Towards solving hierarchy problem with asymptotically safe gravity - - PowerPoint PPT Presentation

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Towards solving hierarchy problem with asymptotically safe gravity

Masatoshi Yamada

(Kanazawa Univ. → Kyoto Univ. → Heidelberg Univ.)

with Kin-ya Oda (Osaka Univ.) and Yuta Hamada (KEK & Wisconsin Univ.)

ERG2016@Trieste

SLIDE 2

LHC

• Discovery of Higgs boson with 125 GeV
• The SM well describes the physics up to TeV.

The ATLAS and CMS collaborations, JHEP 08, 045

coupling to Higgs Particle mass

gi = 1 hhimi

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Nothing else…

• No new particles appear yet.
• Diphoton (750 GeV) Olympic was held and finished.
• More than 400 papers entry.
• 17 MeV?? (not LHC)

taken from resonaances

• A. Krasznahorkay et al, Phys. Rev. Lett. 116 (2016) no. 4, 042501
• cf. J. L. Feng, arXiv: 1604.07411, 1608.03591

SLIDE 4

There are mysteries.

• Neutrino mass
• Dark matter
• Baryon number
• Quantum gravity
• Hierarchy problem
• Origin of electroweak scale
• Quantization of charge
• Flavor structure etc…

What can we do at present? How to approach to them?

SLIDE 5

Purpose of study

• Quantum gravity must exist.
• Asymptotically safe gravity is one of

possible candidates.

• Attack to the hierarchy problem.
• Can we establish a new paradigm?
• Symmetry? Principle?

SLIDE 6

Plan

• Revisit Hierarchy problem
• Asymptotically safe gravity
• Higgs-Yukawa model non-minimally coupled

to gravity

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Hierarchy problem

• Renormalized Higgs mass
• O(102 GeV)2 = O(1019 GeV)2 -O(1019 GeV)2

✖ ✖ = + +・・・

λ

Z Λ d4p 1 p2 ∼ Λ2

m2

Λ

m2

R = m2 Λ + Λ2

16π2 (λ + · · · )

m2

R ⌧ m2 Λ

SLIDE 8

In viewpoint of Wilson RG

Γk = Z d4x 1 2(∂µφ)2 − m2

k

2 φ2 − λk 4 φ4

• λk

m2

k

k2 m2

k < 0

m2

k > 0

m2

k

k2 = − C 8π2 λk

SLIDE 9

In viewpoint of Wilson RG

λk m2

k

k2

m2

R = m2 Λ + CΛ2

8π2 λk=0

m2

R

(m2

Λ, λΛ)

λΛ ' λk=0

m2

k

k2 = − C 8π2 λk

SLIDE 10

In viewpoint of Wilson RG

λk m2

k

k2

m2

R = m2 Λ + CΛ2

8π2 λk=0

(m2

Λ, λΛ)

λΛ ' λk=0 = 0

m2

k

k2 = − C 8π2 λk

m2

Λ = −CΛ2

8π2 λΛ

m2

R

SLIDE 11

In viewpoint of Wilson RG

• Λ

2 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 the

phase boundary.

λk m2

k

k2 m2

k < 0

m2

k > 0

m2

k

k2 = − C 8π2 λk

SLIDE 12

In viewpoint of Wilson RG

Hierarchy problem = Criticality problem Why is the Higgs close to critical?

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Comment

• Λ

2

is spurious?

• The position of phase boundary is physically meaningless.
• The distance between the flow and the boundary is physically meaningful.
• In perturbation theory, Λ

2

is always subtracted by the counter term or dimensional regularization.

• Rotation of coordinate. ➡
• Hierarchy problem ⇄ The bare theory of Higgs is almost scale (conformally) invariant.
• If mΛ=0, mR=0 is realized.
• Idea of classical scale (conformal) invariance
• How to generate the scale related to vEW?
• Dimensional transmutation or Dynamical symmetry breaking with TeV scale.

C = 0

µdm2 dµ = m2 16π2 (12λ + · · · )

• cf. RG eq. of m in perturbation

λk

m2

k

k2 m2

k

k2 = 0

λ0

k

m2

k

k2

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

m2

k

k2 = − C 8π2 λk

m2

k

k2 = − C 8π2 λk

m2

k

k2 = 0

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Summary so far

• Hierarchy problem is criticality problem.
• Higgs have to be close to the phase

boundary.

• Λ2 is physically meaningful or not.
• Classical scale (conformal) invariance?
• Gravitational effect?

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Plan

• Revisit Hierarchy problem
• Asymptotically safe gravity
• Higgs-Yukawa model non-minimally coupled

to gravity

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Asymptotically safe gravity

• Suggested by S. Weinberg
• Existence of UV fixed point
• Continuum limit k→∞.
• UV critical surface (UV complete theory) is defined

by relevant operators.

