# Evolution in opening angle combining DGLAP and BFKL logarithms

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## Abstract

We present an evolution equation which simultaneously sums the leading BFKL and DGLAP logarithms for the *integrated* gluon distribution in terms of a single variable, namely the *emission angle* of the gluon. This form of evolution is appropriate for Monte Carlo simulations of events of high energy \(pp\) (and \(p\bar{p}\)) interactions, particularly where small \(x\) events are sampled.

## Keywords

Anomalous Dimension BFKL Equation DGLAP Evolution Unintegrated Gluon Gluon Reggeisation## 1 Introduction

The aim is to devise an evolution equation for PDFs in the low \(x\) region which simultaneously incorporates, at the same level, both the DGLAP and BFKL leading logarithms. There has been attempts in this direction, which, however, have not been very convenient [1, 2]. In the Gribov et al. [1] paper the result was written in terms of an integral over Mellin moments and anomalous dimensions, while Marchesini [2] attempted to improve the CCFM equation by working in terms of highly unintegrated distributions which depended on six arguments.

Procedures to combine BFKL and DGLAP effects, based on CCFM, were implemented in the ‘Small \(x\)’ Monte Carlo [3] and in the ‘CASCADE’ Monte Carlo [4, 5]. These Monte Carlos were written in terms of an ‘effective’ transverse momentum, labelled \(q'\) and \(\bar{q}\), respectively, both variables being proportional to the square root of the gluon emission angle. However, in [3] the finite terms in the DGLAP gluon–gluon splitting function were neglected; and in [4, 5] there was no possibility to include the *full* DGLAP contribution, which is included in the evolution equation proposed here.

Another possibility to unify the BFKL and DGLAP equations was proposed by Kwiecinski et al. [6], where the role of the BFKL contribution was studied for the deep inelastic structure function \(F_2\). However, there, an integral equation was proposed for the unintegrated parton distribution. The equation was written in terms of the usual \(x,k_\mathrm{t}\) variables, and was not converted into the form of an evolution equation. It was already noted by Ciafaloni [7] that ordering in emission angle, provided by the coherence effect, plays an important role. Indeed this angular ordering was the basis of the CCFM integral equation. However, evolution in terms of the opening angle was not discussed.

Here we start with the integral equation analogous to that in [6], and based on this equation, we show how it is possible to obtain an expression which describes the evolution in angle of the emitted parton with respect to the initial proton direction (in the infinite momentum frame). The momentum of the parton transverse to the direction of the proton is denoted by \(k_\mathrm{t}\). A good feature of this evolution is that angular ordering of successive emissions is naturally provided by coherence effects. Therefore already at LO the results should be closer to experimental application. Another point is that the angular variable, \(\theta =k_\mathrm{t}/xp\), accounts for both DGLAP and BFKL large logarithmic intervals; log \(k_\mathrm{t}\) in DGLAP and log\((1/x)\) in BFKL. The evolution equation for PDFs is thus written, in terms of only two arguments—the emission angle \(\theta \) and the momentum fraction \(x\). In this sense its form is very close to the conventional evolution equations. So it should be straightforward to implement.

In the present paper we consider only LO evolution; that is the simultaneous summation of LO BFKL and LO DGLAP logarithms. However, it should be possible to follow the same logic so as to include the known NLO BFKL and DGLAP effects.

## 2 Unified BFKL–DGLAP evolution

*unintegrated*gluon distribution, \(f(x,k_\mathrm{t})\), written in integral form:

Strictly speaking, the BFKL kernel, \(\overline{\mathcal{K}}\), depends on the azimuthal angle^{1} \(\phi \) between \(k_\mathrm{t}\) and \(k_\mathrm{t}'\). However, here, for simplicity, in order not to introduce another variable, we have already integrated over \(\phi \) assuming a flat \(\phi \) dependence of \(f\). That is, we consider only the zero harmonic, which corresponds to the rightmost intercept.^{2}

