Abstract
We use supertwistor space to construct scattering amplitudes of maximal superconformal theories in three and six dimensions. In both cases, the constraints of super-conformal invariance and rationality imply that the three-point amplitude vanishes on-shell, which constrains the four-point amplitude to have vanishing residues in all channels. In three dimensions, we find a unique solution for the four-point amplitude and demonstrate that it agrees with the component result in the BLG theory. This suggests that BLG is the unique three-dimensional theory with classical OSp(8∣4) symmetry that admits a Lagrangian description. We also show that one can derive the four-point amplitude of the ABJM theory from our \( \mathcal{N} = 8 \) result by reducing the supersymmetry, which implies that the tree-level Yangian symmetry recently found in ABJM is also present in BLG. In six dimensions, we find that the consistency conditions imply that all tree-level amplitudes vanish. This leads us to conjecture that an interacting six-dimensional theory with classical OSp(8∣4) symmetry does not have a Lagrangian description, local or nonlocal, unless the (2, 0) tensor multiplets are supplemented by additional degrees of freedom.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Z. Bern, L.J. Dixon and V.A. Smirnov, Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond, Phys. Rev. D 72 (2005) 085001 [hep-th/0505205] [SPIRES].
L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [SPIRES].
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].
N. Berkovits and J. Maldacena, Fermionic T-duality, dual superconformal symmetry and the amplitude/Wilson loop connection, JHEP 09 (2008) 062 [arXiv:0807.3196] [SPIRES].
N. Beisert, R. Ricci, A.A. Tseytlin and M. Wolf, Dull superconformal symmetry from AdS 5 × S 5 superstring integrability, Phys. Rev. D 78 (2008) 126004 [arXiv:0807.3228] [SPIRES].
E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252 (2004) 189 [hep-th/0312171] [SPIRES].
F. Cachazo, P. Svrček and E. Witten, MHV vertices and tree amplitudes in gauge theory, JHEP 09 (2004) 006 [hep-th/0403047] [SPIRES].
H. Elvang, D.Z. Freedman and M. Kiermaier, Recursion relations, generating functions and unitarity sums in N =4 SYM theory, JHEP 04 (2009) 009 [arXiv:0808.1720] [SPIRES].
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].
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].
Z. Bern, J.J.M. Carrasco and H. Johansson, New relations for gauge-theory amplitudes, Phys. Rev. D 78 (2008) 085011 [arXiv:0805.3993] [SPIRES].
L. Mason and D. Skinner, Scattering amplitudes and BCFW recursion in twistor space, arXiv:0903.2083 [SPIRES].
N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, A duality for the S matrix, JHEP 03 (2010) 020 [arXiv:0907.5418] [SPIRES].
T. Kugo and P.K. Townsend, Supersymmetry and the division algebras, Nucl. Phys. B 221 (1983) 357 [SPIRES].
R. Penrose and W. Rindler, Spinors and space-time. Vol. 2: Spinor and twistor methods in space time geometry, Cambridge University Press, Cambridge U.K. (1986) pg. 501 [SPIRES].
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [SPIRES].
A. Agarwal, N. Beisert and T. McLoughlin, Scattering in mass-deformed N>=4 Chern-Simons models, JHEP 06 (2009) 045 [arXiv:0812.3367] [SPIRES].
C. Cheung and D. O’Connell, Amplitudes and spinor-helicity in six dimensions, JHEP 07 (2009) 075 [arXiv:0902.0981] [SPIRES].
T. Dennen, Y.-t. Huang and W. Siegel, Supertwistor space for 6D maximal super Yang-Mills, JHEP 04 (2010) 127 [arXiv:0910.2688] [SPIRES].
J. Bagger and N. Lambert, Gauge symmetry and supersymmetry of multiple M2-branes, Phys. Rev. D 77 (2008) 065008 [arXiv:0711.0955] [SPIRES].
J. Bagger and N. Lambert, Comments on multiple M2-branes, JHEP 02 (2008) 105 [arXiv:0712.3738] [SPIRES].
A. Gustavsson, Algebraic structures on parallel M2-branes, Nucl. Phys. B 811 (2009) 66 [arXiv:0709.1260] [SPIRES].
O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [SPIRES].
