Abstract
We argue that in many physical calculations where the “elliptic” sectors are involved, one can express the results via iterated integrals with almost all weights being rational. Our method is based on the existence of 𝜖-regular basis, which is akin to the 𝜖-finite basis defined in ref. [1]. As a demonstration of our technique, we calculate the photon contribution to the total cross section of the production of two Q\( \overline{Q} \) electron-positron collisions.
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K.G. Chetyrkin, M. Faisst, C. Sturm and M. Tentyukov, 𝜖-finite basis of master integrals for the integration-by-parts method, Nucl. Phys.B 742 (2006) 208 [hep-ph/0601165] [INSPIRE].
K.G. Chetyrkin and F.V. Tkachov, Integration by parts: the algorithm to calculate β-functions in 4 loops, Nucl. Phys.B 192 (1981) 159 [INSPIRE].
F.V. Tkachov, A theorem on analytical calculability of four loop renormalization group functions, Phys. Lett.B 100 (1981) 65 [INSPIRE].
A.V. Kotikov, Differential equation method: the calculation of N point Feynman diagrams, Phys. Lett.B 267 (1991) 123 [Erratum ibid.B 295 (1992) 409] [INSPIRE].
E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cim.A 110 (1997) 1435 [hep-th/9711188] [INSPIRE].
A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett.5 (1998) 497 [arXiv:1105.2076] [INSPIRE].
E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys.A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].
J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett.110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].
R.N. Lee, Reducing differential equations for multiloop master integrals, JHEP04 (2015) 108 [arXiv:1411.0911] [INSPIRE].
R.N. Lee and A.A. Pomeransky, Normalized Fuchsian form on Riemann sphere and differential equations for multiloop integrals, arXiv:1707.07856 [INSPIRE].
S. Bauberger, F.A. Berends, M. Böhm and M. Buza, Analytical and numerical methods for massive two loop selfenergy diagrams, Nucl. Phys.B 434 (1995) 383 [hep-ph/9409388] [INSPIRE].
A. Primo and L. Tancredi, Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph, Nucl. Phys.B 921 (2017) 316 [arXiv:1704.05465] [INSPIRE].
L. Adams, C. Bogner and S. Weinzierl, The iterated structure of the all-order result for the two-loop sunrise integral, J. Math. Phys.57 (2016) 032304 [arXiv:1512.05630] [INSPIRE].
L. Adams and S. Weinzierl, Feynman integrals and iterated integrals of modular forms, Commun. Num. Theor. Phys.12 (2018) 193 [arXiv:1704.08895] [INSPIRE].
J. Broedel, C. Duhr, F. Dulat and L. Tancredi, Elliptic polylogarithms and iterated integrals on elliptic curves II: an application to the sunrise integral, Phys. Rev.D 97 (2018) 116009 [arXiv:1712.07095] [INSPIRE].
J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, Elliptic Feynman integrals and pure functions, JHEP01 (2019) 023 [arXiv:1809.10698] [INSPIRE].
E. Remiddi and L. Tancredi, Differential equations and dispersion relations for Feynman amplitudes. The two-loop massive sunrise and the kite integral, Nucl. Phys.B 907 (2016) 400 [arXiv:1602.01481] [INSPIRE].
E. Remiddi and L. Tancredi, An elliptic generalization of multiple polylogarithms, Nucl. Phys.B 925 (2017) 212 [arXiv:1709.03622] [INSPIRE].
G. Racah, Sulla nascita degli elettroni positivi (in Italian), Nuovo Cim.11 (1934) 477 [INSPIRE].
G. Racah, Sulla nascita di coppie per urti di particelle elettrizzate (in Italian), Il Nuovo Cim.14 (1937) 93.
S. Caron-Huot and J.M. Henn, Iterative structure of finite loop integrals, JHEP06 (2014) 114 [arXiv:1404.2922] [INSPIRE].
R.N. Lee, LiteRed 1.4: a powerful tool for reduction of multiloop integrals, J. Phys. Conf. Ser.523 (2014) 012059 [arXiv:1310.1145] [INSPIRE].
R. Kleiss, W.J. Stirling and S.D. Ellis, A new Monte Carlo treatment of multiparticle phase space at high-energies, Comput. Phys. Commun.40 (1986) 359 [INSPIRE].
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Lee, R., Onishchenko, A. 𝜖-regular basis for non-polylogarithmic multiloop integrals and total cross section of the process e+e−→ 2(Q\( \overline{Q} \)). J. High Energ. Phys. 2019, 84 (2019). https://doi.org/10.1007/JHEP12(2019)084
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DOI: https://doi.org/10.1007/JHEP12(2019)084