1. Effective Theory of Low Energy Gravity Macroscopic Effects of the Trace Anomaly & Dynamical Vacuum Energy E. Mottola, LANL Recent Review: 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) w. P. O. Mazur, Proc. Natl. Acad. Sci. 101, 9545 (2004) arXiv: 1008.5006

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

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

4. Chiral Anomaly in QCD QCD with N f massless quarks has an apparent U(N f )  U ch (N f ) SymmetryQCD with N f massless quarks has an apparent U(N f )  U ch (N f ) Symmetry But U ch (1) Symmetry is Anomalous But U ch (1) Symmetry is Anomalous Effective Lagrangian in Chiral Limit has N f 2 - 1 (not N f 2 ) massless pions at low energies Effective Lagrangian in Chiral Limit has N f 2 - 1 (not N f 2 ) massless pions at low energies Low Energy  0  2  dominated by the anomaly Low Energy  0  2  dominated by the anomaly ~ ~  0  5 q q   j  5 = e 2 N c F  F  /16  2  0  5 q q   j  5 = e 2 N c 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 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 N c = 3 in QCD Measured decay rate verifies N c = 3 in QCD Anomaly Matching of IR ↔ UV

6. 2D Anomaly Action Integrating the anomaly linear in gives Integrating the anomaly linear in  gives  WZ = (c/24  )  d 2 x  g (-    + R  ) This is local but non-covariant. Note kinetic term for This is local but non-covariant. Note kinetic term for  By solving for the WZ action can be also written By solving for  the WZ action can be also written  WZ = S anom [g] S anom [g] Polyakov form of the action is covariant but non-local Polyakov form of the action is covariant but non-local S anom [g] = (-c/96  )  d 2 x  g x  d 2 y  g y R x (  -1 ) xy R y S anom [g] = (-c/96  )  d 2 x  g x  d 2 y  g y R x (  -1 ) xy R y A covariant and local form requires an auxiliary dynamical field  A covariant and local form requires an auxiliary dynamical field  S anom [g;  ] = (-c/96  )  d 2 x  g {(  ) 2 -2R  } S anom [g;  ] = (-c/96  )  d 2 x  g {(  ) 2 -2R  }     

7. Quantum Effects of 2D Anomaly Action Modification of Classical Theory required by Quantum Fluctuations & Covariant Conservation of  T a b  Modification of Classical Theory required by Quantum Fluctuations & Covariant Conservation of  T a b  Metric conformal factor (was constrained)becomes dynamical & itself fluctuates freely (c - 26  c - 25) Metric conformal factor e 2  (was constrained) becomes dynamical & itself fluctuates freely (c - 26  c - 25) Gravitational ‘Dressing’ of critical exponents at 2 nd order phase transitions -- long distance macroscopic physics Gravitational ‘Dressing’ of critical exponents at 2 nd order phase transitions -- long distance macroscopic physics Non-perturbative/non-classical conformal fixed Non-perturbative/non-classical conformal fixed point of 2D gravity: Running of  Additional non-local Infrared Relevant Operator in S EFT Additional non-local Infrared Relevant Operator in S EFT New Massless Scalar Degree of Freedom at low energies

8. Quantum Trace Anomaly in 4D Flat Space Eg. QED in an External EM Field A µ Eg. QED in an External EM Field A µ Triangle One-Loop Amplitude as in Chiral Case Triangle One-Loop Amplitude as in Chiral Case  abcd (p,q) = (k 2 g ab - k a k b ) (g cd p q - q c p d ) F 1 (k 2 ) + ( traceless terms )  abcd (p,q) = (k 2 g ab - k a k b ) (g cd p q - q c p d ) F 1 (k 2 ) + ( traceless terms ) In the limit of massless fermions, F 1 (k 2 ) must have a massless pole: In the limit of massless fermions, F 1 (k 2 ) must have a massless pole:  T ab JcJc JdJd p q k = p + q 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 the two-particle correlated spin-0 state

