Lattice Quantum Chromodynamics Comes of Age Free

In formulation, QCD and QED are strikingly similar. Both are gauge-invariant quantum field theories. The key difference is that photons in QED are neutral; so they can’t interact directly with each other. The gluon is the QCD analog of the photon; it carries the strong force between quarks. But quite unlike photons, gluons do carry color charge, the analog of electric charge. So gluons interact directly with each other as well as with quarks. (See the article by Frank Wilczek in Physics Today, Physics Today 0031-9228 53 8 2000 22  https://doi.org/10.1063/1.1310117 August 2000, page 22 .)

That seemingly innocent change has dramatic consequences for phenomenology. It is the root of QCD’s daunting complexity. Electrons, positrons, and photons can be separated and isolated at macroscopic distances. Quarks, antiquarks, and gluons cannot. This prohibition, called color confinement, assures that all the elementary particles (the hadrons) composed of quarks, antiquarks, and gluons come in precise color-neutral combinations. Loosely speaking, this means that they come either in quark–antiquark pairs (the mesons) or in triplets of quarks (the baryons). Several recently discovered “pentaquark” baryons appear to combine a quark triplet with a quark–antiquark pair (see page 19 of this issue.)

Why only color-neutral combinations? In QCD, quarks can have three colors. Conventionally, they are labeled red, blue, and green, but of course they have nothing to do with optics. Antiquarks have the corresponding anticolors. Triplets of quarks containing equal portions of the three colors are color neutral.

Try to pry loose one of the three valence quarks in a proton. Before going much farther than the radius of the proton (about 1 fm or 10⁻¹³ cm), you’ve done enough work to create a new quark–antiquark pair. Pairs promptly appear, choose new partners, and you find a meson in one hand and a proton or neutron in the other. No isolated quarks!

At distances an order of magnitude smaller than 1 fm or, equivalently, at interaction energies or momentum transfers in the multi-GeV range, α_S, the energy-dependent QCD analog of the fine-structure constant, is effectively weak. In that limited regime, perturbation theory works, and pencil-and-paper methods succeed. But for the larger distances and softer interactions, where confinement is the dominating process, α_S is effectively large and we must resort to computerized numerical simulation.

In 1974, Kenneth Wilson at Cornell University formulated a version of QCD on a discrete spacetime lattice (see the left panel of figure 1) and, with pencil and paper, used it to provide a plausible, but not rigorous, argument for color confinement. ¹ Wilson argued that, on a coarse spacetime lattice, the potential energy of separation of a quark and an antiquark must rise linearly with distance. In 1979 at Brookhaven National Laboratory, Michael Creutz, Laurence Jacobs, and Claudio Rebbi demonstrated the feasibility of doing meaningful numerical simulations with Wilson’s formulation on a Control Data Corp 7600 computer. ² Shortly thereafter, Creutz obtained numerical results for the confinement potential that supported Wilson’s conclusions. That success launched a new branch of computational physics, called lattice gauge theory or lattice QCD. ³ The right panel of figure 1 shows a modern lattice-QCD result for the quark–antiquark potential. ⁴  

For two decades after Creutz’s pioneering 1979 calculations, refinements in algorithms and computing power brought steady gains in precision and consistency. But only in the past four years have powerful algorithmic and theoretical improvements launched us into the age of high-precision lattice QCD—at least for some key hadronic quantities.

By the standards of the strong interactions, “high precision” means 1 or 2%. The impact of this new precision extends beyond the strong interactions. Determining key features of the weak interactions of hadrons—for example, the Cabibbo-Kobayashi-Maskawa (CKM) parameters—requires correcting measured weak decay rates for strong-interaction effects (see box 1). The uncertainties in our knowledge of such fundamental parameters limits the precision with which the standard model of the elementary particles can be tested and probed for new realms of physics.

The most important theoretical advance in recent years has been the development of improved actions, that is, improved methods of formulating QCD on the lattice. As in classical field theory, the QCD action is the integral, over space and time, of the Lagrangian density. In lattice calculations, this four-dimensional integral is approximated by summing over discrete lattice points in spacetime.

