JHEP12(2018)107 Springer July 10, 2018 : , December 1, 2018 December 18, 2018 November 23, 2018 : : : Received Revised Accepted Published [email protected] Published for SISSA by , https://doi.org/10.1007/JHEP12(2018)107 [email protected] , [email protected] , [email protected] , . 3 1806.09570 The Authors. c QCD Phenomenology, Jets

We present a new method for the local subtraction of infrared divergences , [email protected] Sezione di Torino, Via P.INFN, Giuria Sezione 1, di I-10125 Torino, Torino,Via Italy P. Giuria 1, I-10125E-mail: Torino, Italy [email protected] Dipartimento di Fisica and Arnold-Regge Center, Universit`adi Torino, Open Access Article funded by SCOAP Keywords: ArXiv ePrint: to final-state massless particles.subtraction at We NLO, show we how deduceNNLO, our in and method we detail provide the compactly a general organisesat proof structure of NNLO. infrared of principle with the a subtraction complete terms application at to a simple process Abstract: at next-to-next-to-leading orderOur (NNLO) method in attempts QCD, toa for conjugate generic sector the minimal infrared-safe partition localanalytic observables. of counterterm integration the of structure the arising radiation counterterms. from phase In this space first with implementation, the the method simplifications applies following from L. Magnea, E. Maina, G. Pelliccioli, C. Signorile-Signorile, P. Torrielli and S. Uccirati Local analytic sector subtraction at NNLO JHEP12(2018)107 28 ] for a review of re- 31 1 47 50 24 41 42 15 4 – 1 – 6 19 26 23 33 33 11 37 34 34 5 16 4 15 1 38 3.5.1 Integration3.5.2 of the mixed double-unresolved Integration counterterm of the pure double-unresolved counterterm 3.4.1 Integration of the single-unresolved counterterm 4.2 Local4.3 subtraction Collection of results 3.6 Real-virtual counterterm 4.1 Matrix elements 3.2 Sector functions 3.3 Definition of3.4 local counterterms Single-unresolved counterterm 3.5 Double-unresolved counterterm 2.2 Sector functions 2.3 Definition of2.4 local counterterms Counterterm integration 3.1 Generalities 2.1 Generalities the development of aamplitudes number to of high novel orders, techniques,determination for for of the precision parton study distribution calculations of functionscent of (see, final-state developments). scattering for hadronic In example, jets, particular, ref. andexperimental a [ for data consequence is the of the accurate the fact current that and the expected next-to-next-to-leading precision perturbative of order (NNLO) 1 Introduction The increasing precision(LHC), of together with experimental the complexity measurementssions, of constitute at the a final the severe states challenge currently Large probed for Hadron in theoretical hadronic calculations. Collider colli- This challenge has driven B Soft and collinear limitsC of sector Composite functions IR limits ofD the Results double-real for matrix the element mixed double-unresolved counterterm 5 Conclusions A Commutation of soft and collinear limits at NLO 4 Proof-of-concept calculation 3 Local analytic sector subtraction at NNLO Contents 1 Introduction 2 Local analytic sector subtraction at NLO JHEP12(2018)107 ] ]. 31 , 29 – 30 25 as well [ 3 S α ]. These functions are 11 – 2 ], so that the ‘subtraction 42 – 34 – 2 – ]. Real-radiation matrix elements have also been 18 ]. General theorems then ensure that, when consider- 13 , ], with partial information available at 12 24 – 19 [ 2 S ] subtraction methods, based on the idea of introducing local coun- α 33 [ = 4, and all infrared singularities must be cancelled before this stage of CS d ], or be factored into the definition of parton distribution functions, in the 17 – ] and 14 32 [ dimensions. Integration over unresolved radiation must therefore be performed nu- At NNLO, numerical and conceptual challenges related to the proliferation of over- At NLO, the first fully differential results for jet cross sections were obtained [ Even with this detailed knowledge of the relevant theoretical ingredients, the practical In principle, the problem is well understood. Infrared singularities (soft and collinear) d problem’ can be considered solved to this accuracy. lapping singular regions becomeof much several more different significant. methods,collider This which has processes. have led been to successfully NNLO the applied differential development to distributions a for number hadronic of final simple states in electron- usually described as ‘slicing’).FKS Subsequently, two generalterterms algorithms for were all developed, singular the regionsto of achieve phase space, the and cancellation thendescribed of integrating as them poles ‘subtraction’ exactly without in inin need order a full of strict sense). slicing generality parameters in These (which algorithms fast is are and usually currently efficient implemented NLO generators [ the real-radiation matrix elements inreal-radiation the phase singular space regions, to and match requires the a remapping Born-level of configurations. the by isolating singular phase-space regions andcancellation treating by them integrating approximate separately, performing matrix the elements pole within those regions (a procedure reconstruction algorithms as wellmatrix as elements complex become increasingly kinematic intricate,in cuts; and furthermore, they real-radiation cannot bemerically analytically in integrated the calculation is reached. This cancellation involves a careful use of approximations to problem of constructing efficientties and for general generic algorithmsbe infrared-safe for observables highly handling beyond non-trivial. infrared next-to-leadingcollider singulari- observables order The have a (NLO) origin complicated proves of phase-space to structure, the nearly difficulty always involving lies jet- in the fact that typical hadron- with singularities arising fromation the [ phase-space integrationcase of of final-state collinear unresolved initial-state radiation radi- shown [ to factorise infully soft known and at collinear order limits, and the corresponding splitting kernels are arise in virtual correctionsknown to as factorise poles from in scatteringgeneral amplitudes dimensional definitions in regularisation, in terms terms of and universal ofin all functions, gauge-invariant turn matrix which such determined elements admit poles by [ a are fully small known set up of anomalous to dimensions threeing which, loops infrared-safe in [ the cross massless sections, case, virtual are infrared poles must either cancel, when combined at LHC. Acuracy crucial is ingredient the for treatmentto the of the calculation relevant infrared scattering of singularities, amplitudes,real differential and which radiation. distributions from arise to the both this phase-space in integration ac- of virtual unresolved corrections in QCD is rapidly becoming the required accuracy standard for fixed-order predictions JHEP12(2018)107 ]. ], and 58 – Stripper 62 56 ], currently ap- method [ 61 – LO processes have 3 59 ] within the subtraction [ 46 , 45 ]. New ideas are also being in- Antenna framework [ 64 , Projection to Born 63 [ ], while among the first hadronic pro- ], and 55 44 – , 52 43 – 3 – ]. A number of hadronic processes with up to two CoLoRFulNNLO LO. The need for improved and efficient subtrac- slicing [ 3 51 T – q 48 ]. A crucial ingredient is then the choice of ‘sector functions’ used to 32 Nested Soft-Collinear subtractions ], and the first limited applications to differential N ]. slicing technique [ ], and the associated production of a Higgs boson and a jet, achieved with 69 66 – , 47 67 65 ], we present a new approach to the subtraction problem beyond NLO, which at- 70 subtraction [ In order to achieve the desired simplicity, we attempt to take maximal advantage of In this paper, and in a companion paper devoted to the underlying factorisation frame- There are several reasons to surmise that existing methods for NNLO subtraction can N-Jettiness FKS build the desired partition:plify these the functions analytic must integration obeyA of a counterterms second set when crucial of sectors ingredient summomenta are is rules within appropriately in each the recombined. order sector, availability to allowing of sim- for a simple flexible mappings family to of Born parametrisations configurations of in dif- of massless final-state coloured particles. the available freedom inextending the ideas definition that of the haveelement been local of our infrared successfully approach counterterms, implementedstrained is exploiting at to the and NLO. contain partition a of In phase minimal particular, space subset a in of key sectors, soft each and of collinear which is singularities, con- in the spirit of dence from ‘slicing’ parameters identifyingof singular analytic regions information of phase incourse, space; the computational maximal efficiency construction usage of and thearching numerical integration implementation. goals, of These and the are, infuture clearly, counterterms; over- this more and, general paper of implementations. we In present particular, the we first focus basic for the tools moment that on we the hope case to use in tempts to re-examine theadvantage fundamental of building all blocks available of information,help and the to build subtraction streamline a and procedure, simplify minimal take futureview, structure applications. which should The will aim ideal hopefully subtraction toobservables; algorithm, achieve exact in our the locality of following infrared goals: counterterms in complete the generality radiative across phase infrared-safe space; indepen- the technique of troduced [ appeared [ work [ of more than twonext jets, order and in it perturbation willtion similarly theory, algorithms be N is in usefulof fact to existing leading compute ones: to simple the processes examplesplied development include at to of the processes the other with methods, electroweak or initial the states, the refinement with various approaches, including be generalised and improved:tionally on very demanding, the either one in termsthe hand, of large-scale current the numerical applications analytic effort calculations have required; involved,predictions been or on will because computa- the of soon other hand, be it needed is for clear more that precise complicated NNLO processes, such as the production cesses involving coloured final-statewere particles the to production be offramework studied top-antitop [ differentially quark at pairs, NNLOthe achieved there in [ final-state coloured particles at Born level have since been studied at the differential level positron annihilation were first computed in [ JHEP12(2018)107 , i k (2.5) (2.1) (2.2) (2.3) (2.4) partons n , i (1) n A . ∗ ) (0) n X ( ..., A h by revisiting the NLO sub- +1 n ) + ..., 2 i Re k + R δ ( collinear singularities. (2) n +1 = 2 , we use the results to complete a n A and 4 constructs a complete NLO subtrac- NNLO final-state partons with momenta Φ dX d 2 dσ n ) + i Z k + ,V ( , 2 ) ) +

(1) n NLO are schematically written as X X – 4 – A dX +1 ( ( dσ (0) n n n X denote the Born, real, and virtual contributions, A ) + +

i B δ V δ k V ( = n n LO Φ Φ (0) n dX dσ d d , outlining the status of our method and the forthcoming A , and = Z Z 5 observable R ,R ) = = = 1 i , we attack the NNLO problem, displaying the general struc- dσ 2 dX k

(IR) to indicate collectively soft ( B LO 3 n NLO (0) n dX dX ), dσ A A dσ = 0, are expanded in perturbation theory as

2.4 2 i infrared k = ]. B 70 ) and ( , with describing the Born process. Correspondingly, differential cross sections with 2.3 (0) n A , . . . , n With this general strategy in mind, we begin in section We use the term 1 = 1 In eqs. ( respectively, with where, up to NLO, with respect to any infrared-safe appearing in the finalwe state allow at for Born colouredour level. and arguments. colourless We Scattering assume particlesi amplitudes a in involving colour-singlet the initial final state, state, and the latter not affecting 2 Local analytic sector2.1 subtraction at NLO Generalities We restrict our analysis to reactions featuring only massless particles, with for a specific subsetproof-of-concept calculation of of singularities, NNLO subtraction and,pairs. for in the We leptonic section conclude production in ofsteps two section needed quark to constructtechnical a details. competitive algorithm. Four appendices contain a number of local counterterms and appropriateon parametrisations, the and unresolved phase we space.tion integrate algorithm the Effectively, for section counterterms massless finalintegrations. states, which In stands out section ture for of the subtractions simplicity of incounterterms the our relevant required approach, for defining massless sector final functions, states. and We then constructing perform all the local relevant integrations structure of factorised kernels infactorised multiple singular structure limits, of which scattering followspresented amplitudes: in in general [ a from the detailed analysis of this structuretraction will problem. be We define sector functions satisfying our requirements, we introduce ferent unresolved regions. Finally, it is necessary to take maximal advantage of the simple JHEP12(2018)107 +1 n = 4, FKS (2.7) (2.8) (2.9) (2.6) Φ (2.12) (2.10) (2.11) d d dimen- dimen- d d , we may n Φ /d is infrared safe, +1 , n X , b Φ d  = 1 ) , inspired by the = ab ij X ( ) W W n ij rad ( MS scheme. Furthermore, = 4 and do not present any π b Φ K δ X ∈ . d d ) ab +1 X n ij ) in subtracted form as ( b Φ n C d 2.4 − K δ space-time dimensions, the virtual con- ) K, ]. , ) a counterterm )  +1 ) X n ( 2 sector functions X rad 16 2.4 ij ( , b Φ b ( Φ , while the real contribution, finite in n +1 − d  d π δ n 15 a , – 5 –  ∈ Z Z I R δ 6= = 4 = i ab / = + d +1 ∀ ∀ I ct n V

Φ  d n , which cancel those of virtual origin if NLO   , , , , Φ dX d representing the observable under consideration, computed dσ Z i = 1 = 0 = 0 = 1 Z must reproduce all singular limits of the real-radiation contri- + X ij ik ab ab = K W W W W i i i +1 ij NLO 6= n 6= S dX k X b C X Φ ), with i, j dσ i i d S X , and must be sufficiently simple to be analytically integrated in R − ], and satisfying the following properties +1 X ( n in the counterterm, under the assumption that the two coincide in all singular δ 32 Φ d +1 ≡ -body kinematics. n i ) b The NLO-subtraction procedure avoids analytic integration of the full real-radiation In dimensional regularisation, in Φ d X ( i Our first step inof setting the up real-radiation the phase subtractionmethod space formalism [ at by NLO means is of to introduce a partition where the first andphase-space the singularities, second allowing lines an are efficient separately numerical finite integration. in 2.2 Sector functions and rewrite identically the NLO cross section in eq. ( to limits. Defining nowintroduce the the (single) integrated radiation counterterm phase space as The combination bution sions. Note that we allow for the possibility of simplifying the phase-space measure the radiation phase space.sions results The in phase-space explicit integration polesensuring of in the such finiteness singularities of the in cross section [ amplitudes by adding and subtracting to eq. ( δ with tribution features up tois double characterised IR by poles up in to two overlapping singular limits of soft and collinear nature in where the virtual correction has been renormalised in the JHEP12(2018)107 i = and and µ local q i i (2.13) (2.14) (2.15) , only S ij functions as + 1 final-state , ij candidate b n e W b + e a e . ) ja + 1) the ij δ w 1 ib i ), it is easy to verify that all δ e , ) express the fact that a given . + j = , . . . , n k 2.14 qj jb · ij ij s δ 2.11 combinations of indices associated i σ = 1 ia k qi s s δ i s ]) ( all = 2 = µ 39 i = ( k ) ensure that, in a given sector ij ij ab in this sector can thus be built collecting all be the squared centre-of-mass energy, with W ). In order to provide an explicit definition ) and eq. ( 2.11 s method, and as we will further discuss in the R ij – 6 – C , s 2.12 2.10 i k FKS , w , · ) and ( q qi , s kl ) to ( s are projection operators on the limits in which parton σ al ij 2.10 k ij one soft and one collinear singular configurations, 2.9 = 2 = σ ab 6= ) are satisfied, and in particular one finds that i /w C P qi e 1 k, l ) runs over all massless final-state partons, including those /w s a 1 only 2.12 6= = P l and 2.14 i ij ia S δ ) is a normalisation condition that recognises the W . ) to ( } = limits (as well as their product) act non-trivially. A selects 2.9 2.9 ab ij ij ) imply that, upon summing over for the real matrix element W C ij, ji W i ij { S 2.12 K ) = ij ( and the π , let us introduce some notation: let i ij 0 ) the centre-of-mass four-momentum, and ~ There is ample freedom in the choice of sector functions, the only requirement being S , respectively, among all those present in the real-radiation matrix element. The sum s, W ij become collinear (i.e. their relative transverse momentum approaches zero), respectively: √ 2.3 Definition of localAs counterterms discussed above, propertiesthe ( counterterm from which the desired properties follow. The double sum inthat eq. are not ( associatedextensions, with as singular detailed limits. below.properties With This in the choice eqs. definition is in ( made eq. in ( order to ease NNLO We now define NLO sector functions as (see also [ ( momenta of the radiative amplitude. We set following; analytic counterterm integration in turnthe makes correctness it of possible the to show singularity in structure closed of form the subtractionthat terms. they satisfy theof relations ( sector function C rules in eq. ( to sectors that survive inreduce a to given soft unity. orcounterterms, collinear This as limit, fact the is proves corresponding well sector crucial known functions in for the the analytic integration of the subtraction becomes soft (i.e. all componentsj of its four-momentum approachthe zero), and action partons of these operatorsdetail on below. matrix Eq. elements anda ( sector unitary functions partition will of be phase described space. in Eq. ( where JHEP12(2018)107 , j k FKS is the (2.20) (2.21) (2.18) (2.19) (2.16) (2.17) and } i k ) is equal is the set quantities k { ), the term i / } , 2.17 all k
2.16 { i / } R k ij {  i is defined as C , and can be deduced 1 , k i B j / by removing
) i N / i . In eq. ( ji } ( } jr 2.4 k k I B W { { j ij , f  + ij W C µν ij 1 W R B N W ) , R µν ij ) , ij  ij 1 Q ( ,
 C ij = 2 i / C i E L ; when a function takes the argument h  } +  i γ S , and not on π k e k  ≡ , k , k 4 { A 2 − j / j / R. the Euler-Mascheroni constant; µ i i X / , k ij / i, j>i ij j / ij } } lm  i / E W k k C + C } S B γ { { – 7 – i k removes the soft-collinear singularity, which is R ) +  R  i { S ( lm πα i ) i i B  ij µν I S S ij ij ( i B ij momenta obtained from B C i i is structurally similar to, and as simple as, the i = 8 C 6= i P W 6= ; the order in which the projectors act is arbitrary, as ij ij i 6= n l by removing 1 i µν X S m ij ij ij P S ij X S W N i, j h P 1 } K i C C − ), the content of each square bracket in eq. ( k ). As will be detailed in section 1 1 1 i S ij ij ij = { + ij s s s i N N N S −N A i ij h 6= that are singular in such soft and collinear limits, and taking C 2.12 j i S X ≡ K 6= ij , however it has the advantage of being defined without any h i + ) = ) = ) = i X 6= } } } i,j ij W i k k k S X i, j X { { { − R ( ( ( = = = ( R R R i ij K ij ij S K C C i + 1 final-state momenta in the radiative amplitude, while S n is the renormalisation scale and the soft limit is meant to act only on ), it depends on the set of µ B , k j / momenta obtained from ) i / Our candidate counterterm A } n i k { S where set of the of ( where we introduced several notations. Specifically, the prefactor the universal soft and collinearin the NLO singular kernels which limits. factorise We from write the radiative amplitude they commute (see appendix from the sum rules into eqs. 1 ( upon summation over sectors, a crucial propertycounterterm for for counterterm integration. sector explicit parametrisation of the soft and collinear limits. Its constituent building blocks are Here and in theto following, their projection right, unless operators( explicitly are separated understood byfeaturing to parentheses: act the for on composite instancedouble-counted operator in in the the sum expression care of correcting for the double counting of the soft-collinear region. We define therefore terms in the product JHEP12(2018)107 = 0 , and g (2.28) (2.25) (2.26) (2.27) (2.23) (2.24) (2.22) i ¯ k f δ , ! i x , .  − j  2 j x i = 0 = 1 x i − x j 1 x µ j 2 x ˜ k 1+ colour generators, and , − is a gluon, and + +

1 µ a r i i , in a flavour-symmetric k µ F i  T ij ˜ k C jr R P } s T ¯ q , ij } q, from the Born matrix element + ¯ = 0. We now write a Sudakov q s µ r { q j 2 k k ir f }{ ¯ k s δ r j (0) n g f k i i 2 a . · A − , f f , ˜ k ) = 1 if parton k { δ µ µ r δ 2 m g k k + im i = 0 + f s r T a lm  , as r δ 1 is the Born-level squared matrix element ≡ 2 · k s  x j il k ¯ · k k j l s µ ), so that · k B x x with respect to the collinear direction ¯ k − , as they become collinear. We introduce a T g i a i ( µ j − µ x  µ a f a ˜ k i, j a ∗ k δ – 8 – ˜ k ˜ 2 i k + x along = x 6= j (0) n i j + = , so that , satisfy x − a x ¯ k x i r A ) µ and x i · 1+ a ( ¯ lm k , x + i, j ; finally, 2 a k = µ i I  i a j j · is a massless reference vector (for example one of the ˜ k k , k jr = x k x x k F µ s r j lm a C  k k , with  + = ar B + g } s i A j ), as µ a + − k f k ir C k { δ µ i µ s , for 2 } i, j k k = ¯ q a g , and a j ˜ k = q, = k f ≡ ij { x i δ r s r a f µ g k ( i − δ k · k f · = a a µ a δ + k k k j k , defining the collinear direction, using , relevant for soft-gluon emissions, is given by ), while k µ ) ) = = = · i ¯ k ( j lm i 2.5 a µ a I k x ,x ˜ k i x ( = 2 ij indicates the flavour of parton 2 i P k f = is the spin-connected Born-level squared matrix element, obtained by stripping the The NLO soft and collinear kernels are of course well known. In our notation, the ij P µν The transverse momenta We can now writenotation, the as spin-averaged Altarelli-Parisi kernels where we defined the transversethe momenta longitudinal momentum fractions where on-shell momenta of theparametrisation set of where otherwise. In orderparametrisation for to the write momenta themassless vector collinear kernels, we begin by introducing a Sudakov and from its complex conjugate. eikonal kernel defined in eq. ( is the colour-connected Born-level squaredB matrix element, with spin polarisation vectors of the particle with momentum and inserting their sum JHEP12(2018)107 + ¯ q , as j ), or f j δ (2.30) (2.29) (2.31) 2. The q i / , x f i δ x ( = 1 = . commute ij ) momentum i } R , P ¯ . ) gauge group q x T q c ) , k i j , )  j / ( + 1)-body phase jr  }{ N is soft-finite). As i x / ij j i I -body momentum } − n f P  − x j i n k µν ij 1 f f j 2  { { x (1 C Q δ i R − F x T 1 2 C } = 2 ¯ q } q ¯ q , which can be obtained by − , and , ij q, }{ ¯ q ij 1 { j P ij f # j f (indeed,  s i δ f i ν i f δ S R 2 ˜ i k { ij g + is necessary in order to achieve a i i ˜ δ k µ T i P j f ), and similarly for the azimuthal x ˜ k } fq δ x ¯ + q µν ij δ 2 + 1 momenta in the radiative ampli- q i j 2) + Q f n 2.28 }{ x = j i j ⇒ C − f } x x g i ¯ = q d j limit. The Born-level squared amplitudes ) f A f q,  { { ) are not yet suited for a proper subtraction δ δ ij C f + ( − ): in other words, the two limits δ 2 − C + , g j µν (1 j – 9 – 2.20 j defined above contains all phase-space singular- i g x f x F ir jr i x 2.23 δ momenta that do not satisfy s s 2 − ij , consistent with the normalisation C x g c j j " i g f f K n A f limit, and, similarly, in the set ( j N ij δ f C C C i ) and ( δ g Q − 2 = i } 2 S -body) and radiation phase spaces, thereby allowing one ¯ f q . Subtracting from the collinear kernels their soft limits, . Importantly, the same soft-collinear kernel is obtained g on the collinear kernel g i δ n A } q, j f i ¯ 2.19 q { f A . C δ ) = ) = i q δ − S f j j { ji g j δ i ), ( = ij f f Q , x , x δ , with no double counting, the kinematic dependences on the P δ i i i + j is a set of x x ij x = F x . The Casimir eigenvalues relevant for the SU( ) and ( ( 2.18 ≡ = i / j c C f ij can be written as W ij , one gets } µν ) ij µν ij N j k C Q + Q P R Q { A µν ij 2 (2 , g , x i j = / g ji Q = f i x f δ P 1) ( ij δ µν ij A Q hc − = ij = Q -body kinematics for all choices of the C 2 c and for P ij n is off-shell away from the exact ij = N P ) to denote the collinear kernels of eq. ( j = P µν ij j k i jr f Q hc = ( S ij . In the following we will use interchangeably the notations + C , s q P i F subtraction of phase-space singularities. The collinear kernels satisfy the symmetry i ir f k The final ingredient is the soft-collinear kernel for sector s C δ ( appearing in the counterterm must instead feature valid (i.e. on-shell and momentum ¯ q = i ij f B conserving) tude. A kinematic mappingspace is into thus the necessary, in product order ofto to Born integrate factorise ( the the counterterms ( only in the latter. Although the candidate counterterm ities of the product right-hand sides of eqs. ( algorithm. Indeed, conservation away from the exact k also by taking the collineardiscussed limit in of detail eq. in ( one appendix gets the hard-collinear kernels detailed in appendix where We note that the presencelocal of the azimuthal kernels properties acting with the soft projector δ P kernels are azimuthal kernels where we defined the flavour delta functions JHEP12(2018)107 and , we ], as i ij (2.38) (2.35) (2.37) (2.32) (2.34) (2.36) (2.33) 33 W R ) ij C : partons be the on-shell i r in the sum over S c R k = , light-like momenta − a, b, c, c ij ), is the ‘spectator’. c ij k n 6= C , C i i bc algorithm, in any given
2.24 s  ,( if ) ji , and , ij j abc + . We then define the local each term W s  ilm . , as , ) ( = m ac FKS i } + ) s  o k b ijr ) ¯ k ) c ( = , ij { abc k i } = = ( ijr ], we treat separately the soft and abc  c ( ¯ k W ) ) ( } c + } = { ¯ ¯ k k 70 lm . Now one can construct an on-shell, ¯ b k  abc abc { ij a , ( ( i c { k B ) B ¯ ¯ k k C  ) a, b ) i h + , and i abc ( lm l ( ( b jr µν ) as a 6= I ¯ k I k B i R, c i = j , , X 6= i, j>i f c , introduced in eq. ( / 6= = ij c 2.32 l µν b b / r ij X m C k / a ) + , C P 1 1 } i – 10 – i bc 1 k in different sectors and limits. Consistently with abc R ) are built in terms of Mandelstam invariants, and N ij S ( c s { i = s N ¯ k −N i S n ab + of eq. ( , ij a s i + 2.20 c = a ac ) ij W ) = ) = ) = 2 a, b, c ) s 6= } } } )–( ij W abc k k k m ( b abc i − { { { ( C o ¯ k ( ( ( b S , and ) } i , we choose to map differently 2.18 k b i ¯ k S R R R ij , 6= abc { h , and in particular the condition i ( singular limit. In order to take advantage of this freedom, we j + m i X a ij ij ¯ S bc W k 6= a h be two final-state on-shell momenta, and let s -tuple of massless momenta C C n k X R i,j i n i b i + S = = k X − S ) ) ac = s abc abc ( ( b the same and + K ), with assignments } ¯ k ¯ a k ab { k s 2.18 = can also be expressed as abc ) s in eq. ( Since the kernels in eqs. ( abc ( } specify the collinear sector, while parton ¯ k where the barred projectors selectof soft on-shell and momenta collinear to limits, the and kernels. assign Explicitly the appropriate set l, m counterterm as the general structure of factorisedthe virtual hard-collinear amplitudes limits. [ For thechoose hard-collinear to kernel assign in sector the labels j For the soft kernel, ensures momentum conservation.{ Note that the collection of the Next, we select different values of where introduce now a generic Catani-Seymourfollows. final-state mapping Let and parametrisationmomentum of [ another (massless) parton,momentum with conserving have not yet been parametrisedappropriate at kinematic this mapping stage, there into is order follow. still to full In freedom maximally particular,sector to simplify one at choose the can the variance employ most analytic with differentcontributions what integrations mappings to done for different in singular the limits, or even for different JHEP12(2018)107 ), , . ) y R 2.12 , each (2.41) (2.44) (2.40) (2.42) (2.43) (2.39) ij ij − C K + 1)-body (1  n − . To obtain , , i ab )  z pairs, with the abc )). The form of W s , − ij ) ij -independent, and ij y A.4 j . The expression in C (1 y, z, φ z ij W − ; 2 ) = ); in the hard-collinear ) R y
)(1 abc ab ( bc z ij ¯ s 2.12 − R. W  C − in our definition of
i ij (1 j ) and eq. ( j y S S rad C , h Φ − = (1 − ↔ A.3 ), as bc d dz i s 1 i ij bc 0 S ≡ ac + Z C , and s 2.32 ) : − ac ab + ij dy 1 abc 1 s i ( rad 0 W W S , s ij Z Φ i = R S φ C  abc = 2 s = = 1 (see eq. ( − , exploited the collinear sum rule in eq. ( ) ij – 11 – j X y i, j>i ab ji K sin , z − + ↔ W W ) is a sum of terms, each factorised into a matrix i ab i (1 dφ R abc ij , d s S π i s z ) 0 C 2.39 . This operation matches the fact that the integrated S Z j = =  abc can be chosen differently for different S ( rad 1. We use these variables to parametrise the ( i − y ac 1 , Φ X FKS s = ≤ d ) is now suitable for analytic phase-space integration. ij ) i, j ) should be chosen for all permutations of = z  ij ( K , exploiting the soft sum rule in eq. ( r abc i j 6= ≤ K W ( n 2.40 N , s 6= Φ ij r X i,j d ≡ C abc ) i = = S y s K 1 and 0 +1 = n y, z, φ ≤ under the interchange Φ ; ab d y s s ij ( ≤ C ) can be rewritten in terms of a sum over sectors of local counterterms ) we have used the symmetry under exchange rad Φ d 2.35 2.39 We start by introducing the Catani-Seymour parameters Upon summation over sectors, the integrand becomes independent of sector functions. Z leading to the explicit expression so that 0 phase space, consistently with the mappings in eq. ( the counterterm in eq. ( which satisfy In the soft termperformed the we sum have over consideredcontribution that we the have kinematic usedoperator mapping the symmetry is ofand the the kinematic fact mapping that and of the collinear obviously is not split into sectors. In fact The counterterm defined inelement with eq. Born-level ( kinematics,The analytic multiplying integration a of the kernel latterover in with the all radiation real-radiation phase kinematics. sectors, space proceedscounterterm as by first must summing done eventually in cancel the singularities of the virtual contribution, which where it is understood thatas the that action of of un-barred barredeq. projectors ones, ( on namely sector functions is the same 2.4 Counterterm integration constraint that the same eq. ( containing all the singularities of the product where we stress that JHEP12(2018)107 , (see , and # ) (2.49) (2.50) (2.46) (2.47) (2.45) (2.48) , . The j } ) ¯ ) q ijr  q ( j = ( ijr is the az- ( jr ¯ k }{ ), taken as b ¯ j s O c φ f , i y k ~ , , i , as the latter f +  , ) { . . b  # δ = ij µν pr k ) ~ = abc η   ( Q  abc ) ij 2 a ijr R } s ( ln ¯ , s pr ¯ T k ijr } − η ( ) { = 2 ¯ − k } } 3 ) { ln ¯ ¯ k k 8 3  { { + abc − (  ( c + µν  ¯ k R g  1 µν B ), while · i + 2 f
) B  j δ µν  1 } f -body final state. We note that S ¯ abc 2.26 q µν ij hc ( b k  ij N q, ¯ k − Q { with a sum over ‘parent’ partons P R j F i 2 2 f T 1 + S C ij δ 6 ≡ s N  } i, j − ) ¯ q + )  + 4 ) 1 q, g ijr { j abc ( A p ( bc f X ijr i, j>i } ) f ¯ ( δ s C ij ¯ δ k z } } ¯ g q = { C ¯ k p + − q,  f { ) { defined in eq. ( R δ i 1 – 12 –  B f " y i
ijr δ ) B ( rad z, j x hc   is parametrised assigning labels (  ij ) S Φ symmetry factor), carrying momentum  ) P d hc − F , ij  hc ijr 2 − n ,  ( 3 C ) P = i ) Z }  of the hard-collinear counterterm K /ς ) S ¯ ). We have k −  + ijr ) )Γ(2 { − ( jr 2 +1 hc  − g ¯ s  X n − j 2 I i, j>i ijr  ς 1  f ( / 2.33 Γ(2 − B ) δ } y, z, φ  g π hc n +1 ¯ k ; i ij and an arbitrary three-momentum (other than ij ς p f n s Γ(1 { J (4 ς Γ(1 δ ( a X . By defining C  π  k  ~  2 -body phase space for partons with momenta = 2 rad µν √  − A n  X 2 Φ i, j>i  B X ) − hc s i, j>i C s ≡ d µ π I 3 µν ) n +1 =   ς " (4 Z n hc ( S ς ij π s 1 hc − is the × α 2 P 1 N ) K ≡ = − N ) abc ( n = = indicates the symmetry factor associated to the Φ s,  k hc d ( ς I , as detailed below eq. ( hc ij r J (which has absorbed the coincides with the collinear fraction = where in thep last step we replaced the sum over one finds analytic integration of the countertermexactly is therefore to straightforward, all and orders can in be carried out where the integral does not receiveintegrate any contribution to from zero the in azimuthal the kernels z radiation phase space. In our chosen parametrisation, the variable Each term in thec double sum in We first consider the integral where imuthal angle between reference direction, and we have set where JHEP12(2018)107 pole, (2.53) (2.51) (2.52) (2.54) / , is defined as , pole, coming ) )  pr  ( / η ) O −  3 . + ) i − ( lm : the soft counterterm )Γ(2 has absorbed the sym- ) I !#  . By defining, for each ) g  Γ(2 i lm ilm − f ( lm 2 η term, accounting for gluon ilm δ ¯ s  ( rad i A y ln ¯ Φ  Γ(1 ) d C ! 1 2 P = 2 2 ζ ilm − Z il − (   s 7 2 ) }  s ) ¯ k π − { + 2 pairs. The invariant ¯ (4 ilm  (  1 ) ), obtaining ¯ q , and } R. lm = } + 6 q

i ¯ k k /z B { z  S 2 { ) 2.33 lm  ( z  η − i yz + , and the sum R lm light − X 1 } ,  2 ) correctly features a double 1 i ln ¯ ) 1 ¯ B k  – 13 – ) f S =
{ i i

} N s ilm 6= ( lm 6= = (1 2.54 ¯ k l rad
¯ X s m { K } Φ  of the soft counterterm y, z, φ im ¯ k i d s ; s { . Notice that the result contains only a single 1 lm X s /s J I ( p Z B i i B lm l 6= n +1 l 6= i s 6= rad ς l 6= f n X m X ς r Φ 2 Bose-symmetry factor in the C g X / d 1 i l, m f n +1 l δ ς n Z + X ς −N s 1 i , as detailed below eq. ( " , with X = = . Note that eq. (  m ≡ /s n s  ) n +1 I = /ς 2 ς ijr n s µ s c ς s,  +1 1 (  n = s ς S π J and α 2 −N /s l ) = = )), we included a 1 = ijr ( jr s b s I , 2.32 Next we turn to the integral i = ¯ = pr ¯ where in the secondsame step Born-level we kinematic configuration have remappedmetry all factor identical soft-gluonfrom contributions soft-collinear on configurations. the we get the simple result term of the eikonal sum, In our chosen parametrisation can then be analytically integrated, once again to all orders in We parametrise ita by assigning different labels to each term in the eikonal sum, with indistinguishability, and we considered η consistently with the fact that soft singularities are excluded. eq. ( JHEP12(2018)107 ) FKS 2.35 (2.56) (2.55) ! #
kl kr η η  ln ¯ 8 3 ), which is our in the singular , while treating  1 + ln ¯ − ij
2 rad 2.39 } , W kr ζ ¯ k Φ η 7 { d R #) ln ¯ − kl   . ), which in what we have 10 B kl f F η k N F C 6= 2 2.40 ln ¯ 3 2 R C X k, l T } 1 2 } ¯ q ¯ q 6 + q, q, − { { + 4 k 2 k ! f f k A δ  δ  γ C + kl + g + η k f  f only on the product k 2 ln ¯ δ 2 f N ζ  ]), which provides a check of validity of the ij
R

C 7 2 4 } T C

¯ k − 4 k – 14 – { 6 X 6 k . In other words, the counterterm phase space − ) successfully reproduces the pole structure of the
X  and kl } A
rad . Second, in the counterterm definition in eq. ( i A ¯ k B } } C S {
Φ 2.55 C ¯ k ¯ k k } d { g ) are analytically equivalent, but they underpin different 11 { 6= ¯ k k B { g f X k " δ k, l B f 2.40 δ hc + + + has been treated as a constant throughout the computation. A (" I = S  α k +  γ
2 } s µ , provided they correctly reproduce the exact ¯ k {  ) and eq. ( rad S s π b Φ α 2 I d 2.39 = = , which is a direct consequence of having optimally adapted term by term the
 } ¯ k { ) corrections, I  ( Third, eq. ( We conclude this section with three considerations on the structure of the counterterm. We can finally combine soft and hard-collinear integrated counterterms, obtaining, up O philosophies in the implementationpreferred of choice, the subtraction subtraction is scheme. seena as In minimal the singularity eq. incoherent structure ( sum and of ismethod terms, separately but, each optimisable, we of in believe, the which same features presented featuring spirit enhanced is of flexibility. the employed Eq. only ( for analytic integration, represents a single local subtraction limits. In thecalculation, massless no final-state computational advantage case would result detailed fromthe in such an latter this approximation, may however article, become asthe relevant evident counterterm in from phase more the spacenumerical complicated above could implementation. cases. be We leave applied Analogously, these restrictions possibilities in open on order for to future improve studies. the convergence of a we have chosen to applyexactly projectors the phase-spaceis measure exact, andthis coincides feature with is that notspace of crucial measures to the our real-radiation method: matrix one element. could as We well stress consider that approximate phase- kinematic mapping and parametrisation. First, the strong coupling dynamical scale for the couplingit can simply with be the reinstated Born-level in kinematics the counterterm by evaluating The integrated counterterm in eq.virtual ( NLO contribution (seesubtraction for method. example [ Moreover,all we orders note in the simplicity of the integrated counterterms to where we introduced the spin-dependent one-loop collinear anomalous dimension to JHEP12(2018)107 , ) ) 1 , 1 ( ( ) RV (3.3) (3.1) (3.2) K X K ( n +2 δ n , ) is finite in b Φ ) d X RV ( ( RR K ; +2 will be described  n ) +1 n 12 ( b Φ RR δ , d ) associated with K i +2 Z X n ( +1 Φ (1) n and , +1 d ) which correspond to configu- A ) n ) by adding and subtracting 2 δ ( ∗ Z X +1 ( 3.1 K (0) n RR n δ . ) + A h +2 i  X n ) ( Φ (2) n Re 12 d +1 ( A n ∗ K = 2 (0) n + A ) RV δ h 2 ( – 15 – +1 n K Re  Φ d ,RV + 2 +2 features up to a quadruple IR pole in 2 Z

n 2 and to the KLN theorem. It is however clear that the contains all singularities stemming from kinematic con-

b Φ , originating from its one-loop nature, on top of a double +2
d  (0) n (1) n X ) ) + VV ). The distinction between ) A A 12 X Z

( ( 12 n ( = = K are the double-real, the UV-renormalised double-virtual, and the , K ) + + X ) RR V V δ VV ) ( RV 2 singularities of the real matrix element, hence it has the same essence 2 ( n ( +1 Φ K n d K all δ ) , and Z 1 can be schematically written as +2 ( n = VV b K X Φ , d ) with ( +2 X n subtraction, but with much simpler counterterms. Our method at NLO represents NNLO RR ( dX b Φ n d δ Subtraction at NNLO amounts to modifying eq. ( dσ CS = 4, but it is affected by up to four singularities in the double-radiation phase space, Z The sum figurations where exactly twopossible phase-space partons singularities. become We unresolved.these note two that counterterms At the mirror their Dirac NNLO, physical deltaand this meaning, functions with exhausts associated with all and which can be characterisedfeatures as the follows. subset The of single-unresolved phase-spacerations counterterm singularities where only of one parton becomes unresolved, analogously to what happens at NLO. dimension at NNLO is significantlysubtraction more procedure. severe than at the NLO, hence the necessitythree of sets a of counterterms:we write single-unresolved, as double-unresolved, and real-virtual, which stemming from configurations thathas feature up up to to a twosingularity soft double in and/or IR collinear the pole emissions; single-radiation in finite phase due space. to thedifficulty The IR of sum safety evaluating of of and these integrating three complete contributions is radiative matrix elements in arbitrary In dimensional regularisation, d where UV-renormalised real-virtual corrections, respectively, with 3.1 Generalities The NNLO contribution toobservable the differential cross section with respect to a generic IR-safe as thus a bridge between thesevirtues two of long-known both, subtraction and methods, not aiming being at retaining limited the by the mutual3 suboptimal features. Local analytic sector subtraction at NNLO term containing JHEP12(2018)107 . . a  are ) , up (3.6) (3.4) d , the X n ( n RV Φ  ) is both δ ) /d  , and as X 3.5 ) features the c ( +1  n ) , 12 differ from n ( b δ . , and the local b Φ d  RV  K d locally subtracts ( ) I + ) , , 12 ) ( = ) + ) 2 ) = 4, and their sum I ( 2 RV , and ( 2 ( ( c RV − K d ( rad cancels the phase-space ) I K , K  b b Φ ) 2 K , all different d RV ( +2 RV rad rad n ( K b b b Φ Φ Φ K  d d d , and − +1 +1 Z Z n ) ), the sum n n Φ b b Φ Φ X = = ( d 3.5 d ) ) /d 2 − ( +1 +2 ) features the same poles in RV n ( n cancels those poles; furthermore, the finite δ ) X ) b ) Φ 1 ( 1 ( d ( 12 I ( +1 , i, j, k, l K ) (3.5) n I = = 4. The counterterm ,I δ – 16 – ,I X ) 2 +2 ijkl d  ( , ) n ) 1 12 n 1 ( ( W b δ Φ ( rad d I  b K K Φ ; it contains however explicit poles in ) − d 1 1 , , ) +1 n RV RV , , ( X rad rad b Φ ( I b b Φ Φ d +1 ), all terms are separately finite in + d d +2 n ijkj + ) n b 2 Φ 3.5 ( Z Z W . I /d RV , are = = + RRδ . The third subtraction term, , with as many indices as the maximum number of partons that can ) ) RV +2 +1 , up to a sign, making the Born-like contribution finite and integrable. 1 n n ( 12 +2 , b ( +1 Φ is such that the integral I VV n Φ abcd 3.3 . In total, the sum of the four terms in the second line of eq. ( n d I ) VV d )  Φ Φ W  d ijjk n 12 12 features phase space singularities, and these must be cancelled by the finite d as ( = (   topologies Φ )  W I d 1 1 K , ( Z Z − I Z ) + + = 4 and integrable in the single-radiation phase space, making this contribution rad + d b = Φ RV d ( RV K Denoting the corresponding phase-space-integrated counterterms with NNLO dX dσ nature, they can involve threeall or different, four or different partons, two of hencethe either them fourth indices are indices equal. to be Withoutwe loss refer always different, of to so generality as that we the choose allowed the combinations third of and indices, that selects the singularities stemming fromsector an functions identified subset ofsimultaneously partons. be We involved thus in introduce indices an for NNLO-singular singularities configuration. ofAs single-unresolved We far type, reserve as implying the double-unresolved that first configurations two are concerned, in particular those of collinear numerically tractable. Finally, in the firstsame poles line of in eq. ( 3.2 Sector functions As in the NLO case, we start by partitioning the phase space in sectors, each of which the phase-space singularities of counterterm sum sum finite in In the thirdis line finite of in eq. theintegrable ( double-radiation numerically. phase In space,to the making a second this sign, line, contribution so fully that regular their sum and is finite in where subtracted NNLO cross section can be identically rewritten as singularities of in detail in section JHEP12(2018)107 , , 2  ab C (3.9) (3.7) (3.8) double- , . and ) )) c . a , S b, c )) = 1 . We call it . ) and c, d 1 abcd and ( a, b , , , a W   )) ) ) a a,c f f 6= ) and ( 6= ( ( c X constant d a ab , a S a, b, c a, b  6= triple-collinear → C ))  X bc c a, b bc a, b C S ), the list of singular limits acting + 1)-body kinematics, is split into /w n ac 3.8 ) are genuinely double-unresolved, ⇒ ) = ) = . Analogously to the NLO case, we , α > β > = f f ( ( . This implies that, roughly speaking, 3.9 , w ) cd abcd bc 1 abc abc w ( σ ) I /w a SC CS e ab 1 ) and eq. ( abcd bc σ δ – 17 – , w constant 3.7 a + ac , , a,c 6= c 6= 0 0 c X → is sometimes referred to as e d /w ( a constant with the required properties is ab , as any term with ( cd → → β abc 6= ) C c X /w bc → ab a, b RV abcd /e w = ab b , w σ ( 1 /w ab 0 a /e α ) a a , w → e ( . Since , e , w 0 , σ bc , e 0 0 = 0 → 3.1 , w → → σ abcd ], in order to consistently specify the type of configuration as being double-unresolved, ac cd → σ c abcd 71 bc b σ , w , w , e = , w , e ab ab ab a a w w e w e (uniform double-soft configuration of partons ( (uniform double-collinear configuration of partons ( (uniform double-collinear configuration of partons ( (ordered soft (first) and collinear configuration of partons (ordered collinear (first) and soft configuration of partons ( abcd : : : : : W ab , following [ abc abc abc S abcd must be combined with the real-virtual contribution, to cancel its explicit poles in C A possible expression for C In the literature the configuration ) CS SC 2 1 ( the double-real matrix elements;single-unresolved the limits remaining when three acting on configurations matrix are elements, compositions but of collinear not when theyrather than are indicating applied the number of partons that become collinear. Notice that onlynamely the they first cannot be two reduced limits to of compositions of the single-unresolved limits list when ( acting on already considered at NLO, as well as the following double-unresolved limits: With the sector functions definednon-trivially in eq. in ( each NNLO sector includes the single-unresolved projectors NLO-type sectors, the samesector must functions be with true foursingle-unresolved for indices limits, must in order factorise for sectorNLO-sector. the functions cancellation with of two poles indices to in take place the NLO-sector by design them in suchgiven a sector. way as Inrespect to addition, to minimise at NLO, the NNLO and number relatedI there of to is the IR another fact limits property thatas that detailed the to in contribute integrated be section single-unresolved to required, counterterm a new with phase space, they can be defined as ratios of the type There is a certain freedom in the definition of Since the sector functions must add up to 1, in order to represent a unitary partition of JHEP12(2018)107 ) , , , ) ) ) 3.7 αβ αβ αβ ( ( ij ij ( (3.15) (3.12) (3.13) (3.14) (3.11) (3.10) ij W W W ij ij ij C C C i i i , , S S S , and similarly jk kj kl = 1 = 1 (11) ij W W W
 , . ; W = = = ) cdab jikd ijk = ) commute when acting ijk αβ ) ( ijkl ab ijjk ijkj W W , σ ij αβ W + CS W W ( CS ij a j,k 3.10 W σ ij 6= ij ij 6= = 1 X , , ; d P abcd C C C  a, b + 1)-particle phase space with i i i + ). One may verify that ikl ). ijk W ijk satisfy all the requirements that S n ) S S ) = kbid ) ij kl 3.7 ) ijkd ( ( SC , , a, b SC SC π W π αβ ) ) ( αβ , ij W X ∈ ∈ , ( ij , ) , αβ αβ k,i ( ( ab W cd ij ij i,k 6= W αβ X 6= d ijk ijkl ( ijk ij X ijkl W W d ) satisfy k C C C C ij ij ) are given by  W 6= b X 3.8 ij C C , , , ] ijk + k ] C ij ik ik 2.14 ij [ – 18 – ij S S S [ k kl CS ibkd , W W W , , , W β , , ) ij ij ij = = = i,k ab C C C ) and eq. ( ), therefore they have to be introduced as independent = 1 = 1 6= X w ijkl d ijjk ijkj 1 (

, , , ] of the collinear pair ( i 3.7 α 3.7 i i i W W W 6= ) ab b S S S iblk X a abcb ij ij ij e  ( W C W C C ik : : : + + = , , , S ) ) ) ) ibkl ijkl we show that, among the single- and double-unresolved limits that ijjk ijkj abbc αβ αβ αβ αβ ( ( ij ij ( ( ab ij W W B W W W σ ) W W W i i i i ijk 6= S S ( b S X π we also show that all the limits reported in eq. ( jk kj kl is the NLO sector function defined in the ( X ∈ ikl ) under the action of single-unresolved limits. As noted above, in these con- B c W W W ] . One easily verifies that the functions abc ab = = = SC 3.8 ijk [ ), and which stem from their definition in eq. ( (11) ab C W σ ijkl ijkj ijjk 2.12 Finally, the NNLO sector functions satisfy sum rules analogous to the NLO ones in It is now necessary to study the properties of the sector functions defined in eq. ( = W W W i i i ab S S S eq. ( where respect to the parent parton [ σ must apply to NLOsector sector functions functions. defined in It eq. is ( now straightforward to verify that the NNLO so that the NLO sector functions in eq. ( in general be different. and eq. ( figurations the NNLO sectorfunctions. functions To this must end, factorise let into us define products of NLO-type sector In appendix on the sectorsingle- functions, and and double-unresolved that configurationsdepends in the on each combinations our sector. of choice We of these stress sector that limits functions; this with exhaust structure other all functions possible the surviving limits would limits. In appendix we are considering, onlyThey a are subset give a non-zero contribution in the various topologies. to the sector functions in eq. ( JHEP12(2018)107 ), . ,
.
T ijk 2.16 (3.16) (3.17) (3.18) ijkl W C ijkl C ijkj − ), will be W − W 1 RR , which we 1 )
2
( i 3.15 RR RR ik ) ik

S K S )–( SC − ijk , ijk − 2 1 1 ( T ijjk 3.13 CS i L CS i
W , defined in eq. (
, ), as ) ), in full analogy with − + −
1 ijk ) ikl 1 ( 1 ij 2 RR . Sum rules for composite ( ijk T L 2.16
3.15
}
SC SC L C , the single-unresolved coun- ikl )  − ijk ijk − )–( − 1 ) ( 1 1 1 , , 1 , ( SC ij SC SC K
3.13 , defined for the various topologies L ijk ijk − ij ) ijkj ijjk − ijkl − 1 S − 1 1 W W W SC CS CS − , 


) 2 1 + + ( T RR RR RR ijkl ijk ijk L SC ijk ikl ,   ijk C C C  2 + ) ) ( T ) ) SC SC − − − L 2 SC SC SC h h ( SC T 1 1 1 , , – 19 – , + 2 2 T 2 L sub = = = ijkj ( ijjk (

( ijkl )
we show that this property is also respected by ) ) ) 2 L L ). Candidate double-real local counterterms for the L ik ij ik ( T RR S S S B ijk, ikj, jik, jki, kij, kji SC SC SC L − − − , , , { 3.7 2 2 2  − − − − ) ) ) ( ijjk ( ijkj ( ijkl 2 2 2 1 1 1 T ( ( ( L L L + ijjk ijkj ijkl ) L L L


W 1 , where we describe the structure of the double-unresolved ( ij ij ij ij − − − L C C C , 1 1 1 RR h , , 3.5

− − −
), a limited number of products of IR projectors is sufficient to = = ik       ik ij 1 1 1 ) ) ) ) S S S 1 1 1 2 ( ( (


( ij ij ij can thus be defined, in analogy with eq. ( − T i i i 3.10 − by the expressions − 1 L L L S S S 1 1 K T − − − − − − + 1 1 1 ijk ijkl ijk ) we denote the set 1 1 1 )    C C C 12 ( ijk T = ≡ = ≡ = ≡ + + + ( K ij ik ik π S S S sub sub sub ijkl ijkj ijjk + ) = = = 1 ) ( ijjk, ijkj, ijkl RR ) ) RR RR T 2 2 2 ( ijkl ( ( ijjk ijkj K = L L L The different contributions arecontaining only naturally single-unresolved split limits accordingterterm; are to assigned terms to their containing kinematics. only double-unresolved All limits terms are assigned to The order with whichas the all various limits operators are commute.the sector applied functions In to defined matrix appendix invarious elements topologies eq. is ( irrelevant, where we separated the actionfrom of that of the the single-unresolved double-unresolvedT limits ones subtracting these products from thethe matrix finite element, expressions one gets, for the different topologies, countertem integration. Thisthe feature, feasibility distinctive of of the our analytic integration method of at3.3 counterterms. NNLO, is crucial Definition for ofAs local reported counterterms in eq.collect ( all singular configurations of the double-real matrix elements in each sector. By double-unresolved limits, thatfurther follow detailed from in those section counterterm. reported in We eqs. stressthe ( NLO that case, the allowdouble-unresolved properties one singular in limits, to eqs. eliminating perform the ( sums corresponding over sector all functions the prior sectors to that share a given set of where by JHEP12(2018)107 , . , ) ) ) 1 )– ( 20 12 ( RV K ( (3.23) (3.22) (3.19) (3.20) (3.21) 3.19 . This K ) K of RV , ( playing the , multiplied    K  j / i / RR } k {  -body contributions, , n cd T ). This correspondence  i B . ) , k T ij ; once this is achieved, its / k ( 2.16 cd 1 j / , W I i / } rad + k , b RR Φ {  T d  j /  i / ) W } µν k B SC { , ) with eq. ( RR  2 ( T µν ijk double-unresolved counterterm. A direct  ; the single-unresolved structure in L ) Q 3.19 cdef , + + , SC B VV )  , , giving rise to up to four poles in ) T  2 2 2 j , mixed ( ( T T ( , k ef – 20 – + 1 particles at Born level, with W , k / L k I L rad j / / k must compensate the explicit poles in ) n  i / j / i b + that of virtual contribution; in this scenario, Φ i ) ( / cd } ) ) d RR } I 1 k 2 12 ( ) ]. Here we just write symbolically k ij ( { ( T 1 ) are very intuitive and compact. First, notice that the { i,j i,j (  L RV ij L 70  6= K 6= X L  − e f µν B 3.21  B = = = ) ) ) ijk i,j i,j µν )–( 1 2 ijk P 6= ( ( ]. We write therefore, for each topology 6= 12 X T T c h P d ( T 70 K K 2 2 2 1 1 1 3.19 K 2 2 ijk 2 ijk N N N s s , which we refer to as ) = = ≡ 12 (

