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

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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 am- plitudes 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 di- mensions (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 approxima- tion 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 calcu- lated 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 anoma- lous 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 anoma- lous 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 M¨obius 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 + + + + .
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