Gauge hierarchy problem inasymptotically safe gravity‒the resurgence mechanism
Masatoshi Yamada(Ruprecht-Karls-Universität Heidelberg)with Christof Wetterich
Based on C. Wetterich and M.Y., Phys. Lett. B 770, 268

PPP2017@Yukawa InstituteFirst of all…
The “resurgence” mechanism
diﬀers froma method to resume perturbative series.
(Misumi sanʼs talk)Whatʼs resurgence?
According to “Oxford Dictionary of English”,

an increase or revival after a period oflittle activity, popularity

Plan•

Revisit gauge hierarchy problem•

Asymptotically safe gravity•The resurgence mechanism

Gauge hierarchy problem•

Z ⇤Renormalized Higgs massm2⇤✖

=✖

+d4 p12⇠⇤p2+・・・2⇤22mR = m⇤ +( + ···)
216⇡•

O(102 GeV)2 = -O(1019 GeV)2 +O(1019 GeV)222mR ⌧ m⇤

In viewpoint of Wilson RGk =Z

m2k=2k

m2k2

142d x (@µ )2C8⇡ 22k44kk

m2k < 0m2k > 00

m2kk2

In viewpoint of Wilson RG
m2k=2kC8⇡ 2(m2⇤ ,kk⇤)2mRm2kk2

02C⇤
22
mR = m⇤ +8⇡ 2k=0⇤ 'k=0In viewpoint of Wilson RG
m2k=2kC
8⇡ 2(m2⇤ ,m2⇤ =kk⇤)C⇤28⇡ 2⇤m2kk22mR02C⇤
22mR = m⇤ +8⇡ 2k=0=0⇤ 'k=0In viewpoint of Wilson RG
m2k=2km2k < 0C
8⇡ 2kkm2k > 0
m2kk2

02
• Λ determines the position of phase boundary (critical line).
•The phase boundary corresponds to the massless (critical)
theory.

•To obtain small mR, put the bare parameters close to thephase boundary.

In viewpoint of Wilson RG
Picture from web

Gauge hierarchy problem= Criticality problem

Why is the Higgs close to critical?

m2k=k2

C8⇡ 2kk

m2kk2
•

2

Λ is spurious?
••••Discussion

m2k 0=02k0k

m2k 0k2

W. A. Bardeen, FERMILAB-CONF-95-391-T
H. Aoki, S, Iso, Phys. Rev. D86, 013001

The position of phase boundary is physically meaningless.

The distance between the ﬂow and the boundary is physicallymeaningful.
2In perturbation theory, Λ is always subtracted by the counter term orK-I Aoki et. al., hep-th/0002231
dimensional regularization.
•Rotation of coordinate. ➡ C = 0•

Choice of renormalization scheme

m2k=k2C8⇡ 2k

Choice of coordinate of theory space

Gauge hierarchy problem ⇄ The bare theory of Higgs is almost scale(conformally) invariant.

m2k 0=02k

Gauge hierarchy problem•

RG equation for dimensionless scalar mass(deviation from phase boundary)canonical dim.

From observations

anomalous dim.

eﬀective dim.

RG ﬂow of scalar massComment•If m0=0, mR=0 is realized.
••

The theory exactly on the phase boundary.

Idea of classical scale (conformal) invariance•

How to generate the EW scale? (scalegenesis)•Dimensional transmutation or Dynamical
symmetry breaking with TeV scale.
(Next Matsuzaki sanʼs talk)Summary so far•

Gauge hierarchy problem is criticality problem.
••

Λ2 may be not physically meaningful.
••

Why is Higgs close to the phase boundary?

The deviation from the phase boundary is physical.

The positive canonical dimension 2 causes gauge
hierarchy problem.

Plan•Revisit gauge hierarchy problem•

Asymptotically safe gravity•

The resurgence mechanism

Asymptotic safety
Picture from Wiki•

Suggested by S. Weinberg
S. Weinberg, Chap 16 in General Relativity••

Existence of UV ﬁxed point•Continuum limit k→ .

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

•

Dimension of UV critical surface = number of freeparameters.