the candidate NLO local counterterm. As for the double-unresolved con- double-unresolved counterterm; all remaining terms, containing overlaps is to be integrated in } } K k k ) 2 { { ( pure ), as will be further detailed below. K RR RR exactly ij 3.12 ijk S C ) can be derived from soft and collinear limits of scattering amplitudes, which are The double-unresolved kernels appearing in the counterterm definitions of eqs. ( ]. General expressions for the kernels can also be derived starting from the factorisation 3.21 of soft and collinearwill poles be discussed in in virtual detail corrections in ref. to [ fixed-angle scattering amplitudes, as in eq. ( ( universal, and for the massless22 case relevant to this article they were computed in refs. [ on the other hand,double-unresolved makes projectors it naturally suitable becomeparton single-unresolved for projectors which integration originated for in the the firstis parent splitting, necessary, since thus the reproducing integral theThis of structure cancellation of also relies on the factorisation properties of sector functions, presented NLO computation for the processrole with of single-real correction,becomes and tributions, by Born-like matrix elements, analogously to The definitions in eqs. ( candidate single-unresolved counterterm hasterterm, the as very one can sameis deduce structure strict: by as comparing indeed, the eq. iffor NLO ( one instance coun- imagines by means removing of from phase-space a cuts, given the process original all NNLO computation reduces to the of single- and double-unresolved limits,are while assigned still to featuring double-unresolvedcharacterisation kinematics, of mixed double-unresolved countertermswill in be terms discussed of in factorisation ref. kernels [ refer to as JHEP12(2018)107 . ), k k case, , and + ¯ q 2.23 (3.27) (3.28) (3.25) (3.26) (3.24) k q j k partons k , j n + k i . , , , by stripping k i 2 in the k / cd = R . = 0 = 1 µνρσ T B
k k µ B k } ), is such that the z ˜ k k { ρσ kl + + and for the azimuthal Q j 2.29 In the non-factorisable µ j z . removing , ˜ k d µν ij RR 3 ijk , can be easily extracted ). } case. Furthermore, in order to k + # ]. k Q P + k i ν a jki { gg µ i 22 2.28 2 ˜ a k + ; the corresponding kernel is , l ˜ k ˜ k µ with a ) refers to a set of k  CS ˜ k ρσ j have been defined in eq. ( ], multiplied times p , k kl , and + B / k = ) j 2) 22 i j / k , k ( features the doubly-spin-correlated ab i / , 2 in the  ρσ kl k − i I ij } / , and Q jk k A d , c , k = { ijkl C of k ij l µ r / , k / k − / k k P ot mean symmetry under flavour exchange, but C + ( 4 kl j / j / n i / i r k / + , and inserting their sum 2 } with k } µν k · k k i g k k k µν p { { does − ,   B – 21 – j "  k , and , z k a ) cd kl j µν z a k ( ijk P features a colour- and spin-correlated Born contri- kr B µνρσ , and s j − ) Q i + B with µν ij k , and ( cd a ijk i + j 2 , I 2 Q k q ρσ k i kl , i · i,j,k corresponds to eq. (96) of [ k i ], multiplied times ar , jr k collinear splitting at NLO in eq. ( P i ), X 6= + k ) = SC s s = ]; the eikonal kernels k a 22 ij X ), identical to the NLO one in eq. ( (  is the doubly-colour-connected Born matrix element, defined cd c,d qg B µν + ij I 22 ij = 2.26 P − kl k µν → ir , with a kinematics obtained from jk , the set of momenta ( with s P µ 3.26 q cdef kl P µν ijk 2 1 k 1 s ijk q ij B 1 can always be cast in the form Q = a µνρσ ij jk are defined in eqs. (96) and (110) of [ N P C z N s s r B ) r µν ijk k = − k ij · by removing = = · ( cd Q a µ a k I

k } k } } k µνρσ k k { = = { { B a µ a , all symmetric under permutations ], but the expressions are long and therefore will not be reproduced here. We z ˜ k ρσ , obtained from the colour-correlated Born matrix element kl P µν ijk RR RR cd µν 22 , is a light-like vector which specifies how the collinear limit is approached. The Q B µν ijk , one should replace ij ijkl ) 20 µ r P ij k C ( cd SC According to our conventions, Symmetry under permutations of I 3 4 , and inserting the sums l while it corresponds toget eq. (110) of [ only that kernels andwhat flavour happens Kronecker in delta the symbols case combine of in a a symmetric way: this is analogous to Finally, the soft-collinear limit bution external spin polarisation vectors. Hence, once more, the analytickernels. integration of The the counterterms factorisable involvesBorn only double-collinear matrix spin-averaged limit element k defined as and Lorentz structure in eq.radiation-phase-space ( integral of the double-collinear azimuthal terms vanishes identically. where, in analogy with eq. ( from [ note however that while the kernels double-collinear limit obtained from The expressions for thekernels double-collinear spin-averaged kernels In the double-soft limit, for instance in eq. (113) of [ JHEP12(2018)107 , ) B 12 ( RR (3.31) (3.32) (3.33) (3.34) (3.29) (3.30) K icd is applied SC ij limits are ap- C ijk , CS can feature at most . , ) ) , and , RR ) = 0 . ijk , T ijkl T ik icd C C S W W ijkd ibcd − − − RR. SC RR, W W i RR RR, RR ij (1 S ) ) (1 ij ) (1  C 2 ) ( ik C RR RR cd T ik k ijk S S L kl S C SC icd ijk , C −  2 CS − = = ( T ) = . A similar simplification occurs in the 1 L SC CS ) ( ij ikl ) are quite natural, they contain a certain ) (1 ) (1 L + RR RR = = RR SC , ) k ijk hold when the ijk SC − kl 2 cd 2 3.21 S ( T ( T 1 C – 22 – ibcd C ijkd L ij CS L  CS i ij ) = )–( not ) W W S C 1 C + ik + = ( ij ), the commutation relations discussed in appendix S ) = = = L 3.19 2 RR ijk ikl RR (  T 3.7 − RR , where one can exploit the relations icd K − ijk RR RR SC SC ) (1 + becomes = ijkl icd ijk SC SC CS ) ), the candidate mixed double-unresolved counterterm = ( = ( ijk , ) ) C i 2 2 ) ) 12 ( ijkl ( ij 12 S 5 T ( SC CS ( T T L CS 3.30 C SC SC K . This ultimately stems from the factorisable nature of , , K K 2 2 ( ijkj ( ijkl + ijk ijk has been applied to the double-real matrix element, further action on and ) L L has a factorisable nature when applied on the double-real matrix element, ) 12 SC CS ), carry independent information on its singularity structure. These ( icd T ijkl does not produce any effect, and analogously if the limit SC , namely , K C i 2 SC 3.10 ( ijkj S RR L ijk ). CS We now note that, while eqs. ( 3.30 Also the limit 5 however, in thiseq. case, ( the relevant commutation relations are not sufficient to obtain the analogue of valid both on matrix elements and on sector functions, to rewrite free of any contributiondefinition from of the limit and the sum of As a consequence of eq.simplifies ( to Even if this factorisation propertyplied does to the sector functions ofare eq. sufficient ( to prove that the latter by after the action of and of degree of redundancy.four In phase-space fact, singularities, hence the notlisted double-real all in matrix of the eq. element projectorsredundancies ( can relevant to be a eliminated given byinstance, exploiting topology, once the idempotency of projection operators: for JHEP12(2018)107 : ). , , 3.35 (3.37) (3.38) (3.36) (3.35) ijkl ijkj W W RR , RR i ijjk, ijkj, ijkl i  , ) ) = ijkl ijkl ijk ijjk T C W C . , W − − . , and will be sketched in i,j,k ijkl ijkj RR X  6= l i ) (1 ) (1 W W ) RV ik ik + ijkl S S ijk i RR W i → RR i ) , − C − ) ijkj o ) o ik ) ) ik R ij − i,j,k W S S S ijjk X ) (1 6= ) (1 ijk l ijkl + − − W − ) (1 ijk C ijk C + ij (1 (1 (1 − ijjk − S CS RR CS ijkj W o − ijk ijkl ijk + + ) (1 , is to apply kinematic mappings to eq. ( )  ) (1 W C C C (1 ik ik ikl i,j ijk – 23 – + ijk S + + + S 2.3 , 6= ijk X C ) we can proceed as done at NLO. We define the k − ij ik ik T SC − SC ijjk − S S S SC W RR W 3.12 , ) (1 ih ih ih ) (1 )  ) + ( ) (1 ) ) ) s) ) + ( i ) + i i i RR ijk , ijk ij ik S i ik i,j 1 S S S ij where, as usual, all remapped quantities will be denoted ( S S ) 6= S ), with the replacement S X i − CS − − − k CS − K − S − − 3.5 + (1 + (1 (1 (1 + 2.39 (1 − (1 RR (1 (1 ij ij ij ij i ijk ikl (1 hc) and C S ijk C C C , ijk ijkl ijk i i 1 ij SC SC ( 6= 6= + + + C C C SC C 3.4 i i i X X i, j i, j K + + + S S S + ( + ( + + = = = ij ik ik i nh nh nh ) S S S S s) 1 , − h h h − − h ( hc) 1 , ( 1 . ======K ( ) ) ) ) ) K ) ) 1 2 K 2 2 ( ( 12 12 12 ( ( 3.6 T ijkl ijkj ijjk ( ( ( ijkl ijkj ijjk K K K K K K K Using the factorisation propertiesappropriate ( counterterms with remapped kinematics, where in this case barred projectors unresolved counterterm: NLO counterterm of eq.section ( 3.4 Single-unresolved counterterm We start by separating the hard-collinear and the soft contributions to the candidate single- There is ample freedompings in can the be employed choice forkernel. of different The these kernels, detailed or mappings, definition even and ofis for in the different given kinematic contributions principle in mappings to different sections we thewith map- employ same for a each bar. counterterm Finally, the real-virtual counterterm has formally the same structure as the The final step forpens the in construction the of NLO case the discussed NNLO in counterterms, section analogously to what hap- tion of the candidate local counterterms for all contributing topologies After the simplifications just discussed, we are finally in a position to write down the defini- JHEP12(2018)107 , and  j (3.46) (3.44) (3.45) (3.40) (3.41) (3.42) (3.43) (3.39) kl 6= W should be k
r RR ij C , i )). ) , ; if both . S j ) kl , , ) iab ( ( kl , the same )  = , π abc kl ) j ) abc ( ij ) ) ( qj l ijr W ¯ ( kl ¯ s w W abc = αβ 1 ij  ( ij ijr ( ) ) ij i ( ( kl ) using their NLO sum rule, ¯ s k W π s W ) abc abc ( iab  W ( qi ( i ( ) ij RR π ¯ ¯ e s αβ }  (
) ij C ¯ ijr k j ( i = = { S } ijr W S ) )  ( ¯ k }  − { ab , abc abc ¯ k i ( (  ij ij − R { kl S ¯ ¯ σ ) w R  i ) kl ( ab i W − ( jr µν I . W is as follows: if 1
I R i i j kl
6= 6= f ij b a µν X ij RR i, j C W i – 24 – C P 1 RR 1 S
h 6= 1 , ij N ij i r s i i,k N ) 2 6= C i,k RR 6= 6= k X i l 6= k ) X abc ≡ ≡ ≡ −N l S ( ij  , , and analogously for the case ) ¯ σ abc kl kl kl ) i ( kl i  X αβ i, j>i X ijl ) ¯ σ 6= ( s ij W W W abc qi ( P = = αβ ¯


s i, j W ( ij s) ij , = = W hc) RR RR RR 1 , i ) ) C ( i 1 ) the choice of S ( ij ij  S abc abc K   ( i ( kl C C K i i ¯ e i are the colour- and spin-correlated real matrix elements and i,k i,k 3.42 6= 6= W S 6= 6= k k X X should be chosen for all permutations in l l µν i i r R 6= 6= X X i, j i, j ) and ( and = = s) ab 3.41 , hc) R 1 , ( 1 , the same ( j K K 6= 3.4.1 Integration of theAs single-unresolved done counterterm at NLO, we nowspace. integrate the single-unresolved We counterterm first inobtaining its get radiation rid phase of the NLO sector functions In eqs. ( chosen for all permutationsl of where The kinematic mapping of sectorallows functions, to once the factorise integrated countertermintegrate the is analytically structure considered, only of single-unresolved kernels. NLO Explicitly sectors out of the radiation phase space, and apply not only to matrix elements, but also to sector functions: JHEP12(2018)107 , , hc) ) , ,  1 ( ) ( # #)  (3.47) (3.48) ( O ab K   η O + ab  pr # ln ¯ 2 η η + ζ   1 # ln ¯ ln ¯ 7 2
pr 1 2  − } similarly yields η − ¯ k 8 3 6 ab − s) ln ¯ { η 2 ,  as the ones shown + 1 − )  ( A   ln ¯ 1 ab  C iab 2 1 2 ab  R K ( kl +2 g ζ η a a f  1 f − 7 2 W 6= ln ¯ δ N  reads  X
R − + ) a,b ) ) F } T + 2 2 }  ¯ + k 6 C iab k abc  1 ( { 8 3 ( rad { } + 6  ) } ¯ +4 ( q  a Φ − ¯ k   2 q, A ab ijr γ d { { ( kl ar ab C p R ) (3.49)  + RR has absorbed the symmetry factor + η f η = } g a W δ ab
a 2 p ¯ ) k 1 g 1 6= 2 ln ¯ f j ,   f ln ¯ i { + R  ) f δ ) S X ( C  ). The integral of abc
 δ a,b )  " } ( rad } ijr f 1 −  i k (
 ( + i Φ ¯ k ,  hq ) { } } N ) I d { S ( a 2.50 P ! ¯ ¯ k k h R ijr X
{ { iab − ( 6= T ( ab
ab }  1 6 ar ¯ s } RR X ¯ k η R – 25 – R R ) exhibits the same poles in  h,q ¯ i k { +4 s a a ij { S f   = A 6= J 1 C , C +ln ¯
R , C 3.49 R i i X ) 2 ) 1 } ) a, b , g 6= 6= ζ ¯ k rad a a b a X 7 ijr ijr ijr f { (" + X ( jr ( kl Φ ( rad δ i ¯ s − d i,k " Φ hq 6=

