Modeling pions on the lattice

In Quantum Chromodynamics (QCD), the pions are the lightest bound states. Current lattice QCD calculations are not able to study pions at realistic masses due to algorithmic difficulties. Instead, lattice studies are limited to unphysically large pion masses, and Chiral Perturbation Theory (ChPT) is often relied upon to extrapolate lattice results to the phenomenological regime and to the chiral limit, where quarks are massless. One of the outstanding problems in the field is to determine the range of quark masses where ChPT is valid and to understand the nonperturbative physics that may cause ChPT to break down. Given the difficulty of studying QCD, it is interesting and useful to construct a lattice field theory model of pions, which would allow a direct lattice calculation without the need for chiral extrapolations. This model can be used to evaluate the reliability of chiral extrapolations as applied to lattice data in the context of a lattice field theory that is exactly solvable numerically even at small quark masses and in the chiral limit.;In this light, to create a model of pions of two-flavor Quantum Chromodynamics (QCD), a lattice field theory involving two flavors of staggered quarks interacting strongly with Abelian gauge fields is constructed. In the chiral limit, this theory exhibits a SUL(2) x SU R(2) x UA(1) symmetry. The UA(1) symmetry can be broken by introducing a four-fermion term into the action, thereby incorporating the physics of the QCD anomaly. To qualify as a meaningful model of QCD, this lattice model must exhibit spontaneous chiral symmetry breaking and confinement and must have a continuum limit. An interesting mechanism is introduced to address the continuum limit. In particular, an extra dimension allows one to tune a fictitious temperature in order to access a phase of broken symmetry and to find a range where the pion decay constant is much smaller than the lattice cutoff, i.e. Fpi ≪1a . Unlike lattice QCD, a major advantage of this model is that a novel class of remarkably efficient algorithms called Directed Path Algorithms (DPA) can be used to compute a variety of quantities in the chiral limit and close to it. An important accomplishment of this dissertation is the design and implementation of this algorithm. The DPA allows this model of pions to be studied in the chiral limit with lattices sizes as large as L ∼ 13 fm. Calculations in QCD at the moment are only able to work at lattice sizes of a few fm, and thus unlike QCD, this model allows a thorough investigation at large lattices in and near the chiral limit.;This ability to study the chiral limit thoroughly is used to test the predictions of ChPT in the epsilon-regime where MpiL ≪ 1 for the pion mass Mpi. Also, by adding a small quark mass, one can test the predictions of ChPT in the p -regime where MpiL ≳ 1. The results from lattice calculations are shown to be in nice agreement with the finite size scaling predictions of ChPT in these two regimes. This agreement confirms that the model is a reasonable model of low energy pions and allows the low energy constants to be extracted reliably and with controlled errors. In particular, it is shown that the low energy constants extracted from the epsilon-regime match nicely with those extracted from the p-regime.;In ChPT, quantities are expanded as a power series in the dimensionless parameter xi = M2p 16p2F2p . This parameter controls xi the convergence of ChPT. Interestingly, however, in the model studied in this dissertation, xi ≲ 0.002 is necessary before one-loop ChPT describes the data within 1% accuracy. For xi > 0.0035 the data deviate appreciably from the one-loop chiral perturbation theory predictions. It is argued that this qualitative change is the result of a light and narrow sigma-resonance. This property is different from what is expected in QCD, where the sigma resonance is expected to be heavy and broad. However, the investigations in this dissertation suggest that one-loop ChPT may be applicable only at realistic quark masses.