# On the different forms of the kinematical constraint in BFKL

## Abstract

We perform a detailed analysis of the different forms of the kinematical constraint imposed on the low *x* evolution that appear in the literature. We find that all of them generate the same leading anti-collinear poles in Mellin space which agree with BFKL up to NLL order and up to NNLL in \(N=4\) sYM. The coefficients of subleading poles vanish up to NNLL order for all constraints and we prove that this property should be satisfied to all orders. We then demonstrate that the kinematical constraints differ at further subleading orders of poles. We quantify the differences between the different forms of the constraints by performing numerical analysis both in Mellin space and in momentum space. It can be shown that in all three cases BFKL equation can be recast into the differential form, with the argument of the longitudinal variable shifted by the combination of the transverse coordinates.

## 1 Introduction

The sharp increase of the structure function \(F_2\) in Deep Inelastic Scattering process with decreasing value of Bjorken variable *x* was one of the biggest discoveries at HERA collider [1, 2]. This rise of \(F_2\) is being driven in QCD by the increase in the gluon density due to dominance of gluon splitting when the parton’s longitudinal momentum fraction is small. Understanding the behavior of the parton densities at small *x* is not only crucial for the DIS processes but also for the high energy hadronic collisions in accelerators and in the cosmic ray interactions. The LHC opened up the kinematic regime which is particularly sensitive to small *x* values and thus the region where the gluon density dominates. The increase of the gluon density towards small *x* translates then into the rise of the hadronic cross sections for variety of observables with the increasing center-of-mass energy. Within the framework of collinear factorization [3] the evolution of the gluon density is governed by the DGLAP [4, 5, 6, 7, 8] evolution equations which lead to the scaling violations. Although the DIS data are well described by the fits based on the DGLAP evolution, there are hints of novel physics phenomena that can be present at very small values of *x* [9]. The alternative approach to the description of the parton evolution stems from the analysis of the Regge limit in QCD, where the energy of the interaction is assumed to be large \(s\rightarrow \infty \), or in the case of the DIS process, when \(x\rightarrow 0\), while the scale \(Q^2\) is perturbative but fixed. In this limit the cross section is dominated by the exchange of the so-called hard Pomeron which is the solution to the famous BFKL evolution equation [10, 11, 12, 13, 14, 15], see [16] for a review. The two notable differences between DGLAP and BFKL evolution equations are that in the latter the solution exhibits the power like behaviour, \(x^{-\lambda }\) at small values of *x* and the transverse momenta in the gluon cascade are not ordered. The solution thus depends on the transverse momenta of the exchanged reggeized gluons. BFKL equation is known up to NLL order in QCD [17, 18] and up to NNLL order in \(N=4\) sYM theory [19, 20, 21]. A known issue with the BFKL at the NLL order is that the corrections are large compared to the LL result, which renders the result unstable. Thus it was early realized that higher order corrections need to be resummed. In the pioneering works [22, 23, 24] it was pointed out that there are kinematical constraints which need to be included in the evolution. Formally of higher order, when viewed from the perspective of the Regge limit, nevertheless such corrections are numerically very large as was established in [24]. Kinematical constraint was later included in the formalism that combined DGLAP and BFKL equations, and it was demonstrated that very good description of the HERA data was obtained [25]. Various elaborated resummation schemes for the small *x* evolution were later developed [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]. More recently the collinear resummation was applied to the non-linear evolution equations in transverse coordinate space [41, 42, 43, 44].

Given the increased precision of the experimental data and progress in theoretical computations, especially the information about the higher orders in \(N=4\) sYM theory, it is worth to revisit the assumptions in the resummed schemes. Even though the kinematical constraint is well motivated, different approximations, which are used are usually done based on general grounds (i.e. for example low *z* approximation) but without detailed quantitative analysis. The differences between different forms of constraints may be of subleading order but it could happen that they are quite large numerically. In context of the BFKL equation, which utilizes the kinematical constraint as an ingredient, usually one uses a particular form of this constraint, namely the one which limits the transverse momenta of the exchanged gluons. Such choice is also dictated by practical applications, since in this form the angular variable in the evolution equation is integrated over which reduces the number of variables and makes it easier to solve the equation. On the other hand, there are different forms of the kinematical constraint that appear in the literature, which put limit on the transverse momenta of the emitted gluons. Actually in the CCFM equation which receives contribution from both large and low *x* it is a natural choice [45].^{1}

The main goal of this paper is to perform a systematic semi-analytic and numerical comparison of the small *x* evolution equation with different forms of the kinematical constraint, which appear in the literature.

