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Particles and Fields
Eur. Phys. J. C 27, 115-134 (2003)
DOI: 10.1140/epjc/s2002-01098-9

Chiral order and fluctuations in multi-flavour QCD

S. Descotes-Genon1, L. Girlanda2 and J. Stern3

1  Department of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, UK
2  Dipartimento di Fisica, Università di Padova and INFN, Via Marzolo 8, 35131 Padova, Italy
3  Groupe de Physique Théorique, Institut de Physique Nucléaire, 91406 Orsay-Cedex, France

(Received: 31 July 2002 / Revised version: 6 November 2002 / Published online: 15 January 2003 )

Multi-flavour $(N_f \ge 3)$ chiral perturbation theory ( $\chi$PT) may exhibit instabilities due to vacuum fluctuations of sea $\bar
q q$ pairs. Keeping the fluctuations small would require a very precise fine tuning of the low-energy constants $L_4(\mu)$ and $L_6(\mu)$ to $L_4^{\mathrm {crit}}(M_\rho) = - 0.51 \cdot 10^{-3}$, $L_6^{\mathrm {crit}}(M_\rho) = - 0.26 \cdot 10^{-3}$. A small deviation from these critical values - like the one suggested by the phenomenology of OZI-rule violation in the scalar channel - is amplified by huge numerical factors inducing large effects of vacuum fluctuations. This would lead in particular to a strong Nf dependence of chiral symmetry breaking ( $\chi$SB) and a suppression of the multi-flavour chiral order parameters. A simple resummation is shown to cure the instability of $N_f\ge 3$ $\chi$PT, but it modifies the standard expressions of some O(p2) and O(p4) low-energy parameters in terms of observables. On the other hand, for r= ms/m > 15, the two-flavour condensate is not suppressed, due to the contribution induced by massive vacuum $\bar ss$ pairs. Thanks to the latter, the standard two-flavour $\chi$PT is protected from multi-flavour instabilities and could provide a well-defined expansion scheme in powers of non-strange quark masses.

© Società Italiana di Fisica, Springer-Verlag 2003