Iterating QCD scattering amplitudes in the high-energy limit

We analyse the high-energy limit of the gluon-gluon scattering amplitude in QCD, and display an intriguing relation between the finite parts of the one-loop gluon impact factor and the finite parts of the two-loop Regge trajectory.

Further, we know that the picture of high-energy factorisation based on one-reggeisedgluon exchange breaks down at NNLO accuracy [47]. The violation can be explained through infrared factorisation by showing that the real part of the amplitudes becomes nondiagonal in the t-channel-exchange basis [24,25]. Accordingly, it can be predicted how the violation propagates to higher loops, and the three-loop prediction for the violation [27,28], which has NNLL accuracy, has been confirmed by the explicit computation of the threeloop four-point function of N = 4 SYM [51]. In the high-energy factorisation picture, that violation is due to the contribution of the three-Reggeised-gluon exchange [29,52]. Thus, although a study of the BFKL ladder and firstly of its building blocks at NNLL accuracy is yet to be undertaken, we already have a clear picture of how the factorisation violations occur at NNLL accuracy.
While we have a precise knowledge of how the infrared poles occur and what they mean in the loop corrections to impact factors and Regge trajectory, the finite parts of those corrections are treated as free parameters, which neither infrared nor high-energy factorisation can constrain. At NLL accuracy, they are fixed either by direct computation [37, 38, 40, 43-46, 48, 49] or by extracting them from the knowledge of the scattering amplitudes in the exact kinematics [41,42,47,50]. The goal of this short note is to alert the reader to the possibility that there might be some organised structure in the finite parts of impact factors and Regge trajectory as well. We shall discuss amplitudes at NLL accuracy, so we shall not be concerned with factorisation violations here.
In section 2, we focus on the finite part of the one-loop gluon impact factor, having the finite part of the two-loop Regge trajectory as a target. In section 3, we discuss exponentiation patterns that might help in understanding the findings of section 2. In appendix A, we recall the coefficients of the cusp and collinear anomalous dimensions, used throughout the note. In appendix B, we provide for completeness also the one-loop quark impact factor.

