Consistency and Advantage of Loop Regularization Method Merging with Bjorken-Drell's Analogy Between Feynman Diagrams and Electrical Circuits

Details Consistency and Advantage of Loop Regularization Method Merging with Bjorken-Drell's Analogy Between Feynman Diagrams and Electrical Circuits Consistency and Advantage of Loop Regularization Method Merging with Bjorken-Drell's Analogy Between Feynman Diagrams and Electrical Circuits

2011-08-18

arxiv.org/abs/1108.3603v3

Physics

High Energy Physics

High Energy Physics - Phenomenology

V3: 49 pages, 16 figures, we add a direct proof in Sec. II for the consistency condition of gauge invariance and address a nec

Scientific paper

The consistency of loop regularization (LORE) method is explored in multiloop calculations. A key concept of the LORE method is the introduction of irreducible loop integrals (ILIs) which are evaluated from the Feynman diagrams by adopting the Feynman parametrization and ultraviolet-divergence-preserving(UVDP) parametrization. It is then inevitable for the ILIs to encounter the divergences in the UVDP-parameter space due to the generic overlapping divergences in the 4-dimensional momentum space. By computing the so-called $\alpha\beta\gamma$ integrals arising from two loop Feynman diagrams, we show how to deal with the divergences in the parameter space by applying for the LORE method. By identifying the divergences in the UVDP-parameter space to those in the subdiagrams of two loop diagrams, we arrive at the Bjorken-Drell's analogy between Feynman diagrams and electrical circuits, where the UVDP parameters are associated with the conductance or resistance in the electrical circuits. In particular, the sets of conditions required to eliminate the overlapping momentum integrals for obtaining the ILIs are found to be associated with the conservations of electric voltages in any loop of the circuit, and the momentum conservations correspond to the conservations of electrical currents at each vertex, which are known as the Kirchhoff's laws in the electrical circuits. As a practical application, we carry out a calculation for one-loop and two-loop Feynman diagrams in the massive scalar $\phi^4$ theory and illustrate the advantage and general procedure of applying the LORE method to the multiloop calculations of Feynman diagrams when merging with the Bjorken-Drell's analogy, which allows us to obtain the consistent power-law running of the scalar mass at two loop level.

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