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 25 March Frascati

PART I: ULTRAVIOLET DIVERGENCES VIA DOUBLE COPY 25 March Frascati

 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 UV Divergences in Supergravity 25 March Frascati

 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 UV Divergences in Supergravity Cremmer, Julia (1978) Das (1977); Cremmer, Scherk, Ferrara (1978) 25 March Frascati

 1970’s-1980’s: Supersymmetry delays UV divergences until three loops in all 4D pure supergravity theories  Expected counterterm is R 4  In N=8, SUSY and duality symmetry rule out couterterms until 7 loops  Expected counterterm is D 8 R 4  See talk from Johansson  I will discuss N = 4 supergravity in this talk  See also talks from Bossard, Carrasco, Vanhove Arguments about Finiteness 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) 25 March Frascati

 Analogs of E 7(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 Duality Symmetries N=8: E 7(7) N=6: SO(12) N=5: SU(5,1) N=4: SU(4) x SU(1,1) Bossard, Howe, Stelle, Vanhove (2010) Beisert, Elvang, Freedman, Kiermaier, Morales, Stieberger (2010) 25 March Frascati

 N=8 Supergravity  Fantastic progress  See Johansson’s talk  N=4 Supergravity  Four points, L = 3  Unexpected cancellation of R 4 counterterm  Counterterm appears valid under all known symmetries See Bossard’s talk for latest developments  Four points, L = 2, D = 5  Valid non-BPS counterterm D 2 R 4 does not appear Recent Field Theory Calculations Bern, Carrasco, Dixon, Johansson, Kosower, Roiban (2007) Bern, Carrasco, Dixon, Johansson, Roiban (2009) Carrasco, Johansson (2011) Bern, Davies, Dennen, Huang 25 March Frascati

 But how are the calculations done? 1.Find a representation of SYM that satisfies color-kinematics duality. 2.Construct the integrand for a gravity amplitude using the double copy method. 3.Extract the ultraviolet divergences from the integrals. Recent Field Theory Calculations 25 March Frascati

 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) Color-Kinematics Duality Bern, Carrasco, Johansson (2008) 25 March Frascati

 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! Example: Four Gluons 25 March Frascati

 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. Five gluons and more 25 March Frascati

 How are the calculations done? 1.Find a representation of SYM that satisfies color-kinematics duality. 2.Construct the integrand for a gravity amplitude using the double copy method. 3.Extract the ultraviolet divergences from the integrals. Recent Field Theory Calculations 25 March Frascati

 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. Gravity from Double Copy Bern, Carrasco, Johansson (2008) Bern, Dennen, Huang, Kiermaier (2010) 25 March Frascati

 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 Gravity from Double Copy 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) Damgaard, Huang, Sondergaard, Zhang (2012) Carrasco, Chiodaroli, Gunaydin, Roiban (2013) 25 March Frascati

 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 Loop Level Bern, Carrasco, Johansson (2010) Just replace c with n 25 March Frascati

 How are the calculations done? 1.Find a representation of SYM that satisfies color-kinematics duality. 2.Construct the integrand for a gravity amplitude using the double copy method. 3.Extract the ultraviolet divergences from the integrals. Recent Field Theory Calculations 25 March Frascati

PART II: OBTAINING COLOR-DUAL NUMERATORS 25 March Frascati

Known Color-Dual Numerators N = 4 SYM1 Loop2 Loops3 Loops4 LoopsL Loops 4 pointtrivial ansatz 5 pointconstructionansatz 6 pointconstruction 7 pointconstruction n pointconstruction Pure YM1 Loop2 LoopsL Loops 4 pointansatz(All-plus) n point(All-plus and single minus) Boels, Isermann, Monteiro, O’Connell (2013) Bern, Davies, Dennen, Huang, Nohle (2013) Bern, Carrasco, Johansson (2010) Carrasco, Johansson (2011) Bern, Carrasco, Dixon, Johansson, Roiban (2012) Yuan (2012) Bjerrum-Bohr, Dennen, Monteiro, O’Connell (2013) 25 March Frascati

