A first next-to-next-to-leading order study of three-jet production at the LHC

Multi-jet rates at hadron colliders provide a unique possibility for probing Quantum Chromodynamics (QCD), the theory of strong interactions. By comparing theory predictions with collider data, one can directly test perturbative QCD, extract fundamental parameters like the strong coupling $\alpha_s$ and search for physics beyond the Standard Model. In this work we calculate, for the first time, the next-to-next-to-leading (NNLO) QCD corrections to typical three-jet observables and to differential three-to-two jet ratios. We demonstrate that the inclusion of the NNLO corrections significantly reduces the dependence of those observables on the factorization and renormalization scales. Besides its phenomenological value, this proof-of-principle computation represents a milestone in perturbative QCD.


I. INTRODUCTION
The production of highly energetic sprays of particles, also known as jets, is a dominant process at hadron colliders. At high energies, where perturbation theory is expected to hold, this process offers the possibility for studying QCD in great detail. The theory-data comparison of differential multi-jet rates provides essential information about perturbative QCD and the modeling of jet production. The precision of these predictions is typically limited by their dependence on unphysical parameters -such as the renormalization and factorization scales -but it can be systematically increased by including higher-order corrections.
Three-jet production at the Large Hadron Collider (LHC) has been studied in great detail by experimental collaborations, see for example ref. [1][2][3][4][5][6]. Typical observables are jet transverse momenta, angular correlations and, more generally, event-shape observables. A particular type of observable suited for perturbative QCD is the ratio R 3/2 of three-to-two jet rates [7]. These ratios are directly sensitive to parton splittings and are, therefore, proportional to the strong coupling constant α s . This provides an opportunity for measuring the value of α s at the LHC. Cross section ratios have the additional advantage that some systematic uncertainties of experimental and theoretical nature cancel out. A prime example is the dependence on parton distribution functions (pdf).
There is extensive literature on theoretical predictions for multi-jet production through NLO in perturbative QCD [8][9][10][11][12][13], including NLO electroweak corrections [14][15][16]. NLO computations have also been matched to parton-showers [17,18] and are generally present in multi-purpose event generators [19][20][21]. Higher-order predictions for two-jet and single-inclusive jet production have seen extensive development in the past decade and are implemented through NNLO in QCD [22][23][24][25]. The feasibility of NNLO QCD predictions for higher jet multiplicity is limited by the availability of two-loop virtual amplitudes and by the efficient treatment of real radiation contributions. The three-jet two-loop amplitudes have recently been made public in the leading-color approximation [26,27], leaving the real radiation as the last obstacle to predictions accurate at second order in α S .
The aim of this article is twofold. Firstly, it presents NNLO QCD predictions for the production of three jets and R 3/2 ratios at the LHC at 13 TeV. Secondly, it demonstrates the technical ability to treat the NNLO real radiation contributions for processes with five colored partons at the Born level. The completion of the second order corrections to three jet production is a milestone in perturbative QCD computations since, judging by its infra-red structure, it is among the most complicated two-to-three processes at the LHC. This paper is organized as follows: in section II we discuss the technical details of our computation. Section III contains the phenomenological results and their analysis. We conclude with a summary and outlook on future applications in section IV.

II. CALCULATION DETAILS
The computation has been performed within the sector-improved residue subtraction scheme formalism [28,29] which has already been successfully applied to single inclusive jet production [24] and many other hadron collider processes, see refs. [30][31][32]. We work in five-flavour massless QCD without the top quark. Treelevel matrix elements have been taken from the AvH library [33,34] where s 12 = (p 1 +p 2 ) 2 the invariant mass of the incoming partons, in the following way where R (2)l.c. (s 12 ) denotes its leading-colour approximation. It is taken from the C++ implementation provided in ref. [27]. Eq. (2) above is the only approximation made in the present computation. We have checked that the overall contribution of R (2)l.c. (s 12 ) is about 10% and we expect the missing pure virtual contributions beyond the leading-colour approximations to be further suppressed.
We consider production of two and three jets at the LHC with a center of mass energy of 13 TeV with jet requirements adapted from experimental phase space definitions like, for example, ref. [6]. Jets are clustered using the anti-k T algorithm [36] with a radius of R = 0.4 and required to have transverse momentum p T (j) of at least 60 GeV and rapidity y(j) fulfilling |y(j)| < 4.4. All jets passing this requirement are sorted and labeled according to their p T from largest to smallest. Among those jets we require the two leading jets to fulfill p T (j 1 )+p T (j 2 ) > 250 GeV in order to avoid large higher-order corrections in two-jet production close to the phase space boundary. We denote by dσ n the differential cross section for at least n jets fulfilling the above criteria. Its expansion in α S reads We quantify the size of (N)NLO corrections with the help of the following ratios of differential cross sections The pdf set NNPDF31 nnlo as 0118 is used for all perturbative orders. The renormalization µ R and factorization µ F scales are set equal µ R = µ F = µ 0 . The central scale µ 0 is chosen asĤ T /n for n = 1, 2, wherê The sum in the above equation is over all final state partons, irrespective of the jet requirements. Previous studies of perturbative convergence in jet production support this event-based dynamic scale [37,38]. Unless stated otherwise, uncertainties from missing higher orders in perturbation theory are estimated by variation of µ F = µ R by a factor of 2 around the central scale µ 0 . The calculation of the three-jet production cross sections is computationally and technically challenging. The main bottlenecks are the double real radiation corrections and the corresponding integrated subtraction terms, due to large numerical cancellation between individual contributions. The numerical evaluation of the complex double-virtual amplitudes is fast due to the efficient representation presented in [27].