With substantial computational resources at NSF and DOE national centers during the past three years, lattice gauge theorists have used an improved “staggered fermion” (ISF) action to generate, and make publicly available, a large set of gauge-field configurations (see box 2). ⁵ Staggered fermion actions, introduced by John Kogut and Leonard Susskind in 1976, are so called because the algorithm spreads the fermion spins over adjacent lattice points.

The newly available gauge-field configurations include the vacuum-polarization (quark loop) effects of u, d, and s quarks. Several lattice-QCD collaborations, working together, ⁶ have recently used these configurations to determine a variety of hadronic quantities to an unprecedented accuracy of 3%. All of those quantities had been measured previously in the laboratory. Figure 2 plots the ratio of the simulated value to the experimental one for each observable. The only inputs were a few experimentally known hadron masses that were used to determine the lattice spacing and the masses of five of the quark flavors. The t quark is too heavy to contribute. The rest is pure prediction.

The left panel shows the result from a widely used quenched approximation (omitting quark-loop effects). The right panel shows what happens when quark loops are included in the calculations. The results show that quark-loop effects are essential; when they are included, the agreement with experiment is encouraging.

The quantities shown in figure 2 demonstrate the predictive capabilities of lattice QCD. The two hadronic decay parameters f _π and f _K, which describe the strong-interaction contribution to the weak decays of the π and K mesons, measure their quark–antiquark wavefunctions at the origin. The particular mass-difference combinations shown in the figure—involving the nucleon, the doubly strange Ξ baryon, and the ground states of the B_S (strange and bottom flavored) and γ meson families—were chosen because those linear combinations are rather insensitive to a variety of systematic errors.

The flavor-neutral Ψ (charmonium) and γ (bottomonium) meson families, quark analogs of positronium, are, respectively, cc and bb bound states. The figure shows level splittings between different orbital states of these “quarkonium” families. The c and b quarks, with masses of a few GeV, are a thousand times heavier than the u and d.

For the hadronic quantities in figure 2, extrapolation to the physical u and d masses is well under control. But many other important quantities, such as the nucleon mass itself, present greater difficulties. We expect, however, to achieve comparable precision with the nucleon mass, for example, once we have developed the extrapolation procedure for it to the same level of sophistication we already have for the π and K mesons.

As a byproduct of these calculations, one gets a new value for the color fine-structure coupling α_S by combining a nonperturbative lattice determination of energy scales with lattice perturbation theory. This quantity is traditionally calculated at very high energies, where perturbation theory applies. Although the starting energy scale of the new determination is two orders of magnitude lower than that of the perturbative calculation, the two values turn out to agree reassuringly well. For α_S at 91 GeV (the mass of the Z⁰ weak boson, the conventional point of comparison), the current lattice QCD calculation gives 0.121 ± 0.003. The world average from other determinations is 0.117 ± 0.002.

▸CKM matrix elements. An immediate scientific objective is to determine the weak decay constants f _D for the charm-flavored D mesons and f _B for the analogous bottom-flavored B mesons. One needs those constants to extract the CKM matrix elements (box 1) from the experimentally measured meson decay rates. As we have done with f _π and f _K, we expect to determine the heavy-flavored meson decay constants to high precision.

Another important strong-interaction parameter, B _K, characterizes the influence of the strong interactions on the remarkable and well-known quantum mixing phenomenon in which a neutral kaon oscillates between the positive strangeness K⁰ and negative strangeness K ⁰ states. The analogous but less well-measured process involving the B⁰ and B ⁰ meson states is under current study at the SLAC BaBar collider and the Belle collider at KEK in Japan (see Physics Today, Physics Today 0031-9228 54 5 2001 17  https://doi.org/10.1063/1.1381091 May 2001, page 17 ). Both measurements also determine elements of the CKM matrix. Study of B-meson mixing and decay is an important part of the Fermilab Tevatron program.

▸The quark–gluon plasma. The cores of sufficiently dense stars are expected to contain an intriguing phase of matter, predicted by QCD, in which hadrons are so crowded that quarks and gluons are “liberated” and move as if they were in a kind of deconfined plasma. The universe was very likely such a quark–gluon plasma (QGP) for a brief moment after the Big Bang. Such a phase is anticipated to manifest itself at extremely high temperature or density. Experiments under way at Brookhaven’s Relativistic Heavy Ion Collider (RHIC) and at CERN bring heavy nuclei into high-energy collision in hopes of creating the QGP for brief instants (see the article by Thomas Ludlam and Larry McLerran in Physics Today, Physics Today 0031-9228 56 10 2003 48  https://doi.org/10.1063/1.1629004 October 2003, page 48 ).

