5-loops in supergravity.

Transcription

1 5-loops in supergravity. I P h T QCD meets Gravity 2017 s a c l a y Supported by ERC-STG preqft: Strategic Predictions for Quantum Field Theories

2 Gentle talk introducing: Unitarity methods (Generalized) Double Copy X 2 2

3 The scattering amplitudes of many relativistic theories admit a: Double-copy Numerator Algebra This points to previously hidden structure in many theories. Structure yet to be generally understood at the level of the action.

4 N=50 N=60 2 N = 60 Planck TT+lowP Planck TT+lowP+BKP +lensing+ext r0.002 Convex Concave 0.10 = = 25 = n s Many theories amplitudes are double copy!

5 Key Point: MANY Theories are Double Copies Bi-Adjoint Scalar: (S)YM ( (S)QCD ): (S)Gr ( (S)Einstein-YM ): NLSM / Chiral Lagrangian: (S)Born-Infeld: Open String: Closed String: color color color color Bern de Freitas Wong ( 99); Bern Dennen Huang; Du Feng Fu; Bjerrum-Bohr Damgaard Monteiro O Connell Special Galileon: spin-1 spin-1 spin-1 BCJ ( 08) Bjerrum-Bohr Damgaard Vanhove; Steiberger; Feng et al; Mafra Schlotterer ( 08-11); Johansson Ochirov even-spin-0 spin-1 KLT( 86); BCJ ( 08); Chiodaroli Gunaydin Johansson Roiban; Johansson Ochirov; Johansson Kälin Mogull Chen Du 13 Cachazo He Yuan 14 Cachazo He Yuan 14 Cachazo He Yuan 14 Cheung Shen 16 Broedel Schlotterer Stieberger; Mafra Schlotterer Broedel Schlotterer Stieberger; Cheung Shen 16 Z-theory: Broedel Schlotterer Stieberger; JJMC Mafra Schlotterer 0 spin-1 0 even-spin-0 even-spin-0 even-spin-0 spin-1 0 corrected spin-1 color

6 Key Point: MANY Theories are Double Copies Ingredients: color 0 spin 01/21 For all these theories: Bi-Adjoint Scalar (S)YM ( (S)QCD ) (S)Gr ( (S)Einstein-YM ) NLSM (S)Born-Infeld Special Galileon Z-theory Open String Closed String

7 Key Point: MANY Theories are Double Copies Why do such (generalized) gauge choices exist? How can we exploit the benefits in all circumstances?

8 Consider an Amplitude

9 Bern JJMC Dixon Kosower Johansson Roiban 07 Original solution of three-loop four-point N=4 sym and N=8 sugra

10 Bern JJMC Dixon Kosower Johansson Roiban 07 Original solution of three-loop four-point N=4 sym and N=8 sugra

11 Bern JJMC Dixon Kosower Johansson Roiban 07 Original solution of three-loop four-point N=4 sym and N=8 sugra

12 Bern JJMC Dixon Kosower Johansson Roiban 07 Original solution of three-loop four-point N=4 sym and N=8 sugra

13 Bern JJMC Johansson 10

14 Bern JJMC Johansson 10

15 Bern JJMC Johansson 10

16 Bern JJMC Johansson ( 08 10) Color and Kinematics dance together. Solving Yang-Mills theories means solving Gravity theories.

17 Bern JJMC Johansson ( 08 10) Generic D-dimensional YM theories have a fascinating structure at tree-level A tree m = X Color factors and numerator factors satisfy similar lie algebra properties G2cubic c(g)n(g) = - D(G) Vertex Antisymmetry = + Jacobi Color-Kinematic Duality!

18 Bern JJMC Johansson ( 08 10) Generic D-dimensional YM theories have a fascinating structure at tree-level A tree m = X G2cubic c(g)n(g) D(G) YM s Color-Kinematic duality makes manifest gravitational double copy structure: im tree n = X G2cubic n(g)ñ(g) D(G)

19 a geometric guide to color-kinematics Physics = Geometry (the best polytopes are graphs of graphs!)