III. RESULTS
We begin by discussing typical jet observables at hadron colliders. In fig. 1 we show differential cross sections for three-jet production with respect to the transverse momentum p T (j i ) of the ith leading jet. In all histograms the outer bins do not include over-or under-flow events.
The NNLO K-factor of the p T (j 1 ) distribution is not flat: at small p T (j 1 ) one observes small positive NNLO corrections, while at large p T (j 1 ) the corrections tend to be about 10% and positive. The change in scale dependence for this observable when going from NLO to NNLO is also dependent on p T (j 1 ). One observes a small reduc-tion at large p T (j 1 ) (from about 7% at NLO to about 5% at NNLO) while at small p T (j 1 ), where the K-factor is smallest, the scale dependence decreases significantly (from about 4% at NLO to about 1% at NNLO). Interestingly, the scale dependence at NLO and NNLO behaves similarily: it steadily increases with p T (j 1 ). Throughout this work we define the scale dependence as one half of the width of the scale uncertainty band. This is relevant for cases where the scale variation is asymmetric, as for example is the case of p T (j 1 ) at NLO.
The p T (j 2 ) distribution has a different pattern of NNLO corrections: relative to NLO they are negative at low p T (j 2 ), and steadily increase towards larger p T (j 2 ) values. At both NLO and NNLO the scale dependence of p T (j 2 ) is similar to that of p T (j 1 ). On the technical side, the convergence of the numerical integration for the p T (j 2 ) spectrum has been significantly slower than for the other p T observables, which results in increased Monte Carlo uncertainty. To compensate for this, a larger bin size has been used for p T (j 2 ). Independently of its slower numerical convergence, the p T (j 2 ) spectrum shows good perturbative convergence. Such a behavior is in contrast to the two-jet case where the sub-leading p T spectrum is known to get large perturbative corrections due to the strict back-to-back tree-level kinematics [38].
The p T (j 3 ) distribution is well-behaved: it has a flat K-factor and fairly symmetric uncertainty band at both NLO and NNLO. The scale variation is almost independent of p T (j 3 ), about the 5% at NNLO, which is significantly smaller than in the NLO case.
The fact that this observable shows such good convergence and perturbative stability is somewhat remarkable. Naively, one may suspect that the scale used here may not perform very well for this distribution because the scale is based on the kinematics of the full event which, in turn, is dominated by the leading jet(s). Since this finding may be of relevance for the extraction of the strong coupling constant from three-jet events, it may be worth investigating this behavior in more detail. This is outside the scope of the present work.
Next we discuss the observable H T , defined as where the sum is over all jets that pass the jet requirements. We show this observable in fig. 2 for the two-jet process and in fig. 3 for the three-jet process. Both figures are subdivided in two panels showing the same observable but for a different central scale choice: the upper panels for µ 0 =Ĥ T and the lower panels for µ 0 =Ĥ T /2. production of three jets we find that the two central scale choices µ 0 =Ĥ T and µ 0 =Ĥ T /2 produce comparable results, albeit µ 0 =Ĥ T /2 leads to NNLO QCD corrections which are not captured by the NLO scale band. The scale dependence and K NNLO is similar compared to the twojet case. These findings indicate that a central scale of µ 0 =Ĥ leads to slightly better perturbative convergence and, thus, better approximates the actual energy scale relevant for this observable. We have checked that even lower scales, like µ 0 =Ĥ T /4, spoil perturbative convergence.
As a first application of the NNLO-accurate three-jet rates computed in this work we consider ratios between three-jet and two-jet rates. The ratios are defined as where X is some observable of interest. The (N)NLO ratio is defined in such a way that the numerator and denominator on the right hand side are evaluated at the matching order. The scale dependence of the differential cross sections is shown explicitly to emphasize that the scale choices in the numerator and denominator are correlated.
In the upper two panels of fig. 4 we show the ratio R 3/2 (p T (j 1 )). The ratio changes drastically when going from LO to NLO mostly due to the change in the two-jet cross section. The NNLO correction stabilizes the ratio and leads to a very small scale dependence. The K NNLO factor slightly decreases for large momenta, however, it is always fully contained within the NLO scale band. An important observation is that the NNLO scale band is very small in comparison to NLO, reducing it from about 10% down to 3%.
Next we consider the lower two panels in fig. 4, where we show the ratio R 3/2 (H T ) for a central scale µ 0 = H T /2. This observable behaves similarly to R 3/2 (p T (j 1 )) albeit with a slightly larger scale dependence. The reduction in the scale uncertainty when going from NLO to NNLO is of particular importance since this observable is used experimentally for measurements of α S [5]. The leading source of perturbative uncertainty in this data-theory comparison is the scale dependence. The pdf dependence, which is not computed in this work, is expected to be reduced in the ratio.
Jet rates are typically measured in slices of jet rapidity. To demonstrate how our calculation performs in this situation, we divide the phase space in slices of the rapidity difference between the two leading jets and define the ratio of the two-and three-jet rates as The NNLO prediction for this cross section ratio can be found in fig. 5 . The prediction is shown relative to the NLO one. The NNLO correction is negative across the full kinematic range and, overall, behaves very similarly to the one for the rapidity-inclusive ratio R 3/2 (H T ). This remains the case as y * increases, at least in the range of rapidities considered here.

