Leading-Order Determination of the Gluon Polarization from high-p_{T} Hadron Electroproduction

Leading-Order Determination of the Gluon Polarization from high- Hadron Electroproduction

The Hermes Collaboration
Abstract:

Longitudinal double-spin asymmetries of charged hadrons with high transverse momentum have been measured in electroproduction using the Hermes detector at Hera. Processes involving gluons in the nucleon have been enhanced relative to others by selecting hadrons with typically above 1 GeV. In this kinematic domain the gluon polarization has been extracted in leading order making use of the model embedded in the Monte Carlo Generator Pythia 6.2. The gluon polarization obtained from single inclusive hadrons in the range 1 GeV 2.5 GeV using a deuterium target is at a scale and . For different final states and kinematic domains, consistent values of  have been found within statistical uncertainties using hydrogen and deuterium targets.

Lepton-Nucleon Scattering, Deep Inelastic Scattering, QCD, gluon polarization
preprint: July 19, 2019

The HERMES Collaboration

A. Airapetian, N. Akopov, Z. Akopov, E.C. Aschenauer111Now at: Brookhaven National Laboratory, Upton, New York 11772-5000, USA, W. Augustyniak, R. Avakian, A. Avetissian, E. Avetisyan, S. Belostotski, N. Bianchi, H.P. Blok, H. Böttcher, C. Bonomo, A. Borissov, V. Bryzgalov, M. Capiluppi, G.P. Capitani, E. Cisbani, M. Contalbrigo, P.F. Dalpiaz, W. Deconinck222Now at:Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA, R. De Leo, M. Demey L. De Nardo, E. De Sanctis, M. Diefenthaler, P. Di Nezza, J. Dreschler M. Düren, M. Ehrenfried G. Elbakian, F. Ellinghaus333Now at:Institut für Physik, Universität Mainz, 55128 Mainz, Germany, U. Elschenbroich R. Fabbri, A. Fantoni, L. Felawka, S. Frullani, D. Gabbert, G. Gapienko, V. Gapienko, F. Garibaldi, G. Gavrilov, V. Gharibyan, F. Giordano, S. Gliske, H. Guler C. Hadjidakis444Now at: IPN (UMR 8608) CNRS/IN2P3 - Université Paris-Sud, 91406 Orsay, France, M. Hartig555Now at: Institut für Kernphysik, Universität Frankfurt a.M., 60438 Frankfurt a.M., Germany, D. Hasch, T. Hasegawa, G. Hill, A. Hillenbrand, M. Hoek, Y. Holler, B. Hommez, I. Hristova, A. Ivanilov, H.E. Jackson, R. Kaiser, T. Keri, E. Kinney, A. Kisselev, M. Kopytin, V. Korotkov, P. Kravchenko, L. Lagamba, R. Lamb, L. Lapikás, I. Lehmann, P. Lenisa, P. Liebing666Now at: Institute of Environmental Physics and Remote Sensing, University of Bremen, 28334 Bremen, Germany, L.A. Linden-Levy, W. Lorenzon, X.-R. Lu B. Maiheu, N.C.R. Makins, B. Marianski, H. Marukyan, V. Mexner, C.A. Miller, Y. Miyachi, V. Muccifora, M. Murray, A. Mussgiller, E. Nappi, Y. Naryshkin, A. Nass, M. Negodaev, W.-D. Nowak, L.L. Pappalardo, R. Perez-Benito, N. Pickert M. Raithel, D. Reggiani P.E. Reimer, A. Reischl A.R. Reolon, C. Riedl, K. Rith, S.E. Rock777Present affiliation: SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA, G. Rosner, A. Rostomyan, J. Rubin, Y. Salomatin, A. Schäfer, G. Schnell, K.P. Schüler, B. Seitz, C. Shearer T.-A. Shibata, V. Shutov, M. Stancari, M. Statera, J.J.M. Steijger, J. Stewart F. Stinzing, S. Taroian, B. Tchuiko, A. Trzcinski, M. Tytgat, A. Vandenbroucke P.B. van der Nat G. van der Steenhoven Y. Van Haarlem888Now at: Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA, C. Van Hulse, M. Varanda D. Veretennikov, I. Vilardi C. Vogel S. Wang, S. Yaschenko, H. Ye, Z. Ye W. Yu, D. Zeiler, B. Zihlmann P. Zupranski


Physics Division, Argonne National Laboratory, Argonne, Illinois 60439-4843, USA

Istituto Nazionale di Fisica Nucleare, Sezione di Bari, 70124 Bari, Italy

School of Physics, Peking University, Beijing 100871, China

Nuclear Physics Laboratory, University of Colorado, Boulder, Colorado 80309-0390, USA

DESY, 22603 Hamburg, Germany

DESY, 15738 Zeuthen, Germany

Joint Institute for Nuclear Research, 141980 Dubna, Russia

Physikalisches Institut, Universität Erlangen-Nürnberg, 91058 Erlangen, Germany

Istituto Nazionale di Fisica Nucleare, Sezione di Ferrara and Dipartimento di Fisica, Università di Ferrara, 44100 Ferrara, Italy

Istituto Nazionale di Fisica Nucleare, Laboratori Nazionali di Frascati, 00044 Frascati, Italy

Department of Subatomic and Radiation Physics, University of Gent, 9000 Gent, Belgium

Physikalisches Institut, Universität Gießen, 35392 Gießen, Germany

Department of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom

Department of Physics, University of Illinois, Urbana, Illinois 61801-3080, USA

Randall Laboratory of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA

National Institute for Subatomic Physics (Nikhef), 1009 DB Amsterdam, The Netherlands

