Contents
###### Abstract

The Spin Asymmetries of the Nucleon Experiment investigated the spin structure of the proton via inclusive electron scattering at the Continuous Electron Beam Accelerator Facility at Jefferson Laboratory in Newport News, VA. A double–polarization measurement of polarized asymmetries was performed using the University of Virginia solid polarized ammonia target with target polarization aligned longitudinal and near transverse to the electron beam, allowing the extraction of the spin asymmetries and , and spin structure functions and . Polarized electrons of energies of 4.7 and 5.9 GeV were scattered to be viewed by a novel, non-magnetic array of detectors observing a four-momentum transfer range of 2 to 6 GeV. This document addresses the extraction of the spin asymmetries and spin structure functions, with a focus on spin structure function , which we have measured as a function of and in four bins.

Probing Proton Spin Structure: a Measurement of

at Four-momentum Transfer of 2 to 6 GeV

James Davis Maxwell

Poquoson, Virginia

B.S. Physics, Mathematics, University of Virginia, 2004

M.A. Physics, University of Virginia, 2010

A Dissertation presented to the Graduate Faculty

of the University of Virginia in Candidacy for the Degree of

Doctor of Philosophy

Department of Physics

University of Virginia

December, 2011

James Davis Maxwell

December 2011

### Acknowledgments

It feels facile and inadequate to distill my gratitude into a few words on a page quickly skirted in a document so long. Nevertheless, I hope that the many people who helped make this work possible, a list too large to recount entirely here, know the depth of my appreciation for their support and contribution.

This experiment was a trying one, and all those involved deserve many thanks for persevering in the face of such challenges. All the SANE collaborators and JLab staff who contributed their expertise and equipment to the planning and execution of the experiment, as well as those who gave their time during shifts, deserve great credit for this work. In particular, I thank the experiment’s spokespersons, O. Rondon, S. Choi, Z. Meziani and M. Jones for tireless work to make SANE possible. My sincere thanks also go out to the JLab target group, lead by C. Keith, who were indefatigable in the installation and continued repair of the target during the experiment.

After 10 years, the Polarized Target Group at UVa seems almost a second home to me. I thank my graduate student colleagues, especially J. Mulholland who labored beside me in SANE, and J. Pierce and N. Fomin who I rightly consider mentors. I cannot thank the professors of the Target Group enough; I have benefited immeasurably from the guidance of D. Crabb and O. Rondon. Most of all, I thank my advisor, D. Day, my longtime mentor and steadfast advocate.

Finally, I’d like to thank all those who have shaped me as a scientist and a person; my friends, teachers, and family. It should go without saying that I owe all the success I meet to my parents, grandparents and the rest of my family; how can a grateful child ever repay his family? Lastly, I thank my wonderful wife, Ginny, who has been my best friend and vital support all through my graduate career.

Again, thank you all.

## Chapter 1 Introduction

The investigation of our world naturally leads us to seek the most basic building blocks of creation and to uncover how they interact with one another. While the early flights of fancy of Democritus and his school struck eerily close to home, it would be another 2,300 years before J.J. Thompson’s discovery of the electron[1] made the first entry into today’s roll of elementary particles. Cataloging these particles warrants the compilation of their intrinsic qualities, so we have endeavored to measure their mass and charges—the magnitudes of their interaction via the known forces. The measurements of Stern and Gerlach[2] in the 1920s, lead to the addition of spin to this list of fundamental properties.

The concept of spin is aptly, if perhaps misleadingly, named. In the electron, we observe a magnetic moment equivalent to that of a rotating charged particle, but how can a particle of no spatial extent rotate? Spin looks identical to angular momentum, but with the startling caveat that it is unrelated to any motion of the particle in space. We must abandon our intuition and accept spin as an fundamental quality; the electron is a spin- particle.

In 1927, Dennison established that the proton was also a spin- entity. When Stern and Estermann approached the measurement of the proton’s magnetic moment in 1933[3], the study of spin offered a seminal insight. The proton was observed to have an anomalous magnetic moment which was far larger than could be expected for a point particle of spin-. This was the first clue to the internal structure of nucleons—protons and neutrons—and began the inquiry into the nature and behavior of their constituents that continues today.

