Abstract
We explore the on-shell recursion for tree-level scattering amplitudes with massive spinning particles. Based on the factorization structure encoded in the same way by two different recursion relations, we conjecture an all-multiplicity formula for two gauged massive particles of arbitrary spin and any number of identical-helicity gluons. Specializing to quantum chromodynamics (QCD), we solve the on-shell recursion relations in the presence of two pairs of massive quarks and an arbitrary number of identical-helicity gluons. We find closed-form expressions for the two distinct families of color-ordered four-quark amplitudes, in which all gluons comprise a single color-adjacent set. We compare the efficiency of the numerical evaluation of the two resulting analytic formulae against a numerical implementation of the off-shell Berends-Giele recursion. We find the formulae for both amplitude families to be faster for large multiplicities, while the simpler of the two is actually faster for any number of external legs. Our analytic results are provided in a computer-readable format as two files in the supplementary material.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
F. A. Berends, R. Kleiss, P. De Causmaecker, R. Gastmans and T. T. Wu, Single Bremsstrahlung Processes in Gauge Theories, Phys. Lett. B 103 (1981) 124 [INSPIRE].
P. De Causmaecker, R. Gastmans, W. Troost and T. T. Wu, Multiple Bremsstrahlung in Gauge Theories at High-Energies. 1. General Formalism for Quantum Electrodynamics, Nucl. Phys. B 206 (1982) 53 [INSPIRE].
J. F. Gunion and Z. Kunszt, Improved analytic techniques for tree graph calculations and the ggqqℓℓ subprocess, Phys. Lett. B 161 (1985) 333 [INSPIRE].
R. Kleiss and W. J. Stirling, Spinor Techniques for Calculating \( p\overline{p} \) → W±/Z0 + Jets, Nucl. Phys. B 262 (1985) 235 [INSPIRE].
Z. Xu, D.-H. Zhang and L. Chang, Helicity Amplitudes for Multiple Bremsstrahlung in Massless Nonabelian Gauge Theories, Nucl. Phys. B 291 (1987) 392 [INSPIRE].
R. Gastmans and T. Wu, The Ubiquitous photon: Helicity method for QED and QCD, Int. Ser. Monogr. Phys. 80 (1990) 1.
S. J. Parke and T. R. Taylor, An Amplitude for n Gluon Scattering, Phys. Rev. Lett. 56 (1986) 2459 [INSPIRE].
L. J. Dixon, Calculating scattering amplitudes efficiently, in Proceedings of Theoretical Advanced Study Institute in Elementary Particle Physics (TASI 95): QCD and Beyond, Boulder U.S.A. (1996), pg. 539 [hep-ph/9601359] [INSPIRE].
J. M. Drummond and J. M. Henn, All tree-level amplitudes in N = 4 SYM, JHEP 04 (2009) 018 [arXiv:0808.2475] [INSPIRE].
L. J. Dixon, J. M. Henn, J. Plefka and T. Schuster, All tree-level amplitudes in massless QCD, JHEP 01 (2011) 035 [arXiv:1010.3991] [INSPIRE].
R. Britto, F. Cachazo and B. Feng, New recursion relations for tree amplitudes of gluons, Nucl. Phys. B 715 (2005) 499 [hep-th/0412308] [INSPIRE].
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] [INSPIRE].
N. Arkani-Hamed, T.-C. Huang and Y.-t. Huang, Scattering amplitudes for all masses and spins, JHEP 11 (2021) 070 [arXiv:1709.04891] [INSPIRE].
R. Kleiss and W. J. Stirling, Cross-sections for the production of an arbitrary number of photons in electron-positron annihilation, Phys. Lett. B 179 (1986) 159 [INSPIRE].
S. Dittmaier, Weyl-van der Waerden formalism for helicity amplitudes of massive particles, Phys. Rev. D 59 (1998) 016007 [hep-ph/9805445] [INSPIRE].
C. Schwinn and S. Weinzierl, Scalar diagrammatic rules for Born amplitudes in QCD, JHEP 05 (2005) 006 [hep-th/0503015] [INSPIRE].
