Nonperturbative Mellin Amplitudes: Existence, Properties, - - PowerPoint PPT Presentation

Nonperturbative Mellin Amplitudes: Existence, Properties, Applications

• A. Zhiboedov (CERN)

SISSA, Trieste

work with J. Penedones and J. Silva

Introduction

1 lD−2

P l

Z dDx√−g (R − 2Λ + ...)

AdS/CFT

=

+...

AdS/CFT

1 lD−2

P l

Z dDx√−g (R − 2Λ + ...)

AdS/CFT

=

+...

AdS/CFT

nonperturbative perturbative

1 lD−2

P l

Z dDx√−g (R − 2Λ + ...)

AdS/CFT

=

+...

AdS/CFT

nonperturbative perturbative

coarse-grained holography: assume a UV completion and derive rules of consistent AdS EFTs (Hofman-Maldacena type bounds)

(easy)

1 lD−2

P l

Z dDx√−g (R − 2Λ + ...)

AdS/CFT

=

+...

AdS/CFT

nonperturbative perturbative

coarse-grained holography: assume a UV completion and derive rules of consistent AdS EFTs (Hofman-Maldacena type bounds)

(easy)

fine-grained holography: given a consistent gravitational EFT that satisfies all the known bounds construct a UV completion.

(hard)

Any EFT! Only String Theory!

1 lD−2

P l

Z dDx√−g (R − 2Λ + ...)

AdS/CFT

=

+...

AdS/CFT

nonperturbative perturbative

coarse-grained holography: assume a UV completion and derive rules of consistent AdS EFTs (Hofman-Maldacena type bounds)

(easy)

fine-grained holography: given a consistent gravitational EFT that satisfies all the known bounds construct a UV completion.

(hard)

Bootstrap

Is it possible to use bootstrap to shed some light upon the matter?

Tentative CFT data

NO MAYBE

[Rattazzi, Rychkov, Tonni, Vichi ’08]

Maybe No

Bootstrap

Is it possible to use bootstrap to shed some light upon the matter?

Tentative CFT data

NO MAYBE

[Rattazzi, Rychkov, Tonni, Vichi ’08]

Maybe No

no surprising bounds for large N theories yet

Bootstrap

“one is never sure to have completely exploited all the axioms of field theory.” Old wisdom (S-matrix bootstrap): Is it possible to use bootstrap to shed some light upon the matter?

Tentative CFT data

NO MAYBE

[Rattazzi, Rychkov, Tonni, Vichi ’08]

Maybe No

no surprising bounds for large N theories yet

Mellin Representation

Mellin Space Coordinate space Bootstrap

[Mack]

Perturbative Nonperturbative

What goes into the Mellin representation?

?

[Penedones][Paulos] [Fitzpatrick, Kaplan, Penedones, Raju, van Rees] [Aharony, Alday, Bissi, Perlmutter] [Gopakumar, Kaviraj, Sen, Sinha] [Rastelli, Zhou][Caron-Huot] … [Rattazzi, Rychkov, Tonni, Vichi] … [Dodelson, Ooguri] [Yuan] [Poland, Rychkov, Vichi] [Simmons-Duffin] [Hogervorst, van Rees] [Mazac, Paulos]

(Holographic)

Mellin Representation

Mack’s Mellin Ansatz: CFT correlation functions admit the following representation

[Mack ’09]

hO1(x1) . . . On(xn)i = Z [dγij] M(γij)

n

Y

i<j

Γ(γij) |xi xj|2γij

n

X

j=1

γij = 0 , γij = γji , γii = −∆i .

Mellin Representation

Mack’s Mellin Ansatz: CFT correlation functions admit the following representation

[Mack ’09]

hO1(x1) . . . On(xn)i = Z [dγij] M(γij)

n

Y

i<j

Γ(γij) |xi xj|2γij

n

X

j=1

γij = 0 , γij = γji , γii = −∆i .

