Pomeron in the N=4 Supersymmetric Gauge Model at Strong Couplings

A.V.Kotikov, JINR, Dubna

(in collab. with L.N.Lipatov PNPI,Gatchina,S’Petersburg) International Workshop “Supersymmetries and Quantum Symmetries”,

July 29 – August 3, 2013, Dubna

Pomeron in the =4 supersymmetric gauge model N at strong couplings OUTLINE 1. Introduction 2. Results 3. Conclusions. The BFKL Pomeron intercept at =4 super-symmetric gauge N theory in the form of the inverse coupling expansion j =2 2λ 1/2 λ 1 +1/4 λ 3/2 +2(1+3ζ )λ 2 + O(λ 5/2) 0 − − − − − 3 − − is found with the use of the AdS/CFT correspondence in terms of string energies calculated recently. Introduction Pomeron is the Regge singularity of the t-channel partial wave (G.F.Chew and S.C.Frautschi, 1961), (V.N.Gribov, 1962) responsible for the approximate equality of total cross-sections for high energy particle-particle and particle-antiparticle interactions valid in an accordance with the Pomeranchuck theorem (I.Ya.Pomeranchuk, 1958), (L.B.Okun and I.Ya.Pomeranchukand , 1956) In QCD the Pomeron is a colorless object, constructed from reggeized gluons (I.I.Balitsky, V.S.Fadin, E.A.Kuraev and L.N.Lipatov, 1975–1979) The investigation of the high energy behavior of scattering amplitudes in the = 4 Supersymmetric Yang-Mills (SYM) model N (A.V.K., L.N.Lipatov, 2000, 2003) is important for our understand- ing of the Regge processes in QCD. Indeed, this conformal model can be considered as a simpliﬁed ver- sion of QCD, in which the next-to-leading order (NLO) corrections (V.S.Fadin and L.N.Lipatov, 1986) to the Balitsky-Fadin-Kuraev- Lipatov (BFKL) equation are comparatively simple and numerically small. The eigenvalue of the BFKL kernel for this model has the remark- able property of the maximal transcendentality (A.V.K., L.N.Lipatov, 2003)

This property gave a possibility to calculate the anomalous dimensions (AD) γ of the twist-2 Wilson operators in one (L.Lipatov, 2001), (F.A.Dolan and H.Osborn. 2002), two (A.V.K., L.N.Lipatov, 2003), three (A.V.K., L.N.Lipatov, A.I.Onishchenko and V.N.Velizhanin, 2004), four (A.V.K., L.N.Lipatov, A.Rej, M.Staudacher and V.N.Velizhanin, 2007), (Z.Bajnok, R.A.Janik and T.Lukowski,2008), and ﬁve (T.Lukowski, A.Rej and V.N.Velizhanin, 2010) loops using the QCD results (S.Moch, J.A.M.Vermaseren and A.Vogt, 2004) and the asymptotic Bethe ansatz (N.Beisert and M.Staudacher, 2005) improved with wrapping corrections (Z.Bajnok, R.A.Janik and T.Lukowski,2008). On the other hand, due to the AdS/CFT-correspondence (J.Maldacena, 1998), (S.S.Gubser, I.R.Klebanov and A.M.Polyakov, 1998), (E.Witten, 1998), in = 4 SYM some physical quantities can be also com- N puted at large couplings. In particular, for AD of the large spin operators Beisert, Eden and Staudacher constructed the integral equation with the use the asymptotic Bethe-ansatz. This equation reproduced the known results at small coupling constants and it is in a full agreement (M.K.Benna, S.Benvenuti, I.R.Klebanov and A.Scardicchio, 2007), (AVK and L.N.Lipatov, 2007) with large coupling predictions (S.S.Gubser, I.R.Klebanov and A.M.Polyakov, 2002), (S.Frolov and A.A.Tseytlin, 2007), (R.Roiban, A.Tirziu and A.A.Tseytli, 2007). With the use of the BFKL equation in a diﬀusion approximation strong coupling results for AD (S.S.Gubser, I.R.Klebanov and A.M.Polyakov, 2002) and the pomeron-graviton duality (J.Polchinski and M.J.Strassler, 2002, 2003) the Pomeron intercept was calculated at the leading order in the inverse coupling constant (AVK, L.N.Lipatov, A.I.Onishchenko and V.N.Velizhani, 2006), (R.C.Brower, J.Polchinski, M.J.Strassler and C.I.Tan, 2007): j =2 2λ 1/2. 0 − −

