Gluon Polarization In Nucleon

In the context of the so-called valon model, we calculate $\frac{\delta g}{g}$ and show that although it is small and compatible with the measured values, the gluon contribution to the spin of nucleon can be sizable. The smallness of $\frac{\delta g}{g}$ in the measured kinematical region should not be interpreted as $\delta g$ being small. In fact, $\delta g$ itself at small x, and the first moment of the polarized gluon distribution in the nucleon, $\Delta g(Q^2)$, are large


INTRODUCTION
The decomposition of nucleon spin in terms of its constituents have been an active topic both from theoretical and experimental point of views. It is established that the quark contribution, ∆Σ, to the nucleon spin is a small fraction of the nucleon spin. Other sources that might contribute to the nucleon spin come from gluon spin and the overall orbital angular momentum of the partons. Thus, one can write the following spin sum rule for a nucleon: where ∆Σ is the quarks and anti-quarks contribution to the nucleon spin, ∆g is the gluon contribution and L q,g represents the overall orbital angular momentum contribution of the partons.
In deep inelastic scattering, the gluon spin content of the nucleon can be calculated from the Q 2 dependence of the polarized structure function g 1 . Experimentally, it is possible to use semi-inclusive deep inelastic scattering processes to measure δg g from helicity asymmetry in photon-gluon fusion, γ * g → qq process. The COMPASS collaboration [1] have used this method and find a rather small value for δg g = 0.024 ± 0.080 ± 0.057. The smallness of δg g cannot by itself rule out the possibility of a large value for the first moment, ∆g, of the gluon polarization. In fact, when the singlet axial matrix element a 0 was found to be much smaller than the contribution expected from quark-parton model, it was suggested that the deference could be accounted for by a large contribution from the gluon spin: ∆g. This would require a value of ∆g ∼ 3 at Q 2 = 3 GeV 2 in order to obtain the expected value of ∆Σ. Moreover, Altarelli and Ross [2] and Effremov et.al [3] have shown that polarized gluon makes a scaling contribution to the first moment of the polarized structure function, g 1 , which means that it must be large at higher momentum scales.
The total quark spin contribution ∆Σ to the nucleon spin is fairly well determined and gives a value around 0.4. There is no direct determination of orbital angular momentum component, and one is not expected in the near future. In contrast to ∆Σ, knowledge about gluon polarization is limited. The existing and the emerging data on δg(x,Q 2 ) g(x,Q 2 ) cannot rule out the negative and/or zero polarization for gluon, including a possible sign change.
There are mainly three method to access gluon polarization: (1) polarized deep inelastic scattering, in which one would parameterize quark and gluon densities and fit them to the data on polarized structure function g 1 (x, Q 2 ). Gluon enters into the analysis through the Q 2 evolution, but the limited range of Q 2 leads to not so precise determination of δg(x).
Recent data suggests that global fits with positive, negative, zero, and sign changing δg(x) provide equally good agreement. (2) Using cc production in semi-inclusive deep inelastic processes by γ − g fusion. (3) via single particle production in polarized p-p collision.
In this paper we determine the gluon polarization in the polarized proton using the so called valon model, as described bellow.

