Fall and rise of the gluon splitting function (at small x)

Transcription

1 Fall and rie of the gluon plitting function (at mall ) Gavin Salam LPTHE Univ. Pari VI & VII and CNRS In collaboration with M. Ciafaloni, D. Colferai and A. Staśto 8 th workhop on non-perturbative QCD Pari, 7 11 June 24 Fall and rie of the gluon plitting function(at mall ) p.1/18

2 Introduction At high energie ( ) mall, cro ection are uppoed to rie rapidly domain of BFKL phyic reummation of logarithm of (or ): σ n= α n ln n 4 ln 2 α N C π.5 Balitky, Fadin, Kuraev & Lipatov 76 Fall and rie of the gluon plitting function(at mall ) p.2/18

3 Introduction At high energie ( ) mall, cro ection are uppoed to rie rapidly domain of BFKL phyic reummation of logarithm of (or ): σ n= α n ln n 4 ln 2 α N C π.5 Balitky, Fadin, Kuraev & Lipatov 76 Calculation & meaurement of the power growth i holy grail (but only a fraction of the tory) in tudie of high-energy limit of QCD. Fall and rie of the gluon plitting function(at mall ) p.2/18

4 Introduction At high energie ( ) mall, cro ection are uppoed to rie rapidly domain of BFKL phyic reummation of logarithm of (or ): σ n= α n ln n 4 ln 2 α N C π.5 Balitky, Fadin, Kuraev & Lipatov 76 Calculation & meaurement of the power growth i holy grail (but only a fraction of the tory) in tudie of high-energy limit of QCD. But perturbative calculation hold only for purely perturbative problem (e.g. γ γ cattering), correponding to rare kinematical configuration. Fall and rie of the gluon plitting function(at mall ) p.2/18

5 Introduction At high energie ( ) mall, cro ection are uppoed to rie rapidly domain of BFKL phyic reummation of logarithm of (or ): σ n= α n ln n 4 ln 2 α N C π.5 Balitky, Fadin, Kuraev & Lipatov 76 Calculation & meaurement of the power growth i holy grail (but only a fraction of the tory) in tudie of high-energy limit of QCD. But perturbative calculation hold only for purely perturbative problem (e.g. γ γ cattering), correponding to rare kinematical configuration. Hard to meaure eperimentally & of limited wider relevance Fall and rie of the gluon plitting function(at mall ) p.2/18

6 Intro: emi-perturbative tudie Proton tructure function, F 2 (, Q 2 ), i mot widely-tudied high-energy quantity ( Bjorken, Q 2 photon virtuality). Fall and rie of the gluon plitting function(at mall ) p.3/18

7 Intro: emi-perturbative tudie Proton tructure function, F 2 (, Q 2 ), i mot widely-tudied high-energy quantity ( Bjorken, Q 2 photon virtuality). Etenively tudied at HERA Important for LHC & high-energy ν cattering Fall and rie of the gluon plitting function(at mall ) p.3/18

8 Intro: emi-perturbative tudie Proton tructure function, F 2 (, Q 2 ), i mot widely-tudied high-energy quantity ( Bjorken, Q 2 photon virtuality). Etenively tudied at HERA Important for LHC & high-energy ν cattering -dependence i non-perturbative, but Q 2 dependence i predicted by DGLAP equation, in term of quark (q(, Q 2 )) and gluon (g(, Q 2 )) ditribution: F 2 = C 2q q + C 2g g ln Q 2q = P qq q + P qg g ln Q 2g = P gq q + P gg g Coefficient (C 2i ) and plitting (P ij ) function are perturbative. Fall and rie of the gluon plitting function(at mall ) p.3/18

