Status of asymptotic safety in gravity-matter

systems

Masatoshi Yamada (Ruprecht-Karls-Universität Heidelberg)

KEK Theory workshop 2019

General relativity• Einstein theory

• Well describes observed facts:

• Mercury perihelion

• Gravitational wave

• etc.

Towards quantum gravity• The quantized Einstein-Hilbert action is not

perturbatively renormalizable.

• Higher derivative gravity

• Perturbatively renormalizable

• Ghost (unitarity) problem

G. ’t Hooft and M. Veltman,  Annales Poincare Phys.Theor.,A20,69

Stelle, K.S. Phys.Rev. D16 (1977) 953-969

In this talk• We introduce quantum gravity based on

asymptotic safety.

• Pure gravity case will be presented by Prof. Ohta.

• We focus on AS for gravity-matter systems.

• The key word is the anomalous dimension induced by quantum gravity effects.

Contents• What is Asymptotic Safety (AS)?

• AS for the standard model and gravity

• Prediction for the Higgs mass, ~125 GeV

• Prediction for top-quark mass, ~170 GeV

• AS for beyond the standard model and gravity

• The gauge hierarchy problem

• Dark matter physics (Higgs portal type)

Asymptotic safety• Suggested by S. Weinberg

• Existence of non-trivial UV fixed point

• Continuum limit k→∞.

• UV critical surface (UV complete theory) is spanned by relevant operators.

• Dimension of UV critical surface = number of free parameters.

• Generalization of asymptotic free

• Non-perturbatively renormalizable gravity

S. Weinberg, Chap 16 in General Relativity

Fig. from A.Eichhorn, Front.Astron.Space Sci. 5 (2019) 47

Asymptotic freedom• Asymptotic freedom

Asymptotic safety• Asymptotic safety

Functional renormalization group

k@k�k =1

2Str[(�(2)

k +Rk)�1k@kRk]

g1

g2

gi

�k =

Zd4x[g1O1 + g2O2 + · · ·+ giOi + · · · ]

�k '

Zd4x[g1O1 + g2O2]

S = �⇤

� = �k=0

exact flow

truncated flow

projection

Wetterich equation

Critical exponent

• RG eq. around FP g*

• Solution of RG eq. negative eigenvalue k ! 0

✓i > 0

✓i < 0

relevant

irrelevant

Relevant: θ> 0• Free parameter

Irrelevant θ< 0• Predictable parameter

Landau pole

Irrelevant θ< 0• Predictable parameter

Landau pole

PredictionUV complete

(no Landau pole)

No dangerous divergence =Safe!

RG flow of g (dimensionless Newton constant)

g

Irrelevant at Gaussian FP

Relevant at non-trivial FP

Found.Phys. 48 (2018) no.10, 1407-1429

Earlier studies• Truncated system for pure gravity

Earlier studies• Truncated system for pure gravity

Einstein-Hilbert truncation e.g. M. Reuter, F. Saueressig, Phys.Rev. D65 (2002) 065016

Earlier studies• Truncated system for pure gravity

f(R) truncation e.g. K. Falls, D. Litim, J. Schröder, Phys.Rev. D99 (2019) no.12, 126015G.Brito, N.Ohta, A. Pereira, A.Tomaz, M.Y., Phys.Rev. D98 (2018) no.2, 026027

R71

Earlier studies• Truncated system for pure gravity

Higher derivative truncation Ie.g. D. Benedetti et al. Mod.Phys.Lett. A24 (2009) 2233-2241Y.Hamada, M.Y., JHEP 1708 (2017) 070

Earlier studies• Truncated system for pure gravity

Higher derivative truncation IIL.Bosma, B.Knorr, F.Saueressig, Phys.Rev.Lett. 123 (2019) no.10, 101301

Earlier studies• Truncated system for pure gravity

Higher derivative truncation IIIB.Knorr, C.Ripken, F.Saueressig, Class.Quant.Grav. 36 (2019) no.23, 234001

Earlier studies• These studies have shown the finite number of relevant directions.

• There are 3 relevant directions (?)

• which means 3 free parameters

• For details, listen Prof. Ohta’s talk.

Open questions• What is degrees of freedom associated to the non-trivial (Reuter) fixed point?

• Unitarity problem (or ghost problem)

• Robustness of number of relevant operators.

• Scheme-independent calculations.

