Orbital dynamics from double copy and EFT Talk at Radcor - PowerPoint PPT Presentation

Orbital dynamics from double copy and EFT Talk at Radcor Conference, Avignon, France, 11 Sep 2019, with Zvi Bern, Clifford Cheung, Radu Roiban, Chia-Hsien Shen, Mikhail P. Solon, arXiv:1901.04424 (PRL), arXiv:1908.01493 Mao Zeng 11 Sep, 2019 Institute for Theoretical Physics, ETH Zürich 1

Outline 1. Introduction 2. Two-loop amplitudes 3. Effective potential between black holes 4. Discussions 2

Introduction

BIRTH OF AN ERA • LIGO / VIRGO detected gravitational waves: BH-BH (2015), BH-NS (2017), NS-NS (2019?) LIGO & VIRGO, arXiv:1602.03837 LIGO & VIRGO, arXiv:1710.05832 • Next-gen. experiments (LISA, CE, ET…): high S-N ratio, • Precision predictions necessary. 3 dominated by theory uncertainty .

ANATOMY OF GRAVITATIONAL WAVE SIGNAL [Picture: Antelis, Moreno, 1610.03567] Inspiral Post-Newtonian / Post-Minkowskian / EOB Merger Numerical relativity / EOB resummation Ringdown Perturbative quasi-normal modes 4

POST-NEWTONIAN EXPANSION Potential in post Newtonian expansion, e.g. in c.o.m. frame: [Talks by Christian Sturm, Andreas Maier] [Holstein, Ross, ’08; Neill, Rothstein, ’13] 5PN approximate [Bini, Damour, Geralico, ’19] Jaranowski, Schaefer, ’01] 4PN [Damour, Jaranowski, Schäfer, Bernard, Blanchet, Bohe, Schaefer, ’97; Damour, Jaranowski, Schaefer, ’97; Blanchet, Faye, ’00; Damour, 5 1PN, ∼ Gv 2 0PN, ∼ G ( ) ) V = − Gm 1 m 2 ⃗ p 2 1 + 3 ( m 1 + m 2 ) 2 ( 1 + r m 1 m 2 2 m 1 m 2 + G 2 m 1 m 2 ( m 1 + m 2 ) m 1 m 2 ( ) 1 + + . . . 2 r 2 ( m 1 + m 2 ) 2 1PN, ∼ G 2 1PN [Einstein, Infeld, Hoffman ’38] . 2PN [Ohta et al. , ’73] . 3PN [Jaranowski, Faye, Marsat, Marchand, Foffa, Sturani, Mastrolia, Sturm, Porto, Rothstein…] 5PN static [Foffa, Mastrolia, Sturani, Sturm, Bodabilla, ’19; Blümlein, Maier, Marquard, ’19]

POST-MINKOWSKIAN EXPANSION Our new 3PM result: [Bern, Cheung, Roiban, Shen, Solon, MZ, PRL ’19; See also talk by Mikhail Solon 1908.01493 (long paper)] 6 Damour, Deruelle, Ibanez, Martin, Ledvinka, Schaefer, Bicak…] Bound orbit: GM / r ∼ v 2 . Hyperbolic orbit / scattering: expand with GM / r ≤ v ∼ c . [Bertotti, Kerr, Plebanski, Portilla, Westpfahl, Goller, Bel, 0PN 1PN 2PN 3PN 4PN 5PN 6PN 7PN ( 1 + v 2 + v 4 + v 6 + v 8 + v 10 + v 12 + v 14 + . . . ) G 1 1PM ( 1 + v 2 + v 4 + v 6 + v 8 + v 10 + v 12 + . . . ) G 2 2PM ( 1 + v 2 + v 4 + v 6 + v 8 + v 10 + . . . ) G 3 3PM ( 1 + v 2 + v 4 + v 6 + v 8 + . . . ) G 4 4PM ( 1 + v 2 + v 4 + v 6 + . . . ) G 5 5PM ( 1 + v 2 + v 4 + . . . ) G 6 6PM . . .

HOW QFT HELPS Donoghue, ’94; Holstein, Donoghue, ’04; Neill, Rothstein, ’13; Vaidya, ’14; Kosower, ? Maybee, O’Connell, ’18 …] 7 [picture: LIGO] formulation: Damgaard, Haddad, Helset, ’19] Hierarchy of scales in bound state systems: effective field theory : [NRGR: Goldberger, Rothstein, ’04; Porto, ’06; relativistic 2 3 5 6 1 4 Manifest gauge invariance through scattering amplitudes , with carefully defined classical limit [Iwasaki ’71; Gupta, Radford, ’79;

