Abstract
We present new formulas for n-particle tree-level scattering amplitudes of six-dimensional \( \mathcal{N}=\left(1,1\right) \) super Yang-Mills (SYM) and \( \mathcal{N}=\left(2,2\right) \) supergravity (SUGRA). They are written as integrals over the moduli space of certain rational maps localized on the (n − 3)! solutions of the scattering equations. Due to the properties of spinor-helicity variables in six dimensions, the even-n and odd-n formulas are quite different and have to be treated separately. We first propose a manifestly supersymmetric expression for the even-n amplitudes of \( \mathcal{N}=\left(1,1\right) \) SYM theory and perform various consistency checks. By considering soft-gluon limits of the even-n amplitudes, we deduce the form of the rational maps and the integrand for n odd. The odd-n formulas obtained in this way have a new redundancy that is intertwined with the usual SL(2, ℂ) invariance on the Riemann sphere. We also propose an alternative form of the formulas, analogous to the Witten-RSV formulation, and explore its relationship with the symplectic (or Lagrangian) Grassmannian. Since the amplitudes are formulated in a way that manifests double-copy properties, formulas for the six-dimensional \( \mathcal{N}=\left(2,2\right) \) SUGRA amplitudes follow. These six-dimensional results allow us to deduce new formulas for five-dimensional SYM and SUGRA amplitudes, as well as massive amplitudes of four-dimensional \( \mathcal{N}=4 \) SYM on the Coulomb branch.
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References
E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252 (2004) 189 [hep-th/0312171] [INSPIRE].
R. Roiban, M. Spradlin and A. Volovich, On the tree level S matrix of Yang-Mills theory, Phys. Rev. D 70 (2004) 026009 [hep-th/0403190] [INSPIRE].
F. Cachazo and Y. Geyer, A ‘twistor string’ inspired formula for tree-level scattering amplitudes in N = 8 SUGRA, arXiv:1206.6511 [INSPIRE].
F. Cachazo and D. Skinner, Gravity from rational curves in twistor space, Phys. Rev. Lett. 110 (2013) 161301 [arXiv:1207.0741] [INSPIRE].
F. Cachazo, L. Mason and D. Skinner, Gravity in twistor space and its Grassmannian formulation, SIGMA 10 (2014) 051 [arXiv:1207.4712] [INSPIRE].
C. Cheung and D. O’Connell, Amplitudes and spinor-helicity in six dimensions, JHEP 07 (2009) 075 [arXiv:0902.0981] [INSPIRE].
Y.-T. Huang, Non-chiral S-matrix of N = 4 super Yang-Mills, arXiv:1104.2021 [INSPIRE].
Z. Bern, J.J. Carrasco, T. Dennen, Y.-T. Huang and H. Ita, Generalized unitarity and six-dimensional helicity, Phys. Rev. D 83 (2011) 085022 [arXiv:1010.0494] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering equations and Kawai-Lewellen-Tye orthogonality, Phys. Rev. D 90 (2014) 065001 [arXiv:1306.6575] [INSPIRE].
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. Mason and D. Skinner, Ambitwistor strings and the scattering equations, JHEP 07 (2014) 048 [arXiv:1311.2564] [INSPIRE].
Y. Geyer, A.E. Lipstein and L.J. Mason, Ambitwistor strings in four dimensions, Phys. Rev. Lett. 113 (2014) 081602 [arXiv:1404.6219] [INSPIRE].
E. Casali, Y. Geyer, L. Mason, R. Monteiro and K.A. Roehrig, New ambitwistor string theories, JHEP 11 (2015) 038 [arXiv:1506.08771] [INSPIRE].
Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, Loop integrands for scattering amplitudes from the Riemann sphere, Phys. Rev. Lett. 115 (2015) 121603 [arXiv:1507.00321] [INSPIRE].
Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, One-loop amplitudes on the Riemann sphere, JHEP 03 (2016) 114 [arXiv:1511.06315] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, One-loop corrections from higher dimensional tree amplitudes, JHEP 08 (2016) 008 [arXiv:1512.05001] [INSPIRE].
Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, Two-loop scattering amplitudes from the Riemann sphere, Phys. Rev. D 94 (2016) 125029 [arXiv:1607.08887] [INSPIRE].
Y. Geyer and R. Monteiro, Two-loop scattering amplitudes from ambitwistor strings: from genus two to the nodal Riemann sphere, arXiv:1805.05344 [INSPIRE].
Y. Geyer, Ambitwistor strings: worldsheet approaches to perturbative quantum field theories, Ph.D. thesis, Inst. Math., Oxford U., Oxford, U.K., (2016) [arXiv:1610.04525] [INSPIRE].
M. Heydeman, J.H. Schwarz and C. Wen, M5-brane and D-brane scattering amplitudes, JHEP 12 (2017) 003 [arXiv:1710.02170] [INSPIRE].
J.M. Drummond, Review of AdS/CFT integrability, chapter V.2: dual superconformal symmetry, Lett. Math. Phys. 99 (2012) 481 [arXiv:1012.4002] [INSPIRE].
M. Perry and J.H. Schwarz, Interacting chiral gauge fields in six-dimensions and Born-Infeld theory, Nucl. Phys. B 489 (1997) 47 [hep-th/9611065] [INSPIRE].
M. Aganagic, J. Park, C. Popescu and J.H. Schwarz, World volume action of the M-theory five-brane, Nucl. Phys. B 496 (1997) 191 [hep-th/9701166] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering equations and matrices: from Einstein to Yang-Mills, DBI and NLSM, JHEP 07 (2015) 149 [arXiv:1412.3479] [INSPIRE].
F. Cachazo, P. Cha and S. Mizera, Extensions of theories from soft limits, JHEP 06 (2016) 170 [arXiv:1604.03893] [INSPIRE].
T. Dennen, Y.-T. Huang and W. Siegel, Supertwistor space for 6D maximal super Yang-Mills, JHEP 04 (2010) 127 [arXiv:0910.2688] [INSPIRE].
Y.-T. Huang and A.E. Lipstein, Amplitudes of 3D and 6D maximal superconformal theories in supertwistor space, JHEP 10 (2010) 007 [arXiv:1004.4735] [INSPIRE].
T. Dennen and Y.-T. Huang, Dual conformal properties of six-dimensional maximal super Yang-Mills amplitudes, JHEP 01 (2011) 140 [arXiv:1010.5874] [INSPIRE].
A. Brandhuber, D. Korres, D. Koschade and G. Travaglini, One-loop amplitudes in six-dimensional (1, 1) theories from generalised unitarity, JHEP 02 (2011) 077 [arXiv:1010.1515] [INSPIRE].
B. Czech, Y.-T. Huang and M. Rozali, Chiral three-point interactions in 5 and 6 dimensions, JHEP 10 (2012) 143 [arXiv:1110.2791] [INSPIRE].
J. Plefka, T. Schuster and V. Verschinin, From six to four and more: massless and massive maximal super Yang-Mills amplitudes in 6d and 4d and their hidden symmetries, JHEP 01 (2015) 098 [arXiv:1405.7248] [INSPIRE].
C. Wen, M5-brane and D-brane scattering amplitudes, talk at the QCD meets gravity workshop, Bhaumik Institute, UCLA, Los Angeles, CA, U.S.A., (2017).
S.J. Parke and T.R. Taylor, An amplitude for n gluon scattering, Phys. Rev. Lett. 56 (1986) 2459 [INSPIRE].
V.P. Nair, A current algebra for some gauge theory amplitudes, Phys. Lett. B 214 (1988) 215 [INSPIRE].
S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140 (1965) B516 [INSPIRE].
