Abstract
We study perturbative aspects of recently proposed integrated four-point correlators in \( \mathcal{N} \) = 4 supersymmetric Yang-Mills with all classical gauge groups using standard Feynman diagram computations. We argue that perturbative contributions of the integrated correlators are given by linear combinations of periods of certain conformal Feynman graphs, which were originally introduced for the construction of perturbative loop integrands of the un-integrated correlator. This observation allows us to evaluate the integrated correlators to high loop orders. We explicitly compute one of the integrated correlators up to four loops in the planar limit, and up to three loops for the other integrated correlator, and find agreement with the results obtained from supersymmetric localisation. The identification between the integrated correlators and certain periods also implies non-trivial relations among these periods, given that one may compute the integrated correlators using localisation. We illustrate this idea by considering one of the integrated correlators at five loops in the planar limit, where the localisation result leads to a prediction for the period of a certain six-loop integral.
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D.J. Binder, S.M. Chester, S.S. Pufu and Y. Wang, N = 4 super-Yang-Mills correlators at strong coupling from string theory and localization, JHEP 12 (2019) 119 [arXiv:1902.06263] [INSPIRE].
S.M. Chester, Genus-2 holographic correlator on AdS5 × S5 from localization, JHEP 04 (2020) 193 [arXiv:1908.05247] [INSPIRE].
S.M. Chester and S.S. Pufu, Far beyond the planar limit in strongly-coupled N = 4 SYM, JHEP 01 (2021) 103 [arXiv:2003.08412] [INSPIRE].
S.M. Chester, M.B. Green, S.S. Pufu, Y. Wang and C. Wen, Modular invariance in superstring theory from N = 4 super-Yang-Mills, JHEP 11 (2020) 016 [arXiv:1912.13365] [INSPIRE].
S.M. Chester, M.B. Green, S.S. Pufu, Y. Wang and C. Wen, New modular invariants in N = 4 super-Yang-Mills theory, JHEP 04 (2021) 212 [arXiv:2008.02713] [INSPIRE].
D. Dorigoni, M.B. Green and C. Wen, Novel representation of an integrated correlator in N = 4 supersymmetric Yang-Mills theory, Phys. Rev. Lett. 126 (2021) 161601 [arXiv:2102.08305] [INSPIRE].
D. Dorigoni, M.B. Green and C. Wen, Exact properties of an integrated correlator in N = 4 SU(N) SYM, JHEP 05 (2021) 089 [arXiv:2102.09537] [INSPIRE].
S.M. Chester, R. Dempsey and S.S. Pufu, Bootstrapping N = 4 super-Yang-Mills on the conformal manifold, arXiv:2111.07989 [INSPIRE].
S. Collier and E. Perlmutter, Harnessing S-duality in N = 4 SYM & supergravity as SL(2, Z)-averaged strings, arXiv:2201.05093 [INSPIRE].
L.F. Alday, S.M. Chester and T. Hansen, Modular invariant holographic correlators for N = 4 SYM with general gauge group, JHEP 12 (2021) 159 [arXiv:2110.13106] [INSPIRE].
D. Dorigoni, M.B. Green and C. Wen, Exact results for duality-covariant integrated correlators in N = 4 SYM with general classical gauge groups, arXiv:2202.05784 [INSPIRE].
M.B. Green and C. Wen, Maximal U(1)Y-violating n-point correlators in N = 4 super-Yang-Mills theory, JHEP 02 (2021) 042 [arXiv:2009.01211] [INSPIRE].
D. Dorigoni, M.B. Green and C. Wen, Exact expressions for n-point maximal U(1)Y-violating integrated correlators in SU(N) N = 4 SYM, JHEP 11 (2021) 132 [arXiv:2109.08086] [INSPIRE].
R.H. Boels, Maximal R-symmetry violating amplitudes in type IIB superstring theory, Phys. Rev. Lett. 109 (2012) 081602 [arXiv:1204.4208] [INSPIRE].
