×

The two-loop five-point amplitude in \( \mathcal{N}=8\) supergravity. (English) Zbl 1414.83094

Summary: We compute the symbol of the two-loop five-point amplitude in \( \mathcal{N}=8\) supergravity. We write an ansatz for the amplitude whose rational prefactors are based on not only 4-dimensional leading singularities, but also \(d\)-dimensional ones, as the former are insufficient. Our novel \(d\)-dimensional unitarity-based approach to the systematic construction of an amplitude’s rational structures is likely to have broader applications, for example to analogous QCD calculations. We fix parameters in the ansatz by performing numerical integration-by-parts reduction of the known integrand. We find that the two-loop five-point \( \mathcal{N}=8\) supergravity amplitude is uniformly transcendental. We then verify the soft and collinear limits of the amplitude. There is considerable similarity with the corresponding amplitude for \( \mathcal{N}=4\) super-Yang-Mills theory: all the rational prefactors are double copies of the Yang-Mills ones and the transcendental functions overlap to a large degree. As a byproduct, we find new relations between color-ordered loop amplitudes in \( \mathcal{N}=4 \) super-Yang-Mills theory.

MSC:

83E50 Supergravity
81T60 Supersymmetric field theories in quantum mechanics
81U20 \(S\)-matrix theory, etc. in quantum theory
70S15 Yang-Mills and other gauge theories in mechanics of particles and systems

Software:

Kira; Azurite; HyperInt
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Drummond, JM; Henn, J.; Smirnov, VA; Sokatchev, E., Magic identities for conformal four-point integrals, JHEP, 01, 064, (2007)
[2] Bern, Z.; Czakon, M.; Dixon, LJ; Kosower, DA; Smirnov, VA, The Four-Loop Planar Amplitude and Cusp Anomalous Dimension in Maximally Supersymmetric Yang-Mills Theory, Phys. Rev., D 75, (2007)
[3] Alday, LF; Maldacena, JM, Gluon scattering amplitudes at strong coupling, JHEP, 06, 064, (2007)
[4] Drummond, JM; Henn, J.; Korchemsky, GP; Sokatchev, E., Dual superconformal symmetry of scattering amplitudes in N = 4 super-Yang-Mills theory, Nucl. Phys., B 828, 317, (2010) · Zbl 1203.81112
[5] Goncharov, AB; Spradlin, M.; Vergu, C.; Volovich, A., Classical Polylogarithms for Amplitudes and Wilson Loops, Phys. Rev. Lett., 105, 151605, (2010)
[6] Duhr, C.; Gangl, H.; Rhodes, JR, From polygons and symbols to polylogarithmic functions, JHEP, 10, 075, (2012) · Zbl 1397.81355
[7] Duhr, C., Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes, JHEP, 08, 043, (2012) · Zbl 1397.16028
[8] 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]. · Zbl 1049.81644
[9] Arkani-Hamed, N.; Bourjaily, JL; Cachazo, F.; Caron-Huot, S.; Trnka, J., The All-Loop Integrand For Scattering Amplitudes in Planar N = 4 SYM, JHEP, 01, 041, (2011) · Zbl 1214.81141
[10] Arkani-Hamed, N.; Bourjaily, JL; Cachazo, F.; Trnka, J., Local Integrals for Planar Scattering Amplitudes, JHEP, 06, 125, (2012) · Zbl 1397.81428
[11] Bourjaily, JL; Herrmann, E.; Trnka, J., Prescriptive Unitarity, JHEP, 06, 059, (2017) · Zbl 1380.81388
[12] Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, Fusing gauge theory tree amplitudes into loop amplitudes, Nucl. Phys.B 435 (1995) 59 [hep-ph/9409265] [INSPIRE]. · Zbl 1049.81644
[13] Britto, R.; Cachazo, F.; Feng, B., Generalized unitarity and one-loop amplitudes in N = 4 super-Yang-Mills, Nucl. Phys., B 725, 275, (2005) · Zbl 1178.81202
[14] Bern, Z.; Carrasco, JJM; Johansson, H.; Kosower, DA, Maximally supersymmetric planar Yang-Mills amplitudes at five loops, Phys. Rev., D 76, 125020, (2007)
[15] A. Postnikov, Total positivity, Grassmannians, and networks, math/0609764.
