N=4 SYM Quantum Spectral Curve in BFKL regime
PPrepared for submission to JHEP N = 4 SYM Quantum Spectral Curve in BFKL regime
Mikhail Alfimov , , , Nikolay Gromov , , and Vladimir Kazakov National Research University Higher School of Economics, ul. Usacheva, d. 6, Moscow, RussianFederation 119048 National Research Nuclear University Moscow Engineering Physical Institute, Kashirskoe sh., d.31, Moscow, Russia 115409 P.N. Lebedev Physical Institute of the Russian Academy of Sciences, Leninskiy pr., d. 53, Moscow,Russia 119991 Mathematics Department, King’s College London, The Strand, London WC2R 2LS, UK St.Petersburg INP, Gatchina, 188 300, St.Petersburg, Russia Laboratoire de Physique de l’ ´Ecole Normale Sup´erieure, ENS, Universit´e PSL, CNRS, SorbonneUniversit´e, Universit´e de Paris, F-75005 Paris, France
E-mail: [email protected] , [email protected] , [email protected]
Abstract:
We review the applications of the Quantum Spectral Curve (QSC) method to theRegge (BFKL) limit in N = 4 supersymmetric Yang-Mills theory. QSC, based on quantumintegrability of the AdS /CFT duality, was initially developed as a tool for the study ofthe spectrum of anomalous dimensions of local operators in the N = 4 SYM in the planar, N c → ∞ limit. We explain how to apply the QSC for the BFKL limit, which requiresnon-trivial analytic continuation in spin S and extends the initial construction to non-locallight-ray operators. We give a brief review of high precision non-perturbative numericalsolutions and analytic perturbative data resulting from this approach. We also describe asa simple example of the QSC construction the leading order in the BFKL limit. We showthat the QSC substantially simplifies in this limit and reduces to the Faddeev-KorchemskyBaxter equation for Q-functions. Finally, we review recent results for the Fishnet CFT,which carries a number of similarities with the Lipatov’s integrable spin chain for interactingreggeized gluons. a r X i v : . [ h e p - t h ] M a r ontents N = 4 SYM 3 N = 4 SYM 72.4 Numerical solution 132.5 Analytic results from QSC 15 The Balitsky-Fadin-Kuraev-Lipatov (BFKL) approximation [1, 2] in Quantum Chro-modynamics (QCD) has marked the beginning of a new era in the study of the propertiesof hadron collisions at high energy. Among various important results following BFKL ap-proach, the Pomeron spectrum was calculated in the leading [1–3], and then, 20 years later,in the next-to-leading [4] logarithmic approximations (LO and NLO). The complexity ofthe conventional Feynman perturbation theory increases dramatically at each order andit is unlikely that the NNLO calculation in QCD is reachable with the currently availabletechniques.The history of integrability in gauge theories starts from the renowned Lev Lipatov’swork [5], where the effective Lagrangian for reggeized gluons in QCD was shown to be in-tegrable. Lipatov was also the first to notice the equivalence of this system to the SL (2 , C )integrable quantum spin chain [6]. Following the seminal paper [7], the integrability ap-proach to Lipatov’s model has been substantially advanced in [8–11].The BFKL approximation was then successfully applied to the study of the spectrumof anomalous dimensions of certain operators in N = 4 SYM theory [12]. The resultof this calculation appears to be slightly less complex than the original Regge limit inQCD, possibly due to the super-symmetry. As it was noticed in [12, 13] the result followsthe so-called maximal transcendentality selection principle – only the most complicatedin transcedentality parts of the QCD result are present in its N = 4 SYM counterpart,– 1 –ith exactly the same coefficients . Even in this, maximally super-symmetric theory thecomplexity of the traditional Feynman perturbation technique increases factorially at eachorder. However, a powerful alternative method is available in this theory: due to thequantum integrability of planar N = 4 SYM theory it became possible to compute thespectrum of the planar N = 4 SYM theory at any value of the ’t Hooft coupling λ =16 π g = g Y M N c ! This integrability was discovered at one loop in the seminal Minahan-Zarembo paper [15] , then it was demonstrated at two-loops [18], then at strong couplingin [19–21], via AdS/CFT correspondence, then generalized up to wrapping order in [22–25], and finally to all loops, first in terms of the exact Y-system [26], and then of theTBA equations [27–29] for the simplest Konishi-like operators. A few years ago, this lineof research resulted in the final formulation of the solution of the spectral problem – theQuantum Spectral Curve (QSC) of the AdS /CFT duality [30, 31].The QSC equations can be formulated as a finite system of Riemann-Hilbert-type equa-tions on 4+4 Baxter Q-functions. The QSC formalism has led to the tremendous progressin high precision numerical and multi-loop computations in a multitude of particular phys-ical quantities (see the reviews [32, 33] and citations therein). Notably, the dimension ∆ oftwist-2 operator of the type tr( Z ∇ S + Z ) was computed for virtually arbitrary coupling andarbitrary complex values of spin S [34] (see the Figure 1 and Figure 2). The particularcorner of this picture, when S (cid:39) − O ( g ) at weak coupling g →
0, corresponds to theBFKL regime. The loop expansion of S can be written as follows S = − g χ LO (∆) + g χ NLO (∆) + g χ NNLO (∆) + g χ NNNLO (∆) + O ( g ) , (1.1)where χ N k LO (∆) is the BFKL kernel eigenvalue at the ( k + 1)-st loop. The BFKL limit ofQSC was first explored in [35] and the LO BFKL spectrum χ LO (∆) with zero conformalspin was reproduced there. The coefficient χ NLO (∆) calculated in [12] was successfully re-produced by the QSC calculations in [36]. Then, significant progress was made in the samework [36], where QSC allowed to derive the NNLO BFKL kernel eigenvalue χ NNLO (∆),which was later confirmed by the other method in [37]. It should be noted that all theBFKL kernel eigenvalues up to NNLO included can be written in terms of nested harmonicsums and multiple zeta values. Moreover, the high precision QSC numerical technique [34]is currently adapted in [38]: for any order χ N k LO (∆) can be analyzed by the QSC nu-merical algorithm. Moreover, the QSC approach allowed to compute anomalous dimen-sions of the length-2 operators with both non-zero spins, including the conformal one n ,tr( Z ∇ S + ∇ n ⊥ Z ) [38].In this short review, we give a concise formulation of the QSC and its reduction inthe case of BFKL limit. We will briefly review the ideas of numerical and perturbative See also [14]. The first glimpses on such integrability can be found in L.Lipatov’s talks [16, 17]. In the case of non-zero conformal spin