Perspectives on the double copy: Outlook and Summary Henrik - PowerPoint PPT Presentation

Perspectives on the double copy: Outlook and Summary Henrik Johansson Uppsala U. & Nordita Dec 13, 2019 UCLA QCD Meets Gravity

Double copy Gauge theory à Gravity Gauge symmetry à Diffeomorphism symmetry Driven by technical simplicity (KLT, BCJ, CHY, …) We saw many instances of this during this workshop: Marco Chiadoroli, Eduardo Casali, Gustav Mogull, Mikhail Solon, Emil Bjerrum-Bohr, Fei Teng, Ricardo Monteiro, Laurentiu Rodina, Lance Dixon, Andres Luna, Silvia Nagy, Lionel Mason Many interesting open problems to work on…

Op Opportunities of the double copy Hidden structures in gauge theory à BCJ relations � Casali, Tourkine, Rodina à kinematic Lie algebra (CK duality) � Mason Unified framework à web of theories (theory classification) � Chiodaroli à recycling of calculations � Mogull, Dixon, Teng Generality of the DC à amplitudes (& form factors in gauge theory) à BH solutions � Monteiro à classical scattering/radiation � Solon, Luna, Ochirov à curved backgrounds, off shell ? � Adamo, Nagy Deeper physical understanding of DC ?

Textbook perturbative gravity is complicated de Donder = gauge = After symmetrization 100 terms ! � Bjerrum-Bohr higher order vertices… 10 3 terms complicated diagrams: 10 4 terms 10 7 terms 10 21 terms 10 31 terms

On-she On shell si simplifications ns Graviton plane wave: Yang-Mills polarization On-shell 3-graviton vertex: = Yang-Mills vertex Gravity scattering amplitude: Yang-Mills amplitude tree (1 , 2 , 3 , 4) = st M GR u A YM tree (1 , 2 , 3 , 4) ⊗ A YM tree (1 , 2 , 3 , 4) Kawai, Lewellen, Tye Gravity processes = “squares” of gauge theory ones - entire S-matrix Bern, Carrasco, HJ

Squaring Squ ng of YM the heory Gravity processes = squares of gauge theory ones - entire S-matrix Yang-Mills Gravity → squared numerators (BCJ double copy) → pure Yang-Mills Einstein gravity + dilaton + axion E.g. → N =4 super-YM N =8 supergravity → � Dixon, Parra-Martinez, Teng, Herrmann

Wh Why/ y/How double copy y works ? 3 ingredients are often sufficient: Spectrum (2pt) + à consistent amplitudes Interactions (3pt) + Gauge symmetry (4pt) Additionally, for consistency one should check that properties such as factorization, crossing symmetry and unitarity are preserved by double copy

Spectrum � Chiodaroli, Nagy States are obtained by tensoring Lorentz reps. (dilaton) (axion) ( ε h ) ij ε (( i µ ε j )) = (graviton) µ ν ν More precisely ( ε B ) ij ε [ i µ ε j ] = ( B -field) , µ ν ν in D dim: ε i µ ε j ν δ ij ( ε φ ) µ ν = (dilaton) . D − 2

Matter spectrum � Chiodaroli global local Matter spin 1 Matter spin 3/2 Tensors: Spin 1/2: Spin 0:

Sp Spectrum – br broken symmetry Chiodaroli, Gunaydin, HJ, Roiban Spontaneous symmetry breaking: broken local broken global spin 1: (massive) Spin 3/2: (massive)

In Interactions on-shell: a c a ca c a a i a c i

In Interactions - ma matter on-shell: a c a c a c a c many more examples….

Hi Higher-de deriv ivativ ive in interactio ions Gauss-Bonnet: a c a c cubic Riemann: a c a c Dixon, Brödel; Bern, Edison, Kosower, Parra-Martinez; Nohle, HJ; … � Rodina

Gauge & diffeo invariance BCJ cubic diagram form: gauge invariance: Jacobi identity − = diffeo invariance: kinematic Jacobi Id � Chiodaroli, Nagy CK duality à kinematic Lie algebra ?

Example: axion-dilaton gravity Consider double copy of D -dimensional pure YM: ( ε h ) ij ε (( i µ ε j )) = (graviton) µ ν ν ( ε B ) ij ε [ i µ ε j ] = ( B -field) , States: µ ν ν ε i µ ε j ν δ ij ( ε φ ) µ ν = (dilaton) . D − 2 Amplitudes consistent with the theory: C D ⁄ d D x √− g − 1 2( D − 2) ∂ µ φ∂ µ φ + 1 1 6 e − 4 φ/ ( D − 2) H λµν H λµν S = 2 R + In 4D this is axion-dilaton gravity: Symmetry allows for consistent truncation of scalars

Ex Example: pure GR HJ, Ochirov Pure 4D Einstein gravity: Does not match YM 2 spectrum: Deform YM theories with massless fundamental quarks Anti-align the spins of the quarks à gives scalars in GR e.g. become ghosts if � Nagy

Ex Examp mple: YM-Ei Einstein th theory GR+YM amplitudes are “heterotic” double copies Chiodaroli, Gunaydin, GR + YM = YM ⊗ (YM + φ 3 ) HJ, Roiban A µ ⊗ A ν h µ ν ∼ A µ ⊗ φ a A µa ∼ N = 0,1,2,4 YM-E φ 3 N = 0,1,2,4 SYM YM + supergravity Note: N =0,1,2 YM-E are contaminated by axion-dilaton states

