The Jacobi Polynomials QCD analysis for the polarized structure function
Abstract:
We present the results of our QCD analysis for polarized quark distribution and structure function . We use very recently experimental data to parameterize our model. New parameterizations are derived for the quark and gluon distributions for the kinematic range , GeV. The analysis is based on the Jacobi polynomials expansion of the polarized structure functions. Our calculations for polarized parton distribution functions based on the Jacobi polynomials method are in good agreement with the other theoretical models. The values of and are determined.
1 Introduction
The nature of the shortdistance structure of polarized nucleons is one of the central questions of present day hadron physics. For more than sixteen years, polarized inclusive deep inelastic scattering has been the main source of information on how the individual partons in the nucleon are polarized at very short distances.
The theoretical and experimental status on the spin structure of the nucleon has been discussed in great detail in several recent reviews (see, e.g., Refs. [1, 2, 3, 4]. During the recent years several comprehensive analysis of the polarized deep inelastic scattering (DIS) data, based on nexttoleadingorder quantum chromodynamics, have appeared in Refs.[529]. In these analysis the polarized parton density functions (PPDFs) are either written in terms of the wellknown parameterizations of the unpolarized PDFs or parameterized independently, and the unknown parameters are determined by fitting the polarized DIS data.
Determination of parton distributions in a nucleon in the framework of quantum chromodynamics (QCD) always involves some modeldependent procedure. Instead of relying on mathematical simplicity as a guide, we take a viewpoint in which the physical picture of the nucleon structure is emphasized. That is, we consider the model for the nucleon which is compatible with the description of the bound state problem in terms of three constituent quarks. We adopt the view that these constituent quarks in the scattering problems should be regarded as the valence quark clusters rather than pointlike objects. They have been referred to as . In the valon model, the proton consists of two “up” and one “down” valons. These valons thus, carry the quantum numbers of the respective valence quarks. Hwa and et al. [3038] found evidence for the valons in the deep inelastic neutrino scattering data, suggested their existence and applied it to a variety of phenomena. Hwa [39] has also successfully formulated a treatment of the low reactions based on a structural analysis of the valons. In [40] Hwa and Yang refined the idea of the valon model and extracted new results for the valon distributions. In [41, 42, 43] unpolarized PDFs and hadronic structure functions in the NLO approximation were extracted.
In Ref. [25] the polarized valon model is applied to determine the quark helicity distributions and polarized proton structure functions in the NLO approximation by using the Bernstein polynomial approach.
The extraction of the quark helicity distributions is one of the main tasks of the semiinclusive deep inelastic scattering (SIDIS) experiments (HERMES [44], COMPASS [45], SMC[46]) with the polarized beam and target. Recently in Ref. [28] the polarized valon model was applied and analyzed the flavorbroken light sea quark helicity distributions with the help of a Pauliblocking ansatz. The results reported of this paper is good agreement with the HERMES experimental data for the quark helicity distributions in the nucleon for up, down, and strange quarks from semiinclusive deepinelastic scattering [44]. The reported results of Ref. [44] is based on the Bernstein polynomial expansion method in the polarized valon model framework.
Since very recently experimental data are available from the HERMES
collaboration [47] for the spin structure function
, therefore there is enough motivation to study and utilize the
spin structure and quark helicity distributions extracted
via the phenomenological model. Since these recently experimental data
are just for different value of and not fixed , one can not use the
the Bernstein polynomial expansion method. Because in this method we need to use
experimental data for each bin of separately. So it seems suitable, to use the Jacobi
polynomial expansion method.
In this paper we use the idea of the polarized valon model to obtain
the PPDFs in the LO and NLO approximations. The results of the present analysis is based on
the Jacobi polynomials expansion of the polarized structure function.
The plan of the paper is to give an introduction to the polarized
valon distributions in Section 2. Parametrization of parton
densities are written down in this section. In Section 3 we
present a brief review of the theoretical background of the QCD
analysis in two loops. The method of the QCD analysis of polarized
structure function, based on Jacobi polynomials are written down
in Section 4. A description of the procedure of the QCD fit of
data are illustrated in Section 5. Section 6 contains a
results and conclusions of the QCD analysis.