some data connected to the NNNLO BFKL kernel eigenvalueswere extracted from QSC in [38], namely: the intercept function (BFKL kernel eigenvalue at ∆ = 0)for different values of conformal spin, slope-to-intercept function (BFKL kernel eigenvalue derivative withrespect to conformal spin) and curvature function (BFKL kernel eigenvalue 2nd derivative with respect to∆), both calculated at the special BPS point of dimension 0 and conformal spin 1. – 2 –pproaches to the solution of the QSC equations. We also shortly describe the mostimportant numerical and perturbative analytic results for the BFKL limit of N = 4 SYMtheory.The BFKL also gave an inspiration for a number of other research directions over theyears. In particular, we will explain here how the BFKL limit influenced the currently beingactively developed so-called fishnet conformal field theory (FCFT) proposed in [39] as adouble scaling limit combining weak coupling and strong gamma -deformation of N = 4SYM theory. Fishnet CFT attracted much of attention in the last few years [40 ? –58].It was generalized to any dimension D [46], and for D = 2 and a specific representationfor conformal spins this model can be identified (for certain physical quantities) with theLipatov’s SL (2 , R ) spin chain for LO approximation of BFKL. As Fishnet CFT has asignificant interplay with the initial ideas of BFKL integrability, we briefly review here theconstruction and integrability properties of fishnet CFT. N = 4 SYM
In the present Section we briefly review the integrable structure of N = 4 SYM. Weconnect the BFKL kernel eigenvalues to the dimensions of the twist-2 operators follow-ing [12], which allows us to apply the QSC to the study of the BFKL spectrum. Thisstep requires certain generalisation of the gluing condition in QSC for the local operators,allowing for non-integer spins. As an example we demonstrate how to reproduce the Bax-ter equation, which initially appeared in the problem of diagonalization of the LO BFKLkernal. Let us start from briefly describing the relation of the BFKL spectrum in N = 4 SYMto those in QCD. The key statement governing this connection is the so-called principle ofmaximal transcendentality [12, 13]. One can find a detailed description of this principle in[59]. Below we demonstrate this principle in application to the BFKL kernel eigenvalues.In what follows we use the extension of the formula (1.1) for the case of non-zero conformalspin n S = − g χ LO (∆ , n ) + g χ NLO (∆ , n ) + (2.1)+ g χ NNLO (∆ , n ) + g χ NNNLO (∆ , n ) + O ( g ) . Consider first the LO BFKL Pomeron kernel eigenvalues for QCD and N = 4 SYMwhich can be found in [1, 2] and [12] respectively. We see that the answers are identicaland are given by χ LO (∆ , n ) = − (cid:18) ψ (cid:18) n + ∆2 (cid:19) + ψ (cid:18) n − ∆2 (cid:19) − ψ (1) (cid:19) , (2.2)where ∆ is the dimension of the state and n is the conformal spin. Moreover, in the LOthe BFKL kernel eigenvalues coincide for all 4D gauge theories [12].– 3 –or the next order in perturbation theory the situation is more subtle. The NLOBFKL kernel eigenvalue for QCD was calculated in [4] χ QCDNLO (∆ , n ) = 4 (cid:20) − (cid:18) n, − ∆2 (cid:19) − (cid:18) n, (cid:19) + 6 ζ (3)+ (2.3)+ (cid:18) − ζ (2) − n f N c (cid:19) χ LO (∆ , n )4 + ψ (cid:48)(cid:48) (cid:18) n − ∆2 (cid:19) + ψ (cid:48)(cid:48) (cid:18) n + ∆2 (cid:19) −− (cid:18) − n f N c (cid:19) (cid:18) χ (∆ , n )16 − ψ (cid:48) (cid:18) n + ∆2 (cid:19) + ψ (cid:48) (cid:18) n − ∆2 (cid:19)(cid:19) ++ π cos π (1+∆)2 ∆ sin π (1+∆)2 (cid:18)(cid:18) (cid:18) n f N c (cid:19) − − (cid:19) δ n − (cid:18) n f N c (cid:19) ∆ − − δ n (cid:19)(cid:21) , where N c is the number of colors and n f is the number of flavours of the fermions in theconsidered version of QCD. The same quantity in N = 4 SYM was first presented in [12].We write it down here from [13] χ N =4NLO (∆ , n ) = 4 (cid:20) − (cid:18) n, − ∆2 (cid:19) − (cid:18) n, (cid:19) + 6 ζ (3)+ (2.4)+ 14 (cid:18) − ζ (2) (cid:19) χ LO (∆ , n ) + ψ (cid:48)(cid:48) (cid:18) n − ∆2 (cid:19) + ψ (cid:48)(cid:48) (cid:18) n + ∆2 (cid:19)(cid:21) . The function Φ( n, x ) in the eigenvalues (2.3) and (2.4) is equal toΦ( n, x ) = ∞ (cid:88) k =0 ( − k +1 k + x + n/ (cid:104) ψ (cid:48) ( k + n + 1) − ψ (cid:48) ( k + 1) + ( − k +1 ( β (cid:48) ( k + n + 1)++ β (cid:48) ( k + 1)) + 1 k + x + n/ ψ ( k + n + 1) − ψ ( k + 1)) (cid:21) (2.5)and β (cid:48) ( z ) = 14 (cid:20) ψ (cid:48) (cid:18) z + 12 (cid:19) − ψ (cid:48) (cid:16) z (cid:17)(cid:21) = + ∞ (cid:88) k =0 ( − k +1 ( z + k ) . (2.6)Now we are going to explain the maximal transcendentality principle with the exampleof NLO BFKL kernel eigenvalue. There is a way to rewrite both χ LO (∆ , n ) (2.2) and χ NLO (∆ , n ) for both QCD (2.4) and N = 4 SYM (2.4) in terms of nested harmonic sums.In the LO we get simple result χ LO (∆ , n ) = − (cid:18) S (cid:18) n − (cid:19) + S (cid:18) n − ∆2 − (cid:19)(cid:19) , (2.7)from which we see that the LO BFKL kernel eigenvalues in QCD and N = 4 SYM havetranscendentality 1, where the transcendentality of a harmonic sum S n ,...,n m is defined to be | n | + · · · + | n m | , transcendentality of a product is given by a sum of the transcendentalitiesof the multipliers. In the same way one defines the transcendentality of the MZV ζ n ,...,n m ,which is given by the sum n + · · · + n m . In particular, π and log 2 have transcedentality 1. Our identification of the variable γ from [12] with our parameters is: γ = (∆ − S ) /
2, where ∆ is thedimension and S is the spin. – 4 –o express (2.3) and (2.4) through nested harmonic sums we need to introduce someadditional formulas. In [60] it was shown, that for n = 0 we can represent the NLO BFKLkernel eigenvalue for N = 4 SYM as follows χ N =4NLO (∆ ,
0) = F (cid:18) (cid:19) + F (cid:18) − ∆2 (cid:19) , (2.8)where F ( x ) = 4 (cid:18) − ζ (3) + π log 2 + π S ( x − π S − ( x −
