On kinematical constraints in boson-fermion systems

The strong interactions of fundamental particles have been successfully described by Quantum ChromoDynamics (QCD). It is a non-Abelian gauge theory, based on the SU(3) gauge group. QCD involves the gauge fields and the fields of spin 1/2 particles as gluons and quarks, respectively. Quarks are fermions in the fundamental representation of the color SU(3) gauge group. Gluons are boson fields in the adjoint (octet) representation. Moreover, quarks carry flavour degrees of freedom which are independent of the color. A distinguished feature of QCD is asymptotic freedom. Politzer, Gross and Wilczek, and 't Hooft discovered the property of asymptotic freedom in non-Abelian gauge field theories. It allows QCD to be treated perturbatively at high energies. On the other hand, the strong interaction becomes nonperturbative at low energies, because the coupling constant of QCD increases rapidly in that regime. This problem can be overcome in the framework of effective field theories. A remarkably successful effective Lagrangian approach to low-energy QCD is that of Chiral Perturbation Theory (χPT). The effective degrees of freedom of χPT are hadrons rather than quarks and gluons. χPT has been applicable in the flavour SU(2) sector of low-energy QCD. This effective field theory is based on the simple observation that QCD is chirally symmetric in the limit where the up and down current quark masses vanish. This implies that the handedness of quarks is a conserved property in that limit. However, the QCD vacuum is spontaneously broken. Since the pion masses (~140 MeV) are much smaller than the nucleon masses (~939 MeV), the pions can be identified to be the Goldstone bosons of the spontaneously broken chiral symmetry. The SU(2) χPT relies on the principles of quantum field theory and on the symmetries of QCD. A generalization of the chiral SU(2) scheme to the SU(3) flavour group which includes the strangeness sector is mathematically straightforward and has been done. Though the mass of the strange quark is much larger than the up and down quark masses, it is still small on the typical chiral scale of 1 GeV. The required approximate Goldstone boson octet is readily found with the pions, the kaons, and the eta-meson. But the domain of validity of χPT is restricted to a small neighbourhood of threshold energies. A generalization to higher energies is desired. We study the on-shell scattering processes for two-body systems involving bosons with negative parity and spin 0 or 1 and fermions with positive parity and spin 1/2 or 3/2. A problem may be caused from the fact that helicity partial-wave scattering amplitudes are kinematically constrained. We solve such a problem by derivation of suitable transformations which eliminate all kinematical constraints. Resulting amplitudes are useful for partial-wave analysis and construction of effective field theories based on unitarity and micro causality. The procedure requires a parameterization of scattering amplitudes in terms of invariant functions free of kinematical singularities. We develop a novel algebra, so that it leads to the efficient computation of such functions by virtue of computer algebra codes. We examine the hadrogenesis conjecture. It gives a systematic framework that various resonances can be conjectured to be a result of coupled-channel dynamics. It relies on a selection of a few fundamental hadronic degrees of freedom. The selection is guided by properties of QCD in the large-Nc limit. Starting with the relativistic chiral SU(3) Lagrangian, coupled-channel interactions are analytically extrapolated to higher energies by means of unitarity and micro causality. Our calculation contains not only the parameter-free term, i.e. the Weinberg-Tomozawa interaction, but also the s-, t-, and u-channel exchanges. At present, we restrict ourselves to a fermion with a spin-one-half in the calculation. We argue that further computation involving a spin-three-half fermion is required, because it is part of the baryon ground state multiplet which arises in the large-Nc limit of QCD

On kinematical constraints in the hadrogenesis conjecture for the baryon resonance spectrum

We consider the reaction dynamics of bosons with negative parity and spin $0$ or $1$ and fermions with positive parity and spin $\frac{1}{2}$ or $\frac{3}{2}$. Such systems are of central importance for the computation of the baryon resonance spectrum in the hadrogenesis conjecture. Based on a chiral Lagrangian the coupled-channel partial-wave scattering amplitudes have to be computed. We study the generic properties of such amplitudes. A decomposition of the various scattering amplitudes into suitable sets of invariant functions expected to satisfy Mandelstam's dispersion-integral representation is presented. Sets are identified that are free from kinematical constraints and that can be computed efficiently in terms of a novel projection algebra. From such a representation one can deduce the analytic structure of the partial-wave amplitudes. The helicity and the conventional angular-momentum partial-wave amplitudes are kinematically constrained at the Kibble conditions. Therefore an application of a dispersion-integral representation is prohibitively cumbersome. We derive covariant partial-wave amplitudes that are free from kinematical constraints at the Kibble conditions. They correspond to specific polynomials in the 4-momenta and Dirac matrices that solve the various Bethe-Salpeter equations in the presence of short-range interactions analytically.Comment: 18 page

