Burkhardt-Cottingham Sum Rule Research Articles

Transverse-momentum-dependent parton distribution functions for spin-1 hadrons

We show transverse-momentum-dependent parton distribution functions (TMDs) for spin-1 hadrons including twist-3 and 4 functions by taking the decomposition of a quark correlation function in the Lorentz-invariant way with the conditions of Hermiticity and parity invariance. We found 30 new TMDs in the tensor-polarized spin-1 hadron at twists 3 and 4 in addition to 10 TMDs at twist 2. Since time-reversal-odd terms of the collinear correlation function should vanish after integrals over the partonic transverse momentum, we obtained new sum rules for the time-reversal-odd structure functions, {\int d^2 k_T g_{LT} = \int d^2 k_T h_{LL} = \int d^2 k_T h_{3LL} =0}∫d2kTgLT=∫d2kThLL=∫d2kTh3LL=0, at twists 3 and 4. We also indicated that transverse-momentum-dependent fragmentation functions exist in tensor-polarized spin-1 hadrons. The TMDs can probe color degrees of freedom, so that they are valuable in providing unique opportunities for creating interdisciplinary physics fields such as gluon condensate, color Aharonov-Bohm effect, and color entanglement. We also found three new collinear PDFs at twists 3 and 4, and a twist-2 relation and a sum rule were derived in analogy to the Wandzura-Wilczek relation and the Burkhardt-Cottingham sum rule on the structure function {g_2}g2.

Burkhardt-Cottingham-type sum rules for light-cone and quasi-PDFs

The Burkhardt-Cottingham (BC) sum rule connects the twist-3 light-cone parton distribution function (PDF) $g_{T}(x)$ to the twist-2 helicity PDF $g_{1}(x)$. The chiral-odd counterpart of the BC sum rule relates the twist-3 light-cone PDF $h_{L}(x)$ to the twist-2 transversity PDF $h_{1}(x)$. These BC-type sum rules can also be derived for the corresponding quasi-PDFs. We perform a perturbative check of the BC-type sum rules in the quark target model and the Yukawa model, by going beyond the ultra-violet (UV) divergent terms. We employ dimensional regularization (DR) and cut-off schemes to regulate UV divergences, and show that the BC-type sum rules hold for DR, while they are generally violated when using a cut-off. This violation can be traced back to the breaking of rotational invariance. We find corresponding results for the sum rule relating the mass of the target to the twist-3 PDF $e(x)$. Moreover, we supplement our analytical results with numerical calculations.

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.

Accessing the nucleon transverse structure in inclusive deep inelastic scattering

We revisit the standard analysis of inclusive Deep Inelastic Scattering off nucleons taking into account the fact that on-shell quarks cannot be present in the final state, but they rather decay into hadrons – a process that can be described in terms of suitable “jet” correlators. As a consequence, a spin-flip term associated with the invariant mass of the produced hadrons is generated nonperturbatively and couples to the target's transversity distribution function. In inclusive cross sections, this provides an hitherto neglected and large contribution to the twist-3 part of the g2 structure function, that can explain the discrepancy between recent calculations and fits of this quantity. It also provides an extension of the Burkhardt–Cottingham sum rule, providing new information on the transversity function, as well as an extension of the Efremov–Teryaev–Leader sum rule, suggesting a novel way to measure the tensor charge of the proton.

Forward two-photon exchange in elastic lepton\u2013proton scattering and hyperfine-splitting correction

We relate the forward two-photon exchange (TPE) amplitudes to integrals of the inclusive lepton–proton scattering cross sections. These relations yield an alternative way for the evaluation of the TPE correction to hyperfine-splitting (HFS) in the hydrogen-like atoms with an equivalent to the standard approach (Iddings, Drell and Sullivan) result implying the Burkhardt–Cottingham sum rule. For evaluation of the individual effects (e.g., elastic contribution) our approach yields a distinct result. We compare both methods numerically on examples of the elastic contribution and the full TPE correction to HFS in electronic and muonic hydrogen.

