Renormalization in Classical Effective Field Theory (CLEFT)

1. Renormalization in Classical Effective Field Theory (CLEFT) Barak Kol Hebrew University - Jerusalem Jun 2009, Crete • Outline • Definition & Domain of applicability • Review of results (caged, EIH) • Standing puzzles • Renormalization (in progress) • Based on BK and M. Smolkin • 0712.2822 (PRD) – caged • 0712.4116 (CQG) – PN • In progress

2. Domain of applicabilityGeneral condition Consider a field theory with two widely separated scales r0<<L Seek solutions perturbatively in r0/L.

3. Binary system • The search for Gravitational waves is on: LIGO (US), VIRGO (Italy), GEO (Hannover), TAMA (Japan) • Sources: binary system (steady), collapse, collision • Dim’less parameters For periodic motion the latter two are comparable – virial theorem

4. Matched Asymptotic Expansion (MAE) Two zones. Bdry cond. come from matching over overlap. Near: r0 finite, L invisible. Far: L finite, r0 point-like. Effective Field Theory (EFT) Replace the near zone by effective interactions of a point particle Two (equivalent) methods

5. Applications • Born-Oppenheimer • Caged BHs • Binary system • Post Newtonian (PN) • Extreme Mass Ratio (EMR) • BHs in Higher dimensions • Non-gravitational

6. Post-Newtonian Small parameter v2 Far zone Validity always initially, never at merger Extreme Mass Ratio m/M if initially, then throughout

7. Non-gravitational • Electro-statics of conducting spheres • Scattering of long λ waves • Boundary layers in fluid dynamics • More…

8. Engages the deep concepts of quantum field theory including: Action rather than EOM approach Feynman diagrams Loops Divergences Regularization including dimensional reg. Renormalization and counter-terms The historical hurdles of Quantum Field Theory (1926-1948-1970s) could have been met and overcome in classical physics. Theoretical aspects

9. Brief review of results • Goldberger & Rothstein (9.2004) – Post-Newtonian (PN) including 1PN=Einstein-Infeld-Hoffmann (EIH) • Goldberger & Rothstein (11.2005) BH absorption incorporated through effective BH degrees of freedom • Chu, Goldberger & Rothstein (2.2006) caged black holes – asymptotic charges

10. r0 L Near Far Caged Black Holes Effective interaction: field quadrupole at hole’s location induces a deformation and mass quadrupole

11. Definition of ADM mass in terms of a 0-pt function, rather than 1-pt function as in CGR CGR US • Rotating black holes

12. First Post-Newtonian ≡ Einstein-Infeld-Hoffmann • Newtonian two-body action • Add corrections in v/c • Expect contributions from • Kinetic energy • Potential energy • Retardation

13. Post-Newtonian approximation: v<<c – slow motion (CLEFT domain) Start with Stationary case (see caged BHs) Technically – KK reduction over time “Non-Relativistic Gravitation” - NRG fields The Post-Newtonian action 0712.4116 BK, Smolkin

14. Physical interpretation of fields Φ – Newtonian potential A – Gravito-magnetic vector potential

15. Feynman rules x EIH in CLEFT Action φ Ai

16. Feynman diagrams PN2 in CLEFT: Gilmore, Ross 0810

17. The black hole metric Comments The static limit a=0. Uniqueness Holds all information including: horizon, ergoregion, singularity. Black Hole Effective Action

18. Motion through curved background • Problem: Determine the motion through slowly curving background r0<<L (CLEFT domain) • Physical expectations • Geodesic motion • Spin is parallel transported • Finite size effects (including tidal) • backreaction

19. Matched Asymptotic expansion (MAE) approach. • “Near zone”. • Need Non-Asymptotically flat BH solutions.

20. Replace MAE by EFT approach Replace the BH metric by a black hole effective action Recall that Hawking replacedthe black hole by a black body We shallreplace theblack holeby ablack box. EFT approach

21. Std definition by integrating out Saddle point approximation Stresses that we can integrate out only given sufficient boundary conditions CLEFT Definition of Eff Action 0712.2822 BK, Smolkin

22. Goal:Compute the Black hole effective action Comments • Universality • Perturbative (in background fields, ∂kg|x) • Non-perturbative • Issue: regularize the action, subtract reference background

23. First terms • Point particle • Spin (in flat space) • Finite size effects, e.g. “Love numbers”, Damour and collab; Poisson • Black hole stereotyping

24. What is the Full Result?

25. (Reminder) Post-Newtonian approximation: v<<c – slow motion (CLEFT domain) Start with Stationary case (see caged BHs) Technically – KK reduction over time “Non-Relativistic Gravitation” - NRG fields The Post-Newtonian action 0712.4116 BK, Smolkin

26. Adding time back • Generalize the (NRG) field re-definition • Choosing an optimal gauge (especially for t dependent gauge). Optimize for bulk action. • Possibly eliminating redundant terms (proportional to EOM) by field re-definition

27. Goal:Obtain the gauge-fixed action allowing for time dependence - Make Newton happy…

28. Quadratic levelΦ, A sector Proceed to Cubic sector and onward…

29. What is the full Non-Linear Result?

30. k Renormalization Before considering gravity let us consider Take β=0. The renormalized point charge q(k) or q(r) is defined through

31. An integral equation • q(k) satisfies Comments: • The equation can be solved iteratively, reproducing the diagrammatic expansion of q(k). • The equation is classically polynomial for polynomial action

32. Relation with Φ(r) • Φ(r) is defined to be the field due to a point charge • It is directly related to q(r) through • While q(r) satsifies the above integral equation, Φ(r) satisfies a differential equation – • namely, the equation of motion

33. Re-organizing the PN expansion These ideas can be applied to PN. For instance at 2PN Can be interpreted through mass renormalization

34. Comment:The beta function equation

35. Recap • Theory which combines Einstein’s gravity, (Quantum) Field Theory and experiment. • Ripe • caged black holes • 1PN (Einstein-Infeld-Hoffmann) • Black hole effective action • Post-Newtonian action • Renormalization

36. ΕΦΧΑΡΙΣΤΟ! Thank you! Darkness and Light in our region