Abstract
Correlators of Wilson-line operators in non-abelian gauge theories are known to exponentiate, and their logarithms can be organised in terms of collections of Feynman diagrams called webs. In [1] we introduced the concept of Cweb, or correlator web, which is a set of skeleton diagrams built with connected gluon correlators, and we computed the mixing matrices for all Cwebs connecting four or five Wilson lines at four loops. Here we complete the evaluation of four-loop mixing matrices, presenting the results for all Cwebs connecting two and three Wilson lines. We observe that the conjuctured column sum rule is obeyed by all the mixing matrices that appear at four-loops. We also show how low-dimensional mixing matrices can be uniquely determined from their known combinatorial properties, and provide some all-order results for selected classes of mixing matrices. Our results complete the required colour building blocks for the calculation of the soft anomalous dimension matrix at four-loop order.
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N. Agarwal, A. Danish, L. Magnea, S. Pal and A. Tripathi, Multiparton webs beyond three loops, JHEP 05 (2020) 128 [arXiv:2003.09714] [INSPIRE].
F. Bloch and A. Nordsieck, Note on the radiation field of the electron, Phys. Rev. 52 (1937) 54 [INSPIRE].
V.V. Sudakov, Vertex parts at very high-energies in quantum electrodynamics, Sov. Phys. JETP 3 (1956) 65 [Zh. Eksp. Teor. Fiz. 30 (1956) 87] [INSPIRE].
D.R. Yennie, S.C. Frautschi and H. Suura, The infrared divergence phenomena and high-energy processes, Annals Phys. 13 (1961) 379 [INSPIRE].
T. Kinoshita, Mass singularities of Feynman amplitudes, J. Math. Phys. 3 (1962) 650 [INSPIRE].
T.D. Lee and M. Nauenberg, Degenerate systems and mass singularities, Phys. Rev. 133 (1964) B1549 [INSPIRE].
G. Grammer Jr. and D.R. Yennie, Improved treatment for the infrared divergence problem in quantum electrodynamics, Phys. Rev. D 8 (1973) 4332 [INSPIRE].
A.H. Mueller, On the asymptotic behavior of the Sudakov form-factor, Phys. Rev. D 20 (1979) 2037 [INSPIRE].
J.C. Collins, Algorithm to compute corrections to the Sudakov form-factor, Phys. Rev. D 22 (1980) 1478 [INSPIRE].
A. Sen, Asymptotic behavior of the Sudakov form-factor in QCD, Phys. Rev. D 24 (1981) 3281 [INSPIRE].
A. Sen, Asymptotic behavior of the wide angle on-shell quark scattering amplitudes in non-Abelian gauge theories, Phys. Rev. D 28 (1983) 860 [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, Sudakov form-factor in QCD, Phys. Lett. B 220 (1989) 629 [INSPIRE].
L. Magnea and G.F. Sterman, Analytic continuation of the Sudakov form-factor in QCD, Phys. Rev. D 42 (1990) 4222 [INSPIRE].
L.J. Dixon, L. Magnea and G.F. Sterman, Universal structure of subleading infrared poles in gauge theory amplitudes, JHEP 08 (2008) 022 [arXiv:0805.3515] [INSPIRE].
E. Gardi and L. Magnea, Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes, JHEP 03 (2009) 079 [arXiv:0901.1091] [INSPIRE].
T. Becher and M. Neubert, On the structure of infrared singularities of gauge-theory amplitudes, JHEP 06 (2009) 081 [Erratum ibid. 11 (2013) 024] [arXiv:0903.1126] [INSPIRE].
I. Feige and M.D. Schwartz, Hard-soft-collinear factorization to all orders, Phys. Rev. D 90 (2014) 105020 [arXiv:1403.6472] [INSPIRE].
G.F. Sterman, Partons, factorization and resummation, TASI ’95, in Theoretical Advanced Study Institute in Elementary Particle Physics (TASI 95): QCD and beyond, (1995), pg. 327 [hep-ph/9606312] [INSPIRE].
E. Laenen, Resummation for observables at TeV colliders, Pramana 63 (2004) 1225 [INSPIRE].
G. Luisoni and S. Marzani, QCD resummation for hadronic final states, J. Phys. G 42 (2015) 103101 [arXiv:1505.04084] [INSPIRE].
