Dijet and photon-jet correlations in proton-proton collisions at RHIC
We discuss correlations in azimuthal angle as well as correlations in two-dimensional space of transverse momenta of two jets as well as photon and jet. 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. The collinear NLO contributions dominate over -factorization cross section at small relative azimuthal angles as well as for asymmetric transverse momentum configurations.
The subject of jet correlations is interesting in the context of recent detailed studies of hadron-hadron correlations in nucleus-nucleus (1) and proton-proton (2) collisions. Effects of geometrical jet structure were discussed recently in Ref.(3). 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 induced by the QCD radiation. Here we discuss the case of elementary hadronic collisions. Our analysis is a first step towards the nuclear case.
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 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 diagrams included in our calculations which illustrate the physics situation. The soft emissions, not explicit in our calculation, are hidden in model unintegrated parton (gluon) distribution functions (UPDF,UGDF). In our calculation UGDFs or UPDFs are assumed to be given and are taken from the literature.
The -factorization was originally proposed for heavy quark production (4). In recent years it was used to describe several high-energy processes, such as total cross section in virtual photon - proton scattering (5), heavy quark inclusive production (6); (7), heavy quark – heavy antiquark correlations (8); (9), inclusive photon production (10); (11), inclusive pion production (12); (13), Higgs boson (14) or gauge boson (15) production and dijet correlations in photoproduction (16) and hadroproduction (17).
Up to now no theoretical calculation for photon-jet were presented in the literature, even for elementary collisions. In leading-order collinear-factorization approach the photon and the associated jet are produced back-to-back. If transverse momenta of partons entering the hard process are included, the transverse momenta of the photon and the jet are no longer balanced and finite (non-zero) correlations in a broad range of relative azimuthal angle and/or in lengths of transverse momenta of the photon and the jet are obtained. The finite correlations can be also obtained in higher-order collinear-factorization approach (18).
Here we discuss the region of relatively semi-hard jets/photons, i.e. the region related to the recently measured hadron-hadron correlations at RHIC and photon-hadron correlations being analysed (19). Here the resummation effects may be expected to be important. The resummation physics is addressed in our case through the -factorization approach.
It is known that at high energies, at midrapidities and not too large transverse momenta of the jet (or photon) production is dominated by (sub)processes initiated by gluons. In this paper we concentrate on such processes. In this presentation we discuss mainly -factorization approach. Some aspects of the standard collinear approach are discussed in Refs.(20); (21).
In the -factorization approach the cross section for the production of a pair of partons or photon and parton (k,l) can be written as
and and are so-called transverse masses defined as , where is the mass of a parton. In the case of photon-jet correlations there is no sum over (). In the following we shall assume that all partons are massless. The objects denoted by and in the equation 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.
3.1 Dijet correlations
In Fig.3 we show two-dimensional
maps of the cross section in for processes
shown in Fig.1.
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
For completeness in Fig.4 we show azimuthal angle dependence of the cross section for all four components. There is no sizable 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 effects, such as primordial distribution of partons in the nucleon,
the evolution scale responsible for the soft resummation effects.
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 . This is demonstrated in Fig.5 for the subprocess. The smaller the bigger decorrelation in azimuthal angle can be observed. In Fig.5 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 characteristic for the nonperturbative physics. 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.6 we show azimuthal-angle correlations for the component (dominant at midrapidities) for different UGDFs from the literature. Rather different results are obtained for different UGDFs. In principle, experimental results could select the “best” UGDF. We do not need to mention that such measurements are not easy at RHIC and hadron correlations are studied instead of jet correlations.
3.2 Photon-jet correlations
Let us start from presenting our results on the plane. In Fig.7 we show the maps for different UPDFs used in the -factorization approach as well as for NLO collinear-factorization approach for GeV and at the Tevatron energy GeV. In the case of the Kwieciński distribution we have taken = 1 GeV for the exponential nonperturbative form factor and the scale parameter = 100 GeV. Rather similar distributions are obtained for different UPDFs. The distribution obtained in the NLO approach differs qualitatively from those obtained in the -factorization approach. First of all, one can see a sharp ridge along the diagonal . This ridge corresponds to a soft singularity when the unobserved parton has very small transverse momentum . At the same time this corresponds to the azimuthal angle between the photon and the jet being . Obviously this is a region which cannot be reliably calculated in collinear pQCD. There are different practical possibilities to exclude this region from the calculations (21).
