Is Perturbative N = 8 Supergravity Finite?

Is Perturbative N = 8 Supergravity Finite? Radu Roiban Pennsylvania State University Z. Bern, L. Dixon, R.R. hep-th/06086 Z. Bern, J.J. Carrasco, L.Dixon H. Johansson, D. Kosower, R.R., hep-th/07022

Outline Discuss evidence that N = 8 may be perturbatively finite Conventional wisdom on UV divergences in (super)gravity generalities and explicit calculations; direct and indirect information from string theory; kinematics and other observations hinting at good UV behaviour Higher loop calculations in N = 8 supergravity reduce supergravity to N = SYM KLT relations generalized unitarity method All-order arguments for finitness 3-loop -graviton amplitude and its N = -like finitness Outlook

Introduction G N /M 2 P l is dimensionfull: Standard statement: quantization of gravity following usual rules of point-like quantum field theory leads to a nonrenormalizable theory Requires UV completion or Different rules String theory: new length scale extended objects infinitely many fields LQG nonperturbative effects

Introduction G N /M 2 P l is dimensionfull: Standard statement: quantization of gravity following usual rules of point-like quantum field theory leads to a nonrenormalizable theory Requires UV completion or Different rules String theory: new length scale extended objects infinitely many fields LQG nonperturbative effects Is it possible to remain within standard field theory framework? symmetries: supersymmetry can make nongravitational theories perturbatively finite. Can it do the same for gravity? Most symmetric of supergravities: N = 8 (32 supercharges)

N = 8 supergravity dewit, Freedman (977) Cremmer, Julia,+Scherk (978, 979) largest amount of supersymmetry and spins smaller than 2 classically unique; smallest representation of the symmetry algebra contains the graviton spectrum: 2 8 = 256 massless fields helicity 2 3/2 /2 0 +/2 + +3/2 +2 number of fields 8 28 56 70 56 58 8 names h ψi vij χ ijk s ijkl χ + ijk v ij + ψ i + h + 32 supercharges, SU(8) R-symmetry, E 7 duality symmetry, add your name here symmetry spectrum = tensor product of 2 N = abelian vector multiplets helicity /2 0 +/2 + number of fields 6 names v χ i s ij χ + i v +

Opinions over the years (compilation by Lance Dixon) 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, (982) Howe, Stelle (98) Marcus, Sagnotti (985) Howe, Stelle (989) Our cut calculations indicate, but do not yet prove, that there is no three-loop counterterm for N = 8 supergravity, contrary to the expectations from superspace powercounting bounds. On the other hand... we infer a counterterm at five-loops with nonvanishing coefficient. Bern, Dixon, Dunbar, Perelstein, Rozowsky (998) The new estimates are in agreement with recent results derived from unitarity calculations... in five and six dimensions. For N=8 supergravity in four dimensions, we speculate that the onset of divergences may... occur at the six-loop level. Howe, Stelle (2002)

...it is striking that these arguments suggest that maximally extended supergravity has no ultraviolet divergences when reduced to four dimensions Green, Russo, Vanhove (2006)... we discussed evidence that four-dimensional N = 8 supergravity may be ultraviolet finite. Bern, Dixon, RR (2006)... recently discovered nonrenormalization properties of... the fourgraviton amplitude in type II superstring theory [Berkovits] lead to the absence of ultraviolet divergences in the four graviton amplitude of N = 8 supergravity up to at least eight loops. Green, Russo, Vanhove (2006)

What do we know about perturbative gravity and supergravity? Pure gravity covariance: on-shell counterterms are built out of Riemann tensors together with the condition R µν = 0 and R = 0 dimensional analysis: [G N ] = [R] = 2 #R s = +#loops -loop: total derivative in d: R ijkl R ijkl 2R ij R ij + R 2 t Hooft, Veltman 2-loops: Feynman diagram calculation nontrivial counterterm S 2 loops ct = c 2 R ij klr kl pqr pq ij with c 2 0 Goroff, Sangotti; van de Ven

N = 8 supergravity General reasoning: R 3 cannot be supersymmetrized Grisaru; Tomboulis susy completion of R valid counterterm at 3-loops Deser, Kay, Stelle; Kallosh, Howe, Stelle, Townsend; Gross, Witten R stum tree Explicit calculations; improved general arguments -loop, -, 5-, 6-, (7-)points 2-loops -point Bern, Dixon, Dunbar, Perelstein, Rozowski Bjerrum-Bohr, Dunbar, Ita, etc Bern, Dixon, Dunbar, Perelstein, Rozowski

String theory: direct and indirect information: perturbative -loop - and 5-points; n-point expression 2-loops -points Vanishing theorems (no D 2L R above L 5) Green, Schwarz Montag d Hoker, Phong d Hoker, Phong Berkovits Potential issues: Closed strings: built-in cutoff τ [ 2, 2 ], τ 2 [πα τ 2, ) Limits: low energy string UV α 0, κ 0 = g s α, Λ-fixed α, g s -fixed, Λ Not immediatey clear that the two limits commute Quantum level: lim string loops = loops of α 0 α 0 string theory Obviously true if low energy limit is a finite theory checked at and 2 loops by explicit calculations; no all-loop argument mass of string states same as cutoff M 2 Λ 2 α

String theory: direct and indirect information: non-perturbative: use duality symmetries T-duality (R /R), SL(2, Z), string theory/m-theory duality Green, Vanhove, Russo, h. Kwon conjectured exact -graviton effective action of string theory from - and 2-loop d supergravity calculations Potential issues: d physics from effective action on R,9 n T n with n 5 relate regulators relate higher and lower-dimensional fields possible divergences that break higher dim. symmetries possible dimension-dependent cancellations

When may divergences show up? (mostly) Field theory reasoning 3 loops superspace powercounting 5 loops partial unitarity cuts Green, Schwarz, Brink; Howe, Stelle; Marcus, Sagnotti Bern, Dixon, Dunbar Perelstein, Rozowsky 5 loops Manifest N = 6 supespace Howe, Stelle 6 loops Manifest N = 7 supespace Howe, Stelle 7 loops Manifest N = 8 supespace Grisaru, Siegel 8 loops Identification of invariants at nonlinear level (N -extended susy N -loop ct.) 9 loops naive translation of Berkovits vanishing theorems to d = Kallosh; Howe, Lindstrom Green, Russo, Vanhove Superpotential-type arguments No proof that divergences actually appear

Field theory calculations: methods, techniques, etc Main tools: unitarity method and the KLT relations (Generalized) Unitarity method: Feynman diagrammatics: Bern, Dixon, Dunbar, Kosower way too general applies to any field theory allows off-shell external legs (gauge) symmetries restored after summation over diagrams symmetry-based cancellations appear miraculous hides simplicity e.g. four graviton scattering: M tree (,2,3 +, + ) = κ2 s 2 [ 2 2 23 3 ] [ 2 2 2 3 3 ]

Field theory calculations: methods, techniques, etc Main tools: unitarity method and the KLT relations (Generalized) Unitarity method: Feynman diagramatics: Bern, Dixon, Dunbar, Kosower way too general applies to any field theory allows off-shell external legs (gauge) symmetries restored after summation over diagrams symmetry-based cancellations appear miraculous hides simplicity Solution: stay on-shell l = SS 2IT = T T Unitarity: relation between discontinuity of amplitude at some loop order and lower loop amplitudes loop : 2IT loop = l l 2 = d LIPS on shell 2 loops: 2IT 2 loops = l l 2 2 l 2 l 3 l3 l l2

Field theory calculations: methods, techniques, etc Main tools: unitarity method and the KLT relations (Generalized) Unitarity method: Bern, Dixon, Dunbar, Kosower Interpret a cut as requiring that the cut propagators are present can cut several times the same amplitude on each side of a generalized cut there is an amplitude reconstruct full amplitude from its generalized cuts need D-dimensional cuts to ensure no terms are missing Key technical point: The (generalized) unitarity method reduces the calculation of on-shell loop amplitudes to knowledge of on-shell tree amplitudes and their products

