Non-planar cusp and transcendental anomalous dimension at four loops in N=4 supersymmetric Yang-Mills theory

We compute the non-planar contribution to the universal anomalous dimension of the SU(4)-singlet twist-two operators in N=4 supersymmetric Yang-Mills theory at four loops through Lorentz spin eighteen. From this, we numerically evaluate the non-planar contribution to the four-loop light-like cusp anomalous dimension and derive the transcendental zeta(3) and zeta(5) parts of the universal anomalous dimension for arbitrary Lorentz spin in analytic form.

The anti-de Sitter/conformal field theory (AdS/CFT) correspondence [1][2][3], also known as holographic duality, has been one of the most active and tantalizing research topics in high-energy theory over the past two decades. This implies that quantum gravity in anti-de Sitter space, with constant negative curvature, is equivalent to a lower-dimensional non-gravitational quantum field theory of conformal type, N = 4 supersymmetric Yang-Mills (SYM) theory, living on the boundary of that gravitational space. The AdS/CFT correspondence has led to a plethora of intriguing physical insights and powerful novel methods of calculation [4][5][6][7][8][9][10][11][12][13][14][15][16]. The latter allow us to solve longstanding problems not only in supersymmetric toy models, but also in real theories of nature, such as quantum chromodynamics (QCD) [17][18][19].
So far, investigations of the AdS/CFT correspondence have largely been confined to the planar limit, in which Feynman diagrams of planar topologies contribute, while non-planar topologies are far more difficult to tackle. It is obviously of paramount interest to go beyond the planar limit, as this will allow us to significantly deepen and consolidate our understanding of the AdS/CFT correspondence and to access as-yet unexplored regions of it.
Quantities of key interest include the anomalous dimensions of the operators, composed of the quantum fields of N = 4 SYM theory, that are of leading twist, twist two, and are singlets under the internal symmetry group SU(4). These operators are sorted by their Lorentz spin j, which counts the covariant derivatives, and are multiplicatively renormalized, sharing the same, universal anomalous dimension γ uni (j), which just depends on j. Non-planar contributions to the latter can be obtained by directly computing, by means of advanced computerized methods, the relevant Feynman diagrams in perturbation theory in powers of the gauge coupling g.
The study of the renormalization of composite operators in N = 4 SYM theory has led to the discovery of the relation of this problem with exactly solvable models [20]. The integrability in the planar limit was intensively studied and established from both sides of the AdS/CFT correspondence (see Ref. [6] for a review and Refs. [7,21] for the recently developed Quantum Spectral Curve approach). In the non-planar case, integrability-based methods have been considered in general in Refs. [22,23]. Non-planar contributions to anomalous dimensions serve as a welcome laboratory for stringent tests of the ideas and models thus proposed. This provides a strong motivation for our present work.
Once a general result for the universal anomalous dimension is established, it is interesting to study its analytical properties and particular limits. The most interesting one, j → ∞, yields the light-like cusp anomalous dimension γ cusp [24], which can be computed by alternative methods, too. The planar part of γ cusp has been found to all orders a long time ago, through the asymptotic Bethe-ansatz equation [5]. Recently, its non-planar part has been established through four loops, at O(g 8 ), via the Sudakov form factor, numerically in Ref. [25,26] and analytically in Ref. [27], and via the light-like polygonal Wilson loops, again analytically, in Ref. [28]. At four loops in QCD, at O(α 4 s ) in the strong-coupling constant α s , the quark cusp anomalous dimension in the planar limit has been found via the quark form factor in Ref. [29], its contribution with quartic fundamental color factor has been obtained, again via the quark form factor, in Ref. [30], and the complete quark and gluon cusp anomalous dimensions have been established via the massless quark and gluon form factors just recently in Ref. [31].