1) + 2 S ( x − − S − , ( x − (cid:1) . Utilizing (2.8) and (2.9), we are able to rewrite (2.4) in the following way χ N =4NLO (∆ , n ) = 12 (cid:18) F (cid:18) n (cid:19) + F (cid:18) − ∆ − n (cid:19) + F (cid:18) − n (cid:19) ++ F (cid:18) − ∆ + n (cid:19)(cid:19) + 4 (cid:18) R n (cid:18) − ∆2 (cid:19) + R n (cid:18) (cid:19)(cid:19) , (2.10)where R n ( γ ) = − (cid:18) S − (cid:16) γ + n − (cid:17) + π (cid:19) (cid:16) S (cid:16) γ + n − (cid:17) − S (cid:16) γ − n − (cid:17)(cid:17) . (2.11)Therefore, from (2.7), (2.8), (2.9), (2.10) and (2.11) we see that all the contributions to(2.10) have transcendentality 3. Furthermore, we see that all the terms of the QCD resultwith the maximal transcendentality 3 coincide exactly with the result of N = 4 SYM !In the next part we will explain the relation of the BFKL kernel eigenvalues to thespectrum of length-2 (twist-2 in the case of zero conformal spin) operators in N = 4 SYM.Then we will explain how to adapt the QSC for the local operators to describe non-integerspin and conformal spin, which would then lead to the BFKL kernel eigenvalue. In this part we describe the analytic structure of the length-2 operators with conformalspin in N = 4 SYM. Namely, we restrict ourselves to the states with the quantum numbers J = 2, J = J = 0. Let us first consider the case of zero conformal spin S = 0, whichare called the twist-2 sl (2) operators O = tr ZD S + Z + (permutations) . (2.12)In the gauge theory the physical operators should have even number of derivatives to beconformal primaries. The dimensions of these physical operators ∆ were calculated forarbitrary even integer S up to several loops order [13, 17, 61]. In [62] it was understoodhow to describe these operators with integer spins within the QSC approach to the leadingorder in the perturbation theory. This technique was then generalised to the higher ordersin [63–67]. – 5 –s we explain below in order to make a connection with the Regge regime one needsto be able to analytically continue in S . In the works [68, 69] it was understood how toremove the constraint of integer even spin S from the integrable description, leading to theanalytic continuation of the anomalous dimension of the twist-2 sl (2) operators. At the1-loop order one gets the following result [13, 70]∆ = 2 + S + 8 g ( ψ ( S + 1) − ψ (1)) + O ( g ) . (2.13)As we can see the formula above (2.13) is regular for all S > −
1, but has a simple poleat S = −
1. In fact as now both S and ∆ are allowed to be non-integer we can invertthe function for the range where it is regular and instead consider S (∆) which is a moreconvenient function [13, 60, 71–73]. In particular (2.13) leads to S ∆ > (∆) = ∆ − − g ( ψ (∆ − − ψ (1)) + O ( g ) . (2.14)Below we describe the method which would allow us to compute this function in N = 4SYM at finite coupling (see the Figure 1 for n = 0). At the moment we only need to knowits main features. We see that S (∆) is an even function of ∆. At finite g it has a smoothparabolic shape, however, at weak coupling it becomes piece-wise linear: S = | ∆ | − | ∆ | > S = − | ∆ | <
1. Thus the g → S does not commute with the g → S (∆) is a smooth function at finite g , it generates two different series expansionsin g , depending on the range of ∆. Within this picture the BFKL expansion is the seriesexpansion of S (∆) for | ∆ | < g . Note that all physical operators belong tothe opposite region ∆ > Z )), and thus the equation (2.14) is only valid for ∆ >
1. At the same time for | ∆ | < S | ∆ | < (∆) = − g (cid:18) − ψ (cid:18) (cid:19) − ψ (cid:18) − ∆2 (cid:19) + 2 ψ (1) (cid:19) + O ( g ) . (2.15)To understand how (2.14) and (2.15) are related to each other we have to complexify ∆and S . We explain below how this can be done using QSC at finite g , and the resultingfunction Re S (∆) is presented at the Figure 2. We see that the function S (∆) has severalsheets, which are connected through a branch cut. When coupling goes to zero the branchpoints collide forming the singularity in the function S (∆). The domains | ∆ | < > S | ∆ | < (∆) + S ∆ > (∆)should be regular around ∆ = 1 at any order in g , as in this combination the branchcut cancels. Indeed, one can check that the simple poles in the both functions around∆ = 1 has residues opposite in sign and disappear in the sum at order g . At the higherorders the poles at ∆ = 1 become more and more severe, but this cancellation also can– 6 – -1 |n|+1-|n|-1 -|n|-2 Figure 1 . Trajectory of the length-2 operator for conformal spin n = S as a function of the fulldimension ∆. The dots correspond to the physical operators with S + n ∈ Z ≥ . be verified explicitly to all known orders. This requirement gives non-trivial relationsbetween the functions S ∆ > (∆), obtained perturbatively as an analytic continuation fromthe dimensions of physical operators, and the BFKL kernel eigenvalue S | ∆ | < (∆).Now we are ready to turn to non-zero conformal spin. Namely, non-zero conformalspin adds the derivative in the orthogonal direction to the operators (2.12) O = tr ZD S + ∂ S ⊥ Z + (permutations) . (2.16)The physical states now correspond to non-negative integer S and S , whose sum is even.We follow the same strategy for (2.16) as for the case of zero conformal spin. Analogously,having the anomalous dimensions for the physical operators, we can build the analyticcontinuation in the spins S and S and identify them with the spin S and conformal spin n respectively. This analytic continuation is illustrated with the Figure 1. The physicaloperators are designated with the dots on the operator trajectory. As in the case of zeroconformal spin exchange of the roles of ∆ and S = S allows us to reach the BFKL regime.Having these analytic properties in mind, we understand how to apply the QSC tothe study of the BFKL spectrum: we need to analytically continue to non-integer spin S and conformal spin n not only the anomalous dimensions, but the QSC itself: Q-functions,their asymptotics and analytic structure etc. In the next section we are going to startfrom the brief description of the QSC basics. Then we will use this setup to consider thecalculation of the LO BFKL kernel eigenvalue with non-zero conformal spin by the QSCmethod. N = 4 SYM
We are going to present here the formulation of the QSC in terms of the Q-systemand gluing conditions, the details of which can be found in [30, 32, 38]. Algebraic part ofthe QSC framework is described as follows. For the N = 4 SYM we have the system of 2 – 7 – igure 2 . Riemann surface of the function S (∆) for twist-2 operators. Q-functions, which are denoted as Q a ,...,a n | i ,...,i m ( u ) , ≤ n, m ≤ . (2.17)The Q-functions (2.17) have two groups of indices: a ’s are called “bosonic” and i ’s arecalled “fermionic”. They are antisymmetric with respect to the exchange of any pair ofindices in these two groups. Not all of the 256 Q-functions in question are independent.They are subject to the set of the so-called Pl¨ucker’s QQ-relations, which are written in[30] Q A | I Q Aab | I = Q + Aa | I Q − Ab | I − Q − Aa | I Q + Ab | I , (2.18) Q A | I Q A | Iij = Q + A | Ii Q − A | Ij − Q − Aa | Ii Q + Ab | Ij ,Q Aa | I Q A | Ii = Q + Aa | Ii Q − Ab | I − Q − Aa | Ii Q + A | I , where A and I are multi-indices from the set { , , , } .The standard normalization is chosen to be Q ∅|∅ = 1 . (2.19)Then the structure of the QQ-relations allows to express the whole Q-system in terms ofthe “basic” set of 8 Q-functions Q a |∅ , a = 1 , . . . , Q ∅| i , i = 