N =8 Supergravity at Five Loops Henrik Johansson Uppsala U. & - PowerPoint PPT Presentation

N =8 Supergravity at Five Loops Henrik Johansson Uppsala U. & Nordita Amplitudes in the LHC era GGI Florence, Oct 31, 2018 Based on recent work: 1701.02519, 1708.06807, 1804.09311 w/ Zvi Bern, John Joseph Carrasco, Wei-Ming Chen, Alex Edison, Julio Parra-Martinez, Radu Roiban, Mao Zeng and older work: 0702112, 0905.2326, 1008.3327, 1201.5366 w/ Zvi Bern, John Joseph Carrasco, Lance Dixon, David Kosower, Radu Roiban

Outline Ou Motivation & Review: Status of N =8 SUGRA UV behavior Previous 3,4 loop results Key steps in calculation Generalized double copy for gravity ampl. Controlling UV behavior of N =4 SYM Improved UV integration, IBP & vacuum diag. Results at 5 loops The critical UV behavior at 5 loops Simplicity in pattern of diagrams Conclusion

SUGRA status on one page Known facts: Ferrara, Zumino, Deser, Kay, Stelle, Howe, Lindström, Susy forbids 1,2 loop div. R 2 , R 3 Green, Schwarz, Brink, Marcus, Sagnotti Pure gravity 1-loop finite, 2-loop divergent Goroff & Sagnotti With matter: 1-loop divergent ‘t Hooft & Veltman N=8 SG Naively susy allows 3-loop div. R 4 N =8 SG and N =4 SG 3-loop finite! Bern, Carrasco, Dixon, HJ, Kosower, Roiban, Davies, Dennen, Huang N =8 SG: no divergence before 7 loops D>4 d ivergences @ L =2,3,4 UFinite? Marcus, Sagnotti, Bern, Dixon, Dunbar, Perelstein, Rozowsky, Carrasco, HJ, Kosower, Roiban Only known D=4 SG divergence: Bern, Davies, Dennen, Smirnov 2 N =4 @ 4 loops ( à more questions than answers) 7-loop D =4 calculation difficult instead work out 5 loops in D =24/5 à this talk

Why is it interesting ? If N =8 SG is perturbatively finite, why is it interesting ? It might be finite for a good reason! hidden new symmetry Other mechanism or structure à open a host of possibilities Any indication of hidden structures yet? Gravity is a double copy of gauge theories Gravity Color-Kinematics: kinematics = Lie algebra Bern, Carrasco, HJ Kallosh et al., Nicolai, Constraints from E-M duality ? Roiban, Freedman Hidden superconformal symmetry ? Ferrara, Kallosh, Van Proeyen; Loebbert, Mojaza, Plefka; HJ, Mogull, Teng; Caron-Huot, Trinh, … Extended N =4 superspace ? Bossard, Howe, Stelle Symmetry? Exceptional field theory Bossard, Kleinschmidt

UV problem = basic power counting Naively expect gravity to behave worse than Yang-Mills d 4 L p. . . ( κ p µ p ν ) . . . Z Gravity: non-renormalizable ∼ dimensionful coupling p 2 1 p 2 2 p 2 3 . . . p 2 n d 4 L p . . . ( gp µ ) . . . Z Yang-Mills: renormalizable ∼ dimensionless coupling p 2 1 p 2 2 p 2 3 . . . p 2 n ( p µ ) 2 L → ( k µ ) 2 L For finite gravity à vast cancellations needed ∼ seems implausible, but exists for N =8 SG in all known ampl’s. external momenta

Textbook perturbative gravity is complicated ! de Donder = gauge = After symmetrization 100 terms ! higher order vertices… 10 3 terms complicated diagrams: 10 4 terms 10 7 terms 10 21 terms 10 31 terms