[16] N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A.B. Goncharov, A. Postnikov and J. Trnka, Grassmannian Geometry of Scattering Amplitudes, Cambridge University Press (2016) https://doi.org/10.1017/CBO9781316091548 [arXiv:1212.5605] [INSPIRE]. · Zbl 1365.81004
[17] Arkani-Hamed, N.; Trnka, J., The Amplituhedron, JHEP, 10, 030, (2014) · Zbl 1388.81166
[18] Bern, Z.; Carrasco, JJM; Johansson, H., New Relations for Gauge-Theory Amplitudes, Phys. Rev., D 78, (2008)
[19] Bjerrum-Bohr, NEJ; Damgaard, PH; Vanhove, P., Minimal Basis for Gauge Theory Amplitudes, Phys. Rev. Lett., 103, 161602, (2009)
[20] S. Stieberger, Open & Closed vs. Pure Open String Disk Amplitudes, arXiv:0907.2211 [INSPIRE]. · Zbl 1284.81245
[21] Bern, Z.; Dennen, T.; Huang, Y-t; Kiermaier, M., Gravity as the Square of Gauge Theory, Phys. Rev., D 82, (2010)
[22] Bern, Z.; Carrasco, JJM; Johansson, H., Perturbative Quantum Gravity as a Double Copy of Gauge Theory, Phys. Rev. Lett., 105, (2010)
[23] Bern, Z.; Carrasco, JJM; Dixon, LJ; Johansson, H.; Roiban, R., Simplifying Multiloop Integrands and Ultraviolet Divergences of Gauge Theory and Gravity Amplitudes, Phys. Rev., D 85, 105014, (2012)
[24] Carrasco, JJ; Johansson, H., Five-Point Amplitudes in N = 4 Super-Yang-Mills Theory and N = 8 Supergravity, Phys. Rev., D 85, (2012)
[25] Mafra, CR; Schlotterer, O., Two-loop five-point amplitudes of super Yang-Mills and supergravity in pure spinor superspace, JHEP, 10, 124, (2015) · Zbl 1388.83860
[26] Bern, Z.; Carrasco, JJ; Chen, W-M; Johansson, H.; Roiban, R., Gravity Amplitudes as Generalized Double Copies of Gauge-Theory Amplitudes, Phys. Rev. Lett., 118, 181602, (2017)
[27] Bern, Z.; Carrasco, JJM; Chen, W-M; Johansson, H.; Roiban, R.; Zeng, M., Five-loop four-point integrand of N = 8 supergravity as a generalized double copy, Phys. Rev., D 96, 126012, (2017)
[28] Bern, Z.; etal., Ultraviolet Properties of \( \mathcal{N} \) = 8 Supergravity at Five Loops, Phys. Rev., D 98, (2018)
[29] Z. Bern, J.J. Carrasco, M. Chiodaroli, H. Johansson and R. Roiban, The Duality Between Color and Kinematics and its Applications, to appear (2019).
[30] Dixon, LJ; Drummond, JM; Henn, JM, Bootstrapping the three-loop hexagon, JHEP, 11, 023, (2011) · Zbl 1306.81092
[31] Caron-Huot, S.; Dixon, LJ; McLeod, A.; Hippel, M., Bootstrapping a Five-Loop Amplitude Using Steinmann Relations, Phys. Rev. Lett., 117, 241601, (2016)
[32] Dixon, LJ; Drummond, J.; Harrington, T.; McLeod, AJ; Papathanasiou, G.; Spradlin, M., Heptagons from the Steinmann Cluster Bootstrap, JHEP, 02, 137, (2017) · Zbl 1377.81197
[33] J. Drummond, J. Foster, Ö. Gürdoğan and G. Papathanasiou, Cluster adjacency and the four-loop NMHV heptagon, arXiv:1812.04640 [INSPIRE].
