Infrared structure of e+e- --> 3 jets at NNLO
A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover, G. Heinrich
aa r X i v : . [ h e p - ph ] N ov Preprint typeset in JHEP style - PAPER VERSION
ZU-TH 18/07, IPPP/07/62, Edinburgh 2007-26
Infrared structure of e + e − → jets at NNLO A. Gehrmann–De Ridder
Institute for Theoretical Physics, ETH, CH-8093 Z¨urich, SwitzerlandE-mail: [email protected]
T. Gehrmann
Institut f¨ur Theoretische Physik, Universit¨at Z¨urich, Winterthurerstrasse 190,CH-8057 Z¨urich, SwitzerlandE-mail: [email protected]
E.W.N. Glover
Institute of Particle Physics Phenomenology, Department of Physics,University of Durham, Durham, DH1 3LE, UKE-mail: [email protected]
G. Heinrich
School of Physics, The University of Edinburgh, Edinburgh EH9 3JZ, ScotlandE-mail: [email protected]
Abstract:
We describe the calculation of the next-to-next-to-leading order (NNLO) QCDcorrections to three-jet production and related event shape observables in electron-positronannihilation. Infrared singularities due to double real radiation at tree level and single realradiation at one loop are subtracted from the full QCD matrix elements using antennafunctions, which are then integrated analytically and added to the two loop contribu-tion. Using this antenna subtraction method, we obtain numerically finite contributionsfrom five-parton and four-parton processes, and observe an explicit analytic cancellationof infrared poles in the four-parton and three-parton contributions. All contributions areimplemented in a flexible parton-level event generator programme, allowing the numeri-cal computation of any infrared-safe observable related to three-jet final states to NNLOaccuracy.
Keywords:
QCD, Jets, LEP and ILC Physics, NLO and NNLO Computations. ontents
1. Introduction 22. Perturbative calculation of jet observables in e + e − -annihilation 3
3. Phase space mappings 10
4. Parton-level contributions to e + e − → jets up to NNLO 23
5. Construction of the NLO subtraction term 336. Construction of the N colour factor 34
7. Construction of the N colour factor 38
8. Construction of the /N colour factor 41
9. Construction of the N F N colour factor 43
0. Construction of the N F /N colour factor 46
11. Construction of the N F colour factor 48
12. Construction of the N F,γ colour factor 49
13. Numerical implementation 5014. Thrust distribution as an example 5215. Conclusions and Outlook 57
1. Introduction
Among jet observables, the three-jet production rate in electron–positron annihilation playsa very prominent role. The initial experimental observation of three-jet events at PE-TRA [1], in agreement with the theoretical prediction [2], provided first evidence for thegluon, and thus strong support for the theory of Quantum Chromodynamics (QCD) [3].Subsequently the three-jet rate and related event shape observables were used for the pre-cise determination of the QCD coupling constant α s (see [4] for a review). Especially atLEP, three-jet observables were measured to a very high precision and the error on the ex-traction of α s from these data is dominated by the uncertainty inherent in the theoreticaldescription of the jet observables. This description is at present based on a next-to-leadingorder (NLO) calculation [5–10], combined with next–to-leading logarithmic (NLL) resum-mation [11, 12] and inclusion of power corrections [13]. The calculation of next-to-next-to-leading order (NNLO), i.e. O ( α s ), corrections to the three-jet rate in e + e − annihilationhas therefore been high on the list of priorities for a long time [14].Besides its phenomenological importance, the three-jet rate has also served as a theo-retical testing ground for the development of new techniques for higher order calculationsin QCD: both the subtraction [5, 15, 16] and the phase-space slicing [7] methods for theextraction of infrared singularities from NLO real radiation processes were developed inthe context of the first three-jet calculations. The systematic formulation of phase-spaceslicing [9] as well as the dipole subtraction [10] method were also first demonstrated forthree-jet observables, before being applied to other processes.– 2 –ver the past years, many of the ingredients necessary for NNLO calculations of jetobservables have become available: two-loop corrections to all phenomenologically relevantmassless 2 → → → → e + e − → j [24], Higgs pro-duction [25] and vector boson production [26] at hadron colliders, as well as the QEDcorrections to muon decay [27].Furthermore, exploiting the specific kinematic features of the observable under con-sideration, exclusive NNLO results were derived for the forward-backward asymmetry in e + e − annihilation [28], for e + e − → j [29, 30] and most recently for Higgs production athadron colliders [31].In the present paper, we employ a recently developed general technique for the treat-ment of infrared singularities, antenna subtraction [32], to derive the NNLO corrections tothree-jet production in electron-positron annihilation. The first phenomenological applica-tions of our results to the thrust distribution were documented earlier in [33].The paper is structured as follows: in section 2, we outline the perturbative calculationof jet observables and summarise the antenna subtraction method used here. The imple-mentation of this method requires phase space mappings, which are described in Section 3.All relevant tree-level, one-loop and two-loop matrix elements are listed in section 4. Sec-tion 5 briefly summarises how the NLO corrections are implemented using antenna sub-traction. Sections 6–12 contain the subtraction terms constructed for all colour factorsrelevant in this calculation. The numerical implementation of all terms into a parton-levelevent generator is described in Section 13. As a first example of the implementation, wediscuss the NNLO corrections to the thrust distribution in section 14. A summary and anoutlook on applications is given in Section 15.
2. Perturbative calculation of jet observables in e + e − -annihilation To obtain the perturbative corrections to a jet observable at a given order, all partonic mul-tiplicity channels contributing to that order have to be summed. In general, each partonicchannel contains both ultraviolet and infrared (soft and collinear) singularities. The ul-traviolet poles are removed by renormalisation, however for suitably inclusive observables,the soft and collinear infrared poles cancel among each other when all partonic channelsare summed over [34].While infrared singularities from purely virtual corrections are obtained immediatelyafter integration over the loop momenta, their extraction is more involved for real emission(or mixed real-virtual) contributions. Here, the infrared singularities only become explicitafter integrating the real radiation matrix elements over the phase space appropriate to the– 3 –et observable under consideration. In general, this integration involves the (often iterative)definition of the jet observable, such that an analytic integration is not feasible (and alsonot appropriate). Instead, one would like to have a flexible method that can be easilyadapted to different jet observables or jet definitions.Three types of approaches for this task have been developed so far. Phase-space slicingtechniques [7, 9, 35] decompose the final state phase space into resolved regions, which areintegrated numerically and unresolved regions, which are integrated analytically. The sec-tor decomposition approach [22, 23] divides the integration region into sectors containing asingle type of singularity each. Subsequently, the phase space integration is expanded intodistributions. In this approach, the coefficients of all infrared divergent terms, as well asthe finite remainder, can be computed numerically. Finally, subtraction methods [5, 15, 16]extract infrared singularities of the real radiation contributions using infrared subtractionterms. These terms are constructed such that they approximate the full real radiation ma-trix elements in all singular limits while still being simple enough to integrate analytically.To specify the notation, we define the tree-level n -parton contribution to the m -jetcross section (for tree level cross sections n = m ; we leave n = m for later reference) in d dimensions by,d σ B = N X n dΦ n ( p , . . . , p n ; q ) 1 S n |M n ( p , . . . , p n ) | J ( n ) m ( p , . . . , p n ) . (2.1)the normalisation factor N includes all QCD-independent factors as well as the dependenceon the renormalised QCD coupling constant α s , P n denotes the sum over all configurationswith n partons, dΦ n is the phase space for an n -parton final state with total four-momentum q µ in d = 4 − ǫ space-time dimensions,dΦ n ( p , . . . , p n ; q ) = d d − p E (2 π ) d − . . . d d − p n E n (2 π ) d − (2 π ) d δ d ( q − p − . . . − p n ) , (2.2)while S n is a symmetry factor for identical partons in the final state. The jet function J ( n ) m defines the procedure for building m jets out of n partons. The main property of J ( n ) m isthat the jet observable defined above is collinear and infrared safe as explained in [10, 38].In general J ( n ) m contains θ and δ -functions. J ( n ) m can also represent the definition of the n -parton contribution to an event shape observable related to m -jet final states. |M n | denotes a squared, colour ordered tree-level n -parton matrix element. Contri-butions to the squared matrix element which are subleading in the number of colours canequally be treated in the same context, noting that these subleading terms yield configura-tions where a certain number of essentially non-interacting particles are emitted betweena pair of hard radiators. By carrying out the colour algebra, it becomes evident that non-ordered gluon emission inside a colour-ordered system is equivalent to photon emission offthe outside legs of the system [36,37]. For simplicity, these subleading colour contributionsare also denoted as squared matrix elements |M m | , although they often correspond purelyto interference terms between different amplitudes. The precise definition depends on thenumber and types of particles involved in the process. However, all colour orderings aresummed over in P m with the appropriate colour weighting.– 4 –rom (2.1), one obtains the leading order approximation to the m -jet cross section byintegration over the appropriate phase space.d σ LO = Z dΦ m d σ B . (2.3)Depending on the jet function used, this cross section can still be differential in certainkinematic quantities. At NLO, we consider the following m -jet cross section,d σ NLO = Z dΦ m +1 (cid:0) d σ RNLO − d σ SNLO (cid:1) + "Z dΦ m +1 d σ SNLO + Z dΦ m d σ VNLO . (2.4)The cross section d σ RNLO has the same expression as the Born cross section d σ B (2.1) aboveexcept that n → m + 1, while d σ VNLO is the one-loop virtual correction to the m -partonBorn cross section d σ B . The cross section d σ SNLO is a (preferably local) counter-term ford σ RNLO . It has the same unintegrated singular behaviour as d σ RNLO in all appropriate limits.Their difference is free of divergences and can be integrated over the ( m + 1)-parton phasespace numerically. The subtraction term d σ SNLO has to be integrated analytically over allsingular regions of the ( m + 1)-parton phase space. The resulting cross section added tothe virtual contribution yields an infrared finite result.Several methods for constructing NLO subtraction terms systematically were proposedin the literature [10, 15, 16, 39–41]. For some of these methods, extension to NNLO wasdiscussed [32,42,43] and partly worked out. Up to now, the only method worked out in fulldetail to NNLO is antenna subtraction [39, 40]. In our calculation of NNLO corrections tothree-jet observables, we used this method, which we briefly outline in the following. Thedetails of the method, and a full definition of the notation, can be found in [32].The basic idea of the antenna subtraction approach is to construct the subtractionterms from antenna functions which encapsulate all singular limits due to the emissionof unresolved partons between two colour-connected hard partons. This construction ex-ploits the universal factorisation of both phase space and squared matrix elements in allunresolved limits. The full antenna subtraction term is then constructed by summingproducts of antenna functions with reduced matrix elements over all possible unresolvedconfigurations.At NLO, the antenna subtraction term thus reads:d σ SNLO = N X m +1 dΦ m +1 ( p , . . . , p m +1 ; q ) 1 S m +1 × X j X ijk |M m ( p , . . . , ˜ p I , ˜ p K , . . . , p m +1 ) | J ( m ) m ( p , . . . , ˜ p I , ˜ p K , . . . , p m +1 ) . (2.5)The key ingredient is the phase space mapping which relates the original momenta p i , p j , p k describing the two hard radiator partons i and k and the emitted parton j to a– 5 –
1i jk I i jkIm+1 m+1K K
Figure 1:
Illustration of NLO antenna factorisation representing the factorisation of both thesquared matrix elements and the ( m + 1)-particle phase space. The term in square brackets repre-sents both the antenna function X ijk and the antenna phase space dΦ X ijk . redefined on-shell set ˜ p I , ˜ p K which are linear combinations of p i , p j , p k . The phase spacemapping yielding this redefinition is described in detail in section 3 below. With thismapping,the phase space factorises,dΦ m +1 ( p , . . . , p m +1 ; q ) = dΦ m ( p , . . . , ˜ p I , ˜ p K , . . . , p m +1 ; q ) · dΦ X ijk ( p i , p j , p k ; ˜ p I + ˜ p K ) . (2.6)The other elements of the subtraction term also depend on either the original mo-menta p i , p j , p k or the redefined on-shell momenta ˜ p I , ˜ p K but not both. This enables thesubtraction term to completely factorise.To be more specific, both the m -parton amplitude and the jet function J ( m ) m dependonly on p , . . . , ˜ p I , ˜ p K , . . . , p m +1 i.e. on the redefined on-shell momenta ˜ p I , ˜ p K . On theother hand, the tree-level three-parton antenna function X ijk depends only on p i , p j , p k . X ijk describes all of the configurations (for this colour ordered amplitude) where parton j is unresolved. It can be obtained from appropriately normalised tree-level three-partonsquared matrix elements. The antenna factorisation of squared matrix element and phasespace can be illustrated pictorially, as displayed in Figure 1. Together particles i and k form a colour connected hard antenna that radiates particle j . In doing so, the momentaof the radiators change to form particles I and K . The type of particle may also change.One can therefore carry out the integration over the antenna phase space appropriateto p i , p j and p k analytically, exploiting the factorisation of the phase space of eq. (2.6).The NLO antenna phase space dΦ X ijk is proportional to the three-particle phase space, ascan be seen by using m = 2 in the above formula. For the analytic integration, we can use(2.6) to rewrite each of the subtraction terms in the form, |M m | J ( m ) m dΦ m Z dΦ X ijk X ijk = |M m | J ( m ) m dΦ m X ijk where |M m | , J ( m ) m and dΦ m depend only on p , , . . . , ˜ p I , ˜ p K , . . . , p m +1 and dΦ X ijk and X ijk depend only on p i , p j , p k . This integration is performed analytically in d dimensions, yield-ing the integrated three-parton antenna function X ijk , to make the infrared singularitiesexplicit and added directly to the one-loop m -particle contributions.– 6 –
1i I iIm+2 m+2Ll lLjjk k
Figure 2:
Illustration of NNLO antenna factorisation representing the factorisation of both thesquared matrix elements and the ( m + 2)-particle phase space when the unresolved particles arecolour connected. At NNLO, the m -jet production is induced by final states containing up to ( m + 2) partons,including the one-loop virtual corrections to ( m + 1)-parton final states. As at NLO, onehas to introduce subtraction terms for the ( m + 1)- and ( m + 2)-parton contributions.Schematically the NNLO m -jet cross section reads,d σ NNLO = Z dΦ m +2 (cid:0) d σ RNNLO − d σ SNNLO (cid:1) + Z dΦ m +2 d σ SNNLO + Z dΦ m +1 (cid:16) d σ V, NNLO − d σ V S, NNLO (cid:17) + Z dΦ m +1 d σ V S, NNLO + Z dΦ m d σ V, NNLO , (2.7)where d σ SNNLO denotes the real radiation subtraction term coinciding with the ( m + 2)-parton tree level cross section d σ RNNLO in all singular limits. Likewise, d σ V S, NNLO is theone-loop virtual subtraction term coinciding with the one-loop ( m + 1)-parton cross sectiond σ V, NNLO in all singular limits. Finally, the two-loop correction to the m -parton cross sectionis denoted by d σ V, NNLO .At NNLO, individual antenna functions are obtained from normalised four-parton tree-level and three-parton one-loop matrix elements. The full antenna subtraction term is thenconstructed by summing products of antenna functions with reduced matrix elements overall possible unresolved configurations.In d σ SNNLO , we have to distinguish four different types of unresolved configurations:(a) One unresolved parton but the experimental observable selects only m jets;(b) Two colour-connected unresolved partons (colour-connected);(c) Two unresolved partons that are not colour connected but share a common radiator(almost colour-unconnected); – 7 –d) Two unresolved partons that are well separated from each other in the colour chain(colour-unconnected).Among those, configuration (a) is properly accounted for by a single tree-level three-partonantenna function like used already at NLO. Configuration (b) requires a tree-level four-parton antenna function (two unresolved partons emitted between a pair of hard partons)as shown in Figure 2, while (c) and (d) are accounted for by products of two tree-levelthree-parton antenna functions. The subtraction terms for these configurations read:d σ S,aNNLO = N X m +2 dΦ m +2 ( p , . . . , p m +2 ; q ) 1 S m +2 × " X j X ijk |M m +1 ( p , . . . , ˜ p I , ˜ p K , . . . , p m +2 ) | × J ( m +1) m ( p , . . . , ˜ p I , ˜ p K , . . . , p m +2 ) , (2.8)d σ S,bNNLO = N X m +2 dΦ m +2 ( p , . . . , p m +2 ; q ) 1 S m +2 × " X jk (cid:0) X ijkl − X ijk X IKl − X jkl X iJL (cid:1) × |M m ( p , . . . , ˜ p I , ˜ p L , . . . , p m +2 ) | J ( m ) m ( p , . . . , ˜ p I , ˜ p L , . . . , p m +2 ) , (2.9)d σ S,cNNLO = −N X m +2 dΦ m +2 ( p , . . . , p m +2 ; q ) 1 S m +2 × " X j,l X ijk x mlK |M m ( p , . . . , ˜ p I , ee p K , ee p M , . . . , p m +2 ) | × J ( m ) m ( p , . . . , ˜ p I , ee p K , ee p M , . . . , p m +2 )+ X j,l X klm x ijK |M m ( p , . . . , ee p I , ee p K , ˜ p M , . . . , p m +2 ) | × J ( m ) m ( p , . . . , ee p I , ee p K , ˜ p M , . . . , p m +2 ) , (2.10)d σ S,dNNLO = −N X m +2 dΦ m +2 ( p , . . . , p m +2 ; q ) 1 S m +2 × " X j,o X ijk X nop |M m ( p , . . . , ˜ p I , ˜ p K , . . . , ˜ p N , ˜ p P , . . . , p m +2 ) | × J ( m ) m ( p , . . . , ˜ p I , ˜ p K , . . . , ˜ p N , ˜ p P , . . . , p m +2 ) . (2.11)Again, the original momenta of the ( m +2)-parton phase space are denoted by i, j, . . . , while– 8 – i jkm+1 1 I i jkIm+1K K 1 I i jkIm+1K K Figure 3:
Illustration of NNLO antenna factorisation representing the factorisation of both theone-loop “squared” matrix elements (represented by the white blob) when the unresolved particlesare colour connected. the combined momenta obtained from a phase space mapping are labelled by
I, J, . . . . Onlythe combined momenta appear in the jet function. The phase space mappings appropriateto the different cases are described in detail in section 3 below. X ijkl is a four-partonantenna function, containing all configurations where partons j and k are unresolved,while x ijk is a three-parton sub-antenna function containing only limits where parton j is unresolved with respect to parton i , but not limits where parton j is unresolved withrespect to parton k . The factorisation of the phase space is analogous to the factorisationat NLO (2.6), such that integration of these antenna functions over the antenna phasespace amounts to inclusive three-particle or four-particle integrals [22].In single unresolved limits, the one-loop cross section d σ V, NNLO is described by the sumof two terms [44]: a tree-level splitting function times a one-loop cross section and a one-loop splitting function times a tree-level cross section. Consequently, the one-loop singleunresolved subtraction term d σ V S, NNLO is constructed from tree-level and one-loop three-parton antenna functions, as sketched in Figure 3. Several other terms in d σ V S, NNLO cancelwith the results from the integration of terms in the double real radiation subtractionterm d σ SNNLO over the phase space appropriate to one of the unresolved partons, thusensuring the cancellation of all explicit infrared poles in the difference d σ V, NNLO − d σ V S, NNLO .Explicitly, the one-loop single unresolved subtraction term is given by the sum of the threefollowing contributions:d σ V S, ,aNNLO = N X m +1 dΦ m +1 ( p , . . . , p m +1 ; q ) 1 S m +1 × " X ik −X ijk ( s ik ) |M m +1 ( p , . . . , p i , p k , . . . , p m +1 ) | × J ( m +1) m ( p , . . . , p i , p k , . . . , p m +1 ) , (2.12)d σ V S, ,bNNLO = N X m +1 dΦ m +1 ( p , . . . , p m +1 ; q ) 1 S m +1 × X j " X ijk |M m ( p , . . . , ˜ p I , ˜ p K , . . . , p m +1 ) | J ( m ) m ( p , . . . , ˜ p I , ˜ p K , . . . , p m +1 )– 9 – X ijk |M m ( p , . . . , ˜ p I , ˜ p K , . . . , p m +1 ) | J ( m ) m ( p , . . . , ˜ p I , ˜ p K , . . . , p m +1 ) , (2.13)d σ V S, ,cNNLO = N X m +1 dΦ m +1 ( p , . . . , p m +1 ; q ) 1 S m +1 × " X ik X ijk ( s ik ) X o X nop |M m ( p , . . . , p i , p k , . . . , ˜ p N , ˜ p P , . . . , p m +1 ) | × J ( m ) m ( p , . . . , p i , p k , . . . , ˜ p N , ˜ p P , . . . , p m +1 ) , (2.14)In here, X ijk denotes a one-loop three-parton antenna function.Finally, all remaining terms in d σ SNNLO and d σ V S, NNLO have to be integrated over thefour-parton and three-parton antenna phase spaces. After integration, the infrared polesare rendered explicit and cancel with the infrared pole terms in the two-loop squared matrixelement d σ V, NNLO .The subtraction terms d σ SNLO , d σ SNNLO and d σ V S, NNLO require three different types ofantenna functions corresponding to the different pairs of hard partons forming the antenna:quark-antiquark, quark-gluon and gluon-gluon antenna functions. In the past [39,40], NLOantenna functions were constructed by imposing definite properties in all single unresolvedlimits (two collinear limits and one soft limit for each antenna). This procedure turns outto be impractical at NNLO, where each antenna function must have definite behaviours ina large number of single and double unresolved limits. Instead, we derived these antennafunctions in a systematic manner from physical matrix elements known to possess thecorrect limits. The quark-antiquark antenna functions can be obtained directly from the e + e − → j real radiation corrections at NLO and NNLO [29]. For quark-gluon and gluon-gluon antenna functions, effective Lagrangians [45,46] are used to obtain tree-level processesyielding a quark-gluon or gluon-gluon final state. The antenna functions are then derivedfrom the real radiation corrections to these processes. Quark-gluon antenna functions werederived [47] from the purely QCD (i.e. non-supersymmetric) NLO and NNLO correctionsto the decay of a heavy neutralino into a massless gluino plus partons [45], while gluon-gluon antenna functions [48] result from the QCD corrections to Higgs boson decay intopartons [46].All tree-level three-parton and four-parton antenna functions and three-parton one-loop antenna functions are listed in [32]. Their integration over the antenna phase spaceamounts to performing inclusive infrared-divergent three-parton and four-parton phasespace integrals. Techniques for evaluating these integrals are described in [22, 49], and allintegrated antenna functions are documented in [32].
