A general algorithm to build mixed real and virtual antenna functions for higher-order calculations

The antenna-subtraction technique has demonstrated remarkable effectiveness in providing next-to-next-to-leading order in αs (NNLO) predictions for a wide range of processes relevant for the Large Hadron Collider. In a previous paper [1], we demonstrated how to build real-radiation antenna functions for any number of real emissions directly from a specified list of unresolved limits. Here, we extend this procedure to the mixed case of real and virtual radiation, for any number of real and virtual emissions. A novel feature of the algorithm is the requirement to match the antenna constructed with the correct unresolved limits to the other elements of the subtraction scheme. We discuss how this can be achieved and provide a full set of real-virtual NNLO antenna functions (together with their integration over the final-final unresolved phase space). We demonstrate that these antennae can be combined with the real-radiation antennae of ref. [1] to form a consistent NNLO subtraction scheme that cancels all explicit and implicit singularities at NNLO. We anticipate that the improved antenna functions should be more amenable to automation, thereby making the construction of subtraction terms for more complicated processes simpler at NNLO.


Introduction
The Large Hadron Collider (LHC) offers an unprecedented opportunity to scrutinise a wide range of observables involving Higgs bosons, electroweak bosons, top quarks, and hadronic jets with remarkable accuracy.Through precise experimental measurements, we can directly investigate the fundamental interactions of elementary particles at short distances, pushing the boundaries of our knowledge and providing valuable insights into the fundamental interactions that govern the universe.The exploration of LHC physics, particularly in the absence of new particle discoveries, holds immense significance.By scrutinising the LHC data with high precision, even the slightest deviations from the predictions of the Standard Model (SM) can have profound implications for our understanding of the natural world.Such small deviations in measurements have the potential to revolutionise our knowledge and guide us towards physics beyond the Standard Model.Hence, precision phenomenology emerges as a crucial component in the quest for new physics.
With the anticipated dataset from the High-Luminosity LHC, the statistical uncertainties on many observables will be negligible and percent-level accuracy is likely to be achieved experimentally.Achieving similar percent-level accuracy for theoretical predictions requires advancements in fixed-order calculations, parton distribution functions, parton showers, and the modeling of non-perturbative effects.Ongoing progress is being made in all these areas.In the realm of perturbative Quantum Chromodynamics (QCD), reaching the desired level of refinement typically involves extending fixed-order calculations to at least next-to-nextto-leading order (NNLO) in the strong-coupling expansion.
However, higher-order calculations demand special attention due to the intricate interplay between real and virtual corrections across different-multiplicity phase spaces [2,3].Implicit infrared divergences arise from unresolved real radiation, such as soft or collinear emissions, and are ultimately cancelled by explicit poles in the virtual matrix elements.This cancellation takes place through integration over the relevant unresolved phase space.Subtraction schemes are currently regarded as the most elegant solution to address these complexities.
At NNLO, the situation is less advanced.Despite recent progress, two-loop matrix elements represent significant challenges often requiring bespoke integral reduction relations and the evaluation of new master integrals.At the same time, the pattern of cancellation of infrared divergences across the different-multiplicity final states is much more complicated.Several subtraction schemes have been devised [16][17][18][19][20][21][22] and the implementation of these methods is currently done one process at a time.They do not straightforwardly scale to higher multiplicities.
In the antenna subtraction scheme, antenna functions are used to subtract specific sets of unresolved singularities, so that a typical subtraction term has the form where X ℓ n+2 represents an ℓ-loop, (n + 2)-particle antenna, i h 1 and i h 2 represent the hard radiators, and i 3 to i n+2 denote the n unresolved particles.As the hard radiators may either be in the initial or in the final state, final-final (FF), initial-final (IF), and initial-initial (II) configurations need to be considered in general.M is the reduced matrix element, with n fewer particles and where I h 1 and I h 2 represent the particles obtained through an appropriate mapping, with p µ i representing the four-momentum of particle i.At NLO antennae have n = 1 and ℓ = 0, at NNLO one needs antennae with n = 2, ℓ = 0 and with n = 1, ℓ = 1, while at N 3 LO, one needs antennae with n = 3, ℓ = 0, with n = 2, ℓ = 1 and with n = 1, ℓ = 2.
In the original formulation of the antenna scheme, the antennae were based on matrix elements describing radiation from processes with two coloured particles: γ * → q q, χ → gg and H → gg, covering the cases where the coloured particles are massless quarks and gluons.The corresponding X 0 4 , X 1 3 and X 0 3 antennae are therefore perfect subtraction terms for the NNLO contributions to processes with two coloured particles.It was straightforward to utilise these matrix-element-based antennae for processes with three coloured particles, such as e + e − → 3 jets, pp → V +jet, pp → H+jet, and for the leading colour contributions to four coloured particle processes like pp → 2 jets.Pushing to the next step, the full colour pp → 2 jets required significant additional work [78].Going beyond the current state of the art with the matrix-element-based antenna approach is a formidable task.This is because the complexity associated with the subtraction terms becomes increasingly challenging as the particle multiplicity grows.This complexity stems from two primary reasons.
Firstly, the double-real-radiation antenna functions obtained from matrix elements do not always indicate which particles act as the hard radiators.This is particularly the case for antennae involving gluons.To address this issue, sub-antenna functions are introduced.However, constructing these sub-antenna functions at NNLO is an arduous task and often involves introducing unphysical denominators that complicate the analytic integration of the subtraction term.Additionally, analytic integrals are usually known only for the complete antenna functions.As a result, the assembly of antenna-subtraction terms requires careful manipulation to ensure that the sub-antenna functions combine appropriately to form the full antenna functions before integration.
Secondly, NNLO antenna functions can exhibit spurious limits that need to be eliminated through explicit counter terms.However, these counter terms can introduce further spurious limits themselves.Consequently, this can initiate a complex chain of interdependent subtraction terms that do not necessarily reflect the actual singularity structure of the underlying process.
Both of these issues are obstacles to a full automation of the antenna-subtraction scheme at NNLO [79].In a recent paper [1], we addressed these issues.We introduced a general algorithm for building antenna functions directly from a specified set of desired infrared limits with a uniform template, in a way that simplifies the construction of subtraction terms in general, while still being straightforwardly analytically integrable.We then constructed a general algorithm to construct real-radiation antenna functions following strictly these design principles and applied it to the case of single-real and double-real radiation, required for NLO and NNLO calculations.The technique makes use of an iterative procedure to remove overlaps between different singular factors that are subsequently projected into the full phase space.As the technique produces only denominators that match physical propagators, all antenna functions could straightforwardly be integrated analytically, which is a cornerstone of the antenna-subtraction method.
In this paper, we extend the general algorithm of Ref. [1] to the construction of antennae with ℓ ̸ = 0. Unlike in the solely real-radiation case, the mixed real and virtual antenna functions contain both explicit and implicit singularities.To illustrate the algorithm, we construct the real-virtual antennae (n = 1, ℓ = 1) explicitly.The real-virtual antenna functions are built directly from the relevant one-loop limits, properly accounting for the overlap between different limits.The universal factorisation properties of multi-particle loop amplitudes, when one or more particles are unresolved, have been well studied in the literature [80][81][82][83] and serve as an input to the algorithm.
In addition to building a full set of new X 1 3 antennae with both hard radiators in the final state, we demonstrate that the new antenna functions (along with the X 0 3 and X 0 4 of Ref. [1]) form a complete NNLO subtraction scheme in which the subtraction terms cancel the explicit singularities in the one-and two-loop matrix elements, without leftover infrared singularities hiding in the matrix elements (either by undercounting or overcounting).This means that the new antenna functions have to satisfy particular constraints.First, the cancellation of poles at the real-virtual level (RV) means that the explicit poles in the X 1   3   antenna have to cancel against other RV subtraction terms.In the antenna scheme, these explicit poles are proportional to X 0 3 antennae.Therefore, the X 1 3 must have a particular pole structure multiplying an X 0 3 antenna function.At the double-virtual level (VV), the combinations of integrated antennae coming from the double-real (RR) and RV levels must match the explicit pole structure of the two-loop matrix elements.In the antenna scheme, this is encoded through a combination of the J (2) 2 and J (1) 2 operators in colour space [36].Provided that the pole structure from the relevant combination of J is unchanged, the subtraction terms will cancel the explicit poles in the two-loop matrix elements.
The current approach to automation of antenna subtraction [79] involves a reformulation of the colour-ordered antenna subtraction technique in colour space.This method, known as 'colourful antenna subtraction', offers a systematic way to construct antenna subtraction terms by working upwards from the most virtual layer, rather than starting from the maximally real layer and working down.By translating infrared poles of virtual corrections captured by J (2) 2 and J (1) 2 into real-radiation dipole insertions in colour space, the method efficiently constitutes subtraction terms for single-real radiation up to one-loop level and for double-real radiation at the tree level.One of the key advantages of this approach is the avoidance of directly handling the divergent behavior of real-emission corrections.This feature represents a significant simplification at NNLO.The double-real subtraction term can be obtained as the final step of a fully automatable procedure, eliminating the need to deal with the involved infrared structure of double-real radiation matrix elements.The completion of a consistent set of improved antenna functions for (double-)real and real-virtual radiation presented here will further reduce the complexity of the subtraction terms, because they avoid the need to subtract spurious limits, and therefore reduce the computational overhead associated with precision calculations.
The paper is structured as follows.We outline the design principles for constructing general X ℓ n+2 antenna functions in Section 2 as well as the principles for matching to the other elements of an antenna-subtraction scheme.We describe the general construction algorithm in Section 3 and give the specific details for the construction of final-final X 1 3 in Section 4. The full set of one-loop unresolved limits and target poles for the X 1  3 is given in Section 5. Using the previous sections, we illustrate the algorithm by explicitly constructing a full set of X 1 3 real-virtual antenna functions for hard radiators in the final state in Section 6.Finally, we define the J operators in this NNLO antennasubtraction scheme (out of the new {X 0 3 , X 0 4 , X 1 3 }) and compare their pole structure to the generic VV pole structures in Section 7.This demonstrates that the new subtraction terms will cancel the explicit poles in the two-loop matrix elements and form a complete NNLO subtraction scheme.We conclude and give an outlook on further work in Section 8.For the sake of completeness, we also enclose appendices listing the tree-level single-unresolved limits, details of the analytic integration over the final-state antenna phase space, and a list of integrated X 1  3 antenna that are based on the X 0 3 antenna of Ref. [24].

Design principles
Within the antenna-subtraction framework, subtraction terms are constructed using antenna functions that describe the unresolved partonic radiation (both soft and collinear) emitted from a pair of hard radiator partons.The construction of an antenna-subtraction term typically involves the following elements: • antennae composed of two hard radiators that accurately capture the infrared singularities arising from the emission of n unresolved partons; • an on-shell momentum mapping that ensures that the invariant mass of the antenna is preserved while producing the on-shell momenta that appear in the "reduced" matrix element; and • a colour factor associated with the specific process and antenna.
The latter two items on this list have been solved for general processes, while the first is subject of the current paper and our previous paper [1].
In the following, we will describe the design principles we impose upon a general idealised X ℓ n+2 antenna function, with at least one loop.As opposed to the ℓ = 0 case, antenna functions with additional virtual elements contain explicit poles in the dimensionalregularisation parameter, ϵ.We therefore impose two different sets of design principles: the generic design principles discussed in Section 2.1; and the antenna-scheme-dependent design principles discussed in Section 2.2.The former principles ensure that the antenna function has the correct infrared limits, but does not fix these unambiguously.This ambiguity is resolved by the latter principles which match the explicit singularity structure of the new antenna functions onto a specific antenna-subtraction scheme.

Generic design principles
The generic design principles outlined in Ref. [1] are sufficient to ensure that the antenna has the correct infrared limits.Specifically, we impose the following requirements: I. each antenna function has exactly two hard particles ("radiators") which cannot become unresolved; II. each antenna function captures all (multi-)soft limits of its unresolved particles; III. where appropriate (multi-)collinear and mixed soft and collinear limits are decomposed over "neighbouring" antennae; IV. antenna functions do not contain any spurious (unphysical) limits; V. antenna functions only contain singular factors corresponding to physical propagators; and VI.where appropriate, antenna functions obey physical symmetry relations (such as line reversal).
As mentioned earlier, the original NNLO antenna functions derived in [16,23,24] do not obey all of these requirements, as they typically violate (some of) these principles.This is particularly the case for quark-gluon or gluon-gluon antennae because the matrix elements they are derived from will inevitably have a divergent limit when one of the gluonic radiators becomes soft (thereby violating principle I).These principles will form the core of the algorithm for constructing X ℓ n+2 antennae with the desired infrared limits.

Antenna-scheme-dependent design principles
The generic principles are sufficient to produce compact analytic expressions that correctly capture the unresolved behaviour of ℓ-loop matrix elements in the (multi-)soft and (multi-) collinear limits.Unlike the ℓ = 0 case, these unresolved limits have explicit singularities, and therefore the X ℓ n+2 antennae constructed from them will also carry explicit ϵ-poles.However, it is straightforward to find terms that contain explicit singularities but which do not contribute in any of the unresolved regions.Such terms, can be added to X ℓ n+2 without violating any of the generic principles.However, doing so will clearly change the explicit pole structure.This means that the generic principles alone lead to an inherent ambiguity in defining the X ℓ n+2 antennae.If one wishes to design a full subtraction scheme, the real-virtual antenna must both have the correct unresolved limits and have an explicit pole structure of the correct form that cancels against other terms in the subtraction scheme.Therefore, we need to resolve the ambiguity in the explicit ϵ singularities by matching onto a set of target ϵ-pole structures that ensure that the subtraction terms in each multiplicity layer (a) correctly describe the unresolved limits of the matrix elements, and (b) precisely cancel the ϵ singularities of the matrix elements.
To match onto a particular antenna-subtraction scheme, we therefore introduce one further principle: VII.where appropriate, combinations of terms that are not singular in the relevant unresolved regions can be added to match onto "target poles", T (i h 1 , i 3 , ..., i n+2 , i h 2 ).
To illustrate this principle for the case of the X 1 3 , "target poles", T (i h , j, k h ), take the following schematic form within the NNLO antenna-subtraction scheme: In order to match onto such "target poles", we are free to add certain combinations of terms.
An example of such a combination of terms is, which is not divergent in the soft j, collinear ij and collinear jk limits.It therefore does not affect the behaviour of the antenna function in those limits.However, adding such a term clearly affects the explicit poles in the X 1 3 antenna as can be seen from the expansion in ϵ, This allows us to match the pole structure of the X 1 3 antenna to the other subtraction terms in a way that cancels the explicit poles at the RV level.
These seven principles are sufficient to devise an algorithm for constructing a general X ℓ n+2 antenna function and here we will apply it to the construction of X 1 3 antenna functions with final-final kinematics.We will build the X 1 3 antenna functions from the infrared limits and match them to the NNLO antenna-subtraction scheme.The new X 1  3 antenna functions form the final ingredients for improved final-final antenna-subtraction at NNLO (along with the results of Ref. [1]).To test for the consistency of these ingredients, one has to integrate the real-virtual antenna over the antenna phase space, and combine all the various integrated implicit singularities to cancel the explicit singularities of the two-loop matrix elements.This is detailed in full in Section 7.

The algorithm
In Ref. [1], we proposed a general algorithm to build (multiple-)real radiation antenna functions at tree-level.In this paper, we extend this algorithm to the construction of X ℓ n+2 antenna functions, where ℓ ̸ = 0.
Unlike the algorithm for real-radiation antenna functions, the algorithm for X ℓ n+2 antenna functions has two distinct stages: Stage 1 In this step we ensure that the antenna function has the correct infrared singular limits.This stage closely follows the algorithm for real-radiation antennae in Ref. [1].We systematically start from the most singular limit, and build the list of target functions, {L i }, from relevant (multi-)soft, (multi-)collinear, and soft-collinear limits.
As in Ref. [1], we define a down-(P ↓ i ) and up-projector (P ↑ i ) for each unresolved limit (L i ) to be included.A down-projector P ↓ i maps the invariants of the full phase space to the relevant subspace.An associated up-projector P ↑ i restores the full phase space by re-expressing all variables valid in the sub-space in terms of invariants valid in the full phase space.It is to be emphasised that down-projectors P ↓ i and up-projectors P ↑ i are typically not inverse to each other, as down-projectors destroy information about less-singular and finite pieces.
The down-projectors are necessary to identify the overlapping region between the antenna function developed so far and the target function associated with the unresolved limit under consideration.Conversely, up-projectors express the argument in terms of antenna invariants.Furthermore, through careful selection of the up-projectors, the antenna function can be exclusively represented using invariants corresponding to physical propagators.
The set of target functions provides a clear definition of the antenna function's behavior in all unresolved limits specific to the particular antenna being considered.In each unresolved limit, the antenna function must approach the corresponding target function to accurately capture the singular behavior exhibited by the squared matrix element.Additionally, the antenna function must remain finite in all limits not explicitly described by a target function.This crucial aspect guarantees the absence of spurious singularities (unlike antenna functions extracted directly from physical matrix elements).
As explained in Ref. [1], the algorithm, which ensures the above characteristics and meets the generic design principles, can be written as where X ℓ n+2;N is the output of Stage 1.

