Quark-Hadron Duality

TL;DR

This review formalizes quark–hadron duality in QCD through Wilson’s operator product expansion and analyzes deviations via two complementary models: instanton-based and resonance-based. It explains how duality violations arise from nonperturbative effects, yielding exponential behavior in Euclidean space and oscillatory remnants in Minkowski space, with distinct scaling in various processes. The instanton model predicts oscillatory tails in spectral densities and specific power-law suppressions for widths and decays, while the resonance model (large $N_c$) produces a comb-like spectrum smoothed by widths, leading to oscillations damped exponentially at high energy. Comparisons with hadronic $\tau$ data and 't Hooft-model simulations illustrate qualitative agreement and highlight the need for precise spectral measurements to constrain the magnitude and energy onset of duality violations, ultimately improving determinations of QCD parameters such as $\alpha_s$.

Abstract

I review the notion of the quark-hadron duality from the modern perspective. Both, the theoretical foundation and practical applications are discussed. The proper theoretical framework in which the problem can be formulated and treated is Wilson's operator product expansion (OPE). Two models developed for the description of duality violations are considered in some detail: one is instanton-based, another resonance-based. The mechanisms they represent are complementary. Although both models are rather primitive (their largest virtue is their simplicity) they hopefully capture important features of the phenomenon. Being open for improvements, they can be used "as is" for orientation in the studies of duality violations in the processes of practical interest.

Figures (14)

• Figure 1: The one-loop graph determining the polarization operator and the spectral density in the leading (parton) approximation. The "photon" momentum is denoted by $q$.
• Figure 2: The one-gluon correction in the the polarization operator. The gluon momentum is denoted by $k$.
• Figure 3: Transmitting a large external momentum through a soft field.
• Figure 4: The transition operator $\hat{T}$ determining the hadronic $\tau$ width, $\Gamma_{\rm hadr} (\tau ) = M_\tau^{-1}\,{\rm Im}\, \langle\tau |\hat{T} |\tau\rangle$.
• Figure 5: The transition operator $\hat{T}$ determining the total semileptonic width of $B$ mesons.