Exploring minijets beyond the leading power

Exploring minijets beyond the leading power

Abstract

The crucial parameter of the current Monte Carlo models of high energy hadron-hadron interaction is the transverse momentum cutoff for parton-parton interactions which slowly grows with energy and regularizes the cross section. This modification of the collinear factorization formula goes beyond the leading power and thus a natural question arises if such cutoff can be extracted from a formalism which takes into account power corrections. In this work, we consider the High Energy Factorization (HEF) valid at small and a new model, based on a similar principle to HEF, which in addition has a limit respecting the Dokshitzer-Dyakonov-Troyan formula for the dijet momentum disbalance spectrum. Minijet cross section and its suppression is then analyzed in two ways. First, we study minijets directly in the low- region, and demonstrate that higher twist corrections do generate suppression of the inclusive jet production cross section though these effects are not leading to the increase of the cutoff with incident energy. Second, we consider hard inclusive dijet production where Multi Parton Interactions (MPIs) with minijets produce power corrections. We introduce an observable constructed from differential cross section in the ratio of dijet disbalance to the average dijet and demonstrate that the region is sensitive to the cutoff in the MPI minijet models. The energy dependence of the cutoff is reflected in the energy dependence of the bimodality coefficient of the distribution. We compare calculated from , where one can conveniently control MPIs by the program parameters, and HEF for a few unintegrated gluon distributions (UGDs). We find that the energy dependence of is very sensitive to the particular choice of UGD and in some models it resembles predictions of the Monte Carlo models.

1 Introduction

The rise of the total cross section in hadron-hadron collisions with energy is driven by minijets, i.e. jets with relatively low transverse momenta , of the order of a few GeV. From the QCD point of view, this growth is attributed to the rise of the parton density inside a hadron with decreasing value of longitudinal momentum (or increasing CM energy of the collision). At leading order (LO) the colliding partons (mostly gluons at high energies) produce two final state partons and give rise to two jets. The problem is, however, that the resulting QCD expression is divergent when . This is of course not a paradox, simply the very low region is out of the applicability of the formalism operating on partons (i.e. collinear factorization theorem [[1]], see Section 2.1). Thus one has to introduce a cutoff, , above which the formula makes sense [[2]]. This is the starting point for so-called minijet models and models including Multi Parton Interactions (MPIs) which are at the heart of modern event generators like [[3]] or [[4]]. The basic idea is that since the minijet cross section can easily exceed the total cross section (for low values of ), the ratio , with being a non-diffractive inelastic component of the total cross section, gives an average number of hard binary collisions per event, i.e. MPI events [[5]]. The is a free parameter of the model. Typically, one does not implement the sharp cutoff but rather a smooth transition regulated by another parameter . Comparison of the models with MPI with the data indicates that hadron production at small impact parameters grows in these models too fast with increase of . Also the cross section of the interaction at large impact parameters grows faster than indicated by the data on profile function of the interaction leading to cross section much larger than the experimental one [[6], [7]]. The typical resolution is to let the parameter to be energy dependent , slowly growing with .

We see that there are two general features of the minijet models: (i) an existence of a scale above which perturbative collinear factorization applies and (ii) the MPI-type events. Let us note, that in a typical minijet model these features are related in the sense that the MPI models require the property (i), which in turn, on itself, can be viewed as a consequence of color confinement [[5]] and is independent on MPIs. However at the LHC energies one needs a cutoff on the scale of and growing with , making it unlikely that the cutoff could be solely non-perturbative effect. On the other hand, the MPIs became a separate branch of high energy physics, not necessarily related to minijets. For example one of the typical direct MPI signals is expected to be a four-jet hard event with back-to-back dijets [[8]]. On the theory side the MPI physics is a very complicated subject and most often is restricted to the double parton scattering (DPS), see [[9]] for a comprehensive review. So far no proof exists of the QCD factorization theorem for DPS, although recently a progress has been made towards the proof of DPS in the double Drell-Yan process [[10]].

