\thechapter Abstract
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Scuola di Dottorato in Fisica, Astrofisica e Fisica Applicata

Dipartimento di Fisica

Corso di Dottorato in Fisica, Astrofisica e Fisica Applicata

Ciclo XXVI


Unbiased spin-dependent
Parton Distribution Functions

Settore Scientifico Disciplinare FIS/02






Supervisore:
Professor Stefano FORTE
Co-Supervisore:
Dottor Juan ROJO
Coordinatore:
Professor Marco BERSANELLI
Tesi di Dottorato di:
Emanuele R. NOCERA

Anno Accademico 2013


Commission of the final examination:

External Referee:

Prof. Richard D. BALL

External Members:

Prof. Mauro ANSELMINO

Prof. Giovanni RIDOLFI

Final examination:

February 28, 2014

Università degli Studi di Milano, Dipartimento di Fisica, Milano, Italy

MIUR subjects:

FIS/02 - Fisica Teorica, Modelli e Metodi Matematici

PACS:

02.70.Uu, 07.05.Mh, 12.38.-t, 13.60.Hb, 13.88.+e

Keywords:

Spin, Parton Distribution Functions (PDF), Neural Networks, NNPDF, High-energy Physics

Internal illustrations:

Emanuele R. Nocera, made with ROOT v5.34/09 and feynMF v1.5

Fig. 2: courtesy of Mauro Anselmino

Fig. 4: taken from Ref. [1]

Fig. 5: courtesy of Stefano Carrazza

Template design:

Anna Lisa Varri, Ph.D.

Typeset by Emanuele R. Nocera using TeXLive 2013 on Ubuntu 12.04 LTS

To my family

Chapter \thechapter Abstract

We present the first unbiased determination of spin-dependent, or polarized, Parton Distribution Functions (PDFs) of the proton. A statistically sound representation of the corresponding uncertainties is achieved by means of the NNPDF methodology: this was formerly developed for unpolarized distributions and is now generalized to the polarized here for the first time. The features of the procedure, based on robust statistical tools (Monte Carlo sampling for error propagation, neural networks for PDF parameterization, genetic algorithm for their minimization, and possibly reweighting for including new data samples without refitting), are illustrated in detail. Different sets of polarized PDFs are obtained at next-to-leading order accuracy in perturbative quantum chromodynamics, based on both fixed-target inclusive deeply-inelastic scattering data and the most recent polarized collider data. A quantitative appraisal on the potential role of future measurements at an Electron-Ion Collider is also presented. We study the stability of our results upon the variation of several theoretical and methodological assumptions and we present a detailed investigation of the first moments of our polarized PDFs, compared to other recent analyses. We find that the uncertainty on the gluon distribution from available data was substantially underestimated in previous determinations; in particular, we emphasize that a large contribution to the gluon may arise from the unmeasured small-x region, against the common belief that this is actually rather small. We demonstrate that an Electron-Ion Collider would provide evidence for a possible large gluon contribution to the nucleon spin, though with a sizable residual uncertainty.

Chapter \thechapter List of Publications

Refereed publications

Publications in preparation

  • R. D. Ball et al., A first unbiased global extraction of polarized parton distributions

Publications in conference proceedings

Chapter \thechapter Acknowledgements

I would like to thank all those people who have supported me in the course of my work as a graduate student towards my Ph.D., particularly during the completion of this Thesis. I apologize in advance for missing some of them.

First of all, I am grateful to my supervisor, prof. Stefano Forte, who set an example to me of a scientist and a teacher. In particular, I acknowledge his wide expertise in physics, his patient willingness, his inexhaustible enthusiasm and his illuminating ideas, from which I had the opportunity to benefit (and learn) almost every day in the last three years.

Second, I am in debt to dr. Juan Rojo, my co-supervisor, for having introduced me to the NNPDF methodology, for continuous assistance with issues about code writing and for encouragement in pursuing my research with dedication. I owe my gratitude to him also for the opportunity he will give me to spend a couple of months in Oxford soon. To both Stefano and Juan I give my thanks for their training not only in undertaking scientific research, but also in presenting results: in particular, I very much appreciated (and benefitted from) their advices on talks and proceedings prepared for conferences in which I took part as a speaker.

Besides, I would like to thank prof. Richard D. Ball, who accepted to be the referee of this Thesis, and had carefully reviewed the manuscript: it has much improved thanks to his corrections and suggestions. I thank as well the other two external members of the final examination committee, prof. Mauro Anselmino and prof. Giovanni Ridolfi.

Additional thanks are due to: the Ph.D. school board, for extra financial support provided for housing costs during my stay in Milan; the Laboratorio di Calcolo & Multimedia (LCM) staff, for the computational resources provided to undertake the numerical analyses presented in this Thesis; dr. Francesco Caravaglios, who has allowed for sharing with me not only his office, but also his personal ideas on physics, mathematics, metaphysics, biology, chemistry: though looking weird at first, they finally reveal to me his genius.

I thank all those people I met during the last three years who also taught me a lot with their experience and knowledge. I would like to mention the members of the Department of Physics at the University of Milan, particularly those of the Theoretical Division who participated in the weekly journal club: Daniele Bettinelli, Giuseppe Bozzi, Giancarlo Ferrera, Alessandro Vicini. I also thank Mario Raciti for giving me the opportunity to work as a teaching assistant to his course in General Physics for bachelor students in Comunicazione Digitale.

I aknowledge several physicists I met at conferences and workshops, who raised me in the spin physics community: in particular Alessandro Bacchetta, Elena Boglione, Aurore Courtoy, Isabella Garzia, Francesca Giordano, Delia Hasch, Stefano Melis, Barbara Pasquini, Alexei Prokudin, Marco Radici, Ignazio Scimemi.

I very much appreciated the friendly and cheerful environment in the Milan Ph.D. school, thanks to all my colleagues fellow students. I just mention Alberto, Alice, Elena, Elisa, Rosa, Sofia. Special thanks are deserved by Stefano Carrazza, who has rapidly become for me an example of efficiency and hard work and a friend. I really enjoyed our discussions not only about physics and NNPDF code, but also about motorbikes and trekking. I will never forget our journey from Milan to Marseille to attend DIS2013 nor our excursion to Montenvers, Grand Balcon Nord and Plan de l’Aiguille.

