Conformal Bootstrap
in Mellin Space
Rajesh Gopakumar
ICTS-TIFR, Bengaluru, India
Strings 2016, YMSC- Tsinghua, Beijing,
Aug. 3rd, 2016
Based on: R. G., A. Kaviraj, J. Penedones, K. Sen and
A. Sinha (arXiv:160y.xxxxx) - to appear
The Bootstrap Rebooted
Successful revival of the conformal bootstrap
program in recent years [Ratazzi-Rychkov-Tonni-Vichi….].
Here: describe a new (old) approach.
Calculationally effective - reproduce existing + new
analytic results for operator dim. and OPE coeffs. :
A. ϵ -expansion for Wilson-Fisher fixed point.
B. Large spin limit in any dimension.
Conceptually suggestive - hints of a dual AdS
description.
Two Ingredients
1.
Go back to the original approach of Polyakov - now in a
modern incarnation. Use a new set of blocks (built from Witten
diagrams) instead of conformal blocks - conceptually
suggestive.
2.
Natural to implement in Mellin space [Mack, Penedones, .…].
Exploit meromorphy and analogy to scattering amplitudes in
momentum space - calculationally effective.
A Sampling of Results
A. ϵ -expansion for Wilson-Fisher fixed point (d = 4 − ϵ ).
19 2
2
1 2
109 3
ϵ
ϵ + O(ϵ3 ) [Wilson-Kogut]
∆ φ2 = 2 − ϵ +
+
ϵ +
ϵ + O(ϵ4 )
2 108
11664
3
162
!
"
2
ϵ
6
[109ℓ3 (ℓ + 2) + 373ℓ2 − 816ℓ − 756] − 432ℓ(ℓ + 1)Hℓ−1
(3)
3 (3)
δℓ =
γℓ =
1−
+ ϵ δℓ
5832ℓ2 (1 + ℓ)2
54
ℓ(ℓ + 1)
;
∆φ = 1 −
(
Cφφφ2
Cℓ
Cℓf ree
(3)
C2
(3)
C8
17 2
1
= 1 − ϵ − ϵ + O(ϵ3 )
3
81
(ℓ(ℓ + 1) − 1) (H2ℓ − Hℓ−1 ) 2
(3) 3
ϵ
+
C
=1+
ℓ ϵ
2
2
9ℓ (1 + ℓ)
1019357
3872826169
509
(3)
(3)
C4 =
C6 =
=
17496
114307200
871363785600
54561737953
58967348085478139
(3)
=
C10 =
.
20195722939392
32190271338864038400
)
A Sampling of Results (Contd.)
Large spin limit (in any dimension).
•
large twist gap (δτ
gap
log(ℓ) ≫ 1
[Fitzpatrick et.al., Komargodski-
# $τ m
Γ (∆φ ) Γ (2ℓm + τm ) 1
Cmφφ 2
γℓ − 2γφ = −
!
" !
"
τm 2
τm 2
ℓ
Γ ∆φ −
Γ ℓm +
2
2−ℓm
2
•
)
small twist gap ( δτ
2
gap
Also OPE Coeff.
)
log(ℓ) ≪ 1
d−2
d−2
(1)
+ γφ =
+ gδφ + O(g 2 )
∆φ =
2
2
[Alday-Zhiboedov]
(1)
∆φ2 = d − 2 + gδφ2 + O(g 2 )
α0 (g) + α1 (g) log ℓ + α2 (g)(log ℓ)2 + · · ·
γℓ − 2γφ =
ℓd−2
αk = −2d−3 (−g)k+2 Cφφφ2
Γ
!d
2
Zhiboedov]
" !
" (1)
(1)
(1)
− 1 Γ d2 − 12 (δφ2 )k (δφ2 − 2δφ )2
√
+ O(g k+3 )
k! π
The New Old….
Expand the four pt. function in terms of a new set of building
blocks - a new basis of expansion.
sum over physical
spectrum
A(u, v) = ⟨O(1)O(2)O(3)O(4)⟩
=
!
(s)
(t)
(u)
c∆,ℓ (W∆,ℓ (u, v) + W∆,ℓ (u, v) + W∆,ℓ (u, v))
∆,ℓ
Demand of these blocks that they satisfy
1. Conformal invariance.
2. Consistency with unitarity - factorisation on physical
operators with right residues.
