An Alternative Approach to the Extraction of Structure Functions in Deep Inelastic e-p Scattering at 5 to 20 GeV

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1 Commun. Theor. Phys. (Beijing, China 39 (2003 pp c International Academic Publishers Vol. 39, No. 5, May 15, 2003 An Alternative Approach to the Extraction of Structure Functions in Deep Inelastic e-p Scattering at 5 to 20 GeV N. Ghahramany and M. Nouri Department of Physics, Shiraz University, Shiraz 71548, Iran (Received July 4, 2002; Revised October 8, 2002 Abstract Two structure functions W 1(x, Q 2 and W 2(x, Q 2 are determined by using the cross sections measured in the deep inelastic electron-proton scattering experiments at Stanford Linac in the energy range of 5 to 20 GeV. In this paper an alternative mathematical approach have been used in such determination, resulting in a larger number of points in the graphs of the structure functions. PACS numbers: Fb Key words: structure function, inelastic scattering, cross section, alternative approach 1 Introduction One of the theoretical models used for interpretation of the Bjorken scaling is parton model. In this model a nucleon consists of partons (point-like objects. At high momentum transfers, the elastic form factor is very small, and the inelastic scattering of the incident electron is much more probable than the elastic scattering. In a general inelastic scattering processes, there is an extra variable in addition to Q 2, because the space and time components of the momentum transfer Q are no longer related by Q 2 = 2Mν, where ν is defined in terms of the incident and the emergent electron energies, and E and E are as follows: E = E ν = E Q 2 2M nucleon and x = Q2 2Mν. In terms of Q 2 and x, the inclusive inelastic cross section is given by Ref. [1], d 2 σ ( dωde = α 2 1 E sin 2 (θ/2 4 [2W 1(Q 2, xsin 2 (θ/2 + W 2 (Q 2, xcos 2 (θ/2], (1 where W 1 and W 2 are the structure functions. In order to study the inner structure of the proton, W s should be extracted from the above equation. To do this it is more convenient to change Eq. (1 as follows: [2] where and d 2 σ dωde = Γ[σ T (x, Q 2 + ɛσ L (x, Q 2 ], (2 Γ = α E ( 2 (E 4π 2 Q 2 E Q2 E 1 ɛ 2M (3 1 ɛ = (1 + (E E 2, (4 Q 2 tan 2 (θ/2 σ T and σ L are related to the structure functions by W 1 (x, Q 2 = 1 (E 4π 2 E Q2 σ T (x, Q 2, (5 α 2M W 2 (x, Q 2 = 1 ( (E 4π 2 E Q2 Q 2 α 2M Q 2 + (E E 2 [σ T (x, Q 2 + σ L (x, Q 2 ]. A separate determination of the two inelastic structure functions W 1 and W 2 (or equivalently, σ L and σ T requires the values of the differential cross sections at several values of the angle θ for fixed x and Q 2. According to Eq. (2, σ L is the slope and σ T is the intercept of a linear fit to Σ = 1 Γ d 2 σ dωde (θ, x, Q2. (6 The structure functions are then directly calculable from Eqs. (5. 2 Method of Calculation and Data Analysis The deep inelastic scattering cross section data and their corresponding errors used here, in spite of being old, are the most recent available ones taken from Ref. [3], a sample of which is printed in Table 1. They are in the unit of barn/str GeV, and are given for several values of θ, E, and E. As mentioned before, to find the point of the graph of W 1 (x, Q 2 and W 2 (x, Q 2 versus Q 2 the following strategy is pursued: Since each point will be extracted from a linear fit to Σ versus ɛ, all points of Σ ɛ curve should be calculated at fixed x and fixed Q 2, but still with varying ɛ. Therefore ɛ should be changed so that x and Q 2 remain fixed, and this is a technically formidable task. There is another method, however, easier and more practical, which is as follows. First, x is determined for each line of data and those lines with x closed to 0.25 (the value of x at which we are to find the curves of W s as functions of Q 2 are selected.

