A Tour of the S-Matrix: QCD, Super-Yang-Mills and Supergravity. Continuous Advances in QCD 2006 Zvi Bern, UCLA

Transcription

1 A Tour of the S-Matrix: QCD, Super-Yang-Mills and Supergravity Continuous Advances in QCD 2006 Zvi Bern, UCLA 1 1

2 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 an amazing advance in our ability to compute scattering amplitudes The call of the LHC: multi-parton scattering. Talks from Carola Berger, Andreas Brandhuber, Darren Forde, Peter Svrcek, Gabriele Travaglini and Diana Vaman. Witten s remarkable duality to topological string theory. Talks from Radu Roiban, Marcus Spradlin, Peter Svrcek, Andreas Brandhuber, Gabriele Travaglini Can we resum (planar) N = 4 super-yang-mills theory? Talk from Marcus Spradlin The structure of perturbative quantum gravity. Reexamine standard wisdom on quantum gravity. Talk from Harald Ita 2 2

3 LHC Physics The LHC will start operations in We will have lots of multi-particle processes. Want reliable predictions. 3 3

4 Example: Susy Search Gianatti and Mangano Early ATLAS TDR studies using PYTHIA overly optimistic. ALPGEN vs PYTHIA 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 4

5 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 5

6 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 6 6

7 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 7 7

8 More Realistic NLO Wishlist Les Houches 2005 Still in our dreams. Bold action required! 8 8

9 Proposals for Dealing with Wishlist Numerical or semi-numerical approaches: 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 On-shell methods: unitarity method, twistor inpiration, on-shell recursion, bootstrap approach: 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 9 9

10 Why are Feynman diagrams clumsy for high-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

11 Spinors and Twistors Spinor helicity for gluon polarizations in QCD: Penrose Twistor Transform: Witten s remarkable twistor-space link: QCD scattering amplitudes Early work from Nair Witten; Roiban, Spradlin and Volovich Topological String Theory See Spradlin s Sunday talk 11 11

12 Amazing Simplicity Witten conjectured that in twistor space gauge theory amplitudes have delta-function support on curves of degree: See Roiban s talk Connected picture See Svrcek s talk Structures imply an amazing simplicity in the scattering amplitudes. Amplitudes are much much simpler than anyone believed. Disconnected picture Witten Roiban, Spradlin and Volovich Cachazo, Svrcek and Witten Gukov, Motl and Neitzke Bena Bern and Kosower The simplicity survives addition of masses or loops 12 12

13 Parke and Taylor (1984) MHV Amplitudes At tree level Parke and Taylor conjectured a very simple form for n-gluon amplitudes Amazingly, this simplicity continues to loops and to general helicities. Bern, Dixon, Dunbar, Kosower Cachazo, Svrcek, Witten; Bern, Dixon, Kosower Brandhuber, Spence and Travaglini As Peter Svrcek will explain, these MHV amplitudes can be thought of as vertices for building new amplitudes. Cachazo, Svrcek and Witten See talks from Svrcek, Brandhuber and Travaglini Proven by Berends and Giele 13 13

14 A Remarkable Twistor String Formula The following formula encapsulates the entire tree-level S-matrix of N = 4 super-yang-mills: Integral over the Moduli and curves Witten Roiban, Spradlin and Volovich Degree d polynomial in the moduli a k Strange formula from Feynman diagram viewpoint. But it s true: impressive checks by Roiban, Spradlin and Volovich See Radu Roiban s talk 14 14

15 On-shell Formalisms Curiously, a practical on-shell formalism was constructed at loop level prior to trees: unitarity method. (1994) Bern, Dixon, Dunbar, Kosower Solution at tree-level had to await Witten s twistor inspiration. (2004) -- MHV vertices -- on-shell recursion Britto, Cachazo, Feng, Witten Cachazo, Svrcek Witten; Brandhuber, Spence, Travaglini Combining unitarity method with on-shell recursion gives loop-level on-shell bootstrap. (2006) Berger, Bern, Dixon, Forde, Kosower 15 15

16 Unitarity Method Bern, Dixon, Dunbar and Kosower Two-particle cut: Three- particle cut: Generalized unitarity: Bern, Dixon and Kosower As observed by Britto, Cachazo and Feng quadruple cut freezes integral: Coefficients of box integrals always easy. 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 16

17 Early On-Shell Bootstrap Bern, Dixon, Kosower (1997) Early Approach: Use Unitarity Method with D = 4 helicity states. Efficient means for obtaining logs and polylogs. Use factorization properties to find rational function contributions. Key problems preventing widespread applications: Difficult to find rational functions with desired factorization properties. Systematization unclear key problem

18 Tree-Level On-Shell Recursion New representations of tree amplitudes from IR consistency of oneloop amplitudes in N = 4 super-yang-mills theory. Using intuition from twistors and generalized unitarity: Bern, Del Duca, Dixon, Kosower; Roiban, Spradlin, Volovich Britto, Cachazo, Feng BCF + Witten A n A n-k+1 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 Applies as well to massive theories See Forde s talk See Svrcek s talk. See Vaman s talk for connection to Lagragian Britto, Cachazo, Feng and Witten 18 Badger, Glover, Khoze and Svrcek 18