9 Intro: emi-perturbative tudie Proton tructure function, F 2 (, Q 2 ), i mot widely-tudied high-energy quantity ( Bjorken, Q 2 photon virtuality). Etenively tudied at HERA Important for LHC & high-energy ν cattering -dependence i non-perturbative, but Q 2 dependence i predicted by DGLAP equation, in term of quark (q(, Q 2 )) and gluon (g(, Q 2 )) ditribution: F 2 = C 2q q + C 2g g ln Q 2q = P qq q + P qg g ln Q 2g = P gq q + P gg g Coefficient (C 2i ) and plitting (P ij ) function are perturbative. P ij and C 2i both have mall- enhancement, (α ln 1/) n, at all order poor perturbative convergence need BFKL reummation Fall and rie of the gluon plitting function(at mall ) p.3/18

10 Intro: emi-perturbative tudie Proton tructure function, F 2 (, Q 2 ), i mot widely-tudied high-energy quantity ( Bjorken, Q 2 photon virtuality). Etenively tudied at HERA Important for LHC & high-energy ν cattering -dependence i non-perturbative, but Q 2 dependence i predicted by DGLAP equation, in term of quark (q(, Q 2 )) and gluon (g(, Q 2 )) ditribution: F 2 = C 2q q + C 2g g ln Q 2q = P qq q + P qg g ln Q 2g = P gq q + P gg g Coefficient (C 2i ) and plitting (P ij ) function are perturbative. P ij and C 2i both have mall- enhancement, (α ln 1/) n, at all order poor perturbative convergence need BFKL reummation P gg () 4 ln 2 α N C π + Fall and rie of the gluon plitting function(at mall ) p.3/18

11 Perturbative tructure Small- gluon plitting function ha logarithmic enhancement: P gg () = n=1 α n lnn1 1 + n=2 α n ln n Fall and rie of the gluon plitting function(at mall ) p.4/18

12 Perturbative tructure Small- gluon plitting function ha logarithmic enhancement: P gg () = n=1 α n lnn1 1 Leading Log (LL): ᾱ + ζ(3) 3 ᾱ4 ln ζ(5) 6 ᾱ6 ln n=2 α n ln n Fall and rie of the gluon plitting function(at mall ) p.4/18

13 Perturbative tructure Small- gluon plitting function ha logarithmic enhancement: P gg () = n=1 + n=2 α n lnn1 1 α n ln n Leading Log (LL): ᾱ + ζ(3) 3 ᾱ4 ln ζ(5) 6 ᾱ6 ln Net-to-Leading Log (NLL): A 2 ᾱ 2 + A 31 ᾱ 3 ln 1 + A 42ᾱ 4 ln Fadin & Lipatov 98 Camici & Ciafaloni 98 Fall and rie of the gluon plitting function(at mall ) p.4/18

14 Perturbative tructure Small- gluon plitting function ha logarithmic enhancement: P gg () = n=1 + n=2 α n lnn1 1 α n ln n Leading Log (LL): ᾱ + ζ(3) 3 ᾱ4 ln ζ(5) 6 ᾱ6 ln Net-to-Leading Log (NLL): A 2 ᾱ 2 + A 31 ᾱ 3 ln 1 + A 42ᾱ 4 ln Fadin & Lipatov 98 Camici & Ciafaloni 98 Fall and rie of the gluon plitting function(at mall ) p.4/18

15 Perturbative tructure Small- gluon plitting function ha logarithmic enhancement: P gg () = n=1 + n=2 α n lnn1 1 α n ln n NNLO (α 3 ): firt mall- enhancement in gluon plitting function. Moch, Vermaeren & Vogt, 4 Leading Log (LL): ᾱ + ζ(3) 3 ᾱ4 ln ζ(5) 6 ᾱ6 ln Net-to-Leading Log (NLL): A 2 ᾱ 2 + A 31 ᾱ 3 ln 1 + A 42ᾱ 4 ln Fadin & Lipatov 98 Camici & Ciafaloni 98 Fall and rie of the gluon plitting function(at mall ) p.4/18

16 Perturbative tructure Small- gluon plitting function ha logarithmic enhancement: P gg () = n=1 α n lnn LO NLO NNLO + n=2 α n ln n P gg ().3.2 α (Q 2 ) =.225 NNLO (α 3 ): firt mall- enhancement in gluon plitting function. Moch, Vermaeren & Vogt, 4.1 1/2 < µ 2 /Q 2 < Fall and rie of the gluon plitting function(at mall ) p.4/18

