Multi-loop scattering amplitudes in maximally supersymmetric gauge and gravity theories.

1. Multi-loop scattering amplitudes in maximally supersymmetric gauge and gravity theories. Twistors, Strings and Scattering Amplitudes DurhamAugust 24, 2007 Zvi Bern, UCLA C. Anastasiou, ZB, L. Dixon, and D. Kosower, hep-th/0309040 ZB, L. Dixon and V. Smirnov, hep-th/0505205 ZB, M. Czakon, L. Dixon, D. Kosower and V. Smirnov, hep-th/0610248 ZB, N.E.J. Bjerrum-Bohr, D. Dunbar, hep-th/0501137 ZB, L. Dixon , R. Roiban, hep-th/0611086 ZB, J.J. Carrasco, L. Dixon, H. Johansson, D. Kosower, R. Roiban, hep-th/0702112 ZB, J.J. Carrasco, H. Johansson , D. Kosower arXiv:0705.1864 [hep-th] ZB, J.J. Carrasco, D. Forde, H. Ita and H. Johansson, arXiv:0707.1035 [hep-th]

2. Outline Since the “twistor revolution” of a few years ago we have seen a very significant advance in our ability to compute scattering amplitudes. • QCD: multi-parton scattering for LHC – not discussed here. • Supersymmetric gauge theory: resummation of planar N = 4 super-Yang-Mills scattering amplitudes to all loop orders. • Quantum gravity: reexamination of standard wisdom on ultraviolet properties of quantum gravity.

3. Maximal Supersymmetry In this talk we will discuss high loop orders of scattering amplitudes in maximally supersymmetric gauge and gravity theories. • N = 4 super-Yang-Mills theory is most promising D = 4 gauge theory that we will likely be able to solve completely. • Maximally supersymmetric gravity theory is the most promising theory which may be UV finite. • Scattering amplitudes provide a powerful way to explore and confirm the AdS/CFT correspondence.

4. Twistors Expose Amazing Simplicity Penrose twistor transform: Earlywork from Nair Witten conjectured that in twistor–space gauge theory amplitudes have delta-function support on curves of degree: Leads to MHV rules Connected picture Disconnected picture Witten Roiban, Spradlin and Volovich Cachazo, Svrcek and Witten Gukov, Motl and Neitzke Bena Bern and Kosower Structures imply an amazing simplicity in the scattering amplitudes. This simplicity gives us good reason to believe that there is much more structure to uncover at higher loops.

5. Twistor Structure at One Loop 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 connected with loop-level simplicity. Complete Amplitudes?? Higher loops??

6. Bern, Dixon, Dunbar and Kosower Onwards to Loops: Unitarity Method Two-particle cut: Three- particle cut: Generalized unitarity: Bern, Dixon and Kosower Generalized cut interpreted as cut propagators not canceling. A number of recent improvements to method Bern, Dixon and Kosower; Britto, Buchbinder, Cachazo and Feng; Berger, Bern, Dixon, Forde and Kosower; Britto, Feng and Mastrolia; Anastasiou, Britto, Feng; Kunszt, Mastrolia; ZB, Carasco, Johanson, Kosower; Forde

7. Why are Feynman diagrams clumsy for loop or 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. • Need on-shell formalism.

8. N = 4 Super-Yang-Mills to All Loops Since ‘t Hooft’s paper thirty years ago on the planar limit of QCD we have dreamed of solving QCD in this limit. This is too hard. N = 4 sYM is much more promising. • Heuristically, we expect magical simplicity in the • scattering amplitude 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 amplitudes to all loop orders. As we heard at this conference we are well on our way to achieving this goal. Talks from Alday, Volovich and Travaglini

9. N = 4 Multi-loop Amplitude 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!

10. Loop Iteration of the Amplitude Four-point one-loop, N = 4 amplitude: To check for iteration use evaluation of two-loop integrals. Planar contributions. Obtained via unitarity method. Bern, Rozowsky, Yan Integrals known and involve 4th order polylogarithms. V. Smirnov

11. Loop Iteration of the Amplitude The planar four-point 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. Confirmation directly on integrands. Cachazo, Spradlin and Volovich

12. D = 4 – 2 Generalization to n Points Anastasiou, Bern, Dixon, Kosower Can we guess the n-point result? Expect simple structure. Trick: use collinear behavior for guess Bern, Dixon, Kosower Have calculated two-loop splitting amplitudes. Following ansatz satisfies all collinear constraints Valid for planar MHV amplitudes Confirmed by direct computation at five points! Cachazo, Spradlin and VolovichBern, Czakon, Kosower, Roiban, Smirnov

13. Three-loopGeneralization 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 c1 and c2. Five-point calculation would determine these.