2 Polarized valon distributions
The idea of nucleon as a bound state of three quarks was presented for the first time in Ref. [48]. On the other hand the similar idea, which is called valon model was presented in Ref. [37]. According to the valon model framework, one can assume a simple form for the exclusive valon distribution which facilitate the phenomenological analysis as follows
(1) 
where is the momentum fraction of the ’th valon. The and type inclusive valon distributions can be obtained by double integration over the unspecified variables
The normalization parameter has been fixed by
(4) 
and is equal to , where
is the EulerBeta function. R.C. Hwa and C. B. Yang
[40] have recalculated the unpolarized valon
distribution in the proton with a new set of parameters. The new
values of , are found to be and
.
To describe the quark distribution in the valon model, one can try to relate the polarized quark distribution functions or to the corresponding valon distributions and . The polarized valon can still have the valence and sea quarks that are polarized in various directions, so long as the net polarization is that of the valon. When we have only one distribution to analyze, it is sensible to use the convolution in the valon model to describe the proton structure in terms of the valons. In the case that we have two quantities, unpolarized and polarized distributions, there is a choice of which linear combination exhibits more physical contents. Therefore, in our calculations we assume a linear combination of and to determine respectively the unpolarized () and polarized () valon distributions .
According to unpolarized and polarized valon model framework we have [25]
(5) 
and
(6) 
where and and also and . As we can similarly see for the unpolarized case, the polarized quark distribution can be related to a polarized valon distribution in Eq. (6).
The polarized parton distribution of the polarized constituent quarks in the Eq. (6) is dependent on a probe [25]. It is assumed in more detail that the polarized constituent quarks distribution, that is, in the scale of is equal to . From Eq. (6) now is obvious that the polarized quark distribution in the scale of is equal to polarized valon distribution.
Here we want to use the polarized valon distributions which introduced in [25]. According to improve polarized valon picture the nonsinglet polarized valon distribution functions are as following
(7) 
where
(8) 
the subscript refers to and valons. The motivation for choosing this functional form is that the term controls the low behavior valon densities, and terms the large values. The remaining polynomial factor accounts for the additional medium values. The normalization constants for and valons are as following
(9)  
and
(10)  
These quantities are chosen such that the are the first moments of , . Here is the Euler Beta–function being related to the –function.
In the present approach the QCD–evolution equations are solved in
Mellin– space as will describe in section 3. The Mellin–transform of the valon densities is performed and Mellin–
moments are calculated for complex arguments by
.
As seen from Eq. (7) there are five parameters for each valon distribution. To meet both the quality of the present data and the reliability of the fitting program, the number of parameters has to be reduced. Assuming flavor symmetry and a flavor symmetric sea one only has to derive one general polarized sea–quark distribution. The first moments of the polarized valon distributions and can be fixed by the parameters and as measured in neutron and hyperon –decays according to the relations :
(11) 
The factor 2 in Eq. (11) is due to the existence of two type valons. A reevaluation of and was performed in Ref. [24] on the basis of updated –decay constants [49] leading to
(12) 
This choice reduces the number of parameters to be fitted for each
nonsinglet valon density to four.
On the other hand the singlet polarized valon distribution which is defined in [25] is as following
3 The theoretical background of the QCD analysis
Let us define the Mellin moments for the polarized structure function :
(15) 
In the QCDimproved quark parton model (QPM), i.e., at leading twist, and to leading logarithmic order in the running strong coupling constant of QuantumChromodynamics (LO QCD), the deepinelastic scattering off the nucleon can be interpreted as the incoherent superposition of virtualphoton interactions with quarks of any flavor . By angular momentum conservation, a spin parton can absorb a hard photon only when their spin orientations are opposite. The spin structure function has then a probabilistic interpretation, which for the proton reads [50]
(16)  
Here, the quantity is the squared fourmomentum transferred by the virtual photon, is the order of moments, is the charge, in units of the elementary charge , of quarks of flavor , is the average squared charge of the active quark flavors, and is the quark helicity distribution for quarks of flavor . Correspondingly, is antiquark helicity distributions. Moreover the flavor singlet and flavor nonsinglet quark helicity distributions are defined as
(17)  
and
(18) 
For the analysis presented in this paper, only the three lightest
quark flavors, , are taken into account and the number of
active quark flavors is equal to three.
The twist–2 contributions to the structure function can be represented in terms of the polarized parton densities and the coefficient functions in the Mellin N space by [2]
(19) 
in this equation the NLO running coupling constant is given by
(20) 
where and . In the above equations, we choose as a fixed
parameter and is an unknown parameter which can be
obtained by fitting
to experimental data.