Field theory calculations: methods, techniques, etc KLT relations: Kawai, Lewellen, Tye Derived from string theory tree amplitudes and the observation that closed string states are created by bilinears in operators creating open string states; additional factors due to zero modes no issues with low energy limit and reduction to d relate N = SYM and N = 8 sugra tree amplitudes for any states M tr (, 2, 3, ) = is 2A tr (, 2, 3, )Atr (, 2,, 3) M tr 5 (, 2, 3,, 5) = is 2s 3 A tr 5 (, 2, 3,, 5)Atr 5 (2,,, 3, 5) + (2 3) M tr 6 = 2 terms of the type s3 A 6 A 6 Captures spectrum decomposition [N = 8] = [N = ] [N = ] Key technical point: in conjunction with the unitarity method, reduce supergravity cuts to SYM cuts!

KLT+unitarity [ gravity sts SYM sts M n (...l, l 2, l 3...)M n2 (...l, l 2, l 3...) = (s ij factors) ][ A n (...l, l 2, l 3...)A n2 (...l, l 2, l 3...) SYM sts KLT terms ] A n (...l, l 2, l 3...)A n 2 (...l, l 2, l 3...) such a decomposition is not manifest at the level of Lagrangian Construction of gravity amplitudes is a -step process 0. identify the cuts that uniquely specify the amplitude of interest. construct the color-stripped SYM cuts required by the KLT 2. construct the gravity cuts 3. identify the functions whose generalized cuts reproduce point 2. Is there an echo of the finiteness of N = SYM?

Example: -loop SYM vs. -loop supergravity 2 l l 2 q q 2 3 -loop N = super-yang-mills Bern, Dixon, Dunbar, Kosower N = l l 2 = is 2 s 23 2 l l 2 3 = i s 2 s 23 A tree (, 2, 3, ) (2l k 2 )(2l 2 k ) N =8 -loop N = 8 supergravity (M (, 2, 3, ) = s 2 A (, 2, 3, )A (, 2,, 3)) M tree (, 2, l, l 2 )M tree (l 2, l, 3, ) = (s 2 s 23 ) 2s 2A tree (, 2, 3, ) s 2 A tree (2,,, 3) (2l k 2 )(2l 2 k ) (2l k )(2l 2 k 3 ) [ 2 = s 2 s 3 s 23 M tree (, 2, 3, ) 3 + 2 3 + + l l l l 3 2 2 ] 3

SYM: A loop (,2,3,) = is 2 s 23 A tree (,2,3,) Supergravity: 2 3 M loop (,2,3,) = [ s 2 s 23 A tree (,2,3,) ] [ 2 2 + + 3 3 2 ] prefactor is square of SYM one 2-particle cuts iterate to all orders Bern, Dixon, Dunbar, Perelstein, Rozowski same degree of divergence/convergence as N = SYM!

This pattern persists at 2-loops Bern, Dixon, Dunbar, Perelstein, Rozowski N = SYM Bern, Rozowski, Yan Bern, Dixon, Dunbar, Perelstein, Rozowski = i 2 s 2 s 23 c s 2 +c 2 s 2 + perm s N = 8 supergravity: strip off color and square factors! Bern, Dixon, Dunbar, Perelstein, Rozowski = s 2 s 23 2 s 2 2 + s 2 2 + perm s same integrals in SYM and sugra same UV properties Can this continue?

