SPONTANEOUS GENERATION OF GEOMETRY IN 4D

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1 SPONTANEOUS GENERATION OF GEOMETRY IN 4D Dani Puigdomènch Dual year Russia-Spain: Particle Physics, Nuclear Physics and Astroparticle Physics 10/11/11 Based on arxiv: and wor on progress. by J. Alfaro, D. Espriu, D.P.

2 Outline Motivation The model: - Structure - Equations of motion - Possible counterterms Perturbative calculation Effective action and universal constants Conclusions

3 Motivation The chiral lagrangian L = f πtr µ U µ U + α 1 Tr µ U µ U ν U ν U + α Tr µ U ν U µ U ν U +... U exp i π/f π, π π a τ a / L = O(p )+O(p 4 )+O(p 6 )+... L QCD = i ψ Dψ = i ψ L Dψ L + i ψ R Dψ R SU() L SU() R SU() V

4 What does gravity have to do with chiral theory? L = M P gr + Lmatter L = M P gr + α1 gr + α g(rµν ) + α 3 g(rµναβ ) It is a non-renormalizable theory - Most relevant operator has two derivatives (ignoring the cosmological constant) - It has a dimensionfull constant ( M P ) - It is non linear - It describes massless quanta - Which is the equivalent of here? L QCD - This question was addressed in D in arxiv:

5 The Model Structure - No a priori notion of metric should exist, only an affine connection defining parallel transport of tangent vectors manifold. - The lagrangian should be manifestly independent of the field g µν (x). - The spontaneous breaing of a suitable global symmetry should be triggered by a fermion condensate. - The graviton field should appear as the Goldstone boson of that broen global symmetry. - Is the Weinberg-Witten theorem an obstruction? No. v a on a

6 - Symmetry pattern and Lagrangian: SO(D) GL(D) (Global) L = i ψ a γ a µ χ µ + i χ µ γ a µ ψ a µ χ µ = µ χ µ + ω ab µ σ ab χ µ L = i ψ a γ a µ χ µ + i χ µ γ a µ ψ a + ib a µ( ψ a χ µ + χ µ ψ a )+c det(b a µ) if < ψ a χ µ + χ µ ψ a >= 0 SO(D) GL(D) SO(D)

7 Equations of motion L = i ψ a γ a µ χ µ + i χ µ γ a µ ψ a + ib a µ( ψ a χ µ + χ µ ψ a )+c det(b a µ) - Varying w.r.t B a µ < ψ a χ µ + χ µ ψ a >= ic 1 (D 1)! aa...a D µµ...µ D B a µ...b a D µd =0 Bµ a = 0 - Vacuum of the theory (gap equation for wµ ab =0) V eff = c det(b a µ) d D c D D! aa...a D µµ...µ D B a µ...b a D µd iδ µ a tr (π) D tr(log(γa µ + ib a µ)) If there is spontaneous breaing. B a µ = Me a µ; M = 0 d D (π) D (γa µ + ib a µ) 1 =0 e a µ can be interpreted as the vierbein geometry is generated!

8 - All vacua are equivalent: Bµ a = Mδµ a - In 4D: cδ a µ M d D M M δµ a (π) D + M =0 cδ a µ M 3 M 3 π δµ a log M 4πµ γ +1 =0 M = µ e π c(µ) ; µ dc dµ = 1 π - Propagator in the broen phase: 1 () i j = i δ i M j γi ( im) j + M

9 General perturbation w ab µ = 0 B a µ = Me σ 1 (x) Me σ (x) Me σ 3 (x) Me σ 4 (x) Possible counterterms and their e.o.m. S D = 1 B a 4! µbνb b ρb c σ d abcd µνρσ d 4 x S R = 1 R [µν]ab B a ρbσ b µνρσ d 4 x where R [µν]ab =[ µac, νcb ]. There is a third possible counterterm, the Gauss-Bonet term, but in 4D it is topological and we do not expect to see it in the perturbative calculation.

