[PPT] - Chiral critical behavior of Dirac materials: Gross-Neveu-Yukawa PowerPoint Presentation

SLIDE 1

Chiral critical behavior of Dirac materials: Gross-Neveu-Yukawa model at three loops

arXiv:1703.08801 Luminita N. Mihaila, Nikolai Zerf, BI, Igor F. Herbut, Michael M. Scherer [Mih+17]

Bernhard Ihrig July 5, 2017

Institute For Theoretical Physics - University of Heidelberg

SLIDE 2

Outline

Motivation Part 1: Graphene Motivation Part 2: Critical Phenomena & Universality Renormalization group in critical phenomena Tools and Technique Resummation Results and Conclusion

1

SLIDE 3

Motivation Part 1: Graphene

SLIDE 4

Graphene: Electrons on the Honeycomb Lattice Focus on 2D Dirac material graphene

Tight-binding Hamiltonian

H0 = −t

R,i,σ
u†

σ(

R)vσ( R + δi) + h.c.

Fourier-transform into momentum space

uR 1 N

k

u

k

e ik R

δ1 δ2 δ3

ky kx K K

Retain only Fourier-components at K K

H0 K 3ta 2

q K

uq vq iqx qy iqx qy uq vq

At low energies quasirelativistic linear dispersion

Dirac Cones E q vF q semi metallic phase (SM)

2

SLIDE 5

Graphene: Electrons on the Honeycomb Lattice Focus on 2D Dirac material graphene

Tight-binding Hamiltonian

H0 = −t

R,i,σ
u†

σ(

R)vσ( R + δi) + h.c.

Fourier-transform into momentum space

u†

R,σ =

1 √ N

k

u†

k,σe i

k· R δ1 δ2 δ3

ky kx

K
K′
Retain only Fourier-components at K K

H0 K 3ta 2

q K

uq vq iqx qy iqx qy uq vq

At low energies quasirelativistic linear dispersion

Dirac Cones E q vF q semi metallic phase (SM)

2

SLIDE 6

Graphene: Electrons on the Honeycomb Lattice Focus on 2D Dirac material graphene

Tight-binding Hamiltonian

H0 = −t

R,i,σ
u†

σ(

R)vσ( R + δi) + h.c.

Fourier-transform into momentum space

u†

R,σ =

1 √ N

k

u†

k,σe i

k· R δ1 δ2 δ3

ky kx

K
K′
Retain only Fourier-components at K K

H0 K 3ta 2

q K

uq vq iqx qy iqx qy uq vq

At low energies quasirelativistic linear dispersion

Dirac Cones E q vF q semi metallic phase (SM)

kx ky ✏

ky kx ✏

2

SLIDE 7

Graphene: Electrons on the Honeycomb Lattice Focus on 2D Dirac material graphene

Tight-binding Hamiltonian

H0 = −t

R,i,σ
u†

σ(

R)vσ( R + δi) + h.c.

Fourier-transform into momentum space

u†

R,σ =

1 √ N

k

u†

k,σe i

k· R δ1 δ2 δ3

ky kx

K
K′
Retain only Fourier-components at

K, K′

H0,

K = 3ta

2

q≃

K,σ

(u†

q,σ, v†
q,σ)
−iqx + qy

iqx + qy u

q,σ

v

q,σ
At low energies quasirelativistic linear dispersion

Dirac Cones E q vF q semi metallic phase (SM)

kx ky ✏

ky kx ✏

2

SLIDE 8

Graphene: Electrons on the Honeycomb Lattice Focus on 2D Dirac material graphene

Tight-binding Hamiltonian

H0 = −t

R,i,σ
u†

σ(

R)vσ( R + δi) + h.c.

Fourier-transform into momentum space

u†

R,σ =

1 √ N

k

u†

k,σe i

k· R δ1 δ2 δ3

ky kx

K
K′
Retain only Fourier-components at

K, K′

H0,

K = 3ta

2

q≃

K,σ

(u†

q,σ, v†
q,σ)
−iqx + qy

iqx + qy u

q,σ

v

q,σ
At low energies quasirelativistic linear dispersion

Dirac Cones E±( q) ≈ ±vF| q| ⇒ semi metallic phase (SM)

kx ky ✏

ky kx ✏

2

SLIDE 9

Graphene: Electrons on the Honeycomb Lattice Write it as a fjeld theory

Action for low energies in D = 2 + 1 from Z(β) = Tre−βH =
D[ ¯

ψ, ψ]e−S S = β dτd x L0 =

dτd

x ¯ ψ( x, τ)(I2 ⊗ γµ)∂µψ( x, τ) , ¯ ψ = ψ†(I2 ⊗ γ0) with 8-component spinor

x T

n

dq 2 a 2 ei

n iq x u

K q

n

v K q

n

u K q

n

v K q

n

Same structure as in QED but with an Euclidean Clifford Algebra!

I2

z 1 z y 2

I2

x

2

Lacking a mass term there is a chiral symmetry

e i 5 e i 5 with

5 y y 5

3

SLIDE 10

Graphene: Electrons on the Honeycomb Lattice Write it as a fjeld theory

Action for low energies in D = 2 + 1 from Z(β) = Tre−βH =
D[ ¯

ψ, ψ]e−S S = β dτd x L0 =

dτd

x ¯ ψ( x, τ)(I2 ⊗ γµ)∂µψ( x, τ) , ¯ ψ = ψ†(I2 ⊗ γ0) with 8-component spinor ψ† = (ψ†

↑, ψ† ↓)

ψ†

σ(

x, τ) = T

ωn

Λ d q (2πa)2 eiωn+i

q·
x[u†

σ(

K + q, ωn), v†

σ(

K + q, ωn), u†

σ(−

K + q, ωn), v†

σ(−

K + q, ωn)]

Same structure as in QED but with an Euclidean Clifford Algebra!

I2

z 1 z y 2

I2

x

2

Lacking a mass term there is a chiral symmetry

e i 5 e i 5 with

5 y y 5

3

SLIDE 11

Graphene: Electrons on the Honeycomb Lattice Write it as a fjeld theory

Action for low energies in D = 2 + 1 from Z(β) = Tre−βH =
D[ ¯

ψ, ψ]e−S S = β dτd x L0 =

dτd

x ¯ ψ( x, τ)(I2 ⊗ γµ)∂µψ( x, τ) , ¯ ψ = ψ†(I2 ⊗ γ0) with 8-component spinor ψ† = (ψ†

↑, ψ† ↓)

ψ†

σ(

x, τ) = T

ωn

Λ d q (2πa)2 eiωn+i

q·
x[u†

σ(

K + q, ωn), v†

σ(

K + q, ωn), u†

σ(−

K + q, ωn), v†

σ(−

K + q, ωn)]

Same structure as in QED but with an Euclidean Clifford Algebra!

γ0 = I2 ⊗ σz, γ1 = σz ⊗ σy, γ2 = I2 ⊗ σx ⇒ {γµ, γν} = 2δµν

Lacking a mass term there is a chiral symmetry

e i 5 e i 5 with

5 y y 5

3

SLIDE 12

Graphene: Electrons on the Honeycomb Lattice Write it as a fjeld theory

Action for low energies in D = 2 + 1 from Z(β) = Tre−βH =
D[ ¯

ψ, ψ]e−S S = β dτd x L0 =

dτd

x ¯ ψ( x, τ)(I2 ⊗ γµ)∂µψ( x, τ) , ¯ ψ = ψ†(I2 ⊗ γ0) with 8-component spinor ψ† = (ψ†

↑, ψ† ↓)

ψ†

σ(

x, τ) = T

ωn

Λ d q (2πa)2 eiωn+i

q·
x[u†

σ(

K + q, ωn), v†

σ(

K + q, ωn), u†

σ(−

K + q, ωn), v†

σ(−

K + q, ωn)]

Same structure as in QED but with an Euclidean Clifford Algebra!

γ0 = I2 ⊗ σz, γ1 = σz ⊗ σy, γ2 = I2 ⊗ σx ⇒ {γµ, γν} = 2δµν

Lacking a mass term there is a chiral symmetry

ψ → e iγ5ψ, ¯ ψ → ¯ ψe iγ5 with γ5 = σy ⊗ σy , {γ5, γµ} = 0

3

SLIDE 13

Graphene: Electron-Electron-Interactions Focus on 2D Dirac material graphene:

Electron-electron Interaction [Herbut ’06]

H1 =

X,

Y,σ,σ′

nσ( X)

Uδ

X, Y +

e2(1 − δ

X, Y)

4π| X − Y|

nσ′(

Y) = U

i

ni,↑ni,↓ + V1

i,j,σ,σ′

ni,σnj,σ′ + V2

i,j,σ,σ′

ni,σnj,σ′ + . . . [Raghu et al. ’07]

Critical interaction strength for transition to

charge density wave (CDW) u u v v

r spin density wave (SDW)

I4 u u v v

Dynamical mass gap generation spontaneously

breaks the chiral symmetry

4

SLIDE 14

Graphene: Electron-Electron-Interactions Focus on 2D Dirac material graphene:

Electron-electron Interaction [Herbut ’06]

H1 =

X,

Y,σ,σ′

nσ( X)

Uδ

X, Y +

e2(1 − δ

X, Y)

4π| X − Y|

nσ′(

Y) = U

i

ni,↑ni,↓ + V1

i,j,σ,σ′

ni,σnj,σ′ + V2

i,j,σ,σ′

ni,σnj,σ′ + . . .

2 4 1 2 2 4 CDW U QSH CDW SM V1 QSH SDW SDW V2

[Raghu et al. ’07]

Critical interaction strength for transition to

charge density wave (CDW) u u v v

r spin density wave (SDW)

I4 u u v v

Dynamical mass gap generation spontaneously

breaks the chiral symmetry

4

SLIDE 15

Graphene: Electron-Electron-Interactions Focus on 2D Dirac material graphene:

Electron-electron Interaction [Herbut ’06]

H1 =

X,

Y,σ,σ′

nσ( X)

Uδ

X, Y +

e2(1 − δ

X, Y)

4π| X − Y|

nσ′(

Y) = U

i

ni,↑ni,↓ + V1

i,j,σ,σ′

ni,σnj,σ′ + V2

i,j,σ,σ′

ni,σnj,σ′ + . . .

2 4 1 2 2 4 CDW U QSH CDW SM V1 QSH SDW SDW V2

[Raghu et al. ’07]

Critical interaction strength for transition to

charge density wave (CDW) φ = ¯ ψψ ∝ u†

σuσ − v† σvσ

r spin density wave (SDW)

I4 u u v v

Dynamical mass gap generation spontaneously

breaks the chiral symmetry

4

SLIDE 16

Graphene: Electron-Electron-Interactions Focus on 2D Dirac material graphene:

Electron-electron Interaction [Herbut ’06]

H1 =

X,

Y,σ,σ′

nσ( X)

Uδ

X, Y +

e2(1 − δ

X, Y)

4π| X − Y|

nσ′(

Y) = U

i

ni,↑ni,↓ + V1

i,j,σ,σ′

ni,σnj,σ′ + V2

i,j,σ,σ′

ni,σnj,σ′ + . . .

2 4 1 2 2 4 CDW U QSH CDW SM V1 QSH SDW SDW V2

[Raghu et al. ’07]

Critical interaction strength for transition to

charge density wave (CDW) φ = ¯ ψψ ∝ u†

σuσ − v† σvσ

r spin density wave (SDW)
φ = ¯

ψ( σ ⊗ I4)ψ ∝ u†

τ

σττ′uτ′ − v†

τ

σττ′vτ′

Dynamical mass gap generation spontaneously

breaks the chiral symmetry

4

SLIDE 17

Graphene: Electron-Electron-Interactions Focus on 2D Dirac material graphene:

Electron-electron Interaction [Herbut ’06]

H1 =

X,

Y,σ,σ′

nσ( X)

Uδ

X, Y +

e2(1 − δ

X, Y)

4π| X − Y|

nσ′(

Y) = U

i

ni,↑ni,↓ + V1

i,j,σ,σ′

ni,σnj,σ′ + V2

i,j,σ,σ′

ni,σnj,σ′ + . . .

