Introductiontothelarge chargeexpansion Domenico Orlando - - PowerPoint PPT Presentation

Introductiontothelarge chargeexpansion

Domenico Orlando

INFN | Torino

15 September 2020 | Crete Center for Theoretical 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, arXiv:2003.08396, arXiv:2005.03021, arXiv:2008.03308 and more to

come…

Domenico Orlando Introduction to the large charge expansion

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

Why are we here? Conformal fjeld theories

extrema of the RG fmow critical phenomena

• 1.0
• 0.5

quantum gravity string theory

Domenico Orlando Introduction to the large charge expansion

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 theory where anomalous dimension and OPE coeffjcients simplify.

Domenico Orlando Introduction to the large charge expansion

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. (However, arXiv:1710.11161).

Domenico Orlando Introduction to the large charge expansion

8 Introduction

Concrete results

We consider the O(N) vector model in three dimensions. In the IR it fmows to a conformal fjxed point Wilson & Fisher. We fjnd an explicit formula for the dimension of the lowest primary at fjxed charge:

ΔQ = c3/2

2√

πQ3/2 + 2√ πc1/2Q1/2 − 0.094 + O

• Q−1/2

Domenico Orlando Introduction to the large charge expansion

9 Introduction

Summary of the results: O(2)

2 4 6 8 10 12 14 2 4 6 8 10 D(Q) Q MC data fit

• u

r p r e d i c t i

• n

Domenico Orlando Introduction to the large charge expansion

10 Introduction

Scales

We want to write a Wilsonian effective action. Choose a cutoff Λ, separate the fjelds into high and low frequency

φH, φL and do the path integral over the high-frequency part:

eiSΛ(φL)=

• DφH eiS(φH,φL)

Domenico Orlando Introduction to the large charge expansion

10 Introduction

Scales

We want to write a Wilsonian effective action. Choose a cutoff Λ, separate the fjelds into high and low frequency

φH, φL and do the path integral over the high-frequency part:

eiSΛ(φL)=

• DφH eiS(φH,φL)

t

• h

a r d

Domenico Orlando Introduction to the large charge expansion

11 Introduction

Scales

• We look at a fjnite box of typical length R
• The U(1) charge Q fjxes a second scale ρ1/2 ∼ Q1/2/R

1 R ≪ Λ ≪ ρ1/2 ∼ Q1/2 R ≪ ΛUV For Λ ≪ ρ1/2 the effective action is weakly coupled and under perturbative control in powers of ρ−1.

Domenico Orlando Introduction to the large charge expansion

12 Introduction

Wilsonian action

The Wilsonian action is fundamentally useless because it contains infjnite terms. At best: a cute qualitative picture; might allow you to get the anomalies right; something that helps you organize perturbative calculations, if your system is already weakly-coupled for some reason; maybe a convergent expansion in derivatives.

s u p e r s t i t i

• n

Domenico Orlando Introduction to the large charge expansion

12 Introduction

Wilsonian action

The Wilsonian action is fundamentally useless because it contains infjnite terms. At best:

• a cute qualitative picture;
• might allow you to get the anomalies right;
• something that helps you organize perturbative calculations, if

your system is already weakly-coupled for some reason;

• maybe a convergent expansion in derivatives.

s u p e r s t i t i

• n

Domenico Orlando Introduction to the large charge expansion

12 Introduction

Wilsonian action

The Wilsonian action is fundamentally useless because it contains infjnite terms. At best:

• a cute qualitative picture;
• might allow you to get the anomalies right;
• something that helps you organize perturbative calculations, if

your system is already weakly-coupled for some reason;

• maybe a convergent expansion in derivatives.

s u p e r s t i t i

• n

Domenico Orlando Introduction to the large charge expansion

13 Introduction

Too good to be true?

2 4 6 8 10 12 14 2 4 6 8 10 D(Q) Q MC data fit

Domenico Orlando Introduction to the large charge expansion

14 Introduction

Too good to be true?

Think of Regge trajectories. The prediction of the theory is m2 ∝ J

• 1 + O
• J−1

but experimentally everything works so well at small J that String Theory was invented.

Domenico Orlando Introduction to the large charge expansion

15 Introduction

Too good to be true?

The unreasonable effectiveness

• f the large charge expansion.

Domenico Orlando Introduction to the large charge expansion

16 Introduction

Today’s talk

The EFT for the O(2) model in 2 + 1 dimensions 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

16 Introduction

Today’s talk

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.

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

16 Introduction

Today’s talk

The EFT for the O(2) model in 2 + 1 dimensions Justify and prove all my claims from fjrst principles

• 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

Use the large-charge expansion together with supersymmetry. Discuss some phenomenological applications

Domenico Orlando Introduction to the large charge expansion

16 Introduction

Today’s talk

The EFT for the O(2) model in 2 + 1 dimensions Justify and prove all my claims from fjrst principles Use the large-charge expansion together with supersymmetry.

• qualitatively different behavior
• compute three-point functions
• resum the large-charge expansion
• see explicitly the next saddle in the partition function

Discuss some phenomenological applications

Domenico Orlando Introduction to the large charge expansion

16 Introduction

Today’s talk

The EFT for the O(2) model in 2 + 1 dimensions 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

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

The O(2) model

The simplest example is the Wilson–Fisher (WF) point of the O(2) model in three dimensions.

• Non-trivial fjxed point of the φ4 action

LUV = ∂μφ∗ ∂μφ − u(φ∗φ)2

• Strongly coupled
• In nature: 4He.
• Simplest example of spontaneous symmetry breaking.
• Not accessible in perturbation theory. Not accessible in 4 − ε.

Not accessible in large N.

• Lattice. Bootstrap.

Domenico Orlando Introduction to the large charge expansion

20 An EFT for a CFT

Charge fjxing

We assume that the O(2) symmetry is not accidental. We consider a subsector of fjxed charge Q. Generically, the classical solution at fjxed charge breaks spontaneously U(1) → ∅. We have one Goldstone boson χ.

Domenico Orlando Introduction to the large charge expansion

21 An EFT for a CFT

An action for χ

Start with two derivatives: L[χ] = fπ 2 ∂μχ ∂μχ − C3 (χ is a Goldstone so it is dimensionless.) We want to describe a CFT: we can dress with a dilaton L σ χ fπe

2fσ

2

μχ μχ

e

6fσC3

e

2fσ

2

μσ μσ

ξR

f2 The fmuctuations of χ give the Goldstone for the broken U 1 , the fmuctuations of σ give the (massive) Goldstone for the broken conformal invariance.

Domenico Orlando Introduction to the large charge expansion

21 An EFT for a CFT

An action for χ

Start with two derivatives: L[χ] = fπ 2 ∂μχ ∂μχ − C3 (χ is a Goldstone so it is dimensionless.) We want to describe a CFT: we can dress with a dilaton L[σ, χ] = fπe−2fσ 2 ∂μχ ∂μχ − e−6fσC3 + e−2fσ 2

• ∂μσ ∂μσ − ξR

f2

• The fmuctuations of χ give the Goldstone for the broken U(1), the

fmuctuations of σ give the (massive) Goldstone for the broken conformal invariance.

Domenico Orlando Introduction to the large charge expansion

22 An EFT for a CFT

Linear sigma model

We can put together the two fjelds as

Σ = σ + ifπχ

and rewrite the action in terms of a complex scalar

ϕ =

1 √ 2f e−fΣ We get L[ϕ] = ∂μϕ∗ ∂μϕ − ξRϕ∗ϕ − u(ϕ∗ϕ)3 Only depends on dimensionless quantities b = f 2fπ and u = 3(Cf 2)3. Scale invariance is manifest. The fjeld ϕ is some complicated function of the original φ.

