Abstract
Through a very detailed investigation involving a set of simple perturbative amplitudes we show that the answer for the question put in the title of the present work is: undoubtedly NO! We are not restricting the sentence to the amplitudes which are considered as anomalous. The referred investigation is performed by using a procedure alternative to the traditional regularization methods. In the context of such a strategy the amplitudes are not modified in intermediary steps of the calculation, like traditional regularization procedures do, and only the validity of the linearity in the integration operation is assumed in the operations made in Feynman integrals typical of the perturbative calculations. The central point of the investigation is the question related to the consistent interpretation of the amplitudes. For these purposes, in all amplitudes having power counting indicating the possibility of divergences, the relations among Green functions, Ward identities and low energy limits are analyzed, in a model having different species of massive 1/2 spin fermions coupled to spin 0 and 1 (even and odd parity) boson fields, formulated in a space-time dimension D=1+1. We show that the maintenance of the linearity in operations involving Feynman integrals excludes the possibility of an anomalous term in the Ward identity relating the axial-vector and the pseudo-scalar-vector two point functions amplitudes. In addition, we show that it is not possible the maintenance of Ward identities and low energy limits in a consistent way if the amplitudes of the perturbative calculations are quantities to be regularized, just because there is no regularization method which is capable to give acceptable results for the divergent objects present in the calculations. This conclusion includes the Dimensional Regularization method which cannot give unique results if the linearity and the symmetric integration are simultaneously required in Feynman integrals. The qualitative conclusions apply in an equally-way in other space-time dimensions having, therefore, implications in many phenomenological consequences of quantum field theories stated through perturbative solutions.
Highlights
It remains no doubt about the fact that the quantum field theory represents the adequate tool for the description of relativistic interactions of fundamental particles
In the context of such a strategy the amplitudes are not modified in intermediary steps of the calculation, like traditional regularization procedures do, and only the validity of the linearity in the integration operation is assumed in the operations made in Feynman integrals typical of the perturbative calculations
The dimensional regularization [9] [10] [11], strictly speaking, cannot be applied since odd tensors do not admit extensions to continuum and complex dimensions. Due to this reason the discussions about explicit calculations of the triangle anomalies are made in a scenario where the amplitudes are admitted to be ambiguous quantities and the violations in symmetry relations are associated to their divergent character [4]: it is not possible to choose the arbitrariness involved such that the four expected symmetry properties can be satisfied [12] [13] [14]
Summary
It remains no doubt about the fact that the quantum field theory represents the adequate tool for the description of relativistic interactions of fundamental particles. The dimensional regularization [9] [10] [11], strictly speaking, cannot be applied since odd tensors (the Levi-Civita tensor) do not admit extensions to continuum and complex dimensions Due to this reason the discussions about explicit (perturbative) calculations of the triangle anomalies are made in a scenario where the amplitudes are admitted to be ambiguous quantities and the violations in symmetry relations are associated to their divergent character [4]: it is not possible to choose the arbitrariness involved such that the four expected symmetry properties can be satisfied [12] [13] [14]. A general discussion about all presented results is performed in the Section 10
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