Three-loop universal anomalous dimension of the Wilson operators in N=4 SUSY Yang-Mills model

We present results for the three-loop universal anomalous dimension of Wilson twist-2 operators in the N=4 Supersymmetric Yang-Mills model. These results are obtained by extracting the most complicated contributions from the three loop non-singlet anomalous dimensions in QCD which were calculated recently. Their singularities at j=1 agree with the predictions obtained from the BFKL equation for N=4 SYM in the next-to-leading order. The asymptotics of universal anomalous dimension at large j is in an agreement with the expectations based on an interpolation between weak and strong coupling regimes in the framework of the AdS/CFT correspondence.


Introduction
The anomalous dimensions (AD) of the twist-two Wilson operators govern the Bjorken scaling violation for parton distributions in a framework of Quantum Chromodynamics (QCD) [1]. These quantities are expressed through the Mellin transformation of the splitting kernels W b→a (x) for the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equation [2] which relates the parton densities f a (x, Q 2 ) (hereafter a = λ, g, φ for the spinor, vector and scalar particles, respectively) with different values of Q 2 as follows The anomalous dimensions and splitting kernels in QCD are well known up to the next-toleading order (NLO) of the perturbation theory [1]. The QCD expressions for AD can be transformed to the case of the N = 1 Supersymmetric Yang-Mills theories (SYM) if one will use for the Casimir operators C A , C F , T f the following values C A = C F = N c , T f = N c /2 (the last substitution follows from the fact, that each gluino λ i being a Majorana particle gives a half of the contribution for the Dirac spinor). For extended supersymmetric theories the anomalous dimensions cannot be obtained in this simple way, because additional contributions coming from scalar particles should be also taken into account [3]. Recently these anomalous dimensions were calculated in the next-to-leading approximation [4] for the N = 4 Supersymmetric Yang-Mills theory.
It turns out, that the expressions for eigenvalues of the AD matrix in the N = 4 SYM can be derived directly from the QCD anomalous dimensions without tedious calculations by using a number of plausible arguments. The method elaborated in Ref. [5] for this purpose is based on special properties of solutions of the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation [6,7] in this model and a new relation between the BFKL and DGLAP equations (see [3]). In the NLO approximation this method gives the correct results for AD eigenvalues, which was checked by direct calculations in [4]. Its properties will be reviewed below only shortly and a more extended discussion can be found in [5].
Next-to-next-to-leading order (NNLO) corrections to AD in QCD were calculated recently [8] * in the nonsinglet case. Using these results and the method of Ref. [5] we derive in this paper the eigenvalues of the anomalous dimension matrix for the N = 4 SYM in the NNLO approximation.