Domenico Orlando Introduction to the large charge expansion

23 An EFT for a CFT

Centrifugal barrier

The O(2) symmetry acts as a shift on χ. Fixing the charge is the same as adding a centrifugal term ∝

1 |ϕ|2 .

|φ

2

V

• r

i g i n a l |

ϕ

|6 centrifugal barrier n e w v a c u u m

Domenico Orlando Introduction to the large charge expansion

24 An EFT for a CFT

Ground state

We can fjnd a fjxed-charge solution of the type

χ(t, x) = μt σ(t, x) = 1

f log(v) = const., where

μ ∝ Q1/2 + . . .

v ∝ 1 Q1/2 The classical energy is E = c3/2VQ3/2 + c1/2RVQ1/2 + O

• Q−1/2

Domenico Orlando Introduction to the large charge expansion

25 An EFT for a CFT

Fluctuations

The fmuctuations over this ground state are described by two modes.

• A universal “conformal Goldstone”. It comes from the breaking
• f the U(1).

ω =

1 √ 2 p

• The massive dilaton. It controls the magnitude of the quantum
• fmuctuations. All quantum effects are controled by 1/Q.

ω = 2μ + p2

2μ (This is a heavy fmuctuation around the semiclassical state. It has nothing to do with a light dilaton in the full theory)

Domenico Orlando Introduction to the large charge expansion

26 An EFT for a CFT

Non-linear sigma model

Since σ is heavy we can integrate it out and write a non-linear sigma model (NLSM) for χ alone. L[χ] = k3/2(∂μχ ∂μχ)3/2 + k1/2R(∂μχ ∂μχ)1/2 + . . . These are the leading terms in the expansion around the classical solution χ = μt. All other terms are suppressed by powers of 1/Q.

Domenico Orlando Introduction to the large charge expansion

27 An EFT for a CFT

State-operator correspondence

The anomalous dimension on Rd is the energy in the cylinder frame.

Δ

Sd−1 Rd H R × Sd−1 Sd−1 Protected by conformal invariance: a well-defjned quantity.

Domenico Orlando Introduction to the large charge expansion

28 An EFT for a CFT

Conformal dimensions

We know the energy of the ground state. The leading quantum effect is the Casimir energy of the conformal Goldstone. EG = 1 2 √ 2

ζ(− 1

2|S2) = −0.0937 . . .

This is the unique contribution of order Q0. Final result: the conformal dimension of the lowest operator of charge Q in the O(2) model has the form

ΔQ = c3/2

2√

πQ3/2 + 2√ πc1/2Q1/2 − 0.094 + O

• Q−1/2

Domenico Orlando Introduction to the large charge expansion

29 An EFT for a CFT

The O(2N) model

Next step: O(2N). We take 2N fjelds and an action that is invariant under

φa → Ma

bφb,

MTM = 1. The conserved current associated to the global O(2n) symmetry is now matrix-valued and has the form (jμ)ab = (φa ∂μφb − φb ∂μφa). we can only fjx the rank(O(2n)) coeffjcients in the directions of the mutually commuting Cartan generators HI. qI = 1 2 ⟨QHI⟩ ,

• HI, HJ

= 0, ⟨HIHJ⟩ = 2δIJ.

Domenico Orlando Introduction to the large charge expansion

30 An EFT for a CFT

The O(2N) model

The qI transform under the action of O(2n), while the spectrum of the system is invariant. The energy of a state of fjxed charge Q can only depend on the conjugacy class of Q. There exists a homogeneous ground state. There is always an O(2n) transformation M such that MQM−1 =

n

∑

I=1

ˆ qIHI =   

ˆ q − ˆ q 0 0 0 0 0

...   . where ˆ q = q1 + . . . qN

Domenico Orlando Introduction to the large charge expansion

31 An EFT for a CFT

The O(2N) model

The ground-state energy only depends on the sum of the charges ˆ q = q1 + · · · + qN and takes the same form E = c3/2(N) 2√

π

ˆ q3/2 + 2√

πc1/2(N) ˆ

q1/2 + O

• ˆ

q−1/2 The coeffjcients depend on N and cannot be computed in the EFT (but e.g. in large-N).

Domenico Orlando Introduction to the large charge expansion

32 An EFT for a CFT

Fluctuations

The symmetry breaking pattern is O(2N)

exp.

− → U(N)

spont.

− → U(N − 1) and there are dim(U(N)/U(N − 1)) = 2N − 1 degrees of freedom (DOF).

• One singlet, the universal conformal Goldstone ω =

1 √ 2p

• One vector of U(N − 1), with quadratic dispersion ω = p2

2μ + . . .

We have singled out the time. The system is non-relativistic. antiferromagnet ω ∝ p ferromagnet ω ∝ p2 (count double)

Domenico Orlando Introduction to the large charge expansion

33 An EFT for a CFT

Type II Goldstones

The inverse propagator for the type-II is D−1 = 1

2(∇2 − ∂2 0)

μ ∂0

−μ ∂0

1 2(∇2 − ∂2 0)

• and the dispersion relation

ω =

• p2 + μ2 ± μ.

Each type-II Goldstone counts for two DOF: 1 + 2 × (N − 1) = 2N − 1. Only the type-I has a Q0 contribution: it is universal.

Domenico Orlando Introduction to the large charge expansion

34 An EFT for a CFT

O(4) on the lattice

2 4 6 8 10 12 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 D(j, j) j Δj = c3/2

2√

π(2j)3/2 + 2√ πc1/2(2j)1/2 − 0.094 + O

• j−1/2

Domenico Orlando Introduction to the large charge expansion

35 An EFT for a CFT

What happened?

We started from a CFT. There is no mass gap, there are no particles, there is no Lagrangian. We picked a sector. In this sector the physics is described by a semiclassical confjguration plus massless fmuctuations. The full theory has no small parameters but we can study this sector with a simple EFT. We are in a strongly coupled regime but we can compute physical

• bservables using perturbation theory.

Domenico Orlando Introduction to the large charge expansion

36 Large N vs. Large Charge

Large N vs. Large Charge

Domenico Orlando Introduction to the large charge expansion

37 Large N vs. Large Charge

The model

φ4 model on R × Σ for N complex fjelds

Sθ[ϕi] =

N

∑

i=1

• dt dΣ
• gμν(∂μϕi)∗(∂νϕi) + rϕ∗

i ϕi + u

2(ϕ∗

i ϕi)2

It fmows to the WF in the IR limit u → ∞ when r is fjne-tuned. We compute the partition function at fjxed charge Z(Q1, . . . , QN) = Tr

• e−βH

N

∏

i=1

δ( ˆ

Qi − Qi)

• where

ˆ Qi =

• dΣ j0

i = i

• dΣ [ ˙

ϕ∗

i ϕi − ϕ∗ i ˙

ϕi].

Dimensions of operators of fjxed charge Q on R3 (state/operator):

Δ(Q) = − 1 β log ZS2(Q).

Domenico Orlando Introduction to the large charge expansion

38 Large N vs. Large Charge

Fix the charge

Explicitly Z =

π

−π N

∏

i=1

dθi 2π

N

∏

i=1

eiθiQi Tr

• e−βH

N

∏

i=1

e−iθi ˆ

Qi

• .