[34] Bern, Z.; Dixon, LJ; Perelstein, M.; Rozowsky, JS, Multileg one loop gravity amplitudes from gauge theory, Nucl. Phys., B 546, 423, (1999) · Zbl 0953.83006
[35] Bern, Z.; Dixon, LJ; Dunbar, DC; Perelstein, M.; Rozowsky, JS, On the relationship between Yang-Mills theory and gravity and its implication for ultraviolet divergences, Nucl. Phys., B 530, 401, (1998)
[36] Naculich, SG; Nastase, H.; Schnitzer, HJ, Two-loop graviton scattering relation and IR behavior in N = 8 supergravity, Nucl. Phys., B 805, 40, (2008) · Zbl 1190.83096
[37] Brandhuber, A.; Heslop, P.; Nasti, A.; Spence, B.; Travaglini, G., Four-point Amplitudes in N =8 Supergravity and Wilson Loops, Nucl. Phys., B 807, 290, (2009) · Zbl 1192.83064
[38] Boucher-Veronneau, C.; Dixon, LJ, \( \mathcal{N} \) ≥ 4 Supergravity Amplitudes from Gauge Theory at Two Loops, JHEP, 12, 046, (2011) · Zbl 1306.81078
[39] Bern, Z.; Cheung, C.; Chi, H-H; Davies, S.; Dixon, L.; Nohle, J., Evanescent Effects Can Alter Ultraviolet Divergences in Quantum Gravity without Physical Consequences, Phys. Rev. Lett., 115, 211301, (2015)
[40] Bern, Z.; Chi, H-H; Dixon, L.; Edison, A., Two-Loop Renormalization of Quantum Gravity Simplified, Phys. Rev., D 95, (2017)
[41] Dunbar, DC; Jehu, GR; Perkins, WB, Two-Loop Gravity amplitudes from four dimensional Unitarity, Phys. Rev., D 95, (2017)
[42] Tkachov, FV, A Theorem on Analytical Calculability of Four Loop Renormalization Group Functions, Phys. Lett., 100B, 65, (1981)
[43] Chetyrkin, KG; Tkachov, FV, Integration by Parts: The Algorithm to Calculate β-functions in 4 Loops, Nucl. Phys., B 192, 159, (1981)
[44] Gluza, J.; Kajda, K.; Kosower, DA, Towards a Basis for Planar Two-Loop Integrals, Phys. Rev., D 83, (2011)
[45] Ita, H., Two-loop Integrand Decomposition into Master Integrals and Surface Terms, Phys. Rev., D 94, 116015, (2016)
[46] Larsen, KJ; Zhang, Y., Integration-by-parts reductions from unitarity cuts and algebraic geometry, Phys. Rev., D 93, (2016)
[47] Böhm, J.; Georgoudis, A.; Larsen, KJ; Schönemann, H.; Zhang, Y., Complete integration-by-parts reductions of the non-planar hexagon-box via module intersections, JHEP, 09, 024, (2018) · Zbl 1398.81264
[48] Abreu, S.; Febres Cordero, F.; Ita, H.; Page, B.; Zeng, M., Planar Two-Loop Five-Gluon Amplitudes from Numerical Unitarity, Phys. Rev., D 97, 116014, (2018)
[49] Kosower, DA, Direct Solution of Integration-by-Parts Systems, Phys. Rev., D 98, (2018)
[50] Manteuffel, A.; Schabinger, RM, A novel approach to integration by parts reduction, Phys. Lett., B 744, 101, (2015) · Zbl 1330.81151
[51] Peraro, T., Scattering amplitudes over finite fields and multivariate functional reconstruction, JHEP, 12, 030, (2016) · Zbl 1390.81631
[52] Maierhöfer, P.; Usovitsch, J.; Uwer, P., Kira — A Feynman integral reduction program, Comput. Phys. Commun., 230, 99, (2018)
[53] A.V. Smirnov and F.S. Chuharev, FIRE6: Feynman Integral REduction with Modular Arithmetic, arXiv:1901.07808 [INSPIRE].
[54] Kotikov, AV, Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett., B 254, 158, (1991)
[55] Z. Bern, L.J. Dixon and D.A. Kosower, Dimensionally regulated one loop integrals, Phys. Lett.B 302 (1993) 299 [Erratum ibid.B 318 (1993) 649] [hep-ph/9212308] [INSPIRE]. · Zbl 1007.81512
[56] T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys.B 580 (2000) 485 [hep-ph/9912329] [INSPIRE]. · Zbl 1071.81089
[57] Henn, JM, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett., 110, 251601, (2013)
[58] 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].
