Prospects for the Future: The Message from Analyticity

1. Prospects for the Future:The Message from Analyticity High Precision for Hard Processes at the LHC Zurich, 2006 Zvi Bern, UCLA

2. LHC Physics The LHC will start operations in 2007! We are all extremely excited by this.

3. Outline This will not be a traditional summary talk. • A few comments on the impressive progress we • have heard about in this conference. • Mostly I wish to discuss scattering amplitudes and • the hope the newly uncovered structures and simplicity • gives for the future. • The challenges awaiting us.

4. Summary of Recent Progress Areas of progress we heard about at this conference: • NLO • -- Explicit results. Applications to QCD and Electroweak • -- new theoretical understanding • Parton showering Monte Carlos and resummation • -- merging with matrix elements • NNLO • -- progress towards general differential cross sections • -- splitting functions and DIS • Theoretical issues • -- Consistent treatment of decay widths • -- Applications to gravity

5. NLO • Traditional analytic with various improvements: • talks from Denner, Oleari, Spira, Uwer and Zeppenfeld • Semi-numerical • talks from Binoth, Dittmaier, Pittau, Weinzierl, Zanderighi • Pure numerical • talk from Catani, Passarino and Soper • On-shell methods • talks from Brandhuber, Britto, Dunbar, Feng, Forde, Glover, • Kosower, Travaglini • Hybrid methods • talk from Zhu and Papadopoulos • Construction of Cross Sections • talk from Catani “Best” approach probably will combine a number of these ideas plus new ones. New milestones: In QCD, one-loop six gluon amplitudes with all helicities and n gluon amplitudes with 2 and 3 negative helicities; In electroweak

6. Which is the best bottle? Numerical Passarino Veltman On Shell Semi- Numerical Improved Traditional New Ideas? To open a bottle perform complete six and seven point calculations.

7. NNLO Impressive progress for fully differential processes: • Subtraction or Antenna approach. • Drell-Yan and jets discussed. • talks from Gehrmann, Kilgore, Del Duca, Somogyi • Sector decomposition approach. • no talk but lots of progress: Binoth, Heinrich; Anastasiou, Melnikov and Petriello. PDF’s and DIS • NNLO PDF’s and DIS Coefficient Functions • Talks from Marchesini, Moch, Stirling Improved understanding of IR singularities Talks from Dixon and Mitov

8. Parton Showering or Resummation For realistic prediction need parton showering or resummation Need to import NLO matrix elements into this. Talks from de Florian, Giele, Grazzini, Nason, Webber W+ W-: MC@NLO vs Resummation W+ W-: Parton Shower vs NLO Grazzini’s talk Webber’s talk MC@NLO: Good features of parton showers merged with good features of NLO. Heard new ideas for merging NLO with parton showers.

9. The quest for simplicity We all know that loop computations are a serious and complicated business. Might amplitudes in general be a lot simpler than we thought? This seems hard to believe given the factorial explosions we normally face. In this talk I want to argue that in general amplitudes are much simpler than we thought, though more needs to be done to fully expose the simplicity.

10. NLO QCD: where we are now We are only now starting to achieve six point calculations Typicalexamples of current calculations: Bern, Dixon and Kosower Dixon, Kunszt and Signer Campbell and Ellis Dittmaier, Uwer, Weinzierl Six gluon amplitudes ZB, Dixon, Dunbar and Kosower Britto, Cachazo, Feng Bidder, Bjerrum-Bohr, Dixon, Dunbar Bedford, Brandhuber, Spence and Travaglini ZB, Bjerrum-Bohr, Dunbar, Ita Ellis, Giele and Zanderighi Xiao, Yang and Zhu Electroweak 4 f cross-sections Denner, Dittmaier, Roth and Wieder

11. Les Houches NLO Wishlist Les Houches 2005 Bold action required! Talks from Dissertori and Huston

12. Example: Susy Search ALPGEN vs PYTHIA Early ATLAS TDR studies using PYTHIA overly optimistic. • ALPGEN is based on LO • matrix elements and much • better at modeling hard jets. • What will disagreement between • ALPGEN and data mean? Hard • to tell. Need NLO. An important task for people in this room is to figure out how to actually compute this.

13. What we need • Numerical stability. • Scalable to 6 and 7 external partons. • A general solution that applies to any process. • Automation. What we’re dreaming of • Modest growth in complexity with increasing number of legs. • “Compact” analytic expressions. We heard a number of analytic and numerical proposals for trying to do so.

14. Where is the simplicity? This is from a 4-point evaluation which is very simple to do by today’s standards.

15. All steps should be in terms of gauge invariant • on-shell states. • To fully expose the simplicity a radical rewriting of perturbative • QCD is needed. Why are Feynman diagrams clumsy for high-multiplicity processes? • Vertices and propagators involve gauge-dependent off-shell states. Origin of the complexity.

16. “One of the most remarkable discoveries in elementary particle physics has been that of the existence of the complex plane.” J. Schwinger in “Particles, Sources and Fields” Vol 1 • We saw a beautiful application of this comment in Denner’s • talk – complex mass scheme for unstable particles. • A second application are on-shell methods.

17. Spinors and Twistors Spinor helicity for gluon polarizations in QCD: Penrose Twistor Transform: Witten Earlywork from Nair Witten’s remarkable twistor-space link: Witten; Roiban, Spradlin and Volovich Scattering amplitudes Topological String Theory

18. Remarkable Simplicity Witten conjectured that in twistor–space gauge theory amplitudes have delta-function support on curves of degree: Connected picture Disconnected picture Witten Roiban, Spradlin and Volovich Cachazo, Svrcek and Witten Gukov, Motl and Neitzke Bena Bern and Kosower Structures imply an amazing simplicity in the scattering amplitudes. Massless gauge theory tree amplitudes are much simpler than anyone imagined.

19. MHV Amplitudes Parke and Taylor (1984) At tree level Parke and Taylor conjectured a very simple form for n-gluon scattering. + + + + Proven by Berends and Giele In twistor space this is represented by a straight line

20. Cachazo, Svrcek and Witten MHV Vertices twistor space momentum space Define Equivalently Kosower These MHV amplitudes can be thought of as vertices for building new amplitudes.

21. Six gluonexample Kunszt via susy A “nifty” or “numerically ready” calculation

22. “Nifty” Calculations Suppose you know all the amplitudes up to n-1 points, if you can obtain the n-point amplitude by drawing some pictures and writing down the answer we define it to be a “nifty” calculation. (This is not the same as the maximally efficient calculation.) Weinziel’s talk “Nifty” calculations exhibit the newly uncovered structures and simplicity. Very nice, but: What about higher points? What about masses? What about one loop? What about massive loops? What about higher loops? We shall consider each of these questions in turn.

23. Tree-Level On-Shell Recursion New representations of tree amplitudes from IR consistency of one-loop amplitudes in N = 4 super-Yang-Mills theory. Bern, Del Duca, Dixon, Kosower; Roiban, Spradlin, Volovich Using intuition from twistors and generalized unitarity: Britto, Cachazo, Feng BCF + Witten An-k+1 An Ak+1 On-shell conditions maintained by shift. • Proof relies on so little. Power comes from generality • Cauchy’s theorem • Basic field theory factorization properties

24. Tree-Level On-Shell Recursion Helicities or states Partitions of legs separating legs j and l Frozen value of the shift All you do write down the term corresponding to the diagram and numerically insert the shifted (complex) momenta -- “nifty” calculation. • Simple processes with masses looked at – no problem in principle for going on. • If you wish to go more than one level in the recursion then • should clean up before inserting in the next level. Badger, Glover, Khoze and Svrcek; Forde and Kosower Talk from Forde

25. One Loop At one loop there are only a limited number of nearly “nifty” complete calculations of amplitudes. However all one-loop calculations have important pieces that can be calculated in a “nifty” way: Near D=4 any one-loop amplitude can be expressed as a sum over box, triangle, bubble and rational function contributions + rational Box coefficients can be obtained in a “nifty” way. I want to discussbox coefficients and then show you a “nearly nifty” calculation of a full amplitude.

26. Bern, Dixon, Dunbar and Kosower Unitarity Method The statement that box coefficients are simple is best understood in the context of the unitarity method. “Unitarity method” turns unitarity into a practical method for obtaining complete amplitudes with an arbitrary number of kinematic variables. Completely equivalent to Feynman diagram results. Two-particle cut: Three- particle cut: Application of generalized unitarity: Generalized cut interpreted as cut propagators not canceling. Used to obtain (Matrix elements implemented in MCFM) Bern, Dixon and Kosower A number of recent refinements to method discussed in this conference: Talks from Britto, Feng, Forde, Kosower, Pittau

27. Quadruple Cut Consider massless case: As observed by Britto, Cachazo and Feng quadruple cut freezes integral: hep-th/0412104 Solve system of equations: 4 integrals and 4 delta functions } etc Box coefficient nifty: Works very nicely even when some Ki2 vanish. What about bubbles and triangles and rational terms? Not “nifty” but singificant movement in this direction. Talks from Britto, Feng, Forde, Kosower and Pittau

28. One Loop At one-loop the coefficients of all box integral functions in any massless gauge theories have beautiful twistor space interpretation Box integral Twistor space support Bern, Del Duca, Dixon and Kosower Britto, Cachazo and Feng Three negative helicities Profound implication restricting box coefficients Four negative helicities The existence of such twistor structures connected with loop-level simplicity of box coefficients in massless case • Twistor structure of complete amplitudes not mapped out. • Higher loops not mapped out at all.

