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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/0411338 1.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

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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)

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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

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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 = 0.40 +/- 0.01 GeV² and = 0.09 +/- 0.01 [ = 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 H1 96-97 data)

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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

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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

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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?

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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)

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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…

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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

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