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

Regular Article - Theoretical Physics

## 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 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 four-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 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 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.

### Keywords

Feynman Diagram Circuit Analogy Dimensional Regularization Regularization Scheme Loop Momentum

## Notes

### Acknowledgements

The authors would like to thank J.W. Cui and Y.B. Yang for useful discussions and L.F. Li for helpfully reading the manuscript. This work was supported in part by the National Science Foundation of China (NSFC) under Grant #Nos. 10821504, 10975170 and the key project of the Chinese Academy of Science.

### References

1. 1.
G. ’t Hooft, M.J.G. Veltman, Nucl. Phys. B 44, 189 (1972)
2. 2.
G. Bonneau, PAR-LPTHE-88-52a, May 23, 1989, unpublished Google Scholar
3. 3.
Y.L. Wu, Int. J. Mod. Phys. A 18, 5363 (2003). arXiv:hep-th/0209021
4. 4.
Y.L. Wu, Mod. Phys. Lett. A 19, 2191 (2004). arXiv:hep-th/0311082
5. 5.
J.W. Cui, Y.L. Wu, Int. J. Mod. Phys. A 23, 2861 (2008). arXiv:0801.2199 [hep-ph]
6. 6.
J.W. Cui, Y. Tang, Y.L. Wu, Phys. Rev. D 79, 125008 (2009). arXiv:0812.0892 [hep-ph]
7. 7.
Y.L. Ma, Y.L. Wu, Int. J. Mod. Phys. A 21, 6383 (2006). arXiv:hep-ph/0509083
8. 8.
Y.L. Ma, Y.L. Wu, Phys. Lett. B 647, 427 (2007). arXiv:hep-ph/0611199
9. 9.
J.W. Cui, Y.L. Ma, Y.L. Wu, Phys. Rev. D 84, 025020 (2011). arXiv:1103.2026 [hep-ph]
10. 10.
Y.B. Dai, Y.L. Wu, Eur. Phys. J. C 39, S1 (2004). arXiv:hep-ph/0304075
11. 11.
D. Huang, Y.L. Wu, e-Print. arXiv:1110.4491 [hep-ph]
12. 12.
Y. Tang, Y.L. Wu, Commun. Theor. Phys. 54, 1040 (2010). arXiv:0807.0331 [hep-ph]
13. 13.
Y. Tang, Y.L. Wu, Commun. Theor. Phys. 57, 629 (2012). arXiv:1012.0626 [hep-ph]
14. 14.
Y. Tang, Y.L. Wu, J. High Energy Phys. 1111, 073 (2011). arXiv:1109.4001 [hep-ph]
15. 15.
J. Bjorken, S. Drell, Relativistic Quantum Fields (McGraw-Hill, New York, 1965), p. 220. 396 p.
16. 16.
M.E. Peskin, D.V. Schroeder, An Introduction to Quantum Field Theory (Addison-Wesley, Reading, 1995), 842 p. Google Scholar
17. 17.
C. Itzykson, J.B. Zuber, Quantum Field Theory. International Series in Pure and Applied Physics (McGraw-Hill, New York, 1980), 705 p. Google Scholar
18. 18.
E. Brezin, J.C. Le Guillou, J. Zinn-Justin, Phys. Rev. D 9, 1121 (1974)
19. 19.
K.G. Chetyrkin, S.G. Gorishny, S.A. Larin, F.V. Tkachov, Phys. Lett. B 132, 351 (1983)
20. 20.
F.M. Dittes, Yu.A. Kubyshin, O.V. Tarasov, Theor. Math. Phys. 37, 879 (1979). Teor. Mat. Fiz. 37, 66 (1978)
21. 21.
D.I. Kazakov, O.V. Tarasov, A.A. Vladimirov, JINR-E2-12249. Dubna, JINR. Feb 1979. 23 pp. Published in Sov. Phys. JETP 50, 521 (1979), Zh. Eksp. Teor. Fiz. 77, 1035 (1979) Google Scholar
22. 22.
D. Huang, Y.L. Wu, in preparation Google Scholar
23. 23.
D. Huang, Y.L. Wu, work in progress Google Scholar