Conformal Field Theories in the

Epsilon and 1/N Expansions

Lin Fei

A Dissertation

Presented to the Faculty

of Princeton University

in Candidacy for the Degree

of Doctor of Philosophy

Recommended for Acceptance

by the Department of

Physics

Adviser: Simone Giombi

September 2017

c© Copyright by Lin Fei, 2017.

All rights reserved.

Abstract

In this thesis, we study various conformal eld theories in two dierent approximation

schemes - the ε-expansion in dimensional continuation, and the large N expansion.

We rst propose a cubic theory in d = 6 − ε as the UV completion of the quartic

scalar O(N) theory in d > 4. We study this theory to three-loop order and show

that various operator dimensions are consistent with large-N results. This theory

possesses an IR stable xed point at real couplings for N > 1038, suggesting the

existence of a perturbatively unitary interacting O(N) symmetric CFT in d = 5. Ex-

tending this model to Sp(N) symmetric theories, we nd an interacting non-unitary

CFT in d = 5. For the special case of Sp(2), the IR xed point possesses an enhanced

symmetry given by the supergroup OSp(1|2). We also observe that various operator

dimensions of the Sp(2) theory match those from the 0-state Potts model. We provide

a graph theoretic proof showing that the zero, two, and three-point functions in the

Sp(2) model and the 0-state Potts model indeed match to all orders in perturbation

theory, strongly suggesting their equivalence. We then study two fermionic theories

in d = 2 + ε - the Gross-Neveu model and the Nambu-Jona-Lasinio model, together

with their UV completions in d = 4 − ε given by the Gross-Neveu-Yukawa and the

Nambu-Jona-Lasinio-Yukawa theories. We compute their sphere free energy and cer-

tain operator dimensions, passing all checks against large-N results. We use two

sided Padé approximations with our ε-expansion results to obtain estimates of vari-

ous quantities in the physical dimension d = 3. Finally, we provide evidence that the

N = 1 Gross-Neveu-Yukawa model which contains a 2-component Majorana fermion,

and the N = 2 Nambu-Jona-Lasinion-Yukawa model which contains a 2-component

Dirac fermion, both have emergent supersymmetry.

iii

Acknowledgements

I would like to express my sincerest gratitude to my advisor Simone Giombi, and Igor

Klebanov for their guidance, insights, and encouragement throughout the years. I

would also like to thank my colleague Grigory Tarnopolsky for all the help he has

given me across many projects throughout my PhD studies.

My wonderful experience at Princeton wouldn't be possible without all of my fel-

low graduate students, especially Mykola Dedushenko, Kenan Diab, Po-Shen Hsin,

Vladimir Kirilin, Lauren McGough, Sarthak Parikh, and Grigory Tarnopolsky. Of

course it also wouldn't be possible without the unwavering support of my loving

family.

Additionally, I would also like to thank:

Princeton University for providing me with such an excellent environment to do

research.

The National Science Foundation for providing funding for my work as well as

theoretical physics as a whole.

Wolfram Research for developing and maintaining Mathematica, a pivotal tool in

almost everything I do.

iv

To Helen and Neil

v

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

1 Introduction 1

1.1 Universality and Renormalization . . . . . . . . . . . . . . . . . . . . 1

1.2 Review of Wilson-Fisher ε-expansion . . . . . . . . . . . . . . . . . . 5

1.3 Review of large-N expansion . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 The Gross-Neveu Model and its UV completion . . . . . . . . . . . . 10

1.5 Overview of chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Critical O(N) Models in 6− ε Dimensions 17

2.1 Introduction and Summary . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Summary of large N results for the critical O(N) CFT . . . . . . . . 21

2.3 The IR xed point of the cubic theory in d = 6− ε . . . . . . . . . . 25

2.4 Analysis of xed points at nite N . . . . . . . . . . . . . . . . . . . 32

2.5 Operator mixing and anomalous dimensions . . . . . . . . . . . . . . 37

2.5.1 The mixing of σ2 and φiφi operators . . . . . . . . . . . . . . 37

2.5.2 The mixing of σ3 and σφiφi operators . . . . . . . . . . . . . . 42

2.6 Comments on CT and 5-d F theorem . . . . . . . . . . . . . . . . . . 45

2.7 Gross-Neveu-Yukawa model and a test of 3-d F-theorem . . . . . . . 49

vi

3 Three-Loop Analysis of the Critical O(N) Models in 6−ε Dimensions 53

3.1 Introduction and Summary . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2 Three-loop β-functions in d = 6− ε . . . . . . . . . . . . . . . . . . . 55

3.3 The IR xed point . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.3.1 Dimensions of φ and σ . . . . . . . . . . . . . . . . . . . . . . 62

