The Double Copy of a Point Charge Ricardo Monteiro Queen Mary University of London QCD meets Gravity UCLA, 12 December 2019 Based on arXiv:1912.02177 with Kwangeon Kim, Kanghoon Lee, Isobel Nicholson, David Peinador Veiga Ricardo Monteiro (Queen Mary) Double Copy of a Point Charge 1 / 16
Motivation Simple gauge theory solution: Coulomb. Schwarzschild is natural double copy of Coulomb. Full story? Dilaton? Ricardo Monteiro (Queen Mary) Double Copy of a Point Charge 2 / 16
Motivation Simple gauge theory solution: Coulomb. Schwarzschild is natural double copy of Coulomb. Full story? Dilaton? Double-copy structure of Einstein equations? double copy double field theory doubled geometry ( x µ , ˜ x µ ) Gravity = YM × YM Ricardo Monteiro (Queen Mary) Double Copy of a Point Charge 2 / 16
Double copy of Coulomb: perturbative approach Ricardo Monteiro (Queen Mary) Double Copy of a Point Charge 3 / 16
Gravity ∼ (Yang-Mills) 2 Scattering amplitudes [Kawai, Lewellen, Tye ’86; Bern, Carrasco, Johansson ’08; . . . ] µ = e ik · x ǫ µ T a , A a YM states: ǫ µ has D − 2 dof. Ricardo Monteiro (Queen Mary) Double Copy of a Point Charge 4 / 16
Gravity ∼ (Yang-Mills) 2 Scattering amplitudes [Kawai, Lewellen, Tye ’86; Bern, Carrasco, Johansson ’08; . . . ] µ = e ik · x ǫ µ T a , A a YM states: ǫ µ has D − 2 dof. NS-NS gravity states: H µν = e ik · x ε µν , ε µν = ǫ µ ˜ ǫ ν or linear comb. ( D − 2 ) 2 dof: graviton h µν + dilaton φ + B-field B µν . Ricardo Monteiro (Queen Mary) Double Copy of a Point Charge 4 / 16
Gravity ∼ (Yang-Mills) 2 Scattering amplitudes [Kawai, Lewellen, Tye ’86; Bern, Carrasco, Johansson ’08; . . . ] µ = e ik · x ǫ µ T a , A a YM states: ǫ µ has D − 2 dof. NS-NS gravity states: H µν = e ik · x ε µν , ε µν = ǫ µ ˜ ǫ ν or linear comb. ( D − 2 ) 2 dof: graviton h µν + dilaton φ + B-field B µν . A grav ( ε µν ) = A YM ( ǫ µ ǫ µ i ) ⊗ dc A YM (˜ i ) Interactions i Ricardo Monteiro (Queen Mary) Double Copy of a Point Charge 4 / 16
Gravity ∼ (Yang-Mills) 2 Scattering amplitudes [Kawai, Lewellen, Tye ’86; Bern, Carrasco, Johansson ’08; . . . ] µ = e ik · x ǫ µ T a , A a YM states: ǫ µ has D − 2 dof. NS-NS gravity states: H µν = e ik · x ε µν , ε µν = ǫ µ ˜ ǫ ν or linear comb. ( D − 2 ) 2 dof: graviton h µν + dilaton φ + B-field B µν . A grav ( ε µν ) = A YM ( ǫ µ ǫ µ i ) ⊗ dc A YM (˜ i ) Interactions i Strings insight: closed string ∼ (‘left’ open string) × (‘right’ open string) Ricardo Monteiro (Queen Mary) Double Copy of a Point Charge 4 / 16
Gravity ∼ (Yang-Mills) 2 Scattering amplitudes [Kawai, Lewellen, Tye ’86; Bern, Carrasco, Johansson ’08; . . . ] µ = e ik · x ǫ µ T a , A a YM states: ǫ µ has D − 2 dof. NS-NS gravity states: H µν = e ik · x ε µν , ε µν = ǫ µ ˜ ǫ ν or linear comb. ( D − 2 ) 2 dof: graviton h µν + dilaton φ + B-field B µν . A grav ( ε µν ) = A YM ( ǫ µ ǫ µ i ) ⊗ dc A YM (˜ i ) Interactions i Strings insight: closed string ∼ (‘left’ open string) × (‘right’ open string) Perturbative classical solutions First map free solutions (linear). Then correct solutions in double-copy-ish perturbation theory. × ∼ Ricardo Monteiro (Queen Mary) Double Copy of a Point Charge 4 / 16
