A U L (3) ⊗ U R (3) low-energy effective lagrangian for the nonet of pseudogoldstone bosons that appear in the large N c limit of QCD is presented including terms up to four derivatives and explicit symmetry breaking terms up to quadratic in the quark masses. The one-loop renormalization of the couplings is worked out using the heat-kernel technique and dimensional renormalization. The calculation is carried through for U L (n l ) ⊗ U R (n l ), thus allowing for a generic number n l of light quark flavours. The crucial advantages that the expansion in powers of 1/N c bring about are discussed. Special emphasis is put in pointing out what features are at variance with the SU L ⊗SU R results when the singlet η ′ is included in the theory.The pattern of the lowest-lying states in the spectrum of strong interactions uncovers an approximate continuous symmetry of nature, the so-called chiral symmetry, which is spontaneously broken. The octet of pseudoscalar particles -π, K, and η -, with masses much smaller than those of the next excited states -the octet of vector particles ρ, ω and K * , the baryons-, are the accepted candidates for pseudo-goldstone bosons associated to the spontaneous breaking of the symmetry.This approximate symmetry is well incorporated in QCD as three of the quarks happen to be light. In the (chiral) limit of vanishing m u , m d , m s the QCD lagrangian has the symmetry freedom of arbitrarily rotating with unitary matrices the quark field components in the space of flavours (u,d,s), independently in the Left and the Right sectors of chirality eigenstates. The symmetry group is U L (3) ⊗ U R (3) and is explicitly broken by the light quark masses; if it had not, the lightest mesons would indeed have been massless particles. This breaking is small, though, since the light quark masses are much smaller than the typical hadronic scale of a few hundred MeV. (This is certainly so for the u and d quarks ( 2 < m u < 8 and 5 < m d < 15 MeV) and still approximately verified for the heavier s-quark ( 100 < m s < 300 MeV), [1]).Empirically, however, only the SU L (3) ⊗ SU R (3) symmetry subgroup, spontaneously broken to SU L+R (3), is manifest: instead of a nonet of light pseudoscalars only an octet is observed. Of the remaining U L (1) ⊗ U R (1), the vector part provides the conserved baryon number current, whereas the axial U A (1) does not seem reflected at all in the spectrum, either as a conserved quantum number or as a goldstone boson. The first possibility would imply that all massive hadrons would appear in parity doublets and this is not what is observed. On the other hand, by its quantum numbers the η ′ would be the ninth candidate for goldstone boson; but if the U A (1) were realized in the Goldstone mode and explicitly broken only by the same quark mass terms that break the SU L (3) ⊗ SU R (3) one would expect that the ninth pseudo-goldstone boson would have a mass similar to the pion (to the masses in the octet): actually, a singlet pseudoscalar meson ought to exist with a mass smaller than √ 3m...

We obtain explicit expressions for the η and η ′ masses and decay constants using U L (3) ⊗ U R (3) chiral perturbation theory at next-to-leading order in a combined expansion in p 2 and 1/N c . A numerical fit of the parameters appearing to this order of the expansion is also discussed. *

The heavy singlet field is integrated out from the U L (3) ⊗ U R (3) Chiral Perturbation Theory and it is shown how its effects on the low-energy dynamics are reduced to effective vertices for the light mesons. The results are matched against the standard SU L (3) ⊗ SU R (3) Chiral Perturbation Theory in order to establish the relations between the coupling constants from both theories to one-loop level accuracy.

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