10 Conformally flat vierbein: B a µ = Me a µ = Mδ a µe σ ; gµν = e σ e σ e σ e σ 1 R [µν]abb a ρb b σ µνρσ = 1 ( µw µν ν ν w µν µ + w µc µ w cν w βγ α S D (on shell) =M 4 S R (on shell) =M ν w ν µc w cν µ )e σ M = 1 ( β σδ γ α γ σδ β α) w ab µ = e a ν µ E νb + e a νe ρb Γ ν µρ =M d 4 x g = M 4 d 4 xe σ d 4 x gr = M d 4 x 3 σ 1 µσ µ σ d 4 x 3 (σ)(1 σ + σ σ ) e σ

11 S D (on shell) =M 4 S R (on shell) =M d 4 x g = M 4 d 4 xe i σ i d 4 x gr = M d 4 x 3σ 4 + σ 4 + 1σ 4 + 4σ 3 + σ 3 + 1σ 3 + 4σ + 3σ + 1σ + 4σ 1 + 3σ 1 + σ 1 1 ( 3σ 1 3 σ + 4 σ 1 4 σ + σ 1 σ σ 1 4 σ 3 + σ 1 σ σ 1 3 σ σ 1 σ σ 4 σ σ 1 σ σ 3 σ σ 3 1 σ 4 + σ 3 σ 4 ) +O(σ 3 )

12 The Weinberg-Witten theorem Let p and p be one-particle, spin- massless states with the same helicity ±. If p T ab p are Lorentz covariant, then T µ ν lim p p p T ab p =0 = i ψ µ γ a ν ψ a + i ψ a γ a ν ψ µ δ µ ν L Our stress-energy tensor has no tangent indices. It is not a ran- Lorentz covariant tensor. In the broen phase, a vierbein is generated and the two indices can be related. Then one is in the same situation of General Relativity.

13 The model in a nutshell L =i ψ a γ a µ χ µ + i χ µ γ a µ ψ a + ib a µ( ψ a χ µ + χ µ ψ a )+c det(b a µ)+a 1 R [µν]abb a ρb b σ µνρσ a) < ψ a χ µ + χ µ ψ a > B a µ b) w ab µ =0 B a µ = Mδ a µ c) w ab µ = 0 B a µ = Me a µ = Me σ 1 (x) Me σ (x) Me σ 3 (x) Me σ 4 (x) d) w µab = 1 (δ µb a σ µ δ µa b σ µ ) w ab µ = e a ν µ E νb + e a νe ρb Γ ν µρ

14 Perturbative Calculation Feynman rules σ i i 1 Mδi µ σ i i 1 8 Mδi µ σ i σ i σ i i 1 48 Mδi µ σ i w bc µ iγ a σ bc = γa 4 [γ b, γ c ]

15 + p + p σ l + σ l = = 4 j=1 4 j=1 4 l=1 4 l =1 l = j σ l!! Tr σ l M 4 16π + σ l (p j + p l )M 48π 4 + j=1 + M 4 3π + M p 3π d D (π) D imδl µ 1 () µ j imδj ν 1 ( + p) ν l M log 4πµ M log 4πµ M p j 3π γ + 3 γ 1 σ l p M + O(p 4 ). 48π M log 4πµ γ 1 3 Correspond to gr expanded to second order in the fields

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17 The divergent parts M 4 e 8π i σ i log M µ For the conformally flat metric the previous diagrams add up to the closed expression (including the finite contributions): M 4 e σ 8π log M e σ 4πµ γ + 3

18 + p + p w µ + w µ = 0; w µ + w µ = 0 + p + p w µ w ν + w µ w ν =! d D Tr! (π) D iγa σ bc 1 () µ d iγd σ ef 1 ( + p) ν a M = log 4πµ γ + 1 M 4 4π D µνbcef + M 16π E µνbcef + 1 F µνbcef (p )+O(p ) D µνbcef wµ bc wν ef =4wµ νb w ν µb w be µ w be µ ; E µνbcef w bc µ w ef ν =0 D µνbcef w bc µ w ef ν (on shell) = 0; F µνbcef w bc µ w ef ν (on shell) =0

19 Effective action for a conformally flat metric S eff = d 4 x c M 4 e σ + M 4 e σ 8π log M e σ µ + A σe σ M + σe σ M 3π log M e σ µ Both for a conformally flat and a general diagonal perturbation: S eff = d 4 x M P 3π = Λ M P 3π = c (µ)+ 1 M 8π log µ M 4 g + A (µ)+ 1 c (µ)+ 1 8π log M 48π log µ M µ M 4 M Expansion parameter of the effective theory: A (µ)+ 1 M 48π log µ M gr µ da dµ = 1 48π µ dc dµ = 1 8π p M p M if perturbative breadown +...

20 Conclusions It is possible to construct a 4D model where a spontaneous symmetry breaing generates the metric degrees of freedom. No WW theorem obstruction. The number of local and covariant counterterms before the breaing is very limited. This constrains the amount of divergences from the quantum corrections. When the e.o.m are used: - The low energy effective theory obtained corresponds to GR with a Cosmological term plus higher order operators. - It is renormalizable. There are two free parameters one can adjust to fit the physical values of the parameters of nature.

21 THANK YOU