Generalization of asymptotic freeAsymptotic safety•

Asymptotic freeAsymptotic safety•

Asymptotic safetyFunctional renormalization group=gik=0exact ﬂowk =Z

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

g2g1

S=

⇤truncated ﬂowk 'Zd4 x[g1 O1 + g2 O2 ]Wetterich equation

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

Critical exponent•

Classiﬁcation of ﬂow around FP•

RG eq. around FP g*
irrelevant✓i < 0•

Solution of RG eq.

eigenvalue

k!0✓i > 0relevant

Earlier studies
Cf.
O. Lauscher, M. Reuter, Phys. Rev. D66, 025016•Pure gravity

K. Falls, et. al., arXiv: 1301.4191
D. Benedeti, et. al,. Mod. Phys. Lett. A24, 2233

↵R + R2 + Rµ⌫ Rµ⌫ , etc.

•

Truncation: f (R),

•There exists a UV ﬁxed point in gravity sector•

Number of relevant operators: 3

3 free parameters
Cf.

•

With matters

R. Percacci, D. Perini, Phys. Rev. D67, 081503
P. Dona, et. al., Phys. Rev. D89, 084035•stability of FP

J. Meibohm, et. al., Phys.Rev. D93, 084035
R. Percacci, D. Perini, Phys. Rev. D68, 044018
G. Narain, R. Percacci, CQG 27, 075001•

scalar-gravity system

R. Percalli et. al, Phys. Lett. B689, 90
G. P. Vacca, O. Zanusso, Rhys. Rev. Lett. 105, 231601
J. Daum et. al., JHEP 01, 084•

Higgs-Yukawa system

U. Harst, M. Reuter, JHEP 05, 119
A. Eichhorn, H. Gies, New Phys. J., 113, 125012
A. Eichhorn, Phys. Rev. D86, 105021•

gauge ﬁeld system•

Fermionic systemetc.

Many possibilities•

Prediction of Higgs mass
M. Shaposhinikov, C. Wetterich, Phys. Lett. B683 (2010), 196•

Prediction of top-quark massA. Eichhorn, A Held, arXiv:1707.01107•U(1) Landau pole problem
N. Christiansen, A. Eichhorn, Phys.Lett. B770 (2017) 154•

Cosmological constant problem
C. Wetterich, arXiv:1704.08040•Gauge hierarchy problem
C. Wetterich, M.Y., Phys. Lett. B 770 (2017), 268

Plan
•

Revisit gauge hierarchy problem•

Asymptotically safe gravity•

The resurgence mechanism

Corrections to scalar mass•

RG equation for scalar mass•

Contributions from graviton ﬂuctuation

graviton loop

Quantum Scalingof scalar mass•

Critical exponent (eﬀective dimension)•

At Gaussian-matter ﬁxed point:Quantum Scalingof scalar mass•

Critical exponent•

If we have a large anomalous dimension

RG ﬂow of scalar mass

RG ﬂow of scalar massresurgence mechanism

self-tune criticality

Possibility of θ<0•Higgs-Yukawa model
K. oda and M.Y., Class.Quant.Grav. 33 (2016) no.12, 125011•

We ﬁndwhich yields.

Comment on“Classical” scale invariance•

Gaussian-matter ﬁxed point: Matterʼs FP is trivial.

•In continuum limit k→ , matter coupling constants converge
to the trivial ﬁxed point.

•

The scalar couplings keep vanishing up to below the Planckscale.

•The “classical” scale invariance is realized at the Planckscale. In particular, the ﬂatland is realized.

•

What is the origin of the EW scale?

Summary•

Gauge hierarchy problem is criticality problem.

•

Asymptotically safe gravity is one of strong candidatesof quantum gravity.

•Graviton ﬂuctuation could induce large anomalousdimension.

•

Gauge hierarchy problem could be solved by theresurgence mechanism.

•We need new physics.

Prospects•

Dependence of critical exponent of scalar mass on
number of particles.

Working with J. M. Pawlowski, M. Reichert, C. Wetterich•Higgs portal dark matter

Working with A. Eichhorn, Y. Hamada, J. Lumma•

Neutrino mass (Majorana mass)
Working with A. Eichhorn, A. Held. S. Lippold, Y. Hamada•

Strong CP problem?

•Weak gravity conjecture•

Relation with string theory

Working with T. Denz, A. Eichhorn, S. Lippold,J. M. Pawlowski, M. Reichert, F. Versteegen