Playing with finite terms
In order to illustrate the issue, we shall consider the tree-level amplitude for gluon-gluon scattering g a g b → g a g b , which may be written as [3], where a, a , b, b represent the colours of the scattering gluons, and G represents the adjoint representation of SU (N ), with (T c G ) ab = if acb , with the colour matrices normalised in the fundamental representation as tr(T c F T d F ) = δ cd /2. The coefficient functions C g(0) , which yield the leading-order impact factors, are given in Ref. [3] in terms of their spin structure and in Ref. [53] at fixed helicities of the external partons. Their specific form, though, is immaterial here, since we shall factor the tree amplitude away.
Note that to leading power of s/t, amplitudes are dominated by gluon exchange in the t channel, i.e. only t-channel gluon ladders contribute to scattering amplitudes in the high-energy limit. Note also that the colour component of the amplitude (2.1) is odd under s ↔ u exchange, and it defines the exchange of an antisymmetric octet in the t channel. However, also the kinematic component of Eq. (2.1) is odd under s ↔ u exchange, due to the overall factor of s/t and since in the high-energy limit u −s. Thus, the amplitude (2.1) is even under s ↔ u exchange, which is a general feature when only t-channel gluon ladders contribute.
In the loop corrections to a scattering amplitude in the high-energy limit, two or more gluons characterise the ladder exchange in the t-channel exchange. Colour-wise, the gluon ladder is then described by the 8 ⊗ 8 colour representation, which is decomposed where the term in curly brackets in the direct sum is even under s ↔ u exchange, while the term in square brackets is odd. However, to LL accuracy in ln(s/|t|), only the antisymmetric octet contributes. In fact, the virtual radiative corrections to eq. (2.1) are obtained, to all orders in α S , by replacing [3], is the one-loop Regge trajectory, which is related to a one-loop transverse-momentum integration. In dimensional regularization, it can be written as with C A = N c , the number of colours, and The reggeization of the gluon is the fact that higher order corrections to gluon exchange in the t channel can be accounted for by dressing the gluon propagator with the exponential of Eq. (2.2). Thus, at LL accuracy in ln(s/t), i.e. considering the corrections of O(α n S ln n (s/t)), the amplitude is real, and only the antisymmetric octet contributes to it. At NLL accuracy in ln(s/t), i.e. considering the corrections of O(α n S ln n−1 (s/t)), the amplitude develops an imaginary part, however one can show, either within the highenergy limit [54] or through infrared factorisation [25], that only the antisymmetric octet contributes to the real part of the amplitude. That is the reggeization of the gluon at NLL accuracy. However, because also virtual corrections to the impact factors contribute to the scattering amplitude at NLL accuracy, one needs a prescription to disentangle the virtual corrections to the impact factors in Eq. (2.1) from the ones that reggeize the gluon (2.2).
For the antisymmetric octet at NLL accuracy, that prescription is provided by [38] M aa bb (2.5) where we analytically continue ln(−s) = ln(s) − iπ, for s > 0, and where the impact factors are real and have the loop expansion, The Regge trajectory is loop-expanded as, (2.8) Then we write the amplitude (2.5) as a loop expansion proportional to the tree amplitude, The real part of the one-loop term of Eqs. (2.5) and (2.9) is The Regge trajectory, α (1) (ε) = 2C A /ε, is in fact independent of the type of parton undergoing the high-energy scattering process. It is also independent of the infrared regularisation scheme which is used. Conversely, the one-loop impact factors are process and infrared-scheme dependent. They were computed in conventional dimensional regularization (CDR)/'t-Hooft-Veltman (HV) schemes in Ref. [37][38][39][40][41][42], and in the dimensional reduction (DR)/ four dimensional helicity (FDH) schemes in Ref. [41,42]. To all orders of ε, the real part of the unrenormalised one-loop amplitude can be written as [42], , (2.11) with N f the number of light quark flavours, and where, like in Eq. (2.9), we have factored outg S 2 M (0)aa bb gg→gg . In Eq. (2.11), we have used the regularisation parameter, δ R = 1 in CDR/HV schemes, δ R = 0 in the DR/FDH schemes [55]. Note also that in the first line of Eq. (2.11) the transcendental functions come from one-loop box diagrams, while the rational functions in the second line come from one-loop bubble diagrams [42]. Using Eq. (2.10), we can write the one-loop gluon impact factor to all orders of ε as, . (2.12) Expanding Eq. (2.12) in ε, its ε poles are accounted for by the cusp anomalous dimension and by the gluon collinear anomalous dimension [28] 2 , so through O(ε) the one-loop gluon impact factor is K . Further, we can introduce the quantity, AD 3 , defined as, and we write Eq. (2.13) as, Since we are interested in exploring the relation of Eq. (2.13) to the two-loop Regge trajectory, out of the two-loop term in Eqs. (2.5) and (2.9), we only need the real part of the coefficient of the single logarithm, where α (2) (ε) is the two-loop Regge trajectory [43][44][45][46][47]. In CDR/HV, the unrenormalised two-loop trajectory reads, where in the second line we have used Eq. (2.14) and factored out the one-loop trajectory α (1) (ε) = 2C A /ε, and rewritten it as 2 α (1) (2ε) for later convenience. A few comments are in order, when we compare Eqs. (2.15) and (2.17): as we said, in infrared and in high-energy factorisations, the finite part of the one-loop gluon impact factor and the finite part of the two-loop Regge trajectory are treated as free parameters, i.e. quantities to be determined by an explicit computation. However, besides the fact that the O(ε 0 ) term of the one-loop gluon impact factor (2.15) is related to the two-loop cusp anomalous dimension, up to the ζ 2 term with a different coefficient 4 , we observe the even more surprising fact that the O(ε) term of one-loop gluon impact factor (2.15) exactly reproduces the O(ε 0 ) term of the two-loop trajectory (2.17). In fact, apart for the double pole and the different coefficient of the ζ 2 term, one obtains the two-loop trajectory (2.17) by changing sign to Eq. (2.15) and multiplying it by 2C A /ε, i.e. by the one-loop trajectory. Such an iterative structure looks surprising: a parton-species dependent one-loop quantity predicts, albeit not fully, a process independent two-loop quantity 5 .

