Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati , , Based on work with: Zvi Bern, John Joseph Carrasco, Lance Dixon, Michael Douglas, Radu Roiban, Matt von Hippel Towards Determining the UV Behavior of Maximal Supergravity

H. Johansson, Frascati 2013 SUGRA status in one slide After 35 years of supergravity, we can only now make very precise statements about the D=4 ultraviolet structure. No D=4 divergence of pure SG has been found to date. Susy forbids 1,2 loop div., R 2, R 3 c.t. incompatible with susy Pure gravity 1-loop finite, 2-loop divergent Goroff & Sagnotti With matter: 1-loop divergent ‘t Hooft & Veltman Naively susy allows 3-loop div. R 4 N =8 SG and N =4 SG 3-loop finite! N =8 SG: no divergence before 7 loops 7-loop div. in D=4 implies a 5-loop div. in D=24/5 -- calculation in progress! UFinite? N=8 SG

H. Johansson, Frascati 2013 Why is it interesting ? If N =8 SG is perturbatively finite, why is it interesting ? It better be finite for a good reason! Hidden new symmetry, for example Understanding the mechanism might open a host of possibilities Any indication of hidden structures yet? Gravity is a double copy of gauge theories Color-Kinematics: kinematics = Lie algebra Constraints from E-M duality Kallosh,…. Hidden superconformal N=4 SUGRA ? Symmetry? Gravity Bern, Carrasco, HJ Ferrara, Kallosh, Van Proeyen

Henrik Johansson Gauge Theory Analogy Gauge theory in D>4 have same problem as D=4 gravity Non-renormalizable due to dimensionful coupling However, D=5 SYM has a UV completion: (2,0) theory in D=6 Is D=5 SYM perturbatively UV finite ? Douglas; Lambert et al. If yes, how does it work ? If no, what do we need to add ? Solitons, KK modes ? Douglas; Lambert et al. Understanding D=5 SYM might (or might not) give clues to how to understand D=4 gravity. (2,0) theory ? D=5 SYM

H. Johansson, Frascati 2013 Review UV status N=8 SUGRA and N=4 SYM 4pt amplitudes and UV divergences 3,4-loop N=8 SUGRA & N=4 SYM 5-loop nonplanar SYM 6-loop planar D=5 SYM 5pt amplitudes and UV divergences 1,2,3-loop N=8 SUGRA & N=4 SYM Current 5-loop progress Conclusion Outline

H. Johansson, Frascati 2013 UV properties N =8 SG N =8 SG: conventional superspace power counting forbids L=1,2 divergences Deser, Kay, Stelle; Howe and Lindström; Green, Schwarz, Brink; Howe, Stelle; Marcus, Sagnotti Three-loop divergence ruled out by calculation: Bern, Carrasco, Dixon, HJ, Kosower, Roiban, (2007), Bern, Carrasco, Dixon, HJ, Roiban (2008 ) L<7 loop divergences ruled out by counterterm analysis, using E 7(7) symmetry and other methods, but a L=7 divergence is still possible Beisert, Elvang, Freedman, Kiermaier, Morales, Stieberger; Björnsson, Green, Bossard, Howe, Stelle, Vanhove, Kallosh, Ramond, Lindström, Berkovits, Grisaru, Siegel, Russo, and more…. In D=4 dimensions: In D>4 dimensions: Through four loops N =8 SG and N =4 SYM diverge in exactly the same dimension: Marcus and Sagnotti; Bern, Dixon, Dunbar, Perelstein, Rozowsky; Bern, Carrasco, Dixon, HJ, Kosower, Roiban

H. Johansson, Frascati 2013 UV divergence trend Plot of critical dimensions of N = 8 SUGRA and N = 4 SYM Known bound for N = 4 Bern, Dixon, Dunbar, Rozowsky, Perelstein; Howe, Stelle current trend for N = 8 If N = 8 div. at L=7 calculations: 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, Siegel, Russo, and more…. 1-2 loops: Green, Schwarz, Brink; Marcus and Sagnotti 3-5 loops: Bern, Carrasco, Dixon, HJ, Kosower, Roiban 6 loops: Bern, Carrasco, Dixon, Douglas, HJ, von Hippel Finite ? Divergent

