DGLAP and BFKL Equations in $\Mathcal {N}= 4$ SYM: from Weak to Strong Coupling

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This paper deals with the review of the known properties of the Balitsky-Fadin-Kuraev- Lipatov (BFKL) [1] and Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) [7] equations in = 4 Supersymmetric Yang-Mills (SYM) theory [12, 13]. Lev LipatovN is one of the main contributors to the discovery and subsequent study of both these well known equations. The BFKL and DGLAP equations resum, respectively, 2 2 the most important contributions proportional to αs ln(1/x) and αs ln(Q /Λ ) in two different kinematical regions of the Bjorken parameter∼ x and the “mass”∼ Q2 of the virtual photon for the process of deep inelastic lepton-hadron scattering (DIS) and as such, they are among the main ingredients in the analysis and description of experimental data for lepton-nucleon and nucleon-nucleon scattering. In the case of supersymmeric theories these equations simplify drastically. Moreover, in the case of = 4 SYM they become related with each other for the nonphysical values of Mellin momentsN j as it was proposed by Lev Lipatov in [14]. The first hint towards the simplification of BFKL and DGLAP equations in supersymmetric theories came from the study of quasi-partonic operators in = 1 SYM in leading order (LO) approximation [15, 16]. It was shown, that quasi-partonicN operators in this theory are unified in supermultiplets and their anomalous dimensions could be described in terms of single universal anomalous dimension by shifting the argument of the latter by some integer number. Calculations in = 4 SYM, where the coupling constant is not renormalized, gave even more remarkableN results. Namely, it turned out, that all twist-2 operators in this theory belong to the same supermultiplet with their anomalous dimension matrix fixed completely by the superconformal invariance. The LO universal anomalous dimension in = 4 SYM was found to be proportional to Ψ(j 1) Ψ(1), N − − which implies that DGLAP evolution equations for the matrix elements of quasi-partonic operators are equivalent to the Schrödinger equation for the integrable Heisenberg spin model [17, 18]. In QCD the integrability remains only in a small sector of these operators [19,20] (see also [21,22]). On the other hand, in the case of = 4 SYM the equations for other sets of operators are also integrable [23–25]. N Similar results related to integrability of BFKL and Bartels-Kwiecinski-Praszalowicz (BKP) [26] equations in the multi-colour limit of QCD were obtained earlier in [28]. Later it was also shown [14], that in the case of = 4 SYM there is a deep relation N between the BFKL and DGLAP evolution equations. Namely, the j-plane singularities of the anomalous dimensions of Wilson twist-2 operators in this case can be obtained from the eigenvalues of the BFKL kernel by analytic continuation. The next-to-leader (NLO) calculations in = 4 SYM demonstrated [14], that some of these relations are valid also in higher ordersN of perturbation theory. In particular, the BFKL equation has the property of the hermitian separability and the eigenvalues of the anomalous dimension matrix may be expressed in terms of the universal function γuni(j). In what follows, in Section 2 we discuss the BFKL equation and its Pomeron solution

2 both at weak and strong coupling values. Section 3 is devoted to DGLAP equation and related anomalous dimensions of Wilson twist-2 operators. Here, as in a case of BFKL equation we are not limiting ourselves by weak coupling regime and discuss also the behavior of anomalous dimensions at strong coupling. The strong coupling study both in the case of BFKL and DGLAP equations was made possible by an extensive use of both integrability and AdS/CFT correspondence approaches. Finally, we come with a summary in Conclusion.

2 BFKL equation and Pomeron

The behavior of scattering amplitudes in high energy or Regge limit may be conveniently described by the positions of singularities of their partial wave amplitudes in complex angular momentum plane. In particular, the behavior of the total cross-section for the scattering of colorless particles at high energies is related to the Regge pole with quantum numbers of the vacuum and even parity - the so called Pomeron. Within perturbation theory the latter is given by the bound state of two reggeized gluons and is described by now famous BFKL equation [1]. In what follows we are going to describe the properties of BFKL equation and its Pomeron solution in the case of maximally supersymmetric = 4 Yang-Mills theory. N 2.1 BFKL equation and perturbation theory In the high-energy limit (s t) the total cross-section σ(s) for the scattering of colour- less particles A and B takes≫ the − following factorized form d2qd2q′ a+i∞ dω s ω σ(s) = Φ (q) Φ (q′) G (q, q′), s = q q′ , (2.1) (2π)2 q2 q′2 A B 2πi s ω 0 | || | Z Za−i∞ 0 where Φi(qi) are impact factors of the colliding particles with momenta pA, pB, s =2pApB ′ is their squared invariant mass and Gω(q, q ) is the t-channel partial wave for the reggeized gluon-gluon scattering. The latter depends on two transverse momenta of the reggeized gluons in the t channel q and q′1 and is given by the solution to BFKL equation. Using the dimensional regularization and the MS-scheme to deal with ultraviolet and ′ infrared divergences in the intermediate expressions, the BFKL equation for Gω(q, q ) can be written in the form

ωG (q, q ) = δD−2(q q )+ dD−2q K(q, q ) G (q , q ) , (2.2) ω 1 − 1 2 2 ω 2 1 Z where K(q , q ) = 2 ω(q ) δD−2(q q )+ K (q , q ) (2.3) 1 2 1 1 − 2 r 1 2 1Here and below we omit arrows in the notation of transverse momenta and simply write q and q′ instead of ~q and q~′.

3 and the space-time dimension D = 4 2ε for ε 0. Here, ω(q) is the gluon Regge − → trajectory and the integral kernel Kr(q1, q2) is related to the real particle production in t-channel. In = 4 SYM the description of BFKL equation follows closely the one in the N case of QCD. Moreover, in leading logarithmic approximation the expressions for BFKL integral kernel in QCD and = 4 SYM coincide and the diﬀerence starts at the level of N NLO corrections. Initially NLO radiative corrections to BFKL integral kernel at t = 0 were calculated in the case of QCD in [29] 2 and later were generalized to the case of = 4 SYM in [32]. N As it was shown in [29, 32], a complete and orthogonal set of eigenfunctions of the 1 homogeneous BFKL equation in LO is given by (γBFKL = 2 + iν):

q2 γBFKL−1 G (q2/q′2, θ) = einθ , (2.4) n,γ q′2 where integer conformal spin n and real ν enter the parametrization of conformal weights m andm ˜ of the principal series of unitary M¨obius group representation 1+ n 1 n m = + iν , m˜ = − + iν (2.5) 2 2 The BFKL kernel in this representation is diagonalized up to the eﬀects related with 2 the running coupling constant as(q ) and up to NLO in the case of QCD we have

