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SIGMA 10 (2014), 051, 28 pages arXiv:1207.4712
https://doi.org/10.3842/SIGMA.2014.051
Contribution to the Special Issue on Progress in Twistor Theory
Gravity in Twistor Space and its Grassmannian Formulation
Freddy Cachazo a, Lionel Mason b and David Skinner c
a) Perimeter Institute for Theoretical Physics, 31 Caroline St., Waterloo, Ontario N2L 2Y5, Canada
b) The Mathematical Institute, 24-29 St. Giles', Oxford OX1 3LB, UK
c) DAMTP, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
Received November 21, 2013, in final form April 23, 2014; Published online May 01, 2014
Abstract
We prove the formula for the complete tree-level $S$-matrix of $\mathcal{N}=8$ supergravity recently conjectured by two of the authors.
The proof proceeds by showing that the new formula satisfies the same BCFW recursion relations that physical amplitudes are known to satisfy,
with the same initial conditions.
As part of the proof, the behavior of the new formula under large BCFW deformations is studied.
An unexpected bonus of the analysis is a very straightforward proof of the enigmatic $1/z^2$ behavior of gravity.
In addition, we provide a description of gravity amplitudes as a multidimensional contour integral over a Grassmannian.
The Grassmannian formulation has a very simple structure; in the N$^{k-2}$MHV sector the integrand is essentially the product of that of an MHV
and an $\overline{{\rm MHV}}$ amplitude, with $k+1$ and $n-k-1$ particles respectively.
Key words:
twistor theory; scattering amplitudes; gravity.
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References
- Arkani-Hamed N., Bourjaily J., Cachazo F., Caron-Huot S., Trnka J., The
all-loop integrand for scattering amplitudes in planar ${\mathcal N}=4$
SYM, J. High Energy Phys. 2011 (2011), no. 1, 041,
46 pages, arXiv:1008.2958.
- Arkani-Hamed N., Bourjaily J., Cachazo F., Trnka J., Unification of residues
and Grassmannian dualities, J. High Energy Phys. 2011
(2011), no. 1, 049, 45 pages, arXiv:0912.4912.
- Arkani-Hamed N., Cachazo F., Cheung C., Kaplan J., A duality for the $S$
matrix, J. High Energy Phys. 2010 (2010), no. 3, 020,
69 pages, arXiv:0907.5418.
- Arkani-Hamed N., Cachazo F., Cheung C., Kaplan J., The $S$-matrix in twistor
space, J. High Energy Phys. 2010 (2010), no. 3, 110,
47 pages, arXiv:0903.2110.
- Arkani-Hamed N., Cachazo F., Kaplan J., What is the simplest quantum field
theory?, J. High Energy Phys. 2010 (2010), no. 9, 016,
90 pages, arXiv:0808.1446.
- Arkani-Hamed N., Kaplan J., On tree amplitudes in gauge theory and gravity,
J. High Energy Phys. 2008 (2008), no. 4, 076, 21 pages,
arXiv:0801.2385.
- Bedford J., Brandhuber A., Spence W., Travaglini G., A recursion relation for
gravity amplitudes, Nuclear Phys. B 721 (2005), 98-110,
hep-th/0502146.
- Benincasa P., Boucher-Veronneau C., Cachazo F., Taming tree amplitudes in
general relativity, J. High Energy Phys. 2007 (2007),
no. 11, 057, 25 pages, hep-th/0702032.
- Berkovits N., Alternative string theory in twistor space for ${\mathcal N}=4$
super-Yang-Mills theory, Phys. Rev. Lett. 93 (2004),
011601, 3 pages, hep-th/0402045.
- Bourjaily J.L., Trnka J., Volovich A., Wen C., The Grassmannian and the
twistor string: connecting all trees in ${\mathcal N}=4$ SYM,
J. High Energy Phys. 2011 (2011), no. 1, 038, 19 pages,
arXiv:1006.1899.
- Britto R., Cachazo F., Feng B., New recursion relations for tree amplitudes of
gluons, Nuclear Phys. B 715 (2005), 499-522,
hep-th/0412308.
- Britto R., Cachazo F., Feng B., Witten E., Direct proof of the tree-level
scattering amplitude recursion relation in Yang-Mills theory,
Phys. Rev. Lett. 94 (2005), 181602, 4 pages,
hep-th/0501052.
