Fitting two nucleons inside a box: exponentially suppressed corrections to the Luscher's formula

Scattering observables can be computed in lattice field theory by measuring the volume dependence of energy levels of two particle states. The dominant volume dependence, proportional to inverse powers of the volume, is determined by the phase shifts. This universal relation (\Lu's formula) between energy levels and phase shifts is distorted by corrections which, in the large volume limit, are exponentially suppressed. They may be sizable, however, for the volumes used in practice and they set a limit on how small the lattice can be in these studies. We estimate these corrections, mostly in the case of two nucleons. Qualitatively, we find that the exponentially suppressed corrections are proportional to the {\it square} of the potential (or to terms suppressed in the chiral expansion) and the effect due to pions going ``around the world'' vanishes. Quantitatively, the size of the lattice should be greater than $\approx(5 {fm})^3$ in order to keep finite volume corrections to the phase less than $1^\circ$ for realistic pion mass.Comment: 18 pages, 5 figures, 6 figure

Adversarial Transformations for Semi-Supervised Learning

We propose a Regularization framework based on Adversarial Transformations (RAT) for semi-supervised learning. RAT is designed to enhance robustness of the output distribution of class prediction for a given data against input perturbation. RAT is an extension of Virtual Adversarial Training (VAT) in such a way that RAT adversarialy transforms data along the underlying data distribution by a rich set of data transformation functions that leave class label invariant, whereas VAT simply produces adversarial additive noises. In addition, we verified that a technique of gradually increasing of perturbation region further improve the robustness. In experiments, we show that RAT significantly improves classification performance on CIFAR-10 and SVHN compared to existing regularization methods under standard semi-supervised image classification settings.Comment: Accepted by AAAI 202

Lattice QCD Simulations of Baryon Spectra and Development of Improved Interpolating Field Operators

Large sets of baryon interpolating field operators are developed for use in lattice QCD studies of baryons with zero momentum. Because of the cubical discretization of space, the continuum rotational group is broken down to a finite point group. Operators are classified according to the irreducible representations of the double octahedral group. At first, three-quark quasi-local operators are constructed for each isospin and strangeness with suitable symmetry of Dirac indices. Nonlocal baryon operators are formulated in a second step as direct products of the quasi-local spinor structures together with lattice displacements. Appropriate Clebsch-Gordan coefficients of the octahedral group are used to form linear combinations of such direct products. The construction maintains maximal overlap with the continuum SU(2) group in order to provide a physically interpretable basis. Nonlocal operators provide direct couplings to states that have nonzero orbital angular momentum. Monte Carlo simulations of nucleon and delta baryon spectra are carried out with anisotropic lattices of anisotropy 3.0 with $\beta=6.1$. Gauge configurations are generated by the Wilson gauge action in quenched approximation with space-time volumes (1.6\,\mbox{fm})^3\times 2.1\,\mbox{fm} and (2.4\,\mbox{fm})^3\times 2.1\,\mbox{fm}. The Wilson fermion action is used with pion mass \simeq 500\,\mbox{MeV}. The variational method is applied to matrices of correlation functions constructed using improved operators in order to extract mass eigenstates including excited states. Stability of the obtained masses is confirmed by varying the dimensions of the matrices. The pattern of masses for the low-lying states that we compute is consistent with the pattern that is observed in nature. Ordering of masses is consistent for positive-parity excited states, but mass splittings are considerably larger than the physical values. Baryon masses for spin $S \ge 5/2$ states are obtained in these simulations. Hyperfine mass splittings are studied for both parities. No significant finite volume effect is seen at the quark mass that is used

Finite volume corrections to pi-pi scattering

Lattice QCD studies of hadron-hadron interactions are performed by computing the energy levels of the system in a finite box. The shifts in energy levels proportional to inverse powers of the volume are related to scattering parameters in a model independent way. In addition, there are non-universal exponentially suppressed corrections that distort this relation. These terms are proportional to exp(-m_pi L) and become relevant as the chiral limit is approached. In this paper we report on a one-loop chiral perturbation theory calculation of the leading exponential corrections in the case of I=2 pi-pi scattering near threshold.Comment: 17 pages, 2 figures, 1 table. Version published in PR