ωQCD =4a (q2) χ(n, γ )+ δQCD(n, γ )a (q2) , (2.6) MS s BFKL MS BFKL s where LO and NLO eigenvalues of BFKL kernel using formulae of [32] are given by n n χ(n, γ) = 2Ψ(1) Ψ γ + Ψ 1 γ + (2.7) − 2 − − 2 67 10 n n n δQCD(n, γ) = 2ζ(2) f χ(n, γ)+6ζ(3)+Ψ′′ γ + +Ψ′′ 1 γ + MS 9 − − 9 N 2 − 2 c 11 2 n 1 2Φ(n, γ) 2Φ(n, 1 γ) f χ2(n, γ) − − − − 3 − 3 N 2 c π2 cos(πγ) n˜ γ(1 γ) + 1+ f − δ2 sin2(πγ)(1 2γ) N 3 2(3 2γ)(1+2γ) · n − ( c − n˜ 2+3γ(1 γ) 3+ 1+ f − δ0 , (2.8) − N 3 (3 2γ)(1+2γ) · n c − ) 2The t = 0 case can be found in the recent papers [31]. 6

4 m ′ ′′ Here, δn is the Kroneker symbol, and Ψ(z), Ψ (z) and Ψ (z) are the Euler Ψ -function and its derivatives. The function Φ(n, γ) is given by

∞ ( 1)k+1 Φ(n, γ) = − Ψ′(k + n + 1) Ψ′(k + 1) k + γ + n/2" − Xk=0 1 +( 1)k+1 β′(k + n +1)+ β′(k + 1) Ψ(k + n + 1) Ψ(k + 1) − − k + γ + n/2 − # (2.9)

and 1 z +1 z β′(z)= Ψ′ Ψ′ 4" 2 − 2 # To obtain the corresponding results in = 4 SYM we should account for contributions of scalars and fermions transforming in theN adjoint representation of the gauge group. This way in DR scheme [33] we get n n δN=4(n, γ) = 6ζ(3)+Ψ′′ γ + +Ψ′′ 1 γ + DR 2 − 2 2Φ(n, γ) 2Φ(n, 1 γ) 2ζ(2)χ(n, γ), (2.10) − − − − where the DR coupling constanta ˆs is related to MS one as as [34] 1 aˆ = a + a2. (2.11) s s 3 s It should be noted, that the sum Φ(n, γ)+Φ(n, 1 γ) can be further rewritten (see [14]) as − a combination of functions with arguments dependent on γ+n/2 M and 1 γ+n/2 M˜ only. Indeed, we have ≡ − ≡

Φ(n, γ)+Φ(n, 1 γ)= χ(n, γ) β′(M)+ β′(1 M) − − +Φ (M) β′(M) [Ψ(1) Ψ(M)]+Φ (1 M) β′(1 M) Ψ(1) Ψ(1 M) , 2 − − 2 − − f − − − h i where χ(n, γ) is given by Eq. (2.7) and f f f

∞ β′(k +1)+( 1)kΨ′(k + 1) ∞ ( 1)k (Ψ(k + 1) Ψ(1)) Φ (M)= − − − . (2.12) 2 k + M − (k + M)2 k=0 k=0 X X From the latter property we see, that BFKL equation in = 4 SYM satisﬁes the property of hermitian separability (see Ref. [14] and discussionsN therein). Moreover, the NLO eigenvalue of BFKL kernel in = 4 SYM could be further rewritten in terms of analyti- N cally continued harmonic sums, see for example appendix C in [37]. This way we ﬁrst get

5 the property of maximal transcedentality3 [14] manifest directly4 for the eigenvalues of BFKL kernel expressed in terms of harmonic sums with well deﬁned weights and second - the expressions for Pomeron intercept in = 4 SYM as a function of conformal spin n N take the form [37]: n 1 ωLO =4S − , (2.13) 0 1 2 n 1 n 1 4π2 n 1 ωNLO =8S − +8S − + S − , (2.14) 0 2,1 2 3 2 3 1 2 where ω0 = ω(n, 1/2) and

M M M + j S (M)=( 1)M ( 1)j S (j) . (2.15) i1,...,ik − − j j i1,...,ik j=1 X are binomial harmonic sums [38].

2.2 AdS/CFT correspondence and Pomeron at strong coupling AdS/CFT-correspondence [39–41] provides us with a unique possibility to study BFKL equation and its Pomeron solution at strong coupling. In the framework of AdS/CFT correspondence the BFKL Pomeron is equivalent to the reggeized graviton [42], what in particular allows us to construct the Pomeron interaction model as a generally covariant eﬀective theory for the reggeized gravitons [43]. The Pomeron intercept for zero conformal spin n = 0 and at the leading order in the inverse coupling constant could be easily obtained using diﬀusive approximation for BFKL equation [31,32], strong coupling results for anomalous dimensions [44–46] and the mentioned Pomeron-graviton duality [42, 47]. This is the approach, which was used in5 [48], see also [49] for similar results. Moreover, it turns out that the approach of [48] could be also extended to include higher order corrections in inverse coupling constant [50]. First, one should note that due to symmetry of BFKL kernel eigenvalue in = 4 SYM under substitution γBFKL 1 γBFKL the latter is an even function of ν:N → − ∞ j 1= ω = ω + ( 1)m D ν2m , (2.16) − 0 − m m=1 X 2 where (λ = g Nc is t’Hooft coupling constant) λ λ ω = 4 ln 2 1 c + O(λ3) , (2.17) 0 4π2 − 1 16π2 3See the discussion for the case of anomalous dimensions in the next section 4Actually it could be already inferred from the expression in Eq. (2.10), see discussion in [14] 5See Erratum to that paper

6 λ δ(2m)(1/2) λ2 D = 2 22m+1 1 ζ + + O(λ3) . (2.18) m − 2m+1 4π2 (2m)! 64π4 and according to [14] we have

x 1 π 2 y c1 = 2ζ2+ 11ζ3 32Ls3 14πζ2 7.5812 , Ls3(x)= ln 2 sin dy . 2 ln 2 − 2 − ≈ − 0 2 Z (2.19) Second, the M¨obius invariance and hermicity of BFKL kernel suggest that the expansion (2.16) should be also valid at strong coupling. Next, to make the connection with the results for anomalous dimensions of local composite operators6 (string energies in the language of AdS/CFT correspondence) γ at strong coupling we use the relation of the latter to BFKL anomalous dimension γBFKL [29, 51, 52]: ω j γ = γ + = + iν . (2.20) BFKL 2 2 Now, due to the energy-momentum conservation γ(j = 2) = 0 we have

ν(j =2)= i (2.21)

and the small ν expansion for the eigenvalues of the BFKL kernel for reggeized graviton takes the form ∞ j 2= D ( ν2)m 1 , (2.22) − m − − m=1 X where ν2 is related to γ according to Eq. (2.20)

j 2 ν2 = γ . (2.23) − 2 − On the other hand, due to the ADS/CFT correspondence the string energies E in dimen- sionless units are related to the anomalous dimensions γ of twist-two operators as7 [40,41]

