Contribution of Orbital Angular Momentum to the Nucleon Spin

TL;DR

This paper calculates the orbital angular momentum contributions of quarks and gluons to the nucleon spin using the valon model at next-to-leading order, revealing small positive quark and large negative gluon contributions.

Contribution

It provides the first detailed calculation of quark and gluon orbital angular momentum in the nucleon at NLO within the valon model, including their Q^2 dependence.

Findings

Quark orbital angular momentum is positive but small.

Gluon orbital angular momentum is negative and large.

Partonic angular momentum varies with Q^2.

Abstract

We have calculated the Orbital Angular Momentum of quarks and gluons in the nucleon. The calculations are carried out in the next to leading order utilizing the so-called valon model. It is found that the average quark orbital angular momentum is positive, but small, and the average gluon orbital angular momentum is negative and large. We also report on some regularities about the total angular momentum of the quarks and the gluon, as well as on the orbital angular momentum of the separate partons. We have also provided partonic angular momentum, $L^{q,g}$ as a function of $Q^2$.

Equations

$$\frac{1}{2}=\frac{1}{2}\Delta \Sigma +\Delta G+L_{q,g}$$

$$\frac{1}{2}=\frac{1}{2}\sum _q\Delta q+\sum _q{\mathcal{L}}_q^z+\Delta G+{\mathcal{L}}_g^z$$

$$\frac{1}{2}=\frac{1}{2}\sum _q\Delta q+\sum _q{L}_q^z+J_g^z$$

$$L^q=\int dx\int d^{2}b(x{H}^q(x,b)+x{E}^q(x,b)-{\tilde{H}}^q(x,b))$$

$$J_{q,g}^z=\frac{1}{2}[A_{q,g}(0)+B_{q,g}(0)]$$

$$B(0)=\sum _i{B}_i(0)=0$$

$$J_q^z(x)=\frac{1}{2}x[<q(x)>+B_q(0)],J_g^z(x)=\frac{1}{2}x[<g(x)>+B_g(0)].$$

$$F_2^h(x,Q^2)=\sum _{valon}{\int }_x^{1}dy{G}_{valon}^h(y)F_2^{valon}(\frac{x}{y},Q^2)$$

$$g_1^h(x,Q^2)=\sum _{valon}{\int }_x^{1}dy\delta G_{valon}^h(y)g_1^{valon}(\frac{x}{y},Q^2)$$

$$\delta q^{NS}(z,Q_0^2)=\delta q^S(z,Q_0^2)=\delta (z-1)$$

$$\delta g(z,Q_0^2)=0$$

Full text

1 INTRODUCTION

Polarized deep inelastic scattering processes is the most direct tool to probe the spin content of the nucleon. In such experiments detailed information can be extracted on the shape and the magnitude of the spin dependent parton distributions, $\Delta q_{f}(x,Q^{2})$. Deep inelastic scattering reveals that the nucleon is a rather complicated object consisting of an infinite number of quarks, anti-quarks, and gluons. It is a common belief that other strongly interacting particles also exhibit similar internal structure.

The decomposition of nucleon spin in terms of its constituents has been a challenging and an active topic in hadron physics, both from theoretical and experimental points of view. It is now established that quarks carry a small fraction of the nucleon spin. Other sources that might contribute to the nucleon spin are gluon spin and the overall orbital angular momentum of the partons. Thus, it is common to write the following spin sum rule for a nucleon.

[TABLE]

Over the past few years we have studied the first two terms of the above sum rule within the framework of the so called valon model [1] in the next to leading order. The valon model is a phenomenological model for hadrons, introduced first by R.C. Hwa about thirty years ago [2]. The model has been quite successful in describing a variety of hadronic phenomena [3, 4]. In the polarized deep inelastic scattering domain, the model has successfully reproduced the existing data on $g_{1}^{p,n,d}$, and individual parton contributions, $\Delta q_{f}(x,Q^{2})$, to the nucleon spin, while predicting new results yet to be tested. Among other things, the model has predicted that the sea quark polarization is negligible, which is now proven to be the case[5]. This finding is because the valons are generated by perturbative dressing in QCD. In such processes with massless quarks, helicity is conserved and therefore, the hard gluons can not produced the sea quark polarization perturbatively. So, it turns out that sea polarization is consistent with zero. We have also shown that although $\delta g(x,Q^{2})$ is small, but its first moment, $\Delta G$ is large and grows as $Q^{2}$ increases [6]. This is consistent with QCD and with available experimental data [7],[8],[9],[10], [11],[12],[13],[14],[15] . Elsewhere, we have reported that with a fixed and almost $Q^{2}$ independent value for $\Delta\Sigma$ and with the growing $\Delta G$ it is not possible to achieve $s_{z}=\frac{1}{2}$ for the nucleon spin. Therefore, there should be other contributing element in order to arrive at spin $\frac{1}{2}$ of the nucleon. The only possible place would be the orbital angular momentum of the partons. In [16] we used Eq. (1) and estimated, but not independently calculated, the magnitude of the overall orbital angular momentum of the partons inside the nucleon. Our conclusion was that overall orbital angular momentum of partons is negative and decreases as $Q^{2}$ increases. In fact, almost twenty five years ago, P. G. Ratcliffe [17] suggested that a consistent interpretation of the Dokshitze-Gribov-Altarelli-Parisi evolution equation of helicity weighted parton distributions requires the partons to carry a sizable orbital angular momentum. Moreover, he concluded that for large $Q^{2}$, the average orbital angular momentum will be negative.