29. Massive Loops What about massive loops? Much less is understood about the analytic structure of this case. Nevertheless, quadruple cut is frozen. 3 Again solve the system and plug into product of trees 2 4 1 See hep-th/0506068 for a simple example with a uniform mass in the loop. Brandhuber, McNamara, Spence and Travaglini Pittau’s talk discussed a reformulation of this in a more traditional framework.

30. Example at One Loop Can we any compute complete amplitudes at one loop in a nifty way? If all loop momentum dependent poles are unmodified by the z shift an on-shell recursion determines the coefficients rather straightforwardly. ZB, Bjerrum-Bohr, Dunbar, Ita Used to determine all log terms for split helicity configuration • Consider 6 point: Scalar loop All coefficients of boxes, triangles and bubbles obtainable in a “nifty” way for QCD split helicity configuration.

31. The rational function can also be obtained from on-shell recursion Talks from Kosower and Forde Berger, ZB, Dixon, Forde and Kosower Not quite “nifty”. Also overlap contributions obtained from extracting residues. All terms “nearly nifty” There is still much to do to get to generic “nearly nifty” calculations, but we heard significant progress from Binoth, Britto, Brandhuber, Dunbar, Feng, Kosower, Forde, Glover, Papadopoulos, Pittau, Travaglini and Zhu.

32. Two Loops: Where is the simplicity? Typical example of 2 loop amplitude: From ZB, De Freitas and Dixon Similar results from Anastasiou, Glover, Oleari and Tejeda-Yeomans This definitely does not look simple. + another 15 pages

33. Where is the simplicity? We have a long way to go to uncover structures and “simplicity” at two loops and beyond. • What can twistor space tell us? I don’t have much • to say here, but someone should study this. • 2) Are there any hints of simplicity? • 3) Trouble with the integrals.

34. q p p q p q A Two-Loop Hint Consider the four gluon all-positive helicity amplitude in QCD. This is the simplest example. If we can’t find simplicity here there is no hope for any other QCD amplitudes. + + + + If you expand it in polylogs it is some moderate mess. Instead let’s write it in a special basis of integrals planar Bern, Dixon, Kosower hep-th/0001001 non-planar Why do the planar and non-planar double boxes look the same? I believe this is a clue.

35. Integration At two-loops and beyond there are serious roadblocks to fully unraveling the structure of amplitudes. • Laporta algorithm can solve for a given set of integrals, • but we don’t know ahead of time what the basis of master • integrals will be. Output basis can look a very • different depending on choices. (Smirnov)2 have a proposal • for systematizing the basis choice – Groebner basis. • Fantastic new tool: publicly available MB package from • Czakon for (numerically) evaluating loop integrals. • (Similar program from Anastasiou and Daleo, but not public.) • Rather non-trivial issues with construction of cross-sections. • talks from Del Duca, Gehrmann, Kilgore, Somogyi

36. The Road Ahead • Attack NLO calculations on experimenters’ wishlist. • Automation for general processes. • Assembly of full cross-sections, e.g., Catani-Seymour formalism or improved versions • Inclusion of more matrix elements into MC@NLO, Vincia, Powheg, etc. • NNLO: Finish jets. jets. jet. Improved PDF’s and fragmentation. This is a difficult road What do we need to do to make this happen?

37. Galileo Galilei Institute: Physics Challenge See the organizers: Brandhuber, Del Duca, Glover, Kosower, Passarino, Spence, Travaglini, Zeppenfeld Aug 27-Oct 26, 2007 GGI Physics Challenges: • Jet observable at NNLO; • Average thrust for jets. • All one loop amplitudes for pp 4 jets • ppW + 3 jets. • Full one-loop top production with decays • folded in. Unstable particles within loop. • Evaluator for higher loop integrals. • Program or web page where you feed • kinematics, get back a number. • Compilation of known results.

38. at one loop. • Parton showers. • At LO merge with W + 3 partons • At NLO merge with W + 1 and W + 2 partons • Produce automated program to construct IR subtraction • counterterms for evaluating cross sections. • Plug in color ordered amplitudes out comes cross sections. • Plug and play with standard interface. • Make it flexible to add new physics. • Electroweak corrections to W + jet production • Two-loop renormalization of electroweak Lagrangian • in the complex pole (mass) scheme More GGI Challenges

39. Summary • We need energetic action to provide the • full range of precision calculations for the LHC. • Lots of great progress described in this conference. • Many concrete new results. • Many important issues remain. GGI Challenge. Get the job done in time for LHC data