Three-loop helicity amplitudes for quark-gluon scattering in QCD

We compute the three-loop helicity amplitudes for $q\bar{q} \to gg$ and its crossed partonic channels, in massless QCD. Our analytical results provide a non-trivial check of the color quadrupole contribution to the infrared poles for external states in different color representations. At high energies, the $qg \to qg$ amplitude shows the predicted factorized form from Regge theory and confirms previous results for the gluon Regge trajectory extracted from $qq' \to qq'$ and $gg \to gg$ scattering.


I. INTRODUCTION
The computation of multiloop scattering amplitudes in Quantum Chromodynamics (QCD) plays a fundamental role for the Standard Model (SM) precision program carried out at particle colliders such as the Large Hadron Collider (LHC) at CERN. Suitably combined with real-radiation contributions, they provide a powerful tool to generate predictions for a variety of collider observables, allowing for precise comparisons with experimental data [1].
In fact, matching the shrinking experimental errors with correspondingly precise theory predictions allows one to discover even subtle signals from possible physics scenarios beyond the SM.
In addition to their phenomenological significance, analytic computations of scattering amplitudes enable investigations of general properties of perturbative Quantum Field Theories (QFT), including comparative studies of QCD amplitudes with their supersymmetric counterparts. The more loops, external legs, or particle masses one is considering for a scattering amplitude, the more challenging its computation becomes. In recent years, significant progress has been achieved for the reduction of loop integrals to master integrals and their analytical evaluation, resulting in the calculation of previously inaccessible multiloop amplitudes. At two loops, various QCD amplitudes became available for 2 → 3 scattering processes involving mostly massless particles , paving the way for the first Next-to-Next-to-Leading-Order (NNLO) studies at LHC [28][29][30]. At three loops, first QCD amplitudes were computed for 2 → 2 scattering processes [31][32][33][34]. At four loops, 2 → 1 form factors were obtained in full-color QCD [35][36][37].
Analytical results for multiloop scattering amplitudes can also provide non-trivial information about all-order results in QCD. An interesting case is the so-called Regge limit [38] of large collision energy, where universal factorization properties can be observed in QCD amplitudes. The BFKL formalism [39,40] allows one to describe all-order structures in QCD through the exchange of so-called "Reggeized gluons", which resum leading contributions of the quark and gluon interactions at high energies. With the recent determination of the three-loop Regge trajectory [33,41], the last missing ingredient for next-to-next-to-leadinglogarithmic analysis became available. This paper concludes our analytical calculation of all four-parton scattering amplitudes in three-loop QCD. Previously, we presented the helicity amplitudes for the process qq → q q and crossed channels [32] and for the process gg → gg [33]. In this work, we provide the helicity amplitudes for qq → gg scattering and crossed channels in full-color, massless QCD. Our calculation checks the predicted quadrupole contribution to the infrared poles for a process with external legs in different color representation [42,43]. By analyzing the high-energy limit of the qg → qg amplitude, we check the universality of the predicted factorization and the three-loop expression for the Regge trajectory [33,41].
The rest of this paper is organized as follows. In section II, we set up our notation and describe the color and Lorentz decomposition of the scattering amplitude. In section III we discuss our computation of the bare helicity amplitudes employing the tensor decomposition provided in the previous section and analytical solutions for the master integrals [34,44].
In section IV, we describe the UV renormalization and give details for the subtraction of IR poles up to three loops. In section V we present our final results and enumerate the checks we have performed to verify their correctness. Finally, in section VI we discuss the high energy (Regge) limit of the qg → qg amplitudes. We draw our conclusions in section VII. We reserve the appendices for lengthy formulas with explicit results for all the relevant anomalous dimensions (appendix A) and for the impact factors and the gluon Regge trajectory (appendix B).

