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,

dominated by theory uncertainty.

• Precision predictions necessary.

3

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:

V = −Gm1m2 r 0PN, ∼ G ( 1 + ⃗ p2 m1m2 1PN, ∼ Gv2 ( 1 + 3(m1 + m2)2 2m1m2 ) ) + G2m1m2(m1 + m2) 2r2 1PN, ∼ G2 ( 1 + m1m2 (m1 + m2)2 ) + . . . 1PN [Einstein, Infeld, Hoffman ’38]. 2PN [Ohta et al., ’73]. 3PN [Jaranowski,

Schaefer, ’97; Damour, Jaranowski, Schaefer, ’97; Blanchet, Faye, ’00; Damour, Jaranowski, Schaefer, ’01] 4PN [Damour, Jaranowski, Schäfer, Bernard, Blanchet, Bohe, Faye, Marsat, Marchand, Foffa, Sturani, Mastrolia, Sturm, Porto, Rothstein…] 5PN

static [Foffa, Mastrolia, Sturani, Sturm, Bodabilla, ’19; Blümlein, Maier, Marquard, ’19] 5PN approximate [Bini, Damour, Geralico, ’19]

[Holstein, Ross, ’08; Neill, Rothstein, ’13]

[Talks by Christian Sturm, Andreas Maier]

5

POST-MINKOWSKIAN EXPANSION

Bound orbit: GM/r ∼ v2. Hyperbolic orbit / scattering: expand with GM/r ≤ v ∼ c. [Bertotti, Kerr, Plebanski, Portilla, Westpfahl, Goller, Bel,

Damour, Deruelle, Ibanez, Martin, Ledvinka, Schaefer, Bicak…] 0PN 1PN 2PN 3PN 4PN 5PN 6PN 7PN 1PM ( 1 + v2 + v4 + v6 + v8 + v10 + v12 + v14 + . . . ) G1 2PM ( 1 + v2 + v4 + v6 + v8 + v10 + v12 + . . . ) G2 3PM ( 1 + v2 + v4 + v6 + v8 + v10 + . . . ) G3 4PM ( 1 + v2 + v4 + v6 + v8 + . . . ) G4 5PM ( 1 + v2 + v4 + v6 + . . . ) G5 6PM ( 1 + v2 + v4 + . . . ) G6 . . .

Our new 3PM result: [Bern, Cheung, Roiban, Shen, Solon, MZ, PRL ’19;

1908.01493 (long paper)] See also talk by Mikhail Solon 6

HOW QFT HELPS

Hierarchy of scales in bound state systems: effective field theory: [NRGR: Goldberger, Rothstein, ’04; Porto, ’06; relativistic

formulation: Damgaard, Haddad, Helset, ’19] [picture: LIGO]

1 2 3 4 5 6

Manifest gauge invariance through scattering amplitudes, with carefully defined classical limit [Iwasaki ’71; Gupta, Radford, ’79;

Donoghue, ’94; Holstein, Donoghue, ’04; Neill, Rothstein, ’13; Vaidya, ’14; Kosower, Maybee, O’Connell, ’18 …]

?

7

Two-loop amplitude

GRAVITY = (YANG MILLS)2

Infinite tower, even 3-graviton vertex has ∼ 100 terms. A3(1−2−3+) = ⟨12⟩3 ⟨23⟩⟨31⟩, M3(1−2−3+) = ⟨12⟩6 ⟨23⟩2⟨31⟩2 Simplification from “square” / double copy! Kawai-Lewellen-Tye (KLT) relations from string theory: Mtree

4

(1, 2, 3, 4) = −is12 Atree

4

(1, 2, 3, 4) Atree

4

(1, 2, 4, 3) Mtree

5

(1, 2, 3, 4, 5) = is12s34 Atree

5

(1, 2, 3, 4, 5) Atree

5

(2, 1, 4, 3, 5) + is13s24 Atree

5

(1, 3, 2, 4, 5) Atree

5

(3, 1, 4, 2, 5) 4 dimensions: g± ⊗ g± ∼ h±±.

