CCFM Evolution with Unitarity Corrections

CCFM Evolution with Unitarity Corrections

Emil Avsar, Edmond Iancu Emil.Avsar@cea.fr, Edmond.Iancu@cea.fr Institut de Physique Théorique de Saclay, F-91191 Gif-sur-Yvette, France
July 13, 2019
Abstract

We considerably extend our previous analysis of the implementation of an absorptive boundary condition, which mimics saturation effects, on the linear CCFM evolution. We present detailed results for the evolution of the unintegrated gluon density in the presence of saturation and extract the energy dependence of the emerging saturation momentum. We show that CCFM and BFKL evolution lead to almost identical predictions after including the effects of gluon saturation and of the running of the coupling. We moreover elucidate several important and subtle aspects of the CCFM formalism, such as its relation to BFKL, the structure of the angular ordered cascade, and the derivation of more inclusive versions of CCFM. We also propose non–leading modifications of the standard CCFM evolution which may play an important role for phenomenological studies.

1 Introduction

In a previous work, Ref. (Avsar:2009pv, ), we have proposed a method for effectively implementing saturation and unitarity within a generic linear evolution equation for the unintegrated gluon distribution, so like the BFKL (BFKL, ) and the CCFM (Ciafaloni:1987ur, ; Catani:1989sg, ; Catani:1989yc, ) equations. The method is based on enforcing an absorptive boundary condition at low transverse momenta which prevents the gluon phase–space occupation numbers to grow beyond their physical values at saturation. Our method is the extension of a strategy originally introduced in relation with analytic studies of the BFKL evolution in the presence of saturation (Iancu:2002tr, ; Mueller:2002zm, ), whose deeper justification (Munier:2003vc, ) lies in the correspondence between high–energy evolution in QCD and the reaction–diffusion process in statistical physics (Iancu:2004es, ). This correspondence is however limited to asymptotically high energies and to a fixed coupling (Dumitru:2007ew, ), whereas our analysis in Ref. (Avsar:2009pv, ) shows that the absorptive boundary method is in fact more general and also very powerful. After reformulating this method in such a way to be suitable for numerical simulations, we have demonstrated its efficiency by comparing the numerical solutions to the BFKL equation with absorptive boundary condition against those of the proper non–linear generalization of BFKL which includes saturation — the Balitsky–Kovchegov (BK) equation (Balitsky:1995ub, ; Kovchegov:1999yj, ). We have thus shown that the absorptive boundary method successfully reproduces the results of the BK equation for both fixed and running coupling, and for all the energies, and not only the asymptotic ones. This success, together with its relative simplicity, makes this method a very compelling tool for phenomenological studies at LHC and, in particular, for implementing saturation within Monte Carlo based event generators, such as CASCADE (Jung:2000hk, ) and LDCMC (Kharraziha:1997dn, ). Most importantly, our effective method can be also implemented within formalisms whose non–linear generalizations are not known, so like the CCFM formalism (Ciafaloni:1987ur, ; Catani:1989sg, ; Catani:1989yc, ) which lies at the basis of the above mentioned generators. It is our main purpose in this paper to provide an extensive numerical study of the CCFM evolution supplemented with the absorptive boundary condition, and thus demonstrate the role of the saturation effects in that context. Some preliminary results in that sense were already presented in Ref. (Avsar:2009pv, ), but out present analysis will considerably enlarge that previous analysis, in particular by exploring the CCFM formalism in much more detail.

Based on our current theoretical understanding and on extrapolations from the phenomenology at HERA and RHIC, we expect at LHC a considerably larger phase space where saturation effects should be important. The characteristic transverse momentum scale for the onset of unitarity corrections is the saturation momentum , and is expected to grow quite fast with increasing energy. Next–to–leading order BFKL calculations (Triantafyllopoulos:2002nz, ) suggest a power law with , which appears to be supported by the HERA data at small (Stasto:2000er, ; Iancu:2003ge, ; Soyez:2007kg, ). For forward jet production in proton–proton collisions at LHC, one thus expects in the ballpark of 2 to 3 GeV. Even higher values could be reached in nucleus–nucleus collisions, or in some rare events, like Mueller–Navelet jets (Iancu:2008kb, ).

A priori, it seems natural to look for saturation effects in the underlying event, that is, in the bulk of the particle production at relatively low momenta, where the saturation effects (also viewed as multiple scattering (Acosta:2004wqa, ; Field:2002vt, )) are clearly important. However, the underlying event at LHC will be extremely complex and difficult to study. Besides, for such low momenta, it may be difficult to separate saturation physics from the genuinely non–perturbative physics in the soft sector of QCD. (A similar ambiguity occurs in the interpretation of the small– data at HERA.) It is therefore useful to recall at this point that saturation effects can make themselves felt even at relatively large momenta , well above , via phenomena like the “geometric scaling” observed at HERA (Stasto:2000er, ; Marquet:2006jb, ; Gelis:2006bs, ). Such phenomena, which reflect the change in the unintegrated gluon distribution at high due to saturation at low (Iancu:2002tr, ; Mueller:2002zm, ; Triantafyllopoulos:2002nz, ; Munier:2003vc, ), are particularly interesting in that they represent signatures of saturation in the high– domain which was traditionally believed to fully lie within the realm of the “standard” pQCD formalism — the DGLAP evolution (DGLAP, ) of the parton distributions together with the collinear factorization of the hadronic cross–sections.