In this work we have undertaken an attempt to understand the origin of the cutoff and the low suppression within the perturbative QCD. As we will discuss later, the application of the cutoff to the collinear factorization formula extends it beyond the leading power. Thus, any approach which aims to explain the cutoff has to incorporate higher twists. Non-negligible power corrections may be generated by large transverse momenta of incoming partons entering the hard collision, as compared to the hard scale of the process. These features are naturally incorporated in the High Energy Factorization (HEF) (or -factorization) approaches [[11], [12], [13], [14], [15]]. There, the transverse momentum of the dijet pair is no longer zero, but equals to the sum of the transverse momenta of the incoming off-shell gluons. The distribution of these gluons in longitudinal and transverse momenta is given by so-called Unintegrated Gluon Distribution (UGD). Thus, in principle, the cutoff on the jet is related to the behavior of UGDs in transverse momentum which, in the low limit, is given by Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation [[16], [17], [18]] or some BFKL-type evolution. Furthermore, the gluon emissions with small transverse momenta are suppressed by the Sudakov form factor. In fact, for some UGD models [[19], [20]] the transverse momentum of the gluons is generated by the Sudakov form factor and the standard Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution. This is somewhat similar to the soft gluon resummation [[21]] technique which was used in [[22], [23]] to build an eikonal minijet model which does not require a cutoff (but it is suitable only for the total cross section).

The strategy for our paper is as follows. Using the HEF for inclusive dijet production we shall perform two independent studies of the cutoff:

Study 1.

A direct study, where we calculate the spectrum for and see if there is a suppression and determine its energy dependence.

Study 2.

An indirect study, where we analyze the hard dijet production with and look for an observable which is sensitive to power corrections which would come from MPIs in minijet models.

As for the study 1, the issue of a direct access to the cutoff within an approach involving an internal is actually known in the literature. In [[24]] it was shown that indeed such approach can produce suppression, which has roughly the correct energy dependence. There is however an important difference to our study 1. We use HEF of [[12], [13], [14], [15]] which factorizes the cross section into UGD and a genuine off-shell hard process which extends collinear minijet formula beyond leading power. In [[24]] the minijet production was considered in the sense of a chain of emissions which does not have hard process. It is rather suitable for constructing shower-like Monte Carlo program [[25]] that can be used to study particle production [[26]]. More precisely, in [[24]] the authors considered a modification of the Catani-Ciafaloni-Fiorani-Marchesini (CCFM) [[27], [28], [29]] evolution, so-called linked dipole chain model [[30]], in which any emission in the chain can contribute a minijet (the emissions are unordered in transverse momenta and thus following this logic any sub-collision in the chain can be considered as ‘hard’). On the contrary, in HEF, we require that the large enough hard scale is present that distinguishes the hard process from the chain of remaining emissions. Since this hard scale is identified with the jet the two directly emitted partons should be actually considered as hard jets, not the minijets. In the first approximation hard jets are produced back-to-back and described by the leading power collinear approach, which does not feature any suppression factor. We will see this feature in our calculations when we compute the spectra in the small region from HEF. That is, we will find no suppression in the spectra of the type present in the minijet models. Nonetheless, it does not mean that there are no minijets in HEF. In fact, HEF takes into account additional emissions visible as the jet imbalance, and thus as power corrections.

The above motivates the study 2, which concentrates on the indirect access to minijets in HEF. We introduce an observable related to the dijet imbalance , which is sensitive to minijets. Specifically, we shall consider the cross section differential in the ratio , of to the dijet average . We will check actual sensitivity of this observable on minijets, in particular on cutoff, using and then we shall compare them to similar calculations in HEF models. Next, we introduce bimodality coefficient which characterizes the spectrum. We observe that the energy dependence of this coefficient is very sensitive to the particular minijet model. We will see that some of the UGDs used in HEF give energy dependence similar to the one coming from the minijet models in . This would then indirectly confirm the statement from [[24]], but in a way that can be confirmed experimentally when such observable is measured.

Our work is organized as follows. In Section 2 we systematically review theory behind minijets. First, in Subsection 2.1 we review the collinear factorization for the minijet production and then, in Subsection 2.2, we describe in details how the cutoff is introduced. In Subsection 2.3 we review the HEF and discuss its relevance to minijet cross section. In particular, we shall explain that the leading twist limit of HEF does not reproduce the result of Dokshitzer-Dyakonov-Troyan (DDT) [[31]] for the dijet momentum disbalance. Therefore, in the next Subsection 2.4 we construct a model similar to HEF but having the DDT limit. In the following sections we will turn to numerical simulations. First, in Section 3 we shall describe in some details the process under consideration, kinematic cuts, etc. in order to unambiguously define the observables. Later, in Section 4 we will analyze the inclusive dijet spectra in the low region in order to see whether the suppression is produced in HEF and the DDT-based model we constructed in Subsection 2.4 (study 1). Finally, in Section 5 we will turn to hard inclusive dijets and study the minijets as a power correction (study 2). We will summarize and make our conclusions in Section 6.

