Simons 2017
Conformal Bootstrap At Large Charge
Alexander Zhiboedov, Harvard U Simons Foundation, 2017
with Daniel Jafferis and Baur Mukhametzhanov
Approaches to Bootstrap This talk: bootstrap for heavy operators of low spin (high twist) CFT d>2
Approaches to Bootstrap This talk: bootstrap for heavy operators of low spin (high twist) CFT d>2
⌧=
J
Comment on Heavy States
Comment on Heavy States What do we expect for heavy states?
Comment on Heavy States What do we expect for heavy states? •
Generic heavy states look thermal for few-body observables (Eigenstate Thermalization Hypothesis) [Deutsch ’91, Srednicki ’94] [Dymarsky, Lashkari, Liu ’16]
Comment on Heavy States What do we expect for heavy states? •
Generic heavy states look thermal for few-body observables (Eigenstate Thermalization Hypothesis) [Deutsch ’91, Srednicki ’94] [Dymarsky, Lashkari, Liu ’16]
[at strong coupling these are black black hole microstates]
Comment on Heavy States What do we expect for heavy states? •
Generic heavy states look thermal for few-body observables (Eigenstate Thermalization Hypothesis) [Deutsch ’91, Srednicki ’94] [Dymarsky, Lashkari, Liu ’16]
[at strong coupling these are black black hole microstates] •
In this talk we consider a special heavy state that is not thermal and does not correspond to a black hole in the bulk.
Comment on Heavy States What do we expect for heavy states? •
Generic heavy states look thermal for few-body observables (Eigenstate Thermalization Hypothesis) [Deutsch ’91, Srednicki ’94] [Dymarsky, Lashkari, Liu ’16]
[at strong coupling these are black black hole microstates]
In this talk we consider a special heavy state that is not thermal and does not correspond to a black hole in the bulk.
•
They are particularly simple from the bootstrap point of view.
Setup •
Consider a CFT with continuous global symmetries in d>2
Setup •
Consider a CFT with continuous global symmetries in d>2
•
For simplicity we can focus on U(1)
Setup •
Consider a CFT with continuous global symmetries in d>2
•
For simplicity we can focus on U(1)
•
We are interested in the lightest state of large charge Q OQ ,
min (Q)
Q
1
Setup •
Consider a CFT with continuous global symmetries in d>2
•
For simplicity we can focus on U(1)
•
We are interested in the lightest state of large charge Q OQ ,
min (Q)
Q
How do we describe such a state?
1
Setup •
Consider a CFT with continuous global symmetries in d>2
•
For simplicity we can focus on U(1)
•
We are interested in the lightest state of large charge Q OQ ,
min (Q)
Q
1
How do we describe such a state? Not much is known about such states in a generic CFT.
Large Q limit
Large Q limit Basic set of observables is GQ (zi , z¯i ) ⌘ hOQ (0)Oq1 (z1 , z¯1 ) . . . Oqn (zn , z¯n )Oqn+1 (1, 1)O
Q (1)i
Large Q limit Basic set of observables is GQ (zi , z¯i ) ⌘ hOQ (0)Oq1 (z1 , z¯1 ) . . . Oqn (zn , z¯n )Oqn+1 (1, 1)O
In the large Q limit we have lim
Q!1
min (Q)
!1
Q (1)i
Large Q limit Basic set of observables is GQ (zi , z¯i ) ⌘ hOQ (0)Oq1 (z1 , z¯1 ) . . . Oqn (zn , z¯n )Oqn+1 (1, 1)O
Q (1)i
In the large Q limit we have lim
min (Q)
Q!1
!1
The correlator is dominated by the ``vacuum’’+excitations Q
G (zi , z¯i ) ⇠ e
Pn
i=1
Pi min (Q+ k=1 qk )|⌧i+1 ⌧i |
Large Q limit Basic set of observables is GQ (zi , z¯i ) ⌘ hOQ (0)Oq1 (z1 , z¯1 ) . . . Oqn (zn , z¯n )Oqn+1 (1, 1)O
Q (1)i
In the large Q limit we have lim
min (Q)
Q!1
!1
The correlator is dominated by the ``vacuum’’+excitations Q
G (zi , z¯i ) ⇠ e
Pn
i=1
Assume: Large Q limit exists
Pi min (Q+ k=1 qk )|⌧i+1 ⌧i |
Large Q limit Basic set of observables is GQ (zi , z¯i ) ⌘ hOQ (0)Oq1 (z1 , z¯1 ) . . . Oqn (zn , z¯n )Oqn+1 (1, 1)O
Q (1)i
In the large Q limit we have lim
min (Q)
Q!1
!1
The correlator is dominated by the ``vacuum’’+excitations Q
G (zi , z¯i ) ⇠ e
Pn
i=1
Assume: Large Q limit exists
Pi min (Q+ k=1 qk )|⌧i+1 ⌧i | [Caron-Huot ’17] [Alday, Caron-Huot ’17]
Part I: Effective Field Theory ( O(2) WF ) [Hellerman, Orlando, Reffert, Watanabe ’15]
Part I: Effective Field Theory ( O(2) WF ) [Hellerman, Orlando, Reffert, Watanabe ’15]
A self-consistent proposal to solve crossing equations perturbatively to arbitrary order in 1/Q.
Part I: Effective Field Theory ( O(2) WF ) [Hellerman, Orlando, Reffert, Watanabe ’15]
A self-consistent proposal to solve crossing equations perturbatively to arbitrary order in 1/Q. (similar story in AdS) [Heemskerk, Penedones, Polchinski, Sully ’09]
Part I: Effective Field Theory ( O(2) WF ) [Hellerman, Orlando, Reffert, Watanabe ’15]
A self-consistent proposal to solve crossing equations perturbatively to arbitrary order in 1/Q. (similar story in AdS) [Heemskerk, Penedones, Polchinski, Sully ’09]
Part II: Bootstrap •
other solutions
•
uniqueness
Effective Field Theory
Effective Field Theory for O(2) WF-model [Hellerman, Orlando, Reffert, Watanabe 15’]
Large Q limit of O(2) WF model is controlled by a conformally invariant effective Lagrangian for a Goldstone boson
hOQ Oq1 . . . Oqn O
Qi
=
Z
D Oq 1 . . . Oq n e
SEF T
Effective Field Theory for O(2) WF-model [Hellerman, Orlando, Reffert, Watanabe 15’]
Large Q limit of O(2) WF model is controlled by a conformally invariant effective Lagrangian for a Goldstone boson
hOQ Oq1 . . . Oqn O
Qi
=
Z
D Oq 1 . . . Oq n e
SEF T
Upshot: One can systematically compute correlators in the 1/Q expansion.
Effective Field Theory for O(2) WF-model [Hellerman, Orlando, Reffert, Watanabe 15’]
Large Q limit of O(2) WF model is controlled by a conformally invariant effective Lagrangian for a Goldstone boson
hOQ Oq1 . . . Oqn O
Qi
=
Z
D Oq 1 . . . Oq n e
SEF T
Upshot: One can systematically compute correlators in the 1/Q expansion. Bootstrap: Generates a nontrivial solution to crossing (large N, AdS actions)
Effective Field Theory for O(2) WF-model [Hellerman, Orlando, Reffert, Watanabe 15’]
Large Q limit of O(2) WF model is controlled by a conformally invariant effective Lagrangian for a Goldstone boson
hOQ Oq1 . . . Oqn O
Qi
=
Z
action
D Oq 1 . . . Oq n e
SEF T
Upshot: One can systematically compute correlators in the 1/Q expansion. Bootstrap: Generates a nontrivial solution to crossing (large N, AdS actions)
Effective Field Theory for O(2) WF-model [Hellerman, Orlando, Reffert, Watanabe 15’]
Large Q limit of O(2) WF model is controlled by a conformally invariant effective Lagrangian for a Goldstone boson
hOQ Oq1 . . . Oqn O
Qi
=
Z
action
D Oq 1 . . . Oq n e
SEF T light operators
Upshot: One can systematically compute correlators in the 1/Q expansion. Bootstrap: Generates a nontrivial solution to crossing (large N, AdS actions)
Effective Field Theory for O(2) WF-model [Hellerman, Orlando, Reffert, Watanabe 15’]
Large Q limit of O(2) WF model is controlled by a conformally invariant effective Lagrangian for a Goldstone boson
hOQ Oq1 . . . Oqn O
Qi
=
Z
action
D Oq 1 . . . Oq n e integral
SEF T light operators
Upshot: One can systematically compute correlators in the 1/Q expansion. Bootstrap: Generates a nontrivial solution to crossing (large N, AdS actions)
Effective Field Theory for O(2) WF-model [Hellerman, Orlando, Reffert, Watanabe 15’]
Effective Field Theory for O(2) WF-model [Hellerman, Orlando, Reffert, Watanabe 15’]
This EFT captures physics for distances r 1 Q p ⌧⇤⌧ ⇢= R 4⇡R2 and is conformally invariant.
Effective Field Theory for O(2) WF-model [Hellerman, Orlando, Reffert, Watanabe 15’]
This EFT captures physics for distances r 1 Q p ⌧⇤⌧ ⇢= R 4⇡R2 and is conformally invariant. (Certain region in the space of cross ratios)
Effective Field Theory [Hellerman, Orlando, Reffert, Watanabe ’15]
Effective Field Theory [Hellerman, Orlando, Reffert, Watanabe ’15]
The leading order terms takes the form [ ]=0
U (1) :
!
+c
|@ | ⌘ ( g µ⌫ @µ @⌫ )1/2
Effective Field Theory [Hellerman, Orlando, Reffert, Watanabe ’15]
The leading order terms takes the form [ ]=0 SE =
Z
3
p
d x g
✓
U (1) : 1 3 |@ | 12⇡↵2
!
+c ✓
|@ | ⌘ ( g µ⌫ @µ @⌫ )1/2 µ
(@µ |@ |)(@ |@ |) |@ | R + 2 8⇡↵ |@ |2 Q ⇢= 4⇡R2
◆
+ i⇢
Z
+ ... 3
◆
p
+
d x g˙
Effective Field Theory [Hellerman, Orlando, Reffert, Watanabe ’15]
The leading order terms takes the form [ ]=0 SE =
Z
3
p
d x g
✓
U (1) : 1 3 |@ | 12⇡↵2
!
+c
|@ | ⌘ ( g µ⌫ @µ @⌫ )1/2
✓
µ
(@µ |@ |)(@ |@ |) |@ | R + 2 8⇡↵ |@ |2 Q ⇢= 4⇡R2
◆
+ i⇢
Z
+ ... 3
¯=
3/2
p
+
d x g˙
The saddle point equation takes the form iµ⌧ + 0 p µR = ↵ Q + p + O(Q 2 Q
◆
)
Effective Field Theory
Effective Field Theory We introduce the Goldstone field (x) =
and compute the action.
↵ 1 iµ⌧ + p ⇡(x) 2 µ
Effective Field Theory We introduce the Goldstone field (x) =
and compute the action. SE =
↵ 1 iµ⌧ + p ⇡(x) 2 µ
Q
(⌧out ⌧in ) + S⇡ , R Z ✓ ◆ 1 1 3 p 2 2 S⇡ = d x g ⇡˙ + (@i ⇡) + 16⇡ 2 ✓ ◆ Z i 1 3 3 p 3 2 p + d x g ⇡ ˙ + ⇡(@ ˙ ⇡) + O(Q i 2 96⇡ ↵ Q3/4 p 2 3/2 1/2 + Q + C + O(Q ) Q = ↵Q 3
1
),
Effective Field Theory We introduce the Goldstone field (x) =
and compute the action. SE =
↵ 1 iµ⌧ + p ⇡(x) 2 µ
Q
(⌧out ⌧in ) + S⇡ , R Z ✓ ◆ 1 1 3 p 2 2 S⇡ = d x g ⇡˙ + (@i ⇡) + 16⇡ 2 ✓ ◆ Z i 1 3 3 p 3 2 p + d x g ⇡ ˙ + ⇡(@ ˙ ⇡) + O(Q i 2 96⇡ ↵ Q3/4 p 2 3/2 1/2 + Q + C + O(Q ) Q = ↵Q 3 vacuum energy
one-loop determinant
1
),
Effective Field Theory [Monin, Pirtskhalava, Rattazzi, Seibold 16’]
Effective Field Theory [Monin, Pirtskhalava, Rattazzi, Seibold 16’]
For the light operators we write Oq = cq |@ | = cq µ
q
e
µq⌧
q
eiq + cR q R|@ | iq↵ 1 1 + p ⇡(⌧, n) + 2 µ 2
✓
q
2 iq
iq↵ p 2 µ
e
+ ...
◆2
!
⇡ 2 (⌧, n)
Effective Field Theory [Monin, Pirtskhalava, Rattazzi, Seibold 16’]
For the light operators we write Oq = cq |@ | = cq µ
q
e
µq⌧
Now let’s compute!
q
eiq + cR q R|@ | iq↵ 1 1 + p ⇡(⌧, n) + 2 µ 2
✓
q
2 iq
iq↵ p 2 µ
e
+ ...
◆2
!
⇡ 2 (⌧, n)
Effective Field Theory [Monin, Pirtskhalava, Rattazzi, Seibold 16’]
Effective Field Theory [Monin, Pirtskhalava, Rattazzi, Seibold 16’]
The four-point function takes the form G(z, z¯) ⌘ hOQ (0)O gq (z, z¯) ⌘ (z z¯)
1 2
q
¯)Oq (1)O Q (1)i q (z, z
G(z, z¯)
Effective Field Theory [Monin, Pirtskhalava, Rattazzi, Seibold 16’]
The four-point function takes the form G(z, z¯) ⌘ hOQ (0)O gq (z, z¯) ⌘ (z z¯)
1 2
q
¯)Oq (1)O Q (1)i q (z, z
G(z, z¯)
Crossing becomes gq (z, z¯) = g
q
✓
1 1 , z z¯
◆
Effective Field Theory [Monin, Pirtskhalava, Rattazzi, Seibold 16’]
The four-point function takes the form G(z, z¯) ⌘ hOQ (0)O gq (z, z¯) ⌘ (z z¯)
1 2
q
¯)Oq (1)O Q (1)i q (z, z
G(z, z¯)
Crossing becomes gq (z, z¯) = g
2⌧
z z¯ = e ,
q
z + z¯ p = cos ✓ ⌘ x 2 z z¯
✓
1 1 , z z¯
◆
Effective Field Theory The result to leading order takes the form gq (z, z¯) = cq c
q↵
2
q
Q
q
e
p
↵q Q⌧
D(⌧, x) ⌘ h⇡(0, n2 )⇡(⌧, n1 )i
✓
2
1
q ↵ q p ⌧ + p D(⌧, x) 2 Q 4 Q
◆
Effective Field Theory The result to leading order takes the form gq (z, z¯) = cq c
q↵
2
q
Q
q
e
p
↵q Q⌧
✓
2
1
q ↵ q p ⌧ + p D(⌧, x) 2 Q 4 Q
D(⌧, x) ⌘ h⇡(0, n2 )⇡(⌧, n1 )i ✓
D(⌧, x) =
@⌧2
1 + 4S 2 2
◆
D(⌧, x) =
1 X 2J + 1 |⌧ | + e ⌦J J=1
⌦J |⌧ |
4 (⌧ ) (1
PJ (x)
x)
⌦J =
r
J(J + 1) 2
◆
Effective Field Theory The result to leading order takes the form gq (z, z¯) = cq c
q↵
2
q
Q
q
p
↵q Q⌧
e
✓
2
1
q ↵ q p ⌧ + p D(⌧, x) 2 Q 4 Q
◆
D(⌧, x) ⌘ h⇡(0, n2 )⇡(⌧, n1 )i ✓
D(⌧, x) =
@⌧2
1 + 4S 2 2
◆
D(⌧, x) =
1 X 2J + 1 |⌧ | + e ⌦J J=1
Speed of sound
2 cs
=
1 d
1
⌦J |⌧ |
4 (⌧ ) (1
PJ (x)
x)
⌦J =
r
J(J + 1) 2
is fixed by conformal invariance
Crossing
gq (z, z¯) = cq c
q↵
2
q
Q
q
e
p
↵q Q⌧
✓
2
1
q ↵ q p ⌧ + p D(⌧, x) 2 Q 4 Q
◆
Crossing
gq (z, z¯) = cq c
q↵
2
q
Q
q
e
p
↵q Q⌧
✓
2
1
q ↵ q p ⌧ + p D(⌧, x) 2 Q 4 Q
Crossing transformation acts as follows ⌧! D(⌧, x) =
q!
⌧
1 X 2J + 1 |⌧ | + e ⌦J J=1
q ⌦J |⌧ |
PJ (x)
◆
Crossing
gq (z, z¯) = cq c
q↵
2
q
Q
q
e
p
↵q Q⌧
✓
2
1
q ↵ q p ⌧ + p D(⌧, x) 2 Q 4 Q
Crossing transformation acts as follows ⌧! D(⌧, x) =
q!
⌧
1 X 2J + 1 |⌧ | + e ⌦J
q ⌦J |⌧ |
J=1
Analyticity at ✓
2 @⌧
⌧ =0
1 + 4S 2 2
◆
D(⌧, x) =
4 (⌧ ) (1
x)
PJ (x)
◆
OPE
OPE The leading correction takes the form gqEF T (z, z¯)
= cq c
q↵
2
q
(z z¯)
↵ 2q
p
Q
✓
1 1+ p 4 Q
✓
↵q q+ 2 !
