New particlelike solutions in modified gravity: the Higgs monopoles André Füzfa Massimiliano Rinaldi & Sandrine Schlögel Namur Center for Complex Systems - naXys University of Namur Eventuellement suivi de « A n’ouvrir qu’en cas de Prix Nobel » Part One SOME PARTICLELIKE SOLUTIONS IN CLASSICAL FIELD THEORY Why particlelike solutions? ! Original motivation: → Temptative models of elementary particles such as electron in terms of solutions to classical field equations → Naive view since no quantum mechanical aspects… Regularization of singularities in classical field theory for point sources (spherical symmetry) ! Alternatives to black holes ! → Black holes: singular solutions → classical regularisation of gravitational singularities → new candidates for compact objects (high-energy astrophysical sources such as quasars, galactic centers, etc.) What makes a particlelike solution? ! Spherical symmetry ! Static (Time-independent) equilibrium configurations ! Globally regular (no singularities at r=0 and r->∞) ! Asymptotically flat: vanishing forces at spatial infinity ! Finite-energy/mass ! Stable under perturbations (when possible) An example in electromagnetism ! Electron: some electrical charge distribution, pointlike at distance ! Which electric potential? => Maxwell equations Naive classical vision! Quantum Mechanics! Electric potential → For static potential : Poisson equation V’’r+2V’=0 ; (.)’=d(.)/dr The solution V(r) ~ 1/r is singular! Particlelike solution Inner region, regularity at r=0 Electron radius Outer region: fast decrease when r->∞ So that total energy is bounded Distance 1) Einstein-Maxwell equations ! Motivation: → can gravity renormalise singularity in the electric field of a point source? (Without quantum effects…) ! No! Reissner-Nordström solution (1918) → Einstein equations for the gravitational (metric) field → Covariant Maxwell equations for the electromagnetic field → Singular electric field : E(r)~1/r2 → Singular metric fields: True singularity at r=0 Coordinate singularity (gtt=0 Event Horizon) 2) Einstein-Yang-Mills equations ! Non-abelian gauge fields → Yang-Mills (YM) equations: generalization of Maxwell for nonabelian gauge groups ! No particlelike solutions for → gravitation alone (e.g. Schwarzschild black hole solution) → Yang-Mills (YM) equations ! Particlelike solution for Einstein-Yang-Mills (BartnikMcKinnon, 1988): → Balance between gravitational attraction and YM repulsive force → Discrete family of particlelike solutions (globally regular, asymptotically flat, finite-energy) → Non-perturbative effect due to intricate non-linearities, first shown numerically → Existence of stable solutions 2) Einstein-Yang-Mills equations Building a particlelike solution Two examples Toward +∞ Variation of initial conditions Toward -∞ The parameter tuned corresponds to central value of gtt 3) Schrodinger-Newton equations ! Penrose’s programme for quantum decoherence: → motivation collapse of the wavefunction due to gravitational interactions → Schrodinger equation with newtonian gravitational potential → Poisson equation with matter density proportional to wavefunction probability density ! Particlelike solutions (Moroz, Penrose, Todd, 1998): → Numerically: discrete family of damped oscillatory wavefunctions Ψ labelled by number of zeros → Proof of uniqueness and global regularity → Localized state since <Ψ|QΨ> now finite (better asymptotics than free particle) → Ground state (n=0) proved stable 3) Schrodinger-Newton equations Building a particlelike solution with n=0 Solutions with n zeroes (n=1,2,3) Toward +∞ Variation of initial conditions Ψ(r=0) Toward -∞ Only 15 digits for Ψ(r=0)!! 4) Boson stars in general relativity ! Einstein-Klein-Gordon equations: → compact objects made of scalar field (boson) → Theorem: only a complex scalar field with oscillating phase can do the job ! Particlelike solutions: boson stars → Continuous family of solutions → Alternatives to compact objects such as neutron stars ; candidates for active galactic nuclei or quasars → Possible formation in the early universe → Critical phenomena in their gravitational collapse Ruffini & Bonazzola 1969; Colpi, Shapiro & Wasserman 1986 4) Boson stars in general relativity Building a boson star Toward +∞ Variation of phase velocity ω Φ=σ(r)eiωt Toward -∞ Some examples of boson stars Summary ! Non-perturbative regular solutions to non-linear ordinary differential equations with good asymptotics ! Related complex features: critical phenomena, universality laws, etc. ! Interesting candidates for astrophysical purposes, though not of quantum nature (like black holes) ! Particlelike solutions beyond general relativity? → Let us look