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
LHC The ATLAS and CMS collaborations, JHEP 08, 045 coupling to Higgs Particle mass • Discovery of Higgs boson with 125 GeV • The SM well describes the physics up to TeV. 1 g i = h h i m i
Nothing else… taken from resonaances A. Krasznahorkay et al, Phys. Rev. Lett. 116 (2016) no. 4, 042501 cf. J. L. Feng, arXiv: 1604.07411, 1608.03591 • No new particles appear yet. • Diphoton (750 GeV) Olympic was held and finished. • More than 400 papers entry. • 17 MeV?? (not LHC)
There are mysteries. What can we do at present? How to approach to them? • Neutrino mass • Dark matter • Baryon number • Quantum gravity • Hierarchy problem • Origin of electroweak scale • Quantization of charge • Flavor structure etc…
Purpose of study possible candidates. • Quantum gravity must exist. • Asymptotically safe gravity is one of • Attack to the hierarchy problem. • Can we establish a new paradigm? • Symmetry? Principle?
Plan to gravity • Revisit Hierarchy problem • Asymptotically safe gravity • Higgs-Yukawa model non-minimally coupled
Hierarchy problem = +・・・ + Z Λ d 4 p 1 p 2 ∼ Λ 2 • Renormalized Higgs mass m 2 Λ ✖ ✖ λ Λ + Λ 2 m 2 R = m 2 16 π 2 ( λ + · · · ) • O(10 2 GeV) 2 = O(10 19 GeV) 2 -O(10 19 GeV) 2 m 2 R ⌧ m 2 Λ
In viewpoint of Wilson RG 2( ∂ µ φ ) 2 − m 2 1 � Z 2 φ 2 − λ k d 4 x k 4 φ 4 Γ k = m 2 k 2 = − C λ k k 8 π 2 λ k m 2 m 2 k < 0 k > 0 m 2 k k 2 0
In viewpoint of Wilson RG m 2 k 2 = − C k 8 π 2 λ k λ k ( m 2 Λ , λ Λ ) m 2 m 2 R k k 2 0 Λ + C Λ 2 m 2 R = m 2 8 π 2 λ k =0 λ Λ ' λ k =0
In viewpoint of Wilson RG m 2 k 2 = − C k 8 π 2 λ k λ k ( m 2 Λ , λ Λ ) Λ = − C Λ 2 m 2 8 π 2 λ Λ m 2 m 2 k R k 2 0 Λ + C Λ 2 m 2 R = m 2 = 0 8 π 2 λ k =0 λ Λ ' λ k =0
In viewpoint of Wilson RG 2 determines the position of phase boundary (critical line). theory. phase boundary. m 2 k 2 = − C λ k k 8 π 2 λ k m 2 m 2 k < 0 k > 0 m 2 k k 2 0 • Λ • The phase boundary corresponds to the massless (critical) • To obtain small m R , put the bare parameters close to the
In viewpoint of Wilson RG Hierarchy problem = Criticality problem Why is the Higgs close to critical?
Comment 2 is spurious? 2 is always subtracted by the counter term or dimensional regularization. H. Aoki, S, Iso, Phys. Rev. D86, 013001 W. A. Bardeen, FERMILAB-CONF-95-391-T cf. RG eq. of m in perturbation m 2 k 2 = − C k 8 π 2 λ k λ 0 λ k m 2 0 k k k 2 = 0 m 2 0 m 2 k k k 2 k 2 • Λ • The position of phase boundary is physically meaningless. • The distance between the flow and the boundary is physically meaningful. • In perturbation theory, Λ m 2 0 m 2 k 2 = − C C = 0 k k 8 π 2 λ k k 2 = 0 • Rotation of coordinate. ➡ • Hierarchy problem ⇄ The bare theory of Higgs is almost scale (conformally) invariant. • If m Λ =0, m R =0 is realized. µdm 2 dµ = m 2 16 π 2 (12 λ + · · · ) • Idea of classical scale (conformal) invariance • How to generate the scale related to v EW ? • Dimensional transmutation or Dynamical symmetry breaking with TeV scale.
