Abstract
We recently showed how nonhomogeneous second-order difference equations that appear in describing the ABJM quantum spectral curve can be solved using a Mellin space technique. In particular, we provided explicit results for anomalous dimensions of twist-1 operators in the sl(2) sector at arbitrary spin values up to the four-loop order. We showed that the obtained results can be expressed in terms of harmonic sums with additional factors in the form of a fourth root of unity, and the maximum transcendentality principle therefore holds. Here, we show that the same result can also be obtained by directly solving the mentioned difference equations in the space of the spectral parameter u. The solution involves new highly nontrivial identities between hypergeometric functions, which can have various applications. We expect that this method can be generalized both to higher loop orders and to other theories, such as the N=4 supersymmetric Yang–Mills theory.
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References
G. ’t Hooft, “A planar diagram theory for strong interactions,” Nucl. Phys. B, 72, 461–473 (1974).
J. M. Maldacena, “The large N limit of superconformal field theories and supergravity,” Internat. J. Theor. Phys., 38, 1113–1133 (1999); arXiv:hep–th/9711200v3 (1997); “The large N limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys., 2, 231–252 (1998).
S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, “Gauge theory correlators from non–critical string theory,” Phys. Lett. B, 428, 105–114 (1998); arXiv:hep–th/9802109v2 (1998).
E. Witten, “Anti–de Sitter space and holography,” Adv. Theor. Math. Phys., 2, 253–291 (1998); arXiv:hepth/9802150v2 (1998).
N. Beisert et al., “Review of AdS/CFT integrability: An overview,” Lett. Math. Phys., 99, 3–32 (2012); arXiv:1012.3982v5 [hep–th] (2010).
D. Bombardelli, A. Cagnazzo, R. Frassek, F. Levkovich–Maslyuk, F. Loebbert, S. Negro, I. M. Szécsényi, A. Sfondrini, S. J. van Tongeren, and A. Torrielli, “An integrability primer for the gauge–gravity correspondence: An introduction,” J. Phys. A: Math. Theor., 49, 320301 (2016); arXiv:1606.02945v2 [hep–th] (2016).
S. J. van Tongeren, “Integrability of the AdS5×S5 superstring and its deformations,” J. Phys. A: Math. Theor., 47, 433001 (2014); arXiv:1310.4854v3 [hep–th] (2013).
M. de Leeuw, A. C. Ipsen, C. Kristjansen, and M. Wilhelm, “Introduction to integrability and one–point functions in N=4 SYM and its defect cousin,” in: Integrability: From Statistical Systems to Gauge Theory, Les Houches School of Physics, Les Houches, France (2017); arXiv:1708.02525v1 [hep–th] (2017).
N. Gromov, “Introduction to the spectrum of N=4 SYM and the quantum spectral curve,” arXiv:1708.03648v1 [hep–th] (2017).
S. Komatsu, “Lectures on three–point functions in N = 4 supersymmetric Yang–Mills theory,” arXiv: 1710.03853v2 [hep–th] (2017).
V. Kazakov, “Quantum spectral curve of γ–twisted N=4 SYM theory and fishnet CFT,” Rev. Math. Phys., 30, 1840010 (2018); arXiv:1802.02160v1 [hep–th] (2018)
M.–L. Ge, A. J. Niemi, K. K. Phua, and L. A. Takhtajan, eds., Ludwig Faddeev Memorial Volume: A Life in Mathematical Physics, World Scientific, Singapore (2018).
O. Aharony, O. Bergman, D. L. Jafferis, and J. Maldacena, “N=6 superconformal Chern–Simons–matter theories M2–branes and their gravity duals,” JHEP, 0810, 091 (2008); arXiv:0806.1218v4 [hep–th] (2008).
M. Staudacher, “The factorized S–matrix of CFT/AdS,” JHEP, 0505, 054 (2005); arXiv:hep–th/0412188v1 (2004).
G. Arutyunov, S. Frolov, and M. Staudacher, “Bethe ansatz for quantum strings,” JHEP, 0410, 016 (2004); arXiv:hep–th/0406256v3 (2004).
