Abstract
We present the complete formula for the cusp anomalous dimension at four loops in QCD and in maximally supersymmetric Yang-Mills. In the latter theory it is given by \( {\left.{\Gamma}_{\mathrm{cusp},\mathrm{A}}\right|}_{\alpha_s^4}=-{\left(\frac{\alpha_sN}{\pi}\right)}^4\left[\frac{73{\pi}^6}{20160}+\frac{\zeta_3^2}{8}+\frac{1}{N^2}\left(\frac{31{\pi}^6}{5040}+\frac{9{\zeta}_3^2}{4}\right)\right]. \) Our approach is based on computing the correlation function of a rectangular light-like Wilson loop with a Lagrangian insertion, normalized by the expectation value of the Wilson loop. In maximally supersymmetric Yang-Mills theory, this ratio is a finite function of a cross-ratio and the coupling constant. We compute it to three loops, including the full colour dependence. Integrating over the position of the Lagrangian insertion gives the four-loop Wilson loop. We extract its leading divergence, which determines the four-loop cusp anomalous dimension. Finally, we employ a supersymmetric decomposition to derive the last missing ingredient in the corresponding QCD result.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A.M. Polyakov, Gauge Fields as Rings of Glue, Nucl. Phys.B 164 (1980) 171 [INSPIRE].
G.P. Korchemsky and A.V. Radyushkin, Renormalization of the Wilson Loops Beyond the Leading Order, Nucl. Phys.B 283 (1987) 342 [INSPIRE].
G.P. Korchemsky and A.V. Radyushkin, Loop Space Formalism and Renormalization Group for the Infrared Asymptotics of QCD, Phys. Lett.B 171 (1986) 459 [INSPIRE].
J.C. Collins, D.E. Soper and G.F. Sterman, Factorization of Hard Processes in QCD, Adv. Ser. Direct. High Energy Phys.5 (1989) 1 [hep-ph/0409313] [INSPIRE].
J.C. Collins, Sudakov form-factors, Adv. Ser. Direct. High Energy Phys.5 (1989) 573 [hep-ph/0312336] [INSPIRE].
G.P. Korchemsky, Asymptotics of the Altarelli-Parisi-Lipatov Evolution Kernels of Parton Distributions, Mod. Phys. Lett.A 4 (1989) 1257 [INSPIRE].
G.P. Korchemsky and G. Marchesini, Structure function for large x and renormalization of Wilson loop, Nucl. Phys.B 406 (1993) 225 [hep-ph/9210281] [INSPIRE].
G.F. Sterman, Summation of Large Corrections to Short Distance Hadronic Cross-Sections, Nucl. Phys.B 281 (1987) 310 [INSPIRE].
S. Catani and L. Trentadue, Resummation of the QCD Perturbative Series for Hard Processes, Nucl. Phys.B 327 (1989) 323 [INSPIRE].
G.P. Korchemsky and G. Marchesini, Resummation of large infrared corrections using Wilson loops, Phys. Lett.B 313 (1993) 433 [INSPIRE].
Ø. Almelid, C. Duhr and E. Gardi, Three-loop corrections to the soft anomalous dimension in multileg scattering, Phys. Rev. Lett.117 (2016) 172002 [arXiv:1507.00047] [INSPIRE].
Ø. Almelid, C. Duhr, E. Gardi, A. McLeod and C.D. White, Bootstrapping the QCD soft anomalous dimension, JHEP09 (2017) 073 [arXiv:1706.10162] [INSPIRE].
T. Becher and M. Neubert, Infrared singularities of scattering amplitudes and N3LL resummation for n-jet processes, JHEP01 (2020) 025 [arXiv:1908.11379] [INSPIRE].
S. Caron-Huot, E. Gardi, J. Reichel and L. Vernazza, Infrared singularities of QCD scattering amplitudes in the Regge limit to all orders, JHEP03 (2018) 098 [arXiv:1711.04850] [INSPIRE].
B. Mistlberger, Higgs boson production at hadron colliders at N3LO in QCD, JHEP05 (2018) 028 [arXiv:1802.00833] [INSPIRE].
C. Anastasiou, C. Duhr, F. Dulat, F. Herzog and B. Mistlberger, Higgs Boson Gluon-Fusion Production in QCD at Three Loops, Phys. Rev. Lett.114 (2015) 212001 [arXiv:1503.06056] [INSPIRE].
T. Gehrmann, E.W.N. Glover, T. Huber, N. Ikizlerli and C. Studerus, Calculation of the quark and gluon form factors to three loops in QCD, JHEP06 (2010) 094 [arXiv:1004.3653] [INSPIRE].