• Its dimension = number of free parameters.
• Generalization of asymptotically free
• S. Weinberg, Chap 16 in General Relativity

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Functional renormalization group

k∂kΓk = 1 2Str[(Γ(2)

k

+ Rk)−1k∂kRk] g1 g2 gi Γk = Z d4x[g1O1 + g2O2 + · · · + giOi + · · · ] Γk ' Z d4x[g1O1 + g2O2]

S = ΓΛ

Γ = Γk=0

exact flow truncated flow

skippable!

釈迦に説法 (preaching to the experts)

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Critical exponent

• Classification of flow around FP
• RG eq. around FP g*
• Solution of RG eq.

gi(k) = g∗

i + N

X

j

ζi

j

✓Λ k ◆θj

θi > 0

θi < 0

relevant irrelevant

eigenvalue k → 0

∂t = −k∂k

∂tgi = βi(g∗) + ∂βi ∂gj

• g=g∗(gj − g∗

j ) + · · ·

SLIDE 19

Earlier studies

• Pure gravity
• Truncation:
• Number of relevant operators: 3
• With matters
• stability of FP
• scalar-gravity system
• Higgs-Yukawa system
• gauge field system
• Fermionic system
• Prediction of Higgs mass

f(R), αR + βR2 + γRµνRµν, etc.

Cf.

• O. Lauscher, M. Reuter, Phys. Rev. D66, 025016
• K. Falls, et. al., arXiv: 1301.4191
• D. Benedeti, et. al,. Mod. Phys. Lett. A24, 2233

Cf.

• R. Percacci, D. Perini, Phys. Rev. D67, 081503
• P. Dona, et. al., Phys. Rev. D89, 084035
• J. Meibohm, et. al., Phys.Rev. D93, 084035
• R. Percacci, D. Perini, Phys. Rev. D68, 044018
• G. Narain, R. Percacci, CQG 27, 075001
• 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
• U. Harst, M. Reuter, JHEP 05, 119
• A. Eichhorn, H. Gies, New Phys. J., 113, 125012
• A. Eichhorn, Phys. Rev. D86, 105021

etc.

• M. Shaposhinikov, C. Wetterich, Phys. Lett. B683, 196

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Hierarchy problem for Λcc

• Why is Λcc small?

Why is the universe critical?

Taken from Wiki M, Reuter, F. Saueressing, Phys. Rev. D65, 065016

Λcc << 10-120

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Plan

• Revisit Hierarchy problem
• Asymptotically safe gravity
• Higgs-Yukawa model non-minimally coupled

to gravity

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Higgs-Yukawa model

• Effective action
• Potentials
• Toy model of Higgs-inflation (mentioned in latter)

V (φ2) = Λcc + m2φ2 + λφ4 + · · · F(φ2) = M 2

pl + ξφ2 + · · ·

Γk = Z d4xpg 1 2gµν∂µφ∂νφ + V (φ2) F(φ2)R + ¯ ψ / rψ + yφ ¯ ψψ

• + Sgf + Sgh

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Set-up

• Background field method
• de-Sitter metric is used.
• de-Donder (Landau) gauge
• Cutoff function: Litim cutoff;

gµν = ¯ gµν + hµν Rk(z) = (k2 − z)θ(k2 − z) Rk(z − R/4) = (k2 − (R/4))θ(k2 − (z − /R/4))

• P. Dona, R. Percacci, Phys. Rev. D87, 045002

scalar and gravity fermion

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Without fermion

• Scalar-gravity system
• 5 dimensional theory space

{Mpl2, Λcc, m2, ξ, λ}

• Gaussian-matter FP:
• Critical exponents:

M 2

pl, Λcc

m2, ξ

λ

2.143 ± 2.879i

0.143 ± 2.879i

−2.627

θi =

¯ M 2

pl ∗ = 2.38 × 10−2

¯ Λ∗

cc = 8.82 × 10−3

• R. Percacci, D. Perini, Phys. Rev. D68, 044018
• G. Narain, R. Percacci, CQG 27, 075001

¯ m2∗ = ¯ ξ∗ = ¯ λ∗ = 0

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With a fermion

• Higgs-Yukawa system
• 6 dimensional theory space

{Mpl2, Λcc, m2, ξ, λ, y}

• Gaussian-matter FP:
• Critical exponents:

M 2

pl, Λcc

m2, ξ

λ θi =

−0.4909 ± 2.461i 1.509 ± 2.4615i −2.6069 −1.464

y

¯ Λ∗

cc = 3.72 × 10−3

¯ M 2

pl ∗ = 1.63 × 10−2

Without non-minimal coupling:

• R. Percacci et. al, Phys. Lett. B689, 90
• G. P. Vacca, O. Zanusso, Phys. Rev. Lett. 105, 231601
• K. Oda, M. Y., CQG 33, 125011

¯ m2∗ = ¯ ξ∗ = ¯ λ∗ = ¯ y∗ = 0

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Result

• Fermionic effect makes m2 and ξ irrelevant.
• m2, ξ are not free parameters.
• Mpl2, Λcc determine low energy physics.
• m2, ξ, λ are generated.

