One-loop triple collinear splitting amplitudes in QCD

We study the factorisation properties of one-loop scattering amplitudes in the triple collinear limit and extract the universal splitting amplitudes for processes initiated by a gluon. The splitting amplitudes are derived from the analytic Higgs plus four partons amplitudes. We present compact results for primitive helicity splitting amplitudes making use of super-symmetric decompositions. The universality of the collinear factorisation is checked numerically against the full colour six parton squared matrix elements.


Introduction
A full understanding of the infrared structure of QCD matrix elements is an unavoidable step towards making precise predictions of Standard Model backgrounds at hadron colliders. In order to make finite predictions for cross-sections we must cancel infrared singularities between unresolved real radiation corrections and singularities in the virtual (loop) corrections. The study of infrared properties of perturbative gauge theories have a broader scope beyond this application since the universal behaviour provides a strong constraint on the structure of scattering amplitudes.
The soft and collinear infrared limits at next-to-leading order (NLO) have been understood long ago and general algorithms (e.g. Catani-Seymour [1] or Frixone-Kunszt-Signer [2]) for the computation of infrared finite cross-sections form the core of the current generation of precision tools used to make theoretical predictions for the LHC experiments.
In the last ten years or so a lot of effort has been put into generalising these techniques to next-to-next-leading order (NNLO) and a variety of different techniques now exist with the ability to make finite predictions for important LHC observables (e.g. references [3][4][5][6][7][8]). All of these methods rely on knowledge of the underlying factorisation properties of QCD amplitudes in the double unresolved limits at tree-level [9] and single unresolved limits at one-loop [10][11][12][13][14].
The first step at next-to-next-to-next-to-leading order (N 3 LO) has been taken recently through the complete calculation of fully inclusive Higgs production at hadron colliders up to O(α 5 s ) in the large top quark mass limit [15]. This calculation has been performed in a number of different stages building expansions around the soft limit [16][17][18] and using the reverse unitarity method to obtain each component of the triple-virtual [19][20][21], squared real-virtual [22,23], double-virtual-real [24][25][26][27], double-real-virtual [28,29] and triple-real radiation [30] as an expansion in the dimensional regularisation parameter. The poles of these separate contributions cancel analytically when summed together and combined with the counter-terms for UV poles [31][32][33][34] and initial state infrared singularities [35][36][37][38][39].
Further steps are required to extend these techniques to fully differential observables in an analogous way to the NLO and NNLO cases. Many of the infrared regions that must be accounted for in such a procedure are now fully understood. The missing ingredients that remain are the one-loop triple collinear splitting functions involving gluons. Though the factorisation of the squared matrix elements are sufficient for the construction of infrared finite cross sections, factorisation at the amplitude level [40] can yield much more compact expressions leading to a more efficient construction of the factorised squared matrix element, especially when considering spin-correlations. Figure 1 shows the real and virtual contributions to a cross section up to N 3 LO and the primary singular limits which are either multiple soft, S i 1 ...im , or multiple collinear, C i 1 ...im . The factorisation amplitudes have been computed in all cases [9,24,26,[41][42][43][44][45][46] except for the triple-collinear and double-soft limits of the double-real-virtual. The triple collinear limit at one-loop has been considered at the squared amplitude level for q → qQQ [47] and for the mixed QCD+QED cases of q → qγγ, q → qgγ, g →qqγ, γ →qqγ and γ →qqg [48,49].
The structure of the article is as follows. We first introduce the notation for the amplitudes and the squared amplitudes together with their respective colour decompositions and collinear limits. In section 3 we describe a parametrisation of the multi-collinear limit using spinor-helicity variables which we will use to compute the splitting amplitudes. We then present the g → ggg and g →qqg splitting amplitudes and describe the symmetries and super-symmetric decompositions used to obtain a compact representation. We then check the universality of the new splitting amplitudes by taking a numerical limit of the gg → gggg and gg →qqgg in NJet before reaching our conclusions. Figure 1: The contributions to perturbative cross sections up to N 3 LO. This consists of virtual (V) corrections up to three loops and real radiation (R) corrections with up to 3 additional unresolved legs. In the real radiation contributions the primary infrared limits of soft (S) and collinear (C) should be removed from the matrix elements and re-combined with the virtual corrections to obtain an infrared finite result.

Notation
A general QCD amplitude can be decomposed into a basis of SU (N c ) colour factors and ordered partial amplitudes which depend only on the momenta and helicities of the external legs. For an n-point L-loop amplitude this can be represented as, where a i , λ i and p i are colour indices (adjoint or fundamental), helicity and momenta of the i th leg. Unless explicitly indicated otherwise, we understand that the index i runs from 1 to n, e.g.
For cross-section computations we are required to square these amplitudes and sum over the colour indices. This sum can be represented as, Figure 2: Factorisation of tree and one-loop amplitudes in the multi-collinear limit.
where the matrix C (L,L ) n is a function of N c defined by p which further decompose colour and flavour structure due to the internal loops, where X runs over the independent primitive topologies at L loops and p runs over permutations of the n external legs. Eq. (2.3) can thus be equivalently written as n is a vector of primitive amplitudes A In the limit where m of the external legs become simultaneously collinear, the amplitudes factorise into a product of lower multiplicity amplitudes and splitting amplitudes which contain all the infrared divergences: n and Sp (L) n can either be primitive or partial n-point amplitudes and splitting amplitudes respectively, while and P ≡ p 1 + · · · + p m . A schematic representation of this factorisation is shown in Figure 2. The sum of internal helicity states λ P leads to spin correlations in the factorized squared amplitude M (L,L ) , where we can define in terms of partial amplitudes or equivalently in terms of primitive amplitudes. In the colour matrix C and similar for C (L,L ) Sp,n . For brevity, the results presented in this paper will often omit the subscript indicating the number of partons involved in an amplitude, since this can be deduced by its arguments, i.e.

A spinor parametrisation of the multi-collinear limit
We define the multiple collinear limit using a parametrisation of the full kinematics in term of a parameter δ, such that the collinear limit in Eq. (2.8) is identified as the leading term as δ → 0, i.e.
The parametrisation is defined by, where z i = (p i · η)/(P · η) are the momentum fractions of the unresolved partons, η is an arbitrary light-like momentum andP is the massless projection of P = m i=1 p i , The vectors k µ T,i are orthogonal to P ,P and η Momentum conservation implies that: The function K µ i is a generic map that keeps the factorized momenta m + 1, . . . , n on-shell as well as absorbing the recoil P 2 /(2P · η)η µ , and it satisfies K µ i → p µ i as δ → 0. The exact form is not important for our purpose of explicitly taking the limit and various mappings have been considered in the literature (for example in the Catani-Seymour subtraction [1] or Kosower's antenna [57]). When implementing the collinear phase-space numerically we employed the Catani-Seymour map as described in Appendix A.
Since we are working at the amplitude level, we would like to have a parametrisation of the limit valid for the spinors of p i as well. This can be achieved using an appropriate choice of the transverse vectors k T,i , In the above we use the notation where the spinor variables z i and [z i ] differ by a phase from the usual parametrisation which uses √ z i . It is worth to notice that both ω i and [ω i ] are O(δ) in the collinear limit. The spinors parametrisation then reads, We find that this is a convenient way to take the limit at the amplitude level since the spinor variables z i obey Schouten identities: ijk cyclic z i jk = 0, (3.12) as well as momentum conservation, For the triple collinear splitting amplitudes this means we have the kinematics of a fivepoint function event though the colour space is that of a four-point function.

Example: the tree-level MHV multi-collinear splitting amplitude
The result for the multi-collinear limit of the maximal-helicity-violating (MHV) amplitude has been known for a long time. More recently the general helicity cases were also examined through use of the MHV rules [45,46]. This case is incredibly straightforward and serves as a useful example of the general treatment introduced in the previous section.
We start with the Parke-Taylor MHV amplitude with particles 1 and r > m having negative helicities and all others positive helicity, where the product in the denominator is considered modulo n. The limit is simply taken by applying eq. (3.11) where we have used eq. (3.9) to perform the power counting. For i, j ∈ [1, m] this can be seen explicitly, One can clearly arrive at this final result without being so explicit about the parametrisation, yet it is convenient to have one in a generic implementation.

One-loop basis functions for pp → H + 2j in the triple collinear limit
The analytic H + 4 parton amplitudes have been computed using unitarity cuts and expressed in terms of the universal infrared poles plus finite logarithmic and di-logarithmic functions as well as rational terms. Taking the triple collinear limit of the infrared poles, rational terms and logarithms as above presents no difficulties. Dealing with the dilogarithmic parts requires some minor effort to ensure the arguments are in the appropriate region so the limit will converge. Polylogarithmic identities are well known and understood in huge detail (see Ref. [58] for a recent review) -way beyond the simple structures appearing here. Nevertheless we collect some potentially useful identities here to aid the reader, One function requiring a bit more thought is the three mass triangle which has square roots appearing in the arguments of the di-logarithms [59][60][61][62][63]: One other minor issue with the results available in the literature is that the NMHV expressions have been presented using Forde's method for triple cuts [64]. This method gives the coefficients as the sum over solutions to the on-shell equations. To aid our computation we performed this sum explicitly to write the coefficients in terms of the usual spinor products of the external momenta.

Colour structure and primitive amplitude decomposition
In the section we will suppress all helicity superscripts and the function arguments are taken to represent both momenta and helicity. The tree-level colour decomposition can be written as, where tr(a 1 , . . . , a n ) = T a 1 ji 1 T a 2 i 1 i 2 . . . T an i n−1 j in terms of the fundamental generators of SU (N c ) andf abc = i √ 2f abc in terms of the adjoint structure constants. The relation between the two representations can be shown to hold using the Kleiss-Kuijf relations [65] for the splitting amplitudes, The one-loop colour decomposition is 1 , where the partial amplitudes are composed of primitive amplitudes as follows: The primitive amplitudes for the gluon and fermion loops obey line-reversal symmetry, and so in all we have three independent gluon loop primitive amplitudes, three fermion loop primitive amplitudes and two tree-level primitive amplitudes. The colour summed Born and virtual corrections can then be written according to (2.13) using: We also choose to present the results using the super-symmetric decomposition: since this yields particularly compact expressions. We also include the scheme dependence for both the FDH (δ R = 0) and CDR (δ R = 1) schemes.

Results
We define the following phase-free quantities, Since there can be no repeated index in either α ijk and β ijk each can be uniquely specified by the two first labels.
The integral functions are defined using the following basis, We express the infrared poles and associated logarithms as described by Catani's formula [47], All results in this section are presented unrenormalized.

Colour structure and primitive amplitude decomposition
The colour structure of the tree-level splitting amplitudes is where T (a 1 , . . . , a n ) i = T a 1 k 1 T a 1 k 1 k 2 . . . T an k n−1 i . Note that charge conjugation symmetry allows us to write Sp (0) (−P ; 2 q , 1q, 3) = Sp (0) (−P ; 2q, 1 q , 3). At one-loop we have three colour structures, The quark primitive splitting amplitudes also have a useful super-symmetric decomposition [11]. In this case we can write the complicated "left-moving" amplitudes in terms of simpler ones built using the N = 4 super-multiplet,

Results
As before all results in this section are presented unrenormalized. The non-vanishing independent tree-level splitting amplitudes g →qqg are 12 23 (5.15) and the others are obtained by conjugation using the relation .