2 4 1 2 2 4 CDW U QSH CDW SM V1 QSH SDW SDW V2

[Raghu et al. ’07]

Critical interaction strength for transition to

charge density wave (CDW) φ = ¯ ψψ ∝ u†

σuσ − v† σvσ

r spin density wave (SDW)
φ = ¯

ψ( σ ⊗ I4)ψ ∝ u†

τ

σττ′uτ′ − v†

τ

σττ′vτ′

Dynamical mass gap generation spontaneously

breaks the chiral symmetry

4

SLIDE 18

Graphene: Effective fjeld theory

Incorporate order parameter fjelds with an semiphenomenological ansatz

LB = 1 2φa(m2 − ∂µ)φa + λ 4! (φaφa)2 → Dynamics and self-interaction develop below certain energy scale

Couple fjelds to the Fermion by a Yukawa interaction

Y

g I2 I4 (in CDW)

Y

g

a a

I4 (in SDW)

This can be generalized to arbitrary number of Fermion fmavors Nf (graphene Nf

2) I2 I4 I2 I2Nf

a

I4

a

I2Nf

Full action is then the Gross-Neveu-Yukawa model (GNY)

Y B

g 1 2 m2

2

4

5

SLIDE 19

Graphene: Effective fjeld theory

Incorporate order parameter fjelds with an semiphenomenological ansatz

LB = 1 2φa(m2 − ∂µ)φa + λ 4! (φaφa)2 → Dynamics and self-interaction develop below certain energy scale

Couple fjelds to the Fermion by a Yukawa interaction

LY = gφ ¯ ψ(I2 ⊗ I4)ψ (in CDW) LY = gφa ¯ ψ(σa ⊗ I4)ψ (in SDW)

This can be generalized to arbitrary number of Fermion fmavors Nf (graphene Nf

2) I2 I4 I2 I2Nf

a

I4

a

I2Nf

Full action is then the Gross-Neveu-Yukawa model (GNY)

Y B

g 1 2 m2

2

4

5

SLIDE 20

Graphene: Effective fjeld theory

Incorporate order parameter fjelds with an semiphenomenological ansatz

LB = 1 2φa(m2 − ∂µ)φa + λ 4! (φaφa)2 → Dynamics and self-interaction develop below certain energy scale

Couple fjelds to the Fermion by a Yukawa interaction

LY = gφ ¯ ψ(I2 ⊗ I4)ψ (in CDW) LY = gφa ¯ ψ(σa ⊗ I4)ψ (in SDW)

This can be generalized to arbitrary number of Fermion fmavors Nf (graphene Nf = 2)

¯ ψ(I2 ⊗ I4)ψ → ¯ ψ(I2 ⊗ I2Nf )ψ ¯ ψ(σa ⊗ I4)ψ → ¯ ψ(σa ⊗ I2Nf )ψ

Full action is then the Gross-Neveu-Yukawa model (GNY)

Y B

g 1 2 m2

2

4

5

SLIDE 21

Graphene: Effective fjeld theory

Incorporate order parameter fjelds with an semiphenomenological ansatz

LB = 1 2φa(m2 − ∂µ)φa + λ 4! (φaφa)2 → Dynamics and self-interaction develop below certain energy scale

Couple fjelds to the Fermion by a Yukawa interaction

LY = gφ ¯ ψ(I2 ⊗ I4)ψ (in CDW) LY = gφa ¯ ψ(σa ⊗ I4)ψ (in SDW)

This can be generalized to arbitrary number of Fermion fmavors Nf (graphene Nf = 2)

¯ ψ(I2 ⊗ I4)ψ → ¯ ψ(I2 ⊗ I2Nf )ψ ¯ ψ(σa ⊗ I4)ψ → ¯ ψ(σa ⊗ I2Nf )ψ

Full action is then the Gross-Neveu-Yukawa model (GNY)

L = L0 + LY + LB = ¯ ψ/ ∂ψ + gφ ¯ ψψ + 1 2 φ(m2 − ∂2

µ)φ + λ

4! φ4

5

SLIDE 22

Motivation Part 2: Critical Phenomena & Universality

SLIDE 23

Critical Phenomena

Critical phenomena deal with continous (second order) phase transitions in

macroscopic systems

Characterized by collective behavior on all length scales near the transition

∼

[Chialvo ’10]

Correlation-length diverges with a power law

t 1 C t with t T Tc T

More power laws with critical exponents

assume for correlator G r t r rd

2

Here quantum phase transition at T

0 but with critical interaction strength T Tc

6

SLIDE 24

Critical Phenomena

Critical phenomena deal with continous (second order) phase transitions in

macroscopic systems

Characterized by collective behavior on all length scales near the transition

T < Tc T > Tc T ∼Tc Subcritical Critical Supercritical

[Chialvo ’10]

Correlation-length diverges with a power law

t 1 C t with t T Tc T

More power laws with critical exponents

assume for correlator G r t r rd

2

Here quantum phase transition at T

0 but with critical interaction strength T Tc

6

SLIDE 25

Critical Phenomena

Critical phenomena deal with continous (second order) phase transitions in

macroscopic systems

Characterized by collective behavior on all length scales near the transition

T < Tc T > Tc T ∼Tc Subcritical Critical Supercritical

[Chialvo ’10]

Correlation-length diverges with a power law

t 1 C t with t T Tc T

More power laws with critical exponents

assume for correlator G r t r rd

2

Here quantum phase transition at T

0 but with critical interaction strength ξ T Tc

6

SLIDE 26

Critical Phenomena

Critical phenomena deal with continous (second order) phase transitions in

macroscopic systems

Characterized by collective behavior on all length scales near the transition

T < Tc T > Tc T ∼Tc Subcritical Critical Supercritical

[Chialvo ’10]

Correlation-length diverges with a power law

ξ ∼ |t|−ν (1 + C|t|ω + . . . ) with t ≡ T − Tc T

More power laws with critical exponents

assume for correlator G r t r rd

2

Here quantum phase transition at T

0 but with critical interaction strength ξ T Tc

6

SLIDE 27

Critical Phenomena

Critical phenomena deal with continous (second order) phase transitions in

macroscopic systems

Characterized by collective behavior on all length scales near the transition

T < Tc T > Tc T ∼Tc Subcritical Critical Supercritical

[Chialvo ’10]

Correlation-length diverges with a power law

ξ ∼ |t|−ν (1 + C|t|ω + . . . ) with t ≡ T − Tc T

More power laws with critical exponents α, β, γ, δ, . . .

assume for correlator G(r, t) = Φ±(r/ξ) rd−2+η

Here quantum phase transition at T

0 but with critical interaction strength ξ T Tc

6

SLIDE 28

Critical Phenomena

Critical phenomena deal with continous (second order) phase transitions in

macroscopic systems

Characterized by collective behavior on all length scales near the transition

T < Tc T > Tc T ∼Tc Subcritical Critical Supercritical

[Chialvo ’10]

Correlation-length diverges with a power law

ξ ∼ |t|−ν (1 + C|t|ω + . . . ) with t ≡ T − Tc T

More power laws with critical exponents α, β, γ, δ, . . .

assume for correlator G(r, t) = Φ±(r/ξ) rd−2+η

Here quantum phase transition at T = 0 but with

critical interaction strength ξ T Tc

6

SLIDE 29

Universality What is universality and how well do we understand it?

Universality can be defjned as existence of an IR-fjxed point in the RG
Theories that fmow in the same IR-FP are in the same universality class
Universality classes differ by their critical exponents (

)

Consider the O N model in the 3D Ising universality class

[El-Showk et al. ’14] 1 2

a m2 a

4

a a 2

Ref. Year Method Guida, Zinn-Justin 1998

expansion

0 63050 250 0 03650 500 0 814 18 Guida, Zinn-Justin 1998 3D exp 0 63040 130 0 03350 250 0 799 11 Campostrini et al. 2002 HT 0 63012 16 0 03639 15 0 825 50 Deng, Blöte 2003 MC 0 63020 12 0 03680 20 0 821 5 Hasenbusch 2010 MC 0 63002 10 0 03627 10 0 832 6 El-Showk et al. 2014 cBS 0 62999 5 0 03631 3 0 8303 18 Question: Can we achieve a similar precision and agreement of methods ?

7

SLIDE 30

Universality What is universality and how well do we understand it?

Universality can be defjned as existence of an IR-fjxed point in the RG
Theories that fmow in the same IR-FP are in the same universality class
Universality classes differ by their critical exponents (

)

Consider the O N model in the 3D Ising universality class

[El-Showk et al. ’14] 1 2

a m2 a

4

a a 2

Ref. Year Method Guida, Zinn-Justin 1998

expansion

0 63050 250 0 03650 500 0 814 18 Guida, Zinn-Justin 1998 3D exp 0 63040 130 0 03350 250 0 799 11 Campostrini et al. 2002 HT 0 63012 16 0 03639 15 0 825 50 Deng, Blöte 2003 MC 0 63020 12 0 03680 20 0 821 5 Hasenbusch 2010 MC 0 63002 10 0 03627 10 0 832 6 El-Showk et al. 2014 cBS 0 62999 5 0 03631 3 0 8303 18 Question: Can we achieve a similar precision and agreement of methods ?

7

SLIDE 31

Universality What is universality and how well do we understand it?

Universality can be defjned as existence of an IR-fjxed point in the RG
Theories that fmow in the same IR-FP are in the same universality class
Universality classes differ by their critical exponents (ν, η, . . . )
Consider the O N model in the 3D Ising universality class

[El-Showk et al. ’14] 1 2

a m2 a

4

a a 2

Ref. Year Method Guida, Zinn-Justin 1998

expansion

0 63050 250 0 03650 500 0 814 18 Guida, Zinn-Justin 1998 3D exp 0 63040 130 0 03350 250 0 799 11 Campostrini et al. 2002 HT 0 63012 16 0 03639 15 0 825 50 Deng, Blöte 2003 MC 0 63020 12 0 03680 20 0 821 5 Hasenbusch 2010 MC 0 63002 10 0 03627 10 0 832 6 El-Showk et al. 2014 cBS 0 62999 5 0 03631 3 0 8303 18 Question: Can we achieve a similar precision and agreement of methods ?

7

SLIDE 32

Universality What is universality and how well do we understand it?

Universality can be defjned as existence of an IR-fjxed point in the RG
Theories that fmow in the same IR-FP are in the same universality class
Universality classes differ by their critical exponents (ν, η, . . . )
Consider the O(N) model in the 3D Ising universality class

[El-Showk et al. ’14] L = 1 2 φa(m2 − ∂µ)φa + λ 4! (φaφa)2 Ref. Year Method Guida, Zinn-Justin 1998

expansion

0 63050 250 0 03650 500 0 814 18 Guida, Zinn-Justin 1998 3D exp 0 63040 130 0 03350 250 0 799 11 Campostrini et al. 2002 HT 0 63012 16 0 03639 15 0 825 50 Deng, Blöte 2003 MC 0 63020 12 0 03680 20 0 821 5 Hasenbusch 2010 MC 0 63002 10 0 03627 10 0 832 6 El-Showk et al. 2014 cBS 0 62999 5 0 03631 3 0 8303 18 Question: Can we achieve a similar precision and agreement of methods ?

7

SLIDE 33

Universality What is universality and how well do we understand it?

Universality can be defjned as existence of an IR-fjxed point in the RG
Theories that fmow in the same IR-FP are in the same universality class
Universality classes differ by their critical exponents (ν, η, . . . )
Consider the O(N) model in the 3D Ising universality class

[El-Showk et al. ’14] L = 1 2 φa(m2 − ∂µ)φa + λ 4! (φaφa)2 Ref. Year Method ν η ω Guida, Zinn-Justin 1998 ǫ-expansion 0.63050(250) 0.03650(500) 0.814(18) Guida, Zinn-Justin 1998 3D exp 0.63040(130) 0.03350(250) 0.799(11) Campostrini et al. 2002 HT 0.63012(16) 0.03639(15) 0.825(50) Deng, Blöte 2003 MC 0.63020(12) 0.03680(20) 0.821(5) Hasenbusch 2010 MC 0.63002(10) 0.03627(10) 0.832(6) El-Showk et al. 2014 cBS 0.62999(5) 0.03631(3) 0.8303(18) Question: Can we achieve a similar precision and agreement of methods ?

7

SLIDE 34

Universality What is universality and how well do we understand it?

Universality can be defjned as existence of an IR-fjxed point in the RG
Theories that fmow in the same IR-FP are in the same universality class
Universality classes differ by their critical exponents (ν, η, . . . )
Consider the O(N) model in the 3D Ising universality class

[El-Showk et al. ’14] L = 1 2 φa(m2 − ∂µ)φa + λ 4! (φaφa)2 Ref. Year Method ν η ω Guida, Zinn-Justin 1998 ǫ-expansion 0.63050(250) 0.03650(500) 0.814(18) Guida, Zinn-Justin 1998 3D exp 0.63040(130) 0.03350(250) 0.799(11) Campostrini et al. 2002 HT 0.63012(16) 0.03639(15) 0.825(50) Deng, Blöte 2003 MC 0.63020(12) 0.03680(20) 0.821(5) Hasenbusch 2010 MC 0.63002(10) 0.03627(10) 0.832(6) El-Showk et al. 2014 cBS 0.62999(5) 0.03631(3) 0.8303(18) ⇒ Question: Can we achieve a similar precision and agreement of methods ?

