Scattering of Spinning Black Holes from Amplitudes based on work with Alfredo Guevara and Justin Vines arXiv:1812.06895, 1906.10071 [hep-th] Alexander Ochirov ETH Z¨ urich QCD Meets Gravity 2019, UCLA, December 12 1 / 28
Motivation 2 / 28
“photo” by Event Horizon Telescope Collaboration ’19
Artist’s impression of BH merger. Credit: SXS
Motivation ◮ BH merger GW150914 seen by LIGO+Virgo Hanford, Washington (H1) Livingston, Louisiana (L1) 1,0 0,5 0,0 -0,5 Streckung (10 -21 ) -1,0 L1 gemessen H1 gemessen H1 gemessen (verschoben, invertiert) 1,0 0,5 0,0 -0,5 -1,0 Numerisch (Relativitätstheorie) Numerisch (Relativitätstheorie) Rekonstruiert (Elementarwelle) Rekonstruiert (Elementarwelle) Rekonstruiert (Vorlage) Rekonstruiert (Vorlage) 0,5 ◮ EOB Hamiltonian from PM scattering instead of from PN 2-body bound-state dynamics Buonanno, Damour ’98 → Damour ’16 ◮ On-shell amplitude methods: quantum gravity scattering easier than GR dynamics e.g. 3PM 0-spin Hamiltonian by Bern, Cheung, Roiban, Shen, Solon, Zeng ’19 talks by Shen, Solon and Zeng 5 / 28
Motivation ◮ BH merger GW150914 seen by LIGO+Virgo Hanford, Washington (H1) Livingston, Louisiana (L1) 1,0 0,5 0,0 -0,5 Streckung (10 -21 ) -1,0 L1 gemessen H1 gemessen H1 gemessen (verschoben, invertiert) 1,0 0,5 0,0 -0,5 -1,0 Numerisch (Relativitätstheorie) Numerisch (Relativitätstheorie) Rekonstruiert (Elementarwelle) Rekonstruiert (Elementarwelle) Rekonstruiert (Vorlage) Rekonstruiert (Vorlage) 0,5 ◮ EOB Hamiltonian from PM scattering instead of from PN 2-body bound-state dynamics Buonanno, Damour ’98 → Damour ’16 ◮ On-shell amplitude methods: quantum gravity scattering easier than GR dynamics e.g. 3PM 0-spin Hamiltonian by Bern, Cheung, Roiban, Shen, Solon, Zeng ’19 talks by Shen, Solon and Zeng This talk: ◮ 1PM and 2PM BH scattering with spin from amplitudes 5 / 28
Outline 1. Spin exponentiation from minimal coupling 2. 1PM with general spin dependence 3. Aligned-spin results at 2PM 4. Summary & outlook 6 / 28
Spin exponentiation from minimal coupling 7 / 28
Spin exponentiation from minimal coupling Want: extract classical spin dependence ( S µ ∈ R 4 ) from quantum spin amplitudes ( s ∈ Z + ) 7 / 28
Minimal-coupling 3-pt amplitudes Arkani-Hamed, Huang, Huang ’17 k � 12 � ⊙ 2 s p 2 = − κ √ M ( s, +) m 2 s − 2 x 2 , m ( p 1 · ε + ) 2 x = − 3 2 � √ � − 1 [12] ⊙ 2 s = − κ m ( p 1 · ε − ) 2 M ( s, − ) = m 2 s − 2 x − 2 , 3 2 p 1 e.g. M (0 , ± ) = − κ ( p 1 · ε ± ) 2 3 8 / 28
