1. From Weak to Strong Coupling at Maximal Supersymmetry David A. Kosower with Z. Bern, M. Czakon, L. Dixon, & V. Smirnov 2007 Itzykson Meeting: Integrability in Gauge and String Theory June 22, 2007

2. From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007 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. From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007 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 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

4. From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007 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)

5. From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007 Simple structure in twistor space, completely unexpected from the Lagrangian

6. From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007 Novel relation between different orders in perturbation theory Compute leading-twist anomalous dimension (↔ spinning-string mass) Eden & Staudacher (spring 2006) analyzed the integral equation emerging from the spin chain and conjectured an all-orders expression for f, – Want to check four-loop term Use on-shell methods not conventional Feynman diagrams

7. From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007 Unitarity-Based Calculations Bern, Dixon, Dunbar, & DAK (1994)

8. From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007 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 (in N = 4, a smaller set of boxes) Combine with use of complex momenta to determine box coeffs directly in terms of tree amplitudes Britto, Cachazo, & Feng (2004)

9. From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007 Generalized Cuts Require presence of multiple propagators at higher loops too

10. From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007 At one loop, Green, Schwarz, & Brink (1982) Two-particle cuts iterate to all orders Bern, Rozowsky, & Yan (1997) Three-particle cuts give no new information for the four-point amplitude N =4 Cuts at Two Loops = =  for all helicities one-loop scalar box

11. From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007 Two-Loop Four-Point Result Integrand known Bern, Rozowsky, & Yan (1997) Integrals known by 2000  could have just evaluated Singular structure is an excellent guide Sterman & Magnea (1990); Catani (1998); Sterman & Tejeda-Yeomans (2002)

12. From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007 Two-loop Double Box Smirnov (1999) Physics is 90% mental, the other half is hard work — Yogi Berra

13. From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007 Transcendentality Also called ‘polylog weight’ N = 4 SUSY has maximal transcendentality = 2  loop order QCD has mixed transcendentality: from 0 to maximal

14. From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007 Infrared-singular structure is an excellent guide Sterman & Magnea (1990); Catani (1998); Sterman & Tejeda-Yeomans (2002) IR-singular terms exponentiate Soft or “cusp” anomalous dimension: large-spin limit of trace- operator anomalous dimension Korchemsky & Marchesini (1993) True for all gauge theories

15. From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007 Iteration Relation in N = 4 Look at corrections to MHV amplitudes, at leading order in N c Including finite terms Anastasiou, Bern, Dixon, & DAK (2003) Conjectured to all orders Requires non-trivial cancellations not predicted by pure supersymmetry or superspace arguments

16. From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007 N =4 Integrand at Higher Loops Bern, Rozowsky, & Yan (1997)

17. From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007 Iteration Relation Continued Confirmed at three loops Bern, Dixon, & Smirnov (2005) Can extract 3-loop anomalous dimension & compare to Kotikov, Lipatov, Onishchenko and Velizhanin “extraction” from Moch, Vermaseren & Vogt result highest polylog weight

18. From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007 All-Loop Form Bern, Dixon, & Smirnov (2005) Connection to a kind of conformal invariance? Drummond, Korchemsky, & Sokatchev

19. From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007 Amplitudes at Strong Coupling Alday & Maldacena (5/2007) Introduce D-brane as regulator Study open-string scattering on it: fixed-angle high-momentum Use saddle-point (classical solution) approach Gross & Mende (1988) Replace D-brane with dimensional regulator, take brane to IR First strong-coupling computation of G (“collinear”) anomalous dim

20. From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007 Calculation Dick [Feynman]'s method is this. You write down the problem. You think very hard. Then you write down the answer. — Murray Gell-Mann Integral set Unitarity Calculating integrals Computation of only needs O ( ² −2 ) terms

21. From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007 Integrals Start with all four-point four-loop integrals with no bubble or triangle subgraphs (expected to cancel in N =4)  7 master topologies (only three-point vertices)  25 potential integrals (others obtained by canceling propagators)

22. From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007

24. 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

25. From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007 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

26. From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007 A Posteriori, Conformal Properties Consider candidate integrals with external legs taken off shell Require that they be conformally invariant Easiest to analyze using dual diagrams Drummond, Henn, Smirnov, & Sokatchev (2006) Require that they be no worse than logarithmically divergent   10 pseudo-conformal integrals, including all 8 that contribute to amplitude ( Sokatchev : two others divergent off shell )

27. From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007 59 ints Bern, Carrasco, Johansson, DAK (5/2007)

28. From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007 Mellin–Barnes Technique Introduce Feynman parameters (best choice is still an art) & perform loop integrals Use identity to create an m -fold representation Singularities are hiding in Γ functions Move contours to expose these singularities (all poles in ² ) Expand Γ functions to obtain Laurent expansion with functions of invariants as coefficients Done automatically by Mathematica package MB (Czakon) Compute integrals numerically or analytically

29. From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007 Result Computers are useless. They can only give you answers. — Pablo Picasso Further refinement Cachazo, Spradlin, & Volovich (12/2006)

30. From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007 Extrapolation to Strong Coupling Use KLV approach Kotikov, Lipatov & Velizhanin (2003) Constrain at large â: solve Predict two leading strong-coupling coefficients

31. From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007 Known strong-coupling expansion Gubser, Klebanov, Polyakov; Kruczenski; Makeenko (2002) Frolov & Tseytlin (2002) Two loops predicts leading coefficient to 10%, subleading to factor of 2 Four-loop value predicts leading coefficient to 2.6%, subleading to 5%!

32. From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007 This suggests the ‘exact’ answer should be obtained by flipping signs of odd-ζ terms in original Eden–Staudacher formula ( Bern, Czakon, Dixon, DAK, Smirnov ) The same series is obtained by consistency of analytic continuations ( Beisert, Eden, & Staudacher ) leading to a ζ-dependent ‘dressing’ factor for the spin-chain/string world-sheet S -matrix and a modified Eden–Staudacher equation The equation can be integrated numerically ( Benna, Benvenuti, Klebanov, Scardicchio ) and reproduces known strong-coupling terms The equation can be studied analytically in the strong-coupling region ( Kotikov & Lipatov; Alday, Arutyunov, Benna, Eden, Klebanov; Kostov, Serban, Volin; Beccaria, DeAngelis, Forini ) and reproduces the leading coefficient; equivalent equation from string Bethe ansatz ( Casteill & Kristjansen ) Approximation reproduces ‘exact’ answer to better than 0.2%!

33. From Weak to Strong Coupling at Maximal Supersymmetry, Itzykson Meeting, June 22, 2007 Conclusions and Perspectives “The time has come,” the Walrus said, “To talk of many things: Of AdS–and CFT–and scaling- limits– Of spin-chains–and N=4– Whether it is truly summable– And what this is all good for.” First evidence for the ‘dressing factor’ in anomalous dimensions Transition from weak to strong coupling is smooth here N =4 is a guide to uncovering more structure in gauge theories; can it help us understand the strong-coupling regime in QCD? Unitarity is the method of choice for performing the explicit calculations needed to make progress and ultimately answer the questions What is the connection of the iteration relation to integrability or other structures?