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
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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,
|λ |2 h i ∆M = − top Λ2 + ··· H 8π2 UV
Wide range of elementary particles’ masses,
m m e ≈ 10−6, H ≈ 10−17 mτ mP SM :
Effective field theory ⇒ Limited validity at high UV (quantum triviality problem)
No widely accepted implementation of quantum gravity
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Statement of our Idea
"Another" approach to the hierarchy problem
Potential Expansion :
Break U(1)-symmetry to a Zn-symmetry
Goldstone boson → pseudo-Goldstone boson w w mT w/ non-vanishing mass 1 depending on n mL Z
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
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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 = √ (φ1 + iφ2) 2 α α α Projecting {φ1, φ2} onto the unit vectors e : ψ ≡ ei φi
Z3 Z6 Z12 Invariants constructed as power sum symmetric polynomials
X α k Pk = (ψ ) α
We choose the linear combination,
∗ ∗n n n+1 n ρ = φ φ , σn = (φ + φ ) + (−1) 2 ρ 2
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∗ ∗n n n+1 n ρ = φ φ , σn = (φ + φ ) + (−1) 2 ρ 2
Expanding the potential, λ n 4 2 λ2 = 0 SSB regime UZ [ρ, σn; x] = λ (ρ − κ) + (ρ − κ) + gnσn + ··· 2 2 κ = 0 SYM regime
Z3 Z6 Z12 Zn→∞ ∼ U(1)
W/ Longitudinal and Transverse masses {in SSB}
2 2 2 n −1 mL = 2λ4κ , mT = n κ 2 gn
κ : needs fine-tuning, λ4 : runs logarithmically, gn : (n > 4) faster than logarithmically
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Microscopic (UV) Theory −→ Macroscopic (IR) Theory : The Functional Renormalization Group
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The Functional Renormalization Group
Wilson’s Idea : Integrate out modes along infinitesimal momentum shells.
Full Effective action Γ[φ] ⇒ Effective Average Action Γk [φ] | {z } | {z } (Legendre transform of exp{Z[J}]) (Obtained by adding a regulator term),
Adding to the action, 1 Z ∆Sk [φ] = φ(−q)Rk (q)φ(q) 2 q Where,
lim Rk (q) > 0 q2/k 2 →0
lim Rk (q) = 0 k 2/q2 →0
lim Rk (q) → ∞ k2→Λ→∞
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The Functional Renormalization Group
k Given, t = ln Λ , ∂t = −k∂k
The Wetterich (flow) Eq : 1 (2) −1 ∂t Γ = STr Γ + R (∂t R ) k 2 k 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 Rk . We use the Litim regulator,
2 2 2 2 Rk ≈ Zφ,k (k − q )Θ(k − q )
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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,
U(1) Zn SPYS = Sφ + Sφ + Sψ + Sψφ,
Giving the effective average action,
Z Z h i φ,k 2 2 Zn / Γk = (∂µφ1) + (∂µφ2) + Uk + Zψ,k ψj ∂ψj + hk ψj (φ1 + iγ5φ2) ψj x 2
Where, j ∈ {1, NF }
With the potential expansion,
λ λ Zn 4 2 6 3 U [ρ, σn; x] = λ (ρ − κ) + (ρ − κ) + (ρ − κ) + gnσn k 2 2 3!
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Generated Masses
Theory’s masses :
Eigen-values of Hessian of Γk ,
m2 = 2λ κ L 4 Z3
2 2 n/2−1 mT = n κ gn
2 2 mψ = h κ
Z∞
For numerical analysis, the choices of {d = 4 & n = 6} are made !
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Flow Equations
1 ∂t u = −du + d − 2 + ηφ 2ρu ;ρ + nσnu ;σ k k 2 k k " # 4Ω η 1 η 1 η 1 + d 1 − φ + 1 − φ − d 1 − ψ 2 2 γ 2 d d + 2 1 + mT d + 2 1 + mL d + 1 1 + mψ
β-functions :
! ! m m ∂t ∂ uk = ∂t λ2m, ∂t ∂ uk = ∂t gm ρ˜ ρ˜=κ σ˜n ρ˜=κ σ˜n=0 σ˜n=0
The Yukawa coupling & Anomalous dimensions :
h2 h2 1 η = , η = φ SYM 2 ψ SYM 2 2 8π 16π (1 + λ2) h2 1 1 ∂ h2 = η + 2η h2 + + t SYM φ ψ 2 2 48π (1 + λ2) (1 + λ2)
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The Running Couplings
d = 4, dγ = 4, n = 6, tUV = 15.
Ignoring ηϕ, ηψ in quantum corrections
Inspiration from the SM values
vev ≈ 246, mt ≈ 173, mH ≈ 125 GeV
Initial conditions estimates
2 λ2 ≈ 0.0055, λ4 ≈ 0.087, h ≈ 0.48
2 0 2 0 −2 λ2 ∝ k , λ4 ∝ k , h ∝ k , g6 ∝ k
⇒ Fine-tune λ2 && choose g6 ≈ 0.10
tc ≈ 7.1 of SSB
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The Running Couplings
d = 4, dγ = 4, n = 6, tUV = 15.
