Particular and Unique solutions of DGLAP evolution equations in Next- to-next-to-leading Order and Structure Functions at low-x

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1 Inernionl Journl o ienii Reserh Puliions Volume Issue ovemer Priulr Unique soluions o DP evoluion equions in e- o-e-o-eing Orer ruure unions low- sr- We resen riulr unique soluions o single non-single Doshizer-riov-iov- lrelli-prisi DP evoluion equions in ne-o-ne-o-leing orer O low- We oin -evoluions o eueron roon neuron ierene rio o roon neuron sruure unions low- rom DP evoluion equions he resuls o -evoluions re omre wih HER lo- low-q We lso omre our resul o -evoluion o roon sruure union wih reen glol rmeerizion P o: 8-6H Ine erms- Priulr soluion omlee soluion Unique soluion lrelli-prisi equion ruure union ow- hysis Rsn Rjhow Physis Dermen H ollege Jmugurih oniur ssm Ini non-single sruure unions in leing orer O neo-leing orer O hene -evoluion o eueron roon neuron ierene rio o roon neuron - evoluion o eueron in O O low- hve een reore he sme ehnique n e lie o he DP evoluion equions in ne-o-ne-o-leing orer O or single non-single sruure unions o oin -evoluions o eueron roon neuron ierene rio o roon neuron sruure unions hese O resuls re omre wih he HER H [] [] low- low-q we lso omre our resuls o -evoluion o roon sruure unions wih reen glol rmeerizion [] II HEORY he DP evoluion equions wih sliing unions or I IRODUIO single non-single sruure unions in O re in he n our erlier wors [-] we oin riulr soluion o he sr orms [-] IDoshizer-riov-iov- lrelli-prisi DP evoluion equions [5-8] or -evoluions o single s [ { } I s I s I s [ { } I s I s I I { } where I qq qg I Pqq Pqg 5 I 6 I 7

2 Inernionl Journl o ienii Reserh Puliions Volume Issue ovemer I P Here Q qg P P [ P is he QD u o rmeer lso P qg [ P qg 8 P R P 67 π π 8 P 5 P P 56 8 qq 6 5 qg π qg 8 P qg P qg P qg qq qq P z z qg P = = = R = ½ qq P PP { } z z

3 Inernionl Journl o ienii Reserh Puliions Volume Issue ovemer P P P qg wih = = - Here resuls re rom ire -se evoluion P is lule using orrn ge [5] Ee or vlues very lose o zero o P his rmeerizions evie rom he e eressions y less hn one r in hous whih n e onsier s suiienly ure or miml ury or he onvoluions wih qur ensiies sligh jusmen shoul one using low ineger momens [6] he srong ouling onsn Q α is rele wih he -union s [7] 6 6 log s s α π α π α π Q Q α α where e us now inroue he vrile u = -w noe h [8] u u w he ove series is onvergen or u < ine <w< so <u<- hene he onvergene rierion is sisie ow using ylor ension meho we n rewrie w s

4 Inernionl Journl o ienii Reserh Puliions Volume Issue ovemer u w u u whih overs he whole rnge o u <u<- ine is smll in our region o isussion he erms onining higher owers o n e neglee s our irs roimion s isusse in our erlier wors [-] w n e roime or smll- s u w imilrly w w n e roime or smll- s u w u w Using equions in equions 5 erorming u-inegrions we ge s s s where - 5 qq qg qg qg here

5 Inernionl Journl o ienii Reserh Puliions Volume Issue ovemer 5 [ [ e us ssume or simliiy [-] = K where K is union o In his onneion erlier we onsiere [-] K = e - where re onsns greemen o he resuls wih eerimenl is oun o e eellen or = 5 = 5 = = 5 = or low- in leing orer = = 6 = 5 = -8 in ne-o-leing orer lso we n onsier wo numeril rmeers suh h = = = where y suile hoie o we n reue he error o minimum Hene equion eomes P Q 5 where P Q [ K K K K K K K K K eonly using equions in equions erorming u-inegrion equion eomes P Q 6 where

