The gravitational Higgs phenomenon GraviGUT The main question NCG On the phase structure of quantum gravity Roberto Percacci 1 1 SISSA, Trieste, Italy Quantum gravity in Paris March 23, 2017
The gravitational Higgs phenomenon GraviGUT The main question NCG Outline The gravitational Higgs phenomenon 1 GraviGUT 2 3 The main question 4 NCG
The gravitational Higgs phenomenon GraviGUT The main question NCG Basic question Why is the gravitational connection not dynamical?
The gravitational Higgs phenomenon GraviGUT The main question NCG Higgs mechanism I Certain gauge fields are experimentally seen to be massive. How to reconcile a mass with gauge invariance? Higgs field φ with values in some vectorspace V W a G -invariant potential with minima in G / H ⊂ V Adapted coordinates in V : φ = ( ρ, σ ) Dynamics gives � φ 2 � = ρ 2 0 Can choose “unitary gauge” φ = ( ρ, σ 0 )
The gravitational Higgs phenomenon GraviGUT The main question NCG Higgs mechanism II Decompose A = A | L ( H ) + A | P where L ( G ) = L ( H ) ⊕ P D φ = ∂φ + A φ = ( D ρ, D σ ) where D σ = ∂σ + A i K i ( σ ) D ρ = ∂ ρ and in particular D σ 0 = A i | P K i ( σ 0 ) in unitary gauge ( D φ ) 2 �→ ( ∂ρ ) 2 + ρ 2 0 ( A | P ) 2
The gravitational Higgs phenomenon GraviGUT The main question NCG Higgs mechanism III only Goldstone bosons are necessary for Higgs mechanism Higgs particle “only” necessary for perturbative renormalizability
The gravitational Higgs phenomenon GraviGUT The main question NCG Higgsless Higgs mechanism Higgs field h = ρ − ρ 0 also has mass ≈ λρ 0 4 ( φ 2 − ρ 2 W = λ 0 ) 2 lim λ →∞ W with ρ 0 =const ρ decouples in the limit, leaves gauged nonlinear sigma model T. Appelquist, C.W. Bernard, Phys.Rev.D22:200,1980. A.C. Longhitano, Phys.Rev.D22:1166,1980.
The gravitational Higgs phenomenon GraviGUT The main question NCG Low Energy EFT For p 2 ≪ m 2 h , ρ = ρ 0 For p 2 ≪ m 2 A , A | P = 0 or D σ = 0 Example: Meissner effect in superconductivity
The gravitational Higgs phenomenon GraviGUT The main question NCG Gravity with more Variables Spacetime manifold M , dim M = 4 frame field (a.k.a. soldering form) θ a µ , det θ � = 0 pseudo-fiber metric, γ ab signature + , + , + , − linear connection, A µ ab (structure group GL ( 4 ) ) First two carry nonlinear realizations of GL ( 4 )
The gravitational Higgs phenomenon GraviGUT The main question NCG Induced structures in TM If we think of θ : TM → E E real vectorbundle with fiber dimension 4 local bases { ∂ µ } in TM and { e a } in E then g µν = θ a µ θ b ν γ ab Γ λµν = θ − 1 a µ A λ ab θ b ν + θ − 1 µ ∂ λ θ a ν a
The gravitational Higgs phenomenon GraviGUT The main question NCG Torsion and Nonmetricity Θ µ a ν = ∂ µ θ a ν − ∂ ν θ a µ + A µ ab θ b ν − A ν ab θ b µ ∆ λ ab = − ∂ λ γ ab + A λ ca γ cb + A λ cb γ ac