Schemes based on the NLO DGLAP evolution have been quite successful at HERA (at least for not too low ) (Klein:2008di, ), but at LHC one will probe much smaller values of Bjorken , and this even for relatively hard (e.g., in the forward kinematics). The jets to be measured at LHC will carry relatively large transverse momenta  GeV, but because of the high–energy kinematics, their description may require –factorization (KTFACT, ) together with the BFKL, or CCFM, evolution of the unintegrated gluon distribution. Moreover, saturation effects, like geometric scaling, could manifest themselves in hard observables at LHC, so like the cross–section for forward jet production (Iancu:2008kb, ). Thus LHC will for the first time allow us to study saturation physics in the kinematical regime where this physics lies in the realm of perturbative QCD.

The BFKL formalism, properly generalized to include the non–linear effects responsible for gluon saturation (Balitsky:1995ub, ; Kovchegov:1999yj, ; JKLW, ; CGC, ; CGCreviews, ), is specially tailored to describe the evolution of the unintegrated gluon distribution with increasing energy and its approach towards saturation. As such, this is well–suited to study the high–energy evolution of inclusive cross–sections, and it is able to accommodate important phenomena, like the geometric scaling at HERA (Stasto:2000er, ; Marquet:2006jb, ; Gelis:2006bs, ), or the turnover in the DIS structure function at semi–hard (Iancu:2003ge, ; Soyez:2007kg, ). However, this is not the right formalism to also describe exclusive final states, because it misses the quantum coherence between successive gluon emissions in the process of high–energy evolution. Besides, there are additional problems, to be discussed at the end of section 2.3, which make the BFKL formalism unsuitable for studying exclusive final states. All such problems are corrected in the CCFM formalism (Ciafaloni:1987ur, ; Catani:1989sg, ; Catani:1989yc, ), which has also the advantage to apply within a wider kinematical region, interpolating between the high energy evolution (the realm of BFKL) and the evolution with increasing virtuality (the realm of DGLAP). A similar formalism derived out of CCFM but using a different physical picture for the evolution is the Linked Dipole Chain (LDC) model (Andersson:1995ju, ) which covers the same kinematical region as CCFM, and our method of implementing saturation can be equally well applied also to this formalism. Infact one of the main equations to which we apply our method is equivalent to the master equation in LDC. This will be discussed more below.

Like BFKL, the CCFM formalism is based on the –factorization. This makes it possible to take into account some of the NLO corrections in the collinear approach by simply treating the kinematics of the scattering more accurately (Andersson:2002cf, ; Andersen:2003xj, ; Andersen:2006pg, ). Still like BFKL, the CCFM evolution resums all powers of which are accompanied by large energy logarithms , with the center of mass energy. In fact, the CCFM and BFKL evolutions yield identical predictions for the dominant behaviour in the formal high–energy limit . But this formal limit is conceptually unrealistic and phenomenologically irrelevant, since it violates unitarity. What is relevant, is the approach towards the unitarity limit and gluon saturation with increasing energy, which is a priori different in the two formalisms. In the recent years, this approach has been extensively studied within the BFKL evolution, by using its non–linear generalizations: the BK (Balitsky:1995ub, ; Kovchegov:1999yj, ) and the JIMWLK equations (JKLW, ; CGC, ). However, no such a study was performed within the context of the CCFM evolution prior to our recent work (Avsar:2009pv, ). As already stated, such a study is the main objective of the present work.

The present study should be viewed as a step towards a systematic inclusion of the effects of saturation in the description of exclusive final states. Most studies have so far concentrated on more inclusive observables, for which a good description can generally be obtained (for sufficiently high ) also with linear evolution equations alone. A number of papers emphasized the need and importance of studying saturation in exclusive final states (Avsar:2005iz, ; Avsar:2006jy, ; Avsar:2007xg, ; Flensburg:2008ag, ), but so far the explicit studies were mostly confined to inclusive observables. The possibility of looking at more exclusive observables will undoubtedly make it easier to distinguish the predictions of linear and non–linear evolutions.

But before we consider the effects of saturation, we shall dedicate a large part of this paper to a detailed presentation of the respective linear evolution, in order to clarify several non–trivial aspects of the CCFM formalism which are important for our purposes. In doing so, we will try to carefully motivate the steps leading to the final evolution equations (simplified versions of the CCFM formalism) on which we shall apply our saturation boundary. Among the different aspects of CCFM to be discussed here, there are parts which have been already presented in previous papers that we shall refer to, but there are also parts which to our knowledge have never been presented before. In this paper, we shall try to give a unified presentation of all the relevant aspects, using a intuitive geometrical representation for the phase–space of the CCFM evolution. To facilitate the reading of the paper, we have moved some of the most technical developments to appendices, and kept only the important results in the main text. Here is a summary of the topics to be covered in what follows and also of our main conclusions:

Sect. 2 will introduce the basics of the CCFM formalism, that the rest of the work will rely on. In particular, in Sect. 2.2 we shall clarify the relation between the phase–space of the CCFM evolution and that of the BFKL evolution, thus recovering previous results in Ref. (Salam:1999ft, ), but from a different perspective. In Sect. 2.3 we shall present the standard version of the CCFM evolution as an integral equation, and on this occasion we shall explain the approximations which are involved in this rewriting and which are often kept implicit in the literature. In Sect. 2.4 and Appendix A we discuss some subtle aspects, and correct some mistakes in the literature, concerning the virtual (‘non–Sudakov’) form factors which express the probability for not emitting gluon in the course of the evolution. The careful derivation of these form factors, as outlined in Sects 2.3 and 2.4 and in Appendix A, will also allow us to better understand their physical origin and thus propose some improved expressions for them, which treat more accurately the kinematics (by including effects of recoils in the energy). The new form factors, to be presented in appendix B, differ from the standard ones by terms which are formally of higher order, but which may be numerically important111We shall not include these new form factors in our present numerical analysis since their structure is such that they can be efficiently implemented only within Monte Carlo simulations. and thus interesting for the phenomenology. Another aspect of the form factor is its appropriate form in the formal high energy limit, which is relevant again for the comparison to BFKL and this will be discussed in appendix C.

Sections 3 and 4 are the key sections in this paper. In Sect 3 we successively simplify the CCFM formalism and reduce it to a set of simpler, integral and differential, equations, which are more suitable for numerical simulations. The most general equation, the integral equation (3.1), explicitely preserves all the hard and soft gluon emissions from the -channel propagators. This is the equation at the basis of the CASCADE Monte Carlo event generator (Jung:2000hk, ), and it can be generalized to include saturation effects, as we shall explain in Sect. 4.1. But for the present purposes, it is preferable to work with simpler versions of the CCFM evolution, namely Eqs. (3.11) and (3.12), which are more ‘inclusive’ — in the sense that some of the virtual form factors are used to cancel the ‘soft’ gluon emissions. Such cancellations are not exact, but rather require some additional kinematical approximations, which are however in the spirit of the CCFM formalism. These approximations are also similar to those underlying the ‘Linked Dipole Chain’ (LDC) model (Andersson:1995ju, ). And indeed, our Eq. (3.11) appears to be equivalent to the master equation in Ref. (Andersson:1995ju, ), although our respective approaches are quite different. Intriguingly, it turns out that if one performs the same type of approximations starting with the BFKL formalism, one is again led to the same two equations (3.11) and (3.12) (see the discussion in Sect. 3.4). In our opinion, this points out towards a deep similarity between the BFKL and CCFM formalisms, which when applied to inclusive observables differ only in the accuracy to which they treat the kinematics.

In Sect. 4 we introduce the absorptive boundary method which effectively implements saturation within a linear evolution equation, so like BFKL or CCFM. First, in Sect. 4.1, we describe this method on the example of the BFKL equation, where the comparison with the non–linear BK equation will allow us to demonstrate the success of the method. Then, in Sect. 4.2, we analytically study the high–energy behaviour of the approximate CCFM equations (3.11) and (3.12) and thus determine the energy–dependence of the associated saturation momenta, in the case of a fixed coupling. Our analysis shows that the CCFM evolution is somewhat faster than the BFKL one: the respective saturation exponent is slightly larger. On the other hand, the characteristic functions which determine the momentum–dependent anomalous dimension, are very similar to the BFKL one, for both Eq. (3.11) and Eq. (3.12). In Appendix D, we relax some of the approximations used in deriving Eq. (3.11) and thus obtain a more accurate equation, whose high–energy predictions are even closer to those of the BFKL equation.

Finally, Section 5 presents a systematic numerical analysis of the various evolution equations — the BFKL equation and the simplified versions, Eqs. (3.11) and (3.12), of the CCFM evolution — with the purpose to illustrate the role of the saturation boundary and also of the running coupling effects. We first demonstrate that the respective linear equations (no saturation boundary) lead to rather different evolutions, which are moreover infrared unstable in the case of a running coupling. Then we show that the infrared instability is cured after including the saturation boundary (the saturation momentum effectively acts as a hard infrared cutoff which increases with the energy) and moreover the respective predictions of the various equations remain very close to each other, up to astronomically high energies. Hence, in so far as the unintegrated gluon distribution is concerned, the BFKL and CCFM evolutions properly corrected for saturation and endowed with a running coupling are rather similar to each other.

2 Review of CCFM

Our aim in this section is to review the CCFM approach, mainly the work in (Catani:1989sg, ). This will prepare us in understanding our subsequent strategy for simplifying this formalism and most efficiently complete it with a saturation boundary. The original formulation in (Catani:1989sg, ) is very careful and complete, but also quite technical and not always transparent. We shall therefore try to mostly give a geometrical representation of the equations to be presented here. In this process, we will derive to some interesting results that we were not aware of, and also clarify some points which are often confusing in the literature, such as the precise form of the “non–Sudakov” form factor .

2.1 Kinematics and Basics

Figure 1: Kinematics of the gluon radiation. The last t-channel propagator is denoted by , while the real -channel gluons are represented by the horizontal lines.

We use Fig. 1 to define the kinematics and schematically introduce the physical picture. This figure represents a gluon ladder as produced by the CCFM evolution; we denote by the transverse momenta of the real, –channel, gluons, and with the transverse momenta of the space–like, –channel, propagators, which are not explicitly shown in the figure. The incoming virtual gluon has zero transverse momentum (that is, this gluon is taken to be collinear with the parent hadron, not shown here), and hence one has . We will use and to denote the energy fractions of the –channel and –channel gluons, respectively, measured with respect to the energy of the incoming proton. In Fig. 1, the –channel (or ‘real’) gluons are enumerated according to their energy:

(2.1)

but this ordering is not necessarily the same as that of the gluon emissions along the ladder (i.e., it is not assumed that the gluon with energy fraction is emitted out right after that with fraction , etc.). Rather, as we shall shortly see, the real gluon emissions in the CCFM ladder are ordered according to their angles . This ordering issue is potentially confusing, since e.g. the relation would be strictly true if the labels attached to gluons in Fig. 1 were also indicating their order of emission, i.e., if the real gluons emissions were ordered according to their energy (which they are not). The resolution of this puzzle, to be explained in detail later on, is that the actual emissions which do not obey energy ordering are also soft in the sense of carrying little transverse momenta (‘–conserving’), and hence they do not affect the momenta of the –channel gluons: the latter are fully determined by the ‘hard’ emissions which are simultaneously ordered in angle and in energy. (See Sect. 2.3 for details.)

The (integrated) CCFM gluon structure function can be written as

(2.2)

where , and and are the virtual corrections associated with the eikonal and the non-eikonal vertices respectively. The theta function is a consequence of the quantum coherence between successive emission which implies that the emission angle must increase when moving upwards along the ladder (i.e., towards the hard scattering). Notice that one can have any angular ordering for the given energy ordering. (We shall later relabel the real gluons according to the angular ordering.) The angle in the argument of is the maximum angle allowed by coherence and is determined by the kinematics of the hard scattering; roughly, with the virtuality of the incoming photon. In this case, the structure function (2.2) gives the gluon distribution, (Marchesini:1994wr, ).

The virtual form factors in (2.2) are given by

(2.3)
(2.4)

where and the integral over the transverse momentum has been written in terms of the associated angular variable . Notice that the exponent in the non–eikonal form factor is positive. The apparent divergence in and in the real emission density is regulated by a collinear momentum cut (which also regulates the divergence in ).

One should be aware that (2.2) does not correspond to an exclusive final state. In (2.2) one has already inclusively summed over all subsequent emissions from the outgoing gluons . Each such gluon can further radiate within a cone of half opening angle . These final emissions are such that the real emission probability is exactly compensated by virtual corrections of the type , and they are therefore not visible in the expression (2.2) for the (inclusive) gluon distribution. For the study of exclusive final states, however, one needs to include all the emissions.

2.2 The phase space and relation to BFKL

In this subsection we shall describe in detail the phase–space for real and virtual gluon emissions within the CCFM ladder and then argue that, up to subleading effects, this is essentially the same as the phase–space for the BFKL evolution. This will allow us to conclude that the CCFM and BFKL evolutions become identical with each other in the high energy limit. (See also Ref. (Salam:1999ft, ) for similar considerations.)

Let us start with the virtual emissions and rewrite the eikonal form factor in (2.3) as

(2.5)

with . Let us now consider the two emissions in Fig. 2. In this figure the horizontal axis222These type of diagrams have been widely used for example in (Andersson:1995ju, ), but notice that there the horizontal is taken as instead of . Our choice is the same as the one in (Salam:1999ft, ). is ln while the vertical axis is ln (from now on we shall for simplicity omit using ). A black dot in such figures denotes a real gluon with the respective values for the energy () and the transverse momentum (). Since , the emission angle will be constant along diagonal lines in the figure. The diagonal line shown in figure 2 denotes the maximum angle, determined by . The diagonal lines parallel to this line and which pass through the gluons will indicate the angle of the gluons (see e.g. Fig. 5).

Figure 2: Geometrical representation of two emissions (fat dots) in CCFM shown in the -phase space. The total phase space is bounded by the diagonal line which indicates the maximal angle allowed by coherence. The shaded regions show the phase regions over which (left) and (right) are integrated over.

The shaded regions in the two figures indicate the phase space over which the non–eikonal and eikonal form factors are integrated over. The rightmost figure, representing , is easier to understand. Here the integral is bounded by and , that is we integrate over the region between the two vertical lines, and the integral goes up to , as can be seen in the figure. To understand the leftmost figure, note that in defined in (2.4), the –integral is the same as in while the –integral (or, equivalently, that over ) is integrated from up. The horizontal line bounding the shaded region in that figure corresponds to . Since the exponent is negative in and is positive in , we get a complete cancelation in the region of overlap. It is then easy to see from Fig. 2 that the region which is left is the one shown in Fig. 3, and that the net exponent is negative.

Figure 3: The shaded region is the region left over where there is a net contribution from the virtual form factors. It is bounded from above by . The real emissions are represented by fat dots.

Now, remember that in the BFKL evolution, the virtual corrections are contained in the “non–Sudakov” (or “non–eikonal”) factor defined by

(2.6)

and we see that this corresponds to the shaded area in Fig. 3. Thus we find

(2.7)

Figure 4: A complete set of emissions in the CCFM phase space. Real emissions are represented by fat dots. The shaded regions are the regions over which we have contributions from the virtual form factors. We have here ignored the constraint from the maximal angle.

Consider now a complete set of emissions in the initial chain as shown in Fig. 4. In the figure the gluons are obviously enumerated according to their energy. The explanation of the phase space in figure is the following. The are determined by the relation . For example, since , we have (in the following all the momenta denote the norm of the transverse components). Since we have , where the small recoil is neglected in the figure. Then the subsequent real gluons have small momenta so that up to gluon 6 which has large momentum, and therefore and so on. In the end we get an integral over the total shaded region in Fig. 4, and again this is exactly the same that we would have in BFKL.

Of course in CCFM the total phase space is determined by so in Fig. 4 we have assumed the constraint not to cut the shaded regions. Hence, a difference between the CCFM and BFKL ladder appears towards the end of the chain. This difference however is not enhanced by , and hence it is subleading from the viewpoint of the BFKL resummation. Moreover, the CCFM real–gluon emissions are ordered according to their angle, and not to the energy. However, in the ensuing gluon distribution (2.2) there is a sum over all the possible angular orderings for a given energy ordering. Therefore the phase space for real emissions is also the same, up to the subleading difference mentioned above.

