Alargecharge torulestrongcoupling Domenico Orlando Introduction - PowerPoint PPT Presentation

INFN | Torino 8 Oct 2019 | Bridging perturbative and nonperturbative physics arXiv:1505.01537 , arXiv:1610.04495 , arXiv:1707.00711 , arXiv:1804.01535 , arXiv:1902.09542 , arXiv:1905.00026 , arXiv:1909.02571 , arXiv:1909.08642 and more to come… Domenico Orlando A large charge to rule strong coupling Alargecharge torulestrongcoupling 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 A large charge to rule strong coupling

3 Introduction Domenico Orlando string theory quantum gravity A large charge to rule strong coupling 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 But 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 A large charge to rule strong coupling

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 A large charge to rule strong coupling 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 A large charge to rule strong coupling

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 A large charge to rule strong coupling (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 A large charge to rule strong coupling 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 A large charge to rule strong coupling 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 A large charge to rule strong coupling 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 A large charge to rule strong coupling 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 A large charge to rule strong coupling • 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 A large charge to rule strong coupling

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 A large charge to rule strong coupling

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; At best: infjnite terms. The Wilsonian action is fundamentally useless because it contains Wilsonian action A large charge to rule strong coupling • a cute qualitative picture;

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 At best: infjnite terms. The Wilsonian action is fundamentally useless because it contains Wilsonian action A large charge to rule strong coupling • 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 12 Introduction 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 A large charge to rule strong coupling • a cute qualitative picture; • might allow you to get the anomalies right; • something that helps you organize perturbative calculations, if

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 A large charge to rule strong coupling • 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 A large charge to rule strong coupling 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 A large charge to rule strong coupling 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 A large charge to rule strong coupling ( ( J − 1 )) m 2 ∝ J 1 + O

15 Introduction Too good to be true? The unreasonable effectiveness of the large charge expansion . Domenico Orlando A large charge to rule strong coupling

16 Introduction Today’s talk An asymptotically safe theory at large charge A nearly critical theory at large charge The signature of a light dilaton Domenico Orlando A large charge to rule strong coupling The effective fjeld theory (EFT) for the O ( 2 ) model in d = 3 • An EFT for a CFT. • The physics at the saddle. • State/operator correspondence for anomalous dimensions. • A QCD-like theory • Conformal dimensions • Decoupling • An EFT for walking theories

17 Introduction Domenico Orlando A large charge to rule strong coupling

18 An EFT for a CFT An EFT for a CFT Domenico Orlando A large charge to rule strong coupling

19 An EFT for a CFT model in three dimensions. Not accessible in large N . Domenico Orlando A large charge to rule strong coupling 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, fjxing the charge breaks it. We have one Goldstone boson χ . Domenico Orlando A large charge to rule strong coupling We assume that the O ( 2 ) symmetry is not accidental. We consider a subsector of fjxed charge Q . It will look like a spontaneous breaking 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 A large charge to rule strong coupling L [ χ ] = f π 2 ∂ μ χ ∂ μ χ − C 3