Wandzura-Wilczek Relation Research Articles

Twist-three cross-sections in deeply virtual Compton scattering

We study the deeply virtual Compton scattering process with both twist-two and twist-three Compton form factors and present our cross-sections formulas with all polarization configurations. While the twist-three contributions are generally assumed to be negligible in the literature due to the kinematical suppression, we compare them with the twist-two ones at typical JLab 6 GeV and 12 GeV kinematics as well as EIC kinematics and show their kinematical suppression explicitly, justifying the leading-twist approximation made in the literature. In addition, we also estimate the twist-three Compton form factors using Wandzura-Wilczek relations and inputs of twist-two generalized parton distributions based on a reggeized spectator model. With those estimated Compton form factors, we analyze the kinematical behavior of twist-two and twist-three cross-sections in a wide range of kinematics, and discuss the optimal regions for separating the leading-twist effects from the higher-twist ones.

Twist-2 relation and sum rule for tensor-polarized parton distribution functions of spin-1 hadrons

Sum rules for structure functions and their twist-2 relations have important roles in constraining their magnitudes and x dependencies and in studying higher-twist effects. The Wandzura-Wilczek (WW) relation and the Burkhardt-Cottingham (BC) sum rule are such examples for the polarized structure functions g1 and g2. Recently, new twist-3 and twist-4 parton distribution functions were proposed for spin-1 hadrons, so that it became possible to investigate spin-1 structure functions including higher-twist ones. We show in this work that an analogous twist-2 relation and a sum rule exist for the tensor-polarized parton distribution functions f1LL and fLT, where f1LL is a twist-2 function and fLT is a twist-3 one. Namely, the twist-2 part of fLT is expressed by an integral of f1LL (or b1) and the integral of the function f2LT = (2/3)fLT− f1LL over x vanishes. If the parton-model sum rule for f1LL (b1) is applied by assuming vanishing tensor-polarized antiquark distributions, another sum rule also exists for fLT itself. These relations should be valuable for studying tensor-polarized distribution functions of spin-1 hadrons and for separating twist-2 components from higher-twist terms, as the WW relation and BC sum rule have been used for investigating x dependence and higher-twist effects in g2. In deriving these relations, we indicate that four twist-3 multiparton distribution functions FLT, GLT, {H}_{LL}^{perp } , and HTT exist for tensor-polarized spin-1 hadrons. These multiparton distribution functions are also interesting to probe multiparton correlations in spin-1 hadrons. In the near future, we expect that physics of spin-1 hadrons will become a popular topic, since there are experimental projects to investigate spin structure of the spin-1 deuteron at the Jefferson Laboratory, the Fermilab, the nuclotron-based ion collider facility, the electron-ion colliders in US and China in 2020’s and 2030’s.

The polarized two-loop massive pure singlet Wilson coefficient for deep-inelastic scattering

We calculate the polarized massive two–loop pure singlet Wilson coefficient contributing to the structure functions g1(x,Q2) analytically in the whole kinematic region. The Wilson coefficient contains Kummer–elliptic integrals. We derive the representation in the asymptotic region Q2≫m2, retaining power corrections, and in the threshold region. The massless Wilson coefficient is recalculated. The corresponding twist–2 corrections to the structure function g2(x,Q2) are obtained by the Wandzura–Wilczek relation. Numerical results are presented.

Moments of the neutrong2structure function at intermediateQ2

We present new experimental results of the $^3$He spin structure function $g_2$ in the resonance region at $Q^2$ values between 1.2 and 3.0 (GeV/c)$^2$. Spin dependent moments of the neutron were then extracted. Our main result, the resonance contribution to the neutron $d_2$ matrix element, was found to be small at $<Q^2>$=2.4 (GeV/c)$^2$ and in agreement with the Lattice QCD calculation. The Burkhardt-Cottingham sum rule for $^3$He and the neutron was tested with the measured data and using the Wandzura-Wilczek relation for the low $x$ unmeasured region. A small deviation was observed at $Q^2$ values between 0.5 and 1.2 (GeV/c)$^2$ for the neutron.

Transverse Spin Structure Function

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

Truncated Mellin moments: Useful relations and implications for the spin structure function $g_2$

We review our previous studies of truncated Mellin moments of parton distributions. We show in detail the derivation of the evolution equation for double truncated moments. The obtained splitting function has the same rescaled form as in a case of the single truncated moments. We apply the truncated moments formalism to QCD analyses of the spin structure functions of the nucleon, $g_1$ and $g_2$. We generalize the Wandzura-Wilczek relation in terms of the truncated moments and find new sum rules. We derive the DGLAP-like evolution equation for the twist-2 part of $g_2$ and solve it numerically. We find also useful relations between the truncated and untruncated moments.

Nucleon isovector structure functions in (2+1)-flavor QCD with domain wall fermions

We report on numerical lattice QCD calculations of some of the low moments of the nucleon structure functions. The calculations are carried out with gauge configurations generated by the RBC and UKQCD collaborations with (2+1)-flavors of dynamical domain wall fermions and the Iwasaki gauge action ($\beta = 2.13$). The inverse lattice spacing is $a^{-1} = 1.73$ GeV, and two spatial volumes of ((2.7{\rm fm})^3) and ((1.8 {\rm fm})^3) are used. The up and down quark masses are varied so the pion mass lies between 0.33 and 0.67 GeV while the strange mass is about 12 % heavier than the physical one. The structure function moments we present include fully non-perturbatively renormalized iso-vector quark momentum fraction, (< x >_{u-d}), helicity fraction, (< x >_{\Delta u - \Delta d}), and transversity, (< 1 >_{\delta u - \delta d}), as well as an unrenormalized twist-3 coefficient, (d_1). The ratio of the momentum to helicity fractions, (< x >_{u-d}/< x >_{\Delta u - \Delta d}), does not show dependence on the light quark mass and agrees well with the value obtained from experiment. Their respective absolute values, fully renormalized, show interesting trends toward their respective experimental values at the lightest quark mass. A prediction for the transversity, (0.7 < < 1 >_{\delta u -\delta d} < 1.1), in the (\bar{\rm MS}) scheme at 2 GeV is obtained. The twist-3 coefficient, (d_1), though yet to be renormalized, supports the perturbative Wandzura-Wilczek relation.

Importance of Lorentz structure in the parton model: Target mass corrections, transverse momentum dependence, positivity bounds

We show that respecting the underlying Lorentz structure in the parton model has very strong consequences. Failure to insist on the correct Lorentz covariance is responsible for the existence of contradictory results in the literature for the polarized structure function ${g}_{2}(x)$, whereas with the correct imposition we are able to derive the Wandzura-Wilczek relation for ${g}_{2}(x)$ and the target-mass corrections for polarized deep inelastic scattering without recourse to the operator product expansion. We comment briefly on the problem of threshold behavior in the presence of target-mass corrections. Careful attention to the Lorentz structure has also profound implications for the structure of the transverse momentum dependent parton densities often used in parton model treatments of hadron production, allowing the ${\mathbit{k}}_{T}$ dependence to be derived explicitly. It also leads to stronger positivity and Soffer-type bounds than usually utilized for the collinear densities.

What can break the Wandzura-Wilczek relation?

We analyze the breaking of the Wandzura-Wilczek relation for the g_2 structure function, emphasizing its connection with transverse momentum dependent parton distribution functions. We find that the relation is broken by two distinct twist-3 terms, and clarify how these can be separated in measurements of double-spin asymmetries in semi-inclusive deep inelastic scattering. Through a quantitative analysis of available g_2 data we also show that the breaking of the Wandzura-Wilczek relation can be as large as 15-30% of the size of g_2.

Nucleon structure with two flavors of dynamical domain-wall fermions

We present a numerical lattice quantum chromodynamics calculation of isovector form factors and the first few moments of the isovector structure functions of the nucleon. The calculation employs two degenerate dynamical flavors of domain-wall fermions, resulting in good control of chiral symmetry breaking. Nonperturbative renormalization of the relevant quark currents is performed where necessary. The DBW2 gauge action is used to further improve the chiral behavior while maintaining a reasonable physical lattice volume. The inverse lattice spacing, a{sup -1}, is approximately 1.7 GeV. Degenerate up and down dynamical quark masses of approximately 1, 3/4 and 1/2 times the strange quark mass are used. The physical volume of the lattice is about (1.9 fm){sup 3}. The ratio of the isovector vector to axial charges, g{sub A}/g{sub V}, tends to a lower value than the experimental value as the quark mass is reduced toward the physical point. Momentum-transfer dependences of the isovector vector, axial, induced-tensor and induced-pseudoscalar form factors are calculated. The Goldberger-Treiman relation holds at low momentum transfer and yields an estimation of the pion-nucleon coupling, g{sub {pi}}{sub NN}=15.5(1.4), where the quoted error is only statistical. We find that the flavor nonsinglet quark momentum fraction {sub u-d} and quark helicity fractionmore » {sub {delta}}{sub u-{delta}}{sub d} overshoot their experimental values after linear chiral extrapolation. We discuss possible systematic errors for this discrepancy. An estimate for transversity, {sub {delta}}{sub u-{delta}}{sub d}=0.93(6) in MS at 2 GeV, is obtained and a twist-3 polarized moment, d{sub 1}, appears small, suggesting that the Wandzura-Wilczek relation holds approximately. We discuss in detail the systematic errors in the calculation, with particular attention paid to finite volume, excited-state contamination, and chiral extrapolations.« less

Target mass effects in deep-inelastic scattering on the deuteron

We calculate the effects of the target mass on deep-inelastic electron scattering on targets of spin one such as the deuteron, focusing on the novel structure functions that enter. It is important to understand these effects given the recent Hermes measurements of the b1(x,Q2) structure function of the deuteron. We also derive the spin-one analogues of the Wandzura–Wilczek relations and discuss possible calculations in lattice QCD.

Target mass corrections for virtual Compton scattering at twist-2 and generalized, nonforward Wandzura–Wilczek and Callan–Gross relations

The off-cone Compton operator of twist-2 is Fourier transformed using a general procedure which is applicable, in principle, to any QCD tensor operator of definite (geometric) twist. That method allows, after taking the nonforward matrix elements, to separate quite effectively their imaginary part and to reveal some hidden structure in terms of appropriately defined variables, including generalized Nachtmann variables. In this way, without using the equations of motion, generalizations of the Wandzura–Wilczek relation and of the mass-corrected Callan–Gross relation to the nonforward scattering, having the same shape as in the forward case, are obtained. In addition, new relations for those structure functions which vanish in the forward case are derived. All the structure functions are expressed in terms of iterated generalized parton distributions of nth order. In addition, we showed that the absorptive part of twist-2 virtual Compton amplitude is determined by the nonforward extensions of g 1 , W 1 and W 2 only.

Helicity-dependent twist-two and twist-three generalized parton distributions in light-front QCD

We investigate the helicity dependent twist-two and twist-three generalized parton distributions in light-front Hamiltonian QCD for a massive dressed quark target. Working in the kinematical region $\xi< x <1$, we obtain the splitting functions for the evolutions of twist-two quark and gluon distributions in a straightforward way. For the twist-three distribution, we find that all contributions are proportional to the quark mass and thus the twist-three distribution is directly related to the chiral symmetry breaking dynamics in light-front QCD. We also show that the off-forward Wandzura-Wilczek relation is violated in perturbative QCD for a massive dressed quark.We calculate the quark mass correction to the WW relation in the off-forward case and show that it is related to $h_1(x)$ in the forward limit. We extract the 'genuine twist three' part of the matrix element in the forward case and verify the Burkhardt Cottingham and Efremov-Leader-Teryaev sum rules.

Polarized deep inelastic diffractiveepscattering: Operator approach

Polarized inclusive deep-inelastic diffractive scattering is dealt with in a quantum field theoretic approach. The process can be described in the general framework of non-forward scattering processes using the light-cone expansion in the generalized Bjorken region applying the generalized optical theorem. The diffractive structure functions $g_1^{D(3)}$ and $g_2^{D(3)}$ are calculated in the twist-2 approximation and are expressed by diffractive parton distributions, which are derived from pseudo-scalar two-variable operator expectation values. In this approximation the structure functions $g_2^{D(3)}$ is obtained from $g_1^{D(3)}$ by a Wandzura-Wilczek relation similar as for deep inelastic scattering. The evolution equations are given. We also comment on the higher twist contributions in the light-cone expansion.

A note on Wandzura–Wilczek relations

Wandzura-Wilczek (WW) relations between matrix-elements of bilocal light-ray operators have recently regained interest in connection with off-forward scattering processes. Originally derived for matrix elements over leading-twist operators, their generalization to off-forward and exclusive processes gets complicated by the presence of higher-twist operators that are total derivatives of leading-twist ones and do not contribute to forward-scattering. We demonstrate that, for exclusive matrix-elements, the inclusion of these operators into WW-relations is essential for fulfilling constraints imposed by the conformal symmetry of massless QCD.

An analysis of the next-to-leading order corrections to the gT(=g1+g2) scaling function

Abstract We present a general method for obtaining the quantum chromodynamical radiative corrections to the higher-twist (power-suppressed) contributions to inclusive deep-inelastic scattering in terms of light-cone correlation functions of the fundamental fields of quantum chromodynamics. Using this procedure, we calculate the previously unknown O (α s ) corrections to the twist-three part of the spin scaling function gT(xB,Q2)(=g1(xB,Q2)+g2(xB,Q2)) and the corresponding forward Compton amplitude ST(ν,Q2). Expanding our result about the unphysical point xB=∞, we arrive at an operator product expansion of the nonlocal product of two electromagnetic current operators involving twist-two and -three operators valid to O (α s ) for forward matrix elements. We find that the Wandzura–Wilczek relation between g1(xB,Q2) and the twist-two part of gT(xB,Q2) is respected in both the singlet and non-singlet sectors at this order, and argue its validity to all orders. The large-Nc limit does not appreciably simplify the twist-three Wilson coefficients.

Wandzura–Wilczek approximation from generalized rotational invariance

Abstract We present the alternative way of derivation of the Wandzura–Wilczek relations between the kinematical twist-3 and twist-2 functions, parameterizing hadronic matrix element in two-photon processes γ ★ π → γπ and γ ★ γ → ππ . The new equations, providing the independence of the physical cross-sections on the choice of the light-cone direction, are suggested and explored. The amplitude of γ ★ γ → ππ up to genuine twist-3 accuracy is found.

Modified Wandzura-Wilczek relation with the Nachtmann variable

If one retains ${M}^{2}{/Q}^{2}$ terms in the kinematics, the Nachtmann variable $\ensuremath{\xi}$ seems to be more appropriate to describe deep inelastic lepton-nucleon scattering. Up to the first power of ${M}^{2}{/Q}^{2},$ a modified Wandzura-Wilczek relation with respect to $\ensuremath{\xi}$ was derived. Kinematical correction factors are given as functions of $\ensuremath{\xi}$ and ${Q}^{2}.$ A comparison of the modified ${g}_{2}^{\mathrm{WW}}(\ensuremath{\xi})$ and the original ${g}_{2}^{\mathrm{WW}}(x)$ with the most recent ${g}_{2}$ data is shown.

Small-xbehavior of the nonsinglet polarized structure functiong2in the double logarithmic approximation

The nonsinglet spin dependent structure function $g_2(x,Q^2)$ is studied at small x within the double logarithmic approximation of perturbative QCD. Both positive and negative signature contributions are considered, and we find a power-like growth in $1/x$. We discuss how our result fits into the Wandzura-Wilczek relation.

Quantum field theoretic treatment of the non-forward compton amplitude in the generalized bjorken region

A quantum field theoretic treatment of the leading light-cone part of the virtual Compton amplitude is presented. The twist-decomposition of the operators is performed by a group-theoretic procedure respecting the Lorentz group O(3,1). The twist-2 contributions to the Compton amplitude are calculated and it is shown that the electromagnetic current is conserved for these terms. Relations between the amplitude functions associated to the symmetric and asymmetric part of the Compton amplitude are derived. These relations generalize the Calln-Gross and Wandzura-Wilczek relations of forward scattering for the non-forward Compton amplitude.