Unitary conformal
bootstrap
Aninda Sinha
CHEP, Indian Institute of Science
1
Saturday, 23 January 16
2
Saturday, 23 January 16
2
Saturday, 23 January 16
dictionary meaning
Saturday, 23 January 16
dictionary meaning
Saturday, 23 January 16
General philosophy
• No Lagrangian
• No Feynman diagrams
• No regularization, RG etc.
• Only conformal symmetry and some
physics inputs.
Saturday, 23 January 16
Outline
Contents
• Epsilon expansion review
• Modern bootstrap review (small sampling)
• A new approach
d>2, no supersymmetry
Saturday, 23 January 16
• Based on 1502.01437, 1504.0072 with
Apratim Kaviraj and Kallol Sen
• 1510.07770 with Kallol Sen
• 16xx.xxxxx with Rajesh Gopakumar,
Apratim Kaviraj, Joao Penedones and Kallol
Sen
Saturday, 23 January 16
Epsilon expansion: Review
Epsilon-expansion; Wilson;Wilson-Fisher;Polyakov;.Mack.....Rychkov, Tan
O(N) model
Z
d
4 ✏
⇥
i 2
x (@µ ) + (
i i 2
)
⇤
• Wilson-Fisher fixed point.
• 3d Ising model (critical point of boiling
water) N = 1, ✏ = 1
•
•
1
N = 1, ✏ = 2
2d Ising model c=2
XY model N = 2, ✏ = 1
7
Saturday, 23 January 16
8
Saturday, 23 January 16
• State of the art is epsilon^5 by Kleinert et
al.
• Needs ~135 diagrams. Possibility of
mistakes.
• Any way the series is asymptotic.
9
Saturday, 23 January 16
hep-th/9503230
cf: Critical properties of phi4
theories by Kleinert et al. 539
pages.
10
Saturday, 23 January 16
=
d
2
2

N +2 2
N +2
3N + 14
+
✏ +
6
2
2
4(N + 8)
4(N + 8)
(N + 8)2
Ising
d = 2 ! 0.11
1
actual = = 0.125
8
N=1
d = 3 ! 0.519
numerics ⇡ 0.518
XY
N=2
d = 3 ! 0.52,
1 3
✏
4
experiments ⇡ 0.521
expt ⇡ 0.506
Numerical results are from bootstrap based on methods pioneered by Rattazzi,
Rychkov, Vichi and Tonni and used by El Showk et al. Often quoted as most
accurate numerical estimates for the 3d Ising model at criticality.
11
Saturday, 23 January 16
2
=d
Ising
N +2
N +2
2
2+
✏+
(13N + 44)✏
3
N +8
2(N + 8)
d = 2 ! 1.136
actual = 1
d = 3 ! 1.45
XY
seems good
but!!
d = 3 ! 1.54
↵=2
5
!
O(✏ ) !
Saturday, 23 January 16
d
d
numerics ⇡ 1.41
expts ⇡ 1.41
expt ⇡ 1.51
2
0.055
0.004
12
expt ⇡
0.013
International
space station
superfluid He
experiment
discrepancy!
“most precise” 1403.4545, El Showk et al
13
Saturday, 23 January 16
Numerics are not yet
accurate to resolve the
disagreement with
experiments
Kos, Poland, Simmons-Duffin,Vichi, 2015
14
Saturday, 23 January 16
Numerics are not yet
accurate to resolve the
disagreement with
experiments
Kos, Poland, Simmons-Duffin,Vichi, 2015
14
Saturday, 23 January 16
Higher spin operators: Wilson-Kogut
`
( r )=d
N +2 2
2+`+
2✏ (1
2(N + 8)
2
✏
N =1! ( r )=
(1
54
`
6
)
`(` + 1)
6
)
`(` + 1)
For large spin, we can use analytic
bootstrap since the blocks are known in
this limit for any dimension [Kaviraj, Sen, AS, 2015]!
Find precise agreement with this!