E2 =(j + Γ)2 4, Γ= 2γ (2.24) − − and therefore we can obtain from Eq. (2.23) the relation between the parameter ν of the conformal weights for the principal series of unitary representations of the Möbius group and the string energy E E2 ν2 = +1 . (2.25) − 4 6These are anomalous dimensions of twist 2 operators contributing to DGLAP equation [7] 7Note that our expression (2.24) for the string energy E differs from a definition, in which E is equal to 2 2 the scaling dimension ∆sc. Still the equation (2.24) is correct as it can be written as E = (∆sc 2) 4 and coincides with Eqs. (45) and (3.44) from Refs. [40] and [41], respectively. − −

7 This expression for ν2 can be inserted in the r.h.s. of Eq. (2.22) and gives us the following expression for the Regge trajectory of the graviton in the AdS space

∞ E2 m j 2= D +1 1 . (2.26) − m 4 − m=1 X Next, we assume that Eq. (2.26) is also valid both at large j and large λ in the region 1 j √λ, where the calculations of string energies at strong coupling were performed [53,≪ 54].≪ These energies can be presented in the form 8

2 2 E S S S 7/2 = √λ h0(λ)+ h1(λ) + h2(λ) + O S , (2.27) 4 2 √λ λ where ai1 ai2 ai3 ai2 hi(λ) = ai0 + + + + . (2.28) √λ λ √λ3 λ2 The contribution proportional to S can be extracted from the result of Basso [55, 56] taking Jan = 2 according to [57]:

I3(√λ) 2 I1(√λ) 2 h0(λ)= + = , (2.29) I2(√λ) √λ I2(√λ) − √λ

where Ik(√λ) is the modiﬁed Bessel functions. This gives us the following values of coeﬃcients a0i 1 15 135 a = 1, a = , a = a = , a = (2.30) 00 01 − 2 02 03 8 04 128

Next, the coeﬃcients a10 and a20 come from the consideration of the classical energy of the folded spinning string conﬁgurations corresponding to the twist-two operators (see, for example, [54]) 3 3 a = , a = . (2.31) 10 4 20 − 16

Finally, the one-loop coefficient a11 was found in [57] considering different asymptotical regimes and accounting for Basso result [55] 3 a = 1 ζ , (2.32) 11 16 − 3 Now, comparing both sides of Eq. (2.26) at large j values gives us the desired values of coefficients Dm (see Appendix A in [50]) and the value for Pomeron intercept j0 at zero

8 Here we put S = j 2, which in particular is related to the use of the angular momentum Jan =2 in calculations of Refs [53,− 54].

8 conformal spin in the limit λ takes the form 9 →∞ 2 1 1 1 145 9 1 1 j = 2 1+ 1+3ζ + 2a ζ + O . 0 −λ1/2 2λ1/2 − 8λ − 3 λ3/2 12 − 128 − 2 3 λ2 λ5/2 (2.33) 2 with unknown coefficient a12 in 1/λ correction in Eq. (2.33). Later the strong coupling expansion of Pomeron intercept for zero conformal spin was calculated even further within quantum spectral curve (QSC) approach with the result given by [59]: 2 1 1 1 j = = 2 + + (6ζ + 2) (2.34) 0 − λ1/2 − λ 4 λ3/2 3 λ2 361 1 511 1 1 + 18 ζ + + 39 ζ + + . (2.35) 3 64 λ5/2 3 32 λ3 O λ7/2 2.3 BFKL equation and integrability The most intriguing property of BFKL equation in = 4 SYM is its all-loop integrability tightly connected to all-loop integrability of anomalousN dimensions of local operators in = 4 SYM, see [60] for a review. Initially, the integrability of multicolor QCD was N discovered in LO by Lev Lipatov precisely in the study of Regge limit of high-energy scattering [28] and only latter in the study of anomalous dimensions of composite operators [17, 18]. Calculations in the maximally supersymmetric = 4 SYM, where the coupling constant is not renormalized and all twist-2 operators entNer in the same multiplet, showed that their anomalous dimension matrix is fixed completely by the super-conformal invariance and its entries are expressed in terms of universal anomalous dimension, which in LO is proportional to Ψ(S 1) Ψ(1) (see the discussion in next section). The latter − − property means, that the evolution equations for the matrix elements of quasi-partonic operators in the multicolour limit Nc are equivalent to the Schrödinger equation for an integrable Heisenberg spin model [17,18].→∞ Subsequent study of these and other operators both at leading and higher orders, starting from the works of [62,63] showed that the discovered integrability property in = 4 SYM may be extended to all loops. It turned out, that a lot of methods developedN for two-dimensional integrable systems, such as sigma- model and spin-chain S-matrices [64–71], Asymptotic Bethe Ansatz (ABA) [62,63,68,72], Thermodynamic Bethe Ansatz (TBA) [73–76] as well as Y and T -systems [77–80] could be also used for computation of anomalous dimensions of = 4 SYM operators not only at weak but also strong and in general finite coupling constant.N The most advanced framework for such spectral problem calculations at the moment is offered by quantum spectral curve (QSC) method. The latter is an alternative reformulation of TBA equations as a finite set of algebraic relations for Baxter type Q-functions together with analyticity and Riemann-Hilbert monodromy conditions for the latter [81–85]. Within the quantum spectral curve formulation one can relatively easy obtain numerical solution for any coupling

9Using a similar approach, the coeﬃcients λ−1 and λ−3/2 were calculated also in the paper [58]. ∼ ∼ 9 and state [86,87]. Also, QSC formulation allowed to construct iterative perturbative solutions for these theories at weak coupling up to, in principle, arbitrary loop order [85,88]. What is more important QSC allows also for the study of BFKL regime [37, 89, 90]. The integrability of LO BFKL equation (in leading logarithmic approximation) could be conveniently studied in the impact parameter space ~ρ, where the BFKL equation has the Schro¨odinger like form [28]:

Ef( ~ρ1, ~ρ2)= H12f( ~ρ1, ~ρ2) (2.36)

where in leading logarithmic approximation hamiltonian H12 takes holomorphically sep- arable form

∗ H12 = h12 + h12 1 1 h12 = ln(p1p2)+ (ln ρ12)p1 + (ln ρ12)p2 +2γE . (2.37) p1 p2

Here impact parameters are complex coordinates ρk = xk + iyk, ρ12 = ρ1 ρ2 and ∗ − pk, pk play the role of their canonically conjugated momenta. Moreover, it turns out that hamiltonian H12 coincides with the hamiltonian of Heisenberg (XXX) SL(2,C) spin chain [28], where SL(2,C) (Möbius) group generators are given by M 3 = ρ ∂ , M + = ∂ , M − = ρ2∂ (2.38) k k k k k k − k k and ∂k = ∂/(∂ρk). The eigenfunctions of LO BFKL hamiltonian are easy to find with the use of Möbius invariance. This way we get wave functions labeled by two conformal weights (2.5) [91]:

ρ m ρ∗ m˜ f (~ρ , ~ρ ; ~ρ ) = 12 12 , (2.39) m,m˜ 1 2 0 ρ ρ ρ∗ ρ∗ 10 20 10 20 with eigenvalues 1+ n E = 4Re Ψ + ıν 4Ψ(1) . (2.40) m,m˜ 2 − The Pomeron intercept as a function of conformal spin n is then given by10 λ ω0 = Em,m˜ (2.41) −8π2 ν=0

Having established a connection of LO BFKL hamiltonian with integrable SL(2,C) spin chain, we may write the generating function for its integrals of motion qr in terms of spin chain transfer matrix T (u) [28]:

n n−r T (u)= Tr(L1(u)L2(u) ...Ln(u)) = u qr , (2.42) r=0 X 10 2 λ = g Nc is again t’Hooft coupling constant.

