Abstract:
We discuss the generalization of Wigner’s causality bounds and Bethe’s integral formula for the e ective range parameter to arbitrary dimension and arbitrary angular momentum. We consider the impact of these constraints on the separation of low- and high-momentum scales and universality in low-energy scattering.

Abstract:
Universal low-energy behaviour ${2 m c}\over{\ln |s-4m^2|}$ of the scattering function of particles of positive mass m near the threshold $s=4m^2$, and ${\pi} \over {\ln |s-4m^2|}$ for the corresponding S-wave phase-shift, is established for weakly coupled field theory models with a positive mass m in space-time dimension 3; c is a numerical constant independent of the model and couplings. This result is a non-perturbative property based on an exact analysis of the scattering function in terms of a two-particle irreducible (or Bethe-Salpeter) structure function. It also appears as generic by the same analysis in the framework of general relativistic quantum field theory.

Abstract:
We prove that the entanglement created in the low-energy scattering of two particles in two dimensions is given by a universal coefficient that is independent of the interaction potential. This is strikingly different from the three dimensional case, where it is proportional to the total scattering cross section. Before the collision the state is a product of two normalized Gaussians. We take the purity as the measure of the entanglement after the scattering. We give a rigorous computation, with error bound, of the leading order of the purity at low-energy. For a large class of potentials, that are not assumed to be spherically symmetric, we prove that the low-energy behaviour of the purity, $\mathcal P$, is universal. It is given by $\mathcal P= 1- \frac{1}{(\ln (\sigma/\hbar))^2} \mathcal E$, where $\sigma$ is the variance of the Gaussians and the entanglement coefficient, $\mathcal E$, depends only on the masses of the particles and not on the interaction potential. The entanglement depends strongly in the difference of the masses. It takes its minimum when the masses are equal, and it increases rapidly with the difference of the masses.

Abstract:
For a very large class of potentials, $V(\vec{x})$, $\vec{x}\in R^2$, we prove the universality of the low energy scattering amplitude, $f(\vec{k}', \vec{k})$. The result is $f=\sqrt{\frac{\pi}{2}}\{1/log k)+O(1/(log k)^2)$. The only exceptions occur if $V$ happens to have a zero energy bound state. Our new result includes as a special subclass the case of rotationally symmetric potentials, $V(|\vec{x}|)$.

Abstract:
In the framework of quantum field theory (QFT) on noncommutative (NC) space-time with the symmetry group $O(1,1)\times SO(2)$, we prove that the Jost-Lehmann-Dyson representation, based on the causality condition taken in connection with this symmetry, leads to the mere impossibility of drawing any conclusion on the analyticity of the $2\to 2$-scattering amplitude in $\cos\Theta$, $\Theta$ being the scattering angle. Discussions on the possible ways of obtaining high-energy bounds analogous to the Froissart-Martin bound on the total cross-section are also presented.

Abstract:
We analyze low-energy scattering for arbitrary short-range interactions plus an attractive 1/r^6 tail. We derive the constraints of causality and unitarity and find that the van der Waals length scale dominates over parameters characterizing the short-distance physics of the interaction. This separation of scales suggests a separate universality class for physics characterizing interactions with an attractive 1/r^6 tail. We argue that a similar universality class exists for any attractive potential 1/r^{alpha} for alpha >= 2. We also discuss the extension to multi-channel systems near a magnetic Feshbach resonance. We discuss the implications for effective field theory with attractive singular power law tails.

Abstract:
We prove that, in (2+1) dimensions, the S-wave phase shift, $ \delta_0(k)$, k being the c.m. momentum, vanishes as either $\delta_0 \to {c\over \ln (k/m)} or \delta_0 \to O(k^2)$ as $k\to 0$. The constant $c$ is universal and $c=\pi/2$. This result is established first in the framework of the Schr\"odinger equation for a large class of potentials, second for a massive field theory from proved analyticity and unitarity, and, finally, we look at perturbation theory in $\phi_3^4$ and study its relation to our non-perturbative result. The remarkable fact here is that in n-th order the perturbative amplitude diverges like $(\ln k)^n$ as $k\to 0$, while the full amplitude vanishes as $(\ln k)^{-1}$. We show how these two facts can be reconciled.

Abstract:
An effective field theory developed for systems interacting through short-range interactions can be applied to systems of cold atoms with a large scattering length and to nucleons at low energies. It is therefore the ideal tool to analyze the universal properties associated with the Efimov effect in three- and four-body systems. In this "progress report", we will discuss recent results obtained within this framework and report on progress regarding the inclusion of higher order corrections associated with the finite range of the underlying interaction.

Abstract:
We analyze statistical probability distributions of intensities collected by diffraction techniques like Low-Energy Electron Diffraction. A simple theoretical model based in hard-sphere potentials and LEED formalism is investigated for different values of relevant parameters: energy, angle of incidence, muffin-tin potential radius, maximum spherical component $l_{max}$, number of stacked layers, and full multiple-scattering or kinematic model. Given a complex enough system (e.g., including multiple scattering by at least two Bravais lattices), the computed probability distributions agree rather well with a $\chi^{2}_{2}$ one, characteristic of the Gaussian Unitary Ensemble universality class associated to quantum chaos. A hypothesis on the possible impact of the chaoticity of wavefunctions on correlation factors is tested against the behaviour of the Pendry R-factor and the Root Mean Squared Deviation factor.

Abstract:
At long distances interactions between neutral ground state atoms can be described by the Van der Waals potential V(r) =-C6/r^6-C8/r^8 - ... . In the ultra-cold regime atom-atom scattering is dominated by s-waves phase shifts given by an effective range expansion p cot d0 (p) = -1/a0 + r0 p^2/2 + ... in terms of the scattering length a0 and the effective range r0. We show that while for these potentials the scattering length cannot be predicted, the effective range is given by the universal low energy theorem r0 = A + B/a0+ C/a0^2 where A,B and C depend on the dispersion coefficients Cn and the reduced di-atom mass. We confront this formula to about a hundred determinations of r0 and a0 and show why the result is dominated by the leading dispersion coefficient C6. Universality and scaling extends much beyond naive dimensional analysis estimates.