Generalized Wilson-Fisher critical points from conformal

Generalized Wilson-Fisher critical points from conformal symmetry

Tassos Petkou (Aristotle University of Thessaloniki)

INFN Bologna, 3 May 2017

Based on arXiv:1604.07310 (JHEP), 1611.10344 (PRL) and 1702.03938 (JHEP) with F. Gliozzi, A. Guerrieri & C. Wen

OVERVIEW

▸ Motivation: The Atlas of Conformal Field Theories.

▸ State-of-the-art evidence: Vector-models in general dimensions.

▸ A concrete framework: Generalized Free CFTs.

▸ Deformed GFCFTs: Generalized Wilson-Fisher like critical points.

▸ Examples: multi critical points, O(N) models, multiple deformations.

▸ Outlook

MOTIVATION: THE ATLAS OF CONFORMAL FIELD THEORIES

2

d

3 4 5 6 7 · · ·8

MINIMAL MODELS,

LIOUVILLE, WZW MODELS, SUSY, LEE-YANG,…

N=1,2,4 SCFTS,…(2,0) TENSOR THEORY,…

UPPER LIMIT FOR

SCFTS

O(N) VECTOR MODELS

O(N) VECTOR MODELS

GENERALIZED FREE CFTS

STATE-OF-THE-ART EVIDENCE: VECTOR MODELS IN GENERAL DIMENSIONS

▸ Spectrum of normal free CFTs in any d:

▸ Interacting CFTs (Large-N Vector Models)

Scalar � : �2,�4, ...Tµ⌫ ...

Fermions , ̄ : ̄ , ...Tµ⌫ ...

ALWAYS THERE

e.g. scalars in d-dimensions - the mother theory

S =1

2

Z�a(�@2)�a +

Z�(�a�a � N

g) , a = 1, 2, ..N

Lagrange multiplier


▸ The “gap equation” implies:

▸ A critical point is reached when can kill the UV divergence, allowing a regular limit i.e. in .

▸ An equivalent, and suggestive form of the gap eq. is

for some “RG” parameter .

: �a�a :

N⇡ 1

g=

�(1� d2 )

2d⇡d/2�

d2�1 + d⇤

d�22F1

✓1,

d

2;d

2+ 1;� �

⇤2

◆

1/g

⇤ ! 1 d = 3

⇤6�d�(t): �a�a :

N= �2

t = ⇤2/�

UV cutoff

For this has a finite number of poles as .d > 6 t ! 0


▸ and can be calibrated to coincide at . They have a relative “weight” in .

▸ A similar story holds for fermions.

▸ The role of and was studied in an ancient version of the conformal bootstrap, that allowed a scan in . [T.P. (93-94)]

▸ Scalar and fermionic CFTs were studied for general dimensions, and results analytic in were obtained. [See also related work by A. M. Vasiliev et. al (80s) and J. Gracey. (90-93)]

� � ⌘ �2 d = 6

d > 6

N

⇤2�d�̃(t̃) ̄a a

NTrI = �1 , t̃ =⇤2

�21

�2 �1

d

d


▸ In contrast to CFTs with gauge fields, all critical quantities are proportional to the anomalous dimensions of and . �

3 4 5 6 7 8

-1.5

-1.0

-0.5

3 4 5 6 7 8

-1

1

2

3

4

5d

d

�� =d

2� 1 +

1

N��(d) � =

d

2� 1

2+

1

NTrI� (d)

Unitarity regions


▸ The dimensions of the σ-fields are

�1 = 1 +1

NTrI2

d� 2� (d)

3 4 5 6 7 8

-60

-40

-20

3 4 5 6 7 8

-0.4

-0.2

0.2

0.4

0.6

0.8

1.0

�2 = 2 +1

N

4(d� 1)(d� 2)

d� 4��(d)

d

d


▸ However, the “central charges” are given by

▸ Quite remarkably for even dimensions, they are non zero:

CbT (d) = C

(0)T (d)

"N +

cT (d)

C(0)T (d)

+O(1/N)

#

CfT (d) = C(0)

T (d)

"N +

c̃T (d)

C(0)T (d)

+O(1/N)

#C(0)T (d) =

d

2

1

S2d

TrI

cT (d) =(�1)

d2+1d(d� 4)(d� 2)!

