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Using Mathematica 3.0, the Schrödinger 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.

We study the accuracy of the bound-state parameters obtained with the method of dispersive sum rules, one of the most popular theoretical approaches in nonperturbative QCD and hadron physics. We make use of a quantum-mechanical potential model since it provides the only possibility to probe the reliability and the accuracy of this method: one obtains the bound-state parameters from sum rules and compares these results with the exact values calculated from the Schrödinger equation. We investigate various possibilities to fix the crucial ingredient of the method of sum rules -the effective continuum threshold -and propose modifications which lead to a remarkable improvement of the accuracy of the extracted ground-state parameters compared to the standard procedures adopted in the method. Although the rigorous control of systematic uncertainties in the method of sum rules remains unfeasible, the application of the proposed procedures in QCD promises a considerable increase of the actual accuracy of the extracted hadron parameters.

Masses of tetraquarks with two heavy quarks and open charm and bottom are calculated in the framework of the diquark-antidiquark picture in the relativistic quark model. All model parameters were regarded as fixed by previous considerations of various properties of mesons and baryons. The light quarks and diquarks are treated completely relativistically. The c quark is assumed to be heavy enough to make the diquark configurations dominating. The diquarks are considered not to be pointlike but to have an internal structure which is taken into account by the calculated diquark form factor entering the diquark-gluon interaction. It is found that all the (cc)(qq ′ ) tetraquarks have masses above the thresholds for decays into open charm mesons. Only the I(J P ) = 0(1 + ) state of (bb)(ūd) lies below the BB * threshold and is predicted to be narrow.

We study the pion form factor in a broad range of spacelike momentum transfers within the local-duality version of QCD sum rules. We make use of the recently calculated two-loop double spectral density of the AV A correlator including O(1) and O(αs) terms, which allows us to give predictions for the pion form factor and to study the interplay between the nonperturbative and perturbative contributions to the pion form factor without any reference to the pion distribution amplitude. Our results demonstrate the dominance of the nonperturbative contribution to the form factor up to relatively large values of the momentum transfer: namely, the nonperturbative O(1) term, which provides the 1/Q 4 power correction, gives more than half of the pion form factor in the region Q 2 ≤ 20 GeV 2 .

We study the uncertainties of the determination of the ground-state parameters from ShifmanVainshtein-Zakharov (SVZ) sum rules, making use of the harmonic-oscillator potential model as an example. In this case, one knows the exact solution for the polarization operator Π(µ), which allows one to obtain both the OPE to any order and the spectrum of states. We start with the OPE for Π(µ) and analyze the extraction of the square of the ground-state wave function, R ∝ |Ψ0( r = 0)| 2 , from an SVZ sum rule, setting the mass of the ground state E0 equal to its known value and treating the effective continuum threshold as a fit parameter. We show that in a limited "fiducial" range of the Borel parameter there exists a solution for the effective threshold which precisely reproduces the exact Π(µ) for any value of R within the range 0.7 ≤ R/R0 ≤ 1.15 (R0 is the known exact value). Thus, the value of R extracted from the sum rule is determined to a great extent by the contribution of the hadron continuum. Our main finding is that in the cases where the hadron continuum is not known and is modeled by an effective continuum threshold, the systematic uncertainties of the sum-rule procedure cannot be controlled.

Discussing four-point Green functions of bilinear quark currents in large-Nc QCD, we formulate rigorous criteria for selecting diagrams appropriate for the analysis of potential tetraquark poles. We find that both flavor-exotic and cryptoexotic (i.e., flavor-nonexotic) tetraquarks, if such poles exist, have a width of order O(1/N 2 c ), so they are parametrically narrower compared to the ordinaryqq mesons, which have a width of order O(1/Nc). Moreover, for flavor-exotic states, the consistency of the large-Nc behavior of "direct" and "recombination" Green functions requires two narrow flavorexotic states, each coupling dominantly to one specific meson-meson channel.PACS numbers: 11.15. Pg, 12.38.Lg, 12.39.Mk, 14.40.Rt MOTIVATIONTo provide the theoretical understanding of exotic tetraquark mesons, many candidates for which have been reported in the recent years (see [1,2]), it is conventional to refer to QCD with a large number of colors N c [i.e., SU(N c ) gauge theory for large N c ] with a simultaneously decreasing coupling α s ∼ 1/N c [3,4]: at N c -leading order, large-N c QCD Green functions have only non-interacting mesons as intermediate states; tetraquark bound states may emerge only in N c -subleading diagrams [5]. For many years, this fact was believed to provide the theoretical explanation of the non-existence of exotic tetraquarks. However, as emphasized in [6], even if the exotic tetraquark bound states appear only in subleading diagrams, the crucial question is the width of these objects: if narrow, they might be well observed in nature. The conclusion of [6] was that, if tetraquark states exist, they may be as narrow as the ordinary mesons, i.e., have a width ∼ 1/N c . This issue has been further addressed in [7], discussing the dependence of the width of tetraquark mesons on their flavor structure. Finally, [8] reported for the cryptoexotic tetraquarks an even smaller width of order N −3 c . Before drawing a conclusion about the width of a potential tetraquark pole in large-N c QCD, it is mandatory to formulate rigorous criteria for selecting those QCD diagrams that may lead to the appearance of this pole: the crucial property of such sequence of diagrams is the presence of four-quark intermediate states and the corresponding cuts in the variable s if one expects to observe a tetraquark pole in s [9]. The presence of such a four-particle s-cut should be established on the basis of the Landau equations [10]. It turns out that some of the diagrams attributed to the tetraquark pole in previous analyses are lacking the necessary four-particle cut and thus may not be related to the tetraquark properties. According to our findings, the tetraquark width at large N c does not depend on its flavor structure: both flavor-exotic and flavor-nonexotic tetraquarks have the same width of order 1/N 2 c . We analyse four-point Green functions of bilinear quark currents of various flavor content; any such function depends on six kinematical variables: the four momenta squared of the external currents,, and the two...