Abstract
We calculate three-loop photon spectral density in QED with N different species of electrons. The obtained results were expressed in terms of iterated integrals, which can be either reduced to Goncharov’s polylogarithms or written in terms of one-fold integrals of harmonic polylogarithms and complete elliptic integrals. In addition, we provide threshold and high-energy asymptotics of the calculated spectral density. It is shown that the use of the obtained spectral density correctly reproduces separately calculated moments of corresponding photon polarization operator.
References
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[14] D. J. Broadhurst, N. Gray, K. Schilcher, Gauge-invariant on-shellZ2in QED, QCD and theeffective field theory of a static quark, Zeitschrift f ̈ur Physik C 52 (1991) 111–122.doi:10.1007/BF01412333.
[15] K. Melnikov, T. van Ritbergen, The three loop on-shell renormalization of QCD and QED, NuclearPhysics B 591 (2000) 515–546.arXiv:hep-ph/0005131,doi:10.1016/S0550-3213(00)00526-5.13
[2] G. K ̈all ́en, A. Sabry, Fourth order vacuum polarization, Matematisk-Fysiske Meddelelser KongeligeDanske Videnskabernes Selskab 29 (17) (1955) 556.
[3] P. A. Baikov, D. J. Broadhurst, Three loop QED vacuum polarization and the four loop muonanomalous magnetic moment, in: Proceedings of the 4th International Workshop on SoftwareEngineering and Artificial Intelligence for High-Energy and Nuclear Physics, 1995, pp. 0167–172.arXiv:hep-ph/9504398.12
[4] T. Kinoshita, M. Nio, Sixth-order vacuum-polarization contribution to the lamb shift of muonichydrogen, Physical Review Letters 82 (16) (1999) 3240.
[5] T. Kinoshita, M. Nio, Accuracy of calculations involvingα3vacuum-polarization diagrams:Muonic hydrogen lamb shift and muon g – 2, Physical Review D 60 (5) (1999) 053008.
[6] T. Kinoshita, W. Lindquist, Parametric formula for the sixth-order vacuum polarization contri-bution in quantum electrodynamics, Physical Review D 27 (4) (1983) 853.
[7] T. Hahn, Generating Feynman diagrams and amplitudes with FeynArts 3, Computer PhysicsCommunications 140 (2001) 418–431.arXiv:hep-ph/0012260,doi:10.1016/S0010-4655(01)00290-9.
[8] K. G. Chetyrkin, F. V. Tkachov, Integration by parts: The algorithm to calculateβ-functions in4 loops, Nuclear Physics B 192 (1981) 159.
[9] F. V. Tkachov, A theorem on analytical calculability of 4-loop renormalization group functions,Physics Letters B 100 (1) (1981) 65–68, available athttp://www.sciencedirect.com/science/article/B6TVN-46YSNNV-109/1/886c25adc81acf1d171b80d7b1e7cb7f.
[10] R. N. Lee, A. I. Onishchenko, Master integrals for bipartite cuts of three-loop photon self energy,Journal of High Energy Physics 04 (2021) 177.arXiv:2012.04230,doi:10.1007/JHEP04(2021)177.
[11] R. N. Lee, A. I. Onishchenko,ε-regular basis for non-polylogarithmic multiloop integrals andtotal cross section of the processe+e−→2(Q ̄Q), Journal of High Energy Physics 12 (2019) 084.arXiv:1909.07710,doi:10.1007/JHEP12(2019)084.
[12] H. Ferguson, D. Bailey, A polynomial time, numerically stable integer relation algorithm, RNRTechnical Report RNR-91-032 (1992).
[13] N. Gray, D. J. Broadhurst, W. Grafe, K. Schilcher, Three-loop relation of quarkMSand polemasses, Zeitschrift f ̈ur Physik C 48 (1990) 673–679.doi:10.1007/BF01614703.
[14] D. J. Broadhurst, N. Gray, K. Schilcher, Gauge-invariant on-shellZ2in QED, QCD and theeffective field theory of a static quark, Zeitschrift f ̈ur Physik C 52 (1991) 111–122.doi:10.1007/BF01412333.
[15] K. Melnikov, T. van Ritbergen, The three loop on-shell renormalization of QCD and QED, NuclearPhysics B 591 (2000) 515–546.arXiv:hep-ph/0005131,doi:10.1016/S0550-3213(00)00526-5.13