Dimensional effects in Efimov physics

Efimov physics is drastically affected by the change of spatial dimensions. Efimov states occur in a tridimensional (3D) environment, but disappear in two (2D) and one (1D) dimensions. In this paper, dedicated to the memory of Prof. Faddeev, we will review some recent theoretical advances related to the effect of dimensionality in the Efimov phenomenon considering three-boson systems interacting by a zero-range potential. We will start with a very ideal case with no physical scales, passing to a system with finite energies in the Born-Oppenheimer (BO) approximation and finishing with a general system. The physical reason for the appearance of the Efimov effect is given essentially by two reasons which can be revealed by the BO approximation - the form of the effective potential is proportional to $1/R^2$ ($R$ is the relative distance between the heavy particles) and its strength is smaller than the critical value given by $-(D-2)^2/4$, where $D$ is the effective dimension

Hyperfinite-Dimensional Representations of Canonical Commutation Relation

This paper presents some methods of representing canonical commutation relations in terms of hyperfinite-dimensional matrices, which are constructed by nonstandard analysis. The first method uses representations of a nonstandard extension of finite Heisenberg group, called hyperfinite Heisenberg group. The second is based on hyperfinite-dimensional representations of so(3). Then, the cases of infinite degree of freedom are argued in terms of the algebra of hyperfinite parafermi oscillators, which is mathematically equivalent to a hyperfinite-dimensional representation of so(n).Comment: 18 pages, LaTe

The nrMSSU(5) and universality of soft masses

We discuss the problem of universality of the soft, supersymmetry-breaking terms in the minimal supersymmetric SU(5) model (MSSU(5)) completed with flavor-dependent nonrenormalizable operators (NROs), or nrMSSU(5). These are exploited to correct the wrong fermion spectrum and to slow down the too-fast decay rate of the proton that the MSSU(5) model predicts. In general, the presence of such operators in the superpotential and K\"ahler potential gives rise to tree-level flavor- and CP-violating entries in the sfermion mass matrices at the cutoff scale, even when the mediation of supersymmetry (SUSY) breaking is generation and field-type independent. We identify the conditions under which such terms can be avoided.Comment: 4 pages, LateX, to appear in the proceedings of SUSY09, Boston, MA, USA, 5-10 June 200

Radii in weakly-bound light halo nuclei

A systematic study of the root-mean-square distance between the constituents of weakly-bound nuclei consisting of two halo neutrons and a core is performed using a renormalized zero-range model. The radii are obtained from a universal scaling function that depends on the mass ratio of the neutron and the core, as well as on the nature of the subsystems, bound or virtual. Our calculations are qualitatively consistent with recent data for the neutron-neutron root-mean-square distance in the halo of $^{11}$Li and $^{14}$Be nuclei

Comment on "Efimov States and their Fano Resonances in a Neutron-Rich Nucleus"

By introducing a mass asymmetry in a non-Borromean three-body system, without changing the energy relations, the virtual state pole cannot move from the negative real axis of the complex energy plane (with nonzero width) and become a resonance, because the analytical structure of the unitarity cuts remains the same.Comment: To be published in PR