1 DE SITTER BREAKING FROM GRAVITONS AT THE

˚ 1˚ DE SITTER BREAKING FROM GRAVITONS AT THE ONE LOOP LEVEL Tomislav Prokopec, ITP Utrecht University D. Glavan, S. P. Miao, T. Prokopec and R. P. Woodard, ``Graviton Loop Corrections to Vacuum Polarization in de Sitter in a General Covariant Gauge, ” arxiv: 1504. 00894, CQG (2015? ) D. Glavan S. P. Miao, T. Prokopec and R. P. Woodard, Class. Quant. Grav 31 (2014) 175002, [ar. Xiv: 1308. 3453 [gr-qc]] Mainz, 25 Jun 2015

CONTENTS 1) QUANTUM EFFECTS ON DE SITTER 2) REVIEW OF THE MASSLESS SCALAR FIELD 3) WHEN STABILITY FAILS. . YUKAWA THEORY 4) STABLE THEORY: SCALAR ELECTRODYNAMICS 5) PERT QUANTUM GRAVITY ON DE SITTER: VACUUM POLARIZATION 6) CONCLUSIONS AND OPEN PROBLEMS ˚ 2˚

˚ 3˚ QUANTUM EFFECTS ON DE SITTER: SCALARS

QUANTUM FIELDS ON DE SITTER ˚ 4˚ ● ONE CAN QUANTIZE: SCALAR, FERMIONIC, VECTOR AND TENSOR (GRAVITON) FIELDS ON DE SITTER, AND CONSIDER THEIR (NON-) PERTURBATIVE EVOLUTION AND BACKREACTION ON d. S BACKGROUND ● EXAMPLE 1: MASSLESS SCALAR ON DE SITTER (D=4): ACTION: EOM for : In a d. S invariant state, the propagator must be a function of d. S inv distance l(x; x’): then the propagator obeys a d. S inv equation: This equation has no solution! Allen, Folacci, PRD 35 (1987)

MASSLESS SCALAR ON DE SITTER ● MASSLESS MINIMALLY COUPLED SCALAR (MMCS) D=4: ● THE NAIVE SOLUTION (in D=4) is (up to an irrelevant constant): This UNIQUE d. S inv solution solves the wrong equation: Here denotes the antipodal point of : From it follows that the terms in 2 nd square brackets [. ] source the second -function in Eq. (**). The solution (**) follows from: ˚ 5˚

MASSLESS SCALAR ON DE SITTER 2 ˚ 6˚ ● CONCLUSION: MMCS cannot be quatized in a d. S inv way. ● ONE POSSIBLE SOLUTION (that breaks d. S sym, respects spatial homogeneity, but solves the right equation) is: This d. S breaking solution solves the right equation, but (when dim reg is applied) the coincident propagator grows (linearly) with time. This has physical consequences. For example, when one introduces a quartic interaction at one loop one generates a Hartree mass: In a self-interacting scalar theory, the mass squared thus grows linearly in time. This is a consequence of abundant particle creation in d. S. Precisely this type of particle creation generates scalar cosmological perturbations, which in turn induce CMB temperature fluctuations and seed the Universe’s large scale structure. ☺ QUESTION: IS THERE A d. S INV SCALAR MASS?

˚ 7˚ MASSIVE SCALAR FIELD PROPAGATOR ● d. S INVARIANT SCALAR FIELD PROPAGATOR Chernikov, Tagirov, Annales Poincaré Phys. Theor. A 9 (1968) ● COINCIDENT SCALAR PROPAGATOR The m→ 0 LIMIT IS SINGULAR; EXPLAINS WHY THERE IS NO d. S INV SOLUTION.

˚ 8˚ SCALAR FIELD: MASS GENERATION ● in a self-interacting scalar theory, scalar has a growing mass. Is there a d. S inv late time limit? ● RESUMMATION: SELF-CONSISTENT HARTREE: ● ACTION ● GAP EQUATION ● SOLUTION Ford, Vilenkin, PRD 26 (1982); Serreau; Garbrecht, Rigopoulos Prokopec; Prokopec, Lazzari (2012+) (2011+) ● THERE ARE STUDIES OF 2 and 3 LOOPS (primarily to check Stochastic inflation; also w<-1 found) Kahya, Onemli, Woodard (2010); Onemli, Woodard (2004)

