Confronting NLO BFKL Kernels with proton structure function data L. Schoeffel (CEA/SPP) Work done in collaboration with R. Peschanski (CEA/SPhT) and C. Royon (CEA/SPP) hep-ph/ Introduction to BFKL equation 2.LO BFKL fit to F2(x,Q²) [H1 data 96/97] 3.NLO case (Kernels + fits) 4.Direct study of the resummation schemes needed in the expression of BFKL Kernels 40th Rencontres de Moriond

F2 well described by DGLAP fits What happens if  S Ln(Q²/Q 0 ²) <<  S Ln(1/x) ? => Needs a resummation of  S Ln(1/x) to all orders (by keeping the full Q² dependence) => Relax the strong ordering of k T ² => We need an integration over the full k T space xG(x,Q²) =  dk T ²/k T ² f(x,k T ²) BFKL equation relates f n and f n-1 (f n = K  f n-1 ) => f(x, k T ²) ~ x -  k T diffusion term => increase of F2(x,Q²) at low x… ** p Introduction to BFKL equation (in DIS)

After a Mellin transform in x and Q², F2 can be written as F2(x,Q²) =  d  d  /(2i  )² (Q²/Q 0 ²)  x -  F2( ,  ) At LO F2( ,  ) = H( ,  ) / [  -   LO (  )] with  =  S 3/   LO (  ) is the BFKL Kernel ~ 1/  + 1/(1-  ) For example DGLAP at LO would give  (  )~1/  H( ,  ) is a regular function and the pole contribution  =   LO (  ) leads to a unique Mellin transform in . Then, a saddle point approx. at low x => L = Ln(Q 2 /Q 0 2 ) et Y = Ln(1/x) F 2 (x,Q 2 ) = N exp{ ½ L+Y LO (½)-½L 2 /(’’ LO (½)Y)} ~ Q/Q 0 x -=  (½) F2 expression from the BFKL Kernel at LO Note :  c = ½ is the saddle point

F 2 (x,Q 2 ) = N exp{ ½ L+Y LO (½)-½L 2 /(’’ LO (½)Y)} ~ Q/Q 0 x -=  (½) L = Ln(Q 2 /Q 0 2 ) et Y = Ln(1/x) Very good description of F2 at low x<0.01 with a 3 parameters fit // global QCD fit of H1… We get : Q 0 ² = / GeV² and  = / [  =  S 3/  ] => Much lower than the typical value expected here  ~ 0.25 => Higher orders (NLO) corr. needed with  S running (RGE)… Results at LO F2 (measured by H data)

Consistency condition at NLO  NLO ( , ,  RGE ) verifies the relation :  =  p =  RGE  NLO ( ,  p,  RGE ) // LO condition  =   LO (  ) Numerically =>  p ( ,  RGE ) Then we get :  NLO ( ,  p,  RGE )   eff ( ,  RGE ) BFKL Kernel(s) at NLO Calculations at NLO => singularities => resummation required by consistency with the RGE (different schemes aviable) NLO BFKL Kernels  ( , ,  ) = LO case

F 2 (x,Q 2 ) = N exp{ c L+ RGE Y eff ( c,  RGE )- ½L 2 /( RGE ’’ eff ( c, RGE )Y)} L = Ln(Q 2 /Q 0 2 ) et Y = Ln(1/x) Saddle point approximation in  (// LO case) : with  c =  saddle such that  ’ eff (  c,…) = 0 NLO and LO values of the intercept are compatible For a reasonnable value of  RGE Deriving F2(x,Q²) from BFKL at NLO

Results at NLO F2 compared with LO predictions and 2 schemes at NLO… Sizeable differences are visible between the two resummation schemes at NLO The LO fit (with 3 param.) gives a much better description than NLO fit (2 param.) for Q²<8.5 GeV² => Pb with the saddle pt approx? => Pb with the NLO Kernels?

From F2( ,Q²) =  d  /(2i  ) (Q²/Q0²)  f( ,  ) => ∂lnF 2 (,Q 2 )/∂lnQ 2 = *(,Q 2 ) Then we can determine *(,Q 2 )~ saddle from parametrisations of F2 data(x,Q²) -> F 2 (,Q 2 ) ->  saddle Then, we will study the consistency relation  =  RGE (Q 2 )  NLO (  *(), ,  RGE ) (reminiscent from the similar relation at LO) Study of the consistency relation at NLO Determination of  saddle ( ,Q²)

From  *(, ) => we determine  NLO (*,, RGE ) Consistency relation and verify the relation  NLO (  *, ,  RGE )=  /  RGE ( ) ? In black :  NLO (  *, ,Q²) In Red :  /  RGE ( )  The consistency relation does not hold exactly BUT  NLO (  *, ,Q²) is linear in  and does not diverge => spurious singularities properly resummed! (it is not the case for all schemes…) In practice we have :  NLO (  *, ,  RGE ) ~  /  OUT Note : Recalculating  eff with this relation does not change the results on the F2(x,Q²) fits…

Effective approximation of the BFKL Kernels at NLO => 2 parameters formula for F2(x,Q²) at low x<0.01 => Reasonnable description of the F2 data => Sensitivity to the resum. schemes + pb with the 2 lowest Q² bins… Direct studies of the resum. schemes in Mellin space => The consistency relation holds approximately => Discrimination of the different schemes / existence of spurious singularities Further studies : * Beyond the saddle point approximation : unknown aspects of prefactors could play a role (NLO impact factors…) * New resum. schemes… Summary