A major goal of contemporary nuclear physics is to explain the structure of atomic nuclei and their reactions in a microscopic way. That approach, also known as the ab initio method, starts from two- and many-body forces between protons and neutrons and applies quantum many-body theory to deal with the multitude of nucleons. Many methods, including Faddeev‒Yakubovsky equations, the no-core shell model, coupled cluster methods, many-body perturbation theory, and quantum Monte Carlo, have been designed to treat such strongly interacting multiparticle systems.
The methods listed above use continuous spacetime coordinates. A recent alternative approach instead represents a physical system in a finite volume with discrete spacetime coordinates subjected to sampling by Monte Carlo methods. That sounds simple and straightforward, but it is not, as the new book by Timo Lähde and Ulf-G. Meißner, Nuclear Lattice Effective Field Theory: An Introduction, explains thoroughly. Meißner, a professor of physics at the University of Bonn in Germany, is well known for his research on effective field theories in nuclear and particle physics. Lähde, a staff member at the Research Center Jülich in Germany, has been heavily involved in lattice calculations in condensed-matter and nuclear physics.
The text’s first two chapters provide helpful background information not specifically about lattice calculations. One offers a general introduction to effective field theories (EFTs), and the other introduces nuclear forces in chiral EFT. The authors give several beautiful examples that demonstrate the basic ideas underlying an EFT. The section entitled “A Short Recipe for the Construction of an EFT” is a remarkably concise and clear explanation of the concept. The book’s introduction to chiral EFT-based nuclear forces, in which nucleons and pions are the typical active degrees of freedom rather than quarks and gluons, is also clear and accessible.
The main body of the book consists of six chapters that present lattice methods in a systematic way. The first of those chapters introduces the basic mathematical concepts of the lattice approach, such as Grassmann fields, transfer matrices, and auxiliary fields. The next chapter addresses chiral nuclear forces, with a special focus on how they are put on the lattice. The discussion evolves from leading order to next-to-leading order and finally next-to-next-to-leading order. Each order is presented in small steps that make the chapter easy for the unacquainted reader to follow.
Subsequent chapters are devoted to the characteristic problems that arise with increasing numbers of nucleons. The authors discuss at length methods for how to extract phase shifts on the lattice for two and three nucleons, and they carefully weigh the advantages and disadvantages of the Luescher, spherical wall, and auxiliary potential methods. Increasing the number of nucleons beyond three, as the authors show, creates the need for Monte Carlo sampling of lattice calculations. The last chapters move up to medium-mass nuclei and to chiral forces up to next-to-next-to-next-to-leading order. The authors report on some of the first attempts to solve these thorny problems, but they acknowledge that actual solutions are still in the future.
Lattice methods are technical and beset by a cumbersome mathematical apparatus. Thus any explanation of them could be difficult to read for outsiders or newcomers to the field. However, Lähde and Meißner manage to get the complex material across in a digestible way by breaking it up into small steps and knowing when repetition is helpful. They also introduce from scratch such basic concepts as phase shifts, without referring the reader to another book. Such care makes Nuclear Lattice Effective Field Theory self-contained, which is a great practical advantage to the reader. The six appendices further enhance the self-contained character of the text by summarizing notations and conventions and providing more details about useful mathematical functions, nuclear force properties, and the application of Monte Carlo methods.
Nuclear Lattice Effective Field Theory offers interesting insight into the endless problems that the pioneers of lattice methods had to confront during the past 20–40 years, such as rotational symmetry breaking, the sign problem, and the unfavorable scaling of Monte Carlo algorithms. Solutions have been worked out for most hurdles, and thus we can say that nuclear lattice EFT has come of age; the field is ready for more comprehensive and systematic applications in nuclear theory.
Apart from work by some of the authors’ collaborators, the method has not yet been applied widely in nuclear theory. Lähde and Meißner’s helpful primer has the potential to stimulate increased efforts by serving newcomers as an essential guide to the field.
Ruprecht Machleidt is a professor of physics at the University of Idaho. His research interests include the theory of nuclear forces, for which he has applied meson theory (Bonn potential) and chiral effective field theory.
