Natural Mass Hierarchy in Potts-Yukawa Systems & Its - - PowerPoint PPT Presentation

▶
natural mass hierarchy in potts yukawa systems
SMART_READER_LITE
LIVE PREVIEW

Natural Mass Hierarchy in Potts-Yukawa Systems & Its - - PowerPoint PPT Presentation

Sep 28, 2022 144 likes •534 views

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

slide-1
SLIDE 1

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

slide-2
SLIDE 2

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

slide-3
SLIDE 3

Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References

Motivation

2 / 24

slide-4
SLIDE 4

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, ∆MH = − |λtop|2 8π2

  • Λ2

UV + · · ·

  • Wide range of elementary particles’ masses,

me mτ ≈ 10−6, mH mP ≈ 10−17 SM : Effective field theory ⇒ Limited validity at high UV (quantum triviality problem) No widely accepted implementation of quantum gravity

3 / 24

slide-5
SLIDE 5

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 Zn-symmetry Goldstone boson → pseudo-Goldstone boson w/ non-vanishing mass mT

mL ≪ 1 depending on Zn

  • 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

slide-6
SLIDE 6

Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References

The Potts-Yukawa System (PYS)

4 / 24

slide-7
SLIDE 7

Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References

Scalar Invariants

We start w/ a complex field φ =

1 √ 2 (φ1 + iφ2)

Projecting {φ1, φ2} onto the unit vectors eα : ψα ≡ eα

i φi

Z3 Z6 Z12 Invariants constructed as power sum symmetric polynomials Pk =

  • α

(ψα)k We choose the linear combination, ρ = φ∗φ , σn = (φ∗n + φn) + (−1)n+1 2 ρ

n 2 5 / 24

slide-8
SLIDE 8

Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References

ρ = φ∗φ , σn = (φ∗n + φn) + (−1)n+1 2 ρ

n 2

Expanding the potential, UZn[ρ, σn; x] = λ2(ρ − κ) + λ4 2 (ρ − κ)2 + gnσn + · · ·

  • λ2 = 0

SSB regime κ = 0 SYM regime

Z3 Z6 Z12 Zn→∞ ∼ U(1) W/ Longitudinal and Transverse masses {in SSB} m2

L = 2λ4κ

, m2

T = n2κ

n 2 −1gn

κ : needs fine-tuning, λ4 : runs logarithmically, gn : (n > 4) faster than logarithmically

6 / 24

slide-9
SLIDE 9

Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References

Microscopic (UV) Theory − → Macroscopic (IR) Theory : The Functional Renormalization Group

6 / 24

slide-10
SLIDE 10

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]

  • (Legendre transform of exp{Z[J }])

⇒ Effective Average Action Γk[φ]

  • (Obtained by adding a regulator term),

Adding to the action, ∆Sk[φ] = 1 2

  • q

φ(−q)Rk(q)φ(q) Where, lim

q2/ k2→0 Rk(q) > 0

lim

k2/ q2→0 Rk(q) = 0

lim

k2→Λ→∞

Rk(q) → ∞

7 / 24

slide-11
SLIDE 11

Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References

The Functional Renormalization Group

Given, t = ln

  • k

Λ

  • ,

∂t = −k∂k The Wetterich (flow) Eq : ∂tΓk = 1 2 STr

  • Γ(2)

k

+ Rk −1 (∂tRk)

  • 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, Rk ≈ Zφ,k(k2 − q2)Θ(k2 − q2)

8 / 24

slide-12
SLIDE 12

Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References

PYS Flow

8 / 24

slide-13
SLIDE 13

Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References

The Potts-Yukawa System Flow

Our microscopic action becomes, SPYS = SU(1)

φ

+ SZn

φ + Sψ + Sψφ,

Giving the effective average action, Γk =

  • x

Zφ,k 2

  • (∂µφ1)2 + (∂µφ2)2

+ UZn

k

+ Zψ,kψj / ∂ψj + hkψj (φ1 + iγ5φ2) ψj

  • Where, j ∈ {1, NF }

With the potential expansion, UZn

k [ρ, σn; x] = λ2(ρ − κ) + λ4

2 (ρ − κ)2 + λ6 3! (ρ − κ)3 + gnσn

9 / 24

slide-14
SLIDE 14

Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References

Generated Masses

Theory’s masses : Eigen-values of Hessian of Γk, m2

L = 2λ4κ

m2

T = n2κ

n/ 2−1gn

m2

ψ = h2κ

Z3 Z∞ For numerical analysis, the choices of {d = 4 & n = 6} are made!