### 1.1 Leptons, Quarks and Bosons

The Standard Model provides only three types of elementary particles, two of which have corresponding antiparticles. There are six known leptons: the electron, muon and tauon, and their corresponding neutrinos; six known quarks: the up, down, charm, strange, top and bottom; and five known bosons: the gluon of the strong force, and the photon, Z and W of the electroweak force. We model the interactions of the spin- quarks and leptons which form matter via dynamical rules involving the exchange of the spin-1 mediating bosons.

Quantum Electrodynamics (QED) describes the interaction of all electromagnetically charged particles via the photon. Codified by Feynman, Schwinger and Tomanaga, QED has produced startlingly accurate predictions and represents the crowning achievement of modern Physics. Measurements of the electron’s anomalous magnetic moment agree with QED beyond 10 significant digits[4].

Quantum Chromodynamics (QCD) is the attempt to extend the rules and success of QED towards the description of the interaction of gluons and quarks. Quarks and gluons carry “color” charge; the electromagnetic charges of QED become six charges under QCD: red, anti-red, blue, anti-blue, green and anti-green. QCD is based upon an SU(3) symmetry group of the three colors, which form a “color octet” of gluons and a “color singlet” gluon which is not observed in our world[5]. These 8 gluons are superpositions of color and anti-color charges; for example, a red quark could exchange a red–anti-blue gluon to become blue.

QCD exhibits two related properties which make it quite different from QED: confinement and asymptotic freedom. Confinement requires that naturally occurring particles be colorless. This explains why we don’t observe free quarks, only combinations of two (mesons) or three (baryons) in which the colors of the quarks add up to white—as in red–anti-red or red–green–blue, for example.

Asymptotic freedom arises from the fact that gluons carry color charge and can thus couple to themselves. In QED, we observe “charge-screening” in which particle–antiparticle pair loops produced in the vacuum around an electron, for instance, serve to lessen the apparent charge of the electron as the distance from the electron increases. But in QCD, we have not only particle–antiparticle loops, but also gluon loops.

Since the gluon itself carries color charge, a red charge will beget more red charge in the vacuum around it, creating an anti-screening. As the distance from a color charge increases, the charge appears larger. Thus color charges in close proximity have a low coupling constant and are essentially free, but as they move away the coupling strength becomes greater and greater. As we will see later, this vanishing coupling strength at short distances enables a perturbative description of quark–gluon interactions at high energies.

### 1.2 Scattering Experiments

Scattering experiments have been the mainstay of elementary particle studies beginning with Rutherford’s seminal experiments in 1911. Rutherford, Geiger and Marsden[6, 7] scattered alpha particles through thin gold foil, and were able to discern the nucleus of the atom as a compact entity with a charge a multiple of the electron charge. The advance of experimental technology continues to expand the reach of scattering probes of nuclear structure.

The fundamental measured quantity in scattering experiments is the cross section. We first define two quantities, seen in figure 1.1111A note on the diagrams in this document. Unless otherwise noted, they are my own, most produced as vector graphics in Inkscape. They are available for free use with attribution.: for an incoming particle approaching a target particle, the distance by which it would have missed the target had it continued on its original path is called the impact parameter , and the angle of the final trajectory from the initial is the scattering angle . More generally, for a infinitesimal area around , , the particle will scatter into a solid angle around , . We will see that we can use the ratio to connect experimental observation of scattering processes to theoretical prediction.

#### 1.2.1 Variables

Before embarking on a discussion of the formalism of scattering processes, we will quickly establish a lexicon of commonly used variables. For an electron of four momentum interacting with a target particle of four momentum , as in figure 1.2, a single virtual photon is exchanged at leading order, scattering the electron at angle and resulting in final state four momenta of the particles and . The virtual photon four momentum is , which for a space-like virtual photon has , and includes an energy component , the energy loss of the electron. We thus define , the four-momentum transfer squared of the process.

It is useful in inclusive experiments, where only the final electron state is observed, to define the invariant mass of the final state , as well as the invariant scalar , whose significance will be explained later. In the laboratory frame, where , we have the following kinematic relations222We will be using natural units, in which , unless otherwise noted.:

 ν=E−E′Q2=4EE′sin2θ2W2=M2+2Mν−Q2x=Q22Mν. (1.1)

### 1.3 Inclusive Electron Scattering

We can construct the transition probability of a particular process using the invariant amplitude, or so-called “matrix element,” for the process, and the differential phase space available:

 transition rate=2πℏ|M|2×(% phase space) (1.2)

This is known as Fermi’s “Golden Rule.” The amplitude contains the dynamical information on the process, which we build using the Feynman calculus, while the phase space is simply the kinematical “room to maneuver” from the initial to final states.