1 , . . . , • Imposing the so-called unimodularity condition Q | = 1 , (2.20)– 8 –e are able to introduce the system of Hodge-dual Q-functions Q a ...a n | i ...i m ≡ ( − nm (cid:15) b n +1 ...b a ...a n (cid:15) j m +1 ...j i ...i m Q b n +1 ...b | j m +1 ...j , (2.21)where there is no summation over the repeated indices and which satisfy the sameQQ-relations (2.18). The condition (2.20) allows to write down the following relationsbetween the Q-functions with the lower and upper indices Q a | i Q b | j = − δ ij , Q a | i Q b | i = − δ ab , (2.22) Q a |∅ = ( Q a | i ) + Q ∅| i , Q ∅| i = ( Q a | i ) + Q a |∅ ,Q a |∅ Q a |∅ = 0 , Q ∅| i Q ∅| i = 0 . • Another symmetry is called the H-symmetry. Its general form is given by Q A | I → (cid:88) | B | = | A | , | J | = | I | ( H [ | A |−| I | ] b ) BA ( H [ | A |−| I | ] f ) JI Q B | J , (2.23)where the sum goes over the repeated multi-indices. The definition of H JI is thefollowing: ( H f ) JI ≡ ( H F ( u )) j i ( H F ( u )) j i . . . ( H F ( u )) j | I | i | I | and the same for ( H b ) BA and( H B ( u )) b k a k , k = 1 , . . . , | A | with H B,F ( u ) being 4 × i -periodic matrices. The uni-modularity condition leads us to the restrictiondet H B ( u ) det H F ( u ) = 1 . (2.24)To proceed with the calculation of the spectrum of N = 4 SYM we have to endowthe described Q-system with the analytic structure. To start with, we designate the basicQ-functions with this structure as P a (same as Q a |∅ ), P a ( Q a |∅ ), Q i ( Q ∅| i ) and Q i ( Q ∅| i ).One can find the asymptotics of the Q-functions in [32] P a (cid:39) A a u − ˜ M a , P a (cid:39) A a u ˜ M a − , Q i (cid:39) B i u ˆ M i − , Q i (cid:39) B i u − ˆ M i , (2.25)where ˜ M a , a = 1 , . . . , M i , i = 1 , . . . , , |
4) (i.e. quantum numbers of the states)˜ M a = (cid:26) J + J − J , J − J + J , − J + J + J , − J − J − J (cid:27) , ˆ M i = (cid:26)
12 (∆ − S + + 2) ,
12 (∆ + S + ) ,
12 ( − ∆ − S − + 2) ,
12 ( − ∆ + S − ) (cid:27) , (2.26)where S ± = S ± S .As the Q-system is generated by the set of 8 basic Q-functions, we first ascribe themthe analytic structure dictated by the classical limit of the Q-functions [32]. The minimalchoice of the cut structure consistent with the asymptotics (2.25) is presented on the Figure3. Then, according to the (2.18) we can generate two versions of the Q-system: upper halfplane and lower half plane analytic (UHPA and LHPA respectively). In what follows wewill define by Q a | i the UHPA solution of the one of the QQ-relations from (2.18) Q + a | i − Q − a | i = P a Q i (2.27)– 9 – a P a − g g − g g Q i Q i Figure 3 . Analytic structure of the P - and Q -functions on their defining sheet. with the large u asymptotic Q a | i (cid:39) − i A a B i − ˜ M a + ˆ M i u − ˜ M a + ˆ M i . (2.28)Substitution of the formulas Q i = −Q + a | i P a or P a = −Q a | i Q i into (2.27) allows to fix theproducts of the coefficients of the leading asymptotics of the P - and Q -functions and for a , i = 1 , . . . , A a A a = i (cid:81) j =1 (cid:16) ˜ M a − ˆ M j (cid:17)(cid:81) b =1 b (cid:54) = a (cid:16) ˜ M a − ˜ M b (cid:17) , B i B i = − i (cid:81) a =1 (cid:16) ˆ M i − ˜ M a (cid:17) (cid:81) j =1 j (cid:54) = i (cid:16) ˆ M i − ˆ M j (cid:17) , (2.29)where there is no summation over the repeated indices a and i .An important consequence of the formula Q i = −Q a | i P a combined with the formula(2.27) is the existence of a 4th order Baxter equation for the functions Q i , i = 1 , . . . , Q [+4] i − Q [+2] i (cid:104) D − P [+2] a P a [+4] D (cid:105) + Q i (cid:104) D − Q a P a [+2] D + P a P a [+4] D (cid:105) −− Q [ − i (cid:104) ¯ D + P [ − a P a [ − ¯ D (cid:105) + Q [ − i = 0 , (2.30)where the functions D , D and D are given by D = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P P P P P P P P P − P − P − P − P − P − P − P − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , D = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( P P P P P P P P P − P − P − P − P − P − P − P − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,D = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P P P P P P P P P − P − P − P − P − P − P − P − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (2.31)while the bars over ¯ D and ¯ D are understood as the complex conjugation ¯ f ( u ) = f (¯ u ) ofthe functions defined above. The same equation for Q i , i = 1 , . . . , P a into P a for a = 1 , . . . , P - and Q -functions do not– 10 –ontradict each other. But as we approach the real axis, this is no longer the case and wehave to cross the cut. This makes the QQ-relations ambiguous in the vicinity of the cuts,as the shifts by ± i/ u ± i/ u or, in other words, which branch of the multi-valued function touse. The way to resolve the analytic continuation ambiguity is to interpret the values ofthe Q -function in the upper half plane, far enough from the cuts, as a Q -function withthe upper index Q i and in the lower half plane the same Q -function should obey the QQ-relations as if it was a function with lower index Q ↑ i . I.e. in the lower half plane thisfunction satisfies (2.30), whereas in the upper half plane it satisfies the same equation with P a interchanged with P a . The ↑ is added to indicate that Q ↑ i does not have cuts in thelower half plane. To close the system of equations we notice that there is yet another wayto build the set of functions Q ↑ i , satisfying (2.30) and having no cuts in the lower halfplane. One can, starting from Q i , which has no cuts in the upper half plane and P a – andbuild Q ↓ i using QQ-relations. Then the complex conjugate of Q ↓ i will satisfy (2.30) andwill have no cuts in the lower half plane.Indeed, due to the property, valid for real S , S and ∆ we can always assume that¯ P a = C ba P b , ¯ P a = − C ab P b , C = diag { , , − , − } (2.32)due to the H-symmetry. Thus if we complex conjugate the Baxter equation (2.30), we getthe same equation for ¯ Q j . This implies that Q ↓ i ( u ) is a linear combination of ¯ Q j withregular coefficients Q i ( u ) = M ij ( u ) ¯ Q j ( u ) , Q i ( u ) = (cid:0) M − t (cid:1) ij ( u ) ¯ Q j ( u ) , (2.33)where − t denotes the inversion and transposition of the matrix. Technically it will be moreconvenient to work with short cuts [ − g, g ]. We can connect the branch points of Q theway we like, but this would create an infinite ladder of cuts in the lower half plane. It willalso modify the notion of the conjugate function, as the conjugation now will also involvethe analytic continuation under the cut. So we conclude that in the short cut conventionsthe gluing condition reads as˜ Q i ( u ) = M ij ( u ) ¯ Q j ( u ) , ˜ Q i ( u ) = (cid:0) M − t (cid:1) ij ( u ) ¯ Q j ( u ) , (2.34)which we will use further. Let us now list the properties of M ij ( u ), which is called thegluing matrix. The details of the derivation of these properties can be found in [38]. Theyare: • M ij ( u ) is an i -periodic matrix of H-transformation. • M ij ( u ) is analytic in the whole complex plane. • M ij ( u ) is hermitian as a function¯ M ij ( u ) = M ji ( u ) . (2.35)– 11 –here exist another additional symmetry of the P -functions. The P -functions of thestates with the Cartan charges˜ M a = { , , , − } , (2.36)ˆ M i = (cid:26)