Two-loop amplitude

GRAVITY = (YANG MILLS) 2 4 5 5 5 5 5 4 4 Kawai-Lewellen-Tye (KLT) relations from string theory: 8 Simplification from “square” / double copy! Infinite tower, even 3-graviton vertex has ∼ 100 terms. ⟨ 12 ⟩ 3 ⟨ 12 ⟩ 6 A 3 ( 1 − 2 − 3 + ) = M 3 ( 1 − 2 − 3 + ) = ⟨ 23 ⟩⟨ 31 ⟩ , ⟨ 23 ⟩ 2 ⟨ 31 ⟩ 2 M tree ( 1 , 2 , 3 , 4 ) = − is 12 A tree ( 1 , 2 , 3 , 4 ) A tree ( 1 , 2 , 4 , 3 ) M tree ( 1 , 2 , 3 , 4 , 5 ) = is 12 s 34 A tree ( 1 , 2 , 3 , 4 , 5 ) A tree ( 2 , 1 , 4 , 3 , 5 ) + is 13 s 24 A tree ( 1 , 3 , 2 , 4 , 5 ) A tree ( 3 , 1 , 4 , 2 , 5 ) 4 dimensions: g ± ⊗ g ± ∼ h ±± .

GRAVITY = (YANG MILLS) 2 , in “BCJ form” if 9 duality , i.e. double-copy construction. [Bern, Carrasco, Johansson, ’08] D dimensions: more convenient to use color-kinematics c j n j = g m − 2 ∑ Gauge theory amplitude: A tree m D j j n j satisfies same Jacobi identities as c j . n j n j ˜ ∑ Then gravity amplitude M tree = i , from c j → n j . m D j j

TWO-LOOP CUTS [Bern, Cheung, Roiban, Shen, Solon, MZ, PRL ’19; 1908.01493 (long paper)] Very compact expression at 2 loops. Higher loops within reach! 1 1 4 4 4 10 2   d D x √ g ∫ S = 2 R + 1 ∑ D µ ϕ i D µ ϕ i − m i ϕ 2 ( )  − 1 i  i = 1 , 2 9 2 3 8 9 8 2 3 2 3 8 7 6 6 5 7 5 7 9 1 4 5 6 1 4 10 1 4 (c) (a) (b) From KLT: M tree ( 1 , 2 , 3 , 4 ) = − is 12 A tree ( 1 , 2 , 3 , 4 ) A tree ( 1 , 2 , 4 , 3 ) . { ]} ( ) C ( c ) [ GR = − i 2 t 2 m 4 1 m 4 Tr [( / 7 / 2 / 8 / 6 / 1 / 2 + 1 5 ) 4 ] + ( 7 ↔ 8 ) ( k 5 − k 8 ) 2 + t 6 ( k 6 + k 8 ) 2

TWO-LOOP INTEGRAND classical effects. 11 Cuts merged into an integrand with diagrams & numerators: 2 3 2 3 2 3 2 3 5 6 7 1 4 1 4 1 4 1 4 ( 1 ) ( 2 ) ( 3 ) ( 4 ) 2 3 2 3 2 3 2 3 1 4 1 4 1 4 1 4 ( 5 ) ( 6 ) ( 7 ) ( 8 ) Diagram symmetries imposed. ∼ 90KB file. 4 and D dimensional results agree, up to “ µ ” terms with no

INTEGRATING THE AMPLITUDE 12 • m 1 ̸ = m 2 , even planar master integrals unknown! Smirnov, '01; Lower topologies: Bianchi, Leoni, 1612.05609 Henn & Smirnov, '13; Duhr, Amplitudes '18; Heller, von Manteu ff el, Schabinger, '19 • Simplification 1: expand in small q ∼ ℏ / R ≤ m i , √ s . • Simplification 2: Expand in v ≪ 1 from potential d 4 ℓ ) localized on + ve energy matter poles. ∫ region. (

NR INTEGRATION / VELOCITY EXPANSION 1 1 1 1 1 Plan: Series expansion around static limit, then resum by 1 13 1 matching to simple functions. 1 1 Step 1: determine integrand in potential region I T = ( E 1 + ω ) 2 − ( p + l ) 2 − m 2 × ω 2 − l 2 ω 2 − ( l + q ) 2 = 2 m 1 l 2 ( l + q ) 2 + . . . ω − ω P 1 = ( ω − ω P 1 )( ω − ω A 1 ) , ( E 1 + ω ) 2 − ( p + l ) 2 − m 2 √ 1 + 2 pl + l 2 . ω P 1 , ω A 1 = − E 1 ± E 2 ω P 1 ≪ | l | , ω A 1 ≈ − 2 m 1

NR INTEGRATION / VELOCITY EXPANSION 2 in Mikhail Solon’s talk Only need 3D propagator integrals. More 1 2 Step 3: Spatial integration 2 1 1 14 1 2 energy matter poles. 1 Step 2: Energy integration, keeping only residues from + ve ∫ ∫ [ ] d ω 1 d ω 2 δ ( ω 1 + ω 2 ) 1 ω 1 − ω P 1 + i 0 + ω 1 ↔ ω 2 2 π = − i ∫ [ ] d ω 1 = 1 ω − ω P 1 + i 0 + − ω − ω P 1 + i 0 2 π d 3 l − i i ( ) ∫ 2 m 1 l 2 ( l + q ) 2 = − 32 m 1 | q | . ( 2 π ) 3