1 X 1 ↵ q 2J + 1 + p (z z¯) 2 ⌦J PJ (x) + O Q 4 Q ⌦J J=1 2
2
1
◆
log(z z¯)
OPE The leading correction takes the form leading primary
gqEF T (z, z¯)
= cq c
q↵
2
q
(z z¯)
↵ 2q
p
Q
✓
1 1+ p 4 Q
✓
↵q q+ 2 !
1 X 1 ↵ q 2J + 1 + p (z z¯) 2 ⌦J PJ (x) + O Q 4 Q ⌦J J=1 2
2
1
◆
log(z z¯)
OPE The leading correction takes the form leading primary
gqEF T (z, z¯)
= cq c
q↵
2
q
(z z¯)
↵ 2q
p
Q
✓
anomalous dimension
1 1+ p 4 Q
✓
↵q q+ 2 !
1 X 1 ↵ q 2J + 1 + p (z z¯) 2 ⌦J PJ (x) + O Q 4 Q ⌦J J=1 2
2
1
◆
log(z z¯)
OPE The leading correction takes the form leading primary
gqEF T (z, z¯)
= cq c
q↵
2
q
(z z¯)
↵ 2q
p
Q
✓
anomalous dimension
1 1+ p 4 Q
✓
↵q q+ 2 !
1 X 1 ↵ q 2J + 1 + p (z z¯) 2 ⌦J PJ (x) + O Q 4 Q ⌦J J=1 2
2
1
descendant and new primaries
◆
log(z z¯)
OPE The leading correction takes the form leading primary
gqEF T (z, z¯)
= cq c
q↵
2
q
(z z¯)
↵ 2q
p
Q
✓
anomalous dimension
1 1+ p 4 Q
✓
↵q q+ 2 !
1 X 1 ↵ q 2J + 1 + p (z z¯) 2 ⌦J PJ (x) + O Q 4 Q ⌦J J=1 2
2
◆
1
descendant and new primaries
EFT field
Regge trajectory
log(z z¯)
OPE The leading correction takes the form leading primary
gqEF T (z, z¯)
= cq c
q↵
2
q
(z z¯)
↵ 2q
p
Q
✓
anomalous dimension
1 1+ p 4 Q
✓
↵q q+ 2 !
1 X 1 ↵ q 2J + 1 + p (z z¯) 2 ⌦J PJ (x) + O Q 4 Q ⌦J J=1 2
2
◆
1
descendant and new primaries
EFT field
Regge trajectory
More generally, the spectrum contains operators r 1 X J(J + 1) ~ n nJ ⌦J , ⌦J = Q+ Q = 2 J=1
log(z z¯)
Bootstrap
Setup We consider heavy-light-light-heavy four-point function
G(z, z¯) ⌘ hOQ (0)O gq (z, z¯) ⌘ (z z¯)
1 2
q
¯)Oq (1)O Q (1)i q (z, z
G(z, z¯)
Setup We consider heavy-light-light-heavy four-point function
G(z, z¯) ⌘ hOQ (0)O gq (z, z¯) ⌘ (z z¯)
1 2
q
¯)Oq (1)O Q (1)i q (z, z
G(z, z¯)
s: 12-34 t: 13-24 u: 14-23
Lessons from EFT From the EFT answer we see:
Lessons from EFT From the EFT answer we see: Finite number of Regge trajectories contribute
Lessons from EFT From the EFT answer we see: Finite number of Regge trajectories contribute s- and u-channel OPE do not have an overlapping convergence region (within EFT)
Lessons from EFT From the EFT answer we see: Finite number of Regge trajectories contribute s- and u-channel OPE do not have an overlapping convergence region (within EFT) t-channel is dominated by heavy operators (EFT does not describe short distance physics)
z-plane G(z, z¯) ⌘ hOQ (0)O
¯)Oq (1)O Q (1)i q (z, z
Bootstrap at Large Charge Correspondingly we have:
Bootstrap at Large Charge Correspondingly we have: Finite number of Regge trajectories contribute (assumption)
Bootstrap at Large Charge Correspondingly we have: Finite number of Regge trajectories contribute (assumption) s- and u-channel match smoothly (s=u crossing)
Bootstrap at Large Charge Correspondingly we have: Finite number of Regge trajectories contribute (assumption) s- and u-channel match smoothly (s=u crossing) macroscopic limit of correlators exist (alternative to t-channel)
Macroscopic Limit of Correlators
Macroscopic Limit of Correlators Consider a state on the cylinder created by an operator of dimension and charge Q
Macroscopic Limit of Correlators Consider a state on the cylinder created by an operator of dimension and charge Q We denote the sphere radius by R
Macroscopic Limit of Correlators Consider a state on the cylinder created by an operator of dimension and charge Q We denote the sphere radius by R Macroscopic limit: R!1 !1
Macroscopic Limit of Correlators Consider a state on the cylinder created by an operator of dimension and charge Q We denote the sphere radius by R Macroscopic limit: R!1 !1
such that correlation functions of light operators in some finite region L stay finite.
Thermodynamic Limit of Correlators
Thermodynamic Limit of Correlators
The usual thermodynamic limit corresponds to energy and charge density kept fixed lim
!1,R!1
Rd
=✏
Q lim = q Q!1,R!1 Rd 1 [Dymarsky, Lashkari, Liu]
Thermodynamic Limit of Correlators
The usual thermodynamic limit corresponds to energy and charge density kept fixed lim
!1,R!1
Rd
E=
=✏
Q lim = q Q!1,R!1 Rd 1 [Dymarsky, Lashkari, Liu]
R
Thermodynamic Limit of Correlators
The usual thermodynamic limit corresponds to energy and charge density kept fixed lim
!1,R!1
Rd
E=
=✏
Q lim = q Q!1,R!1 Rd 1 [Dymarsky, Lashkari, Liu]
R
This limit is expected to exist generically, but in some very special cases it definitely does not exist. (CFTs with moduli space of vacua)
Macroscopic Limit of Correlators
Macroscopic Limit of Correlators A more familiar way to put it is as follows GQ (zi , z¯i ) ⌘ hOQ (0)Oq1 (z1 , z¯1 ) . . . Oqn (zn , z¯n )Oqn+1 (1, 1)O
zi = 1
wi 1 d
H
,
z¯i = 1
w ¯i 1 d
H
Q (1)i
Macroscopic Limit of Correlators A more familiar way to put it is as follows GQ (zi , z¯i ) ⌘ hOQ (0)Oq1 (z1 , z¯1 ) . . . Oqn (zn , z¯n )Oqn+1 (1, 1)O
zi = 1
wi 1 d
,
z¯i = 1
H
✏
G (wi , w ¯i ) ⌘
lim
H !1
1 d
H
Q (1)i
w ¯i 1 d
H
Pn+1 i=1
Li
G 1
wi 1/d H
,1
w ¯i 1/d H
!
Macroscopic Limit of Correlators A more familiar way to put it is as follows GQ (zi , z¯i ) ⌘ hOQ (0)Oq1 (z1 , z¯1 ) . . . Oqn (zn , z¯n )Oqn+1 (1, 1)O
zi = 1
wi 1 d
,
z¯i = 1
H
✏
G (wi , w ¯i ) ⌘
lim
H !1
1 d
H
Q (1)i
w ¯i 1 d
H
Pn+1 i=1
Li
G 1
wi 1/d H
,1
w ¯i 1/d H
!
Existence of this limit is a nontrivial property of the OPE
Macroscopic Limit of Correlators A more familiar way to put it is as follows GQ (zi , z¯i ) ⌘ hOQ (0)Oq1 (z1 , z¯1 ) . . . Oqn (zn , z¯n )Oqn+1 (1, 1)O
zi = 1
wi 1 d
,
w ¯i
z¯i = 1
1 d
H
✏
G (wi , w ¯i ) ⌘
lim
H !1
1 d
H
Q (1)i
H
Pn+1 i=1
Li
G 1
wi 1/d H
,1
w ¯i 1/d H
!
Existence of this limit is a nontrivial property of the OPE
cHH † L c0
L d
H
Macroscopic Limit of Correlators For the four-point function existence of the macroscopic limit constrains short distance behavior of the large Q correlator ✓ ◆ w w ¯ ✏,q q G (w, w) ¯ = lim Q G 1 p ,1 p Q!1 Q Q z=1
w p Q
G(z, z¯) ⌘ hOQ (0)O
z=1
w ¯ p Q
¯)Oq (1)O Q (1)i q (z, z
Macroscopic Limit of Correlators For the four-point function existence of the macroscopic limit constrains short distance behavior of the large Q correlator ✓ ◆ w w ¯ ✏,q q G (w, w) ¯ = lim Q G 1 p ,1 p Q!1 Q Q z=1
w p Q
G(z, z¯) ⌘ hOQ (0)O
z=1
w ¯ p Q
¯)Oq (1)O Q (1)i q (z, z
We combine this with the s- and u- channel OPE, smoothness of the correlator and finite number of Regge trajectories assumption.
