QCD modeling of hadron physics

Abstract

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:

πandρmesons, and their diquark partners, from a contact interaction

“…[20,22]. With this alternative prescription, m G = 0.132 GeV corresponds to α ir /π = 0.93, a magnitude commensurate with contemporary estimates of the zero-momentum value of a running-coupling in QCD [75][76][77][78].…”

“…5 We do that by performing a least-squares fit in the chiral limit to [44,46,[73][74][75] Table I. We remark that, in fitting, the same weight was given to each quantity because they are equally important.…”

Baryons as relativistic three-quark bound states

“…While this does not speak for its broad applicability in the meson sector, the same features also hold for their structure properties such as form factors, decays and more [188,395]; thus for processes involving pions, kaons, ρ mesons etc. rainbow-ladder can be considered reliable.…”

“…In principle the problem can be rigorously solved by calculating the amplitudes directly in the moving frame, so that instead of one complex variable z f or z i one solves for two real variables in the domain [−1, 1]. However, this is a numerically expensive procedure especially for baryons and so far it has only been implemented for the pion [395].…”

Charmed mesons at finite temperature and chemical potential

“…Great progress has been achieved in recent years in the study of their properties with lattice QCD methods, both in vacuum and at finite temperature [2]. In the continuum, however, although in principle the complex of the Schwinger-Dyson (DS) and Bethe-Salpeter (BS) equations provides an adequate framework, tremendous challenges still remain in describing simultaneously light-and heavy-flavored mesons in vacuum within a single model interaction-truncation scheme [3][4][5]. On the other hand, there is pressing need for different pieces of information on properties of such mesons both in vacuum and at finite T and µ for guiding experimental proposals at existing and forthcoming facilities.…”