高能轻子核子深度非弹性散射过程末态粒子的方位角不对称
|
摘要
核子结构是现代物理学的重要课题。在理论和实验两方面推动下,核子结构的研究在近三十年内取得了长足进步,已经从领头扭度部分子分布函数推广到高扭度部分子关联函数(其贡献是Λ/Q幂次压低的),从横动量积分的一维部分子分布/关联函数q(x)推广到横动量依赖的(TMD)三维部分子分布/关联函数q(x,k→)。高能轻子核子深度非弹性散射(DIS)过程是核子结构研究的重要手段,而观测量包含末态喷注(强子)的半单举轻子核子深度非弹性散射(SIDIS)过程能够探测核子结构的更多信息。半单举深度非弹过程末态粒子方位角分布不对称是一类重要的观测量,是国际上如HERMES、COMPASS、JLab等定位于核子结构研究的大型实验合作组的重要观测量之一。在中低横动量区域,高扭度对末态粒子方位角不对称的贡献不可忽略,利用系统的方法处理高扭度贡献对于核子结构研究和QCD理论都具有重要的意义。
计算高扭度对反应截面贡献的标准方法是由Ellis、Furmanski、Petronzio、邱建伟和Sterman等发展的共线展开技术。共线展开技术的基本步骤是：首先,将部分子硬散射过程对部分子动量在共线动量方向k=xp附近作泰勒展开,同时将胶子场分解为共线分量和非共线分量；其次,利用Ward恒等式联系不同阶部分子散射矩阵；再次,将全部项加和、整理,得到规范不变的部分子关联矩阵；最后,把部分子关联矩阵用7-矩阵展开,得到强子张量和截面,并示用QCD运动方程将所有矩阵元约化到一个完备集。利用上述共线展开技术,人们得到了扭度-4部分子关联函数对单举轻子核子深度非弹过程的贡献。我们组梁作堂和王新年将上述共线展开技术推广到半单举轻子核子深度非弹过程eN→eqX中。他们发现,如果考虑末态喷注产生这种相对于强子产生更为简单的情况,半单举深度非弹(SIDIS)过程和单举深度非弹(DIS)过程的差别只是运动学因子δ2(k→'-k→),这个运动学因子不影响共线展开技术的应用,但是它会将分离出来的分布/关联函数变成横动量依赖的。以此发现为基础,他们得到轻子不极化、核子横向极化情况下扭度-3层次上eN→eqX过程的截面和方位角不对称。
本论文的第一个工作,就是在我们组工作的基础上,计算轻子、核子各种极化组合下eN→eqX过程反应截面和方位角不对称的形式,进而将共线展开技术推广到扭度-4层次上非极化半单举轻子核子深度非弹过程eN→eqX中,得到截面和方位角不对称的形式。结果表明,方位角不对称正比于高扭度部分子关联函数与领头扭度部分子分布函数的比,两者都是规范不变的物理量。末态喷注的方位角不对称和的测量是研究核子高扭度部分子关联函数的有效手段。
研究方位角不对称的核依赖,是本论文的另一个工作。我们组提出的一个描写核环境对部分子分布函数影响的模型指出,产生规范链的胶子线可以连接到原子核内部的不同核子上,使得规范链中包含了原子核内部核子分布的信息。在双胶子关联近似下,这个模型会导致横动量分布出现简单的高斯展宽,但进一步的分析却表明这个模型也会导致方位角不对称的核抑制效应。本文在此工作的基础上,利用这个模型去研究更高阶的方位角不对称。结果表明,方位角不对称(cos2Φ>存在更强的核抑制效应。最终的物理结论是：原子核对部分子横动量分布的影响是让它更胖(横动量分布的展宽效应)更圆(方位角不对称的核抑制效应)。
初步建立从现有实验数据抽取高扭度部分子关联函数参数化形式的方法,是本论文的第三个工作。高扭度部分子关联函数和领头扭度部分子分布函数一样,都是非微扰QCD动力学决定的物理量,人类目前尚无良方去从第一原理出发给出它们的解析形式,而只能从实验数据中抽取它们的参数化形式。本论文总结了当前半单举轻子核子深度非弹过程末态粒子方位角不对称的实验数据,以及目前人们处理这个过程高扭度对方位角不对称贡献的做法,强调高扭度关联函数的参数化对于Boer-Mulders函数的抽取也具有重要意义。论文只考虑高扭度部分子关联函数造成的影响,在末态简单的放入领头扭度碎裂函数。作者独立编写了以微分演化算法为核心的Fortran拟合程序,并根据微分演化算法的特性给出了快速确定参数不确定范围的方法。论文利用ZEUS组和EMC组的部分数据,对高扭度关联函数的参数化形式做了初步拟合,结果具有一定的参考价值。
Nucleon structure is an important topic of modern physics. With the boost from theory and experiments, for the last 30 years the research on nucleon structure have made great progress, evolving from the leading twist parton distribution functions to higher twist parton correlation functions (givingΛ/Q power suppressed contributions), from transverse-momentum-integrated one dimensional parton distribution/correlation functions q(x) to transverse-momentum-dependent three dimensional parton distribu-tion/correlation functions q(x, k⊥). High energy deep inelastic lepton-nucleon scatter-ing(DIS) is an important tool of research on nucleon structure, and the semi-inclusive DIS process with one final state jet(hadron) observed probe more information on nu-cleon structure. The azimuthal asymmetries of final state particles in SIDIS process are one important kind of observables, and measurements of azimuthal asymmetries are important goals of large international experimental collaborations such as HERMES, COMPASS, JLab. In intermediate and small transverse momentum region, higher twist contributions to azimuthal asymmetries are not negligible, and it is important to calcu-late higher twist contributions with systematic methods.