Since ˆ Q depends on the momenta, the integration is not trivial but well understood. ZΣ(Q) =

π

−π

dθ 2π e−iθQ

• ϕ(2πβ)=eiθϕ(0)

Dϕi e−S[ϕ] =

π

−π

dθ 2π e−iθQ

• ϕ(2πβ)=ϕ(0)

Dϕi e−Sθ[ϕ]

Domenico Orlando Introduction to the large charge expansion

38 Large N vs. Large Charge

Fix the charge

Explicitly Z =

π

−π N

∏

i=1

dθi 2π

N

∏

i=1

eiθiQi Tr

• e−βH

N

∏

i=1

e−iθi ˆ

Qi

• .

Since ˆ Q depends on the momenta, the integration is not trivial but well understood. ZΣ(Q) =

π

−π

dθ 2π e−iθQ

• ϕ(2πβ)=eiθϕ(0)

Dϕi e−S[ϕ] =

π

−π

dθ 2π e−iθQ

• ϕ(2πβ)=ϕ(0)

Dϕi e−Sθ[ϕ] boundary condition

Domenico Orlando Introduction to the large charge expansion

38 Large N vs. Large Charge

Fix the charge

Explicitly Z =

π

−π N

∏

i=1

dθi 2π

N

∏

i=1

eiθiQi Tr

• e−βH

N

∏

i=1

e−iθi ˆ

Qi

• .

Since ˆ Q depends on the momenta, the integration is not trivial but well understood. ZΣ(Q) =

π

−π

dθ 2π e−iθQ

• ϕ(2πβ)=eiθϕ(0)

Dϕi e−S[ϕ] =

π

−π

dθ 2π e−iθQ

• ϕ(2πβ)=ϕ(0)

Dϕi e−Sθ[ϕ] covariant derivative

Domenico Orlando Introduction to the large charge expansion

39 Large N vs. Large Charge

Effective actions

The covariant derivative approach: Sθ[ϕ] =

N

∑

i=1

• dt dΣ
• (Dμϕi)∗(Dμϕi) + R

8 ϕ∗

i ϕi + 2u(ϕ∗ i ϕi)2

• where
• D0ϕ = ∂0ϕ + i θ

βϕ

Diϕ = ∂iϕ Stratonovich transformation: introduce Lagrange multiplier λ and rewrite the action as SQ =

N

∑

i=1

• −iθiQi +
• dt dΣ
• Di

μϕi

∗ Di

μϕi

• + (r + λ)ϕ∗

i ϕi

• Expand around the VEV

ϕi =

1 √ 2 Ai + ui,

λ =

• m2 − r
• + ˆ

λ

Domenico Orlando Introduction to the large charge expansion

40 Large N vs. Large Charge

Saddle point equations

With some massaging, we fjnd the fjnal equations

• FΣ (Q) = mQ + Nζ(− 1

2|Σ, m),

mζ( 1

2|Σ, m) = − Q N.

The control parameter is actually Q/N.

Domenico Orlando Introduction to the large charge expansion

41 Large N vs. Large Charge

Large Q/N

If Q/N ≫ 1 we can use Weyl’s asymptotic expansion. Tr

• eΔΣt =

∞

∑

n=0

Kntn/2−1. The zeta function is written in terms of the geometry of Σ (heat kernel coeffjcients) mΣ =

• 4π

V Q 2N 1/2 + R 24

• V

4π Q 2N −1/2 + . . . FΣ 2N = 2 3

• 4π

V Q 2N 3/2 + R 12

• V

4π Q 2N 1/2 + . . .

Domenico Orlando Introduction to the large charge expansion

42 Large N vs. Large Charge

Order N

FS2(Q) = 4N 3 Q 2N 3/2 + N 3 Q 2N 1/2 − 7N 360 Q 2N −1/2 − 71N 90720 Q 2N −3/2 + O

• e−√

Q/(2N)

Domenico Orlando Introduction to the large charge expansion

42 Large N vs. Large Charge

Order N

FS2(Q) = 4N 3 Q 2N 3/2 + N 3 Q 2N 1/2 − 7N 360 Q 2N −1/2 − 71N 90720 Q 2N −3/2 + O

• e−√

Q/(2N)

leading Q3/2

Domenico Orlando Introduction to the large charge expansion

42 Large N vs. Large Charge

Order N

FS2(Q) = 4N 3 Q 2N 3/2 + N 3 Q 2N 1/2 − 7N 360 Q 2N −1/2 − 71N 90720 Q 2N −3/2 + O

• e−√

Q/(2N)

1/Q expansion

Domenico Orlando Introduction to the large charge expansion

42 Large N vs. Large Charge

Order N

FS2(Q) = 4N 3 Q 2N 3/2 + N 3 Q 2N 1/2 − 7N 360 Q 2N −1/2 − 71N 90720 Q 2N −3/2 + O

• e−√

Q/(2N)

EFT coeffjcients EFT coeffjcients

Domenico Orlando Introduction to the large charge expansion

42 Large N vs. Large Charge

Order N

FS2(Q) = 4N 3 Q 2N 3/2 + N 3 Q 2N 1/2 − 7N 360 Q 2N −1/2 − 71N 90720 Q 2N −3/2 + O

• e−√

Q/(2N)

asymptotic expansion

Domenico Orlando Introduction to the large charge expansion

43 Large N vs. Large Charge

Where is the universal Goldstone?

Domenico Orlando Introduction to the large charge expansion

44 Large N vs. Large Charge

Was it worth it?

Domenico Orlando Introduction to the large charge expansion

45 Large N vs. Large Charge

Final result

Δ(Q) =

4N 3 + O(1) Q 2N 3/2 + N 3 + O(1) Q 2N 1/2 + . . . − 0.0937 . . .

 

would you like to know more? Domenico Orlando Introduction to the large charge expansion

45 Large N vs. Large Charge

Final result

Δ(Q) =

4N 3 + O(1) Q 2N 3/2 + N 3 + O(1) Q 2N 1/2 + . . . − 0.0937 . . .

2 4 6 8 10 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 N c32 2 4 6 8 0.0 0.1 0.2 0.3 0.4 N c12

would you like to know more? Domenico Orlando Introduction to the large charge expansion

46 Large charge and supersymmetry

Large charge and supersymmetry

Domenico Orlando Introduction to the large charge expansion

47 Large charge and supersymmetry

And Now for Something Completely Different

All the models that you have seen have something in common: isolated vacuum. No moduli space. What happens when there is a fmat direction? Many known examples of (non-Lagrangian) N ≥ 2 SCFT in four dimensions. Coulomb branch with a dimension-one moduli space: all the physics is encoded in a single operator O and every chiral operator is just On. We will write an effective action for a canonically-normalized dimension-one vector multiplet Φ.

