Edinburgh 2017/11, IPPP/17/33, DCPT/17/66, MCnet179 Higgs Boson Plus Dijets: Higher Order Corrections
Abstract
The gluon fusion component of Higgsboson production in association with dijets is of particular interest because it both (a) allows for a study of the structure of the Higgsboson couplings to gluons, and (b) provides a background to the otherwise clean study of Higgsboson production through vectorboson fusion. The degree to which this background can be controlled, and the structure of the gluonHiggs coupling extracted, both depend on the successful description of the perturbative corrections to the gluonfusion process.
High Energy Jets (HEJ) provides allorder, perturbative predictions for multijet processes at hadron colliders at a fully exclusive, partonic level. We develop the framework of HEJ to include the process of Higgsboson production in association with at least two jets. We discuss the logarithmic accuracy obtained in the underlying allorder results, and calculate the first nexttoleading corrections to the framework of HEJ, thereby significantly reducing the corrections which arise by matching to and merging fixedorder results.
Finally, we compare predictions for relevant observables obtained with NLO and HEJ. We observe that the selection criteria commonly used for isolating the vectorboson fusion component suppresses the gluonfusion component even further than predicted at NLO.
Contents:
 1 Introduction

2 The Formal Accuracy of High Energy Jets
 2.1 Logarithmic Corrections and Logarithmic Accuracy
 2.2 FadinKuraevLipatov Amplitudes
 2.3 Construction of the Simplest HEJ Amplitude
 2.4 Regularisation and Leading Logarithmic AllOrder Cross Sections
 2.5 The First Set of SubLeading Corrections
 2.6 High Energy Corrections to Higgs Boson Production with Jets
 2.7 Matching and Merging of Fixed Order Samples and Final Results
 3 Analysis of Results
 4 Conclusions
 A TreeLevel Amplitudes for
1 Introduction
Immediately after the observation[1, 2] at the CERN LHC of the fundamental Higgslike boson, attention turned to measuring the strength and properties of its couplings to other SM particles, and its intrinsic properties. Initially, these measurements were performed by studying inclusive Higgs boson production in the Higgs boson decay channels and [3, 4, 5, 6, 7, 8, 9, 10, 11]. As the inclusive Higgs boson production is dominated by gluonfusion Higgs boson production, any measurement of the strength of the coupling of the Higgs boson to e.g. will involve a product of this coupling with the coupling for the production of the Higgs boson through gluon fusion, mediated by heavy (top and bottom) quark loops.
A precise measurement of the coupling of the Higgs boson to the electroweak bosons is obviously important to determine if indeed a single fundamental Higgs boson is fully responsible for the massgeneration of fundamental particles and electroweak symmetry breaking, as in the Standard Model. In this respect, it is interesting to study Higgs boson production directly through weak boson fusion. At the LHC, this process would occur perturbatively in the process of Higgs boson production in association with at least two hard jets. This process is of interest then not just as a perturbative correction (at order ) to the inclusive Higgs boson production through gluon fusion, but also as a Born level process that allows for a direct measurement of the strength of the coupling between the Higgs boson and the weak bosons. Since the quantum interference between the two contributing production channels of socalled vectorboson fusion (VBF) (involving a direct coupling between the Higgs and the weak bosons) and gluon fusion (GF) is insignificant[12, 13, 14], it is justified to discuss the processes separately. The study of weak boson fusion production of Higgs bosons then allows for a measurement of the higgs boson to weak boson coupling without relying on a knowledge of the loopinduced coupling strength between gluons and the Higgs boson.
The final analyses of data after RunI[15, 16, 11] allowed for the Higgs boson production to be studied for small numbers of coproduced jets, in particular also for the production in association with two or more jets. These measurements, therefore, start probing directly the VBF production mechanism, where the Bornlevel process involves quarks only scattering by the exchange of a weak boson. This is dominated by valence quarks, and hence the resulting jets will carry a significantly larger part of the lightcone proton momenta than what is the case of the gluonfusion production mechanism, where the cross section contribution for inclusive cuts is dominated by the component. The distinctive topology for VBF allows for event selection cuts on e.g. a large invariant mass and/or rapidity separation between the dijets in order to suppress background. This also suppresses the contribution from the gluonfusion process relative to VBF. While the inclusive GF cross section is dominated by the component, the component dominates[17, 18] after a large invariant mass between the dijets is required.
