[PPT] - Playing with multi-critical systems: RG and CFT approaches in the PowerPoint Presentation

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Playing with multi-critical systems: RG and CFT approaches in the ε-expansion in d>2

Gian Paolo Vacca INFN - Bologna

Cagliari , 11 Ottobre 2017

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Based on

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A. Codello, M. Safari, G.P.V., O. Zanusso
M. Safari, G.P.V.
JHEP 1704 (2017) 127 [arXiv:1703.04830]
[arXiv:1705.05558]
Phys. Rev. D98 (2017) 081701 [arXiv:1706.06887]
[arXiv:1708.09795]

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Outline

Introduction: universal data of critical theories from
A new non trivial example: higher derivative multicritical theories
Functional Perturbative RG
Conclusions

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CFT
Multicritical theories

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Introduction

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Physical systems, very different at microscopic level, can show phases characterized by the same Universal behavior when the correlation length diverges (2nd order phase transition). Most famous example: 3D Ising universality class (Magnetic systems, Water) in a Landau-Ginzburg description as a scalar QFT,

S = J ∑

hiji

sisj + B ∑

i

si si = ±1

Critical phenomena are conveniently described by Quantum and Statistical Field Theories.

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5

Critical theories

Theory space The critical theories are points in a suitable theory space characterized by scale invariance. If there is Poincare’ invariance it is

ften lifted to conformal invariance

(fields and symmetries)

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Fundamental physics in a QFT description require renormalizability conditions

which in the most general case goes under the name of Asymptotic Safety: existence of a fixed point with a finite number of UV attractive directions. Asymptotic freedom is a particular case with a gaussian fixed point. In a Renormalization Group description critical field theories are associated to fixed points of the flow, where scale invariance is realized.

These fixed points may control the IR behavior of the theories.

(example: Wilson-Fisher fixed point)

RG

Perturbation theory in presence of small parameters,

e.g. ε-expansion below the critical dimension

Wilsonian non perturbative, exact equations but not solvable in
practice. (Polchinski and Wetterich/Morris equations)

Formulations:

Wilson (1971), Wilson and Fisher (1972)

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CFT

Critical theories often show an enhanced conformal symmetry In d=2 it is a infinite dimensional Virasoro symmetry, but also in d>2 one can take advantage of the SO(d+1,1) symmetry group. Recently the old proposal of Polyakov was pushed forward in what is called Conformal Bootstrap, based on the consistency of conformal block expansions of the 4 point correlators (in s,t channels) Also in CFT the perturbative ε-expansion is very useful and several different aproaches are available. Conformal data: a CFT is fixed by the scaling dimensions of the primary

perators and by the structure constants defining their 3 point correlators.

hOa(x)Ob(y)i = ca ab |x y|2∆a

hOa(x)Ob(y)Oc(z)i = Cabc |x y|∆a+∆b−∆c|y z|∆b+∆c−∆a|z x|∆c+∆a−∆b

El-Showk, Paulos, Poland, Rychkov, Simmons-Duffin and Vichi (2012)

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The main constraints are given by the field content and the symmetries, but this leaves still too many possible theories for a generic dimension d. It is therefore useful to start from some kind of Landau-Ginzburg description to single out some possible solutions.

This is the starting point for an RG analysis.
In a CFT this leads to include the Schwinger-Dyson Equations (SDE) which

force a recombination in multiplet of composite operators (in particular changing the nature from primary to descendant).

Lagrangian description

⌧S (x) O1(y) O2(z) . . .

= 0

Rychkov, Tan (2015) Nii (2016) Hasegawa and Nakayama (2017) Codello, Safari, G.P.V., Zanusso (2017) Basu, Krishnan (2015)

Ignore contact terms S = Z ddx X

i

giOi(φ)

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9

(Functional) perturbative RG: systematic expansion but resummation needed Functional nonperturbative RG: very powerful but no fully systematic way to organize corrections available. RG is generally affected by scheme dependence but it is very powerful CFT: using the full machinery at analytic level is in general very complicated. CFT is not scheme dependent! Conformal bootstrap is hard to apply for more complicated models Perturbative approaches share the convergence problems with RG

RG and CFT : pro et contra at criticality

Can we obtain in some approximation the same results in the two approaches?

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It can be partially done in the perturbative ε-expansion approximation using

the universal beta function coefficients, e.g. in a massless scheme How to get in an RG framework informations on the critical theory? If conformal, the so called conformal data?

Universal data and RG

g MS

Critical quantities are encoded in the expansion coefficients describing the flow around the scale invariant point:

βk(g∗ + δg) = ∑

i

Mk

i δgi + ∑ i,j

Nk

ij δgi δgj + O(δg3) ,

βi(g∗) = 0 .