RELATIVISTIC INTEGRATION matter poles IBP done by Kira [Maierhoefer, Usovitsch, Uwer, ’17] . 9 master integrals (H & 15 Conservative contribution: localize on cannot both collapose 1 − m 2 2 − m 2 ∫ ∫ δ ( ℓ 2 1 ) δ ( ℓ 2 2 ) N i d d ℓ 1 d d ℓ 2 I i = ∏ 7 i = 3 ℓ 2 i Exact v result from differential equations ∂ I i /∂ s = M ij I j . N topology) rescaled to O ( t 0 ) . Take t → 0 limit in M ij . Boundary condition: regular at threshold s = ( m 1 + m 2 ) 2 . Scalar result: H + xH = log( − t ) arcsinh ( | p | / m 1 ) + arcsinh ( | p | / m 2 ) . | p | ( E 1 + E 2 ) 128 π 2

COMPARISON WITH UNEXPANDED INTEGRAL 16 m 1 = m 2 results in [Bianchi, Leoni, 1612.05609] , thanks to Loopedia.org . 25 master integrals x π 2 log x + log 3 x ( ) ( I H ) 3 log( − t ) + (non-singular in t ) � ϵ 0 = − 4 � 1 − x 2 3 log( − t ) − x ( π 2 log( − x ) + log 3 ( − x ) ) ( I xH ) + (non-singular in t ) . � ϵ 0 = − 4 � 1 − x 2 ϵ 0 = 4 π 2 log( − t ) log( − x ) + imaginary log( x ) → log( − x ) + i π, ( I H + I xH ) � �

Effective potential between black holes

PM POTENTIAL AS EFT MATCHING COEFFICIENT energy pole. Alternative QM treatment: [Cristofoli, Bjerrum-Bohr, Damgaard, Vanhove, ’19] Uncalculated IR divergent integrals cancel in matching. [From Cheung, Rothstein, Solon, ’18 PRL] 17 i ( k 0 , k ) = , √ • kinetic term with only + ve k 2 + m 2 k 0 − A,B + i 0 k k ′ = − iV ( k , k ′ ) , • k 0 -independent vertex. - k ′ - k Matching at L loops gives V ( k , k ′ ) with smooth ℏ → 0 limit. p k 1 k L p ′ M L -loop = M L -loop · · · = EFT full - p - k 1 - k L - p ′

RESULT: 3PM CONSERVATIVE POTENTIAL 3 EFT matching coefficients 2 12 1 [Bern, Cheung, Roiban, Shen, Solon, MZ, PRL ’19; 1908.01493 (long paper)] 4 3 18 √ √ p 2 + m 2 p 2 + m 2 H 3PM ( p , r ) = 2 + V 3PM ( p , r ) 1 + ( G ) n V 3PM ( p , r ) = ∑ c n ( p 2 ) | r | n = 1 m = m 1 + m 2 , ν = m 1 m 2 , E = E 1 + E 2 , ξ = E 1 E 2 E 2 , γ = E m , σ = p 1 · p 2 m 2 m 1 m 2 [ ] c 1 = ν 2 m 2 c 2 = ν 2 m 3 − 4 νσ ( 1 − 2 σ 2 ) ν 2 ( 1 − ξ ) ( 1 − 2 σ 2 ) 2 ( 1 − 2 σ 2 ) ( 1 − 5 σ 2 ) , , − γ 2 ξ γ 2 ξ γξ 2 γ 3 ξ 2 √ 4 ν ( 3 + 12 σ 2 − 4 σ 4 ) arcsinh [ σ − 1 c 3 = ν 2 m 4 ( 3 − 6 ν + 206 νσ − 54 σ 2 + 108 νσ 2 + 4 νσ 3 ) − γ 2 ξ √ σ 2 − 1 − 3 νγ ( 1 − 2 σ 2 ) ( 1 − 5 σ 2 ) − 3 νσ ( 7 − 20 σ 2 ) + 2 ν 3 ( 3 − 4 ξ ) σ ( 1 − 2 σ 2 ) 2 2 ( 1 + γ )( 1 + σ ) 2 γξ γ 4 ξ 3 ν 2 ( 3 + 8 γ − 3 ξ − 15 σ 2 − 80 γσ 2 + 15 ξσ 2 ) ( 1 − 2 σ 2 ) ] ν 4 ( 1 − 2 ξ ) ( 1 − 2 σ 2 ) 3 + . − 4 γ 3 ξ 2 2 γ 6 ξ 4