E. Conde and A. Marzolla, Lorentz constraints on massive three-point amplitudes, JHEP 09 (2016) 041 [arXiv:1601.08113] [INSPIRE].
E. Conde, E. Joung and K. Mkrtchyan, Spinor-helicity three-point amplitudes from local cubic interactions, JHEP 08 (2016) 040 [arXiv:1605.07402] [INSPIRE].
A. Ochirov, Helicity amplitudes for QCD with massive quarks, JHEP 04 (2018) 089 [arXiv:1802.06730] [INSPIRE].
F. A. Berends and W. T. Giele, Recursive Calculations for Processes with n Gluons, Nucl. Phys. B 306 (1988) 759 [INSPIRE].
M. Dinsdale, M. Ternick and S. Weinzierl, A Comparison of efficient methods for the computation of Born gluon amplitudes, JHEP 03 (2006) 056 [hep-ph/0602204] [INSPIRE].
C. Duhr, S. Hoeche and F. Maltoni, Color-dressed recursive relations for multi-parton amplitudes, JHEP 08 (2006) 062 [hep-ph/0607057] [INSPIRE].
W. T. Giele and G. Zanderighi, On the Numerical Evaluation of One-Loop Amplitudes: The Gluonic Case, JHEP 06 (2008) 038 [arXiv:0805.2152] [INSPIRE].
R. K. Ellis, W. T. Giele, Z. Kunszt, K. Melnikov and G. Zanderighi, One-loop amplitudes for W + 3 jet production in hadron collisions, JHEP 01 (2009) 012 [arXiv:0810.2762] [INSPIRE].
A. Lazopoulos, Multi-gluon one-loop amplitudes numerically, arXiv:0812.2998 [INSPIRE].
S. Badger, B. Biedermann and P. Uwer, NGluon: A Package to Calculate One-loop Multi-gluon Amplitudes, Comput. Phys. Commun. 182 (2011) 1674 [arXiv:1011.2900] [INSPIRE].
S. Badger, B. Biedermann, L. Hackl, J. Plefka, T. Schuster and P. Uwer, Comparing efficient computation methods for massless QCD tree amplitudes: Closed analytic formulas versus Berends-Giele recursion, Phys. Rev. D 87 (2013) 034011 [arXiv:1206.2381] [INSPIRE].
D. Forde and D. A. Kosower, All-multiplicity amplitudes with massive scalars, Phys. Rev. D 73 (2006) 065007 [hep-th/0507292] [INSPIRE].
P. Ferrario, G. Rodrigo and P. Talavera, Compact multigluonic scattering amplitudes with heavy scalars and fermions, Phys. Rev. Lett. 96 (2006) 182001 [hep-th/0602043] [INSPIRE].
C. Schwinn and S. Weinzierl, On-shell recursion relations for all Born QCD amplitudes, JHEP 04 (2007) 072 [hep-ph/0703021] [INSPIRE].
A. Ochirov, Spinning massive particles and black holes, Lectures at 2nd SAGEX training school at Humboldt University of Berlin, Berlin Germany (2020), http://www.youtube.com/watch?v=anQwlWTQauM.
R. Aoude, K. Haddad and A. Helset, On-shell heavy particle effective theories, JHEP 05 (2020) 051 [arXiv:2001.09164] [INSPIRE].
S. Ballav and A. Manna, Recursion relations for scattering amplitudes with massive particles II: massive vector bosons, arXiv:2109.06546 [INSPIRE].
H. Johansson and A. Ochirov, Double copy for massive quantum particles with spin, JHEP 09 (2019) 040 [arXiv:1906.12292] [INSPIRE].
V. Del Duca, L. J. Dixon and F. Maltoni, New color decompositions for gauge amplitudes at tree and loop level, Nucl. Phys. B 571 (2000) 51 [hep-ph/9910563] [INSPIRE].
H. Kawai, D. C. Lewellen and S. H. H. Tye, A Relation Between Tree Amplitudes of Closed and Open Strings, Nucl. Phys. B 269 (1986) 1 [INSPIRE].