Stage 2
The output of Stage 1 guarantees that X ℓ n+2;N has the chosen unresolved limits {L i }.However, as discussed above this does not uniquely determine the mixed realvirtual antenna since one can construct a term (which we will denote by TPoles) which contains poles in ϵ but does not contribute in any of the unresolved limits.
One is therefore at liberty to define different antenna-subtraction schemes that differ by explicit ϵ-singular terms that do not affect the unresolved singular limits of the antenna.
We therefore add an antenna-scheme-dependent Stage 2 that ensures that the X ℓ n+2 antenna has the correct explicit poles to match onto the other types of subtraction terms in the desired antenna-subtraction scheme.We fix the scheme by specifying that the explicit ϵ-poles, match certain defined "target poles", These target poles must be selected such that the constructed X ℓ n+2 is more convenient for use in a wider N n+ℓ LO subtraction scheme.Different schemes would entail different choices for T .
As in Stage 1, we introduce certain projectors P ↓ T , P ↑ T (at the relevant perturbative order) to identify these additional ϵ-singular contributions which meet all the design principles.Schematically, we can write this final step of the algorithm as and we require For later convenience we define the contribution from Stage 2 to be, Taking into account both Stage 1 and Stage 2, the constructed mixed real-virtual antenna for a given set of infrared limits {L i } and matched to a scheme in which the required ϵ-poles are defined by T , will satisfy Table 1.Identification of X 1 3 antenna according to the particle type and colour-structures.These antennae only contain singular limits when particle b (or equivalently momentum j) is unresolved, in addition to explicit ϵ poles.Antennae are classified as quark-antiquark, quark-gluon and gluongluon according to the particle type of the parents (i.e. after the antenna mapping).

Construction of real-virtual antenna functions
The above design principles and algorithm have been set-out for the construction of a general X ℓ n+2 antenna function.Now we specialise to the case of constructing real-virtual X 1  3 antenna functions.Together with the new X 0 3 and X 0 4 of Ref. [1], the X 1 3 functions complete the re-formulation of all antenna functions necessary for NNLO calculations, which now meet the design principles.
We demonstrate the construction of real-virtual antenna functions X 1 3 (i h a , j b , k h c ), where the particle types are denoted by a, b, and c, which carry four-momenta i, j, and k, respectively.Particles a and c should be hard, and the antenna functions must have the correct limits when particle b is unresolved.Frequently, we drop explicit reference to the particle labels in favour of a specific choice of X according to Table 1.
For the specific case of X 1 3 (i h a , j b , k h c ) there are three such limits (meeting the generic design principles), corresponding to particle b becoming soft, particles a and b becoming collinear, or particles c and b becoming collinear, so that the list of target functions is, ab are well known and we organise them in our notation in Section 5.
In addition, in order to match onto a particular antenna-subtraction scheme, we require a target pole structure, T ≡ T (i h , j, k h ).We want to match the constructed X 1  3 to the full NNLO antenna-subtraction scheme, we therefore require the ϵ-poles to have a similar ϵpole structure to Ref. [16].This means a collection of ϵ-poles multiplying X 0 3 antennae.By removing the contribution to the poles from the renormalisation term, we write the full set of target poles, T (i h , j, k h ), for the unrenormalised X 1 3 as, T (i h g , j g , k h g ) = 0, (4.12) Here R ϵ is an overall factor defined as This factor ensures that the X 1 3 antennae derived here have the same overall normalisation as those in Ref. [16].We have also introduced the convenient notation to separate the loop-type structures from the unresolved-type structures: where ∼ reflects the fact that the LHS multiplies D 0,OLD 3 while the RHS multiplies D 0 3 .

Template antennae
For convenience, we define a general unrenormalised real-virtual antenna function in terms of the contributions produced by the various steps of the algorithm.At leading-colour, we have, while the corresponding sub-leading-colour expression is, (k h , j; i h ) (4.20) and the quark-loop contribution is, (k h , j; i h ) (4.21) since the one-loop soft factor is only non-zero at leading-colour.
and similarly for TPoles(i h , j, k h ) and TPoles(i h , j, k h ).

Stage 1
All X 1 3 (i h , j, k h ) antenna functions are defined over the full three-particle phase space, whereas each unresolved limit lives on a restricted part of phase space: the j soft limit, the ij collinear limit, and the jk collinear limit.
We define the soft down-projector by its action on integer powers of invariants as S ↓ j : and keep only the terms proportional to λ −2 .
For the corresponding up-projector S ↑ j we choose a trivial mapping which leaves all variables unchanged.The collinear down-projector acts on integer powers of invariants and is defined in analogy to Eq. (4.16) of Ref. [1], but keeps only terms of order λ −1 .The corresponding up-projector is the same as in Eq. (4.17) of [1], This up-projector ensures the presence of s ijk denominators, which are present in matrix elements corresponding to physical propagators and means that the same integration tools for one-loop matrix elements can be used in the integration of the constructed X 1 3 over its Lorentz-invariant antenna phase space.
The subtracted single-unresolved one-loop factors are built from unrenormalised colourordered limits and are given by Ssoft (1) b (i h , j, k h ) , (4.28) Scol (1) and analogously for the sub-leading colour and quark-flavour contributions.The subscripts a, b, c represent the particle types which carry momenta i, j, k respectively.The unrenormalised one-loop single-unresolved limits are listed in full in Section 5. We have used the feature that the only overlap between the ij-and jk-collinear limits occurs in Ssoft (1) so that which was also observed in Ref. [1].

Stage 2
We now turn to the construction of TPoles(i h , j, k h ), which does not contribute to any unresolved limit but does contain explicit ϵ poles.In the language of section 3, then schematically We observe that each of the target pole structures in Eqs.(4.2)-(4.15) is of the form, where Poles is combination of ϵ-poles and factors like s −ϵ ij .We therefore choose to achieve Stage 2 through two iterative steps (rather than one), adding a projector for each step: Step 1 We introduce a projector P ↑ X (and the trivial projector P ↓ X ) to ensure that the ϵ-poles are proportional to X 0 3 (i h , j, k h ); and Step 2 We introduce projectors P ↑ ϵ and P ↓ ϵ to adjust the pole structure multiplying X 0 3 to match T (i h , j, k h ).
Step 1 We define the projector P ↑ X such that, P ↑ X : The inverse projector P ↓ X is simply unity.We define TPoles X (i h , j, k h ) to be the contribution arising from the action of P ↑ X such that, TPoles X (i h , j, k h ) = (P ↑ X − 1)(Ssoft (1) (i h , j, k h ) + Scol (1) (i h , j; k h ) + Scol (1) Ssoft (1) (i h , j, k h ) does not contain splitting functions, so the action of P ↑ X on Ssoft is trivial, which guarantees, Furthermore, because of the structure of the one-loop splitting functions, we also have We will explain more clearly how this is achieved in the specific example of A 1 3 in Section 4.5.We also note that following the iterative structure of Eq. (3.1), we define the fourth step of the algorithm to be where X 1 3;3 (i h , j, k h ) is given in Eq. (4.32).
Step 2 The operators P ↓ ϵ and P ↑ ϵ are defined as follows.P ↓ ϵ is defined by Laurent expanding the argument in ϵ and discarding terms of O ϵ 0 and higher.
P ↑ ϵ is defined by extending the argument to an all-orders expression in ϵ, which agrees with the argument up to O ϵ 0 .This is not a unique action.For the case of the X 1 3 we choose, where possible, for P ↑ ϵ to result in linear combinations of {s −ϵ ik , s −ϵ ijk , (s ik + s jk ) −ϵ } (which are simply-integrable objects) multiplied by simple ϵ-poles and X 0 3 .Only two structures appear in the construction of the X 1 3 : where with S ik etc defined as in Eq. (4.17).We define TPoles ϵ (i h , j, k h ) to be the contribution arising from the action of P ↑ ϵ and P ↓ ϵ such that, +Scol (1) (i h , j; k h ) + Scol (1) (k h , j; i h ) + TPoles X (i h , j, k h ) .
These contributions typically contain a factor which suppresses all the unresolved limits in the X 0 3 to which it multiplies.Λ 1 suppresses any contributions to the soft-j limit or the collinear-ij or collinear-jk limits.Λ 2 suppresses contributions to the soft-j or collinear-jk limits so that We will explain more clearly how this works in detail in the specific example of A 1 3 in section 4.5.
In the iterative language of Eq. (3.1) the fifth and final step of the algorithm is thus where we define the complete constructed antenna function, It is convenient to combine the contributions from Eqs. (4.39) and (4.49), to obtain a single contribution (as in the antenna templates of Eqs.(4.19), (4.20) and (4.21)), and we define TPoles(i h , j, k h ) ≡ TPoles X (i h , j, k h ) + TPoles ϵ (i h , j, k h ). (4.53) It is to be emphasised again that TPoles(i h , j, k h ) does not contribute in any unresolved limits, but does carry explicit poles in ϵ.Indeed, using Eqs.(4.41), (4.42) and (4.50), it is straightforward to see that Finally, the algorithm of this paper ensures that ab (i h , j) , (4.56) and