Mi

j ≡ ∂βi

∂gj

∗

Ni

jk ≡ 1

2 ∂2βi ∂gj∂gk

∗

Moving to a diagonal basis in the linear sector

∗

∑

i,j

Sai Mij (S−1)j

b = −θaδa b .

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ne can extract not only the scaling dimensions, but also, reversing an argument

from Cardy for a CFT, some OPE coefficients (structure constants) at order O(ε)

Universal data and RG

a = (d ∆a)a + X

b,c

˜ Cabc bc + O(3) .

S = S∗ + X

a

µθaa Z ddx Oa(x) + O(2) .

RG flow seen along the eigendirections around the fixed point up to second order

hOa(x) Ob(y) · · · i = X

c

1 |x y|∆a+∆b−∆c Ccab hOc(x) · · · i

˜ Ca

bc = ∑ i,j,k

Sa

i Ni jk (S1)j b (S1)kc ,

θa = d − ∆a

Take home message

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Scheme dependence

RG scheme changes correspond to a coupling reparameterization

¯ gi = ¯ gi(g)

¯ bi( ¯ g) = ∂ ¯ gi ∂gj bj(g) .

Linear term coefficients transform homogeneously

¯ Mi

j = ∂ ¯

gi ∂gl Ml

k

∂gk ∂ ¯ gj .

Quadratic term coefficients transform inhomogeneously

¯ Ni

jk = ∂ ¯

gi ∂gc n Nc

ab + 1

2 Mc

d

∂2gd ∂ ¯ gl∂ ¯ gm ∂ ¯ gl ∂ga ∂ ¯ gm ∂gb − 1 2 Mda ∂2gc ∂ ¯ gl∂ ¯ gm ∂ ¯ gl ∂gb ∂ ¯ gm ∂gd

− 1

2 Md

b

∂2gc ∂ ¯ gl∂ ¯ gm ∂ ¯ gl ∂ga ∂ ¯ gm ∂gd

∂ga

∂ ¯ gk ∂gb ∂ ¯ gj . ¯ ˜ Cc

ab = ˜

Cc

ab + 1

2 (θc − θa − θb) ∂2gc ∂ ¯ gl∂ ¯ gm ∂ ¯ gl ∂ga ∂ ¯ gm ∂gb ,

⇒ ⇒

¯ θa = θa ,

Take home message

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Scheme dependence

Invariance condition Condition for scheme independence In general not fulfilled. But it can be at the critical dimension.

¯ ˜ Cc

ab = ˜

Cc

ab +⇔ 1

2 (θc − θa − θb) ∂2gc ∂ ¯ gl∂ ¯ gm ∂ ¯ gl ∂ga ∂ ¯ gm ∂gb ,= 0

θc − θa − θb = 0

g MS

Employing the ε-expansion and scheme dimensionless OPE coefficients are less sensitive to scheme changes

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Ising Universality Class

Two fixed points:

ε-expansion below d=4 for the LG critical model

L = 1 2(∂φ)2 + gφ4

12gφ2 12gφ2

Leading counterterms in perturbation theory at order , dim reg

g2

UV gaussian IR Wilson-Fisher Anomalous dimension:

η = 2˜ γ1 = 96g2

Rescaling the coupling: g → (4π)2g

g∗ = ✏ 72 g∗ = 0 ⌘ = ✏ 54

is a universal quantity, independent from any coupling reparameterization!

η

Lc.t. = 1 ✏ 1 2(4⇡)2 (12g)24 − 1 ✏ 1 6(4⇡)4 (4!g)2(@)2

beta function:

g MS

g = −✏g + 72g2

d = 4−✏

4! gφ 4! gφ

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Anomalous dimension from scale invariance and SDE Interacting 2-point function at criticality:

∆ = δ + γ1

hφxφyi = c |xy|2∆

EOM:

h2xφx2yφyi = 16g2hφ3

xφ3 yi ' 16g23!

c3 |xy|6

2x2yhφxφyi = c2∆(2∆ + 2)(2∆ + 2 d)(2∆ + 4 d) |xy|2∆+4 ' 32cγ1 |xy|6

c = 1 4π2

δ = d 2 −1

⇒

Rescaling the coupling as before: g → (4π)2g

⇒

In agreement with the 2-loop result! At leading order γ1 = 3g2c2 + O(g3) γ1 = 48g2 + O(g3)

2φ = 4gφ3

Ising Universality Class

d = 4−✏

Take home message

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Assuming conformal invariance and the SDE in ε-expansion

First partial studies for Ising (also O(N)),

Tricritical and Lee-Yang UC

Leading CFT constraints on multicritical theories

Systematic full study for all single scalar field multicritical models

Rychkov, Tan (2015) Nii (2016) Hasegawa and Nakayama (2017) Basu, Krishnan (2015)

A. Codello, M. Safari, G.P.V., O. Zanusso JHEP 1704 (2017) 127

Landau-Ginzburg lagrangian

S[] = Z ddx n1 2(@)2 + µ( m

2 −1)✏ g

m!mo

dm = 2m m − 2 .