3. Crossing symmetry - built in by summing over channels.
But now not guaranteed that expansion is consistent with OPE
in, say, s-channel.
….vs The Old New
Generically will have spurious powers
u∆φ as
well as
u∆φlog(u).
Demanding that these terms cancel gives constraints on op. dim.
and OPE coeff. (Note: log terms not from anomalous dim.)
Contrast with the “conventional” bootstrap which has:
1. Conformal Invariance of conf. blocks
(s)
G∆,ℓ (u, v)
2. Consistency with unitarity - factorisation on physical
operators with right residues.
3. Consistency with OPE - only physical operators exchanged.
But now not guaranteed that crossing symmetry is satisfied.
Demanding gives nontrivial constraints on op. dim. etc.
The New-Old Building Blocks
Polyakov’s building blocks are, essentially, exchange Witten
diagrams - better behaved compared to conformal blocks.
(s)
W∆,ℓ (u, v)
=
And similarly for t- and
u-channels
These are a) conformally invariant b) obey factorisation and
are c) by construction, crossing invariant.
But Witten diagrams are complicated in position space. Easier
to deal with in Mellin space. Properties like factorisation as
well as asymptotic behaviour more transparent.
Migrating to Mellin Space
A(u, v) =
!
s t 2
2
2
dsdtu v Γ (−t)Γ (s + t)Γ (∆φ − s)M(s, t)
Conformal Blocks:
(s)
(s)
(identical ext. scalars)
Mack Polynomials
G∆,ℓ (u, v) → B∆,ℓ (s, t)
∆−ℓ
2h−∆−ℓ
"
Γ(
−
s)Γ(
− s)
(s)
iπ(2s+∆+ℓ−2h)
2
2
B∆,ℓ (s, t) = e
−1
P∆,ℓ (s, t)
2
Γ(∆φ − s)
!
Cancels out “shadow” pole
External scalar
Exponential behaviour for large s.
Residue - 3-point fn.
Factorises on poles.
(s)
B∆,ℓ (s, t)
∞
!
Rm (t)
=
+ ...
2s − ∆ + ℓ − 2m
m=0
Descendant poles
[Mack,
Penedones,
FitzpatrickKaplan,….]
Mellin Space (Contd.)
Witten Diagram Blocks:
(s)
(s)
W∆,ℓ (u, v) → M∆,ℓ (s, t)
Choose as meromorphic piece of conformal block.
(s)
M∆,ℓ (s, t)
∞
!
Rm (t)
=
2s
−
∆
+
ℓ
−
2m
m=0
Could add polynomial (of degree ℓ ) - ambiguity.
No exponentially growing behaviour at infinity.
For scalar exchange (ℓ=0), explicitly given by
∆−d)
2
Γ
(∆
+
)
1
φ
[Penedones, Paulos….]
(s)
2
M∆,0 (s, t) =
2s − ∆ Γ(1 + ∆ − h)
(h=d/2)
∆
∆ ∆
∆
, 1 − ∆φ + , − s; 1 + − s, 1 + ∆ − h; 1)
3 F2 (1 − ∆φ +
2
2 2
2
Consistency
Now sum over (s,t,u)-channel contributions.
M(s, t) =
!
"
(s)
(t)
(u)
c∆,ℓ M∆,ℓ (s, t) + M∆,ℓ (s, t) + M∆,ℓ (s, t)
#
∆,ℓ
∆φ
Requiring cancellation of spurious powers u and u∆φlog(u)
equivalent to cancelling spurious single and double poles:
(1)
(2)
qtot (t)
qtot (t)
+
+ (physical)+ (spurious descendants)
(measure) x M(s, t) =
2
(s − ∆φ )
s − ∆φ
Easier condition to implement - nett residue identically
vanishes as a function of t.
A natural decomposition into partial wave orth. polynomials
qtot (t) =
!
∆,ℓ
(s)
(q∆,ℓ
+
(t)
q∆,ℓ
+
(u)
q∆,ℓ )Qℓ,0 (t)
=0⇒
!