N. Beisert, “The SU(22) dynamic S–matrix,” Adv. Theor. Math. Phys., 12, 945–979 (2008); arXiv:hep–th/0511082v4 (2005).
N. Beisert, “The analytic Bethe ansatz for a chain with centrally extended su(22) symmetry,” J. Stat. Mech., 0701, P01017 (2007); arXiv:nlin/0610017v2 (2006).
N. Beisert, B. Eden, and M. Staudacher, “Transcendentality and crossing,” J. Stat. Mech., 0701, P01021 (2007); arXiv:hep–th/0610251v2 (2006).
R. A. Janik, “The AdS5×S5 superstring worldsheet S–matrix and crossing symmetry,” Phys. Rev. D, 73, 086006 (2006); arXiv:hep–th/0603038v2 (2006).
G. Arutyunov and S. Frolov, “On AdS5×S5 string S–matrix,” Phys. Lett. B, 639, 378–382 (2006); arXiv:hepth/0604043v2 (2006).
G. Arutyunov, S. Frolov, and M. Zamaklar, “The Zamolodchikov–Faddeev algebra for AdS5×S5 superstring,” JHEP, 0704, 002 (2007); arXiv:hep–th/0612229v3 (2006).
C. Ahn and R. I. Nepomechie, “N =6 super Chern–Simons theory S–matrix and allloop Bethe ansatz equations,” JHEP, 0809, 010 (2008); arXiv:0807.1924v2 [hep–th] (2008).
J. A. Minahan and K. Zarembo, “The Bethe ansatz for N=4 super Yang–Mills,” JHEP, 0303, 013 (2003); arXiv:hep–th/0212208v3 (2002).
N. Beisert and M. Staudacher, “The N=4 SYM integrable super spin chain,” Nucl. Phys. B, 670, 439–463 (2003); arXiv:hep–th/0307042v3 (2003).
N. Beisert and M. Staudacher, “Long–range psu(2; 24) Bethe ansatze for gauge theory and strings,” Nucl. Phys. B, 727, 1–62 (2005); arXiv:hep–th/0504190v3 (2005).
J. A. Minahan and K. Zarembo, “The Bethe ansatz for superconformal Chern–Simons,” JHEP, 0809, 040 (2008); arXiv:0806.3951v6 [hep–th] (2008).
D. Gaiotto, S. Giombi, and X. Yin, “Spin chains in N=6 superconformal Chern–Simons–matter theory,” JHEP, 0904, 066 (2009); arXiv:0806.4589v2 [hep–th] (2008).
N. Gromov and P. Vieira, “The all loop AdS4/CFT3 Bethe ansatz,” JHEP, 0901, 016 (2009); arXiv: 0807.0777v2 [hep–th] (2008).
N. Gromov, V. Kazakov, and P. Vieira, “Exact spectrum of anomalous dimensions of planar N =4 supersymmetric Yang–Mills theory,” Phys. Rev. Lett., 103, 131601 (2009); arXiv:0901.3753v3 [hep–th] (2009).
D. Bombardelli, D. Fioravanti, and R. Tateo, “Thermodynamic Bethe ansatz for planar AdS/CFT: A proposal,” J. Phys. A: Math. Theor., 42, 375401 (2009); arXiv:0902.3930v2 [hep–th] (2009).
N. Gromov, V. Kazakov, A. Kozak, and P. Vieira, “Exact spectrum of anomalous dimensions of planar N=4 supersymmetric Yang–Mills theory: TBA and excited states,” Lett. Math. Phys., 91, 265–287 (2010); arXiv: 0902.4458v4 [hep–th] (2009).
G. Arutyunov and S. Frolov, “Thermodynamic Bethe ansatz for the AdS5×S5 mirror model,” JHEP, 0905, 068 (2009); arXiv:0903.0141v3 [hep–th] (2009).
A. Cavaglia, D. Fioravanti, and R. Tateo, “Extended Y–system for the AdS5/CFT4 correspondence,” Nucl. Phys. B, 843, 302–343 (2011); arXiv:1005.3016v3 [hep–th] (2010).
J. Balog and Á. Hegedus, “AdS5×S5 mirror TBA equations from Y–system and discontinuity relations,” JHEP, 1108, 095 (2011); arXiv:1104.4054v2 [hep–th] (2011).