In the context of scattering, we want to develop an expression for the differential cross section to relate to measured scattering angles and energies:

 dσ=|M|2F×dQ (1.3)

for Lorentz invariant phase space and a flux factor [8].

To build an invariant amplitude for a scattering process such as the one shown in figure 1.2 for lepton–lepton scattering, the Feynman calculus333See [8] table 6.2 or [5] section 7.5. prescribes the factors to collect based on features of our diagram444Feynman diagrams in this document will generally show space-time proceeding from left to right.. For each line leaving the diagram, we include an external line factor such as or for an incoming or outgoing electron. This , and its adjoint , represent solutions to the momentum space Dirac equation . Each vertex adds a , with representing the coupling strength of the vertex, here the charge of the electron . We then need factors for internal line propagation, which in this case is a photon: .

After including delta function factors to ensure conservation of momentum, we have an integral over internal momenta

 (2π)4∫[¯u(k′)γμu(k)]igμνq2[¯u(p′)γμu(p)]δ4(k−k′−q)δ4(p−p′+q)d4q, (1.4)

which we integrate and cancel the delta functions to reach the matrix element

 M=−g2e(k−k′)2[¯u(k′)γμu(k)][¯u(p′)γμu(p)]. (1.5)

#### 1.3.1 Electron–Muon Scattering

The matrix element we have achieved in equation 1.5 applies directly to scattering. By proceeding with this example, we illustrate a procedure which will carry over naturally to the case of elastic electron–proton scattering.

For the time being, we will assume no knowledge of the spin degrees of freedom; to find such a scattering amplitude we need to average over all spin states of to get , which we can compare with measurement.

Squaring our matrix element we have:

 |M|2=e4(k−k′)4[¯u(k′)γμu(k)][¯u(p′)γμu(p)][¯u(k′)γνu(k)]∗[¯u(p′)γνu(p)]∗. (1.6)

As we produce the spin average, it is convenient to separate the sums over the electron and muon spins such that

 ¯¯¯¯¯¯¯¯¯¯¯|M|2=e4q4LμνeLmuonμν, (1.7)

with the electron tensor

 Lμνe=12∑spins[¯u(k′)γμu(k)][¯u(k′)γνu(k)]∗, (1.8)

and a similar muon tensor. Using “Casimir’s trick” we can turn these sums over spins into traces of matrices, which we then apply trace theorems555See [8] sections 6.3 and 6.4 or [5] section 7.7. to simplify and remove the bilinear covariants of the Dirac equation:

 Lμνe=12Tr((⧸k′+m)γμ(⧸k+m)γν)=2(k′μkν+k′νkμ−(k′⋅k−m2)gμν). (1.9)

Now plugging these electron and muon tensor expressions back into 1.7, we have the following expression, with the mass of the electron, and of the muon:

 ¯¯¯¯¯¯¯¯¯¯¯|M|2=8e4q4[(k′⋅p′)(k⋅p)+(k′⋅p)(k⋅p′)−m2p′⋅p−M2k′⋅k+2m2M2]. (1.10)

Armed with this expression, we can construct a differential cross section for scattering in the laboratory frame. For a stationary muon as shown in figure 1.3, and neglecting the electron mass, we recall the relations of section 1.2.1 to get

 ¯¯¯¯¯¯¯¯¯¯¯|M|2=8e4q4[−12q2M(E−E′)+2EE′M2+12M2q2]=8e4q42M2EE′{cos2θ2−q22M2sin2θ2}. (1.11)

Now we apply the golden rule to build a differential cross section, still neglecting the electron mass:

 dσ=14ME¯¯¯¯¯¯¯¯¯¯¯|M|24π212E′dE′dΩd3p′2p′0δ4(p+q−p′) (1.12)

Finally, we arrive at a result, combining equations 1.11 and 1.12:

 (dσdΩ)lab=⎛⎝α24E2sin4θ2⎞⎠E′E{cos2θ2−q22M2sin2θ2}, (1.13)

with the factor arising from the target’s recoil, and the fine structure constant .

If we have a condition where the mass of the target particle is much larger than the scattering energy in equation 1.13, we recognize a familiar result from experiment—the Mott cross section of spin coupled Coulomb scattering:

 (dσdΩ)Mott=α24E2⎛⎝cos2θ2sin4θ2⎞⎠E′E. (1.14)

#### 1.3.2 Elastic Electron–Proton Scattering

Were the proton a point charge with Dirac magnetic moment , we would have reached our goal at equation 1.13. For a proton with internal structure, we need to adjust our matrix element accordingly. The key is that we can keep our electron tensor as is, carrying over what we know well from quantum electrodynamics and addressing the proton tensor separately:

 ¯¯¯¯¯¯¯¯¯¯¯|M|2=e4q4LμνelectronWprotonμν. (1.15)

Taking a step back, we change the matrix element from equation 1.5 accordingly; the of a spin- point particle doesn’t apply to the proton:

 M=−g2e(k−k′)2[¯u(k′)γμu(k)][¯u(p′)[ ? ]u(p)]. (1.16)

To fill those square brackets which have taken the place of a , we look for a four-vector to fit between our Dirac spinors. We naively build a four-vector out of , , and bilinear covariants, except which is ruled out by parity conservation. Following section 8.2 of [8], without loss of generality, we can insert

 [f1(q2)γμ+κ2Mf2(q2)iσμνqν], (1.17)

where we have introduced two independent form factors, and , and the anomalous magnet moment . These two form factors parametrize the unknown behavior shown by the open circle in figure 1.4. In practice, these form factors are written so that no interference terms appear in the cross section:

 GE≡f1+κq24M2f2GM≡f1+κf2 (1.18)

Now, for elastic scattering, equation 1.13 becomes

 (dσdΩ)lab=⎛⎝α24E2sin4θ2⎞⎠E′E{G2E+τG2M1+τcos2θ2+2τG2Msin2θ2} (1.19)

with . This is the Rosenbluth cross section, with the Sachs form factors and . We can think of the form factors as the extent of the electric and magnetic charge, and are rightly the Fourier transforms of the charge distributions. Differences between the ratios of these form factors from measurements using polarization transfer and Rosenbluth separation techniques continue to prompt inquiry[9, 10]. An overview of these electromagnetic form factors can be found in reference [11].

#### 1.3.3 Deep Inelastic Electron–Proton Scattering

As we peer deeper into the proton using a virtual photon of smaller wavelength, the increased energy of the scattering interaction will tear apart the proton. In elastic scattering , the final state of the proton could be represented by the Dirac entry into the matrix element. As we break up the proton , shown in figure 1.5, we need a new formalism for the final state.

In inelastic scattering, the invariant mass of the final state , or the “missing” mass in inclusive scattering, becomes a quantity of interest. With increasing , peaks emerge in the spectrum of versus the missing mass . The first, at equal to the proton mass, is the elastic peak in which the proton does not break up. At higher are resonance peaks in which the target is excited into resonant baryon states, such as the at mass 1232 MeV (see figure 1.6). Beyond the resonances is the smooth curve made up of the many complicated multi-particle states of deep inelastic scattering.

As in the case of elastic scattering, to proceed to form an expression for this scattering we separate the matrix element into an electron tensor and a proton tensor:

 d2σdΩdE′=α22Mq4E′ELμνeWμν. (1.20)

We recognize the electron tensor, now dealing with the spins explicitly:

 Lμνe=12∑spins¯u(k,s)γμu(k′,s′)¯u(k′,s′)γνu(k,s)=k′μkν+k′νkμ−gμνk⋅k′+[iϵμνλσqλsσ], (1.21)

after summing over spins, where here we have enclosed the part which is antisymmetric under interchange in brackets, which includes the spin vector for the electron .

As we look to the proton tensor , we must be even more general in our formulation than in the elastic case as we can’t even rely on Dirac . Taking into account parity conservation, Lorentz invariance, gauge invariance, and standard discrete symmetries of the strong force, we can maintain generality while parameterizing in four dimensionless structure functions[14], two symmetric in , interchange (superscript ) and two antisymmetric (superscript ):

 Wμν(q;p,S)=W(S)μν(q;p)+iW(A)μν(q;p,S) (1.22)

with

 12MW(S)μν(q;p)=(−gμν+qμqνq2)W1(p⋅q,q2)+(pμ−p⋅qq2qμ)(pν−p⋅qq2qν)W2(p⋅q,q2)M212MW(A)μν(q;p,S)=ϵμναβqαMSβG1(p⋅q,q2)+ϵμναβqα[(p⋅q)Sβ+(S⋅q)pβ]G2(p⋅q,q2)M. (1.23)

Here we have used the proton spin vector . We notice the symmetric portion of the hadronic tensor consists of two spin-independent structure functions, and , while the spin-dependent, antisymmetric portion gives us structure functions, and .

As we measure experimental cross sections, we access different structure functions depending on our control of the spin degrees of freedom[15]. For instance, unpolarized electron–proton scattering results in a cross section which is proportional to the symmetric terms:

 d2σunpoldΩdE′(k,p;k′)=α2Mq4E′EL(S)μνWμν(S). (1.24)

Or, if we take a difference of cross sections of opposite target spin polarizations, still summing over electron spins, we can measure the antisymmetric terms:

 ∑s′[d2σdΩdE′(k,s,p,−S;k′,s′)−d2σdΩdE′(k,s,p,S;k′,s′)]=2α2MQ4E′EL(A)μνWμν(A). (1.25)

We will present explicit expressions for the structure functions in terms of cross sections of different spin orientations in section 2.4. We can now focus our interest in these structure functions to continue our investigation of the structure of the nucleon.

### 1.4 Bjorken Scaling

We have seen that as we increase the momentum transfer of our scattering interaction, the proton ceases to behave like a point particle, revealing internal structure. At yet higher , we begin to suspect the presence of point particles, or partons, inside the proton (figure 1.7) as the first two proton structure functions simplify to

 2mWpoint1(ν,Q2)=Q22mνδ(1−Q22mν)νWpoint2(ν,Q2)=δ(1−Q22mν). (1.26)

Here we notice these functions depend only on the dimensionless ratio , where mass is of that of the constituent particle inside the proton [8].

With this in mind we define the deep inelastic regime in the Bjorken limit:

 −q2≡Q2→large,ν=E−E′→∞,x=Q22p⋅q=Q22Mνconstant. (1.27)

In the Bjorken limit, the proton structure functions, which depend on and , become dependent only upon the dimensionless Bjorken , a sign that the partons themselves have no internal structure. Figure 1.8 shows an example of scaling behavior for .

Thus, in the Bjorken limit we can give the structure functions as

 MW1(ν,Q2)≡F1(x,Q2)−−−−−→large Q2F1(x),νW2(ν,Q2)≡F2(x,Q2)−−−−−→large Q2F2(x),(p⋅q)2νG1(ν,Q2)≡g1(x,Q2)−−−−−→large Q2g1(x),ν(p⋅q)G2(ν,Q2)≡g2(x,Q2)−−−−−→large Q2g2(x). (1.28)

We have now bundled up all the inner workings of the proton into these four scaling structure functions which are functions only of in the Bjorken limit. Bjorken can be thought of as the fraction of the proton’s momentum which was carried by the struck constituent particle. Obviously, in the lab frame the proton is stationary; this definition applies in the Breit frame of reference, where the outgoing momentum of the proton is equal but opposite the incoming momentum, shown in figure 1.9.

From equations 1.26 and 1.28, we also see a useful relation between the unpolarized structure functions:

 F2(x)=2xF1(x), (1.29)

known as the Callan-Gross relation. Looking at figure 1.8, the scaling behavior falls off at high and low , hinting at the effects of the constituents’ interactions. The change, or so-called “evolution”, of the structure functions in is described by the Dokshitzer–Gribov-âLipatov-âAltarelli-âParisi (DGLAP) equations[17, 18].

### 1.5 Compton Scattering & Inclusive ep→eX

Before moving on to a deeper discussion of the spin structure functions, it is worthwhile to take a brief aside to show another way to look at the hadronic tensor and thus , , , and . As the hadronic tensor deals with the virtual photon’s intersection with the proton, the connection with virtual Compton scattering is not entirely unintuitive.

If we consider virtual () forward () Compton scattering seen in figure 1.10, we can express the scattering amplitude in terms of the electromagnetic current as

 Tμν(q;p,s)=i∫d4zeiq⋅z⟨p,s|T(Jμ(z)Jν(0))|p,s⟩ (1.30)

with the time ordering operator [19, 15].

The hadronic tensor can be similarly expressed as the Fourier transform of the matrix elements of the commutator of electromagnetic currents in inclusive scattering:

 Wμν(q;p,s)=12π∫d4zeiq⋅z⟨p,s|[(Jμ(z),Jν(0)]|p,s⟩ (1.31)

With equations 1.30 and 1.31, the relation between the forward virtual Compton tensor and the inclusive hadronic tensor, properly a result of the optical theorem, is apparent:

 Wμν(ν,Q2)=1πImTμν(ν,Q2). (1.32)

The hadronic tensor is proportional to the imaginary (or absorptive) part of the forward virtual Compton tensor[15, 20].

