Poles at Infinity in Gravity

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1 Poles at Infinity in Gravity Enrico Herrmann QCD Meets Gravity UCLA Based on: arxiv: , , in collaboration with:. Zvi Bern, Sean Litsey, James Stankowicz, Jaroslav Trnka. work in progress with: Zvi Bern, Michael Enciso, Julio Parra Martinez, Jaroslav Trnka December 7, 2016 Enrico Herrmann (Caltech) Poles at Infinity in Gravity December 7, / 20

2 ... Disclaimer: Not a complete story yet!...mmmmmm. Comments and Discussion welcome Enrico Herrmann (Caltech) Poles at Infinity in Gravity December 7, / 20

3 Motivation Motivation - Gravity in the UV open problem: N = 8 SUGRA divergence at 7-loops? BCJ at maximal cuts: N (l 1 l 2 ) 8 power counting: integral diverges I (d4 l) 7 (l l) 8 (l 2 ) 22 log Λ enhanced cancellations in gravity between diagrams [Bern,Davies,Dennen] Does divergence cancel? What is the origin of the cancellations? Is there an integrand understanding of the UV-cancellations analog to IR-cancellations? see later Enrico Herrmann (Caltech) Poles at Infinity in Gravity December 7, / 20

4 Motivation Motivation How to expose and understand the cancellations? practical approach: half-max sugra (N = 4sYM N = 0YM) practical approach: in D = 2 loops! [Bern,Enciso,Kosower,Parra-Martinez,Zeng] integrand based cancellation between diagrams Heavy use of IBP technology and integrand decomposition into: 1 UV-divergent pieces 2 UV fininte pieces QCD meets Gravity! Mao s talk! 3 total derivatives Comments: Magic clever basis choice! How does M know IBPs? prior knowledge of full integrand necessary 7 loops? Enrico Herrmann (Caltech) Poles at Infinity in Gravity December 7, / 20

5 Motivation Motivation Generalized Unitarity understanding of the cancellations? 1 Loop Cancellations: [Bern,Carrasco,Forde,Ita,Johansson] M = i d i + i c i + i b i gravity tree amplitudes good large z behavior under BCFW-shift... N 5 sugra: use Forde s formalism to demonstrate that bubble and N 5 sugra: triangle coefficients vanish. good large z behavior of cut no pole at infinity c i = b i = 0! can compute 7-loop cuts easily Enrico Herrmann (Caltech) Poles at Infinity in Gravity December 7, / 20

6 Motivation Outline 1 Poles at Infinity 2 N = 4 super Yang Mills 3 Gravity 4 Conclusion-Outlook Enrico Herrmann (Caltech) Poles at Infinity in Gravity December 7, / 20

7 Poles at Infinity Poles at Infinity Integrals in momentum twistor space: Significance of infinity twistor in momentum space: abcdefghijklml = αλ 1 λ1 + βλ 2 λ2 + γλ 1 λ2 + δλ 2 2 λ1 l 3 d 4 ls 4 = l 2 (l+1) 2 (l 2) 2 = dα dβ dγ dδ (αβ γδ)(αβ γδ+β)(αβ γδ α) 1 δ= αβ γ dα dβ dγ α β γ poles at infinity ABI Enrico Herrmann (Caltech) Poles at Infinity in Gravity December 7, / 20

8 Poles at Infinity Poles at Infinity and UV structure Triangle Integral UV finite! abcmel = αλ 1 λ1 + βλ 2 λ2 + γλ 1 λ2 + δλ 2 λ1 2 l 3 d 4 ls 4 = l 2 (l+1) 2 (l 2) 2 = dα dβ dγ dδ (αβ γδ)(αβ γδ+β)(αβ γδ α) 1 δ= αβ γ dα dβ dγ α β γ single pole at infinity Bubble Integral UV divergent! l 2 3 d 4 l 1 4 = l 2 (l+1+2) 2 = dα dβ dγ dδ (αβ γδ)(αβ γδ+α+β +1) δ= αβ γ dβ dγ dα (α+β+1) γ double pole at infinity What is the precise relation between poles at infinity and UV-divergence? Enrico Herrmann (Caltech) Poles at Infinity in Gravity December 7, / 20

9 N = 4 super Yang Mills Absence of Poles at infinity in N = 4sYM (a) planar N = 4 sym: Dual Conformal Invariance Absence of infinity twistor no poles at infinity in momentum space (b) beyond the planar limit: [Arkani-Hamed, Bourjaily, Cachazo, Trnka, Bern, EH, Litsey, Stankowicz] Basic difficulty for non-planar amplitudes no global variables L-loop amplitude in local integral expansion di k : A(p i ) = ˆ di k (p i, l j ) k absence of poles l, 2- and 3-loops p q p q q 6 5 p kkabsence of poles at infinity UV-finiteness term-by-term (a) (b) Enrico Herrmann (Caltech) Poles at Infinity in Gravity December 7, / 20

10 N = 4 super Yang Mills From YM to gravity: Celebrating BCJ BCJ double copy link between YM and gravity [Bern, Carrasco, Johansson] amplitude level double copy: A L n ˆ ci n i (l) M L n ˆ ni (l)ñ i (l) D i (l) D i (l) cubic graphs cubic graphs if BCJ numerator l-independent N = 8 SUGRA same UV structure as N = 4 sym (e.g. 2loop 4pt) Poles at infinity are present in N = 8 SUGRA look at cut dz z Pole at z l(z) higher poles at higher loops detailed understanding of poles at infinity needed Enrico Herrmann (Caltech) Poles at Infinity in Gravity December 7, / 20