Ignoring ηϕ, ηψ in quantum corrections
Inspiration from the SM values
vev ≈ 246, mt ≈ 173, mH ≈ 125 GeV
Initial conditions estimates
2 λ2 ≈ 0.0055, λ4 ≈ 0.087, h ≈ 0.48
2 0 2 0 −2 λ2 ∝ k , λ4 ∝ k , h ∝ k , g6 ∝ k
⇒ Fine-tune λ2 && choose g6 ≈ 0.10
tc ≈ 7.1 of SSB
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The Running Couplings
λ4 :
Dashed : Global U(1) (g6 = 0)
λ4 runs logarithmically to IR
Solid : Z6 symmetry (finite g6) 2 λ4 freeze out below scales ∼ κ g6 Goldstone mode gains mass
g6 : Dies off towards IR.
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Generated Mass Hierarchy
At t IR = −20,
−2 vev ≈ 241 GeV, mH ≈ 127 GeV, mT ≈ 10 GeV, mt ≈ 170 GeV
With a hierarchy,
m H ≈ 0.75 ∼ 1 mt m T ≈ 7.9 · 10−5 1 mH
Hence we have,
2 Flow to IR O(h ) ∼ O (λ4) ∼ O (g6) ======⇒ O(mt ) ∼ O(mH ) O(mT ) 0.5 0.1 0.1
14 / 24 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
Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References
Asymptotic Safety Inclusion
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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
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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.
2 3 βλ = b0 + b1 + gfλ(g, Λ) λ + b2λ + b3λ + ···
Sign of b1 + gfλ(g, Λ) determine if FP is UV/IR attractive.
2 If b0 = 0. Non trivial solution : Must have a loop diagram ∝ λ or higher
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Asymptotic Safety Inclusion
Current Model
- For n > 4 : No one-loop diagram ⇒ No non-trivial UV FP - For n = 3 or 4 :
Possible Non-trivial FP ⇒ No mass hierarchy ! gn marginal, runs logarithmically to IR Extension : a complex field χ transforming under an independent U(1)
Include a Z3-symmetric χ-flavored term.
Microscopic action becomes
g U(1) U(1) Z6 Z3 SPYS = Sϕ + Sχ + Sϕ + Sχ + Sϕχ + Sψ + Sψϕ, Ansatz for the effective action Z n ∗ ∗ Z3,6 ∗ ∗ Γk = Zφ,k ∂µφ ∂µφ + Zχ,k ∂µχ ∂µχ + Uk [φ, φ , χ, χ ; x] x ∗ o +iZψ,k ψ∂ψ/ + hk ψ [(1 − γ5)φ − (1 + γ5)φ ] ψ
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Implementation
Extend to 2 complex fields,
1 1 φ = √ (φ1 + iφ2) , χ = √ (χ1 + iχ2) 2 2 W/ charges s.t. 2πi 8πi φ → e 6 φ , χ → e 6 χ Obtaining the invariants :
∗ ) ρφ = φ φ ∗ U(1) - symmetric ρχ = χ χ
∗3 3 τ = χ + χ Z3 - symmetric
∗6 6 σ = φ + φ Z6 - symmetric ∗2 ∗ 2 ζ1 = φ χ + φ χ ∗4 4 ∗ ζ2 = φ χ + φ χ i.a terms ∗2 ∗4 2 4 ζ3 = φ χ + φ χ ∗2 2 2 ∗2 ζ4 = φ χ + φ χ
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Implementation
A closer look at the term φ2χ + φ∗2χ∗
2 2 Featuring : φ1χ2 , φ1χ2 , φ1φ2χ1
Charges running through loop arising from the Z3,6 symmetries.
SYM : ! 3 - Charges = 0 ⇒ No λ term under Z3,6
- Including a scale-dependent ηφ/χ can induce a λ3 term via :
SSB : Acquired vev in φ sector - φ∗/φ , φ/φ i.a. can arise ⇒ Possibility to generate λ3 term.
