Dijet correlations in $pp$ collisions at RHIC
We compare results of -factorization approach and next-to-leading order collinear-factorization approach for dijet correlations in proton-proton collisions at RHIC energies. We discuss correlations in azimuthal angle as well as correlations in two-dimensional space of transverse momenta of two jets. Some -factorization subprocesses are included for the first time in the literature. Different unintegrated gluon/parton distributions are used in the -factorization approach. The results depend on UGDF/UPDF used. Limitations due to leading jet condition are discussed.
Dijet correlations in collisions at RHIC
\FullConferenceHigh-pT physics at LHC
March 23-27 2007
University of Jyväskylä, Jyväskylä, Finland
The jet correlations are interesting in the context of recent detailed studies of hadron-hadron correlations in nucleus-nucleus  and proton-proton  collisions. Those studies provide interesting information on the dynamics of nuclear and elementary collisions. Effects of geometrical jet structure were discussed recently in Ref.. No QCD calculation of parton radiation was performed up to now in this context. Before going into hadron-hadron correlations it seems indispensable to better understand correlations between jets due to the QCD radiation. In this paper we address the case of elementary hadronic collisions in order to avoid complicated and not yet well understood nuclear effects. Our analysis should be considered as a first step in order to understand the nuclear case in the future. In leading-order collinear-factorization approach jets are produced back-to-back. These leading-order jets are therefore not included into correlation function, although they contribute a big () fraction to the inclusive cross section.
The truly internal momentum distribution of partons in hadrons due to Fermi motion (usually neglected in the literature) and/or any soft emission would lead to a decorrelation from the simple kinematical back-to-back configuration. In the fixed-order collinear approach only next-to-leading order terms lead to nonvanishing cross sections at and/or (moduli of transverse momenta of outgoing partons). In the -factorization approach, where transverse momenta of gluons entering the hard process are included explicitly, the decorrelations come naturally in a relatively easy to calculate way. In Fig.1 we show -factorization processes discussed up to now in the literature [5, 6, 7]. The soft emissions, not explicit in our calculation, are hidden in model unintegrated gluon distribution functions (UGDF). In our calculation the last objects are assumed to be given and are taken from the literature [8, 9, 10].
In addition we include two new processes (see Fig.2), not discussed up to now in the context of -factorization approach. We shall discuss their role at the RHIC energy = 200 GeV.
Furthermore we compare results obtained within the -factorization approach and results obtained in the NLO collinear-factorization. Here we wish to address the problem of the relation between both approaches. We shall identify the regions of the phase space where the hard processes, not explicitly included in the leading-order -factorization approach, dominate over the contributions calculated with UGDFs. We shall show how the results depend on UGDFs used.
We shall concentrate on the region of relatively semi-hard jets, i.e. on the region related to the recently measured hadron-hadron correlations at RHIC.
The cross section for the production of a pair of partons (k,l) can be written as
and and are so-called transverse masses defined as , where is the mass of a parton. In the following we shall assume that all partons are massless. The objects denoted by and in the equation (1) above are the unintegrated parton distributions in hadron and , respectively. They are functions of longitudinal momentum fraction and transverse momentum of the incoming (virtual) parton.
After some simple algebra one obtains a handy formula:
This 5-dimensional integral is now calculated for each point on the map .
Up to now we have considered only processes with two explicit hard partons. Now we shall discuss also processes with three explicit hard partons. In Fig.3 we show a typical process and kinematical variables needed in the description of the process. We select the particle 1 and 2 as those which correlations are studied. This is only formal as all possible combinations are considered in real calculations.
The cross section for can be calculated according to the standard parton model formula:
where the longitudinal momentum fractions are evaluated as
After a simple algebra  we get finally:
where is restricted to the interval . The last formula is very useful in calculating the cross section for particle 1 and particle 2 correlations.
Let us concentrate first on processes calculated within -factorization approach.
In Fig.4 we show two-dimensional maps in for all -factorization processes shown in Fig.1 and Fig.2. Only very few approaches in the literature include both gluons and quarks and antiquarks. In the calculation above we have used Kwieciński UPDFs with exponential nonperturbative form factor ( = 1 GeV) and the factorization scale = 100 GeV.
In Fig.5 we show fractional contributions (individual component to the sum of all four components) of the above four processes on the two-dimensional map . One point here requires a better clarification. Experimentally it is not possible to distinguish gluon and quark/antiquark jets. Therefore in our calculation of the dependence one has to symmetrize the cross section (not the amplitude) with respect to gluon – quark/antiquark exchange (). While at midrapidities the contribution of diagram + is comparable to the diagram , at larger rapidities the contributions of diagrams of the type B dominate. The contribution of diagram is relatively small in the whole phase space. When calculating the contributions of the diagram and one has to be careful about collinear singularity which leads to a significant enhancement of the cross section at =0 and , i.e. in the one jet case, when both partons are emitted in the same direction. This is particularly important for the matrix elements obtained by the naive analytic continuation from the formula for on-shell initial partons. The effect can be, however, easily eliminated with the jet-cone separation algorithm .
