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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.

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Perturbative Ultraviolet Calculations in Supergravity

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  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

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