A. Gehrmann-De Ridder, T. Gehrmann and E.W.N. Glover, Antenna subtraction at NNLO, JHEP 09 (2005) 056 [hep-ph/0505111] [INSPIRE].
G. Somogyi, Z. Trócsányi and V. Del Duca, Matching of singly- and doubly-unresolved limits of tree-level QCD squared matrix elements, JHEP 06 (2005) 024 [hep-ph/0502226] [INSPIRE].
S. Catani and M. Grazzini, An NNLO subtraction formalism in hadron collisions and its application to Higgs boson production at the LHC, Phys. Rev. Lett. 98 (2007) 222002 [hep-ph/0703012] [INSPIRE].
M. Czakon, A novel subtraction scheme for double-real radiation at NNLO, Phys. Lett. B 693 (2010) 259 [arXiv:1005.0274] [INSPIRE].
R. Boughezal, C. Focke, X. Liu and F. Petriello, W-boson production in association with a jet at next-to-next-to-leading order in perturbative QCD, Phys. Rev. Lett. 115 (2015) 062002 [arXiv:1504.02131] [INSPIRE].
G.F.R. Sborlini, F. Driencourt-Mangin and G. Rodrigo, Four-dimensional unsubtraction with massive particles, JHEP 10 (2016) 162 [arXiv:1608.01584] [INSPIRE].
F. Caola, K. Melnikov and R. Röntsch, Nested soft-collinear subtractions in NNLO QCD computations, Eur. Phys. J. C 77 (2017) 248 [arXiv:1702.01352] [INSPIRE].
F. Herzog, Geometric IR subtraction for final state real radiation, JHEP 08 (2018) 006 [arXiv:1804.07949] [INSPIRE].
L. Magnea, E. Maina, G. Pelliccioli, C. Signorile-Signorile, P. Torrielli and S. Uccirati, Local analytic sector subtraction at NNLO, JHEP 12 (2018) 107 [Erratum ibid. 06 (2019) 013] [arXiv:1806.09570] [INSPIRE].
L. Magnea, E. Maina, G. Pelliccioli, C. Signorile-Signorile, P. Torrielli and S. Uccirati, Factorisation and subtraction beyond NLO, JHEP 12 (2018) 062 [arXiv:1809.05444] [INSPIRE].
Z. Capatti, V. Hirschi, A. Pelloni and B. Ruijl, Local unitarity: a representation of differential cross-sections that is locally free of infrared singularities at any order, arXiv:2010.01068 [INSPIRE].
E. Gardi and L. Magnea, Infrared singularities in QCD amplitudes, Nuovo Cim. C 32N5-6 (2009) 137 [Frascati Phys. Ser. 50 (2010) 137] [arXiv:0908.3273] [INSPIRE].
A.V. Manohar and M.B. Wise, Heavy quark physics, Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 10 (2000) 1 [INSPIRE].
N. Brambilla, A. Pineda, J. Soto and A. Vairo, Effective field theories for heavy quarkonium, Rev. Mod. Phys. 77 (2005) 1423 [hep-ph/0410047] [INSPIRE].
T. Becher, A. Broggio and A. Ferroglia, Introduction to soft-collinear effective theory, Springer, Cham, Switzerland (2015) [arXiv:1410.1892] [INSPIRE].
A. Mitov, G.F. Sterman and I. Sung, Computation of the soft anomalous dimension matrix in coordinate space, Phys. Rev. D 82 (2010) 034020 [arXiv:1005.4646] [INSPIRE].
J.M. Henn and T. Huber, The four-loop cusp anomalous dimension in N = 4 super Yang-Mills and analytic integration techniques for Wilson line integrals, JHEP 09 (2013) 147 [arXiv:1304.6418] [INSPIRE].
E. Gardi, From webs to polylogarithms, JHEP 04 (2014) 044 [arXiv:1310.5268] [INSPIRE].
G. Falcioni, E. Gardi, M. Harley, L. Magnea and C.D. White, Multiple gluon exchange webs, JHEP 10 (2014) 010 [arXiv:1407.3477] [INSPIRE].
