All tree-level amplitudes in massless QCD. (English) Zbl 1214.81297
Summary: We derive compact analytical formulae for all tree-level color-ordered gauge theory amplitudes involving any number of external gluons and up to four massless quarkanti-quark pairs. A general formula is presented based on the combinatorics of paths along a rooted tree and associated determinants. Explicit expressions are displayed for the next-to-maximally helicity violating (NMHV) and next-to-next-to-maximally helicity violating (NNMHV) gauge theory amplitudes. Our results are obtained by projecting the previously-found expressions for the super-amplitudes of the maximally supersymmetric super Yang-Mills theory (\({\mathcal N} = 4\) SYM) onto the relevant components yielding all gluon-gluino tree amplitudes in \({\mathcal N} = 4\) SYM. We show how these results carry over to the corresponding QCD amplitudes, including massless quarks of different avors as well as a single electroweak vector boson. The public Mathematica package GGT is described, which encodes the results of this work and yields analytical formulae for all \({\mathcal N} = 4\) SYM gluon-gluino trees. These in turn yield all QCD trees with up to four external arbitrary-flavored massless quark-anti-quark pairs.
MSC:
| 81V05 | Strong interaction, including quantum chromodynamics |
| 81T13 | Yang-Mills and other gauge theories in quantum field theory |
| 81T60 | Supersymmetric field theories in quantum mechanics |
| 05C05 | Trees |
Keywords:
QCD phenomenologySoftware:
WHIZARD; MadGraph; Comix; Axodraw; CutTools; HELAC; PHEGAS; AMEGIC++; O'Mega; CompHep; JaxoDraw; BlackHat; S@M; MathematicaReferences:
| [1] | T. Stelzer and W.F. Long, Automatic generation of tree level helicity amplitudes, Comput. Phys. Commun.81 (1994) 357 [hep-ph/9401258] [SPIRES]. · doi:10.1016/0010-4655(94)90084-1 |
| [2] | J. Alwall et al., MadGraph/MadEvent v4: The New Web Generation, JHEP09 (2007) 028 [arXiv:0706.2334] [SPIRES]. · doi:10.1088/1126-6708/2007/09/028 |
| [3] | A. Pukhov et al., CompHEP: A package for evaluation of Feynman diagrams and integration over multi-particle phase space. User’s manual for version 33, hep-ph/9908288 [SPIRES]. |
| [4] | F. Krauss, R. Kuhn and G. Soff, AMEGIC++ 1.0: A Matrix element generator in C++, JHEP02 (2002) 044 [hep-ph/0109036] [SPIRES]. · doi:10.1088/1126-6708/2002/02/044 |
| [5] | F.A. Berends and W.T. Giele, Recursive Calculations for Processes with n Gluons, Nucl. Phys.B 306 (1988) 759 [SPIRES]. · doi:10.1016/0550-3213(88)90442-7 |
| [6] | T. Gleisberg and S. Hoeche, Comix, a new matrix element generator, JHEP12 (2008) 039 [arXiv:0808.3674] [SPIRES]. · doi:10.1088/1126-6708/2008/12/039 |
| [7] | F. Caravaglios and M. Moretti, An algorithm to compute Born scattering amplitudes without Feynman graphs, Phys. Lett.B 358 (1995) 332 [hep-ph/9507237] [SPIRES]. |
| [8] | F. Caravaglios, M.L. Mangano, M. Moretti and R. Pittau, A new approach to multi-jet calculations in hadron collisions, Nucl. Phys.B 539 (1999) 215 [hep-ph/9807570] [SPIRES]. · doi:10.1016/S0550-3213(98)00739-1 |
| [9] | A. Kanaki and C.G. Papadopoulos, HELAC: A package to compute electroweak helicity amplitudes, Comput. Phys. Commun.132 (2000) 306 [hep-ph/0002082] [SPIRES]. · Zbl 1031.81507 · doi:10.1016/S0010-4655(00)00151-X |