In general, however, the energy weighting of the real and virtual emissions is different for the BFKL and CCFM evolutions. For the latter, this is encoded in the splitting functions in Eqs. (2.2) and (2.3)–(2.4), which show that the (real and virtual) gluon emissions are distributed logarithmically in — the rapidity of the –channel gluons. The BFKL evolution, on the other hand, retains those diagrams which resum all orders in : these are gluon ladders in which the –channel gluons are strongly ordered in longitudinal momentum () and distributed with the logarithmic weight . Clearly, strong ordering in the –channel implies a similar ordering in the –channel — from , it follows that —, so that the positions (in energy) of the t-channel propagators uniquely determine the positions of the real gluons. This is true in the strict high energy limit, where at fixed , and is very small. Beyond this, however, it is important that the CCFM configurations are generated according to the rule of quantum coherence, and hence they represent realistic final states (at least, up to further emissions from the real CCFM gluon, as explained as the end of Sect. 2.1). Thus, although the two types of evolution provide identical results for the (inclusive) gluon distribution in the formal high–energy limit the CCFM evolution is more appropriate for describing actual final states (see the discussion at the end of section 2.3). Moreover, this formal high–energy limit becomes meaningless in the presence of saturation, as we shall later explain, and when this limit is properly taken (see in Sect. 4), differences are expected to appear already in inclusive quantities, so like the gluon distribution.

2.3 The structure of the angular ordered cascade

Returning to Eq. (2.2), this can be further simplified to obtain a more familiar expression for the gluon distribution. To that aim, we shall divide the initial state radiation into two classes, “soft” and “hard” (or “fast”) gluons (Catani:1989sg, ). This is done as follows. Consider the set of initial gluons shown in Fig. 5. The “hard” subset of gluons are those which are not in angle followed by a gluon with more energy, i.e. with higher . All other gluons are defined as being “soft” gluons. In Fig. 5, the gluons marked by 1, 2, 3, 4, 5 and 6 are hard gluons, since as compared to them, there are no other gluons with larger angle and higher energy. (Recall that a larger angle would mean a gluon above the respective diagonal line, while a higher energy means a gluon to the left of the vertical line through the gluon.) The gluons marked by and are soft gluons. For example, is in angle followed by 1 and 1 has larger energy than since it is located to the left of it. Gluon is followed by which has larger energy, which itself is followed by 4 in angle which has also larger energy than . The hard gluons are thus ordered both in angle and in energy.

Figure 5: Dividing the CCFM radiation into two subsets: hard emissions (enumerated emissions) and soft emissions (indexed by small letters) as explained in the text. All real emissions are represented by fat dots.

The soft gluons can furthermore be divided into clusters. Define the cluster as consisting of those soft gluons which have their angle between and , and which have energies less than . Thus in Fig. 5, belongs to cluster since () and , and belong to since and so on. In this example there are no gluons in . The phase spaces for the clusters are shown in Fig. 6. We now see that the “soft” gluons which belong to the cluster are indeed soft in the sense of having lower energies than the “hard” gluon which defines the cluster.

Figure 6: The representation of the hard emissions, enumerated fat dots, together with the clusters . For each hard emission , the cluster contains all the soft emissions associated with that hard emission.

Figure 7: The hard and soft emissions, represented by fat dots, in CCFM together with the phase space regions for the “non-Sudakov” (regions ) and the “Sudakov” (regions ) form factor. The Sudakov cancels the real soft emissions.

The advantage of this separation is that it allows us to rewrite the distribution (2.2) in a simpler way. To that aim, notice that the soft gluons in have all transverse momenta smaller than . This is obvious from the figures and more formally follows from the fact that333We denote by and the angle and energy fraction for a soft gluon belonging to cluster . and implies . Then one can write

(2.8)

where denotes the set of all emissions while denotes the subset of hard emissions, and the phase–space for a hard gluon in is the whole respective cluster , as shown in Fig. 6. The gluon distribution (2.2) can then be written as

(2.9)

where , and we now index the gluons by using the angular ordering. The meaning of this equation is simple: it says that we can construct each chain by adding an arbitrary number of soft gluons between each pair of hard emissions. We can now further simplify this expression, by using cancelations between real and virtual contributions to the emission of the soft gluons.

To that purpose, we refer to Fig. 7 where we have identified the hard and soft emissions from Fig. 5, and the numerated emissions are the hard ones. For these we also indicate the angles. The soft gluons are marked by small letters. We now define the new “non–Sudakov” form factor by

(2.10)

Similarly we define the “Sudakov” factors by

(2.11)

Thus we have

(2.12)

Now, the summation over the real soft emissions in each cluster in (2.9) exponentiates,

By the definition of the Sudakov in (2.11) we thus have

(2.14)

i.e. the real soft emissions are exactly compensated by the Sudakov form factors. After thus inclusively summing over all soft emissions, the structure function can be finally written as

This expression involves the hard gluons alone.

Before leaving this section, two comments are in order:

(i) In Fig. 7 the horizontal lines between the real emissions represent the momenta of the –channel gluons. As obvious from the figure, these momenta are determined solely by the hard subset of emissions (c.f. Eq. (2.8)) which are ordered in both energy and angle. This a posteriori explains why the condition is approximately true irrespective whether the labels refer to energy ordering, or to the angular ordering (which is the actual order of the gluon emissions). This argument shows that there is an implicit approximation in the CCFM formalism — the fact that soft gluon emissions are assumed not to change the virtual transverse momenta. Hence, without further loss of accuracy, we will later use similar approximations to simplify the expression of the gluon distribution even further (see Sect. 3).