Below we use recent calculations (N.Gromov, D.Serban, I.Shenderovich and D.Volin, 2011), (B.Basso, 2011), (N.Gromov and S.Valatka, 2011), (R.Roiban and A.A.Tseytlin, 2011) of string energies to ﬁnd the strong coupling corrections to the Pomeron intercept j =2 ∆ 0 − in next orders. We discuss also the relation between the Pomeron intercept and the slope of the anomalous dimension at j =2. BFKL equation at small coupling constant The eigenvalue of the BFKL equation in = 4 SYM model: N (AVK., L.N.Lipatov, 2000, 2003) λ λ 2 j 1= ω = [χ(γ )+ δ(γ ) ], λ = g Nc, − 4π2 BFKL BFKL 16π2 where λ is the t’Hooft coupling constant and 1 γ = + iν BFKL 2 and χ(γ) = 2Ψ(1) Ψ(γ) Ψ(1 γ), − − − δ(γ)=Ψ′′(γ)+Ψ′′(1 γ)+6ζ 2ζ χ(γ) 2Φ(γ) 2Φ(1 γ) . − 3 − 2 − − − Here Ψ(z) and Ψ′(z), Ψ′′(z) are the Euler Ψ -function and its derivatives. The function Φ(γ) is deﬁned as follows 1 1 z +1 z Φ(γ)=2 ∞ β′(k + 1) , β′(z)= [Ψ′( ) Ψ′( )] . k=0 k + γ 4 2 − 2 Due to the symmetry γ 1 γ , ω is an even BFKL → − BFKL function of ν m 2m ω = ω0 + ∞ ( 1) Dm ν , (1) m=1 − where λ λ 3 ω = 4ln2 1 c  + O(λ ) , 0 2  1 2    4π  − 16π    (2m) 2 2m+1 λ δ (1/2) λ 3 Dm = 2 2 1 ζ2m+1 2 + 4 + O(λ ) .  −  4π (2m)! 64π and 1 π c1 = 2ζ2 + 11ζ3 32Ls3( ) 14πζ2 7.5812 ,   2ln2  − 2 −  ≈ where x 2 y Ls (x)= ln 2 sin( ) dy. 3 0 − 2 Thus, the rightmost Pomeron singularity of the partial wave fj(t) in the perturbation theory is situated at (at ν = o)

λ λ 3 j =1+ ω =1+4ln2 1 c  + O(λ ) (2) 0 0 2  1 2    4π  − 16π    for small values of coupling λ.

Due to the M¨obius invariance and hermicity of the BFKL hamil- tonian in =4 SUSY expansion (1) is valid also at large coupling N constants. In the framework of the AdS/CFT correspondence the BFKL Pomeron is equivalent to the reggeized graviton (J.Polchinski and M.J.Strassler, 2002, 2003). AdS/CFT correspondence Due to the energy-momentum conservation, the universal anomalous dimension of the stress tensor Tµν should be zero, γ(j =2)=0. It is important, that the anomalous dimension γ contributing to the DGLAP equation ( (V.N.Gribov and L.N.Lipatov, 1972), (L.N.Lipatov, 1975), (G.Altarelli and G.Parisi, 1977), (Yu.L. Dok- shitzer,1977) does not coincide with γBFKL appearing in the BFKL equation. They are related as follows (V.S.Fadin and L.N.Lipatov, 1998), (G.P.Salam, 1998) ω j γ = γ + = + iν , BFKL 2 2 where the additional contribution ω/2 is responsible in particular for the cancelation of the singular terms 1/γ3 obtained from the ∼ NLO corrections to the eigenvalue of the BFKL kernel. Using above relations one obtains ν(j =2)= i. As a result, for the Pomeron intercept we derive the following representation for the correction ∆ to the graviton spin 2 (j = 2, j =2 ∆)(remember 0 − m 2m j = j0 + ∞ ( 1) Dm ν (3) m=1 − !!!)

