The Algebraic Structure of the KLT Relations for Gauge and Gravity Tree Amplitudes

Hadleigh Frost

Symmetry, integrability and geometry: methods and applications, Tome 17 (2021)

Résumé

We study the Kawai–Lewellen–Tye (KLT) relations for quantum field theory by reformulating it as an isomorphism between two Lie algebras. We also show how explicit formulas for KLT relations arise when studying rational functions on ${\mathcal M}_{0,n}$, and prove identities that allow for arbitrary rational functions to be expanded in any given basis. Via the Cachazo–He–Yuan formulas for, these identities also lead to new formulas for gauge and gravity tree amplitudes, including formulas for so-called Bern–Carrasco–Johansson numerators, in the case of non-linear sigma model and maximal-helicity-violating Yang–Mills amplitudes.

Détail

Keywords: perturbative gauge theory, double copy, binary trees, Lie polynomials.
Mots-clés : Lie coalgebras

Bibliographie

[1] Adamo T., Casali E., Roehrig K. A., Skinner D., “On tree amplitudes of supersymmetric Einstein–Yang–Mills theory”, J. High Energy Phys., 2015:12 (2015), 177, 14 pp., arXiv: 1507.02207 | DOI

[2] Bedford J., Brandhuber A., Spence B., Travaglini G., “A recursion relation for gravity amplitudes”, Nuclear Phys. B, 721 (2005), 98–110, arXiv: hep-th/0502146 | DOI

[3] Berends F. A., Giele W. T., “Recursive calculations for processes with $n$ gluons”, Nuclear Phys. B, 306 (1988), 759–808 | DOI

[4] Bern Z., Carrasco J. J.M., Johansson H., “New relations for gauge-theory amplitudes”, Phys. Rev. D, 78 (2008), 085011, 19 pp., arXiv: 0805.3993 | DOI

[5] Bjerrum-Bohr N. E.J., Bourjaily J. L., Damgaard P. H., Feng B., “Manifesting color-kinematics duality in the scattering equation formalism”, J. High Energy Phys., 2016:9 (2016), 094, 17 pp., arXiv: 1608.00006 | DOI

[6] Bjerrum-Bohr N. E.J., Damgaard P. H., Søndergaard T., Vanhove P., “The momentum kernel of gauge and gravity theories”, J. High Energy Phys., 2011:1 (2011), 001, 18 pp., arXiv: 1010.3933 | DOI

[7] Bjerrum-Bohr N. E.J., Damgaard P. H., Vanhove P., “Minimal basis for gauge theory amplitudes”, Phys. Rev. Lett., 103 (2009), 161602, 4 pp., arXiv: 0907.1425 | DOI

[8] Bridges E., Mafra C. R., “Algorithmic construction of SYM multiparticle superfields in the BCJ gauge”, J. High Energy Phys., 2019:10 (2019), 022, 35 pp., arXiv: 1906.12252 | DOI

[9] Brion M., Vergne M., “Arrangement of hyperplanes. I Rational functions and Jeffrey–Kirwan residue”, Ann. Sci. École Norm. Sup. (4), 32 (1999), 715–741, arXiv: math.DG/9903178 | DOI

[10] Brown F., Dupont C., “Single-valued integration and superstring amplitudes in genus zero”, Comm. Math. Phys., 382 (2021), 815–874, arXiv: 1910.01107 | DOI

[11] Cachazo F., Fundamental BCJ relation in ${\mathcal N}=4$ SYM from the connected formulation, arXiv: 1206.5970

[12] Cachazo F., He S., Yuan E. Y., “Scattering of massless Particles in arbitrary dimension”, Phys. Rev. Lett., 113 (2014), 171601, 4 pp., arXiv: 1307.2199 | DOI

[13] Cachazo F., He S., Yuan E. Y., “Scattering equations and matrices: from Einstein to Yang–Mills, DBI and NLSM”, J. High Energy Phys., 2015:7 (2015), 149, 43 pp., arXiv: 1412.3479 | DOI

[14] Cachazo F., Mason L., Skinner D., “Gravity in twistor space and its Grassmannian formulation”, SIGMA, 10 (2014), 051, 28 pp., arXiv: 1207.4712 | DOI

[15] Cachazo F., Skinner D., “Gravity from rational curves in twistor space”, Phys. Rev. Lett., 110 (2013), 161301, 4 pp., arXiv: 1207.0741 | DOI

[16] Carrasco J. J.M., Mafra C. R., Schlotterer O., “Abelian $Z$-theory: NLSM amplitudes and $\alpha'$-corrections from the open string”, J. High Energy Phys., 2017:6 (2017), 093, 31 pp., arXiv: 1608.02569 | DOI

[17] Chaiken S., Kleitman D. J., “Matrix tree theorems”, J. Combinatorial Theory Ser. A, 24 (1978), 377–381 | DOI

[18] Chen G., Johansson H., Teng F., Wang T., “Next-to-MHV Yang–Mills kinematic algebra”, J. High Energy Phys., 2021:10 (2021), 042, 54 pp., arXiv: 2104.12726 | DOI

[19] Cheung C., Shen C.-H., “Symmetry for flavor-kinematics duality from an action”, Phys. Rev. Lett., 118 (2017), 121601, 5 pp., arXiv: 1612.00868 | DOI