17 NLO DGLAP veru data F p 2+c i () =.2(i=24) =.32 =.5 =.8 =.13 (i=2) =.2 =.32 NMC BCDMS SLAC H1 =.5 =.8 =.13 =.2 =.32 H1 96 Preliminary (ISR) H1 97 Preliminary (low Q 2 ) H Preliminary (high Q 2 ) =.5 =.8 =.13 (i=1) =.2 NLO QCD Fit H1 Preliminary c i ()=.6 (i()-.4) =.32 =.5 =.8 =.13 =.18 NLO DGLAP fit give good decription of data So do preliminary NNLO DGLAP fit Evidence of ome problem for very mall 1 3 intabilitie from NLO to NNLO negative gluon 2 =.25 =.4 =.65(i=1) Q 2 /GeV 2 Fall and rie of the gluon plitting function(at mall ) p.5/18

18 LL, NLL? Reummation tatu LL term rie very fat, P gg ().5. Incompatible with data. Ball & Forte 95 NLL term go negative very fat. No one even tried fitting the data! [NB: Taking NLL term of P gg i almot the wort poible epanion] P gg () α (Q2 ) = LL NLL LO DGLAP Fall and rie of the gluon plitting function(at mall ) p.6/18

19 LL, NLL? Reummation tatu LL term rie very fat, P gg ().5. Incompatible with data. Ball & Forte 95 NLL term go negative very fat. No one even tried fitting the data! [NB: Taking NLL term of P gg i almot the wort poible epanion] P gg () α (Q2 ) = LL NLL LO, NLO, NNLO DGLAP Fall and rie of the gluon plitting function(at mall ) p.6/18

20 Improving on NLL? Start with kernel... BFKL α + α 2 ln Q 2 Q 2 DGLAP α + α 2 ln Q2 Q 2 Q 2 + Q 2 Q 2 anti-dglap Fall and rie of the gluon plitting function(at mall ) p.7/18

21 Improving on NLL? Start with kernel... BFKL α + α 2 ln ln Q 2 ln 2 Q 2 Q 2 Q 2 DGLAP α Q 2 ln + α 2 ln ln Q2 Q 2 + Q 2 Q 2 anti-dglap Fall and rie of the gluon plitting function(at mall ) p.7/18

22 Improving on NLL? Start with kernel... BFKL GPS, Ciafaloni, Colferai ibid. + Staśto 3 α + α 2 ln ln Q 2 ln 2 Q 2 Q 2 Q 2 DGLAP α Q 2 ln + α 2 ln ln Q2 Q 2 combined BFKL+DGLAP kernel K (beware double counting) + Q 2 Q 2 anti-dglap Fall and rie of the gluon plitting function(at mall ) p.7/18

23 Building up the kernel α (Q2 ) =.215 Build up characteritic function, i.e. the Mellin tranform of kernel (fied coupling) α χ(γ) = N LL BFKL ᾱ χ(γ) = ( ) dk 2 k 2 γ = K(k, k ) k 2 k 2 Height of minimum i BFKL power γ Fall and rie of the gluon plitting function(at mall ) p.8/18

24 Building up the kernel α (Q2 ) =.215 Build up characteritic function, i.e. the Mellin tranform of kernel (fied coupling) α χ(γ) = N LL BFKL ᾱ χ(γ) = ( ) dk 2 k 2 γ = K(k, k ) k 2 k 2 Height of minimum i BFKL power -.5 LL + NLL BFKL γ Fall and rie of the gluon plitting function(at mall ) p.8/18

25 Building up the kernel α (Q2 ) =.215 Build up characteritic function, i.e. the Mellin tranform of kernel (fied coupling) α χ(γ) = N LL BFKL DGLAP LL + NLL BFKL γ ᾱ χ(γ) = ( ) dk 2 k 2 γ = K(k, k ) k 2 k 2 Height of minimum i BFKL power NB: DGLAP = rotated plot of γ(n) Fall and rie of the gluon plitting function(at mall ) p.8/18