9. Triangle Amplitude in QED Triangle Amplitude in QED Determining the Amplitude by Symmetries and Its Finite Parts Determining the Amplitude by Symmetries and Its Finite Parts M. Giannotti & E. M. Phys. Rev. D 79, 045014 (2009) M. Giannotti & E. M. Phys. Rev. D 79, 045014 (2009)  abcd : Mass Dimension 2 Use low energy symmetries: 2. By current conservation: p c t i abcd (p,q) = 0 = q d t i abcd (p,q) All (but one) of these 13 tensors are dimension ≥ 4, so dim(F i ) ≤ -2 so these scalar F i (k 2 ; p 2,q 2 ) are completely UV Convergent  T ab JcJc JdJd p q k = p + q 1. By Lorentz invariance, can be expanded in a complete set of 13 tensors t i abcd (p,q), i =1, …13:  abcd (p,q) = Σ i F i t i abcd (p,q)

10. Triangle Amplitude in QED Triangle Amplitude in QED Ward Identities Ward Identities 3. By stress tensor conservation Ward Identity:  b  T ab  A = eF ab  J b   4. Bose exchange symmetry:  abcd (p,q) =  abdc (q,p) Finally all 13 scalar functions F i (k 2 ; p 2, q 2 ) can be found in terms of Finally all 13 scalar functions F i (k 2 ; p 2, q 2 ) can be found in terms of finite (IR) Feynman parameter integrals and the polarization, finite (IR) Feynman parameter integrals and the polarization,  ab (p) = (p 2 g ab - p a p b )  (p 2 )  abcd (p,q) = (k 2 g ab - k a k b ) (g cd p q - q c p d ) F 1 (k 2 ; p 2, q 2 ) + …  abcd (p,q) = (k 2 g ab - k a k b ) (g cd p q - q c p d ) F 1 (k 2 ; p 2, q 2 ) + … (12 other terms, 11 traceless, and 1 with zero trace when m=0) (12 other terms, 11 traceless, and 1 with zero trace when m=0)Result: with D = (p 2 x + q 2 y)(1-x-y) + xy k 2 + m 2 with D = (p 2 x + q 2 y)(1-x-y) + xy k 2 + m 2 UV Regularization Independent

11. Triangle Amplitude in QED Triangle Amplitude in QED Spectral Representation and Sum Rule Im F 1 (k 2 = -s): Non-anomalous,vanishes when m=0 Numerator & Denominator cancel here obeys a finite sum rule independent of p 2, q 2, m 2 and as p 2, q 2, m 2  0 + Massless scalar intermediate two-particle state analogous to the pion in chiral limit of QCD

12. Massless Anomaly Pole For p 2 = q 2 = 0 (both photons on shell) and m e = 0 the pole at k 2 = 0 describes a massless e + e - pair moving at v=c 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  a massless scalar 0 + state which couples to gravity Effective vertex h  (g   -    )  ´  F  F  h  (g   -    )  ´  F  F  Effective Action special case Effective Action special case of general of general form form

13. Scalar Pole in Gravitational Scattering In Einstein’s Theory only transverse, tracefree polarized waves (spin-2) are emitted/absorbed In Einstein’s Theory only transverse, tracefree polarized waves (spin-2) are emitted/absorbed and propagate between sources and propagate between sources T´  and T  The scalar parts give only non-progagating The scalar parts give only non-progagating constrained interaction (like Coulomb field in E&M) But for m e = 0 there is a scalar pole in the But for m e = 0 there is a scalar pole in the triangle amplitude coupling to photons  TJJ  triangle amplitude coupling to photons This scalar wave propagates in gravitational This scalar wave propagates in gravitational scattering between sources scattering between sources T´  and T  Couples to trace Couples to trace T´   masslessphotons  TTT  triangle of massless photons has similar pole New scalar degrees of freedom in EFT New scalar degrees of freedom in EFT

