particle physics, precisely

improved algorithms and faster computers enable first high-precision calculations of hadron properties

Now, with faster computers, improved algorithms that employ fewer simplifications of physical processes, and better-performing codes, four QCD collaborations involving 26 researchers have reported calculations of nine different hadron masses, covering the entire range of the hadron spectrum, with an error rate of 3% or less. This report, published in the journal Physical Review Letters under the title “High-precision lattice QCD confronts experiment,” [1] marks the first time that lattice QCD calculations have achieved such precise results for such diverse physical quantities using the same QCD parameters.

QCD is the theory of the “strong force,” by which gluons bind quarks together to form hadrons. QCD theorists use computer simulations to determine the most probable arrangements of gluons and quarks inside a particle and then use these configurations to determine the particle’s properties. Calculating all the possible positions of quarks and gluons would be impossible, so theorists simplify the problem by imagining space and time not as a continuum, but as a lattice — a four-dimensional grid of discrete points at which quarks and gluons can reside. This approach, called lattice QCD, transforms an impossible problem into one that is extremely difficult but, in principle, solvable.

Lattice QCD has already explained why quarks do not exist as separate particles, explored the symmetries of the strong force, and predicted the temperature at which protons and neutrons melt. Soon theorists expect they will be able to make predictions that can be tested experimentally in particle accelerators, probing the limits of the Standard Model of particle interactions.

“Lattice QCD simulations are needed to obtain a quantitative understanding of the physical phenomena controlled by the strong interactions, to determine a number of the basic parameters of the Standard Model, and to make precise tests of the Standard Model’s range of validity,” explains Doug Toussaint, Professor of Physics at the University of Arizona and a member of the MILC Collaboration, one of the four teams that co-authored the “QCD confronts experiment” paper.

“Despite the many successes of the Standard Model,” Toussaint says, “high energy physicists believe that to understand physics at the shortest distances a more general theory will be required, which unifies all four of the fundamental forces of nature.” Those four forces are the strong force; the weak force, which causes a type of radioactive decay; electromagnetism, which ties electrons to atomic nuclei; and gravity, which theorists have found difficult to integrate with the other three forces.

Figure 1. The decay rate of a heavy bottom quark in a B meson, a process controlled by the weak force, is obscured by the binding effect of the strong force in a dynamic “sea” of quark-antiquark and gluon activity. Accurate lattice QCD calculations that include this activity are needed in order to correct the uncertainties in experimentally measured decay rates.

“The Standard Model is expected to be a limiting case of this more general theory, just as classical mechanics is a limiting case of the more general quantum mechanics,” Toussaint says. “A central objective of the experimental program in high energy physics, and of lattice QCD simulations, is to determine the range of validity of the Standard Model, and to search for new physics beyond it. Thus, QCD simulations play an important role in efforts to obtain a deeper understanding of the fundamental laws of physics.”

One way of determining the validity of the Standard Model is to tabulate the rates at which heavy quarks decay into lighter ones in an array of numbers known as the Cabibbo-Kobayashi-Maskawa (CKM) matrix. The CKM matrix elements — some of which are still unknown — define the intersection between the strong and weak forces in events that involve both. If the Standard Model is complete, then certain combinations of these rates will add up to 100%. If they add up to something else, then new particles or forces must await discovery.

Unfortunately, it is impossible to directly measure the decay of an individual heavy quark into a lighter one, because every quark is bound by a swarm of gluons to other quarks or to an antiquark. Experimenters have to measure the decays of composite particles, such as B mesons (Figure 1), and then try to calculate the CKM numbers by subtracting the effects of the extra quarks and gluons. That subtraction is easier said than done, because within each hadron is a dynamic “sea” of gluons exchanging gluons among themselves, as well as quark-antiquark pairs popping in and out of existence. The effect of all this frenetic strong-force activity is called “quark vacuum polarization,” and it has been one of the most difficult quantities to include in lattice QCD calculations.

Figure 2. Lattice QCD calculations divided by experimental measurements for nine different hadron masses, without quark vacuum polarization (left) and with quark vacuum polarization (right). A value of 1 represents perfect agreement.

Quark vacuum polarization is the most computationally expensive ingredient in a QCD simulation because moving quarks from real space into lattice space increases the number of quarks by a factor of 16. In the past, most QCD simulations have either omitted quark vacuum polarization altogether (“quenched QCD”) or inflated the lightest quark masses by a factor of 10 to 20, reducing both the computational effort and the accuracy of the results.

