XXXIII International Conference of Theoretical Physics MATTER TO THE DEEPEST: Recent Developments in Physics of Fundamental Interactions, USTROŃ'09 Ustron, Poland September 11-16, 2009

Hamzeh Khanpour In collaboration with: Ali Khorramian, Shahin Atashbar Tehrani Semnan University and School of Particles and Accelerators, IPM (Institute for Studies in Theoretical Physics and Mathematics), Tehran, IRAN Determination of valence quark densities up to higher order of perturbative QCD Hamzeh Khanpour (IPM & Semnan University) USTRON 2009

Outline Abstract Introduction DIS Kinematics Parton distributions and hard processes (1,2) What we need for 4-loop calculations? Splitting functions and Coefficient functions Pade’ approximate The method of the QCD analysis of Structure Function The procedure of the QCD fits of F 2 data Results and Conclusion Hamzeh Khanpour (IPM & Semnan University) USTRON 2009

Abstract We present the results of our QCD analysis for non-singlet unpolarized quark distributions and structure function F 2 (x,Q 2 ) up to N 3 LO. The analysis is based on the Jacobi polynomials technique of reconstruction of the structure functions from its Mellin moments. The fit results for the non-singlet parton distribution function, their correlated errors and evolution are presented. Our results on and strong coupling constant up to N 3 LO are presented and compared with those obtained from deep inelastic scattering processes. In the N 3 LO analysis, the QCD scale and the strong coupling constant were determined with the help of Pade-approximation. Hamzeh Khanpour (IPM & Semnan University) USTRON 2009

Introduction Complete knowledge of the parton distributions in the nucleon is essential for calculating deep inelastic processes at high energies. All calculations of high energy processes with initial hadrons, whether within the standard model or exploring new physics, require parton distribution functions (PDF's) as an essential input. The assessment of PDF's, their uncertainties and extrapolation to the kinematics relevant for future colliders such as the LHC is an important challenge to high energy physics in recent years. Parton distributions form indispensable ingredients for the analysis of all hard-scattering processes involving initial-state hadrons. Presently the next-to-leading order is the standard approximation for most important processes but the N 2 LO and N 3 LO corrections need to be included, however, in order to arrive at quantitatively reliable predictions for hard processes at present and future high-energy colliders. For quantitatively reliable predictions of DIS and hard hadronic scattering processes, perturbative QCD corrections beyond the next-to-leading order, N 2 LO and N 3 LO, need to be taken into account. For at least the next ten years, proton (anti-) proton colliders will continue to form the high- energy frontier in particle physics. At such machines, many quantitative studies of hard (high mass/scale) standard-model and new-physics processes require a precise understanding of the parton structure of the proton. Complete knowledge of the parton distributions in the nucleon is essential for calculating deep inelastic processes at high energies. All calculations of high energy processes with initial hadrons, whether within the standard model or exploring new physics, require parton distribution functions (PDF's) as an essential input. The assessment of PDF's, their uncertainties and extrapolation to the kinematics relevant for future colliders such as the LHC is an important challenge to high energy physics in recent years. Parton distributions form indispensable ingredients for the analysis of all hard-scattering processes involving initial-state hadrons. Presently the next-to-leading order is the standard approximation for most important processes but the N 2 LO and N 3 LO corrections need to be included, however, in order to arrive at quantitatively reliable predictions for hard processes at present and future high-energy colliders. For quantitatively reliable predictions of DIS and hard hadronic scattering processes, perturbative QCD corrections beyond the next-to-leading order, N 2 LO and N 3 LO, need to be taken into account. For at least the next ten years, proton (anti-) proton colliders will continue to form the high- energy frontier in particle physics. At such machines, many quantitative studies of hard (high mass/scale) standard-model and new-physics processes require a precise understanding of the parton structure of the proton. Hamzeh Khanpour (IPM & Semnan University) USTRON 2009

The deep-inelastic lepton-nucleon scattering is the source of important information about the nucleons structure. DIS Kinematics Deep inelastic scattering in the quark parton model. The negative four momentum transfer squared at the electron vertex The Bjorken scale variable x and the inelasticity y Hamzeh Khanpour (IPM & Semnan University) USTRON 2009