20 Convenient language: graphs of graphs JJMC û ˆt c(g) n(g) Color-Kinematics

21 Graphs contributing to an ordered tree (color-stripped) generate the 1-skeleton of Stasheff polytopes joined only by ˆt JJMC 5pt example ˆt Note: same color-order! (these polytopes are also called associahedra)

22 You might think you need (m-2)! of these color-ordered amplitudes to capture everything because this is what is required to touch every vertex at least once: JJMC

23 You might think you need (m-2)! of these color-ordered amplitudes to capture everything because this is what is required to touch every vertex at least once: JJMC In fact such a choice is the KK-basis proven sufficient by Del Duca Dixon and Maltoni

24 But notice because of color-kinematics only (m-2)! nodes are needed to specify both the color factors and numerator factors of everyone JJMC

25 But notice because of color-kinematics only (m-2)! nodes are needed to specify both the color factors and numerator factors of everyone JJMC

26 But notice because of color-kinematics only (m-2)! nodes are needed to specify both the color factors and numerator factors of everyone JJMC This reduces the set of necessary color-ordered amplitudes (associahedra) to (m-3)! : BCJ relations

27 At every multiplicity the masters can be chosen to form the 1-skeleton of a polytope related by û on every internal edge of the relevant scattering graphs JJMC û 321 û û 231 x y z 312 û û û 132 (these polytopes are called permutahedra)

28 Can linearly solve for the (m-2)! numerators of the masters in terms of the (m-3)! BCJ independent color-ordered amplitudes. In fact you get (m-3)! numerators in terms of the color-ordered amplitudes and (m-3)(m-3)! free functions. JJMC (generalized gauge freedom)

29 JJMC Building blocks at 6-points: color-ordered amplitude set of masters associahedron permutahedron

30 105 JJMC cubic graphs at 6 pt

31 JJMC set of masters full amplitude w 6 x y z 5 masters fixed by 6

32 JJMC TREE-LEVEL SUMMARY 1. Gauge invariant building blocks that speak to the theory: color-ordered amplitudes associahedra 2. CK means only need to specify the boundary data: the master graphs given by the relevant permutahedron 3. Can solve for the full amplitude efficiently in terms of the (n-3)! independent associohedra = f( ) (linear) physics < > geometry

33 Can generalize c/k numerators to off-shell multi-loop integrands: By introducing ansatze. BCJ; BCDJR; CJ; Bern Davies Dennen Huang Nohle; Johansson Ochirov; Mogull O Connell; Johanson Kälin Mogull;... Yang (FIRST 5-loop N=8 SG Calc: Form Factor) By introducing massive over-redundancy in graphs: By exploiting BRST invariance of pure-spinor superstrings: JJMC Mafra Schlotterer By recycling forward limits & CHY formalism: He Schlotterer Zhang Can generalize BCJ amp relns at loops: Vanhove Tourkine; Hohenegger Stieberger; He Schlotterer; Boels Isermann Can take CHY tree-rep to loop integrand via ambitwistor string: Adamo Casali Skinner; Geyer Monteiro Mason Tourkine; He Yuan; Baadsgaard Bjerrum-Bohr Bourjaily Damgaard Feng; Geyer Monteiro

34 But let s say you don t want to do any of that. Given a generic (non color-dual) representation for a gauge amplitude and all you want is the related gravity amplitudes. Is there a simple path forward?

35 YES.

36 The idea is natural: take all non-vanishing kinematic-jacobi combinations (the triangles) double-copy them with each other use this information to define off-shell contact graphs in the double-copy theory. = X = 1 6 X i J i1 J 0 i2 + J i2 J 0 i1 d (1) i1 i2 with Bern Chen Johansson Roiban (2017)

37 How does this come together for a full integrand?

38 Unitarity Bern Dixon Dunbar and Kosower ( 94 95) Bern Dixon and Kosower ( 96) Britto Cachazo and Feng ( 04)

39 SPANNING CUTS Ø < leads to notion of a Minimal Spanning Set EASY VERIFICATION

40 EASY VERIFICATION

41 EASY VERIFICATION NATURAL CONSTRUCTION METHOD OF MAXIMAL CUTS On Shell Bern JJMC Kosower Johansson (`07) Ø Ø ( 8 exposed propagators p 2 =0) Ø ÿ ÿ ÿø HFinal Answer no cut conditions!l

42 Ful-loop Example

43 3-loop cubic graphs Graph N = 4 sym numerators. (a)-(d) s 2 (e)-(g) s(p τ 45 ) (h) s(τ 26 + τ 36 ) t(τ 17 + τ 27 )+st (i) s(p τ 45 ) t(p τ 56 + p 2 6) (s t)p 2 6/3 Bern JJMC Chen Johansson Roiban

44 ASSIGN square of 3-loop cubic graphs to N=8 SG Graph (a)-(d) (e)-(g) (h) (i) N = 8 SG cubic numerators. i 2 hs 2 h s( ) h i 2 s(p ) t( )+st h s(p ) t(p p 2 6) (s t)p 2 6/3 i 2 i 2 Bern JJMC Chen Johansson Roiban

