A Tree-Level Amplitudes for qg\to qg

# Higgs Boson Plus Dijets: Higher Order Corrections

## Abstract

The gluon fusion component of Higgs-boson production in association with dijets is of particular interest because it both (a) allows for a study of the -structure of the Higgs-boson couplings to gluons, and (b) provides a background to the otherwise clean study of Higgs-boson production through vector-boson fusion. The degree to which this background can be controlled, and the -structure of the gluon-Higgs coupling extracted, both depend on the successful description of the perturbative corrections to the gluon-fusion process.

High Energy Jets (HEJ) provides all-order, perturbative predictions for multi-jet processes at hadron colliders at a fully exclusive, partonic level. We develop the framework of HEJ to include the process of Higgs-boson production in association with at least two jets. We discuss the logarithmic accuracy obtained in the underlying all-order results, and calculate the first next-to-leading corrections to the framework of HEJ, thereby significantly reducing the corrections which arise by matching to and merging fixed-order results.

Finally, we compare predictions for relevant observables obtained with NLO and HEJ. We observe that the selection criteria commonly used for isolating the vector-boson fusion component suppresses the gluon-fusion component even further than predicted at NLO.

## 1 Introduction

Immediately after the observation[1, 2] at the CERN LHC of the fundamental Higgs-like 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 gluon-fusion 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 mass-generation 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 so-called vector-boson 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 loop-induced coupling strength between gluons and the Higgs boson.

The final analyses of data after Run-I[15, 16, 11] allowed for the Higgs boson production to be studied for small numbers of co-produced 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 Born-level 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 light-cone proton momenta than what is the case of the gluon-fusion 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 gluon-fusion 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 gluon-fusion contribution over weak-boson fusion, but for a slightly less restrictive cut on the invariant mass, which allows more gluon-fusion events in the sample, it is possible to study the -structure of the gluon-Higgs 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 gluon-fusion contribution in the region of phase space with a significant invariant mass between the dijets is more challenging than is the case for weak-boson fusion. The reason is that the gluon-fusion component allows for a colour-octet exchange between the dijets, whereas the weak-boson fusion component obviously has no colour exchange between the jets. This leads to a different radiation pattern for the two processes[21], where the gluon-fusion component will radiate more hard, observable jets in the rapidity region spanned by the colour octet exchange than the weak-boson 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 colour-octet exchange emphasise the contribution from real-emission, higher-order perturbative corrections, but it is also accompanied by a tower of logarithms from virtual corrections. Both sources of perturbative corrections are included in the BFKL-equation[25, 26, 27, 28], which captures the dominant logarithms ) which govern the high-energy limit of the on-shell 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 Born-level 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 on-shell 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 on-shell 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 infra-red 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 leading-logarithmic 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 four-momenta of each of the particles. The framework merges fixed-order (currently leading order), high-multiplicity matrix-elements with an all-order 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 Z-boson or virtual photon in association with two jets[34]. These studies indicate that in the large-invariant mass, and the large rapidity difference-region, 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 gluon-fusion 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 gluon-fusion contribution to Higgs boson+dijets when studying weak-boson fusion.

This paper presents the impact on the physics analyses, and the implementation of High Energy Jets for the gluon-fusion contribution to Higgs-boson 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 sub-leading 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 dijet-production, and QCD production of W+dijets, Z/+dijets (both starting at order ), and Higgs boson+dijets (starting at order ).

Underpinning HEJ is an all-order approximation to the on-shell, hard-scattering 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 all-order approximations are supplemented by corrections using the fixed-order (so far just tree-level) predictions for several jet multiplicities. As such, HEJ provides an alternative procedure for merging fixed-order samples of various jet multiplicities to that of CKKW-L[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 fixed-order 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 Multi-Regge kinematics (MRK), defined by a large center-of-mass 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 multi-particle production) indicates that in this limit the on-shell scattering amplitude should scale as [39]

 M∼ sα1(t1)1H sα2(t2)2H γ(t1,t2,s12/(s1Hs2H)). (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 spin-1 gluonic colour-octet exchange. If the flavour assignment of a sub-process 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 (colour-octet) exchanges between each jet are called the Fadin-Kuraev-Lipatov (FKL) configurations. These will form the leading contribution in to the given jet configuration.

