1. QCD at the LHC: What needs to be done? West Coast LHC Meeting Zvi Bern, UCLA Part 2: Higher Order QCD

2. Outline General overview Examples of importance of higher order QCD Experimenters’ wish lists What is the problem? Evils of unphysical formalisms. The S-matrix reloaded: unitarity, twistors, and recursion. Recent calculations and promise for the future What needs to be done.

3. Overview QCD at hadron collider involves a number of complex issues: Parton distribution functions Parton Showers Monte Carlos Underlying Events Hadronization Resummation Higher order QCD very definite calculations need to be done. Steve Ellis’ talk

4. Example:Higgs + 2 jets from Weak Boson Fusion Example: Higgs + 2 jets from Weak Boson Fusion Purpose: After discovery of Higgs Boson measure HWW coupling Background uncertainty can be reduced with an NLO calculation.

5. Example: Susy Search Early studies using PYTHIA over optimistic. ALPGEN vs PYTHIA PYTHIA does not properly model hard jets. ALPGEN is based on LO matrix elements and is better at modeling hard jets. What will disagreement between ALPGEN and data mean? Hard to tell. Need NLO.

6. Merging NLO with Parton Showers It is important to merge NLO with parton showering. Soft and collinear emission properly treated with parton showers. Standard tool for experimenters. Hard emission treated properly by NLO. Standard tool for theorists MC@NLO First example of merging NLO with shower Monte Carlo See Dave Soper

7. The Gold Standard: NNLO Drell-Yan Rapidity Distributions Amazingly good stabilty Theoretical uncertainties less than 1%

8. What needs to be done at NLO? Experimenters to theorists:“Please calculate the following at NLO” Theorists to experimenters: “In your dreams”

9. More Realistic Experimenter’s Wish List Les Houches 2005 Bold action is required even for this

10. State- of-the-Art NLO QCD Five point is still state-of-the art in QCD: Typical examples: Brute force calculations give GB expressions – numerical stability? Amusing numbers: 6g: 10,860 diagrams, 7g: 168,925 diagrams Much worse difficulty: integral reduction generates nasty determinants

11. It Is Time to Dream To attack the wish list need new ideas: Numerical approaches. Promising recent progress. Analytic on-shell methods: unitarity method, on-shell recursion, bootstrap approach Binoth and Heinrich Kaur; Giele, Glover, Zanderighi Binoth, Guillet, Heinrich, Pilon, Schubert; Soper and Nagy; Ellis, Giele and Zanderighi; Anastasiou and Daleo; Czakon; Binoth, Heinrich and Ciccolini Bern, Dixon, Dunbar, Kosower; Bern and Morgan; Cachazo, Svrcek and Witten; Bern, Dixon, Kosower; Bedford, Brandhuber, Spence, Travaglini; Bern, Dixon, Del Duca and Kosower; Britto, Cachazo, Feng and Witten; Berger, Bern, Dixon, Kosower, Forde

12. Why are Feynman diagrams clumsy for multi-parton processes? The vertices and propagators involve gauge-dependent off-shell states. This is the origin of the complexity. To solve the problem we should rewrite perturbative quantum field theory. All steps should be in term of gauge invariant on-shell states. Radical rewriting of perturbative expansion needed.

13. On-shell Formalisms Curiously, an on-shell formalism was constructed at loop level prior to trees: unitarity method. (1994) Solution at tree-level had to await Witten’s twistor inspiration. (2004) -- MHV vertices -- on-shell recursion Combining both give one-loop on-shell bootstrap (2005) Britto, Cachazo, Feng, Witten Cachazo, Svrcek Witten Bern, Dixon, Kosower Forde and Kosower; Berger, Bern, Dixon, Forde amd Kosower Bern, Dixon, Dunbar, Kosower Bern and Morgan

14. Spinors and Twistors Spinor helicity for gluon polarizations in QCD: Penrose Twistor Transform: Witten’s remarkable twistor-space link: QCD scattering amplitudes Topological String Theory Early work from Nair Witten; Roiban, Spradlin and Volovich Key implication: There are simple structure in gauge theory amplitudes

15. Amazing Simplicity Witten Conjectured that in twistor –space gauge theory amplitudes should be supported on curves of degree: Connected picture Disconnected picture These structures imply an amazing simplicity in the scattering amplitudes. MHV vertices for building amplitudes Cachazo, Svrcek and Witten

16. Loop Amplitudes Bern, Dixon, Dunbar and Kosower (1994) Bern and Morgran (1995) Summary of results from early papers: Key result: Any massless loop amplitude in any theory is fully determined from D-dimensional tree amplitudes and unitarity to all loop orders. Off-shell formulations are unnecessary. Four-dimensional cut constructibility: At one-loop, any amplitude in a massless susy gauge theory is fully constructible from four- dimensional tree amplitudes (even in presence of IR and UV divergences). Use helicity. One-loop QCD: If we use spinor helicity for the tree amplitudes we drop rational functions in loop amplitudes, but logs and polylogs all constructed correctly.