Evolution equation in N = 4 SYM
The reason to investigate the BFKL and DGLAP equations in the case of supersymmetric theories is based on a common belief, that the high symmetry may significantly simplify the structure of these equations. Indeed, it was found in the leading logarithmic approximation (LLA) [10], that the so-called quasi-partonic operators in N = 1 SYM are unified in supermultiplets with anomalous dimensions obtained from universal anomalous dimensions γ uni (j) by shifting its arguments by an integer number. Further, the anomalous dimension matrices for twist-2 operators are fixed by the superconformal invariance [10]. Calculations in the maximally extended N = 4 SYM, where the coupling constant is not renormalized, give even more remarkable results. Namely, it turns out, that here all twist-2 operators enter in the same multiplet, their anomalous dimension matrix is fixed completely by the super-conformal invariance and its universal anomalous dimension in LLA is proportional to Ψ(j − 1) − Ψ(1), which means, that the evolution equations for the matrix elements of quasi-partonic operators in the multicolour limit N c → ∞ are equivalent to the Schrödinger equation for an integrable Heisenberg spin model [11,12]. In QCD the integrability remains only in a small sector of the quasi-partonic operators [13]. In the case of N = 4 SYM the equations for other sets of operators are also integrable [14,15,16]. Evolution equations for quasi-partonic operators are written in an explicitly super-conformal form in Ref. [17].
Similar results related to the integrability of the multi-colour QCD were obtained earlier in the Regge limit [18]. Moreover, it was shown [3], that in the N = 4 SYM there is a deep relation between BFKL and DGLAP evolution equations. Namely, the j-plane singularities of AD of the Wilson twist-2 operators in this case can be obtained from the eigenvalues of the BFKL kernel by their analytic continuation. The NLO calculations in N = 4 SYM demonstrated [5], that some of these relations are valid also in higher orders of perturbation theory. In particular, the BFKL equation has the property of the hermitian separability, the linear combinations of the multiplicatively renormalized operators do not depend on the coupling constant, the eigenvalues of the anomalous dimension matrix are expressed in terms of the universal function γ uni (j) which can be obtained also from the BFKL equation [5]. The results for γ uni (j) were checked by the direct calculations in Ref. [4] In the N = 4 SYM theory [19] we have the following field content: one gluon g, four Majorana fermions λ and three complex scalars φ. All particles belong to the adjoint representation of the gauge group SU(N c ). This model possesses an internal SU(4) symmetry. In the N = 4 SYM theory one can introduce the following colour and SU(4) singlet local Wilson twist-2 operators [20,10,21,22]: where D µ are covariant derivatives. The spinors λ i and field tensor G ρµ describe gluinos and gluons, respectively, and φ r are the complex scalar fields appearing in the N = 4 supersym-metric model. Indices i = 1, · · · , 4 and r = 1, · · · , 6 refer to SU(4) and SO(6) ≃ SU(4) groups of inner symmetry, respectively. The symbolŜ implies a symmetrization of each tensor in the Lorentz indices µ 1 , ..., µ j and a subtraction of its traces. The anomalous dimension matrices can be written for unpolarized and polarized cases, respectively, as follows Note, that in the super-multiplet of twist-2 operators there are also operators with fermion quantum numbers and operators anti-symmetric in two Lorentz indices [10,22]. For the case N = 4 the multiplicatively renormalized operators were found in an explicit way and their universality properties for all orders of perturbation theory were formulated in Refs. [12,5].
After their diagonalization, the new unpolarized γ and polarizedγ AD matrices have the following form which corresponds to AD matrices for multiplicatively renormalizable linear combinations of operators (1)- (5). Here, the matrices V, V −1 ,Ṽ andṼ −1 were calculated in [5] and in LO we have γ lm (j) = 0,γ lm (j) = 0 for l, m = +, 0, −. In NLO the AD matrices become triangle [4] due to superconformal invariance breaking [23], similar to the case of N = 1 SYM [21]. The eigenvalues γ l (j) andγ l (j) govern the power-like violation of the Bjorken scaling for the parton distributions.
Due to the fact that all twist-2 operators belong to the same supermultiplet the anomalous dimensions γ l (j) andγ l (j) (l = +, 0, −) have the properties [12,5] where γ uni (j) is the universal anomalous dimension.

Method of obtaining AD eigenvalues in N = 4 SYM
As it was already pointed out in Introduction, the universal anomalous dimension can be extracted directly from the QCD results without finding the scalar particle contribution. This possibility is based on deep relation between DGLAP and BFKL dynamics in the N = 4 SYM [3,5].
To begin with, the eigenvalues of the BFKL kernel turn out to be analytic functions of the conformal spin |n| at least in two first orders of perturbation theory [5]. Further, in the framework of the DR-scheme [24] one can obtain from the BFKL equation (see [3]), that there is no mixing among the special functions of different transcendentality levels i † , i.e. all special functions at the NLO correction contain only sums of the terms ∼ 1/j i (i = 3). More precisely, if we introduce the transcendentality level for the eigenvalues of integral kernels of the BFKL equations as functions of γ and appearing in the perturbation theory in an accordance with the complexity of the terms in the corresponding sums then for the BFKL kernel in the leading order (LO) and in NLO the corresponding levels are i = 1 and i = 3, respectively.
Because in N = 4 SYM there is a relation between the BFKL and DGLAP equations (see [3,5]), the similar properties should be valid for the anomalous dimensions themselves, i.e. the basic functions γ uni (j) and γ (2) uni (j) are assumed to be of the types ∼ 1/j i with the 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. Such contributions could be generated and/or removed by an approximate finite renormalization of the coupling constant. But these terms do not appear in the DR-scheme.
It is known, that at the LO and NLO approximations the most complicated contributions (with i = 1 and i = 3, respectively) are the same for all LO and NLO anomalous dimensions in QCD [1] and for the LO and NLO scalar-scalar anomalous dimensions [4]. This property allows one to find the universal anomalous dimensions γ (0) uni (j) and γ (1) uni (j) without knowing all elements of the anomalous dimension matrix [5], which was verified by the exact calculations in [4].
Using above arguments, we conclude, that at the NNLO level there is only one possible candidate for γ (2) uni (j). Namely, it is the most complicated part of the nonsinglet QCD anomalous dimension matrix (with the SUSY relation for the QCD color factors C F = C A = N c ). Indeed, after the diagonalization of the anomalous dimension matrix the eigenvalues γ l (j) andγ l (j) in Eq. (7) should have this most complicated part as a common contribution because they differ each from others only by a shift of the argument (see Eq. (8)) and their differences are constructed from less complicated terms. The non-diagonal matrix elements of γ ab (j) in Eq. (7) contain also only less complicated terms (see, for example, AD exact expressions at LO and NLO approximations in Refs. [1] for QCD and [4] for N = 4 SYM) and therefore they cannot generate the most complicated contributions to γ l (j) andγ l (j).
Thus, the most complicated part of the nonsinglet NNLO QCD anomalous dimension should coincide (up to color factors) with the universal anomalous dimension γ (2) uni (j).

