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Towards Determining the UV Behavior of Maximal Supergravity

Towards Determining the UV Behavior of Maximal Supergravity. Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati. 1201.5366, 1207.6666, 1210.7709 Based on work with: Zvi Bern, John Joseph Carrasco, Lance Dixon, Michael Douglas, Radu Roiban , Matt von Hippel.

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Towards Determining the UV Behavior of Maximal Supergravity

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  1. Towards Determining the UV Behavior of Maximal Supergravity Henrik Johansson CERN March 26, 2013 BUDS workshop INFN Frascati 1201.5366, 1207.6666, 1210.7709 Based on work with: Zvi Bern, John Joseph Carrasco, Lance Dixon, Michael Douglas, RaduRoiban, Matt von Hippel

  2. N=8 SG UFinite? 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., R2, R3c.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. R4 • 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! H. Johansson, Frascati 2013

  3. Gravity Symmetry? 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 ? • Bern, Carrasco, HJ Ferrara, Kallosh, Van Proeyen H. Johansson, Frascati 2013

  4. D=5 SYM (2,0) theory ? 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. Henrik Johansson

  5. Outline • Review UV status N=8 SUGRAand 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 H. Johansson, Frascati 2013

  6. UV properties N=8 SG In D=4 dimensions: • 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 E7(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: 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

  7. UV divergence trend Plot of critical dimensions of N = 8 SUGRA and N = 4 SYM 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 calculations: 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, VanhoveKallosh, Ramond, Lindström, Berkovits, Grisaru, Siegel, Russo, and more…. H. Johansson, Frascati 2013

  8. UV divergence trend Plot of critical dimensions of N = 8 SUGRA and N = 4 SYM 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 calculations: 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, VanhoveKallosh, Ramond, Lindström, Berkovits, Grisaru, Siegel, Russo, and more…. H. Johansson, Frascati 2013

  9. N=8 Amplitude and Counter Term Structure divergence occurs in 4pt amplitude form (any dimension) Loop order Counter term D = 8 1 2 D = 7 3 D = 6 4 D = 5.5 ? 5 ? D = 24/5 ? If amplitude for L  4 has at least 8 derivatives then by dimensional analysis: no divergence before L = 7 ! H. Johansson, Frascati 2013

  10. N=8 Amplitude and Counter Term Structure divergence occurs in 4pt amplitude form (any dimension) Loop order Counter term D = 8 1 2 D = 7 3 D = 6 4 D = 5.5 5 ? ? D = 26/5 ? If amplitude for L  5 has at least 10 derivatives then by dimensional analysis: no divergence before L = 8 ! H. Johansson, Frascati 2013

  11. Earliest appearance of N= 8 Divergence H. Johansson, Frascati 2013

  12. 3, 4, 5, 6-Loop Amplitudes H. Johansson, Frascati 2013

  13. 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

  14. 4-loops: 85 integral types

  15. 4-loops N =4 SYM and N =8 SG Bern, Carrasco, Dixon, HJ, Roiban1201.5366 • 85 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, QMUL 2013

  16. N=4 SYM 5-loop Amplitude N=4 SYM important stepping stone to N=8 SG 1207.6666 [hep-th] Bern, Carrasco, HJ, Roiban • 416 integral topologies: • Used maximal cut method • Bern, Carrasco, HJ, Kosower • Maximal cuts: 410 • Next-to-MC: 2473 • N2MC: 7917 • N3MC: 15156 Unitarity cuts done in D dimensions...integrated UV div. in D=26/5 H. Johansson, Frascati 2013

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

  18. Testing D=5 intercept Plot of critical dimensions of N = 8 SUGRA and N = 4 SYM 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 calculations: Known bound for N= 4 Bern, Dixon, Dunbar, Rozowsky, Perelstein; Howe, Stelle current trend for N= 8 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, VanhoveKallosh, Ramond, Lindström, Berkovits, Grisaru, Siegel, Russo, and more…. H. Johansson, Frascati 2013

  19. 6-Loop Planar D=5 SYM Bern, Carrasco, Dixon, Douglas, HJ, von Hippel • 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 H. Johansson, Frascati 2013

  20. 6-Loop Planar D=5 SYM The Parking Spot Escalation our secret collaborator…

  21. 6-Loop D=5 SYM divergence Bern, Carrasco, Dixon, Douglas, HJ, von Hippel • 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.

  22. 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 ?

  23. 5pt N=8 SUGRA calculations H. Johansson, Frascati 2013

  24. 2 2 3 3 1 1 4 4 C-K amplitudes at 1loop Duality-satisfying loop amplitudes: Green, Schwarz, Brink (1982) N=4 SYM: All-plus QCD: N=4 SYM and All-plus QCD: 1106.4711 [hep-th] Carrasco, HJ H. Johansson, Frascati 2013

  25. 1-loop 5-pts UV divergences Carrasco, HJ 1106.4711 [hep-th] SYM UV div in D=8: SG UV div in D=8: SU(8) violating SG UV div in D=8: counterterms:

  26. 2-loop 5-pts N =4 SYM and N =8 SG Carrasco, HJ 1106.4711 [hep-th] The 2-loop 5-point amplitude with Color-Kin. duality N = 8 SG obtained from numerator double copies H. Johansson, Frascati 2013

  27. 2-loop 5-pts UV divergences Carrasco, HJ 1106.4711 [hep-th] SYM UV div in D=7: SG UV div in D=7: SU(8) violating SG UV div in D=8:

  28. 3-loop 5-point SYM and N =8 SG Non-planar D-dimensional amplitude with manifest color-kinematics duality Carrasco, HJ (to appear) (“ladder-like” diagrams) (N = 8 SG obtained from squaring the numerators) H. Johansson, Frascati 2013

  29. 3-loop 5-point SYM and N =8 SG some “Mercedes-like” diagrams… Carrasco, HJ (to appear) H. Johansson, Frascati 2013

  30. 3-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 2013

  31. 3-loop 5-pts UV divergences H. Johansson, Frascati 2013

  32. N=8 SG 5-loop Status Bern, Carrasco, HJ, Roiban (in progress) Working on reorganizing 5-loop N=4 SYM • 416 + 336 = 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

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

  34. 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…

  35. THANK YOU!

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