On DIS Wilson coefficients in N = 4 super Yang-Mills theory

In this note we evaluate Wilson coefficients for"deep inelastic scattering"(DIS) in N=4 SYM theory at NLO in perturbation theory, using as a probe an R-symmetry conserved current. They exhibit uniform transcendentality and coincide with the piece of highest transcendentality in the corresponding QCD Wilson coefficients. We extract from the QCD result a NNLO prediction for the N=4 SYM Wilson coefficient, and comment on the features of its Regge limit asymptotics.

form factors [21,22,23] and correlation functions [24,25] 6 . Significantly, the as yet heuristic nature of the observations and the presence of some exception to the rule of "direct extraction" from QCD results 7 make the detection of such generalized MTP an always necessary, preliminary check to set its predictive power in the context at hand.
We have here evaluated the simplest non-trivial (NLO) contribution to "deep inelastic scattering" Wilson coefficients for N = 4 SYM, using as external hard probe the current conserved under the internal SU (4) R-symmetry of the theory. Although in a conformal theory boundstate "hadrons" do not form, one can imagine to redefine the unphysical asymptotic states 8 via quark/gluon distributions, governed by the N = 4 SYM DGLAP equations [1], in which all emerging collinear divergencies are factorized out 9 . Looking at the final result (2.7)-(2.9), of immediate evidence are a) its extreme simplicity (all rational terms in z cancel and only logarithms of z and 1 − z survive), b) its uniform transcendentality-degree two, evident in its Mellin transform (2.11)- (2.13), and c) the fact that it coincides with the highest transcendental part of the QCD result (2.4)-(2.5), provided the appropriate set of splitting functions reflecting the field content of the theories is taken into account and with the color factor prescription mentioned above. These are not a priori obvious observations. At least in the case of anomalous dimensions for leading(two)twist operators -which govern the IR-divergent O(1/ǫ) 10 part of the inclusive DIS cross-section -the maximal transcendentality property seems to be deeply connected to the maximal supersymmetry of the gauge theory (MTP is broken already in the N = 2 case 11 ). While in this sense the 1/2 BPS [32] current (2.6) is the best candidate to gain properties a) and b) for the O(1/ǫ) contribution 12 it is non-trivial that they extend to the IR-finite part. It is also non-obvious that the maximal transcendentality observed for N = 4 SYM form factors [22] would be maintained in the cross-sections here analyzed. Interestingly, using the same basis (in Mellin space) which diagonalizes the twist-two anomalous dimensions of N = 4 SYM [1], the result simplifies further and can be written -in total analogy with the anomalous dimension case -in terms of a universal function C uni (j) expressed as combination of harmonic sums and appearing with shifted arguments in the first two entries of the Wilson coefficient vector (3.4). Even more interesting are the vanishing of the third vector component, and a shift in the first entry which cancels the singular behavior in the Regge limit. Elaborating on these features and extrapolating them to the next order in perturbation theory we propose -extracting it from the QCD result [33] -the DIS Wilson coefficient for N = 4 SYM at NNLO, verifying its self-consistency with the analysis of its Regge (j → 1) and quasi-elastic (j → ∞) asymptotics .
Since it usually requires two non-trivial orders to reliably identify the general structure of a perturbative computation, it would be important to extend the calculations to NNLO and confirm the predictions (3.10) and (3.22), and with them the properties described above. Furthermore, one could proceed generalizing this computation to crossing-related processes such as the supersymmetric generalization of the e + e − annihilation and Drell-Yan lepton-pair production [34]. 6 See also the field-theory related structure observed for superstring amplitudes [26]. 7 In the case of scattering amplitudes, see the mismatch [27], mentioned in [18], between the terms with highest transcendentality in the leading-color two-loop QCD amplitude of [28] and the corresponding N = 4 SYM amplitude. 8 See also [29]. 9 In an operative sense, this logic is not different from what one experiences in QCD, where the true, physical, asymptotic states are not reachable via perturbation theory and are actually defined via partonic distribution functions. 10 We use the FDH dimensional reduction scheme of [30,31]. 11 Private communication by L. Lipatov. 12 It is not difficult to check that using the R-symmetry singlet, non conserved, currentψ γ µ ψ would add to the formulas obtained, together with a UV divergent part due to bubble diagrams, also terms of transcendentality one and zero. This note proceeds with the presentation of the main result in N = 4 SYM, its comparison to QCD (Section 2) and the extraction from QCD of an NNLO prediction (Section 3). Appendices A, B and C recall, respectively, the LO expressions for splitting functions appearing in our analysis, a convenient basis for expressing our result and our prediction and basic definitions of harmonic sums.

