Pomeron in the N=4 supersymmetric gauge model at strong couplings

We find the BFKL Pomeron intercept at N=4 supersymmetric gauge theory in the form of the inverse coupling expansion j0=2-2/lambda**(1/2)-1/lambda + 1/4 1/lambda**(3/2) + 2(1+3zeta3)/lambda**(2) + O(1/lambda**(5/2)) with the use of the AdS/CFT correspondence in terms of string energies calculated recently. The corresponding slope gamma'(2) of the anomalous dimension calculated directly up to the fifth order of perturbation theory turns out to be in an agreement with the closed expression obtained from the recent Basso results.


BFKL equation at small coupling constant
The eigenvalue of the BFKL equation in N = 4 SYM model has the following perturbative expansion [4,5] (see also Ref. [6]) where λ is the t'Hooft coupling constant. The quantities χ and δ are functions of the conformal weights m and m of the principal series of unitary Möbius group representations, but for the conformal spin n = m − m = 0 they depend only on the BFKL anomalous dimension γ BF KL = m + m 2 = 1 2 + iν (2) and are presented below [4,5] Here Ψ(z) and Ψ ′ (z), Ψ ′′ (z) are the Euler Ψ -function and its derivatives. The function Φ(γ) is defined as follows where Due to the symmetry of ω to the substitution γ BF KL → 1 − γ BF KL expression (1) is an even function of ν where ω 0 = 16 ln 2 λ 4π 2 1 + c 1 According to Ref. [5] we have where (see [36]) Thus, the rightmost Pomeron singularity of the partial wave f j (t) in the perturbation theory is situated at for small values of coupling λ. In turn, the anomalous dimension γ also has the square root singularity in this point, which means, that the convergency radius of the perturbation series in λ for the anomalous dimension γ = γ(ω, λ) at small ω is given by the expression It will be interesting to find higher order corrections to the BFKL intercept j 0 from the investigation of convergency of the perturbation theory using the analytic results for the anomalous dimensions obtained recently [5,12,13,14,15]. Note, that the BFKL singularity for positive ω is situated at positive λ = λ cr . But it is expected, that with growing ω the nearest singularity, responsible for the perturbation theory divergency will be at negative λ. Positions of both singularities can be found from the perturbative expansion of γ with the possible use of appropriate resummation methods (cf. [12]). Due to the Möbius invariance and hermicity of the BFKL hamiltonian in N = 4 SUSY expansion (7) is valid also at large coupling constants. In the framework of the AdS/CFT correspondence the BFKL Pomeron is equivalent to the reggeized graviton [27]. In particular, in the strong coupling regime λ → ∞ where the leading contribution ∆ = 2/ √ λ was calculated in Refs. [28,29,30]. Below we find next-to-leading terms in the strong coupling expansion of the Pomeron intercept. In the next section the simple approach to the intercept estimates discussed shortly in Ref. [28] will be reviewed.

3
AdS/CFT correspondence Due to the energy-momentum conservation, the universal anomalous dimension of the stress tensor T µν should be zero, i.e., γ(j = 2) = 0.
It is important, that the anomalous dimension γ contributing to the DGLAP equation [37] does not coincide with γ BF KL appearing in the BFKL equation. They are related as follows [7] (see also [38]) where the additional contribution ω/2 is responsible in particular for the cancelation of the singular terms ∼ 1/γ 3 obtained from the NLO corrections (1) to the eigenvalue of the BFKL kernel [7]. Using above relations one obtains As a result, from eq. (7) for the Pomeron intercept we derive the following representation for the correction ∆ (13) to the graviton spin 2 In the diffusion approximation, where D m = 0 for m ≥ 2, one obtains from (17) the relation between the diffusion coefficient D 1 and ∆ (see [28]) This relation was also obtained in Ref. [39]. According to (13) and (17), we have the following small-ν expansion for the eigenvalue of the BFKL kernel where ν 2 is related to γ according to eq. (15) On the other hand, due to the ADS/CFT correspondence the string energies E in dimensionless units are related to the anomalous dimensions γ of the local operators as follows [19,25] and therefore we can obtain from (20) the relation between the parameter ν for the principal series of unitary representations of the Möbius group and the string energy E This expression for ν 2 can be inserted in the r.h.s. of Eq. (19) leading to the following expression for the Regge trajectory of the graviton in the anti-de-Sitter space Note [28], that due to (22) expression (7) for the eigenvalue of the BFKL kernel in the diffusion approximation (18) is equivalent to the linear graviton Regge trajectory where its slope α ′ and the Mandelstam invariant t, defined in the 10-dimensional space, equal and R is the radius of the anti-de-Sitter space. Now we return to the eq. (23) in general case. We assume below, that it is valid also at large j and large λ in the region where the strong coupling calculations of energies were performed [32,35]. Comparing the l.h.s. and r.h.s. of (23) at large j values gives us the coefficients D m and ∆ (see Appendix A). 3