12 (∆ − S + + 2) ,
12 (∆ + S + ) ,
12 ( − ∆ − S − + 2) ,
12 ( − ∆ + S − ) (cid:27) , where S ± ≡ S ± S , have the certain parity P a ( − u ) = ( − a +1 P a ( u ) , P a ( − u ) = ( − a P a ( u ) . (2.37)The conjugation symmetry and the symmetry above (2.37) lead to two additional con-straints [38] on the gluing matrix. Let us first concentrate on the physical states, when S and S are integer and have the same parity . As it follows from the power-like large u asymptotics of the Q -functions in this case, the only possible ansatz for the gluing matrixis a constant matrix. In [30, 38] it was shown that for the physical length-2 states with thecharges (2.36) from the abovementioned properties of the gluing matrix and two additionalconstraints mentioned earlier it follows that M ij (cid:16) e iπ ( ˆ M i − ˆ M j ) + 1 (cid:17) = 0 . (2.38)From (2.38), we immediately see that only when the difference between the charges isan odd integer, M ij is non-zero. It is the case only forˆ M − ˆ M = − S − S + 1 , (2.39)ˆ M − ˆ M = − S + S + 1 . Therefore, taking into account the hermiticity of the gluing matrix, for integer S and S ,that have the same parity (i.e. for the physical states) the equations (2.38) lead to thegluing matrix M ij = M M M M . (2.40)In the case of at least one of the spins S and S being non-integer, we see, that becauseall differences ˆ M i − ˆ M j are non-integer the equations (2.38) can have only zero matrix as asolution, if this matrix is assumed to be constant. This leads us to the conclusion that thegluing matrix cannot be constant anymore for non-integer spins. The minimal way to dothis keeping it i -periodic would be to add exponential contributions and get the following This is dictated by the cyclicity condition for the states in the sl (2) Heisenberg spin chain. At one loop this is dictated by the cyclicity condition for the states in the sl (2) Heisenberg spin chain,appearing in the perturbation theory. – 12 –luing matrix M = M M M M ¯ M M M M ¯ M M M ++ M M M M e πu + M M M M e − πu . (2.41)To summarize, the key observation here is that if we want to consider non-physicalstates with S or S being non-integer, this inevitably requires modification of the gluingconditions. From now on let us use the notations for the spins coming from BFKL physics S = S and S = n .In the next Subsection we will briefly describe how the QQ-relations together with thegluing condition (2.34) can be used to compute numerically the function S (∆ , n ). The QQ-system together with the gluing conditions allows for the efficient numericalalgorithm, developed in [34]. This method was described in details with the
Mathematica code attached in a recent review [32]. Here we describe the main steps very briefly.First, one notices that the P a and P a functions, a = 1 , . . . ,
4, which have only oneshort cut [ − g, g ] on the main sheet of the Riemann surface, can be parametrized veryefficiently as rapidly converging series expansions in powers of 1 /x ( u ), where x ( u ) = ( u + √ u − g √ u + 2 g ) / (2 g ). After that one can recover the whole Q-system in terms of theseexpansion coefficients. For that one first uses Q + a | i − Q − a | i = −Q + b | i P a P b , (2.42)which follows directly from (2.27) and Q i = −Q a | i P i , to solve for Q a | i . After that onefinds Q i and ˜ Q i from Q i = −Q + a | i P a , ˜ Q i = −Q + a | i ˜ P a . (2.43)Note that this involves ˜ P a , which is the same series as P a , but with x → /x for u ∈ [ − g, g ]. In this way we reconstruct both Q i and ˜ Q i in terms of the expansion coefficientsin P a and P a . Finally, one fixes these coefficients from the gluing condition (2.34).In practice for the numerical purposes one truncates the series expansion in P . Inthis case the gluing condition cannot be satisfied exactly. The strategy is to minimizethe discrepancy in the gluing condition by adjusting the coefficients numerically with avariation of a Newton method.Application of the numerical procedure allows to calculate the dependence of the inter-cept function j = S (∆ = 0 , n = 0) + 2 for zero conformal spin [74] on the ‘t Hooft coupling– 13 –
10 100 1000 λ j Figure 4 . The dependence of the intercept function j on the ‘t Hooft coupling λ in the logarithmicscale. Data obtained from the numerical procedure are depicted in red. It represents a perfectinterpolation between the weak [13, 71] and strong [60, 73–75] coupling results (blue dashes). λ , which is drawn on the Figure 4. By fitting the polynomial dependence of this quantityon the coupling, we obtain several first coefficients j ( λ ) = 1 + 0 . λ − . λ (2.44) − . λ + 0 . λ + O ( λ ) . Comparing the first two coefficients of (2.44), we find complete agreement with the LO(2.2) and NLO BFKL (2.4) kernel eigenvalues calculated at ∆ = 0 and n = 0 .Moreover, we can use our numerical algorithm to calculate not only the intercept func-tions, but also the BFKL kernel eigenvalues at different orders in g . Namely, in [36] fromfitting the numerical values of the spin S (∆ , n ) for ∆ = 0 .
45 and n = 0 for different valuesof the coupling constant one was able to extract the numerical values of the coefficients χ N k LO (∆ = 0 .
45) from (1.1), which are listed in the Table 1.
Table 1 . Numerical values of the BFKL kernel eigenvalue at different orders.Order Value ErrorN LO 10774 . − N LO − . − N LO 1 . × − N LO − . × − After renormalizing these results according to the change of the expansions parameter from g to λ . – 14 – .2 0.4 0.6 0.8 1.0 g - - - - S n = = = = Figure 5 . The dependence of the numerical values of the spin S (0 , n ) on the coupling constant g , calculated for the conformal spins n = 0, n = 3 / n = 2 and n = 3 (dots). The dashed linesrepresent weak coupling expansion and the continuous lines represent the strong coupling expansionof the spin obtained in [38]. The first line of the Table 1 contains the numerical value of the NNLO BFKL kerneleigenvalue for ∆ = 0 .
45, which was computed analytically for the first time in [36] by theQSC method. In that work there was shown that the calculated numerical value in theTable 1 coincides with the value of the analytic result for ∆ = 0 .
