Abstract
We construct a prescriptive, bubble power-counting basis of one-loop integrands suitable for representing amplitude integrands in less-supersymmetric (1 ≤ \( \mathcal{N} \) ≤ 4) Yang-Mills theory. With the exception of massless bubbles, all integrands have unambiguous, leading singularities as coefficients defined in field theory; for the massless bubbles on external legs, we find two natural choices which lead to different integrands that highlight distinct aspects of field theory. For concreteness, we give the all-multiplicity integrands for MHV amplitudes, and the split-helicity amplitude integrand for six-particle NMHV. The basis we construct is mostly pure and is divided into to separately UV- and IR-finite sectors of fixed transcendental weight, resulting in UV- and IR-finite ratio functions of n-particle helicity amplitudes.
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References
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].
J. M. Drummond, J. Henn, V. A. Smirnov and E. Sokatchev, Magic identities for conformal four-point integrals, JHEP 01 (2007) 064 [hep-th/0607160] [INSPIRE].
L. F. Alday and J. M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [INSPIRE].
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] [INSPIRE].
J. M. Drummond, J. M. Henn and J. Plefka, Yangian symmetry of scattering amplitudes in N = 4 super Yang-Mills theory, JHEP 05 (2009) 046 [arXiv:0902.2987] [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].
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].
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].
R. Britto, F. Cachazo and B. Feng, Generalized unitarity and one-loop amplitudes in N = 4 super-Yang-Mills, Nucl. Phys. B 725 (2005) 275 [hep-th/0412103] [INSPIRE].
Z. Bern, N. E. J. Bjerrum-Bohr and D. C. Dunbar, Inherited twistor-space structure of gravity loop amplitudes, JHEP 05 (2005) 056 [hep-th/0501137] [INSPIRE].
N. E. J. Bjerrum-Bohr, D. C. Dunbar, H. Ita, W. B. Perkins and K. Risager, The no-triangle hypothesis for N = 8 supergravity, JHEP 12 (2006) 072 [hep-th/0610043] [INSPIRE].
Z. Bern, J. J. Carrasco, D. Forde, H. Ita and H. Johansson, Unexpected cancellations in gravity theories, Phys. Rev. D 77 (2008) 025010 [arXiv:0707.1035] [INSPIRE].
N. E. J. Bjerrum-Bohr and P. Vanhove, Absence of triangles in maximal supergravity amplitudes, JHEP 10 (2008) 006 [arXiv:0805.3682] [INSPIRE].
J. L. Bourjaily, E. Herrmann, C. Langer and J. Trnka, Building bases of loop integrands, JHEP 11 (2020) 116 [arXiv:2007.13905] [INSPIRE].
G. Passarino and M. J. G. Veltman, One loop corrections for e+ e− annihilation into μ+ μ− in the Weinberg model, Nucl. Phys. B 160 (1979) 151 [INSPIRE].
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] [INSPIRE].
P. Mastrolia, G. Ossola, T. Reiter and F. Tramontano, Scattering amplitudes from unitarity-based reduction algorithm at the integrand-level, JHEP 08 (2010) 080 [arXiv:1006.0710] [INSPIRE].
J. L. Bourjaily, E. Herrmann and J. Trnka, Prescriptive unitarity, JHEP 06 (2017) 059 [arXiv:1704.05460] [INSPIRE].
H. Elvang, Y.-T. Huang and C. Peng, On-shell superamplitudes in N < 4 SYM, JHEP 09 (2011) 031 [arXiv:1102.4843] [INSPIRE].
A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP 05 (2013) 135 [arXiv:0905.1473] [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].
P. Benincasa, On-shell diagrammatics and the perturbative structure of planar gauge theories, arXiv:1510.03642 [INSPIRE].
T. Hahn, Generating and calculating one loop Feynman diagrams with FeynArts, FormCalc, and LoopTools, in 6th international workshop on new computing techniques in physics research: software engineering, artificial intelligence neural nets, genetic algorithms, symbolic algebra, automatic calculation, (1999) [hep-ph/9905354] [INSPIRE].
G. Ossola, C. G. Papadopoulos and R. Pittau, CutTools: a program implementing the OPP reduction method to compute one-loop amplitudes, JHEP 03 (2008) 042 [arXiv:0711.3596] [INSPIRE].