Petersburg Nuclear Physics Institute, Gatchina, Leningrad region 188300, Russia

Institute for High Energy Physics, Protvino, Moscow region 142281, Russia

Institut für Theoretische Physik, Universität Regensburg, 93040 Regensburg, Germany

Istituto Nazionale di Fisica Nucleare, Sezione Roma 1, Gruppo Sanità and Physics Laboratory, Istituto Superiore di Sanità, 00161 Roma, Italy

TRIUMF, Vancouver, British Columbia V6T 2A3, Canada

Department of Physics, Tokyo Institute of Technology, Tokyo 152, Japan

Department of Physics, VU University, 1081 HV Amsterdam, The Netherlands

Andrzej Soltan Institute for Nuclear Studies, 00-689 Warsaw, Poland

Yerevan Physics Institute, 375036 Yerevan, Armenia

1 Introduction

In recent years a major goal in the study of Quantum Chromo-Dynamics (QCD) has been the detailed investigation of the spin structure of the nucleon and the determination of the partonic composition of its spin projection [1]

(1)

Here is the contribution of the quark and anti-quark helicities, is the contribution of the gluon helicity, and and are the quark and gluon orbital angular momenta, respectively, in a reference system where the nucleon has very large longitudinal momentum. The individual terms in the sum depend on the scale and the renormalization scheme. Recent results from experiments [2, 3] and fits in next-to-leading order (NLO) QCD [4, 5, 6, 7] to helicity-dependent inclusive Deep-Inelastic Scattering (DIS) data [2, 3, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] yield a value of at GeV in the  scheme [19]. In contrast to the quark helicity distributions, the knowledge of the gluon helicity distribution function is still limited. There are no direct experimental determinations of parton orbital angular momenta. Most of the existing knowledge about originates from next-to-leading order perturbative QCD (pQCD) fits to the helicity-dependent structure function of the nucleon, where is the Bjorken scaling variable, which is in leading-order (LO) identified with the longitudinal parton momentum fraction in the nucleon. In DIS the renormalization and factorization scales are set equal to the photon virtuality . Because the virtual photon does not couple directly to gluons (see Fig. 1b), is only weakly sensitive to gluons through the DGLAP evolution [20, 21, 22] of the helicity-dependent Parton Distribution Functions (PDFs). At next-to-leading order pQCD, additional sensitivity to gluons arises from the Photon-Gluon Fusion (PGF) subprocess (see Fig. 1b). However, the limitations on the precision and kinematic range in and of the measurements result in large experimental and theoretical uncertainties on the determination of the gluon helicity distribution function . Results for from recent pQCD fits to inclusive DIS data [4, 5, 6, 7] are typically of order 0.5 with uncertainties up to .

An alternative constraint on the extraction of in NLO pQCD fits comes from the measurements of double-spin asymmetries in production of inclusive mesons or jets with high transverse momentum in polarized proton-(anti-)proton scattering. First measurements were performed by E704 [23] and more recent data were obtained by Phenix [24] and Star [25] at Rhic. The inclusion of the RHIC-data in recent NLO pQCD fits [6] improves the accuracy on significantly. One finds smaller than 0.1, with a possible node in the distribution. This is driven mainly by the RHIC data, which constrain the magnitude of for , but cannot determine its sign as they mainly probe the product of the gluon helicity distribution at two values.

Figure 1: Feynman diagrams for hard subprocesses: a) DIS, b) Photon-Gluon Fusion, and c) QCD Compton scattering.

In order to increase the sensitivity to in lepton-nucleon scattering, other observables besides the inclusive helicity-dependent structure function have been studied. These observables are expected to include a direct contribution from gluons. For example, in hadron leptoproduction this gluonic contribution can be relatively enhanced by detecting charmed hadrons, or inclusive hadrons or hadron pairs at high transverse momenta .

Charmed hadron electroproduction is a suitable channel because it is dominated by the PGF subprocess [26] and a hard scale is introduced by the mass of the charm-quark pair, which makes pQCD calculations of this process possible. For light final state quarks, the selection of hadrons with high enhances the relative contribution of the gluon subprocesses and the relevant transverse momenta provide the scale (see Sect. 5.3). In the high domain other calculable hard pQCD subprocesses such as QCD Compton (QCDC) scattering (see Fig. 1c) are relatively enhanced as well, whereas soft, non-perturbative processes are suppressed. Charm electroproduction is being investigated by Compass [26, 27]. Inclusive single hadron leptoproduction was studied by E155 [28]. Hadron-pair leptoproduction at high was studied by Hermes [29], SMC [30] and Compass [31, 32]. For these experiments high is in the range from one to a few GeV.

Throughout this paper, the term “LO” is applied to all leading order subprocesses contributing to hadron production at nonzero . These are the tree level processes at , but also the quark scattering process (DIS) at . While the former processes involve hard gluons, and can therefore involve substantial parton transverse momentum in the hard scattering, in the latter process is equal to zero, but hadrons acquire from soft initial and final state radiation. This paper presents the LO extraction of the gluon polarization  from longitudinal double-spin asymmetries of charged inclusive hadrons measured in electroproduction using a deuterium target by Hermes at Hera. The contributions of signal and background have been determined by a Pythia Monte Carlo simulation, which includes LO pQCD as well as non-perturbative subprocesses. Consistency checks have been performed for different kinematic regions, different final states and using data from a hydrogen target. The data taken with the deuterium target correspond to an integrated luminosity three times larger than that taken with the hydrogen target, see table 1. Compared to the previous Hermes publication [29], which used measurements of hadron pairs of opposite charge from a hydrogen target, this analysis includes a much larger sample of single hadrons, and a significantly more comprehensive treatment of the underlying physics processes [33, 34].

