SANE long paper v2

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 (citation not found: E80) (citation not found: E130) (citation not found: EMC) (citation not found: E142) (citation not found: E154) (citation not found: HERMESa) for parallel beam and target spins, or even for parallel and orthogonal configurations (citation not found: E143) (citation not found: SMC) (citation not found: E155D) (citation not found: E155) (citation not found: E155x), to semi-inclusive measurements with detection of a \(\pi\) or \(K\) meson in coincidence with the scattered electron (citation not found: hermessidis) (citation not found: smcsidis) and the investigation of the gluon polarization (citation not found: compassg) (citation not found: 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 (citation not found: hermessidis) (citation not found: clas) (citation not found: halla).

The modern description of nucleon structure is done in terms of transverse momentum dependent quark distributions functions (citation not found: Mulders:1995dh) defined in terms of quark-quark (\(qq\)) and quark-gluon-quark(\(qgq\)) correlations in the nucleon. Two of the leading twist distributions from \(qq\) correlations translate, after integration 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 \(Q^2=-q_\mu^2\). The quark helicity distribution \(\Delta q(x)\) (or \(g_{1L}\)) is related to the spin SF \(g_1(x,Q^2)\). 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, 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 \(g_{1T}(x,k^2_T)\), \(g_T(x)\) is given by (citation not found: Kotzinian:1995cz) (citation not found: Tangerman:1994bb) \[\begin{aligned} g_{T}(x) = \int d^2 k_T \frac{k_T^2}{2M^2}\frac{g_{1T}(x,k^2_T)}{x} + \frac{m}{M}\frac{h_1(x)}{x} + \tilde g_T(x),\nonumber\\\end{aligned}\] 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 In fact, carrying out the integration of \(g_{1T}\) expressed in terms of Lorentz invariant amplitudes (citation not found: Tangerman:1994bb) one can obtain \[\begin{aligned} \lefteqn{ g_{T}(x) = \int^1_x dy \frac{g_1(y)}{y}{}}\nonumber\\ & {}\!\!\! \displaystyle{+ \frac{m}{M}\Bigl[\frac{h_1(x)}{x} - \int^1_x dy \frac{h_1(y)}{y}\Bigr] + \tilde g_T(x) -\int^1_x dy \frac{\tilde g_T(y)}{y}} &. \label{eq:gtb}\end{aligned}\] 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 \[\begin{aligned} \lefteqn{ \Delta\sigma = {{4 \alpha^2 E'^2} \over {M E(E-E')Q^2}} {}} \nonumber \\ & {}\!\!\! \displaystyle{\sin\theta \cos\phi \Bigl(g_1(x ,Q^2) + \frac{2E}{E-E'}~ g_2(x,Q^2)\Bigr)}& \] This unique feature of transverse polarized scattering allows direct access to sub-leading, twist-3 processes in a direct measurement (citation not found: Jaffe:1989xx). In terms of the \(g_1\) and \(g_2\) SF’s, \(g_T\) can be written simply as \[\begin{aligned} g_T(x,Q^2) = g_1(x,Q^2) + g_2(x,Q^2)\end{aligned}\] The result for the twist-2 part of \(g_2\) found by Wandzura and Wilczeck (citation not found: wand) \[\begin{aligned} \nonumber g_2^{WW}(x,Q^2)= & -\gql + \displaystyle{\int_x^1 g_1(y,Q^2) {dy\over y}}\\ \] 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 virtual Compton scattering spin asymmetry (SA) \((x,Q^2) = \sigma_{LT}/\sigma_T\), 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 , \(g_T\) can be expressed as \[g_T(x,Q^2) = \frac{E-E'}{\sqrt{ Q^2}} F_1(x,Q^2) A_2(x,Q^2)\] \(g_T\) can then be understood as being a measure of the polarization of quarks with spins perpendicular to the virtual photon helicity.

\(g_T\) 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 \(\sigma_L\).

With the suppression of \(h_1\) by the ratio \(m/M\)(citation not found: artru) (citation not found: ralston) (citation not found: 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 \(d_2\) \[\begin{aligned} \overline{g}_2 (x) = \tilde g_T(x) - \int^1_x dy \frac{\tilde g_T(y)}{y}\nonumber \\ \int_0^1 x^2 \overline{g}_2(x,Q^2)dx =\frac{1}{3} d_2(Q^2), \label{eq:OPE}\end{aligned}\] which can be calculated in lattice QCD (citation not found: goeckeler). However, it should be kept in mind that since \(h_1\) is a leading twist quantity (comparable in magnitude to ), even if the ratio \(m/M\) were of the order of \(\sim 1\%\), \(h_1\) could represent a significant contribution to \(\overline{g}_2\). Only a handful of measurements of \(d_2\) exist to date, from SLAC (citation not found: E143) (citation not found: E155) (citation not found: E155x), and from \(RSS\) (citation not found: 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 (citation not found: stein) (citation not found: balit) (citation not found: ehre), bag (citation not found: strat) and chiral quark models (citation not found: soliton) (citation not found: waka) can also be tested by comparing their predictions to the measured moments of . Moreover, gives access to the polarizabilities of the color fields (citation not found: 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 and in consequence, to the Bjorken sum rule (citation not found: phil) \[\int_0^1 g_1(x,Q^2)dx =\frac{1}{2} a_0 + \frac{M^2}{9Q^2}\bigl(a_2 + 4d_2 + 4f_2\bigr) + O \Bigl(\frac{M^4}{Q^4}\Bigr). \label{eq:twist}\] 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 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 directly. Therefore, it is easier to measure the parallel and perpendicular asymmetries which are related to the spin asymmetries and by \[\begin{aligned} \nonumber A_1 =& \displaystyle{\frac{1}{(E+E^\prime)D^\prime}\Bigl((E-E^\prime\cos\theta) A_\parallel - \frac{E^\prime \sin\theta} {\cos\phi}A_\perp\Bigr)}\\ A_2 =& \displaystyle{\frac{\sqrt{Q^2}}{2ED^\prime}\Bigl(A_\parallel + \frac{E-E^\prime\cos\theta} {E^\prime\sin\theta\cos\phi}A_\perp\Bigr)} \label{eq:ntup} \end{aligned}\] 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.

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