Squeezed gluon vacuum and the global colour model of QCD

We discuss how the vacuum model of Celenza and Shakin with a squeezed gluon condensate can explain the existence of an infrared singular gluon propagator frequently used in calculations within the global colour model. In particular, it reproduces a recently proposed QCD-motivated model where low energy chiral parameters were computed as a function of a dynamically generated gluon mass. We show how the strength of the confining interaction of this gluon propagator and the value of the physical gluon condensate may be connected. ∗e-mail: [email protected] 1 The Global Colour Model (CGM) of Quantum Chromodynamics (QCD), whose main aspects have been reviewed in the last years [1] [3], is a quark-gluon quantum field theory that very successfully models QCD for low energy hadronic processes. In this approach an effective gluon correlator models the interaction between quark currents, and quark and gluon confinement may appear via the criterion of absence of real q poles for the propagators [4,5]. This is, for instance, the case of an effective gluon propagator with an infrared singularity like a delta function δ(k) at low energy [5]. There are many recent calculations exemplifying the remarkable success of this procedure [6,7]. It relates the hadronic properties to the Schwinger functions of quarks and gluons, therefore, when comparing the theoretical calculations to some low energy data, as pseudoscalars masses and decay constants or other chiral parameters, we are learning how is the infrared behavior of the quark and gluon propagators. As the time goes on this semi phenomenological tool may reveal to be even more successful than the relativistic quark model or the bag model. However, the question for the mechanism which leads to the infrared singularities present in the gluon propagator in such calculations remains open. The infrared enhancement of the gluon propagator due to the nonAbelian character of the theory and in particular due to the gluon-gluon self coupling, in principle could be understood in a rigorous study of the QCD vacuum. Unfortunately the simple perturbative vacuum is unstable [8], and there is no stable (gauge invariant) coherent vacuum in Minkowski space [9]. On the other hand in the context of the construction of a gauge invariant, stable QCD vacuum in Minkowski space, the squeezed condensate of gluons has become a topic of interest to uncover the underlying dynamics of the theory [10] [12]. Within this class of vacuum models, Pavel et al. [13] recently proposed a phenomenological vacuum based on Abelian QCD which leads exactly to the infrared singularity in the gluon propagator as considered in Ref. [5]. On the other hand calculations within the GCM suggest that the best fit of chiral symmetry breaking parameters are obtained with a propagator containing a delta function plus a propagator that behaves as 1/k in the ultraviolet (consistent with QCD) but damped at k = 0 [7,14]. Unfortunately, this is not the case of Ref. [13] where a 2 simple 1/k is obtained together with the delta function. Even if a model of the squeezed QCD vacuum is not determined from first principles, its properties may be very representative if they lead to a consistent phenomenology, indicating the path to the actual vacuum. In this note we show that the model of the QCD vacuum proposed by Celenza and Shakin [10,11] reproduces completely the gluon propagator of Ref. [14], i.e. it gives a singular part à la Munczek and Nemirovsky [5] as well as a piece containing a dynamically generated mass [15,16]. The effective dynamical gluon mass is the unique parameter in the model, and the gluon propagator obtained according to Ref. [10,11] is totally compatible with the idea of the GCM. We will briefly discuss the CGM and justify a gluon correlator which reproduces many aspects of the chiral symmetry breaking phenomenology. Then, we show that this gluon correlator naturally appears in the squeezed vacuum model of Ref. [10,11], verifying that the parameters involved in the model (the strength of the confining interaction, the dynamical gluon mass and the value of the physical gluon condensate) are consistent with the ones in the literature. The coincidence between the models may indeed suggest an interesting role for the Celenza and Shakin vacuum model. The action of the GCM can be obtained from the QCD generating functional through the standard method presented in Ref. [1]: