Three-Jet Production in Electron-Positron Collisions at Next-to-Next-to-Leading Order Accuracy

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1 Three-Jet roduction in Electron-ositron Collisions at Next-to-Next-to-Leading Order Accuracy Vittorio Del Duca,,* Claude Duhr, 2,3, Adam Kardos, 4 Gábor Somogyi, 4 and oltán Trócsányi 4 Institute for Theoretical hysics, ETH ürich, 8093 ürich, Switzerland 2 CERN Theory Division, 2 Geneva 23, Switzerland 3 Center for Cosmology, article hysics and henomenology (C3), Université Catholique de Louvain, 348 Louvain-La-Neuve, Belgium 4 University of Debrecen and MTA-DE article hysics Research Grou, H-400 Debrecen, O Box 05, Hungary (Received 7 Aril 206; ublished 7 October 206) We introduce a comletely local subtraction method for fully differential redictions at next-to-next-toleading order (NNLO) accuracy for jet cross sections and use it to comute event shaes in three-jet roduction in electron-ositron collisions. We validate our method on two event shaes, thrust and C arameter, which are already known in the literature at NNLO accuracy and comute for the first time oblateness and the energy-energy correlation at the same accuracy. DOI: 0.03/hysRevLett One of the most imortant fundamental arameters in the standard model is the strong couling α s. A clean environment for determining α s is the study of event shae distributions in e e collisions []. Indeed, while at leading order (LO) the roduction of two jets is a urely electroweak rocess, the dominant contribution to the roduction rate of every additional jet in the final state is directly roortional to the strong couling. Since the initial state does not involve colored artons, nonerturbative QCD corrections are restricted to hadronization and ower corrections affecting the final state configuration. These corrections can be determined either by extracting them from data by comarison to Monte Carlo redictions or by using analytic models. The recision of the theoretical redictions is thus mostly limited by the accuracy in the erturbative exansion in the strong couling. Currently, the state of the art includes next-to-leading order (NLO) redictions for the roduction of u to five jets [2 4] (and u to seven jets in the leading color aroximation [5]), and next-to-next-to-leading order (NNLO) redictions for the roduction of three jets [6,7]. Moreover, fixed-order redictions can be matched to resummation calculations (see examles in Ref. [8]), which take into account classes of logarithmically enhanced contributions to all orders in erturbation theory. The goal of this Letter is twofold. First, we resent a framework to comute fully differential redictions at NNLO accuracy for rocesses with a colorless initial state and involving any number of colored massless articles in the final state. Second, we aly our method to event shae ublished by the American hysical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the ublished article s title, journal citation, and DOI. observables with at least three hard final-state artons. As our framework allows for a fully differential descrition of the final state, it uts no restriction beyond infrared safety on the class of observables that we can consider. The cornerstone of our framework is the comletely local subtractions for fully differential redictions at next-tonext-to-leading order (CoLoRFulNNLO) method to regularize infrared divergences [9]. The method is based on the universal factorization of QCD matrix elements in soft and collinear limits. It takes into account all sin and colorcorrelations among the final-state articles and as a result the subtractions are comletely local. redictions at NNLO in erturbative QCD generically require the comutation of two-loo corrections to the Born rocess, as well as one-loo and tree-level contributions to the rocesses with one or two additional artons in the final state. The two-loo matrix elements for γ = q qg have been comuted in Ref. [0]. The one-loo corrections to four-jet roduction have been comuted in Ref. [], and the tree-level matrix elements for the roduction of five jets were obtained for first time in Ref. [2]. The sum of these contributions is finite for infrared-safe observables, but taken searately, they all exhibit exlicit divergences coming from loo integrations and/or imlicit divergences when one or more artons in the final state are unresolved. Thus each contribution needs to be searately rendered finite in four dimensions before any numerical comutation can be erformed. The CoLoRFulNNLO method deals with this issue by using universal counterterms to redistribute divergences between different final-state multilicities. In every singular region of hase sace we subtract the corresonding infrared divergence through a suitably constructed aroximate matrix element. As different singular regions overla, a careful bookkeeing is required in order to avoid double =6=7(5)=52004(5) ublished by the American hysical Society

