Calculation of the Gluon Distribution Function Using Alternative Method for the Proton Structure Function

Commun. Theor. Phys. (Beijing, China 40 (2003 pp. 551 557 c International Academic Publishers Vol. 40, No. 5, November 15, 2003 Calculation of the Gluon Distribution Function Using Alternative Method for the Proton Structure Function N. Ghahramany and G.R. Boroun Physics Department, Shiraz University, Shiraz 71454, Iran (Received March 4, 2003 Abstract A calculation of the proton structure function F 2(, Q 2 is reported with an approimation method that relates the reduced cross section derivative and the F 2(, Q 2 scaling violation at low by using quadratic form for the structure function. This quadratic form approimation method can be used to determine the structure function F 2(, Q 2 from the HERA reduced cross section data taken at low. This new approach can determine the structure functions F 2(, Q 2 with reasonable precision even for low values which have not been investigated. We observe that the Q 2 dependence is quadratic over the full kinematic covered range. To test the validity of our new determined structure functions, we find the gluon distribution function in the leading order approimation with our new calculation for the structure functions and compare them with the QCD parton distribution functions. PACS numbers: 13.60.Hb Key words: proton structure function, gluon, QCD, deep inelastic scattering 1 Introduction Deep-inelastic lepton-nucleon scattering (DIS has been pivotal in the development of the understanding of strong interaction dynamics. Measurements of the inclusive DIS cross section have been essential for testing QCD. Previous fied target DIS eperiments have observed scaling violation, i.e. the variation of the structure functions with Q 2, the squared four-momentum transfer between lepton and nucleon, for fied values of Bjorken, which are well described by perturbative QCD. The Q 2 evolution of the proton structure function F 2 (, Q 2 is related to the gluon momentum distribution in the proton, g(, Q 2, and to the strong interaction coupling constant, α s. These can be determined with precise deep-inelastic scattering cross section data measured over a wide range of Bjorken and Q 2. There have been two main strands of interest in eperiments on DIS since the initial observation of Bjorken scaling. Firstly, they are used to investigate the theory of the strong interaction, and secondly, they are used to determine the momentum distributions of the partons within the nucleon. The observation of the Bjorken scaling established that the quark-parton model (QPM is a valid framework to interpret the data and thus that the deep inelastic structure functions can be used to measure parton distributions in the nucleon. [1,2] The quark-parton model grew out of the attempt by Feynman to provide a simple physical picture of the scaling that had been predicted by Bjorken and had been observed in the first high energy deep inelastic electron scattering eperiments at SLAC, where F 2 was observed to be independent of Q 2 for values around 0.3. The model states that the nucleon is full of pointlike noninteracting scattering centers known as partons. The lepton-hadron reaction cross-section is approimated by an incoherent sum of elastic lepton-parton scattering cross-sections (see Fig. 1. In the infinite momentum frame it is then easy to show that the variable is identified with the fraction of the nucleon s momentum involved in the hard scattering. [2] The measurement of the proton structure function by DIS processes in the low region, where is the Bjorken variable, has opened a new era in parton density measurements. However, from eperiments one can measure only the structure function but not the gluon or quark densities. Hence the direct relations between F 2 (, Q 2 and the gluon distribution G(, Q 2 are etremely important, for the eperimental values of G(, Q 2 can be etracted using the data on F 2 (, Q 2. But the etracted F 2 from the cross sections depends on the value of R, and it is determined using the standard DGLAP QCD fit to the H 1 data. [3 5] The structure functions F 2 and F L are related to the cross sections σ T and σ L for interaction of transversely and longitudinally polarized virtual photons with protons. [2] In the quark parton model F L is predicted to be zero for spin 1/2 partons, while in QCD, F L acquires a nonzero value due to gluon radiation, which is proportional to the strong coupling constant α s. [6] Due to the positivity of the cross sections for longitudinally and transversely polarized photon s scattering off protons, the two proton structure functions F 2 and F L obey the relation E-mail: ghahramany@physics.susc.ac.ir