A Measurement of g2p at Low Q2

Jefferson Lab has been at the forefront of a program to study the polarized structure of nucleons using electron scattering. Measurements of the spin dependent structure functions, [Formula: see text] and [Formula: see text], have proven to be powerful tools in testing and understanding QCD. The neutron structure function [Formula: see text] has been measured extensively in Hall A at Jefferson Lab over a wide range of [Formula: see text], but data for [Formula: see text] remains scarce. This docment will discuss the [Formula: see text] experiment, which ran in Hall A at Jefferson Lab in the spring of 2012, and will provide the first measurement of [Formula: see text] in the resonance region; covering [Formula: see text] GeV2. The 0[Formula: see text] moment of [Formula: see text] provides a test of the Burkhardt-Cottingham sum rule, which states that the integral of [Formula: see text] over the Bjorken scaling variable [Formula: see text] goes to zero. This sum rule, valid for all values of [Formula: see text], has been satisfied for the neutron, but a violation is suggested for the proton at high [Formula: see text]. The 2[Formula: see text] moment allows for a benchmark test of [Formula: see text]PT at low [Formula: see text]. Specifically, the behavior of the longitudinally-transverse spin polarizability ([Formula: see text]), as [Formula: see text]PT calculations of this quantity deviate significantly from the measured neutron data. This document will discuss the current status of the analysis along with preliminary results.

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.

Measurement of the virtual-photon asymmetry A 2 and the spin-structure function g 2 of the proton

A measurement of the virtual-photon asymmetry A_2(x,Q^2) and of the spin-structure function g_2(x,Q^2) of the proton are presented for the kinematic range 0.004 < x < 0.9 and 0.18 GeV^2 < Q^2 < 20 GeV^2. The data were collected by the HERMES experiment at the HERA storage ring at DESY while studying inclusive deep-inelastic scattering of 27.6 GeV longitudinally polarized leptons off a transversely polarized hydrogen gas target. The results are consistent with previous experimental data from CERN and SLAC. For the x-range covered, the measured integral of g_2(x) converges to the null result of the Burkhardt-Cottingham sum rule. The x^2 moment of the twist-3 contribution to g_2(x) is found to be compatible with zero.

Deep inelastic processes and the equations of motion

We show that the Politzer theorem on the equations of motion implies approximateconstraints on the quark correlator. These, in turn, restrict considerably, for sufficientlylarge Q2, the number of independent distribution functions that characterize the internalstructure of the nucleon, and of independent fragmentation functions. This result leadsus to suggesting an alternative method for determining transversity. Moreover, ourapproach implies predictions on the Q2 -dependence of some azimuthal asymmetries,like Sivers, Qiu-Sterman and Collins asymmetry. Lastly, we discuss some implicationson the Burkhardt-Cottingham sum rule. Key words: Equations of motion, deep inelastic.

MOMENTS OF SPIN STRUCTURE FUNCTIONS: SUM RULES AND POLARIZABILITIES

Nucleon structure study is one of the most important research areas in modern physics and has challenged us for decades. Spin has played an essential role and often brought surprises and puzzles to the investigation of the nucleon structure and the strong interaction. New experimental data on nucleon spin structure at low to intermediate momentum transfers combined with existing high momentum transfer data offer a comprehensive picture in the strong region of the interaction and of the transition region from the strong to the asymptotic-free region. Insight into some aspects of the theory for the strong interaction, Quantum Chromodynamics (QCD), is gained by exploring lower moments of spin structure functions and their corresponding sum rules (i.e., the Bjorken, Burkhardt–Cottingham, Gerasimov–Drell–Hearn (GDH), and the generalized GDH). These moments are expressed in terms of an operator-product expansion using quark and gluon degrees of freedom at moderately large momentum transfers. The higher-twist contributions have been examined through the evolution of these moments as the momentum transfer varies from higher to lower values. Furthermore, QCD-inspired low-energy effective theories, which explicitly include chiral symmetry breaking, are tested at low momentum transfers. The validity of these theories is further examined as the momentum transfer increases to moderate values. It is found that chiral perturbation theory calculations agree reasonably well with the first moment of the spin structure function g1 at low momentum transfer of 0.05–0.1 GeV2 but fail to reproduce some of the higher moments, noticeably, the neutron data in the case of the generalized polarizability δLT. The Burkhardt–Cottingham sum rule has been verified with good accuracy in a wide range of Q2 assuming that no singular behavior of the structure functions is present at very high excitation energies.

