METHOD OF CHARACTERISTICS AND GLUON DISTRIBUTION FUNCTION

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1 METHOD OF CHARACTERISTICS AND GLUON DISTRIBUTION FUNCTION Saiful Islam and D. K. Choudhury Dep. Of Physics Gauhai Universiy, Guwahai, Assam, India. saiful.66@rediffmail.com ; dkc_phys@yahoo.co.in

2 Absrac The applicaion of mehod of characerisics o obain analyical soluion of Dokshizer, Gribov, Lipaov, Alarelli, Parisi (DGLAP evoluion equaions is relaively new. In he presen paper, we solve DGLAP equaions by using his mehod and obain an analyical form of gluon disribuion funcion a small-. Comparisons wih eac resuls as well as wih daa are repored.

3 Inroducion : Dokshizer, Gribov, Lipaov, Alarelli, Parisi (DGLAP evoluion equaions have been playing very imporan role in undersanding he dynamics of evoluions of quark and gluons. In his paper, we solve DGLAP equaions in leading order by using mehod of characerisics and obain an analyical form of gluon disribuion funcion a small-.

4 Before enering he main conen of his paper, Le us very briefly inroduce - 7.Deep Inelasic Scaering ( DIS. Mehod of Characerisics wih QCD Phase Diagram

5 Deep Inelasic Scaering ( DIS : e p scaering e - X q e - p The naure of scaering depends on q - A Small - ( large Q : The probe resolves he srucure of he arge. The scaering is inelasic. Muli-paricle producion in he hadron final sae.

6 QCD Phase Diagram -

7 Mehod of Characerisics : Mehod of characerisics an imporan echnique for solving iniial value problems of firs order Parial Differenial Equaion ( PDE. In his mehod, he coordinaes (, are ransferred o an appropriae new se of coordinaes (s, so ha he PDE reduces o ODE (ordinary differenial equaion which can be solved by he sandard mehods.

8 . n 3 -, 4 (, ln(, w ]...(, ( ( 9, ( } ( ( { (, (, (, ( } ln( 8 [{ ( 3, ( f Q here z F z z z G z z z z dz G z zg dz G n G s s f s DGLAP equaion for Gluon Disribuion have he sandard form :

9 To evaluae he inegrals of equaion (, we inroduce a variable u as : u = -z ( And epress /z approimaely for small values of as : /z = /(-u = (-u - = + u (3 Using Tailor Epansion for F s (/z, and G (/z, a small- and performing he inegraions w.r.. z of equaion (, we ge, Where, a small- s G(, G(, s F P( G(, Q( R( F (, S(,......( 4 P( Q( R( { 68 S( ln(/ ( } 6 {4ln(/ (4 3} ( ( ( (8

10 A reasonable approimae relaionship beween F s (, and G(,, represening he relaive srengh of gluon o single disribuion, can be aken as : Using eqn.(9 in (4, F s (, = k G(,.. (9 wih, < k <. J ( G(, G(, H ( G(, ( Where, H( P( kr( ( J( Q( ks( (

11 Equaion ( is a Firs Order PDE, which can be solved by using Mehod of Characerisics : Le us inroduce wo variables S and as follows : d ds d ds J ( ( (4 Use of eqns. (3 & (4 in eqn. ( gives, dg U ( S, G( S,...(5 ds which is an ODE in new coordinaes (S,. Here, U(S, H ( P( kr( [{ ln( k {4 ln(/ ( 9 ln(/ (3 ( 6 ] ( }...(6

12 Coordinae ransformaion equaions :. d ds ln S C (7 Boundary condiion This gives, a S =, = and =. S ln( / (8

13 ..( ( ln 9 68 ( ( C S k k ds d J ds d Using same Boundary Condiion, finally we obain,...( ( 9 68 ( k

14 In erms of S and, he ODE (5 now becomes, dg( S, G( S, Inegraing, 4 [( 4 4k ( 3 k 6 4k ( 3 ep{ ( 68 3( 9 k S 68 k S} ds 9 4k ( 3...( 3 ln lng( S, 4 [( 4 4k ln ( 3 k S 4k ( 3 S 3 ( 4k ( k ( 9 ep{ k S ( 68 k S}] C ( Boundary condiion : when S =, G(S, = G(.