15
Saturday, 23 January 16
Numerical bounds in fractional
dimensions
1309.5089, El-Showk et al
16
Saturday, 23 January 16
Unitarity based approach
• Polyakov in 1974 suggested a Lagrangian
free approach to criticality based on
unitarity. This approach has not been
examined carefully in the literature (at all,
although it keeps getting cited in modern
times).
• This approach gave the correct leading
order (in epsilon) anomalous dimensions
for certain operators.
• The general equations proved too hard to
solve and this program was abandoned.
Saturday, 23 January 16
• In fact the bootstrap approach pioneered
by Migdal, Polyakov received quite a bit of
criticism from Wilson himself who
overwhelmingly favoured his RG based
approach which of course had several
physics advantages.
Saturday, 23 January 16
Wilson-Nobel lecture 1982
19
Saturday, 23 January 16
• However, the epsilon expansion approach
which was quite successful faces issues.
• The series is asymptotic and resummation
methods are needed.
• Let’s see what Polyakov has to say about
the RG!
Saturday, 23 January 16
Polyakov interview 2003.
Source: Rychkov, 2011
21
Saturday, 23 January 16
• Clearly the question arises: Can we do
better? cf classification like Minimal models
for 2d CFTs.
Saturday, 23 January 16
• In fact recently we have shown [Sen, AS
2015] that using Polyakov’s approach one
can easily reproduce the epsilon^2 results
without calculating any Feynman diagrams.
• We are developing this further and
hopefully the full power of this method will
be elucidated soon.
Saturday, 23 January 16
• Let me briefly review what can be done
analytically using existing techniques.
Saturday, 23 January 16
Quick review of modern bootstrap
“directchannel”
Saturday, 23 January 16
“crossedchannel”
Quick review of modern bootstrap
“directchannel”
Saturday, 23 January 16
“crossedchannel”
Quick review of modern bootstrap
“directchannel”
Saturday, 23 January 16
“crossedchannel”
even spin
Quick review of modern bootstrap
“directchannel”
X
u
1+
P⌧,` g⌧,` (u, v) = ( )
v
⌧,`
Saturday, 23 January 16
“crossedeven spin
✓channel”
◆
X
1+
P⌧,` g⌧,` (v, u)
⌧,`
Quick review of modern bootstrap
Can only be
reproduced upon
considering large
spin operators on
the RHS
“directchannel”
X
u
1+
P⌧,` g⌧,` (u, v) = ( )
v
⌧,`
Saturday, 23 January 16
“crossedeven spin
✓channel”
◆
X
1+
P⌧,` g⌧,` (v, u)
⌧,`
Quick review of modern bootstrap
Can only be
reproduced upon
considering large
spin operators on
the RHS
“directchannel”
X
u
1+
P⌧,` g⌧,` (u, v) = ( )
v
⌧,`
u=
Saturday, 23 January 16
2 2
x12 x34
x224 x213
,
v=
“crossedeven spin
✓channel”
◆
X
1+
P⌧,` g⌧,` (v, u)
⌧,`
2 2
x14 x23
x224 x213
Conformal
cross ratios
Quick review of modern bootstrap
Crossing
Can only be
reproduced upon
considering large
spin operators on
the RHS
“directchannel”
X
u
1+
P⌧,` g⌧,` (u, v) = ( )
v
⌧,`
u=
Saturday, 23 January 16
2 2
x12 x34
x224 x213
,
v=
u$v
“crossedeven spin
✓channel”
◆
X
1+
P⌧,` g⌧,` (v, u)
⌧,`
2 2
x14 x23
x224 x213
Conformal
cross ratios
Quick review of modern bootstrap
Crossing
Can only be
reproduced upon
considering large
spin operators on
the RHS
“directchannel”
X
u
1+
P⌧,` g⌧,` (u, v) = ( )
v
⌧,`
u=
Twist
⌧=
Saturday, 23 January 16
`
2 2
x12 x34
x224 x213
,
v=
u$v
“crossedeven spin
✓channel”
◆
X
1+
P⌧,` g⌧,` (v, u)
⌧,`
2 2
x14 x23
x224 x213
Conformal
cross ratios
Quick review of modern bootstrap
Crossing
Can only be
reproduced upon
considering large
spin operators on
the RHS
“directchannel”
X
u
1+
P⌧,` g⌧,` (u, v) = ( )
v
⌧,`
u=
Twist
⌧=
Saturday, 23 January 16
`
2 2
x12 x34
x224 x213
,
P⌧,`
v=
u$v
“crossedeven spin
✓channel”
◆
X
1+
P⌧,` g⌧,` (v, u)
⌧,`
2 2
x14 x23
x224 x213
Conformal
cross ratios
OPE x OPE
Quick review of modern bootstrap
Crossing
Can only be
reproduced upon
considering large
spin operators on
the RHS
“directchannel”
X
u
1+
P⌧,` g⌧,` (u, v) = ( )
v
⌧,`
u=
Twist
⌧=
`
2 2
x12 x34
x224 x213
,
v=
“crossedeven spin
✓channel”
◆
X
1+
P⌧,` g⌧,` (v, u)
⌧,`
2 2
x14 x23
x224 x213
Conformal
cross ratios
OPE x OPE
P⌧,`
g⌧,` (u, v)
Saturday, 23 January 16
u$v
Dolan, Osborn;
Blocks
Saturday, 23 January 16
Closed form expressions for conformal
blocks are known only in even dimensions.
Saturday, 23 January 16
Dolan, Osborn;
Closed form expressions for conformal
blocks are known only in even dimensions.
However, simplifications occur in certain limits
Saturday, 23 January 16
Dolan, Osborn;
Fitzpatrick et al;
Komargodski,
Zhiboedov
Closed form expressions for conformal
blocks are known only in even dimensions.
However, simplifications occur in certain limits
`
1
u ⌧ 1, v < 1
Saturday, 23 January 16
In the crossed
channel we
interchange u, v
Dolan, Osborn;
Fitzpatrick et al;
Komargodski,
Zhiboedov
Closed form expressions for conformal
blocks are known only in even dimensions.
Dolan, Osborn;
However, simplifications occur in certain limits
`
1
u ⌧ 1, v < 1
(d)
g⌧,` (u, v)
= u (1
“factorizes”
Saturday, 23 January 16
⌧
2
In the crossed
channel we
interchange u, v
⌧
⌧
v) 2 F1 ( + `, + `, ⌧ + 2`, 1
2
2
`
Fitzpatrick et al;
Komargodski,
Zhiboedov
v)F (d) (⌧, u)
(twist,spin,v) x (twist, u, d)
Closed form expressions for conformal
blocks are known only in even dimensions.
Dolan, Osborn;
However, simplifications occur in certain limits
`
1
u ⌧ 1, v < 1
(d)
g⌧,` (u, v)
= u (1
“factorizes”
Saturday, 23 January 16
⌧
2
In the crossed
channel we
interchange u, v
⌧
⌧
v) 2 F1 ( + `, + `, ⌧ + 2`, 1
2
2
`
Fitzpatrick et al;
Komargodski,
Zhiboedov
v)F (d) (⌧, u)
(twist,spin,v) x (twist, u, d)
Closed form expressions for conformal
blocks are known only in even dimensions.
Dolan, Osborn;
However, simplifications occur in certain limits
`
1
u ⌧ 1, v < 1
(d)
g⌧,` (u, v)
= u (1
“factorizes”
Saturday, 23 January 16
⌧
2
In the crossed
channel we
interchange u, v
⌧
⌧
v) 2 F1 ( + `, + `, ⌧ + 2`, 1
2
2
`
Fitzpatrick et al;
Komargodski,
Zhiboedov
v)F (d) (⌧, u)
(twist,spin,v) x (twist, u, d)
Saturday, 23 January 16
Recursion relations for blocks in any dimension
Saturday, 23 January 16
F (d) (⌧, v) =
2
(1
v)
d 2
2
2 F1
1
(⌧
2
1
d + 2), (⌧
2
d + 2), ⌧
d + 2, v .