 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 Numerators by Ansatz 25 March Frascati

 Jacobi identity moves one leg past another  Strategy: Keep moving leg 1 around the loop  Obtain a finite difference equation for pentagon One-Loop Construction Depends on the leg we cycle around Yuan (2012) Bjerrum-Bohr, Dennen, Monteiro, O’Connell (2013) 25 March Frascati

 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 One-Loop Construction 25 March Frascati

 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 One-Loop Construction 25 March Frascati

 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: Integrand Reduction 25 March Frascati

 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 Integrand Reduction Box cut coefficientFactors from integrand reduction 25 March Frascati

 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 Higher Multiplicity 25 March Frascati

 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 Prospects for Higher Loops 25 March Frascati

 How are the calculations done? 1.Find a representation of SYM that satisfies color-kinematics duality. 2.Construct the integrand for a gravity amplitude using the double copy method. 3.Extract the ultraviolet divergences from the integrals. Recent Field Theory Calculations 25 March Frascati

PART III: EXTRACTING ULTRAVIOLET DIVERGENCES 25 March Frascati

Bern, Davies, Dennen, Huang (2012)  N = 4 SYM copy  Use BCJ representation  Pure YM copy  Use Feynman diagrams in Feynman gauge  Only one copy needs to satisfy the duality  Double copy gives N = 4 supergravity  Power counting gives linear divergence  Valid counterterm under all known symmetries Three Loop Construction 25 March Frascati

 Numerators satisfy BCJ duality  Factor of pulls out of every graph  Graphs with triangle subdiagrams have vanishing numerators N = 4 SYM Copy Bern, Carrasco, Johansson (2010) 25 March Frascati

 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 Pure YM Copy 25 March Frascati

 To extract ultraviolet divergences from integrals: 1.Series expand the integrand and select the logarithmic terms 2.Reduce all the tensors in the integrand 3.Regulate infrared divergences (uniform mass) 4.Subtract subdivergences 5.Evaluate vacuum integrals Ultraviolet Analysis 25 March Frascati

 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. 1. Series Expansion 25 March Frascati

 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 1. Series Expansion 25 March Frascati

 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 2. Tensor Reduction 25 March Frascati

 Integrals have infrared divergences (in 4 dimensions)  One strategy: Use dimensional regulator for both IR and UV  Subdivergences will cancel automatically, because there 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! 3. Infrared Regulator Marcus, Sagnotti (1984) Vladimirov 25 March Frascati

 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 4. Subdivergences Marcus, Sagnotti (1984) Vladimirov Regulator dependent Regulator independent Reparametrize subintegral 25 March Frascati

 Get about 600 vacuum integrals containing UV information  Evaluation:  MB: Mellin Barnes integration  FIESTA: Sector decomposition  FIRE: Integral reduction using integration by parts identities 5. Vacuum Integrals cancelled propagator doubled propagator Czakon A.V. Smirnov & Tentyukov A.V. Smirnov 25 March Frascati

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! Three Loop Result 25 March Frascati

 If R 4 counterterm is allowed by supersymmetry, why is it not present?  R 4 nonrenormalization from heterotic string  Violates Noether-Gaillard-Zumino current conservation  Hidden superconformal symmetry  Existence of off-shell superspace formalism  Different proposals lead to different expectations for four loops Opinions on the Result Tourkine, Vanhove (2012) Kallosh (2012) Bossard, Howe, Stelle (2012) Kallosh, Ferrara, Van Proeyen (2012) 25 March Frascati

 Allowed counterterm D 2 R 4, 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 Four Loop Setup Bern, Davies, Dennen, A. V. Smirnov, V. A. Smirnov (in progress) Bern, Carrasco, Dixon, Johansson, Roiban (2012) 25 March Frascati

 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. Vast Simplifications? 25 March Frascati

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! Four Loop Status Czakon (2004) 25 March Frascati

 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 Summary 25 March Frascati