Abstract
Using Slavnov’s scalar product of a Bethe eigenstate and a generic state in closed XXZ spin-\(\frac{1}{2}\) chains, with possibly twisted boundary conditions, we obtain determinant expressions for tree-level structure constants in 1-loop conformally-invariant sectors in various planar (super) Yang-Mills theories. When certain rapidity variables are allowed to be free rather than satisfy Bethe equations, these determinants become discrete KP τ-functions.
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- 1.
Superstrings with tree-level interactions only, and no spacetime loops.
- 2.
The limit in which the number of colours N c →∞, the gauge coupling g YM →0, while the ’t Hooft coupling \(\lambda= g^{2}_{\mathit{YM}} N_{c}\) remains finite.
- 3.
Further highlights of integrability in modern quantum field theory and in string theory include (1) Classical integrable hierarchies in matrix models of non-critical strings, from the late 1980’s [10], (2) Finite gap solutions in Seiberg-Witten theory of low-energy SYM2 in the mid 1990’s [11–14], (3) Integrability in QCD scattering amplitudes in the mid 1990’s [8, 15–17], (4) Free fermion methods in works of Nekrasov, Okounkov, Nakatsu, Takasaki and others on Seiberg-Witten theory, in the 2000’s [18, 19], (5) Integrable spin chains in works of Nekrasov, Shatashvili and others on SYM2, in the 2000’s [20], (6) Integrable structures, particularly the Yangian, that appear in recent studies of SYM4 scattering amplitudes [21, 22]. There are many more.
- 4.
In this work, we restrict our attention to this class of local composite operators. In particular, we do not consider descendants or operators with non-zero spin, for which the 2-point and 3-point functions are different.
- 5.
Three operators \(\mathcal{O}_{i}\), of length L i , i∈{1,2,3}, are non-extremal if l ij =L i +L j −L k >0.
- 6.
The SYM4 expression of [29] is a special case of the general expression obtained here.
- 7.
There are definitely more gauge theories that are conformally-invariant at 1-loop or more, with SU(2) sectors that map to states in spin-\(\frac{1}{2}\) chains. Here we consider only samples of theories with different supersymmetries and operator content.
- 8.
XXX spin-\(\frac{1}{2}\) chains are XXZ spin-\(\frac{1}{2}\) chains with an anisotropy parameter Δ=1.
- 9.
The fact that the structure constants in these two types of theories should be handled differently was pointed out to us by C. Ahn and R. Nepomechie.
- 10.
In [29], S[L,N 1,N 2] was denoted by S[L,{N}].
- 11.
Minahan and Zarembo obtained their remarkable result in the context of the complete scalar sector of SYM4. The relevant spin chain in that case is SO(6) symmetric. Here we focus our attention on the restriction of their result to the SU(2) scalar subsector.
- 12.
We are interested in local single-trace composite operators that consist of many fundamental fields. These fields are interacting. In a weakly-interacting quantum field theory, one can consistently choose to ignore all interactions beyond a chosen order in perturbation theory. In the planar theory under consideration, perturbation theory can be arranged according to the number of loops in Feynman diagrams computed. In a 1-loop approximation, one keeps only 1-loop diagrams.
- 13.
We use β in two different ways. 1. To indicate the deformation parameter in \(\mathrm{SYM}_{4}^{\beta}\) theories, and 2. To indicate that a certain state is a Bethe eigenstate of the spin-chain Hamiltonian. There should be no confusion with 1, in which β is a parameter but never a subscript, while in 2 it is always a subscript.
- 14.
- 15.
- 16.
For visual clarity, we have allowed for a gap between the B-lines and the C-lines in Fig. 5. There is also a gap between the N 3-th and (N 3+1)-th vertical lines, where N 3=3 in the example shown, that indicates separate portions of the lattice that will be relevant shortly. The reader should ignore this at this stage.
- 17.
The following result does not require that any set of rapidities satisfy Bethe equations.
- 18.
The conclusion that, in order to obtain a determinant formula, one of the single-trace operators should be BPS-like, was obtained in discussions with C. Ahn and R. Nepomechie.
