1. “NNLO Logarithmic Expansions and High Precision
Determinations of the QCD background at the LHC:
The case of the Z resonance”
Marco Guzzi
Department of Physics
University of Salento and I.N.F.N. Lecce, Italy
In collaboration with C. Corianò and A. Cafarella
Martina Franca June 2007

2. Summary
QCD has entered its precision era with the advent of the LHC.
For this reason we need to determine the QCD partonometry
using some “golden plated” modes.
This will allow to improve considerably
our knowledge of the parton densities.
In particular, one of the first searches that will be performed
at the LHC will be study of the rapidity and invariant mass
distributions in Drell-Yan on the Z resonance and at larger invariant mass.
The study of the evolution plays a crucial role in this context, in fact we show
that it is the evolution that “drives” the NLO predictions
toward a NNLO suppression of the cross section, at least in Drell-Yan.
There are subtle issues concerning the “resummation”
implicit in the choice of the solution - aspect that we address in our
study - and that that introduce a theoretical indetermination of the prediction.
This indetermination is comparable in size to the change that one gets
when going from NLO to NNLO in the hard scatterings.

3. specifically
We present a next-to-next to leading order determination
with the respective errors on the Pdf’s of the Drell-Yan
invariant mass distributions and the rapidity distributions of
the lepton pair production at typical LHC energy.
(A.Cafarella C. Corianò and M.G. hep-ph/ )
We quantify the impact of all the scale dependences in the hard
scattering and in the DGLAP evolution up to the same perturbative
order by using the PDF evolution code CANDIA.
(A.Cafarella C. Corianò and M.G. Nucl.Phys.B748: ,2006.)
This analysis is useful for studying the case of extra neutral currents in this channel in extensions of the SM.
(C. Corianò A. Faraggi and M.G. arXiv:  [hep-ph],
C. Corianò, N. Irges, S. Morelli hep-ph/ ,
C. Corianò, N. Irges, S. Morelli hep-ph/ ).

4. In the study of new Physics in selected processes at the LHC
we need precision (NNLO QCD) but….
it is unlikely that many processes will be computed at NNLO
in the near future (too difficult…)
but the hard scatterings are known, for some
inclusive and less inclusive processes
dσ/dM, dσ/dM dY.
(Hamberg. Matsuura and Van Neerven (91))
(C. Anastasiou, L. Dixon, K.Melnikov, F. Petriello Phys. Rev. D )
These studies have been performed before that the analytical
NNLO PDGLAP were computed (Vogt, Vermaseren, Moch.).
The NNLO evolution is crucial for a consistent extraction of NNLO
PDF’s and for determining the cross sections.

5. LO, 70’s
Gribov-Lipatov
Altarelli Parisi
Dokshitzer
NLO, 80’s
Floratos, Ross, Sachrajda,
Curci,
Furmanski
Petronzio

6. QCD at work……. One of the 4 pieces NNLO
Moch, Vermaseren and Vogt, 2004
QCD at work…….

7. NLO NLO The issues that we are going to address are already
encountered in the RGE’s for the running coupling.
Example:
One can obtain exact or “truncated” solutions of this equation
NLO
NLO

8. Moving to NNLO…
There no exact solutions of this equation, but one can
find truncated ones, expressed in terms of another scale
We can proceed to see the implications of this reasoning
To the case of the NNLO pdf’s

9. Solved by
The equation has summed the leading logs

10. The logarithmic ansatz at LO
L.E. Gordon, C. Corianò (1995)
For the photon pdf’s
Da Luz Vieira, Storrow,
1991
Inserted in the DGLAP
Exact solution

11. A logarithmic ansatz “captures” the exact solution in this case,
The issue is: how does the story changes once we move to
NLO?
A similar ansatz had been proposed by C.Corianò and L.E.
Gordon. Now this older ansatz is understood as a “first truncated
solution”. The approach presented here generalizes the CG ansatz

12. The NLO CG ansatz gave recursion relations of the form
But no formal proof of its validity was available. We can go one step further and “solve the recursion relation in terms of the initial conditions (A0 , B0), showing its correctness.
To do so we take the moments of the recursion relations.
Solutions of the RR

13. The solution associated to the ansatz takes the form
This is a solution of the truncated equation where we have expanded the r.h.s ( P/ )
One can solve similarly
Using a NLO ansatz but
of higher accuracy

14. Having in mind how to attach the problem we can move up to NNLO
some Benchmarks are available for the evolution
(Les Houches hep-ph/ )
The benchmarks have been obtained using the exact splitting functions
(hep-ph/ , hep-ph/ ).
Agreement between “brute force code” and the “Mellin method”
It has been developed a new method of evolution which is
general, not “brute force”, but that can reproduce the exact solutions
of the “brute force”
(A. Cafarella, C. Corianò and M. Guzzi, Nucl.Phys.B748: ,2006. )

15. The benchmarks for the LHC
New Physics
A general Analysis of the Z’ Models (Claudio’s talk)
Errors on the PDF’s
Predictions: Drell-Yan, rapidity distributions
The benchmarks for the LHC
Precise determinations of some observables at the LHC:
reduction of the μR / μF dependence
Control on the “accuracy” of the DGLAP solution:
“truncated” or “exact” solutions.
Accuracy in the kinematical region 10^(-5)<x<1,
sensitive to the LHC.

