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The S-Matrix Reloaded: Twistors, Unitarity, Gauge Theories and Gravity

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The S-Matrix Reloaded: Twistors, Unitarity, Gauge Theories and Gravity Ch F-X ang PD e w Click to buy NOW! w m o w c .d k. ocu-trac TheThe SS--MatrixMatrix Reloaded:Reloaded: TwistorsTwistors,, UnitarityUnitarity,, GaugeGauge TheoriesTheories andand GravityGravity Potsdam 2006: Integrability in Gauge andand StringString Theory Zvi Bern, UCLA Based on papers with: I. Bena, C. Berger, N.E.J Bjerrum-Bohr, M. Czakon, L. Dixon, D. Dunbar, D. Forde, H. Ita, D. Kosower, R. Roiban and V. Smirnov. 1 Ch F-X ang PD e w Click to buy NOW! w m o w c .d k. ocu-trac Outline “A method is more important that a discovery, since the right method can lead to new and even more important discoveries” -- L.D. Landau The past two years have seen a significant advance in our ability to compute scattering amplitudes. •The call of the LHC: multi-parton scattering at loop level. •Can we resum (planar) N = 4 super-Yang- Mills theory? •The structure of perturbative quantum gravity. Reexamine standard wisdom on quantum gravity. 2 Ch F-X ang PD e w Click to buy NOW! w m o w c .d k. ocu-trac LHCLHC PhysicsPhysics The LHC will start operations in 2007. We will have lots of multi-particle processes. Want reliable predictions. 3 Ch F-X ang PD e w Click to buy NOW! w m o w c .d k. ocu-trac Example:Example: SusySusy SearchSearch Gianatti and Mangano Early ATLAS TDR studies using ALPGEN vs PYTHIA 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. Such a calculation is well beyond anything that has been done using Feynman diagrams 4 Ch F-X ang PD e w Click to buy NOW! w m o w c .d k. ocu-trac State-of-the-Art NLO QCD Five point is still state-of-the art for QCD cross-sections: 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 dets. “Grim”determinant 5 Ch F-X ang PD e w Click to buy NOW! w m o w c .d k. ocu-trac What needs to be done at NLO? Experimenters to theorists: “Please calculate the following at NLO” Theorists to experimenters: “In your dreams” A key theoretical problem for LHC is NLO 6 Ch F-X ang PD e w Click to buy NOW! w m o w c .d k. ocu-trac MoreMore RealisticRealistic NLONLO WishlistWishlist Les Houches 2005 Bold action required! 7 Ch F-X ang PD e w Click to buy NOW! w m o w c .d k. ocu-trac Consider an integral Evaluate this integral via Passarino-Veltman reduction. Result is … 8 Ch F-X ang PD e w Click to buy NOW! w m o w c .d k. ocu-trac Result of performing the integration Numerical stability is a key issue. Clearly, there should be a better way 9 Ch F-X ang PD e w Click to buy NOW! w m o w c .d k. ocu-trac Why are Feynman diagrams clumsy for high loop or multiplicity processes? • Vertices and propagators involve gauge-dependent off-shell states. Origin of the complexity. • To get at root cause of the trouble we must rewrite perturbative quantum field theory. •All steps should be in terms of gauge invariant on-shell states. On shell formalism. •Radical rewrite of gauge theory needed. 10 Ch F-X ang PD e w Click to buy NOW! w m o w c .d k. ocu-trac “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 11 Ch F-X ang PD e w Click to buy NOW! w m o w c .d k. ocu-trac OnOn--shellshell FormalismsFormalisms With on-shell formalisms we can exploit analytic properties • Curiously, a practical on-shell formalism was constructed at loop level prior to tree level: unitarity method. Bern, Dixon, Dunbar, Kosower (1994) • Solution at tree-level had to await Witten’stwistor inspiration. (2004) -- MHV vertices Cachazo, Svrcek Witten; Brandhuber, Spence, Travaglini -- On-shell recursion Britto, Cachazo, Feng, Witten • Combining unitarity method with on-shell recursion gives loop-level on-shell bootstrap. (2006) Berger, Bern, Dixon, Forde, Kosower 12 Ch F-X ang PD e w Click to buy NOW! w m o w c .d k. ocu-trac SpinorsSpinors andand TwistorsTwistors Spinor helicity for gluon polarizations in QCD: Penrose twistor transform: Early work from Nair Witten’sremarkable twistor-space link: Witten; Roiban, Spradlin and Volovich QCD scattering amplitudes Topological String Theory 13 Ch F-X ang PD e w Click to buy NOW! w m o w c .d k. ocu-trac Amazing 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 Structures imply an amazing simplicity Cachazo, Svrcek and Witten in the scattering amplitudes. Amplitudes Gukov, Motl and Neitzke are much much simpler than anyone Bena Bern and Kosower imagined. 