Graviton Regge trajectory and Pomeron intercept
The coefficients D 1 and D 2 at large λ can be written as follows 4 where a 01 and a 10 are calculated in Appendix A As a result, we find eigenvalue (23) of the BFKL kernel at large λ in the form of the nonlinear Regge trajectory of the graviton in the anti-de-Sitter space Note, that the perturbation theory for the BFKL equation gives this trajectory at small ω = j − 1 (see eq. (1)), where However the energy-momentum constraint (14), leading to ω = 1 at E = 0, is not fulfilled in the perturbation theory, because at γ → 0 the right-hand side of (1) contains the pole singularities which should be cancelled after an appropriate resummation of all orders. Neglecting the term D 2 E 2 /2 ∼ E 2 /λ 3/2 at λ → ∞ in comparison with a larger correction a 01 E 2 /λ, we obtain the graviton trajectory (30) in the form Solving this quadratic equation, one can derive with the same accuracy (see [32,35]) On the other hand, due to (21) this relation can be written as follows and for In particular, for j = 4 one obtains the anomalous dimension for the Konishi operator γ = γ K [32] (see also Appendix B) in an agreement with eq. (28). For the anomalous dimension at j−2 ∼ 1/ √ λ from (34) we obtain the square root singularity similar to that appearing at small j − 1 = ω 0 (8) where D 1 (28) is equal to the correction ∆ to the graviton trajectory intercept with our accuracy Note, that in the region j − 2 < −∆, the anomalous dimesnion is complex similar to it in the perturbative regime at j − 1 < ω 0 (8). Moreover, the position of the BFKL singularity of γ at large coupling constants can be found from the calculation of the radius of the divergency of the perturbation theory in 1/ √ λ at small j − 2.

Anomalous dimension near j = 2
At j = 2, the universal anomalous dimension is zero (14), but its derivative γ ′ (2) (the slope of γ) has a nonzero value in the perturbative theory as it follows from exact three-loop calculations [12,28]. Two last terms were calculated by V. Velizhanin [42] from the explicit results for γ in five loops [15]. To find the slope γ ′ (2) at large values of the coupling constant we calculate the derivatives of the l.h.s. and r.h.s. of eq. (19) written in the form in the variable j for j = 2 using γ(2) = 0: where D 1 is found in Appendix A in the expansion up to m = 3 (see (A.8)). So we obtain explicitly Here is the leading contribution to the string energy in the limit j << √ λ, as it is shown in (A.1) and (A.2)).
It is important, that now there is an explicit expression for h 0 (λ) (see (A.3)). It was obtained from the Basso result [33] by taking the value of the angular momentum J an equal to two (see [34]). Substituting (A. 3) in (41), we have the closed form for the slope γ ′ (2) which is in full agreement with predictions (38) of perturbation theory. Note, that in [12] with the use of two first terms of perturbation theory we suggested the simple quadratic equation to resum γ ′ (2) in all orders It turns out, that the solution of this quadratic equation indeed interpolates γ ′ (2) between week and strong coupling regimes rather well (see ref. [28]). In particular, for large λ values, the equation (44) leads to whereas the strong coupling expansion of the explicit result (43) is given below 6 Numerical analysis of the Pomeron intercept j 0 (λ) Let us obtain an unified expression for the position of the Pomeron singularity j 0 = 1 + ω 0 for arbitrary values of λ, using an interpolation between weak and strong coupling regimes.
It is convenient to replace ω 0 with the new variable t as follows This variable has the asymptotic behavior t 0 ∼ λ at λ → 0 and t 0 ∼ √ λ/2 at λ → ∞ similar to the case of the cusp anomalous dimension (see, for example, [11]). So, following the method of Refs. [11,12,41], we shall write a simple algebraic equation for t 0 = t 0 (λ) whose solution will interpolate ω 0 for the full λ range.
On Fig. 1, we plot the pomeron intercept j 0 as a function of the coupling constant λ. The behavior of the pomeron intercept j 0 shown in Fig.1 is similar to that found in QCD with some additional assumptions (see ref. [30]).

