Abstract
We identify a diffeomorphism Lie algebra in the self-dual sector of Yang-Mills theory, and show that it determines the kinematic numerators of tree-level MHV amplitudes in the full theory. These amplitudes can be computed off-shell from Feynman diagrams with only cubic vertices, which are dressed with the structure constants of both the Yang-Mills colour algebra and the diffeomorphism algebra. Therefore, the latter algebra is the dual of the colour algebra, in the sense suggested by the work of Bern, Carrasco and Johansson. We further study perturbative gravity, both in the self-dual and in the MHV sectors, finding that the kinematic numerators of the theory are the BCJ squares of the Yang-Mills numerators.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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 [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].
N.E.J. Bjerrum-Bohr, P.H. Damgaard and P. Vanhove, Minimal basis for gauge theory amplitudes, Phys. Rev. Lett. 103 (2009) 161602 [arXiv:0907.1425] [SPIRES].
S. Stieberger, Open & closed vs. pure open string disk amplitudes, arXiv:0907.2211 [SPIRES].
B. Feng, R. Huang and Y. Jia, Gauge amplitude identities by on-shell recursion relation in S-matrix program, Phys. Lett. B 695 (2011) 350 [arXiv:1004.3417] [SPIRES].
Z. Bern, T. Dennen, Y.-t. Huang and M. Kiermaier, Gravity as the square of gauge theory, Phys. Rev. D 82 (2010) 065003 [arXiv:1004.0693] [SPIRES].
M. Kiermaier, Gravity as the square of gauge theory, Prog. Theor. Phys. Suppl. 188 (2011) 177 [SPIRES].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, T. Sondergaard and P. Vanhove, The momentum kernel of gauge and gravity theories, JHEP 01 (2011) 001 [arXiv:1010.3933] [SPIRES].
C.R. Mafra, O. Schlotterer and S. Stieberger, Explicit BCJ numerators from pure spinors, arXiv:1104.5224 [SPIRES].
Z. Bern, J.J.M. Carrasco and H. Johansson, Perturbative quantum gravity as a double copy of gauge theory, Phys. Rev. Lett. 105 (2010) 061602 [arXiv:1004.0476] [SPIRES].
T. Sondergaard, New relations for gauge-theory amplitudes with matter, Nucl. Phys. B 821 (2009) 417 [arXiv:0903.5453] [SPIRES].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, T. Sondergaard and P. Vanhove, Monodromy and Jacobi-like relations for color-ordered amplitudes, JHEP 06 (2010) 003 [arXiv:1003.2403] [SPIRES].
S.H. Henry Tye and Y. Zhang, Dual identities inside the gluon and the graviton scattering amplitudes, JHEP 06 (2010) 071 [arXiv:1003.1732] [SPIRES].
Y. Jia, R. Huang and C.-Y. Liu, U(1)-decoupling, KK and BCJ relations in \( \mathcal{N} = 4 \) SYM, Phys. Rev. D 82 (2010) 065001 [arXiv:1005.1821] [SPIRES].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, B. Feng and T. Sondergaard, Gravity and Yang-Mills amplitude relations, Phys. Rev. D 82 (2010) 107702 [arXiv:1005.4367] [SPIRES].
H. Tye and Y. Zhang, Comment on the identities of the gluon tree amplitudes, arXiv:1007.0597 [SPIRES].
D. Vaman and Y.-P. Yao, Constraints and generalized gauge transformations on tree-level gluon and graviton amplitudes, JHEP 11 (2010) 028 [arXiv:1007.3475] [SPIRES].
Y.-X. Chen, Y.-J. Du and B. Feng, On tree amplitudes with gluons coupled to gravitons, JHEP 01 (2011) 081 [arXiv:1011.1953] [SPIRES].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, B. Feng and T. Sondergaard, Unusual identities for QCD at tree-level, J. Phys. Conf. Ser. 287 (2011) 012030 [arXiv:1101.5555] [SPIRES].
Z. Bern and T. Dennen, A color dual form for gauge-theory amplitudes, arXiv:1103.0312 [SPIRES].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, H. Johansson and T. Sondergaard, Monodromy–like relations for finite loop amplitudes, JHEP 05 (2011) 039 [arXiv:1103.6190] [SPIRES].
O. Hohm, On factorizations in perturbative quantum gravity, JHEP 04 (2011) 103 [arXiv:1103.0032] [SPIRES].
D.-p. Zhu, Zeros in scattering amplitudes and the structure of nonabelian gauge theories, Phys. Rev. D 22 (1980) 2266 [SPIRES].
