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The Double Copy of a Point ChargeRicardo MonteiroQueen Mary University of LondonQCD meets GravityUCLA, 12 December 2019Based on arXiv:1912.02177 withKwangeon Kim, Kanghoon Lee, Isobel Nicholson, David Peinador VeigaRicardo Monteiro (Queen Mary)Double Copy of a Point Charge1 / 16

MotivationSimple gauge theory solution: Coulomb.Schwarzschild is natural double copy of Coulomb.Full story? Dilaton?Ricardo Monteiro (Queen Mary)Double Copy of a Point Charge2 / 16

MotivationSimple gauge theory solution: Coulomb.Schwarzschild is natural double copy of Coulomb.Full story? Dilaton?Double-copy structure of Einstein equations?double copyGravity YM YMRicardo Monteiro (Queen Mary)double field theorydoubled geometryDouble Copy of a Point Charge(x µ , x̃µ )2 / 16

Double copy of Coulomb:perturbative approachRicardo Monteiro (Queen Mary)Double Copy of a Point Charge3 / 16

Gravity (Yang-Mills) 2Scattering amplitudes[Kawai, Lewellen, Tye ’86; Bern, Carrasco, Johansson ’08; . . . ]YM states: A aµ eik ·x µ T a ,Ricardo Monteiro (Queen Mary) µ has D 2 dof.Double Copy of a Point Charge4 / 16

Gravity (Yang-Mills) 2Scattering amplitudes[Kawai, Lewellen, Tye ’86; Bern, Carrasco, Johansson ’08; . . . ]YM states: A aµ eik ·x µ T a , µ has D 2 dof.NS-NS gravity states: Hµν eik ·x εµν ,εµν µ νor linear comb.(D 2)2 dof: graviton hµν dilaton φ B-field Bµν .InteractionsAgrav (εiµν ) AYM ( iµ ) dc AYM ( iµ )Strings insight: closed string (‘left’ open string) (‘right’ open string)Ricardo Monteiro (Queen Mary)Double Copy of a Point Charge4 / 16

Gravity (Yang-Mills) 2Scattering amplitudes[Kawai, Lewellen, Tye ’86; Bern, Carrasco, Johansson ’08; . . . ]YM states: A aµ eik ·x µ T a , µ has D 2 dof.NS-NS gravity states: Hµν eik ·x εµν ,εµν µ νor linear comb.(D 2)2 dof: graviton hµν dilaton φ B-field Bµν .InteractionsAgrav (εiµν ) AYM ( iµ ) dc AYM ( iµ )4Strings insight: closed string (‘left’ open string) (‘right’ open string)Agauge( µ) Agauge( µ) Agravity("µ µ )Perturbative classical solutionsFirst map free solutions (linear). Then correct solutions in double-copy-ish perturbation theory. Ricardo Monteiro (Queen Mary) Double Copy of a Point Charge4 / 16

Double copy for Coulomb?Linearised “fat graviton”:[Luna, RM, Nicholson, Ochirov, O’Connell, White, Westerberg 16] 1Hµν hµν h Pµν [h] Bµν Pµν [φ](graviton B-field dilaton)2(P is coord. space projector)µνAaµ Coulomb usual gauge:Natural double copy:Hµν [also Goldberger, Ridgway 16]Coulomb different gauge:Natural double copy:[RM, O’Connell, White 14]Ricardo Monteiro (Queen Mary)qauµrMuµ uνrAaµ hµν qakµr2Mkµ kνru µ (1, 0, 0, 0) , µ q a 0 .both graviton and dilaton.k dt dr ,k2 0 .exact Schwarzschild!Kerr-Schild double copy.Double Copy of a Point Charge5 / 16

Double copy for Coulomb?Linearised “fat graviton”:[Luna, RM, Nicholson, Ochirov, O’Connell, White, Westerberg 16] 1Hµν hµν h Pµν [h] Bµν Pµν [φ](graviton B-field dilaton)2(P is coord. space projector)µνAaµ Coulomb usual gauge:Natural double copy:Hµν [also Goldberger, Ridgway 16]Coulomb different gauge:Natural double copy:Muµ uνrAaµ hµν [RM, O’Connell, White 14]qauµrqakµr2Mkµ kνru µ (1, 0, 0, 0) , µ q a 0 .both graviton and dilaton.k dt dr ,k2 0 .exact Schwarzschild!Kerr-Schild double copy.Both consistent with ‘convolution’ idea [Anastasiou, Borsten, Duff, Hughes, Nagy 14] :Aaµ inv(Φ)aȧ AȧνRicardo Monteiro (Queen Mary)(1/r ) inv(1/r ) (1/r ) (1/r )Double Copy of a Point Charge5 / 16

