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Pomeron in the = 4 supersymmetric gauge model at N strong couplings A. V. Kotikova, L. N. Lipatovb,c, a Bogoliubov Laboratory of Theoretical Physics 3 Joint Institute for Nuclear Research 1 0 141980 Dubna, Russia 2 n b II. Institut fu¨r Theoretische Physik, a J Universität Hamburg 5 Luruper Chaussee 149, 22761 Hamburg, Germany ] h t - c Theoretical Physics Department, p e Petersburg Nuclear Physics Institute h Orlova Rosha, Gatchina, [ 188300, St. Petersburg, Russia 1 v 2 8 8 0 . Abstract 1 0 3 We find the BFKL Pomeron intercept at N = 4 super-symmetric gauge theory in the form 1 v: of the inverse coupling expansion j0 = 2−2λ−1/2−λ−1+1/4λ−3/2+2(1+3ζ3)λ−2+O(λ−5/2) with the use of the AdS/CFT correspondence in terms of string energies calculated recently. i X The corresponding slope γ (2) of the anomalous dimension calculated directly up to the fifth ′ r a order of perturbation theory turns out to be in an agreement with the closed expression obtained from the recent Basso results. PACS: 12.38.Bx 1 Introduction PomeronistheReggesingularity ofthet-channel partialwave [1] responsible fortheapprox- imate equality of total cross-sections for high energy particle-particle and particle-antiparticle interactions valid in an accordance with the Pomeranchuck theorem[2]. In QCD the Pomeron is a colorless object, constructed from reggeized gluons [3]. The investigation of the high energy behavior of scattering amplitudes in the = 4 Super- N symmetric Yang-Mills (SYM) model [4, 5, 6] is important for our understanding of the Regge processes in QCD. Indeed, this conformal model can be considered as a simplified version of QCD, in which the next-to-leading order (NLO) corrections [7] to the Balitsky-Fadin-Kuraev- Lipatov (BFKL) equation [3] are comparatively simple and numerically small. In the N = 4 SYM the equations for composite states of several reggeized gluons and for anomalous dimensions of quasi-partonic operators turn out to be integrable at the leading logarithmic approximation [8, 9]. Further, the eigenvalue of the BFKL kernel for this model has the remarkable property of the maximal transcendentality [5]. This property gave a possibility to calculate the anomalous dimensions (AD) γ of the twist-2 Wilson operatorsin one [10], two [5, 11], three [12], four [13, 14] and five [15] loops using the QCD results [16] and the asymptotic Bethe ansatz [17] improved with wrapping corrections [14] 1 in an agreement with the BFKL predictions [4, 5]. On the other hand, due to the AdS/CFT-correspondence [19, 20, 21], in = 4 SYM some N physical quantities can be also computed at large couplings. In particular, for AD of the large spin operators Beisert, Eden and Staudacher constructed the integral equation [22] with the use the asymptotic Bethe-ansatz. This equation reproduced the known results at small coupling constants and is in a full agreement (see [23, 24]) with large coupling predictions [25, 26]. With the use of the BFKL equation in a diffusion approximation [3, 4, 6], strong coupling results for AD [25] and the pomeron-graviton duality [27] the Pomeron intercept was calculated at the leading order in the inverse coupling constant (see the Erratum[28] to the paper [12]). 2 Similar results in the = 4 SYM and QCD were obtained in Refs. [29] and [30]. The N Pomeron-graviton duality in the = 4 SYM gives a possibility to construct the Pomeron N interaction model as a generally covariant effective theory for the reggeized gravitons [31]. Below we use recent calculations [32, 33, 34, 35] of string energies to find the strong coupling corrections to the Pomeron intercept j = 2 ∆ in next orders. We discuss also the relation 0 − between the Pomeron intercept and the slope of the anomalous dimension at j = 2. 