8

GRAVITY = (YANG MILLS)2

D dimensions: more convenient to use color-kinematics duality, i.e. double-copy construction. [Bern, Carrasco, Johansson, ’08] Gauge theory amplitude: Atree

m

= gm−2 ∑

j

cjnj Dj , in “BCJ form” if nj satisfies same Jacobi identities as cj. Then gravity amplitude Mtree

m

= i ∑

j

˜ nj nj Dj , from cj → nj.

9

TWO-LOOP CUTS

[Bern, Cheung, Roiban, Shen, Solon, MZ, PRL ’19; 1908.01493 (long paper)]

S = ∫ dDx √g  − 1 2R + 1 2 ∑

i=1,2

( DµϕiDµϕi − miϕ2

i

)  

(a)

1 3 4 6 5 7 2 8 9

(b)

9 8 1 3 4 6 5 7 2

(c)

1 3 4 5 6 8 7 2 9 10

From KLT: Mtree

4

(1, 2, 3, 4) = −is12Atree

4

(1, 2, 3, 4)Atree

4

(1, 2, 4, 3).

C(c)

GR = −i

{ 2t2m4

1m4 2 + 1

t6 [ Tr[(/ 7/ 2/ 8/ 6/ 1/ 5)4] + (7 ↔ 8) ]} ( 1 (k5 − k8)2 + 1 (k6 + k8)2 )

Very compact expression at 2 loops. Higher loops within reach!

10

TWO-LOOP INTEGRAND

Cuts merged into an integrand with diagrams & numerators:

(2)

1 2 3 4

(1)

1 2 3 4 5 6 7

(3)

1 2 3 4

(4)

1 2 4 3

(5)

1 2 3 4

(6)

2 1 3 4

(8) (7)

1 2 3 4 1 2 3 4

Diagram symmetries imposed. ∼ 90KB file. 4 and D dimensional results agree, up to “µ” terms with no classical effects.

11

INTEGRATING THE AMPLITUDE

• m1 ̸= m2, even planar master integrals unknown!

Smirnov, '01; Lower topologies: Henn & Smirnov, '13; Duhr, Amplitudes '18; Heller, von Manteuffel, Schabinger, '19 Bianchi, Leoni, 1612.05609

• Simplification 1: expand in small q ∼ ℏ/R ≤ mi, √s.
• Simplification 2: Expand in v ≪ 1 from potential
• region. (

∫ d4ℓ) localized on +ve energy matter poles.

12

NR INTEGRATION / VELOCITY EXPANSION

Plan: Series expansion around static limit, then resum by matching to simple functions. Step 1: determine integrand in potential region

IT = 1 (E1 + ω)2 − (p + l)2 − m2

1

× 1 ω2 − l2 1 ω2 − (l + q)2 = 1 ω − ωP1 1 2m1 l2(l + q)2 + . . . 1 (E1 + ω)2 − (p + l)2 − m2

1

= 1 (ω − ωP1)(ω − ωA1), ωP1, ωA1 = −E1 ± √ E2

1 + 2pl + l2 .

ωP1 ≪ |l|, ωA1 ≈ −2m1

13

NR INTEGRATION / VELOCITY EXPANSION

Step 2: Energy integration, keeping only residues from +ve energy matter poles. 1 2π ∫ dω1 ∫ dω2 δ(ω1 + ω2) 1 2 [ 1 ω1 − ωP1 + i0 + ω1 ↔ ω2 ] = 1 2π ∫ dω 1 2 [ 1 ω − ωP1 + i0 + 1 −ω − ωP1 + i0 ] = − i 2 Step 3: Spatial integration

∫ d3l (2π)3 ( − i 2 ) 1 2m1 l2(l + q)2 = − i 32m1|q| . Only need 3D propagator integrals. More in Mikhail Solon’s talk

14

RELATIVISTIC INTEGRATION

cannot both collapose

Conservative contribution: localize on matter poles Ii = ∫ ddℓ1 ∫ ddℓ2 δ(ℓ2

1 − m2 1)δ(ℓ2 2 − m2 2) Ni

∏7

i=3 ℓ2 i

Exact v result from differential equations ∂Ii/∂s = Mij Ij.