J. Henn, A.V. Smirnov, V.A. Smirnov, M. Steinhauser and R.N. Lee, Four-loop photon quark form factor and cusp anomalous dimension in the large-Nclimit of QCD, JHEP03 (2017) 139 [arXiv:1612.04389] [INSPIRE].
J.M. Henn, A.V. Smirnov, V.A. Smirnov and M. Steinhauser, A planar four-loop form factor and cusp anomalous dimension in QCD, JHEP05 (2016) 066 [arXiv:1604.03126] [INSPIRE].
A. von Manteuffel and R.M. Schabinger, Quark and gluon form factors in four loop QCD: The \( {N}_f^2 \)and NqγNfcontributions, Phys. Rev.D 99 (2019) 094014 [arXiv:1902.08208] [INSPIRE].
A. von Manteuffel and R.M. Schabinger, Planar master integrals for four-loop form factors, JHEP05 (2019) 073 [arXiv:1903.06171] [INSPIRE].
R. Abbate, M. Fickinger, A.H. Hoang, V. Mateu and I.W. Stewart, Thrust at N3LL with Power Corrections and a Precision Global Fit for αs (mZ ), Phys. Rev.D 83 (2011) 074021 [arXiv:1006.3080] [INSPIRE].
T. Becher and M.D. Schwartz, A precise determination of αsfrom LEP thrust data using effective field theory, JHEP07 (2008) 034 [arXiv:0803.0342] [INSPIRE].
W. Bizoń, P.F. Monni, E. Re, L. Rottoli and P. Torrielli, Momentum-space resummation for transverse observables and the Higgs p⊥at N3LL + NNLO, JHEP02 (2018) 108 [arXiv:1705.09127] [INSPIRE].
W. Bizoń et al., Fiducial distributions in Higgs and Drell-Yan production at N3LL + NNLO, JHEP12 (2018) 132 [arXiv:1805.05916] [INSPIRE].
W. Bizoń et al., The transverse momentum spectrum of weak gauge bosons at N3LL + NNLO, Eur. Phys. J.C 79 (2019) 868 [arXiv:1905.05171] [INSPIRE].
X. Chen et al., Precise QCD Description of the Higgs Boson Transverse Momentum Spectrum, Phys. Lett.B 788 (2019) 425 [arXiv:1805.00736] [INSPIRE].
N. Beisert et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys.99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
N. Beisert, B. Eden and M. Staudacher, Transcendentality and Crossing, J. Stat. Mech.0701 (2007) P01021 [hep-th/0610251] [INSPIRE].
S. Moch, J.A.M. Vermaseren and A. Vogt, The Three loop splitting functions in QCD: The Nonsinglet case, Nucl. Phys.B 688 (2004) 101 [hep-ph/0403192] [INSPIRE].
J.G.M. Gatheral, Exponentiation of Eikonal Cross-sections in Nonabelian Gauge Theories, Phys. Lett.B 133 (1983) 90 [INSPIRE].
J. Frenkel and J.C. Taylor, Nonabelian Eikonal Exponentiation, Nucl. Phys.B 246 (1984) 231 [INSPIRE].
G.P. Korchemsky, Instanton effects in correlation functions on the light-cone, JHEP12 (2017) 093 [arXiv:1704.00448] [INSPIRE].
I.A. Korchemskaya and G.P. Korchemsky, On lightlike Wilson loops, Phys. Lett.B 287 (1992) 169 [INSPIRE].
J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Conformal Ward identities for Wilson loops and a test of the duality with gluon amplitudes, Nucl. Phys.B 826 (2010) 337 [arXiv:0712.1223] [INSPIRE].
R.N. Lee, A.V. Smirnov, V.A. Smirnov and M. Steinhauser, Four-loop quark form factor with quartic fundamental colour factor, JHEP02 (2019) 172 [arXiv:1901.02898] [INSPIRE].
J.M. Henn, T. Peraro, M. Stahlhofen and P. Wasser, Matter dependence of the four-loop cusp anomalous dimension, Phys. Rev. Lett.122 (2019) 201602 [arXiv:1901.03693] [INSPIRE].
R.H. Boels, T. Huber and G. Yang, Four-Loop Nonplanar Cusp Anomalous Dimension in N = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett.119 (2017) 201601 [arXiv:1705.03444] [INSPIRE].