(s)
(t)
(u)
(q∆,ℓ + q∆,ℓ + q∆,ℓ ) = 0
∆
specialisation of mack polynomials
∀ℓ
Devil in the Details
Use a spectral representation for
(s)
M∆,ℓ (s, t) =
spectral weight
(s)
µ∆,ℓ (ν)
!
(s)
M∆,ℓ (s, t)
etc.
Mack Polynomials
i∞
(s)
(s)
(s)
dν µ∆,ℓ (ν)Ων,ℓ (s)Pν,ℓ (s, t)
−i∞
h−ν−ℓ
2
Γ2 (∆φ − h+ν−ℓ
)Γ
(∆
−
)
φ
2
2
= 2
(ν − (∆ − h)2 )Γ(ν)Γ(−ν)(h + ν − 1)ℓ (h − ν − 1)ℓ
(s)
Ων,ℓ (s)
Γ( h+ν−ℓ
− s)Γ( h−ν−ℓ
− s) 2 h + ν + ℓ 2 h − ν + ℓ
2
2
=
Γ(
)Γ (
)
Γ2 (∆φ − s)
2
2
Arises from the “split” representation of Witten exchange
diagrams - bulk to bulk propagators in terms of bulk to bdry.
Spectral weight exhibits poles at operator (+ shadow) as well as
“double trace” of external ops - gives rise to spurious poles.
Useful for picking out the specific contributions that behave as
(2,s)
(1,s)
q∆,ℓ (t) q∆,ℓ (t)
,
(s − ∆φ )2 s − ∆φ
Devil in the Details (Contd.)
At these poles the residue simplifies. Mack Polynomials reduce
(s)
Pν,ℓ (s, t)
→
2∆φ +ℓ
Qℓ,0 (t)
Q’s are a nice set of orthogonal polynomials (continuous Hahn)
- can be written in terms of 3F2(−ℓ, ...; 1) hypergeometric functions.
(2,s)
(1,s)
q∆,ℓ (t)
q∆,ℓ (t)
(s)
(measure) x M∆,ℓ (s, t) =
+
+ ...
2
(s − ∆φ )
s − ∆φ
Where
(2,s)
q∆,ℓ
(2,s)
(2,s)
2∆ +ℓ
q∆,ℓ (t) = q∆,ℓ Qℓ,0 φ (t)
and
Γ(2∆φ + ℓ − h)
= − 2ℓ−2
2
(ℓ − ∆ + 2∆φ )(ℓ + ∆ + 2∆φ − 2h)
(1,s)
(1,s)
2∆ +ℓ
q∆,ℓ (t) = q∆,ℓ Qℓ,0 φ (t)
(1,s)
q∆,ℓ =
with
Γ(2∆φ + ℓ − h + 1)
22ℓ−4 (ℓ − ∆ + 2∆φ )2 (ℓ + ∆ + 2∆φ − 2h)2
Thus in s-channel can explicitly write the contribution to
spurious poles from each op. with definite ℓ .
Way Too Much Detail
Need to add in the t- and u-channel spurious pole contributions.
A fixed spin in t-channel gets contributions from all ℓ
(t)
M∆,ℓ′ (s, t)
=
!
2∆ +ℓ
(t)
q∆,ℓ′ (s, ℓ)Qℓ,0 φ (t)
ℓ
Use orthogonality to pick out nett contribution
(t)
qℓ (s) =
!
∆,ℓ′
c∆,ℓ′
"
2∆ +ℓ
(t)
dtM∆,ℓ′ (s, t)Qℓ,0 φ (t)Γ2 (s + t)Γ2 (−t)
Determines pole pieces:
(t)
(2,t)
qℓ
(t)
(1,t)
= qℓ (s)|s=∆φ , qℓ
dq (s)
|s=∆φ
= ℓ
ds
Convenient to also add disconnected (identity op.) piece
(1,t)
q
separately ℓ,disc . Also u-channel gives identical contribution.
Thus final constraint eqns:
(
!
∆
(i,s)
q∆,ℓ
+
(i,t)
2qℓ )
=0
; ∀ℓ ; (i=1,2)
The Epsilon Expansion
Implement this schema for the W-F fixed point in ( d = 4 − ϵ ) as
alternative to Feynman diagrams. [Polyakov, Rychkov-Tan, K. Sen-Sinha]
Exploit that (∆ = d) for the stress tensor and look at ℓ =2
channel. Only s-channel contributes till O(ϵ ).