3. Phase space mappings
The subtraction terms for single and double unresolved configurations described in theprevious section involve the mapping of the momenta appearing in the antenna functions– 10 –nto combined momenta, which appear in the reduced matrix elements and in the jetfunction.At NLO, one needs momentum mappings from three partons to two partons, F (3 → : { p i , p j , p k } → { ˜ p I , ˜ p K } . At NNLO, several further mappings are needed. First and foremost, one needs momentummappings from four partons to two partons, F (4 → : { p i , p j , p k , p l } → { ˜ p I , ˜ p L } . In addition, repeated mappings from three partons to two partons are also required.For the subtraction and phase space factorisation to work correctly, all mappings mustsatisfy the following requirements (specified here for the example of F (4 → ):1. momentum conservation: ˜ p I + ˜ p L = p i + p j + p k + p l .2. the new momenta should be on-shell: ˜ p I = 0, ˜ p L = 0.3. the new momenta should reduce to the appropriate original momenta in the exactsingular limits, e.g. ˜ p I = p i + p j + p k , ˜ p L = p l in the ( i, j, k ) triple collinear limit.4. the mapping should not introduce spurious singularities.The momentum mappings we use follow largely those worked out in [40, 43]. Thedifferent types of mappings needed for our calculation are described in detail in the followingsubsections. In the single unresolved limit where parton j becomes unresolved and i, k are the hardradiators, the momenta of the partons i, j, k are mapped to ˜ p I = g ( ij ) and ˜ p K = g ( kj ) in thefollowing way: ˜ p I = x p i + r p j + z p k ˜ p K = (1 − x ) p i + (1 − r ) p j + (1 − z ) p k , where x = 12( s ij + s ik ) h (1 + ρ ) s ijk − r s jk i z = 12( s jk + s ik ) h (1 − ρ ) s ijk − r s ij i ρ = 1 + 4 r (1 − r ) s ij s jk s ijk s ik . (3.1)The parameter r can be chosen conveniently, we use [40] r = s jk s ij + s jk . – 11 – .2 Mappings from five partons to three partons In the construction of the subtraction terms for the five-parton channel, one encounters,both the four-parton antenna functions (double unresolved limits), and products of twothree-parton antenna functions, which compensate for the single unresolved limits of thedouble unresolved subtraction terms. The former require a 4 → → In the double unresolved limit where partons i and i become unresolved and i , i arethe hard radiators, the partons i , . . . , i are mapped to the partons j , j with momenta˜ p j ≡ ^ ( i i i ) , ˜ p j ≡ ^ ( i i i ) . (3.2)We will use the shorthand notation (3.2) extensively in the following, keeping in mindhowever that ˜ p j and ˜ p j are linear combinations of all four original momenta with amapping F (4 → : { p i , p i , p i , p i } → { ˜ p j , ˜ p j } given by:˜ p j = x p i + r p i + r p i + z p i ˜ p j = (1 − x ) p i + (1 − r ) p i + (1 − r ) p i + (1 − z ) p i . (3.3)Defining s kl = ( p i k + p i l ) , the coefficients are given by [43] r = s + s s + s + s r = s s + s + s x = 12( s + s + s ) h (1 + ρ ) s − r ( s + 2 s ) − r ( s + 2 s )+( r − r ) s s − s s s i z = 12( s + s + s ) h (1 − ρ ) s − r ( s + 2 s ) − r ( s + 2 s ) − ( r − r ) s s − s s s i ρ = h r − r ) s s λ ( s s , s s , s s )+ 1 s s n r (1 − r ) + r (1 − r ))( s s + s s − s s )+ 4 r (1 − r ) s s + 4 r (1 − r ) s s oi ,λ ( u, v, w ) = u + v + w − uv + uw + vw ) . – 12 – i i i i = ⇒ i unresolved i , i radiators X ( i , i , i ) l = g i i l = i l = g i i l = i = ⇒ l unresolved l , l radiators Y ( l , l , l ) j = g l l j = g l l j = l Figure 4: F (5 → → ( i , i , i , i , i ; j , j , j ): both combined partons ( l and l ) are radiators inthe second step, original parton i becomes unresolved. i i i i i = ⇒ i unresolved i , i radiators X ( i , i , i ) l = g i i l = g i i l = i l = i = ⇒ l unresolved l , l radiators Y ( l , l , l ) j = g l l j = g l l j = l Figure 5: F (5 → → ( i , i , i , i , i ; j , j , j ): one of the combined partons (here l ) becomes un-resolved between l and l in the second step. The four-particle antennae X ijkl contain by construction all colour-connected double un-resolved limits of the ( m + 2)-parton matrix element where partons j and k become un-resolved. However, X ijkl can also become singular in single unresolved limits, where itdoes not coincide with limits of the matrix element. These limits have to be subtracted asindicated in eq. 2.9 and as described in section 2.3 of [32]. In the limit where j becomesunresolved between i and k , the four-particle antenna collapses to the three-particle an-tenna X ( i, j, k ) and a three-particle “remainder matrix element” X ( I, K, l ), which alsohas the form of a three-particle antenna. If one of the partons
I, K, l becomes unresolved ina second single emission, the resulting momenta in these limits coincide with those from adouble unresolved configuration as defined in subsection 3.2.1. Therefore we need momen-tum mappings corresponding to such a “two-step” emission in order to be able to subtractthe spurious singularities of the four-particle antennae X ijkl .As explained in section 2.2 above and in [32], we have to distinguish between colour-connected unresolved partons and almost colour-unconnected unresolved partons. If theunresolved partons are colour-connected, we use the momentum mappings F (5 → → , C : { p i , p i , p i , p i , p i } → { p l , p l , p l , p l } → { ˜ p j , ˜ p j , ˜ p j } , where the different types B,Care described in more detail below. These two-step mappings first map the four partons i , . . . , i making up the four-particle antenna to three “intermediate” partons l , l , l , andthen map the three intermediate partons to two partons j , j . The fifth parton i = l = j only acts as a spectator. An additional type of mapping, denoted by F (5 → → , is neededfor subtraction terms of two almost colour-unconnected unresolved partons, as defined ineq. 2.10 and in [32], and involves redefinitions of all five initial partons. All three mappingsare depicted in Figures 4–6.In the first step, one of the partons { i , i , i , i } becomes unresolved. The momentum– 13 – i i i i = ⇒ i unresolved i , i radiators X ( i , i , i ) l = g i i l = g i i l = i l = i = ⇒ l unresolved l , l radiators Y ( l , l , l ) j = l j = g l l j = g l l Figure 6: F (5 → → ( i , i , i , i , i ; j , j , j ): i is the shared hard radiator, i = l becomesunresolved between l and l in the second step. i = l is always unaffected by the mapping. The precise definition of the resulting momenta l , l , l depends on the mapping. In the first step of the two-step mapping F (5 → → wehave l = x i + r i + z i ≡ g i i l = i l = (1 − x ) i + (1 − r ) i + (1 − z ) i ≡ g i i l = i , (3.4)while for the mappings F (5 → → and F (5 → → , the first step is of the form l = x i + r i + z i ≡ g i i l = i l = (1 − x ) i + (1 − r ) i + (1 − z ) i ≡ g i i l = i . (3.5)The momentum i is always the unresolved one, denoted generically by i u in the following.The two hard radiators denoted by i a and i b depend on the mapping. In F (5 → → , i and i are the hard radiators, while in F (5 → → and F (5 → → , i and i are the hardradiators.Note that mappings of type C apply to all configurations where one of the combinedpartons becomes unresolved in the second step, so they also include the cases where theroles of l and l are interchanged.The coefficients r , x , z appearing in eqs. (3.4) and (3.5) are given by eq. (3.1), r = s ub s au + s ub x = 12( s au + s ab ) h (1 + ρ ) s aub − r s ub i z = 12( s ub + s ab ) h (1 − ρ ) s aub − r s au i ρ = h r (1 − r ) s au s ub s ab s aub i (3.6)where now s ub = ( i u + i b ) etc. – 14 –irst step Second stepMapping type i u i a , i b i s l u l a , l b l s B i i , i i l l , l l C i i , i i l or l ( l or l ), l l K i i , i i l l , l l Table 1:
Identification of the unresolved, radiator and spectator momenta for both steps of themomentum mappings F (5 → → , C , K . In the second step, one of the intermediate partons { l , l , l } becomes unresolved. Theresulting momenta j , j , j are defined as j = x l a + r l u + z l b ≡ g l a l u (3.7) j = (1 − x ) l a + (1 − r ) l u + (1 − z ) l b ≡ g l b l u (3.8) j = l r , where again l u denotes the momentum which is unresolved in the second step, l a , l b arethe radiators and l r does not take part in the second recombination step. The coefficients r , x , z are defined analogously to eq. (3.6), where now s ub = ( l u + l b ) etc.The combination of antenna functions associated with the repeated unresolved singu-larity is thus X ( i a , i u , i b ) Y ( l a , l u , l b ) (3.9)as indicated in figures 4–6 where X and Y generically stand for three-particle tree an-tenna functions. The identification of the radiator and unresolved momenta i a , . . . , l b forthe different mappings F (5 → → , C , K can be read off from Table 1 and is also illustrated inFigures 4–6. The antenna phase space mappings require two uniquely identified hard radiator momentaand an ordered emission of the unresolved partons.With the antenna functions of [32], it is not always possible to uniquely identify thehard momenta, especially if more than one final state parton is a gluon. Moreover, in thefour-parton antenna functions (containing two unresolved partons) at subleading colour,the emission is not colour-ordered. It was already outlined in [32] that in both these cases,a further decomposition of the antenna functions into different sub-antennae configurationsis required.The tree-level three-parton antenna functions D (1 q , g , g ) (quark-gluon-gluon) and F (1 g , g , g ) (gluon-gluon-gluon) contain more than one antenna configuration, since each– 15 –luon can become unresolved. Their decomposition was discussed in [32], it reads: D (1 , ,
4) = d (1 , ,
4) + d (1 , , , (3.10) F (1 , ,
3) = f (1 , ,
2) + f (3 , ,
1) + f (2 , , , (3.11)with d (1 , ,
4) = 1 s s s s s + s s + s s + s s s + 52 s + 12 s ! + O ( ǫ ) , (3.12) f (1 , ,
2) = 1 s s s s s + s s s + s s s + 83 s ! + O ( ǫ ) . (3.13)With this decomposition, the sub-antennae d ( i, j, k ) and f ( i, j, k ) contain only a softsingularity associated with gluon j , and collinear singularities i k j and k k j , such that i and k can be identified as hard radiators. Soft singularities associated with i or k andthe collinear singularity i k k , which were present in the full antenna functions, are nowcontained in different sub-antennae, obtained by permutations of the momenta. Therefore,each sub-antenna can have a unique phase space mapping ( i, j, k ) → ( e ij, f kj ).The one-loop three-parton antenna functions D (1 q , g , g ) (quark-gluon-gluon) and F (1 g , g , g ) (gluon-gluon-gluon) can be decomposed according to the same pattern, ex-ploiting the fact that each can be written as a function proportional to its tree-level coun-terpart plus a function which is not singular in any unresolved limit.The decomposition of tree-level four-parton antenna functions is more involved, espe-cially since both single and double unresolved limits have to be accounted for properly. Itturns out to be very useful to introduce the following combinations of three-parton antennafunctions: Q (1 , ,
2) = d (1 , , − A (1 , , ,R (1 , ,
2) = Q (1 , , − Q (1 , , S (1 , ,
2) = Q (1 , ,
2) + Q (1 , ,
3) + E (1 , , . (3.14)None of these contains any soft limit or collinear 1 k i limit. Only Q and R contain a2 k S is also finite in this limit, owing to the N = 1 supersymmetry relationbetween the tree-level splitting functions, P q ¯ q → G ( z ) + P gg → G ( z ) = P qg → Q ( z ) + P qg → Q (1 − z ) . (3.15)Among the tree-level four-parton antenna functions, only ˜ A (1 q , g , g , ¯ q ) (quark-gluon-gluon-antiquark at subleading colour), D (1 q , g , g , g ) (quark-gluon-gluon-gluon), E (1 q , q ′ , ¯ q ′ , g ) (quark-quark-antiquark-gluon at leading colour), F (1 g , g , g , g ) (gluon-gluon-gluon-gluon) as well as G (1 g , q , ¯ q , g ) and ˜ G (1 g , q , ¯ q , g ) (gluon-quark-antiquark-gluon at leading and subleading colour) must be decomposed into sub-antennae. In thecontext of the three-jet calculation discussed here, F , G and ˜ G do not contribute andwill not be discussed further. – 16 –he decomposition of ˜ A (1 q , g , g , ¯ q ) is needed because both gluons can becomecollinear either with quark 1 q or with antiquark 2 ¯ q . The two possible phase space mappingsare of the F (5 → type described in section 3.2.1(a): (1 , , , → ( g , g , , , → ( g , g A (1 , , ,
2) = ˜ A ,a (1 , , ,
2) + ˜ A ,b (1 , , , , (3.16)with ˜ A ,a (1 , , ,
2) = ˜ a (1 , , ,
2) + ˜ a (2 , , , , ˜ A ,b (1 , , ,
2) = ˜ A ,a (1 , , , , (3.17)where ˜ a ( i, j, k, l ) contains only singularities for i k j or k k l and was defined in [32]. Withthis decomposition, ˜ A ,a (1 , , ,
2) contains only 1 k k , , , → ( g , g E (1 q , q ′ , ¯ q ′ , g )contains limits where either the quark-antiquark pair (3 q ′ , ¯ q ′ ) or the gluon 5 g can becomesoft. Since these limits yield different hard radiator partons, they can not be accounted forin a single phase space mapping, but require two separate F (5 → mappings:(a): (1 , , , → ( g , g , (b): (1 , , , → ( g , g . By analysing the different triple and single collinear limits of E , one finds the followingdecomposition: E ,a (1 , , ,
5) = B (1 , , ,
5) + E (5 , , Q (1 , g (34) , g (54)) , (3.18) E ,b (1 , , ,
5) = E (1 , , , − E ,a (1 , , , . (3.19)After this decomposition, E ,a can be used with mapping (a) and E ,b with mapping (b).The above decomposition also ensures a well-defined behaviour in all double and singleunresolved limits (see [32] for a definition of the splitting factors): E ,a (1 , , , q ′ → , ¯ q ′ → −→ S (3 , ,E ,b (1 , , , q ′ k ¯ q ′ , g → −→ S ( z ) 1 s P q ¯ q → G ( z ) ,E ,a (1 , , , q k q ′ k ¯ q ′ −→ P non − ident . → Q ( w, x, y ) ,E ,a (1 , , ,
5) + E ,b (1 , , , q ′ k ¯ q ′ k g −→ P → G ( w, x, y ) ,E ,b (1 , , , q k g , q ′ k ¯ q ′ −→ s s P qg → Q ( z ) P q ¯ q → G ( y ) , (3.20)– 17 – ,b (1 , , , g → −→ S E (1 , , ,E ,a (1 , , , q ′ k ¯ q ′ −→ s P q ¯ q → G ( z ) d (1 , (34) ,
5) + ang . ,E ,b (1 , , , q ′ k ¯ q ′ −→ s P q ¯ q → G ( z ) d (1 , , (34)) + ang . ,E ,b (1 , , , q k g −→ s P qg → Q ( z ) E ((15) , , ,E ,b (1 , , , ¯ q ′ k g −→ s P qg → Q ( z ) E (1 , , (45)) , (3.21)while all other limits are zero. It can be seen that only the triple collinear 3 k k k q ′ , ¯ q ′ ) → g limits, which belong to different mappings.The decomposition of the quark-gluon-gluon-gluon antenna function D (1 q , g , g , g ),is more involved, since any pair of two gluons can become soft. We consider four different F (5 → mappings:(a): (1 , , , → ( g , g , (b): (1 , , , → ( g , g , (c): (1 , , , → ( g , g , (d): (1 , , , → ( g , g . The numerous different double and single unresolved limits of this antenna function can bedisentangled very elegantly by repeatedly exploiting the N = 1 supersymmetry relation [37]among the different triple collinear splitting functions [35, 37, 50]. Using this relation, onecan show that the following left-over combination is finite in all single unresolved anddouble unresolved limits: D ,l (1 , , ,
5) = D (1 , , , − " A (1 , , ,
5) + A (1 , , , − (cid:16) ˜ E (1 , , ,
4) + ˜ E (1 , , , (cid:17) + ˜ A (1 , , , − E (1 , , ,
3) + B (1 , , ,
3) + C (1 , , , − E (1 , , ,
5) + B (1 , , ,
5) + C (1 , , , A (1 , , S ( g (13) , g (43) ,
5) + A (1 , , S ( g (15) , g (45) , A (3 , , S (1 , g (54) , g (34)) . (3.22)Starting from the terms in this expression, the following sub-antennae can be constructed: D ,a (1 , , ,
5) = 12 D ,l (1 , , ,
5) + A (1 , , , −
12 ˜ E (1 , , , A (1 , , S ( g (13) , g (43) , A (3 , , (cid:16) S (1 , g (54) , g (34)) − R (1 , g (54) , g (34)) (cid:17) − E (5 , , Q (1 , g (34) , g (54)) − A (1 , , E ( g (13) , g (43) , A (1 , , Q ( g (13) , , g (43)) ,D ,b (1 , , ,
5) = D ,a (1 , , , ,D ,c (1 , , ,
5) = ˜ A ,a (1 , , , − E (1 , , ,
3) + B (1 , , ,
3) + C (1 , , , E (3 , , Q (1 , g (54) , g (34)) + A (1 , , E ( g (13) , g (43) , a (1 , , Q ( g (13) , , g (43)) + a (4 , , Q ( g (15) , , g (45)) ,D ,d (1 , , ,
5) = D ,c (1 , , , . (3.23)With this decomposition, each D ,i contains only singularities appropriate to phase spacemapping (i). The sum of the D ,i adds to D : D ,a + D ,b + D ,c + D ,d = D , (3.24)such that only D must be integrated analytically over the antenna phase space.The above decomposition disentangles the different double and single unresolved limits: D ,a (1 , , , g → , g → −→ S ,D ,b (1 , , , g → , g → −→ S ,D ,c (1 , , ,
5) + D ,d (1 , , , g → , g → −→ S S ,D ,c (1 , , ,
5) + D ,d (1 , , , q k g , g → −→ S ( z ) 1 s P qg → Q (1 − z ) ,D ,a (1 , , ,
5) + D ,c (1 , , ,
5) + D ,d (1 , , , g k g , g → −→ S ( z ) 1 s P gg → G ( z ) ,D ,a (1 , , , q k g , g → −→ S ( z ) 1 s P qg → Q ( z ) ,D ,b (1 , , , q k g , g → −→ S ( z ) 1 s P qg → Q ( z ) ,D ,c (1 , , ,
5) + D ,d (1 , , , q k g , g → −→ S ( z ) 1 s P qg → Q (1 − z ) ,D ,b (1 , , ,
5) + D ,c (1 , , ,
5) + D ,d (1 , , , g k g , g → −→ S ( z ) 1 s P gg → G ( z ) ,D ,a (1 , , , q k g k g −→ P → Q ( w, x, y ) ,D ,b (1 , , , q k g k g −→ P → Q ( w, x, y ) ,D ,c (1 , , ,
5) + D ,d (1 , , , q k g k g −→ ˜ P → Q ( w, x, y ) ,D (1 , , , g k g k g −→ P → G ( w, x, y ) ,D ,a (1 , , ,
5) + D ,c (1 , , , q k g , g k g −→ s s P qg → Q ( z ) P gg → G ( y ) ,D ,b (1 , , ,
5) + D ,d (1 , , , q k g , g k g −→ s s P qg → Q ( z ) P gg → G ( y ) , (3.25) D ,a (1 , , , g → −→ S d (1 , , , – 19 – ,c (1 , , ,
5) + D ,d (1 , , , g → −→ S d (1 , , ,D ,a (1 , , , g → −→ S d (1 , , ,D ,b (1 , , , g → −→ S d (1 , , ,D ,b (1 , , , g → −→ S d (1 , , ,D ,c (1 , , ,
5) + D ,d (1 , , , g → −→ S d (1 , , ,D ,a (1 , , , q k g −→ s P qg → Q ( z ) d ((13) , , ,D ,c (1 , , , q k g −→ s P qg → Q ( z ) d ((13) , , ,D ,b (1 , , , q k g −→ s P qg → Q ( z ) d ((15) , , ,D ,d (1 , , , q k g −→ s P qg → Q ( z ) d ((15) , , ,D ,a (1 , , , g k g −→ s P gg → G ( z ) d (1 , (34) ,
5) + ang . ,D ,b (1 , , ,
5) + D ,d (1 , , , g k g −→ s P gg → G ( z ) d (1 , , (34)) + ang . ,D ,b (1 , , , g k g −→ s P gg → G ( z ) d (1 , (45) ,
3) + ang . ,D ,a (1 , , ,
5) + D ,c (1 , , , g k g −→ s P gg → G ( z ) d (1 , , (45)) + ang . . (3.26)All other limits are vanishing. It can be seen that certain limits are shared among severalantenna functions, which can be largely understood due to two reasons:1. in a gluon-gluon collinear splitting, either gluon can become soft, and the gluon-gluonsplitting function is always shared between two sub-antennae, as in (3.10), (3.11) todisentangle the two soft limits.2. the unresolved emission of gluons 3 g and 5 g is shared between the mappings (c) and(d) according to the decomposition of the non-ordered antenna function ˜ A , whichdistributes the soft limit of either gluon between both mappings. The angular terms in the single unresolved limits are associated with a gluon splitting intotwo gluons or into a quark-antiquark pair. They average to zero after integration over theantenna phase space. To ensure numerical stability and reliability, this average has to takeplace within each phase space mapping. We have checked this to be the case for the abovedecompositions of E and D . The angular average in single collinear limits can be madeusing the standard momentum parametrisation [10, 51] for the i k j limit: p µi = zp µ + k µ ⊥ − k ⊥ z n µ p · n , p µj = (1 − z ) p µ − k µ ⊥ − k ⊥ − z n µ p · n , with 2 p i · p j = − k ⊥ z (1 − z ) , p = n = 0 . – 20 –n this p µ denotes the collinear momentum direction, and n µ is an auxiliary vector. Thecollinear limit is approached by k ⊥ → i k j limit of the four-parton antenna functions X ( i, j, k, l ),one chooses n = p k to be one of the non-collinear momenta, such that the antenna functioncan be expressed in terms of p , n , k ⊥ and p l . Expanding in k µ ⊥ yields only non-vanishingscalar products of the form p l · k ⊥ . Expressing the integral over the antenna phase spacein the ( p, n ) centre-of-mass frame, the angular average can be carried out as12 π Z π d φ ( p l · k ⊥ ) = 0 , π Z π d φ ( p l · k ⊥ ) = − k ⊥ p · p l n · p l p · n . (3.27)Higher powers of k µ ⊥ are not sufficiently singular to contribute to the collinear limit. Usingthe above average, we could analytically verify the cancellation of angular terms withineach single phase space mapping, which is independent on the choice of the reference vector n µ . The remainder of this section has been modified compared to the original version of thepaper:
In the N and N colour factor, the angular averaging is not sufficient to cancelthe 1 /ǫ poles in the four-parton one loop subtraction terms [B]. In either of these colourfactors, the difference d σ V, NNLO − d σ V S, NNLO contains left-over poles of the form1 ǫ X (1 , i, Y ( e i, j, e i ) J (3)3 ( e i, j, e i ) n s − ǫ e ij + s − ǫ e ij − s − ǫ j − s − ǫ j − s − ǫ i + s − ǫ o , (3.28)where X and Y are tree-level three-parton antenna functions. Contrary to statementsmade in [B], these terms do not appear in the colour factor N F N in our implementation.Furthermore, for these two colour factors the five-parton subtraction terms themselvesdo introduce spurious limits from large angle soft radiation. The single soft limit of i or k in (6.3) is non-vanishing. Instead, it yields (soft i ):+ 12 d (2 , k, j ) A (1 , f jk, f k ) h S i g ( jk ) + S i g (2 k ) − S g (2 k ) i g ( jk ) − S ij − S i + S ij i + (1 ↔ − A (1 , k, A ( g (1 k ) , j, g (2 k )) h S g (1 k ) ij + S g (2 k ) ij − S g (1 k ) i g (2 k ) − S ij − S ij + S i i (3.29)with S abc = 2 s ac s ab s bc (3.30)To account for this large angle soft radiation, a new subtraction term d σ ANNLO is intro-duced. This term is added to the five-parton subtraction term d σ SNNLO , and its integratedform is subtracted from the four-parton subtraction term d σ V S, NNLO , cancelling the left -over 1 /ǫ terms and adding new finite contributions to the four-parton and the five-partonsubtraction term.The new subtraction term d σ ANNLO contributes only in the N and N colour factors.Its contribution to N reads:d σ ANNLO,N = N N dΦ ( p , . . . , p ; q ) 13! X ( i,j,k ) ∈ P C (3 , , ( – 21 – 12 (cid:16) S ^ ((1 i ) k ) i ^ (( ji ) k ) − S g (1 i ) i g ( ji ) − S i ^ (( ji ) k ) + S i g ( ji ) − S i ^ ((1 i ) k ) + S i g (1 i ) (cid:17) × d ( g (1 i ) q , k g , g ( ji ) g ) A ( ^ ((1 i ) k ) q , ^ (( ji ) k ) g , ¯ q ) J (3)3 ( ] p (1 i ) k , ] p ( ji ) k , p )+ 12 (cid:16) S ^ ((1 k ) i ) k ^ (( jk ) i ) − S g (1 k ) k g ( jk ) − S k ^ (( jk ) i ) + S k g ( jk ) − S k ^ ((1 k ) i ) + S k g (1 k ) (cid:17) × d ( g (1 k ) ¯ q , i g , g ( jk ) g ) A ( ^ ((1 k ) i ) q , ^ (( jk ) i ) g , ¯ q ) J (3)3 ( ] p (1 k ) i , ] p ( jk ) i , p )+ 12 (cid:16) S ^ ((2 i ) k ) i ^ (( ji ) k ) − S g (2 i ) i g ( ji ) − S i ^ (( ji ) k ) + S i g ( ji ) − S i ^ ((2 i ) k ) + S i g (2 i ) (cid:17) × d ( g (2 i ) ¯ q , k g , g ( ji ) g ) A (1 q , ^ (( ji ) k ) g , ^ ((2 i ) k ) ¯ q ) J (3)3 ( p , ] p ( ji ) k , ] p (2 i ) k )+ 12 (cid:16) S ^ ((2 k ) i ) k ^ (( jk ) i ) − S g (2 k ) k g ( jk ) − S k ^ (( jk ) i ) + S k g ( jk ) − S k ^ ((2 k ) i ) + S k g (2 k ) (cid:17) × d ( g (2 k ) ¯ q , i g , g ( jk ) g ) A (1 q , ^ (( jk ) i ) g , ^ ((2 k ) i ) ¯ q ) J (3)3 ( p , ] p ( jk ) i , ] p (2 k ) i ) − (cid:16) S ^ ((1 i ) k ) i ^ ((2 i ) k ) − S ^ ((1 i ) k ) ij − S ^ ((2 i ) k ) ij + S g (1 i ) ij + S g (2 i ) ij − S g (1 i ) i g (2 i ) (cid:17) × A ( g (1 i ) q , k g , g (2 i ) ¯ q ) A ( ^ ((1 i ) k ) q , j g , ^ ((2 i ) k ) ¯ q ) J (3)3 ( ] p (1 i ) k , p j , ] p (2 i ) k ) − (cid:16) S ^ ((1 k ) i ) k ^ ((2 k ) i ) − S ^ ((1 k ) i ) kj − S ^ ((2 k ) i ) kj + S g (1 k ) kj + S g (2 k ) kj − S g (1 k ) k g (2 k ) (cid:17) × A ( g (1 k ) q , i g , g (2 k ) ¯ q ) A ( ^ ((1 k ) i ) q , j g , ^ ((2 k ) i ) ¯ q ) J (3)3 ( ] p (1 k ) i , p j , ] p (2 k ) i ) ) (3.31)The new contribution to the N five-parton subtraction term is:d σ ANNLO,N = N N dΦ ( p , . . . , p ; q )12 X ( i,j ) ∈ (3 , (cid:16) S ^ ((1 i )5) ij + S ^ ((2 i )5) ij − S ^ ((1 i )5) i ^ ((2 i )5) − S g (1 i ) ij − S g (2 i ) ij + S g (1 i ) i g (2 i ) (cid:17) × A ( g (1 i ) q , g , g (2 i ) ¯ q ) A ( ^ ((1 i )5) q , j g , ^ ((2 i )5) ¯ q ) J (3)3 ( ] p (1 i )5 , ] p (2 i )5 , p j ) (3.32)These large-angle soft subtraction terms contain soft antenna functions of the form S ajc which is simply the eikonal factor for a soft gluon j emitted between hard partons a and c .Those soft factors are associated with an antenna phase space mapping ( i, j, k ) → ( I, K ).The hard momenta a , c do not need to be equal to the hard momenta i , k in the antennaphase space - they can be arbitrary on-shell momenta.The integral of each of these soft antenna functions over the antenna phase space canbe written as S ac ; ik = Z dΦ X ijk S ajc = ( s IK ) − ǫ Γ (1 − ǫ ) e ǫγ Γ(1 − ǫ ) (cid:18) − ǫ (cid:19) (cid:20) − ǫ + ln ( x ac,IK ) + ǫ Li (cid:18) − − x ac,IK x ac,IK (cid:19)(cid:21) , (3.33)where we have defined x ac,IK = s ac s IK ( s aI + s aK )( s cI + s cK ) . (3.34)– 22 –O γ ∗ → q ¯ qg tree levelNLO γ ∗ → q ¯ qg one loop γ ∗ → q ¯ q gg tree level γ ∗ → q ¯ q q ¯ q tree levelNNLO γ ∗ → q ¯ qg two loop γ ∗ → q ¯ q gg one loop γ ∗ → q ¯ q q ¯ q one loop γ ∗ → q ¯ q q ¯ q g tree level γ ∗ → q ¯ q g g g tree level γ ∗ → ggg (one loop) Table 2:
The partonic channels contributing to e + e − → So that the integration of the new N subtraction terms reads Z dΦ X ijk d σ ANNLO,N = N N (cid:16) α s π (cid:17) dΦ ( p , . . . , p ; q ) × X ( i,j ) ∈ (3 , ( (cid:16) S g (1 i ) g ( ji );1 j − S j ;1 j − S g ( ji );1 j + S j ;1 j − S g (1 i );1 j + S j (cid:17) d (1 q , i g , j g ) A ( g (1 i ) q , g ( ji ) g , ¯ q ) J (3)3 ( f p i , f p ji , p ) + (1 ↔ − (cid:16) S g (1 i ) g (2 i );12 − S − S g (2 i ) j ;12 + S j ;12 − S g (1 i ) j ;12 + S j ;12 (cid:17) A (1 q , i g , ¯ q ) A ( g (1 i ) q , j g , g (2 i ) ¯ q ) J (3)3 ( f p i , p j , f p i ) ) (3.35)while for the N term the integration of the 5-parton contribution over the antenna phasespace yields Z dΦ X ijk d σ ANNLO,N = N N (cid:16) α s π (cid:17) dΦ ( p , p , p , p ; q )12 X ( i,j ) ∈ (3 , (cid:16) S g (1 i ) j ;12 + S g (2 i ) j ;12 − S g (1 i ) g (2 i );12 − S j ;12 − S j ;12 + S (cid:17) (3.36)
4. Parton-level contributions to e + e − → jets up to NNLO Three-jet production at tree-level is induced by the decay of a virtual photon (or otherneutral gauge boson) into a quark-antiquark-gluon final state. At higher orders, this processreceives corrections from extra real or virtual particles. The individual partonic channelsthat contribute through to NNLO are shown in Table 4.According to the structure of the coupling to the external vector boson, one distin-guishes non-singlet and singlet contributions. The non-singlet contributions arise from– 23 –he interference of amplitudes where the external gauge boson couples to the same quarklines, while the pure singlet contribution is due to the interference of amplitudes wherethe external gauge boson couples to different quark lines. Up to NLO, only non-singletcontributions appear. It is only at NNLO, that the first non-vanishing singlet terms areallowed. These appear in the tree-level γ ∗ → q ¯ q q ¯ q g process, the one-loop γ ∗ → q ¯ q gg and γ ∗ → q ¯ q q ¯ q processes and the two-loop γ ∗ → q ¯ qg process. All these processes yieldboth non-singlet and singlet contributions. The γ ∗ → ggg process, which is mediated by aclosed quark loop, is entirely a singlet contribution. In four-jet observables at O ( α s ), thesinglet contributions were found to be extremely small [52]. Also, the singlet contributionfrom three-gluon final states to three-jet observables was found to be negligible [53].Matrix elements and subtraction terms at NLO and NNLO can be naturally decom-posed according to their colour structure. The cross section at NLO receives contributionsfrom three different colour factors:d σ NLO = d σ NLO,N + d σ NLO, /N + d σ NLO,N F . (4.1)The NNLO contribution to the cross section receives contributions from seven differentcolour factors:d σ NNLO = d σ NNLO,N + d σ NNLO,N + d σ NNLO, /N +d σ NNLO,N F N + d σ NNLO,N F /N + d σ NNLO,N F + d σ NNLO,N