10 / 24

slide-15
SLIDE 15

Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References

Flow Equations

∂tuk = −duk + 1 2

  • d − 2 + ηφ

2ρuk;ρ + nσnuk;σ

  • + 4Ωd

d

  • 1 −

ηφ d + 2

  • 1

1 + m2

T

+

  • 1 −

ηφ d + 2

  • 1

1 + m2

L

− dγ

  • 1 −

ηψ d + 1

  • 1

1 + m2

ψ

  • β-functions :

∂t

  • ∂m

˜ ρ uk

  • ˜

ρ=κ ˜ σn=0

  • = ∂tλ2m,

∂t

  • ∂m

˜ σnuk

  • ˜

ρ=κ ˜ σn=0

  • = ∂tgm

The Yukawa coupling & Anomalous dimensions : ηφ

  • SYM = h2

8π2 , ηψ

  • SYM =

h2 16π2 1 (1 + λ2)2 ∂th2

  • SYM =
  • ηφ + 2ηψ
  • h2 +

h2 48π2

  • 1

(1 + λ2)2 + 1 (1 + λ2)

  • 11 / 24
slide-16
SLIDE 16

Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References

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 λ2 ≈ 0.0055, λ4 ≈ 0.087, h2 ≈ 0.48 λ2 ∝ k2, λ4 ∝ k0, h2 ∝ k0, g6 ∝ k−2 ⇒ Fine-tune λ2 && choose g6 ≈ 0.10 tc ≈ 7.1 of SSB

11 / 24

slide-17
SLIDE 17

Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References

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 λ2 ≈ 0.0055, λ4 ≈ 0.087, h2 ≈ 0.48 λ2 ∝ k2, λ4 ∝ k0, h2 ∝ k0, g6 ∝ k−2 ⇒ Fine-tune λ2 && choose g6 ≈ 0.10 tc ≈ 7.1 of SSB

12 / 24

slide-18
SLIDE 18

Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References

The Running Couplings

λ4 : Dashed : Global U(1) (g6 = 0) λ4 runs logarithmically to IR Solid : Z6 symmetry (finite g6) λ4 freeze out below scales ∼ κ2g6 Goldstone mode gains mass g6 : Dies off towards IR.

13 / 24

slide-19
SLIDE 19

Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References

Generated Mass Hierarchy

At t IR = −20, vev ≈ 241 GeV, mH ≈ 127 GeV, mT ≈ 10−2 GeV, mt ≈ 170 GeV With a hierarchy, mH mt ≈ 0.75 ∼ 1 mT mH ≈ 7.9 · 10−5 ≪ 1 Hence we have, O(h2)

0.5

∼ O (λ4)

0.1

∼ O (g6)

0.1 Flow to IR

= = = = = = = = ⇒ O(mt) ∼ O(mH) ≫ O(mT )

14 / 24

slide-20
SLIDE 20

Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References

Asymptotic Safety Inclusion

15 / 24

slide-21
SLIDE 21

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

slide-22
SLIDE 22

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. βλ = b0 +

  • b1 + gfλ(g, Λ)
  • λ + b2λ2 + b3λ3 + · · ·

Sign of

  • b1 + gfλ(g, Λ)
  • determine if FP is UV/IR attractive.

If b0 = 0. Non trivial solution : Must have a loop diagram ∝ λ2 or higher

16 / 24

slide-23
SLIDE 23

Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References

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 gn marginal, runs logarithmically to IR ⇒ No mass hierarchy! Extension : a complex field χ transforming under an independent U(1) Include a Z3-symmetric χ-flavored term. Microscopic action becomes Sg

PYS = SU(1) ϕ

+ SU(1)