[59] T. Gehrmann, J.M. Henn and N.A. Lo Presti, Analytic form of the two-loop planar five-gluon all-plus-helicity amplitude in QCD, Phys. Rev. Lett.116 (2016) 062001 [Erratum ibid.116 (2016) 189903] [arXiv:1511.05409] [INSPIRE]. · Zbl 1356.81169
[60] Papadopoulos, CG; Tommasini, D.; Wever, C., The Pentabox Master Integrals with the Simplified Differential Equations approach, JHEP, 04, 078, (2016)
[61] Gehrmann, T.; Henn, JM; Lo Presti, NA, Pentagon functions for massless planar scattering amplitudes, JHEP, 10, 103, (2018) · Zbl 1402.81256
[62] T. Gehrmann and E. Remiddi, Two loop master integrals for γ* → 3 jets: The Nonplanar topologies, Nucl. Phys.B 601 (2001) 287 [hep-ph/0101124] [INSPIRE].
[63] Chicherin, D.; Henn, J.; Mitev, V., Bootstrapping pentagon functions, JHEP, 05, 164, (2018)
[64] Abreu, S.; Page, B.; Zeng, M., Differential equations from unitarity cuts: nonplanar hexa-box integrals, JHEP, 01, 006, (2019) · Zbl 1409.81157
[65] S. Abreu, L.J. Dixon, E. Herrmann, B. Page and M. Zeng, The two-loop five-point amplitude in\( \mathcal{N} \) = 4 super-Yang-Mills theory, arXiv:1812.08941 [INSPIRE].
[66] Chicherin, D.; Gehrmann, T.; Henn, JM; Lo Presti, NA; Mitev, V.; Wasser, P., Analytic result for the nonplanar hexa-box integrals, JHEP, 03, 042, (2019)
[67] D. Chicherin, T. Gehrmann, J.M. Henn, P. Wasser, Y. Zhang and S. Zoia, All master integrals for three-jet production at NNLO, arXiv:1812.11160 [INSPIRE].
[68] Badger, S.; Brønnum-Hansen, C.; Hartanto, HB; Peraro, T., First look at two-loop five-gluon scattering in QCD, Phys. Rev. Lett., 120, (2018) · Zbl 1409.81155
[69] S. Badger et al., Applications of integrand reduction to two-loop five-point scattering amplitudes in QCD, PoS(LL2018)006 (2018) [arXiv:1807.09709] [INSPIRE].
[70] Abreu, S.; Febres Cordero, F.; Ita, H.; Page, B.; Sotnikov, V., Planar Two-Loop Five-Parton Amplitudes from Numerical Unitarity, JHEP, 11, 116, (2018)
[71] Badger, S.; Brønnum-Hansen, C.; Hartanto, HB; Peraro, T., Analytic helicity amplitudes for two-loop five-gluon scattering: the single-minus case, JHEP, 01, 186, (2019) · Zbl 1409.81155
[72] Abreu, S.; Dormans, J.; Febres Cordero, F.; Ita, H.; Page, B., Analytic Form of the Planar Two-Loop Five-Gluon Scattering Amplitudes in QCD, Phys. Rev. Lett., 122, (2019)
[73] D. Chicherin, J.M. Henn, P. Wasser, T. Gehrmann, Y. Zhang and S. Zoia, Analytic result for a two-loop five-particle amplitude, arXiv:1812.11057 [INSPIRE].
[74] F. Cachazo, Sharpening The Leading Singularity, arXiv:0803.1988 [INSPIRE].
[75] Arkani-Hamed, N.; Bourjaily, JL; Cachazo, F.; Postnikov, A.; Trnka, J., On-Shell Structures of MHV Amplitudes Beyond the Planar Limit, JHEP, 06, 179, (2015) · Zbl 1388.81272
[76] Herrmann, E.; Trnka, J., Gravity On-shell Diagrams, JHEP, 11, 136, (2016) · Zbl 1390.83402
[77] Heslop, P.; Lipstein, AE, On-shell diagrams for \( \mathcal{N} \) = 8 supergravity amplitudes, JHEP, 06, 069, (2016) · Zbl 1388.83822
[78] Herrmann, E.; Trnka, J., UV cancellations in gravity loop integrands, JHEP, 02, 084, (2019) · Zbl 1411.83135
[79] J.L. Bourjaily, E. Herrmann and J. Trnka, Amplitudes at Infinity, arXiv:1812.11185 [INSPIRE].