N. Arkani-Hamed, T.-C. Huang and Y.-T. Huang, Scattering amplitudes for all masses and spins, arXiv:1709.04891 [INSPIRE].
S. He, Z. Liu and J.-B. Wu, Scattering equations, twistor-string formulas and double-soft limits in four dimensions, JHEP 07 (2016) 060 [arXiv:1604.02834] [INSPIRE].
Y. Zhang, CHY formulae in 4d, JHEP 07 (2017) 069 [arXiv:1610.05205] [INSPIRE].
I.M. Gelfand, M.M. Kapranov and A.V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Modern Birkhäuser Classics, Birkhäuser, Boston, MA, U.S.A., (1994).
F. Cachazo, Resultants and gravity amplitudes, arXiv:1301.3970 [INSPIRE].
F. Cachazo and G. Zhang, Minimal basis in four dimensions and scalar blocks, arXiv:1601.06305 [INSPIRE].
M. Spradlin and A. Volovich, From twistor string theory to recursion relations, Phys. Rev. D 80 (2009) 085022 [arXiv:0909.0229] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering in three dimensions from rational maps, JHEP 10 (2013) 141 [arXiv:1306.2962] [INSPIRE].
E. Witten, New ‘gauge’ theories in six-dimensions, Adv. Theor. Math. Phys. 2 (1998) 61 [JHEP 01 (1998) 001] [hep-th/9710065] [INSPIRE].
N. Arkani-Hamed, J. Bourjaily, F. Cachazo, A. 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, J. Bourjaily, F. Cachazo and J. Trnka, Unification of residues and Grassmannian dualities, JHEP 01 (2011) 049 [arXiv:0912.4912] [INSPIRE].
E. Witten, Parity invariance for strings in twistor space, Adv. Theor. Math. Phys. 8 (2004) 779 [hep-th/0403199] [INSPIRE].
J.L. Bourjaily, J. Trnka, A. Volovich and C. Wen, The Grassmannian and the twistor string: connecting all trees in N = 4 SYM, JHEP 01 (2011) 038 [arXiv:1006.1899] [INSPIRE].
N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, A duality for the S matrix, JHEP 03 (2010) 020 [arXiv:0907.5418] [INSPIRE].
T. Banks and N. Seiberg, Symmetries and strings in field theory and gravity, Phys. Rev. D 83 (2011) 084019 [arXiv:1011.5120] [INSPIRE].
H. Kawai, D.C. Lewellen and S.-H. Henry Tye, A relation between tree amplitudes of closed and open strings, Nucl. Phys. B 269 (1986) 1 [INSPIRE].
D.V. Volkov and V.P. Akulov, Is the neutrino a Goldstone particle?, Phys. Lett. B 46 (1973) 109 [INSPIRE].
R. Kallosh, Volkov-Akulov theory and D-branes, hep-th/9705118 [INSPIRE].
J.J.M. Carrasco, C.R. Mafra and O. Schlotterer, Semi-Abelian Z-theory: NLSM+ \( \phi \) 3 from the open string, JHEP 08 (2017) 135 [arXiv:1612.06446] [INSPIRE].
C. Cheung, C.-H. Shen and C. Wen, Unifying relations for scattering amplitudes, JHEP 02 (2018) 095 [arXiv:1705.03025] [INSPIRE].
C. Cheung, G.N. Remmen, C.-H. Shen and C. Wen, Pions as gluons in higher dimensions, JHEP 04 (2018) 129 [arXiv:1709.04932] [INSPIRE].
C. Cheung and C.-H. Shen, Symmetry for flavor-kinematics duality from an action, Phys. Rev. Lett. 118 (2017) 121601 [arXiv:1612.00868] [INSPIRE].
I. Low and Z. Yin, Ward identity and scattering amplitudes for nonlinear σ-models, Phys. Rev. Lett. 120 (2018) 061601 [arXiv:1709.08639] [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].