26 Building up the kernel α (Q2 ) =.215 Build up characteritic function, i.e. the Mellin tranform of kernel (fied coupling) α χ(γ) = N LL BFKL anti-dglap DGLAP LL + NLL BFKL γ ᾱ χ(γ) = ( ) dk 2 k 2 γ = K(k, k ) k 2 k 2 Height of minimum i BFKL power NB: DGLAP = rotated plot of γ(n) Fall and rie of the gluon plitting function(at mall ) p.8/18

27 Building up the kernel α (Q2 ) =.215 Build up characteritic function, i.e. the Mellin tranform of kernel (fied coupling) α χ(γ) = N LL BFKL combined anti-dglap DGLAP LL + NLL BFKL γ ᾱ χ(γ) = ( ) dk 2 k 2 γ = K(k, k ) k 2 k 2 Height of minimum i BFKL power NB: DGLAP = rotated plot of γ(n) Fall and rie of the gluon plitting function(at mall ) p.8/18

28 Eamine BFKL power a a function of α Ω LL BFKL NLL BFKL Ω epanion cheme A cheme B Combining BFKL + DGLAP give ignificant tabiliation of power. With ame logic, other theorit find imilar reult! Forhaw, Ro & Sabio Vera 99 Altarelli, Ball, Forte, 4 prelim. Power i roughly conitent with eperiment Good tarting point for phenomenology Α Fall and rie of the gluon plitting function(at mall ) p.9/18

29 Eamine BFKL power a a function of α Ω LL BFKL NLL BFKL Ω epanion cheme A cheme B Α Combining BFKL + DGLAP give ignificant tabiliation of power. With ame logic, other theorit find imilar reult! Forhaw, Ro & Sabio Vera 99 Altarelli, Ball, Forte, 4 prelim. Power i roughly conitent with eperiment Good tarting point for phenomenology NB: power hown here i property of kernel, not of cro ection... Fall and rie of the gluon plitting function(at mall ) p.9/18

30 Iteration of kernel Green function, Q 2 ( ) Green function: G ln ; Q, Q, Q 2 Fall and rie of the gluon plitting function(at mall ) p.1/18

31 Iteration of kernel Green function, Q 2 ( ) Green function: G ln ; Q, Q 2π k 2 G(Y; k + ε, k ) 1 1 (a) LL cheme A cheme B k = 2 GeV, Q Y Fall and rie of the gluon plitting function(at mall ) p.1/18

32 Iteration of kernel Green function, Q 2 ( ) Green function: G ln ; Q, Q 2π k 2 G(Y; k + ε, k ) 1 1 (b) NLL α (q 2 ) NLL α (k 2 ) cheme B k = 2 GeV, Q Y Fall and rie of the gluon plitting function(at mall ) p.1/18

33 Green function effective DGLAP plitting function Contruct a gluon denity from Green function (take k k ): g(, Q 2 ) Q d 2 k G (ν =k 2) (ln 1/, k, k ) Fall and rie of the gluon plitting function(at mall ) p.11/18

34 Green function effective DGLAP plitting function Contruct a gluon denity from Green function (take k k ): g(, Q 2 ) Q d 2 k G (ν =k 2) (ln 1/, k, k ) Numerically olve equation for effective plitting function, P gg,eff (z, Q 2 ) : dg(, Q 2 ) dz ( ) = d ln Q 2 z P gg,eff(z, Q 2 ) g z, Q2 Fall and rie of the gluon plitting function(at mall ) p.11/18

35 Green function effective DGLAP plitting function Contruct a gluon denity from Green function (take k k ): g(, Q 2 ) Q d 2 k G (ν =k 2) (ln 1/, k, k ) Numerically olve equation for effective plitting function, P gg,eff (z, Q 2 ) : dg(, Q 2 ) dz ( ) = d ln Q 2 z P gg,eff(z, Q 2 ) g z, Q2 Factoriation Splitting function: red path Green function: all path k 2 g(,q 2 ) 2 P gg Evolution path in,k g(,q 1) factorized (nonperturbative) g Q 2 Q 1 Fall and rie of the gluon plitting function(at mall ) p.11/18