| [10] | A. Cafarella, C.G. Papadopoulos and M. Worek, Helac-Phegas: a generator for all parton level processes, Comput. Phys. Commun.180 (2009) 1941 [arXiv:0710.2427] [SPIRES]. · doi:10.1016/j.cpc.2009.04.023 |
| [11] | M. Moretti, T. Ohl and J. Reuter, O’Mega: An optimizing matrix element generator, hep-ph/0102195 [SPIRES]. |
| [12] | W. Kilian, T. Ohl and J. Reuter, WHIZARD: Simulating Multi-Particle Processes at LHC and ILC, arXiv:0708.4233 [SPIRES]. |
| [13] | S.J. Parke and T.R. Taylor, Perturbative QCD Utilizing Extended Supersymmetry, Phys. Lett.B 157 (1985) 81 [SPIRES]. |
| [14] | Z. Kunszt, Combined Use of the Calkul Method and N = 1 Supersymmetry to Calculate QCD Six Parton Processes, Nucl. Phys.B 271 (1986) 333 [SPIRES]. |
| [15] | M.T. Grisaru, H.N. Pendleton and P. van Nieuwenhuizen, Supergravity and the S Matrix, Phys. Rev.D 15 (1977) 996 [SPIRES]. |
| [16] | M.T. Grisaru and H.N. Pendleton, Some Properties of Scattering Amplitudes in Supersymmetric Theories, Nucl. Phys.B 124 (1977) 81 [SPIRES]. · doi:10.1016/0550-3213(77)90277-2 |
| [17] | S.J. Parke and T.R. Taylor, An Amplitude for n Gluon Scattering, Phys. Rev. Lett.56 (1986) 2459 [SPIRES]. · doi:10.1103/PhysRevLett.56.2459 |
| [18] | V.P. Nair, A current algebra for some gauge theory amplitudes, Phys. Lett.B 214 (1988) 215 [SPIRES]. |
| [19] | 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] [SPIRES]. · Zbl 1049.81644 · doi:10.1016/0550-3213(94)90179-1 |
| [20] | 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] [SPIRES]. · doi:10.1016/0550-3213(94)00488-Z |
| [21] | Z. Bern, L.J. Dixon and D.A. Kosower, On-Shell Methods in Perturbative QCD, Annals Phys.322 (2007) 1587 [arXiv:0704.2798] [SPIRES]. · Zbl 1122.81077 · doi:10.1016/j.aop.2007.04.014 |
| [22] | C.F. Berger and D. Forde, Multi-Parton Scattering Amplitudes via On-Shell Methods, arXiv:0912.3534 [SPIRES]. |
| [23] | E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys.252 (2004) 189 [hep-th/0312171] [SPIRES]. · Zbl 1105.81061 · doi:10.1007/s00220-004-1187-3 |
| [24] | R. Britto, F. Cachazo and B. Feng, New Recursion Relations for Tree Amplitudes of Gluons, Nucl. Phys.B 715 (2005) 499 [hep-th/0412308] [SPIRES]. · Zbl 1207.81088 · doi:10.1016/j.nuclphysb.2005.02.030 |
| [25] | R. Britto, F. Cachazo, B. Feng and E. Witten, Direct Proof Of Tree-Level Recursion Relation In Yang-Mills Theory, Phys. Rev. Lett.94 (2005) 181602 [hep-th/0501052] [SPIRES]. · doi:10.1103/PhysRevLett.94.181602 |
| [26] | R. Britto, B. Feng, R. Roiban, M. Spradlin and A. Volovich, All split helicity tree-level gluon amplitudes, Phys. Rev.D 71 (2005) 105017 [hep-th/0503198] [SPIRES]. |
| [27] | N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, A Duality For The S Matrix, JHEP03 (2010) 020 [arXiv:0907.5418] [SPIRES]. · Zbl 1271.81098 · doi:10.1007/JHEP03(2010)020 |
| [28] | M. Bianchi, H. Elvang and D.Z. Freedman, Generating Tree Amplitudes in N = 4 SYM and N = 8 SG, JHEP09 (2008) 063 [arXiv:0805.0757] [SPIRES]. · Zbl 1245.81083 · doi:10.1088/1126-6708/2008/09/063 |
| [29] | J.M. Drummond and J.M. Henn, All tree-level amplitudes in N = 4 SYM, JHEP04 (2009) 018 [arXiv:0808.2475] [SPIRES]. · doi:10.1088/1126-6708/2009/04/018 |