N. Kidonakis, G. Oderda and G.F. Sterman, Evolution of color exchange in QCD hard scattering, Nucl. Phys. B 531 (1998) 365 [hep-ph/9803241] [INSPIRE].
I.A. Korchemskaya and G.P. Korchemsky, High-energy scattering in QCD and cross singularities of Wilson loops, Nucl. Phys. B 437 (1995) 127 [hep-ph/9409446] [INSPIRE].
S.M. Aybat, L.J. Dixon and G.F. Sterman, The two-loop anomalous dimension matrix for soft gluon exchange, Phys. Rev. Lett. 97 (2006) 072001 [hep-ph/0606254] [INSPIRE].
S.M. Aybat, L.J. Dixon and G.F. Sterman, The two-loop soft anomalous dimension matrix and resummation at next-to-next-to leading pole, Phys. Rev. D 74 (2006) 074004 [hep-ph/0607309] [INSPIRE].
A. Mitov, G.F. Sterman and I. Sung, The massive soft anomalous dimension matrix at two loops, Phys. Rev. D 79 (2009) 094015 [arXiv:0903.3241] [INSPIRE].
A. Ferroglia, M. Neubert, B.D. Pecjak and L.L. Yang, Two-loop divergences of scattering amplitudes with massive partons, Phys. Rev. Lett. 103 (2009) 201601 [arXiv:0907.4791] [INSPIRE].
A. Ferroglia, M. Neubert, B.D. Pecjak and L.L. Yang, Two-loop divergences of massive scattering amplitudes in non-Abelian gauge theories, JHEP 11 (2009) 062 [arXiv:0908.3676] [INSPIRE].
N. Kidonakis, Two-loop soft anomalous dimensions and NNLL resummation for heavy quark production, Phys. Rev. Lett. 102 (2009) 232003 [arXiv:0903.2561] [INSPIRE].
Y.-T. Chien, M.D. Schwartz, D. Simmons-Duffin and I.W. Stewart, Jet physics from static charges in AdS, Phys. Rev. D 85 (2012) 045010 [arXiv:1109.6010] [INSPIRE].
O. 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].
O. Almelid, C. Duhr, E. Gardi, A. McLeod and C.D. White, Bootstrapping the QCD soft anomalous dimension, JHEP 09 (2017) 073 [arXiv:1706.10162] [INSPIRE].
T. Becher and M. Neubert, Infrared singularities of scattering amplitudes and N3LL resummation for n-jet processes, JHEP 01 (2020) 025 [arXiv:1908.11379] [INSPIRE].
G. Falcioni, E. Gardi, C. Milloy and L. Vernazza, Climbing three-Reggeon ladders: four-loop amplitudes in the high-energy limit in full colour, arXiv:2012.00613 [INSPIRE].
G.F. Sterman, Infrared divergences in perturbative QCD, AIP Conf. Proc. 74 (1981) 22 [INSPIRE].
J.G.M. Gatheral, Exponentiation of eikonal cross-sections in non-Abelian gauge theories, Phys. Lett. B 133 (1983) 90 [INSPIRE].
J. Frenkel and J.C. Taylor, Non-Abelian eikonal exponentiation, Nucl. Phys. B 246 (1984) 231 [INSPIRE].
A. Mitov, G. Sterman and I. Sung, Diagrammatic exponentiation for products of Wilson lines, Phys. Rev. D 82 (2010) 096010 [arXiv:1008.0099] [INSPIRE].
E. Gardi, E. Laenen, G. Stavenga and C.D. White, Webs in multiparton scattering using the replica trick, JHEP 11 (2010) 155 [arXiv:1008.0098] [INSPIRE].
E. Gardi, J.M. Smillie and C.D. White, On the renormalization of multiparton webs, JHEP 09 (2011) 114 [arXiv:1108.1357] [INSPIRE].
T. Becher and M. Neubert, Infrared singularities of scattering amplitudes in perturbative QCD, Phys. Rev. Lett. 102 (2009) 162001 [Erratum ibid. 111 (2013) 199905] [arXiv:0901.0722] [INSPIRE].