One of the results of this relation is the connection between virtual photon absorption asymmetries and , and the structure functions. Asymmetries and are defined in terms of virtual photon absorption cross sections for polarized photons and nucleons; these 4 cross sections are labeled by the spin sum, anti-parallel or parallel , and L or T for a longitudinal or transverse photon[21].

 A1=σT1/2−σT3/2σT1/2+σT3/2A2=2σTL1/2σT1/2+σT3/2 (1.33)

The spin structure functions are expressed in terms of these asymmetries and the structure function as

 (1.34)

for .

## Chapter 2 Proton Spin Structure

In the previous chapter we established a framework for studying nucleon structure through lepton scattering experiments, parameterizing the proton’s unknown behavior in four structure functions. In this chapter we will endeavor to interpret physical meaning from these structure functions, detail a methodology to measure them, and review existing measurements. We take advantage of excellent review papers on the study of nucleon spin structure in this chapter, references [22, 21, 15, 20, 19, 23, 24, 25, 26].

### 2.1 Partons

Faced with Bjorken scaling, we look for a model of the proton with point particle constituents. The parton model put forward by Feynman in 1969 [27] does just this, describing a nucleon made up of different kinds of point particles, partons, which were later recognized as quarks and gluons.

In this model, we consider the constituent partons to be semi-free and point-like. We can begin to put together a picture of how the spin of these partons might contribute to the spin of the proton, as in this non-relativistic wave function for a proton made of up () and down () quarks [22]:

 |p↑⟩=1√6(2|u↑u↑d↓⟩−|u↑u↓d↑⟩−|u↓u↑d↑⟩), (2.1)

where the superscript arrows represent the spin state of the quarks as aligned or anti-aligned with the proton spin. Here the quarks carry all of the proton’s spin.

#### 2.1.1 Structure Functions in the Parton Model

Armed with a model of a proton made of semi-free partons, we return to deep inelastic electron–proton scattering to formulate our structure functions, recalling the hadronic tensor . Following references [28, 15], if we let be the number of partons with charge , momentum fraction , and spin vector , inside a nucleon of momentum and spin vector , we can express our hadronic tensor as

 Wμν(q;P,S)=W(S)μν(q;P)+iW(A)μν(q;P,S)=∑q,se2q12P⋅q∫10dx′x′δ(x′−x)nq(x′,s;S)wμν(x′,q,s). (2.2)

The sum goes over all quarks and anti-quarks. Here the can been seen as the analogue of the hadronic tensor for the case of photon interacting with a “free” parton.

As we see in figure 2.1, we have now simplified the photon–proton interaction to a photon–parton vertex with the parton as a point, charged fermion. We can thus calculate using QED, leaving the strong interaction dynamics in the number density function. Treating as we did , but with replacements and , we have:

 wμν=w(S)μν+iw(A)μν (2.3)

with

 w(S)μν=2[2x2PμPν+xPμqν+xqμPν−x(p⋅q)gμν]w(A)μν=2mqϵμναβsαqβ. (2.4)

Before we move forward, we condense our notation so that the parton number densities are

 qλ≡P∫d2p⊥nq(p,λ;Λ=1/2), (2.5)

so that represents the number density of quarks with momentum , helicity in a proton of momentum and helicity . We can now create the unpolarized number density and difference of spin-dependent quark distribution functions :

 q(x)=q+(x)+q−(x),Δq(x)=q+(x)−q−(x). (2.6)

Integrating over the assumed small transverse momentum and comparing with equation 1.23, we combine with the above equations to arrive at predictions for our structure functions in this quark-parton interpretation:

 2xF1(x,Q2)=F2(x,Q2)=12∑qe2qq(x), (2.7)

where are the charges of these quark flavors and we have used the Callan–Gross relation of equation 1.29. Likewise, plugging in gives us expressions of the spin structure functions

 g1(x,Q2)=12∑qe2qΔq(x),g2(x,Q2)=0. (2.8)

In the zero result for , we begin to see cracks in the so-called naive quark-parton model. The hard-photon, free-quark interaction is not sensitive to in which transverse spin is important. Non-zero values of can be obtained by adding transverse momentum to the model, which we have neglected above, but these formulations have an extreme sensitivity to the quark mass. To access we abandon our simplistic model in favor of the more robust formulation of QCD in DIS.