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] [INSPIRE].
S. Abreu et al., Caravel: a C++ framework for the computation of multi-loop amplitudes with numerical unitarity, Comput. Phys. Commun. 267 (2021) 108069 [arXiv:2009.11957] [INSPIRE].
S. J. Bidder, N. E. J. Bjerrum-Bohr, D. C. Dunbar and W. B. Perkins, One-loop gluon scattering amplitudes in theories with N < 4 supersymmetries, Phys. Lett. B 612 (2005) 75 [hep-th/0502028] [INSPIRE].
R. Britto, E. Buchbinder, F. Cachazo and B. Feng, One-loop amplitudes of gluons in SQCD, Phys. Rev. D 72 (2005) 065012 [hep-ph/0503132] [INSPIRE].
D. C. Dunbar, W. B. Perkins and E. Warrick, The unitarity method using a canonical basis approach, JHEP 06 (2009) 056 [arXiv:0903.1751] [INSPIRE].
A. Ochirov, All one-loop NMHV gluon amplitudes in N = 1 SYM, JHEP 12 (2013) 080 [arXiv:1311.1491] [INSPIRE].
E. W. Nigel Glover and C. Williams, One-loop gluonic amplitudes from single unitarity cuts, JHEP 12 (2008) 067 [arXiv:0810.2964] [INSPIRE].
R. Britto, Loop amplitudes in gauge theories: modern analytic approaches, J. Phys. A 44 (2011) 454006 [arXiv:1012.4493] [INSPIRE].
R. Runkel, Z. Szőr, J. P. Vesga and S. Weinzierl, Integrands of loop amplitudes within loop-tree duality, Phys. Rev. D 101 (2020) 116014 [arXiv:1906.02218] [INSPIRE].
H. Johansson and A. Ochirov, Pure gravities via color-kinematics duality for fundamental matter, JHEP 11 (2015) 046 [arXiv:1407.4772] [INSPIRE].
J. Nohle, Color-kinematics duality in one-loop four-gluon amplitudes with matter, Phys. Rev. D 90 (2014) 025020 [arXiv:1309.7416] [INSPIRE].
F. Cachazo, Sharpening the leading singularity, arXiv:0803.1988 [INSPIRE].
J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, Elliptic Feynman integrals and pure functions, JHEP 01 (2019) 023 [arXiv:1809.10698] [INSPIRE].
J. M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].
D. Forde, Direct extraction of one-loop integral coefficients, Phys. Rev. D 75 (2007) 125019 [arXiv:0704.1835] [INSPIRE].
R. Britto and B. Feng, Integral coefficients for one-loop amplitudes, JHEP 02 (2008) 095 [arXiv:0711.4284] [INSPIRE].
Z. Bern, E. Herrmann, S. Litsey, J. Stankowicz and J. Trnka, Evidence for a nonplanar amplituhedron, JHEP 06 (2016) 098 [arXiv:1512.08591] [INSPIRE].
J. L. Bourjaily, C. Langer and K. Patatoukos, Locally-finite quantities in SYM, JHEP 04 (2021) 298 [arXiv:2102.02821] [INSPIRE].
J. L. Bourjaily, S. Caron-Huot and J. Trnka, Dual-conformal regularization of infrared loop divergences and the chiral box expansion, JHEP 01 (2015) 001 [arXiv:1303.4734] [INSPIRE].
N. Arkani-Hamed and E. Y. Yuan, One-loop integrals from spherical projections of planes and quadrics, arXiv:1712.09991 [INSPIRE].
E. Herrmann and J. Parra-Martinez, Logarithmic forms and differential equations for Feynman integrals, JHEP 02 (2020) 099 [arXiv:1909.04777] [INSPIRE].
J. L. Bourjaily, E. Gardi, A. J. McLeod and C. Vergu, All-mass n-gon integrals in n dimensions, JHEP 08 (2020) 029 [arXiv:1912.11067] [INSPIRE].
N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo and J. Trnka, Singularity structure of maximally supersymmetric scattering amplitudes, Phys. Rev. Lett. 113 (2014) 261603 [arXiv:1410.0354] [INSPIRE].
Z. Bern, E. Herrmann, S. Litsey, J. Stankowicz and J. Trnka, Logarithmic singularities and maximally supersymmetric amplitudes, JHEP 06 (2015) 202 [arXiv:1412.8584] [INSPIRE].