(5.16)
The sub-leading colour tree-level splitting amplitudes g →qgq are not independent because they can be expressed in terms of (5.15) using the KK relation (4.4) re-written with the quark labels, A sample of two representative tree-level splitting amplitudes g →qgq is The non-zero independent one-loop splitting amplitudes g →qqg are The expressions for the non-zero independent one loop splitting amplitudes g →qgq are we could evaluate as close to the precise limit as possible, we implemented the checks in octuple precision using the qd and OneLoop [70] packages. We check the validity of Eq. (2.9) by computing the ratio between the two sides of the equation summed over the external helicities where M n,s and P n,s in the denominator are defined from M n and P n by summing over the external helicities: Eq. (2.9) obviously implies It is worth observing that the finite one-loop all-plus and all-minus four-gluon helicity amplitudes, while giving no contribution to the NLO squared matrix element, they give instead a finite contribution to r collinear123 because of spin correlations. In Fig. 5 we plot r collinear123 − 1 as a function of the invariant mass s 123 of the three collinear partons. More in detail we verify the validity of Eq. (7.2) in double, doubledouble and double-quadruple precision for both gluon (on the left) and quark (on the right) splitting functions. As one can see, going to higher precision allowed us to make stronger checks on phase-space space points which are closer to the limit, where the numerical evaluation is highly unstable at lower floating-point precision.
Similarly, we also numerically verified Eq. (2.8) for each primitive amplitude and all the helicity configurations, although all of these already contribute to the check described above.
As well as the numerical checks we have also verified that all splitting functions factorise correctly in the iterated collinear limit, where the scale s 12 s 123 and P 123 =P 12 + p 3 . All di-logarithms drop out in this limit though some care should be taken to ensure the hierarchy of scales is imposed correctly.

Conclusions
In this article we have computed the one-loop triple collinear splitting amplitudes in QCD initiated by a gluon. These functions are one of the last remaining ingredients to complete the classification of universal infrared limits relevant at N 3 LO. Some effort has been taken to ensure the splitting amplitudes have compact analytic forms. We made use of the spinor-helicity formalism and super-symmetric decompositions and related the pure gluonic amplitudes to the ones containing a quark anti-quark pair. The primitive amplitude colour decomposition was also a useful tool to express full colour and helicity summed splitting functions which were all checked explicitly against the numerical matrix elements for 2 → 4 scattering in NJet. In the course of these checks we made use of the high precision numerical evaluation available with up to 64 digits via the qd package. This allowed us to probe deep into the collinear limit and verify that all parts of the computation behaved correctly. This was particularly important for the spin correlated and sub-leading colour corrections which are significantly suppressed.
There are still some missing ingredients needed for the constructions of a fully differential N 3 LO subtraction scheme. Firstly, the quark initiated channels are still unavailablethey are not directly accessible from the H + 2j amplitudes since they have been computed in the effective theory where the Higgs couples only to gluons. The necessary splitting amplitudes could be extracted from the vector boson plus four parton one-loop amplitudes [71].
Secondly when integrating the splitting functions over the unresolved phase space the expansion of the limit may be required to higher order in the dimensional regularisation parameter . This would require a new computation of the one-loop matrix elements valid in D = 4 − 2 dimensions which is quite feasible using modern unitarity methods. The appearance of the one-loop pentagon function in the full D-dimensional amplitude may complicate this part of the computation even if it is only required in the triple collinear limit.
We hope that the expressions presented here will be of use in future high precision QCD computations.

Acknowledgments
We are grateful to Franz Herzog, Tom Melia and Einan Gardi for useful discussions. The work of S.B. is supported an STFC Rutherford Fellowship ST/L004925/1.

A Generation of collinear phase space points
In this Appedix we illustrate a practical way to generate a set of on-shell n-particle phasespace points where the first m particles approach the collinear limit 1|| · · · ||m. The limit is approached by varying a single free parameter δ as δ → 0 and it is based on the parametrisation presented in Section 3. This has been used for the numerical checks we discussed in Section 7.
As a first step we generate an on-shell (n − m + 1)-particle phase space point defining the set of momenta {P , p m+1 (0), p m+2 (0), . . . , p n (0)} (A.1) where, as suggested by the notation, p i (0) for i ≥ m + 1 are the momenta of the noncollinear particles at δ = 0, whileP is the sum of the collinear momenta in the limit. We then define the exact collinear limit as the set of momenta where z i are randomly generated real numbers satisfying Eq. (3.6). In order to avoid regions with soft kinematics (which would introduce other kinds of singularities) one can generate a set of random numbers between, for example, 1 and 3 and divide them by their sum. In order to define the orthogonal direction we must specify the reference vector η appearing in Eq. (3.2). A particular convenient choice is one of the non-collinear vectors, i.e. η µ = p µ m+1 (0