During my stay in Milan I was housed in Centro Giovanile Pavoniano: I would like to thank the directorate, in particular Fr. Giorgio, for his kind hospitality, as well as all other guests who contributed to a friendly and enjoyable atmosphere.

Finally, I thank my family, in particular my parents, for their support: even though they stopped understanding what I am doing long ago, they have given me the chance to look at the world with curiosity and love.

Emanuele R. Nocera

Contents

*

\mainmatter

Chapter \thechapter Introduction

The investigation of the internal structure of nucleons is an old and intriguing problem which dates back to almost fifty years ago. For the past few decades, physicists have been able to describe with increasing details the fundamental particles that constitute protons and neutrons, which actually make up all nuclei and hence most of the visible matter in the Universe. This understanding is encapsulated in the Standard Model, supplemented with pertubartive Quantum Chromodynamics (QCD), the field theory which currently describes the strong interaction between the nucleon’s fundamental constituents, quarks and gluons. It is a remarkable property of QCD, known as confinement, that these are not seen in isolation, but only bound to singlet states of the their respective strong color charge.

Protons and neutrons are spin one-half bound states. Spin is one of the most fundamental concepts in physics, deeply rooted in Poincaré invariance and hence in the structure of space-time itself. The elementary constituents of the nucleon carry spin, quarks are spin one-half particles and gluons are spin-one particles. It is worth recalling that the discovery of the fact that the proton has structure - and hence really the birth of strong interaction physics - was due to spin, through the measurement of a very unexpected anomalous magnetic moment of the proton by O. Stern and collaborators in 1933 [2]. After decades of ever more detailed studies of nucleon structure, the understanding of the observed spin of the nucleon in terms of their constituents is a major challenge, far from being succesfully achieved.

1 Historical overview on the nucleon structure

The current picture of the nucleon structure is the result of more than half a century of theoretical and experimental efforts from physicists around the world. Even though a detailed historical overview is beyond the scope of this introduction, we find it useful to summarize the main steps in the building of our knowledge on the proton structure, with some emphasis on its spin.

Quarks were originally introduced in 1963 by Gell-Mann, Ne’eman and Zweig, simply based on symmetry considerations [3, 4, 5, 6], in an attempt to bring order into the large array of strongly-interacting particles observed in experiment. In a few words, they recognised that the known hadrons could be associated to some representations of the special unitary group. This led to the concept of quarks as the building blocks of hadrons. Mesons were expected to be quark-antiquark bound states, while baryons were interpreted as bound states of three quarks. In Nature there are no indications of the existence of other multiquark states: in order to explain this fundamental evidence and to satisfy the Pauli exclusion principle for baryons, such as the or the which are made up of three quarks of the same flavor, the spin-one-half quarks had to carry a new quantum number [7], later termed colour. The modern version of this constituent quark model still successfully describes most of the qualitative features of the baryon spectroscopy.

A modern realization of Rutherford’s experiment has shown us that quarks are real. This experiment is the deeply-inelastic scattering (DIS) of electrons (and, later, other leptons, including positrons, muons and neutrinos) off the nucleon, a program that was started in the late 1960’s at SLAC [8] (for a review see also Ref. [9]). A high-energy lepton interacts with the nucleon, via exchange of a highly virtual gauge boson. For a virtuality of GeV, distances shorter than fm are probed in the proton. The early DIS results led to an interpretation as elastic scattering of the lepton off pointlike, spin-one-half, constituents of the nucleon [10, 11, 12, 13], called partons. At first, this was understood in the so-called parton model: in this model, the nucleon is observed in the so-called infinite momentum frame, a Lorentz frame in which it is moving with large four-momentum: partons are assumed to move collinearly to the parent hadron, hence their transverse momenta and masses can be neglected. Lepton-nucleon scattering is then described in the impulse approximation, i.e. partons are treated as free particles and all partons’ self-interactions are neglected. In the impulse approximation, lepton-nucleon scattering is simply the incoherent sum of lepton interactions with the individual partons in the nucleon, which are carrying a fraction of its four-momentum. These interactions can be computed in perturbation theory, and have to be weighted with the probability that the nucleon contains a parton with the proper value of . This probability, denoted as , encodes the momentum density of any parton species , with longitudinal fraction , in a nucleon , and is called Parton Distribution Function (PDF). This cannot be computed using perturbative theory, since it depends on the non-perturbative process that determines the structure of the nucleon; hence, it has to be determined from the experiment.

Partons carrying fractional electric charge were subsequently identified with the quarks. The existence of gluons was proved indirectly from a missing () contribution [14, 15] to the proton momentum not accounted for by the quarks. Later on, direct evidence for gluons was found in three-jet production in electron-positron annihilation [16, 17, 18]. From the observed angular distributions of the jets it became clear that gluons have spin one [19, 20].

The successful parton interpretation of DIS assumed that partons are almost free (i.e., non-interacting) on the short time scales set by the high virtuality of the exchanged photon. This implied that the underlying theory of the strong interactions must actually be relatively weak on short time or, equivalently, distance scales [21]. In a groundbreaking development, Gross, Wilczek and Politzer showed in 1973 that the non-abelian theory of quarks and gluons, QCD, which had just been developed a few months earlier [22, 23, 24], possessed this remarkable feature of asymptotic freedom [25, 26], a discovery for which they were awarded the 2004 Nobel Prize for Physics. The interactions of partons at short distances, while weak in QCD, were then predicted to lead to visible effects in the experimentally measured DIS structure functions known as scaling violations [27, 28]. These essentially describe the response of the partonic structure of the proton to the resolving power of the virtual photon, set by its virtuality . The greatest triumph of QCD is arguably the prediction of scaling violations, which have been observed experimentally and verified with great precision. Deeply-inelastic scattering thus paved the way for QCD.