## 3 Evolution in \(\theta \)

*integrated*gluon distribution, \(g(x,\theta )\), which contains both BFKL and DGLAP logarithms, in terms of the single variable–the gluon emission angle

^{3}\(\theta \). That is, a ‘unified’ evolution equation for \(\mathrm{d}g(x,\theta )/\mathrm{d}\ln \theta \). The relation between the (conventional) integrated gluon distribution, \(g\), and the distribution, \(f\), unintegrated over its transverse momentum is

When we change the limit of integration in (7) to \(\theta _1=\theta +\mathrm{d}\theta \) we have the usual DGLAP contribution, equivalent to the replacement \(\ln (k^2_{1t})=\ln (k_\mathrm{t}^2)+2\mathrm{d}\theta /\theta \), plus the contribution from the BFKL part arising from the increase of the available \(\ln (1/x')\) interval; \(\mathrm{d}\ln (1/x')=\mathrm{d}\ln (\theta )\). Indeed, for a relatively large \(k_\mathrm{t}\), the condition \(\theta ' < \theta \) in (7) limits the part of the \(x'\) domain in (1).

^{4}The lower limit of the \(z\) integration in the DGLAP part is given by

Since the LO contribution is now written in terms of an integral over \(\mathrm{d}\theta '/\theta '\), it appears that we may be able to find an evolution equation in the usual derivative form for the *integrated* distribution \(g(x,\theta )\). That is, it seems that we may be able to obtain an evolution equation for \(\mathrm{d}g(x,\theta )/ \mathrm{d}\mathrm{ln}\theta ^2\). But first we have some points we must investigate.

### 3.1 Ensuring the evolution is for an integrated distribution

This is an advantage of the evolution in terms of \(\theta \) in comparison with the conventional evolution in terms \(k_\mathrm{t}\) (or \(k^2\)). In the latter (\(k_\mathrm{t}\)) case, we face a contribution from \(k'_\mathrm{t}>k_\mathrm{t}\) in the BFKL part.^{5}

Using the extrapolation (13, 14), one may perform a new global parton analysis. For input we need to parametrise the DGLAP-like parton distribution at \(k_\mathrm{t}=k_0\) *only* in some limited interval of \(1>x>x_0\). Then the DGLAP part of the evolution will provide the input for the BFKL part at \(x=x_0\) at all \(k_\mathrm{t}>k_0\), while the contribution for \(k_\mathrm{t}<k_0\) will be given, say, by (13). Now all the energy- (i.e. \(1/x\)-) dependence at small \(x<x_0\) will be driven by the BFKL part of the equation, and not by the input distribution as in conventional DGLAP evolution.

Finally, we should mention that since the infrared domain is limited by the value of \(k_\mathrm{t}<k_0\), and not defined in terms of the angle \(\theta \), the evolution (12) should be considered only in the region of \(\theta >k_0/xp\), and not at some \(\theta >\theta _0\) domain with \(\theta _0=const\). Of course, formally, in infinite momentum frame the initial momentum \(p\rightarrow \infty \); so any \(\theta _0=const\) is acceptable. Nevertheless, it would be better to bear in mind the realistic condition \(\theta >k_0/xp\).^{6}

### 3.2 Energy-momentum conservation

While the DGLAP evolution conserves the energy (and the flavour) of system of partons this is not true for the LO BFKL equation. Formally in the leading \(\ln (1/x)\) approximation an additional energy of the new partons is negligibly small (\(\sim 1/\ln (1/x)\)), but numerically this maybe is not a negligible effect.

A problem is that in the second term of (16) we now sample the region \(\theta '>\theta \), since the function \(f(x,k'_\mathrm{t})\) depends on \(x\) and not on \(x/z\). Recall, however, that after the subtraction of the leading double-logarithmic term (which was included in the DGLAP part) the violation of energy conservation in the remaining BFKL part is rather small, and is caused only by *next-to-leading corrections*. Thus formally, at LO level, we may neglect the second term of (16); that is, the term which restores energy conservation. However, since the integral over \(k'_\mathrm{t}\) is well convergent for \(k'_\mathrm{t}>k_\mathrm{t}\), it is sufficient in the second term of (16), just to take a simple extrapolation into the \(\theta '>\theta \) domain using, at each value of \(x\), the ‘frozen’ anomalous dimension of the unintegrated gluon density, \(f(x,k'_\mathrm{t})\). To be more precise, we may in fact ensure exact energy-momentum conservation by performing a few iterations; where the previous iteration provides the values of \(f(x,k'_\mathrm{t})\) for \(\theta '>\theta \).