[Penedones ’10]

Outline Mellin transform (in CFTs) Existence Properties Applications

Review of Mellin transform

Mellin Transform

Mellin transformation is defined as follows

[Mellin ’1896]

Φ(x) : Rn

+ → C

M[Φ](z) = Z

Rn

+

dx xz−1Φ(x) M −1[F](x) = 1 (2πi)n Z

a+iRn dz x−zF(z)

What are natural classes of functions associated with it?

M-space

Space of functions holomorphic in a sectorial domain , which are polynomially bounded with powers in an open domain U.

Θ

M U

Θ :

Θ ∈ Rn, U ∈ Rn, 0 ∈ Θ Φ(x) ∈ M U

Θ

x = (r1eiθ1, ..., rneiθn)

θ ∈ Θ r ∈ Rn

+

|Φ(x)| < C(a)|x−a|, x = (r, θ) ∈ SΘ, a ∈ U

θ

U

W-space

Similarly, we define a space of functions holomorphic in a tube and decaying exponentially fast in the imaginary direction

W Θ

U :

F(z), z ∈ U + iRn |F(u + iv)| ≤ K(u)e−supθ∈Θθ·v u ∈ U U

θ

Mellin Inversion Theorems

[Antipova ’07]

Theorem I: Given , the Mellin transform exists and . We also have .

Φ(x) ∈ MU

Θ

M[Φ] ∈ WΘ

U

M−1M[Φ] = Φ

Theorem II: Given , the inverse Mellin transform exist and . We also have .

F(z) ∈ WΘ

U

M−1[F] ∈ MU

Θ

MM−1[F] = F

θ

U Φ < poly F < exp

Examples

e−u = Z cu+i∞

cu−i∞

dγ12 2πi Γ(γ12)u−γ12, cu > 0, |arg[u]| < π 2 .

Γ(γ12) ∼ e− π

2 |Im[γ12]|

The Cahen-Mellin integral:

Examples

e−u = Z cu+i∞

cu−i∞

dγ12 2πi Γ(γ12)u−γ12, cu > 0, |arg[u]| < π 2 .

Γ(γ12) ∼ e− π

2 |Im[γ12]|

The Cahen-Mellin integral:

Γ(2γ12)Γ(α − 2γ12) Γ(α) ∼ e−2π|Im[γ12]|

1 (1 + √u)α = Z cu+i∞

cu−i∞

dγ12 2πi Γ(2γ12)Γ(α − 2γ12) Γ(α) u−γ12, 0 < cu < α 2 , |arg[u]| < 2π

Mellin transform in CFTs

Are analyticity in a sectorial domain and polynomial boundedness properties of the physical correlators?

Four-point Function

Let us consider a four-point function of identical scalar primary operators

hO(x1)O(x2)O(x3)O(x4)i = F(u, v) (x2

13x2 24)∆ ,

u = x2

12x2 34

x2

13x2 24

, v = x2

14x2 23

x2

13x2 24

F(u, v) = F(v, u) = v−∆F ✓u v , 1 v ◆ ˆ M(γ12, γ14) ≡ Z ∞ dudv uv uγ12vγ14F(u, v)

(crossing)

We can try to define the Mellin transform as follows

Region of Integration

(principal Euclidean sheet)

Mellin transform = Integral over the principal sheet

π

π 2

−π φ t

(1, 0) (0, 1) u v

Analyticity Domain

Claim: Any CFT correlator is analytic in a sectorial domain .

F(u, v) ΘCFT = (arg[u], arg[v])

arg(v) arg(u) 2π −2π 2π −2π

(Reduced) Mellin Amplitude

ˆ M(γ12, γ14) = [Γ(γ12)Γ(γ13)Γ(γ14)]2 M(γ12, γ14)

It is common in the field to define the following reduced Mellin amplitude

[Γ(γ12)Γ(γ14)Γ(∆ − γ12 − γ14)]2 ∼ e−π(|Im[γ12]|+|Im[γ14]|+|Im[γ12+γ14]|) = e−supΘCF T (arg[u]Im[γ12]+arg[v]Im[γ14])

The pre-factor correctly captures analytic properties of the correlator!

