Double Copy for Classical Gravity

Home , Dilaton

Alexander Ochirov ETH Zurich

QCD Meets Gravity, Bhaumik Institute, UCLA, Dec 13, 2017

1 / 25 Invitation: double copy at a glance

Bern, Carrasco, Johansson (2008)

Amplitudes for quantum scattering:

 X njcj  YM = BCJ duality  A Dj j cubic for gauge theory: ∈  c c = c n n = n i − j k ⇔ i − j k

Double copy X njn0 Gravity = i j for gravity: M Dj j cubic ∈ zoo of BCJ/CHY-related theories

Want to use for classical Einstein gravity!

2 / 25 Outline

1. Nice special cases 2. Product graviton (a.k.a. fat) 3. Scattering in fat gravity 4. Towards Einstein gravity

5. Outlook

Focus: simplest nontrivial examples

3 / 25 Nice special cases

4 / 25 I Extensions possible and welcome Taub-NUT spacetimes by Luna, Monteiro, O’Connell, White (2015) Stress tensors, energy conditions by Ridgway, Wise (2015) Accelerating BHs by Luna, Monteiro, Nicholson, O’Connell, White (2016) Curved Kerr-Schild spacetimes by Bahjat-Abbas, Luna, White (2017) and Carrillo-Gonzalez, Penco, Trodden (2017) I Other similar solutions yet to be found I Schematic understanding summarized as a ˜b Hµν = Aµ ? Φab ? Aν Anastasiou, Borsten, Duﬀ, Hughes, Nagy (2014) also Silvia’s talk I Need to understand from more general framework

Nice special cases Monteiro, O’Connell (2011) Self-dual gravity vs Self-dual Yang-Mills h+w ¯ = ∂w∂ φ A+ = ∂wφ − 2 − − hw¯w¯ = ∂ φ Aw¯ = ∂ φ − − − − Monteiro, O’Connell, White (2014) Kerr-Schild metrics vs Abelian gauge ﬁeld 2 gµν = ηµν + kµkνφ, k = 0 Aµ = kµφ | {z } κhµν

5 / 25 Nice special cases Monteiro, O’Connell (2011) Self-dual gravity vs Self-dual Yang-Mills h+w ¯ = ∂w∂ φ A+ = ∂wφ − 2 − − hw¯w¯ = ∂ φ Aw¯ = ∂ φ − − − − Monteiro, O’Connell, White (2014) Kerr-Schild metrics vs Abelian gauge ﬁeld 2 gµν = ηµν + kµkνφ, k = 0 Aµ = kµφ | {z } κhµν I Extensions possible and welcome Taub-NUT spacetimes by Luna, Monteiro, O’Connell, White (2015) Stress tensors, energy conditions by Ridgway, Wise (2015) Accelerating BHs by Luna, Monteiro, Nicholson, O’Connell, White (2016) Curved Kerr-Schild spacetimes by Bahjat-Abbas, Luna, White (2017) and Carrillo-Gonzalez, Penco, Trodden (2017) I Other similar solutions yet to be found I Schematic understanding summarized as a ˜b Hµν = Aµ ? Φab ? Aν Anastasiou, Borsten, Duﬀ, Hughes, Nagy (2014) also Silvia’s talk I Need to understand from more general framework 5 / 25 Product graviton

6 / 25 Perturbative classical double copy

Luna, Monteiro, Nicholson, AO, O’Connell, Westerberg, White (2016)

2 (0) 2 µν κ µν ∂ A = Jµ ∂ H = T X µ ⇔ | − 2 ⇔

X 2 | 2 ∂2A(1) A(0) ∂2H(1) H(0) µ ∝ ⇔ µν ∝ ⇔ X | X | 2 (2) (0)3 2 (2) (0)3 ∂ Aµ A ∂ Hµν H ∝ ⇔ | ∝ ⇔ X | X

Natural ﬁeld Hµν — “product” graviton (a.k.a. “fat”)

Luna, Nicholson, O’Connell, White (2017)

7 / 25 Perturbations for simplest source Luna, Monteiro, Nicholson, AO, O’Connell, Westerberg, White (2016)

µν µ ν (3) Stationary mass at origin:∗ T = Mu u δ (x)

(0)µν κ 1 µν κ M µ ν H = T = u u X 2 p2 2 4πr X (1)µν 1 YM YMµν,µ2ν2,µ3ν3 (0) (0) H = 2 V3 V3 Hµ2ν2 Hµ3ν3 −2p ⊗ X κ2 M 2 = [simple calculation] = rˆµrˆν − 2 4(4πr)2

| YM i V3 µ µ µ = √ ηµ µ (p1−p2)µ +ηµ µ (p2−p3)µ +ηµ µ (p3−p1)µ = 1 2 3 2 1 2 3 2 3 1 3 1 2 |

∗uµ = (1, 0),r ˆµ = (0, x/r) and r = |x| 8 / 25 Perturbations for simplest source Luna, Monteiro, Nicholson, AO, O’Connell, Westerberg, White (2016)