Discussion
We have no explanation for the intriguing effect we have observed in Eq. (2.15), which hints at much more structure in the finite parts of scattering amplitudes in the high-energy limit than ever thought of so far. Although we ignore the physical mechanism which underpins it, it is worth making a few general remarks about a possible exponentiated form and iterative structure of the QCD scattering amplitudes in the high-energy limit, including their finite parts. However, since infrared factorisation treats the finite parts of impact factor and Regge trajectory as free parameters, thus offering no guidance to how they might exponentiate, the remarks below are purely heuristic.
We recall that, on other grounds, in Ref. [28] it was hinted that the impact factors, including their finite contributions, might exponentiate. Compounded with the already known exponentiation of the Regge trajectory, that amounts, to NLL accuracy, to an exponentiation of the (real part of the) QCD one-loop amplitude, M (1) gg→gg . However, at two loops, although higher orders terms through O(ε 2 ) would be present, the square of the one-loop amplitude would yield merely the 2C g(1) α (1) term of Eq. (2.16), which is subtracted when we compute the two-loop trajectory. Thus, one would need additional one-loop terms, beyond simply the one-loop amplitude.
We note that there is an exponentiation pattern for which the finite part of the Regge trajectory is not a free parameter. That occurs in planar N = 4 SYM, where the BDS ansatz [58,59] prescribes that the four-point scattering amplitude be written as, In a likely related context, SCET in the high-energy limit, the two-loop cusp anomalous dimension occurs as the finite term of a one-loop soft exchange [57]. where a is the 't-Hooft gauge coupling, and with f (l) (ε) an l-loop dependent second-order polynomial in ε, Accordingly, in the high-energy limit of the planar N = 4 SYM amplitudes [60][61][62][63], the l-loop Regge trajectory can be written as [62], For the QCD amplitudes in the high-energy limit, it is tempting to conjecture an exponentiation pattern like in Eq. (3.1), even more so given that in Eq. (3.1) the dependence of M (1) n on 2ε at two loops stems from the general infrared structure of the QCD two-loop amplitudes [64,65]. However, let us also note the differences of the QCD amplitudes in the high-energy limit with respect to planar N = 4 SYM and/or to the BDS ansatz. Firstly, the finite part of the Regge trajectory (2.17) agrees with the QCD gluon collinear anomalous dimension only at the level of the terms of highest transcendentality. Secondly, the fact that the O(ε) term of the impact factor (2.15) agrees fully with the O(ε 0 ) term of the Regge trajectory (2.17), although present also in planar N = 4 SYM, is unaccounted for by the BDS ansatz, which is accurate through O(ε 0 ).
Finally, if a similar pattern holds at three-loop level, it certainly does in a more intricate way, since the finite part of the three-loop Regge trajectory in planar N = 4 SYM [62] does not occur as such in the higher order terms in ε of the lower-loop impact factor and Regge trajectory.
That concludes our heuristic exploration of possible exponentiation patterns. As regards the underpinning physical mechanism, it is conceivable that since the high-energy limit is characterised by two large and disparate hard scales s −t Λ 2 , the infrared factorisation formulae need be revisited to account for it, with the possible outcome that also some finite parts of the amplitudes exponentiate, beyond what was envisaged in Ref. [28]. Exploring it is left to future analyses.

Acknowledgements
The author would like to thank Lance Dixon, Claude Duhr, Giulio Falcioni, Einan Gardi, Lorenzo Magnea, Bernhard Mistlberger for stimulating discussions and for a critical reading of the manuscript.

A Anomalous dimensions
The perturbative expansion of the cusp anomalous dimension [66,67], divided by the relevant quadratic Casimir factor C i , is The perturbative expansion of the collinear anomalous dimension is

B Quark impact factor
In CDR/HV, to all orders of ε the one-loop quark impact factor is [39] C q (1) .
(B.1) Expanding Eq. (2.12) in ε, its ε poles are accounted for by the cusp anomalous dimension and by the quark collinear anomalous dimension, so through O(ε) the one-loop quark impact factor is q the one-loop coefficient of the quark collinear anomalous dimension (A.4). Eq. (B.2) can be also written as, K + AD ε which shows that the same structure found in Eqs. (2.15) and (2.17) can be distinguished also in the one-loop quark impact factor, although polluted by additional contributions.