New particlelike solutions in modified gravity: the Higgs

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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
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