Z. Bern, L. J. Dixon, M. Perelstein and J. S. Rozowsky, Multileg one loop gravity amplitudes from gauge theory, Nucl. Phys. B 546 (1999) 423 [hep-th/9811140] [INSPIRE].
H. Johansson and A. Ochirov, Color-Kinematics Duality for QCD Amplitudes, JHEP 01 (2016) 170 [arXiv:1507.00332] [INSPIRE].
L. de la Cruz, A. Kniss and S. Weinzierl, Double Copies of Fermions as Matter that Interacts Only Gravitationally, Phys. Rev. Lett. 116 (2016) 201601 [arXiv:1601.04523] [INSPIRE].
R. W. Brown and S. G. Naculich, KLT-type relations for QCD and bicolor amplitudes from color-factor symmetry, JHEP 03 (2018) 057 [arXiv:1802.01620] [INSPIRE].
S. D. Badger, E. W. N. Glover and V. V. Khoze, Recursion relations for gauge theory amplitudes with massive vector bosons and fermions, JHEP 01 (2006) 066 [hep-th/0507161] [INSPIRE].
R. Britto and A. Ochirov, On-shell recursion for massive fermion currents, JHEP 01 (2013) 002 [arXiv:1210.1755] [INSPIRE].
S. D. Badger, E. W. N. Glover, V. V. Khoze and P. Svrček, Recursion relations for gauge theory amplitudes with massive particles, JHEP 07 (2005) 025 [hep-th/0504159] [INSPIRE].
N. Arkani-Hamed and J. Kaplan, On Tree Amplitudes in Gauge Theory and Gravity, JHEP 04 (2008) 076 [arXiv:0801.2385] [INSPIRE].
M.-Z. Chung, Y.-T. Huang, J.-W. Kim and S. Lee, The simplest massive S-matrix: from minimal coupling to Black Holes, JHEP 04 (2019) 156 [arXiv:1812.08752] [INSPIRE].
A. Falkowski and C. S. Machado, Soft Matters, or the Recursions with Massive Spinors, JHEP 05 (2021) 238 [arXiv:2005.08981] [INSPIRE].
M. Chiodaroli, H. Johansson and P. Pichini, Compton Black-Hole Scattering for s ≤ 5/2, arXiv:2107.14779 [INSPIRE].
A. Herderschee, S. Koren and T. Trott, Constructing 𝒩 = 4 Coulomb branch superamplitudes, JHEP 08 (2019) 107 [arXiv:1902.07205] [INSPIRE].
R. Aoude and C. S. Machado, The Rise of SMEFT On-shell Amplitudes, JHEP 12 (2019) 058 [arXiv:1905.11433] [INSPIRE].
R. Franken and C. Schwinn, On-shell constructibility of Born amplitudes in spontaneously broken gauge theories, JHEP 02 (2020) 073 [arXiv:1910.13407] [INSPIRE].
S. Ballav and A. Manna, Recursion relations for scattering amplitudes with massive particles, JHEP 03 (2021) 295 [arXiv:2010.14139] [INSPIRE].
A. Ochirov and B. Page, Multi-Quark Colour Decompositions from Unitarity, JHEP 10 (2019) 058 [arXiv:1908.02695] [INSPIRE].
T. Melia, Dyck words and multiquark primitive amplitudes, Phys. Rev. D 88 (2013) 014020 [arXiv:1304.7809] [INSPIRE].
T. Melia, Getting more flavor out of one-flavor QCD, Phys. Rev. D 89 (2014) 074012 [arXiv:1312.0599] [INSPIRE].
R. Kleiss and H. Kuijf, Multi-gluon cross-sections and five jet production at hadron colliders, Nucl. Phys. B 312 (1989) 616 [INSPIRE].
T. Melia, Proof of a new colour decomposition for QCD amplitudes, JHEP 12 (2015) 107 [arXiv:1509.03297] [INSPIRE].
Z. Bern, J. J. M. Carrasco and H. Johansson, New Relations for Gauge-Theory Amplitudes, Phys. Rev. D 78 (2008) 085011 [arXiv:0805.3993] [INSPIRE].