19 Loop-Level On-Shell Bootstrap 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 19 19

20 6 points 7 points Numerical Results for n Gluons Choose specific points in phase-space see hep-ph/ Scalar loop contributions Naive diagram count 8 points Method for obtaining this: Berger s talk n-point analytical expressions: Forde s talk + 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 20 20

21 Future Steps Attack items on experimenters wishlist. Construct automation for general processes. Assembly of full cross-sections, e.g., Catani-Seymour formalism. Massive loops -- tree recursion understood. First principles derivation of formalism: Large z behavior of loop amplitudes. General understanding of factorization with complex momenta. Loop-level connection to Lagrangian

22 N = 4 Super-Yang-Mills to All Loops In 1974 t Hooft proposed that we can solve QCD in planar ( t Hooft) limit. This is too hard. N = 4 sym is much more promising. N = 4 sym is a conformal field theory (CFT). UV finite. It is the CFT appearing in Maldacena s AdS/CFT correspondence. 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 entire perturbative series

23 Twistor Space Hint At one-loop the coefficients of all integral functions have beautiful twistor space interpretations Box integral Twistor space support Three negative helicities Bern, Dixon and Kosower Britto, Cachazo and Feng Four negative helicities The existence of such twistor structures implies looplevel simplicity. Supports notion that we should be able to evaluate amplitudes to all loop orders

24 Loop Iteration of the Amplitude Four-point one-loop, N = 4 amplitude: To check for iteration we should evaluate two-loop integrals Planar contributions Obtained via unitarity method Integrals known and involve 4 th order polylogarithms. 24 V. Smirnov 24

25 Loop Iteration of the Amplitude The planar two-loop amplitude undergoes fantastic simplification. Anastasiou, Bern, Dixon, Kosower is universal function related to IR singularities Thus we have succeeded to express two-loop four point planar amplitude as iteration of one-loop amplitude

26 Generalization to n Points Anastasiou, Bern, Dixon, Kosower Can we guess the n-point result? Expect simple structure. Trick: use collinear behavior for guess Have calculated two-loop splitting amplitudes. Following ansatz satisfies all collinear constraints Valid for planar MHV amplitudes Spradlin s Saturday talk for non-trivial evidence and slick tricks 26 26

27 Three-loop Generalization From unitarity method we get three-loop planar integrand: Bern, Rozowsky, Yan Use Mellin-Barnes integration technology and apply hundreds of harmonic polylog identities: V. Smirnov Vermaseren and Remiddi Bern, Dixon, Smirnov Answer actually does not actually depend on c 1 and c 2. Five-point calculation would 27 determine these. 27

28 All legs and All Loop Generalization Why not be bold and guess scattering amplitudes for all loop all legs (at least for MHV amplitudes)? Remarkable formula from Magnea and Sterman tells us IR singularities to all loop orders. Guides construction. Collinear limits gives us the key analytic information, at least for MHV amplitudes. After subtracting IR singularities finite remainder is: All loop resummation of finite remainder An unnamed constant Soft anomalous dimension Or leading twist high spin anomalous dimension Or cusp anomalous dimension Or high moment limit of Altarelli-Parisi splitting kernel One- loop finite remainder. Complicated function of kinematic variables See Spradlin s Saturday talk for 28 further discussion of N = 4 multi-loop amplitudes 28

29 Finite Remainder All loop resummation of a one-loop amplitude in planar limit. In QCD this type of function contributes to physical quantites such as jet rates. IR divergences cancel against similar divergences from real emission diagrams

30 Link to Integrability It is strongly suspected that N = 4 super-yang-mills is integrable in the planar limit. Minhan and Zarembo; Beisert, Krisjansen, Staudacher and many others Recent proposal for soft/cusp anomalous dimension in N = 4 SYM to all perturbative orders, based on integrability. Eden, Staudacher, hep-ph/ Generating function for If Staudacher and Eden are right, we know finite remainder of all-loop MHV planar amplitudes, up to overall constant! 30 30

31 Connection of Gravity and Gauge Theory See Harald Ita s talk At tree level Kawai, Lewellen and Tye have presented a relationship between closed and open string amplitudes. In field theory limit relationship is between gravity and gauge theory Gravity amplitude where we have stripped all coupling constants Color stripped gauge theory amplitude Full gauge theory amplitude Holds for any external states. See review: gr-qc/ Progress in gauge theory can be imported into gravity theories 31 31

32 Divergences in Supergravity Conventional wisdom states that it impossible to construct a finite quantum field theory of gravity Serious flaw with all previous studies of divergences. Rely on powercounting, taking into account only supersymmetry. We now have a much deeper understanding: hidden structures, dualities, twistors, connection to sym and KLT. Perturbative N = 8 supergravity inherits its property from N = 4 sym. Is it finite, contrary to prevailing wisdom? Suppose we wanted to check this with Feynman diagrams: First potential divergence is at 5 loops This single diagram has terms prior to evaluating any integrals. Impossible to evaluate! 32 32

