Goldstone theorem and diquark confinement beyond rainbow-ladder approximation

Bender, Axel; Roberts, Craig D.; Smekal, Lorenz von

doi:10.1016/0370-2693(96)00372-3

Cited by 406 publications

(720 citation statements)

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“…The use of DSEs to study meson phenomena is empowered by the existence noted above of a systematic, nonperturbative and symmetry-preserving truncation scheme [41,42]. It means that exact results, such as those outlined heretofore, can be illustrated and predictions for experiment made with readily quantifiable errors.…”

Section: Meson Spectroscopymentioning

confidence: 99%

Hadron properties and Dyson–Schwinger equations

Roberts

2008

Progress in Particle and Nuclear Physics

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186

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An overview of the theory and phenomenology of hadrons and QCD is provided from a DysonSchwinger equation viewpoint. Following a discussion of the definition and realisation of light-quark confinement, the nonperturbative nature of the running mass in QCD and inferences from the gap equation relating to the radius of convergence for expansions of observables in the current-quark mass are described. Some exact results for pseudoscalar mesons are also highlighted, with details relating to the U A (1) problem, and calculated masses of the lightest J = 0, 1 states are discussed. Studies of nucleon properties are recapitulated upon and illustrated: through a comparison of the ln-weighted ratios of Pauli and Dirac form factors for the neutron and proton; and a perspective on the contribution of quark orbital angular momentum to the spin of a nucleon at rest. Comments on prospects for the future of the study of quarks in hadrons and nuclei round out the contribution.

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Section: Meson Spectroscopymentioning

confidence: 99%

Hadron properties and Dyson–Schwinger equations

Roberts

2008

Progress in Particle and Nuclear Physics

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186

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“…The notation k stands for Λ d 4 k/(2π) 4 . For divergent integrals a translationally invariant regularization is necessary; the regularization scale Λ is to be removed at the end of all calculations, after renormalization, and will be suppressed henceforth.…”

Section: Quark Propagatormentioning

confidence: 99%

“…It satisfies the homogeneous Bethe-Salpeter equation [BSE] Γ(p o , p i ; P ) = (4) with Γ the Bethe-Salpeter amplitude [BSA] Γ(k o , k i ; P ) = S(k o ) −1 χ(k o , k i ; P ) S(k i ) −1 . (5) The kernel K is the qq irreducible quarkantiquark scattering kernel.…”

Section: Mesonsmentioning

confidence: 99%

“…There is a known constructive scheme [4] that defines a diagrammatic expansion of the BSE kernel corresponding to any diagrammatic expansion of the quark self-energy such that the vector and axial-vector WTIs are preserved. Among other things, this guarantees the Goldstone boson nature of the flavor non-singlet pseudoscalars independently of model details [5].…”

Section: Effect On Meson Observablesmentioning

confidence: 99%

“…For example, following the general procedure of Ref. [4], one-loop dressing of the quark-gluon vertex in the DSE requires 5 additions to the ladder BSE kernel to preserve the relevant WTIs, see Fig. 15.…”

Section: Current Conservationmentioning

confidence: 99%

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QCD modeling of hadron physics

Maris

Tandy

2006

Nuclear Physics B - Proceedings Supplements

103

120

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We review recent developments in the understanding of meson properties as solutions of the Bethe-Salpeter equation in rainbow-ladder truncation. Included are recent results for the pseudoscalar and vector meson masses and leptonic decay constants, ranging from pions up to cc bound states; extrapolation to bb states is explored. We also present a new and improved calculation of Fπ(Q 2 ) and an analysis of the πγγ transition form factor for both π(140) and π(1330). Lattice-QCD results for propagators and the quark-gluon vertex are analyzed, and the effects of quark-gluon vertex dressing and the three-gluon coupling upon meson masses are considered. DYSON-SCHWINGER EQUATIONS OF QCDThe Dyson-Schwinger equations [DSEs] are the equations of motion of a quantum field theory. They form an infinite hierarchy of coupled integral equations for the Green's functions (n-point functions) of the theory. Bound states (mesons, baryons) appear as poles in the Green's functions. Thus, a study of the poles in n-point functions using the set of DSEs will tell us something about hadrons. For recent reviews on the DSEs and their use in hadron physics, see Refs. [1,2,3]. Quark propagatorThe exact DSE for the quark propagator is 1where D µν (q = k − p) is the renormalized dressed gluon propagator, and Γ i ν (k, p) is the renormalized dressed quark-gluon vertex. The notation k stands for Λ d 4 k/(2π) 4 . For divergent integrals a translationally invariant regularization is necessary; the regularization scale Λ is to be removed at the end of all calculations, after renormalization, and will be suppressed henceforth.1 We use Euclidean metric {γµ, γν } = 2δµν , γ † µ = γµ andThe solution of Eq. (1) can be written asrenormalized according to S(p) −1 = i / p + m(µ) at a sufficiently large spacelike µ 2 , with m(µ) the current quark mass at the scale µ. Both the propagator, S(p), and the vertex, Γ i µ depend on the quark flavor, although we have not indicated this explicitly. The renormalization constants Z 2 and Z 4 depend on the renormalization point and on the regularization mass-scale, but not on flavor: in our analysis we employ a flavor-independent renormalization scheme. MesonsBound states correspond to poles in n-point functions: for example a meson appears as a pole in the 2-quark, 2-antiquark Green's function G (4) = 0|q 1 q 2q1q2 |0 . In the vicinity of a meson, i.e. in the neighborhood of P 2 = −M 2 with M being the meson mass, such a Green's function behaves likewhere P is the total 4-momentum of the meson, p o and p i are the momenta of the outgoing quark and incoming quark respectively, and similarly for k i and k o . Momentum conservation relates these momenta:

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