Choudhury, D K

Deshamukhya, Atri

IACS Institutional Repository

Indian J. Phys. 7SA (5). 573-577 (2001)U P  Aan international journalAsymptotics and corrections to asymptotics of non-singlet structure function at low xAtri D esham  ukhya a i^  D  K Choudhury*Department of Physics| Gauhati University,Guwahati 781 Olf, Assam, IndiaE-mail : dkc@^phys cjb.nct R e c e iv e d  J4  D e c e m b e r  1999, a c c e p te d  2H J u n e  2 0 0 JAbstract : We comment on the uniqueness of r-evolution {t = log (Q^ / A*)) of non-singlet structure functions at low a obtained from DGLAP equations Dependence of asymptotics on boundary conditions are also discussed.Keywords : structure functions, low x , asymptoticsPACS No. : l2 .38.-tIn recent years, an approxim ate m ethod o f  so lv in g  D G L A P  equations [ 1 -4] at lo w  x has been pursued [ 5 ,6]. In that approach, we expressed the D okshitzer-G ribov-L ipatov-A ltarelli-Parisi (DGLAP) equations as partial differential equations in x (the Bjorken variable x = Q^ f2p,q)  and / ( /  s= In g^  /  A^) using  the Taylor series expansion  and assum ing its validity at low  x. One o f the lim itation s o f  the approach is that the so lutions  reported are not unique [ 5 ,6]. T hey are selected  as the sim plest ones with a sin g le  boundary condition  ~  the non-perturbative x distribution at so m e initial point t =  H ow ever, com plete  solution o f  D G L A P  equations w ith tw o differential variables in general, need tw o  boundary cond itions [7, 8].The aim  o f  the present note is to explore the uniqueness o f  the solution w hen it sa tisfies som e given  physically  appropriate boundary con d ition s. T h e D G L A P  equation for non-singlet structure function  w h ich  e v o lv e  independent o f  sing let and gluon distributions [1 -4 ] isdFdtNS ^—  = - ^ [ ( 3 + 4  log (1 -  X)) (X, t)( 1)^Corresponding Authorw h ere  t =  log iQ ' and A, =*a 2 j ' / 3 3 ^ 2A/, Nj  b e in g  thenumber o f  quark flavours.Let us introduce the variable m =  I -  r. and note thatl~ w P )This .series (2) is convergent for m < 1. Since x < 2 < 1, so  0  < M <  1 -  jc and hcncc the convergence condition is satisfied. U sing (2), we wite in (1)d x ‘(3)-^5| X , , | = r/=l '  • \k = \  Jw hich covers the w hole range o f  u, 0 <  « <  1 -N on-sing let structure functions are expected  to be w ell- behaved in the entire x range, unlike the gluon or singlet structure functions w hich m ight d iverge for or ->  0  as in B alitsky-Fadin- K uraev-Lipatov (B F K L ) inspired m od els [9 ,1 0 ] . It is thereforejustified  if  the higher order derivatives i.e. —— j— f o r />  1 arcd  Xneglected  in (3). T his is  m ore justifiab le for sm all x(x «  1 ), yieldingkdF'^^{x,t), ,  j  -  f ( j t ,  f) + jc £  «*dx (4)©2001 lACS576 Atri Deshamukhya and D K Choudhuryasymptotic scaling [18] holds. Besides at very high jc (jc ->  1), it should also conform to the boundary conditions as given by (25) instead of asymptotic limit (26).This then yields=P  (43)and= P (44)instead o f (27) and (28).Using (43) and (44) one finally obtains1-to to X(JC)| Jl-*11-to X { x ) \y(jr)- F ( x  t  il’ o)U ^ >  Yix) +  F(AT,ro)| Jt->J 'nx)\K(:r)(45)FIgara 1. h(x) va x u defined in (33) of the text.L et us n o w  m ak e a fe w  co m m e n ts . A s  h as been  mentioned e a r lie r , th e  D G L A P  e q u a t io n s  h a v e  b e e n  s o lv e d  earlier a n a ly tica lly  and that to o  e x a c tly  in th e low -jr reg im e . Extensive  n u m erica l a n a ly s is  o f  th e  e x a c t so lu tio n s  o f  th o se  equations m low -jr reg io n  has a lso  b een  d o n e  in  the la st fe w  y ea rs w ith  Next to  lea d in g  order (N L O ) accu ra cy  [1 9 -2 1 ] . Q u es tio n s  about the sc h e m e  d e p e n d e n c e  o f  su ch  a n a ly s ise s  has ju s t  