Towards Determining the UV Behavior of Maximal Supergravity

Transcription

1 Towards Determining the UV Behavior of Maximal Supergravity Henrik Johansson CERN Aug 9, 0 SUSY 0, ICTP Trieste!! Based on work with: Zvi Bern, John Joseph Carrasco, Lance Dixon, Radu Roiban

2 Text-book: perturbative gravity is complicated! de Donder gauge: = = After symmetrization ~ 00 terms! higher order vertices ~0 terms H. Johansson SUSY 0

3 On-shell simplifications Graviton plane wave: On-shell -graviton vertex: Yang-Mills polarization = Gravity scattering amplitude: Yang-Mills vertex Yang-Mills amplitude M GR tree(,,, ) = st u AYM tree(,,, ) A YM tree(,,, ) Gravity processes = squares of gauge theory ones - entire S-matrix Bern, Carrasco, HJ [BCJ] H. Johansson SUSY 0

4 Gravity should be simple off shell Yang-Mills cubic very schematically: Einstein gravity cubic very schematically: And gravity should be a double copy of a YM theory: Bern, Carrasco, HJ H. Johansson SUSY 0

5 UV problem = basic power counting Naively expect gravity to behave worse than Yang-Mills Gravity: non-renormalizable dimensionful coupling Z d L p...(applepµ p )... p p p...p n Yang-Mills: renormalizable dimensionless coupling Z d L p...(gpµ )... p p p...p n For finite gravity à vast cancellations needed seems implausible, but exists for N= SG in all known ampl s. H. Johansson SUSY 0 (p µ ) L! (k µ ) L external momenta

6 Outline! UV status of N= SUGRA! Duality between color and kinematics! Double-copy structure of gravity! Ability to calculate! Amplitude UV behavior from duality! Current -loop SUGRA progress! Conclusion H. Johansson SUSY 0

7 Facts: SUGRA status on one page! No D= divergence of pure SG has been found to date.! Susy forbids, loop div. R, R Ferrara, Zumino, Deser, Kay, Stelle, Howe, Lindström, Green, Schwarz, Brink, Marcus, Sagnotti! Pure gravity -loop finite, -loop divergent Goroff & Sagnotti! With matter: -loop divergent t Hooft & Veltman! Naively susy allows -loop div. R! N= SG and N= SG -loop finite! Bern, Carrasco, Dixon, HJ, Kosower, Roiban, Davies, Dennen, Huang! N= SG: no divergence before loops! D> divergences obey: Marcus, Sagnotti, Bern, Dixon, Dunbar, Perelstein, Rozowsky, Carrasco, HJ, Kosower, Roiban! -loop div. in D= implies a -loop div. in D=/ -- calculation in progress! N= SG UFinite? H. Johansson SUSY 0

8 Why is it interesting?! If N= 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! Gravity is a double copy of gauge theories! Color-Kinematics: kinematics = Lie algebra Bern, Carrasco, HJ! Constraints from E-M duality? Kallosh,.! Hidden superconformal N= SUGRA? Ferrara, Kallosh, Van Proeyen! Extended N= superspace? Bossard, Howe, Stelle H. Johansson SUSY 0 Symmetry?

9 Known UV divergences in D> Plot of critical dimensions of N = SUGRA and N = SYM Finite Divergent? Known bound for N = Bern, Dixon, Dunbar, Rozowsky, Perelstein; Howe, Stelle current trend for N = If N = div. at L= H. Johansson SUSY 0

10 Known UV divergences in D> Plot of critical dimensions of N = SUGRA and N = SYM calculations: - loops: Green, Schwarz, Brink; Marcus and Sagnotti - loops: Bern, Carrasco, Dixon, HJ, Kosower, Roiban loops: Bern, Carrasco, Dixon, Douglas, HJ, von Hippel Finite Divergent? Known bound for N = Bern, Dixon, Dunbar, Rozowsky, Perelstein; Howe, Stelle current trend for N = If N = div. at L= L = lowest loop order for possible D = divergence Beisert, Elvang, Freedman, Kiermaier, Morales, Stieberger; Björnsson, Green, Bossard, Howe, Stelle, Vanhove Kallosh, Ramond, Lindström, Berkovits, Grisaru, Siegel, Russo, Cederwall, Karlsson, H. and Johansson more. SUSY 0

