1. Could N=8 Supergravity be a finite theory of quantum gravity? Z. Bern, L.D., R. Roiban, PLB644:265 [hep-th/0611086] Z. Bern, J.J. Carrasco, L.D., H. Johansson, D. Kosower, R. Roiban, PRL98:161303 [hep-th/0702112] Lance Dixon (SLAC) CIFAR/LindeFest meeting at Stanford March 7, 2008

2. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 2 Introduction Quantum gravity is nonrenormalizable by power counting, because the coupling, Newton’s constant, G N = 1/M Pl 2 is dimensionful String theory cures the divergences of quantum gravity by introducing a new length scale, the string tension, at which particles are no longer pointlike. Is this necessary? Or could enough symmetry, e.g. supersymmetry, allow a point particle theory of quantum gravity to be perturbatively ultraviolet finite? If the latter is true, even if in a “toy model”, it would have a big impact on how we think about quantum gravity.

3. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 3 Is (N=8) supergravity finite? A question that has been asked many times, over many years. Reports of the death of supergravity are exaggerations. One year everyone believed that supergravity was finite. The next year the fashion changed and everyone said that supergravity was bound to have divergences even though none had actually been found. S. Hawking (1994)

4. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 4 Since has mass dimension 2, and the loop-counting parameter G N = 1/M Pl 2 has mass dimension -2, every additional requires another loop, by dimensional analysis Counterterms in gravity On-shell counterterms in gravity should be generally covariant, composed from contractions of Riemann tensor. Terms containing Ricci tensor and scalar removable by nonlinear field redefinition in Einstein action One-loop  However, is Gauss-Bonnet term, total derivative in four dimensions. So pure gravity is UV finite at one loop (but not with matter) ‘t Hooft, Veltman (1974)

5. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 5 Pure gravity diverges at two loops Relevant counterterm, is nontrivial. By explicit Feynman diagram calculation it appears with a nonzero coefficient at two loops Goroff, Sagnotti (1986); van de Ven (1992)

6. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 6 Pure supergravity (N ≥ 1): Divergences deferred to at least three loops However, at three loops, there is a perfectly acceptable counterterm, even for N=8 supergravity: The square of the Bel-Robinson tensor, abbreviated, plus (many) other terms containing other fields in the N=8 multiplet. Deser, Kay, Stelle (1977); Kallosh (1981); Howe, Stelle, Townsend (1981) produces first subleading term in low-energy limit of 4-graviton scattering in type II string theory: Gross, Witten (1986) 4-graviton amplitude in (super)gravity cannot be supersymmetrized – it produces a helicity amplitude (-+++) forbidden by supersymmetry Grisaru (1977); Tomboulis (1977)

7. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 7 Maximal supergravity Most supersymmetry allowed, for maximum particle spin of 2 Theory has 2 8 = 256 massless states. Multiplicity of states, vs. helicity, from coefficients in binomial expansion of (x+y) 8 – 8 th row of Pascal’s triangle SUSY charges Q a, a=1,2,…,8 shift helicity by 1/2 DeWit, Freedman (1977); Cremmer, Julia, Scherk (1978); Cremmer, Julia (1978,1979) Ungauged theory, in flat spacetime

8. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 8 Study higher-dimensional versions of N=8 supergravity to see what critical dimension D c they begin to diverge in, as a function of loop number L Compare with analogous results for N=4 super-Yang-Mills theory (a finite theory in D = 4). Supergravity scattering amplitudes: vs. Key technical ideas: Kawai-Lewellen-Tye (KLT) (1986) relations to express N=8 supergravity tree amplitudes in terms of simpler N=4 super-Yang-Mills tree amplitudes Unitarity to reduce multi-loop amplitudes to products of trees Bern, LD, Dunbar, Perelstein, Rozowsky (1998) Bern, LD, Dunbar, Kosower (1994)

9. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 9 Compare spectra 2 8 = 256 massless states, ~ expansion of (x+y) 8 SUSY 2 4 = 16 states ~ expansion of (x+y) 4

10. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 10 Kawai-Lewellen-Tye relations Derive from relation between open & closed string amplitudes. Low-energy limit gives N=8 supergravity amplitudes as quadratic combinations of N=4 SYM amplitudes, consistent with product structure of Fock space, KLT, 1986

11. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 11 AdS/CFT vs. KLT AdS = CFT gravity = gauge theory weak strong KLT gravity = (gauge theory) 2 weak

12. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 12 Amplitudes via perturbative unitarity Very efficient due to simple structure of tree helicity amplitudes S-matrix is a unitary operator between in and out states  Unitarity relations (cutting rules) for amplitudes Generalized unitarity (more propagators open) necessary to reduce everything to trees (in order to apply KLT relations)

13. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 13 Generalized unitarity at 3 loops 3-particle cuts contain a 5-point 1-loop amplitude. Chop loop amplitude further, into (4-point tree) x (5-point tree) in all inequivalent ways: Using KLT, each product of 3 supergravity trees decomposes into pairs of products of 3 (twisted, nonplanar) SYM trees. To evaluate them, need the full 3-loop N=4 SYM amplitude

14. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 14 Many higher-loop contributions follow from a simple “recycling” property of the 2-particle cuts at one loop The “Rung Rule” Bern, Rozowsky, Yan (1997), BDDPR (1998); also Cachazo, Skinner (2008) N=4 SYM rung rule N=8 SUGRA rung rule 2 +

15. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 15 1 loop: Resulting simplicity at 1 and 2 Loops “color dresses kinematics” Bern, Rozowsky, Yan (1997); Bern, LD, Dunbar, Perelstein, Rozowsky (1998) 2 loops: Green, Schwarz, Brink; Grisaru, Siegel (1981) N=8 supergravity: just remove color, square prefactors!

16. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 16 In N=4 SYM Ladder diagrams (Regge-like) In N=8 supergravity Extra in gravity from “charge” = energy Schnitzer, hep-th/0701217

17. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 17 N=4 SYM More UV divergent diagrams N=8 supergravity Integral in D dimensions scales as  Critical dimension D c for log divergence (if no cancellations) obeys N=8 N=4 SYM BDDPR (1998)

18. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 18 But: Look at one loop, lots of legs evidence that it is better generic gravity (spin 2) N=8 supergravity generic gauge theory (spin 1) N=4 SYM

19. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 19 No triangles found! a pentagon linear in  scalar box with no triangle a generic pentagon quadratic in  linear box  scalar triangle But all N=8 amplitudes inspected so far, with 5,6,~7,… legs, contain no triangles  more like than N=4 SYM, Bjerrum-Bohr et al., hep-th/0610043. Bern, Carrasco, Forde, Ita, Johansson, 0707.1035[th] (pure gravity) ; Kallosh, 0711.2108 [hep-th]; Bjerrum-Bohr, Vanhove, 0802.0868 [th]

20. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 20 A key L-loop topology 2-particle cut exposes Regge-like ladder topology, containing numerator factor of L-particle cut exposes one-loop (L+2)-point amplitude – but would (heavily) violate the no-triangle hypothesis Implies additional cancellations in the left loop Bern, LD, Roiban (2006)

21. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 21 Three-loop case  Evaluate all the cuts – 3-particle and 4-particle cuts as well as 2-particle. Bern, Carrasco, L.D., H. Johansson, D. Kosower, R. Roiban, hep-th/0702112 Something must cancel the bad “left-loop” behavior of this contribution. But what? First order for which N=4 SYM and N=8 supergravity might have had a different D c (loop momenta in numerator)

22. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 22 3 loops: 2-particle cuts  rung-rule integrals +  These numerators are precise squares of corresponding N=4 SYM numerators From 3- and 4-particle cuts, find 2 additional integral topologies and numerators:

23. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 23 Non-rung-rule N=4 SYM at 3 loops

24. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 24 Non-rung-rule N=8 SUGRA at 3 loops

25. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 25 Leading UV behavior Obtain by neglecting all dependence on external momenta, Each four-point integral becomes a vacuum integral For example, graph (e) becomes where denotes a doubled propagator

26. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 26 Sum of vacuum integrals (e) (f) (g) (h) (i) Total 4 0 8 -4 -8 0 0 4 0 -8 4 0 0 0 0 -4 0 -4 0 0 0 0 8 8 - 8 = 0 0 0 0 -2 0 -2 + 2 = 0 leading UV divergence cancels perfectly! Cancellations beyond no-triangle hypothesis! linearly dependent

27. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 27 N=8 SUGRA still no worse than N=4 SYM in UV Leading behavior of individual topologies in N=8 SUGRA is But leading behavior cancels after summing over topologies  Lorentz invariance  no cubic divergence In fact there is a manifest representation – same behavior as N=4 SYM Evaluation of integrals  no cancellation at this level  At 3 loops, D c = 6 for N=8 SUGRA as well as N=4 SYM

28. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 28 # of loops explicit 1, 2 and 3 loop computations Same power count as N=4 super-Yang-Mills UV behavior unknown # of contributions inferred from 2- and 3-loop 4-point, via cuts 4 loop calculation in progress: 52 cubic topologies Beyond three loops higher-loop cancellations inferred from 1-loop n-point (no triangle hypothesis) …

29. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 29 Puzzle N=8 SUGRA has more symmetry than type II string theory. It has a continuous, noncompact E 7(7) ; also SU(8) < E 7 Symmetry is not present in massive string spectrum. Suppose N=8 SUGRA is finite, order by order in perturbation theory. True coupling is s/M Pl 2 Series is unlikely to converge, rather be asymptotic. To define it nonperturbatively, need new states/instantons/... Can these come from string theory? Coefficient of every term in N=8 SUGRA amplitude is independent of E 7 /SU(8) moduli, so lack of convergence is also independent. How could stringy states/effects remedy the series, if they don’t have the right symmetries? conversations with Banks, Berenstein, Shenker, Susskind,...; also Green, Ooguri, Schwarz, 0704.0777 [hep-th]

30. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 30 Summary Old power-counting formula from iterated 2-particle cuts predicted At 3 loops, new terms found from 3- and 4-particle cuts reduce the overall degree of divergence, so that, not only is N=8 finite at 3 loops, but D c = 6 at L=3, same as for N=4 SYM! Will the same happen at higher loops, so that the formula continues to be obeyed by N=8 supergravity as well? If so, N=8 supergravity would represent a perturbatively finite, pointlike theory of quantum gravity Not of direct phenomenological relevance, but could it point the way to other, more relevant, finite theories? N=8 N=4 SYM

31. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 31 Extra slides

32. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 32 Old power counting potential counterterm N=8 at every loop order D=4 divergence at five loops Superspace-based speculation that D=4 case diverges only at L=6, not L=5 Multi-loop string calculations seem not to allow past L=2. String/M duality arguments with similar conclusions, suggesting possibility of finiteness No triangle hypothesis for 1-loop amplitudes Howe, Stelle, hep-th/0211279 Reasons to reexamine whether it might be too conservative: Berkovits, hep-th/0609006; Green, Russo, Vanhove, hep-th/0611273 Green, Russo, Vanhove, hep-th/0610299 Bjerrum-Bohr et al., hep-th/0610043

33. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 33 M theory duality N=1 supergravity in D=11 is low-energy limit of M theory Compactify on two torus with complex parameter  Use invariance under combined with threshold behavior of amplitude in limits leading to type IIA and IIB superstring theory Conclude that at L loops, effective action is If all dualities hold, and if this result survives the compactification to lower D – i.e. there are no cancellations between massless modes and Kaluza-Klein excitations – then it implies – same as in N=4 SYM – for all L Green, Vanhove, hep-th/9910055, hep-th/0510027; Green, Russo, Vanhove, hep-th/0610299  1

34. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 34 Zero-mode counting in string theory New pure spinor formalism for type II superstring theory – spacetime supersymmetry manifest Zero mode analysis of multi-loop 4-graviton amplitude in string theory implies:  At L loops, for L < 7, effective action is  For L = 7 and higher, run out of zero modes, and arguments gives If result survives both the low-energy limit,  ’  0, and compactification to D=4 – i.e., no cancellations between massless modes and either stringy or Kaluza-Klein excitations – then it implies finiteness through 8 loops Berkovits, hep-th/0609006; Green, Russo, Vanhove, hep-th/0611273

35. Could N=8 Supergravity Be Finite? L. Dixon CIFAR/LindeFest, March 7 '08 35 An important – but simple – subset of the full 4-particle cuts, of the form  4-particle cuts Full 4-particle cuts involve 72 different twisted SYM configurations. They work, with the above integrals (a)-(i), confirming our representation of the 3-loop N=8 supergravity amplitude. can be used to determine the term in (h):