Non-abelian factorisation for next-to-leading-power threshold logarithms

Soft and collinear radiation is responsible for large corrections to many hadronic cross sections, near thresholds for the production of heavy final states. There is much interest in extending our understanding of this radiation to next-to-leading power (NLP) in the threshold expansion. In this paper, we generalise a previously proposed all-order NLP factorisation formula to include non-abelian corrections. We define a non-abelian radiative jet function, organising collinear enhancements at NLP, and compute it for quark jets at one loop. We discuss in detail the issue of double counting between soft and collinear regions. Finally, we verify our prescription by reproducing all NLP logarithms in Drell-Yan production up to NNLO, including those associated with double real emission. Our results constitute an important step in the development of a fully general resummation formalism for NLP threshold effects.


Introduction
The properties of QCD radiation near partonic thresholds for the production of heavy final states have a significant impact on a wide range of phenomenologically relevant collider observables. Typically, if ξ is a dimensionless kinematic variable vanishing at the threshold, differential QCD cross-sections will contain terms of the form nm log m ξ ξ + + c (δ) nm δ(ξ) + c (0) nm log m ξ + . . . , (1.1) where the ellipsis denotes terms suppressed by further powers of ξ. The first set of terms, at leading power in ξ, originates from the singularities associated with soft and collinear gluon emission. These singularities are universal and factorising, which leads to the possibility of resumming the resulting logarithms to all orders in perturbation theory. The formalism to perform this resummation is well-known, and it has been extensively applied to a plethora of collider observables (see, for example, [1][2][3][4][5][6][7][8][9]). The second set of terms in Eq. (1.1), which are localised at threshold, originates mostly from singular virtual corrections to the production amplitude. These terms can also be organised to all orders for processes which are electroweak at tree level [10][11][12], albeit with reduced predictive power. The vast amount and increased precision of LHC data, together with the lack of any striking signature for new physics, make the third set of terms in Eq. (1.1), at next-to-leading power in the threshold parameter ξ, potentially relevant for precision Standard Model studies. Indeed, quite a large body of work has already been devoted to this problem. The fact that at least some NLP contributions can be understood to all orders is well known, as a consequence of the LBKD theorem [13][14][15]. Further evidence for a non-trivial relation between LP and NLP logarithms came from the analysis of DGLAP splitting functions in Ref. [16]. Since then, and following early studies in [17,18], several groups have attempted to construct a systematic formalism for understanding NLP logarithms, using a variety of methods, ranging from path integral techniques [19], to diagrammatic approaches [20], physical evolution kernels [21][22][23][24][25][26], effective field theories [27,28], and other techniques [29][30][31]. Interestingly, the study of next-to-soft contributions to scattering amplitudes in both gauge theory and gravity from a more formal point of view, based on asymptotic symmetries of the S matrix, has also received a great deal of attention (see for example [32][33][34][35][36][37][38]).
Recently, building on the results of [15], in Ref. [39] we proposed a factorisation formula for the Drell-Yan scattering amplitude, valid at the accuracy needed to generate NLP logarithms in the cross section. This formula generalises the factorisation of soft and collinear divergences by including NLP effects, and contains the same universal functions as the leading-power factorisation, together with a new universal radiative jet function, responsible for next-to-soft emission from a collinearly enhanced configuration. Ref. [39] evaluated this quantity up to one-loop order for an external quark, and, using as a guideline the calculation performed with the method of regions [40][41][42] in Ref. [43], succeeded in reproducing a set of NLP terms in the Drell-Yan cross section at NNLO, originally computed in [44,45].
The factorisation formula proposed in Refs. [15,39] was, however, appropriate for an abelian theory, and could only reproduce abelian-like QCD contributions, proportional to the color factor C n F at O(α n s ). The aim of this paper is to provide a fully non-abelian NLP factorisation formula, a generalisation from previous results which is non-trivial for a number of reasons.