Requiring a significant invariant mass between dijets is interesting not just as a tool to suppress the gluonfusion contribution over weakboson fusion, but for a slightly less restrictive cut on the invariant mass, which allows more gluonfusion events in the sample, it is possible to study the structure of the gluonHiggs couplings[19, 20]. In particular, such analyses of the sample allow for an extraction of mixing parameters in scenarios with violation in the Higgs sector. However, the correct description of the gluonfusion contribution in the region of phase space with a significant invariant mass between the dijets is more challenging than is the case for weakboson fusion. The reason is that the gluonfusion component allows for a colouroctet exchange between the dijets, whereas the weakboson fusion component obviously has no colour exchange between the jets. This leads to a different radiation pattern for the two processes[21], where the gluonfusion component will radiate more hard, observable jets in the rapidity region spanned by the colour octet exchange than the weakboson fusion process. This again leads to an increase in the expected number of hard jets in the event as the rapidity span is increased. This behaviour is universal for all processes allowing for a colour octet exchange between jets, and has already been observed in both pure dijet production[22, 23] and the production of W+dijets[24]. Not just does the colouroctet exchange emphasise the contribution from realemission, higherorder perturbative corrections, but it is also accompanied by a tower of logarithms from virtual corrections. Both sources of perturbative corrections are included in the BFKLequation[25, 26, 27, 28], which captures the dominant logarithms ) which govern the highenergy limit of the onshell scattering matrix elements.
However, such logarithms are not systematically included in the standard perturbative methods for obtaining predictions for LHC observables. Analyses of e.g. W production in association with dijets for both D0[24] (at the 1.96 TeV Tevatron) and ATLAS[29] (at the 7 TeV LHC) consistently reveal a tension between data and a standard set of predictions in the region of phase space of large dijet invariant mass or rapidity separation. This is true for the differential cross section depending on just the Bornlevel momenta, and for observables describing additional jet activity. This tension between data and the predictions of the standard tools is therefore present for the observables and the region of phase space that is of direct relevance for the study of Higgs boson production in association with dijets.
The dominant logarithms of are, however, systematically included in the calculations of the onshell partonic scattering amplitudes within the framework of High Energy Jets[30, 31, 32, 33, 34]. The framework is based on an approximation to the body onshell scattering matrix element. Within this approximation, both real and virtual corrections are included to all orders in perturbation theory. The virtual corrections not only cancel the infrared poles from the real corrections, but also contribute to the finite part of the matrix element. In fact, this finite contribution is instrumental in achieving leadinglogarithmic accuracy. This is in contrast to the standard formulation of a parton shower, where the assumed Sudakov form of the virtual corrections keeps the shower unitary, allowing for a probabilistic interpretation of emission.
In High Energy Jets, the sum over and the integration over each body phase space is performed explicitly using Monte Carlo sampling, and as such the predictions are made at the partonic level with direct access to the fourmomenta of each of the particles. The framework merges fixedorder (currently leading order), highmultiplicity matrixelements with an allorder description of the dominant logarithms. The formalism has been implemented for several processes, and compares favourably to data for e.g. dijet (or more) production[22, 35, 23], the production of a W boson in association with two jets[24, 29] and the production of a Zboson or virtual photon in association with two jets[34]. These studies indicate that in the largeinvariant mass, and the large rapidity differenceregion, the logarithms of HEJ are important, and their inclusion improves the theoretical prediction.
The experimental studies of dijets and W+dijets therefore also indicate that High Energy Jets should be relevant for a successful description of the gluonfusion production of a Higgs boson in association with dijets, in particular in the region of interest for the study of properties, and for understanding how to use the radiation pattern to successfully suppress the gluonfusion contribution to Higgs boson+dijets when studying weakboson fusion.
This paper presents the impact on the physics analyses, and the implementation of High Energy Jets for the gluonfusion contribution to Higgsboson production in association with dijets. The earlier application of High Energy Jets included the leading logarithms in only. In Section 2 we discuss the first systematic inclusion of part of the subleading contributions within the framework of High Energy Jets. The resulting predictions for several of the observables measured in Higgs boson+dijet production are presented in Section 3, and the conclusion discussed in Section 4.
2 The Formal Accuracy of High Energy Jets
In this section we will present the procedure used for obtaining predictions within High Energy Jets (HEJ). HEJ is concerned with the description of processes involving a channel colour exchange between two jets, such as dijetproduction, and QCD production of W+dijets, Z/+dijets (both starting at order ), and Higgs boson+dijets (starting at order ).