M.A. Bandres, A.E. Lipstein and J.H. Schwarz, Studies of the ABJM theory in a formulation with manifest SU(4) R-symmetry, JHEP 09 (2008) 027 [arXiv:0807.0880] [SPIRES].
D. Gaiotto, N=2 dualities, arXiv:0904.2715 [SPIRES].
D. Gaiotto and J. Maldacena, The gravity duals of N =2 superconformal field theories, arXiv:0904.4466 [SPIRES].
N. Seiberg, Non-trivial fixed pointd of the renormalization group in six dimensions, Phys. Lett. B 390 (1997) 169 [hep-th/9609161] [SPIRES].
C.G. Callan Jr., J.A. Harvey and A. Strominger, Worldbrane actions for string solitons, Nucl. Phys. B 367 (1991) 60 [SPIRES].
K. Becker and M. Becker, Boundaries in M-theory, Nucl. Phys. B 472 (1996) 221 [hep-th/9602071] [SPIRES].
D.M. Kaplan and J. Michelson, Zero modes for the D = 11 membrane and five-brane, Phys. Rev. D 53 (1996) 3474 [hep-th/9510053] [SPIRES].
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] [SPIRES].
P. Pasti, D.P. Sorokin and M. Tonin, On Lorentz invariant actions for chiral p-forms, Phys. Rev. D 55 (1997) 6292 [hep-th/9611100] [SPIRES].
A. Gustavsson and S.-J. Rey, Enhanced N =8 supersymmetry of ABJM theory on R(8) and R(8)/Z(2), arXiv:0906.3568 [SPIRES].
H. Samtleben and R. Wimmer, N=8 superspace constraints for three-dimensional gauge theories, JHEP 02 (2010) 070 [arXiv:0912.1358] [SPIRES].
T. Bargheer, N. Beisert, W. Galleas, F. Loebbert and T. McLoughlin, Exacting N =4 superconformal symmetry, JHEP 11 (2009) 056 [arXiv:0905.3738] [SPIRES].
G.P. Korchemsky and E. Sokatchev, Symmetries and analytic properties of scattering amplitudes in N =4 SYM theory, Nucl. Phys. B 832 (2010) 1 [arXiv:0906.1737] [SPIRES].
G.P. Korchemsky and E. Sokatchev, Superconformal invariants for scattering amplitudes in N =4 SYM theory, Nucl. Phys. B 839 (2010) 377 [arXiv:1002.4625] [SPIRES].
T. Bargheer, F. Loebbert and C. Meneghelli, Symmetries of tree-level scattering amplitudes in N =6 superconformal Chern-Simons theory, Phys. Rev. D 82 (2010) 045016 [arXiv:1003.6120] [SPIRES].
A.S. Galperin, E.A. Ivanov and V.I. Ogievetsky, Grassmann analyticity and extended supersymmetries, JETP Lett. 33 (1981) 168 [Pisma Zh. Eksp. Teor. Fiz. 33 (1981) 176] [SPIRES].
A. Galperin, E. Ivanov, S. Kalitsyn, V. Ogievetsky and E. Sokatchev, Unconstrained N =2 matter, Yang-Mills and supergravity theories in harmonic superspace, Class. Quant. Grav. 1 (1984) 469 [SPIRES].
S. Ferrara and E. Sokatchev, Conformal superfields and BPS states in AdS(4/7) geometries, Int. J. Mod. Phys. B 14 (2000) 2315 [hep-th/0007058] [SPIRES].
Z. Bern and D.A. Kosower, Color decomposition of one loop amplitudes in gauge theories, Nucl. Phys. B 362 (1991) 389 [SPIRES].
A. Gustavsson, One-loop corrections to Bagger-Lambert theory, Nucl. Phys. B 807 (2009) 315 [arXiv:0805.4443] [SPIRES].
Z. Bern, J.J.M. Carrasco, H. Ita, H. Johansson and R. Roiban, On the structure of supersymmetric sums in multi-loop unitarity cuts, Phys. Rev. D 80 (2009) 065029 [arXiv:0903.5348] [SPIRES].
Z. Bern, J.J.M. Carrasco, T. Dennen and Y.t. Huang, to be published.
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] [SPIRES].