N. Gromov, V. Kazakov, S. Leurent, and Z. Tsuboi, “Wronskian solution for AdS/CFT Y–system,” JHEP, 1101, 155 (2011); arXiv:1010.2720v2 [hep–th] (2010).
N. Gromov, V. Kazakov, S. Leurent, and D. Volin, “Solving the AdS/CFT Y–system,” JHEP, 1207, 023 (2012); arXiv:1110.0562v3 [hep–th] (2011).
D. Bombardelli, D. Fioravanti, and R. Tateo, “TBA and Y–system for planar AdS4/CFT3,” Nucl. Phys. B, 834, 543–561 (2010); arXiv:0912.4715v1 [hep–th] (2009).
N. Gromov and F. Levkovich–Maslyuk, “Y–system, TBA, and quasi–classical strings in AdS4 × CP3,” JHEP, 1006, 088 (2010); arXiv:0912.4911v4 [hep–th] (2009).
A. Cavaglià, D. Fioravanti, and R. Tateo, “Discontinuity relations for the AdS4/CFT3 correspondence,” Nucl. Phys. B, 877, 852–884 (2013); arXiv:1307.7587v1 [hep–th] (2013).
D. Correa, J. Maldacena, and A. Sever, “The quark anti–quark potential and the cusp anomalous dimension from a TBA equation,” JHEP, 1208, 134 (2012); arXiv:1203.1913v2 [hep–th (2012).
N. Drukker, “Integrable Wilson loops,” JHEP, 1310, 135 (2013); arXiv:1203.1617v2 [hep–th] (2012).
N. Gromov and F. Levkovich–Maslyuk, “Quantum spectral curve for a cusped Wilson line in N=4SYM,” JHEP, 1604, 134 (2016); arXiv:1510.02098v1 [hep–th] (2015).
N. Gromov and F. Levkovich–Maslyuk, “Quark–anti–quark potential in N=4SYM,” JHEP, 1612, 122 (2016); arXiv:1601.05679v2 [hep–th] (2016).
L. F. Alday, D. Gaiotto, and J. Maldacena, “Thermodynamic bubble ansatz,” JHEP, 1109, 032 (2011); arXiv: 0911.4708v3 [hep–th] (2009).
L. F. Alday, J. Maldacena, A. Sever, and P. Vieira, “Y–system for scattering amplitudes,” J. Phys. A: Math. Theor., 43, 485401 (2010); arXiv:1002.2459v2 [hep–th] (2010).
L. F. Alday, D. Gaiotto, J. Maldacena, A. Sever, and P. Vieira, “An operator product expansion for polygonal null Wilson loops,” JHEP, 1104, 088 (2011); arXiv:1006.2788v2 [hep–th] (2010).
B. Basso, A. Sever, and P. Vieira, “Spacetime and flux tube S–matrices at finite coupling for N =4 supersymmetric Yang–Mills theory,” Phys. Rev. Lett., 111, 091602 (2013); arXiv:1303.1396v1 [hep–th] (2013).
B. Basso, J. Caetano, L. Cordova, A. Sever, and P. Vieira, “OPE for all helicity amplitudes,” JHEP, 1508, 018 (2015); arXiv:1412.1132v1 [hep–th] (2014).
D. Fioravanti, S. Piscaglia, and M. Rossi, “Asymptotic bethe ansatz on the GKP vacuum as a defect spin chain: Scattering particles and minimal area Wilson loops,” Nucl. Phys. B, 898, 301–400 (2015); arXiv:1503.08795v2 [hep–th] (2015).
B. Basso, S. Caron–Huot, and A. Sever, “Adjoint BFKL at finite coupling: A shortcut from the collinear limit,” JHEP, 1501, 027 (2015); arXiv:1407.3766v2 [hep–th] (2014).
M. Alfimov, N. Gromov, and V. Kazakov, “QCD pomeron from AdS/CFT quantum spectral curve,” JHEP, 1507, 164 (2015); arXiv:1408.2530v4 [hep–th] (2014).