∆ = ∞ Dm. m=1 In the diffusion approximation, where Dm = 0 for m 2, ≥ (A.V.Kotikov, L.N.Lipatov, A.I.Onishchenko and V.N.Velizhanin, 2006) D ∆ . 1 ≈ So ,we have the following small-ν expansion for the eigenvalue of the BFKL kernel (basic equation) 2 m j 2= ∞ Dm ( ν ) 1 , (4) − m=1  − −  where ν2 is related to γ as 2 2 j ν =  γ  .     −2 −    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 twist-two operators as follows (J.Maldacena, 1998), (S.S.Gubser, I.R.Klebanov and A.M.Polyakov, 1998) E2 = (j + Γ)2 4, Γ= 2γ (5) − − and therefore we can obtain from (5) the relation between the parameter ν for the principal series of unitary representations of the Möbius group and the string energy E 2 2 E  ν =  +1 . (6)     −  4      This expression for ν2 can be inserted in the r.h.s. of Eq. (4) leading to the following expression for the Regge trajectory of the graviton in the anti-de-Sitter space (another form of the basic equation) m E2 ∞    j 2= Dm  +1 1 . (7)       − m=1  4  −        Note, that due to (6) the eigenvalue of the BFKL kernel in the diffusion approximation j = j ∆ν2 = 2 ∆(ν2 + 1) , 0 − − is equivalent to the linear graviton Regge trajectory α E2 j =2+ ′t, α t = ∆ , 2 ′ 2 where its slope α′ and the Mandelstam invariant t, defined in the 10-dimensional space, equal R2 E2 α = ∆ , t = ′ 2 R2 and R is the radius of the anti-de-Sitter space. Now we return to the eq. (7), i.e. m E2 ∞    j 2= Dm  +1 1 .       − m=1  4  −        in general case. We assume below, that it is valid also at large j and large λ in the region 1 j √λ, (8) ≪ ≪ where there are the strong coupling calculations of energies. Comparing the l.h.s. and r.h.s. of (7) at large j values gives us the coefficients Dm and ∆ Graviton Regge trajectory and Pomeron intercept I. String energy at 1 << j << √λ The recent results for the string energies (N.Gromov and S.Valatka, 2011) in the region restricted by inequalities (8) can be presented in the form: (Here we put S = j 2, which in particular is related to the use of − the angular momentum Jan =2 in calculations of Refs (N.Gromov and S.Valatka, 2011), (R.Roiban and A.A.Tseytlin, 2011))

E2 S S S2 √   7/2 = λ h (λ)+ h (λ) + h (λ)  + O(S ), (9)  0 1 2   √  4 2  λ λ      where ai1 ai2 ai3 ai4 hi(λ) = ai0 + + + + . √λ λ √λ3 λ2 The contribution S can be extracted directly from the Basso re- ∼ sult (B. Basso, 2011) according to (N.Gromov, D.Serban, I.Shenderovich and D.Volin, 2011)

I3(√λ) 2 I1(√λ) 2 h0(λ)= + = , (10) I2(√λ) √λ I2(√λ) − √λ where Ik(√λ) is the modified Bessel functions. It leads to the following values of coefficients a0i 1 15 135 a = 1, a = , a = a = , a = 00 01 − 2 02 03 8 04 128 The coefficients a10 and a20 come from considerations of the classical part of the folded spinning string corresponding to the twist-two operators (R.Roiban and A.A.Tseytlin, 2011)) 3 3 a = , a = . (11) 10 4 20 − 16 The one-loop coefficient a11 is found recently in (N.Gromov and S.Valatka, 2011) considering different asymptotical regimes with taking into account the Basso result (B. Basso, 2011) 3 a = (1 ζ ), (12) 11 16 − 3 where ζ3 is the Euler ζ-function. II. Equations for coefficients Dm and the Pomeron intercept 2 ∆ − Thus, from expression (9) we obtain the following expansions of even powers of E in the small parameter j/√λ 2 2 2 2 3 3 E  S S E  S  2  3/2 3   = λ h (λ)+2h h (λ) ,   = λ h (λ) .    0 0 1    0    √     4  4  λ  4  8           Comparing the coefficients in the front of S, S2 and S3 in the l.h.s. and r.h.s of (7), i.e m E2 ∞    j 2= Dm  +1 1 .       − m=1  4  −        we derive the equations √λ 1 = h D , D = (D +2D +3D ) , 2 0 1 1 1 2 3 1 λ 0 = h D + h2 D , D = (D +3D ) , 2 1 1 4 0 2 2 2 3 1 √λ λ3/2 0 = h D + h h D + h3 D . 2√λ 2 1 4 0 1 2 8 0 3 Their perturbative solution leads is given below 2 1 2 h1 4 h1 D1 = , D2 = 2 D1 = 3/2 3, √λ h0 − λ h0 − λ h0 2 2 4 2h1 h0h2 8 2h1 h2h0 D3 = 2 −4 D1 = 5/2 −5 . λ h0 λ h0 and, correspondingly, D = D 3D , D = D 2D +3D . 2 2 − 3 1 1 − 2 3 Finally, we obtain the correction ∆ to the Pomeron intercept in the form