First, it is necessary to include the emission of colour-correlated gluons from outside the hard interaction. These diagrams were called next-to-eikonal webs in Refs. [19,20], where they were shown to be described by generalised Wilson line operators, obeying exponentiation properties similar to their leading-power counterparts. Second, the definition of the radiative jet function must be generalised to cope with a non-abelian operator insertion for the additional gluon. Third, one must address double counting of soft and collinear regions at NLP level, a problem which occurs also at leading power, or in effective field theory approaches [27,28], but was easily circumvented in the abelian limit.
The structure of our paper is as follows. In Section 2, we introduce a new, non-abelian definition of the radiative jet function, in terms of a non-abelian conserved current, and we use it to derive a complete factorisation formula for the Drell-Yan amplitude, with the required accuracy to reproduce all NLP effects. In Section 3, we compute the new radiative jet function to one loop, which is sufficient to generate all NLP logarithms in the cross section at NNLO. In Section 4, we use these results to check that the known non-abelian terms in the Drell-Yan K-factor up to NNLO are indeed reproduced. In Section 5, we briefly describe how our methods can also reproduce double-real emission contributions at NNLO. In Section 6 we present our conclusion and outline future work towards an effective NLP resummation formalism.
2 The non-abelian NLP factorisation formula

Leading power factorisation
Anticipating the Drell-Yan application of Section 4, we consider a quark scattering amplitude involving two partons with momenta p 1 and p 2 , which we write as so that A has the external fermion wave functions removed. In massless QCD, A is affected by infrared and collinear divergences, which however factorise to all orders in the form [46] A(p 1 , In Eq. (2.2), J(p i , n i ) is a jet function, collecting collinear singularities associated with parton i: it depends on an auxiliary vector n i , as described below; this dependence cancels with the other factors in Eq. (2.2), so that the full scattering amplitude is independent of n i , as expected. For a quark, the jet function is given by 1 where the fermion field ψ(x) absorbs the external parton of momentum p, and Φ n (∞, 0) is a Wilson line in the direction of the auxiliary four vector n µ , guaranteeing gauge invariance, and defined according to The soft function S(β i ) collects soft divergences, and is a correlator of Wilson lines directed along the classical trajectories of the hard emitting particles: in fact, β i is a dimensionless vector proportional to the four-velocity of parton i according to p i = Qβ i , with Q a hard scale. We define then The final ingredient of Eq. (2.2), the eikonal jet function J (p i , n i ), is responsible for subtracting the double counting of soft-collinear configurations, which contribute to both the jets and the soft function. The eikonal jet is obtained by replacing the hard line of momentum p µ in the partonic jet with a Wilson line with four-velocity β µ , yielding After factorising all singular (and universal) contributions, the matching to the exact amplitude order by order yields H, an infrared-finite, but process-dependent hard function.
Eq. (2.2) forms the starting point for describing radiation at leading power in the threshold expansion, where the soft function corresponds to dressing the hard function with virtual gluons of 4-momentum k µ → 0, whose infrared singularities cancel those associated with real emissions. In what follows, considering real radiation up to next-to-soft level, it will be convenient to generalise the soft function to include sub-leading powers of momentum in the propagators and emission vertices for virtual gluons. This can be done by defining a next-to-soft function as where F p is a generalised Wilson line operator, constructed in Ref. [19], which generates the required next-to-soft emission vertices. A general definition of such an operator for generic trajectories can be given in a coordinate-space representation, and is presented in [19]. For straight semi-infinite trajectories one can easily transform to a momentum-space representation, given by . (2.8) Note that we have given the result for a fermion, where Σ µν = i 4 [γ µ , γ ν ] is the appropriate Lorentz-group spin generator. The subscript on the RHS of Eq. (2.7) indicates than one should truncate the resulting expression to include at most one next-to-soft vertex. Correspondingly, one may define a next-to-soft jet function With these definitions, the non-radiative amplitude reads as schematically depicted in Fig. 1(a). Just as in Eq. (2.2), H is obtained by matching to the full amplitude on the left-hand side. It differs from the function H appearing in the factorisation formula in Eq. (2.2), as next-to-soft effects have now been explicitly factored out.