Upper critical dimension (unitary: e.g. Ising, Tricritical,…) (non unitary: e.g. Lee-Yang, Blume-Capel,…)

d = dm−✏

m = 2n m = 2n+1

even

dd

dc = 4, 3, · · · dc = 6, 10 3 , · · ·

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Leading CFT constraints on multicritical theories

EOM: 2φ = g (m − 1)!φm−1 Primary operators ( )

[φi]

i 6= m1 Composite operators Scaling dimensions

∆i = iδ + γi = d 2 −1 = m− ✏ 2 δm = 2 m−2

Free theory

hφk(x)φl(y)i free = δkl k!

ck

|x y|2kδm , c = 1 4π Γ(δm) πδm = 1 (dm2)Sdm .

hφn1(x1)φn2(x2)φn3(x3)i free =

Cfree

n1,n2,n3

|x1x2|δm(n1+n2n3)|x2x3|δm(n2+n3n1)|x3x1|δm(n3+n1n2) ,

(A.7)

Cfree

n1,n2,n3 =

n1! n2! n3! ⇣

n1+n2n3 2

⌘ ! ⇣

n2+n3n1 2

⌘ ! ⇣

n3+n1n2 2

⌘ ! c

n1+n2+n3 2

Together with the constraints on 2 and 3-points CFT correlators one can compute the leading non trivial values for the scaling dimensions and families of structure constants

∆a

Cabc

∆m1 = ∆1 + 2

= )

gm1 = g1 + (m2)e 2 .

Note that for ✏ 6= 0

, ni+nj−nk ≥ 0

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Leading CFT constraints on multicritical theories

Some examples: m = 2n

2x2y hf(x)f(y)i

| |

LO

=

16n

(n1)2 g1

c

|x y|4+

2 n1

,

)i LO =

g2

(2n1)!

c2n1

|x y|4+

2 n1

g1 = 2(n1)2

(2n)!

Γ ⇣

1 n1

⌘2(n1) g2

(4p)2n + O(g3) ,

Itzykson, Drouffe (1989)

2x2y hf(x)f(y)f2(z)i

g2 = g

(4p)2 + O(g2) ,

n = 2 . g2 = 8(n+1)(n1)3 (n2)(2n)! Γ ⇣

1 n 1

⌘

2(n1)

g2

(4p)2n + O(g3) ,

n > 2 .

2x hf(x)fk(y)fk+1(z)i

gk = 2(n1) n!2 k!

(kn)!Γ

⇣

1 n1

⌘n1 g

(4p)n + O(g2) ,

k n ,

J. O’Dwyer, H. Osborn (2008)

The last relation is valid, in particular, for , so that

k = 2n−1

) =

n!3

(2n)!(4p)nΓ

✓ 1 n1 ◆1n e + O(e2) .

g

(II.6), g2n1 = g1 + (n1)e,

descendant operator

2n

Anomalous dimensions
2x hf(x)f(y)f2(z)i

Take home message

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Leading CFT constraints on multicritical theories

Structure constants

⇤x ⇤xhφ(x) φ2k(y) φ2l1(z)i

C1,2k,2l1 = n!3

(2n)! (n1)2 (k l)(k l + 1) (2k)!(2l1)! (n+lk1)!(k+nl)!(k+ln)!e + O(e2) ,

namely k + l n, 1n (l k)  n, l k 6= 0, 1,

2x2y hφ(x)φ(y)φ2k(z)i



C1,1,2k =

(n1)4

k(k1)(kn)(kn+1) n!6

(2n)!2 (2k)!

k!2(2nk1)!e2 + O(e3) ,

for k 6= n1, n and 2  k  2n 1.

note, however, that for comparison with

The results for as functions of the coupling extend to the case of odd theories, for which one has also

γ1, γ2, C1,k,l, C1,1,2k

2x2y2z hφ(x)φ(y)φ(z)i

C111 =

(2n−1)6

28n(n−1)n!3(c

n− 1

2

dd g)3 + O(g4) .

codd = 1 4p Γ

2

2n 1

p

2 2n1

.