Underpinning HEJ is an allorder approximation to the onshell, hardscattering matrix elements, explicit in the momenta of all particles, and for each multiplicity. The cancellation of IR singularities between real and virtual corrections is organised with subtraction terms, which are sufficiently simple to allow the explicit summation over multiplicities, and the integration over phase space to be performed using Monte Carlo techniques. The approximation to the hard scattering matrix element ensures a certain logarithmic accuracy of the predictions, which will be detailed in Section 2.1. As further discussed in Section 2.7, the allorder approximations are supplemented by corrections using the fixedorder (so far just treelevel) predictions for several jet multiplicities. As such, HEJ provides an alternative procedure for merging fixedorder samples of various jet multiplicities to that of CKKWL[36, 37], which is based on the logarithmic accuracy achieved in a parton shower. Instead, the merging procedure of HEJ maintains both the logarithmic accuracy at large invariant mass between jets (as discussed in the next session) and the fixedorder accuracy of the merged samples.
2.1 Logarithmic Corrections and Logarithmic Accuracy
In this section we will first identify the leading contribution to Higgs boson production in association with dijets when these dijets have a large invariant mass. We then identify a source of systematic and logarithmically (in the invariant mass) enhanced perturbative corrections both for real emissions and virtual corrections, and discuss how these logarithmic corrections can be summed to all orders using the formalism of High Energy Jets.
Leading Contributions at Large Invariant Mass
Consider for illustration the production of a Higgs boson in association with dijets, with the rapidity of the Higgs boson between that of the jets. We label final momenta as shown in Fig. 1, such that the rapidities satisfy and the incoming momentum () is in the backward (forward) direction. In the following, we will be frequently interested in amplitudes in the limit of MultiRegge kinematics (MRK), defined by a large centerofmass energy , large invariant masses between all outgoing momenta, and fixed channel momenta. For our current example, we introduce the channel momenta of the system as and consider large , keeping and fixed. An analysis of the analytic properties of scattering amplitudes [38] (e.g. from Regge theory for multiparticle production) indicates that in this limit the onshell scattering amplitude should scale as [39]
(1) 
Here, is the spin of the particle that can be exchanged in the channel between the particle of jet 1 and the Higgs boson, is the equivalent for the channel between the Higgs boson and the particle of jet 2 and is a function of transverse scales only. For a given momentum configuration of the jets and the Higgs boson, the leading contribution to production therefore comes from the subprocesses with a parton flavour assignment to the jets which allows for the particle of the largest possible spin to be connecting the jets. For QCD this is the spin1 gluonic colouroctet exchange. If the flavour assignment of a subprocess is such that a quark exchange is mandated, then the contribution to the jet cross section (proportional to the square of the matrix element) from this subprocess is suppressed by the invariant mass of the dijet pair, as compared to the subprocess where a gluon exchange is possible. For a given momentum configuration of the jets, the flavour assignments of the incoming states and of the corresponding jets which can proceed through gluon (colouroctet) exchanges between each jet are called the FadinKuraevLipatov (FKL) configurations. These will form the leading contribution in to the given jet configuration.
{subfigure}[t]0.4  {subfigure}[t]0.4 
{subfigure}[t]0.4  {subfigure}[t]0.4 
We illustrate this by continuing the example above. Consider first the case where both the incoming and the outgoing partons making up the jets are gluons as shown in Fig. 6. At Born level, the spin of all exchanged particles is 1 (since they are all gluons), and therefore the amplitude must scale as , where in the MRK limit , such that depends on transverse scales only. This scaling is indeed demonstrated in Fig. 7. This plot shows , where the square of the Born level matrix element (extracted from Madgraph5_aMC@NLO[40]) is evaluated in the phase space configurations of increasing rapidity separation between all particles. In particular, the 4momenta of the two jets and the Higgs boson are parametrised in terms of their transverse momenta, azimuthal angle and rapidity as
(2) 
The specific choices for angles and transverse momenta are irrelevant for the conclusion, but here the phase space points used in the plot were GeV, , , , , and where is increasing along the axis. The matrix element exhibits the expected MultiRegge scaling according to Eq. (1), for spin1 (gluon) exchanges, as tends to a constant as increases.