(d� 1)�d2 + 1

�!�d2 � 1

�!S2

d

, c̃T (d) =(�1)

d2 d(d� 2)(d� 2)!

2�d2 + 1

�!�d2 � 1

�!S2

d

C(0)T (d) =

d

(d� 1)S2d

Sd =2⇡d/2

�(d/2)

d = 6, 8, 10, 12, ... d = 4, 6, 8, 10, ...

[The fermonic results first appeared in Diab et. al. arXiv:1601.07198]


▸ Moreover, the ratios and are integersc̃T (d)

C(0)T (d)

cT (d)

C(0)T (d)

4 6 8 10

-5

5

10

15

20 FERMIONSSCALARS

Integer shifts in the “central charge” for even dimensions


▸ An atlas of Vector Model CFTs for all dimensions

d2 3 4 5 6 7 8

N FREE SCALARS

N FREE FERMIONS

Non trivial CFT

The free σCFTs


▸ We suggest the existence of free CFTs - the σCFTs - for all even dimensions and that are decoupled from the normal free CFTs for special dimensions: d=even.

▸ They go beyond the older notion of Generalized Free CFTs [e.g

Papadodimas & El Showk arXiv:1101.4163] that they are non-unitary since

▸ The can be described by higher-derivative Lagrangians

▸ They are intimately and universally related to the normal free CFTs: and are universal marginal couplings.

d > 6 d > 4

�2,�1 <d

2� 1 , d > 6, d > 4

L2 =1

2�2(�@2)

d2�2�2 , L1 =

1

2�1(�@2)

d2�1�1

�2�2 �1 ̄

A CONCRETE FRAMEWORK: GENERALIZED FREE CFTS

▸ We studied these CFTs with just the help of the OPE unveiling a rich and intriguing spectrum [T.P et. al. arXiv:1604:07310]

▸ An elementary scalar conformal field has 2-pt function.

▸ It also satisfies the “elementariness” condition

▸ For we had obtained [T.P. hep-th:9410093].

which vanishes for all even .

h�(x1)�(x2)i =1

x

2��

12

�2

h�(x1)�(x2)�(x3)i = g�1

(x212x

223x

213)

��/3⌘ 0

g�2(d) = 2(d� 3)g⇤(d) , g2⇤(d) =2�(d� 2)

��3� d

2

��3

�d2 � 1

�

d > 6


▸ GFCFTs are fully determined by the 2-pt functions of their elementary fields. E.g. the 4-pt function is

▸ The conformal OPE statement is

h�(x1)�(x2)�(x3)�(x4)i =1

(x212x

234)

��

1 + v

��

✓1 +

1

(1� Y )��

◆�

u =x

212x

234

x

213x

224

, v =x

212x

234

x

214x

223

, Y = 1� u

v

h�(x1)...�(x4)i = 1

(x212x

234)

��

"1 +

g

2O2

CO2

H�O2+

1X

k=1

g

2T2,2k

CT2,2k

H�T2,2k

#

The CPWs (conf. blocks)

3-pt function coefficients 2-pt function coefficients


▸ We wish to be totally agnostic regarding the spectrum.

▸ The generic CPW for a scalar is

▸ We learn that if is nor related to , the CPW entails a natural expansion

H� = v�2

"F 0�(Y ) +

1X

n=1

vn

n!

�12�

�4n

(�)2n(�+ 1� d2 )n

Fn�(Y )

#

F ka (Y ) ⌘ 2F1

⇣a2+ k,

a

2+ k; a+ 2k;Y

⌘, F k

a+2(Y ) = F k+1a (Y ) , k = 0, 1, 2, 3..

� d1/d

H� = v�2

"F 0�(Y ) +

1X

n=1

⇣vd

⌘nH

(n)�

#


▸ A similar results holds for all higher-spin CPWs

▸ The free 4-pt function involved just the single power but infinite powers of .