NNLO anomalous dimension for N = 4 SYM
The final three-loop result ‡ for the universal anomalous dimension γ uni (j) for N = 4 SYM ‡ Note, that in an accordance with Ref. [7] our normalization of γ(j) contains the extra factor −1/2 in comparison with the standard normalization (see [1]) and differs by sign in comparison with Vermaseren-Moch-Vogt one [8]. and S −a (j) = The expression (14) is defined for all integer values of arguments (see [26,5]) but can be easily analytically continued to real and complex j by the method of Refs. [27,5].
The limit j → 1 The limit j → 1 is important for the investigation of the small-x behavior of parton distributions (see review [31] and references therein). Especially it became popular recently because there are new experimental data at small x produced by the H1 and ZEUS collaborations in HERA [32].
Using asymptotic expressions for harmonic sums at j = 1+ω → 1 (see formulae in Appendix at r = −1) we obtain for the N = 4 universal anomalous dimension γ uni (j) in Eq. (9) in an agreement with the predictions for γ (0) uni (1 + ω) and also for the first term of γ (2) uni (1 + ω) coming from an investigation of BFKL equation at NLO accuracy in [3] § .
The limit j → −r, r ≥ 0 In the case of N = 4 SYM the eigenvalue of the BFKL kernel [3] is analytic in the conformal spin |n|, which allows us to continue it to the negative values of |n| [3]. It gives a possibility to find the singular contributions to anomalous dimensions of the twist-2 operators not only at j = 1 but also at other integer negative points j = 0, −1, −2, · · ·. In the Born approximation for the universal anomalous dimension of the supermultiplet of the twist-2 operators one can obtain in such way γ uni = 4â (Ψ(1) − Ψ(j − 1)) [3], which coincides with the result of the direct calculations (see [11,12] and Eq. (10)).
In the NLO approximation the study of the relation between BFKL and DGLAP equation at j → −r, r ≥ 0 was done in [5]. Its extension to the NNLO level is a subject of future investigations. Here we present only the AD results in this limit.
Using formulae presented in Appendix for the harmonic sums calculated in the limit j = ω − r → −r, r ≥ 0 we can find singularities of γ uni (j) (9) at small ω γ (2) uni (j) = 4 where Unfortunately, the results of Refs. [3,5] contain a misprint. Namely, the coefficient in front ofâ 3 obtained in the limit j → 1 in Eq. (39) of Ref. [5] should be multiplied by a factor 4.
Here, to shorten notation when presenting resulting expression for singularities of γ (2) uni (j) we dropped argument r + 1 in all harmonic sums. So, the double-logarithmic poles ∼ ω −3 and ∼ ω −5 appear in the anomalous dimensions γ (1) uni (j) and γ (2) uni (j), respectively, only in the case of even r values, that is in an agreement with the predictions from the BFKL equation [5].