A 1 3 construction in full detail
To make the construction explicit, we work through the construction of A 1  3 as an example before describing the full set of improved real-virtual antenna functions in Section 6.A 1 3 (i h q , j g , k h q ) is the leading-colour antenna function with quark and antiquark hard radiators, which encapsulates the one-loop limits when the gluon becomes unresolved.The relevant unresolved limits are S (1)  g (i h , j, k h ), P (1)  qg (i h q , j g ), P (1)  qg (k q, j g ), which are given in Section 5. Additionally, we choose a target for the ϵ-poles before renormalisation, which is consistent with the above limits but matches the ϵ-pole structures appearing in the antenna-subtraction scheme.The target pole structure for A 1 3 is given by We want to match the constructed X 1 3 to the full NNLO antenna-subtraction-scheme, we therefore require the ϵ-poles to have a similar ϵ-pole structure to Ref. [16].
We choose to simplify our notation by introducing the following structure where w = s jk /s ik .Note that in the w → 0 limit, G(w, ϵ) vanishes.Before renormalisation, A 1 3 is built iteratively in pieces in the following order: A 1 3 (i h q , j g , k h q ) = Ssoft (1) (i h q , j g , k h q ) + Scol (1) (i h q , j g ; k h q ) + Scol (1) (k h q , j g ; i h q ) (4.66) +TPoles(i h q , j g , k h q ) , with the TPoles contribution constructed in two steps as in Section 4.3 according to Eqs. (4.39) and (4.49), TPoles(i h q , j g , k h q ) = TPoles X (i h q , j g , k h q ) + TPoles ϵ (i h q , j g , k h q ).(4.67) The first contribution is simply the one-loop soft factor, where S (0) g is the tree-level eikonal factor given in Appendix A. The second piece is given by the overlap of the one-loop splittting function P (1) qg (i h , j) and the one-loop soft factor Ssoft (1) in the ij collinear limit, projected-up into the full phase-space: Here we use the short-hand notation to indicate an n-loop splitting function up-projected into the full phase space of the antenna and the tree-level splitting functions, P ab are given in Appendix A. The third contribution is given by Scol (1) (k h q , j g ; i h q ) = C ↑ kj P (1)  qg (k h q , j g ) − C ↓ kj Ssoft (1) (i h q , j g , k h q ) + Scol (1) ≡ P (0) qg (i h q , j g ; it is straightforward to see that the terms proportional to S ij S jk /S ik in Eqs.(4.68), (4.69) and (4.71), combine to give a term which factorises onto A 0 3 (i h q , j g , k h q ) such that Ssoft (1) (i h q , j g , k h q ) + Scol (1) (i h q , j g ; k h q ) + Scol (1) (k h q , j g ; i This combination completes Stage 1 of the algorithm and is to some extent a complete construction of A 1 3 , in the sense that it encapsulates the fundamental one-loop unresolved limits we require of it.
Stage 2 of the algorithm preserves the unresolved limits but includes explicit poles that do not contribute in these limits.The next piece TPoles X ensures that the explicit pole structure of the A 1 3 factors onto A 0 3 : TPoles X (i h q , j g , k h q ) = (P ↑ X − 1) Ssoft (1) (i h q , j g , k h q ) + Scol (1) (i h q , j g ; k h q ) + Scol (1)

.74)
This term vanishes in the unresolved region for the following reason.The first term in the final line appears to have a singularity in the jk collinear limit due to the 1/s jk factor.However, in this limit the hypergeometric function G(s jk /s ijk , ϵ) approaches zero and this behaviour therefore suppresses the singularity due to the 1/s jk factor.A similar argument holds for the second term.As such, neither term in Eq. (4.74) contributes to any unresolved limit, although they evidently do contribute explicit ϵ poles.In summary, The running total for A 1 3 is given by Ssoft (1) (i h q , j g , k h q ) + Scol (1) (i h q , j g ; k h q ) + Scol (1) (k h q , j g ; i h q ) + TPoles X (i h q , j g , k h . Effectively, the tree-level splitting functions in Eq. (4.73) have been promoted to full A 0 3 antenna functions.
The next contribution, TPoles ϵ , is also part of antenna-scheme matching, for which we have the target pole structure proportional to A 0 3 , given in Eq. (4.64).The resulting expression is given by TPoles +Scol (1) (i h q , j g ; k h q ) + Scol (1) (k h q , j g ; i h q ) + TPoles X (i h q , j g , k h q ) , As discussed earlier, the logarithmic structure of Λ 1 suppresses all the unresolved limits present in the A 0 3 antenna at every order in ϵ.This structure also carries a 1/ϵ 2 factor, so TPoles ϵ (i h q , j g , k h q ) contains explicit ϵ poles (which are important for antenna-scheme matching) but does not contribute in the unresolved limits.In summary, Finally, including the renormalisation term and combining terms together we find a compact expression for A 1 3 given by We see that qg (i h , j) , (4.79) and 5 One-loop single unresolved limits The universal soft and collinear factorisation properties of multiparticle real-virtual amplitudes have been well studied in the literature [80][81][82][83].In this section, we list the unrenormalised, colour-ordered unresolved factors at one loop in conventional dimensional regularisation (CDR), which are consistent with the formulations in [16,[82][83][84][85][86][87].The overall factor R ϵ ensures that the antennae constructed here have the same normalisation as those in Ref. [24].The full-colour (unrenormalised) one-loop soft factor is given by where S (1)  g (i h , j, k h ) = 0 , (5.3) S (1)  g (i h , j, k h ) = 0 , (5.4) and formally we define any soft factor, where particle b with momentum j is not a gluon, as zero.The tree-level single-unresolved limits are given in Appendix A.
In general, the full-colour (unrenormalised) one-loop splitting function is decomposed by ab (i, j) + N F P ab (i, j) . (5.5) As at leading-order (see Appendix A), we organise the splitting functions according to which particle is a hard-radiator.This means that P ab (i h , j) is not singular in the limit where the hard radiator a becomes soft and is directly related to the usual spin-averaged one-loop splitting functions given in terms of the momentum fraction carried by particle j (x j ), defined with reference to the third particle in the antenna.

Real-virtual antenna functions
In this section, we give compact expressions for the full set of real-virtual antennae.In deriving these antennae, we have made use of MAPLE, hypexp [88,89] and FORM [90,91].