▸ In fact it holds

H�s = v�s�s

2

"Y sF 0

�s+s(Y ) +1X

n=0

⇣vd

⌘nH

(n)�s

#

v��

Y

1 +1

(1� Y )��=

1X

s=0,2,4,...

↵s(��)YsF 0

2��+2s(Y )

↵s(��) ⌘g2T2,s

CT2,s

=(��)s/2(��)s

2s�1s!�� + s�1

2

�s/2

Contribution of leading twist fields with �s = 2�� + s , s = 0, 2, 4, ..


▸ But the CPWs contain an infinity of additional terms. Where are they?

▸ Terms of order , without factors arise from the scalar and spin-2 CPWs. They yield

▸ This gives

▸ To make it vanish one option is to require

v��+1 Y

v��+1

"g22C2

�3�

8(�� + 1)(2�� + 2� d)F 12��

(Y )�g2T2,2

CT2,2

4

dF 02��+2(Y )

#

v��+12�2

�(d� 2� 2��)

d(4�� + 2� d)F 12��

(Y )

�� =d

2� 1


▸ This implies that the higher-spin fields have dimensions

We find the normal free CFT whose 4-pt function is constructed just from higher-spin conserved currents.

▸ When we have some other options to “kill” the unwanted term from the OPE:

▸ An intriguing one is to fix and take the limit. This produces the same OPE as the normal free CFT.

▸ This is reminiscent of a similar limit in spin-glass systems. It is also consistent with the perception that is the mean-field theory limit.

�s = d� 2 + s HS conserved currents

�� 6= d/2� 1

�� d ! 1

d ! 1


▸ The other way to deal with is to enhance the OPE. For example, the term could be cancelled by the CPW of the scalar

v��+1

�� 6= d/2� 1

O4(x) =1

2@µ�(x)@µ�(x)� ��

2 + 2�� d

�(x)@2�(x)

hO4(x1)O4(x2)i = C4

(x212)

2��+2, h�(x1)�(x2)O4(x3)i = g4

x

212

(x213x

223)

��+1

C4 =2d�2

�(2 + 4�� d)

(2 + 2�� d), g4 = 2�2

�


▸ But then, the bag of Aeolus opens up and an infinite number of higher-twist higher-spin towers need to be included in the unitary case when .

▸ To actually evaluate one can used the modern bootstrap, or even the “old” skeleton expansion. In both cases one departs from free CFTs.

▸ However, when the twist towers terminate for even . But this requires that we enter into the realm of non-unitary CFTs.

�� > d/2� 1

��

�� = d/2� 1� ⌧ , ⌧ = 2, 4, . . .

d


▸ For the 4-pt functions of and we find:

This means that is a ghost for .

▸ Morever, for it becomes null, despite the fact that its 3-pt function with the σ’s is non-zero.

▸ What happens is that because the e.o.m. is high-derivative, one can construct a tower of higher-twist conformal primaries, until the HS current towers is reached.

�2 �1

O2(x) d = 8 , d = 6

d = 10 , d = 8

C4

��2

=8d(10� d)

6� d, C4

��1

=2d(8� d)

4� d


▸ A caricature of the OPE in free σCFTs

▸ For e.g. all fields with spins contribute to the e.m. tensor.

h�(x1)..�(x4)i ⇠ 1 + [2��]0 + [2�� + 2]2 + [2�� + 4]4 + · · ·+ [2�� + 2]0 + [2�� + 2 + 2]2 + · · ·+ · · ·+ [d� 2]0 + [d]2 + · · ·

spin-4 CPW

HS conserved currentse.m. tensor

�2 s = 2, 4, ..., d� 4

DEFORMED GFCFTS: GENERALIZED WILSON-FISHER LIKE CRITICAL POINTS.

▸ Going over to non-unitary Generalized Free CFTs gives an unexpected bonus: the analytic structure of the free OPE becomes interesting.

▸ Up to an overall factor we have (I now switch to the “modern” notation..)