Fixes leading correction to
∆φ and Cℓ=2 = CφφT.
Now include t-channel but only φ2 contributes to low orders.
Harmonic
1 2
109 3
19 2
ϵ
2
4
3
ϵ +
ϵ + O(ϵ )
ϵ + O(ϵ )
∆φ = 1 − +
∆ φ2 = 2 − ϵ +
number
2 108
11664
3
162
!
"
3
2
ϵ2
6
[109ℓ
(ℓ
+
2)
+
373ℓ
− 816ℓ − 756] − 432ℓ(ℓ + 1)Hℓ−1
(3)
(3)
3
γℓ =
1−
+ ϵ δℓ
δℓ =
54
ℓ(ℓ + 1)
5832ℓ2 (1 + ℓ)2
Cφφφ2
17
1
= 1 − ϵ − ϵ2 + O(ϵ3 )
3
81
Cℓ
Cℓf ree
=1+
(ℓ(ℓ + 1) − 1) (H2ℓ − Hℓ−1 ) 2
(3) 3
ϵ
+
C
ℓ ϵ
2
2
9ℓ (1 + ℓ)
Higher Spins and the Large ℓ Limit
Another analytically tractable limit is that of large spins (double) light cone expansion. [Fitzpatrick et.al. ,Komargodski/Alday-Zhiboedov]
∆
Q
Here, simplify t-channel by using ℓ,0 (t) for large ℓ.
For “strong coupling” ( δτgap log(ℓ) ≫ 1) with minimal twist op.
Cmφφ 22−ℓm Γ (∆φ )2 Γ (2ℓm + τm )
γℓ − 2γφ = −
!
" !
"
τm 2
τm 2
Γ ∆φ − 2 Γ ℓm + 2
# $τ m
1
ℓ
Also OPE Coeff.
For “weak coupling” (δτgap log(ℓ) ≪ 1 ) with a whole tower of
minimal twist ops. but with a perturbative parameter “g”.
α0 (g) + α1 (g) log ℓ + α2 (g)(log ℓ)2 + · · ·
γℓ − 2γφ =
ℓd−2
αk = −2
d−3
(−g)
k+2
Cφφφ2
Γ
!d
2
" ! d 1 " (1) k (1)
(1)
− 1 Γ 2 − 2 (δφ2 ) (δφ2 − 2δφ )2
√
+ O(g k+3 )
k! π
The 3d Ising Model
Can combine the results of the ϵ -expansion and large ℓ limit.
Compare with numerical results on 3d Ising Model [El-Showk-PaulosPoland-Rychkov-SimmonsDuffin-Vichi…].
Dimensions of
coefficients.
Central charge
φ → σ , φ2 → ε , φ∂ ℓ φ
d2 ∆2φ
cT =
(d − 1)2 C2
2
⇒
dim
Ising model
1.4126
✏-expansion known results 1.45061
✏-expansion new results
–
Abs % deviation (old)
2.69
Abs % deviation (new)
–
and, for the first time, OPE
cT
cf ree
5ϵ2
233ϵ3
=1−
−
+ O(ϵ4 )
324
8748
dim
0.51815
0.518604
–
0.078
–
C 2
1.0518
1.3333
0.914
26.77
13.14
cT /cf ree
0.9465
0.98457
0.9579
4.02
1.20
[Hathrell,
Jack-Osborn]
` = 4 dim
5.0208
5.01296
5.02198
0.156
0.02
Laundry List
• Some Extensions (“Can this method be applied to X?” - Yes):
A. Non-identical scalars (easy).
B. Spinning external operators.
[Costa-Penedones-Poland-Rychkov]
C. N-component scalars [P.Dey-A. Kaviraj-A. Sinha].
D. Gross-Neveu-Yukawa like theories, 3d QED … [cf. Igor/Sergei talks]
E. More constraints from descendant spurious poles - systematic
procedure for solving constraint equations.
F. Numerical implementation (underway).
G. Nontrivial CFTs in dimensions greater than four (baby steps).
Thank You