6 Inernionl Journl o ienii Reserh Puliions Volume Issue ovemer 6 ] [ P ] [ Q he generl soluions [ ] o equions 5 is U V = where is n rirry union U = V = where re onsn hey orm soluion o equions Q P 7 We oserve h he grnge s uiliry sysem o orinry ierenil equions [ ] ourre in he ormlism n no e solve wihou he iionl ssumion o linerizion equion inrouion wo numeril rmeers hese rmeers oes no ee in he resuls o - evoluion o sruure unions olving equion 7 we oin e U e[ V where [ K K K K K K K K K K K K Priulr oluions I U V re wo ineenen soluions o equion 7 i α re rirry onsns hen V = αu + my e en s omlee soluion o equion 5 hen he omlee soluion [ ] α e e[ 8 is wo-rmeer mily o lnes he one rmeer mily eermine y ing = α hs equion e e[ α α Diereniing equion wih rese o α we oin e α Puing he vlue o α gin in equion we oin enveloe

7 Inernionl Journl o ienii Reserh Puliions Volume Issue ovemer 7 e e[ hereore e whih is merely riulr soluion o he generl soluion ow eining e = where = Q Λ ny lower vlue Q = Q we ge rom equion e whih gives he -evoluion o single sruure union in O or α Proeeing ely in he sme wy eining e where we ge e whih gives he -evoluion o non-single sruure union in O or = α We oserve h i ens o zero hen equions ens o O equion [] i ens o zero hen hese equions ens o O equion [-] Physilly ens o zero mens numer o lvours is high gin eining

8 Inernionl Journl o ienii Reserh Puliions Volume Issue ovemer 8 e we oin rom equion e whih gives he -evoluion o single sruure union in O or = α imilrly eining e We ge e whih gives he -evoluion o non-single sruure union in O or = α Deueron roon neuron sruure unions mesure in ee inelsi elero-rouion n e wrien in erms o single non-single qur isriuion unions [] s = 5 5 = n = n = 8 ow using equions in equion 5 we will ge -evoluion o eueron sruure union low- in O s e e where he inu unions re 5 5 imilrly using equions in equions we ge he evoluions o roon neuron ierene rio o roon neuron sruure unions low- in O s

9 Inernionl Journl o ienii Reserh Puliions Volume Issue ovemer e e n n e [ n R n n where R is onsn or ie- n he inu unions re n n or he omlee soluion o equion 5 we e = α in equion 8 We oserve h i we e = α in equion 8 ierenie wih rese o α s eore we n no eermine he vlue o α In generl i we e = α y we ge in he soluions he owers o he o-eiien o in eonenil r in - evoluions o eueron roon neuron ierene o roon neuron sruure unions e yy- he numerors o he

10 Inernionl Journl o ienii Reserh Puliions Volume Issue ovemer irs erm insie he inegrl sign e yy- or -evoluions in O Hene i y vries rom minimum = o mimum = hen yy- vries rom o hus y his mehoology inse o hving single soluion we rrive o soluions o ourse he rnge or hese soluions is resonly nrrow Unique oluions Due o onservion o he eleromgnei urren mus vnish s Q goes o zero [ ] lso R in his limi Here R inies rio o longiuinl rnsverse ross-seions o virul hoon in DI roess his imlies h sling shoul no e vli one in he region o very low-q he ehnge hoon is hen lmos rel he lose similriy o rel hooni hroni inerions jusiies he use o he Veor eson Dominne VD one [-] or he esriion o In he lnguge o erurion heory his one is equivlen o semen h hysil hoon sens r o is ime s re oin-lie hoon r s virul hron [] he ower euy o elining sling violions wih iel heorei mehos ie riive orreions in QD remins however unhllenge in s muh s hey rovie us wih rmewor or he whole -region wih essenilly only one ree rmeer Λ [5] or Q vlues muh lrger hn Λ he eeive ouling is smll erurive esriion in erms o qurs gluons inering wely mes sense or Q o orer Λ he eeive ouling is ininie we nno me suh iure sine qurs gluons will rrnge hemselves ino srongly oun lusers nmely hrons [] so he erurion series res own smll-q [] hus i n e hough o Λ s mring he ounry eween worl o qusi-ree qurs gluons he worl o ions roons so on he vlue o Λ is no reie y he heory; i is ree rmeer o e eermine rom eerimen I shoul ee h i is o he orer o yil hroni mss [] ine he vlue o Λ is so smll we n e Q = Λ = ue o onservion o he eleromgnei urren [] his ynmil reiion grees wih mos ho rmeerizions wih he [5] Using his ounry oniion in equion 8 we ge = e 5 ow eining e = we ge rom equion 5 e 6 whih gives he -evoluion o single sruure union in O Proeeing ely in he sme wy we ge e 7 whih gives he -evoluion o non-single sruure union in O gin eining