Domenico Orlando Introduction to the large charge expansion

48 Large charge and supersymmetry

Effective action

We have a single vector multiplet. The kinetic term is just Lk =

• d4θ Φ2 + c.c. = |∂φ|2 + fermions + gauge fjelds

There will also be a WZ term for the Weyl symmetry and U 1 charge. Because of 2, everything else is a D-term and does not contribute to protected quantities. LEFT LK

αLWZ

The coeffjcient α fjxes the a-anomaly of the EFT. It has to match the anomaly in the UV. Claim: at large R-charge this action is all you need for any 2 theory (with one-dimensional moduli space). from anomaly

Domenico Orlando Introduction to the large charge expansion

48 Large charge and supersymmetry

Effective action

We have a single vector multiplet. The kinetic term is just Lk =

• d4θ Φ2 + c.c. = |∂φ|2 + fermions + gauge fjelds

There will also be a WZ term for the Weyl symmetry and U(1) charge. Because of 2, everything else is a D-term and does not contribute to protected quantities. LEFT LK

αLWZ

The coeffjcient α fjxes the a-anomaly of the EFT. It has to match the anomaly in the UV. Claim: at large R-charge this action is all you need for any 2 theory (with one-dimensional moduli space). from anomaly

Domenico Orlando Introduction to the large charge expansion

48 Large charge and supersymmetry

Effective action

We have a single vector multiplet. The kinetic term is just Lk =

• d4θ Φ2 + c.c. = |∂φ|2 + fermions + gauge fjelds

There will also be a WZ term for the Weyl symmetry and U(1) charge. Because of N = 2, everything else is a D-term and does not contribute to protected quantities. LEFT = LK + αLWZ The coeffjcient α fjxes the a-anomaly of the EFT. It has to match the anomaly in the UV. Claim: at large R-charge this action is all you need for any 2 theory (with one-dimensional moduli space). from anomaly

Domenico Orlando Introduction to the large charge expansion

48 Large charge and supersymmetry

Effective action

We have a single vector multiplet. The kinetic term is just Lk =

• d4θ Φ2 + c.c. = |∂φ|2 + fermions + gauge fjelds

There will also be a WZ term for the Weyl symmetry and U(1) charge. Because of N = 2, everything else is a D-term and does not contribute to protected quantities. LEFT = LK + αLWZ The coeffjcient α fjxes the a-anomaly of the EFT. It has to match the anomaly in the UV. Claim: at large R-charge this action is all you need for any N = 2 theory (with one-dimensional moduli space). from anomaly

Domenico Orlando Introduction to the large charge expansion

49 Large charge and supersymmetry

Observables

Three-point function of the Coulomb branch operators

• Φn1(x1)Φn2(x2) ¯

Φn1+n2(x3)

• =

Cn1,n2,n1+n2 |x1 − x3|2n1Δ|x2 − x3|2n2Δ The OPE of Φ with itself is regular, so we can set x2 x1 and the three-point function is actually a two-point function. Cn n

n n

x1 x2

2nΔ

Φn x1 Φn x2

Q nΔ is the controlling parameter (it’s the R-charge) The coeffjcients satisfy a Toda lattice equation that can be solved using as boundary condition the one loop EFT computation.

Domenico Orlando Introduction to the large charge expansion

49 Large charge and supersymmetry

Observables

Three-point function of the Coulomb branch operators

• Φn1(x1)Φn2(x2) ¯

Φn1+n2(x3)

• =

Cn1,n2,n1+n2 |x1 − x3|2n1Δ|x2 − x3|2n2Δ The OPE of Φ with itself is regular, so we can set x2 = x1 and the three-point function is actually a two-point function. Cn′,n−n′,n = |x1 − x2|2nΔ

Φn(x1) ¯ Φn(x2)

• Q = nΔ is the controlling parameter (it’s the R-charge)

The coeffjcients satisfy a Toda lattice equation that can be solved using as boundary condition the one loop EFT computation.

Domenico Orlando Introduction to the large charge expansion

49 Large charge and supersymmetry

Observables

Three-point function of the Coulomb branch operators

• Φn1(x1)Φn2(x2) ¯

Φn1+n2(x3)

• =

Cn1,n2,n1+n2 |x1 − x3|2n1Δ|x2 − x3|2n2Δ The OPE of Φ with itself is regular, so we can set x2 = x1 and the three-point function is actually a two-point function. Cn′,n−n′,n = |x1 − x2|2nΔ

Φn(x1) ¯ Φn(x2)

• Q = nΔ is the controlling parameter (it’s the R-charge)

The coeffjcients satisfy a Toda lattice equation that can be solved using as boundary condition the one loop EFT computation.

Domenico Orlando Introduction to the large charge expansion

50 Large charge and supersymmetry

Final result

The fjnal result for the generator O of the Coulomb branch is: On(x1) ¯ On(x2) = Cn(τ, ¯

τ)Γ(2nΔ + α + 1)

|x1 − x2|2nΔ The coeffjcient Cn is scheme-dependent. The gamma term is universal, only depends on α. This result is valid for any rank-one theory, Lagrangian or not. We have completely resummed the 1/Q expansion.

Domenico Orlando Introduction to the large charge expansion

51 Large charge and supersymmetry

Comparison with localization

How well does this work? For the special case of SU(2) SQCD with Nf = 4 we can compare with

• localization. arXiv:1602.05971

1 5 10 15 20 25 30 40 50 5 10 15 20 25 30 35 n

• 6
• 4
• 2

2 4

Domenico Orlando Introduction to the large charge expansion

52 Large charge and supersymmetry

Comparison with boostrap

For strongly coupled theories one can use bootstrap to place bounds

• n the three-point coeffjcients with n = 1.

This is the worst possible situation for us. And still…

1.0 1.5 2.0 2.5 3.0 ∆ϕ 1 2 3 4 λ2

ϕ2

H0 H1 H2 SU(N) MFT

Taken from arXiv:2006.01847

would you like to know more? Domenico Orlando Introduction to the large charge expansion

53 Asymptotically safe QFT

An asymptotically safe QFT

Domenico Orlando Introduction to the large charge expansion

54 Asymptotically safe QFT

IR vs. UV

We have discussed an IR fjxed point. The fjxed charge induces a scale ΛQ = Q1/d

r .

We need a hierarchy for the scale Λ of the EFT 1 r ≪ Λ ≪ ΛQ ≪ ΛUV The situation improves if we consider a ultraviolet (UV) fjxed point. 1 r ≪ ΛUV ≪ Λ ≪ ΛQ and we can take the charge as large as we like.

Domenico Orlando Introduction to the large charge expansion

55 Asymptotically safe QFT

An asymptotically safe theory

L = − 1 2 Tr(FμνFμν) + Tr ¯ Qi/ DQ + y Tr

• ¯

QLHQR + ¯ QRH†QL

• + Tr
• ∂μH† ∂μH
• − u Tr
• H†H

2 − v(Tr H†H)2 − R 6 Tr

• H†H
• .

In the Veneziano limit of NF → ∞, NC → ∞ with the ratio NF/NC fjxed, this theory is asymptotically safe. Perturbatively-controlled UV fjxed point

α∗

g = 26

57ε,

α∗

y = 4

19ε,

α∗

h =

√ 23 − 1 19

ε, α∗

v = −0.13ε.

Domenico Orlando Introduction to the large charge expansion

56 Asymptotically safe QFT

An asymptotically safe theory

New features from our point of view

• H is a matrix. There is a large non-Abelian global symmetry
• there are fermions
• there are gluons
• it’s a four-dimensional system
• we have a trustable effective action

Domenico Orlando Introduction to the large charge expansion

57 Asymptotically safe QFT

The scalar sector

Inspired by the O(2) model we use a homogeneous ansatz H0 = e2iMtB, and the equations of motion (EOM) reduce to 2M2 = uB2 + v Tr

• B2

− R 12. For simplicity QL = −QR = J 1 − 1

• ,

where 1 is the NF/2 × NF/2 identity matrix. The ground state is M = μ 1 − 1

• ,

B = b 1 1

• .