Y. Wang and X. Yin, Supervertices and non-renormalization conditions in maximal supergravity theories, arXiv:1505.05861 [INSPIRE].
L.F. Alday, J.M. Henn, J. Plefka and T. Schuster, Scattering into the fifth dimension of N = 4 super Yang-Mills, JHEP 01 (2010) 077[arXiv:0908.0684] [INSPIRE].
J.M. Henn, S.G. Naculich, H.J. Schnitzer and M. Spradlin, Higgs-regularized three-loop four-gluon amplitude in N = 4 SYM: exponentiation and Regge limits, JHEP 04 (2010) 038 [arXiv:1001.1358] [INSPIRE].
J.M. Henn, S.G. Naculich, H.J. Schnitzer and M. Spradlin, More loops and legs in Higgs-regulated N = 4 SYM amplitudes, JHEP 08 (2010) 002 [arXiv:1004.5381] [INSPIRE].
W.-M. Chen, Y.-T. Huang and C. Wen, Exact coefficients for higher dimensional operators with sixteen supersymmetries, JHEP 09 (2015) 098 [arXiv:1505.07093] [INSPIRE].
M. Bianchi, A.L. Guerrieri, Y.-T. Huang, C.-J. Lee and C. Wen, Exploring soft constraints on effective actions, JHEP 10 (2016) 036 [arXiv:1605.08697] [INSPIRE].
N. Craig, H. Elvang, M. Kiermaier and T. Slatyer, Massive amplitudes on the Coulomb branch of N = 4 SYM, JHEP 12 (2011) 097 [arXiv:1104.2050] [INSPIRE].
S. He and E.Y. Yuan, One-loop scattering equations and amplitudes from forward limit, Phys. Rev. D 92 (2015) 105004 [arXiv:1508.06027] [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].
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].
N. Arkani-Hamed, F. Cachazo and C. Cheung, The Grassmannian origin of dual superconformal invariance, JHEP 03 (2010) 036 [arXiv:0909.0483] [INSPIRE].
S. Caron-Huot and D. O’Connell, Spinor helicity and dual conformal symmetry in ten dimensions, JHEP 08 (2011) 014 [arXiv:1010.5487] [INSPIRE].
I. Bandos, Britto-Cachazo-Feng-Witten-type recurrent relations for tree amplitudes of D = 11 supergravity, Phys. Rev. Lett. 118 (2017) 031601 [arXiv:1605.00036] [INSPIRE].
I. Bandos, Spinor frame formalism for amplitudes and constrained superamplitudes of 10D SYM and 11D supergravity, arXiv:1711.00914 [INSPIRE].
C.M. Hull, Strongly coupled gravity and duality, Nucl. Phys. B 583 (2000) 237 [hep-th/0004195] [INSPIRE].
M. Henneaux, V. Lekeu and A. Leonard, The action of the (free) (4, 0)-theory, JHEP 01 (2018) 114 [Erratum ibid. 05 (2018) 105] [arXiv:1711.07448] [INSPIRE].
M. Henneaux, V. Lekeu, J. Matulich and S. Prohazka, The action of the (free) N = (3, 1) theory in six spacetime dimensions, arXiv:1804.10125 [INSPIRE].
M. Chiodaroli, M. Günaydin and R. Roiban, Superconformal symmetry and maximal supergravity in various dimensions, JHEP 03 (2012) 093 [arXiv:1108.3085] [INSPIRE].
L. Borsten, D = 6, N = (2, 0) and N = (4, 0) theories, Phys. Rev. D 97 (2018) 066014 [arXiv:1708.02573] [INSPIRE].
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Cachazo, F., Guevara, A., Heydeman, M. et al. The S matrix of 6D super Yang-Mills and maximal supergravity from rational maps. J. High Energ. Phys. 2018, 125 (2018). https://doi.org/10.1007/JHEP09(2018)125
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DOI: https://doi.org/10.1007/JHEP09(2018)125