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Superfiniteness of N = 8 Supergravity at Three Loops and Beyond

Superfiniteness of N = 8 Supergravity at Three Loops and Beyond. Julius Wess Memorial November 6, 2008 Zvi Bern, UCLA. Based on following papers: ZB, N.E.J. Bjerrum-Bohr, D.C. Dunbar, hep-th/0501137 ZB, L. Dixon , R. Roiban, hep-th/0611086

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Superfiniteness of N = 8 Supergravity at Three Loops and Beyond

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  1. Superfiniteness of N = 8 Supergravity at Three Loops and Beyond Julius Wess Memorial November 6, 2008 Zvi Bern, UCLA Based on following papers: ZB, N.E.J. Bjerrum-Bohr, D.C. Dunbar, hep-th/0501137 ZB, L. Dixon , R. Roiban, hep-th/0611086 ZB, J.J. Carrasco, H. Johansson and D. Kosower, arXiv:0705.1864 [hep-th] ZB, J.J. Carrasco, D. Forde, H. Ita and H. Johansson, arXiv:0707.1035 [hep-th] ZB, J.J. Carrasco, H. Johansson, arXiv:0805.3993 [hep-ph] ZB, J.J. Carrasco, L.J. Dixon, H. Johansson, R. Roiban , arXiv:0808.4112 [hep-th]

  2. Outline Will present concrete evidence for non-trivial UV cancellations in N = 8 supergravity, and perhaps UV finiteness. • Review of conventional wisdom on UV divergences in quantum gravity. • Remarkable simplicity of gravity amplitudes. • Calculational method – reduce gravity to gauge theory: (a) Kawai-Lewellen-Tye tree-level relations. (b) Modern unitarity method (instead of Feynman diagrams). • All-loop arguments for UV finiteness of N = 8 supergravity. • Explicit three-loop calculation and “superfiniteness”. • Progress on four-loop calculation. • Origin of cancellation -- generic to all gravity theories.

  3. N = 8 Supergravity The most supersymmetry allowed for maximum particle spin of 2 is N = 8. Eight times the susy of N = 1 theory of Ferrara, Freedman and van Nieuwenhuizen We consider the N = 8 theory of Cremmer and Julia. 256 massless states • Reasons to focus on this theory: • With more susy expect better UV properties. • High symmetry implies technical simplicity. • Recently conjectured by Arkani-Hamed, Cachazo • and Kaplan to be “simplest” quantum field theory.

  4. Finiteness of N = 8 Supergravity? We are interested in UV finiteness of N = 8 supergravity because it would imply a new symmetry or non-trivial dynamical mechanism. The discovery of either should have a fundamental impact on our understanding of gravity. • Non-perturbative issues and viable models of Nature • are not the goal for now. • Here we only focus on order-by-order UV finiteness • and to identify the mechanism behind them.

  5. Power Counting at High Loop Orders Dimensionful coupling Gravity: Gauge theory: Extra powers of loop momenta in numerator means integrals are badly behaved in the UV. Non-renormalizable by power counting. Much more sophisticated power counting in supersymmetric theories but this is the basic idea.

  6. 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. squared Bel-Robinson tensor expected counterterm Deser, Kay, Stelle (1977); Kaku, Townsend, van Nieuwenhuizen (1977), Ferrara, Zumino (1978)

  7. Opinions from the 80’s 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) 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) The idea that allD = 4 supergravity theories diverge at 3 loops has been the accepted wisdom for over 25 years

  8. 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:Ifa 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. Also need to extrapolate to higher loops. Superspace gets here with additional speculations. Stelle (2006) 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) Where are the N = 8 Divergences? Depends on whom you ask and when you ask. Note: none of these are based on demonstrating a divergence. They are based on arguing susy protection runs out after some point.