N. Gromov, F. Levkovich–Maslyuk, and G. Sizov, “Pomeron eigenvalue at three loops in N=4 supersymmetric Yang–Mills theory,” Phys. Rev. Lett., 115, 251601 (2015); arXiv:1507.04010v2 [hep–th] (2015).
M. Alfimov, N. Gromov, and G. Sizov, “BFKL spectrum of N=4 SYM: non–zero conformal spin,” JHEP, 1807, 181 (2018); arXiv:1802.06908v4 [hep–th] (2018).
B. Basso, S. Komatsu, and P. Vieira, “Structure constants and integrable bootstrap in planar N=4 SYM theory,” arXiv:1505.06745v1 [hep–th] (2015).
B. Basso, V. Gonçalves, and S. Komatsu, “Structure constants at wrapping order,” JHEP, 1705, 124 (2017); arXiv:1702.02154v1 [hep–th] (2017).
Y. Jiang, S. Komatsu, I. Kostov, and D. Serban, “Clustering and the three–point function,” J. Phys. A: Math. Theor., 49, 454003 (2016); arXiv:1604.03575v2 [hep–th] (2016).
I. Balitsky, V. Kazakov, and E. Sobko, “Structure constant of twist–2 light–ray operators in the Regge limit,” Phys. Rev. D, 93, 061701 (2016); arXiv:1506.02038v2 [hep–th] (2015).
A. Cavaglià, N. Gromov, and F. Levkovich–Maslyuk, “Quantum spectral curve and structure constants in N=4 SYM: Cusps in the ladder limit,” JHEP, 1810, 060 (2018); arXiv:1802.04237v3 [hep–th] (2018).
B. Eden and A. Sfondrini, “Tessellating cushions: Four–point functions in N=4 SYM,” JHEP, 1710, 098 (2017); arXiv:1611.05436v1 [hep–th] (2016).
B. Eden, Y. Jiang, D. le Plat, and A. Sfondrini, “Colour–dressed hexagon tessellations for correlation functions and non–planar corrections,” JHEP, 1802, 170 (2018); arXiv:1710.10212v2 [hep–th] (2017).
T. Bargheer, J. Caetano, T. Fleury, S. Komatsu, and P. Vieira, “Handling handles I: Nonplanar integrability,” arXiv:1711.05326v1 [hep–th] (2017).
T. Fleury and S. Komatsu, “Hexagonalization of correlation functions II: Two–particle contributions,” JHEP, 1802, 177 (2018); arXiv:1711.05327v1 [hep–th] (2017).
S. Giombi and S. Komatsu, “Exact correlators on the Wilson loop in N=4 SYM: Localization defect CFT and integrability,” JHEP, 1805, 109 (2018); arXiv:1802.05201v3 [hep–th] (2018).
B. Eden, Y. Jiang, M. de Leeuw, T. Meier, D. le Plat, and A. Sfondrini, “Positivity of hexagon perturbation theory,” arXiv:1806.06051v2 [hep–th] (2018).
M. de Leeuw, C. Kristjansen, and K. Zarembo, “One–point functions in defect CFT and integrability,” JHEP, 1508, 098 (2015); arXiv:1506.06958v2 [hep–th] (2015).
I. Buhl–Mortensen, M. de Leeuw, C. Kristjansen, and K. Zarembo, “One–point functions in AdS/dCFT from matrix product states,” JHEP, 1602, 052 (2016); arXiv:1512.02532v2 [hep–th] (2015).
I. Buhl–Mortensen, M. de Leeuw, A. C. Ipsen, C. Kristjansen, and M. Wilhelm, “One–loop one–point functions in gauge–gravity dualities with defects,” Phys. Rev. Lett., 117, 231603 (2016); arXiv:1606.01886v3 [hep–th] (2016).
T. Harmark and M. Wilhelm, “Hagedorn temperature of AdS5/CFT4 via integrability,” Phys. Rev. Lett., 120, 071605 (2018); arXiv:1706.03074v3 [hep–th] (2017).
T. Harmark and M. Wilhelm, “The Hagedorn temperature of AdS5/CFT4 at finite coupling via the quantum spectral curve,” Phys. Lett. B, 786, 53–58 (2018); arXiv:1803.04416v1 [hep–th] (2016).
