nonperturbative mellin amplitudes: existence, properties, … · 2020. 1. 31. · nonperturbative...
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![Page 1: Nonperturbative Mellin Amplitudes: Existence, Properties, … · 2020. 1. 31. · Nonperturbative Mellin Amplitudes: Existence, Properties, Applications A. Zhiboedov (CERN) SISSA,](https://reader035.vdocuments.mx/reader035/viewer/2022070211/61002b9760ffc72f653c0f66/html5/thumbnails/1.jpg)
Nonperturbative Mellin Amplitudes: Existence, Properties, Applications
A. Zhiboedov (CERN) SISSA, Trieste
work with J. Penedones and J. Silva
![Page 2: Nonperturbative Mellin Amplitudes: Existence, Properties, … · 2020. 1. 31. · Nonperturbative Mellin Amplitudes: Existence, Properties, Applications A. Zhiboedov (CERN) SISSA,](https://reader035.vdocuments.mx/reader035/viewer/2022070211/61002b9760ffc72f653c0f66/html5/thumbnails/2.jpg)
Introduction
![Page 3: Nonperturbative Mellin Amplitudes: Existence, Properties, … · 2020. 1. 31. · Nonperturbative Mellin Amplitudes: Existence, Properties, Applications A. Zhiboedov (CERN) SISSA,](https://reader035.vdocuments.mx/reader035/viewer/2022070211/61002b9760ffc72f653c0f66/html5/thumbnails/3.jpg)
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AdS/CFT
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AdS/CFT
nonperturbative
perturbative
![Page 5: Nonperturbative Mellin Amplitudes: Existence, Properties, … · 2020. 1. 31. · Nonperturbative Mellin Amplitudes: Existence, Properties, Applications A. Zhiboedov (CERN) SISSA,](https://reader035.vdocuments.mx/reader035/viewer/2022070211/61002b9760ffc72f653c0f66/html5/thumbnails/5.jpg)
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AdS/CFT
nonperturbative
perturbative
coarse-grained holography: assume a UV completion and derive rules of consistent AdS EFTs (Hofman-Maldacena type bounds) (easy)
![Page 6: Nonperturbative Mellin Amplitudes: Existence, Properties, … · 2020. 1. 31. · Nonperturbative Mellin Amplitudes: Existence, Properties, Applications A. Zhiboedov (CERN) SISSA,](https://reader035.vdocuments.mx/reader035/viewer/2022070211/61002b9760ffc72f653c0f66/html5/thumbnails/6.jpg)
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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)
![Page 7: Nonperturbative Mellin Amplitudes: Existence, Properties, … · 2020. 1. 31. · Nonperturbative Mellin Amplitudes: Existence, Properties, Applications A. Zhiboedov (CERN) SISSA,](https://reader035.vdocuments.mx/reader035/viewer/2022070211/61002b9760ffc72f653c0f66/html5/thumbnails/7.jpg)
Any EFT! Only String Theory!
1
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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)
![Page 8: Nonperturbative Mellin Amplitudes: Existence, Properties, … · 2020. 1. 31. · Nonperturbative Mellin Amplitudes: Existence, Properties, Applications A. Zhiboedov (CERN) SISSA,](https://reader035.vdocuments.mx/reader035/viewer/2022070211/61002b9760ffc72f653c0f66/html5/thumbnails/8.jpg)
BootstrapIs it possible to use bootstrap to shed some light upon the matter?
Tentative CFT data
NOMAYBE
[Rattazzi, Rychkov, Tonni, Vichi ’08]
Maybe
No
![Page 9: Nonperturbative Mellin Amplitudes: Existence, Properties, … · 2020. 1. 31. · Nonperturbative Mellin Amplitudes: Existence, Properties, Applications A. Zhiboedov (CERN) SISSA,](https://reader035.vdocuments.mx/reader035/viewer/2022070211/61002b9760ffc72f653c0f66/html5/thumbnails/9.jpg)
BootstrapIs it possible to use bootstrap to shed some light upon the matter?