In the analysis performed in this work we found that the constrains all generate the same structure in the Mellin space for the leading and first subleading anti-collinear poles up to NLL order in \(\ln 1/x\) in QCD and in \(N=4\) sYM and up to NNLL in \(N=4\) sYM. As such they are all consistent up to the highest known order of perturbation theory. The calculations were performed both in the Mellin space and also by performing the direct solution of the BFKL with kinematical constraint in the momentum space. To this aim we have developed new numerical method for the solution of the BFKL equation with kinematical constraint which is based on reformulating this equation in the differential form. It turns out that all the constraints can be recast as effective shift of the longitudinal momentum fraction in the argument of the unintegrated gluon distribution. We see that, as expected the solutions are substantially reduced with respect to the LL case, but the differences between the different forms of the kinematical constraints are non-negligible and can reach up to 10–15 % for the value of the intercept. This can have important effect for the phenomenology.

## 2 Kinematical constraints

*x*was first considered in works [22, 23, 46].

*x*unintegrated gluon density, \(\mathbf{k}_{{T}}\) and \(\mathbf{k}_{{T}}'=\mathbf{k}_{{T}}+\mathbf{q}_{{T}}\) are the transverse momenta of the exchanged gluons and \(\mathbf{q}_{{T}}\) is the transverse momentum of the gluon emitted. The longitudinal momentum fractions of the exchanged gluons are

*x*and \({x \over z}\) respectively. The flow of the momenta in the BFKL cascade is illustrated in diagram Fig. 1. We will also use the notation \({k}_{{T}}^2\equiv \mathbf{k}_{{T}}^2\) for the squared transverse momenta for the rest of the paper. The rescaled coupling constant is defined as \(\bar{\alpha }_s\equiv \frac{\alpha _s N_c}{\pi }\). In the leading logarithmic approximation the integration over \({\mathbf{q}_{{T}}}\) in the Eq. (1), is not constrained by an upper limit thus violating the energy-momentum conservation. Thus for example in the context of the DIS process in principle there should be a limit on the integration over the transverse momentum

*x*is the Bjorken variable and \(W^2\) is the c.m.s energy squared of the photon–proton system in DIS.

There is a stronger constraint arising however from the requirement that in the low *x* formalism the exchanged gluons have off-shellness dominated by the transverse components, i.e. one keeps terms that obey \(|k^2| \simeq {k}_{{T}}^2\). The derivation of the kinematical constraint here follows [24].

*z*limit (9) can be approximated to

*x*approximation to the CCFM evolution in [24]. The lower bound on

*z*, i.e. \(z>x\) results in the upper bound on \({q}_{{T}}^2<{k}_{{T}}^2/x\) providing local condition for energy-momentum conservation.

## 3 Comparison of improved kernel with results in \(N=4\) sYM

Since the works of [23, 24] it is known that the kinematical constraint provides very important contribution to the NLL order and beyond in BFKL. Even though formally of subleading order, numerically this correction is very large and leads to the strong reduction of the intercept, aligning the BFKL equation more with phenomenology, for selected early works which implement kinematical corrections in BFKL see for example [25, 54, 55, 56]. At the NLL order the kinematical constraint, when viewed as a correction in the Mellin space gives rise to the cubic poles in the Mellin conjugated variable to the transverse momentum in the BFKL eigenvalue. The \(N=4\) sYM eigenvalue at LL is identical to the QCD case, and at NLL the same leading cubic poles appear in both theories [57]. Since the constraint is originating from the kinematics specific to the Regge limit of the cascade, one could expect it to be universal in both theories. It is useful to explore whether or not the kinematical constraint correctly reproduces the terms in the higher order NNLL known only in the supersymmetric case [19, 20, 21] and what are the differences between the different forms of the kinematical constraint discussed in the literature.

In this section we shall perform a detailed comparison of leading and subleading poles in the Mellin space originating from the kinematical constraint with the results in \(N=4\) sYM case up to NNLL. First, we shall perform the comparison on the example of the constraint (12). Later on, in Sect. 4, we shall check that the results are consistent for different forms of the kinematical constraints up to certain order of poles, and we shall identify this order. In that way we can quantify the level of differences between different forms of the constraints.

### 3.1 NLL and NNLL of N = 4 sYM and \(\omega \) expansion

Let us begin with recalling the definition of the BFKL eigenvalue and the Mellin transform.

