Abstract
We compute the fermionic contribution to the photon-quark form factor to four-loop order in QCD in the planar limit in analytic form. From the divergent part of the latter the cusp and collinear anomalous dimensions are extracted. Results are also presented for the finite contribution. We briefly describe our method to compute all planar master integrals at four-loop order.
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References
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].
L. Magnea and G.F. Sterman, Analytic continuation of the Sudakov form-factor in QCD, Phys. Rev. D 42 (1990) 4222 [INSPIRE].
I.A. Korchemskaya and G.P. Korchemsky, On lightlike Wilson loops, Phys. Lett. B 287 (1992) 169 [INSPIRE].
G.F. Sterman and M.E. Tejeda-Yeomans, Multiloop amplitudes and resummation, Phys. Lett. B 552 (2003) 48 [hep-ph/0210130] [INSPIRE].
G. Kramer and B. Lampe, Two Jet Cross-Section in e + e − Annihilation, Z. Phys. C 34 (1987) 497 [Erratum ibid. C 42 (1989) 504] [INSPIRE].
T. Matsuura and W.L. van Neerven, Second Order Logarithmic Corrections to the Drell-Yan Cross-section, Z. Phys. C 38 (1988) 623 [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 [INSPIRE].
T. Gehrmann, T. Huber and D. Maître, Two-loop quark and gluon form-factors in dimensional regularisation, Phys. Lett. B 622 (2005) 295 [hep-ph/0507061] [INSPIRE].
P.A. Baikov, K.G. Chetyrkin, A.V. Smirnov, V.A. Smirnov and M. Steinhauser, Quark and gluon form factors to three loops, Phys. Rev. Lett. 102 (2009) 212002 [arXiv:0902.3519] [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, JHEP 06 (2010) 094 [arXiv:1004.3653] [INSPIRE].
R.N. Lee and V.A. Smirnov, Analytic ϵ-expansions of Master Integrals Corresponding to Massless Three-Loop Form Factors and Three-Loop g-2 up to Four-Loop Transcendentality Weight, JHEP 02 (2011) 102 [arXiv:1010.1334] [INSPIRE].
T. Gehrmann, E.W.N. Glover, T. Huber, N. Ikizlerli and C. Studerus, The quark and gluon form factors to three loops in QCD through to O(ϵ2), JHEP 11 (2010) 102 [arXiv:1010.4478] [INSPIRE].
K.G. Chetyrkin and F.V. Tkachov, Integration by Parts: The Algorithm to Calculate β-functions in 4 Loops, Nucl. Phys. B 192 (1981) 159 [INSPIRE].
A.V. Smirnov, Algorithm FIRE — Feynman Integral REduction, JHEP 10 (2008) 107 [arXiv:0807.3243] [INSPIRE].
A.V. Smirnov and V.A. Smirnov, FIRE4, LiteRed and accompanying tools to solve integration by parts relations, Comput. Phys. Commun. 184 (2013) 2820 [arXiv:1302.5885] [INSPIRE].
A.V. Smirnov, FIRE5: a C++ implementation of Feynman Integral REduction, Comput. Phys. Commun. 189 (2015) 182 [arXiv:1408.2372] [INSPIRE].
J.M. Henn, A.V. Smirnov and V.A. Smirnov, Evaluating single-scale and/or non-planar diagrams by differential equations, JHEP 03 (2014) 088 [arXiv:1312.2588] [INSPIRE].
A.V. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett. B 254 (1991) 158 [INSPIRE].
Z. Bern, L.J. Dixon and D.A. Kosower, Dimensionally regulated pentagon integrals, Nucl. Phys. B 412 (1994) 751 [hep-ph/9306240] [INSPIRE].
E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cim. A 110 (1997) 1435 [hep-th/9711188] [INSPIRE].
T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].
J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].
F. Cachazo, Sharpening The Leading Singularity, arXiv:0803.1988 [INSPIRE].
T. Gehrmann, J.M. Henn and T. Huber, The three-loop form factor in \( \mathcal{N}=4 \) super Yang-Mills, JHEP 03 (2012) 101 [arXiv:1112.4524] [INSPIRE].
R.H. Boels, B.A. Kniehl, O.V. Tarasov and G. Yang, Color-kinematic Duality for Form Factors, JHEP 02 (2013) 063 [arXiv:1211.7028] [INSPIRE].
R. Boels, B.A. Kniehl and G. Yang, Master integrals for the four-loop Sudakov form factor, Nucl. Phys. B 902 (2016) 387 [arXiv:1508.03717] [INSPIRE].
A. von Manteuffel, E. Panzer and R.M. Schabinger, On the Computation of Form Factors in Massless QCD with Finite Master Integrals, arXiv:1510.06758 [INSPIRE].
P. Nogueira, Automatic Feynman graph generation, J. Comput. Phys. 105 (1993) 279 [INSPIRE].
J.A.M. Vermaseren, New features of FORM, math-ph/0010025 [INSPIRE].
J. Kuipers, T. Ueda, J.A.M. Vermaseren and J. Vollinga, FORM version 4.0, Comput. Phys. Commun. 184 (2013) 1453 [arXiv:1203.6543] [INSPIRE].