L. de la Cruz, A. Kniss and S. Weinzierl, Proof of the fundamental BCJ relations for QCD amplitudes, JHEP 09 (2015) 197 [arXiv:1508.01432] [INSPIRE].
M.-x. Luo and C.-k. Wen, Recursion relations for tree amplitudes in super gauge theories, JHEP 03 (2005) 004 [hep-th/0501121] [INSPIRE].
A. Kanaki and C. G. Papadopoulos, HELAC: A Package to compute electroweak helicity amplitudes, Comput. Phys. Commun. 132 (2000) 306 [hep-ph/0002082] [INSPIRE].
P. H. Damgaard, K. Haddad and A. Helset, Heavy Black Hole Effective Theory, JHEP 11 (2019) 070 [arXiv:1908.10308] [INSPIRE].
A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP 05 (2013) 135 [arXiv:0905.1473] [INSPIRE].
N. Arkani-Hamed, J. Bourjaily, F. Cachazo and J. Trnka, Local Spacetime Physics from the Grassmannian, JHEP 01 (2011) 108 [arXiv:0912.3249] [INSPIRE].
N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, S. Caron-Huot and J. Trnka, The All-Loop Integrand For Scattering Amplitudes in Planar N = 4 SYM, JHEP 01 (2011) 041 [arXiv:1008.2958] [INSPIRE].
N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, A. B. Goncharov, A. Postnikov and J. Trnka, Grassmannian Geometry of Scattering Amplitudes, Cambridge University Press, Cambridge U.K. (2016), [arXiv:1212.5605] [INSPIRE].
N. Arkani-Hamed and J. Trnka, The Amplituhedron, JHEP 10 (2014) 030 [arXiv:1312.2007] [INSPIRE].
L. Mason and D. Skinner, Ambitwistor strings and the scattering equations, JHEP 07 (2014) 048 [arXiv:1311.2564] [INSPIRE].
G. Albonico, Y. Geyer and L. Mason, Massive ambitwistor-string models, to appear.
F. Cachazo, S. He and E. Y. Yuan, Scattering of Massless Particles in Arbitrary Dimensions, Phys. Rev. Lett. 113 (2014) 171601 [arXiv:1307.2199] [INSPIRE].
F. Cachazo, S. He and E. Y. Yuan, Scattering of Massless Particles: Scalars, Gluons and Gravitons, JHEP 07 (2014) 033 [arXiv:1309.0885] [INSPIRE].
L. Dolan and P. Goddard, Proof of the Formula of Cachazo, He and Yuan for Yang-Mills Tree Amplitudes in Arbitrary Dimension, JHEP 05 (2014) 010 [arXiv:1311.5200] [INSPIRE].
L. Dolan and P. Goddard, The Polynomial Form of the Scattering Equations, JHEP 07 (2014) 029 [arXiv:1402.7374] [INSPIRE].
S. G. Naculich, Scattering equations and BCJ relations for gauge and gravitational amplitudes with massive scalar particles, JHEP 09 (2014) 029 [arXiv:1407.7836] [INSPIRE].
S. G. Naculich, Amplitudes for massive vector and scalar bosons in spontaneously-broken gauge theory from the CHY representation, JHEP 09 (2015) 122 [arXiv:1506.06134] [INSPIRE].
S. G. Naculich, CHY representations for gauge theory and gravity amplitudes with up to three massive particles, JHEP 05 (2015) 050 [arXiv:1501.03500] [INSPIRE].
L. de la Cruz, A. Kniss and S. Weinzierl, The CHY representation of tree-level primitive QCD amplitudes, JHEP 11 (2015) 217 [arXiv:1508.06557] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2111.06847
Supplementary Information
ESM 1
(TGZ 3 kb)
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Lazopoulos, A., Ochirov, A. & Shi, C. All-multiplicity amplitudes with four massive quarks and identical-helicity gluons. J. High Energ. Phys. 2022, 9 (2022). https://doi.org/10.1007/JHEP03(2022)009
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2022)009