2 BFKL equation at small coupling constant The eigenvalue of the BFKL equation in = 4 SYM model has the following perturbative N expansion [4, 5] (see also Ref. [6]) λ λ j 1 = ω = χ(γ )+δ(γ ) , λ = g2N , (1) − 4π2 BFKL BFKL 16π2 c (cid:20) (cid:21) 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 1The anomalous dimensions up to four loops were calculated also with the use of the Baxter equation [18]. f 2The value of this intercept was estimated earlier in Ref.[11]. 1 conformal spin n = m m = 0 they depend only on the BFKL anomalous dimension − m+m 1 f γ = = +iν (2) BFKL 2 2 f and are presented below [4, 5] χ(γ) = 2Ψ(1) Ψ(γ) Ψ(1 γ), (3) − − − ′′ ′′ δ(γ) = Ψ (γ)+Ψ (1 γ)+6ζ 2ζ χ(γ) 2Φ(γ) 2Ψ(1 γ). (4) 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)k+1 ∞ Φ(γ) = 2 − β (k +1), (5) ′ k +γ k=0 X where 1 z +1 z β (z) = Ψ Ψ . (6) ′ ′ ′ 4" 2 − 2 # (cid:16) (cid:17) (cid:16) (cid:17) Due to the symmetry of ω to the substitution γ 1 γ expression (1) is an even BFKL BFKL → − function of ν ω = ω + ∞ ( 1)mD ν2m, (7) 0 m − m=1 X where λ λ ω = 16ln2 1+c +O(λ3), (8) 0 4π2 " 116π2# λ δ(2m)(1/2) λ2 D = 8 22m+1 1 ζ + +O(λ3). (9) m − 2m+14π2 (2m)! 64π4 (cid:16) (cid:17) According to Ref. [5] we have 1 π 165 c = 11ζ 32Ls πζ 7.5912, (10) 1 3 3 2 − 2ln2 − 2 − 16 ≈ − (cid:18) (cid:16) (cid:17) (cid:19) where (see [36]) x y Ls (x) = ln2 2sin dy. (11) 3 − 2 Z0 (cid:12)(cid:12) (cid:16) (cid:17)(cid:12)(cid:12) Thus, therightmost Pomeron singularity ofthe(cid:12)partial w(cid:12)ave f (t) in theperturbationtheory (cid:12) (cid:12) j is situated at λ λ j = 1+ω = 1+16ln2 1+c +O(λ3) (12) 0 0 4π2 " 116π2# 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 π2ω ω λ = 1 c +O ω3 . cr 1 4ln2 − 64ln2 (cid:18) (cid:19) (cid:16) (cid:17) 2 It will be interesting to find higher order corrections to the BFKL intercept j from the inves- 0 tigation 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 . But it is expected, that with growing ω the nearest singu- cr larity, 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 → ∞ j = 2 ∆, (13) 0 − 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 Dueto the energy-momentum conservation, the universal anomalous dimension of the stress tensor T should be zero, i.e., µν γ(j = 2) = 0. (14) It is important, that the anomalous dimension γ contributing to the DGLAP equation [37] does not coincide with γ appearing in the BFKL equation. They are related as follows [7] BFKL (see also [38]) ω j γ = γ + = +iν, (15) 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 (1) to the eigenvalue of the BFKL ∼ kernel [7]. Using above relations one obtains ν(j = 2) = i. (16) As a result, from eq. (7) for the Pomeron intercept we derive the following representation for the correction ∆ (13) to the graviton spin 2 ∞ ∆ = D . (17) m m=1 X In the diffusion approximation, where D = 0 for m 2, one obtains from (17) the relation m ≥ between the diffusion coefficient D and ∆ (see [28]) 1 D ∆. (18) 1 ≈ 3 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 j 2 = ∞ D ( ν2)m 1 , (19) m − − − mX=1 (cid:16) (cid:17) where ν2 is related to γ according to eq. (15) j 2 ν2 = γ . (20) − 2 − (cid:18) (cid:19) 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 the local operatorsas follows [19, 25] E2 = (j +Γ)2 4, Γ = 2γ (21) − − 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 E2 ν2 = +1 . (22) − 4 ! Thisexpressionforν2 canbeinsertedinther.h.s. ofEq. (19)leadingtothefollowingexpression for the Regge trajectory of the graviton in the anti-de-Sitter space E2 m ∞ j 2 = D +1 1 . (23) m − " 4 ! − # m=1 X Note [28], that due to (22) expression (7) for the eigenvalue of the BFKL kernel in the diffusion approximation (18) j = j ∆ν2 = 2 ∆ ν2 +1 , (24) 0 − − (cid:16) (cid:17) is equivalent to the linear graviton Regge trajectory α j = 2+ ′t, αt = ∆E2/2, (25) ′ 2 where its slope α and the Mandelstam invariant t, defined in the 10-dimensional space, equal ′ R2 α = , t = E2/R2 (26) ′ 2 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 1 j √λ, (27) ≪ ≪ 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 and ∆ (see Appendix A). 