Kinematical constraint effects in the evolution equations based on angular ordering

We study effects of imposing various forms of the kinematical constraints on the full form of the CCFM equation and its non-linear extension. We find that imposing the constraint in its complete form modifies significantly way the shape of gluon density as compared to forms of the constraint used in the numerical calculations and phenomenological applications. In particular the resulting gluon density is suppressed for large values of hard scale related parameter and kT of gluon. This result might be important in description studies of jet correlations at Large Hadron Collider within CCFM approach.


Introduction
In the high energy limit of hadron scattering, in the regime where the center of mass energy is larger than any other available scale, perturbative approach to processes with high momentum transfer allows factorization of the cross section into a hard matrix element with initial off-shell gluons and an unintegrated gluon density [1,2]. The unintegrated gluon density is a function of the longitudinal momentum fraction x and transverse momentum k T of a gluon. After taking into account formally subleading corrections coming from coherence of gluon emissions, one is lead to the CCFM set of equations [3][4][5] which introduce gluon density dependent on hard scale related to the probe as well as one introduces unintegrated quark densities. The status of phenomenological relevance of CCFM framework is not fully satisfactory. In principle, it is a set of equations that should be the ideal framework for application to final states at high energies and covering DGLAP and BFKL domains. It has been implemented in the Monte Carlo event generator [6]. However, so far good agreement with high precision data has been successfully achieved only in rather inclusive processes like F 2 and Drell-Yan [7]. It is known that on the theory side the CCFM physics is still to be completed. Below we outline the main points. For instance: • the CCFM system of equations has been so far solved in decoupled approximation in the, gluonic (for short the equation for gluon density is usually called CCFM) and nonsinglet channels neglecting correlations between partons [8]. This might result in improper q x, k T p Figure 1: The plot visualizing kinematical variables used in the CCFM equation. The black blob represents hard process while the gray oval represents hadron.
treatment of the kinematical region where the gluon degrees of freedom are less dominant than sea quarks. Since gluon might be artificially dominating over quarks where it should actually be suppressed. Furthermore the corrections coming from inclusion of transversal momentum dependence to splitting functions P gq and P qq are not known and in order to have complete picture they should be calculated [8].
• the impact of kinematical effects introducing energy conservation in the CCFM evolution have not been investigated in all detail. As it turns out from our study it is necessary to revisit the inclusion of the so called kinematical constraint [9] into the CCFM equation.
The orgin of the kinematical constraint follows from refining the assumption that the tchannel off-shell gluon's 4-momentum is dominated by k 2 T . The kinematical constraint causes suppression of the gluon density and even overrides the angular ordering in regions of large k T .
• only recently the CCFM has been promoted to non-linear equation [10][11][12] therefore allowing for the possibility to investigate interplay of coherence effects and saturation [1] in exclusive processes like dijet production at the LHC [13][14][15][16][17][18][19][20]. In particular the important question is what is the role of the angular ordering and kinematical effect in the evolution at the non-linear level. The optimal form of initial conditions is also not known.
This publication is a continuation of the work done in [21,22]. We compare numerically forms of non-Sudakov form factor used in the literature and solve the full CCFM equation and its non-linear extension -including the 1/(1 − z) pole and kinematical constraint in the kernel of the equation i.e. not only in the non-Sudakov form factor. However we keep α s constant in order to have clear picture of the role of the kinematical effects.