Notice from Fig. 1 that to obtain a PDF at small \(x\) using DGLAP evolution we have to start evolving from an input distribution at rather low \(x\) from the beginning. Analogously, in the BFKL case, to obtain a large \(k_\mathrm{t}\) gluon PDF, we need to start evolving from large \(k_\mathrm{t}\). Of course, both DGLAP and BFKL contain the double-log terms which allow DGLAP to evolve from large \(x\) (and BFKL to evolve from low \(k_\mathrm{t}\)). However, for example in the DGLAP case, if we start from large \(x\), then we will generate a PDF \(\propto \exp (\sqrt{(4\alpha _s N_c/\pi )\ln (1/x) \ln Q^2})\), but never containing a power of \(x\), that is, never^{7} one of the form \(x^{-\lambda }\). The evolution in \(\theta \) will be more physical, since it starts from a region of relatively large \(x\) and low \(k_\mathrm{t}\). This is more natural for an input PDF, which is driven by physics at large distances \(({\sim }0.5\) fm), corresponding to a parton confined inside a proton.

### 3.3 The Sudakov \(T\)-factor

*(which will change the final values of*\(k_\mathrm{t}\)

*and*\(x\)) are produced during the DGLAP evolution from scale \(k_\mathrm{t}\) up to the hard scale \(\mu \). This probability is given by so-called \(T\)-factor

### 3.4 The quark contributions

Since the splitting function corresponding to the quark to gluon transition, \(P_{gq}(z)\), contains a \(1/z\) singularity (analogous to that in \(P_{gg}\)) we have to consider a possible ‘BFKL’ contribution to this \(q\rightarrow g\) transition. Recall, however, that there is no high energy (\(\ln (1/x)\)) leading log BFKL term for quark exchange. Therefore within our LO approximation, in the quark cell we have to keep only the logarithmic \(dk^{'2}_\mathrm{t}/k^{'2}_\mathrm{t}\) (DGLAP-like) contribution with \(k'_\mathrm{t}<<k_\mathrm{t}\). Then the only possible form of the BFKL kernel \(K_{qg}(k,k')\) is again pure logarithmic \(1/k^2_\mathrm{t}\) [21, 22], which should be subtracted to avoid double counting. In other words, at LO level, the whole \(q\rightarrow g\) splitting is completely described by the usual DGLAP term.

## 4 Discussion

It is relevant to mention how the present approach compares with that of Refs. [23, 24, 25] and the references therein. In Ref. [23] a small \(x\) resummation of the BFKL contributions was performed for the DGLAP splitting functions, that is, for the anomalous dimension. However, the small-\(x\) power behaviour is still controlled by the input distribution, and not generated by the BFKL part of the evolution. Recall that the BFKL effects go beyond the anomalous dimension, and involve higher-twist effects. In Refs. [24, 25] the DGLAP-induced contributions were resummed to obtain the correction to the BFKL-Pomeron intercept. This achieved stability of the (next-to-leading-order) BFKL intercept by resumming a major part of the higher-order contributions. The procedure is very recursive equation, taking contributions from a large region of the phase space. The improved BFKL equation was not written in terms of the evolution of integrated parton densities. Again, it was claimed that the small \(x\) power behaviour is mainly controlled by the input distribution. In both approaches it was not shown that the angle is a good variable, which brings uniformity to the different contributions to the equation.