Polynomial Boundedness

We consider a subtracted correlator

Fsub(u, v) = F(u, v) − (1 + u−∆ + v−∆) − X

τgap≤τ<τsub Jmax

X

J=0 h τsub−τ

2

i

X

m=0

C2

OOOτ,J

⇣ u−∆+ τ

2 +mg(m)

τ,J (v) + v−∆+ τ

2 +mg(m)

τ,J (u) + v− τ

2 −mg(m)

τ,J (u

v ) ⌘

crossing symmetric same analyticity properties improved behavior in the OPE limits

Dangerous Regions

π

π 2

−π φ t

(1, 0) (0, 1) u v

Double light-cone limit

u, v → 0

[Alday, Eden, Korchemsky, Maldacena, Sokatchev ’10] [Alday, AZ ’15]

Polynomial Boundedness

Claim:

Saturated in minimal models and perturbative field theories

|Fsub(u, v)| ≤ C(cu, cv) 1 |u|cu 1 |v|cv , (u, v) ∈ ΘCF T , (cu, cv) ∈ U

UCFT

Mellin Amplitude

Fsub(u, v) = Z dγ12 2πi dγ14 2πi ˆ M(γ12, γ14)u−γ12v−γ14

UCFT

UCFT :

3d Ising As an example consider the spin operator in the 3d Ising model

σ ⟨σσσσ⟩

F Ising

3d

(u, v) =

• 1 + u−∆ + v−∆

+ Z

C

dγ12 2πi dγ14 2πi ˆ M(γ12, γ14)u−γ12v−γ14,

C : ∆σ − 1 2 < Re(γ12), Re(γ14), Re(γ12) + Re(γ14) < 1 2 . This representation converges in .

ΘCFT

Given that CFT correlators in a general CFT admit the Mellin representation:

What are the properties of Mellin amplitudes?

Analyticity of Polynomial boundedness of Subtractions = Deformed contour

F(u, v) F(u, v)

Properties of the CFT Mellin Amplitudes

Mellin vs OPE

How is the Mellin representation consistent with the OPE?

=

O(x1) O(x2) O(x3) O(x4) O(x1) O(x2) O(x3) O(x4)

Oi Oi

X

i

X

i

Reproduce the OPE (unitarity) Cancel the disconnected piece (Polyakov conditions)

F(u, v) =

• 1 + u−∆ + v−∆

+ Z

C

dγ12 2πi dγ14 2πi [Γ(γ12)Γ(γ14)Γ(∆ − γ12 − γ14)]2 M(γ12, γ14)u−γ12v−γ14

OPE

Let us review in more detail the unitarity properties of Mellin amplitudes

t = 2∆ − 2γ12, s = 2(γ12 + γ14 − ∆) = −2γ13

M(s, t) ' C2

∆∆τQτ,d J,m(s)

t (τ + 2m) + ... , m = 0, 1, 2, ... , Qτ,d

J,m(s) = K(∆, J, m) Qτ,d J,m(s)

The residues are controlled by the so-called Mack polynomials

OPE

Let us review in more detail the unitarity properties of Mellin amplitudes

t = 2∆ − 2γ12, s = 2(γ12 + γ14 − ∆) = −2γ13

M(s, t) ' C2

∆∆τQτ,d J,m(s)

t (τ + 2m) + ... , m = 0, 1, 2, ... , Qτ,d

J,m(s) = K(∆, J, m) Qτ,d J,m(s)

The residues are controlled by the so-called Mack polynomials

Maximal Mellin Analyticity Conjecture: Mellin amplitudes have only poles located at twists

• f the physical operators (except the identity).

