Perturbative ambiguities in compactified spacetime and resurgence - PowerPoint PPT Presentation

Perturbative ambiguities in compactified spacetime and resurgence structure Okuto Morikawa ( 森 川 億 人 ) Kyushu University 2020/9/24 Resurgence 2020 in YITP K. Ishikawa, O.M., K. Shibata and H. Suzuki, PTEP 2020 (2020) 063B02 [arXiv:2001.07302 [hep-th]] O.M. and H. Takaura, PLB 807 (2020) 135570 [arXiv:2003.04759 [hep-th]] M. Ashie, O.M., H. Suzuki and H. Takaura, PTEP 2020 (2020) 093B02 [arXiv:2005.07407 [hep-th]] Perturbative ambiguities in R d − 1 × S 1 Okuto Morikawa (Kyushu U.) 2020/9/24 Resurgence 2020 1 / 27

Contents Introduction 1 Factorial growth in QFT Resurgence structure in R d − 1 × S 1 PFD-type ambiguity on circle compactification 2 Enhancement mechanism and bion Vacuum energy of SUSY C P N model Renormalon on circle compactification 3 ∄ Renormalon ambiguities Subtlety of large N limit on circle compactification “Renormalon precursor” and decompactification Summary 4 Perturbative ambiguities in R d − 1 × S 1 Okuto Morikawa (Kyushu U.) 2020/9/24 Resurgence 2020 2 / 27

Contents Introduction 1 Factorial growth in QFT Resurgence structure in R d − 1 × S 1 PFD-type ambiguity on circle compactification 2 Enhancement mechanism and bion Vacuum energy of SUSY C P N model Renormalon on circle compactification 3 ∄ Renormalon ambiguities Subtlety of large N limit on circle compactification “Renormalon precursor” and decompactification Summary 4 Perturbative ambiguities in R d − 1 × S 1 Okuto Morikawa (Kyushu U.) 2020/9/24 Resurgence 2020 3 / 27

Factorial growth in QFT Two sources of factorial growth of perturbative coefficients: Proliferation of Feynman diagrams (PFD) 1 Ground state energy in QM ∼ ( − 1) k (2 k )! Zeeman effect ∼ (2 k )! Stark effect V ( φ ) ∼ φ 3 ∼ Γ( k + 1 / 2) V ( φ ) ∼ φ 4 ∼ ( − 1) k Γ( k + 1 / 2) ∼ k ! Double well ∼ k ! Periodic cosine Renormalon [’t Hooft ’79] 2 ∼ β k 0 k ! k p ( β 0 : one-loop coefficient of the beta function) Perturbative ambiguities in R d − 1 × S 1 Okuto Morikawa (Kyushu U.) 2020/9/24 Resurgence 2020 4 / 27

Resurgence theory and renormalon problem Perturbative ambiguities (Borel resummation) � � g 2 / 16 π 2 � k � − 16 π 2 / g 2 � PFD : k ! ⇒ δ ∼ exp k � � g 2 / 16 π 2 � k � � β k − 16 π 2 / ( β 0 g 2 ) ⇒ δ ∼ exp Renormalon : 0 k ! k Resurgence structure [Bogomolny ’80, Zinn-Justin ’81] � − 16 π 2 / g 2 � δ PFD ∼ exp = exp ( − 2 S I ) � Cancellation (Instanton action S I = 8 π 2 / g 2 ) δ Instanton ∼ exp ( − 2 S I ) But, what cancels the renormalon ambiguity δ Renormalon ? ◮ Non-trivial configuration with S = S I /β 0 ( δ NP ∼ e − 2 S I /β 0 )? ◮ Not known (cf. cancellation between terms of OPE [David ’82]) ◮ Higher-order perturbation theory? Perturbative ambiguities in R d − 1 × S 1 Okuto Morikawa (Kyushu U.) 2020/9/24 Resurgence 2020 5 / 27