Explicit knowledge of γ uni (j) for general value of j would unfold the non-planar anatomy of the anomalous dimensions in N = 4 SYM theory. A possible avenue to this goal is to evaluate γ uni (j) for as many values of j as possible and to try and extract from this the general result. In N = 4 SYM theory, non-planarity appears for the first time at O(g 8 ). In Refs. [32][33][34], the non-planar contributions to γ uni (j) at O(g 8 ) were analytically calculated for the first three nontrivial values j = 4, 6, 8. Recently, the non-planar anomalous dimensions for the twist-two operators in N = 4 SYM theory were computed by means of the asymptotic expansions of the four-point function of length-two half-BPS operators [35], confirmed our results and provided new result for j = 10. The purpose of this Letter is to push this endeavor as far as possible, which turns out to be through j = 18, thanks to cutting-edge technology and computing power. We will thus be able to reconstruct the general coefficients of ζ 3 and ζ 5 and to obtain an independent numerical result for γ cusp at O(g 8 ). The former is new, and the latter confirm previous findings in Ref. [32] and Refs. [25][26][27][28], respectively.
Specifically, the set of local, gauge-invariant, SU(4)singlet, twist-two operators of definite Lorentz spin j in N = 4 SYM theory reads: where the spinors λ i refer to the gauginos, the field strength tensor G µν to the gauge fields, φ r are the complex scalar fields of extended supersymmetry, and D µ are covariant derivatives. The indices i = 1, . . . , 4 and r = 1, . . . , 3 refer to the SU(4) and SO(6) ≃ SU(4) groups of internal symmetry, respectively. The symbol S implies a symmetrization of the respective tensor in the Lorentz indices µ 1 , ..., µ j and a subtraction of all its possible traces. As mentioned above, these operators form the multiplicatively renormalized operators, whose anomalous dimensions are expressed through the so-called universal anomalous dimension up to integer argument shifts [36], where λ = g 2 YM N c is the 't Hooft coupling constant. In the planar limit, γ uni (j) is analytically known for arbitrary j through seven loops [37][38][39][40][41][42] and for special values of j even through ten loops [43][44][45][46][47][48][49], e.g. for j = 4 corresponding to the Konishi operator. In the latter case, we quote the result through four loops [43][44][45] here: As for the non-planar contributions to γ uni (j) at O(g 8 ), the state of the art is given by [32][33][34] γ uni,np (6) = where we have pulled out common factors. If such a factorization were preserved for the higher j values, this could considerably simplify the procedure of finding the general form of γ (3) uni,np (j). In this Letter, we extend Eqs. (6)-(8) by the next five terms. Our computational procedure is similar to Refs. [32][33][34]. We work in the programming language FORM [50]. Specifically, we generate all the contributing Feynman diagrams with DIANA [51] based on QGRAF [52], evaluate the color traces with COLOR [53], reduce the occuring scalar integrals to the master integrals of Ref. [54] with the custom-made program package BAMBA based on the Laporta algorithm [55], and reduce the propagator-type diagrams to fully massive tadpole diagrams using infrared rearrangement (IRR) [56] (see also Refs. [54,57,58] for details). Typical Feynman diagrams are depicted in Fig. 1. This setup allows us to proceed to j = 10 only. Further progress is enabled by the recently developed package FORCER [59] based on FORM [50], which was used for the computations of the anomalous dimension of twist-two operators in QCD [60,61]. Altogether, we have  where our result for j = 10 coincides with corresponding result from Ref. [35]. In contrast to Eqs. (6)-(8), it is not possible to extract common factors in Eqs. (9)-(13), so that our expectations regarding factorization have to be dropped. Equipped with the information contained in Eqs. (6)-(13), we now try to reconstruct the general form of γ (3) uni,np (j), i.e. to determine the j dependence of the coefficients of ζ 5 and ζ 3 and the rational reminder in the ansatz For this purpose, we adopt a powerful method based on number theory which has been proposed in Ref. [62] and successfully applied to the reconstruction of anomalous dimensions in N = 4 SYM theory [41,42,63] and QCD [60,64,65]. This method is based on the assumption that γ uni,np,ζ5 (j), γ uni,np,ζ3 (j), and γ uni,np,rational (j) in Eq. (14) are linear combinations of certain basis functions with certain coefficients. As for the basis functions and coefficients, we are guided by several heuristic observations.