µν µ ν (3) Stationary mass at origin:† T = Mu u δ (x)

(0)µν κ 1 µν κ M µ ν H = T = u u X 2 p2 2 4πr X (1)µν 1 YM YMµν,µ2ν2,µ3ν3 (0) (0) H = 2 V3 V3 Hµ2ν2 Hµ3ν3 −2p ⊗ X κ2 M 2 = [simple calculation] = rˆµrˆν − 2 4(4πr)2

Orders H(0)µν, H(1)µν and H(2)µν together:

κ M κ3 M 2 κ5 M 3 Hµν = uµuν rˆµrˆν uµuν + O(κ7) 2 4πr − 2 4(4πr)2 − 2 6(4πr)3

†uµ = (1, 0),r ˆµ = (0, x/r) and r = |x| 9 / 25 Harmonic coordinate condition = de Donder gauge: µν µν µν µν µν √ g g η κ h ∂µ √ g g = 0 ∂µh = 0 − ≡ − − ⇔ Luna, Monteiro, Nicholson, AO, O’Connell, Westerberg, White (2016) I Perturbative map between Hµν to hµν, φ and Bµν: (i) (i) q (i) q (i) (i) (i) Hµν = hµν Pµν h + Pµν φ + Bµν + µν − T q 1 qµ∂ν + qν ∂µ P = ηµν µν D 2 − (q ∂) − · (0) I maps harmonic to “BCJ” coordinates, µν = 0 Tµν T κ M κ3 1 Result: hµν = uµuν + [6uµuν + 2ˆrµrˆν ] + O(κ5) 2 4πr 2 8(4πr)2

κ M 5 φ = + O(κ ),Bµν = 0 − 2 4πr

10 / 25 Luna, Monteiro, Nicholson, AO, O’Connell, Westerberg, White (2016) I Perturbative map between Hµν to hµν, φ and Bµν: (i) (i) q (i) q (i) (i) (i) Hµν = hµν Pµν h + Pµν φ + Bµν + µν − T q 1 qµ∂ν + qν ∂µ P = ηµν µν D 2 − (q ∂) − · (0) I maps harmonic to “BCJ” coordinates, µν = 0 Tµν T κ M κ3 1 Result: hµν = uµuν + [6uµuν + 2ˆrµrˆν ] + O(κ5) 2 4πr 2 8(4πr)2

κ M 5 φ = + O(κ ),Bµν = 0 − 2 4πr

10 / 25 κ M κ3 1 Result: hµν = uµuν + [6uµuν + 2ˆrµrˆν ] + O(κ5) 2 4πr 2 8(4πr)2

κ M 5 φ = + O(κ ),Bµν = 0 − 2 4πr

Interpretation of results Theory in question: Fat gravity — “ = 0 supergravity” or “NS-NS gravity” N stringy name e.g. in Geyer, Monteiro (2017) Z D 2 1 µ 1 −2κφ/(D−2) λµν SN = d x√ g R ∂µφ∂ φ e Hλµν H =0 − κ2 − 2(D 2) − 6 − Harmonic coordinate condition = de Donder gauge: µν µν µν µν µν √ g g η κ h ∂µ √ g g = 0 ∂µh = 0 − ≡ − − ⇔ Luna, Monteiro, Nicholson, AO, O’Connell, Westerberg, White (2016) I Perturbative map between Hµν to hµν, φ and Bµν: (i) (i) q (i) q (i) (i) (i) Hµν = hµν Pµν h + Pµν φ + Bµν + µν − T q 1 qµ∂ν + qν ∂µ P = ηµν µν D 2 − (q ∂) − · (0) I maps harmonic to “BCJ” coordinates, µν = 0 Tµν T

10 / 25 Interpretation of results Theory in question: Fat gravity — “ = 0 supergravity” or “NS-NS gravity” N stringy name e.g. in Geyer, Monteiro (2017) Z D 2 1 µ 1 −2κφ/(D−2) λµν SN = d x√ g R ∂µφ∂ φ e Hλµν H =0 − κ2 − 2(D 2) − 6 − Harmonic coordinate condition = de Donder gauge: µν µν µν µν µν √ g g η κ h ∂µ √ g g = 0 ∂µh = 0 − ≡ − − ⇔ Luna, Monteiro, Nicholson, AO, O’Connell, Westerberg, White (2016) I Perturbative map between Hµν to hµν, φ and Bµν: (i) (i) q (i) q (i) (i) (i) Hµν = hµν Pµν h + Pµν φ + Bµν + µν − T q 1 qµ∂ν + qν ∂µ P = ηµν µν D 2 − (q ∂) − · (0) I maps harmonic to “BCJ” coordinates, µν = 0 Tµν T κ M κ3 1 Result: hµν = uµuν + [6uµuν + 2ˆrµrˆν ] + O(κ5) 2 4πr 2 8(4πr)2

κ M 5 φ = + O(κ ),Bµν = 0 − 2 4πr 10 / 25 JNW solution

Fisher (1948), Buchdahl (1959), Bronnikov (1973), Wyman (1981) Janis, Newman, Winicour (1968) Schwarzschild coordinates:

γ γ 1 γ R R− R − ds2 = 1 dt2 + 1 dρ2 + 1 ρ2dΩ2 − − ρ − ρ − ρ κ Y R φ = ln 1 2 4πR − ρ

2 solution parameters (R, γ) (M,Y ): ⇔ p κ2 √M 2 + Y 2 M R = 2G M 2 + Y 2 = , γ = 2 4π √M 2 + Y 2

I To harmonic coordinates via r = ρ R/2 −

11 / 25 So far: I Used harmonic coordinates to jumpstart product graviton I Identiﬁed one-body solution as JNW naked singularity I Struggled with gauge dependence to obtain metric

JNW match

12 / 25 JNW match

JNW to harmonic coordinates via r = ρ R/2: − κ M κ3 1 hµν = uµuν + (7M 2 Y 2)uµuν + (M 2 +Y 2)ˆrµrˆν + O(κ5) JNW 2 4πr 2 8(4πr)2 − κ Y φ = + O(κ5) JNW − 2 4πr vs κ M κ3 1 hµν = uµuν + [6uµuν + 2ˆrµrˆν ] + O(κ5) 2 4πr 2 8(4πr)2 κ M φ = + O(κ5) − 2 4πr Match for M = Y ! So far: I Used harmonic coordinates to jumpstart product graviton I Identiﬁed one-body solution as JNW naked singularity I Struggled with gauge dependence to obtain metric 12 / 25 Scattering in fat gravity

13 / 25 Perturbative setup: µ (0)µ (1)µ (0)µ µ µ xj (τ) = xj (τ) + xj (τ) + . . . , xj (τ) = bj + vj τ a (0)a (1)a (0)a a cj (τ) = cj (τ) + cj (τ) + . . . , cj (τ) = cj | a (0)a (1)a Aµ(x) = Aµ (x) + Aµ (x) + ... = | + + ...

Classical scattering from Yang-Mills to gravity

Goldberger, Ridgway (2016)

2 colored point particles classically: Z µ a a X (d) a D G = J = g dτ δ x xj(τ) c (τ)vjν (τ) YM e.o.m. µν ν − j j=1,2

dvjµ(τ) a a ν mj = gG c (τ)v (τ) Lorentz force dτ µν j j dca(τ) j = gf abcAb cc(τ)vµ(τ) current conservation dτ µ j j

14 / 25 Classical scattering from Yang-Mills to gravity

Goldberger, Ridgway (2016)

2 colored point particles classically: Z µ a a X (d) a D G = J = g dτ δ x xj(τ) c (τ)vjν (τ) YM e.o.m. µν ν − j j=1,2

dvjµ(τ) a a ν mj = gG c (τ)v (τ) Lorentz force dτ µν j j dca(τ) j = gf abcAb cc(τ)vµ(τ) current conservation dτ µ j j

Perturbative setup: µ (0)µ (1)µ (0)µ µ µ xj (τ) = xj (τ) + xj (τ) + . . . , xj (τ) = bj + vj τ a (0)a (1)a (0)a a cj (τ) = cj (τ) + cj (τ) + . . . , cj (τ) = cj | a (0)a (1)a Aµ(x) = Aµ (x) + Aµ (x) + ... = | + + ...

| 14 / 25 Result: κ 3 Z k2H(1) = − m m d−q d−q −δ(k−q −q )−δ(q ·v )ei(q1·b1)−δ(q ·v )ei(q2·b2) µν 2 1 2 1 2 1 2 1 1 2 2 µ ν 2 µ ν µ ν P12P12 v1 ·v2 (µ ν) (v1 ·v2) Q12Q12 P12P12 × 2 2 + 2 2 P12 Q12 + 2 2 − 2 2 , q1 q2 q1 q2 4 q1 q2 (k·v2) (k·v2) 2 µ 2 µ µ µ µ µ µ q1 v1 q2 v2 P12 =(k·v1)v2 − (k·v2)v1 ,Q12 = (q1 − q2) − + (k·v1) (k·v2)

form from Luna, Nicholson, O’Connell, White (2017) Intepretation: radiation from scattering of JNW singularities

Classical scattering in fat gravity

Goldberger, Ridgway (2016)

| a (0)a (1)a Aµ(x) = Aµ (x) + Aµ (x) + ... = | + + ...

| X (1)a (1) Classical double copy: Aµ (x) Hµν (x) = → X

15 / 25 Classical scattering in fat gravity

Goldberger, Ridgway (2016)

| a (0)a (1)a Aµ(x) = Aµ (x) + Aµ (x) + ... = | + + ...