10 where Lax Lk - operators are given by

u + ρ p p u 0 1 L (u)= k k k = + ρ 1 p . (2.43) k ρ2 p u ρ p 0 u ρ k k − k k − k k − k To study BFKL equation within integrability approach beyond LO it is convenient to use quantum spectral curve method [81–85]. Besides reproducing LO BFKL eigenvalues [89] authors of [90] were able to deduce the next-to-next-to-leading order (NNLO) corrections for BFKL eigenvalues at zero conformal spin in agreement with [92,93]. In addition, the authors of [37] obtained NNLO corrections for Pomeron intercept as a function of conformal spin n:

16π2 32π4 ωNNLO = 32(S S S S 2S ) S S , (2.44) 0 1,4 − 3,2 − 1,2,2 − 2,2,1 − 2,3 − 3 3 − 45 1 where the arguments of the binomial sums (2.15) are (n 1)/2. There are also partial − results for the next-to-next-to-next-to-leading order (NNNLO) intercept [37]. The range of applicability of QSC approach is not limited by weak coupling expansions but could be also used to obtain results at strong coupling. This way for Pomeron intercept as a function of conformal spin n we get [37]:

(n 1)(n + 2) j =2 n + − 0 − λ1/2 − (n 1)(n + 2)(2n 1) (n 1)(n + 2)(7n2 9n 1) 1 − − + − − − + , (2.45) − 2λ 8λ3/2 O λ2 There are also analytical results at ﬁnite values of coupling constant for slope to intercept and curvature functions [37]. Finally, it should be mentioned that numerically QSC approach allows study of BFKL eigenvalues at arbitrary values of coupling constant without restrictions to weak or strong coupling regime considered above, see [37] for more details.

3 Anomalous dimensions and DGLAP equation

Phenomenologically, Bjorken scaling violation for parton distributions11 in a framework of QCD is governed by the anomalous dimensions matrix γab(j) (here as = αs/(4π))

1 ∞ j−1 (k) k+1 γab(j) = dx x Wb→a(x) = γab (j)as , (3.46) 0 Z Xk=0 11Here we consider only the spin-average case. The results for the spin-dependent case can be found in [14].

11 related to Mellin moments of splitting kernels Wb→a(x) entering DGLAP evolution equa- 2 tions [7] for parton distribution densities fa(x, Q ) (hereafter a = λ, g, φ stands for the spinor, vector and scalar particles respectively):

d 1 dy f (x, Q2) = W (x/y) f (y, Q2) , d ln Q2 a y b→a b Zx b 1 X d ˜ 2 dy ˜ ˜ 2 2 fa(x, Q ) = Wb→a(x/y) fb(y, Q ) . (3.47) d ln Q x y Z Xb In QCD the anomalous dimensions and splitting kernels are completely known up to NNLO of the perturbation theory (see [94,95] and references therein). The similar results in the case of extended SYM may be obtained from the corresponding QCD results N with the help of Casimir substitutions: CA = CF = Nc, Tf nf = Nc/2. In addition, in the case of = 2 and = 4 SYM we should also account forN contributions of scalar N N particles, see [14, 96] for the case of = 4 SYM. It turns out, however, that the latter calculations accounting for scalar contributionsN could be completely abandoned in the case of = 4 SYM. N Indeed, the expressions for eigenvalues of the anomalous dimension matrix in the = 4 SYM can be derived directly from the QCD anomalous dimensions without tedious N calculations by using a number of plausible arguments. The method elaborated in Ref. [14] for this purpose is based on special properties of the integral kernel for the BFKL equation [1, 29, 32] in this model and a new relation between the BFKL and DGLAP equations (see [14]). First, it turns out that the expression for BFKL kernel in = 4 SYM posses a property of maximal transcedentality12. Next, using the mentionedN relation of BFKL and DGLAP equations in this model [14] we may conjecture the similar property of maximal transcedentality for anomalous dimensions [14]. Moreover, it turns out that eigenvalues of anomalous dimensions matrix in the case of = 4 SYM are given by N the maximal trancedentality part of eigenvalues for QCD anomalous dimension matrix. In NLO approximation this method gives the correct results for anomalous dimensions eigenvalues, which were checked by direct calculations in Ref. [96]. Using the results for the NNLO corrections to anomalous dimensions in QCD [94, 95] and the method of Ref. [14] we have also derived the NNLO eigenvalues of the anomalous dimension matrix in =4 N SYM [48], which were later conﬁrmed by the integrability predictions in [64]. Starting from four loops, i.e. above existing QCD results, the corresponding results for the anomalous dimensions in = 4 SYM can be obtained (see [97–99]) for example from N the long-range asymptotic Bethe ansatz equations together with some additional terms, so-called ﬁnite size or wrapping corrections, coming in agreement with Luscher approach. 13

12See discussion later in this section 13The three- and four-loop results for the universal anomalous dimension have been also reproduced (see [100, 101]) by solution of so-called asymptotic long-range Baxter equation.