Over the following two decades or so, studies of nucleon structure became ever more detailed and precise. This was partly due to increased luminosities and energies of lepton machines, eventually culminating in the HERA electron-proton collider [29]. Also, hadron colliders entered the scene. It was realized, again thanks to asymptotic freedom and factorization, which follows from it, that the partonic structure of the nucleon seen in DIS is universal, in the sense that a variety of sufficiently inclusive hadron collider processes, characterized by a large scale, admit a factorized description [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41]. This offered the possibility of learning about other aspects of nucleon structure (and hence, QCD), for instance about its gluon content which is not primarily accessed in DIS. Being known with more precision, nucleon structure also allowed for new physics studies at hadron colliders, the outstanding example perhaps being the discovery of the and bosons at CERN’s collider [42, 43, 44]. The Tevatron and the Large Hadron Collider (LHC) are the most recent continuations of this line of research, which has culminated in the discovery of the Higgs boson [45, 46], announced by ATLAS and CMS collaborations on the July 2012.

Concerning spin physics, a milestone in the study of the nucleon was the advent of polarized electron beams in the early seventies [47]. This later on allowed for DIS measurements with polarized lepton beams and nucleon targets [48], and offered the possibility of studying whether quarks and antiquarks show on average preferred spin directions inside a polarized nucleon. The program of polarized DIS has been continuing ever since and it is now a successful branch of particle physics. Its most important result is the finding that quark and antiquark spins provide an anomalously small - only about - amount of the proton spin [49, 50], firstly observed by the EMC experiment in the late 1980’s. This finding, which opened a spin crisis in the understanding of the nucleon structure [51], has raised the interest of physicists in clarifying the potential role played by new candidates to the nucleon’s spin, like gluons’ polarizations and partons’ orbital angular momenta. In parallel, there also was a very important line of research on polarization phenomena in hadron-hadron reactions in fixed-target kinematics. In particular, unexpectedly large single-transverse spin asymmetries were seen [52, 53, 54, 55, 56].

In the last decade, the advent of the Relativistic Heavy Ion Collider (RHIC), the first machine to collide polarized proton beams, started to probe the proton spin in new profound ways [57], complementary, but independent, to polarized DIS. In particular, more knowledge on the polarization of gluons in the proton and details of the flavor structure of the polarized quarks and antiquarks has been recently achieved, as we will discuss in detail in this Thesis. However, despite a flurry of experimental and theoretical activity, a complete and satisfactory understanding of the so-called spin puzzle is still lacking.

2 Compelling questions in spin physics

The information on the proton spin structure is encoded in spin-dependent, or polarized, Parton Distribution Functions (PDFs) of quarks, antiquarks and gluons

(1)

which are the momentum densities of partons with helicity along () or opposite () the polarization direction of the parent nucleon . The dependence of the parton distributions, known as evolution [58], is quantitatively predictable in perturbative QCD, thanks to asymptotic freedom. Physically, it may be thought of the consequence of the fact that partons are observed with higher resolution when they are probed at higher scales; hence it is more likely that a struck quark has radiated one or more gluons so that it is effectively resolved into several partons, each with lower momentum fraction. Similarly, a struck quark may have originated from a gluon splitting into a quark-antiquark pair.

Polarized inclusive, neutral-current, DIS allows one to only access the flavor combinations , and the gluon polarization, though the latter is mostly determined indirectly by scaling violations. Of particular interest is the singlet quark antiquark combination , since its integral, known as the singlet axial charge, yields the average of all quark and antiquark contributions to the proton spin:

(2)

The anomalously small value observed experimentally for this quantity, from almost three decades of DIS measurement after the EMC result, strenghten the common belief that only about a quarter of the proton spin is carried by quarks and antiquarks. The EMC result was followed by an intense scrutiny of the basis of the corresponding theoretical framework, which led to the realization [59, 60] that the perturbative behavior of polarized PDFs deviates from parton model expectations, according to which gluons decouple at large energy scale. The almost vanishing value measured by EMC for the singlet axial charge can be explained as a cancellation between a reasonably large quark spin contribution, e.g. , as expected intuitively, and an anomalous gluon contribution, altering Eq. (2). A large value of the gluon contribution to the proton spin is required to achieve such a cancellation, and QCD predicts that this contribution grows with the energy scale. Despite some experimental evidence has suggested that the gluon polarization in the nucleon may be rather small, we emphasize that it is instead still largely uncertain, as we will carefully demonstrate in this Thesis. Other candidates for carrying the nucleon spin can come from quark and gluon orbital angular momenta [61, 62, 63] (for a recent discussion on the spin decomposition see also Ref. [64]).

In any case, the results from polarized inclusive DIS clearly call for further investigations: we summarize in the following some of the outstanding questions to be aswered in spin physics.

  • With which accuracy do we know spin-dependent parton distributions? The assessment of the singlet and gluon contributions to the proton spin requires in turn a determination of polarized parton distributions from available experimental data. In the last decade, several such determinations have been performed at next-to-leading order (NLO) in QCD [65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80], which is the current state-of-the-art accuracy for polarized fits, mostly based on DIS. Some of them also include a significant amount of data other than DIS, namely from semi-inclusive DIS (SIDIS) with identified hadrons in final states [77, 75, 80] or from polarized proton-proton collisions [75].

    However, we notice that they are all based on the standard Hessian methodology for PDF fitting and uncertainty estimation. This approach is known [81] to potentially lead to an underestimation of PDF uncertainties, due to the limitations in the linear propagation of errors and, more importantly, to PDF parametrization in terms of fixed functional forms. These issues are especially delicate when the experimental information is scarce, like in the case of polarized data. In particular, in this Thesis we will clearly demonstrate that a more flexible PDF parametrization is better suited to analyse polarized experimental data without prejudice. This will lead to larger, but more faithful, estimates of PDF uncertainties than those obtained in the other available analyses. At least, one should conclude that our knowledge of parton’s contribution to the nucleon spin is much more uncertain than commonly believed, unless one is willing to make some a priori assumptions on their behavior in the unmeasured kinematic regions. The two following questions address in much detail some issues related to quarks and gluons separately.

  • How do gluons contribute to the proton spin? The interest in an accurate determination of the gluon polarization is of particular interest for both phenomenological and theoretical reasons.

    On the phenomenological side, inclusive DIS allows for an indirect determination of the gluon distribution, through scaling violations. Since experimental data have a rather limited lever arm, it follows that is only weakly constrained. Processes other than inclusive DIS, which receive leading contributions from gluon initiated subprocesses, are better suited to provide direct information on the gluon distribution. In particular, these include open-charm production in fixed-target experiments and jet or semi-inclusive production in proton-proton collisions. However, the kinematic coverage of these data is limited: hence the integral of the gluon distribution can receive large contributions from the unmeasured region, in particular from the small- region.