This paper is organized as follows: In Sect. 2 the experimental method is described, in Sect. 3 the asymmetry results are given, in Sect. 4 the determination of  with a description of the physics model of the reactions is discussed, in Sect. 5 the Pythia Monte Carlo simulation is described, in Sect. 6 the determination of  is explained, and in Sect. 7 the summary and conclusions are given.

2 The Hermes Experiment

Positrons of momentum 27.6 GeV were stored in the Hera lepton ring at Desy. The initially unpolarized beam was transversely polarised by an asymmetry in the emission of synchrotron radiation associated with a spin flip (Sokolov-Ternov mechanism [35]). The polarization was rotated to the longitudinal direction for passage through an gaseous internal fixed target of longitudinally nuclear-polarized atoms. The scattered positron and hadrons produced were detected in a forward magnetic spectrometer. The beam helicity was reversed periodically. The beam polarization was measured continuously by two independent polarimeters using Compton backscattering of circularly polarized laser light [36, 37]. The average beam polarization for the data used in this analysis is shown in Tab. 1. The target [38] consisted of longitudinally nuclear-polarized pure atomic hydrogen or deuterium gas in an open-ended 40 cm long storage cell. The cell was fed by an atomic-beam source based on Stern-Gerlach separation combined with radio-frequency transitions of atomic hyperfine states [39]. The sign of the nuclear polarization of the atoms was chosen randomly every 60 s (90 s) for the hydrogen (deuterium) target. The polarization and the atomic fraction inside the target cell were continuously measured [40, 41]. The average values of the target polarization for both hydrogen and deuterium data are shown in Tab. 1. The luminosity was measured by detecting pairs originating from Bhabha scattering of the beam positrons off electrons in the target atoms, and also pairs from annihilations [42].

Year Target Luminosity Average Beam Average Target
pb Polarization Polarization
1996 H 12.6
1997 H 37.3
2000 D 138.7
Table 1: Integrated luminosities, average beam and target polarizations for the data used in this analysis.

The Hermes spectrometer [43] consisted of two identical halves separated by a horizontal flux diversion plate, which limited the minimum detected angle. The geometrical acceptance was in the horizontal (bending) plane and between in the vertical plane resulting in a range of polar angles between and . Each half was instrumented with 3 planes of hodoscopes, 36 planes of drift chambers, and 9 planes of proportional chambers. The particle identification system consisted of an electromagnetic calorimeter, a pre-shower hodoscope, a transition-radiation detector, and a Čerenkov detector. Detailed descriptions of these components can be found in Refs.  [43, 44, 45, 46, 47, 48]. Positrons within the acceptance could be separated from hadrons with an efficiency exceeding 98% and a hadron contamination of less than 1%.

The main Hermes physics trigger was formed by a coincidence of hits in the hodoscopes in the front and back regions of the spectrometer with the requirement of an energy deposit above 1.4 GeV in the calorimeter. This trigger was almost 100% efficient for positrons with energies above threshold. Events with no positron in the acceptance were recorded using a mixture of the main trigger and another trigger formed by a coincidence between the hodoscopes and two tracking planes, requiring that there is at least one charged track. The influence of trigger efficiencies on the analysis has been studied in [33].

3 Experimental Results

The ratio  of helicity-dependent to helicity-averaged gluon distributions, i.e. the gluon polarization, is determined by measuring the double-spin cross section asymmetry of one or two high- inclusive hadrons produced in the scattering of longitudinally polarized positrons incident on the longitudinally polarized target. The definitions of the kinematic variables in electroproduction used in this paper are shown in Tab. 2. The longitudinal double-spin cross section asymmetry is given by the ratio of helicity-dependent to helicity-averaged cross sections =, where = ( )/2, , and the single (double) arrows denote the relative alignment of longitudinal spins of the lepton (nucleon) with respect to the lepton beam direction.

The data for this analysis were collected in 1996, 1997, and 2000 (see Tab. 1). The analysis presented in this paper includes all (unidentified) charged hadrons. Separate asymmetries are given for each charge, target, and event category.

, 4–momenta of the initial and final state leptons
Polar and azimuthal angle of the scattered positron
Mass of the initial target nucleon
4–momentum of the virtual photon
Negative squared 4-momentum transfer
Energy of the virtual photon
Bjorken scaling variable
Fractional energy of the virtual photon
       Squared invariant mass of the virtual-photon nucleon system
4-momentum of a hadron in the final state
Transverse momentum of a hadron
with respect to the virtual photon
with respect to the incoming positron
Transverse hadron momentum from fragmentation
For two hadrons: 
Fractional energy of the final state hadron
Parton momentum fraction
Mandelstam variable for partonic process
Mandelstam variable for partonic process
Mandelstam variable for partonic process
pQCD scale
  for ) Transverse momentum of final state partons
in the CM-system of the hard subprocess
Intrinsic transverse momentum of partons
in the nucleon and photon
Table 2: Definition of kinematic variables.