[80] Parke, SJ; Taylor, TR, An Amplitude for n Gluon Scattering, Phys. Rev. Lett., 56, 2459, (1986)
[81] Bern, Z.; Dixon, LJ; Smirnov, VA, Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond, Phys. Rev., D 72, (2005)
[82] Kotikov, AV; Lipatov, LN, On the highest transcendentality in N = 4 SUSY, Nucl. Phys., B 769, 217, (2007) · Zbl 1117.81103
[83] D. Chicherin, T. Gehrmann, J.M. Henn, P. Wasser, Y. Zhang and S. Zoia, The two-loop five-particle amplitude in\( \mathcal{N} \) = 8 supergravity, arXiv:1901.05932 [INSPIRE].
[84] Berends, FA; Giele, WT; Kuijf, H., On relations between multi-gluon and multigraviton scattering, Phys. Lett., B 211, 91, (1988)
[85] Bern, Z.; Herrmann, E.; Litsey, S.; Stankowicz, J.; Trnka, J., Logarithmic Singularities and Maximally Supersymmetric Amplitudes, JHEP, 06, 202, (2015) · Zbl 1388.81136
[86] Arkani-Hamed, N.; Bourjaily, JL; Cachazo, F.; Trnka, J., Singularity Structure of Maximally Supersymmetric Scattering Amplitudes, Phys. Rev. Lett., 113, 261603, (2014)
[87] Bern, Z.; Davies, S.; Dennen, T., Enhanced ultraviolet cancellations in \( \mathcal{N} \) = 5 supergravity at four loops, Phys. Rev., D 90, 105011, (2014)
[88] Weinberg, S., Infrared photons and gravitons, Phys. Rev., 140, b516, (1965)
[89] Akhoury, R.; Saotome, R.; Sterman, G., Collinear and Soft Divergences in Perturbative Quantum Gravity, Phys. Rev., D 84, 104040, (2011)
[90] Hodges, A., Eliminating spurious poles from gauge-theoretic amplitudes, JHEP, 05, 135, (2013) · Zbl 1342.81291
[91] Badger, S.; Frellesvig, H.; Zhang, Y., A Two-Loop Five-Gluon Helicity Amplitude in QCD, JHEP, 12, 045, (2013)
[92] Drummond, J.; Duhr, C.; Eden, B.; Heslop, P.; Pennington, J.; Smirnov, VA, Leading singularities and off-shell conformal integrals, JHEP, 08, 133, (2013) · Zbl 1342.81574
[93] Abreu, S.; Britto, R.; Duhr, C.; Gardi, E., Cuts from residues: the one-loop case, JHEP, 06, 114, (2017) · Zbl 1380.81421
[94] P.A. Baikov, Explicit solutions of the three loop vacuum integral recurrence relations, Phys. Lett.B 385 (1996) 404 [hep-ph/9603267] [INSPIRE].
[95] P.A. Baikov, Explicit solutions of the multiloop integral recurrence relations and its application, Nucl. Instrum. Meth.A 389 (1997) 347 [hep-ph/9611449] [INSPIRE].
