Outline Introduction and motivation Gauge-fermion theories - PowerPoint PPT Presentation

Beta functions at large N f Anders Eller Thomsen aethomsen@cp3.sdu.dk CP 3 -Origins, University of Southern Denmark ITP, Heidelberg University, 17th July 2018 Anders Eller Thomsen (CP 3 -Origins) Beta functions at large N f Heidelberg ’18 1 / 27

Outline Introduction and motivation Gauge-fermion theories Gauge-Yukawa theories Summary and outlook Ka´ ca Bradonji´ c Based on: Oleg Antipin , Nicola Andrea Dondi , Francesco Sannino , AET, and Zhi-Wei Wang [arXiv:1803.09770], to appear in PRD Anders Eller Thomsen (CP 3 -Origins) Beta functions at large N f Heidelberg ’18 2 / 27

Outline Introduction and motivation 1 Gauge-fermion theories 2 Gauge-Yukawa theories 3 Summary and outlook 4 Anders Eller Thomsen (CP 3 -Origins) Beta functions at large N f Heidelberg ’18 3 / 27

Renormalization group flow 4 F 0 ,µν F µν L = − 1 + i Ψ 0 γ µ ( ∂ µ − ig 0 A 0 ,µ )Ψ 0 0 Quantum corrections give infinite contributions � � � d 4 k Tr γ µ ( / k − / p ) γ ν / k = − g 2 = ∞ 0 (2 π ) 4 k 2 ( k − p ) 2 Calculability is recovered using dimensional regularization, d = 4 − ǫ � � � d d k Tr γ µ ( / k − / p ) γ ν / k = − g 2 0 (2 π ) d k 2 ( k − p ) 2 � 2 � �� ≃ − i g 2 ǫ + 5 − 4 π 12 π 2 ( p 2 g µν − p µ p ν ) 0 3 − γ E + log p 2 Anders Eller Thomsen (CP 3 -Origins) Beta functions at large N f Heidelberg ’18 3 / 27

Renormalization group flow 4 F 0 ,µν F µν L = − 1 + i Ψ 0 γ µ ( ∂ µ − ig 0 A 0 ,µ )Ψ 0 0 Quantum corrections give infinite contributions � � � d 4 k Tr γ µ ( / k − / p ) γ ν / k = − g 2 = ∞ 0 (2 π ) 4 k 2 ( k − p ) 2 Calculability is recovered using dimensional regularization, d = 4 − ǫ � � � d d k Tr γ µ ( / k − / p ) γ ν / k = − g 2 0 (2 π ) d k 2 ( k − p ) 2 � 2 � �� ≃ − i g 2 ǫ + 5 − 4 π 12 π 2 ( p 2 g µν − p µ p ν ) 0 3 − γ E + log p 2 Anders Eller Thomsen (CP 3 -Origins) Beta functions at large N f Heidelberg ’18 3 / 27

Renormalization group flow 4 F 0 ,µν F µν L = − 1 + i Ψ 0 γ µ ( ∂ µ − ig 0 A 0 ,µ )Ψ 0 0 Quantum corrections give infinite contributions � � � d 4 k Tr γ µ ( / k − / p ) γ ν / k = − g 2 = ∞ 0 (2 π ) 4 k 2 ( k − p ) 2 Calculability is recovered using dimensional regularization, d = 4 − ǫ � � � d d k Tr γ µ ( / k − / p ) γ ν / k = − g 2 0 (2 π ) d k 2 ( k − p ) 2 � 2 � �� ≃ − i g 2 ǫ + 5 − 4 π 12 π 2 ( p 2 g µν − p µ p ν ) 0 3 − γ E + log p 2 Anders Eller Thomsen (CP 3 -Origins) Beta functions at large N f Heidelberg ’18 3 / 27

