New particlelike solutions in modified gravity: the Higgs

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
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,
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
!   Stable under perturbations (when possible)
An example in electromagnetism
!   Electron: some electrical charge distribution, pointlike at
! Which electric potential? => Maxwell equations
Electric potential
→ For static potential : Poisson equation
V’’r+2V’=0 ; (.)’=d(.)/dr
The solution V(r) ~ 1/r is singular!
Particlelike solution
at r=0
Electron radius
Outer region:
fast decrease when r->∞
So that total energy is bounded
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:
at r=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
→  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
→  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
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
Toward -∞
Some examples of boson stars
!   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)
Also seen on La Libre Belgique 13/09/2013
Part Two
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
!   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 !
Inflation, dark energy,
modified gravity
Particle !
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
Spherical symmetry
in Schwarzschild gauge
Analogy with Newton’s second law!
H ó position
()’ ó d()/dt
H’(r=0)=0 (regularity)
« Acceleration »
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!
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
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
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
→  Formation during inflation? As a result of gravitational instabilty?
→  generalisation beyond the unitary gauge and with matter directly
coupled to the Higgs
Troisième partie
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
!   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
!   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
→  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
→ ☐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
→  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
→  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
Application: Higgs inflation
!   Solution ? Inflaton = Higgs field non-minimally coupled to
!   Einstein frame dynamics
Einstein frame potential
Jordan frame potential * (coupling function )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 ,
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