In this chapter, we have presented an NLO calculation of single-inclusive jet production in longitudinally polarized proton-proton collisions. We have shown in some detail how to compute jet cross sections on a largely analytical level within the framework of the small-cone approximation, making use as much as possible of the previously calculated single-inclusive hadron production cross sections discussed in Chap. 4. By a comparison to the results of the Monte-Carlo jet code of [113] it has been demonstrated that this approximation is applicable for jet cone sizes up toR'0.7. Details of the algorithm used to define the jet do not affect the experimentally relevant spin asymmetries significantly.
Our code has the advantage of being numerically stable, fast, and extremely efficient, since the use of the SCA allowed us to perform the phase space integration of all partonic cross sections analytically and to cancel poles associated with collinear configurations in the final state explicitly. Singularities stemming from the initial state as well as ultraviolet and infrared divergencies of the partonic matrix elements are treated in complete analogy to the case of single-inclusive hadron production, discussed in Chap. 4, by a factorization
5.4 Summary and Conclusions 103
and a renormalization procedure, respectively. They give rise to a residual dependence of the hadronic cross sections on unphysical scales. We have shown, however, that this source of theoretical uncertainty is significantly reduced when including NLO corrections to the Born cross sections. In addition to the mild scale dependence, single-inclusive jet observables are insensitive to final state hadronization mechanisms and thus provide a particularly clean tool for studying the spin structure of the nucleon in polarized pp-collisions.
We have demonstrated that the double-spin asymmetries for large-pT jet production at RHIC are sensitive to ∆g even for rather moderate beam polarizations and integrated luminosities. Therefore, first data on jet production, which are expected to be released soon by the Star collaboration, will shed some light on the gluon polarization ∆g. On the long run, single-inclusive jet production will contribute invaluable information to a quantitative determination of the spin-dependent parton distributions of the nucleon.
NLO Corrections to
Longitudinally Polarized
Photoproduction of Inclusive Hadrons
In the foregoing chapters we have studied single-inclusive hadron and jet production in polarized proton-proton collisions as a tool for gaining information on the spin structure of the proton. Alternatively, the parton content of the nucleon can be accessed in photopro-duction of inclusive hadrons – either on a fixed target, as in the already running COMPASS experiment at CERN [27], or at a future lepton-hadron collider, for instance the planned eRHIC facility at BNL [28]. It was found in a first exploratory LO analysis [126] that in-vestigating photoproduction reactions in polarized electron-proton collisions additionally allows studying the inner (spin) structure of real photons which is completely unknown so far. Motivated by these twofold prospects, in this chapter we will focus on the process
l(Pl) +p(Pp)→l0(Pl0) +π(Pπ) +X , (6.1) where a lepton scatters elastically off a proton via the exchange of a (quasi-) real photon to produce a hadron in the final state. The by-productsX emerging from the reaction are not detected and will not be of any concern to us.
Throughout this chapter we will put special emphasis on aspects of photoproduction reactions we have not encountered in hadroproduction processes, since these have been discussed in some detail in Chap. 4. After a brief discussion of our current knowledge of the photon structure we will start with the calculation of the partonic matrix elements relevant for the various channels contributing to (6.1). It is vital to take into account that the photon can interact either directly as an elementary particle, or resolve into partonic constituents which in turn interact with the partons of the proton. We adopt the partonic matrix elements of Chap. 4 and Ref. [31] for the resolved contributions. The direct components are known for some time in the polarized [127] and unpolarized case [128, 129], but we recalculate them both. Again, we aim to perform the entire calculation on a largely
6.1 The Parton Structure of the Photon 105
analytical level and implement the partonic matrix elements in a fast computer code. The numerical results we thereby obtain will show sensitivity to the parton distributions of the photon as well as the gluon polarization of the proton.
6.1 The Parton Structure of the Photon
What makes the photon special in comparison to all other kinds of hadrons and partons is its two-fold occurrence as an elementary particle and also as a hadron-like conglomerate of quarks, antiquarks and gluons. Whereas the parton distributions of the structureless
“direct” photon are trivial,
(∆)γγ(x, µ)∼δ(1−x), (6.2)
in the polarized case the parton content of the “resolved” photon, ∆fγ with f = q,q, g,¯ is completely unknown. The spin-averaged fγ have been determined from data of DIS off a quasi-real photon target in e+e−collisions [130]. The evolution of the spin-dependent – and likewise the unpolarized – parton densities is governed by inhomogeneous integro-differential equations [131-133], where we have denoted the convolution of the parton distributions with appropriately defined splitting functions by
The spin-dependent parton-to-parton ∆Pij and photon-to-parton splitting functions ∆ki and their unpolarized counterparts can be calculated perturbatively and are known up to two loops, i.e., NLO accuracy [67-70, 131, 133],
∆ki(x, µ) = αem Altogether, the (∆)fγ are of order αem/αs. In contrast to the homogeneous DGLAP equations (2.46), describing the scale dependence of the parton densities of the proton, Eqs. (6.3) contain inhomogeneous terms which account for a photon-to-parton splitting.
They give rise to a so-called “pointlike” solution ∆fplγ in addition to the homogeneous, hadronic contribution ∆fhadγ ,
∆fγ(x, µ) = ∆fplγ(x, µ) + ∆fhadγ (x, µ). (6.6)
The pointlike part can be calculated perturbatively. It depends only on the boundary conditions, but not on the non-perturbative input, which resides solely in the ∆fhadγ and has to be extracted from experiment. In the absence of any data the latter is fixed by vector meson dominance (VMD) like assumptions, stating that the photon tends to fluctuate into states of identical quantum numbers.
Beyond the leading order inαsthe decomposition of the parton densities into pointlike and hadronic components depends on the factorization scheme chosen. In the analysis of unpolarized e+e− data two factorization schemes have been used. Contrary to the conventional MS factorization, the so-called DISγscheme [132] absorbs the NLOγ?γ →qq¯ coefficient functionCγ inF2γ, which diverges asx→1, into the definition of the photonic parton densities. This allows for choosing very similar, VMD-inspired, inputs in LO and NLO analyses. A similar scheme has been proposed in the polarized case [133]. Technically, the MS and the DISγscheme are related via a simple factorization scheme transformation, which will be discussed in Sec. 6.2.
Since the polarized parton densities ∆fγ are completely unknown, usually two ex-treme scenarios are employed to study the sensitivity of physical observables to the spin-dependent parton content of the photon. Figure 6.1. shows ∆uγ and ∆gγ at a scale µ = √
10 GeV at LO and NLO, determined in the DISγ scheme for these parametriza-tions. Whereas for the “minimal” set a vanishing hadronic input is imposed,
∆fhadγ,min(x, µ0) = 0, (6.7)
in the “maximal” scenario the maximal hadronic input compatible with the positivity bound |∆fγ(x, µ)| ≤fγ(x, µ) is used,
∆fhadγ,max(x, µ0) =fhadγ (x, µ0), (6.8) with the unpolarized parton densitiesfγ of Ref. [130]. The pointlike contribution is chosen to vanish at the input scaleµ0. Note that Eq. (6.7) yields non-vanishing parton densities forµ > µ0 due to the inhomogeneous nature of the evolution equations (6.3), in contrast to the homogeneous evolution of hadronic parton densities, Eq. (2.46). From Fig. 6.1 it can also be seen that the quark distributions exhibit a characteristic bump as x → 1, which is caused by the pointlike part of the solution (6.6). As we will see in Sec. 6.3, in experiments probing large values of x the ∆fhadγ can therefore be hardly accessed.
Of course, the parametrizations (6.7) and (6.8) illustrate only the extreme scenarios for the parton distributions of the photon, derived solely on the basis of physically moti-vated model assumptions. Definite knowledge of the hadronic components of the photon can only be obtained by an analysis of data from future experiments, for instance, at eRHIC [28]. However, the very concept of photonic quark and gluon densities has firmly been established by extensive studies at the unpolarized HERA and LEP colliders [134], which revealed that resolved contributions to photoproduction cross sections are sizeable in certain kinematic regions. Above that, it has been shown that spin-averagede+e− and ep data can simultaneously be described by the same set of photonic parton densities, which marks an important test for their universality and therefore success of the concept of a composite photon.