A Parton Shower for High Energy Jets

We present a method to match the multi-parton states generated by the High Energy Jets Monte Carlo with parton showers generated by the Ariadne program using the colour dipole model. The High Energy Jets program already includes a full resummation of soft divergences. Hence, in the matching it is important that the corresponding divergences in the parton shower are subtracted, keeping only the collinear parts. We present a novel, shower-independent method for achieving this, enabling us to generate fully exclusive and hadronized events with multiple hard jets, in hadronic collisions. We discuss in detail the arising description of the soft, collinear and hard regions by examples in pure QCD jet-production.


Introduction
The CERN Large Hadron Collider is testing not only several suggested extensions of the Standard Model, but is simultaneously testing the theoretical description of particle collisions within the Standard Model. With the increased energy at the LHC over earlier colliders, the accurate theoretical description of several observables will necessitate the inclusion of several additional jets of hardness similar to that of the relevant lowest-order process being studied.
The all-order perturbative description included in the general-purpose Monte Carlo programs [1][2][3][4] relies on properties of soft-and collinear radiation from the lowest-order hard process, and therefore underestimates the amount of radiation of similar hardness. The shower-description of radiative corrections can be corrected above a chosen mergingscale with the full tree-level matrix elements with, e.g., the CKKW(-L) [5,6] or MLM [7] approach. The multiplicity, to which the tree-level matrix elements are required to be evaluated, will depend on the chosen merging scale. A low merging scale will correct the parton shower in much of phase space, but the multiplicity to which the tree-level matrix elements must be evaluated is then very large (and in practice the maximum multiplicity is limited).
Several alternative approaches have been developed in order to respond to the need for the description of semi-hard emissions to all orders, not just those which can be reached by matching from the shower-formalism. Some of these approaches, like Cascade [8,9] and ARIADNE [6], originate from the study of specific phase space regions at earlier colliders, where the semi-hard radiative corrections were already deemed relevant. In the formulation of Cascade, the semi-hard emissions are generated from the PDF-evolution according to the CCFM equation [10][11][12][13], while ARIADNE implements the Colour Dipole Model [14][15][16][17] of the evolutions of colour dipoles after the hard scattering.
The current study uses the new approach of High Energy Jets (HEJ ) [18][19][20], which is built on standard collinear factorisation between hard scattering matrix elements and PDFs, but calculates the hard scattering matrix elements to all orders. This is achieved within an approximation, which becomes exact in the limit of large invariant mass between all particles. This is the exact opposite limit of collinear emissions. The close similarity between the formulation of the resummation in HEJ and the normal perturbative expansion of a fixed-order calculation allows for a simple procedure [20] for matching the HEJ resummation to full tree-level accuracy, similar to the merging of a parton shower and matrix elements in a CKKW-L or MLM merging procedure.
The multi-jet predictions from HEJ are in the form of partonic final states, with no collinear enhancement of radiation. While HEJ describes the jet count and topology, jet shapes are completely ignored. The jet cones are mostly empty, except for the single parton taking up all the jet momentum, not unlike the situation of a tree-level generator.
In order to arrive at a more realistic description of the final state of the particle scattering, first a resummation of the (soft 1 and) collinear emissions of a parton shower is necessary, in order to secondly add a hadronisation step. While the description of jet profiles within HEJ is not unlike the situation at tree-level, the challenge of matching the description to a shower is completely unlike that at tree-level solved in the approach of CKKW [5,6], MLM [7] or Vincia [21]. This is because HEJ is summing its own tower of real corrections to all orders and, on top of that, also includes the leading virtual corrections in the limit of hard, wide angle emissions. The challenge of matching HEJ with a shower is to avoid double counting between the two all-order approaches.
We will present a shower-independent subtraction algorithm for matching HEJ and a parton shower. Furthermore, this paper contains a study of the effects of the parton shower and hadronisation on the predictions arising from HEJ . This is obtained for an implementation of the specific evolution of the final state according to ARIADNE, with the string hadronisation as implemented in PYTHIA [22,23]. In Section 2 we present the ingredients of HEJ necessary for the further discussion, and in Section 3 we do the same for ARIADNE. Section 4 presents the matching of the two all-order perturbative approaches. Section 5 discusses the impact of the addition of the shower and hadronisation on various classes of observables, both some very sensitive to the description of collinear radiation (the shower profile), and some which will turn out to be modified only slightly (e.g. vetos of hard jets). The addition of a parton shower on top of HEJ should extend the validity of the prediction to regions with a large ratio also of transverse scales.