11 Known UV divergences in D> Plot of critical dimensions of N = SUGRA and N = SYM calculations: - loops: Green, Schwarz, Brink; Marcus and Sagnotti - loops: Bern, Carrasco, Dixon, HJ, Kosower, Roiban loops: Bern, Carrasco, Dixon, Douglas, HJ, von Hippel / or /? Divergent Finite? Known bound for N = Bern, Dixon, Dunbar, Rozowsky, Perelstein; Howe, Stelle current trend for N = If N = div. at L= L = lowest loop order for possible D = divergence Beisert, Elvang, Freedman, Kiermaier, Morales, Stieberger; Björnsson, Green, Bossard, Howe, Stelle, Vanhove Kallosh, Ramond, Lindström, Berkovits, Grisaru, Siegel, Russo, Cederwall, Karlsson, H. and Johansson more. SUSY 0

12 Historical record where is the N = div.? loops Conventional superspace power counting Green, Schwarz, Brink (9) Howe and Stelle (99) Marcus and Sagnotti (9) loops Partial analysis of unitarity cuts; If N = harmonic superspace exists; algebraic renormalisation! Bern, Dixon, Dunbar, Perelstein, Rozowsky (99) Howe and Stelle (00,009) loops If N = harmonic superspace exists Howe and Stelle (00) loops If N = harmonic superspace exists; string theory U-duality analysis; lightcone gauge locality arguments; E () analysis, unique / BPS candidate loops Explicit identification of potential susy invariant counterterm with full non-linear susy 9 loops Assume Berkovits superstring non-renormalization theorems can be carried over to N = supergravity Finite Identified cancellations in multiloop amplitudes; lightcone gauge locality and E (), inherited from hidden N= SC gravity Grisaru and Siegel (9); Green, Russo, Vanhove; Kallosh; Beisert, Elvang, Freedman, Kiermaier, Morales, Stieberger; Bossard, Howe, Stelle, Vanhove Howe and Lindström; Kallosh (9) Green, Russo, Vanhove (00) Bern, Dixon, Roiban (00), Kallosh (009 ), Ferrara, Kallosh, Van Proeyen (0) note: above arguments/proofs/speculation are only lower bounds à only an explicit calculation can prove the existence of a divergence!

13 Color-Kinematics Duality H. Johansson SUSY 0

14 Color-Kinematics Duality Yang-Mills theories are controlled by a kinematic Lie algebra Amplitude represented by cubic graphs: numerators color factors propagators Color & kinematic numerators satisfy same relations: Jacobi identity f bac = f abc Duality: color kinematics Bern, Carrasco, HJ antisymmetry H. Johansson SUSY 0

15 Some details of color-kinematics duality Bern, Carrasco, HJ can be checked for pt on-shell ampl. using Feynman rules Example with two quarks:. (A µ ) contact interactions absorbed into cubic graphs by hand =s/s or by auxiliary field B (A µ ). Beyond -pts duality not automatic è Lagrangian reorganization. Known to work at tree level: all-n example Kiermaier; Bjerrum-Bohr et al.. Enforces (BCJ) relations on partial amplitudes è (n-)! basis. Same/similar relations control string theory S-matrix Bjerrum-Bohr, Damgaard, Vanhove; Stieberger H. Johansson SUSY 0

16 Gravity is a double copy Gravity amplitudes obtained by replacing color with kinematics BCJ The two numerators can belong to different theories: (N =) (N =) N = sugra (N =) (N =) N = sugra (N =) (N =0) N = sugra (N =) (matter) N = matter (N =0) (N =0) Einstein gravity + axion+ dilaton similar to Kawai- Lewellen-Tye but works at loop level

17 UV calculations using C-K duality H. Johansson SUSY 0

18 C-K duality unifies,-loop UV behavior, loops N=,,, SG are particularly easy to understand M SG = X i n i (SYM Integral) i Bern, Boucher-Veronneau, Dixon, HJ no loop momenta N= SYM numerator: 0 0 N=0,,, SYM integral: finite div. div. β-fn vanish gauge theory renormalizablity But N= matter diverges: Fischler; Fradkin, Tseytlin div Bern, Davies, Dennen, Huang 09. [hep-th]