Conclusion
We found the intercept of the BFKL pomeron at weak and strong coupling regimes in the N = 4 Super-symmetric Yang-Mills model. At large couplings λ → ∞, the correction ∆ for the Pomeron intercept j 0 = 2 − ∆ has the form (see Appendix (A.21)) The anomalous dimension has a square-root singularity at the value of the BFKL intercept both in the weak and strong coupling regimes. This value is related to the radius of convergency of perturbation theory in λ and 1/ √ λ near the points j 0 = 1 and j 0 = 2, respectively. The fourth corrections in (62) contain unknown coefficient a 12 , which will be obtained after the evaluation of spinning folded string on the two-loop level. Some estimations were given in Section 6.
The slope of the universal anomalous dimension at j = 2 known by the direct calculations [42] up to the fifth order of perturbation theory can be written as follows according to the well known Basso result [33] for local operators of an arbitrary twist.

A Appendix
Here we discuss coefficients D m and the Pomeron intercept 2 − ∆ using expression (23) at comparatively large j in the region j << √ λ.
A.1 String energy at 1 << j << √ λ The recent results for the string energies [34] in the region restricted by inequalities (27) can be presented in the form 5 The contribution ∼ √ S can be extracted directly from the Basso result [33] taking J an = 2 according to [34]: The coefficients a 10 and a 20 come from considerations of the classical part of the folded spinning string corresponding to the twist-two operators 6 (see, for example, [35]) The one-loop coefficient a 11 is found recently in the paper [34] (see also [43]), considering different asymptotical regimes with taking into account the Basso result [33] where ζ 3 is the Euler ζ-function.
All calculations were performed for nonzero values of the angular momentum J an (really, J an = 2 was used) and are applicable also to the finite S values. 7 Moreover, all these coefficients are in a full agreement with numerical Y -system predictions (see [45,32] and references therein).
A.2 Equations for coefficients D m and the Pomeron intercept 2 − ∆ Thus, from expression (A.1) we obtain the following expansions of even powers of E in the small parameter j/ √ λ Comparing the coefficients in the front of S, S 2 and S 3 in the l.h.s. and r.h.s of (23), we derive the equations Their perturbative solution leads is given below (A.11) .12) and, correspondingly, Finally, we obtain the correction ∆ to the Pomeron intercept in the form where the λ-dependence of parameters h i is given in Eqs.  Here a 02 , a 12 , a 03 and a 04 are parameters which should be calculated in future at two, three and four loops of the string perturbation theory. It is important, that the coefficients D k tend to zero at large λ as λ −n+1/2 Analogously, we can obtain expressions for D 2 , D 1 and ∆: Using a similar approach, the coefficientsd 1 andd 2 were found recently in the paper [40]. The corresponding coefficients c 2,0 and c 3,0 in [40] coincide with ourd 1 andd 2 but in the expression for the Pomeron intercept they contributed with an opposite sign. Further, in the talk of Miguel S. Costa "Conformal Regge Theory" on IFT Workshop "Scattering Amplitudes in the Multi-Regge limit" (Universidad Autonoma de Madrid, 24 -26 Oct 2012) (see http://www.ift.uam.es/en/node/3985) the sign of these contributions to the Pomeron intercept was correct but there is a misprint the definition of the parameter of expansion. Note, however, that we have the next termd 3 in the strong coupling expansion.

B Appendix
We apply Eqs. (19) and (20) with j = 4 (and/or S = 2) and D i (i = 1, 2, 3) obtained in Appendix A, to find the large λ asymptotics of the anomalous dimension of the Konishi operator. So, it obeys to the equation 1. It is convenient to consider firstly the particular case, when D 2 = D 3 = 0 and, thus, D 1 = D 1 = 2/ √ λh 0 . So, we have 2 = D 1 (x − 1) (B2) and x = 2 where h 0 has the closed form (A.3). So, the anomalous dimension γ K can be represented as For the case of the classic string, where h 0 = 1, i.e. a 00 = 1 and a 0i = 0 (i ≥ 1), we reconstruct well-known results 8 .

2.
In the case when all D i (i = 1, 2, 3) are nonzero, it is convenient to represent the solution of the equation (B1) in the following form Expanding D i in the inverse series of √ λ and compare the coefficients in the front of λ 0 and 1/ √ λ, we have We would like to note that our coefficient in the front of λ −1/4 is equal to 1, which in an agreement with calculations performed in [45,32,35]. Further, the coefficient in front of λ −3/4 agrees with the results of [34] (see also Refs. [43] and [46]).