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Integrands for QCD rational terms and \( \mathcal{N} = {4} \) SYM from massive CSW rules. (English) Zbl 1397.81432

Summary: We use massive CSW rules to derive explicit compact expressions for integrands of rational terms in QCD with any number of external legs. Specifically, we present all-\(n\) integrands for the one-loop all-plus and one-minus gluon amplitudes in QCD. We extract the finite part of spurious external-bubble contributions systematically; this is crucial for the application of integrand-level CSW rules in theories without supersymmetry. Our approach yields integrands that are independent of the choice of CSW reference spinor even before integration. Furthermore, we present a recursive derivation of the recently proposed massive CSW-style vertex expansion for massive tree amplitudes and loop integrands on the Coulomb-branch of \( \mathcal{N} = {4} \) SYM. The derivation requires a careful study of boundary terms in all-line shift recursion relations, and provides a rigorous (albeit indirect) proof of the recently proposed construction of massive amplitudes from soft-limits of massless on-shell amplitudes. We show that the massive vertex expansion manifestly preserves all holomorphic and half of the anti-holomorphic supercharges, diagram-by-diagram, even off-shell.

MSC:

81V05 Strong interaction, including quantum chromodynamics
81S40 Path integrals in quantum mechanics
81T70 Quantization in field theory; cohomological methods
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[1] Cachazo, F.; Svrček, P.; Witten, E., Gauge theory amplitudes in twistor space and holomorphic anomaly, JHEP, 10, 077, (2004) · doi:10.1088/1126-6708/2004/10/077
[2] Brandhuber, A.; Spence, BJ; Travaglini, G., One-loop gauge theory amplitudes in N = 4 super Yang-Mills from MHV vertices, Nucl. Phys., B 706, 150, (2005) · Zbl 1119.81363 · doi:10.1016/j.nuclphysb.2004.11.023
[3] Risager, K., A direct proof of the CSW rules, JHEP, 12, 003, (2005) · doi:10.1088/1126-6708/2005/12/003
[4] Elvang, H.; Freedman, DZ; Kiermaier, M., Recursion relations, generating functions and unitarity sums in N = 4 SYM theory, JHEP, 04, 009, (2009) · doi:10.1088/1126-6708/2009/04/009
[5] Elvang, H.; Freedman, DZ; Kiermaier, M., Proof of the MHV vertex expansion for all tree amplitudes in N = 4 SYM theory, JHEP, 06, 068, (2009) · doi:10.1088/1126-6708/2009/06/068
[6] Bullimore, M.; Mason, L.; Skinner, D., MHV diagrams in momentum twistor space, JHEP, 12, 032, (2010) · Zbl 1294.81094 · doi:10.1007/JHEP12(2010)032
[7] Bullimore, M., MHV diagrams from an all-line recursion relation, JHEP, 08, 107, (2011) · Zbl 1298.81162 · doi:10.1007/JHEP08(2011)107
[8] Mason, L.; Skinner, D., The complete planar S-matrix of N = 4 SYM as a Wilson loop in twistor space, JHEP, 12, 018, (2010) · Zbl 1294.81122 · doi:10.1007/JHEP12(2010)018
[9] Cohen, T.; Elvang, H.; Kiermaier, M., On-shell constructibility of tree amplitudes in general field theories, JHEP, 04, 053, (2011) · Zbl 1250.81072 · doi:10.1007/JHEP04(2011)053
[10] Badger, S., Direct extraction of one loop rational terms, JHEP, 01, 049, (2009) · Zbl 1243.81219 · doi:10.1088/1126-6708/2009/01/049
[11] Ettle, JH; Fu, C-H; Fudger, JP; Mansfield, PR; Morris, TR, S-matrix equivalence theorem evasion and dimensional regularisation with the canonical MHV Lagrangian, JHEP, 05, 011, (2007) · doi:10.1088/1126-6708/2007/05/011
[12] Brandhuber, A.; Spence, B.; Travaglini, G.; Zoubos, K., One-loop MHV rules and pure Yang-Mills, JHEP, 07, 002, (2007) · doi:10.1088/1126-6708/2007/07/002
[13] Caron-Huot, S., Loops and trees, JHEP, 05, 080, (2011) · Zbl 1296.81128 · doi:10.1007/JHEP05(2011)080
[14] 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 · doi:10.1007/JHEP01(2011)041
[15] Boels, R.; Schwinn, C., CSW rules for a massive scalar, Phys. Lett., B 662, 80, (2008) · Zbl 1282.81136
[16] Nigel Glover, E.; Williams, C., One-loop gluonic amplitudes from single unitarity cuts, JHEP, 12, 067, (2008) · Zbl 1329.81283 · doi:10.1088/1126-6708/2008/12/067
[17] Craig, N.; Elvang, H.; Kiermaier, M.; Slatyer, T., Massive amplitudes on the Coulomb branch of N = 4 SYM, JHEP, 12, 097, (2011) · Zbl 1306.81088 · doi:10.1007/JHEP12(2011)097
[18] M. Kiermaier, The Coulomb-branch S-matrix from massless amplitudes, arXiv:1105.5385 [INSPIRE].