14. All-Leg All-Loop Generalization Why not be bold and guess scattering amplitudes for all loop and all legs (at least for MHV amplitudes)? • Remarkable formula from Magnea and Sterman tells us • IR singularities to all loop orders. Checks construction. • Collinear limits gives us the key analytic information, at • least for MHV amplitudes. One-loop constant All loops • Soft anomalous dimension • Or leading twist high spin anomalous dimension • Or cusp anomalous dimension

15. All-loop Resummation in N = 4 Super-YM Theory For MHV amplitudes: ZB, Dixon, Smirnov constant independent of kinematics. IR divergences finite part of one-loop amplitude cusp anomalous dimension all-loop resummed amplitude Wilson loop Gives a definite prediction for all values of coupling given the Beisert, Eden, Staudacher integral equation for the cusp anomalous dimension. See Roiban’s talk In a beautiful paper Alday and Maldacena confirmed this conjecture at strong coupling from an AdS string computation. See Alday’s and Volovoch’s talks Very suggestive link to Wilson loops even at weak coupling Drummond, Korchemsky, Sokatchev ; Brandhuber, Heslop, and Travaglini

16. Some Open Problems • What is the multi-loop structure of non-MHV amplitudes? • Very likely there is an iteration and exponentiation. Input from • strong coupling would be very helpful. • What is precice multi-loop twistor-space structure? Knowing • this would be extremely helpful. • What is the twistor-space structure at strong coupling? • Is there an iteration and exponentiation for subleading color? • What happens for CFT cases with less susy? • Can we extract the spectrum of physical states from the • exponentiated scattering amplitudes?

17. Quantum Gravity at High Loop Orders A key unsolved question is whether a finite point-like quantum gravity theory is possible. • Gravity is non-renormalizable by power counting. • Every loop gains mass dimension - 2. • At each loop order potential counterterm gains extra • As loop order increases potential counterterms must have • either more R’s or more derivatives Dimensionful coupling

18. Any supergravity: is not a valid supersymmetric counterterm. Produces a helicity amplitude forbidden by susy. Divergences in Gravity Vanish on shell One loop: Pure gravity 1-loop finite (but not with matter) vanishes by Gauss-Bonnet theorem ‘t Hooft, Veltman (1974) Two loop: Pure gravity counterterm has non-zero coefficient: Goroff, Sagnotti (1986); van de Ven (1992) Grisaru (1977); Tomboulis (1977) The first divergence in any supergravity theory can be no earlier than three loops.

19. Why N = 8 Supergravity? • UV finiteness of N = 8 supergravity would imply a new • symmetry or non-trivial dynamical mechanism. • The discovery of either would have a fundamental impact on • our understanding of gravity. • High degree of supersymmetry makes this the most promising • theory to investigate. • By N = 8 we mean ungauged Cremmer-Julia supergravity. No known superspace or supersymmetry argument prevents divergences from appearing at some loop order. Potential counterterm predicted by susy power counting Deser, Kay, Stelle (1977); Kaku, Townsend, van Nieuwenhuizen (1977); Deser and Lindtrom(1980); Kallosh (1981); Howe, Stelle, Townsend (1981). A three loop divergence was the widely accepted wisdom coming from the 1980’s.