The standard method of calculating higher twist contributions is collnear expan-sion technique developed by Ellis, Furmanski, Petronzio, Jian-Wei Qiu and Sterman et.al. The procedures of collinear expansion technique are as follows:first, perform a Taylor expansion of the parton hard scattering process over parton momentum near their collinear to the nucleon momentum direction k=xp, in the mean time decom-pose the gluon field into a collinear component and non-collinear components; second, employ Ward identities to relate parton scattering matrices of different orders to each other; third, add all terms together and recombine them, then we obtain many gauge invariant parton correlation matrices; finally, expand parton correlation matrices withγ-matrices, take trace and simplify the final results with QCD equation of motion. With above collinear expansion technique they obtained the twist-4 contributions to deep inelastic scattering process. Members of our group, Zuo-tang Liang and Xin-Nian Wang, extended the previous collinear expansion technique to semi-inclusive DIS process eN→eqX. They found that, if the jet production process which is simpler with respect to hadron production is considered, the only difference between semi-inclusive DIS and inclusive DIS is that the former contains an extra kinematic factorδ2(k⊥'-k⊥), which will not affect the application of the collinear expansion technique, while this kinematic factor will render the extracted parton distribution/correlation func-tions transverse-momentum-dependent. With this discovery, they obtained twist-3 con-tributions to the cross section and (cos 0) azimuthal asymmetry with unpolarized lepton and transversely polarized nucleon.
The first work of current thesis is to calculate the cross section and azimuthal asym-metries of the process eN eN→eqX with lepton and nucleon taking any polarization combinations, and furthur extend collinear expansion to unpolarized semi-inclusive DIS process eN→eqX at twist-4 level, obtaining the cross section and (cos 20) azimuthal asymmetry. The result indicates that (cos 20) azimuthal asymmetry is propotional to the ratio of higher twist parton correaltion functions to leading twist parton distribu-tion functions, both of which are gauge invariant physical quantities. Measurement of the azimuthal asymmetries (cos 0) and (cos 20) of jet production is an effective way of studying higher twist parton correlation functions.
Studing the nuclear dependence of azimuthal asymmetries is another work of cur-rent thesis. In recent years our group members proposed a model which describes the effects of nuclear environment on parton distribution functions. The model states that gluon lines that produce gauge link can connect to different nucleons in a large nucleus, thus the gauge link contains the information of nucleon distribution in a large nucleus. Under "Maximal Two-Gluon Correlation Apprximation" they find a simple gaussian kT broadening, while furthur analysis showed that it generate a clear nuclear suppression effect on (cos 0) azimuthal asymmetry. Current thesis also employ this model to study even higher order azimuthal asymmetries, and find that (cos 20) azimuthal asymmetry show a stronger nuclear suppression effect. The final physical conclusion is:nuclear environment have such an effect on the parton transverse momentum distribution that it makes it more fat (transverse momentum broadening effect) and more round (suppres-sion effect of azimuthal asymmetries).
Developing the method to extract the parameterization form of higher twist parton correlation functions from experimental data, is the third work of current thesis. Higher twist parton correlation functions, like leading twist parton distribution functions, are physical quantities decided by non-perturbative QCD dynamics, whose analytical forms are not currently obtainable from ab initio calculations. The only way to them is to extract their parameterization from experimental data. Current thesis review the exper-iments on azimuthal asymmetries in SIDIS and how the theorist deal with higher twist contributions to azimuthal asymmetries, and stress that parameterization of higher twist correlation functions is also very important to the extraction of Boer-Mulders functions. The thesis just take contributions from higher twist parton correaltion functions into account, and convulute them with leading twist fragmentation functions. The author independently developed Fortran code employing differential evolution algorithm, and give a fast way of obtaining uncertainties of parameters according to the charactoristics of differential evolution. With data from ZEUS and EMC collaborations, current the-sis perform an preliminary fit to the parameterization form of higher twist correlation functions. The results are of some reference value.
