1 / 44

Perturbative Ultraviolet Calculations in Supergravity

Perturbative Ultraviolet Calculations in Supergravity. Tristan Dennen (NBIA) Based on work with: Bjerrum -Bohr, Monteiro , O’Connell Bern, Davies, Huang, A. V. Smirnov, V. A. Smirnov. Part I: Ultraviolet divergences Via double copy . UV Divergences in Supergravity.

keefe
Télécharger la présentation

Perturbative Ultraviolet Calculations in Supergravity

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Perturbative Ultraviolet Calculations in Supergravity Frascati Tristan Dennen (NBIA) Based on work with: Bjerrum-Bohr, Monteiro, O’Connell Bern, Davies, Huang, A. V. Smirnov, V. A. Smirnov

  2. Frascati Part I: Ultraviolet divergences Via double copy

  3. UV Divergences in Supergravity • Naively, two derivative coupling in gravity makes it badly ultraviolet divergent • Non-renormalizable by power counting • But: extra symmetry enforces extra cancellations • Supersymmetry to the rescue? • Added benefit: makes calculations simpler Frascati

  4. UV Divergences in Supergravity • Naturally, the theory with the most symmetry is the best bet for ultraviolet finiteness • N = 8 supergravity • Half-maximal supergravity has also attracted attention recently • N = 4 supergravity • This is the primary focus of my talk Cremmer, Julia (1978) Das (1977); Cremmer, Scherk, Ferrara (1978) Frascati

  5. Arguments about Finiteness • 1970’s-1980’s: Supersymmetry delays UV divergences until three loops in all 4D pure supergravity theories • Expected counterterm is R4 • In N=8, SUSY and duality symmetry rule out couterterms until 7 loops • Expected counterterm is D8R4 • See talk from Johansson • I will discuss N = 4 supergravity in this talk • See also talks from Bossard, Carrasco, Vanhove Grisaru; Tomboulis; Deser, Kay, Stelle; Ferrara, Zumino; Green, Schwarz, Brink; Howe, Stelle; Marcus, Sagnotti; etc. Bern, Dixon, Dunbar; Perelstein, Rozowsky (1998); Howe and Stelle (2003, 2009); Grisaru and Siegel (1982); Howe, Stelle and Bossard (2009); Vanhove; Bjornsson, Green (2010); Kiermaier, Elvang, Freedman (2010); Ramond, Kallosh (2010); Beisert et al (2010); Kallosh; Howe and Lindström (1981); Green, Russo, Vanhove (2006) Bern, Carrasco, Dixon, Johansson, Roiban (2010) Beisert, Elvang, Freedman, Kiermaier, Morales, Stieberger (2010) Frascati

  6. Duality Symmetries • Analogs of E7(7) for lower supersymmetry • Can help UV divergences in these theories • Still have candidate counterterms at L = N - 1(1/N BPS) • Nice analysis for N = 8 counterterms N=8: E7(7) N=6: SO(12) N=5: SU(5,1) N=4:SU(4) x SU(1,1) Frascati Bossard, Howe, Stelle, Vanhove (2010) Beisert, Elvang, Freedman, Kiermaier, Morales, Stieberger (2010)

  7. Recent Field Theory Calculations • N=8 Supergravity • Fantastic progress • See Johansson’s talk • N=4Supergravity • Four points, L = 3 • Unexpected cancellation of R4counterterm • Counterterm appears valid under all known symmetries See Bossard’s talk for latest developments • Four points, L = 2, D = 5 • Valid non-BPS countertermD2R4 does not appear Bern, Carrasco, Dixon, Johansson, Kosower, Roiban (2007) Bern, Carrasco, Dixon, Johansson, Roiban(2009) Carrasco, Johansson (2011) Bern, Davies, Dennen, Huang Frascati

  8. Recent Field Theory Calculations • But how are the calculations done? • Find a representation of SYM that satisfies color-kinematics duality. • Construct the integrand for a gravity amplitude using the double copy method. • Extract the ultraviolet divergences from the integrals. Frascati