*z*dependence of the kernel

*K*due to the kinematical constraint. Of course the fixed order LL and NLL kernels do not have the \(\omega \) dependence, it is only the resummed kernel, in this case the kernel with the kinematical constraint that obtains this additional dependence. As is well known, see for example [24, 25, 26, 34, 51], such dependence in turn generates the series in \(\alpha _s\) and therefore resums the contributions from all orders in the strong coupling. The result with the constraint (12) is well known [23]

*x*Bjorken is defined as \(x=Q^2/s\) with \(Q^2\) the (minus) virtuality of the photon. For the case of the different process, like for example Mueller-Navelet jets with comparable transverse momenta, the appropriate variable would be \(QQ_0/s\), where \(Q \simeq Q_0\) are the scales of the order of transverse momenta of the jets. The eigenvalue in this case would be different from (16) as it should correspond to the symmetric choice of scales. Therefore one needs to perform the scale changing transformation which generates terms starting at an appropriate order. Up to NLL order this was discussed in [17, 18, 26]. We shall recall in detail the scale changing transformation, and what terms it generates up to NNLL in the next subsection, here we shall directly start from the symmetric counterpart of the eigenvalue (16) which has the following form

^{2}\(\omega _1=\bar{\alpha }_s\chi _0+\bar{\alpha }_s^2 \chi _1\), substituting again into (21) and extracting terms proportional to \(\bar{\alpha }_s^2\) which results in

### 3.2 Scale changing transformation at NNLL

In this section we shall recall the scale changing transformation and compute it to NNLL order. The scale changing transformations were discussed in [18], see also [34, 51].

*Q*,\({Q}_{{0}}\) are hard scales for this process. For example in Mueller-Navelet process of production of two jets with comparable transverse momenta \(Q\approx {Q}_{{0}}\). On the other hand in DIS \(Q\gg {Q}_{{0}}\). The equation for

*G*as a function of rapidity is

### 3.3 Leading and subleading poles in expansion of kernel with kinematical constraint

Now, we can get back to expression (54). Knowing that the derivative \(\chi ^{(i)}\) has the leading pole \(\sim {\gamma ^{-(1+i)}}\), and a vanishing subleading pole, we can conclude that the leading pole in \(\chi _{k+1}\) is \(\sim \gamma ^{-(2k+3)}\), and its \(\gamma ^{-(2k+2)}\) pole vanishes.

## 4 Properties of the kernel in Mellin space

In this section we shall investigate in detail the differences between the different constraints on the level of the Mellin transform.

*u*to 1 /

*u*. The angular integral can be performed, resulting in the hypergeometric function,

*z*approximation, is given by [24]

The first feature of the effective kernels in all cases is that they are strongly modified in the large \(\gamma \) region. The results from the constraint (9) and (12) are very close in the \(\gamma \simeq 1\) region. Comparing Eqs. (66) and (16) we found that they are in fact identical at \(\gamma =1\) for any value of \(\omega \). One can observe that for \(z\rightarrow 1\) both (12) and (9) lead to a strong suppression of the anti-collinear phase-space, i.e. \({k}_{{T}}^2 < {k}_{{T}}^{'2}\), whereas even for \(z\simeq 1\) the constraint (8) leaves some phase space for integration. On the other hand the constraints (10) and (12) are very close in the small \(\gamma \) region while the (9) is lower in that range. This can be understood from the fact that in the collinear region, i.e. for \({k}_{{T}}^2 > {k}_{{T}}^{\prime 2}\) the latter constraint is stronger, particularly if *z* is large.

Next, we have checked numerically the coefficients in front of the poles in \(\gamma \) and found agreement between different forms of constraints in the leading \(\sim 1/(1-\gamma )^{2k+1}\) and subleading poles \(\sim 1/(1-\gamma )^{2k}\) poles when the expressions for the kernels are expanded out to NLL and NNLL.

There is one notable difference between (9) and the other two constraints, in that in the first one the collinear pole in \(\gamma \) is absent. This can be understood as for large values of \(z>1/2\) constraint (9) implies the restriction of the integration over collinear region as well.

## 5 Differential form of the evolution equation with kinematical constraints

The evolution equation in the presence of the kinematical constraint becomes an integral equation, over the longitudinal variable *z* as is evident from Eq. (60). Equations in this form were solved in the momentum space via different techniques in the past. One technique utilized in the past in Ref. [25] was based on the extension of the method introduced in [59]. This method incorporates the interpolation in two variables *x* and \({k}_{{T}}^2\) with orthogonal polynomials. By doing this the integral equations can be transformed into the set of linear algebraic equations and solved by simply inverting the large matrix. An alternative method was used in [34] and it was checked that it coincides with the one proposed in [25].

*x*is motivated by the reduction of computational complexity of the integral on the right hand side of the equation. After performing the derivative we find, that no matter what is the form of the kinematical constraint the resulting equation can be written in the differential form and has the same form containing two terms - real emission term with shifted argument

*x*and virtual correction term - under the remaining integral over the emitted momentum \(\mathbf{q}\). The argument of the real emission term is shifted such, that if the kinematical constraint takes the form \(\theta \left( k_C\left( \mathbf{q},\mathbf{k}\right) -z\right) \), then the argument of the unintegrated gluon density function in the real emission term changes in following way:

*x*the equation with kinematical constraint in the low

*z*approximation, Eq. (10). The equation formally differentiated in

*x*reads

*x*results in

## 6 Numerical method and results

### 6.1 Numerical results

*x*so that the integral over the kernel can be extended to large

*x*values. We introduce the lower cutoff onto the transverse momenta \(Q_0=0.1 \; \mathrm{GeV}\). The results were obtained for the fixed value of the coupling constant \(\bar{\alpha }_s=0.2\).

