taming the -expansion with the analytic bootstrap › s3fs-pu… · 4 starting point of the...

Taming the ε−expansion with the Analytic Bootstrap

Luis Fernando Alday

University of Oxford

OIST Symposium - March 2018

Based on work with Henriksson and van Loon

Luis Fernando Alday Taming the ε−expansion with the Analytic Bootstrap

What will this talk be about?

We will study conformal field theories in D > 2

Very relevant for Physics.

Interesting interplay with Mathematics.

Ubiquitous in dualities in string and gauge theory.

Unfortunately, studying CFT in D > 2 is not so easy...

Symmetries are less powerful than in D = 2...

In general they do not have a Lagrangian description...

In a Lagrangian theory we can use Feynman diagrams:

A(g) = A(0) + gA(1) + · · ·

But generic CFTs don’t have a small coupling constant!

In spite of all this, progress can be made!


Conformal bootstrap

Conformal bootstrap: resort to consistency conditions!

Conformal symmetryProperties of the OPEUnitarityCrossing symmetry

Successfully applied to 2d in the eighties... 25 years later it wasfinally implemented in D > 2! [Rattazzi, Rychkov, Tonni, Vichi ’08]

The starting point of a numerical revolution.

Today: Analytic results for CFTs in the spirit of the conformalbootstrap!


Analytic conformal bootstrap

Which kind of analytic results will you get today?

Analytic bootstrap (or the light-cone bootstrap on esteroids)

Results for operators with spin in a generic CFT!

O` ∼ ϕ∂µ1 · · · ∂µ`ϕ

Study their scaling dimension ∆ for large values of the spin `:

∆(`) = `+ c0 +c1

`+

c2

`2+ · · · , double-twist

Obtain analytic results to all orders in 1/` resorting only toconsistency conditions.

These results will actually be valid even for finite values of the spin!

Other analytic approaches to be discussed across the week!


CFT - Ingredients

Main ingredient:

Conformal Primary local operators: O∆,`

(x), [Kµ,O(0)] = 0

@@@I

��

Dimension Lorentz spin

In addition we have descendants Pµk ...Pµ1O∆,`= ∂µk ...∂µ1O∆,`

.

Operators form an algebra (OPE)

Oi (x)Oj(0) =∑

k∈prim.

Cijk |x |∆k−∆i−∆j

Ok(0) +xµ∂µOk(0) + · · ·︸︷︷︸all fixed

CFT data: The set ∆i and Cijk characterizes the CFT.


CFT - Basics

The observable: Four-point function of identical operators.

〈O(x1)O(x2)O(x3)O(x4)〉 =G(u, v)

x2∆O12 x2∆O

34

, u =x2

12x234

x213x

224

, v =x2

14x223

x213x

224

It admits a decomposition in conformal blocks:

C∆,` C∆,`

2 3

1 4

O∆,`∑∆,`

〈OOOO〉 =

G(u, v) = 1 +∑∆,`

C 2∆,`G∆,`(u, v)

��>

6 ZZZ}

Identity operator Conformal primaries Conformal blocksLuis Fernando Alday Taming the ε−expansion with the Analytic Bootstrap

Conformal bootstrap

It is crossing symmetric: v∆OG(u, v) =︸︷︷︸x1↔x3

u∆OG(v , u)

Combining the two...

C∆,` C∆,`

2 3

1 4

O∆,`∑∆,`

=∑∆,`

C∆,`

C∆,`

O∆,`

2 3

1 4

A remarkable...but hard equation!

v∆O

1 +∑∆,`

C 2∆,`G∆,`(u, v)

︸︷︷︸

Easy to expand around u = 0, v = 1

= u∆O

1 +∑∆,`

C 2∆,`G∆,`(v , u)

︸︷︷︸

Easy to expand around u = 1, v = 0


Numerical vs Analytic bootstrap

Study this equation in different regions, u = zz̄ , v = (1− z)(1− z̄)

z̄

z1

1

In the Euclidean regime z̄ = z∗.

We can study crossing aroundu = v = 1

4

Starting point of the numericalbootstrap.