II. COLOR AND LORENTZ DECOMPOSITION
We consider the quark-gluon scattering process in massless QCD, where the momenta satisfy The kinematics of the process eq. (1) can be parametrized in terms of the usual Mandelstam invariants with u = −t − s. We find it convenient to introduce the dimensionless variable to parametrize our results.
The primary physical scattering process considered in this paper is which can be obtained from the process (1) by a crossing of external legs with p 3,4 → −p 3,4 .
For this process, the physical region of the phase space is given by Results for other physical scattering processes will subsequently be derived from the result for process (5) by considering further crossings. The bare amplitude for process (5) can be decomposed in three different color structures C i , Here, i 1 and i 2 are the fundamental color indices of the external quarks with momenta p 1 and p 2 , and a 3 and a 4 are the adjoint color indices of the external gluons with momenta p 3 and p 4 , respectively. Further, α s,b is the bare strong coupling. In eq. (7) we also introduced the notation [i] to indicate a color component index of the amplitude. The three color structures are where we work in QCD with color group SU (N c ) and n f massless quark flavors. The matrices (T a ) i 2 i 1 are the generators of SU (N c ) in the fundamental representation. We use Tr[T a T b ] = 1 2 δ ab and denote the quadratic Casimir operators in the fundamental and adjoint representation by C F and C A , respectively.
The amplitude coefficients A [i] can be decomposed further into Lorentz-covariant struc- where the F j are scalar form factors. To regulate ultraviolet and infrared divergences, we employ dimensional regularization and use d = 4 − 2 for the number of space-time dimensions. We denote the external gluon polarization vectors as (p i ) = i with the transversality condition for the external gluon momenta (p i ) · p i = 0 (i = 3, 4). To simplify the Lorentz decomposition, we also fix the gauge of the external gluons such that 3 · p 2 = 4 · p 1 = 0, which leads to the following gluon polarization sums pol µ 3 Since we are ultimately interested in computing the helicity amplitudes for this process in the 't Hooft-Veltman scheme (tHV) scheme, we use the Lorentz structures [31,45,46] and introduce projection operators P i which extract the form factors from the amplitude, In eq. (12), we introduced the short-hand notation P i · A which implies a sum over the polarizations of the external particles. By introducing the matrix the projectors can be compactly defined as where We stress that in conventional dimensional regularization there is a fifth Lorentz structure which would need to be taken into account in eq. (9). In the tHV scheme we take internal momenta in d = 4 − 2 dimensions and keep external momenta and polarizations in four dimensions. As explained in refs. [45,46], this allows us to essentially ignore this fifth evanescent structure completely and work with just the four structures (11), which are linearly independent in four space-time dimensions. We also point out that the decompositions of eqs. (7) and (9), as well as the explicit form of the projectors (13), hold to any orders in perturbation theory.

III. HELICITY AMPLITUDES
From the form factors F j one can construct amplitudes for definite helicities of the external particles. We denote the helicity of the incoming quark as λ q ; the helicity of the incoming anti-quark λq is then automatically fixed due to helicity conservation along the massless quark line. We refer to the quark line helicity with the symbol λ qq = {λ q λq} which can take which allow us to perform a consistency check on our calculation. Results for right-handed quarks can subsequently be obtained by a parity transformation. We write for the lefthanded spinors u L (p 2 ) = 2|, u L (p 1 ) = |1], and for the polarization vector of the gluons Inserting these equations into the Lorentz structures T j (11) gives the helicity amplitudes where the little group scaling is captured by the overall spinor factors , s L−+ = 2 24 [13] 23 [24] , s L+− = 2 23 [41] 24 [32] , and we have defined the scalar helicity amplitudes The amplitudes for right-handed quarks are related to those for left-handed quarks by By exchanging the two outgoing gluons, we find that Bose symmetry implies the relations We also note that These identities will serve as an important check of our calculations.
We expand the helicity amplitudes inᾱ s,b ≡ α s,b /(4π), We employ Qgraf [47] to produce Feynman diagrams and find 3 diagrams at tree level, 30 diagrams at one loop, 595 diagrams at two loops and 14971 at three loops. We give a few representative samples of the three-loop diagrams contributing to the process in figure 1.
We use Form [48] to apply the Lorentz projectors of eq. (11) to the diagrams and to perform the Dirac and color algebra. In this way, we obtain the form factors as linear combinations of a large number (∼ 10 7 ) of scalar Feynman integrals with rational coefficients.
We parametrize the corresponding -loop Feynman integrals according to where γ E ≈ 0.5772 is Euler's constant, µ 0 is the scale of dimensional regularization, and the denominators D j are inverse propagators for the respective integral family "top". More details on the integral families can be found in ref. [32]. Using Reduze 2 [49,50] and Finred, an in-house implementation of the Laporta algorithm [51] based on finite field arithmetic [52][53][54][55] and syzygy algorithms [56][57][58][59][60][61], we reduced these integrals to a linear combination of 486 master integrals. Upon insertion of the recently computed solutions for the master integrals [34,44] we arrive at an analytical result for the helicity amplitudes in terms of harmonic polylogarithms.

