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

• Dynamical mass gap generation spontaneously Graphene: Electron-Electron-Interactions i breaks the chiral symmetry or spin density wave (SDW) charge density wave (CDW) [Raghu et al. ’07] Focus on 2D Dirac material graphene: 4 4 QSH QSH • Electron-electron Interaction [Herbut ’06] � � V 2 2 CDW e 2 ( 1 − δ � Y ) X ,� SDW � n σ ( � n σ ′ ( � H 1 = X ) U δ � Y + Y ) X ,� 4 π | � X − � SM Y | 0 � X ,� Y ,σ,σ ′ SDW 0 CDW 0 1 2 2 4 � � � = U n i , ↑ n i , ↓ + V 1 n i ,σ n j ,σ ′ + V 2 n i ,σ n j ,σ ′ + . . . U V 1 � i , j � ,σ,σ ′ �� i , j �� ,σ,σ ′ • Critical interaction strength for transition to φ = � ¯ ψψ � ∝ u † σ u σ − v † σ v σ � φ = � ¯ σ ⊗ I 4 ) ψ � ∝ u † σ ττ ′ u τ ′ − v † ψ ( � τ � τ � σ ττ ′ v τ ′

Graphene: Electron-Electron-Interactions i breaks the chiral symmetry or spin density wave (SDW) charge density wave (CDW) [Raghu et al. ’07] Focus on 2D Dirac material graphene: 4 4 QSH QSH • Electron-electron Interaction [Herbut ’06] � � V 2 2 CDW e 2 ( 1 − δ � Y ) X ,� SDW � n σ ( � n σ ′ ( � H 1 = X ) U δ � Y + Y ) X ,� 4 π | � X − � SM Y | 0 � X ,� Y ,σ,σ ′ SDW 0 CDW 0 1 2 2 4 � � � = U n i , ↑ n i , ↓ + V 1 n i ,σ n j ,σ ′ + V 2 n i ,σ n j ,σ ′ + . . . U V 1 � i , j � ,σ,σ ′ �� i , j �� ,σ,σ ′ • Critical interaction strength for transition to φ = � ¯ ψψ � ∝ u † σ u σ − v † σ v σ � φ = � ¯ σ ⊗ I 4 ) ψ � ∝ u † σ ττ ′ u τ ′ − v † ψ ( � τ � τ � σ ττ ′ v τ ′ • Dynamical mass gap generation spontaneously

• Couple fjelds to the Fermion by a Yukawa interaction • This can be generalized to arbitrary number of Fermion fmavors N f (graphene N f • Full action is then the Gross-Neveu-Yukawa model (GNY) Graphene: Effective fjeld theory Y a I 4 a I 2 N f 0 B I 2 g 1 2 m 2 2 4 4 I 2 N f I 2 I 4 Y Y g I 2 I 4 (in CDW) g a a I 4 (in SDW) 2) 5 • Incorporate order parameter fjelds with an semiphenomenological ansatz 2 φ a ( m 2 − ∂ µ ) φ a + λ L B = 1 4 ! ( φ a φ a ) 2 → Dynamics and self-interaction develop below certain energy scale

• This can be generalized to arbitrary number of Fermion fmavors N f (graphene N f • Full action is then the Gross-Neveu-Yukawa model (GNY) Graphene: Effective fjeld theory B a I 2 N f 0 Y g a 1 2 m 2 2 4 4 I 4 I 2 I 2 N f 2) (in SDW) (in CDW) I 2 I 4 5 • Incorporate order parameter fjelds with an semiphenomenological ansatz 2 φ a ( m 2 − ∂ µ ) φ a + λ L B = 1 4 ! ( φ a φ a ) 2 → Dynamics and self-interaction develop below certain energy scale • Couple fjelds to the Fermion by a Yukawa interaction L Y = g φ ¯ ψ ( I 2 ⊗ I 4 ) ψ ψ ( σ a ⊗ I 4 ) ψ L Y = g φ a ¯

• Full action is then the Gross-Neveu-Yukawa model (GNY) Graphene: Effective fjeld theory 4 4 2 m 2 2 1 g B Y 0 5 (in CDW) (in SDW) • Incorporate order parameter fjelds with an semiphenomenological ansatz 2 φ a ( m 2 − ∂ µ ) φ a + λ L B = 1 4 ! ( φ a φ a ) 2 → Dynamics and self-interaction develop below certain energy scale • Couple fjelds to the Fermion by a Yukawa interaction L Y = g φ ¯ ψ ( I 2 ⊗ I 4 ) ψ ψ ( σ a ⊗ I 4 ) ψ L Y = g φ a ¯ • This can be generalized to arbitrary number of Fermion fmavors N f (graphene N f = 2) ¯ ¯ ψ ( I 2 ⊗ I 4 ) ψ → ψ ( I 2 ⊗ I 2 N f ) ψ ψ ( σ a ⊗ I 4 ) ψ ψ ( σ a ⊗ I 2 N f ) ψ ¯ ¯ →

Graphene: Effective fjeld theory (in CDW) (in SDW) 5 • Incorporate order parameter fjelds with an semiphenomenological ansatz 2 φ a ( m 2 − ∂ µ ) φ a + λ L B = 1 4 ! ( φ a φ a ) 2 → Dynamics and self-interaction develop below certain energy scale • Couple fjelds to the Fermion by a Yukawa interaction L Y = g φ ¯ ψ ( I 2 ⊗ I 4 ) ψ ψ ( σ a ⊗ I 4 ) ψ L Y = g φ a ¯ • This can be generalized to arbitrary number of Fermion fmavors N f (graphene N f = 2) ¯ ¯ ψ ( I 2 ⊗ I 4 ) ψ → ψ ( I 2 ⊗ I 2 N f ) ψ ψ ( σ a ⊗ I 4 ) ψ ψ ( σ a ⊗ I 2 N f ) ψ ¯ ¯ → • Full action is then the Gross-Neveu-Yukawa model (GNY) µ ) φ + λ 2 φ ( m 2 − ∂ 2 L = L 0 + L Y + L B = ¯ ∂ψ + g φ ¯ ψ/ ψψ + 1 4 ! φ 4

Motivation Part 2: Critical Phenomena & Universality

• Characterized by collective behavior on all length scales near the transition ∼ • Correlation-length diverges with a power law • More power laws with critical exponents • Here quantum phase transition at T Critical Phenomena T c T critical interaction strength 0 but with 2 r d r assume for correlator G r t T c T T t with C t 1 t [Chialvo ’10] macroscopic systems 6 • Critical phenomena deal with continous (second order) phase transitions in

• Correlation-length diverges with a power law • More power laws with critical exponents • Here quantum phase transition at T Critical Phenomena t T c T critical interaction strength 0 but with 2 r d r assume for correlator G r t T T c T C t with 1 t [Chialvo ’10] macroscopic systems 6 • Critical phenomena deal with continous (second order) phase transitions in • Characterized by collective behavior on all length scales near the transition T ∼ T c T > T c T < T c Subcritical Critical Supercritical

• Correlation-length diverges with a power law • More power laws with critical exponents • Here quantum phase transition at T Critical Phenomena t T c T critical interaction strength 0 but with 2 r d r assume for correlator G r t T T c T with 6 C t 1 t [Chialvo ’10] macroscopic systems • Critical phenomena deal with continous (second order) phase transitions in • Characterized by collective behavior on all length scales near the transition T ∼ T c T > T c T < T c Subcritical Critical Supercritical ξ

• More power laws with critical exponents • Here quantum phase transition at T Critical Phenomena T T c T critical interaction strength 0 but with 2 r d r assume for correlator G r t with 6 macroscopic systems [Chialvo ’10] • Critical phenomena deal with continous (second order) phase transitions in • Characterized by collective behavior on all length scales near the transition T ∼ T c T > T c T < T c Subcritical Critical Supercritical ξ • Correlation-length diverges with a power law t ≡ T − T c ξ ∼ | t | − ν ( 1 + C | t | ω + . . . )

• Here quantum phase transition at T Critical Phenomena [Chialvo ’10] T c T critical interaction strength 0 but with T with 6 macroscopic systems • Critical phenomena deal with continous (second order) phase transitions in • Characterized by collective behavior on all length scales near the transition T ∼ T c T > T c T < T c Subcritical Critical Supercritical ξ • Correlation-length diverges with a power law t ≡ T − T c ξ ∼ | t | − ν ( 1 + C | t | ω + . . . ) • More power laws with critical exponents α, β, γ, δ, . . . assume for correlator G ( r , t ) = Φ ± ( r /ξ ) r d − 2 + η

Critical Phenomena [Chialvo ’10] T c T critical interaction strength T with 6 macroscopic systems • Critical phenomena deal with continous (second order) phase transitions in • Characterized by collective behavior on all length scales near the transition T ∼ T c T > T c T < T c Subcritical Critical Supercritical ξ • Correlation-length diverges with a power law t ≡ T − T c ξ ∼ | t | − ν ( 1 + C | t | ω + . . . ) • More power laws with critical exponents α, β, γ, δ, . . . assume for correlator G ( r , t ) = Φ ± ( r /ξ ) r d − 2 + η • Here quantum phase transition at T = 0 but with