IV. CONCLUSIONS
In this work we present for the first time NNLOaccurate predictions for three-jet rates at the LHC. We compute differential distributions for typical jet observables like H T and the transverse momentum of the ith leading jet, i = 1, 2, 3, as well as differential three-to-two jet ratios. Scale dependence is the main source of theoretical uncertainty for this process at NLO, and it gets significantly reduced after the inclusion of the NNLO QCD corrections. Notably, the three-to-two jet ratios stabilize once the second-order QCD corrections are accounted for. A central goal of the present work is to demonstrate the feasibility of three-jet hadron collider computations with NNLO precision. With this proof-of-principle goal attained, one can now turn one's attention to the broad landscape of phenomenological applications for threejet production at the LHC. Examples include studies of event-shapes [6,39,40], determination of the running of the strong coupling constant α s through TeV scales and resolving the question of scale setting in multi-jet production. Another major benefit from having NNLOaccurate predictions is the reliability of the theory uncertainty estimates.
On the technical side, the enormous computational cost of the present calculation (∼ 10 6 CPUh) makes it clear that further refinements in the handling of real radiation contributions to NNLO calculations are desirable.

V. NOTE ADDED: ERRATUM TO THE PUBLISHED VERSION
In the original publication, we evaluated the two-loop finite remainder function R (2)l.c. (s 12 ) defined in equation (2) with an incorrect colour factor. This oversight was due to a missing conversion factor between the conventions for the colour generator T a ij used by the authors of ref. [27] (see [41], section 2 before equation 2.3) and our convention (see ref. [29], appendix A). By convention, the generators in ref. [27] are normalised such that Tr T a T b = δ ab . In our convention we use Tr T a T b = 1 2 δ ab , which implies a factor of √ 2 per appearing colour generator T a ij . The following table lists the colour factors and the conversion coefficient for the square of a colour factor as it appears in the squared matrix element for each partonic channel: Channel Colour factor C (|C| 2 ) our /(|C| 2 ) ref. [1] These conversion factors should have been included in our original calculation, and we include them now in this erratum. These factors are sizable and have implications on the phenomenology. In this version of the document, we provide the corrected plots of the original publication. The NNLO prediction increases flatly by about ≈ 10%. This implies that the double virtual contribution is about ≈ 10% of the total NNLO cross-section in contrast to our previous findings of ≈ 2%. With this, the naive estimate for corrections from sub-leading colour terms would correspond to 1% corrections of the NNLO QCD prediction.