Abstract
In this talk we discuss a class of Feynman integrals, which can be expressed to all orders in the dimensional regularisation parameter as iterated integrals of modular forms. We review the mathematical prerequisites related to elliptic curves and modular forms. Feynman integrals, which evaluate to iterated integrals of modular forms go beyond the class of multiple polylogarithms. Nevertheless, we may bring for all examples considered the associated system of differential equations by a non-algebraic transformation to an \(\varepsilon \)-form, which makes a solution in terms of iterated integrals immediate.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
D.J. Broadhurst, J. Fleischer, O. Tarasov, Z. Phys. C 60, 287 (1993), arXiv:hep-ph/9304303
F.A. Berends, M. Buza, M. Böhm, R. Scharf, Z. Phys. C 63, 227 (1994)
S. Bauberger, M. Böhm, G. Weiglein, F.A. Berends, M. Buza, Nucl. Phys. Proc. Suppl. 37B, 95 (1994), arXiv:hep-ph/9406404
S. Bauberger, F.A. Berends, M. Böhm, M. Buza, Nucl. Phys. B 434, 383 (1995), arXiv:hep-ph/9409388
S. Bauberger, M. Böhm, Nucl. Phys. B 445, 25 (1995), arXiv:hep-ph/9501201
M. Caffo, H. Czyz, S. Laporta, E. Remiddi, Nuovo Cim. A 111, 365 (1998), arXiv:hep-th/9805118
S. Laporta, E. Remiddi, Nucl. Phys. B 704, 349 (2005), arXiv:hep-ph/0406160
B.A. Kniehl, A.V. Kotikov, A. Onishchenko, O. Veretin, Nucl. Phys. B 738, 306 (2006), arXiv:hep-ph/0510235
S. Groote, J.G. Körner, A.A. Pivovarov, Ann. Phys. 322, 2374 (2007), arXiv:hep-ph/0506286
S. Groote, J. Körner, A. Pivovarov, Eur. Phys. J. C 72, 2085 (2012), arXiv:1204.0694
D.H. Bailey, J.M. Borwein, D. Broadhurst, M.L. Glasser, J. Phys. A 41, 205203 (2008), arXiv:0801.0891
S. Müller-Stach, S. Weinzierl, R. Zayadeh, Commun. Number Theor. Phys. 6, 203 (2012), arXiv:1112.4360
L. Adams, C. Bogner, S. Weinzierl, J. Math. Phys. 54, 052303 (2013), arXiv:1302.7004
S. Bloch, P. Vanhove, J. Number Theory 148, 328 (2015), arXiv:1309.5865
L. Adams, C. Bogner, S. Weinzierl, J. Math. Phys. 55, 102301 (2014), arXiv:1405.5640
L. Adams, C. Bogner, S. Weinzierl, J. Math. Phys. 56, 072303 (2015), arXiv:1504.03255
L. Adams, C. Bogner, S. Weinzierl, J. Math. Phys. 57, 032304 (2016), arXiv:1512.05630
E. Remiddi, L. Tancredi, Nucl. Phys. B 880, 343 (2014), arXiv:1311.3342
S. Bloch, M. Kerr, P. Vanhove, Adv. Theor. Math. Phys. 21, 1373 (2017), arXiv:1601.08181
S. Groote, J.G. Körner (2018), arXiv:1804.10570
S. Bloch, M. Kerr, P. Vanhove, Compos. Math. 151, 2329 (2015), arXiv:1406.2664
E. Remiddi, L. Tancredi, Nucl. Phys. B 907, 400 (2016), arXiv:1602.01481
L. Adams, C. Bogner, A. Schweitzer, S. Weinzierl, J. Math. Phys. 57, 122302 (2016), arXiv:1607.01571
L. Adams, S. Weinzierl, Commun. Number Theor. Phys. 12, 193 (2018), arXiv:1704.08895
C. Bogner, A. Schweitzer, S. Weinzierl, Nucl. Phys. B 922, 528 (2017), arXiv:1705.08952
L. Adams, S. Weinzierl, Phys. Lett. B 781, 270 (2018), arXiv:1802.05020