Renormalon cancellation in R d − 1 × S 1 ? A possible candidate [Argyres–¨ Unsal ’12, Dunne–¨ Unsal ’12, . . . ]: Bion [¨ Unsal ’07] = fractional instanton/anti-instanton pair on R d − 1 × S 1 with Z N -twisted boundary conditions (BC) � A / N for A � = N ϕ A ( x , x d + 2 π R ) = e 2 π im A R ϕ A ( x , x d ) , m A R = 0 for A = N � �� � N -component field S B = S I / N ← similar N dependence! ( β 0 = 11 3 N for SU ( N ) GT) Resurgence on S 1 compactified spacetime? PT Semi-classical object δ Renormalon ∼ e − 16 π 2 / ( β 0 g 2 ) ❩ ✚ R 4 ? ✚ ❩ R 3 × S 1 δ Bion ∼ e − 2 S B = e − 2 S I / N δ Renormalon ? = QFT on R 3 × S 1 in R → ∞ ◮ QFT on R 4 Def. Perturbative ambiguities in R d − 1 × S 1 Okuto Morikawa (Kyushu U.) 2020/9/24 Resurgence 2020 6 / 27

Resurgence structure in R d − 1 × S 1 No renormalons in SU (2) and SU (3) QCD(adj.) on R 3 × S 1 [Anber–Sulejmanpasic ’14] Questions: What cancels the bion ambiguity? (PFD?) 1 Are there no renormalons in other theories? ( SU ( ∀ N )?) 2 How does δ Renormalon on R 4 emerge in R → ∞ ? 3 Answers [O.M.–Takaura, Ashie–O.M.–Suzuki–Takaura]: Enhancement of PFD due to Z N twisted BC: k ! → N k k ! 1 PT Semi-classical object δ PFD ∼ e − 16 π 2 / g 2 R 4 δ Instanton ∼ e − 2 S I R 3 × S 1 PFD ∼ e − 16 π 2 / ( Ng 2 ) δ ′ δ Bion ∼ e − 2 S I / N No renormalons in SU ( ∀ N ) QCD(adj.) on R 3 × S 1 2 “Renormalon precursor” R →∞ Renormalon on R 4 → 3 Helpful in giving a unified understanding on resurgence in QFT Perturbative ambiguities in R d − 1 × S 1 Okuto Morikawa (Kyushu U.) 2020/9/24 Resurgence 2020 7 / 27

Contents Introduction 1 Factorial growth in QFT Resurgence structure in R d − 1 × S 1 PFD-type ambiguity on circle compactification 2 Enhancement mechanism and bion Vacuum energy of SUSY C P N model Renormalon on circle compactification 3 ∄ Renormalon ambiguities Subtlety of large N limit on circle compactification “Renormalon precursor” and decompactification Summary 4 Perturbative ambiguities in R d − 1 × S 1 Okuto Morikawa (Kyushu U.) 2020/9/24 Resurgence 2020 8 / 27

Enhancement of PFD ambiguity Enhanced PFD-type ambiguities cancel bion ambiguities PFD: k ! ⇒ N k k ! δ PFD ∼ e − 16 π 2 / g 2 ⇒ δ ′ PFD ∼ e − 16 π 2 / ( Ng 2 ) ↔ δ bion | R 3 × S 1 Mechanism ( ∼ Linde problem [’80] in finite-temperature QFT): S 1 compactification → IR divergences 1 � � 1 � d d − 1 p d d p · · · = finite → · · · = ∞ at IR 2 π R p d ∈ Z / R twisted BC → twist angles as IR regulators 2 � g 2 � k 1 1 Amplitude ( Ng 2 ) k p 2 → = ⇒ = p 2 + ( p d + m A ) 2 m A R p d =0 , A =1 � �� � Enhancement! Let us study the C P N model on R × S 1 ◮ Bion calculus [Fujimori-Kamata-Misumi-Nitta-Sakai ’18] Perturbative ambiguities in R d − 1 × S 1 Okuto Morikawa (Kyushu U.) 2020/9/24 Resurgence 2020 9 / 27

2-dimensional C P N model with twisted BC 2D C P N − 1 model with Z N -twisted BC ( A = 1, 2, . . . , N ) � S = 1 z A ∂ µ z A − j µ j µ + f (¯ z A z A − 1) � � d 2 x ∂ µ ¯ g 2 z A ← → z A z A = 1) ∂ µ z A , f is an auxiliary field (¯ where j µ = (1 / 2 i )¯ # of vacuum bubble diagrams, T k ( j µ → ¯ zz , propagator → 1) � δ � 2 k 1 1 δ zz ) 2 � k ∼ 4 k Γ( k + 1 / 2) � T k = (¯ (2 k )! δ ¯ z δ z k ! Amplitude of ( k + 1)-loop connected diagram � � k +1 � V 2 ( g 2 ) k F 2 k ( p i , 1 , m A ) dp i , 1 � p 2 =0 , A =1 − → � 2 k (2 π R ) k +1 2 π i =1 [ q 2 i , 1 + m 2 A + f 0 ] i =1 ( F 2 k : 2 k th-order polynomial, q : linear combination of { p i } , f 0 = � f � ) � � IR divergence in massless limit m 2 d 2 p → A + f 0 → 0 ( dp ) Perturbative ambiguities in R d − 1 × S 1 Okuto Morikawa (Kyushu U.) 2020/9/24 Resurgence 2020 10 / 27

IR structure and enhancement mechanism A + f 0 = 1 / ( NR ) 2 + f 0 works as an IR regulator m 2 ◮ m 2 A ≫ f 0 � g 2 � k � Ng 2 � k ∼ V 2 1 → V 2 1 A =1 − Enhancement! R 2 ( m A R ) k − 1 R 2 4 π N 4 π ◮ m 2 A ≪ f 0 � g 2 � k ( m A R ) α ∼ V 2 ❤❤❤❤❤❤ ✭ � ✭✭✭✭✭✭ Enhancement ❤ R 2 ( f 0 R 2 ) ( k + α − 1) / 2 4 π α ≥ 0 Dependence on NR Λ (Λ = µ e − 2 π/ ( β 0 g 2 ) : dynamical scale) ◮ NR Λ ≪ 1 (Bion calculus is valid) f 0 R ∼ g 2 � 4 π ⇒ [ m 2 A = O ( g 0 )] ≫ [ f 0 = O ( g 4 )] Enhancement! ◮ NR Λ ≫ 1 (“Large N ”) m 2 1 ✭✭✭✭✭✭ ❤❤❤❤❤❤ ✭ f 0 ∼ Λ 2 A ⇒ ( NR Λ) 2 ≪ 1 = Enhancement ❤ f 0 Perturbative ambiguities in R d − 1 × S 1 Okuto Morikawa (Kyushu U.) 2020/9/24 Resurgence 2020 11 / 27

Discussions on enhancement mechanism PT Semi-classical object δ PFD ∼ e − 16 π 2 / g 2 R 4 δ Instanton ∼ e − 2 S I R 3 × S 1 PFD ∼ e − 16 π 2 / ( Ng 2 ) δ ′ δ Bion ∼ e − 2 S I / N Enhancement phenomenon is consistent with bion Borel singularities at m A R = A / N = 1 / N , 2 / N , . . . ���� ���� 1-bion 2-bion Disconnected diagram ( n connected) is suppressed by 1 / N n − 1 �� k T k ( g 2 ) k � ◮ # of connected diagrams ∼ ln Sum over flavor indices gives rise to further N factor Vacuum energy of 2D SUSY C P N model (following slides) ◮ NR Λ ≫ 1 [Ishikawa–O.M.–Shibata–Suzuki] ◮ NR Λ ≪ 1 [Fujimori–Kamata–Misumi–Nitta–Sakai ’18] Perturbative ambiguities in R d − 1 × S 1 Okuto Morikawa (Kyushu U.) 2020/9/24 Resurgence 2020 12 / 27

Vacuum energy of 2D SUSY C P N − 1 model 2D SUSY C P N − 1 model with SUSY breaking term � � σ P + + σ P − ) χ A � S = N z A D µ D µ z A + ¯ d 2 x χ A ( / − ¯ σσ + ¯ D + ¯ λ � � � d 2 x δǫ z A z A − 1 � + m A ¯ π R N A where D µ = ∂ µ + iA µ , γ 5 = − i γ x γ y , P ± = (1 ± γ 5 ) / 2, z A z A = 1, ¯ z A χ A = 0 and ¯ z A χ A = 0. and we impose ¯ Vacuum energy as a function of δǫ + E (1) δǫ + E (2) δǫ 2 + . . . E ( δǫ ) = E (0) ���� =0 Vacuum energy at NR Λ ≪ 1 [Fujimori-Kamata-Misumi-Nitta-Sakai] N − 1 b � ( − 1) b ( b !) 2 ( NR Λ) 2 b − 1 E Bion = 2Λ b =1 � δǫ + [ − 2 γ E − 2 ln (4 π b /λ R ) ∓ π i ] δǫ 2 + . . . � × Perturbative ambiguities in R d − 1 × S 1 Okuto Morikawa (Kyushu U.) 2020/9/24 Resurgence 2020 13 / 27