The “Symanzik-improved staggered-quark discretization” of the lattice, recently developed by the MILC Collaboration and others, includes the vacuum polarization effects of all three light quark flavors (up, down, and strange) and allows QCD simulations with much smaller and more realistic quark masses than previously possible. This algorithm improves on an older method called “staggering” — spreading each quark over four neighboring lattice points, and then eliminating many components to reduce the number of redundant quarks coming from the lattice. Redundant quarks are further reduced by rescaling their effect on the system’s evolution. In addition, unphysical interactions produced by staggering have been minimized.

Including quark vacuum polarization has dramatically improved the accuracy of lattice QCD calculations, as Figure 2 shows. These results validate the accuracy of the improved staggered-quark discretization. “In this study, we chose to calculate quantities whose values are well known experimentally,” says MILC collaborator Bob Sugar of the University of California, Santa Barbara. “This allowed us to verify that we have control over all sources of systematic errors.” Now that lattice QCD has accurately reproduced a range of well known quantities, researchers will have more confidence in the calculation of other quantities that experiments cannot determine.

Describing the MILC Collaboration’s work at NERSC, Sugar says, “The bulk of our allocation goes into the generation of lattices — snapshots of the systems we are simulating — using the improved discretization. Seaborg is one of only two machines available to us that can efficiently handle the large, low-temperature lattices. These lattices are saved, and then used to calculate a wide variety of physical quantities. In order to maximize the physics output from the large investment in the generation of these lattices, we are making them available to other theorists for their research.” The MILC collaborators produced the gluon configurations, as well as the raw simulation data for pions and kaons, that were used in the “QCD confronts experiment” paper.

Table 1: Impact of improved lattices on the determination of CKM matrix elements.

Figure 3. Constraints on the Standard Model parameters ρ and η, the least well known elements of the CKM matrix. For the Standard Model to be correct, they must be restricted to the region of overlap of all the solidly colored bands. The top figure shows the constraints as they exist today. The bottom figure shows the constraints as they would exist with no improvement in the experimental errors, but with all lattice gauge theory uncertainties reduced to 3%. [Image by R. Patterson from “The SCaLeS Report.” [3]]

The impact of improved lattice calculations on constraints to the Standard Model is summarized in Table 1, taken from a report by the Lattice QCD Executive Committee [2]. The first column indicates the quantity measured experimentally, the second column the effected CKM matrix element, and the third column the hadronic matrix element which must be calculated on the lattice. The fourth column shows the current non-lattice errors, which are primarily experimental, and the fifth the current lattice errors. The column labeled Lattice Errors Phase I shows the expected lattice errors once the analysis has been completed on existing lattices, and the column labeled Lattice Errors Phase II shows the expected lattice errors once analysis has been completed on the lattices currently being generated or planned for the near future. Note that in all cases for which experiments have been performed, the current lattice error is significantly larger than the experimental one. However, once the Phase II analysis has been completed, the lattice errors will be less than or comparable to the experimental ones. Figure 3 graphically depicts the impact of improved lattice calculations on constraints to the Standard Model.

QCD theory and computation are now poised to fulfill their role as an equal partners with experiment. Sugar comments, “The importance of this can be seen from the fact that a significant fraction of the $750 million per year that the United States spends on experimental high energy physics is devoted to the study of the weak decays of strongly interacting particles. To fully capitalize on this investment, the lattice calculations must keep pace with the experimental measurements.”

In addition to Toussaint and Sugar, the MILC Collaboration includes Christopher Aubin and Claude Bernard of Washington University, Tommy Burch of the University of Regensburg, Carleton DeTar and James Osborn of the University of Utah, Steven Gottlieb of Indiana University, Eric Gregory of the University of Arizona, Urs Heller of Florida State University, and James Hetrick of the University of the Pacific.

Research funding: HEP, SciDAC, NSF, PPARC

1. C. T. H. Davies et al. (HPQCD, UKQCD, MILC, and Fermilab Lattice Collaborations), “High-precision lattice QCD confronts experiment,” Phys. Rev. Lett. 92, 022001 (2004).

2. Lattice QCD Executive Committee, “Overview of Lattice Gauge Theory Infrastructure Project” (2004), http://www.physics.ucsb.edu/~sugar/hepap.pdf.

3. Phillip Colella, Thom H. Dunning, Jr., William D. Gropp, and David E. Keyes, eds., “A Science-Based Case for Large-Scale Simulation” (Washington, D.C.: DOE Office of Science, July 30, 2003).