552 N. Ghahramany and G.R. Boroun Vol. 40 0 F L F 2. [7] Fig. 1 Schematic diagram of lepton-hadron scattering in the quark-parton model. In this paper we obtain an approimate relation between the reduced cross section derivative and the F 2 (, Q 2 scaling violation with a quadratic structure function form. Using this alternative method we can determine the structure functions F 2 (, Q 2 with high accuracy without prior knowledge of R(, Q 2. 2 Theory In the one-photon echange approimation, the neutral current double differential cross section, d 2 σ/ddq 2, is given by the epression d 2 σ ddq 2 = 2πα2 Y + Q 4 σ r, (1 we observe that the reduced cross section is linear to F 2 (, Q 2 and F L (, Q 2, σ r F 2 (, Q 2 y2 Y + F L (, Q 2, (2 where Y + = 1 + (1 y 2. Here Q 2 is the squared fourmomentum transfer and y = Q 2 /s is the inelasticity with s the e-p center of mass energy squared. [3] The contribution of the longitudinal structure function F L to the cross section can be sizeable only at large values of the inelasticity y, and in most of the kinematic range the relation σ r F 2 holds to a very good approimation. For calculation of the structure function F 2, we need the structure function ratio, R = F L /(F 2 F L, which has not yet been measured at HERA. But it was calculated by using the QCD parton parametrization [4,5] with the NLO strong coupling constant [4] and the GRV and MRS parton parametrization. [2] However, from eperiments one can measure only the structure function but not F L (, Q 2, which is proportional to the strong coupling constant α s. [6,7] About half of the proton s momentum is carried by gluons. Despite this, the determination of the density g(, Q 2 of gluons in the proton as a function of Bjorken and the momentum transfer squared, Q 2, has turned out to be a difficult task. The principal difficulty in measuring the gluon density in deep inelastic scattering is that the gluons do not contribute to the cross section in the zeroth order QCD (Quark-Parton Model, where the structure functions scale does not depend on Q 2. In the net order (leading order: LO scaling violations occur through gluon bremsstrahlung from quarks and quark pair creation from gluons. At small, < 10 2, the latter process dominates the scaling violations. Hence the direct relations between F 2 (, Q 2 and the gluon distribution G(, Q 2 or the longitudinal structure function F L (, Q 2 are etremely important, for the eperimental values of G(, Q 2 and F L (, Q 2 can be etracted using the data on F 2 (, Q 2. [8] Here G(, Q 2 = g(, Q 2 is the gluon momentum density and g(, Q 2 is the gluon number density of the proton. In perturbative QCD, the longitudinal structure function F L (, Q 2 is proportional to α s. It is given to leading order, by an integral over the quark and gluon distributions, [2,9] F L (, Q 2 = α s(q 2 [ 4 π 3 1 dy +2c y ( y 2 ( 1 y 1 dy ( 2F2 (y, Q 2 y y ] yg(y, Q 2. (3 At small ( 10 3, the right-hand side is dominated by the gluon contribution. [10] In fact, it can be shown that (at n f = 4, F L (, Q 2 2α s 10 g(2.5, Q 2. (4 π 9 5.9 The longitudinal structure function is therefore a very clean probe for the small gluon distribution. [11] On the other hand, by taking advantage of the Prytz LO method, [12] the gluon momentum density is obtained, g(, Q 2 df 2(/2, Q 2 /d ln Q 2 (40/27(α s /4π. (5 Now, using Eqs. (4 and (5, F L (, Q 2 is found directly from the slope of F 2 (, Q 2 at 1.25, namely F L (, Q 2 6 df 2 (1.25, Q 2 5.9 d ln Q 2. (6 Substituting this leading order relation in Eq. (2 for each constant value of, an approimate method is presented, from which the structure function F 2 (, Q 2 is obtained without the prior knowledge of parton distribution function. On this basis we get y 2 σ r = F 2 (, Q 2 6 df 2 (1.25, Q 2 5.9 Y + d ln Q 2. (7 In order to obtain the structure functions and solve Eq. (7, the reduced cross section data of Tables 9 12 in Ref. [16] are used. Considering the relationship between the reduced cross sections at and 1.25, shows the similar relation between structure functions within corresponding error. Thus, we have σ r = F 2 (, Q 2 β y2 Y + df 2 (, Q 2 d ln Q 2, (8