3.1 Event categories

Simulations indicate that subprocesses involving hard gluons are relatively enhanced by measuring hadrons with high with respect to the virtual photon direction (). Correlations between hadrons in an event may also enhance the signal. Events are categorized by the number of hadrons observed in an event and whether kinematic information on the scattered positron is available or not. Each possible combination of two hadrons is counted as a separate event in the pairs category. The categories are defined in detail as follows:

  • ‘anti-tagged’ single inclusive hadrons: Events with leptons in the acceptance were not included in this category. The hadron transverse momentum was measured with respect to the beam direction as the direction of the virtual photon is unknown. In most cases, the undetected positron had a small scattering angle (and hence is small) and stayed inside the beam pipe. The difference between and is then very small. However, the positron could also escape the detector acceptance because of a large scattering angle, in which case was large. The large angle of the virtual photon with respect to the beam axis results in a significantly larger than of the hadron. Although these events with large are rare, they can account for a significant fraction of the hadrons at high . For  GeV the deuteron (proton) data sample in this category contains 1272k (419k) hadrons.

  • ‘tagged’ single inclusive hadrons: The scattered positron has been detected with , , and . The hadron transverse momentum is measured with respect to the virtual photon direction. For  GeV this deuteron (proton) data sample contains 53k (19k) hadrons.

  • inclusive pairs of hadrons: The hadron pair sample consists of all pairs of charged hadrons with GeV. The transverse momentum is measured with respect to the beam direction, because only in 10% of the events the positron was detected. With the additional requirement  GeV the deuteron (proton) data sample contains 60k (20k) hadron pairs. With this requirement applied, 6% of the anti-tagged inclusive hadrons with GeV are contained within the pairs sample.

For all three categories, Hermes data are available for various combinations of target and/or hadron charge detected. As the samples differ in the hard subprocess and final state kinematics and fractions of contributing subprocesses, the corresponding results for the gluon polarization  provide a measure of the consistency of the extraction. The final result for  is obtained from the anti-tagged inclusive hadrons originating from a deuterium target. The other data samples have too small a statistical power to justify carrying out the extensive analysis needed to obtain the systematic uncertainties.

3.2 Asymmetry results

The double-spin asymmetry measured is given by

(2)

Here () is the number of hadrons or hadron pairs for target spin orientation parallel (anti-parallel) to the beam spin orientation, () is the corresponding integrated luminosity, and () is the integrated luminosity weighted by the live-time fraction and the absolute values of beam and target polarizations. There is a small background () from positrons misidentified as hadrons (and vice versa). In the tagged category a correction was applied for an approximately contribution of positrons originating from charge-symmetric processes.

The asymmetries for the anti-tagged and tagged categories are shown as a function of transverse momentum in Figs. 2 and 3, respectively, and listed in tables 9 - 10. The asymmetry of the pairs is presented as a function of the minimum requirement, , in Fig. 4 and in table 11. The considerably different values of the asymmetries in the different categories, charges and targets are due to the different underlying mixtures of subprocesses and of quark content, as discussed in Sect. 5.3.

Figure 2: Measured asymmetry for the anti-tagged category of events for positive (left) and negative (right) inclusive hadrons from hydrogen (top) and deuterium (bottom) targets as a function of The uncertainties are statistical only. There is an overall normalization uncertainty of 5.2% (3.9%) for hydrogen (deuterium). The curves show the Monte Carlo asymmetries for three different fixed values assumed for the gluon polarization.
Figure 3: Measured asymmetry for the tagged category of events for positive (left) and negative (right) inclusive hadrons from hydrogen (top) and deuterium (bottom) targets as a function of The uncertainties are statistical only. There is an overall experimental normalization uncertainty of 5.2% (3.9%) for hydrogen (deuterium). The curves show the Monte Carlo asymmetries for three different fixed values assumed for the gluon polarization.
Figure 4: Measured asymmetry for hadron pairs produced from hydrogen (left) and deuterium (right) targets as a function of the minimum value of . The uncertainties are statistical only. There is an overall experimental normalization uncertainty of 5.2% (3.9%) for hydrogen (deuterium). The curves show the Monte Carlo asymmetries for three different values assumed for the gluon polarization.

The curves in Figs. 2, 3, and 4 show the asymmetries calculated by the procedure discussed in Sect. 4 using the values  = , , (from top to bottom). They illustrate the sensitivity of the Hermes data to . The data are close to the central curve indicating small average values of .

4 Physics Model

4.1 Subprocesses

Both the helicity-averaged and helicity-dependent cross sections include contributions from hard subprocesses that can be calculated using pQCD and from soft subprocesses such as those described by the Vector-Meson Dominance (VMD) model (see Fig. 1). A smooth transition from soft subprocesses to hard subprocesses is regulated by a set of cutoff parameters (for details [33, 49, 50, 51]). The measured asymmetry is the weighted sum of the asymmetries of all subprocesses. When it is impossible to reliably separate the subprocesses experimentally, as in fixed-target experiments, the fractions of events originating from the different subprocesses must be modeled. In the analysis described in this paper, this is done using the spin independent Monte Carlo (MC) program Pythia 6.2 [49, 50, 51].

The various subprocesses are classified in terms of the model used in Pythia. In this model, the wave function of the incoming photon has three components, a “VMD”, a “direct” and an “anomalous” one. The generic photon processes following from this decomposition are depicted in Fig. 5. The direct photon interacts as a point-like particle with the partons of the nucleon, while the VMD and anomalous components interact through their hadronic structure.

Figure 5b shows an example of a direct process. The direct pQCD subprocesses studied in this analysis are the DIS process (Fig. 1a), the processes PGF (Fig. 1b), and QCDC (Fig. 1c).