12 3.1 Maximal transcedentality principle As we have already mentioned above in = 4 SYM there is a way to deduce eigenvalues N of anomalous dimensions matrix from QCD results without computing contributions of scalar particles. This possibility is based on the deep relation between the DGLAP and BFKL dynamics in the = 4 SYM [14, 32]. Indeed, the eigenvaluesN of the BFKL kernel in = 4 SYM are analytic functions of the conformal spin n at least in first two orders ofN perturbation theory (see Eqs. (2.6), (2.7) and (2.10)). Moreover, a detailed inspection of equations (2.7) and (2.10) shows that there is no mixing among special functions of different transcendentality levels i 14. Indeed, in the framework of DR-scheme [33], all special functions entering expression for NLO correction contain only sums of the terms 1/γi (i = 3). More precisely, introducing the transcendentality level i for eigenvalues ω(γ∼) of BFKL integral kernel at zero conformal spin in an accordance with the complexity of the terms in the corresponding sums Ψ 1/γ, Ψ′ β′ ζ(2) 1/γ2, Ψ′′ β′′ Φ ζ(3) 1/γ3, ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ we see that transcedentality levels of LO and NLO BFKL kernel eigenvalues are given by i = 1 and i = 3, respectively. Now, taking into account the relation between the BFKL and DGLAP equations in = 4 SYM (see [14,32]) we may conjecture that similar properties should be also valid for N (0) (1) the anomalous dimensions themselves, i.e. universal anomalous dimensions γuni(j), γuni(j) (2) and γuni(j) are assumed to have trancedentality levels i = 1, i = 3 and i = 5, respectively. An exception could be for the terms appearing at a given order from previous orders of the perturbation theory and having a lesser trancedentality level. Such contributions could be removed by an approximate finite renormalization of the coupling constant. Moreover, these terms do not appear in the DR-scheme. It is known, that in LO and NLO approximations (accounting for SUSY relation for QCD color factors CF = CA = Nc) the most complicated contributions (with i = 1 and i = 3, respectively) are the same for all LO and NLO eigenvalues of QCD anomalous dimensions matrix [94, 95] and also for LO and NLO scalar-scalar anomalous dimensions in = 4 SYM [96]. This property allows one to find = 4 SYM universal anomalous N (0) (1) N dimensions γuni(j) and γuni(j) without knowing all elements of the anomalous dimensions matrix [14], which was verified by the exact calculations in [96]. Using above arguments it was concluded in [14], that at the NNLO level there is (2) only one possible candidate for γuni(j). Namely, it is the most complicated part of QCD anomalous dimensions matrix (accounting again for SUSY relation for QCD color factors CF = CA = Nc). Indeed, after the diagonalization of QCD anomalous dimensions matrix the most complicated parts (having maximum transcedentality level) of its eigenvalues

14 Similar arguments were used also in [102, 103] to obtain analytic results for contributions of some complicated massive Feynman diagrams without direct calculations. For the relation of diagrams without sub-divergences and transcedentalities (knots) see [104, 105].

13 differ from each other by a shift in their arguments (Lorentz spin) and the differences are constructed from less complicated terms. The non-diagonal matrix elements of the anomalous dimensions matrix also contain only less complicated terms (see, for example exact expressions for LO and NLO QCD anomalous dimensions in Refs. [94,95] and [96] for the case of = 4 SYM) and as such they cannot contribute to the most complicated parts N of eigenvalues of anomalous dimensions matrix. Summarizing, the most complicated part of NNLO eigenvalues of QCD anomalous dimensions should coincide (up to color factors) (2) with = 4 SYM universal anomalous dimension γuni(j). All these arguments in general applyN to singlet anomalous dimensions. Nevertheless, as was shown in [48] the universal anomalous dimension in = 4 SYM could be also obtained from corresponding QCD results for nonsinglet anomalousN dimensions available at that moment. We would like to note that the transcedentaly was either used or have been observed also in the study of other quantities. In the same way as above, in [106] the universal coefficient function of deep inelastic scattering in the framework of = 4 SYM was N obtained from the most complicated part of the corresponding QCD Wilson coefficients [107]. For more recent application of the transcendentality principle see for example the discussion of energy -energy correlations [108] and collinear anomalous dimension in [109] in the case of QCD and = 4 SYM. The maximal transcedentality property was also observed in the study ofN scattering amplitudes (see [110] and references therein) and formfactors [111].

3.2 Universal anomalous dimensions in =4 SYM N Let us consider universal15 anomalous dimensions of the following color and SU(4) singlet Wilson twist-2 operators g ˆ a a = SG ... − G , (3.48) Oµ1,...,µj ρµ1 Dµ2 Dµ3 Dµj 1 ρµj λ = Sˆλ¯aγ ... λai , (3.49) Oµ1,...,µj i µ1 Dµ2 Dµj φ = Sˆφ¯a ... φa , (3.50) Oµ1,...,µj r Dµ1 Dµ2 Dµj r where are covariant derivatives. The spinors λ and gauge field tensor G describe Dµ i ρµ gluino and gluon field respectively, while φr stand for complex scalar fields. For all operators in Eqs. (3.48)-(3.50) Sˆ denotes symmetrization of the tensors in the Lorentz indices µ1, ..., µj and subtraction of their traces. The elements of the LO anomalous dimension matrix in the = 4 SYM are given by N (see [18]): 2 1 1 γ(0)(j) = 4 Ψ(1) Ψ(j 1) + , gg − − − j j +1 − j +2 15The considered operators belong to the same = 4 SYM superconformal multiplet and as such share the same universal anomalous dimension. N

14 1 2 2 1 1 γ(0)(j) = 8 + , γ(0)(j) = 12 , λg j − j +1 j +2 ϕg j +1 − j +2 2 2 1 8 γ(0)(j) = 2 + , γ(0)(j) = , gλ j 1 − j j +1 qϕ j − 1 2 6 γ(0)(j) = 4 Ψ(1) Ψ(j)+ , γ(0)(j) = , λλ − j − j +1 ϕλ j +1 1 1 γ(0)(j) = 4(Ψ(1) Ψ(j + 1)) , γ(0)(j) = 4 , (3.51) ϕϕ − gϕ j 1 − j − This anomalous dimension matrix could be easily diagonalized and we get simple expression [14, 18]:

4S1(j 2) 0 0 −1 − − DΓD = 0 4S1(j) 0 , 0− 0 4S (j + 2) − 1 where

j 1 S (j) Ψ(j + 1) Ψ(1) . 1 ≡ − ≡ r r=1 X From this we conclude that at LO the universal anomalous dimension for the considered operator supermultiplet is given by

γ(0) (j) = 4S (j 2) uni − 1 − The universal anomalous dimensions at NLO and NNLO could be obtained using the method described in subsection 3.1. This way up to three loops we get 16 [14, 48]:

αN γ(j) γ (j) =aγ ˆ (0) (j)+â2γ(1) (j)+â3γ(2) (j)+ ..., aˆ = c , (3.52) ≡ uni uni uni uni 4π where 1 γ(0) (j +2) = S , (3.53) 4 uni − 1 1 γ(1) (j +2) = S + S 2 S +2 S S + S , (3.54) 8 uni 3 −3 − −2,1 1 2 −2 1 γ(2) (j +2) = 2 S S S 2 S S 3 S + 24 S 32 uni −3 2 − 5 − −2 3 − −5 −2,1,1,1 16 Note, that in an accordance with Ref. [29] our normalization of γ(j) contains the extra factor 1/2 in comparison with the standard normalization (see [14]) and differs by sign in comparison with one− from Ref. [94, 95].