IV. UV AND IR SUBTRACTIONS
The bare helicity amplitudes (24) contain UV and IR divergences, which appear as poles in the Laurent expansion in . The MS renormalized strong α s (µ) is defined through whereᾱ s = α s (µ)/(4π), µ is the renormalization scale and The β-function coefficients are defined in the standard way through We also recall the values of the standard quadratic Casimir constants for a SU (N c ) gauge group: With this, up to third order of the perturbative expansion, we have In the following, we use boldface symbols to denote vectors in colour space, that is, we define H = H [1] , H [2] , for the decomposition of the amplitude with respect to the basis C i . Using the expansion of (24), we collect theᾱ s coefficients of the UV finite, but IR divergent, amplitudes as so that the renormalized helicity amplitudes can be written as The IR singularity structure of QCD amplitudes has been studied at two loops in ref. [62] and was extended up to three loops in refs. [42,[63][64][65][66][67][68][69][70]. The IR divergences can be subtracted from our renormalized amplitudes multiplicatively: Here Z is a color matrix acting on the space spanned by the C i basis vectors (8) and H λ, fin are finite remainders, also called hard scattering functions. The matrix Z can be written as where P denotes the path-ordering of color operators [67] in increasing values of µ from left to right. It can be omitted up to three loops, since to this order [Γ(µ), Γ(µ )] = 0. The color-space correlation structure at three-loops allows one to decompose the soft anomalous dimension operator Γ into so-called dipole (Γ dipole ) and quadrupole (∆ 4 ) contributions according to The dipole term Γ dipole can be written as where γ K (ᾱ s ) is the cusp anomalous dimension [71][72][73][74][75][76] and γ i the quark (gluon) collinear anomalous dimension [77][78][79][80] of the i-th external particle, which are given in our notation in appendix A. Further, T a i represents the color generator of the i-th parton in the scattering amplitude, for a final(initial)-state quark (anti-quark), for a final(initial)-state anti-quark (quark), The quadrupole term ∆ 4 contributes for the first time at three loops. It can be written in the kinematical region (6) as [32,33,42,43] ∆ where C = ζ 5 + 2ζ 2 ζ 3 and D 1 (x), D 2 (x) are linear combinations of harmonic polylogarithms as [32,33,42,44]. They read Here the argument x has been suppressed, and for the HPLs we used a compact notation similar to [81,82]:  In terms of the color vector space introduced in (8) and of the quantities we have just defined we find the explicit form where N ± c = (N 2 c ±1)/2 and C = ζ 5 +2ζ 2 ζ 3 . Unlike Γ dipole , ∆ with where the Z n are the coefficients of the expansion of Z inᾱ s and explicitly read [32,67]: Above we have used with the last equal sign giving the definition of the perturbative coefficients Γ .
The explicit expression for the perturbative expansions of the cusp anomalous dimension and of the quark (gluon) collinear anomalous dimensions are given in the appendix.

