Research Journal of Recent Sciences _________________________________________________ ISSN 2277-2502 Vol. 3(3), 1-3, March (2014) Res.J.Recent Sci. International Science Congress Association 1 Parameterization of Proton Structure Function using Fractal Inspired ModelGogoi Rupjyoti Department of Physics, Tezpur University, Napaam-784028, Assam, INDIAAvailable online at: www.isca.in , www.isca.me Received 25th August 2012, revised 20th August 2013, accepted 27th September 2013Abstract In recent years, applicability of fractal geometry in the structure of nucleons is getting importance. This work reports a fractal inspired parameterization for the structure function of proton. In some earlier work by the author, a fractal model of proton structure was proposed with few fitted parameters. Later on this model was applied to few other aspects of nucleon structure. This paper reports a new parameterization applicable in a wide range of x and Q2 of deep inelastic scattering in the light of recent data. The calculated chi square value proves the goodness of fit and therefore the model is expected to have wider range of applicability. Keywords: Deep inelastic scattering, structure function, self-similar dimension. Introduction Measurement of nucleon structure functions has attracted the attention of theoretical and experimental high energy physicist since decades. Experimentally, the electron proton collider HERA (Hadron Electron Ring Accelerator) explores the proton structure at high centre of mass energy, , where , being the lepton beam energy and is the proton beam energy1,2,3. Theoretically, the structure functions of proton are determined by QCD fits. In the method of parameterization, a certain functional form containing a set of parameters for the structure function is assumed. The model is then fitted to experimental data at different energy scales. Fractal objects are characterised by scaling laws or power laws, reflecting a deep internal symmetry4,5. They are characterised by a self-similar dimension () and magnification factor () related by . In some earlier work, parameterization of structure function of proton was reported on the basis of HERA data6-8. A statistical model for proton structure analysing fractal characteristics was proposed by S.N. Banerjee et al9-11The fractal inspired model can be applied to deep inelastic neutrino-nucleon scattering, gluon distribution function inside the nucleon, ultra high energy neutrino nucleon scattering cross sections, longitudinal structure function of the nucleon and to other related areas12-15. In this paper the author reanalyses the fractal inspired model with recent data16. The motivation for this reanalysis is the applicability of the earlier model in a limited range of data and also the limitation of earlier model to describe neutrino-nucleon scattering and ultra high energy neutrino nucleon interaction cross section12,14. MethodologySelf-similar objects are described by a self-similar dimension and a magnification factor. Deep inside the proton structure, more gluon-gluon interactions are observed. So a scaling described by a power law, which is a characteristic of self-similar objects, is expected to exist in the proton structure. The un-integrated u-quark density exhibits a linear behavior as a function of at fixed and as a function of at fixed x, where Qis thefour momentum transfer squared and x is thefraction of proton momentum carriedaway by a constituent parton in deep inelastic scattering. Such linear behavior suggests that and or their suitable functions can be considered as the magnification factors for fractal proton. Considering 1/x and as the magnification factors, the following form for un-integrated parton density can be suggested, (1) Now, the model needs to identify the four parameters best fitted to latest experimental data. In equation (1), is the dimensional correlation relating the two magnification factors 1/x and , whereas and are the fractal dimensions associated with 1/x and respectively, being the normalization constant. Results and Discussion In this work, the author has reported a parameterization of proton structure function (x,Q with recent data reported by H1 and ZEUS collaboration16. Out of the four parameters (where =0-3), and should be positive and non zero since they are identified as the self-similar dimensions4-5. By Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 3(3), 1-3, March (2014) Res. J. Recent Sci. International Science Congress Association 2 imposing a positivity constraint on the two parameters and , a set of parameters as given in table-1 was obtained. (x,Q is then calculated within the fractal approach using equation (1). Figure-1 shows the plot of estimated (x,Qwithin the fractal inspired model and (x,Q obtained from HERA as function of x for different values of the four momentum transfer squared16 The chi square analysis is done to observe the reliability of the fit. The values of standard deviation and chi square are also recorded in table-1. The chi square value ensures the reliability of parameter estimation. Table-1 The fitted parameters with 2 value for the fractal inspired model D D D Q2 Standard deviation dof 1.03038 -0.07292 1.025473 0.76741 0.07939 0.0525 1.004 Figure-1 The plot of (x,Q as a function of in bins of . The lines represent estimated values from fractal inspired model of this work and the symbols represent the data presented by H1 and ZEUS collaboration16. Research Journal of Recent Sciences ______________________________________________________________ ISSN 2277-2502Vol. 3(3), 1-3, March (2014) Res. J. Recent Sci. International Science Congress Association 3 Conclusion In this parameterization of proton structure with recent data, the goodness of parameter estimation is reflected in the graphs and also in chi square analysis. Two of the parameters ( and ) are identified as the self-similar dimensions of proton. The fractal characteristics of hadrons like proton as pursued in some references within a statistical quark model are found to be in good agreement with recent experimental data9-11. Our result compliments such idea in deep inelastic region. The fractal inspired model will be tested for a wider range of and available from experiments and also for other measures of proton structure which will provide a detailed analysis of the model. Acknowledgement The author acknowledges financial grant from Tezpur University in the form of start up grant. References 1.Adloff C et al: H1 collaboration, Deep Inelastic Inclusive ep Scattering at Low and a Determination of , Euro. Phys. J.,C22, 33 hep-ex/0012053 (2002)2.Brietweg J et al: ZEUS collaboration, Measurement of the Proton Structure Function at very Low at HERA, Phys Lett.,B487, 53 hep-ex/0005018 (2000)3.Cooper-Sarkar Amanda, What did HERA teach us about the structure of the proton?, J. Phys. G,39, 093001 (2012)4.Mandelbrot B.B., Fractal Geometry of Nature, W.H. Freeman, N.Y. (1983)5.Barnsley M.F., Fractal Everywhere, Academic Press, N.Y. (1993) 6.Lastovicka T, Self-similar Properties of the Proton Structure at Low , Euro Phys JC24, 529 (2002) 7.Choudhury D.K. and Gogoi Rupjyoti, Some Comments on Fractality of Proton at Small , Ind. J. Phys., 80, 659 (2006)8.Choudhury D.K. and Gogoi Rupjyoti, Self-similarity and a Parameterization of Proton Structure Function at Small ”, Ind. J. Phys., 80, 823 (2006) 9.Banerjee S N, Bhattacharya A, Chakraborti B and Banerjee S, The Properties of a Hadron, Int. J. Mod. Phys., A16, 201 (2001) 10.Banerjee S N, Bhattacharya A, Chakraborti B and Banerjee S, On Some Properties of Meson, Int. J. Mod. Phys. A17,4939 (2002)11.Bhattacharya A, Banerjee S N, Chakraborti B and Banerjee S, A Study on the Structure of the Proton, Nucl. Phys., B142, 13 (2005)12.Choudhury D.K. and Gogoi Rupjyoti, Deep Inelastic Neutrino Scattering and Fractal Models of Nucleon Structure Functions at Small , Ind. J. 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