S. Moch, B. Ruijl, T. Ueda, J.A.M. Vermaseren and A. Vogt, Four-Loop Non-Singlet Splitting Functions in the Planar Limit and Beyond, JHEP10 (2017) 041 [arXiv:1707.08315] [INSPIRE].
S. Moch, B. Ruijl, T. Ueda, J.A.M. Vermaseren and A. Vogt, On quartic colour factors in splitting functions and the gluon cusp anomalous dimension, Phys. Lett.B 782 (2018) 627 [arXiv:1805.09638] [INSPIRE].
Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, One loop n point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys.B 425 (1994) 217 [hep-ph/9403226] [INSPIRE].
O. Erdoğan and G. Sterman, Gauge Theory Webs and Surfaces, Phys. Rev.D 91 (2015) 016003 [arXiv:1112.4564] [INSPIRE].
L.F. Alday, E.I. Buchbinder and A.A. Tseytlin, Correlation function of null polygonal Wilson loops with local operators, JHEP09 (2011) 034 [arXiv:1107.5702] [INSPIRE].
L.F. Alday, P. Heslop and J. Sikorowski, Perturbative correlation functions of null Wilson loops and local operators, JHEP03 (2013) 074 [arXiv:1207.4316] [INSPIRE].
L.F. Alday, J.M. Henn and J. Sikorowski, Higher loop mixed correlators in N = 4 SYM, JHEP03 (2013) 058 [arXiv:1301.0149] [INSPIRE].
O.T. Engelund and R. Roiban, On correlation functions of Wilson loops, local and non-local operators, JHEP05 (2012) 158 [arXiv:1110.0758] [INSPIRE].
O.T. Engelund and R. Roiban, Correlation functions of local composite operators from generalized unitarity, JHEP03 (2013) 172 [arXiv:1209.0227] [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].
T. Fleury and R. Pereira, Non-planar data of \( \mathcal{N} \) = 4 SYM, arXiv:1910.09428 [INSPIRE].
D. Chicherin et al., Correlation functions of the chiral stress-tensor multiplet in \( \mathcal{N} \) = 4 SYM, JHEP06 (2015) 198 [arXiv:1412.8718] [INSPIRE].
L.F. Alday, B. Eden, G.P. Korchemsky, J. Maldacena and E. Sokatchev, From correlation functions to Wilson loops, JHEP09 (2011) 123 [arXiv:1007.3243] [INSPIRE].
J.M. Henn, A.V. Smirnov and V.A. Smirnov, Analytic results for planar three-loop four-point integrals from a Knizhnik-Zamolodchikov equation, JHEP07 (2013) 128 [arXiv:1306.2799] [INSPIRE].
J.M. Henn, A.V. Smirnov and V.A. Smirnov, Evaluating single-scale and/or non-planar diagrams by differential equations, JHEP03 (2014) 088 [arXiv:1312.2588] [INSPIRE].
J.M. Henn and B. Mistlberger, Four-Gluon Scattering at Three Loops, Infrared Structure and the Regge Limit, Phys. Rev. Lett.117 (2016) 171601 [arXiv:1608.00850] [INSPIRE].
T. Gehrmann and E. Remiddi, Two loop master integrals for γ∗ → 3 jets: The Planar topologies, Nucl. Phys.B 601 (2001) 248 [hep-ph/0008287] [INSPIRE].
A.V. Smirnov, FIRE5: a C++ implementation of Feynman Integral REduction, Comput. Phys. Commun.189 (2015) 182 [arXiv:1408.2372] [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo and J. Trnka, Local Integrals for Planar Scattering Amplitudes, JHEP06 (2012) 125 [arXiv:1012.6032] [INSPIRE].
J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett.110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].
P. Wasser, Analytic properties of Feynman integrals for scattering amplitudes, MSc Thesis, Institut für Physik, Johannes Gutenberg-Universität, Mainz Germany (2016) and online at https://publications.ub.uni-mainz.de/theses/frontdoor.php?sourceopus=100001967.
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo and J. Trnka, Singularity Structure of Maximally Supersymmetric Scattering Amplitudes, Phys. Rev. Lett.113 (2014) 261603 [arXiv:1410.0354] [INSPIRE].
Z. Bern, E. Herrmann, S. Litsey, J. Stankowicz and J. Trnka, Evidence for a Nonplanar Amplituhedron, JHEP06 (2016) 098 [arXiv:1512.08591] [INSPIRE].
M. Argeri and P. Mastrolia, Feynman Diagrams and Differential Equations, Int. J. Mod. Phys.A 22 (2007) 4375 [arXiv:0707.4037] [INSPIRE].
