1. Testing BFKL evolution with Mueller-Navelet jets
Cyrille Marquet
RIKEN BNL Research Center
based on C. Marquet and C. Royon, arXiv:
ISMD 2007, Berkeley, USA

2. Contents
Introduction: BFKL evolution high-energy evolution with fixed hard-momentum scale in pQCD
Mueller-Navelet jets: h h → Jet X Jet a jet in each of the forward directions and a large rapidity interval in between  large values of x probed in both hadrons, pdfs under control  focus on the (BFKL) resummation of large logarithms ln(s/k1k2)
BFKL ressumation: from LL to NLL a collinear-improved kernel is needed for meaningful predictions
The observable: correlations in azimuthal angle they provide clean experimental probes and are theoretically under control

3. Introduction : BFKL evolution
linear pQCD evolutions:
weakly-coupled regime
DGLAP evolution towards larger momentum scale kT
well known, works very well to describe hard processes in hadronic collisions
- BFKL evolution towards larger center-of-mass energy s strong hints for the need of BFKL in forward jets are HERA
Kepka, Marquet, Peschanski and Royon (2007)
idea: study the BFKL evolution with azimuthal correlations of Mueller-Navelet jets
BFKL resummation of
large αS ln(x1 x2 s / k1k2)
x1 ~ x2 close to 1: large values of x probed in both hadrons, the pdf’s are under control, no small-x effects
a jet in each of the forward directions
and a large rapidity interval in between
Balitsky, Fadin, Kuraev and Lipatov (2007) J X

4. in practice, NLL-BFKL is needed
Why large logarithms ?
consider 2 to 2 scattering with (Regge limit):
the exchanged particle has a very small
longidudinal momentum
the final-state particles are separated by a large rapidity interval
their contribution goes as and is as large
next-order diagrams:
with n gluons:
need ressumation
of leading logs (LL)
 BFKL equation
in practice, NLL-BFKL is needed

5. The observable: Mueller-Navelet jets
Mueller and Navelet (1994)
moderate values of x1, typically 0.05
k1 >> QCD  collinear factorization of
y1 ~ 4
h+h  Jet+X+Jet
large rapidity
interval Δη ~ 8
y2 ~ - 4
k2 >> QCD  collinear factorization of
moderate values of x2, typically 0.05
pQCD: need ressumation of powers of αS Δη ~ 1 in the partonic cross-section gg → JXJ

6. kT factorization and BFKL evolution
- from hh → JXJ to gg → JXJ: collinear factorization of the pdf’s
- from gg → JXJ to gg → X: kT factorization
of the BFKL Green function
final-state
particles
leading-order
impact factors (I. F.)
kT factorization is also proved at NLL
but there are many complications
Green function, this is what
resums the powers of αS Δη
this simple formula holds at LL
Fadin et al. ( )

7. Going to NLL-BFKL
- it took ~ 10 years to compute the NLL Green function
and the NLO impact factors are also very difficult to compute
Fadin and Lipatov (1998) Ciafaloni (1998)
for instance, with the photon I.F. (relevant in DIS) will probably take ~ 10 years
the jet impact factors are known
in progress, Bartels et al.
Bartels, Colferai and Vacca (2002)
- for jet production, everything is known, but at NLO there are difficulties
in merging kT and collinear factorizations
after 5 years, still no numerical results
- it has been argued that a complete NLL-BFKL calculation would be flawed:
the truncation of the perturbative series is spurious, it contains singularities
which would not be there with all orders
for double vector meson production in γ*-γ* scattering there is a full computation
and the result is unstable when varying the renormalization scheme
Salam (1998), Ciafaloni, Colferai and Salam (1999)
Ivanov and Papa (2006)
- the problem comes from Green function, and it can be cured
we will use Salam’s schemes, the only ones used so far for phenomenological studies
Ciafaloni, Colferai, Salam, Stasto, Altarelli, Ball, Forte, Brodsky, Lipatov, Fadin, … (1999-now)
Peschanski, Royon and Schoeffel (2005), Kepka, Marquet, Peschansi and Royon (2007)

8. BFKL Green functions
the integral kernel of the LL-BFKL equation is conformal invariant:
Lipatov (1986)
this allows to solve the equation and obtain the LL Green function
discrete index
called conformal spin
Mellin transformation
now running coupling
(with symmetric scale)
the NLL Green function (in our approach with LO impact factors)
from to :
the ω integration leads to the implicit equation
effective kernel

9. Salam’s regularisation schemes
has spurious singularities in Mellin (γ) space, they lead to unphysical
results, this is an artefact of the truncation of the perturbative series
- to produce meaningful NLL-BFKL results, one has to add to the higher order
corrections which are responsible for the canceling the singularities
in momemtum space, the poles correspond to the known DGLAP limits k1 >> k2 and
k1 << k2 , this gives information/constraints on what add to the next-leading kernel
regularisation
Strategy:
implicit equation
there is some arbitrary:
different schemes S3, S4, …
in practice, each value k1k2
leads to a different effective kernel
in this work, we extended those schemes to p ≠ 0 :
with

10. the scale-invariant part of :
Some formulae
the scale-invariant part of :
with
only the scale-invariant part of the NLL-BFKL kernel contributes to our observable:
it is symmetric under the sustitution , the scale-dependent part of the kernel
is antisymmetric and does not contribute

11. The ΔΦ distribution we used the following variables
and we studied the normalized ΔΦ distribution
|y| < 0.5 for a symetric
situation y1 ~ - y2
given in terms of the coefficients
ET cut = 20 GeV (Tevatron)
50 GeV (LHC)
important piece: contains the Δη and R dependencies
 parameter-free predictions

12. Some results for (1/σ) dσ/dΔΦ
fixed R = 1 and several Δη
fixed Δη = 10 and several R
LL-BFKL
NLL-BFKL
the decorrelation increases with
the rapidity interval Δη
the decorrelation increases
when R deviates from 1

13. Scale/Scheme dependence
scale dependence
Tevatron
LHC
almost no scheme dependence
scale uncertainty quite large
we also noticed that the uncertainty due to pdfs is negligible
we suspect very little sensitivity to NLO impact factor
 very interesting observable

14. Conclusions
- the correlation in azimuthal angle between two jets gets weaker as their separation in rapidity increases
- we obtained parameter free predictions in the BFKL framework at next-leading accuracy, valid for large enough rapidity intervals
- there is some data from the D0 collaboration at the Tevatron, but for rapidity intervals Δη smaller than 5
- our predictions underestimate the correlation while predictions overestimate it
see also Sabio Vera and Schwenssen (2007)
feasibility study in collaboration with Christophe Royon (D0/Atlas) and Ramiro Debbe (Star/Atlas)
the CDF miniplugs cannot measure pT well but are suited for azimuthal angle measurements
with new CDF detectors called miniplugs
possibilities for Δη =12 with ET cut = 5 GeV
prospects for future measurements:
- at the LHC
- at the Tevatron
predictions for future
measurements at CDF