45 ASSIGN square of 3-loop cubic graphs to N=8 SG This is just the starting point. Graph (a)-(d) (e)-(g) (h) (i) N = 8 SG cubic numerators. i 2 hs 2 h s( ) h i 2 s(p ) t( )+st h s(p ) t(p p 2 6) (s t)p 2 6/3 i 2 i 2 Bern JJMC Chen Johansson Roiban

46 Those cubic grav dressings automatically satisfies all of these cuts N 0 cut Bern JJMC Chen Johansson Roiban

47 Those cubic grav dressings automatically satisfies all of these cuts too N 1 cut Bern JJMC Chen Johansson Roiban

48 Those cubic grav dressings satisfy most of these cuts! N 2 cut Bern JJMC Chen Johansson Roiban

49 Only 4 nonvanishing cuts N 2 cut Need to add 4 contact contributions Bern JJMC Chen Johansson Roiban

50 N 2 cut Bern JJMC Chen Johansson Roiban

51 N 2 cut Bern JJMC Chen Johansson Roiban

52 N 2 cut 0 l k l k 3 l 5 0 l 1 0 l 5 l l 5 0 l4 l 1 l 2 0 l4 0 l 6 l 5 l Bern JJMC Chen Johansson Roiban

53 N 2 cut 0 l k l 6 0 l 5 0 l l 5 Bern JJMC Chen Johansson Roiban

54 N 2 cut in(1) in(1) + in(1) Bern JJMC Chen Johansson Roiban

55 N 2 cut in(1) in(1) + in(1) in(1) +- in(1) +- in(1) Bern JJMC Chen Johansson Roiban

56 N 2 cut in(1) +- in(1) +- in(1) Bern JJMC Chen Johansson Roiban

57 N 2 cut in(1) +- in(1) +- in(1) ( )( )+ ( )( ) Bern JJMC Chen Johansson Roiban

58 N 2 cut in(1) +- in(1) +- in(1) ( )( )+ ( )( ) Bern JJMC Chen Johansson Roiban

59 N 2 cut 0 l k l k 3 l 5 0 l 1 0 l 5 l l 5 0 l4 l 1 l 2 0 l4 0 l 6 l 5 l Bern JJMC Chen Johansson Roiban

60 N 2 cut k l 3 3 l 0 4 l k l k 3 l 5 0 l 1 0 l 5 l l 5 0 l4 l 1 l 2 0 l4 0 l 6 l 5 l Bern JJMC Chen Johansson Roiban

61 N 2 cut 0 l k k l 3 3 l 5 0 l 1 0 l 5 l l 5 0 l4 k l 3 3 l 1 l 2 0 l4 0 l 5 l Bern JJMC Chen Johansson Roiban

62 N 2 cut 0 l k k l 3 3 l 5 0 l 1 0 l 5 l l 5 0 l4 k l 3 3 l 1 l 2 0 l4 0 l 5 l Bern JJMC Chen Johansson Roiban

63 N 2 cut - 0 l k k l 3 3 l 5 0 l 1 0 l 5 l l 5 0 l4 k l 3 3 l 1 l 2 0 l4 0 l 5 l Bern JJMC Chen Johansson Roiban

64 Need to add 4 contacts.but you just write them down = 1 9 X i J i1 J 0 i2 + J i2j 0 i1 d (1) i1 i2 Bern JJMC Chen Johansson Roiban

65 Graph (a)-(d) (e)-(g) (h) (i) N = 8 SG cubic numerators. i 2 hs 2 h s( ) h i 2 s(p ) t( )+st h s(p ) t(p p 2 6) (s t)p 2 6/3 i 2 i 2 Complete 3-loop integrand for N=8 SG Bern JJMC Chen Johansson Roiban

66 Some more examples

67 Some 5-loop examples

68 5-loops a potential N2 contact This is serious. (5-loops is definitely not a joke)

69 5-loops potential N2 contact Contact / Missing Information you can just write down: X = 8 pages local 5-loops isn t for the faint of heart.

70 5-loops potential N2 contact Note: very non-planar far from the ladder graphs. (cubic graphs)

71 5-loops potential N2 contact sym Numerator

72 5-loops potential N2 contact truth cut = X g2cut n g 2 d g KLT! non-local c/k numerators (truth) 26 pages non-local

73 5-loops potential N2 contact X g2cut n g 2 d g 26 pages (double copy of cubic sym)

74 5-loops potential N2 contact X Contact / Missing Information = you just write it down!