    On the theoretical side, it is a remarkable feature of QCD that the gluon contribution to the nucleon spin may well be significant even at large momentum scales. The reason is that the integral of evolves as  [59], that is, it rises logarithmically with . This peculiar evolution pattern is a very deep prediction of QCD, related to the so-called axial anomaly. It has inspired ideas that a reason for the smallness of the quark spin contribution should be sought in a shielding of the quark spins due to a particular perturbative part of the DIS process  [59]. The associated contributions arise only at order ; however, the peculiar evolution of the first moment of the polarized gluon distribution would compensate this suppression. To be of any practical relevance, a large positive gluon spin contribution, , would be required even at low hadronic scales of a GeV or so. A very large polarization of the confining fields inside a nucleon, even though suggested by some nucleon models [82, 83, 84, 85], would be a very puzzling phenomenon and would once again challenge our picture of the nucleon.

  • What are the patterns of up, down, and strange quark and antiquark polarizations? Inclusive DIS provides information only on the total flavor combinations , . Nevertheless, in order to understand the proton helicity structure in detail, one needs to learn about the various quark and antiquark densities, , , , and , separately. This also provides an important additional test of the smallness of the quark spin contribution, and could reveal genuine flavor asymmetry in the proton sea, claimed by some models of nucleon structure [86, 87]. These predictions are often related to fundamental concepts such as the Pauli principle: since the proton has two valence- quarks which primarily spin along with the proton spin direction, pairs in the sea will tend to have the quark polarized opposite to the proton. Hence, if such pairs are in a spin singlet, one expects and, by the same reasoning, . Such questions become all the more exciting due to the fact that rather large unpolarized asymmetries have been observed in DIS and Drell-Yan measurements [88, 89, 90]. Further fundamental questions concern the strange quark polarization. The polarized DIS measurements point to a sizable negative polarization of strange quarks, in line with other observations of significant strange quark effects in nucleon structure.

  • What orbital angular momenta do partons carry? Quark and gluon orbital angular momenta are the other candidates for the carriers of the proton spin. Consequently, theoretical work focused also on these in the years after the spin crisis was announced. A conceptual breakthrough was made in the mid 1990s when it was realized [62] that a particular class of off-forward nucleon matrix elements, in which the nucleon has different momentum in the initial and final states, measure total parton angular momentum. Simply stated, orbital angular momentum is , where the operator can be viewed in Quantum Mechanics as a derivative with respect to momentum transfer. Thus, in analogy with the measurement of the Pauli form factor, it takes a finite momentum transfer on the nucleon to access matrix elements with operators containing a factor . It was also shown how these off-forward distributions, referred to as generalized parton distribution functions (GPDs), may be experimentally determined from certain exclusive processes in lepton-nucleon scattering, the prime example being Deeply-Virtual Compton Scattering (DVCS)  [62]. A major emphasis in current and future experimental activities in lepton scattering is on the DVCS and related reactions.

  • What is the role of transverse spin in QCD? So far, we have only considered the helicity structure of the nucleon, that is, the partonic structure we find when we probe the nucleon when its spin is aligned with its momentum. Experimental probes with transversely polarized nucleons could also be studied, both at fixed-target and collider facilities, and it has been known for a long time now that very interesting spin effects are associated with this in QCD. Partly, this is known from theoretical studies which revealed that besides the helicity distributions discussed above, for transverse polarization there is a new set of parton densities, called transversity [91, 92]. They are defined analogously to Eq. (1), but now for transversely polarized partons polarized along or opposite to the transversely polarized proton. Furthermore, if we allow quarks to have an intrinsic Fermi motion in the nucleon, they can be interpreted in light of more fundamental objects, the so-called Transverse Momentum Dependent parton distribution functions (TMDs) [93], in which the dependence on the intrinsic transverse momentum is made explicit. We refer to [94] for a comprehensive review on TMDs and the transverse spin structure of the proton. Here, we only mention that the present knowledge of TMDs is comparable to that of PDFs in the early 1970’s and very little is known about transversity. An intensive experimental campaign is ongoing to take data in polarized SIDIS and to provide a better determination of these distributions [95, 96, 97, 98].

3 Outline of the thesis

This Thesis addresses the three first questions in the above list, presenting a determination of spin-dependent parton distributions for the proton. In particular, two sets are obtained, the first based on inclusive DIS data only, the second also including the most recent data from polarized proton-proton collisions. In comparison to other recent analyses, our study is performed within the NNPDF methodology, which makes use of robust statistical tools, including Monte Carlo sampling for error propagation and parametrization of PDFs in terms of neural networks. The methodology has been succesfully applied to the unpolarized case [99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111] with the goal of providing a faithful representation of the PDF underlying probability distribution. This is particularly relevant with polarized data, which are rather scarce and, in general, affected by larger uncertainties than those of their unpolarized counterparts. Unlike all other standard fits, our parton sets do not suffer from the theoretical bias introduced either by a fixed functional form for PDF parametrization or by quadratic approximation in the Hessian propagation of errors. For this reason, we consider our unbiased determination to be crucial for investigating to which extent the common belief that about a quarter of the nucleon spin is carried by quarks and antiquarks, while the gluon contribution is even much smaller, actually holds.

Our parton determinations are publicly released together with computational tools to use them, including FORTRAN, C++ and Mathematica interfaces. Hence they could be used for any phenomenological study of hard scattering processes involving polarized hadrons in initial states. We should notice that, in addition to the investigation of the nucleon spin structure, such studies have recently included probes of different beyond-standard-model (BSM) scenarios [112] and possibly the determination of the Higgs boson spin in the diphoton decay channel, by means of the linear polarization of gluons in an unpolarized proton [113].

In this Thesis, we will not address the study of either TMDs or GPDs, but we notice that the methods ilustrated here apply to the determination of any non-perturbative object from experimental data: hence they could be used to determine such new distributions in the future, in so far as experimental data will reach more and more abundance and accuracy. Also, we do not describe either the apparatus which had to be developed to carry out spin physics experiments or the related technical challenges which had to be faced. A complete survey on these aspects can be found in Refs. [94, 114, 115].

The outline of this Thesis is as follows.

  • Chapter Document. Polarized Deeply-Inelastic Scattering. We review the theoretical formalism for the description of DIS with both polarized lepton beams and nuclear targets. In particular we derive the expressions for the differential cross-section of the process in terms of polarized structure functions. We will restrict our discussion to the contribution arising from the exchange of a virtual photon between the lepton and the nucleon. This is indeed sufficient to describe currently available esperimental data, whose energy does not exceed a few hundreds of GeV and none of which come from neutrino beams: hence we do not include either the (suppressed) contribution to neutral-current DIS mediated by a boson or charged-current DIS mediated by a boson. Then, we present the parton model expectations for DIS spin asymmetries introducing helicity-dependent, or polarized, PDFs and we discuss how they are modified in the framework of perturbative QCD. We complete our theoretical overview on polarized DIS with a sketch of spin sum rules, a summary of the relevant phenomenological relations between structure functions and measured observables, and the formalism adopted to take into account kinematic target mass corrections.

  • Chapter Document. Phenomenology of polarized Parton Distributions. We review how a set of PDFs is usually determined from a global fit to experimental data. First, we sketch the general strategy for PDF determination and its main theoretical and methodological issues, focusing on those which are peculiar to the polarized case. Second, we summarize how some of these problems are addressed within the NNPDF methodology, with the goal of providing a statistically sound determination of PDFs and their uncertainties. Finally, we provide an overview on available polarized PDF sets.

  • Chapter Document. Unbiased polarized PDFs from inclusive DIS. We present the first determination of polarized PDFs based on the NNPDF methodology, NNPDFpol1.0. This analysis includes all available data from inclusive, neutral-current, polarized DIS and aims at an unbiased extraction of total quark-antiquark and gluon distributions at NLO accuracy. We discuss how the statistical distribution of experimental data is sampled with Monte Carlo generation of pseudodata. We provide the details of the QCD analysis and discuss the PDF parametrization in terms of neural networks; we describe the minimization strategy and the peculiarities in the polarized case. We present the NNPDFpol1.0 parton set, illustrating its statistical features and its stability upon the variation of several theoretical and methodological assumptions. We also compare our results to other recent polarized PDF sets. Finally, we discuss phenomenological implications for the spin content of the proton and the test of the Bjorken sum rule. The analysis presented in this Chapter has been published by the NNPDF collaboration as a refereed paper [116].

  • Chapter Document. Polarized PDFs at an Electron-Ion Collider. We investigate the potential impact of inclusive DIS data from a future Electron-Ion Collider (EIC) on the determination of polarized PDFs. After briefly motivating our study, we illustrate which EIC pseudodata sets we use in our analysis and how the fitting procedure needs to be optimized. Resulting PDFs are compared to NNPDFpol1.0 throughout. Finally, we reassess their first moments and we give an estimate of the charm contribution to the structure function of the proton at an EIC. The analysis presented in this Chapter has been published by the NNPDF Collaboration as a refereed paper [117].

  • Chapter Document. Global determination of unbiased polarized PDFs. We extend the analysis presented in Chap. Document in order to include in our parton set, on top of inclusive DIS data, also recent measurements of open-charm production in fixed-target DIS, and of jet and production in polarized proton-proton collisions. Hence, we present the first global determination of polarized PDFs based on the NNPDF methodology: NNPDFpol1.1. After motivating our analysis, we review the theoretical description of the new processes and present the features of the relative experimental data we include in our study. We then turn to a detailed discussion of the way the NNPDFpol1.1 parton set is obtained via Bayesian reweighting of prior PDF Monte Carlo ensembles, followed by unweighting. We also present its main features in comparison to NNPDFpol1.0. Finally, we discuss some phenomenological implications for the spin content of the proton, based on our new polarized parton set. The analysis discussed in this Chapter has been presented in preliminary form in Refs. [118, 119].

  • Chapter Document. Conclusions and outlook. We will draw our conclusions, highlighting the main results presented in this Thesis. We also provide an outlook on future possible developments in the determination of polarized parton distributions within the NNPDF methodology.

  • Appendix Document. Statistical estimators. We collect the definitions of the statistical estimators used in the NNPDF analyses presented in Chaps. Document-Document-Document. Despite they were already described in Refs. [120, 99, 104], we find it useful to give them for completeness and ease of reference here.

  • Appendix Document. A Mathematica interface to NNPDF parton sets. We present a package for handling both unpolarized and polarized NNPDF parton sets within a Mathematica notebook file. This allows for performig PDF manipulations easily and quickly, thanks to the powerful features of the Mathematica software. The package was tailored to the users who are not familiar with FORTRAN or C++ programming codes, on which the standard available PDF interface, LHAPDF [121, 122], is based. However, since our Mathematica package includes all the features available in the LHAPDF interface, any user can benefit from the interactive usage of PDFs within Mathematica. The Mathematica interface to NNPDF parton sets appeared as a contribution to conference proceedings in Ref. [123].

  • Appendix Document. The FONLL scheme for up to . We collect the relevant explicit formulae for the practical computation of the polarized DIS structure function of the proton, , within the FONLL approach [124] up to . In particular, we will restrict to the heavy charm quark contribution to the polarized proton structure function , which might be of interest for studies at an Electron-Ion Collider in the future, as mentioned in Chap. Document.

Chapter \thechapter Polarized Deeply-Inelastic Scattering

This Chapter is devoted to a detailed discussion of Deeply-Inelastic Scattering (DIS) with both polarized lepton beams and nuclear targets. In particular, we will focus on neutral-current DIS, limited to the kinematic regime in which the exchange of a virtual photon between the lepton and the nucleon provides the leading contribution to the process. In Sec. 4, we rederive the expression for the differential cross-section of polarized DIS in terms of polarized structure functions. We present the naive parton model expectations for spin asymmetries in Sec. 5.1 and we discuss how they should be modified in the framework of QCD in Sec. 5.2. We complete our theoretical overview on polarized DIS with a sketch of spin sum rules in Sec. 6. Finally, we summarize the relevant phenomenological relations between structure functions and measured observables in Sec. 7, and the formalism adopted to take into account kinematic target mass corrections in Sec. 8.