计算高扭度对反应截面贡献的标准方法是由Ellis、Furmanski、Petronzio、邱建伟和Sterman等发展的共线展开技术。共线展开技术的基本步骤是：首先,将部分子硬散射过程对部分子动量在共线动量方向k=xp附近作泰勒展开,同时将胶子场分解为共线分量和非共线分量；其次,利用Ward恒等式联系不同阶部分子散射矩阵；再次,将全部项加和、整理,得到规范不变的部分子关联矩阵；最后,把部分子关联矩阵用7-矩阵展开,得到强子张量和截面,并示用QCD运动方程将所有矩阵元约化到一个完备集。利用上述共线展开技术,人们得到了扭度-4部分子关联函数对单举轻子核子深度非弹过程的贡献。我们组梁作堂和王新年将上述共线展开技术推广到半单举轻子核子深度非弹过程eN→eqX中。他们发现,如果考虑末态喷注产生这种相对于强子产生更为简单的情况,半单举深度非弹(SIDIS)过程和单举深度非弹(DIS)过程的差别只是运动学因子δ2(k→'-k→),这个运动学因子不影响共线展开技术的应用,但是它会将分离出来的分布/关联函数变成横动量依赖的。以此发现为基础,他们得到轻子不极化、核子横向极化情况下扭度-3层次上eN→eqX过程的截面和方位角不对称
本论文的第一个工作,就是在我们组工作的基础上,计算轻子、核子各种极化组合下eN→eqX过程反应截面和方位角不对称的形式,进而将共线展开技术推广到扭度-4层次上非极化半单举轻子核子深度非弹过程eN→eqX中,得到截面和方位角不对称
研究方位角不对称的核依赖,是本论文的另一个工作。我们组提出的一个描写核环境对部分子分布函数影响的模型指出,产生规范链的胶子线可以连接到原子核内部的不同核子上,使得规范链中包含了原子核内部核子分布的信息。在双胶子关联近似下,这个模型会导致横动量分布出现简单的高斯展宽,但进一步的分析却表明这个模型也会导致方位角不对称
初步建立从现有实验数据抽取高扭度部分子关联函数参数化形式的方法,是本论文的第三个工作。高扭度部分子关联函数和领头扭度部分子分布函数一样,都是非微扰QCD动力学决定的物理量,人类目前尚无良方去从第一原理出发给出它们的解析形式,而只能从实验数据中抽取它们的参数化形式。本论文总结了当前半单举轻子核子深度非弹过程末态粒子方位角不对称的实验数据,以及目前人们处理这个过程高扭度对方位角不对称贡献的做法,强调高扭度关联函数的参数化对于Boer-Mulders函数的抽取也具有重要意义。论文只考虑高扭度部分子关联函数造成的影响,在末态简单的放入领头扭度碎裂函数。作者独立编写了以微分演化算法为核心的Fortran拟合程序,并根据微分演化算法的特性给出了快速确定参数不确定范围的方法。论文利用ZEUS组和EMC组的部分数据,对高扭度关联函数的参数化形式做了初步拟合,结果具有一定的参考价值。
Nucleon structure is an important topic of modern physics. With the boost from theory and experiments, for the last 30 years the research on nucleon structure have made great progress, evolving from the leading twist parton distribution functions to higher twist parton correlation functions (givingΛ/Q power suppressed contributions), from transverse-momentum-integrated one dimensional parton distribution/correlation functions q(x) to transverse-momentum-dependent three dimensional parton distribu-tion/correlation functions q(x, k⊥). High energy deep inelastic lepton-nucleon scatter-ing(DIS) is an important tool of research on nucleon structure, and the semi-inclusive DIS process with one final state jet(hadron) observed probe more information on nu-cleon structure. The azimuthal asymmetries of final state particles in SIDIS process are one important kind of observables, and measurements of azimuthal asymmetries are important goals of large international experimental collaborations such as HERMES, COMPASS, JLab. In intermediate and small transverse momentum region, higher twist contributions to azimuthal asymmetries are not negligible, and it is important to calcu-late higher twist contributions with systematic methods.
The standard method of calculating higher twist contributions is collnear expan-sion technique developed by Ellis, Furmanski, Petronzio, Jian-Wei Qiu and Sterman et.al. The procedures of collinear expansion technique are as follows:first, perform a Taylor expansion of the parton hard scattering process over parton momentum near their collinear to the nucleon momentum direction k=xp, in the mean time decom-pose the gluon field into a collinear component and non-collinear components; second, employ Ward identities to relate parton scattering matrices of different orders to each other; third, add all terms together and recombine them, then we obtain many gauge invariant parton correlation matrices; finally, expand parton correlation matrices withγ-matrices, take trace and simplify the final results with QCD equation of motion. With above collinear expansion technique they obtained the twist-4 contributions to deep inelastic scattering process. Members of our group, Zuo-tang Liang and Xin-Nian Wang, extended the previous collinear expansion technique to semi-inclusive DIS process eN→eqX. They found that, if the jet production process which is simpler with respect to hadron production is considered, the only difference between semi-inclusive DIS and inclusive DIS is that the former contains an extra kinematic factorδ2(k⊥'-k⊥), which will not affect the application of the collinear expansion technique, while this kinematic factor will render the extracted parton distribution/correlation func-tions transverse-momentum-dependent. With this discovery, they obtained twist-3 con-tributions to the cross section and (cos 0) azimuthal asymmetry with unpolarized lepton and transversely polarized nucleon.
The first work of current thesis is to calculate the cross section and azimuthal asym-metries of the process eN eN→eqX with lepton and nucleon taking any polarization combinations, and furthur extend collinear expansion to unpolarized semi-inclusive DIS process eN→eqX at twist-4 level, obtaining the cross section and (cos 20) azimuthal asymmetry. The result indicates that (cos 20) azimuthal asymmetry is propotional to the ratio of higher twist parton correaltion functions to leading twist parton distribu-tion functions, both of which are gauge invariant physical quantities. Measurement of the azimuthal asymmetries (cos 0) and (cos 20) of jet production is an effective way of studying higher twist parton correlation functions.
