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  • Print publication year: 2015
  • Online publication date: May 2015

7 - Integrability and scattering amplitudes

from Part II - Gauge/Gravity Duality


Solving interacting quantum (field) theories exactly for all values of the coupling constant, and not just for very small coupling constant where perturbation theory is applicable, is a long-standing open problem of theoretical physics. By exactly solving we mean diagonalising the corresponding Hamiltonian, such that both eigenstates and eigenvalues are known explicitly. This is an extraordinarily difficult task: for instance, for QCD this would mean finding the complete mass spectrum from first principles from the QCD Lagrangian using analytical methods, as well as the associated eigenstates. This is certainly impossible at present for such a complicated theory.

The same problem occurs also for quantum mechanics. For most quantum systems, the complete set of eigenstates is not known. However, within quantum mechanics there are some cases where an exact solution is possible: the harmonic oscillator and the hydrogen atom, for instance. These systems should be viewed as toy models since they are ideal approximations to systems realised in nature. For instance, to describe real oscillators in solids, higher order interaction terms have to be added.

In quantum field theory there are also exactly solvable toy models: however, they are defined in low spacetime dimensions, for instance in d = 1 + 1. An example is the Thirring model which is integrable in the sense that it has an infinite number of conserved quantities. Are there any interacting exactly solvable quantum field theories also in 3+1 dimensions? The surprising answer is yes: there is a large amount of evidence that N = 4 Super Yang– Mills theory has an integrable structure, at least in the planar (large N) limit. The evidence for such an integrable structure at strong coupling has been found using the AdS/CFT correspondence. Since N = 4 Super Yang–Mills is a conformal field theory, in the planar limit the theory is characterised completely by the scaling dimensions Δ of the composite local operators built from products of elementary fields.

[1] Bena, Iosif, Polchinski, Joseph, and Roiban, Radu. 2004. Hidden symmetries of the AdS5 × S5 superstring. Phys. Rev., D69, 046002.
[2] Arutyunov, Gleb, and Frolov, Sergey. 2009. Foundations of the AdS5xS5 superstring. Part I. J. Phys., A42, 254003.
[3] Berenstein, David Eliecer, Maldacena, Juan Martin, and Nastase, Horatiu Stefan. 2002. Strings in flat space and pp waves from N = 4 Super Yang-Mills. J. High Energy Phys., 0204, 013.
[4] Mangano, Michelangelo and Parke, S. 1991. Multiparton amplitudes in gauge theories. Phys. Rep., 200, 301.
[5] Alday, Luis F., and Maldacena, Juan. 2008. Gluon scattering amplitudes at strong coupling. J. High Energy Phys., 0706, 064.
[6] Gross, David J., and Mende, Paul F. 1987. The high-energy behavior of string scattering amplitudes. Phys. Lett., B197, 129.
[7] Drummond, J. M., Henn, J., Korchemsky, G. P., and Sokatchev, E. 2008. The hexagon Wilson loop and the BDS ansatz for the six-gluon amplitude. Phys. Lett., B662, 456–460.
[8] Plefka, Jan. 2005. Spinning strings and integrable spin chains in the AdS/CFT correspondence. Living Rev. Relativity, 8, 9.
[9] Beisert, Niklas, Ahn, Changrim, Alday, Luis F., Bajnok, Zoltan, Drummond, James M., et al. 2012. Review of AdS/CFT integrability: an overview. Lett. Math. Phys., 99, 3–32.
[10] Alday, Luis F., and Roiban, Radu. 2008. Scattering amplitudes, Wilson loops and the string/gauge theory correspondence. Phys. Rep., 468, 153–211.
[11] Elvang, Henriette, and Huang, Yu-tin. 2015. Scattering Amplitudes in Gauge Theory and Gravity. Cambridge University Press.
[12] Minahan, J. 2012. Spin chains in N = 4 Super Yang-Mills. Lett. Math. Phys., 99, 33–58.
[13] Minahan, J. A., and Zarembo, K. 2003. The Bethe ansatz for N = 4 Super Yang-Mills. J. High Energy Phys., 0303, 013.
[14] Kazakov, V. A., Marshakov, A., Minahan, J. A., and Zarembo, K. 2004. Classical/quantum integrability in AdS/CFT. J. High Energy Phys., 0405, 024.
[15] Beisert, N., Kristjansen, C., and Staudacher, M. 2003. The dilatation operator of conformal N = 4 Super Yang-Mills theory. Nucl. Phys., B664, 131–184.
[16] Beisert, Niklas. 2004. The dilatation operator of N = 4 Super Yang-Mills theory and integrability. Phys. Rep., 405, 1–202.
[17] Beisert, Niklas. 2004. The su(2|3) dynamic spin chain. Nucl. Phys., B682, 487–520.
[18] Eden, B., Jarczak, C., and Sokatchev, E. 2005. A Three-loop test of the dilatation operator in N = 4 SYM. Nucl. Phys., B712, 157–195.
[19] Beisert, N., Dippel, V., and Staudacher, M. 2004. A novel long range spin chain and planar N = 4 super Yang-Mills. J. High Energy Phys., 0407, 075.
[20] Bern, Zvi, Dixon, Lance J., and Smirnov, Vladimir A. 2005. Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond. Phys. Rev., D72, 085001.
[21] Beisert, Niklas, Eden, Burkhard, and Staudacher, Matthias. 2007. Transcendentality and crossing. J. Stat. Mech., 0701, P01021.
[22] Drummond, J. M., Henn, J., Korchemsky, G. P., and Sokatchev, E. 2010. Dual superconformal symmetry of scattering amplitudes in N = 4 Super Yang-Mills theory. Nucl. Phys., B828, 317–374.
[23] Drummond, J. M., Henn, J., Korchemsky, G. P., and Sokatchev, E. 2013. Generalized unitarity for N = 4 super-amplitudes. Nucl. Phys., B869, 452–492.
[24] Dolan, Louise, Nappi, Chiara R., and Witten, Edward. 2004. Yangian symmetry in D = 4 superconformal Yang-Mills theory. ArXiv:hep-th/0401243.
[25] Drummond, James M., Henn, Johannes M., and Plefka, Jan. 2009. Yangian symmetry of scattering amplitudes in N = 4 Super Yang-Mills theory. J. High Energy Phys., 0905, 046.
[26] Drummond, J. M., and Ferro, L. 2010. Yangians, Grassmannians and T-duality. J. High Energy Phys., 1007, 027.