Finally, in Section 6 we present our conclusions and outline some future improvements of our matching procedure.

The High Energy Jets Monte Carlo
The High Energy Jets ( HEJ) framework [18][19][20] provides a perturbative approximation to the hard scattering matrix elements to jet production to any order in the coupling, which is exact in the limit of large invariant mass between all particles. The formalism is inspired by the high energy factorisation of matrix elements (as pioneered by BFKL [24][25][26][27]), in that only certain partonic configurations are described (namely those leading in the limit of large invariant mass between all partons, also denoted Multi-Regge Kinematics -MRK ). Within the framework of BFKL, a number of kinematic approximations are applied in order to cast the cross section in the form of a two-dimensional integral equation. This also entails a number of approximations to the phase space. These approximations are avoided within HEJ . The advances in computing power allow the calculation of the cross section as an explicit integration over matrix elements of each multiplicity, with the inclusion of (an approximation to the) virtual corrections, and a simple organisation of the cancellation of IR singularities between real and virtual corrections. This allows for the construction of an approximation, which retains the logarithmic accuracy of the BFKL approach, but is more accurate when compared to the full QCD amplitudes in the phase space regions relevant for the LHC. The details are discussed in Ref. [18][19][20]; here we will briefly repeat the discussion of the points which are necessary for constructing the matching to a parton shower.
The all-order treatment in HEJ starts with the approximation to the tree-level scattering amplitude for the scattering process with flavours f 1 f 2 → f 1 g · · · gf 2 , where the final state particles are listed according to their rapidity, and f 1 , f 2 can be quarks, anti-quarks or gluons. We will call these states FKL-configurations (Fadin-Kuraev-Lipatov [25]). The scattering amplitude is approximated at lowest order by the following expression [20] M t f 1 f 2 →f 1 g...gf 2 where S f 1 f 2 →f 1 f 2 2 indicates the square of pure current-current scattering, and K f 1 , K f 2 are flavour-dependent colour-factors (which can depend also on the momentum of the particles of each flavour f 1 , f 2 , see Ref. [20] for more details). The use of flavour-dependent colour factors allows one to display explicitly the complete factorisation at tree-level of all QCD processes f 1 f 2 → f 1 f 2 into contractions of normal currents over a t-channel pole. g 2 = 4πα s is the QCD coupling, and t i is the square of the local t-channel momentum, t = q 2 i , q i = p a − i j=1 p j , where p a momentum of the incoming parton of negative z-momentum, and the momenta p i of the outgoing partons are ordered with increasing rapidity.
The effective vertex for emissions of gluons takes the form [18] This form of the effective vertex is fully gauge invariant; the Ward Identity, p j · V = 0 (j = 2, ..., n − 1) can easily be checked, and is valid for all momenta p j (i.e. not just in the MRK -limit). This allows for a meaningful approximation to the scattering amplitude to be constructed. The virtual corrections are approximated with the Lipatov ansatz [27] for the t-channel gluon propagators (see Ref. [18] for more details). This is obtained by the simple replacement in Eq. (2.1) of whereα is regulated in D = 4 + 2ε dimensions and q 2 i is the Euclidean square of the transverse components of q i . The cancellation of the poles in ε between the real and virtual corrections is organised with a mix of a subtraction and a phase space slicing (basically limiting the phase space region of the subtraction terms), such that the regulated matrix elements used in the resummation of HEJ are given by [20] M reg,f 1 f 2 →f 1 g···gf 2 The all-order dijet cross section is then simply calculated as the phase space integral over any number of gluon emissions from the initial scattering f 1 f 2 → f 1 f 2 . Matching to high-multiplicity tree-level matrix elements is obtained by reweighting the event with the ratio of the square of the tree-level matrix element (evaluated using MadGraph [28]) and the approximation to this in Eq. (2.1), both evaluated on a set of momenta derived from the hard jets only (in order to reduce the multiplicity, and therefore not to exhaust the limited number of available full tree-level matrix elements too quickly). This procedure is summarised in the following formula where n is the partonic multiplicity of the final state. The second line in Eq. (2.6) describes the matching of HEJ to high-multiplicity tree-level by a reweighting of the exclusive mjet rate projected with the operator O e mj with the ratios of the relevant tree-level matrix elements. The last line describes the inclusion of PDFs and the momentum-conserving delta-functional. We define the inclusive two-jet operator O 2j to return one if the final state contains at least two hard (p ⊥ > 35 GeV) jets; in the current study, we use the antikt algorithm with an R = 0.6, and the E-recombination scheme. Furthermore, we require that the extremal partons from HEJ are members of the extremal jets, in order to ensure that the partonic configuration matches the situation for which the HEJ resummation scheme was developed.