The CCFM equation
The CCFM equation for gluon density reads: where k ′ = |k + (1 − z)q| and the moduli of the two dimensional vectors transversal to the collision plane are denoted |k| ≡ k T , |q| ≡ q T , x is gluon's longitudinal momentum fraction and ᾱ s = N c α s /π. Also the rescaled momentum is introduced asq = |q| = q T /(1 − z). The splitting function assumes the form: The Sudakov form-factor evaluated in double logarithmic approximation reads: The non-Sudakov form-factor regularizing 1/z singularity in angular ordered region is: The kinematical constraint The integration overq in the equation (1), although being constrained from below by the soft cutoff Q 0 , is not constrained by an upper limit thus violating the energy-momentum conservation. Moreover in the low x formalism one requires that in the denominator of the off-shell gluon propagator one keeps terms that obey |k 2 | = k 2 T . In order to be consistent as argued in [4] the non-Sudakov form-factor should be accompanied by a kinematical constraint limiting the above integration overq. In approximated form it reads After including it in the (4) we have: The condition (5) at z ≪ 1 guaranties that |k 2 | ≃ k 2 T . In [9] it has been extended to region including also the case when z ∼ 1. Below we re-obtain the full form of kinematical constraint emphasizing the its role in conservation of energy. Having k = z p + +zp − + k ⊥ with p + and p − being the initial state gluon momenta + and − components, we can write the expression for k 2 Note, that for the full propagator |k 2 | > k 2 T , thus the full propagator causes stronger suppression of the amplitude which has to be taken into account when using |k 2 | ≃ k 2 T . A cut-off like (7) is a simple implementation of the suppression. The condition |k 2 | ≃ k 2 T translates approximately to Using the identity q 2 =z(1 − z)ŝ − q T 2 = 0, we can expresszŝ = q T 2 /(1 − z) and insert it into (8) to obtain: The lower bound on z > x results in the upper bound on q 2 T < k 2 T /x ≃ŝ providing local condition  (6), the brown line corresponds to form-factor given by formula (11) while blue line to form-factor given by formula (12). The form-factor given by (11) is multiplied by the function for energy-momentum conservation. The CCFM with the kinematical constraint included reads [9]: and the non-Sudakov form-factor after inclusion of the full form of the kinematical constraint assumes form Please note the presence of the function θ At the level of the kernel of the BFKL equation obtained directly from Feynmann diagrams (virtual and real corrections are kept apart) one imposes the kinematical constraint at the so called "unresummed" level on real emissions. However in [9] it has been observed, at a level of BFKL equation, that imposing the kinematical constraint on unresummed equation and performing algebraic transformations leading to resummed equation one obtains a θ-function affecting the full kernel. The starting point of the CCFM is a resummed kernel, therefore this result suggested how the complete form of the kinematical constraint should be imposed on the kernel of CCFM. The authors of [9] solve the CCFM at small z limit therefore the function θ k 2 T (1−z)q 2 − z is neglected and in most of the phenomenological and theoretical applications of the CCFM this term is neglected [7,15,[23][24][25][26][27][28] as well. The following form of non-Sudakov form factor is usually being used: and the discussed above the θ-function is not taken into account. As shown in the Fig. 2 the simplified non-Sudakov form-factor (6) approximates well the exact one (11) in regions of z where the θ-function is not affecting the kernel while the (12) is quite different. In our analysis we consider following equations: where the splitting function includes non-Sudakov given by formula (11) and where the non-Sudakov given by formula (12).

Saturation effects and kinematical constraint combined
To account for gluon recombination at large gluon densities the CCFM equation has been promoted to non-linear equation by including a quadratic term [10][11][12] which reads: where we also included the kinematical constraint of the form (11) in the kernel. Simpler versions of the equation above have been already analyzed in [22] and it has been observed that • the equation leads to phenomenon called saturation at the saturation scale [22,26] • the saturation strongly suppresses the gluon density at low x and low k T The natural question arises how are these results modified when some of the approximations are not taken and how are they modified if the kinematical effect is imposed in the full form.

Numerical results and the discussion
We present the numerical results withᾱ S = 0.2 and the parameter Q 0 set to Q 0 = 1 GeV. To study the variation of the results depending on Q 0 we produce a solution also for Q 0 = 0.    (14), black (solid) -equation (15), blue (dotted) -equation (16).

