Ultraviolet Dynamics of Fermions and Gravity Cold Quantum Cofgee on June 26, 2018 Marc Schifger , Heidelberg University with A. Eichhorn and S. Lippoldt arXiv 18xx.xxxxx with A. Eichhorn, S. Lippoldt, J. M. Pawlowski and M. Reichert arXiv 1807.xxxxx
Motivation for Quantum Gravity
• Simplest version of a black hole: • For E G N : Why quantum gravity? theory [LIGO Collaboration, 2016] QG efgects are expected c M PL . GR is not a fundamental General Relativity: physical singularity at r Schwarzschild black hole collaboration Existence of black holes by LIGO 2 • several high precision tests • latest confjrmation:
• For E G N : Why quantum gravity? theory Source: Northern Arizona University QG efgects are expected c M PL 2 General Relativity: Schwarzschild black hole collaboration Existence of black holes by LIGO • several high precision tests • latest confjrmation: • Simplest version of a black hole: physical singularity at r = 0 . ⇒ GR is not a fundamental
Why quantum gravity? Schwarzschild black hole Source: Northern Arizona University QG efgects are expected theory General Relativity: 2 collaboration Existence of black holes by LIGO • several high precision tests • latest confjrmation: • Simplest version of a black hole: physical singularity at r = 0 . ⇒ GR is not a fundamental √ � c • For E > M PL = G N :
• Triviality problem in scalar [M. Gockeler et al., 1997] , • Singularities might carry over to [D. Buttazzo, 2013] QG might cure Landau poles breakdown beyond Planck scale SM is only efgective theory SM Why quantum gravity with matter? [H. Gies and J. Jäckel, 2004] Standard Model: [J. Fröhlich, 1982] abelian gauge theories and 3 • well tested low energy model • formulated as QFT
• Singularities might carry over to Why quantum gravity with matter? Standard Model: abelian gauge theories [J. Fröhlich, 1982] [H. Gies and J. Jäckel, 2004] SM SM is only efgective theory breakdown beyond Planck scale QG might cure Landau poles Image credits: Fleur Versteegen 3 • well tested low energy model • formulated as QFT • Triviality problem in scalar ϕ 4 and [M. Gockeler et al., 1997] ,
Why quantum gravity with matter? Standard Model: abelian gauge theories [J. Fröhlich, 1982] [H. Gies and J. Jäckel, 2004] SM Image credits: Fleur Versteegen 3 • well tested low energy model • formulated as QFT • Triviality problem in scalar ϕ 4 and [M. Gockeler et al., 1997] , • Singularities might carry over to → SM is only efgective theory → breakdown beyond Planck scale → QG might cure Landau poles
Why quantum gravity with matter? [M. Gockeler et al., 1997] , compatibility with SM in IR provides test for quantum theory of gravity Image credits: Fleur Versteegen SM Standard Model: [H. Gies and J. Jäckel, 2004] [J. Fröhlich, 1982] abelian gauge theories 3 • well tested low energy model • formulated as QFT • Triviality problem in scalar ϕ 4 and • Singularities might carry over to → SM is only efgective theory → breakdown beyond Planck scale → QG might cure Landau poles
Outline Motivation for Quantum Gravity Asymptotically Safe Quantum Gravity Efgective Universality for Gravity and Matter Induced Couplings at UV Fixed Point 4
Asymptotically Safe Quantum Gravity
• Efgective fjeld theory approach: Loss of predictivity at M PL How to quantize Gravity? [G. ’t Hooft and M. J. G. Veltman, 1974] [M. H. Gorofg and A. Sagnotti, 1986] [J. F. Donoghue and B. R. Holstein, 2015] 6 • [ G N ] = 2 − d ⇒ GR is perturbatively non-renormalizable in d = 4
How to quantize Gravity? [G. ’t Hooft and M. J. G. Veltman, 1974] [M. H. Gorofg and A. Sagnotti, 1986] [J. F. Donoghue and B. R. Holstein, 2015] 6 • [ G N ] = 2 − d ⇒ GR is perturbatively non-renormalizable in d = 4 • Efgective fjeld theory approach: Loss of predictivity at M PL
• Asymptotic safety Asymptotic safety [S. Weinberg, 1979] all dimensionless couplings enter a scale invariant regime interacting theory in the UV non-perturbative renormalizability CMS Collaboration, 2017 7 • Asymptotic freedom ◮ all couplings vanish in the UV ◮ perturbative renormalizability