For completeness in Fig.6 we show azimuthal angle dependence of the cross section for all four components. There is no sizeable difference in the shape of azimuthal distribution for different components.
The Kwieciński approach allows to separate the unknown perturbative effects incorporated via nonperturbative form factors and the genuine effects of QCD evolution. The Kwieciński distributions have two external parameters:
the parameter responsible for nonperturbative effect (for details see ),
the evolution scale (for details see ).
While the latter can be identified physically with characteristic kinematical quantities in the process , the first one is of nonperturbative origin and cannot be calculated from first principles. The shapes of distributions depends, however, strongly on the value of the parameter in which the inital momentum distribution is encoded. This is demonstrated in Fig.7 where we show angular correlations in azimuth for the subprocess. The smaller the bigger decorrelation in azimuthal angle can be observed. In Fig.7 we show also the role of the evolution scale in the Kwieciński distributions. The QCD evolution embedded in the Kwieciński evolution equations populate larger transverse momenta of partons entering the hard process. This significantly increases the initial (nonperturbative) decorrelation in azimuth. For transverse momenta of the order of 10 GeV the effect of evolution is of the same order of magnitude as the effect due to the nonperturbative physics of hadron confinement. For larger scales of the order of 100 GeV, more adequate for jet production, the initial condition is of minor importance and the effect of decorrelation is dominated by the evolution. Asymptotically (infinite scales) there is no dependence on the initial condition provided reasonable initial conditions are taken.
In Fig.8 we show the maps for different UGDFs and for processes in the broad range of transverse momenta 5 GeV 20 GeV for the RHIC energy W = 200 GeV. In this calculation we have not imposed any particular cuts on rapidities. We have not imposed also any cut on the transverse momentum of the unobserved third jet in the case of calculation. The small transverse momenta of the third jet contribute to the sharp ridge along the diagonal . Naturally this is therefore very difficult to distinguish these three-parton states from standard two jet events. In principle, the ridge can be eliminated by imposing a cut on the transverse momentum of the third (unobserved) parton . There are also other methods to eliminate the ridge and underlying soft processes which is discussed in Ref..
When calculating dijet correlations
in the standard NLO () approach we have taken all possible dijet
combinations. This is different from what is usually taken in
experiments , where correlation between leading jets
are studied. In our notation this means and
. When imposing such extra condition on our NLO
calculation we get the dash-dotted curve in
In this case for .
This vanishing of the cross section is of purely kinematical origin.
Since in the -factorization calculation only two jets are explicit,
there is no such an effect in this case.
This means that the region of should
be useful to test models of UGDFs.
For completeness in Fig.10 we show a
two-dimensional plot with imposing the leading-jet
Surprisingly the leading-jet condition removes a big part of
the two-dimensional space. In particular, regions with
and cannot be populated via
Motivated by the recent experimental results of hadron-hadron correlations at RHIC we have discussed dijet correlations in proton-proton collisions. We have considered and compared results obtained with collinear next-to-leading order approach and leading-order -factorization approach.
In comparison to recent works in the framework of -factorization approach, we have included two new mechanisms based on and hard subprocesses. This was done based on the Kwieciński unintegrated parton distributions. We find that the new terms give significant contribution at RHIC energies. In general, the results of the -factorization approach depend on UGDFs/UPDFs used, i.e. on approximation and assumptions made in their derivation.
The results obtained in the standard NLO approach depend significantly whether we consider correlations of any jets or correlations of only leading jets. In the NLO approach one obtains = 0 if for leading jets as a result of a kinematical constraint. Similarly = 0 if or .
There is no such a constraint in the -factorization approach which gives a nonvanishing cross section at small relative azimuthal angles between leading jets and transverse-momentum asymmetric configurations. We conclude that in these regions the -factorization approach is a good and efficient tool for the description of leading-jet correlations. Rather different results are obtained with different UGDFs which opens a possibility to verify them experimentally. Alternatively, the NLO-forbidden configurations can be described only by higher-order (NNLO and higher-order) terms. We do not need to mention that this is a rather difficult and technically involved computation.
What are consequences for particle-particle correlations measured recently at RHIC requires a separate dedicated analysis. Here the so-called leading particles may come both from leading and non-leading jets. This requires taking into account the jet fragmentation process.
Acknowledgments This work is partially supported by the grant of the Polish Ministry of Scientific Research and Information Technology number 1 P03B 028 28.
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