| X (1)a (1) Classical double copy: Aµ (x) Hµν (x) = → X Result: κ 3 Z k2H(1) = − m m d−q d−q −δ(k−q −q )−δ(q ·v )ei(q1·b1)−δ(q ·v )ei(q2·b2) µν 2 1 2 1 2 1 2 1 1 2 2 µ ν 2 µ ν µ ν P12P12 v1 ·v2 (µ ν) (v1 ·v2) Q12Q12 P12P12 × 2 2 + 2 2 P12 Q12 + 2 2 − 2 2 , q1 q2 q1 q2 4 q1 q2 (k·v2) (k·v2) 2 µ 2 µ µ µ µ µ µ q1 v1 q2 v2 P12 =(k·v1)v2 − (k·v2)v1 ,Q12 = (q1 − q2) − + (k·v1) (k·v2)

form from Luna, Nicholson, O’Connell, White (2017) Intepretation: radiation from scattering of JNW singularities

15 / 25 Connection to scattering amplitude

Luna, Nicholson, O’Connell, White (2017)

Start from Yang-Mills again: µ 2 = (D Φ)†(D Φ) m Φ†Φ Lscalar µ ¯ı i − ¯ı i µ µ 2 µ = ig (∂ Φ†)A Φ Φ†A (∂ Φ) + g Φ†A A Φ Lint − µ − µ µ

Feynman diagrams are

2,¯ 4,l 2,¯ 4,l 2,¯ 4,l 2,¯ 4,l 2,¯ 4,l = 5,a + 5,a + 5,a + 5,a + 5,a A 1,¯ı 3,k 1,¯ı 3,k 1,¯ı 3,k 1,¯ı 3,k 1,¯ı 3,k 2,¯ 4,l 2,¯ 4,l 2,¯ 4,l 2,¯ 4,l + 5,a + 5,a + 5,a + 5,a 1,¯ı 3,k 1,¯ı 3,k 1,¯ı 3,k 1,¯ı 3,k

16 / 25 Scalar scattering amplitude with gluon radiation AO (2014, PhD Thesis) Luna, Nicholson, O’Connell, White (2017)

 p2 p q p2 p q  2 − 2 2 − 2 i n1 = n k + k = √ 2(p1 ·ε)((p1 −q1)·(2p2 −q2)) 2 p1 p1 q1 p1 p1 q1 − − + (p1 ·k)(ε·(2p2 −q2))

 p2 p q p2 p q  2 − 2 2 − 2 i n3 = n k + k = √ 2((p1 −q1)·ε)(p1 ·(2p2 −q2)) 2 p1 p1 q1 p1 p1 q1 − − − ((p1 −q1)·k)(ε·(2p2 −q2)) i  p2 p2 q2  √ (q1 ·ε)((2p1 −q1)·(2p2 −q2)) − 2 n5 = n k =   +((2p2 −q2)·ε)((2p1 −q1)·k) p1 p1 q1 − −((2p1 −q1)·ε)((2p2 −q2)·k)

Color-kinematic duality: c c = c n n = n 1 − 3 5 1 − 3 5 17 / 25 Result after double copy:

3 2 2 = iκ m m εµεν MCl − 1 2 µ ν 2 µ ν µ ν P P v v µ ν (v v ) Q Q P P 12 12 + 1 · 2 P ( Q ) + 1 · 2 12 12 12 12 , × q2q2 q2q2 12 12 4 q2q2 − (k v )2(k v )2 1 2 1 2 1 2 · 2 · 2 q2vµ q2vµ P µ =(k v )vµ (k v )vµ,Qµ = (q q )µ 1 1 + 2 2 12 · 1 2 − · 2 1 12 1 − 2 − (k v ) (k v ) · 1 · 2

Classical limit — large-mass expansion Luna, Nicholson, O’Connell, White (2017) Reinstate ~: 2 i µ m ig S[A, Φ], scalar = (DµΦ)¯ı†(D Φ)i Φ¯ı†Φi,Dµ = ∂µ Aµ ~ L − ~2 − ~ inhomogeneous in m large-mass expansion A j ⇒

e.g. n = 4i m2m (v v )(v ε) + O(m2), etc. 1 √2 1 2 1 · 2 1 ·

18 / 25 Classical limit — large-mass expansion Luna, Nicholson, O’Connell, White (2017) Reinstate ~: 2 i µ m ig S[A, Φ], scalar = (DµΦ)¯ı†(D Φ)i Φ¯ı†Φi,Dµ = ∂µ Aµ ~ L − ~2 − ~ inhomogeneous in m large-mass expansion A j ⇒

e.g. n = 4i m2m (v v )(v ε) + O(m2), etc. 1 √2 1 2 1 · 2 1 ·

Result after double copy:

Compare results: Goldberger, Ridgway (2016) κ 3 Z k2H(1) = − m m d−q d−q −δ(k−q −q )−δ(q ·v )ei(q1·b1)−δ(q ·v )ei(q2·b2) µν 2 1 2 1 2 1 2 1 1 2 2 µ ν 2 µ ν µ ν P12P12 v1 ·v2 (µ ν) (v1 ·v2) Q12Q12 P12P12 × 2 2 + 2 2 P12 Q12 + 2 2 − 2 2 q1 q2 q1 q2 4 q1 q2 (k·v2) (k·v2) 3 2 2 MCl = − iκ m1m2εµεν µ ν 2 µ ν µ ν P12P12 v1 ·v2 (µ ν) (v1 ·v2) Q12Q12 P12P12 × 2 2 + 2 2 P12 Q12 + 2 2 − 2 2 q1 q2 q1 q2 4 q1 q2 (k·v2) (k·v2)

Luna, Nicholson, O’Connell, White (2017) Product graviton radiation from scattering amplitude: Z 2 (1) i i(q1 b1) i(q2 b2) amp. k Hµν = − d−q1d−q2−δ(k q1 q2)−δ(q1 v1)e · −δ(q2 v2)e · Cl 8m1m2 − − · · M

Integration from spacetime wave packets of intermediate particles: Z − − iqj ·(bj −x) ψj x = d qjδ(qj vj)e qj | i · | i

19 / 25 Towards Einstein gravity

20 / 25 q q εµε˜ν = ε(µε˜ν) (ε ε˜)Pµν + (ε ε˜)Pµν + ε[µε˜ν] | −{z · } | · {z } | {z } graviton dilaton axion 1 k q + k q P q = η µ ν ν µ µν D 2 µν − (k q) − · Massive scalar couples to gravitons and dilaton: 2iκ = 2iκpµpνε , = − m2, = 0 Mgraviton − µν Mdilaton D 2 Maxion −

Appearance of dilaton and axion Double copy of massive scalar vertex: k,a = i√2g(p ε)T a p k p · − k µ ν = 2iκ(p ε)(p ε˜) = 2iκp p εµε˜ν p k p − · · − −

21 / 25 Massive scalar couples to gravitons and dilaton: 2iκ = 2iκpµpνε , = − m2, = 0 Mgraviton − µν Mdilaton D 2 Maxion −

Appearance of dilaton and axion Double copy of massive scalar vertex: k,a = i√2g(p ε)T a p k p · − k µ ν = 2iκ(p ε)(p ε˜) = 2iκp p εµε˜ν p k p − · · − −

q q εµε˜ν = ε(µε˜ν) (ε ε˜)Pµν + (ε ε˜)Pµν + ε[µε˜ν] | −{z · } | · {z } | {z } graviton dilaton axion 1 k q + k q P q = η µ ν ν µ µν D 2 µν − (k q) − ·

21 / 25 Appearance of dilaton and axion Double copy of massive scalar vertex: k,a = i√2g(p ε)T a p k p · − k µ ν = 2iκ(p ε)(p ε˜) = 2iκp p εµε˜ν p k p − · · − −

q q εµε˜ν = ε(µε˜ν) (ε ε˜)Pµν + (ε ε˜)Pµν + ε[µε˜ν] | −{z · } | · {z } | {z } graviton dilaton axion 1 k q + k q P q = η µ ν ν µ µν D 2 µν − (k q) − · Massive scalar couples to gravitons and dilaton: 2iκ = 2iκpµpνε , = − m2, = 0 Mgraviton − µν Mdilaton D 2 Maxion − 21 / 25 Fundamental double copy + chiral fermions: Johansson, AO (2014) + + + + (ψ ψ−) (ψ− ψ ) φ − φ− = φ a ⊗ ⊕ ⊗ ⇒ ⊕ ⊕ + + + + + + 2 − 3− 2 3− 2− 3 2 3− 2− 3

= + × ×

+ + + + + + 1− 4 − 1− 4 1 4− 1− 4 1 4−

Gravity states in 4d

Graviton states: Scalar states:

+ + + + dilaton axion + εµν = εµ εν εµν− = εµν + εµν = εµ εν− + dilaton axion + ε− = ε−ε− ε− = ε ε = ε−ε µν µ ν µν µν − µν µ ν Symbolically: µ ν µν + + A A h φ − φ− ⊗ ⇒ ⊕ ⊕

22 / 25 Gravity states in 4d

Graviton states: Scalar states:

Fundamental double copy + chiral fermions: Johansson, AO (2014) + + + + (ψ ψ−) (ψ− ψ ) φ − φ− = φ a ⊗ ⊕ ⊗ ⇒ ⊕ ⊕ + + + + + + 2 − 3− 2 3− 2− 3 2 3− 2− 3

= + × ×

+ + + + + + 1− 4 − 1− 4 1 4− 1− 4 1 4−

22 / 25 Pure gravities via fundamental double copy

Johansson, AO (2014)