Parton distributions and hard processes (1) inclusive photon-exchange deep-inelastic scattering (DIS) At zeroth order in the strong coupling constant the hard coefficient functions c a,i are trivial, and the momentum fraction carried by the struck quark i is equal to Bjorken- x if mass effects are neglected. In general, the structure functions are given by Hamzeh Khanpour (IPM & Semnan University) USTRON 2009

Parton distributions and hard processes (2) Parton distributions f i : evolution equations ( ⊗ = Mellin convolution) The initial conditions are not calculable in perturbative QCD. The task of determining these distributions can be divided in two steps. The first is the determination of the non-perturbative initial distributions at some (usually rather low) scale Q 0. The second is the perturbative calculation of their scale dependence (evolution) to obtain the results at the hard scales Q. The initial distributions have to be fitted using a suitable set of hard- scattering observables. Hamzeh Khanpour (IPM & Semnan University) USTRON 2009

Determination of the parton density functions from experimental data is according to the following procedure: The parton density functions are parameterised by smooth analytical functions at a low starting scale Q 0 2 as a function of x with a certain number of free parameters. They are evolved in Q 2 using the DGLAP equations. Afterwards predictions for structure functions and cross-sections are calculated. The free parameters are determined by performing a fit to the data. Several constraints are imposed during this procedure. For an accurate determination of the parton density functions the coefficient and splitting functions have to be precisely known as they govern the calculation of the structure functions and the evolution, respectively. Since 2005 all of them are known to next-to-next-to-leading order (N 2 LO). Procedure Hamzeh Khanpour (IPM & Semnan University) USTRON 2009

what we need for 4-loop calculations? The splitting functions P (n) and the process-dependent hard coefficient functions c a admit expansions in powers of strong coupling constant The first n+1 terms define the N n LO approximation. N 2 LO: N 3 LO: Hamzeh Khanpour (IPM & Semnan University) USTRON 2009

In spite of the unknown 4–loop splitting functions one may wonder whether any statement can be made on the non–singlet parton distributions and at the 4–loop level !!!!!!! Splitting functions and Coefficient functions The complete NNLO splitting functions and coefficient functions are known [J. A. M. Vermaseren, A. Vogt and S. Moch, Nucl. Phys. B 724 (2005) 3] [S. Moch, J. A. M. Vermaseren and A. Vogt, Nucl. Phys. B 688 (2004) 101] [J. A. M. Vermaseren, A. Vogt and S. Moch, Nucl. Phys. B 724 (2005) 3] The 3-loop coefficient functions are known Hamzeh Khanpour (IPM & Semnan University) USTRON 2009

The answer is: Pade’- approximate Hamzeh Khanpour (IPM & Semnan University) USTRON 2009 [J. Ellis, E. Gardi, M. Karliner and M.A. Samuel, Phys. Rev. D54 (1996) ] [J. Ellis, E. Gardi, M. Karliner and M.A. Samuel, Phys. Lett. B366 (1996) 268. ] [M.A. Samuel, J. Ellis and M. Karliner, Phys. Rev. Lett. 74 (1995) ]

Padé estimates for the large-N behaviour of the three-loop contributions to the non-singlet coefficient function C 2,ns (N). The three-loop approximants are compared with exact results. Coefficient functions Pade’ [0/2] Pade’ [1/1] Hamzeh Khanpour (IPM & Semnan University) USTRON 2009

The perturbative expansion of the non-singlet N-space coefficient functions Hamzeh Khanpour (IPM & Semnan University) USTRON 2009

4-loop Splitting functions Pade’ [0/2] Pade’ [1/1] Hamzeh Khanpour (IPM & Semnan University) USTRON 2009

The perturbative expansion of the anomalous dimension Hamzeh Khanpour (IPM & Semnan University) USTRON 2009

The method of the QCD analysis of SF One of the simplest and fastest possibilities in the structure function reconstruction from the QCD predictions for its Mellin moments is Jacobi polynomials expansion. [ A. N. Khorramian and S. A. Tehrani, Phys. Rev. D 78 (2008) ] Hamzeh Khanpour (IPM & Semnan University) USTRON 2009 [ S. Atashbar Tehrani and A. N. Khorramian, JHEP 0707 (2007) 048 ]