[19] Bern, Z.; Dixon, LJ; Dunbar, DC; Kosower, DA, One loop n point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys., B 425, 217, (1994) · Zbl 1049.81644 · doi:10.1016/0550-3213(94)90179-1
[20] Bern, Z.; Dixon, LJ; Dunbar, DC; Kosower, DA, Fusing gauge theory tree amplitudes into loop amplitudes, Nucl. Phys., B 435, 59, (1995) · doi:10.1016/0550-3213(94)00488-Z
[21] Bern, Z.; Chalmers, G.; Dixon, LJ; Kosower, DA, One loop N gluon amplitudes with maximal helicity violation via collinear limits, Phys. Rev. Lett., 72, 2134, (1994) · doi:10.1103/PhysRevLett.72.2134
[22] Mahlon, G., Multi - gluon helicity amplitudes involving a quark loop, Phys. Rev., D 49, 4438, (1994)
[23] Bern, Z.; Morgan, A., Massive loop amplitudes from unitarity, Nucl. Phys., B 467, 479, (1996) · doi:10.1016/0550-3213(96)00078-8
[24] Bern, Z.; Dixon, LJ; Kosower, DA, On-shell recurrence relations for one-loop QCD amplitudes, Phys. Rev., D 71, 105013, (2005)
[25] Bern, Z.; Dixon, LJ; Kosower, DA, The last of the finite loop amplitudes in QCD, Phys. Rev., D 72, 125003, (2005)
[26] Brandhuber, A.; McNamara, S.; Spence, B.; Travaglini, G., Recursion relations for one-loop gravity amplitudes, JHEP, 03, 029, (2007) · doi:10.1088/1126-6708/2007/03/029
[27] Giele, WT; Kunszt, Z.; Melnikov, K., Full one-loop amplitudes from tree amplitudes, JHEP, 04, 049, (2008) · Zbl 1246.81170 · doi:10.1088/1126-6708/2008/04/049
[28] Bern, Z.; Dixon, LJ; Dunbar, DC; Kosower, DA, One loop selfdual and N = 4 super Yang-Mills, Phys. Lett., B 394, 105, (1997)
[29] Boels, RH, On BCFW shifts of integrands and integrals, JHEP, 11, 113, (2010) · Zbl 1294.81089 · doi:10.1007/JHEP11(2010)113
[30] Britto, R.; Mirabella, E., External leg corrections in the unitarity method, JHEP, 01, 045, (2012) · Zbl 1306.81328 · doi:10.1007/JHEP01(2012)045
[31] Alday, LF; Henn, JM; Plefka, J.; Schuster, T., Scattering into the fifth dimension of N =4 super Yang-Mills, JHEP, 01, 077, (2010) · Zbl 1269.81079 · doi:10.1007/JHEP01(2010)077
[32] A. Sever and P. Vieira, Symmetries of the N = 4 SYM S-matrix, arXiv:0908.2437 [INSPIRE].