Double copy for Coulomb? Linearised “fat graviton”: [Luna, RM, Nicholson, Ochirov, O’Connell, White, Westerberg 16] � h µν − 1 � H µν = 2 h + P µν [ h ] + B µν + P µν [ φ ] ( graviton + B-field + dilaton ) ( P µν is coord. space projector) Ricardo Monteiro (Queen Mary) Double Copy of a Point Charge 5 / 16
Double copy for Coulomb? Linearised “fat graviton”: [Luna, RM, Nicholson, Ochirov, O’Connell, White, Westerberg 16] � h µν − 1 � H µν = 2 h + P µν [ h ] + B µν + P µν [ φ ] ( graviton + B-field + dilaton ) ( P µν is coord. space projector) µ = − q a u µ = ( 1 , 0 , 0 , 0 ) , ∂ µ q a = 0 . A a Coulomb usual gauge: r u µ H µν = M Natural double copy: r u µ u ν both graviton and dilaton. [also Goldberger, Ridgway 16] Ricardo Monteiro (Queen Mary) Double Copy of a Point Charge 5 / 16
Double copy for Coulomb? Linearised “fat graviton”: [Luna, RM, Nicholson, Ochirov, O’Connell, White, Westerberg 16] � h µν − 1 � H µν = 2 h + P µν [ h ] + B µν + P µν [ φ ] ( graviton + B-field + dilaton ) ( P µν is coord. space projector) µ = − q a u µ = ( 1 , 0 , 0 , 0 ) , ∂ µ q a = 0 . A a Coulomb usual gauge: r u µ H µν = M Natural double copy: r u µ u ν both graviton and dilaton. [also Goldberger, Ridgway 16] µ = q a k 2 = 0 . A a Coulomb different gauge: r k µ k = dt + dr , h µν = 2 M Natural double copy: k µ k ν exact Schwarzschild! r [RM, O’Connell, White 14] Kerr-Schild double copy. Ricardo Monteiro (Queen Mary) Double Copy of a Point Charge 5 / 16
Double copy for Coulomb? Linearised “fat graviton”: [Luna, RM, Nicholson, Ochirov, O’Connell, White, Westerberg 16] � h µν − 1 � H µν = 2 h + P µν [ h ] + B µν + P µν [ φ ] ( graviton + B-field + dilaton ) ( P µν is coord. space projector) µ = − q a u µ = ( 1 , 0 , 0 , 0 ) , ∂ µ q a = 0 . A a Coulomb usual gauge: r u µ H µν = M Natural double copy: r u µ u ν both graviton and dilaton. [also Goldberger, Ridgway 16] µ = q a k 2 = 0 . A a Coulomb different gauge: r k µ k = dt + dr , h µν = 2 M Natural double copy: k µ k ν exact Schwarzschild! r [RM, O’Connell, White 14] Kerr-Schild double copy. Both consistent with ‘convolution’ idea [Anastasiou, Borsten, Duff, Hughes, Nagy 14] : A a a ∗ A ˙ a µ ∗ inv (Φ) a ˙ ( 1 / r ) ∗ inv ( 1 / r ) ∗ ( 1 / r ) = ( 1 / r ) � ν Ricardo Monteiro (Queen Mary) Double Copy of a Point Charge 5 / 16
Double copy for Coulomb: JNW solution Clue from momentum states: take polarisations ǫ µ , ˜ ǫ µ . ǫ · k = ˜ ǫ · k = 0 Simplest double copy: ε µν = ǫ µ ˜ ǫ ν . Ricardo Monteiro (Queen Mary) Double Copy of a Point Charge 6 / 16
Double copy for Coulomb: JNW solution Clue from momentum states: take polarisations ǫ µ , ˜ ǫ µ . ǫ · k = ˜ ǫ · k = 0 ǫ · q = q 2 = 0 Simplest double copy: ε µν = ǫ µ ˜ ǫ ν . ǫ · q = ˜ ∆ µν = η µν − k µ q ν + k ν q µ Why not ǫ ( µ ˜ ǫ ν ) , ǫ [ µ ˜ ǫ ν ] , ǫ · ˜ ǫ ∆ µν ? k · q Ricardo Monteiro (Queen Mary) Double Copy of a Point Charge 6 / 16
