Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References Natural Mass Hierarchy in Potts-Yukawa Systems & Its Implementations in Asymptotic Safety Passant Ali University of Cologne / University of Bonn 07.07.2020 1 / 24
Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References Outline Motivation Statement of our Idea Introducing the Potts-Yukawa System (PYS) Inclusion of Gravitational Effects Discussion 2 / 24
Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References Motivation 2 / 24
Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References Standard Model Limitations Hierarchy problems Loop corrections to the higgs mass e.g, ∆ M H = − | λ top | 2 � � Λ 2 UV + · · · 8 π 2 Wide range of elementary particles’ masses, m e m H ≈ 10 − 6 , ≈ 10 − 17 m τ m P SM : Effective field theory ⇒ Limited validity at high UV (quantum triviality problem) No widely accepted implementation of quantum gravity 3 / 24
Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References Statement of our Idea "Another" approach to the hierarchy problem Potential Expansion : Break U ( 1 ) -symmetry to a Z n -symmetry Goldstone boson → pseudo-Goldstone boson � � � � � m T � w/ non-vanishing mass m L ≪ 1 depending on Z n � � Using FRG : flow from a UV-cutoff to IR. Why FRG? - Inclusion of canonically irrelevant couplings - Dynamical generation of masses in the flow Is it possible to Extend this mechanism to arbitrarily high scales? ⇒ Modify theory & search for an interactive, UV fixed point 4 / 24
Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References The Potts-Yukawa System (PYS) 4 / 24
Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References Scalar Invariants 1 We start w/ a complex field φ = 2 ( φ 1 + i φ 2 ) √ Projecting { φ 1 , φ 2 } onto the unit vectors e α : ψ α ≡ e α i φ i Z 6 Z 12 Z 3 Invariants constructed as power sum symmetric polynomials � ( ψ α ) k P k = α We choose the linear combination, n σ n = ( φ ∗ n + φ n ) + ( − 1 ) n + 1 2 ρ ρ = φ ∗ φ , 2 5 / 24
Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References σ n = ( φ ∗ n + φ n ) + ( − 1 ) n + 1 2 ρ n ρ = φ ∗ φ , 2 Expanding the potential, � U Z n [ ρ, σ n ; x ] = λ 2 ( ρ − κ ) + λ 4 λ 2 = 0 SSB regime 2 ( ρ − κ ) 2 + g n σ n + · · · κ = 0 SYM regime Z n →∞ ∼ U ( 1 ) Z 3 Z 6 Z 12 W/ Longitudinal and Transverse masses {in SSB} n m 2 m 2 T = n 2 κ 2 − 1 g n L = 2 λ 4 κ , κ : needs fine-tuning, λ 4 : runs logarithmically, g n : ( n > 4 ) faster than logarithmically 6 / 24
Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References Microscopic (UV) Theory − → Macroscopic (IR) Theory : The Functional Renormalization Group 6 / 24
Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References The Functional Renormalization Group Wilson’s Idea : Integrate out modes along infinitesimal momentum shells. Full Effective action Γ[ φ ] ⇒ Effective Average Action Γ k [ φ ] � �� � � �� � (Legendre transform of exp {Z [ J } ]) (Obtained by adding a regulator term), Adding to the action, ∆ S k [ φ ] = 1 � φ ( − q ) R k ( q ) φ ( q ) 2 q Where, k 2 → 0 R k ( q ) > 0 lim q 2 / q 2 → 0 R k ( q ) = 0 lim k 2 / lim R k ( q ) → ∞ k 2 → Λ →∞ 7 / 24
Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References The Functional Renormalization Group � � Given, t = ln k , ∂ t = − k ∂ k Λ The Wetterich (flow) Eq : �� � ∂ t Γ k = 1 � − 1 Γ ( 2 ) 2 STr + R k ( ∂ t R k ) k Solution : trajectory in Theory space. End-points are : The bare action (UV-limit) The full effective action (IR-limit) . Precise trajectory depends on R k . We use the Litim regulator, R k ≈ Z φ, k ( k 2 − q 2 )Θ( k 2 − q 2 ) 8 / 24
Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References PYS Flow 8 / 24
Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References The Potts-Yukawa System Flow Our microscopic action becomes, S PYS = S U ( 1 ) + S Z n φ + S ψ + S ψφ , φ Giving the effective average action, � Z φ, k � � � ( ∂ µ φ 1 ) 2 + ( ∂ µ φ 2 ) 2 � + U Z n Γ k = + Z ψ, k ψ j / ∂ψ j + h k ψ j ( φ 1 + i γ 5 φ 2 ) ψ j 2 k x Where, j ∈ { 1 , N F } With the potential expansion, k [ ρ, σ n ; x ] = λ 2 ( ρ − κ ) + λ 4 2 ( ρ − κ ) 2 + λ 6 3 ! ( ρ − κ ) 3 + g n σ n U Z n 9 / 24
Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References Generated Masses Theory’s masses : Eigen-values of Hessian of Γ k , m 2 L = 2 λ 4 κ Z 3 m 2 T = n 2 κ n / 2 − 1 g n m 2 ψ = h 2 κ Z ∞ For numerical analysis, the choices of { d = 4 & n = 6 } are made! 10 / 24
Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References Flow Equations ∂ t u k = − du k + 1 � � � � d − 2 + η φ 2 ρ u k ; ρ + n σ n u k ; σ 2 �� � � � � � � + 4 Ω d 1 1 1 η φ η φ η ψ 1 − 1 − 1 − + − d γ d + 2 d + 2 d + 1 1 + m 2 1 + m 2 1 + m 2 d T L ψ β -functions : � � � � � � � � ∂ m ∂ m ∂ t ρ u k = ∂ t λ 2 m , ∂ t σ n u k = ∂ t g m � � ˜ � ˜ ρ = κ ˜ � ˜ ρ = κ σ n = 0 σ n = 0 ˜ ˜ The Yukawa coupling & Anomalous dimensions : SYM = h 2 h 2 1 � � η φ , η ψ SYM = � � 8 π 2 16 π 2 ( 1 + λ 2 ) 2 h 2 � � 1 1 ∂ t h 2 � � � h 2 + η φ + 2 η ψ SYM = ( 1 + λ 2 ) 2 + � 48 π 2 ( 1 + λ 2 ) 11 / 24
Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References The Running Couplings d = 4 , d γ = 4 , n = 6 , t UV = 15. Ignoring η ϕ , η ψ in quantum corrections Inspiration from the SM values vev ≈ 246 , m t ≈ 173 , m H ≈ 125 GeV Initial conditions estimates h 2 ≈ 0 . 48 λ 2 ≈ 0 . 0055 , λ 4 ≈ 0 . 087 , λ 2 ∝ k 2 , λ 4 ∝ k 0 , h 2 ∝ k 0 , g 6 ∝ k − 2 ⇒ Fine-tune λ 2 && choose g 6 ≈ 0 . 10 t c ≈ 7 . 1 of SSB 11 / 24
Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References The Running Couplings d = 4 , d γ = 4 , n = 6 , t UV = 15. Ignoring η ϕ , η ψ in quantum corrections Inspiration from the SM values vev ≈ 246 , m t ≈ 173 , m H ≈ 125 GeV Initial conditions estimates h 2 ≈ 0 . 48 λ 2 ≈ 0 . 0055 , λ 4 ≈ 0 . 087 , h 2 ∝ k 0 , λ 2 ∝ k 2 , λ 4 ∝ k 0 , g 6 ∝ k − 2 ⇒ Fine-tune λ 2 && choose g 6 ≈ 0 . 10 t c ≈ 7 . 1 of SSB 12 / 24
Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References The Running Couplings λ 4 : Dashed : Global U ( 1 ) ( g 6 = 0) λ 4 runs logarithmically to IR Solid : Z 6 symmetry (finite g 6 ) λ 4 freeze out below scales ∼ κ 2 g 6 Goldstone mode gains mass g 6 : Dies off towards IR. 13 / 24
Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References Generated Mass Hierarchy At t IR = − 20, m T ≈ 10 − 2 GeV , vev ≈ 241 GeV , m H ≈ 127 GeV , m t ≈ 170 GeV With a hierarchy, m H ≈ 0 . 75 ∼ 1 m t m T ≈ 7 . 9 · 10 − 5 ≪ 1 m H Hence we have, Flow to IR O ( h 2 ) ∼ O ( λ 4 ) ∼ O ( g 6 ) = = = = = = = = ⇒ O ( m t ) ∼ O ( m H ) ≫ O ( m T ) 0 . 5 0 . 1 0 . 1 14 / 24
Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References Asymptotic Safety Inclusion 15 / 24
Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References Asymptotic Safety Inclusion Bypass the UV cutoff : Extend to arbitrarily high scales Combine our mechanism of mass-hierarchy w/ asymptotically-safe gravity In asymptotic safety : We can thus have a non-trivial theory at arbitrarily high energies, w/ guaranteed non-divergence Provide a solid, non-perturbative approach to quantum gravity 15 / 24
Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References Asymptotic Safety Inclusion Require our theory to have UV fixed point (FP) Observables in IR are governed by the FP If FP is interactive & UV-attractive, ⇒ A predictive, asymptotically-safe theory is established. [15] General form of β -functions : w/ g , Λ : gravitational couplings. � � λ + b 2 λ 2 + b 3 λ 3 + · · · β λ = b 0 + b 1 + gf λ ( g , Λ) � � Sign of determine if FP is UV/IR attractive. b 1 + gf λ ( g , Λ) If b 0 = 0. Non trivial solution : Must have a loop diagram ∝ λ 2 or higher 16 / 24
Recommend
More recommend