• 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 a 2 a m 2 Universality 0 821 5 0 03680 20 0 63020 12 MC Deng, Blöte 2003 2010 0 825 50 0 03639 15 0 63012 16 Hasenbusch 0 03627 10 MC 0 63002 10 2002 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 ? HT 0 03350 250 Campostrini et al. Method ) [El-Showk et al. ’14] 1 2 a 4 a Ref. Year Guida, Zinn-Justin 0 799 11 1998 -expansion 0 63050 250 0 03650 500 0 814 18 Guida, Zinn-Justin 1998 3D exp 0 63040 130 What is universality and how well do we understand it? 7 • Universality can be defjned as existence of an IR-fjxed point in the RG

• Universality classes differ by their critical exponents ( • Consider the O N model in the 3D Ising universality class a 2 a m 2 2010 0 63012 16 0 03639 15 0 825 50 Deng, Blöte 2003 MC 0 63020 12 0 03680 20 0 821 5 Hasenbusch Universality MC 0 63002 10 2002 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 ? HT 0 03350 250 Campostrini et al. 0 799 11 ) [El-Showk et al. ’14] 1 2 a 4 a 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 What is universality and how well do we understand it? 7 • 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

• Consider the O N model in the 3D Ising universality class a 2 a m 2 Universality 2010 0 63012 16 0 03639 15 0 825 50 Deng, Blöte 2003 MC 0 63020 12 0 03680 20 0 821 5 Hasenbusch 0 63002 10 MC 2002 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 ? HT 0 03350 250 Campostrini et al. Guida, Zinn-Justin [El-Showk et al. ’14] 1 2 a 4 a Ref. Year 0 799 11 Method 1998 -expansion 0 63050 250 0 03650 500 0 814 18 Guida, Zinn-Justin 1998 3D exp 0 63040 130 What is universality and how well do we understand it? 7 • 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, . . . )

Universality 2010 0 63012 16 0 03639 15 0 825 50 Deng, Blöte 2003 MC 0 63020 12 0 03680 20 0 821 5 Hasenbusch MC 2002 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 ? HT Campostrini et al. What is universality and how well do we understand it? -expansion [El-Showk et al. ’14] Ref. Year Method 0 799 11 1998 Guida, Zinn-Justin 0 63050 250 0 03650 500 0 814 18 Guida, Zinn-Justin 1998 3D exp 0 63040 130 0 03350 250 7 • 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 2 φ a ( m 2 − ∂ µ ) φ a + λ L = 1 4 ! ( φ a φ a ) 2

Universality 2003 3D exp What is universality and how well do we understand it? Campostrini et al. 2002 HT Deng, Blöte MC Guida, Zinn-Justin Hasenbusch 2010 MC El-Showk et al. 2014 cBS Question: Can we achieve a similar precision and agreement of methods ? 1998 7 Ref. Year 1998 Guida, Zinn-Justin [El-Showk et al. ’14] Method • 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 2 φ a ( m 2 − ∂ µ ) φ a + λ L = 1 4 ! ( φ a φ a ) 2 ν η ω ǫ -expansion 0 . 63050 ( 250 ) 0 . 03650 ( 500 ) 0 . 814 ( 18 ) 0 . 63040 ( 130 ) 0 . 03350 ( 250 ) 0 . 799 ( 11 ) 0 . 63012 ( 16 ) 0 . 03639 ( 15 ) 0 . 825 ( 50 ) 0 . 63020 ( 12 ) 0 . 03680 ( 20 ) 0 . 821 ( 5 ) 0 . 63002 ( 10 ) 0 . 03627 ( 10 ) 0 . 832 ( 6 ) 0 . 62999 ( 5 ) 0 . 03631 ( 3 ) 0 . 8303 ( 18 )

Universality 2003 3D exp What is universality and how well do we understand it? Campostrini et al. 2002 HT Deng, Blöte MC Guida, Zinn-Justin Hasenbusch 2010 MC El-Showk et al. 2014 cBS 1998 7 Ref. 1998 Guida, Zinn-Justin Method Year [El-Showk et al. ’14] • 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 2 φ a ( m 2 − ∂ µ ) φ a + λ L = 1 4 ! ( φ a φ a ) 2 ν η ω ǫ -expansion 0 . 63050 ( 250 ) 0 . 03650 ( 500 ) 0 . 814 ( 18 ) 0 . 63040 ( 130 ) 0 . 03350 ( 250 ) 0 . 799 ( 11 ) 0 . 63012 ( 16 ) 0 . 03639 ( 15 ) 0 . 825 ( 50 ) 0 . 63020 ( 12 ) 0 . 03680 ( 20 ) 0 . 821 ( 5 ) 0 . 63002 ( 10 ) 0 . 03627 ( 10 ) 0 . 832 ( 6 ) 0 . 62999 ( 5 ) 0 . 03631 ( 3 ) 0 . 8303 ( 18 ) ⇒ Question: Can we achieve a similar precision and agreement of methods ?

Renormalization group in critical phenomena

• Simplest extension of the O N model by a massless Fermion • New universality classes named after the broken symmetries 1 i 5 • Gaussian FP: both couplings relevant • Wilson-Fisher FP: only • Non-Gaussian FP: both couplings irrelevant 0 O 2 1 (work in progress) • -functions ( g 2 ) d d The Gross-Neveu-Yukawa model d d T a fjxed point Three physical fjxed points (FP): irrelevant NGFP NGFP WF 0 chiral XY SC chiral Heisenberg chiral Ising CDW T a The Gross-Neveu-Yukawa model (GNY) 2 1 1 SDW T a 1 2 a O 3 O 2 8 µ ) φ a + λ 2 φ a ( m 2 − ∂ 2 L = ¯ ∂ψ + g φ a ¯ ψ/ ψ T a ψ + 1 4 ! ( φ a φ a ) 2

• New universality classes named after the broken symmetries 1 i 5 • Gaussian FP: both couplings relevant • Wilson-Fisher FP: only • Non-Gaussian FP: both couplings irrelevant 0 O 2 1 (work in progress) • -functions ( g 2 ) d d The Gross-Neveu-Yukawa model d d T a fjxed point Three physical fjxed points (FP): irrelevant NGFP NGFP WF 0 chiral XY SC chiral Heisenberg chiral Ising CDW T a The Gross-Neveu-Yukawa model (GNY) 2 1 1 SDW T a 1 2 a O 3 O 2 8 µ ) φ a + λ 2 φ a ( m 2 − ∂ 2 L = ¯ ∂ψ + g φ a ¯ ψ/ ψ T a ψ + 1 4 ! ( φ a φ a ) 2 • Simplest extension of the O ( N ) model by a massless Fermion

• Gaussian FP: both couplings relevant • Wilson-Fisher FP: only • Non-Gaussian FP: both couplings irrelevant The Gross-Neveu-Yukawa model d • -functions ( g 2 ) d d 0 fjxed point d 0 The Gross-Neveu-Yukawa model (GNY) Three physical fjxed points (FP): irrelevant NGFP NGFP WF (work in progress) 8 SC SDW CDW chiral Heisenberg chiral Ising chiral XY µ ) φ a + λ 2 φ a ( m 2 − ∂ 2 L = ¯ ∂ψ + g φ a ¯ ψ/ ψ T a ψ + 1 4 ! ( φ a φ a ) 2 • Simplest extension of the O ( N ) model by a massless Fermion • New universality classes named after the broken symmetries T a = 1 Z 2 → 1 T a = 1 2 σ a O ( 3 ) → O ( 2 ) T a = 1 , i γ 5 O ( 2 ) → 1

• Gaussian FP: both couplings relevant • Wilson-Fisher FP: only • Non-Gaussian FP: both couplings irrelevant The Gross-Neveu-Yukawa model SDW WF NGFP NGFP irrelevant Three physical fjxed points (FP): The Gross-Neveu-Yukawa model (GNY) (work in progress) SC chiral XY 8 chiral Heisenberg chiral Ising CDW µ ) φ a + λ 2 φ a ( m 2 − ∂ 2 L = ¯ ∂ψ + g φ a ¯ ψ/ ψ T a ψ + 1 4 ! ( φ a φ a ) 2 • Simplest extension of the O ( N ) model by a massless Fermion • New universality classes named after the broken symmetries T a = 1 Z 2 → 1 T a = 1 2 σ a O ( 3 ) → O ( 2 ) T a = 1 , i γ 5 O ( 2 ) → 1 • β -functions ( α ≡ g 2 ) d α  ! β α = = 0   d ln µ  fjxed point ( α ∗ , λ ∗ ) d λ !  β λ = = 0   d ln µ

The Gross-Neveu-Yukawa model CDW WF NGFP Three physical fjxed points (FP): The Gross-Neveu-Yukawa model (GNY) (work in progress) SC chiral XY SDW chiral Heisenberg 8 chiral Ising µ ) φ a + λ 2 φ a ( m 2 − ∂ 2 L = ¯ ∂ψ + g φ a ¯ ψ/ ψ T a ψ + 1 4 ! ( φ a φ a ) 2 • Simplest extension of the O ( N ) model by a massless Fermion • New universality classes named after the broken symmetries T a = 1 Z 2 → 1 T a = 1 2 σ a O ( 3 ) → O ( 2 ) T a = 1 , i γ 5 O ( 2 ) → 1 • β -functions ( α ≡ g 2 ) d α  ! β α = = 0   d ln µ  fjxed point ( α ∗ , λ ∗ ) d λ !  β λ = = 0   d ln µ α • Gaussian FP: both couplings relevant • Wilson-Fisher FP: only λ irrelevant NGFP ∗ • Non-Gaussian FP: both couplings irrelevant λ

