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
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
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
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/Q0 x-= (½)
L = Ln(Q2/Q02) et Y = Ln(1/x)
Note : c = ½ is the saddle point
Results at LO
F2(x,Q2) = N exp{½L+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 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
F2 (measured by H1
96-97 data)
=> Higher orders (NLO) corr.
needed with S running (RGE)…
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)
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
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?
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)
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…
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…