The purpose of this paper is to report the results that we have obtained for the orbital angular momentum contribution of quarks and gluons to the proton spin. Our calculations are carried out in the next to leading order in perturbation theory and are based on the valon representation of the nucleon. It is important to mention that one could, as well, use other polarized and unpolarized parton distributions, such as those obtained from the available global fits to carry out the same calculation. The main reason for us in using the valon model is that, first, it was handy and secondly, it’s outcomes and the predictions have proven to be consistent with all the experimental data that are available. Hence, providing a reasonable confidence in its physical validity.

2 The Experimental data

Over the past two decades theoretical framework for the understanding of the spin structure of nucleon has been developed and numerous experiments were carried out. We now have a fairly good understanding of the first term in the sum rule above. Total quark contribution to the nucleon spin is about $\Delta\Sigma=0.4$. Some data are also emerged on the gluon polarization [18][19] showing that $\frac{\delta g(x,Q^{2})}{g(x,Q^{2})}$ is small. These data are reported at individual kinematics, i.e. at separate $(x,Q^{2})$ points, and lack the same level of precision achieved for the quark sector. The data from COMPASS collaboration [20] may even be able to rule out a negative value for $\frac{\delta g(x,Q^{2})}{g(x,Q^{2})}$, which has been a controversy over the past few years. Nevertheless, the smallness of $\frac{\delta g}{g}$ by itself cannot rule out the possibility of a large value for the first moment, $\Delta G$, of the gluon polarization. In Figure 1 we present $\frac{\delta g(x,Q^{2})}{g(x,Q^{2})}$ that is obtained from our model. We have calculated $\frac{\delta g(x,Q^{2})}{g(x,Q^{2})}$ at each kinematical point for which the data exists. The apparent wide band in the figure is actually several closely packed curves corresponding to the several values of $Q^{2}$ at which data points are measured. The details of this calculation can be found in [6]. In reference [21] a new method is suggested for measuring the gluon polarization. In Figure 2 we also show our results on the first moments of the quark, $\Delta\Sigma$, and the gluon, $\Delta G$, polarization in the nucleon. Substituting these valued in Equation 1, gives a measure of the total overall angular momentum of the partons inside the nucleon, which is also shown in figure 2.

3 Orbital Angular Momentum

Considering that the spin of quarks accounts for only a part of the nucleon spin, and that of the gluon is still unclear, a substantial fraction of the nucleon spin must be due to the orbital angular momentum. Unfortunately, in the gauge theories there is no unique decomposition of the nucleon spin into contributions due to spin and the orbital angular momentum of quarks and gluons. For example, Jaffe and Manohar [22] have used a light-like hypersurface, employed the light-cone framework and light-cone gauge and arrived at the following decomposition

[TABLE]

In this decomposition each term is defined as the matrix element of the corresponding term in $+12$ component of the orbital angular momentum tensor. The first and the third terms are interpreted as the quark and gluon spin, respectively. The second and the forth terms are identified as the quark and gluon orbital angular momentum. In this decomposition, except for the first term, individual terms are not separately gauge invariant. An alternative decomposition is provided by Ji [23]

[TABLE]

where each term is separately gauge invariant. However, the gluon total angular momentum is not decomposed, in a gauge invariant way, into its spin and the orbital angular momentum. In Ji’s decomposition, The total spin of quarks, $J_{q}$, and that of the gluons, $J_{g}$, are related to the generalized parton distribution (GPD) at twist-two level. Other decompositions have also been proposed [24][25]. A thorough analysis of these decompositions is given in [26]. Briefly, it has been established that there are only two types of complete decompositions of the nucleon spin. The first one is the decomposition of canonical type, while the other is the decomposition of mechanical (or kinetic) type. The famous Jaffe-Manohar decomposition belong to the former, while another complete decomposition proposed in [25] is of the mechanical type. Since these two quark orbital angular momenta (OAMs) are apparently different, the gluon OAMs are also different in the two types of nucleon spin decomposition.