14 Ch F-X ang PD e w Click to buy NOW! w m o w c .d k. ocu-trac Parke and Taylor (1984) MHV Amplitudes At tree level Parke and Taylor conjectured a very simple form for n-gluon scattering. + + + + Proven by Berends and Giele Amazingly, this simplicity continues to loops Bern, Dixon, Dunbar, Kosower and to general helicities. Cachazo, Svrcek, Witten; Bern, Dixon, Kosower Brandhuber, Spence and Travaglini These MHV amplitudes can be thought of as vertices for building new amplitudes. Cachazo, Svrcek and Witten 15 Ch F-X ang PD e w Bern, Dixon, Dunbar and Kosower Click to buy NOW! w m o w c .d k. ocu-trac UnitarityUnitarity MethodMethod Two-particle cut: Three- particle cut: Generalized As observed by Britto, Cachazo and Feng unitarity: quadruple cut freezes integral: Coefficients of box integrals always easy. Bern, Dixon and Kosower Generalized cut interpreted as cut propagators not canceling. Recent improvements for bubble and triangle contributions 16 Britto, Buchbinder, Cachazo and Feng; Britto, Feng and Mastrolia Ch F-X ang PD e w Click to buy NOW! w m o w c .d k. ocu-trac TreeTree--LevelLevel OnOn--ShellShell RecursionRecursion 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 Britto, Cachazo, Feng and Witten •Basic field theory factorization properties •Applies as well to massive theories Badger, Glover, Khoze and Svrcek 17 Ch F-X ang PD e w Click to buy NOW! w m o w c .d k. ocu-trac LoopLoop--LevelLevel OnOn--ShellShell BootstrapBootstrap Berger, Bern, Dixon, Forde and Kosower Shifted amplitude function of a complex parameter Shift maintains on-shellness and momentum conservation Use unitarity method (in special cases on-shell recursion) Use on-shell recursion Use auxiliary on-shell recursion in another variable 18 See David Kosower’stalk Ch F-X ang PD e w Click to buy NOW! w m o w c .d k. ocu-trac NumericalNumerical ResultsResults forfor nn GluonsGluons Choose specific points in phase-space –see hep-ph/0604195 Scalar loop contributions Naive diagram count 6 points 7 points 8 points + 3,017,489 other diagrams Modest growth in complexity as number of legs increases At 6 points these agree with numerical results of Ellis, Giele and Zanderighi 19 Ch F-X ang PD e w Click to buy NOW! w m o w c .d k. ocu-trac NN == 44 SuperSuper--YangYang--MillsMills toto AllAll LoopsLoops In 1974 ‘t Hooftproposed that we can solve QCD in planar (‘t Hooft) limit. This is too hard. N = 4 sYM is much more promising. •Heuristically, we expect magical simplicity especially in planar limit with large ‘t Hooft coupling –dual to weakly coupled gravity in AdS space. Can we solve (planar) N = 4 super-Yang-Mills theory? Initial Goal: Resum amplitude to all loop orders. 20 Ch F-X ang PD e w Click to buy NOW! w m o w c .d k. ocu-trac What we need 1. Intuition and bold guesses. 2. Sufficiently powerful methods for confirming and guiding guesses. (a) Unitarity method. Bern, Dixon, Dunbar and Kosower (b) A loop integration package: MB. Czakon 3. Faith and optimism. 21 Ch F-X ang PD e w Click to buy NOW! w m o w c .d k. ocu-trac Example: Twistor Space Hint At one-loop the coefficients of all integral functions have beautiful twistor space interpretations Box integral Twistor space support Three negative Bern, Dixon and Kosower helicities Britto, Cachazo and Feng Four negative helicities The existence of such twistor structures implies loop- level simplicity. Supports notion that we should be able to evaluate amplitudes to all loop orders. 22 Ch F-X ang PD e w Click to buy NOW! w m o w c .d k. ocu-trac NN == 44 MultiMulti--looploop AmplitudeAmplitude Bern, Rozowsky and Yan Consider one-loop in N = 4: The basic D-dimensional two-particle sewing equation Applying this at one-loop gives Agrees with known result of Green, Schwarz and Brink. The two-particle cuts algebra recycles to all loop orders! 23 Ch F-X ang PD e w Click to buy NOW! w m o w c .d k.
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