Real radiation up to NLP order
Let us now consider adding the radiation of an additional (next-to-)soft gluon to the amplitude in Eq. (2.10). The emission of the extra gluon can be assigned to different factors in the nonradiative amplitude, as shown in Fig. 1(b,c,d). Proceeding by analogy with the treatment of where µ and a are the Lorentz and the color adjoint indices of the emitted gluon, respectively, and we have introduced the radiative jet function, defined, as in Refs. [15,39], by The crucial issue in generalising the radiative jet function to the non-abelian theory is the definition of the non-abelian gauge current j µ,a (x). First of all, it must be a conserved current, ∂ µ j µ a = 0, in order for the radiative jet to obey the Ward identity which is the natural generalisation of the abelian case, and is a necessary ingredient for the proof of our factorisation formula. Furthermore, we must require that the matrix element in Eq. (2.12) should fully reproduce the relevant terms in the amplitude when the (next-to-)soft gluon is radiated from virtual gluons inside the jet. It turns out that the standard, textbook definition of the non-abelian Noether current (see, for example, Ref. [47]) does not have this property. One must however keep in mind that Noether currents are not uniquely defined: in general, it is possible to add improvement terms (see, for example, Ref. [48]) which do not spoil charge conservation but may improve other symmetry properties of the operator. In our case, we have found that an improvement term indeed exists which reproduces all relevant terms in diagrams where the (next-to-)soft gluon is emitted though a three-gluon vertex. Our choice for the non-abelian current is then which is indeed conserved, as one can readily verify. Note that the last term in Eq. (2.14) (the 'improvement') does not contribute to ∂ µ j µ a . Our next step is to define an operator matrix element describing (next-to-)soft real gluon radiation from the soft factor of the non-radiative amplitude, as depicted in Fig. 1 This definition is similar to the non-radiative function of Eq. (2.7), but one of the gluons is now real rather than virtual. A similar quantity is defined at leading power in Refs. [49][50][51][52][53]. The radiative next-to-soft function obeys the Ward identity Following Ref. [19], the radiative next-to-soft function can be shown to exponentiate, and it can be evaluated using next-to-soft webs, generalising the methods used at leading power. These webs can connect all hard partons in the process, thus they are not captured by the emission of gluons from inside single-parton jets. Nevertheless, there is clearly a double counting of contributions between the jets and the next-to-soft function, which is directly analogous to the double counting of soft and collinear contributions in Eq. (2.2). We may correct for this by subtracting from the jet emission contributions defined in Eq. (2.11) their next-to-soft expansion, according to where the subtraction term on the right-hand side is simply defined as the next-to-soft approximation to the full radiative jet function. Given that the overlap between the soft and jet functions must be separately gauge-invariant, the Ward identity of Eq. (2.16) implies Having precisely defined the jet and next-to-soft contributions to the radiative amplitude in terms of operator matrix elements, and having taken care to subtract double counted contributions, the emissions from the hard sub-process, which we denote by A H µ,a and depict in Fig. 1(c), are defined by matching to the full radiative amplitude A µ,a . We will discuss their properties in the following subsection.

