Abstract
We consider examples of two-loop two- and three-point Feynman integrals for which the calculation results have the property of maximal transcendentality.
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References
A. V. Kotikov and L. N. Lipatov, “DGLAP and BFKL equations in the N=4 supersymmetric gauge theory,” Nucl. Phys. B, 661, 19–61 (2003); Erratum, 685, 405–407 (2004); arXiv:hep-ph/0112346v1 (2001).
L. N. Lipatov, “Reggeization of the vector meson and the vacuum singularity in nonabelian gauge theories,” Sov. J. Nucl. Phys., 23, 338–345 (1976)
V. S. Fadin, E. A. Kuraev, and L. N. Lipatov, “On the Pomeranchuk singularity in asymptotically free theories,” Phys. Lett. B, 60, 50–52 (1975)
E. A. Kuraev, L. N. Lipatov, and V. S. Fadin, “Multiregge processes in the Yang–Mills theory,” JETP, 44, 443–451; “The Pomeranchuk singularity in nonabelian gauge theories,” JETP, 45, 199–204 (1977)
I. I. Balitsky and L. N. Lipatov, “The Pomeranchuk singularity in quantum chromodynamics,” Sov. J. Nucl. Phys., 28, 822–829 (1978)
Ya. Ya. Balitskii and L. N. Lipatov, “Calculation of the meson–meson interaction cross section in quantum chromodynamics,” JETP Lett., 30, 355–358 (1979).
V. S. Fadin and L. N. Lipatov, “BFKL pomeron in the next-to-leading approximation,” Phys. Lett. B, 429, 127–134 (1998)
M. Ciafaloni and G. Camici, “Energy scale(s) and next-to-leading BFKL equation,” Phys. Lett. B, 430, 349–354 (1998).
L. Brink, J. H. Schwarz, and J. Scherk, “Supersymmetric Yang–Mills theories,” Nucl. Phys. B, 121, 77–92 (1977)
F. Gliozzi, J. Scherk, and D. I. Olive, “Supersymmetry, supergravity theories, and the dual spinor model,” Nucl. Phys. B, 122, 253–290 (1977).
A. V. Kotikov, L. N. Lipatov, and V. N. Velizhanin, “Anomalous dimensions of Wilson operators in N=4 SYM theory,” Phys. Lett. B, 557, 114–120 (2003).
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, 521–529 (2004).
L. Bianchi, V. Forini, and A. V. Kotikov, “On DIS Wilson coefficients in N=4 super Yang–Mills theory,” Phys. Lett. B, 725, 394–401 (2013).
S. Moch, J. A. M. Vermaseren, and A. Vogt, “The three-loop splitting functions in QCD: The non-singlet case,” Nucl. Phys. B, 688, 101–134 (2004)
J. A. M. Vermaseren, A. Vogt, and S. Moch, “The third-order QCD corrections to deep-inelastic scattering by photon exchange,” Nucl. Phys. B, 724, 3–182 (2005).
D. J. Broadhurst, “The master two loop diagram with masses,” Z. Phys. C, 47, 115–124 (1990).
J. Fleischer, A. V. Kotikov, and O. L. Veretin, “Analytic two-loop results for self-energy- and vertex-type diagrams with one non-zero mass,” Nucl. Phys. B, 547, 343–374 (1999); “Applications of the large mass expansion,” Acta Phys. Polon. B, 29, 2611–2625 (1998).
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, 193–231 (2012)
L. J. Dixon, “Scattering amplitudes: The most perfect microscopic structures in the universe,” J. Phys. A, 44, 454001 (2011)
L. J. Dixon, J. M. Drummond, and J. M. Henn, “Analytic result for the two-loop six-point NMHV amplitude in N=4 super Yang–Mills theory,” JHEP, 01, 024 (2012)
T. Gehrmann, J. M. Henn, and T. Huber, “The three-loop form factor in N=4 super Yang–Mills,” JHEP, 03, 101 (2012)
A. Brandhuber, G. Travaglini, and G. Yang, “Analytic two-loop form factors in N=4 SYM,” JHEP, 1205, 082 (2012)
J. M. Henn, S. Moch, and S. G. Naculich, “Form factors and scattering amplitudes in N=4 SYM in dimensional and massive regularizations,” JHEP, 1112, 024 (2011).