H. Johansson, Frascati 2013 UV divergence trend Plot of critical dimensions of N = 8 SUGRA and N = 4 SYM Known bound for N = 4 Bern, Dixon, Dunbar, Rozowsky, Perelstein; Howe, Stelle current trend for N = 8 If N = 8 div. at L=7 calculations: 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, Siegel, Russo, and more…. 1-2 loops: Green, Schwarz, Brink; Marcus and Sagnotti 3-5 loops: Bern, Carrasco, Dixon, HJ, Kosower, Roiban 6 loops: Bern, Carrasco, Dixon, Douglas, HJ, von Hippel 26/5 or 24/5 ? Finite ? Divergent

H. Johansson, Frascati 2013 N =8 Amplitude and Counter Term Structure 4pt amplitude form (any dimension) divergence occurs in Counter term D = 8 D = 6 D = 7 D = 5.5 Loop order If amplitude for L  4 has at least 8 derivatives then by dimensional analysis: no divergence before L = 7 ! D = 24/5 ? 5 ? ?

H. Johansson, Frascati 2013 N =8 Amplitude and Counter Term Structure 4pt amplitude form (any dimension) divergence occurs in Counter term D = 8 D = 6 D = 7 D = 5.5 Loop order If amplitude for L  5 has at least 10 derivatives then by dimensional analysis: no divergence before L = 8 ! D = 26/5 ? 5 ? ?

H. Johansson, Frascati 2013 Earliest appearance of N = 8 Divergence 3 loops Conventional superspace power counting Green, Schwarz, Brink (1982) Howe and Stelle (1989) Marcus and Sagnotti (1985) 5 loops Partial analysis of unitarity cuts; If N = 6 harmonic superspace exists; algebraic renormalisation Bern, Dixon, Dunbar, Perelstein, Rozowsky (1998) Howe and Stelle (2003,2009) 6 loops If N = 7 harmonic superspace exists Howe and Stelle (2003) 7 loops If N = 8 harmonic superspace exists; string theory U-duality analysis; lightcone gauge locality arguments; E 7(7) analysis, unique 1/8 BPS candidate Grisaru and Siegel (1982); Green, Russo, Vanhove; Kallosh; Beisert, Elvang, Freedman, Kiermaier, Morales, Stieberger; Bossard, Howe, Stelle, Vanhove 8 loops Explicit identification of potential susy invariant counterterm with full non-linear susy Howe and Lindström; Kallosh (1981) 9 loops Assume Berkovits’ superstring non-renormalization theorems can be carried over to N = 8 supergravity Green, Russo, Vanhove (2006) Finite Identified cancellations in multiloop amplitudes; lightcone gauge locality and E 7(7), inherited from hidden N=4 SC gravity Bern, Dixon, Roiban (2006), Kallosh (2009–12), Ferrara, Kallosh, Van Proeyen (2012)

H. Johansson, Frascati , 4, 5, 6-Loop Amplitudes

3-loop N =8 SG & N =4 SYM Color-kinematics dual form: Bern, Carrasco, HJ UV divergent in D=6: Bern, Carrasco, Dixon, HJ, Roiban

4-loops: 85 integral types

H. Johansson, QMUL loops N =4 SYM and N =8 SG Bern, Carrasco, Dixon, HJ, Roiban diagrams Power counting manifest both N =4 and N =8 Both diverge in D=11/2 up to overall factor, divergence same as for N =4 SYM part

H. Johansson, Frascati 2013 N =4 SYM 5-loop Amplitude N =4 SYM important stepping stone to N =8 SG [hep-th] Bern, Carrasco, HJ, Roiban 416 integral topologies: Used maximal cut method Bern, Carrasco, HJ, Kosower Maximal cuts: 410 Next-to-MC: 2473 N 2 MC: 7917 N 3 MC: Unitarity cuts done in D dimensions...integrated UV div. in D=26/5

H. Johansson, Frascati 2013 N =4 SYM 5-loop UV divergence Non-Planar UV divergence in D=26/5: Double traces and single-trace NNLC finite in D=26/5, only single-trace LC and NLC are divergent Vanish! Bern, Carrasco, HJ, Roiban

H. Johansson, Frascati 2013 Testing D=5 intercept Plot of critical dimensions of N = 8 SUGRA and N = 4 SYM Known bound for N = 4 Bern, Dixon, Dunbar, Rozowsky, Perelstein; Howe, Stelle current trend for N = 8 If N = 8 div. at L=7 calculations: 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, Siegel, Russo, and more…. 1-2 loops: Green, Schwarz, Brink; Marcus and Sagnotti 3-5 loops: Bern, Carrasco, Dixon, HJ, Kosower, Roiban 6 loops: Bern, Carrasco, Dixon, Douglas, HJ, von Hippel