The valon model description of nucleon
Deep inelastic scattering reveals that the nucleon has a complicated internal structure.
Other strongly interacting particles also exhibit similar structure. However, under certain conditions, hadrons behave as consisting of three (or two) constituents. Therefore, it seems to make sense to decompose a nucleon into three constituent quarks called U and D. We identify them as valons. A valon has its own internal structure, consisting of a valence quark and a host of qq pairs and gluons. The structure of a valon emerges from the dressing of a valence quark with qq pairs and gluons in perturbative QCD. We take the view that when a nucleon is probed with high Q 2 it is the internal structure of the valon that is resolved. The valon concept was first developed by R. C. Hwa [4], and in Refs. [5,6,7] it was utilized to calculate unpolarized structure functions of a number of hadrons. This representation is also used to calculate the polarized structure of nucleon.
The details can be found in [8,9] .
We have worked in M S scheme with Λ QCD = 0.22 GeV and Q 2 0 = 0.283 GeV 2 . The polarized and unpolarized structure of a valon is calculated in the framework of Next-to-Leading order in QCD. Then, the polarized (unpolarized) structure function of the nucleon is obtained by the convolution of the valon structure with the valon distribution in the hosting nucleon: where δG h valon (y) is the helicity distribution of the valon in the hosting hadron and is the polarized structure function of the valon. A similar relation can also be written for the unpolarized structure function, F 2 . We maintain the results of Ref. [8] for the polarized structure function, but re-analyze the unpolarized case. This is necessary in order to arrive at a consistent conclusion on δg g . In the moment space the initial densities for both polarized and unpolarized densities of the partons in a valon are taken to be as follows, The above initial densities means that if Q 2 is small enough, at some point we may identify g valon 1 ( x y , Q 2 ) and f valon 2 ( x y , Q 2 ) as δ(z−1), for the reason that we cannot resolve its internal structure at such Q 2 value. Here f valon 2 ( x y , Q 2 ) is the unpolarized structure function of the valon.
In figure (1) a sample of results for the unpolarized structure function, F 2 , is presented.
The data points are from [10,11]. Similar results are also obtained at different kinematics [5]. of a valon is obtained simply from QCD processes via DGLAP evolution [18,19,20].
Our main concern here is to determine the polarized and unpolarized gluon distributions and hence, the ratio δg(x,Q 2 ) g(x,Q 2 ) . The gluon is a component of the singlet sector of the evolution kernel. Their moments are given as where L ≡ α s (Q 2 )/α s (Q 2 0 ), and δP (0)n is 2 × 2 singlet matrix of splitting functions, given δP (0)n lm are the n th moments of the polarized splitting functions and U accounts for the 2-loop contributions as an extension to the leading order. The explicit forms of these functions are given in [21] in the next-to-leading order. Now it is straightforward to calculate preferably to radiate a gluon with helicity parallel to the quark polarization. Since the net quark spin in a valon is positive, it follows that perturbatively radiated gluon from quarks must have ∆g > 0. We also note that the growth rate of ∆g is especially fast for the relatively low Q 2 . In order to satisfy the sum rule in Equation (1) it requires that the orbital angular momentum component to be negative and decreasing as Q 2 increases [9].
Figures (1) and (2) demonstrate that the model can accommodate the experimental data shown the results at Q 2 = 5 GeV 2 and compared it with the global fits from [22,23,24].
The unpolarized gluon distribution is also shown in figure (5) and is compared with the results obtained from various global fits [25,26,27,28] .
Our results for the sea quark polarization is consistent with zero, and yields a positive value for the first moment of gluon, ∆g(Q 2 ), which increases with Q 2 reaching a value of around 0.5 at Q 2 = 10 GeV 2 as can be seen from figure (3).
It is now straight forward to calculate the ratio δg(x) g(x) in the proton. This ratio is calculated and shown in figure (6). The calculation is done for the proton at each value of COMPASS open charm data disagree with our results. However, these two data points are the least accurate ones with very large error bars. In contrast, our results are in good agreement with the remaining experimental points, including the very recent one from HERMES and COMPASS [29,30,31,32,33,34,35].

Conclusion
We calculated gluon polarization in a polarized proton in the valon model and compared it with the existing data, including the most recent one from HERMES collaboration [30].
Since the experimental data are obtained at different Q 2 values, the calculations are also carried out at the corresponding Q 2 , individually. It is evident from the results that the polarized valon model of nucleon not only agrees with the existing data on g 1 but also provides a clear resolution for the spin problem. We maintain the view that ∆g(Q 2 ) is positive and increases with Q 2 . The growth of ∆g(Q 2 ) in part is compensated by a negative and large orbital angular momentum, L q,g . Although, we have not calculated L q and L g individually, but the overall L q,g is given in [8]. g(x,Q 2 ) calculated in the valon model and compared with the exist experimental data. The apparent wide band in the figure is actually seven closely packed curves corresponding to the seven values of Q 2 s at which the data points are measured . The data are from Refs. [29,30,31,32,33,34,35].
This suggests that even if δg(x, Q 2 ) maybe small at relatively large x, but the first moment of gluon polarization in the proton is sizable. In fact, δg(x, Q 2 ) is quiet large at small x.