回路正则化法与Bjorken-Drell类比法在费曼图与电路中的一致性及优越性
Consistency and Advantage of Loop Regularization Method Merging with Bjorken-Drell's Analogy Between Feynman Diagrams and Electrical Circuits
摘要：
探讨了循环正则化（LORE）方法在多循环计算中的一致性。LORE方法的一个关键概念是引入不可约回路积分（ILI），这些积分通过采用Feynman参数化和紫外线发散保持（UVDP）参数化从Feynman图中进行计算。这是不可避免的ILI遇到的UVDP参数空间中的分歧，由于在四维动量空间的一般重叠的分歧。通过计算由两圈Feynman图产生的所谓的$αβγ$积分，我们展示了如何用LORE方法处理参数空间中的发散。通过识别UVDP参数空间中的子图中的发散，我们得出了费曼图和电路之间的Bjorken-Drell类比。UVDP参数对应于电路中的电导或电阻，费曼图中的发散归因于无限大的电导或零电阻。特别是，所需的条件，以消除重叠的动量积分获得ILI被发现与保守的电压，和动量守恒对应的电流，这是已知的基尔霍夫定律在电路类比。作为对质量标量$ϕ^4$理论的应用，它使我们能够得到著名的耦合常数的对数运行和标量质量在两个回路水平上的一致幂律运行。特别是结合比约克-德雷尔电路类比的优点，明确演示了将LORE方法应用于费恩曼图多回路计算的一般过程。
英文摘要：
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 $αβγ$ 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 an application to the massive scalar $ϕ^4$ theory, it 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.
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