Peter Uwer *)

1. Graduiertenkolleg “Physik an Hadronbeschleunigern”, Freiburg 07.11.07 ppttj and ppWWj at next-to-leading order in QCD Peter Uwer*) Universität Karlsruhe Work in collaboration with S.Dittmaier, S. Kallweit and S.Weinzierl *) Financed through Heisenberg fellowship and SFB-TR09

2. Contents • Introduction • Methods • Results • Conclusion / Outlook

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

4. Why do we need to go beyond the Born approximation ?

5. Residual scale dependence Quantum corrections lead to scale dependence of the coupling constants, i.e:  Large residual scale dependence of the Born approximation In particular, if we have high powers of as:

6. Scale dependence 40 % 30 % 20 % 10 % 1 3 2 4 n 22 QCD 23 QCD 24 QCD For m≈ mt and a variation of a factor 2 up and down: In addition we have also the factorization scale... Born approximation gives only crude estimate!  Need loop corrections to make quantitative predictions

7. Corrections are not small... Top-quark pair production at LHC: [Dawson, Ellis, Nason ’89, Beenakker et al ’89,’91, Bernreuther, Brandenburg, Si, P.U. ‘04] ~30-40% m/mt Scale independent corrections are also important !

8. ...and difficult to estimate WW production via gluon fusion: [Duhrssen, Jakobs, van der Bij, Marquard 05Binoth, Ciccolini,Kauer Krämer 05,06] tot = no cuts, std = standard LHC cuts, bkg = Higgs search cuts 30 % enhancement due to an “NNLO” effect (as2)

9. To summarize: NLO corrections are needed because • Large scale dependence of LO predictions… • 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

10. Shall we calculate NLO corrections for everything ?

11. WW + 1 Jet ― Motivation Higgs search: • For 155 GeV < mh < 185 GeV, H  WW is important channel • In mass range 130 ―190 GeV, VBF dominates over ggH [Han, Valencia, Willenbrock 92 Figy, Oleari, Zeppenfeld 03, Berger,Campbell 04, …] NLO corrections for VBF known Signal: two forward tagging jets + Higgs Background reactions: WW + 2 Jets, WW + 1 Jet Top of the Les Houches list 07 NLO corrections unknown If only leptonic decay of W´s and 1 Jet is demanded (improved signal significance)

12. t t + 1 Jet ― Motivation LHC is as top quark factory • 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)

13. Methods

14. Next-to leading order corrections 1 1 1 n n n+1 *) * 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 *) For hadronic initial state additional term from factorization…

15. Ingredients for NLO 1 1 1 1 1 1 n n n n n+1 n+1 Many diagrams, complicated structure, Loop integrals (scalar and tonsorial) divergent (soft and mass sing.) * Combination procedure to add virtual and real corrections + Many diagrams, divergent (after phase space integ.)

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

17. Dipole subtraction method (1) [Frixione,Kunszt,Signer ´95, Catani,Seymour ´96, Nason,Oleari 98, Phaf, Weinzierl, Catani,Dittmaier,Seymour, Trocsanyi ´02] 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 Recently: NNLO algorithm [Daleo, Gehrmann, Gehrmann-de Ridder, Glover, Heinrich, Maitre]

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

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

20. 1 n+1 LO amplitudes enter in many places…

21. Leading order amplitudes ― techniques Many different methods to calculate LO amplitudes exist (Tools: Alpgen [MLM et al], Madgraph [Maltoni, Stelzer], O’mega/Whizard [Kilian,Ohl,Reuter],…) We used: • Berends-Giele recurrence relations • Feynman-diagramatic approach • Madgraph based code Helicity bases Issues: Speed and numerical stability

22. 1 1 * n n

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

24. Passarino-Veltman reduction [Passarino, Veltman 79] ?

25. Reduction of tensor integrals — what we did… Four and lower-point tensor integrals: Reduction à la Passarino-Veltman, withspecial reductionformulae insingular regions,  two complete independent implementations ! Five-point tensor integrals: • Apply4-dimensional reductionscheme, 5-point tensor integrals are reduced to 4-point tensor integrals  No dangerous Gram determinants! [Denner, Dittmaier 02] Based on the fact that in 4 dimension 5-point integrals can be reduced to 4 point integrals [Melrose ´65, v. Neerven, Vermaseren 84] • 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

26. 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

27. Results

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

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

30. 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 ! Feynarts 1.0 — Mathematica — Fortran77 Virtual corrections: QGraf — Form3 — C,C++

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

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

33. Differential distributions Preliminary *) *) Virtual correction cross checked, real corrections underway

34. pTdistribution of the additional jet LHC Tevtron Corrections of the oder of 10-20 %, again scale dependence is improved

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

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

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

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

39. Scale dependence WW+1jet [Dittmaier, Kallweit, Uwer 07] Cross section defined as in tt + 1 Jet [NLO corrections have been calculated also by Ellis,Campbell, Zanderighi t0+1d]

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

41. Conclusions General lesson: • 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

42. 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

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

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

45. The End