Applications of chiral perturbation theory to lattice QCD

These notes contain the written version of lectures given at the 2009 Les Houches Summer School “Modern perspectives in lattice QCD: Quantum field theory and high performance computing.” The goal is to provide a pedagogical introduction to the subject, and not a comprehensive review. Topics covered include a general introduction, the inclusion of scaling violations in chiral perturbation theory, partial quenching and mixed actions, chiral perturbation theory with heavy kaons, and the effects of finite volume, both in the pand ǫ-regimes. I. WHY CHIRAL PERTURBATION THEORY FOR LATTICE QCD? It is often claimed that lattice QCD provides a tool for computing hadronic quantities numerically, with fully controlled systematic errors, from first principles. This assertion is of course based on the fact that lattice QCD provides a nonperturbative definition of QCD (in fact, the only one to date). So, why not choose the parameters (in particular, the quark masses) at, or very near, their physical values, and compute all quantities of interest? There are at least two major obstacles to this. The first obstacle is that lattice QCD is formulated in euclidean space, rather than in Minkowski space. This is a necessary restriction if one wants to use Monte Carlo methods in order to evaluate expectation values of operators from which one extracts physical quantities. In euclidean space correlation functions do not give direct access to physical scattering amplitudes; they first need to be continued to Minkowski space. If one is interested in physics at some scale for which an effective field theory (EFT) is available, one can use this EFT in order to match to euclidean lattice correlation functions in the regime of validity of the EFT. In the case of the chiral EFT for QCD, the form of the correlation functions is predicted in terms of a finite number of coupling constants to a given finite order in a momentum expansion. Once the value of these is known from lattice QCD computations, the EFT can then be used to continue to Minkowski space. Chiral perturbation theory (ChPT) provides this EFT framework for the low-energy physics of the (pseudo-) Nambu–Goldstone bosons of QCD. A second obstacle is that in practice lattice QCD computations are carried out at values of the up and down quark mass larger than those observed in nature, because of limits on the size of the physical volumes that can be handled with presently available computers, and because of the rapid increase in algorithmic cost with decreasing quark masses. If the quark masses are nevertheless still small enough for ChPT to be applicable, it can be used to extrapolate to the physical values for these light quark masses, again using ChPT to connect an “unphysical” lattice computation (namely, at values of the light quark masses larger than their real-world values) to physical quantities (those observed in nature). It should be said that lattice QCD is moving toward the physical point, i.e. , that numerical computations are being done with the light (up and down) quark masses at or very near the physical values. If lattice quark masses are very near the physical values, simple smooth extrapolations are in principle enough to obtain hadronic quantities of interest at the physical point, and the use of ChPT may become less important in this respect. I will return to this point below. Both of the obstacles described above are examples of how ChPT (and, in general, EFTs) can be used to connect “unphysical” computations with physical quantities. This is particularly helpful in the case of lattice QCD, where often computations are easier, or even only possible, in some unphysical regime. The use of ChPT is by no means limited to extrapolations in quark masses, or continuation from euclidean space. Other examples of the use of ChPT to bridge the gap between the lattice and the real world are, in increasing order of boldness: 1. Nonzero lattice spacing. While in state-of-the-art lattice computations the lattice spacing is small (in units of ΛQCD), scaling violations are still significant. It is possible 1 The strange quark mass can be taken at its physical value, although it is theoretically interesting to see what happens if one varies the strange quark mass as well.