Studing the nuclear dependence of azimuthal asymmetries is another work of cur-rent thesis. In recent years our group members proposed a model which describes the effects of nuclear environment on parton distribution functions. The model states that gluon lines that produce gauge link can connect to different nucleons in a large nucleus, thus the gauge link contains the information of nucleon distribution in a large nucleus. Under "Maximal Two-Gluon Correlation Apprximation" they find a simple gaussian kT broadening, while furthur analysis showed that it generate a clear nuclear suppression effect on (cos 0) azimuthal asymmetry. Current thesis also employ this model to study even higher order azimuthal asymmetries, and find that (cos 20) azimuthal asymmetry show a stronger nuclear suppression effect. The final physical conclusion is:nuclear environment have such an effect on the parton transverse momentum distribution that it makes it more fat (transverse momentum broadening effect) and more round (suppres-sion effect of azimuthal asymmetries).
Developing the method to extract the parameterization form of higher twist parton correlation functions from experimental data, is the third work of current thesis. Higher twist parton correlation functions, like leading twist parton distribution functions, are physical quantities decided by non-perturbative QCD dynamics, whose analytical forms are not currently obtainable from ab initio calculations. The only way to them is to extract their parameterization from experimental data. Current thesis review the exper-iments on azimuthal asymmetries in SIDIS and how the theorist deal with higher twist contributions to azimuthal asymmetries, and stress that parameterization of higher twist correlation functions is also very important to the extraction of Boer-Mulders functions. The thesis just take contributions from higher twist parton correaltion functions into account, and convulute them with leading twist fragmentation functions. The author independently developed Fortran code employing differential evolution algorithm, and give a fast way of obtaining uncertainties of parameters according to the charactoristics of differential evolution. With data from ZEUS and EMC collaborations, current the-sis perform an preliminary fit to the parameterization form of higher twist correlation functions. The results are of some reference value.
引文
[1]F. Halzen and A. Martin, Quarks & Leptons:An introductory course in modern particle physics (John Wiley & Sons, New York, USA,1984).
[2]K. G. Wilson, Phys. Rev.179,1499 (1969).
[3]T. Muta, World Sci. Lect. Notes Phys.57,1 (1998).
[4]E. D. Bloom et al., Phys. Rev. Lett.23,930 (1969).
[5]J. I. Friedman and H. W. Kendall, Ann. Rev. Nucl. Part. Sci.22,203 (1972).
[6]J. D. Bjorken, Phys. Rev.179,1547 (1969).
[7]R. P. Feynman, Phys. Rev. Lett.23,1415 (1969).
[8]J. D. Bjorken and E. A. Paschos, Phys. Rev.185,1975 (1969).
[9]R. P. Feynman, Photon-hadron interactions (Addison-Wesley,1972) reading 1972, 282p.
[10]R. Brock et al. (CTEQ), Rev. Mod. Phys.67,157 (1995).
[11]J. C. Collins and D. E. Soper, Nucl. Phys. B194,445 (1982).
[12]R. K. Ellis, W. Furmanski, and R. Petronzio, Nucl. Phys. B212,29 (1983).
[13]R. K. Ellis, W. Furmanski, and R. Petronzio, Nucl. Phys. B207,1 (1982).
[14]J.-W.Qiu, Phys. Rev. D42,30 (1990).
[15]J.-w. Qiu and G. F. Sterman, Nucl. Phys. B353,105 (1991).
[16]J.-w. Qiu and G. F. Sterman, Nucl. Phys. B353,137 (1991).
[17]M. E. Peskin and D. V. Schroeder, Reading, USA:Addison-Wesley (1995) 842 p.
[18]M. Gell-Mann, Phys. Lett.8,214 (1964).
[19]G.Zweig, (1964),CERN-TH-401.
[20]G.Zweig, (1964), CERN-TH-412.
[21]H. Georgi and H. D. Politzer, Phys. Rev. Lett.40,3 (1978).
[22]R. N. Cahn, Phys. Lett. B78,269 (1978).
[23]R. N. Cahn, Phys. Rev. D40,3107 (1989).
[24]G. Altarelli, R. K. Ellis, and G. Martinelli, Nucl. Phys. B143,521 (1978).
[25]M. Gluck and E. Reya, Nucl. Phys. B145,24 (1978).
[26]E. Leader, A. V. Sidorov, and D. B. Stamenov, Phys. Rev. D75,074027 (2007), arXiv:hep-ph/0612360.
[27]J.-w. Qiu and G. F. Sterman, Phys. Rev. Lett.67,2264 (1991).
[28]J.-w. Qiu and G. F. Sterman, Nucl. Phys. B378,52 (1992).
[29]J.-w. Qiu and G. F. Sterman, Phys. Rev. D59,014004 (1999), arXiv:hep-ph/9806356.
[30]X. Ji, J.-W. Qiu, W. Vogelsang, and F. Yuan, Phys. Lett. B638,178 (2006), arXiv:hep-ph/0604128.
[31]X. Ji, J.-W. Qiu, W. Vogelsang, and F. Yuan, Phys. Rev. Lett.97,082002 (2006), arXiv:hep-ph/0602239.
[32]X. Ji, J.-w. Qiu, W. Vogelsang, and F. Yuan, Phys. Rev. D73,094017 (2006), arXiv:hep-ph/0604023.
[33]J. Zhou, F. Yuan, and Z.-T. Liang, Phys. Rev. D78,114008 (2008), arXiv:0808.3629 [hep-ph].
[34]F. Yuan and J. Zhou, Phys. Lett. B668,216 (2008), arXiv:0806.1932 [hep-ph].
[35]J. Zhou, F. Yuan, and Z.-T. Liang, Phys. Rev. D79,114022 (2009), arXiv:0812.4484 [hep-ph].
