Octagon at finite coupling
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Springer
Received: May 15, 2020 Accepted: June 30, 2020 Published: July 29, 2020
Octagon at finite coupling
a
Department of Physics, Arizona State University, Tempe, AZ 85287-1504, U.S.A. b Institut de Physique Th´eorique,1 Universit´e Paris Saclay, CNRS, CEA, 91191 Gif-sur-Yvette, France
E-mail: [email protected], [email protected] Abstract: We study a special class of four-point correlation functions of infinitely heavy half-BPS operators in planar N = 4 SYM which admit factorization into a product of two octagon form factors. We demonstrate that these functions satisfy a system of nonlinear integro-differential equations which are powerful enough to fully determine their dependence on the ’t Hooft coupling and two cross ratios. At weak coupling, solution to these equations yields a known series representation of the octagon in terms of ladder integrals. At strong coupling, we develop a systematic expansion of the octagon in the inverse powers of the coupling constant and calculate accompanying expansion coefficients analytically. We examine the strong coupling expansion of the correlation function in various kinematical regions and observe a perfect agreement both with the expected asymptotic behavior dictated by the OPE and with results of numerical evaluation. We find that, surprisingly enough, the strong coupling expansion is Borel summable. Applying the Borel-Pad´e summation method, we show that the strong coupling expansion correctly describes the correlation function over a wide region of the ’t Hooft coupling. Keywords: Integrable Field Theories, Conformal Field Theory ArXiv ePrint: 2003.01121
1
Unit´e Mixte de Recherche 3681 du CNRS.
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP07(2020)219
JHEP07(2020)219
A.V. Belitskya,b and G.P. Korchemskyb
Contents 1 Introduction
2 4 4 5 6 8 9 10
3 Octagon at weak coupling
11
4 Octagon at strong coupling 4.1 Leading order 4.2 Beyond leading order
14 15 16
5 Strong coupling expansion 5.1 Next-to-leading order 5.2 Solution at finite z 5.2.1 Quantization conditions 5.2.2 Profile function
17 17 19 21 22
6 Properties of strong coupling expansion 6.1 Improved expansion 6.2 Kinematical limits
23 23 24
7 Numerical checks 7.1 Borel-Pad´e improvement 7.2 Order of limits phenomenon
29 29 30
8 Conclusions
31
A Analytical regularization
33
B Quantization condition from zeroth moment
35
C Matrix representation
37
D Profile function in the null limit
38
–1–
JHEP07(2020)219
2 Octagon as a Fredholm determinant 2.1 Kinematical limits 2.2 Determinant representation of the octagon 2.3 Similarity transformation 2.4 Modified Bessel kernel 2.5 Method of differential equations 2.6 Moments
1
Introduction
G4 = hO1 (x1 )O2 (x2 )O3 (x3 )O1 (x4 )i =
G(z, z¯)
(x212 x213 x224 x234 )K/2
,
(1.1)
where the four operators are built out of two complex scalars Z and X and their complex ¯ K/2 ) + permutations, O2 = tr(X K ) and O3 = tr(Z¯ K ). conjugate partners, O1 = tr(Z K/2 X Here G(z, z¯) is a function of
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