### 2.2 pQCD and Duality

Quantum Chromodynamics moves beyond the naive model of semi-free partons to tackle the color charge interactions between the quarks via mediating gluons. However, the study of semi-free quarks was not entirely wasted. Due to the property of asymptotic freedom discussed in section 1.1, quarks in the nucleon actually do appear to be nearly free at small enough distance scales. This means at high we can treat the processes perturabtively, in what is aptly named perturbative QCD, or pQCD.

At large , pQCD describes experimental findings quite well. pQCD correctly predicts the logarithmic violations of Bjorken scaling in the structure function , which comes from gluon production and quark–anti-quark pair creation. Due to pQCD, we can expect the structure function expression in terms of parton distribution functions from section 2.1.1 to hold at high . However at low , as the interactions between quarks and gluons become important, pQCD predictions should break down.

At lower , approaching the region where resonance production dominates the cross section, a peculiar property was discovered which extends the usefulness of pQCD. In 1970, Bloom and Gilman [29, 30] saw that when the structure function was measured in the resonance region, it roughly averaged out to the value of expected from the scaling limit.

Defining the Nachtmann scaling variable

 ξ=2x1+√1+4M2x2Q2 −−−−→Q2→∞ x (2.9)

attempts to generalize Bjorken to take into account target mass corrections, counteracting the troublesome sensitivity to the quark mass. Plotting gives a convincing view of duality. As increases, the resonance peaks can be seen sliding along the curve of at high , as seen in figure 2.2. When an individual resonance follows duality in a given region, we call it “local” duality. In “global” duality, this averaging is satisfied over all resonances. Duality thus extends the results of pQCD into regions of far lower than might be expected, for certain quantities[23].

### 2.3 Moments and Twist

When evaluating the behavior of structure functions as they evolve in , it is useful to define moments, or -weighted integrals, of the structure functions. We define the th moment of and as

 M(n)1(Q2)=∫10dxxn−1F1(x,Q2)M(n)2(Q2)=∫10dxxn−2F2(x,Q2). (2.10)

These are the Cornwall–Norton moments [32]. For of we have an effective count of quark charges, while of gives the momentum sum rule. Likewise, the spin structure function moments are

 Γ(n)1(Q2)=∫10dxxn−1g1(x,Q2)Γ(n)2(Q2)=∫10dxxn−2g2(x,Q2). (2.11)

#### 2.3.1 Operator Product Expansion

To describe quark–hadron duality, as well as the spin structure function in QCD, we turn to the operator product expansion. The “OPE” was introduced in 1968 by K. Wilson [33] as a way to understand the behavior of moments in DIS, and remains useful after the formulation of QCD to evaluate calculations outside the perturbative region. In the case of inclusive DIS, the OPE lets us express the products of operators in the asymptotic limit. The operators we are interested in are the electromagnetic currents as discussed in section 1.5.

In the OPE, as the spatial four-vector goes to zero, the product of operators and can be expressed as the series

 limz→0Oa(z)Ob(0)=∑kCabk(z)Ok(0) (2.12)

The key here is that the so-called Wilson coefficients contain all the spatial dependence in the sum. The equivalence holds as long as the external states of the process have momenta which are small compared to the separation . Since our coupling constant in QCD is small at short distances due to asymptotic freedom, we can calculate the coefficients in the perturbative range[26]. Thus pQCD calculations can be used to understand our operators in other regimes.

To apply the OPE for the spin structure functions, we start with the expression for the hadronic tensor in terms of the commutator of electromagnetic currents (equation 1.31):

 Wμν(q;p,s)=12π∫d4zeiq⋅z⟨p,s|[(Jμ(z),Jν(0)]|p,s⟩. (2.13)

Taking the Fourier transform of 2.12 gives us the momentum space version of the OPE, which we can apply to 2.13:

 limz→0∫d4zeiq⋅zOa(z)Ob(0)=∑kCabk(q)Ok(0). (2.14)

The product of our electromagnetic currents in equation 2.13 can now be expanded as a sum of local operators times coefficients which are functions of . These expansion operators are quark and gluon operators with arbitrary dimension and spin . The contribution of any operators to is of order

 (p⋅q)n(QM)2+n−d=(p⋅q)n(QM)2−τ, (2.15)

where we now define the twist of the operator as .