[36]J. Zhou, F. Yuan, and Z.-T. Liang, Phys. Lett. B678,264 (2009), arXiv:0901.3601 [hep-ph].
[37]F. Yuan and J. Zhou, Phys. Rev. Lett.103,052001 (2009), arXiv:0903.4680 [hep-ph].
[38]J. Zhou, F. Yuan, and Z.-T. Liang, AIP Conf. Proc.1149,503 (2009).
[39]J. Zhou, F. Yuan, and Z.-T. Liang, Phys. Rev. D81,054008 (2010), arXiv:0909.2238 [hep-ph].
[40]Z.-t. Liang and X.-N. Wang, Phys. Rev. D75,094002 (2007), arXiv:hep-ph/0609225.
[41]Z.-t. Liang, X.-N. Wang, and J. Zhou, Phys. Rev. D77,125010 (2008), arXiv:0801.0434 [hep-ph].
[42]J.-H. Gao, Z.-t. Liang, and X.-N. Wang, Phys. Rev. C81,065211 (2010), arXiv:1001.3146 [hep-ph].
[43]R. L. Jaffe and M. Soldate, Presented at 1981 Conf. on Perturbative QCD, Talla-hasse, Fla., Mar 25-28,1981.
[44]R. L. Jaffe and M. Soldate, Phys. Lett. B105,467 (1981).
[45]R. L. Jaffe and M. Soldate, Phys. Rev. D26,49 (1982).
[46]D. W. Sivers, Phys. Rev. D41,83 (1990).
[47]J. C. Collins, Nucl. Phys. B396,161 (1993), arXiv:hep-ph/9208213.
[48]X.-d. Ji, J.-P. Ma, and F. Yuan, Phys. Lett. B597,299 (2004), arXiv:hep-ph/0405085.
[49]X.-d. Ji, J.-p. Ma, and F. Yuan, Phys. Rev. D71,034005 (2005), arXiv:hep-ph/0404183.
[50]X.-d. Ji, J.-P. Ma, and F. Yuan, JHEP 07,020 (2005), arXiv:hep-ph/0503015.
[51]J. C. Collins, Phys. Lett. B536,43 (2002), arXiv:hep-ph/0204004.
[52]X.-d. Ji and F. Yuan, Phys. Lett. B543,66 (2002), arXiv:hep-ph/0206057.
[53]A. V. Belitsky, X. Ji, and F. Yuan, Nucl. Phys. B656,165 (2003), arXiv:hep-ph/0208038.
[54]J.-H. Gao, Phys. Rev. D82,014018 (2010), arXiv:1005.4305 [hep-ph].
[55]O. Nachtmann, Nucl. Phys. B63,237 (1973).
[56]H. Georgi and H. D. Politzer, Phys. Rev. D14,1829 (1976).
[57]J. J. Aubert et al. (European Muon), Phys. Lett. B130,118 (1983).
[58]M. Arneodo et al. (European Muon), Z. Phys. C34,277 (1987).
[59]M. R. Adams et al. (E665), Phys. Rev. D48,5057 (1993).
[60]J. Breitweg et al. (ZEUS), Phys. Lett. B481,199 (2000), arXiv:hep-ex/0003017.
[61]S. Chekanov et al. (ZEUS), Phys. Lett. B551,226 (2003), arXiv:hep-ex/0210064.
[62]M. Osipenko et al. (CLAS), Phys. Rev. D80,032004 (2009), arXiv:0809.1153 [hep-ex].
[63]A. Bressan (COMPASS), (2009), arXiv:0907.5511 [hep-ex].
[64]H. Avakian (CLAS), TMD2010 (2010).
[65]F. Giordano and R. Lamb (HERMES), PoS DIS2010,106 (2010).
[66]P. J. Mulders and R. D. Tangerman, Nucl. Phys. B461,197 (1996), arXiv:hep-ph/9510301
[67]D. Boer and P. J. Mulders, Phys. Rev. D57,5780 (1998), arXiv:hep-ph/9711485.
[68]K. A. Oganesian, H. R. Avakian, N. Bianchi, and P. Di Nezza, Eur. Phys. J. C5, 681 (1998), arXiv:hep-ph/9709342.
[69]V. Barone, Z. Lu, and B.-Q. Ma, Phys. Lett. B632,277 (2006), arXiv:hep-ph/0512145.
[70]V. Barone, A. Prokudin, and B.-Q. Ma, Phys. Rev. D78,045022 (2008), arXiv:0804.3024 [hep-ph].
[71]B. Zhang, Z. Lu, B.-Q. Ma, and I. Schmidt, Phys. Rev. D78,034035 (2008), arXiv:0807.0503 [hep-ph].
[72]V. Barone, S. Melis, and A. Prokudin, Phys. Rev. D81,114026 (2010), arXiv:0912.5194 [hep-ph].
[73]P. Schweitzer, T. Teckentrup, and A. Metz, Phys. Rev. D81,094019 (2010), arXiv:1003.2190 [hep-ph].
[74]J.-g. Chay, S. D. Ellis, and W. J. Stirling, Phys. Rev. D45,46 (1992).
[75]A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn, and V. F. Weisskopf, Phys. Rev. D9,3471 (1974).
[76]R. L. Jaffe, Phys. Rev. D11,1953 (1975).
[77]F. Yuan, Phys. Lett. B575,45 (2003), arXiv:hep-ph/0308157.