2 counting when subtracting the divergences. Moreover, within our framework beyond NLO, we also need to consider iterated singular limits where artons become successively unresolved. Finally, sin correlations in gluon decay are retained, and the counterterms are fully local in hase sace. In the CoLoRFulNNLO framework the distribution of the NNLO correction to an observable J can be written as a sum of three contributions, each being searately finite in d ¼ 4 dimensions, σ NNLO ½JŠ ¼ where ¼fdσ RR J dσ RR;A 2 J m ¼ m ; m ðþ ½dσ RR;A J dσ RR;A 2 J m Šg d¼4 ; ð2þ dσ RV dσ RV;A m ¼ dσ VV m dσ RR;A dσ RR;A J ½dσ RR;A 2 dσrr;a 2 2 dσ RV;A dσ RR;A A J m Š A ; ð3þ d¼4 J m : d¼4 ð4þ J n denotes the value of the infrared-safe observable J evaluated on a final state with n resolved artons. The various subtraction terms have been defined exlicitly and their integrals over the factorized hase sace of the unresolved artons have been obtained in Ref. [9]. Equation (2) includes the double-real (RR) contribution that exhibits singularities whenever one or two artons become unresolved. We subtract from this an aroximate cross section dσ RR;A 2 which has the same singularities in the doubly unresolved limits as the RR matrix element. The difference is still singular in singly unresolved regions and an additional counterterm is needed. To that effect, we subtract the quantity dσ RR;A, and comensate for the overla between the singly and doubly unresolved regions through the term dσ RR;A 2. Similarly, Eq. (3) describes the emission of one additional arton at one loo, the real-virtual (RV) contribution. In addition to exlicit infrared oles coming from the loo integration, the RV contribution has kinematic singularities when this additional arton is soft or collinear to another colored article. These are regularized by the aroximate one-loo cross section dσ RV;A. The difference is now free of imlicit singularities, but the exlicit infrared singularities are still resent. These oles are the same as the singly unresolved singularities in dσ RR;A in integrated form, and they cancel once the corresonding term is added back. The integral of dσ RR;A, however, still has one-arton singularities, which are regularized by the last term in Eq. (3). The last contribution to the NNLO distribution, shown in Eq. (4), includes two-loo virtual (VV) corrections to the LO rocess. The two-loo integrations lead to exlicit infrared singularities. In Eqs. (2) and (3) we hadintroduced five counterterms, but so far only dσ RR;A has been added back in integrated form. The exlicit two-loo singularities cancel against the hase-sace singularities of the remaining four counterterms, which are shown exlicitly in integrated forms in Eq. (4). We comuted all terms in these integrated forms that become singular in d ¼ 4 dimensions and demonstrated the cancellation of these divergences analytically. We also comuted analytically the logarithmic terms in the finite art of the integrated subtractions that become singular on the edges of the hase sace, while we evaluated the rest of the finite art of the integrated subtractions numerically. We add the uncertainty of these numerical evaluations to the uncertainty of the Monte Carlo integration of the n-arton integrations in Eq. () in quadrature. CoLoRFulNNLO has already been successfully alied to comute NNLO corrections to differential distributions describing the decay of a Higgs boson into a air of b quarks [3]. Here we aly this framework for the first time to the comutation of NNLO observables with more than two colored artons in the final state. In articular, we consider event-shae observables in e e γ 3 jets and we study NNLO corrections to them. IfOdenotes a generic event shae observable, we write dσ σ 0 do ¼ α s 2π AðOÞ 2BðOÞ αs 3CðOÞOðα 4 2π 2π sþ; αs ð5þ where σ 0 is the leading-order rediction for the rocess e e hadrons in erturbation theory. In this Letter we concentrate on four event shaes. The first two, thrust [4,5] and C arameter [6,7], have already been studied at NNLO accuracy [6,7] and serve as a validation of our method. The other two, oblateness and energy-energy correlation, have never been resented at NNLO accuracy and constitute our main henomenological results. Thrust is defined as T ¼ max ij~n ~ i j ; ð6þ ~n ij~ i j where ~ i denote the three-momenta of the artons and ~n defines the direction of the thrust axis, ~n T, by maximizing