N. Gromov, V. Kazakov, S. Leurent, and D. Volin, “Quantum spectral curve for planar N=4 super–Yang–Mills theory,” Phys. Rev. Lett., 112, 011602 (2014); arXiv:1305.1939v2 [hep–th] (2013).
N. Gromov, V. Kazakov, S. Leurent, and D. Volin, “Quantum spectral curve for arbitrary state/operator in AdS5/CFT4,” JHEP, 1509, 187 (2015); arXiv:1405.4857v3 [hep–th] (2014).
V. Kazakov, S. Leurent, and D. Volin, “T–system on T–hook: Grassmannian solution and twisted quantum spectral curve,” JHEP, 1612, 044 (2016); arXiv:1510.02100v2 [hep–th] (2015).
C. Marboe and D. Volin, “The full spectrum of AdS5/CFT4 I: Representation theory and one–loop Q–system,” J. Phys. A: Math. Theor., 51, 165401 (2018); arXiv:1701.03704v3 [hep–th] (2017).
A. Cavaglià, D. Fioravanti, N. Gromov, and R. Tateo, “Quantum spectral curve of the N=6 supersymmetric Chern–Simons theory,” Phys. Rev. Lett., 113, 021601 (2014); arXiv:1403.1859v2 [hep–th] (2014).
D. Bombardelli, A. Cavaglià, D. Fioravanti, N. Gromov, and R. Tateo, “The full quantum spectral curve for AdS4/CFT3,” JHEP, 1709, 140 (2017); arXiv:1701.00473v4 [hep–th] (2017).
R. Klabbers and S. J. van Tongeren, “Quantum spectral curve for the eta–deformed AdS5xS5 superstring,” Nucl. Phys. B, 925, 252–318 (2017); arXiv:1708.02894v2 [hep–th] (2017).
N. Gromov, F. Levkovich–Maslyuk, and G. Sizov, “Quantum spectral curve and the numerical solution of the spectral problem in AdS5/CFT4,” JHEP, 1606, 036 (2016); arXiv:1504.06640v3 [hep–th] (2015).
Á. Heged˝us and J. Konczer, “Strong coupling results in the AdS5/CFT4 correspondence from the numerical solution of the quantum spectral curve,” JHEP, 1608, 061 (2016); arXiv:1604.02346v1 [hep–th] (2016).
D. Bombardelli, A. Cavaglià, R. Conti, and R. Tateo, “Exploring the spectrum of planar AdS4/CFT3 at finite coupling,” JHEP, 1804, 117 (2018); arXiv:1803.04748v2 [hep–th] (2018).
C. Marboe and D. Volin, “Quantum spectral curve as a tool for a perturbative quantum field theory,” Nucl. Phys. B, 899, 810–847 (2015); arXiv:1411.4758v2 [hep–th] (2014).
L. Anselmetti, D. Bombardelli, A. Cavaglià, and R. Tateo, “12 loops and triple wrapping in ABJM theory from integrability,” JHEP, 1510, 117 (2015); arXiv:1506.09089v2 [hep–th] (2015).
R. N. Lee and A. I. Onishchenko, “ABJM quantum spectral curve and Mellin transform,” JHEP, 1805, 179 (2018); arXiv:1712.00412v2 [hep–th] (2017).
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, 0809, 027 (2008); arXiv:0807.0880v2 [hep–th] (2008).
T. Klose, “Review of AdS/CFT integrability chapter IV.3: N=6 Chern–Simons and STRINGS on AdS4×CP3,” Lett. Math. Phys., 99, 401–423 (2012); arXiv:1012.3999v5 [hep–th] (2010).
G. Grignani, T. Harmark, and M. Orselli, “The SU(2)×SU(2) sector in the string dual of N =6 superconformal Chern–Simons theory,” Nucl. Phys. B, 810, 115–134 (2009); arXiv:0806.4959v4 [hep–th] (2008).
A. V. Kotikov and L. N. Lipatov, “DGLAP and BFKL equations in the N=4 supersymmetric gauge theory,” Nucl. Phys. B, 661, 19–61 (2003); Erratum, 685, 405–407 (2004); arXiv:hep–ph/0208220v3 (2002).