We can illustrate the suppression introduced when one requires a quark exchange in the channel by considering the squared matrixelements for nonFKL configurations versus a corresponding FKL configuration. We will consider the three rapidity orderings of the flavour content in the process shown in panels (b) to (d) of Fig. 6. The rapidityordering can proceed through colouroctet exchanges between each of the jetpairs , and (and the Higgs boson) and hence is an FKL configuration. The square of the matrix element for the cross section then scales as (where depends on transverse scales only). If now the parton content of and is swapped, the previous possibility of a gluon exchange between jets 1 and 2 is replaced by a quark exchange. Therefore, the scatteringprocess will scale as (where depends on transverse scales only), which is therefore suppressed by one power of with respect to the FKL configuration. The third configuration we consider is . Like the second configuration, this only allows a quark exchange between jets 1 and 2, now with the Higgs boson in between in rapidity, and hence scales as (where depends on transverse scales only).
We illustrate the behaviour of these matrix elements in Fig. 8. The left plot clearly shows the resulting suppression of the square of the matrix elements for the nonFKL configurations ( (blue) and (green)) compared to the FKL ordering (red). The latter tends to a constant times while the first two exhibit an exponential suppression for large (corresponding to a powersuppression in ). The suppression is indeed verified to be on the righthand plot in Fig. 8. Here, the squared matrix elements divided by has been multiplied by and tends to a constant for large in both cases.
Leading Contribution from Perturbative QCD
An alternative derivation of the dominance of the FKL configurations can be found by considering which
of all the possible colour connections will dominate in the MultiReggeKinematic (MRK) limit. As the Higgs boson is
colourneutral and irrelevant for the arguments, we restrict here the
discussion to amplitudes involving just quarks and gluons, and follow the treatment of Ref. [41]. We begin by
considering the process
By explicit calculation one quickly finds (see appendix A) that the treelevel result for the initial states fixed as the gluon incoming with positive lightcone momentum and the quark with negative lightcone momentum, the amplitude for the two rapidity orderings of the final state in the MRK limit scale as
(3) 
in agreement with Eq. (1) and hence the dominant flavourconfiguration in the MRK limit is given by the momentum configuration with . As illustrated in the figures, this is the configuration where a colour octet (two colour lines) is exchanged, when particles are drawn ordered in rapidity.
Dominant Contributions at Arbitrary Multiplicities
The result of the previous section in fact generalises beyond the simple process. In Ref. [41], the compact ParkeTaylor expression[42] for the maximally helicity violating (MHV) amplitudes for allgluon processes was used to show that for an arbitrary number of gluons, the colour connections which dominate kinematically in the MRK limit are those which can be represented on a socalled twosided plot. An example of such a plot is shown in Fig. 11. The momentum of the incoming particles are labelled (negative momentum), and (positive momentum), and the outgoing particles are ordered in rapidity from left to right.
The colour connections which dominate in the MRK limit are found[41] to be precisely all those which may be drawn without any crossed lines. Furthermore, these colour connections all contribute with the same kinematic factor in the MRK limit.
The colour factor arising from these planar colour connections coincides with the colour factor from a single diagram with maximal tchannel gluon exchanges. In other words, for gluons, the single colour factor of the FKL amplitude would be
(4) 
where , , … are the colour indices of the rapidityordered external gluons and the are the repeated indices of tchannel gluons. All other independent permutations of the indices multiply kinematic factors which are suppressed in the MRK limit. The final result for the limit of the colour summedandaveraged square of the scattering amplitude agrees with that of the highenergy limit of QCD derived by FadinKuraevLipatov (FKL)[26].
The multiRegge kinematic limit of the kinematic part of the ParkeTaylor amplitudes is found[43] to be such that the full colour summed and averaged square of the scattering amplitude receives a factor
(5) 
for each final state gluon beyond the first two. For example, the MRK limit of the colour and spin summed and averaged matrix element for is
(6) 
Similarly, the MRK limit of the colour and spin summed and averaged matrix element for is
(7) 
Up to this multiplicity, only MHV configurations contribute to the amplitude. The above expressions Eqs. (6) and (7) therefore already cover the most general case.
In the following, we consider the partons extremal in rapidity (i.e. partons 1 and 2 for the Born process, 1 and 3 for the scattering and 1 and in the general scattering) to be hard in the perturbative sense. Additional partons emitted inbetween in rapidity are then considered part of the radiative corrections to the process.