J. L. Bourjaily, E. Herrmann, C. Langer, A. J. McLeod and J. Trnka, Prescriptive unitarity for non-planar six-particle amplitudes at two loops, JHEP 12 (2019) 073 [arXiv:1909.09131] [INSPIRE].
J. L. Bourjaily, E. Herrmann, C. Langer, A. J. McLeod and J. Trnka, All-multiplicity nonplanar amplitude integrands in maximally supersymmetric Yang-Mills theory at two loops, Phys. Rev. Lett. 124 (2020) 111603 [arXiv:1911.09106] [INSPIRE].
S. Caron-Huot and K. J. Larsen, Uniqueness of two-loop master contours, JHEP 10 (2012) 026 [arXiv:1205.0801] [INSPIRE].
G. Kälin, G. Mogull and A. Ochirov, Two-loop N = 2 SQCD amplitudes with external matter from iterated cuts, JHEP 07 (2019) 120 [arXiv:1811.09604] [INSPIRE].
S. Badger, G. Mogull and T. Peraro, Local integrands for two-loop all-plus Yang-Mills amplitudes, JHEP 08 (2016) 063 [arXiv:1606.02244] [INSPIRE].
Z. Bern and A. G. Morgan, Massive loop amplitudes from unitarity, Nucl. Phys. B 467 (1996) 479 [hep-ph/9511336] [INSPIRE].
N. Arkani-Hamed, F. Cachazo and J. Kaplan, What is the simplest quantum field theory?, JHEP 09 (2010) 016 [arXiv:0808.1446] [INSPIRE].
M. Berg, I. Buchberger and O. Schlotterer, String-motivated one-loop amplitudes in gauge theories with half-maximal supersymmetry, JHEP 07 (2017) 138 [arXiv:1611.03459] [INSPIRE].
Z. Bern, L. J. Dixon and D. A. Kosower, Two-loop g → gg splitting amplitudes in QCD, JHEP 08 (2004) 012 [hep-ph/0404293] [INSPIRE].
R. Britto and E. Mirabella, Single cut integration, JHEP 01 (2011) 135 [arXiv:1011.2344] [INSPIRE].
B. Feng, T. Li and X. Li, Analytic tadpole coefficients of one-loop integrals, JHEP 09 (2021) 081 [arXiv:2107.03744] [INSPIRE].
R. Baumeister, D. Mediger, J. Pečovnik and S. Weinzierl, Vanishing of certain cuts or residues of loop integrals with higher powers of the propagators, Phys. Rev. D 99 (2019) 096023 [arXiv:1903.02286] [INSPIRE].
G. F. Sterman, An introduction to quantum field theory, Cambridge University Press, Cambridge, U.K. (1993).
T. Kinoshita, Mass singularities of Feynman amplitudes, J. Math. Phys. 3 (1962) 650 [INSPIRE].
T. D. Lee and M. Nauenberg, Degenerate systems and mass singularities, Phys. Rev. 133 (1964) B1549 [INSPIRE].
C. L. Basham, L. S. Brown, S. D. Ellis and S. T. Love, Energy correlations in electron-positron annihilation: testing QCD, Phys. Rev. Lett. 41 (1978) 1585 [INSPIRE].
C. L. Basham, L. S. Brown, S. D. Ellis and S. T. Love, Energy correlations in electron-positron annihilation in quantum chromodynamics: asymptotically free perturbation theory, Phys. Rev. D 19 (1979) 2018 [INSPIRE].
A. Strominger, Asymptotic symmetries of Yang-Mills theory, JHEP 07 (2014) 151 [arXiv:1308.0589] [INSPIRE].
A. Strominger, Lectures on the infrared structure of gravity and gauge theory, arXiv:1703.05448 [INSPIRE].
W. T. Giele, E. W. N. Glover and D. A. Kosower, Higher order corrections to jet cross-sections in hadron colliders, Nucl. Phys. B 403 (1993) 633 [hep-ph/9302225] [INSPIRE].
S. Frixione, Z. Kunszt and A. Signer, Three jet cross-sections to next-to-leading order, Nucl. Phys. B 467 (1996) 399 [hep-ph/9512328] [INSPIRE].