36 P gg (z) plitting function reult 1 Q = 4.5 GeV α (Q2 ) =.215 LL (fied α ) LO DGLAP z P(z) z Fall and rie of the gluon plitting function(at mall ) p.12/18

37 P gg (z) plitting function reult 1 Q = 4.5 GeV α (Q2 ) =.215 LL (fied α ) LL (α (q2 )) LO DGLAP z P(z) z Fall and rie of the gluon plitting function(at mall ) p.12/18

38 P gg (z) plitting function reult 1 Q = 4.5 GeV α (Q2 ) =.215 LL (fied α ) LL (α (q2 )) NLL B LO DGLAP z P(z) z Fall and rie of the gluon plitting function(at mall ) p.12/18

39 P gg (z) plitting function reult 1.8 ω-epanion (1999) NLL B (23) LO DGLAP z P gg (z).6.4 Q = 4.5 GeV α (Q2 ) = /4 < µ 2 /Q 2 < z Fall and rie of the gluon plitting function(at mall ) p.12/18

40 Dominant phenomenological tructure i dip Rapid rie in P gg i not for today energie! Main feature i a dip at ω-epanion (1999) NLL B (23) LO DGLAP z P gg (z).3.2 Q = 4.5 GeV α (Q2 ) = /4 < µ 2 /Q 2 < z Fall and rie of the gluon plitting function(at mall ) p.13/18

41 Dominant phenomenological tructure i dip Rapid rie in P gg i not for today energie! Main feature i a dip at 1 3 Quetion:.5.4 ω-epanion (1999) NLL B (23) LO DGLAP Variou dip have been een Thorne 99, 1 (running α, NLL) ABF 99 3 (fit, running α ) CCSS 1, 3 (running α, NLL B ) z P gg (z).3.2 Q = 4.5 GeV α (Q2 ) =.215 I it alway the ame dip?.1 1/4 < µ 2 /Q 2 < z Fall and rie of the gluon plitting function(at mall ) p.13/18

42 Dominant phenomenological tructure i dip Rapid rie in P gg i not for today energie! Main feature i a dip at 1 3 Quetion:.5.4 ω-epanion (1999) NLL B (23) LO DGLAP Variou dip have been een Thorne 99, 1 (running α, NLL) ABF 99 3 (fit, running α ) CCSS 1, 3 (running α, NLL B ) z P gg (z).3.2 Q = 4.5 GeV α (Q2 ) =.215 I it alway the ame dip? I the dip a rigorou prediction?.1 1/4 < µ 2 /Q 2 < z Fall and rie of the gluon plitting function(at mall ) p.13/18

43 Dominant phenomenological tructure i dip Rapid rie in P gg i not for today energie! Main feature i a dip at 1 3 Quetion: Variou dip have been een Thorne 99, 1 (running α, NLL) ABF 99 3 (fit, running α ) CCSS 1, 3 (running α, NLL B ) I it alway the ame dip? I the dip a rigorou prediction? What i it origin? Running α, momentum um rule...? z P gg (z) ω-epanion (1999) NLL B (23) LO DGLAP Q = 4.5 GeV α (Q2 ) = z 1/4 < µ 2 /Q 2 < 4 Fall and rie of the gluon plitting function(at mall ) p.13/18

44 Dominant phenomenological tructure i dip Rapid rie in P gg i not for today energie! Main feature i a dip at 1 3 Quetion: Variou dip have been een Thorne 99, 1 (running α, NLL) ABF 99 3 (fit, running α ) CCSS 1, 3 (running α, NLL B ) I it alway the ame dip? I the dip a rigorou prediction? What i it origin? Running α, momentum um rule...? NNLO DGLAP give a clue ᾱ 3 ln 1 z P gg (z) ω-epanion (1999) NLL B (23) LO DGLAP NNLO DGLAP Q = 4.5 GeV α (Q2 ) = z 1/4 < µ 2 /Q 2 < 4 Fall and rie of the gluon plitting function(at mall ) p.13/18