IBP done by Kira [Maierhoefer, Usovitsch, Uwer, ’17]. 9 master integrals (H & N topology) rescaled to O(t0). Take t → 0 limit in Mij. Boundary condition: regular at threshold s = (m1 + m2)2. Scalar result: H + xH = log(−t) 128π2 arcsinh (|p|/m1) + arcsinh (|p|/m2) |p| (E1 + E2) .

15

COMPARISON WITH UNEXPANDED INTEGRAL

m1 = m2 results in [Bianchi, Leoni, 1612.05609], thanks to Loopedia.org.

25 master integrals

(IH)

• ϵ0 = −4

3 log(−t) x 1 − x2 ( π2 log x + log3 x ) + (non-singular in t) (IxH)

• ϵ0 = −4

3 log(−t) −x 1 − x2 ( π2 log(−x) + log3(−x) ) + (non-singular in t) . log(x) → log(−x) + iπ, (IH + IxH)

• ϵ0 = 4π2 log(−t) log(−x) + imaginary

16

Effective potential between black holes

PM POTENTIAL AS EFT MATCHING COEFFICIENT

[From Cheung, Rothstein, Solon, ’18 PRL]

(k0, k) = i k0 − √ k2 + m2

A,B + i0

, k k′

• k′
• k

= −iV (k, k′) ,

• kinetic term with only +ve

energy pole.

• k0-independent vertex.

Matching at L loops gives V(k, k′) with smooth ℏ → 0 limit. Uncalculated IR divergent integrals cancel in matching.

ML-loop

EFT

=

· · ·

p

• p

k1

• k1

kL

• kL

p′

• p′

= ML-loop

full

Alternative QM treatment: [Cristofoli, Bjerrum-Bohr, Damgaard, Vanhove, ’19]

17

RESULT: 3PM CONSERVATIVE POTENTIAL

[Bern, Cheung, Roiban, Shen, Solon, MZ, PRL ’19; 1908.01493 (long paper)]

H3PM(p, r) = √ p2 + m2

1 +

√ p2 + m2

2 + V3PM(p, r)

V3PM(p, r) =

3

∑

n=1

( G |r| )n cn(p2)

m = m1 + m2, ν = m1m2 m2 , E = E1 + E2, ξ = E1E2 E2 , γ = E m , σ = p1 · p2 m1m2

c1 = ν2m2 γ2ξ ( 1 − 2σ2) , c2 = ν2m3 γ2ξ [ 3 4 ( 1 − 5σ2) − 4νσ (1 − 2σ2) γξ − ν2(1 − ξ) (1 − 2σ2)2 2γ3ξ2 ] , c3 = ν2m4 γ2ξ [ 1 12 ( 3 − 6ν + 206νσ − 54σ2 + 108νσ2 + 4νσ3) − 4ν (3 + 12σ2 − 4σ4) arcsinh √

σ−1 2

√ σ2 − 1 − 3νγ (1 − 2σ2) (1 − 5σ2) 2(1 + γ)(1 + σ) − 3νσ (7 − 20σ2) 2γξ + 2ν3(3 − 4ξ)σ (1 − 2σ2)2 γ4ξ3 − ν2 (3 + 8γ − 3ξ − 15σ2 − 80γσ2 + 15ξσ2) (1 − 2σ2) 4γ3ξ2 + ν4(1 − 2ξ) (1 − 2σ2)3 2γ6ξ4 ] .

EFT matching coefficients 18

VALIDATION

• 4PN part of Hamiltonian agrees with [Jaranowski, Schäfer,

1508.01016]. New 5PN part subsequently verified [Bini, Damour, Geralico, ’19].

• Equivalence shown by:
• 1. Canonical transformations of Hamiltonian.
• 2. Gauge-invariant EFT amplitudes.
• 3. Scattering angles from classical equations of motion.
• 4. Binding-energy of quasi-circular orbit.
• “ ” terms have no classical effects, by dimension shifting.
• Apparent collinear divergence due to expansion in small

q m s. Small-q limit does not commute with small-m regge limit of [Amati, Ciafaloni, Veneziano, ’90].