The VMD component is characterized by small-scale, non-perturbative fluctuations of the photon into a pair existing long enough to evolve into a hadronic state before the interaction with the nucleon. This process can be described in the framework of the VMD model, where the hadronic state is treated as a vector meson (e.g., ) with the same quantum numbers as the photon. Higher-mass and non-resonant states are added in the Generalized VMD (GVMD) model. The (G)VMD hadronic states can undergo all the interactions with the nucleon allowed in hadronic physics, i.e., elastic and diffractive as well as inelastic non-diffractive reactions. The latter can be either soft (“low-”) processes or hard QCD processes. A generic example of a VMD process is shown in Fig. 5a.

Figure 5: Contributions to hard p interactions: (a) VMD, (b) direct, and (c) anomalous. Only the basic graphs are illustrated; additional partonic activity is allowed in all three processes. The presence of spectator jets has been indicated by dashed lines, while full lines show partons that (may) give rise to high-p jets. The gray ovals represent multiparton wave functions. Anomalous states are built up from a perturbatively given fluctuation, while VMD fluctuations allow no simple perturbative representation; hence the difference in the placing of the ovals [51].

The anomalous photon is characterized by sufficiently large-scale, perturbative fluctuations of the photon into a pair. The allowed processes are the same pQCD processes as in the hard VMD case, with the difference that for the anomalous component the parton distributions of the photon are relevant, whereas for the description of the hard VMD component those of the vector meson are used. Both hard VMD and anomalous components are usually referred to as “resolved” photons. Depending on whether a quark or a gluon in the nucleon is struck by a resolved photon the corresponding hard subprocesses are labeled by a ‘q’ or ‘g’ in this paper. A generic example of an anomalous process is shown in Fig. 5c. The resolved-photon processes are of with a hidden term in the evolution of the photon’s parton distributions canceling the additional vertex [52].

For hard subprocesses the nucleon is described by helicity-averaged (helicity-dependent) PDFs, which are the average (difference) of the number densities of partons of type f whose spins are aligned, , those whose spins are anti-aligned, , with respect to the nucleon spin: = (= ), where = , , , or . The integral over , , gives the total spin contribution of the respective partons to the nucleon spin, as used in Eq. 1. The hard part of the single-inclusive differential helicity-dependent cross section for the process can be expressed as an integral over the parton distribution functions, the hard partonic cross sections for the subprocesses , and the fragmentation functions. It can be written schematically as

(3)

and correspondingly for the helicity-averaged cross section and distributions. Here is the fraction of the nucleon momentum carried by parton and () is the corresponding nucleon PDF. Similarly is the fraction of the photon momentum carried by parton , and () is the corresponding photon PDF. For the direct-photon processes equals and () reduces to . The fragmentation function describes the hadronization of a parton into a hadron with a momentum . The hard partonic cross section () depends on the subprocess kinematics, the renormalization and factorization scales, and on in case of the direct-photon processes. Here, and are the Mandelstam variables for the partonic interaction, which are related to and . More information on the kinematic variables is given in table 2. The cross section for hadron pairs () can be obtained analogously to Eq. 3.

The cross sections and asymmetries of the soft VMD interactions can only be modeled phenomenologically. The Pythia model incorporates the total and hadronic cross section parameterizations of Donnachie and Landshoff [53] together with quark counting rules [54, 55]. This model successfully describes the measured total, elastic, and diffractive cross sections over a wide energy range. The non-diffractive cross section is modeled in Pythia as the difference of the total cross section and the summed elastic and diffractive cross sections; the corresponding subprocess is called “low-”. The Pythia model provides a smooth transition from real to virtual photons and is applicable from very small to large values of . It uses a number of cutoff, scale, and suppression parameters together with several possible prescriptions on how to use them to select the underlying subprocess of an event. The default prescriptions and the cutoff and scale parameters were developed and tuned to match high energy data. In this application to the lower energy of Hermes  the influence of various prescriptions and parameter values has been carefully studied (see Sects. 5 and  6.4).

Subprocess # Class Description Name
soft VMD
91 background elastic VMD exclusive VMD
92 background single-diffractive VMD
93 background single-diffractive VMD
94 background double-diffractive VMD
95 background soft non-diffractive VMD low-
RESOLVED (hard VMD and anomalous)
11 background QCD QCD
12 background . .
13 background . .
28 background . .
28 signal . QCD
53 signal . .
68 signal . .
DIRECT
99 background LO DIS DIS
131 background (transverse) QCDC QCDC
132 background (longitudinal) QCDC .
135 signal (transverse) PGF PGF
136 signal (longitudinal) PGF .
Table 3: Description of the subprocesses used in this paper. Columns from left to right: subprocess, Pythia subprocess number, classification as signal or background, description, and name used in this paper. A vector meson is denoted by .

Table 3 shows a compilation of the modeled reactions, the corresponding Pythia subprocess numbers, their classification, description, and name used in this paper.

4.2 Signal and Background Asymmetries

In the simulation, the cross section is considered to arise from an incoherent superposition of all contributing subprocess amplitudes. The kinematic selection criteria (e.g., event category and hadron ) for the Monte Carlo are the same as those for the data. Pythia events are generated independent of helicity, therefore the MC asymmetry is calculated by weighting each selected MC generated hadron with the calculated event asymmetry . The average of these weights is , the asymmetry for subprocess

(4)

where is the number of entries. The event-by-event weighting method guarantees the correct integration over the subprocess kinematics, and all partons in the nucleon and the photon (where applicable). The Monte Carlo asymmetry is the sum of the asymmetries from signal () and background () subprocesses weighted by their fraction of entries and . It is given by

(5)

where is the fraction of entries from the subprocess calculated in the PYTHIA simulation. Background processes are all subprocesses that do not involve a hard gluon from the initial nucleon. These include all soft processes, the direct processes DIS and QCDC, and all resolved pQCD processes, which involve a quark or antiquark in the nucleon, i.e., QCD . They are listed in Tab. 3. All subprocesses involving a hard gluon of the nucleon in the initial state are considered to be signal processes, i.e., PGF and the hard processes.