33 Gravity In Harald Ita s talk you will hear about a rather different strategy to gravity calculations mainly for tree and one loop, but similar strategy works at higher loops as well. Strategy already used to demonstrate that N = 8 sugra is less divergent than previously thought. First potential divergence will be at least 5 loops! Bern, Dixon, Dunbar, Rozowsky, and Yan; Howe and Stelle Related twistor structures exist in gravity as in gauge theory. Bern, Bjerrum-Bohr, Dunbar MHV rules for gravity. Bjerrum-Bohr, Ita, Perkins, Risager On-shell recursion for gravity. Bedford, Brandhuber, Spence and Travaglini Cachazo and Svrcek 33 33

34 Summary Remarkable duality to twistor string theory. Inspiration from twistor space: amazing simplicity On-shell methods unitarity and factorization. New twistor inspired ideas: MHV vertices and on-shell recursion, loop-level on-shell recursion. Applications of on-shell methods: (a) LHC phenomenology. (b) Resummation of planar super-yang-mills theory. (c) Structure of perturbative quantum gravity. These developments discussed further in session B 34 34

35 Extra Transparancies 35 35

36 Other theories Khoze, hep-th/ Two classes of (large N c ) conformal gauge theories inherit the same large N c perturbative amplitude properties from N=4 SYM: 36 36

37 Other theories Khoze, hep-th/ Two classes of (large N c ) conformal gauge theories inherit the same large N c perturbative amplitude properties from N=4 SYM: 1. Theories obtained by orbifold projection product groups, matter in particular bi-fundamental rep s Bershadsky, Johansen, hep-th/

38 Other theories Khoze, hep-th/ Two classes of (large N c ) conformal gauge theories inherit the same large N c perturbative amplitude properties from N=4 SYM: 1. Theories obtained by orbifold projection product groups, matter in particular bi-fundamental rep s Bershadsky, Johansen, hep-th/ The N=1 supersymmetric beta-deformed conformal theory same field content as N=4 SYM, but superpotential is modified: Leigh, Strassler, hep-th/

39 Other theories Khoze, hep-th/ Two classes of (large N c ) conformal gauge theories inherit the same large N c perturbative amplitude properties from N=4 SYM: 1. Theories obtained by orbifold projection product groups, matter in particular bi-fundamental rep s Bershadsky, Johansen, hep-th/ The N=1 supersymmetric beta-deformed conformal theory same field content as N=4 SYM, but superpotential is modified: Leigh, Strassler, hep-th/ Supergravity dual known for this case, deformation of AdS 5 x S 5 Lunin, Maldacena, hep-th/

40 Other theories Khoze, hep-th/ Two classes of (large N c ) conformal gauge theories inherit the same large N c perturbative amplitude properties from N=4 SYM: 1. Theories obtained by orbifold projection product groups, matter in particular bi-fundamental rep s Bershadsky, Johansen, hep-th/ The N=1 supersymmetric beta-deformed conformal theory same field content as N=4 SYM, but superpotential is modified: Leigh, Strassler, hep-th/ Supergravity dual known for this case, deformation of AdS 5 x S 5 Lunin, Maldacena, hep-th/ Breakdown of inheritance at five loops (!?) for more general marginal perturbations of N=4 SYM? Khoze, hep-th/

41 Beyond three loops Recent proposal for soft/cusp anomalous dimension in N=4 SYM to all perturbative orders (!), based on integrability. Eden, Staudacher, hep-ph/ where is the solution to an integral equation with Bessel-function kernel 37 37

42 Beyond three loops Recent proposal for soft/cusp anomalous dimension in N=4 SYM to all perturbative orders (!), based on integrability. Eden, Staudacher, hep-ph/ where is the solution to an integral equation with Bessel-function kernel Perturbative expansion: 37 37

43 Beyond three loops Recent proposal for soft/cusp anomalous dimension in N=4 SYM to all perturbative orders (!), based on integrability. Eden, Staudacher, hep-ph/ where is the solution to an integral equation with Bessel-function kernel Perturbative expansion:? 37 37

44 Progress Towards the Dream Results with on-shell methods: Complete QCD amplitudes with legs. see David Darren Forde s talk Logarithmic contributions via on-shell recursion. Improved ways to obtain logarithmic contributions via unitarity method. All six-gluon helicities. Key Feature: Modest growth in complexity as n increases

45 Loop-Level Recursion z New Features: Presence of branch cuts. Unreal poles poles appear with complex momenta. Double poles. Pure phase for real momenta Spurious singularities that cancel only against polylogs. Double count between cuts and recursion. See Carola Berger s and Darren Forde s talks 39 39

46 The Gold Standard: NNLO Drell-Yan Rapidity Distributions Amazingly good stabilty Theoretical uncertainties less than 1% 40 40

47 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