• …

Potential solution to the ghost problem

• Action for asymptotically safe gravity

What is their pole structure? L.Bosma, B.Knorr, F.Saueressig, Phys.Rev.Lett. 123 (2019) no.10, 101301B.Knorr, C.Ripken, F.Saueressig, Class.Quant.Grav. 36 (2019) no.23, 234001

Contents• What is Asymptotic Safety (AS)?

• AS for the standard model and gravity

• Prediction for the Higgs mass, ~125 GeV

• Prediction for top-quark mass, ~170 GeV

• AS for beyond the standard model and gravity

• The gauge hierarchy problem

• Dark matter physics (Higgs portal type)

The SM and gravity• Working assumption:

• Consider the system where the SM is coupled to gravity.

• No new matter.

• Einstein-Hilbert truncation

Beta function• For a matter coupling α

• γα is the anomalous dimension induced by quantum gravity effects.

Prediction of Higgs mass• Prediction of quartic coupling constant

• RG equation

• We find the Gaussian FP, λ* =0.

• Critical exponent (anomalous dimension)J.Pawlowski, M.Reichert, C.Wetterich, M.Y.,Phys.Rev. D99 (2019) no.8, 086010

RG flow of quartic coupling

QG decoupledIrrelevant

Landau pole

Irrelevant Landau pole

The red trajectory is the prediction.

RG flow of quartic coupling

QG decoupled

⭐

The top-Yukawa induces positive λ.

Predicted point

Irrelevant Landau pole

Irrelevant Landau pole

The red trajectory is the prediction.

Top quark mass vs.

Higgs mass

• For mt=171.3 GeV, mH=126.5 GeV

• For mt=230 GeV, mH=233 GeV

• Current experimental results (LHC)

• mt=170.5±0.7 GeV, mH=125.10±0.14 GeVarXiv: 1904.05237; PDG

Prediction of Higgs mass = Prediction of top mass

M.Shaposhnikov, C.Wetterich, Phys.Lett. B683 (2010) 196-200

RG flow of YukawaQG decoupled

Irrelevant Landau pole

Irrelevant Asymptotically safe

relevant Asymptotically free

The red trajectory is the prediction.

A. Eichhorn, A.Held, Phys.Lett. B777 (2018) 217-221

RG flow of YukawaQG decoupled

⭐

Irrelevant Landau pole

Predicted point

Irrelevant Asymptotically safe

relevant Asymptotically free

The red trajectory is the prediction.

The SM

A. Eichhorn, A.Held, Phys.Lett. B777 (2018) 217-221

Prediction of top mass

FP value of Newton constant

FP value of Cosmological constant

A. Eichhorn, A.Held, Phys.Lett. B777 (2018) 217-221

Contents• What is Asymptotic Safety (AS)?

• AS for the standard model and gravity

• Prediction for the Higgs mass, ~125 GeV

• Prediction for top-quark mass, ~170 GeV

• AS for beyond the standard model and gravity

• The gauge hierarchy problem

• Dark matter physics (Higgs portal type)

Gravitational corrections to scalar mass parameter

• RG equations

• Anomalous dimension

• Graviton induced anomalous dimensionJ.Pawlowski, M.Reichert, C.Wetterich, M.Y.,Phys.Rev. D99 (2019) no.8, 086010

0

RG flow of scalar mass

“Classical” scale invarianceQG decoupled

Resurgence mechanismC.Wetterich, M.Y., Phys.Lett. B770 (2017) 268-271

W.Bardeen, FERMILAB-CONF-95-391-TC.Wetterich, M.Y., Phys.Lett. B770 (2017) 268-271

Higgs portal interaction• An additional scalar field

• We find the Gaussian FP at which the couplings become irrelevant.

S

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

Possible extension of the SM• The boundary condition at the Planck scale

• To generate finite values in low energy

• Additional fermion and U(1) gauge field χ Xμ

at

Kinetic mixing

RG flow of scalar couplings

The additional fermion is stable. Dark matter candidate

Y.Hamada, K.Tsumura, M.Y.,Working in progress C.f. M.Hashimoto, S.Iso, Y.Orikasa, Phys.Rev. D89 (2014) no.1, 016019

Realize the Coleman-Weinberg mechanism

Summary• Asymptotically safe gravity is a possible quantum gravity.

• Irrelevant couplings are predictable.

• Higgs mass and top-quark mass

• Conditions for extensions of the SM.

• What I could not talk

• RG flow of U(1) gauge coupling

• Mass hierarchy in the quark sector