M. Benna, I. Klebanov, T. Klose and M. Smedback, Superconformal Chern-Simons theories and AdS 4 /CFT 3 correspondence, JHEP 09 (2008) 072 [arXiv:0806.1519] [SPIRES].
R. Boels, Covariant representation theory of the Poincaré algebra and some of its extensions, JHEP 01 (2010) 010 [arXiv:0908.0738] [SPIRES].
T. Chern, Superconformal field theory in six dimensions and supertwistor, arXiv:0906.0657 [SPIRES].
M. Hatsuda, Y.-t. Huang and W. Siegel, First-quantized N =4 Yang-Mills, JHEP 04 (2009) 058 [arXiv:0812.4569] [SPIRES].
S. Ferrara and E. Sokatchev, Representations of (1,0) and (2,0) superconformal algebras in six dimensions: Massless and short superfields, Lett. Math. Phys. 51 (2000) 55 [hep-th/0001178] [SPIRES].
P. Benincasa and F. Cachazo, Consistency conditions on the S-matrix of massless particles, arXiv:0705.4305 [SPIRES].
A. Strominger, Open p-branes, Phys. Lett. B 383 (1996) 44 [hep-th/9512059] [SPIRES].
E. Witten, Some comments on string dynamics, hep-th/9507121 [SPIRES].
R. Dijkgraaf, E.P. Verlinde and H.L. Verlinde, BPS spectrum of the five-brane and black hole entropy, Nucl. Phys. B 486 (1997) 77 [hep-th/9603126] [SPIRES].
J.H. Schwarz, Self-dual superstring in six dimensions, hep-th/9604171 [SPIRES].
O.J. Ganor, Six-dimensional tensionless strings in the large-N limit, Nucl. Phys. B 489 (1997) 95 [hep-th/9605201] [SPIRES].
G. Dall’Agata, K. Lechner and M. Tonin, Large covariant actions for N =1, D =6 supergravity theories with chiral bosons, Nucl. Phys. B 512 (1998) 179 [hep-th/9710127] [SPIRES].
O.J. Ganor and A. Hanany, Small E 8 instantons and tensionless non-critical strings, Nucl. Phys. B 474 (1996) 122 [hep-th/9602120] [SPIRES].
N. Seiberg and E. Witten, Comments on string dynamics in six dimensions, Nucl. Phys. B 471 (1996) 121 [hep-th/9603003] [SPIRES].
E. Witten, Phase transitions in M-theory And F-theory, Nucl. Phys. B 471 (1996) 195 [hep-th/9603150] [SPIRES].
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].
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].
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].
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] [SPIRES].
I.A. Bandos et al., Covariant action for the super-five-brane of M-theory, Phys. Rev. Lett. 78 (1997) 4332 [hep-th/9701149] [SPIRES].
P. Claus, R. Kallosh and A. Van Proeyen, M 5-brane and superconformal (0,2) tensor multiplet in 6 dimensions, Nucl. Phys. B 518 (1998) 117 [hep-th/9711161] [SPIRES].
P.-M. Ho, Y. Imamura, Y. Matsuo and S. Shiba, M5-brane in three-form flux and multiple M2-branes, JHEP 08 (2008) 014 [arXiv:0805.2898] [SPIRES].
M.A. Bandres, A.E. Lipstein and J.H. Schwarz, N = 8 superconformal Chern-Simons theories, JHEP 05 (2008) 025 [arXiv:0803.3242] [SPIRES].
J.P. Gauntlett and J.B. Gutowski, Constraining maximally supersymmetric membrane actions, JHEP 06 (2008) 053 [arXiv:0804.3078] [SPIRES].
J. Bagger and N. Lambert, Three-algebras and N =6 Chern-Simons gauge theories, Phys. Rev. D 79 (2009) 025002 [arXiv:0807.0163] [SPIRES].
M. Van Raamsdonk, Comments on the Bagger-Lambert theory and multiple M2-branes, JHEP 05 (2008) 105 [arXiv:0803.3803] [SPIRES].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1004.4735
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Huang, Yt., Lipstein, A.E. Amplitudes of 3D and 6D maximal superconformal theories in supertwistor space. J. High Energ. Phys. 2010, 7 (2010). https://doi.org/10.1007/JHEP10(2010)007
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2010)007