17. Unitarity Method Two-particle cut: Generalized triple cut: Three- particle cut: Unitarity method combines very effectively with twistor-inspired ideas. Should be interpreted as demanding that cut propagators do not cancel.

18. On-Shell Bootstrap Bern, Dixon, Kosower hep-ph/9708239 Difficult to find rational functions with desired factorization properties. Unclear how to automate. Use Unitarity Method with D = 4 helicity states. Efficient means for obtaining logs and polylogs. Build from on-shell tree amplitudes. Use factorization properties to find rational function part. Check numerically against Feynman diagrams Early Approach: Key problems preventing widespread applications:

19. Britto, Cachazo, Feng On rhs only on-shell tree amplitudes with fewer legs appear. Evaluate with momenta shifted by a complex amount AnAn A k+1 A n-k+1 Tree-Level On-Shell Recursion New representations of tree amplitudes from IR consistency of one- loop amplitudes in N = 4 super-Yang-Mills theory. With intution from twistors and generalized unitarity: Bern, Del Duca, Dixon, Kosower; Roiban, Spradlin, Volovich on-shell recursion

20. Simple Proof of On-Shell Recursion Proof relies on so little: Cauchy’s theorem Basic field theory factorization properties Consider shifted amplitude : Britto, Cachazo, Feng and Witten At tree level we know all the residues:

21. Merging Unitarity With Loop-Level Recursion New Features: Presence of branch cuts. unreal poles – poles which appear only for complex momenta. double poles – S-matrices in general have double poles Spurious singularities that cancel only against polylogs. Add rational functions to remove these. Double counts between cuts and recursion. These result in overlap diagrams. Bern, Dixon Kosower; Forde and Kosower; Berger, Bern, Dixon, Forde, Kosower, Pure phase for real momenta

22. Five-point example Only one non-vanishing recursive diagram: Assume we already have log terms computed from D = 4 cuts. The most challenging part was rational function terms. Only two overlap diagrams: The rational function terms are as easy to get as at tree level!

23. Six-Point Example

24. What needs to be done? Firmer theoretical basis for formalism is needed: Large z behavior of loop amplitudes. General understanding of unreal poles. Complex factorization of amplitudes. Attack experimenters’ wishlist. Massive loops -- tree recursion understood. Connection to Lagrangian – Space-cone gauge. Improved evaluation of triangles or bubble integrals would be helpful. Assembly of full cross-sections. Catani-Seymour dipole method. Automation for general processes. Chalmers and Siegel; Vaman and Yao Carola Berger’s presentation Badger, Glover, Khoze, Svrcek Many theoretical and practical aspects

25. Other Applications A method is even more important than a discovery, since the right method will lead to new and even more important discoveries. — L.D. Landau On-shell method have been applied to a variety of problems. Examples: Resummation of MHV N = 4 super-Yang-Mills amplitudes to all loop orders. UV Finiteness properties of N = 8 supergravity — Definitely less divergent than people had thought. — Is it finite, contrary the accepted wisdom? We have the technology to find out! Bern, Rozowsky, Yan; Anastasiou, Bern, Dixon, Kosower; Bern, Dixon, Smirnov; Buchbinder and Cacazho; Cachazo, Spradlin,Volovich Bern, Dixon,Dunbar,Perelstein,Rozowsky; Howe and Stelle; Bern, Bjerrum-Bohr, Dunbar

26. Summary Analysis of LHC experiments involve complex issues in QCD. Higher order QCD is a key issue facing us. Conventional approaches have failed to provide full range of desired calculations – time for bold action at NLO. New methods (a) numerical approaches (b) on-shell methods – reformulation of quantum field theory New results for six partons and n-partons Much more needs to be done to set up formalism, automation, and construction of physical cross-sections for comparison to data. Experimenters’ wish list awaits us.