SUGRA tensoring vector states ghosts = matter matter ⊗ µ ν µν (φ φ) (φ φ) 2φ or =0+0 A A h ,φ,a + ⊗ ⊕ ⊗ +→ N ⊗ → (ψ ψ−) (ψ− ψ ) φ, a ⊗ ⊕ ⊗ → µ (Φ =1 φ) (Φ =1 φ) Φ′ =2 or N N N =1+0 =1 A G =1, Φ =2 ⊗ ⊕ ⊗ +→ N VN ⊗ → N N (Φ =1 ψ−) (Φ =1 ψ ) Φ =2 N ⊗ ⊕ N ⊗ → N µ + =2+0 =2 A G =2, =2 (Φ =2 ψ−) (Φ =2 ψ ) =2 N VN ⊗ → N VN N ⊗ ⊕ N ⊗ →VN =1+1 =1 =1 G =2, 2Φ =2 (Φ =1 Φ =1) (Φ =1 Φ =1) 2Φ =2 N VN ⊗VN → N N N ⊗ N ⊕ N ⊗ N → N + =2+1 =2 =1 G =3, =4 (Φ =2 Φ =1) (Φ =2 Φ =1) =4 N VN ⊗VN → N VN N ⊗ N ⊕ N ⊗ N →VN =2+2 =2 =2 G =4, 2 =4 (Φ =2 Φ =2) (Φ =2 Φ =2) 2 =4 N VN ⊗VN → N VN N ⊗ N ⊕ N ⊗ N → VN 2-loop with N = 2 SUSY by Johansson, K¨alin,Mogull (2017)

23 / 25 Cancelling dilaton and axion

I Ghost double copy of chiral fermions Johansson, AO (2014) I Scalar ghost double copy of scalar (for dilaton only) Luna, Nicholson, O’Connell, White (2017) I Projector-based product graviton (classical) Luna, Monteiro, Nicholson, AO, O’Connell, Westerberg, White (2016) I Double copy of Faddeev-Popov ghosts `ala Siegel Silvia’s talk I Double copy to pure gravity directly? Projector on internal edges e.g. Bern, Cheung, Chi, Davies, Dixon, Nohle (2015) Alfredo’s talk

24 / 25 Outlook Lots of recent progress on interface between ﬁelds

I From unbound to bound orbits via classical double copy Goldberger, Ridgway (2016), Goldberger, Prabhu, Thompson (2017) Goldberger, Ridgway (2017) Walter’s and Alec’s talks I Black holes in supersymmetric gravities N = 2 BPS black hole by Cardoso, Nagy, Nampuri (2016) Simon’s talk I Kerr-Schild double copy with curved background Bahjat-Abbas, Luna, White (2017) Carrillo-Gonzalez, Penco, Trodden (2017) I EFT+amplitudes techniques for scattering potentials etc. Bjerrum-Bohr, Donoghue, Holstein, Planté,Vanhove (2014-16) Foffa, Mastrolia, Sturani, Sturm (2016) Pierpaolo’s talk I Effective one-body Hamiltonian for 2 spinning BHs Bini, Damour (2017) Connection to amplitudes in Damour (2017) Challenge: 2-loop scattering amplitude of scalar masses

I Leading singularities for classical limit of gravity scattering Cachazo, Guevara (2017), Guevara (2017) Alfredo’s talk 25 / 25 Thank you!

26 / 25 Backup slides

27 / 25 Self-dual and Kerr-Schild cases

p 4 gµν = ηµν + κhµν (Exactly) κ = 32πGNewtonc−

Monteiro, O’Connell (2011) Self-dual gravity vs Self-dual Yang-Mills i στ i νρ Rλµνρ = 2 λµστ R νρ Fλµ = 2 λµνρF h+w ¯ = ∂w∂ φ A+ = ∂wφ − 2 − − hw¯w¯ = ∂ φ Aw¯ = ∂ φ 2 − − 2 − − ∂ φ + κ ∂wφ, ∂ φ = 0 ∂ φ + ig[∂wφ, ∂ φ]= 0 { − } −

Monteiro, O’Connell, White (2014) Kerr-Schild metrics vs Abelian gauge ﬁeld 2 gµν = ηµν + kµkνφ, k = 0 Aµ = kµφ | {z } κhµν Einstein equations linearize Maxwell equations µ κ ρ µ µ µ µ µ Rν = ∂ ∂ h + ∂ hρ ∂ hν ∂ F = ∂ ∂ A ∂ A 2 νρ ν − ρ µν µ ν − ν µ

28 / 25 Perturbative metrics and harmonic coordinates

g = η + κ h vs. √ g gµν = ηµν κ hµν µν µν µν − −

κ κ2 hµν = h(0)µν + h(1)µν + h(2)µν + 2 2 ··· κ κ2 Hµν = H(0)µν + H(1)µν + H(2)µν + 2 2 ···

Harmonic coordinate condition = de Donder gauge:

∂ √ g gµν = 0 ∂ h(i)µν = 0 µ − ⇔ µ

Simply related at lowest order: h hµν = hµν ηµν + O(κ) − 2

29 / 25 Linear order κ ∂2hµν ∂µ∂ hαν ∂ν∂ hµα + η ∂ ∂ hαβ = T µν − α − α µν α β − 2 κ de Donder gauge graviton ∂2hµν = T µν − 2 κ product graviton ∂2Hµν = T µν − 2 In momentum space:

(0)µν 1 κ µν H = T = X p2 2

Simplest source — stationary mass at origin:‡ κ M T µν = Muµuνδ(3)(x) H(0)µν = uµuν ⇒ 2 4πr

‡uµ = (1, 0) and r = |x| 30 / 25 First nonlinear order

iκ BCJ/KLT double copy at 3 points: = A A˜ M3 2 3 3 iκ V = V YM V YM, ⇒ 3 2 3 ⊗ 3

YM i V µ µ µ = ηµ1µ2 (p1 p2)µ3 + ηµ2µ3 (p2 p3)µ1 + ηµ3µ1 (p3 p1)µ2 3 1 2 3 √2 − − −

1 H(1)µν = V YM V YMµν H(0)µ2ν2 H(0)µ3ν3 −2p2 3 ⊗ 3 µ2ν2,µ3ν3 X = X

31 / 25 Linearized ﬁelds in = 0 gravity N Z D 2 1 µ 1 −2κφ/(D−2) λµν SN = d x√ g R ∂µφ∂ φ e Hλµν H , =0 − κ2 − 2(D 2) − 6 − where Hλµν = ∂λBµν + ∂µBνλ + ∂ν Bλµ Linearized equations of motion: 2 α α α β ∂ hµν ∂µ∂ hαν ∂ν ∂ hµα + ηµν ∂ ∂ hαβ = 0 − − 2 α α ∂ Bµν ∂µ∂ Bαν ∂ν ∂ Bµα = 0 − − ∂2φ = 0 Gauge freedom xµ xµ κ ξµ: → − 0 hµν h = hµν + ∂µξν + ∂ν ξµ ηµν (∂ ξ) → µν − · 0 Bµν B = Bµν + ∂µζν ∂ν ζµ → µν − First guess at product graviton Hµν = hµν + Bµν : 2 α α α β ∂ Hµν ∂µ∂ Hαν ∂ν ∂ Hµα + ηµν ∂ ∂ Hαβ = 0 − 0 − Hµν H = Hµν + ∂µ(ξν + ζν ) + ∂ν (ξµ ζµ) ηµν (∂ ξ) → µν − − · 32 / 25 Linear waves

2 ip x ∂ Hµν = 0 Hµν = Aµνe · ⇒  2 ip x  ∂ h = 0 h = a e ·  µν ⇒ µν µν 2 2 ip x p = 0 ∂ Bµν = 0 Bµν = bµνe ·  ⇒  2 ip x  ∂ φ = 0 φ = c e ·  ⇒

de Donder gauge residual freedom,

2 h h0 = h + ∂ ξ + ∂ ξ η (∂ ξ), ∂ ξ = 0 µν → µν µν µ ν ν µ − µν · ν tr a tr a0 = tr a i(D 2)(p ξ) → − − · µν can be used to ﬁx tr a = η aµν = 0

33 / 25 Linear waves Given complete set of (D 2) polarization vectors, − µ i µ i p εµ = q εµ = 0 p q + q p εi εi = η µ ν µ µ µ ν µν − (p q) · can decompose D D matrices into (D 2) (D 2) ones: × − × −  ip x i j ip x Hµν = Aµνe · = fijεµενe ·   ip x tr i j ip x  hµν = aµνe · = f 6 ε ε e · p2 = 0 ij µ ν ip x i j ip x  B = b e · = f˜ ε ε e ·  µν µν ij µ ν  ip x φ = c e ·

tr c δij f = f 6 + f˜ + ij ij ij D 2 ⇔ − φ p q + q p H = h + B + η µ ν µ µ µν µν µν D 2 µν − (p q) − · 34 / 25 Linear product graviton Luna, Monteiro, Nicholson, AO, O’Connell, Westerberg, White (2016) Linear waves: h = 0 q Hµν = hµν + Bµν + Pµνφ

General linear order: H = h P q h + B + P q φ µν µν − µν µν µν NB! simplicity due to using hµν and de Donder gauge Inverse:  1  φ = HB = (H H )  µν 2 µν − νµ 1 q q  h0 (H + H ) P H = h P h µν ≡ 2 µν νµ − µν µν − µν 0 hµν , hµν and H(µν) are equivalent up to gauge:

0 0 0 0 0 qµ h = hµν + ∂µξ + ∂ν ξ ηµν (∂ ξ ), ξ = h µν ν µ − · µ (D 2)(q ∂) − · qµ H = hµν + ∂µξν + ∂ν ξµ ηµν (∂ ξ), ξµ = (h φ) (µν) − · (D 2)(q ∂) − − · 35 / 25 Nonlinear product graviton Luna, Monteiro, Nicholson, AO, O’Connell, Westerberg, White (2016) Product graviton: H(0) = h(0) P q h(0) + B(0) + P q φ(0) µν µν − µν µν µν H(1) = h(1) P q h(1) + B(1) + P q φ(1) + (1) µν µν − µν µν µν Tµν

