# Transverse Spin Structure Functiong2x,Q2

Authors: H. Bagherpour; M. Saeedhosseini
Journal: ISRN High Energy Physics
 (2013)

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## Abstract

The transverse spin-dependent nucleon structure functiong2x,Q2 at large Q2 is computed within the Wandzura-Wilczek relation, using NLO g1 fit. Also we investigate the twist-3 contribution of g2x,Q2 via calculating d2. It turns out that although the twist-3 part of g2x,Q2 can be negligible for DIS processes, it has a significant value in resonance region.


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## Body

## 1. Introduction

Over the past 30 years, significant progress has been made in understanding the spin structure of the nucleon through measurements using polarized deep inelastic lepton scattering (DIS). Most of these experiments were focused on precise measurements of the spin structure functiong1. In the naive quark parton model (QPM), g1 is directly related to contributions of the individual quark flavors to the overall spin of the nucleon. Sum rules based on this simple model have provided fertile ground for understanding the origin of the nucleon spin in terms of quark degrees of freedom. In addition, next-to-leading-order (NLO) analyses of the world g1 data (see, e.g., [1–5]) have provided indirect information about the role of gluons in the nucleon’s spin. Polarized DIS also provides information about a second spin structure function, g2, which is identically zero in the naive QPM [6]. Interest in g2 arises because, unlike g1, contributions from certain nonperturbative QCD processes such as quark-gluon correlations enter at the same order in Q2 as asymptotically free contributions [7].


## 2. Theoretical Background

### 2.1. Theg2 Structure Function

The nucleon’s second polarized structure functiong2(x,Q2) had never been measured till 1990, and there had been a few theoretical studies of it. If we define qf(x) and q-f(x) as the expectation value for the number of quarks and antiquarks of flavor f in the hadron whose momentum fraction lies in the interval [x;x+dx], then in the parton model it can be shown that
(1)F1(x)=12∑fef2[qf(x)+q-f(x)],g1(x)=12∑fef2[qf(x)-q-f(x)],
where the quark charge ef enters due to the fact that the cross section is proportional to the squared charge of the target. The Callan-Gross [8] relation shows that F2 can be defined entirely in terms of F1, but there is no such simple physical interpretation of g2. This spin-dependent structure function is determined by the x-dependence of the quarks transverse momenta and the off-shellness, both of which are unknown in the parton model [9]. Ignoring quark mass effect of order 𝒪(mq/∧QCD), g2 can be separated into leading twist and higher-twist components as follows:
(2)g2(x,Q2)=g2WW(x,Q2)+g-2(x,Q2),
where
(3)g2(x,Q2)=-∫x1∂∂y[mqMhT(y,Q2)+ζ(y,Q2)]dyy.To twist-3, there are three contributions tog2 [10]:g2ww: the leading twist-2 term, which depends only on g1,hT: arises from the quark transverse polarization distribution. Also twist-2, this term, is suppressed by the smallness of the quark mass,ζ: the twist-3 part which arises from quark-gluon interactions,Using the fact thatg1 and g2 contain the same twist-2 operator, Wandzura-Wilczek [11] derived the following expression for the asymptotically free contribution to g2, in terms of g1:
(4)g2ww(x,Q2)=-g1(x,Q2)+∫x1g1(x,Q2)xdx.Also this relation can be extracted directly from twist-2 operators of OPE.


### 2.2. Operator Product Expansion

An appropriate formalism to studyg2 beyond the simple QPM is the operator product expansion (OPE) [13, 14] which is the model-independent approach based directly on QCD. The OPE allows us to write the hadronic matrix element in deep inelastic scattering (DIS) in terms of a series of normalized operators of increasing twist. The leading contribution is twist-2, with higher-twist terms suppressed by powers of 1/Q2. Keeping only terms up to twist-3, the moments of g1 and g2 at fixed Q2 can be related to the twist-2 and twist-3 reduced matrix elements, an and dn [14],
(5)∫01xng1(x,Q2)dx=an2,n=0,2,4,…,∫01xng2(x,Q2)dx=12nn+1(dn-an),n=2,4….In the expression above,dn directly appears in the equation for g2allowing us to study the higher-twist structure of the nucleon at leading order. One obtains
(6)∫01xn[g1(x,Q2)+n+1ng2(x,Q2)]dx=12dn(Q2),n=2,4,6….Hence, forn=2,
(7)d2=2∫01x2[g1(x,Q2)+32g2(x,Q2)]dx.At largeQ2, the dn matrix element is related to the color polarization, which describes how the color electric and magnetic fields respond to the nucleon spin. At lower momentum transfer, dn(Q2) provides a mean to study the transition from perturbative to nonperturbative behavior and to quantify higher-twist effects. And although dn is a higher-twist OPE object, the definition holds for all Q2.


