A novel strategy to handle divergences typical of perturbative calculations is implemented for the Nambu-Jona-Lasinio model and its phenomenological consequences investigated. The central idea of the method is to avoid the critical step involved in the regularization process, namely, the explicit evaluation of divergent integrals. This goal is achieved by assuming a regularization distribution in an implicit way and making use, in intermediary steps, only of very general properties of such regularization. The finite parts are separated from the divergent ones and integrated free from effects of the regularization. The divergent parts are organized in terms of standard objects, which are independent of the (arbitrary) momenta running in internal lines of loop graphs. Through the analysis of symmetry relations, a set of properties for the divergent objects are identified, which we denominate consistency relations, reducing the number of divergent objects to only a few. The calculational strategy eliminates unphysical dependencies of the arbitrary choices for the routing of internal momenta, leading to ambiguity-free, and symmetry-preserving physical amplitudes. We show that the imposition of scale properties for the basic divergent objects leads to a critical condition for the constituent quark mass such that the remaining arbitrariness is removed. The model becomes predictive in the sense that its phenomenological consequences do not depend on possible choices made in intermediary steps. Numerical results are obtained for physical quantities at the one-loop level for the pion and sigma masses and pion-quark and sigmaquark coupling constants.

The correlation between the neutral electromagnetic pion decay, the Sutherland-Veltman paradox and the AV V triangle anomaly phenomenon is discussed within the framework of an alternative strategy to handle the divergences involved in the perturbative evaluation of the associated physical amplitudes. We show that the general characteristic of the adopted strategy allows us to recover the traditional treatment for the problem as well as allows us to construct an alternative way to look at the problem where the ambiguities play no relevant role.

We implement a cutoff-independent regularization of four-fermion interactions to calculate the colorsuperconducting gap parameter in quark matter. The traditional cutoff regularization has difficulties for chemical potentials µ of the order of the cutoff , predicting in particular a vanishing gap at µ ∼ . The proposed cutoff-independent regularization predicts a finite gap at high densities and indicates a smooth matching with the weak coupling QCD prediction for the gap at asymptotically high densities. The study of the properties of high-density quark matter has attracted great interest recently-for reviews and extensive lists of references see Refs. [1]. The earlier studies [2] on this subject found energy gaps of the order of a few MeVs. Since gaps of this order are much too small to have observable consequences, not much attention was given to the subject until recently, when it was shown [3,4] within the context of instanton-motivated four-fermion interactions that gaps of the order of 100 MeV could be obtained. The possibility of gaps of this order were corroborated by subsequent study [5] using weak coupling renormalization group techniques for QCD. The result of Ref.[5] for the two-flavor spin-0 superconducting gap can be written aswhere µ is the chemical potential and g = g(µ) is the QCD coupling constant. This result is clearly nonperturbative but was derived by assuming weak coupling, an assumption likely to be valid only at very high densities. Although inapplicable for densities typically found in the interiors of neutron stars, it seems to be a sound prediction for the color superconducting gap at asymptotically high baryon number densities. In particular, using the one-loop running of g(µ) with µ, Eq. (1) predicts that (µ) is an increasing function of µ. Elaborations and corrections [6] to Eq. (1) do not change this behavior. In view of the inapplicability of weak coupling techniques at densities of phenomenological interest and the fact that nonperturbative lattice techniques are not yet sufficiently developed to deal with such problems, the use of phenomenological models seem to be necessary to make progress in the field. In this context, models with nonrenormalizable four-fermion interactions have been extensively used to study different aspects of dynamical chiral symmetry breaking (Dχ SB) and high-density quark matter [7]. However, four-fermion models at the one-loop level predict vanishing superconducting gaps at high densities, a feature that is caused by the use of a regularizing momentum cutoff of the divergent one-loop integrals [8][9][10]. Since the phenomenon of superconductivity involves momenta of the order of the Fermi momentum k F , for baryon number densities such that k F ∼ the cutoff regularization clearly becomes inadequate and the vanishing of the gap at high densities might not be a physical feature of the problem. Although the QCD prediction of Eq.(1) of a nonzero gap is valid only at very high densities, there seems to be no physical motivation for expecting vanishing gaps...

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A very general calculational strategy is applied to the evaluation of the divergent physical amplitudes which are typical of perturbative calculations. With this approach in the final results all the intrinsic arbitrariness of the calculations due to the divergent character is still present. We show that by using the symmetry properties as a guide to search for the (compulsory) choices in such a way as to avoid ambiguities, a deep and clear understanding of the role of regularization methods emerges. Requiring then an universal point of view for the problem, as allowed by our approach, very interesting conclusions can be stated about the possible justifications of most intriguing aspect of the perturbative calculations in quantum field theory: the triangle anomalies.

The one-loop radiative correction arising from the Lorentz and CPT non-invariant fermionic sector of an extended version of quantum electrodynamics (QED) is explicitly evaluated by using a general calculational method. The possibility of Lorentz and CPT violations, of the Chern-Simons form, being radiatively induced are investigated. We show that from our results it is possible to recover the (ambiguous) value previously obtained by other authors. However, if we choose the interpretations for the divergent integrals in such a way that the usual QED U(1) gauge invariance is maintained in the calculated amplitudes, to be consistent with renormalizability, rigorous constraints emerge for the radiatively induced Chern-Simons term and consequently, a non-vanishing non-ambiguous value can be obtained.

Traditional cutoff regularization schemes of the Nambu-Jona-Lasinio model limit the applicability of the model to energy-momentum scales much below the value of the regularizing cutoff. In particular, the model cannot be used to study quark matter with Fermi momenta larger than the cutoff. In the present work, an extension of the model to high temperatures and densities recently proposed by Casalbuoni, Gatto, Nardulli, and Ruggieri is used in connection with an implicit regularization scheme. This is done by making use of scaling relations of the divergent one-loop integrals that relate these integrals at different energy-momentum scales. Fixing the pion decay constant at the chiral symmetry breaking scale in the vacuum, the scaling relations predict a running coupling constant that decreases as the regularization scale increases, implementing in a schematic way the property of asymptotic freedom of quantum chromodynamics. If the regularization scale is allowed to increase with density and temperature, the coupling will decrease with density and temperature, extending in this way the applicability of the model to high densities and temperatures. These results are obtained without specifying an explicit regularization. As an illustration of the formalism, numerical results are obtained for the finite density and finite temperature quark condensate and applied to the problem of color superconductivity at high quark densities and finite temperature.