The partonic configurations not conforming to the ordering described above are included in HEJ by simply adding the contributions order-by-order (again using MadGraph [28]), but no all-order summation is performed of these non-FKL configurations. In the current study, we will focus on the FKL-configurations, since this is where special all-order attention is needed in order to avoid double counting in the subsequent shower. The non-FKL configurations could possibly be added to the combined HEJ+shower sample through e.g. a vetoed CKKW-L-procedure (vetoing any FKL-configuration which might arise).
The matching of HEJ to high-multiplicity tree-level accuracy is currently performed with up to four jets in the final state, limited by the time taken to evaluate the full expressions. Importantly, the description in HEJ goes beyond approximating leadingorder high-multiplicity matrix elements. As discussed, the Lipatov ansatz [27] is used to give an approximation to the virtual corrections at all orders in addition. This resums to all-orders the leading logarithmic virtual corrections to the t-channel poles.
By construction, the HEJ framework therefore provides a description directed particularly at hard, wide-angle QCD radiation. The transverse momenta of gluons emitted in-between (in rapidity) the two extremal partons can take on any value, however the subtraction suppresses the contribution from emissions with a transverse momentum less than λ. We note that although Eq. (2.5) contains the exponential of the logarithm of a momentum, it is not directly related to a Sudakov factor of a normal parton shower -firstly, the logarithm is not of the emitted momentum, secondly, the resummation is not formulated as a unitary evolution (i.e. of constant total cross section). In order to perform a matching to a parton shower, we have to define the relation between the emissions of HEJ and those of the shower. Since the emissions of HEJ populate all of phase space (in-between in rapidity of the scatterers of flavour f 1 and f 2 ), we do not want to define specific regions to populate with HEJ and with the shower. Rather, we will let both formalisms populate their respective phase spaces, but define a subtraction term for the shower Sudakov, such that double counting is avoided by reducing the probability of a certain emission from the shower by the probability that HEJ had already performed the given emission. A parton shower framework, such as ARIADNE [6], is necessary in order to evolve the partonic state of HEJ to the state of hadronisation, primarily by populating the partonic state with further soft and collinear radiation. As the shower, and also the subsequent string hadronization, relies on having well-defined colour connection between partons, we first need to briefly discuss how these are obtained from HEJ .

The Colour Connections of High Energy Jets
The colour-ordered Parke-Taylor amplitudes [29] for tree-level gg → g · · · g-scattering allow for a very neat analysis [30,31] of the dominant colour configurations in the limit of widely separated, hard gluons. The conclusion, as presented in Ref. [30,31], is that the leading contribution in this limit of Multi-Regge-Kinematics (MRK ) is provided by the colour configurations which can be untwisted to two non-crossing ladders, connecting the rapidityordered gluons. Fig. 1 (left) contains an example of a configuration contributing in the MRK -limit, and one (right) which does not. The numbering of the final state partons is according to their rapidity.
The colour connections in Fig. 1 (left) can be summarised as a134b2a, and if the point for particle 2 is moved to the left side of the same plot, then no colour lines cross. This is always possible if in the colour connection string (like a134b2a), the particles entering between the two initial state gluons are ordered in rapidity, as in the case of a134b2a.
The colour connections in Fig. 1 (right) can be summarised as a1324ba, which contain an un-ordered string between a and b, and as long as rapidity ordering of the particles is reflected in the vertical position of the particles in Fig. 1 then one cannot avoid crossing colour lines by moving the points from the left to the right side of the plot. This configuration is suppressed in the MRK -limit. Furthermore, the study of Ref. [30,31] shows that all the leading configurations each have the same limit in the MRK -limit, and indeed lead to the colour traces resulting in a colour factor C A for every final state gluon. The limit agrees with that predicted by the amplitudes of Fadin-Kuraev-Lipatov (FKL) [25].