2 Minijets in selected approaches

2.1 Collinear factorization and soft gluon resummation

The starting point for a typical minijet model is the collinear factorization formula, which however has to be modified. In this introductory section we review this issue in a more quantitative way.

The QCD collinear factorization theorem (see e.g. [[1]] for a review) expresses the cross section for hard dijet production as

(1)

where , are integrated parton distribution functions (PDFs) for a parton inside a hadron , and is a partonic, fully differential, cross section which can be calculated order by order in perturbation theory. In general the partons can be quarks and gluons, including heavy quarks. The phase space cuts necessary to define a jet cross section (i.e. a suitable jet algorithm) are hidden inside the partonic cross section. The hard scale is the largest scale in the problem and is typically taken to be the average transverse momentum of the jets, . The remainder, i.e. the higher ‘twist’ corrections in (1) are suppressed by the powers of the ratio , where is the largest of some other scales present in the problem, e.g. heavy quark masses, dijet disbalance, etc.

Since the purpose of this work is to study minijets, let us restrict to the semi-hard jets having transverse momenta . In addition, we are interested in the total CM energies being much larger then this scale. For such regime the factorization theorem (1) starts to fail. Two major sources for this are various large logs (containing ratios of very different scales) and power corrections which are no longer small.

Certainly, the formula (1) would be perfectly valid for fixed and , but obviously this is not the case for minijets. In order to illustrate the problems more quantitatively, let us consider a cross section (1) as a function of the disbalance between the jets, , when . To leading logarithmic accuracy it is given by the formula due to Dokshitzer-Dyakonov-Troyan, the so-called ‘DDT formula’1 [[31]]

(2)

where is a ‘Sudakov’ form factor for a parton (for the original Sudakov’s form factor in QED see [[32]]). It can be thought of as a probability for the parton to evolve between the scales and without any resolvable emissions. We shall give the explicit formula later (see Subsection 2.4, Eq. (33)), for now let us just mention that

(3)

for being the lowest scale in our problem. Let us remark, that the relevant DDT formula in [[31]] was actually derived for a production of hadrons in hadron-hadron collision, and thus it contained fragmentation functions which accompanied the form factors , in (2). For the purpose of this paper we have adjusted that formula for dijets by setting the fragmentation functions to be the delta functions. Let us note, that due to the listed properties of the Sudakov form factors, this formula reduces to (1) when integrated over the jet disbalance . Since the appearance of the DDT formula a lot of effort has been put into improving the accuracy of perturbative predictions for such semi-inclusive observables. In particular so-called Transverse Momentum Dependent (TMD) factorization theorem has been established for certain processes [[1]]. We shall discuss these at the end of this section and for the purpose of the present discussion we shall stick to the leading-log formula (2).

In case of minijets, the formula (2) looses its accuracy as now can be easily of the order of (which is the average of the jets). This can be seen by inspecting the derivative in (2) as a function of , for example in the pure gluonic channel:

(4)

This distribution is plotted in Fig. 1 as a function of for fixed and , and two values of (note that for simplicity we have used the same values of entering both PDFs in (4)). In this presentation we use the leading order GRV98 PDF set [[33]] (we explain the reason for using this PDF set in Section 3). We see that the characteristic , let us call it , generated by the density is large comparing to the average of minijets so that (left plot in Fig. 1; may be defined for example as the value for which the distribution has a maximum, although median would probably be a more realistic estimate). For comparison, we plot the same distribution for hard jets (right plot in Fig. 1) with . For the latter, the ratio becomes much smaller than the unity and the situation improves with increasing scale. To summarize, the power corrections cannot be neglected for minijets and one has to necessarily venture beyond leading ‘twist’ to account for minijets. Let us remind, that the formula (2) is a more ‘exclusive’ version of (1) and the condition that we can neglect the power corrections is actually a condition necessary to obtain (1) when the integral over is performed.

Figure 1: The density entering the DDT formula for two different values of and two fixed values of (left) and (right). The PDF set used here was GRV98 [[33]].