20. Where are the N = 8 Divergences? Depends on who you ask and when you ask. Howe and Lindstrom (1981) Green, Schwarz and Brink (1982) Howe and Stelle (1989) Marcus and Sagnotti (1985) 3 loops:Conventional superspace power counting. 5 loops: Partial analysis of unitarity cuts. If harmonic superspace with N = 6 susy manifest exists 6 loops: If harmonic superspace with N = 7 susy manifest exists 7 loops: If a superspace with N = 8 susy manifest were to exist. 8 loops: Explicit identification of potential susy invariant counterterm with full non-linear susy. 9 loops: Assume Berkovits’ superstring non-renormalization theorems can be naively carried over to N = 8 supergravity. Naïve extrapolation from 6 loops needed. ZB, Dixon, Dunbar, Perelstein, and Rozowsky (1998) Howe and Stelle (2003) Howe and Stelle (2003) Grisaru and Siegel (1982) Kallosh; Howe and Lindstrom (1981) Green, Vanhove, Russo (2006) Note: none of these are based on demonstrating a divergence. They are based on arguing susy protection runs out after some point.

21. Reasons to Reexamine This • The number of established counterterms in any supergravity • theory is zero. • 2)Discovery of remarkable cancellations at 1 loop – • the “no-triangle hypothesis”.ZB, Dixon, Perelstein, Rozowsky • ZB, Bjerrum-Bohr and Dunbar; Bjerrum-Bohr, Dunbar, Ita, Perkins and Risager • 3) Every explicit loop calculation to date finds N = 8 supergravity • has identical power counting as in N = 4 super-Yang-Mills theory, • which is UV finite.Green, Schwarz and Brink; ZB, Dixon, Dunbar, Perelstein, Rozowsky; • Bjerrum-Bohr, Dunbar, Ita, PerkinsRisager; ZB, Carrasco, Dixon, Johanson, Kosower, Roiban. • 4) Very interesting hint from string dualities. Chalmers; Green, Vanhove, Russo • – Dualities restrict form of effective action. May prevent • divergences from appearing in D = 4 supergravity. • – Difficulties with decoupling of towers of massive states. • 5) Gravity twistor-space structure similar to gravity. • Derivative of delta function support See Green’s talk Witten; ZB, Bjerrum-Bohr, Dunbar

22. Gravity Feynman Rules Propagator in de Donder gauge: Three vertex has about 100 terms: An infinite number of other messy vertices Gravity looks to be a hopeless mess

23. Feynman Diagrams for Gravity Suppose we want to put an end to the speculations by explicitly calculating to see what is true and what is false: Suppose we wanted to check superspace claims with Feynman diagrams: If we attack this directly get terms in diagram. There is a reason why this hasn’t been evaluated previously. In 1998 we suggested that five loops is where the divergence is: This single diagram has terms prior to evaluating any integrals. More terms than atoms in your brain.

24. N = 8 Supergravity Loop Amplitudes ZB, Dixon, Dunbar, Perelstein and Rozowsky (1998) Basic Strategy N = 8 Supergravity Tree Amplitudes Unitarity N = 4 Super-Yang-Mills Tree Amplitudes KLT Divergences • Kawai-Lewellen-Tye relations: sum of products of gauge • theory tree amplitudes gives gravity tree amplitudes. • Unitarity method: efficient formalism for perturbatively • quantizing gauge and gravity theories. Loop amplitudes • from tree amplitudes. ZB, Dixon, Dunbar, Kosower (1994) Key features of this approach: • Gravity calculations mapped into much simpler gauge • theory calculations. • Only on-shell states appear.

25. KLT Relations At tree level Kawai, Lewellen and Tye presented a relationship between closed and open string amplitudes. In field theory limit, relationship is between gravity and gauge theory Gravityamplitude Color stripped gauge theory amplitude where we have stripped all coupling constants Full gauge theory amplitude Holds for any external states. See review: gr-qc/0206071 Progress in gauge theory can be imported into gravity theories

26. In D = 4 diverges for L 5. L is number of loops. N = 8 Power Counting To All Loop Orders From ’98 paper: • Assumed iterated 2 particle cuts give • the generic UV behavior. • Assumed no cancellations with other • uncalculated terms. No evidence was found that more than 12 powers of loop momenta come out of the integrals. Result from ’98 paper Elementary power counting gave finiteness condition: counterterm was expected in D = 4, for

