Effective Theory of Low Energy Gravity

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Effective Theory of Low Energy Gravity

• Macroscopic Effects of the Trace Anomaly
• Dynamical Vacuum Energy
• E. Mottola, LANL
• Recent Review
• w. R. Vaulin, Phys. Rev. D 74, 064004 (2006)
• w. P. Anderson R. Vaulin, Phys. Rev. D 76,
  024018 (2007)
• Review Article w. I. Antoniadis Mazur, N.
  Jour. Phys. 9, 11 (2007)
• w. M. Giannotti, Phys. Rev. D 79, 045014 (2009)
• w. P. Anderson C. Molina-Paris, Phys. Rev. D
  80, 084005 (2009)
• w. P. O. Mazur,
  Proc. Natl. Acad. Sci. 101, 9545 (2004)

arXiv 1008.5006
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Outline

• Effective Field Theory Anomalies
• Massless Scalar Poles in Anomaly Amplitudes
• Effective Theory of Low Energy Gravity
• New Scalar Degrees of Freedom from the Trace
  Anomaly
• Conformal Phase Transition RG Running of ?
• IR Conformal Fixed Point Scaling Exponents
• Horizon Effects Gravitational Condensate Stars
• Cosmological Dark Energy as Macroscopic
  Dynamical Condensate

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Effective Field Theory Quantum Anomalies

• EFT Expansion of Effective Action in Local
  Invariants
• Assumes Decoupling of Short (UV) from Long
  Distance (IR)
• But Massless Modes do not decouple
• Massless Chiral, Conformal Symmetries are
  Anomalous
• Macroscopic Effects of Short Distance physics
• Special Non-Local Terms Must be Added to Low
  Energy EFT
• IR Sensitivity to UV degrees of freedom
• Important on horizons because of large
  blueshift/redshift

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Chiral Anomaly in QCD

• QCD with Nf massless quarks has an apparent
  U(Nf) Ä Uch(Nf) Symmetry
• But Uch(1) Symmetry is Anomalous
• Effective Lagrangian in Chiral Limit has Nf 2 -
  1 (not Nf2 ) massless pions at low energies
  
• Low Energy p0 2 g dominated by the anomaly
• p0 ?5 q q ?? j ?5 e2 Nc
  F?? F ??/16?2
• q
• No Local Action in chiral limit in terms of F??
  but Non-local IR Relevant Operator that
  violates naïve decoupling of UV
• Measured decay rate verifies Nc 3 in QCD
• Anomaly Matching of IR ? UV

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2D Anomaly Action

• Integrating the anomaly linear in ? gives
• ?WZ (c/24?) ?d2x ?g (-??? R?)
• This is local but non-covariant. Note kinetic
  term for ?
• By solving for ? the WZ action can be also
  written
• ?WZ Sanomg Sanomg
• Polyakov form of the action is covariant but
  non-local
• Sanomg (-c/96?) ?d2x?gx ?d2y?gy
  Rx(?-1)xyRy
• A covariant and local form requires an auxiliary
  dynamical field ?
• Sanomg ? (-c/96?) ?d2x ?g (??)2
  -2R?

?
?
?
?
?
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Quantum Effects of 2D Anomaly Action

• Modification of Classical Theory required by
  Quantum Fluctuations Covariant Conservation
  of ?Tab?
• Metric conformal factor e2? (was constrained)
  becomes dynamical itself fluctuates freely (c
  - 26 ? c - 25)
• Gravitational Dressing of critical exponents
  at 2nd order phase transitions -- long distance
  macroscopic physics
• Non-perturbative/non-classical conformal fixed
• point of 2D gravity Running of ?
• Additional non-local Infrared Relevant Operator
  in SEFT

New Massless Scalar Degree of Freedom at low
energies
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Quantum Trace Anomaly in 4D Flat Space

• Eg. QED in an External EM Field Aµ
• Triangle One-Loop Amplitude as in Chiral
  Case
• ?abcd (p,q) (k2 gab - ka k b) (gcd pq - qc
  pd) F1(k2) (traceless terms)
• In the limit of massless fermions, F1(k2) must
  have a massless pole

Jc
p
Tab
?
k p q
q
Jd
Corresponding Imag. Part Spectral Fn. has a ?
fn This is a new massless scalar degree of
freedom in the two-particle correlated
spin-0 state
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ltTJJgt Triangle Amplitude in QED

• Determining the Amplitude by Symmetries and
  Its Finite Parts
• M. Giannotti E. M. Phys. Rev. D 79, 045014
  (2009)
• ?abcd Mass Dimension 2 Use low
  energy symmetries
• 2. By current conservation pctiabcd(p,q) 0
  qdtiabcd(p,q)
• All (but one) of these 13 tensors are dimension
  4, so dim(Fi) -2 so
• these scalar Fi(k2 p2,q2) are completely UV
  Convergent

