Could N=8 Supergravity be a finite theory of quantum gravity?

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. Introduction • Quantum gravity is nonrenormalizable by power counting, because the coupling, Newton’s constant, GN = 1/MPl2is 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. Could N=8 Supergravity Be Finite? L. Dixon

3. 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) Is (N=8) supergravity finite? A question that has been asked many times, over many years. Could N=8 Supergravity Be Finite? L. Dixon

4. 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 Since has mass dimension 2, and the loop-counting parameter GN = 1/MPl2has mass dimension -2, every additional requires another loop, by dimensional analysis 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) Could N=8 Supergravity Be Finite? L. Dixon

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) Could N=8 Supergravity Be Finite? L. Dixon

6. cannot be supersymmetrized – it produces a helicity amplitude (-+++) forbidden by supersymmetry Grisaru (1977); Tomboulis (1977) 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 Could N=8 Supergravity Be Finite? L. Dixon

7. SUSY charges Qa, a=1,2,…,8 shift helicity by 1/2 Maximal supergravity DeWit, Freedman (1977); Cremmer, Julia, Scherk (1978); Cremmer, Julia (1978,1979) • Most supersymmetry allowed, for maximum particle spin of 2 • Theory has 28 = 256 massless states. • Multiplicity of states, vs. helicity, from coefficients in binomial expansion of (x+y)8– 8th row of Pascal’s triangle • Ungauged theory, in flat spacetime Could N=8 Supergravity Be Finite? L. Dixon

8. Supergravity scattering amplitudes: vs. • Study higher-dimensional versions of N=8 supergravity to see what critical dimension Dc 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). Bern, LD, Dunbar, Perelstein, Rozowsky (1998) • 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, Kosower (1994) Could N=8 Supergravity Be Finite? L. Dixon

9. 28 = 256 massless states, ~ expansion of (x+y)8 SUSY 24 = 16 states ~ expansion of (x+y)4 Compare spectra Could N=8 Supergravity Be Finite? L. Dixon

10. 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, Kawai-Lewellen-Tye relations KLT, 1986 Could N=8 Supergravity Be Finite? L. Dixon

11. AdS = CFT gravity = gauge theory weak strong KLT gravity = (gauge theory)2 weak weak AdS/CFT vs. KLT Could N=8 Supergravity Be Finite? L. Dixon

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

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 Could N=8 Supergravity Be Finite? L. Dixon

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

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

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

17. Integral in D dimensions scales as  Critical dimensionDc for log divergence (if no cancellations) obeys N=8 N=4 SYM BDDPR (1998) More UV divergent diagrams N=4 SYM N=8 supergravity Could N=8 Supergravity Be Finite? L. Dixon

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

19. No triangles found! N=4 SYM, 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 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] Could N=8 Supergravity Be Finite? L. Dixon

20. 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 A key L-loop topology • Implies additional cancellations in the left loop • Bern, LD, Roiban (2006) Could N=8 Supergravity Be Finite? L. Dixon

21. Three-loop case • First order for which N=4 SYM and N=8 supergravity • might have had a different Dc (loop momenta in numerator) Something must cancel the bad “left-loop” behavior of this contribution. But what? • 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 Could N=8 Supergravity Be Finite? L. Dixon

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: Could N=8 Supergravity Be Finite? L. Dixon

23. Non-rung-rule N=4 SYM at 3 loops Could N=8 Supergravity Be Finite? L. Dixon

24. Non-rung-rule N=8 SUGRA at 3 loops Could N=8 Supergravity Be Finite? L. Dixon

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

26. Cancellations beyond no-triangle hypothesis! leading UV divergence cancels perfectly! 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 linearly dependent 0 0 0 0 8 8 - 8 = 0 0 0 0 -2 0 -2 + 2 = 0 Could N=8 Supergravity Be Finite? L. Dixon

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, Dc = 6 for N=8 SUGRA as well as N=4 SYM Could N=8 Supergravity Be Finite? L. Dixon

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

29. Puzzle conversations with Banks, Berenstein, Shenker, Susskind, ...; also Green, Ooguri, Schwarz, 0704.0777 [hep-th] • N=8 SUGRA has more symmetry than type II string theory. • It has a continuous, noncompact E7(7) ; also SU(8) < E7 • Symmetry is not present in massive string spectrum. • Suppose N=8 SUGRA is finite, order by order in perturbation theory. • True coupling is s/MPl2 • 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 E7 /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? Could N=8 Supergravity Be Finite? L. Dixon

30. N=8 N=4 SYM 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 Dc = 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? Could N=8 Supergravity Be Finite? L. Dixon

31. Extra slides Could N=8 Supergravity Be Finite? L. Dixon

32. Reasons to reexamine whether it might be too conservative: at every loop order • 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 Berkovits, hep-th/0609006; Green, Russo, Vanhove, hep-th/0611273 N=8 Green, Russo, Vanhove, hep-th/0610299 Bjerrum-Bohr et al., hep-th/0610043 Old power counting potential counterterm D=4 divergence at five loops Could N=8 Supergravity Be Finite? L. Dixon

33. M theory duality Green, Vanhove, hep-th/9910055, hep-th/0510027; Green, Russo, Vanhove, hep-th/0610299 • N=1 supergravity in D=11 is low-energy limit of M theory • Compactify on two torus with complex parameter W • 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 W 1 Could N=8 Supergravity Be Finite? L. Dixon

34. Zero-mode counting in string theory Berkovits, hep-th/0609006; Green, Russo, Vanhove, hep-th/0611273 • 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, a’  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 Could N=8 Supergravity Be Finite? L. Dixon

35. 4-particle cuts An important – but simple – subset of the full 4-particle cuts, of the form  can be used to determine the term in (h): • 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. Could N=8 Supergravity Be Finite? L. Dixon