Consistency and advantage of loop regularization method merging with Bjorken–Drell’s analogy between Feynman diagrams and electrical circuits

Abstract

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 ultravioletdivergence-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 αβγ integrals arising from two loop Feynman diagrams, we show how to deal with the divergences in the parameter space with the LORE method. By identifying the divergences in the UVDP parameter space to those in the subdiagrams, we arrive at the Bjorken-Drell's analogy between Feynman diagrams and electrical circuits. The UVDP parameters are shown to correspond to the conductance or resistance in the electrical circuits, and the divergence in Feynman diagrams is ascribed to the infinite conductance or zero resistance. 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, and the momentum conservations correspond to the conservations of electrical currents, which are known as the Kirchhoff's laws in the electrical circuits analogy. As a practical application, we carry out a detailed calculation for one-loop and two-loop Feynman diagrams in the massive scalar φ 4 theory, which enables us to obtain the well-known logarithmic running of the coupling constant and the consistent power-law running of the scalar mass at two loop level. Especially, we present an explicit demonstration on the general procedure of applying the LORE method to the multiloop calculations of Feynman diagrams when merging with the advantage of Bjorken-Drell's circuit analogy.

Quantum Structure of Field Theory and Standard Model Based on Infinity-Free Loop Regularization/Renormalization

“…It has been shown in ref. [50] that there does exist an exact correspondence between the UVDP parameter integrals and those from the original loop momenta.…”

“…It shows the advantage of LORE method as the quadratic structure is involved for scalar or Higgs interactions [50] and plays an important role for effective field theories with dynamically generated spontaneous symmetry breaking [43].…”

“…For that, it turns out to be very useful to introduce the so-called UVDP parametrization in evaluating the ILIs from Feynman diagrams [18,50].…”

“…The LORE method has also been applied to clarify the issue raised in [48] for the process H → γγ through a W-boson loop in the unitary gauge, and show that even for the finite amplitudes of Feynman diagrams, it still needs to adopt a consistent regularization for ensuring the cancellation between tensor-type and scalar-type divergent integrals [49] in order to obtain the finite amplitudes. Recently, it has been shown that the LORE method enables us to demonstrate consistently the general divergence structure of QFTs by high-loop order calculations [50,51]. In particular, it is interesting to observe in [50] that the evaluation of ILIs from Feynman integrals by adopting the so-called ultraviolet divergence-preserving (UVDP) parametrization naturally leads to the Bjorken-Drell's circuit analogy between Feynman diagrams and electric circuits [52], which allows us to show the one-to-one correspondence between the divergences of the UVDP parameters and the subdiagrams of Feynman diagrams.…”

“…Recently, it has been shown that the LORE method enables us to demonstrate consistently the general divergence structure of QFTs by high-loop order calculations [50,51]. In particular, it is interesting to observe in [50] that the evaluation of ILIs from Feynman integrals by adopting the so-called ultraviolet divergence-preserving (UVDP) parametrization naturally leads to the Bjorken-Drell's circuit analogy between Feynman diagrams and electric circuits [52], which allows us to show the one-to-one correspondence between the divergences of the UVDP parameters and the subdiagrams of Feynman diagrams.…”