∆ = D1 + D2 + D3 = D1 D2 + D3 − 2 2 1 4 h1 8 2h1 h2h0 = 2 + 3/2 3 + 5/2 −5 . √λ h0 λ h0 λ h0 III. Strong coupling expansions of Dm and ∆ Using expressions (11)-(12) we have 8r3 1 4 c3 c4 1 D = + O   , D = c + + + O   , 3   2  2   5/2  7/2  3/2  1/2  3/2  λ λ  − λ  λ λ λ       2 d1 d2 d3 d4 1 D = 1+ + + + + O   , 1   1/2  1/2 3/2 2  5/2   λ λ   λ  λ λ λ     where 3 3 21 c = a = , c = a 3a a = (7 8ζ ), r =2a2 a = , 2 10 4 3 11 − 10 01 16 − 3 3 10 − 20 16 9 c = a +3a (2a2 a ) 3a a = a (5+4ζ ) 4 12 10 01 − 02 − 11 01 12 − 16 3 and 1 13 29 d = 2a = , d =2a2 a = , d =2a a a3 a = , 1 − 01 2 2 01 − 02 − 8 3 01 02 − 01 − 03 − 8 97 d = a4 3a2 a +2a a + a2 a = . 4 01 − 01 02 01 03 02 − 04 −128 Here a02, a12, a03 and a04 are parameters which should be calculated in future at two, three and four loops of the string perturbation theory. Analogously, we can obtain expressions for D2, D1 and ∆: 4 c3 c4 1 D = c + + + O   , 2  2   3/2  1/2  3/2  −λ  λ λ λ     2 d1 d2 d3 d4 1 D = 1+ + + + + O   , 1  2   1/2  1/2 3/2  5/2  λ  λ λ λ λ λ   ˆ ˆ ˆ ˆ   2  d d d d 1  1 2 3 4   ∆ = 1+ + + + + O  ,  2   1/2  1/2 3/2  5/2  λ  λ λ λ λ λ       where

c2 = c2, c3 = c3, c4 = c4 +6r3, d1 = d1 = dˆ1 , d2 = d2 +4c2, d3 = d3 +4c3, d4 = d4 +4c4 + 12r3 , ˆ ˆ ˆ d2 = d2 +2c2, d3 = d3 +2c3, d4 = d4 +2c4 +4r3 So, we have 1 1 145 9 dˆ = , dˆ = , dˆ = 1 3ζ , dˆ = 2a ζ . 1 2 2 − 8 3 − − 3 4 12 − 128 − 2 3 Using a similar approach, the coefficients dˆ1 and dˆ2 were found recently in the paper (M.S.Costa, V.Goncalves and J.Penedones, 2012) The corresponding coefficients c2,0 and c3,0 coincide with our dˆ1 and dˆ2 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 term dˆ3 in the strong coupling expansion. Anomalous dimension near j =2 At j = 2, the universal anomalous dimension is zero, but its derivative γ′(2) (the slope of γ) has a nonzero value in the perturbative theory 2 3 4 5 λ 1 λ 2 λ 7 λ 11 λ 6 γ (2) = +     +     + O(λ ) , ′                 −24 224 − 524 2024 − 3524         as it follows from exact three-loop calculations (AVK, L.N.Lipatov, A.I.Onishchenko and V.N.Velizhanin, 2004, 2006). Two last terms were calculated by V. Velizhanin from the explicit results for γ in five loops (T.Lukowski, A.Rej and V.N.Velizhanin, 2010). 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. (4) written in the form j 2m j 2= D  γ  1 m      − m=1 2 −  −        in the variable j for j =2 using γ(2) = 0:

1=(1 2γ′(2)) mDm (1 2γ′(2)) D1 . − m=1 ≡ − So we obtain explicitly √λ 1 2γ (2) = h (λ) . − ′ 2 0 and √λI3(√λ) γ′(2) = , − 4 I2(√λ) which is in full agreement with predictions of perturbation theory. Anomalous dimension of the Konishi operator We apply Eqs. (4), i.e. j 2m j 2= D  γ  1 m      − m=1 2 −  −        with j = 4 (and/or S = 2) and Di (i = 1, 2, 3) obtained earlier, to ﬁnd the large λ asymptotics of the anomalous dimension of the Konishi operator. So, it obeys to the equation m 2 2 = Dm (x 1) , x (2 γk) (13) m=1 − ≡ − 1. It is convenient to consider ﬁrstly the particular case, when D2 = D3 =0 and, thus, D1 = D1 =2/√λh0. So, we have 2 = D (x 1) 1 − and 2 x = +1 = √λh0 +1 , D1 where h0 has the closed form (10). i.e.