• Writing for the renormalization constants Z • Regularize infjnite diagrams in D • MS-Scheme: renormalization constants are the -poles Z n g Z i 2 m 2 m 2 2 Perturbative RG procedure Z i g new mass scale and render couplings massless by introducing 4 Z Z 2 4 Z 1 n 1 n n g 2 g 2 i n with g 2 fjnite i n 1 n 1 i 1 with div. 9 div. m 2 1 Z the infjnities are then absorbed by the Zs and determined by i p Z div. fjnite p 2 Z 4 Z 2 m 2 fjnite div. Z Z g Z 1 2 Z with with fjnite Z g Z g Z m Z 2 Z • Introduce renormalized fjelds and couplings ψ 0 = Z 1 / 2 φ 0 = Z 1 / 2 ψ ψ, φ φ, g 0 = Z g g , λ 0 = Z λ λ, 0 = Z m m 2

• Regularize infjnite diagrams in D • MS-Scheme: renormalization constants are the -poles Z n Perturbative RG procedure g 2 m 2 m 2 2 Z i 1 Z i 1 n 1 i new mass scale 1 n 1 n i g 2 n with n i g 2 g 2 n g and render couplings massless by introducing 9 with m 2 4 with with • Introduce renormalized fjelds and couplings ψ 0 = Z 1 / 2 φ 0 = Z 1 / 2 ψ ψ, φ φ, g 0 = Z g g , λ 0 = Z λ λ, 0 = Z m m 2 • Writing for the renormalization constants Z = 1 + δ Z the infjnities are then absorbed by the δ Zs and determined by p ( δ Z ψ + � ¯ i / ψψ � div. ) = fjnite p 2 · δ Z φ + m 2 · δ Z φ 2 + � φφ � div. = fjnite Z φ 2 ≡ Z m Z φ ψψ ≡ Z g Z 1 / 2 ψψ + � g ¯ g · δ Z φ ¯ ψψ � div. = fjnite Z φ ¯ φ Z ψ λ · δ Z φ 4 + � φφφφ � div. = fjnite Z φ 4 ≡ Z λ Z 2 φ

• MS-Scheme: renormalization constants are the -poles Z n n Z i 1 Z i 1 n 1 i 1 Perturbative RG procedure with n i g 2 n with n i g 2 g 2 n 1 9 with with m 2 • Introduce renormalized fjelds and couplings ψ 0 = Z 1 / 2 φ 0 = Z 1 / 2 ψ ψ, φ φ, g 0 = Z g g , λ 0 = Z λ λ, 0 = Z m m 2 • Writing for the renormalization constants Z = 1 + δ Z the infjnities are then absorbed by the δ Zs and determined by p ( δ Z ψ + � ¯ i / ψψ � div. ) = fjnite p 2 · δ Z φ + m 2 · δ Z φ 2 + � φφ � div. = fjnite Z φ 2 ≡ Z m Z φ ψψ ≡ Z g Z 1 / 2 ψψ + � g ¯ g · δ Z φ ¯ ψψ � div. = fjnite Z φ ¯ φ Z ψ λ · δ Z φ 4 + � φφφφ � div. = fjnite Z φ 4 ≡ Z λ Z 2 φ • Regularize infjnite diagrams in D = 4 − ǫ and render couplings massless by introducing new mass scale µ m 2 → m 2 µ − 2 g → g µ ǫ/ 2 , λ → λµ ǫ ,

Perturbative RG procedure with i with i i with with 9 m 2 • Introduce renormalized fjelds and couplings ψ 0 = Z 1 / 2 φ 0 = Z 1 / 2 ψ ψ, φ φ, g 0 = Z g g , λ 0 = Z λ λ, 0 = Z m m 2 • Writing for the renormalization constants Z = 1 + δ Z the infjnities are then absorbed by the δ Zs and determined by p ( δ Z ψ + � ¯ i / ψψ � div. ) = fjnite p 2 · δ Z φ + m 2 · δ Z φ 2 + � φφ � div. = fjnite Z φ 2 ≡ Z m Z φ ψψ ≡ Z g Z 1 / 2 ψψ + � g ¯ g · δ Z φ ¯ ψψ � div. = fjnite Z φ ¯ φ Z ψ λ · δ Z φ 4 + � φφφφ � div. = fjnite Z φ 4 ≡ Z λ Z 2 φ • Regularize infjnite diagrams in D = 4 − ǫ and render couplings massless by introducing new mass scale µ m 2 → m 2 µ − 2 g → g µ ǫ/ 2 , λ → λµ ǫ , • MS-Scheme: renormalization constants are the ǫ -poles ∞ ∞ α ( n ) ( g 2 , λ ) δ Z ( n ) α ( n ) ( g 2 , λ ) = O (( g 2 + λ ) n ) � � Z i = 1 + δ Z i = 1 + = 1 + ǫ n n = 1 n = 1

• For the Boson propagator • For the Yukawa vertex g 4 vertex • For the Diagrams to compute 1574 more div. 153 more div. 40 more div. 10 • For the Fermion propagator � ¯ ψψ � div. φ e φ e φ φ φ e e e e e e + 32 more φ e e e e e e e e e φ φ e e e e φ e

• For the Yukawa vertex g 4 vertex • For the Diagrams to compute 1574 more div. 153 more div. 10 • For the Fermion propagator � ¯ ψψ � div. φ e φ e φ φ φ e e e e e e + 32 more φ e e e e e e e e e φ φ e e e e φ e • For the Boson propagator � φφ � div. φ e e e e e e φ φ φ φ φ + 40 more e φ e e e φ φ φ φ e e e e e e e e φ φ

4 vertex • For the Diagrams to compute 1574 more div. 10 • For the Fermion propagator � ¯ ψψ � div. φ e φ e φ φ φ e e e e e e + 32 more φ e e e e e e e e e φ φ e e e e φ e • For the Boson propagator � φφ � div. φ e e e e e e φ φ φ φ φ + 40 more e φ e e e φ φ φ φ e e e e e e e e φ φ • For the Yukawa vertex � g ¯ ψψ � div. φ e e e e e e φ e φ e e e φ φ φ φ + 153 more φ φ e e e e e e φ e φ φ e e e e e φ e e e e

Diagrams to compute 10 • For the Fermion propagator � ¯ ψψ � div. φ e φ e φ φ φ e e e e e e + 32 more φ e e e e e e e e e φ φ e e e e φ e • For the Boson propagator � φφ � div. φ e e e e e e φ φ φ φ φ + 40 more e φ e e e φ φ φ φ e e e e e e e e φ φ • For the Yukawa vertex � g ¯ ψψ � div. φ e e e e e e φ e φ e e e φ φ φ φ + 153 more φ φ e e e e e e φ e φ φ e e e e e φ e e e e • For the φ 4 vertex � φφφφ � div. φ φ φ e e φ φ e φ φ φ e φ φ φ φ e e e e + 1574 more φ e e e e φ φ e φ e e φ φ e e φ φ φ φ φ φ e φ

Tools and Technique

2 m 2 F n • Problem: • Goal: Reduce the number of integrals to so-called master integrals • Idea: In dimensional regularization the D dimensional boundary integrals provide k f k • By explicitly performing the differentiations one obtains recurrence relations connecting • Take a look at a simple example m 2 n k k D k m 2 n m 2 n D k 2 m 2 n m 2 n m 2 n d D k 2 1 2 k n k 2 2 nm 2 F n 1 d D k 1 2 D 2 n 2 n F n k 2 2 d D k 2 D 2 nm 2 D k 2 D Basics of computing Feynman diagrams d D k k 2 20,000 integrals to compute! d D k 2 D 0 a complicated Feynman integral to several simpler ones master integrals F n d D k 2 D 1 IBP k 2 F n D 2 n 2 2 n 1 0 d D k 2 D How to treat a Feynman diagram? 1 11

2 m 2 F n • Goal: Reduce the number of integrals to so-called master integrals • Idea: In dimensional regularization the D dimensional boundary integrals provide k f k • By explicitly performing the differentiations one obtains recurrence relations connecting • Take a look at a simple example m 2 n k k D k m 2 n m 2 n D k 2 m 2 n m 2 n m 2 n d D k 2 1 2 k n k 2 2 nm 2 F n 1 d D k 1 2 D 2 n 2 n F n k 2 2 d D k 2 D 2 nm 2 D k 2 D Basics of computing Feynman diagrams d D k k 2 d D k 2 D 0 a complicated Feynman integral to several simpler ones master integrals F n d D k 2 D 1 IBP k 2 F n D 2 n 2 2 n 1 0 d D k 2 D How to treat a Feynman diagram? 1 11 • Problem: ∼ 20,000 integrals to compute!

2 m 2 F n • Idea: In dimensional regularization the D dimensional boundary integrals provide k f k • By explicitly performing the differentiations one obtains recurrence relations connecting • Take a look at a simple example m 2 n k k D k m 2 n m 2 n D k 2 m 2 n m 2 n m 2 n 2 d D k 2 2 k n k 2 1 d D k k 2 D D 2 n d D k 2 D 2 nm 2 k 2 1 D 2 n F n 2 nm 2 F n 1 2 Basics of computing Feynman diagrams d D k k 2 d D k 2 D 0 a complicated Feynman integral to several simpler ones master integrals F n d D k 2 D 1 k 2 IBP F n D 2 n 2 2 n 1 0 d D k 2 D How to treat a Feynman diagram? 1 11 • Problem: ∼ 20,000 integrals to compute! • Goal: Reduce the number of integrals to so-called master integrals

2 m 2 F n • By explicitly performing the differentiations one obtains recurrence relations connecting • Take a look at a simple example m 2 n k k D k m 2 n m 2 n D k 2 m 2 n m 2 n m 2 n d D k 2 2 k n k 2 2 1 d D k 2 2 n D D k 2 d D k 2 D 2 nm 2 k 2 1 D 2 n F n 2 nm 2 F n 1 d D k Basics of computing Feynman diagrams k 2 IBP d D k a complicated Feynman integral to several simpler ones master integrals F n d D k 2 D 1 k 2 1 F n d D k How to treat a Feynman diagram? D 2 D 0 1 2 n 2 2 n 11 • 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 � ∂ ∂ k µ f ( k , . . . ) = 0 ( 2 π ) D

2 m 2 F n • Take a look at a simple example m 2 n k k D k m 2 n m 2 n D k 2 m 2 n m 2 n m 2 n d D k 2 2 k n k 2 2 1 d D k D 2 D 2 n k 2 d D k 2 D 2 nm 2 k 2 1 D 2 n F n 2 nm 2 F n 1 d D k Basics of computing Feynman diagrams k 2 F n d D k a complicated Feynman integral to several simpler ones F n d D k 2 1 1 k 2 IBP D D 2 How to treat a Feynman diagram? 2 n D 11 d D k 0 1 2 n 2 • 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 � ∂ ∂ k µ f ( k , . . . ) = 0 ( 2 π ) D • By explicitly performing the differentiations one obtains recurrence relations connecting → master integrals

2 m 2 F n k k D k m 2 n m 2 n D k 2 m 2 n m 2 n m 2 n d D k 2 d D k d D k 2 2 k n k 2 1 Basics of computing Feynman diagrams 2 D 2 n k 2 d D k 2 D 2 nm 2 k 2 1 D 2 n F n 2 nm 2 F n 1 D 1 k 2 D d D k a complicated Feynman integral to several simpler ones How to treat a Feynman diagram? 1 IBP F n d D k 2 n d D k D 2 2 11 1 2 n 0 • 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 � ∂ ∂ k µ f ( k , . . . ) = 0 ( 2 π ) D • By explicitly performing the differentiations one obtains recurrence relations connecting → master integrals • Take a look at a simple example � F ( n ) = ( k 2 + m 2 ) n ( 2 π ) D

2 m 2 F n Basics of computing Feynman diagrams How to treat a Feynman diagram? D 2 n 2 2 n 1 d D k IBP 1 d D k d D k d D k d D k 2 nm 2 F n 11 1 a complicated Feynman integral to several simpler ones d D k d D k • 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 � ∂ ∂ k µ f ( k , . . . ) = 0 ( 2 π ) D • By explicitly performing the differentiations one obtains recurrence relations connecting → master integrals • Take a look at a simple example � F ( n ) = ( k 2 + m 2 ) n ( 2 π ) D δ µ − 2 k µ n � ∂ � � µ ∂ k µ k µ 0 = ( k 2 + m 2 ) n = ( k 2 + m 2 ) n + ( 2 π ) D k µ ( k 2 + m 2 ) n + 1 ( 2 π ) D ( 2 π ) D D − 2 n � � = ( k 2 + m 2 ) n + ( k 2 + m 2 ) n + 1 = ( D − 2 n ) F ( n ) + 2 nm 2 F ( n + 1 ) ( 2 π ) D ( 2 π ) D

Basics of computing Feynman diagrams d D k 2 nm 2 d D k d D k d D k d D k 1 How to treat a Feynman diagram? d D k IBP 1 11 d D k a complicated Feynman integral to several simpler ones • 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 � ∂ ∂ k µ f ( k , . . . ) = 0 ( 2 π ) D • By explicitly performing the differentiations one obtains recurrence relations connecting → master integrals • Take a look at a simple example F ( n ) = − D − 2 n + 2 � F ( n ) = = ⇒ ( 2 n − 2 ) m 2 F ( n − 1 ) ( k 2 + m 2 ) n ( 2 π ) D δ µ − 2 k µ n � ∂ � � µ ∂ k µ k µ 0 = ( k 2 + m 2 ) n = ( k 2 + m 2 ) n + ( 2 π ) D k µ ( k 2 + m 2 ) n + 1 ( 2 π ) D ( 2 π ) D D − 2 n � � = ( k 2 + m 2 ) n + ( k 2 + m 2 ) n + 1 = ( D − 2 n ) F ( n ) + 2 nm 2 F ( n + 1 ) ( 2 π ) D ( 2 π ) D

• Why use FORM ? • Much built-in functions (integration, differ- • Very limited functions (calculus • Small and fast • Big and slow • Limited by disc space • Limited by RAM • Does only what you ask for • Tries to do everything Tools with tensors, ...) FORM entiation, solving equations ...) Use a fully automized toolchain? Mathematica stupid but fast (from the FORM tutorial) Computing in FORM [Vermaseren ’84] 12 • 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]

• Why use FORM ? • Much built-in functions (integration, differ- • Very limited functions (calculus • Small and fast • Big and slow • Limited by disc space • Limited by RAM • Does only what you ask for • Tries to do everything Tools with tensors, ...) FORM entiation, solving equations ...) Use a fully automized toolchain? Mathematica stupid but fast (from the FORM tutorial) Computing in FORM [Vermaseren ’84] 12 • 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]

• Much built-in functions (integration, differ- • Very limited functions (calculus • Small and fast • Big and slow • Limited by disc space • Limited by RAM • Does only what you ask for • Tries to do everything Tools with tensors, ...) FORM entiation, solving equations ...) Use a fully automized toolchain? Mathematica stupid but fast (from the FORM tutorial) Computing in FORM [Vermaseren ’84] 12 • 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] • Why use FORM ?

• Much built-in functions (integration, differ- • Very limited functions (calculus • Small and fast • Big and slow • Limited by disc space • Limited by RAM • Does only what you ask for • Tries to do everything Tools with tensors, ...) FORM entiation, solving equations ...) Use a fully automized toolchain? Mathematica Computing in FORM [Vermaseren ’84] 12 • 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] • Why use FORM ? ⇒ stupid but fast (from the FORM tutorial)

• Very limited functions (calculus • Small and fast • Limited by disc space • Does only what you ask for Tools Use a fully automized toolchain? with tensors, ...) FORM entiation, solving equations ...) Computing in FORM [Vermaseren ’84] Mathematica 12 • 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] • Why use FORM ? ⇒ stupid but fast (from the FORM tutorial) • Much built-in functions (integration, differ- • Big and slow • Limited by RAM • Tries to do everything

Tools Mathematica with tensors, ...) FORM Use a fully automized toolchain? entiation, solving equations ...) 12 Computing in FORM [Vermaseren ’84] • 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] • Why use FORM ? ⇒ stupid but fast (from the FORM tutorial) • Much built-in functions (integration, differ- • Very limited functions (calculus • Small and fast • Big and slow • Limited by disc space • Limited by RAM • Does only what you ask for • Tries to do everything

FT 1 i 1 i 2 Q k 1 Ta i 2 i 3 FT 1 i 3 i 4 k 2 Denab Q FT ij p Den ab p m inpo1 EXP Insert explicit expressions (propagators, vertices, traces, expansions ...) & perform integration with IBP d Dk 1 2 D d Dk 2 2 Toolchain inpt1 D Insert hardcoded masterintegrals D3 D6 DN B4 E3 MATAD Collect renormalization constants Zs in FORM and proceed for - and -functions in Mathematica tad2l inpl inpla Apply the rules and topologies Generate diagrams with its symmetry factors QGRAF Assign Feyman rules and mass hiearchy Q q m Ta ij dabcd k 1 tad1l k 2 m Q2E Asymptotic expansion 1 Map on topologies Defjne Lagrangian 13

FT 1 i 1 i 2 Q k 1 Ta i 2 i 3 FT 1 i 3 i 4 k 2 Denab Q FT ij p Den ab p m 1 Asymptotic expansion Q2E m k 2 k 1 Toolchain inpl Apply the rules and topologies dabcd ij Ta q m Assign Feyman rules and mass hiearchy Q Map on topologies tad1l Defjne Lagrangian 2 Mathematica -functions in - and Collect renormalization constants Zs in FORM and proceed for MATAD D6 DN B4 E3 Insert hardcoded masterintegrals D3 D d Dk 2 inpla D 2 d Dk 1 Insert explicit expressions (propagators, vertices, traces, expansions ...) & perform integration with IBP EXP inpo1 tad2l inpt1 QGRAF 13 Generate diagrams with its symmetry factors e e e e e e e e e φ φ φ e e φ φ φ φ φ φ e φ e φ φ e e e e e e e φ φ φ e φ e e e e e φ φ e e e e φ e φ e e e φ φ e φ φ φ e e φ φ φ φ φ φ φ φ e e e e φ e e e e φ e e φ e e e e e e e e e e e e e e φ φ φ φ φ φ φ e e