[96] Grozin, AG, Integration by parts: An Introduction, Int. J. Mod. Phys., A 26, 2807, (2011) · Zbl 1247.81138
[97] Frellesvig, H.; Papadopoulos, CG, Cuts of Feynman Integrals in Baikov representation, JHEP, 04, 083, (2017) · Zbl 1378.81039
[98] Kosower, DA; Larsen, KJ, Maximal Unitarity at Two Loops, Phys. Rev., D 85, (2012)
[99] Bern, Z.; Herrmann, E.; Litsey, S.; Stankowicz, J.; Trnka, J., Evidence for a Nonplanar Amplituhedron, JHEP, 06, 098, (2016) · Zbl 1388.81908
[100] Bern, Z.; Enciso, M.; Shen, C-H; Zeng, M., Dual Conformal Structure Beyond the Planar Limit, Phys. Rev. Lett., 121, 121603, (2018)
[101] Bern, Z.; Enciso, M.; Ita, H.; Zeng, M., Dual Conformal Symmetry, Integration-by-Parts Reduction, Differential Equations and the Nonplanar Sector, Phys. Rev., D 96, (2017)
[102] Chicherin, D.; Henn, JM; Sokatchev, E., Implications of nonplanar dual conformal symmetry, JHEP, 09, 012, (2018) · Zbl 1398.81098
[103] Z. Bern, L.J. Dixon and D.A. Kosower, Dimensionally regulated pentagon integrals, Nucl. Phys.B 412 (1994) 751 [hep-ph/9306240] [INSPIRE]. · Zbl 1007.81512
[104] Tarasov, OV, Connection between Feynman integrals having different values of the space-time dimension, Phys. Rev., D 54, 6479, (1996) · Zbl 0925.81121
[105] Lee, RN, 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, 474, (2010) · Zbl 1203.83051
[106] Georgoudis, A.; Larsen, KJ; Zhang, Y., Azurite: An algebraic geometry based package for finding bases of loop integrals, Comput. Phys. Commun., 221, 203, (2017)
[107] Gaiotto, D.; Maldacena, J.; Sever, A.; Vieira, P., Pulling the straps of polygons, JHEP, 12, 011, (2011) · Zbl 1306.81153
[108] Schabinger, RM, A New Algorithm For The Generation Of Unitarity-Compatible Integration By Parts Relations, JHEP, 01, 077, (2012) · Zbl 1306.81359
[109] Y. Zhang, Lecture Notes on Multi-loop Integral Reduction and Applied Algebraic Geometry, 2016, arXiv:1612.02249 [INSPIRE].
[110] Bern, Z.; Carrasco, JJ; Dixon, LJ; Johansson, H.; Kosower, DA; Roiban, R., Three-Loop Superfiniteness of N = 8 Supergravity, Phys. Rev. Lett., 98, 161303, (2007)
[111] Bern, Z.; Carrasco, JJM; Dixon, LJ; Johansson, H.; Roiban, R., Manifest Ultraviolet Behavior for the Three-Loop Four-Point Amplitude of N = 8 Supergravity, Phys. Rev., D 78, 105019, (2008)
[112] Bern, Z.; Carrasco, JJ; Dixon, LJ; Johansson, H.; Roiban, R., The Ultraviolet Behavior of N = 8 Supergravity at Four Loops, Phys. Rev. Lett., 103, (2009)
[113] Green, MB; Russo, JG; Vanhove, P., String theory dualities and supergravity divergences, JHEP, 06, 075, (2010) · Zbl 1288.81107
[114] Bossard, G.; Howe, PS; Stelle, KS, On duality symmetries of supergravity invariants, JHEP, 01, 020, (2011) · Zbl 1214.83041
[115] Beisert, N.; Elvang, H.; Freedman, DZ; Kiermaier, M.; Morales, A.; Stieberger, S., E_{7(7)} constraints on counterterms in N = 8 supergravity, Phys. Lett. B, 694, 265, (2011)
[116] P. Vanhove, The Critical ultraviolet behaviour of N = 8 supergravity amplitudes, arXiv:1004.1392 [INSPIRE].
[117] Björnsson, J.; Green, MB, 5 loops in 24/5 dimensions, JHEP, 08, 132, (2010) · Zbl 1290.81144
[118] Björnsson, J., Multi-loop amplitudes in maximally supersymmetric pure spinor field theory, JHEP, 01, 002, (2011) · Zbl 1214.81134
[119] Bossard, G.; Howe, PS; Stelle, KS; Vanhove, P., The vanishing volume of D = 4 superspace, Class. Quant. Grav., 28, 215005, (2011) · Zbl 1230.83091
[120] Beneke, M.; Kirilin, G., Soft-collinear gravity, JHEP, 09, 066, (2012) · Zbl 1398.83031
[121] Dunbar, DC; Norridge, PS, Infinities within graviton scattering amplitudes, Class. Quant. Grav., 14, 351, (1997) · Zbl 0925.83030
[122] Naculich, SG; Schnitzer, HJ, Eikonal methods applied to gravitational scattering amplitudes, JHEP, 05, 087, (2011) · Zbl 1296.83028
[123] White, CD, Factorization Properties of Soft Graviton Amplitudes, JHEP, 05, 060, (2011) · Zbl 1296.83030
[124] Abreu, S.; Britto, R.; Duhr, C.; Gardi, E., Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case, JHEP, 12, 090, (2017) · Zbl 1383.81321
[125] Abreu, S.; Britto, R.; Duhr, C.; Gardi, E., Algebraic Structure of Cut Feynman Integrals and the Diagrammatic Coaction, Phys. Rev. Lett., 119, (2017) · Zbl 1383.81321
[126] Panzer, E., Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals, Comput. Phys. Commun., 188, 148, (2015) · Zbl 1344.81024
[127] L.J. Dixon, E. Herrmann, K. Yan and H.X. Zhu, Soft emission function at two loops, to appear (2019).