It is now shown that at the twist-three level, once the generalized parton distribution are integrated over $x$ , both decompositions given in eq. (2) and Eq. (3) can be obtained. [27], [28].

As mentioned, Ji’s decomposition are related to twist-two generalized parton distributions, which can be measured in deeply virtual compton scattering. The quark orbital angular momentum in terms of GPDs is given by [29]

[TABLE]

The GPDs describe the dynamics of partons in the transverse plane in position space. Complementary information on the dynamics of partons in the transverse plane, but in the momentum space, is obtained from Transverse Momentum Dependent parton distributions (TMD-PDF)[30] [31]. Therefore, one naturally expects that TMD and GPDs will teach us about partonic orbital angular momentum.

The orbital angular momentum of partons play an important role in hadron physics. It is well known that in order to have a non-zero anomalous magnetic moment, the light cone wave function of nucleon must have components with $L_{z}\neq 0$ [32].

In the local limit, GPDs reduce to form factors, which are obtained from the matrix elements of the energy momentum tensor $\Theta^{\mu\nu}$. Since one can define $\Theta^{\mu\nu}$ for each parton, one can identify the momentum fraction and contributions to the orbital angular momentum of each quark flavor and gluon in a hadron. Spin flip form factor $B(q^{2})$ which is the analog of the Pauli form factor $F_{2}(Q^{2})$ of the nucleon provides a measure of the orbital angular momentum carried by each quark and gluon constituent of the nucleon at $q^{2}=0$. Similarly, the spin conserving form factor $A(q^{2})$, the analog of Dirac form factor $F_{1}(q^{2})$, allows one to measure the momentum fraction carried by each constituent. This is the underlying physics of Ji’s sum rule [29]:

[TABLE]

where, $B_{q,g}$ are the second moments of unpolarized spin-flip GPD in the forward limit. It is subject to the constraint that

[TABLE]

that is, when summed over all partons, spin flip form factor vanishes. For composite systems, this has been proven by Brodsky, et al. in the light cone representation[33]. In fact, it is a consequence of equivalence principle.

For the quark and the gluon sectors, the above equation translates into

[TABLE]

Based on some lattice calculations and the model dependent analysis [34] it is expected that $B_{q,g}$ to be small. In fact, lattice calculations show that the valence quarks give a value between $-0.077$ and $0.015$ [35]. Excluding the unlikely possibility for large value due to strange quark and anti-quark, we find that the sum of the contributions from the sea quarks and the gluons must be small. We do realize the possibility that gluon and the sea quark contributions could be large, but with opposite sign. Also, they can be large, but have nodes such that their second moments become small. Lattice calculations [36] have verified that indeed the total anomalous gravitomagnetic moment of the nucleon is zero. A more Recent lattice calculation [47] have shown that $B_{q,g}=0.00(6)$. They have used a different notation, namely $T_{2}(0)$, and presented their results in Table III of their paper.

Therefore, in the following analysis we will set $B_{q,g}=0$. With a zero value for $B_{q,g}$, the ”mechanical orbital angular momentum” of partons, $L_{q,g}$, can be determined entirely from polarized and unpolarized parton distributions. Moreover, with such an assumption the evolution equation for the angular momentum distributions $J_{q,g}$ is exactly the same as that for the unpolarized quark and gluon distributions [37]. These distributions are evaluated by us in the valon model with good accuracy in a wide range of kinematics $Q^{2}=[0.4,10^{6}]$ $GeV^{2}$ and $x=[10^{-6},0.95]$ and will be utilized here. The details and the functional form of the unpolarized parton distributions in the valon representation can be found in [4]. The table .1 show the numerical results for $\Delta\Sigma,\Delta g,L_{q},L_{g}$ in some values of $Q^{2}$.

In Figure 3 we show the behavior of $L_{q}(Q^{2})$, and $L_{g}(Q^{2})$ at several $Q^{2}$ values. It is apparent that while the quark orbital angular momentum is small and positive, the gluon orbital angular momentum is negative and decreases as $Q^{2}$ increases.

We have checked to make sure that if our results reproduces $J^{p}=\frac{1}{2}=J_{q}+J_{g}$. The results are shown in Figure 4. Evidently, this is the case.