No. 5 Calculation of the Gluon Distribution Function Using Alternative Method for the Proton Structure Function 553 where β = 0.95 ± 0.03. In order to find F 2 (, Q 2, the derivative method [13] which is based upon the cross section derivative, (dσ r /d ln y is used. Taking the derivative of Eq. (7 with respect to ln y for each value of constant, we get ( dσr d ln y = ( df2 d ln y 2βy 2 2 y Y 2 + ( df2 d ln y ( β y2 d 2 F 2 Y + d(ln y 2. (9 Although a study of the derivative (df 2 /d ln Q 2 at low was presented previously by ZEUS [14,15] and H 1 collaboration, [13] where F 2 (, Q 2 was assumed to depend linearly on ln Q 2. We calculate all derivatives σ r as a function of ln y at each constant value of, without such assumption. In these calculations we used the inclusive cross sections accompanied by the total errors for the deep inelastic-scattering of positrons off protons for momentum transfers squared Q 2 = 8.5, 12, 15, 20, 22.5, 25, 35 GeV 2 using data collected by the H 1 detector at HERA in 1996 and 1997. [16] 3 Calculation and Result Due to the contribution of the longitudinal structure function to the DIS cross section, equation (2 is proportional to y 2, the F 2 (, Q 2 term dominates at y 0.04, and the relation σ r = F 2 holds to a very good approimation. Thus the contribution of the second term of the right-hand side of Eq. (9 can be sizeable only at y > 0.04. Therefore, the data all used in this paper have y 0.04, i.e. the structure function depends on inelasticity y. Hence, in Eq. (9 we need derivative of σ r with respect to ln y at each constant value. In order to obtain that, we use the reduced cross sections data [16] with their total errors. For each constant value, in order to obtain the precise variations of σ r as a function of ln y, every two adjacent values of have been averaged. It is observed that these derivative are quadratic. Therefore the derivatives of F 2 (, Q 2 with respect to ln Q 2 is also quadratic. Then by substituting (dσ r /d ln y in Eq. (9, (df 2 /d ln y or (df 2 /d ln Q 2 and (d 2 F 2 /d(ln y 2 or (d 2 F 2 /d(ln Q 2 2 are obtained. We see the ln Q 2 dependence of F 2 is observed to be non-linear. It can be well described by a quadratic epression with the following form: F 2 (, Q 2 = a( + b( ln Q 2 + c((ln Q 2 2. Now, knowing the slope of F 2 and σ r with respect to ln y at each y or corresponding Q 2, and then substituting in Eq. (8, the value of structure function F 2 (, Q 2 is obtained with the corresponding overall errors and tabulated in Table 1. In this table we compared our obtained structure function values with the eisting data, indicating the fact that the structure function F 2 (, Q 2 can be determined with reasonable precision. In addition, the undetermined structure functions have also been calculated. Fig. 2 Calculation of the structure function F 2(, Q 2, plotted as a function of for Q 2 = 8.5 GeV 2. The error bars are due to the total reduced cross section errors and the model. Fig. 3 As Fig. 2 but for Q 2 = 12 GeV 2. Fig. 4 As Fig. 2 but for the Q 2 = 15 GeV 2. In Figs. 2 7 we draw the structure functions obtained at each value of Q 2 in and observe a continuous rise towards low. In Figs. 8 13 we compare our results with the data in Ref. [16]. In order to test the structure function quadratic form into ln Q 2, we plot this quadratic function into Q 2 at each constant of. Hence there is a good agreement between this quadratic function and the data published in Ref. [16], i.e. the structure function is a quadratic function of ln Q 2.

