The proton spin and the Wigner rotation

Abstract

It is shown that in both the gluonic and strange sea explanations of the Ellis-Jaffe sum rule violation discovered by the European Muon Collaboration (EMC), the spin of the proton, when viewed in its rest reference frame, could be fully provided by quarks and antiquarks within a simple quark model picture, taken into account the relativistic effect from the Wigner rotation.To be published in Z. Phys. C * Fellow of Alexander von Humboldt Foundation. 1The European Muon Collaboration (EMC) experiment [1] on deep inelastic scattering of polarized leptons from polarized protons has received attention by the particle physics community recently [2]. The reason is that the result of the integrated spin-dependent structure function data is significantly smaller than that expected from the Ellis-Jaffe sum rule [3,4], and that in a naive interpretation of this small result one is led to the conclusion [1,5] that the fraction of the proton spin carried by quarks and antiquarks is smaller than expected. There are essentially two theoretical aspects for understanding this startling EMC result. The first concerns the mechanism for the violation of the Ellis-Jaffe sum rule (EJSR), which is obtained [3,4] by using the experimental values of G A /G V measured in weak semileptonic decays together with flavor SU(3) symmetry results and an additional assumption of a negligible strange sea contribution to the proton spin. Many theoretical speculations have been proposed to explain the EJSR violation, such as to attribute the small EMC result to gluonic contributions due to the U(1) axial anomaly [4,6], non-vanishing strange quark contributions [7], consequences of the QCD anomaly and the spontaneous breaking of the chiral symmetry [8], the unjustified x → 0 extrapolation of the data[9], flavor SU(3) symmetry breaking [10], and non-perturbative QCD effects at low Q 2 [4, 11, 12] et al.. This paper is not intended to discuss the aspect concerning the EJSR violation. Our attention is focused on the second issue concerning the EMC result: the small data triggered the proton "spin crisis" [5], i.e., the intriguing question of how the spin of the proton is distributed among its quark spin, gluon spin and orbital angular momentum. At present it is commonly taken for granted that the EMC result implies that there must be some contribution due to gluon polarization or orbital angular momentum to the proton spin. For example, in the gluonic [6] and strange sea [7] explanations of the EJSR breaking, the proton spin carried by the spin of quarks and antiquarks was estimated to be of about 70% in the former [13] and negligible in the latter [7]. We will show, however, that the above understandings are not in contradiction with a simple quark model picture in which the spin of the proton, when viewed in its rest ref-2 erence frame, is fully provided by the vector sum of the spin of quarks and antiquarks.The key points for understanding the proton spin puzzle lie in the facts that the vector sum of the constituent spin for a composite syste...

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“…We then consider the Melosh-Wigner rotation effect in the calculation of the parton densities [31][32][33], and apply the valence quark distribution functions calculated in the light-cone SU(6) quark-spectator-diquark model [34] to estimate the probability of a valence quark directly fragmenting to a hadron. This can be realized through the phenomenology Gribov-Lipatov relation [35][36][37][38],…”

“…The free parameters are reduced to only a few numbers, which can be referred to in Table I. Table 1 The quark distribution functions of octet baryons in the light-cone SU(6) quark-diquark model [39] Baryon The polarized quark distributions are obtained by introducing the Melosh-Wigner correction factor [32,33] ∆q(x) =c…”

Quark-antiquark asymmetry of helicity distributions in the nucleon sea

“…(6) and fitting results of ∆l u (x) and ∆l d (x) (see Table II) together, a simplification can be made by fitting two parameters [l u (x) andl d (x)] instead of four with the method declared in Eq. (9). We getl u andl d by fitting experimental data of single-spin asymmetries A W ± L at RHIC [23].…”

“…However, it was pointed out in Refs. [8][9][10][11] that the quark helicity observed in polarized deep inelastic scattering (DIS) is actually the quark spin defined in light-form dynamics [12] and it is different from that in the quark model, which is defined in instant-form dynamics. Therefore the small helicity sum observed by the experiments is not in conflict with the quark model due to the reduction of the light-cone spin relative to the instant-form spin by the Melosh-Wigner rotation [13][14][15], which is a relativistic effect of quark transversal motions.…”

Quark Angular Momentum and Gravitational Form Factors

“…As a relativistic effect, even if the proton in the rest frame is an S-wave system as described in the naive quark model, the orbital angular momentum (OAM) is nonnegligible in the infinite momentum frame or the light-cone formalism where the parton model is well established. 6,7 Besides, the gluon can also contribute a large fraction.…”