15 +6 S + S + S 12 S + S + S −4,1 −3,2 −2,3 − −3,1,1 −2,1,2 −2,2,1 2 S +2 S2 3 S + S 2 S S 8 S + S − 2 1 −3 3 − −2,1 − 1 −4 −2 +4 S S +2 S2 +3 S 12 S 10 S + 16 S (3.55) 2 −2 2 4 − −3,1 − −2,2 −2,1,1 and S S (j), S S (j), S S (j) are harmonic sums a ≡ a a,b ≡ a,b a,b,c ≡ a,b,c j j ( 1)r ( 1)r S (j)= ± , S (j)= ± S (r) (3.56) ±a ra ±a,±b,... ra ±b,... r=1 r=1 X X and

S (j) = ( 1)j S (j)+ S ( ) 1 ( 1)j . (3.57) −a,b,c,··· − −a,b,c,... −a,b,c,··· ∞ − − The expression (3.57) is deﬁned for positive integer values of arguments (see [14, 112, 113]) but can be easily analytically continued to real and complex j by the method of Refs. [112–114]. It is interesting to study various limits of the obtained expressions for universal anomalous dimension.

3.2.1 The limit j 1 → The limit j 1 is important for the investigation of the small-x behavior of parton distributions→ (see review [115] and references therein). Especially it became popular recently due to new experimental data for high energy processes studied at LHC. In addition, there are very precise combined experimental data at small x produced by H1 and ZEUS collaborations at HERA [116]. Using asymptotic expressions for harmonic sums at j =1+ ω 1 (see [14, 48]) the → universal anomalous dimension γuni(j) in Eq. (3.52) takes the form 4 γ(0) (1 + ω) = + ω1 , (3.58) uni ω O (1) γ (1 + ω) = 32 ζ + ω1 , (3.59) uni − 3 O 1 1 γ(2) (1 + ω) = 32ζ 232 ζ 1120ζ + 256ζ ζ + ω1 (3.60) uni 3 ω2 − 4 ω − 5 3 2 O in the agreement with the predictions coming from BFKL equation at NLO accuracy [32].

3.2.2 The limit j 4 → This limit corresponds to Konishi operator [117], which was used extensively in integrability [60] and AdS-CFT correspondence [39–41] studies recently. The anomalous dimension

16 for Konishi supermultiplet coincides with our expression (3.52) for j =4 γ (j) = 6â +24â2 168â3 , (3.61) uni j=4 − s s − s

which was also confirmed by direct calculation at two [96, 118–120] and three-loop [121] orders. Now. there are results for four, five, six, seven and up to eleven loop corrections to its anomalous dimension [122–124], [125–128], [129,130], [131,132] and [85,88]. Moreover, there are analytical results at strong coupling [53, 57, 59] and numerical calculations [86, 87, 133, 134] of these anomalous dimensions at any finite coupling.

3.2.3 The limit j →∞ This limit is related to the study of the asymptotics of structure functions and cross- sections at x 1 corresponding to the quasi-elastic kinematics of the deep-inelastic ep → scattering. In the limit j the expressions for anomalous dimensions (3.53)-(3.55) simplify signiﬁcantly and takes→ ∞ the form

γ(0) (j) = 4 ln j + γ + j−1 , (3.62) uni − E O (1) γ (j) = 8ζ ln j + γ + 12ζ + j−1 , (3.63) uni 2 E 3 O (2) γ (j) = 88ζ ln j + γ 16ζ ζ 80ζ + j−1 , (3.64) uni − 4 E − 2 3 − 5 O where γE is Euler constant. The strong coupling expansion in this limit could be again studied with the use of AdS/CFT correspondence [39–41]. The latter provides us with very interesting strong coupling prediction [44] (see also [135–137]) for large-j behavior of anomalous dimensions of twist-2 operators: αN γ(j)= a(z) ln j , z = c = 4â , (3.65) π s where (see Ref. [45] for asymptotic corrections) 3 ln 2 lim a = z1/2 + + z−1/2 . (3.66) z→∞ − 8π O On the other hand, the results for γuni(j) in Eqs. (3.52) and (3.62)–(3.64) allow us to find three first terms of the small-z expansion of the coefficient a(z) π2 11 lim a = z + z2 π4z3 + ... . (3.67) z→0 − 12 − 720 To resum this series Lev Lipatov suggested the following equation for the approximate functiona ˜ [96] π2 z = a + a2 , (3.68) − 12 e17 e Solving the latter we may find its weak-coupling expansion

π2 1 a˜ = z + z2 π4z3 + (z4) (3.69) − 12 − 72 O and strong-coupling asymptotics

2√3 6 a˜ = z1/2 + + z−1/2 1.1026 z1/2 +0.6079 + z−1/2 . (3.70) − π π2 O ≈ − O It is remarkable, that the predictions for NNLO corrections based on the above simple equation and the NNLO results for universal anomalous dimension (3.55) and (3.64)) are agree with each other with the accuracy 10%. It means, that this extrapolation seems to be good for all values of z17. ∼ The limit j could be also studied with the help of Beisert-Eden-Staudacher (BES) integral equation→ ∞ [139] for some function f(x, z) related to a(z) in (3.65) at x = 0, i.e. f(0, z)= a(z). At weak coupling z this equation gives a possibility to easily calculate as many coeﬃcients cm in the expansion

m f(0, z) = cm z . m=0 X

as needed. The coefficients cm respect the principle of maximal transcedentality, i.e. cm ζ(2m) for m> 0 (or products of ζ-functions with the sum of indices equal to 2m). Moreover,∼ up to 4-loop order these coefficients are in agreement with the ones, obtained directly from the calculation of Feynman diagrams [138, 140]. The analysis of strong coupling limit z of BES equation is less straightforward. Still, Gupser-Klebanov-Polyakov strong coupling→∞ asymptotics z1/2 (see the r.h.s. of ∼ (3.66)) was first reproduced from BES equation numerically in [141] and then analytically in [142]. Later the study of strong coupling limit of the function f(0, z) continued in Refs. [143–148]. In particular, the authors of [147] found a way to evaluate subleading coefficientsc ˜m of the expansion

(1−m)/2 f(0, z) = c˜m z m=0 X The ﬁrst three coeﬃcients are in agreement with string side calculations performed in [44], [45] and [46], respectively. Moreover, the results of [147] are also in agreement with maximum transcendentality principe:c ˜ ln 2 andc ˜ ζ(m) for m> 1 (or products of 1 ∼ m ∼ ζ-function with the sum of indexes equal to m).

17Some improvement of (3.68) can be found in [138].

