INFN | Torino 5 January 2018 | Rikkyo University [arXiv:1707.00711] and more to come… Domenico Orlando The unreasonable effectiveness of the large charge expansion Theunreasonable effectivenessofthe largechargeexpansion Domenico Orlando based on [arXiv:1505.01537] , [arXiv:1610.04495] , [arXiv:1707.00710] ,
Introduction Effective action from classical scale invariance Quantum analysis Who’s who 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 The unreasonable effectiveness of the large charge expansion
Introduction Effective action from classical scale invariance Quantum analysis Outline Introduction Effective action from classical scale invariance Quantum analysis Domenico Orlando The unreasonable effectiveness of the large charge expansion
Introduction Effective action from classical scale invariance Quantum analysis Outline Introduction Effective action from classical scale invariance Quantum analysis Domenico Orlando The unreasonable effectiveness of the large charge expansion
Introduction Effective action from classical scale invariance Domenico Orlando string theory quantum gravity The unreasonable effectiveness of the large charge expansion 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 . coeffjcients simplify. Domenico Orlando The unreasonable effectiveness of the large charge expansion In presence of a symmetry there can be sectors of the theory where anomalous dimension and OPE
Introduction Effective action from classical scale invariance Quantum analysis The idea Study subsectors of the theory with fjxed quantum number Q . In each sector, a large Q is the controlling parameter in a perturbative expansion . Domenico Orlando The unreasonable effectiveness of the large charge expansion
Introduction Effective action from classical scale invariance Quantum analysis no bootstrap here! This approach is orthogonal to bootstrap . We will use an effective action. We will access sectors that are diffjcult to reach with bootstrap. Domenico Orlando The unreasonable effectiveness of the large charge expansion But see [arXiv:1710.11161]
Introduction Effective action from classical scale invariance Quantum analysis Concrete results fmows to a conformal fjxed point [Wilson & Fisher] . We fjnd an explicit formula for the dimension of the lowest primary at fjxed charge : Domenico Orlando The unreasonable effectiveness of the large charge expansion We consider the O ( N ) vector model in three dimensions . In the IR it π Q 3 / 2 + 2 √ π c 1 / 2 Q 1 / 2 − 0 . 094 + O ( Q − 1 / 2 ) Δ Q = c 3 / 2 2 √
Introduction Effective action from classical scale invariance Domenico Orlando The unreasonable effectiveness of the large charge expansion Summary of the results: O(2) Quantum analysis 14 12 10 o u r p r e d 8 i c t D(Q) i o n 6 4 2 MC data fit 0 2 4 6 8 10 Q
Introduction Effective action from classical scale invariance Domenico Orlando [Hasenbusch and Vicari] The unreasonable effectiveness of the large charge expansion Quantum analysis Summary of the results: O(N) 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 Q 1 2 3 4
Introduction Effective action from classical scale invariance Quantum analysis Scales We want to write a Wilsonian effective action . Choose a cutoff Λ , separate the fjelds into high and low frequency Domenico Orlando The unreasonable effectiveness of the large charge expansion 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 Choose a cutoff Λ , separate the fjelds into high and low frequency Domenico Orlando Effective action from classical scale invariance The unreasonable effectiveness of the large charge expansion We want to write a Wilsonian effective action . Scales Quantum analysis d r a h o φ H , φ L and do the path integral over the high-frequency part: o t ∫ e iS Λ ( φ L ) = D φ H e iS ( φ H , φ L )
Introduction Effective action from classical scale invariance Quantum analysis Scales 1 R Domenico Orlando The unreasonable effectiveness of the large charge expansion ▶ 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 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 The unreasonable effectiveness of the large charge expansion
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 The unreasonable effectiveness of the large charge expansion 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 The unreasonable effectiveness of the large charge expansion 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 The unreasonable effectiveness of the large charge expansion
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 The unreasonable effectiveness of the large charge expansion
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 The unreasonable effectiveness of the large charge expansion ▶ 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 The unreasonable effectiveness of the large charge expansion ▶ a cute qualitative picture; ▶ might allow you to get the anomalies right;
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; At best: infjnite terms. The Wilsonian action is fundamentally useless because it contains Wilsonian action Quantum analysis The unreasonable effectiveness of the large charge expansion ▶ a cute qualitative picture; ▶ might allow you to get the anomalies right; ▶ something that helps you organize perturbative calculations, if
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