From Scattering Amplitudes to Classical Observables

1. From Scattering Amplitudes to Classical Observables David A. KosowerInstitut de Physique Théorique, CEA–Saclay work with Ben Maybee and Donal O’Connell (Edinburgh) [arXiv:1811.10950];by Ben Maybee, Donal O’Connell, and Justin Vines [1906.09260] @ Amplitudes 2019, Trinity College, Dublin— July 2, 2019

2. Gravitational Waves • The dawn of gravitational-wave astronomy: LIGO & VIRGO • 25 events since Sept 2015, 15 in 2019 (to June 30) gracedb.ligo.org/latest/ • 2+ neutron-star pair mergers, 18 black hole pair mergers, rest ? • Dramatic direct confirmation starting at the centennial of GR This past May also marked the 100th anniversary of the solar eclipse which yielded the first observational confirmation of General Relativity

3. Gravitational Waves • Can expect dramatic discoveries in • Cosmology: presence and distribution of heavy objects, limits [1906.08000] • Astrophysics: neutron-star equation of state—GW170817; S190425z; S190510g? • Need theoretical input for waveforms • Higher-order terms in the Post-Newtonian expansion still await • General relativists have worked hard for many results • Using an Effective One-Body formalism Buonanno, Pan, Taracchini, Barausse, Bohé, Cotesta, Shao, Hinderer, Steinhoff, Vines; Damour, Nagar, Bernuzzi, Bini, Balmelli, Messina; Iyer, Sathyaprakash; Jaranowski, Schaefer • Numerical RelativityPretorius; Campanelliet al.; Baker et al. • Joined by Effective Field Theorists Goldberger, Rothstein; Goldberger, Li, Prabhu, Thompson; Chester; Porto,… Kol; Levi,…

4. Theorists’ Role • Waveform templates • For detection • For measurements • Three Phases: • Inspiral: a weak-field perturbative approach works • Merge: strong-field, numerics needed • Ringdown: normal modes

5. Approaches • Traditional: solve General Relativity perturbatively • Effective Field Theory: use separation of scales to compute better in General Relativity • An idea: use scattering amplitudes • Calculate only what’s needed for physical quantities • Double copy: Gravity ~ (Yang–Mills)2 Kawai, Lewellen, Tye; Bern, Carrasco, Johansson

6. Direct Route to Predictions • Compute Potential • Compute effective-field theory scattering amplitude from parametrized & match amplitude to EFT Cheung, Rothstein, Solon; Bern, Cheung, Roiban, Shen, Solon, Zeng  Zvi Bern’s talk • Extract potential from form of terms in scattering amplitude Bjerrum-Bohr, Damgaard, Festuccia, Planté, Vanhove; Foffa, Mastrolia, Sturani, Sturm Chung, Huang, Kim, Lee Sangmin Lee’s talk Guevara, Ochirov, Vines  Alexander Ochirov’s talk • Feed potential into Effective-One-Body formalism • Compute Effective Action Plefka, Shi, Steinhoff, Wang Jan Plefka’s talk • Other connections to classical scattering Goldberger, Ridgway; Goldberger, Ridgway, Li, Prabhu; Shen

7. Our Strategy: Take the Scenic Route • Pick well-defined observables in the quantum theory That are also relevant classically • Express them in terms of scattering amplitudes in the quantum theory • Understand how to take the classical limit efficiently

8. Set-up • Scatter two massive particles • Look at three observables: • Change in momentum (‘impulse’) of one of them • Momentum radiated during the scattering • Spin kick We all love scattering amplitudes But they aren’t the final goal or physically meaningful on their own

9. Wave Packets • Point particles: localized positions and momenta • Wavefunction • Initial state: integral over on-shell phase space Notation tidies up s

10. Classical Limit, part 1 • Classical limit requires : restore • We’re still relativistic field theorists: keep • Dimensional analysis [Ampln] • in couplings: ; • Distinguish wavenumber from momentum: • Net: n-point, L-loop amplitude in scalar QED scales as Not the whole story, of course

11. Momentum Insert a complete set of states and rewrite, Think of as final-state momenta: connection to scattering amplitudes

12. Impulse • aka time integral of change in momentum Write and use unitarity Expression holds to all orders in perturbation theory

13. Impulse Diagrammatically ( is momentum mismatch, momentum transfer) First term is linear in amplitude +

14. Radiated Momentum • Expectation of messenger (photon or graviton) momentum Insert complete set of states Expressible as scattering amplitude squared or cut of amplitude Valid to all orders

15. Classical Limit, part 2 • Three scales • : Compton wavelength • : wavefunction spread • : impact parameter • Particles localized: • Well-separated wave packets: More careful analysis confirms this ‘Goldilocks’ condition

16. Classical Limit, part 2 • In-state wavefunctions and both represent particle • Should be sharply peaked • Overlap should be • Angular-averaged on-shell functions transmit constraint to • Shrinking on : • Fixed on : • Natural integration variables for messenger (massless) momenta are wavenumbers, not momenta: • mismatch , • transfer (from analysis of outgoing states), • loop (from unitarity) • More factors of to extract

17. Impulse at LO • O() • Only first term contributes, with tree-level amplitude Integrate over wavefunctions under , take

18. Scalar QED Impulse at LO • 2  2 amplitude • Impulse (use ) • Evaluate the integral

19. (Dilaton) Gravity Impulse at LO • Squaring pure Yang–Mills gives gravity + dilaton • Impulse • Evaluate integral (same as QED) • Remove the dilaton to obtain the classic GR result Portilla; Westpfahl, Goller; Ledvinka, Schäfer, Bičák; Damour

20. Impulse at NLO • O(): both terms contribute • First term, with one-loop amplitude • Second term, with tree amplitudes

21. Impulse at NLO • Massless loops are purely quantum • as are vertex, wavefn, propagator corrections—after renormalization • Loops with masses are not purely quantum • Left with triangles, boxes, and cut boxes • Technical note: summing over gives functions • Example: triangle contribution • Boxes and cut boxes individually singular as • Cancel between contributions • Need to Laurent expand • Technical note: need to retain terms in s until after expansion • General proof of cancellation lacking • Result agrees with direct classical calculation

22. Radiation at LO • Leading contribution is O() • Five-point tree amplitude • Scalar QED matches classical result • Momentum conservation automatic, no need to add Abraham–Lorentz–Dirac force as in the classical theory

23. Spin kick Maybee, O’Connell, Vines • Spin using Pauli–Lubanski vector • A few subtleties

24. Summary • Gravitational-wave astronomy makes new demands on theoretical calculations in GR • Opportunity for scattering amplitudes, as the double copy offers a simpler avenue to such calculations • Study observables valid in both quantum and classical theories • Organize formalism to take limit in a simple way • Count s • Momenta for massive particles, wavenumbers for massless • Examples: impulse, radiated momentum, spin kick