[78]H. Avakian et al., Mod. Phys. Lett. A24,2995 (2009), arXiv:0910.3181 [hep-ph].
[79]H. Avakian, A. V. Efremov, P. Schweitzer, and F. Yuan, Phys. Rev. D81,074035 (2010), arXiv:1001.5467 [hep-ph].
[80]S. J. Brodsky, H.-C. Pauli, and S. S. Pinsky, Phys. Rept.301,299 (1998), arXiv:hep-ph/9705477.
[81]B. Pasquini, S. Cazzaniga, and S. Boffi, Phys. Rev. D78,034025 (2008), arXiv:0806.2298 [hep-ph].
[82]B. Pasquini and F. Yuan, Phys. Rev. D81,114013 (2010), arXiv:1001.5398 [hep-ph].
[83]S. J. Brodsky, D. S. Hwang, and I. Schmidt, Phys. Lett. B530,99 (2002), arXiv:hep-ph/0201296.
[84]S. J. Brodsky, D. S. Hwang, and I. Schmidt, Nucl. Phys. B642,344 (2002), arXiv:hep-ph/0206259.
[85]A. Bacchetta, A. Schaefer, and J.-J. Yang, Phys. Lett. B578,109 (2004), arXiv:hep-ph/0309246.
[86]B. Andersson, G. Gustafson, G. Ingelman, and T. Sjostrand, Phys.Rept.97,31 (1983).
[87]X. Artru, J. Czyzewski, and H. Yabuki, Z. Phys. C73,527 (1997), arXiv:hep-ph/9508239.
[88]P. M. Nadolsky et al., Phys. Rev. D78,013004 (2008), arXiv:0802.0007 [hep-ph]
[89]S. Kretzer, Phys. Rev. D62,054001 (2000), arXiv:hep-ph/0003177.
[90]M. Anselmino et al., Eur. Phys. J. A39,89 (2009), arXiv:0805.2677 [hep-ph].
[91]D. de Florian, R. Sassot, M. Stratmann, and W. Vogelsang, Phys. Rev. D80, 034030 (2009), arXiv:0904.3821 [hep-ph].
[92]J. Pumplin et al., Phys. Rev. D65,014013 (2001), arXiv:hep-ph/0101032.
[93]R. Storn and K. Price, J. of Global Optimization 11,341 (1997).
[94]M. Anselmino et al., Eur. Phys. J. A47,35 (2011), arXiv:1101.4199 [hep-ex].
[1]Liang Z t and Wang X N 2007 Phys. Rev. D75 094002 (Preprint hep-ph/0609225)
[2]Gao J H, Liang Z t and Wang X N 2010 Phys. Rev. C81 065211 (Preprint 1001.3146)
[3]Song Y k, Gao J h, Liang Z t and Wang X N 2011 Phys. Rev. D83 054010 (Preprint 1012.4179)
[4]Cahn R N 1978 Phys. Lett. B78 269
[5]Ellis R K, Furmanski W and Petronzio R 1982 Nucl. Phys. B207 1
[6]Ellis R K, Furmanski W and Petronzio R 1983 Nucl. Phys. B212 29
[7]Qiu J W 1990 Phys. Rev. D42 30-44
[8]Qiu J w and Sterman G F 1991 Nucl. Phys. B353 105-136
[9]Qiu J w and Sterman G F 1991 Nucl. Phys. B353 137-164
[10]Liang Z t, Wang X N and Zhou J 2008 Phys. Rev. D77 125010 (Preprint 0801.0434)
[2]K. G. Wilson, Phys. Rev.179,1499 (1969).
[3]T. Muta, World Sci. Lect. Notes Phys.57,1 (1998).
[4]E. D. Bloom et al., Phys. Rev. Lett.23,930 (1969).
[5]J. I. Friedman and H. W. Kendall, Ann. Rev. Nucl. Part. Sci.22,203 (1972).
[6]J. D. Bjorken, Phys. Rev.179,1547 (1969).
[7]R. P. Feynman, Phys. Rev. Lett.23,1415 (1969).
[8]J. D. Bjorken and E. A. Paschos, Phys. Rev.185,1975 (1969).
[9]R. P. Feynman, Photon-hadron interactions (Addison-Wesley,1972) reading 1972, 282p.
[10]R. Brock et al. (CTEQ), Rev. Mod. Phys.67,157 (1995).
[11]J. C. Collins and D. E. Soper, Nucl. Phys. B194,445 (1982).
[12]R. K. Ellis, W. Furmanski, and R. Petronzio, Nucl. Phys. B212,29 (1983).
[13]R. K. Ellis, W. Furmanski, and R. Petronzio, Nucl. Phys. B207,1 (1982).
[14]J.-W.Qiu, Phys. Rev. D42,30 (1990).
[15]J.-w. Qiu and G. F. Sterman, Nucl. Phys. B353,105 (1991).
[16]J.-w. Qiu and G. F. Sterman, Nucl. Phys. B353,137 (1991).
[17]M. E. Peskin and D. V. Schroeder, Reading, USA:Addison-Wesley (1995) 842 p.
[18]M. Gell-Mann, Phys. Lett.8,214 (1964).
[19]G.Zweig, (1964),CERN-TH-401.
[20]G.Zweig, (1964), CERN-TH-412.
[21]H. Georgi and H. D. Politzer, Phys. Rev. Lett.40,3 (1978).
