INFN | Torino 17 June 2020 | SNS arXiv:1505.01537 , arXiv:1610.04495 , arXiv:1707.00711 , arXiv:1804.01535 , arXiv:1902.09542 , arXiv:1905.00026 , arXiv:1909.02571 , arXiv:1909.08642 , arXiv:2003.08396 , arXiv:2005.03021 and more to come… Domenico Orlando Introduction to the large charge expansion Introductiontothelarge chargeexpansion Domenico Orlando
Introduction Who’s who S. Reffert (AEC Bern); L. Alvarez Gaumé (CERN and SCGP); F . Sannino (CP3-Origins); D. Banerjee (DESY); S. Chandrasekharan (Duke); S. Hellerman (IPMU); M. Watanabe (Weizmann). Domenico Orlando Introduction to the large charge expansion
3 Introduction Domenico Orlando string theory quantum gravity Introduction to the large charge expansion extrema of the RG fmow critical phenomena Why are we here? Conformal fjeld theories 1.0 0.5 0.0 - 0.5 - 1.0 0.0 0.2 0.4 0.6 0.8
4 Introduction 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 . Domenico Orlando Introduction to the large charge expansion
5 Introduction Why are we here? Conformal fjeld theories are hard In presence of a symmetry there can be sectors of the coeffjcients simplify. Domenico Orlando Introduction to the large charge expansion theory where anomalous dimension and OPE
6 Introduction 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 Introduction to the large charge expansion
7 Introduction 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 Introduction to the large charge expansion (However, arXiv:1710.11161 ).
8 Introduction 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 Introduction to 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 √
9 Introduction Domenico Orlando Introduction to the large charge expansion Summary of the results: O(2) 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
10 Introduction Scales We want to write a Wilsonian effective action . Choose a cutoff Λ , separate the fjelds into high and low frequency Domenico Orlando Introduction to 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 )
10 Introduction Domenico Orlando Introduction to the large charge expansion Choose a cutoff Λ , separate the fjelds into high and low frequency We want to write a Wilsonian effective action . Scales 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 )
11 Introduction Scales 1 R Domenico Orlando Introduction to 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 .
n o i t i t s r e p u s Domenico Orlando 12 Introduction 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 Introduction to the large charge expansion
n o i t i t s r e p u s Domenico Orlando 12 Introduction your system is already weakly-coupled for some reason; At best: infjnite terms. The Wilsonian action is fundamentally useless because it contains Wilsonian action Introduction to the large charge expansion • a cute qualitative picture; • might allow you to get the anomalies right; • something that helps you organize perturbative calculations, if • maybe a convergent expansion in derivatives.
12 your system is already weakly-coupled for some reason; Domenico Orlando Introduction Introduction to the large charge expansion At best: Wilsonian action infjnite terms. The Wilsonian action is fundamentally useless because it contains n o i t i t s r e p • a cute qualitative picture; u s • might allow you to get the anomalies right; • something that helps you organize perturbative calculations, if • maybe a convergent expansion in derivatives.
13 Introduction Domenico Orlando Introduction to the large charge expansion Too good to be true? 14 12 10 8 D(Q) 6 4 2 MC data fit 0 2 4 6 8 10 Q
14 Introduction Too good to be true? Think of Regge trajectories . The prediction of the theory is but experimentally everything works so well at small J that String Theory was invented. Domenico Orlando Introduction to the large charge expansion m 2 ∝ J � � J − 1 �� 1 + O
15 Introduction Too good to be true? The unreasonable effectiveness of the large charge expansion . Domenico Orlando Introduction to the large charge expansion
16 Introduction Today’s talk Justify and prove all my claims from fjrst principles Use the large-charge expansion together with supersymmetry. Discuss some phenomenological applications asymptotically safe theories walking dynamics Domenico Orlando Introduction to the large charge expansion The EFT for the O ( 2 ) model in 2 + 1 dimensions
16 Introduction Today’s talk Justify and prove all my claims from fjrst principles Use the large-charge expansion together with supersymmetry. Discuss some phenomenological applications asymptotically safe theories walking dynamics Domenico Orlando Introduction to the large charge expansion The EFT for the O ( 2 ) model in 2 + 1 dimensions • An effective fjeld theory (EFT) for a CFT. • The physics at the saddle. • State/operator correspondence for anomalous dimensions.
16 Introduction Today’s talk Justify and prove all my claims from fjrst principles Use the large-charge expansion together with supersymmetry. Discuss some phenomenological applications asymptotically safe theories walking dynamics Domenico Orlando Introduction to the large charge expansion The EFT for the O ( 2 ) model in 2 + 1 dimensions • well-defjned asymptotic expansion (in the technical sense) • justify why the expansion works at small charge • compute the coeffjcients in the effective action in large- N
16 Introduction Today’s talk Justify and prove all my claims from fjrst principles Use the large-charge expansion together with supersymmetry. Discuss some phenomenological applications asymptotically safe theories walking dynamics Domenico Orlando Introduction to the large charge expansion The EFT for the O ( 2 ) model in 2 + 1 dimensions • qualitatively different behavior • compute three-point functions • resum the large-charge expansion • see explicitly the next saddle in the partition function
16 Introduction Today’s talk Justify and prove all my claims from fjrst principles Use the large-charge expansion together with supersymmetry. Discuss some phenomenological applications Domenico Orlando Introduction to the large charge expansion The EFT for the O ( 2 ) model in 2 + 1 dimensions • asymptotically safe theories • walking dynamics
17 Introduction Domenico Orlando Introduction to the large charge expansion
18 An EFT for a CFT An EFT for a CFT Domenico Orlando Introduction to the large charge expansion
19 An EFT for a CFT model in three dimensions. Not accessible in large N . Domenico Orlando Introduction to the large charge expansion The O ( 2 ) model The simplest example is the Wilson–Fisher (WF) point of the O ( 2 ) • Non-trivial fjxed point of the φ 4 action L UV = ∂ μ φ ∗ ∂ μ φ − u ( φ ∗ φ ) 2 • Strongly coupled • In nature: 4 He . • Simplest example of spontaneous symmetry breaking. • Not accessible in perturbation theory. Not accessible in 4 − ε . • Lattice. Bootstrap.
20 An EFT for a CFT Charge fjxing Generically, the classical solution at fjxed charge breaks We have one Goldstone boson χ . Domenico Orlando Introduction to the large charge expansion We assume that the O ( 2 ) symmetry is not accidental. We consider a subsector of fjxed charge Q . spontaneously U ( 1 ) → ∅ .
We want to describe a CFT: we can dress with a dilaton 21 6 f σ C 3 Domenico Orlando conformal invariance. fmuctuations of σ give the (massive) Goldstone for the broken The fmuctuations of χ give the Goldstone for the broken U 1 , the f 2 ξ R μ σ μ σ 2 2 f σ e e An EFT for a CFT μ χ μ χ 2 2 f σ f π e L σ χ ( χ is a Goldstone so it is dimensionless.) Start with two derivatives: An action for χ Introduction to the large charge expansion 2 ∂ μ χ ∂ μ χ − C 3 L [ χ ] = f π
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