7

SLIDE 35

Renormalization group in critical phenomena

SLIDE 36

The Gross-Neveu-Yukawa model The Gross-Neveu-Yukawa model (GNY)

L = ¯ ψ/ ∂ψ + gφa ¯ ψ Taψ + 1 2 φa(m2 − ∂2

µ)φa + λ

4! (φaφa)2

Simplest extension of the O N model by a massless Fermion
New universality classes named after the broken symmetries

chiral Ising CDW Ta 1

2

1 chiral Heisenberg SDW Ta

1 2 a

O 3 O 2 chiral XY SC Ta 1 i 5 O 2 1 (work in progress)

functions (

g2) d d d d fjxed point Three physical fjxed points (FP):

Gaussian FP: both couplings relevant
Wilson-Fisher FP: only

irrelevant

Non-Gaussian FP: both couplings irrelevant

NGFP NGFP WF 8

SLIDE 37

The Gross-Neveu-Yukawa model The Gross-Neveu-Yukawa model (GNY)

L = ¯ ψ/ ∂ψ + gφa ¯ ψ Taψ + 1 2 φa(m2 − ∂2

µ)φa + λ

4! (φaφa)2

Simplest extension of the O(N) model by a massless Fermion
New universality classes named after the broken symmetries

chiral Ising CDW Ta 1

2

1 chiral Heisenberg SDW Ta

1 2 a

O 3 O 2 chiral XY SC Ta 1 i 5 O 2 1 (work in progress)

functions (

g2) d d d d fjxed point Three physical fjxed points (FP):

Gaussian FP: both couplings relevant
Wilson-Fisher FP: only

irrelevant

Non-Gaussian FP: both couplings irrelevant

NGFP NGFP WF 8

SLIDE 38

The Gross-Neveu-Yukawa model The Gross-Neveu-Yukawa model (GNY)

L = ¯ ψ/ ∂ψ + gφa ¯ ψ Taψ + 1 2 φa(m2 − ∂2

µ)φa + λ

4! (φaφa)2

Simplest extension of the O(N) model by a massless Fermion
New universality classes named after the broken symmetries

chiral Ising CDW Ta = 1 Z2 → 1 chiral Heisenberg SDW Ta = 1

2 σa

O(3) → O(2) chiral XY SC Ta = 1, iγ5 O(2) → 1 (work in progress)

functions (

g2) d d d d fjxed point Three physical fjxed points (FP):

Gaussian FP: both couplings relevant
Wilson-Fisher FP: only

irrelevant

Non-Gaussian FP: both couplings irrelevant

NGFP NGFP WF 8

SLIDE 39

The Gross-Neveu-Yukawa model The Gross-Neveu-Yukawa model (GNY)

L = ¯ ψ/ ∂ψ + gφa ¯ ψ Taψ + 1 2 φa(m2 − ∂2

µ)φa + λ

4! (φaφa)2

Simplest extension of the O(N) model by a massless Fermion
New universality classes named after the broken symmetries

chiral Ising CDW Ta = 1 Z2 → 1 chiral Heisenberg SDW Ta = 1

2 σa

O(3) → O(2) chiral XY SC Ta = 1, iγ5 O(2) → 1 (work in progress)

β-functions (α ≡ g2)

βα = dα d ln µ

!

= 0 βλ = dλ d ln µ

!

= 0        fjxed point (α∗, λ∗) Three physical fjxed points (FP):

Gaussian FP: both couplings relevant
Wilson-Fisher FP: only

irrelevant

Non-Gaussian FP: both couplings irrelevant

NGFP NGFP WF 8

SLIDE 40

The Gross-Neveu-Yukawa model The Gross-Neveu-Yukawa model (GNY)

L = ¯ ψ/ ∂ψ + gφa ¯ ψ Taψ + 1 2 φa(m2 − ∂2

µ)φa + λ

4! (φaφa)2

Simplest extension of the O(N) model by a massless Fermion
New universality classes named after the broken symmetries

chiral Ising CDW Ta = 1 Z2 → 1 chiral Heisenberg SDW Ta = 1

2 σa

O(3) → O(2) chiral XY SC Ta = 1, iγ5 O(2) → 1 (work in progress)

β-functions (α ≡ g2)

βα = dα d ln µ

!

= 0 βλ = dλ d ln µ

!

= 0        fjxed point (α∗, λ∗) Three physical fjxed points (FP):

Gaussian FP: both couplings relevant
Wilson-Fisher FP: only λ irrelevant
Non-Gaussian FP: both couplings irrelevant

λ α

NGFP NGFP∗ WF 8

SLIDE 41

Perturbative RG procedure

Introduce renormalized fjelds and couplings

ψ0 = Z1/2

ψ ψ,

φ0 = Z1/2

φ φ,

g0 = Zgg, λ0 = Zλλ, m2

0 = Zmm2

Writing for the renormalization constants Z

1 Z the infjnities are then absorbed by the Zs and determined by ip Z

div.

fjnite p2 Z m2 Z

2

div.

fjnite with Z

2

ZmZ g Z g

div.

fjnite with Z ZgZ1 2Z Z

4

div.

fjnite with Z

4

Z Z2

Regularize infjnite diagrams in D

4 and render couplings massless by introducing new mass scale g g

2

m2 m2

2

MS-Scheme: renormalization constants are the -poles

Zi 1 Zi 1

n 1

Z n

i

1

n 1 n i

g2

n

with

n i

g2 g2

n

9

SLIDE 42

Perturbative RG procedure

Introduce renormalized fjelds and couplings

ψ0 = Z1/2

ψ ψ,

φ0 = Z1/2

φ φ,

g0 = Zgg, λ0 = Zλλ, m2

0 = Zmm2

Writing for the renormalization constants Z = 1 + δZ the infjnities are then absorbed by

the δZs and determined by i/ p(δZψ + ¯ ψψdiv.) = fjnite p2 · δZφ + m2 · δZφ2 + φφdiv. = fjnite with Zφ2 ≡ ZmZφ g · δZφ ¯

ψψ + g ¯

ψψdiv. = fjnite with Zφ ¯

ψψ ≡ ZgZ1/2 φ Zψ

λ · δZφ4 + φφφφdiv. = fjnite with Zφ4 ≡ ZλZ2

φ

Regularize infjnite diagrams in D

4 and render couplings massless by introducing new mass scale g g

2

m2 m2

2

MS-Scheme: renormalization constants are the -poles

Zi 1 Zi 1

n 1

Z n

i

1

n 1 n i

g2

n

with

n i

g2 g2

n

9

SLIDE 43

Perturbative RG procedure

Introduce renormalized fjelds and couplings

ψ0 = Z1/2

ψ ψ,

φ0 = Z1/2

φ φ,

g0 = Zgg, λ0 = Zλλ, m2

0 = Zmm2

Writing for the renormalization constants Z = 1 + δZ the infjnities are then absorbed by

the δZs and determined by i/ p(δZψ + ¯ ψψdiv.) = fjnite p2 · δZφ + m2 · δZφ2 + φφdiv. = fjnite with Zφ2 ≡ ZmZφ g · δZφ ¯

ψψ + g ¯

ψψdiv. = fjnite with Zφ ¯

ψψ ≡ ZgZ1/2 φ Zψ

λ · δZφ4 + φφφφdiv. = fjnite with Zφ4 ≡ ZλZ2

φ

Regularize infjnite diagrams in D = 4 − ǫ and render couplings massless by introducing

new mass scale µ g → gµǫ/2, λ → λµǫ, m2 → m2µ−2

MS-Scheme: renormalization constants are the -poles

Zi 1 Zi 1

n 1

Z n

i

1

n 1 n i

g2

n

with

n i

g2 g2

n

9

SLIDE 44

Perturbative RG procedure

Introduce renormalized fjelds and couplings

ψ0 = Z1/2

ψ ψ,

φ0 = Z1/2

φ φ,

g0 = Zgg, λ0 = Zλλ, m2

0 = Zmm2

Writing for the renormalization constants Z = 1 + δZ the infjnities are then absorbed by

the δZs and determined by i/ p(δZψ + ¯ ψψdiv.) = fjnite p2 · δZφ + m2 · δZφ2 + φφdiv. = fjnite with Zφ2 ≡ ZmZφ g · δZφ ¯

ψψ + g ¯

ψψdiv. = fjnite with Zφ ¯

ψψ ≡ ZgZ1/2 φ Zψ

λ · δZφ4 + φφφφdiv. = fjnite with Zφ4 ≡ ZλZ2

φ

Regularize infjnite diagrams in D = 4 − ǫ and render couplings massless by introducing

new mass scale µ g → gµǫ/2, λ → λµǫ, m2 → m2µ−2

MS-Scheme: renormalization constants are the ǫ-poles

Zi = 1 + δZi = 1 +

∞

n=1

δZ(n)

i

= 1 +

∞

n=1

α(n)

i

(g2, λ) ǫn with α(n)

i

(g2, λ) = O((g2 + λ)n)

9

SLIDE 45

Diagrams to compute

For the Fermion propagator ¯

ψψdiv.

e e e φ e e e φ φ e e e e e φ e φ e φ e e e e e φ e φ φ e e e

+ 32 more

For the Boson propagator

div.

40 more

For the Yukawa vertex g

div.

153 more

For the

4 vertex div.

1574 more

10

SLIDE 46

Diagrams to compute

For the Fermion propagator ¯

ψψdiv.

e e e φ e e e φ φ e e e e e φ e φ e φ e e e e e φ e φ φ e e e

+ 32 more

For the Boson propagator φφdiv.

φ φ e e φ φ e e e e φ φ φ e e e e e φ φ e φ φ e e e e φ φ e e

+ 40 more

For the Yukawa vertex g

div.

153 more

For the

4 vertex div.

1574 more

10

SLIDE 47

Diagrams to compute

For the Fermion propagator ¯

ψψdiv.

e e e φ e e e φ φ e e e e e φ e φ e φ e e e e e φ e φ φ e e e

+ 32 more

For the Boson propagator φφdiv.

φ φ e e φ φ e e e e φ φ φ e e e e e φ φ e φ φ e e e e φ φ e e

+ 40 more

For the Yukawa vertex g ¯

ψψdiv.

e e φ φ e e e e φ e φ e φ e e e e φ e φ φ e e e φ e e e e φ e φ e φ e e φ e e

+ 153 more

For the

4 vertex div.

1574 more

10

SLIDE 48

Diagrams to compute

For the Fermion propagator ¯

ψψdiv.

e e e φ e e e φ φ e e e e e φ e φ e φ e e e e e φ e φ φ e e e

+ 32 more

For the Boson propagator φφdiv.

φ φ e e φ φ e e e e φ φ φ e e e e e φ φ e φ φ e e e e φ φ e e

+ 40 more

For the Yukawa vertex g ¯

ψψdiv.

e e φ φ e e e e φ e φ e φ e e e e φ e φ φ e e e φ e e e e φ e φ e φ e e φ e e

+ 153 more

For the φ4 vertex φφφφdiv.

φ φ φ φ e e e e φ φ φ φ e e e φ φ e φ φ φ φ φ φ φ φ φ e e φ φ φ φ e e e e e e φ φ e e

+ 1574 more

10

SLIDE 49

Tools and Technique

SLIDE 50

Basics of computing Feynman diagrams How to treat a Feynman diagram?

Problem:

20,000 integrals to compute!

Goal: Reduce the number of integrals to so-called master integrals
Idea: In dimensional regularization the D dimensional boundary integrals provide

dDk 2

D

k f k

By explicitly performing the differentiations one obtains recurrence relations connecting

a complicated Feynman integral to several simpler ones master integrals

Take a look at a simple example

F n dDk 2

D

1 k2 m2 n

IBP

F n D 2n 2 2n 2 m2 F n 1 dDk 2

D

k k 1 k2 m2 n dDk 2

D k2

m2 n dDk 2

D k

2k n k2 m2 n

1

dDk 2

D

D 2n k2 m2 n dDk 2

D

2nm2 k2 m2 n

1

D 2n F n 2nm2F n 1

11

SLIDE 51

Basics of computing Feynman diagrams How to treat a Feynman diagram?

Problem: ∼ 20,000 integrals to compute!
Goal: Reduce the number of integrals to so-called master integrals
Idea: In dimensional regularization the D dimensional boundary integrals provide

dDk 2

D

k f k

By explicitly performing the differentiations one obtains recurrence relations connecting

a complicated Feynman integral to several simpler ones master integrals

Take a look at a simple example

F n dDk 2

D

1 k2 m2 n

IBP

F n D 2n 2 2n 2 m2 F n 1 dDk 2

D

k k 1 k2 m2 n dDk 2

D k2

m2 n dDk 2

D k

2k n k2 m2 n

1

dDk 2

D

D 2n k2 m2 n dDk 2

D

2nm2 k2 m2 n

1

D 2n F n 2nm2F n 1

11

SLIDE 52

Basics of computing Feynman diagrams How to treat a Feynman diagram?

Problem: ∼ 20,000 integrals to compute!
Goal: Reduce the number of integrals to so-called master integrals
Idea: In dimensional regularization the D dimensional boundary integrals provide

dDk 2

D

k f k

By explicitly performing the differentiations one obtains recurrence relations connecting

a complicated Feynman integral to several simpler ones master integrals

Take a look at a simple example

F n dDk 2

D

1 k2 m2 n

IBP

F n D 2n 2 2n 2 m2 F n 1 dDk 2

D

k k 1 k2 m2 n dDk 2

D k2

m2 n dDk 2

D k

2k n k2 m2 n

1

dDk 2

D

D 2n k2 m2 n dDk 2

D

2nm2 k2 m2 n

1

D 2n F n 2nm2F n 1

11

SLIDE 53

Basics of computing Feynman diagrams How to treat a Feynman diagram?

Problem: ∼ 20,000 integrals to compute!
Goal: Reduce the number of integrals to so-called master integrals
Idea: In dimensional regularization the D dimensional boundary integrals provide
dDk

(2π)D ∂ ∂kµ f(k, . . . ) = 0

By explicitly performing the differentiations one obtains recurrence relations connecting

a complicated Feynman integral to several simpler ones master integrals

Take a look at a simple example

F n dDk 2

D

1 k2 m2 n

IBP

F n D 2n 2 2n 2 m2 F n 1 dDk 2

D

k k 1 k2 m2 n dDk 2

D k2

m2 n dDk 2

D k

2k n k2 m2 n

1

dDk 2

D

D 2n k2 m2 n dDk 2

D

2nm2 k2 m2 n

1

D 2n F n 2nm2F n 1

11

SLIDE 54

Basics of computing Feynman diagrams How to treat a Feynman diagram?

Problem: ∼ 20,000 integrals to compute!
Goal: Reduce the number of integrals to so-called master integrals
Idea: In dimensional regularization the D dimensional boundary integrals provide
dDk

(2π)D ∂ ∂kµ f(k, . . . ) = 0

By explicitly performing the differentiations one obtains recurrence relations connecting

a complicated Feynman integral to several simpler ones → master integrals

Take a look at a simple example

F n dDk 2

D

1 k2 m2 n

IBP

F n D 2n 2 2n 2 m2 F n 1 dDk 2

D

k k 1 k2 m2 n dDk 2

D k2

m2 n dDk 2

D k

2k n k2 m2 n

1

dDk 2

D

D 2n k2 m2 n dDk 2

D

2nm2 k2 m2 n

1

D 2n F n 2nm2F n 1

11

SLIDE 55

Basics of computing Feynman diagrams How to treat a Feynman diagram?

Problem: ∼ 20,000 integrals to compute!
Goal: Reduce the number of integrals to so-called master integrals
Idea: In dimensional regularization the D dimensional boundary integrals provide
dDk

(2π)D ∂ ∂kµ f(k, . . . ) = 0

By explicitly performing the differentiations one obtains recurrence relations connecting

a complicated Feynman integral to several simpler ones → master integrals

Take a look at a simple example

F(n) =

dDk

(2π)D 1 (k2 + m2)n

IBP

F n D 2n 2 2n 2 m2 F n 1 dDk 2

D

k k 1 k2 m2 n dDk 2

D k2

m2 n dDk 2

D k

2k n k2 m2 n

1

dDk 2

D

D 2n k2 m2 n dDk 2

D

2nm2 k2 m2 n

1

D 2n F n 2nm2F n 1

11

SLIDE 56

Basics of computing Feynman diagrams How to treat a Feynman diagram?

Problem: ∼ 20,000 integrals to compute!
Goal: Reduce the number of integrals to so-called master integrals
Idea: In dimensional regularization the D dimensional boundary integrals provide
dDk

(2π)D ∂ ∂kµ f(k, . . . ) = 0

By explicitly performing the differentiations one obtains recurrence relations connecting

a complicated Feynman integral to several simpler ones → master integrals

Take a look at a simple example

F(n) =

dDk

(2π)D 1 (k2 + m2)n

IBP

F n D 2n 2 2n 2 m2 F n 1 0 =

dDk

(2π)D ∂ ∂kµ kµ 1 (k2 + m2)n =

dDk

(2π)D δµ

µ

(k2 + m2)n +

dDk

(2π)D kµ −2kµn (k2 + m2)n+1 =

dDk

(2π)D D − 2n (k2 + m2)n +

dDk

(2π)D 2nm2 (k2 + m2)n+1 = (D − 2n)F(n) + 2nm2F(n + 1)

11

SLIDE 57

Basics of computing Feynman diagrams How to treat a Feynman diagram?

Problem: ∼ 20,000 integrals to compute!
Goal: Reduce the number of integrals to so-called master integrals
Idea: In dimensional regularization the D dimensional boundary integrals provide
dDk

(2π)D ∂ ∂kµ f(k, . . . ) = 0

By explicitly performing the differentiations one obtains recurrence relations connecting

a complicated Feynman integral to several simpler ones → master integrals

Take a look at a simple example

F(n) =

dDk

(2π)D 1 (k2 + m2)n

IBP

= ⇒ F(n) = − D − 2n + 2 (2n − 2)m2 F(n − 1) 0 =

dDk

(2π)D ∂ ∂kµ kµ 1 (k2 + m2)n =

dDk

(2π)D δµ

µ

(k2 + m2)n +

dDk

(2π)D kµ −2kµn (k2 + m2)n+1 =

dDk

(2π)D D − 2n (k2 + m2)n +

dDk

(2π)D 2nm2 (k2 + m2)n+1 = (D − 2n)F(n) + 2nm2F(n + 1)

11

SLIDE 58

Tools Use a fully automized toolchain?

Generate diagrams with QGRAF → very fast [Nogueira ’93]
Interface with Q2E for the asymptotic expansion [Seidensticker ’99]
Asymptotic expansion with EXP [Seidensticker ’97]
Apply reccurence relations with MATAD and MINCER [Steinhauser ’99]

Computing in FORM [Vermaseren ’84]

Why use FORM ?

stupid but fast (from the FORM tutorial) Mathematica

Much built-in functions (integration, differ-

entiation, solving equations ...)

Big and slow
Limited by RAM
Tries to do everything

FORM

Very limited functions (calculus

with tensors, ...)

Small and fast
Limited by disc space
Does only what you ask for

12

SLIDE 59

Tools Use a fully automized toolchain?

Generate diagrams with QGRAF → very fast [Nogueira ’93]
Interface with Q2E for the asymptotic expansion [Seidensticker ’99]
Asymptotic expansion with EXP [Seidensticker ’97]
Apply reccurence relations with MATAD and MINCER [Steinhauser ’99]

Computing in FORM [Vermaseren ’84]

Why use FORM ?

stupid but fast (from the FORM tutorial) Mathematica

Much built-in functions (integration, differ-

entiation, solving equations ...)

Big and slow
Limited by RAM
Tries to do everything

FORM

Very limited functions (calculus

with tensors, ...)

Small and fast
Limited by disc space
Does only what you ask for

12

SLIDE 60

Tools Use a fully automized toolchain?

Generate diagrams with QGRAF → very fast [Nogueira ’93]
Interface with Q2E for the asymptotic expansion [Seidensticker ’99]
Asymptotic expansion with EXP [Seidensticker ’97]
Apply reccurence relations with MATAD and MINCER [Steinhauser ’99]

Computing in FORM [Vermaseren ’84]

Why use FORM ?

stupid but fast (from the FORM tutorial) Mathematica

Much built-in functions (integration, differ-

entiation, solving equations ...)

Big and slow
Limited by RAM
Tries to do everything

FORM

Very limited functions (calculus

with tensors, ...)

Small and fast
Limited by disc space
Does only what you ask for

12

SLIDE 61

Tools Use a fully automized toolchain?

Generate diagrams with QGRAF → very fast [Nogueira ’93]
Interface with Q2E for the asymptotic expansion [Seidensticker ’99]
Asymptotic expansion with EXP [Seidensticker ’97]
Apply reccurence relations with MATAD and MINCER [Steinhauser ’99]

Computing in FORM [Vermaseren ’84]

Why use FORM ? ⇒ stupid but fast (from the FORM tutorial)

Mathematica

Much built-in functions (integration, differ-

entiation, solving equations ...)

Big and slow
Limited by RAM
Tries to do everything

FORM

Very limited functions (calculus

with tensors, ...)

Small and fast
Limited by disc space
Does only what you ask for

12

SLIDE 62

Tools Use a fully automized toolchain?

Generate diagrams with QGRAF → very fast [Nogueira ’93]
Interface with Q2E for the asymptotic expansion [Seidensticker ’99]
Asymptotic expansion with EXP [Seidensticker ’97]
Apply reccurence relations with MATAD and MINCER [Steinhauser ’99]

Computing in FORM [Vermaseren ’84]

Why use FORM ? ⇒ stupid but fast (from the FORM tutorial)

Mathematica

Much built-in functions (integration, differ-

entiation, solving equations ...)

Big and slow
Limited by RAM
Tries to do everything

FORM

Very limited functions (calculus

with tensors, ...)

Small and fast
Limited by disc space
Does only what you ask for

12

SLIDE 63

Tools Use a fully automized toolchain?

Generate diagrams with QGRAF → very fast [Nogueira ’93]
Interface with Q2E for the asymptotic expansion [Seidensticker ’99]
Asymptotic expansion with EXP [Seidensticker ’97]
Apply reccurence relations with MATAD and MINCER [Steinhauser ’99]

Computing in FORM [Vermaseren ’84]

Why use FORM ? ⇒ stupid but fast (from the FORM tutorial)

Mathematica

Much built-in functions (integration, differ-

entiation, solving equations ...)

Big and slow
Limited by RAM
Tries to do everything

FORM

Very limited functions (calculus

with tensors, ...)

Small and fast
Limited by disc space
Does only what you ask for

12

SLIDE 64

Toolchain

Defjne Lagrangian Generate diagrams with its symmetry factors

QGRAF

Assign Feyman rules and mass hiearchy Q q m FTij p Denab p m Ta ij dabcd Apply the rules and topologies FT1i1i2 Q k1 Ta i2i3 FT1i3i4 k2 Denab Q k1 k2 m

Q2E

Asymptotic expansion 1 Map on topologies inpl tad1l inpla inpt1 tad2l inpo1

EXP

Insert explicit expressions (propagators, vertices, traces, expansions ...) & perform integration with IBP dDk1 2 D dDk2 2 D Insert hardcoded masterintegrals D3 D6 DN B4 E3

MATAD Collect renormalization constants Zs in FORM and proceed for

and
functions in

Mathematica

13

SLIDE 65

Toolchain

Defjne Lagrangian Generate diagrams with its symmetry factors

φ φ e e φ φ e e e e φ φ φ e e e e φ φ e e e e φ e φ e φ e e e e φ e φ e φ e e φ e e e e φ e φ φ e e e e e φ e e e φ e e e φ φ e e e e e φ e φ e φ e e φ φ φ φ e e e e φ φ φ φ e e e φ φ e φ φ φ φ φ φ e e e e e e φ

QGRAF

Assign Feyman rules and mass hiearchy Q q m FTij p Denab p m Ta ij dabcd Apply the rules and topologies FT1i1i2 Q k1 Ta i2i3 FT1i3i4 k2 Denab Q k1 k2 m

Q2E

Asymptotic expansion 1 Map on topologies inpl tad1l inpla inpt1 tad2l inpo1

EXP

Insert explicit expressions (propagators, vertices, traces, expansions ...) & perform integration with IBP dDk1 2 D dDk2 2 D Insert hardcoded masterintegrals D3 D6 DN B4 E3

MATAD Collect renormalization constants Zs in FORM and proceed for

and
functions in

Mathematica

13

SLIDE 66

Toolchain

Defjne Lagrangian Generate diagrams with its symmetry factors

φ φ e e φ φ e e e e φ φ φ e e e e φ φ e e e e φ e φ e φ e e e e φ e φ e φ e e φ e e e e φ e φ φ e e e e e φ e e e φ e e e φ φ e e e e e φ e φ e φ e e φ φ φ φ e e e e φ φ φ φ e e e φ φ e φ φ φ φ φ φ e e e e e e φ

QGRAF

Assign Feyman rules and mass hiearchy Q ≫ q, m = FTij(p) = Denab(p, m) = Ta ij = dabcd Apply the rules and topologies FT1i1i2 Q k1 Ta i2i3 FT1i3i4 k2 Denab Q k1 k2 m

Q2E

Asymptotic expansion 1 Map on topologies inpl tad1l inpla inpt1 tad2l inpo1

EXP

Insert explicit expressions (propagators, vertices, traces, expansions ...) & perform integration with IBP dDk1 2 D dDk2 2 D Insert hardcoded masterintegrals D3 D6 DN B4 E3

MATAD Collect renormalization constants Zs in FORM and proceed for

and
functions in

Mathematica

13

SLIDE 67

Toolchain

Defjne Lagrangian Generate diagrams with its symmetry factors

φ φ e e φ φ e e e e φ φ φ e e e e φ φ e e e e φ e φ e φ e e e e φ e φ e φ e e φ e e e e φ e φ φ e e e e e φ e e e φ e e e φ φ e e e e e φ e φ e φ e e φ φ φ φ e e e e φ φ φ φ e e e φ φ e φ φ φ φ φ φ e e e e e e φ

QGRAF

Assign Feyman rules and mass hiearchy Q ≫ q, m = FTij(p) = Denab(p, m) = Ta ij = dabcd Apply the rules and topologies

Q Q Q+k1 k2 k1

Q+k2

Q+k1-k2

= FT1i1i2(Q+k1)Ta i2i3 FT1i3i4(k2)Denab(Q+k1 −k2, m) . . .

Q2E

Asymptotic expansion 1 Map on topologies inpl tad1l inpla inpt1 tad2l inpo1

EXP

Insert explicit expressions (propagators, vertices, traces, expansions ...) & perform integration with IBP dDk1 2 D dDk2 2 D Insert hardcoded masterintegrals D3 D6 DN B4 E3

MATAD Collect renormalization constants Zs in FORM and proceed for

and
functions in

Mathematica

13

SLIDE 68

Toolchain

Defjne Lagrangian Generate diagrams with its symmetry factors

φ φ e e φ φ e e e e φ φ φ e e e e φ φ e e e e φ e φ e φ e e e e φ e φ e φ e e φ e e e e φ e φ φ e e e e e φ e e e φ e e e φ φ e e e e e φ e φ e φ e e φ φ φ φ e e e e φ φ φ φ e e e φ φ e φ φ φ φ φ φ e e e e e e φ

QGRAF

Assign Feyman rules and mass hiearchy Q ≫ q, m = FTij(p) = Denab(p, m) = Ta ij = dabcd Apply the rules and topologies

Q Q Q+k1 k2 k1

Q+k2

Q+k1-k2

= FT1i1i2(Q+k1)Ta i2i3 FT1i3i4(k2)Denab(Q+k1 −k2, m) . . .