3.3.2 Dimensions of quadratic and cubic operators . . . . . . . . . . 63

3.4 Analysis of critical N as a function of ε . . . . . . . . . . . . . . . . . 67

3.4.1 Unitary xed points for all positive N . . . . . . . . . . . . . 72

3.5 Non-unitary theories . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.6 Appendix A. Summary of three-loop results . . . . . . . . . . . . . . 77

3.6.1 Counterterms . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4 Critical Sp(N) Models in 6− ε Dimensions and Higher Spin dS/CFT 80

4.1 Introduction and Summary . . . . . . . . . . . . . . . . . . . . . . . . 80

4.2 The IR xed points of the cubic Sp(N) theory . . . . . . . . . . . . . 85

4.2.1 Estimating operator dimensions with Padé approximants . . . 89

4.2.2 Sphere free energies . . . . . . . . . . . . . . . . . . . . . . . . 90

4.3 Symmetry enhancement for N = 2 . . . . . . . . . . . . . . . . . . . 92

5 Equivalence between the Sp(2) model and the 0-state Potts model 95

5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.2 Review of Potts model tensor calculation . . . . . . . . . . . . . . . . 97

5.3 Prove the equivalence for two-point functions . . . . . . . . . . . . . . 100

5.3.1 Some general observations about the diagrams . . . . . . . . . 100

5.3.2 Dening three sets of graphs: GC , GF , GA . . . . . . . . . . . 102

5.3.3 Relation between GA and GC . . . . . . . . . . . . . . . . . . 105

5.3.4 Relation between GA and GF . . . . . . . . . . . . . . . . . . 107

5.4 three-point functions and zero point functions . . . . . . . . . . . . . 109

vii

6 Yukawa CFTs and Emergent Supersymmetry 111

6.1 Introduction and Summary . . . . . . . . . . . . . . . . . . . . . . . . 111

6.2 The Gross-Neveu-Yukawa Model . . . . . . . . . . . . . . . . . . . . . 116

6.2.1 Free energy on S4−ε . . . . . . . . . . . . . . . . . . . . . . . . 119

6.2.2 2 + ε expansions . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.2.3 Large N expansions . . . . . . . . . . . . . . . . . . . . . . . . 127

6.2.4 Padé approximants . . . . . . . . . . . . . . . . . . . . . . . . 129

6.3 The Nambu-Jona-Lasinio-Yukawa Model . . . . . . . . . . . . . . . . 133

6.3.1 Free energy on S4−ε . . . . . . . . . . . . . . . . . . . . . . . . 135

6.3.2 2 + ε expansions . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.3.3 Large N expansions . . . . . . . . . . . . . . . . . . . . . . . . 140

6.3.4 Padé approximants . . . . . . . . . . . . . . . . . . . . . . . . 142

6.4 Models with Emergent Supersymmetry . . . . . . . . . . . . . . . . . 144

6.4.1 Theory with 4 supercharges . . . . . . . . . . . . . . . . . . . 144

6.4.2 Theory with 2 supercharges . . . . . . . . . . . . . . . . . . . 148

A Useful integrals for one-loop calculation 157

B Summary of three-loop results 160

B.1 Counterterms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

C Sample diagram calculations 163

C.1 Some useful integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

C.2 Example of a two point function diagram . . . . . . . . . . . . . . . . 165

C.3 Example of a three point function diagram . . . . . . . . . . . . . . . 166

D Beta functions and anomalous dimensions for general Yukawa theo-

ries 169

Bibliography 176

viii

Chapter 1

Introduction

1.1 Universality and Renormalization

Conformal Field Theories (CFT) have important applications in many areas from

string theory to condensed matter physics. They are Quantum Field Theories (QFT)

which are invariant under conformal transformations - transformations that preserve

angles locally. In particular, scaling invariance is a part of conformal invariance. At

the critical points of statistical systems, the correlation length scale ξ diverges. In the

continuum theory that describes the physics across distances that are large compared

to the lattice spacing, the correlation functions are insensitive to the microscopic de-

tails of the model and become scale-invariant. Thus, CFTs play a primary role in the

understanding of critical points in statistical mechanics. Remarkably, a wide range

of seemingly unrelated statistical systems share some universal large scale proper-

ties despite their microscopic dierences. In order to understand the origin of these

universal properties, it is necessary to introduce the notion of renormalization.

A simple example of universality is provided by the central limit theorem. If we

are summing over a large number of random variables with arbitrary distribution

(provided that large deviations with respect to the mean decrease fast enough), the

1

asymptotic behavior is always a Gaussian, regardless of the distribution of the orig-

inal random variables. One can understand this using a renormalization strategy by

recursively averaging over pairs of random variables. In general, the operation of

averaging over two identical random variables with distribution ρ results in a random

variable with a transformed distribution T ρ. However, the limiting distribution ρ∗

is necessarily a xed point of this operation T ρ∗ = ρ∗. One nds that the Gaussian

distribution is an attractive xed point of these operations. Thus, no matter what

the initial distribution is, after repeated application of the averaging operation, all of

them will tend towards the Gaussian xed point.