O. Schlotterer and S. Stieberger, “Motivic multiple zeta values and superstring amplitudes,” J. Phys. A, 46, 475401 (2013)
J. Broedel, O. Schlotterer, and S. Stieberger, “Polylogarithms, multiple zeta values, and superstring amplitudes,” Fortsch. Phys., 61, 812–870 (2013)
S. Stieberger and T. R. Taylor, “Maximally helicity violating disk amplitudes, twistors, and transcendental integrals,” Phys. Lett. B, 716, 236–239 (2012).
B. Eden, “Three-loop universal structure constants in N=4 SYSY Yang–Mills theory,” arXiv:1207.3112v1 [hepth] (2012)
R. G. Ambrosio, B. Eden, T. Goddard, P. Heslop, and C. Taylor, “Local integrands for the five-point amplitude in planar N=4 SYM up to five loops,” JHEP, 1501, 116 (2015)
D. Chicherin, R. Doobary, B. Eden, P. Heslop, G. P. Korchemsky, and E. Sokatchev, “Bootstrapping correlation functions in N=4 SYM,” JHEP, 1603, 031 (2016);arXiv:1506.04983v1 [hep-th] (2015)
B. Eden and A. Sfondrini, “Three-point functions in N=4 SYM: The hexagon proposal at three loops,” JHEP, 1602, 165 (2016);arXiv:1510.01242v3 [hep-th] (2015).
A. V. Kotikov, “The property of maximal transcendentality in the N=4 supersymmetric Yang–Mills,” in: Subtleties in Quantum Field Theory: Lev Lipatov Festschrift (D. Diakonov, ed.), Petersburg Nucl. Phys. Inst., Gatchina (2010), pp. 150–174; arXiv:1005.5029v1 [hep-th] (2010); “The property of maximal transcendentality: Calculation of anomalous dimensions in the N=4 SYM and master integrals,” Phys. Part. Nucl., 44, 374–385 (2013).
D. I. Kazakov and A. V. Kotikov, “Uniqueness method: Multiloop calculations in QCD,” Theor. Math. Phys., 73, 1264–1274 (1987); “Total as correction to the deep-inelastic scattering cross-sections ratio R = sL/sT in QCD,” Nucl. Phys. B, 307, 721–762 (1988); Erratum, 345, 299–300 (1990).
A. V. Kotikov, “Method of calculating the moments of the structure functions of deep inelastic scattering in quantum chromodynamics,” Theor. Math. Phys., 78, 134–143 (1989).
A. V. Kotikov, “The Gegenbauer polynomial technique: The evaluation of a class of Feynman diagrams,” Phys. Lett. B, 375, 240–248 (1996).
D. I. Kazakov, “Analytical methods for multiloop calculations: Two lectures on the method of uniqueness,” Preprint JINR E2-84-410, Joint Inst. Nucl. Res., Dubna (1984).
K. G. Chetyrkin and F. V. Tkachov, “Integration by parts: The algorithm to calculate ß-functions in 4 loops,” Nucl. Phys. B, 192, 159–204 (1981)
F. V. Tkachov, “A theorem on analytical calculability of 4-loop renormalization group functions,” Phys. Lett. B, 100, 65–68 (1981)
A. N. Vasil’ev, Yu. M. Pis’mak, and Yu. R. Khonkonen, “1/n Expansion: Calculation of the exponents in the order 1/n2 for arbitrary number of dimensions,” Theor. Math. Phys., 47, 465–475 (1981).
D. I. Kazakov and A. V. Kotikov, “On the value of the as-correction to the Callan–Gross relation,” Phys. Lett. B, 291, 171–176 (1992).
A. V. Kotikov, “New method of massive Feynman diagrams calculation,” Modern Phys. Lett. A, 6, 677–692 (1991); “New method of massive Feynman diagrams calculation: Vertex type functions,” Internat. J. Modern Phys. A, 7, 1977–1991 (1992).
J. M. Henn and J. C. Plefka, Scattering Amplitudes in Gauge Theories (Lect. Notes Phys., Vol. 883), Springer, Berlin (2014).
A. V. Kotikov, “Differential equations method: New technique for massive Feynman diagrams calculation,” Phys. Lett. B, 254, 158–164 (1991); “Differential equations method: The calculation of vertex type Feynman diagrams,” Phys. Lett. B, 259, 314–322 (1991); “Differential equation method: The calculation of N point Feynman diagrams,” Phys. Lett. B, 267, 123–127 (1991); “New method of massive N point Feynman diagrams calculation,” Modern Phys. Lett. A, 6, 3133–3141 (1991)
E. Remiddi, “Differential equations for Feynman graph amplitudes,” Nuovo Cimento A, 110, 1435–1452 (1997).
B. A. Kniehl, A. V. Kotikov, A. I. Onishchenko, and O. L. Veretin, “Two-loop sunset diagrams with three massive lines,” Nucl. Phys. B, 738, 306–316 (2006)
B. A. Kniehl and A. V. Kotikov, “Calculating four-loop tadpoles with one non-zero mass,” Phys. Lett. B, 638, 531–537 (2006); “Counting master integrals: Integration-by-parts procedure with effective mass,” Phys. Lett. B, 712, 233–234 (2012).