Derivation of the non-abelian factorisation formula
Combining the above ingredients gives a total radiative amplitude where we defined In order to characterise the radiative jet functions it is convenient, as in Refs. [15,39] to introduce polarisation tensors [54] so that the total radiative amplitude is given by the sum of "K-polarised" and "G-polarised" gluons. Considering first the K-projection of the jet contribution to the amplitude, and using Eqs. (2.11, 2.13, 2.18), we find where we have again Taylor expanded in k. The emission of a G-gluon from a jet, on the other hand, is given by where we set k → 0 in the hard function, retaining only the leading term in its Taylor expansion, owing to the fact that the G tensor acts on terms proportional to p µ i to make them O(k) [15,39]. Combining this with the K-gluon emissions, with the G-gluon contribution from the subtraction term, and with emissions from the hard function, as given by Eq. (2.23), we find that the total amplitude, to the required accuracy, becomes 2 where the factor of 1/2 in the first term is due to the fact that we have placed this inside the sum over hard particles, for brevity. Reconstructing the expression for the non-radiative amplitude given in Eq. (2.10), we may express Eq. (2.27) as where we have used Eq. (2.10) to replace derivatives of the hard interaction with those acting on the full non-radiative amplitude and jet functions. Eq. (2.28) is our final non-abelian factorisation formula, capturing all NLP contributions near threshold. A few comments are in order.
• The first two terms in Eq. (2.28) contain next-to-eikonal webs, composed of generalised Wilson lines [19], dressing the non-radiative amplitude, together with a derivative operator. These terms provide a non-abelian version of the original Low's theorem (in the absence of collinear enhancements).
• The remaining terms in square brackets organise emissions from jet functions, generalising to the non-abelian theory the results of [15,39].
• The last term in Eq. (2.28) corrects the radiative jet factors for the double counting of contributions between the radiative jets and next-to-soft functions.
• Eq. (2.28) describes the amplitude stripped of external wave functions for the hard partons. To build a cross section, these must be reinstated, as in Eq. (2.1), noting that the derivative in Eq. (2.28) does not act on the wave functions.
As discussed in Refs. [39,43], considerable simplifications occur in Eq. (2.28) upon choosing the auxiliary vectors n i to be null, n 2 i = 0. In this case, one may work in a renormalisation scheme such that the non-radiative soft and jet functions are unity, to all orders in perturbation theory. Eq. (2.28) then becomes As in Refs. [39,43], in the detailed calculations below we will further make the specific choice of reference vectors n 1 = p 2 , n 2 = p 1 , (2.30) whose interpretation is that n i is the anti-collinear direction associated with p i . This is physically motivated by the fact that p 1 and p 2 are the only momenta in the problem at hand, and it allows to make direct contact with the method-of-regions calculation of Ref. [43].

The non-abelian radiative jet function
Before testing Eq. (2.29) in Drell-Yan production, we must first calculate the non-abelian radiative quark jet function, defined in Eq. (2.12). To perform a test at NNLO, we need to compute J µ,a at one loop, which we do for null n, in order to use the result in Eq. (2.29). Relevant Feynman diagrams are shown in Fig. 2. Defining the perturbative coefficients of J µ,a via the diagram of Fig. 2(a) gives One-loop diagrams are shown in Fig. 2(b)-(h). Notice that we are computing the bare J µ,a , as required in Eq. (2.29), following the discussion in Ref. [39]. We can then use the fact that the integral for (h) is scaleless for null n and thus vanishing. Similarly, we have omitted external-leg vacuum polarisations dressing the tree-level diagram. The result can be cast in the form Defining the kinematic variables t = −2p · k , n µ p = n µ 2n · p , n µ k = n µ 2n · k , r = n · k n · p , (3.4) we find that the coefficients in Eq. (3.3) can be written as Figure 3: Example diagrams for the next-to-soft radiaive jet function, where the p leg has been replaced by a generalised Wilson line, and • denotes a next-to-soft emission vertex, arising from Eq. (2.8).
The first two terms in Eq. (3.3) are accompanied by a factor (2p · k) − , corresponding to the collinear scale associated with radiation from a jet [39,43]. The third term in Eq. (3.3), on the other hand, contains a different ratio of scales involving the auxiliary vector n. Note that for the choices in Eq. (2.30) the ratio for both jets becomes This is the same dependence arising in (next-to-)soft webs connecting both external partons (shown for example in Fig. 1(d)). Terms with this scale dependence thus constitute the double counting of overlapping (next-to-)soft and collinear regions for the virtual gluon momentum, to be removed by the subtraction term A J µ,a . In our present calculation, one may interpret this overlap diagrammatically by defining a next-to-soft radiative jet function J µ,a (p, k, n). This function appears in the subtraction term A J µ,a instead of the full radiative jet function used in the definition of A J i µ,a , Eq. (2.11). By analogy with Eq. (2.11), we then write J µ,a (p 1 , n 1 , k) J(p 2 , n 2 ) . (3.7) The function J µ,a can be obtained from the diagrams for the full radiative jet, by replacing the emission vertices on the p leg with the soft or next-to-soft Feynman rules arising from Eq. (2.8), and including at most one next-to-soft vertex. At tree-level (using the normalisation of Eq. (3.1)) one simply finds J and one finds that J so that the next-to-soft radiative jet function reproduces precisely the third term in Eq. (3.3): subtracting it from the full jet leaves only collinear contributions, as required. According to Eq. (2.29), for the complete result one also needs the radiative next-to-soft function S µ at one-loop. The relevant diagrams are similar to those entering the next-to-soft radiative jet function. The leading power soft diagrams can be obtained simply by relabelling p → p 1 , n → p 2 . For the next-to-soft contribution, there are two sets of diagrams: those where the next-to-soft emission vertex is on leg p 1 , and those where it is on leg p 2 . One may then write where the subscripts E and NE refer to eikonal and next-to-eikonal contributions respectively.

Application to Drell-Yan production
We now have all ingredients to verify Eq. (2.29) in the Drell-Yan process where q andq denote a quark and antiquark respectively, V * an off-shell vector boson, and arguments label 4-momenta. At cross-section level, all LP and NLP threshold logarithms are associated with real emission of soft or next-to-soft gluons; virtual gluons, however, can be hard and collinear, thus loop corrections test all ingredients in Eq. (2.28). As usual, one defines the threshold variable z = Q 2 /s, representing the fraction of available energy carried by the final state vector boson; the threshold limit then corresponds to z → 1. The K-factor at fixed order in perturbation theory is defined by with σ (n) the n-loop cross section. As was the case in [39,43], the first non-trivial test of Eq. (2.28) is to reproduce the real-virtual contribution to the NNLO K-factor. To do so, we need the treelevel and one-loop amplitudes with one real emission. Using Eq. (2.28), we find where A (0) is the leading order non-radiative amplitude, stripped of external spinors, and we have used the tree-level radiative jet function in Eq. (3.2), as well as the physical polarisation condition k µ µ (k) = 0. We have also defined colour generators T a 1,2 acting on the p 1 , p 2 legs respectively. For the one-loop amplitude, we use Eq. (2.29), which gives 4) where in the second term we have used the fact that the derivative of the non-radiative tree-level amplitude vanishes [39]. Eq. (4.4) can be further simplified by noting that, at tree level, the nextto-soft function contribution precisely cancels the next-to-soft radiative jet contribution.The only missing ingredient at this point is the derivative of the one-loop non-radiative amplitude, which was already derived in Ref. [39]. For example, the contribution from the p 1 leg is given by It is straightforward to assemble all the ingredients 3 , to compute the full real-virtual contribution NNLO K-factor. We find where For comparison with the exact two-loop calculation, we note that the real-virtual contribution to the NNLO K-factor is not separately available in the literature [55,56]. We have performed an independent calculation of this result, similar to the one carried out for the abelian-like contributions in Ref. [39]. We find that Eq. (4.6) reproduces exactly the full NNLO result, when the latter is truncated to NLP in (1 − z), including non-logarithmic contributions.

Double real emission contributions
In Section 4, we have focused on a single additional gluon emission dressing the non-radiative amplitude. Although a full factorisation formula for multiple emissions is beyond the scope of this paper, we can nevertheless obtain the double-real emission contributions to Drell-Yan production at NNLO by noting that all purely real-emission near-threshold contributions are (next-to-)soft in nature, with no hard collinear terms. This fact was already exploited in Ref. [20], where next-to-soft Feynman rules were employed to compute the abelian part of the NNLO K-factor for double real emission. That calculation can easily be reproduced and generalised to the full non-abelian theory in the present framework. In essence, all relevant terms can be obtained by dressing the Born amplitude with (next-to-)soft webs. Formally, by analogy with Eq. (2.15), we may define a double radiative next-to-soft function according to * A sampling of soft and next-to-soft diagrams resulting from this definition are shown in Fig. 4. We have evaluated all diagrams using the next-to-soft Feynman rules arising from Eq. (2.8), and we have integrated over the three-body phase space as in Refs. [20,55]. The result for the double real emission contribution to the NNLO K factor is Figure 4: Sample diagrams contributing to the double radiative next-to-soft function, where • denotes a next-to-eikonal Feynman rule, and all other couplings to the external lines are eikonal.
which again agrees with an exact calculation, including non-logarithmic NLP terms. It is not in fact surprising that this happens: one may derive the next-to-soft Feynman rules by systematically expanding the exact unintegrated amplitude in the emitted gluon momenta (following the diagrammatic approach of Ref. [20]). Thus, the effective approach and the full calculation agree at the amplitude level by construction. We have also checked that, upon combining our results for double real emission and real-virtual corrections with the well-known two-loop virtual corrections, and with mass factorisation, the complete result of Refs. [55,56] is reproduced, to the expected accuracy.

Conclusion
In this paper, we have derived an all-order factorisation formula, Eq. (2.28), organising at the amplitude level all contributions which give rise to threshold logarithms up to next-to leading power. The formula has been derived for the Drell-Yan process, but we expect it to apply, with minor modifications, for all processes involving the annihilation of QCD partons into electroweak final states, such as (multiple) Higgs production via gluon fusion or multiple vector boson production. Eq. (2.28) generalises the well-known leading power soft-collinear factorisation formula described in Ref. [46], as well as previous formulae that included only abelian-like contributions [15,39]. It contains similar universal functions, namely the leading-power soft and jet functions, together with a radiative jet function. We have generalised the definition of the latter to a non-abelian theory, and calculated this quantity at one-loop order for quark jets.
We have verified our formula by reproducing known threshold logarithms at NNLO in Drell-Yan production, which is a non-trivial check at loop-level since both collinear and soft momentum regions are tested. We discussed how to remove the double counting of next-to-soft and collinear contributions via a subtraction term, by defining a next-to-soft radiative jet function. We note that a more general definition of this function, in particular for general values of the auxiliary vector n, deserves further study, which we postpone to future work. As was the case for the abelian-like contributions in previous work, we find that there is a non-vanishing loop-level contribution to NLP logarithms from hard collinear configurations of virtual gluons: this leads to a breaking of the next-to-soft theorems discussed for example in Refs. [33,34], at loop level.
A new feature of the present work is the role of next-to-soft web diagrams, describing the correlated emission of gluons external to the hard interaction. Using next-to-soft webs, we reproduce the double real emission contributions in Drell-Yan production, which are dominated near threshold by radiation which is always (next-to-)soft, whether or not it is collinear. We note that, as discussed in detail in Refs. [19,20], the web language is potentially much more powerful, implying formal exponentiation of next-to-soft contributions in a much more general context, as discussed for example in Refs. [57][58][59][60][61][62][63][64][65][66][67]. We conclude that Eq. (2.28) is an important step towards a general resummation procedure for NLP threshold logarithms. Further necessary ingredients include the calculation of radiative jet functions for external gluon jets, and the elucidation of subleading collinear effects in processes with final state parton jets (see, for example [68]). These developments will be the subject of future work.