Renormalization group flow The 1 /ǫ poles are absorbed into the bare couplings and fields Z g = 1 + 1 + 1 ǫ Z (1) ǫ 2 Z (2) g 0 = µ ǫ/ 2 Z g g , where + . . . g g As a result the renormalized coupling g ( µ ) gets a running � � g 3 d g − 1 + g ∂ 2 Z (1) 12 π 2 + O ( g 5 ) d ln µ = 1 ( − 1 β g = g g ) = 2 ∂ g Computing the beta functions have all the usual difficulties of perturbation theory Anders Eller Thomsen (CP 3 -Origins) Beta functions at large N f Heidelberg ’18 4 / 27

Renormalization group flow The 1 /ǫ poles are absorbed into the bare couplings and fields Z g = 1 + 1 + 1 ǫ Z (1) ǫ 2 Z (2) g 0 = µ ǫ/ 2 Z g g , where + . . . g g As a result the renormalized coupling g ( µ ) gets a running � � g 3 d g − 1 + g ∂ 2 Z (1) 12 π 2 + O ( g 5 ) d ln µ = 1 ( − 1 β g = g g ) = 2 ∂ g Computing the beta functions have all the usual difficulties of perturbation theory Anders Eller Thomsen (CP 3 -Origins) Beta functions at large N f Heidelberg ’18 4 / 27

Gauge-fermion theories at 1-loop QED: Landau Pole g β g g ln µ A fundamental theory must reach a FP in the UV QCD: Asymptotic freedom g β g g ln µ Anders Eller Thomsen (CP 3 -Origins) Beta functions at large N f Heidelberg ’18 5 / 27

Gauge-fermion theories at 1-loop QED: Landau Pole g β g g ln µ A fundamental theory must reach a FP in the UV QCD: Asymptotic freedom g β g g ln µ Anders Eller Thomsen (CP 3 -Origins) Beta functions at large N f Heidelberg ’18 5 / 27

Perturbative UVFP In a gauge-Yukawa theory D.F. Litim and F. Sannino [1406.2337] � N f � β α g = 4 − 11 α 2 2 + f ( α g , α y , α λ ) g 3 N c A perturbative FP can be reached at N f , N c → ∞ β α α α ln µ A non-vanishing α g in the UV can tame the other couplings, e.g. β α y ≃ α y (13 α y − 6 α g ) Anders Eller Thomsen (CP 3 -Origins) Beta functions at large N f Heidelberg ’18 6 / 27

Perturbative UVFP In a gauge-Yukawa theory D.F. Litim and F. Sannino [1406.2337] � N f � β α g = 4 − 11 α 2 2 + f ( α g , α y , α λ ) g 3 N c A perturbative FP can be reached at N f , N c → ∞ β α α α ln µ A non-vanishing α g in the UV can tame the other couplings, e.g. β α y ≃ α y (13 α y − 6 α g ) Anders Eller Thomsen (CP 3 -Origins) Beta functions at large N f Heidelberg ’18 6 / 27

Perturbative UVFP In a gauge-Yukawa theory D.F. Litim and F. Sannino [1406.2337] � N f � β α g = 4 − 11 α 2 2 + f ( α g , α y , α λ ) g 3 N c A perturbative FP can be reached at N f , N c → ∞ β α α α ln µ A non-vanishing α g in the UV can tame the other couplings, e.g. β α y ≃ α y (13 α y − 6 α g ) Anders Eller Thomsen (CP 3 -Origins) Beta functions at large N f Heidelberg ’18 6 / 27

Where the large N f fits in Idea: Organize the the computation as an expansion in 1 / N f Computational control in a limit of QFT A new non-vanishing zero of the beta function β α α α ln µ A tool for model building S. Abel and F. Sannino [1707.06638] , E. Molinaro, F. Sannino and Z.W. Wang [1807.03669] , R.B. Mann et al. [1707.02942] Anders Eller Thomsen (CP 3 -Origins) Beta functions at large N f Heidelberg ’18 7 / 27

Where the large N f fits in Idea: Organize the the computation as an expansion in 1 / N f Computational control in a limit of QFT A new non-vanishing zero of the beta function β α α α ln µ A tool for model building S. Abel and F. Sannino [1707.06638] , E. Molinaro, F. Sannino and Z.W. Wang [1807.03669] , R.B. Mann et al. [1707.02942] Anders Eller Thomsen (CP 3 -Origins) Beta functions at large N f Heidelberg ’18 7 / 27

