MCWS, Frascati 22-24 May 2006

1. “NNLO evolution of the pdf's and their errors: applications to the Drell-Yan and Higgs cross sections” Marco Guzzi Department of Physics University of Lecce, INFN Lecce, Italy MCWS, Frascati 22-24 May 2006 M. Guzzi MCWS Frascati

2. OUTLINE In the study of new Physics in selected processes at the LHC we need precision (NNLO QCD) It is unlikely that many processes will be computed in the near future, but DY@NNLO is known and people have been able to extract dσ/dM, dσ/dM dY. Vrap is available. (C. Anastasiou, L. Dixon, K.Melnikov, F. Petriello Phys. Rev. D 69 094008) These studies have been performed before that the analytical expansion of the NNLO PDGLAP was computed. At that time the complete NNLO evolution kernels needed for a consistent extraction of NNLO PDF’s were not known. perturbative evolutions based upon the known moments of the required DGLAP eqns. fastest MRST programs contains slowest M. Guzzi MCWS Frascati

3. Therefore improvements are needed. PEGASUS has been released by A. Vogt and we have some benchmark (Les Houches hep-ph/0204316) (A. Vogt Comput. Phys. Commun. 170, 65 2005) The benchmarks have been uploaded with the exact splitting functions (hep-ph/0204316, hep-ph/0511119). Agreement between “brute force code” and the “Mellin method” is difficult to test. The issue of the benchmarks in the PDF’s evolution has therefore to be resolved quite independently It has been developed a new method of evolution, not “brute force”, but that can reproduce the exact solution of “brute force” (A. Cafarella, C. Corianò and M. Guzzi, hep-ph/0512358 accepted for a pubblication on Nucl. Phys. B) M. Guzzi MCWS Frascati

4. A general Analysis of the Z’ Models (Claudio’s talk) New Physics Errors on the PDF’s Predictions: Higgs and Drell--Yan INTRODUCTION Precise determinations of some observables at the LHC require benchmarks for the parton distribution functions which can be matched only at next-to-next-to leading order in the strong coupling constant in order to reduce the factorization and renormalization scale dependence of the perturbative series. At this order the solution of the RGE's involve non trivial resummations of the corresponding logarithms which should be very accurate in order to provide exact predictions in the range 10^(-5) < x < 1. We formulate alternative expansions, which are generic for both forward and non- forward twist-2 operators, that converge to the solutions of the RGE's with very high precision. The Higgs case is illustrated in detail. In the Drell-Yan case these analysis can be used to test with high accuracy most of the Z's models. Work done in collaboration with Alessandro Cafarella (CRETE) and Claudio Corianò (LECCE). M. Guzzi MCWS Frascati

5. DGLAP EQUATIONS: A GARDEN OF SOLUTIONS In the resolution of the DGLAP Eqns. we can classify different kinds of solutions EXACT SOLUTIONS @ 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 M. Guzzi MCWS Frascati

6. NNLO: Non Singlet Case “Exact eqn.” @ NNLO Exact solution where we defined M. Guzzi MCWS Frascati

7. 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 M. Guzzi MCWS Frascati

8. Solving the recursion relations with the condition which gives in x-space By a Mellin-transform of this soution one can argue that it is the exact solution obtained solving DGLAP in the Mellin space M. Guzzi MCWS Frascati

9. TRUNCATED SOLUTIONS: Non Singlet Case κ-th truncated equation being the Rκcoefficientdependent 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 M. Guzzi MCWS Frascati

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

11. 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 all orders 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 to all orders of the DGLAP equation (singlet/non singlet). This form holds both for “exact” solutions and for accurate solutions. “Exact” solutions of the truncated equation are obtained from this expression by sending κ’ to . M. Guzzi MCWS Frascati

12. NNLO Truncated Solutions: The Singlet Case NNLO Truncated vector eqn whose generic solution can be written as (Buras Rev. Mod. Phys.52; 159, 1980) The U1 and U2 operators are defined the chain of commutators The logarithmically expanded ansatz will be M. Guzzi MCWS Frascati

13. The property Ô = Ô++ + Ô+- + Ô-+ + Ô-- holds Solving the Rec.Rel. for the Bn coeff. we obtain Solving the rec. rel. for the Cn coeff. we obtain Cn--, Cn++, Cn-+, Cn+- C++=C--/. {- ---> +, + ---> -}; C-+=C+-/. {- ---> +, + ---> -} M. Guzzi MCWS Frascati

14. The ++projected piece of the vector solution reads Formal solution in the x-space There is an exact analytical agreement between these solutions and the ones obtained by Vogt in the Mellin space. M. Guzzi MCWS Frascati