Sum Rule We write condition for smoothness of the correlator. The contribution of one Regge trajectory to the correlator takes the form 1 X ✏J |⌧ | cJ e PJ (x) gq (⌧, x) = J=0
s/u-channel conformal block in the large Q limit
For ⌧ > 0 this is the s-channel expansion For ⌧ < 0 this is the u-channel expansion
Sum Rule We write condition for smoothness of the correlator. The contribution of one Regge trajectory to the correlator takes the form 1 X ✏J |⌧ | cJ e PJ (x) gq (⌧, x) = J=0
s/u-channel conformal block in the large Q limit
For ⌧ > 0 this is the s-channel expansion For ⌧ < 0 this is the u-channel expansion Crossing is invariance under ✓ ◆ 1 1 gq (z, z¯) = g q , z z¯
⌧!
⌧ 2⌧
z z¯ = e ,
z + z¯ p = cos ✓ ⌘ x 2 z z¯
Sum Rule The correlator of this form, however, looks singular at ⌧ =0 1 X ✏J |⌧ | c e PJ (x) gq (⌧, x) = J J=0
Sum Rule The correlator of this form, however, looks singular at ⌧ =0 1 X ✏J |⌧ | c e PJ (x) gq (⌧, x) = J J=0
Physically, nothing special happens at non-coincident points. Non-analyticity, therefore, should cancel in the physical correlator.
Sum Rule The correlator of this form, however, looks singular at ⌧ =0 1 X ✏J |⌧ | c e PJ (x) gq (⌧, x) = J J=0
Physically, nothing special happens at non-coincident points. Non-analyticity, therefore, should cancel in the physical correlator. It is convenient to express it as follows cJ ✏2n+1 J
=
(2J + 1) lim
✏!0
Z
⇡
d✓ sin ✓ 0
⇥
@⌧2n+1
gq (✏, ✓)
@⌧2n+1
g
q(
⇤
✏, ✓) PJ (cos ✓)
z-plane G(z, z¯) ⌘ hOQ (0)O
¯)Oq (1)O Q (1)i q (z, z
Sum Rule cJ ✏2n+1 J
=
(2J + 1) lim
✏!0
Z
⇡
d✓ sin ✓ 0
⇥
@⌧2n+1
gq (✏, ✓)
@⌧2n+1
g
q(
⇤
✏, ✓) PJ (cos ✓)
The crucial observation is that nontrivial contribution could only come from short distances ✓ ⌧ 1 In this region the behavior of the correlator is dictated by existence of the macroscopic limit. [Pappadopulo, Rychkov, Espin, Rattazzi]
Sum Rule cJ ✏2n+1 J
=
(2J + 1) lim
✏!0
Z
⇡
d✓ sin ✓ 0
⇥
@⌧2n+1
gq (✏, ✓)
@⌧2n+1
g
q(
⇤
✏, ✓) PJ (cos ✓)
The crucial observation is that nontrivial contribution could only come from short distances ✓ ⌧ 1 In this region the behavior of the correlator is dictated by existence of the macroscopic limit. [Pappadopulo, Rychkov, Espin, Rattazzi]
For N Regge trajectories we would have in the LHS N X i=1
2n+1 cJ,i ✏J,i
Bootstrap Equations
As a result we get an infinite set of bootstrap equations n,0 J,0
+3
J,1
+
N X
2n+1 cJ,i ✏J,i
= (2J +
i=1
⌦2J
2 1)Wn (⌦J )
J(J + 1) ⌘ 2
n = 0, 1, 2, ...
Bootstrap Equations
As a result we get an infinite set of bootstrap equations n,0 J,0
+3
J,1
+
N X
2n+1 cJ,i ✏J,i
= (2J +
i=1
⌦2J
2 1)Wn (⌦J )
J(J + 1) ⌘ 2
n = 0, 1, 2, ... degree n polynomial
(expansion of Legendre polynomials)
Bootstrap Equations
As a result we get an infinite set of bootstrap equations n,0 J,0
+3
J,1
+
N X
2n+1 cJ,i ✏J,i
= (2J +
i=1
descendant
⌦2J
2 1)Wn (⌦J )
J(J + 1) ⌘ 2
n = 0, 1, 2, ... degree n polynomial
(expansion of Legendre polynomials)
Bootstrap Equations
As a result we get an infinite set of bootstrap equations n,0 J,0
+3
J,1
+
N X
2n+1 cJ,i ✏J,i
= (2J +
i=1
anomalous dimension of the heavy operator
descendant
⌦2J
2 1)Wn (⌦J )
J(J + 1) ⌘ 2
n = 0, 1, 2, ... degree n polynomial
(expansion of Legendre polynomials)
Bootstrap Equations
As a result we get an infinite set of bootstrap equations n,0 J,0
+3
J,1
+
N X
2n+1 cJ,i ✏J,i
= (2J +
i=1
anomalous dimension of the heavy operator
descendant
⌦2J
2 1)Wn (⌦J )
J(J + 1) ⌘ 2
n = 0, 1, 2, ... degree n polynomial
(expansion of Legendre polynomials)
Unknowns: •
LHS: anomalous dimensions and three-point couplings
•
RHS: coefficients of the polynomial
Solutions
Solutions For one Regge trajectory the solution is unique and coincides with the EFT result ✏J = ⌦J 2J + 1 cJ = ⌦J
Solutions For one Regge trajectory the solution is unique and coincides with the EFT result ✏J = ⌦J 2J + 1 cJ = ⌦J
For N Regge trajectories the solution is given in terms of roots of a certain polynomial N Y
i=1
(x
✏i (z)) = xN
P1 (z)xN
deg(Pk ) = k
1
+ P2 (z)xN
2
+ · · · + ( 1)N PN (z) = 0
z ⌘ ⌦2J
and similarly a simple formula for three-point functions.
Solutions For one Regge trajectory the solution is unique and coincides with the EFT result ✏J = ⌦J 2J + 1 cJ = ⌦J
For N Regge trajectories the solution is given in terms of roots of a certain polynomial N Y
i=1
(x
✏i (z)) = xN
P1 (z)xN
deg(Pk ) = k
1
+ P2 (z)xN
z ⌘ ⌦2J
2
+ · · · + ( 1)N PN (z) = 0 [Girard, Newton 17th century]
and similarly a simple formula for three-point functions.
Example, N=2 Regge Trajectories
z⌘
2 ⌦J
J(J + 1) = 2
✏1 (z) = az + b + ✏2 (z) = az + b
n ✏1 (z)
+
n ✏2 (z)
p p
=
(cz + d)(ez + f ), (cz + d)(ez + f ).
n X
k=0
ak z
k
Solutions • Some of the solutions look like weakly coupled EFTs with extra particles
2 ✏J
=
2 2 c s ⌦J
+m
2
Solutions • Some of the solutions look like weakly coupled EFTs with extra particles
2 ✏J
=
2 2 c s ⌦J
+m
2
• Other solutions do not admit simple weakly coupled Lagrangian interpretation (for N>4 could not be generically written in terms of radicals) [Abel-Ruffini]
Conclusions
At large global charge Q crossing equations admit a non-trivial and solvable truncation that involves finite number of Regge trajectories •
At large global charge Q crossing equations admit a non-trivial and solvable truncation that involves finite number of Regge trajectories •
We classified the solutions to leading order. The OPE data is encoded in roots of a certain polynomial •
At large global charge Q crossing equations admit a non-trivial and solvable truncation that involves finite number of Regge trajectories •
We classified the solutions to leading order. The OPE data is encoded in roots of a certain polynomial •
Some solutions admit weakly coupled EFT description, while others do not (Regge trajectory = EFT field) •
At large global charge Q crossing equations admit a non-trivial and solvable truncation that involves finite number of Regge trajectories •
We classified the solutions to leading order. The OPE data is encoded in roots of a certain polynomial •
Some solutions admit weakly coupled EFT description, while others do not (Regge trajectory = EFT field) •
Goldstone EFT is the unique solution for one Regge trajectory •
•
What happens in real CFTs?
(help from numerics is needed; how large is ``large charge’’?)
•
What happens in real CFTs?
(help from numerics is needed; how large is ``large charge’’?)
•
Which solutions survive to arbitrary order in 1/Q?
•
What happens in real CFTs?
(help from numerics is needed; how large is ``large charge’’?)
•
Which solutions survive to arbitrary order in 1/Q?
•
Are there non-trivial examples beyond our assumptions?