▸ This contains terms like that can be expanded as

Conformal block

h�1(x1)...�4(x4)i ⇠ g(v, u)

u�

u� =1X

⌧=0

1X

`=0

ca,b⌧,` Ga,b2�+2⌧+`,`(u, v)


▸ The OPE coefficients are calculated as

▸

▸ with

▸ Setting these have poles at

ca,b⌧,` =(�1)`(2⌫)`

⌧ !`!(⌫)`(⌫ + `+ 1)⌧

(a+ �)`+⌧ (b+ �)`+⌧ (a+ � � ⌫)⌧ (b+ � � ⌫)⌧(2� + `+ 2⌧ � 1)`(2� � ⌫ + `+ ⌧ � 1)⌧ (2� � 2⌫ + ⌧ � 1)⌧

⌫ = d�22 , a = � 1

2�12 and b = 12�34 with �ij = �i ��j .

�` = 2� + `+ 2⌧

�? = 1� `+ k (k = 1, 2, . . . , `) ,

�? = 1 + ⌫ + k (k = 1, 2, . . . , ⌧) ,

�? = 1 + `+ 2⌫ + k (k = 1, 2, . . . , ⌧)


▸ Since the expanded function is regular, the poles must cancel.

▸ This can be done by conformal blocks like that must be present in the OPE.

▸ Indeed, generic conformal blocks have poles for certain values of dimensions and spin [Kos et. al. arXiv:1406.4858]

▸ Notice that and are different.

Ga,b2�+2⌧ 0+`0,`0(u, v)

Ga,b�0,`0(u, v) ⇠

R

�0 ��?0Ga,b

�?,`(u, v)

�⇤0 �⇤


▸ The deep reason for this cancellation is the fact that the conformal blocks with with and have the same eigenvalues wrt to the quadratic and quartic Casimirs.

�⇤ �⇤0

C2(�?, `) = C2(�?0 , `0) , C4(�?, `) = C4(�?0 , `0)

C2(�, `) = �(�� d) + `(`+ d� 2) ,

C4(�, `) = �2(�� d)2 +1

2d(d� 1)�(�� d)

+ `2(`+ d� 2)2 +1

2(d� 1)(d� 4)`(`+ d� 2) .


▸ Solving the Casimir equations, we get three series of poles

▸ are the poles found in [Kos et. al. arXiv:1406.4858]

▸ are the poles of our OPE coefficients.

▸ Then, schematically it holds

�?0 �? `1� `0 � k 1� `+ k `0 + k k = 1, 2, . . .

d2 � k d

2 + k `0 k = 1, 2, . . .d+ `0 � k � 1 d+ `+ k � 1 `0 � k k = 1, 2, . . . , `

�⇤0

�⇤

c(� ! �⇤) =R̃

��⇤) R̃+ c(�⇤0)R = 0


▸ We learn that the spectrum of Generalized Free CFTs is actually very rich.

▸ Nevertheless, their 4pt function look very simple, due to some highly nontrivial cancellations.

▸ In CFT language (Zamolodchikov), a descendant becomes a null state, and it is cancelled out.

▸ But in an interacting CFT descendants may become quasi primaries. Then it is tempting to try to “guess” the deformed 4pt function by matching it smoothly to the known free 4pt function as [Rychkov & Tan, 1505.00963, Nakayama unpublished notes]

lim�!�⇤,d!d⇤

gint(v, u) = gfree(v, u)

EXAMPLES: MULTI CRITICAL POINTS, O(N) MODELS, MULTIPLE DEFORMATIONS

▸ Example at . Consider the normal free CFT d = 3� ✏

h�2(x1)�3(x2)�

2(x3)�3(x4)i ⇠ gfree(v, u)

gf (u, v) = 3Gaf ,bf��f

,0(u, v) + 10Gaf ,bf��5

f,0(u, v) + spinning blocks

gI(u, v) = (3 +O(✏))Ga,b��,0

(u, v) + . . .

a = b = 12 (��3 ��2), and ��i = ��i

f+ �(1)

i ✏+ �(2)i ✏2 + . . .


▸ The conformal block of the interacting theory has a pole as

▸ This should reproduce the free OPE, hence

▸ This means that and we find

Ga,b��,0

(u, v) =(�(1)

3 � �(1)2 )2✏2

12(�� f )G

af ,bf��5

f,0(u, v) + . . .