In Figure 5 we present the gluon spin, $\Delta g$, the gluon orbital angular momentum, $L_{g}$ and the total angular momentum as a function of $Q^{2}$. This figure indicates that $J_{g}$ is independent of $Q^{2}$ and contributes an amount of about $0.22$ to the nucleon spin.

in Figure 6 the total angular momentum of individual quark flavors, $J_{u}$ and $J_{d}$, are presented. In this figure we also compare our results for $J_{u,d}$ with those of Ref. [35, 38, 39, 40, 41, 42, 43, 44]. We further found that the orbital angular momentum of u-quark, $L_{u}$, and d-quark, $L_{d}$, have opposite signs and largely cancel each other. Our results indicate that $L_{d}$ is positive and $L_{u}$ is negative. This is shown in Figure 7 in comparison with those from [35, 38] . Their difference is shown in Figure 8 and seems that the dependence on $Q^{2}$ is marginal. In an interesting paper [38] authors have derived a sum rule for spin-1 system through which they have obtained the total and the orbital angular momenta for $u$ and $d$ quarks in the proton at $Q^{2}=4$ $GeV^{2}$. Our results on total angular momentum carried by quarks, that is, $J_{q}=J_{u}+J_{d}$ agrees with the findings of Ref. [38], amounting to 0.26 at $Q^{2}=4$ $GeV^{2}$. This is interesting, because the two approaches are quite different. The two approaches also agree on the sign of $L_{u}$ and within the errors, the numerical values are also compatible. Our findings, however, is different from those of [38] on the total and the orbital angular momenta of the $d$ quark. Yet, both approaches produce compatible values for the spin component of the the $d$ quark. The total quark orbital angular momentum, $L^{Q}=L_{u}+L_{d}$ in our model gives a value of $0.08$ at $Q^{2}=4$ $GeV^{2}$, whereas, the result of the Ref. [35] is $-0.016\pm 0.084$. An earlier estimate of $L^{Q}=0.05-0.15$ is also given by Ji and Tang [29]. It is evident that within the quoted errors, the two values are not far apart. In fact, except for $L_{d}$, within the errors, our results are fairly close to those of Ref. [35]. We find no crossover between $L_{u}$ and $L_{d}$ when $Q^{2}$ is varied. We also find that $L_{u}-L_{d}$ remains large and negative and this finding is in nice agreement with [48].

Finally, we note that $Q^{2}$ dependence of $J_{g}$ is marginal, and the interplay is between $\Delta g$ and $L_{g}$, the former increases with $Q^{2}$, while the latter decreases. This is evident from Figure 5. It is also interesting to mention that above $Q^{2}$ around $5$ $GeV^{2}$ or so, the total angular momentum of quarks and the gluons seems to approach to an identical value, indicating that they equally share the spin of the nucleon. This observation is manifestly apparent from Figure 4.

4 conclusion

Although the valon representation of hadrons is a simple phenomenological model, it describes the structure of nucleon rather nicely. within this model, we have investigated the orbital angular momentum contribution of quarks and gluons to the nucleon spin. It shows that the quarks orbital angular momentum contribution to the total angular momentum of the nucleon is positive and relatively small. However, the gluon orbital angular momentum contributes substantially. Thus, we conclude that Gluon is a major player in describing the spin structure of nucleon. On the one hand, while $\frac{\delta g}{g}$ is small, first moment of the gluon polarization, $\Delta G$, is large and increases as $Q^{2}$ grows. On the other hand, its orbital angular momentum is large and negative, thus compensating the growth of $\Delta G$.

Some regularities also have emerged from our study: both orbital and total angular momenta of the u-quark and the d-quark seems to be independent of $Q^{2}$, though, some $Q^{2}$ dependence for $J_{u}$ is observed at low $Q^{2}$, but rapidly disappears. The same is true for the total angular momentum of the gluon, However, its orbital angular momentum varies. These are evident from figures 7 and 5, respectively. Finally, we notice that our calculations seems to be compatible with those that are obtained in [35]. We have also presented various orbital angular momenta components as a function of $Q^{2}$ which may be utilized to gain information on some generalized parton distributions.

5 Appendix: The Valon Model

Our understanding of hadron structure comes from the deep inelastic data and the hadron spectroscopy. In the latter picture, hadrons are bound states of massive particles, loosely called ”quarks” or ”constituent quarks”. The bound states of those entities describe the static properties of the hadrons. On the other hand, the interpretation of the deep inelastic data relies on the quarks of the QCD Lagrangian with a very small mass. The hadronic structure in this picture is intimately connected with the presence of a large number of partons (quarks and gluons). The quarks that participate in the bound state problem and the quarks of the QCD Lagrangian differ in other important properties as well. The very obvious example is the color charge of quark field in QCD Lagrangian, which is not gauge invariant and, thus, ill defined; reflecting the color of gluons in an interacting theory. Whereas, color associated with the quarks of a bound state (constituent quark) is a well defined entity.