14. Constructing the EFT of Gravity Equivalence Principle (Symmetry) Assume Equivalence Principle (Symmetry) Metric Order Parameter Field g ab Metric Order Parameter Field g ab Only two strictly relevant operators (R,  ) Only two strictly relevant operators (R,  ) Einstein’s General Relativity is an EFT Einstein’s General Relativity is an EFT But EFT = General Relativity + Quantum Corrections But EFT = General Relativity + Quantum Corrections Semi-classical Einstein Eqs. (k << M pl ): Semi-classical Einstein Eqs. (k << M pl ): G ab +  g ab = 8  G  T ab  But there is also a quantum (trace) anomaly: But there is also a quantum (trace) anomaly:  T a a  = b F + b' (E - 3  R ) + b"  R  T a a  = b F + b' (E - 3  R ) + b"  R Massless Poles  New (marginally) relevant operator(s) needed Massless Poles  New (marginally) relevant operator(s) needed 2 E=R abcd R abcd - 4R ab R ab + R 2 F=C abcd C abcd

16. Effective Action for the Trace Anomaly Local 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, auxiliary fields -- leads to the equations of motion,

17. IR Relevant Term in the Action 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 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

18. Dynamical Vacuum Energy Conformal part of the metric, g ab = e 2  g ab Conformal part of the metric, g ab = e 2  g ab constrained --frozen--by trace of Einstein’s eq. R=4  constrained --frozen--by trace of Einstein’s eq. R=4  becomes dynamical and can fluctuate due to ,  becomes dynamical and can fluctuate due to ,  Fluctuations of ,  describe a conformally invariant phase of gravity in 4D  non-Gaussian statistics of CMB 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 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  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  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)

20. 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 ,  Depends upon the global topology of spacetimes and its boundaries, horizons

21. Schwarzschild Spacetime solves homogeneous  4  = 0 Timelike Killing field (Non-local Invariant) Timelike Killing field (Non-local Invariant) K a = (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

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

23. Stress-Energy Tensor in Boulware Vacuum – Radial Component Spin 0 field Dots – Direct Numerical Evaluation of (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) Diverges on horizon— Large macroscopic effect

26. A Simple Model Proc. Natl. Acad. Sci., 101, 9545 (2004)

27. Analog to quantum BEC transition near the classical horizon Can now check with full EFT of Low Energy Gravity

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

29. New Horizons in Gravity Einstein’s classical theory receives Quantum Corrections relevant at macroscopic Distances & near Event Horizons Einstein’s 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 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 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 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’ 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  eff is a dynamical condensate which can change in the phase transition & remove ‘black hole’ interior singularity

30. Gravitational Condensate Stars resolve all ‘black hole’ Gravitational Condensate Stars resolve all ‘black hole’ paradoxes  Astrophysics of gravastars testable The cosmological dark energy of our Universe may be a 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

31. Exact Effective Action &Wilson Effective Action Integrating out Matter + … Fields in Fixed Gravitational Background gives the Exact Quantum Effective Action Integrating out Matter + … Fields in Fixed Gravitational Background gives the Exact Quantum Effective Action The possible terms in S exact [g] can be classified according to their repsonse to local Weyl rescalings g  e 2  g The possible terms in S exact [g] can be classified according to their repsonse to local Weyl rescalings g  e 2  g S exact [g] = S local [g] + S anom [g] + S Weyl [g] S exact [g] = S local [g] + S anom [g] + S Weyl [g] S local [g] = (1/16  G)  d 4 x  g (R - 2  ) +  n≥4 M Pl 4-n S (n) local [g] S local [g] = (1/16  G)  d 4 x  g (R - 2  ) +  n≥4 M Pl 4-n S (n) local [g] Ascending series of higher derivative local terms, n>4 irrelevant Ascending series of higher derivative local terms, n>4 irrelevant Non-local but Weyl-invariant (neutral under rescalings) Non-local but Weyl-invariant (neutral under rescalings) S Weyl [g] = S Weyl [e 2  g] S Weyl [g] = S Weyl [e 2  g] S anom [g] special non-local terms that scale linearly with , logarithmically with distance, representatives of non-trivial cohomology under Weyl group S anom [g] 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 Wilson effective action captures all IR physics S eff [g] = S HE [g] + S anom [g]