There is yet another source of errors in the DDT formula, namely the sub-leading logs. Actually, in its original formulation the DDT formula was written for processes with only two hadrons such as for instance the Drell-Yan process [[34]]. Assuming strong ordering in the transverse momenta of emitted gluons one obtains a formula similar to (2) but with two Sudakov form factors instead of four (and of course with an appropriate hard partonic cross section relevant to Drell-Yan process). However, the strong ordering in transverse momenta for the soft gluons is a too strong assumption and gives a non-physical suppression in the low limit. The improved approach for Drell-Yan pairs was proposed in [[21], [35], [36]] which resums the soft gluons (thus the approach is often called the ‘soft gluon resummation’) using the impact parameter space conjugate to transverse momenta. As a result one finds a flat distribution at small rather than an exponentially suppressed cross section. Soft gluon resummation extends beyond the leading logarithm, but still more general approach exists, so called Transverse Momentum Dependent (TMD) factorization (see e.g. [[1]]). It is a rigorous factorization theorem of QCD and is valid to leading power in the hard scale. It is important to note, that this theorem is valid for processes with at most two hadrons. Thus the most complicated processes are Drell-Yan process [[37]] and semi-inclusive deep inelastic scattering [[38]]. The theorem is violated when more hadrons are present [[39]], thus it fails for example for jet production in hadron-hardon collision. However, although the TMD factorization is not a strict leading-power theorem holding to all orders in , it has been shown in [[40]] that it holds to next-to-logarithmic accuracy for the latter case.

Before discussing the power corrections to (2) let us make some general comments. The twist corrections to deep inelastic structure functions, i.e. the corrections with being the photon virtuality, were studied long time ago in the context of the operator product expansion (OPE) [[41]] and using Feynman diagrams [[42]]. While OPE is very general, it becomes very complicated for more exclusive processes (see e.g. [[43]] and [[44]]). As for jet production in hadron-hadron collisions no higher twist factorization exists (see also a discussion of power corrections coming from heavy quarks in the end of this subsection). On the other hand there are approaches which take into account all power corrections of a certain class. At very large energies the logs of the form , where is a fraction a hadron longitudinal momentum carried by the parton, become large and can be resummed by means of the BFKL equation. Let us note, however, that it is often arguable if such logs should be resummed at currently achievable energies, as most of the observables measured at LHC can be explained using collinear factorization supplemented by the DGLAP-type parton showers. Nevertheless, the BFKL formulation leads to HEF, which as mentioned in the Introduction resums the power corrections of the form . We shall describe HEF in more details in Subsection 2.3.

For completeness let us discuss a special case when . For the case of the Drell-Yan process this kinematic region was studied in [[45], [46]], before the DDT formula was established. The corrections of this type can be obtained calculating explicitly additional emission by means of process, away from the singular (soft and/or collinear) region. In particular, the HEF partially recovers this perturbative limit for certain UGDs.

Finally, let us make some comments on the power corrections coming from the heavy quark masses. Actually, they can be explicitly taken into account in the hard cross section, order-by-order. The problem is however, that by doing so the cross section becomes infra-red unsafe for large , i.e. we shall encounter logs of the type where is the mass of a heavy quark . This problem can typically be addressed by so-called general-mass scheme, which supplements the hard cross section with a proper subtraction terms (see [[47]] for a general proof and [[48]] for a formulation for jets in DIS at NLO). However, for jets in hadron-hadron collisions there is a problem with the cancellation of soft singularities when incoming lines are massive [[49]] and thus the power corrections are unlikely to be controlled using the general-mass schemes. We shall ignore all these complications as we will be focused on pure gluonic contributions, which should dominate at high energies.

2.2 Singularity and soft cutoff

In this section we shall discuss in detail the concept of the soft transverse momentum cutoff. We shall restrict our considerations to gluons only. This is done for two major reasons. First, the gluons dominate at high energies and this is sufficient to illustrate all the effects we analyze in the paper (we do not aim at giving any predictions or comparisons with data). Second, later on we shall make comparisons across models including HEF, which is basically restricted only to gluons dominating at high energies. In principle one could consider off-shell quarks, but the subject is still poorly developed and would unnecessarily complicate our study (see [[50], [51]] for selected recent results).

