Attackingstrongcoupling withlargecharge Domenico Orlando based on - PowerPoint PPT Presentation

Albert Einstein Center for Fundamental Physics University of Bern 8 June 2017 | Torino in collaboration with: O. Loukas, S. Reffert (AEC Bern); L. Alvarez Gaumé (CERN and SCGP); D. Banerjee (DESY); S. Chandrasekharan (Duke); S. Hellerman, M. Watanabe (IPMU). Domenico Orlando Attacking strong coupling with large charge Attackingstrongcoupling withlargecharge Domenico Orlando based on [arXiv:1505.01537] , [arXiv:1610.04495] and more to come…

Introduction Effective action from classical scale invariance Quantum analysis Who’s who Domenico Orlando Attacking strong coupling with large charge

Introduction Effective action from classical scale invariance Quantum analysis Outline Introduction Effective action from classical scale invariance Quantum analysis Domenico Orlando Attacking strong coupling with large charge

Introduction Effective action from classical scale invariance Quantum analysis Outline Introduction Effective action from classical scale invariance Quantum analysis Domenico Orlando Attacking strong coupling with large charge

Introduction Effective action from classical scale invariance Domenico Orlando string theory quantum gravity Attacking strong coupling with large charge extrema of the RG fmow critical phenomena Why are we here? Conformal fjeld theories Quantum analysis 1.0 0.5 0.0 - 0.5 - 1.0 0.0 0.2 0.4 0.6 0.8

Introduction Effective action from classical scale invariance Quantum analysis Why are we here? Conformal fjeld theories are hard Most conformal fjeld theories (CFTs) lack nice limits where they become simple and solvable. No parameter of the theory can be dialed to a simplifying limit . dimension and OPE coeffjcients simplify. IDEA: study subsectors of the theory with fjxed quantum number Q . expansion. Domenico Orlando Attacking strong coupling with large charge There are sometimes sectors of the theory where anomalous Typically this happens in presence of a symmetry . A large Q becomes the controlling parameter in a perturbative

Introduction Effective action from classical scale invariance Quantum analysis no bootstrap here! This approach is to bootstrap . We will use an effective action. We will access sectors bootstrap. Domenico Orlando Attacking strong coupling with large charge completely orthogonal that are exponentially diffjcult to reach with

Introduction primary at fjxed charge : Domenico Orlando The very same formula describes the large-R-charge sector of a Effective action from classical scale invariance Attacking strong coupling with large charge A very concrete example where this type of simplifjcation happens. Summary of the results Quantum analysis We consider the O ( N ) vector model in three dimensions . In the IR it fmows to a conformal fjxed point [Wilson & Fisher] . We will fjnd an explicit formula for the dimension of the lowest π Q 3 / 2 + 2 √ π c 1 / 2 Q 1 / 2 − 0 . 093 + O ( Q − 1 / 2 ) Δ Q = c 3 / 2 2 √ supersymmetric N = 2 , d = 3 model .

Introduction Effective action from classical scale invariance Domenico Orlando Attacking strong coupling with large charge Quantum analysis Summary of the results Conformal dimensions in the O ( 2 ) model Δ o 12 u r p r e d i c 10 t i o n prediction 8 lattice 6 4 2 J 2 4 6 8 10

Introduction Effective action from classical scale invariance Domenico Orlando [Hasenbusch and Vicari] Attacking strong coupling with large charge Quantum analysis Summary of the results Δ 3.5 3.0 o u r p r 2.5 e O ( 2 ) d i c t i o n O ( 3 ) 2.0 O ( 4 ) 1.5 O ( 5 ) 1.0 0.5 J 1 2 3 4

Q 1 2 R ρ 1 2 the effective action is weakly coupled and under perturbative control in powers of ρ Introduction R Domenico Orlando 1 . Claim: For Λ 1 Q g 2 Λ UV R Q 1 2 ρ 1 2 Λ 1 Effective action from classical scale invariance with We think of the CFT as a Wilsonian effective action at the fjxed point The U 1 charge Q fjxes a second scale ρ 1 2 We look at a fjnite box of typical length R We need to understand the scales. Choose a cutoff Λ , separate the fjelds into high and low frequency We want to write a Wilsonian effective action . Scales Quantum analysis Attacking strong coupling with large charge φ H , φ L and do the path integral over the high-frequency part: ∫ e iS Λ ( φ L ) = D φ H e iS ( φ H , φ L )