L. Adams, E. Chaubey, S. Weinzierl (2018), arXiv:1804.11144
L. Adams, E. Chaubey, S. Weinzierl (2018), arXiv:1806.04981
M. Sgaard, Y. Zhang, Phys. Rev. D 91, 081701 (2015), arXiv:1412.5577
R. Bonciani et al., JHEP 12, 096 (2016), arXiv:1609.06685
A. von Manteuffel, L. Tancredi, JHEP 06, 127 (2017), arXiv:1701.05905
A. Primo, L. Tancredi, Nucl. Phys. B 921, 316 (2017), arXiv:1704.05465
J. Ablinger, J. Blümlein, A. De Freitas, M. van Hoeij, E. Imamoglu, C.G. Raab, C.S. Radu, C. Schneider, J. Math. Phys. 59(6), 062305 (2018), arXiv:1706.01299 [hep-th]
J.L. Bourjaily, A.J. McLeod, M. Spradlin, M. von Hippel, M. Wilhelm, Phys. Rev. Lett. 120, 121603 (2018), arXiv:1712.02785
M. Hidding, F. Moriello (2017), arXiv:1712.04441
G. Passarino, Eur. Phys. J. C 77, 77 (2017), arXiv:1610.06207
E. Remiddi, L. Tancredi, Nucl. Phys. B 925, 212 (2017), arXiv:1709.03622
J. Broedel, C. Duhr, F. Dulat, L. Tancredi, JHEP 05, 093 (2018), arXiv:1712.07089
J. Broedel, C. Duhr, F. Dulat, L. Tancredi, Phys. Rev. D 97, 116009 (2018), arXiv:1712.07095
J. Broedel, C. Duhr, F. Dulat, B. Penante, L. Tancredi (2018), arXiv:1803.10256
R.N. Lee, A.V. Smirnov, V.A. Smirnov, JHEP 03, 008 (2018), arXiv:1709.07525
R.N. Lee, A.V. Smirnov, V.A. Smirnov (2018), arXiv:1805.00227
J. Broedel, C.R. Mafra, N. Matthes, O. Schlotterer, JHEP 07, 112 (2015), arXiv:1412.5535
J. Broedel, N. Matthes, O. Schlotterer, J. Phys. A 49, 155203 (2016), arXiv:1507.02254
J. Broedel, N. Matthes, G. Richter, O. Schlotterer, J. Phys. A 51, 285401 (2018), arXiv:1704.03449
E. D’Hoker, M.B. Green, Ö. Gürdogan, P. Vanhove, Commun. Number Theor. Phys. 11, 165 (2017), arXiv:1512.06779
S. Hohenegger, S. Stieberger, Nucl. Phys. B 925, 63 (2017), arXiv:1702.04963
J. Broedel, O. Schlotterer, F. Zerbini (2018), arXiv:1803.00527
J.A.M. Vermaseren, Int. J. Mod. Phys. A 14, 2037 (1999), arXiv:hep-ph/9806280
E. Remiddi, J.A.M. Vermaseren, Int. J. Mod. Phys. A 15, 725 (2000), arXiv:hep-ph/9905237
A. Beilinson, A. Levin, in Motives, ed. by U. Jannsen, S. Kleiman, J.-P. Serre, Proceedings of Symposia in Pure Mathematics, vol. 55, Part 2 (AMS, 1994), pp. 123–190
A. Levin, Comput. Math. 106, 267 (1997)
A. Levin, G. Racinet (2007), arXiv:math/0703237
B. Enriquez, Selecta Math. 20, 491 (2014), arXiv:1003.1012
F. Brown, A. Levin (2011), arXiv:1110.6917
J. Wildeshaus, Lecture Notes in Mathematics, vol. 1650 (Springer, Berlin, 1997)
A.V. Kotikov, Phys. Lett. B 254, 158 (1991)
A.V. Kotikov, Phys. Lett. B 267, 123 (1991)
E. Remiddi, Nuovo Cim. A 110, 1435 (1997), arXiv:hep-th/9711188
T. Gehrmann, E. Remiddi, Nucl. Phys. B 580, 485 (2000), arXiv:hep-ph/9912329
M. Argeri, P. Mastrolia, Int. J. Mod. Phys. A 22, 4375 (2007), arXiv:0707.4037
S. Müller-Stach, S. Weinzierl, R. Zayadeh, Commun. Math. Phys. 326, 237 (2014), arXiv:1212.4389
J.M. Henn, Phys. Rev. Lett. 110, 251601 (2013), arXiv:1304.1806