F,γ . (4.2)The first six terms in this equation are non-singlet contributions, the last term is thenumerically unimportant singlet contribution.In the following, we list the matrix elements for the contributing partonic channelsshown in Table 4 and discuss their structure. The tree-level amplitude M q ¯ q ( n − g for a virtual photon to produce a quark-antiquark pairand ( n − γ ∗ ( q ) → q ( p )¯ q ( p ) g ( p ) . . . g ( p n )can be expressed as sum over the permutations of the colour ordered amplitude M A,n ofthe possible orderings for the gluon colour indices M q ¯ q ( n − g = ie ( √ g ) n − X ( i,...,k ) ∈ P (3 ,...,n ) ( T a i · · · T a n ) i i M A,n ( p , p , . . . , p n , p ) . (4.3)The squared matrix elements for n = 3 , . . . ,
5, summed over gluon polarisations, butexcluding symmetry factors for identical particles, are given by, (cid:12)(cid:12) M q ¯ qg (cid:12)(cid:12) = N A (1 q , g , ¯ q ) , (4.4) (cid:12)(cid:12) M q ¯ qgg (cid:12)(cid:12) = N X ( i,j ) ∈ P (3 , N A (1 q , i g , j g , ¯ q ) − N ˜ A (1 q , g , g , ¯ q ) , (4.5)– 24 – (cid:12) M q ¯ qggg (cid:12)(cid:12) = N " X ( i,j,k ) ∈ P (3 , , (cid:16) N A (1 q , i g , j g , k g , ¯ q ) − ˜ A (1 q , i g , j g , k g , ¯ q ) (cid:17) + (cid:18) N + 1 N (cid:19) ¯ A (1 q , g , g , g , ¯ q ) , (4.6)where, N n = 4 πα X q e q (cid:0) g (cid:1) ( n − (cid:0) N − (cid:1) (cid:12)(cid:12) M q ¯ q (cid:12)(cid:12) , (4.7)and (cid:12)(cid:12) M q ¯ q (cid:12)(cid:12) = 4(1 − ǫ ) q . (4.8)The squared colour-ordered matrix elements A , A and ˜ A are given in [32]. For the fiveparton case [54], A (1 q , i g , j g , k g , ¯ q ) (cid:12)(cid:12) M q ¯ q (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) M A, ( p , p i , p j , p k , p ) (cid:12)(cid:12)(cid:12)(cid:12) (4.9)˜ A (1 q , i g , j g , k g , ¯ q ) (cid:12)(cid:12) M q ¯ q (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) M A, ( p , p i , p j , p k , p ) + M A, ( p , p i , p k , p j , p ) + M A, ( p , p k , p i , p j , p ) (cid:12)(cid:12)(cid:12)(cid:12) , (4.10)¯ A (1 q , i g , j g , k g , ¯ q ) (cid:12)(cid:12) M q ¯ q (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ( i,j,k ) ∈ P (3 ,..., M A, ( p , p i , p j , p k , p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (4.11)In the subleading colour contribution ˜ A , gluon k is effectively photon-like, while in the sub-subleading colour contribution (also called Abelian contribution), ¯ A , all three gluons areeffectively photon-like. Photon-like gluons do not couple to three- and four-gluon vertices,and there are no simple collinear limits as any two photon-like gluons become collinear. Asa consequence, the only colour-connected pair in ¯ A are the quark and antiquark.The tree-level amplitude for γ ∗ ( q ) → q ( p )¯ q ( p ) q ′ ( p )¯ q ′ ( p )is given by M q ¯ qq ′ ¯ q ′ = ie g δ q q δ q q (cid:18) δ i i δ i i − N δ i i δ i i (cid:19) M B, ( p , p , p , p )+(1 ↔ , ↔ , (4.12)where δ q q δ q q indicates the quark flavours. The amplitude M B, ( p , p , p , p ) thus de-notes the contribution from the q ¯ q –pair coupling to the vector boson. The identical quarkamplitude is obtained M q ¯ qq ¯ q = M q ¯ qq ′ ¯ q ′ − (2 ↔ . (4.13)– 25 –he resulting four-quark squared matrix elements, summed over final state quarkflavours and including symmetry factors are given by (cid:12)(cid:12) M q (cid:12)(cid:12) = X q,q ′ (cid:12)(cid:12) M q ¯ qq ′ ¯ q ′ (cid:12)(cid:12) + X q | M q ¯ qq ¯ q | = N " N F B (1 q , q , ¯ q , ¯ q ) − N (cid:0) C (1 q , q , ¯ q , ¯ q ) + C (2 ¯ q , ¯ q , q , q ) (cid:1) + N F,γ ˆ B (1 q , q , ¯ q , ¯ q ) , (4.14)where B (1 q , q , ¯ q , ¯ q ) (cid:12)(cid:12) M q ¯ q (cid:12)(cid:12) = (cid:12)(cid:12) M B, ( p , p , p , p ) (cid:12)(cid:12) ,C (1 q , q , ¯ q , ¯ q ) (cid:12)(cid:12) M q ¯ q (cid:12)(cid:12) = − Re (cid:18) M B, ( p , p , p , p ) M , † B, ( p , p , p , p ) (cid:19) , (4.15)ˆ B (1 q , q , ¯ q , ¯ q ) (cid:12)(cid:12) M q ¯ q (cid:12)(cid:12) = Re (cid:18) M B, ( p , p , p , p ) M , † B, ( p , p , p , p ) (cid:19) . (4.16)Explicit expressions for B and C are given in [32]. The last term, ˆ B , is proportional tothe charge weighted sum of the quark flavours, N F,γ , which for electromagnetic interactionsis given by, N F,γ = ( P q e q ) P q e q . (4.17)It is relevant only for observables where the final state quark charge can be determined.There are four colour structures in the tree-level amplitude for γ ∗ ( q ) → q ( p )¯ q ( p ) q ′ ( p )¯ q ′ ( p ) g ( p )which reads M q ¯ qq ′ ¯ q ′ g = ie g √ δ q q δ q q × (cid:20) T a i i δ i i M ,aB, ( p , p , p , p , p ) − N T a i i δ i i M ,cB, ( p , p , p , p , p )+ T a i i δ i i M ,bB, ( p , p , p , p , p ) − N T a i i δ i i M ,dB, ( p , p , p , p , p ) (cid:21) +(1 ↔ , ↔ . (4.18)The amplitude M ,xB, ( p , p , p , p , p ) for x = a, . . . , d denotes the contribution from the q ¯ q –pair coupling to the vector boson. Due to the colour decomposition, the followingrelation holds between the leading and subleading colour amplitudes: M ,eB, ( p , p , p , p , p ) = M ,aB, ( p , p , p , p , p ) + M ,bB, ( p , p , p , p , p )= M ,cB, ( p , p , p , p , p ) + M ,dB, ( p , p , p , p , p ) . (4.19)– 26 –s before, the identical quark matrix element is obtained by permuting the antiquarkmomenta, M q ¯ qq ¯ qg = M q ¯ qq ′ ¯ q ′ g − (2 ↔ . (4.20)The squared matrix element, summed over flavours and including symmetry factors isgiven by, (cid:12)(cid:12) M qg (cid:12)(cid:12) = X q,q ′ (cid:12)(cid:12) M q ¯ qq ′ ¯ q ′ g (cid:12)(cid:12) + X q | M q ¯ qq ¯ qg | = N " N N F (cid:16) B ,a (1 q , g , ¯ q ′ ; 3 q ′ , ¯ q ) + B ,b (1 q , ¯ q ′ ; 3 q ′ , g , ¯ q ) (cid:17) + N F N (cid:16) B ,c (1 q , g , ¯ q ; 3 q ′ , ¯ q ′ ) + B ,d (1 q , ¯ q ; 3 q ′ , g , ¯ q ′ ) − B ,e (1 q , ¯ q ; 3 q ′ , ¯ q ′ ; 5 g ) (cid:17) − C (1 q , q , ¯ q , g , ¯ q ) + (cid:18) N + 1 N (cid:19) (cid:16) ˜ C (1 q , q , ¯ q , g , ¯ q ) + ˜ C (2 ¯ q , ¯ q , q , g , q ) (cid:17) − N N
F,γ (cid:16) ˆ B ,a (1 q , g , ¯ q ′ ; 3 q ′ , ¯ q ) + ˆ B ,b (1 q , ¯ q ′ ; 3 q ′ , g , ¯ q ) − ˆ B ,e (1 q , ¯ q ′ ; 3 q ′ , ¯ q , g ) (cid:17) + N F,γ N (cid:16) ˆ B ,c (1 q , g , ¯ q ; 3 q ′ , ¯ q ′ ) + ˆ B ,d (1 q , ¯ q ; 3 q ′ , g , ¯ q ′ ) + ˆ B ,e (1 q , ¯ q ; 3 q ′ , ¯ q ′ ; 5 g ) (cid:17) , (4.21)where for x = a, . . . , eB ,x ( . . . ) (cid:12)(cid:12) M q ¯ q (cid:12)(cid:12) = |M ,xB, ( p , p , p , p , p ) | , (4.22)ˆ B ,x ( . . . ) (cid:12)(cid:12) M q ¯ q (cid:12)(cid:12) = Re (cid:16) M ,xB, ( p , p , p , p , p ) M ,x, † B, ( p , p , p , p , p ) (cid:17) , (4.23)and C (1 q , q , ¯ q , g , ¯ q ) (cid:12)(cid:12) M q ¯ q (cid:12)(cid:12) = − (cid:18) M ,aB, ( p , p , p , p , p ) M ,c, † B, ( p , p , p , p , p )+ M ,bB, ( p , p , p , p , p ) M ,d, † B, ( p , p , p , p , p )+ M ,aB, ( p , p , p , p , p ) M ,d, † B, ( p , p , p , p , p )+ M ,bB, ( p , p , p , p , p ) M ,c, † B, ( p , p , p , p , p ) (cid:19) , (4.24)˜ C (1 q , q , ¯ q , g , ¯ q ) (cid:12)(cid:12) M q ¯ q (cid:12)(cid:12) = − Re (cid:18) M ,eB, ( p , p , p , p , p ) M ,e, † B, ( p , p , p , p , p ) (cid:19) . (4.25) The renormalised one-loop amplitude M q ¯ qg for a virtual photon to produce a quark-antiquark pair together with a single gluon, γ ∗ ( q ) → q ( p )¯ q ( p ) g ( p )– 27 –ontains a single colour structure such that M q ¯ qg = ie √ g (cid:18) g π (cid:19) T a i i M A, ( p , p , p ) . (4.26)Unless stated otherwise, the renormalisation scale is set to µ = q .The interference of the one-loop amplitude with the three-parton tree-level amplitude(4.3) is given by 2Re (cid:16) M , † q ¯ qg M q ¯ qg (cid:17) = N (cid:16) α s π (cid:17) A (1 × (1 q , g , ¯ q ) , (4.27)where A (1 × (1 q , g , ¯ q ) = (cid:18) N (cid:2) A (1 q , g , ¯ q ) + A ( s ) A (1 q , g , ¯ q ) (cid:3) − N h ˜ A (1 q , g , ¯ q ) + A ( s ) A (1 q , g , ¯ q ) i + N F ˆ A (1 q , g , ¯ q ) (cid:19) , (4.28)where A , ˜ A and ˆ A are given up to O ( ǫ ) in [32].Moreover, the one-loop process γ ∗ → ggg also yields three-jet final states. Since thisprocess has no tree-level counterpart, it does only contribute at NNLO. Its amplitude canbe denoted as [53] M ggg = i X q e q √ g (cid:18) g π (cid:19) d a a a M C, ( p , p , p ) . (4.29)The one-loop corrections to γ ∗ → γ ∗ ( q ) → q ( p )¯ q ( p ) g ( p ) g ( p )contains two colour structures, M q ¯ qgg = ie g (cid:18) g π (cid:19) × " X ( i,j ) ∈ P (3 , ( T a i T a j ) i i (cid:18) N M ,aA, ( p , p i , p j , p ) − N M ,bA, ( p , p i , p j , p )+ N F M ,cA, ( p , p i , p j , p ) + P q e q e M ,eA, ( p , p i , p j , p ) (cid:19) + 12 δ a i a j δ i i M ,dA, ( p , p , p , p ) , (4.30)where M ,dA, ( p , p , p , p ) = M ,dA, ( p , p , p , p ) . (4.31)– 28 –he “squared” matrix element is the interference between the tree-level and one-loopamplitudes,2 (cid:12)(cid:12)(cid:12) M , † q ¯ qgg M q ¯ qgg (cid:12)(cid:12)(cid:12) = N (cid:16) α s π (cid:17) × " X ( i,j ) ∈ P (3 , (cid:18) N A ,a (1 q , i g , j g , ¯ q ) − A ,b (1 q , i g , j g , ¯ q ) + N N F A ,c (1 q , i g , j g , ¯ q )+ N N
F,γ A ,e (1 q , i g , j g , ¯ q ) (cid:19) − ˜ A ,a (1 q , g , g , ¯ q ) − ˜ A ,d (1 q , g , g , ¯ q ) − N ˜ A ,b (1 q , g , g , ¯ q )+ N F N ˜ A ,c (1 q , g , g , ¯ q ) + N F,γ N ˜ A ,e (1 q , g , g , ¯ q ) ! , (4.32)where for x = a, . . . , d , A ,x (1 q , i g , j g , ¯ q ) (cid:12)(cid:12) M q ¯ q (cid:12)(cid:12) = Re (cid:16) M ,xA, ( p , p i , p j , p ) M , † A, ( p , p i , p j , p ) (cid:17) , (4.33)˜ A ,x (1 q , g , g , ¯ q ) (cid:12)(cid:12) M q ¯ q (cid:12)(cid:12) = Re (cid:16) ˜ M ,xA, ( p , p , p , p ) ˜ M , † A, ( p , p , p , p ) (cid:17) , (4.34)and ˜ M ,xA, ( p , p , p , p ) = M ,xA, ( p , p , p , p ) + M ,xA, ( p , p , p , p ) . (4.35)The renormalised singularity structure of the various contributions can be easily writ-ten in terms of the tree-level squared matrix elements multiplied by combinations of infraredsingularity operators [55], for which we use the notation defined in [32]. Explicitly, we find P oles ( A ,a (1 q , i g , j g , ¯ q )) = 2 (cid:16) I (1) qg ( ǫ, s i ) + I (1) gg ( ǫ, s ij ) + I (1) g ¯ q ( ǫ, s j ) (cid:17) A (1 q , i g , j g , ¯ q ) , (4.36) P oles ( A ,b (1 q , i g , j g , ¯ q )) = 2 I (1) q ¯ q ( ǫ, s ) A (1 q , i g , j g , ¯ q ) , (4.37) P oles ( A ,c (1 q , i g , j g , ¯ q )) = 2 (cid:16) I (1) qg,F ( ǫ, s i ) + I (1) gg,F ( ǫ, s ij ) + I (1) g ¯ q,F ( ǫ, s j ) (cid:17) A (1 q , i g , j g , ¯ q ) , (4.38) P oles ( ˜ A ,a (1 q , g , g , ¯ q )) = (cid:16) I (1) gg ( ǫ, s ) + I (1) qg ( ǫ, s ) + I (1) g ¯ q ( ǫ, s ) + I (1) qg ( ǫ, s ) + I (1) g ¯ q ( ǫ, s ) (cid:17) ˜ A (1 q , g , g , ¯ q ) , (4.39) P oles ( ˜ A ,b (1 q , g , g , ¯ q )) = 2 I (1) q ¯ q ( ǫ, s ) ˜ A (1 q , g , g , ¯ q ) , (4.40) P oles ( ˜ A ,c (1 q , g , g , ¯ q )) = (cid:16) I (1) gg,F ( ǫ, s ) + I (1) qg,F ( ǫ, s ) + I (1) g ¯ q,F ( ǫ, s ) + I (1) qg,F ( ǫ, s ) + I (1) g ¯ q,F ( ǫ, s ) (cid:17) × ˜ A (1 q , g , g , ¯ q ) , (4.41) P oles ( ˜ A ,d (1 q , g , g , ¯ q )) = (cid:16) I (1) q ¯ q ( ǫ, s ) + 2 I (1) gg ( ǫ, s ) − I (1) qg ( ǫ, s ) − I (1) g ¯ q ( ǫ, s ) − I (1) qg ( ǫ, s ) − I (1) g ¯ q ( ǫ, s ) (cid:17) × ˜ A (1 q , g , g , ¯ q ) . (4.42)– 29 –s at tree-level, the one-loop amplitude for γ ∗ ( q ) → q ( p )¯ q ( p ) q ′ ( p )¯ q ′ ( p )contains two colour structures, M q ¯ qq ′ ¯ q ′ = ie g (cid:18) g π (cid:19) δ q q δ q q × " δ i i δ i i (cid:18) N M ,aB, ( p , p , p , p ) − N M ,bB, ( p , p , p , p ) + N F M ,cB, ( p , p , p , p ) (cid:19) − N δ i i δ i i N M ,dB, ( p , p , p , p ) − N M ,eB, ( p , p , p , p )+ N F M ,fB, ( p , p , p , p ) ! + (1 ↔ , ↔ , (4.43)where M ,aB, ( p , p , p , p ) + M ,eB, ( p , p , p , p ) = M ,bB, ( p , p , p , p ) + M ,dB, ( p , p , p , p ) . (4.44)As before, the identical quark matrix element is obtained by permuting the antiquarkmomenta, M q ¯ qq ¯ q = M q ¯ qq ′ ¯ q ′ − (2 ↔ . (4.45)Summing over flavours and including symmetry factors, we find that the “squared”matrix element, is given by2 (cid:12)(cid:12)(cid:12) M , † q M q (cid:12)(cid:12)(cid:12) = X q,q ′ (cid:12)(cid:12)(cid:12) M , † q ¯ qq ′ ¯ q ′ M q ¯ qq ′ ¯ q ′ (cid:12)(cid:12)(cid:12) + X q (cid:12)(cid:12)(cid:12) M , † q ¯ qq ¯ q M q ¯ qq ¯ q (cid:12)(cid:12)(cid:12) = N (cid:16) α s π (cid:17) " N N F B ,a (1 q , q , ¯ q , ¯ q ) − N F N B ,b (1 q , q , ¯ q , ¯ q ) + N F B ,c (1 q , q , ¯ q , ¯ q ) − C ,d (1 q , q , ¯ q , ¯ q ) + 1 N C ,e (1 q , q , ¯ q , ¯ q ) − N F N C ,f (1 q , q , ¯ q , ¯ q ) − C ,d (2 ¯ q , ¯ q , q , q ) + 1 N C ,e (2 ¯ q , ¯ q , q , q ) − N F N C ,f (2 ¯ q , ¯ q , q , q )+ N N