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 4-momenta of the two jets and the Higgs boson are parametrised in terms of their transverse momenta, azimuthal angle and rapidity as

 pj1=p⊥1(coshy1,cosϕ1,sinϕ1,sinhy1)pjH=(m⊥HcoshyH,p⊥HcosϕH,p⊥HsinϕH,m⊥HsinhyH)pj2=p⊥2(coshy2,cosϕ2,sinϕ2,sinhy2). (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 Multi-Regge scaling according to Eq. (1), for spin-1 (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 matrix-elements for non-FKL 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 rapidity-ordering can proceed through colour-octet exchanges between each of the jet-pairs , 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 scattering-process 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 non-FKL 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 power-suppression in ). The suppression is indeed verified to be on the right-hand 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 Multi-Regge-Kinematic (MRK) limit. As the Higgs boson is colour-neutral 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 1. Without loss of generality we take the backward incoming parton to be the quark. For the outgoing quark and gluon, there are obviously two possible rapidity-orderings : and . These are shown in diagrams with rapidity-ordered final states in Figs. 9 and 10, together with the corresponding planar colour connections.

By explicit calculation one quickly finds (see appendix A) that the tree-level result for the initial states fixed as the gluon incoming with positive light-cone momentum and the quark with negative light-cone momentum, the amplitude for the two rapidity orderings of the final state in the MRK limit scale as

 |M(yq≪yg)|∼s γ1 and|M(yq≫yg)|∼√s γ2, (3)

in agreement with Eq. (1) and hence the dominant flavour-configuration 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 Parke-Taylor expression[42] for the maximally helicity violating (MHV) amplitudes for all-gluon 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 so-called two-sided 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 t-channel gluon exchanges. In other words, for gluons, the single colour factor of the FKL amplitude would be

 fcac1d1fd1c2d2...fdn−1cncb, (4)

where , , … are the colour indices of the rapidity-ordered external gluons and the are the repeated indices of t-channel 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 summed-and-averaged square of the scattering amplitude agrees with that of the high-energy limit of QCD derived by Fadin-Kuraev-Lipatov (FKL)[26].

The multi-Regge kinematic limit of the kinematic part of the Parke-Taylor amplitudes is found[43] to be such that the full colour summed and averaged square of the scattering amplitude receives a factor

 4g2CAk2i,⊥ (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

 ¯¯¯¯¯¯¯¯¯¯¯¯|M|2⟶4^s2(N2C−1) g2CAk21⊥ g2CAk22⊥. (6)

Similarly, the MRK limit of the colour and spin summed and averaged matrix element for is

 ¯¯¯¯¯¯¯¯¯¯¯¯|M|2⟶4^s2(N2C−1) g2CAk21⊥ 4g2CAk22⊥g2CAk23⊥. (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 in-between 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

 ∫d2k2⊥(2π)2∫y3y1 dy24π4g2CAk22⊥=y3−y14π∫d2k2⊥(2π)24g2CAk22⊥=(y3−y1) 4αsCA ∫d2k2⊥(2π)21k22⊥. (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 Born-level scattering introduce corrections proportional to the rapidity separation between the extremal (Born-level) 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 tree-level matrix elements for any number of final-state gluons, i.e. the real corrections to the Born level. The virtual corrections are encoded at all-orders through simple factors multiplying the -channel poles and hence the colour discussion above generalises immediately to these cases too.

At higher multiplicities, also non-MHV 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 Parke-Taylor amplitudes [41]). Flipping the helicity of any gluon emitted in-between 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 Parke-Taylor formula.

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 Fadin-Kuraev-Lipatov (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 t-channel. 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 left-hand-side 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:

 Mf1(pa)f2(pb)→f1(p1)f2(p2)→^s C(pa,p1) 1(pa−p1)2 C(pb,p2), (9)

where indicates an impact factor, which depends on the two momenta along the same direction on the light-cone only (i.e.  are the parton momenta each with the maximum positive light-cone momentum, have the largest negative light-cone 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

 |Mf1(pa)f2(pb)→f1(p1)f2(p2)|2→^s2 |C(pa,p1)|2(pa−p1)2 |C(pb,p2)|2(pb−p2)2. (10)