Q2E

Asymptotic expansion = ⋆ + 1 ⋆ + ⋆ + ⋆ Map on topologies inpl tad1l inpla inpt1 tad2l inpo1

EXP

Insert explicit expressions (propagators, vertices, traces, expansions ...) & perform integration with IBP dDk1 2 D dDk2 2 D Insert hardcoded masterintegrals D3 D6 DN B4 E3

MATAD Collect renormalization constants Zs in FORM and proceed for

and
functions in

Mathematica

13

SLIDE 69

Toolchain

Defjne Lagrangian Generate diagrams with its symmetry factors

φ φ e e φ φ e e e e φ φ φ e e e e φ φ e e e e φ e φ e φ e e e e φ e φ e φ e e φ e e e e φ e φ φ e e e e e φ e e e φ e e e φ φ e e e e e φ e φ e φ e e φ φ φ φ e e e e φ φ φ φ e e e φ φ e φ φ φ φ φ φ e e e e e e φ

QGRAF

Assign Feyman rules and mass hiearchy Q ≫ q, m = FTij(p) = Denab(p, m) = Ta ij = dabcd Apply the rules and topologies

Q Q Q+k1 k2 k1

Q+k2

Q+k1-k2

= FT1i1i2(Q+k1)Ta i2i3 FT1i3i4(k2)Denab(Q+k1 −k2, m) . . .

Q2E

Asymptotic expansion = ⋆ + 1 ⋆ + ⋆ + ⋆ Map on topologies inpl tad1l inpla . . . inpt1 tad2l inpo1 . . .

EXP

Insert explicit expressions (propagators, vertices, traces, expansions ...) & perform integration with IBP dDk1 2 D dDk2 2 D Insert hardcoded masterintegrals D3 D6 DN B4 E3

MATAD Collect renormalization constants Zs in FORM and proceed for

and
functions in

Mathematica

13

SLIDE 70

Toolchain

Defjne Lagrangian Generate diagrams with its symmetry factors

φ φ e e φ φ e e e e φ φ φ e e e e φ φ e e e e φ e φ e φ e e e e φ e φ e φ e e φ e e e e φ e φ φ e e e e e φ e e e φ e e e φ φ e e e e e φ e φ e φ e e φ φ φ φ e e e e φ φ φ φ e e e φ φ e φ φ φ φ φ φ e e e e e e φ

QGRAF

Assign Feyman rules and mass hiearchy Q ≫ q, m = FTij(p) = Denab(p, m) = Ta ij = dabcd Apply the rules and topologies

Q Q Q+k1 k2 k1

Q+k2

Q+k1-k2

= FT1i1i2(Q+k1)Ta i2i3 FT1i3i4(k2)Denab(Q+k1 −k2, m) . . .

Q2E

Asymptotic expansion = ⋆ + 1 ⋆ + ⋆ + ⋆ Map on topologies inpl tad1l inpla . . . inpt1 tad2l inpo1 . . .

EXP

Insert explicit expressions (propagators, vertices, traces, expansions ...) & perform integration with IBP = dDk1 (2π)D dDk2 (2π)D . . . ⇒ Insert hardcoded masterintegrals D3, . . . , D6, DN, B4, E3, . . .

MATAD Collect renormalization constants Zs in FORM and proceed for

and
functions in

Mathematica

13

SLIDE 71

Toolchain

Defjne Lagrangian Generate diagrams with its symmetry factors

φ φ e e φ φ e e e e φ φ φ e e e e φ φ e e e e φ e φ e φ e e e e φ e φ e φ e e φ e e e e φ e φ φ e e e e e φ e e e φ e e e φ φ e e e e e φ e φ e φ e e φ φ φ φ e e e e φ φ φ φ e e e φ φ e φ φ φ φ φ φ e e e e e e φ

QGRAF

Assign Feyman rules and mass hiearchy Q ≫ q, m = FTij(p) = Denab(p, m) = Ta ij = dabcd Apply the rules and topologies

Q Q Q+k1 k2 k1

Q+k2

Q+k1-k2

= FT1i1i2(Q+k1)Ta i2i3 FT1i3i4(k2)Denab(Q+k1 −k2, m) . . .

Q2E

Asymptotic expansion = ⋆ + 1 ⋆ + ⋆ + ⋆ Map on topologies inpl tad1l inpla . . . inpt1 tad2l inpo1 . . .

EXP

Insert explicit expressions (propagators, vertices, traces, expansions ...) & perform integration with IBP = dDk1 (2π)D dDk2 (2π)D . . . ⇒ Insert hardcoded masterintegrals D3, . . . , D6, DN, B4, E3, . . .

MATAD ⇒ Collect renormalization constants Zs in FORM and proceed for β- and γ-functions in Mathematica

13

SLIDE 72

β-functions and renormalization functions

β- and γx = d ln Zx/d ln µ functions for chiral Ising [Mih+17]

βα = −ǫ α +

3 + 2Nf
α2 +
24λ2 −

9 8 + 6Nf

α2 − 24αλ
α + 1

64

(−697 + 912ζ3 + 2Nf (67

+ 112Nf + 432ζ3))α4 + 1152(7 + 5Nf )α3λ + 192(91 − 30Nf )α2λ2 − 13824αλ3 βλ = −ǫλ +

36λ2 + 4Nf αλ −Nf α2

+

4Nf α3 + 7Nf α2λ − 72Nf αλ2 − 816λ3

+ 1 32

Nf α4(5 − 628Nf − 384ζ3) + 2Nf α3(1736Nf − 4395 − 1872ζ3)λ

− 48Nf α2(72Nf − 361− 648ζ3)λ2+ 49536Nf αλ3 + 6912(145 + 96ζ3)λ4 γψ = α 2 −

α2

16 + 3Nf α2 4

+
α3(48ζ3 − 15 + 4(47 − 12Nf )Nf ) + 768α2λ − 2112αλ2

128 γφ = 2Nf α +

24λ2 − 5Nf α2

2

+
Nf α3(21 + 200Nf + 48ζ3) + 960Nf α2λ − 2880Nf αλ2 − 6912λ3

32 γφ2 = 12λ + 2

Nf α2 − 12Nf αλ − 72λ2

+ 4Nf α3(−9 + 4Nf + 3ζ3) + 3 2 Nf α2(11 − 24Nf + 120ζ3)λ + 72

4Nf αλ2 + 87λ3

36

2

4Nf Nf

2

... ...

14

SLIDE 73

β-functions and renormalization functions

β- and γx = d ln Zx/d ln µ functions for chiral Ising [Mih+17]

βα = −ǫ α +

3 + 2Nf
α2 +
24λ2 −

9 8 + 6Nf

α2 − 24αλ
α + 1

64

(−697 + 912ζ3 + 2Nf (67

+ 112Nf + 432ζ3))α4 + 1152(7 + 5Nf )α3λ + 192(91 − 30Nf )α2λ2 − 13824αλ3 βλ = −ǫλ +

36λ2 + 4Nf αλ −Nf α2

+

4Nf α3 + 7Nf α2λ − 72Nf αλ2 − 816λ3

+ 1 32

Nf α4(5 − 628Nf − 384ζ3) + 2Nf α3(1736Nf − 4395 − 1872ζ3)λ

− 48Nf α2(72Nf − 361− 648ζ3)λ2+ 49536Nf αλ3 + 6912(145 + 96ζ3)λ4 γψ = α 2 −

α2

16 + 3Nf α2 4

+
α3(48ζ3 − 15 + 4(47 − 12Nf )Nf ) + 768α2λ − 2112αλ2

128 γφ = 2Nf α +

24λ2 − 5Nf α2

2

+
Nf α3(21 + 200Nf + 48ζ3) + 960Nf α2λ − 2880Nf αλ2 − 6912λ3

32 γφ2 = 12λ + 2

Nf α2 − 12Nf αλ − 72λ2

+ 4Nf α3(−9 + 4Nf + 3ζ3) + 3 2 Nf α2(11 − 24Nf + 120ζ3)λ + 72

4Nf αλ2 + 87λ3

βλ = −ǫλ +

36λ2 + 4Nfαλ −Nfα2

+ . . .

φ φ φ φ φ φ φ φ φ φ φ φ

...

φ φ e e φ φ φ φ e e e e φ φ φ φ e e e e

...

14

SLIDE 74

Critical exponents

Correlation-length exponent and anomlous dimensions found at the NGFP (α∗, λ∗) ηψ = γψ(α∗, λ∗) = d ln Zψ d ln µ (α∗, λ∗) (Fermion anomalous dim.) ηφ = γφ(α∗, λ∗) = d ln Zφ d ln µ (α∗, λ∗) (Boson anomalous dim.) and from the dimensionless mass term

function

m2

2

2 m2

1

d m2 dm2 2

2

Recall: G r t r rd

2

E.g. for the chiral Ising universality class [Mih+17]

1

2 3 10Nf s 6 3 2Nf 513 7587Nf 666N2

f

5264N3

f

96N4

f

s 171 510Nf 436N2

f

48N3

f

108 3 2Nf

3s 2

2 3 2Nf 180 33s Nf 3 328Nf 2s 216 3 2Nf

3 2

2Nf 3 2Nf 27 594Nf 2916N2

f

88N3

f

9 57Nf 208N2

f s

36 3 2Nf

3s 2 15

SLIDE 75

Critical exponents

Correlation-length exponent and anomlous dimensions found at the NGFP (α∗, λ∗) ηψ = γψ(α∗, λ∗) = d ln Zψ d ln µ (α∗, λ∗) (Fermion anomalous dim.) ηφ = γφ(α∗, λ∗) = d ln Zφ d ln µ (α∗, λ∗) (Boson anomalous dim.) and from the dimensionless mass term β-function β ˜

m2 = (−2 + γφ − γφ2) ˜

m2 ν−1 = − dβ ˜

m2

d ˜ m2

(α∗,λ∗)

= 2 − ηφ + ηφ2 Recall: G(r, t) = Φ±(r/ξ) rd−2+η E.g. for the chiral Ising universality class [Mih+17]

1

2 3 10Nf s 6 3 2Nf 513 7587Nf 666N2

f

5264N3

f

96N4

f

s 171 510Nf 436N2

f

48N3

f

108 3 2Nf

3s 2

2 3 2Nf 180 33s Nf 3 328Nf 2s 216 3 2Nf

3 2

2Nf 3 2Nf 27 594Nf 2916N2

f

88N3

f

9 57Nf 208N2

f s

36 3 2Nf

3s 2 15

SLIDE 76

Critical exponents

Correlation-length exponent and anomlous dimensions found at the NGFP (α∗, λ∗) ηψ = γψ(α∗, λ∗) = d ln Zψ d ln µ (α∗, λ∗) (Fermion anomalous dim.) ηφ = γφ(α∗, λ∗) = d ln Zφ d ln µ (α∗, λ∗) (Boson anomalous dim.) and from the dimensionless mass term β-function β ˜

m2 = (−2 + γφ − γφ2) ˜

m2 ν−1 = − dβ ˜

m2

d ˜ m2

(α∗,λ∗)

= 2 − ηφ + ηφ2 Recall: G(r, t) = Φ±(r/ξ) rd−2+η E.g. for the chiral Ising universality class [Mih+17]

ν−1 =2− (3+10Nf +s)ǫ 6(3 + 2Nf ) − 513−7587Nf −666N2

f − 5264N3 f −96N4 f + s(171+510Nf +436N2 f +48N3 f )

108(3 + 2Nf )3s ǫ2 + . . . ηψ = ǫ 2(3 + 2Nf ) + 180 + 33s + Nf (3 − 328Nf + 2s) 216(3 + 2Nf )3 ǫ2 + . . . ηφ = 2Nf ǫ 3 + 2Nf + 27 + 594Nf + 2916N2

f + 88N3 f + (9 − 57Nf + 208N2 f )s

36(3 + 2Nf )3s ǫ2 + . . .

15

SLIDE 77

Asymptotic series

Take a closer look for graphene Nf = 2 in the chiral Ising case ν−1 ≈ 2 −0.952ǫ +0.00723ǫ2 −0.0949ǫ3 coeffjcients grow! ηψ ≈ 0.0714ǫ −0.00671ǫ2 −0.0243ǫ3 coeffjcients grow! ηφ ≈ 0.571ǫ +0.124ǫ2 −0.0278ǫ3 The series diverge asymptotically! (Remember we work in D 4 )

1 2 3 4 5 6 7 8 9 10 0.0 0.2 0.4 0.6 0.8 1.0 1.2 from 1 2 3 4 5 6 7 8 9 10 0.00 0.05 0.10 0.15 0.20 from 1 2 3 4 5 6 7 8 9 10 0.0 0.2 0.4 0.6 0.8 1.0 from

For D 3 the direct substitution 1 does not provide reliable values...