M.B. Green and C. Wen, Modular forms and SL(2, Z)-covariance of type IIB superstring theory, JHEP 06 (2019) 087 [arXiv:1904.13394] [INSPIRE].
B. Eden, P. Heslop, G.P. Korchemsky and E. Sokatchev, Hidden symmetry of four-point correlation functions and amplitudes in N = 4 SYM, Nucl. Phys. B 862 (2012) 193 [arXiv:1108.3557] [INSPIRE].
B. Eden, P. Heslop, G.P. Korchemsky and E. Sokatchev, Constructing the correlation function of four stress-tensor multiplets and the four-particle amplitude in N = 4 SYM, Nucl. Phys. B 862 (2012) 450 [arXiv:1201.5329] [INSPIRE].
J.L. Bourjaily, P. Heslop and V.-V. Tran, Perturbation theory at eight loops: novel structures and the breakdown of manifest conformality in N = 4 supersymmetric Yang-Mills theory, Phys. Rev. Lett. 116 (2016) 191602 [arXiv:1512.07912] [INSPIRE].
J.L. Bourjaily, P. Heslop and V.-V. Tran, Amplitudes and correlators to ten loops using simple, graphical bootstraps, JHEP 11 (2016) 125 [arXiv:1609.00007] [INSPIRE].
F. Gonzalez-Rey, I.Y. Park and K. Schalm, A note on four point functions of conformal operators in N = 4 super Yang-Mills, Phys. Lett. B 448 (1999) 37 [hep-th/9811155] [INSPIRE].
B. Eden, P.S. Howe, C. Schubert, E. Sokatchev and P.C. West, Four point functions in N = 4 supersymmetric Yang-Mills theory at two loops, Nucl. Phys. B 557 (1999) 355 [hep-th/9811172] [INSPIRE].
B. Eden, P.S. Howe, C. Schubert, E. Sokatchev and P.C. West, Simplifications of four point functions in N = 4 supersymmetric Yang-Mills theory at two loops, Phys. Lett. B 466 (1999) 20 [hep-th/9906051] [INSPIRE].
B. Eden, C. Schubert and E. Sokatchev, Three loop four point correlator in N = 4 SYM, Phys. Lett. B 482 (2000) 309 [hep-th/0003096] [INSPIRE].
M. Bianchi, S. Kovacs, G. Rossi and Y.S. Stanev, Anomalous dimensions in N = 4 SYM theory at order g4, Nucl. Phys. B 584 (2000) 216 [hep-th/0003203] [INSPIRE].
J. Drummond, C. Duhr, B. Eden, P. Heslop, J. Pennington and V.A. Smirnov, Leading singularities and off-shell conformal integrals, JHEP 08 (2013) 133 [arXiv:1303.6909] [INSPIRE].
D.J. Broadhurst and D. Kreimer, Knots and numbers in ϕ4 theory to 7 loops and beyond, Int. J. Mod. Phys. C 6 (1995) 519 [hep-ph/9504352] [INSPIRE].
O. Schnetz, Quantum periods: a census of ϕ4-transcendentals, Commun. Num. Theor. Phys. 4 (2010) 1 [arXiv:0801.2856] [INSPIRE].
F.C.S. Brown, On the periods of some Feynman integrals, arXiv:0910.0114 [INSPIRE].
O. Schnetz, Graphical functions and single-valued multiple polylogarithms, Commun. Num. Theor. Phys. 08 (2014) 589 [arXiv:1302.6445] [INSPIRE].
E. Panzer, Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals, Comput. Phys. Commun. 188 (2015) 148 [arXiv:1403.3385] [INSPIRE].
O. Schnetz, Numbers and functions in quantum field theory, Phys. Rev. D 97 (2018) 085018 [arXiv:1606.08598] [INSPIRE].
E. Panzer and O. Schnetz, The Galois coaction on ϕ4 periods, Commun. Num. Theor. Phys. 11 (2017) 657 [arXiv:1603.04289] [INSPIRE].