3 the sum on the right-hand side over all directions of the final-state articles. In order to define oblateness, we need two variants of this definition, thrust major and thrust minor. Thrust major is given by T M ¼ max ij~n ~ i j ; ð7þ ~n ~n T ¼0 ij~ i j where ~n defines the direction of the thrust-major axis, ~n TM, by maximizing the sum on the right-hand side over all directions orthogonal to the thrust axis. Similarly, T m ¼ ij~n Tm ~ i j ; with ~n ij~ i j Tm ¼ ~n T ~n TM ; ð8þ defines thrust minor, where the thrust-minor axis, ~n Tm,is orthogonal to both the thrust and thrust-major axes. Oblateness O is then the difference of thrust major and thrust minor [8], O ¼ T M T m : ð9þ The value of the C arameter for massless final-state articles is C ar ¼ 3 i;jj~ i jj~ j jsin 2 θ ij 2 ð ij~ i jþ 2 ; ð0þ where θ ij is the angle between ~ i and ~ j. Finally, energy-energy correlation [9] is the normalized energy-weighted cross section defined in terms of the angle between two articles i and j in an event, EECðχÞ ¼ σ had X i;j Ei E j Q 2 dσ e e ijxδðcos χ cos θ ij Þ; ðþ where Q 2 is the squared center-of-mass energy, E i and E j are the article energies, θ ij ¼ π χ is the angle between the two articles, and σ had is the total hadronic cross section. Exerience shows that comuting radiative corrections to the distributions of C arameter, oblateness, and energy-energy correlations is numerically more challenging than for other three-jet event shaes. As a validation of our method, we show in Figs. and 2 the third-order coefficient in Eq. (5) foro ¼ τ T and O ¼ C ar. We observe a very good numerical convergence of our method at NNLO: the absolute uncertainties of the integrations are shown as shaded narrow bands around the solid line on the uer anels (hardly visible) and the relative ones around the lines at one on the lower anels of Figs. and 2. We comare our results to the redictions of Refs. [6,7] and we find agreement over a large range of τ and C arameter. As the redictions ublished in Ref. [7] are affected by an issue with the generation of hase sace in the code used to comute those results [20], we make FIG.. The NNLO coefficient of the weighted τ ¼ T distribution. The lower anels show the redictions of Ref. [7], denoted as SW, (middle anel) and those of Ref. [6], denoted as GGGH, (lower anel) normalized to ours, as well as the relative uncertainties of the numerical integrations (shaded band around the line at one). Also shown in the middle anel are the redictions of MadGrah5_aMC@NLO, denoted as MG5, for τ > =3. comarisons to udated but unublished results rovided to us by S. Weinzierl. We observe statistically significant differences beyond the kinematical limits (τ ¼ =3 and C ar ¼ 3=4) where the three-article final states vanish and the event shaes are determined by a four-jet final state. In these regions the CðOÞ coefficients are determined by the NLO corrections to four-jet roduction, which have been known for a long time [2,3] and can also be comuted with modern automated tools, such as MadGrah5_aMC@NLO [2]. We have checked that our redictions are in comlete agreement with those of Ref. [3] for the C arameter and also with MadGrah5_aMC@NLO for both FIG. 2. The same as Fig. for the C arameter, with the middle anel showing also the redictions of Ref. [3], denoted as NT, for C ar > 3=

4 FIG. 3. Weighted distributions of oblateness O at LO, NLO, and NNLO accuracy in erturbative QCD. The bands reresent the deendence on the renormalization scale varied in the range ξ R μ=μ 0 ½0.5; 2Š around the default scale μ 0 ¼ ffiffiffiffiffiffi Q 2. The lower anel shows the relative scale deendence (band) at NNLO accuracy and the relative uncertainty of the integrations (error bars). thrust and C arameter. This can be seen on the middle anels of Figs. and 2 where we show the results of MadGrah5_aMC@NLO for τ > =3 and those of Ref. [3] for C ar > 3=4 normalized to ours. We resent redictions for the distributions of oblateness O and energy-energy correlation EEC at NNLO ffiffiffiffiffiffi accuracy in erturbative QCD for collider energy Q 2 ¼ 9.2 GeV in Figs. 3 and 4. The bands reresent the deendence of the redictions on the renormalization scale varied in the range [0.5, 2] times our default scale: the total center-of-mass energy. We use α s ¼ 0.8 for the central value and the three-loo running of the strong couling for comuting the scale variations. The lower anels show the relative scale deendence of the NNLO redictions and the relative uncertainties of the integrations. Both oblateness and energy-energy correlation are known to vanish in the dijet limit. Moreover, oblateness is exected to vanish also for FIG. 5. Distribution of the NNLO coefficient for oblateness O. The error bars reresent the statistical uncertainty of the Monte Carlo integrations. cylindrically symmetric final states, while for three-arton events one has 0 O = ffiffiffi 3. Indications of these features are visible in Figs. 3 and 4. We observe that the NNLO corrections slightly lower, and also slightly modify the shae of the O distribution comared to NLO, while the NNLO corrections enhance the EEC distribution almost uniformly. The changes in the shaes of the distributions due to the NNLO corrections can be areciated better by looking at the distributions of the NNLO coefficients directly, as shown in Figs. 5 and 6. Also for these distributions, we observe good numerical convergence of our code. We conclude by commenting on the behavior of the distributions corresonding to small values of the event shaes. Those regions are dominated by kinematical configurations where one of the three artons is unresolved, resulting in logarithmically enhanced contributions. In order to obtain reliable redictions the large logarithms must be resummed to all orders in erturbation theory, which is beyond the scoe of the resent study. In this Letter we have introduced the CoLoRFulNNLO method to comute NNLO radiative corrections for rocesses with colorless initial states. We have alied it to obtain recise redictions for event shae distributions in three-jet roduction in e e collisions. We observe very good numerical convergence of our redictions over the whole range of values of the event shaes. We emhasize FIG. 4. Distributions of energy-energy correlation EEC at LO, NLO, and NNLO accuracy in erturbative QCD. The bands and the lower anel are like in Fig. 3. FIG. 6. Same as Fig. 5 for energy-energy correlation

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