16

SLIDE 78

Asymptotic series

Take a closer look for graphene Nf = 2 in the chiral Ising case ν−1 ≈ 2 −0.952ǫ +0.00723ǫ2 −0.0949ǫ3 ← coeffjcients grow! ηψ ≈ 0.0714ǫ −0.00671ǫ2 −0.0243ǫ3 ← coeffjcients grow! ηφ ≈ 0.571ǫ +0.124ǫ2 −0.0278ǫ3 The series diverge asymptotically! (Remember we work in D 4 )

1 2 3 4 5 6 7 8 9 10 0.0 0.2 0.4 0.6 0.8 1.0 1.2 from 1 2 3 4 5 6 7 8 9 10 0.00 0.05 0.10 0.15 0.20 from 1 2 3 4 5 6 7 8 9 10 0.0 0.2 0.4 0.6 0.8 1.0 from

For D 3 the direct substitution 1 does not provide reliable values...

16

SLIDE 79

Asymptotic series

Take a closer look for graphene Nf = 2 in the chiral Ising case ν−1 ≈ 2 −0.952ǫ +0.00723ǫ2 −0.0949ǫ3 ← coeffjcients grow! ηψ ≈ 0.0714ǫ −0.00671ǫ2 −0.0243ǫ3 ← coeffjcients grow! ηφ ≈ 0.571ǫ +0.124ǫ2 −0.0278ǫ3 ⇒ The series diverge asymptotically! (Remember we work in D = 4 − ǫ)

1 2 3 4 5 6 7 8 9 10 0.0 0.2 0.4 0.6 0.8 1.0 1.2 from 1 2 3 4 5 6 7 8 9 10 0.00 0.05 0.10 0.15 0.20 from 1 2 3 4 5 6 7 8 9 10 0.0 0.2 0.4 0.6 0.8 1.0 from

For D 3 the direct substitution 1 does not provide reliable values...

16

SLIDE 80

Asymptotic series

Take a closer look for graphene Nf = 2 in the chiral Ising case ν−1 ≈ 2 −0.952ǫ +0.00723ǫ2 −0.0949ǫ3 ← coeffjcients grow! ηψ ≈ 0.0714ǫ −0.00671ǫ2 −0.0243ǫ3 ← coeffjcients grow! ηφ ≈ 0.571ǫ +0.124ǫ2 −0.0278ǫ3 ⇒ The series diverge asymptotically! (Remember we work in D = 4 − ǫ)

1 2 3 4 5 6 7 8 9 10 Nf 0.0 0.2 0.4 0.6 0.8 1.0 1.2 ai(Nf ) from 1/ν |a1| O(ϵ1) |a2| O(ϵ2) |a3| O(ϵ3) 1 2 3 4 5 6 7 8 9 10 Nf 0.00 0.05 0.10 0.15 0.20 ai(Nf ) from ηψ |a1| O(ϵ1) |a2| O(ϵ2) |a3| O(ϵ3) 1 2 3 4 5 6 7 8 9 10 Nf 0.0 0.2 0.4 0.6 0.8 1.0 ai(Nf ) from ηφ |a1| O(ϵ1) |a2| O(ϵ2) |a3| O(ϵ3)

For D 3 the direct substitution 1 does not provide reliable values...

16

SLIDE 81

Asymptotic series

Take a closer look for graphene Nf = 2 in the chiral Ising case ν−1 ≈ 2 −0.952ǫ +0.00723ǫ2 −0.0949ǫ3 ← coeffjcients grow! ηψ ≈ 0.0714ǫ −0.00671ǫ2 −0.0243ǫ3 ← coeffjcients grow! ηφ ≈ 0.571ǫ +0.124ǫ2 −0.0278ǫ3 ⇒ The series diverge asymptotically! (Remember we work in D = 4 − ǫ)

1 2 3 4 5 6 7 8 9 10 Nf 0.0 0.2 0.4 0.6 0.8 1.0 1.2 ai(Nf ) from 1/ν |a1| O(ϵ1) |a2| O(ϵ2) |a3| O(ϵ3) 1 2 3 4 5 6 7 8 9 10 Nf 0.00 0.05 0.10 0.15 0.20 ai(Nf ) from ηψ |a1| O(ϵ1) |a2| O(ϵ2) |a3| O(ϵ3) 1 2 3 4 5 6 7 8 9 10 Nf 0.0 0.2 0.4 0.6 0.8 1.0 ai(Nf ) from ηφ |a1| O(ϵ1) |a2| O(ϵ2) |a3| O(ϵ3)

For D = 3 the direct substitution ǫ = 1 does not provide reliable values...

16

SLIDE 82

Resummation

SLIDE 83

Taming an asymptotic series Why do asymptotic series occur in perturbation theory?

As an easy example consider

f(g) =

dx e−x2+gx4
Recall perturbative QFT:

j e

1 2

D

1 4

j

0.0 0.2 0.4 0.6 0.8 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Expand integrand w.r.t. g and interchange sum

and integration asymptotic series f L g

L k

fkgk

L k

gk k dx x4ke

x2

0.0 0.2 0.4 0.6 0.8 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

17

SLIDE 84

Taming an asymptotic series Why do asymptotic series occur in perturbation theory?

As an easy example consider

f(g) =

dx e−x2+gx4
Recall perturbative QFT:

Z[j] =

Dφe− 1

2 φ·D−1φ+λφ4+j·φ

0.0 0.2 0.4 0.6 0.8 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Expand integrand w.r.t. g and interchange sum

and integration asymptotic series f L g

L k

fkgk

L k

gk k dx x4ke

x2

0.0 0.2 0.4 0.6 0.8 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

17

SLIDE 85

Taming an asymptotic series Why do asymptotic series occur in perturbation theory?

As an easy example consider

f(g) =

dx e−x2+gx4
Recall perturbative QFT:

Z[j] =

Dφe− 1

2 φ·D−1φ+λφ4+j·φ

0.0 0.2 0.4 0.6 0.8 1.0 g 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 f(g)

Expand integrand w.r.t. g and interchange sum

and integration asymptotic series f L g

L k

fkgk

L k

gk k dx x4ke

x2

0.0 0.2 0.4 0.6 0.8 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

17

SLIDE 86

Taming an asymptotic series Why do asymptotic series occur in perturbation theory?

As an easy example consider

f(g) =

dx e−x2+gx4
Recall perturbative QFT:

Z[j] =

Dφe− 1

2 φ·D−1φ+λφ4+j·φ

0.0 0.2 0.4 0.6 0.8 1.0 g 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 f(g)

Expand integrand w.r.t. g and interchange sum

and integration ⇒ asymptotic series f [L](g) =

L

k=0

fkgk =

L

k=0

gk k!

dx x4ke−x2

0.0 0.2 0.4 0.6 0.8 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

17

SLIDE 87

Taming an asymptotic series Why do asymptotic series occur in perturbation theory?

As an easy example consider

f(g) =

dx e−x2+gx4
Recall perturbative QFT:

Z[j] =

Dφe− 1

2 φ·D−1φ+λφ4+j·φ

0.0 0.2 0.4 0.6 0.8 1.0 g 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 f(g)

Expand integrand w.r.t. g and interchange sum

and integration ⇒ asymptotic series f [L](g) =

L

k=0

fkgk =

L

k=0

gk k!

dx x4ke−x2

0.0 0.2 0.4 0.6 0.8 1.0 g 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 f(g) f(g) f [1](g) f [2](g) f [3](g)

17

SLIDE 88

Taming an asymptotic series: Padé Approximation How can we extract reliable estimates also for large g?

Padé Approximation

[M/N]f = a0 + a1g + · · · + aMgM 1 + b1g + · · · + bNgN

No knowledge about large-order behavior of

f L g but already a good approximation

0.0 0.2 0.4 0.6 0.8 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

For a better approximation we need knowledge about large-order behavior of f L g which

is of the form fk

k

k 1

kk k

The parameters

and can be extracted from f g by a saddle-point approximation

18

SLIDE 89

Taming an asymptotic series: Padé Approximation How can we extract reliable estimates also for large g?

Padé Approximation

[M/N]f = a0 + a1g + · · · + aMgM 1 + b1g + · · · + bNgN

No knowledge about large-order behavior of

f [L](g) but already a good approximation

0.0 0.2 0.4 0.6 0.8 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

For a better approximation we need knowledge about large-order behavior of f L g which

is of the form fk

k

k 1

kk k

The parameters

and can be extracted from f g by a saddle-point approximation

18

SLIDE 90

Taming an asymptotic series: Padé Approximation How can we extract reliable estimates also for large g?

Padé Approximation

[M/N]f = a0 + a1g + · · · + aMgM 1 + b1g + · · · + bNgN

No knowledge about large-order behavior of

f [L](g) but already a good approximation

0.0 0.2 0.4 0.6 0.8 1.0 g 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 f(g) f(g) [1/1]f (g) [1/2]f (g) [2/1]f (g)

For a better approximation we need knowledge about large-order behavior of f L g which

is of the form fk

k

k 1

kk k

The parameters

and can be extracted from f g by a saddle-point approximation

18

SLIDE 91

Taming an asymptotic series: Padé Approximation How can we extract reliable estimates also for large g?

Padé Approximation

[M/N]f = a0 + a1g + · · · + aMgM 1 + b1g + · · · + bNgN

No knowledge about large-order behavior of

f [L](g) but already a good approximation

0.0 0.2 0.4 0.6 0.8 1.0 g 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 f(g) f(g) [1/1]f (g) [1/2]f (g) [2/1]f (g)

For a better approximation we need knowledge about large-order behavior of f [L](g) which

is of the form fk ∼ (−α)kΓ(k + β + 1) ≈ (−α)kk!kβ

The parameters

and can be extracted from f g by a saddle-point approximation

18

SLIDE 92

Taming an asymptotic series: Padé Approximation How can we extract reliable estimates also for large g?

Padé Approximation

[M/N]f = a0 + a1g + · · · + aMgM 1 + b1g + · · · + bNgN

No knowledge about large-order behavior of

f [L](g) but already a good approximation

0.0 0.2 0.4 0.6 0.8 1.0 g 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 f(g) f(g) [1/1]f (g) [1/2]f (g) [2/1]f (g)

For a better approximation we need knowledge about large-order behavior of f [L](g) which

is of the form fk ∼ (−α)kΓ(k + β + 1) ≈ (−α)kk!kβ

The parameters α and β can be extracted from f(g) by a saddle-point approximation

18

SLIDE 93

Taming an asymptotic series: Padé Approximation How can we extract reliable estimates also for large g?

Factorial grow ⇒ Borel-Leroy-summable

Bf (g) =

k

fk Γ(k + β + 1) gk =

k

Bkgk

Borel-Transformation

f g

dt e

tBf gt

0.0 0.2 0.4 0.6 0.8 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

For fjnite L this transformation just reinserts

k 1 . To avoid this one have to fjnd a nonpolynomial function which Taylor expansion coincides with Bf g

Use a Padé approximant for this, giving the so-called Padé-Borel Transformation

M N f

g dt e

t M N Bf gt

This is still ignoring the detailed large-order behavior fk

kk k

19

SLIDE 94

Taming an asymptotic series: Padé Approximation How can we extract reliable estimates also for large g?

Factorial grow ⇒ Borel-Leroy-summable

Bf (g) =

k

fk Γ(k + β + 1) gk =

k

Bkgk

Borel-Transformation

Bf (g) = ∞ dt e−tBf (gt) .

0.0 0.2 0.4 0.6 0.8 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

For fjnite L this transformation just reinserts

k 1 . To avoid this one have to fjnd a nonpolynomial function which Taylor expansion coincides with Bf g

Use a Padé approximant for this, giving the so-called Padé-Borel Transformation

M N f

g dt e

t M N Bf gt

This is still ignoring the detailed large-order behavior fk

kk k

19

SLIDE 95

Taming an asymptotic series: Padé Approximation How can we extract reliable estimates also for large g?

Factorial grow ⇒ Borel-Leroy-summable

Bf (g) =

k

fk Γ(k + β + 1) gk =

k

Bkgk

Borel-Transformation

Bf (g) = ∞ dt e−tBf (gt) .

0.0 0.2 0.4 0.6 0.8 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

For fjnite L this transformation just reinserts Γ(k + β + 1). To avoid this one have to

fjnd a nonpolynomial function which Taylor expansion coincides with Bf (g)

Use a Padé approximant for this, giving the so-called Padé-Borel Transformation

M N f

g dt e

t M N Bf gt

This is still ignoring the detailed large-order behavior fk

kk k

19

SLIDE 96

Taming an asymptotic series: Padé Approximation How can we extract reliable estimates also for large g?

Factorial grow ⇒ Borel-Leroy-summable

Bf (g) =

k

fk Γ(k + β + 1) gk =

k

Bkgk

Borel-Transformation

Bf (g) = ∞ dt e−tBf (gt) .