Summary so far boundary. • Hierarchy problem is criticality problem. • Higgs have to be close to the phase • Λ 2 is physically meaningful or not. • Classical scale (conformal) invariance? • Gravitational effect?
Plan to gravity • Revisit Hierarchy problem • Asymptotically safe gravity • Higgs-Yukawa model non-minimally coupled
Asymptotically safe gravity by relevant operators. S. Weinberg, Chap 16 in General Relativity • Suggested by S. Weinberg • Existence of UV fixed point • Continuum limit k→∞. • UV critical surface (UV complete theory) is defined • Its dimension = number of free parameters. • Generalization of asymptotically free
Functional renormalization group exact flow 釈迦に説法 skippable! truncated flow (preaching to the experts) Γ = Γ k =0 g i g 2 S = Γ Λ g 1 k ∂ k Γ k = 1 2Str[( Γ (2) + R k ) − 1 k ∂ k R k ] k Z d 4 x [ g 1 O 1 + g 2 O 2 + · · · + g i O i + · · · ] Γ k = Z d 4 x [ g 1 O 1 + g 2 O 2 ] Γ k '
eigenvalue Critical exponent irrelevant relevant • Classification of flow around FP ∂ t = − k ∂ k • RG eq. around FP g * � ∂ t g i = β i ( g ∗ ) + ∂β i � g = g ∗ ( g j − g ∗ j ) + · · · � ∂ g j � θ i < 0 • Solution of RG eq. k → 0 ◆ θ j N ✓ Λ X ζ i g i ( k ) = g ∗ i + j k j θ i > 0
Earlier studies Cf. etc. A. Eichhorn, Phys. Rev. D86, 105021 U. Harst, M. Reuter, JHEP 05, 119 J. Daum et. al., JHEP 01, 084 G. P. Vacca, O. Zanusso, Rhys. Rev. Lett. 105, 231601 R. Percalli et. al, Phys. Lett. B689, 90 G. Narain, R. Percacci, CQG 27, 075001 R. Percacci, D. Perini, Phys. Rev. D68, 044018 J. Meibohm, et. al., Phys.Rev. D93, 084035 P. Dona, et. al., Phys. Rev. D89, 084035 R. Percacci, D. Perini, Phys. Rev. D67, 081503 D. Benedeti, et. al,. Mod. Phys. Lett. A24, 2233 K. Falls, et. al., arXiv: 1301.4191 O. Lauscher, M. Reuter, Phys. Rev. D66, 025016 Cf. M. Shaposhinikov, C. Wetterich, Phys. Lett. B683, 196 • Pure gravity f ( R ) , α R + β R 2 + γ R µ ν R µ ν , etc. • Truncation: • Number of relevant operators: 3 • With matters • stability of FP • scalar-gravity system • Higgs-Yukawa system • gauge field system A. Eichhorn, H. Gies, New Phys. J., 113, 125012 • Fermionic system • Prediction of Higgs mass
Hierarchy problem for Λcc Why is the universe critical? Taken from Wiki M, Reuter, F. Saueressing, Phys. Rev. D65, 065016 Λcc << 10 -120 • Why is Λcc small?