[33] Henn, JM; Naculich, SG; Schnitzer, HJ; Spradlin, M., Higgs-regularized three-loop four-gluon amplitude in N = 4 SYM: exponentiation and Regge limits, JHEP, 04, 038, (2010) · Zbl 1272.81117 · doi:10.1007/JHEP04(2010)038
[34] Henn, JM; Naculich, SG; Schnitzer, HJ; Spradlin, M., More loops and legs in Higgs-regulated N = 4 SYM amplitudes, JHEP, 08, 002, (2010) · Zbl 1291.81254 · doi:10.1007/JHEP08(2010)002
[35] Henn, JM, Dual conformal symmetry at loop level: massive regularization, J. Phys., A 44, 454011, (2011) · Zbl 1270.81137
[36] Bern, Z.; Rozowsky, J.; Yan, B., Two loop four gluon amplitudes in N = 4 super Yang-Mills, Phys. Lett., B 401, 273, (1997)
[37] 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, 085001, (2005)
[38] 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, 085010, (2007)
[39] Bern, Z.; Carrasco, J.; Johansson, H.; Kosower, D., Maximally supersymmetric planar Yang-Mills amplitudes at five loops, Phys. Rev., D 76, 125020, (2007)
[40] Bern, Z.; Carrasco, JJ; Dennen, T.; Huang, Y-t; Ita, H., Generalized unitarity and six-dimensional helicity, Phys. Rev., D 83, 085022, (2011)
[41] Brandhuber, A.; Korres, D.; Koschade, D.; Travaglini, G., One-loop amplitudes in six-dimensional (1,1) theories from generalised unitarity, JHEP, 02, 077, (2011) · Zbl 1294.81091 · doi:10.1007/JHEP02(2011)077
[42] Dennen, T.; Huang, Y-t, Dual conformal properties of six-dimensional maximal super Yang-Mills amplitudes, JHEP, 01, 140, (2011) · Zbl 1214.81150 · doi:10.1007/JHEP01(2011)140
[43] Hatsuda, M.; Huang, Y-t; Siegel, W., First-quantized N = 4 Yang-Mills, JHEP, 04, 058, (2009) · doi:10.1088/1126-6708/2009/04/058
[44] Elvang, H.; Huang, Y-t; Peng, C.; Huang, Y-t; Peng, C., On-shell superamplitudes in N <4 SYM, JHEP, 09, 031, (2011) · Zbl 1301.81120 · doi:10.1007/JHEP09(2011)031
[45] Y.-t. Huang, Non-Chiral S-matrix of N = 4 Super Yang-Mills, arXiv:1104.2021 [INSPIRE].
[46] R.M. Schabinger, Scattering on the Moduli Space of N = 4 Super Yang-Mills, arXiv:0801.1542 [INSPIRE]. · Zbl 1270.81141
[47] Boels, RH, No triangles on the moduli space of maximally supersymmetric gauge theory, JHEP, 05, 046, (2010) · Zbl 1288.81075 · doi:10.1007/JHEP05(2010)046
[48] Boels, RH; Schwinn, C., On-shell supersymmetry for massive multiplets, Phys. Rev., D 84, 065006, (2011)
[49] Kiermaier, M.; Naculich, SG, A super MHV vertex expansion for N = 4 SYM theory, JHEP, 05, 072, (2009) · doi:10.1088/1126-6708/2009/05/072
[50] Parke, SJ; Taylor, T., An amplitude for n gluon scattering, Phys. Rev. Lett., 56, 2459, (1986) · doi:10.1103/PhysRevLett.56.2459
[51] Ferrario, P.; Rodrigo, G.; Talavera, P., Compact multigluonic scattering amplitudes with heavy scalars and fermions, Phys. Rev. Lett., 96, 182001, (2006) · doi:10.1103/PhysRevLett.96.182001
[52] Forde, D.; Kosower, DA, All-multiplicity amplitudes with massive scalars, Phys. Rev., D 73, 065007, (2006)
[53] Rodrigo, G., Multigluonic scattering amplitudes of heavy quarks, JHEP, 09, 079, (2005) · doi:10.1088/1126-6708/2005/09/079
[54] Kleiss, R.; Stirling, WJ, Spinor techniques for calculating pp → W \^{}{±}/Z\^{}{0} + jets, Nucl. Phys., B 262, 235, (1985) · doi:10.1016/0550-3213(85)90285-8
[55] Dittmaier, S., Weyl-Van der Waerden formalism for helicity amplitudes of massive particles, Phys. Rev., D 59, 016007, (1998)
[56] Bianchi, M.; Elvang, H.; Freedman, DZ, Generating tree amplitudes in N = 4 SYM and N = 8 SG, JHEP, 09, 063, (2008) · Zbl 1245.81083 · doi:10.1088/1126-6708/2008/09/063
[57] Schwinn, C.; Weinzierl, S., SUSY Ward identities for multi-gluon helicity amplitudes with massive quarks, JHEP, 03, 030, (2006) · Zbl 1226.81142 · doi:10.1088/1126-6708/2006/03/030
[58] Feng, B.; Wang, J.; Wang, Y.; Zhang, Z., BCFW recursion relation with nonzero boundary contribution, JHEP, 01, 019, (2010) · Zbl 1269.81084 · doi:10.1007/JHEP01(2010)019
[59] Feng, B.; Liu, C-Y, A note on the boundary contribution with bad deformation in gauge theory, JHEP, 07, 093, (2010) · Zbl 1290.81168 · doi:10.1007/JHEP07(2010)093
[60] Feng, B.; Zhang, Z., Boundary contributions using fermion pair deformation, JHEP, 12, 057, (2011) · Zbl 1306.81101 · doi:10.1007/JHEP12(2011)057
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