V. CHECKS AND EXACT RESULTS
First, we have checked that our results for the lower loop amplitudes are consistent with the literature. In particular, we have compared our tree-level, one-loop and two-loop results for the bare helicity amplitudes for qq → gg in the helicity configurations (15) against the results provided in the ancillary files of ref. [83] and find analytical agreement through to weight six. We have also checked that our one-loop expressions for qq → gg and qg → qg match results obtained with the automated one-loop generator OpenLoops [84,85]. At the three-loop level, we have verified that the IR singularities of our results for the renormalized helicity amplitudes in eq. (32) match the pattern predicted by eqs. (33)- (44), which provides a highly non-trivial check. From the high energy limit of our amplitudes we extract the quark and gluon impact factors and find that they are consistent with previous results, which tests lower loop contributions to the renormalized amplitude up to weight six. Moreover, we extract the gluon Regge trajectory and find agreement with previous results, which provides a stringent check of the finite contributions to the three-loop amplitudes presented in this paper. The high energy limit will be described in more detail in the next section.
Our analytic results for the three-loop finite remainders H λ, fin are expressed in terms of harmonic polylogarithms with transcendental weight up to six. Alternatively, these can be converted to a functional basis of logarithms, classical polylogarithms and a few multiple polylogarithms with at most three-fold nested sums [31]. We provide a general conversion table for harmonic polylogarithms up to weight six in the ancillary files of the arXiv submission of this article.
From our results for the process qq → gg we also derive explicit expressions for the helicity amplitudes for qg → qg scattering, which requires a non-trivial analytical continuation.
Details for this procedure are given in ref. [32]. The remaining partonic channels gg → qq and gq → gq are not provided explicitly, since they can be obtained by a simple crossing of external legs without any non-trivial analytic continuation. While our results are relatively compact, of the order of 1 megabyte per partonic channel, they are too lengthy to be presented here. We include them in computer-readable format in the ancillary files on arXiv.
In figure 3 we show the finite remainder of the amplitude at different loop orders interfered with the tree-level amplitude for the processes qq → gg and qg → qg. The interferences are averaged (summed) over polarizations and color in the initial (final) state. Additionally, since with the results of this paper all 2 → 2 partonic channels are now available in threeloop massless QCD, we find it useful to compare virtual corrections for the processes qq → gg, qg → qg, gg → gg and qq →QQ. In figure 4, we show the contributions to the squared amplitude at different orders inᾱ s , normalized by the respective tree-level squared amplitude. Again, we average (sum) over polarization and color in the initial (final) states.
Below we define more in detail the quantities we present in the plots.
We rewrite the finite amplitude as a vector in color and helicity space and define the contraction between different elements in this vector space as  where the factor 4πα s in eq. (47) replicates the overall normalization of eq. (7). N is the initial-state color and polarization averaging factor, which depends on the process and takes the following values: for qg → qg, The initial and final state polarization sum runs over all helicity configurations. The color factors C i and the spinor factors s λ are different for the various processes: for qq → gg they are given in eqs. (8) and (19), while for qg → qg they are obtained by applying the transformation p 2 ↔ p 3 to those of qq → gg. For the other two channels gg → gg and qq →QQ, they can be found in refs. [33] and [32] respectively.  We expand the squared amplitude normalized by the tree-level contribution inᾱ s according to with Finally, for the numerical evaluation, we have set µ 2 = s = m 2 Z , α s (µ) = 0.118, n f = 5 and N c = 3.

