way to solve the theory exactly from first principles, without any assumptions, to the desired precision. However, in practice the calculation power is limited, which requires a smart use of the available resources. One needs to choose an action which gives the best physical description of the system, with minimum errors, using the available computational power. The limited computer resources force one to use approximate physical constants which are different from their true physical values:

The lattice discretization means approximating continuous and infinite space-time by a finite lattice spacing and size. The smaller the lattice, and the bigger the gap between nodes, the bigger the error. Limited resources commonly force the use of smaller physical lattices and larger lattice spacing than wanted, leading to larger errors than wanted.
The quark masses are also approximated. Quark masses are larger than experimentally measured. These have been steadily approaching their physical values, and since the 2000s a few collaborations have used nearly physical values to extrapolate down to physical values.

Limitations

The method suffers from a few limitations:

Currently there is no formulation of lattice QCD that allows us to simulate the real-time dynamics of a quark-gluon system such as quark–gluon plasma.
It is computationally intensive, with the bottleneck not being flops but the bandwidth of memory access.
Computations of observables at nonzero baryon density suffer from a sign problem, preventing direct computations of thermodynamic quantities.

References

External links

Gupta - Introduction to Lattice QCD
Lombardo - Lattice QCD at Finite Temperature and Density
Chandrasekharan, Wiese - An Introduction to Chiral Symmetry on the Lattice
Kuti, Julius - Lattice QCD and String Theory
The FermiQCD Library for Lattice Field theory
Flavour Lattice Averaging Group