χ

+ SZ6

ϕ + SZ3 χ + Sϕχ + Sψ + Sψϕ,

Ansatz for the effective action Γk =

  • x
  • Zφ,k ∂µφ∗∂µφ + Zχ,k ∂µχ∗∂µχ + U

Z3,6 k

[φ, φ∗, χ, χ∗; x] +iZψ,kψ/ ∂ψ + hkψ [(1 − γ5)φ − (1 + γ5)φ∗] ψ

  • 17 / 24
slide-24
SLIDE 24

Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References

Implementation

Extend to 2 complex fields, φ = 1 √ 2 (φ1 + iφ2) , χ = 1 √ 2 (χ1 + iχ2) W/ charges s.t. φ → e

2πi 6 φ

, χ → e

8πi 6 χ

Obtaining the invariants : ρφ = φ∗φ ρχ = χ∗χ

  • U(1) - symmetric

τ =

  • χ∗3 + χ3

Z3 - symmetric σ =

  • φ∗6 + φ6

Z6 - symmetric ζ1 =

  • φ∗2χ∗ + φ2χ
  • ζ2 =
  • φ∗4χ + φ4χ∗

ζ3 =

  • φ∗2χ∗4 + φ2χ4

ζ4 =

  • φ∗2χ2 + φ2χ∗2

           i.a terms

18 / 24

slide-25
SLIDE 25

Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References

Implementation

A closer look at the term

  • φ2χ + φ∗2χ∗

Featuring :

  • φ2

1χ2 , φ2 1χ2 , φ1φ2χ1

  • Charges running through loop arising from the Z3,6

symmetries. SYM :

  • Charges !

= 0 ⇒ No λ3 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.

19 / 24

slide-26
SLIDE 26

Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References

Selected β-Functions : SYM

ηϕ = g2

2,1

4π2 λ0,2 + 1 2 λ2,0 + 1 2 + h2 8π2 ηχ = 9g2

0,3

8π2 λ0,2 + 1 4 + g2

2,1

8π2 λ2,0 + 1 4 βg2,1 = g2,1

  • fλ + ηϕ + ηχ

2 − 1

  • +

g2,1λ2,2 8 √ 2π2 λ0,2 + 1 2 λ2,0 + 1 + O

  • g2,1
  • βg0,3 = g0,3
  • fλ + 3

2 ηχ − 1

  • +

1 16π2

  • −

g2,3a

  • λ2,0 + 1

2 + 4g2,1g2,2

  • λ2,0 + 1

3 + 6g0,3λ0,4

  • λ0,2 + 1

3

  • Indeed we see both ηφ & ηχ supplying βg2,1 with g3

2,1 contributions

20 / 24

slide-27
SLIDE 27

Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References

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

21 / 24

slide-28
SLIDE 28

Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References

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

22 / 24

slide-29
SLIDE 29

Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References

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
slide-30
SLIDE 30

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

slide-31
SLIDE 31

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

slide-32
SLIDE 32

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

slide-33
SLIDE 33

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

slide-34
SLIDE 34

Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References

Thanks For Your Attention

24 / 24

slide-35
SLIDE 35

Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References

References

[1] Bertrand DELAMOTTE. “An Introduction to the Nonperturbative Renormalization Group”. In : Lecture Notes in Physics (2012), p. 49-132. ISSN : 1616-6361.

DOI : 10.1007/978-3-642-27320-9_2. URL :

http://dx.doi.org/10.1007/978-3-642-27320-9_2. [2] Astrid EICHHORN. An asymptotically safe guide to quantum gravity and

  • matter. 2018. arXiv : 1810.07615 [hep-th].

[3] Astrid EICHHORN et Aaron HELD. “Viability of quantum-gravity induced ultraviolet completions for matter”. In : Physical Review D 96.8 (oct. 2017). ISSN : 2470-0029. DOI : 10.1103/physrevd.96.086025. URL : http://dx.doi.org/10.1103/PhysRevD.96.086025. [4] Astrid EICHHORN, Aaron HELD et Jan M. PAWLOWSKI. “Quantum-gravity effects

  • n a Higgs-Yukawa model”. In : Phys. Rev. D 94 (10 nov. 2016), p. 104027. DOI :

10.1103/PhysRevD.94.104027. URL : https://link.aps.org/doi/10.1103/PhysRevD.94.104027. [5] Astrid EICHHORN et al. “Quantum gravity fluctuations flatten the Planck-scale Higgs potential”. In : Physical Review D 97.8 (avr. 2018). ISSN : 2470-0029.