554 N. Ghahramany and G.R. Boroun Vol. 40 Table 1 The values determined structure functions F 2(, Q 2 as a quadratic function with total errors based on the values σ r and total errors of Ref. [16]. Q 2 y σ r δ total F 2 (publi. F 2 (deter. δ total 8.5 0.00020 0.4700 1.152 0.029 1.193 1.244 0.032 8.5 0.00032 0.2930 1.080 0.025 1.092 1.107 0.026 8.5 0.00041 0.2405 1.036 0.024 1.052 0.024 8.5 0.00050 0.1880 0.992 0.023 0.996 1.004 0.023 8.5 0.00080 0.1180 0.893 0.025 0.894 0.895 0.025 8.5 0.00130 0.0720 0.797 0.028 0.797 0.798 0.028 12 0.00032 0.4150 1.217 0.020 1.249 1.270 0.022 12 0.00041 0.3405 1.181 0.019 1.215 0.020 12 0.00050 0.2660 1.146 0.018 1.156 1.165 0.019 12 0.00080 0.1660 0.986 0.027 0.989 0.993 0.027 12 0.00130 0.1020 0.878 0.023 0.879 0.880 0.023 12 0.00200 0.0660 0.825 0.027 0.825 0.825 0.027 12 0.00320 0.0410 0.725 0.029 0.725 0.725 0.029 15 0.00032 0.5190 1.283 0.024 1.342 1.434 0.029 15 0.00041 0.4255 1.255 0.021 1.308 0.023 15 0.00050 0.3320 1.228 0.019 1.247 1.249 0.020 15 0.00080 0.2080 1.115 0.017 1.121 1.125 0.017 15 0.00130 0.1270 0.969 0.029 0.971 0.972 0.029 15 0.00200 0.0830 0.865 0.025 0.866 0.866 0.025 15 0.00320 0.0520 0.774 0.025 0.774 0.774 0.025 20 0.00041 0.5640 1.336 0.023 1.378 0.024 20 0.00050 0.4430 1.285 0.020 1.324 1.349 0.022 20 0.00080 0.2770 1.178 0.018 1.190 1.192 0.018 20 0.00130 0.1700 1.059 0.018 1.062 1.063 0.018 20 0.00200 0.1110 0.939 0.028 0.940 0.940 0.028 20 0.00320 0.0690 0.819 0.023 0.819 0.819 0.023 22.5 0.00041 0.6200 1.354 0.024 1.852 0.040 22.5 0.00050 0.4980 1.315 0.022 1.440 0.026 22.5 0.00080 0.3120 1.210 0.018 1.237 0.019 22.5 0.00130 0.1910 1.075 0.018 1.078 0.018 22.5 0.00200 0.1245 0.962 0.028 0.964 0.029 22.5 0.00320 0.0775 0.849 0.025 0.850 0.025 25 0.00080 0.3460 1.242 0.019 1.263 1.273 0.020 25 0.00130 0.2130 1.091 0.018 1.097 1.097 0.018 25 0.00200 0.1380 0.985 0.029 0.987 0.987 0.029 25 0.00320 0.0860 0.879 0.028 0.880 0.880 0.028 35 0.00130 0.2980 1.181 0.018 1.195 1.205 0.019 35 0.00200 0.1940 1.031 0.018 1.035 1.036 0.018 35 0.00320 0.1210 0.935 0.031 0.936 0.936 0.031 Therefore in order to test the validity of our obtained quadratic structure functions, we calculate the gluon distribution functions by using LO Prytz method to compared with the theoretical predictions. In doing so, the structure function data given in Table 1 are used. At each constant value of, we can calculate (df 2 /d ln Q 2, hence to etract the gluon distribution function at Q 2 = 20 GeV 2. The values of (df 2 /d ln Q 2 are found and the value of α s is assumed to be α s = 0.203±0.01. [13,16] The values of gluon structure function obtained at low- can be observed in Table 2. Table 2 The gluon distribution function G(2 at Q 2 = 20 GeV 2, based on our structure functions data of Table 1. The error bars are total errors. df 2 /d ln Q 2 δ total G(2 δ total 0.0005 0.430 0.017 17.96 1.13 0.0008 0.342 0.002 14.28 0.71 0.0013 0.286 0.007 11.94 0.66 0.0020 0.206 0.006 8.60 0.49 0.0032 0.198 0.002 8.27 0.41

No. 5 Calculation of the Gluon Distribution Function Using Alternative Method for the Proton Structure Function 555 As can be seen in Fig. 14, the values of G(2, Q 2 = 20 GeV 2 increase as decreases. Via comparison of these values with different QCD parton distribution functions, [17 20] (which are based upon values of our structure function quadratic form, we observe that these gluon distribution function values are within QCD parton distributions, and this illustrates the correctness and preciseness of our calculations (see Fig. 14. Fig. 5 As Fig. 2 but for Q 2 = 20 GeV 2. Fig. 6 As Fig. 2 but for Q 2 = 25 GeV 2. Fig. 7 As Fig. 2 but for Q 2 = 35 GeV 2. Fig. 8 Comparing of our structure functions to the data of Ref. [16], plotted as a function of Q 2 for = 0.0005. The solid line represents fits to F 2 in bin of according to a polynomial F 2(, Q 2 = a( + b( ln Q 2 + c((ln Q 2 2. Fig. 9 As Fig. 8 but for = 0.00032. Fig. 10 As Fig. 8 but for = 0.0008.