˚ 9˚ SCALAR FIELD: MASS GENERATION ● STOCHASTIC INFLATION RESULT Starobinsky, Yokoyama 1996 QUALITATIVELY THE SAME AS THE GAP EQUATION, BUT QUANTITATIVE DIFFERENCE:

˚ 10˚ WHEN STABILITY FAILS: YUKAWA THEORY

FERMIONS IN YUKAWA ● AT ONE LOOP IN YUKAWA THEORY PHOTONS ACQUIRE A MASS: ˚ 11˚ Garbrecht, Prokopec, PRD 73 (2006) ► m=scalar mass; y=Yukawa, ξ=nonminimal coupling, R=Ricci scalar STOCHASTIC THEORY IS OBTAINED BY INTEGRATING OUT THE FERMIONS. ONE OBTAINS THE FOLLOWING EFFECTIVE POTENTIAL: Miao, Woodard, PRD (2006), gr-qc/0602110 . . AND THE CONTRIBUTION TO THE ENERGY DENSITY IS:

˚ 12˚ EFFECTIVE POTENTIAL IS UNSTABLE ● EFFECTIVE POTENTIAL CONTRIBUTION TO THE ENERGY DENSITY unstable evolution TWO DIFFERENT VALUES OF V 0

˚ 13˚ STABLE THEORY: SCALAR ELECTRODYNAMICS

SCALAR ELECTRODYNAMICS ˚ 14˚ ● (BARE) LAGRANGIAN ● ONE RELEASES SYSTEM FROM A FREE VACUUM STATE Initial boundary divergences lead to exponentially decaying tails, can be removed by chosing a suitable (pertubative) interacting state ● THE PHOTON GAINS A (PERTURBATIVE) MASS Prokopec, Tornkvist, Woodard (2002, 2003); Prokopec, Puchwein, (2004) ● THE SCALAR CAN BE KEPT (PERTURBATIVELY) MASSLESS ● QUANTUM BACKREACTION: 2 LOOP STRESS-ENERGY TENSOR Prokopec, Tsamis, Woodard (2007, 2008)

● SOME OF THE CONTRIBUTING DIAGRAMS ˚ 15˚

˚ 16˚ WHAT HAPPENS AT LATE TIMES? DOES ONE REACH A DE SITTER INVARIANT STATE? IF YES, WHICH ONE? TO ANSWER THESE QUESTIONS WE NEED NON-PERTURBATIVE METHOD STOCHASTIC INFLATION

STOCHASTIC SCALAR QED ˚ 17˚ Prokopec, Tsamis, Woodard, Ann. Phys. 323 (2008) [0707. 0847 [gr-qc]] STRATEGY: identify active and passive fields. Active fields are amplified in IR and can produce leading log(a). Passive fields (gauge fields, fermions): NEED TO BE INTEGRATED OUT Stochasticize the resulting EFFECTIVE SCALAR THEORY, with Veff( ). ☺ RESULTS: ♣ the scalar and photon acquire a (nonperturbative) mass: NB: AGREE WITH ♣ fluctuations give constant contributions to the density invariants & c. c. : NB: These results are fully nonperturbative (not suppressed by the coupling const. e) CONCLUSION: IN SQED ASYMPTOTICALLY ONE REACHES A d. S INVARIANT, STABLE STATE OF LOWER ENERGY

AFTER INTEGRATING THE (PASSIVE) PHOTONS ONE GETS THE EFFECTIVE SCALAR POTENTIAL Veff SMALL FIELD ˚ 18˚

CONTRIBUTION TO ENERGY DENSITY FROM THE EFFECTIVE SCALAR THEORY ˚ 19˚

˚ 20˚ QUANTUM GRAVITY ON DE SITTER: STABLE OR UNSTABLE?

GRAVITONS ON DE SITTER NON-COVARIANT APPROACH ˚ 21˚ ● WHEN ONE FULLY FIXES THE GAUGE, GRAVITONS ON d. S CAN BE THOUGHT AS TWO POLARIZATIONS OF A MMCS: EOM for GRAVITONS : ● IDENTICAL PROBLEM AS MMCS: NO d. S INV PROPAGATOR!? LHS: THE SAME OPERATOR AS FOR MCMS! no solution! BUT the RHS PROJECTORS Pij BREAK d. S INV! No general proof as yet whether d. S inv propagator exist! Domazet, Prokopec, in prograss We do have now graviton propagator in a covariant exact (de Donder) gauge! Mora, Tsamis, Woodard, J. Math. Phys. 53(2012); Miao, Tsamis, Woodard, J. Math. Phys. 52 (2011) This propagator cannot be used to study de Sitter breaking (gauge dependence? ) of physical quantities.