J.M. Henn, Lectures on differential equations for Feynman integrals, J. Phys.A 48 (2015) 153001 [arXiv:1412.2296] [INSPIRE].
D. Chicherin, T. Gehrmann, J.M. Henn, P. Wasser, Y. Zhang and S. Zoia, All Master Integrals for Three-Jet Production at Next-to-Next-to-Leading Order, Phys. Rev. Lett.123 (2019) 041603 [arXiv:1812.11160] [INSPIRE].
E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys.A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].
D. Maître, HPL, a mathematica implementation of the harmonic polylogarithms, Comput. Phys. Commun.174 (2006) 222 [hep-ph/0507152] [INSPIRE].
M. Beneke and V.M. Braun, Power corrections and renormalons in Drell-Yan production, Nucl. Phys.B 454 (1995) 253 [hep-ph/9506452] [INSPIRE].
J. Davies, A. Vogt, B. Ruijl, T. Ueda and J.A.M. Vermaseren, Large-Nfcontributions to the four-loop splitting functions in QCD, Nucl. Phys.B 915 (2017) 335 [arXiv:1610.07477] [INSPIRE].
R. Brüser, A. Grozin, J.M. Henn and M. Stahlhofen, Matter dependence of the four-loop QCD cusp anomalous dimension: from small angles to all angles, JHEP05 (2019) 186 [arXiv:1902.05076] [INSPIRE].
J.A. Gracey, Anomalous dimension of nonsinglet Wilson operators at O(1/Nf) in deep inelastic scattering, Phys. Lett.B 322 (1994) 141 [hep-ph/9401214] [INSPIRE].
A. Grozin, Four-loop cusp anomalous dimension in QED, JHEP06 (2018) 073 [arXiv:1805.05050] [INSPIRE].
T. van Ritbergen, J.A.M. Vermaseren and S.A. Larin, The Four loop β-function in quantum chromodynamics, Phys. Lett.B 400 (1997) 379 [hep-ph/9701390] [INSPIRE].
T. van Ritbergen, A.N. Schellekens and J.A.M. Vermaseren, Group theory factors for Feynman diagrams, Int. J. Mod. Phys.A 14 (1999) 41 [hep-ph/9802376] [INSPIRE].
A.V. Kotikov, L.N. Lipatov, A.I. Onishchenko and V.N. Velizhanin, Three loop universal anomalous dimension of the Wilson operators in N = 4 SUSY Yang-Mills model, Phys. Lett.B 595 (2004) 521 [Erratum ibid.B 632 (2006) 754] [hep-th/0404092] [INSPIRE].
L.J. Dixon, The Principle of Maximal Transcendentality and the Four-Loop Collinear Anomalous Dimension, JHEP01 (2018) 075 [arXiv:1712.07274] [INSPIRE].
Z. Bern, M. Czakon, L.J. Dixon, D.A. Kosower and V.A. Smirnov, The Four-Loop Planar Amplitude and Cusp Anomalous Dimension in Maximally Supersymmetric Yang-Mills Theory, Phys. Rev.D 75 (2007) 085010 [hep-th/0610248] [INSPIRE].
V.N. Velizhanin, The Non-planar contribution to the four-loop universal anomalous dimension in N = 4 Supersymmetric Yang-Mills theory, JETP Lett.89 (2009) 593 [arXiv:0902.4646] [INSPIRE].
V.N. Velizhanin, The Non-planar contribution to the four-loop anomalous dimension of twist-2 operators: First moments in N = 4 SYM and non-singlet QCD, Nucl. Phys.B 846 (2011) 137 [arXiv:1008.2752] [INSPIRE].
V.N. Velizhanin, Non-planar anomalous dimension of twist-2 operators: higher moments at four loops, Nucl. Phys.B 885 (2014) 772 [arXiv:1404.7107] [INSPIRE].
T. Huber, A. von Manteuffel, E. Panzer, R.M. Schabinger and G. Yang, The Four-Loop Cusp Anomalous Dimension from the \( \mathcal{N} \) = 4 Sudakov Form Factor, arXiv:1912.13459 [INSPIRE].
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
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1911.10174
Electronic supplementary material
ESM 1
(TGZ 2 kb)
Rights and permissions
This article is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.
About this article
Cite this article
Henn, J.M., Korchemsky, G.P. & Mistlberger, B. The full four-loop cusp anomalous dimension in \( \mathcal{N} \) = 4 super Yang-Mills and QCD. J. High Energ. Phys. 2020, 18 (2020). https://doi.org/10.1007/JHEP04(2020)018
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2020)018