0.0 0.2 0.4 0.6 0.8 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

For fjnite L this transformation just reinserts Γ(k + β + 1). To avoid this one have to

fjnd a nonpolynomial function which Taylor expansion coincides with Bf (g)

Use a Padé approximant for this, giving the so-called Padé-Borel Transformation

PB[M/N]

f

(g) = ∞ dt e−t[M/N]Bf (gt)

This is still ignoring the detailed large-order behavior fk

kk k

19

SLIDE 97

Taming an asymptotic series: Padé Approximation How can we extract reliable estimates also for large g?

Factorial grow ⇒ Borel-Leroy-summable

Bf (g) =

k

fk Γ(k + β + 1) gk =

k

Bkgk

Borel-Transformation

Bf (g) = ∞ dt e−tBf (gt) .

0.0 0.2 0.4 0.6 0.8 1.0 g 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 f(g) f(g) PB[1/1]

f

(g) PB[1/2]

f

(g) PB[2/1]

f

(g)

For fjnite L this transformation just reinserts Γ(k + β + 1). To avoid this one have to

fjnd a nonpolynomial function which Taylor expansion coincides with Bf (g)

Use a Padé approximant for this, giving the so-called Padé-Borel Transformation

PB[M/N]

f

(g) = ∞ dt e−t[M/N]Bf (gt)

This is still ignoring the detailed large-order behavior fk

kk k

19

SLIDE 98

Taming an asymptotic series: Padé Approximation How can we extract reliable estimates also for large g?

Factorial grow ⇒ Borel-Leroy-summable

Bf (g) =

k

fk Γ(k + β + 1) gk =

k

Bkgk

Borel-Transformation

Bf (g) = ∞ dt e−tBf (gt) .

0.0 0.2 0.4 0.6 0.8 1.0 g 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 f(g) f(g) PB[1/1]

f

(g) PB[1/2]

f

(g) PB[2/1]

f

(g)

For fjnite L this transformation just reinserts Γ(k + β + 1). To avoid this one have to

fjnd a nonpolynomial function which Taylor expansion coincides with Bf (g)

Use a Padé approximant for this, giving the so-called Padé-Borel Transformation

PB[M/N]

f

(g) = ∞ dt e−t[M/N]Bf (gt)

This is still ignoring the detailed large-order behavior fk ∼ (−α)kk!kβ

19

SLIDE 99

Taming an asymptotic series: Conformal mapping How to incorporate the full large-order behavior?

Borel-Leroy sum for large k

Bf (g) ∼

k large

k

(−α)kgk = 1 1 + αg

Conformal mapping regarding with singularity

g 1 w g 1 ag 1 1 ag 1 g 4 a w 1 w 2

0.0 0.2 0.4 0.6 0.8 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 0.96 0.98 1.00 1.36 1.38 1.40

Implies that W w

Bf g w has a convergent Taylor expansion for w 1

Reexpand the Borel-Leroy sum in this new variable w g

B L

f

g

L k

Bkgk

L k

Wk w g

k

Borel-Transformation with this for all g analytical series gives

L f

g

L k

Wk dt e

t w gt k

20

SLIDE 100

Taming an asymptotic series: Conformal mapping How to incorporate the full large-order behavior?

Borel-Leroy sum for large k

Bf (g) ∼

k large

k

(−α)kgk = 1 1 + αg

Conformal mapping regarding with singularity

g = −1/α w(g) = √1 + ag − 1 √1 + ag + 1 , g = 4 a w (1 − w)2

0.0 0.2 0.4 0.6 0.8 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 0.96 0.98 1.00 1.36 1.38 1.40

Implies that W w

Bf g w has a convergent Taylor expansion for w 1

Reexpand the Borel-Leroy sum in this new variable w g

B L

f

g

L k

Bkgk

L k

Wk w g

k

Borel-Transformation with this for all g analytical series gives

L f

g

L k

Wk dt e

t w gt k

20

SLIDE 101

Taming an asymptotic series: Conformal mapping How to incorporate the full large-order behavior?

Borel-Leroy sum for large k

Bf (g) ∼

k large

k

(−α)kgk = 1 1 + αg

Conformal mapping regarding with singularity

g = −1/α w(g) = √1 + ag − 1 √1 + ag + 1 , g = 4 a w (1 − w)2

0.0 0.2 0.4 0.6 0.8 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 0.96 0.98 1.00 1.36 1.38 1.40

Implies that W(w) = Bf (g(w)) has a convergent Taylor expansion for |w| < 1
Reexpand the Borel-Leroy sum in this new variable w g

B L

f

g

L k

Bkgk

L k

Wk w g

k

Borel-Transformation with this for all g analytical series gives

L f

g

L k

Wk dt e

t w gt k

20

SLIDE 102

Taming an asymptotic series: Conformal mapping How to incorporate the full large-order behavior?

Borel-Leroy sum for large k

Bf (g) ∼

k large

k

(−α)kgk = 1 1 + αg

Conformal mapping regarding with singularity

g = −1/α w(g) = √1 + ag − 1 √1 + ag + 1 , g = 4 a w (1 − w)2

0.0 0.2 0.4 0.6 0.8 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 0.96 0.98 1.00 1.36 1.38 1.40

Implies that W(w) = Bf (g(w)) has a convergent Taylor expansion for |w| < 1
Reexpand the Borel-Leroy sum in this new variable w(g)

B[L]

f (g) = L

k=0

Bkgk =

L

k=0

Wk[w(g)]k

Borel-Transformation with this for all g analytical series gives

L f

g

L k

Wk dt e

t w gt k

20

SLIDE 103

Taming an asymptotic series: Conformal mapping How to incorporate the full large-order behavior?

Borel-Leroy sum for large k

Bf (g) ∼

k large

k

(−α)kgk = 1 1 + αg

Conformal mapping regarding with singularity

g = −1/α w(g) = √1 + ag − 1 √1 + ag + 1 , g = 4 a w (1 − w)2

0.0 0.2 0.4 0.6 0.8 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 0.96 0.98 1.00 1.36 1.38 1.40

Implies that W(w) = Bf (g(w)) has a convergent Taylor expansion for |w| < 1
Reexpand the Borel-Leroy sum in this new variable w(g)

B[L]

f (g) = L

k=0

Bkgk =

L

k=0

Wk[w(g)]k

Borel-Transformation with this for all g analytical series gives

CM[L]

f (g) = L

k=0

Wk ∞ dt e−t[w(gt)]k

20

SLIDE 104

Taming an asymptotic series: Conformal mapping How to incorporate the full large-order behavior?

Borel-Leroy sum for large k

Bf (g) ∼

k large

k

(−α)kgk = 1 1 + αg

Conformal mapping regarding with singularity

g = −1/α w(g) = √1 + ag − 1 √1 + ag + 1 , g = 4 a w (1 − w)2

0.0 0.2 0.4 0.6 0.8 1.0 g 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 f(g) f(g) CM[1]

f (g)

CM[2]

f (g)

CM[3]

f (g)

0.96 0.98 1.00 1.36 1.38 1.40

Implies that W(w) = Bf (g(w)) has a convergent Taylor expansion for |w| < 1
Reexpand the Borel-Leroy sum in this new variable w(g)

B[L]

f (g) = L

k=0

Bkgk =

L

k=0

Wk[w(g)]k

Borel-Transformation with this for all g analytical series gives

CM[L]

f (g) = L

k=0

Wk ∞ dt e−t[w(gt)]k

20

SLIDE 105

Results and Conclusion

SLIDE 106

Results fpor chiral Ising and Nf = 2

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 D 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 1/ν (4 − ϵ) (1st order) (4 − ϵ) (2nd order) (4 − ϵ) (3rd order) 2.8 3.0 3.2 0.8 1.0 1.2

21

SLIDE 107

Results fpor chiral Ising and Nf = 2

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 D 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 1/ν (4 − ϵ) (1st order) (4 − ϵ) (2nd order) (4 − ϵ) (3rd order) (2 + ϵ) (4th order) 2.8 3.0 3.2 0.8 1.0 1.2

21

SLIDE 108

Results fpor chiral Ising and Nf = 2

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 D 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 1/ν (4 − ϵ) (1st order) (4 − ϵ) (2nd order) (4 − ϵ) (3rd order) (2 + ϵ) (4th order) (4 − ϵ) Padé [1/1] (4 − ϵ) Padé [1/2] (4 − ϵ) Padé [2/1] 2.8 3.0 3.2 0.8 1.0 1.2

21

SLIDE 109

Results fpor chiral Ising and Nf = 2

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 D 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 1/ν (4 − ϵ) (1st order) (4 − ϵ) (2nd order) (4 − ϵ) (3rd order) (2 + ϵ) (4th order) (4 − ϵ) Padé [1/1] (4 − ϵ) Padé [1/2] (4 − ϵ) Padé [2/1] PB9/2

[1/1](D)

PB9/2

[2/1](D)

2.8 3.0 3.2 0.8 1.0 1.2

21

SLIDE 110

Results fpor chiral Ising and Nf = 2

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 D 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 1/ν (4 − ϵ) (1st order) (4 − ϵ) (2nd order) (4 − ϵ) (3rd order) (2 + ϵ) (4th order) (4 − ϵ) Padé [1/1] (4 − ϵ) Padé [1/2] (4 − ϵ) Padé [2/1] PB9/2

[1/1](D)

PB9/2

[2/1](D)

conf. map.

2.8 3.0 3.2 0.8 1.0 1.2

21

SLIDE 111

Results fpor chiral Ising and Nf = 2

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 D 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 1/ν (4 − ϵ) (1st order) (4 − ϵ) (2nd order) (4 − ϵ) (3rd order) (2 + ϵ) (4th order) (4 − ϵ) Padé [1/1] (4 − ϵ) Padé [1/2] (4 − ϵ) Padé [2/1] PB9/2

[1/1](D)

PB9/2

[2/1](D)

conf. map.

P4,3(D) [4/4](D) 2.8 3.0 3.2 0.8 1.0 1.2

21

SLIDE 112

Results fpor chiral Ising and Nf = 2

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 D 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 1/ν (4 − ϵ) (1st order) (4 − ϵ) (2nd order) (4 − ϵ) (3rd order) (2 + ϵ) (4th order) (4 − ϵ) Padé [1/1] (4 − ϵ) Padé [1/2] (4 − ϵ) Padé [2/1] PB9/2

[1/1](D)

PB9/2

[2/1](D)

conf. map.

P4,3(D) [4/4](D) Monte Carlo 2.8 3.0 3.2 0.8 1.0 1.2

21

SLIDE 113

Results fpor chiral Ising and Nf = 2

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 D 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 1/ν (4 − ϵ) (1st order) (4 − ϵ) (2nd order) (4 − ϵ) (3rd order) (2 + ϵ) (4th order) (4 − ϵ) Padé [1/1] (4 − ϵ) Padé [1/2] (4 − ϵ) Padé [2/1] PB9/2

[1/1](D)

PB9/2

[2/1](D)

conf. map.

P4,3(D) [4/4](D) Monte Carlo

conf. bootstrap

2.8 3.0 3.2 0.8 1.0 1.2

21

SLIDE 114

Results fpor chiral Ising and Nf = 2

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 D 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 1/ν (4 − ϵ) (1st order) (4 − ϵ) (2nd order) (4 − ϵ) (3rd order) (2 + ϵ) (4th order) (4 − ϵ) Padé [1/1] (4 − ϵ) Padé [1/2] (4 − ϵ) Padé [2/1] PB9/2

[1/1](D)

PB9/2

[2/1](D)

conf. map.

P4,3(D) [4/4](D) Monte Carlo

conf. bootstrap

FRG 2.8 3.0 3.2 0.8 1.0 1.2

21

SLIDE 115

Results for chiral Ising and Nf = 2

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 D 0.00 0.02 0.04 0.06 0.08 0.10 ηψ (4 − ϵ) (1st order) (4 − ϵ) (2nd order) (4 − ϵ) (3rd order)

22

SLIDE 116

Results for chiral Ising and Nf = 2

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 D 0.00 0.02 0.04 0.06 0.08 0.10 ηψ (4 − ϵ) (1st order) (4 − ϵ) (2nd order) (4 − ϵ) (3rd order) (2 + ϵ) (3rd order) (2 + ϵ) (4th order)

22

SLIDE 117

Results for chiral Ising and Nf = 2

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 D 0.00 0.02 0.04 0.06 0.08 0.10 ηψ (4 − ϵ) (1st order) (4 − ϵ) (2nd order) (4 − ϵ) (3rd order) (2 + ϵ) (3rd order) (2 + ϵ) (4th order) (4 − ϵ) Padé [1/1] (4 − ϵ) Padé [1/2]

22

SLIDE 118

Results for chiral Ising and Nf = 2

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 D 0.00 0.02 0.04 0.06 0.08 0.10 ηψ (4 − ϵ) (1st order) (4 − ϵ) (2nd order) (4 − ϵ) (3rd order) (2 + ϵ) (3rd order) (2 + ϵ) (4th order) (4 − ϵ) Padé [1/1] (4 − ϵ) Padé [1/2] PB7/2

[1/1](D)

PB7/2

[1/2](D)

22

SLIDE 119

Results for chiral Ising and Nf = 2

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 D 0.00 0.02 0.04 0.06 0.08 0.10 ηψ (4 − ϵ) (1st order) (4 − ϵ) (2nd order) (4 − ϵ) (3rd order) (2 + ϵ) (3rd order) (2 + ϵ) (4th order) (4 − ϵ) Padé [1/1] (4 − ϵ) Padé [1/2] PB7/2

[1/1](D)

PB7/2

[1/2](D)

conf. map.