Our aim is different. We wish to determine an evolution equation for an *integrated* gluon distribution, which simultaneously sums both the leading BFKL and the DGLAP logarithms, in terms of a *single* variable. We have shown that the appropriate variable is the emission angle, \(\theta \), of the emitted gluon; giving an evolution equation for \(\partial g(x,\theta )/ \partial \mathrm{ln}\theta ^2\). This novel equation is given by (17) (or (20), when the quark contribution is included). It brings uniformity to the two different contributions to the equation. A crucial observation is that, although the right-hand side depends on \(g(x',\theta ')\), this does not pose a problem, since the contribution comes from \(\theta '<\theta \) where \(g(x',\theta ')\) is known from the previous evolution.

Recall that the inequality \(\theta '<\theta \) is provided by the kinematical constraint \(k^{'2}_\mathrm{t}<k^2_\mathrm{t}/z\) of (4), which simultaneously accounts for the major part of the higher-order BFKL next-to-leading contribution [8]. Besides this, we add to the BFKL part of our equation the next-to-leading term which provides energy-momentum conservation.

The evolution in \(\theta \) for the *integrated* gluon distribution, \(g(x,\theta )\), is in contrast to the conventional BFKL equation, which is written for the *unintegrated* gluon distribution, \(f(x,k_\mathrm{t}^2)\). In this case there is diffusion in log\(k_\mathrm{t}^2\) to larger values of \(k_\mathrm{t}\), as well as smaller \(k_\mathrm{t}\) and in terms of \(k'_\mathrm{t}\) integrals we have the contribution from \(k'_\mathrm{t}>k_\mathrm{t}\). Rather, \(\theta \) in the natural variable for evolution of an integrated distribution.

This form of ‘integrated’ evolution in terms of a *single* variable should be convenient for implementation in Monte Carlo simulations of events for high energy \(pp\) (and \(p\bar{p}\)) collisions, particularly where small \(x\) events are sampled. For instance, it would be useful to have the possibility to implement in a Monte Carlo generator the PDFs obtained independently from a global parton analysis, based on the angular evolution proposed here. Instead, for example, the gluon PDF used by CASCADE [4, 5] is evolved and fitted by the same CASCADE Monte Carlo description of a limited set of data.

## Footnotes

- 1.
For the DGLAP contribution we have a flat \(\phi \) dependence from the beginning, due to strong \(k_\mathrm{t}\) ordering.

- 2.
It was demonstrated in [10] that the full BFKL amplitude is well approximated by the sum of the leading ‘zero’ harmonic contribution and simple two-reggeised-gluon exchange.

- 3.
Recall that some Monte Carlo generators actually make use of the angular variable. However, while the HERWIG Monte Carlo [11, 12, 13] accounts for DGLAP evolution, it neglects the BFKL contribution (and the higher-twist BFKL effects), whereas the CASCADE Monte Carlo [4, 5] does not include the

*full*DGLAP splittings. - 4.
The condition \(z<1\) means that for low \(k'_\mathrm{t}<k_\mathrm{t}\), the upper limit of \(\theta '\) in the BFKL part is not \(\theta \), but it is \(\theta '_\mathrm{max}=\theta k'_\mathrm{t}/k_\mathrm{t}\).

- 5.
An alternative way to see that the evolution in \(\theta \) can be written in terms of integrated densities is to take the integral ‘by parts’, based on the relation \(\mathrm{d}(u)v=\mathrm{d}(uv)-u\mathrm{d}(v)\); see [14].

- 6.
At first sight, it appears that working in terms of \(\theta \) we get a result which depends explicitly on the incoming proton momentum \(p\). This is not completely true. For a very large \(p\) the logarithm of angle (\(\ln \theta \)) plays the role of (pseudo)rapidity, and under variation of \(p\) the argument \(\ln \theta ^2\) is simply shifted by a constant value.

- 7.
We could put \(x^{-\lambda }\) in the input distribution, but then \(\lambda \) is arbitrary, and not generated by BFKL dynamics.

## Notes

### Acknowledgments

MGR thanks the IPPP at the University of Durham for hospitality. This work was supported by the Federal Program of the Russian State RSGSS-4801.2012.2.

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