Although lattice-gauge simulations do not describe the kinematics of a heavy-ion collision, they can predict the equilibrium properties of the QGP: the phase diagram, the equation of state, and fluctuations in particle number. Figure 3 contains the result of a recent lattice-gauge calculation that shows how the fluctuation of net strangeness density increases with temperature in a hot, flavor-neutral ensemble of quarks and antiquarks. ⁷ The inflection point near 180 MeV in the figure is taken to indicate the transition to the QGP from a lower-temperature phase of quarks and antiquarks confined in hadrons. Such calculations provide crucial input for phenomenological models of QGP formation and decay.

In a flavor-neutral ensemble, with equal populations of quarks and antiquarks, the net baryon number vanishes. (A quark carries baryon number +⅓. For an antiquark, it’s −⅓). A long-standing algorithmic challenge has been to find a way of simulating QCD at nonzero net baryon-number density—for example, in stellar interiors or the “nuclear fragmentation” regions of phase space in relativistic heavy-ion collisions.

Introducing a nonzero chemical potential to push the baryon density away from zero makes the determinant of the quark-action matrix (box 2) complex. Then the weighting factor in the first equation in box 2 can no longer be interpreted as a probability, and the usual importance-sampling techniques lose their effectiveness. (A similar complication plagues condensed-matter physics calculations for systems in which the conduction-electron occupancy deviates from half filling.) Good progress has been made recently in obtaining results for small net baryon density. ⁸  

▸Hadronic structure. The internal structure of the nucleon is fundamental to nuclear physics. A principal scientific goal is to determine quantitatively how quarks and gluons produce the binding and spin of the nucleon. To that end, vigorous experimental programs are in progress at MIT, Thomas Jefferson National Accelerator Facility, SLAC, Fermilab, DESY, and CERN. (See page 9 of this issue.) RHIC will soon be joining the enterprise. The nucleon is easy to study by lattice QCD, because it is the lightest of the three-quark baryons. And determining the static properties of hadrons is natural for lattice QCD. We hope to understand the nucleon’s structure through a combination of numerical simulation and experiment.

Lattice-gauge theorists use the Feynman path-integral technique to quantize the field theory (see box 2). The Feynman approach actually leads us to carry out calculations with an imaginary time coordinate. That feature is standard in statistical quantum mechanics. In fact, the Feynman integration over alternative paths determines the partition function for an ensemble of interacting gluons, quarks, and antiquarks in thermal equilibrium. In lattice-QCD calculations, the temperature is inversely proportional to the duration of the whole lattice volume in imaginary time.

If we keep the imaginary-time duration small, we can study high-temperature features such as the QGP. But if we make the duration large enough, we’re simulating a temperature close to zero. Quantities of interest at zero temperature include the masses of a wide variety of hadrons, their decay amplitudes, the quark–antiquark potential, and various static properties of hadrons, such as the internal distributions of charge and magnetization.

Lattice QCD near zero temperature also addresses the complicated structure of the vacuum. The vacuum state of QCD, its zero-temperature ground state, is remarkably rich in structure. ⁹ The gluon field fluctuates with twists and turns, tracing out topological knots called instantons. Understanding the ground state is fundamental to understanding QCD.

We cannot, however, calculate everything. Because of its close relationship to statistical thermodynamics, lattice QCD in its current formulation is unsuited for simulating real-time processes such as multiparticle scattering and the nonequilibrium behavior of the QGP. For such processes we rely on phenomenological models to extrapolate from the domain where lattice QCD does work.