Take home message

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Functional perturbative RG example: Ising UC

How to study deformations around the Wilson-Fisher fixed point?

d = 4 − ✏

Dimensionful beta functions (global rescaling as before)

L = 1 2(@)2 + g1 + g22 + g33 + g44

1 = 12 g2g3 108 g3

3 288 g2g3g4 + 48 g1g2 4

2 = 24 g4g2 + 18 g2

3 1080 g2 3g4 480 g2g2 4

3 = 72 g4g3 3312 g3g2

4

4 = 72 g2

4 3264 g3 4

Couplings: Functions: 1 loop 2 loop

a = 1 2 b = −1 2

L = 1 2Z()(@)2 + V ()

2 loop

c = −1 6

V = 1 2⌘V (1) + a(V (2))2 (4⇡)2 + bV (2)(V (3))2 (4⇡)4 + · · · Z = ⌘Z + 1 2⌘Z(1) + c(V (4))2 (4⇡)4 + · · ·

Field independent Z anomalous dimension Take home message

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FPRG for multicritical models

RG, even RG, odd

Codello, Safari, G.P.V., Zanusso arXiv:1705:05558 Codello, Safari, G.P.V., Zanusso Phys. Rev. D98 (2017) 081701 O’Dwyer, Osborn (2008)

We limit to a truncation

L = 1 2Z()(@)2 + V ()

Rescaling functions and fields to dimensionless quantities v(ϕ), z(ϕ)

bv = d v(j) + d 2 + h 2 j v0(j) + n 1 n! cn1 4 v(n)(j)2

n 1

48 c2n2 Γ(dn) ∑

r+s+t=2n r, s, t 6= n

Kn

rst

r!s!t! v(r+s)(j) v(s+t)(j) v(t+r)(j)

(n 1)2

16 n! c2n2 ∑

s+t=n

n 1 + Ln

st

s!t! v(n)(j) v(n+s)(j) v(n+t)(j) , Kn

rst =

Γ nr

n1

Γ

ns

n1

Γ

nt

n1

Γ
r

n1

Γ
s

n1

Γ
t

n1

, Ln

st = y(dn) y(sdn) y(tdn) + y(1) ,

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FPRG for multicritical models

V (r) r V (r)

r s t V (r+s) V (r+t) V (s+t)

V (s+t) s V (s) V (t) t V (r)

c.t.

r V (r)

Contributions from multi-loop diagrams: LO NLO

z = ⌘ z(') + d − 2 + ⌘ 2 ' z0(') − (n − 1)2 (2n)! c2n2 4 v(2n)(')2 +n − 1 n! cn1 2 h z(n)(') v(n)(') + z(n1)(') v(n+1)(') i

Z(r)(∂φ)2 r V (r) Z(r−1)∂φ Z(r−2)

First one finds the fixed point for the critical coupling

(2n)!2

4 n!3 cn1g = e n n1h + n!4

(2n)!

1 3 Γ(dn) n!2 ∑

r+s+t=2n r, s, t 6= n

Kn

rst

(r!s!t!)2 + (n1) ∑

s+t=n

n1 + Ln

st

s!2t!2

e2 .

Then one expands in all the couplings associated to all operators

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FPRG for multicritical models

General pattern of mixing

V : 1 φ · · · φ2n−1 φ2n

· · ·

φ4n−3 φ4n−2

· · ·

Z :

(∂φ)2 · · ·

φ2n−3(∂φ)2 φ2n−2(∂φ)2 · · · W1 : φ⇤2φ

· · ·

W2 :

(∂µ∂νφ)2 · · ·

W3 :

(⇤φ)2 · · ·

At leading order the stability matrix M is triangular.

≥

˜ gi = 2(n − 1)n!

(2n)!

i!

(i − n)! e

˜ wi = 2(n − 1)n!

(2n)! (i + 1)! (i − n + 1)! e .

B B B @ M(0) M(2) M(4) ... 1 C C C A

˜ gi = ih 2 + (n 1)i!

(i n)!

2 n!

(2n)!

 e n n 1 h

+ 2n h d2n

i

+ (n 1)i!n!6 (2n)!2

Γ(dn) ∑

r+s+t=2n r, s, t 6= n

Kn

rst

(r!s!t!)2

 2n! 3(i n)! r!

(i 2n + r)!

e2

+ (n 1)2i!n!5 (2n)!2

∑

s+t=n

n 1 + Ln

st

(s!t!)2

 1

(i n)!

2s! n!(i 2n + s)!

e2.