On chiral excitations with exotic quantum numbers

We consider the flavour sextet of charmed meson resonances with J^P= 1^+ quantum numbers that is predicted by the leading order chiral Lagrangian with up, down and strange quarks. The effect of chiral correction terms as determined previously from QCD lattice data is worked out. Pole masses in the complex energy plane are derived. The most promising signal from such states accessible in experiments like Belle, LHCb and PANDA are foreseen in the s-wave pi D* phase shift and the eta D* invariant mass distribution. For physical quark masses a rapid variation of the phase shift in between the eta D* and the bar K Ds* thresholds is predicted.Comment: 14 pages, three figures - published versio

A coupled-channel system with anomalous thresholds and unitarity

We consider the isospin one-half example system, with $D \,\pi , D\,\eta, D_s \bar K, D^* \pi , D^*\eta, D^*\bar K$ coupled channels in the $J^P = 1^-$ partial wave, chosen such that various phenomena that come with the opening of an anomalous threshold can be illustrated in a step-wise procedure by a suitable variation of up, down and strange quark masses. We use a set of LEC in the chiral Lagrangian that were adjusted to a large set of Lattice QCD results. The six phase shifts and inelasticity parameters are presented for various choices of the pion mass. For a pion mass of 150 MeV there are no anomalous thresholds encountered. The small change from 150 MeV to 145 MeV pion mass causes a dramatic impact of the anomalous threshold on the phase shifts.Comment: 11 pages, 4 figure

On chiral excitations with exotic quantum numbers

We consider the flavour sextet of charmed meson resonances with Jᴾ = 1⁺ quantum numbers that is predicted by the leading order chiral Lagrangian with up, down and strange quarks. The effect of chiral correction terms as determined previously from QCD lattice data is worked out. Pole masses in the complex energy plane are derived. The most promising signal from such states accessible in experiments like Belle, LHCb and PANDA is foreseen in the s-wave π D* phase shift and the η D* invariant mass distribution. For physical quark masses a rapid variation of the phase shift in-between the η D* and the K D¯* s thresholds is predicted

Triangle and box diagrams in coupled-channel systems from the chiral Lagrangian

We perform an analysis of triangle- and box-loop contributions to the generalized potential in the scattering of Goldstone bosons off the J^P= 0^- and 1^- charmed mesons. Particular emphasis is put on the use of on-shell mass parameters in such contributions in terms of a renormalization scheme that ensures the absence of power-counting violating terms. This is achieved with a systematically extended set of Passarino--Veltman basis functions, that leads to manifest power-counting conserving one-loop expressions and avoids the occurrence of superficial kinematical singularities. Compact expressions to chiral order three and four are presented that are particularly useful in coding such coupled-channel systems. Our formal results are generic and prepare analogous computations for other systems, like meson-baryon scattering from the chiral Lagrangian.Comment: 58 pages, 7 figures and 2 tables, minor corrections and extension

On chiral extrapolations of charmed meson masses and coupled-channel reaction dynamics

We perform an analysis of QCD lattice data on charmed-meson masses. The quark-mass dependence of the data set is used to gain information on the size of counterterms of the chiral Lagrangian formulated with opencharm states with JP = 0− and JP = 1− quantum numbers. Of particular interest are those counterterms that are active in the exotic flavor sextet channel. A chiral expansion scheme in which physical masses enter the extrapolation formulas is developed and applied to the lattice data set. Good convergence properties are demonstrated, and an accurate reproduction of the lattice data based on ensembles of PACS-CS, MILC, ETMC, and HSC with pion and kaon masses smaller than 600 MeV is achieved. It is argued that a unique set of lowenergy parameters is obtainable only if additional information from HSC on some scattering observables is included in our global fits. The elastic and inelastic s-wave πD and ηD scattering as considered by HSC is reproduced faithfully. Based on such low-energy parameters, we predict 15 phase shifts and inelasticities at physical quark masses but also for an additional HSC ensemble at smaller pion mass. In addition, we find a clear signal for a member of the exotic flavor sextet states in the ηD channel, below the KD¯ s threshold. For the isospin-violating strong decay width of the D* s0(2317), we obtain the range (104–116) keV