## 2.1. Theg2 Structure Function

The nucleon’s second polarized structure functiong2(x,Q2) had never been measured till 1990, and there had been a few theoretical studies of it. If we define qf(x) and q-f(x) as the expectation value for the number of quarks and antiquarks of flavor f in the hadron whose momentum fraction lies in the interval [x;x+dx], then in the parton model it can be shown that
(1)F1(x)=12∑fef2[qf(x)+q-f(x)],g1(x)=12∑fef2[qf(x)-q-f(x)],
where the quark charge ef enters due to the fact that the cross section is proportional to the squared charge of the target. The Callan-Gross [8] relation shows that F2 can be defined entirely in terms of F1, but there is no such simple physical interpretation of g2. This spin-dependent structure function is determined by the x-dependence of the quarks transverse momenta and the off-shellness, both of which are unknown in the parton model [9]. Ignoring quark mass effect of order 𝒪(mq/∧QCD), g2 can be separated into leading twist and higher-twist components as follows:
(2)g2(x,Q2)=g2WW(x,Q2)+g-2(x,Q2),
where
(3)g2(x,Q2)=-∫x1∂∂y[mqMhT(y,Q2)+ζ(y,Q2)]dyy.To twist-3, there are three contributions tog2 [10]:g2ww: the leading twist-2 term, which depends only on g1,hT: arises from the quark transverse polarization distribution. Also twist-2, this term, is suppressed by the smallness of the quark mass,ζ: the twist-3 part which arises from quark-gluon interactions,Using the fact thatg1 and g2 contain the same twist-2 operator, Wandzura-Wilczek [11] derived the following expression for the asymptotically free contribution to g2, in terms of g1:
(4)g2ww(x,Q2)=-g1(x,Q2)+∫x1g1(x,Q2)xdx.Also this relation can be extracted directly from twist-2 operators of OPE.


## 2.2. Operator Product Expansion

An appropriate formalism to studyg2 beyond the simple QPM is the operator product expansion (OPE) [13, 14] which is the model-independent approach based directly on QCD. The OPE allows us to write the hadronic matrix element in deep inelastic scattering (DIS) in terms of a series of normalized operators of increasing twist. The leading contribution is twist-2, with higher-twist terms suppressed by powers of 1/Q2. Keeping only terms up to twist-3, the moments of g1 and g2 at fixed Q2 can be related to the twist-2 and twist-3 reduced matrix elements, an and dn [14],
(5)∫01xng1(x,Q2)dx=an2,n=0,2,4,…,∫01xng2(x,Q2)dx=12nn+1(dn-an),n=2,4….In the expression above,dn directly appears in the equation for g2allowing us to study the higher-twist structure of the nucleon at leading order. One obtains
(6)∫01xn[g1(x,Q2)+n+1ng2(x,Q2)]dx=12dn(Q2),n=2,4,6….Hence, forn=2,
(7)d2=2∫01x2[g1(x,Q2)+32g2(x,Q2)]dx.At largeQ2, the dn matrix element is related to the color polarization, which describes how the color electric and magnetic fields respond to the nucleon spin. At lower momentum transfer, dn(Q2) provides a mean to study the transition from perturbative to nonperturbative behavior and to quantify higher-twist effects. And although dn is a higher-twist OPE object, the definition holds for all Q2.


## 3. Formalism

We summarize the key features of our QCD analysis here. Knowing the fact that world data forg1 cover a broad range in x and Q2 with relatively high precision motivates us to extract polarized parton distributions from NLO fits to the data. In our calculation, the polarized parton densities are parameterized at a starting scale Q02=4GeV2and are evolved to higher factorization scales using a numerical solution of the polarized NLO DGLAP evolutions [15–17]. We take the QCD coupling as a free parameter which is fitted to the data. The QCD DGLAP equations are solved in the Mellin space.The centerpiece of our approach is the Jacobi polynomial expansion; this method was developed and applied to a variety of different QCD analyses for unpolarized [18, 19] and polarized applications [4, 5]. In this approach, we expand the structure functions in terms of the Jacobi polynomials Θnα,β(8)xg1Nmax=xβ(1-x)α∑n=0NmaxΘnα,β(x)∑j=0ncj(n)(α,β)[xg1,j+2].This relates theg1(x,Q2) structure function to their moments [4].In this analysis we useNmax=9, α=3.0, and β=0.5. The world data are at relatively large Q2, where higher-twist effects should be negligible; therefore the evolved parton distributions allow one to calculate the twist-2 spin-dependent structure functions g1 and g2WW, in most kinematic regions accessible today. Due to the fact that our calculations are performed in DIS region where transverse spin is a suppressed leading twist, we can neglect the twist-3 part of g2, ζ(y,Q2). Also, ignoring the masses of three light quarks is another fact that allows us to have an estimate ofg2, by these approximations, g-2≈0 then g2WW (see (2), (3)). By the last assumption, we can see that the contribution of quark transverse distribution, hT, is omitted too. Precise measurements of g2 at specific values of x and Q2 can be compared to g2WW, providing a unique opportunity to cleanly isolate higher-twist contributions.


## 4. Results

We have performed a QCD analysis of the inclusive polarized DIS data at NLO and extracted the spin structure functiong1(x,Q2). We have used an expansion in the Jacobi polynomials to facilitate the analysis. The Jacobi polynomials help us to produce g1(x,Q2) in an analytical (not numerical) form.For the measured region0.03<x<0.8, proton and deuteron spin structure functions g2WW(x,Q2) are calculated using our parametrization on g1(x,Q2) and the possible contributions of higher-twist effects to g2p,d are quantified using twist-3 reduced matrix element d2 at fixed Q2. By using (7) and substitution g2WW for g2, as discussed, we neglect the contribution from the region 0≤x<0.03 because of the x2 suppression factor. For 0.8<x≤1, we assume that both g1 and g2 behave as (1-x)3. We obtained the value of dp(2)=1.3×10-2 and dd(2)=-8.8×10-3. The computational processes were performed at Q2=5GeV2. The consistency of these values with zero is another reasonable proof for our assumptions in the most common kinematic region. Comparisons of our results for the third moments of g2 with recent data and other model predictions are listed in Table 1.Table 1
Comparison of model results with proton and deuteron data.
Paper
d
p
(
2
)
d
d
(
2
)
This paper
1.3
×
10
-
2
-
8.8
×
10
-
3
c.m. bag model [21]
1.74
×
10
-
2
6.79
×
10
-
3
MIT bag model [22, 23]
1.0
×
10
-
2
5.0
×
10
-
3
QCD sum rule [24]
-
(
0.6
±
0.3
)
×
10
-
2
-
(
17
±
5
)
×
10
-
3
QCD sum rule [25]
-
(
0.3
±
0.3
)
×
10
-
2
-
(
13
±
5
)
×
10
-
3
Data [26]
(
0.32
±
0.17
)
×
10
-
2
(
7.9
±
4.8
)
×
10
-
3
Lattice QCD [27]
-
(
4.8
±
0.5
)
×
10
-
2
-
(
22
±
6
)
×
10
-
3Figure3 shows xg2 for the proton and deuteron as a function of Q2 for selected value of x. Data are for E155 experiment. The error bars shown are the statistical and systematic ones added in quadrature. The curves show xg2WW based on our model (solid) and the BB model (dash-dot), both using NLO g1 fit, which are in good agreement with the data. xg2 from E143 (square) [20] and E155 (circle) [12] experiments together with our twist-2 g2WW at Q2=5GeV2 for xg2p and xg2d are presented in Figures 1 and 2.Figure 1
The polarized structure functionxg2pas a function of Q2 in intervals of x.Figure 2
The polarized structure functionxg2das a function of Q2 in intervals of x.Figure 3
x
g
2 for the proton and deuteron as a function of Q2 for selected value of x. Data are for E155 [12] experiment. The error bars shown are the statistical and systematic ones added in quadrature. The curves show xg2WW based on model (solid) and the BB model (dash-dot) [2].



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*Source: 519842-2013-06-27.xml*