1. Peter Uwer *) Universität Karlsruhe *) Financed through Heisenberg fellowship and SFB-TR09 Workshop on massive particle production at the LHC —— October, 2007, Berlin Production of heavy particles and jets at next-to-leading order in QCD Work in collaboration with S.Dittmaier, S. Kallweit and S.Weinzierl

2. 2 Contents 1. Introduction 2. Methods 3. Results 4. Conclusion / Outlook

3. 3 Preliminaries TechnicalitiesPhysics Experts Non-Experts  Outline of the main problems/issues/challenges with only brief description of methods used

4. 4 Introduction Why do we need next-to-leading order corrections ● LO predictions give in many cases only rough estimate Large uncertainty due to residual scale dependence as a consequence of uncalculated higher orders ● New channels/new kinematics in higher orders can have important impact in particular in the presence of cuts ● Impact of NLO corrections very difficult to predict without actually doing the calculation ?

5. 5 The master equation for the LHC LHC physics = Standard Model + New Physics New Physics = LHC physics - Standard Model LHC is a discovery machine !

6. 6 WW + 1 Jet ― Motivation ● For 155 GeV < m h < 185 GeV, H  WW is important channel ● In mass range 130 ―190 GeV, VBF dominates over gg  H NLO corrections for VBF known [Figy, Oleari, Zeppenfeld 03, Berger,Campbell 04, …] Signal: Background reactions: WW + 2 Jets, WW + 1 Jet If only leptonic decay of W´s and 1 Jet is demanded (  improved signal significance) Higgs search: two forward tagging jets + Higgs NLO corrections unknown Top of the Les Houches list 07

7. 7 t t + 1 Jet ― Motivation  Important signal process - Top quark physics plays important role at LHC - Large fraction of inclusive tt are due to tt+jet - Search for anomalous couplings - Forward-backward charge asymmetry (Tevatron) - Top quark pair production at NNLO ? - New physics ? - Also important as background (H via VBF) LHC is as top quark factory

8. 8 Methods

9. 9 Next-to leading order corrections Experimentally soft and collinear partons cannot be resolved due to finite detector resolution  Real corrections have to be included The inclusion of real corrections also solves the problem of soft and collinear singularities  Regularization needed  dimensional regularisation 1 n 1 n * 1 n+1

10. 10 How to do the cancellation in practice Consider toy example: Phase space slicing method: Subtraction method [Frixione,Kunszt,Signer ´95, Catani,Seymour ´96, Nason,Oleari 98, Phaf, Weinzierl, Catani,Dittmaier,Seymour, Trocsanyi ´02] [Giele,Glover,Kosower]

11. 11 Dipole subtraction method (1) How it works in practise: Requirements: in all single-unresolved regions Due to universality of soft and collinear factorization, general algorithms to construct subtractions exist [Frixione,Kunszt,Signer ´95, Catani,Seymour ´96, Nason,Oleari 98, Phaf, Weinzierl, Catani,Dittmaier,Seymour, Trocsanyi ´02] Recently: NNLO algorithm[Daleo, Gehrmann, Gehrmann-de Ridder, Glover, Heinrich, Maitre]

12. 12 Dipole subtraction method (2) Universal structure: Generic form of individual dipol: Leading-order amplitudes Vector in color space Color charge operators, induce color correlation Spin dependent part, induces spin correlation universal Example gg  ttgg: 6 different colorstructures in LO, 36 (singular) dipoles ! ! Universality of soft and coll. Limits!

13. 13 Dipole subtraction method — implementation LO – amplitude, with colour information, i.e. correlations List of dipoles we want to calculate 0 1 2 3 4 5 reduced kinematics, “tilde momenta” + V ij,k Dipole d i

14. 14 Leading order amplitudes ― techniques Many different methods to calculate LO amplitudes exist We used: ● Berends-Giele recurrence relations ● Feynman-diagramatic approach ● Madgraph based code Issues: Speed and numerical stability Helicity bases

15. 15 Virtual corrections Issues: ● Scalar integrals ● How to derive the decomposition? Traditional approach: Passarino-Veltman reduction Scalar integrals Large expressions  numerical implementation Numerical stability and speed are important

16. 16 Reduction of tensor integrals — what we did… Reduction à la Passarino-Veltman, with special reduction formulae in singular regions,  two complete independent implementations ! Five-point tensor integrals: Four and lower-point tensor integrals: ● Apply 4-dimensional reduction scheme, 5-point tensor integrals are reduced to 4-point tensor integrals Based on the fact that in 4 dimension 5-point integrals can be reduced to 4 point integrals  No dangerous Gram determinants! [Melrose ´65, v. Neerven, Vermaseren 84] [Denner, Dittmaier 02] ● Reduction à la Giele and Glover [Duplancic, Nizic 03, Giele, Glover 04] Use integration-by-parts identities to reduce loop-integrals nice feature: algorithm provides diagnostics and rescue system