Structure function in x space F 2 Structure function as extracted from the DIS ep process can be, up to N 3 LO written as The combinations of parton densities in the non-singlet regime and the valence region x>0.3 for F 2 p and F 2 d in LO is In the region x<0.3 for the difference of the proton and deuteron data we use Hamzeh Khanpour (IPM & Semnan University) USTRON 2009

The evolution equations are solved in Mellin-N space and the Mellin transforms of the above distributions are denoted by respectively. The non–singlet structure functions are given by Here denotes the strong coupling constant and are the non–singlet Wilson coefficients in Hamzeh Khanpour (IPM & Semnan University) USTRON 2009

The solution of the non–singlet evolution equation for the parton densities to 4–loop order reads Here, denote the Mellin transforms of the (k + 1)–loop splitting functions. [J. Blumlein, H. Bottcher and A. Guffanti, Nucl. Phys. B 774, 182 (2007)] [ A. N. Khorramian and S. A. Tehrani, Phys. Rev. D 78 (2008) ]

The procedure of the QCD fits of F 2 data In the present analysis we choose the following parameterization for the valence quark densities By QCD fits of the world data for, we can extract valence quark densities using the Jacobi polynomials method. For the non-singlet QCD analysis presented here we use the structure function data measured in charged lepton proton and deuteron deep-inelastic scattering. [J. Blumlein, H. Bottcher and A. Guffanti, Nucl. Phys. B 774, 182 (2007)] [ A. N. Khorramian and S. A. Tehrani, Phys. Rev. D 78 (2008) ] Hamzeh Khanpour (IPM & Semnan University) USTRON 2009

Results

[J. Blumlein, H. Bottcher and A. Guffanti, Nucl. Phys. B 774, 182 (2007)] [ A. N. Khorramian and S. A. Tehrani, Phys. Rev. D 78 (2008) ] Hamzeh Khanpour (IPM & Semnan University) USTRON 2009

Table: Parameters values of the LO, NLO, NNLO and N 3 LO non-singlet QCD fit at Q 0 2 = 4 GeV 2 Fit results Hamzeh Khanpour (IPM & Semnan University) USTRON 2009

Comparison of the parton densities xu v and xd v at the input scale Q 0 2 = 4 GeV 2 and at different orders in QCD as resulting from the present analysis. Parton densities at the input scale Q 0 2 = 4 GeV 2 Hamzeh Khanpour (IPM & Semnan University) USTRON 2009

The parton densities xu v and xd v at the input scale Q 0 2 = 4 GeV 2 (solid line) compared to results obtained from N 3 LO analysis by BBG (dashed line). Hamzeh Khanpour (IPM & Semnan University) USTRON 2009 [J. Blumlein, H. Bottcher and A. Guffanti, Nucl. Phys. B 774, 182 (2007)]

Hamzeh Khanpour (IPM & Semnan University) USTRON 2009 The parton densities xu v (x,Q 2 ) and xd v (x,Q 2 ) at N 3 LO evolved up to Q 2 = GeV 2 (solid line) compared to results obtained by BBG (dashed line).

The structure functions F 2 p and F 2 d as function of Q 2 i n intervals of x. Shown are the pure QCD fit in N 3 LO (solid line) and the contributions from target mass corrections TMC (dashed line) and higher twist HT (dashed–dotted line). The arrows indicate the regions with W 2 >12.5 GeV 2. The shaded areas represent the fully correlated 1σ statistical error bands.

The structure function F 2 NS as function of Q 2 in intervals of x. Shown is the pure QCD fit in N 3 LO (solid line).

Conclusion We performed a QCD analysis of the non–singlet world data up to N 3 LO and determined the valence quark densities xu v (x,Q 2 ) and xd v (x,Q 2 ) with correlated errors. Parameterizations of these parton distribution functions and their errors were derived in a wide range of x and Q 2 as fit results at LO, NLO, NNLO, and N 3 LO. In the analysis the QCD scale and the strong coupling constant α s (M 2 Z ), were determined up to N 3 LO. In spite of the unknown 4–loop splitting functions we applied Pade’- approximate for determination of non–singlet parton distributions and QCD at the 4–loop level. Our calculations for non-singlet unpolarized quark distribution functions based on the Jacobi polynomials method are in good agreement with the other theoretical models. Hamzeh Khanpour (IPM & Semnan University) USTRON 2009

Thanks to attention !! Hamzeh Khanpour (IPM & Semnan University) USTRON 2009