Quark-antiquark antennae
As shown in Table 1, there are three one-loop three-parton antennae with quark-antiquark parents that describe the emission of a gluon, organised by colour structure: A 1 3 , A This expansion differs from A 1, OLD 3 in Eq. (5.18) of Ref. [16], starting from the rational part at O (1/ϵ).In a similar way to the constructed A 0 4 in Ref. [1], this is simply because the A 0 3 given in Ref. [1] differs at O (ϵ) from A 0, OLD 3 of Ref. [16].The choice of A 0 3 impacts the ϵ poles of A 1  3 at both the unintegrated and integrated levels, because A 0 3 factorises onto explicit 1/ϵ 2 poles in Eq. (6.4).If instead the original A 0, OLD 3 , of Ref. [16], is used in Eq. (6.4), the integrated antenna in Eq. (C.1) contains exactly the same poles as A 1, OLD 3 in Eq. (5.18) of Ref. [16] and differs only at O ϵ 0 .
Similarly, for the sub-leading-colour q q antenna, and after integration we find the expression, This expansion only differs from A 1, OLD 3 in Eq. (5.19) of Ref. [16] at O ϵ 0 .In this case, the choice of A 0 3 does not impact the ϵ poles of A 1 3 because they are at most 1/ϵ and the A 0 3 given in Ref. [1] differs only at O (ϵ) from A 0, OLD 3 of Ref. [16].For the quark-loop q q antenna, there are no unrenormalised unresolved limits and so the antenna is simply a renormalisation term: The integrated version is given by which only differs from A 1, OLD 1 in Eq. (5.20) of Ref. [16] at O ϵ 0 .In this case, the choice of A 0 3 does not impact the ϵ poles of A 1 3 because they are at most 1/ϵ and the A 0 3 given in Ref. [1] differs only at O (ϵ) from A 0, OLD 3 of Ref. [16].

Quark-gluon antennae
As shown in Table 1, there are six one-loop three-parton antennae with quark-gluon parents organised by colour structure: , and E 1 3 .The antenna functions constructed here are directly related to the antenna functions given in Ref. [16] by Note that D 1, OLD 3 was extracted from an effective Lagrangian describing heavy neutralino decay into a gluino-gluon pair, where the gluino plays the role of the quark [24].Firstly, D 1, OLD 3 contains unresolved configurations where either of the gluons can be soft, so this is decomposed here such that only one gluon can be soft.Secondly, the extracted antennae D 1, OLD 3 (and D 0, OLD 4 ) contain both leading-colour and sub-leading colour limits and they receive special treatment in the antenna scheme.In Ref. [1], we effectively split D 0, OLD 4 into a combination of D 0 4 and D 0 4 antennae and we perform a similar decomposition here of D 1, OLD 3 into D 1  3 and D 1 3 .Due to the absence of a sub-leading colour D 1, OLD 3 antenna, we only have target poles, T (i h q , j g , k h g ), for the combination of D 1 3 (i h q , j g , k h g ) + D 1 3 (i h q , j g , k h g ).We choose to place the resulting TPoles ϵ term in the formula for D 1 3 .To recap, the combination of D 1 3 (i h q , j g , k h g ) + D 1 3 (i h q , j g , k h g ) have been used to match ϵ poles in the existing antenna-subtraction-scheme, while D 1 3 (i h q , j g , k h g ) contains the leading-colour limits when j is unresolved and D 1 3 (i h q , j g , k h g ) contains the sub-leading-colour limits when j is unresolved.This means the two antennae D 1  3 , D 1 3 could in principle be used independently in subtraction terms to cancel relevant one-loop unresolved limits, but the ϵ-pole cancellation may require specific attention.
The D 1 3 formula is given by ) and after integration we find the expression The D 1 3 formula is given by and after integration we find the expression The combination of 2(D because the D 0 3 given in Ref. [1] differs at O ϵ 0 from d 0, OLD 3 of Ref. [16].The choice of D 0 3 impacts the ϵ poles of D 1 3 and D 1 3 at both the unintegrated and integrated levels because D 0 3 factorises onto explicit 1/ϵ 2 poles in Eq. (6.15) and Eq.(6.17).
The quark-loop qg antenna function is given by and after integration we find the expression This expansion differs from D 1, OLD 3 /2 in Eq. (6.23) of Ref. [16], starting from the rational part at O (1/ϵ).In the D 1  3 formula, the poles are at most 1/ϵ and they only appear in the renormalisation term.The finite difference between D 0 3 and d 0, OLD 3 from Ref. [16] therefore only impacts the O (1/ϵ) poles.
The E 1 3 -type antennae contain contributions to only one limit -the jk collinear limit, when the QQ pair become collinear and as such they are simpler expressions than the others.The first antenna is given by where the Λ 2 /ϵ 2 term suppresses the only limit in the E 0 3 to which it factorises (the jk collinear limit) and thus only affects the ϵ pole structure of E 1 3 .After integration we find the expression which differs from E 1, OLD 3 in Eq. (6.34) of Ref. [16], starting from the rational part at O (1/ϵ).In the E 1  3 formula, the poles are at most 1/ϵ.The finite difference between E 0 3 and E 0, OLD 3 from Ref. [16] therefore only impacts the O (1/ϵ) poles.The sub-leading-colour antenna is given by and after integration we find the expression which differs from E 1, OLD 3 in Eq. (6.35) of Ref. [16], starting from the rational part at O 1/ϵ 2 .In the E 1  3 formula, the poles are at most 1/ϵ 2 .The finite difference between E 0 3 and E 0, OLD 3 from Ref. [16] therefore impacts the O 1/ϵ 2 poles.
The quark-loop antenna is given by and after integration we find the expression which differs from E 1, OLD 3 in Eq. (6.36) of Ref. [16], starting from the rational part at O (1/ϵ).In the E 1  3 formula, the poles are at most 1/ϵ.The finite difference between E 0 3 and E 0, OLD 3 from Ref. [16] therefore impacts the O (1/ϵ) poles of E 1 3 , although these are the deepest poles in this case.This is because of cancellations at O 1/ϵ 2 between the integrals of the first and second terms in Eq. (6.25).