17. 17 What about twistor inspired techniques ? ● For tree amplitudes no advantage compared to Berends-Giele like techniques (numerical solution!) ● In one-loop many open questions – Spurious poles – exceptional momentum configurations – speed My opinion: ● For tree amplitudes tune Berends-Giele for stability and speed taking into account the CPU architecture of the LHC periode: x86_64 ● For one-loop amplitudes have a look at cut inspired methods

18. 18 Results

19. 19 tt + 1-Jet production Sample diagrams (LO): Partonic processes: related by crossing One-loop diagrams (~ 350 (100) for gg (qq)): Most complicated 1-loop diagrams pentagons of the type:

20. 20 Leading-order results — some features Observable: ● Assume top quarks as always tagged ● To resolve additional jet demand minimum k t of 20 GeV Note: ● Strong scale dependence of LO result ● No dependence on jet algorithm ● Cross section is NOT small LHC Tevatron

21. 21 Checks of the NLO calculation ● Leading-order amplitudes checked with Madgraph ● Subtractions checked in singular regions ● Structure of UV singularities checked ● Structure of IR singularities checked Most important: ● Two complete independent programs using a complete different tool chain and different algorithms, complete numerics done twice ! Virtual corrections: QGraf — Form3 — C,C++ Feynarts 1.0 — Mathematica — Fortran77

22. 22 Top-quark pair + 1 Jet Production at NLO [Dittmaier, P.U., Weinzierl PRL 98:262002, ’07] ● Scale dependence is improved ● Sensitivity to the jet algorithm ● Corrections are moderate in size ● Arbitrary (IR-safe) obserables calculable Tevtron LHC  work in progress

23. 23 Forward-backward charge asymmetry (Tevatron) ● Numerics more involved due to cancellations, easy to improve ● Large corrections, LO asymmetry almost washed out ● Refined definition (larger cut, different jet algorithm…) ? Effect appears already in top quark pair production [Kühn, Rodrigo] [Dittmaier, P.U., Weinzierl PRL 98:262002, ’07]

24. 24 Differential distributions Prelimina ry *) *) Virtual correction cross checked, real corrections underway

25. 25 p T distribution of the additional jet Corrections of the oder of 10-20 %, again scale dependence is improved LHC Tevtron

26. 26 Pseudo-Rapidity distribution LHC Tevtron  Asymmetry is washed out by the NLO corrections

27. 27 Top quark p t distribution The K-factor is not a constant!  Phase space dependence, dependence on the observable Tevtron

28. 28 WW + 1 Jet Leading-order – sample diagrams Next-to-leading order – sample diagrams Many different channels!

29. 29 Checks Similar to those made in tt + 1 Jet Main difference: Virtual corrections were cross checked using LoopTools [T.Hahn]

30. 30 Scale dependence WW+1jet Cross section defined as in tt + 1 Jet [Dittmaier, Kallweit, Uwer 07] [NLO corrections have been calculated also by Ellis,Campbell, Zanderighi t 0 +1d, and Binoth, Guillet, Karg, Sanguinetti]

31. 31 Cut dependence [Dittmaier, Kallweit, Uwer 07] Note: shown results independent from the decay of the W´s

32. 32 Conclusions ● NLO calculations are important for the success of LHC ● After more than 30 years (QCD) they are still difficult ● Active field, many new methods proposed recently! ● Many new results General lesson:

33. 33 Conclusions Top quark pair + 1-Jet production at NLO: ● Two complete independent calculations ● Methods used work very well ● Cross section corrections are under control ● Further investigations for the FB-charge asymmetry necessary (Tevatron) ● Preliminary results for distributions

34. 34 Conclusions WW + 1-Jet production at NLO: ● Two complete independent calculations ● Scale dependence is improved (  LHC jet-veto) ● Corrections are important [Gudrun Heinrich ]

35. 35 Outlook ● Proper definition of FB-charge asymmetry ● Further improvements possible (remove redundancy, further tuning, except. momenta,…) ● Distributions ● Include decay ● Apply tools to other processes

36. 36 The End

37. 37 Numerical precisision Process with light quarks (Berends-Giele recursion)

38. 38 Rapidity versus Pseudo-Rapidity Tevtron

39. 39 Introduction — Top quark pair production + 1 Jet Higgs searches at LHC [Atlas `03] The WBF process is important over a wide Higgs mass range Important backgrounds:  Precise predictions for pp  tt + jet are necessary [Alves, Eboli, Plehn, Rainwater ’04] WW