45 with numerical accuracy10 − .In addition, the numerical algorithm can be used to explore the spectrum for non-zerovalues of conformal spin n (to see the algorithm for different values of ∆, n and g at workone can use the Mathematica file code_for_arxiv.nb from the [38] arXiv submission).One can find the plots of the spin S (0 , n ), which differs from the intercept function j ( n )by the additive constant 2, for different (even non-integer) values of the conformal spin n on the Figure 5.To conclude, the numerical method of [34] allowed to obtain the BFKL kernel eigen-value non-perturbatively with huge precision.A number of analytic methods was developed to solve the QSC at small g [36, 63,64, 66, 67]. Unfortunately, those methods rely on a particular basis of η -functions, whichworks very well for the local operators of for BFKL eigenvalues at some integer values of ∆,but are not sufficient in general. In the next section we will use an alternative method toobtain the LO BFKL kernel eigenvalue analytically. Recently, new very promising methodswere developed in [76–78], based on the Mellin transformations, which could allow for asystematic analytic calculation of S (∆) order by order in g for generic ∆. Apart from the numerical results mentioned in the previous Section, a number of ana-lytic results related to the BFKL spectrum was obtained recently using the QSC methods– 15 –n [31, 35, 36, 38, 74, 79]. To mention a few: • NNLO BFKL kernel eigenvalue for the conformal spin n = 0. • Intercept function for arbitrary conformal spin n up to NNLO order and partial resultat NNNLO order. • Strong coupling expansion of the intercept function for arbitrary value of the confor-mal spin n . • Slope-to-slope function in the BPS point ∆ = 2, S = 0 and n = 0 at all loops. • Slope-to-intercept dS (∆ , n ) /dn and curvature d S (∆ , n ) /d ∆ functions in the BPSpoint ∆ = 0, S = − n = 1 at all loops.Here we demonstrate the power of the QSC method deriving the leading order Faddeev-Korchemsky Baxter equation with non-zero conformal spin for Lipatov spin chain analyt-ically.As explained above, we need to study the regime of the anomalous dimensions of thetwist-2 operators, when the coupling constant g →
0, such that S ≡ S (∆ , g ) becomes − O ( g ). According to the discussion in the previous section this is the case for | ∆ | < | n | . In other words one can say that we keep the ratio Λ ≡ g / ( S + 1) finite, whereasthe combination w ≡ S + 1 = O ( g ) can be used as a small expansion parameter. Thesecond spin S = n plays the role of a parameter.In order to reproduce the Faddeev-Korchemsky Baxter equation we are going to utilisethe subset of the QQ-relations known as P µ -system. The P µ -system is represented by thefunctions P a ( u ), P a ( u ), which we introduced before and of an anti-symmetric matrix µ ab ( u )(see [30, 62] for the detailed description). They satisfy the following equations˜ µ ab − µ ab = P a ˜ P b − P b ˜ P a , ˜ P a = µ ab P b , (2.45)˜ µ ab − µ ab = P a ˜ P b − P b ˜ P a , ˜ P a = µ ab P b , P a P a = 0 , µ ab µ bc = δ ca , ˜ µ ab = µ ++ ab , ˜ µ ab = µ ab ++ . Before proceeding we remind a couple of notations. Our notation for the BFKL scalingparameter is w = S + 1. It is also convenient to use the notation Λ = g /w .To start solving the P µ -system in the BFKL regime we have to determine the scalingof the P -, Q - and µ -functions in the small w limit. In what follows we are going to usethe arguments from [35], thus as from (2.29) for the length-2 state in question (2.16) inthe BFKL limit A a A a = O ( w ) for a = 1 , . . . , B i B i = O ( w ), these functions can bechosen to scale as w P a = P (0) a + O ( w ) , P a = P (0) a + O ( w ) (2.46) Q i = Q (0) i + O ( w ) , Q i = Q (0) i + O ( w ) (2.47)and the µ -functions scale as w − µ ab = w − (cid:16) µ (0) ab + O ( w ) (cid:17) , µ ab = w − (cid:16) µ (0) ab + O ( w ) (cid:17) . (2.48)– 16 –dditionally, as all the P -functions for the length-2 states being considered possess thecertain parity from the P µ -system equations (2.45) we can conclude that the functions µ + ab ( u ) have the certain parity.Let us restrict ourselves from now on in this Section to the case of integer conformalspin S = n . We know that the P -functions have only one cut on one of the sheets, thereforethey can be written as a series in the Zhukovsky variable x ( u ) = ( u + √ u − w ) / (2 √ Λ w ) .Then, we are also allowed to apply the certain H-transformation to the P -functions, whichdo not alter their asymptotics and parity. Applying this transformation, we can set thecoefficients A = A = − A = A = 1 and some other coeficients in the series in x ( u ) to 0.Thus, we arrive to the formulas in the LO P (0)1 = 1 u , P (0)2 = 1 u , P (0)3 = A (0)3 , P (0)4 = A (0)4 u + c (1)4 , Λ u , (2.49) P (0)1 = A u + c , Λ u , P (0)2 = A , P (0)3 = − u , P (0)4 = 1 u , where A (0)3 = ((∆ − n ) − n ) − i , (2.50) A (0)4 = ((∆ + n ) − − n ) − i originate from the expansion of (2.29) at small w and c (1)4 , and c , are some yet unknowncoefficients.The situation with the asymptotics of the µ -functions is more subtle. For non-zero S it appears that the asymptotics of the µ cannot be power-like anymore [80] and theminimal modification of the leading asymptotics of the µ -functions with the lower indicesat u → ±∞ is µ ab ∼ ( u − S − , u − S , u − S +1 , u − S +1 , u − S +2 , u − S +3 ) e π | u | , (2.51)while the µ -functions with the upper indices have the same asymptotics but in the reverseorder. The fact that the asymptotics should be modified as (2.51) can be derived fromthe fact that the gluing matrix (2.41) asymptotic receives additional exponential terms. Inthe work [38] it was shown, that the correct ansatz for the µ -functions in the LO wouldbe polynomials of the powers consistent with the asymptotics (2.51) for S = − w and w close to 0. We do not write this ansatz explicitly as it is not important for furtherdiscussion.Substituting (2.49) and the ansatz for the µ -functions into (2.45) in the LO and fromthe analyticity properties of the P - and µ -functions we fix the constants c , and c (1)4 , c , = − c (1)4 , , c (1)4 , = − i Λ96 (cid:16)(cid:0) ∆ − (cid:1) − (cid:0) ∆ + 1 (cid:1) n + n (cid:17) (2.52) Here we rewrote the Zhukovsky variable using g = Λ w . – 17 –nd the µ -functions in the LO up to an overall constant (we do not write them explicitlyas they are not relevant for further discussion).After the substitution of the LO P -functions (2.49) together with (2.52) into (2.30) inthe LO (for the same result with the zero conformal spin see [35]), one can derive, thatthis 4th order Baxter equation factorizes (cid:2) ( u + 2 i ) D + ( u − i ) D − − u − − (∆ + n ) (cid:21) × (2.53) × (cid:20) D + D − − − − (∆ − n ) u (cid:21) Q (0) j = 0 , where D = e i∂ u is the shift operator (the same equation with n replaced by − n is true for Q (0) j , j = 1 , . . . , th orderBaxter equation can be found by soliving the second order Baxter equation Q (0)++ + Q (0) −− + (cid:18) − − n ) − u (cid:19) Q (0) = 0 . (2.54)Redefining the Q-function to be Q = Q j u we exactly reproduce the Baxter equationfor the SL (2 , C ) spin chain [7–11] from the QSC for N = 4 SYM which is a highlynon-trivial test of the all-loop integrability of this theory. This derivation established thedesired connection between the integrability in the 4D gauge theory in the BFKL limitand integrability of the N = 4 SYM. Having this result, one can go further and by usingQSC explore many more quantities, such as NNLO BFKL kernel eigenvalue, numericaltwist-2 and length-2 operator trajectories, intercept function, slope, curvature and slope-to-intercept function etc. (see [34, 36, 38] and references therein). Above we described how