27. Additional Cancellations at One Loop Crucial hint of additional cancellation comes from one loop. Surprising cancellations not explained by any known susy mechanism are found beyond four points One derivative coupling Two derivative coupling One loop Two derivative coupling means N = 8 should have a worse power counting relative to N = 4 super-Yang-Mills theory. • Cancellations observed in MHV amplitudes. • “No-triangle hypothesis” — cancellations in all other amplitudes. • Confirmed by explicit calculations at 6,7 points. ZB, Dixon, Perelstein Rozowsky (1999) ZB, Bjerrum-Bohr and Dunbar (2006) Bjerrum-Bohr, Dunbar, Ita, Perkins, Risager (2006)

28. No-Triangle Hypothesis One-loop D = 4 theorem: Any one loop massless amplitude is a linear combination of scalar box, triangle and bubble integrals with rational coefficients: • In N = 4 Yang-Mills only box integrals appear. No • triangle integrals and no bubble integrals. • The “no-triangle hypothesis” is the statement that • same holds in N = 8 supergravity.

29. 2 3 .. 1 4 L-Loop Observation ZB, Dixon, Roiban numerator factor From 2 particle cut: 1 in N = 4 YM Using generalized unitarity and no-triangle hypothesis all one-loop subamplitudes should have power counting of N = 4 Yang-Mills numerator factor From L-particle cut: Above numerator violates no-triangle hypothesis. Too many powers of loop momentum. There must be additional cancellation with other contributions!

30. Complete Three-Loop Calculation ZB, Carrasco, Dixon, Johansson, Kosower, Roiban Besides iterated two-particle cuts need following cuts: reduces everything to product of tree amplitudes For first cut have: Use KLT supergravity super-Yang-Mills N = 8 supergravity cuts are sums of products of N = 4 super-Yang-Mills cuts

31. Complete three -loop result ZB, Carrasco, Dixon, Johansson, Kosower, Roiban; hep-th/0702112 All obtainable from iterated two-particle cuts, except (h), (i), which are new.

32. Cancellation of Leading Behavior To check leading UV behavior we can expand in external momenta keeping only leading term. Get vacuum type diagrams: Doubled propagator Violates NTH Does not violate NTH but bad power counting After combining contributions: The leading UV behavior cancels!!

33. Finiteness Conditions Through L = 3 loops the correct finiteness condition is (L > 1): “superfinite” in D = 4 • same as N = 4 super-Yang-Mills • bound saturated at L = 3 notthe weaker result from iteratedtwo-particle cuts: finite in D = 4 for L = 3,4 (’98 prediction) Beyond L = 3, as already explained, from special cuts we have good reason to believe that the cancellations continue. All one-loop subamplitudes should have same UV power-counting as N = 4 super-Yang-Mills theory.

34. Origin of Cancellations? There does not appear to be a supersymmetry explanation for observed cancellations, especially if they hold to all loop orders, as we have argued. If it is not supersymmetry what might it be?

35. How does behave as ? Proof of BCFW recursion requires Tree Cancellations in Pure Gravity Unitarity method implies all loop cancellations come from tree amplitudes. Can we find tree cancellations? You don’t need to look far: proof of BCFW tree-level on-shell recursion relations in gravity relies on the existence such cancellations! Britto, Cachazo, Feng and Witten; Bedford, Brandhuber, Spence and Travaglini Cachazo and Svrcek; Benincasa, Boucher-Veronneau and Cachazo Susy not required Consider the shifted tree amplitude:

36. n legs Loop Cancellations in Pure Gravity ZB, Carrasco, Forde, Ita, Johansson Powerful new one-loop integration method due to Forde makes it much easier to track the cancellations. Allows us to link one-loop cancellations to tree-level cancellations. Observation: Most of the one-loop cancellations observed in N = 8 supergravity leading to “no-triangle hypothesis” are already present in non-supersymmetric gravity. Susy cancellations are on top of these. Maximum powers of Loop momenta Cancellation generic to Einstein gravity Cancellation from N = 8 susy Proposal: This continues to higher loops, so that most of the observed N = 8 multi-loop cancellations are notdue to susy but in fact are generic to gravity theories!