Similarly, the correct MRK limit of the scattering amplitude for the three-particle final state on the right-hand side may be written

 |Mf1(pa)f2(pb)→f1(p1)g(p2)f2(p3)|2→^s2|C(pa,p1)|2(pa−p1)2|VL(p2)|2(pa−p1)2(pb−p3)2|C(pb,p3)|2(pb−p3)2=^s2|C(pa,p1)|2t1|VL(p2)|2t1 t2|C(pb,p3)|2t2 (11)

where is a so-called Lipatov vertex. The only difference to the form of the two-particle 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 colour-connections 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 gauge-independent 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

 |VL(p2)|2t1 t2→4g2CAk22⊥. (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 leading-order contribution to the colour-and-spin summed-and-averaged square of the scattering amplitude is given by

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯|MFKLgg→g1⋯gn|2=2^s24(N2C−1)n∏1g2CAk2i⊥. (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 colour-octet exchanges. This is the prescription for including the all-order virtual corrections to the Born-level colour octet exchange by making the following substitution in Eq. (10):

 1ti−1 → 1ti−1 exp[^α(qi)(yi−1−yi)] (14)

where

 ^α(qi) =−g2 CA Γ(1−ε)(4π)2+ε2ε(q2i⊥/μ2)ε, (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 sub-leading level[44, 39, 45, 46], which leads to a perturbative correction to leading-logarithmic 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 multi-regge-kinematic limit has no dependence on the rapidities of the final-state particles (in essence because the limit of infinite rapidity-separation has been applied). The poles in in the dimensionally regulated inclusion of the virtual corrections through the Lipatov ansatz turn out to cancel order-by-order 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 centre-of-mass 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 sub-leading) 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 large-rapidity 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, all-order 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 all-order results also for some sub-leading corrections, namely quark-initiated processes with one gluon emitted outside the FKL-ordered phase space. These configurations correspond to the suppressed contributions studied in Figure 8. The leading logarithmic corrections to these processes constitute the first sub-leading logarithmic corrections included in HEJ. The configurations constitute the largest part of the sub-leading cross-section, which previously was included through the naïve addition of fixed-order samples. The inclusion of these sub-leading (and their matching to fixed-order 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 multi-particle amplitudes, we can both allow these to have a more complicated kinematic dependence than the of the FKL-amplitudes, and account for the longitudinal momentum-conservation 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

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯|MQCDqQ→qQ|2→¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯|MFKLqQ→qQ|2=2^s24(N2C−1)g2CFk21⊥g2CFk22⊥=(g2CF)24(N2C−1)2^s2k21⊥k22⊥. (16)

The leading-order QCD result is given by

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯|MQCDqQ→qQ|2=(g2CF)24(N2C−1)^s2+^u2^t2. (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 many-particle (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 exchange2;

• gauge-invariant for any additional gluon emitted, i.e. the Ward Identity is fulfilled (not just asymptotically in the MRK-limit, as for FKL-amplitudes, 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 tree-level 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 multi-particle 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 multi-particle amplitudes, which are valid even before the Multi-Regge-Kinematic limit is applied.

Since we will be evaluating the amplitudes numerically in the Monte-Carlo integration, there is no problem in keeping the full kinematic dependence on the -channel propagator-momentum in Eq. (17) rather than performing the MRK-approximation . 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

 |jμ±(p1,pa) j±μ(p2,pb)|2=s2,|jμ±(p1,pa) j∓μ(p2,pb)|2=u2. (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 building-blocks as

 |Mq−g+→q−g+|2=1N2C−1 |⟨b|ρ|2⟩⟨1|ρ|a⟩|2⋅(g2s CF 1t1)⋅(g2s [12 1+z2z (CA−1CA)+1CA] 1t2), (19)

where (and again ). This is written for the case of a backward moving incoming gluon; for a forward-moving 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 all-order 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, process-independent. 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 FKL-ordered gluon emissions. The simplest process to study is . It is necessary to sum the contributions from all five tree-level diagrams. After some manipulation in the high-energy limit this yields[30]

 MHEJtree qQ→qgQ=−g2sTa1i1iaTa2i3ib SqQ(p1,p3,pa,pb)q21q22×igsfa1b2a2 εν2(p2)Vν2L(q1,q2), (20)

where is still a contraction of currents:

 SqQ(p1,p3,pa,pb)=jμ(p1,pa) jμ(p3,pb), (21)

and is a Lipatov-type vertex for gluon emission, which is given by:

 VνL(qi,qi+1)=−(qi+qi+1)ν+pνa2(q2ipi+1⋅pa+pi+1⋅pbpa⋅pb+pi+1⋅pnpa⋅pn)+pa↔p1−pνb2(q2i+1pi+1⋅pb+pi+1⋅papb⋅pa+pi+1⋅p1pb⋅p1)−pb↔pn. (22)

This form is slightly more involved than the standard Lipatov (or Reggeon-Reggeon-particle-) 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 sub-leading, but crucial for obtaining analytic results in BFKL. Conversely, the full form of Eq. (22) unsurprisingly gives a more accurate description of the sub-asymptotic region of phase space, and thus leads to smaller matching-corrections. In choosing to perform the phase space integrations numerically, we are free to choose the numerically more accurate form.

Now the power of the high-energy 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

 MHEJtree=−g2sTa1i1iaTan−1inib SqQ(p1,pn,pa,pb)√q21q2jq2j+1q2n−1×n−1∏k=2igsfak−1bkak ενk(pk)VνkL(qk−1,qk)√q2k−1q2k. (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 All-Order Cross Sections

In sections 2.12.3 we identified the leading contributions for jet production in the multi-Regge-kinematic limit, and showed how to obtain an accurate approximation to the Born-level 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 colour-octet exchanges of the rapidity-ordered final state, and these singularities of the Born-level 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 Born-level 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 tree-level 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:-

 1ti → 1ti exp[^α(qi)(yi+1−yi)] (24)

with

 ^α(qi) =−g2 CA Γ(1−ε)(4π)2+ε2ε(q2i⊥/μ2)ε. (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 sub-leading level [44, 39, 45, 46] and explicitly checked against the two-loop 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 sub-leading processes) as explicit sums of -body 4-dimensional 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 Born-level 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, four-dimensional amplitudes, which can be summed and integrated explicitly using Monte-Carlo 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

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯∣∣MHEJε f1f2→f1⋅g⋅f2∣∣2=14 (N2C−1) ∥∥Sf1f2→f1f2∥∥2⋅ (g2s Kf1 1t1)⋅ (g2s Kf2 1tn−1)⋅n−2∏i=1(−g2sCAtiti+1 Vμ(qi,qi+1)Vμ(qi,qi+1))⋅n−1∏j=1exp[2^α(qj)(yj+1−yj)]. (26)

The colour factors are if particle is a quark and if it is a gluon. The matrix element above describes the leading-logarithmic 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 phase-space 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 ,

 Vμ(qk−1,qk)Vμ(qk−1,qk)tk−1tk→−4p2k⊥. (27)

We therefore have

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯∣∣MHEJtree,ε f1f2→f1(n−2)gf2({pi})∣∣2⟶pk→0(4g2sCAp2k⊥) ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯∣∣MHEJtree,ε f1f2→f1(n−3)gf2({pi}∖pk)∣∣2. (28)

The set of particle momenta on the right-hand 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 real-subtraction term to soft momenta, i.e. . The integral of the real-emission subtraction term is then found as

 μ−2ε∫softd3+2εpk(2π)3+2ε2Ek 4g2sCA|pk⊥|2=μ−2ε∫λ0d2+2εpk⊥(2π)2+2ε∫yk+1yk−1dyk4π 4g2sCA|pk⊥|2=μ−2ε4g2sCA(2π)2+2ε(4π)(yk+1−yk−1)∫λ0d2+2εpk⊥|pk⊥|2=g2sCAπ(2π)2+2ε (yk+1−yk−1) 1επ1+εΓ(1+ε)(λ2μ2)ε. (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

 Missing or unrecognized delimiter for \left (30)

Combining this with the contribution from the integral of the real-emission subtraction term in Eq. (29) and expanding in , the pole in and (the dependence on ) cancels exactly. This is in fact true order-by-order in , and the finite correction which remains can be absorbed into the regularised trajectory

 ω0(q2⊥) =g2sCA4π2 log(λ2q2⊥)=αsCAπ log(λ2q2⊥). (31)

We can repeat this for each real emission between the extremal partons, which yields the following all-order description of dijet production:

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯∣∣MHEJε f1f2→f1⋅g⋅f2∣∣2=14 (N2C−1) ∥∥Sf1f2→f1f2∥∥2⋅ (g2s Kf1 1t1)⋅