[22]R. N. Cahn, Phys. Lett. B78,269 (1978).
[23]R. N. Cahn, Phys. Rev. D40,3107 (1989).
[24]G. Altarelli, R. K. Ellis, and G. Martinelli, Nucl. Phys. B143,521 (1978).
[25]M. Gluck and E. Reya, Nucl. Phys. B145,24 (1978).
[26]E. Leader, A. V. Sidorov, and D. B. Stamenov, Phys. Rev. D75,074027 (2007), arXiv:hep-ph/0612360.
[27]J.-w. Qiu and G. F. Sterman, Phys. Rev. Lett.67,2264 (1991).
[28]J.-w. Qiu and G. F. Sterman, Nucl. Phys. B378,52 (1992).
[29]J.-w. Qiu and G. F. Sterman, Phys. Rev. D59,014004 (1999), arXiv:hep-ph/9806356.
[30]X. Ji, J.-W. Qiu, W. Vogelsang, and F. Yuan, Phys. Lett. B638,178 (2006), arXiv:hep-ph/0604128.
[31]X. Ji, J.-W. Qiu, W. Vogelsang, and F. Yuan, Phys. Rev. Lett.97,082002 (2006), arXiv:hep-ph/0602239.
[32]X. Ji, J.-w. Qiu, W. Vogelsang, and F. Yuan, Phys. Rev. D73,094017 (2006), arXiv:hep-ph/0604023.
[33]J. Zhou, F. Yuan, and Z.-T. Liang, Phys. Rev. D78,114008 (2008), arXiv:0808.3629 [hep-ph].
[34]F. Yuan and J. Zhou, Phys. Lett. B668,216 (2008), arXiv:0806.1932 [hep-ph].
[35]J. Zhou, F. Yuan, and Z.-T. Liang, Phys. Rev. D79,114022 (2009), arXiv:0812.4484 [hep-ph].
[36]J. Zhou, F. Yuan, and Z.-T. Liang, Phys. Lett. B678,264 (2009), arXiv:0901.3601 [hep-ph].
[37]F. Yuan and J. Zhou, Phys. Rev. Lett.103,052001 (2009), arXiv:0903.4680 [hep-ph].
[38]J. Zhou, F. Yuan, and Z.-T. Liang, AIP Conf. Proc.1149,503 (2009).
[39]J. Zhou, F. Yuan, and Z.-T. Liang, Phys. Rev. D81,054008 (2010), arXiv:0909.2238 [hep-ph].
[40]Z.-t. Liang and X.-N. Wang, Phys. Rev. D75,094002 (2007), arXiv:hep-ph/0609225.
[41]Z.-t. Liang, X.-N. Wang, and J. Zhou, Phys. Rev. D77,125010 (2008), arXiv:0801.0434 [hep-ph].
[42]J.-H. Gao, Z.-t. Liang, and X.-N. Wang, Phys. Rev. C81,065211 (2010), arXiv:1001.3146 [hep-ph].
[43]R. L. Jaffe and M. Soldate, Presented at 1981 Conf. on Perturbative QCD, Talla-hasse, Fla., Mar 25-28,1981.
[44]R. L. Jaffe and M. Soldate, Phys. Lett. B105,467 (1981).
[45]R. L. Jaffe and M. Soldate, Phys. Rev. D26,49 (1982).
[46]D. W. Sivers, Phys. Rev. D41,83 (1990).
[47]J. C. Collins, Nucl. Phys. B396,161 (1993), arXiv:hep-ph/9208213.
[48]X.-d. Ji, J.-P. Ma, and F. Yuan, Phys. Lett. B597,299 (2004), arXiv:hep-ph/0405085.
[49]X.-d. Ji, J.-p. Ma, and F. Yuan, Phys. Rev. D71,034005 (2005), arXiv:hep-ph/0404183.
[50]X.-d. Ji, J.-P. Ma, and F. Yuan, JHEP 07,020 (2005), arXiv:hep-ph/0503015.
[51]J. C. Collins, Phys. Lett. B536,43 (2002), arXiv:hep-ph/0204004.
[52]X.-d. Ji and F. Yuan, Phys. Lett. B543,66 (2002), arXiv:hep-ph/0206057.
[53]A. V. Belitsky, X. Ji, and F. Yuan, Nucl. Phys. B656,165 (2003), arXiv:hep-ph/0208038.
[54]J.-H. Gao, Phys. Rev. D82,014018 (2010), arXiv:1005.4305 [hep-ph].
[55]O. Nachtmann, Nucl. Phys. B63,237 (1973).
[56]H. Georgi and H. D. Politzer, Phys. Rev. D14,1829 (1976).
[57]J. J. Aubert et al. (European Muon), Phys. Lett. B130,118 (1983).
[58]M. Arneodo et al. (European Muon), Z. Phys. C34,277 (1987).
[59]M. R. Adams et al. (E665), Phys. Rev. D48,5057 (1993).
[60]J. Breitweg et al. (ZEUS), Phys. Lett. B481,199 (2000), arXiv:hep-ex/0003017.
[61]S. Chekanov et al. (ZEUS), Phys. Lett. B551,226 (2003), arXiv:hep-ex/0210064.
[62]M. Osipenko et al. (CLAS), Phys. Rev. D80,032004 (2009), arXiv:0809.1153 [hep-ex].
