Example of resonance saturation at one loop

Abstract

We argue that the large-N c expansion of QCD can be used to treat a Lagrangian of resonances in a perturbative way. As an illustration of this we compute the L 10 coupling of the chiral Lagrangian by integrating out resonance fields at one loop. Given a Lagrangian and a renormalization scheme, this is how in principle one can answer in a concrete and unambiguous manner questions such as at what scale resonance saturation takes place.Ever since the early times of vector meson dominance ͓1͔ there has been constant phenomenological evidence for the lowest vector and axial vector states to essentially saturate hadronic observables whenever their contribution is allowed by quantum number conservation. In the context of chiral perturbation theory ͓2,3͔ resonance saturation was suggested to generalize also to the scalar and pseudoscalar sectors ͓4͔, and indeed all the O(p 4 )L i couplings were obtained by means of integrating out the appropriate resonance fields. 1 However, this integration was carried out at the tree level; i.e., the Lagrangian was effectively treated only as classical.Specifically Ref. ͓4͔ made the choice to represent vector and axial-vector particles by antisymmetric tensor fields and wrote down a Lagrangian with SU 3 L ϫSU 3 R -symmetric interactions of the form 2where V, A, S, and S 1 stand for the octet vector, axial-vector, scalar and singlet scalar resonance fields, respectively, and U is the exponential of the Goldstone fields. Other terms appearing in the Lagrangian of Ref. ͓4͔ will be of no relevance for the discussion that follows and are not considered in Eq. ͑1͒.As is well known the field representation is not unique and, for instance, in the case of spin-one particles different authors have chosen different representations to describe them ͑i.e., an antisymmetric tensor, Yang-Mills field, hiddensymmetry field, etc. ͓7,8͔͒. As a consequence of this, it was seen that ambiguities in physical observables may occur. In Ref. ͓6͔ these ambiguities were resolved by imposing shortdistance matching onto the QCD operator product expansion of certain Green's functions. As a matter of fact, it was shown later on in Ref. ͓9͔ that all the above choices in the representation were actually field redefinitions of the particular Lagrangian of Eq. ͑1͒.Let us take the case of L 10 as an example. Integrating the vector and axial-vector fields in the Lagrangian ͑1͒ at the tree level leads to the low-energy chiral Lagrangian of Eq. ͑3͒ ͑see below͒ with equations relating couplings below and above threshold, such asHere L 10 () stands for the O(p 4 ) coupling in the lowenergy Lagrangian after the V and A resonance fields have been integrated out, i.e.,whereas L 10 R () is the akin coupling, but at the level of the resonance Lagrangian ͑1͒. The other couplings L 1Ϫ9 complete the list at O(p 4 ) ͓3͔. The statement of resonance saturation is then tantamount to the equationand expresses the fact that the whole low-energy coupling L 10 is directly ''produced'' in the process of integrating the resonance field. The res...

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