3 m 3 When this paper was almost prepared for publication, we found the article [40] containing some of our results (see discussions in Appendix A). 4 4 Graviton Regge trajectory and Pomeron intercept The coefficients D and D at large λ can be written as follows 4 1 2 2 2a 8a 01 10 D = 1 , D = , , (28) 1 √λ − √λ ! 2 − λ3/2 where a and a are calculated in Appendix A 01 10 1 3 a = , a = . (29) 01 10 −4 8 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 E2 E2 2 E2 j 2 = D +D + , E2 = 2αt. (30) 1 2 ′ − 4  4 ! 2    Note, that the perturbation theory for the BFKL equation gives this trajectory at small ω = j 1 (see eq. (1)), where − 1 E2 γ = +iν, ν2 = +1 . (31) BFKL 2 − 4 ! However theenergy-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 E2/2 E2/λ3/2 at λ in comparison with a larger correction 2 ∼ → ∞ a E2/λ, we obtain the graviton trajectory (30) in the form 01 2 2a E2 8a E2 2 01 10 j 2 = 1 . (32) − √λ − √λ ! 4 − λ3/2 4 ! Solving this quadratic equation, one can derive with the same accuracy (see [32, 35]) 2 E2 a +a (j 2) 01 10 = j 2 1+2 − . (33) √λ 4 − √λ ! (cid:16) (cid:17) On the other hand, due to (21) this relation can be written as follows 1 2 2 a01 +a10(j 2) j 2γ = + j 2 1+2 − (34) 2√λ − √λ − √λ ! (cid:16) (cid:17) (cid:16) (cid:17) and for j 2 >> 1/√λ we have − 1 1 j 2γ = 2(j 2)λ1/4 1+ +a +a (j 2) . (35) 01 10 − − " j 2 − ! √λ# q − 4 Hereweconsideronlythe calculationofthe λ−1 correctiontoPomeronintercept. Moregeneralresultsare presented in Appendix A. 5 In particular, for j = 4 one obtains the anomalous dimension for the Konishi operator γ = γ [32] (see also Appendix B) K 1 1 1 2 γ = λ1/4 1+ +a +2a = λ1/4 1+ (36) K 01 10 − " (cid:18)2 (cid:19) √λ# " √λ# in an agreement with eq. (28). Fortheanomalousdimensionatj 2 1/√λfrom(34)weobtainthesquarerootsingularity − ∼ similar to that appearing at small j 1 = ω (8) 0 − λ1/4 a 01 γ = 1+ D +j 2 D , (37) − √2 √λ! 1 − − 1 (cid:16)q q (cid:17) whereD (28)isequal tothecorrection∆tothegravitontrajectoryintercept withouraccuracy 1 2 1 ∆ = D 1+ . 1 ≈ √λ 2√λ! Note, that in the region j 2 < ∆, the anomalous dimesnion is complex similar to it in the − − perturbative regime at j 1 < ω (8). Moreover, the position of the BFKL singularity of γ at 0 − 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. − 5 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 2 3 4 5 λ 1 λ 2 λ 7 λ 11 λ γ (2) = + + +O(λ6), (38) ′ −24 2 24! − 5 24! 20 24! − 35 24! 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 j 2m j 2 = D γ 1 (39) m − m=1 "(cid:18)2 − (cid:19) − # X in the variable j for j = 2 using γ(2) = 0: 1 = 1 2γ (2) mD 1 2γ (2) D , (40) ′ m ′ 1 − ≡ − (cid:16) (cid:17) mX=1 (cid:16) (cid:17) where D is found in Appendix A in the expansion up to m = 3 (see (A.8)). So we obtain 1 explicitly √λ 1 2γ (2) = h (λ). (41) ′ 0 − 2 6 Here a ∞ 0i h (λ) = 1+ (42) 0 (λ)i/2 i=1 X is the leading contribution to the string energy in the limit j << √λ, as it is shown in (A.1) and (A.2)). Itisimportant, thatnowthereisanexplicit expression forh (λ)(see(A.3)). Itwasobtained 0 from the Basso result [33] by taking the value of the angular momentum J equal to two (see an [34]). Substituting (A.3) in (41), we have the closed form for the slope γ (2) ′ √λI (√λ) 3 γ (2) = , (43) ′ − 4 I (√λ) 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 ′ λ 1 2 = γ (2)+ (γ (2)) . (44) ′ ′ 24 − 2 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 √λ 1 γ (2) = +1+O , (45) ′ −2√3 √λ! whereas the strong coupling expansion of the explicit result (43) is given below √λ 5 1 151 15 1 1 γ (2) = 1 + + +O . (46) ′ − 4 − 2√λ 8 λ 8 λ3/2 (cid:18)λ2(cid:19)! 