22

SLIDE 120

Results for chiral Ising and Nf = 2

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 D 0.00 0.02 0.04 0.06 0.08 0.10 ηψ (4 − ϵ) (1st order) (4 − ϵ) (2nd order) (4 − ϵ) (3rd order) (2 + ϵ) (3rd order) (2 + ϵ) (4th order) (4 − ϵ) Padé [1/1] (4 − ϵ) Padé [1/2] PB7/2

[1/1](D)

PB7/2

[1/2](D)

conf. map.

P4,3(D)

22

SLIDE 121

Results for chiral Ising and Nf = 2

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 D 0.00 0.02 0.04 0.06 0.08 0.10 ηψ (4 − ϵ) (1st order) (4 − ϵ) (2nd order) (4 − ϵ) (3rd order) (2 + ϵ) (3rd order) (2 + ϵ) (4th order) (4 − ϵ) Padé [1/1] (4 − ϵ) Padé [1/2] PB7/2

[1/1](D)

PB7/2

[1/2](D)

conf. map.

P4,3(D)

conf. bootstrap

22

SLIDE 122

Results for chiral Ising and Nf = 2

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 D 0.00 0.02 0.04 0.06 0.08 0.10 ηψ (4 − ϵ) (1st order) (4 − ϵ) (2nd order) (4 − ϵ) (3rd order) (2 + ϵ) (3rd order) (2 + ϵ) (4th order) (4 − ϵ) Padé [1/1] (4 − ϵ) Padé [1/2] PB7/2

[1/1](D)

PB7/2

[1/2](D)

conf. map.

P4,3(D)

conf. bootstrap

FRG

22

SLIDE 123

Results for chiral Ising and Nf = 2

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 D 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 ηφ (4 − ϵ) (1st order) (4 − ϵ) (2nd order) (4 − ϵ) (3rd order) 2.9 3.0 3.1 0.4 0.6 0.8

23

SLIDE 124

Results for chiral Ising and Nf = 2

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 D 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 ηφ (4 − ϵ) (1st order) (4 − ϵ) (2nd order) (4 − ϵ) (3rd order) (2 + ϵ) (3rd order) (2 + ϵ) (4th order) 2.9 3.0 3.1 0.4 0.6 0.8

23

SLIDE 125

Results for chiral Ising and Nf = 2

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 D 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 ηφ (4 − ϵ) (1st order) (4 − ϵ) (2nd order) (4 − ϵ) (3rd order) (2 + ϵ) (3rd order) (2 + ϵ) (4th order) (4 − ϵ) Padé [1/1] (4 − ϵ) Padé [1/2] (4 − ϵ) Padé [2/1] 2.9 3.0 3.1 0.4 0.6 0.8

23

SLIDE 126

Results for chiral Ising and Nf = 2

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 D 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 ηφ (4 − ϵ) (1st order) (4 − ϵ) (2nd order) (4 − ϵ) (3rd order) (2 + ϵ) (3rd order) (2 + ϵ) (4th order) (4 − ϵ) Padé [1/1] (4 − ϵ) Padé [1/2] (4 − ϵ) Padé [2/1] PB7/2

[1/2](D)

PB7/2

[2/1](D)

2.9 3.0 3.1 0.4 0.6 0.8

23

SLIDE 127

Results for chiral Ising and Nf = 2

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 D 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 ηφ (4 − ϵ) (1st order) (4 − ϵ) (2nd order) (4 − ϵ) (3rd order) (2 + ϵ) (3rd order) (2 + ϵ) (4th order) (4 − ϵ) Padé [1/1] (4 − ϵ) Padé [1/2] (4 − ϵ) Padé [2/1] PB7/2

[1/2](D)

PB7/2

[2/1](D)

conf. map.

2.9 3.0 3.1 0.4 0.6 0.8

23

SLIDE 128

Results for chiral Ising and Nf = 2

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 D 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 ηφ (4 − ϵ) (1st order) (4 − ϵ) (2nd order) (4 − ϵ) (3rd order) (2 + ϵ) (3rd order) (2 + ϵ) (4th order) (4 − ϵ) Padé [1/1] (4 − ϵ) Padé [1/2] (4 − ϵ) Padé [2/1] PB7/2

[1/2](D)

PB7/2

[2/1](D)

conf. map.

P4,3(D) [4/4]ηφ(D) 2.9 3.0 3.1 0.4 0.6 0.8

23

SLIDE 129

Results for chiral Ising and Nf = 2

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 D 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 ηφ (4 − ϵ) (1st order) (4 − ϵ) (2nd order) (4 − ϵ) (3rd order) (2 + ϵ) (3rd order) (2 + ϵ) (4th order) (4 − ϵ) Padé [1/1] (4 − ϵ) Padé [1/2] (4 − ϵ) Padé [2/1] PB7/2

[1/2](D)

PB7/2

[2/1](D)

conf. map.

P4,3(D) [4/4]ηφ(D) Monte Carlo 2.9 3.0 3.1 0.4 0.6 0.8

23

SLIDE 130

Results for chiral Ising and Nf = 2

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 D 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 ηφ (4 − ϵ) (1st order) (4 − ϵ) (2nd order) (4 − ϵ) (3rd order) (2 + ϵ) (3rd order) (2 + ϵ) (4th order) (4 − ϵ) Padé [1/1] (4 − ϵ) Padé [1/2] (4 − ϵ) Padé [2/1] PB7/2

[1/2](D)

PB7/2

[2/1](D)

conf. map.

P4,3(D) [4/4]ηφ(D) Monte Carlo

conf. bootstrap

2.9 3.0 3.1 0.4 0.6 0.8

23

SLIDE 131

Results for chiral Ising and Nf = 2

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 D 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 ηφ (4 − ϵ) (1st order) (4 − ϵ) (2nd order) (4 − ϵ) (3rd order) (2 + ϵ) (3rd order) (2 + ϵ) (4th order) (4 − ϵ) Padé [1/1] (4 − ϵ) Padé [1/2] (4 − ϵ) Padé [2/1] PB7/2

[1/2](D)

PB7/2

[2/1](D)

conf. map.

P4,3(D) [4/4]ηφ(D) Monte Carlo

conf. bootstrap

FRG 2.9 3.0 3.1 0.4 0.6 0.8

23

SLIDE 132

Results for chiral Ising

Results for chiral Ising at Nf = 2 and D = 3

Universality class Method 1/ν ηψ ηφ Chiral Ising (4 − ǫ) 1st order [Mih+17] 1.048 0.0714 0.571 (4 − ǫ) 2nd order [Mih+17] 1.055 0.0647 0.695 (4 − ǫ) 3rd order [Mih+17] 0.960 0.0404 0.667 (4 − ǫ) Padé [1/1] [Mih+17] 1.055 0.0653 0.729 (4 − ǫ) Padé [1/2] [Mih+17] 0.919 0.0495 0.650 (2 + ǫ) 4th order [GLS16] 1.377 0.0817 0.874 PBβ

[1/1] [Mih+17]

1.055 0.0654 – PBβ

[1/2] [Mih+17]

– 0.0517 0.644 PBβ

[2/1] [Mih+17]

1.048 – 0.673 conformal mapping CM[3]

β [Mih+17]

1.108 0.0602 0.609 (4 − ǫ) Padé [2/1] [Mih+17] 1.048 – 0.672 polynomial interpolation P4,3 [GLS16]; [Mih+17] 0.977 0.0451 0.723 Padé-interpolation [4/4] [GLS16]; [Mih+17] 1.026 – 0.734 Monte Carlo [Kär+94] 1.00(4) – 0.754(8) Monte Carlo [CL13] 1.20(1) 0.38(1) 0.62(1) Conformal Bootstrap [Ili+17] 0.88 0.044 0.742 Functional RG [Kno16] 0.994(2) 0.0276 0.7765

24

SLIDE 133

Results for chiral Ising

Results for chiral Ising at Nf = 2 and D = 3

Universality class Method 1/ν ηψ ηφ Chiral Ising (4 − ǫ) 1st order [Mih+17] 1.048 0.0714 0.571 (4 − ǫ) 2nd order [Mih+17] 1.055 0.0647 0.695 (4 − ǫ) 3rd order [Mih+17] 0.960 0.0404 0.667 (4 − ǫ) Padé [1/1] [Mih+17] 1.055 0.0653 0.729 (4 − ǫ) Padé [1/2] [Mih+17] 0.919 0.0495 0.650 (2 + ǫ) 4th order [GLS16] 1.377 0.0817 0.874 PBβ

[1/1] [Mih+17]

1.055 0.0654 – PBβ

[1/2] [Mih+17]

– 0.0517 0.644 PBβ

[2/1] [Mih+17]

1.048 – 0.673 conformal mapping → CM[3]

β [Mih+17]

1.108 0.0602 0.609 (4 − ǫ) Padé [2/1] [Mih+17] 1.048 – 0.672 polynomial interpolation P4,3 [GLS16]; [Mih+17] 0.977 0.0451 0.723 Padé-interpolation [4/4] [GLS16]; [Mih+17] 1.026 – 0.734 Monte Carlo [Kär+94] 1.00(4) – 0.754(8) Monte Carlo [CL13] 1.20(1) 0.38(1) 0.62(1) Conformal Bootstrap [Ili+17] 0.88 0.044 0.742 Functional RG [Kno16] 0.994(2) 0.0276 0.7765

24

SLIDE 134

Results for chiral Ising

Results for chiral Ising at Nf = 2 and D = 3

Universality class Method 1/ν ηψ ηφ Chiral Ising (4 − ǫ) 1st order [Mih+17] 1.048 0.0714 0.571 (4 − ǫ) 2nd order [Mih+17] 1.055 0.0647 0.695 (4 − ǫ) 3rd order [Mih+17] 0.960 0.0404 0.667 (4 − ǫ) Padé [1/1] [Mih+17] 1.055 0.0653 0.729 (4 − ǫ) Padé [1/2] [Mih+17] 0.919 0.0495 0.650 (2 + ǫ) 4th order [GLS16] 1.377 0.0817 0.874 PBβ

[1/1] [Mih+17]

1.055 0.0654 – PBβ

[1/2] [Mih+17]

– 0.0517 0.644 PBβ

[2/1] [Mih+17]

1.048 – 0.673 conformal mapping → CM[3]

β [Mih+17]

1.108 0.0602 0.609 (4 − ǫ) Padé [2/1] [Mih+17] 1.048 – 0.672 polynomial interpolation → P4,3 [GLS16]; [Mih+17] 0.977 0.0451 0.723 Padé-interpolation [4/4] [GLS16]; [Mih+17] 1.026 – 0.734 Monte Carlo [Kär+94] 1.00(4) – 0.754(8) Monte Carlo [CL13] 1.20(1) 0.38(1) 0.62(1) Conformal Bootstrap [Ili+17] 0.88 0.044 0.742 Functional RG [Kno16] 0.994(2) 0.0276 0.7765

24

SLIDE 135

Results for chiral Ising

Results for chiral Ising at Nf = 2 and D = 3

Universality class Method 1/ν ηψ ηφ Chiral Ising (4 − ǫ) 1st order [Mih+17] 1.048 0.0714 0.571 (4 − ǫ) 2nd order [Mih+17] 1.055 0.0647 0.695 (4 − ǫ) 3rd order [Mih+17] 0.960 0.0404 0.667 (4 − ǫ) Padé [1/1] [Mih+17] 1.055 0.0653 0.729 (4 − ǫ) Padé [1/2] [Mih+17] 0.919 0.0495 0.650 (2 + ǫ) 4th order [GLS16] 1.377 0.0817 0.874 PBβ

[1/1] [Mih+17]

1.055 0.0654 – PBβ

[1/2] [Mih+17]

– 0.0517 0.644 PBβ

[2/1] [Mih+17]

1.048 – 0.673 conformal mapping → CM[3]

β [Mih+17]

1.108 0.0602 0.609 (4 − ǫ) Padé [2/1] [Mih+17] 1.048 – 0.672 polynomial interpolation → P4,3 [GLS16]; [Mih+17] 0.977 0.0451 0.723 Padé-interpolation → [4/4] [GLS16]; [Mih+17] 1.026 – 0.734 Monte Carlo [Kär+94] 1.00(4) – 0.754(8) Monte Carlo [CL13] 1.20(1) 0.38(1) 0.62(1) Conformal Bootstrap [Ili+17] 0.88 0.044 0.742 Functional RG [Kno16] 0.994(2) 0.0276 0.7765

24

SLIDE 136

Conclusion Conclusion

GNY model paradigmatic example for interacting QFTs and essential for comprehensive

understanding

QFTs are effective theories also found in condensed matter systems
Adding massless Fermions to the O(N) model give rise to new universality classes
Lots of activity (MC, cBS, FRG) to gain agreement

Outlook

Extend computer algebra to 4 loops (in progress)
Consider also chiral XY (in progress)
Find large-order behavior of the GNY model (ideas ???)

⇒ GOAL: Gain a more comprehensive understanding of QFTs and critical phenomena!

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