Functional renormalizationgroup approach toasymptotically safe gravity
Masatoshi Yamada
Jilin University (吉林大学)
Phys.Lett.B 770 (2017) 268-271;
Phys.Rev.D 97 (2018) 8, 086004;
Phys.Rev.D 99 (2019) no.8, 086010;
Phys.Rev.D 100 (2019) no.6, 066017;
Eur.Phys.J.C 80 (2020) 5, 368;
Phys.Lett.B 813 (2021) 135975;
JHEP 03 (2022) 130

Functional Renormalization Group@RIKEN

Quantum gravity•One says•

Quantum gravity based on the Einstein-Hilbert
action is not renormalizable.

•Because the Newton constant is massdimension -2.Quantum gravity•

One says•

Quantum gravity based on the Einstein-Hilbert
action is not renormalizable.
perturbatively•

Because the Newton constant is massdimension -2.

Contents
Asymptotically safe gravity:••

Universality class at xed point••

non-perturbatively renormalizable quantum gravity

Duality between asymptotically safe and free theories

Asymptotic safety for scalar-gravity systemfi•Asymptotic freedom•

Asymptotic freedom

Asymptotic safety•

Asymptotic safety

What is asymptotic safety?
•In UV limit (Λ→ )•Asymptotic freedom: asymptotically reach to a freetheory (Gaussian FP). Perturbative•

Asymptotic safety: asymptotically reach to aninteracting theory (Non-trivial FP). Non-perturbative

Asymptotically safe theories•

D=3 non-linear σ model (Scalar theory)•

D=3 Gross-Neveu model (Fermionic theory)•

D=5 Yang-Mills theory???

•

D=4 gravity???
Does there exist asymptotically safe gauge theory?

Functional renormalization group
=gi

k=0

exact owk =Z

d4 x[g1 O1 + g2 O2 + · · · + gi Oi + · · · ]projection

g2g1

S=

⇤

truncated ow

k 'Z

d4 x[g1 O1 + g2 O2 ]

flfl

1(2)
1
Wetterich equation k@k k = Str[( k + Rk ) k@k Rk ]2

Functional renormalization group=gi

k=0exact ow
k =Z

d4 x[g1 O1 + g2 O2 + · · · + gi Oi + · · · ]projection

g2
g1

S=

⇤

truncated ow

k 'Z

d4 x[g1 O1 + g2 O2 ]1(2)1
Wetterich equation k@k k = Str[( k + Rk ) k@k Rk ]
2

fl

fl

Fixed point

Critical exponent•RG eq. around FP g*irrelevant✓i < 0•

Solution of RG eq.

- eigenvalue

k!0✓i > 0relevantRelevant: θ> 0•Free parameter

Irrelevant θ< 0•Predictable parameter
Landau pole

Irrelevant θ< 0•Predictable parameter
Landau pole

Irrelevant θ< 0•Predictable parameter
Landau pole

UV complete(no Landau pole)

Prediction

No dangerous divergence=Safe!

Gravitational systemff•

E ective action for pure gravity

Gravitational system•E ective action for pure gravity
Truncate

Einstein-Hilbert truncation

e.g. scholarpedia

ffwith anomalous dimension induced by quantum gravity

Gravitational system•

Fixed point•

Critical exponents

Gravitational system

Gravitational system

Higher derivative truncation :f(R) truncation•f(R) truncation (pure gravity)

Phys.Rev. D99 (2019) no.12, 126015

Up to R71

Finite number ofrelevant couplings=Renormalizalbe!
3 free parameters?

Higher derivative truncation :22
R +C truncation
Inclusion of higher dimensional operators
S.Saswato, C. Wetterich, MY, JHEP 03 (2022) 130•

Asymptotically safe xed point

Graviton propagator:

Asymptotically free xed point

Graviton propagator:

fi

•fi

•

Higher derivative truncation :22R +C truncation
Inclusion of higher dimensional operators
S.Saswato, C. Wetterich, MY, JHEP 03 (2022) 130

Asymptotically safe xed point•

Asymptotically free xed point

fi

•fi

•

Contents
Asymptotically safe gravity:••

Universality class at xed point••

non-perturbatively renormalizable quantum gravity

Duality between asymptotically safe and free theories

Asymptotic safety for scalar-gravity systemfi•Is asymptotically safe
theory ultimate?

An Example ofasymptotically safe theories
Non-linear σ model in 3 dim.
•

Scalar theory with a eld constraint•

Symmetry breaking O(N) →O(N-1) in the linear σ model.

•Describes dynamics of massless NG bosons (pions).