C.J. Goebel, F. Halzen and J.P. Leveille, Angular zeros of Brown, Mikaelian, Sahdev and Samuel and the factorization of tree amplitudes in gauge theories, Phys. Rev. D 23 (1981) 2682 [SPIRES].
G. Chalmers and W. Siegel, Simplifying algebra in Feynman graphs. II: Spinor helicity from the spacecone, Phys. Rev. D 59 (1999) 045013 [hep-ph/9801220] [SPIRES].
D.G. Boulware and L.S. Brown, Tree graphs and classical fields, Phys. Rev. 172 (1968) 1628 [SPIRES].
F.A. Berends and W.T. Giele, Recursive calculations for processes with n gluons, Nucl. Phys. B 306 (1988) 759 [SPIRES].
L.S. Brown, Summing tree graphs at threshold, Phys. Rev. D 46 (1992) 4125 [hep-ph/9209203] [SPIRES].
W.A. Bardeen, Self-dual Yang-Mills theory, integrability and multi-parton amplitudes, Yukawa International Seminar 1995: from the standard model to grand unified theories, Kyoto Japan, 21–25 Aug 1995, FERMILAB-CONF-95-379-T [SPIRES].
D. Cangemi, Self-dual Yang-Mills theory and one-loop maximally helicity violating multi-gluon amplitudes, Nucl. Phys. B 484 (1997) 521 [hep-th/9605208] [SPIRES].
G. Chalmers and W. Siegel, The self-dual sector of QCD amplitudes, Phys. Rev. D 54 (1996) 7628 [hep-th/9606061] [SPIRES].
L.J. Mason and D. Skinner, Gravity, twistors and the MHV formalism, Commun. Math. Phys. 294 (2010) 827 [arXiv:0808.3907] [SPIRES].
J.F. Plebanski, Some solutions of complex Einstein equations, J. Math. Phys. 16 (1975) 2395 [SPIRES].
J. Hoppe, Quantum theory of a massless relativistic surface and a two-dimensional bound state problem, Ph.D. Thesis, M.I.T., Boston U.S.A. (1982) [http://dspace.mit.edu/handle/1721.1/15717].
S. Ananth, L. Brink, R. Heise and H.G. Svendsen, The N = 8 supergravity Hamiltonian as a quadratic form, Nucl. Phys. B 753 (2006) 195 [hep-th/0607019] [SPIRES].
S. Ananth and S. Theisen, KLT relations from the Einstein-Hilbert Lagrangian, Phys. Lett. B 652 (2007) 128 [arXiv:0706.1778] [SPIRES].
S. Ananth, The quintic interaction vertex in light-cone gravity, Phys. Lett. B 664 (2008) 219 [arXiv:0803.1494] [SPIRES].
L.J. Mason and E.T. Newman, A connection between the Einstein and Yang-Mills equations, Commun. Math. Phys. 121 (1989) 659 [SPIRES].
M. Dunajski and L.J. Mason, Hyper-Kähler hierarchies and their twistor theory, Commun. Math. Phys. 213 (2000) 641 [math/0001008]. = MATH/0001008;
J. Broedel and R. Kallosh, From lightcone actions to maximally supersymmetric amplitudes, JHEP 06 (2011) 024 [arXiv:1103.0322] [SPIRES].
A.D. Popov, M. Bordemann and H. Romer, Symmetries, currents and conservation laws of self-dual gravity, Phys. Lett. B 385 (1996) 63 [hep-th/9606077] [SPIRES].
A.D. Popov, Self-dual Yang-Mills: symmetries and moduli space, Rev. Math. Phys. 11 (1999) 1091 [hep-th/9803183] [SPIRES].
M. Wolf, On hidden symmetries of a super gauge theory and twistor string theory, JHEP 02 (2005) 018 [hep-th/0412163] [SPIRES].
A.D. Popov and M. Wolf, Hidden symmetries and integrable hierarchy of the N = 4 supersymmetric Yang-Mills equations, Commun. Math. Phys. 275 (2007) 685 [hep-th/0608225] [SPIRES].
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Monteiro, R., O’Connell, D. The kinematic algebra from the self-dual sector. J. High Energ. Phys. 2011, 7 (2011). https://doi.org/10.1007/JHEP07(2011)007
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2011)007