(2.6)
The general recursion relation, relating the conformal blocks for d dimensions to those in d
2
dimensions are worked out in section 5 of [1]. We write down the relation for the crossed conformal
Recursion
relations for blocks in any dimension
blocks for convenience,
✓
z̄ z
(1 z)(1
z̄)
◆2
g
(d)
,` (v, u)
=g
4(` 2)(d + ` 3)
(d 2)
g 2,` (v, u)
(d + 2` 4)(d + 2` 2)

4(d
3)(d
2)
( + `)2
(d 2)
g ,`+2 (v, u)
d 2
2)(d 2 )
16( + ` 1)( + ` + 1)
(d + ` 4)(d + ` 3)(d + `
2)2
(d 2)
g ,` (v, u) .
4(d + 2` 4)(d + 2` 2)(d + `
3)(d + `
1)
(2.7)
(d 2)
2,` (v, u)
In the limit when ` ! 1 at fixed ⌧ =
relation simplifies to,
✓
1
v
v
◆2
(d)
`, and for z ! 0 and z̄ = 1
(d 2)
4,`+2 (v, u)
g⌧,` (v, u) = g⌧
(d 2)
2,` (v, u)
g⌧
+
2
Saturday, 23 January 16
(d
16(d ⌧
v + O(z), the above
1 (d 2)
g
(v, u)
16 ⌧ 2,`+2
⌧ 2)2
(d
g⌧,`
3)(d ⌧ 1)
(2.8)
2)
(v, u) .
F (d) (⌧, v) =
2
(1
v)
d 2
2
2 F1
1
(⌧
2
1
d + 2), (⌧
2
d + 2), ⌧
d + 2, v .
(2.6)
The general recursion relation, relating the conformal blocks for d dimensions to those in d
2
dimensions are worked out in section 5 of [1]. We write down the relation for the crossed conformal
Recursion
relations for blocks in any dimension
blocks for convenience,
✓
z̄ z
(1 z)(1
z̄)
◆2
g
(d)
,` (v, u)
Crossed
channel
=g
4(` 2)(d + ` 3)
(d 2)
g 2,` (v, u)
(d + 2` 4)(d + 2` 2)

4(d
3)(d
2)
( + `)2
(d 2)
g ,`+2 (v, u)
d 2
2)(d 2 )
16( + ` 1)( + ` + 1)
(d + ` 4)(d + ` 3)(d + `
2)2
(d 2)
g ,` (v, u) .
4(d + 2` 4)(d + 2` 2)(d + `
3)(d + `
1)
(2.7)
(d 2)
2,` (v, u)
In the limit when ` ! 1 at fixed ⌧ =
relation simplifies to,
✓
1
v
v
◆2
(d)
`, and for z ! 0 and z̄ = 1
(d 2)
4,`+2 (v, u)
g⌧,` (v, u) = g⌧
(d 2)
2,` (v, u)
g⌧
+
2
Saturday, 23 January 16
(d
16(d ⌧
v + O(z), the above
1 (d 2)
g
(v, u)
16 ⌧ 2,`+2
⌧ 2)2
(d
g⌧,`
3)(d ⌧ 1)
(2.8)
2)
(v, u) .
Dolan, Osborn;
F (d) (⌧, v) =
2
(1
v)
d 2
2
2 F1
1
(⌧
2
1
d + 2), (⌧
2
d + 2), ⌧
d + 2, v .
(2.6)
The general recursion relation, relating the conformal blocks for d dimensions to those in d
2
dimensions are worked out in section 5 of [1]. We write down the relation for the crossed conformal
Recursion
relations for blocks in any dimension
blocks for convenience,
✓
z̄ z
(1 z)(1
z̄)
◆2
g
(d)
,` (v, u)
Crossed
channel
=g
4(` 2)(d + ` 3)
(d 2)
g 2,` (v, u)
(d + 2` 4)(d + 2` 2)

4(d
3)(d
2)
( + `)2
(d 2)
g ,`+2 (v, u)
d 2
2)(d 2 )
16( + ` 1)( + ` + 1)
(d + ` 4)(d + ` 3)(d + `
2)2
(d 2)
g ,` (v, u) .
4(d + 2` 4)(d + 2` 2)(d + `
3)(d + `
1)
(2.7)
(d 2)
2,` (v, u)
Solution to recursion relations
`, and for zin
! 0closed
and z̄ = 1 form
v + O(z), the above
relation
simplifies
to, in even d.
known
only
In the limit when ` ! 1 at fixed ⌧ =
✓
1
v
v
◆2
(d)
(d 2)
4,`+2 (v, u)
g⌧,` (v, u) = g⌧
(d 2)
2,` (v, u)
g⌧
+
2
Saturday, 23 January 16
(d
16(d ⌧
1 (d 2)
g
(v, u)
16 ⌧ 2,`+2
⌧ 2)2
(d
g⌧,`
3)(d ⌧ 1)
(2.8)
2)
(v, u) .
Dolan, Osborn;
F (d) (⌧, v) =
2
(1
v)
d 2
2
2 F1
1
(⌧
2
1
d + 2), (⌧
2
d + 2), ⌧
d + 2, v .
(2.6)
The general recursion relation, relating the conformal blocks for d dimensions to those in d
2
dimensions are worked out in section 5 of [1]. We write down the relation for the crossed conformal
Recursion
relations for blocks in any dimension
blocks for convenience,
✓
z̄ z
(1 z)(1
z̄)
◆2
g
(d)
,` (v, u)
Crossed
channel
=g
4(` 2)(d + ` 3)
(d 2)
g 2,` (v, u)
(d + 2` 4)(d + 2` 2)

4(d
3)(d
2)
( + `)2
(d 2)
g ,`+2 (v, u)
d 2
2)(d 2 )
16( + ` 1)( + ` + 1)
(d + ` 4)(d + ` 3)(d + `
2)2
(d 2)
g ,` (v, u) .
4(d + 2` 4)(d + 2` 2)(d + `
3)(d + `
1)
(2.7)
(d 2)
2,` (v, u)
Solution to recursion relations
`, and for zin
! 0closed
and z̄ = 1 form
v + O(z), the above
relation
simplifies
to, in even d.
known
only
In the limit when ` ! 1 at fixed ⌧ =
✓
1
v
◆2
(d)
(d 2)
4,`+2 (v, u)
(d 2)
2,` (v, u)
v
In the large
spin limit and u ⌧ 1, v
(d
+
relation simplifies.
16(d ⌧
g⌧,` (v, u) = g⌧
g⌧
2
Saturday, 23 January 16
1 (d 2)
16
<g⌧12,`+2(v, u)
⌧ 2)2
(d
g⌧,`
3)(d ⌧ 1)
the recursion(2.8)
2)
(v, u) .
Dolan, Osborn;
F (d) (⌧, v) =
2
(1
v)
d 2
2
2 F1
1
(⌧
2
1
d + 2), (⌧
2
d + 2), ⌧
d + 2, v .
(2.6)
The general recursion relation, relating the conformal blocks for d dimensions to those in d
2
dimensions are worked out in section 5 of [1]. We write down the relation for the crossed conformal
Recursion
relations for blocks in any dimension
blocks for convenience,
✓
z̄ z
(1 z)(1
z̄)
◆2
g
(d)
,` (v, u)
=g
4(` 2)(d + ` 3)
(d 2)
g 2,` (v, u)
(d + 2` 4)(d + 2` 2)

4(d
3)(d
2)
( + `)2
(d 2)
g ,`+2 (v, u)
d 2
2)(d 2 )
16( + ` 1)( + ` + 1)
(d + ` 4)(d + ` 3)(d + `
2)2
(d 2)
g ,` (v, u) .
4(d + 2` 4)(d + 2` 2)(d + `
3)(d + `
1)
(2.7)
(d 2)
2,` (v, u)
Crossed
channel
Dolan, Osborn;
Solution to recursion relations in closed form
relation
simplifies
to, in even d.
known
only
In thethe
limitansatz
when `for
!1
fixed ⌧ = form
`, and
for zconformal
! 0 and z̄ blocks
= 1 v at
+ O(z),
above by,
Inserting
theatfactorized
of the
largethe
` given
at `
✓
1
v
◆2
(d)
g⌧,` (v, u)
(d)
g(d⌧,`2)(v, u)
= k(d2` (1
2)
⌧
2
u)v 1 F(d(d)2)(⌧, v) ,
In the large spin limit and
(dand
⌧ noticing
2)
1 and u ⌧ 1 into the above recursion +relation
that
k(v,2(`+2)
(1
g
u)
.
relation simplifies.