Minimal-coupling 3-pt amplitudes Arkani-Hamed, Huang, Huang ’17 k � 12 � ⊙ 2 s p 2 = − κ √ M ( s, +) m 2 s − 2 x 2 , m ( p 1 · ε + ) 2 x = − 3 2 � √ � − 1 [12] ⊙ 2 s = − κ m ( p 1 · ε − ) 2 M ( s, − ) = m 2 s − 2 x − 2 , 3 2 p 1 e.g. M (0 , ± ) = − κ ( p 1 · ε ± ) 2 3 Angular-momentum structure inside: = M (0 , +) � 12 � ⊙ 2 s − ik µ ε + σ µν � � ν ¯ M ( s, +) = M (0 , +) 3 [2 | ⊙ 2 s exp | 1] ⊙ 2 s 3 3 m 2 s m 2 s p 1 · ε + = M (0 , − ) [12] ⊙ 2 s − ik µ ε − ν σ µν � � M ( s, − ) = M (0 , − ) 3 � 2 | ⊙ 2 s exp | 1 � ⊙ 2 s 3 3 m 2 s m 2 s p 1 · ε − Guevara, AO, Vines ’18 inspired by soft theorems, e.g. Cachazo, Strominger ’14 8 / 28
Angular-momentum exponential of Kerr Vines ’17 Stress-energy tensor (eff. source) for lin. Kerr BH: * BH ( x ) = 1 � dτ p ( µ exp( a ∗ ∂ ) ν ) p µ = mu µ T µν ρ p ρ δ (4) ( x − uτ ) , m δ ( p · k ) p ( µ exp( − ia ∗ k ) ν ) ρ p ρ , S µ = ma µ T µν BH ( k ) = ˆ * Hat notation absorbs straightforward powers of 2 π . 9 / 28
Angular-momentum exponential of Kerr Vines ’17 Stress-energy tensor (eff. source) for lin. Kerr BH: * BH ( x ) = 1 � dτ p ( µ exp( a ∗ ∂ ) ν ) p µ = mu µ T µν ρ p ρ δ (4) ( x − uτ ) , m δ ( p · k ) p ( µ exp( − ia ∗ k ) ν ) ρ p ρ , S µ = ma µ T µν BH ( k ) = ˆ Couple to on-shell graviton h µν ( k ) → ˆ δ ( k 2 ) ε µ ε ν : − ik µ ε ν S µν � � δ ( p · k )( p · ε ) 2 exp h µν ( k ) T µν BH ( − k ) = ˆ δ ( k 2 )ˆ , p · ε S µν = ǫ µνρσ p ρ a σ where * Hat notation absorbs straightforward powers of 2 π . 9 / 28
Kerr ⇐ minimal coupling to gravity Guevara, AO, Vines ’18 − ik µ ε ν S µν � � δ ( p · k )( p · ε ) 2 exp h µν ( k ) T µν BH ( − k ) = ˆ δ ( k 2 )ˆ p · ε 10 / 28
Kerr ⇐ minimal coupling to gravity Guevara, AO, Vines ’18 − ik µ ε ν S µν � � δ ( p · k )( p · ε ) 2 exp h µν ( k ) T µν BH ( − k ) = ˆ δ ( k 2 )ˆ p · ε Compare to = M (0 , +) − ik µ ε + σ µν k � ν ¯ � p 2 M ( s, +) 3 [2 | ⊙ 2 s exp | 1] ⊙ 2 s 3 m 2 s p 1 · ε + = M (0 , − ) − ik µ ε − ν σ µν � � M ( s, − ) 3 � 2 | ⊙ 2 s exp | 1 � ⊙ 2 s 3 m 2 s p 1 · ε − p 1 10 / 28
Kerr ⇐ minimal coupling to gravity Guevara, AO, Vines ’18 − ik µ ε ν S µν � � δ ( p · k )( p · ε ) 2 exp h µν ( k ) T µν BH ( − k ) = ˆ δ ( k 2 )ˆ p · ε Compare to = M (0 , +) − ik µ ε + σ µν k � ν ¯ � p 2 M ( s, +) 3 [2 | ⊙ 2 s exp | 1] ⊙ 2 s 3 m 2 s p 1 · ε + = M (0 , − ) − ik µ ε − ν σ µν � � M ( s, − ) 3 � 2 | ⊙ 2 s exp | 1 � ⊙ 2 s 3 m 2 s p 1 · ε − p 1 Matching spin-induced multipole structure! complementary picture: 1-body EFT of Kerr by Levi, Steinhoff ’15 match to Wilson coeffs by Chung, Huang, Kim, Lee ’18 10 / 28
Spin exponentiation in covariant form Covariant formulation: Bautista, Guevara ’19 − ik µ ε ν Σ µν � � M ( s ) = M (0) 3 ε 2 · exp · ε 1 3 p 1 · ε Lorentz generators: (Σ µν ) σ 1 ...σ s τ 1 ...τ s = Σ µν,σ 1 τ 1 δ σ 2 τ 2 . . . δ σ s τ s Σ µν,σ τ = i [ η µσ δ ν τ − η νσ δ µ τ ] + . . . + δ σ 1 τ 1 . . . δ σ s − 1 τ s − 1 Σ µν,σ s τ s , Polarization tensors: Guevara, AO, Vines ’18, Chung, Huang, Kim, Lee ’18 pµ = i � p ( a | σ µ | p b ) ] ε a 1 ...a 2 s pµ 1 ...µ s = ε ( a 1 a 2 . . . ε a 2 s − 1 a 2 s ) ε ab , √ pµ 1 pµ s 2 m 11 / 28