I3(√λ) 2 I1(√λ) 2 h0(λ)= + = . I2(√λ) √λ I2(√λ) − √λ So, the anomalous dimension γK can be represented as

 1 1 1  1/2 1/4   2 γ = (√λh + 1) λ  h + + O  . K 0  0   3/2  2  − ≈  2√λ√h − λ   0 8λh    0     For the case of the classic string, where h0 =1, i.e. a00 =1 and a =0 (i 1), we reconstruct well-known results 0i ≥ 1/4 1 1 1 2 γ λ 1+ + O   . K     √  3/2  − ≈  2 λ − 8λ λ     For the exact values of h0, we have 2  1+ a 1  (1 + a )  1  1/4 01 01   2 γ λ 1+ + a  + O  K   02     √    3/2  − ≈  2 λ 2λ  − 4  λ           1/4 1 29 1 = λ 1+ + + O   .     √  3/2   4 λ 32λ λ     2. In the case when all Di (i =1, 2, 3) are nonzero, it is convenient to represent the solution of the equation (13). i.e. m 2 2 = Dm (x 1) , x (2 γk) m=1 − ≡ − in the following form x x = √λh +1+ x + 2 . 0 1 √λ

Expanding Di in the inverse series of √λ and compare the coef- ﬁcients in the front of λ0 and 1/√λ, we have

x1 = 2a10, x2 = 2a11 +4a20 .

So, the solution with the coefficients a10, a11, a20 has the form a +1+2a 2 γ λ1/4(1 + 01 10 − K ≈ 2√λ 2 1 (1 + a01 +2a10) 1 + [a +2a +4a ]+ O  ) . 02 11 20    3/2  2λ − 4 λ    Using Eq.s (11)-(12) the exact values of aij, we have 1/4 1 1 1 2 γ λ 1+ + [1 6ζ ]+ O   K  3    √  3/2  − ≈  λ 4λ − λ     1/4 We would like to note that our coefficient in the front of λ− is equal to 1, which in an agreement with calculations performed in (N.Gromov, V.Kazakov and P.Vieira, 2009), (N.Gromov, D.Serban, I.Shenderovich and D.Volin, 2011), (R.Roiban and A.A.Tseytlin, 3/4 2011). Further, the coefficient in front of λ− agrees with (N.Gromov and S.Valatka, 2011), (S. Frolov, 2012). Numerical analysis of the Pomeron intercept j0(λ) Let us obtain an unified expression for the position of the Pomeron singularity j0 =1+ ω0 for arbitrary values of λ, using an interpolation between weak and strong coupling regimes. It is convenient to replace ω0 with the new variable t as follows ω t t = 0 , ω = 0 . 0 1 ω 0 1+ t − 0 0 This variable has the asymptotic behavior t λ at λ 0 and 0 ∼ → t √λ/2 at λ similar to the case of the cusp anoma- 0 ∼ → ∞ lous dimension (AVK, L.N.Lipatov and V.N.Velizhanin, 2003) So, following to (AVK, L.N.Lipatov and V.N.Velizhanin, 2003), (AVK, L.N.Lipatov, A.I.Onishchenko and V.N.Velizhanin, 2004), (Z.Bern, M.Czakon, L.J.Dixon, D.A.Kosower and V.A.Smirnov, 2007) we shall write a simple algebraic equation for t0 = t0(λ) whose solution will interpolate ω0 for the full λ range. We choose the equation of the form 2 k0(λ) = k1(λ)t0 + k2(λ)t0 , (14) where the following anzatz for the coefficinets k0, k1 and k2 is used: 2 1 k0(λ) = β0λ + α0λ , k1(λ) = β1 + α1λ, k2(λ) = γ2λ− + β2 + β2λ.