 W 6= k d X . Eq. ( kl hc) l 10 , and the sum Z k , W a hc } g a ij 1 6= i Z W ( h kl ¯ k X J F f X 6= 2 { I 6= δ
kl k k,l i C W } r i,k 6= X + } 6= ¯ k h,q W i ¯ q i p 6= k
X X l i,k  { k, l i 6= X q, X } run over the NLO multiplicity, barred momenta and invariants refer i,k 6= k { X   6= ¯ k  l a q 6= k R X 2 { f l   s +2 +1 X " δ µ i,j>i 2 i 2 n n s s s) ς ς µ X + + µ  , and +2 +1 X 1 S  i,j>i 1  π ( n n h ς ς α 2 S +2 +1 I π S π n n +1 +2 α 2 1 α ς ς 2 −N = n n ς ς N − ) = . The combination of hard-collinear and soft contributions is straightforward, as = = = } = = = ¯ k +1 s) { n , ( 1 hc) ) ( /ς , 1 1 ( I ( +2 I I n where indices to NLO kinematics, and the same realς kinematics in the NLO case, yielding where, in the last step, all identical soft-gluon contributions have been remapped on fully analogous to its NLO counterpart in eq. ( in the single-unresolved radiation phase space two expressions which are suitable for analytic integration. Indeed, the integral of JHEP12(2018)107 ), 3.15 (3.50) (3.51) (3.52) (3.53) appear ) )–( RR. 12  ( 3.13 K poles between ijkl . is factorised in ), ( W / hq i and ) ) = 1 W ) 2.12 ik 2 S ( ijjk klij − W K W which must be integrated i (1 ) ) + ij 12 ijkl ( S , C ijkl − K + W (1 ( ik = 1 RR, S ijk  h ) RR, ik ijkl C S αβ C we see that, while their integration has + i,j,k ( ik X ij ijkl 6= ij ik l W S C C + S 3.1 + 1)-body phase space. . It is important to note, however, that in h ) + = = n  1 ) ( ijkj -body kinematics consists of two parts: the pure i,j I – 26 – αβ W , ( n 6= ij RR RR X i k + ). Owing to the simplifications discussed at the end ) ) W ik ik ij  ijkl is much simpler than the two terms taken separately, S , which must be integrated in the double-radiation in the ( 3.5 ) S ) = 1 ) RV C C 2 ij ( − ijk 12 − ij C ( i K C akbk (1 C i S K )(1 W S ijk i + S C + ) , the full structure of NLO sector functions 2 − + ( (1 ik abbk RV i K S 6= h W ) sector by sector ), together with ( X ), due to the single-unresolved nature of the involved projectors. Such i,j + ij = 4 of the sum ( π = d 3.12 ) X ∈ 2.55 12 ab , analogously to what done for the single-unresolved counterterm, the double- occurs ( , the sum ijk ) ab K 1 C ( W 3.3 + ij I ), as well as in ) S 2 ( 3.49 and K Introducing remapped kinematics forfunctions the double-real matrix element and for the sector and to as well as one is able tosector recast functions the that above add expression up in to a 1, according form to that the explicitly sum features rules only of sums eqs. of ( Exploiting eqs. ( and it reads phase space, and the mixed double-unresolvedin counterterm a single-radiation phase space.to From be section performed independently,only the combined in non-integrated the counterterms last lineof of section eq. ( 3.5 Double-unresolved counterterm The double-unresolved counterterm with double-unresolved counterterm poles are identical (up to athe sign) to finiteness the in ones of theeq. real-virtual ( matrix element, thus showing front of the integrated singularities, whichRV means that the cancellation of 1 at NLO in eq. ( JHEP12(2018)107 ij ) is S ij (3.55) (3.54) 3.54 C i RR S
 jk ijk RR ijk W C ijk j C  ij , C i S S  S ijk ij  RR ij i ) +  0 C ) C
C i ij ijk RR
kj ijkl
ijcc S ( C  jk C kj } W ij icd,jef RR ij ¯ k ( ac ijk W + C W { RR } C i S  RR ijk ¯ C k i jk jk + ik S  { cc S C
ik S  C
jk W i B ijkl S j  RR ij )
jk i  S W S ij C cdab cdef ( kj S jk cc ijk  W  I
B
W
RR. C − kjil jk C W ) jk h i,j kj ij + ik  C W + 6= C
icd X S ) h S cdab c akbk ( jf W  j i jk i,k ¯ s ij abcd S icd W  ) S + W 6= jk ( ef abcb X )  l C W ¯
s + W  + k icd C  )
− W ( je αβ ) kl 6= k ( ik ¯ s – 27 – jk jk j X + S g αβ abcd ijcd W abbk ijkl ( j W ( C ji + abcd W f k } h j W C W δ + − abbc C S ) ) W ¯ k ) )  ) S i ijkl { ik i ij kl ( ij cd W + ac ( ( ijk  )  ( ) αβ S ) I ) π π W ( π S ij + i cd C αβ X ∈ ∈  X ∈ i,j i,j ijk αβ ( S αβ ij ) ) i,k ( B ( W ij ( ij ij 6= 6= ab cd ) ab
ij X ijkl π kl 6=  e X f S ( ( W l ij X ijk ∈ W i kl W π π ( i C ijkl cd i,j i,j ij i  S ijk X ∈ ∈
C 6= I 6= 6= C W 6= abc j X X C
ij c d kl j C ij ab cd  X i ijk i,j kl  i i  i,j,c jk j j C S S 6= W S C S X   6= C 6= 6= ik 2 h c l l l>k l>k 1  X X  d  W k kl 2 S i j j i,j i,j i,j i N  S + jk 6= C i,k 6= 6= 6= 6= 6= 6= X X X k k>i k>i k>j k>j k k X k k X k X X 6= k X C l i i = j 6= 6= X i,j,k i,k>i i i,j,k S X X X X X X X X 6= X i,j X i,j>i i,j i,j>i i,j>i i,j>i i,j>i 6= l = RR l ) − − − − + + + + − + − ij 12 S ( K + ) 2 ( K We stress that in each contributiongoes the kinematics a of different the mapping double-real onto matrixof the element the Born under- integrands to one, the so kinematic asThe invariants that to explicit naturally maximally definition appear adapt of in the the the respective parametrisation barred kernels. limits appearing in the first three lines of eq. ( unresolved counterterm finally reads JHEP12(2018)107 )). . kl , C ( (3.60) (3.61) (3.62) (3.56) (3.58) (3.57) (3.59) singu- ) π ) , and we acd . / ( f ij , , d ( ¯ k a,b,e,f , ) satisfy  ) k ) ijk ) ) π , ( 6= ij kl cd π ( ( acd s ijkr ( ef  3.61 ) π π ( a, b, c, d , , s  } ) + 0 ¯  k , n acd ∈ ∈ 6= +¯ ) ( bef 0 ) ) { bd ¯ s .  abcd ijcc s ) ( ab cd s acd acd B } ( bf ( n +  ¯ s ¯ k ¯ k , n ) ijr,klr abcd ( { , ( ad n ij  ) and eq. ( } = = ( kk s } k  ¯ ) ) k ) cc I ¯ k 0 { ) are listed in appendix { = = B  + 3.58 ) ) ) = + ij 3.54 ) acd,bef acd,bef ijr,klr (  µνρσ rk ( ( ( n f ) abcd abcd ( ( n d I } ¯ ¯ k k B ¯ ¯ k k ¯ k -body kinematics but, peculiarly, it 2 ijcd {  n ( )  abc,bcd − } ( 0 ) ¯ k B } , ijr ( lr ) { ¯ k ij ) ¯ ( s  rr { 0 , ) I , ijr acd ) ) cd ( cr ( f d 0 0 ijr ¯ s B ¯ + k ( kl k ijr ijr ) , ( dr ( kr ¯ ) s k ¯ ¯ s s i,j,c , – 28 – f  cd ijr ) ) X ( cd  6= s acd C ¯ d s ( ef appearing in eq. ( ) is ) ) ) would vanish by color conservation in the absence s d j ρσ ijkr  + kl f ( ( abcd rk ) should be chosen for all permutations of acd ( +¯ P } should be chosen for all permutations in I ( be C ij bd ) } abc 3.54 ( RR ¯ k cc ¯ ) s s ) 3.57 g ¯ i k s { I c ( rk { acd ,  jr f + ( bf I δ ¯ i, j, k s i,k,l i,j,k , s µν = 8 ) a,b ad X a 6=  ) ir − ( ij B 6= 6= s br 6= c , k ) s s 0 h I f ( r + r b µν − o ijk C f acd a,b ) c ( e µν P ij  C 6= k ¯ acd,bcd k h ( 2 2 P 2 and 1 1 1 } + 2 o 2 ijk + 2 counterterm, and, at the same time, it features the same phase-space ) ) 1 ¯ k N s N N b ) acd,bef { i,j . ( h k N ) acd ¯ = = k ( b abcd 1 6= RV ( + h ( = = 4 ¯ n k ( , the same r ¯ k a I K = = n k RR RR ) ) RR RR = = i, j, c ijk ijk ) ) 6= C C ijkl ijkl 0 acd,bef acd,bef ij C C c abcd abcd ( ( e ( ( c S } ¯ k } ¯ ac k ¯ k ¯ k S { { The mixed double-unresolved countertermneeds features to beoperation integrated is analytically necessary only to in showlarities as that the the such phase an space integralsingularities of features is the a same single explicit radiation. 1 This The remaining composite limits of 3.5.1 Integration of the mixed double-unresolved counterterm and it can be easily shown that the two remappings in eq. ( where the same We have introduced the remapping Notice that the second lineof in phase-space eq. mappings: ( its rolethe is proper to limits ensure also that the inin double-unresolved the the counterterm presence fourth fulfil of and the fifth mappings. lines of The eq. definition ( of the barred limits where have introduced the mapping JHEP12(2018)107 . ) ), )).  ; if ,  j ) kl , the 3.35 ( . j ijkl (3.67) (3.68) (3.63) (3.64) (3.65) (3.66) ijjk . The = ijr π ( jr l  ) W = s W , hc) (¯ , ) ij k 1 ijkl ( ijkj hc ( ij RR ijr RR π ( kl I J W ( W   ) π , ); if ) W are given by ,  ) RR ijkj RR  ijkl 0 ijk  ) ) R  , which have to be  3.42 . W ) C C ) ) ijr (  ) − − 12 0 } ( ijk . Following eqs. ( ijr,klr ijkl RR ¯ ) k ( ijr,kcd I ( C { )(1 }  C )(1 } 12  ¯ k ) ( ik ) and ( − ij ¯ k − { ijr,klr terms: these cancel out in R S { ( S  and ijk K i  } )(1 ijjk − ) − )(1
¯ k µν C 3.41 cd ijk 2 ), it is clear that, as desired, k ik { ( ik (1 W B (1 − B  S S I S  ) CS B ) − ijk − 3.47 l ijk − )(1 RR f 1 show that the phase-space integral ijr 0 ) ( kd ijr (1 i ik (1 C ( lr ¯ s CS ) D SC ijr ¯ s ) S kl , and analogously for the case ) cd ( , ijk ij + 0 ikl ¯ s + ) − C ijr i ijr S ) ijl 0 ( kc  ( kr ) ¯ + s ijr (1 ¯ 0 s ) CS − ( kr SC ijr ik ) k g ( lr ¯ s ij k + S (1 ¯ S s + ). The result is f  ijk ), and the barred limits on S ijr i ( kl δ  − ) ¯ s − h ) ijk µν k k kl – 29 – ik g SC (1 3.35 ik , 6= C 2.49 6= k (1 P S ) c d f X + S + δ ) − ijkl 1  ijr ijk − 1 1 ij ( jr ) ijr C (1 ¯ s C S ( kl N N (1 ik ¯  s h should be chosen for all permutations in −N + counterterm, which is obtained combining the soft con- + S ) ijk 0 i hc ik ij ij r s) = 2 = = ijkl − S C , J S S C    h (1 ) ) ) i − +  12 should be the same as in eqs. ( ) i,k ( i i + 6= ik ijr ijr ijr (1 6= k X S ijk S ( ( ( l ik S K ij } } } i, j h S C − ¯ ¯ ¯ ) with the second line of eq. ( k k k ) C   is defined in eq. ( i { { { 6= + X (1 i i,j>i , but do contribute to the integrals     S ) i,j r ik S , the same ij hc R R R ij 3.64 6= j − i,j S  ) X +2 +1 12 k J C k ( 6= kl kl n n  i X  (1 ς ς 6= S k i 6= i C C K l i,j,k ij S 1 3.67 k 6= X X 6= i,j  i,j,k l + C S X X i,j ) 6=  integral contains all non-integrable phase-space singularities of − + )–( + l −N 2 and ( − + + = = j hc) = K s) of the hard-collinear contribution can be recast in the simple form 3.65 , should be chosen for all permutations of , 6= hc) 0 12 , hc) 12 r ( k , ( hc) We now consider the The explicit computations reported in appendix We start by considering the hard-collinear contribution to , 12 I ( 12 K ( 12 I ( K do not hamper numerical integrability. tributions of the last three equations of ( In eqs. ( same both By comparing eq. ( the leftover integrable logarithmic singularities, contained in the integral kernel where the integral I We stress that inthe the sum last expression weevaluated separately. have kept the we have JHEP12(2018)107 , , ) in ) #) (3.73) (3.70) (3.72) (3.69) (3.71) . 3.5  iab ( kl hq ab η W  W  ln ¯ 2 ) , ζ 1 2 7 2  iab , ) ( in turn features − ,  } − ) ) 2 also contains the 0  ¯ k 6 1 ) 0 (  ) {   I iab,kcd ab 12 A ( ( ab η } iab,klr C I ( R ¯ k iab,klr g # ln ¯ ( } i a {
} f , and ¯ ab k 
. The same considerations ) } δ ¯ η k { k 1 ¯ k  { lk ( S + {  ln ¯ abcd I ab µν  − .  1 ab B ) a 1 ab 8 3 B
) B } l } R  ¯ − k f 6= ) a ¯ k iab kl ) { ( kd and for C { 6= r 0 ar ( ¯ s hq iab C ) is η ) iab ) ( lr show that the phase-space integral X ( cd a,b kl ¯ 0 s ) ab + 12 ¯ s W ln ¯ , iab ( ) iab + k hq , while barred momenta and invariants ( kr ) D ( kc R i 12 I  ¯ 0 s ¯ ( S s 0 ij
! a iab ) h iab ( lr
f η I h g 6= ( kr ¯ 6= s k h ¯ s S iab N ar f X ( kl X a,b η  δ , ¯ s R h,q − ), and the limits in this case are defined by ) 1 T + k k g µν kl – 30 – 6 = k iab 6= 6= ! f P ( ab +ln ¯
c d X 2.53 δ ¯ a s hq +4 2 } )  1 ζ γ 1 ¯  k 1 A C s 7 { iab N C ( kl + N + J − ¯ s g −N a i i h , it follows by construction that a 2 f 6= f ). Combining soft and hard-collinear contributions, the 6= hc)  10 S ) hold in this case as well, referring now to the comparison = 2 = = , b a C δ X h should be chosen for RV i h   

12 0

F ) ) ) i,k ( 2 6= 6= r 3.48 3.65 C 6= k I X a l a iab iab iab X ( ( ( } X h,q X g + ¯ q } } } i


, as necessary in order to compute the second line of eq. ( f q, ¯ ¯ ¯ k k k } ) } } δ {  { { { ¯ k a ¯ k ¯ k    2 is defined in eq. ( f poles as { { s { i RV δ µ s ab ab ab ( X + / R  R R R J R s) K , ) and eq. ( " S k +1 +2 π kl kl n n α 12 S 2 + (" collects the same phase-space singularities as ς ς ( C C ) , and the same − I × 3.69 1 k 12 S = = N ( k, l
I poles as = } 6= ¯ k s) 0 of the soft contribution can be recast as , / { r s) 12 ) Since ( , I 12 12 ( = 4. We stress that these considerations hold separately in each NLO sector ( I refer to NLO kinematics, and finally one must choose the same explicit 1 same 1 d where the soft andsector collinear functions, limits but are not meant on to the logarithms be ln applied ¯ on matrix elements and on where that were applied below eq. ( between eq. ( final expression for the integrated counterterm where the integral I The explicit computations reported in appendix JHEP12(2018)107 . (3.75) (3.76) (3.74)  ) is thus ijkl RR, W , . , , , 3.76  ijkj . Their contri- RR  = 1 = 1 = 1 ijjk i W ) = 1 ) = 1 ijkl ) W abcd iljk C ijkl ijkl ijkd C ijlk jikl C ijkl RR W W W RR i W W C i,j,k ) i ik ik X
i,j,k 6= i,k − ) + + l S S X jl 6= ijk 6= −
l X d ijk S − C )(1 jk ijkl ijkl ijkl ijkl C ijk ik − ik − S ijk C C W W S − C S il ( ( − C S )(1 − ). Next, we consider all terms in − ijk ijk ijk ik )(1 ik − 1 − ijkl ijkl S ij S )(1 1  SC CS CS S 3.57 C C jk − − 
ijk S −
ij jk ikl )–( ijk ik − )(1 (1 S S CS S ik SC CS − − ijk ijk + 3.55 S − 1 ik ikl ij − CS S SC – 31 – S 1 , , , , , + ijk − SC )+ − 1 C ijk ij 1 ), together with = 1 = 1 ijkl S )+( ) = 1 ) = 1 ) = 1 SC C − ijk ik k>j X ijk 3.53 ibjk ijkj S j (1 jiki iaba ikkj 6= CS − )+( l W W l>k j>i X X SC W W W ijk i ik (1 j ik i i,j 6= + + S C + 6= + b 6= S X 6= k>i X k X k k>j ij k X − + ijkl i ij S ijk ij ijkj C (1 iaab ijkj 6= S S j>i j>i j C X X X + W W ) W h ijk j>i ( X ( ijk ( + + + ik  ijk jk C ( S ( ik π ijk i,j h + SC S X ∈ ), as well as eq. ( CS 6= C i X ik k ab X i S ) and reads i,j,k ijk ijk 6= ijk X h 6= 3.15 C l = C X i,j + + ) 3.35 CS )–( 2 ijk = ijk ) containing the four-particle double-collinear barred limits ) have already been defined in eqs. ( ( ) 2 K ( SC 3.13 SC 3.76 3.76 K which is manifestly freesuitable of for NNLO sector analytic functions. integrationof over The the the counterterm barred double-unresolved in limits. phaseeq. eq. ( ( space, First, upon we definition eq. note ( that the barred limits appearing in the first line of Introducing remapped kinematicsunresolved for counterterm the can be double-real finally matrix cast element, in the the form pure double- We work oneqs. this ( expression by symmetrising indices, and exploiting the sum rules in The candidate pure double-unresolvedfrom counterterm, eq. summed ( over NNLO sectors, follows 3.5.2 Integration of the pure double-unresolved counterterm JHEP12(2018)107 . ,  ) !  ) ) 0 RR. (3.78) (3.79) (3.77) (3.80) 0 ijr ) ( lr 0 . ¯ s ijr,kcd ijkl ijr (  ) )  ( kr ) 0 } icd,jkr 0 ¯ s C ( ¯ ijr k ( kl } {     ¯ s ¯ k   , of the subset { g ijr,klr l ijr,klr  cd ( µν jk 4 f ( ) becomes } δ , can be explicitly S } B cd µν ¯ k ¯ k k ) − ) { f B {  CS 3.77  ik ijr  C ijr ( kd ) ( cd S ¯ s µν 0 ¯ s + ) icd − µνρσ g B ( kr and ) k 1 ¯ ijr s 0 B   f ( kc ,  ijr δ ) ¯ ) s ) ( kr at NNLO.   ¯ SC ) 0 0 s ) ) ijk ijr ¯ ) icd q icd ( lr 0 0 ( jr ( jk q jr ijr ijr ¯ s ¯ s ¯ s ( lr ( lr ijr ( kl CS ¯ ¯ s ,s s  ¯ s → , , ir ) ) ) ) µν g s ij 0 0 − i,j,k k ( s ijr X ijr e hc ijr f ijr 6= jk ( kr ( kr ( kl l ( kl δ + ¯ ¯ s s ¯ µν s ¯ s P l e  i,j f  ) hc i  ij 6= ( C cd X ρσ µν k jl P I

S hc hc ) kl kl j>i − X i,j,k i,j,k i,j,k i,j,k P P jr il X X 6= 6= 6= 6= – 32 – − ) c c d d  , s S )  2 2 jr 1 1 j ir ), involving the limits − il ( ir s ij , s ( S I s jk i ir −N −N f − S 3.76 s µν ij contribution to C ( = = ik s − hc S ij F + ik µν P ) − S C i RR RR hc ( " 1 jr

ij R − I  T P 1 j i,j ijk ijk f  6= ikl l>k X l C C i,j C j  i,j 6= l>k X − − l 6= SC 6= l>k X ki 6= 6= k l>k X X l k>i k X ik X jk i, j>i i,j S S only involves simple combinations of soft and collinear kernels, all remapped 2 1 6= j>i ) X X − X − k i, j>i 2 N i (     ij ik 2 2 1 6= S S j X K i N − − − X − 1 1 = ≡ ) ) 2 2 ijk ( ijk ( cc4 cc4 K K SC CS in an optimal manner so asThe to make complete their analytic integration integration asdetails of straightforward as of possible. the the implementation, pure willlimit be double-unresolved ourselves, presented counterterm, for in a the along forthcoming sakeof with publication. of the other Here illustration, terms we to that will the enter the computation, in section Note that Finally, the remaining terms in eq.defined ( as Defining the barred limits in terms of soft and collinear kernels, eq. ( bution can be rewritten as JHEP12(2018)107 (4.1) ) are (3.81) (3.82) 3.81 , ), it is defined as # poles dictated by . )   ij ( ) denotes the collec- 2.39  G W ( + G RV kl η  . ij 0 ln ¯ q 0 C  1 i ; we limit ourselves to stressing ¯ ¯ ¯ q g , q q q , S kl ) 2 contains all phase-space singulari- q q q R ( − ) ) is slightly subtler: it can be further I k  ij ( 6= → → → RV C ( X G -singular contributions in eq. ( k, l  − − − K + e e e + i + + + S ! e e e 0 limit, encoding process-specific information.  k i  γ 6= → – 33 – features a structure of explicit  X + i, j k : : : 2 =  C RV )