J.M. Henn, G.P. Korchemsky and B. Mistlberger, The full four-loop cusp anomalous dimension in N = 4 super Yang-Mills and QCD, JHEP 04 (2020) 018 [arXiv:1911.10174] [INSPIRE].
A. von Manteuffel, E. Panzer and R.M. Schabinger, Cusp and collinear anomalous dimensions in four-loop QCD from form factors, Phys. Rev. Lett. 124 (2020) 162001 [arXiv:2002.04617] [INSPIRE].
A.A. Vladimirov, Generating function for web diagrams, Phys. Rev. D 90 (2014) 066007 [arXiv:1406.6253] [INSPIRE].
A.A. Vladimirov, Exponentiation for products of Wilson lines within the generating function approach, JHEP 06 (2015) 120 [arXiv:1501.03316] [INSPIRE].
E. Gardi and C.D. White, General properties of multiparton webs: proofs from combinatorics, JHEP 03 (2011) 079 [arXiv:1102.0756] [INSPIRE].
M. Dukes, E. Gardi, E. Steingrimsson and C.D. White, Web worlds, web-colouring matrices, and web-mixing matrices, J. Comb. Theor. A 120 (2013) 1012 [arXiv:1301.6576] [INSPIRE].
E. Gardi, J.M. Smillie and C.D. White, The non-Abelian exponentiation theorem for multiple Wilson lines, JHEP 06 (2013) 088 [arXiv:1304.7040] [INSPIRE].
M. Dukes, E. Gardi, H. McAslan, D.J. Scott and C.D. White, Webs and posets, JHEP 01 (2014) 024 [arXiv:1310.3127] [INSPIRE].
M. Dukes and C.D. White, Web matrices: structural properties and generating combinatorial identities, arXiv:1603.01589 [INSPIRE].
O. Erdoğan and G. Sterman, Gauge theory webs and surfaces, Phys. Rev. D 91 (2015) 016003 [arXiv:1112.4564] [INSPIRE].
G. Falcioni, E. Gardi and C. Milloy, Relating amplitude and PDF factorisation through Wilson-line geometries, JHEP 11 (2019) 100 [arXiv:1909.00697] [INSPIRE].
M. Mézard, G. Parisi and M. Virasoro, Spin glass theory and beyond, World Scientific, Singapore (1986).
E. Laenen, G. Stavenga and C.D. White, Path integral approach to eikonal and next-to-eikonal exponentiation, JHEP 03 (2009) 054 [arXiv:0811.2067] [INSPIRE].
On-Line Encyclopedia of Integer Sequences, A000670 Fubini numbers: number of preferential arrangements of n labeled elements; or number of weak orders on n labeled elements; or number of ordered partitions of [n], https://oeis.org/A000670.
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].
K.G. Chetyrkin, G. Falcioni, F. Herzog and J.A.M. Vermaseren, Five-loop renormalisation of QCD in covariant gauges, JHEP 10 (2017) 179 [Addendum ibid. 12 (2017) 006] [arXiv:1709.08541] [INSPIRE].
P. Cvitanovic, Group theory: birdtracks, Lie’s and exceptional groups, Princeton Univ. Pr., Princeton, NJ, U.S.A. (2008).
V. Ahrens, M. Neubert and L. Vernazza, Structure of infrared singularities of gauge-theory amplitudes at three and four loops, JHEP 09 (2012) 138 [arXiv:1208.4847] [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, JHEP 10 (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].
S. Catani, D. Colferai and A. Torrini, Triple (and quadruple) soft-gluon radiation in QCD hard scattering, JHEP 01 (2020) 118 [arXiv:1908.01616] [INSPIRE].
A. Vladimirov, Structure of rapidity divergences in multi-parton scattering soft factors, JHEP 04 (2018) 045 [arXiv:1707.07606] [INSPIRE].
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Agarwal, N., Magnea, L., Pal, S. et al. Cwebs beyond three loops in multiparton amplitudes. J. High Energ. Phys. 2021, 188 (2021). https://doi.org/10.1007/JHEP03(2021)188
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DOI: https://doi.org/10.1007/JHEP03(2021)188