75 5-loops potential N3 contact N 3 contact = o shell =o shell 20 4@truth 20 4@ X g2cut n g 2 d g X g2cut n g 2 d g X g2cut n g 2 d g X g2n 2 contacts X N g d g g2n 2 contacts 1 A cut N g d g A cut 3 5

76 5-loops potential N3 contact Contact / Missing Information you just write down: X 2 2 =

77 5-loops potential N3 contact 105 cubic graphs contribute to the cut. Note: very non-planar far from the ladder graphs. (cubic graphs)

78 5-loops potential N3 contact truth cut = X n g 2 d g2cut g KLT! non-local c/k numerators (truth) +30 pages non-local

79 5-loops potential N3 contact X n g 2 g2cut d g 35 pages (double copy of cubic YM)

80 5-loops potential N3 contact X N g g2n 2 contacts d g 25 N2 contacts (Necessary N2 contacts)

81 5-loops potential N3 contact Contact / Missing Information X 2 2 =

82 Summary c/k + gen. gauge transforms can directly double-copy non-c/k representations resulting in add l local higher-point contact terms (something you can figure out more or less from tree-level considerations) = X

83 Gen. Double Copy Summary Control through 5-pt => all N^2 cuts Control through 6-pt => all N^3 cuts and so on Multiplicity and loop-order independent! works for any double-copy theory b/c of single-copy properties (sym/nlsm/z-theory/ ) provides a simple path forward for tough to crack multi loop double-copy constructions

84

85 We have the five loop N=8 SG integrand. Bern JJMC Chen Johansson Roiban Zeng: arxiv:

86 Integrand has passed many non-trivial tests Bern JJMC Chen Johansson Roiban Zeng: arxiv: N^7 cuts verified in *independent* checks no missing data D=22/5 top-level UV-finite (expected by everyone. anything else would ve likely meant glitch in the calculation) What about D=24/5? Reminder: D=24/5 at 5-loops is the first potential critical dimension challenging N=8 SG having the same perturbative UV behavior as N=4 sym D N =4 SYM c (L) =4+6/L D N =8 SG c (5) =???

87 Technical Slide re: progress since August Skeleton vacuum integral relevant in the UV:

88 Technical Slide re: progress since August Skeleton vacuum integral relevant in the UV: Assign dots" to indicate squaring of propagators.

89 Technical Slide re: progress since August Skeleton vacuum integral relevant in the UV: Assign dots" to indicate squaring of propagators. Original rep in aug in D=24/5 possibly meant considering 8 dots.

90 Technical Slide re: progress since August Skeleton vacuum integral relevant in the UV: Assign dots" to indicate squaring of propagators. Original rep in aug in D=24/5 possibly meant considering 8 dots. 8 dots => TRILLIONS of IBP relations.

91 Technical Slide re: progress since August Skeleton vacuum integral relevant in the UV: Assign dots" to indicate squaring of propagators. Original rep in aug in D=24/5 possibly meant considering 8 dots. 8 dots => TRILLIONS of IBP relations. Now constructed manifest power counting N=4 sym.

92 Technical Slide re: progress since August Skeleton vacuum integral relevant in the UV: Assign dots" to indicate squaring of propagators. Original rep in aug in D=24/5 possibly meant considering 8 dots. 8 dots => TRILLIONS of IBP relations. Now constructed manifest power counting N=4 sym. LAST WEEEK: rebuilt a new N=8 SG integrand; passes all cuts.

93 Technical Slide re: progress since August Skeleton vacuum integral relevant in the UV: Assign dots" to indicate squaring of propagators. Original rep in aug in D=24/5 possibly meant considering 8 dots. 8 dots => TRILLIONS of IBP relations. Now constructed manifest power counting N=4 sym. LAST WEEEK: rebuilt a new N=8 SG integrand; passes all cuts. Looks like at most 4 dots per skeleton => < 1 million IBP relations.

94 Technical Slide re: progress since August Skeleton vacuum integral relevant in the UV: Assign dots" to indicate squaring of propagators. Original rep in aug in D=24/5 possibly meant considering 8 dots. 8 dots => TRILLIONS of IBP relations. Now constructed manifest power counting N=4 sym. LAST WEEEK: rebuilt a new N=8 SG integrand; passes all cuts. Looks like at most 4 dots per skeleton => < 1 million IBP relations. Alex Edison and Julio Para Martinez have joined up on the integration expansion front. Stay tuned for D=24/5!