Where the large N f fits in Idea: Organize the the computation as an expansion in 1 / N f Computational control in a limit of QFT A new non-vanishing zero of the beta function β α α α ln µ A tool for model building S. Abel and F. Sannino [1707.06638] , E. Molinaro, F. Sannino and Z.W. Wang [1807.03669] , R.B. Mann et al. [1707.02942] Anders Eller Thomsen (CP 3 -Origins) Beta functions at large N f Heidelberg ’18 7 / 27

Where the large N f fits in Idea: Organize the the computation as an expansion in 1 / N f Computational control in a limit of QFT A new non-vanishing zero of the beta function β α α α ln µ A tool for model building S. Abel and F. Sannino [1707.06638] , E. Molinaro, F. Sannino and Z.W. Wang [1807.03669] , R.B. Mann et al. [1707.02942] Anders Eller Thomsen (CP 3 -Origins) Beta functions at large N f Heidelberg ’18 7 / 27

Outline Introduction and motivation 1 Gauge-fermion theories 2 Gauge-Yukawa theories 3 Summary and outlook 4 Anders Eller Thomsen (CP 3 -Origins) Beta functions at large N f Heidelberg ’18 8 / 27

1 / N f counting The goal is to expand the a gauge theory in 1 / N f ; N f � 4 F µν F µν + L = − 1 i Ψ I γ µ ( ∂ µ − ig A µ )Ψ I I =1 It is insufficient to take N f → ∞ : β g = d g 1 12 π 2 g 3 N f : d t = Introduce a ’t Hooft-like coupling K = g 2 N f 4 π 2 � 1 / N f ≪ 1 limit: g ∼ K = cst. 1 / N f Anders Eller Thomsen (CP 3 -Origins) Beta functions at large N f Heidelberg ’18 8 / 27

1 / N f counting The goal is to expand the a gauge theory in 1 / N f ; N f � 4 F µν F µν + L = − 1 i Ψ I γ µ ( ∂ µ − ig A µ )Ψ I I =1 It is insufficient to take N f → ∞ : β g = d g 1 12 π 2 g 3 N f : d t = Introduce a ’t Hooft-like coupling K = g 2 N f 4 π 2 � 1 / N f ≪ 1 limit: g ∼ K = cst. 1 / N f Anders Eller Thomsen (CP 3 -Origins) Beta functions at large N f Heidelberg ’18 8 / 27

1 / N f counting The goal is to expand the a gauge theory in 1 / N f ; N f � 4 F µν F µν + L = − 1 i Ψ I γ µ ( ∂ µ − ig A µ )Ψ I I =1 It is insufficient to take N f → ∞ : β g = d g 1 12 π 2 g 3 N f : d t = Introduce a ’t Hooft-like coupling K = g 2 N f 4 π 2 � 1 / N f ≪ 1 limit: g ∼ K = cst. 1 / N f Anders Eller Thomsen (CP 3 -Origins) Beta functions at large N f Heidelberg ’18 8 / 27

What diagrams should we include? Diagrams contain fermion loops with n ≥ 2 gauge insertions; µ n µ 3 � 1 � ( n − 2) / 2 µ 2 ∼ g n N f = O N f µ 1 Loops with n = 2 are “free”; internal gauge lines must be dressed as = O (1) Loops with n ≥ 3 decreases the order of a diagram. Anders Eller Thomsen (CP 3 -Origins) Beta functions at large N f Heidelberg ’18 9 / 27

What diagrams should we include? Diagrams contain fermion loops with n ≥ 2 gauge insertions; µ n µ 3 � 1 � ( n − 2) / 2 µ 2 ∼ g n N f = O N f µ 1 Loops with n = 2 are “free”; internal gauge lines must be dressed as = O (1) Loops with n ≥ 3 decreases the order of a diagram. Anders Eller Thomsen (CP 3 -Origins) Beta functions at large N f Heidelberg ’18 9 / 27