The lowest, or leading twist, twist-2, contributes the largest in the Bjorken limit, with higher twist contributions suppressed by powers of . Using dispersion relations, we can apply the OPE to equation 2.13 to arrive at expressions for the odd moments of our structure functions. Ignoring contributions beyond twist-3, we have

 ∫10xn−1g1(x,Q2)dx=12an−1;for n=1,3,5...∫10xn−1g2(x,Q2)dx=n−12n(dn−1−an−1);for n=3,5... (2.16)

where and are matrix elements of the quark and gluon operators for twist-2 and twist-3, respectively.

#### 2.3.2 Burkhardt–Cottingham Sum Rule

The OPE has nothing to say about the term of the expression in equation 2.16, but the Burkhardt–Cottingham sum, which addresses the first moment of , is not entirely unexpected [34]:

 Γ2(Q2)=∫10dxg2(x,Q2)=0. (2.17)

This result was first derived from the asymptotic behavior of the virtual Compton helicity amplitude which is proportional to .

If this B.C. sum rule is violated, it is likely due to one of two circumstances, according to reference [35]:

1. is so singular that does not exist.

2. has a delta function singularity at .

#### 2.3.3 Wandzura–Wilczek Relation

By combining the two equations in 2.16, we can cancel the leading twist terms to achieve an expression for and :

 ∫10xn−1dx(g1(x,Q2)+nn+1g2(x,Q2))=dn2 (2.18)

for an integer greater or equal to 3. After performing Mellin transforms, which relate the product of moments of two functions to the moment of their convolution, we arrive at the following result:

 gWW2(x,Q2)+g1(x,Q2)=∫10dyyg1(y,Q2) (2.19)

where here we have set the twist-3 terms to zero. We’ve labeled in this equation as to designate that this expression ignores higher twist terms. As it stands, this expression, known as the Wandzura–Wilczek relation[36], allows us to determine the leading twist portion of using knowledge of , which in turn allows its expression in terms of the parton model. It should be noted that the OPE does not cover the term of the expansion, so this definition assumes validity of the Burkhardt–Cottingham sum rule.

With our definition of , we have relegated the higher twist contribution to into the portion here called :

 g2(x,Q2)=gWW2(x,Q2)+¯g2(x,Q2) (2.20)

While Wandzura and Wilczek went further to hazard that is zero, we can think of it as the interesting part of [24]. The moments

 ∫10dxxn¯g2(x,Q2)=n4(n+1)dn(Q2) (2.21)

are of twist–3 and thus access quark–gluon correlations[37].

This can itself be split into multiple terms, following [38]:

 (2.22)

where we introduce , the twist–3 contribution, and , the “transversity” distribution from transverse quark polarization, which is a twist–2 term suppressed by the ratio of the quark to target nucleon mass[20].

#### 2.3.4 Twist–Three and g2

While the operator product expansion has given us a foundation to express in the form of higher-order twist, with twist we are left with a mathematical construct from which it is difficult to draw physical meaning. To understand higher-twist, we must consider parton correlations initially present in the participating hadrons.

Higher-twist processes can be thought of as involving more than one parton of the hadron in the scattering process, such as in the example in figure 2.3. We can see the influence of other partons through helicity exchange which is necessary to allow the process. This exchange can happen in two ways in QCD: through single quark scattering in which the quark carries angular momentum though its transverse axis; or through quark scattering with a transverse-polarized gluon from the hadron [22].

Twist–3 represents the first of the higher-order terms, and therefore gives the greatest contribution to and , after leading-order, of course. In twist–3 we see quark–gluon–quark correlations; instead of viewing only a bare quark we are beginning to probe how the quarks and gluons interact in the context of the nucleon! With this in mind, , which offers the most direct view of these correlations, becomes an attractive quantity to measure.

### 2.4 Measuring Spin Structure Functions

As we asserted in section 1.3.3, we can access the antisymmetric portion of the hadronic tensor via deep inelastic electron–proton scattering by taking a difference of cross sections of opposite polarizations. In this section, we’ll develop expressions to obtain the structure functions and using measurements of asymmetries of cross sections, from a polarized electron beam upon a polarized proton target, anticipating the measurements of SANE.

To save space, we define the difference of cross sections and expand it following the steps of section 1.3.3:

 Δσ=∑s′[d2σdΩdE′(k,s,p,−S;k′,s