On-she On shell si simplifications ns Graviton plane wave: Yang-Mills polarization On-shell 3-graviton vertex: = Yang-Mills vertex Gravity scattering amplitude: Yang-Mills amplitude tree (1 , 2 , 3 , 4) = st M GR u A YM tree (1 , 2 , 3 , 4) ⊗ A YM tree (1 , 2 , 3 , 4) Kawai, Lewellen, Tye Gravity processes = “squares” of gauge theory ones - entire S-matrix Bern, Carrasco, HJ

Historical record – where is the N = 8 div. ? Conventional superspace power counting 3 loops Green, Schwarz, Brink (1982) Howe and Stelle (1989) Marcus and Sagnotti (1985) Partial analysis of unitarity cuts; If N = 6 harmonic 5 loops Bern, Dixon, Dunbar, Perelstein, Rozowsky (1998) superspace exists; algebraic renormalisation Howe and Stelle (2003,2009) 6 loops If N = 7 harmonic superspace exists Howe and Stelle (2003) If N = 8 harmonic superspace exists; Grisaru and Siegel (1982); 7 loops Green, Russo, Vanhove; Kallosh; string theory U-duality analysis; Beisert, Elvang, Freedman, lightcone gauge locality arguments; Kiermaier, Morales, Stieberger; E 7(7) analysis, unique 1/8 BPS candidate Bossard, Howe, Stelle, Vanhove Explicit identification of potential susy invariant Howe and Lindström; 8 loops Kallosh (1981) counterterm with full non-linear susy Assume Berkovits ’ superstring non-renormalization 9 loops Green, Russo, Vanhove (2006) theorems can be carried over to N = 8 supergravity Identified cancellations in multiloop amplitudes; Finite Bern, Dixon, Roiban (2006), lightcone gauge locality and E 7(7) , Kallosh (2009–12), Ferrara, Kallosh, Van Proeyen (2012) inherited from hidden N=4 SC gravity note: above arguments/proofs/speculation are only lower bounds à only an explicit calculation can prove the existence of a divergence!

N =8 Amplitude and Counter Term Structure divergence Loop 4pt amplitude form Counter term first occurs in order (any dimension) 1 D c = 8 Green, Schwarz, Brink Bern, Dixon, Dunbar, D c = 7 2 Perelstein, Rozowsky Bern, Carrasco, Dixon, 3 D c = 6 HJ, Kosower, Roiban Bern, Carrasco, 4 D c = 5.5 Dixon, HJ, Roiban ? ? D c = 24/5 ? 5 ? ? ∼ ∂ 10 R 4 D c = 26/5 ? ∼ ∂ 10 R 4 The critical dimension divergence tells us how many derivatives are pulled out of the integral à counter term structure @

Known UV divergences in D >4 Plot of critical dimensions of N = 8 SUGRA and N = 4 SYM 1-2 2 loops: Green, Schwarz, Brink; Marcus and Sagnotti 3-5 5 loops: Bern, Carrasco, Dixon, HJ, Kosower, Roiban calculations: 6 loops: Bern, Carrasco, Dixon, Douglas, HJ, von Hippel 6 26/5 or 24/5 ? Divergent Known bound for N = 4 Bern, Dixon, Dunbar, Rozowsky, ? Perelstein; Howe, Stelle current trend for N = 8 Finite If N = 8 div. at L =7 L = 7 lowest loop order for possible D = 4 divergence Beisert, Elvang, Freedman, Kiermaier, Morales, Stieberger; Björnsson, Green, Bossard, Howe, Stelle, Vanhove Kallosh, Ramond, Lindström, Berkovits, Grisaru, H. Johansson 2013 Siegel, Russo, Cederwall, Karlsson, and more….