Domenico Orlando Introduction to the large charge expansion

58 Asymptotically safe QFT

Ground state energy and fmuctuations

The ground state has energy E = 3 2 N2

F

αh + αv

2π V 1/3 J 4/3 + R 36 V 2π2 2/3 J 2/3 − 1 144 R 6 2 V 2π2 4/3 J 0 + O

• J −2/3

which is a natural expansion in J = 2Jαh + αv NF ≫ 1 We have again an expansion in powers of the charge. The leading exponent is 4/3 because we are in four dimensions.

Domenico Orlando Introduction to the large charge expansion

59 Asymptotically safe QFT

Goldstones

The symmetry-breaking pattern is quite involved SU(NF) × SU(NF) × U(1)

exp.

− → C(M) × SU(NF)

spont.

− → C(M). where C(M) = SU(NF/2) × SU(NF/2) × U(1)2. Type-I and type-II Goldstones.

• One conformal Goldstone ω =

p √ 3, which is a singlet of C(M)

• One bifundamental with ω = p2

2μ

• One fjeld in the (Adj, 1) and one in the (1, Adj) with

ω =

• αh

3αh+2αv p

Total count: 1 + 2 × (NF/2)2 + 2 × (N2

F/4 − 1) = N2 F − 1 = dim(SU(NF))

Domenico Orlando Introduction to the large charge expansion

60 Asymptotically safe QFT

Summing it up

• We can use the large-charge expansion for asymptotically safe

theories

• Being in the UV, the large-charge condition is more natural
• For the QCD-inspired model that we have considered:
• Fermions and gluons decouple.
• 1/J expansion of the anomalous dimensions, starting at J 4/3
• Rich spectrum of Goldstone modes, with linear and quadratic

dispersions.

would you like to know more? Domenico Orlando Introduction to the large charge expansion

61 Conclusions

In conclusion

• With the large-charge approach we can study strongly-coupled

systems perturbatively.

• Select a sector and we write a controllable effective theory.
• The strongly-coupled physics is (for the most part) subsumed in a

semiclassical state.

• Compute the CFT data.
• Very good agreement with lattice (supersymmetry, large N).
• Precise and testable predictions.

Domenico Orlando Introduction to the large charge expansion

62 Large N vs. Large Charge

Large N vs. Large Charge

Domenico Orlando Introduction to the large charge expansion

63 Large N vs. Large Charge

The model

φ4 model on R × Σ for N complex fjelds

Sθ[ϕi] =

N

∑

i=1

• dt dΣ
• gμν(∂μϕi)∗(∂νϕi) + rϕ∗

i ϕi + u

2(ϕ∗

i ϕi)2

It fmows to the WF in the IR limit u → ∞ when r is fjne-tuned. We compute the partition function at fjxed charge Z(Q1, . . . , QN) = Tr

• e−βH

N

∏

i=1

δ( ˆ

Qi − Qi)

• where

ˆ Qi =

• dΣ j0

i = i

• dΣ [ ˙

ϕ∗

i ϕi − ϕ∗ i ˙

ϕi].

Dimensions of operators of fjxed charge Q on R3 (state/operator):

Δ(Q) = − 1 β log ZS2(Q).

Domenico Orlando Introduction to the large charge expansion

64 Large N vs. Large Charge

Fix the charge

Explicitly Z =

π

−π N

∏

i=1

dθi 2π

N

∏

i=1

eiθiQi Tr

• e−βH

N

∏

i=1

e−iθi ˆ

Qi

• .

Since ˆ Q depends on the momenta, the integration is not trivial but well understood. ZΣ(Q) =

π

−π

dθ 2π e−iθQ

• ϕ(2πβ)=eiθϕ(0)

Dϕi e−S[ϕ] =

π

−π

dθ 2π e−iθQ

• ϕ(2πβ)=ϕ(0)

Dϕi e−Sθ[ϕ]

Domenico Orlando Introduction to the large charge expansion

64 Large N vs. Large Charge

Fix the charge

Explicitly Z =

π

−π N

∏

i=1

dθi 2π

N

∏

i=1

eiθiQi Tr

• e−βH

N

∏

i=1

e−iθi ˆ

Qi

• .

Since ˆ Q depends on the momenta, the integration is not trivial but well understood. ZΣ(Q) =

π

−π

dθ 2π e−iθQ

• ϕ(2πβ)=eiθϕ(0)

Dϕi e−S[ϕ] =

π

−π

dθ 2π e−iθQ

• ϕ(2πβ)=ϕ(0)

Dϕi e−Sθ[ϕ] boundary condition

Domenico Orlando Introduction to the large charge expansion

64 Large N vs. Large Charge

Fix the charge

Explicitly Z =

π

−π N

∏

i=1

dθi 2π

N

∏

i=1

eiθiQi Tr

• e−βH

N

∏

i=1

e−iθi ˆ

Qi

• .

Since ˆ Q depends on the momenta, the integration is not trivial but well understood. ZΣ(Q) =

π

−π

dθ 2π e−iθQ

• ϕ(2πβ)=eiθϕ(0)

Dϕi e−S[ϕ] =

π

−π

dθ 2π e−iθQ

• ϕ(2πβ)=ϕ(0)

Dϕi e−Sθ[ϕ] covariant derivative

Domenico Orlando Introduction to the large charge expansion

65 Large N vs. Large Charge

Effective actions

The covariant derivative approach: Sθ[ϕ] =

N

∑

i=1

• dt dΣ
• (Dμϕi)∗(Dμϕi) + R

8 ϕ∗

i ϕi + 2u(ϕ∗ i ϕi)2

• where
• D0ϕ = ∂0ϕ + i θ

βϕ

Diϕ = ∂iϕ Stratonovich transformation: introduce Lagrange multiplier λ and rewrite the action as SQ =

N

∑

i=1

• −iθiQi +
• dt dΣ
• Di

μϕi

∗ Di

μϕi

• + (r + λ)ϕ∗

i ϕi

• Expand around the VEV

ϕi =

1 √ 2 Ai + ui,

λ =

• m2 − r
• + ˆ

λ

Domenico Orlando Introduction to the large charge expansion

66 Large N vs. Large Charge

Effective action for ˆ

λ

We can now integrate out the ui and get an effective action for ˆ

λ

alone Sθ[ ˆ

λ] =

N

∑

i=1

• Vβ
• θ2

i

β2 + m2

• A2

i

2 + Tr

• log
• −Di

μDi μ + m2 + ˆ

λ

• − A2

i

2 Tr ˆ

λΔ ˆ λ

• .

This is a non-local action for ˆ

λ, that can be expanded order-by-order

in 1/N. Today we will only look at the leading order (saddle point).

Domenico Orlando Introduction to the large charge expansion

67 Large N vs. Large Charge

Saddle point equations

                     ∂SQ ∂m2 =

N

∑

i=1

Vβ 2 A2

i + ζ(1|θi, Σ, m)

• = 0,

∂SQ ∂θi = −iQ + θi

β VA2

i + 1

s ∂ ∂θi

ζ(s|θi, Σ, m)

• s=0

= 0 ∂SQ ∂Ai = Vβ

• θ2

i

β2 + m2

• Ai = 0.

where

ζ(s|θ, Σ, m) = ∑

n∈Z∑ p

2πn

β

+ θ

β

2 + E(p)2 + m2 −s .

Domenico Orlando Introduction to the large charge expansion

68 Large N vs. Large Charge

Saddle point equations

With some massaging, we fjnd the fjnal equations

• FΣ (Q) = mQ + Nζ(− 1

2|Σ, m),

mζ( 1

2|Σ, m) = − Q N.

