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Violation of quark–hadron duality

The missing oscillation in the OPE

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Abstract

The origin of quark–hadron duality violations (DVs) can be related to the singularities of the Laplace transform of the spectral function. With the help of rather generic properties of the large-\(N_c\) approximation and a generalized form for the radial trajectories found in Regge Theory, we may locate these singularities in the complex plane and obtain an expression for the DVs which turns out to agree with general expectations. Using the two-point vector correlator as a test laboratory, we show how the usual dispersion relation may give rise to perturbation theory, the power corrections from the condensate expansion and DVs.

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Notes

  1. In what follows, we will consider the perturbative expansion as the OPE contribution from the unit operator.

  2. We will simplify our world by taking the chiral limit.

  3. For simplicity, we are normalizing the parton model contribution to unity.

  4. Even if we consider the log corrections in the \(b_n\) coefficients, at high enough \(q^2\), a finite number of these power corrections cannot give rise to the oscillations commonly seen in the spectral data.

  5. In fact, to fulfill the reality property \(\mathcal {A}({q^2}^*)=\mathcal {A}(q^2)^*\), one should rotate \(\sigma \) clockwise for \(\mathrm {Im}\, q^2>0\) and anti-clockwise for \(\mathrm {Im}\, q^2<0\).

  6. We will assume that \(e^{\sigma q^2}\sigma \, \mathcal {B}^{[\rho ]}(\sigma )\) always falls off for to zero for \(|q^2|\rightarrow \infty \).

  7. In this context, \(N_c=3\) is large enough.

  8. We will use these units for the rest of this article.

  9. Expressions such as (14) are known as a Dirichlet series [17]. To our knowledge the first paper to connect these series to large-\(N_c\) QCD was Ref. [18].

  10. This expression says that perturbation theory is the part of the large-\(q^2\) expansion which depends solely on powers of \(\log (-q^2)\).

  11. Recall that our momentum is in units of \(\Lambda _{QCD}\).

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Acknowledgements

I am very grateful to D. Boito, I. Caprini, M. Golterman and K. Maltman for innumerable discussions on DVs and a pleasant collaboration. Work supported by CICYTFEDER-FPA2017-86989-P and by Grant No. 2017 SGR 1069. IFAE is partially funded by the CERCA program of the Generalitat de Catalunya.

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  1. Grup de Fisica Teorica, Departamento de Fisica, Universitat Autonoma de Barcelona, Bellaterra, 08193, Barcelona, Spain

    Santiago Peris

  2. Institut de Fisica d’Altes Energies (IFAE) and The Barcelona Institute of Science and Technology, Campus UAB, Bellaterra, 08193, (Barcelona), Spain

    Santiago Peris

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Peris, S. Violation of quark–hadron duality. Eur. Phys. J. Spec. Top. 230, 2691–2698 (2021). https://doi.org/10.1140/epjs/s11734-021-00255-1

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