Let us start by writing LO contribution to (1). We parametrize the momenta of hadrons as

(5)

where and is the CM energy squared. The kinematics of the hard subprocess is

(6)
(7)

with momentum conservation . Obviously , are directly related to rapidities in the following way

(8)

with . Due to the transverse momentum conservation both outgoing jets have exactly the same transverse momentum . In what follows we shall simply use notation for brevity. In the above kinematics, the cross section can be calculated as

(9)

where the amplitude squared and averaged/summed over spin and color reads

(10)

Typically, as the hard scale one chooses the of the jets. From (9) we see that the cross section diverges like

(11)

In the pioneering work [[5]] the MPI model was constructed with modified to remove this singularity by defining

(12)

where is the model parameter we have briefly discussed in the Introduction. For example in version 8.1 of it is defined as

(13)

for standard settings (including pre-determined PDF sets to be used by default). Let us mention, that the MPI model and the entire event generation procedure in is very complex, much more then the simple Eq. (12). Nevertheless Eq. (12) constitutes one of the core building blocks of this powerful program.

The spectrum of minijets within the presented model should exhibit a strong suppression for small , slowly growing with energy. It is interesting to ask if such a suppression could be directly observed. Putting this question aside, we will simply calculate (see Section 4) the inclusive dijet production in the small region using and compare with the minijet spectrum . There are a few interesting features of this calculation (thought to be more realistic than (12)) which will be discussed later.

2.3 High Energy Factorization

Let us now discuss how the power corrections in (2) can be taken into account in -factorization (we use the terms ‘high energy factorization’ and ’-factorization’ interchangeably in the present work, although both terms have different origin).

In -factorization the cross section is calculated as a convolution of so-called unintegrated gluon distributions (UGDs) and an off-shell matrix element. UGDs depend not only on longitudinal momentum fractions , but also on the transverse momenta of the gluons – a feature neglected in the collinear factorization due to the power counting. For the first time -factorization was used in [[11]] for inclusive jet production at high energies using basically process . Let us note that the process does not exist when the incoming partons are on-shell and collinear, but it appears at lowest order in the factorization approach. Later, a similar idea (originally called HEF) was used to compute heavy quark production [[12], [13], [14], [15]] by means of a gauge invariant matrix element for which was extracted from the Green function utilizing suitable eikonal projectors. The UGDs were assumed to undergo BFKL evolution. A natural step forward was to adopt the HEF to account for jet production processes at high energy. Thus, the HEF has been extended to all channels [[52]], including gluons. At small the forward jets are especially interesting. They can be treated in a limiting case of HEF, where one of the gluons becomes on-shell [[53], [54], [55]]. In this approximation, this gluon is treated as a ’large-’ gluon and is assigned a standard collinear PDF. In the Color Glass Condensate (CGC) approach [[56]] a similar idea was used to study forward particle production in saturation domain and exists under the name of the ‘hybrid’ formalism [[57]]. In fact, the hybrid version of HEF can be derived from CGC in the dilute limit [[58], [59]]. Several observables relevant for LHC have been calculated within the hybrid HEF, see Refs. [[60], [61], [62], [63], [51]]. In the present work we are not concerned with forward jets thus we shall not use the hybrid version of HEF, but the original one with two off-shell incoming particles.

A)

B)

Figure 2: A) Schematic representation of the factorization formula (14) B) The hard gauge invariant tree level off-shell process expressed in terms of a matrix element of straight infinite Wilson lines, with the slopes being (top) and (bottom). The blue blob on the r.h.s. denotes a standard QCD contribution with four and triple gluon vertices. Only planar (color-ordered) diagrams are shown.

The factorization formula for HEF reads (including only gluons)

(14)

where , are UGDs for hadrons and is the partonic cross section build up from the gauge invariant amplitude (Fig. 2A). The momenta of the off-shell gluons have the following form relevant to the high energy approximation:

(15)