4 General formalism

Let us consider the inclusive, neutral-current, inelastic scattering of a polarized lepton (electron or muon) beam off a polarized nucleon target,

(3)

where the four-momenta of the incoming (outgoing) lepton (), the nucleon target and the undetected final hadronic system are labelled as (), , and respectively. If the momentum transfer involved in the reaction is much smaller than the boson mass, as it is customary at polarized DIS facilities, the only sizable contribution to the process is given by the exchange of a virtual photon, see Fig. 1.

{fmffile}

simple \fmfframe(0,10)(0,20) {fmfgraph*}(200,125) \fmfleftli,P1 \fmfrightlo,X1 \fmflabelP1 \fmflabelli \fmflabello \fmflabelX1 \fmffermion,tension=2,label=,l.side=leftli,v2 \fmffermion,tension=2,label=,l.side=leftv2,lo \fmfphoton,label=,l.side=rightv1,v2 \fmffermion,label=,l.side=rightP1,v1 \fmffermion,label=,l.side=rightv1,X1 \fmfblob.15wv2 \fmffreeze\fmfiplain,tension=2vpath(__li,__v2) shifted (thick*(+0.0,+1.7)) \fmfiplain,tension=2vpath(__li,__v2) shifted (thick*(+0.5,-1.5)) \fmfiplain,tension=2vpath(__v2,__lo) shifted (thick*(+0.0,+1.7)) \fmfiplain,tension=2vpath(__v2,__lo) shifted (thick*(-0.5,-1.5)) \captionnamefont \captiontitlefont

Figure 1: Virtual-photon-exchange contribution to neutral-current DIS.

In order to work out the kinematics, we denote the nucleon mass, , the lepton mass, , the covariant spin four-vector of the incoming (outgoing) lepton () and the spin four-vector of the nucleon, . In the target rest frame, we define the four-momenta to be

(4)

The deeply inelastic regime is identified by the invariant mass of the final hadronic system to be much larger than the nucleon mass, namely

(5)

This allows us to neglect all masses and to use the approximation

(6)

Based on these assumptions, only two kinematical variables (besides the centre of mass energy or, alternatively, the lepton beam energy ) are needed to describe the process in Eq. (3). They can be chosen among the following invariants:

the laboratory-frame photon square momentum,
the laboratory-frame photon energy,
the Bjorken scaling variable,
the energy fraction lost by the incoming lepton ,

where is the scattering angle between the incoming and the outgoing lepton beams.

The differential cross-section for lepton-nucleon scattering then reads

(11)

where

(12)

is the phase-space factor for the unmeasured hadronic system and

(13)

is the squared amplitude including the Fourier transform of the quark electromagnetic current flowing through the hadronic vertex. Since we are describing the scattering of polarized leptons on a polarized target, with no measurement of the outgoing lepton polarization nor of the final hadronic system, in Eq. (11) we must sum over the final lepton spin and over all final hadrons , but must not average over the initial lepton spin, nor sum over the nucleon spin.

It is customary to define the leptonic tensor

(14)

and the hadronic tensor

(15)

in order to rewrite Eq. (11) as

(16)

or, in the target rest frame, where , and considering , ,

(17)

This is the differential cross-section for finding the scattered lepton in solid angle with energy usually quoted in the literature (see for example Ref. [125]). In Eq. (17), is the fine-structure electromagnetic constant, while is the azimuthal angle of the outgoing lepton. The variables and are natural ones, in that they are measured in the laboratory frame, by detecting the scattered lepton. However, it is more convenient to perform the variable transformation and to express the differential cross-section in terms of the latter quantities as

(18)

since these are gauge-invariant and dimensionless.

4.1 Leptonic tensor

In a completely general way, the leptonic tensor can be decomposed into a symmetric and an antisymmetric part under interchange

(19)

Recalling the identity satisfied by the spinor , for a fermion with polarization vector ,

(20)

and summing only on , the leptonic tensor reads

(21)

Trace computation via Dirac algebra finally leads to (retaining lepton masses)

(22)
(23)

If the incoming lepton is longitudinally polarized, its spin vector can be expressed as

(24)

i.e. it is parallel () or antiparallel () to the direction of motion ( is twice the lepton helicity). Then, Eq. (23) reads

(25)

Notice that the lepton mass appearing in Eq. (23) has been cancelled by the denominator in Eq. (24), which refers to a longitudinally polarized lepton. In contrast, if it is transversely polarized, that is, , no such cancellation occurs and the corresponding contribution is suppressed by a factor .

4.2 Hadronic tensor

The hadronic tensor allows for a decomposition analogous to Eq. (19), that is

(26)

where the symmetric and antisymmetric parts can be expressed in terms of two pairs of structure functions, , and , , as

(27)
(28)

It is customary to introduce the dimensionless structure functions

(29)
(30)

and to rewrite the symmetric and antisymmetric parts of the hadronic tensor as

(31)
(32)

These expressions give the most general gauge-invariant decompositions of the hadronic tensor for pure electromagnetic, parity conserving, interaction, see e.g [126] and references therein. A theoretical description of both neutral- and charged-current DIS at energies of the order of the weak boson masses (or higher) must include parity violating terms in the decomposition of the hadronic tensor. Because of them, one no longer has the correspondence that its symmetric part, Eq. (31), is spin independent and its antisymmetric part, Eq. (32), is spin dependent. Actually, the spin-dependent part of the hadronic tensor becomes a superposition of symmetric and antisymmetric pieces. Four more independent structure functions appear in this case, usually called , , and : the first multiplies a term independent from the lepton or nucleon spin four-vector, while the three latter do not. Such a general decomposition of the hadronic tensor in DIS, with particular emphasis on the polarized case, can be found in Ref. [127].

Since experimental data on polarized DIS are taken with electron or muon beams at momentum transfer values not exceeding GeV, we can safely neglect the contribution from a weak boson exchange to describe them properly. However, the most general decomposition will be needed in the future to handle neutral-current DIS at high energies, as it may be performed at an Electron-Ion Collider [128, 129], or charged-current DIS with neutrino beams, as it might be available at a neutrino factory [130].

4.3 Polarized cross-sections differences

Insertion of Eqs. (19) and (26) into Eq. (18) yields the expression

(33)

and differences of cross-sections with opposite target helicity states single out the tensor antisymmetric parts

(34)

In the target rest frame, using the notation of Fig. 2, we parametrize the nucleon spin four-vector as

(35)

where we have assumed . Taking the direction of the incoming lepton to be along the -axis, we also have

(36)
(37)
\captionnamefont\captiontitlefont
Figure 2: Azimuthal and polar angles of the final lepton momentum, , and the nucleon polarization vector, . The initial lepton moves along the positive -axis. Often one defines the lepton plane as the plane.