OPE

Mack polynomials have a few interesting properties

Qτ,d

J,m(s) = (−1)JQτ,d J,m(−s − τ − 2m)

QJ,0(s) = 2J τ

2

2

J

(τ + J − 1)J

3F2(−J, J + τ − 1, −s

2; τ 2, τ 2; 1) lim

s→∞ Q∆,τ,d J,m (s) = sJ + O(sJ−1)

Qτ,d

J,m(s) = m

J 2 (m + τ) J 2 C

( d−2

2

) J τ 2 + m + s

p m(m + τ) ! .

Higher m’s can be computed recursively.

(high energy) (flat space limit)

lim

J,s→∞ Q∆,τ,d J,m (s) = 21+J−mJ2J+s+τ+m−1e−2J

π Γ( τ+2m+s

2

)2 sm + ...

(AdS effects)

Twist Spectrum of CFTs

Unitarity The twist spectrum in a CFT is very complex and contains an infinite number of accumulation points

lim

J→∞ τ (n) 1,2 (J) = τ1 + τ2 + 2n

lim

J→∞ τ (m) τ (n)

1,2 (J0),τ3(J) = τ (n)

1,2 (J0) + τ3 + 2m

τ ≥ d − 2

[Komargodski, AZ 12’] [Fitzpatrick, Kaplan, Poland, Simmons-Duffin 12’] [Callan, Gross] [Parisi]

τ1 τ2 τ1 + τ2

Bound on Regge

The reduced Mellin amplitude M is polynomially bounded. The precise power follows from the bound on the Regge limit

[Maldacena, Shenker, Stanford] [Caron-Huot] x+ x− x1 x2 x4 x3

Conformal Regge Theory

[Costa, Goncalves, Penedones ’12]

In Mellin space the Regge limit becomes

lim

s→∞ M(s, t) '

Z ∞

−∞

dν β(ν)ων,j(ν)(t)sj(ν) + (s)j(ν) sin (πj(ν)) + ...

jfull(0) ≤ 1 jplanar(0) ≤ 2 lim

|s|→∞ |Mplanar(s, t)| < c|s|2+✏

lim

|s|→∞ |Mfull(s, t)| ≤ c|s|

Re[t] < d − 2

Extrapolated away from the imaginary axis.

Polyakov Conditions

In the full nonperturbative correlator we have to make sure that we do not have double counting as we close the contour

F(u, v) =

• 1 + u−∆ + v−∆

+ Z

C

dγ12 2πi dγ14 2πi [Γ(γ12)Γ(γ14)Γ(∆ − γ12 − γ14)]2 M(γ12, γ14)u−γ12v−γ14

These are the Polyakov conditions. Absent in the planar theory.

[cf. Gopakumar, Kaviraj, Sen, Sinha]

The subtlety is related to the accumulation points in the twist space.

Polyakov Conditions

As we close the contour we get the divergent sum

− 1 2 (CGFF

τ=2Δ,J)2Qτ,d J,0(γ14)Γ2(γ14)Γ2

( τ 2 − γ14) Γ2 (Δ − τ 2 )

= 4 Γ2(Δ) 1 J (J2(Δ−γ14)Γ2(γ14) + J2γ14Γ2(Δ − γ14) + . . . ) , J → ∞ , τ → 2Δ ,

thus avoiding the double counting problem.

Polyakov Conditions

As we close the contour we get the divergent sum

− 1 2 (CGFF

τ=2Δ,J)2Qτ,d J,0(γ14)Γ2(γ14)Γ2

( τ 2 − γ14) Γ2 (Δ − τ 2 )

= 4 Γ2(Δ) 1 J (J2(Δ−γ14)Γ2(γ14) + J2γ14Γ2(Δ − γ14) + . . . ) , J → ∞ , τ → 2Δ ,

thus avoiding the double counting problem. A simple way to fix it is to consider

−u∂uF(u, v) = Δu−Δ + ∫C dγ12 2πi dγ14 2πi γ12 ̂ M(γ12, γ14)u−γ12v−γ14 .

solves both convergence and double counting problems.