The event-by-event weight for hard subprocesses is given by

(6)

where for , i.e., direct photon processes. The hard subprocess asymmetry is . The lowest order equations for important hard subprocess asymmetries are compiled in appendix A. The VMD and GVMD diffractive subprocesses may have small asymmetries at Hermes energies [56, 57, 58, 59]. The asymmetry of the low- process was estimated from the measured asymmetries and found to be non-zero (see Sect. 5.3). In both cases, the virtual-photon depolarization factor (see Eq. 28) has to be applied to the weight in order to account for the transformation of the virtual-photon nucleon asymmetry into a lepton-nucleon asymmetry. The asymmetry from signal subprocesses depends on the unknown  averaged over the subprocess kinematics in the specified range. It can be written as

(7)

where is the number of entries from all signal processes. The extraction of the quantity of interest, , is based on Eq. 7 replacing the unknown asymmetry by

(8)

In Sect. 6 methods will be described to extract  from the right hand side of Eq. 7.

5 Monte Carlo simulation

The relevant subprocess cross sections have been modeled by the Pythia Monte Carlo program, which uses Jetset [60] for describing the fragmentation process. The standard helicity-averaged input PDFs used are CTEQ5L [61] for the nucleon and Schuler and Sjöstrand [62] for the photon. The scale of the subprocesses is defined to be (also commonly referred to as ). Electromagnetic radiative effects [63, 64] have been added to Pythia  and they constitute a relatively small correction for hadron production at Hermes kinematics [33]. Events generated by Pythia are passed through a complete Geant 3 [65] simulation of the Hermes spectrometer.

5.1 Monte Carlo tuning

In order to account for the relatively low center-of-mass energy of the Hermes experiment several parameters in the event generation were adjusted and the model describing exclusive vector meson production was improved [33]. This was done in the kinematic region of the tagged events because more kinematic variables are measured for this category than for the anti-tagged category. The tuning of the fragmentation parameters [66] was performed using a subsample with GeV and GeV where the DIS process (Fig. 1a) is dominant and NLO corrections are small. The values of the adjusted parameters, shown in Tab. 7 in appendix B, are used for all event categories.

Figure 6 shows the measured and the simulated cross sections as a function of , , and for the tagged category of events using a deuterium target. Both the simulated and measured cross sections are not corrected for acceptance effects. These cross sections vary over more than three orders of magnitude. The data and MC simulation agree to within 15% for , where most of the data reside for the tagged event category. Thus in this region the modified Pythia 6.2 program with the adjusted parameters gives a good representation of the cross section at Hermes energies.

Figure 6: Top panels: Measured cross section in the Hermes acceptance for tagged hadrons as a function of (left), (middle), and (right) for positive (full points) and negative hadrons (open points) using a deuterium target. The lines show the tuned Pythia 6.2 calculation. Bottom panels: The corresponding ratios of the Pythia calculation to the measured cross section.

The description of the kinematic dependences of the tuned Monte Carlo code for the individual subprocesses must be consistent with independent LO pQCD calculations [67]. Such calculations presently exist only for inclusive production and only in the collinear approach, where the intrinsic transverse momentum of the partons in the nucleon and in the virtual photon, and also the transverse momentum arising from the fragmentation process are set to zero.

For a comparison of Pythia with these LO pQCD calculations a special simulation with and was performed, by replacing the string fragmentation performed by JETSET with weights obtained from the fragmentation functions of Ref. [68]. The resulting transverse momentum of the is calculated according to . Both this simulation and the pQCD calculation are performed in the Hermes kinematics for inclusive production at GeV, , disregarding the detector acceptance. Figure 7 compares the resulting cross sections for resolved photon, QCDC, and PGF processes from the simulation and the pQCD calculation. In the collinear approach the DIS subprocess is not included, because the of the final state hadron is zero, and also for low ( GeV) it does not result in a sizable .

Figure 7: Cross sections for inclusive production from resolved photon, QCDC, and PGF processes simulated using Pythia (solid lines) compared to the LO pQCD calculations from Ref. [69] (full points). Simulation and calculation are done in the collinear approach at GeV, . Green/Grey lines: subprocess cross sections after varying the renormalization and factorization scales by factors of and in the simulation.

The agreement between the simulated cross sections for the individual subprocesses and the calculations is well within the scale uncertainty () of the simulation (the dashed lines in Fig. 7). The LO pQCD calculations show a similar dependence on the variation of the renormalization and factorization scales (), see Fig. 11 in Ref. [67].