Both in SYM and sugra 2-particle cuts iterate to all orders The rung rule: A rung connecting two existing lines (SYM) or 2 (sugra) factors of the square of the sum of momenta of the two connected lines some uncertainty in sugra factor Bern, Rozowski, Yan Bern, Dixon, Dunbar, Perelstein, Rozowski l l [ ( l l + l 2 ) 2 ] 2 2 l 2 generates all 2-particle cut constructible diagrams Examples: [ s (L ) ] 2 2 2 l l 2 L loops 3 [ s 23 (l + l 2 ) 2(L 2) ] 2 l (L ) loops l2

Consequences for UV behaviour: [ s 23 (l +l 2 ) 2(L 2) ] 2 l (L ) loops l2 s 2 23 (s 2s 3 s 23 M tree D R ) Amplitude scaling convergence: DL 2 d DL l (l2 ) n(l 2) (l 2 ) 3L+ + n(l 2) < 3L + counterterm of the type D R { n = SYM n = 2 supergravity 2PCC: first divergence appears at L = 5 in D = But this is only the beginning D c = 2 + 0 L N = 8 D c = + 6 L N = Only 2PCC diagrams included: there could be further terms Speculations that first divergence at L = 6 Howe, Stelle String theory hints: higher loop counterterm structure different late 2006: Berkovits; Green, Russo, Vanhove

-loop revisited: no-triangles and all-loop cancellations N = 8 -loop amplitudes analyzed explicitly: 5-, 6- and 7-point: Bjerrum-Bohr, Dunbar, Ita, Perkins, Risager all expressed as sum of box integrals (no triangles or bubbles) good arguments that it holds for all -loop amplitudes formalized in the no-triangle hypothesis Bern, Bjerrum-Bohr, Dunbar Bjerrum-Bohr, Dunbar, Ita, Perkins, Risager implications for our story follow through integral reduction: d D l (2l k ) d D l 2 (l k ) 2... = l (l 2 (l k ) 2 + k 2) l 2 (l k ) 2... 2 l. k k n k l l k k 2 k n k l k k 2 k n l k k2 k 2 k n k l l k k 2 one factor of l µ in numerator one less propagator N-gon diagram: less that (N ) loop momentum numerator factors

A crucial observation [s 25 (l + k ) 2(L 2) ] 2 l l l 2 (L+2) gon subdiagram (L ) loops L-particle cut with (L + 2)-gon diagram on one side and (L 2) factors of loop momentum! contrast with explicit calculation for 5- and 6-point amplitudes contrast with no-triangle hypothesis to any loop order more cancellations must take place at any loop order better UV behavior must be expected too all loop orders! 2 3 No triangles

A crucial observation [s 25 (l + k ) 2(L 2) ] 2 l l l 2 (L+2) gon subdiagram (L ) loops L-particle cut with (L + 2)-gon diagram on one side and (L 2) factors of loop momentum! contrast with explicit calculation for 5- and 6-point amplitudes contrast with no-triangle hypothesis to any loop order More general: any -loop subdiagram should not contain triangle integrals! No triangles

First place to test our arguments: 3-loops Dc N = Dc N =8 3-particle cut of ladder diagram: -loop 5- point amplitude w/ (l+k ) in the numerator against explicit calculation/no- restriction 2 l 3 Possible sources of major improvement: nonplanar contributions (still 2-particle cut constructible) diagrams not constructable from 2-particle cuts Warning: no-triangle restriction is not the universal cure ; bad overall powercounting 2 l 3 consistent with no-triangle behavior Only one way to be sure: explicitly construct the 3-loop integrand

... and so we did Bern, Carrasco, Dixon, Johansson, Kosower, RR Need to analyze 2-, 3- and -particle cuts 2-particle cut constructible diagrams s 2 s 2 s 2 s 2 s 2 2 [ ( l+k ) 2 ] 2 s 2 2 [ ( l+k ) 2 ] 2 s 2 2 [ ( l+k ) 2 ] 2 l l l

