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%Format:latex  %\documentclass[12pt,epsf]{article}  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  % Glen's definitions  \newcommand{\versiondate}{\mbox{2006}}  \newcommand{\solidangle}{\mbox{194 msr}}  \newcommand{\caldrift}{\mbox{325 cm}}  \newcommand{\Apar}{\mbox{$A_{\parallel}$}}  \newcommand{\Aperp}{\mbox{$A_{\perp}$}}  \newcommand{\Aon}{\mbox{$A_1^n$}}  \newcommand{\Aop}{\mbox{$A_1^p$}}  \newcommand{\Atn}{\mbox{$A_2^n$}}  \newcommand{\Atp}{\mbox{$A_2^p$}}  \newcommand{\Hethree}{\mbox{$^3$He}}  \newcommand{\NHthree}{\mbox{NH$_3$}}  \newcommand{\x}{\mbox{$x$}}  \newcommand{\Qsqr}{\mbox{$Q^2$}}  \newcommand{\Gep}{\mbox{$G_{Ep}$}}  \newcommand{\Gen}{\mbox{$G_{En}$}}  \newcommand{\degrees}{\mbox{$^\circ$}}  \newcommand{\GeVc}{\mbox{GeV\hspace{-0.08cm}/c}}  \newcommand{\GeVcsqr}{\mbox{(GeV\hspace{-0.08cm}/c)$^2$}}  \newcommand{\pionz}{\mbox{$\pi^0$}}  \newcommand{\pionp}{\mbox{$\pi^+$}}  \newcommand{\gtwo}{\mbox{$g_2$}}  \newcommand{\gone}{\mbox{$g_1$}}  \newcommand{\gon}{\mbox{$g_1^n$}}  \newcommand{\gop}{\mbox{$g_1^p$}}  \newcommand{\gtp}{\mbox{$g_2^p$}}  \newcommand{\thetascat}{\mbox{$\theta_{scat}$}}  \newcommand{\phiscat}{\mbox{$\phi_{scat}$}}  \newcommand{\etal}{\mbox{\it et al.}}  \newcommand{\Aone}{\mbox{$A_1$}}  \newcommand{\Atwo}{\mbox {$A_2$}}  %\newcommand{\Cerenkov}{\mbox{\v{C}erenkov}}  \newcommand{\JLab}{\mbox{JLab}}  \newcommand{\calwidth}{\mbox{120}}  \newcommand{\calheight}{\mbox{218}}  \newcommand{\xgoestoone}{\mbox{$\x \rightarrow 1$}}  \newcommand{\gtww}{\mbox{$g_2^{WW}$}}  \newcommand{\gtbar}{\mbox{$\bar{g}_2$}}  \newcommand{\dtwo}{\mbox{$d_2$}}  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  % Dave's Definitions  \newcommand{\SANEbeamrequesthours}{\mbox{654}}  \newcommand{\SANEbeamrequestdays}{\mbox{27}}  \newcommand{\SANEluminosity}{\mbox{$8.5\cdot10^{34}$}}  \newcommand{\xDISmax}{\mbox{0.63}}  \newcommand{\xRESONANCEmax}{\mbox{0.80}}  %\newcommand{\A1n}{\mbox{A_1^n}}  %\newcommand{\A1p}{\mbox{A_1^p}}  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  % Oscar's Definitions  \def\al{$A_1$}  \def\at{$A_2$}  %\def\qsq{$Q^2$}  %\def\deg{^\circ}  %\def\bul{$\bullet$}  %\def\apt{$A^p_2$}  \def\gt{$g_2$}  %\def\gtp{$g^p_2\ $}  \def\et{{\it et al.}}  \def\gl{$g_1$}  \def\gql{g_1(x,Q^2)}  \def\gqt{g_2(x,Q^2)}  \def\gww{$g_2^{WW}$}  %\def\glp{$g^p_1\ $}  %\def\gln{$g^n_1\ $}  %\def\gld{$g^d_1\ $}  \def\apar{$A_{\parallel}$}  \def\aper{$A_\perp$}  %\def\alp{$A_1^p$}  %\def\atp{$A_2^p$}  \hyphenation{author another created financial paper re-commend-ed symbol}  %\markright{SANE - DRAFT - \versiondate}  %\markright{SANE}  %\pagestyle{myheadings}  % moved title, authors, abstract to end  %\section{\bf Brief Review of the Status of the Nucleon Spin Structure}  After more than 30 years of experimental and theoretical work, the study of the   nucleon spin structure has entered a mature stage, extending beyond the  exploration of the properties of the polarized structure functions in the scaling  regime into the region of the Bjorken scaling variable $x$  near its unity upper limit. Moreover, the experimental techniques have expanded   from the original simple approach of measuring   double spin asymmetries in inclusive deep   inelastic scattering - DIS~\cite{E80,E130,EMC,E142,E154,HERMESa}   % {\tt(note new HERMES longpaper)}   for parallel beam and target spins, or even for parallel and orthogonal  configurations~\cite{E143,SMC,E155D,E155,E155x}, to semi-inclusive measurements with   detection of a $\pi$ or $K$ meson in coincidence with the scattered   electron~\cite{hermessidis,smcsidis} and the investigation of the gluon   polarization~\cite{compassg,hermesg}. From the inclusive measurements in DIS it   has been established that the quarks carry only about 25\% of the nucleon spin,  and   from the inclusive and semi-inclusive measurements, the quark polarization by   flavor has been determined~\cite{hermessidis,clas,halla}.  The modern description of nucleon structure is done in terms of transverse   momentum dependent quark distributions functions~\cite{Mulders:1995dh} defined in   terms of quark-quark ($qq$) and quark-gluon-quark($qgq$) correlations in the   nucleon.  % of the Bjorkenscaling variable $x$ and   %the four-momentum transfer squared  %$Q^2=-q_\mu^2$.   %three of which are leading twist.   Two of the leading twist   distributions from $qq$ correlations translate,  after integration   %of the $q(x,k^2_T)$ distribution (also known as $f_1$)   over the transverse momentum $\vec k_\perp$,   into the more familiar  structure functions (SF) measured in DIS. The longitudinal momentum  distribution $q(x,k^2_T)$ (also  known as $f_1$) leads to the unpolarized SF $F_1(x,Q^2)$, which is a function of  the Bjorken scaling variable $x$ and the four-momentum transfer squared  $Q^2=-q_\mu^2$.  % after integration over $\vec k_\perp$ and t  The quark helicity distribution $\Delta q(x)$ (or $g_{1L}$) is related to the spin   SF $g_1(x,Q^2)$.  %; and transversity $\delta(x)$ (or $h_T$).   %The others involve both longitudinal and transverse components: $g_{1T}$;  %Collins $h_{1T}^\perp$; Sivers  %$f_{1T}^\perp$; $h_{1L}^\perp$; and $h_1^\perp$.   These distributions have quark flavor indices associated with them and the  nucleon structure functions are linear combinations of all active flavors,  weighted by their charges squared.  At subleading twist-3, there are two $k_T$-integrated distributions related to   $qq$ correlations, %that can be measured in DIS,  namely $g_{T}(x)$ and $h_{L}(x)$. In   addition, at the same twist-3 ${\cal O} (1/Q)$, three-particle $qgq$ correlations   lead to the corresponding distributions $\tilde g_{T}(x)$ and $\tilde h_{L}(x)$.  The transverse distribution $g_{T}(x)$ is of particular interest, because it can be   measured in inclusive double polarized DIS with target polarization transverse   to the beam helicity. In terms of the $k_T$ dependent distribution   $g_{1T}(x,k^2_T)$, %The distribution is especially interesting because after   %integrated over $k_T$ with a $k_T^2/2M^2$ weight,   %it is directly connected to the $g_2^q(x)$ distributionfunction  $g_T(x)$ is given by~\cite{Kotzinian:1995cz,Tangerman:1994bb}  \begin{eqnarray}  %g_2^q(x) = \frac{d}{dx}g^{q}_{1T}(x)\\  %g_{1T}(x) = \frac{1}{2}\sum_q e^2_q g^q_{1T}(x)  g_{T}(x) = \int d^2 k_T \frac{k_T^2}{2M^2}\frac{g_{1T}(x,k^2_T)}{x}  + \frac{m}{M}\frac{h_1(x)}{x} + \tilde g_T(x),\nonumber\\  \end{eqnarray}  where the $h_1(x)$ term represents the contribution of the transversity   distribution (net transverse quark spin in a transversely polarized nucleon),   that is suppressed in DIS by the ratio of the quark to nucleon masses, $m/M$.   This expression highlights the importance of transverse quark momentum even in   inclusive measurements: $g_T$ would be negligibly different from the  $qgq$-correlations dependent $\tilde g_T$ without transverse momentum  %On the other hand, a non-zero value of $g_T$ must include   In fact, carrying out the integration of $g_{1T}$ expressed in terms of Lorentz   invariant amplitudes~\cite{Tangerman:1994bb} one can obtain  \begin{eqnarray}  \lefteqn{  g_{T}(x) = \int^1_x dy \frac{g_1(y)}{y}{}}\nonumber\\   & {}\!\!\! \displaystyle{+ \frac{m}{M}\Bigl[\frac{h_1(x)}{x} - \int^1_x dy \frac{h_1(y)}{y}\Bigr] + \tilde g_T(x) -\int^1_x dy \frac{\tilde g_T(y)}{y}} &.  \label{eq:gtb}  \end{eqnarray}  where the first term depends only on the twist-2 quark helicity distribution   $g_1$, which is definitely not zero.  The mixed twist (2 and 3) nature of $gT$ arises from the contribution of the   $\tilde g_T$ terms. As it would be expected, the same terms contribute to the   $g_2(x,Q^2)$ SF, which dominates the difference of cross sections in DIS  with polarized beams on a transversely polarized target  %For orthogonal spins %the difference is%  %$\cos\alpha = \sin\theta \cos\phi$, so both $G_1$ and $G_2$ contribute  \begin{eqnarray}  %\lefteqn{{{d^2\sigma^{\uparrow \rightarrow}} \over {d\Omega dE'}} -  %{{d^2\sigma^{\downarrow\rightarrow}}\over {d\Omega dE'}}   %\Delta\sigma = {{4 \alpha^2 E'} \over {Q^2 E}} % {}} \nonumber \\  %%& {}\!\!\! %~ {{4 \alpha^2 E'} \over {Q^2 E}}   % E' \sin\theta \cos\phi \bigl(M G_1(\nu ,Q^2) + 2 E G_2(\nu ,Q^2)\bigr)   %\nonumber %&  \lefteqn{  \Delta\sigma = {{4 \alpha^2 E'^2} \over {M E(E-E')Q^2}} {}} \nonumber \\  & {}\!\!\! %~  \displaystyle{\sin\theta \cos\phi \Bigl(g_1(x ,Q^2) + \frac{2E}{E-E'}~ g_2(x,Q^2)\Bigr)}&  %\nonumber %&   \end{eqnarray}  This unique feature of transverse polarized scattering allows direct access to sub-leading,  twist-3 processes in a direct measurement~\cite{Jaffe:1989xx}.   %,Jaffe:1996zw}.  %Measuring $g_2$ opening a window on the confinement of quarks and gluons inside  % nucleons.% and other hadrons.  %, which is related to increasing  %interactions as the separations between hadronic constituents increase at low energy.   In terms of the $g_1$ and $g_2$ SF's, $g_T$ can be written simply as   \begin{eqnarray}  g_T(x,Q^2) = g_1(x,Q^2) + g_2(x,Q^2)  \end{eqnarray}  The result for the twist-2 part of $g_2$ found by Wandzura and   Wilczeck~\cite{wand}  %and a mixed twist-2/twist-3 part $\overline{g}_2$~\cite{ralston,jaft}  \begin{eqnarray}  %\nonumber\gqt = & g^{WW}_2(x,Q^2) + \overline{g}_2(x,Q^2)\\  \nonumber g_2^{WW}(x,Q^2)= & -\gql + \displaystyle{\int_x^1 g_1(y,Q^2)  {dy\over y}}\\   %\overline{g}_2(x,Q^2) = & \displaystyle{-\int_x^1 {\partial\over\partial y   %}\Bigl({m\over M}{h_T(y,Q^2) y} + \xi(y,Q^2)\Bigr){dy\over y}}  %\label{eq:eqtot}  \end{eqnarray}   corresponds to the first term of $g_T$ in eq. (\ref{eq:gtb}).  The structure of the nucleon can also be described in  terms of forward virtual Compton scattering.  % The SF's $F_1$, $g_1$ and $g_T$ % resulting from integrated over $\vec  % k_\perp$ can be investigated with inclusive measurements, all other  %distributions require semi-inclusive experiments.   %The mixed twist   %$g_T$ measures the polarization of quarks with spins   %perpendicular to the virtual photon spin. It is related to   The virtual Compton scattering spin asymmetry (SA) \at$(x,Q^2) =   \sigma_{LT}/\sigma_T$,  is formed from the longitudinal-transverse interference cross section  $\sigma_{LT}$ and the transverse cross section $\sigma_T$ for the scattering   of polarized electrons on polarized nucleons. In terms of \at, $g_T$ can be  expressed  as   \begin{equation}  g_T(x,Q^2) = \frac{E-E'}{\sqrt{ Q^2}} F_1(x,Q^2) A_2(x,Q^2)  \end{equation}  %, where $\nu = E - E^\prime$ represents the energy loss of a lepton with  %initial energy $E$.   $g_T$ can then be understood as being a measure of the polarization of quarks   with spins perpendicular to the virtual photon helicity.   %and $Q^2=-q_\mu^2$ is the four-momentum transfer squared.  $g_T$ can also be identified as the polarized partner of the unpolarized   longitudinal $F_L(x,Q^2) = 2 x F_1 R$, which has a similar form in terms of   $F_1$ and the ratio of the longitudinal to transverse virtual photon cross  sections $R = \sigma_L/\sigma_T$. $F_L$ is zero at leading twist but becomes  non-zero through higher twist effects resulting from non-zero parton transverse  momentum, which give rise to finite values of $\sigma_L$.   %although in DIS this contribution is suppressed by the ratio  %$m/M$\cite{artru,ralston,jaft}.   With the suppression of $h_1$ by the ratio $m/M$\cite{artru,ralston,jaft}.   the third moment of the interaction dependent part $\tilde gT$  can be related by the operator product expansion  (OPE) to the reduced twist-3 quark matrix element $d_2$  %, if the small twist-2 quark mass dependent term is neglected,  \begin{eqnarray}  \overline{g}_2 (x) = \tilde g_T(x) - \int^1_x dy \frac{\tilde g_T(y)}{y}\nonumber \\  \int_0^1 x^2 \overline{g}_2(x,Q^2)dx =\frac{1}{3} d_2(Q^2),  \label{eq:OPE}  \end{eqnarray}  which can be calculated in lattice QCD~\cite{goeckeler}.   However, it should be kept in mind that since $h_1$ is a leading twist   quantity (comparable in magnitude to \gl), even if the ratio   $m/M$  were of the order of $\sim 1\%$, $h_1$ could represent a significant contribution   to $\overline{g}_2$.  %, since the pure twist-3 piece $\xi$ might be considerably smaller than \gl.   Only a handful of measurements of $d_2$ exist to date, from  SLAC~\cite{E143,E155,E155x}, and from $RSS$~\cite{Slifer:2008xu} at Jefferson Lab. The SLAC  measurements have been combined into   a single number for the proton $d_2(Q^2 = 5~ {\rm GeV}^2) = 0.0032 \pm 0.0017$. The  lattice QCD result at the same $Q^2$ is $d_2 = 0.004\pm0.005$. The  $RSS$ proton result   covers a wide range of $x$ from 0.29 to 0.84, corresponding to the region of  the resonances from $W$ = 1.91 GeV to the pion production threshold. Using Nachtmann moments, which are required to correct for the target recoil at low $Q^2$, the $RSS$ result including the elastic contribution is   $d_2(Q^2 = 1.3~ {\rm GeV}^2) = 0.0104\pm 0.0014$ (total error).  In addition to lattice QCD, QCD sum   rules~\cite{stein,balit,ehre}, bag~\cite{strat} and chiral quark   models~\cite{soliton,waka} can  also be tested by comparing their predictions to the measured moments of \gt.  Moreover, \gt\ gives access to the polarizabilities of the color   fields~\cite{Ji:1995qe} (with  additional knowledge of the twist-4 matrix element $f_2$). The   magnetic and electric polarizabilities are $\chi_B = (4d_2 + f_2)/3$ and   $\chi_E = (2d_2 -f_2)/2$, respectively. Knowledge of these properties  of the color fields is an important step in understanding QCD.  The twist-4 $f_2$ matrix element represents quark-quark  interactions, and reflects the higher twist corrections to the  individual proton and neutron moments of \gl \ and in consequence, to the  Bjorken sum rule~\cite{phil}  \begin{equation}  \int_0^1 g_1(x,Q^2)dx =\frac{1}{2} a_0 + \frac{M^2}{9Q^2}\bigl(a_2 + 4d_2 +   4f_2\bigr) + O \Bigl(\frac{M^4}{Q^4}\Bigr).  \label{eq:twist}  \end{equation}  These matrix elements are related to the higher moments of the SSF's, which  have a strong dependence on the high $x$ contributions.  From an experimental point of view, the measurement of \at \ is simpler than  that of the absolute cross section difference for scattering of longitudinally   polarized electrons on transversely polarized nucleons, which is required to   access \gt \ directly. Therefore, it is easier  to measure the parallel \apar \ and perpendicular \aper \ asymmetries which  are related to the spin asymmetries \al \ and \at \   by \begin{eqnarray}  \nonumber A_1 =&   \displaystyle{\frac{1}{(E+E^\prime)D^\prime}\Bigl((E-E^\prime\cos\theta)  A_\parallel - \frac{E^\prime \sin\theta} {\cos\phi}A_\perp\Bigr)}\\  A_2 =& \displaystyle{\frac{\sqrt{Q^2}}{2ED^\prime}\Bigl(A_\parallel +   \frac{E-E^\prime\cos\theta} {E^\prime\sin\theta\cos\phi}A_\perp\Bigr)}  \label{eq:ntup}  \end{eqnarray}  where all quantities ( $\theta$ and $\phi$ are the scattered lepton's polar  and azimuthal angles, respectively) are measured in the same experiment,   with the exception  of the small contribution from the unpolarized structure function $R(Q^2,W) =   \sigma_L/\sigma_T$ to the virtual photon depolarization  $\displaystyle{D^\prime= (1-\varepsilon)/(1+\varepsilon R)}$. Here  $\varepsilon = 1/(1+2(1+\nu^2/Q^2)\tan^2(\theta/2))$ is the well   known longitudinal polarization of the virtual photon. These expressions are  suitably modified for the case when the beam and target spins aren't exactly  perpendicular.  %\input{intro-bib.tex}