The second important point is the relation to BFKL mentioned earlier and which deserves some clarifications. To compare to BFKL, it is convenient to define by . This implies ), and therefore

(2.16)

Thus in the CCFM ladder one can distinguish two vertices contributing to the splitting . The one corresponding to the small– pole is dubbed the “non–eikonal” vertex as it comes from the piece of the three gluon vertex in which the polarization of the parent gluon is inherited by the real gluon . The opposite pole is dubbed the “eikonal” vertex, and in this case the polarization, together with most of the energy, is inherited by the –channel propagator . The hard emissions previously identified are associated with the pole in Eq. (2.16), while the soft emissions with the pole (Catani:1989sg, ). In contrast, in a BFKL ladder, only the pole would be present, and the energy ordering coincides with the actual sequence of emissions along the cascade. In that case the typical gluons are such that and hence , as anticipated at the end of Sect. 2.2. Thus in the corresponding phase–space integrals, like Eq. (2.6), one can replace the measure by . We are now prepared for our second comment:

(ii) Despite the formal equivalence between the CCFM and BFKL evolutions (in the high energy limit), the latter cannot be used to generate the final state, not only because it does not obey the condition of angular ordering (as required by quantum coherence), but also because the Regge kinematics cannot be ensured in practice when trying to generate a BFKL ladder. The reason is as follows: the BFKL emission probabilities for real and, respectively, virtual gluons are separately infrared divergent (see e.g. Eq. (2.6)) and thus require an infrared regulator . Although the dependence upon formally cancels in the complete result, the introduction of this soft momentum cutoff will falsify the condition that in the intermediate steps. Indeed, when a new value for is randomly generated with probability law , the typical value value is such that

(2.17)

which in principle should be of for the Regge kinematics to apply, but in reality becomes of (meaning as well) whenever is taken to be small enough.

Within the CCFM evolution, this problem is avoided due to the presence of both types of poles, and , and to the angular ordering. In that context, the dependence on is present in the Sudakov form factor, as the areas in Fig. 7 are cut from below by . In the limit, successive emissions will become very close to each other in angle. Indeed, an emission typically occurs when the corresponding region (that we now define separately for each emission, either hard, or soft; see Fig. 8) has an area of ,

(2.18)

which implies when . In that case one can identify two limiting cases: (i) and more or less similar to the momentum of the previous emission, or (ii) and either of the order of, or much smaller than, the of the previous emission. These two possibilities are precisely the type of emissions already present in CCFM and therefore the structure of the cascade is not altered by , unlike what happens in BFKL.

Of course, in the presence of saturation the dependence on any soft momentum disappears naturally, as the dynamically generated saturation momentum, which grows rapidly in the course of the evolution, provides a natural cutoff (see the discussion in Sect. 4.1).

2.4 The virtual form factors

In this subsection we shall display some more explicit formulæ for the non–Sudakov and Sudakov form factors and , whose detailed derivation is presented in appendix A. Although such formulæ were already presented in the original work (Catani:1989sg, ), it turns out that they are often written in a wrong way in the literature (see the discussion in appendix A). To avoid such errors, and also in order to be able to generalize these form factors by including non–leading effects — a task that we address in appendix B —, it is essential to have a proper understanding of the derivation of the corresponding formulæ. This is briefly discussed here and then in more detail in appendix A.

Let us first recall that for the hard emissions, , while for the soft emissions . Then one can as a first approximation set for the hard emissions (except in the pole), and for the soft emissions (except in the pole). Therefore the region has the transverse momentum bounded from below by , and from above by , while the respective integral is bounded between and . Thus we approximately have

(2.19)

Defining , and switching from to by using one gets

(2.20)

The reason we did not set in the lower limit of the integral is because the CCFM equation is usually written in terms of the so–called rescaled momenta defined by so that the factor is absorbed into the definition of . Then

(2.21)

which is the form used in (Catani:1989sg, ). Equation 2.21 for is, however, usually written in a different way in the literature (see Eq. (A.1) in the appendix). In appendix A, we give a more careful derivation of the non-Sudakov form factor, and we demonstrate that the correct form is indeed given by 2.21.

Figure 8: Hard and soft emissions, with the Sudakov (regions ) defined for each individual emission. To leading order there is no non-Sudakov (regions ) associated with the soft emissions. Real emissions are again represented by fat dots, while the and energy fraction of the virtual propagators are denoted by crosses. In this example, emission number 2 is a soft emission.

To write down the Sudakov one first needs to define it for each, hard and soft, individual emission. In Fig. 8 we show an explicit chain of hard and soft emissions, where the real emissions are indexed according to their angular ordering, and where we also explicitely show the virtual t-channel propagators by crosses. The individual Sudakov form factors are then defined as the integrals over the regions in the figure (note that these are not the same regions as before, now we define such a region for each emission, not just for the hard ones).

We see that the region is to the left bounded by the energy of , while there is no lower limit for the energy444Eventually there will be a limit coming from the soft momentum cut, so that the region does not extend to infinite size. In momenta it is bounded between the angles of the real gluons and . Therefore we may write the Sudakov as

(2.22)

where represents the soft cutoff. Since we must have we get and . Defining , we have and

(2.23)

where . Now if one wishes one can let and then the usual form for the Sudakov form factor is obtained.