As discussed in Ref.(22) the Kwieciński distributions are very useful to treat both the nonperturbative (intrinsic nonperturbative transverse momenta) and the perturbative (QCD broadening due to parton emission) effects on the same footing. In Fig.8 we show the effect of the scale evolution of the Kwieciński UPDFs on the azimuthal angle correlations between the photon and the associated jet. We show results for different initial conditions ( = 0.5, 1.0, 2.0 GeV). At the initial scale (fixed here as in the original GRV (23) to be = 0.25 GeV) there is a sizable difference of the results for different . The difference becomes less and less pronounced when the scale increases. At = 100 GeV the differences practically disappear. This is due to the fact that the QCD-evolution broadening of the initial parton transverse momentum distribution is much bigger than the typical initial nonperturbative transverse momentum scale.
In Fig.9 we show azimuthal angular correlations for RHIC. In this case integration is made over transverse momenta GeV and rapidities . The standard NLO collinear cross section grows somewhat faster with energy than the -result with unintegrated Kwieciński distribution. This is partially due to approximation made in calculation of the off-shell matrix elements.
Let us consider now some aspect of the standard NLO approach.
Here 3 jets with transverse momenta and
Motivated by the recent experimental results of hadron-hadron correlations at RHIC we have discussed jet-jet and photon-jet correlations.
In comparison to recent works on dijet production in the framework of -factorization approach, we have included two new mechanisms based on and hard subprocesses. This was done using 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.
An interesting observation has been made for azimuthal angle correlations. At relatively small transverse momenta ( 5–10 GeV) the subprocesses, not contributing to the correlation function in the collinear approach, dominate over components. The latter dominate only at larger transverse momenta, i.e. in the traditional jet region.
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 . In this presentation we have discussed explicitly only a similar case of photon-jet correlations.
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.
On the contrary, in the case of correlations of any unrestricted jets (all possible dijet combinations) the NLO cross section exceeds the cross section obtained in the -factorization approach with different UGDFs. This is therefore a domain of the standard fixed-order pQCD. We recommend such an analysis as an alternative to study leading-jet correlations.
Consequences for particle-particle correlations, measured recently at RHIC, require a separate dedicated analysis.
We have discussed also photon-jet correlation observables. Up to now such correlations have not been studied experimentally. As for the dijet case we have concentrated on the region of small transverse momenta (semi-hard region) where the -factorization approach seems to be the most efficient and theoretically justified tool. We have calculated correlation observables for different unintegrated parton distributions from the literature. Our previous analysis of inclusive spectra of direct photons suggests that the Kwieciński distributions give the best description at low and intermediate energies. We have discussed the role of the evolution scale of the Kwieciński UPDFs on the azimuthal correlations. In general, the bigger the scale the bigger decorrelation in azimuth is observed. When the scale (photon) (associated jet) (for the kinematics chosen 100 GeV) is assumed, much bigger decorrelations can be observed than from the standard Gaussian smearing prescription often used in phenomenological studies.
The correlation function depends strongly on whether it is the correlation of the photon and any jet or the correlation of the photon and the leading-jet which is considered. In the last case there are regions in azimuth and/or in the two-dimensional () space which cannot be populated in the standard next-to-leading order approach. In the latter case the -factorization seems to be a useful and efficient tool.
At RHIC one can measure jet-hadron correlations only for not too high transverse momenta of the trigger photon and of the associated hadron. This is precisely the semihard region discussed here. In this case the theoretical calculations would require inclusion of the fragmentation process. This can be done easily assuming independent parton fragmentation method.
Acknowledgments. This presentation is devoted to the memory of Jozsó Zimányi, fantastic physicist, extraordinary man and very good friend of the Polish Nuclear and Particle Physics Community. A.S. congratulates the Budapest group for organizing the fantastic memorial workshop. We are also indebted to Jan Rak from the PHENIX collaboration for the discussion of recent correlation results from RHIC. This work was partially supported by the grant of the Polish Ministry of Scientific Research and Information Technology number 1 P03B 028 28.
- multiplies UPDFs in the impact parameter space and is responsible for nonperturbative effects included in addition to perturbative effects embedded in the Kwieciński evolution equations (for more details see e.g. (13)).
- Jet 1 (with ) and jet 2 (with ) are those which correlations are studied.
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