The off-shell partonic cross section is defined by a reduction of the Green’s function, where the off-shell legs and are contracted with eikonal projectors proportional to and . Unlike the amplitude used in original HEF, the gluonic off-shell hard process cannot be just calculated from the standard Feynman diagrams in a gauge invariant way. There are number of ways this can be done in consistency with the high energy approximation used to define the hard process. First, one can include the bremsstrahlung from the lines to which the hard process is attached. At high energies those lines are eikonal. Such idea was used in [[52]] to calculate and later in [[64]] a general method for helicity amplitudes as well as numerical algorithm for any number of partons was developed. Second, in the approximation used to derive (14) the gauge invariant amplitude for is equivalent to the Lipatov’s vertex [[65], [66]] in the quasi-multi-regge kinematics. A more general approach is to consider matrix element of straight infinite Wilson line operators with the polarization of the off-shell gluon identified as the Wilson line slope [[67]]. This method can be also used beyond the high energy approximation [[68], [69]]. Finally, a method generalizing the BCFW recursion [[70], [71]] to the off-shell case is also available [[72], [73]]. Although the Lagrangian method of [[65]] is the most general, in practical computations, especially for multiple external legs, the other mentioned methods are more efficient. For the hybrid version of (14) a very efficient method of calculating helicity amplitudes for was found in [[74]]. Some other applications and different ways of calculating were given e.g. in [[52], [75]]. Moreover, many other studies have been done using -factorization, see for example [[76], [77], [78], [79], [80], [81], [51]].

The partonic cross section in (14) is defined as

(16)

where is the two-particle phase space while is the amplitude squared for the gauge invariant off-shell process discussed above. Using the method of [[67]] it can be calculated as follows. First, the amplitude is decomposed into the color-ordered amplitudes [[82]]. For the one particular ordering of the external lines the color-ordered amplitude is given by the planar diagrams displayed in Fig. 2B in Feynman gauge. The double lines on the top and the bottom correspond to the Wilson line propagators. Calculation of these diagrams (with proper normalization) gives the following result for the square of the amplitude

(17)

where

(18)

Above we have used abbreviations . The standard and auxiliary Mandelstam invariants read

(19)
(20)

They satisfy and . The order of arguments in (17) corresponds to the order of the external legs (see Fig. 2B). The color dressed amplitude is obtained by summing over all noncyclic permutations of the external legs (minus equivalent permutations due to the relations like , etc.)

(21)

The factor constitutes the helicity average for the off-shell gluons as their ‘polarization’ vectors can be thought of to be ‘continuous’. It is because one can show that these polarizations are and which depend on the transverse angle spanning between and .

Using the same kinematics as for the collinear case (but now with (15) for initial states) we can write the cross section as

(22)

where

(23)

The invariants in (17) can be easily expressed in terms of integration variables in (22). Comparing this with the collinear expression (9) we see that the singularity can be potentially regularized by a nonzero . Let us note, however, that can be zero even if , generated in UGDs are nonzero. In fact due to the transverse momentum conservation whenever the jets are back-to-back and the singularity remains bare. For nonzero , the depends on relative orientation of the vectors , . Since UGDs do not generally depend on angles, the only correlations can be hidden inside the matrix element. Moreover, the expression (22) has to be integrated over transverse variables to be actually compared with the collinear expression. We shall later perform a detailed numerical study and see whether the modification of factor due to can produce a cutoff similar to minijet models. This in principle would be possible, as one can check that the median of the transverse momenta given by UGDs grows with decrease of . Anticipating the result, however, let us recall, that actually (22) should be used in the hard scattering regime, that is for large. This can be also understood by realizing that the main contribution to comes from the collinear region. In fact it can be shown that

(24)

where , are the angles on the transverse plane of the vectors , and the first term on the r.h.s. is the collinear matrix element. By using the above expression and expanding in powers of one can find systematically power expansion of the cross section. The UGDs are typically peaked for small values of , thus the collinear contribution is the dominant one (the leading power contribution). Therefore one should expect that the applicability of (22) is in the high domain.

Let us make now a few comments about the HEF. First, concerns the collinear limit of (22). One would expect that for large the cross section calculated converges to the collinear one (9). Performing the expansion (24) and retaining the first collinear contribution only we are left with integrals in (22) of the type

(25)

where is the upper bound on which in practice is constrained by the grid size of the UGDs or specific kinematic cuts. The point is that the function is in general not exactly a collinear gluon PDF, which is defined as

(26)

Thus, we will overshoot the collinear result if the hard scale is not too large and the UGDs do not fall very rapidly with . In other words, the convergence to the collinear result for finite is rather weak. The remedy could be to set , but this not inherent part of the HEF. Let us illustrate the above with a concrete and practical example. According to the Kimber-Martin-Ryskin (KMR) prescription [[19], [20]] (actually its commonly used simplified form), an UGD can be constructed from a collinear PDF as follows:

(27)

where is the Sudakov form factor. We see that the has to be equal to in order to recover upon integration over .