Polyakov Conditions

Convergence of the sum over spins leads to the following condition

τgap 2 > Re[γ14] > Δ − τgap 2

So that we get the following condition on the nonperturbative Mellin amplitude

lim

|γ12|→0, arg[γ12]≠0 M(γ12, γ14) = 0,

τgap 2 > Re[γ14] > Δ − τgap 2

In principle there are more conditions to explore. Mellin amplitude reggeizes close to the accumulation points.

Summary

CFT connected correlators admit a deformed contour Mellin representations

Fconn(u, v) ≡ F(u, v) − (1 + u−∆ + v−∆) = Z Z

C

dγ12 2πi dγ14 2πi [Γ(γ12)Γ(γ14)Γ(∆ − γ12 − γ14)]2 M(γ12, γ14)u−γ12v−γ14

Reduced Mellin amplitude M satisfies: Maximal Mellin Analyticity (Unitarity) Crossing Symmetry Polynomial Boundedness Polyakov conditions

Applications

Minimal Models

All these general properties can be explicitly checked for minimal models. This is the only known example of Mellin amplitudes of fully non-perturbative correlators.

[Furlan, Petkova ’89] [Lowe ’16] [Alday, AZ ’15] [Yuan ’18] [Dotsenko, Fateev]

F 2d Ising

σσσσ

(u, v) = p√u + √v + 1 √ 2 8 √uv

ˆ M(γ12, γ14) = − r 2 π Γ ✓ 2γ12 − 1 4 ◆ Γ ✓ 2γ14 − 1 4 ◆ Γ(−2γ12 − 2γ14)

Minimal Models

[Furlan, Petkova ’89] [Lowe ’16] [Alday, AZ ’15] [Yuan ’18] [Dotsenko, Fateev]

ˆ M(γ12, γ14) = C0 Z [dξ12] Z [dξ15] Z [dξ24] Z [dξ34] Z [dξ35]Γ(ξ12)Γ(ξ15) Γ(−2αα+ + γ12 − ξ12)Γ(ξ24)Γ(ξ15 − ξ24 − ξ34 + 1)Γ

• −2α2

+ + 2αα+ + γ12 − ξ34 + 2

• Γ(ξ34)Γ(−2αα+ − ξ12 − ξ15 + ξ34 − 1)Γ(−2αα+ + γ14 − ξ15 + ξ24 + ξ34 − 1)

Γ

• −2α2

+ + 2αα+ + γ14 + ξ12 + ξ24 − ξ35 + 1

• Γ
• 2α2

+ − ξ15 + ξ24 − ξ35 − 1

• Γ
• ξ12 − ξ34 − ξ35 + 2αα+ − 2α2

+ + 1

• Γ(ξ35)Γ(−ξ12 − ξ24 + ξ35 + 1)

Γ

• −γ12 − γ14 + ξ15 − ξ24 + ξ35 + 4α2

+ − 1

• Γ
• 2α2

+ + ξ15

• Γ(−4αα+ − ξ12 − ξ15 + ξ34 − 1)

Γ(−γ12 − γ14 − ξ24) Γ

• 2α2

+ + ξ35

• Γ
• −4α2

+ + 4αα+ + ξ12 − ξ34 − ξ35 + 3

.

hΦ1,3OΦ1,3Oi

Lightray Operators

We used it compute energy correlators in N=4 SYM

hO†(p)E(~ n1)E(~ n2)O(p)i

[Belitsky, Henn, Hohenegger, Korchemsky, Sokatchev, Yan, AZ ’13-’19]

E ∼ ANEC = Z dy−T−−

It is trivial to compute Wightman functions using Mellin amplitudes

hO1(x1) . . . On(xn)i = Z [dij] M(ij)

n

Y

i<j

Γ(ij) (x2

ij + i✏x0 ij)γij

[Kologlu, Kravchuk, Simmons-Duffin, AZ]