When we pass an event from High Energy Jets to ARIADNE, we choose a colour configuration at random from the set of colour connections which are leading in the MRK -limit, and pass the event using an interface conforming to the Les Houches accord [32].

The ARIADNE Dipole Cascade
The ARIADNE program [6] is based on the colour dipole model developed by the Lund group [14][15][16][17], where gluon emissions are modelled as coherent radiation from two colourconnected partons. The general idea is best described in e + e − -annihilation into jets, where the inclusive probability of emitting a gluon from the original qq-pair is given by the well known matrix element where x 1 and x 3 are the final-state energy fractions of the quark and anti-quark respectively after the emission. A subsequent gluon emission will then come either from the dipole between the quark and the gluon or from the one between the gluon and the anti-quark, with a trivial generalization to further emissions. The splitting function in eq. (3.1) is modified slightly in the case of dipoles between gluons, giving e.g. for a gluon-gluon dipole, We note that in the soft and collinear limits we have e.g.
where z and Q 2 are the standard energy fraction and virtuality splitting variables, which gives Since a given gluon splitting gets contributions from two dipoles we obtain and we recover the standard gluon splitting function. Subsequent emissions are ordered in a Lorentz-invariant transverse momentum defined as (where s ij is the squared invariant mass of partons i and j) which, together with a conveniently defined rapidity, results in a dipole splitting function which can be approximated by The emissions are then made exclusive by introducing no-emission probabilities, or Sudakov form factors, giving the probability that there were no emissions beteen two scales, giving rise to the standard (next-to-leading) logarithmic resummation of soft and collinear divergences.
In the final state radiation, ARIADNE also includes the g → qq splitting, but in this paper we will only concern ourselves with gluon emissions and we will therefore not go into further details.
ARIADNE has a fairly unique way of handling radiation in collisions where there are incoming hadrons. In a normal parton shower one would apply a backwards evolution of initial-state splittings, and in more recent dipole shower implementations such as those in PYTHIA [1] and SHERPA [4], dipoles are defined between incoming and outgoing partons in the hard interaction. The ARIADNE program, in contrast, uses the so-called Soft Radiation Model [16], where there are dipoles between the hadron remnants and the partons from the hard interactions.
Here we will rely on the HEJ program to generate the initial-state emissions, and we will therefore not go into details of this Soft Radiation Model. Instead we will go back to the dipole splitting function in Eq. (3.7) to see how it relates to the matrix elements generated in HEJ .
The standard g → gg splitting function can be derived from ratios of matrix elements (see e.g. chapter 5 in [33]) as Looking at the dipole splitting function in Eq. (3.7), we can associate the emitted gluon to one or the other emitter, depending on which is closer. We can define e.g. 1−z = x 1 /(2−x 3 ) and in the limit where parton 3 retains most of its energy we have dy ≈ dz/z giving in the end This relates the matrix elements to splitting functions, and we will utilise this to define a subtraction term for the shower, based on the matrix elements used in HEJ . With this we can proceed with subtracting the radiation probabilities arising in HEJ from the ARIADNE dipole splittings.

The Subtracted Shower
The idea for combining the all-order resummations of HEJ and ARIADNE is to first let HEJ generate an event according to Eq. (2.6), and then let ARIADNE shower these events using a splitting function, which has been modified to subtract the effects of the resummation already included in HEJ .
The events from HEJ will consist of parton configurations where the two partons extremal in rapidity are required to be members of hard jets (where the scale of hardness is chosen in the analysis) with absolute rapidities less than a cutoff y max of, say, 5.5. The minimum transverse momenta of these extremal partons can be required to be larger than a scale not much smaller than the jet scale used in the analysis [20]. The phase space in-between these two extremal partons is populated by gluons with transverse momenta above some small cut-off λ (of order 1 GeV), below which a subtraction is applied, in order to organise the cancellation of IR divergences between real and virtual corrections. The HEJ events therefore consist of partons of any transverse momentum, and with absolute rapidities less than y max , with the transverse momentum of the two extremal partons larger than the minimum jet transverse scale.