In order to address another possible issue of HEF, let us consider the cross section as a function of the jet disbalance , . It can be calculated within HEF using (22). Let us now find the collinear limit of . It is easy to see, that it will not converge to the DDT formula (2). This is not necessarily a problem, as the natural domains of applicability of HEF formula and DDT are very different. Nevertheless, it would be interesting to have a formula which includes subleading powers of while possessing the leading twist limit given by (2). We shall construct such a formula in the next subsection.

Finally, let us mention that in practical applications it is convenient to use Monte Carlo programs to generate various observables for jets, instead of using the formulae like e.g. (22). Thus in our study we use an implementation of HEF in a computer program [[83]] which relies on the adaptive Monte Carlo [[84]]. It allows to generate partonic events (‘weighted’ or ‘unweighted’), store them and make further analysis in a convenient way. No parton shower or hadronization is done in the current version. Let us however mention that the dependence of gluon distributions acts much like the initial state parton shower (see e.g. [[61], [85]]).

2.4 Extension of DDT beyond leading power

In order to make our analysis as complete as possible, we will construct now a version of HEF which in the leading power limit reduces to the DDT formula (2) for the dijet disbalance spectrum. The goal of doing this is to use broad spectrum of models with internal gluon . In HEF described in the previous subsection the dependent UGDs take into account two ladders of initial state emissions for each colliding hadron; it is most transparent when UGDs are considered within the KMR approach (27) (see Fig. 2A). There are no final state emission ladders in HEF, whilst the DDT formula (2) has a one ladder attached to each leg of the hard process, including the final state lines (Fig. 2B) [[31]]. Of course, the DDT formula is the leading twist expression, on the contrary to HEF. Below, we shall construct a HEF-based model which has a similar philosophy to the DDT, but includes power corrections.

A)


B)

Figure 3: A) In HEF with the KMR prescription (27) the of initial state gluons on both sides is produced by the gluon PDF and the Sudakov form factor B) In the leading twist DDT formula of Eq. (2) one ladder of emissions is associated with each leg of the hard process.

Let us first write (2) as

(28)

where we have used the symmetry with respect to exchange of hadrons (this gives a factor of ). Basing on the above and using (27) we now define

(29)

where the ‘initial state’ contribution is

(30)

while the ‘final state’ contribution is

(31)

We have defined the final state transverse momentum distribution as

(32)

The Sudakov form factor we use is given by the following formula [[31]]:

(33)

where

(34)
(35)

The cutoff parameter is taken to be . We note that there are various forms of the cutoff parameter in the literature, see for example [[19]]. The partonic cross section is calculated in the exact same way as in the hybrid HEF described before, taking into account the gauge invariant off-shell amplitude with only one leg being off-shell

(36)

where was calculated for instance in [[53]] and using helicity amplitudes in [[74]]. It reads

(37)

with the invariants defined in (19),(20), but now . In the above form the on-shell limit is visible right away: when we have , , and we get the known collinear result.

The partonic cross section with final state off-shell is a new construction and to our knowledge does not exist in the literature. It is constructed from the gauge invariant off-shell amplitude with the final state particle taken off-shell

(38)

where is the two-particle phase space to produce a spacelike state with mass . Let us now explain, how the amplitude is calculated, as it differs from the standard way the HEF amplitudes are obtained.

A)

B)

Figure 4: A) Momentum assignment in the final state contribution to (29); the final state momentum is off-shell. B) Diagrams contributing to the gauge invariant final state off-shell process; the Wilson line slope is given by the vector so that .

First consider the kinematics involved in (31), see Fig. 4A. The idea is that first the two states are produced: an on-shell gluon and the off-shell one with momentum , . Next, this off-shell dressed gluon undergoes emissions described by defined in (32) and becomes on-shell , . The first stage happens via the off-shell gauge invariant process calculated from diagrams depicted in Fig. 4B according to the prescription of [[67]]. As the Wilson line slope we take here the momentum (not the eikonal vectors , as it was the case for HEF), so that

(39)

The result reads

(40)

It looks basically the same as (37) but now

(41)
(42)

It is important to mention that, by construction, the maximal allowed value of is