As for the basis functions, in the case of anomalous dimensions of twist-two operators in N = 4 SYM theory, these are known to be generalized harmonic sums, defined as [66,67] S a1,...,an (M ) = M j=1 (sign(a 1 )) j j |a1| S a2,...,an (j) , (15) where the indices a 1 , . . . , a n may take all (positive and negative) integer values, except for −1. The weight or transcendentality ℓ of the sum S a1,...,an is defined as the sum of the absolute values of its indices, ℓ = |a 1 | + . . . + |a n |, and the weight of a product of generalized harmonic sums is equal to the sum of their weights. For twist-two operators, there is an additional simplification, thanks to the so-called generalized Gribov-Lipatov reciprocity [68][69][70][71], which reflects the symmetry of the underlying processes under the crossing of scattering channels. As a consequence, the harmonic sums can enter the anomalous dimensions only in the form of special combinations satisfying the above-mentioned property by themselves. In practice, this allows us to impose restrictions on the choice of basis functions leaving us with a smaller numbers of so-called binomial harmonic sums, defined as [66] S a1,...,an (N ) =  There are 2 ℓ−1 binomial harmonic sums at transcendentality ℓ.
As for the coefficients in front of the basis functions, inspection of the expressions of the j-dependent anomalous dimensions that are already known reveals that they are usually small integer numbers. So, in general, we obtain a system of Diophantine equations. If the number of equations is equal to the number of variables, then we can solve such a system exactly. However, this requires the knowledge of the anomalous dimensions for a large number of fixed j values. Fortunately, the system of Diophantine equations can be solved with the help of special methods from number theory even if the number of equations is less than the number of variables. In fact, we may then apply the Lenstra-Lenstra-Lovasz algorithm [72], which allows us to reduce the matrix obtained from the system of Diophantine equations to a form in which the rows are the solutions of the system with the minimal Euclidean norm.
Nevertheless, we may exploit the information encoded in Eqs. (6)- (13) to numerically recover the non-planar contribution to the cusp anomalous dimension with useful precision. To this end, we proceed along the lines of Refs. [60,61] and approximately reconstruct the fourloop splitting function. We recall that the n-loop splitting function P (n) (x) is related to the anomalous dimension of the respective twist-two spin-j operator, with j = 2, 4, 6, . . ., by a Mellin transformation, where the negative sign is a standard convention. In QCD, the diagonal splitting functions at n loops in general assume the following form in the limit x → 1 [68]: where k = q, g. A (n) q and A (n) g are the n-loop quark and gluon cusp anomalous dimensions, respectively [24]. In N = 4 SYM theory, the splitting functions, being related to the anomalous dimensions through the Mellin transformation in Eq. (19), satisfy the maximal-transcedentality principle [36], and we may use Eq. (20), with kk replaced by np. Since C (n) k and D (n) k can be predicted from lowerorder information [68] and non-planarity appears for the first time at n = 4, we have C To summarize, using modern computational techniques, we have considerably advanced our knowledge of the non-planar sector of N = 4 SYM theory by studying the universal anomalous dimension of the local, gaugeinvariant, SU(4)-singlet, twist-two operators of definite Lorentz spin j. Specifically, we have pushed the state of the art from j = 8 [32][33][34] [73] to j = 18. The five new terms for j = 10, . . . , 18 are all in agreement with the generic coefficient of ζ 5 already derived in Ref. [32]. The new information allowed us to uniquely determine also the generic coefficient of ζ 3 , but it does not yet suffice to pin down the generic expression of the rational term. However, we managed to find a rather precise numerical result for the j → ∞ limit for the universal anomalous dimension through consideration of the x → 1 limit of the corresponding splitting function, following Ref. [60,61], which gives the numerical result for the cusp anomalous dimension. This result agrees with previous determinations based on very different approaches [25][26][27][28]31].