FT 1 i 1 i 2 Q k 1 Ta i 2 i 3 FT 1 i 3 i 4 k 2 Denab Q Toolchain Q2E tad1l inpl Map on topologies 1 Asymptotic expansion k 2 m inpt1 k 1 Apply the rules and topologies ij Ta inpla tad2l Defjne Lagrangian Insert hardcoded masterintegrals D3 Mathematica -functions in - and Collect renormalization constants Zs in FORM and proceed for MATAD D6 DN B4 E3 D inpo1 2 d Dk 2 D 2 d Dk 1 Insert explicit expressions (propagators, vertices, traces, expansions ...) & perform integration with IBP EXP QGRAF 13 Generate diagrams with its symmetry factors e e e e e e e e e φ φ φ e e φ φ φ φ φ φ e φ e φ φ e e e e e e e φ φ φ e φ e e e e e φ φ e e e e φ e φ e e e φ φ e φ φ φ e e φ φ φ φ φ φ φ φ e e e e φ e e e e φ e e φ e e e e e e e e e e e e e e φ φ φ φ φ φ φ e e Assign Feyman rules and mass hiearchy Q ≫ q , m = FT ij ( p ) = Den ab ( p , m ) = = dabcd

Toolchain EXP Ta ij Apply the rules and topologies Q2E Asymptotic expansion 1 Map on topologies inpl tad1l inpla inpt1 tad2l inpo1 Insert explicit expressions (propagators, vertices, traces, expansions ...) & perform integration with IBP Defjne Lagrangian d Dk 1 2 D d Dk 2 2 D Insert hardcoded masterintegrals D3 D6 DN B4 E3 MATAD Collect renormalization constants Zs in FORM and proceed for - and -functions in Mathematica QGRAF 13 Generate diagrams with its symmetry factors e e e e e e e e e φ φ φ e e φ φ φ φ φ φ e φ e φ φ e e e e e e e φ φ φ e φ e e e e e φ φ e e e e φ e φ e e e φ φ e φ φ φ e e φ φ φ φ φ φ φ φ e e e e φ e e e e φ e e φ e e e e e e e e e e e e e e φ φ φ φ φ φ φ e e Assign Feyman rules and mass hiearchy Q ≫ q , m = FT ij ( p ) = Den ab ( p , m ) Q+k 1 k 2 Q Q Q+k 1 -k 2 = FT 1 i 1 i 2 ( Q + k 1 ) Ta i 2 i 3 FT 1 i 3 i 4 ( k 2 ) Denab ( Q + k 1 − k 2 , m ) . . . = = dabcd k 1 -Q+k 2

Toolchain EXP Ta ij Apply the rules and topologies Q2E Asymptotic expansion Map on topologies inpl tad1l inpla inpt1 tad2l inpo1 Insert explicit expressions (propagators, vertices, traces, expansions ...) & perform integration with IBP Defjne Lagrangian d Dk 1 2 D d Dk 2 2 D Insert hardcoded masterintegrals D3 D6 DN B4 E3 MATAD Collect renormalization constants Zs in FORM and proceed for - and -functions in Mathematica QGRAF 13 Generate diagrams with its symmetry factors e e e e e e e e e φ φ φ e e φ φ φ φ φ φ e φ e φ φ e e e e e e e φ φ φ e φ e e e e e φ φ e e e e φ e φ e e e φ φ e φ φ φ e e φ φ φ φ φ φ φ φ e e e e φ e e e e φ e e φ e e e e e e e e e e e e e e φ φ φ φ φ φ φ e e Assign Feyman rules and mass hiearchy Q ≫ q , m = FT ij ( p ) = Den ab ( p , m ) Q+k 1 k 2 Q Q Q+k 1 -k 2 = FT 1 i 1 i 2 ( Q + k 1 ) Ta i 2 i 3 FT 1 i 3 i 4 ( k 2 ) Denab ( Q + k 1 − k 2 , m ) . . . = = dabcd k 1 -Q+k 2 = ⋆ + 1 ⋆ + ⋆ + ⋆

Toolchain EXP Ta ij Apply the rules and topologies Q2E Asymptotic expansion Map on topologies inpl tad1l inpla inpt1 tad2l inpo1 Insert explicit expressions (propagators, vertices, traces, expansions ...) & perform integration with IBP Defjne Lagrangian d Dk 1 2 D d Dk 2 2 D Insert hardcoded masterintegrals D3 D6 DN B4 E3 MATAD Collect renormalization constants Zs in FORM and proceed for - and -functions in Mathematica QGRAF 13 Generate diagrams with its symmetry factors e e e e e e e e e φ φ φ e e φ φ φ φ φ φ e φ e φ φ e e e e e e e φ φ φ e φ e e e e e φ φ e e e e φ e φ e e e φ φ e φ φ φ e e φ φ φ φ φ φ φ φ e e e e φ e e e e φ e e φ e e e e e e e e e e e e e e φ φ φ φ φ φ φ e e Assign Feyman rules and mass hiearchy Q ≫ q , m = FT ij ( p ) = Den ab ( p , m ) Q+k 1 k 2 Q Q Q+k 1 -k 2 = FT 1 i 1 i 2 ( Q + k 1 ) Ta i 2 i 3 FT 1 i 3 i 4 ( k 2 ) Denab ( Q + k 1 − k 2 , m ) . . . = = dabcd k 1 -Q+k 2 = ⋆ + 1 ⋆ . . . . . . + ⋆ + ⋆

Toolchain inpla Ta ij Apply the rules and topologies Q2E Asymptotic expansion Map on topologies inpl tad1l inpt1 Defjne Lagrangian tad2l inpo1 EXP Insert explicit expressions (propagators, vertices, traces, expansions ...) & perform integration with IBP MATAD Collect renormalization constants Zs in FORM and proceed for - and -functions in Mathematica QGRAF 13 Generate diagrams with its symmetry factors e e e e e e e e e φ φ φ e e φ φ φ φ φ φ e φ e φ φ e e e e e e e φ φ φ e φ e e e e e φ φ e e e e φ e φ e e e φ φ e φ φ φ e e φ φ φ φ φ φ φ φ e e e e φ e e e e φ e e φ e e e e e e e e e e e e e e φ φ φ φ φ φ φ e e Assign Feyman rules and mass hiearchy Q ≫ q , m = FT ij ( p ) = Den ab ( p , m ) Q+k 1 k 2 Q Q Q+k 1 -k 2 = FT 1 i 1 i 2 ( Q + k 1 ) Ta i 2 i 3 FT 1 i 3 i 4 ( k 2 ) Denab ( Q + k 1 − k 2 , m ) . . . = = dabcd k 1 -Q+k 2 = ⋆ + 1 ⋆ . . . . . . + ⋆ + ⋆ � d Dk 1 � d Dk 2 = ( 2 π ) D . . . ( 2 π ) D ⇒ Insert hardcoded masterintegrals D3 , . . . , D6 , DN , B4 , E3 , . . .

Toolchain tad1l Ta ij Apply the rules and topologies Q2E Asymptotic expansion Map on topologies inpl inpla Defjne Lagrangian inpt1 tad2l inpo1 EXP Insert explicit expressions (propagators, vertices, traces, expansions ...) & perform integration with IBP MATAD Mathematica QGRAF 13 Generate diagrams with its symmetry factors e e e e e e e e e φ φ φ e e φ φ φ φ φ φ e φ e φ φ e e e e e e e φ φ φ e φ e e e e e φ φ e e e e φ e φ e e e φ φ e φ φ φ e e φ φ φ φ φ φ φ φ e e e e φ e e e e φ e e φ e e e e e e e e e e e e e e φ φ φ φ φ φ φ e e Assign Feyman rules and mass hiearchy Q ≫ q , m = FT ij ( p ) = Den ab ( p , m ) Q+k 1 k 2 Q Q Q+k 1 -k 2 = FT 1 i 1 i 2 ( Q + k 1 ) Ta i 2 i 3 FT 1 i 3 i 4 ( k 2 ) Denab ( Q + k 1 − k 2 , m ) . . . = = dabcd k 1 -Q+k 2 = ⋆ + 1 ⋆ . . . . . . + ⋆ + ⋆ � d Dk 1 � d Dk 2 = ( 2 π ) D . . . ( 2 π ) D ⇒ Insert hardcoded masterintegrals D3 , . . . , D6 , DN , B4 , E3 , . . . ⇒ Collect renormalization constants Zs in FORM and proceed for β - and γ -functions in