  9. Color-Kinematics Duality • Color-kinematics duality provides a construction of gravity amplitudes from knowledge of Yang-Mills amplitudes • In general, Yang-Mills amplitudes can be written as a sum over trivalent graphs • Color factors • Kinematic factors • Duality rearranges the amplitude so color and kinematics satisfy the same identities (Jacobi) Bern, Carrasco, Johansson (2008) Frascati

  10. Example: Four Gluons • Four Feynman diagrams • Color factors based on a Lie algebra • Color factors satisfy Jacobi identity: • Numerator factors satisfy similar identity: • Color and kinematics satisfy the same identity! Frascati

  11. Five gluons and more • At higher multiplicity, rearrangement is nontrivial • But still possible • Claim: We can always find a rearrangement so color and kinematics satisfy the same Jacobi constraint equations. Frascati

  12. Recent Field Theory Calculations • How are the calculations done? • Find a representation of SYM that satisfies color-kinematics duality. • Construct the integrand for a gravity amplitude using the double copy method. • Extract the ultraviolet divergences from the integrals. Frascati

  13. Gravity from Double Copy • Once numerators are in color-dual form,“square” to construct a gravity amplitude • Gravity numerators are a double copy of gauge theory ones! • Proved using BCFW on-shell recursion • The two copies of gauge theory don’t have to be the same theory. Bern, Carrasco, Johansson (2008) Bern, Dennen, Huang, Kiermaier (2010) Frascati

  14. Gravity from Double Copy • The two copies of gauge theory don’t have to be the same theory. • Supergravity states are a tensor product of Yang-Mills states • And more! • Relatively compact expressions for gravity amplitudes • Inherited from Yang-Mills simplicity • In a number of interesting theories N = 8 sugra: (N = 4 SYM) x (N= 4 SYM) N = 6 sugra: (N = 4 SYM) x (N = 2 SYM) N= 4 sugra: (N = 4 SYM) x (N = 0 SYM) N = 0 sugra: (N = 0 SYM) x (N = 0 SYM) Frascati Damgaard, Huang, Sondergaard, Zhang (2012) Carrasco, Chiodaroli, Gunaydin, Roiban (2013)

  15. Loop Level • What we really want is multiloop gravity amplitudes • Color-kinematics duality at loop level • Consistent loop labeling between three diagrams • Non-trivial to find duality-satisfying sets of numerators • Double copy gives gravity Just replace c with n Frascati Bern, Carrasco, Johansson (2010)

  16. Recent Field Theory Calculations • How are the calculations done? • Find a representation of SYM that satisfies color-kinematics duality. • Construct the integrand for a gravity amplitude using the double copy method. • Extract the ultraviolet divergences from the integrals. Frascati

  17. Frascati Part ii: obtaining color-dual numerators

  18. Known Color-Dual Numerators Bern, Carrasco, Johansson (2010) Carrasco, Johansson (2011) Bern, Carrasco, Dixon, Johansson, Roiban (2012) Yuan (2012) Bjerrum-Bohr, Dennen, Monteiro, O’Connell (2013) Frascati Boels, Isermann, Monteiro, O’Connell (2013) Bern, Davies, Dennen, Huang, Nohle (2013)

  19. Numerators by Ansatz • Strategy: • Write down an ansatz for a master numerator • All possible terms • Subject to power counting assumptions • Symmetries of the graph • Take unitarity cuts of the ansatz and match against the known amplitude • Gives a set of constraints • Very powerful, but relies on having a good ansatz • Not always possible to write down all possible terms Frascati

  20. One-Loop Construction • Jacobi identity moves one leg past another • Strategy: Keep moving leg 1 around the loop • Obtain a finite difference equation for pentagon Yuan (2012) Bjerrum-Bohr, Dennen, Monteiro, O’Connell (2013) Frascati Depends on the leg we cycle around