OPE coefficients are read off the quadratic expansion of the beta functions

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The Blume-Capel or Ticritical Lee-Yang

A nontrivial (non unitary) UC in Dimensionless beta functions

d = 3

d = 10 3 −✏

bv = 10 3 v + 2 3 jv0 + e ✓ v 1 2 jv0 ◆

+ h

2 jv0

+ 1

3v(2)(v(4))2 3 2(v(3))2v(4) , bz = 2 3 jz0 + h ✓ z + 1 2 jz0 ◆

1

30(v(5))2 .

L = 1 2(∂φ)2 + gφ5

qi = 10 3 − 2i 3 + e ⇣

−1 + i

2 ⌘

− ˜

gi ˜ gi = e 153 ⇣52 5 i − 139 12 i2 − 1 2i3 + 19 12i4 − di,5 ⌘ ,

⌘ = 2˜ 1 = − ✏ 765

Critical exponents

⌫ = ✓−1

2

= 1 2 − 7✏ 1020

OPE coefficients: overlap with CFT +SDE in

are

˜ C115

√ ˜

g1 = 4

√

15 + O(e) ,

˜ C124

√ ˜

g1 = 32

√

15 + O(e) ,

˜ C133

√ ˜

g1 = −108

√

15 + O(e) , (4.2)

g(e) =

pe

60

p

102 .

Fixed point: CFT results can be completed using

g(✏)

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HIgher derivative multicritical scalar theories

Higher derivative models are getting some attention in the last years. They are non unitary, but can be physically relevant.

Brust, Hinterbichler (2017) Gracey (2017) Gliozzi, Guerrieri, Petkou, Wen (2017) Nakayama (2016)

Motivations: Theory of elasticity (e.g. Riva-Cardy model) Quantum Gravity (possibly related in various ways) Theoretically some correspond to new families of non unitary CFT also in higher dimensions. Recent works Free theories Multicritical in CFT O(N) quartic in RG Physics of polymers (isotropic Lifshitz theories)

Schwahn et. al. (1999) Osborn, Stergiou (2016)

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We consider the symmetric theories, with a kinetic term S =

Z ddx 1

2φ (−⇤)kφ

dc = 2nk n − 1 Z2

The critical theory at can have marginal interactions characterized by symmetric operators with derivatives and fields if (dimensional analysis)

d = dc−✏

ε-expansion RG analysis We want to compare to (and go beyond) recent CFT results. At the critical dimension the theory is free

dc

[φ] = δ = d 2 − k

Field dimension Free propagator

Z2

with integer and an even integer. Examples:

φ2n φ2n φn−1(∂φ)2

c = 1 (4π)kΓ(k) Γ(δ) πδ

G0(x) = c |x|2δ

2l α

l

α

α k n − 1 + 2l = 2nk n − 1

k = 1 → (l = 0, α = 2n) k = 2 → (l = 0, α = 2n) k = 2 → (l = 1, α = n + 1) → n = 1 + 2m

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Therefore in a perturbative framework there is a pattern of mixing of operators with a different number of derivatives (relevant, marginal or irrelevant) We find that a recent analysis ( ) done in a CFT framework is valid only for a subset of the theories such that the pair of integers are coprime, so that they correspond to the case

f a pure interaction operator at the fixed point, with one critical

coupling (theories of first kind)

F. Gliozzi, A. Guerrieri, A. C. Petkou and C. Wen (2017)

φ2n

(k, n−1)

In the other cases (theories of second kind) the pattern increases of complexity with since the critical theory can be characterized by several couplings.

k

We shall address here only the family of theories with

which are characterized by two critical couplings. Novel pattern.

(k = 2, n = 1+2m)

Improving the present picture

We reproduce these results obtaining some higher order results
M. Safari and G.P.V.

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28

v ! v [(4⇡)kΓ(k)]n Γ(n)/Γ()n

Kn,k

rst ⌘ Γ ((n r)) Γ ((n s)) Γ ((n t))

Γ (r) Γ (s) Γ (t) ,

Jn,k

st

⌘ (n) (s) (t) + (1)

βv = − dv + d − 2k + η 2 φ v0 + 1 n! v(n) 2 −Γ(nδn) 1 3 X

r+s+t=2n r, s, t 6= n

Kn,k

rst

r!s!t! v(r+s)v(s+t)v(t+r) − 1 n! X

s+t=n

Jn,k

st

s!t! v(n)v(n+s)v(n+t)

(2n)!2 n!3 g = (n 1)✏ n ⌘ + n!4(n 1)2 (2n)! 1 3 Γ(nn) n!2 X

r+s+t=2n r, s, t 6= n

Kn,k

rst

(r!s!t!)2 + X

s+t=n

Jn,k

st

s!2t!2

✏2

φ = ⌘ 2 = (−1)k+1 n(n)k k(n + k)k 2(n − 1)2n!6 (2n)!3 ✏2

Working with dimensionless quantitites with convenient rescalings

coprime (first kind)