3 More inclusive versions of CCFM

In section 2.3 we have shown that one can use the Sudakov form factors to cancel the real soft emissions, and this resulted in a simplified expression for the gluon distribution, Eq. (LABEL:eq:ccfmstruct2). This distribution is more “inclusive” than the original one, Eq. (2.2), which explicitly includes all soft emissions, yet it is equivalent to it for the calculation of the gluon distribution. In this section, we shall construct other, even more inclusive, versions of the CCFM evolution, which are better adapted for numerical calculations. In these constructions, we shall exploit the flexibility which exists in defining the CCFM evolution, as associated with the various approximations involved in its derivation. Note that “getting more inclusive” is not the only possibility for deriving different versions of CCFM. In appendix B we shall derive yet another version by including non–leading effects related to recoils. That version could be implemented in a Monte Carlo simulation, so like CASCADE.

3.1 Integral equations

From now on we shall work with the ‘unintegrated’ gluon distribution, i.e., the number of gluons with a given longitudinal momentum fraction and a given transverse momentum , which is obtained by undoing the integral over the last angular variable in Eq. (2.2) (or (2.9)) and replacing . It will be also convenient to replace the maximal angle by a corresponding momentum variable , via the substitution ; as explained after Eq. (2.2), one has roughly (the virtuality of the space–like photon exchanged in DIS). The integral equation satisfied by is easy to derive from Eq. (2.9), and reads

(3.1)

where we are using rescaled momenta within the integrand: and . The third argument of the gluon distribution inside the integrand, i.e. , truly means that the maximal angle corresponding to this distribution is the angle of the emitted real gluon, that is, , with . Hence, this equation can be read as follows: the final –channel gluon with energy fraction and transverse momentum (and for a maximum emission angle measured by ) is generated via the splitting of a previous –channel gluon with energy fraction and transverse momentum (and for a maximum emission angle measured by and ). This is the most exclusive version of the integral equation and includes all the hard and soft emissions. This equation is implemented in the CASCADE event generator (Jung:2000hk, ).

In the more inclusive version we can sum over all soft emissions (in the regions in Fig. 6) so that the Sudakov factors disappear and we are left with the gluon distribution in Eq. (LABEL:eq:ccfmstruct2). This version gives rise to the following integral equation

(3.2)

This equation is simpler to solve than (3.1) but is still not very easy to deal with, not even numerically. Notice that one might as well use Eq. (3.2) as the basis for an event generator, but then all soft emissions must later be included as final state radiation. In what follows, however, we shall be concentrating on the small– part of the gluon distribution, where the small– values () are dominating. In that case one can replace all rescaled momenta with regular momenta, and rewrite in the argument of within the integrand; also the energy ordering becomes automatic. Hence the equation becomes

(3.3)

3.2 Geometrical representation of the real vs. virtual cancelations

Starting with Eq. (3.3), we shall later derive the most inclusive, and also the simplest, version of the CCFM equation. To that aim it is important to understand in depth the structure of the virtual corrections encoded in the ‘non–Sudakov’ form factor , Eq. (2.10). Note first that, despite its name, this form factor is essentially not different from a genuine Sudakov one, since it also represents the negative exponent of an area in the phase space, namely the areas in figures 7 and 8. It can therefore be used to cancel the real emissions confined to this phase space, as was first noted in (Andersson:1995ju, ). As we review in appendix A however, this is strictly true only if the additional kinematical constraint , which ensures that the squared 4–momenta of the –channel propagators are dominated by their transverse part (Catani:1989sg, ; Kwiecinski:1996td, ), is enforced within (3.3).

Figure 9: The phase space for . The region contributes with negative weight, and it is seen that . The shaded region is thus the region left over, which is also the region to which the (real) -conserving emissions are confined. To the right we show symbolically the insertion of -conserving emissions between a pair of non--conserving emissions. The propagator is unchanged by this insertion.

Assuming from now on, we can distinguish between two cases: and . When , the regions in figures 7 and 8 correspond to real emissions with and is guaranteed to be smaller than 1. Consider now the situation in Fig. 9 where . As we show in appendix A, we have

(3.4)

where the regions (shaded region), and are shown in Fig. 9. The first observation is that the triangular regions and have the same area, and hence they cancel in the exponent of Eq. (3.4). The upper diagonal line in Fig. 9 indicates the line through which the kinematical constraint limit holds, while the lower diagonal line indicates the angle of . Now, the emissions lying below are, in the spirit of the approximations made in section 2.3 , - conserving. When the kinematical constraint is included, these emissions are confined to the shaded region (region ) in figure 9. This can be understood as follows: due to angular ordering and the fact that we are looking at -conserving emissions, all subsequent radiation must lie in the region or in . However, if we had a real emission in region , then because that emission is -conserving, the -channel propagator emitting this gluon would have transverse momentum approximately equal to and it would necessarily have bigger energy than the real gluon. Therefore we see that it must be located under the upper diagonal line in Fig. 9. This, however, would violate the kinematical constraint and such an emission is therefore not possible. Thus we are left with the fact that all real -conserving emissions are confined to region . Therefore the inclusive summation over all the real emissions, which are inserted in between two non–-conserving emissions, leads to a factor

(3.5)

To conclude (recall that )

(3.6)

which shows that the “non-Sudakov” factor cancels the -conserving emissions. This was first noticed in (Andersson:1995ju, ), and it was later used in (Salam:1999ft, ) as well555 In (Salam:1999ft, ), however, only the possibilities and were considered. In that case there is obviously no need for the kinematical constraint.. One is then left with a much simpler formula for which can be derived from Eq. (3.2) after including the constraint , and the fact that we are left only with the non–-conserving emissions. This last constraint, however, can be enforced in various ways, which are all consistent with each other within the present approximations. Therefore there is no unique equation that one can derive. In what follows we shall consider two different possibilities and then study the ensuing equations.

