QCD pomeron from AdS/CFT Quantum Spectral Curve
JHE
The Strand 1 2 4 8 9 10 11
London WC 1 2 4 8 9 10 11
R 1 2 4 8 9 10 11
LS 1 2 4 8 9 10 11
U.K. 1 2 4 8 9 10 11
Open Access 1 2 4 8 9 10 11
c The Authors. 1 2 4 8 9 10 11
0 St. Petersburg INP
1 24 , rue Lhomond, 75005 Paris , France
2 Orme des Merisiers, CEA Saclay 91191 Gif-sur-Yvette Cedex , France
3 Moscow Institute of Physics and Technology
4 Mikhail Alfimov
5 Mathematics Department, King's College London
6 Institut de Physique Theorique
7 P.N. Lebedev Physical Institute
8 Place Jussieu , 75005 Paris , France
9 Gatchina , 188 300, St. Petersburg , Russia
10 Institutskiy per. 9, 141700 Dolgoprudny , Russia
11 Leninskiy pr. 53, 119991 Moscow , Russia
Using the methods of the recently proposed Quantum Spectral Curve (QSC) originating from integrability of N = 4 Super-Yang-Mills theory we analytically continue the scaling dimensions of twist-2 operators and reproduce the so-called pomeron eigenvalue of the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation. Furthermore, we recovered the Faddeev-Korchemsky Baxter equation for Lipatov's spin chain and also found its generalization for the next-to-leading order in the BFKL scaling. Our results provide a non-trivial test of QSC describing the exact spectrum in planar N = 4 SYM at infinitely many loops for a highly nontrivial non-BPS quantity and also opens a way for a systematic expansion in the BFKL regime.
Quantum; Spectral; Integrable Field Theories; AdS-CFT Correspondence; QCD
1 Introduction
Motivation from weak coupling
3 Quantum spectral curve — generalities
4 Quantum spectral curve for twist-2 operators 4.1 4.2 4.3
Leading order solution for Pµ -system
Next-to-Leading-Order solution
Going to the next sheet
4.4 LO BFKL dimension
5 Discussion
A Derivation of the 4-th order equation (3.2) for Q
B Computation of Pj in the NLO
C NLO solution for Q
scribed by the recently proposed in [1, 2] Quantum Spectral Curve (QSC) equations,
following a long and successful study of this problem during the last decade [3]. The
paper [2] generalizes [1] to an arbitrary operator/state of the theory and reveals its
general mathematical structure in terms of the analytic Q-system. The QSC approach has
already a history of a few non-trivial tests and applications. In the weak coupling limit,
reproduced [1] and then the method was applied for calculating the dimension of Konishi
operator at 10 loops [4]. For the small S expansion of anomalous dimension of twist-2
in [6]. The results for the cusp anomalous dimension at small angle of the cusp, known
from localization [7] and TBA [6, 8, 9], we reproduced in an elementary way from QSC
in [1]. The QSC method was recently generalized to the case of ABJM theory [10] which
allowed the efficient calculation of the ABJM slope function and helped to identify the
mysterious interpolating function fixing the dependence of dispersion relation on the ’t Hooft
for this model.
Here we demonstrate another application of the QSC to an important problem —
belonging to the sl2 sector in the BFKL limit, corresponding to a double scaling regime
of small ’t Hooft constant g ≡
dimension, obtained in [13–15] from the direct re-summation of Feynman graphs:
2 − 2
Mills theory in the planar limit since only the gluons appear inside the Feynman diagrams
characteristic function (1.1) was shown. This formula was a result of a long and remarkable
history of applications of the BFKL method to the study of Regge limit of high energy
22–29]. The effective action for the high-energy processes in nonabelian gauge theories was
derived in [30]. Recently, certain scattering amplitudes describing the adjoint sector (single
reggeized gluon) were computed by means of all loop integrability in the BFKL limit [31]
in the integrable polygonal Wilson loop formalism [32].
To recover the formula (1.1) from our QSC approach, we will have to compute certain
quantities not only in the LO, but also in the NLO. In particular, we extract from the
analytic Q-system describing QSC the Baxter-Faddeev-Korchemsky equation for the pomeron
wave function [21, 33–37] in the LO and generalize it to the NLO. Some other ingredients
the LO or even up to NLO. These calculations lay out a good basis for the construction
known only up to NLO correction to (1.1) from the direct computation of [15, 38, 39].
Our method, designed here for the case of pomeron singularity (a bound state of two
reggeized gluons) should be applicable to the study of a bound state of L reggeized gluons
Let us stress that the main result of this paper — the correct reproduction of this
formula from the QSC — is a very non-trivial test of the QSC as well as of the whole
integrability approach to planar AdS/CFT spectrum. It sums up an infinite number of
the so-called wrapping corrections [40, 41] providing a test for infinitely many loops for a
highly nontrivial non-BPS quantity.
1Recently tested by a heroic strong coupling two loop calculation in [12].
To get an idea of the QSC it is instructive to start from t (...truncated)