J.M. Henn, J. Phys. A 48, 153001 (2015), arXiv:1412.2296
J. Ablinger, A. Behring, J. Blümlein, A. De Freitas, A. von Manteuffel, C. Schneider, Comput. Phys. Commun. 202, 33 (2016), arXiv:1509.08324 [hep-ph]
L. Adams, E. Chaubey, S. Weinzierl, Phys. Rev. Lett. 118, 141602 (2017), arXiv:1702.04279
J. Bosma, K.J. Larsen, Y. Zhang, Phys. Rev. D 97, 105014 (2018), arXiv:1712.03760
M. Kontsevich, D. Zagier, in Mathematics Unlimited - 2001 and Beyond, ed. by B. Engquist, W. Schmid (2001), p. 771
K.-T. Chen, Bull. Am. Math. Soc. 83, 831 (1977)
A.B. Goncharov, Math. Res. Lett. 5, 497 (1998)
A.B. Goncharov (2001), arXiv:math.AG/0103059
J.M. Borwein, D.M. Bradley, D.J. Broadhurst, P. Lisonek, Trans. Am. Math. Soc. 353:3, 907 (2001), arXiv:math.CA/9910045
S. Moch, P. Uwer, S. Weinzierl, J. Math. Phys. 43, 3363 (2002), arXiv:hep-ph/0110083
J. Vollinga, S. Weinzierl, Comput. Phys. Commun. 167, 177 (2005), arXiv:hep-ph/0410259
F. Brown (2014), arXiv:1407.5167
O.V. Tarasov, Phys. Rev. D 54, 6479 (1996), arXiv:hep-th/9606018
O.V. Tarasov, Nucl. Phys. B 502, 455 (1997), arXiv:hep-ph/9703319
C. Bogner, S. Weinzierl, Int. J. Mod. Phys. A 25, 2585 (2010), arXiv:1002.3458
C. Bogner, S. Weinzierl, J. Math. Phys. 50, 042302 (2009), arXiv:0711.4863
F.V. Tkachov, Phys. Lett. B 100, 65 (1981)
K.G. Chetyrkin, F.V. Tkachov, Nucl. Phys. B 192, 159 (1981)
M. van Hoeij, J. Symb. Comput. 24, 537 (1997)
P.A. Baikov, Nucl. Instrum. Methods A389, 347 (1997), arXiv:hep-ph/9611449
R.N. Lee, Nucl. Phys. B 830, 474 (2010), arXiv:0911.0252
D.A. Kosower, K.J. Larsen, Phys. Rev. D 85, 045017 (2012), arXiv:1108.1180
S. Caron-Huot, K.J. Larsen, JHEP 1210, 026 (2012), arXiv:1205.0801
H. Frellesvig, C.G. Papadopoulos, JHEP 04, 083 (2017), arXiv:1701.07356
J. Bosma, M. Sogaard, Y. Zhang, JHEP 08, 051 (2017), arXiv:1704.04255
M. Harley, F. Moriello, R.M. Schabinger, JHEP 06, 049 (2017), arXiv:1705.03478
C. Meyer, JHEP 04, 006 (2017), arXiv:1611.01087
C. Meyer, Comput. Phys. Commun. 222, 295 (2018), arXiv:1705.06252
W.A. Stein, Modular Forms, A Computational Approach (American Mathematical Society, Providence, 2007)
C. Bogner, A. Schweitzer, S. Weinzierl, these proceedings
Acknowledgements
S.W. would like to thank the organisers and KMPB for the organisation of the inspiring conference.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Adams, L., Weinzierl, S. (2019). On a Class of Feynman Integrals Evaluating to Iterated Integrals of Modular Forms. In: Blümlein, J., Schneider, C., Paule, P. (eds) Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory. Texts & Monographs in Symbolic Computation. Springer, Cham. https://doi.org/10.1007/978-3-030-04480-0_2
Download citation
DOI: https://doi.org/10.1007/978-3-030-04480-0_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-04479-4
Online ISBN: 978-3-030-04480-0
eBook Packages: Computer ScienceComputer Science (R0)