Gravitational instantons and anomalous chiral symmetry breaking Yu Hamada (Kyoto Univ.) Based on [arXiv:2009.08728] w/ Jan M. Pawlowski (Heidelberg U.) Masatoshi Yamada (Heidelberg U.) Exact Renormalization Group 2020 (3Nov. 2020) 1 Asymptotic safety of gravity [Weinberg ’76] • Existence of non-trivial UV fixed point for gravity Status of the asymptotic safety paradigm for quantum gravity and matter 3 g(k) ≡ Gk2 • eg.) Einstein-Hilbert truncation: [arXiv:1709.03696] [Reuter ’98] [Souma ’99] 1.5 g* 1 d4x g R Γk = ( − 2Λ) ] 1.0 16πG ∫ [ g(k) 0.5 • We can take continuum limit: k → ∞. 0.0 105 109 1013 1017 1021 1025 • UV-complete theory space k/GeV M = critical surface of the fixed pt. pl • Fixed pt. still exists even when other operators such as n μν R , RμνR , … are included. → non-perturbatively renormalizable quantum theory for gravity 2 Fig. 1 Beta functions and the corresponding running for a UV attractive free/interacting fixed point (lower/upper panels). As the fixed points are UV attractive, the value of the coupling at one reference scale is an initial condition that can be chosen freely. Upper panels: beta function and scale dependence of the Newton coupling according to Eq. (20). the scaling of couplings is determined by the sum of canonical scaling and quantum scaling which is a consequence of loop e↵ects. In fact, a fixed point can arise from a balance between canonical and quantum scaling, as β = d g + ⌘ (g ), (2) gi − g¯i i i i where ⌘i is an anomalous scaling dimension that arises as a consequence of quantum fluctuations. An interacting fixed point lies at gi⇤ = ⌘i/dg¯i . (3) 4 For instance, in d =4 ✏,theλ4φ theory has a fixed point at λ4⇤ = 16⇡2✏/3, the Wilson-Fisher− fixed point [7] playing an important role in sta- tistical physics. In that setting, interacting fixed points encode the scaling exponents near a continuous (second or higher order) phase transition, where scale invariance is due to a diverging correlation length at criticality. QFTs live in theory space, which is the infinite-dimensional space of all couplings that are compatible with the symmetries of the model. Asymptotic safety/freedom is the existence of a fixed point in this space, i.e., for a model to become asymptotically safe, all couplings have to reach a scale-invariant fixed point. Thus, determining whether a model can become asymptotically safe/free, requires us to explore the RG flow of all infinitely many couplings. On the other hand, in perturbatively renormalizable models with asymptotic freedom one typically does not think about higher-order couplings, but restrict the setting to the perturbatively renormalizable ones, thus seemingly working PhenomenologicalStatus of the asymptotic implication? safety paradigm for quantum gravity and matter 3 [arXiv:1709.03696] g • Gravity is strong coupling for k ≳ Mpl 1.5 * ] 1.0 [ • could cause chiral symmetry breaking g(k) of fermions: 0.5 ψψ 0.0 ⟨ ¯ ⟩ ≠ 0 105 109 1013 1017 1021 1025 k/GeV → All fermions acquire dynamical masses ∼ Mpl → inconsistent with light fermions in our world → phenomenologically exclude asymptotic safety scenario [Eichhorn-Gies, ’11] [Eichhorn, ’12] [Meibohm-Pawlowski, ’16] [Eichhorn-Lippoldt, ’16] [Eichhorn-Held, ’17] This talk: GravitationalFig. 1 Beta functions instantons and the can corresponding trigger chiral running sym. for a breaking! UV attractive free/interacting fixed point (lower/upper panels). As the fixed points are UV attractive, the value of the coupling at one reference scale is an initial condition that can be chosen freely. Upper 3 panels: beta function and scale dependence of the Newton coupling according to Eq. (20). the scaling of couplings is determined by the sum of canonical scaling and quantum scaling which is a consequence of loop e↵ects. In fact, a fixed point can arise from a balance between canonical and quantum scaling, as β = d g + ⌘ (g ), (2) gi − g¯i i i i where ⌘i is an anomalous scaling dimension that arises as a consequence of quantum fluctuations. An interacting fixed point lies at gi⇤ = ⌘i/dg¯i . (3) 4 For instance, in d =4 ✏,theλ4φ theory has a fixed point at λ4⇤ = 16⇡2✏/3, the Wilson-Fisher− fixed point [7] playing an important role in sta- tistical physics. In that setting, interacting fixed points encode the scaling exponents near a continuous (second or higher order) phase transition, where scale invariance is due to a diverging correlation length at criticality. QFTs live in theory space, which is the infinite-dimensional space of all couplings that are compatible with the symmetries of the model. Asymptotic safety/freedom is the existence of a fixed point in this space, i.e., for a model to become asymptotically safe, all