Double copy for Coulomb: JNW solutionClue from momentum states: take polarisations µ , µ . · k · k 0Simplest double copy: εµν µ ν .Ricardo Monteiro (Queen Mary)Double Copy of a Point Charge6 / 16

Double copy for Coulomb: JNW solutionClue from momentum states: take polarisations µ , µ .Simplest double copy: εµν µ ν .Why not (µ ν) , [µ ν] ,Ricardo Monteiro (Queen Mary) · µν ?Double Copy of a Point Charge µν · k · k 0 · q · q q 2 0kµ qν kν qµ ηµν k ·q6 / 16

Double copy for Coulomb: JNW solutionClue from momentum states: take polarisations µ , µ .Simplest double copy: εµν µ ν .Why not (µ ν) , [µ ν] , · µν ? µν · k · k 0 · q · q q 2 0kµ qν kν qµ ηµν k ·qGeneral: graviton B-field dilaton. µν µνεµν C (h) (µ ν) · C (B) [µ ν] C (φ) · .D 2D 2Ricardo Monteiro (Queen Mary)Double Copy of a Point Charge6 / 16

Double copy for Coulomb: JNW solution · k · k 0Clue from momentum states: take polarisations µ , µ .Simplest double copy: εµν µ ν .Why not (µ ν) , [µ ν] , · µν ? µν · q · q q 2 0kµ qν kν qµ ηµν k ·qGeneral: graviton B-field dilaton. µν µνεµν C (h) (µ ν) · C (B) [µ ν] C (φ) · .D 2D 2Linearised (Coulomb)2 : no B-field, M C (h) graviton, Y C (φ) dilaton. Hµν M 2 2 uµ uνuu Pµν Y PµνrrrPµνhu2ri 1(ηµν2r qµ lν qν lµ )Y 0: linearised Schwarzschild solution.Any Y : linearised JNW solution [Janis, Newman, Winicour ’68].Ricardo Monteiro (Queen Mary)Double Copy of a Point Charge6 / 16

Perturbative construction(0)(1)Starting point: Hµν is linearised solution, Hµν is first non-linear correction.H (0)H(1) H (0)Gauge theory field AaµA(1)aµ ( p1 ) i2p12Zd D p2 d D p3 δ D (p1 p2 p3 )f abc V µβγ(0)bAβ(0)c(p2 )Aγ(p3 )YM vertex V (p1 , p2 , p3 )µβγ (p1 p2 )γ η µβ (p2 p3 )µ η βγ (p3 p1 )β η γµGravity field Hµν graviton dilaton B-field0H (1)µµ ( p1 ) 14p12Zd D p2 d D p3 δ D (p1 p2 p3 )Simplification: index factorisation.Ricardo Monteiro (Queen Mary)0 0 0V µβγ V µβ γ(0)(0)Hββ 0 (p2 )Hγγ 0 (p3 )[analogous to Bern, Grant 99; Hohm 11; Cheung, Remmen 16]Double Copy of a Point Charge7 / 16

Perturbative construction(0)(1)Starting point: Hµν is linearised solution, Hµν is first non-linear correction.H (0)H(1) H (0)Gauge theory field AaµA(1)aµ ( p1 ) i2p12Zd D p2 d D p3 δ D (p1 p2 p3 )f abc V µβγ(0)bAβ(0)c(p2 )Aγ(p3 )YM vertex V (p1 , p2 , p3 )µβγ (p1 p2 )γ η µβ (p2 p3 )µ η βγ (p3 p1 )β η γµGravity field Hµν graviton dilaton B-field0H (1)µµ ( p1 ) 14p12Zd D p2 d D p3 δ D (p1 p2 p3 )Simplification: index factorisation.Case Y MCase Y 6 M0 0 0V µβγ V µβ γ(0)(0)Hββ 0 (p2 )Hγγ 0 (p3 )[analogous to Bern, Grant 99; Hohm 11; Cheung, Remmen 16].[Luna, RM, Nicholson, Ochirov, O’Connell, White, Westerberg 16].[Kim, Lee, RM, Nicholson, Veiga 19]Comparison to exact solution messy (gauge choices, field redefinitions).Ricardo Monteiro (Queen Mary)Double Copy of a Point Charge7 / 16

