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Confronting NLO BFKL Kernels with proton structure function data

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. 40th Rencontres de Moriond.

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Confronting NLO BFKL Kernels with proton structure function data

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  1. 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 40th Rencontres de Moriond • Introduction to BFKL equation • LO BFKL fit to F2(x,Q²) [H1 data 96/97] • NLO case (Kernels + fits) • Direct study of the resummation schemes • needed in the expression of BFKL Kernels

  2. Introduction to BFKL equation (in DIS) F2 well described by DGLAP fits What happens if S Ln(Q²/Q0²) << 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 kT² => We need an integration over the full kT space xG(x,Q²) =  dkT²/kT² f(x,kT²) * BFKL equation relates fn and fn-1 (fn = K  fn-1) => f(x, kT²) ~ x-  kT diffusion term => increase of F2(x,Q²) at low x… p

  3. F2 expression from the BFKL Kernel at LO After a Mellin transform in x and Q², F2 can be written as F2(x,Q²) =  dd /(2i)² (Q²/Q0²) 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 => F2(x,Q2) = N exp{½L+YLO(½)-½L2/(’’LO(½)Y)} ~ Q/Q0x-= (½) L = Ln(Q2/Q02) et Y = Ln(1/x) Note : c = ½ is the saddle point

  4. Results at LO F2(x,Q2) = N exp{½L+YLO(½)-½L2/(’’LO(½)Y)} ~ Q/Q0x-= (½) L = Ln(Q2/Q02) 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 : Q0² = 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)… F2 (measured by H1 96-97 data)

  5. BFKL Kernel(s) at NLO = LO case NLO BFKL Kernels (,,) Calculations at NLO => singularities => resummation required by consistency with the RGE (different schemes aviable) • 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)

  6. Deriving F2(x,Q²) from BFKL at NLO Saddle point approximation in  (// LO case) : F2(x,Q2) = N exp{cL+RGEYeff(c, RGE)-½L2/(RGE’’eff(c,RGE)Y)} L = Ln(Q2/Q02) et Y = Ln(1/x) with c =saddle such that ’eff(c,…) = 0 NLO and LO values of the intercept are compatible For a reasonnable value of RGE

  7. 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?

  8. Study of the consistency relation at NLO Determination of saddle(,Q²) From F2(,Q²) =  d/(2i) (Q²/Q0²)f(,) => ∂lnF2(,Q2)/∂lnQ2 = *(,Q2) Then we can determine *(,Q2)~saddle from parametrisations of F2 data(x,Q²) -> F2(,Q2) -> saddle Then, we will study the consistency relation = RGE(Q2) NLO(*(), ,RGE) (reminiscent from the similar relation at LO)

  9. Consistency relation From  *(, <Q2>) => we determine NLO(*,,RGE) • and verify the relation • NLO(*,,RGE)=/RGE(<Q2>) ? • In black : NLO(*,,Q²) • In Red :  /RGE(<Q2>) • 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…

  10. Summary • 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…

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