11 Inernionl Journl o ienii Reserh Puliions Volume Issue ovemer e we oin rom equion 5 e 8 whih gives he -evoluion o single sruure union in O imilrly e we ge e whih gives he -evoluion o non-single sruure union in O hereore orresoning resuls or -evoluion o eueron roon neuron ierene rio o roon neuron sruure unions re e e e n n

12 Inernionl Journl o ienii Reserh Puliions Volume Issue ovemer n [ R n n gin -evoluion o eueron sruure union in O is e e lrey we hve menione [-] h he eerminion o -evoluions o roon neuron sruure unions lie h o eueron sruure union is no suile y his mehoology I is o e noe h unique soluions o evoluion equions o ieren sruure unions re sme wih riulr soluions or y mimum y = in = α y relion III REU D DIUIO In he resen er we omre our resuls o -evoluion o eueron roon neuron ierene rio o roon 5 neuron sruure unions wih he HER [] [] low- low-q In se o HER [] roon neuron sruure unions re mesure in he rnge Q 5 ev oreover here P ev where P is he rnsverse momenum o he inl se ryon In se o roon eueron sruure unions re mesure in he rnge 75Q 7eV We onsier numer o lvours = We lso omre our resuls o -evoluion o roon sruure unions wih reen glol rmeerizion [] his rmeerizion inlues rom H-6 \ ZEU- 67X8 E665 ig: Resuls o -evoluions o eueron sruure unions or he reresenive vlues o given in he igures or y = soli lines y mimum she lines in = α y relion he she line lso reresens resuls o unique soluion or onveniene vlue o eh oin is inrese y ing i where i = re he numerings o

13 Inernionl Journl o ienii Reserh Puliions Volume Issue ovemer urves ouning rom he oom o he lowermos urve s he - h orer D oins lowes-q vlues in he igures re en s inu o es he evoluion equion In ig we resen our resuls o -evoluions o eueron sruure unions or he reresenive vlues o given in he igures or y = soli lines y mimum she lines in = α y relion he she line lso reresens resuls o unique soluion D oins lowes-q vlues in he igures re en s inu o es he evoluion equion greemen wih he [] is oun o e goo O resuls or y = re o eer greemen wih eerimenl in generl ig: Resuls o -evoluions o roon sruure unions or he reresenive vlues o given in he igures or y = soli lines y mimum she lines in = α y relion he she line lso reresens resuls o unique soluion or onveniene vlue o eh oin is inrese y ing i where i = re he numerings o urves ouning rom he oom o he lowermos urve s he -h orer D oins lowes-q vlues in he igures re en s inu o es he evoluion equion In ig we resen our resuls o -evoluions o roon sruure unions or he reresenive vlues o given in he igures or y = soli lines y mimum she lines in = α y relion he she line lso reresens resuls o unique soluion D oins lowes-q vlues in he igures re en s inu o es he evoluion equion greemen wih he [] is oun o e goo O resuls or y = re o eer greemen wih eerimenl in generl In ig we omre our resuls o -evoluions o roon sruure unions wih reen glol rmeerizion [] long she lines or he reresenive vlues o given in he igures or y = soli lines y mimum she lines in = α y relion he she line lso reresens resuls o unique soluion D oins lowes-q vlues in he igures re en s inu o es he evoluion equion greemen wih he resuls is oun o e goo In ig we resen our resuls o -evoluions o neuron sruure unions or he reresenive vlues o given in he igures or y = soli lines y mimum she lines in = α y relion he she line lso reresens resuls o unique soluion D oins lowes-q vlues in he igures re en s inu o es he evoluion equion greemen wih he [] is oun o e goo O resuls or y = re o eer greemen wih eerimenl in generl

14 Inernionl Journl o ienii Reserh Puliions Volume Issue ovemer ig: Resuls o -evoluions o roon sruure unions wih reen glol rmrizion long she lines or he reresenive vlues o given in he igures or y = soli lines y mimum she lines in = α y relion he she line lso reresens resuls o unique soluion D oins lowes-q vlues in he igures re en s inu or onveniene vlue o eh oin is inrese y ing 5i where i = re he numerings o urves ouning rom he oom o he lowermos urve s he -h orer ig: Resuls o -evoluions o neuron sruure unions or he reresenive vlues o given in he igures or y = soli lines y mimum she lines in = α y relion he she line lso reresens resuls o unique soluion or onveniene vlue o eh oin is inrese y ing i where i = re he numerings o urves ouning rom he oom o he lowermos urve s he -h orer D

