Review of lattice results concerning low-energy particle physics

Abstract

We review lattice results related to pion, kaon, D-and B-meson physics with the aim of making them easily accessible to the particle-physics community. More specifically, we report on the determination of the lightquark masses, the form factor f + (0), arising in semileptonic K → π transition at zero momentum transfer, as well as the decay-constant ratio f K / f π of decay constants and its consequences for the CKM matrix elements V us and V ud . Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of SU(2) L × SU(2) R and SU(3) L ×SU(3) R Chiral Perturbation Theory and review the determination of the B K parameter of neutral kaon mixa e-mail: gilberto@itp.unibe.ch ing. The inclusion of heavy-quark quantities significantly expands the FLAG scope with respect to the previous review. Therefore, we focus here on D-and B-meson decay constants, form factors, and mixing parameters, since these are most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. In addition we review the status of lattice determinations of the strong coupling constant α s .

Baryons as relativistic three-quark bound states

“…In the course of this review we summarise aspects of lattice QCD that are useful for the later discussion of the baryon spectra and form factors. Naturally, in a non-expert review we skip over most technical details and refer the interested reader to specialised lattice reviews such as [156][157][158][159].…”

“…with m π = 138 MeV, have been performed in the past years for some selected observables, see e.g. [157][158][159]257] for reviews and results. Many other quantities, however, are just too cost-intensive and will be discussed later also for a range of heavier pion masses, some even as large as m π = 600 MeV.…”

Topological susceptibility on the lattice and the three-flavour quark condensate

“…In this context, it is interesting to consider the topological susceptibility as a way to extract the quark condensate, taking into account the possibility that it is not the order parameter triggering three-flavour chiral symmetry breaking. Indeed various analyses of lattice results [15,[37][38][39][40][41][42] suggest an overall good agreement between lattice simulations and χPT concerning chiral series obtained as an expansion in m u and m d only (N f = 2 χPT), but significant disagreements concerning the chiral expansions in powers of m u , m d , m s (N f = 3 χPT). In some cases, small values of the N f = 3 quark condensate and pseudoscalar decay constant fail to saturate the LO chiral expansions.…”

“…A first avenue consists in computing the spectrum and/or the dynamics of light pseudoscalar mesons in order to extract the low-energy constants encoding the pattern of chiral symmetry breaking in the two chiral limits (for a recent overview, see ref. [15]). Two other alternative strategies are often used in lattice simulations to determine the three-flavour quark condensate, both being related to the topological properties of QCD on the lattice.…”

“…It turns out that, in the N f = 2 chiral limit (m u = m d = 0) [13], the breakdown of chiral symmetry (responsible for the presence of pseudoscalar Goldstone bosons corresponding to pions) is triggered by the quark condensate Σ(2) = − lim mu,m d →0 ūu . The situation is not so clear for the pattern of chiral symmetry breaking in the N f = 3 chiral limit where m s also vanishes and kaons and η become also Goldstone bosons [14,15]. In particular, one would expect chiral order parameters (such as the quark condensate) to decrease as the number of massless flavours in the theory increases.…”