• 1. By Lorentz invariance, can be
• expanded in a complete set of
• 13 tensors tiabcd(p,q), i 1, 13
• ?abcd (p,q) Si Fi tiabcd(p,q)

Jc
p
Tab
?
k p q
q
Jd
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ltTJJgt Triangle Amplitude in QED

• Ward Identities
• 3. By stress tensor conservation Ward Identity
  ?b?Tab?A eFab ?Jb? ?
• 4. Bose exchange symmetry ?abcd (p,q)
  ?abdc (q,p)
• Finally all 13 scalar functions Fi(k2 p2, q2)
  can be found in terms of
• finite (IR) Feynman parameter integrals
  and the polarization,
• ?ab(p) (p2gab - papb) ?(p2)
• ?abcd (p,q) (k2 gab - ka k b) (gcd pq -
  qc pd) F1(k2 p2, q2)
• (12 other terms, 11 traceless, and 1 with zero
  trace when m0)
• Result
• with D (p2 x q2 y)(1-x-y) xy k2
  m2
• UV Regularization Independent

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ltTJJgt Triangle Amplitude in QED
Spectral Representation and Sum Rule
Numerator Denominator cancel here
Im F1(k2 -s) Non-anomalous,vanishes when m0
obeys a finite sum rule independent of p2, q2, m2
and as p2, q2 , m2 ? 0
Massless scalar intermediate two-particle state
analogous to the pion in chiral limit of QCD
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Massless Anomaly Pole

• For p2 q2 0 (both photons on shell) and me
  0 the pole at k2 0 describes a massless e e -
  pair moving at vc collinearly, with opposite
  helicities in a total spin-0 state (relativistic
  Cooper pair in QFT vacuum)
• ? a massless scalar 0 state which couples
  to gravity
• Effective vertex
• h?? (g?? ? - ????)? ? F??F??
• Effective Action special case
• of general
• form

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Scalar Pole in Gravitational Scattering

• In Einsteins Theory only transverse, tracefree
  polarized waves (spin-2) are emitted/absorbed
• and propagate between sources T?? and T??
• The scalar parts give only non-progagating
• constrained interaction (like Coulomb field in
  EM)

• But for me 0 there is a scalar pole in the
• ?TJJ? triangle amplitude coupling to photons
• This scalar wave propagates in gravitational
• scattering between sources T?? and T??

• Couples to trace T??
• ?TTT? triangle of massless photons has similar
  pole
• New scalar degrees of freedom in EFT

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Constructing the EFT of Gravity

• Assume Equivalence Principle (Symmetry)
• Metric Order Parameter Field gab
• Only two strictly relevant operators (R, ?)
• Einsteins General Relativity is an EFT
• But EFT General Relativity Quantum
  Corrections
• Semi-classical Einstein Eqs. (k ltlt Mpl)
• Gab ? gab 8p G ? Tab?
• But there is also a quantum (trace) anomaly
• ? Taa? b F b' (E - 3 ?R )
  b" ?R
• Massless Poles ?New (marginally) relevant
  operator(s) needed

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ERabcdRabcd - 4RabRab R2
FCabcdCabcd
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Effective Action for the Trace AnomalyLocal
Auxiliary Field Form

• Two New Scalar Auxiliary Degrees of Freedom
• Variation of the action with respect to ?, ? --
  the
• auxiliary fields -- leads to the equations of
  motion,

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IR Relevant Term in the Action
The effective action for the trace anomaly scales
logarithmically with distance and therefore
should be included in the low energy
macroscopic EFT description of gravity Not
given in powers of Local Curvature
This is a non-trivial modification of classical
General Relativity required by quantum effects in
the Std. Model
Fluctuations of new scalar degrees of freedom
allow ?eff to vary dynamically, and can generate
a Quantum Conformal Phase of 4D Gravity where
?eff ? 0
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Dynamical Vacuum Energy

• Conformal part of the metric, gab e2?
  gab
• constrained --frozen--by trace of
  Einsteins eq. R4?
• becomes dynamical and can fluctuate due to
  ?, ?
• Fluctuations of ?, ? describe a conformally
  invariant phase of gravity in 4D ?
  non-Gaussian statistics of CMB
• In this conformal phase G-1 and ? flow to zero
  fixed point
• The Quantum Phase Transition to this phase
  characterized by the melting of the scalar
  condensate ?
• ? a dynamical state dependent condensate
  generated by SSB of global Conformal Invariance