5.2 Effects of intrinsic and fragmentation transverse momenta

While the effect of intrinsic and fragmentation transverse momenta cannot yet be studied in LO pQCD calculations, a Pythia simulation can be used. For the standard simulations presented in this analysis a Gaussian distribution with a 0.4 GeV width is used for both and  [66]. These values are consistent with those obtained in Ref. [70]. Both intrinsic and fragmentation transverse momenta alter the relationships of to , from to , and hence the distribution of and . This in turn influences the dependence of the cross section on . The effects on the cross section for inclusive production from the PGF subprocess, of first adding nonzero GeV and secondly using Jetset with GeV are shown in Fig. 8. Including only in the simulation decreases from 1.9 GeV to 1.6 GeV and from 0.32 to 0.28, and increases the cross section by a factor of two. Including both and further decreases to 1.1 GeV and to 0.22, and increases the cross section by another factor of 10. These studies show that at fixed-target kinematics, like at Hermes, intrinsic and fragmentation transverse momenta cannot be neglected in pQCDC calculations. Similar conclusions were drawn in Ref. [71]. Perhaps resummation techniques [72], which account for initial and final state radiation effects, can help to achieve more realistic calculations.

Figure 8: The simulated cross section for inclusive production from the PGF subprocess vs. , , and for GeV, . The simulations are done using the collinear approach ((green/gray) solid line), the collinear approach together with intrinsic for the partons in nucleon and photon (dashed line), and with intrinsic together with fragmentation transverse momenta (solid line). For the first two simulations fragmentation is modeled using the KKP-fragmentation functions [68], for the third one Jetset with the standard settings listed in Table 7 is used.

5.3 Analysis of Monte Carlo events

Pythia events are used to calculate cross sections, individual subprocess fractions and event weights within the Hermes acceptance. The event weights for the pQCD processes (Eq. 6) are obtained using the hard subprocess asymmetries (see appendix A) and GRSV (standard scenario) [73] helicity-dependent PDFs in conjunction with the GRV98 [74] helicity-averaged PDFs to calculate for the nucleon. In order to calculate for the photon the averages of the maximal and minimal scenarios of the GRS [75, 76] helicity-dependent PDFs are used in conjunction with the GRS [77] helicity-averaged PDFs.

For elastic and diffractive VMD processes the asymmetry is set to zero [59]. For the low- process two alternative assumptions for the asymmetry have been investigated: and where is a parameterization of the photon-nucleon asymmetry in inclusive DIS. The resulting MC asymmetries and the measured asymmetry are shown in Fig. 9 (left) vs. for the tagged category and a hydrogen target. The corresponding deuterium data are not shown because for this target both assumptions are indistinguishable and match the data. For the anti-tagged category (Fig. 9 (right)) the dependence of the double-spin inclusive asymmetry is shown for both targets. The model matches the data better than in the kinematic domains where is large (low- for tagged and the lowest for anti-tagged categories, respectively) and contributions of hard QCD processes are negligible. The standard asymmetry for the low- process was chosen to be , because of this agreement and because the semi-inclusive asymmetry for all charged hadrons is approximately equal to the measured inclusive asymmetry. The world data on have been parameterized by for , and extrapolated to the smaller -values () typical for the anti-tagged sample.

Figure 9: Left panels: The measured double-spin inclusive asymmetry (full points) and two MC asymmetries (solid and dashed lines) based on different assumptions for the low- subprocess asymmetry (top) and the process fractions vs. (bottom) for tagged events on a hydrogen target. Right panels: Double-spin asymmetry vs. for anti-tagged hadrons on a hydrogen (top) and deuterium (bottom) target.

To avoid any bias from the experimental trigger to the results presented, the MC events received an additional weight to account for trigger inefficiencies, if measured and simulated cross sections are compared. The dependences of the cross sections, individual subprocess fractions , average event weights , and weighted asymmetries for the three event categories are shown in Figs. 10 (anti-tagged), 11 (tagged), and 12 (pairs).

All three categories have in common that:

  • The cross sections span four orders of magnitude, decreasing rapidly with ;

  • Reasonable agreement between data and Monte Carlo is observed for low transverse momenta. With increasing the Monte Carlo description becomes worse, underestimating the data by up to a factor of four at the largest ;

  • The fractions and decrease with increasing and the corresponding asymmetries are very small or zero, respectively;

  • In general the contributions from hard QCD subprocesses increase with increasing . At high subprocesses involving quarks in the nucleon contribute less than the signal processes;

  • The asymmetries for the two signal subprocesses, QCD and PGF have opposite sign. For a positive gluon polarization like that of GRSV, this results in a sizable negative asymmetry for PGF, and positive asymmetries for the processes;

  • Some asymmetries and fractions depend on the charge of the hadron.

Even though soft effects from initial and final state radiation and additional nonperturbative processes are taken into account in the Pythia simulation, the Monte Carlo simulation still fails to describe the cross sections at GeV. This shortcoming may be explained by missing large higher order corrections to the hard processes. These corrections have been evaluated for the next to leading order (NLO) cross section in Ref. [67], in the collinear approach for GeV and GeV, for all hard processes (QCD , PGF, QCDC) contributing in this region. The similar kinematics of hard processes in the pQCD-calculation and the PYTHIA simulation allows one to approximate the effect of NLO corrections to the Monte Carlo cross section. A -factor, i.e., the ratio of LO to NLO cross sections is applied as a weight to each hadron originating from a hard process. The -factors from Ref. [67] are very large (almost 5) at GeV and decrease with to about 2.5 at GeV. For the reweighting of the Monte Carlo events they have been extrapolated down to GeV, and it was assumed that can be approximated by (see the discussion in Sect. 5.1 about the collinear approximation). The results shown in the cross section ratio of Fig. 10 indicate that the inclusion of NLO effects to the Monte Carlo could significantly improve the description of the cross section. Effects of similar size may exist for the other categories, but NLO calculations for those are not yet available. The -factors for the asymmetry have also been calculated in [67] and are approximately 2 in the experimental range. Unfortunately it is not possible to consistently take into account -factors in the extraction of , therefore the result will essentially be a LO result subject to potentially large NLO corrections.