H. Johansson, Frascati Loop Planar D=5 SYM 68 planar diagrams Given by dual conformal invariance (up to integer 0,1,-1,2,-2... prefactors) Independently constructed by: Eden, Heslop, Korchemsky, Sokatchev; Bourjaily, DiRe, Shaikh, Spradlin, Volovich Bern, Carrasco, Dixon, Douglas, HJ, von Hippel

6-Loop Planar D=5 SYM The Parking Spot Escalation our secret collaborator…

6-Loop D=5 SYM divergence Using integration by parts identities, div. simplifies to 3 integrals Numerical integration – modified version of FIESTA 1000 node cluster at Stony Brook Result: divergence is nonzero. What cancels this divergence ? Solitons/KK modes ? Douglas; Lambert et al. Bern, Carrasco, Dixon, Douglas, HJ, von Hippel

Divergence to all loop orders ? Bern, Carrasco, Dixon, Douglas, HJ, von Hippel Intriguing pattern of UV divergence in critical dimension of maximal susy YM Accurate to <1% Why do the UV divergences follow this approximate curve? Is it asymptotically exact ?

H. Johansson, Frascati pt N=8 SUGRA calculations

H. Johansson, Frascati 2013 C-K amplitudes at 1 loop Green, Schwarz, Brink (1982) Duality-satisfying loop amplitudes: N=4 SYM: All-plus QCD: N=4 SYM and All-plus QCD: [hep-th] Carrasco, HJ

SYM UV div in D=8: Carrasco, HJ [hep-th] SU(8) violating SG UV div in D=8: SG UV div in D=8: 1-loop 5-pts UV divergences counterterms:

H. Johansson, Frascati loop 5-pts N =4 SYM and N =8 SG The 2-loop 5-point amplitude with Color-Kin. duality N = 8 SG obtained from numerator double copies Carrasco, HJ [hep-th]

2-loop 5-pts UV divergences SYM UV div in D=7: SG UV div in D=7: Carrasco, HJ [hep-th] SU(8) violating SG UV div in D=8:

H. Johansson, Frascati loop 5-point SYM and N =8 SG ( “ ladder-like ” diagrams) Carrasco, HJ (to appear) Non-planar D-dimensional amplitude with manifest color-kinematics duality ( N = 8 SG obtained from squaring the numerators)

H. Johansson, Frascati loop 5-point SYM and N =8 SG some “ Mercedes-like ” diagrams … Carrasco, HJ (to appear)

H. Johansson, Frascati loop 5-point SYM and N =8 SG Carrasco, HJ (to appear) … in total 42 diagrams. For SYM the UV divergent diagrams (in D=6) are very simple: (for SG the UV div. comes from the other diagrams as well)

H. Johansson, Frascati loop 5-pts UV divergences

N =8 SG 5-loop Status Working on reorganizing 5-loop N =4 SYM Bern, Carrasco, HJ, Roiban (in progress) = 752 integral topologies BCJ: 2500 functional Jacobi eqns Relaxing ansatz: Non-manifest crossing symmetry Allow for non-local numerators Gauge variant numerators Relax power counting Allow for triangle diagrams … Once we have integrand, integration will take ~ 1 day: No subdivergences in D=24/5 No IR divergences since D>4, and absence of bubbles No more difficult than IBP:s for N =4 SYM < 1 day If needed, numerical integration ~ few days

SYM SG ? 3 loops, D=6: 4 loops, D=11/2: SYM SG H. Johansson, Frascati 2013 Fortune-telling from pattern 5 loops, D=26/5: related to diagrams in the quartic Casimir

Summary Explicit calculations in N = 8 SUGRA up to four loops show that the power counting exactly follows that of N = 4 SYM -- a finite theory 5 loop calculation in D=24/5 probes the potential 7-loop D=4 counterterm -- will provide critical input to the N = 8 question ! D=5 SYM have a 6-loop UV divergence, showing that the standard perturbative expansion misses some of the (2,0) theory contributions. Color-Kinematics duality allows for gravity calculations for multiloop multipoint amplitudes -- greatly facilitating UV analysis in gravity. Numbers in UV divergences of N =8 SUGRA and N =4 SYM coincide, suggesting a deeper connection between the theories Stay tuned for the 5-loop SUGRA result…