Q 1 2 R ρ 1 2 the effective action is weakly coupled and under perturbative control in powers of ρ Introduction Q 1 2 with 1 R Λ ρ 1 2 g 2 R Λ UV Q 1 Claim: For Λ 1 . Domenico Orlando We think of the CFT as a Wilsonian effective action at the fjxed point The U 1 charge Q fjxes a second scale ρ 1 2 Effective action from classical scale invariance We look at a fjnite box of typical length R Quantum analysis Scales We want to write a Wilsonian effective action . Choose a cutoff Λ , separate the fjelds into high and low frequency Attacking strong coupling with large charge We need to understand the scales. d r a h o o t φ H , φ L and do the path integral over the high-frequency part: ∫ e iS Λ ( φ L ) = D φ H e iS ( φ H , φ L )

Introduction Effective action from classical scale invariance Domenico Orlando R 1 with We think of the CFT as a Wilsonian effective action at the fjxed point We need to understand the scales. Attacking strong coupling with large charge Scales Quantum analysis Choose a cutoff Λ , separate the fjelds into high and low frequency We want to write a Wilsonian effective action . d r a h o o t φ H , φ L and do the path integral over the high-frequency part: ∫ e iS Λ ( φ L ) = D φ H e iS ( φ H , φ L ) ▶ We look at a fjnite box of typical length R ▶ The U ( 1 ) charge Q fjxes a second scale ρ 1 / 2 ∼ Q 1 / 2 / R R ≪ Λ ≪ ρ 1 / 2 ∼ Q 1 / 2 ≪ Λ UV = g 2 ⇒ Q ≫ 1 Claim: For Λ ≪ ρ 1 / 2 the effective action is weakly coupled and under perturbative control in powers of ρ − 1 .

Introduction Effective action from classical scale invariance Quantum analysis Outline Introduction Effective action from classical scale invariance Quantum analysis Domenico Orlando Attacking strong coupling with large charge

Introduction Wilson and Fisher showed that this fmows to a conformal IR fjxed point. Domenico Orlando action for this universality class. Effective action from classical scale invariance Attacking strong coupling with large charge Quantum analysis The O ( N ) model The UV Lagrangian of the O ( N ) vector model is of the form L UV = ∂ μ φ a ∂ μ φ a − g 2 ( φ a φ a ) 2 , − − − − → IR conformal fjxed point . UV theory RG fmow The idea is to make use of this fact to write an effective Wilsonian

Introduction Wilson and Fisher showed that this fmows to a conformal IR fjxed point. Domenico Orlando action for this universality class. Effective action from classical scale invariance Attacking strong coupling with large charge Quantum analysis The O ( N ) model forget this The UV Lagrangian of the O ( N ) vector model is of the form L UV = ∂ μ φ a ∂ μ φ a − g 2 ( φ a φ a ) 2 , − − − − → IR conformal fjxed point . UV theory RG fmow The idea is to make use of this fact to write an effective Wilsonian

n o i t i t s r e p u s Domenico Orlando Introduction Effective action from classical scale invariance maybe a convergent expansion in derivatives. your system is already weakly-coupled for some reason; something that helps you organize perturbative calculations, if might allow you to get the anomalies right; a cute qualitative picture; At best: infjnite terms. The Wilsonian action is fundamentally useless because it contains Wilsonian action Quantum analysis Attacking strong coupling with large charge

n o i t i t s r e p u s Domenico Orlando Introduction Effective action from classical scale invariance maybe a convergent expansion in derivatives. your system is already weakly-coupled for some reason; something that helps you organize perturbative calculations, if might allow you to get the anomalies right; a cute qualitative picture; At best: infjnite terms. The Wilsonian action is fundamentally useless because it contains Wilsonian action Quantum analysis Attacking strong coupling with large charge

n o i t i t s r e p u s Domenico Orlando Introduction Effective action from classical scale invariance maybe a convergent expansion in derivatives. your system is already weakly-coupled for some reason; something that helps you organize perturbative calculations, if might allow you to get the anomalies right; At best: infjnite terms. The Wilsonian action is fundamentally useless because it contains Wilsonian action Quantum analysis Attacking strong coupling with large charge ▶ a cute qualitative picture;

n o i t i t s r e p u s Domenico Orlando Introduction Effective action from classical scale invariance maybe a convergent expansion in derivatives. your system is already weakly-coupled for some reason; something that helps you organize perturbative calculations, if At best: infjnite terms. The Wilsonian action is fundamentally useless because it contains Wilsonian action Quantum analysis Attacking strong coupling with large charge ▶ a cute qualitative picture; ▶ might allow you to get the anomalies right;