In Eq. (19), , and are moments of the polarized parton distributions in a proton. , are also the th moments of spindependent Wilson coefficients given by
(21) 
and
(22) 
with defined as in Ref. [2].
According to improved polarized valon model framework, determination of the moments of parton distributions in a proton can be done strictly through the moments of the polarized valon distributions.
The moments of PPDFs are denoted respectively by: , , and . Therefore, the moments of the polarized and valence quark in a proton are convolutions of two moments:
(23) 
(24) 
In the above equation is the moment of distribution, i.e. .
The moment of the polarized singlet distribution () is as follows:
(25) 
here is the moment of distribution, i.e. . For the gluon distribution we have
(26) 
where in
Eqs. (23,24), in
Eq. (25) and also which is
the quarktogluon evolution
function, are given in Ref. [25]
It is obvious
that the final form for involves the total of 15
unknown parameters. If the parameters can be obtained then the
computation of all moments of the PPDFs and the polarized
structure
function (PSF), , are possible.
4 The method of the QCD analysis of PSF
The evolution equations allow one to calculate the dependence of the PPD’s provided at a certain reference point . These distributions are usually parameterized on the basis of plausible theoretical assumptions concerning their behavior near the end points .
In the phenomenological investigations of the polarized and unpolarized structure functions, for example or for a given value of , only a limited number of experimental points, covering a partial range of values of , are available. Therefore, one cannot directly determine the moments. A method devised to deal with this situation is to take averages of the structure function weighted by suitable polynomials.
The evolution equation can be solved and QCD predictions for PSF obtained with the help of various methods. For example we can use the Bernstein polynomial to determine PPD’s in the NLO approximation to obtain some unknown parameters to parameterize PPD’s at . In this way, we can compare theoretical predictions with the experimental results for the Bernstein averages just in Mellin space. To obtain these experimental averages from the E143 and SMC data [51, 52], we need to fit for each bin in separately [25].
If we want to take into account very recent experimental data [47] for all range value of to determine PPDFs, it is convenience to apply Jacobi polynomials expansion and not the Bernstein one. The advantage of application for all data points, especially very recently HERMES experimental data [47], and not just a series of data points, is our motivation to use Jacobi polynomials and not Bernstein polynomial to study spin dependent of parton distribution function.
One of the simplest and fastest possibilities in the PSF reconstruction from the QCD predictions for its Mellin moments is Jacobi polynomials expansion. The Jacobi polynomials are especially suited for this purpose since they allow one to factor out an essential part of the dependence of the SF into the weight function [53]. Thus, given the Jacobi moments , a structure function may be reconstructed in a form of the series [54][58]
(27) 
where is the number of polynomials and are the Jacobi polynomials of order ,
(28) 
where are the coefficients that expressed through functions and satisfy the orthogonality relation with the weight as following
(29) 
For the moments, we note that the dependence is entirely contained in the Jacobi moments
(30)  
obtained by inverting Eq. 27, using Eqs. (28,
29) and also definition of moments,
.
Using Eqs. (2730) now, one can relate the PSF with
its Mellin moments
(31) 
where are the moments determined by Eqs.(16,19). , and have to be chosen so as to achieve the fastest convergence of the series on the R.H.S. of Eq. (31) and to reconstruct with the required accuracy. In our analysis we use , and . The same method has been applied to calculate the nonsinglet structure function from their moments [5963].
Obviously the dependence of the polarized structure function is defined by the dependence of the moments.
5 The procedure of the QCD fit of data
The remarkable growth of experimental data on inclusive polarized
deep inelastic scattering of leptons off nucleons over the last
years allows to perform refined QCD analyzes of polarized
structure functions in order to reveal the spin–dependent
partonic structure of the nucleon. For the QCD analysis presented
in the present paper the following data sets are used: the HERMES
proton data [64, 47], the SMC
proton data [65], the E143 proton data
[51], the EMC proton data
[66, 67]. The number of the published
data points above for the different data sets
are summarized in Table 1 for data on together with the
and –ranges for different experiments.
Experiment  –range 

number of data points  Ref.  