RR -finite remainder RV  ij ( k X K R,RV i R 6= ) involve single-soft and single-collinear kernels of NLO-level " B,V,VV  X i, j ). Analogously to what done at NLO in eq. ( run over real-radiation multiplicities, and  l 3.82 = 2 s µ ) 3.81  and RV , which will instead be detailed in a subsequent publication, together S ( π ) k α 2 K RV − ( I = . The production channels available in this case are RV F C The corresponding real-virtual counterterm R T to 4 Proof-of-concept calculation In order to demonstrateapply the it to validity of di-jet productionradiative our corrections in local up electron-positron to subtraction NNLO, annihilation, restricting method, as our analysis a in to test this the case. contributions section proportional We we consider singular term. Theintegrands IR of limits a higher of complexity thein than latter, full the NLO generality. in ones, We particular, but leave involve still the can kernels completion be which of handled these represent analytically contributions to future work. computational challenges. Indeed, asconcerned, far they as are the proportionaltheir to IR real limits or incomplexity. colour-connected The real eq. structure matrix of ( the elements,split hence into the sum of a process-specific regular contribution, plus a universal phase-space- In this papercounterterm we do not aimwith the at completion giving of athat the such final integrals an expression contributing analytic forthat to integration of the is the pure integrated of double-unresolved real-virtual counterterms, a hence comparable it does or not lower pose complexity any new with significant respect to ties appearing in eq. ( where the indices tion of terms that are non-singular in the The real-virtual NNLO contribution its nature of virtual one-loop matrix element, namely 3.6 Real-virtual counterterm JHEP12(2018)107 , ). , 134 and 4.1 (4.7) (4.5) (4.6) (4.2) (4.3) (4.4)  C q . The  ) they , 2 , 0 Φ  34 , 324 ) ( 7541 S /d 324 O + 4 11753   2 , is Φ π 3 − d 6 32 2 109 3.5 77 27 π − = RR. + 2 i − 2 2 , 27 77 π 3
. π 7 6 ζ + 7 6 rad 34 3  9 + 26 S Φ ζ 2 − 3 s d ζ 3 µ − − 9 ζ 9  134 1 28 ln 9 +   100 − , and  + + 3 3 − , 234 ζ  )  Φ  C 54 2 353 , 3 + /d 16 54 + + 4 abcd 407 + 4 (  rad
2 − Φ − 2 Φ 2 π 3 11 d 34 19 2 d π S π ) − 11 18 7 9 + = 2  − − 18 11  1 π , abcd F 1   1 (  n 7 9 C  1 rad  1 Φ + – 34 – − R d Φ 2 , according to the process definitions in eq. ( + 2 +  134 0 T d  2 q 2 =  2 1 , C   − 14 2  14 9 9 + R 3 + S +2 Φ π (single-unresolved), where labels 1 and 2 refer to  4 2 + n − α 2 2 and ¯ 3  34 3 rad /d 3 0 Φ 34   1 3  − 1 S + ) coefficient of the total cross section is thus q Φ d F F F 3 3 C  2 S ), involving four parton indices, is required. We introduce, Φ h d 3 LO C C C    d 4 α 2 −  σ , ( 3 R R R  Z 2  T T T  2 =  3.58 O =  R s rad  2 2 2 2 µ 2 T Φ    s   µ  rad 2 3 d S S S 2 2 π π π s s  + Φ α α α 2 2 2 µ µ  1 NNLO Z d σ ) matrix elements are known analytically, and up to      ( S π 2 S = α 2 B B B × × × α ) ( 2 = = = = ( O I ] RR RV VV 2 , 74 rad – rad Φ contribution to the 72 d Φ (double-unresolved), and d F Z C Z 234 R , while labels 3 and 4 refer to ¯ In the case weonly are the considering, parametrisation thanks ( totherefore, the the phase-space simple measure singularity structure of the process, C q The integrated pure double-unresolved counterterm, according to section We now proceed to computeular and process. integrate the local counterterms relevant for this4.2 partic- Local subtraction The non-zero double-real singular limits for the process we are considering are T where, in this case, The relevant yield [ 4.1 Matrix elements JHEP12(2018)107 , (4.9) (4.8) ) z (4.10) (4.11) ) lm , , jm z s )+ s . z ij −  dz 1 s # + − − abcd abcd 0 (1 ) i s s 0 Z  − (1 ) ( im y i i 0  z s jl ) ) y dy O s 1 z z ) )( − 0 0 + 27 232 im Z jl z − − 2 0 s  s (1  − − y 2 z (1 (1 2 2 + + − 2 (1 y z z π ) 81 ) 0 il φ 2948 ) ) y jm z s 7 0 0 18 ( s z z −
− il 0  2 ij 2 coincides with the squared s (sin − − s  π 1 ,x ) (1 0 ) 2 (1 (1 + 2 dφ y ,z 0 0 54 , π 0 2 131 , abcd z z 
0 ( cd 0 0 ijlm 0 17 ) s 9 ( rad Z + y y z y 0 3 depends explicitly on the invariant are two massless partons, other than Φ − ζ p p = ¯ d − ) d − dz 3


2 1 9  0 1 , 1 0 0 38 0 y,z,φ,y Z 3 (1 x x Z ; abcd 0  , x 2 2 ( rad abcd  0 and 0 y s z − ) s + Φ c "
− − 2 , z dy 2 0 d 0 , 1
 1 1 y 0 0 2 ijlm ( y Z rad ). Different terms in the eikonal sum can be  − } 0 2 – 35 – 2 Φ − ¯ k 1 s d µ 4.6 { + 2 − dx 1 1  2 z z  0 y, z, φ, y 0 Z 0 / Z
lm 1 z F  y ; 0 , 2 2 8 − z s B C
− 0 2 R F
 ) + abcd − x =1 ) abcd 0 2 T s z C , s 1 x X 2 − l,m R ) − 2  abcd , 1 − z R S s abcd π 1 T (1 0 α ) + 2 BT rad − )( 2
x , z 2 0 1 y s 0   Φ 1 . In this parametrisation, all integrations for the process we are 4 ( x z d N
− s 2 h  N B 0 ) (1 abcd − z y Z N = × × = = (1 , according to eq. ( 1 0 − − y s } = = z 0 0 1
) , RR RR y y 0 34 (1 are the Catani-Seymour variables relative to the secondary-radiation phase 2 h h parametrises the azimuth between subsequent emissions. In the chosen { y ,

ij ij 0 and can be parametrised as 0 0 ) ) are the unresolved partons, while 0 abcd z S S y y − y y abcd ) ( b rad = x 2 2 , 1 , Φ − − − − y s } d abcd 0 0 1 1 and rad ), the double-radiation phase space ( cd rad and ij (which in the present case of course exhaust the list of final-state particles). Using y z 0 s Φ { Φ Z y a b d d = (1 = (1 = = = = ] = ¯ 3.58 Z bc bd cd ab ac Z 22 ad s s s s s and s abcd where remapped to the sameresult Born is kinematics, and, performing the relevant colour algebra, the considering are straightforward. For theis case [ of double-soft radiation the relevant integral where, for the processcentre-of-mass energy at hand, the invariant where space, and parametrisation, four outwhile of the the remaining two six involve square involvedexpressions roots are related binary to invariants azimuthal dependence. have simple The explicit expressions, a eq. ( s where JHEP12(2018)107 ) = ,  ( } and # = 1, O ) h (4.15) (4.12) (4.13) (4.14) (4.17) (4.16)  h ijk ( + { .    O ) hq  ik = 4, sector ijk ( + s s and 889 108 W d O  } − R − + , k 2 ) . Adding up all  z = 2 when 134  π r 648 ( 3 8 {  1 2 r + 23941 O i + =  z − r + and }  1  324 2 12149 ) π k ) must be combined with hq [34] +  ijk η 2 = 1 or { − 2 31 12 W . RR,  2 r ln ¯ − 4.16 31 R ij ik + π 18 s . − S 3  ) ζ 74 27 k k 2 −  1 ,  8 3 z z + (1 ( 9 3 80  + 2  rad + 1 = 3 particles, the indices 1 − + + ik O ) 3 3 R  i i s Φ r n k ζ T z z + d z 1 + − ijk 3 2 k 9 [34] s + − hq z " Z 122  η 2 i  ) in the case of two soft quarks, in a process ij   z W + = 2 ) ln ¯ s 2 2 i , s k  k + R z µ involves − z + ( z 2 ijkr 3.57 s ( rad   RR j – 36 – µ   1 − + R z Φ S 8 3 π  i 4  , and we can choose d ijk α 54 2 kj z 425 } F R + s C + i Z T − C r z − 2 ij [34] 2 3 F R 34 2 µ ijk hq S  s T C π s 2 2 ) and eq. (  ik,j ≡ ) applies to the configurations , 2 = 2  W t 2 R ik ln  11 18 2 3 s  s rad R S ,  µ π  2 3.55 ik,j 2 4.12 Φ α BT s 2 ). The subtraction of the double-counted soft-collinear limit is  − 1 t R  µ ,  1 d 2  T  1 1 R S  π 4.6 { + 2 3 T α Z 2 B N × ], we have set F in the other cases. The result in eq. ( 2 2 3 S − π C  = = h α 14 2 22 S 9 π , R = α 2 − − T ) − 20 ), and restricting our analysis to the case in which only the single-collinear RR 1 2 ( = = 3 hq   1 ) I = 3 ijk S 3 1 π 3.49 ( r hq α 2 C − I )   2 , + B × contribution, and we can explicitly check that their sum is finite in ijkr ( rad hq = , as seen from eq. ( Φ ) } d W RV 2 ( take values in the set Z = 2, while I 234 RV by sector in the NLO phase space. Indeed where the real-radiation matrixq element q the Next, we consider the integration offormula, the single-unresolved eq. counterterm, ( applying the general limit is non-zero. We find contributions to the pure double-unresolved integrated counterterm, we get very simple in this case, since one has as can be deduced fromfeaturing eq. ( only two partons at Born level, identified here with where, following [ Note that the result in{ eq. ( be similarly computed, and it yields The double-collinear contribution (before the subtraction of the soft-collinear region) can JHEP12(2018)107 , ) )   ( (4.21) (4.18) (4.19) (4.20) ( O O . + + )  ( hq  O W 20 . + ) − R  2 i ( hq
π R O h 7 6 ), as a sum of three W i ¯ S +
+ R − 3.5 3 hq , i ζ 1 ), considered in the par- [34]
 W h S ) 9 hq ¯ S R 100 − 2.55 12 i . ( C 1  −
) I + h 1 − 12 h ¯ S − (  ¯ S ) 2[34] − hq h 3 K 1 20 C C  RV − ( , − + 8 3 ) + ) hq 2
K 2 hq + π ( h C  r RV 7 9 ¯ S [34] W ( + K h − S I  h [34] R   − ¯  − 1 S η i + ) ) 8 3 h 1
1 ) – 37 – 1 − ln ¯ ( h ( 2  it reads 2 ( + I ¯ S 2 −  8 3 I r K  1 − + 1[34] hq + + 1 + −  [34] r 3 C 2  4 R η 34 3 µ + RV hq T s  VV  RR ln ¯ 2 3 C  [34] ln  − = = = ) with the real-virtual local counterterm in the same NLO +  S   2 h h  1 =2 s 2 R ¯ µ S sub sub sub s ,n  µ T h = 4. Indeed we find that 4.18 ). In sector rad  F R 2 3  1  d Φ C F RR RV VV T

S S d R π π C I 3.73 3 2 α α 2 2 T R Z ×  2 3 − T − R R  2 T T = = 2 denotes the NLO counterterm given in eq. (  s ) µ   1 1 S π 12 α 2  2 3 2 3 =2 ( hq  S S S I π , n π π α 2 α α 2 2 F B − ) C

= = = = I ) ) RV ( hq The cross section is constructed in general, as shown in eq. ( 12 RV ( hq K ( I I struct numerically the total crossflavours. section for We emphasise the production howeverand that of local: two the quark formalism a pairs we detailed ofmassless constructed numerical different partons is implementation is completely for being general all developed processes and involving will only be final presentedfinite state in and forthcoming integrable work. contributions, given by 4.3 Collection ofThe results heart of the subtractionical procedure integration of is the the finite combinationphysical remainder of distributions of analytic and the results cross real-radiation with squared sections. numer- matrix For element, this to get proof of concept, we will simply recon- where ticular case of two non-gluonNNLO final-state subtraction partons for at Born thenumerical level. process consistency All at check. required hand ingredients for are now assembled, and we can proceed to a The final ingredient forpresent subtraction case, is it the is integral given of by the real-virtual counterterm. In the sector must be finite in general formula, eq. ( The combination of eq. ( The next ingredient is the mixed double-unresolved contribution, which can be read off the JHEP12(2018)107 = 4. (4.22) (4.24) (4.25) (4.23) d ], or can 70 35. For real . , . = 0

 /s 2 2 s , µ µ . 00011 00038 . . ln 0 0 4 3 00040   . − 0 9 44  40787186 . . − 90635 29491 . . 2 0 = 1 π − + 2 40806 1 9  . 2 s − µ × × 3 = 1 01949914 . ζ ln F F 0 8 3 LO C C − ×  σ 3 that also the logarithmic renormalisation-scale R R – 38 – ζ F F F T T 1 C C 2 2 C   + 4 R R R S S NNLO π π T T T 2 α α σ 2 2 11 2 2 2
    − S π S S π π B B 2 α  α α 2 2 = =   = ≡ B B sub sub . . = = an NNLO num NNLO RR RV K sub 1 1 K Φ Φ d d ), obtaining VV Z Z 4.21 Our local IR counterterms are defined through a unitary partition of the phase space out of a collection ofbe universal defined kernels, in written terms in ofbe terms gauge-invariant of obtained operator kinematic as matrix invariants, elements, limits which asfigurations. of can detailed radiative Overlapping in matrix singularities [ are elements fully inof taken the such into dominant singular account by soft limits, suitable and with compositions collinear no con- need to resort to sector-decomposition techniques. space, and working towardsobservable an cross section. optimal organisation of the numericalinto integration sectors, of in the suchsingularities, a associated way with soft as and to(up collinear configurations isolate to of in two an at each identified set NLO, sector of and a partons up minimal to number four of at phase-space NNLO). In each sector, the counterterms are built infrared divergences up toapplied NNLO to in processes QCD. featuring only Thethe massless method initial partons, state, and has not as for involvingprocedure a now coloured is first been partons conceived significant in developed with step and the towardsIR aim a counterterms, of general minimising aiming formulation. complexity for in Our their the subtraction definition complete of analytic the local integration in the unresolved phase dependence is correctly reproduced with the same accuracy. 5 Conclusions In this work we have presented a new scheme to perform local analytic subtraction of to be compared with the analytical result For completeness, we also show in figure The rescaled NNLO correction, evaluated numericallyis by then means of the subtraction method, where, for definiteness, inradiation, the we second have line written we aterms have Monte randomly in Carlo chosen eq. code ( to integrate numerically the remaining two In this case, it is given by The subtracted double-virtual contribution is computed analytically, and is finite in JHEP12(2018)107 �� �� � � �������� � � � � ��������� � � � � ���� ���� � � – 39 – � � � � � �

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�������� �� ������ ������ ������ ����� ������ � � � � � � Figure 1 In this article we have integrated all needed counterterms over the exact phase-space The kinematic mappings necessary for phase-space factorisation, as well as the The sector functions that realise the phase-space partition are engineered in such a erm contributions in the samenaturally sector. adapted to This allows theyielding us kinematic simple to invariants integrands employ that to parametrisations be appear that evaluated in are analytically. each singularmeasures, contribution, without exploring the possibility of approximating the latter in the relevant soft allowing to subtract,NNLO sector contributions by featuring sector NLO kinematics. in the NLO phaseparametrisations of space, the the radiation phaseare singularities space devised of over by which the maximally the exploitingonly counterterms the chosen are freedom integrated, differently one for has different in their sectors, definition. but They also, are importantly, not for different countert- A number of sum rules,recombine stemming various from subsets the of definition sectors,ally of prior yielding the to integrands sector performing that in functions, countertermand all allow integration, cases one collinear eventu- are to kernels. solely madereproduce Moreover, up the by through sums complete factorisation of structure elementary relations, infrared of NNLO NLO sector sectors functions in all relevant single-unresolved limits, way as to satisfy fundamental relations that allow to achieve the main goals of the method. JHEP12(2018)107 contribution to F C , as simply as in the R  T poles and phase-space singularities of , which bears witness to the extreme   – 40 – ]), and they are not expected to pose an obstacle for a general 76 , 75 at NNLO, which has been detailed explicitly. ¯ q q → To summarize, this article represents a first step towards the formulation of a general, While in this paper we have concentrated on the general structure of our method, At NNLO, we have deduced the structure of the subtraction scheme in full generality for At NLO, we have shown that the proposed subtraction method works in the general − e + the extension to theimplementing massive the case, subtraction as method well in a asthat fully the the differential completion present framework. work of We lays believe an however a efficient solid computer foundation code for these future developments. problems are expected to arise. local, analytic, and minimal subtraction scheme,processes relevant for at generic NNLO multi-particle in hadronic QCD.taken, To reach including this the goal, a analytic number integrationfor of of important final-state steps the still processes, remaining need double-unresolved to the be counterterms generalisation to include initial-state massless partons, and corresponding integrals are therefore(see, closely for related example to [ application integrals of known the in method.more the work, The literature in inclusion order ofspace to initial design mappings, state and as radiation test well is appropriate as expected sector implementing functions to and collinear require dedicated factorization, phase- but no new conceptual in particular concerning sectoronly a functions simple and example phase-space offurther mappings, implementation, technical we difficulties and emphasize for we thatstates the have we at extension do given NNLO: of not indeed, our expect ancounterterms algorithm significant are important essentially to advantage combinations of a of our general the (re-mapped) method massless NNLO is final splitting that kernels. the The required local of the scheme, and of theAs delicate organisation for of different double-unresolved contributions to counterterms,performed the we cross the section. have relevant deduced integrationse their in a structure proof-of-concept in case, general, the and massless QCD final states. All single-unresolvedof and double-real mixed origin double-unresolved counterterms have beenNLO integrated case, analytically and to the all properties of orderscorrectly sector in functions reproduce, have allowed sector us to by showreal-virtual that sector, contributions. these the integrals We explicit stress that this is a highly non-trivial test of the consistency case of massless QCDcally final the states, full with structure theof of integrated the virtual counterterms method, one-loop reproducing we singularities. analyti- exactly have to Moreover, shown as all that a orderssimplicity the test in of NLO of the the counterterm the integrands dimensional integration involved. power regulator can be performed fication in the casesreactions considered (for here, example this when mightmassive including case). instead initial-state be This partons, the possibility orbeyond case will for the for be a scope general investigated generalisation of in hadronic the to dedicated present the paper. future studies, which are and collinear limits. While this possibility would not have resulted in any analytic simpli- JHEP12(2018)107 (A.6) (A.3) (A.4) (A.5) (A.1) (A.2) ) we find . ) ..., i , ij 2.19 + ( 6= 0 in the soft limit. π 2  on the physical real- ∈ ) → , , ) i jr on ratios of elementary ij a, b, c ab / e s jr which do not contain a C ∀ ∀ s ij = 1 = 1 . + i ij C i + ij ij S ir P s µν ir W W ( s ..., i B ( / and ij a , S + / i µν ij jr C ij ), with the result s ir S i 2 j Q  s gives rise to a singular contribution x C S i i S x j 2.26 1 + 1 + + = F F l B ⇒ ⇒ C C ij = = } } ), is not singular in the soft limit for parton ¯ ) we have noted that q in eq. ( P ¯ q , q, i q, j { – 41 – { S j x A.4 j f 2.29 h f = 0 = independent of δ δ 1 il g ij g j i bc ia ia ij ja s N i and f e s s s s f /w δ j i δ i 1 /w = e + x ij i S + 1 i + 6= C P l e R i j ir jr ij x s x s = = ). The action of operators C A A ij ij i , and, in the process, deduce the form of the soft-collinear kernel C C S 2 2 W W ij , , 2.20 i g g , defined in eq. ( is given by ij j j C S f f µν ij C δ δ ij 6= 0 = 0 s Q g g i i and ib ia ij f f . We now rewrite the remaining contributions in terms of Mandelstam ab ) we used the fact that only i = 0. The same happens for all terms in s s s i δ δ s i S /x ij = = A.3 S ij µν C ij Q i P S appearing in eq. ( in the collinear limit, while in eq. ( Next, we consider the action of the composite projector ij il P i /w , hence We now note that i denominator 1 invariants, using the definition of radiation amplitude squared, where,pendences without in loss the of real generality, and we Born-like drop matrix all elements. kinematic Starting de- from eq. ( where in eq. ( 1 We start by verifying that the sequentialdoes action of not the depend singular projectors on on their sector functions ordering. To this end note that massless invariants A Commutation of soft andIn collinear this limits appendix, atcollinear as NLO limits an example,S we explicitly show the commutation of the soft and We would like touseful thank discussions. S. The Frixione, work F. ofprogramme PT Herzog, for was V. funded research by and Hirschi the innovation and European under2020 Union N. the researchers: Seventh Marie Deutschmann Framework Train Curie for to grant Move very agreement (T2M). N. 609402- Acknowledgments JHEP12(2018)107 : ) l 3.7 (A.7) (A.8) (A.9) . (A.10) (A.11) ij , are non- )), in these ). We begin abc correspond to 3.7 , . CS ), together with 3.11   not are singular in the A.8 is independent of jr jr j ir ir s s il , and s s 2 2 = , /s (see eq. ( does jr abc l j ir B F F . To this end, we start i jl s i j s x σ s x C C becomes soft. Taking now 2 f S 2 or SC } } ijkl i C . ¯ ¯ F q q . j , F ) and eq. ( q, q, C W jl ir jr kl C { { = s } s j j B } ¯ q B f f abcd ¯ q k δ δ ij ) q, 2 il jl i q, A.11 C { s and g g ( s kl s { j the ratio i i , j 1 f I f f i f i δ δ δ ij δ N 6= 6= abc g l X ijkj g i C + + i f i ,l C = f X δ ij W , 6= δ ir ir jr jr k jl , s s C s s ab + + B ij S A A ij i i 2 j ir jr , C ijjk s s x C C s x 6= l 1 X ab 1 2 2 – 42 – W A A g g C ir j j jr C C f f , s −N s −N 2 2 δ δ a 0 (the two definitions differ by subleading soft terms). g g g g = S ij j j = 2 i i f f s f f → δ δ . The equality of eq. ( δ δ R R g ), we get } i i g g i   ¯ q i i ir f S q f f S δ s 1 1 { δ δ ij ij A.1 j 1 ij ij s s N N f = = δ C C ), in the collinear limit and get F −N B B ij C r P ), shows the desired commutation of limits in each sector A.2 = = = i + = S g R R l A.4 j i f ij S δ ). In particular, we note that the soft limit C A ij i C C S 2.30 ) and ( = 0, rather to taking j f A.3 → C i ), including their relation to the NLO-like functions defined in eq. ( x limit according to eq. ( i 3.8 S In this appendix we explore theand properties of ( the NNLO sector functionsby defined in establishing eqs. which ( limits, among vanishing in the threeby analysing sector the topologies behaviour of the sector-function denominator operator relations ( B Soft and collinear limits of sector functions where in the last two steps we have used colour algebra, and the definition of the Casimir According to property ( we can therefore set Among all the terms incollinear the limit, double hence sum, only those with We can now verify commutationWe find by considering the two singular limits in reversed order. which is eq. ( taking The soft-collinear limit is thus the where the ellipses denote terms that remain regular as parton JHEP12(2018)107 ,
kj (B.2) (B.1) σ + , . This is σ ) jk σ kj , gives a non- , , σ  i L kd ) cd + σ lkji σ αβ σ jk ( ik j,k σ σ ( i,j,c + 6= X d ) X 6= + d ) then proves that the ) ) αβ , lkij , ( ib αβ i,j σ , αβ ) ( ji σ B.1 ( ij 6= X σ i c kj cd , σ + , kjij 6= h σ σ σ b + + X jiki jd cd d σ σ + i,c σ klji ] + kd , 6= σ ) = ij X σ + [ i,j d i,c jk ) = ) i σ 6= + σ 6= X lk X kjji i,k d ( d , only if the numerator of the latter, 6= ikjk σ σ , beginning with those involving the ibkj c . ) ) 6= jiik i,j i X X σ d ) commute when acting on σ σ ) σ klij 6= 6= X + + ) αβ αβ c kd d X abcd , σ + ( ( ib ib and we have used the definition of the + αβ σ + ) kl σ ( σ ib αβ + W + ( ij 3.10 j σ i i kiji σ ] αβ i,k jicd jbid ( σ ( ij ikkj 6= 6= i σ ij ibjk 6= σ ) σ ikjk b b [ X X X σ σ 6= d c σ jilk = σ b + X ( ) . Inspection of eq. ( αβ σ j,i j,c σ and + = ( = ij i σ 6= 6= + – 43 – X X αβ = σ i ) = S i,j d d + ( ij kiij jikd L i 6= j j X σ σ ijkj ibjd = c σ ijcd 6= 6= 6= ibkj ikkj b σ ). Now we note that a singular limit σ σ ibcd b c X jikl X X  = + σ σ i i j,k σ σ i + ijlk i + ) S = + S 6= X + + d σ + S 3.11 σ αβ ijkd i,j i,c jkik ( ji + + ijjk σ i,c σ 6= 6= ibjd ijcd X X σ ibjk ijkj d d ijlk σ ijk 6= σ σ X ( σ i i σ d + i,k σ + ( i ijkl i ijkd 6= 6= i,c i,j 6= i X ) b c + S σ SC d X X + ) exhaust the surviving ones in each sector. σ 6= 6= 6= X X S c d d X jkki αβ ======( i i i i ij i,k σ ijjk ijkl σ 6= 6= 6= 6= 6= 3.10 σ σ σ σ σ σ X b b b c h X σ X X d σ X + i i i i i i S S S S S S ======. ij ij i σ σ σ σ σ σ σ ijk ijk ijk S ijkl i S C ij ij C S ijk ijk ijk C = S ijkl = SC CS C = C = C σ σ = = SC CS σ ij σ ij ] denotes the parent parton of σ σ S C ijk i ij i ijkl ijk ijk S C S i C , appears as one of the addends of Next, we show that all of the limits in eq. ( i S SC CS S i i abcd S S on squared matrix elements,method, sector and functions commutation are of aordered limits crucial limits, is but acting non-trivial artificial on insingle-soft ingredient the limit this of denominator case. our function We list below all relevant limits reported in eq. ( a crucial step for ourof method, independent since configurations commutation ofwhile one limits commutation needs drastically can reduces to be the understood explore. number from physical Furthermore, considerations one when limits must are note taken that, where [ NLO-like sector functions inzero eq. result, ( when applied toσ the sector functions limits. We find JHEP12(2018)107 , .
kd lk . (B.5) (B.6) (B.4) (B.3) σ , σ
kd j,k ] + σ 6= X ij d [ kl ) , i,k k σ
σ 6= X αβ d ( lk ji ) + ) σ σ k ] αβ ( αβ ij + + ij ( ij , [ σ kl σ kd , and not consid- d σ ] σ σ ij = ij = [ i i i,k C ) σ ) 6= , X ijlk αβ d i,j ) αβ ijkd ( ji σ ij ( , ji , 6= . σ σ X
σ d σ αβ  , and not considered above, , ) + ( ij i + i,k
+ ) ab kd σ ) ijkj ijkl αβ 6= , ) ikkj X , , ( kl σ S kj d σ σ ijkl =
αβ σ αβ σ σ ( ji αβ ( ij σ jk , ( ij = kj = = satisfy σ , we get σ jkik + σ +
i,j,k + σ σ h = jikd ) σ σ X h + σ σ 6= lk jk kl σ i ) d ijkl ijk + σ = σ αβ σ + = σ ikjk S ( i σ ib ikl ijk + ) C j,k αβ + σ ] i ) jk ( ij σ S 6= CS X jiki ij σ S i αβ σ ijk kl d jilk [ ( ikjk αβ + SC ij SC σ h k 6= σ ( ij σ + σ σ b ijkl and X i σ σ ijk + ) ) CS + ijk  and ijk C , + = ) = ijkj . ) C = ijkd αβ αβ ijk = σ ( ( ik jiik ij ) = = σ jikl αβ CS = ijk CS C σ – 44 – σ σ ( jiid ij jiki σ jiik jikl σ + klij σ ibjk i,k σ σ σ σ σ σ = + σ ijkj = = + + σ 6= X SC ijk + σ d ) i = + σ + + σ + ijjk ijkl ijk S + ijkj = ijlk ijlk σ αβ C C ( CS ij ijjd σ i ijk σ σ ijk σ ijkd i i i i ijkj ijkl ijjk ijkl σ σ 6= + ijjk σ + + b ( S S h S σ σ σ σ X S ), where the limits are applied to the sector-function de- ijk σ CS CS ( i,j i,k ======i ijjk ijkl ijkl 6= 6= X ikl X B.6 d d σ σ S σ CS ijk σ σ σ σ σ σ σ σ ij ij ======ik ik )–( SC ijk ijk SC ijkl ijkl S S S S σ σ σ σ σ σ σ C C C C ij ij ij ij ij ij ij B.2 ijk ijk ijk ijkl C C C C C C C ijk ijk ikl ijk C C ij ik SC CS ikl ijk ijk ijk = S ijkl S SC CS SC = CS C = = C σ = = SC = = σ SC CS = = σ = σ σ σ = σ σ = ijk σ = = σ σ , are sufficient to prove that all non-vanishing limits in the different topologies ij σ ik ijkl ijk ijk σ σ C σ σ S S ijk ijk ikl C ijk ijk ij ij ikl ijkl ij ijk ijk SC CS C ik S C CS SC SC C C CS S ij ij ik SC ij CS SC S S C ijk ijk ij ijkl ij ij C ijkl C C C C C C C Finally, the mixed soft-collinear limits The relations in eqs.nominator ( commute when acting on the sector functions. The same commutation relations hold when Coming to double-collinear limits of type we find Moving on to ordered limits involving the double-soft limit ered above. Next, we list ordered limits involving the single-collinear limit JHEP12(2018)107 . ik ): C . (B.8) (B.9) (B.7) j ; L 3.10 j . L jk C . of a uniform ] and ij i σ abcd C . , which could be L i (first) and are generic combi- ik W λ 0 i ijkl C , one has for example L L C L = [ jk ; ] = ; C kl , and in sector ij L C . ijkj 6 ijjk C , i ij W σ , a generic set of soft and collinear W cd C i LL ), the action on [ L C automatically also involve the limit , L ] = s of the limits ij 3.8 i jk L 7 C C ] ][ , LL c , and jk i uniform composition of L ab L [ C and ijk 0 C ik C ij L for topology for topology , ordered composition of C C c 1 in eq. ( , = S ] ] = = [ ⇐⇒ – 45 – i , σ ijk ijk jk a ij L C LL S C C C ] ⇐⇒ s ij jk L ] = [ C C i c [ 0 0 L const. 0 α > β > [ LL 0 , and , and i , → → L ] ik L → → ij [ , ij j j S S j j ij C , S , λ , λ , /λ /λ jk . i i i i jk jk C λ λ λ λ A which can be directly identified with the limits given in eq. ( C ] = C ik j ( ( 7 C : S , , i ]: i ij ij j S L [ ] = [ C L C j i ij L ) are collections of soft (collinear) limits, while L , , C c j k L jk S S ( C ] = [ ) exhaust all single- and double-unresolved configurations in each sector. In other [ s ) have been applied. We start by denoting with i , , i i L L S S j 3.10 3.10 Owing to the prescription The next step in our analysis is to prove that the compositions of the limits given in L Repeated limits can in all cases be readily simplified. Given a generic limit Note that compositions of limits involving both • • [ 7 6 Indeed We conclude that all possibleobtained single- as and ordered double-unresolved compositions singular without configurations repetition can be where nations of limits. Thea remaining pair uniform of compositions soft involve either limits, a pair of collinear or generated by combining inlected all by possible the ways sector the functions, namely single-soft and single-collinearcomposition limits involving se- soft andcomposition collinear where limits the is soft equivalent limits to act the first: corresponding ordered All single- and double-unresolved configurations in each sector can then be systematically limits, corresponding to configurationscollections where of some energies, physical or quantities angles,such or limits similar, can approach be zero. either Compositions ‘uniform’ of or two ‘ordered’, (or with more) the two cases being defined as was done in appendix eq. ( words, there are no leftovereq. singular ( phase-space regions after all combinations of limits in applied to the physical double-real matrix elements, as can be proved analogously to what JHEP12(2018)107 , ). j S 3.10 (B.15) (B.12) (B.13) (B.14) (B.10) (B.11) can be , and it j ij S S , ij S satisfies k S , and with from the list of limits, , , , and satisfies ijkj , , jk , , , ijk . . ijk W ijjk ijkj C C C L ijkj ijkj ijk W W ijkj ijjk ijjk ijkl W W L L W W ijk W W SC is not present at the right of L ij ij L L L , L i , and i i CS C C S S ik ijk . ijk ijk S ijk S ik ijk ijk ijk ij C S CS . If C C SC C SC S = 0 , 0 0 j . 0 0 0 0 0 0 . , , the single-soft limit S L L ij L L L L L L S ijjk ijk = = = 0 . At this point one can use , = = = 0 = = 0 = = = i for topology C W ij S limit commutes with all other limits in the list j C ijkj ijkj ijkj ijjk j – 46 – ijkj ijkj ijjk ijjk ijkj ijkj ijjk , ijkl S W W W LS i W W W W W W W W , and C k S L L L ik L L L L L , to remove jk jk S i i 0 i i : ij 0 S ij ij S S C C L S S , ijk L ), the 0 0 C j k C C jk k L L jk jk S S ijjk 0 0 jk jk . Consequently, one can remove S CS B.6 C C C and , and 0 W L L 0 0 , which commutes with C C jk ik ijk L 0 0 L either gives zero or can be replaced by that of L L )–( jk S , we note that it commutes with C L L j 0 C jk SC S , B.2 L C kl or , one can remove it from the list of contributing limits. A similar C in its stead. The list of singular limits is thus reduced to the first line ijk ijk , can either be moved to the rightmost position, where it gives zero, or ijk C ijkj ijjk ij ), CS jk C . Therefore, when appearing in a generic composition of limits, it can be W W can be either moved to the rightmost position, where it gives zero, or replaced i SC C or S , k 3.10 k S ik S S , i Since by statement holds for Topology Besides commuting with so that replaced by and add of eq. ( satisfies Since the action of simply removed from the list. Considering now moved to the right until it encounters valid for generic limits the latter can be moved to the rightmost position, where it vanishes: Topology According to eqs. ( except S • • • To conclude, we reduce this list of limits, topology by topology, to that given in eq. ( JHEP12(2018)107 ). kl can ), ν C 0 (C.4) (C.1) (C.2) (C.3) 2 (B.16) (B.17) ˜ k 0 0 j 3.10 µ ijkj ˜ k 0 x , ˜ k } W ¯ q  ) q,
{  0 k k ) f x 0 j δ . x ij ,  2 ) in the case of ) ijr,jkr Q  ( ) g ijr } . . F ijr ), as k ( kr ikl ( r ¯ f k ¯ s C ¯ k δ { , ijk ijk ) ) 2  SC A 2.26 ijr,jkr ( 2 ijr C ijr − µν CS CS ( jr } ( kr or ¯ ) s d s ¯ ij k B , , . {   Q ijr + ¯ ) −  ν ). First, we consider composing ikl ( jk ijkl ijk jk ) ¯ 2 s ˜ k −  B ijr C ˜ k µ ) Q ( kr )  ijr SC ) , ˜ k SC s ( jr 3.54 2 k ijr , ¯ s ijr = ) , 0 j ( jk 0 ( jk ijr 2 + ¯ S 0 k  0 ), ) ( kr
x ˜ k ) Q ) x ¯ Q ˜ 0 j s k 2 ijk · ijkl ijr x 2 ) ijr ( jk g + ijr ˜ k C ( 2 C ˜ k ( jr . These limits can either be moved to the k 3.10 ij ) } in eq. ( Q , ¯ f s , (2 ¯ − k δ Q ijr ik { ik ab 0 j ijkj ( jk 0 k 1 = A S +  S x x P 0 k C W W , , – 47 – ij , k  µν limits holds unchanged with respect to the one − . The list of singular limits in sector f ij ij ) P ij ) R  ( C . The remappings described in the following apply Q ) ij kl ijk in the case of C C ijr  jk ij − , ( Q k , C ijr  i , x kr ( ¯ i k C Q 3.5 ijk SC ¯ s + S ) 0 j 2) S ) , 2 x ) ij ) are defined in analogy to eq. ( : or : ijr and CS − jr 2 ( kr − P − ijr ) s d in topology ( k jr ( , s d ) or C.1 can either be moved to the rightmost position, where it gives ¯ s ijk  S ijkj ijkl ijr ir + ¯ ij − +( ( jr kj  ijr s s C ik ( ) ), including the detailed dependence on the remapped phase- ¯ s ( j jk ( W W C S ) ¯ k jk ijr C ( jr 0 k µν P ij ijr 2 ¯ s 1 3.54 x ( jk P ¯ s = and N 1 = = ) ij k 0 s 0 j N S ˜ ijr k ijkl x jk ( = = , and we introduced the shorthand notations W P in eq. ( ) RR 12 i, j, k ( ijk 6= K C r The discussion of the relevant for rightmost position, where theypresent yield in zero, the or list, ( beThe replaced final by list limits of contributing that limits are thus already coincides with the third line of eq. ( zero, or replaced by thus be reduced to the second line of eq. ( Topology As a consequence, + ij ) • C 2 ( where The primed variables in eq. ( squared matrix element neededK for the evaluationspace of variables the described double-unresolved in section also counterterm to the corresponding sectora functions double-collinear limit and a collinear limit. We find C Composite IR limitsIn of this appendix the we double-real list the matrix composite element soft and collinear limits of the double real-radiation JHEP12(2018)107  , )), .  )  0 ) # (C.9) (C.5) (C.6) (C.7) (C.8) ) (C.10) 3.54 , ) ijr # ( ijr,jcc d ( ijr,jkr ) ) ¯ k ijr ( } ( jd · } ¯ ¯ k s ijr ijr ¯ ( ra ( ja k ˜ k { ¯ ¯ s s {  2 ,  (recall that the ) cc )  2    B B ) ijr ) ) ijr ˜ k 0 ( jr  ) . Composition of a ( c ¯ s ijk , ¯ k ij ) ijr ijr ! ( c ( jc ( d , and we have defined · 2  ijk ¯ 2 s , ¯ k ) ) ijr ) Q ) ˜ k i ( jd 2 ijr,jcc 0 · j 0  ( 2 ¯ s x i,j ˜ RR k ijr ) x ˜ k ˜ } k ( jd · i ) 6= ij 2 X 2 ¯ ¯ k s − − c ˜ k x ) ˜ k { S ijr,jkr ijr 2 j  ( ( cd (2 +   x ijr } s ijr,jkr cc ( jc ) ) ( ) + 2 ( 2 2¯ ¯ k ¯ s } B { 4   ijr ijr ) − ijr ) − ) ¯ k ij ( rd ( jd ) ( c  d ij ¯ ¯ { s s s ) ¯ k ( ijr cc ijr ( jc  · µν  − ( ) ¯ s ijr + J } ( a 1 ˜ R k B B ijr,jcd ) ) ¯ ¯ k k ( i,j 2 ijr T ) 