556 N. Ghahramany and G.R. Boroun Vol. 40 Fig. 11 As Fig. 8 but for = 0.0013. Fig. 12 As Fig. 8 but for = 0.002. Fig. 13 As Fig. 8 but for = 0.0032. Fig. 14 The solid circles represent our gluon prediction using the polynomial structure functions F 2(, Q 2 obtained in our method at Q 2 = 20 GeV 2. The error bars show total errors. The solid curve shows the gluon prediction from the obtained equation, compared with the parton parametrizations: MRSD (dashed curve, MRSG (dotted curve, GRV(98 & MRS (98 (dashed-dotted curve, MRSD (solid curve: using data in Table 1. Hence with this result, we can say that the structure function is a quadratic epression in ln Q 2 and unlike Refs. [14] and [15] it is non-linear. Therefore, this approimated method which is used here to find the structure function F 2 (, Q 2 with a quadratic form, from which the gluon distribution function values have been found, is in good agreement with the results of

No. 5 Calculation of the Gluon Distribution Function Using Alternative Method for the Proton Structure Function 557 different QCD parton distribution functions. 4 Conclusion Based upon the variations of the reduced cross section in the low region, an approimate method for the calculation of the structure function F 2 (, Q 2 is presented. The structure function is observed to be a quadratic function of ln Q 2. In this method, the structure function F 2 (, Q 2 is determined without the prior knowledge of the longitudinal structure function F L (, Q 2. Careful investigation of our results shows a good agreement with the previous published structure functions F 2 (, Q 2. In addition, the undetermined structure functions have also been calculated (shown in Table 1. By using these quadratic structure function to calculate the gluon distribution function and by comparing with the QCD parton distribution functions, one concludes that these quadratic structure functions are more realistic, because the calculated gluon distribution functions are within QCD parton distributions. References [1] W. Buchmuller and G. Ingelman, eds., Proc. Workshop on Physics at HERA, Hamburg (1991. [2] A.M. Cooper-Sarkar, R.C.E. Derenish, and A.De Roeck, Int. J. Mod. Phys. A13 (1998 3385. [3] H 1 (C. Adloff, et al., Phys. Lett. B393 (1997 452. [4] W.J. Marciano, Phys. Rev. D29 (1984 5801. [5] H 1 (S. Aid, et al., Nucl. Phys. B470 (1996 3. [6] A. Zee, F. Wilczek, and S.B. Treiman, Phys. Rev. D10 (1974 2881. [7] E.B. Zijlstra and W.Van Neerven, Nucl. Phys. B383 (1992 525. [8] K. Bora and D.K. Choudhury, Phys. Lett. B354 (1995 151. [9] R.G. Roberts, The Structure of the Proton, Cambridge University Press, Cambridge (1990. [10] A.M. Cooper-Sarkar, et al., Z. Phys. C39 (1988 281; A.M. Cooper-Sarkar, et al., Proc. Workshop on Physics at HERA, (DESY, 1994, eds. W. Buchmüller and G. Ingelman, Vol.I, p. 155. [11] Lynne H. Orr and W.J. Stirling, Phys. Rev. Lett. 66 (1991 (13. [12] K. Prytz, Phys. Lett. B311 (1993 2861. [13] H 1 (I. Abt, et al., Phys. Lett. B321 (1994 161. [14] ZEUS (M. Drrick, et al., Phys. Lett. B345 (1995 576. [15] ZEUS (J. Breitweg, et al., Eur. Phys. J. C7 (1999 609. [16] H 1 (C. Adloff, et al., Eur. Phys. J. C21 (2001 33. [17] A.D. Martin, W.J. Stirling, and R.G. Roberts, Phys. Rev. D47 (1993 867. [18] A.D. Martin, W.J. Stirling, and R.G. Roberts, Phys. Lett. B354 (1995 155 [19] M. Gluck, E. Reya, and A. Vogt, Eur. Phys. J. C5 (1998 461. [20] A.D. Martin, W.J. Stirling, and R.G. Roberts, Phys. Lett. B306 (1993 145.