These events are then given to ARIADNE. When ARIADNE performs a trial emission, we subtract from the ARIADNE splitting function the effective splitting function as obtained from HEJ for the one extra ARIADNE-emission, as indicated in Eq. (3.10) where the matrix elements are evaluated using the tree-level HEJ -formalism of Eq. (2.1) (with matching to high-multiplicity full tree-level matrix element, if appropriate). The ratio of the square of matrix elements in Eq. (4.1) is given by the factors included in the brackets on the last line of Eq. (2.1) -up to the effects of the momentum reshuffling in order to keep the incoming momenta on-shell after the trial emission of the shower (see below). However, in order to take properly into account the effects of the reshuffling, we evaluate the full matrix element (not just the extra factors of V µ V µ ) on the respective momenta. The square of the matrix elements is Lorentz-invariant, and the value of z is retained from ARIADNE in the proper dipole frame. This method therefore avoids the need for any Jacobians from a change of reference system. The n-particle configuration is the original HEJ -event, and the ARIADNE trial emission is added to the final state configuration for the evaluation of the matrix element for the n + 1-particle configuration with a recoil strategy, which keeps the incoming partons with zero transverse momentum: Various algorithms were devised for dividing the subtraction of the momentum of the trial emission onto the original HEJ partons, each giving very similar results for all the distributions and observables tested. We also tested the subtraction using both the tree-level and the fully regulated formalism for HEJ , and the differences are minimal (as expected, since the difference is higher order effects). We therefore choose to use just the tree-level matrix elements, since they evaluate slightly faster (and one avoids complications from the reshuffling moving momenta across the regulating parameter λ).
No subtraction is applied if the ARIADNE trial emission is outside the phase space where HEJ would emit gluons, e.g. if the trial emission would result in a configuration where the most forward or backward gluon is softer than the jet scale. This region of emissions collinear to the incoming beam is a domain which is populated purely by the shower.
The described procedure ensures that all the emissions performed by ARIADNE have been properly subtracted to avoid double-counting. But it should also be noted that the veto algorithm 2 used to order the emissions in ARIADNE, automatically ensures that only the subtracted splitting kernel is exponentiated in the Sudakov form factors.
The ordering of emissions in (invariant) transverse momentum in ARIADNE is of some concern. As HEJ produces also some very soft gluons (∼ 1 GeV), there may arise situations where a soft gluon ends up fairly close to a hard one. In this situation ARIADNE would normally first emit some hard, collinearly enhanced, emissions before the softer ones, but now, the dipole between the soft and hard gluon from HEJ becomes too small to allow the full range of collinearly-enhanced radiation. The effect is that the very hard jets in our matching procedure will become a bit too narrow. This issue will be discussed further when we look at the resulting jet shapes in the next section.

The Algorithm
What follows here is a step-by-step description of the algorithm as implemented in the HEJ /ARIADNE interface.
1. Generate a partonic state with HEJ , using a given cutoff in transverse momentum for the extremal partons, and a small transverse momentum cutoff for soft gluons.
2. This state is then sent via PYTHIA to ARIADNE using the Les Houches interface.
3. ARIADNE then sets up its internal dipole event record and starts the dipole cascade starting from a scale given by the largest transverse momentum of any of the HEJ partons.
• Calculate the kinematics of the gluon and the approximate ratio of matrix elements corresponding to the splitting function used r A = |M n+1 | 2 / |M n | 2 according to Eq. (3.10).
• This gluon is then sent to HEJ using a call-back function where it is inserted in the original partonic state and calculate the ratio of the corresponding matrix element and the original matrix element, r H = M t n+1 2 / M t n 2 .
• Veto the emission with probability r h /r a . This is equivalent to using the subtracted splitting function D(p 2 ⊥ , y) − D subt (p 2 ⊥ , y) in the shower. In addition as this is done inside the veto algorithm, it is the subtracted splitting function which is resummed in the Sudakov form factor.
If the emission is outside the extremal jets, it is kept only if it has a transverse momentum below the cutoff for extremal jets. Furthermore if the emission is a finalstate splitting of a gluon into a qq pair, it is simply kept as such emissions cannot be produced by HEJ . Note also that any gluon emission with a transverse momentum below the phase space slicing parameter λ in HEJ will be kept.

After the dipole cascade in ARIADNE, the partonic final state is hadronized by
PYTHIA, where also the decay of unstable hadrons is performed.
A few notes are in order.
• First it is clear that we avoid double-counting all over phase space. For each emission in the shower, only those which could not have been produced in HEJ are kept without change. For those which do correspond to a HEJ emission, the corresponding emission probability is subtracted, basically only retaining the collinear pole, while the soft gluons (above the HEJ cutoff) are effectively vetoed by the same subtraction algorithm.