Double copy for Coulomb: JNW solution Clue from momentum states: take polarisations ǫ µ , ˜ ǫ µ . ǫ · k = ˜ ǫ · k = 0 ǫ · q = q 2 = 0 Simplest double copy: ε µν = ǫ µ ˜ ǫ ν . ǫ · q = ˜ ∆ µν = η µν − k µ q ν + k ν q µ Why not ǫ ( µ ˜ ǫ ν ) , ǫ [ µ ˜ ǫ ν ] , ǫ · ˜ ǫ ∆ µν ? k · q General: graviton + B-field + dilaton. � ǫ ν ) − ∆ µν � ǫ ν ] + C ( φ ) ∆ µν + C ( B ) ǫ [ µ ˜ ε µν = C ( h ) ǫ ( µ ˜ D − 2 ǫ · ˜ ǫ D − 2 ǫ · ˜ ǫ . Ricardo Monteiro (Queen Mary) Double Copy of a Point Charge 6 / 16
Double copy for Coulomb: JNW solution Clue from momentum states: take polarisations ǫ µ , ˜ ǫ µ . ǫ · k = ˜ ǫ · k = 0 ǫ · q = q 2 = 0 Simplest double copy: ε µν = ǫ µ ˜ ǫ ν . ǫ · q = ˜ ∆ µν = η µν − k µ q ν + k ν q µ Why not ǫ ( µ ˜ ǫ ν ) , ǫ [ µ ˜ ǫ ν ] , ǫ · ˜ ǫ ∆ µν ? k · q General: graviton + B-field + dilaton. � ǫ ν ) − ∆ µν � ǫ ν ] + C ( φ ) ∆ µν + C ( B ) ǫ [ µ ˜ ε µν = C ( h ) ǫ ( µ ˜ D − 2 ǫ · ˜ ǫ D − 2 ǫ · ˜ ǫ . Linearised (Coulomb) 2 : no B-field, M ∼ C ( h ) graviton, Y ∼ C ( φ ) dilaton. � u 2 � u 2 � u µ u ν �� � � u 2 � = − 1 H µν = M − P µν + Y P µν P µν 2 r ( η µν − q µ l ν − q ν l µ ) r r r r Y = 0: linearised Schwarzschild solution. Any Y : linearised JNW solution [Janis, Newman, Winicour ’68] . Ricardo Monteiro (Queen Mary) Double Copy of a Point Charge 6 / 16
Perturbative construction Starting point: H ( 0 ) µν is linearised solution, H ( 1 ) µν is first non-linear correction. H ( 0 ) H ( 1 ) = H ( 0 ) field A a Gauge theory µ i � f abc V µβγ A ( 0 ) b ( p 2 ) A ( 0 ) c A ( 1 ) a µ ( − p 1 ) = d D p 2 d D p 3 δ D ( p 1 + p 2 + p 3 ) ( p 3 ) γ β 2 p 2 1 V ( p 1 , p 2 , p 3 ) µβγ = ( p 1 − p 2 ) γ η µβ + ( p 2 − p 3 ) µ η βγ + ( p 3 − p 1 ) β η γµ YM vertex Gravity field H µν ∼ graviton + dilaton + B-field 1 � V µβγ V µ ′ β ′ γ ′ H ( 1 ) µµ ′ ( − p 1 ) = H ( 0 ) ββ ′ ( p 2 ) H ( 0 ) d D p 2 d D p 3 δ D ( p 1 + p 2 + p 3 ) γγ ′ ( p 3 ) 4 p 2 1 Ricardo Monteiro (Queen Mary) Double Copy of a Point Charge 7 / 16
Perturbative construction Starting point: H ( 0 ) µν is linearised solution, H ( 1 ) µν is first non-linear correction. H ( 0 ) H ( 1 ) = H ( 0 ) field A a Gauge theory µ i � f abc V µβγ A ( 0 ) b ( p 2 ) A ( 0 ) c A ( 1 ) a µ ( − p 1 ) = d D p 2 d D p 3 δ D ( p 1 + p 2 + p 3 ) ( p 3 ) γ β 2 p 2 1 V ( p 1 , p 2 , p 3 ) µβγ = ( p 1 − p 2 ) γ η µβ + ( p 2 − p 3 ) µ η βγ + ( p 3 − p 1 ) β η γµ YM vertex Gravity field H µν ∼ graviton + dilaton + B-field 1 � V µβγ V µ ′ β ′ γ ′ H ( 1 ) µµ ′ ( − p 1 ) = H ( 0 ) ββ ′ ( p 2 ) H ( 0 ) d D p 2 d D p 3 δ D ( p 1 + p 2 + p 3 ) γγ ′ ( p 3 ) 4 p 2 1 Simplification: index factorisation. [analogous to Bern, Grant 99; Hohm 11; Cheung, Remmen 16] Ricardo Monteiro (Queen Mary) Double Copy of a Point Charge 7 / 16
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