E143(p)  0.031 – 0.749  1.27 – 9.52  28  [51]  
E143(p)  0.031 – 0.749  2, 3, 5 (Fixed)  84  [51]  
HERMES(p)  0.028 – 0.660  1.01 – 7.36  19  [64]  
HERMES(p)  0.023 – 0.660  2.5 (Fixed)  20  [64]  
HERMES(p)  0.026 – 0.731  1.1214.29  62  [47]  
SMC(p)  0.005 – 0.480  1.30 – 58.0  12  [65]  
SMC(p) 
0.005 – 0.480  10 (Fixed)  12  [65]  
EMC(p)  0.015 – 0.466  3.50 – 29.5  10  [66]  
EMC(p)  0.015 – 0.466  3.50 – 29.5  10  [67]  
proton  257 
Table 1: Published data points above GeV.
In the fitting procedure we started with the 15 parameters selected, i.e. 4 parameters for each nonsinglet polarized valon distribution, 6 parameters for singlet polarized valon distribution and to be determined. For this set of parameters the sea–quark distribution was assumed to be described according to flavor symmetry.
In the further procedure we fixed 2 parameters at their values obtained in the first minimization and chose the first moment of polarized Nonsinglet valon distributions in the polarized case [25]. The lack of constraining power of the present data on the polarized parton densities has to be stressed, however. Since only more precise data can improve the situation, so we add the recent experimental data achieved from HERMES group[47].
The final minimization was carried out under the above conditions and determined the remaining 15 parameters. The values and errors of these parameters along with those parameters fixed in the parametrization, Sec. 2, are summarized in LO and NLO in table 2. The results on are discussed separately in section 6. The starting scale of the evolution was chosen as GeV.
Using the CERN subroutine MINUIT [68], we defined a global for all the experimental data points and found an acceptable fit with minimum in the LO case and in the NLO case. In this table we compare the results reported in [25] which are based on Bernstein approach and the results of the present analysis. Also we obtain the uncertainties of the parameters in Jacobi Approach which are not calculated in Bernstein approach. We should notice that the Jacobi and Bernstein polynomials are merely used as a tool in the fitting procedure and the results are independent of it.
Bernstein Approach [25]  Jacobi Approach  
LO  NLO  LO  NLO  
value  value  value  value  
MeV  203  235  201110  24558 
0.0020  0.0038  0.0015(fixed)  0.0018(fixed)  
2.3789  2.1501  2.44890.024  2.35900.028  
1.7518  0.8859  1.90500.196  0.9683 0.206  
11.0804  10.6537  13.54200.378  22.6095 0.418  
1.4629  0.1548  1.83390.120  1.9778 0.150  
0.005  0.0046  0.0029(fixed)  0.0026(fixed)  
1.5465  1.5859  1.60590.029  1.66380.034  
1.8776  1.5835  2.20090.353  1.6214 0.453  
8.5042  9.6205  10.23710.829  13.1966 0.942  
0.8608  0.8410  0.87510.091  0.84830.142  
0.0004  0.0025  0.0003 10  0.005210  
0.2954  3.1148  0.26110.191  3.00340.241  
6.9134  15.8114  6.54290.174  14.8696 0.214  
30.9851  21.1500  29.69990.475  18.8057 0.051  
39.7383  10.5025  37.93790.202  8.4598 0.298  
16.4605  0.9162  15.58790.041  0.34100.050  
154.98/123=1.260  115.62/123=0.940  236.577/242=0.978  225.920/242=0.933 
Table 2: Parameter values in LO and NLO of the parameter fit for Bernstein and Jacobi approaches.
6 Results and conclusions of the QCD analysis
We have performed a QCD analysis of the inclusive polarized deep–inelastic charged lepton–nucleon scattering data to next–to–leading order and derived parameterizations of polarized valon distributions at a starting scale together with the QCD–scale in the polarized valon model framework.
The analysis was performed using the Jacobi polynomials–method to determine the parameters of the problem in a fit to the data.
A new aspect in comparison with previous analyzes is that we determine the parton densities and the QCD scale in leading and next–to–leading order by using Jacobi polynomial expansion method.
Detailed comparisons were performed to the results obtained in other recent parameterizations [24, 69, 70, 71, 72]. The previous results are widely compatible with the present parameterizations. These distributions can be used in the numerical calculations for polarized high–energy scattering processes at hadron– and –colliders.
Looking at the dependence of the structure function
in intervals of gives insight to the scaling
violations in the spin sector. As in the unpolarized case the
presence of scaling violations are expected to manifest in a slope
changing with . The proton data on have been
plotted in such a way in Figure 1 and confronted with the QCD NLO
curves of the present analysis. Corresponding curves of the
parameterizations [24, 69, 70, 71, 72] are also shown.
Slight but nonsignificant differences between the different
analyzes are observed in the intervals at low values of .