(k, n−1)

The relevant diagrams are similar to the case k=1. We go up to 2(n-1) loops. Beta functions constructed looking at the poles in scheme

1/✏

g MS

Fixed point relation: Anomalous dimension: with beta functional

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29

Examples (first kind)

k = 2, n = 2

k = 3, n = 2

k = 3, n = 3

dc = 8

dc = 12

dc = 9

v = −dv + d − 4 + ⌘ 2 'v(1) + 1 2 ⇣ v(2)⌘2 + 1 12v(2) ⇣ v(3)⌘2 v = −dv + d − 6 + ⌘ 2 'v(1) + 1 2 ⇣ v(2)⌘2 + 43 120v(2) ⇣ v(3)⌘2

v = −dv + d − 6 + ⌘ 2 'v(1) + 1 6 ⇣ v(3)⌘2 − 35⇡2 8192 ⇣ v(4)⌘3 + 31 1260v(3)v(4)v(5) − 7 144v(2) ⇣ v(5)⌘2

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30

Agreement with CFT looking at correlators of composite operators .

φi

From CFT known

nly up to order O(ε)!

Results

OPE coefficients Anomalous dimensions for composite operators

F. Gliozzi, A. Guerrieri,
A. C. Petkou and C. Wen (2017)

X ˜ i = ⌘ 2i + (n 1)i! (i n)! 2n! (2n)!  ✏ n n 1 ⌘

+(n 1)2i!n!6

(2n)!2 Γ(nn) X

r+s+t=2n r, s, t 6= n

Kn,k

rst

(r!s!t!)2  2n! 3(i n)! r! (i 2n + r)!

✏2

+(n 1)2i!n!5 (2n)!2 X

s+t=n

Jn,k

st

s!2t!2  1 (i n)! 2s! n!(i 2n + s)!

✏2

˜ Ck

ij = 1

n! i! (i − n)! j! (j − n)! − Γ(nn)(n − 1)n!3 (2n)! X

r+s+t=2n r, s, t 6= n

Kn,k

rst

r!s!t!2 j! (j − s − t)! i! (i + s − 2n)! ✏ − (n − 1)n!2 (2n)! X

s+t=n

Jn,k

st

s!t!  1 n! j! (j − n − s)! i! (i − n − t)! + 1 s! i! (i − n)! j! (j − n − s)! + 1 s! j! (j − n)! i! (i − n − s)!

✏

SLIDE 31

31

CFT : derivation (without computing loops)

For this critical theory the SDE are given

2kφ = (−1)k−1 g (2n−1)!φ2n−1

∆i = ✓d 2 −k ◆ i + γi

Scaling dimensions of :

φi

2k

x2k yhφxφyi =

g2 (2n1)!2 hφ2n

− 1 x

φ2n

− 1 y

i

Anomalous dimension starting from

γ1

hφxφyi = c |x y|2∆1

(−1)k−124kkΓ(k)2 ✓ k n−1 ◆

2k

c γ1 = g2 (2n−1)!c2n−1

2k

xhφxφm y φm + 1 z

i = g(1)k

− 1

(2n1)! hφ2n

− 1 x

φm

y φm + 1 z

i

Recurrence relation for from

γm+1−γm

One can find the structure constants starting from

C1,2m,2l−1

2k

xhφxφ2m y φ2l − 1 z

i = g(1)k

− 1

(2n1)! hφ2n

− 1 x

φ2m

y φ2l − 1 z

i

All the methods give the same perturbative results.

Fixed point coupling imposing descendant condition on

g(✏) γ2n−1 Evaluate r.h.s. at tree level

SLIDE 32

32

The case (second kind)

(k = 2, n = 1+2m)

L = 1

2⇤2 + V + 1 2Z(@)2 + 1 2W1⇤2 + 1 2W2(@)2⇤ + 1 2W3(@)4

Several diagrams contribute to quadratic and even more to cubic terms in the potentials (couplings). In perturbation theory we compute the counter-terms as poles in the scheme, at functional level. For example

1/✏

g MS

c = k n − 1 = 1 m, dc = 2nk n − 1 = 4m + 2 m = 4 + 2 m, Ul,c.t. = 1 2 r! 1 (4⇡)(r−1)(k+δc)Γr(k) Γr(c) Γ(rc) (−1)l l! 2 (r − 1)✏ V (r)()(−⇤)lV (r)()

V 2 counterterms

Similarly at quadratic order one has and counterterms

V Z

Z2

Cubic c.t. can also be computed. Non local divergent terms are present in separate diagrams but cancel in the sum. Critical theory Including some deformations