The control parameter is actually Q/N.

Domenico Orlando Introduction to the large charge expansion

69 Large N vs. Large Charge

Small Q/N

The zeta function can be expanded in perturbatively in small Q/N. Result:

Δ(Q)

Q = 1 2 + 4

π2

Q N + 16

• π2 − 12
• Q2

3π4N2 + . . .

• Expansion of a closed expression
• Start with the engineering dimension 1/2
• Reproduce an infjnite number of diagrams from a fjxed-charge
• ne-loop calculation

Domenico Orlando Introduction to the large charge expansion

70 Large N vs. Large Charge

Large Q/N

If Q/N ≫ 1 we can use Weyl’s asymptotic expansion. Tr

• eΔΣt =

∞

∑

n=0

Kntn/2−1. The zeta function is written in terms of the geometry of Σ (heat kernel coeffjcients) mΣ =

• 4π

V Q 2N 1/2 + R 24

• V

4π Q 2N −1/2 + . . . FΣ 2N = 2 3

• 4π

V Q 2N 3/2 + R 12

• V

4π Q 2N 1/2 + . . .

Domenico Orlando Introduction to the large charge expansion

71 Large N vs. Large Charge

Order N

FS2(Q) = 4N 3 Q 2N 3/2 + N 3 Q 2N 1/2 − 7N 360 Q 2N −1/2 − 71N 90720 Q 2N −3/2 + O

• e−√

Q/(2N)

Domenico Orlando Introduction to the large charge expansion

71 Large N vs. Large Charge

Order N

FS2(Q) = 4N 3 Q 2N 3/2 + N 3 Q 2N 1/2 − 7N 360 Q 2N −1/2 − 71N 90720 Q 2N −3/2 + O

• e−√

Q/(2N)

leading Q3/2

Domenico Orlando Introduction to the large charge expansion

71 Large N vs. Large Charge

Order N

FS2(Q) = 4N 3 Q 2N 3/2 + N 3 Q 2N 1/2 − 7N 360 Q 2N −1/2 − 71N 90720 Q 2N −3/2 + O

• e−√

Q/(2N)

1/Q expansion

Domenico Orlando Introduction to the large charge expansion

71 Large N vs. Large Charge

Order N

FS2(Q) = 4N 3 Q 2N 3/2 + N 3 Q 2N 1/2 − 7N 360 Q 2N −1/2 − 71N 90720 Q 2N −3/2 + O

• e−√

Q/(2N)

EFT coeffjcients EFT coeffjcients

Domenico Orlando Introduction to the large charge expansion

71 Large N vs. Large Charge

Order N

FS2(Q) = 4N 3 Q 2N 3/2 + N 3 Q 2N 1/2 − 7N 360 Q 2N −1/2 − 71N 90720 Q 2N −3/2 + O

• e−√

Q/(2N)

asymptotic expansion

Domenico Orlando Introduction to the large charge expansion

72 Large N vs. Large Charge

Universal term: integrate all but one

Domenico Orlando Introduction to the large charge expansion

73 Large N vs. Large Charge

Order N0

The order N0 terms are Sθ[ ˆ

σ, ˆ λ] =

• dt dΣ
• (Dμ ˆ

σ)∗(Dμ ˆ σ) + (m2 + ˆ λ) ˆ σ∗ ˆ σ

+ ˆ

λv( ˆ σ + ˆ σ∗)

(N − 1)1/2

• + 1

2

• dx1 dx2 ˆ

λ(x1) ˆ λ(x2)D(x1 − x2)2

where D(x − y) is the propagator (DμDμ + m2)−1. At low energies we can approximate the non-local term as

• dt dΣ ˆ

λ(x)2ζ(2|θ, Σ, m) ≈ V

2m

• dt dΣ ˆ

λ(x)2

and we can integrate ˆ

λ out.

Domenico Orlando Introduction to the large charge expansion

74 Large N vs. Large Charge

Order N0

The inverse propagator for σ is 1/2(ω2 + p2 + 4m2) mω −mω 1/2(ω2 + p2)

• It describes a massive mode and a massless mode with dispersion

ω2 + 1

2p2 + . . . = 0

ω2 + 8m2 + 3

2p2 + . . . = 0 This is the conformal Goldstone that we have seen in the EFT. Its contribution to the partition function is EG = 1 2 1 √ 2

ζ(1/2|S2) = −0.0937 . . .

This is universal. Does not depend on N or Q.

Domenico Orlando Introduction to the large charge expansion

74 Large N vs. Large Charge

Order N0

The inverse propagator for σ is 1/2(ω2 + p2 + 4m2) mω −mω 1/2(ω2 + p2)

• It describes a massive mode and a massless mode with dispersion

ω2 + 1

2p2 + . . . = 0

ω2 + 8m2 + 3

2p2 + . . . = 0 This is the conformal Goldstone that we have seen in the EFT. Its contribution to the partition function is EG = 1 2 1 √ 2

ζ(1/2|S2) = −0.0937 . . .

This is universal. Does not depend on N or Q. determinant

Domenico Orlando Introduction to the large charge expansion

74 Large N vs. Large Charge

Order N0

The inverse propagator for σ is 1/2(ω2 + p2 + 4m2) mω −mω 1/2(ω2 + p2)

• It describes a massive mode and a massless mode with dispersion

ω2 + 1

2p2 + . . . = 0

ω2 + 8m2 + 3

2p2 + . . . = 0 This is the conformal Goldstone that we have seen in the EFT. Its contribution to the partition function is EG = 1 2 1 √ 2

ζ(1/2|S2) = −0.0937 . . .

This is universal. Does not depend on N or Q. speed of sound

Domenico Orlando Introduction to the large charge expansion

75 Large N vs. Large Charge

Was it worth it?

Domenico Orlando Introduction to the large charge expansion

76 Large N vs. Large Charge

Final result

Δ(Q) =

4N 3 + O(1) Q 2N 3/2 + N 3 + O(1) Q 2N 1/2 + . . . − 0.0937 . . .

 

Domenico Orlando Introduction to the large charge expansion

76 Large N vs. Large Charge

Final result

Δ(Q) =

4N 3 + O(1) Q 2N 3/2 + N 3 + O(1) Q 2N 1/2 + . . . − 0.0937 . . .

2 4 6 8 10 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 N c32 2 4 6 8 0.0 0.1 0.2 0.3 0.4 N c12

Domenico Orlando Introduction to the large charge expansion

77 Large charge and supersymmetry

Large charge and supersymmetry

Domenico Orlando Introduction to the large charge expansion

78 Large charge and supersymmetry

And Now for Something Completely Different

All the models that you have seen have something in common: isolated vacuum. No moduli space. What happens when there is a fmat direction? Many known examples of (non-Lagrangian) N ≥ 2 SCFT in four dimensions. Coulomb branch with a dimension-one moduli space: all the physics is encoded in a single operator O and every chiral operator is just On. We will write an effective action for a canonically-normalized dimension-one vector multiplet Φ.