Plan to gravity • Revisit Hierarchy problem • Asymptotically safe gravity • Higgs-Yukawa model non-minimally coupled
Higgs-Yukawa model • Effective action 1 � Z d 4 x p g 2 g µ ν ∂ µ φ∂ ν φ + V ( φ 2 ) � F ( φ 2 ) R + ¯ r ψ + y φ ¯ Γ k = ψ / ψψ + S gf + S gh • Potentials V ( φ 2 ) = Λ cc + m 2 φ 2 + λφ 4 + · · · pl + ξφ 2 + · · · F ( φ 2 ) = M 2 • Toy model of Higgs-inflation (mentioned in latter)
Set-up P. Dona, R. Percacci, Phys. Rev. D87, 045002 scalar and gravity fermion g µ ν = ¯ g µ ν + h µ ν • Background field method • de-Sitter metric is used. • de-Donder (Landau) gauge • Cutoff function: Litim cutoff; R k ( z ) = ( k 2 − z ) θ ( k 2 − z ) R k ( z − R/ 4) = ( k 2 − ( R/ 4)) θ ( k 2 − ( z − /R/ 4))
Without fermion {M pl2 , Λ cc , m 2 , ξ, λ} G. Narain, R. Percacci, CQG 27, 075001 R. Percacci, D. Perini, Phys. Rev. D68, 044018 • Scalar-gravity system • 5 dimensional theory space ∗ = 2 . 38 × 10 − 2 ¯ M 2 pl ¯ cc = 8 . 82 × 10 − 3 Λ ∗ • Gaussian-matter FP: m 2 ∗ = ¯ ξ ∗ = ¯ λ ∗ = 0 ¯ • Critical exponents: m 2 , ξ M 2 pl , Λ cc λ θ i = − 2 . 627 2 . 143 ± 2 . 879 i 0 . 143 ± 2 . 879 i
With a fermion {M pl2 , Λ cc , m 2 , ξ, λ, y} K. Oda, M. Y. , CQG 33, 125011 G. P. Vacca, O. Zanusso, Phys. Rev. Lett. 105, 231601 R. Percacci et. al, Phys. Lett. B689, 90 Without non-minimal coupling: • Higgs-Yukawa system • 6 dimensional theory space ∗ = 1 . 63 × 10 − 2 ¯ M 2 pl ¯ cc = 3 . 72 × 10 − 3 Λ ∗ • Gaussian-matter FP: m 2 ∗ = ¯ ξ ∗ = ¯ λ ∗ = ¯ y ∗ = 0 ¯ • Critical exponents: m 2 , ξ M 2 λ y pl , Λ cc θ i = 1 . 509 ± 2 . 4615 i − 0 . 4909 ± 2 . 461 i − 2 . 6069 − 1 . 464
Result • Fermionic effect makes m 2 and ξ irrelevant. • m 2 , ξ are not free parameters. • M pl2 , Λ cc determine low energy physics. • m 2 , ξ, λ are generated.
Is criticality of m 2 solved?
Is criticality of m 2 solved? No
Flow of scalar mass 2 is generated, m 2 grows up due to the canonical scaling (2m 2 ). 2 , Λ cc is still needed. • RG eq. 9¯ � 2 � 2 1 + 2¯ 2¯ Λ cc − ¯ 1 + 2¯ 1 + 2¯ � � � � � � Λ cc ξ � 2 − 9 ξ 9 ξ 1 M pl 18 λ � m 2 = 2 ¯ m 2 − ∂ t ¯ m 2 ) 2 � ¯ � ¯ � ¯ � 2 − � − M pl − ¯ m 2 ) 2 48 π 2 M pl − ¯ M pl − ¯ 2 (1 + 2 ¯ Λ cc (1 + 2 ¯ 2 2 (1 + 2 ¯ m 2 ) Λ cc Λ cc � 2 1 + 2¯ 1 + 2¯ + 3 ¯ 3 ¯ + ∂ t ¯ M pl − 2 ¯ − 2¯ � � � ξ ξ � ξ M pl M pl M pl � ¯ � ¯ 96 π 2 ¯ ¯ � 2 − � 2 M pl − ¯ M pl − ¯ M pl M pl 2 2 (1 + 2 ¯ m 2 ) Λ cc Λ cc 6 ¯ 1 + 2¯ ∂ t ¯ 3 ¯ � � − y 2 1 ξ � ξ M pl M pl � ¯ + 2 − + 8 π 2 , ∂ t = − k ∂ k ¯ M pl − ¯ ¯ M pl − ¯ � 96 π 2 Λ cc (1 + 2 ¯ m 2 ) M pl Λ cc m 2 ∼ finite k → ∞ , m 2 → 0 m 2 k ∼ M pl • Once m • Fine-tuning of M pl
Higgs mass Gravitational couplings Criticality of Higgs mass ⇄ Criticality of the universe m 2 { M pl , Λ cc } • Higgs is controlled by the gravitational effect. • But, why are they located at the critical place?
Comment on irrelevant ξ • ξφ 2 R becomes also irrelevant. • ξ plays crucial role in Higgs-inflation.
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