J. Fleischer, M. Yu. Kalmykov, and A. V. Kotikov, “Two-loop self-energy master integrals on shell,” Phys. Lett. B, 462, 169–177 (1999).
J. Fleischer, A. V. Kotikov, and O. L. Veretin, “The differential equation method: Calculation of vertex-type diagrams with one non-zero mass,” Phys. Lett. B, 417, 163–172 (1998)
A. Kotikov, J. H. KÜhn, and O. Veretin, “Two-loop formfactors in theories with mass gap and Z-boson production,” Nucl. Phys. B, 788, 47–62 (2008).
A. V. Kotikov, “The property of maximal transcendentality: Calculation of master integrals,” Theor. Math. Phys., 176, 913–921 (2013); arXiv:1212.3732v1 [hep-ph] (2012).
B. A. Kniehl, A. V. Kotikov, Z. V. Merebashvili, and O. L. Veretin, “Strong-coupling constant with flavor thresholds at five loops in the modified minimal-subtraction scheme,” Phys. Rev. Lett., 97, 042001 (2006);“Heavy-quark pair production in polarized photon-photon collisions at next-to-leading order: Fully integrated total cross sections,” Phys. Rev. D, 79, 114032 (2009)
B. A. Kniehl, A. V. Kotikov, and O. L. Veretin, “Orthopositronium lifetime: Analytic results in O(a) and O(a3 log(a)),” Phys. Rev. Lett., 101, 193401 (2008);“Orthopositronium lifetime at O(a) and O(a3 log(a)) in closed form,” Phys. Rev. A, 80, 052501 (2009).
T. Gehrmann, J. M. Henn, and T. Huber, “The three-loop form factor in N=4 super Yang–Mills,” JHEP, 1203, 101 (2012).
J. M. Henn, “Multiloop integrals in dimensional regularization made simple,” Phys. Rev. Lett., 110, 251601 (2013);Erratum}, 111, 039902 (2013).
J. M. Henn, “Lectures on differential equations for Feynman integrals,” J. Phys. A, 48, 153001 (2015).
C. Duhr, “Mathematical aspects of scattering amplitudes,” arXiv:1411.7538v1 [hep-ph] (2014).
A. Devoto and D. W. Duke, “Table of integrals and formulae for Feynman diagram calculations,” Riv. Nuovo Cimento, 7, 1–39 (1984).
E. Remiddi and J. A. M. Vermaseren, “Harmonic polylogarithms,” Internat. J. Modern Phys. A, 15, 725–754 (2000).
A. I. Davydychev and M. Yu. Kalmykov, “Massive Feynman diagrams and inverse binomial sums,” Nucl. Phys. B, 699, 3–64 (2004).
V. N. Gribov and L. N. Lipatov, “Deep inelastic ep scattering in perturbation theory,” Sov. J. Nucl. Phys., 15, 438–450 (1972)
L. N. Lipatov, “The parton model and perturbation theory,” Sov. J. Nucl. Phys., 20, 94–102 (1975)
G. Altarelli and G. Parisi, “Asymptotic freedom in parton language,” Nucl. Phys. B, 126, 298–318 (1977)
Yu. L. Dokshitzer, “Calculation of structure functions of deep-inelastic scattering and e+e- annihilation by perturbation theory in quantum chromodynamics,” Sov. Phys. JETP, 46, 641 (1977).
A. V. Kotikov and L. N. Lipatov, “NLO corrections to the BFKL equation in QCD and in supersymmetric gauge theories,” Nucl. Phys. B, 582, 19–43 (2000).
L. N. Lipatov, “Next-to-leading corrections to the BFKL equation and the effective action for high energy processes in QCD,” Nucl. Phys. Proc. Suppl. A, 99, 175–179 (2001).
A. V. Kotikov, “Deep inelastic scattering: Q2 dependence of structure functions,” Phys. Part. Nucl., 38, 1–40 (2007)
A. V. Kotikov and V. N. Velizhanin, “Analytic continuation of the Mellin moments of deep inelastic structure functions,” arXiv:hep-ph/0501274v2 (2005).