3.3 Deriving the differential equations

In Ref. (Avsar:2009pv, ) we have the restriction to non–-conserving emissions by introducing the theta function into the r.h.s. of the integral Eq. (3.3). This was also the prescription originally used in Ref. (Andersson:1995ju, ), and the equations derived in (Avsar:2009pv, ) and (Andersson:1995ju, ) are indeed equivalent. After inserting this constraint together with the ‘kinematical’ one and removing , Eq. (3.3) becomes

Since for all cases of physical interest (recall that in DIS), we further have . Therefore the angular ordering is automatic and can be neglected. This means that the dependence on the third variable drops out, at least in the l.h.s. But a similar argument holds also for the function for under the integral, because we have (indeed, we are left only with emissions satisfying or ). Dropping then the dependence of upon its third variable, we obtain

(3.8)

The next step is to perform the integration over the azimuthal angle . To that aim, it is convenient to replace by , which is allowed within the current approximations666Indeed, we have either , in which case the replacement is obviously correct, or , in which case both the first and the second theta function can be replaced by 1., and then switch the integration variables from to and, respectively, from to (which we rename as ). Making this replacement and doing the integral one then gets

(3.9)

where

(3.10)

and . Differentiating w.r.t to we finally deduce the following differential equation for the unintegrated gluon distribution in the CCFM formalism

(3.11)

As mentioned above, (3.11) is equivalent to the master equation in the LDC formalism (Andersson:1995ju, ) (up to some trivial redefinitions: the distribution in (Andersson:1995ju, ) corresponds to our times the proton radius and some constants).

Yet another way to implement the restriction to non–-conserving emissions is to switch the integration variable in Eq. (3.3) from to and then replace with . Note that this constraint is more restrictive that the theta function in Eq. (LABEL:eq:firststep). In this case the angular integration in (LABEL:eq:firststep) becomes trivial. Also the replacement is now exact, and so is also the requirement (which, we recall, allows one to ignore the dependence of upon its third, ‘maximal angle’, variable). We thus deduce

(3.12)

3.4 More on the relation to BFKL

Before moving on to discuss the issues of unitarity and saturation, we would like discuss the effects of the kinematic constraint on the BFKL equation, and we would like to show that the same approximations used above in deriving the more inclusive equations (3.11) and (3.12) applied on BFKL, with the kinematical constraint, leads exactly to the same equations. In section 5 we will present numerical results of the BFKL equation with the kinematic constraint included, with and without the saturation boundary to be described in the next section.

The BFKL equation can be written

(3.13)

where we have included the kinematical constraint (with ). The function is the non-Sudakov form factor modified to include the kinematical constraint. The correct form for can be found by requiring that once more it can be used to exactly compensate the -conserving emissions which are now also modified by the kinematical constraint. The phase space for is illustrated in Fig. 10. The region we are looking for is the shaded one, region . Region which was allowed before is now excluded due to the kinematical constraint. Indeed the real -conserving emissions are confined to region , since just like for CCFM, any real emission in would create a -channel propagator below the diagonal line, violating the kinematical constraint. It is then seen that (see also (Kwiecinski:1996td, ))

(3.14)

where is the usual non-Sudakov, and the second factor is just (again if the kinematical constraint is automatic).

Figure 10: The phase space for the modified BFKL non-Sudakov form factor when the kinematical constraint is taken into account. Region which was previously included is now excluded. Real -conserving emissions are confined to region .

Next we inclusively sum over the -conserving emissions in (3.13), canceling . If we use the explicit constraint , we get

and this leads to an equation which is exactly identical to (3.9) and (3.11). We can also use the constraint in which case we obtain (3.12). We should also mention that we could repeat the arguments in appendix D for BFKL with the kinematical constraint. In that case the region left over by the real-virtual cancellations is exactly equal to the region in (D.2).

4 CCFM evolution with saturation boundary

The CCFM evolution, so like any other linear evolution in perturbative QCD, predicts an unlimited growth of the gluon distribution with increasing , thus leading to unitarity violations in the high energy limit. This is so since the linear evolution equations miss the non–linear phenomena responsible for unitarization, which are gluon saturation and multiple scattering. It turns out, however, that the phenomenon of gluon saturation merely acts as a kind of cutoff, which limits the growth of the gluon distribution, but does not modify the mechanism responsible for this growth. This makes it possible to mimic the effects of saturation by appropriately implementing this cutoff on the linear evolution equations, without a detailed understanding of the underlying non–linear phenomena. In this section we shall motivate and describe the implementation of this cutoff — actually, an absorptive boundary condition —, which then will be used, in Sect. 5, within numerical simulations of the CCFM and BFKL evolutions in the presence of unitarity corrections.

4.1 Unitarity and Saturation momentum

In what follows we shall explain our method for effectively implementing saturation on the example of the BFKL equation (BFKL, ). This is interesting since the corresponding non–linear generalization which obeys unitarity is also known — this is the Balitsky–Kovchegov (BK) equation (Balitsky:1995ub, ; Kovchegov:1999yj, ) —, and thus it can be used to test our method. The derivation of the BK equation has been recently pushed to next–to–leading order accuracy (Balitsky:2006wa, ; Kovchegov:2006vj, ; Balitsky:2008zz, ; Balitsky:2009xg, ), but here we shall limit ourselves to its leading–order version, which is also the accuracy of the CCFM formalism. However, this LO version will be eventually extended to include a running coupling (both for BFKL and for CCFM), since the running coupling effects modify in an essential way the high energy evolution — these are the only NLO corrections which remain important for asymptotically high energy.

The (leading–order) BK equation for the unintegrated gluon distribution reads