For moments of leptoproduction structure functions we show that all dependence on the renormalization and factorization scales disappears provided that all the ultraviolet logarithms involving the physical energy scale Q are completely resummed. The approach is closely related to Grunberg's method of Effective Charges. A direct and simple method for extracting Λ M S from experimental data is advocated. 1 ) C.J.Maxwell@durham.ac.uk 2 ) Abolfazl.Mirjalili@durham.ac.uk
An analytical solution based on the Laplace transformation technique for the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi DGLAP evolution equations is presented at next-to-leading order accuracy in perturbative QCD. This technique is also applied to extract the analytical solution for the proton structure function, F p 2 (x, Q 2 ), in the Laplace s-space. We present the results for the separate parton distributions for all parton species, including valence quark densities, the anti-quark and strange sea parton distribution functions (PDFs), and the gluon distribution. We successfully compare the obtained parton distribution functions and the proton structure function with the results from GJR08 [Eur. Phys. J C 53 (2008) 355-366] and KKT12 [J. Phys. G 40 (2013) 045002] parametrization models as well as the x-space results using QCDnum code. Our calculations show a very good agreement with the available theoretical models as well as the deep inelastic scattering (DIS) experimental data throughout the small and large values of x. The use of our analytical solution to extract the parton densities and the proton structure function is discussed in detail to justify the analysis method considering the accuracy and speed of calculations. Overall, the accuracy we obtain from the analytical solution using the inverse Laplace transform technique is found to be better than 1 part in 10 4 to 10 5 . We also present a detailed QCD analysis of non-singlet structure functions using all available DIS data to perform global QCD fits. In this regard we employ the Jacobi polynomial approach to convert the results from Laplace s space to Bjorken x space. The extracted valence quark densities are also presented and compared to the JR14, MMHT14, NNPDF and CJ15 PDFs sets. We evaluate the numerical effects of target mass corrections (TMCs) and higher twist (HT) terms on various structure functions, and compare fits to data with and without these corrections.
The feature of parton densities inside the nucleon at low Q 2 plays an essential role in particle physics. To calculate these densities there are different types of models, among which the chiral quark model (χ QM) is the famous one. In this model, baryons are made from the ground states of constituent quarks, which are initially bare and finally surrounded by pseudo-scalar mesonic clouds. Here, in addition to mesonic clouds, we also take into account the gluonic clouds. The required bare quarks are obtained using the solution of the Dirac equation under a squared radial potential. The wavefunctions which are primarily in momentum space can be converted to a distribution in the x-Bjorken space by two different methods. Using the modified χ QM, we extract the parton distributions inside the proton at low Q 2 values and compare them with some experimental and phenomenological groups. The distributions can be evolved to high-energy scales, using the DGLAP evolution equations. The results are in good agreement with available experimental data and some phenomenological groups.
Polarized parton distributions and structure functions of the nucleon are analyzed in the improved valon model. The valon representation provides a model to represent hadrons in terms of quarks, providing a unified description of bound state and scattering properties of hadrons. Polarized valon distributions are seen to play an important role in describing the spin dependence of parton distributions in the leading order (LO) and next-to-leading order (NLO) approximations. In the polarized case, a convolution integral is derived in the framework of the valon model. The Polarized valon distribution in a proton and the polarized parton distributions inside the valon are necessary to obtain the polarized parton distributions in a proton. Bernstein polynomial averages are used to extract the unknown parameters of the polarized valon distributions by fitting to the available experimental data. The predictions for the NLO calculations of the polarized parton distributions and proton structure functions are compared with the LO approximation. It is shown that the results of the calculations for the proton structure function, xg p 1 , and its first moment, Γ p 1 , are in good agreement with the experimental data for a range of values of Q 2 . Finally the spin contribution of the valons to the proton is calculated.
We perform an all-orders resummation of the QCD Adler D-function for the vector correlator, in which the portion of perturbative coefficients containing the leading power of b, the first beta-function coefficient, is resummed. To avoid a renormalization scale dependence when we match the resummation to the exactly known next-to-leading order (NLO), and next-NLO (NNLO) results, we employ the Complete Renormalization Group Improvement (CORGI) approach in which all RG-predictable ultra-violet logarithms are resummed to all-orders, removing all dependence on the renormalization scale. We can also obtain fixed-order CORGI results. Including suitable weight-functions we can numerically integrate these results for the Dfunction in the complex energy plane to obtain so-called "contour-improved" results for the ratio R and its tau decay analogue R τ . We use the difference 1 between the all-orders and fixed-order (NNLO) results to estimate the uncertainty in α s (M 2 Z ) extracted from R τ measurements, and find α s (M 2 Z ) = 0.120±0.002. We also estimate the corresponding uncertainty in α(M 2 Z ) arising from hadronic corrections by considering the uncertainty in R(s), in the low-energy region, and compare with other estimates. Analogous resummations are also given for the scalar correlator. As an adjunct to these studies we show how fixed-order contour-improved results can be obtained analytically in closed form at the two-loop level in terms of the Lambert W -function and hypergeometric functions.
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