16(d ⌧ 3)(d ⌧ 1)
(d)
v
= g⌧
4,`+2 (v, u)
(v, u)
2,` (v, u)
u⌧
1, v 16
<g⌧12,`+2
the recursion(2.8)
g⌧
2
(d 2)
⌧,`
we arrive at the following recursion relation satisfied by the functions F
2
(1 v) F
(d)
(⌧, v) = 16F
(d 2)
(⌧
(2.9)
(d 2)
4, v) 2vF 2
(⌧
(d
2, v)+
16(d ⌧
u) = 24 k2` (1 u),
(⌧, v) given in [2],
⌧ 2)2
3)(d ⌧
1)
v 2 F (d
2)
(⌧, v) .
(2.10)
As we have explicitly checked, the solutions in (2.6) satisfy these recursion relations for general d
dimensions.
Saturday, 23 January 16
New results from bootstrap
Saturday, 23 January 16
New results from bootstrap
Gauss Hypergeometric
Saturday, 23 January 16
New results from bootstrap
Gauss Hypergeometric
F
(d)
(⌧, v) =
2
(1
⌧
v)
d
2
2
2 F1
✓
1
(⌧
2
1
d + 2), (⌧
2
d + 2), (⌧
d + 2), v
Kaviraj, Sen, AS
Saturday, 23 January 16
◆
New results from bootstrap
Gauss Hypergeometric
F
(d)
(⌧, v) =
2
(1
⌧
v)
d
2
2
2 F1
✓
1
(⌧
2
1
d + 2), (⌧
2
d + 2), (⌧
d + 2), v
Kaviraj, Sen, AS
Bootstrap equation demands at leading order
Saturday, 23 January 16
◆
New results from bootstrap
Gauss Hypergeometric
F
(d)
(⌧, v) =
2
(1
⌧
v)
d
2
2
2 F1
✓
1
(⌧
2
1
d + 2), (⌧
2
d + 2), (⌧
d + 2), v
Kaviraj, Sen, AS
Bootstrap equation demands at leading order
1 ⇡ (function of u) ⇥ v ⌧ /2
Saturday, 23 January 16
(1
v)
F (d) (⌧, v)
◆
New results from bootstrap
Gauss Hypergeometric
F
(d)
(⌧, v) =
2
(1
⌧
v)
d
2
2
2 F1
✓
1
(⌧
2
1
d + 2), (⌧
2
d + 2), (⌧
d + 2), v
Kaviraj, Sen, AS
Bootstrap equation demands at leading order
1 ⇡ (function of u) ⇥ v ⌧ /2
Saturday, 23 January 16
(1
v)
F (d) (⌧, v)
Needs large
◆
New results from bootstrap
Gauss Hypergeometric
F
(d)
(⌧, v) =
2
(1
⌧
v)
d
2
2
2 F1
✓
1
(⌧
2
1
d + 2), (⌧
2
d + 2), (⌧
d + 2), v
Kaviraj, Sen, AS
Bootstrap equation demands at leading order
1 ⇡ (function of u) ⇥ v ⌧ /2
To match powers of v, we
must have
Saturday, 23 January 16
(1
v)
F (d) (⌧, v)
Needs large
◆
New results from bootstrap
Gauss Hypergeometric
F
(d)
(⌧, v) =
2
(1
⌧
v)
d
2
2 F1
2
✓
1
(⌧
2
1
d + 2), (⌧
2
d + 2), (⌧
d + 2), v
Kaviraj, Sen, AS
Bootstrap equation demands at leading order
1 ⇡ (function of u) ⇥ v ⌧ /2
To match powers of v, we
must have
⌧ =2
Fitzpatrick et al; Komargodski,
Zhiboedov
Saturday, 23 January 16
+ 2n
(1
v)
F (d) (⌧, v)
Needs large
◆
New results from bootstrap
Gauss Hypergeometric
F
(d)
(⌧, v) =
2
(1
⌧
v)
d
2
2 F1
2
✓
1
(⌧
2
1
d + 2), (⌧
2
d + 2), (⌧
d + 2), v
Kaviraj, Sen, AS
Bootstrap equation demands at leading order
1 ⇡ (function of u) ⇥ v ⌧ /2
To match powers of v, we
must have
⌧ =2
Fitzpatrick et al; Komargodski,
Zhiboedov
Saturday, 23 January 16
+ 2n
(1
v)
F (d) (⌧, v)
Needs large
Same as what appears in
MFT. OPE’s known.