The principal computational challenges faced by lattice QCD are reducing discretization errors and extrapolating down to the small physical masses of the u and d quarks:

▸Discretization errors. Representing spacetime by a regular grid of discrete points introduces artifacts that become small as the lattice spacing a is decreased. However, the computational cost (the number of requisite computer operations) grows very steeply with decreasing a—something like a ⁻¹ or a ⁻⁸—when the computation includes quark-loop effects. In the end, one must extrapolate to zero lattice spacing. To improve the accuracy of that extrapolation, we place a high premium on finding improved algorithms that reduce discretization artifacts. Currently, large-scale computations with improved action algorithms work with a lattice spacing as small as 0.09 fm.

▸Light-quark masses. Computational cost, at fixed a, grows approximately as the inverse square of the quark masses in question. That makes it too expensive to let the u and d quarks be as light as they are in nature. (The proton is a hundred times heavier than the sum of its three valence quarks.) In the limit of vanishing quark masses, QCD has a special “chiral” symmetry from which one can define a perturbative expansion in the small quark mass and thus anchor the extrapolation down from the unphysically high masses used in the lattice calculations. So if we can simulate QCD in the regime where chiral perturbation theory applies, we can extrapolate from lattice QCD to the small u and d masses with some confidence. In current large-scale lattice simulations, it is feasible to take the u and d masses as low as three times their physical values. That’s well within the range of chiral perturbation theory and well below the quark-mass values that had to be used in lattice calculations before the ISF improvement.

The key to the recent advances in lattice-gauge theory has been the development of improved lattice actions for describing the motion and interaction of quarks and gluons. The improvements refine the discretization of the quark action. The ISF action is the most extensively exploited of these algorithmic improvements. ⁵  

Let us examine one of the steps in the improvement process. In a lattice simulation, the simplest term describing the interaction between quarks and gluons involves the inner product of an antiquark field at one lattice site and the quark field at a neighboring site. To maintain gauge invariance, however, one also has to include in that product the gluon matrix U on the link joining the two lattice sites.

The improvement schemes use not only the shortest path to connect adjacent sites but also a combination of products of gluon matrices along longer, more circuitous paths between immediate and more distant neighbors, as shown in figure 4. With the correct linear combination of such paths, we can reduce discretization errors, in effect, by smoothing the sharp corners of the lattice. We remove errors proportional to a ². So we’re left with errors that scale like a ⁴ and a ² α _S. The added complexity of the circuitous paths increases the computational cost by a factor of two or three, but the greater cost is handsomely repaid in better accuracy at modest lattice spacing.

One way to measure the effectiveness of the improved formulation is to observe how calculated quantities vary with lattice spacing. For example, figure 5 compares the lattice-spacing dependence of the ISF-action calculation of the nucleon mass with calculations that use older action algorithms. ¹⁰ To avoid the added complication of extrapolation, all of these lattice calculations used unrealistically heavy quarks. Therefore the nucleon mass, in the limit of vanishing a, comes out about 300 MeV too heavy. But what matters in this comparison of quark-action algorithms is the sensitivity to lattice spacing. The smaller the slope, the better the improvement. Clearly, the ISF action does better in this test.

Given sufficient computational resources (see box 3), the future prospects for high-precision lattice calculations are excellent. The ISF action is only one of several improved actions currently being investigated. ¹¹ Others make even further improvements. ¹² Their formulation is more complicated; their importance-sampling algorithms are not yet ready for large-scale computation. But we expect significant gains in the near future.

An altogether different approach emphasizes a more accurate treatment of chiral symmetry at nonzero lattice spacing that provides much better control of the extrapolation to the physical u and d quark masses. ^{13   14} That approach is considerably more expensive than the ISF action, but some promising work is in progress. It’s important to pursue alternatives to ISF for two reasons. First, the implications of some of the approximations used in the ISF simulation are not completely understood. Second, systematic uncertainties in the chirally accurate methods and the ISF methods are sufficiently different that they provide good cross checks.

A host of interesting fundamental physics questions await application of the new tools. These include stringent tests of the standard model of particle physics, full characterization of the quark—gluon plasma, quantitative determination of the structure of the nucleon, and the prediction of masses and decay channels for observed and conjectured exotic hadronic states, including pentaquarks, purely gluonic particles called glueballs, and quark—gluon hybrids (see Physics Today, September 2003, page 19).

Carleton DeTar is a professor of physics at the University of Utah in Salt Lake City.

Steven Gottlieb is a professor of physics at Indiana University in Bloomington.