Diapositive 1

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) ltlt ?S Ln(1/x) ?
• gt Needs a resummation of ?S Ln(1/x) to all
  orders
• (by keeping the full Q² dependence)
• gt Relax the strong ordering of kT²
• gt 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)
gt f(x, kT²) x-? ? kT diffusion term gt
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 gt
F2(x,Q2) N exp½L?Y?LO(½)-½L2/(??LO(½)Y)
Q/Q0 x-?? ?(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 exp½L?Y?LO(½)-½L2/(??LO(½)Y)
Q/Q0 x-?? ?(½)
L Ln(Q2/Q02) et Y Ln(1/x)

• Very good description of F2 at
• low xlt0.01 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/?
• gt Much lower than the typical
• value expected here ? 0.25
• gt 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
• gt singularities
• gt 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 gt ?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 Q²lt8.5 GeV²

gt Pb with the saddle pt approx? gt 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(?,?) gt
?lnF2(?,Q2)/?lnQ2 ?(?,Q2) Then we can
determine ?(?,Q2)?saddle from
parametrisations of F2 data(x,Q²) -gt F2(?,Q2) -gt
?saddle Then, we will study the consistency
relation ? ?RGE(Q2) ?NLO(?(),
?,?RGE) (reminiscent from the similar relation
at LO)
9
Consistency relation
From ? (?, ltQ2gt) gt we determine ?NLO(?,?,?RGE)

• and verify the relation
• ?NLO(?,?,?RGE)? /?RGE(ltQ2gt) ?
• In black ?NLO(?,?,Q²)
• In Red ? /?RGE(ltQ2gt)
• The consistency relation does not
• hold exactly BUT ?NLO(?,?,Q²)
• is linear in ? and does not diverge
• gt 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
• gt 2 parameters formula for F2(x,Q²) at low
  xlt0.01
• gt Reasonnable description of the F2 data
• gt Sensitivity to the resum. schemes pb with
  the 2 lowest Q² bins
• Direct studies of the resum. schemes in Mellin
  space
• gt The consistency relation holds approximately
• gt 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