  9. Reasons to Reexamine This • The number of established UV divergences for any pure supergravity theory in D = 4 is zero! • 2)Discovery of novel cancellations at 1 loop – • the “no-triangle integral property”.ZB, Dixon, Perelstein, Rozowsky; • ZB, Bjerrum-Bohr, Dunbar; Bjerrum-Bohr, Dunbar, Ita, Perkins, Risager; Bjerrum-Bohr, Vanhove • Arkani-Hamed, Cachazo, Kaplan • 3) Every explicit loop calculation to date finds N = 8 supergravity • has identical power counting as N = 4 super-Yang-Mills theory, • which is UV finite.Green, Schwarz and Brink; ZB, Dixon, Dunbar, Perelstein, Rozowsky; • Bjerrum-Bohr, Dunbar, Ita, Perkins Risager; ZB, Carrasco, Dixon, Johanson, Kosower, Roiban. • 4) Interesting hint from string dualities. Chalmers; Green, Russo, Vanhove • – Dualities restrict form of effective action. May prevent • divergences from appearing in D = 4 supergravity, although • issues with decoupling of towers of massive states.

  10. Gravity Feynman Rules Propagator in de Donder gauge: Three vertex: About 100 terms in three vertex An infinite number of other messy vertices. Naive conclusion: Gravity is a nasty mess.

  11. … + Gravity vs Gauge Theory Consider the gravity Lagrangian Infinite number of complicated interactions Compare to Yang-Mills Lagrangian Only three and four point interactions Gravity seems so much more complicated than gauge theory. Multiloop calculations appear impossible.

  12. Standard Off-Shell Formalisms In graduate school you learned that scattering amplitudes need to be calculated using unphysical gauge dependent quantities: off-shell Green functions Standard machinery: –Fadeev-Popov procedure for gauge fixing. – Taylor-Slavnov Identities. – BRST. – Gauge fixed Feynman rules. – Batalin-Fradkin-Vilkovisky quantization for gravity. – Off-shell constrained superspaces. For all this machinery relatively few calculations in quantum gravity to check assertions on UV properties. Explicit calculations from ‘t Hooft and Veltman; Goroff and Sagnotti; van de Ven

  13. Why are Feynman diagrams clumsy for high loop calculations? • Vertices and propagators involve gauge-dependent off-shell states. Origin of the complexity. • To get at root cause of the trouble we need to do things differently. unphysical states propagate • All steps should be in terms of gauge invariant • on-shell states. On-shell formalism. • Radical rewrite of quantum field theory needed.

  14. Simplicity of Gravity Amplitudes On-shell three vertices contains all information: gauge theory: “square” of Yang-Mills vertex. gravity: Any gravity scattering amplitude constructible solely from on-shell 3 vertex. • BCFW on-shell recursion for tree amplitudes. • Unitarity method for loops. Britto, Cachazo, Feng and Witten; Brandhuber, Travaglini, Spence; Cachazo, Svrcek; Benincasa, Boucher-Veronneau, Cachazo; Arkani-Hamed and Kaplan, Hall ZB, Dixon, Dunbar and Kosower; ZB, Dixon, Kosower; Buchbinder and Cachazo; ZB, Carrasco, Johansson, Kosower; Cachzo and Skinner.

  15. If • Remarkably, gravity is as well behaved at as gauge theory. • We just need three vertex to start the recursion! On-Shell Recursion Britto, Cachazo, Feng and Witten Consider tree amplitude under complex deformation of the momenta. complex momenta A(z) is amplitude with shifted momenta Sum over residues gives the on-shell recursion relation Poles in z come from kinematic poles in amplitude.

  16. KLT Relations A remarkable relation between gauge and gravity amplitudes exist at tree level which we exploit. At tree level Kawai, Lewellen and Tye derived 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 Strongly suggests a unified description of gravity and gauge theory.

  17. Gravity and Gauge Theory Amplitudes Berends, Giele, Kuijf; ZB, De Freitas, Wong gravity gauge theory • Holds for all states appearing in a string theory. • Holds for all states of N = 8 supergravity.

  18. Bern, Dixon, Dunbar and Kosower Onwards to Loops: Unitarity Method Two-particle cut: Three-particle cut: Generalized unitarity: Allows us to systematically Construct loop amplitudes from on-shell tree amplitudes. Bern, Dixon and Kosower Generalized cut interpreted as cut propagators not canceling. A number of recent improvements to method Britto, Cachazo, Feng; Buchbinber, Cachazo; ZB, Carrasco, Johansson, Kosower; Cachazo and Skinner; Ossola, Papadopoulos, Pittau; Forde; Berger, ZB, Dixon, Forde, Kosower.

  19. … + no Gravity seems so much more complicated than gauge theory. Gravity vs Gauge Theory Consider the gravity Lagrangian Infinite number of irrelevant (for S matrix) interactions! Only three-point interactions needed for on-shell recursion Compare to Yang-Mills Lagrangian Only three-point interactions needed. Multiloop calculations appear impossible.