Probing Quark-Gluon Interactions with Transverse Polarized Scattering

We have extracted QCD matrix elements from our data on doubly polarized inelastic scattering of electrons on nuclei. We find the higher twist matrix element d˜2, which arises strictly from quark-gluon interactions, to be unambiguously nonzero. The data also reveal an isospin dependence of higher twist effects if we assume that the Burkhardt-Cottingham sum rule is valid. The fundamental Bjorken sum rule obtained from the a0 matrix element is satisfied at our low momentum transfer.

LORENTZ INVARIANCE RELATIONS AND WANDZURA–WILCZEK APPROXIMATION

A complete list of the so-called Lorentz invariance relations between parton distribution functions is given and some of their consequences are discussed, such as the Burkhardt-Cottingham sum rule. The violation of these relations is considered in a model independent way. It is shown that several Lorentz invariance relations are not violated in a generalized Wandzura–Wilczek approximation, indicating that numerically their violation may be small.

Polarized deep inelastic and elastic scattering from gauge/string duality

In this paper, we investigate deep inelastic and elastic scattering on a polarized spin-$\frac{1}{2}$ hadron using gauge/string duality. This spin-$\frac{1}{2}$ hadron corresponds to a supergravity mode of the dilatino. The polarized deep inelastic structure functions are computed in the supergravity approximation at large t' Hooft coupling $\ensuremath{\lambda}$ and finite $x$ with ${\ensuremath{\lambda}}^{\ensuremath{-}1/2}\ensuremath{\ll}xl1$. Furthermore, we discuss the moments of all structure functions, and propose an interesting sum rule ${\ensuremath{\int}}_{0}^{1}dx{g}_{2}(x,{q}^{2})=0$ for the ${g}_{2}$ structure function which is known as the Burkhardt-Cottingham sum rule in QCD. In the end, the elastic scattering is studied and elastic form factors of the spin-$\frac{1}{2}$ hadron are calculated within the same framework.

Q2 evolution of 3He spin structure functions moments

Sum rules offer the opportunity to test our understanding of the structure of a system on general grounds. In this paper we investigate the 3He nucleus using the Gerasimov-Drell-Hearn and Burkhardt-Cottingham sum rules. We have previously used the polarized 3He target measurements to extract the neutron spin structure, our goal here is to investigate rather the 3He nucleus. We have determined for the first time the Q2 evolution of $\Gamma_1(Q^2)=\int_0^{1} g_1(x,Q^2) dx$ and $\Gamma_2(Q^2)=\int_0^1 g_2(x,Q^2) dx$ for 3He in the range 0.1 GeV2 ≤ Q2 ≤ 0.9 GeV2 with good precision. Γ1(Q2) displays a smooth variation from high to low Q2. However, we do not see yet its expected turnover toward the real photon limit value. On the other hand the Burkhardt-Cottingham sum rule holds at all measured Q2s within uncertainties due mainly to cancellations between the elastic, quasielastic and the Δ(1232) resonance contributions.

QCD radiative and power corrections and generalized Gerasimov-Drell-Hearn sum rules

We extend the earlier suggested QCD-motivated model for the $Q^2$-dependence of the generalized Gerasimov-Drell-Hearn (GDH) sum rule which assumes the smooth dependence of the structure function $g_T$, while the sharp dependence is due to the $g_2$ contribution and is described by the elastic part of the Burkhardt-Cottingham sum rule. The model successfully predicts the low crossing point for the proton GDH integral, but is at variance with the recent very accurate JLAB data. We show that, at this level of accuracy, one should include the previously neglected radiative and power QCD corrections, as boundary values for the model. We stress that the GDH integral, when measured with such a high accuracy achieved by the recent JLAB data, is very sensitive to QCD power corrections. We estimate the value of these power corrections from the JLAB data at $Q^2 \sim 1 {GeV}^2$. The inclusion of all QCD corrections leads to a good description of proton, neutron and deuteron data at all $Q^2$.

Q2 evolution of the neutron spin structure moments using a 3He target.

We have measured the spin structure functions g(1) and g(2) of 3He in a double-spin experiment by inclusively scattering polarized electrons at energies ranging from 0.862 to 5.058 GeV off a polarized 3He target at a 15.5 degrees scattering angle. Excitation energies covered the resonance and the onset of the deep inelastic regions. We have determined for the first time the Q2 evolution of Gamma(1)(Q2)= integral (1)(0)g(1)(x,Q2)dx, Gamma(2)(Q2)= integral (1)(0)g(2)(x,Q2)dx, and d(2)(Q2)= integral (1)(0)x(2)[2g(1)(x,Q2)+3g(2)(x,Q2)]dx for the neutron in the range 0.1< or =Q2< or =0.9 GeV2 with good precision. Gamma(1)(Q2) displays a smooth variation from high to low Q2. The Burkhardt-Cottingham sum rule holds within uncertainties and d(2) is nonzero over the measured range.

Spin structure of the nucleon at low energies

The spin structure of the nucleon is analyzed in the framework of a Lorentz-invariant formulation of baryon chiral perturbation theory. The structure functions of doubly virtual Compton scattering are calculated to one-loop accuracy (fourth order in the chiral expansion). We discuss the generalization of the Gerasimov-Drell-Hearn sum rule, the Burkhardt-Cottingham sum rule and other moments of the nucleon spin structure functions. We give predictions for the forward and the longitudinal-transverse spin polarizabilities of the proton and the neutron at zero and finite photon virtuality. A detailed comparison to results obtained in heavy baryon chiral perturbation theory is also given.

Burkhardt-Cottingham sum rule and forward spin polarizabilities in heavy baryon chiral perturbation theory

We study spin-dependent sum rules for forward virtual Compton scattering (VVCS) off the nucleon in heavy baryon chiral perturbation theory (HBChPT) at order $\mathcal{O}{(p}^{4}).$ We show how these sum rules can be evaluated from low energy expansions (in the virtual photon energy) of the forward VVCS amplitude. We study in particular the Burkhardt-Cottingham sum rule in HBChPT and the higher terms in the low energy expansion, which can be related to the generalized forward spin polarizabilities of the nucleon. The dependence of these observables on the photon virtuality ${Q}^{2}$ can be accessed, at small and intermediate ${Q}^{2}$ values, from existing and forthcoming data at Jefferson Lab.

Precision measurement of the proton and deuteron spin structure functions g2 and asymmetries A2

We have measured the spin structure functions g2p and g2d and the virtual photon asymmetries A2p and A2d over the kinematic range 0.02⩽x⩽0.8 and 0.7⩽Q2⩽20GeV2 by scattering 29.1 and 32.3 GeV longitudinally polarized electrons from transversely polarized NH3 and 6LiD targets. Our measured g2 approximately follows the twist-2 Wandzura–Wilczek calculation. The twist-3 reduced matrix elements d2p and d2n are less than two standard deviations from zero. The data are inconsistent with the Burkhardt–Cottingham sum rule if there is no pathological behavior as x→0. The Efremov–Leader–Teryaev integral is consistent with zero within our measured kinematic range. The absolute value of A2 is significantly smaller than the A2<R(1+A1)/2 limit.

Bjorken sum rule at low Q2

A description of the generalized Gerasimov–Drell–Hearn sum rules for proton and neutron is suggested, using their relation to the Bjorken sum rule. The results support an earlier conjecture, that the structure function gT features a smooth Q2-dependence, while the structure function g2 is changing rapidly, due to the elastic contribution to the Burkhardt–Cottingham sum rule. A possible violation of this later sum rule is briefly discussed.