_

• I. Antoniadis, E. M., Phys. Rev. D45 (1992) 2013
• I. Antoniadis, P. O. Mazur, E. M., Phys. Rev. D
  55 (1997) 4756, 4770
• Phys. Rev. Lett. 79 (1997) 14 Phys. Lett. B444
  (1998), 284 N. Jour. Phys. 9, 11 (2007)

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Stress Tensor of the Anomaly
Variation of the Effective Action with respect to
the metric gives stress-energy tensor

• Quantum Vacuum Polarization in Terms of
• (Semi-) Classical Auxiliary potentials
• ?, ? Depends upon the global topology of
  spacetimes and its boundaries, horizons

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Schwarzschild Spacetime

• solves homogeneous D4? 0
• Timelike Killing field (Non-local
  Invariant)
• Ka (1, 0, 0, 0)
• Energy density scales like e-4? f-2
• Auxiliary Scalar Potentials give Geometric
  (Coordinate Invariant) Meaning to Non-Local
  Quantum correlations becoming Large on Horizon

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Anomaly Scalars in Schwarzschild Space

• General solution of ?, ? equations as functions
  of r are easily found in Schwarzschild case
• q, cH, c? are integration constants, q
  topological charge
• Similar solution for ? with q', cH, c?
• Linear time dependence (p, p') can be added
• Only way to have vanishing ? as r ? ? is c? q
  0
• But only way to have finiteness on the horizon
  is
• cH 0, q 2
• Topological obstruction to finiteness vs.
  falloff of stress tensor
• Five conditions on 8 integration constants for
  horizon finiteness

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Stress-Energy Tensor in Boulware Vacuum Radial
Component
Dots Direct Numerical Evaluation of ltTabgt
(Jensen et. al. 1992) Solid Stress Tensor from
the Auxiliary Fields of the Anomaly (E.M. R. V.
2006) Dashed Page, Brown and Ottewill
approximation (1982-1986)
Spin 0 field
Diverges on horizonLarge macroscopic effect
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A Simple Model
Proc. Natl. Acad. Sci., 101, 9545 (2004)
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Analog to quantum BEC transition near the
classical horizon Can now check with full
EFT of Low Energy Gravity
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Gravitational Vacuum Condensate StarsGravastars
as Astrophysical Objects

• Cold, Dark, Compact, Arbitrary M, J
• Accrete Matter just like a black hole
• But matter does not disappear down a hole
• Relativistic Surface Layer can re-emit radiation
• Can support Electric Currents, Large Magnetic
  Fields
• Possibly more efficient central engine for Gamma
  Ray Bursters, Jets, UHE Cosmic Rays
• Formation should be a violent phase transition
  converting gravitational energy and baryons
  into HE leptons and entropy
• Gravitational Wave Signatures
• Dark Energy as Condensate Core -- Finite Size
  Casimir effect
• of boundary conditions at the horizon

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New Horizons in Gravity

• Einsteins classical theory receives Quantum
  Corrections relevant at macroscopic Distances
  near Event Horizons
• These arise from new scalar degrees of freedom in
  the EFT of Gravity required by the
  Conformal/Trace Anomaly
• EFT of Gravity is fine provided these anomaly
  degrees of freedom are taken into account
• Their Fluctuations allow ? to flow to zero at an
  IR conformal fixed point (can/should be checked
  by ERG)
• Their Fluctuations can induce a Quantum Phase
  Transition at the horizon of a black hole
• ?eff is a dynamical condensate which can change
  in the phase transition remove black hole
  interior singularity

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• Gravitational Condensate Stars resolve all
  black hole
• paradoxes ? Astrophysics of gravastars testable
• The cosmological dark energy of our Universe may
  be a
• macroscopic finite size effect whose value
  depends not
• on microphysics but on the cosmological horizon
  scale

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Exact Effective Action Wilson Effective Action

• Integrating out Matter Fields in Fixed
  Gravitational Background gives the Exact Quantum
  Effective Action
• The possible terms in Sexactg can be classified
  according to their repsonse to local Weyl
  rescalings g ? e2? g
• Sexactg Slocalg Sanomg
  SWeylg
• Slocalg (1/16?G) ? d4x ?g (R - 2 ?) ?n4
  MPl4-n S(n)localg
• Ascending series of higher derivative
  local terms, ngt4 irrelevant
• Non-local but Weyl-invariant (neutral under
  rescalings)
• SWeylg SWeyle2?g
• Sanomg special non-local terms that scale
  linearly with ?, logarithmically with
  distance, representatives of non-trivial
  cohomology under Weyl group
• Wilson effective action captures all IR physics

Seffg SHEg Sanomg
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