Spin exponentiation in covariant form Covariant formulation: Bautista, Guevara ’19 − ik µ ε ν Σ µν � � M ( s ) = M (0) 3 ε 2 · exp · ε 1 3 p 1 · ε Lorentz generators: (Σ µν ) σ 1 ...σ s τ 1 ...τ s = Σ µν,σ 1 τ 1 δ σ 2 τ 2 . . . δ σ s τ s Σ µν,σ τ = i [ η µσ δ ν τ − η νσ δ µ τ ] + . . . + δ σ 1 τ 1 . . . δ σ s − 1 τ s − 1 Σ µν,σ s τ s , Polarization tensors: Guevara, AO, Vines ’18, Chung, Huang, Kim, Lee ’18 pµ = i � p ( a | σ µ | p b ) ] ε a 1 ...a 2 s pµ 1 ...µ s = ε ( a 1 a 2 . . . ε a 2 s − 1 a 2 s ) ε ab , √ pµ 1 pµ s 2 m Spinor-helicity formulation: Guevara, AO, Vines ’19 − ik µ ε + σ µν � ν ¯ � = M (0) | 1] ⊙ 2 s = M (0) M ( s, +) m 2 s [2 | ⊙ 2 s exp m 2 s [2 | ⊙ 2 s exp( − 2 k · a ) | 1] ⊙ 2 s 3 3 3 p 1 · ε + − ik µ ε − ν σ µν � � = M (0) | 1 � ⊙ 2 s = M (0) M ( s, − ) m 2 s � 2 | ⊙ 2 s exp m 2 s � 2 | ⊙ 2 s exp(2 k · a ) | 1 � ⊙ 2 s 3 3 3 p 1 · ε − 1 1 a µ, ˙ α a µ, β 2 m 2 ǫ µνρσ p a ν σ β 2 m 2 ǫ µνρσ p a ν ¯ α ˙ = ρσ,α , β = σ ρσ, ˙ α ˙ β σ µν = i σ µν = i 2 σ [ µ ¯ σ ν ] , σ [ µ σ ν ] ¯ 2 ¯ (and tensor generalizations) 11 / 28
Spin quantization 1 2 mǫ λµνρ Σ µν p ρ Define Pauli-Lubanski vector operator Σ λ = Its 1-particle matrix elements are S { a }{ b } = ( − 1) s ε { a } · Σ µ · ε { b } pµ p p = − s � p ( a 1 | σ µ | p ( b 1 ] + [ p ( a 1 | ¯ σ µ | p ( b 1 � ǫ a 2 b 2 . . . ǫ a 2 s ) b 2 s ) � � 2 m 12 / 28
Spin quantization 1 2 mǫ λµνρ Σ µν p ρ Define Pauli-Lubanski vector operator Σ λ = Its 1-particle matrix elements are S { a }{ b } = ( − 1) s ε { a } · Σ µ · ε { b } pµ p p = − s � p ( a 1 | σ µ | p ( b 1 ] + [ p ( a 1 | ¯ σ µ | p ( b 1 � ǫ a 2 b 2 . . . ǫ a 2 s ) b 2 s ) � � 2 m Spin quantized explicitly: ss µ p , a 1 = . . . = a 2 s = 1 , � 2 s ( s − 1) s µ p , j =1 a j = 2 s + 1 , ε p { a } · Σ µ · ε { a } p � 2 s = ( s − 2) s µ p , j =1 a j = 2 s + 2 , ε p { a } · ε { a } p . . . − ss µ p , a 1 = . . . = a 2 s = 2 , in terms of unit spin vector p = − 1 s µ � � p 1 | σ µ | p 1 ] + [ p 1 | ¯ σ µ | p 1 � � p · s p = 0 2 m 1 p = − 1 s 2 p = − 1 u p 1 γ µ γ 5 u 1 u p 2 γ µ γ 5 u 2 = 2 m ¯ 2 m ¯ p 12 / 28
Spin asymmetry of chiral reps Puzzle: = M (0) m 2 s � 21 � ⊙ 2 s = M (0) two reps of M ( s, +) m 2 s [2 | ⊙ 2 s e − 2 k · a | 1] ⊙ 2 s 3 3 3 seem to depend differently on a µ 13 / 28
Spin asymmetry of chiral reps Puzzle: = M (0) m 2 s � 21 � ⊙ 2 s = M (0) two reps of M ( s, +) m 2 s [2 | ⊙ 2 s e − 2 k · a | 1] ⊙ 2 s 3 3 3 seem to depend differently on a µ Fix: Guevara, AO, Vines ’18 “divide” by m 2 s � 2 | ⊙ 2 s e k · a | 1 � ⊙ 2 s = lim 1 m 2 s [2 | ⊙ 2 s e − k · a | 1] ⊙ 2 s 1 s →∞ ε 2 · ε 1 = lim lim s →∞ s →∞ 13 / 28
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