(large degeneracy = WGC, extremal RN BHs)
Bootstrap for generic heavy operators (macroscopic limits/ETH) •
Bootstrap for generic heavy operators (macroscopic limits/ETH) •
•
CFTs with moduli spaces
Bootstrap for generic heavy operators (macroscopic limits/ETH) •
•
CFTs with moduli spaces
•
Analytic bootstrap in the macroscopic limit
Alexander Zhiboedov, Harvard U Simons Foundation, 2017
with Daniel Jafferis and Baur Mukhametzhanov
Approaches to Bootstrap This talk: bootstrap for heavy operators of low spin (high twist) CFT d>2
Approaches to Bootstrap This talk: bootstrap for heavy operators of low spin (high twist) CFT d>2
⌧=
J
Comment on Heavy States
Comment on Heavy States What do we expect for heavy states?
Comment on Heavy States What do we expect for heavy states? •
Generic heavy states look thermal for few-body observables (Eigenstate Thermalization Hypothesis) [Deutsch ’91, Srednicki ’94] [Dymarsky, Lashkari, Liu ’16]
Comment on Heavy States What do we expect for heavy states? •
Generic heavy states look thermal for few-body observables (Eigenstate Thermalization Hypothesis) [Deutsch ’91, Srednicki ’94] [Dymarsky, Lashkari, Liu ’16]
[at strong coupling these are black black hole microstates]
Comment on Heavy States What do we expect for heavy states? •
Generic heavy states look thermal for few-body observables (Eigenstate Thermalization Hypothesis) [Deutsch ’91, Srednicki ’94] [Dymarsky, Lashkari, Liu ’16]
[at strong coupling these are black black hole microstates] •
In this talk we consider a special heavy state that is not thermal and does not correspond to a black hole in the bulk.
Comment on Heavy States What do we expect for heavy states? •
Generic heavy states look thermal for few-body observables (Eigenstate Thermalization Hypothesis) [Deutsch ’91, Srednicki ’94] [Dymarsky, Lashkari, Liu ’16]
[at strong coupling these are black black hole microstates]
In this talk we consider a special heavy state that is not thermal and does not correspond to a black hole in the bulk.
•
They are particularly simple from the bootstrap point of view.
Setup •
Consider a CFT with continuous global symmetries in d>2
Setup •
Consider a CFT with continuous global symmetries in d>2
•
For simplicity we can focus on U(1)
Setup •
Consider a CFT with continuous global symmetries in d>2
•
For simplicity we can focus on U(1)
•
We are interested in the lightest state of large charge Q OQ ,
min (Q)
Q
1
Setup •
Consider a CFT with continuous global symmetries in d>2
•
For simplicity we can focus on U(1)
•
We are interested in the lightest state of large charge Q OQ ,
min (Q)
Q
How do we describe such a state?
1
Setup •
Consider a CFT with continuous global symmetries in d>2
•
For simplicity we can focus on U(1)
•
We are interested in the lightest state of large charge Q OQ ,
min (Q)
Q
1
How do we describe such a state? Not much is known about such states in a generic CFT.
Large Q limit
Large Q limit Basic set of observables is GQ (zi , z¯i ) ⌘ hOQ (0)Oq1 (z1 , z¯1 ) . . . Oqn (zn , z¯n )Oqn+1 (1, 1)O
Q (1)i
Large Q limit Basic set of observables is GQ (zi , z¯i ) ⌘ hOQ (0)Oq1 (z1 , z¯1 ) . . . Oqn (zn , z¯n )Oqn+1 (1, 1)O
In the large Q limit we have lim
Q!1
min (Q)
!1
Q (1)i
Large Q limit Basic set of observables is GQ (zi , z¯i ) ⌘ hOQ (0)Oq1 (z1 , z¯1 ) . . . Oqn (zn , z¯n )Oqn+1 (1, 1)O
Q (1)i
In the large Q limit we have lim
min (Q)
Q!1
!1
The correlator is dominated by the ``vacuum’’+excitations Q
G (zi , z¯i ) ⇠ e
Pn
i=1
Pi min (Q+ k=1 qk )|⌧i+1 ⌧i |
Large Q limit Basic set of observables is GQ (zi , z¯i ) ⌘ hOQ (0)Oq1 (z1 , z¯1 ) . . . Oqn (zn , z¯n )Oqn+1 (1, 1)O
Q (1)i
In the large Q limit we have lim
min (Q)
Q!1
!1
The correlator is dominated by the ``vacuum’’+excitations Q
G (zi , z¯i ) ⇠ e
Pn
i=1
Assume: Large Q limit exists
Pi min (Q+ k=1 qk )|⌧i+1 ⌧i |
Large Q limit Basic set of observables is GQ (zi , z¯i ) ⌘ hOQ (0)Oq1 (z1 , z¯1 ) . . . Oqn (zn , z¯n )Oqn+1 (1, 1)O
Q (1)i
In the large Q limit we have lim
min (Q)
Q!1
!1
The correlator is dominated by the ``vacuum’’+excitations Q
G (zi , z¯i ) ⇠ e
Pn
i=1
Assume: Large Q limit exists
Pi min (Q+ k=1 qk )|⌧i+1 ⌧i | [Caron-Huot ’17] [Alday, Caron-Huot ’17]
Part I: Effective Field Theory ( O(2) WF ) [Hellerman, Orlando, Reffert, Watanabe ’15]
Part I: Effective Field Theory ( O(2) WF ) [Hellerman, Orlando, Reffert, Watanabe ’15]
A self-consistent proposal to solve crossing equations perturbatively to arbitrary order in 1/Q.
Part I: Effective Field Theory ( O(2) WF ) [Hellerman, Orlando, Reffert, Watanabe ’15]
A self-consistent proposal to solve crossing equations perturbatively to arbitrary order in 1/Q. (similar story in AdS) [Heemskerk, Penedones, Polchinski, Sully ’09]
Part I: Effective Field Theory ( O(2) WF ) [Hellerman, Orlando, Reffert, Watanabe ’15]
A self-consistent proposal to solve crossing equations perturbatively to arbitrary order in 1/Q. (similar story in AdS) [Heemskerk, Penedones, Polchinski, Sully ’09]
Part II: Bootstrap •
other solutions
•
uniqueness
Effective Field Theory
Effective Field Theory for O(2) WF-model [Hellerman, Orlando, Reffert, Watanabe 15’]
Large Q limit of O(2) WF model is controlled by a conformally invariant effective Lagrangian for a Goldstone boson
hOQ Oq1 . . . Oqn O
Qi
=
Z
D Oq 1 . . . Oq n e
SEF T
Effective Field Theory for O(2) WF-model [Hellerman, Orlando, Reffert, Watanabe 15’]
Large Q limit of O(2) WF model is controlled by a conformally invariant effective Lagrangian for a Goldstone boson
hOQ Oq1 . . . Oqn O
Qi
=
Z
D Oq 1 . . . Oq n e
SEF T
Upshot: One can systematically compute correlators in the 1/Q expansion.
Effective Field Theory for O(2) WF-model [Hellerman, Orlando, Reffert, Watanabe 15’]
Large Q limit of O(2) WF model is controlled by a conformally invariant effective Lagrangian for a Goldstone boson
hOQ Oq1 . . . Oqn O
Qi
=
Z
D Oq 1 . . . Oq n e
SEF T
Upshot: One can systematically compute correlators in the 1/Q expansion. Bootstrap: Generates a nontrivial solution to crossing (large N, AdS actions)
Effective Field Theory for O(2) WF-model [Hellerman, Orlando, Reffert, Watanabe 15’]
Large Q limit of O(2) WF model is controlled by a conformally invariant effective Lagrangian for a Goldstone boson
hOQ Oq1 . . . Oqn O
Qi
=
Z
action
D Oq 1 . . . Oq n e
SEF T
Upshot: One can systematically compute correlators in the 1/Q expansion. Bootstrap: Generates a nontrivial solution to crossing (large N, AdS actions)
Effective Field Theory for O(2) WF-model [Hellerman, Orlando, Reffert, Watanabe 15’]
Large Q limit of O(2) WF model is controlled by a conformally invariant effective Lagrangian for a Goldstone boson
hOQ Oq1 . . . Oqn O
Qi
=
Z
action
D Oq 1 . . . Oq n e
SEF T light operators
Upshot: One can systematically compute correlators in the 1/Q expansion. Bootstrap: Generates a nontrivial solution to crossing (large N, AdS actions)
Effective Field Theory for O(2) WF-model [Hellerman, Orlando, Reffert, Watanabe 15’]
Large Q limit of O(2) WF model is controlled by a conformally invariant effective Lagrangian for a Goldstone boson
hOQ Oq1 . . . Oqn O
Qi
=
Z
action
D Oq 1 . . . Oq n e integral
SEF T light operators
Upshot: One can systematically compute correlators in the 1/Q expansion. Bootstrap: Generates a nontrivial solution to crossing (large N, AdS actions)
Effective Field Theory for O(2) WF-model [Hellerman, Orlando, Reffert, Watanabe 15’]
Effective Field Theory for O(2) WF-model [Hellerman, Orlando, Reffert, Watanabe 15’]
This EFT captures physics for distances r 1 Q p ⌧⇤⌧ ⇢= R 4⇡R2 and is conformally invariant.