( ja  { · } ) 6= } ) ¯ X s ijr ijr c ¯   ij ¯  k q ijr ( rc ( jc ˜ ) k  q ( jr ijr { ¯ ¯ i s s ) ijr 2 Q ( ¯ s +  ( kr µν x ijr }{ ) } i ¯  s ( kr j   ) + ijr cd  ¯ R k , ) ¯ f ) ( s x 2 − ) ) i ijr ij { } B ( jk f ijr j jk 2 ijr 2 ( – 48 –  ¯ ¯ { − P k s ( jk ijr ijr  x ˜ } k δ ( jr { ( jr  ) ¯ j C ) s R i ¯ ¯ ¯ k s s )  j ( x ijr,jcd jr ij + + { ( ( d  jk I S 2  ijr R } ) j ) ( jr + k Q  ) i ¯ k C µν x ¯ f j s jk µν i j jk i j ijr { x ) j jr x ( kr + P x C x  x x R i C S ¯ s 2 i ) j ) − ijr " ,s j x ) ) i cd f ( jk i i x ( ij + jr 2 j ( ( ) ir S ¯ jr jr s ( c ij B C I " x i s j s ) I I ) k jr g ( ( j Q x ) x j j f j ij f jr  f f ( f , s cd ij − jr C should be chosen for all permutations of should be chosen for all permutations of µν  C δ ij C C s ir , s J 2 2 2 2 i,j P A , s 1 1 1 1 2 i,j,c s ir ˜ 2 k ij 6= ( 1 ˜ 1 k X ir 6= C ij i,j s c N N N N i, j s i,j,c d s ( s 2 ij ij N 6= N X ( 6= i, j, k s c  g d Q 6= j µν ij = = 2 = 2 = 2 = 4 = ij 2 f  6= 1 δ P 2 2) r Q 2 g r N 1 i 1 2 RR RR RR f − 2) N δ N + d  − ijk ijk ijk ( d = = C C C ( − − must be the same that was used in the definition of ij ij ij − appears in the sector function associated with these contributions in eq. ( = S S RR C = ) k i ij ij ij ) ij S i, j, c S ( cd C C ij ( a i ij J 6= S Q 0 C and index c the quantities where the same where the same double-soft limit and a collinear limit (on the same pair of particles) yields Acting with further soft limits leads to JHEP12(2018)107 )). )). kl kl ( ( (C.15) (C.13) (C.14) (C.11) (C.12) π π ) ) ij ij ( ( . Next, we π . π , ( ( , ijk   , π π ) # )  0 )   ) ) 0 . ijcd ijr,jcd ( ijr,klr ( ( icd, jef }  } ( ijr,klr } ) ¯ k ( ¯ k i } ¯ k ). Further applying the ( { cd } { ¯ k {   ¯ k  { {  cd cd  − I µν C.10 B B ) . i B µν o ) ( jd cdef with a soft limit. We get . ) B 0, of the kernel in eq. (111) of I  B 0 ) s ijr ) ) ( jd ijr  ) ) ij + ( kr ¯ s ijkl ) ( cd → ijr ijr ) ¯ 0 s ) ) and ( ( ( cd I ijr 0 i icd  ) ) C ) ) ( lr ( ijr ¯ ( jf s } jc  ) ( lr ijr j ¯ s ¯ s ¯ k I ( jc i,j ijr ¯ s icd  ) k icd ) C.9 ( kl ¯ ( ef s { ( ,  ) 6= ijr ¯ s ¯  s ) ( } X icd ijr ) 0 } ) ( je i,j c,d ijr ¯ i,j  k ( R i ( kl ijr ¯ ¯ s k ( jr { } 6= 6= ( kr + ¯ i s icd X ) { kl c  d ¯ ¯ I ( jd k s g k  j ¯ s l { )  cd C f icd ) f i  R ( cd ( δ jr leads to k R ) would vanish by color conservation in the ab- ¯ s ) C R µν I i icd kl kl – 49 – S j ( ( jc j cd j ik j ) f P ¯ s S f C I i S C.8 S ( ) jr C ) g ) i ) C i i j I i ( jr ( g should be chose for all permutations in ( jr i,j i,j cd ( f g jr k j I j I δ I 6= should be chosen for all permutations in 6= I f f j X f e i f j δ f should be chosen for all permutations of j δ A C f 6= f 2 C 2 1 1 i,j C i,j C 1 k, l i, k, l C i d 2 6= 6= 2 1 1 X N N i, j X 1 N 6= c d 6= 6= c N N 2 N 0 " 0 1 6= r 2 r − = 2 = 4 ≡ − 1 r 2 . = 2 = 2 N = 2 = −N o . and and RR = = ) s RR ij RR ( cd i, j i, j ijkl ij I RR ijkl C S 6= 6= ij C r ik ij r S i is the strongly-ordered limit, ( ), the second line in eq. ( S ), the same i C S . i leads to i S o . S i S C.8 3.57 ) s S ij ( cd ], after an appropriate remapping, defined by I 22 Taking successively a double-soft limit and a single-soft limit, we get where ref. [ where the same where the same Acting with a further double soft limit consider the composition of the double-collinear limit As in eq. ( sence of phase-space mappings: itsfulfils role is the to proper ensure that limits theyields also double-unresolved the counterterm in slightly the cumbersome presence expressionssoft of in limit the eqs. ( mappings. Enforcing this requirement As in eq. ( JHEP12(2018)107 ij RR (D.1) S  (C.16) (C.17) (C.18) (C.19) ij ik C ijkl S i C , S ijk ,
 ijk ) jk  CS br ) ), and introduc- ijk δ
CS W j kl C aj
RR. δ iab,jkr S ijk kl ij ( 3.51  W  iab,jkr } ( k C C )( k ¯ W k f + } i S ij ijkl { ¯ k C kl
S  { ijk C yields 3.12 C kj  − ij i,k . C µν j ik k j 6= B X C ) l f B ij W S S )
bj C
+ C +  ) ) δ ijk i and iab jk )  ij ) + ( jk jk S ak i i,j,k iab iab ¯ s ] C ( ( kr δ iab W i X ) iab 6= CS i ( kr W ¯ kj s } l ( [
¯ s ) ,
S jk ¯ + f } k iab ) − ( jr kj { ¯ k kl C jk C iab ¯ s  bk { ( jk ijk iab  j ijk δ )  W ( jr ¯ s C R W ), using eqs. ( i S ¯ C s ( h ab R aj C + + kl I jk δ ij ( jk jk − br k  C S µν ijk 3.63 C jk f δ  k ij  C
j P j W C ) S ak S f – 50 – S ) ) 2 jk CS i δ i ( C αβ ( ij ab ) ab g
 ( i ji jk j I W ( I j ab f C − f kj i,j,k I C δ W r r
jk ab X k ab h C 6= r f l W r + ab C jk ab C C  ) C − j − i C C jk ) W k S αβ − f j,k ( C j ij ijk j,k j,k,r ] αβ = j,k,r k j,k C S ( X ij = = j,k − j,k,r a X C W jk = b = j,k,r a S [ i X b = = a f X +  W 2 ik = b a S + 1 1 ] b C ij S 2 ij − 1 1 jk N N  [ C ijkl ijkl C f N N ijk ijk 2 1 i  = = C C C S = =  i,j −

h CS CS 6= X kl kl k ≡ RR

i,j RR 6= kl kj r W W X ab k ijk i kl kl X C W i,j>i ijk W 6= C k C C i C − X jk i,j S S ij C = + S k i,j,k i,j,k i,k i S X X hc) 6= 6= 6= X , S l l l , the hard-collinear contribution to the mixed double-unresolved counterterm can be 12 + − + + ( ab K W cast in the form In this appendixunresolved we counterterm show organisethe explicitly themselves NLO how in phase the theing space. remapped terms form kinematics Starting of in for from single-unresolved the the eq. double-real limits integrated ( matrix in mixed element and double- for the sector functions which completes our list of relevant nested limits. D Results for the mixed double-unresolved counterterm Finally, inserting a further double soft limit on partons where the color structure is given by the combinations Composing a double-collinear limit and a single-soft limit, we get JHEP12(2018)107 .  µν j R give, S (D.6) (D.3) (D.4) (D.5) (D.2) ) ij − ) ) z ) i S z z z S ) − ij ) − ) RR. ) − − ijkl 1 − C ijr 1 ijr jr ( 1 1 ) ijr ( jr C ijr 1 ¯ s z, ( jr ( jr ¯ s z, (  ¯  s z, ¯ s z,
( y ( y  ( y hc y jk kl R. ijk hc ij R. ij hc R. hc S ij R, ijk ij P ), and similarly for C kl P W k jk j P P 1 C ) jr − S , 1 ) C S kl , C 1 ) 1 ) ijk , , ij , s ijr ik   C ijr , and similarly for  ( rad  S ijr ( rad ijr ir S
( rad ( rad ,  Φ ,  ) s CS ,  ,  Φ ij ) ) ) ( Φ d ) Φ − d
d ijr C d i,j,k ijr hc ( 1 ijr ijr ), and we exploited the fact ijr ij ( jr ( jr X Z kj Z ( jr } ( jr 6=
 ¯ Z s ¯ l s ¯ P s ¯ Z s ¯ k  R  R jk  W  {  + R R 2.49 hc kl hc k hc jk ij hc for jk ij j W ijkl ij ij ijk S J R J C S J C J C C jk hc C ij k ≡ C +2 +1 +2 +1 +1 +2 +1 +2 P S ij ijk +2 +1 +2 +1 +2 +1 +2 +1 j n n n n n n n n n n n n n n n n R ς ς ς ς ς ς ς ς C ς ς ς ς S ς ς ς ς 1 1 − 1 1 CS 1 1  1 1 N N

N N N N − N N ijk kl kj = = ij ======S – 51 – W W CS ij kl
RR + RR is defined in eq. ( RR RR C kl C    jk  j
k j hc j contributes jk ij W contributes S S S jk S W S J k − S − − ijkl ijk W − i i i j jk ), as well as their phase-space integrals, are explicitly com- C C i,j,k S ik S S S X i,k C 6= S ij ij l − 6=  X − − l D.2 C C  1 + − − 1 1   1 (   +  ij i,j ijk ijkl S 6= X ijk k C C ij ij ij C CS C C 1 X 1 i, j>i , , 1 1 , , − rad rad contributes to the integrated counterterm rad rad Φ Φ = d Φ Φ d ijk d d Z hc) Z Z , Z CS 12 , while the real matrix element is written as ( µν ij K The nested collinear limit The nested collinear limit where the hard-collinear integral that azimuthal terms integratelimit to zero in the radiation phase space. The soft-collinear anteeing the hard-collinear characterupon of integration the counterterm. Terms containing The singular limits inputed eq. in ( the following.we For do brevity, in not all displayQ kinematic contributions to dependences, the writing hard-collinearWe counterterm note that all limits are accompanied by single- and double-soft subtractions, guar- becomes Using the NLO sector-function sum rules, and appropriate symmetrisations, the latter JHEP12(2018)107 ) ) ) z z z (D.9) (D.7) (D.8) k (D.11) (D.10) ) ) − S − ) − 1 1 ijr ijr  1 jr ( ijr ( jr ) ( jr ¯ s z, ¯ s z, ¯ s z, jk ( ( ijr y ( y ( kl y hc C hc R. ij R. hc ij R. ij W P P kl P jk  i k jk 1 ) 1 ) ) , is , C 1 ) S C , C k ijr jk  ijr k ijr j ( rad ijr ( rad ( kj S ( rad hc) S C S , Φ Φ i,k Φ ) with the four-particle W  k d d  6= X  d 12 l S ( ,  + ,  , ) ,  Z Z ) I ), together with D.8 + Z ) )  )  j R ijr R ) ijr ( jr R ijr ijr ( jr S ( jr ( jk ijr ¯ s kl ¯ s ( kj 3.12 jk ijr ¯ s jk (    C  W ) } C W C hc k  hc ¯ k ij k hc ij j ij ijr RR, { J S jk ( jk S J S J  jk C ). W R C ijkl +2 +1 +2 +1 k +2 +1 +2 +1 j h +2 +1 +2 +1 n n n n n n n n S  C ς ς ς ς n n n n S ς ς ς ς ς ς ς ς i 3.64   kl  1 1 1 1 S 1 1  C − N N ). Using eq. ( + N N N N k = ,  jk ) = = kl S = = = = C 3.68 C ijr – 52 –  ( jr RR j ) ¯ s RR  RR S ) RR  ijr  ( kl  ijkl   ijr hc ) j ij hc) jk ( kl jk C , W J S S S ijr W ( jk 12 kl ikl ( − − i,j − kl C i W 6= X K ik k k S ik C SC 1 ) S jk S , S  −  C ijr − − 1 ( rad j X i, j>i 1 i,k i,j,k  1 Φ S X  6= 6= X  d l l  ij +2 +1 S n n + − − Z ijk ς ς ijkl ) with a double-soft limit we get ) with a three-particle double-collinear limit one finds 1 ijk C C +2 +1 C D.5 n n ijk ijk ς ς D.4 −N ij C = = CS CS 1 , 1 1 , , hc) , rad rad rad Φ 12 Φ Φ ( d d d I We next turn to the soft term in eq. ( Z Z Z and introducing, as usual, remapped kinematics for the sector functions and for the limits which can be straightforwardly rewritten as eq. ( Collecting all the above integrated terms, the resulting integral Finally, replacing the three-particle double-collinear limitone in we eq. ( get the result Acting on eq. ( Combining eq. ( JHEP12(2018)107 z i (D.16) (D.14) (D.15) (D.12) (D.13)
, which − yz . 1 RR ijk 1 ) cd , RR  C z R ik icd z
 ( rad RR. − − S yz jk Φ − RR. yz ijk 1 . 1  i unless explicitly d , C ikl 1 i 1 C )
cd ,
 ijk
1 cd ) ) Z , kj ijk R icd − , C ) SC R ( rad ) ijk icd iab 1
( rad k ( jk W
Φ 1 icd C SC } icd ( cd S Φ d kl C ( cd ijk ¯ ¯ + k s d ¯ s  ijk j − + { Z W  S cd jk 1 ,  s Z ) SC ) kl ijk  R SC ) ab J W 1 icd − C icd i C i ,  R ( cd jk ijk + ( cd 1 icd i ) i ¯ s 6= k 6= ( cd ¯ s c C jk d S ¯ X s S ≡ S  icd i i ijk g cd ( cd s SC h C i ¯ 6= cd s 6=  ab h f R C J c d
i X − + δ  i RR R ik R s 6= jk jk S i k S   J ikl ijk i,d  +2 +1 +2 +1 i
S
C X W i  n n n n 6= 6= j ) C ς ς ς ς c kj 6= ijk i SC jk S 1 1 i,d g αβ i i,d X S ). Next, we note that collinear contributions i C ( W
ik 6= X C f 6= 6= c h kl c δ g − jk W −N −N i i,d we get ij X f + 2.53 1 W C 6= – 53 – S δ = = + c +2 +1 +2 +1 ij k ij ) kl n n n n 
S ς ς ς ς S ) S ijk +2 +1 +2 +1 C jk αβ i RR 1 1 ( n n n n ij αβ i S ς ς ς ς ( ij −
W SC W + 1 1
−N −N ij W ), as well as their phase-space integrals, are explicitly  jk ijk jk + ik i S = = C S C 6= −N −N
ijk W j j i X ijk − j jk i D.13 C = = S S 1 C i S S RR
i W  S is defined in eq. ( S − ik kl  ijk  s jk h S i i,k W i J 6= C i,j 6= ij X ), and renaming indices, we finally get l k k>j k S X SC j 6= i S X i 1 S k S , + 6= 1 6= i , 3.52 6= X X rad i, k i, j  ijk − rad X i, j Φ C − Φ × − d i d − S = ) are proportional to the combination Z h = Z s) 1 , , s) , D.13 12 rad ( 12 Φ ( K d K Z Further acting with a double-soft limit to eq. ( integrates to where the soft integral The singular limitscomputed in below. eq. For brevity, ( instated the otherwise. following Let we us set beginlimit. by We considering find the iteration of a soft limit and a double-soft By means of eq. ( of the matrix element, we obtain the expression JHEP12(2018)107 , D  ) (D.17) iab ( B 427 ] } Phys. Lett. ¯ k C32N5-6 , {  Phys. Rev. ab , ]. ]. R (1989) 573 i ]. 5 kl Phys. Lett. , C SPIRE QCD and beyond. SPIRE ]. k Nuovo Cim. IN IN , S SPIRE [ ][ arXiv:0903.1126 , in IN  ) ][ (2013) 199905] SPIRE iab ( kl IN [ W 111 kl C ). 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