  20. 1 2 2 1 4 3 4 3 4 4 3 3 2 1 2 1 N = 8 Supergravity from N = 4 Super-Yang-Mills Using unitarity method and KLT we express cuts of N = 8 supergravity amplitudes in terms of N = 4 amplitudes. Key formula for N = 4 Yang-Mills two-particle cuts: Key formula for N = 8 supergravity two-particle cuts: Note recursive structure! Generates all contributions with s-channel cuts.

  21. Iterated Two-Particle Cuts to All Loop Orders ZB, Rozowsky, Yan (1997); ZB, Dixon, Dunbar, Perelstein, Rozowsky (1998) constructible from iterated 2 particle cuts not constructible from iterated 2 particle cuts Rung rule for iterated two-particle cuts N = 4 super-Yang-Mills N = 8 supergravity

  22. counterterm expected in D = 4, for Power Counting To All Loop Orders From ’98 paper: • Assumed rung-rule contributions give • the generic UV behavior. • Assumed no cancellations with other • uncalculated terms. Elementary power counting from rung rule gives finiteness condition: In D = 4 finite for L < 5. L is number of loops.

  23. No-Triangle Property ZB, Dixon, Perelstein, Rozowsky; ZB, Bjerrum-Bohr and Dunbar; Bjerrum-Bohr, Dunbar, Ita, Perkins, Risager. One-loop D = 4 theorem: Any one loop amplitude is a linear combination of scalar box, triangle and bubble integrals with rational coefficients: Passarino and Veltman, etc • In N = 4 Yang-Mills only box integrals appear. No • triangle integrals and no bubble integrals. • The “no-triangle property” is the statement that • same holds in N = 8 supergravity. • Recent proofs by Bjerrum-Bohr and Vanhove; Arkani-Hamed, Cachazo and Kaplan

  24. 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 property all one-loop subamplitudes should have power counting of N = 4 Yang-Mills numerator factor From L-particle cut: Above numerator violates no-triangle property. Too many powers of loop momentum in one-loop subamplitude. There must be additional cancellation with other contributions!

  25. Full Three-Loop Calculation ZB, Carrasco, Dixon, Johansson, Kosower, Roiban Besides iterated 2 particle cuts need: reduces everything to product of tree amplitudes For second cut have: Use KLT supergravity super-Yang-Mills N = 8 supergravity cuts are sums of products of N = 4 super-Yang-Mills cuts

  26. Complete Three Loop Result ZB, Carrasco, Dixon, Johansson, Kosower, Roiban; hep-th/0702112 ZB, Carrasco, Dixon, Johansson, Roiban arXiv:0808.4112 [hep-th] N = 8 supergravity amplitude manifestly has diagram-by-diagram power counting of N = 4 sYM! By integrating this we have demonstrated D = 6 divergence. Superfinite: UV cancellations beyond those needed for finiteness

  27. 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 is saturated at L = 3 not the weaker result from iteratedtwo-particle cuts: finite in D = 4 for L = 3,4 (old prediction) Beyond L = 3, as already explained, from special cuts we have strong evidence that cancellations continue to all loop orders. All one-loop subdiagrams should have same UV power-counting as N = 4 super-Yang-Mills theory. No known susy argument explains all-loop cancellations

  28. N = 8 Four-Loop Calculation in Progress ZB, Carrasco, Dixon,Johansson, Roiban Some N = 4 YM contributions: 50 planar and non-planar diagrammatic topologies N = 4 super-Yang-Mills case is complete. N = 8 supergravity still in progress – so far looks good. • Four loops will teach us a lot: • Direct challenge to a potential superspace explanation: • existence of N = 6 superspace suggested by Stelle. • 2. Study of cancellations will lead to better understanding. • 3. Need 16 not 14 powers of loop momenta to come out • of integrals to get power counting of N = 4 sYM

  29. Origin of Cancellations? There does not appear to be a supersymmetry explanation for observed cancellations, especially as the loop order continues to increase. If it is not supersymmetry what might it be?