R. Harlander, T. Seidensticker and M. Steinhauser, Complete corrections of \( \mathcal{O}\left(\alpha {\alpha}_s\right) \) to the decay of the Z boson into bottom quarks, Phys. Lett. B 426 (1998) 125 [hep-ph/9712228] [INSPIRE].
T. Seidensticker, Automatic application of successive asymptotic expansions of Feynman diagrams, hep-ph/9905298 [INSPIRE].
R.N. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685 [INSPIRE].
R.N. Lee, LiteRed 1.4: a powerful tool for reduction of multiloop integrals, J. Phys. Conf. Ser. 523 (2014) 012059 [arXiv:1310.1145] [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].
A. Vogt, Next-to-next-to-leading logarithmic threshold resummation for deep inelastic scattering and the Drell-Yan process, Phys. Lett. B 497 (2001) 228 [hep-ph/0010146] [INSPIRE].
C.F. Berger, Higher orders in A(α s )/[1 − x]+ of nonsinglet partonic splitting functions, Phys. Rev. D 66 (2002) 116002 [hep-ph/0209107] [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].
S. Moch, J.A.M. Vermaseren and A. Vogt, Three-loop results for quark and gluon form-factors, Phys. Lett. B 625 (2005) 245 [hep-ph/0508055] [INSPIRE].
J.A. Gracey, Anomalous dimension of nonsinglet Wilson operators at O(1/N f ) in deep inelastic scattering, Phys. Lett. B 322 (1994) 141 [hep-ph/9401214] [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].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo and J. Trnka, Local Integrals for Planar Scattering Amplitudes, JHEP 06 (2012) 125 [arXiv:1012.6032] [INSPIRE].
A.E. Lipstein and L. Mason, From the holomorphic Wilson loop to ‘d log’ loop-integrands for super-Yang-Mills amplitudes, JHEP 05 (2013) 106 [arXiv:1212.6228] [INSPIRE].
N. Arkani-Hamed et al., Scattering amplitudes and the positive Grassmannian, Cambridge University Press (2016), arXiv:1212.5605.
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, Logarithmic Singularities and Maximally Supersymmetric Amplitudes, JHEP 06 (2015) 202 [arXiv:1412.8584] [INSPIRE].
F. Caola, J.M. Henn, K. Melnikov and V.A. Smirnov, Non-planar master integrals for the production of two off-shell vector bosons in collisions of massless partons, JHEP 09 (2014) 043 [arXiv:1404.5590] [INSPIRE].
O.V. Tarasov, Connection between Feynman integrals having different values of the space-time dimension, Phys. Rev. D 54 (1996) 6479 [hep-th/9606018] [INSPIRE].
R.N. Lee, Space-time dimensionality D as complex variable: Calculating loop integrals using dimensional recurrence relation and analytical properties with respect to D, Nucl. Phys. B 830 (2010) 474 [arXiv:0911.0252] [INSPIRE].
J.M. Henn, Lectures on differential equations for Feynman integrals, J. Phys. A 48 (2015) 153001 [arXiv:1412.2296] [INSPIRE].
D. Maître, HPL, a mathematica implementation of the harmonic polylogarithms, Comput. Phys. Commun. 174 (2006) 222 [hep-ph/0507152] [INSPIRE].
V.A. Smirnov, Analytic tools for Feynman integrals, Springer Tracts Mod. Phys. 250 (2012) 1.
P.A. Baikov and K.G. Chetyrkin, Four Loop Massless Propagators: An Algebraic Evaluation of All Master Integrals, Nucl. Phys. B 837 (2010) 186 [arXiv:1004.1153] [INSPIRE].
R.N. Lee, A.V. Smirnov and V.A. Smirnov, Master Integrals for Four-Loop Massless Propagators up to Transcendentality Weight Twelve, Nucl. Phys. B 856 (2012) 95 [arXiv:1108.0732] [INSPIRE].
E. Panzer, Feynman integrals and hyperlogarithms, Ph.D. Thesis, Humboldt U., Berlin, Inst. Math. (2015), arXiv:1506.07243 [INSPIRE].
E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].
W. Wasow, Asymptotic expansions for ordinary differential equations, Pure and Applied Mathematics, Vol. XIV, Interscience Publishers John Wiley & Sons, Inc., New York, London, Sydney (1965).
M. Beneke and V.A. Smirnov, Asymptotic expansion of Feynman integrals near threshold, Nucl. Phys. B 522 (1998) 321 [hep-ph/9711391] [INSPIRE].
V.A. Smirnov and E.R. Rakhmetov, The strategy of regions for asymptotic expansion of two loop vertex Feynman diagrams, Theor. Math. Phys. 120 (1999) 870 [hep-ph/9812529] [INSPIRE].
V.A. Smirnov, Applied asymptotic expansions in momenta and masses, Springer Tracts Mod. Phys. 177 (2002) 1.
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Henn, J.M., Smirnov, A.V., Smirnov, V.A. et al. A planar four-loop form factor and cusp anomalous dimension in QCD. J. High Energ. Phys. 2016, 66 (2016). https://doi.org/10.1007/JHEP05(2016)066
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DOI: https://doi.org/10.1007/JHEP05(2016)066