19

VALIDATION

• 4PN part of Hamiltonian agrees with [Jaranowski, Schäfer,

1508.01016]. New 5PN part subsequently verified [Bini, Damour, Geralico, ’19].

• Equivalence shown by:
• 1. Canonical transformations of Hamiltonian.
• 2. Gauge-invariant EFT amplitudes.
• 3. Scattering angles from classical equations of motion.
• 4. Binding-energy of quasi-circular orbit.
• “ ” terms have no classical effects, by dimension shifting.
• Apparent collinear divergence due to expansion in small

q m s. Small-q limit does not commute with small-m regge limit of [Amati, Ciafaloni, Veneziano, ’90].

19

VALIDATION

• 4PN part of Hamiltonian agrees with [Jaranowski, Schäfer,

1508.01016]. New 5PN part subsequently verified [Bini, Damour, Geralico, ’19].

• Equivalence shown by:
• 1. Canonical transformations of Hamiltonian.
• 2. Gauge-invariant EFT amplitudes.
• 3. Scattering angles from classical equations of motion.
• 4. Binding-energy of quasi-circular orbit.
• “ ” terms have no classical effects, by dimension shifting.
• Apparent collinear divergence due to expansion in small

q m s. Small-q limit does not commute with small-m regge limit of [Amati, Ciafaloni, Veneziano, ’90].

19

VALIDATION

• 4PN part of Hamiltonian agrees with [Jaranowski, Schäfer,

1508.01016]. New 5PN part subsequently verified [Bini, Damour, Geralico, ’19].

• Equivalence shown by:
• 1. Canonical transformations of Hamiltonian.
• 2. Gauge-invariant EFT amplitudes.
• 3. Scattering angles from classical equations of motion.
• 4. Binding-energy of quasi-circular orbit.
• “ ” terms have no classical effects, by dimension shifting.
• Apparent collinear divergence due to expansion in small

q m s. Small-q limit does not commute with small-m regge limit of [Amati, Ciafaloni, Veneziano, ’90].

19

VALIDATION

• 4PN part of Hamiltonian agrees with [Jaranowski, Schäfer,

1508.01016]. New 5PN part subsequently verified [Bini, Damour, Geralico, ’19].

• Equivalence shown by:
• 1. Canonical transformations of Hamiltonian.
• 2. Gauge-invariant EFT amplitudes.
• 3. Scattering angles from classical equations of motion.
• 4. Binding-energy of quasi-circular orbit.
• “ ” terms have no classical effects, by dimension shifting.
• Apparent collinear divergence due to expansion in small

q m s. Small-q limit does not commute with small-m regge limit of [Amati, Ciafaloni, Veneziano, ’90].

19

VALIDATION

• 4PN part of Hamiltonian agrees with [Jaranowski, Schäfer,

1508.01016]. New 5PN part subsequently verified [Bini, Damour, Geralico, ’19].

• Equivalence shown by:
• 1. Canonical transformations of Hamiltonian.
• 2. Gauge-invariant EFT amplitudes.
• 3. Scattering angles from classical equations of motion.
• 4. Binding-energy of quasi-circular orbit.
• “ ” terms have no classical effects, by dimension shifting.
• Apparent collinear divergence due to expansion in small

q m s. Small-q limit does not commute with small-m regge limit of [Amati, Ciafaloni, Veneziano, ’90].

19

VALIDATION

• 4PN part of Hamiltonian agrees with [Jaranowski, Schäfer,

1508.01016]. New 5PN part subsequently verified [Bini, Damour, Geralico, ’19].

• Equivalence shown by:
• 1. Canonical transformations of Hamiltonian.
• 2. Gauge-invariant EFT amplitudes.
• 3. Scattering angles from classical equations of motion.
• 4. Binding-energy of quasi-circular orbit.
• “µ” terms have no classical effects, by dimension shifting.
• Apparent collinear divergence due to expansion in small

q m s. Small-q limit does not commute with small-m regge limit of [Amati, Ciafaloni, Veneziano, ’90].

19

VALIDATION

• 4PN part of Hamiltonian agrees with [Jaranowski, Schäfer,

1508.01016]. New 5PN part subsequently verified [Bini, Damour, Geralico, ’19].