A. V. Kotikov, L. N. Lipatov, A. Rej, M. Staudacher, and V. N. Velizhanin, “Dressing and wrapping,” J. Stat. Mech., 2007, P10003 (2007)
Z. Bajnok, R. A. Janik, and T. Lukowski, “Four loop twist two, BFKL, wrapping, and strings,” Nucl. Phys. B, 816, 376–398 (2009).
A. V. Kotikov, A. Rej, and S. Zieme, “Analytic three-loop solutions for image N=4 SYM twist operators,” Nucl. Phys. B, 813, 460–483 (2009)
M. Beccaria, A. V. Belitsky, A. V. Kotikov, and S. Zieme, “Analytic solution of the multiloop Baxter equation,” Nucl. Phys. B, 827, 565–606 (2010).
T. Lukowski, A. Rej, and V. N. Velizhanin, “Five-loop anomalous dimension of twist-two operators,” Nucl. Phys. B, 831, 105–132 (2010).
C. Marboe, V. Velizhanin, and D. Volin, “Six-loop anomalous dimension of twist-two operators in planar N=4 SYM theory,” JHEP, 1507, 084 (2015).
M. Staudacher, JHEP, 0505, 054 (2005)
N. Beisert and M. Staudacher, “Long-range psu(2, 24) image Bethe ansätze for gauge theory and strings,” Nucl. Phys. B, 727, 1–62 (2005).
M. Beccaria, “Three loop anomalous dimensions of twist-3 gauge operators in N=4 SYM,” JHEP, 0709, 023 (2007)
M. Beccaria, V. Forini, T. Lukowski, and S. Zieme, “Twist-three at five loops, Bethe ansatz and wrapping,” JHEP, 0903, 129 (2009)
V. N. Velizhanin, “Six-loop anomalous dimension of twist-three operators in N=4 SYM,” JHEP, 1011, 129 (2010).
W. Siegel, “Supersymmetric dimensional regularization via dimensional reduction,” Phys. Lett. B, 84, 193–196 (1979).
J. M. Maldacena, “The large-N limit of superconformal field theories and supergravity,” Internat. J. Theoret. Phys., 38, 1113–1133 (1999); “The large N limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys., 2, 231–252 (1998)
S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, “Gauge theory correlators from non-critical string theory,” Phys. Lett. B, 428, 105–114 (1998)
E. Witten, “Anti de Sitter space and holography,” Adv. Theor. Math. Phys., 2, 253–291 (1998).
A. V. Kotikov and L. N. Lipatov, “On the highest transcendentality in N=4 SUSY,” Nucl. Phys. B, 769, 217–255 (2007)
M. K. Benna, S. Benvenuti, I. R. Klebanov, and A. Scardicchio, “A test of the AdS/CFT correspondence using high-spin operators,” Phys. Rev. Lett., 98, 131603 (2007).
B. Basso, G. P. Korchemsky, and J. Kotanski, “Cusp anomalous dimension in maximally supersymmetric Yang–Mills theory at strong coupling,” Phys. Rev. Lett., 100, 091601 (2008);“Embedding nonlinear O(6) sigma model into N=4 super-Yang–Mills theory,” Nucl. Phys. B, 807, 397–423 (2009).
N. Beisert, B. Eden, and M. Staudacher, “Transcendentality and crossing,” J. Stat. Mech., 2007, P01021 (2007); arXiv:hep-th/0610251v2 (2006).
R. C. Brower, J. Polchinski, M. J. Strassler, and C. I. Tan, “The Pomeron and gauge/string duality,” JHEP, 0712, 005 (2007).
M. S. Costa, V. Goncalves, and J. Penedones, “Conformal Regge theory,” JHEP, 1212, 091 (2012)
A. V. Kotikov and L. N. Lipatov, “Pomeron in the N=4 supersymmetric gauge model at strong couplings,” Nucl. Phys. B, 874, 889–904 (2013)
N. Gromov, F. Levkovich-Maslyuk, G. Sizov, and S. Valatka, “Quantum spectral curve at work: From small spin to strong coupling in N=4 SYM,” JHEP, 1407, 156 (2014).
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This research was supported by the Russian Foundation for Basic Research (Grant No. 16-02-00790_a).
Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 190, No. 3, pp. 455–467, March, 2017.
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Kotikov, A.V. The property of maximal transcendentality: Calculation of Feynman integrals. Theor Math Phys 190, 391–401 (2017). https://doi.org/10.1134/S0040577917030084
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DOI: https://doi.org/10.1134/S0040577917030084