LFP = 1 222 + 1 2h2m(@)2 + g2(2m+1)

SLIDE 33

33

Beta functionals at quadratic order (dimensionless) from melon diagrams

βv = dv+ d4 + η 2 ϕv(1) + v(m

+ 1)z(m

1)

(m+1)! + (v(2m

+ 1))2

(2m+1)! βz = 2z+ d4 + η 2 ϕz(1)+2v(2m+1)z(2m+1) (2m+1)! + z(m+1)z(m1) (m+1)! + 3m+2 2(2m+1) (z(m))2 (m + 1)! 2(m + 1) (2m + 1) (v(3m+2))2 (3m+1)! βw1 = ηw1+d4 + η 2 ϕw(1)

1 +(m+1)2Γ(δc)

m4Γ(4+δc) (v(4m+2))2 (4m+1)! (m+1)Γ(δc) m3Γ(3 + δc) z(3m) v(3m+2) (3m+1)!

Γ(δc)

m2Γ(2+δc) (z(2m))2 2(3m+1)(2m)!

defines the anomalous dimension at FP needed if looking for properties of some irrelevant operators.

βw1

η = −βw1(ϕ = 0)

βw2,3

(k = 2, n = 1+2m)

m 2m -loops 2m m m 3m -loops 4m 3m 2m -loops

SLIDE 34

34

Pattern of mixing

1 φ · · · φ2m+1 φ2(m+1) · · · φ4m+1 φ2(2m+1) · · · φ6m+1 · · · 1 · · · φ2m−1 φ2m · · · φ4m−1 · · · ×(∂φ)2 1 · · · φ2m−1 · · · ×(∂2φ)2

(k = 2, n = 1+2m)

Fixed points

2m✏g = (2(2m + 1))!(2m)! (3m + 1)!(m + 1)!2 g h + (2(2m + 1))!2 (2m + 1)!3 g2 m✏h =  3m + 2 2(2m + 1) + m m + 1

(2m)!2

(m + 1)!m!2 h2 − 2(m + 1) 2m + 1 (2(2m + 1))!2 (3m + 1)!m!2 g2

Anomalous dimensions (from triangular part of the stability matrix)

˜ i = i! (i − m − 1)! (2m)! (m + 1)!2 h + i! (i − 2m − 1)! 2(2(2m + 1))! (2m + 1)!2 g, i = m + 1, · · · , 3m + 1

˜ 1 = Γ(c) m2Γ(2 + c) (2m)! 4(3m + 1)h2 − Γ(c) m4Γ(4 + c)2(m + 1)2(2m + 1)2(4m + 1)! g2 ˜ ωm−1 = (2m)! (m + 1)!h

(first non zero)

Others have non trivial mixing giving complicated expressions.

SLIDE 35

35

One fixed point is simple: pure derivative interaction

(k = 2, n = 1+2m)

Anomalous dimensions

h = 2m(m + 1)(2m + 1) 2 + 7m(m + 1) (m + 1)!m!2 (2m)!2 ✏

g = 0

⌘ = Γ(c) Γ(2+c) 2(m+1)2(2m+1)2 (3m+1)(2+7m(m+1))2 (m+1)!2m!4 (2m)!3 ✏2 ˜ i = i! (i−m−1)! 2m(2m + 1) 2 + 7m(m + 1) m! (2m)! ✏ .

i > m

˜ !i = (i2m2)! 3m + 2 2m + 1 m + 1 (i3m2)! + 1 (i3m3)! + m(m + 1) (i 3m 1)!

2m(2m + 1)

2 + 7m(m + 1) m! (2m)!✏

i > 3m

At order they are associated to all operators in V and Z terms.

O(✏)

SLIDE 36

36

g = 0 g =

3

p 138 13

✏

11100 g =

13 + 3

p 138

✏

11100 h = 3✏ 8 h = 1 185 ⇣ 42 4 p 138 ⌘ ✏ h = 2 185 ⇣ 21 + 2 p 138 ⌘ ✏

There are 3 non trivial fixed points

Some explicit results for k=2, n=3

Anomalous dimensions

˜ 1 = ⌘ 2, ˜ 2n−1 = (n − 1)✏ − ⌘ 2 .