Figure 10: The cross sections and subprocess contributions, in the Hermes acceptance as a function of for the anti-tagged category of events and a deuterium target (left: positively charged hadrons, right: negatively charged hadrons). Top: The measured cross section and that generated by Pythia. Second row: The ratio of these two cross sections. Also shown is the effect of the -factor based on Ref. [67] (see text). Third row: The subprocess fractions from Pythia. Bottom two rows: The asymmetries and the asymmetries weighted with the subprocess fractions for each subprocess using Refs. [73] and [74] for the gluon PDFs.
Figure 11: The cross sections and subprocess contributions, in the Hermes acceptance, as a function of for the tagged category of events and a deuterium target (left: positively charged hadrons, right: negatively charged hadrons). Top: The measured cross section and that generated by Pythia. Second row: The ratio of these two cross sections. Third row: The subprocess fractions from Pythia. Bottom two rows: The asymmetries and the asymmetries weighted with the subprocess fractions for each subprocess using Refs. [73] and [74] for the gluon PDFs.
Figure 12: The cross sections and subprocess contributions, in the Hermes acceptance, as a function of the minimum value of for the production of inclusive hadron pairs on a deuterium target. Top: The measured cross section and that generated by Pythia. Second row: The ratio of these two cross sections. Third row: The subprocess fractions from Pythia. Bottom two rows: The asymmetries and the asymmetries weighted with the subprocess fractions for each subprocess using Refs. [73] and [74] for the gluon PDFs.

For the anti-tagged category the LO DIS fraction dominates the yield of positive hadrons at high . This is due to the subsample of events with the positron having a large scattering angle and missing the Hermes acceptance. The subprocess fractions for LO DIS and QCDC are larger for positive hadrons because of -quark dominance. Both signal subprocesses contribute approximately 20% to the cross section at high . The pairs category has a larger signal fraction than the other categories, but a much smaller number of events. The mixture of the background processes and their contribution to the background asymmetry is different for each category.

6 Determination of the gluon polarization

6.1 Kinematic considerations and requirements

The average value of  in a range is determined directly from Eq. 9 (see Sect. 6.2). However, as shown in Fig. 13, there is a large range of spanned by the data for each range. In order to circumvent this difficulty, the value of  and the appropriate value of is determined through a minimization procedure using a functional form for  (see Sect. 6.3). The scale dependence of  is neglected because almost all pQCD models are monotonic and vary slowly as a function of over the relatively small relevant range.

Figure 13: The range of generated for different values of calculated by Pythia for all signal processes, for the anti-tagged category of events and a deuterium target.

In order to optimize the accuracy of the following criteria that maximize the sensitivity of the MC asymmetry to , are applied to the individual data samples:

1.0 GeV 2.0 GeV (tagged);
1.0 GeV 2.5 GeV (anti-tagged);
2.0 GeV (pairs).

These requirements balance the statistical accuracy of the measured asymmetries (decreasing with , as shown in Figs. 2-4) against the signal process fractions (increasing with , as shown in Figs. 10-12). For the events within these limits it is observed that:

  • The Pythia simulations displayed in Fig. 14 show a strong correlation between the hard scattering transverse momentum of the signal subprocesses and the measured hadronic ();

  • For larger values of , there is greater sensitivity to the hard processes involving the gluon (see Figs. 2-4), which leads to reduced systematic uncertainties due to corrections for background asymmetries.

Figure 14: The correlation of the average hard scattering of all signal subprocesses as calculated by Pythia with the hadron for inclusive hadrons as calculated for the experimental data for the deuterium target. Left: tagged category; Center: anti-tagged category; Right: hadron pairs category. The dotted line goes along () and the vertical dashed line shows the minimum () used for the analysis.

The gluon polarization  is determined using Eqs. 5, 7, and 8. The anti-tagged category has sufficient statistics to allow extraction of in four bins (1.0 - 1.2 - 1.5 - 1.8 - 2.5 GeV), which are obtained by combining the bins shown in Fig. 2 and table 9. The other categories are represented by a single range in .

6.2 dependence of

If the dependence of on and is weak in the limited kinematic range of the experiment,  can be factored from the r.h.s of Eqs. 7, so that together with Eqs. 8 we obtain for the gluon polarization averaged over the covered and ranges

(9)

where the subprocess fractions and kinematics are determined using Pythia. As is shown in Fig. 13, different ranges in correspond to different ranges and distributions in . It is intrinsic to this method that there is no knowledge on the dependence of on , therefore no meaningful value of the average can be determined by this method, which nevertheless can be used as a consistency check between the different independent data sets.

The results for different event categories, targets and hadron charges are listed in table 4 and shown in Fig. 15 as a function of . The results for the pairs category are displayed at the average , and those for the tagged category at the average . Each of these data sets has a somewhat different mixture of background and signal processes as a function of , as seen in Figs. 10-12. The measured values of  should be equal for both targets and both hadron charges because of the same range in and . The values shown in Fig. 15 within each category and for each bin indeed agree in general within the statistical uncertainties. This is a strong indication that Pythia provides a consistent description of the underlying physics. The systematic charge dependence is accounted for by assigning a systematic uncertainty to the value of the (Pythia parameter PARJ(21)).

In [29] the kinematic selections for the hadron pairs used to calculate the asymmetry to extract  was GeV and GeV. These events are mostly contained in the event sample used to calculate the asymmetry in the left panel of fig. 4 if