•

Perturbatively non-renormalizablefi•

An Example ofasymptotically safe theories•

Linear σ model•

Non-linear σ modelPhase diagram of 3d linear σ model
Arrows: From UV to IR

Wilson-Fisher (IR) FP(non-perturbative)

Broken phase

Symmetric phase

Gaussian (UV) FP(perturbative)(dimensionless mass)

Phase diagram of 3d linear σ model
Arrows: From UV to IR

Wilson-Fisher (IR) FP(non-perturbative)

Broken phaseσ becomes massive anddecouples from the dynamics

Regime described by
Non-linear sigma modelPhase diagram of 3d linear σ model
Arrows: From UV to IR

Wilson-Fisher (IR) FP(non-perturbative)
Broken phaseσ becomes massive anddecouples from the dynamics

Regime described by
Non-linear sigma model

Non-trivial UV FP(non-perturbative)Phase diagram of 3d linear σ model
Arrows: From UV to IR

Wilson-Fisher (IR) FP(non-perturbative)
linear σ model

Broken phase
Same universality classnon-linear σ model

Non-trivial UV FP(non-perturbative)To summarize
Non-linear σ model in 3 dimO(N-1)

Asymptotically safe gravity•

Perturbatively non-renormalizable•

Perturbatively non-renormalizable•

Asymptotically safe (UV FP)•

Asymptotically safe (UV FP)•

Constraint on elds•

Constraint on elds

Same universality class
Perturbatively renormalizable•

Unitary (at Gaussian FP)•

Asymptotically free (Gaussian FP)•IR xed point (Wilson-Fisher FP)fi•fifi

O(N) linear σ model in 3 dim?

Contents
Asymptotically safe gravity:••

Universality class at xed point••

non-perturbatively renormalizable quantum gravity

Duality between asymptotically safe and free theories

Asymptotic safety for scalar-gravity systemfi•Scalar-gravity systemff•

E ective action•

Einstein-Hilbert truncation: linear in R•

Minimal coupling•

Local potential approximation

Critical exponent of scalarinteractions•

E ective potential

canonically relevant

Gaussian FP:

ffNon-trivial FP:

canonically marginalcanonically irrelevant

Gravity-induced anomalousdimension•

RG for scalar couplings•

Anomalous dimension
Pawlowski, Reichert, Wetterich, MY, Phys.Rev. D99 (2019) no.8, 086010

Critical exponent=1/(8πGN*)

Gravitational interaction tends tomake scalar couplings irrelevant!

=1/(8πGN*)

Irrelevant θ< 0•Predictable
Landau pole

Prediction

UV completeRG ow
Irrelevantfl

QG decoupledRG ow
Irrelevant
QG decoupledflg

RG ow
Irrelevantfl

QG decoupledRG owRelevant(due to yt)

Irrelevant
QG decoupled

Predictedpoint⭐

flThe top-Yukawa induces positive λ.

Top quark mass
vs.
Higgs mass•

For mt=171.3 GeV, mH=126.5 GeV
Shaposhnikov, Wetterich, Phys.Lett. B683 (2010) 196-200•

For mt=230 GeV, mH=233 GeV•

Current experimental results (LHC)
Eur. Phys. J. C 80 (2020) 658; PDG•

mt=170.5 0.7 GeV, mH=125.10 0.14 GeV

Higgs portal interaction•SM extension including new scalar S•

All couplings irrelevant:

Eichhorn, Yuta.Hamada, Lumma, M.Y., Phys.Rev. D97 (2018) no.8, 086004

at
The potential is at above the Planck scale. (Flatland)•This becomes strong constrains on extensions of theSM.
fl•Possible extension of the SM•

The boundary condition at the Planck scale:atTo generate nite values in low energy
Additional Majorana fermions χs and U(1) gauge eld XμKinetic mixingfi•fi•

mH=125 GeV

Negative Higgs mass term

Coleman-Weinberg for <S>

FlatlandStrong prediction•

7 parameters (λH, λHS, λS, yL, yR, gX, gmix)⇨ 1 parameter•

6 constraints:•Higgs mass: mH=125 GeV•Electroweak vacuum: vH=246 GeV•

Majorana fermions as dark matters: ΩDMh2=0.12•

Flatland condition:atPrediction to Dark matter physics(Majorana fermionic DM)

Summary••

Asymptotically safe gravity:•

Non-perturbatively renormalizable•

Universality class•

Indicate the existence of new degrees of freedom.

Strong constrains on extensions of the SM.

Appendix

Asymptotic safety•

Suggested by S. Weinberg
S. Weinberg, Chap 16 in General Relativity•Continuum limit k→ .

