1. Unitarity and Amplitudes at Maximal Supersymmetry David A. Kosower with Z. Bern, J.J. Carrasco, M. Czakon, L. Dixon, D. Dunbar, H. Johansson, R. Roiban, M. Spradlin, V. Smirnov, C. Vergu, & A. Volovich Jussieu FRIF Workshop Dec 12–13, 2008

2. Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008 QCD Nature’s gift: a fully consistent physical theory Only now, thirty years after the discovery of asymptotic freedom, are we approaching a detailed and explicit understanding of how to do precision theory around zero coupling Can compute some static strong-coupling quantities via lattice Otherwise, only limited exploration of high-density and hot regimes To understand the theory quantitatively in all regimes, we seek additional structure String theory returning to its roots

3. Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008 An Old Dream: Planar Limit in Gauge Theories ‘t Hooft (1974) Consider large-N gauge theories, g 2 N ~ 1, use double-line notation Planar diagrams dominate Sum over all diagrams  surface or string diagram

4. Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008 How Can We Pursue the Dream? We want a story that starts out with an earthquake and works its way up to a climax. — Samuel Goldwyn Study N = 4 large- N gauge theories: maximal supersymmetry as a laboratory for learning about less-symmetric theories Easier to perform explicit calculations Several representations of the theory

5. Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008 Descriptions of N =4 SUSY Gauge Theory A Feynman path integral Boundary CFT of IIB string theory on AdS 5  S 5 Maldacena (1997); Gubser, Klebanov, & Polyakov; Witten (1998) Spin-chain model Minahan & Zarembo (2002); Staudacher, Beisert, Kristjansen, Eden, … (2003–2006) Twistor-space topological string B model Nair (1988); Witten (2003) Roiban, Spradlin, & Volovich (2004); Berkovits & Motl (2004)

6. Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008 Is there any structure in the perturbation expansion hinting at ‘solvability’? Explicit higher-loop computations are hard, but they’re the only way to really learn something about the theory

7. Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008 Recent Revelations Iteration relation: four- and five-point amplitudes may be expressed to all orders solely in terms of the one-loop amplitudes Cusp anomalous dimension to all orders: BES equation & hints of integrability  Basso’s talk Role of ‘dual’ conformal symmetry But the iteration relation doesn’t hold for the six-point amplitude Structure beyond the iteration relation: yet to be understood

8. Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008 Traditional technology: Feynman Diagrams

9. Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008 Feynman Diagrams Won’t Get You There Huge number of diagrams in calculations of interest — factorial growth 8 gluons (just QCD): 34300 tree diagrams, ~ 2.5 ∙ 10 7 terms ~2.9 ∙ 10 6 1-loop diagrams, ~ 1.9 ∙ 10 10 terms But answers often turn out to be very simple Vertices and propagators involve gauge-variant off-shell states Each diagram is not gauge invariant — huge cancellations of gauge-noninvariant, redundant, parts in the sum over diagrams Simple results should have a simple derivation — Feynman (attr) Is there an approach in terms of physical states only?

10. Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008 How Can We Do Better? Dick [Feynman]'s method is this. You write down the problem. You think very hard. Then you write down the answer. — Murray Gell-Mann

11. Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008 New Technologies: On-Shell Methods Use only information from physical states Use properties of amplitudes as calculational tools – Unitarity → unitarity method – Underlying field theory → integral basis Formalism for N = 4 SUSY Integral basis: Unitarity

12. Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008 Unitarity: Prehistory General property of scattering amplitudes in field theories Understood in ’60s at the level of single diagrams in terms of Cutkosky rules – obtain absorptive part of a one-loop diagram by integrating tree diagrams over phase space – obtain dispersive part by doing a dispersion integral In principle, could be used as a tool for computing 2 → 2 processes No understanding – of how to do processes with more channels – of how to handle massless particles – of how to combine it with field theory: false gods of S -matrix theory

13. Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008 Unitarity as a Practical Tool Bern, Dixon, Dunbar, & DAK (1994) Compute cuts in a set of channels Compute required tree amplitudes Reconstruct corresponding Feynman integrals Perform algebra to identify coefficients of master integrals Assemble the answer, merging results from different channels

14. Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008 One-loop all-multiplicity MHV amplitude in N = 4

15. Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008 Generalized Unitarity Can sew together more than two tree amplitudes Corresponds to ‘leading singularities’ Isolates contributions of a smaller set of integrals: only integrals with propagators corresponding to cuts will show up Bern, Dixon, DAK (1997) Example: in triple cut, only boxes and triangles will contribute  Vanhove’s talk Combine with use of complex momenta to determine box coeffs directly in terms of tree amplitudes Britto, Cachazo, & Feng (2004) No integral reductions needed

16. Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008 Generalized Cuts Require presence of multiple propagators at higher loops too

17. Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008 Cuts Compute a set of six cuts, including multiple cuts to determine which integrals are actually present, and with which numerator factors Do cuts in D dimensions

18. Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008 8 integrals present 6 given by ‘rung rule’; 2 are new UV divergent in D = (vs 7, 6 for L = 2, 3) Integrals in the Amplitude

19. Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008 Dual Conformal Invariance Amplitudes appear to have a kind of conformal invariance in momentum space Drummond, Henn, Sokatchev, Smirnov (2006) All integrals in four-loop four-point calculation turn out to be pseudo-conformal: dually conformally invariant when taken off shell (require finiteness as well, and no worse than logarithmically divergent in on-shell limit) Dual variables k i = x i+1 – x i Conformal invariance in x i

20. Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008 Easiest to analyze using dual diagrams Drummond, Henn, Smirnov & Sokatchev (2006) All coefficients are ±1 in four-point (through five loops) and parity-even part of five-point amplitude (through two loops)

21. Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008 59 ints Bern, Carrasco, Johansson, DAK (5/2007)

22. Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008 A Mysterious Connection to Wilson Loops Motivated by Alday–Maldacena strong-coupling calculation, look at a ‘dual’ Wilson loop at weak coupling: at one loop, amplitude is equal to the Wilson loop for any number of legs (up to addititve constant) Drummond, Korchemsky, Sokatchev (2007) Brandhuber, Heslop, & Travaglini (2007) Equality also holds for four- and five-point amplitudes at two loops Drummond, Henn, Korchemsky, Sokatchev (2007–8)

23. Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008 Conformal Ward Identity Drummond, Henn, Korchemsky, Sokatchev (2007) In four dimensions, Wilson loop would be invariant under the dual conformal invariance Slightly broken by dimensional regularization Additional terms in Ward identity are determined only by divergent terms, which are universal Four- and five-point Wilson loops determined completely Equal to corresponding amplitudes! Beyond that, functions of cross ratios

24. Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008 Open Questions What happens beyond five external legs? Does the amplitude still exponentiate as suggested by the iteration relation? Suspicions of breakdown from Alday–Maldacena investigations If so, at how many external legs? Is the connection between amplitudes and Wilson loops “accidental”, or is it maintained beyond the five-point case at two loops? Compute six-point amplitude at two loops

25. Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008

26. Basic Integrals

27. Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008 Decorated Integrals

28. Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008 Result Take the kinematical point and look at the remainder (test of the iteration relation) u i — independent conformal cross ratios

29. Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008 Comparison to Wilson Loop Calculation With thanks to Drummond, Henn, Korchemsky, & Sokatchev Constants in M differ: compare differences with respect to a standard kinematic point Wilson Loop = Amplitude!

30. Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008 Questions Answered Does the exponentiation ansatz break down? Yes Does the six-point amplitude still obey the dual conformal symmetry? Almost certainly Is the Wilson loop equal to the amplitude at six points? Very likely

31. Unitarity and Amplitudes at Maximal Supersymmetry, Jussieu Workshop, Dec 12, 2008 Questions Unanswered What is the remainder function? Can one show analytically that the amplitude and Wilson-loop remainder functions are identical? How does it generalize to higher-point amplitudes? Can integrability predict it? What is the origin of the dual conformal symmetry? What happens for non-MHV amplitudes?