1. Multi-Parton QCD Amplitudes For The LHC. Harald Ita, UCLA Based on: arXiv:0803.4180 ; arXiv:0807.3705; arXiv:0808.0941 In collaboration with: Carola Berger, Zvi Bern, Fernando Febres Cordero, Lance Dixon, Darren Forde, Daniel Maitre and David Kosower. Recently joined by Tanju Gleisberg

2. Content: 1.Simulation of scattering processes. 2.LO versus NLO. 3.The Wish-List. 4.On-Shell Methods. 5.Example technicalities. 6.Amplitudes from BlackHat. 7.Conclusions.

3. Scattering processes at hadron colliders: A multi-layered problem Tevatron graphic taken from Rick Field

4. …+ multi-parton scattering +... proton remnants. hard scattering quark/gluon. Tevatron graphic taken from Rick Field

5. Understanding the underlying event. Control over parton distribution functions. The hard scattering: LO NLO (Virtual <- BlackHat, Real) Beyond Showering (matching). Hadronization. Layers of simulation:

6. Some random set of tools. 1.Parton-level LO matrix element generators: MADGRAPH; Maltoni, Stelzer … 2.LO ME + shower MCs + …: ALPGEN; Mangano, Moretti, Piccinini, Pittau, Polosa HERWIG; Marchesini, Webber, Abbiendi, Corcella, Knowles, Moretti, Odagiri, Richardson, Seymour, Stanco PYTHIA; Sjostrand, Mrenna, Skands SHERPA; Gleisberg, Hoeche, Krauss, Schoenherr, Schumann, Siegert, Winter … 3.NLO PL matrix element generators: MCFM; Campbell, Ellis; (max 6 partons) (BlackHat …,6,7,..(?) partons) … 4.MC@NLO,POWHEG-method; Frixione, Webber; +Nason, Ridolfi, Oleari high-multiplicities + automation

7. Leading Order versus Next to Leading Order. QCD for :

8. LO versus NLO. Tree-level (LO) predictions qualitative, due to poor convergence of expansion in strong coupling, large NLO corrections can be 30% - 80% of LO. implies scale uncertainty: 1 jet about 12% 2 jets about 24% 3 jets about 36% K-factors ?

9. Tevatron: Single Top Production single top production Matrix element method uses full information of LO matrix elements to pull the signal out of background. It should be possible to do better by using NLO matrix elements (about 20% gain in significance). T. Aaltonen et al. [CDF Collaboration], arXiv:08092581

10. the LHC: an example of discovery [ATLAS Collaboration] S. Vahsen M L Mangano [arXiv:0809.1567] Producing heavy colored particles Backgrounds: Irreducible: Z(->neutrinos)+4 jets Reducible: W(->tau+neutrino)+3 jets W(->undetected leptons) +4 jets top pairs Instrumental: Multijets

11. How good are our tools? SMPR-model: Mrena, Richardson, 2004. MLM-model: Alwall et al. arXiv:0706.2569. LO, MCFM; parton level; including Bern, Dixon, Kosower, Weinzierl matrix elements. NLO. Wanted: LHC studies with extra jets: T. Aaltonen et al. [CDF Collaboration], arXiv:0711.4044 +...

12. Wish-List.

13. QCD: “Experimenters’ Wish List” Les Houches 2007 Five-particle processes under good control with Feynman diagram based approaches. Six-particle processes still difficult challenge.

14. Most physics results done from Feynman diagram approach: –QCD corrections to vector boson pair production ( W + W -, W ± Z & ZZ ) via vector boson fusion (VBF). (Jager, Oleari, Zeppenfeld)+(Bozzi) –QCD and EW corrections to Higgs production via VBF. (Ciccolini, Denner, Dittmaier) –pp → Higgs +2 jets. (via gluon fusion Campbell, Ellis, Zanderighi), (via weak interactions Ciccolini, Denner, Dittmaier). pp → Higgs +3 jets (leading contribution) (Figy, Hankele, Zeppenfeld). –pp→. (Beenakker, Dittmaier, Krämer, Plümper, Spira, Zerwas), (Dawson, Jackson, Reina, Wackeroth) –pp → ZZZ, (Lazopoulos, Petriello, Melnikov) pp → +(McElmurry) –pp → WWZ, WWW (Hankele, Zeppenfeld, Campanario, Oleari, Prestel) –pp→WW+j+X. (Campbell, Ellis, Zanderighi). (Dittmaier, Kallweit, Uwer) –pp →W/Z (Febres Cordero, Reina, Wackeroth), –pp → +jet (Dittmaier,Uwer,Weinzierl), – (Bredenstein,Denner,Dittmaier,Pozzorini), What Has Been Done?

15. What we need. Scalable to larger numbers of external partons. A general solution that applies to any process. Numerical stability and efficiency. Automation.

16. On-Shell Methods.

17. Calculating using Feynman diagrams is Hard! A Factorial growth in the number of terms, particularly bad for large number of gluons. Example of difficulty: 6 gluons ~10,000 diagrams. 7 gluons ~150,000 diagrams. Gauge dependant quantities, large cancellations between terms. Economic techniques? Note: this is trivial on modern computer. Non-trivial for larger numbers of external particles.

18. Result of performing the integration gives: Calculations explode for larger numbers of particles or loops. There should be a better way.

19. Calculated ON-SHELL, amplitudes much simpler than expected. For example: some tree level all-multiplicity gluon amplitudes can fit on a page: On-shell simplifications. Park, Taylor

20. Vertices and propagators involve unphysical gauge-dependent off-shell states. Feynman diagram loops have to be off-shell because they encode the uncertainty principle. Keep particles on-shell in intermediate steps of calculation, not in final results. Think off-shell, work on-shell! Bern, Dixon, Dunbar, Kosower Fact: Off-shellness is essential for getting the correct answer. Would like to reconstruct amplitude using only on-shell information.

21. Exploit universal physical properties for decomposition in terms of on-shell objects: 1. Unitarity relation. 2. Universal Factorization. 3. Complex momenta through spinor variables. On-shell methods: opening a gate to new computational possibilities...

22. Past year progress using unitarity and related techniques, –gg → gggg amplitude. (Bern,Dixon,Kosower), (Britto,Feng,Mastrolia), (Bern,Bjerrum-Bohr,Dunbar,H.I.), (Berger,Bern,Dixon,Forde,Kosower), (Bedford,Brandhuber,Spence,Travaglini) (Xiao,Yang,Zhu),(Berger,Bern,Dixon,Forde,Kosower), ( Giele,Kunszt,Melnikov ) –Lots of gluons (Giele,Zanderighi), (Berger, Bern, Dixon, Febres Cordero, Forde,H.I., Kosower, Maître) –6 photons (Nagy, Soper), (Ossola, Papadopoulos, Pittau), (Binoth, Heinrich, Gehrmann, Mastrolia) –pp → ZZZ, WZZ, WWZ, ZZZ (Binoth, Ossola, Papadopoulos, Pittau), –gg → using D-Dimensional Unitarity ( Ellis,Giele,Kunszt,Melnikov ) Numerical packages under construction: –BlackHat Berger, Bern, Dixon, Febres Cordero, Forde, H.I., Kosower, Maître –CutTools Ossola, Papadopoulos, Pittau –Rocket Ellis, Giele, Kunszt, Melnikov, Zanderighi What Has Been Done?

23. Example technicalities.

24. The result: one-loop basis. All external momenta in D=4, loop momenta in D=4-2ε (dimensional regularization). Cut Part from unitarity cuts in 4 dimensions. Rational part from on-shell recurrence relations. See Bern, Dixon, Dunbar, Kosower, hep-ph/9212308. Rational partCut part Process dependent D=4 rational integral coefficients