Effective Field Theory for O(2) WF-model [Hellerman, Orlando, Reffert, Watanabe 15’]
This EFT captures physics for distances r 1 Q p ⌧⇤⌧ ⇢= R 4⇡R2 and is conformally invariant. (Certain region in the space of cross ratios)
Effective Field Theory [Hellerman, Orlando, Reffert, Watanabe ’15]
Effective Field Theory [Hellerman, Orlando, Reffert, Watanabe ’15]
The leading order terms takes the form [ ]=0
U (1) :
!
+c
|@ | ⌘ ( g µ⌫ @µ @⌫ )1/2
Effective Field Theory [Hellerman, Orlando, Reffert, Watanabe ’15]
The leading order terms takes the form [ ]=0 SE =
Z
3
p
d x g
✓
U (1) : 1 3 |@ | 12⇡↵2
!
+c ✓
|@ | ⌘ ( g µ⌫ @µ @⌫ )1/2 µ
(@µ |@ |)(@ |@ |) |@ | R + 2 8⇡↵ |@ |2 Q ⇢= 4⇡R2
◆
+ i⇢
Z
+ ... 3
◆
p
+
d x g˙
Effective Field Theory [Hellerman, Orlando, Reffert, Watanabe ’15]
The leading order terms takes the form [ ]=0 SE =
Z
3
p
d x g
✓
U (1) : 1 3 |@ | 12⇡↵2
!
+c
|@ | ⌘ ( g µ⌫ @µ @⌫ )1/2
✓
µ
(@µ |@ |)(@ |@ |) |@ | R + 2 8⇡↵ |@ |2 Q ⇢= 4⇡R2
◆
+ i⇢
Z
+ ... 3
¯=
3/2
p
+
d x g˙
The saddle point equation takes the form iµ⌧ + 0 p µR = ↵ Q + p + O(Q 2 Q
◆
)
Effective Field Theory
Effective Field Theory We introduce the Goldstone field (x) =
and compute the action.
↵ 1 iµ⌧ + p ⇡(x) 2 µ
Effective Field Theory We introduce the Goldstone field (x) =
and compute the action. SE =
↵ 1 iµ⌧ + p ⇡(x) 2 µ
Q
(⌧out ⌧in ) + S⇡ , R Z ✓ ◆ 1 1 3 p 2 2 S⇡ = d x g ⇡˙ + (@i ⇡) + 16⇡ 2 ✓ ◆ Z i 1 3 3 p 3 2 p + d x g ⇡ ˙ + ⇡(@ ˙ ⇡) + O(Q i 2 96⇡ ↵ Q3/4 p 2 3/2 1/2 + Q + C + O(Q ) Q = ↵Q 3
1
),
Effective Field Theory We introduce the Goldstone field (x) =
and compute the action. SE =
↵ 1 iµ⌧ + p ⇡(x) 2 µ
Q
(⌧out ⌧in ) + S⇡ , R Z ✓ ◆ 1 1 3 p 2 2 S⇡ = d x g ⇡˙ + (@i ⇡) + 16⇡ 2 ✓ ◆ Z i 1 3 3 p 3 2 p + d x g ⇡ ˙ + ⇡(@ ˙ ⇡) + O(Q i 2 96⇡ ↵ Q3/4 p 2 3/2 1/2 + Q + C + O(Q ) Q = ↵Q 3 vacuum energy
one-loop determinant
1
),
Effective Field Theory [Monin, Pirtskhalava, Rattazzi, Seibold 16’]
Effective Field Theory [Monin, Pirtskhalava, Rattazzi, Seibold 16’]
For the light operators we write Oq = cq |@ | = cq µ
q
e
µq⌧
q
eiq + cR q R|@ | iq↵ 1 1 + p ⇡(⌧, n) + 2 µ 2
✓
q
2 iq
iq↵ p 2 µ
e
+ ...
◆2
!
⇡ 2 (⌧, n)
Effective Field Theory [Monin, Pirtskhalava, Rattazzi, Seibold 16’]
For the light operators we write Oq = cq |@ | = cq µ
q
e
µq⌧
Now let’s compute!
q
eiq + cR q R|@ | iq↵ 1 1 + p ⇡(⌧, n) + 2 µ 2
✓
q
2 iq
iq↵ p 2 µ
e
+ ...
◆2
!
⇡ 2 (⌧, n)
Effective Field Theory [Monin, Pirtskhalava, Rattazzi, Seibold 16’]
Effective Field Theory [Monin, Pirtskhalava, Rattazzi, Seibold 16’]
The four-point function takes the form G(z, z¯) ⌘ hOQ (0)O gq (z, z¯) ⌘ (z z¯)
1 2
q
¯)Oq (1)O Q (1)i q (z, z
G(z, z¯)
Effective Field Theory [Monin, Pirtskhalava, Rattazzi, Seibold 16’]
The four-point function takes the form G(z, z¯) ⌘ hOQ (0)O gq (z, z¯) ⌘ (z z¯)
1 2
q
¯)Oq (1)O Q (1)i q (z, z
G(z, z¯)
Crossing becomes gq (z, z¯) = g
q
✓
1 1 , z z¯
◆
Effective Field Theory [Monin, Pirtskhalava, Rattazzi, Seibold 16’]
The four-point function takes the form G(z, z¯) ⌘ hOQ (0)O gq (z, z¯) ⌘ (z z¯)
1 2
q
¯)Oq (1)O Q (1)i q (z, z
G(z, z¯)
Crossing becomes gq (z, z¯) = g
2⌧
z z¯ = e ,
q
z + z¯ p = cos ✓ ⌘ x 2 z z¯
✓
1 1 , z z¯
◆
Effective Field Theory The result to leading order takes the form gq (z, z¯) = cq c
q↵
2
q
Q
q
e
p
↵q Q⌧
D(⌧, x) ⌘ h⇡(0, n2 )⇡(⌧, n1 )i
✓
2
1
q ↵ q p ⌧ + p D(⌧, x) 2 Q 4 Q
◆
Effective Field Theory The result to leading order takes the form gq (z, z¯) = cq c
q↵
2
q
Q
q
e
p
↵q Q⌧
✓
2
1
q ↵ q p ⌧ + p D(⌧, x) 2 Q 4 Q
D(⌧, x) ⌘ h⇡(0, n2 )⇡(⌧, n1 )i ✓
D(⌧, x) =
@⌧2
1 + 4S 2 2
◆
D(⌧, x) =
1 X 2J + 1 |⌧ | + e ⌦J J=1
⌦J |⌧ |
4 (⌧ ) (1
PJ (x)
x)
⌦J =
r
J(J + 1) 2
◆
Effective Field Theory The result to leading order takes the form gq (z, z¯) = cq c
q↵
2
q
Q
q
p
↵q Q⌧
e
✓
2
1
q ↵ q p ⌧ + p D(⌧, x) 2 Q 4 Q
◆
D(⌧, x) ⌘ h⇡(0, n2 )⇡(⌧, n1 )i ✓
D(⌧, x) =
@⌧2
1 + 4S 2 2
◆
D(⌧, x) =
1 X 2J + 1 |⌧ | + e ⌦J J=1
Speed of sound
2 cs
=
1 d
1
⌦J |⌧ |
4 (⌧ ) (1
PJ (x)
x)
⌦J =
r
J(J + 1) 2
is fixed by conformal invariance
Crossing
gq (z, z¯) = cq c
q↵
2
q
Q
q
e
p
↵q Q⌧
✓
2
1
q ↵ q p ⌧ + p D(⌧, x) 2 Q 4 Q
◆
Crossing
gq (z, z¯) = cq c
q↵
2
q
Q
q
e
p
↵q Q⌧
✓
2
1
q ↵ q p ⌧ + p D(⌧, x) 2 Q 4 Q
Crossing transformation acts as follows ⌧! D(⌧, x) =
q!
⌧
1 X 2J + 1 |⌧ | + e ⌦J J=1
q ⌦J |⌧ |
PJ (x)
◆
Crossing
gq (z, z¯) = cq c
q↵
2
q
Q
q
e
p
↵q Q⌧
✓
2
1
q ↵ q p ⌧ + p D(⌧, x) 2 Q 4 Q
Crossing transformation acts as follows ⌧! D(⌧, x) =
q!
⌧
1 X 2J + 1 |⌧ | + e ⌦J
q ⌦J |⌧ |
J=1
Analyticity at ✓
2 @⌧
⌧ =0
1 + 4S 2 2
◆
D(⌧, x) =
4 (⌧ ) (1
x)
PJ (x)
◆
OPE
OPE The leading correction takes the form gqEF T (z, z¯)
= cq c
q↵
2
q
(z z¯)
↵ 2q
p
Q
✓
1 1+ p 4 Q
✓
↵q q+ 2 !