16. Drell-Yan process : : parton distribution functions
: partonic cross section

17. LO : NLO : virtual : real: qg : Drell,Yan (’70)
Altarelli,Ellis,Martinelli(’78,’79);
Kubar-Andre’,Paige(’79);
Harada,Kaneko,Sakai(’79)
virtual :
real:
qg :

18. DGLAP EQUATIONS: A GARDEN OF SOLUTIONS
In the resolution of the DGLAP Eqns. we can classify different kinds of solutions
EXACT fixed (l): solutions in a closed form, achieved
by solving DGLAP at a fixed perturbative order (l), without expanding
around αs =0 the quantity P(αs)/β(αs). Only in Non-Singlet case
TRUNCATED SOLUTIONS: solutions of theκ-th truncated equation
which are expanded around (αs,α0)=(0,0) with O(αs^κ) accuracy.
Non-Singlet and Singlet case
HIGHER ORDER TRUNCATED SOLUTIONS: solutions of theκ-th
truncated equation which are expanded around (αs,α0)=(0,0)
with O(αs^(κ+m) ) accuracy.
Non-Singlet and Singlet case

19. NNLO: Non Singlet Case “Exact eqn.” @ NNLO Exact solution
where we have defined

20. This solution is exactly reproduced using the x-space ansatz
where the coeff. Ds,t,n(x) is
A chain of recursion
relations is generated

21. NLO pattern
NNLO pattern

22. Solving the recursion relations with the condition
which gives in x-space
By a Mellin-transform of this soution one can see that it is the
exact solution obtained solving DGLAP in the Mellin space.

23. A complicated logarithmic resummation is going on!
where we have defined:

24. TRUNCATED SOLUTIONS: Non Singlet Case
κ-th truncated equation
being the Rκ coefficient dependent on P^(0), P^(1),…,P^(κ).
The solution of the truncated eqn is expanded in order to obtain
the NNLO (2-th truncated) solution in the Mellin space, which reads

25. The NNLO “truncated solution” is exactly reproduced by the x-space ansatz
Initial conditions
B0=0, C0=0
In Mellin space
it gives

26. Higher Order Truncated Solutions: Non Singlet Case
where the coefficients Rκdepend only on P^(0), P^(1) and P^(2). Expanding
its solution we obtain a higher order truncated solution with O(αs^κ) accuracy
The higher order truncated solutions of evolution equations of DGLAP type can be organized
in the following form
where k’ can be taken as large as we want.
We claim that this is the solution expanded
at all orders of the DGLAP equation
(singlet/non singlet).

27. x
Behaviour of the ĸ truncated solutions vs the asymptotic one. Non-Singlet case

28. Numerical results: CANDIA vs LesHouhes @NNLO
M. Guzzi 2007 Martina Franca

29. The truncated cros sections vs the aymptotic ones

30. Truncated vs Asymptotic: Cross sections for MRST 2001 input
For k=2, the differences around the Z peak are less than 1%
but they grow up to 4% when we change Q.

31. The Drell-Yan factorized cross section
General form of the
Q-differential cross section:
Hamberg, Matzura, Van Neerven,
Nucl. Phys. B 359, 343 (1991)
Hadronic structure function:
Parton luminosity
Hard scattering

32. K-Factors Analysis
We have a growth
which is more
than 10%

33. K-Factors Analysis: going from NLO up to NNLO
2.7%
4.4%
1.5%
Going from NLO up to NNLO we have a reduction which is
compatible with the errors on the cross sections due to the
PDFs which are around 3%

34. Cross section in the Z peak region
Some plots of the Drell-Yan cross section calculated with Candia
in the Z peak region and in the fast falling region.

35. NNLO Z’ Cross section at the LHC in the peak region
Carena et al. model for extra U(1)_{B-L}

36. Z channel @ the LHC relevant to test extensions of the SM
(R.Armillis, C.Corianò, M.G. forthcoming paper)

37. Renormalization-Factorization scale dependence of the cross section
The pdf’s take a µR dependence from the DGLAP splitting functions

38. µR/µF dependence of the cross section:numerical results
Moving to higher order the scale dependence
reduces

39. Errors on the PDF’s Experimental errors: Theoretical errors:
Errors on the global fit analysis on a wide range of experimental data of DIS
Theoretical errors:
Errors due to the change of perturbative order, logarithmic effects, higher twists contributions.
Once we know the uncertainties on the PDFs we generate different sets
of cross sections (Martin, Roberts, Thorne and Stirling Eur. Phys. J. C. 28, ,
S. Alekhin Phys. Rev. D 68, )
The error on a generic observable (i.e. cross sections and K-factors)
has been calculated by the standard linear propagation of the errors

40. Cross sections with the Errors on the PDF’s

41. Alekhin’s errors on the cross sections
Errors on the cross sections are around 4%

42. DRELL-YAN: RAPIDITY DISTIBUTIONS
Radipity cross section
(C.Anastasiou,L. Dixon, K. Melnikov, F. Petriello, Phys.Rev.D69:094008,2004 ) ×
Factorization Formula

43. Rapidity distribution with errors on the Pdfs

44. Rapidity distribution with µF scale dependence

45. ∆σ = ∆ + ∆ We need accuracy in QCD evolution!
The QCD evolution drives the NNLO
cross section to an overall reduction in the
region that we have analyzed!
∆σ = ∆ + ∆

46. CONCLUSIONS
CANDIA and CANDIAdy:
we need precise determination of the pdf’s for precise determination of the cross sections.
The PDFs evolution drives the cross section.
In the case of SM extensions we can have many U(1) models, and we need to search for the correct one
(if any!!!).
This is a tough task. Requires critical information on the
SM/QCD background.
For some special processes, such as DY we can do an
excellent job through NNLO.

47. CANDIA & CANDIAdy The Candia evolution code documentation will be
available very soon for numerical applications at the LHC.
candia +
candiady
C. Corianò. A. Cafarella and M.G. (forthcoming paper)

48. Back up slides