COVARIANT GRAVITON PROPAGATOR ˚ 22˚ Mora, Tsamis, Woodard, J. Math. Phys. 53(2012); Miao, Tsamis, Woodard, J. Math. Phys. 52 (2011) THE GRAVITON PROPAGATOR CONTAINS SPIN 2 AND SPIN 0 PARTS ● EXACT COVARIANT DE DONDER GAUGE (operator): {b=gauge parameter) ● SPIN 2 PART (TT on both index groups) MMCS: EXPECT d. S BREAKING

˚ 23˚ COVARIANT GRAVITON PROPAGATOR 2 ● GAUGE PARAMETER ● SPIN 0 PART (spin 0 projectors) TACHYONIC WHEN <D (b<2, D>1) TACHYONIC WHEN Re[D]>0 • FOR SIMPLICITY WE CONSIDER A CLASS OF GAUGES b>2, ( >D) FOR WHICH THE N-PROPAGATOR IS d. S INVARIANT ● APPLICATIONS: coinc. graviton propagator (in de Donder gauge): d. S breaking Kahya, Miao Woodard (2011), 1112. 4442

ONE LOOP VACUUM POLARIZATION ˚ 24˚ Glavan, Miao, TP, Woodard (2015) WE USED THE COVARIANT GRAVITON AND PHOTON PROPAGATORS DIAGRAMS MIAO & WOODARD CALCULATED SPIN=2 CONTRIBUTION (in a heroic calculation they included the d. S breaking contribution) GLAVAN & TP CALCULATED THE SPIN=0 CONTRIBUTION (sheepishly they studied only the b>2 ( >D) CASE, when the gauge dependent part of the s=0 propagator is d. S invariant) Miao & Woodard found a v=dependence in vac polarisation v=ln[a( )/a( ’)] not present in the vac pol from Leondard & Woodard paper. Suggests a gauge dependence.

VACUUM POLARIZATION ON d. S ˚ 25˚ Glavan, Miao, TP, Woodard (2015) USE THE COVARIANT GRAVITON AND PHOTON PROPAGATORS TO CALC GRAVITON INDUCED 1 LOOP VAC POL: + COUNTER TERMS USEFUL (F-G/EL-MAG) REPRESENTATION: Prokopec, Tornkvist, Woodard; Leonard, Prokopec, Woodard 1304. 7265, 1210. 6968 ON MINKOWSKI: LORENTZ INVARIANT (G=0) BUT GAUGE DEPENDENT RESULT NEGATIVE SEMIDEFINITE FOR b>2:

VACUUM POLARIZATION: SPIN 2 ˚ 26˚ SPIN 2 PART: CONTAINS QUALITATIVELY DIFFERENT TERMS when compared with the vac pol of Leonard & Woodard obtained in a fixed (de Donder) gauge.

VACUUM POLARIZATION: SPIN 0 ˚ 27˚ SPIN 0 PART: COMPLICATED FUNCTIONS CONTAINING GENERALISED HYPERGEOMETRIC FUNCTIONS AND THEIR DERIVATIVES YOU DO NOT WANT TO SEE THEM! THEY STRONGLY DEPEND ON b.

RENORMALIZATION ˚ 28˚ WE USE DIM REG (RESPECTS ALL SYMMETRIES) • SURPRISING RESULT: NEED A NON-COVARIANT COUNTER TERM C (THAT ALSO VIOLATES d. S SYMMETRY) FROM FLAT SPACE RESULT:

RENORMALIZATION 2 THE NONCOVARIANT COUNTER TERM: ˚ 29˚

THE ORIGIN OF NON-COVARIANT RENORMALIZATION ˚ 30˚ THE ORIGIN OF THE NONCOVARIANT (d. S BREAKING) COUNTER TERM: ☼ GRAVITATIONAL INTERACTIONS HAVE TWO DERIVATIVES: - increases superficial degree of divergence (gravity non-renormalizable) ☼ THE COINCIDENT GRAVITON PROPAGATOR DIVERGES ON d. S AS: 1/(D-4) [fluctuations at all scales contribute: primarily UV effect] ☼ IN LORENTZIAN SIGNATURE: INTERACTIONS TIME ORDERED, [exemplified by the (t-t’) and (t’-t) functions in the propagator] IS THERE ANY WAY OF GETTING RID OF IT? ☼ INCOMPLETE (albeit covariant) GAUGE FIXING, thus an average gauge. SINCE THE PHOTON AND GRAVITON ARE GAUGE INVARIANT, WRITING THE ACTION FOR A GAUGE INV COMBINATION MIGHT CANCEL THE NON-COV CT. CRITICISM: CAN BE REALLY DONE ONLY AT LINEAR ORDER IN GRAVITATIONAL PERTURBATIONS.

PHYSICAL EFFECTS OF GRAVITONS ˚ 31˚ LOOP EFFECTS ON PHOTONS ON d. S: 1 PI EOM NB: THERE IS ALSO 2 PI: EFFECTS ON THE PHOTON CORRELATOR CAN STUDY EFFECTS WITH CLASSICAL SOURCES: point charge, dipoles OR ON DYNAMICAL PHOTONS (such as vacuum fluctuations) NB: SO FAR THE EFFECTS WERE STUDIED BY USING THE Leonard, Woodard VAC POL calculated in a fixed (de Donder gauge): NOT (YET) POSSIBLE TO STUDY GAUGE (IN-)DEPENDENCE. Leonard, Woodard 1202. 5080, 1304. 7265 (2012, ’ 13) NB 2: TO GET A GAUGE INDEPENDENT ANSWER, ONE WOULD HAVE TO DO AN ANALOGOUS CALCULATION AS DONE BY BJERRUM BOHR. HARD!

˚ 32˚ PHYSICAL EFFECTS: POINT CHARGE Glavan, Prokopec, Miao, Woodard 1308. 3453(2013) RESPONSE TO POINT CHARGE q (at the origin) ON d. S CLASSICAL ONE LOOP NB: SAME RESULTS OBTAINED IN PHYSICAL RADIAL COORDINATES NB 2: EFFECT GROWS WITH DISTANCE PHYSICAL DISTANCE/RH NB 3: POINT MAGNETIC DIPOLE: GIVES ANALOGOUS RESULTS

˚ 33˚ PHYSICAL EFFECT: DYNAMICAL PHOTON Wang, Woodard 1408. 1448 (2014) 1 LOOP MODIFICATION OF DYNAMICAL PHOTON CLASSICAL ONE LOOP

˚ 33˚ GRAVITON SELF-ENERGY: SCALARS Park, Woodard, PRD, ar. Xiv: 1101. 5804, 1109. 4187 (2011) Leonard, Park, Prokopec, Woodard, PRD, 1403. 0896 (2014) COUNTER TERMS USE MMC SCALAR PROPAGATOR AND 3 and 4 GRAVITON-SCALAR VERTICES AT 1 LOOP SCALARS DO NOT AFFECT GRAVITONS SIGNIFICANTLY ON d. S, i. e. NO TERMS THAT GROW SECULARLY IN TIME FOR DETAILS SEE Leonard, Park, Prokopec, Woodard, PRD, 1403. 0896 (2014) NEWTONIAN (BARDEEN) POTENTIALS on d. S COULD BE AFFECTED AS

DISCUSSION ˚ 35˚ PHYSICS OF DE SITTER IS ESSENTIALLY NONPERTURBATIVE (for massless and light scalars, gravitons, and other fields that couple to them). WHILE IT IS NOT KNOWN HOW TO STUDY EFFECTS OF QUANTUM GRAVITON FLUCTUATIONS, PROGRESS HAS BEEN MADE FOR SCALAR, VECTOR AND FERMIONIC FIELDS (both using perturbative methods and stochastic inflation). SCALAR AND VECTOR FIELDS ACQUIRE/YIELD COMPUTABLE CORRECTIONS DURING INFLATION; FERMIONIC FIELDS TEND TO DESTABILIZE DE SITTER. QUANTUM FLUCTUATIONS OF SCALARS ARE STRONG ENOUGH TO RESTORE A BROKEN SYMMETRY IN THE CASE OF REAL AND N-COMPONENT SCALAR FIELD WITH AN O(N) SYMMETRY. LATE TIME STATE CAN BE COMPUTED: FLUCTUATIONS ARE LIGHT AROUND THE ORIGIN, HEAVY AT LARGE FIELD VALUES.