3,4,5-loop calculations

3-loop N =8 SG & N =4 SYM Using color-kinematics duality: Bern, Carrasco, HJ Be Bern, Carrasco, Dixon, HJ, , Ko Kosower, , Ro Roiban UV divergent in D =6: Be Bern, , Carrasco, , Di Dixon, , HJ HJ, , Ro Roiban A (3) � c V (A) + 12 N c ( V (A) + 3 V (B) )) × ( u Tr[ T a 1 T a 2 T a 3 T a 4 ] + perms) pole = 2 g 8 stA tree ( N 3 � � V (A) + 3 V (B) = ζ 3 ⌘ 8 M (3) � ⇣ κ ( stu ) 2 M tree ( V (A) + 3 V (B) ) 6 pole = 10 � 2 �

4-loops: 85 diagrams, 2 masters 7 3 7 8 6 3 8 7 2 3 8 5 5 1 2 5 6 4 7 4 1 1 5 6 2 3 6 8 4 5 4 7 8 1 6 2 6 5 3 4 4 5 8 7 7 8 1 7 6 6 2 3 2 4 3 4 1 5 7 5 8 8 5 8 2 7 7 4 5 3 6 6 1 8 6 3 8 7 3 4 5 2 8 1 4 5 6 7 8 7 3 2 2 7 1 5 4 6 6 5 3 8 7 3 8 3 3 3 1 3 4 3 2 5 5 4 7 4 6 6 6 7 7 7 1 6 3 7 2 2 2 2 6 2 8 7 7 4 3 3 5 8 8 5 4 4 8 8 8 5 6 5 6 5 1 6 1 4 1 1 8 6 1 4 5 8 4 5 7 5 5 6 8 6 5 6 2 4 1 2 7 8 (78) (77) 8 1 4 8 7 7 8 3 3 3 2 3 2 3 2 3 2 2 4 8 6 8 5 3 4 1 5 7 1 5 7 4 1 5 7 4 1 5 7 4 5 7 1 5 7 2 5 6 6 6 6 6 6 1 6 7

4-loop N =8 SG and N =4 SYM Be Bern, Carrasco, Dixon, HJ, J, Ro Roiban 1201.5366 •85 diagrams •Power counting manifest •N =4 & N =8 diverge in D =11/2 up to overall factor, divergence same as for N =4 SYM part

5 loops à 752 cubic graphs à 3 masters à Ansätze ~ 500k almost work à Back to the drawing board!

5-loop N =4 SYM the traditional way 1207.6666 [hep-th] N =4 SYM important stepping stone to N =8 SG Bern, Carrasco, HJ, Roiban • 416 nonvanishing integral topologies: (335) (370) (404) (410) • Used maximal cut method Bern, Carrasco, HJ, Kosower • Maximal cuts: 410 N 2 MC N 3 MC MC NMC • Next-to-MC: 2473 Unitarity cuts done in D dimensions • N 2 MC: 7917 • N 3 MC: 15156 integrated UV div. in D =26/5 Non-Planar UV divergence in D =26/5: � div = − 144 � � + 12( V (a) + 2 V (b) + V (c) ) � A (5) 5 g 12 stA tree N 3 N 2 c V (a) � 4 4 c � × Tr[ T a 1 T a 2 T a 3 T a 4 ]

Key methods for 5 loops

Double copy is necessary Unitarity & Ansätze possible way forward? Pessimistic counting: • Works for 5-loop N =4 SYM n SYM ∼ 8000 terms • 5-loop SG seems too difficult n SG ∼ (8000) 2 / 2 (ansatz: billions of terms) ∼ 30 000 000 terms Only way: use some form of double copy • On maximal cuts à naïve double copy works à square SYM numerators • On non-maximal cuts à KLT works in principle, but not in practice • KLT relations are non-local, non-crossing symmetric à bad for loops • Need something better than KLT, and less constraining than BCJ Generalized double copy --- when color-kinematics duality is non-manifest

Generalized Double copy Bern, Carrasco, Chen, HJ, Roiban Consider 4pt tree-level as warm-up: Assume: not BCJ numerators YM Gravity Contact terms have to to vanish if numerator Jacobi relation holds Note: example too simple since all 4pt tree numerators obey BCJ

Generalized Double copy Bern, Carrasco, Chen, HJ, Roiban Consider two 4pt trees in a unitarity cut: YM L R GR sum rows or columns In fact, the contact is given by independent of i and j Jacobi à contact terms are bilinears in the Jacobi discrepancies Jacobi à appears to work for general cuts