S. Catani and M. H. Seymour, A general algorithm for calculating jet cross-sections in NLO QCD, Nucl. Phys. B 485 (1997) 291 [Erratum ibid. 510 (1998) 503] [hep-ph/9605323] [INSPIRE].
A. Gehrmann-De Ridder, T. Gehrmann and E. W. N. Glover, Antenna subtraction at NNLO, JHEP 09 (2005) 056 [hep-ph/0505111] [INSPIRE].
G. Somogyi, Z. Trócsányi and V. Del Duca, A subtraction scheme for computing QCD jet cross sections at NNLO: regularization of doubly-real emissions, JHEP 01 (2007) 070 [hep-ph/0609042] [INSPIRE].
S. Catani, The singular behavior of QCD amplitudes at two loop order, Phys. Lett. B 427 (1998) 161 [hep-ph/9802439] [INSPIRE].
N. Kidonakis, G. Oderda and G. F. Sterman, Evolution of color exchange in QCD hard scattering, Nucl. Phys. B 531 (1998) 365 [hep-ph/9803241] [INSPIRE].
S. M. Aybat, L. J. Dixon and G. F. Sterman, The two-loop soft anomalous dimension matrix and resummation at next-to-next-to leading pole, Phys. Rev. D 74 (2006) 074004 [hep-ph/0607309] [INSPIRE].
L. J. Dixon, L. Magnea and G. F. Sterman, Universal structure of subleading infrared poles in gauge theory amplitudes, JHEP 08 (2008) 022 [arXiv:0805.3515] [INSPIRE].
E. Gardi and L. Magnea, Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes, JHEP 03 (2009) 079 [arXiv:0901.1091] [INSPIRE].
T. Becher and M. Neubert, Infrared singularities of scattering amplitudes in perturbative QCD, Phys. Rev. Lett. 102 (2009) 162001 [Erratum ibid. 111 (2013) 199905] [arXiv:0901.0722] [INSPIRE].
O. Almelid, C. Duhr and E. Gardi, Three-loop corrections to the soft anomalous dimension in multileg scattering, Phys. Rev. Lett. 117 (2016) 172002 [arXiv:1507.00047] [INSPIRE].
G. F. Sterman, Partons, factorization and resummation, in Theoretical Advanced Study Institute in Elementary Particle Physics (TASI′95): QCD and beyond, (1995), p. 327 [hep-ph/9606312] [INSPIRE].
Z. Bern, M. Czakon, L. J. Dixon, D. A. Kosower and V. A. Smirnov, The four-loop planar amplitude and cusp anomalous dimension in maximally supersymmetric Yang-Mills theory, Phys. Rev. D 75 (2007) 085010 [hep-th/0610248] [INSPIRE].
Z. Bern, J. J. M. Carrasco, H. Johansson and D. A. Kosower, Maximally supersymmetric planar Yang-Mills amplitudes at five loops, Phys. Rev. D 76 (2007) 125020 [arXiv:0705.1864] [INSPIRE].
L. J. Dixon, J. M. Drummond and J. M. Henn, Bootstrapping the three-loop hexagon, JHEP 11 (2011) 023 [arXiv:1108.4461] [INSPIRE].
S. Caron-Huot, L. J. Dixon, F. Dulat, M. von Hippel, A. J. McLeod and G. Papathanasiou, Six-gluon amplitudes in planar N = 4 super-Yang-Mills theory at six and seven loops, JHEP 08 (2019) 016 [arXiv:1903.10890] [INSPIRE].
L. J. Dixon and Y.-T. Liu, Lifting heptagon symbols to functions, JHEP 10 (2020) 031 [arXiv:2007.12966] [INSPIRE].
P. Benincasa and D. Gordo, On-shell diagrams and the geometry of planar N < 4 SYM theories, JHEP 11 (2017) 192 [arXiv:1609.01923] [INSPIRE].
N. Arkani-Hamed and J. Trnka, The amplituhedron, JHEP 10 (2014) 030 [arXiv:1312.2007] [INSPIRE].
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Bourjaily, J.L., Herrmann, E., Langer, C. et al. Integrands of less-supersymmetric Yang-Mills at one loop. J. High Energ. Phys. 2022, 126 (2022). https://doi.org/10.1007/JHEP03(2022)126
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DOI: https://doi.org/10.1007/JHEP03(2022)126