Lightray Operators

We get the following expressions Compare this with the OPE expression

hE(~ n1)E(~ n2)i = (p0)2 8⇡2 "X

i

p∆i 4⇡4Γ(∆i 2) Γ( ∆i1

2

)3Γ( 3∆i

2

)f 4,4

∆i (⇣) + 1

4(2(⇣) 0(⇣)) #

ζ = 1 − cos χ 2

f ∆1,∆2

∆

(ζ) = ζ

∆−∆1−∆2+1 2

2F1

✓∆ − 1 + ∆1 − ∆2 2 , ∆ − 1 − ∆1 + ∆2 2 , ∆ + 1 − d 2, ζ ◆

• Sum is over Regge trajectories

[Kologlu, Kravchuk, Simmons-Duffin, AZ] [Belitsky, Hohenegger, Korchemsky, Sokatchev, AZ]

AdS/EFT

Consider an effective field theory in AdS

S = Z dd+1x√−g(−∂µφ∂µφ + µ1φ4 + µ2(∂µφ∂µφ)2 + ...)

[Adams, Arkani-Hamed, Dubovsky, Nicolis, Rattazzi ’06]

Using dispersion relations we get

µ2 = 2 π Z dssσ(s) s3 > 0.

[Hartman, Jain, Kundu ’16]

Can we use similar arguments in Mellin space?

Positivity of Mack Polynomials

Mack polynomials share some positivity properties with Legendre polynomials

Qτ,d

J,m(s) ≥ 0,

s ≥ 0. ∂n

s Qτ,d J,m(s)|s≥0 ≥ 0 .

We also observed evidence for more complicated EFThedron-like identities for Legendre polynomials.

[Arkani-Hamed, Huang, to appear]

Positive geometry for conformal bootstrap?

[Arkani-Hamed, Huang, Shao ’18] [Sen, Sinha, Zahed '19]

Dispersion Relations

Therefore, we can run an identical dispersion relation argument

µAdS

2

= 1 2πi I dt0 (t0)3 M(0, t0) ≥ 0

Similarly, higher order couplings receive the contribution suppressed by an extra power of the mass

• f the particle that we integrated out

μAdS

n

(∂μϕ∂μϕ)n

µAdS

n

= 1 2πi I dt0 (t0)n M(0, t0)

[Caron-Huot '17] [Alday, Bissi ’16]

Bootstrap in Mellin Space

We can also do bootstrap in Mellin space. Consider dispersion relations

M 3d(γ12, γ13, γ14) (γ12 − ∆σ

3 )(γ13 − ∆σ 3 )(γ14 − ∆σ 3 ) = −M 3d( ∆σ 3 , γ13, 2∆σ 3

− γ13) (γ13 − ∆σ

3 )2

1 γ12 − ∆σ

3

+ 1 γ14 − ∆σ

3

!

+1 2 X

τ,J,m

C2

τ,JQτ,d J,m(γ13)

(∆σ − τ

2 − m − ∆σ 3 )(γ13 − ∆σ 3 )(∆σ − γ13 + (m + τ 2 − ∆σ) − ∆σ 3 )

× ✓ 1 γ12 − ∆σ + τ

2 + m +

1 γ14 − ∆σ + τ

2 + m

◆

Bootstrap in Mellin Space

We now write Polyakov conditions

M(0, γ13, ∆σ − γ13) = 0

The result takes the following form

X

τ,J,m

C2

τ,Jατ,J,m = 0

ατ,J,m = 1 (τ − 2∆σ

3

+ 2m)(τ − 4∆σ

3

+ 2m) (τ + 2m − ∆σ)Qτ,d

J,m( ∆σ 3 )

(τ − 2∆σ

3

+ 2m)(τ − 4∆σ

3

+ 2m) − ∆σ 3 Qτ,d

J,m( ∆σ 3 )0

τ + 2m − 2∆σ ! .

positive by unitarity

Bootstrap in Mellin Space

The alpha functionals have remarkable properties

1 2 3 4 5 6 7 0.02 0.04 0.06 0.08 0.10

twist

X

τ,J,m

C2

τ,Jατ,J,m = 0

scalar operators

Bootstrap in Mellin Space

The alpha functionals have remarkable properties

twist

X

τ,J,m

C2

τ,Jατ,J,m = 0

spinning operators

2 4 6 8 10 12 14 0.0 0.5 1.0 1.5

2

• 1.02

1.04 1.06 1.08 1.10

• 0.1

0.1 0.2

[Carmi, Caron-Huot ’19] [Mazac, Rastelli, Zhou ’19]

They are extremal functionals!