DOI : 10.1103/physrevd.97.086004. URL :

http://dx.doi.org/10.1103/PhysRevD.97.086004. [6] Lin FEI et al. Yukawa CFTs and Emergent Supersymmetry. 2016. arXiv : 1607.05316 [hep-th].

24 / 24

slide-36
SLIDE 36

Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References

References

[7] Holger GIES. “Introduction to the Functional RG and Applications to Gauge Theories”. In : Lecture Notes in Physics (2012), p. 287-348. ISSN : 1616-6361.

DOI : 10.1007/978-3-642-27320-9_6. URL :

http://dx.doi.org/10.1007/978-3-642-27320-9_6. [8] Holger GIES et Michael M. SCHERER. “Asymptotic safety of simple Yukawa systems”. In : The European Physical Journal C 66.3-4 (fév. 2010), p. 387-402.

ISSN : 1434-6052. DOI : 10.1140/epjc/s10052-010-1256-z. URL :

http://dx.doi.org/10.1140/epjc/s10052-010-1256-z. [9] Clemens GNEITING. “Higgs Mass Bounds from Renormalization Flow”. Mém. de

  • mast. Philosophenweg 16, 69120 Heidelberg, DE : Faculty of Physics et

Astronomy, University of Heidelberg, 2005. [10] Geoffrey R GOLNER. “Investigation of the potts model using renormalization-group techniques”. In : Physical Review B 8.7 (1973), p. 3419. [11] WK74 Wilson KG et JB KOGUT. “The renormalization group and the\({{\epsilon}}\)-expansion”. In : Phys. Rept 12 (1974), p. 75-200. [12]

  • F. LÉONARD, B. DELAMOTTE et N. WSCHEBOR. Naturally light scalar particles :

a generic and simple mechanism. 2018. arXiv : 1802.09418 [hep-ph].

24 / 24

slide-37
SLIDE 37

Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References

References

[13] F LÉONARD, B DELAMOTTE et Nicolás WSCHEBOR. “Naturally light scalar particles : a generic and simple mechanism”. In : arXiv preprint arXiv :1802.09418 (2018). [14] Daniel F . LITIM. “Optimized renormalization group flows”. In : Physical Review D 64.10 (oct. 2001). ISSN : 1089-4918. DOI : 10.1103/physrevd.64.105007. URL : http://dx.doi.org/10.1103/PhysRevD.64.105007. [15] Andreas NINK. UV Critical Surface. Scholarpedia, 8(7) :31015. 2013. URL : http://www.scholarpedia.org/article/File:UVCriticalSurface.png. [16] Stefan RECHENBERGER. “Asymptotic Safety of Yukawa Systems”. Mém. de

  • mast. Max-Wien-Platz 1, 07743 Jena, DE : Physikalisch-Astronomische Fakultät

Friedrich Schiller Universiät Jena, 2010. [17] Michael SCHERER. “Introduction to Renormalization with Applications in Condensed-Matter and High-Energy Physics”. In : (2018). [18] Emilio TORRES et al. “Fermion-induced quantum criticality with two length scales in Dirac systems”. In : Physical Review B 97.12 (mar. 2018). ISSN : 2469-9969.

DOI : 10.1103/physrevb.97.125137. URL :

http://dx.doi.org/10.1103/PhysRevB.97.125137. [19] Christof WETTERICH. “Exact evolution equation for the effective potential”. In : Physics Letters B 301.1 (1993), p. 90-94.

24 / 24

slide-38
SLIDE 38

Motivation Potts-Yukawa System Functional RG PYS Flow Asymptotic Safety Discussion References

References

[20] Fa-Yueh WU. “The potts model”. In : Reviews of modern physics 54.1 (1982),

  • p. 235.

[21] RKP ZIA et DJ WALLACE. “Critical behaviour of the continuous n-component Potts model”. In : Journal of Physics A : Mathematical and General 8.9 (1975), p. 1495. [22] Riccardo Ben Alı ZINATI et Alessandro CODELLO. “Functional RG approach to the Potts model”. In : Journal of Statistical Mechanics : Theory and Experiment 2018.1 (2018), p. 013206.

24 / 24