15 Inernionl Journl o ienii Reserh Puliions Volume Issue ovemer 5 oins lowes-q vlues in he igures re en s inu o es he evoluion equion ig5: Resuls o -evoluions o ierene o roon neuron sruure unions or he reresenive vlues o given in he igures or y = soli lines y mimum she lines in = α y relion he she line lso reresens resuls o unique soluion or onveniene vlue o eh oin is inrese y ing i where i = re he numerings o urves ouning rom he oom o he lowermos urve s he -h orer D oins lowes-q vlues in he igures re en s inu o es he evoluion equion

16 Inernionl Journl o ienii Reserh Puliions Volume Issue ovemer 6 ig6: Resuls o -evoluions o rio o roon neuron sruure unions n soli lines or he reresenive vlues o given in he igures D oins lowes-q vlues in he igures re en s inu In ig5 we resen our resuls o -evoluions o ierene o roon neuron sruure unions or he reresenive vlues o given in he igures or y = soli lines y mimum she lines in = α y relion he she line lso reresens resuls o unique soluion D oins lowes-q vlues in he igures re en s inu o es he evoluion equion greemen wih he [] is oun o e goo O resuls or y mimum re o eer greemen wih eerimenl in generl In ig6 we resen our resuls o -evoluions o rio o roon neuron sruure unions n soli lines or he reresenive vlues o given in he igures hough oring o our heory he rio shoul e ineenen o ue o he l o suiien moun o ue o lrge error rs ler u onlusion n no e rwn hough we omre our resuls whih y = y mimum in = α y relion wih greemen o he resul wih eerimenl is oun o e eellen wih y = or - evoluion in ne-o-ne-o leing orer In ig7 we lo soli line oe line soli line oe line where = α s π gins Q in he Q rnge 75 Q 5 ev hough he elii vlues o re no neessry in luling - evoluion o ye we oserve h or = 8 = 5 errors eome minimum in he Q rnge 5 Q 5 ev Q ev 5 ig7: soli line oe line soli line oe linewhere = α s π gins Q in he Q rnge 75 Q 5 ev REEREE [] R Rjhow J K rm Inin J Phys [] R Rjhow J K rm Inin J Phys 78 7 [] R Rjhow J K rm Inin J Phys [] R Rjhow U Jmil J K rm he-h58 [5] lrelli Prisi ul Phys [6] lrelli Phys Re 8 8 [7] V riov iov ov J ul Phys 75 [8] Y Doshizer ov Phys JEP [] lo H ollorion DEY 8-6 he-e8 8 [] rneoo he-e 6 ul Phys 8 7 [] D rin e l he-h5 [] W vn eerven Vog ul Phys heh65; ul Phys he-h77; Phys e he-h76 [] oh J Vermseren Vog ul Phys Pro ul 6 he- h6; ul Phys 688 he-h [] oh J Vermseren ul Phys heh55 [5] ehrmnn E Remii omu Phys ommun 6 he-h77 [6] oh J Vermseren Vog ul Phys Pro ul 6 heh6; ul Phys 688 he-h [7] rell e l ul Phys he-h 558 [8] I rsheyn I Ryzhi les o Inegrls eries Prous e len Jerey emi Press ew Yor 65 [] yres Jr Dierenil Equions hum's Ouline eries rw- Hill 5 [] H iller Pril Dierenil Equion John Willey 6 [] Hlzen Drin Qurs eons: n Inrouory ourse in oern Prile Physis John Wiley 8 [] ele e l mll- Physis DEY - Ooer [] rmmer J D ullivn Eleromgnei Inerions o Hrons es Donnhie

17 Inernionl Journl o ienii Reserh Puliions Volume Issue ovemer 7 hw Plenum Press 78 [] H uer e l Rev o Phys [5] E Rey Perurive Qunum hromoynmis DEY 788 DO-H 7 7 UHOR irs uhor Dr Rsn Rjhow Ph D ssisn Proessor Physis Dermen H ollege Jmugurih oniur ssm Ini E-mil:rsnrjhow@gmilom