F,γ ˆ B ,a (1 q , q , ¯ q , ¯ q ) − N F,γ N ˆ B ,b (1 q , q , ¯ q , ¯ q ) + N F N F,γ ˆ B ,c (1 q , q , ¯ q , ¯ q ) , (4.46)where for x = a, b, cB ,x (1 q , q , ¯ q , ¯ q ) (cid:12)(cid:12) M q ¯ q (cid:12)(cid:12) = Re (cid:16) M ,xB, ( p , p , p , p ) M , † B, ( p , p , p , p ) (cid:17) , (4.47)ˆ B ,x (1 q , q , ¯ q , ¯ q ) (cid:12)(cid:12) M q ¯ q (cid:12)(cid:12) = Re (cid:16) M ,xB, ( p , p , p , p ) M , † B, ( p , p , p , p ) (cid:17) , (4.48)and for x = d, e, fC ,x (1 q , q , ¯ q , ¯ q ) (cid:12)(cid:12) M q ¯ q (cid:12)(cid:12) = − Re (cid:16) M ,xB, ( p , p , p , p ) M , † B, ( p , p , p , p ) (cid:17) . (4.49)– 30 –sing the infrared singularity operators of [55], we can extract the singular contribu-tions of the renormalised one-loop contribution as, P oles ( B ,a (1 q , q , ¯ q , ¯ q )) = 2 (cid:16) I (1) q ¯ q ( ǫ, s ) + I (1) q ¯ q ( ǫ, s ) (cid:17) B (1 q , q , ¯ q , ¯ q ) , (4.50) P oles ( B ,b (1 q , q , ¯ q , ¯ q )) =2 (cid:16) I (1) q ¯ q ( ǫ, s ) − I (1) q ¯ q ( ǫ, s ) + 2 I (1) q ¯ q ( ǫ, s ) − I (1) q ¯ q ( ǫ, s ) + I (1) q ¯ q ( ǫ, s ) + I (1) q ¯ q ( ǫ, s ) (cid:17) × B (1 q , q , ¯ q , ¯ q ) , (4.51) P oles ( C ,d (1 q , q , ¯ q , ¯ q )) = 2 (cid:16) I (1) q ¯ q ( ǫ, s ) + I (1) q ¯ q ( ǫ, s ) (cid:17) C (1 q , q , ¯ q , ¯ q ) , (4.52) P oles ( C ,e (1 q , q , ¯ q , ¯ q )) =2 (cid:16) I (1) q ¯ q ( ǫ, s ) + I (1) q ¯ q ( ǫ, s ) + I (1) q ¯ q ( ǫ, s ) + I (1) q ¯ q ( ǫ, s ) − I (1) q ¯ q ( ǫ, s ) − I (1) q ¯ q ( ǫ, s ) (cid:17) × C (1 q , q , ¯ q , ¯ q ) , (4.53) P oles ( ˆ B ,a (1 q , q , ¯ q , ¯ q )) = 2 (cid:16) I (1) q ¯ q ( ǫ, s ) + I (1) q ¯ q ( ǫ, s ) (cid:17) ˆ B (1 q , q , ¯ q , ¯ q ) , (4.54) P oles ( ˆ B ,b (1 q , q , ¯ q , ¯ q )) =2 (cid:16) I (1) q ¯ q ( ǫ, s ) − I (1) q ¯ q ( ǫ, s ) + 2 I (1) q ¯ q ( ǫ, s ) − I (1) q ¯ q ( ǫ, s ) + I (1) q ¯ q ( ǫ, s ) + I (1) q ¯ q ( ǫ, s ) (cid:17) × ˆ B (1 q , q , ¯ q , ¯ q ) . (4.55) The renormalised two-loop amplitude M q ¯ qg for a virtual photon to produce a quark-antiquark pair together with a single gluon, γ ∗ ( q ) → q ( p )¯ q ( p ) g ( p )contains a single colour structure such that M q ¯ qg = ie √ g (cid:18) g π (cid:19) T a i i M A, ( p , p , p ) . (4.56)At NNLO, there are two contributions. One from the interference of the two-loop andtree-level amplitudes (4.3), the other from the square of the one-loop amplitudes givenin (4.26). These terms were computed in [18] by reducing the large number of two-loopFeynman integrals to a small number of master integrals, using integration-by-parts [56]and Lorentz-invariance [57] identities, solved using the Laporta algorithm [58]. The relevantmaster integrals (two-loop four-point functions with one off-shell leg) were then derived [59]from their differential equations [57, 60, 61].The resulting virtual three-parton contributions are given by,2Re (cid:16) M , † q ¯ qg M q ¯ qg (cid:17) = N (cid:16) α s π (cid:17) A (2 × (1 q , g , ¯ q ) , (4.57)Re (cid:16) M , † q ¯ qg M q ¯ qg (cid:17) = N (cid:16) α s π (cid:17) A (1 × (1 q , g , ¯ q ) . (4.58)– 31 –ollowing [55], we organise the infrared pole structure of the NNLO contributions renor-malised in the MS scheme in terms of the tree and renormalised one-loop amplitudes suchthat, P oles (cid:16) A (2 × (1 q , g , ¯ q ) + A (1 × (1 q , g , ¯ q ) (cid:17) = 2 (cid:20) − (cid:16) I (1) q ¯ qg ( ǫ ) (cid:17) − β ǫ I (1) q ¯ qg ( ǫ )+ e − ǫγ Γ(1 − ǫ )Γ(1 − ǫ ) (cid:18) β ǫ + K (cid:19) I (1) q ¯ qg (2 ǫ ) + H (2) q ¯ qg (cid:21) A (1 q , g , ¯ q )+2 I (1) q ¯ qg ( ǫ ) A (1 × (1 q , g , ¯ q ) . (4.59)Here, I (1) q ¯ qg ( ǫ ) = N (cid:16) I (1) qg ( ǫ, s ) + I (1) qg ( ǫ, s ) (cid:17) − N I (1) q ¯ q ( ǫ, s )+ N F (cid:16) I (1) qg,F ( ǫ, s ) + I (1) qg,F ( ǫ, s ) (cid:17) , (4.60)with the individual I (1) ij defined in [32] and H (2) q ¯ qg = e ǫγ ǫ Γ(1 − ǫ ) " (cid:18) ζ + 589432 − π (cid:19) N + (cid:18) − ζ − − π (cid:19) + (cid:18) − ζ −
316 + π (cid:19) N + (cid:18) − π (cid:19) N N F + (cid:18) − − π (cid:19) N F N + 527 N F . . (4.61)We denote the finite contributions as, F inite ( A (2 × (1 q , g , ¯ q )) = N A (2 × ,finite ,N + A (2 × ,finite , + 1 N A (2 × ,finite , /N + N N F A (2 × ,finite ,NN F + N F N A (2 × ,finite ,N F /N + N F A (2 × ,finite ,N F + N F,γ (cid:18) N − N (cid:19) A (2 × ,finite ,N F,γ , (4.62) F inite ( A (1 × (1 q , g , ¯ q )) = N A (1 × ,finite ,N + A (1 × ,finite , + 1 N A (1 × ,finite , /N + N N F A (1 × ,finite ,NN F + N F N A (1 × ,finite ,N F /N + N F A (1 × ,finite ,N F . (4.63)Explicit formulae for the finite remainders have been given in [18]. These are expressedin terms of one-dimensional and two-dimensional harmonic polylogarithms (HPLs and2dHPLs) [59, 62], which are generalisations of the well-known Nielsen polylogarithms [63].A numerical implementation, which is required for all practical applications, is availablefor HPLs and 2dHPLs [64]. – 32 –inally, a finite NNLO contribution arises from the squared one-loop amplitude (4.29)for γ ∗ → ggg : C (1 × (1 g , g , g ) = N F,γ (cid:18) N − N (cid:19) C (1 × ,finite ,N F,γ . (4.64)
5. Construction of the NLO subtraction term
Three-jet production at the leading order is given by:d σ RLO = N dΦ ( p , p , p ; q ) A (1 q , g , ¯ q ) J (3)3 ( p , p , p ) . (5.1)This leading order cross section defines the normalisation for all higher order correctionsdiscussed in the following.At NLO, the tree-level four-parton processes γ ∗ → q ¯ qgg , γ ∗ → q ¯ qq ′ ¯ q ′ (non-identicalquarks) and γ ∗ → q ¯ qq ¯ q (identical quarks) yield three-jet final states. Only the two formerprocesses require subtraction, since the third process is infrared finite.The four-parton real radiation contribution to the NLO cross section isd σ RNLO = N dΦ ( p , . . . , p ; q ) × ( N X ( i,j ) ∈ P (3 , A (1 q , i g , j g , ¯ q ) − N ˜ A (1 q , g , g , ¯ q ) + N F B (1 q , q , ¯ q , ¯ q ) − N (cid:0) C (1 q , q , ¯ q , ¯ q ) + C (2 ¯ q , ¯ q , q , q ) (cid:1) ) J (4)3 ( p , . . . , p ) . (5.2)The antenna subtraction term is then constructed as:d σ SNLO = N dΦ ( p , . . . , p ; q ) × ( X ( i,j ) ∈ P (3 , (cid:20) N d (1 q , i g , j g ) A ( g (1 i ) q , g ( ji ) g , ¯ q ) J (3)3 ( f p i , f p ji , p )+ N d (2 ¯ q , i g , j g ) A (1 q , g ( ji ) g , g (2 i ) ¯ q ) J (3)3 ( p , f p ji , f p i ) − N A (1 q , i g , ¯ q ) A ( g (1 i ) q , j g , g (2 i ) ¯ q ) J (3)3 ( f p i , p j , f p i ) (cid:21) + N F (cid:20) E (1 q , q ′ , ¯ q ′ ) A ( g (13) q , g (43) g , ¯ q ) J (3)3 ( f p , f p , p )+ E (2 ¯ q , q ′ , ¯ q ′ ) A (1 q , g (34) g , g (24) ¯ q ) J (3)3 ( p , f p , f p ) (cid:21)) . (5.3)Integration of this subtraction term over the antenna phase spaces yields:d σ SNLO = N (cid:16) α s π (cid:17) dΦ ( p , . . . , p ; q ) × (cid:26) N (cid:2) D ( s ) + D ( s ) (cid:3) − N A ( s ) + N F (cid:2) E ( s ) + E ( s ) (cid:3)(cid:27) × A (1 q , g , ¯ q ) J (3)3 ( p , p , p ) (5.4)– 33 –ogether with the virtual one-loop contribution to γ ∗ → q ¯ qg ,d σ VNLO = N (cid:16) α s π (cid:17) dΦ ( p , . . . , p ; q ) J (3)3 ( p , p , p ) × (cid:18) N (cid:2) A (1 q , g , ¯ q ) + A ( s ) A (1 q , g , ¯ q ) (cid:3) − N h ˜ A (1 q , g , ¯ q ) + A ( s ) A (1 q , g , ¯ q ) i + N F ˆ A (1 q , g , ¯ q ) (cid:19) , (5.5)one obtains P oles (cid:0) d σ SNLO (cid:1) + P oles (cid:0) d σ VNLO (cid:1) = 0 , (5.6)thus yielding an infrared-finite result.
6. Construction of the N colour factor The N colour factor receives contributions from five-parton tree-level γ ∗ → q ¯ qggg , four-parton one-loop γ ∗ → q ¯ qgg and tree-level two-loop γ ∗ → q ¯ qg . The multiple gluon emissionsare colour-ordered, and the squared matrix elements do not contain interference amplitudesbetween different orderings. In the loop contributions to this colour factor, non-planarmomentum arrangements are absent. At leading colour, the five parton contribution to three-jet final states arises from the colour-ordered emission of three gluons in γ ∗ → q ¯ qggg . The matrix element for one ordering is: (cid:12)(cid:12) M q ¯ q g (cid:12)(cid:12) = N N A (1 q , i g , j g , k g , ¯ q ) . (6.1)The real radiation contribution to the cross section is obtained by averaging over all possiblesix orderings:d σ RNNLO,N = N N dΦ ( p , . . . , p ; q ) 13! X ( i,j,k ) ∈ P (3 , , A (1 q , i g , j g , k g , ¯ q ) J (5)3 ( p , . . . , p )= N N dΦ ( p , . . . , p ; q ) 13! X ( i,j,k ) ∈ P C (3 , , (cid:2) A (1 q , i g , j g , k g , ¯ q ) + A (1 q , k g , j g , i g , ¯ q ) (cid:3) J (5)3 ( p , . . . , p ) , (6.2)where the second expression is obtained by restricting the summation to the three cyclicpermutations of the gluon momenta, while making the corresponding three non-cyclic per-mutations explicit. The cyclic form (6.2) is more appropriate for the construction of thereal radiation subtraction term, since this form matches onto the full quark-gluon antennafunctions of [32], which have a cyclic ambiguity in their momentum arrangements.– 34 –he real radiation subtraction term for this colour factor readsd σ SNNLO,N = N N dΦ ( p , . . . , p ; q ) 13! X ( i,j,k ) ∈ P C (3 , , ( d (1 q , i g , j g ) A ( g (1 i ) q , g ( ji ) g , k g , ¯ q ) J (4)3 ( f p i , f p ji , p k , p )+ f ( i g , j g , k g ) A (1 q , g ( ij ) g , g ( kj ) g , ¯ q ) J (4)3 ( p , f p ij , f p kj , p )+ d (2 ¯ q , k g , j g ) A (1 q , i g , g ( jk ) g , g (2 k ) ¯ q ) J (4)3 ( p , p i , f p jk , f p k )+ d (1 q , k g , j g ) A ( g (1 k ) q , g ( jk ) g , i g , ¯ q ) J (4)3 ( f p k , f p jk , p i , p )+ f ( k g , j g , i g ) A (1 q , g ( kj ) g , g ( ij ) g , ¯ q ) J (4)3 ( p , f p kj , f p ij , p )+ d (2 ¯ q , i g , j g ) A (1 q , k g , g ( ji ) g , g (2 i ) ¯ q ) J (4)3 ( p , p k , f p ji , f p i )+ D ,a (1 q , i g , j g , k g ) − d (1 q , i g , j g ) d ( g (1 i ) q , g ( ji ) g , k g ) − f ( i g , j g , k g ) d (1 q , g ( ij ) g , g ( kj ) g ) ! A ( ] (1 ij ) q , ] ( kji ) g , ¯ q ) J (3)3 ( g p ij , g p kji , p )+ D ,b (1 q , i g , j g , k g ) − f ( i g , j g , k g ) d (1 q , g ( kj ) g , g ( ij ) g ) − d (1 q , k g , j g ) d ( g (1 k ) q , g ( jk ) g , i g ) ! A ( ] (1 kj ) q , ] ( ijk ) g , ¯ q ) J (3)3 ( g p kj , g p ijk , p )+ D ,c (1 q , i g , j g , k g ) − d (1 q , i g , j g ) d ( g (1 i ) q , k g , g ( ji ) g ) ! A ( ] (1 ik ) q , ] ( jki ) g , ¯ q ) J (3)3 ( g p ik , g p jki , p )+ D ,d (1 q , i g , j g , k g ) − d (1 q , k g , j g ) d ( g (1 k ) q , i g , g ( jk ) g ) ! A ( ] (1 ki ) q , ] ( jik ) g , ¯ q ) J (3)3 ( g p ki , g p jik , p )+ D ,a (2 q , i g , j g , k g ) − d (2 q , i g , j g ) d ( g (2 i ) q , g ( ji ) g , k g ) − f ( i g , j g , k g ) d (2 q , g ( ij ) g , g ( kj ) g ) ! A ( ] (2 ij ) q , ] ( kji ) g , ¯ q ) J (3)3 ( g p ij , g p kji , p )+ D ,b (2 q , i g , j g , k g ) − f ( i g , j g , k g ) d (2 q , g ( kj ) g , g ( ij ) g ) − d (2 q , k g , j g ) d ( g (2 k ) q , g ( jk ) g , i g ) ! A ( ] (2 kj ) q , ] ( ijk ) g , ¯ q ) J (3)3 ( g p kj , g p ijk , p )– 35 – D ,c (2 q , i g , j g , k g ) − d (2 q , i g , j g ) d ( g (2 i ) q , k g , g ( ji ) g ) ! A ( ] (2 ik ) q , ] ( jki ) g , ¯ q ) J (3)3 ( g p ik , g p jki , p )+ D ,d (2 q , i g , j g , k g ) − d (2 q , k g , j g ) d ( g (2 k ) q , i g , g ( jk ) g ) ! A ( ] (2 ki ) q , ] ( jik ) g , ¯ q ) J (3)3 ( g p ki , g p jik , p ) − ˜ A (1 q , i g , k g , ¯ q ) − A (1 q , i g , ¯ q ) A ( g (1 i ) q , k g , ] (2 i ) ¯ q ) − A (1 q , k g , ¯ q ) A ( g (1 k ) q , i g , ] (2 k ) ¯ q ) ! A ( ] (1 ik ) q , j g , ] (2 ki ) ¯ q ) J (3)3 ( g p ik , p j , g p ki ) − d (1 q , i g , j g ) d (2 ¯ q , k g , g ( ji ) g ) A ( g (1 i ) q , ^ (( ji ) k ) g , g (2 k ) ¯ q ) J (3)3 ( f p i , ] p ( ji ) k , f p k ) − d (2 ¯ q , k g , j g ) d (1 q , i g , g ( jk ) g ) A ( g (1 i ) q , ^ (( jk ) i ) g , g (2 k ) ¯ q ) J (3)3 ( f p i , ] p ( jk ) i , f p k ) − d (1 q , k g , j g ) d (2 ¯ q , i g , g ( jk ) g ) A ( g (1 k ) q , ^ (( jk ) i ) g , g (2 i ) ¯ q ) J (3)3 ( f p k , ] p ( jk ) i , f p i ) − d (2 ¯ q , i g , j g ) d (1 q , k g , g ( ji ) g ) A ( g (1 k ) q , ^ (( ji ) k ) g , g (2 i ) ¯ q ) J (3)3 ( f p k , ] p ( ji ) k , f p i )+ 12 d (1 q , i g , j g ) d ( g (1 i ) q , k g , g ( ji ) g ) A ( ^ ((1 i ) k ) q , ^ (( ji ) k ) g , ¯ q ) J (3)3 ( ] p (1 i ) k , ] p ( ji ) k , p )+ 12 d (1 q , k g , j g ) d ( g (1 k ) q , i g , g ( jk ) g ) A ( ^ ((1 k ) i ) q , ^ (( jk ) i ) g , ¯ q ) J (3)3 ( ] p (1 k ) i , ] p ( jk ) i , p )+ 12 d (2 ¯ q , i g , j g ) d ( g (2 i ) ¯ q , k g , g ( ji ) g ) A (1 q , ^ (( ji ) k ) g , ^ ((2 i ) k ) ¯ q ) J (3)3 ( p , ] p ( ji ) k , ] p (2 i ) k )+ 12 d (2 ¯ q , k g , j g ) d ( g (2 k ) ¯ q , i g , g ( jk ) g ) A (1 q , ^ (( jk ) i ) g , ^ ((2 k ) i ) ¯ q ) J (3)3 ( p , ] p ( jk ) i , ] p (2 k ) i ) − A (1 q , i g , ¯ q ) d ( g (1 i ) q , k g , j g ) A ( ^ ((1 i ) k ) q , g ( jk ) g , g (2 i ) ¯ q ) J (3)3 ( ] p (1 i ) k , f p jk , f p i )+ 12 d (1 q , k g , j g ) A ( g (1 k ) q , i g , ¯ q ) A ( ^ ((1 k ) i ) q , g ( jk ) g , g (2 i ) ¯ q ) J (3)3 ( ] p (1 k ) i , f p jk , f p i ) − A (1 q , k g , ¯ q ) d ( g (1 k ) q , i g , j g ) A ( ^ ((1 k ) i ) q , g ( ji ) g , g (2 k ) ¯ q ) J (3)3 ( ] p (1 k ) i , f p ji , f p k )+ 12 d (1 q , i g , j g ) A ( g (1 i ) q , k g , ¯ q ) A ( ^ ((1 i ) k ) q , g ( ji ) g , g (2 k ) ¯ q ) J (3)3 ( ] p (1 i ) k , f p ji , f p k ) − A (1 q , i g , ¯ q ) d ( g (2 i ) ¯ q , k g , j g ) A ( g (1 i ) q , g ( jk ) g , ^ ((2 i ) k ) ¯ q ) J (3)3 ( f p i , f p jk , ] p (2 i ) k )+ 12 d (2 ¯ q , k g , j g ) A (1 q , i g , g (2 k ) ¯ q ) A ( g (1 i ) q , g ( jk ) g , ^ ((2 k ) i ) ¯ q ) J (3)3 ( f p i , f p jk , ] p (2 k ) i ) − A (1 q , k g , ¯ q ) d ( g (2 k ) ¯ q , i g , j g ) A ( g (1 k ) q , g ( ji ) g , ^ ((2 k ) i ) ¯ q ) J (3)3 ( f p k , f p ji , ] p (2 k ) i )+ 12 d (2 ¯ q , i g , j g ) A (1 q , k g , g (2 i ) ¯ q ) A ( g (1 k ) q , g ( ji ) g , ^ ((2 i ) k ) ¯ q ) J (3)3 ( f p k , f p ji , ] p (2 i ) k )– 36 – A (1 q , k g , ¯ q ) A ( g (1 k ) q , i g , g (2 k ) ¯ q ) A ( ^ ((1 k ) i ) q , j g , ^ ((2 k ) i ) ¯ q ) J (3)3 ( ] p (1 k ) i , p j , ] p (2 k ) i ) − A (1 q , i g , ¯ q ) A ( g (1 i ) q , k g , g (2 i ) ¯ q ) A ( ^ ((1 i ) k ) q , j g , ^ ((2 i ) k ) ¯ q ) J (3)3 ( ] p (1 i ) k , p j , ] p (2 i ) k ) ) . (6.3) The leading colour four-parton contribution comes from the one-loop correction to γ ∗ → q ¯ qgg , where the gluonic emissions are colour-ordered. It readsd σ V, NNLO,N = N N (cid:16) α s π (cid:17) dΦ ( p , . . . , p ; q ) × X ( i,j ) ∈ (3 , A ,a (1 q , i g , j g , ¯ q ) J (4)3 ( p , p , p , p ) , (6.4)The one-loop single unresolved subtraction term for this colour factor isd σ V S, NNLO,N = N N (cid:16) α s π (cid:17) dΦ ( p , . . . , p ; q ) ( − X ( i,j ) ∈ (3 , (cid:20) D ( s i ) + 13 F ( s ij ) + 12 D ( s j ) (cid:21) A (1 q , i g , j g , ¯ q ) J (4)3 ( p , . . . , p )+ ( X ( i,j ) ∈ (3 , (cid:20) d (1 q , i g , j g ) h A ( g (1 i ) q , g ( ji ) g , ¯ q ) + A ( s ) A ( g (1 i ) q , g ( ji ) g , ¯ q ) i + d (1 q , i g , j g ) A ( g (1 i ) q , g ( ji ) g , ¯ q )+ 12 (cid:16) D ( s ij ) + D ( s g ( ji ) ) (cid:17) d (1 q , i g , j g ) A ( g (1 i ) q , g ( ji ) g , ¯ q )+ (cid:18) D ( s i ) + 13 F ( s ij ) + 12 D ( s j ) − D ( s ij ) (cid:19) d (1 q , i g , j g ) A ( g (1 i ) q , g ( ji ) g , ¯ q )+ b log q s ij d (1 q , i g , j g ) A ( g (1 i ) q , g ( ji ) g , ¯ q ) − (cid:18) D ( s g ( ji ) ) − D ( s ij ) + A ( s g (1 i )2 ) − D ( s j ) + 12 D ( s j ) − A ( s ) (cid:19) d (1 q , i g , j g ) A ( g (1 i ) q , g ( ji ) g , ¯ q ) (cid:21) J (3)3 ( f p i , f p ji , p ) + (1 ↔ ) − X ( i,j ) ∈ (3 , (cid:20) ˜ A (1 q , i g , ¯ q ) A ( g (1 i ) q , j g , g (2 i ) ¯ q )+ (cid:0) A ( s ) − A ( s i ) (cid:1) A (1 q , i g , ¯ q ) A ( g (1 i ) q , j g , g (2 i ) ¯ q )+ 12 (cid:18) A ( s i ) − D ( s g (1 i ) j ) − D ( s g (2 i ) j ) − A ( s ) + 12 D ( s j ) + 12 D ( s j ) (cid:19) A (1 q , i g , ¯ q ) A ( g (1 i ) q , j g , g (2 i ) ¯ q ) (cid:21) J (3)3 ( f p i , f p i , p j ) (6.5)– 37 – .3 Three-parton contribution The three-parton contribution consists of the two-loop three-parton matrix element to-gether with the integrated forms of the five-parton and four-parton subtraction terms,d σ SNNLO,N + d σ V S, NNLO,N = N × ((cid:20) (cid:0) D ( s ) + D ( s ) (cid:1) − (cid:0) D ( s ) − D ( s ) (cid:1) − (cid:16) ˜ A ( s ) − A ( s ) A ( s ) (cid:17) + 12 D ( s ) + 12 D ( s ) − ˜ A ( s ) (cid:21) A (1 q , g , ¯ q )+ 12 (cid:0) D ( s ) + D ( s ) (cid:1) A (1 q , g , ¯ q )+ b ǫ (cid:20) D ( s ) (cid:0) ( s ) − ǫ − ( s ) − ǫ (cid:1) + D ( s ) (cid:0) ( s ) − ǫ − ( s ) − ǫ (cid:1) (cid:21) A (1 q , g , ¯ q ) ) d σ , (6.6)where we defined the three-parton normalisation factord σ = N (cid:16) α s π (cid:17) dΦ ( p , p , p ; q ) J (3)3 ( p , p , p ) . (6.7)Combining the infrared poles of this expression with the two loop matrix element, weobtain the cancellation of all infrared poles in this colour factor, P oles (cid:16) d σ SNNLO,N (cid:17) + P oles (cid:16) d σ V S, NNLO,N (cid:17) + P oles (cid:16) d σ V, NNLO,N (cid:17) = 0 . (6.8)