*x*and different forms of the kinematical constraint are shown in plots in Fig. 4 together with the solution to the LL BFKL. The results reflect the trend of the semi-analytical results from the Sect. 4. In plots of \({{k}_{{T}}}^2\) dependence for \(x=10^{-2}\) it is clear that the solution with the full kinematical constraint and the solution of the equation with the kinematical constraint in the \({k}_{{T}}^{\prime 2}<k_T^2/z\) approximation get close to each other for low \({k_T}^2\). On the other hand, for large values of \({k}_{{T}}^2\) it is the two constraints (10) and (9) which come close to each other and (9) has a different behavior. This is related to the different behavior in the collinear limit, i.e. the fact that constraint (9) introduces modification in the collinear limit as well as discussed previously. In any case the solution with the constraint (10) always leads to the results which is largest, consistent with the analysis from the Mellin space. Similar conclusions can be reached from the analysis of the plots with the

*x*dependence for small \({k_T}^2\) shown in Fig. 5. The small

*z*approximation of the kinematical constraint results in the solution being closer to the solution of the equation with the \({q_T}^2\simeq {k^{\prime }_T}^2\) approximation of the kinematical constraint for large \({k_T}^2\). From Fig. 5 we see the onset of the power behaviour in

*x*, and in all versions of the kinematical constraint the intercept is strongly reduced with respect to the LL approximation. The differences between the solutions with different versions of the kinematical constraint are much smaller than between resummed and LL result. Nevertheless, the differences between different versions of the kinematical constraint are non-negligible and can reach up to a factor 2-3 depending on the range of

*x*and \({k}_{{T}}\).

## 7 Conclusions

In the paper we have performed a detailed analysis of the three versions of kinematical constraints imposed onto the momentum space BFKL equation. We used the fixed coupling BFKL equation as a basic equation on which kernel’s the constrains are imposed. This choice allowed us to investigate semi-analytically the properties of the kinematical constrains in the Mellin space and to compare to known results in sYM version of the BFKL equations where the results are known up to NNLL order. In particular, we observed that the leading poles in Mellin space generated by the kinematical constraint obey maximal transcendentality principle. In addition we proved that the subleading poles vanish at NLL and NNLL order both for BFKL and its sYM version, and that such feature can be expected to hold to all orders. Furthermore we observed that the \(\chi _{eff}\) function obtained with the most general version of the constraint i.e. (9) for large values of \(\gamma \) agrees well with the more commonly used version of the constraint i.e (12). However, at small values of \(\gamma \) the \(\chi _{eff}\) function with (12) agrees with (10) and both of them differ from the case when constraint (9) is imposed. In particular, it is important to note that the last one changes the collinear structure of the kernel i.e. it limits *z* to \(z<1/2\). While on formal grounds this effect might be neglected in the small *z* limit it has nonnegligible impact on the resulting gluon density. We demonstrate that by numerical solutions of the momentum space BFKL equation for unintegrated gluon density for all versions of the constraint as well as for the unmodified BFKL equation. The understanding of action of the kinematical constraint across the whole region of *z* will be particularly important for ongoing efforts to unify the BFKL and DGLAP evolution schemes [60, 61, 62] where effects of kinematical constrains may play a crucial role. Furthermore, the inclusion of the kinematical constraint is known to be relevant for BFKL phenomenology [25, 63, 64, 65, 66, 67, 68]. However, so far only the simplest version of it was used in the context of BFKL. It will be interesting to see how different versions of the constraint perform while a fit to \(F_2\) data is done or what are the differences when the description of processes probing gluon density at very low *x* like high energy neutrinos or very forward dijet data is attempted. Furthermore, the limit of infinite strong coupling in N=4 sYM could be investigated for the resummed models with different constraints, which would provide important information on the resummation procedure at high energy.

## Footnotes

## Notes

### Acknowledgements

We thank Stanisław Jadach for comments and discussions. We thank Radek Zlebcik for discussions and collaboration at the initial stages of the project. Krzysztof Kutak acknowledges the support of the COST *Action CA16201 Unraveling new physics at the LHC through the precision frontier*. This work was supported by the Department of Energy Grants No. DE-SC-0002145 and DE-FG02-93ER40771, as well as the National Science centre, Poland, Grant No. 2015/17/B/ST2/01838.

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