References
Maldacena, J.M.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231 (1980). hep-th/9711200
Bena, I., Polchinski, J., Roiban, R.: Hidden symmetries of the AdS(5)×S5 superstring. Phys. Rev. D 69, 046002 (2004). hep-th/0305116
Tseytlin, A.: Review of AdS/CFT integrability, Chapter II.1: classical AdS5×S 5 string solutions. arXiv:1012.3986, and references therein
Minahan, J.A., Zarembo, K.: The Bethe-ansatz for N=4 super Yang-Mills. J. High Energy Phys. 0303, 013 (2003). hep-th/0212208
Beisert, N., Kristjansen, C., Staudacher, M.: The dilatation operator of N=4 super Yang-Mills theory. Nucl. Phys. B 664, 131 (2003). hep-th/0303060
Beisert, N., Staudacher, M.: The \(\mathcal{N} = 4\) SYM integrable super spin chain. Nucl. Phys. B 670, 439 (2003). hep-th/0307042
Belitsky, A.V., Derkachov, S.E., Korchemsky, G.P., Manashov, A.N.: Dilatation operator in (super-)Yang-Mills theories on the light-cone. Nucl. Phys. B 708, 115 (2005). hep-th/0409120
Korchemsky, G.P.: Review of AdS/CFT integrability, Chapter IV.4: integrability in QCD and \(\mathcal{N} < 4 \) SYM. arXiv:1012.4000, and references therein
Beisert, N., et al.: Review of AdS/CFT integrability: an overview. arXiv:1012.3982, and the reviews that it introduces
Ginsparg, P., Moore, G.: Lectures on 2D gravity and 2D string theory (TASI 1992). In: Di Francesco, P., Ginsparg, P., Zinn-Justin, J. (eds.) 2D Gravity and Random Matrices. arXiv:hep-th/9304011, hep-th/9306153
Seiberg, N., Witten, E.: Monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory. Nucl. Phys. B 426, 19–52 (1994). Erratum-ibid B 430, 485–486 (1994). hep-th/9407087
Seiberg, N., Witten, E.: Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD. Nucl. Phys. B 431, 484–550 (1994). hep-th/9408099
Gorsky, A., Krichever, I., Marshakov, A., Mironov, A., Morozov, A.: Integrability and Seiberg-Witten exact solution. Phys. Lett. B 355, 466–474 (1995). hep-th/9505035
Marshakov, A.: Seiberg-Witten Theory and Integrable Systems. World Scientific, Singapore (1999), and references therein
Lipatov, L.N.: High energy asymptotics of multi-colour QCD and exactly solvable lattice models. JETP Lett. 59, 596 (1994). hep-th/9311037
Faddeev, L.D., Korchemsky, G.P.: High energy QCD as a completely integrable model. Phys. Lett. B 342, 311 (1995). hep-th/9404173
Korchemsky, G.P.: Bethe ansatz for QCD pomeron. Nucl. Phys. B 443, 255 (1995). hep-ph/9501232
Nekrasov, N., Okounkov, A.: Seiberg-Witten theory and random partitions. In: Etingof, P., Retakh, V., Singer, I.M. (eds.) The Unity of Mathematics: In Honor of the Ninetieth Birthday of I.M. Gelfand. Progress in Mathematics, vol. 244, pp. 525–596 (2006). hep-th/0306238
Nakatsu, T., Takasaki, K.: Melting crystal, quantum torus and Toda hierarchy. Commun. Math. Phys. 285, 445–468 (2009). arXiv:0710.5339, and references therein
Nekrasov, N., Shatashvili, S.: Quantization of integrable systems and four dimensional gauge theories. arXiv:0908.4052, and references therein
Drummond, J.M.: Dual superconformal symmetry. arXiv:1012.4002
Alday, L.F.: Scattering amplitudes at strong coupling. arXiv:1012.4003
Okuyama, K., Tseng, L.S.: Three-point functions in N=4 SYM theory at one-loop. J. High Energy Phys. 0408, 055 (2004). hep-th/0404190
Roiban, R., Volovich, A.: Yang-Mills correlation functions from integrable spin chains. J. High Energy Phys. 0409, 032 (2004). hep-th/0407140
Alday, L.F., David, J.R., Gava, E., Narain, K.S.: Structure constants of planar N=4 Yang Mills at one loop. J. High Energy Phys. 0509, 070 (2005). hep-th/0502186
Escobedo, J., Gromov, N., Sever, A., Vieira, P.: Tailoring three-point functions and integrability. J. High Energy Phys. 1109, 28 (2011). arXiv:1012.2475
Escobedo, J., Gromov, N., Sever, A., Vieira, P.: Tailoring three-point functions and integrability II. Weak/strong coupling match. J. High Energy Phys. 1109, 29 (2011). arXiv:1104.5501
Gromov, N., Sever, A., Vieira, P.: Tailoring three-point functions and integrability III. Classical tunneling. J. High Energy Phys. 1207, 44 (2012). arXiv:1111.2349
Foda, O.: \(\mathcal{N} = 4\) SYM structure constants as determinants. J. High Energy Phys. 1203, 96 (2012). arXiv:1111.4663
Brink, L., Schwarz, J.H., Scherk, J.: Supersymmetric Yang-Mills theories. Nucl. Phys. B 121, 77 (1977)
