The spin structure of the nucleon

This article reviews our present understanding of QCD spin physics: the proton spin puzzle and new developments aimed at understanding the transverse structure of the nucleon. Present experimental investigations of the nucleon’s internal spin structure, the theoretical interpretation of the different measurements, and the open questions and challenges for future investigation are discussed.

Figure 1

The COMPASS spectrometer; for a description, see the text.

Figure 2

Semi-inclusive DIS studied at COMPASS, HERMES, and JLab.

Figure 3

The PHENIX detector at RHIC as configured for data taking in 2012.

Figure 4

World data on $x{g}_1$ as a function of $x$ for the proton (top), the deuteron (middle), and the neutron (bottom) at the $Q^2$ of the measurement. Only data points for $Q^2>1{\mathrm{GeV}}^2$ and $W>2.5\mathrm{GeV}$ are shown. Error bars are statistical errors only.

Figure 5

World data for $g_1(x,Q^2)$ for the proton with $Q^2>1{\mathrm{GeV}}^2$ and $W>2.5\mathrm{GeV}$. For clarity a constant $c_i=0.28(11.6-i)$ has been added to the $g_1$ values within a particular $x$ bin starting with $i=0$ for $x=0.006$. Error bars are statistical errors only. [Also shown is the QCD fit of 275.]

Figure 6

Convergence of the first moment integral of $g_1$ as a function of the lower integration limit $x_{\min }$ for the Bjorken integral (isospin nonsinglet) and the Ellis-Jaffe integral (isosinglet) from the COMPASS proton and deuteron data at $Q^2=3{\mathrm{GeV}}^2$. The arrows indicate the theoretical expectations. Error bars are statistical errors only.

Figure 7

Semi-inclusive longitudinal double-spin asymmetries for identified pions and kaons from COMPASS (75, 79) and HERMES (51) for the proton (left) and the deuteron (right) as functions of $x$ at the $Q^2$ of the measurements. The error bars and bands indicate the statistical and systematic uncertainties, respectively. Adapted from 75 (left, proton target) and 79 (right, deuteron target).

Figure 8

COMPASS (79) and HERMES (58) results for the strangeness polarization $x\Delta s(x)$ as a function of $x$. The data are obtained in a leading-order analysis of SIDIS asymmetries (including those for charged kaons) and using the DSS fragmentation functions (206). The inner error bar represents the statistical uncertainty; the full bar represents the quadratic sum of statistical and systematic uncertainties.

Figure 9

Jet production from quark-gluon scattering in polarized proton-proton collisions.

Figure 10

Production of a $c\overline{c}$ pair in polarized photon-gluon fusion is being used to measure gluon polarization in the polarized proton.

Figure 11

Gluon polarization $\Delta g/g$ from leading-order analyses of hadron or hadron-pair production as a function of the probed gluon momentum fraction $x_g$. Also shown are NLO fits from 208 and 276. The inner error bar represents the statistical uncertainty; the full bar represents the quadratic sum of statistical and systematic uncertainties. The horizontal bar indicates the $x_g$ range of the measurement. Adapted from 39.

Figure 12

The longitudinal double-spin asymmetry in ${\pi }^0$ production measured by PHENIX (283) and in jet production measured by STAR (214), shown with calculations based on the DSSV polarized parton distributions that were updated to include these results. The relationship between the pion and jet $p_T$ scales is given by the mean $z$ value of $\sim 0.5$ (34). Error bars represent the statistical uncertainty. From 106.

Figure 13

Calculation of the NLO QCD evolution of $J_u$, $L_d$, $J_d$, and $L_u$ in the cloudy bag with model input scale $Q_0=0.4\mathrm{GeV}$. From 334.

Figure 14

The leading DVCS and Bethe-Heitler processes.

Figure 15

The $t$ dependence of the DVCS cross section for several values of $Q^2$ as measured by H1 and ZEUS. The curves are results of fits of the form $e^{-b|t|}$ with $b$ being related to the transverse extension of partons in the proton at a given $x$ and $Q^2$ (see the text). The inner error bar represents the statistical uncertainty; the full bar represents the quadratic sum of statistical and systematic uncertainties.

Figure 16

Overview of all DVCS azimuthal asymmetry amplitudes measured at HERMES with proton and deuterium targets, given at the average kinematics. The inner error bar represents the statistical uncertainty; the full bar represents the quadratic sum of statistical and systematic uncertainties.

Figure 17

The leading beam-spin asymmetry amplitude $a(t)=A_{LU}^{\sin \varphi }$ differential in $t$, $x$, and $Q^2$ as measured by CLAS, from 231. An earlier CLAS measurement (331) is indicated by the square. The open triangles represent the cross-section data from Hall A (180). Error bars are statistical errors only.

Figure 18

Sivers amplitudes for charged pions measured by HERMES with a proton target. The Sivers amplitudes for $K^{+}$ (not shown here) appear to be nearly twice as large as those for ${\pi }^{+}$. The inner error bar represents the statistical uncertainty; the full bar represents the quadratic sum of statistical and systematic uncertainties. From 60.

Figure 19

Sivers amplitudes for unidentified charged hadrons measured by COMPASS with a proton target (37). The hadron yield is dominated by pions. The bands indicate the systematic uncertainties.

Figure 20

Collins asymmetry for the double ratios of like-sign ($L$), unlike-sign ($U$), and any charged ($C$) pion pairs from Belle (322). $A^{UL}$ and $A^{UC}$ are sensitive to different combinations of the favored and unfavored Collins fragmentation functions. The bands indicate the systematic uncertainties.

Figure 21

Collins amplitudes for charged pions measured by HERMES with a proton target. The inner error bar represents the statistical uncertainty; the full bar represents the quadratic sum of statistical and systematic uncertainties. From 62.

Figure 22

Collins amplitudes for unidentified charged hadrons measured by COMPASS with a proton target (36). The hadron yield is dominated by pions. Note that a different definition of the Collins angle results in amplitudes with the opposite sign compared to other measurements. The bands indicate the systematic uncertainties.

Figure 23

The transverse single-spin asymmetry in forward ${\pi }^{\pm }$ production as measured in polarized proton-proton collisions across a range of center-of-mass energies. From left to right, the data are from 262, 17, 84, and 103. Error bars are statistical errors only.