• The emissions which are affected by the Soft Radiation Model in ARIADNE are those related to dipoles connecting the proton remnants, and these will give gluons outside the extremal jets, and are therefore forced to be below the extremal jet cut. Hence, possible effects of the unconventional treatment of initial state radiation in ARIADNE are minimized, and will not lead to extra jets.
• It can be argued that if a gluon emission which is accepted in ARIADNE corresponds to a partonic configuration which is not possible to produce by HEJ , further emissions from the dipoles connected to this gluon should be unrestricted. Clearly if the gluon was collinear to a HEJ -jet, the emissions from the dipole between that jet and the gluon should not be subtracted. On the other hand, emissions from the other dipole should still be subtracted, and the subtraction in the first dipole should be small since it would always correspond to collinear emission. We have tried two scenaria: one where all emissions are subtracted, and one where only emissions from dipoles between original HEJ partons are subtracted (this is the default). The differences turned out to be almost negligible.

Results
In this section we compare first the effective splitting function from HEJ with that of ARIADNE, in both the soft and the collinear regions. We then study in details the effects of the shower on a sample event-configuration from HEJ , before moving on to a study of the resulting shower profiles, which are compared to ATLAS data. Finally, we discuss the impact of the shower on a few observables discussed in e.g. Ref. [20,[35][36][37], which are sensitive to the description of hard, radiative corrections.

Comparison of Splitting Functions
In Fig. 2 we compare the average value of the splitting functions D/z calculated in ARIADNE and with HEJ (Eq. (4.1)), after division with α s (to remove the effects of a different evaluation of α s in ARIADNE and HEJ ), as a function of the transverse momentum of the trial emission. The average is over each bin in the distribution, for an unweighted HEJ event sample of 26 GeV dijets, where at least one reconstructed jet above 30 GeV is required in the post-shower analysis. We have used only the subset of ARIADNE trial splittings where the original dipole consists of partons from HEJ , and where the trial splitting is at least a distance R = 0.5 away from any of the original HEJ partons. This is to test only the soft (but not collinear) description in the two frameworks. Generally, the effective splitting functions for wide-angle emissions are quite similar in HEJ and ARIADNE. However, we see that, on average, the ARIADNE splitting function is larger than the effective splitting function from HEJ only for emissions of transverse momenta less than about 10 GeV, and then only by a small amount. This means that effectively, wide-angle emissions harder than about 10 GeV are automatically vetoed in the subtraction mechanism, since the probability for emissions is larger in HEJ than in ARIADNE. We see by comparing the explicit numbers that the HEJ splitting function (after division by α s ) tends to C A πk 2 ⊥ in the MRK region of semi-hard, wide-angle emissions. This is the result of the full tree-level QCD (and the pure BFKL formalism).
In Fig. 3 we compare the average value of D/z in ARIADNE and that arising from HEJ for a sample of ARIADNE trial emissions (again from original HEJ partons) of harder than 10 GeV in transverse momentum, as a function of the distance r (in (rapidity,φ)) to the nearest HEJ parton. Here we see that at small r, the ARIADNE splitting function is an order of magnitude larger than the subtraction term from HEJ . The HEJ subtraction term is (on average) larger than the (average) ARIADNE splitting function only for r > 0.6, which in this case was also chosen as the jet size parameter in the anti-kt jet clustering. The change in behaviour for the effective HEJ splitting function at the jet-size parameter is because of the use of two different effective emission vertices, depending on whether or not the additional HEJ -emissions are collinear to the partons extremal in rapidity.  Figure 4. Top: A Lego R -plot of the momentum configuration from HEJ, and the average outcome from 10000 showers of the same HEJ-event. Bottom Left: the 4 hard (p ⊥ > 60 GeV) jets resulting from the jet clustering, before and after the showering. The radii of the circles are proportional to the momentum of the jet. Bottom right: The shower profile of two of the jets, as a function of r = ∆y 2 + ∆φ 2 . The dotted lines correspond to the initial partons in each case.