However, the data are well covered within the errors by all
analyzes.
In Figures 2–3 the fitted parton distribution functions, in leading and nexttoleading order for all sets of parameterizations [24, 69, 70, 71] and their errors are presented at the starting scale .
The polarized structure function measured in the interval GeV GeV, Figure 4, using the world asymmetry data is well described by our QCD NLO curve. We also compare to corresponding representations of the parameterizations [71, 70, 69, 24], which are compatible within the present results.
In Figures 5–8 the scaling violations of the individual polarized momentum densities are depicted in the range choosing the NLO distributions. The upvalence distribution , Figure 5, evolves towards smaller values of and the peak around becomes more flat in the evolution from to . Statistically this distribution is constrained best among all others. The downvalence distribution , Figure 6, remains negative in the same range, although it is less constraint by the present data than the upvalence density. Also here the evolution is towards smaller values of and structures at larger flatten out.
The momentum density of the polarized singlet quark , Figure 7, is positive in the kinematic range shown for all and for , but changes sign for lower values of . The maximum of the distribution at around moves to at . At the same time a minimum around moves to .
The momentum density of the polarized gluon , Figure 8, is positive in the kinematic range shown for all . Also in this case the evolution moves the shape towards lower values of and flattens the distribution.
By having polarized parton distributions, the first moments of the polarized parton distributions can be obtain. The first moments of the polarized parton densities in NLO in the scheme at for different sets of recent parton parameterizations are presented in table 3.
Distribution  Model  Ref. [25]  BB [69]  GRSV [70]  AAC [24] 

0.926  0.9206  0.9278  
–0.3391  –0.3313  –0.341  –0.3409  –0.3416  
0.851  0.8593  0.8399  
–0.4207  –0.3937  –0.415  –0.4043  –0.4295  
–0.0817  –0.0624  0.074  –0.0625  –0.0879  
0.6828  0.8076 
Table 3: Comparison
of the first moments of the polarized parton densities in NLO in
the scheme at for different
sets of recent parton parameterizations. The second column (Model)
contains the first moments which is obtained from the valon model
and the Jacobi polynomials expansion method.
We can also obtain the first moment of (the EllisJaffe sum rule) by
(32) 
The results have also been given in table 4. The first moments of polarized parton distributions also shown for some value of .
1  0.9260  0.3410  0.0965  0.0814  0.5850  0.1161 

3  0.9215  0.3393  0.0923  0.0816  0.8651  0.1195 
5  0.9202  0.3389  0.0911  0.0817  0.9837  0.1205 
10  0.9189  0.3384  0.0898  0.0818  1.1383  0.1215 
Table 4: The first moments of polarized parton
distributions, , , ,
, and in the NLO
approximation for some value of .
In the framework
of QCD the spin of the proton can be expressed in terms of the
first moment of the total quark and gluon helicity distributions
and their orbital angular momentum, i.e.
(33) 
where is the total orbital
angular momentum of all the quarks and gluons.
The contribution of addition of and
for typical value of
GeV is around 0.978 in our analysis.
We can also compare this value in NLO with other recent analysis.
The reported value from BB model [69] is 1.096,
AAC model [24] is 0.837 and
also GRSV model [70] is 0.785.
In the QCD analysis we parameterized the strong coupling constant in terms of four massless flavors determining . The LO and NLO results fitting the data, are
(34) 
These results can be expressed in terms of :
(35) 
These values can be compared with results from other QCD analyzes of polarized inclusive deep–inelastic scattering data
(36) 
and with the value of the current world average
(37) 
We hope our results of QCD analysis of structure functions in terms of Jacobi polynomials could be able to describe more complicated hadron structure functions. We also hope to be able to consider the symmetry breaking of polarized sea quarks by using the polarized structure function expansion in the Jacobi polynomials.
7 Acknowledgments
We are especially grateful to G. Altarelli for fruitful suggestions and critical remarks. A.N.K. is grateful to A. L. Kataev for useful discussions and remarks during the visit to ICTP. I am grateful to the staff of this center for providing excellent conditions for work. A.N.K is grateful to CERN for their hospitality whilst he visited there and could amend this paper. We would like to thank M. Ghominejad and Z. Karamloo for reading the manuscript of this paper. We acknowledge the Institute for Studies in Theoretical Physics and Mathematics (IPM) for financially supporting this project. S.A.T. thanks Persian Gulf university for partial financial support of this project.
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