37. What needs to be done? • N = 8 four-loop computation. Can we demonstrate that four- • loop N = 8 amplitude has the same UV power counting as • N = 4 super-Yang-Mills? Certainly feasible. • Can we construct a proof of perturbative UV finiteness of N = 8? • Perhaps possible using unitarity method – formalism is recursive. • Investigate higher-loop pure gravity power counting to • study cancellations. (It does diverge.)Goroff and Sagnotti; van de Ven • Twistor structure of gravity loop amplitudes? Bern, Bjerrum-Bohr, Dunbar • Link to a twistor string description of N = 8? Abou-Zeid, Hull, Mason • Can we find other examples with less susy that may be finite? • Guess:N = 6 supergravity theories will be perturbatively finite.

38. Summary • Twistor-space structures tell us gauge theory and gravity • amplitudes are much simpler than previously anticipated. • Unitarity method gives us a powerful means for constructing • multi-loop amplitudes. • Resummation of N = 4 MHV sYM amplitudes –match to • strong coupling! • At four points through three loops, establishedN = 8 supergravity • has same power counting as N = 4 Yang-Mills. • Proposed that most of the observed N = 8 cancellations are • present in generic gravity theories, with susy cancellations • on top of these. • N = 8 supergravity may be the first example of a unitary point-like • perturbatively UV finite theory of quantum gravity in D = 4. • Proof is an open challenge.

39. Extra Transparencies

40. Non-trivial confirmation Method is designed to give same results as Feynman diagrams. Examples: • Two-loop-four gluon QCD amplitudes ZB, De Frietas, Dixon • – matches results of Glover, Oleari and Tejeda-Yeomans • 2. Four-loop cusp anomalous dimension in N = 4 sYM • – Beisert, Eden and Staudacher equation matches result. see Beisert’s talk • 3. Resummation of 1,2,3 loop calculations to all loop order in • N = 4 super-Yang-Mills theory Anastasiou, ZB, Dixon, Kosower; • ZB, Dixon, Smirnov • –matches the recent beautiful strong coupling construction of • Alday and Maldacena. see Alday’s talk ZB, Czakon, Dixon, Kosower, Smirnov

41. Opinions from the 80’s We have not shown that a three-loop counterterm is not present in N > 4, although it is tempting to conjecture that this may be the case. Howe and Lindstrom (1981) If certain patterns that emerge should persist in the higher orders of perturbation theory, then … N = 8 supergravity in four dimensions would have ultraviolet divergences starting at three loops. Green, Schwarz, Brink (1982) Unfortunately, in the absence of further mechanisms for cancellation, the analogous N = 8 D = 4 supergravity theory would seem set to diverge at the three-loop order. Howe, Stelle (1984) There are no miracles… It is therefore very likely that all supergravity theories will diverge at three loops in four dimensions. … The final word on these issues may have to await further explicit calculations. Marcus, Sagnotti (1985) Widespread agreement that N = 8 supergravity should diverge at 3 loops

42. Comments on Higher Loops Rule of thumb: If we can compute N = 4 Yang-Mills to a given order we can do the same for N = 8 supergravity. We obtained the planar N = 4 YM amplitude at 5 loops: ZB, Carrasco, Johansson, Kosower Origin of claim that even 5 loop N = 8 supergravity is feasible.

43. What’s New? • In the 1960’s unitarity and analyticity widely used. • However, not understood how to use unitarity to • reconstruct complete amplitudes with more than 2 • kinematic variables. Mandelstam representation Double dispersion relation Only 2 to 2 processes. A(s,t ) With unitarity method we can build arbitrary amplitudes at any loop order from tree amplitude. Bern, Dixon, Dunbar and Kosower Unitarity method builds loops from tree ampitudes. A(s1 , s2 , s3 , ...)

44. Other technical difficulties in the 60’s: • Non-convergence of dispersion relations. • Ambiguities or subtractions in the dispersion relations. • Confusion when massless particles present. • Inability to reconstruct rational functions with no • branch cuts. The unitarity method overcomes these difficulties by (a) Using dimension regularization to make everything well defined. (b) Bypassing dispersion relations by writing down Feynman representations giving both real and imaginary parts.