Dyson-Schwinger equations and their application to hadronic physics

Roberts, Craig D.; Williams, Anthony G.

doi:10.1016/0146-6410(94)90049-3

Cited by 1,024 publications

(558 citation statements)

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“…Refs. [1,2,3] and references therein. Provided that the (effective) quark-quark interaction reduces to the perturbative running coupling in the ultraviolet region, it is also straightforward to reproduce the asymptotic behavior of Eqs.…”

Section: Quark Propagatormentioning

confidence: 99%

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QCD modeling of hadron physics

Maris

Tandy

2006

Nuclear Physics B - Proceedings Supplements

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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Section: Quark Propagatormentioning

confidence: 99%

“…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 and…”

Section: Quark Propagatormentioning

confidence: 99%

QCD modeling of hadron physics

Maris

Tandy

2006

Nuclear Physics B - Proceedings Supplements

120

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“…(55). The infrared suppression of Z(p 2 ) and enhancement of M (p 2 ) are longstanding predictions of DSE studies [1]. Indeed, this property of asymptotically free theories was elucidated in Refs.…”

Section: Comparison With Lattice Simulationsmentioning

confidence: 98%

“…The appearance of such a gap breaks the theory's chiral symmetry. This shows that the gap equation can be an important tool for studying DCSB, and it has long been used to explore this phenomenon in both QED and QCD [1].…”

Section: Nonperturbative Truncationmentioning

confidence: 99%

“…The broad framework is that of continuum strong QCD, by which I mean the continuum nonperturbative methods and models that can address these phenomena, especially those where a direct connection with QCD can be established, in one true limit or another. Naturally, everyone has a favourite tool and, in this connection, the Dyson-Schwinger equations (DSEs) are mine [1]. The framework is appropriate here because the last decade has seen a renaissance in its phenomenological application, with studies of phenomena as apparently unconnected as low-energy ππ scattering, B → D * decays and the equation of state for a quark gluon plasma [2,3,4].…”

Section: Introductionmentioning

confidence: 99%

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Unifying Aspects of Light- and Heavy-Systems

Roberts¹

2004

Heavy Quark Physics

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Abstract. Dyson-Schwinger equations furnish a Poincaré covariant framework within which to study hadrons. A particular feature is the existence of a nonperturbative, symmetry preserving truncation that enables the proof of exact results. Key to the DSE's efficacious application is their expression of the materially important momentumdependent dressing of parton propagators at infrared length-scales, which is responsible for the magnitude of constituent-quark masses and the length-scale characterising confinement in bound states. A unified quantitative description of light-and heavy-quark systems is achieved by capitalising on these features.

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