95 9 Tons of exciting stuff I haven t even had a chance to begin to talk about. Beautiful body of work going into Solving and Understanding Properties of Scattering Eqns ; Dolan Goddard; Lam Yao; Bjerrum-Bohr Bourjaily Damgaard Feng; Du Teng Wu; Nandan Pleua Wormsbecher; He Liu Wu; Renewed interest in soft/colinear limits and EFT: ; Cachazo Strominger; Cheung Kampf Novotny Shen Trnka; Nandan Plefka Wormsbecher; Nandan Pleua Wormsbecher; He Liu Wu; Broedel de Leeuw Plefka Rosso; Bern Davies Nohle ; BernDavies Di Vecchia Nohle; Golden Spradlin; Di Vecchia Marotta Mojaza; Duo Luo; Kallosh; Kallosh Karlsson Murli; Nandan Plefka Schlotterer Wen;Klose McLoughlin Nandan Plefka Travaglini; Broedel de Leeuw Plefka Rosso Universality in string interactions: ; Huang Schlotterer; Chepelev Tseytlin; Caron-HuotKomargodski Sever Zhiboedev; Non-planar on-shell diagrams: ; Arkani-Hamed Bourjaily Cachazo Postnikov Trnka ; Herrmann Trnka Bourjaily ; Heslop Lipstein ; Franco Galloni Penante Wen ; Benincasa Gordo;

96 9 Tons of exciting stuff I haven t even had a chance to begin to talk about. Physical Understanding of Integrated Multiloop Gauge Amplitudes ; Dixon Hippel McLeod Trnka Caron-Huot; Dixon McLeod von Hippel; [Combinations of: {Drummond Gloden GoncharovPapathanasiouParkerPaulosSpradlinScherlisVergu Volovich}]Del Duca Duhr Smirnov; Caron-Huot; Dixon Drummond Henn Dixon Drummond Henn; Caron-Huot He; Dixon Drummond von Hippel Pennington; Physical Aspects of Infinities in Gravity: ; Bern Edison Kosower Parra-Martinez; Bern Chi Dixon Edison; Bern Cheung Chi Davies Dixon Nohle; UV Properties & Anomalies in Lower SUSY SG: ; Bern Enciso Parra-Martinez Zeng; Bern Davies Nohle; Freedman Kallosh Murli Van ProeyenYamada; Bern Davies Dennen; Kallosh; Bern Davies Dennen AV. Smirnov VA Smirnov; Bern Davies Tristan Dennen; Bern Davies Dennen Y.T. Huang; JJMC Kallosh Tseytlyn Roiban; Recent Integration innovations: ; Bosma Sogaard Zhang; Gluza Jelinski Kosower; Georgoudis) Larsen Zhang; Kosower; Mastrolia Peraro Primo; Remidd Tancredi; Gehrmann Henn Presti; Baadsgaard Bjerrum-Bohr Bourjaily Damgaard; Henn; Johansson Kosower Larsen Søgaard; Eden VA Smirnov; Feng Chang Chen Gu Zhang; AV Smirnov; von Manteuffel Schabinger; Caron-Huot Henn; Johansson Kosower Larsen; Pierpaolo Mastrolia Mirabella Ossola Peraro ;

97 9 Tons of exciting stuff I haven t even had a chance to begin to talk about. Integrability and Amplitudes and Correlation Functions ; Gromov Kazakov Korchemsky Negro Sizov; Beisert Garus Rosso; Brandhuber Hughes Panerai Spence Travaglini; Aprile Drummond Heslop; Brandhuber Kostacinska Penante Travaglini Young; Korchemsky; Eden Heslop Mason; Bork Onishchenko; Brandhuber Hughes Spence Travaglini; Eden Sfondrini ; Chicherin Heslop Korchemsky Bourjaily Heslop Tran; Eden Paul; Koster Mitev Staudacher; Sokatchev. Alday Korchemsky;Beisert Müller Plefka Vergu; Koster Mitev Staudacher Wilhelm; Müller Münkler Plefka Pollok Zarembo; Kanning Ko Staudacher; Ferro Łukowsk Meneghelli Plefka Staudacher; MZV Polylogs ; Broedel Matthes Richter Schlotterer; Henn AV Smirnov VA Smirnov; D Hoker; Broedel; Green Gurdogan Vanhove; Broedel Sprenger Orjuela; Puhlfürst Stieberger; D Hoker Green Vanhove; Broedel Mafra Matthes Schlotterer; Broedel Schlotterer Stieberger Terasoma; Brown; Amplituhedron ; Arkani-Hamed Thomas Trnka; [[Ferro Lukowski Orta Parisi]] ; Enciso; Dennen Prlina Spradlin Stanojevic Volovich; Ferro Łukowski Staudacher