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Is criticality of m2 solved?

SLIDE 28

Is criticality of m2 solved?

No

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Flow of scalar mass

• RG eq.
• Once m

2 is generated, m 2 grows up due to the canonical scaling

(2m

2).

• Fine-tuning of Mpl

2, Λcc is still needed.

∂t ¯ m2 = 2 ¯ m2 − 1 48π2  9¯ Λcc

• 1 + 2¯

ξ

• 2

¯ Mpl − ¯ Λcc 2 − 9

• 2¯

Λcc − ¯ Mpl 1 + 2¯ ξ 2 2 (1 + 2 ¯ m2) ¯ Mpl − ¯ Λcc 2 − 9

• 1 + 2¯

ξ 2 2 (1 + 2 ¯ m2)2 ¯ Mpl − ¯ Λcc − 18λ (1 + 2 ¯ m2)2

• + ∂t ¯

Mpl − 2 ¯ Mpl 96π2 ¯ Mpl  − 2¯ ξ ¯ Mpl + 3 ¯ Mpl

• 1 + 2¯

ξ

• 2

¯ Mpl − ¯ Λcc 2 − 3 ¯ Mpl

• 1 + 2¯

ξ 2 2 (1 + 2 ¯ m2) ¯ Mpl − ¯ Λcc 2

• +

1 96π2 ∂t ¯ ξ ¯ Mpl  2 − 3 ¯ Mpl ¯ Mpl − ¯ Λcc + 6 ¯ Mpl

• 1 + 2¯

ξ

• (1 + 2 ¯

m2) ¯ Mpl − ¯ Λcc

• − y2

8π2 ,

∂t = −k∂k

m2

k → ∞, m2 → 0

k ∼ Mpl

m2 ∼ finite

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Higgs mass

Gravitational couplings

{Mpl, Λcc} m2

• Higgs is controlled by the gravitational effect.
• But, why are they located at the critical place?

Criticality of Higgs mass ⇄ Criticality of the universe

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Comment on irrelevant ξ

• ξφ2R becomes also irrelevant.
• ξ plays crucial role in Higgs-inflation.

SLIDE 32

Brief review on Higgs-inflation

• Higgs-inflation can explain the Planck obs.
• Action (Jordan frame)
• Conformal transformation(Jordan ➡ Einstein)

SJ = Z d4x√−g  1 + ξ h2 M 2

pl

+ · · · ! M 2

pl

2 R + 1 2(∂µh)2 − V

• h2

1 + ξ h2 M 2

pl

+ · · · ! gµν → gE

µν

V

• h2

→ V

• h2

⇣ 1 + ξ h2

M 2

pl + · · ·

⌘2

• F. Bezrukov, M. Shaposhinikov, Phys. Lett. B659, 703

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Brief review on Higgs-inflation

• To explain the experimental data, need

largeξ

• ξ～105 for λ～0.1
• ξ～10 for λ～λ*
• F. Bezrukov, M. Shaposhinikov, Phys. Lett. B659, 703
• Y. Hamada, et. al., Phys. Rev. Lett., 112, 241301
• J. L. Cook, et. al., Phys. Rev. D89, 103525

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Is large ξ possible?

• RG eq. of ξ (canonical dim. =0)
• Large suppression factor ➡ ξ basically is not large.

∂tξ = − 1 576π2  1 + 2 ¯ m2 ¯ M 2

pl − ¯

Λcc 9 + 39 ¯ M 2

pl

¯ M 2

pl − ¯

Λcc + 60 ¯ M 2

pl 2

• ξ0 − ¯

Λcc 2 ! + 3 (3 + 32ξ) ¯ M 2

pl − ¯

Λcc − 6 ¯ M 2

pl (11 + 2ξ)

⇣ ¯ M 2

pl − ¯

Λcc ⌘2 − 60 ¯ M 2

pl 2 (1 + 2ξ)

⇣ ¯ M 2

pl − ¯

Λcc ⌘3 + 216ξ (1 + 2ξ)2 (1 + 2 ¯ m2)3 ⇣ ¯ M 2

pl − ¯

Λcc ⌘ + 9 h ¯ Λcc (5 − 2ξ) − 2 ¯ M 2

pl (1 + 2ξ)

i (1 + 2ξ) (1 + 2 ¯ m2)

• ξ0 − ¯

Λcc 2 + 27 (1 + 2ξ)