Tentative CFT data
NOMAYBE
[Rattazzi, Rychkov, Tonni, Vichi ’08]
Maybe
No
no surprising bounds for large N theories yet
![Page 10: Nonperturbative Mellin Amplitudes: Existence, Properties, … · 2020. 1. 31. · Nonperturbative Mellin Amplitudes: Existence, Properties, Applications A. Zhiboedov (CERN) SISSA,](https://reader035.vdocuments.mx/reader035/viewer/2022070211/61002b9760ffc72f653c0f66/html5/thumbnails/10.jpg)
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
NOMAYBE
[Rattazzi, Rychkov, Tonni, Vichi ’08]
Maybe
No
no surprising bounds for large N theories yet
![Page 11: Nonperturbative Mellin Amplitudes: Existence, Properties, … · 2020. 1. 31. · Nonperturbative Mellin Amplitudes: Existence, Properties, Applications A. Zhiboedov (CERN) SISSA,](https://reader035.vdocuments.mx/reader035/viewer/2022070211/61002b9760ffc72f653c0f66/html5/thumbnails/11.jpg)
Mellin Representation
Mellin Space Coordinate space
Bootstrap
[Mack]
PerturbativeNonperturbative
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)
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Mellin Representation
Mack’s Mellin Ansatz: CFT correlation functions admit the following representation
[Mack ’09]
hO1(x1) . . .On(xn)i =
Z[d�ij ]M(�ij)
nY
i<j
�(�ij)
|xi � xj |2�ij
nX
j=1
�ij = 0 , �ij = �ji , �ii = ��i .
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Mellin Representation
Mack’s Mellin Ansatz: CFT correlation functions admit the following representation
[Mack ’09]
hO1(x1) . . .On(xn)i =
Z[d�ij ]M(�ij)
nY
i<j
�(�ij)
|xi � xj |2�ij
nX
j=1
�ij = 0 , �ij = �ji , �ii = ��i .
[Penedones ’10]
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Outline
Mellin transform (in CFTs)
Existence
Properties
Applications
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Review of Mellin transform
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Mellin Transform
Mellin transformation is defined as follows[Mellin ’1896]
�(x) : Rn+ ! C
M [�](z) =
Z
Rn+
dx xz�1�(x)
M�1[F ](x) =1
(2⇡i)n
Z
a+iRn
dz x�zF (z)
What are natural classes of functions associated with it?
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M-space
Space of functions holomorphic in a sectorial domain , which are polynomially bounded with powers in an open domain U.
Θ
MU⇥ : ⇥ 2 Rn, U 2 Rn, 0 2 ⇥ �(x) 2 MU
⇥
x = (r1ei✓1 , ..., rne
i✓n) ✓ 2 ⇥ r 2 Rn+
|�(x)| < C(a)|x�a|, x = (r, ✓) 2 S⇥, a 2 U
θU
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W-space
Similarly, we define a space of functions holomorphic in a tube and decaying exponentially fast in the imaginary directionW⇥
U : F (z), z 2 U + iRn
|F (u+ iv)| K(u)e�sup✓2⇥✓·v u 2 U
Uθ
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Mellin Inversion Theorems[Antipova ’07]
Theorem I: Given , the Mellin transform exists and . We also have .
Φ(x) ∈ MUΘ
M[Φ] ∈ WΘU M−1M[Φ] = Φ
Theorem II: Given , the inverse Mellin transform exists and . We also have .
F(z) ∈ WΘU
M−1[F] ∈ MUΘ MM−1[F] = F
θU
� < poly F < exp
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Examples
e�u =
Z cu+i1
cu�i1
d�122⇡i
�(�12)u��12 , cu > 0, |arg[u]| < ⇡
2.
�(�12) ⇠ e�⇡2 |Im[�12]|
The Cahen-Mellin integral:
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Examples
e�u =
Z cu+i1
cu�i1
d�122⇡i
�(�12)u��12 , cu > 0, |arg[u]| < ⇡
2.
�(�12) ⇠ e�⇡2 |Im[�12]|
The Cahen-Mellin integral:
�(2�12)�(↵� 2�12)
�(↵)⇠ e�2⇡|Im[�12]|
1
(1 +pu)↵
=
Z cu+i1
cu�i1
d�122⇡i
�(2�12)�(↵� 2�12)
�(↵)u��12 ,
0 < cu <↵
2, |arg[u]| < 2⇡
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Mellin transform in CFTs
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Are analyticity in a sectorial domain and polynomial boundedness properties of the physical correlators?
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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)
(x213x
224)
� , u =x212x
234
x213x
224
, v =x214x
223
x213x
224
F (u, v) = F (v, u) = v��F
✓u
v,1
v
◆
M̂(�12, �14) ⌘Z 1
0
dudv
uvu�12v�14F (u, v)
(crossing)
We can try to define the Mellin transform as follows
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Region of Integration
(principal Euclidean sheet)
Mellin transform = Integral over the principal sheet
⇡⇡2
0�⇡ �
t
(1, 0)
(0, 1)
u
v
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Analyticity Domain
Claim: Any CFT correlator is analytic in a sectorial domain .
F(u, v)ΘCFT = (arg[u], arg[v])
arg(v)
arg(u)2⇡�2⇡ 0
2⇡
�2⇡
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(Reduced) Mellin Amplitude
M̂(�12, �14) = [�(�12)�(�13)�(�14)]2 M(�12, �14)
It is common in the field to define the following reduced Mellin amplitude
[�(�12)�(�14)�(�� �12 � �14)]2 ⇠ e�⇡(|Im[�12]|+|Im[�14]|+|Im[�12+�14]|)
= e�sup⇥CFT(arg[u]Im[�12]+arg[v]Im[�14])
The pre-factor correctly captures analytic properties of the correlator!
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Polynomial Boundedness
We consider a subtracted correlator
Fsub(u, v) = F (u, v)� (1 + u�� + v��)
�X
⌧gap⌧<⌧sub
JmaxX
J=0
h⌧sub�⌧
2
i
X
m=0
C2OOO⌧,J
⇣u��+ ⌧
2+mg(m)⌧,J (v) + v��+ ⌧
2+mg(m)⌧,J (u) + v�
⌧2�mg(m)
⌧,J (u
v)⌘
crossing symmetric same analyticity properties improved behavior in the OPE limits
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Dangerous Regions
⇡⇡2
0�⇡ �
t
(1, 0)
(0, 1)
u
v
Double light-cone limit u, v ! 0
[Alday, Eden, Korchemsky, Maldacena, Sokatchev ’10]
[Alday, AZ ’15]
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Polynomial Boundedness
Claim:
Saturated in minimal models and perturbative field theories
|Fsub(u, v)| C(cu, cv)1
|u|cu1
|v|cv , (u, v) 2 ⇥CFT , (cu, cv) 2 UUCFT
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Mellin Amplitude
Fsub(u, v) =
Zd�122⇡i
d�142⇡i
M̂(�12, �14)u��12v��14
UCFT
UCFT :
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3d Ising
As an example consider the spin operator in the 3d Ising model
σ⟨σσσσ⟩
F Ising3d (u, v) =
�1 + u�� + v��
�+
Z
C
d�122⇡i
d�142⇡i
M̂(�12, �14)u��12v��14 ,
C : �� � 1
2< Re(�12),Re(�14),
Re(�12) + Re(�14) <1
2.
This representation converges in .ΘCFT
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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)
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Properties of the CFT Mellin Amplitudes
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Mellin vs OPEHow is the Mellin representation consistent with the OPE?
=
O(x1)
O(x2) O(x3)
O(x4)
O(x1)
O(x2) O(x3)
O(x4)
OiOi
X
i
X
i
Reproduce the OPE (unitarity)
Cancel the disconnected piece (Polyakov conditions)
F (u, v) =�1 + u�� + v��
�
+
Z
C
d�122⇡i
d�142⇡i
[�(�12)�(�14)�(�� �12 � �14)]2 M(�12, �14)u
��12v��14
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OPELet us review in more detail the unitarity properties of Mellin amplitudes
t = 2�� 2�12,
s = 2(�12 + �14 ��) = �2�13
M(s, t) ' �C2
��⌧Q⌧,dJ,m(s)
t� (⌧ + 2m)+ ... , m = 0, 1, 2, ... ,
Q⌧,dJ,m(s) = K(�, J,m) Q⌧,d
J,m(s)
The residues are controlled by the so-called Mack polynomials
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OPELet us review in more detail the unitarity properties of Mellin amplitudes
t = 2�� 2�12,
s = 2(�12 + �14 ��) = �2�13
M(s, t) ' �C2
��⌧Q⌧,dJ,m(s)
t� (⌧ + 2m)+ ... , m = 0, 1, 2, ... ,
Q⌧,dJ,m(s) = K(�, J,m) Q⌧,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 of the physical operators (except the identity).
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OPE
Mack polynomials have a few interesting propertiesQ⌧,d
J,m(s) = (�1)JQ⌧,dJ,m(�s� ⌧ � 2m)
QJ,0(s) =2J
�⌧2
�2J
(⌧ + J � 1)J3F2(�J, J + ⌧ � 1,�s
2;⌧
2,⌧
2; 1)
lims!1
Q�,⌧,dJ,m (s) = s
J +O(sJ�1)
Q⌧,dJ,m(s) = m
J2 (m+ ⌧)
J2 C
( d�22 )
J
⌧2 +m+ sp
m(m+ ⌧)
!.
Higher m’s can be computed recursively.
(high energy)
(flat space limit)
limJ,s!1
Q�,⌧,dJ,m (s) = 21+J�mJ2J+s+⌧+m�1e�2J ⇡
�( ⌧+2m+s2 )2
sm + ... (AdS effects)
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Twist Spectrum of CFTsUnitarity
The twist spectrum in a CFT is very complex and contains an infinite number of accumulation points
limJ!1
⌧ (n)1,2 (J) = ⌧1 + ⌧2 + 2n
limJ!1
⌧ (m)
⌧ (n)1,2 (J0),⌧3
(J) = ⌧ (n)1,2 (J0) + ⌧3 + 2m
⌧ � d� 2
[Komargodski, AZ 12’][Fitzpatrick, Kaplan, Poland, Simmons-Duffin 12’][Callan, Gross] [Parisi]
τ1
τ2
τ1 + τ2
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Bound on ReggeThe 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�
x1x2
x4
x3
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Conformal Regge Theory[Costa, Goncalves, Penedones ’12]
In Mellin space the Regge limit becomes
lims!1
M(s, t) 'Z 1
�1d⌫ �(⌫)!⌫,j(⌫)(t)
sj(⌫) + (�s)j(⌫)
sin (⇡j(⌫))+ ...
jfull(0) 1
jplanar(0) 2
lim|s|!1
|Mplanar(s, t)| < c|s|2+✏
lim|s|!1
|Mfull(s, t)| c|s|
Re[t] < d − 2
Extrapolated away from the imaginary axis.
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Polyakov ConditionsIn 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
d�122⇡i
d�142⇡i
[�(�12)�(�14)�(�� �12 � �14)]2 M(�12, �14)u
��12v��14
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.
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Polyakov ConditionsAs we close the contour we get the divergent sum
−12
(CGFFτ=2Δ,J)
2Qτ,dJ,0(γ14)Γ2(γ14)Γ2 ( τ
2− γ14) Γ2 (Δ −
τ2 )
=4
Γ2(Δ)1J (J2(Δ−γ14)Γ2(γ14) + J2γ14Γ2(Δ − γ14) + . . . ) , J → ∞ , τ → 2Δ ,
thus avoiding the double counting problem.
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Polyakov ConditionsAs we close the contour we get the divergent sum
−12
(CGFFτ=2Δ,J)
2Qτ,dJ,0(γ14)Γ2(γ14)Γ2 ( τ
2− γ14) Γ2 (Δ −
τ2 )
=4
Γ2(Δ)1J (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
2πidγ14
2πiγ12M̂(γ12, γ14)u−γ12v−γ14 .
solves both convergence and double counting problems.
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Polyakov ConditionsConvergence 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.
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SummaryCFT connected correlators admit a deformed contour Mellin representationsFconn(u, v) ⌘ F (u, v)� (1 + u�� + v��)
=
Z Z
C
d�122⇡i
d�142⇡i
[�(�12)�(�14)�(�� �12 � �14)]2 M(�12, �14)u
��12v��14
Reduced Mellin amplitude M satisfies:
Maximal Mellin Analyticity (Unitarity)
Crossing Symmetry
Polynomial Boundedness
Polyakov conditions
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Applications
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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���� (u, v) =
ppu+
pv + 1p
2 8puv
M̂(�12, �14) = �r
2
⇡�
✓2�12 �
1
4
◆�
✓2�14 �
1
4
◆�(�2�12 � 2�14)
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Minimal Models [Furlan, Petkova ’89][Lowe ’16][Alday, AZ ’15]
[Yuan ’18]
[Dotsenko, Fateev]
M̂(�12, �14) = C0
Z[d⇠12]
Z[d⇠15]
Z[d⇠24]
Z[d⇠34]
Z[d⇠35]�(⇠12)�(⇠15)
�(�2↵↵+ + �12 � ⇠12)�(⇠24)�(⇠15 � ⇠24 � ⇠34 + 1)���2↵2
+ + 2↵↵+ + �12 � ⇠34 + 2�
�(⇠34)�(�2↵↵+ � ⇠12 � ⇠15 + ⇠34 � 1)�(�2↵↵+ + �14 � ⇠15 + ⇠24 + ⇠34 � 1)
���2↵2
+ + 2↵↵+ + �14 + ⇠12 + ⇠24 � ⇠35 + 1���2↵2
+ � ⇠15 + ⇠24 � ⇠35 � 1�
��⇠12 � ⇠34 � ⇠35 + 2↵↵+ � 2↵2
+ + 1��(⇠35)�(�⇠12 � ⇠24 + ⇠35 + 1)
����12 � �14 + ⇠15 � ⇠24 + ⇠35 + 4↵2
+ � 1�
��2↵2
+ + ⇠15��(�4↵↵+ � ⇠12 � ⇠15 + ⇠34 � 1)
�(��12 � �14 � ⇠24)
��2↵2
+ + ⇠35����4↵2
+ + 4↵↵+ + ⇠12 � ⇠34 � ⇠35 + 3� .
h�1,3O�1,3Oi
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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 =
Zdy�T��
It is trivial to compute Wightman functions using Mellin amplitudes
hO1(x1) . . .On(xn)i =
Z[d�ij ]M(�ij)
nY
i<j
�(�ij)
(x2ij + i✏x0
ij)�ij
[Kologlu, Kravchuk, Simmons-Duffin, AZ]
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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⇡4�(�i � 2)
�(�i�12 )3�( 3��i
2 )f4,4�i
(⇣) +1
4(2�(⇣)� �0(⇣))
#
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⇣ =1� cos�
2<latexit sha1_base64="WN3VzgxIu19GEHmO21XEK7vuyIA=">AAAd13ichZlZcxy3EYDXzuVsDsvJY6IKKjRLkotccSm7rDiOIx4iRYnHipcochgGg8HswpyLAIbL5dRWnpLKa/5B3vOa/Jj8mzQwSwG704q3yOUMvgbQaHT3NIZhkQill5b++8GH3/v+D374o49+3P7JT3/284/vffKLY5WXkvEjlie5PAmp4onI+JEWOuEnheQ0DRP+JrxcM/zNNZdK5NmhHhX8PKX9TMSCUQ1NF/d+E9xyTckfSNUliyRguYKvgSBBDr3I8vji3txSZ8l+SPOiO7mYa00+vYtPfv3PIMpZmfJMs4QqddZdKvR5RaUWLOHjdlAqXlB2Sfv8DC4zmnJ1XtmVjMk8tEQkziX8ZprYVr9HRVOlRmkIkinVAzXLTCPGzkodPz2vRFaUmmesniguE6JzYsxCIiE508kILiiTAnQlbEAlZRqM156fhx9C/hypSA1yqQc0izpKjx6QxUWyAx3AbEZpDkYbRXREYPQF0ucZlzSBQWEWkRYSbGrHYXnECexSREORCD0iQ6EHhEJ7pmDLwXLETUPqCYg1FKGSt+0YpSrt0GbzI5LmkpM8I3rACS0KDppnDFpi22IslidEglVgU82wptkOk3KaiaxPHvJOv0OCU7uMwNgvDKvT8QIRGShEIzNUsJVp3gd7LBAjxFaMcK2N7cFoUq2MH4HGa4NcsFpbsAJ8R2CCIZWRIkaQwxd4YEKKwUgJphbsGApWDNN07qy9UmowwldknV6LiByINAXzLK6XcSwy8nVkWlX0rJ9SkXRYnn5Ta7JNlQZzRODjMOtK2S/hvvt0gSwvdb80Q7eDXc4jdchPNnIJilTbFK6XjWf2YIdExFVv4lD+do/PzACPl54+7j5FVXpg1mY84bwd7POrEhyqN+2YC2TivfZCD9JxuyEqYjBPNiZknli9SJL3BTNLsx+ym2sy6RItkLDUZC/Wi0fGByZDKDKRbaPBgUbTbKPVzW+bqDXV1pe0GAh2A8qCtoHIWFJGfNKqpmfnhdJ5EcW1LDhuPiS2g0lMxrdAgMQi4dMRHaamQ/2Bbgx8SplgAm+OrCfZNAH+eC0oCcJ0qvONgis+nnQO6lsCCUCR+nII0cBJBt7Ao6meZmSpYlVrG0MojYiCrFDor2pPh5s2LHCdQ4aTfHOy5P0y4dVn4yot1Nj8tbuYwYapUprZqIawE4pEOVf13gn9QLXJd3xgDG4WTqVNJCm95CTmoywtyDCXl5A1ZJ27nJesyL5NwIuaXprorlPUAnjsLSdrYDIJ6cC5ScaHEEApeDlMFxSRJORs+ZxUQQwJsAoKk7shWue6Y+9meTye7Slneka2S4TKJkb0iRFNeKxBEKRgBNEfwM2ThngBX2ddKy4fVjDso3FDcZ3cyQwhirVIIm4kYSzJPckgpHIiZx5y5sk5EXNCn5kpN67I2RfnM/tTVeMLI74B38tjq/zDADK9WXCnU8GzDja+ghVUc5+Pfz/3Rb2kRw0ljDc4ZeGu1sEPdPuIJgcmAzHjte/bsK+NWkFCs759vk7P842FEocdC1mU64bBbyzSAh46DZbnhoks1qMGo/zK9IN4uIQHkcxvmvtUwEa9c6TGAJBAIsAZDRPagCkEQAUhaNJJAw5oEteeR7pQurTnScSluAYzXkNWrJ+xUuZD1fBF0MjYSChOOk/5TTAI85vq0z/Z3bVdPh3D8DZrVIu/gy0mTv2mY/+/0aw3fOeQd06wKTm/fLfvM5tHzSbQpBjQpocbFEJh11Svb1Cfwn3DfpsGbaII6pUKvhJsxHXD1ms2g7gZEVK7SPKsoWRSe20aRs0FbBu2PWGzUFqPHjQdIDYAUnEDXMcArqnEmDKdlOgjqx4alKe831Rwz6A9FIGbmQDgiD0uKSATF4gZb82It/We3e3/gXEPKKNAYkPwJHp/DtgnNi5s2bbfzLanHj5t4l0P7zbxmofXmvi1h18juPDxRdEwWM/r32v2X/Hwip8id8pEi8WEa6jRyUaZMZ1L9dheQOJ0tponAyg3YLjRpCCGwIS6roQaIuS2jMgL9ceGSVcim+/t1DKtVqKmagdHUyIHR4jI3rTIXlNkc3tKZHMbGWVa5AAR6VmZdyI9VGZzWgadqpieqsBE4AThi9AM2Taob3zzlbopsw/OfSeTF+9koXncfGzvc4ILI7JbKSq7lY6bShzKqQUfSsRu8ZRIL26KvOISmxGakSnX8ksQbwjbZkT8RZ7608NtU2Ynl74M3CKOBnWb72jwrGnu7WhqLrhtysD5VaRN9W0zor6wfxri0IwIH+dTOsJtU+Z6WubaynhZU5hi5y4xIPXTneCqOUZsWzE/WyhI+SSZNJsjrJkaqmk4Y+Xg1FCCmKIeDutxIpiuCwtzzLZHcmIObRNdzYsDOMANuWRwSp9dRUjv8lpc0eYiw9DhEMHMYYbgyGEkcYXcYYif9vwsjx1HnD3sO9xH8MDhAYKFwwLB3zr8LYIvHb5EcOJwgq0sdRxx7TBzGElqYe5wjuDCYSRvhlcOXyFYOoxEb6gcVgjWDiOpNiwdLhF87fA1gocODxF84/ANgkcOjxB86/Bt/YAnR0XxnqhZccIryFirDq8ieM1hpJIJ1x1eR/Bzh58jeMPhDQRvOryJ4BcOv0DwlsNbCH7p8EsEv3L4FYK3HUZKgnDH4R0E7zqMVI/hnsNI9RP2HEaqv/C1w0hxGe47jNS94YHDBwg+dPgQwUcOI6VdeOzwMYLfOPwGwScOnyD4rcNvEXzq8KmNGTgscGbezHjvrf03GsEqT+BcyjPzNsr0hBMiPP/qd8MBML+0XqOJmLzJg+tCaJq4J+RsNXAXkfX732a1sOpxJCbZmseRoGTrHkeikj33OBKWbMPjSFyyTY8jgcleeByJTLblcSQ02UuPI7HJXnkcCU627XEkOtmOx5HwZLseR+KT7XkcCVDW8zgSoey1x5EQZfseR2KUHXgcCVJ26HEkStmRx5EwZcceR+KUvfE4EqjsxONIpLK3HkdClZ16/NSPsqNMaHLM7bn1fdF1Y14UVuZ1IcQr8mwd+Rx5uN76/BZ5Z+tzpOrIfA4VEXHqP78q6fRJO4h4DLXlXDfg3LyA6ousMpkkG5umLJrcTE8SZKbmyvKsTEM4/rSnDuLBcPt9T6ZgeDCd2/0joDkjNE4c9qAwbl/cm+vO/gezeXG83Ok+6Sy//nzu2erkv5sftX7V+m3rYavb+rL1rPWi1WsdtVjrr61/tf7d+s/9t/f/cv9v9/9ei374waTPL1tTn/v/+B/aYleD</latexit>
f�1,�2
� (⇣) = ⇣���1��2+1
2 2F1
✓�� 1 +�1 ��2
2,�� 1��1 +�2
2,�+ 1� d
2, ⇣
◆
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• Sum is over Regge trajectories
[Kologlu, Kravchuk, Simmons-Duffin, AZ]
[Belitsky, Hohenegger, Korchemsky, Sokatchev, AZ]
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AdS/EFT
Consider an effective field theory in AdS
S =
Zdd+1x
p�g(�@µ�@
µ�+ µ1�4 + µ2(@µ�@
µ�)2 + ...)
[Adams, Arkani-Hamed, Dubovsky, Nicolis, Rattazzi ’06]
Using dispersion relations we get
µ2 =2
⇡
Zds
s�(s)
s3> 0.
[Hartman, Jain, Kundu ’16]
Can we use similar arguments in Mellin space?
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Positivity of Mack Polynomials
Mack polynomials share some positivity properties with Legendre polynomials
Q⌧,dJ,m(s) � 0, s � 0.
@ns Q
⌧,dJ,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]
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Dispersion Relations
Therefore, we can run an identical dispersion relation argument
µAdS2 =
1
2⇡i
Idt0
(t0)3M(0, t0) � 0
Similarly, higher order couplings receive the contribution suppressed by an extra power of the mass of the particle that we integrated out
μAdSn (∂μϕ∂μϕ)n
µAdSn =
1
2⇡i
Idt0
(t0)nM(0, t0)
[Caron-Huot '17][Alday, Bissi ’16]
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Bootstrap in Mellin SpaceWe can also do bootstrap in Mellin space. Consider dispersion relations
M3d(�12, �13, �14)
(�12 � ��3 )(�13 � ��
3 )(�14 � ��3 )
= �M3d(��
3 , �13,2��3 � �13)
(�13 � ��3 )2
1
�12 � ��3
+1
�14 � ��3
!
+1
2
X
⌧,J,m
C2⌧,JQ
⌧,dJ,m(�13)
(�� � ⌧2 �m� ��
3 )(�13 � ��3 )(�� � �13 + (m+ ⌧
2 ���)� ��3 )
⇥✓
1
�12 ��� + ⌧2 +m
+1
�14 ��� + ⌧2 +m
◆
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Bootstrap in Mellin Space
We now write Polyakov conditions
M(0, �13,�� � �13) = 0
The result takes the following formX
⌧,J,m
C2⌧,J↵⌧,J,m = 0
↵⌧,J,m =1
(⌧ � 2��3 + 2m)(⌧ � 4��
3 + 2m)
(⌧ + 2m���)Q⌧,d
J,m(��3 )
(⌧ � 2��3 + 2m)(⌧ � 4��
3 + 2m)� ��
3
Q⌧,dJ,m(��
3 )0
⌧ + 2m� 2��
!.
positive by unitarity
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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
⌧,J,m
C2⌧,J↵⌧,J,m = 0scalar operators
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Bootstrap in Mellin SpaceThe alpha functionals have remarkable properties
twist
X
⌧,J,m
C2⌧,J↵⌧,J,m = 0spinning operators
2 4 6 8 10 12 140.0
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!
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3d Ising
The sum rule takes the form
�X
⌧<2��,J>0,m
C2⌧,J↵⌧,J =
X
⌧>2��,J>0,m
C2⌧,J↵⌧,J +
X
⌧,J=0
C2⌧,J↵⌧,J
Plugging the available numbers in the 3d Ising we get
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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 <34
(d − 2)
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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 <34
(d − 2)
Is there a theory of QG that looks like GR+minimal scalar below the Planck scale?
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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 <34
(d − 2)
Is there a theory of QG that looks like GR+minimal scalar below the Planck scale?
Surprisingly, not always!
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Holographic Functionals
1.05 1.10 1.15 1.20 1.25�
-5
5
beyond HPPS!
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Further comments
Mellin amplitudes are well-defined nonperturbatively
Subtractions (non-Mellin pieces)
“Meromorphic” and polynomially bounded
Positivity of Mack polynomialsDispersion relations in Mellin space (AdS EFT)Matrix elements of light-ray operators
Twist accumulation points
New bootstrap functionals suited for holography
Polyakov conditions
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Further comments
Mellin amplitudes are well-defined nonperturbatively
Subtractions (non-Mellin pieces)
“Meromorphic” and polynomially bounded
Positivity of Mack polynomialsDispersion 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
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Flat Space Limit[Penedones ’10]
Flat space limit naturally emerges in the fixed angle regime
lims,t!1; st�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]
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CFT OPE
Operators can be expanded in the basis of primaries
Oi(x)Oj(0) =X
k
�ijk|x|�i+�j��k (Ok(0) + xµ@µOk(0) + ...)
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CFT OPE
Operators can be expanded in the basis of primaries
Oi(x)Oj(0) =X
k
�ijk|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)
(x212x
234)
�
u =x212x
234
x213x
224
, v =x214x
223
x213x
224
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CFT Bootstrap2-, 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)
OiOi
X
i
X
i
CFT data: (�,J) �ijk