General point charge: JNW solutionUnique static, spherically symmetric, asymptotically flat solutionof Einstein minimally coupled scalar.Two parameters (M, Y ) or (ρ0 , γ). Found by Janis, Newman, Winicour ’68: γ γ 1 γρ0ρ0ρ0ds2 1 dt 2 1 dρ2 1 ρ2 dΩ2ρρρ pYρ0Mφ log 1 ρ0 2 M 2 Y 2γ ρ0ρM2 Y 2Y 0: vacuum gravity Schwarzschild (usual coords)Y 6 0: naked singularity at origin ρ ρ0 , cf. no-hair theoremsRicardo Monteiro (Queen Mary)Double Copy of a Point Charge8 / 16

General point charge: JNW solutionUnique static, spherically symmetric, asymptotically flat solutionof Einstein minimally coupled scalar.Two parameters (M, Y ) or (ρ0 , γ). Found by Janis, Newman, Winicour ’68: γ γ 1 γρ0ρ0ρ0ds2 1 dt 2 1 dρ2 1 ρ2 dΩ2ρρρ pYρ0Mφ log 1 ρ0 2 M 2 Y 2γ ρ0ρM2 Y 2Y 0: vacuum gravity Schwarzschild (usual coords)Y 6 0: naked singularity at origin ρ ρ0 , cf. no-hair theoremsGeneral JNW not Kerr-Schild. Exact double copy map? Yes.Ricardo Monteiro (Queen Mary)Double Copy of a Point Charge8 / 16

General point charge: JNW solutionUnique static, spherically symmetric, asymptotically flat solutionof Einstein minimally coupled scalar.Two parameters (M, Y ) or (ρ0 , γ). Found by Janis, Newman, Winicour ’68: γ γ 1 γρ0ρ0ρ0ds2 1 dt 2 1 dρ2 1 ρ2 dΩ2ρρρ pYρ0Mφ log 1 ρ0 2 M 2 Y 2γ ρ0ρM2 Y 2Y 0: vacuum gravity Schwarzschild (usual coords)Y 6 0: naked singularity at origin ρ ρ0 , cf. no-hair theoremsGeneral JNW not Kerr-Schild. Exact double copy map? Yes.SESolution above is in Einstein frame. In string frame, gµν e2φ gµν.Ricardo Monteiro (Queen Mary)Double Copy of a Point Charge8 / 16

Double copy of Coulomb:exact map with double field theoryRicardo Monteiro (Queen Mary)Double Copy of a Point Charge9 / 16

Classical double copy: exactKerr-Schild double copy“Exact perturbation”[RM, O’Connell, White 14] [with Luna, Nicholson 15-18]gµν ηµν φ kµ kνwhere kµ is null and geodesic wrt ηµν and gµν .(k µ g µν kν η µν kν )Einstein equations linearise:g µν η µν φ k µ k νR µ ν 12 α [ µ (φk α kν ) ν (φk α k µ ) α (φk µ kν )] µ η µν νStationary vacuum case (take k0 1):0 Rµ0 Ricardo Monteiro (Queen Mary)1 ν F µν2for F dADouble Copy of a Point ChargeAµ φ kµ10 / 16

Classical double copy: exactKerr-Schild double copy“Exact perturbation”[RM, O’Connell, White 14] [with Luna, Nicholson 15-18]gµν ηµν φ kµ kνwhere kµ is null and geodesic wrt ηµν and gµν .(k µ g µν kν η µν kν )Einstein equations linearise:g µν η µν φ k µ k νR µ ν 12 α [ µ (φk α kν ) ν (φk α k µ ) α (φk µ kν )] µ η µν νStationary vacuum case (take k0 1):0 Rµ0 1 ν F µν2for F dAAµ φ kµSimplest example: Schwarzschild (Coulomb) (Coulomb)Why abelian? Exact linear EinsteinRicardo Monteiro (Queen Mary)exact linear YM, i.e., Maxwell.Double Copy of a Point Charge10 / 16

Classical double copy: exactKerr-Schild double copy“Exact perturbation”[RM, O’Connell, White 14] [with Luna, Nicholson 15-18]gµν ηµν φ kµ kνwhere kµ is null and geodesic wrt ηµν and gµν .(k µ g µν kν η µν kν )Einstein equations linearise:g µν η µν φ k µ k νR µ ν 12 α [ µ (φk α kν ) ν (φk α k µ ) α (φk µ kν )] µ η µν νStationary vacuum case (take k0 1):0 Rµ0 1 ν F µν2for F dAAµ φ kµSimplest example: Schwarzschild (Coulomb) (Coulomb)Why abelian? Exact linear EinsteinRest of the talkexact linear YM, i.e., Maxwell.Vacuum here. Kerr-Schild-type ansatz for NS-NS gravity?Why is this double copy?Ricardo Monteiro (Queen Mary)Double Copy of a Point Charge10 / 16

Classical double copy: exactDouble Field Theory[Siegel ’93] [Hull, Zwiebach ’09 Hohm ’10]For our purposes: fancy formulation of NS-NS gravity.Motivation: low-energy effective theory of closed string exhibiting T-duality.ΛM NDoubled space XM (x µ , x̃µ ) , dim 2D .String on torus: quantised momenta, winding. Mixed by T-duality.DFT idea: (x µ , x̃µ ) conjugate to (momenta, winding). T-duality: O(D, D). 0 δµ νT O(D, D) : (Λ) (J )(Λ) (J ) . JMN is O(D, D) metric.δµ ν 0T-duality manifest: O(D, D) covariance.Section condition, e.g., / x̃µ 0 : correct dof, breaks covariance.Ricardo Monteiro (Queen Mary)Double Copy of a Point Charge11 / 16

Classical double copy: exactDouble Field Theory[Siegel ’93] [Hull, Zwiebach ’09 Hohm ’10]For our purposes: fancy formulation of NS-NS gravity.Motivation: low-energy effective theory of closed string exhibiting T-duality.ΛM NDoubled space XM (x µ , x̃µ ) , dim 2D .String on torus: quantised momenta, winding. Mixed by T-duality.DFT idea: (x µ , x̃µ ) conjugate to (momenta, winding). T-duality: O(D, D). 0 δµ νT O(D, D) : (Λ) (J )(Λ) (J ) . JMN is O(D, D) metric.δµ ν 0T-duality manifest: O(D, D) covariance.Section condition, e.g., / x̃µ 0 : correct dof, breaks covariance.NS-NS fields packaged as tensor and scalar wrt to O(D, D). µν g g µρ BρνGeneralised metric: HMN O(D, D) .Bµρ g ρν gµν Bµρ g ρσ Bσν DFT dilaton d : e 2d g e 2φ .Ricardo Monteiro (Queen Mary)Double Copy of a Point Charge11 / 16

Classical double copy: exactKerr-Schild-inspired ansatzRecall Kerr-Schild ansatz:DFT version: takegµν ηµν ϕ kµ kνkµ null and geodesic.H0MN HMN (gµν ηµν , Bµν 0) ,[Lee 18] [Cho, Lee 19][Kim, Lee, RM, Nicholson, Veiga 19] 1HMN H0MN ϕ KM K̄N KN K̄M ϕ2 K̄ 2 KM KN2 µ 11kk̄ µ KM K̄ Mνν2 ηµν k2 ηµν k̄where kµ and k̄µ are null and satisfy diff. contraints (or k̄µ not null).Ricardo Monteiro (Queen Mary)Double Copy of a Point Charge12 / 16

Classical double copy: exactKerr-Schild-inspired ansatzRecall Kerr-Schild ansatz:DFT version: takegµν ηµν ϕ kµ kνkµ null and geodesic.H0MN HMN (gµν ηµν , Bµν 0) ,[Lee 18] [Cho, Lee 19][Kim, Lee, RM, Nicholson, Veiga 19] 1HMN H0MN ϕ KM K̄N KN K̄M ϕ2 K̄ 2 KM KN2 µ 11kk̄ µ KM K̄ Mνν2 ηµν k2 ηµν k̄where kµ and k̄µ are null and satisfy diff. contraints (or k̄µ not null).ϕgµν ηµν k(µ k̄ν) ,1 ϕ2 (k · k̄ )ϕBµν k[µ k̄ν] .ϕ1 2 (k · k̄)First examples of exact double copy with dilaton and B-field.JNW solution: fits ansatz, B-field is pure gauge.Ricardo Monteiro (Queen Mary)Double Copy of a Point Charge[Lee 18]12 / 16