In the bound state problem we regard a proton as consisting of three quarks and pion, a quark-antiquark pair. These are the constituent quarks. In deep inelastic scattering a proton is viewed as having valence quarks, sea quarks, and gluons, collectively called partons. In fact, even at $Q^{2}$ as low as a few $GeV^{2}$ the gluons carry nearly half of the nucleon momentum. To reconcile the two pictures of hadron, it is necessary to realize that the quarks probed in deep inelastic scattering are current quarks of the QCD Lagrangian and not the constituent quarks of the bound state problem. The failure to recognize this difference can lead to many mistakes.

By definition, a valon is a structureful object consisting of a valence quark plus its associated cloud of sea quarks and gluons. The cloud, or the structure of the valon is due to the dressing process in QCD. Indeed, it is shown [45][46] that one can dress a QCD Lagrangian field to all orders in perturbation theory and construct such an object (which we called a valon) in conformity with the color confinement. From this point of view, a valon emerges from the dressing of a valence quark with gluons and $q\bar{q}$ pair in QCD. In a bound state problem those processes are virtual and a good approximation for the problem is to regard a valon as an integral unit whose internal structure cannot be resolved. Thus, it may be identified as an indivisible , point-like object. As such, in a bound state problem they interact with each other in a way that is characterized by the valon wave function. On the other hand, in a scattering process, the virtual partons in a valon can be excited and put on mass-shell. They respond independently in an inclusive hard collision with a $Q^{2}$ dependence that can be calculated in QCD at high $Q^{2}$. The point is that the valons play a dual role in hadrons: on the one hand, they are constituents of bound state problem involving the confinement at large distances. On the other hand, they are quasi-particles whose internal structure are probed with high resolution and are related to the short distance problem of current operators. This picture suggests that the structure function of a hadron involves a convolution of two distributions, namely, the valon distribution in the hosting hadron and the parton distribution in the valon. In an unpolarized situation on may write the structure function of hadron $h$ as follows

[TABLE]

where $F_{2}^{valon}(\frac{x}{y},Q^{2})$ is the structure function of the probed valon and can be calculated in perturbative QCD. The function$G_{valon}^{h}(y)$ represents the valon distribution in the hosting hadron carrying momentum fraction $y$ of the hadron.It is $Q^{2}$ independent. These functions are already calculated for a number of hadrons. Details for the proton can be found in [4].

Similarly, for a polarized hadron, we can write the polarized structure function,$g_{1}^{h}$ as

[TABLE]

where, $\delta G_{valon}^{h}(y)$ is the helicity distribution of valon in the hadron with momentum fraction $y$ of the hadron. $g_{1}^{valon}(\frac{x}{y},Q^{2})$ is the polarized structure function of the valon. Again, the detailed calculation of these functions are given in [1].

We have worked in $\overline{MS}$ scheme with $\Lambda_{QCD}=0.22$ $GeV$. The initial scale of energy is $Q_{0}^{2}=0.283$ $GeV^{2}$. The motivation for this initial inputs at $Q_{0}^{2}$ comes from the phenomenological consideration that requires us to choose the initial input densities as $\delta(z-1)$ at $Q_{0}^{2}$. This condition means that the internal structure of the valon can not be resolved at $Q_{0}^{2}$ and at this initial scale, the nucleon can be considered as a bound state of three valence quarks that carry all the momentum and the spin of the nucleon. Increasing the $Q^{2}$ values resolved the other partons in the nucleon. Therefore, our initial input densities to solve the DGLAP equations inside the valon are

[TABLE]

In the valon picture of hadron, the deep inelastic scattering with high enough $Q^{2}$ actually it is the structure of a valon that is probed. At low $Q^{2}$ the valon structure cannot be probed and hence behaves as a quark in the bound state problem. This means that if $Q^{2}$ of the probe is less than a threshold value of $Q_{0}$ then, a valon would appear as a constituent quark. Yet, from the early SLAC days of the deep inelastic scattering on proton, we know that quark distribution in a proton shows precocious scaling for $Q^{2}$ in the range as low as one $GeV^{2}$. That is, $Q^{2}$ evolution has already run the course. For this reason, if $Q^{2}$ is small enough we may identify valon structure,$F^{valon}(z,Q^{2})$ as $\delta(z-1)$ at some point, because we cannot resolve its internal structure at that $Q^{2}$ value.