14 64 ... ... 2 N f 4 N f 2 36 32 2 128 4 32 β -functions and renormalization functions β - and γ x = d ln Z x / d ln µ functions for chiral Ising [Mih+17] � � 9 � � α 2 + 24 λ 2 − α 2 − 24 αλ � � � β α = − ǫ α + 3 + 2 N f 8 + 6 N f α + 1 ( − 697 + 912 ζ 3 + 2 N f ( 67 + 112 N f + 432 ζ 3 )) α 4 + 1152 ( 7 + 5 N f ) α 3 λ + 192 ( 91 − 30 N f ) α 2 λ 2 − 13824 αλ 3 � � 36 λ 2 + 4 N f αλ − N f α 2 � � 4 N f α 3 + 7 N f α 2 λ − 72 N f αλ 2 − 816 λ 3 � β λ = − ǫλ + + � + 1 N f α 4 ( 5 − 628 N f − 384 ζ 3 ) + 2 N f α 3 ( 1736 N f − 4395 − 1872 ζ 3 ) λ − 48 N f α 2 ( 72 N f − 361 − 648 ζ 3 ) λ 2 + 49536 N f αλ 3 + 6912 ( 145 + 96 ζ 3 ) λ 4 � � � � α 3 ( 48 ζ 3 − 15 + 4 ( 47 − 12 N f ) N f ) + 768 α 2 λ − 2112 αλ 2 � γ ψ = α α 2 16 + 3 N f α 2 + 2 − N f α 3 ( 21 + 200 N f + 48 ζ 3 ) + 960 N f α 2 λ − 2880 N f αλ 2 − 6912 λ 3 � � � � 24 λ 2 − 5 N f α 2 γ φ = 2 N f α + + � N f α 2 − 12 N f αλ − 72 λ 2 � γ φ 2 = 12 λ + 2 + 4 N f α 3 ( − 9 + 4 N f + 3 ζ 3 ) 4 N f αλ 2 + 87 λ 3 � � + 3 2 N f α 2 ( 11 − 24 N f + 120 ζ 3 ) λ + 72

14 64 ... ... 32 2 128 4 32 β -functions and renormalization functions β - and γ x = d ln Z x / d ln µ functions for chiral Ising [Mih+17] � � 9 � � α 2 + 24 λ 2 − α 2 − 24 αλ � � � β α = − ǫ α + 3 + 2 N f 8 + 6 N f α + 1 ( − 697 + 912 ζ 3 + 2 N f ( 67 + 112 N f + 432 ζ 3 )) α 4 + 1152 ( 7 + 5 N f ) α 3 λ + 192 ( 91 − 30 N f ) α 2 λ 2 − 13824 αλ 3 � φ φ φ � 36 λ 2 + 4 N f αλ − N f α 2 � � 4 N f α 3 + 7 N f α 2 λ − 72 N f αλ 2 − 816 λ 3 � e φ β λ = − ǫλ + + φ φ φ φ φ φ � φ + 1 N f α 4 ( 5 − 628 N f − 384 ζ 3 ) + 2 N f α 3 ( 1736 N f − 4395 − 1872 ζ 3 ) λ φ e φ φ − 48 N f α 2 ( 72 N f − 361 − 648 ζ 3 ) λ 2 + 49536 N f αλ 3 + 6912 ( 145 + 96 ζ 3 ) λ 4 � 36 λ 2 + 4 N f αλ − N f α 2 � � β λ = − ǫλ + + . . . � � � α 3 ( 48 ζ 3 − 15 + 4 ( 47 − 12 N f ) N f ) + 768 α 2 λ − 2112 αλ 2 � γ ψ = α α 2 16 + 3 N f α 2 + 2 − φ φ φ e e φ N f α 3 ( 21 + 200 N f + 48 ζ 3 ) + 960 N f α 2 λ − 2880 N f αλ 2 − 6912 λ 3 � � � � 24 λ 2 − 5 N f α 2 e e e e γ φ = 2 N f α + + φ e e φ φ φ � N f α 2 − 12 N f αλ − 72 λ 2 � γ φ 2 = 12 λ + 2 + 4 N f α 3 ( − 9 + 4 N f + 3 ζ 3 ) 4 N f αλ 2 + 87 λ 3 � � + 3 2 N f α 2 ( 11 − 24 N f + 120 ζ 3 ) λ + 72

2 m 2 d m 2 E.g. for the chiral Ising universality class [Mih+17] N f 3 f s 2 N f 216 3 2 s 328 N f 33 s Critical exponents 180 3 2 3 2 3 s 2 N f 108 3 f 48 N 3 2 N f 2 N f 2 9 2 3 s 2 N f 36 3 208 N 2 57 N f f 436 N 2 88 N 3 f 2916 N 2 594 N f 27 2 N f 3 f s 171 510 N f 2 (Fermion anomalous dim.) (Boson anomalous dim.) and from the dimensionless mass term -function m 2 2 1 d m 2 2 2 Recall: G r t r r d 1 7587 N f f 96 N 4 f 5264 N 3 f 2 666 N 2 513 2 N f 6 3 s 10 N f 3 15 Correlation-length exponent and anomlous dimensions found at the NGFP ( α ∗ , λ ∗ ) η ψ = γ ψ ( α ∗ , λ ∗ ) = d ln Z ψ d ln µ ( α ∗ , λ ∗ ) η φ = γ φ ( α ∗ , λ ∗ ) = d ln Z φ d ln µ ( α ∗ , λ ∗ )

E.g. for the chiral Ising universality class [Mih+17] N f 3 f s Critical exponents 2 N f 2 s 328 N f 33 s 180 2 2 3 2 N f 3 s 2 N f 108 3 f 48 N 3 f 216 3 2 3 f 2 3 s 2 N f 36 3 208 N 2 57 N f 9 88 N 3 510 N f f 2916 N 2 594 N f 27 2 N f 3 2 N f 436 N 2 f s 171 10 N f (Fermion anomalous dim.) (Boson anomalous dim.) m 2 m 2 m 2 Recall: 1 2 3 15 s 6 3 96 N 4 f 5264 N 3 f 666 N 2 7587 N f 2 N f 513 Correlation-length exponent and anomlous dimensions found at the NGFP ( α ∗ , λ ∗ ) η ψ = γ ψ ( α ∗ , λ ∗ ) = d ln Z ψ d ln µ ( α ∗ , λ ∗ ) η φ = γ φ ( α ∗ , λ ∗ ) = d ln Z φ d ln µ ( α ∗ , λ ∗ ) m 2 = ( − 2 + γ φ − γ φ 2 ) ˜ and from the dimensionless mass term β -function β ˜ � ν − 1 = − d β ˜ G ( r , t ) = Φ ± ( r /ξ ) � = 2 − η φ + η φ 2 � d ˜ r d − 2 + η � ( α ∗ ,λ ∗ )

Critical exponents (Boson anomalous dim.) Recall: m 2 m 2 m 2 15 (Fermion anomalous dim.) Correlation-length exponent and anomlous dimensions found at the NGFP ( α ∗ , λ ∗ ) η ψ = γ ψ ( α ∗ , λ ∗ ) = d ln Z ψ d ln µ ( α ∗ , λ ∗ ) η φ = γ φ ( α ∗ , λ ∗ ) = d ln Z φ d ln µ ( α ∗ , λ ∗ ) m 2 = ( − 2 + γ φ − γ φ 2 ) ˜ and from the dimensionless mass term β -function β ˜ � ν − 1 = − d β ˜ G ( r , t ) = Φ ± ( r /ξ ) � = 2 − η φ + η φ 2 � d ˜ r d − 2 + η � ( α ∗ ,λ ∗ ) E.g. for the chiral Ising universality class [Mih+17] 513 − 7587 N f − 666 N 2 f − 5264 N 3 f − 96 N 4 f + s ( 171 + 510 N f + 436 N 2 f + 48 N 3 f ) ν − 1 = 2 − ( 3 + 10 N f + s ) ǫ ǫ 2 + . . . − 6 ( 3 + 2 N f ) 108 ( 3 + 2 N f ) 3 s 2 ( 3 + 2 N f ) + 180 + 33 s + N f ( 3 − 328 N f + 2 s ) ǫ ǫ 2 + . . . η ψ = 216 ( 3 + 2 N f ) 3 27 + 594 N f + 2916 N 2 f + 88 N 3 f + ( 9 − 57 N f + 208 N 2 f ) s 2 N f ǫ ǫ 2 + . . . η φ = + 3 + 2 N f 36 ( 3 + 2 N f ) 3 s

Asymptotic series 9 from 0.20 0.15 0.10 0.05 0.00 10 8 1 7 6 5 4 3 2 1 0 2 from 0.4 1 does not provide reliable values... 3 the direct substitution For D from 1.0 0.8 0.6 0.2 3 0.0 10 9 8 7 6 5 4 0 1.2 16 4 2 coeffjcients grow! coeffjcients grow! 1.0 4 ) 0 1 2 3 The series diverge asymptotically! (Remember we work in D 5 0.2 0.8 0.6 6 0.4 0.0 9 7 8 10 Take a closer look for graphene N f = 2 in the chiral Ising case ν − 1 ≈ − 0 . 952 ǫ + 0 . 00723 ǫ 2 − 0 . 0949 ǫ 3 η ψ ≈ 0 . 0714 ǫ − 0 . 00671 ǫ 2 − 0 . 0243 ǫ 3 η φ ≈ 0 . 571 ǫ + 0 . 124 ǫ 2 − 0 . 0278 ǫ 3