Domenico Orlando Introduction to the large charge expansion

79 Large charge and supersymmetry

Effective action

We have a single vector multiplet. The kinetic term is just Lk =

• d4θ Φ2 + c.c. = |∂φ|2 + fermions + gauge fjelds

There will also be a WZ term for the Weyl symmetry and U 1 charge. Because of 2, everything else is a D-term and does not contribute to protected quantities. LEFT LK

αLWZ

The coeffjcient α fjxes the a-anomaly of the EFT. It has to match the anomaly in the UV. Claim: at large R-charge this action is all you need for any 2 theory (with one-dimensional moduli space). from anomaly

Domenico Orlando Introduction to the large charge expansion

79 Large charge and supersymmetry

Effective action

We have a single vector multiplet. The kinetic term is just Lk =

• d4θ Φ2 + c.c. = |∂φ|2 + fermions + gauge fjelds

There will also be a WZ term for the Weyl symmetry and U(1) charge. Because of 2, everything else is a D-term and does not contribute to protected quantities. LEFT LK

αLWZ

The coeffjcient α fjxes the a-anomaly of the EFT. It has to match the anomaly in the UV. Claim: at large R-charge this action is all you need for any 2 theory (with one-dimensional moduli space). from anomaly

Domenico Orlando Introduction to the large charge expansion

79 Large charge and supersymmetry

Effective action

We have a single vector multiplet. The kinetic term is just Lk =

• d4θ Φ2 + c.c. = |∂φ|2 + fermions + gauge fjelds

There will also be a WZ term for the Weyl symmetry and U(1) charge. Because of N = 2, everything else is a D-term and does not contribute to protected quantities. LEFT = LK + αLWZ The coeffjcient α fjxes the a-anomaly of the EFT. It has to match the anomaly in the UV. Claim: at large R-charge this action is all you need for any 2 theory (with one-dimensional moduli space). from anomaly

Domenico Orlando Introduction to the large charge expansion

79 Large charge and supersymmetry

Effective action

We have a single vector multiplet. The kinetic term is just Lk =

• d4θ Φ2 + c.c. = |∂φ|2 + fermions + gauge fjelds

There will also be a WZ term for the Weyl symmetry and U(1) charge. Because of N = 2, everything else is a D-term and does not contribute to protected quantities. LEFT = LK + αLWZ The coeffjcient α fjxes the a-anomaly of the EFT. It has to match the anomaly in the UV. Claim: at large R-charge this action is all you need for any N = 2 theory (with one-dimensional moduli space). from anomaly

Domenico Orlando Introduction to the large charge expansion

80 Large charge and supersymmetry

Observables

Three-point function of the Coulomb branch operators

• Φn1(x1)Φn2(x2) ¯

Φn1+n2(x3)

• =

Cn1,n2,n1+n2 |x1 − x3|2n1D|x2 − x3|2n2D The OPE of Φ with itself is regular, so we can set x2 = x1 and the three-point function is actually a two-point function. Cn′,n−n′,n = |x1 − x2|2nD

Φn(x1) ¯ Φn(x2)

= eqn−q0 Q = nD is the controlling parameter (it’s the R-charge)

Domenico Orlando Introduction to the large charge expansion

81 Large charge and supersymmetry

Two-point function

• Φn(x1) ¯

Φn(x2)

=

• Dφ φn(x1) ¯

φn(x2)e−Sk

We can just pull the sources in the action and minimize Sk + Ssources ∝ k0 +

• d4x
• ∂μφ ∂μ ¯

φ

− Q log φδ(x − x1) − Q log ¯

φδ(x − x2)

• At the minimum:

S = k0 + k1Q − Q log Q + 2Q log |x1 − x2| + O

• Q0

so qn = k0 + k1Q +

• Q + 1

2

• log(Q) + O
• Q0

Domenico Orlando Introduction to the large charge expansion

81 Large charge and supersymmetry

Two-point function

• Φn(x1) ¯

Φn(x2)

=

• Dφ φn(x1) ¯

φn(x2)e−Sk

We can just pull the sources in the action and minimize Sk + Ssources ∝ k0 +

• d4x
• ∂μφ ∂μ ¯

φ

− Q log φδ(x − x1) − Q log ¯

φδ(x − x2)

• At the minimum:

S = k0 + k1Q − Q log Q + 2Q log |x1 − x2| + O

• Q0

so qn = k0 + k1Q +

• Q + 1

2

• log(Q) + O
• Q0

EFT parameters

Domenico Orlando Introduction to the large charge expansion

82 Large charge and supersymmetry

Two-point function: tree level

Adding the WZ term gives another contribution qn = k1Q + k0 + Q log(Q) +

• α + 1

2

• log(Q) + O
• Q0

This is the tree-level result. Corrections from quantum fmuctuations in the path integral. No other tree-level terms.

Domenico Orlando Introduction to the large charge expansion

83 Large charge and supersymmetry

Two-point function: quantum corrections

1/Q is the loop-counting parameter because we are expanding around a vacuum expectation value (VEV) that depends on Q. Sum of a ground state piece and a series in 1/Q. qn = k0 + k1Q + Q log(Q) +

• α + 1

2

• log(Q) +

∞

∑

m=1

km(α) Qm Compute order-by-order

• +
• +
• +
• +
• k1 α

1 2

α2 α

1 6

Domenico Orlando Introduction to the large charge expansion

83 Large charge and supersymmetry

Two-point function: quantum corrections

1/Q is the loop-counting parameter because we are expanding around a VEV that depends on Q. Sum of a ground state piece and a series in 1/Q. qn = k0 + k1Q + Q log(Q) +

• α + 1

2

• log(Q) +

∞

∑

m=1

km(α) Qm interactions from WZ Compute order-by-order

• +
• +
• +
• +
• k1 α

1 2

α2 α

1 6

Domenico Orlando Introduction to the large charge expansion

83 Large charge and supersymmetry

Two-point function: quantum corrections

1/Q is the loop-counting parameter because we are expanding around a VEV that depends on Q. Sum of a ground state piece and a series in 1/Q. qn = k0 + k1Q + Q log(Q) +

• α + 1

2

• log(Q) +

∞

∑

m=1

km(α) Qm Compute order-by-order

• +
• +
• +
• +
• k1(α) = 1

2

• α2 + α + 1

6

• Domenico Orlando

Introduction to the large charge expansion

84 Large charge and supersymmetry

Supersymmetry to the rescue

There is a better way. The qn satisfy a Toda lattice equation arXiv:0910.4963 ∂τ∂ ¯

τqn = eqn+1−qn − eqn−qn−1

This is integrable, but it’s hard to fjnd explicit solutions. Unless… …we use the form that follows from the existence of the asymptotic expansion qn k0 τ τ Qf τ τ Q Q

α

1 2

Q

m 1

km α Qm

Domenico Orlando Introduction to the large charge expansion

84 Large charge and supersymmetry

Supersymmetry to the rescue

There is a better way. The qn satisfy a Toda lattice equation arXiv:0910.4963 ∂τ∂ ¯

τqn = eqn+1−qn − eqn−qn−1

This is integrable, but it’s hard to fjnd explicit solutions. Unless… …we use the form that follows from the existence of the asymptotic expansion qn = k0(τ, ¯

τ) + Qf(τ, ¯ τ) + Q log(Q)

+

• α + 1

2

• log(Q) +

∞

∑

m=1

km(α) Qm

Domenico Orlando Introduction to the large charge expansion

84 Large charge and supersymmetry

Supersymmetry to the rescue

There is a better way. The qn satisfy a Toda lattice equation arXiv:0910.4963 ∂τ∂ ¯

τqn = eqn+1−qn − eqn−qn−1

This is integrable, but it’s hard to fjnd explicit solutions. Unless… …we use the form that follows from the existence of the asymptotic expansion qn = k0(τ, ¯

τ) + Qf(τ, ¯ τ) + Q log(Q)

+

• α + 1

2

• log(Q) +

∞

∑

m=1

km(α) Qm n

• τ

Domenico Orlando Introduction to the large charge expansion

85 Large charge and supersymmetry

Recursion relation

We can actually solve the recursion relation, using the value of k1(α) found at one loop. qn = k0(τ, ¯

τ) + Qf(τ, ¯ τ) + log(Γ(2n + α + 1))

The log term is universal, only depends on α. We have completely resummed the 1/Q expansion. In terms of the generator O of the Coulomb branch we have: On(x1) ¯ On(x2) = Cn(τ, ¯

τ)Γ(2nΔ + α + 1)

|x1 − x2|2nΔ The coeffjcient Cn is scheme-dependent.