Asymptotic safety • Asymptotic freedom • all couplings vanish in the UV • perturbative renormalizability [S. Weinberg, 1979] a scale invariant regime renormalizability A. Eichhorn, 2017 7 • Asymptotic safety ◮ all dimensionless couplings enter ◮ interacting theory in the UV ◮ non-perturbative
Search for Asymptotic safety gi dimensional quantum 8 Study RG-fmow of dimensionless couplings g i = ¯ g i k − d ¯ β g i ( ⃗ g ) = k ∂ k g i = − d ¯ + f i ( ⃗ g ) g i g i ⇒ balancing of dimensional term with quantum correction can lead to AS
• Solution to linearized fmow equations g i k Critical Exponents i eig M with j k k j c j V i j g i 9 j • Linearized β -functions � ( ∂β g i ) � ∑ j ) 2 ) � � g = g ∗ ( g j − g ∗ ( ( g j − g ∗ β g i = β g i g = g ∗ + j ) + O � � ∂ g j � �
Critical Exponents j with j c j V i j 9 • Linearized β -functions � ( ∂β g i ) � ∑ j ) 2 ) � � g = g ∗ ( g j − g ∗ ( ( g j − g ∗ β g i = β g i g = g ∗ + j ) + O � � ∂ g j � � • Solution to linearized fmow equations ) − Θ j ( k ∑ g i ( k ) = g ∗ i + − eig ( M ) = Θ i . k 0
Critical Exponents j dimensional i with 9 c j V i j • Solution to linearized fmow equations ) − Θ j ( k ∑ g i ( k ) = g ∗ i + − eig ( M ) = Θ i . k 0 Re (Θ i ) < 0 • irrelevant direction • g i ( k ) k →∞ − − − → g ∗ • c j drop out • g ∗ i is a prediction
Critical Exponents with relevant direction i dimensional i 9 j c j V i j • Solution to linearized fmow equations ) − Θ j ( k ∑ g i ( k ) = g ∗ i + − eig ( M ) = Θ i . k 0 Re (Θ i ) < 0 Re (Θ i ) > 0 • relevant direction • irrelevant direction • g i ( k ) k →∞ − − − → g ∗ • c j remain • adjust c j to reach g ∗ • c j drop out • g ∗ • one free parameter for each i is a prediction
Critical Exponents with test Number of relevant directions determines predictivity relevant direction i dimensional i 9 j j c j V i • Solution to linearized fmow equations ) − Θ j ( k ∑ g i ( k ) = g ∗ i + − eig ( M ) = Θ i . k 0 Re (Θ i ) < 0 Re (Θ i ) > 0 • relevant direction • irrelevant direction • g i ( k ) k →∞ − − − → g ∗ • c j remain • adjust c j to reach g ∗ • c j drop out • g ∗ • one free parameter for each i is a prediction
Critical Exponents [A. Eichhorn, 2017] relevant direction 9 Re (Θ i ) < 0 • irrelevant direction • g ∗ i is a prediction Re (Θ i ) > 0 • relevant direction • one free parameter for each
Tool: Functional Renormalization Group k [H. Gies, 2006] Non-Perturbative Renormalisation Group Equation [Wetterich, 1993], [Reuter, 1996] 10 k ∂ k Γ k = 1 (( ) = 1 ) − 1 Γ (2) + R k k ∂ k R k 2 STr 2 Γ k = scale dependent efgective action R k = IR regulator • exact 1-loop equation • extract β -functions via projection • truncation needed → not closed
Efgective Universality for Gravity and Matter
G h • two difgerent ”avatars” of the Newton coupling Avatars of the Newton coupling N f G h 12 • Einstein-Hilbert gravity minimally coupled to fermions, i.e. 1 ∫ d 4 x √ g ( R − 2Λ) + ∫ d 4 x √ g ¯ ψ i / ∑ S = − ∇ ψ i 16 π G N i =1
Avatars of the Newton coupling N f 12 • Einstein-Hilbert gravity minimally coupled to fermions, i.e. 1 ∫ d 4 x √ g ( R − 2Λ) + ∫ d 4 x √ g ¯ ψ i / ∑ S = − ∇ ψ i 16 π G N i =1 √ √ ∼ ∼ G h ¯ G 3 h ψψ • two difgerent ”avatars” of the Newton coupling
• On the quantum level • Efgective universality: • Compare both avatars on the level of their G h G h Efgective universality G G G G G h G G G G -functions at G h [A. Eichhorn, P. Labus, J. M. Pawlowski and M. Reichert, 2018] Quantitative agreement of difgerent avatars of the Newton coupling set of identities (mSTI’s) relates avatars Gauge fjxing, Regulator [Weinberg, 1995] 2-loop universality is lost G N 13 • Classically: Difgeomorphism invariance ⇒ there should only be one Newton coupling
• Efgective universality: • Compare both avatars on the level of their G h G h G G G G G h G G G G Efgective universality G h -functions at [A. Eichhorn, P. Labus, J. M. Pawlowski and M. Reichert, 2018] Quantitative agreement of difgerent avatars of the Newton coupling set of identities (mSTI’s) relates avatars [Weinberg, 1995] 13 • Classically: Difgeomorphism invariance ⇒ there should only be one Newton coupling • On the quantum level ◮ [ G N ] = − 2 ⇒ 2-loop universality is lost ◮ Gauge fjxing, Regulator
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