  21. One-Loop Construction • Assume we know the box numerators (more on this in a minute) • They are independent of loop momentum • Solution to difference equation is linear in loop momentum • Get projections of linear term • 4 different projections give us the entire linear term in terms of box numerators Frascati

  22. One-Loop Construction • More generally: given all scalar parts, we can rebuild everything else • Two options for scalar parts: • Match unitarity cuts • Guess boxes (and hexagons, etc.),and use reflection identities • Boxes from self-dual YM • Good for MHV amplitudes Frascati

  23. Integrand Reduction • Reduce integrals to pentagons • Partial fraction propagators • Purely algebraic • Doesn’t care about loop momentum in numerators • Key assumption: loop momentum drops out after reducing the amplitude to pentagons • Then reduce pentagons to boxes and match unitarity cuts • Obtain equations like: Frascati

  24. Integrand Reduction • All numerators expressed in terms of hexagon, via Jacobi • This is a linear equation for hexagon numerators • Do the same for all cuts and all permutations • Get 719 equations for 6!=720 unknown numerators • The final unknown is fixed by the condition that loop momentum drops out Frascati Box cut coefficient Factors from integrand reduction

  25. Higher Multiplicity • Seven point amplitudes work the same way • 7! = 5040 heptagon scalar numerators • 5019 equations • 19 extra constraints from our assumptions • 2 degrees of freedom in color-dual form • Works for any R sector • Reconstruct the rest of the numerators • Conjecture: This procedure will work for arbitrary multiplicity at 1 loop Frascati

  26. Prospects for Higher Loops • Some of this might extend to higher loops • Exploit global properties of the system of Jacobi identities • Express loop-dependence in terms of scalar parts of master numerators • Two loops looks promising • Integrand reduction might not extend so easily • Hybrid approach? Use ansatz techniques for scalar parts, then reconstruct loop dependence, match to cuts Frascati

  27. Recent Field Theory Calculations • How are the calculations done? • Find a representation of SYM that satisfies color-kinematics duality. • Construct the integrand for a gravity amplitude using the double copy method. • Extract the ultraviolet divergences from the integrals. Frascati

  28. Frascati Part III: Extracting Ultraviolet Divergences

  29. Three Loop Construction Bern, Davies, Dennen, Huang (2012) • N = 4 SYM copy • Use BCJ representation • Pure YM copy • Use Feynman diagramsin Feynman gauge • Only one copy needs to satisfy the duality • Double copy gives N = 4supergravity • Power counting giveslinear divergence • Valid counterterm under all known symmetries Frascati

  30. N = 4 SYM Copy • Numerators satisfy BCJ duality • Factor of pulls out of every graph • Graphs with triangle subdiagrams have vanishing numerators Bern, Carrasco, Johansson (2010) Frascati

  31. Pure YM Copy • Pros and cons of Feynman diagrams 😃 Straightforward to write down 😃 Analysis is relatively easy to pipeline 😃 D-dimensional 😡 Lots of diagrams 😡 Time and memory constraints • Many of the N=4 SYM BCJ numerators vanish • If one numerator vanishes, the other is irrelevant • Power counting for divergences – can throw away most terms very quickly Frascati

  32. Ultraviolet Analysis • To extract ultraviolet divergences from integrals: • Series expand the integrand and select the logarithmic terms • Reduce all the tensors in the integrand • Regulate infrared divergences (uniform mass) • Subtract subdivergences • Evaluate vacuum integrals Frascati

  33. 1. Series Expansion • Counterterms are polynomial in external kinematics • Count up the degree of the polynomial using dimension operator • Derivatives reduce the dimension of the integral by at least 1. • Apply again to reduce further… all the way down to logarithmic. • Now can drop dependence on external momenta. Frascati

  34. 1. Series Expansion • Simpler: equivalent to series expansion of the integrand • Terms less than logarithmic have no divergence • Terms more than logarithmic vanish as the IR regulator is set to zero • Left with logarithmically divergent terms Frascati