VI. HIGH ENERGY LIMIT
In the high-energy or Regge limit, quantum field theoretic scattering amplitudes become particularly simple and are known to exhibit universal factorization properties. In the following, we consider the process for which t-channel gluon exchanges provide the dominant contribution to the amplitude at high energies. The Regge limit is defined as s → ∞ for fixed scattering angle, that is, (52). For the variable x = −t/s, the Regge limit corresponds to x → 0.
Following the investigation [86,87], we split the renormalized amplitude into the definite The definite-signature amplitudes H qg→qg,+ and H qg→qg,− are referred to as the even and odd amplitudes. We expand them up to third order inᾱ s , where we use for the signature-symmetric logarithm and the color operators [88,89] are Here the T i (i=1,. . . ,4) are assigned according to eq. (37). Explicitly, we find Following ref. [86], one can show that the coefficients H  [41,87,90,91]. Up to next-to-leading logarithmic (NLL) accuracy, the odd signature amplitude is completely determined by the gluon Regge trajectory and by the so-called quark and gluon impact factors, that describe the interaction of the reggeized gluon with external states. The factorization structure for the odd amplitude becomes more complex in the next-to-next-to-leading logarithmic (NNLL) approximation, as both Regge pole and Regge cut [86,88,92,93] contribute at this order. For the even amplitude, only the Regge cut contributes at the NLL level [86] and breaks the simple exponential structure already at this logarithmic order. Starting from NNLL, the odd-signature amplitude receives contributions from both Regge pole and Regge cuts. In ref. [91], a scheme has been proposed to disentangle the two. As in our previous paper [33], we adopt this scheme to study the high-energy behaviour of qg → qg to three loops up to NNLL.
Following the framework outlined in [91], we assume that, by setting the renormalization scale to µ 2 = −t, eq. (54) can be written as where τ g = =1ᾱ s τ is the gluon Regge trajectory and the factors Z q = =0ᾱ s Z ( ) q and Z g = =0ᾱ s Z ( ) g capture the collinear poles of the amplitude [86] for quarks and gluons, respectively. Up to O(ᾱ s ) we have The odd signature color operators O −,( ) k contributing at NNLL [86] are and the even signature ones contributing at NLL [86] are The coefficients B ±,( ) describe the process independent Regge cut contributions [86,91,94] and we report them below for convenience. The odd-signature ones are while for even signature one finds I q and I g are the perturbative expansion coefficients of the quark and gluon impact factors; they can be extracted from the one-and two-loop calculation [83]. The explicit expressions are rather long and are reported to the required orders in in appendix B.
With the perturbative expansion of τ g up to the three-loop order obtained in [33] (and provided in appendix B), we have all the ingredients to fully predict the Regge limit of the process qg → qg through eq. (58), which only requires the tree-level amplitude H qg→qg as an input.
We find by explicit calculation that the high energy limit of our results for the qg → qg three-loop amplitude indeed agrees with this prediction and confirms in particular the literature results [41,86,[95][96][97] for the gluon Regge trajectory as well as quark and gluon impact factors in QCD. This provides a highly non-trivial test of the universality of high energy factorization in QCD.

VII. CONCLUSIONS
In this paper, we have presented the three-loop helicity amplitudes for quark-gluon scattering processes in full-color, massless QCD. To perform this calculation, we have made use of various cutting-edge techniques, in particular to handle the Lorentz decomposition of the scattering amplitude and to solve the highly non-trivial system of integration-by-parts identities required to reduce the amplitude to master integrals.
In addition to our previous calculations for the scattering of four quarks and of four gluons, these latest analytical results confirm predictions for the infrared poles of four-point amplitudes in QCD, also for processes with external states in different color representations.
Moreover, our results have made it possible to verify the factorization properties of partonic amplitudes in the Regge limit. With this work, all three-loop amplitudes for parton-parton scattering processes are publicly available, providing the virtual corrections to dijet production at N 3 LO.

Appendix A: Anomalous dimensions
In this appendix, we list the perturbative expansions of the cusp anomalous dimension and of the quark and gluon collinear anomalous dimensions, The required expansion coefficients of the cusp anomalous dimension read [71][72][73] γ K 0 = 4 , The required expansion coefficients of the quark collinear anomalous dimension are [78] γ q 0 = −3C F , Note that since one can expand τ 1 = K 1 + O( ), the poles of τ g are given exactly by K defined in eq. (B5) (see also ref. [91]).
The expressions above are also provided in electronic format in the arXiv submission of this article.