Supposing now that the incoming lepton is polarized collinearly to its direction of motion, i.e. and , we have

(38)

Note that, owing to currrent conservation, we have and the terms proportional to and in Eq. (31) do not contribute when contracted with the leptonic tensor. Explicit computation of four-momentum products in this equation yields to a new expression for the differential asymmetry (34)

(39)

where is the azimuthal angle between the lepton plane and the plane. Notice that the r.h.s. of this equation is not expressed in terms of the usual invariants and ; to this purpose, let us define the quantity

(40)

and work out a little algebra to obtain the final expression for the differential polarized cross-section difference Eq. (34)

(41)

Results obtained so far need a few comments.

  1. The terms longitudinal and transverse, when speaking about the nucleon polarization, are somewhat ambiguous, insofar as a reference axis is not specified. From an experimental point of view, the longitudinal or transverse nucleon polarizations are defined with respect to the lepton beam axis, thus longitudinal (transverse) indicates the direction parallel (orthogonal) to this axis. We will use the large arrows () to denote these two cases respectively.

  2. Eq. (41) refers to the scattering of longitudinally polarized (positive helicity) leptons off a nucleon with positive or negative polarization along an arbitrary direction . According to Eqs. (23)-(34), the cross-section difference is proportional to , which contains a small factor ; as already noticed, this small factor is cancelled by the factor appearing in the lepton-helicity four-vector, Eq. (24). This would not be the case with transversely polarized leptons, for which one would have , with . Then, transversely polarized leptons lead to tiny cross-section asymmetries of order of .

  3. Eq. (41) can be specialized to particular cases of the nucleon polarization. For longitudinally polarized nucleons, that is , one has and the differential cross-section reads

    (42)

    for nucleons polarized transversely to the lepton direction, one has and the differential cross-section is

    (43)

    In general, the term proportional to is suppressed by a factor , Eq. (40), with respect to the one proportional to : in the Bjorken limit, Eqs. (42)-(43) decouple and only is asymptotically relevant. We emphasize that, in the case of transverse polarizations, both the and structure functions equally contribute, but the whole cross-section difference is suppressed by the overall factor, Eq. (40), of order . In the following, we will mostly concentrate on the longitudinally polarized cross-section difference, Eq. (42).

  4. From Eq. (33), it is straightforward to obtain the unpolarized cross-section for inclusive DIS by averaging over spins of the incoming lepton () and of the nucleon () and by integrating over the azimuthal angle . It reads

    (44)

    Finally, the unpolarized cross-section, expressed in terms of the usual unpolarized structure functions and , when neglecting contributions of order , is

    (45)

5 Factorization of structure functions

In the previous Section, we have parametrized the hadronic tensor, which describes the coupling of the virtual photon to the composite nucleon, in terms of four structure functions, namely , and , , see Eqs. (31)-(32). We have then derived the expression for the differential cross-section asymmetries of longitudinally and transversely polarized nucleons in terms of and , Eqs. (42)-(43). In principle, by performing DIS experiments with nucleons polarized both longitudinally and transversely, one should learn about the structure functions and , as we will discuss in detail in Sec. 7 below. In this Section, we would like to provide a description of DIS in the framework of QCD, and in particular give a factorized expression for the structure function . Actually, even though QCD is asymptotically free, the computation of any cross-section does involve non-perturbative contributions, since the initial and final states are not the fundamental degrees of freedom of the theory, but compound states of quarks and gluons. As we shall see, the factorization theorem allows for the separation of a hard, perturbative and process-dependent part from a low energy, process-independent contribution. The latter is given by the Parton Distribution Functions (PDFs), which parametrize our ignorance on the inner structure of the proton. In order to deal with the factorized expression for the structure function , we will first provide the leading-order (LO) QCD description of polarized DIS, starting from the naive parton model; we will then give a heuristic development of the next-to-leading order (NLO) perturbative QCD corrections to polarized DIS, focusing on their effects on the structure function.

5.1 Naive parton-model expectations

The information on the a priori unknown structure of a polarized nucleon is carried by the structure functions and . As discussed in Sec. 4.2, they can only be functions of and . In the naive parton model [10, 11, 12, 13], they allow for simple expressions, since the cross-section for lepton-nucleon scattering is regarded as the incoherent sum of point-like interactions between the lepton and a free, massless parton

(46)

In this expression, is the fractional charge carried by a parton , is the cross-section for the elementary QED process , and is the PDF, the probability density distribution for the momentum fraction of any parton in a nucleon . In the simple picture provided by the parton model, PDFs do not depend on the scale and the structure functions are observed to obey the scaling law  [10]. This property is related to the assumption that the transverse momentum of the partons is small. In the framework of QCD, however, the radiation of hard gluon from the quarks violates this assumption beyond leading order in pertubation theory, as we will discuss below. Of course, the naive parton model predates QCD, but we find it of great value for its intuitive nature.

If we specialize Eq. (46) to polarized cross-section asymmetries, we should write

(47)

where denotes the elementary cross-section retaining helicity states of both the lepton () and the struck parton (). We also introduced helicity-dependent, or polarized PDFs, , defined as the momentum densities of partons with spin aligned parallel or antiparallel to the longitudinally polarized parent nucleon:

(48)

Explicit expressions for the elementary cross-sections appearing in the r.h.s of Eq. (47) are easily computed at the lowest order in Quantum Electrodynamics (QED):

(49)

Replacing these elementary cross-sections in Eq. (47) leads to the expression

(50)

which can be directly compared to Eq. (42), provided the latter is integrated over the azimuthal angle . Neglecting terms of order , we finally obtain the naive parton model relations between structure functions , and the polarized distributions :

(51)
(52)

These results require a few comments.

  1. The structure function , Eq. (51), can be unambigously expressed in terms of quark and antiquark polarized parton distributions. Assuming the number of flavors to be , we can define the singlet, , and nonsinglet triplet, , and octet, , combinations of polarized quark densities

    (53)
    (54)
    (55)

    where , are the total parton densities. Then, the structure function , Eq (51), can be cast into the form

    (56)

    The structure function does not receive any contribution from gluons, yet we shall see in Sec. 5.2 that it is not true in the framework of QCD beyond Born approximation.