at Higgs gravity… An approach already successful for inflation as recently advocated by PLANCK data A. Füzfa, M. Rinaldi & S. Schlögel, Physical Review Letters 111, 121103 (2013) arXiv:1305.2640 Also seen on La Libre Belgique 13/09/2013 Part Two HIGGS MONOPOLES IN MODIFIED GRAVITY Motivations for Higgs gravity (1/3): ! Mach’s Principle (1872): → Inertia and acceleration are relative → defined with respect to the whole Universe ! Einstein’s inspiring influence*: → Mach’s principle prefigures general covariance → Einstein : Equivalence Principles to impose covariance → General Relativity: mass a geometrical property → Non-locality as a motivation for the cosmological constant ! Inertial mass emerging from interactions → Solid-state physics and Higgs mechanism (effective mass and spontaneous symmetry breaking) *A. Pais, ”Subtle is the Lord...”, 1982 Motivations for Higgs gravity (2/3) ! Beyond Einstein’s general relativity → Brans & Dicke (1961): a running gravitational coupling for a relativistic Mach principle → Tensor-scalar theories of gravitation → Violation of the (Strong) Equivalence principle ! Scalar fields* and cosmology: → cosmic acceleration in the early and late universe (inflation & dark energy) → varying fundamental constants ! Scalar fields* in particle physics → Goldstone boson in symmetry breaking → Standard model Higgs boson: mass generation, cancellations of gauge anomalies, etc. Under Poincaré group Motivations for Higgs gravity (3/3) Gravitation ! &! Cosmology! Φ Inflation, dark energy, modified gravity ? Particle ! Physics! Η Electroweak symmetry Breaking & Standard model physics The Higgs field H rules both the gravitational coupling and the mass of elementary particles Implications for the classical vacuum ! Static configurations of the Higgs field → non-vanishing vacuum expectation value is required to break symmetry Spherical symmetry in Schwarzschild gauge with V(H=v)=0 Analogy with Newton’s second law! H ó position ()’ ó d()/dt H’(r=0)=0 (regularity) « Acceleration » Damping Driving force V(H=0)≠0 Beyond a trivial vacuum ! Starting at rest: → if |Hi|>v : H-> infinity ; infinite ADM mass → if |Hi|< v: H-> 0 with V(H=0)>0 : asymptotically de Sitter ! General relativity : |H(r)|=v → trivial homogeneous solution with finite energy and asymptotically flat spacetime! ! Solution: non-minimal coupling to gravity! H=Hc Effective potential inside matter ! Higgs monopoles: particle-like solutions where H interpolates between Hc and v at spatial infinity Effective potential outside matter Higgs monopoles Globally regular classical distributions of the Higgs field with finite ADM mass ! Isolated gravitational and standard model scalar charges ! Existence due to non-minimal coupling in presence of suffficiently compact objects ! Physical properties of Higgs monopoles ! For a given non-minimal coupling strength, physical parameters are → mass (mainly of matter distribution) → compactness s=rS/R (R = radius of the monopole) → scalar charge and field gradient on the boundary s= 0.1 ξ= 104 ξ= 103 ξ= 102 m~107kg ξ= 10 ξ= 104 Scalar charge amplification ! High coupling: damped oscillatory behavior inside matter ! Precise initial conditions on the boundary are required to reach asymptotic flatness ! Resonances for specific values of mass, compactness and coupling strength and give most interacting monopoles Central field value Boundary field value Conclusions ! Higgs gravity: → combining symmetry breaking in particle physics and violation of the equivalence principle ! Static configuration for classical vacuum → Higgs monopoles: globally regular, asymptotically flat particle-like solutions with finite ADM mass → non-trivial background for quantum perturbations → candidates for weakly interacting dark matter? ! Further studies → Distribution of resonances and physical implications → Higgs monopoles for astrophysical masses → Stability of the monopole under perturbations (time dependent solutions) → Formation during inflation? As a result of gravitational instabilty? Abundance? → generalisation beyond the unitary gauge and with matter directly coupled to the Higgs Troisième partie A N’OUVRIR QU’EN CAS DE PRIX NOBEL… Idées visionnaires ! « Principe » de Mach (1872): → l’inertie et l’accélération ne devraient être que relatives et définies par rapport à l’ensemble de l’Univers → La masse d’inertie d’un corps résulte de l’ensemble des interactions avec tous les autres corps de l’Univers (origine non-locale de la masse d’inertie) ! Inspiration pour la relativité générale → Covariance généralisée → Caractère asymptotique de la gravitation et constante cosmologique ! Visionnaire pour la physique des hautes énergies → La masse via les interactions avec l’environnement (masse effective, supraconductivité et effet Meissner, mécanisme de Brout-Englert-Higgs) La masse en relativité restreinte et générale ! Relativité restreinte: E=mc2 → masse au repos = composante de la quadri-impulsion → Conservation de la norme de la quadri-impulsion et hamiltonien ! Principes d’équivalence de Galilée et d’Einstein → Universalité de la chute libre pour des masses tests → Masse d’inertie équivalente à masse gravitationnelle → Gravitation comme pseudo-force et théorie métrique ! Relativité générale: → Principe d’équivalence d’Einstein implique la covariance → masse reliée aux propriétés géométriques globales de l’espace-temps (exemple : masses ADM, Bondi, cosmologie?, etc.) La masse en mécanique quantique ! Théorème de Wigner: → toute particule élémentaire est une représentation irréductible du groupe de Poincaré dans un espace d’Hilbert donné → Masse: opérateur de Casimir associé aux générateurs infinitésimaux de l’impulsion et commutant avec l’hamiltonien (conservation masse) → Toute particule élémentaire caractérisée uniquement par sa masse, son spin, ses nombres quantiques additifs (nbre baryonique, leptonique, étrangeté, isospin, etc.) ! Equation de Schrodinger et état fondamental → HΨ=E Ψ : énergie au repos (masse) reliée aux termes harmoniques du potentiel ! Equation de Klein-Gordon avec potentiel harmonique → ☐Φ=mΦ: solution en ondes évanescentes (amplitude décroissante exponentiellement) → Masse: terme quadratique d’auto-interaction La masse en physique des particules ! Masses des baryons: → majoritairement due aux interactions fortes des gluons et quarks (>90%) ! Masses des particules élémentaires? → les bosons véhiculant les interactions fondamentales doivent être sans masse (invariance de jauge) → Interaction faible: pourquoi à courte portée? → interaction de jauge: pas de masse pour les fermions ! Masse via de nouvelles interactions? → Mécanisme de Brout-Englert-Higgs → Champ de Higgs chargé sous les interactions: masse des bosons de jauge → Interactions de Yukawa avec les fermions Brisure spontanée de symétrie ! Idée: la symétrie est cachée → solution particulière non symétrique mais symétrie présente au niveau de l’ensemble des solutions possibles ! Exemples: → un crayon en équilibre instable (brisure à basse énergie) → Flambage d’une poutre (brisure à haute énergie) → Ferromagnétisme (brisure à basse énergie) ! Supraconductivité → brisure spontanée de l’invariance de jauge de l’électromagnétisme → apparitions de paires de Cooper (état collectif bosonique) en dessous d’une température critique → Illustration de la masse du photon: effet Meissner → Symétrie cachée et potentiel de Landau-Ginzburg Mécanisme de Brout-Englert-Higgs ! Idée: ajout d’un nouveau champ → scalaire : spin 0 mais chargé sous les interactions électromagnétique et faible → Donc : un doublet de champs complexes en interaction avec les champs de jauge → Si le champ acquiert une « valeur dans le vide » (=valeur classique du champ de Higgs est non nulle), le terme d’interaction donne une masse pour les bosons de jauge → interactions de Yukawa avec les fermions: massesdes particules élémentaires → si un mode survit au niveau quantique: nouvelle particule scalaire (plus tard: boson de Higgs) Interaction électrofaible et boson de Higgs ! Potentiel de Higgs → potentiel effectif ? → obtention d’une valeur classique non nulle par un mécanisme non quantique ! Glashow-Salam-Weinberg → application du mécanisme BEH au groupe électrofaible SU(2)xU(1) (prix Nobel 1979) → Description des couplages aux champs de jauges et à la matière (fermions) et des mécanismes donnant leurs masses → Prédiction des particules W, Z et du boson de Higgs ! ‘t Hooft et Veltman → Obtention d’une théorie renormalisable (prix Nobel 1999) ! Détection du boson de Higgs (Englert, Higgs, prix Nobel 2013) Application: Higgs inflation ! Higgs mechanism assumes Landau-Ginzburg potential: → with λ,v specified by standard model physics Cosmic acceleration when slow-rolling ! Natural candidate for inflaton? No! → required number of efolds is obtained only for unnatural values of λ and initial field amplitude Hi (super-planckian) Application: Higgs inflation ! Solution ? Inflaton = Higgs field non-minimally coupled to gravity ! Einstein frame dynamics Einstein frame potential = Jordan frame potential * (coupling function )4 4 = * ! Non-minimal coupling has to be very high* Add equations for coupling functions & JF to EF, potential in EF, etc. * Fakir & Unruh, PRD 1990, Bezrukov & Shaposhnikov, PLB, 2008. Higgs inflation and Planck data J. Martin, C. Ringeval & V. Vennin Encyclopaedia Inflationaris , arXiv:1303.3787