◆
• Can there be any universal results?
• The answer surprisingly appears to be yes.
• The anomalous dimension of large spin,
large twist operators appears to be
universal.
`
Saturday, 23 January 16
1
General dimensions
Assume minimal twist for stress tensor
exchange d-2
Kaviraj, Sen, AS
With some effort this can be derived
analytically in all d. In terms of cT:
<Stress Stress>~cT
Comment: For Twist 2 Parisi-Callan-Gross theorem/Nachtmann theorem
Saturday, 23 January 16
Universality at
large twist
Saturday, 23 January 16
Universality at
large twist
Plots in diverse spacetime dimensions for
various conformal dimensions of the seed
scalar. Asymptotes indicate same intercept
independent of conformal dimension.
Saturday, 23 January 16
• Do not need large N, not tied with gauge/
gravity duality. Should hold for any CFT.
• Can also reproduce exactly from AdS/CFT
hinting at a universal sector both in gravity
and CFT.
Saturday, 23 January 16
New game ala Polyakov
• The algorithm for the new game is the
following (see Apratim’s talk).
• First we try to derive a spectral function
that gives the “standard” conformal blocks.
• We find that this spectral function leads to
non-convergent behaviour for the 4pt fn at
large complex spectral parameter.
Saturday, 23 January 16
• To fix this and applying Liouville’s theorem
we find that we need to add a factor that is
a square of a gamma function with poles at
the location of double field operators.
• The resulting block is different from the
usual block and leads to anomalous log and
power law singularities.
Saturday, 23 January 16
• Demanding that these vanish gives a set of
algebraic equations.
• Solving the equations in an epsilon
expansion gives readily the epsilon^2
results that we obtain using standard QFT
methods.
• Note that our approach bypasses Feynman
diagrams and usual regularization,
renormalization.
Saturday, 23 January 16
• It will be nice to explore these equations
further and try to get a better analytic
handle.
• One surprise that we had was that the
spectral function approach is similar to
what we would do in AdS/CFT via Witten
diagrams!
• I expect a lot of other surprises to be in
store!
Saturday, 23 January 16
Saturday, 23 January 16
Saturday, 23 January 16
Unitarity approach ala Polyakov
• Due to lack of time, I will explain this
method first in words before showing
equations.
• When we compute CFT four point
functions (of scalar fields for
concreteness), there are two things to keep
in mind.
• First, we can do it using OPEs.
• Second we can use the completeness of
Saturday, 23 January 16
states to compute the imaginary part of the
four point function and then use dispersion
relations. [recall one loop diagrams become imaginary when
intermediate states become onshell]
• It turns out that the unitarity based method
gives terms (log and regular) that are
absent in the OPE (algebraic) approach.
• For consistency we have to cancel these
terms.
• We get algebraic equations involving OPE
and the conformal dimensions. NB: no
cross ratios in these equations!
• Can we solve these equations?
Saturday, 23 January 16
• Why can this be advantageous over
modern methods?
• For starters modern methods do not
provide analytic results that easily--only for
some special sectors. No results on epsilon
expansion using modern techniques.
• As we will see there is a close connection
with Witten diagrams. So there is a lot to
explore.
• May be string theory will be “inevitable.”