For a specific choice of rapidities for the extremal partons in the limit of the matrix element of Eq. (7), the phase space integration of the position of the middle parton will contribute a factor
(8) 
The integral over transverse phase space is IR divergent; the divergence cancels that introduced by the virtual corrections to the scattering. This cancellation is organised by using e.g. dimensional regularisation of the integrals, as will be discussed in more detail later. The point here is that the real (and virtual) corrections to the Bornlevel scattering introduce corrections proportional to the rapidity separation between the extremal (Bornlevel) partons. In the MRK limit, , and so we have sketched the appearance of logarithmic corrections in the perturbative series of the scattering.
This analysis carries through to any order in . One notes that all dependence on the rapidity of the middle partons is absent in the factor in Eq. (5), and in the contribution to the corrections of Eq. (8) . This leads to a simple diffential equation for the cross section in ; this is called the BFKL evolution equation[26, 27, 28].
Above, we have discussed the colour connections present in the MRK limit in the treelevel matrix elements for any number of finalstate gluons, i.e. the real corrections to the Born level. The virtual corrections are encoded at allorders through simple factors multiplying the channel poles and hence the colour discussion above generalises immediately to these cases too.
At higher multiplicities, also nonMHV configurations contribute to the amplitude. In the MRK limit, the dominant configurations all conserve helicity between the incoming gluon and the extremal gluon at the respective end (for MHV configurations, this can be seen directly by considering the numerators in the ParkeTaylor amplitudes [41]). Flipping the helicity of any gluon emitted inbetween the extremal gluons only changes the matrix element by a phase in the MRK limit, so that all helicity configurations which occur in the MRK limit can be related to the ParkeTaylor formula.
2.2 FadinKuraevLipatov Amplitudes
In the previous section, we described the behaviour of QCD amplitudes in the limit of large invariant mass between each particle. Obviously, if the full amplitude is known, the MRK limit of it can be directly obtained. However, the limits can also be derived based on the FadinKuraevLipatov (FKL) amplitudes[25, 26, 27].
QCD scattering amplitudes factorise in the MRK limit into what in the (B)FKL language are called impact factors and Lipatov vertices, which are connected by gluon exchanges in the tchannel. Each of these components of the amplitude depends only on a much reduced subset of momenta and is otherwise independent of the rest of the amplitude. This feature persists after the addition of a Higgs, or boson to the scattering. Two simple examples are shown in Fig. 12.
What is meant by the term “factorisation of the amplitude” is that the correct MRK limit of the amplitude can be obtained from a simple analytic approximation, which consists of factors, each of which depend only on a subset of all the momenta of the process. As an example, in the process on the lefthandside of Fig. 12, the flavour of the external lines may be quark or gluon and in the MRK limit (), the amplitude may be expressed in the form:
(9) 
where indicates an impact factor, which depends on the two momenta along the same direction on the lightcone only (i.e. are the parton momenta each with the maximum positive lightcone momentum, have the largest negative lightcone momentum). The correct MRK limit of the full amplitude would then be obtained with this analytic expression, for any configurations of the transverse momenta. The square of the amplitude is then simply found as
(10) 
Similarly, the correct MRK limit of the scattering amplitude for the threeparticle final state on the righthand side may be written
(11) 
where is a socalled Lipatov vertex. The only difference to the form of the twoparticle final state is the insertion of a vertex and a propagator in the analytic form of the MRK limit, which has a form suggestive of the channel exchange. The channel interpretation of the analytic form of the kinematic part of the amplitude is supported by the colourconnections studied in Sec. 2.1.2, but while the contribution from individual channel Feynman diagrams are obviously gauge dependent, it is important to realise that the MRK limit of the scattering amplitude is a gaugeindependent statement. It just happens to have the analytic form expected from a channel gluon exchange, as expected from the analysis presented in Sec. 2.1.1.
For the impact factors one finds , and in the MRK limit , and one finds[27] that the factor introduced from an additional gluon emission of transverse momentum into the FKL result for the square of the matrix element is simply
(12) 
Therefore, the MRK limit of the QCD amplitudes found in Sec. 2.1.3 are reproduced by the FKL amplitudes[43, 41]. This is true for an arbitrary number of gluons emitted, such that the FKL result for the leadingorder contribution to the colourandspin summedandaveraged square of the scattering amplitude is given by
(13) 
The channel structure of the FKL amplitudes allows for the inclusion of the dimensionally regulated virtual corrections (in dimensions) through the Lipatov Ansatz for the Reggeized channel colouroctet exchanges. This is the prescription for including the allorder virtual corrections to the Bornlevel colour octet exchange by making the following substitution in Eq. (10):
(14) 
where
(15) 
with , such that . This ansatz for the exponentiation of the virtual corrections in the appropriate limit of the parton scattering amplitude has been proved to even the subleading level[44, 39, 45, 46], which leads to a perturbative correction to leadinglogarithmic results for , the Lipatov vertex and the impact factors.