7. Construction of the N colour factor The contribution for the N colour factor to three-jet final states is more involved than allother colour factors. It receives contributions from all partonic subprocesses: γ ∗ → q ¯ qggg and γ ∗ → q ¯ qq ¯ qg at tree-level, γ ∗ → q ¯ qgg and γ ∗ → q ¯ qq ¯ q at one loop and γ ∗ → q ¯ qg at twoloops. All contributions contain a mixture of colour ordered and non-ordered emissions. The subleading colour N contribution of five-parton final states to three jet final states isd σ RNNLO,N = N dΦ ( p , . . . , p ; q ) × " ¯ A (1 q , g , g , g , ¯ q ) − X ( i,j,k ) ∈ P (3 , , ˜ A (1 q , i g , j g , k g , ¯ q ) ! + ˜ C (1 q , q , ¯ q , g , ¯ q ) + ˜ C (2 ¯ q , ¯ q , q , g , q ) − C (1 q , q , ¯ q , g , ¯ q ) J (5)3 ( p , . . . , p )= N dΦ ( p , . . . , p ; q ) × "
13! ¯ A (1 q , g , g , g , ¯ q ) − X ( i,j ) ∈ P (3 , ˜ A (1 q , i g , j g , g , ¯ q )– 38 – ˜ C (1 q , q , ¯ q , g , ¯ q ) + ˜ C (2 ¯ q , ¯ q , q , g , q ) − C (1 q , q , ¯ q , g , ¯ q ) J (5)3 ( p , . . . , p ) , (7.1)where the symmetry factor in front of ¯ A is due to the inherent indistinguishability ofgluons. In ˜ A , gluon (5 g ) is effectively photon-like. It does not participate in any three-gluon or four-gluon vertices, and there are no simple collinear limits as ( i ) g || (5) g and( j ) g || (5) g .The real radiation subtraction term for this colour factor is:d σ SNNLO,N = N N dΦ ( p , . . . , p ; q ) × (cid:26) − X ( i,j ) ∈ (3 , (cid:20) A (1 q , g , ¯ q ) A ( g (15) q , i g , j g , g (25) ¯ q ) J (4)3 ( f p , p i , p j , f p )+ d (1 , i, j ) ˜ A ( g (1 i ) q , g ( ji ) g , g , ¯ q ) J (4)3 ( f p i , f p ji , p , p )+ d (2 , j, i ) ˜ A (1 q , g , g ( ij ) g , g (2 j ) ¯ q ) J (4)3 ( p , p , f p ij , f p j ) (cid:21) + 13! X ( i,j,k ) ∈ P C (3 , , A (1 q , i g , ¯ q ) ˜ A ( g (1 i ) q , j g , k g , g (2 i ) ¯ q ) J (4)3 ( f p i , p j , p k , f p i ) − A (1 q , g , q ) h C ( g (15) q , g (35) q , ¯ q , ¯ q ) + C (2 ¯ q , ¯ q , g (35) q , g (15) q ) i J (4)3 ( f p , f p , p , p ) − A (2 ¯ q , g , ¯ q ) h C (1 q , q , g (45) ¯ q , g (25) ¯ q ) + C ( g (25) ¯ q , g (45) ¯ q , q , q ) i J (4)3 ( p , p , f p , f p ) − X ( i,j ) ∈ (3 , A (1 q , i g , j g , ¯ q ) − d (1 q , i g , j g ) A ( g (1 i ) q , g ( ji ) g , ¯ q ) − d (2 ¯ q , j g , i g ) A (1 q , g ( ij ) g , g (2 j ) ¯ q ) ! A ( ] (1 ij ) q , g , ] (2 ji ) ¯ q ) J (3)3 ( g p ij , p , g p ji ) − X ( i,j ) ∈ (3 , ˜ A (1 q , i g , g , ¯ q ) − A (1 q , i g , ¯ q ) A ( g (1 i ) q , g , g (2 i ) ¯ q ) − A (1 q , g , ¯ q ) A ( g (15) q , i g , g (25) ¯ q ) ! A ( ] (1 i q , j g , ] (2 i ¯ q ) J (3)3 ( g p i , p j , g p i )+ 16 ˜ A (1 q , g , g , ¯ q ) − A (1 q , g , ¯ q ) A ( g (13) q , g , g (23) ¯ q ) − A (1 q , g , ¯ q ) A ( g (14) q , g , g (24) ¯ q ) ! A ( ] (134) q , g , ] (234) ¯ q ) J (3)3 ( g p , p , g p ) − (cid:2) C (1 q , q , ¯ q , ¯ q ) + C (2 ¯ q , ¯ q , q , q ) (cid:3) A ( ] (134) q , g , ] (234) ¯ q ) J (3)3 ( g p , p , g p )+ 12 X ( i,j ) ∈ (3 , d (1 q , i g , j g ) A ( g (1 i ) q , g , ¯ q ) A ( ^ ((1 i )5) q , g ( ji ) g , g (25) ¯ q ) J (3)3 ( ] p (1 i )5 , f p ji , f p )+ d (2 ¯ q , j g , i g ) A (1 q , g , g (2 j ) ¯ q ) A ( g (15) q , g ( ij ) g , ^ ((2 j )5) ¯ q ) J (3)3 ( f p , f p ij , ] p (2 j )5 )– 39 – A (1 q , i g , ¯ q ) A ( g (1 i ) q , g , g (2 i ) ¯ q ) A ( ^ ((1 i )5) q , j g , ^ ((2 i )5) ¯ q ) J (3)3 ( ] p (1 i )5 , ] p (2 i )5 , p j ) !(cid:27) . (7.2) The four-parton contribution comes from the subleading colour one-loop correction to γ ∗ → q ¯ qgg , where the gluonic emissions are colour-ordered, from the leading colour one-loop correction to γ ∗ → q ¯ qgg , where the gluonic emissions are not colour-ordered and fromthe one-loop correction to the identical-flavour process γ ∗ → q ¯ qq ¯ q . It reads:d σ V, NNLO,N = N N (cid:16) α s π (cid:17) dΦ ( p , . . . , p ; q ) × (cid:26) − (cid:18) X ( i,j ) ∈ (3 , A ,b (1 q , i g , j g , ¯ q ) − ˜ A ,a (1 q , g , g , ¯ q ) + ˜ A ,d (1 q , g , g , ¯ q ) (cid:19) − C ,d (1 q , q , ¯ q , ¯ q ) − C ,d (2 ¯ q , ¯ q , q , q ) (cid:27) J (4)3 ( p , p , p , p ) , (7.3)The one-loop single unresolved subtraction term for this colour factor isd σ V S, NNLO,N = N N (cid:16) α s π (cid:17) dΦ ( p , p , p , p ; q ) ( X ( i,j ) ∈ (3 , (cid:20) (cid:18) D ( s i ) + 12 D ( s j ) (cid:19) ˜ A (1 q , i g , j g , ¯ q )+ A ( s ) A (1 q , i g , j g , ¯ q ) (cid:21) J (4)3 ( p , p , p , p ) − A ( s ) ˜ A (1 q , g , g , ¯ q ) J (4)3 ( p , p , p , p )+ (cid:0) A ( s ) + A ( s ) (cid:1) (cid:0) C (1 q , q , ¯ q , ¯ q ) + C (2 ¯ q , ¯ q , q , q ) (cid:1) J (4)3 ( p , p , p , p )+ 12 X ( i,j ) ∈ (3 , (cid:26) − (cid:20) d (1 q , i g , j g ) h ˜ A ( g (1 i ) q , g ( ji ) g , ¯ q ) + A ( s ) A ( g (1 i ) q , g ( ji ) g , ¯ q ) i J (3)3 ( f p i , f p ji , p ) + (1 ↔ (cid:21) − ˜ A (1 q , i g , ¯ q ) A ( g (1 i ) q , j g , g (2 i ) ¯ q ) J (3)3 ( f p i , p j , f p i ) − A (1 q , i g , ¯ q ) h A ( g (1 i ) q , j g , g (2 i ) ¯ q ) + A ( s ) A ( g (1 i ) q , j g , g (2 i ) ¯ q ) i J (3)3 ( f p i , p j , f p i ) − A (1 q , i g , ¯ q ) A ( g (1 i ) q , j g , g (2 i ) ¯ q ) J (3)3 ( f p i , p j , f p i ) − (cid:20) d (1 q , i g , j g ) A ( s g (1 i )2 ) A ( g (1 i ) q , g ( ji ) g , ¯ q ) J (3)3 ( f p i , f p ji , p ) + (1 ↔ (cid:21) − (cid:2) A ( s ) − A ( s i ) (cid:3) A (1 q , i g , ¯ q ) A ( g (1 i ) q , j g , g (2 i ) ¯ q ) J (3)3 ( f p i , p j , f p i ) − h D ( s g (1 i ) j ) + D ( s g (2 i ) j ) i A (1 q , i g , ¯ q ) A ( g (1 i ) q , j g , g (2 i ) ¯ q ) J (3)3 ( f p i , p j , f p i ) − (cid:20) D ( s i ) + 12 D ( s i ) − A ( s i ) (cid:21) A (1 q , i g , ¯ q ) A ( g (1 i ) q , j g , g (2 i ) ¯ q ) J (3)3 ( f p i , p j , f p i )+ (cid:20) D ( s g (1 i ) j ) + 12 D ( s g (2 i ) j ) − A ( s i ) − D ( s j ) − D ( s j ) + A ( s ) (cid:21) – 40 – (1 q , i g , ¯ q ) A ( g (1 i ) q , j g , g (2 i ) ¯ q ) J (3)3 ( f p i , p j , f p i ) − b log q s i A (1 q , i g , ¯ q ) A ( g (1 i ) q , j g , g (2 i ) ¯ q ) J (3)3 ( f p i , p j , f p i ) ) (7.4) The three parton contribution to the N colour factor receives contributions from the three-parton virtual two-loop correction and the integrated five-parton tree-level and four-partonone-loop subtraction terms, which readd σ SNNLO,N + d σ V S, NNLO,N = N × ( − (cid:20) A ( s ) + 12 ˜ A ( s ) + 2 C ( s ) + 12 A ( s ) (cid:0) D ( s ) + D ( s ) (cid:1) −A ( s ) A ( s ) + A ( s ) + ˜ A ( s ) (cid:21) A (1 q , g , ¯ q ) − (cid:0) D ( s ) + D ( s ) (cid:1) ˜ A (1 q , g , ¯ q ) − A ( s ) A (1 q , g , ¯ q ) − b ǫ A ( s ) (cid:0) ( s ) − ǫ − ( s ) − ǫ (cid:1) A (1 q , g , ¯ q ) ) d σ . (7.5)Combining the infrared poles of this expression with the two loop matrix element, weobtain the cancellation of all infrared poles in this colour factor, P oles (cid:16) d σ SNNLO,N (cid:17) + P oles (cid:16) d σ V S, NNLO,N (cid:17) + P oles (cid:16) d σ V, NNLO,N (cid:17) = 0 . (7.6)
8. Construction of the /N colour factor The 1 /N colour factor receives contributions from five-parton tree-level γ ∗ → q ¯ qggg and γ ∗ → q ¯ qq ¯ qg , four-parton one-loop γ ∗ → q ¯ qgg and γ ∗ → q ¯ qq ¯ q as well as tree-level two-loop γ ∗ → q ¯ qg . The gluon emissions are all photon-like, not containing any gluon self-coupling.The four-quark processes contribute through the identical-quark-only terms.The construction of the subtraction terms for this colour factor was discussed in detailin [32, 65]. Two different five-parton final states contribute at 1 /N to three-jet final states at NNLO: γ ∗ → q ¯ qggg and γ ∗ → q ¯ qq ¯ qg with identical quarks.The NNLO radiation term appropriate for the three jet final state is given byd σ RNNLO, /N = N N dΦ ( p , . . . , p ; q ) × (cid:20)
13! ¯ A (1 q , g , g , g , ¯ q ) + 2 ˜ C (1 q , ¯ q , q , ¯ q , g ) (cid:21) J (5)3 ( p , . . . , p ) , (8.1)– 41 –here the symmetry factor in front of ¯ A is due to the inherent indistinguishability ofgluons. The factor 2 in front of ˜ C arises from the fact that two different momentumarrangements contribute to the squared matrix element (4.21). If the quarks and antiquarksare not distinguished by the jet functions, these contribute equally.The real radiation subtraction term for this colour factor is:d σ SNNLO, /N = N N dΦ ( p , . . . , p ; q ) ( X i,j,k ∈ P C (3 , , (cid:20) A (1 q , i g , ¯ q ) ˜ A ( g (1 i ) q , j g , k g , g (2 i ) ¯ q ) J (4)3 ( f p i , p j , p k , f p i )+ (cid:18) ˜ A (1 q , i g , j g , ¯ q ) − A (1 q , i g , ¯ q ) A ( g (1 i ) q , j g , g (2 i ) ¯ q ) − A (1 q , j g , ¯ q ) A ( g (1 j ) q , i g , g (2 j ) ¯ q ) (cid:19) A ( ] (1 ij ) q , k g , ] (2 ij ) ¯ q ) J (3)3 ( g p ij , p k , g p ij ) (cid:21) +2 (cid:20) A (1 q , g , ¯ q ) C ( g (15) q , q , ¯ q , g (25) ¯ q ) J (4)3 ( f p , p , p , f p )+ A (1 q , g , ¯ q ) C ( g (15) q , q , g (45) ¯ q , ¯ q ) J (4)3 ( f p , p , p , f p )+ A (3 q , g , ¯ q ) C (1 q , g (35) q , ¯ q , g (25) ¯ q ) J (4)3 ( f p , p , p , f p )+ A (3 q , g , ¯ q ) C (1 q , g (35) q , g (45) ¯ q , ¯ q ) J (4)3 ( f p , p , p , f p ) − A (1 q , g , q ) C ( g (15) q , g (35) q , ¯ q , ¯ q ) J (4)3 ( f p , p , p , f p ) − A (2 ¯ q , g , ¯ q ) C (1 q , q , g (45) ¯ q , g (25) ¯ q ) J (4)3 ( f p , p , p , f p )+ C (1 q , q , ¯ q , ¯ q ) A ( ] (134) q , g , ] (234) ¯ q ) J (3)3 ( g p , p , g p ) (cid:21)) . (8.2)The sum in the first contribution runs only over the three cyclic permutations of the gluonmomenta to prevent double counting of identical configurations obtained by interchange of j and k . At one-loop, there are two contributions to the colour suppressed contribution proportionalto 1 /N , one from the four quark final state and one from the two quark-two gluon finalstate:d σ V, NNLO, /N = N N (cid:16) α s π (cid:17) dΦ ( p , . . . , p ; q ) (cid:26)
12! ˜ A ,b (1 q , g , g , ¯ q ) + 2 C ,e (1 q , q , ¯ q , ¯ q ) (cid:27) J (4)3 ( p , . . . , p ) , (8.3)where the origin of the symmetry factors is as in the real radiation five-parton contributionsof the previous section.The corresponding subtraction term is:d σ V S, NNLO, /N = N N (cid:16) α s π (cid:17) dΦ ( p , . . . , p ; q )– 42 – ( X i,j ∈ P (3 , (cid:20) − A ( s ) ˜ A (1 q , i g , j g , ¯ q ) J (4)3 ( p , p i , p j , p )+ (cid:18) A (1 q , i g , ¯ q ) h ˜ A ( g (1 i ) q , j g , g (2 i ) ¯ q ) + A ( s ) A ( g (1 i ) q , j g , g (2 i ) ¯ q ) i + ˜ A (1 q , i g , ¯ q ) A ( g (1 i ) q , j g , g (2 i ) ¯ q ) (cid:19) J (3)3 ( f p i , p j , f p i )+ A ( s ) A (1 q , i g , ¯ q ) A ( g (1 i ) q , j g , g (2 i ) ¯ q ) J (3)3 ( f p i , p j , f p i ) (cid:21) − h A ( s ) + A ( s ) + A ( s ) + A ( s ) − A ( s ) − A ( s ) i × C (1 q , q , ¯ q , ¯ q ) J (4)3 ( p , p , p , p ) ) . (8.4) The three-parton contribution consists of the two-loop three-parton matrix element to-gether with the integrated forms of the five-parton and four-parton subtraction terms,d σ SNNLO, /N + d σ V S, NNLO, /N = 1 N ((cid:20)
12 ˜ A ( s ) + 2 C ( s ) + ˜ A ( s ) (cid:21) A (1 q , g , ¯ q )+ A ( s ) ˜ A (1 q , g , ¯ q ) ) d σ . (8.5)Combining the infrared poles of this expression with the two loop matrix element, weobtain the cancellation of all infrared poles in this colour factor, P oles (cid:16) d σ SNNLO, /N (cid:17) + P oles (cid:16) d σ V S, NNLO, /N (cid:17) + P oles (cid:16) d σ V, NNLO, /N (cid:17) = 0 . (8.6)