  35. 2. Tensor Reduction • The tensor integral knows nothing about external vectors • Must be proportional to metric tensor • Contract both sides with metric to get • Generalizes to arbitrary rank – Need rank 8 for 3 and 4 loops Frascati

  36. 3. Infrared Regulator • Integrals have infrared divergences (in 4 dimensions) • One strategy: Use dimensional regulator for both IR and UV • Subdivergences will cancel automatically, becausethere are no 1- or 2-loop divergences in this theory • But, integrals will generally start at • Very difficult to do analytically • Another strategy: Uniform mass regulator for IR • Integrals will start at -- much easier! • Regulator dependence only enters through subleading terms • Now we have a sensible integral! Marcus, Sagnotti (1984) Vladimirov Frascati

  37. 4. Subdivergences • What about higher loops? • At three loops, the integrals have logarithmic subdivergences • Integrals are • Mass regulator can enter subleading terms! • Recursively remove all contributions from divergent subintegrals • Regulator dependence drops out Marcus, Sagnotti (1984) Vladimirov Frascati Regulator dependent Reparametrizesubintegral Regulator independent

  38. 5. Vacuum Integrals • Get about 600 vacuum integrals containing UV information • Evaluation: • MB: Mellin Barnes integration • FIESTA: Sector decomposition • FIRE: Integral reduction using integration by parts identities cancelled propagator doubled propagator Czakon Frascati A.V. Smirnov & Tentyukov A.V. Smirnov

  39. Three Loop Result • Series expand the integrand and select the logarithmic terms • Reduce all the tensors in the integrand • Regulate infrared divergences • Subtract subdivergences • Evaluate vacuum integrals • The sum of all 12 graphs is finite! Frascati

  40. Opinions on the Result • If R4counterterm is allowed by supersymmetry, why is it not present? • R4nonrenormalization from heterotic string • Violates Noether-Gaillard-Zumino current conservation • Hidden superconformalsymmetry • Existence of off-shell superspace formalism • Different proposals lead to different expectations for four loops Tourkine, Vanhove (2012) Kallosh (2012) Frascati Kallosh, Ferrara, Van Proeyen (2012) Bossard, Howe, Stelle (2012)

  41. Four Loop Setup • Allowed countertermD2R4, non-BPS • Will it diverge? • If yes: first example of a divergence in any pure supergravity theory in 4D • If no: hidden symmetry enforcing cancellations? • Same approach as three loops. • N = 4 SYM numerators: 82 nonvanishing (comp. 12) • Pure YM numerators: ~30000 Feynman diagrams (comp. ~1000) • Integrals are generally quadratically divergent • Requires a deeper series expansion Bern, Davies, Dennen, A. V. Smirnov, V. A. Smirnov (in progress) Frascati Bern, Carrasco, Dixon, Johansson, Roiban (2012)

  42. Vast Simplifications? • Four loops is significantly more complicated than three loops. • (unsurprisingly) • Can we avoid subtracting subdivergences? • Ultimately, there are no subdivergences in the theory so far. • Any subdivergences that appear must be a result of our IR regulator. • Observation: in all cases we’ve looked at, subdivergences cancel manifestly with a uniform mass regulator. • 2 loops: D = 4,5,6 • 3 loops: D = 4 • Consistent with four-loop QCD beta function calculations • Analysis of subintegrals for 4 loops suggests the same should be true here. Frascati

  43. Four Loop Status • Series expand the integrand and select the logarithmic terms • Reduce all the tensors in the integrand • Regulate infrared divergences • Subtract subdivergences • Evaluate vacuum integrals • Calculation nearly done • Overall cancellation of and • Now working on showing the necessary cancellation of • Stay tuned! Czakon (2004) Frascati

  44. Summary • Expectations for ultraviolet divergences in supergravity • BCJ color-kinematics duality • Construction of gravity integrand via double copy • How to obtain color-dual numerators • Progress with one-loop construction • UV analysis of gravity integrals • 3 loop, N = 4 supergravity • Status of four loop Frascati

More Related