DIS Wilson coefficients in N = 4 SYM at NLO
In the QCD deep inelastic scattering analysis, the Wilson coefficients C q , C g are the short-distance functions which appear in the IR-safe part of the structure functions, where they multiply respectively the (renormalized) quark and gluon distributions. At order O(α s ) and in MS factorization scheme, the structure function F 2 reads Here µ is the factorization scale [35], the LO non-singlet splitting functions P (0) qq , P qg are reported in Appendix A and C q , C g are given by [36,37] Above, p qq , p qg are the polinomials (A.4)-(A.5) usually introduced in literature and representing the highest transcendental part of the splitting functions. As a result of factorization, both splitting functions and Wilson coefficients C q , C g can be evaluated perturbatively squaring and integrating over the phase space the relevant form factors. At NLO, the processes contributing to C q are of real gluon emission γ * + q → q + g and virtual gluon correction γ * + q → q, while the initial gluon process g + γ * → q +q contributes to C g (see for example [38]).
To mimic the electromagnetic interaction in the N = 4 SYM theory case [39,40] we will substitute the electromagnetic current with the SU (4) R conserved current where a summation over the SU (N c ) and SU (4) R indices is understood. Notice that, although with the choice above we are intentionally not selecting a U (1) subgroup of the internal SU (4) R symmetry so that all the fermions and scalars of the theory are "charged" on equal foot 13 , the absence of truly non-abelian structures in the actual computation makes the effect of the sum over all the 15 generators T I in the vertex with the "SU (4) R photons" practically equivalent to the consideration of 15 U (1) photons.
Together with the presence of many more processes due to scalar contributions both in virtual and real diagrams, the main difference with respect to the QCD setting stays in the necessity of assigning appropriate R-symmetry (flavor) factors to the squared diagrams and interference terms. This is due to the presence, in the Yukawa-type vertices of N = 4 SYM lagrangian, of sigmamatrices connecting the vectorial and spinorial representations of SO(6) R ∼ SU (4) R , under which the N = 4 SYM scalar and fermionic fields, respectively, rotate. In performing the calculation, we have used Feynman rules derived from the N = 4 SYM lagrangian in the notation of [41] and the FDH regularization scheme of [30,31].
The "DIS" Wilson coefficients of N = 4 SYM are then given at NLO (orderâ = αNc 4π ) by the following formulas, extremely simple and symmetric in scalar and fermionic contributions, whereP is defined as the following linear combination of splitting functionŝ Above, the LO splitting functions are those reported in Appendix A, formulas (A.9)-(A.11), C (v) and C (f ) are the quadratic Casimir invariants respectively in the vectorial representation of SO (6) and fundamental of SU (4) (the latter is equivalent to the spinorial representation of SO (6)), C A is the Casimir eigenvalue for the adjoint representation of the gauge group SU (N c ). Already in this form, it is not difficult to see that the first two functions C λ and C g above, with the appropriate truncation of the scalar sector, coincide with the highest transcendental part of their QCD counterpart (2.4)-(2.5) 14 . In Mellin space, formulas (2.7)-(2.9) read Assigning to 1 j n transcendentality degree n, the expressions above exhibit uniform transcendentality of degree 2, which is the maximum degree expected at this order in perturbation theory 15 . In that formulas above are not only given in terms of harmonic sums of fixed degree 2, their uniform, leading transcendentality feature is sometimes referred to as "weak", as opposed to the one in "strong" sense which is the argument of next section.

