Renormalization in

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
• r0ltltL
• 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
• Dimless parameters
• For periodic motion the latter two are comparable
  virial theorem

4
Two (equivalent) methods

• 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

5
Applications

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

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• Post-Newtonian
• Small parameter
• v2
• Far zone
• Validity
• always initially, never at merger

• Extreme Mass Ratio
• m/M
• if initially, then throughout

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Non-gravitational

• Electro-statics of conducting spheres
• Scattering of long ? waves
• Boundary layers in fluid dynamics
• More

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Theoretical aspects

• 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.

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Brief review of results

• Goldberger Rothstein (9.2004) Post-Newtonian
  (PN) including 1PNEinstein-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

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Caged Black Holes
Effective interaction field quadrupole at holes
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

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First Post-Newtonian Einstein-Infeld-Hoffmann

• Newtonian two-body action
• Add corrections in v/c
• Expect contributions from
• Kinetic energy
• Potential energy
• Retardation

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The Post-Newtonian action

• Post-Newtonian approximation vltltc slow motion
  (CLEFT domain)
• Start with Stationary case (see caged BHs)

• Technically KK reduction over time
• Non-Relativistic Gravitation - NRG fields

0712.4116 BK, Smolkin
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• Physical interpretation of fields
• F Newtonian potential
• A Gravito-magnetic vector potential

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EIH in CLEFT

• Feynman rules

Action
f
Ai
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Feynman diagrams
PN2 in CLEFT Gilmore, Ross 0810
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Black Hole Effective Action

• The black hole metric

• Comments
• The static limit a0.
• Uniqueness
• Holds all information including horizon,
  ergoregion, singularity.

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Motion through curved background

• Problem Determine the motion through slowly
  curving background r0ltltL (CLEFT domain)
• Physical expectations
• Geodesic motion
• Spin is parallel transported
• Finite size effects (including tidal)
• backreaction

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Matched Asymptotic expansion (MAE) approach.

• Near zone.
• Need Non-Asymptotically flat BH solutions.

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EFT approach

• Replace MAE by EFT approach
• Replace the BH metric by a black hole effective
  action

• Recall that Hawking replaced the black hole by a
  black body
• We shall replace the black hole by a black box.

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CLEFT Definition of Eff Action
0712.2822 BK, Smolkin

• Std definition by integrating out
• Saddle point approximation

• Stresses that we can integrate out only given
  sufficient boundary conditions

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Goal Compute the Black hole effective action

• Comments
• Universality
• Perturbative (in background fields, ?kgx)
• Non-perturbative
• Issue regularize the action, subtract reference
  background

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First terms

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

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What is the Full Result?
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The Post-Newtonian action

• (Reminder)
• Post-Newtonian approximation vltltc slow motion
  (CLEFT domain)
• Start with Stationary case (see caged BHs)

• Technically KK reduction over time
• Non-Relativistic Gravitation - NRG fields

0712.4116 BK, Smolkin
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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

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Goal Obtain the gauge-fixed action allowing
for time dependence

• - Make Newton happy

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Quadratic levelF, A sector
Proceed to Cubic sector and onward
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What is the full Non-Linear Result?
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Renormalization

• Before considering gravity let us consider

Take ß0. The renormalized point charge q(k) or
q(r) is defined through
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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

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Relation with F(r)

• F(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,
  F(r) satisfies a differential equation
• namely, the equation of motion

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Re-organizing the PN expansion

• These ideas can be applied to PN.
• For instance at 2PN

Can be interpreted through mass renormalization
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CommentThe beta function equation
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Recap

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

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Darkness and Light in our region
?F????S??! Thank you!
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Higher dimensional black objects
Near zone
Higher d ring
Emparan, Harmark, Niarchos, Obers, Rodrigues
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Born-Oppenheimer approximation (1927)

• 01 Field theory
• Compute ?e w. static nuclei
• and derive the effective nuclear interactions.
• In this way the EFT replaces the near zone by
  effective interactions

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Caged BHs and CLEFT
BK Smolkin 12.07

• CLEFT CLassical Effective Field Theory, no is,
  no s
• NRG decompostion (Non Relativistic Gravitation,
  which is the same as temporal KK reduction)

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Post-Newtonian approx.
Damour, Blanchet, Schafer

• NRG decompostion
• terms
• Reconstructed EIH and following
  Cardoso-Dias-Figueras generalized to higher
  dimensions

BK Smolkin 12.07b
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BH degrees of freedom

• Physical origin of eff. deg. of freedom?
• Near horizon fields (notably the metric)
• delocalized through decomposition to spherical
  harmonics