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Quark condensate for various heavy flavors

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Abstract

The quark condensate is calculated within the world-line effective-action formalism, by using for the Wilson loop an ansatz provided by the stochastic vacuum model. Starting with the relation between the quark and the gluon condensates in the heavy-quark limit, we diminish the current quark mass down to the value of the inverse vacuum correlation length, finding in this way a 64 % decrease in the absolute value of the quark condensate. In particular, we find that the conventional formula for the heavy-quark condensate cannot be applied to the c-quark, and that the corrections to this formula can reach 23 % even in the case of the b-quark. We also demonstrate that, for an exponential parametrization of the two-point correlation function of gluonic field strengths, the quark condensate does not depend on the non-confining non-perturbative interactions of the stochastic background Yang–Mills fields.

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Notes

  1. For reviews see [6, 7]

  2. For a review see [10]

  3. For the bosonic case see [12].

  4. The latter formula can be proved by rewriting the double surface integral as

    applying the Stokes’ theorem, which leads to

    and noticing that only the (zz′)-term in (z−z′)2 yields a non-vanishing contribution to the last integral, so that

  5. Rigorously speaking, the correlation functions 〈B μν (0)B μν (y)〉 B and 〈B μν (0)C μν (y)〉 B contain the phase factor \(\exp [i\int_{0}^{y} du_{\mu}\,A_{\mu}(u) ]\). However, the Taylor expansion of such a phase factor would yield correlation functions of more than two B μν ’s. On the other hand, the use of the formfactor F corresponds to accounting for only two B μν ’s. For this reason, we must approximate the said phase factor by unity.

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Acknowledgements

One of us (D.A.) is grateful for the stimulating discussions to O. Nachtmann and M.G. Schmidt. The work of D.A. was supported by the Portuguese Foundation for Science and Technology (FCT, program Ciência-2008) and by the Center for Physics of Fundamental Interactions (CFIF) at Instituto Superior Técnico (IST), Lisbon.

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  1. Departamento de Física and Centro de Física das Interacções Fundamentais, Instituto Superior Técnico, UT Lisboa, Av. Rovisco Pais, 1049-001, Lisbon, Portugal

    Dmitri Antonov & José Emílio F. T. Ribeiro

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  1. Dmitri Antonov
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  2. José Emílio F. T. Ribeiro
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Correspondence to Dmitri Antonov.

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Antonov, D., Ribeiro, J.E.F.T. Quark condensate for various heavy flavors. Eur. Phys. J. C 72, 2179 (2012). https://doi.org/10.1140/epjc/s10052-012-2179-7

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