• Equivalence shown by:
• 1. Canonical transformations of Hamiltonian.
• 2. Gauge-invariant EFT amplitudes.
• 3. Scattering angles from classical equations of motion.
• 4. Binding-energy of quasi-circular orbit.
• “µ” terms have no classical effects, by dimension shifting.
• Apparent collinear divergence due to expansion in small

q ≪ m, s. Small-q limit does not commute with small-m regge limit of [Amati, Ciafaloni, Veneziano, ’90].

19

PREDICTIONS FOR BINDING ENERGY

Comparision with state-of-the-art numerical relativity (“truth”), PN, and EOB approximations. [Antonelli et al., 2019]

• Clear improvement over lower

PM orders.

• Not reaching accuracy of 4PN.
• Hyperbolic orbit comparison

would be more illuminating.

20

PREDICTIONS FOR BINDING ENERGY

Comparision with state-of-the-art numerical relativity (“truth”), PN, and EOB approximations. [Antonelli et al., 2019]

• Clear improvement over lower

PM orders.

• Not reaching accuracy of 4PN.
• Hyperbolic orbit comparison

would be more illuminating.

20

FUTURE OUTLOOK

• Higher orders within reach. Double copy and EFT

integration methods expected to scale well.

• Tail effect, i.e. conservative radiation reaction, to be

treated by ultrasoft modes in EFT.

• Relativistic integration to be further explored to

bypass velocity resummation Canonical differential equations for integrals in soft expansion.

• Spin, finite-size effects in PM expansion [Bini, Damour, ’17;

Vines, ’17, Bini, Damour, ’18; Guevera, Ochirov, Vines, ’18; Vines, Steinhoff, Buonanno, ’18; Chung, Huang, Kim, Lee, ’18; Maybee O’Connell, Vines, ’19; Guevera, Ochirov, Vines, ’19] 21

FUTURE OUTLOOK

• Higher orders within reach. Double copy and EFT

integration methods expected to scale well.

• Tail effect, i.e. conservative radiation reaction, to be

treated by ultrasoft modes in EFT.

• Relativistic integration to be further explored to

bypass velocity resummation Canonical differential equations for integrals in soft expansion.

• Spin, finite-size effects in PM expansion [Bini, Damour, ’17;

Vines, ’17, Bini, Damour, ’18; Guevera, Ochirov, Vines, ’18; Vines, Steinhoff, Buonanno, ’18; Chung, Huang, Kim, Lee, ’18; Maybee O’Connell, Vines, ’19; Guevera, Ochirov, Vines, ’19] 21

FUTURE OUTLOOK

• Higher orders within reach. Double copy and EFT

integration methods expected to scale well.

• Tail effect, i.e. conservative radiation reaction, to be

treated by ultrasoft modes in EFT.

• Relativistic integration to be further explored to

bypass velocity resummation = ⇒ Canonical differential equations for integrals in soft expansion.

• Spin, finite-size effects in PM expansion [Bini, Damour, ’17;

Vines, ’17, Bini, Damour, ’18; Guevera, Ochirov, Vines, ’18; Vines, Steinhoff, Buonanno, ’18; Chung, Huang, Kim, Lee, ’18; Maybee O’Connell, Vines, ’19; Guevera, Ochirov, Vines, ’19] 21

FUTURE OUTLOOK

• Higher orders within reach. Double copy and EFT

integration methods expected to scale well.

• Tail effect, i.e. conservative radiation reaction, to be

treated by ultrasoft modes in EFT.

• Relativistic integration to be further explored to

bypass velocity resummation = ⇒ Canonical differential equations for integrals in soft expansion.

• Spin, finite-size effects in PM expansion [Bini, Damour, ’17;

Vines, ’17, Bini, Damour, ’18; Guevera, Ochirov, Vines, ’18; Vines, Steinhoff, Buonanno, ’18; Chung, Huang, Kim, Lee, ’18; Maybee O’Connell, Vines, ’19; Guevera, Ochirov, Vines, ’19] 21

Thank you!

22