A. Georgoudis, V. Gonçalves, E. Panzer, R. Pereira, A.V. Smirnov and V.A. Smirnov, Glue-and-cut at five loops, JHEP 09 (2021) 098 [arXiv:2104.08272] [INSPIRE].
O. Schnetz, HyperlogProcedures webpage, https://www.math.fau.de/person/oliver-schnetz/.
B. Eden, A.C. Petkou, C. Schubert and E. Sokatchev, Partial nonrenormalization of the stress tensor four point function in N = 4 SYM and AdS/CFT, Nucl. Phys. B 607 (2001) 191 [hep-th/0009106] [INSPIRE].
M. Nirschl and H. Osborn, Superconformal Ward identities and their solution, Nucl. Phys. B 711 (2005) 409 [hep-th/0407060] [INSPIRE].
T. Fleury and R. Pereira, Non-planar data of N = 4 SYM, JHEP 03 (2020) 003 [arXiv:1910.09428] [INSPIRE].
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
U. Naseer and C. Thull, Flavor deformations and supersymmetry enhancement in 4d N = 2 theories, arXiv:2110.09329 [INSPIRE].
P. Cvitanovic, Group theory: birdtracks, Lie’s and exceptional groups, Princeton University Press, Princeton, NJ, U.S.A. (2008).
R.L. Mkrtchian, The equivalence of Sp(2N) and SO(−2N) gauge theories, Phys. Lett. B 105 (1981) 174 [INSPIRE].
P. Cvitanovic and A.D. Kennedy, Spinors in negative dimensions, Phys. Scripta 26 (1982) 5 [INSPIRE].
M. Kontsevich and D. Zagier, Periods, in Mathematics unlimited — 2001 and beyond, Springer, Berlin, Heidelberg, Germany (2001), p. 771.
D. Chicherin, J. Drummond, P. Heslop and E. Sokatchev, All three-loop four-point correlators of half-BPS operators in planar N = 4 SYM, JHEP 08 (2016) 053 [arXiv:1512.02926] [INSPIRE].
D. Chicherin, A. Georgoudis, V. Gonçalves and R. Pereira, All five-loop planar four-point functions of half-BPS operators in N = 4 SYM, JHEP 11 (2018) 069 [arXiv:1809.00551] [INSPIRE].
S. Caron-Huot and F. Coronado, Ten dimensional symmetry of N = 4 SYM correlators, JHEP 03 (2022) 151 [arXiv:2106.03892] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math. 244 (2006) 525 [hep-th/0306238] [INSPIRE].
M. Billò, M. Frau, F. Fucito, A. Lerda and J.F. Morales, S-duality and the prepotential in N = 2* theories. Part I. The ADE algebras, JHEP 11 (2015) 024 [arXiv:1507.07709] [INSPIRE].
M. Billò, M. Frau, F. Fucito, A. Lerda and J.F. Morales, S-duality and the prepotential of N = 2* theories. Part II. The non-simply laced algebras, JHEP 11 (2015) 026 [arXiv:1507.08027] [INSPIRE].
M. Billò, M. Frau, F. Fucito, J.F. Morales and A. Lerda, Resumming instantons in N = 2* theories with arbitrary gauge groups, in 14th Marcel Grossmann meeting on recent developments in theoretical and experimental general relativity, astrophysics, and relativistic field theories, volume 4, World Scientific, Singapore (2017), p. 4139 [arXiv:1602.00273] [INSPIRE].
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Wen, C., Zhang, SQ. Integrated correlators in \( \mathcal{N} \) = 4 super Yang-Mills and periods. J. High Energ. Phys. 2022, 126 (2022). https://doi.org/10.1007/JHEP05(2022)126
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DOI: https://doi.org/10.1007/JHEP05(2022)126