1 X 1 ↵ q 2J + 1 + p (z z¯) 2 ⌦J PJ (x) + O Q 4 Q ⌦J J=1 2
2
1
◆
log(z z¯)
OPE The leading correction takes the form leading primary
gqEF T (z, z¯)
= cq c
q↵
2
q
(z z¯)
↵ 2q
p
Q
✓
1 1+ p 4 Q
✓
↵q q+ 2 !
1 X 1 ↵ q 2J + 1 + p (z z¯) 2 ⌦J PJ (x) + O Q 4 Q ⌦J J=1 2
2
1
◆
log(z z¯)
OPE The leading correction takes the form leading primary
gqEF T (z, z¯)
= cq c
q↵
2
q
(z z¯)
↵ 2q
p
Q
✓
anomalous dimension
1 1+ p 4 Q
✓
↵q q+ 2 !
1 X 1 ↵ q 2J + 1 + p (z z¯) 2 ⌦J PJ (x) + O Q 4 Q ⌦J J=1 2
2
1
◆
log(z z¯)
OPE The leading correction takes the form leading primary
gqEF T (z, z¯)
= cq c
q↵
2
q
(z z¯)
↵ 2q
p
Q
✓
anomalous dimension
1 1+ p 4 Q
✓
↵q q+ 2 !
1 X 1 ↵ q 2J + 1 + p (z z¯) 2 ⌦J PJ (x) + O Q 4 Q ⌦J J=1 2
2
1
descendant and new primaries
◆
log(z z¯)
OPE The leading correction takes the form leading primary
gqEF T (z, z¯)
= cq c
q↵
2
q
(z z¯)
↵ 2q
p
Q
✓
anomalous dimension
1 1+ p 4 Q
✓
↵q q+ 2 !
1 X 1 ↵ q 2J + 1 + p (z z¯) 2 ⌦J PJ (x) + O Q 4 Q ⌦J J=1 2
2
◆
1
descendant and new primaries
EFT field
Regge trajectory
log(z z¯)
OPE The leading correction takes the form leading primary
gqEF T (z, z¯)
= cq c
q↵
2
q
(z z¯)
↵ 2q
p
Q
✓
anomalous dimension
1 1+ p 4 Q
✓
↵q q+ 2 !
1 X 1 ↵ q 2J + 1 + p (z z¯) 2 ⌦J PJ (x) + O Q 4 Q ⌦J J=1 2
2
◆
1
descendant and new primaries
EFT field
Regge trajectory
More generally, the spectrum contains operators r 1 X J(J + 1) ~ n nJ ⌦J , ⌦J = Q+ Q = 2 J=1
log(z z¯)
Bootstrap
Setup We consider heavy-light-light-heavy four-point function
G(z, z¯) ⌘ hOQ (0)O gq (z, z¯) ⌘ (z z¯)
1 2
q
¯)Oq (1)O Q (1)i q (z, z
G(z, z¯)
Setup We consider heavy-light-light-heavy four-point function
G(z, z¯) ⌘ hOQ (0)O gq (z, z¯) ⌘ (z z¯)
1 2
q
¯)Oq (1)O Q (1)i q (z, z
G(z, z¯)
s: 12-34 t: 13-24 u: 14-23
Lessons from EFT From the EFT answer we see:
Lessons from EFT From the EFT answer we see: Finite number of Regge trajectories contribute
Lessons from EFT From the EFT answer we see: Finite number of Regge trajectories contribute s- and u-channel OPE do not have an overlapping convergence region (within EFT)
Lessons from EFT From the EFT answer we see: Finite number of Regge trajectories contribute s- and u-channel OPE do not have an overlapping convergence region (within EFT) t-channel is dominated by heavy operators (EFT does not describe short distance physics)
z-plane G(z, z¯) ⌘ hOQ (0)O
¯)Oq (1)O Q (1)i q (z, z
Bootstrap at Large Charge Correspondingly we have:
Bootstrap at Large Charge Correspondingly we have: Finite number of Regge trajectories contribute (assumption)
Bootstrap at Large Charge Correspondingly we have: Finite number of Regge trajectories contribute (assumption) s- and u-channel match smoothly (s=u crossing)
Bootstrap at Large Charge Correspondingly we have: Finite number of Regge trajectories contribute (assumption) s- and u-channel match smoothly (s=u crossing) macroscopic limit of correlators exist (alternative to t-channel)
Macroscopic Limit of Correlators
Macroscopic Limit of Correlators Consider a state on the cylinder created by an operator of dimension and charge Q
Macroscopic Limit of Correlators Consider a state on the cylinder created by an operator of dimension and charge Q We denote the sphere radius by R
Macroscopic Limit of Correlators Consider a state on the cylinder created by an operator of dimension and charge Q We denote the sphere radius by R Macroscopic limit: R!1 !1
Macroscopic Limit of Correlators Consider a state on the cylinder created by an operator of dimension and charge Q We denote the sphere radius by R Macroscopic limit: R!1 !1
such that correlation functions of light operators in some finite region L stay finite.
Thermodynamic Limit of Correlators
Thermodynamic Limit of Correlators
The usual thermodynamic limit corresponds to energy and charge density kept fixed lim
!1,R!1
Rd
=✏
Q lim = q Q!1,R!1 Rd 1 [Dymarsky, Lashkari, Liu]
Thermodynamic Limit of Correlators
The usual thermodynamic limit corresponds to energy and charge density kept fixed lim
!1,R!1
Rd
E=
=✏
Q lim = q Q!1,R!1 Rd 1 [Dymarsky, Lashkari, Liu]
R
Thermodynamic Limit of Correlators
The usual thermodynamic limit corresponds to energy and charge density kept fixed lim
!1,R!1
Rd
E=
=✏
Q lim = q Q!1,R!1 Rd 1 [Dymarsky, Lashkari, Liu]
R
This limit is expected to exist generically, but in some very special cases it definitely does not exist. (CFTs with moduli space of vacua)
Macroscopic Limit of Correlators
Macroscopic Limit of Correlators A more familiar way to put it is as follows GQ (zi , z¯i ) ⌘ hOQ (0)Oq1 (z1 , z¯1 ) . . . Oqn (zn , z¯n )Oqn+1 (1, 1)O
zi = 1
wi 1 d
H
,
z¯i = 1
w ¯i 1 d
H
Q (1)i
Macroscopic Limit of Correlators A more familiar way to put it is as follows GQ (zi , z¯i ) ⌘ hOQ (0)Oq1 (z1 , z¯1 ) . . . Oqn (zn , z¯n )Oqn+1 (1, 1)O
zi = 1
wi 1 d
,
z¯i = 1
H
✏
G (wi , w ¯i ) ⌘
lim
H !1
1 d
H
Q (1)i
w ¯i 1 d
H
Pn+1 i=1
Li
G 1
wi 1/d H
,1
w ¯i 1/d H
!
Macroscopic Limit of Correlators A more familiar way to put it is as follows GQ (zi , z¯i ) ⌘ hOQ (0)Oq1 (z1 , z¯1 ) . . . Oqn (zn , z¯n )Oqn+1 (1, 1)O
zi = 1
wi 1 d
,
z¯i = 1
H
✏
G (wi , w ¯i ) ⌘
lim
H !1
1 d
H
Q (1)i
w ¯i 1 d
H
Pn+1 i=1
Li
G 1
wi 1/d H
,1
w ¯i 1/d H
!
Existence of this limit is a nontrivial property of the OPE
Macroscopic Limit of Correlators A more familiar way to put it is as follows GQ (zi , z¯i ) ⌘ hOQ (0)Oq1 (z1 , z¯1 ) . . . Oqn (zn , z¯n )Oqn+1 (1, 1)O
zi = 1
wi 1 d
,
w ¯i
z¯i = 1
1 d
H
✏
G (wi , w ¯i ) ⌘
lim
H !1
1 d
H
Q (1)i
H
Pn+1 i=1
Li
G 1
wi 1/d H
,1
w ¯i 1/d H
!
Existence of this limit is a nontrivial property of the OPE
cHH † L c0
L d
H
Macroscopic Limit of Correlators For the four-point function existence of the macroscopic limit constrains short distance behavior of the large Q correlator ✓ ◆ w w ¯ ✏,q q G (w, w) ¯ = lim Q G 1 p ,1 p Q!1 Q Q z=1
w p Q
G(z, z¯) ⌘ hOQ (0)O
z=1
w ¯ p Q
¯)Oq (1)O Q (1)i q (z, z
Macroscopic Limit of Correlators For the four-point function existence of the macroscopic limit constrains short distance behavior of the large Q correlator ✓ ◆ w w ¯ ✏,q q G (w, w) ¯ = lim Q G 1 p ,1 p Q!1 Q Q z=1
w p Q
G(z, z¯) ⌘ hOQ (0)O
z=1
w ¯ p Q
¯)Oq (1)O Q (1)i q (z, z
We combine this with the s- and u- channel OPE, smoothness of the correlator and finite number of Regge trajectories assumption.