Here γ2, αi and βi (i =0, 1, 2) are free parameters, which are ﬁxed using the known asymptotics of ω at λ 0 and λ . 0 → →∞ The solution of quadratic equation (14) is given below

k 4k k 1  0 2  t = 1+ 1 . (15) 0  2    2k2  k −   1    To fix the parameters γ2, αi and βi (i = 0, 1, 2), we use two known coefficients for the weak coupling expansion of ω0: ω =e ˜ λ +˜e λ2 +˜e λ3 + ... (at λ 0) 0 1 2 3 → with ln2 7.5812 e˜ = 0.07023, e˜ = e˜ 0.00337 1 π2 ≈ 2 − 1 16π2 ≈− and first four terms of its strong coupling expansion

2 t˜1 t˜2 t˜3 t˜4 ω = 1 ∆, ∆ = 1+ + + + + ... (at λ ) 0    3/2 2  − √λ  √λ λ λ  →∞  λ    with 1 1 145 9 t˜ = , t˜ = , t˜ = 1 3ζ , t˜ = 2a ζ . 1 2 2 − 8 3 − − 3 4 12 − 128 − 2 3 The coefficients e˜3 and t˜4 are unknown but we estimate them later from the interpolation. Then, for the weak and strong coupling expansions of t one obtains t = e λ + e λ2 + e λ3 + ..., (when λ 0) , 0 1 2 3 → √λ t1 t2 t3 t4 t0 = 2 1 λ 3/2 2  + ..., (when λ ) ,  √λ λ λ   − − − −  →∞ where 5 e =e ˜ , e =e ˜ +˜e2, e =e ˜ +˜e e˜ +˜e3, t = t˜ +2= , 1 1 2 2 1 3 3 1 2 1 1 1 2 3 3 t = t˜ t˜2 = , t = t˜ 2t˜ t˜ + t˜3 = (1+4ζ ), 2 2 − 1 −8 3 3 − 2 1 1 −4 3 39 3 t = t˜ 2t˜ t˜ t˜2 +3t˜ t˜2 t˜4 =2a ζ . 4 4 − 3 1 − 2 2 1 − 1 12 − 128 − 2 3 Comparing the l.h.s. and the r.h.s. of Eq. (14) at λ 0 and → λ , respectively, we derive the following relations →∞ α2 = 4α0, α2 = 10α0, β1 = C1α0, β2 = C2α0, α γ = C α , β = (C 22) 0 2 3 0 0 2 − 4 with the free parameter α0 which disappears in the retionship k1/k2 and k0/k2 and, thus, in the results (15) for t0. Here C 88.60,C 42.41,C 277.0 , 1 ≈ 2 ≈ 3 ≈− which lead to the predictions for the coefficients e3 and t4 2 10e2 +2C2e1e2 +4e1 e3 = 0.00079, − C1 +2C3e1 ≈− 9+16(C 5C +7C ) t = 3 − 1 2 40.5774 4 128 ≈− and, respectively, for the corresponding terms e˜ 0.00066, t˜ 51.0117, a 22.2348 . 3 ≈ − 4 ≈ − 12 ≈ − Note that the results for the coefficients e3, t4, e˜3, t˜4 and a12 do not depend on the free parameter α0. jo - 1 1.0

0.8

0.6

0.4

0.2

z -2 2 4 6 z Figure 1: (color-online). The results for j0 as a function of z (λ = 10 ).

On Fig. 1, we plot the pomeron intercept j0 as a function of the coupling constant λ. The behavior of the pomeron intercept j0 shown in Fig.1 is similar to that found in QCD (A.M.Stasto, 2007). Conclusion We found the intercept of the BFKL pomeron at weak and strong coupling regimes in the =4 Super-symmetric Yang-Mills model. N At large couplings λ , the correction ∆ for the Pomeron → ∞ intercept j =2 ∆ has the form 0 − 2 1 1 1 ∆ = [1 + (1+3ζ3) λ1/2 2λ1/2 − 8λ − λ3/2 145 9 1 1 + 2a ζ  + O  ] .  12 3 2      5/2   − 128 − 2  λ λ      The fourth corrections contain unknown coeﬃcient a12, which will be obtained after the evaluation of spinning folded string on the two-loop level. Some estimations were given in previous Section. The slope of the universal anomalous dimension at j =2 known by the direct calculations (up to λ5 (V.N.Velizhanin)) and can be written as follows √λ I3(√λ) γ′(2) = , − 4 I2(√λ) according to the well known Basso result (B.Basso, 2011).