  30. How does behave as ? Proof of BCFW recursion relies on Tree Cancellations in Pure Gravity Unitarity method implies all loop cancellations come from tree amplitudes. Can we find tree cancellations? ZB, Carrasco, Forde, Ita, Johansson 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; Arkani-Hamed and Kaplan; Arkani-Hamed, Cachazo and Kaplan Susy not required Consider the shifted tree amplitude:

  31. 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 directly link one-loop cancellations to tree-level cancellations. Observation: Most of the one-loop cancellations observed in N = 8 supergravity leading to “no-triangle property” are already present even 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 Key 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! If N = 8 is UV finite suspect also N = 5, 6 is finite.

  32. Schematic Illustration of Status Same power count as N=4 super-Yang-Mills UV behavior unknown from feeding 2 and 3 loop calculations into iterated cuts. No known susy explanation for all-loop cancellations. loops behavior unknown no triangle property. 4-loop calculation in progress. explicit 2- and 3-loop computations terms

  33. Open Issues • Will 4 loops be superfinite? Will be answered soon! • Physical origin of cancellations not understood. Clear • link to high energy behavior of tree amplitude. • Link to N = 4 super-Yang-Mills? So far link is mainly • a technical trick. But KLT relations surely much • deeper. • Can we construct an all orders proof of finiteness? • Can we get a handle on non-perturbative issues?

  34. Summary • Modern unitarity method gives us means to calculate at high • loop orders. Allows us to unravel the UV structure of gravity. • Exploit KLT relations at loop level. Map gravity to gauge theory. • Observed novel cancellations in N = 8 supergravity • – No-triangle property implies cancellations strong enough • for finiteness to all loop orders, but in a limited class of terms. • – At four points three loops, establishedthat cancellations are • complete and N = 8 supergravity has the same power counting • as N = 4 Yang-Mills theory. • – Key cancellations appear to be generic in gravity. • Four-loop N = 8 calculation in progress. N = 8 supergravity may well be the first example of a unitary point-like perturbatively UV finite theory of quantum gravity. Proving this remains a challenge.

  35. Extra transparancies

  36. 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 equivalent to two copies of much • simpler gauge-theory calculations. • Only on-shell states appear.

  37. This single diagram has terms prior to evaluating any integrals. More terms than atoms in your brain! 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 using Feynman diagrams.. In 1998 we suggested that five loops is where the divergence is:

  38. N = 8 All-Orders Cancellations 5-point 1-loop known explicitly must have cancellations between planar and non-planar Using generalized unitarity and no-triangle property any one-loop subamplitude should have power counting of N = 4 Yang-Mills But contributions with bad overall power counting yet no violation of no-triangle property might be possible. Total contribution is worse than for N = 4 Yang-Mills. One-loop hexagon OK

  39. No-Triangle Property Comments • NTP not directly a statement of improved UV behavior. • — Can have excellent UV properties, yet violate NTP. • — NTP can be satisfied, yet have bad UV scaling at • higher loops. • Really just a technical statement on the type • of analytic functions that can appear at one loop. • Used only to demonstrate cancellations of loop momenta • beyond those observed in 1998 paper, otherwise wrong • analytic structure. • ZB, Dixon, Roiban

  40. Method of Maximal Cuts ZB, Carrasco, Johansson, Kosower A refinement of unitarity method for constructing complete higher-loop amplitudes is “Method of Maximal Cuts”. Systematic construction in any theory. To construct the amplitude we use cuts with maximum number of on-shell propagators: tree amplitudes Maximum number of propagator placed on-shell. on-shell Then systematically release cut conditions to obtain contact terms: Fewer propagators placed on-shell. Related to leading singularity method. Cachazo and Skinner; Cachazo; Cachazo, Spradlin, Volovich; Spradlin, Volovich, Wen

  41. ZB, Dixon, Dunbar and Kosower Applications of Unitarity Method • Now the most popular method for pushing • state-of the art one-loop QCD for LHC physics. • 2. Planar N = 4 Super-Yang-Mills • amplitudes to all loop orders. Spectacular • verification by Alday and Maldacena at • four points using string theory. • 3. Study of UV divergences in gravity. Berger, ZB, Dixon, Febres Cordero, Forde, Kosower, Ita, Maitre; Britto and Feng; Ossola, Papadopoulos, Pittau; Ellis, Giele, Kunzst, Melnikov, Zanderighi Anastasiou, ZB, Dixon, Kosower; ZB, Dixon, Smirnov

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