Yangian Gelfand-Zetlin Bases, gl(N)-Jack Polynomials and computation of Dynamical Correlation Functions in the Spin Calogero-Sutherland Model

We consider the gl(N)-invariant Calogero-Sutherland Models with N=1,2,3,... in a unified framework, which is the framework of Symmetric Polynomials. By the framework we mean an isomorphism between the space of states of the gl(N)-invariant Calogero-Sutherland Model and the space of Symmetric Laurent Polynomials. In this framework it becomes apparent that all gl(N)-invariant Calogero-Sutherland Models are manifestations of the same entity, which is the commuting family of Macdonald Operators. Macdonald Operators depend on two parameters $q$ and $t$. The Hamiltonian of gl(N)-invariant Calogero-Sutherland Model belongs to a degeneration of this family in the limit when both $q$ and $t$ approach the N-th elementary root of unity. This is a generalization of the well-known situation in the case of Scalar Calogero-Sutherland Model (N=1). In the limit the commuting family of Macdonald Operators is identified with the maximal commutative sub-algebra in the Yangian action on the space of states of the gl(N)-invariant Calogero-Sutherland Model. The limits of Macdonald Polynomials which we call gl(N)-Jack Polynomials are eigenvectors of this sub-algebra and form Yangian Gelfand-Zetlin bases in irreducible components of the Yangian action. The gl(N)-Jack Polynomials describe the orthogonal eigenbasis of gl(N)-invariant Calogero-Sutherland Model in exactly the same way as Jack Polynomials describe the orthogonal eigenbasis of the Scalar Model (N=1). For each known property of Macdonald Polynomials there is a corresponding property of gl(N)-Jack Polynomials. As a simplest application of these properties we compute two-point Dynamical Spin-Density and Density Correlation Functions in the gl(2)-invariant Calogero-Sutherland Model at integer values of the coupling constant.Comment: 26 pages, AMSLate

Semi-infinite wedges and the conformal limit of the fermionic Calogero-Sutherland Model with spin 12\frac{1}{2}

The conformal limit over an anti-ferromagnetic vacuum of the fermionic spin $\frac{1}{2}$ Calogero-Sutherland Model is derived by using the wedge product formalism. The space of states in the conformal limit is identified with the Fock space of two complex fermions, or, equivalently, with a tensor product of an irreducible level-1 module of \slt and a Fock space module of the Heisenberg algebra.The Hamiltonian and the Yangian generators of the Calogero-Sutherland Model are represented in terms of \slt currents and bosons. At special values of the coupling constant they give rise to the Hamiltonian and the Yangian generators of the conformal limit of the Haldane-Shastry Model acting in an irreducible level-1 module of \slt. At generic values of the coupling constant the space of states is decomposed into irreducible representations of the Yangian.Comment: 26 pages, AMSLaTe

The trigonometric counterpart of the Haldane Shastry Model

The hierarchy of Integrable Spin Chain Hamiltonians, which are trigonometric analogs of the Haldane Shastry Model and of the associated higher conserved charges, is derived by a reduction from the trigonometric Dynamical Models of Bernard-Gaudin-Haldane-Pasquier. The Spin Chain Hamiltonians have the property of $U_q(\hat{gl}_2)$-invariance. The spectrum of the Hamiltonians and the $U_q(\hat{gl}_2)$-representation content of their eigenspaces are found by a descent from the Dynamical Models.Comment: amslatex, 42 pages, discrepancies between amslatex 1.1 and newer versions are rectified, two references are adde

Charged charmonium-like Z+(4430) from rescattering in conventional B decays

In a previous paper we suggested an explanation for the peak designated as $Z(4430)^+$ in the $\psi^\prime\pi^+$ mass spectrum, observed by Belle in ${\bar{\!B}} \to {\psi^\prime} \pi^+ K$ decays, as an effect of $\bar{\!D}{}^{\,*0}D^{\,+} \to \psi^\prime\pi^+$ rescattering in the decays ${\bar{\!B}} \to {D_{s}^{\,\prime -}} {D}$, where the $D_{s}^{\,\prime -}$ is an as-yet unobserved radial excitation of the pseudoscalar ground state $D_{s}^{\,-}$-meson. In this paper, we demonstrate that this hypothesis provides an explanation of the double $Z^+$-like peaking structures, which were studied by LHCb with much higher statistics. While according to our hypothesis, the origin of the peaking structures is purely kinematical, reflecting the presence of a conventional resonance in the hidden intermediate state, the amplitude of the $Z(4430)^+$ peak carries a Breit-Wigner-like complex phase, arising from the intermediate $D_{s}^{\,\prime -}$ resonance. Thus, our hypothesis is entirely consistent with the recent LHCb measurement of the resonant-like amplitude behaviour of the $Z(4430)^+$. We perform a toy fit to the LHCb data, which illustrates that our approach is also consistent with all the observed structure in the LHCb $M({{\psi^\prime}\pi^+})$ spectrum. We suggest a critical test of our hypothesis that can be performed experimentally.Comment: 6 pages, 5 figure