25. Unitarity Method. Unitarity Approach: –Bern, Dixon, Dunbar, Kosower, hep-ph/9403226, hep- th/9409265. –Recent Advances using spinorial integration techniques: Cachazo, Svrcek, Witten; Britto, Feng, Cachazo; Britto, Feng, Mastrolia Generalized Unitarity: –Bern, Dixon, Kosower, hep-ph/9708239, hep-ph/0001001. –Britto, Cachazo, Feng, hep-th/0412103. –Recent Advances: classification of surface terms. del Aguila and Pittau, hep-ph/0404120. Ossola, Papadopoulos and Pittau, hep-ph/0609007. Forde, 0704.1835; Badger, 0806.4600, 0807.1245. Ellis, Giele, Kunszt, 0708.2398; Giele, Kunszt and Melnikov, 0801.2237; Ellis, Giele, Kunszt, Melnikov, 0806.3467;

26. Unitarity: an on-shell method of calculation. Bern, Dixon, Kosower Cutting loops = sewing trees: Sewing: Cutting: 2x And NOT: Equation: Number of diags.

27. Generalized Unitarity: isolate the leading discontinuity. More cuts, more trees, less algebra: Two-particle cut: product of trees contains subset of box-, triangle- and bubble-integrals. (Bern, Dixon, Kosower, Dunbar) Triple-cut: product of three trees contains triangle- and box-integrals. (Bern, Dixon, Kosower) Quadruple-cut: read out single box coefficient. (Britto, Cachazo, Feng) Cutting: n x

28. Boxes: the simplest cuts. Un-physical (=spurious) singularities from parameterization. Have to cancel eventually: role of rational term R. Berger, Bern, Dixon, Febres Cordero, Forde, H.I., Kosower, Maitre 0803.4180; Risager 0804.3310.

29. Amplitudes From BlackHat.

30. BlackHat: A C++ implementation of on- shell techniques for 1-loop amplitudes Portability (standard libraries for unix systems) Modularity (object oriented) Malleability (to accept several routines – numerics and analytics) Numerical precision and efficiency Ready to use with existing Monte Carlo programs –Work in progress with automated real dipole subtraction from Sherpa (with T. Gleisberg)

31. Gluon amplitudes: The Tails. Double-precision numerical computation. Dynamic multi-precision computation. Reference: analytic targets from Bern, Dixon, Dunbar, Kosower, hep-ph/9403226, hep-ph/9409265, hep-ph/0507005. Natural tail Multi- precision Natural tail (III) Multi- precision 100 000 PS points, ET>0.01 s, pseudo rapidity 0.4

32. Watch Instabilities. Monitor using known IR/UV pole structure of amplitudes. Generalization for rational part. (A consistency condition of spurious residues.) Avoid instabilities with analytic tricks: –Use good loop momentum parametrizations & spinor variables.

33. Cross Sections BlackHat : a snapshot… Multiprecision arithmetic gives excellent control over numerical stability… double quadruple double One Loop Helicity Amplitude Rational PartCut Part boxestrianglesbubbles Trees Recursive diagrams Spurious poles One Loop Helicity Amplitude Rational PartCut Part boxestrianglesbubbles Trees Recursive diagrams Spurious poles One Loop Helicity Amplitude Rational PartCut Part boxestrianglesbubbles Trees Recursive diagrams Spurious poles

34. Progress Report.

35. October 08 Z+3jets: Leading Color Stability Study Berger, Bern, Dixon, FFC, Forde, Ita, Kosower, Maître, arXiv:0808.0941[hep-ph] 100 000 PS points, ET>0.01 s, pseudo rapidity 0.4

36. Timing: Z+3 jets. Same order as 6pt gluon amplitudes ~50ms: Quite an improvement compared to other numerical methods! Helicity: +++: 4.01ms/4.06ms 10.6ms/10.7ms (now: ~4ms) -++: 6.41ms/6.64ms 41.7ms/42.9ms ++-: 6.47ms/6.67ms 28.0ms/28.4ms (now~6.5 ms) -+-: 7.70ms/7.92ms 39.2ms/40.5ms Double precisionDynamic multi precision Full amplitude4-D cut-part

37. Conclusions NLO QCD corrections to hard cross sections will be an important tool for LHC analyses. On-shell methods have opened a new gate to computational power in QFTs. BlackHat-implementation has proven good precision properties. We expect first important phenomenological results, for example in V+3jets processes and beyond!