| [30] | J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Dual superconformal symmetry of scattering amplitudes in N = 4 super-Yang-Mills theory, Nucl. Phys.B 828 (2010) 317 [arXiv:0807.1095] [SPIRES]. · Zbl 1203.81112 · doi:10.1016/j.nuclphysb.2009.11.022 |
| [31] | A. Brandhuber, P. Heslop and G. Travaglini, A note on dual superconformal symmetry of the N = 4 super Yang-Mills S-matrix, Phys. Rev.D 78 (2008) 125005 [arXiv:0807.4097] [SPIRES]. |
| [32] | J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Generalized unitarity for N = 4 super-amplitudes, arXiv:0808.0491 [SPIRES]. · Zbl 1262.81195 |
| [33] | J.M. Drummond, J.M. Henn and J. Plefka, Yangian symmetry of scattering amplitudes in N = 4 super Yang-Mills theory, JHEP05 (2009) 046 [arXiv:0902.2987] [SPIRES]. · doi:10.1088/1126-6708/2009/05/046 |
| [34] | J.M. Drummond, Hidden Simplicity of Gauge Theory Amplitudes, Class. Quant. Grav.27 (2010) 214001 [arXiv:1010.2418] [SPIRES]. · Zbl 1204.83001 · doi:10.1088/0264-9381/27/21/214001 |
| [35] | C.F. Berger et al., Precise Predictions for W + 4 Jet Production at the Large Hadron Collider, arXiv:1009.2338 [SPIRES]. |
| [36] | G. Ossola, C.G. Papadopoulos and R. Pittau, Reducing full one-loop amplitudes to scalar integrals at the integrand level, Nucl. Phys.B 763 (2007) 147 [hep-ph/0609007] [SPIRES]. · Zbl 1116.81067 · doi:10.1016/j.nuclphysb.2006.11.012 |
| [37] | G. Ossola, C.G. Papadopoulos and R. Pittau, CutTools: a program implementing the OPP reduction method to compute one-loop amplitudes, JHEP03 (2008) 042 [arXiv:0711.3596] [SPIRES]. · doi:10.1088/1126-6708/2008/03/042 |
| [38] | G. Ossola, C.G. Papadopoulos and R. Pittau, On the Rational Terms of the one-loop amplitudes, JHEP05 (2008) 004 [arXiv:0802.1876] [SPIRES]. · doi:10.1088/1126-6708/2008/05/004 |
| [39] | R.K. Ellis, W.T. Giele and Z. Kunszt, A Numerical Unitarity Formalism for Evaluating One-Loop Amplitudes, JHEP03 (2008) 003 [arXiv:0708.2398] [SPIRES]. · doi:10.1088/1126-6708/2008/03/003 |
| [40] | W.T. Giele, Z. Kunszt and K. Melnikov, Full one-loop amplitudes from tree amplitudes, JHEP04 (2008) 049 [arXiv:0801.2237] [SPIRES]. · Zbl 1246.81170 · doi:10.1088/1126-6708/2008/04/049 |
| [41] | W.T. Giele and G. Zanderighi, On the Numerical Evaluation of One-Loop Amplitudes: The Gluonic Case, JHEP06 (2008) 038 [arXiv:0805.2152] [SPIRES]. · doi:10.1088/1126-6708/2008/06/038 |
| [42] | C.F. Berger et al., An Automated Implementation of On-Shell Methods for One-Loop Amplitudes, Phys. Rev.D 78 (2008) 036003 [arXiv:0803.4180] [SPIRES]. |
| [43] | Z. Bern and A.G. Morgan, Massive Loop Amplitudes from Unitarity, Nucl. Phys.B 467 (1996) 479 [hep-ph/9511336] [SPIRES]. · doi:10.1016/0550-3213(96)00078-8 |
| [44] | Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, One-loop self-dual and N = 4 super Yang-Mills, Phys. Lett.B 394 (1997) 105 [hep-th/9611127] [SPIRES]. |
| [45] | C. Anastasiou, R. Britto, B. Feng, Z. Kunszt and P. Mastrolia, D-dimensional unitarity cut method, Phys. Lett.B 645 (2007) 213 [hep-ph/0609191] [SPIRES]. |
| [46] | R. Britto and B. Feng, Integral Coefficients for One-Loop Amplitudes, JHEP02 (2008) 095 [arXiv:0711.4284] [SPIRES]. · doi:10.1088/1126-6708/2008/02/095 |
| [47] | S.D. Badger, Direct Extraction Of One Loop Rational Terms, JHEP01 (2009) 049 [arXiv:0806.4600] [SPIRES]. · Zbl 1243.81219 · doi:10.1088/1126-6708/2009/01/049 |
