Mellin amplitudes 1D CFTs

Turin 2021

Mellin amplitudes &

1D CFTs

Gabriel J. S. Bliard Based on work with L. Bianchi, D. Bonomi, V. Forini & G. Peveri

[2106.00689]

also [2004.07849]

G. Bliard Mellin amplitudes & 1d CFTs2

Why do we care?CFT1 theories are omnipresent

• Subsector of higher dimensional theories • Boundary of QFT in AdS2 • Defect theories

AdS2 worldsheet excitations Operator insertions on the Wilson line

Mellin amplitude in higher d

• Nice complex analytic structure • Clear links to scattering amplitudes • Redundant variables for 1d


How can we study CFT1?

Localisation [Giombi, Komatsu ’18] [Gorini, Griguolo, Guerrini, Penati, Seminara, Soresina ’20]

Mellin amplitudes [Ferrero, Ghosh, Sinha, Zahed ’18]

Analytic Bootstrap [Liendo, Meneghelli, Mitev ’18] [Mazac , Paulos ’18] [Ferrero, Ghosh, Sinha, Zahed ’18]

Integrability [Grabner, Gromov, Julius ’20] [Drukker Kawamoto ’06]

Feynman diagrams (Weak coupling ) [Cooke, Dekel, Drukker, ’17] [Kyriu, Komatsu ’18] [Barrat, Liendo, Plefka ’20]

Witten diagrams (Strong coupling) [Giombi, Roiban, Tseytlin ’17]

CFT1





Analytic Bootstrap [Liendo, Meneghelli, Mitev ’18] [Mazac , Paulos ’18] [Ferrero, Ghosh, Sinha, Zahed ’18]


Feynman diagrams (Weak coupling ) [Cooke, Dekel, Drukker, ’17] [Kyriu, Komatsu ’18] [Barrat, Liendo, Plefka ’20]

Witten diagrams (Strong coupling) [Giombi, Roiban, Tseytlin ’17]

CFT1


CFT1 4-point Kinematics4-point correlators of scalars:

⟨ϕΔϕ(x1)ϕΔ(x2)ϕΔ(x3)ϕΔ(x4)⟩ =

1(x12x34)2Δϕ

f(z) 0 < z < 1

Block expansion

0 z 1 ∞

f(z) = ( z1 − z )

2Δ

f(1 − z)Crossing equation

f(z) = ∑Δ

c2ΔΔϕΔϕ

zΔ2F1(Δ, Δ; 2Δ; z)

G. Bliard Mellin amplitudes & 1d CFTs

Minimal string surface ending on the line contour (Effective QFT in AdS2)

1D Wilson line

6

AdS2 CFT1 Dictionary



1D Wilson line

Worldsheet excitations in orthogonal directions. Operator insertions

7

AdS2 CFT1

⟨⟨O(x1) . . . O(xn)⟩⟩ = ⟨TrO(x1)W . . . O(xn)W⟩

Dictionary



1D Wilson line

Worldsheet excitations in orthogonal directions. Operator insertions

Large string tension correlators via Witten diagrams [Giombi, Roiban, Tseytlin ’17]

Strong coupling correlators via analytic bootstrap [Liendo, Meneghelli, Mitev ,’18]

8

AdS2 CFT1

⟨⟨O(x1) . . . O(xn)⟩⟩ = ⟨TrO(x1)W . . . O(xn)W⟩

Dictionary


OutlineMellin Transform

Why a 1d Mellin Definition and properties

Results

Example I: QFT in AdS2

Computations and Mellin CFT data

Example II: 1/2 BPS Wilson Line Witten diagrams

The Bootstrap Results and Mellin


Motivation

Is there a formalism that is better suited for these correlators?

We look at higher dimension

Mellin formalism

• Links to scattering amplitudes • Nice complex analytic structure

[Mack ’09] [Penedones ’10] [Fitzpatrick, Kaplan, Penedones, Raju, Van Rees ’11] [Costa, Goncalves,Penedones ’12]

Formalism can be simplified in 1d

11

Mellin formalism in 1D


Results

• Non-Perturbative definition of Mellin amplitude • Non-perturbative sum rules • Closed-form expression for contact diagrams • Perturbative results for higher derivative AdS2 QFT

[Bianchi, GB, Forini, Peveri, ’21]


Definition and Properties

⟨ϕϕϕϕ⟩ =1

(x12x34)2Δf(z)

Ma[g](s) = ∫∞

0dt ( t

1 + t )a

t−1−sf(t)

M−1a [Ma](t) = ∫𝒞

ds2πi

ts ( t1 + t )

−a

Ma(s)

Mellin

Anti-Mellin

Reminder t =z

1 − z=

x12x34

x14x23


Definition and Properties

• is the ‘natural’ variable

• Crossing is manifest:

• The integral converges for

0 < t < ∞

⟨ϕϕϕϕ⟩ =1

(x12x34)2Δf(z)

Ma[g](s) = ∫∞

0dt ( t

1 + t )a

t−1−sf(t)

M−1a [Ma](t) = ∫𝒞

ds2πi

ts ( t1 + t )

−a

Ma(s)

Mellin

Anti-Mellin

Reminder

f(t) = t2Δf ( 1t ) ⇒ M(s) = M(2Δ − s)

2Δϕ − Δ0 < Re(s) < Δ0Ma[g](s) = ∫

∞

0dtt−1−sf(t)

t =z

1 − z=

x12x34

x14x23


How do we deal with ‘Light operators’: ?Δ0 < Δϕ

Improve the behaviour of t → 0 f(t)

f0(t) = f(t) − ( t1 + t )

−2Δ Δϕ

∑Δ+k=Δ0

cΔCΔ,ktΔ+k

ψ0(t) = ∫1

0dtt−1−sf0(t) +

Δϕ

∑Δ+k=Δ0

cΔCΔ,k1

s − Δ − k

Convergence for 2Δϕ − Δ̃ < Re(s) < Δ̃

[Penedones, Silva, Zhiboedov, ’19]


How do we deal with ‘Light operators’: ?Δ0 < Δϕ

Improve the behaviour of t → 0 f(t)

f0(t) = f(t) − ( t1 + t )

−2Δ Δϕ

∑Δ+k=Δ0

cΔCΔ,ktΔ+k

ψ0(t) = ∫1

0dtt−1−sf0(t) +

Δϕ

∑Δ+k=Δ0

cΔCΔ,k1

s − Δ − k

M(s) = ψ0(s) + ψ∞(s)Convergence for 2Δϕ − Δ̃ < Re(s) < Δ̃

Same process for behaviour of to form t → ∞ f(t) ψ∞(s)

The deformed contour keeps the poles of on its right and those of on its left.

𝒞ψ0(s) ψ∞(s)

[Penedones, Silva, Zhiboedov, ’19]


OPE Expansion and CFT data

Doing this for the whole expansion of (sending ) gives a Mellin block expansionf(t) Δ̃ → ∞

M(s) = ∑Δ,k

cΔCΔ,k ( 1s − k − Δ

+1

2Δ − s − k − Δ ) = ∑Δ

cΔ(GΔ(s) + GΔ(2Δϕ − s))

Convergence for 2Δϕ − Δ̃ < Re(s) < Δ̃Reminder


OPE Expansion and CFT data

Doing this for the whole expansion of (sending ) gives a Mellin block expansionf(t) Δ̃ → ∞

M(s) = ∑Δ,k


+1

2Δ − s − k − Δ ) = ∑Δ

cΔ(GΔ(s) + GΔ(2Δϕ − s))

• Poles at the conformal weights of physical operators

• Residue in terms of CFT data

• This allows us to find CFT data from Mellin amplitude

s = Δ + k

Convergence for 2Δϕ − Δ̃ < Re(s) < Δ̃Reminder

CΔ,k =(−1)kΓ(Δ + k)2Γ(2Δ)

Γ(k + 1)Γ(Δ)2Γ(2Δ + k)GΔ(s) = 3F2(Δ, Δ, Δ − s; 2Δ,1 + Δ − s; − 1)

Δ − s

Ress=Δ+k

M(s) = cΔCΔ,k


Perturbation and CFT data

Reminder

Perturbative expansion about generalised free field theory:

We have a block expansion:M(s) = ∑

Δ,k


+1

2Δ − s − k − Δ )

Spectrum: Δ = 2Δϕ + n n ∈ ℕ

Expand in a small parameter : Δ = 2Δϕ + n + ϵ γn

MO(ϵ)(s) = ϵ∑Δ,k

c(1)Δ CΔ,k ( 1

s − k − Δ+

12Δ − s − k − Δ ) + ϵ∑

Δ,k

c(0)Δ γΔCΔ,k ( 1

(s − k − Δ)2−

1(2Δ − s − k − Δ)2 )

Similar strategy for higher loopsdouble poles → γ(1)

nsimple poles → c(1)n


CFT1

Non-SUSY Wilson line

1/2 BPS Wilson line in 4d N=4

1/2 BPS Wilson line in 4d N=2

1/2 BPS Wilson line in ABJM

SYK models

QFT in AdS2

21

Example I: higher derivatives in AdS2


AdS2 computationsKΔ(y, x; xi)

< ϕΔ(x1)ϕΔ(x2)ϕΔ(x2)ϕΔ(x2) > = − λ∫dydx

y2

4

∏i=1 ( y

y2 + (x − xi)2 )Δ

=CΔ

(x12x34)2ΔD̄Δ(z)

−λ

Vertex, propagators, bulk integration:


AdS2 computations

-functions: building blocks of tree level contact diagrams

No closed form expression for general

D̄

Δ

KΔ(y, x; xi)

< ϕΔ(x1)ϕΔ(x2)ϕΔ(x2)ϕΔ(x2) > = − λ∫dydx

y2

4

∏i=1 ( y

y2 + (x − xi)2 )Δ

=CΔ

(x12x34)2ΔD̄Δ(z)

−λ

Vertex, propagators, bulk integration:

D̄1 = − 2log(1 − z)

z− 2

log(z)1 − z

D̄2 = − 2(z2 − z + 1)15(1 − z)2z2

+2z2 − 5z + 515(z − 1)3

log(z) −2z2 + z + 2

15z3log(1 − z)


The setupCFT1 defined as the boundary theory of the QFT in AdS2, with action

S = g∫dydx

y2 ( 12

∂μϕΔ∂μϕΔ +12

Δ(Δ − 1)ϕ2Δ+DLϕ4

Δ)• : theory correlators are the -functions

•Lorentz invariance

•Up to equivalency from equations of motion

•Convenient basis of linearly independent differential operators

D0 = 1 ϕ4 D̄

DL ≃ (∂)2n

DL ≃ (∂)4n

DLϕ4 = (L−1

∏i=0

( 12

∂μ∂μ − (Δ + i)(2Δ + 2i − 1)) ϕ2Δ)

2

ϕ3∂μ∂μϕ = Δ(Δ − 1)ϕ4


The Mellin amplitude

M0Δ(s) = csc(πs)cot(πs)PΔ(s) − csc(πs)

2Δ−1

∑si=1

PΔ(si)s − si

Known closed form expression for D-functions with integer :Δ

MLΔ(s) =

2L

∑k=0

˜(2L

k )M0Δ+L(s + k)

In position space< ϕΔ(x1)ϕΔ(x2)ϕΔ(x3)ϕΔ(x4) > ∝

1(x12x34)2Δ

(1 + (t

1 + t)2L +

1(1 + t)2L

)DΔ+L(t)

PΔ(s) = 3F2(s,2Δϕ − s,12

,1 − Δϕ; 1,1,12

+ Δϕ; 1)

ReminderSint = ∫

dxdyy2 (

L−1

∏i=0

( 12

∂μ∂μ − (Δ + i)(2Δ + 2i − 1)) ϕ2Δ)

2

In Mellin space


The CFT dataSolving explicitly for this higher -derivative interaction theory:γ(1)

n 4L

Reminder

c(0)n γ(1)

n =n

∑p=0

C−1n,p−n Res

s=2Δϕ+p(2Δϕ + p)M(s)

Valid for all L, we computed up to L=8 (32 derivatives)

Ress=2Δϕ+p

(2Δϕ + p)M(s) = ϵ∑n,k

c(0)n γ(1)

n Cn,kδn+k,p

γ(1)L,n(Δϕ) =

Γ(L + Δϕ)4

Γ(2L + 2Δϕ) ∑p,k,l

(−1)lck,lF(Δϕ, L, n, p, k, l)

F(Δϕ, L, n, p, k, l) =(4Δϕ + 2n − 1)p(−2n)p(2L − k − p)l(1 − Δϕ − L)l(2Δϕ + k + p)l(

12 )l

(l!)3(2Δϕ)p( 12 + Δϕ + L)l


Resultsγ(1)L,n(Δϕ) =

4−L (L + 12 )Δϕ

(L + Δϕ)Δϕ

(−L + n + 1)Δϕ−1(L + n + Δϕ + 12 )Δϕ−1

Γ (Δϕ) (Δϕ)3L (2L + Δϕ + 12 )Δϕ−1 (L + 2Δϕ − 1

2 )2L (n + 12 )Δϕ

(n + Δϕ)Δϕ

PL,n(Δϕ)

PL,n(Δϕ) = 1

L=0



4−L (L + 12 )Δϕ

(L + Δϕ)Δϕ

(−L + n + 1)Δϕ−1(L + n + Δϕ + 12 )Δϕ−1

Γ (Δϕ) (Δϕ)3L (2L + Δϕ + 12 )Δϕ−1 (L + 2Δϕ − 1

2 )2L (n + 12 )Δϕ

(n + Δϕ)Δϕ

𝒫L,n(Δϕ)

PL,n(Δϕ)

L=1



4−L (L + 12 )Δϕ

(L + Δϕ)Δϕ

(−L + n + 1)Δϕ−1(L + n + Δϕ + 12 )Δϕ−1

Γ (Δϕ) (Δϕ)3L (2L + Δϕ + 12 )Δϕ−1 (L + 2Δϕ − 1

2 )2L (n + 12 )Δϕ

(n + Δϕ)Δϕ

PL,n(Δϕ)

PL,n(Δϕ)

L=2



4−L (L + 12 )Δϕ

(L + Δϕ)Δϕ

(−L + n + 1)Δϕ−1(L + n + Δϕ + 12 )Δϕ−1

Γ (Δϕ) (Δϕ)3L (2L + Δϕ + 12 )Δϕ−1 (L + 2Δϕ − 1

2 )2L (n + 12 )Δϕ

(n + Δϕ)Δϕ

PL,n(Δϕ)

P0,n(Δϕ) = 1

PL,n(Δϕ)

L=3



4−L (L + 12 )Δϕ

(L + Δϕ)Δϕ

(−L + n + 1)Δϕ−1(L + n + Δϕ + 12 )Δϕ−1

Γ (Δϕ) (Δϕ)3L (2L + Δϕ + 12 )Δϕ−1 (L + 2Δϕ − 1

2 )2L (n + 12 )Δϕ

(n + Δϕ)Δϕ

PL,n(Δϕ)

PL,n(Δϕ) =

L=4

32

Example II: ABJM 1/2 BPS Wilson line


Gauge-fixed bosonic action of type IIA in AdS4 × ℂP3

33

Holography

σ = y τ = x gμν =δμν

y2

SB = T∫ d2σ GAdS4×ℂP3 = T∫ d2σ g (gμν∂μX∂νX̄ + 2 |X |2 + gμν∂μw̄a∂νwa + Lint)


Gauge-fixed bosonic action of type IIA in AdS4 × ℂP3

34

Holography

σ = y τ = x gμν =δμν

y2

SB = T∫ d2σ GAdS4×ℂP3 = T∫ d2σ g (gμν∂μX∂νX̄ + 2 |X |2 + gμν∂μw̄a∂νwa + Lint)

Result of Witten contact diagrams

⟨wa1(x1)w̄a2(x2)wa3(x3)w̄a4

(x4)⟩conn. =

massless fluctuations Remaining directions ℂP3

AdS4

=1T

Cw

(x12x34)2 (δa1a2

δa3a4 (t2(4 + t + log(t)) − (t2 +

4t

+ 3)log(t + 1)) + t2δa1a2

δa3a4 (t → t−1))

bosonic operators Displacement Operator

Δ = 1Δ = 2

wa, w̄a 𝕆a, �̄�aX, X̄ 𝔻, �̄�


The Analytic bootstrap

The dual fields form a chiral supermultiplet:

ℬ− 12

32 ,0,0

𝕆a

𝔻

[Bianchi, GB, Forini, Griguolo, Seminara ’20]


The Analytic bootstrap

The dual fields form a chiral supermultiplet:

Bootstrap order by order in a strong coupling parameter (See also Pietro’s talk)

ϵ

< 𝔽𝔽𝔽𝔽 > =1

x12x34(1 − t +

(1 − t)3

tlog(1 − t) − t(3 − t)log(t))

ℬ− 12

32 ,0,0

1. Use a HyperLogarithm ansatz 2. Impose crossing symmetry and consistent OPE expansion 3. Impose mild behaviour of anomalous dimension 4. Use selection rules to constrain remaining unknowns

𝕆a

𝔻

[Bianchi, GB, Forini, Griguolo, Seminara ’20]


SuperspaceChiral supermutliplet Single chiral superfield

Φ = 𝔽 + θa𝕆a + θaθbϵabcΛc + θaθbθcϵabc𝔻

The block expansion is now in terms of superblocks and superconformal cross ratio:

⟨ΦΦ̄ΦΦ̄⟩ =1

< 12 > < 34 > ∑Δ

CΔtΔ2F1(Δ, Δ + 3,2Δ + 3,t) t =

< 12 > < 34 >< 14 > < 23 >


SuperspaceChiral supermutliplet Single chiral superfield

Φ = 𝔽 + θa𝕆a + θaθbϵabcΛc + θaθbθcϵabc𝔻

The block expansion is now in terms of superblocks and superconformal cross ratio:

⟨ΦΦ̄ΦΦ̄⟩ =1

< 12 > < 34 > ∑Δ

CΔtΔ2F1(Δ, Δ + 3,2Δ + 3,t) t =

< 12 > < 34 >< 14 > < 23 >

All correlators can be found by expanding

⟨𝔽𝔽𝔽𝔽⟩ = ⟨ΦΦ̄ΦΦ̄⟩ |θ=0 =1

x12x34f(t)

⟨𝕆a1�̄�a2𝕆a3

�̄�a4⟩ = (Πi∂θaii)⟨ΦΦ̄ΦΦ̄⟩ |θ=0 =

1(x12x34)2 (δa2

a1δa4

a3( f(t) − tf′ (t) + t2f′ ′ (t)) + δa4

a1δa2

a3(t2f′ (t) + t3f′ ′ (t)))

⟨ΦΦ̄ΦΦ̄⟩ =1

< 12 > < 34 >f(t)


Mixing problem

Exchanged operator will be multiparticle operators:

Dimension 1 2 3 4

Operators …

Δ𝔽𝔽 𝔽∂x𝔽

𝕆a�̄�a

𝔽∂2x𝔽 𝕆a∂x�̄�a

𝕆a�̄�a𝔽𝔽 ΛaΛ̄a

Degenerate operators ( ), not possible to distinguish and find Δa = Δb {Δ, cΔ}

Perturbatively we compute: cΔγΔ∂ΔGΔ = (caγa + cbγb)∂ΔGΔ

Mixing is partially solved: {Φ∂3Φ̄, ΦΦ̄∂Φ∂Φ̄}ΦΦ̄ Φ∂Φ̄ Φ∂2Φ̄

At this order, completely solved (more in Pietro’s talk)γ(1)n = f(Δ)


Mellin

Reminder < Φ(1)Φ̄(2)Φ(3)Φ̄(4) >All four-point correlators are components of

40

< 𝔽𝔽𝔽𝔽 > =1

x12x34(−1 − t − t(3 + t)log(t) +

(1 + t)3

tlog(1 + t))

Given the previous results:

M𝔽(s) = 6Γ(s − 2)Γ(−1 − s)

M𝕆(s) = 6(1 − s)(Γ(−s)Γ(s − 3)δa2a1

δa4a3

− Γ(s − 2)Γ(−1 − s)δa4a1

δa2a3 )

•The crossing symmetry is for bosons & fermions

•The anomalous dimension is found through the Mellin of the SuperBlocks •There is a finite number of double poles •The form is simpler than the constituting parts (D-functions)

M(s) = M(2Δϕ − s)


Conclusions

• a Simple analytic structure • Poles at the weights of the physical operators • Consistent bloc expansion

• The ‘building blocks’ D-functions have a closed-form expression for integer

• Anomalous dimensions in higher 4L-derivative model.

• st order analysis of ABJM 1/2 BPS Wilson line insertions

• Superspace structure solves mixing and relates mellin amplitudes

Δ

1

In the context of the study of CFT1 theories, there is an array of tools to compute the CFT data (Witten diagrams, Analytric bootstrap, Mellin amplitudes). Propose a new tool, the 1d Mellin formalism for 4 point correlators with


Hopes and expectations

• Extend to exchange diagrams and higher loops

• Extend formalism to higher points functions and R-symmetry factors

• Find an ansatz in Mellin space for direct bootstrap

• Links to the S-matrix bootstrap and flat-space scattering amplitudes

• Understand how integrability manifests itself in this setting

43

hank you!


CFT1

x2P x12

I x−112

P x−112 − x−1

14I 1

x−112 − x−1

14

D x−113 − x−1

14

x−112 − x−1

14=

x12x34

x13x24= χ =

tt + 1

In four point functions, we can transform coordinates to one independant variable called conformal cross ratio

Conformal transformations generated by .{P, K, D} 0 K← ϕ(0)K⇋P

∂ϕ(0)K⇋P

. . .Reminder

Primary

Descendants

One can write correlators in terms of the operator product expansion, for example for 4-points

< ϕ(x1)ϕ(x2)ϕ(x3)ϕ(x4) > =1

(x12x34)2Δϕ ∑Δ

cΔχΔ2F1(Δ, Δ,2Δ, χ)


The Sum Rule

Given a Mellin amplitude that

• Is crossing symmetric

• has poles at the position of exchanged operators

• Admits an OPE expansion • Is appropriately bounded at large s

M(s) = M(Δϕ − s)

s = Δ + k

ωF[M̂](s) = ∫𝒞∞

ds2πi

M̂F = 0

|M̂(s) | = |M(s)

Γ(s)Γ(2Δϕ − s)| < Cs−α α > 0

M(s) = ∑Δ

cΔMΔ(s)

46


Flat space limits

General idea: For large enough radius, EAdS2 locally→ ℝ2

R → ∞ R → ∞Δ → ∞

Massless scattering Massive scattering • Mellin formalism • Saddle point • Fourier transform

fixedΔ

•Mellin formalism


⟨𝕆a1(x1)𝕆a2

(x2)𝕆a3(x3)𝕆a4

(x4)⟩ =1

(x12x34)2Δ( f (1)(χ) + ϵf (1)(χ)) + O(ϵ2)





Analytic Bootstrap [Liendo, Meneghelli, Mitev ’18] [Mazac , Paulos] [Ferrero, Ghosh, Sinha, Zahed ’18]


Feynman diagrams

[Cooke, Dekel, Drukker, ’17] [Kyriu, Komatsu ’18] [Barrat, Liendo, Plefka ’20]

Witten diagrams [Giombi, Roiban, Tseytlin ’17]


Superspace v2Superspace structure gives a relation between the different Mellin amplitudes:

M[⟨𝔽𝔽𝔽𝔽⟩] → M[⟨𝔽𝔽𝕆a�̄�a⟩]

Differential equations become linear operations and translations

All correlators will have a simple expression !

g(t) = tf′ (t) − t2f′ ′ (t)

M[g](s) = ∫ dt(t−sf′ (t) − t1−sf′ ′ (t))

M[g](s) = s(2 − s)M[ f ](s)

For a general differential operator g(t) = ∑k,n

ak,ntk∂(n)t f(t)

M[g](s) = ∑k,n

ak,n(−1)k(k − 1 − s)nM[ f ](s + k − n)

Documents

Mellin amplitudes 1D CFTs