The gravitational Higgs phenomenon GraviGUT The main question NCG Gauge invariance G = A ut GL ( 4 ) E b ( x ) θ b ν ( x ) ∂ x ν θ ′ a θ a µ ( x ) µ ( x ′ ) = Λ − 1 a �→ ∂ x ′ µ γ ′ ab ( x ′ ) = Λ ca ( x ) Λ d γ ab ( x ) �→ b ( x ) γ cd ( x ) b ( x ′ ) = ∂ x ν A µ a A ′ a Λ − 1 ac ( x ) A ν c d ( x )Λ d � b ( x ) �→ b ( x ) µ ∂ x ′ µ +Λ − 1 ac ( x ) ∂ ν Λ c � b ( x ) 0 → A ut GL ( 4 ) E → A ut GL ( 4 ) E → D iffM → 0 M is split: θ ∗ : D iffM → A ut GL ( 4 ) E θ ∗ ( f ) = θ ◦ Tf ◦ θ − 1
The gravitational Higgs phenomenon GraviGUT The main question NCG Goldstone Bosons A ut GL ( 4 ) E acts transitively on metric and soldering form γ ( x ) ∈ GL ( 4 ) / SO ( 3 , 1 ) γ ∈ { fiber metrics } ≈ A ut GL ( 4 ) E / A ut SO ( 3 , 1 ) E θ ∈ { isomorphisms TM → E } ≈ A ut GL ( 4 ) E / D iffM
The gravitational Higgs phenomenon GraviGUT The main question NCG Metric gauge θ a µ = δ a µ unbroken group D iffM g µν = γ µν , Γ λµν = A λµν Θ µ a ν = Γ µ a ν − Γ ν a µ
The gravitational Higgs phenomenon GraviGUT The main question NCG Vierbein gauge γ ab = η ab unbroken group A ut SO ( 3 , 1 ) M g µν = θ a µ θ b ν η ab ∆ λ ab = A λ ab + A λ ba
The gravitational Higgs phenomenon GraviGUT The main question NCG Metric and vierbein gauge Not enough freedom to fix both simultaneously In general gauge: connection and two “Goldstone bosons” In either of the two “unitary” gauges: connection and one “Goldstone boson”
The gravitational Higgs phenomenon GraviGUT The main question NCG Low energy action S ( A , γ, θ ) = S G ( A , γ, θ ) + S m ( A , γ, θ ) where � � ν F µν ab + . . . � d 4 x M 2 P θ a µ θ b � S G = | g | � d 4 x � M 2 (Θ · Θ + ∆ · ∆ + Θ · ∆) + . . . � � S m = | g |
The gravitational Higgs phenomenon GraviGUT The main question NCG The Higgs Mechanism v.I flat background: A = 0, θ = 1 , γ = η Θ µ a ν A µ a ν − A ν a µ = ∆ µ ab = A µ ab + A µ ba F µν a ∂ µ A ν a b − ∂ ν A µ a b + A µ ac A ν c b − A ν ac A µ c = b b S contains 1 � | det g | M 2 Q ( A , A ) d 4 x � 2 generically Q is non-degenerate. There is a more general point of view.
The gravitational Higgs phenomenon GraviGUT The main question NCG Levi–Civita Connection given θ , γ , there is a unique ¯ A s.t. ¯ Θ = 0, ¯ ∆ = 0 A abc = 1 + 1 ¯ � � � � E abc + E cab − E bac C abc + C bac − C cab 2 2 where θ − 1 a λ ∂ λ κ bc E abc = µ ∂ µ θ − 1 c λ − θ − 1 c µ ∂ µ θ − 1 γ ad θ d λ θ − 1 λ � � = C abc b b Any connection A can be split uniquely in A = ¯ A + Φ then S ( A , γ, θ ) = S (¯ A ( θ, γ ) + Φ , θ, γ ) = S ′ (Φ , θ, γ )
The gravitational Higgs phenomenon GraviGUT The main question NCG The Higgs Mechanism v.II Θ µ a ν Φ µ a ν − Φ ν a µ = ∆ µ ab = Φ µ ab + Φ µ ba F µν a F µν a ¯ b + ¯ ∇ µ φ ν a b − ¯ ∇ ν φ µ a b + φ µ ac φ ν c b − φ ν ac φ µ c = b b therefore S P ( A , γ, θ ) = S P (¯ A + Φ , γ, θ ) = S H ( γ, θ ) + S Q (Φ , γ, θ ) where S Q ( φ, γ, θ ) = 1 � d 4 x � | det g | M 2 P Q P (Φ , Φ) 2 and S m ( φ, γ, θ ) = 1 � | det g | M 2 Q m (Φ , Φ) d 4 x � 2