the integrability, discovered in the BFKL regime of the QCD,can be understood as a part of a more general integrable structure of N = 4 SYM. In factthe data coming from the BFKL regime has played an essential role in fixing the dress-ing phase of the Beisert-Staudascher equations, which eventually resulted in our currentunderstanding of integrability in this model.Another extremely popular recent topic – Sachdev-Ye-Kitaev (SYK) model also hasmany similarities at the technical level with the problem of BFKL kernel diagonaliza-tion [81–84] (see also a recent review dedicated to this topic [85]).In this section, we will describe yet another example of CFT tightly related to the LOBFKL physics – the Fishnet CFT at any dimension (FCFT D ). We will explain its Feynmangraph content (in planar approximation big graphs have the shape of regular square lattice– at the origin of the name “fishnet”) and reveal the origins of its integrbility, described interms of the conformal SO (1 , D +1) spin chain. We will demonstrate how, in particular caseof D = 2 and zero conformal spin, the problem of computing the anomalous dimensions ofthe simplest operators, described by so-called wheel graphs, boils down to the calculationof the spectrum of Lipatov’s multi-reggeon Hamiltonian.– 18 – .1 Fishnet integrable model The D − dimensional fishnet biscalar conformal field theory (dubbed usually as fishnetCFT, or FCFT D ) is defined by the Lagrangian [46] L d = N c tr[ ¯ X ( − ∂ µ ∂ µ ) D − ω X + ¯ Z ( − ∂ µ ∂ µ ) D + ω Z + (4 π ) D ξ ¯ X ¯ ZXZ ] , (3.1)where X, Z are complex N c × N c matrix fields and ¯ X ≡ X † , ¯ Z ≡ Z † are their Hermitianconjugates. The differential operator in an arbitrary power is defined in a standard way, asan integral operator. The action (3.1) should be supplemented with certain double-tracecounterterms described in [44, 48, 86, 87] (see also [33] and references therein).The FCFT case of the model (in D = 4 dimensions), for the particular “isotropic”case ω = 0, has a local Lagrangian. It was proposed in [39] as a specific double scalinglimit of the γ -deformed N = 4 SYM theory, combining strong (imaginary) deformationand weak coupling. The effective coupling constant is ξ = g N c e − iγ / (4 π ) , where the‘t Hooft coupling λ = g Y M N c → γ → i ∞ . In thislimit, all the fields except for two scalars get decoupled leading to (3.1) with D = 4 and ω = 0, but the model retains the SU ( N c ) global symmetry, which is a remnant of thegauge symmetry of the original N = 4 SYM theory. In the planar limit N c → ∞ , theFCFT is dominated by so-called fishnet Feynman graphs which have the structure of theregular square lattice. Such graphs appear to be integrable [42, 88, 89]. For infinitely longoperators at a critical coupling ξ ∗ those graphs have a continuous limit which is believed tobe described by a specific O (2 ,
4) 2D σ -model [90]. At strong coupling ξ → ∞ the FQFT has a classical description in terms of a dual string-bit fish-chain model as it was derivedin [91]. This exact duality persists at the quantum level too [54].The model has already a rich history of studying various physical quantities, such asexplicit computations of spectra of local operators [39, 40] and 4-point correlation func-tions [44, 48, 49, 92]. The QSC formalism, adopted for the γ -deformed N = 4 SYMtheory [93], appeared in this limit particularly efficient for computations of anomalous di-mensions of operators of the type tr Z L ( ¯ XX ) M + permutations and later was extended toa much wider class of operators in [54]. The FCFT also appears to possess a rich mod-uli space of flat vacua, which are quantum-mechanically stable in spite of the absence ofsupersymmetry in the model [47] (in the planar limit). At the technical level there is a number of places where the study of the spectrum ofthe fishnet models goes along similar steps as the problem of the BFKL spectrum. Thesimilarity of the integrable structure of both models is discussed below. Here we justconsider a simple example of a dimension of operators of the type tr (cid:3) n Z X in D =4 , w = 0, where (cid:3) is the 4D Laplace operator. The dimensions ∆ of these operators canbe obtained by solving the equation [48]1 ξ = ψ (1) (cid:0) (4 − ∆) (cid:1) − ψ (1) (cid:0) (6 − ∆) (cid:1) − ψ (1) (cid:0) ∆4 (cid:1) + ψ (1) (cid:0) (2 + ∆) (cid:1) (∆ − , (3.2)– 19 –hich is reminiscent of the LO BFKL eigenvalue (2.15), written in the form1 g = 4 S + 1 (cid:18) − ψ (cid:18) (cid:19) − ψ (cid:18) − ∆2 (cid:19) + 2 ψ (1) (cid:19) + O (cid:18) g S + 1 (cid:19) . (3.3)The derivation of [94] from a Baxter equation is also very similar, to what is described inthe Section 2.5.Whereas the above analogy is demonstrated only at the structural level, below we givea more precise relation between the BFKL Hamiltonian and the graph-building operatorof the fishnet theory in D = 2 and a particular, spin zero representation for physical spins. We start from the case of general
D, w . Suppose we want to compute the correlator oflocal operators O L ( z ) = tr[ Z ( z )] L in FCFT D K ( z, y ) = (cid:104)O L ( z ) ¯ O L ( y ) (cid:105) . (3.4)Due to the chiral property of the Lagrangian (3.1) the only planar Feynman graphs for thiscorrelator have a “globe” configuration Figure 6, where “parallels” and “meridians” forma regular square-lattice everywhere except for the “south” and “north” poles. The edgesare represented by the propagators in D dimensions given by G h ( x ) = ( x ) − D + ω , G v ( x ) = ( x ) − D − ω (3.5)in horizontal and vertical directions respectively.According to general properties of CFT, such correlator should have the form K ( z, y ) = const | x − y | − O L (3.6)where ∆ is the conformal dimension of this operator.To compute this correlator, it is worth considering a more general operator, of the type O ( z . . . z L ) = tr[ Z ( z ) . . . Z ( z L )] and compute the correlation function K ( z , . . . , z L | y , . . . , y L ) = (cid:104)O ( z . . . z L ) ¯ O ( y . . . y L ) (cid:105) . (3.7)The planar Feynman graphs for such correlator are of cylindrical topology. One shouldopen the ends of meridians converging on the “south” and “north” poles on the Figure 6).If we then amputate the propagators converging at the south pole we obtain the cylindricalconfiguration presented on the Figure 7. Such a graph can be represented at each order ofperturbation theory as the corresponding power of the so-called graph-building operator,acting on the space ( R D ) (cid:78) L , H ( w ) L ( z , z , . . . , z L | z (cid:48) , z (cid:48) , . . . , z (cid:48) L ) = G v ( x (cid:48) ) G v ( x (cid:48) ) . . . G v ( x (cid:48) ) ×× G h ( x (cid:48) (cid:48) ) G h ( x (cid:48) (cid:48) ) . . . G h ( x L (cid:48) (cid:48) ) , (3.8)where we use the notations x ab = x a − x b . Then the above correlator (3.7) can be repre-sented as [32, 39] K ( z , . . . , z L | y , . . . , y L ) = (cid:104) z . . . z L | − ξ L H ( s ) L | y . . . y L (cid:105) , (3.9)– 20 –here the states | y . . . y L (cid:105) are taken in the usual coordinate representation. It is easyto see, already by power counting, or applying the inversion transformation, that eachcylindrical graph entering the perturbative expansion in ξ is conformal. Thus the wholecorrelator will have the coordinate dependence according to the Euclidean D -dimensionalconformal symmetry corresponding to the so (1 , D + 1) algebra. Figure 6 . Planar Fishnet graphs of “globe” typearising in the computation of correlator (3.6).The edges belonging to red “parallels” and theblack “meridians” correspond to propagators fortwo different scalar fields of the Lagrangian (3.1).Due to the chiral properties of vertices of this La-grangian, no other planar diagrams in each looporder exist.