˜ i = 20 i(i − 1)(m − 2) g + 1 2i(i − 1) h + O(coup2), i = 2, 3, 4

˜ 1 = −90 g2 + 1 16 h2

Scaling relation

LF P = 1 2φ 22φ + 1 2hφ2(∂φ)2 + gφ6

˜ 2 = 3✏ 8 ˜ 2 = 1 185 ⇣ 42 4 p 138 ⌘ ✏ ˜ 2 = 2 185 ⇣ 21 + 2 p 138 ⌘ ✏ ˜ 3 = 9✏ 8 ˜ 3 = 2 185 ⇣ 50 3 p 138 ⌘ ✏ ˜ 3 = 2 185 ⇣ 50 + 3 p 138 ⌘ ✏ ˜ 4 = 9✏ 4 ˜ 4 = 4✏ 5 ˜ 4 = 4✏ 5

˜ 1 = 9✏2 1024 ˜ 1 =

8519 762

p 138

✏2

1369000 ˜ 1 =

8519 + 762

p 138

✏2

1369000

˜ 5 = 2✏ ˜ 1

dc = 6

˜ ω0 = h

˜ γ

Anomalous dimensions

SLIDE 37

37

0.002
0.001

0.000 0.001 0.002

0.002
0.001

0.000 0.001 0.002

g h

0.0006
0.0004
0.0002

0.0000 0.0002 0.0004 0.00 0.02 0.04 0.06

g h

Two kind of perturbative phase diagrams Red FP is IR attractive in (g,h) The other two FPs have one more UV attractive direction.

n = 3

n = 2 (even)

(odd) They can also represent new Asympotically Safe theories but non unitary!

Pattern at k=2

First kind Second kind

SLIDE 38

38

CFT

Comparison of the anomalous dimensions Using SDE based on:

γ1

To compare with RG, rescale the couplings accordingly

Evaluate r.h.s. at tree level

and normalize the couplings Solve the recurrence relation with b.c. Agreement between RG and CFT

⇤2= 2(2m+1)g4m+

1+mh2m

1(@)2+h2m⇤

⇤2

x⇤2 yhxyi LO

= 291 c |x y| 2

m 8

3

Y

i=0

(i + 1/m)

h⇤2

xx⇤2 yyi LO

= 4(2m+1)2(4m + 1)!g2c4m+1|xy| 2

m 8 8(2m+1)!m2h2c2m+1|xy| 2 m 8

γi

⇤2

xhxi yi+1 z

i = h ⇥ 2(2m+1)g4m+

1+mh2m

1(@)2+h2m⇤

⇤

x i yi+1 z

i

i+1 i = ✓ i m ◆ (2m)! (m + 1)!h + 4 ✓ i 2m ◆(4m + 1)! (2m)! g

γ1 = O(g2) ' 0

SLIDE 39

Conclusions

In perturbation theory it is possible to obtain leading non trivial results

with renormalization group and with CFT techniques for the conformal universal data. Complete agreement where results overlap.

This approach works both for unitary and non unitary theories.

Tested on some non trivial scalar theories.

39

Among non unitary theories we have identified and studied a non

trivial universality class in 3 dimensions: the Blume-Capel or tri- critical Lee-Yang.

To study higher derivative scalar theories we have employed RG which

relies neither on unitarity nor conformal invariance

Allows to identify scale invariant deformations of higher derivative

free CFTs.

In particular for theories of the “second type” pure potential

deformations are not scale invariant (k=2, n=2m+1)

We have confirmed most of our results using SDE and assuming

conformal symmetry (provides evidence for conformal invariance)

SLIDE 40

Outlook

40

Natural question: can one extend to a functional non perturbative RG

framework some of these ideas?

Can we improve the methods on both RG and CFT sides?

Thank you!

Global symmetries: e.g. O(N) models
Extension to higher derivative theories with odd interactions
Higher order corrections
Higher derivative models of “second type”: alternative CFT

approaches (Conformal Bootstrap or structure of conformal blocks)

Theories for fields with non zero spin content
More geometrical formulation of RG flow of QFTs

SLIDE 41

41

SLIDE 42

42

Non perturbative FRG

What about a non perturbative analysis of these models? Conformal bootstrap may not be easy to apply, due to the lack of unitarity One can proceed with FRG… Let us consider the simpler case k=2 with even n Use LPA with a cutoff:

Rk(p2) = (k4−p4)θ(k4−p4)

Critical dimensions

8, 16 3 , 24 5 , 32 7 , · · ·

0.6
0.5
0.4
0.3
0.2
0.1

0.0 0.1 1 2 3 4 5 6 7 σ ϕmax

0.8
0.6
0.4
0.2

0.0 2 4 6 8 10 σ ϕmax

d=7 d=5

6.0 6.5 7.0 7.5

3.8
3.6
3.4
3.2
3.0
2.8

λ2

6.0 6.5 7.0 7.5 8.0 d 0.0 0.5 1.0 1.5 λ4