• 1 − 10ξ − 16ξ2

(1 + 2 ¯ m2)2 ⇣ ¯ M 2

pl − ¯

Λcc ⌘ + 108 ¯ M 2

plξ (1 + 2ξ)2

(1 + 2 ¯ m2)2 ⇣ ¯ M 2

pl − ¯

Λcc ⌘2 + 72λ4 (1 + 2 ¯ m2)2 1 + 12ξ + 2 ¯ m2 1 + 2 ¯ m2

• +

∂t ¯ M 2

pl − 2 ¯

M 2

pl

1152π2 ¯ M 2

pl

 1 + 2 ¯ m2 ¯ M 2

pl − ¯

Λcc ✓ 3 + 18 ¯ M 2

pl

¯ M 2

pl − ¯

Λcc + 20 ¯ M 2

pl 2

⇣ ¯ M 2

pl − ¯

Λ2

¯ mcc

⌘2 ◆ + 15ξ ¯ M 2

pl

− 6 (1 + ξ) ¯ M 2

pl − ¯

Λcc − 10 ¯ M 2

pl (3 + 4ξ)

⇣ ¯ M 2

pl − ¯

Λcc ⌘2 − 20 ¯ M 2

pl 2 (1 + 2ξ)

⇣ ¯ M 2

pl − ¯

Λcc ⌘3 − 3 h ¯ Λcc − ¯ M 2

pl (5 − 4ξ)

i (1 + 2ξ) (1 + 2 ¯ m2) ⇣ ¯ M 2

pl − ¯

Λcc ⌘2 + 36 ¯ M 2

plξ (1 + 2ξ)2

(1 + 2 ¯ m2)2 ⇣ ¯ M 2

pl − ¯

Λcc ⌘2

• +

∂tξ 1152π2 ¯ M 2

pl

 − 15 + 54 ¯ M 2

pl

¯ M 2

pl − ¯

Λcc + 20 ¯ M 2

pl 2

⇣ ¯ M 2

pl − ¯

Λcc ⌘2 − 6 ¯ M 2

pl (7 + 2ξ)

(1 + 2 ¯ m2) ⇣ ¯ M 2

pl − ¯

Λcc ⌘ − 144 ¯ M 2

plξ (1 + 2ξ)

(1 + 2 ¯ m2) ⇣ ¯ M 2

pl − ¯

Λcc ⌘

• −

y2 48π2 ,

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Summary

• Hierarchy problem = Criticality problem
• Asymptotically safe gravity is candidate of quantum gravity.
• Higgs-Yukawa model
• m

2, ξ becomes irrelevant.

• Unification of Hierarchy problems
• How to fine-tune the relevant parameters?
• Higgs-inflation

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Future works

• Extension of theory space
• More fermions, gauge fields
• Yukawa coupling…
• It should not be irrelevant because φΨΨ by chiral

symmetry: βy ∝ y

• Cf. A. Eichhorn’s talk and A. Eichhorn, A. Held, J. M. Pawlowski: arXiv:

1604.02041

• Higgs-Yukawa with higher derivative gravity

(in progress with Y. Hamada)

_

SLIDE 37

Future aims

• Why do m

2 and Λcc prefer the critical?

• Do we have a relationship that both m

2 and Λcc become

small?

• Can we guarantee it?
• What is the relationship with physical values in low

energy.

• If we believe the classical scale invariance scenario,
• can we guarantee it within asymptotically safe gravity?

SLIDE 38

Thank you for your attention! ご清聴ありがとうございました!

SLIDE 39

Appendix

SLIDE 40

17 MeV

SLIDE 41

17 MeV

SLIDE 42

Diphoton

SLIDE 43

Gauge-fixing and ghost actions

Sgf = 1 α Z d4x√¯ gF

• φ2

¯ gµνΣµΣν

Sgh = Z d4x√¯ g ¯ Cµ  − δρ

µ ¯

D2 − ✓ 1 − 1 + β 2 ◆ ¯ Dµ ¯ Dρ + ¯ Rρ

µ

• Cρ

Σµ = ¯ Dνhνµ − β + 1 4 ¯ Dµh

SLIDE 44

Why m and ξ become irrelevant?

• Effect of fermionic fluctuation
• Matrix

¯ M 2

pl ∗ = 2.38 × 10−2

¯ Λ∗

cc = 8.82 × 10−3

¯ Λ∗

cc = 3.72 × 10−3

¯ M 2

pl ∗ = 1.63 × 10−2

∂βi ∂gj '  

∂βξ ∂ξ ∂βξ ∂m2 ∂βm2 ∂ξ ∂βm2 ∂m2

 

✓2.85544 −6.51993 2.40051 −2.57031 ◆ ✓ 1.6814 −5.39674 1.99718 −2.66334 ◆

θi ' 1 2 ✓∂βξ ∂ξ + ∂βm2 ∂m2 ◆

θi ' 2.855442.57031 2 > 0

θi ' 1.68142.66334 2 < 0