Saturday, 23 January 16
Sen, AS
conf. inv.
fixes 3pt
Unknowns: OPE coefficients, conformal
dimensions
Use dispersion relations to get full
function
Saturday, 23 January 16
• It turns out that cancellation of offending
terms demanding only quadratic scalar
exchange is sufficient to fix the anomalous
dimensions of the exchange operator upto
2nd order in epsilon. [Sen, AS]*
• It involves solving 2 algebraic equations
which fix the OPE coefficients and the
anomalous dimensions.
*
Saturday, 23 January 16
“the paper is very nice, congratulations on being the first people in the world who
actually managed to understand Polyakov’s computation in that paper and to go
further. “.....Slava Rychkov
• What about other exchange operators?
• What about higher orders?
• Is the epsilon expansion an expansion in
terms of the spin of the exchange
operators?
• To answer these questions we found it
more convenient to use the framework of
Witten diagrams and Mellin space.
Saturday, 23 January 16
Mellin space bootstrap
work in progress with A. Kaviraj, J. Penedones and K. Sen
• We can think of CFT correlation functions
as scattering amplitudes in AdS space.
• Mack in 2009 taught us that it is sometimes
simpler to use Mellin space. This was
further explained and emphasised by Joao
Penedones.
• With Apratim, Joao and Kallol we have
reformulated (almost) the unitarity
approach in Mellin space.
Saturday, 23 January 16
Correlation function in Mellin space
Mack, 2009
n
X
ij
=0
ij
=
ji
ii
=
i
i=1
A = h (x1 ) (x2 ) (x3 ) (x4 )i
Saturday, 23 January 16
Unitary amplitudes from Witten diagrams
CPW decomp.
spectral fn
standard blocks
Polyakov
ignores
this piece
spin-J Witten
diagram
Factorization
necessary to
reproduce OPE
1. spectral fn is known. 2. need to fix norm. for F
Saturday, 23 January 16
s channel: spin 0 exchange
Projector
s channel
Symanzik star
formula
Mellin variables
Barnes’ first
lemma
x1 = 0, x2 = r, x3 = R, x4 = R + r
Saturday, 23 January 16
0
Shadow piece
s-channel a’s
NB: no logs as of now.
x1 = 0, x2 = r, x3 = R, x4 = R + r
Saturday, 23 January 16
0
R2
r2 , r02
t/u
channel
log comes from double poles in the Gamma function
The t/u channel result is independent of
spin. We will get a sum over spins.
So we have fixed the F’s.
Saturday, 23 January 16
• It turns out that once we put in the the
spectral function as in the Witten diagram
CPW, s-channel leads to both regular as
well as log terms arising from double poles.
• So now all channels have regular and log
terms which are absent in the algebraic
amplitude.
• DEMAND THAT THESE CANCEL
Saturday, 23 January 16
✏
1 2
=1
+
✏
2 108
= 2 + ✏ + 1 ✏2
fixed by conf. symm.
unknown
+
=0
agrees with 2 loop
Feynman. Can easily
extend to O(n).
Saturday, 23 January 16
Complete set of equations for spin-0 s-channel exchange
OPE coeffs
log term
regular term
d
h=
2
Saturday, 23 January 16
Pole structure for spin-0 exchange in the s-channel
Saturday, 23 January 16
• Observe that in an epsilon expansion 1/
epsilon^3 and 1/epsilon^2 terms in the
equations arise only from quadratic scalar
operator exchange.
• This means that we will get the correct
anomalous dimension for this operator
upto epsilon^2 by solving a pair of
simultaneous equations! Find precise
agreement with Feynman diagram
approach.
Saturday, 23 January 16
• Higher order scalars will contribute in the
equations from 1/epsilon (so will contribute
to the epsilon^3 order in the anom. dim.)
• Non-zero spins will contribute from
epsilon^0 order and hence will contribute
from epsilon^4 order in the anom. dim.)
• A better understanding is in progress.
• Our findings give a different perspective
why the epsilon expansion was useful.
• Where is string theory? Problem for 2016-.
Saturday, 23 January 16
Thank you and seasons greetings!
Saturday, 23 January 16