The hidden-charm pentaquark and tetraquark states

http://cds.cern.ch/record/392684/files/9907240.pdf

Potential NRQCD: an effective theory for heavy quarkonium

Heavy-light diquarks can be the building blocks of a rich spectrum of states which can accommodate some of the newly observed charmoniumlike resonances not fitting a pure $c\overline{c}$ assignment. We examine this possibility for hidden and open charm diquark-antidiquark states deducing spectra from constituent quark masses and spin-spin interactions. Taking the $X(3872)$ as input we predict the existence of a ${2}^{++}$ state that can be associated to the $X(3940)$ observed by Belle and reexamine the state claimed by SELEX, $X(2632)$. The possible assignment of the previously discovered states ${D}_{s}(2317)$ and ${D}_{s}(2457)$ is discussed. We predict $X(3872)$ to be made of two components with a mass difference related to ${m}_{u}\ensuremath{-}{m}_{d}$ and discuss the production of $X(3872)$ and of its charged partner ${X}^{\ifmmode\pm\else\textpm\fi{}}$ in the weak decays of ${B}^{+,0}$.

/pdf/diquark-antidiquark-states-with-hidden-or-open-charm-and-the-5b1xpwe4oh.pdf

Diquark-antidiquark states with hidden or open charm and the nature of X(3872)

We briefly review how nonrelativistic effective field theories give us a definition of the QCD potentials and a coherent field-theory-derived quantum-mechanical scheme to calculate the properties of bound states made by two or more heavy quarks. In this framework heavy quarkonium properties depend only on the QCD parameters (quark masses and αs) and nonpotential corrections are systematically accounted for. The relation between the form of the nonperturbative potentials and the low-energy QCD dynamics is also discussed.

/pdf/effective-field-theories-for-heavy-quarkonium-1h7nywv4aj.pdf

Effective field theories for heavy quarkonium

The one-dimensional Schrodinger equation for the potential x6 + αx2 + l(l + 1)/x2 has many interesting properties. For certain values of the parameters l and α the equation is in turn supersymmetric (Witten) and quasi-exactly solvable (Turbiner), and it also appears in Lipatov's approach to high-energy QCD. In this paper we signal some further curious features of these theories, namely novel spectral equivalences with particular second- and third-order differential equations. These relationships are obtained via a recently observed connection between the theories of ordinary differential equations and integrable models. Generalized supersymmetry transformations acting at the quasi-exactly solvable points are also pointed out, and an efficient numerical procedure for the study of these and related problems is described. Finally we generalize slightly and then prove a conjecture due to Bessis, Zinn-Justin, Bender and Boettcher, concerning the reality of the spectra of certain -symmetric quantum mechanical systems.

/pdf/spectral-equivalences-bethe-ansatz-equations-and-reality-30m2x55lc4.pdf

Spectral equivalences, Bethe ansatz equations, and reality properties in PT-symmetric quantum mechanics.

Bound states of quarks

Using Mathematica 3.0, the Schrodinger equation for bound states is solved. The method of solution is based on a numerical integration procedure together with convexity arguments and the nodal theorem for wave functions. The interaction potential has to be spherically symmetric. The solving procedure is simply defined as some Mathematica function. The output is the energy eigenvalue and the reduced wave function, which is provided as an interpolated function (and can thus be used for the calculation of, e.g., moments by using any Mathematica built-in function) as well as plotted automatically. The corresponding program schroedinger.nb can be obtained from franz.schoeberl@univie.ac.at.

Solving the Schrodinger equation for bound states with Mathematica 3.0

Using Mathematica 3.0, the Schroedinger equation for bound states is solved. The method of solution is based on a numerical integration procedure together with convexity arguments and the nodal theorem for wave functions. The interaction potential has to be spherically symmetric. The solving procedure is simply defined as some Mathematica function. The output is the energy eigenvalue and the reduced wave function, which is provided as an interpolated function (and can thus be used for the calculation of, e.g., moments by using any Mathematica built-in function) as well as plotted automatically.

Solving the Schroedinger equation for bound states with Mathematica 3.0

This review discusses several aspects of the semirelativistic description of bound states by the spinless Salpeter equation (which represents the simplest equation of motion incorporating relativistic effects) and, in particular, presents or recalls some very simple and elementary methods which allow us to derive rigorous statements on the corresponding solutions, that is, on energy levels as well as wave functions.

Franz F. Schöberl

Papers

Bound states of quarks

Solving the Schrodinger equation for bound states with Mathematica 3.0

Solving the Schroedinger equation for bound states with Mathematica 3.0

Semirelativistic treatment of bound states

A Simultaneous and Systematic Study of Meson and Baryon Spectra in the Quark Model