45 Dominant phenomenological tructure i dip Rapid rie in P gg i not for today energie! Main feature i a dip at 1 3 Quetion: Variou dip have been een Thorne 99, 1 (running α, NLL) ABF 99 3 (fit, running α ) CCSS 1, 3 (running α, NLL B ) I it alway the ame dip? I the dip a rigorou prediction? What i it origin? Running α, momentum um rule...? NNLO DGLAP give a clue ᾱ 3 ln 1 z P gg (z) ω-epanion (1999) NLL B (23) LO DGLAP NNLO DGLAP Q = 4.5 GeV α (Q2 ) = z 1/4 < µ 2 /Q 2 < 4 Fall and rie of the gluon plitting function(at mall ) p.13/18

46 Reorganie perturbative erie LL NLL NNLL... α α2 n f α3 α4 cont. α5 ln 1/... ln 3 1/ ln 2 1/ Fall and rie of the gluon plitting function(at mall ) p.14/18

47 Reorganie perturbative erie LL NLL NNLL... α α2 n f α3 α4 cont. α =.5 α5 ln 1/... P gg () ln 3 1/ ln 2 1/ Fall and rie of the gluon plitting function(at mall ) p.14/18

48 Reorganie perturbative erie LL NLL NNLL... At moderately mall, firt term with -dependence are α 1.54 ᾱ 3 ln 1 α2 n f α3 α4 cont. α =.5 α5 ln 1/... P gg () ln 3 1/ ln 2 1/ Fall and rie of the gluon plitting function(at mall ) p.14/18

49 Reorganie perturbative erie LL NLL NNLL... At moderately mall, firt term with -dependence are α 1.54 ᾱ 3 ln ᾱ4 ln3 1 α2 n f α3 α4 cont. α =.5 α5 ln 1/... P gg () ln 3 1/ ln 2 1/ Fall and rie of the gluon plitting function(at mall ) p.14/18

50 α α2 α3 LL NLL NNLL... n f Reorganie perturbative erie At moderately mall, firt term with -dependence are 1.54 ᾱ 3 ln ᾱ4 ln3 1 Minimum when α ln 2 1 ln 1 1 α α4 cont. α =.5 α5 ln 1/ P gg () α 5/2... ln 3 1/ ln 2 1/ Fall and rie of the gluon plitting function(at mall ) p.14/18

51 Sytematic epanion in α α α2 α3 LL NLL NNLL... n f Poition of dip ln 1 min ᾱ Depth of dip d 1.237ᾱ 5/2 α4 cont. α5 ln 1/... ln 3 1/ ln 2 1/ Fall and rie of the gluon plitting function(at mall ) p.15/18

52 Sytematic epanion in α α α2 α3 LL NLL NNLL... n f Poition of dip ln 1 min ᾱ Depth of dip d 1.237ᾱ 5/ ᾱ 3 α4 cont. α5 ln 1/... ln 3 1/ ln 2 1/ Fall and rie of the gluon plitting function(at mall ) p.15/18

53 Sytematic epanion in α α α2 α3 α4 α5... LL NLL NNLL... n f ln 3 1/ cont. ln 1/ ln 2 1/ Poition of dip ln 1 min ᾱ Depth of dip d 1.237ᾱ 5/ ᾱ 3 + NB: convergence i very poor A ever at mall! higher-order term in epanion need NNLL info Fall and rie of the gluon plitting function(at mall ) p.15/18

54 Tet dip propertie v. BFKL+DGLAP reummation ln (1/ min ) (a) Quadratic olution Epanded olution meaured ln(1/ min ) 3/(2 ω c ).1.1 α Tet poition of dip v. α Band i uncertainty due to higher order in α At mall α, good agreement confirmation of dip mechanim At moderate α, normal mall- reummation effect collide with dip ln 1 min 3 2ω c Dip then come from interplay between α 3 ln (NNLO) term and full reummation. [Actually, tory more comple] Fall and rie of the gluon plitting function(at mall ) p.16/18

55 Tet dip propertie v. BFKL+DGLAP reummation (b) Tet depth of dip v. α imilar concluion! depth Quadratic olution Epanded olution meaured depth.1.1 α Fall and rie of the gluon plitting function(at mall ) p.17/18

56 Concluion Proton F 2 data eem conitent with plain fied-order DGLAP Fall and rie of the gluon plitting function(at mall ) p.18/18

57 Concluion Proton F 2 data eem conitent with plain fied-order DGLAP Straightforward mall- (LL, NLL) reummation: Converge very poorly Fall and rie of the gluon plitting function(at mall ) p.18/18

58 Concluion Proton F 2 data eem conitent with plain fied-order DGLAP Straightforward mall- (LL, NLL) reummation: Converge very poorly inconitent with data Fall and rie of the gluon plitting function(at mall ) p.18/18

59 Concluion Proton F 2 data eem conitent with plain fied-order DGLAP Straightforward mall- (LL, NLL) reummation: Converge very poorly inconitent with data Solution to problem look circular (but in t!): Fall and rie of the gluon plitting function(at mall ) p.18/18

60 Concluion Proton F 2 data eem conitent with plain fied-order DGLAP Straightforward mall- (LL, NLL) reummation: Converge very poorly inconitent with data Solution to problem look circular (but in t!): Combine BFKL+DGLAP (+anti-dglap) Fall and rie of the gluon plitting function(at mall ) p.18/18

61 Concluion Proton F 2 data eem conitent with plain fied-order DGLAP Straightforward mall- (LL, NLL) reummation: Converge very poorly inconitent with data Solution to problem look circular (but in t!): Combine BFKL+DGLAP (+anti-dglap) iterate reulting kernel Green function Fall and rie of the gluon plitting function(at mall ) p.18/18

62 Concluion Proton F 2 data eem conitent with plain fied-order DGLAP Straightforward mall- (LL, NLL) reummation: Converge very poorly inconitent with data Solution to problem look circular (but in t!): Combine BFKL+DGLAP (+anti-dglap) iterate reulting kernel Green function factoriation effective DGLAP plitting function Fall and rie of the gluon plitting function(at mall ) p.18/18

63 Concluion Proton F 2 data eem conitent with plain fied-order DGLAP Straightforward mall- (LL, NLL) reummation: Converge very poorly inconitent with data Solution to problem look circular (but in t!): Combine BFKL+DGLAP (+anti-dglap) iterate reulting kernel Green function factoriation effective DGLAP plitting function Reult for P gg plitting function i: Stable and theoretically undertood (e.g. dip) Fall and rie of the gluon plitting function(at mall ) p.18/18

64 Concluion Proton F 2 data eem conitent with plain fied-order DGLAP Straightforward mall- (LL, NLL) reummation: Converge very poorly inconitent with data Solution to problem look circular (but in t!): Combine BFKL+DGLAP (+anti-dglap) iterate reulting kernel Green function factoriation effective DGLAP plitting function Reult for P gg plitting function i: Stable and theoretically undertood (e.g. dip) Similar to NNLO, for 1 3 hould be compatible with HERA data will it olve DGLAP problem for 1 3? Fall and rie of the gluon plitting function(at mall ) p.18/18

65 Concluion Proton F 2 data eem conitent with plain fied-order DGLAP Straightforward mall- (LL, NLL) reummation: Converge very poorly inconitent with data Solution to problem look circular (but in t!): Combine BFKL+DGLAP (+anti-dglap) iterate reulting kernel Green function factoriation effective DGLAP plitting function Reult for P gg plitting function i: Stable and theoretically undertood (e.g. dip) Similar to NNLO, for 1 3 hould be compatible with HERA data will it olve DGLAP problem for 1 3? Work till needed for phenomenology... Fall and rie of the gluon plitting function(at mall ) p.18/18