�

��

In the Lorentzian regime z , z̄ are independent real variables and wecan consider u, v → 0.

Starting point of the analytic (light-cone) bootstrap!


Analytic bootstrap

Analytic bootstrap

Why is this a good idea?

v∆O

1 +∑∆,`

C 2∆,`G∆,`(u, v)

= u∆O

1 +∑∆,`

C 2∆,`G∆,`(v , u)

Direct channel ⇔ Crossed channel

Very complicated interplay between l.h.s. and r.h.s. ... but:

Operators with large spin

Operators with large spin ⇔ Identity operator


Conformal blocks - technicalities

Eigenfunctions of a Casimir operator

CG∆,`(u, v) = J2G∆,`(u, v)

where J2 = (`+ ∆)(`+ ∆− 1) ∼ `2

Small u limit:

G∆,`(u, v) ∼ uτ/2f collτ,` (v), τ = ∆− `

We will introduce the notation

G∆,`(u, v) ≡ uτ/2fτ,`(u, v)

Small v limit:fτ,`(u, v) ∼ log v


Necessity of a large spin sector

Consider the v � 1 limit of the crossing equation: C 2∆,` → aτ,`

v∆O

1 +∑τ,`

aτ,`uτ/2fτ,`(u, v)

= u∆O

1 +∑τ,`

aτ,`vτ/2fτ,`(v , u)

⇓

1 +∑τ,`

aτ,`uτ/2fτ,`(u, v) =

u∆O

v∆O+ subleading terms︸︷︷︸

rest of operators sorted by twist

The r.h.s. is divergent as v → 0.Each term on the l.h.s. diverges as fτ,`(u, v) ∼ log v .In order to reproduce the divergence on the right, we need infiniteoperators, with large spin and whose twist approaches τ = 2∆O(actually τn = 2∆O + 2n)


Necessity of a large spin sector

Consider the v � 1 limit of the crossing equation:

v∆O

1 +∑τ,`

aτ,`uτ/2fτ,`(u, v)

= u∆O

1 +∑τ,`

aτ,`vτ/2fτ,`(v , u)

⇓

1 +∑τ,`


u∆O

v∆O+ subleading terms︸︷︷︸

rest of operators sorted by twist

The r.h.s. is divergent as v → 0.

Each term on the l.h.s. diverges as fτ,`(u, v) ∼ log v .

In order to reproduce the divergence on the right, we need infiniteoperators, with large spin and whose twist approaches τ = 2∆O(actually τn = 2∆O + 2n)


Example: Generalised free fields

Simplest solution: Generalised free fields

G(0)(u, v) = 1 +(uv

)∆O+ u∆O

Intermediate ops: Double twist operators: O�n∂µ1 · · · ∂µ`O

τn,` = 2∆O + 2n

an,` = a(0)n,`

Their OPE coefficients are such that the divergence of a singleconformal block (∼ log v) , as v → 0, is enhanced!

1 +∑τ,`

a(0)τn,`

uτn/2fτn,`(u, v) = 1 +(uv

)∆O︸︷︷︸+u∆O

6

But this divergence is quite universal!


Large spin sector

Additivity property [Fitzpatrick, Kaplan, Poland, Simmons-Duffin; Komargodski, Zhiboedov]

In any CFT with O in the spectrum, crossing symmetry implies theexistence of double twist operators with arbitrarily large spin and

τn,` = 2∆O + 2n +O(

1

`

)an,` = a

(0)n,`

(1 +O

(1

`

))

All CFTs have a large spin sector, for which the operators become”free”!

Can we do perturbations around large spin? YES! We will developa large spin perturbation theory.


Large spin perturbation theory

We would like to exploit the following idea

∑τ,`


u∆O

v∆O+ · · ·

Behaviour at large spin ⇔ Enhanced divergences as v → 0

The presence of the identity on the r.h.s. already led to aremarkable result.Let’s take this to the next level! Solve the following problem:

Find aτ,` such that∑τ,`

aτ,`uτ/2fτ,`(u, v) = Given enhanced singularity︸︷︷︸

6

Power law divergent after applying C finite timesLuis Fernando Alday Taming the ε−expansion with the Analytic Bootstrap


GFF has accumulation points in the twist at

τn = 2∆O + 2n

It is convenient to define ”twist” conformal blocks, thecontribution to the 4pt function from a given twist.