3-particle cuts Use generalized cuts: chop down all the way to tree amplitudes + + The plan:. use KLT to reduce them to planar and nonplanar N = cuts 2. construct the relevant nonplanar N = cuts 3. reassemble the supergravity cuts. identify the integral functions that give these cuts

additional N = diagrams and numerator factors ] [ s 2 (l + l 2 ) 2 l5 2 [ + s 23 (l3 + l ) 2 l6 2 s 2 s 23 ] l 3 l l 6 l 5 l 2 l l s 2 (l + l 2 ) 2 s 23 (l 3 + l ) 2 + 3 (s 2 s 23 ) l 2 5 l l 5 l l 2 l l 3

additional N = 8 supergravity diagrams and numerator factors [ s2 (l + l 2 ) 2 + s 23 (l 3 + l ) 2 s 2 s 23 ] 2 s 2 2 (2((l + l 2 ) 2 s 23 ) + l 2 5 )l2 5 s 2 23 (2((l 3 + l ) 2 s 2 ) + l 2 6 )l2 6 s 2 2 (2l2 7 l2 2 + 2l2 l2 8 + l2 2 l2 8 + l2 l2 7 ) l 3 l 9 l l 6 l 5 l 2 l 7 l 8 l l 0 s 2 23 (2l l2 0 + 2l2 9 l2 + l2 0 l2 + l l2 9 ) +2s 2 s 23 l 2 5 l2 6 ( s2 (l + l 2 ) 2 s 23 (l 3 + l 2 )2) 2 l 5 l ( s 2 2 (l + l 2 ) 2 + s 2 23 (l 3 + l 2 )2 + 3 s 2s 23 s 3 ) l 2 5 Note contact terms l l 2 l l 3

four-particle cuts: easy to draw and hard to evaluate ideally they would be the starting point in present case: powercounting arguments that they bring nothing new higher generalized cuts that support these arguments calculation: the result from 2- and 3-particle cuts is complete M (3) = κ8 [ 2 8stuMtree S 3 + + 2 + 2 + 2 + + 2 + + 2 ]

UV behavior Leading UV divergence leading term at small external momenta all diagrams become vacuum diagrams with doubled propagators exposes triangle subdiagrams For example: s 2 2 [ ( l+k ) 2 ] 2 l = s 2 2 Doubled propagators the diagrams with loop momenta in the numerators determine the leading UV behavior

Vacuum diagrams and their origin 0 8 8 No triangles 0 0 8 unexpected 0 0 0 0 0 0 0 0 8 8 0 0 0 2 0 2

A momentum conservation identity helps: = 2 [ + ] = 2 [ + ] Leading UV divergence cancels algebraically! First subleading term also cancels Cancellations beyond no-triangle behavior First nonzero contribution: only 2 loop momenta in numerator the same as for N = SYM D c consistent with SYM powercounting at 3 loops D c = + 6 L

Superspaces and powercounting An apparent discrepancy: superspace vs. explicit calculations 2-loops: 2 loop momenta external momenta consistent with hypothetical manifest N = 6 superspace Howe, Stelle 3-loops: loop momenta external momenta consistent with a hypothetical manifest N = 7 superspace Higher loop expectations: L-loops: (2L + 8) loop momenta external momenta L = would rule out potential N = 7 explanation L = 5 would rule out any regular superspace explanation

Higher loop expectations and arguments can be tested! -loop integrand certainly constructible: key test step : construct nonplanar SYM integrand; same UV as planar? step 2: reconstruct supergravity integrand from KLT and result at step ; same UV as SYM? If finiteness holds, is it possible to construct a recursive proof? Who/What is responsible: dynamics? an infinite-dim. symmetry? Is there a string theory behind it? Which one? Is N = 8 supergravity unique in exhibiting superfiniteness? perhaps KLT two finite gauge theories?

Summary Main message: N = 8 supergravity is better behaved in the UV than standard superspace arguments suggest generalized unitarity and KLT relations combine into a powerful method for explicit calculations -loop cancellations no-triangle behaviour/hypothesis Arguments that improvement persists at higher loops: the notriangle hypothesis implies all-order cancellations in certain classes of diagrams At 3-loops the cancellations are complete; N = 8 supergravity has the same degree of divergence/convergence as N = SYM Is it finite to all orders?

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