[20] Dolan L., Goddard P., “Proof of the formula of Cachazo, He and Yuan for Yang–Mills tree amplitudes in arbitrary dimension”, J. High Energy Phys., 2014:5 (2014), 010, 24 pp., arXiv: 1311.5200 | DOI

[21] Du Y.-J., Teng F., “BCJ numerators from reduced Pfaffian”, J. High Energy Phys., 2017:4 (2017), 033, 28 pp., arXiv: 1703.05717 | DOI

[22] Elvang H., Freedman D. Z., “Note on graviton MHV amplitudes”, J. High Energy Phys., 2008:5 (2008), 096, 11 pp., arXiv: 0710.1270 | DOI

[23] Frost H., Universal aspects of perturbative gauge theory amplitudes, Ph.D. Thesis, University of Oxford, 2020 https://ora.ox.ac.uk/objects/uuid:2b7fc6f9-ee97-40b0-9c14-1f29dfb916fc

[24] Frost H., Mafra C. R., Mason L., A Lie bracket for the momentum kernel, arXiv: 2012.00519

[25] Geyer Y., Ambitwistor strings: worldsheet approaches to perturbative quantum field theories, Ph.D. Thesis, University of Oxford, 2016 https://ora.ox.ac.uk/objects/uuid:ff543496-58bd-4818-ae0c-cf31eb349c90

[26] Geyer Y., Lipstein A. E., Mason L., “Ambitwistor strings in four dimensions”, Phys. Rev. Lett., 113 (2014), 081602, 5 pp., arXiv: 1404.6219 | DOI

[27] He S., Hou L., Tian J., Zhang Y., “Kinematic numerators from the worldsheet: cubic trees from labelled trees”, J. High Energy Phys., 2021:8 (2021), 118, 24 pp., arXiv: 2103.15810 | DOI

[28] Hodges A., A simple formula for gravitational MHV amplitudes, arXiv: 1204.1930

[29] Kapranov M., The geometry of scattering amplitudes, Talk at Banff Workshop “The Geometry of Scattering Amplitudes”, 2012

[30] Kawai H., Lewellen D. C., Tye S.-H.H., “A relation between tree amplitudes of closed and open strings”, Nuclear Phys. B, 269 (1986), 1–23 | DOI

[31] Kleiss R., Kuijf H., “Multigluon cross sections and 5-jet production at hadron colliders”, Nuclear Phys. B, 312 (1989), 616–644 | DOI

[32] Mafra C. R., Private correspondence, 2020

[33] Mafra C. R., Schlotterer O., “Multiparticle SYM equations of motion and pure spinor BRST blocks”, J. High Energy Phys., 2014:7 (2014), 153, 40 pp., arXiv: 1404.4986 | DOI

[34] Mafra C. R., Schlotterer O., “Berends–Giele recursions and the BCJ duality in superspace and components”, J. High Energy Phys., 2016:3 (2016), 097, 20 pp., arXiv: 1510.08846 | DOI

[35] Mafra C. R., Schlotterer O., “One-loop open-string integrals from differential equations: all-order $\alpha'$-expansions at $n$ points”, J. High Energy Phys., 2020:3 (2020), 007, 79 pp., arXiv: 1908.10830 | DOI

[36] Mason L., Skinner D., “Gravity, twistors and the MHV formalism”, Comm. Math. Phys., 294 (2010), 827–862, arXiv: 0808.3907 | DOI

[37] Mizera S., “Combinatorics and topology of Kawai–Lewellen–Tye relations”, J. High Energy Phys., 2017:8 (2017), 097, 54 pp., arXiv: 1706.08527 | DOI

[38] Mizera S., “Scattering amplitudes from intersection theory”, Phys. Rev. Lett., 120 (2018), 141602, 6 pp., arXiv: 1711.00469 | DOI

[39] Mizera S., Aspects of scattering amplitudes and moduli space localization, Springer Theses, Springer, Cham, 2020, arXiv: 1906.02099 | DOI

[40] Monteiro R., O'Connell D., “The kinematic algebras from the scattering equations”, J. High Energy Phys., 2014:3 (2014), 110, 28 pp., arXiv: 1311.1151 | DOI

[41] Reutenauer C., Free Lie algebras, London Mathematical Society Monographs. New Series, 7, The Clarendon Press, Oxford University Press, New York, 1993

[42] Roiban R., Spradlin M., Volovich A., “Tree-level $S$ matrix of Yang–Mills theory”, Phys. Rev. D, 70 (2004), 026009, 10 pp., arXiv: hep-th/0403190 | DOI

[43] Schechtman V. V., Varchenko A. N., “Arrangements of hyperplanes and Lie algebra homology”, Invent. Math., 106 (1991), 139–194 | DOI

[44] Schocker M., “Lie elements and Knuth relations”, Canad. J. Math., 56 (2004), 871–882, arXiv: math.RA/0209327 | DOI

[45] Stieberger S., Open closed vs. pure open string disk amplitudes, arXiv: 0907.2211

[46] 't Hooft G., “A planar diagram theory for strong interactions”, Nuclear Phys. B, 72 (1974), 461–473 | DOI