  2. The structure function is zero, Eq. (52). However, non-zero values of can be obtained by allowing the quarks to have an intrinsic Fermi motion inside the nucleon. In this case, there is no unambiguous way to calculate in the naive parton model. We will not further investigate this issue in this Thesis; a detailed discussion of the problem can be found in Ref. [126].

  3. The Wilson Operator Product Expansion (OPE) can be applied to the expression of the hadronic tensor in terms of the Fourier transform of the nucleon matrix elements of the elctromagnetic current , Eq. (15). this way, one can give the moments of the structure functions and in terms of hadronic matrix elements of certain operators multiplied by perturbatively calculable Wilson coefficient functions. In particular, it can be shown (see e.g. [126]) that the first moment of the singlet quark density

    (57)

    is related to the matrix element of the flavor singlet axial current. Hence, can be interpreted as the contribution of quarks and antiquarks to the proton’s spin, intuitively twice the expectation value of the sum of the -components of quark and antiquark spins

    (58)

    Uncritically, one should expect , while in a more realistic relativistic model one finds  [61]. In the late 80s, this expectation was found to be in contrast with the anomalously small value measured by the European Muon Collaboration at CERN [49, 50]. This result could be argued to imply that the sum of the spins carried by the quarks in a proton, , was consistent with zero rather than , suggesting a spin crisis in the parton model [51]. This led to an intense scrutiny of the basis of the theoretical calculation of the structure function and the spin crisis was immediately recognized not to be a fundamental problem, but rather an interesting property of spin structure functions to be understood in terms of QCD. We will give a summary of such a description in the following Section.

5.2 QCD corrections and evolution

The parton model predates the formulation of QCD. As soon as QCD is accepted as the theory of strong interactions, with quark and gluon fields as the fundamental fields, one should describe the lepton scattering off partons in the nucleon perturbatively. At Born level, the interaction is described by the Feynman diagram in Fig. 3- as the tree-level scattering of a quark (or antiquark) off the virtual photon . In this case, quarks are free partons and one recovers the parton model expressions for the structure functions, Eqs. (51)-(52). At , several new contributions appear: the emission of a gluon, Fig. 3-, the one-loop correction, Fig. 3-, and the process initiated by a gluon which then splits into a quark-antiquark pair, the so-called photon-gluon fusion (PGF) process, Fig. 3-. The main impact of the QCD interactions is twofold: first, they introduce a mild, calculable, logarithmic dependence in the parton distributions; second, the correction in Fig. 3- generate a contribution to the structure function arising from the polarization of the gluons in the nucleon. We shall describe both these effects in the following.

{fmffile}

QCDcorr \fmfframe(0,0)(0,-70) {fmfgraph*}(80,160) \fmfleftli \fmfrightlo \fmftopgamma \fmffermionli,v2 \fmffermionv2,lo \fmfphoton,tension=2gamma,v2      {fmfgraph*}(80,160) \fmfleftli \fmfrightlo \fmfrightghost \fmftopgamma \fmffermionli,v3,v2 \fmfphotongamma,v2 \fmfvanillav2,v4 \fmffermionv4,lo \fmffreeze\fmfgluonv3,vf \fmfphantom,tension=1vf,ghost      {fmfgraph*}(80,160) \fmfleftli \fmfrightlo \fmftopgamma \fmffermionli,v3,v2 \fmffermionv2,v4,lo \fmfphotongamma,v2 \fmffreeze\fmfgluonv3,v4      {fmfgraph*}(80,160) \fmfleftli \fmfrightlo \fmfrightghost \fmftopgamma \fmfgluonli,v3 \fmffermionv3,v2 \fmfphotongamma,v2 \fmfvanillav2,v4 \fmffermionv4,lo \fmffreeze\fmffermionvf,v3 \fmfphantom,tension=1vf,ghost

\captionnamefont\captiontitlefont
Figure 3: Leading contribution and next-to-leading order corrections to DIS.
5.2.1 Scale dependence of parton distributions

When including NLO corrections, Fig. 3--, problems arise from the so-called collinear singularities linked to the effective masslessness of quarks. The factorization theorem is probed [131] to allow for a separation of the process into a hard and a soft part and for the absorption of the infinity into the soft part (the PDF), which in any case cannot be calculated and has to be determined from experimental data. The scale at which the separation is made is called factorization scale . Schematically, one finds terms of the form , which one splits as follows

(59)

one then absorbs the first term on the r.h.s. of Eq. (59) into the hard part of the process, and the second term into the soft part. The factorization scale can be chosen arbitrarily and, in exact calculations, physical results must not depend on it. In practice, since we never calculate to all orders in perturbation theory, it can make a difference what value we choose, but it turns out that an optimal choice is . Consequently, parton distributions no longer obey exact Bjorken scaling, but develop a slow logarithmic dependence on . Actually, if one keeps only the leading-log terms (proportional to ), one finds that the parton model expressions, Eqs. (51)-(52), still hold, provided the replacement

(60)

to the -dependent PDF is made. We can think of the scale dependence of PDFs within the following picture. As the scale increases, the photon starts to see evidence for the point-like valence quarks within the proton. If the quarks were non-interacting, no further structure would be resolved increasing the resolving scale: the Bjorken scaling would set in, and the naive parton model would be satisfactory. For this reason, we can consider the naive parton model as the approximation of QCD to Born level. However, QCD predicts that on increasing the resolution, one should see that each quark is itself surrounded by a cloud of partons. The number of resolved partons which share the proton’s momentum increases with the scale.

The perturbative dependence of the polarized PDFs on the scale is given by the Altarelli-Parisi evolution equations [58], a set of coupled integro-differential equations. It is customary to write them in the evolution basis, i.e. in terms of linear combinations of the individual parton distributions such that the equations maximally decouple from each other. To this purpose, we define the polarized gluon distribution as in Eq. (48) and the singlet and nonsinglet quark PDF combinations as

(61)
(62)

where and are the scale-dependent quark and antiquark polarized densities of flavor , also defined according to Eq. (48). The evolution equations are coupled for the singlet quark-antiquark combination and the gluon distribution