The Description of Jet Structure
Based on the analysis of the previous subsection, one would expect that the effect of the ARIADNE showering of the HEJ events would be to radiate mostly in the immediate surroundings of the existing HEJ partons, whereas hard wide-angle emissions from ARIADNE should be suppressed. Fig. 4 illustrates the effects of the shower on a sample event from HEJ , where the average outcome of 10,000 showers is plotted along with the initial partons. The top Lego plot illustrates clearly the smearing effect of the shower. The effect on the momentum of the reconstructed hard jets (defined with the anti-k ⊥ algorithm and R = 0.6 and with a transverse momentum larger than 30 GeV) is shown in the bottom left plot, where the radii of the circles is proportional to the momentum of the reconstructed jets. After showering, the jets move only a modest amount in (y, φ) and there is only a small change in the momentum. In this particular sample event, the momenta of three of the jets decreases slightly (from {64, 68, 79} GeV to {57, 63, 76} GeV respectively) while the momentum of the other one increases very slightly from 67 GeV to 69 GeV. Further details can be seen in the jet profile plot in the bottom right of Fig. 4, where the variable plotted is the fraction of the jet's transverse momentum found at a radius r from the jet center: where p ⊥ (r) is the summed transverse momentum in bins of radius r and R is the jet radius used in the anti-k ⊥ algorithm [38].  Figure 5. Jet profiles as measured by ATLAS [39] for different bins of jet transvese momenta (in GeV) compared with our results (for λ = 2 GeV) (full lines). Also shown are the results from PYTHIA Perugia 10 tune with (dashed) and without (dotted) multiple interactions.
In Fig. 5 we present our fully simulated pure QCD inclusive one-jet events (requiring at least two jets of 26 GeV from HEJ, at least one 30 GeV jet after showering) compared to a recent measurement of jet shapes by the ATLAS collaboration [39]. Note that our program does not include any underlying event simulations. We therefore also compare our results with a recent tuning of the PYTHIA program [40] (Perugia 10 [41]) with and without multiple interactions to estimate such effects. As expected, the underlying event mainly contributes to jets with small transverse momenta in the outskirts of the jets.
We find that our jets are very similar to the PYTHIA ones (without multiple interactions) for small transverse momenta, which is a good indication that our matching works as expected. However, at larger p ⊥ our jets seem to be a bit too narrow as compared to data and PYTHIA. The reason for this is the very soft gluons fairly close to the hard ones, which are included in the HEJ resummation. Such situations result in dipole masses which are too small to allow for enough radiation from the hard gluons within ARIADNE.
The problem is that the cascade in ARIADNE becomes somewhat un-ordered -the soft gluons should normally be radiated after the harder collinear ones as noted in the previous section. This is a problem also in other matching procedures, as noted in [42], especially for the so-called MLM [7,43] and PseudoJet [44] algorithms. Indeed, in [39] the ATLAS collaboration see a tendency in the ALPGEN generator, which uses MLM-merging, to produce too narrow jets at large transverse momenta.
We have here run with a lower cutoff (or regularisation parameter) of the transverse momentum in HEJ of 2 GeV, and we have noted that the effect is enhanced if this cutoff is lowered further. Increasing the regularisation parameter further will lead to unacceptably large cancellations between positive and negative weight events. Negative events do not arise in HEJ when the regularisation parameter is allowed to be much smaller. To increase the cutoff above 2 GeV is therefore numerically unacceptable, and in any case it would only reduce the effect, not remove it completely.
One could debate whether the subtraction should be applied to all ARIADNE trial emissions, or only to emissions from ARIADNE dipoles made up of partons from HEJ . In practice this makes very little difference to the results -certainly, the difference between the two choices are inseparable when presented as in Fig. 5 (and Figs. 6-7). We therefore choose to apply the subtraction only in the first emission, in order to limit the amount of momentum reshuffling performed.

Impact on Multi-Jet Observables
Finally, we are ready to study the impact on a few observables of the addition of a shower to the resummation of HEJ . We choose to do so for the observables from a study by ATLAS [45], for which the prediction from HEJ were presented in Ref. [20]. We do not here present a full analysis of the uncertainties from scale and pdf-variation, as performed in Ref. [20], but choose to use as factorisation and renormalisation scale the maximum transverse momentum of any jet, and include the β 0 log-terms, as discussed in Ref. [20,46,47].