The FKL result for the square of the scattering matrix for obtained by using the kinematic approximations valid in the multireggekinematic limit has no dependence on the rapidities of the finalstate particles (in essence because the limit of infinite rapidityseparation has been applied). The poles in in the dimensionally regulated inclusion of the virtual corrections through the Lipatov ansatz turn out to cancel orderbyorder with the poles from the dimensionally regulated integration over the soft phase space of additional emissions (intermediate in rapidity between parton 1 and ) included through the FKL result for the square of the matrix element for . A finite contribution from the virtual corrections is left over. If now the contribution to the centreofmass energy and therefore also to the longitudinal momentum of the incoming partons is ignored from all but the most backward and forward parton, then the sum over the integration over phase space of any parton of intermediate rapidity can be performed analytically. This leads to the much celebrated BFKL equation[28], which captures the leading (and subleading) behaviour in . It is seen that the logarithmic behaviour is the same when using the FKL amplitudes of Eq. (13) and the limit of the full QCD amplitudes as discussed in Sec. 2.1.3. The largerapidity behaviour of the parton amplitudes of full QCD and FKL is the same in terms of powers of , which is sufficient to guarantee the same logarithmic behaviour of the integrated cross section in terms of .
2.3 Construction of the Simplest HEJ Amplitude
In the previous two subsections, we have described how the leading behaviour of scattering amplitudes in QCD arises through the study of channel poles, and how the simple structure in the MRK limit is captured to all orders in by the FKL amplitudes. So far with HEJ, allorder results have been achieved for such FKL configurations only. All other kinematic configurations have been included to fixed order only through a matching and merging procedure described in Sect. 2.7. In this paper, we present for the first time the inclusion of allorder results also for some subleading corrections, namely quarkinitiated processes with one gluon emitted outside the FKLordered phase space. These configurations correspond to the suppressed contributions studied in Figure 8. The leading logarithmic corrections to these processes constitute the first subleading logarithmic corrections included in HEJ. The configurations constitute the largest part of the subleading crosssection, which previously was included through the naïve addition of fixedorder samples. The inclusion of these subleading (and their matching to fixedorder accuracy) therefore gives a much more satisfactory theoretical description of the scattering.
The motivation behind the HEJ framework is to capture the behaviour of amplitudes at large without applying the full tower of approximations necessary for obtaining an analytic answer for the cross section through the BFKL theory. By allowing for numerical integration of multiparticle amplitudes, we can both allow these to have a more complicated kinematic dependence than the of the FKLamplitudes, and account for the longitudinal momentumconservation which is invariably lost in any formulation involving the BFKL equation (at both LL and NLL accuracy).
The simplest of all QCD processes is that of , proceeding through a channel gluon exchange only. The MRK limit of the full QCD result and the FKL approximation of the square of this amplitude is
(16) 
The leadingorder QCD result is given by
(17) 
Two kinematic approximations are necessary to get from the full result to the approximation of FKL: , (where the last equality holds for the simple process). While both of these are valid in the MRK limit, they are easily off by an order of magnitude within the relevant phase space of the LHC.
In constructing a Monte Carlo phase space integrator, which is sufficiently efficient to calculate explicitly the phase space integration over manyparticle (e.g. up to 30) final state phase space, we can seek to build an approximation for the matrix elements, which still captures the leading logarithmic behaviour generated from the channel poles, but which relies on fewer kinematic approximations. In particular, we want the description of the amplitude to be:

exact for the simple process proceeding only through a channel exchange
^{2} ; 
gaugeinvariant for any additional gluon emitted, i.e. the Ward Identity is fulfilled (not just asymptotically in the MRKlimit, as for FKLamplitudes, but exactly, everywhere in phase space), ;

such that the soft divergences of the approximant are cancelled by the terms generated from the Lipatov Ansatz for the virtual corrections to the treelevel results (also for processes); and

sufficiently fast to evaluate such that the numerical integration is feasible.
