Finite volume mass gap and free energy of the SU(N)xSU(N) chiral sigma model

We compute the free energy in the presence of a chemical potential coupled to a conserved charge in the effective SU(N)xSU(N) scalar field theory to third order for asymmetric volumes in general d-dimensions, using dimensional regularization. We also compute the mass gap in a finite box with periodic boundary conditions.

Casimir squared correction to the standard rotator Hamiltonian for the O($n$) sigma-model in the delta-regime

In a previous paper we found that the isospin susceptibility of the O($n$) sigma-model calculated in the standard rotator approximation differs from the next-to-next to leading order chiral perturbation theory result in terms vanishing like $1/ell,,$ for $ell=L_t/Ltoinfty$ and further showed that this deviation could be described by a correction to the rotator spectrum proportional to the square of the quadratic Casimir invariant. Here we confront this expectation with analytic nonperturbative results on the spectrum in 2 dimensions, by Balog and Hegedus for $n=3,4$ and by Gromov, Kazakov and Vieira for $n=4$. We also consider the case of 3 dimensions.

Isospin susceptibility in the O($n$) sigma-model in the delta-regime

We compute the isospin susceptibility in an effective O($n$) scalar field theory (in $d=4$ dimensions), to third order in chiral perturbation theory ($chi$PT) in the delta--regime using the quantum mechanical rotator picture. This is done in the presence of an additional coupling, involving a parameter $eta$, describing the effect of a small explicit symmetry breaking term (quark mass). For the chiral limit $eta=0$ we demonstrate consistency with our previous $chi$PT computations of the finite-volume mass gap and isospin susceptibility. For the massive case by computing the leading mass effect in the susceptibility using $chi$PT with dimensional regularization, we determine the $chi$PT expansion for $eta$ to third order. The behavior of the shape coefficients for long tube geometry obtained here might be of broader interest. The susceptibility calculated from the rotator approximation differs from the $chi$PT result in terms vanishing like $1/ell$ for $ell=L_t/L_stoinfty$. We show that this deviation can be described by a correction to the rotator spectrum proportional to the square of the quadratic Casimir invariant.

Massless sunset diagrams in finite asymmetric volumes

This paper discusses the methods and the results used in an accompanying paper describing the matching of effective chiral Lagrangians in dimensional and lattice regularizations. We present methods to compute 2-loop massless sunset diagrams in finite asymmetric volumes in the framework of these regularizations. We also consider 1-loop sums in both regularizations, extending the results of Hasenfratz and Leutwyler for the case of dimensional regularization and we introduce a new method to calculate precisely the expansion coefficients of the 1-loop lattice sums.

Matching effective chiral Lagrangians with dimensional and lattice regularization

We compute the free energy in the presence of a chemical potential coupled to a conserved charge in effective O($n$) scalar field theory (without explicit symmetry breaking terms) to NNL order for asymmetric volumes in general $d$--dimensions, using dimensional (DR) and lattice regularizations. This yields relations between the 4-derivative couplings appearing in the effective actions for the two regularizations, which in turn allows us to translate results, e.g. the mass gap in a finite periodic box in $d=3+1$ dimensions, from one regularization to the other. Consistency is found with a new direct computation of the mass gap using DR. For the case $n=4, d=4$ the model is the low-energy effective theory of QCD with $N_{rm f}=2$ massless quarks. The results can thus be used to obtain estimates of low energy constants in the effective chiral Lagrangian from measurements of the low energy observables, including the low lying spectrum of $N_{rm f}=2$ QCD in the $delta$--regime using lattice simulations, as proposed by Peter Hasenfratz, or from the susceptibility corresponding to the chemical potential used.

Symanzik effective actions in the large N limit

Symanzik effective actions, conjectured to describe lattice artifacts, are determined for a class of lattice regularizations of the non-linear O(N) sigma model in two dimensions in the leading order of the 1/N-expansion. The class of actions considered includes also ones which do not have the usual classical limit and are not (so far) treatable in the framework of ordinary perturbation theory. The effective actions obtained are shown to reproduce previously computed lattice artifacts of the step scaling functions defined in finite volume, giving further confidence in Symanziks theory of lattice artifacts.

The puzzle of apparent linear lattice artifacts in the 2d non-linear sigma-model and Symanziks solution

Lattice artifacts in the 2d O(n) non-linear sigma-model are expected to be of the form O(a^2), and hence it was (when first observed) disturbing that some quantities in the O(3) model with various actions show parametrically stronger cutoff dependence, apparently O(a), up to very large correlation lengths. In a previous letter we described the solution to this puzzle. Based on the conventional framework of Symanziks effective action, we showed that there are logarithmic corrections to the O(a^2) artifacts which are especially large, (ln(a))^3, for n=3 and that such artifacts are consistent with the data. In this paper we supply the technical details of this computation. Results of Monte Carlo simulations using various lattice actions for O(3) and O(4) are also presented.