Unitarity and Amplitudes at Maximal Supersymmetry

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. 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. An Old Dream: Planar Limit in Gauge Theories ‘t Hooft (1974) • Consider large-N gauge theories, g2N ~ 1, use double-line notation • Planar diagrams dominate • Sum over all diagrams  surface or string diagram

4. 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. Descriptions of N=4 SUSY Gauge Theory • A Feynman path integral • Boundary CFT of IIB string theory on AdS5 S5 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. 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. 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 theiteration relation doesn’t hold for the six-point amplitude • Structure beyond the iteration relation: yet to be understood

8. Traditional technology: Feynman Diagrams

9. 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 ∙ 107 terms ~2.9 ∙ 106 1-loop diagrams, ~ 1.9 ∙ 1010 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. 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. 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: 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 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. One-loop all-multiplicity MHV amplitude in N = 4

15. Generalized Unitarity • Can sew together more than twotree amplitudes • Corresponds to ‘leading singularities’ • Isolates contributions of a smaller setof integrals: only integrals with propagatorscorresponding 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. Generalized Cuts • Require presence of multiple propagators at higher loops too

17. 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. Integrals in the Amplitude • 8 integrals present • 6 given by ‘rung rule’; 2 are new • UV divergent in D = (vs 7, 6 for L = 2, 3)

19. 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 ki = xi+1 – xi • Conformal invariance in xi

20. 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. 59 ints Bern, Carrasco, Johansson, DAK (5/2007)

22. 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. 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. 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

26. Basic Integrals

27. Decorated Integrals

28. Result • Take the kinematical point • and look at the remainder (test of the iteration relation) ui— independent conformal cross ratios

29. 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. 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. 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?