Figure 7 . The cyllinder fishnet configurationresulting from the opening of the ends of propa-gators at the south and north poles of the globe(i.e. separating their coordinates) and ampu-tating the propagators at the south pole. Suchgraphs can be computed by the Bethe-Salpeterprocedure, using the integrability of the graphbuilding operator, as described in this Section.
Introducing the 2D momenta conjugated to the coordinates [ p k , x k ] = i , we can repre-sent the operator (3.8) in a more compact form H ( s ) L = ( p ) − D + ω ( x ) − D − ω ( p ) − D + ω × (3.10) × ( x ) − D − ω ( p ) − D + ω . . . ( p L ) − D + ω ( x L ) − D − ω . The operator H ( w ) L is known to be a conserved charge from the hierarchy of charges ofthe so (1 , D + 1) spin chain encoded into the T-operator [33, 88, 89] T L ( u ) = Tr aux ( ˆ R (cid:48) ( u ) ˆ R (cid:48) ( u ) . . . ˆ R L (cid:48) L ( u )) (3.11)built from L R -matrices acting in the product of principal series representations of so (1 , D + 1)with the weights (∆ , , , . . . ) × (∆ , , , . . . )ˆ R (cid:48) ( u ) = (cid:104) x | R (cid:48) ( u ) | x (cid:48) (cid:105) = (3.12)= 1( x ) − u − D +∆ + ( x (cid:48) ) u + D +∆ − ( x (cid:48) (cid:48) ) − u + D − ∆ + ( x (cid:48) ) u + D − ∆ − , where ∆ ± = ∆ ± ∆ . The matrix multiplication in auxiliary space in (3.11) is understoodas the integration (cid:82) d z (cid:104) x | . . . | z (cid:105)(cid:104) z | . . . | y (cid:105) and the trace Tr aux is understood as Tr aux ( . . . ) = (cid:82) d z (cid:104) z | . . . | z (cid:105) .We can also use the momenta p j to represent the R -matrix in the form R ab ( u, ξ ) = ( x ab ) u + D − ∆ + ( p a ) u +∆ − ( p b ) u − ∆ − ( x ab ) u − D +∆ + . (3.13)– 21 –ndeed, if we take u = − D + ∆ + + (cid:15) and ∆ + = D , then the first factor in (3.12) in thelimit (cid:15) → x (cid:48) (cid:48) ) D ∼ (cid:15) − δ ( D ) ( x (cid:48) (cid:48) ), andit is easy to see that T L ( u ) ∼ (cid:15) L H ( w ) L , where ∆ = D − w, ∆ = D w . (3.14)An interesting particular case of (3.11), mentioned in [89] is the limit u →
0, when∆ = ∆ ≡ ∆, since it is the only way to generate from it the local conserved charge – thespin chain hamiltonian with nearest neighbours interaction. Indeed, the R -matrix becomesin this limit R ab ( u, ξ ) = 1 + uh ab + O ( u ) , (3.15)where h ab (∆) = 2 log( x ab ) + ( x ab ) D − ∆ log( p a p b )( x ab ) − D +∆ = (3.16)= ( p a ) − D +∆ log( x ab )( p a ) D − ∆ + ( p b ) − D +∆ log( x ab )( p b ) D − ∆ + log( p a p b ) , which gives the Heisenberg hamiltonian for non-compact SO (1 , D + 1) spin chain withnearest neighbors interaction of spins (with conformal spin ∆): H ∆ = L (cid:88) a =1 h ab (∆) , (3.17)For the particular dimension D = 2 and conformal spin ∆ = 0, we obtain from here thefamous Lipatov’s SL (2 , C ) spin chain describing the BFKL physics of interacting reggeizedgluons in LO approximation [5] h BF KLab = 2 log( x ab ) + ( x ab ) log( p a p b )( x ab ) − . (3.18)In the particular case of two interacting reggeized gluons (Pomeron state) the spin chainreduces to two-spin hamiltonian H ∆ = 2 h BF KL (3.19)with the spectrum given by the RHS of (3.3).In this way, we see that the BFKL physics in LO approximation emerges as a particularcase of D -dimensional Fishnet CFT, as it was first noticed in [46]. In this short review we made an attempt to briefly explain the studies of the BFKLspectrum in N = 4 SYM, which were inspired by the works of Lev Lipatov on the BFKLintegrability in the 4D gauge theories without and with supersymmetry. His ideas have ledto a very significant progress in understanding the BFKL spectrum of N = 4 SYM, where itwas possible to build the bridge between the integrability of N = 4 SYM and integrabilityof the BFKL limit in the gauge theory. The usage of the Quantum Spectral Curve method– 22 –n this limit has led to a series of new analytic and numerical results mentioned in thisreview.The key feature utilized to link the results (including BFKL spectrum) in N = 4SYM and QCD is the Kotikov-Lipatov’s principle of maximal transcendentality [12, 95].In the Section 2.1 we illustrated this principle with an example of the NLO BFKL kerneleigenvalue in N = 4 SYM and QCD. In general, this principle goes much beyond the BFKLspectrum. Many more observables in N = 4 SYM admit this principle and reproduce themost transcendental part of the corresponding results in QCD. Another key observationby Lipatov [12], which allowed to connect the integrability of N = 4 SYM with the BFKLregime, is the relation between the BFKL regime and the dimensions of the operators in N = 4 SYM explained in the Section 2.2. This enabled the QSC method to be applied forthe BFKL spectrum.We also described here another special limit of N = 4 SYM similar in spirit to theBFKL limit, leading to the Fishnet CFT, currently actively studied in the literature. Weexplained here the generalization of the Fishnet CFT from four to any number D of di-mensions. It appears that, at D = 2 and a special value of spin of the involved fields, theFishnet CFT reduces to Lipatov’s integrable spin chain describing the interacting reggeizedgluons in LO BFKL approximation of QCD. This remarkable correspondence can open newways for the study of BFKL physics.A great deal of the work presented here has been inspired by the ideas of Lev Lipatov.These ideas continue to influence a large community of theoretical physicists working onvarious non-perturbative aspects of quantum gauge theories. They find applications in avariety of domains of theoretical physics, including the integrable field theories. The BFKLspectrum of N = 4 SYM is only one of many such applications, where Lev Lipatov’s ideasgave rise to a great progress and revealed new intriguing problems, which still wait for theirsolution.All three of us had the privilege to know Lev Lipatov personally and had the chance toappreciate, in numerous conversations, his outstanding personality and enormous talent.For one of us (V.K.), Lev Lipatov has been a dear friend over many years. For another(N.G.), Lev Nikolaevich was a dedicated and caring teacher who, over many years, greatlyshaped his taste in research and was always available with deep insight and advice. Andfor the latter of us (M.A.), Lev Lipatov, despite being personally acquainted with him fora short time, left a great heritage of inspiring ideas and will always be an outstandingresearcher to look up to.For us, he will always be a great example of selfless devotion to science. Acknowledgements
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