3d Ising

The sum rule takes the form

− X

τ<2∆σ,J>0,m

C2

τ,Jατ,J =

X

τ>2∆σ,J>0,m

C2

τ,Jατ,J +

X

τ,J=0

C2

τ,Jατ,J

Plugging the available numbers in the 3d Ising we get

Holographic Functionals

The whole formalism is particularly suited for holographic theories due to extra suppression of double trace operators. Let us consider a free scalar in AdS with which we can try to weakly couple to gravity.

Δ < 3 4(d − 2)

Holographic Functionals

The whole formalism is particularly suited for holographic theories due to extra suppression of double trace operators. Let us consider a free scalar in AdS with which we can try to weakly couple to gravity.

Δ < 3 4(d − 2)

Is there a theory of QG that looks like GR+minimal scalar below the Planck scale?

Holographic Functionals

The whole formalism is particularly suited for holographic theories due to extra suppression of double trace operators. Let us consider a free scalar in AdS with which we can try to weakly couple to gravity.

Δ < 3 4(d − 2)

Is there a theory of QG that looks like GR+minimal scalar below the Planck scale? Surprisingly, not always!

Holographic Functionals

1.05 1.10 1.15 1.20 1.25

• 5

5

beyond HPPS!

Further comments

Mellin amplitudes are well-defined nonperturbatively

Subtractions (non-Mellin pieces)

“Meromorphic” and polynomially bounded Positivity of Mack polynomials Dispersion relations in Mellin space (AdS EFT) Matrix elements of light-ray operators Twist accumulation points New bootstrap functionals suited for holography Polyakov conditions

Further comments

Mellin amplitudes are well-defined nonperturbatively

Subtractions (non-Mellin pieces)

“Meromorphic” and polynomially bounded Positivity of Mack polynomials Dispersion relations in Mellin space (AdS EFT) Matrix elements of light-ray operators Twist accumulation points New bootstrap functionals suited for holography

Thank you!

Polyakov conditions

Flat Space Limit

[Penedones ’10]

Flat space limit naturally emerges in the fixed angle regime

lim

s,t→∞; s

t −fixed M(s, t) → A(s, t) [Paulos, Penedones, Toledo, van Rees, Vieira ’16]

Cuts and poles emerge from the condensation of the twist accumulation points.

Why the S-matrix bootstrap is hard?

[from Y.Dokshitzer]

[Maldacena, Simmons-Duffin, AZ] [Gary, Giddings, Penedones]

CFT OPE

Operators can be expanded in the basis of primaries

Oi(x)Oj(0) = X

k

λijk|x|∆i+∆j−∆k (Ok(0) + xµ∂µOk(0) + ...)

CFT OPE

Operators can be expanded in the basis of primaries

Oi(x)Oj(0) = X

k

λijk|x|∆i+∆j−∆k (Ok(0) + xµ∂µOk(0) + ...)

Consider now the four-point function of identical operators: hO(x1)O(x2)O(x3)O(x4)i = G(u, v) (x2

12x2 34)∆

u = x2

12x2 34

x2

13x2 24

, v = x2

14x2 23

x2

13x2 24

CFT Bootstrap

2-, 3-point functions are fixed in terms of the OPE data

=

O(x1) O(x2) O(x3) O(x4) O(x1) O(x2) O(x3) O(x4)

Oi Oi

X

i

X

i

CFT data: (∆, J) λijk