Minahan, J.A.: Review of AdS/CFT integrability, Chapter I.1: spin chains in N=4 Super Yang-Mills. arxiv:1012.3983, and references therein
Leigh, R.G., Strassler, M.J.: Exactly marginal operators and duality in four-dimensional N=1 supersymmetric gauge theory. Nucl. Phys. B 447, 95 (1995). hep-th/9503121
Zoubos, K.: Deformations, orbifolds and open boundaries. arXiv:1012.3998
Berenstein, D., Cherkis, S.A.: Deformations of N=4 SYM and integrable spin chain models. Nucl. Phys. B 702, 49–85 (2004). arXiv:hep-th/0405215
Kachru, S., Silverstein, E.: 4d conformal theories and strings on orbifolds. Phys. Rev. Lett. 80, 4855 (1998). hep-th/9802183
Lawrence, A.E., Nekrasov, N., Vafa, C.: On conformal field theories in four dimensions. Nucl. Phys. B 533, 199 (1998). hep-th/9803015
Ideguchi, K.: Semiclassical strings on AdS(5) x S**5/Z(M) and operators in orbifold field theories. J. High Energy Phys. 0409, 008 (2004). hep-th/0408014
Beisert, N., Roiban, R.: The Bethe ansatz for Z(S) orbifolds of \(\mathcal{N} = 4\) super Yang-Mills theory. J. High Energy Phys. 0511, 037 (2005). hep-th/0510209
Di Vecchia, P., Tanzini, A.: \(\mathcal{N} = 2\) super Yang-Mills and the XXZ spin chain. J. Geom. Phys. 54, 116–130 (2005). hep-th/0405262
Maldacena, J.: The gauge/gravity duality. arXiv:1106.6073
Beisert, N.: Superconformal algebra. arXiv:1012.4004
Kitanine, N., Maillet, J.M., Terras, V.: Form factors of the XXZ Heisenberg spin-\(\frac{1}{2}\) finite chain. Nucl. Phys. B 554, 647–678 (1999). math-ph/9807020
Wheeler, M.: An Izergin–Korepin procedure for calculating scalar products in six-vertex models. Nucl. Phys. B 852, 468–507 (2011). arXiv:1104.2113
Korepin, V.E., Bogoliubov, N.M., Izergin, A.G.: Quantum Inverse Scattering Method and Correlation Functions. Cambridge University Press, Cambridge (1993)
Slavnov, N.A.: Calculation of scalar products of wave functions and form factors in the framework of the algebraic Bethe ansatz. Theor. Math. Phys. 79, 502–508 (1989)
Korepin, V.E.: Calculation of norms of Bethe wave functions. Commun. Math. Phys. 86, 391–418 (1982)
Foda, O., Schrader, G.: XXZ scalar products, Miwa variables and discrete KP. In: Feigin, B., Jimbo, M., Okado, M. (eds.) New Trends in Quantum Integrable Systems, pp. 61–80. World Scientific, Singapore (2010). arXiv:1003.2524
Ohta, Y., Hirota, R., Tsujimoto, S., Inami, T.: Casorati and discrete Gram type determinant representations of solutions to the discrete KP hierarchy. J. Phys. Soc. Jpn. 62, 1872–1886 (1993)
Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, Oxford (1995)
Izergin, A.G.: Partition function of the six-vertex model in a finite volume. Sov. Phys. Dokl. 32, 878–879 (1987)
Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. Dover, New York (2008)
Georgiou, G., Gili, V., Grossardt, A., Plefka, J.: Three-point functions in planar N=4 super Yang-Mills Theory for scalar operators up to length five at the one-loop order. arXiv:1201.0992
Gromov, N., Vieira, P.: Quantum integrability for three-point functions. arXiv:1202.4103
Gromov, N., Vieira, P.: Tailoring three-point functions and integrability IV. Theta-morphism. arXiv:1205.5288
Ahn, C., Foda, O., Nepomechie, R.: OPE in planar QCD from integrability. J. High Energy Phys. 1206, 168 (2012). arXiv:1202.6553
Kostov, I.: Classical limit of the three-point function from integrability. arXiv:1203.6180
Kostov, I.: Three-point function of semiclassical states at weak coupling. arXiv:1205.4412
Acknowledgements
O.F. thanks C. Ahn, N. Gromov, G. Korchemsky, I. Kostov, R. Nepomechie, D. Serban, P. Vieira and K. Zarembo for discussions on the topic of this work and the Inst. H. Poincare for hospitality where it started. Both authors thank the Australian Research Council for financial support, and the anonymous referee for remarks that helped us improve the text.
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Dedicated to Professor M. Jimbo on his 60th birthday.
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Foda, O., Wheeler, M. (2013). Slavnov Determinants, Yang–Mills Structure Constants, and Discrete KP. In: Iohara, K., Morier-Genoud, S., Rémy, B. (eds) Symmetries, Integrable Systems and Representations. Springer Proceedings in Mathematics & Statistics, vol 40. Springer, London. https://doi.org/10.1007/978-1-4471-4863-0_5
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