19 -loop N = SG & N = SYM Color-kinematics dual form: Bern, Carrasco, HJ UV divergent in D=: Bern, Carrasco, Dixon, HJ, Roiban A () pole =g sta tree (N c V (A) + N c (V (A) +V (B) )) (utr[t a T a T a T a ] + perms) apple (stu) M () = 0 M tree (V (A) +V (B) ) pole V (A) +V (B) = H. Johansson SUSY 0

20 -loops: integral types () () () () (0) () (9) () () () (0) () (9) () () () () () () (0) () (9) () () () () () () (0) () (9) () () () () () () () (9) () () () () (9) () () () () (0) () () () () () () () () () () () () () () () (9) () () () () () () () (0) () (9) () () () () () () () (9) () ( () () (9 () () () () ( () () () (9) () () () () (0) () () () () () () () () () () () () () () (0) () () () () () (9) () () () () () () () (0) () (9) () () () () () () () (9) () () (0) (9) () () () () () (9) () () () (0) () (9) () () () () () () () (9) () () () (0) () (9) () () () () () () () (9) () () (0) (9) () () () () () (9) () () () () (9) () () () () (0) () () () ( () () ( () () ( () (0) ( () () () () ( () (9) () () () () (0) () () () () () () (9) () () () () (0) () () () () () () () () () (0) () () () () () () (9) () () () () (0) () () () () () () () () () (9) () () () () (0) () () () () () ()

21 -loops N = SYM and N = SG Bern, Carrasco, Dixon, HJ, Roiban 0. diagrams Power counting manifest N = & N = diverge in D=/ up to overall factor, divergence same as for N= SYM part H. Johansson SUSY 0

22 -loop status H. Johansson SUSY 0

23 N = SYM -loop Amplitude N= SYM important stepping stone to N= SG 0. [hep-th] Bern, Carrasco, HJ, Roiban integral topologies: () (0) (0) (0) Used maximal cut method Bern, Carrasco, HJ, Kosower MC NMC N MC N MC UV divergence in D=/: ( A () = div g sta tree N c ( N c V (a) Tr[T a T a T a T a ] H. Johansson SUSY 0 ) +(V (a) +V (b) + V (c) )

24 UV calc. makes it to Hollywood The Parking Spot Escalation with some help from Sheldon Cooper

25 N = SG -loop Status Construction using only unitarity difficult Works for -loop N= SYM -loop SG too difficult this way (ansatz: billions of terms) Only way: use color-kinematics duality In principle: need only reorganize -loop N= SYM n SYM 000 terms n SG (000) / terms Bern, Carrasco, HJ, Roiban + = integral topologies Minimal ansatz: free parameters 00 functional Jacobi eqns linear eqns Solution exists: 9 free parameters Unfortunately, not all unitarity cuts work, there s some glitch Working with enlarged Ansätze stay tuned for results! H. Johansson SUSY 0

26 UV div. ubiquitous Casimir loops, D=/: SYM $ SG? V (a) +V (b) + V (c) loops, D=/: SYM SG $ V +V + V loops, D=: SYM SG $ V (A) +V (B) = V (A) +V (B) + V (B) V (A) V (B) Pattern à reason to be optimistic About UV behavior at loops! H. Johansson SUSY 0 (d abcd ) related to diagrams in the quartic Casimir

27 Summary! Explicit calculations in N = SUGRA up to four loops show that the power counting exactly follows that of N = SYM -- a finite theory! loop calculation in D=/ probes the potential -loop D= counterterm -- will provide critical input to the N = question!! Color-Kinematics duality allows for gravity calculations for multiloop multipoint amplitudes -- greatly facilitating UV analysis in gravity.! Numbers in UV divergences of N = SUGRA and N = SYM coincide, suggesting a deeper connection between the theories! Stay tuned for the -loop SUGRA result H. Johansson SUSY 0

28