Classical double copy: exactDouble Field Theory versus Double CopyGeneralised metric HM N induces chirality: 11PM N δM N HM N ,P̄M N δM N H M N .22Project into chiral and anti-chiral sectors left and right moving sectors!(pullback to worldsheet)Kerr-Schild-like ansatz HMN H0MN ϕ KM K̄N KN K̄M . . .Satisfy definite chiralities:Double-copy interpretation:Ricardo Monteiro (Queen Mary)(P0 )M N KN KM ,KMAµDouble Copy of a Point Charge(P̄0 )M N K̄N K̄M .K̄MĀµ13 / 16

Classical double copy: exactDouble Field Theory versus Double CopyGeneralised metric HM N induces chirality: 11PM N δM N HM N ,P̄M N δM N H M N .22Project into chiral and anti-chiral sectors left and right moving sectors!(pullback to worldsheet)Kerr-Schild-like ansatz HMN H0MN ϕ KM K̄N KN K̄M . . .Satisfy definite chiralities:Double-copy interpretation:(P0 )M N KN KM ,KMAµ(P̄0 )M N K̄N K̄M .K̄MĀµKLT picture of Kerr-Schild double copy!Ricardo Monteiro (Queen Mary)Double Copy of a Point Charge13 / 16

Classical double copy: exactDouble Field Theory versus Double CopyGeneralised metric HM N induces chirality: 11PM N δM N HM N ,P̄M N δM N H M N .22Project into chiral and anti-chiral sectors left and right moving sectors!(pullback to worldsheet)Kerr-Schild-like ansatz HMN H0MN ϕ KM K̄N KN K̄M . . .(P0 )M N KN KM ,Satisfy definite chiralities:Double-copy interpretation:KM(P̄0 )M N K̄N K̄M .AµK̄MĀµKLT picture of Kerr-Schild double copy!Usual DFT basis: (x µ , 0) , (0, x̃µ ). Mixed by O(D, D).Double-copy basis:1µ2 (xRicardo Monteiro (Queen Mary) x̃ µ , xµ x̃µ ) ,1µ2 (x x̃ µ , x̃µ xµ ) .Double Copy of a Point Charge13 / 16

Classical double copy: exactDFT equations of motion‘Generalised diffeomorphisms’ curvature tensors: RMN , R.With section condition ( / x̃µ 0), NS-NS equations of motion:R(µν) 0R[µν] 0R 0for metric gµνfor B-field Bµνfor DFT dilaton dAs in Kerr-Schild double copy, assume stationarity (Killing 0 ), k0 k̄0 1:4e 2d Rµ0 ν Fνµ 0 ,4e 2d R0µ ν F̄νµ 0 ,F dA ,Aµ e 2d ϕ kµ Cµ ,F̄ d Ā ,Āµ e 2d ϕ k̄µ C̄µ .Cµ , C̄µ absorb non-linearities, non-local.Ricardo Monteiro (Queen Mary)Double Copy of a Point Charge14 / 16

Classical double copy: exactDFT equations of motion‘Generalised diffeomorphisms’ curvature tensors: RMN , R.With section condition ( / x̃µ 0), NS-NS equations of motion:R(µν) 0R[µν] 0R 0for metric gµνfor B-field Bµνfor DFT dilaton dAs in Kerr-Schild double copy, assume stationarity (Killing 0 ), k0 k̄0 1:4e 2d Rµ0 ν Fνµ 0 ,4e 2d R0µ ν F̄νµ 0 ,F dA ,Aµ e 2d ϕ kµ Cµ ,F̄ d Ā ,Āµ e 2d ϕ k̄µ C̄µ .Cµ , C̄µ absorb non-linearities, non-local.Kerr-Schild-like (gµν , Bµν , d) (‘left-moving’ Aµ ) (‘right-moving’ Āµ )JNW (‘left-moving’ Coulomb) (‘right-moving’ Coulomb)Ricardo Monteiro (Queen Mary)Double Copy of a Point Charge14 / 16

ConclusionConclusionRicardo Monteiro (Queen Mary)Double Copy of a Point Charge15 / 16

ConclusionConclusionDouble copy of classical solutions possible.Perturbative double copy: generic but messy.Exact double copy: fully non-linear but not generic.JNW (left-moving Coulomb) (right-moving Coulomb)Double field theory convenient setting for double copy.KLT interpretation of Kerr-Schild-type double copyMuch more to exploreLarger classes of solutions, duality transf., asymptotic symmetries, . . .[Luna et al 15; Luna et al 18] [Godazgar et al; Huang et al; Alawadhi et al; Banerjee et al 19]Aim: general formulation of fully non-linear double copy.Ricardo Monteiro (Queen Mary)Double Copy of a Point Charge16 / 16

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 Mar