9. Construction of the N F N colour factor The colour factor N F N receives contribution from the colour-ordered five-parton tree-levelprocess γ ∗ → q ¯ qq ′ ¯ q ′ g , the four-parton one-loop processes γ ∗ → q ¯ qq ′ ¯ q ′ and γ ∗ → q ¯ qgg atleading colour and from the two-loop three-parton process γ ∗ → q ¯ qg . The NNLO radiation term appropriate for the three jet final state is given byd σ RNNLO,N F N = N N F N dΦ ( p , . . . , p ; q ) × h B ,a (1 q , g , ¯ q ′ ; 3 q ′ , ¯ q ) + B ,b (1 q , ¯ q ′ ; 3 q ′ , g , ¯ q ) i J (5)3 ( p , . . . , p ) , (9.1)The two terms represent the two colour orderings of the leading colour amplitude forthis process. Since the leading colour qq ′ ¯ q ′ g -antenna subtraction terms allows q to representeither a quark or an antiquark, both colour orderings are mixed together. Therefore, it isnot possible to construct a subtraction term for an individual contribution, but only fortheir sum. – 43 –he subtraction term for this contribution isd σ SNNLO,N F N = N N F N dΦ ( p , . . . , p ; q ) × n A (1 q , g , ¯ q ′ ) B ( g (15) q , q ′ , g (45) ¯ q ′ , ¯ q ) J (4)3 ( f p , p , p , f p )+ A (3 q ′ , g , ¯ q ) B (1 q , g (35) q ′ , ¯ q ′ , g (25) ¯ q ) J (4)3 ( p , f p , f p , p )+ G (5 g , q ′ , ¯ q ′ ) A (1 q , g (34) g , g (54) g , ¯ q ) J (4)3 ( p , p , f p , f p )+ G (5 g , q ′ , ¯ q ′ ) A (1 q , g (54) g , g (34) g , ¯ q ) J (4)3 ( p , p , f p , f p )+ (cid:18) E ,a (1 q , q ′ , ¯ q ′ , g ) − G (5 g , q ′ , ¯ q ′ ) d (1 q , g (34) g , g (54) g ) (cid:19) A ( ] (134) q , ] (543) g , ¯ q ) J (3)3 ( g p , p , g p )+ (cid:18) E ,b (1 q , q ′ , ¯ q ′ , g ) − A (1 q , g , ¯ q ′ ) E ( g (15) q , q ′ , g (45) ¯ q ′ ) − G (5 g , q ′ , ¯ q ′ ) d (1 q , g (54) g , g (34) g ) (cid:19) A ( ] (154) q , ] (345) g , ¯ q ) J (3)3 ( g p , p , g p )+ (cid:18) E ,a (2 ¯ q , ¯ q ′ , q ′ , g ) − G (5 g , q ′ , ¯ q ′ ) d (2 ¯ q , g (43) g , g (53) g ) (cid:19) A (1 q , ] (534) g , ] (243) ¯ q ) J (3)3 ( p , g p , g p )+ (cid:18) E ,b (2 ¯ q , ¯ q ′ , q ′ , g ) − A (3 q ′ , g , ¯ q ) E ( g (25) ¯ q , ¯ q ′ , g (35) q ′ ) − G (5 g , q ′ , ¯ q ′ ) d (2 ¯ q , g (53) g , g (43) g ) (cid:19) A (1 q , ] (435) g , ] (253) ¯ q ) J (3)3 ( p , g p , g p ) o . (9.2) The four parton contribution to the N F N colour factor reads:d σ V, NNLO,N F N = N N F N (cid:16) α s π (cid:17) dΦ ( p , . . . , p ; q ) X ( i,j ) ∈ (3 , (cid:16) B ,a (1 q , i q ′ , j ¯ q ′ , ¯ q ) + A ,c (1 q , i g , j g , ¯ q ) (cid:17) J (4)3 ( p , . . . , p ) . (9.3)The average over the permutations of the momenta (3) and (4) has to be made in bothcontributions to this colour factor. In the one-loop correction to the γ ∗ → qq ′ ¯ q ′ ¯ q finalstate B ,a , the secondary quark-antiquark pair has to be symmetrised, since the quark-gluon antenna functions used in the one-loop subtraction terms do not distinguish quarksand antiquarks. The summation over the two colour orderings of the one-loop correctionto the γ ∗ → qgg ¯ q final state A ,c must be kept since the one-loop subtraction functionsappropriate to this term contain both orderings because of their cyclicity.– 44 –he corresponding subtraction term is:d σ V S, NNLO,N F N = N N F N (cid:16) α s π (cid:17) dΦ ( p , . . . , p ; q ) × ( − X ( i,j ) ∈ (3 , (cid:2) A ( s j ) + A ( s i ) (cid:3) B (1 q , i q ′ , j ¯ q ′ , ¯ q ) J (4)3 ( p , . . . , p ) − X ( i,j ) ∈ (3 , G ( s ij ) A (1 q , i g , j g , ¯ q ) J (4)3 ( p , . . . , p )+ ( (cid:18) E (1 q , q ′ , ¯ q ′ ) h A ( g (13) q , g (43) g , ¯ q ) + A ( s ) A ( g (13) q , g (43) g , ¯ q ) i + E (1 q , q ′ , ¯ q ′ ) A ( g (13) q , g (43) g , ¯ q )+ 12 (cid:16) D ( s ) + D ( s g (43) ) (cid:17) E (1 q , q ′ , ¯ q ′ ) A ( g (13) q , g (43) g , ¯ q )+ (cid:0) A ( s ) + A ( s ) − D ( s ) (cid:1) E (1 q , q ′ , ¯ q ′ ) A ( g (13) q , g (43) g , ¯ q )+ b log q s E (1 q , q ′ , ¯ q ′ ) A ( g (13) q , g (43) g , ¯ q ) (cid:19) J (3)3 ( f p , f p , p ) + (1 ↔ ) + ( (cid:18) D (1 q , g , g ) ˆ A ( g (13) q , g (43) g , ¯ q ) + ˆ D (1 q , g , g ) A ( g (13) q , g (43) g , ¯ q )+2 G ( s ) D (1 q , g , g ) A ( g (13) q , g (43) g , ¯ q )+ b ,F log q s D (1 q , g , g ) A ( g (13) q , g (43) g , ¯ q ) (cid:19) J (3)3 ( f p , f p , p ) + (1 ↔ ) . (9.4) The three parton contribution to the N F N colour factor contains the three-parton vir-tual two-loop correction and the integrated five-parton tree-level and four-parton one-loopsubtraction terms, which readd σ SNNLO,N F N + d σ V S, NNLO,N F N = N F N × ((cid:20) E ( s ) + E ( s ) − (cid:0) D ( s ) E ( s ) + D ( s ) E ( s ) (cid:1) + 14 (cid:0) D ( s ) E ( s ) + D ( s ) E ( s ) (cid:1) + 12 ˆ D ( s ) + 12 ˆ D ( s ) + 12 E ( s )+ 12 E ( s ) (cid:21) A (1 q , g , ¯ q ) + 12 (cid:0) E ( s ) + E ( s ) (cid:1) A (1 q , g , ¯ q )+ 12 (cid:0) D ( s ) + D ( s ) (cid:1) ˆ A (1 q , g , ¯ q )+ b ,F ǫ (cid:20) D ( s ) (cid:0) ( s ) − ǫ − ( s ) − ǫ (cid:1) + D ( s ) (cid:0) ( s ) − ǫ − ( s ) − ǫ (cid:1) (cid:21) A (1 q , g , ¯ q )– 45 – b ǫ (cid:20) E ( s ) (cid:0) ( s ) − ǫ − ( s ) − ǫ (cid:1) + E ( s ) (cid:0) ( s ) − ǫ − ( s ) − ǫ (cid:1) (cid:21) A (1 q , g , ¯ q ) ) d σ . (9.5)Combining the infrared poles of this expression with the two loop matrix element, weobtain the cancellation of all infrared poles in this colour factor, P oles (cid:0) d σ SNNLO,N F N (cid:1) + P oles (cid:16) d σ V S, NNLO,N F N (cid:17) + P oles (cid:16) d σ V, NNLO,N F N (cid:17) = 0 . (9.6)
10. Construction of the N F /N colour factor The N F /N colour factor receives contributions from five-parton tree-level γ ∗ → q ¯ qq ′ ¯ q ′ g ,four-parton one-loop γ ∗ → q ¯ qq ′ ¯ q ′ and γ ∗ → q ¯ qgg at subleading colour as well as three-parton two-loop γ ∗ → q ¯ qg . The gluon emissions are all photon-like.This colour factor is part of the QED-type corrections. We described the constructionof the subtraction terms for this colour factor previously in [66]. The NNLO radiation term appropriate for the three jet final state is given byd σ RNNLO,N F /N = N N F N dΦ ( p , . . . , p ; q ) h B ,c (1 q , g , ¯ q ; 3 q ′ , ¯ q ′ )+ B ,d (1 q , ¯ q ; 3 q ′ , g , ¯ q ′ ) − B ,e (1 q , ¯ q ; 3 q ′ , ¯ q ′ ; 5 g ) i J (5)3 ( p , . . . , p )= N N F N dΦ ( p , . . . , p ; q ) 12 X ( i,j ) ∈ (3 , h B ,c (1 q , g , ¯ q ; i q ′ , j ¯ q ′ )+ B ,d (1 q , ¯ q ; i q ′ , g , j ¯ q ′ ) − B ,e (1 q , ¯ q ; i q ′ , j ¯ q ′ ; 5 g ) i J (5)3 ( p , . . . , p ) , (10.1)where the symmetrization over the momenta of the secondary quark-antiquark pair ex-ploits the fact that the jet algorithm does not distinguish quarks and antiquarks. Thissymmetrisation reduces the number of non-vanishing unresolved limits considerably, sincethe interference term in B ,e is odd under this interchange. As a result, the unresolvedstructure of the symmetrised B ,e equals the unresolved structure of B ,c + B ,d .The subtraction term reads:d σ SNNLO,N F /N = N N F N dΦ ( p , . . . , p ; q ) 12 X ( i,j ) ∈ (3 , ( − A (1 q , g , ¯ q ) B ( g (15) q , g (25) ¯ q , i q ′ , j ¯ q ′ ) J (4)3 ( f p , f p , p i , p j ) − A ( i q ′ , g , j ¯ q ′ ) B (1 q , ¯ q , g ( i q ′ , g ( j ¯ q ′ ) J (4)3 ( p , p , f p i , f p j ) − n E (1 q , i q ′ , j ¯ q ′ ) ˜ A ( g (1 j ) q , g ( ij ) g , g , ¯ q ) J (4)3 ( f p j , p , f p ij , p ) + (1 ↔ o − B (1 q , i q ′ , j ¯ q ′ , ¯ q ) − n E (1 q , i q ′ , j ¯ q ′ ) A ( g (1 i ) q , g ( ji ) g , ¯ q ) + (1 ↔ o ! – 46 – A ( ] (1 ij ) q , g , ] (2 ji ) ¯ q ) J (3)3 ( g p ij , g p ji , p ) − ( ˜ E (1 q , i q ′ , j ¯ q ′ , g ) − A ( i q ′ , g , j ¯ q ′ ) E (1 q , g ( i q ′ , g ( j ¯ q ′ ) ! × A ( ] (1 i q , ] ( j i ) g , ¯ q ) J (3)3 ( g p i , p , g p j i ) + (1 ↔ ) + 12 ( E (1 q , i q ′ , j ¯ q ′ ) A ( ] (1 j ) q , g , ¯ q ) A ( ^ ((1 j )5) q , g ( ij ) g , g (25) ¯ q ) J (3)3 ( ] p (1 j )5 , f p , f p ij )+(1 ↔ )) (10.2) The four parton contribution to the N F /N colour factor reads:d σ V, NNLO,N F /N = N N F N (cid:16) α s π (cid:17) dΦ ( p , . . . , p ; q ) ( − X ( i,j ) ∈ (3 , (cid:16) B ,b (1 q , i q ′ , j ¯ q ′ , ¯ q ) + 2 C ,f (1 q , i q , j ¯ q , ¯ q )+ ˜ A ,c (1 q , i g , j g , ¯ q ) (cid:17)) J (4)3 ( p , . . . , p ) . (10.3)Like in the N F N colour factor, the expression is symmetrised over the momenta (3) and(4) to remove terms which are antisymmetric under charge conjugation, and can not beaccounted for properly by the quark-gluon antenna functions.The corresponding subtraction term is:d σ V S, NNLO,N F /N = N N F N (cid:16) α s π (cid:17) dΦ ( p , . . . , p ; q ) × ( (cid:2) A ( s ) + A ( s ) (cid:3) B (1 q , q ′ , ¯ q ′ , ¯ q ) J (4)3 ( p , . . . , p )+ 14 (cid:2) E ( s ) + E ( s ) + E ( s ) + E ( s ) (cid:3) ˜ A (1 q , g , g , ¯ q ) J (4)3 ( p , . . . , p ) − (cid:26)(cid:18) E (1 q , q ′ , ¯ q ′ ) h ˜ A ( g (13) q , g (43) g , ¯ q ) + A ( s ) A ( g (13) q , g (43) g , ¯ q ) i + A ( s ( f ) E (1 q , q ′ , ¯ q ′ ) A ( g (13) q , g (43) g , ¯ q )+ h ˜ E (1 q , q ′ , ¯ q ′ ) + A ( s ) E (1 q , q ′ , ¯ q ′ ) i A ( g (13) q , g (43) g , ¯ q ) (cid:19) J (3)3 ( f p , f p , p )+(1 ↔ (cid:27) − X ( i,j ) ∈ (3 , (cid:18) A (1 q , i g , ¯ q ) ˆ A ( g (1 i ) q , j g , g (2 i ) ¯ q ) + (cid:20) ˆ A (1 q , i g , ¯ q )– 47 – 12 (cid:0) E ( s i ) + E ( s j ) + E ( s i ) + E ( s j ) (cid:1) A (1 q , i g , ¯ q ) (cid:21) A ( g (1 i ) q , j g , g (2 i ) ¯ q )+ b ,F log q s i A (1 q , i g , ¯ q ) A ( g (1 i ) q , j g , g (2 i ) ¯ q ) (cid:19) J (3)3 ( f p i , f p i , p j ) ) (10.4) The three parton contribution to the N F /N colour factor consists of the three-partonvirtual two-loop correction and the integrated five-parton tree-level and four-parton one-loop subtraction terms, which readd σ SNNLO,N F /N + d σ V S, NNLO,N F /N = N F N × ( − (cid:20) B ( s ) + 12 ˜ E ( s ) + 12 ˜ E ( s ) + 12 A ( s ) (cid:0) E ( s ) + E ( s ) (cid:1) + ˆ A ( s ) + 12 ˜ E ( s ) + 12 ˜ E ( s ) (cid:21) A (1 q , g , ¯ q ) − (cid:0) E ( s ) + E ( s ) (cid:1) ˜ A (1 q , g , ¯ q ) −A ( s ) ˆ A (1 q , g , ¯ q ) − b ,F ǫ A ( s ) (cid:0) ( s ) − ǫ − ( s ) − ǫ (cid:1) A (1 q , g , ¯ q ) ) d σ . (10.5)Taking the infrared pole part of this expression, we obtain cancellation of all infraredpoles in this channel: P oles (cid:16) d σ SNNLO,N F /N (cid:17) + P oles (cid:16) d σ V S, NNLO,N F /N (cid:17) + P oles (cid:16) d σ V, NNLO,N F /N (cid:17) = 0 . (10.6)
11. Construction of the N F colour factor The N F colour factor receives contributions only from the four-parton one-loop process γ ∗ → q ¯ qq ′ ¯ q ′ and from the three-parton two-loop process γ ∗ → q ¯ qg .This colour factor is also part of the QED-type corrections, described previously in [66]. The four-parton one-loop contribution to this colour factor isd σ V, NNLO,N F = N N F (cid:16) α s π (cid:17) dΦ ( p , . . . , p ; q ) B ,c (1 q , q ′ , ¯ q ′ , ¯ q ) J (4)3 ( p , . . . , p ) . (11.1)This contribution is free of explicit infrared poles (as can be inferred from the absence ofa five-parton contribution to this colour structure).The subtraction term corresponding to this contribution isd σ V S, NNLO,N F = N N F (cid:16) α s π (cid:17) dΦ ( p , . . . , p ; q ) 12 ((cid:20) (cid:18) ˆ E (1 q , q ′ , ¯ q ′ ) + b ,F log q s E (1 q , q ′ , ¯ q ′ ) (cid:19) A ( g (13) q , g (43) g , ¯ q )– 48 – E (1 q , q ′ , ¯ q ′ ) ˆ A ( g (13) q , g (43) g , ¯ q ) (cid:21) J (3)3 ( f p , f p , p )+ (cid:20) (cid:18) ˆ E (2 ¯ q , q ′ , ¯ q ′ ) + b ,F log q s E (2 ¯ q , q ′ , ¯ q ′ ) (cid:19) A (1 q , g (43) g , g (23) ¯ q )+ E (2 ¯ q , q ′ , ¯ q ′ ) ˆ A (1 q , g (43) g , g (23) ¯ q ) (cid:21) J (3)3 ( p , f p , f p ) ) . (11.2)Although ˆ E and ˆ A contain explicit infrared poles, these cancel in their sum, as can beseen from (5.16) and (6.32) of [32]. d σ V S, NNLO,N F is therefore free of explicit infrared poles. The three parton contribution to the N F colour factor consists of the three-parton virtualtwo-loop correction and the integrated four-parton one-loop subtraction term, which readsd σ V S, NNLO,N F = N F × (cid:20) (cid:16) ˆ E ( s ) + ˆ E ( s ) (cid:17) A (1 q , g , ¯ q ) + (cid:0) E ( s ) + E ( s ) (cid:1) ˆ A (1 q , g , ¯ q )+ b ,F ǫ (cid:2) E ( s ) (cid:0) ( s ) − ǫ − ( s ) − ǫ (cid:1) + E ( s ) (cid:0) ( s ) − ǫ − ( s ) − ǫ (cid:1)(cid:3) A (1 q , g , ¯ q ) (cid:21) d σ . (11.3)Combining the infrared poles of this expression with the two loop matrix element, weobtain the cancellation of all infrared poles in this colour factor, P oles (cid:16) d σ V S, NNLO,N F (cid:17) + P oles (cid:16) d σ V, NNLO,N F (cid:17) = 0 . (11.4)
12. Construction of the N F,γ colour factor
The N F,γ colour factor comes from the interference of amplitudes in which the externalvector boson couples to different quark lines. It receives contributions from five-parton tree-level γ ∗ → q ¯ qq ′ ¯ q ′ g , four-parton one-loop γ ∗ → q ¯ qq ′ ¯ q ′ and γ ∗ → q ¯ qgg as well as three-partontwo-loop γ ∗ → q ¯ qg and γ ∗ → ggg . This colour factor is absent in three-jet production atNLO and four-jet production at LO because of Furry’s theorem [5], and its contributionto four-jet production at NLO is numerically tiny [52]. The numerical magnitude of thisterm in the two-loop corrections to the three-parton channel is equally very small [18, 53].A detailed discussion of this colour factor, and of the effects leading to its numericalsuppression is contained in [52]. All partonic channels contributing to this colour factorare individually finite.In this section, we document this colour factor for completeness. Given that is numer-ical impact can be safely expected to be negligible, we refrain from its implementation.– 49 – The NNLO radiation term appropriate for the three jet final state is given byd σ RNNLO,N
F,γ = N N F,γ dΦ ( p , . . . , p ; q ) × h − N (cid:16) ˆ B ,a (1 q , g , ¯ q ′ ; 3 q ′ , ¯ q ) + ˆ B ,b (1 q , ¯ q ′ ; 3 q ′ , g , ¯ q ) − ˆ B ,e (1 q , ¯ q ′ ; 3 q ′ , ¯ q , g ) (cid:17) + 1 N (cid:16) ˆ B ,c (1 q , g , ¯ q ; 3 q ′ , ¯ q ′ ) + ˆ B ,d (1 q , ¯ q ; 3 q ′ , g , ¯ q ′ ) + ˆ B ,e (1 q , ¯ q ; 3 q ′ , ¯ q ′ ; 5 g ) (cid:17) i × J (5)3 ( p , . . . , p ) . (12.1)Once symmetrised over the quark and antiquark momenta, this term can be integratedsafely without the need for an infrared subtraction. It is free from infrared singularitiesassociated with gluon 5 g unresolved, since the corresponding four-parton tree-level term ˆ B vanishes after symmetrisation over the quark and antiquark momenta. Double unresolvedsingularities can not appear since there is no tree-level three-parton process proportionalto N F,γ . The four parton contribution to the N F,γ colour factor reads:d σ V, NNLO,N
F,γ = N N F,γ (cid:16) α s π (cid:17) dΦ ( p , . . . , p ; q ) × ( N ˆ B ,a (1 q , q , ¯ q , ¯ q ) − N ˆ B ,b (1 q , q , ¯ q , ¯ q )+ N A ,e (1 q , i g , j g , ¯ q ) − N ˜ A ,e (1 q , g , g , ¯ q ) ) J (4)3 ( p , . . . , p ) . (12.2)Terms which vanish under symmetrisation of the quark and antiquark momenta, arisingfrom ˆ B ,c in (4.46) have been omitted here. After this symmetrisation, all explicit infraredpoles present in individual terms in the above expression cancel. Moreover, (12.2) is finitein all single unresolved limits, such that no antenna subtraction is needed. The three-parton contribution to the N F,γ colour factor consists of the three-parton virtualtwo-loop correction to γ ∗ → q ¯ qg [18] and the one-loop squared correction to γ ∗ → ggg [53].Both are individually finite, and were shown to be numerically tiny. Since no subtrac-tions were carried out in the five-parton and four-parton channels, there are no integratedsubtraction terms in the three-parton channel.
