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

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

3 Preliminaries TechnicalitiesPhysics Experts Non-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  s :

6 Scale dependence For  ≈ m t and a variation of a factor 2 up and down: n 20 % 40 % 30 % 10 % 22QCD22QCD 23QCD23QCD 24QCD24QCD In addition we have also the factorization scale...  Need loop corrections to make quantitative predictions Born approximation gives only crude estimate!

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

8...and difficult to estimate WW production via gluon fusion: tot = no cuts, std = standard LHC cuts, bkg = Higgs search cuts 30 % enhancement due to an “NNLO” effect (  s 2 ) [Duhrssen, Jakobs, van der Bij, Marquard 05 Binoth, Ciccolini,Kauer Krämer 05,06]

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 ● 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 [Han, Valencia, Willenbrock 92 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

12 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

13 Methods

14 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 *) For hadronic initial state additional term from factorization… *)

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

16 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]

17 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]

18 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!

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

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

21 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 (Tools: Alpgen [MLM et al], Madgraph [Maltoni, Stelzer], O’mega/Whizard [Kilian,Ohl,Reuter],…)

22 n n * 1 1

23 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

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

25 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

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 diagrams pentagons of the type:

29 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

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

31 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

32 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]

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

34 p T distribution of the additional jet Corrections of the oder of %, again scale dependence is improved LHC Tevtron

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

36 Top quark p t 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 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 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]

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

41 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:

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