couplings have to reach a scale-invariant fixed point. Thus, determining whether a model can become asymptotically safe/free, requires us to explore the RG flow of all infinitely many couplings. On the other hand, in perturbatively renormalizable models with asymptotic freedom one typically does not think about higher-order couplings, but restrict the setting to the perturbatively renormalizable ones, thus seemingly working Gravitational instanton • Gravity is a gauge theory with Lorentz sym: SO(4) ≃ SU(2) × SU(2) μ i ab μ a μ ℒ ⊃ ψ¯ γ ∂μ − σabωμ ψ γ ≡ γ ea ( 2 ) ab i a b σ ≡ [γ , γ ] Spin connection 4 ~ SO(4) gauge field • instanton-like configuration with winding # for the SU(2) sector →gravitational instanton • solution to the vacuum Einstein eq. analogue of Yang-Mills instanton 4 [Hawking ’76] Example of grav. instanton [Eguchi-Hanson ’78] • Eguchi-Hanson metric → not considered in this talk • Taub-NUT metric • K3-surface (Calabi-Yau manifold) → compact and closed manifold → In path integral, it appears as a fluctuation around flat space: [Hebecker-Henkenjohann 1906.07728] K3 worm-hole like throat ρ flat spacetime 5 ’t Hooft vertex [’t Hooft ’86] μ A • axial anomaly of fermions : ⟨∂ Jμ ⟩ ∼ RR˜ → Grav. instanton induces an effective vertex called ’t Hooft vertex. (similarly to QCD instanton) K3 worm-hole like throat ρ ∼ 1tH flat spacetime flat spacetime 1 − γ5 1tH = λtH det ψ¯s ψt + (h . c.) U(1)A operator st 2 This cannot be induced by Taub-NUT, Eguchi-Hanson metrics. (APS-index is 0) 6 ’t Hooft vertex and asymptotically safe gravity • dim. analysis based on the dilute gas approx. −2 1 ⇒ λtH(k) ∼ k exp − ( gN(k) ) ρ ∼ k−1 Status of the asymptotic safety paradigm for quantum gravity and matter 3 • assume k is an energy scale [arXiv:1709.03696] 1.5 g* UV (k ≫ Mpl ) : active (scale invariant) ] 1.0 [ g(k) 0.5 IR (k M ) : exponentially suppress ≪ pl 0.0 105 109 1013 1017 1021 1025 k/GeV What does this affect？→ causes chiral sym. breaking: ⟨ψ¯ ψ⟩ ≠ 0 7 Fig. 1 Beta functions and the corresponding running for a UV attractive free/interacting fixed point (lower/upper panels). As the fixed points are UV attractive, the value of the coupling at one reference scale is an initial condition that can be chosen freely. Upper panels: beta function and scale dependence of the Newton coupling according to Eq. (20). the scaling of couplings is determined by the sum of canonical scaling and quantum scaling which is a consequence of loop e↵ects. In fact, a fixed point can arise from a balance between canonical and quantum scaling, as β = d g + ⌘ (g ), (2) gi − g¯i i i i where ⌘i is an anomalous scaling dimension that arises as a consequence of quantum fluctuations. An interacting fixed point lies at gi⇤ = ⌘i/dg¯i . (3) 4 For instance, in d =4 ✏,theλ4φ theory has a fixed point at λ4⇤ = 16⇡2✏/3, the Wilson-Fisher− fixed point [7] playing an important role in sta- tistical physics. In that setting, interacting fixed points encode the scaling exponents near a continuous (second or higher order) phase transition, where scale invariance is due to a diverging correlation length at criticality. QFTs live in theory space, which is the infinite-dimensional space of all couplings that are compatible with the symmetries of the model. Asymptotic safety/freedom is the existence of a fixed point in this space, i.e., for a model to become asymptotically safe, all couplings have to reach a scale-invariant fixed point. Thus, determining whether a model can become asymptotically safe/free, requires us to explore the RG flow of all infinitely many couplings. On the other hand, in perturbatively renormalizable models with asymptotic freedom one typically does not think about higher-order couplings, but restrict the setting to the perturbatively renormalizable ones, thus seemingly working Nambu-Jona-Lasinio model • 2-flavor NJL model + gluon and gravity Γ = d4x g ψ¯ i∇ψ + λ ℒ + λ ℒ k ∫ [ q q top top] 1 1 + d4x g R + d4x g (Fa )2 2 μν 16πG ∫ 4gs ∫ 2 a 2 σ − π channel : ℒq = (ψ¯ ψ) − (ψ¯ γ5σ ψ) 1 1 − γ5 ’t Hooft vertex : ℒtop = det ψ¯s ψt + (h . c.) 2 st 2 • SU(2)L × SU(2)R symmetry (broken down to SU(2)V ) • RG analysis for running coupling: λtop, λq, gN, gs • Chiral sym. breaking = λq → ∞ 8 RG flow eq. for four-fermion couplings • Functional renormalization group (Wetterich equation): −1 1 (2) ∂tΓk = Tr Γ + Rk ∂tRk 2 [( k ) ] Rk : cutoff function • RGE in the absence of grav. instantons: [Eichhorn-Gies '11] [Braun-Leonhardt-Pospiech '18] 7 2 8 ∂tλ¯q = 2λ¯q − λ¯q − (λ¯q + λ¯ )λ¯ + ⋯ λ¯ , λ¯ :dim. less 2π2 π2 top top q top 13 2 1 ∂tλ¯ = 2λ¯ + λ¯ + 4λ¯ + λ¯q λ¯q + ⋯ top top 3π2 top 6π2 ( top ) We have omitted contributions from graviton and gluon exchanges.