Ex Exception th that t proves th the ru rule… Not all gauge theories obey color-kinematics dualiy Imagine the double copy: ? According to conventional wisdom must be a gravitino and is the number of supersymmetries What goes wrong? The theory only obeys color-kinematics duality if supersymmetric à Kinematic Jacobi Id. à Fierz Id. that enforces SUSY Chiodaroli, Jin, Roiban

We Web of double-co copy co construct ctible theories � Chiodaroli See review 1909.01358 – Bern, Carrasco, Chiodaroli, HJ, Roiban

So Some open problems 20

Gr Gravitational radiation LIGO/VIRGO observations à motivates high-order PN, PM calcs. � many talks: Buonanno, Bjerrum-Bohr, Brustein, Damgaard, Di Vecchia, Guevara, Maia, Maier, Levi, Luna, Kosower, Kälin, Parra-Martinez, Shen, Solon, Steinhoff, Vines, Veneziano, Yang, …. BH gravitational scattering Non-abelian gauge-theory process Using double copy for GW, Goldberger, Ridgway; G, R, Prabhu, Thompson; G, Li, P potentials and observables : Luna, Monteiro, Nicholson, O'Connell, White; Shen; Key problems: Plefka, Steinhoff, Wormsbecher, Shi, Steinhoff, Wang; à scalability Maybee, O'Connell, Vines; � Solon, Zeng, Maier Bern, Cheung, Roiban, Shen, Solon, Zeng; […] à higher-spin extensions � Levi, Steinhoff, Vines, Luna, Ochirov, Yang

Re Removing the dila dilaton ? For massive processes the dilaton couples to mass see e.g. Luna, Nicholson, O’Connell, White; Plefka, Shi, Wang Can be removed by compensating diagrams Luna, Nicholson, O’Connell, White or projectors applied to on-shell states Bern, Cheung, Roiban, Shen, Solon, Zeng � Solon However, methods not completely satisfactory: à What it the most efficient approach? à Is removal complete for all physical processes? à General framework for different theories? (cf. HJ, Ochirov for pure GR)

La Lagrangian fo for massive (QCD)x(QCD) ? HJ, Ochirov [1906.12292] What is the square of QCD ? Gluon: Massive quarks: Lagrangian determined by matching to double copy of QCD ampl. N f ∂ µ ¯ Z )+ κ 2 Z ∂ µ Z L (QCD) 2 = − 2 ⇢ − 1  1 − κ � X 2( Z + ¯ ¯ + m 2 ıµ ⌫ V µ ⌫ ıµ V µ κ 2 R + � 2 + 2 V ∗ ı V ∗ ZZ ı ı 4 ¯ 2 � 1 −  2 ZZ ı =1 Z ) + κ 2  1 − κ � 2( Z + ¯ 16( Z 2 + ¯ Z 2 + 8 ¯ ı ∂ µ ϕ ı − m 2 + ∂ µ ϕ ∗ ZZ ) ı ϕ ∗ ı ϕ ı + i κ h i ıµ ϕ ı ) ∂ µ ( Z − ¯ Z ) − ( Z − ¯ Z )( ∂ µ ϕ ∗ ıµ ∂ µ ϕ ı ) 4 m ı ( ϕ ∗ ı V ıµ + V ∗ ı V ıµ + V ∗ − i κ 2 Z ) + κ 2 i� Z ∂ µ Z − Z ∂ µ ¯ h ıµ ϕ ı )( ¯ ı ϕ ı ∂ µ ¯ Z ∂ µ Z + ¯ ı ∂ µ ϕ ı 4 m ı ( ϕ ∗ ı V ıµ + V ∗ ZZ ∂ µ ϕ ∗ ϕ ∗ 8 N f ⇢ κ 2 h i X | ∂ µ ϕ | − 3 m 2 | ϕ | + 2 m 2 | µ V µ + | V ∗ 8 ϕ ∗ ı ϕ ı ∂ µ ϕ ∗ | ϕ ∗ | ı, | =1 + κ 2 i� h | V ıµ V µ ıµ V ∗ µ | µ V µ + O ( κ 3 ) . 4 m ı m | ϕ ∗ ı ϕ ∗ | + ϕ ı ϕ | V ∗ + 2 ϕ ∗ ı ϕ | V ∗ (1.2) ı | How do we write down all-orders Lagrangians in general DC theories? � Nagy

Wh What i is t the Kinematic A Algebra ? -- If YM numerators obey Jacobi Id. à kinematic algebra should exist! -- Algebra may dramatically simplify GR integrand construction. What is known? Self dual YM in light-cone gauge: Monteiro, O’Connell (’11) � Adamo Generators of diffeomorphism invariance: Lie Algebra: YM vertex Beyond the simplest helicity sectors (NMHV) Chen, HJ, Teng, Wang [1906.10683] vector generator tensor generator

Outlook Ou - A better understanding of gravity perturbation theory and - A bright future for applications to GW calculations. Thanks for a very interesting conference! Th Man Many Than anks to: Zv Zvi, , Clifford, Do Donal, , John Jo Jo Joseph, Ra Radu an and d Ira 25