In Fig. 6 we compare the average number of jets in a dijet-sample (anti-kt, R = 0.6) with a transverse momentum greater than 30 GeV (and rapidity less than 4.5) as a function of the rapidity difference between the most forward and backward hard jet. These two jets are furthermore required to have an average transverse momentum greater than 60 GeV. The black line is the partonic prediction from HEJ (based on just the FKL-configurations discussed in the current paper). The blue dashed line is obtained after the further shower and hadronisation by ARIADNE. The changes (due to the showering) in the number of hard jets is very small indeed, but increasing in significance with the rapidity span and the average number of jets. At a rapidity span of 6 units, the showering and hadronisation  Figure 6. The average number of jets vs. the rapidity difference between the most forward/backward jet as obtained with HEJ, and with HEJ+ARIADNE. The total effects of shower and hadronisation on this observable are minor.
leads to a reduction in the average number of hard jets from roughly 3.6 to slightly above 3.5. For small rapidity spans (∆y < 2), the showering and hadronisation leads to a minute increase in the average number of hard jets, whereas for larger rapidity spans the effect is a larger decrease.
The trend and small effect is found again in the prediction of the gap fraction in Fig. 7, defined here as the exclusive dijet rate over the inclusive dijet rate, as a function of the rapidity span between the most forward/backward pair of hard jets (p ⊥ > 30 GeV). The effects seen in Fig. 6 are repeated here -for small rapidity spans (∆y < 2), the showering and hadronisation leads to a minute decrease in the gap fraction, but for larger rapidity spans, the effect is a small increase.

Outlook
We have here presented a procedure for obtaining exclusive hadronic final states within the framework of HEJ , describing jet production in high-energy hadronic collisions. To do this we have developed a new kind of matching scheme dealing with two separate all-order summations, where the hard jets (and soft contributions) are first generated in HEJ , followed by a final-state shower dressing up the jets with further subtracted soft and collinear radiation. Double-counting is avoided by carefully removing the appropriate soft divergences in the parton shower in a way such that the collinear divergences are still correctly exponentiated.  Figure 7. The gap fraction (exclusive dijet over inclusive dijet rate) as a function of the rapidity span between the most forward/backward hard jet, as obtained with HEJ, and with HEJ+ARIADNE. The total effects of shower and hadronisation on this observable are minor.
We have validated our procedure both technically, by making sure that the subtracted splitting functions look reasonable, and also by comparing the resulting jet shapes with recent measurements by the ATLAS experiment. For the latter we find very good agreement with data. We have also shown that previous predictions published for the HEJ model at parton level are fairly insensitive to the addition of parton showers and hadronization for reasonable choices of jet definitions.
We have, however, found that our results are somewhat sensitive to the very soft gluon emissions also included in HEJ . The reason can be understood from the fact that ARIADNE is forced to work in an un-ordered way. Some soft emissions from HEJ , would normally be emitted by ARIADNE after the emission in the collinear region. This indirectly reduces the phase space available for collinear emissions, giving jets which are a bit too narrow, especially at high transverse momenta. The situation is similar to the problems found in [42] when investigating the so-called MLM [7,43] and PseudoJet [44] merging procedures.
In the future we will investigate different ways of solving this problem, and we have already identified two different solutions. One is based on the CKKW(-L) [5,48] merging procedure. Here the states produced by HEJ would first be reweighted by Sudakov form factors produced by the properly subtracted parton shower, whereafter the shower can be added in the same way as presented in this article. Here we plan to use a recent CKKW-L implementation in the PYTHIA generator [49], which will also allow us to add underlying events from the multiple-interaction model in PYTHIA.
Another option we will pursue is related to the concept of primary or back-bone gluons introduced in [50] and [51]. There it is found that the main real-emission contribution to the cross section according to BFKL is given by gluons which are ordered in both positive and negative light-cone momenta, while other emissions can be conveniently summed over. In our case that would mean that soft gluons produced in HEJ that are not ordered in this fashion would simply be removed in a kind of post-factum resummation. After that the parton shower, this time unsubtracted, can be applied with a simple phase space veto, allowing only gluons emissions which are un-ordered in light-cone momenta, in the same way as has been done in [52] and [53]. Nevertheless, we have found that the effects of the non-ordering of the parton shower are small, and the current implementation is therefore an important step forward in the description of multi-jet events in hadronic collisions. This is the first framework that will properly simulate complete multi-jet final states with proper resummation of also semi-hard emissions at high energies, and will provide an important alternative to the conventional event generator approaches, which are all based on DGLAP resummation only.