13. Numerical implementation
Using the matrix elements and antenna subtraction terms derived in the previous sec-tions, NNLO corrections to any infrared-safe three-jet observable in e + e − annihilation (jet– 50 – partonchannel4 partonchannel3 partonchannel dΦ q ¯ qggg dΦ q ¯ qgg dΦ q ¯ qg Monte CarloPhase Space d σ RNNLO − d σ SNNLO d σ V, NNLO + Z d σ V S, NNLO dΦ X + Z d σ SNNLO dΦ X d σ V, NNLO − d σ V S, NNLO ✲ { p i } ✲ { p i } ✲ { p i } Cross section ✲ { p i } , w ✲ { p i } , w ✲ { p i } , w Definition of Observables5 parton → → → w, { C, S, T } w, { C, S, T } w, { C, S, T } ✲✲✲✲ ⊕ Histograms σ j d σ/ d T d σ/ d S d σ/ d C Figure 7:
Structure of the EERAD3 parton-level Monte Carlo event generator programme. cross section, event shape variable) can be computed numerically. We implemented thisnumerical evaluation into a parton-level event generator program, which we name
EERAD3 .This program is based on the program
EERAD2 [39], which computes four-jet productionat NLO.
EERAD2 contained already the five-parton and four-parton matrix elements relevanthere, as well as the NLO-type subtraction terms d σ S,aNNLO and d σ V S, ,aNNLO .The implementation contains three channels, classified by their partonic multiplicity: • in the five-parton channel, we integrated σ RNNLO − d σ SNNLO . (13.1) • in the four-parton channel, we integrated σ V, NNLO − d σ V S, NNLO . (13.2) • in the three-parton channel, we integrated σ V, NNLO + d σ SNNLO + d σ V S, NNLO . (13.3)The numerical integration over these channels is carried out by Monte Carlo methods usingthe VEGAS [67] implementation. The structure of the programme is displayed in Figure 7.The phase space in the four-parton and five-parton channel is decomposed into wedgeswhich are constructed such that two of the invariants are smaller than any of the otherinvariants. This decomposition allows an optimal generation of phase space points in– 51 –he unresolved limits. The full four-parton phase space is obtained by summing (a) 12wedges with ( s ij , s ik ) smallest, plus (b) 3 wedges with ( s ij , s kl ) smallest. To obtain thefull five-parton phase space, we sum (a) 30 wedges with ( s ij , s ik ) smallest, and (b) 15wedges with ( s ij , s kl ) smallest. The phase space integration in either channel is carriedout by integrating only over a single wedge of type (a) and a single wedge of type (b),while summing the integrands appropriate to all wedges of the given type. In doing thissummation, we combine (in the exact unresolved limits) phase space points which arerelated to each other by a rotation of the system of unresolved partons, thereby largelycancelling the angular-dependent terms. In all colour factors containing angular-dependentterms, the combination of phase space wedges yields a substantial improvement of thenumerical stability of the results.It was already demonstrated above that the integrands in the four-parton and three-parton channel are free of explicit infrared poles. In the five-parton and four-parton chan-nel, we tested the proper implementation of the subtraction by generating trajectoriesof phase space points approaching a given single or double unresolved limit using the RAMBO [68] phase space generator. Along these trajectories, we observe that the antennasubtraction terms converge towards the physical matrix elements, and that the cancella-tions among individual contributions to the subtraction terms take place as expected inthe antenna subtraction method.Moreover, we checked the correctness of the subtraction by introducing a lower cut(slicing parameter) y on all phase space variables, and observing that our results areindependent of this cut (provided it is chosen small enough). This behaviour indicates thatthe subtraction terms ensure that the contribution of potentially singular regions of thefinal state phase space does not contribute to the numerical integrals, but is accounted foranalytically.
14. Thrust distribution as an example
To illustrate the implementation and to study the numerical impact of the individual NNLOcontributions, we consider the thrust distribution. We already reported the NNLO resultson this event shape distribution in a previous paper [33], where the phenomenologicalimplications are discussed in detail.The thrust variable for a hadronic final state in e + e − annihilation is defined as [69] T = max ~n (cid:18) P i | ~p i · ~n | P i | ~p i | (cid:19) , (14.1)where p i denotes the three-momentum of particle i , with the sum running over all particles.The unit vector ~n is varied to find the thrust direction ~n T which maximises the expressionin parentheses on the right hand side.It can be seen that a two-particle final state has fixed T = 1, consequently the thrustdistribution receives its first non-trivial contribution from three-particle final states, which,at order α s , correspond to three-parton final states. Therefore, both theoretically andexperimentally, the thrust distribution is closely related to three-jet production.– 52 – study of the phenomenological implications of the NNLO corrections to the thrustdistribution was presented in [33], illustrating that the NNLO corrections amount to about15% of the total result over the experimentally relevant range 0 . < − T < .
3, and thatinclusion of these corrections results in a considerable stabilization of the renormalisationscale dependence of the theoretical prediction. In the present context, we use the thrustdistribution only as an example to illustrate certain features of our calculation.The three-jet rate and event shapes related to it can be expressed in perturbative QCDby dimensionless coefficients. These coefficients depend, for non-singlet QCD corrections,only on the jet resolution parameter (respectively on the event shape variable). Typically,one denotes these coefficients by
A, B, C, . . . at LO, NLO, NNLO, etc.The perturbative expansion of thrust distribution up to NNLO for renormalisationscale µ = s and α s ≡ α s ( s ) is then given by1 σ had d σ d T = (cid:16) α s π (cid:17) d ¯ A d T + (cid:16) α s π (cid:17) d ¯ B d T + (cid:16) α s π (cid:17) d ¯ C d T . (14.2)Here we define the effective coefficients in terms of the perturbatively calculated coefficients A , B and C , which are all normalised to the tree-level cross section σ = 4 πα s N e q . (14.3)for e + e − → q ¯ q . Using σ had = σ (cid:18) C F (cid:16) α s π (cid:17) + K (cid:16) α s π (cid:17) + O ( α s ) (cid:19) , (14.4)with ( C F = ( N − / (2 N ), C A = N , T R = 1 / N = 3 colours and N F light quarkflavours) K = 14 (cid:20) − C F + C F C A (cid:18) − ζ (cid:19) + C F T R N F ( −
22 + 16 ζ ) (cid:21) , (14.5)we obtain: ¯ A = A , ¯ B = B − C F A , ¯ C = C − C F B + (cid:18) C F − K (cid:19) A . (14.6)These coefficients depend only on the jet resolution parameter or the event shape variableunder consideration, and are independent of electroweak couplings, centre-of-mass energyand renormalisation scale.The above coefficients include only QCD corrections with non-singlet quark couplings.At O ( α s ), these amount to the full corrections, while the O ( α s ) corrections also receive asinglet contribution. As discussed above, this singlet contribution arises from the interfer-ence of diagrams where the external gauge boson couples to different quark lines. In four-jet– 53 – (1-T) d Ad T (1-T) d Bd T Figure 8:
Coefficients of the leading order and next-to-leading order contributions to the thrustdistributions. (1-T) d Cd T
Figure 9:
Coefficient of the next-to-next-to-leading order contribution to the thrust distribution.Solid: corrected for large-angle soft terms; dotted: original result. observables at O ( α s ), these singlet contributions were found to be extremely small [52].Also, the singlet contribution from three-gluon final states to three-jet observables wasfound to be negligible [53].We determine A, B, C from the perturbative contributions to the differential crosssection, normalised to the tree-level hadronic cross section:d A d T = 1 σ d σ LO d T , d B d T = 1 σ d σ NLO d T , d C d T = 1 σ d σ NNLO d T . (14.7)For the determination of the non-singlet coefficients, it is sufficient to consider σ forpure photon exchange, since any electroweak coupling constant cancels out in the aboveratio. The LO and NLO coefficients A ( T ) and B ( T ) were computed in the literature longago [5–8, 10]. They are displayed for comparison in Figure 8.The total NNLO coefficient C ( T ) is displayed in Figure 9. The six different colourfactor contributions to it are shown in Figure 10. It can be seen that the numericallydominant contributions come from the N and N F N colour factors. The contributions of– 54 – N -20000-100000 0 0.1 0.2 0.3 0.4 N F N(1-T) d Cd T -25000250050007500 0 0.1 0.2 0.3 0.4 N -500-2500250500 0 0.1 0.2 0.3 0.4 N F /N -200-1000100 0 0.1 0.2 0.3 0.4 -20000200040006000 0 0.1 0.2 0.3 0.4 N Figure 10:
Different colour factor contributions to NNLO coefficient of the thrust distribution. In N and N colour factors: solid: corrected for large-angle soft terms; dotted: original result. these two colour factors are of opposite sign, with N being of larger absolute magnitude,thus resulting in a total positive result. Contributions at the 10% level of the total comefrom the N F and N colour factors, N F /N amounts to about 5%, while the most subleading1 /N colour factor is below 1%.To illustrate the independence of our results on y , we display the different colour factorcontributions to C ( T ) as function of − ln(1 − T ) in Figure 11 for different values of the phase– 55 – N -10000100020003000x 10 N F N(1-T) d Cd T -80000-60000-40000-20000020000 0 2 4 6 8 10 N -40000-30000-20000-10000010000 0 2 4 6 8 10 N F /N -20000200040006000800010000 0 2 4 6 8 10 -ln(1-T) N y = 10 -7 y = 10 -6 y = 10 -5 -ln(1-T) Figure 11:
Dependence on phase space cut y in different colour factors. We see that the resultsare independent of y for − ln(1 − T ) <
4, but, as explained in the text, differ at larger values of − ln(1 − T ) space cut y = 10 − , − , − . By rescaling all phase space invariants to the total centre-of-mass energy squared, y becomes dimensionless. Since the value of (1 − T ) determines thetypical scale of the smallest resolved invariant, one must require y to be several orders ofmagnitude smaller than (1 − T ) for the cancellation between matrix element and subtraction– 56 –erm to be accurate. Figure 11 shows very clearly that over the phenomenologically relevantrange, i.e. 0 . < − T or equivalently, − ln(1 − T ) <
4, our results do not depend on y .As (1 − T ) approaches y (starting at about (1 − T ) ≈ O (1000) y ), the calculation becomesunreliable as expected. This behaviour can be understood to arise from the fact that thesubtraction terms converge to the full matrix element only once all unresolved invariantsare much smaller than any of the resolved invariants.The numerical convergence of our calculation deteriorates for lower values of y for tworeasons. • the absolute magnitude of matrix elements and subtraction terms increases for de-creasing y both in the five-parton and four-parton channel. Consequently, numericalcancellations between matrix elements and subtraction terms happen over larger or-ders of magnitude, thereby enhancing numerical rounding errors. • the four-parton one-loop matrix elements start themselves to become numericallyunstable because of the presence of inverse Gram determinants, which can becomesingular inside the integration region.Therefore, for all phenomenological applications to the thrust distribution [33], wechoose y = 10 − . For applications to other event shapes, one expects a similar behaviour,and one must first determine the value of y required for reliable predictions in the phe-nomenologically relevant range for that observable. The following has been added compared to the original version of the paper:
The terms ofthe form d σ ANNLO in the five-parton and four-parton contributions to the N and N colourfactors were only implemented in this revised version. They lead to changes in the numericalvalues of the NNLO coefficients which are most pronounced in the approach to the two-jet region. In a recent work, Becher and Schwartz [A] have computed the logarithmicallyenhanced terms which dominate the thrust distribution in the two-jet region using soft-collinear effective theory. They identified a disagreement with our original numerical resultsfor the thrust distribution in the two-jet region for these two colour factors. Our newresults are displayed in Figures 9, 10 and 11, and are now in full agreement with the resultsobtained in [A]. Our numbers also agree with the results obtained in the implementationof [B].In the genuine three-jet region, which is relevant for precision phenomenology, thechanges have a minor numerical impact. The corrections to the NNLO N and N colourfactors also affect all other event shape distributions [C] in a similar manner; minor nu-merical effects in the three-jet region, but more significant effects in the two-jet region.
15. Conclusions and Outlook
In this paper, we provide a detailed description of the calculation of NNLO QCD correctionsto three-jet production and related event shapes in electron-positron annihilation. At thisorder, three-parton, four-parton and five-parton subprocesses contribute. The three-partonand four-parton subprocesses contain explicit infrared singularities from loop corrections.– 57 –our-parton and five-parton subprocesses contain singularities which only become explicitafter integrating the contributions over the phase space relevant to the three-jet final states.Those singularities arise when one or two partons become unresolved (collinear or soft).For an infrared-safe observable, adding together all infrared singularities, one observes acomplete cancellation, resulting in an infrared-finite result.To implement the four-parton and five-parton processes in a numerical programme,one has to devise a procedure for extracting the implicit infrared singularities from them.We extract these singularities using subtraction terms which numerically subtract all in-frared singularities from the five-parton and four-parton channels. The subtraction termsare then integrated analytically and combined with the three-parton channel, where theycancel all explicit infrared poles. The subtraction terms are derived using the antenna sub-traction method, which is based on antenna functions encapsulating all unresolved partonicradiation emitted from a pair of hard radiator partons.Three-jet production at NNLO receives contributions from seven different colour fac-tors. Among these, only the six colour factors of non-singlet configurations require subtrac-tion, while the singlet colour factor is separately finite in all three partonic channels. Wedescribe the construction of the antenna subtraction terms for the six non-singlet colourfactors in detail, and demonstrate the cancellation of infrared poles.All partonic channels have been implemented in a parton-level event generator pro-gramme
EERAD3 , which can be used to compute any infrared-safe observable related tothree-jet final states in e + e − annihilation. We devised various tests of the implementation,demonstrating in particular the correct numerical cancellation between matrix elementsand subtraction terms and the independence on phase space restrictions deep inside thesubtraction regions.We observe that the largest part of the NNLO correction is contained in the twoleading colour factors N and N F N . The remaining four colour factors yield correctionsat or below the ten per cent level.First phenomenological results on the thrust distribution at NNLO were obtainedalready in an earlier paper [33]. In the thrust distribution, the NNLO corrections amountto about 15% of the total result. They are lower in magnitude than the NLO corrections,indicating the perturbative stability of this observable. Inclusion of the NNLO correctionsconsiderably reduces the dependence of the result on the renormalisation scale.At LEP, a wide variety of QCD event shapes was measured to high precision [70].The accurate extraction of the strong coupling constant α s from these data sets was upto now limited by the theoretical uncertainty inherent to the available NLO calculations.We expect that our new NNLO results will improve this situation considerably. Numericalstudies of other event shape variables and of the three-jet rate are ongoing, and will bereported elsewhere.On approaching the two-jet limit ((1 − T ) → e + e − → ep → (2 + 1) jets and pp → V + 1 jet at NNLO without much mod-ification. In these cases, which involve partons in the initial state [41], the same antennafunctions are used with different antenna phase spaces. To accomplish the above-mentionedNNLO calculations therefore still requires the analytical integration of the relevant antennafunctions over the phase spaces relevant to their initial-state kinematics, which appears fea-sible with present technology. NNLO calculations of other exclusive observables at hadroncolliders, such as pp → Acknowledgements
We wish to thank Zoltan Kunszt for many comments and for his continuous encouragementthroughout the whole project. Much of this work has its foundations in earlier projects,carried out in a very pleasant and fruitful collaboration with Ettore Remiddi, whom wewould like to thank for these early contributions, and for many discussions.This research was supported in part by the Swiss National Science Foundation (SNF)under contracts PMPD2-106101 and 200020-109162, by the UK Science and TechnologyFacilities Council and by the European Commission under contract MRTN-2006-035505(Heptools).First steps in this project were taken while the authors attended the KITP Programme“QCD and Collider Physics”, which was supported by the National Science Foundationunder Grant No. PHY99-07949.
Note added:
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