Das, Abhijeet

Saikia, A

English

IACS Institutional Repository

Indian J. Rhys. 82 (6), 789-794 (2008) Quark contribution in the evolution equation of the scaling violation of f 2 Abhijeet Das1* and A Saikia-' depa r tmen t of Phi/sics Dibru College, Dibrugarh - 786 UO'A Assam Indui -Dispur College. Guwaruiti 7S1 006 Assam Inriw E mail abh])eet_dds(r/ yahoo com Abstract : In a number of analyses in the literature, relations connecting the gluon density with the scaling violation of the structure function are reported In all these approaches, the DGLAP equation for the gluon is considered after neglecting the quark term In this work, we calculate the contribution coming from the quark term and estimate the error that enters when the quark contribution is neglecte?d Keywords DGLAP evolution equations, low x, Taylor expansion PACS Nos. 12 38 Bx, 12 38 Aw, 12 38 -t 1. Introduction In the literature, there are a number of approximate relations connecting the gluon density with the scaling violation of the structure function F2 [1-3] These relations are obtained by evaluating the convolution integral involving the gluon density in the DGLAP equation [4] after neglecting the convolution integral involving the singlet structure function i.e. the quarks. In the above methods the gluon convolution integral is evaluated by Taylor expanding the gluon distribution about a chosen point. In this work, we estimate the quark contribution in the DGLAP evolution equation for the singlet structure function using the framework of [5]. We then perform a numerical analysis in which we compare the analytically obtained value of the quark contribution using a test input distribution with the numerically obtained ones. We then determine the amount of error involved in neglecting the quark contribution for a number of standard non-perturbative parametrisations available in the literature [7-12]. Corresponding Author © 2008 IACS 790 Abhijeet Das and A Saikia In Section 2 we describe the formalism and obtain the analytical expression for the quark contribution. In Section 3 we perform the numerical analysis to compare the analytically and numerically obtained values of the quark contribution using a test distribution. In Section 4 we evaluate the amount of error involved in neglecting the quark contribution for various input distributions and in Section 5 we conclude with some discussions. 2. Formalism The DGLAP integrodifferential equation for the singlet structure function is given by a^l).M)[{3 + 4h(1-x)}/j(^) •(1) We consider the following term in the RHS of eq. (1) as the quark contribution QC = [{3 + 4ln(1-x)}F/(x,tf) + 2Ji^{(l + 2 2 )F 2 s ^ > Q 2 l -2F 2 s (x ,0 2 ) (2) Writing z - 1 - u, the integral in (2) becomes ,.j*{w(1-»f}*(£.*)-2j>(**) 0 v J 0 = ' 1 - ' 2 - (3) We evaluate the ^ in (3) as follows. The function F2s(x/1- u% O2) is expanded about a point z = a where 0 < a < 1 - x such that \u-a\<R, R being the radius of convergence. The nth term in the expanded integrand will be Tn^u-a) n_, F"-1 1 + (1-u)2 (n-1)! u (4) where F""1 = ^ - M du . We suppress the O2 dependence of F2! w=o Quark contribution in the evolution equation of the scaling violation ofF„ 791 The series can be integrated term by term and the resulting series converge. Integrating Tn we get at low x 1 - * n-1 4i = }Tndu = ^ar O r=0 M ) ^ 1 2(1-xf' 2(1-* )^ n- r + 1 n- r n - r - 1 n-1 -2> r=0 1_ _ 2 2 n - r + 1 n - r n - r - 1 (5) - _J"ft_L jr""1 where ar=^vF> The completed integrated series can be obtained by evaluating the terms Jv J2, J3 . The second Integral l2 in (3) can be trivially evaluated. The integral I thus being evaluated, the quark contribution becomes, AF*tf-*—a)+LF**) qC~3'* M - « 12 + 1 ^ 45 1-a -a VI 150 tf5) 1-a x 1-a a -a + H F 2 S ( 3 ) 90 ^ 1-a -« 29 60480 £?(6» 1-a • a r«0) p-a -a + a + fl(a) (6) where a is a parameter and fl(a) is given by fl(a) = f-7--aW*> 32 1-a - a ^120 3! F*4) 1-a - a { 11l__a2 240 2! f 11 _ a 4 ^ 8400 4! rf3)f-X -a I 1-a F S ( 5 ) 1-a •a 29 a_ 161280 5! 5 "\ .5(6) 1-a -a + •• (7) Let a0 be the value of a for which R(a) = 0. This gives . = * * ? < 1 > qc~-.2 1-a -CC + SQ 792 Abhijeet Das and A Saikia The function F2S(1) will be sensitive to the variation of x only when x/1-a = a-ai) Hence for x —> 0, a -* SQ and we get qc 3i-2 1-a a (8) where a satisfies the condition (from (7)), [32 -a 7«(2) 1 - a 11 _ a2 \ 240 2! -s(3) 1-a 3 A 1 a 120 3! F r2 <4)( X ) 1-a A 4 > 1_1_ _ a 8400 4! -S(5) 1-a 29 _ a ^ 161280 5! -S(6) 1-a + ••• = 0 0) The value of a is determined by solving (9) with a paramethsation for F2S. 3. Numerical analysis As a test distribution we take £ ( x ) - V ( 1 - A ) ' ( 1 + yx) which at low x F2s(x) = A ^ - (10) Inserting (10) in (9) and considering the first three terms of the series, we get the cubic ( (A + 2)(A+3) A + 2 l 3 (71 A + 2'i 2 1 + - — - a a [23 11(A + 2)' T6 + 240~~ a -(32 7 11(A + 2) (A + 2)(A + 3) , | = 0 32 240 120 For the representative values A = - 0.5, - 0.3, and - 0.08, this gives (11) 7 3 47 2 241 51 _ - a a + a =0 8 32 160 160 (12) 180 3 219 2 3137 4019 _ 5 4 60 300 (13) Quark contribution in the evolution equation of the scaling violation of F 793 1218 3 1007 2 3051 35347 a — a + a 125 80 200 10000 (14) respectively The above cubics are solved numerically by Horner's method [6] to find the value of a Using the value of a the quark contribution from (8) is calculated The quark contribution is also calculated numerically using Simpson's rule from (2) The results are shown in Table 1 Table 1. Comparison of analytical and numerically evaluated values of qc A - 0 5 - 0 3 - 0 08 a 0 27211738583 0 27774612055 0 28345901264 Error in evaluating (/ -5 0 x 1010 60 x 108 4 0 x 107 Value of qc (anal) x x* -1 562816773 -1 0046321418 -0 2898924899 Value of qc (num) x x* -1 574607 -1 026599 -0 2975729 The results clearly show that the analytically derived expression for the quark contribution is in accurate agreement to the numerically computed values 4. Results The quark contribution (8) derived from the DGLAP evolution equation for singlet structure function (1) will be compared to the gluon contribution given by the following term in (1) gc^dz^+^-zf^tf (15) which will allow us to determine the amount of error that enters when the quark contribution is neglected. We consider a number of standard non-perturbative parametnsations available in the literature [7 -12] and calculate the percentage error that enters when the quark contribution is neglected The results, for four quark flavours are shown in Table 2 Table 2. Estimation of error that enters when quark contribution is neglected for various inputs A a qc & gc Actual value Approx value error (x x*) (x x* ) (%) - 03 Aq = 0 411 0 277746 qc = - 1004632 1 96324619 2 37615 17 38 - 0 3 Ag = 0 775 0 58778 gc = 3 066 - 0 04 Aq = 0 92 0 284445 qc = 0 147086 7 74825 7 88357 1 72 - 0 04 Ag = 2 07 0 706696 gc = 3 808488 - 0 26 Aq = 0 222 0 29135 qc = - 1 018093 5 799305 6 0253218 3 75 - 0 33Ag = 1 9 0 50518 gc = 3 171222 Paramete-nsations MRS(A) MRS(R) MRST(01) A Xq = Xg = Xq = Ag = Aq = Ag = 794 Abhijeet Das and A Saikia In the above analysis we have observed that for inputs with positive values of A [8, 10, 12], the values of q.c. are diminishingly small but for the inputs with negative values of A [7, 9, 11], the values of q.c. are considerable. Neglecting the quark contribution thus brings in error particularly when the parton distributions have an x* dependence with negative values of A. 5. Conclusions and discussions We evaluated the convolution integral for the quark sector in the DGLAP equation (1) analytically using the framework of [5] and calculated the quark contribution q c defined in eq. (2). We found that the analytical result for q c is in excellent agreement with the numerically computed value of the same. The fact that the formalism allows us to obtain the point of expansion in terms of the parameter A characterising various parton distributions available in the literature, was taken advantage of to calculate the amount of error entering in the relations like that in Ref. [1 - 3] where the quark contribution is neglected compared to the gluon contribution. We analysed various standard non-perturbative parameterizations in vogue indicating that the error for inputs having positive values of A is diminishingly small while those having negative values of A, the error is considerable. References [1] K Prytz Phys. Lett. B311 286 (1993); Phys. Lett. B332 393 (1994) [2] K Bora and D K Choudhury Phys. Lett. B354 401 (1995) [3] M B Gay Ducati and V P B Goncalves Phys. Lett. B390 401 (1997) [4] Yu L Dokshitzer Sov. Phys. JETP 46 641 (1977); V N Gribov and L N Lipatov Sov. J. Nucl. Phys. 28 822 (1978); G Altarelli and G Parisi Nucl. Phys. B126 298 (1077) [5] A Saikia Pramana - J. Phys. 52 257 (1999) [6] E A Volkov Numerical Methods (Moscow : Mir Publishers) (1982) [7] A D Martin, W J Stirling and R G Roberts Phys. Lett. B306 145 (1993) [8] A D Martin, W J Stirling and R G Roberts Phys. Rev. D50 6734 (1994) [9] A D Martin, W J Stirling and R G Roberts Phys. Lett. B387 419 (1996) [10] A D Martin, R G Roberts, W J Stirling and R S Thorne Eur. Phys. J. C4 463 (1998) [11) A D Martin, R G Roberts, W J Stirling and R S Thorne Eur. Phys. J. C23 73 (2002) [12] A D Martin, R G Roberts, W J Stirling and R S Thorne Eur. Phys. J. C28 455 (2003) 

Quark Contribution in the Evolution Equation of the Scaling Violation of F<SUB>2</SUB>

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Quark Contribution in the Evolution Equation of the Scaling Violation of F<SUB>2</SUB>

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