Resummation of γ uni and the AdS/CFT correspondence
In the limit j → ∞ the AD results (10)-(12) are simplified significantly. Note, that 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.
We obtain the following asymptotics for the N = 4 universal anomalous dimension γ uni (j) in Eq. (9) with Recently there was a great progress in the investigation of the N = 4 SYM theory in a framework of the AdS/CFT correspondence [28] where the strong-coupling limit α s N c → ∞ is described by a classical supergravity in the anti-de Sitter space AdS 5 × S 5 . In particular, a very interesting prediction [29] (see also [30]) was obtained for the large-j behavior of the anomalous dimension for twist-2 operators in the strong coupling regime (see Ref. [33] for asymptotic corrections) ¶ : lim z→∞ a = −z 1/2 + 3 ln 2 8π ¶ Here we took into account, that in our normalization γ(j) contains the extra factor −1/2 in comparison with that in Ref. [29].
On the other hand, all anomalous dimensions γ i (j) andγ i (j) (i = +, 0, −) coincide at large j and our results for γ uni (j) in Eq. (9) allow one to find three first terms of the small-z expansion of the coefficient a(z) lim z→0 a = −z + π 2 12 z 2 − 11 720 π 4 z 3 + ... .
For resummation of this series we suggest the following equation forã [4] z = − a + π 2 12 a 2 (32) interpolating between its weak-coupling expansion up to NNLÕ and strong-coupling asymptotics It is remarkable, that the prediction for NNLO based on the above simple equation is valid with the accuracy ∼ 10%. It means, that this extrapolation seems to be good for all values of z.
Further, for j → 2 let us take into account, that according to the BFKL equation [7] the anomalous dimension of twist-2 operators is quantized in the Regge kinematics: for the principal series of unitary representations of the Möbius group. On the other hand, in the diffusion approximation valid near the leading singularity of the t-channel partial wave the eigenvalue of the BFKL kernel is where ω 0 and D are the Pomeron intercept and diffusion coefficient, respectively. These quantities are functions of the coupling constant. We assume, that for the large coupling constant in N = 4 SUSY the Pomeron intercept approaches the graviton intercept in the AdS 5 × S 5 space [35], which means, that where ∆ is a small number. Further, due to the energy-momentum conservation (γ = 0 for j = 2) the parameters ∆ and D are equal and γ(j) can be expressed near j = 2 only in terms of one parameter Note, that we use the DR-scheme for coupling constant which removes −1/12 from coefficients of a in Eq.(28) of Ref. [4] (see [5,34]). This relation is in an agreement with the prediction of A. Polyakov and other authors [28,29] providing that This number coincides up to 15% with the estimate ∆ = [ √ 3/(2π)]z −1/2 obtained in (43) from the resummation procedure (40). We can expect, that expression (46) with parameter ∆ calculated in (48) gives the anomalous dimension of twist-2 operators for z → ∞ and all j (neglecting the nonlinearity effects).
Recently we were informed by Prof. Chung-I Tan, that the above correction ∆ to the graviton spin j = 2 coincides in form with that obtained in his unpublished work with R. Brower, J. Polchinsky and M. Strassler from the AdS/CFT correspondence. We thank him for helpful discussions.

Conclusion
Thus, in this paper we constructed the anomalous dimension γ uni (j) for the N = 4 supersymmetric gauge theory in the next-to-next-to-leading approximation and verified its selfconsistency in the Regge (j → 1) and quasi-elastic (j → ∞) regimes. Our result for universal anomalous dimension at j = 4 could be used to determine the anomalous dimension of Konishi operator [36] up to 3-loops. It is remarkable, that our results coincide † † with corresponding predictions from dilatation operator approach and integrability [37,38]. The method, developed for this construction, can be applied also to less symmetric cases of N = 1, 2 SYM and QCD, which are very important for phenomenological applications. For the verification of the AdS/CFT correspondence the calculations of the various physical quantities in N = 4 SYM attract a great interest due to a possibility to develop non-perturbative approaches to QCD.
We demonstrated above that the expressions interpolating between the week and strong regime work remarkably well both in limit j → ∞ and j → 2. The integrability of the evolution equations for the quasi-partonic operators in LLA [11,12] is an interesting property of N = 4 SYM which should be verified on NLO and NNLO level. We hope to discuss these problems in our future publications.