Dynamical gravitational corrections to the conformal anomaly Raquel Santos-Garcia Invisibles Workshop 2019 13th of June of 2019 Work in collaboration with Enrique Alvarez and Jesus Anero Idea Can we tell if gravity is dynamical or not by computing the conformal anomaly? Is gravity a fundamental interaction? Existence of gravitons Entropic origin Is gravity a fundamental interaction? Existence of gravitons Entropic origin Einstein’s general Theory for the relativity (GR) gravitational field Fierz-Pauli lagrangian for free spin 2 particles When interactions are GR included in a consistent way Is gravity a fundamental interaction? Existence of gravitons Entropic origin Einstein’s general Theory for the Dictionary between BH laws and relativity (GR) gravitational field thermodymacial laws δS = δA Fierz-Pauli lagrangian for free spin 2 Instead of thinking in Entropic particles the fundamental dof force Jacobson, Padmanabhan, Verlinde,… Einstein equations arise from minimizing the entropy (the area) with a fixed volume When interactions are GR included in a consistent way Conformal anomaly Conformal anomaly Scale In the UV intuitive idea that masses should be unimportant invariance Conformal anomaly Scale In the UV intuitive idea that masses should be unimportant invariance Scale invariance Conformal invariance Weyl invariance (flat spacetime) (flat spacetime) (curved spacetimes) μ μ 2 x → λ x Conformal group gμν → Ω(x) gμν Conformal anomaly Scale In the UV intuitive idea that masses should be unimportant invariance Scale invariance Conformal invariance Weyl invariance (flat spacetime) (flat spacetime) (curved spacetimes) μ μ 2 x → λ x Conformal group gμν → Ω(x) gμν Renormalization procedure, introduces an energy scale Conformal (Weyl) anomaly Conformal anomaly Scale In the UV intuitive idea that masses should be unimportant invariance Scale invariance Conformal invariance Weyl invariance (flat spacetime) (flat spacetime) (curved spacetimes) μ μ 2 x → λ x Conformal group gμν → Ω(x) gμν Renormalization procedure, introduces an energy scale Conformal (Weyl) anomaly Anomaly in the trace of the energy momentum tensor < Tμ > = a Proportional to the one-loop μ 2 counterterm Conformal anomaly Scale In the UV intuitive idea that masses should be unimportant invariance Scale invariance Conformal invariance Weyl invariance (flat spacetime) (flat spacetime) (curved spacetimes) μ μ 2 x → λ x Conformal group gμν → Ω(x) gμν Renormalization procedure, introduces an energy scale Conformal (Weyl) anomaly Anomaly in the trace of the energy momentum tensor < Tμ > = a Proportional to the one-loop μ 2 counterterm If gravity is dynamical or just a background is important in the path integral Sensitive to the graviton a2 ∼ �ϕi �gμν ∫ dynamics Background vs. dynamical gravity We expand the gravitational field in a background and a perturbation gμν = g¯μν + κ hμν Background field Quantum field (constant curvature spaces for simplicity) Background vs. dynamical gravity We expand the gravitational field in a background and a perturbation gμν = g¯μν + κ hμν Background field Quantum field (constant curvature spaces for simplicity) We take for example an scalar field coupled to gravity Lagrangian Anomaly g¯ μ 29 2 g¯ ∇¯ μϕ∇¯ ϕ λ (4π)2 135 1 2 ¯ ¯ μ 2 ¯ g¯ λ g¯ ∇μϕ∇ ϕ + ϕ R − ( 6 ) (4π)2 135 1 g ∇ ϕ∇μϕ + ϕ2R 0 ( μ 6 ) g¯ μ 2 1 2 1078 2 g ∇μϕ∇ ϕ + Rμν − R λ ( 2 ) (4π)2 135 Background vs. dynamical gravity We expand the gravitational field in a background and a perturbation gμν = g¯μν + κ hμν Background field Quantum field (constant curvature spaces for simplicity) We take for example an scalar field coupled to gravity Lagrangian Anomaly g¯ μ 29 2 g¯ ∇¯ μϕ∇¯ ϕ λ (curvature of CCS) (4π)2 135 1 2 ¯ ¯ μ 2 ¯ g¯ λ g¯ ∇μϕ∇ ϕ + ϕ R − ( 6 ) (4π)2 135 Non-dynamical 1 g ∇ ϕ∇μϕ + ϕ2R 0 ( μ 6 ) g¯ μ 2 1 2 1078 2 g ∇μϕ∇ ϕ + Rμν − R λ ( 2 ) (4π)2 135 Background vs. dynamical gravity We expand the gravitational field in a background and a perturbation gμν = g¯μν + κ hμν Background field Quantum field (constant curvature spaces for simplicity) We take for example an scalar field coupled to gravity Lagrangian Anomaly g¯ μ 29 2 g¯ ∇¯ μϕ∇¯ ϕ λ (curvature of CCS) (4π)2 135 1 2 ¯ ¯ μ 2 ¯ g¯ λ g¯ ∇μϕ∇ ϕ + ϕ R − ( 6 ) (4π)2 135 Non-minimal coupling Non-dynamical 1 g ∇ ϕ∇μϕ + ϕ2R 0 ( μ 6 ) g¯ μ 2 1 2 1078 2 g ∇μϕ∇ ϕ + Rμν − R λ ( 2 ) (4π)2 135 Dynamical Quadratic gravity Outlook Can we measure the conformal anomaly and be sensitive to the character of the gravitational field? Outlook Can we measure the conformal anomaly and be sensitive to the character of the gravitational field? Starobinsky’s inflation driven by the trace anomaly of conformally coupled matter fields Precisely models with the coupling of scalar fields to gravity Outlook Can we measure the conformal anomaly and be sensitive to the character of the gravitational field? Starobinsky’s inflation driven by the trace anomaly of conformally coupled matter fields Precisely models with the coupling of scalar fields to gravity Transport effects due to conformal anomaly (similar to the ones due to chiral anomalies) Measurements of scale transport, SEE and SME, in Weyl semimetals Work ongoing regarding same type of computations for fermions Thank you! Backup Let us go back to see the origin of the anomaly Γ0[g¯μν, ϕ¯] = lim Γ[g¯μν, ϕ¯; n] − Γ∞[g¯μν, ϕ¯; n] n→4 { } Renormalized effective action One-loop effective action Divergent piece The full on-shell effective action has the same symmetries as the original theory in any dimension Breitenlohner, Maison μν δ n − 2 δ �Γ[g¯μν, ϕ¯; n] = 2g¯ + ϕ¯ Γ[g¯μν, ϕ¯; n] = 0 [ δg¯μν 2 δϕ¯ ] But we have evanescent operators coming from the counterterm 1 �Γ [g¯ , ϕ¯] = � ℬ[g¯ , ϕ¯] = �[g¯ , ϕ¯] 0 μν [ (n − 4) μν ] μν Anomaly unless we find a proper counterterm .