15. Numerical results: X-SIEVE vs PEGASUS @LO M. Guzzi MCWS Frascati

16. Numerical results: X-SIEVE vs PEGASUS @NLO M. Guzzi MCWS Frascati

17. Numerical results: X-SIEVE vs PEGASUS @NNLO M. Guzzi MCWS Frascati

18. Separation of Factorization/Renormalization scales easily done with X-SIEVE M. Guzzi MCWS Frascati

19. PDF’s ERRORS: Total cross section for the Higgs and DY Higgs production @NNLO A.Cafarella, C.Corianò, M.G., J.Smith, hep-ph/0510179 M. Guzzi MCWS Frascati

20. Errors on the PDF’s Experimental 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, 455 2003, S. Alekhin Phys. Rev. D 68, 014002 2003) The error on a generic observable (i.e. cross sections and K-factors) has been calculated by the standard linear propagation of the errors M. Guzzi MCWS Frascati

21. Total cross section for the Higgs production @ the LHC with Alekhin Inputs M. Guzzi MCWS Frascati

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23. NNLO CASE M. Guzzi MCWS Frascati

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26. Drell-Yan process : : parton distribution functions : partonic cross section Hamberg, Van Neerven, Nucl. Phys. B 359, 343 (1991) ; M. Guzzi MCWS Frascati

27. Alekhin input with μF= μR M. Guzzi MCWS Frascati

28. Alekhin input with μF= μR M. Guzzi MCWS Frascati

29. Alekhin input with μF= μR M. Guzzi MCWS Frascati

30. Alekhin input with μF= μR M. Guzzi MCWS Frascati

31. Alekhin input with μF= μR M. Guzzi MCWS Frascati

32. Alekhin input with μF= μR M. Guzzi MCWS Frascati

33. MRST input M. Guzzi MCWS Frascati

34. MRST input M. Guzzi MCWS Frascati

35. MRST input M. Guzzi MCWS Frascati

36. CONCLUSIONS XSIEVE and PEGASUS: precise determination of the pdf’s for precise determination of the number of events we may be able to detect. 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. M. Guzzi MCWS Frascati

37. BACK-UP SLIDES M. Guzzi MCWS Frascati

38. K-FACTORS Alekhin input with μF= μR M. Guzzi MCWS Frascati

39. MRST input with μF =μR M. Guzzi MCWS Frascati

40. IMPROVEMENTS 1) Resummation at small qT, for instance in Drell-Yan plus jet (NLO) 2) Threshold resummation (x=1) . The fixed order expansion just does not work in some cases. But this has to do with the hard scattering. What about the evolution of the pdf’s ? We are going to show that the logarithmic evolution of the parton densities can be expressed in a form which is very simple and easy to handle. A by-product of this analysis are new benchmarks for the evolution of the pdf's down to very small-x using a software that incorporates this new theoretical approach (Cafarella, Guzzi, C.C.) X-SIEVE: X-Space Solutions by Iterations of Evolution Equation M. Guzzi MCWS Frascati

41. Hunting for Logs in the Evolution of the pdf’s with XSIEVE When we work at a certain fixed order in the strong coupling expansion the accuracy in the solution of the evolution can be either 1) kept at the working order or 2) higher order effects can be also kept into account. Both points of view are acceptable. Do these approaches swamp away the NNLO corrections? 3) Theoretical errors on the pdf’s. These are now given in most of the parameterizations presented so far (at least at NLO). M. Guzzi MCWS Frascati

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43. WHAT DOES IT MEAN “EXACT”? Nothing is exact when we use perturbation theory in the computation of a quantity. We can talk either of Solutions which are ACCURATE at a given order in the αs or 2) we forget about possible higher order corrections in the kernels and solve exactly DGLAP eqns. only with P0 an P1. This second solution is “exact”, but is not ACCURATE. Higher order corrections could, in principle, swamp away the result. Question: can we have a control on this issue? Theorem (Cafarella, Guzzi, C,C.) The all orders solutions of evoltuion equations of DGLAP type can be organized in the following forms Where k’ can be taken as large as we want. We claim that this is the solution to all orders of the DGLAP equation (singlet/non singlet). This form holds both For “exact” solutions and for accurate solutions. “Exact” solutions are obtained from this expression by sending k’ to infinity. M. Guzzi MCWS Frascati

44. Building accurate solutions M. Guzzi MCWS Frascati

45. In the x-space the logarithmic ansatz reads and it gives the recursion relations In the Mellin space convolution products become mutiplications Initial conditions with B0=0 M. Guzzi MCWS Frascati

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47. Higher order truncated solutions M. Guzzi MCWS Frascati

48. Exact Solutions at NLO in Mellin Space can be reproduced by an x-space ansatz of the type where the operators in the x-space act as follow Initial cond M. Guzzi MCWS Frascati

49. The NNLO case: Truncated Solutions κ-th truncated equation NNLO (2-th truncated) solution in the Mellin Space M. Guzzi MCWS Frascati

50. NNLO x-space ansatz Initial conditions M. Guzzi MCWS Frascati