I maps harmonic to BCJ coordinates Tµν Inverse maps:  φ(0) = H(0)   (0) (0) Bµν = H[µν]  (0)  h0(0) = H P q H(0) = h(0) P q h(0) µν (µν) − µν µν − µν  φ(1) = H(1) (1)   − T B(1) = H(1) (1) µν [µν] − T[µν]  (1) (1)  h0(1) = H P q (H(1) (1)) = h(1) P q h(1) µν (µν) − T(µν) − µν − T µν − µν

36 / 25 Symmetric transformation function (1)µν T

Luna, Monteiro, Nicholson, AO, O’Connell, Westerberg, White (2016)

Z (1)µν 1 −D −D −D (0) (0)αβ µ ν ( p1) = 2 d p2d p3δ (p1 + p2 + p3) H2αβH3 p1 p1 T − 4p1 +8pαH(0) H(0)β(µpν) + 8(p p )H(0)µαH(0)ν 2(p p )ηµν H(0) H(0)αβ 2 3αβ 2 2 · 3 2 3 α − 2 · 3 2αβ 3 µν α (0) (0)βγ µν h (0) (0)αβ +4η p H H p γ + P 2(D 6)(p p )H H 2 3αβ 2 3 q − 2 · 3 2αβ 3 α (0) (0)βγ i 4(D 2)p H H p γ − − 2 3αβ 2 3

Obtained equally from Goldberg’s vertex Goldberg (1958) and relaxed Einstein equations in Landau-Lifshitz formalism

Landau, Lifshitz (1941), Poisson, Will (2014)

37 / 25 Linear-order fat graviton for simplest source

(0)µν κ M µ ν H = u u = X 2 4πr Inverse map: κ M φ(0) = H(0) = − 2 4πr κ M κ M h(0) P q h(0) = H(0) P q H(0) = uµuν + P q µν − µν µν − µν 2 4πr µν 2 4πr We ﬁnd κ M κ M h(0) = h(0) = uµuν − 2 4πr µν 2 4πr

38 / 25 First nonlinear-order fat graviton for simplest source X κ2 M 2 H(1)µν = rˆµrˆν = − 2 4(4πr)2 X

κ2 M 2 (1)µν = 3uµuν + 2ˆrµrˆν + 2P µν T − 2 q 4(4πr)2 Inverse map:

φ(1) = H(1) (1) = 0 − T h(1) P q h(1) = H(1) (1) P q (H(1) (1)) µν − µν µν − Tµν − µν − T κ2 M 2 κ2 M 2 = k k + 3uµuν + 2ˆrµrˆν + 2P µν − 2 µ ν 4(4πr)2 2 q 4(4πr)2 We ﬁnd κ2 M 2 κ2 M 2 h(1) = h(1) = [3u u +r ˆµrˆν] − 2 2(4πr)2 µν 2 4(4πr)2 µ ν

39 / 25 Scalar scattering amplitude with gluon radiation AO (2014, PhD Thesis) Feynman to cubic diagrams:

2,¯ 4,l 2,¯ 4,l b a b i Tkm¯ Tm¯ıTl¯ 5,a + 5,a = √ 2 2(p1 ·ε5)(p3 ·(p2 −p4)) 2 (s15 − m1)s24 1,¯ı 3,k 1,¯ı 3,k c1n1 + (p1 ·p5)(ε5 ·(p2 −p4)) = D1

2,¯ 4,l 2,¯ 4,l a b b i Tkm¯ Tm¯ıTl¯ 5,a + 5,a = √ 2 2(p3 ·ε5)(p1 ·(p2 −p4)) 2 (s35 − m1)s24 1,¯ı 3,k 1,¯ı 3,k c3n3 + (p3 ·p5)(ε5 ·(p2 −p4)) = D3 ˜abc b c i f Tk¯ıTl¯ 2,¯ 4,l √ −((p1 +p3)·ε5)((p1 −p3)·(p2 −p4)) 2 s13s24 5,a = +((p2 −p4)·ε5)((p1 −p3)·p5) 1,¯ı 3,k c5n5 −((p1 −p3)·ε5)((p2 −p4)·p5) = D5 Color-kinematic duality: c c = c n n = n 1 − 3 5 1 − 3 5 40 / 25 Large-mass expansion

Luna, Nicholson, O’Connell, White (2017) Reinstate ~: 2 i µ m ig S[A, Φ], scalar = (DµΦ)¯ı†(D Φ)i Φ¯ı†Φi,Dµ = ∂µ Aµ ~ L − ~2 − ~ inhomogeneous in m large-mass expansion A j ⇒

pµ = m vµ, p2 = m2 v2 = 1 j j j j j ⇒ j 2 2 2 qj (pj qj) = mj (vj qj) = − ⇒ · 2mj

n = n(3) + n(2) + ..., e.g. n(3) = n(3) = 4i m2m (v v )(v ε) i i i 1 3 √2 1 2 1 · 2 1 · D = D(1) + D(0) + ..., e.g. D(1) = D(1) = 2m (v k)q2 i i i 1 − 3 − 1 1 · 2

41 / 25