18 3.3 Anomalous dimensions and integrability As we already mention in the section devoted to integrability of BFKL equation in =4 N SYM the DGLAP equation in this theory is also integrable. Moreover, the integrability property of DGLAP equation extends in this theory in planar limit to all loops. First hints that this might be the case came from the calculation of anomalous dimensions of twist 2 operators in = 4 SYM [17,18]. Nevertheless, the real breakthrough in the study of integrability of N= 4 SYM theory started only later with the work of Minahan and N Zarembo [62]. First, Beisert computed the complete one loop dilatation operator in this theory [149] and almost immediately Beisert and Staudacher [63] showed that one-loop dilatation operator in = 4 SYM may be identified with the hamiltonian of integrable N psu(2, 2 4) super spin chain. This observation together with available techniques for integrable spin| chains based on simple Lie algebras [150,151] and superalgebras [152] allowed authors of [63] to write down one-loop Bethe ansatz equations for the discovered = 4 SYM spin chain. Next, after a lot of preliminary work the authors of [72] cameN with the conjecture for all-loop asymptotic Bethe ansatz equations up to so called dressing factor [65], which was fixed later in [68] based on crossing equation proposed in [69]. In what follows we will briefly describe these and subsequents developments in the study of integrability structure of = 4 SYM. N 3.3.1 Asymptotic Bethe-ansatz and Lüscher corrections The problem of calculation of anomalous dimensions could be reduced to problem of diagonalization of dilatation operator, which is one of the generators of superconformal symmetry of = 4 SYM. The latter problems in a language of corresponding psu(2, 2 4) super spin chainN is equivalent to finding a spectrum of spin chain hamiltonian upon| identification of single trace operators with its states. At one-loop order the hamiltonian of psu(2, 2 4) is the one of spin chain with only nearest neighbor interactions. Beyond one loop spin chain| becomes long range with its all-loop asymptotic Bethe ansatz equations given in [72]. The obtained in [72] Bethe ansatz is asymptotic in a sense that it is valid only when the length of spin chain exceeds interaction order. Another novel feature of =4 spin chain is fluctuations of its length, so that the latter is dynamic. The correspondingN = 4 spin chain S-matrix [64, 66] could be decomposed into a product of two su(2 2) factorsN 18 |

Spsu (u ,u )= S (u ,u ) Ssu (u ,u ) Ssu (u ,u ) , (3.71) (2,2|4) 1 2 0 1 2 · (2|2) 1 2 ⊗ (2|2) 1 2 where u1 and u2 are rapidities of scattering magnons and g2 − + 1 + − x (u) x (v) − x (u)x (v) S0(u, v)= + − − g2 exp(2iθ(u, v)) . (3.72) x (u) x (v) 1 − + − − x (u)x (v) 18The choice of a reference state (vacuum) in a spin chain breaks psu(2, 2 4) symmetry down to su(2 2) su(2 2). | | ⊗ | 19 Here θ(u, v) is the BES dressing phase [68] and su(2 2) S-matrices could be ﬁxed completely from the symmetry requirements [66]. The variables| x±(u) are related to magnon rapidities u through Zhukovsky map

2 ± ± ± i u g x (u)= x(u ) , u = u , x(u)= 1+ 1 4 2 , (3.73) ± 2 2 r − u ! In the case of sl(2) sector corresponding to twist 2 operators relevant to DGLAP evolution the long-range asymptotic Bethe ansatz equations for M-magnon state take the form

ˆ x+ 2 M x− x+ (1 g2/x+x− ) M x+ k = k m k m exp (2 i θ(u ,u )) , k =1 .(3.74) − + − − − 2 − + k j − xk xk xm (1 g /xk xm) xk m=1Y,m6=k − − kY=1 The spin length in this case is equal to two and the number of magnons corresponds to the Lorentz spin of twist 2 operators (3.48)-(3.50). Once we know the solution for M Bethe roots uk (magnon rapidities) from the equation above, the asymptotic expression for anomalous dimension (energy of spin chain state) could be found from the formula

M ABA 2 i i γ (g)=2 g + − . (3.75) xk − xk Xk=1 The Bethe ansatz equations in this case can be solved recursively order by order in coupling constant g at arbitrary integer values of M once the one-loop solution for a given state is known. The later is known from the exact solution of Baxter equation [153] and is given by Hahn polynomial. The equations for quantum corrections to the one-loop Bethe roots are linear and thus numerically solvable with high precision. To find the expression for anomalous dimensions valid for arbitrary spin values M we may use the principle of maximal transcendentality [14] and the fact the function basis for anomalous dimensions in = 4 SYM is given by harmonic sums. The reconstruction of universal anomalous dimensionN is then reduced to fixing coefficients in a known basis at a set of fixed integer spin values. Following this procedure for the four-loop universal anomalous dimension we get [97] (M = j + 2) 1 γABA(j +2) = 4 S +6 S + ... + ζ(3)S (S S +2 S ), 256 uni −7 7 − 1 3 − −3 −2,1 where the symbol ... stands for a large set of not shown nested harmonic sums of degree seven. The obtained expression for universal anomalous dimension could be analytically continued also for non-integer spin values19, in particular to points M = r + ω with r - − 19An explanation for how this is done may be found in [113].

20 positive integer. Harmonic sums of degree seven may lead to poles no higher than seventh order in ω. In fact, it is known that none of the sums in r.h.s. of (3.76) can produce 7 such a high-order pole except for the two sums S7 and S−7. Their residues at 1/ω are of opposite sign. Thus, one immediately sees that the sum of the two residues does not cancel. On the other hand, from BFKL studies [14, 32] we know that in the vicinity of the Pomeron pole at M = 1+ ω the four loop anomalous dimensions behaves as − γ (1 + ω) 1/ω4 . (3.76) uni ∼ This latter fact tells us that the asymptotic solution for four-loop anomalous dimension is not compete and one should account for so called wrapping corrections. The first account for wrapping or finite size corrections in the case of = 4 SYM N was done with the help of Lüsher type formula [154]. The latter consist from two types of contributions: F and µ terms. The F -term corresponds to interaction of spin chain particles with virtual particle circulating around cylinder, while the µ-term corresponds to the splitting of the spin chain particle in two other particles, which then recombine after circulating around cylinder. To conjecture Lüscher type formula in the case of = 4 SYM authors of [98, 122] performed large volume analysis of Thermodynamic NBethe Ansatz (TBA) equations for the sinh-Gordon model. The conjectured formula for F -term contains two contributions: the first one accounts for corrections to ABA and as a consequence to Bethe roots, while the second one is due to propagation of virtual particles around cylinder, which directly changes energy of spin chain state with all M particles of type a as [98, 122]:

∞ ∞ dq ∆E(L)= STr Sa2a(q,p )Sa3a(q,p ) ...Sa1a eǫã1 (q)L , (3.77) − 2π a1 a1a 1 a2a 2 aM a Q=1 Z−∞ X where L is spin chain length and the sum is over all possible virtual particles in the theory Q, their polarizations a1 and over all possible intermediate states a2,...,aM . The ca matrix Sba (q,p) describes the scattering of virtual particle of type b and momentum q with the real particle of type a and momentum p, while the exponential factor should be understood as propagator of virtual particle. The µ-term contribution could be obtained by taking residues at the poles of S-matrices, however in N = 4 SYM at weak coupling these contributions are absent. It should be noted, that the same Lüsher type corrections were later much more easily obtained by analyzing large volume limit of TBA directly for = 4 SYM, see for example [73]. N Accounting for leading wrapping or Lüscher corrections [98, 122] the full expression for four-loop universal anomalous dimension takes the form

ABA wr γuni(j +2) = γuni (j +2)+ γuni(j + 2), 1 1 γwr (j +2) = S2 2 S +2 S +4(S S + S 2 S ) 256 uni 2 1 −5 5 4,1 − 3,−2 −2,−3 − −2,−2,1 21 4 S ζ(3) 5 ζ(5) , − −2 − which is in full agreement with BFKL predictions (3.76). Later, using similar technique and a property of Gribov-Lipatov reciprocity (see [155–160] and references therein), the ﬁve-loop corrections fo universal anomalous dimensions have been found in [99].