Domenico Orlando Introduction to the large charge expansion

86 Large charge and supersymmetry

Comparison with localization

How well does this work? For the special case of SU(2) SQCD with Nf = 4 we can compare with

• localization. arXiv:1602.05971

1 5 10 15 20 25 30 40 50 5 10 15 20 25 30 35 n

• 6
• 4
• 2

2 4

Domenico Orlando Introduction to the large charge expansion

87 Large charge and supersymmetry

Comparison with localization

How well does this work? For the special case of SU(2) SQCD with Nf = 4 we can compare with

• localization. arXiv:1602.05971

1 2 5 10 20 35 10 20 30 40 50 Im τ

• 8
• 6
• 4
• 2

2 4

Domenico Orlando Introduction to the large charge expansion

88 Large charge and supersymmetry

A semi-empirical instanton

Domenico Orlando Introduction to the large charge expansion

89 Large charge and supersymmetry

A semi-empirical instanton

We can do better. We have resummed the 1/Q expansion around one vacuum. Exponential corrections coming from the next saddle in the path integral.

5 10 15 20 5.×10-5 1.×10-4 5.×10-4 0.001 0.005

Δ2

n(qloc n

− qEFT

n

) Im(τ)

Δ2

n(a e−b√ n/ Im(τ)), a = 1.8(2), b = 3.2(1) Domenico Orlando Introduction to the large charge expansion

90 Large charge and supersymmetry

Comparison with localization

Once we add the fjrst exponential correction

1 2 5 10 20 35 10 20 30 40 50 Im τ

• 8
• 6
• 4
• 2

2 4

Domenico Orlando Introduction to the large charge expansion

91 Large charge and supersymmetry

Comparison with localization

Once we add the fjrst exponential correction (fjxed τ = 6)

20 40 60 80 100 0.2 0.4 0.6 0.8 1.0 1.2

Δ2

nqn

n

Domenico Orlando Introduction to the large charge expansion

92 Large charge and supersymmetry

Comparison with bootstrap

For strongly coupled theories one can use bootstrap to place bounds

• n the three-point coeffjcients with n = 1.

This is the worst possible situation for us. And still…

Domenico Orlando Introduction to the large charge expansion

93 Large charge and supersymmetry

Comparison with bootstrap

0.0 0.5 1.0 1.5 2.0 2.5 1.90 1.95 2.00 2.05 2.10 2.15 2.20 2.25 2.30 1/c λℰ2*6/5

2

Δϕ=6/5

Domenico Orlando Introduction to the large charge expansion

93 Large charge and supersymmetry

Comparison with bootstrap

0.0 0.5 1.0 1.5 2.0 2.5 1.9 2.0 2.1 2.2 2.3 2.4 2.5 1/c λℰ2*4/3

2

Δϕ=4/3

Domenico Orlando Introduction to the large charge expansion

93 Large charge and supersymmetry

Comparison with bootstrap

0.0 0.5 1.0 1.5 2.0 2.5 1.8 2.0 2.2 2.4 2.6 2.8 3.0 1/c λℰ2*3/2

2

Δϕ=3/2

Domenico Orlando Introduction to the large charge expansion

94 Large charge and supersymmetry

Comparison with boostrap

1.0 1.5 2.0 2.5 3.0 ∆ϕ 1 2 3 4 λ2

ϕ2

H0 H1 H2 SU(N) MFT

Taken from arXiv:2006.01847

Domenico Orlando Introduction to the large charge expansion

95 A light dilaton

Going away from conformality

Domenico Orlando Introduction to the large charge expansion

96 A light dilaton

Going away away from conformality

CFTs are very interesting but very constrained. There is a lot of interesting physics that happens away from conformality. If we don’t go “too far” we can still use large charge effectively. We will fjnd a very distinct signature of new physics associated to a small dilaton mass in the EFT.

Domenico Orlando Introduction to the large charge expansion

97 A light dilaton

Walking dynamics

For example the walking phase when β functions get close to zero remaining very fmat.

λ β(λ)

Domenico Orlando Introduction to the large charge expansion

98 A light dilaton

The EFT

We mimick it adding a small mass for the dilaton. Consider a system with U(1) global symmetry in four dimensions. L[σ, χ] = f 2

πe−2fσ

2 ∂μχ ∂μχ − e−4fσC4 + e−2fσ 2

• ∂μσ ∂μσ − ξR

f2

• − m2

σ

16f 2

• e−4fσ + 4fσ − 1
• mσ is the mass of σ (around σ = 0) that is due to the underlying

(walking) dynamics. It measures the breaking of scale invariance Tμ

μ = m2 σ

f

σ.

Domenico Orlando Introduction to the large charge expansion

99 A light dilaton

What is the dilaton mass?

In the conformal model at fjxed charge the fmuctuations of the dilaton around the classical solution are heavy. Very little to do with mσ, which is a measure of how much the full theory is non-conformal. In the large charge approach it will appear in the semiclassical ground state energy. The semiclassical state resums the quantum effects.

Domenico Orlando Introduction to the large charge expansion

100 A light dilaton

The ground state energy

We just need to solve at fjxed values of the charge. The energy in the cylinder frame has a new, characteristic term r0Ecyl = c4/3 (4π2)1/3 Q4/3 + c2/3Q2/3 −

π2m2

σr4

3f 2 log(Q) + . . . This is the fjrst time that a log(Q) term appears in this game.

Domenico Orlando Introduction to the large charge expansion

101 A light dilaton

The two-point function

Close to the fjxed point, we can still use the state-operator correspondence. The two-point function on R4 for operators of fjxed charge is ⟨OQ(0)O−Q(x)⟩ = 1 |x|2Δ where Δ has a log(Q) correction with respect to the dimension at the fjxed point Δ∗

Δ = Δ∗

• 1 −

m2

σ

24c4/3f 2μ4 log(Q)

• This is a clear signature of a light dilaton in the walking dynamics.

Domenico Orlando Introduction to the large charge expansion

102 A light dilaton

Fluctuations

We can also study the fmuctuations on top of the semiclassical fjxed-charge state. We fjnd again two modes.

• A massless mode, which is not anymore exactly conformal

ω =

1 √ 3

• 1 +

m2

σ

9c4/3f 2μ4

• p
• A massive mode which has essentially the same mass as in the

CFT case

ω = 2μ + p2

2μ This is the mass of the fmuctuation of σ around the VEV.

Domenico Orlando Introduction to the large charge expansion

103 A light dilaton

Summing it up

• The large-charge approach can be used for walking theories.
• We predict a precise signature of a light dilaton in the two-point

functions.

• We have shown the mechanism for the simplest theory.
• The construction can be easily generalized to more realistic

situations (around the conformal window).

Domenico Orlando Introduction to the large charge expansion