began  [221In the lig h t o f  su ch  a p ro g ress , it is  m ea n in g fu l to  conclude w ith  the p h y s ic s  is su e  c lea r ly  b rou gh t o u t in  the present work reported  at L O  l e v e l : w e  h a v e  sh o w n  that the asym p totics ol D G L A P  eq u a tio n  d ep en d  c ru c ia lly  on  the boundary conditions so  m u ch  that a ltern a tiv e  a sy m p to tic s  other than the D L A  one^ ca n  a ls o  b e  o b ta in e d  a s h a v e  b e e n  p u rsu ed  e a r lier  [5 , 6| H o w ev er , i f  the D L A  a sy m p to tic s  are im p o sed  as boundary co n d itio n s, on e can  h a v e  a c lo se d  ex p ress io n  to  take in to account the fin ite-jc c o rrec tio n s  to  su ch  a sy m p to tic s . T h is  feature is fou n d  to  be true at N L O  le v e l to o  [2 3 ] .AcknowledgmentsW e are gratefu l for the support from  the D ep artm en t o f  Science and T ech n o lo g y , G o v ern m en t o f  India .References[1] G AltarclU and G Parisi N u c l .  P h y s , B12 298 (1977)[2] G Altarclli P h y s .  R e p . 811 (1981)[3] V N Gribov and L N Lipatov Sov. J. N u c l . P h y s . 20 94 (197(41 Y L Dokshilzer S o v . P h y s . J E T P  20 94 (1977)[5] D K Choudhury and J K Sarma P r a m a n a  - J . P h y s  38 4S1 (1992); 39 273 (1992)[6] J K Sarma, D K Choudhury and G K Medhi P h y s . U t t . B403 H9 (1997)[7] I Sneddon E l e m e n t s  o f  P a r t i a l  D i f f e r e n t i a l  E q u a t i o n s  (Neu York ; McGraw-Hill) p 50 (1957)[8] F Ayres (Jr) D if fe r e n t ia l  E q u a t io n s  (Singapore : McGraw Hill) p 240 (1980)[9] Y Y Balitsky and L N Lipatov S o v .  J . N u c l .  P h y s . 28 822(1978) ; E A Kuraev, L N Lipatov and V S Fadin Sov. P h y s45 199 (1977)[10] J Kwiecinski. A D Martin and P J Sutton P h y s .  R ev . D46921 (1992)[11] R G Roberts T h e  S t r u c t u r e  o f  P r o to n  (Cambridge : Cambridge Univ. Press) p 30 (1990)[12] F J Yndurain 7Vie T h e o r y  o f  Q u a r k  a n d  G lu o n  In te r a c t io n s (Berlin : Springer-Verlag) p 129 (1992)[13] C Pascaud E  Z o m e r  D E S Y  96-266[14] HI Collabomtion : C Adloff et at Nucl Phys. B497 3 (1997). Hi Collaboration : S Aid et at Nuct. Phys. B4703 (1996) ; ZEllS Collaboration : J Breitweg et at Phys. Lett. B407 432 (1997)[15] CCPR Collaboration : W C Setigman et at Phys. Rev.1213 (1997)which reduces to for x ^ X q , Eq. (45) thus containsfinite -  JC corrections to the DLA result (34) in a closed form.Asymptotics and corrections to asymptotics of non-singlet structure function at low x 577[16] B I Ermolaev. M Greco and S I Troyan CERN-TH/99-155[17] D K Choudhury O n  A p p r o a c h  to  D o u b le  A s y m p to t ic  S c a lin f t  a t  lo w  X I C T P  (Preprint IC/94/323)118] R D Ball and S Forte P h y s .  L e t t . B335 77 (1994) ; P h y s  L e t t  B336 77 (1994) ; HI Collaboration N u c l .  P h y s . B470 3(1996)[19] AD Martin, W J Stirling. R G Roberts and R J Thome (MRST) E ur. P h y s .  I  C4 463 (1998) ; hep-ph/9806404(20] M Gluk, E Reya and A Vogt hep-ph/9806404[21] Coordinated Theoretical / Experimental Project on QCD Phenomenology (CTEQ) Collab : H L Lai e t  a l hep-ph/ 9903282122] R S Thorne, P h y s ic a l  R ev . D60 034031 (1999)[23] A Deshamukhya and D K Choudhury in P r o c e e d m tis  o} th e  2 n d  R e g io n a l  C o n fe re n c e  on  P h y s ic s  R e s e a r c h  in  N o r th  E a s t (Physics Academy of North East, 2000) p34

Asymptotics and Corrections to Asymptotics of Non-Singlet Structure Function at Low x

http://arxiv.iacs.res.in:8080/jspui/bitstream/10821/5959/1/Asymptotics%20and%20Corrections%20to%20Asymptotics_By%20Atri%20Deshamukhya.pdf

Asymptotics and Corrections to Asymptotics of Non-Singlet Structure Function at Low x

Abstract

Similar works

Full text

Available Versions

IACS Institutional Repository