3(�(1)

3 � �(1)2 )2✏2

12(�� f )= 10

�(1)1 = 0

(�(1)3 � �(1)

2 )2

�(2)1

= 40


▸ If we next consider

▸ The interacting OPE has now the pole

h�(x1)�2(x2)�(x3)�

2(x4)i

gf (u, v) = 2Gaf ,bf��f

,0(u, v) + 3Gaf ,bf��3

f,0(u, v) + spinning blocks

gI(u, v) = (2 +O(✏))Ga,b��,0

(u, v) + . . .

a = b =1

2(��2 ��)

Ga,b��,0

(u, v) =(�(1)

2 )2

12�(2)1

Gaf ,bf��5

f,0(u, v) + . . .


▸ But is not in the free OPE, hence we must have leading to

▸ One can do a similar analysis for other correlation functions obtaining the known results

�5f �(1)

2 = 0

(�(1)3 )2 = 40 �(2)

1

�1 =1

1000✏2 +O(✏) , �2 =

3

100✏2 +O(✏)

�n =1

30n(n� 1)(n� 2)✏+O(✏2)


▸ We can play this game for all non-unitary GFCFTs with

▸ We consider OPEs of the form

▸ In the interacting theory, the OPE coefficients are deformed by terms that vanish when

▸ Moreover, the operator becomes a descendant and it should drop out of the spectrum.

▸ Essentially, the above operator becomes the now regular part of the conformal block of

��f =d

2� k, k = 1, 2, 3, ..

�p�p+1 ⇠ �2n�1 + · · · , n = 1, 2, 3...

✏ ! 0

�2n�1

�


▸ The generic conformal block of has a singularity when

▸ Hence, the above mechanism (multiplet recombination) happens when

▸ This is equivalent to having a marginal e.o.m. of the form

�

� !=d

2+ k

d =2nk

n� 1

@2k� ⇠ �2n�1


▸ Our results for the anomalous dimensions are:

▸ and for the OPE coefficients

�(2)1 = (�1)k+12

n⇣

kn�1

⌘

k

k⇣

nkn�1

⌘

k

(n� 1)2(n!)2

(2n)!

�3

�(1)p =

(n� 1)

(n)n(p� n+ 1)n

C1,p,p+2m+1 = ✏2(n!)3

(2n)!

�2(n� 1)2�(p+ 1)�(2m+ p+ 2)

�2(n�m)�2(n+m+ 1)�2(m� n+ p+ 2)

⇥

⇣k

n�1

⌘

k⇣nkn�1

⌘

k

�2(k) ( kn�1 )2kh

( (m+1)kn�1 )k (

�mkn�1 )k

i2


▸ We can study O(N) models the same way….

▸ We can also study models with marginal deformations (like our σCFTs before in ) where the theory is deformed by and (we also assume O(N) symmetry).

▸ Here becomes a descendant of .

▸ However the following remains a primary.

d = 6� ✏�3 �2�

Od = g1�2 + g2�

2 �

Op =1p

2Npg21N + g22

(g1N�2 � g2�2)


▸ We obtain the following results that are consistent with the complicated loop calculations of [Giombi et. al. arXiv: 1404.1094]

�(1)�

�(1)�

=2g21

g21N + g22C2

�� =12 g22

g21N + g22�(1)� ✏

C�i�i�C��

6�(1)� ✏

� 2g1g2g21N + g22

= 0

OUTLOOK

▸ Generalized Free CFTs provide the backbone structure in the web of CFTs in any dimension.

▸ They have a rich analytic structure.

▸ They provide crucial information regarding the theory space near them.

▸ Using this information, we have confirmed all known results for the leading order anomalous dimensions at multicritical points in d=3,4,6 and and also 2<d<3 dimensions, including those for theories with O(N) symmetry.

▸ We have new results for non-unitary theories in d>6.

▸ The holographic duals?

▸ The d—> infinity limit?

▸ The relationship to real systems with long-range interactions?

Documents

Generalized Wilson-Fisher critical points from conformal