Gluon-gluon antennae
As shown in Table 1, there are five one-loop three-parton antennae with gluon-gluon parents organised by colour structure: 3 , and G 1 3 .The antenna functions constructed here are directly related to the antenna functions given in Ref. [16] by ) Note that F 1, OLD 3 was extracted from an effective Lagrangian describing Higgs boson decay into gluons [23].This means that F 1, OLD 3 contains unresolved configurations where any one of the three gluons can be soft, so this is decomposed here such that only one gluon can be soft.The same discussion can be applied to F 1  3 .
The resulting formula for the three-gluon one-loop antenna function at leading-colour is given by and after integration we find the expression This expansion differs from F 1, OLD 3 /3 in Eq. (7.22) of Ref. [16], starting from the rational part at O 1/ϵ 2 .In the F 1  3 formula, the poles are at most 1/ϵ 2 .The finite difference between F 0 3 and f 0, OLD 3 from Ref. [16] therefore impacts the O 1/ϵ 2 poles.
The quark-loop antenna function is given by and after integration we find the expression This expansion differs from F 1, OLD 3 /3 in Eq. (7.23) of Ref. [16], starting from the rational part at O (1/ϵ).In the F 1 3 formula, the poles are at most 1/ϵ and they only appear in the renormalisation term.The finite difference between F 0 3 and f 0, OLD The formula for the one-loop gluon-splitting gg antenna function at leading-colour is given by and after integration we find the expression Firstly, given that E 0 3 = G 0 3 (from Ref. [1]) and that E 1 3 and G 1 3 encapsulate the same limits, these formulae (unintegrated and integrated) are identical for the E 1 3 -and G 1 3 -type antennae: Therefore the discussion for the G 1 3 -type antennae is the same as below Eq. (6.22), Eq. (6.24), and Eq.(6.26), respectively.When the X 0,OLD 3 from Ref. [16] are used, the G 1 3 -and E 1 3type antennae have a different pole structure but the same collinear limits.
The sub-leading-colour antenna function is given by and after integration we find the expression See the discussion for Eq.(6.24), which also applies to Eq. (6.42).The quark-loop antenna function is given by and after integration we find the expression See the discussion for Eq.(6.26), which also applies to Eq. (6.44).

Antenna-subtraction scheme consistency checks
In the antenna subtraction scheme, the virtual (NLO) and double-virtual (NNLO) subtraction terms can be written in terms of integrated dipoles denoted by J (1) 2 and J (2) 2 respectively [36].These integrated dipoles are formed by systematically combining integrated antenna-function contributions from the real and real-virtual layers (together with appropriate mass factorisation terms).The NNLO integrated dipole J 2 naturally emerges from the groups of integrated antenna functions (and mass factorisation kernels) and, together with combinations of J (1) 2 , reproduces and properly subtracts the explicit poles of the double-virtual contribution to the NNLO cross section.The integrated dipoles are therefore intimately related to Catani's IR singularity operators [92] which describe the singularities of virtual matrix elements.It is a non-trivial check of an antenna scheme constructed directly from unresolved limits that the integrated dipoles cancel the explicit poles of the double-virtual contribution.In this section, we write down expressions for J 2 ) and show that they produce the correct pole structure.
However, a residual dependence on the choice of single-real antennae is left in the construction of double-real antenna functions and the real-virtual antenna functions constructed in this paper.The deepest 1/ϵ 4 and 1/ϵ 3 poles correspond to the universal unresolved behaviour and are identical, but the 1/ϵ 2 and 1/ϵ poles are potentially different.This is understood as finite differences in the single-real antennae and should pose no issues in application to the antenna-subtraction scheme, when used with a consistent set of X 0 3 , X 0 4 , and X 1 3 antenna functions.Indeed, this is the case, and we find that Eqs.(7.13)-(7.24)satisfy Eq. (7.5), thereby demonstrating the consistency of the NNLO antenna subtraction scheme based on the antennae derived directly from the desired singular limits presented here and in Ref. [1].unresolved limits are known analytically.The definition of an antenna-subtraction scheme for N 3 LO calculations will require three new types of antenna functions, namely triplereal (RRR), double-real-virtual (RRV), and real-double-virtual (RVV).While the removal of overlapping singularities at N 3 LO may be tedious in practice, we anticipate that our algorithm provides a suitable baseline for these endeavours.
C Integrals of X 1 3 antennae derived using the X 0 3 of Ref. [16] In this appendix, we list the integrals over the antenna phase space of the renormalised X 1 3 antenna constructed using the X 0 3 antennae of Ref. [16].
A For the A-type, E-type and G-type antennae, we find complete agreement with the pole structure of the analogous integrated antennae given in Ref. [16].For the D-type and F -type antennae, we have utilised the X 0 3 sub-antenna given in Eqs.(6.13) and (7.13) of Ref. [16] respectively and therefore the pole structures of the combinations 2 D  respectively, given by Eqs.(6.22), (6.23), (7.22) and (7.23) of Ref. [16] respectively to O ϵ 0 .

( 4 . 1 )
The precise definitions of the one-loop soft factor S (1) b and the one-loop splitting functions P(1)

3
The meanings of the individual terms in Eqs.(4.19)-(4.21)will be made clear in the following subsections, however, we note that in each equation, the first line is produced by Stage 1 of the algorithm, and the second line is added in Stage 2. Therefore, we expect that