Linear equations
We use an initial condition which includes resummed virtual and unresolved contributions, according to [22] and [27], in the form where is the Regge form-factor. The first observation we make from the plots in Fig. 3-5 is that the solutions of equations we study differ significantly. The solutions exhibit also similar features. Solutions of both versions of the kernel with kinematical constraint exhibit a local maximum as functions of k T and p with x and p or k T correspondingly fixed. The positions of local maximum in the plots of p dependence are correlated with the value of k T , with a shift to higher k T for the solution of the equation (15). The peak can be explained by the fact that the contribution of the integral on the right hand side of (14) peaks at around k T ∼ p. To point out: the peak is a result of presence of θ (p − zq) -angular ordering condition. Similar peaks are present also in the plots of k T dependence and resemble Sudakov suppression of k T scales of the order of p in case of   (14), black (solid) -equation (15), blue (dotted) -equation (16).
equation (14) [20]. However, in the case of (15) it seems, that the position of the peak does not depend on the value of p. It seems that the kernel with the kinematical constraint included in an implicit way (15) produces similar effect, though weaker, as an implicit inclusion of the θ-function only in the case of p dependence, but for the k T dependence leads to much smaller suppression. Thus the peak observed in solution of (15) is 'buried' under the result of the evolution. We can conclude that the peak in the k T dependence is a result of an interplay of the inclusion of the explicit θ k 2

T
(1−z)q 2 − z factor and the Sudakov effect.

Non-linear equation
We set the parameter characterizing the strength of the non-linear term R the value R = 1/π GeV in the equation (16). By comparing the CCFM and KGBJS equations we see that the equations give quite similar distributions. This effect (for our choice of the starting scale Q 0 ) comes from the fact that the kinematical constraint suppresses the growth of the gluon so much that the non-linear effects enter only at very low x. Observations made in previous paragraphs are confirmed in  2-dimensional plots ( Fig. 7-8), where we plot absolute relative difference of two amplitudes, solutions of the CCFM and the KGBJS equations, defined by the quantity The function β (x, k T , p), introduced before in [22], can be used to measure the strength of the non-linear effects and to define a saturation scale using the condition: or p-related saturation scale: The conditions above can be seen as equipotential lines in the 2-dimensional plots in Fig. 7-8 where different equipotential lines correspond to different constants on the right-hand side of the equation above. The change in the slope of the β (x, Q s (x, p) , p) at around k T = p reported in [22], apparent in Fig. 7-8, can be understood in the context of the peak at p ∼ k T (Fig. 3-4). For k T > p the contribution of the integral on the right-hand side of the CCFM equation decreases and the gluon density is dominated by the initial condition. By comparing the plots Fig. 7-8 to analogous plots in [22] we see that their main features are very similar. We therefore conclude that the low − x approximation of the KGBJS and CCFM equations taken in [22] does not, at least, modify the relative difference between linear and non-linear equation since the modifications of the kernel did not spoil the saturation pattern visible in the full equation.

Dependence on the starting scale
The starting scale Q 0 dependence is studied in Fig. 9-10 using equations (14) and (16). We observe variation from few up to 25% mostly in the local maximum of the k T and p distributions. There is a peculiarity of the dependence on Q 0 of the k T distributions near the boundary k T = Q 0 which can be explained by the dependence on the initial condition on the parameter Q 0 . Near the point k T = Q 0 the initial condition dominates and since it depends on the parameter the solutions also depend strongly on this parameter. We can tell, though, that for k T ≫ p and p ≫ k T the difference diminishes.

Conclusions and outlook
We have solved linear and non-linear version of the CCFM equation with the full kernel, constant α s and kinematical constraint included in two different ways. Our numerical results make us conclude that the inclusion of the kinematical constraint in its full form into the CCFM equation seems to affect largely the distribution of gluons. In particular inclusion of the kinematical constraint without omitting the θ-function in the kernel of the equation causes strong suppression of the solution for all values of transversal momenta k T . The effect is stronger than the implicit kinematical constraint included via modification of the non-Sudakov form-factor widely used in literature. We expect that the suppression of the distribution will have significant effects on description of decorelations in azimuthal angle of forward-central jets [15,29]. In particular we expect that the inclusion of the θ-function is crucial for suppression of the cross section of emission of gluons in small angles since this configuration corresponds to large values of k T . However, the conclusive comparison of the different approaches presented in this article can be done only after fitting the initial conditions of the considered equations to data, this we, however, leave for future tasks.