•UV critical surface (UV complete theory) is de ned byrelevant operators.

•Dimension of UV critical surface = number of freeparameters.

Generalization of asymptotic freefi•

Existence of UV xed pointfi•

Asymptotic
freedom vs safety
Asymptotic freedom

canonical scaling

anomalous scaling

Asymptotic safety

canonical scaling

anomalous scalingRelevant: θ> 0•

QCD caseIrrelevant θ< 0•

QED and scalar theory
Landau pole

UV complete(no Landau pole)

Prediction

No dangerous divergence,but free theory…”triviality problem”

RG ow ofNewton constantRelevant

Non-trivial FP[gN]=+2large anomalous dimension
inducedIrrelevant[gN]=-2flGaussian FPDegenerate limit•

First-order formalism••

Some components of vierbein degenerate.

•Admit topology-change processes in path integral.
A. Tseytlin,J. Phys. A15 (1982) L105.
G. T. Horowitz, Class. Quant. Grav. 8 (1991) 587.

•Forbids inverse vierbein•

Remove terms divergent in the limit•

Do not use inverse metric.

.

Inverse metric•

Metric••Canonical normalization

Inverse metricModel with null limit•

Including matters, at a certain scale,••••No invariant term•

TopologicalDegenrate limit•

Higgs potential•

In terms of invariance and renormalizability•

Inverse terms diverge for H→0 (symmetric phase)LL gauge theory•

Ordinary YM theory

Ta commutes with γa.
•

LL gauge theoryΣab do not commute with γa.

Make LL gauge eld
Dynamical•

The staring action has no kinetic terms except
for spinor elds.

•Fermionic uctuations make other elds
dynamical.

fifi

fl

fi

fle.g. LL gauge eld in a at spacetime backgroundfi•

Analogy to QCD•

QCD••Nambu-Jona-Lasinio model

Bosonisation

Quark-meson modelNo kinetic term of bosonGravity•

Spinor gravity:

Vierbein and LL gauge eld are composites of spinors

A. Hebecker, C. Wetterich, Phys.Lett. B574 (2003) 269-275•Our modelfi•

Contents
Asymptotically safe gravity:••

Universality class at xed point••

non-perturbatively renormalizable quantum gravity

Duality between asymptotically safe and free theories

Fermion-induced spacetimefi•How to formulate?
•Metric theories are diffeomorphism invariant.

?
•SSB

Diffeomorphism

In this work, we consider local Lorentz SO(1,3):

SO(1,3)local × Diff.

SSBSO(1,3)local

Diffeomorphism

First-order formalism•

Based on SO(1,3) local Lorentz symmetry (and diff.)•Vierbein•

Local-Lorentz (LL) gauge eld

Minimal action (Einstein-Hilbert)fi•Skippable

First-order formalism•

Equation of motion•

Obtain the EH action in the vierbein formalism•

Introducing inverse vierbein breaks SO(1,3)local symmetry.

fi• Kinetic term of LL gaugeeld

Degenerate limit•

Non-linear σ model: O(N-1) invariant•

Constraint on elds
: symmetric phase (O(N) invariant)•Gravity in rst-order formalism•

Constrain on metric: symmetric phase (SO(1,3) invariant).

•More precisely,fi

•

fi

•

Model with degenerate limit
Including matters, at a certain scale,Invariant under SO(1,3)local × diff.

•Only fermions are dynamical!

•No kinetic terms of vierbein, gauge elds, scalar elds.

•These elds would be dynamical via fermion quantum corrections.

fi

•fifi

•

Spontaneous local Lorentz
symmetry breaking•SO(1,3)local × diff.

•

Generation of expectation value of vierbein•

A possible solution would be a at spacetime.

•Effective potential from spinor loop effects:•Precise analysis is in progress.

flDegenerate limit

Skippable

Spontaneous local Lorentz
symmetry breaking
Local Lorentz gauge symmetry is broken.
Symmetric part (metric)(radial modes)
Anti-symmetric part (torsion)(NG modes)

Degrees of freedom (d.o.f.):••

Vierbein: 16 d.o.f. = 10 + 6 d.o.f.
eaten

•

LL gauge eld: 6 d.o.f.

•LL gauge bosons become massive and decouple.

•The symmetry parts (radial modes) are still massless
thanks to diif..
fi•RG ow (ideal)
UV FPIR FP

After LL gauge symmetry breaking

Spontaneous local Lorentz
symmetry breaking

Asymptotically safe gravity regime

flSO(1,3)global × diff.

Proposed modelSO(1,3)local × diff.

????