Asymptotic series 9 from 0.20 0.15 0.10 0.05 0.00 10 8 1 7 6 5 4 3 2 1 0 2 from 0.4 1 does not provide reliable values... 3 the direct substitution For D from 1.0 0.8 0.6 0.2 3 0.0 10 9 8 7 6 5 4 0 1.2 16 5 2 1.0 4 ) 0 1 2 3 4 The series diverge asymptotically! (Remember we work in D 6 7 8 9 10 0.0 0.2 0.4 0.6 0.8 Take a closer look for graphene N f = 2 in the chiral Ising case ν − 1 ≈ − 0 . 952 ǫ + 0 . 00723 ǫ 2 − 0 . 0949 ǫ 3 ← coeffjcients grow! η ψ ≈ 0 . 0714 ǫ − 0 . 00671 ǫ 2 − 0 . 0243 ǫ 3 ← coeffjcients grow! η φ ≈ 0 . 571 ǫ + 0 . 124 ǫ 2 − 0 . 0278 ǫ 3

Asymptotic series 10 0 from 0.20 0.15 0.10 0.05 0.00 9 2 8 7 6 5 4 3 2 1 3 0 0.4 1 does not provide reliable values... 3 the direct substitution For D from 1.0 0.8 0.6 0.2 4 0.0 10 9 8 7 6 5 1 from 16 7 0 1 2 3 4 5 6 8 2 9 10 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Take a closer look for graphene N f = 2 in the chiral Ising case ν − 1 ≈ − 0 . 952 ǫ + 0 . 00723 ǫ 2 − 0 . 0949 ǫ 3 ← coeffjcients grow! η ψ ≈ 0 . 0714 ǫ − 0 . 00671 ǫ 2 − 0 . 0243 ǫ 3 ← coeffjcients grow! η φ ≈ 0 . 571 ǫ + 0 . 124 ǫ 2 − 0 . 0278 ǫ 3 ⇒ The series diverge asymptotically! (Remember we work in D = 4 − ǫ )

Asymptotic series 10 0 0.20 0.15 0.10 0.05 0.00 9 2 8 7 6 5 4 3 2 1 3 1.2 0.4 1 does not provide reliable values... 3 the direct substitution For D 1.0 0.8 0.6 0.2 4 0.0 10 9 8 7 6 5 0 1 1.0 8 2 0 0.8 1 2 3 4 5 6 7 16 9 10 0.0 0.2 0.4 0.6 Take a closer look for graphene N f = 2 in the chiral Ising case ν − 1 ≈ − 0 . 952 ǫ + 0 . 00723 ǫ 2 − 0 . 0949 ǫ 3 ← coeffjcients grow! η ψ ≈ 0 . 0714 ǫ − 0 . 00671 ǫ 2 − 0 . 0243 ǫ 3 ← coeffjcients grow! η φ ≈ 0 . 571 ǫ + 0 . 124 ǫ 2 − 0 . 0278 ǫ 3 ⇒ The series diverge asymptotically! (Remember we work in D = 4 − ǫ ) | a 1 | O ( ϵ 1 ) | a 1 | O ( ϵ 1 ) | a 2 | O ( ϵ 2 ) | a 2 | O ( ϵ 2 ) | a 3 | O ( ϵ 3 ) | a 3 | O ( ϵ 3 ) a i ( N f ) from 1/ ν a i ( N f ) from η ψ a i ( N f ) from η φ | a 1 | O ( ϵ 1 ) | a 2 | O ( ϵ 2 ) | a 3 | O ( ϵ 3 ) N f N f N f

Asymptotic series 7 0.10 0.05 0.00 10 9 8 6 0.20 5 4 3 2 1 1.2 0.15 0 0.8 10 1.0 0.8 0.6 0.4 0.2 0.0 9 1 8 7 6 5 4 3 2 1.0 0 0.6 7 2 0.4 0 1 2 3 4 5 6 16 8 9 10 0.0 0.2 Take a closer look for graphene N f = 2 in the chiral Ising case ν − 1 ≈ − 0 . 952 ǫ + 0 . 00723 ǫ 2 − 0 . 0949 ǫ 3 ← coeffjcients grow! η ψ ≈ 0 . 0714 ǫ − 0 . 00671 ǫ 2 − 0 . 0243 ǫ 3 ← coeffjcients grow! η φ ≈ 0 . 571 ǫ + 0 . 124 ǫ 2 − 0 . 0278 ǫ 3 ⇒ The series diverge asymptotically! (Remember we work in D = 4 − ǫ ) | a 1 | O ( ϵ 1 ) | a 1 | O ( ϵ 1 ) | a 2 | O ( ϵ 2 ) | a 2 | O ( ϵ 2 ) | a 3 | O ( ϵ 3 ) | a 3 | O ( ϵ 3 ) a i ( N f ) from 1/ ν a i ( N f ) from η ψ a i ( N f ) from η φ | a 1 | O ( ϵ 1 ) | a 2 | O ( ϵ 2 ) | a 3 | O ( ϵ 3 ) N f N f N f For D = 3 the direct substitution ǫ = 1 does not provide reliable values...

Resummation

• Recall perturbative QFT: • Expand integrand w.r.t. g and interchange sum f L g 1.8 0.0 L k 0 f k g k L k 0 g k k d x x 4 k e x 2 0.2 1.7 asymptotic series 0.6 0.8 1.0 1.1 1.2 1.3 1.4 1.5 1.9 1.6 0.4 Taming an asymptotic series and integration 0.4 j e 1 2 D 1 4 j 0.0 Why do asymptotic series occur in perturbation theory? 0.2 0.6 1.5 1.9 1.8 1.7 0.8 1.6 1.4 1.3 1.2 1.1 1.0 17 • As an easy example consider � d x e − x 2 + gx 4 f ( g ) =

• Expand integrand w.r.t. g and interchange sum f L g Taming an asymptotic series k 0 f k g k L k 0 g k k d x x 4 k e x 2 0.0 0.4 0.2 0.6 0.8 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 L asymptotic series Why do asymptotic series occur in perturbation theory? 1.1 0.0 0.2 0.4 and integration 0.8 1.0 0.6 1.2 1.8 1.3 1.9 17 1.7 1.6 1.5 1.4 • As an easy example consider � d x e − x 2 + gx 4 f ( g ) = • Recall perturbative QFT: � 2 φ · D − 1 φ + λφ 4 + j · φ D φ e − 1 Z [ j ] =

• Expand integrand w.r.t. g and interchange sum f L g L k 0 f k g k L k 0 g k k d x x 4 k e x 2 0.0 Taming an asymptotic series 0.2 0.6 0.8 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 0.4 and integration asymptotic series 1.2 0.0 0.2 0.4 0.6 Why do asymptotic series occur in perturbation theory? 1.0 1.1 0.8 1.3 1.8 1.4 1.9 17 1.7 1.6 1.5 • As an easy example consider � d x e − x 2 + gx 4 f ( g ) = f ( g ) • Recall perturbative QFT: � 2 φ · D − 1 φ + λφ 4 + j · φ D φ e − 1 Z [ j ] = g

Taming an asymptotic series 0.8 1.9 Why do asymptotic series occur in perturbation theory? L L g k 0.0 0.2 0.4 0.6 1.0 1.7 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.8 17 1.6 1.1 0.2 0.4 0.6 0.8 1.0 1.5 0.0 1.3 1.2 1.4 • As an easy example consider � d x e − x 2 + gx 4 f ( g ) = f ( g ) • Recall perturbative QFT: � 2 φ · D − 1 φ + λφ 4 + j · φ D φ e − 1 Z [ j ] = g • Expand integrand w.r.t. g and interchange sum and integration ⇒ asymptotic series � f k g k = � � f [ L ] ( g ) = d x x 4 k e − x 2 k ! k = 0 k = 0

Taming an asymptotic series 0.8 1.9 L Why do asymptotic series occur in perturbation theory? L g k 0.0 0.2 0.4 0.6 1.0 1.7 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.8 17 1.6 1.1 0.0 0.2 0.4 1.5 0.8 1.0 0.6 1.4 1.2 1.3 • As an easy example consider � d x e − x 2 + gx 4 f ( g ) = f ( g ) • Recall perturbative QFT: � 2 φ · D − 1 φ + λφ 4 + j · φ D φ e − 1 Z [ j ] = g • Expand integrand w.r.t. g and interchange sum f ( g ) and integration ⇒ asymptotic series f [1] ( g ) f [2] ( g ) f [3] ( g ) � f k g k = � � f [ L ] ( g ) = d x x 4 k e − x 2 f ( g ) k ! k = 0 k = 0 g

• No knowledge about large-order behavior of f L g but already a good approximation • For a better approximation we need knowledge about large-order behavior of f L g which • The parameters Taming an asymptotic series: Padé Approximation f k 1.8 1.9 is of the form 1 k k 1.6 k k k and can be extracted from f g by a saddle-point approximation 1.7 1.4 1.5 How can we extract reliable estimates also for large g ? 1.3 1.2 1.1 1.0 0.8 0.6 0.4 0.2 0.0 18 • Padé Approximation [ M / N ] f = a 0 + a 1 g + · · · + a M g M 1 + b 1 g + · · · + b N g N

