Abstract
We generalize the result of our recent paper on the massless single off-shell scalar box integral to the case of two non-adjacent end points off the light cone. An analytic result in d = 4 − 2ε dimensions is established in terms of four Gauss hypergeometric 2F1 functions respectively their single-valued counterparts. This allows for an explicit splitting of real and imaginary parts, as well as an all-order ε-expansion in terms of single-valued polylogarithms.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Haug and F. Wunder, The massless single off-shell scalar box integral — branch cut structure and all-order epsilon expansion, JHEP 02 (2023) 177 [arXiv:2211.14110] [INSPIRE].
Z. Bern, L. Dixon and D.A. Kosower, Dimensionally-regulated pentagon integrals, Nucl. Phys. B 412 (1994) 751 [hep-ph/9306240].
G. Duplančić and B. Nižić, Dimensionally regulated one loop box scalar integrals with massless internal lines, Eur. Phys. J. C 20 (2001) 357 [hep-ph/0006249] [INSPIRE].
R.K. Ellis and G. Zanderighi, Scalar one-loop integrals for QCD, JHEP 02 (2008) 002 [arXiv:0712.1851] [INSPIRE].
O.V. Tarasov, Functional reduction of Feynman integrals, JHEP 02 (2019) 173 [arXiv:1901.09442] [INSPIRE].
T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].
V.A. Smirnov, Analytical result for dimensionally regularized massless on shell double box, Phys. Lett. B 460 (1999) 397 [hep-ph/9905323] [INSPIRE].
C. Anastasiou, E.W.N. Glover and C. Oleari, Application of the negative dimension approach to massless scalar box integrals, Nucl. Phys. B 565 (2000) 445 [hep-ph/9907523] [INSPIRE].
J. Fleischer, F. Jegerlehner and O.V. Tarasov, A new hypergeometric representation of one loop scalar integrals in d dimensions, Nucl. Phys. B 672 (2003) 303 [hep-ph/0307113] [INSPIRE].
O.V. Tarasov, Functional reduction of one-loop Feynman integrals with arbitrary masses, JHEP 06 (2022) 155 [arXiv:2203.00143] [INSPIRE].
J. Ellis, TikZ-Feynman: Feynman diagrams with TikZ, Comput. Phys. Commun. 210 (2017) 103 [arXiv:1601.05437] [INSPIRE].
V.A. Smirnov, Analytic tools for Feynman integrals, Springer (2012) [https://doi.org/10.1007/978-3-642-34886-0] [INSPIRE].
T. Matsuura, S.C. van der Marck and W.L. van Neerven, The calculation of the second order soft and virtual contributions to the Drell-Yan cross section, Nucl. Phys. B 319 (1989) 570.
V.E. Lyubovitskij, F. Wunder and A.S. Zhevlakov, New ideas for handling of loop and angular integrals in D-dimensions in QCD, JHEP 06 (2021) 066 [arXiv:2102.08943] [INSPIRE].
Acknowledgments
We thank the anonymous referee of ref. [1] for inspiring this paper by raising the question whether the result is generalizable to two end points off the light cone. We also thank Oleg V. Tarasov for drawing our attention to the similar hypergeometric representation he obtained for the non-adjacent double off-shell box integral. We are grateful to Werner Vogelsang for helpful comments. This study was supported in part by Deutsche Forschungsgemeinschaft (DFG) through the Research Unit FOR 2926 (Project No. 409651613). J. H. is grateful to the Landesgraduiertenförderung Baden-Württemberg for supporting her re- search.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2302.01956
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Haug, J., Wunder, F. The massless non-adjacent double off-shell scalar box integral — branch cut structure and all-order epsilon expansion. J. High Energ. Phys. 2023, 59 (2023). https://doi.org/10.1007/JHEP05(2023)059
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2023)059