By Professor Terry Draper
With the recent retirement of Professor Keh-Fei Liu (see the Chair’s welcoming remarks in the Spring 2023 newsletter), it is apropos to reflect on some of the history of the lattice QCD group at the University of Kentucky, which formed upon his partnership with yours truly who joined the department’s faculty in 1989. Within the past decade or two, most of our lattice QCD work has been done within the international χQCD collaboration, which has been comprised mostly of our current and former postdocs and students, and their postdocs and students, who are in faculty and postdoc positions across the U.S., Europe, and Asia.
Quantum Chromodynamics (QCD) is the accepted theory of the strong (nuclear) force, which together with the weak and electromagnetic forces is described theoretically within the Standard Model of particle physics, which (despite occasional, exciting, tantalizing, but so far ephemeral exceptions) has been able to account for experimental observation. A major challenge of such tests of the Standard Model and the seeking of experimental signatures of possible beyond Standard Model theories is in the theoretical difficulty of calculating processes involving QCD.
QCD is much like Quantum Electrodynamics (QED), the relativistic quantum field theory of the interaction of (electrically) charged particles, such as electrons, as mediated by the exchange of photons, the quanta of the electromagnetic field. In QED, the coupling constant is small and so QED is amenable to perturbative calculations. In QCD, the fundamental (color) charged particles are quarks and the quanta of the interaction are gluons. In the collision of composite particles such as protons at very high energies, the coupling constant is also small and thus analytic perturbative calculations are reliably made, as with QED. It is the successful comparison of these calculations with experimental measurements which gives the scientific community confidence that QCD is indeed the correct theory of the strong interaction.
But at much lower energies, such as at the scale of the mass of the proton, the QCD coupling constant is not small and perturbative calculations are not reliable or feasible. Lattice QCD provides a nonperturbative approach. The tradeoff is that the continuum of spacetime is approximated with a four-dimensional grid, or lattice, with quark degrees of freedom at the lattice sites, and glue degrees of freedom on the connecting links. Within such a forum, some of the mathematics can be mapped to that of classical statistical mechanics so that estimates can be made stochastically by Monte Carlo sampling with attendant statistical errors. Systematic errors can be estimated, controlled, and reduced as the finite lattice spacing is made smaller, the volume of the lattice universe is made larger, and the masses of the quarks are extrapolated/interpolated to physically appropriate values.
Taking these limits is facilitated in a way that supersedes that available in traditional uses of discretization in physics and engineering; namely, lattice QCD is formulated such that the crucial property of gauge symmetry is preserved at finite lattice spacing. Furthermore, we have focused on a previously created implementation of the lattice theory which, again even at finite lattice spacing, respects the chiral (“handedness”) properties of the continuum action. (Hence the name of our collaboration: χQCD.) This pays remarkable dividends in the clean theoretical interpretation of the lattice results. Although the naïve cost of a chirally symmetric implementation is two orders of magnitude more expensive computationally than conventional lattice approaches, we have done extensive research and invoked or created clever techniques to largely mitigate this increased expense.
Taking these continuum, infinite volume, and physical-mass limits requires enormous computational resources, so much so that within the first two decades after the field was invented some fifty years ago, it seemed overwhelmingly daunting that they could ever be achieved. But exponential growth in computing power has been matched by extraordinary advances in theoretical, algorithmic, and software aspects of the lattice QCD calculations. As a result, for the last decade especially, realistic lattice calculations are not only possible, but sometimes exceed experiment in precision and accuracy, depending on the process studied. For the past twenty years or more, lattice QCD has been regarded as the indispensable third leg of a triad, together with experiment, and traditional analytic theory/modelling.
Early research at UK was enabled by generous support from UK’s Center for Computational Sciences. But the lattice QCD field has matured such that now much work is done by large international collaborations, and the computational resources are provided by grants from federal funding agencies for vast amounts of computer time at national labs. We typically consume tens of millions of CPU-hours (or their GPU equivalent) per year.
A lot of our research has focused on understanding and predicting the properties of protons and neutrons from the behavior of their constituent quarks and gluons, which is one of the primary goals of the Department of Energy’s Nuclear Office. It has garnered some publicity: The work on glue spin in the proton is published in Physical Review Letters (PRL 118 (2017), 10, 102001) as an editors’ suggestion of the issue. It is reported in APS Physics Viewpoint by Steven Bass – “Spinning Gluons in the Proton” (https://physics.aps.org/articles/v10/23). It is also included in Physics of the American Physical Society as one of only eight “Highlights of the Year” for 2017 (https://physics.aps.org/articles/v10/137). The work on the contribution of strange-flavored quarks to the magnetic moment of the proton is published in Physical Review Letters (PRL 118 (2017) 4, 042001) as an editors’ suggestion of the issue. It is reported in Nature by Ross Young – “Strangeness in the proton” (https://www.nature.com/articles/nature21909). The work on proton mass decomposition (i.e., how much of the proton mass is due to quark and glue kinetic energy, etc.) is published in Physical Review Letters (PRL 121, (2018) 21, 212001) as an editors’ suggestion. It is featured in APS Physics Viewpoint by André Walker-Loud – “Dissecting the Mass of the Proton” (https://physics.aps.org/articles/v11/118). See also a review in Science News (https://www.sciencenews.org/article/proton-mass-quarks-calculation?tgt=nr) which wittily describes the work as “a massive undertaking.”
We are particularly excited about our recent research into computing the electric dipole moment (EDM) of the neutron. A nonzero value would establish the violation of Charge-Parity (CP) symmetry. Although a nonzero value is allowed by the Standard Model, the predicted value is orders of magnitude smaller than necessary to explain the observed amount of baryonic matter and the matter-antimatter asymmetry in the universe. It is also negligibly small compared to the designed precision of upcoming high-profile experiments to measure the neutron EDM (nEDM) as an unmistakable signal of beyond Standard Model physics. (Our department’s Professors Brad Plaster, Chris Crawford, and Wolfgang Korsch are involved in the measurement of nEDM at Oak Ridge National Laboratory.)
QCD includes a term which breaks CP symmetry, parametrized by an angle theta. Lattice calculations of nEDM are in units of theta and thus a lattice calculation would constrain values of theta once nEDM is measured experimentally. Lattice calculations of nEDM have been attempted by several groups around the world, but their results are noisy and roughly consistent with zero, that is, no previous lattice calculation has ever seen a nonzero nEDM.
Taking advantage of the special features of the chirally symmetric lattice implementation mentioned above, which are particularly crucial in this context, we find (Phys. Rev. D 108 (2023) 9, 094512) a nonzero result with a statistical error of 10 sigma, compared to the previous world record of 2 sigma. Repeating the calculation on finer and larger lattices to take the continuum, infinite volume, and physical mass limits is ongoing; we aim for a final result with combined statistical and systematic errors at the 5-sigma level.
Professor Liu’s retirement has not slowed down his research and we look forward to more interesting results.