Effective Theory of Low Energy Effective Theory of Low Energy Gravity Gravity

Macroscopic Effects of the Trace Macroscopic Effects of the Trace Anomaly Anomaly

& Dynamical Vacuum Energy& Dynamical Vacuum Energy

E. Mottola, LANLE. Mottola, LANL

Recent Review: Recent Review:

w. w. R. VaulinR. Vaulin, , Phys. Rev. DPhys. Rev. D 74, 064004 (2006) 74, 064004 (2006)

w. w. P. Anderson & R. VaulinP. Anderson & R. Vaulin, , Phys. Rev. DPhys. Rev. D 76, 024018 (2007) 76, 024018 (2007)

Review Article: w. Review Article: w. I. Antoniadis & MazurI. Antoniadis & Mazur, , N. Jour. Phys.N. Jour. Phys. 9, 11 9, 11 (2007)(2007)

w. w. M. GiannottiM. Giannotti, , Phys. Rev. DPhys. Rev. D 79, 045014 (2009) 79, 045014 (2009)w. w. P. Anderson & C. Molina-ParisP. Anderson & C. Molina-Paris, , Phys. Rev. DPhys. Rev. D 80, 80,

084005 (2009)084005 (2009)

w. w. P. O. MazurP. O. Mazur, , Proc. Natl. Acad. Sci. Proc. Natl. Acad. Sci. 101, 101,

9545 (2004)9545 (2004)

arXiv: 1008.5006 arXiv: 1008.5006

OutlineOutline

• Effective Field Theory & AnomaliesEffective Field Theory & Anomalies• Massless Scalar Poles in Anomaly AmplitudesMassless Scalar Poles in Anomaly Amplitudes

• Effective Theory of Low Energy GravityEffective Theory of Low Energy Gravity• New Scalar Degrees of Freedom from the Trace New Scalar Degrees of Freedom from the Trace

AnomalyAnomaly• 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 Horizon Effects & Gravitational Condensate

StarsStars• Cosmological Dark Energy as Macroscopic Cosmological Dark Energy as Macroscopic

Dynamical CondensateDynamical Condensate

Effective Field Theory & Effective Field Theory & Quantum AnomaliesQuantum Anomalies

• EFT = Expansion of Effective Action in EFT = Expansion of Effective Action in LocalLocal InvariantsInvariants

• Assumes Assumes DecouplingDecoupling of Short (of Short (UVUV) from Long ) from Long Distance (Distance (IRIR))

• But But MasslessMassless Modes do Modes do notnot decouple decouple

• Massless Chiral, Conformal Symmetries are Massless Chiral, Conformal Symmetries are AnomalousAnomalous

• MacroscopicMacroscopic Effects of Short Distance physics Effects of Short Distance physics

• Special Special Non-LocalNon-Local Terms Must be Added to Low Terms Must be Added to Low

Energy EFTEnergy EFT

• IRIR Sensitivity to UVUV degrees of freedom

• Important on horizons because of large

blueshift/redshift

Chiral Anomaly in QCDChiral Anomaly in QCD

• QCD with QCD with NNff massless quarks has an massless quarks has an apparentapparent U(NU(Nff) )

UUchch(N(Nff)) SymmetrySymmetry• ButBut U Uchch(1)(1) Symmetry isSymmetry is AnomalousAnomalous• Effective Lagrangian in Chiral Limit hasEffective Lagrangian in Chiral Limit has NNf f

2 2 - 1- 1 ((notnot

NNff2 2 ) ) massless pions at low energiesmassless pions at low energies

• Low EnergyLow Energy 00 2 2 dominateddominated by the anomalyby the anomaly ~~

0 0 55 qq qq j j 55 = e = e22 N Nc c FF F F

/16/1622

qq

• No No LocalLocal Action in chiral limit in terms of Action in chiral limit in terms of FF butbut Non-Non-

locallocal IR IR Relevant OperatorRelevant Operator that violates naïve that violates naïve

decoupling ofdecoupling of UVUV• MeasuredMeasured decay rate verifiesdecay rate verifies NNcc = 3 = 3 in QCDin QCD

Anomaly Matching of Anomaly Matching of IR IR ↔↔ UVUV

2D Anomaly Action2D Anomaly Action

• Integrating the anomaly linear in Integrating the anomaly linear in givesgives

WZ = (c/24) d2x g (- + R)

• This is local but This is local but non-covariantnon-covariant. Note . Note kinetickinetic term term for for • By solving for By solving for the WZ action can be also writtenthe WZ action can be also written

� WZWZ = S= Sanomanom[g] S[g] Sanomanom[g][g]

• Polyakov form of the action is covariant but Polyakov form of the action is covariant but non-non-locallocal

SSanomanom[g] = [g] = (-c/96(-c/96) ) dd22xxggxx dd22yyggyy RRxx((--

11))xyxyRRyy

• A covariant and local form requires an auxiliary A covariant and local form requires an auxiliary dynamicaldynamical field field SSanomanom[g; [g; ] = ] = (-c/96(-c/96) ) dd22x x g {(g {())2 2 --2R2R}}

Quantum Effects of 2D Anomaly Quantum Effects of 2D Anomaly ActionAction

• ModificationModification of Classical Theory required by of Classical Theory required by Quantum Quantum Fluctuations & Covariant Fluctuations & Covariant Conservation of Conservation of TTaa

bb• Metric conformal factor Metric conformal factor e2 (was (was constrained)constrained) becomes becomes dynamicaldynamical & & itself fluctuates freely (itself fluctuates freely (c - 26 c - 26 c - 25 c - 25))• Gravitational ‘Dressing’ of critical Gravitational ‘Dressing’ of critical exponents at 2exponents at 2nd nd order order phase phase transitions --transitions -- long distancelong distance macroscopic macroscopic physicsphysics

• Non-perturbative/non-classical conformal Non-perturbative/non-classical conformal fixed fixed

point of 2D gravity: Running of point of 2D gravity: Running of • Additional Additional non-localnon-local InfraredInfrared Relevant Relevant Operator in Operator in SSEFTEFT

NewNew Massless ScalarMassless Scalar Degree of Freedom at low energies Degree of Freedom at low energies

Quantum Trace Anomaly in 4D Quantum Trace Anomaly in 4D Flat SpaceFlat Space

Eg.Eg. QED in an External EM FieldQED in an External EM Field AAµµ

Triangle One-Loop Amplitude as in Chiral Triangle One-Loop Amplitude as in Chiral CaseCase

abcdabcd (p,q) = (p,q) = (k(k2 2 ggab ab -- kka a kk bb)) (g(gcd cd pp••q - qq - qc c ppdd) F) F11(k(k22)) + +

((traceless termstraceless terms))

In the limit of massless fermions,In the limit of massless fermions, FF11(k(k22)) must have must have

a massless pole:a massless pole:Tab

Jc

Jd

p

qk = p + q

Corresponding Imag. Part Spectral Fn. has a Corresponding Imag. Part Spectral Fn. has a fn fnThis is a new massless scalar degree of freedom in This is a new massless scalar degree of freedom in the two-particle correlated spin-0 statethe two-particle correlated spin-0 state

<TJJ> Triangle Amplitude in QED<TJJ> Triangle Amplitude in QED Determining the Amplitude by Symmetries and Its Finite Determining the Amplitude by Symmetries and Its Finite PartsParts

M. Giannotti & E. M. M. Giannotti & E. M. Phys. Rev. DPhys. Rev. D 7979, 045014 (2009), 045014 (2009)

abcdabcd: : Mass Dimension Mass Dimension 22 Use Use low energylow energy symmetries:symmetries:

2. By 2. By current conservationcurrent conservation: : ppccttiiabcdabcd(p,q) (p,q) = 0 = = 0 = qqddttii

abcdabcd(p,q)(p,q)All (but one) of these 13 tensors are All (but one) of these 13 tensors are dimension ≥ 4dimension ≥ 4, so dim(F, so dim(Fii) ≤ -) ≤ -

2 so2 so

these scalar these scalar FFii(k(k22; p; p22,q,q22)) are completelyare completely UV ConvergentUV Convergent

Tab

Jc

Jd

p

qk = p + q

1. By 1. By Lorentz invariance,Lorentz invariance, can can be be

expanded in a complete set of expanded in a complete set of 13 tensors13 tensors t tii

abcdabcd(p,q), i =1, …13:(p,q), i =1, …13:

abcdabcd (p,q) = (p,q) = ΣΣii F Fi i

ttiiabcdabcd(p,q)(p,q)

<TJJ> Triangle Amplitude in QED<TJJ> Triangle Amplitude in QED Ward IdentitiesWard Identities

3. By 3. By stress tensor conservation Ward Identity:stress tensor conservation Ward Identity: bbTTababAA = eF = eFab ab JJbb

4. 4. Bose exchange symmetry:Bose exchange symmetry: abcdabcd (p,q) = (p,q) = aabdcbdc (q,p) (q,p)

Finally all 13 scalar functions Finally all 13 scalar functions FFii(k(k22; p; p22, q, q22) ) can be found in terms can be found in terms

of of

finitefinite (IR)(IR) Feynman parameter integrals and the polarization,Feynman parameter integrals and the polarization,abab(p) = (p(p) = (p22ggab ab - p- paappbb) ) (p(p22))

abcdabcd (p,q) = (p,q) = (k(k2 2 ggab ab -- kka a kk bb)) (g(gcd cd pp••q - qq - qc c ppdd) F) F11(k(k22; p; p22, q, q22)) + …+ … (12 other terms, 11 traceless, and 1 with zero trace when (12 other terms, 11 traceless, and 1 with zero trace when m=0m=0))

Result:Result:

with with D = (pD = (p22 x + q x + q22 y)(1-x-y) + xy k y)(1-x-y) + xy k22 + m + m22

UVUV Regularization IndependentRegularization Independent

€

kbΓabcd (p,q) = (gac pb −δb

c pa )Π bd (q) + (gadqb −δbdqa )Π bc ( p)

€

F1(k2; p2,q2) =

e2

18π 2k 21− 3m2 dx dy

(1− 4xy)

D0

1−x

∫0

1

∫ ⎧ ⎨ ⎩

⎫ ⎬ ⎭

<TJJ> Triangle Amplitude in <TJJ> Triangle Amplitude in QEDQED

Spectral Representation and Sum Rule

Im F1(k2 = -s): Non-anomalous,vanishes when m=0Numerator & Denominator cancel here

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

Massless Anomaly PoleMassless Anomaly Pole

For For pp22 = q = q2 2 = 0 = 0 (both photons on shell) and(both photons on shell) and m mee = 0 = 0 the pole atthe pole at kk22

= 0= 0 describes a masslessdescribes a massless ee+ + ee - - pair moving at pair moving at v=cv=c collinearlycollinearly, , with opposite helicities in a total spin-0 state (relativistic with opposite helicities in a total spin-0 state (relativistic Cooper pairCooper pair in QFT in QFT vacuumvacuum))

a massless scalar a massless scalar 00++ state which couples to gravitystate which couples to gravity

Effective Effective vertexvertex

hh (g (g - - ))´́ F FFF

Effective Action Effective Action special special casecase

of generalof general formform

Scalar Pole in Gravitational Scalar Pole in Gravitational

ScatteringScattering • In Einstein’s Theory only transverse, In Einstein’s Theory only transverse, tracefree polarized waves (tracefree polarized waves (spin-2spin-2) ) are emitted/absorbed are emitted/absorbed and propagate between sources and propagate between sources T´ and T

• The scalar parts give only The scalar parts give only non-non-

progagatingprogagatingconstrained interaction (like Coulomb constrained interaction (like Coulomb field in E&M)field in E&M)

• But for But for mmee = 0 = 0 there is a scalar pole in the there is a scalar pole in the TJJ triangle amplitude coupling to photons triangle amplitude coupling to photons

• This scalar wave propagates in gravitationalThis scalar wave propagates in gravitational scattering between sources scattering between sources T´ and T

• Couples to trace Couples to trace T´• TTT triangle of masslessmassless photonsphotons has

similar pole

• New New scalarscalar degrees of freedom in degrees of freedom in

EFTEFT

Constructing the EFT of GravityConstructing the EFT of Gravity• Assume Equivalence PrincipleEquivalence Principle (Symmetry)(Symmetry)• Metric Order Parameter Field Metric Order Parameter Field ggabab

• Only two strictly Only two strictly relevantrelevant operators operators (R, (R, )) • Einstein’s General Relativity Einstein’s General Relativity isis an EFT an EFT• But EFT = But EFT = General Relativity General Relativity ++ QuantumQuantum CorrectionsCorrections • Semi-classical Einstein Eqs. Semi-classical Einstein Eqs. (k << M(k << Mplpl):):

GGabab+ + g gabab = 8 = 8 G G TTabab• But there is also a quantum (trace) anomaly:But there is also a quantum (trace) anomaly:

TTaaaa = b F = b F + b' (E+ b' (E - - 33 R )R ) + b" + b" RR

• Massless Poles Massless Poles NewNew (marginally) relevant (marginally) relevant

operator(s) neededoperator(s) needed

22

E=RE=RabcdabcdRRabcd abcd - 4R- 4RababRRab ab

+ R+ R22

F=CF=CabcdabcdCCabcdabcd

Effective Action for the Trace Effective Action for the Trace AnomalyAnomaly

Local Auxiliary Field FormLocal Auxiliary Field Form

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

IR Relevant Term in the ActionIR Relevant Term in the Action

Fluctuations of new scalar degrees of freedom allowFluctuations of new scalar degrees of freedom allow effeff to vary dynamically, and can generate ato vary dynamically, and can generate a

Quantum Conformal Phase of 4D Gravity whereQuantum Conformal Phase of 4D Gravity where effeff 0 0

The effective action for the trace anomaly scales The effective action for the trace anomaly scales logarithmically with distance and therefore logarithmically with distance and therefore

should be included in the low energy should be included in the low energy macroscopic EFT description of gravity—macroscopic EFT description of gravity—

Not given in powers of Local Not given in powers of Local CurvatureCurvature

This is a non-trivial modification of classical General This is a non-trivial modification of classical General Relativity required by quantum effects in the Std. ModelRelativity required by quantum effects in the Std. Model

Dynamical Vacuum Energy Dynamical Vacuum Energy

• Conformal part of the metric, Conformal part of the metric, ggabab = e = e22 g gabab constrained --constrained --frozenfrozen--by trace of Einstein’s --by trace of Einstein’s eq. eq. R=4R=4 becomes becomes dynamicaldynamical and can and can fluctuate fluctuate due due to to , , • Fluctuations of Fluctuations of , , describe a describe a conformally conformally invariant invariant phase phase of gravity in 4D of gravity in 4D non-Gaussian statistics of CMBnon-Gaussian statistics of CMB• In this conformal phase In this conformal phase GG-1-1 and and flow to flow to zero zero fixed pointfixed point• The Quantum Phase Transition to this phase The Quantum Phase Transition to this phase characterized characterized by the by the ‘melting’‘melting’ of the scalar of the scalar condensate condensate • a dynamicala dynamical state dependent condensatestate dependent condensate generated by generated by SSB of global Conformal SSB of global Conformal InvarianceInvariance

_

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)

Stress Tensor of the AnomalyStress Tensor of the Anomaly

Variation of the Effective Action with Variation of the Effective Action with respect to the metric gives stress-respect to the metric gives stress-

energy tensorenergy tensor

• Quantum Vacuum Polarization in Quantum Vacuum Polarization in Terms of Terms of

(Semi-) Classical Auxiliary (Semi-) Classical Auxiliary potentialspotentials• , , Depends upon the global Depends upon the global topology of topology of spacetimes and spacetimes and its boundaries, its boundaries, horizonshorizons

Schwarzschild Spacetime Schwarzschild Spacetime

solvessolves homogeneous homogeneous 44 = 0 = 0

Timelike Killing field (Timelike Killing field (Non-localNon-local Invariant) Invariant)

KKaa = (1, 0, 0, 0) = (1, 0, 0, 0) Energy density scales likeEnergy density scales like e e-4-4 = f = f-2-2

Auxiliary Scalar Potentials give Geometric Auxiliary Scalar Potentials give Geometric ((Coordinate InvariantCoordinate Invariant) Meaning to Non-) Meaning to Non-

Local Quantum correlations becoming Large Local Quantum correlations becoming Large on Horizonon Horizon

22 2 2 22

2(1 )

(1 )Mr M

r

drds dt r d=− − + + Ω

−

Anomaly Scalars in Schwarzschild Anomaly Scalars in Schwarzschild SpaceSpace

• General solution of General solution of , , equations as functions of equations as functions of r are r are easily found in Schwarzschild caseeasily found in Schwarzschild case

• q, cq, cHH, c, c are integration constants, are integration constants, q q topological topological chargecharge• Similar solution for Similar solution for with with q', cq', cHH, c, c

• Linear time dependence (Linear time dependence (p, p'p, p') can be added ) can be added • Only way to have vanishing Only way to have vanishing as as r r is is cc == q = q = 00• But only way to have finiteness on the horizon is But only way to have finiteness on the horizon is

cH = 0, q = 2• Topological obstruction to finiteness vs. falloff of Topological obstruction to finiteness vs. falloff of stress tensor stress tensor • Five conditions on 8 integration constantsFive conditions on 8 integration constants for for horizon finitenesshorizon finiteness

Stress-Energy Tensor in Stress-Energy Tensor in Boulware Vacuum – Radial Boulware Vacuum – Radial

ComponentComponent

Spin 0 field

Dots – Direct Numerical Evaluation of <Tab> (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

A Simple A Simple ModelModel

Proc. Natl. Acad. Sci., 101, 9545 (2004) Proc. Natl. Acad. Sci., 101, 9545 (2004)

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

Gravitational Vacuum Condensate Gravitational Vacuum Condensate

StarsStars

Gravastars as Astrophysical ObjectsGravastars as Astrophysical Objects• Cold, Dark, Compact, ArbitraryCold, Dark, Compact, Arbitrary M, JM, J• AccreteAccrete Matter just like a black hole Matter just like a black hole• But matter does But matter does notnot disappear down a ‘hole’ disappear down a ‘hole’• Relativistic Surface Layer can Relativistic Surface Layer can re-emitre-emit radiation radiation• Can support Electric Currents, Large Can support Electric Currents, Large Magnetic Magnetic

FieldsFields• Possibly more efficient central engine for Possibly more efficient central engine for Gamma Gamma

Ray Ray Bursters, Jets, UHE Cosmic RaysBursters, Jets, UHE Cosmic Rays• Formation should be a violent phase transition Formation should be a violent phase transition

converting gravitational energy and baryons into converting gravitational energy and baryons into HE leptons and entropyHE leptons and entropy

• Gravitational WaveGravitational Wave Signatures Signatures• Dark Energy as CondensateDark Energy as Condensate Core -- Core -- Finite Size Finite Size

Casimir effectCasimir effect of boundary conditions at the horizonof boundary conditions at the horizon

New Horizons in GravityNew Horizons in Gravity

• Einstein’s classical theory receives Quantum Corrections Einstein’s classical theory receives Quantum Corrections relevant at relevant at macroscopicmacroscopic Distances & near Event Horizons Distances & near Event Horizons

• These arise from These arise from new scalar degrees of freedomnew scalar degrees of freedom in the in the EFT of EFT of GravityGravity required by the required by the Conformal/Trace Conformal/Trace AnomalyAnomaly

• EFT of GravityEFT of Gravity is fineis fine providedprovided these anomaly degrees of these anomaly degrees of freedom freedom are taken into accountare taken into account

• Their Fluctuations allow Their Fluctuations allow to flow to zero at an to flow to zero at an IR IR conformal fixed conformal fixed point (can/should be checked by point (can/should be checked by ERGERG))

• Their Fluctuations can induce a Their Fluctuations can induce a Quantum Phase TransitionQuantum Phase Transition at the at the horizon of a ‘black hole’horizon of a ‘black hole’

• effeff is a is a dynamical condensatedynamical condensate which can change in the which can change in the

phase phase transition & remove ‘black hole’ interior transition & remove ‘black hole’ interior singularitysingularity

• Gravitational Condensate StarsGravitational Condensate Stars resolve all resolve all ‘black hole’ ‘black hole’

paradoxesparadoxes Astrophysics of Astrophysics of gravastars gravastars testabletestable• The cosmological dark energy of our The cosmological dark energy of our Universe may be aUniverse may be amacroscopic finite size effectmacroscopic finite size effect whose value whose value depends depends notnot on microphysics but on the on microphysics but on the cosmological cosmological horizon scalehorizon scale

Exact Effective Action &Wilson Exact Effective Action &Wilson Effective ActionEffective Action

• Integrating out Matter + … Fields in Fixed Gravitational Integrating out Matter + … Fields in Fixed Gravitational Background gives Background gives the the ExactExact Quantum Effective Action Quantum Effective Action

• The possible terms in The possible terms in SSexactexact[g][g] can be classified according to their can be classified according to their

repsonse to repsonse to local Weyl rescalings local Weyl rescalings g g e e22 gg

SSexactexact[g] = S[g] = Slocallocal[g] + S[g] + Sanomanom[g] + S[g] + SWeylWeyl[g] [g] • SSlocallocal[g] = (1/16[g] = (1/16G) G) d d44x x g (R - 2 g (R - 2 ) + ) + n≥4n≥4 M MPlPl

4-n 4-n

SS(n)(n)locallocal[g][g]

Ascending series of higher derivative local terms, n>4 Ascending series of higher derivative local terms, n>4 irrelevantirrelevant

• Non-local but Weyl-invariant (neutral under rescalings)Non-local but Weyl-invariant (neutral under rescalings)

SSWeylWeyl[g] = S[g] = SWeylWeyl[e[e22g]g]

• SSanomanom[g] [g] special non-local terms that scale linearly with special non-local terms that scale linearly with , , logarithmically with logarithmically with distance, representatives of non-distance, representatives of non-trivial cohomology under Weyl grouptrivial cohomology under Weyl group

• Wilson effective action captures all Wilson effective action captures all IRIR physics physics SSeffeff[g] = S[g] = SHEHE[g] + S[g] + Sanomanom[g][g]