DISCUSSION ON GRAVITONS ˚ 36˚ PROGRESS ON UNDERSTANDING PERTURBATIVE AND NONPERTURBATIVE EFFECTS IN GRAVITATIONAL SECTOR HAS BEEN SLOW BUT STEADY. WE HAVE THE OLD RESULT OF TSAMIS AND WOODARD ON 2 LOOP GRAVITON QBR ON d. S obtained using cut-off regularization. Need to be checked by using covariant graviton propagator & dim reg. Gauge dependence ill understood. No scattering matrix on d. S available. SOME (IMPARTIAL) RESULTS ON 1 LOOP VACUUM POLARIZATION AVAILABLE. MORE PROGRESS NEEDED TO UNDERSTAND GAUGE DEPENDENCE OF: Coulomb Force, effects on dynamical photons, etc.

˚ 37˚ SYMMETRY RESTORATION ON DE SITTER Prokopec, JCAP 1212 (2012) [ar. Xiv: 1110. 3187 [gr-qc]] Lazzari, Prokopec, ar. Xiv: 1304. 0404 [hep-th] (2013)

MEAN FIELD MASS ˚ 38˚ ● BROKEN SYMMETRY CASE: m²<0: ● MEAN FIELD MASS (self-consistent Hartree approximation) ◙ SOLVING THIS YIELDS (BROKEN SYMMETRY CASE, m²<0): NB: ANALOGOUS ANALYSIS CAN BE DONE FOR O(N) MODEL MASSIVE WOULD-BE GOLDSTONE BOSONS (? ? ) ◙ QUESTION: HOW ACCURATE IS THIS RESULT/METHOD? ◙ ANSWER: ONE CAN CHECK IT BY USING STOCHASTIC THEORY.

EFFECTIVE ACTION APPROACH ˚ 39˚ Lazzari, Prokopec, ar. Xiv: 1304. 0404 [hep-th] (2013) ● USE THE FOKKER PLANCK EQUATION: ◙ ASSUME FOR SIMPLICITY AN INITIAL STATE ◙ USE A LEGENDRE TRANSFORM PARTITION FUNCTION NB: This method integrates out all super-Hubble fluctuations and yields a PDF for deep infrared fields at asymptotically late times. ◙ CONFORMAL DIAGRAM OF d. S: flat and global coordinates

˚ 40˚ PROOF OF SYMMETRY RESTORATION ◙ LEGENDRE TRANSFORM By Cauchy-Schwarz Thm (f= , g=1): (averaging taken w. r. t. PDF ≥ 0; equality holds when f g)

˚ 41˚ ASYMPTOTIC STATE FOR REAL SCALAR ● EFFECTIVE ACTION AND ITS PDF: SYM BREAKING CASE invert labeling! ◙ CONCLUSION: SYMMETRY GETS RESTORED FOR ARBITRARY TREE POTENTIAL! SAME CONCLUSION REACHED IN THE O(N) CASE: GOLDSTONE THM RESPECTED (GOLDSTONE BOSONS REMAIN MASSLESS) ◙ HENCE : quantum fluctuations in d. S strong enough to restore symmetry! ◙ For large (weak ) at ~ 0: a sudden turn in Veff (a `Maxwell construction’)

˚ 42˚ invert labeling!

˚ 43˚ ASYMPTOTIC STATE REAL SCALAR 2 ● ANALYTIC APPROXIMATIONS FOR EFFECTIVE ACTION ♥ FOR SMALL FIELD VALUES: ♥ FOR LARGE FIELD VALUES: FLUCTUATIONS GENERATE A LARGE AMOUNT OF ENERGY ◙ FLUCTUATIONS ARE LIGHT FOR SMALL BACKGROUND FIELD; HEAVY FOR LARGE VALUES OF BACKGROUND FIELD

ASYMPTOTIC STATE SCALAR O(N) ● similar as in O(1) case: symmetry gets restored ♥ FOR SMALL FIELD VALUES: ◙ MASS FOR SMALL/LARGE SELF-COUPLING agrees with the O(1) case (with N 1, /6) ◙ AS IN O(1) MODEL: FLUCTUATIONS ARE LIGHT FOR SMALL BACKGROUND FIELD; HEAVY FOR LARGE VALUES OF BKG FIELD ˚ 44˚