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Selected β-Functions : SYM
g2 h2 η = 2,1 + ϕ 2 2 2 2 4π λ0,2 + 1 λ2,0 + 1 8π 9g2 g2 η = 0,3 + 2,1 χ 2 4 2 4 8π λ0,2 + 1 8π λ2,0 + 1
ηχ g2,1λ2,2 βg = g2,1 fλ + ηϕ + − 1 + √ + O g2,1 2,1 2 2 2 8 2π λ0,2 + 1 λ2,0 + 1
! 3 1 g 4g g 6g λ β = g f + η − 1 + − 2,3a + 2,1 2,2 + 0,3 0,4 g 0,3 0,3 λ χ 2 2 3 3 2 16π λ2,0 + 1 λ2,0 + 1 λ0,2 + 1
3 Indeed we see both ηφ & ηχ supplying βg2,1 with g2,1 contributions
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Fixed Point Solutions
FP 1 initial conditions :
g0,3 → 0.15, g2,1 → 0.30, h → 0.42, fλ → 0.99, fh → −0.0025
FP values Coupling Value Coupling Value
λ2,0 0.00657 λ0,2 0.00774
g0,3 0.147 g2,1 0.331
λ4,0 −0.00505 λ0,4 −0.0108
λ2,2 −0.0145 g2,2 −0.00347
g2,3 0.000358 g2,3a 0.000237
g4,1 0.0000404 g4,1a 0.000169
λ6,0 −0.00249 λ0,6 −0.00812
λ4,2 −0.0103 λ2,4 −0.0147
g6,0 −0.0000000707 g0,6 −0.000000468
g4,2 −0.00000157 g4,2a −0.00165
g2,4 −0.00000225 g2,4a −0.00230 h 0.280
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Fixed Point Solutions
FP 2 initial conditions :
g0,3 → 1.0, g2,1 → 1.3, h → 0.42, fλ → 0.99, fh → −0.0060
FP values Coupling Value Coupling Value
λ2,0 0.0184 λ0,2 0.0414
g0,3 0.362 g2,1 0.544
λ4,0 −0.0549 λ0,4 −0.486
λ2,2 −0.280 g2,2 −0.0573
g2,3 0.0793 g2,3a 0.0538
g4,1 0.00396 g4,1a 0.0194
λ6,0 −0.103 λ0,6 −3.40
λ4,2 −0.747 λ2,4 −2.19
g6,0 −0.0000479 g0,6 −0.00405
g4,2 −0.00232 g4,2a −0.106
g2,4 −0.00706 g2,4a −0.284 h 0.429
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Fixed Point Solutions
As for the Stability Matrices
FP 1
Sorted -ve Eigenvalues = 1.02, 1.00, −0.000759, −0.00444, −0.00783, −0.917, − 0.948, −0.968, −1.00, −1.92, −1.95, −1.98, −2.00, − 2.84, −2.88, −2.90, −2.93, −2.95, −3.00, −3.00, − 3.00, −3.01, −3.01
FP 2
Sorted -ve Eigenvalues = 1.11, 1.01, 0.0572, 0.0189, −0.00637, −0.592, − 0.679, −0.906, −1.02, −1.65, −1.80, −1.95, − 1.99, −2.34, −2.56, −2.67, −2.77, −2.83, − 2.96, −2.99, −3.00, −3.01, −3.02
23 / 24 Extended scalar sector for asymptotic safety inclusion.
Solved for an interactive UV fixed point.
FP values are very sensitive to fh initial condition
FP values signify an unstable potential
In SYM : all ∝ λ3 vertices arise from anomalous dimensions contributions
Next step : Do calculation in SSB to see if problem persists
In SSB : Fermion sector would couple to χ field as well !
Worth noting : Set discrete symmetry precludes gauge fields inclusion !
Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References
Discussion
Broken U(1) global symmetry to a Zn-symmetry Established a mass hierarchy in scalar sector in IR
24 / 24 FP values are very sensitive to fh initial condition
FP values signify an unstable potential
In SYM : all ∝ λ3 vertices arise from anomalous dimensions contributions
Next step : Do calculation in SSB to see if problem persists
In SSB : Fermion sector would couple to χ field as well !
Worth noting : Set discrete symmetry precludes gauge fields inclusion !
Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References
Discussion
Broken U(1) global symmetry to a Zn-symmetry Established a mass hierarchy in scalar sector in IR
Extended scalar sector for asymptotic safety inclusion.
Solved for an interactive UV fixed point.
24 / 24 Next step : Do calculation in SSB to see if problem persists
In SSB : Fermion sector would couple to χ field as well !
Worth noting : Set discrete symmetry precludes gauge fields inclusion !
Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References
Discussion
Broken U(1) global symmetry to a Zn-symmetry Established a mass hierarchy in scalar sector in IR
Extended scalar sector for asymptotic safety inclusion.
Solved for an interactive UV fixed point.
FP values are very sensitive to fh initial condition
FP values signify an unstable potential
In SYM : all ∝ λ3 vertices arise from anomalous dimensions contributions
24 / 24 Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References
Discussion
Broken U(1) global symmetry to a Zn-symmetry Established a mass hierarchy in scalar sector in IR
Extended scalar sector for asymptotic safety inclusion.
Solved for an interactive UV fixed point.
FP values are very sensitive to fh initial condition
FP values signify an unstable potential
In SYM : all ∝ λ3 vertices arise from anomalous dimensions contributions
Next step : Do calculation in SSB to see if problem persists
In SSB : Fermion sector would couple to χ field as well !
Worth noting : Set discrete symmetry precludes gauge fields inclusion !
24 / 24 Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References
Thanks For Your Attention
24 / 24 Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References
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