15 This gives, G( S, 4k 4( G( ep[ 3 ( 4 4k ln ( S k 4 [( 4 k S 4k 4( 6 3 ep{ 68 ( k 9 ( 4k ( ( k 9 68 k S}]...(3 9 S Inpu funcion : Then he inpu funcion becomes, when S =, = and =, G( = G(,.(4

16 Using his inpu and replacing coordinaes (S, by (, applying ransformaion equaions (8 and ( in eqn. (3, we obain, G(, G(, 4k 4( 6 3 ep[ ( 68 ( k 9 4 ( k ln( 4 4 ln{ ( 68 ( k 9 4k }( 3 68 ( k 9 { 4k ( 3 3 ln( ep( 68 3( k 9 ( 68 k ln( 9 {ln( } } ] (4 This is our final soluion for Gluon Disribuion.

17 Sandard DLLA resul for gluon disribuion is given by G(, G(, 48 ep[( ln( ln(.5 ]...(5

18 Resuls and Discussions : In he presen paper, we solve he LO coupled DGLAP equaions in he Bjorken -space by applying he mehod of characerisics and obain a form of gluon disribuion valid o be a small-. For quaniaive analysis, we use MRSTLO inpu a Q o = 4 GeV, QCD cuoff parameer = MeV and n f = 4.

19 We compare our prediced resul wih MRSTLO eac resuls and wih he Daa from H a differen Q values for -5 < < -. A comparison is also made wih sandard DLLA resul. The dependence of our predicion on he values of he arbirary consan k have also been noed. The accepable range of k is found o be < k < -. I is also observed ha he prediced resul is almos independen of k a k -.

20 In figure, we have ploed gluon disribuion G (,, agains -values a a fied Q. Comparison is made wih MRSTLO eac resuls a differen Q = 8,9,, GeV for he same -5 < < - and k =. and a good agreemen is obained.

21 Figure :

22 In figure, he same is done a Q = 6,7 and,5,,5 GeV and i is seen ha disagreemen increases a lower Q as well as a higher Q values.

24 Figure :

25 In figure 3, We compare our prediced resul wih MRSTLO eac resuls, wih he Daa from H and wih sandard DLLA resul a Q = 9 and GeV for same -5 < < - and k =.. Comparison shows more suiabiliy of our prediced resul over sandard DLLA resul. Figure : 3

26 Figure 4 show he dependence of he prediced gluon disribuion on he arbirary consan k.

27 Figure 4 show he dependence of he prediced gluon disribuion on he arbirary consan k.

28 Figure : 4

29 Conclusion : In his paper, he LO coupled DGLAP equaions are solved for gluon disribuion funcions by applying he mehod of characerisics and a form of gluon disribuion valid o be a small- is obained. A good agreemen, wih MRST eac resuls and daa wihin he kinemaical range -5 < < - and 8 < Q < Gev for < k < -, is obained. As he spliing funcions a Ne-o-leading order (NLO and Ne-o-Neo-leading order (NNLO are available in lieraure, i will be ineresing o sudy he gluon disribuion beyond he leading order using he mehod used in his paper.

30 References :. V.N. Gribov and L.N. Lipaov, Sov. J. Nucl. Phys. 5, 438 (97.. L.N. Lipaov, Sov. J. Nucl. Phys., 94 ( G. Alarelli and G. Parisi, Nucl. Phys. B6, 89 ( Yu. L. Dokshizer, Sov. Phys. JETP P46, 64 ( L.F. Abbo, W.B. Awood and R.M. Barne, Phys. Rev. D, 88 ( S.J. Farlow, Parial diffrenial equaions for scieniss and engineers (JohnWilley, 98 p D.K. Choudhury and P.K. Sahariah, Pramana J. Phys. 58, 59 (.

31 8. HERMES Coll., A. Airapeian e al., hep-e/ M.B. Gay Ducai, Brazilian J. Phys. 3, (. 3.G. Soyez, [arxiv:hep-ph/659]. 4.. M.B. Gay Ducai and V.P Goncalves, [arxiv:hep- ph/398].. C. Marque and G. Soyez, [arxiv:hep-ph/548]

Improved Approximate Solutions for Nonlinear Evolutions Equations in Mathematical Physics Using the Reduced Differential Transform Method

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