• For a better approximation we need knowledge about large-order behavior of f L g which • The parameters Taming an asymptotic series: Padé Approximation 1.5 can be extracted from f g by a saddle-point approximation and k k k 1 k k f k is of the form 1.9 1.8 1.7 1.6 1.4 How can we extract reliable estimates also for large g ? 1.3 1.2 1.1 1.0 0.8 0.6 0.4 0.2 0.0 18 • Padé Approximation [ M / N ] f = a 0 + a 1 g + · · · + a M g M 1 + b 1 g + · · · + b N g N • No knowledge about large-order behavior of f [ L ] ( g ) but already a good approximation

• For a better approximation we need knowledge about large-order behavior of f L g which • The parameters Taming an asymptotic series: Padé Approximation 1.5 can be extracted from f g by a saddle-point approximation and k k k 1 k k f k is of the form 1.9 1.8 1.7 How can we extract reliable estimates also for large g ? 1.6 1.4 1.3 0.0 0.2 0.4 0.6 0.8 1.0 1.1 1.2 18 • Padé Approximation f ( g ) [1/1] f ( g ) [1/2] f ( g ) [ M / N ] f = a 0 + a 1 g + · · · + a M g M [2/1] f ( g ) 1 + b 1 g + · · · + b N g N f ( g ) • No knowledge about large-order behavior of f [ L ] ( g ) but already a good approximation g

• The parameters Taming an asymptotic series: Padé Approximation 1.2 can be extracted from f g by a saddle-point approximation and is of the form 1.9 1.8 1.7 1.6 1.5 How can we extract reliable estimates also for large g ? 1.3 1.4 1.1 0.4 0.0 0.2 18 0.6 0.8 1.0 • Padé Approximation f ( g ) [1/1] f ( g ) [1/2] f ( g ) [ M / N ] f = a 0 + a 1 g + · · · + a M g M [2/1] f ( g ) 1 + b 1 g + · · · + b N g N f ( g ) • No knowledge about large-order behavior of f [ L ] ( g ) but already a good approximation g • For a better approximation we need knowledge about large-order behavior of f [ L ] ( g ) which f k ∼ ( − α ) k Γ( k + β + 1 ) ≈ ( − α ) k k ! k β

Taming an asymptotic series: Padé Approximation 1.1 is of the form 1.9 1.8 1.7 1.6 1.5 1.4 How can we extract reliable estimates also for large g ? 1.2 1.3 1.0 0.2 0.0 18 0.4 0.6 0.8 • Padé Approximation f ( g ) [1/1] f ( g ) [1/2] f ( g ) [ M / N ] f = a 0 + a 1 g + · · · + a M g M [2/1] f ( g ) 1 + b 1 g + · · · + b N g N f ( g ) • No knowledge about large-order behavior of f [ L ] ( g ) but already a good approximation g • For a better approximation we need knowledge about large-order behavior of f [ L ] ( g ) which f k ∼ ( − α ) k Γ( k + β + 1 ) ≈ ( − α ) k k ! k β • The parameters α and β can be extracted from f ( g ) by a saddle-point approximation

• Borel-Transformation f g t B f gt • For fjnite L this transformation just reinserts fjnd a nonpolynomial function which Taylor expansion coincides with B f g • Use a Padé approximant for this, giving the so-called Padé-Borel Transformation t M N B f gt • This is still ignoring the detailed large-order behavior f k 1.6 1.7 1.8 1.9 k Taming an asymptotic series: Padé Approximation 1 . To avoid this one have to 1.4 M N f g 0 d t e k k k 1.5 1.1 1.3 d t e k f k k B k g k 1.2 0 0.0 0.2 0.4 0.6 0.8 1.0 How can we extract reliable estimates also for large g ? 19 • Factorial grow ⇒ Borel-Leroy-summable Γ( k + β + 1 ) g k = � � B f ( g ) =

• For fjnite L this transformation just reinserts fjnd a nonpolynomial function which Taylor expansion coincides with B f g • Use a Padé approximant for this, giving the so-called Padé-Borel Transformation t M N B f gt • This is still ignoring the detailed large-order behavior f k k 1.4 1.5 1.6 1.7 1.8 1.9 Taming an asymptotic series: Padé Approximation 1 . To avoid this one have to 1.2 M N f g 0 d t e k k k 1.3 1.1 How can we extract reliable estimates also for large g ? 0 k f k k B k g k 1.0 19 0.0 0.2 0.4 0.6 0.8 • Factorial grow ⇒ Borel-Leroy-summable Γ( k + β + 1 ) g k = � � B f ( g ) = • Borel-Transformation � ∞ d t e − t B f ( gt ) . B f ( g ) =

• Use a Padé approximant for this, giving the so-called Padé-Borel Transformation t M N B f gt • This is still ignoring the detailed large-order behavior f k 1.8 1.2 1.3 1.4 1.5 1.6 1.7 Taming an asymptotic series: Padé Approximation 1.9 How can we extract reliable estimates also for large g ? M N f g 0 d t e k k k 1.1 1.0 0.8 0.6 k f k k B k g k 19 0.0 0 0.2 0.4 • Factorial grow ⇒ Borel-Leroy-summable Γ( k + β + 1 ) g k = � � B f ( g ) = • Borel-Transformation � ∞ d t e − t B f ( gt ) . B 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 B f ( g )

• This is still ignoring the detailed large-order behavior f k Taming an asymptotic series: Padé Approximation 1.5 How can we extract reliable estimates also for large g ? 1.1 1.2 1.3 1.4 1.7 1.6 0.6 1.8 1.9 f 0 k k k 0.8 1.0 0.4 B k g k k 0.2 k f k 19 0.0 0 • Factorial grow ⇒ Borel-Leroy-summable Γ( k + β + 1 ) g k = � � B f ( g ) = • Borel-Transformation � ∞ d t e − t B f ( gt ) . B 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 B f ( g ) • Use a Padé approximant for this, giving the so-called Padé-Borel Transformation � ∞ PB [ M / N ] d t e − t [ M / N ] B f ( gt ) ( g ) =

• This is still ignoring the detailed large-order behavior f k Taming an asymptotic series: Padé Approximation How can we extract reliable estimates also for large g ? 1.0 1.1 1.2 1.3 1.4 1.7 1.6 0.6 1.8 1.9 f 0 k k k 0.8 1.5 0.4 B k g k k 0.2 f k k 19 0 0.0 • Factorial grow ⇒ Borel-Leroy-summable f ( g ) PB [1/1] ( g ) f PB [1/2] Γ( k + β + 1 ) g k = ( g ) � � f B f ( g ) = PB [2/1] ( g ) f f ( g ) • Borel-Transformation � ∞ d t e − t B f ( gt ) . B f ( g ) = 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 B f ( g ) • Use a Padé approximant for this, giving the so-called Padé-Borel Transformation � ∞ PB [ M / N ] d t e − t [ M / N ] B f ( gt ) ( g ) =

Taming an asymptotic series: Padé Approximation 0.4 0 f 1.9 1.8 1.7 1.6 How can we extract reliable estimates also for large g ? 1.4 1.3 1.2 1.1 1.0 0.8 0.6 1.5 0.2 B k g k 0.0 k f k k 19 0 • Factorial grow ⇒ Borel-Leroy-summable f ( g ) PB [1/1] ( g ) f PB [1/2] Γ( k + β + 1 ) g k = ( g ) � � f B f ( g ) = PB [2/1] ( g ) f f ( g ) • Borel-Transformation � ∞ d t e − t B f ( gt ) . B f ( g ) = 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 B f ( g ) • Use a Padé approximant for this, giving the so-called Padé-Borel Transformation � ∞ PB [ M / N ] d t e − t [ M / N ] B f ( gt ) ( g ) = • This is still ignoring the detailed large-order behavior f k ∼ ( − α ) k k ! k β

• Conformal mapping regarding with singularity w 2 • Implies that W w B f g w • Reexpand the Borel-Leroy sum in this new variable w g B L W k w g • Borel-Transformation with this for all g analytical series gives t w gt Taming an asymptotic series: Conformal mapping g f has a convergent Taylor expansion for w 1 k 1.40 1.38 1.36 1.00 0.98 L L 0 L k d t e 0 W k 0 k g B k g k f L k 0 k 1.9 0.96 1.7 1.8 4 k large k 1 g 1 w g 1 ag 1 1 ag How to incorporate the full large-order behavior? g 1 a 1.1 1.6 1.5 1.4 w 1.2 1.3 1.0 0.8 0.6 0.4 0.2 0.0 1 20 • Borel-Leroy sum for large k ( − α ) k g k = � B f ( g ) ∼ 1 + α g

• Implies that W w B f g w • Reexpand the Borel-Leroy sum in this new variable w g B L W k w g • Borel-Transformation with this for all g analytical series gives t w gt Taming an asymptotic series: Conformal mapping L g f 1 0 has a convergent Taylor expansion for w 1.40 1.38 1.36 k k B k g k L 0.98 0 k L f g L k 0 W k 0 d t e k 1.00 0.96 How to incorporate the full large-order behavior? 0.6 k large k 1 a 1.9 0.0 0.2 0.4 w 0.8 1.5 1.8 1.7 1.6 1.0 20 1.4 1.2 1.1 1.3 • Borel-Leroy sum for large k ( − α ) k g k = � B f ( g ) ∼ 1 + α g • Conformal mapping regarding with singularity g = − 1 /α √ 1 + ag − 1 w ( g ) = √ 1 + ag + 1 , g = 4 ( 1 − w ) 2