A. V. Kotikov, L. N. Lipatov, A. I. Onishchenko, and V. N. Velizhanin, “Three loop universal anomalous dimension of the Wilson operators in N=4 SUSY Yang–Mills model,” Phys. Lett. B, 595, 521–529 (2004); Erratum, 632, 754–756 (2006); arXiv:hep–th/0404092v5 (2004).
M. Beccaria and G. Macorini, “QCD properties of twist operators in the N =6 Chern–Simons theory,” JHEP, 0906, 008 (2009); arXiv:0904.2463v3 [hep–th] (2009).
M. Beccaria, F. Levkovich–Maslyuk, and G. Macorini, “On wrapping corrections to GKP–like operators,” JHEP, 1103, 001 (2011); arXiv:1012.2054v2 [hep–th] (2010).
J. Ablinger, J. Blümlein, and C. Schneider, “Analytic and algorithmic aspects of generalized harmonic sums and polylogarithms,” J. Math. Phys., 54, 082301 (2013); arXiv:1302.0378v1 [math–ph] (2013).
J. Ablinger, J. Blümlein, and C. Schneider, “Harmonic sums and polylogarithms generated by cyclotomic polynomials,” J. Math. Phys., 52, 102301 (2011); arXiv:1105.6063v1 [math–ph] (2011).
A. B. Zamolodchikov, “‘Fishing–net’ diagrams as a completely integrable system,” Phys. Lett. B, 97, 63–66 (1980).
D. Chicherin, S. Derkachov, and A. P. Isaev, “Conformal group: R–matrix and star–triangle relation,” JHEP, 1304, 020 (2013); arXiv:1206.4150v2 [math–ph] (2013).
Ö. Gürdoğan and V. Kazakov, “New integrable 4D quantum field theories from strongly deformed planar N=4 supersymmetric Yang–Mills theory,” Phys. Rev. Lett., 117, 201602 (2016); Addendum, 117, 259903 (2016); arXiv:1512.06704v3 [hep–th] (2015).
J. Caetano, O. Gürdoğan, and V. Kazakov, “Chiral limit of N=4 SYM and ABJM and integrable Feynman graphs,” arXiv:1612.05895v3 [hep–th] (2016).
D. Chicherin, V. Kazakov, F. Loebbert, D. Müller, and D.–l. Zhong, “Yangian symmetry for bi–scalar loop amplitudes,” JHEP, 1805, 003 (2017); arXiv:1704.01967v1 [hep–th] (2017).
B. Basso and L. J. Dixon, “Gluing ladder Feynman diagrams into fishnets,” Phys. Rev. Lett., 119, 071601 (2017); arXiv:1705.03545v2 [hep–th] (2017).
N. Gromov, V. Kazakov, G. Korchemsky, S. Negro, and G. Sizov, “Integrability of conformal fishnet theory,” JHEP, 1801, 095 (2017); arXiv:1706.04167v2 [hep–th] (2017).
D. Chicherin, V. Kazakov, F. Loebbert, D. Müller, and D.–l. Zhong, “Yangian symmetry for fishnet Feynman graphs,” Phys. Rev. D, 96, 121901 (2017); arXiv:1708.00007v1 [hep–th] (2017).
D. Grabner, N. Gromov, V. Kazakov, and G. Korchemsky, “Strongly γ–deformed N=4 SYM as an integrable CFT,” Phys. Rev. Lett., 120, 111601 (2018); arXiv:1711.04786v3 [hep–th] (2017).
V. Kazakov and E. Olivucci, “Bi–scalar integrable CFT at any dimension,” Phys. Rev. Lett., 121, 131601 (2018); arXiv:1801.09844v3 [hep–th] (2018).
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Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 198, No. 2, pp. 292–308, February, 2019.
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Lee, R.N., Onishchenko, A.I. Toward an Analytic Perturbative Solution for the Abjm Quantum Spectral Curve. Theor Math Phys 198, 256–270 (2019). https://doi.org/10.1134/S0040577919020077
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DOI: https://doi.org/10.1134/S0040577919020077