The gravitational Higgs phenomenon GraviGUT The main question NCG Gravitational Higgs Phenomenon Generically all components of φ are massive. (Not true for the Palatini action, since Q P has nontrivial kernel) At energy scales p 2 ≪ M 2 P φ = 0 ⇐ ⇒ { Θ = 0 ∆ = 0 } and
The gravitational Higgs phenomenon GraviGUT The main question NCG Gravitational Higgs Phenomenon gravity is a gauge theory of GL ( 4 ) with two Goldstone bosons there are two unitary gauges Higgsless Higgs phenomenon occurs at Planck scale, giving mass to Φ (equivalently A ) at low energy A = ¯ A ( θ, γ ) Θ = 0 and ∆ = 0 means that the theory is in a “Higgs” phase
The gravitational Higgs phenomenon GraviGUT The main question NCG Questions why is the metric nondegenerate? what is the dynamical origin of the Planck scale? does the connection propagate at ultra-Planckian scales?
The gravitational Higgs phenomenon GraviGUT The main question NCG Grand Unification use Higgs phenomenon with G 1 × G 2 ⊂ G to do list: identify GUT group G fit particles in irreps of G write G -invariant action explain symmetry breaking (select order parameter, orbit, potential) check that new particles not seen at low energy have high mass
The gravitational Higgs phenomenon GraviGUT The main question NCG Grand unification: SO ( 10 ) ( e L , ν L , e R , ν R , u r , g , b , d r , g , b , u r , g , b , d r , g , b ) L L R R 16 complex 2 component Weyl spinors of Lorentz 4 doublets and 8 singlets of SU ( 2 ) L Repeat three times. ( n ν = 2 . 984 ± 0 . 008 measured at LEP) Fit exactly in the 16 of SO ( 10 ) ! explains hypercharge assignments
The gravitational Higgs phenomenon GraviGUT The main question NCG A symmetry breaking chain SO ( 10 ) ↓ SO ( 4 ) × SO ( 6 ) ≈ SU ( 2 ) R × SU ( 2 ) L × SU ( 4 ) ↓ SU ( 2 ) R × SU ( 2 ) L × SU ( 3 ) C × U ( 1 ) B − L ↓ U ( 1 ) EM × SU ( 2 ) L × SU ( 3 ) C ↓ U ( 1 ) EM × SU ( 3 ) C requires at least 45 , 16 and 10
The gravitational Higgs phenomenon GraviGUT The main question NCG GraviGUT I use gravitational Higgs phenomenon to construct unified theory of gravity and all other interactions. to do list: identify GraviGUT group G fit particles in irreps of G write G -invariant action explain symmetry breaking (select order parameter, orbit, potential) check that new particles not seen at low energy have high mass
The gravitational Higgs phenomenon GraviGUT The main question NCG GraviGUT II G 1 = SO ( 1 , 3 ) , G 2 = SO ( 10 ) , = ⇒ G = SO ( 1 , 13 ) G = SO ( 3 , 11 ) or keep dim M =4, enlarge fibers of E to have dimension N > 4 order parameter is soldering form � η � � 1 4 � 0 γ = , θ is 4 × N matrix , e . g . � θ � = 0 1 N − 4 0
The gravitational Higgs phenomenon GraviGUT The main question NCG Fermions I F . Nesti, R.P ., Phys. Rev. D 81, 025010 (2010) arXiv:0909.4537 [hep-th] Assume γ ab = η ab , G = SO ( 3 , 11 ) In SO ( 10 ) GUT one family is η ∈ 2 C × 16 C of SO ( 3 , 1 ) × SO ( 10 )
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