2 574 N. Ghahramany and M. Nouri Vol. 39 Then Q 2 is calculated for each line, and those with close values of Q 2 are collected to form separate sets, each to be consequently used for drawing a Σ ɛ curve and taking a point for each W at a specific values of x and Q 2. But even though the values of x and Q 2 for each of these sets are close together, they are still slightly deferent. This will cause a considerable error in the final result. Therefore, it is necessary to modify E and E to fix x at 0.25 and Q 2 at an average value for each set of data. Doing this, we will have a number of sets of data with fixed x and Q 2. But their corresponding new E and E s no longer exist on the table of the data taken from the experiments. That is why we have to interpolate the required inclusive cross sections at these new values of E and E in order to construct Σ from Eq. (6 at fixed values of x and Q 2. After having Σ and calculating ɛ for each line of data, a Σ ɛ graph for each line is obtained. Then by fitting a line to each graph, and finding the intercept (σ T and the slop (σ L of the line, it is possible to calculate W 1 and W 2 from Eq. (5. In order to do this we perform the following steps. Table 1 This table contains respectively from left to right the number of the data line in the original table of data taken from Ref. [3] (Those that have x-value around 0.25, the scattering angle θ in degrees, the electron incident energy E, the electron emergency energy E in GeV, the inclusive inelastic scattering cross section (Σ, and their corresponding errors (DΣ in barn/str GeV. No. θ E E Σ DΣ Table 1 (Continue No. θ E E Σ DΣ First, using a FORTRAN program, x and Q 2 for each line of data on the Table 1 are calculated. Then those lines with x values greater than 0.35 or less than 0.16 are discarded, and the others with corresponding x and Q 2 values are recorded. The gained results at this step are shown in Table 2. Then using a set of programs. Data lines with Q 2 around the suitable values of 1.00, 1.50, 1.90, and so on are collected and saved on separate files. At the next step we have to modify E and E in order to fix x

3 No. 5 An Alternative Approach to the Extraction of Structure Functions in 575 and Q 2 at desired values. This is done using the following two well-known equations: [1,4] Q 2 = 4EE sin 2( θ, (7 2 ( θ 2EE sin 2 x = 2 M(E E. (8 Considering θ fixed and taking total derivatives we have dq 2 ( 1 ( 1 Q 2 = de + E E de, (9 dx ( x = E E(E E But obviously, ( de + E E (E E de. (10 dq 2 Q 2 = Q2 1 Q 2 0 Q 2, (11 0 dx x = x 1 x 0, (12 x 0 where Q 2 0 is the value of Q 2 of each line, Q 2 1 is the average value at which all Q s of each set are to be fixed, x 0 is the x-value of the line, while x 1 = 0.25 everywhere. Putting all of these into Eqs. (9 and (10, we will have two equations with two unknowns, de and de to be determined. After solving the system of equations, the solutions should be added to the previous energy values to get the modified ones, i.e. E 1 = E + de, E 1 = E + de. (13 The results of this step are printed in Table 3. Table 2 (X-selection Table 2 (Continue The next thing to do is to interpolate the values of inclusive cross sections at these new energies. This was performed by using the software Maple. The surfaces used for interpolation are drawn in Fig. 1. From these interpolated values of cross sections and new energies, Σ and ɛ are calculated and drawn. After fitting the lines (Fig. 2 and determining their slopes and intercepts, we can calculate the W s at x = 0.25 and various values of Q 2. These final results can be seen in Fig. 3, where comparison is made with the results of Ref. [3]. The important fact that W 1 and W 2 are independent of Bjorken scaling is clearly illustrated in Fig. 3 regardless of existence of a constant factor multiplied by numerical values of W 1 and W 2.

4 576 N. Ghahramany and M. Nouri Vol. 39 Fig. 1 Sample fitted surfaces for cross section data at θ = 15 and θ = 18 used in the interpolation procedure. Fig. 2 Sample diagrams of Σ versus ɛ, resulted from the interpolated data, and the fitted lines.

5 No. 5 An Alternative Approach to the Extraction of Structure Functions in 577 Fig. 3 The final results gained for the two structure functions W 1 and W 2, showing their approximate constancy which implies the existence of the pint scatterers inside the proton (See the text compared with the results of Ref. [3]. Table 3 Q 2 selection results. In this table the data lines with closed values of Q 2 are found and grouped together as separated sets Table 3 (Continue Conclusion The presented method of data analysis is very straightforward and simple and capable of being made more pre-

6 578 N. Ghahramany and M. Nouri Vol. 39 cise by widening the ranges of Q 2 and x in the fixing processes. By doing this we practically increase the number of data used for making Σ ɛ graphs and naturally will give us more precise results as compared with the previous calculations. Acknowledgments Partial support of Shiraz University Research Council is appreciated. We should also thank Mr. Abbas Shirinifard for his valuable helps in the surface fitting and interpolation processes. References [1] F. Halzen and A.D. Martin, Quarks and Leptons, Wiley, New York (1984. [2] D. Griffiths, Introduction to Elementary Particles, Wiley, New York (1987. [3] A. Bodek, et al., Phys. Rev. D20 ( ; J. Friedman and H. Kendall, Ann. Rev. Nucl. Sci. 22 ( ; G. Miller, et al., Phys. Rev. D5 ( ; E. Bloom, et al., Phys. Rev. Lett. 23 ( , 935. [4] D.H. Perkins, Introduction to High Energy Physics, Addison Wesley, Reading, Mass. (1972; F.E. Close, An Introduction to Quarks and Partons, Academic Press, New York (1979.