Let us first focus on building this simple approximant for the processes. The Lipatov Ansatz can most easily be applied if the analytic structure of the parton amplitude is factorised into a dependence on parton and the parton amplitude (obviously evaluated with the momenta of the parton phase space). It is therefore important to build a good approximant to even the simplest processes, since obviously the multiparticle approximations are built on successive applications of these. We will see that by using helicity amplitudes, we can build such a simple structure for approximants of multiparticle amplitudes, which are valid even before the MultiReggeKinematic limit is applied.
Since we will be evaluating the amplitudes numerically in the MonteCarlo integration, there is no problem in keeping the full kinematic dependence on the channel propagatormomentum in Eq. (17) rather than performing the MRKapproximation . Clearly, the channel poles are described best by maintaining the full dependence on the channel momenta. We now turn to describing the remaining invariants, and . In the full MRK limit, ; in practice, there is a large deviation throughout phase space. By studying the amplitude for , we find that terms proportional to arise from amplitudes where the quarks have identical helicities; while terms proportional to arise from amplitudes where the two quark lines have opposite helicities. Explicitly, in terms of currents , one finds that
(18) 
By working at the helicity amplitude level, we have achieved a description of the amplitude that is exact, and furthermore the analytic form generalises easily to . These components then depend on and separately as in Eq. (9). Hence the product of two scalar impact factors has been expanded to a contraction of vector currents.
In fact, this factorised form also continues when one moves to with the same quark current as above [31]. The gluon current has an additional scalar factor, but it can still be written in a form which depends only on the gluon momenta, and can be found in Eq. (8) of Ref.[32], with the exact amplitude for written in terms of the HEJ buildingblocks as
(19) 
where (and again ). This is written for the case of a backward moving incoming gluon; for a forwardmoving gluon, one would simply define . A similar channel factorised form is found for scattering (in the configuration with scattering of gluons with the same helicity there is of course no unique concept of the channel).
We will later discuss how the scattering amplitude can be extended to capture the allorder leading logarithmic accuracy of the cross section by accounting for the emissions of additional gluons.
The structure of an amplitude approximated by building blocks, each depending only on the momenta of a small subset of the particles is obviously appealing computationally. Not only though are these factors independent of other particle momenta, they are completely independent of the rest of the process and are therefore in that sense, processindependent. So, if particles and are the same flavour in each case (either quark or gluon), the factor of the FKL formalism in Eqs. (9) and (11) will be identical, and so will the currents used in HEJ.
The next building block we need to derive is the Lipatov vertex, , for additional FKLordered gluon emissions. The simplest process to study is . It is necessary to sum the contributions from all five treelevel diagrams. After some manipulation in the highenergy limit this yields[30]
(20) 
where is still a contraction of currents:
(21) 
and is a Lipatovtype vertex for gluon emission, which is given by:
(22) 
This form is slightly more involved than the standard Lipatov (or ReggeonReggeonparticle) vertex of BFKL[47], since it maintains the dependence on each of the 4 quark momenta rather than making the approximation ; the two last terms in each bracket constitutes the eikonal approximation to the emission off each leg. The difference between the form used in BFKL and in HEJ is formally subleading, but crucial for obtaining analytic results in BFKL. Conversely, the full form of Eq. (22) unsurprisingly gives a more accurate description of the subasymptotic region of phase space, and thus leads to smaller matchingcorrections. In choosing to perform the phase space integrations numerically, we are free to choose the numerically more accurate form.
Now the power of the highenergy limit becomes manifest. With only the building blocks derived so far, the leading contribution of the scattering amplitude (in powers of , forming the leading logarithmic contribution to the integrated cross section) for any number of intermediate gluon emissions is described by
(23) 
This structure is shown in Fig. 13, where the Lipatov vertices are shown as grey boxes. The amplitude for the equivalent process with one or two incoming gluons is identical, except for a minor alteration to the function .
2.4 Regularisation and Leading Logarithmic AllOrder Cross Sections
In sections 2.1–2.3 we identified the leading contributions for jet production in the multiReggekinematic limit, and showed how to obtain an accurate approximation to the Bornlevel matrix elements for such processes for any multiplicity of gluon emissions. The only singularities present in this approximation are those arising from the channel propagators in the colouroctet exchanges of the rapidityordered final state, and these singularities of the Bornlevel amplitude are outside the physical region. As discussed in Section 2.1.3, logarithmic corrections in arise in the region of jets widely separated in rapidity. So far, we have discussed Bornlevel results only. In this section, we will discuss the calculation of the cross section to each order in , and the regularisation of the IR singularities. No UV singularities appear at the logarithmic order discussed.