Change of basis and prediction at NNLO
Experience in the calculation of anomalous dimensions [1,2] of leading twist operators in N = 4 SYM suggests that the n-loop anomalous dimension matrix γ ab (j) (a, b = g, λ, φ), after diagonalization, assumes an intriguing form in terms of a single universal function γ uni (j) expressed only through a combination of harmonic sums of constant degree 2n − 1. Explicit expressions for the matrices entering the diagonalization can be found in Appendix B. It is quite natural to ask how Wilson coefficients would appear in this new basis.
Let us first notice that the particular combination of splitting functions (2.10) appearing in formulas (2.7)-(2.9) would also appear as coefficient of log Q 2 µ 2 in the N = 4 analogous of (2.2)-(2.3). In Mellin space it is therefore natural to define the vector with 16γ a (j) = γ Turning to the Wilson coefficients it is natural to define where the three components are given in (2.11)-(2.13) 17 . Rotating again with V −1 given in (B.3) one getsĈ where we defined C uni (j) = 4(S 1,1 (j) − S 2 (j)). (3.5) Comparing this result with the QCD one-loop coefficient functions in Mellin space [33], one realizes that the maximally transcendental part of that result is given precisely by the combination (3.5) of harmonic sums of degree 2. It is therefore for the Wilson coefficients rotated vector that the proper maximum transcendentality principle works, in total analogy with [1]. For the case presented here, however, a couple of new interesting features appear. The main difference with respect to the structure of the diagonal expression (B.2) is the vanishing of the third component for the vectors (3.2) and (3.4). It is quite interesting that the same feature is present in both cases and it would be interesting to check it at NNLO. The other novel feature is the shift of 4 (j−1) 2 in the first component (3.4). It is worthwhile to notice that this additional term is responsible for the cancellation of the singularity appearing in C uni (j − 2) in the Regge limit j → 1.
To better understand the pole structure in the Regge limit, it is interesting to consider the QCD case [45,33,46], where the gluon Wilson coefficientF QCD g (1, j) has the following form 16 The coefficient 4 3 comes from the ratio obtained after collecting C (f ) in the Mellin transform of (2.10). 17 From now on an overall factor C (f ) CA will be neglected.
which, for j → 1, reads with n f the number of active quarks in QCD. Starting from NNLO (n = 2) above, the most singular terms in C (n),QCD g (j) contain (j−1) 1−n contributions and are proportional to the factor n f . Direct inspection of the QCD results shows that, both for anomalous dimensions and coefficient functions, the contributions with a factor n f do not contain terms with maximal transcendentality, 2n, expected at a given loop-order n. So, according to the MTP, the terms ∼ n f , including the singular ones in (3.7), should not contribute to the N = 4 SYM case. Accepting, by analogy with the case of anomalous dimension matrix, that at all orders in perturbation theory there is one universal Wilson coefficient function obeying the MTP, it is easy to see that the only possible poles in its '+' component 18  Another argument supporting the finiteness of the coefficient function in the Regge limit comes from the analysis of BFKL equation for N = 4 SYM. The correspondence between the singularities for j → 1 obtained by the BFKL approach and those appearing in the "+" component of the anomalous dimension (i.e. γ uni (j − 2)) has been checked up to NLO and it is believed to hold at all orders in perturbation theory [1]. In other words, the "+" component of the diagonalized anomalous dimensions matrix in QCD and in N = 4 SYM can be reconstructed from the BFKL approach in the limit j → 1 (see [1,47,3]) 19 . Hence, assuming that the BFKL equation collects all singular terms, for j to1, of the N = 4 SYM structure functionF uni (Q 2 /µ 2 , j), all the singularities should appear only in the anomalous dimension γ uni (j − 2) and not in the Wilson coefficients.
All these arguments suggest that the "+" component of the Wilson coefficient rotated vector should be finite in the Regge limit. It would be interesting to check this conjecture at NNLO by explicit computation.
Following these considerations we can guess the form of C uni (j) at NNLO from the corresponding QCD result [33]. The structure of the result in Mellin space, in analogy with (2.1), should beF where 20 The definition of the nested harmonic sums S ±a,±b,±c,··· (j) is given in Appendix C. The extraction of C uni from the QCD result [46] is rather complicated, and here we will report only the expression for j → ∞. In this limit and up to three loops, one finds 21 Let us comment on the agreement of expressions (3.11)-(3.13) with the expected results coming from exponentiation of the large j logarithms 22 . From general arguments [51,52,53] it is known that in DIS, for a general massless gauge theory, large logarithms for j → ∞ are controlled by the cusp anomalous dimension Γ cusp (â) and an additional anomalous dimension Γ DIS (â). In the case of N = 4 SYM, with vanishing β-function, including finite contributions for j → ∞ along the lines of [54] one gets where g(â) collects all the finite contributions of C uni (j) for j → ∞. For the simple case of N = 4 SYM the function G â, j j 0 can be written as The anomalous dimensions and the function g(â) clearly admit a perturbative expansion in powers ofâ. From our result (3.11)-(3.13) we can read out the first terms of those expansions, getting It is noteworthy that the function Γ cusp (â) coincides up to three loops with the well-known expansion of the cusp anomalous dimension of N = 4 SYM 23 [9,11]. This provides a good compatibility check for our result.
The assumption of finiteness as j → 1 for the "+" component of the vectorĈ leads to the following conjecture for its explicit form at NNLO

A Splitting functions in QCD and in N = 4 SYM
The leading order contributions to the DGLAP splitting functions in QCD read 23 Our convention for the cusp anomalous dimension follows the one used in [9]. The function γ k (a) defined in [11] is given by γ k (a) = 2Γcusp(â) with a = 2â. 24 This is equivalent to study the limit j → 1 of C uni (j − 2), which is the function appearing as the first component of (3.4). 25 As j → −1, nested harmonic sums S ±a,±b,... (j) have a less singular behaviour than standard sums S±a(j) [2].

C Harmonic sums
The basic definition of standard and nested harmonic sums with general indices are  [55,56]), but can be easily analytically continued to real and complex j via the methods of Refs. [1,57,56,58].