3.3.2 Thermodynamic Bethe Ansatz and Quantum Spectral Curve The systematic method to account for wrapping or finite size corrections is offered by Thermodynamic Bethe Ansatz (TBA). First, the idea that TBA approach could be used for = 4 SYM appeared in [161]. Within TBA approach the calculation of spin chain spectrumN with account for finite size corrections reduces to calculation of partition function of the mirror model. The simplification comes from the fact, that mirror model is considered at large volume corresponding to large imaginary times in the original model. As a consequence, its spectrum is well under control and may calculated using mirror Asymptotic Bethe Ansatz (ABA). Finally, the partition function in the mirror model is calculated using saddle point approximation resulting in integral equations for Bethe roots and hole densities corresponding to the mentioned saddle point. The mirror ABA equations for primary excitations may be obtained via analytical continuation of the ABA equations in the original theory. In addition one should determine all bound states (string complexes) of the mirror theory and calculate their dispersion relations together with scattering matrices using bootstrap method, which should allow to write down generic ABA in mirror model for all excitations including bound states. First, mirror ABA for =4 N spin chain was described in [162]. Using the fact that symmetry structure of the scattering matrix su(2 2) (3.71) is the same as that of Hubbard model one can use the TBA solution | of the latter [163, 164]. This resulted in string hypotheses formulated in [165]. Finally, the TBA equations for = 4 SYM were written in [73–76]. The further study of TBA equations continued in theN simplified form of Y and T -systems [77–80] Eventually, the investigation of = 4 SYM TBA equations has led to their simplified alternative formulation in terms ofN quantum spectral curve (QSC), a finite set of functional equations, so called Q-system, together with their analyticity and Riemann-Hilbert monodromy conditions [81–85]. The latter is well suited for both numerical solutions for any coupling and state [86, 87] as well as to analytical calculations both at weak and strong coupling [59, 85, 88, 130, 132, 166–169]. In the sl(2) sector of psu(2, 2 4) spin chain relevant to DGLAP evolution the QSC equations (so called Pµ-system) take| the form [81, 82, 84, 85]:

µ µ˜ = P˜ P P˜ P , (3.78) ab − ab a b − b a ˜ b Pa =(µχ)a Pb , (3.79) [2] µ˜ab = µab , (3.80)

22 where 0 0 0 1 − ab 0 0 +1 0 b cb χ =   , (µχ) µacχ (3.81) 0 1 0 0 a ≡ − +1 0 0 0    and indexes a, b take values 1, 2, 3, 4. µab is antisymmetric matrix constraint by µ µ µ µ + µ µ =1 , µ = µ (3.82) 12 34 − 13 24 14 23 14 23 and all functions Pa and µab are functions of spectral parameter u. The functions Pa have only one Zhukovsky cut (u [ 2g, 2g]) on the defining (physical) sheet, while functions µ have infinitely many branch∈ − points at positions u = 2g+iZ. Next, f [n](u)= f(u+ in ) ab ± 2 and f˜(u) denotes analytic continuation of function f(u) around one of its branch points on the real axis located at 2g and 2g. The boundary conditions for the above Riemann- − Hilbert problem are specified by the large u asymptotics of Pa and µab functions: L+2 L L−2 L P A u− 2 , P A u− 2 , P A u 2 , P A u 2 , 1 ≃ 1 2 ≃ 2 3 ≃ 3 4 ≃ 4 µ u∆−L, µ u∆−1, µ u∆, µ u∆+L, µ u∆+L (3.83) 12 ∼ 13 ∼ 14 ∼ 24 ∼ 34 ∼ with [(L S + 2)2 ∆2][(L + S)2 ∆2] A A = − − − , 1 4 16iL(L + 1) [(L + S 2)2 ∆2][(L S)2 ∆2] A A = − − − − . (3.84) 2 3 16iL(L 1) − Here L is the twist, S - spin and ∆ is conformal dimension of the sl(2) operator under consideration. Choosing appropriate ansatz for Pa functions its possible to solve this Riemann-Hilbert problem up to in principle arbitrary order of perturbation theory [88]. Moreover, using computed values of anomalous dimensions at fixed spin values and the basis of binomial harmonic sums20 it is possible to reconstruct universal anomalous dimension at the required order of perturbation theory. This way the universal anomalous dimensions at six and seven loop order were obtained in Refs. [130] and [132] correspond- ingly. Finally, as we already noted the use of QSC is not limited by weak coupling regime and the authors of [59] where able to obtain analytical results for anomalous dimensions of short sl(2) operators at first four orders in strong coupling expansion.

4 Conclusion

We tried to brieﬂy review the present knowledge of the properties of BFKL and DGLAP equations in the case of = 4 SYM. A lot of current results in = 4 SYM were N N 20The latter is possible with the use of Gribov-Lipatov reciprocity.

23 either obtained with the use of transcendentality principle or confirm the later property of = 4 SYM model. Obtained originally as a property of BFKL kernel eigenvalues atN NLO, the maximum transcedentality property was later observed also for universal anomalous dimensions up to very high order of perturbation theory, scattering amplitudes, form-factors and correlation functions in = 4 SYM. N First, the transcedentality principle was used to extract the results for = 4 SYM anomalous dimensions from the known QCD results at first three orders ofN perturbation theory. Next, the results for four and five loop universal anomalous dimensions were obtained from the asymptotic long-range Bethe equations with account for the so-called wrapping corrections in Lüsher approach. Still, the transcedentality property was used again to construct a basis of harmonic sums, in terms of which the result is written. To get six and seven loop result the quantum spectral curve approach was used instead of asymptotic Bethe equations and Lüsher corrections. AdS/CFT correspondence and all-loop integrability of DGLAP and BFKL equations gave us a unique opportunity to study Pomeron solution and anomalous dimensions of different types of operators at strong and in general at finite coupling. The most suitable approach for later studies at a moment is offered by a variant of Thermodynamic Bethe Ansatz known as Quantum Spectral Curve approach.

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