The reason for developing an approximation to the channel poles of the scattering treelevel matrix elements is that the leading logarithmic contribution to the loop corrections of these processes can still be obtained using the Lipatov ansatz[26], just as discussed for the FKL amplitudes in Section 2.2. This ansatz states that the leading logarithmic contribution to the virtual corrections for amplitudes in the MRK limit can be found to all orders in the coupling by replacing each channel propagator between the two particles of ordered rapidities and () in the amplitudes constructed in section 2.3 as follows:
(24) 
with
(25) 
As mentioned earlier, this ansatz for the exponentiation and factorisation of the virtual corrections in the appropriate limit of the parton scattering amplitude has been proved to hold even at the subleading level [44, 39, 45, 46] and explicitly checked against the twoloop amplitudes for scattering[48].
As demonstrated in e.g. Ref.[32] and below, the poles in cancel exactly between the dimensionally regularised (in dimensions) virtual and real corrections to processes of any multiplicity, when calculated with the constructed amplitudes which ensure the correct leading logarithmic (in ) behaviour of the cross section. This allows for the calculation of the inclusive cross section (for the leading and the included subleading processes) as explicit sums of body 4dimensional phase space integrals of dimensionally regularised particle matrix elements.
The first step in organising the cancellation of the poles in and obtaining the regularised cross sections is to define for each Bornlevel momentum configuration the regions in phase space for which the real corrections for gluon emissions can be calculated to any order in the coupling. It is the phase space region in rapidity delimited by the extremal partons. These partons extremal in rapidity are required to be perturbative (i.e. of a transverse momentum similar to the hard jet scale), since these form parts of the fundamental currents of the formalism, and there is (at LL accuracy) no accompanying virtual corrections to regulate the divergences present as the transverse momenta of these extremal partons tend to zero. However, for the phase space bounded in rapidity by these extremal, hard partons, the soft singularity from the real emission of additional gluons is regulated by the singularity from the virtual corrections to all orders in the coupling (i.e. for any number of emissions into that region of rapidity).
To illustrate the specifics of this procedure, consider for simplicity the process . We will now show how the leading logarithmic perturbative corrections to this process are calculated to all orders through the explicit construction of regulated, fourdimensional amplitudes, which can be summed and integrated explicitly using MonteCarlo techniques.
We will apply dimensional regularisation (working in dimensions) in order to facilitate the cancellation of poles from real and virtual corrections. The colour and spin summed and averaged square of the scattering matrix element for the process (where indicates the possibility of any number of gluons), following from Eq. (23) but extended to dimensions, is
(26) 
The colour factors are if particle is a quark and if it is a gluon. The matrix element above describes the leadinglogarithmic corrections to dijet production at all orders in . These take the form of additional partons in the final state described by the emission vertices and the corresponding exponential factors arising from the virtual contributions to the process. We organise the cancellation of the divergences by means of a phasespace slicing parameter , which separates the “hard” region () from the “soft” region (). The divergences arising from soft emissions arise from the singularities of the emission vertices. Explicitly, in the limit that ,
(27) 
We therefore have
(28) 
The set of particle momenta on the righthand side (the set of momenta obtained by removing ) still satisfies momentum conservation since we are precisely considering the case of . The divergence in the scattering matrix element in the limit is therefore identical to that obtained using the simple factor in Eq. (28). We can therefore organise the cancellation of soft divergences between real and virtual corrections by first subtracting the term in brackets from the square of the Lipatov vertices. Since we only need to regularise the divergence, we will restrict this realsubtraction term to soft momenta, i.e. . The integral of the realemission subtraction term is then found as
(29) 
This contribution will be added to the virtual corrections for the momenta state. These virtual corrections can be found by expanding the exponential factor in the last line of Eq. (26) which spans the rapidity region integrated over in Eq. (29). We therefore find to first order in
(30) 
Combining this with the contribution from the integral of the realemission subtraction term in Eq. (29) and expanding in , the pole in and (the dependence on ) cancels exactly. This is in fact true orderbyorder in , and the finite correction which remains can be absorbed into the regularised trajectory
(31) 
We can repeat this for each real emission between the extremal partons, which yields the following allorder description of dijet production: