nonperturbative mellin amplitudes: existence, properties, … · 2020. 1. 31. · nonperturbative...
TRANSCRIPT
Nonperturbative Mellin Amplitudes: Existence, Properties, Applications
A. Zhiboedov (CERN) SISSA, Trieste
work with J. Penedones and J. Silva
Introduction
1
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ZdDx
p�g (R� 2⇤+ ...)
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AdS/CFT
nonperturbative
perturbative
1
lD�2Pl
ZdDx
p�g (R� 2⇤+ ...)
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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)
1
lD�2Pl
ZdDx
p�g (R� 2⇤+ ...)
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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)
Any EFT! Only String Theory!
1
lD�2Pl
ZdDx
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AdS/CFT=<latexit sha1_base64="Qjzj31T7H7rhI9UeuaTn0LITDWI=">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</latexit>
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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)
BootstrapIs it possible to use bootstrap to shed some light upon the matter?
Tentative CFT data
NOMAYBE
[Rattazzi, Rychkov, Tonni, Vichi ’08]
Maybe
No
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
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
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)
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 .
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]
Outline
Mellin transform (in CFTs)
Existence
Properties
Applications
Review of Mellin transform
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?
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
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
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
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:
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⇡
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)
(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
Region of Integration
(principal Euclidean sheet)
Mellin transform = Integral over the principal sheet
⇡⇡2
0�⇡ �
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)2⇡�2⇡ 0
2⇡
�2⇡
(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!
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
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]
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
Mellin Amplitude
Fsub(u, v) =
Zd�122⇡i
d�142⇡i
M̂(�12, �14)u��12v��14
UCFT
UCFT :
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
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 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
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
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).
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)
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
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
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.
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.
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.
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.
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.
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
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���� (u, v) =
ppu+
pv + 1p
2 8puv
M̂(�12, �14) = �r
2
⇡�
✓2�12 �
1
4
◆�
✓2�14 �
1
4
◆�(�2�12 � 2�14)
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
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]
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=">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</latexit>
f�1,�2
� (⇣) = ⇣���1��2+1
2 2F1
✓�� 1 +�1 ��2
2,�� 1��1 +�2
2,�+ 1� d
2, ⇣
◆
<latexit sha1_base64="BEOaUPRB7wO1rKQZuS7DSuNp0uk=">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</latexit>
• 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 =
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?
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]
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]
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
◆
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
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
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!
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
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)
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?
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!
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 polynomialsDispersion 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 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
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]
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) + ...)
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
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