[63]A. Bressan (COMPASS), (2009), arXiv:0907.5511 [hep-ex].
[64]H. Avakian (CLAS), TMD2010 (2010).
[65]F. Giordano and R. Lamb (HERMES), PoS DIS2010,106 (2010).
[66]P. J. Mulders and R. D. Tangerman, Nucl. Phys. B461,197 (1996), arXiv:hep-ph/9510301
[67]D. Boer and P. J. Mulders, Phys. Rev. D57,5780 (1998), arXiv:hep-ph/9711485.
[68]K. A. Oganesian, H. R. Avakian, N. Bianchi, and P. Di Nezza, Eur. Phys. J. C5, 681 (1998), arXiv:hep-ph/9709342.
[69]V. Barone, Z. Lu, and B.-Q. Ma, Phys. Lett. B632,277 (2006), arXiv:hep-ph/0512145.
[70]V. Barone, A. Prokudin, and B.-Q. Ma, Phys. Rev. D78,045022 (2008), arXiv:0804.3024 [hep-ph].
[71]B. Zhang, Z. Lu, B.-Q. Ma, and I. Schmidt, Phys. Rev. D78,034035 (2008), arXiv:0807.0503 [hep-ph].
[72]V. Barone, S. Melis, and A. Prokudin, Phys. Rev. D81,114026 (2010), arXiv:0912.5194 [hep-ph].
[73]P. Schweitzer, T. Teckentrup, and A. Metz, Phys. Rev. D81,094019 (2010), arXiv:1003.2190 [hep-ph].
[74]J.-g. Chay, S. D. Ellis, and W. J. Stirling, Phys. Rev. D45,46 (1992).
[75]A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn, and V. F. Weisskopf, Phys. Rev. D9,3471 (1974).
[76]R. L. Jaffe, Phys. Rev. D11,1953 (1975).
[77]F. Yuan, Phys. Lett. B575,45 (2003), arXiv:hep-ph/0308157.
[78]H. Avakian et al., Mod. Phys. Lett. A24,2995 (2009), arXiv:0910.3181 [hep-ph].
[79]H. Avakian, A. V. Efremov, P. Schweitzer, and F. Yuan, Phys. Rev. D81,074035 (2010), arXiv:1001.5467 [hep-ph].
[80]S. J. Brodsky, H.-C. Pauli, and S. S. Pinsky, Phys. Rept.301,299 (1998), arXiv:hep-ph/9705477.
[81]B. Pasquini, S. Cazzaniga, and S. Boffi, Phys. Rev. D78,034025 (2008), arXiv:0806.2298 [hep-ph].
[82]B. Pasquini and F. Yuan, Phys. Rev. D81,114013 (2010), arXiv:1001.5398 [hep-ph].
[83]S. J. Brodsky, D. S. Hwang, and I. Schmidt, Phys. Lett. B530,99 (2002), arXiv:hep-ph/0201296.
[84]S. J. Brodsky, D. S. Hwang, and I. Schmidt, Nucl. Phys. B642,344 (2002), arXiv:hep-ph/0206259.
[85]A. Bacchetta, A. Schaefer, and J.-J. Yang, Phys. Lett. B578,109 (2004), arXiv:hep-ph/0309246.
[86]B. Andersson, G. Gustafson, G. Ingelman, and T. Sjostrand, Phys.Rept.97,31 (1983).
[87]X. Artru, J. Czyzewski, and H. Yabuki, Z. Phys. C73,527 (1997), arXiv:hep-ph/9508239.
[88]P. M. Nadolsky et al., Phys. Rev. D78,013004 (2008), arXiv:0802.0007 [hep-ph]
[89]S. Kretzer, Phys. Rev. D62,054001 (2000), arXiv:hep-ph/0003177.
[90]M. Anselmino et al., Eur. Phys. J. A39,89 (2009), arXiv:0805.2677 [hep-ph].
[91]D. de Florian, R. Sassot, M. Stratmann, and W. Vogelsang, Phys. Rev. D80, 034030 (2009), arXiv:0904.3821 [hep-ph].
[92]J. Pumplin et al., Phys. Rev. D65,014013 (2001), arXiv:hep-ph/0101032.
[93]R. Storn and K. Price, J. of Global Optimization 11,341 (1997).
[94]M. Anselmino et al., Eur. Phys. J. A47,35 (2011), arXiv:1101.4199 [hep-ex].
[1]Liang Z t and Wang X N 2007 Phys. Rev. D75 094002 (Preprint hep-ph/0609225)
[2]Gao J H, Liang Z t and Wang X N 2010 Phys. Rev. C81 065211 (Preprint 1001.3146)
[3]Song Y k, Gao J h, Liang Z t and Wang X N 2011 Phys. Rev. D83 054010 (Preprint 1012.4179)
[4]Cahn R N 1978 Phys. Lett. B78 269
[5]Ellis R K, Furmanski W and Petronzio R 1982 Nucl. Phys. B207 1
[6]Ellis R K, Furmanski W and Petronzio R 1983 Nucl. Phys. B212 29
[7]Qiu J W 1990 Phys. Rev. D42 30-44
[8]Qiu J w and Sterman G F 1991 Nucl. Phys. B353 105-136
[9]Qiu J w and Sterman G F 1991 Nucl. Phys. B353 137-164
[10]Liang Z t, Wang X N and Zhou J 2008 Phys. Rev. D77 125010 (Preprint 0801.0434)