[128] Bern, Z.; Dixon, LJ; Perelstein, M.; Rozowsky, JS, One loop n point helicity amplitudes in (selfdual) gravity, Phys. Lett., B 444, 273, (1998)
[129] Kawai, H.; Lewellen, DC; Tye, SHH, A Relation Between Tree Amplitudes of Closed and Open Strings, Nucl. Phys., B 269, 1, (1986)
[130] Britto, R.; Cachazo, F.; Feng, B.; Witten, E., Direct proof of tree-level recursion relation in Yang-Mills theory, Phys. Rev. Lett., 94, 181602, (2005)
[131] Bern, Z.; etal., The Two-Loop Six-Gluon MHV Amplitude in Maximally Supersymmetric Yang-Mills Theory, Phys. Rev., D 78, (2008)
[132] Weinzierl, S., Does one need the \( \mathcal{O} \)(ϵ)- and \( \mathcal{O} \)(ϵ2)-terms of one-loop amplitudes in an NNLO calculation?, Phys. Rev., D 84, (2011)
[133] Green, MB; Schwarz, JH; Brink, L., N = 4 Yang-Mills and N = 8 Supergravity as Limits of String Theories, Nucl. Phys., B 198, 474, (1982)
[134] Z. Bern, L.J. Dixon and D.A. Kosower, One loop corrections to five gluon amplitudes, Phys. Rev. Lett.70 (1993) 2677 [hep-ph/9302280] [INSPIRE].
[135] Bern, Z.; Kosower, DA, Color decomposition of one loop amplitudes in gauge theories, Nucl. Phys., B 362, 389, (1991)
[136] Z. Bern, A. De Freitas and L.J. Dixon, Two loop helicity amplitudes for gluon-gluon scattering in QCD and supersymmetric Yang-Mills theory, JHEP03 (2002) 018 [hep-ph/0201161] [INSPIRE].
[137] Kleiss, R.; Kuijf, H., Multi-Gluon Cross-sections and Five Jet Production at Hadron Colliders, Nucl. Phys., B 312, 616, (1989)
[138] Z. Bern, J.S. Rozowsky and B. Yan, Two loop four gluon amplitudes in N = 4 superYang-Mills, Phys. Lett.B 401 (1997) 273 [hep-ph/9702424] [INSPIRE].
[139] Edison, AC; Naculich, SG, SU(N) group-theory constraints on color-ordered five-point amplitudes at all loop orders, Nucl. Phys., B 858, 488, (2012) · Zbl 1246.81119
[140] Chester, D., Bern-Carrasco-Johansson relations for one-loop QCD integral coefficients, Phys. Rev., D 93, (2016)
[141] Primo, A.; Torres Bobadilla, WJ, BCJ Identities and d-Dimensional Generalized Unitarity, JHEP, 04, 125, (2016)
[142] Arkani-Hamed, N.; Cachazo, F.; Kaplan, J., What is the Simplest Quantum Field Theory?, JHEP, 09, 016, (2010) · Zbl 1291.81356
[143] Bern, Z.; Davies, S.; Dennen, T.; Huang, Y-t, Ultraviolet Cancellations in Half-Maximal Supergravity as a Consequence of the Double-Copy Structure, Phys. Rev., D 86, 105014, (2012)
[144] M. Besier, D. Van Straten and S. Weinzierl, Rationalizing roots: an algorithmic approach, arXiv:1809.10983 [INSPIRE].
[145] J.L. Bourjaily, Efficient Tree-Amplitudes in N = 4: Automatic BCFW Recursion in Mathematica, arXiv:1011.2447 [INSPIRE].
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.