11 Gravity Gravity on-shell functions [EH, Trnka; Heslop, Lipstein] 3pt-amplitudes: squaring relation A 3 = ( 12 4 M 3 = general on-shell diagram (product of 3pt amplitudes) don t square the propagators (YM) 2 = (GR) (φ 3 ) (φ 3 ) factor changes expressions drastically ) 2 Enrico Herrmann (Caltech) Poles at Infinity in Gravity December 7, / 20

12 Gravity Grassmannian Formula for Gravity [EH, Trnka; Heslop, Lipstein] 1 2 Α1 Α4 Α2 Α3 ( ) 1 α1 0 α C = 4 0 α 2 1 α from OS-data, one can discover the gravity formula Yang Mills Ω = dα 1 α 1 dα 2 α 2 dα 3 α 3 dα 4 α 4 δ(c Z) only logarithmic poles all residues correspond to edge removal no l Ω = dα 1 α 3 1 dα 2 α 3 2 dα 3 α 3 3 Gravity dα 4 α 3 4 ( v v) δ(c Z) special numerator v for each vertex collinear properties α 3 i poles l present Enrico Herrmann (Caltech) Poles at Infinity in Gravity December 7, / 20

13 Gravity Grassmannian Formula for Gravity [EH, Trnka] Α 3 Α2 Α 6 Α Α 4 5 Α Ω = l 1 = λ 1 Q 12 λ 3 13, l 2 = λ 5 Q 12 λ 3 35, l 1 1= λ 1 λ 2, l 2 5= λ 5 λ 4, l 1 Q 12 = λ 3 λ 2, l 2 Q 45 = λ 3 λ 4, l 1 Q 123 = λ 3 Q 23 λ 1 13, l 1 +l 2 = λ 3 Q 12 λ 3 [12][23][45] α 1 = 23 13, α 2 = 12 13, α 3 = 45 35, α 4 = 34 35, α 5 = 13 35, α 6 = reduces to single pole if edge is erasable diagram vanishes if momenta in a given vertex become collinear 1 Is there some on-shell picture for residues at infinity? 2 What are the implications for gravity amplitudes? Enrico Herrmann (Caltech) Poles at Infinity in Gravity December 7, / 20

14 Gravity IR: Collinear Behavior of Gravity Amplitudes [EH, Trnka] OS-diagrams suggest special collinear behavior of amplitudes on cut: [l 1 l 2 ] l 1 l 2 For special case of external legs (k 1 k 2 ), known collinear limit of amplitudes (c.f. splitting functions): [Bern, Dixon, Perelstein, Rozowsky] M 12 0 [12] [12] 0 R, M [12] R R, R regular in 12, [12] Test general collinear conjecture on theoretical data! Enrico Herrmann (Caltech) Poles at Infinity in Gravity December 7, / 20

15 Gravity IR: Collinear Behavior of Gravity Amplitudes Labelling issue! 2 1-loop 4pt [Green,Schwarz,Brink] 1 3? [l1] Sum six terms: 4 [l1] Property not manifest term-by-term cancellations required! more impressive cancellation at 2-loops Enrico Herrmann (Caltech) Poles at Infinity in Gravity December 7, / 20

16 Gravity Residues at Infinity more general feature of scattering amplitudes: present at tree level for EFTs in BCFW-recursion [Feng et al.,...] extracting β-function for pure 1-loop with on-shell methods [Arkani-Hamed, Cachazo, Kaplan; also Caron-Huot, Wilhelm] relevant for Forde s method to extract integral 1-loop In gravity: understanding infinity crucial to understand UV-structure infinity would have impact on putative loop recursion initial steps: [Heslop,Lipstein] Enrico Herrmann (Caltech) Poles at Infinity in Gravity December 7, / 20

17 Gravity Residues at Infinity work in prograss w/ Jaroslav Trnka finite poles: infinity: codim 1 residues: OS-diags drop 1-loop + kinematic dressing factor Enrico Herrmann (Caltech) Poles at Infinity in Gravity December 7, / 20

18 Gravity Probing Residues at Infinity Access pole at infinity via BCFW deformation: λ 1 λ 1 + zλ 5, Access pole at infinity via BCFW deformation: λ 5 λ 5 z λ 1 O GR = s 15 z [12][23]([45] z[41]) 2 ( 12 +z 52 )( 13 +z 53 ) infinity: z = & z = pole at z = λ 5 (1+5) λ 3 53, λ λ 3 : remove bridge ( ] 53 [12][23][45] 2 [12][23] ( ) [12][23][42] 2 = ) 2 ( ) Is there a more invariant way to talk about poles at infinity? Enrico Herrmann (Caltech) Poles at Infinity in Gravity December 7, / 20

19 UV poles in amplitudes Gravity Back to the beginning: (work in progress) How to expose and understand the cancellations? half-max sugra (N = 4sYM N = 0YM) in D = 5: Are there cancellations of UV poles at the integrand level, analogous to IR? NO! understand terms that integrate to zero from cuts? Enrico Herrmann (Caltech) Poles at Infinity in Gravity December 7, / 20

20 Conclusion-Outlook Comments-Conclusion Yang-Mills d log, no poles at infinity can be made manifest term-by-term Gravity in the IR collinear vanishing of on-shell diagrams and amplitudes d log structure of finite poles properties are not term-wise manifest in local expansion no integration properties needed Gravity in the UV poles at infinity are present in the cuts enhanced cancellation require more than symmetrization understanding of total derivatives necessary Enrico Herrmann (Caltech) Poles at Infinity in Gravity December 7, / 20