Sum Rule We write condition for smoothness of the correlator. The contribution of one Regge trajectory to the correlator takes the form 1 X ✏J |⌧ | cJ e PJ (x) gq (⌧, x) = J=0
s/u-channel conformal block in the large Q limit
For ⌧ > 0 this is the s-channel expansion For ⌧ < 0 this is the u-channel expansion
Sum Rule We write condition for smoothness of the correlator. The contribution of one Regge trajectory to the correlator takes the form 1 X ✏J |⌧ | cJ e PJ (x) gq (⌧, x) = J=0
s/u-channel conformal block in the large Q limit
For ⌧ > 0 this is the s-channel expansion For ⌧ < 0 this is the u-channel expansion Crossing is invariance under ✓ ◆ 1 1 gq (z, z¯) = g q , z z¯
⌧!
⌧ 2⌧
z z¯ = e ,
z + z¯ p = cos ✓ ⌘ x 2 z z¯
Sum Rule The correlator of this form, however, looks singular at ⌧ =0 1 X ✏J |⌧ | c e PJ (x) gq (⌧, x) = J J=0
Sum Rule The correlator of this form, however, looks singular at ⌧ =0 1 X ✏J |⌧ | c e PJ (x) gq (⌧, x) = J J=0
Physically, nothing special happens at non-coincident points. Non-analyticity, therefore, should cancel in the physical correlator.
Sum Rule The correlator of this form, however, looks singular at ⌧ =0 1 X ✏J |⌧ | c e PJ (x) gq (⌧, x) = J J=0
Physically, nothing special happens at non-coincident points. Non-analyticity, therefore, should cancel in the physical correlator. It is convenient to express it as follows cJ ✏2n+1 J
=
(2J + 1) lim
✏!0
Z
⇡
d✓ sin ✓ 0
⇥
@⌧2n+1
gq (✏, ✓)
@⌧2n+1
g
q(
⇤
✏, ✓) PJ (cos ✓)
z-plane G(z, z¯) ⌘ hOQ (0)O
¯)Oq (1)O Q (1)i q (z, z
Sum Rule cJ ✏2n+1 J
=
(2J + 1) lim
✏!0
Z
⇡
d✓ sin ✓ 0
⇥
@⌧2n+1
gq (✏, ✓)
@⌧2n+1
g
q(
⇤
✏, ✓) PJ (cos ✓)
The crucial observation is that nontrivial contribution could only come from short distances ✓ ⌧ 1 In this region the behavior of the correlator is dictated by existence of the macroscopic limit. [Pappadopulo, Rychkov, Espin, Rattazzi]
Sum Rule cJ ✏2n+1 J
=
(2J + 1) lim
✏!0
Z
⇡
d✓ sin ✓ 0
⇥
@⌧2n+1
gq (✏, ✓)
@⌧2n+1
g
q(
⇤
✏, ✓) PJ (cos ✓)
The crucial observation is that nontrivial contribution could only come from short distances ✓ ⌧ 1 In this region the behavior of the correlator is dictated by existence of the macroscopic limit. [Pappadopulo, Rychkov, Espin, Rattazzi]
For N Regge trajectories we would have in the LHS N X i=1
2n+1 cJ,i ✏J,i
Bootstrap Equations
As a result we get an infinite set of bootstrap equations n,0 J,0
+3
J,1
+
N X
2n+1 cJ,i ✏J,i
= (2J +
i=1
⌦2J
2 1)Wn (⌦J )
J(J + 1) ⌘ 2
n = 0, 1, 2, ...
Bootstrap Equations
As a result we get an infinite set of bootstrap equations n,0 J,0
+3
J,1
+
N X
2n+1 cJ,i ✏J,i
= (2J +
i=1
⌦2J
2 1)Wn (⌦J )
J(J + 1) ⌘ 2
n = 0, 1, 2, ... degree n polynomial
(expansion of Legendre polynomials)
Bootstrap Equations
As a result we get an infinite set of bootstrap equations n,0 J,0
+3
J,1
+
N X
2n+1 cJ,i ✏J,i
= (2J +
i=1
descendant
⌦2J
2 1)Wn (⌦J )
J(J + 1) ⌘ 2
n = 0, 1, 2, ... degree n polynomial
(expansion of Legendre polynomials)
Bootstrap Equations
As a result we get an infinite set of bootstrap equations n,0 J,0
+3
J,1
+
N X
2n+1 cJ,i ✏J,i
= (2J +
i=1
anomalous dimension of the heavy operator
descendant
⌦2J
2 1)Wn (⌦J )
J(J + 1) ⌘ 2
n = 0, 1, 2, ... degree n polynomial
(expansion of Legendre polynomials)
Bootstrap Equations
As a result we get an infinite set of bootstrap equations n,0 J,0
+3
J,1
+
N X
2n+1 cJ,i ✏J,i
= (2J +
i=1
anomalous dimension of the heavy operator
descendant
⌦2J
2 1)Wn (⌦J )
J(J + 1) ⌘ 2
n = 0, 1, 2, ... degree n polynomial
(expansion of Legendre polynomials)
Unknowns: •
LHS: anomalous dimensions and three-point couplings
•
RHS: coefficients of the polynomial
Solutions
Solutions For one Regge trajectory the solution is unique and coincides with the EFT result ✏J = ⌦J 2J + 1 cJ = ⌦J
Solutions For one Regge trajectory the solution is unique and coincides with the EFT result ✏J = ⌦J 2J + 1 cJ = ⌦J
For N Regge trajectories the solution is given in terms of roots of a certain polynomial N Y
i=1
(x
✏i (z)) = xN
P1 (z)xN
deg(Pk ) = k
1
+ P2 (z)xN
2
+ · · · + ( 1)N PN (z) = 0
z ⌘ ⌦2J
and similarly a simple formula for three-point functions.
Solutions For one Regge trajectory the solution is unique and coincides with the EFT result ✏J = ⌦J 2J + 1 cJ = ⌦J
For N Regge trajectories the solution is given in terms of roots of a certain polynomial N Y
i=1
(x
✏i (z)) = xN
P1 (z)xN
deg(Pk ) = k
1
+ P2 (z)xN
z ⌘ ⌦2J
2
+ · · · + ( 1)N PN (z) = 0 [Girard, Newton 17th century]
and similarly a simple formula for three-point functions.
Example, N=2 Regge Trajectories
z⌘
2 ⌦J
J(J + 1) = 2
✏1 (z) = az + b + ✏2 (z) = az + b
n ✏1 (z)
+
n ✏2 (z)
p p
=
(cz + d)(ez + f ), (cz + d)(ez + f ).
n X
k=0
ak z
k
Solutions • Some of the solutions look like weakly coupled EFTs with extra particles
2 ✏J
=
2 2 c s ⌦J
+m
2
Solutions • Some of the solutions look like weakly coupled EFTs with extra particles
2 ✏J
=
2 2 c s ⌦J
+m
2
• Other solutions do not admit simple weakly coupled Lagrangian interpretation (for N>4 could not be generically written in terms of radicals) [Abel-Ruffini]
Conclusions
At large global charge Q crossing equations admit a non-trivial and solvable truncation that involves finite number of Regge trajectories •
At large global charge Q crossing equations admit a non-trivial and solvable truncation that involves finite number of Regge trajectories •
We classified the solutions to leading order. The OPE data is encoded in roots of a certain polynomial •
At large global charge Q crossing equations admit a non-trivial and solvable truncation that involves finite number of Regge trajectories •
We classified the solutions to leading order. The OPE data is encoded in roots of a certain polynomial •
Some solutions admit weakly coupled EFT description, while others do not (Regge trajectory = EFT field) •
At large global charge Q crossing equations admit a non-trivial and solvable truncation that involves finite number of Regge trajectories •
We classified the solutions to leading order. The OPE data is encoded in roots of a certain polynomial •
Some solutions admit weakly coupled EFT description, while others do not (Regge trajectory = EFT field) •
Goldstone EFT is the unique solution for one Regge trajectory •
•
What happens in real CFTs?
(help from numerics is needed; how large is ``large charge’’?)
•
What happens in real CFTs?
(help from numerics is needed; how large is ``large charge’’?)
•
Which solutions survive to arbitrary order in 1/Q?
•
What happens in real CFTs?
(help from numerics is needed; how large is ``large charge’’?)
•
Which solutions survive to arbitrary order in 1/Q?
•
Are there non-trivial examples beyond our assumptions?
(large degeneracy = WGC, extremal RN BHs)
Bootstrap for generic heavy operators (macroscopic limits/ETH) •
Bootstrap for generic heavy operators (macroscopic limits/ETH) •
•
CFTs with moduli spaces
Bootstrap for generic heavy operators (macroscopic limits/ETH) •
•
CFTs with moduli spaces
•
Analytic bootstrap in the macroscopic limit