∑`

a(0)τ,`u

τ/2fτ,`(u, v) ≡ Hτ (u, v)

And twist conformal blocks with insertions:

∑`

a(0)τ,`

J2muτ/2fτ,`(u, v) ≡ H(m)

τ (u, v)



Properties

. Correlator decomposition

G(0)(u, v) =∑n

H(0)τn (u, v)

. Recurrence relation

CH(m+1)τ (u, v) = H(m)

τ (u, v)

. Prescribed behaviour at small u and small v .

H(m)τ (u, v) ∼ uτ/2, small u

H(m)τ (u, v) ∼ v−∆O+m, small v

The functions H(m)τ (u, v) can be systematically constructed!



Back to our problem!∑τ,`

aτ,`uτ/2fτ,`(u, v) = Sing(u, v)

Assume aτ,` admits the following expansion

aτ,` = a(0)τ,`

(bτ,0 +

bτ,1J2

+bτ,2J4

+ · · ·)

⇓∑τ,m

bτ,mH(m)τ (u, v) = Sing(u, v)

Expanding both sides for small u, v we can find all coefficients bτ,m!

The problem has become algebraic!


Relation to inversion formula [Caron-Huot]

The ’enhaced-singularity’ is equivalent to the double-discontinuityaround v = 0:

dDisc[log2 v

]∼ 1, dDisc

[1

vp

]∼ 1

vpsin2 pπ, dDisc [fτ,`(u, v)] = 0

Consider the inversion problem for a given twist∑`

a`fcollτ,` (v) = G (v)

⇓

a` =

∫ 1

0dv K (`, v) dDisc [G (v)]

LSPT implies an equation for K (`, v) and fixes it.

The two methods are equivalent, but the latter is explicitly analyticin the spin!


Wider perspective on CFT

Large spin perturbation theory allows to reconstruct the CFT-datafrom the enhanced singularities, but... the structure of singularitiescan be extremely complicated!

If two operators Oτ1 ,Oτ2 of twists τ1 and τ2 are part of the spectrumthen there is a tower of operators [Oτ1 ,Oτ2 ]n,` of twist

τ[Oτ1 ,Oτ2 ]n,` = τ1 + τ2 + 2n +O(

1

`

)

This should make you happy and sad at the same time!

The spectrum of generic CFTs is hard!

. O is part of the spectrum.

. [O,O]n,` is also part of the spectrum.

. And [[O,O]n1,`1 , [O,O]n2,`2 ]n3,`3too, and so on!


Large spin expansions for non-perturbative CFTs

In non-perturbative CFTs the spectrum is very rich. Hard (but notimpossible!) to apply our idea

∑τ,`

aτ,`uτ/2fτ,`(u, v) = Rich spectrum in the crossed channel

Behaviour at large spin ⇔ complicated divergences as v → 0

If the CFT has a small parameter we are better of, as thisparameter further organises the problem.


LSPT: Strategy

Strategy

1 Use crossing symmetry to determine the enhanced singularities

G(u, v)← G(u, v)|en.sing . =(uv

)∆0

G(v , u)

∣∣∣∣en.sing .

In theories with small parameters the latter follows from CFT-dataat lower orders! (maybe including other correlators)

2 Then use LSPT to reconstruct the CFT-data from the enhancedsingularities.

3 Go to next order and repeat.

This can be turn into an efficient machinery!

. Theories with weakly broken higher spin symmetry.

. Weakly coupled gauge theories.

. 1/N4 corrections to GFF and N = 4 SYM.


CFT with weakly broken HS symmetry

CFT with HS symmetry

The following is part of the spectrumFundamental scalar field ϕ

∂µ∂µϕ = 0→ ∆ϕ =

d − 2

2

Tower of HS conserved currents J(s) = ϕ∂µ1 · · · ∂µsϕ,

conservation→ ∆s = d − 2 + s

Weakly broken HS symmetry

The fundamental field ϕ and HS currents J(s) acquire ananomalous dimensions:

∆ϕ =d − 2

2+ gγϕ + · · ·

∆s = d − 2 + s + gγs + · · ·

Which anomalous dimensions are consistent with crossing symmetry?Luis Fernando Alday Taming the ε−expansion with the Analytic Bootstrap

ε−expansion for WF

We will study the WF-model in d = 4− ε dimensions, where g ∼ ε.