| [48] | Z. Bern, L.J. Dixon and D.A. Kosower, Bootstrapping multi-parton loop amplitudes in QCD, Phys. Rev.D 73 (2006) 065013 [hep-ph/0507005] [SPIRES]. |
| [49] | C.F. Berger, Z. Bern, L.J. Dixon, D. Forde and D.A. Kosower, Bootstrapping one-loop QCD amplitudes with general helicities, Phys. Rev.D 74 (2006) 036009 [hep-ph/0604195] [SPIRES]. |
| [50] | L.F. Alday, J.M. Henn, J. Plefka and T. Schuster, Scattering into the fifth dimension of N = 4 super Yang-Mills, JHEP01 (2010) 077 [arXiv:0908.0684] [SPIRES]. · Zbl 1269.81079 · doi:10.1007/JHEP01(2010)077 |
| [51] | M.L. Mangano and S.J. Parke, Multi-Parton Amplitudes in Gauge Theories, Phys. Rept.200 (1991) 301 [hep-th/0509223] [SPIRES]. · doi:10.1016/0370-1573(91)90091-Y |
| [52] | Z. Bern, L.J. Dixon and D.A. Kosower, One-loop amplitudes for e+e−to four partons, Nucl. Phys.B 513 (1998) 3 [hep-ph/9708239] [SPIRES]. · doi:10.1016/S0550-3213(97)00703-7 |
| [53] | M. Dinsdale, M. Ternick and S. Weinzierl, A comparison of efficient methods for the computation of Born gluon amplitudes, JHEP03 (2006) 056 [hep-ph/0602204] [SPIRES]. · Zbl 1226.81139 · doi:10.1088/1126-6708/2006/03/056 |
| [54] | G. Georgiou, E.W.N. Glover and V.V. Khoze, Non-MHV Tree Amplitudes in Gauge Theory, JHEP07 (2004) 048 [hep-th/0407027] [SPIRES]. · doi:10.1088/1126-6708/2004/07/048 |
| [55] | D. Binosi and L. Theussl, JaxoDraw: A graphical user interface for drawing Feynman diagrams, Comput. Phys. Commun.161 (2004) 76 [hep-ph/0309015] [SPIRES]. · doi:10.1016/j.cpc.2004.05.001 |
| [56] | D. Binosi, J. Collins, C. Kaufhold and L. Theussl, JaxoDraw: A graphical user interface for drawing Feynman diagrams. Version 2.0 release notes, Comput. Phys. Commun.180 (2009) 1709 [arXiv:0811.4113] [SPIRES]. · Zbl 07872412 · doi:10.1016/j.cpc.2009.02.020 |
| [57] | J.A.M. Vermaseren, Axodraw, Comput. Phys. Commun.83 (1994) 45 [SPIRES]. · Zbl 1114.68598 · doi:10.1016/0010-4655(94)90034-5 |
| [58] | D. Maitre and P. Mastrolia, S@M, a Mathematica Implementation of the Spinor-Helicity Formalism, Comput. Phys. Commun.179 (2008) 501 [arXiv:0710.5559] [SPIRES]. · Zbl 1197.83007 · doi:10.1016/j.cpc.2008.05.002 |
| [59] | R.K. Ellis, W.T. Giele and G. Zanderighi, The one-loop amplitude for six-gluon scattering, JHEP05 (2006) 027 [hep-ph/0602185] [SPIRES]. · doi:10.1088/1126-6708/2006/05/027 |
| [60] | J.L. Bourjaily, Efficient Tree-Amplitudes in N = 4: Automatic BCFW Recursion in Mathematica, arXiv:1011.2447 [SPIRES]. |
| [61] | N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, The S-matrix in Twistor Space, JHEP03 (2010) 110 [arXiv:0903.2110] [SPIRES]. · Zbl 1271.81169 · doi:10.1007/JHEP03(2010)110 |
| [62] | J.L. Bourjaily, J. Trnka, A. Volovich and C. Wen, The Grassmannian and the Twistor String: Connecting All Trees in N = 4 SYM, arXiv:1006.1899 [SPIRES]. · Zbl 1214.81194 |
| [63] | A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, arXiv:0905.1473 [SPIRES]. · Zbl 1342.81291 |
| [64] | L. Mason and D. Skinner, Dual Superconformal Invariance, Momentum Twistors and Grassmannians, JHEP11 (2009) 045 [arXiv:0909.0250] [SPIRES]. · doi:10.1088/1126-6708/2009/11/045 |
| [65] | Z. Bern, H. Ita and K. Ozeren, private communication. |
| [66] | P. Uwer and B. Biedermann, private communication. |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