- Bullimore M., New formulae for gravity amplitudes: parity invariance and soft
limits, arXiv:1207.3940.
- Bullimore M., Mason L., Skinner D., Twistor-strings, Grassmannians and
leading singularities, J. High Energy Phys. 2010 (2010),
no. 3, 070, 54 pages, arXiv:0912.0539.
- Cachazo F., Skinner D., Gravity from rational curves in twistor space,
Phys. Rev. Lett. 110 (2013), 161301, 4 pages,
arXiv:1207.0741.
- Cachazo F., Svrcek P., Tree-level recursion relations in general relativity,
hep-th/0502160.
- Cheung C., On-shell recursion relations for generic theories, J. High
Energy Phys. 2010 (2010), no. 3, 098, 19 pages, arXiv:0808.0504.
- Dolan L., Goddard P., Gluon tree amplitudes in open twistor string theory,
J. High Energy Phys. 2009 (2009), no. 12, 032, 32 pages,
arXiv:0909.0499.
- Dolan L., Goddard P., General split helicity gluon tree amplitudes in open
twistor string theory, J. High Energy Phys. 2010 (2010),
no. 5, 044, 27 pages, arXiv:1002.4852.
- Dolan L., Goddard P., Complete equivalence between gluon tree amplitudes in
twistor string theory and in gauge theory, J. High Energy Phys.
2012 (2012), no. 6, 030, 37 pages, arXiv:1111.0950.
- Gukov S., Motl L., Neitzke A., Equivalence of twistor prescriptions for super
Yang-Mills, Adv. Theor. Math. Phys. 11 (2007),
199-231, hep-th/0404085.
- He S., A link representation for gravity amplitudes, J. High Energy
Phys. 2013 (2013), no. 10, 139, 11 pages, arXiv:1207.4064.
- Hodges A., New expressions for gravitational scattering amplitudes,
J. High Energy Phys. 2013 (2013), no. 7, 075, 33 pages,
arXiv:1108.2227.
- Hodges A., A simple formula for gravitational MHV amplitudes,
arXiv:1204.1930.
- Mason L., Skinner D., Dual superconformal invariance, momentum twistors and
Grassmannians, J. High Energy Phys. 2009 (2009), no. 11,
045, 39 pages, arXiv:0909.0250.
- Mason L., Skinner D., Scattering amplitudes and BCFW recursion in twistor
space, J. High Energy Phys. 2010 (2010), no. 1, 064,
65 pages, arXiv:0903.2083.
- Mason L.J., Wolf M., Twistor actions for self-dual supergravities,
Comm. Math. Phys. 288 (2009), 97-123, arXiv:0706.1941.
- Nandan D., Volovich A., Wen C., A Grassmannian étude in NMHV minors,
J. High Energy Phys. 2010 (2010), no. 7, 061, 15 pages,
arXiv:0912.3705.
- Penrose R., The nonlinear graviton, Gen. Relativity Gravitation
7 (1976), 171-176.
- Roiban R., Spradlin M., Volovich A., Tree-level $S$ matrix of Yang-Mills
theory, Phys. Rev. D 70 (2004), 026009, 10 pages,
hep-th/0403190.
- Skinner D., A direct proof of BCFW recursion for twistor-strings,
J. High Energy Phys. 2011 (2011), no. 1, 072, 22 pages,
arXiv:1007.0195.
- Spradlin M., Volovich A., From twistor string theory to recursion relations,
Phys. Rev. D 80 (2009), 085022, 5 pages,
arXiv:0909.0229.
- Spradlin M., Volovich A., Wen C., Three applications of a bonus relation for
gravity amplitudes, Phys. Lett. B 674 (2009), 69-72,
arXiv:0812.4767.
- Vergu C., Factorization of the connected prescription for Yang-Mills
amplitudes, Phys. Rev. D 75 (2007), 025028, 7 pages,
hep-th/0612250.
- Witten E., Parity invariance for strings in twistor space, Adv. Theor.
Math. Phys. 8 (2004), 779-797, hep-th/0403199.
- Witten E., Perturbative gauge theory as a string theory in twistor space,
Comm. Math. Phys. 252 (2004), 189-258,
hep-th/0312171.
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