∆ϕ =d − 2

2+ εγ(1)

ϕ + · · ·

∆s = d − 2 + s + εγ(1)s + · · ·

Carry our program for the correlator 〈ϕϕϕϕ〉

G(u, v) = 1 +(uv

)∆ϕ

+ u∆ϕ︸︷︷︸+εG(1)(u, v) + · · ·

To leading order the intermediate operators are the identity and thetower of conserved currents J(s).



Look at the contribution from the currents to the correlator

G(u, v)|currents =∑`

a`ud−2

2+ε

γ`2 f d−2

2+ε

γ`2,`(u, v)

=∑`

a(0)` u

d−22 f d−2

2,`(u, v) +

ε∑`

ud−2

2

(a

(0)`

γ`2

(log u + ∂τ ) + a(1)`

)f d−2

2,`(u, v)

+ · · ·

Use crossing symmetry to determine the enhanced singularities in aε−expansion.

Use the previous method to compute

a(0)` , a

(0)` γ`, a

(1)` , · · ·



Determining the enhanced singularities in a ε−expansion...

The operator ϕ2 is special because its the only one that gets acorrection to order ε.

G(u, v)|enh.sing . =(uv

)∆ϕ

1︸︷︷︸identity

+ aϕ2v∆ϕ2/2fϕ2(v , u)︸︷︷︸Operator ϕ2

+ · · ·

To order ε3 nothing else contributes a enhanced singularity!

The method we described allows to compute the anomalousdimensions and OPE coefficients of J(s) to this order!



An ugly fact

To order ε2 new, quartic, intermediate operators appear

ϕ2�n∂µ1 · · · ∂µ`ϕ2

This has the potential to make our lives miserable at order ε4...

A pleasant surprice

With our procedure we can compute the OPE coefficient withwhich they appear.

Only operators with n = 0 (approximate twist four) appear![Conjectured some time ago by Liendo, Rastelli and van Rees]

This will simplify our life!



To order ε4 two new sources of enhanced-singularities appear1 Weakly broken currents in the dual channel (I2)

as ∼ ε0, γs ∼ ε2 → asvγs/2 ∼ asγ

2s log2 v ∼ ε4

2 New quadrilinear operators in the dual channel (I4)

as ∼ ε2, γs ∼ ε→ a(q)s vγ

(q)s /2 ∼ asγ

2s log2 v ∼ ε4

The computation of I2 is straightforward. The computation of I4 isvastly simplified by the observation we made in the previous slide!



Structure of enhanced-singularities to order ε4

G(u, v)|enh.sing . =(uv

)∆ϕ(

1 + aϕ2v∆ϕ2/2fϕ2(v , u) + I2(v , u) + I4(v , u))

+· · ·

We can compute the CFT-data to order ε4!

CT

Cfree= 1− 5

324ε2 − 233

8748ε3 −

(100651

3779136− 55

2916ζ3

)ε4 + · · ·


Transcendentality for WF

A transcendentality principle for WF?

Gconn(u, v)|e.s. ∼ ε2 log2 v (2 log(1− v)− log u) +

ε3 log2 v (log u + log u log(1− v) + Li2v + · · · ) +

ε4 log2 v(log2 u log(1− v) + Li3v + · · ·

)+ · · ·

The double discontinuity contributing to the currents is formed bypure transcendental functions!

With this assumption one can go even higher in the ε expansion!


Conclusions

Generic CFTs have a large spin sector, which becomes essentiallyfree. We have shown how to perform a perturbation around thatsector.

This applies to vast families of CFTs and provides an alternative todiagrammatic computations.

Do we have a transcendentality principle for WF?

There are many other analytic approaches to CFT, in the spirit ofthe conformal bootstrap, and the connection to these is afascinating question.


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