Dynamical Gravitational Corrections to the Conformal Anomaly

Home , Conformal anomaly

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

Renormalization procedure, introduces an energy scale Conformal (Weyl) anomaly Conformal anomaly

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

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