6 Numerical analysis of the Pomeron intercept j (λ) 0 Let us obtain an unified expression for the position of the Pomeron singularity j = 1+ω 0 0 for arbitrary values of λ, using an interpolation between weak and strong coupling regimes. It is convenient to replace ω with the new variable t as follows 0 2ω 1+t2 1 t = 0 , ω = 0 − . (47) 0 1 ω2 0 q t − 0 0 This variable has the asymptotic behavior t λ at λ 0 and t √λ/2 at λ 0 0 ∼ → ∼ → ∞ 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 = t (λ) whose 0 0 solution will interpolate ω for the full λ range. 0 We choose the equation of the form k (λ) = k (λ)t +k (λ)t2, (48) 0 1 0 2 0 7 where the following anzatz for the coefficinets k , k and k is used: 0 1 2 k (λ) = β λ+α λ2, k (λ) = β +α λ, k (λ) = γ λ 1 +β +β λ. (49) 0 0 0 1 1 1 2 2 − 2 2 Here γ , α and β (i = 0,1,2)are free parameters, which are fixed using the known asymptotics 2 i i of ω at λ 0 and λ . 0 → → ∞ The solution of quadratic equation (48) is given below k2 +4k k k t = 1 0 2 − 1. (50) 0 q 2k 2 To fix the parameters γ , α and β (i = 0,1,2), we use two known coefficients for the weak 2 i i coupling expansion of ω : 0 ω = e˜ λ+e˜ λ2 +e˜ λ3 +... (at λ 0) (51) 0 1 2 3 → with 4ln2 7.5912 ˜ e˜ = 0.28092, e˜ = d 0.01350 (52) 1 π2 ≈ 2 − 1 16π2 ≈ − and first four terms of its strong coupling expansion 2 t˜ t˜ t˜ t˜ 1 2 3 4 ω = 1 ∆, ∆ = 1+ + + + +... (at λ ) (53) 0 − √λ √λ λ λ3/2 λ2 ! → ∞ with (see below Eq. (62) 1 1 145 9 t˜ = , t˜ = , t˜ = 1 3ζ , t˜ = 2a ζ . (54) 1 2 3 3 4 12 3 2 − 8 − − − 128 − 2 The coefficients e˜ and t˜ are unknown but we estimate them later from the interpolation. 3 4 Then, for the weak and strong coupling expansions of t one obtains t = e λ+e λ2 +e λ3 +..., (when λ 0), (55) 0 1 2 3 → t = √λ 1 t1 t2 t3 t4 +..., (when λ ), (56) 0 2 − √λ − λ − λ3/2 − λ2 → ∞ (cid:16) (cid:17) where 3 11 e = e˜ , e = e˜ , e = e˜ +e˜3, t = t˜ +1 = , t = t˜ t˜2 +1 = , 1 1 2 2 3 3 1 1 1 2 2 2 − 1 − 8 1 t = t˜ 2t˜ t˜ +t˜3 t˜ +1 = 1+12ζ , 3 3 − 2 1 1 − 1 −4 3 (cid:16) (cid:17) (cid:16) (cid:17) 9 3 t = t˜ 2t˜ t˜ t˜2 +3t˜t˜2 t˜4 t˜ 2t˜ +1 = 2a + ζ . (57) 4 4 − 3 1 − 2 2 1 − 1 − 2 − 1 12 128 − 2 3 (cid:16) (cid:17) (cid:16) (cid:17) Comparing the l.h.s. and the r.h.s. of Eq. (48) at λ 0 and λ , respectively, we → → ∞ derive the following relations α = 4α , α = 6α , β = C α , β = C α , γ = C α ,β = C e +C e2 α (58) 2 0 2 0 1 1 0 2 2 0 2 3 0 0 1 1 3 1 0 h i 8 -1 + jo 1 0.8 0.6 0.4 0.2 z -2 2 4 6 Figure 1: (color-online). The results for j as a function of z (λ = 10z). 0 with the free parameter α which disappears in the retionship k /k and k /k and, thus, in 0 1 2 0 2 the results (50) for t . 0 Here C 23.99, C 5.71, C 73.65, (59) 1 2 3 ≈ − ≈ − ≈ which lead to the following predictions for the coefficients e and t in (55) and (56) 3 4 6e +2C e e +4e2 +C e2 121+16(C 3C +5C ) e = 2 2 1 2 1 3 2 0.01676, t = 3 − 1 2 15.5785 3 4 − C +2C e ≈ − 128 ≈ 1 3 1 (60) and, respectively, for the corresponding terms in (51), (53) and (54) e˜ 0.03893, t˜ 11.0192, a 8.1793. (61) 3 4 12 ≈ − ≈ − ≈ Note that the results for the coefficients e , t , e˜ , t˜ and a do not depend on the free 3 4 3 4 12 parameter α . 0 On Fig. 1, we plot the pomeron intercept j as a function of the coupling constant λ. The 0 behavior of the pomeron intercept j shown in Fig.1 is similar to that found in QCD with some 0 additional assumptions (see ref. [30]). 7 Conclusion We found the intercept of the BFKL pomeron at weak and strong coupling regimes in the = 4 Super-symmetric Yang-Mills model. N 9