Renormalization of QFTs follows a similar idea. Let's suppose we have a QFT

described by the Hamiltonian H, and a microscopic scale 1/Λ describing, for exam-

ple, the lattice spacing for a lattice model. Now we recursively integrate out short

distances degrees of freedom to generate an eective Hamiltonian Hλ, with λ ∈ [0, 1],

corresponding to increasing the scale to 1/(λΛ). The various parameters in the new

Hamiltonian Hλ will be dierent, and this leads to a transformation T in the space

of Hamiltonians such that:

λd

dλHλ = T [Hλ] (1.1)

Universality is related to the existence of xed point solutions of the equation:

T [H∗] = 0 (1.2)

If the above equations have an attractive xed point, then all the Hamiltonians

within the basin of attraction will eventually ow to the same xed point - they

belong to the same universality class. The Gaussian xed point is again a stable xed

point in QFTs, but they describe free theories without interactions. However, there

can also be interacting xed points, which are more interesting.

2

The linear partial dierential equations for the correlators of the renormalized

theory, resulting from variations of the parameters under an innitesimal change of

the renormalization scale, are called renormalization group equations. In particular,

using these RG equations, one can compute two functions: the β-function which

describe the rate at which the couplings in H ows, and the anomalous dimension η,

which describes the correction to the classical scaling dimensions of various operators

when interactions are turned on. All of these quantities are computed order by order in

perturbation theory, with higher order corrections coming from higher loop Feynman

diagrams.

For example, the water-vapor phase diagram is described by the three-dimensional

Ising model. Near the critical point, it is described by a Euclidean QFT of a real

scalar eld, φ, with a λφ4 interaction:

S =

∫d3x

(1

2(∂φ)2 +

λ

4!φ4

). (1.3)

The coupling constant λ is the strength of the interaction, which will change with scale

due to renormalization. It turns out, this model is weakly coupled at short distances,

but it becomes strongly coupled at long distances. It is the long distance regime that

is needed for describing the critical behavior, but perturbation theory in λ cannot be

applied to calculate various quantities of interest, such as the scaling dimensions of

various operators. However, theorists have invented ingenious expansion schemes that

have led to good approximations. Two of these include ε-expansion in dimensional

continuation, and large-N expansion.

The idea behind dimensional continuation is that instead of working directly in

d = 3, we treat d as a continuous parameter. Signicant simplication occurs for

d = 4− ε where ε 1. Then the IR stable xed point of the Renormalization Group

occurs for a weak coupling λ of order ε, so that a formal Wilson-Fisher expansion in

3

ε may be developed [1]. The coecients of the rst few terms fall o rapidly, so that

setting ε = 1 provides a rather precise approximation that is in good agreement with

the experimental and numerical results for d = 3 [2].

Another important idea has been the large-N expansion in the O(N) symmetric

QFT of N real scalar elds φi, i = 1, . . . , N , with interaction λ4(φiφi)2. Using a gen-

eralized Hubbard-Stratonovich transformation with an auxiliary eld σ, it is possible

to develop expansion in powers of 1/N (for a comprehensive review, see [3]). The

large N expansion may be developed for a range of d [4, 5, 6, 7, 8, 9, 10, 11, 12] and

compared with the regimes where other perturbative expansions such as dimensional

continuation.

In particular, for d = 4− ε, the Wilson-Fisher ε-expansion may be developed for

any N [2], and the results can be matched with the large N techniques. Also, for

d = 2+ε the large N results match with the perturbative UV xed point of the O(N)

Non-linear Sigma Model (NLσM). Using the rst few terms in the large N expansion

directly in d = 3 provides another approach to estimating the scaling dimensions for

low values of N . Thus, a combination of the large N and ε expansions provides good

approximations for the critical behavior in the entire range 2 < d < 4. One should

note, however, that both expansions are not convergent but rather provide asymptotic

series. There are continued eorts towards obtaining a more rigorous approach to the

O(N) symmetric CFT's using conformal bootstrap ideas [13, 14, 15, 16], and recently

it has led to more precise numerical calculations of the operator scaling dimensions

in three-dimensional CFT's [17, 18]. The bootstrap approach has been successfully

applied not only in the range 2 < d < 4 [19], but also for 4 < d < 6 [20].

4

1.2 Review of Wilson-Fisher ε-expansion

Let us consider the Euclidean eld theory of N real massless scalar elds with an

O(N) invariant quartic interaction

S =

∫ddx

(1

2(∂φi)2 +

λ

4(φiφi)2

). (1.4)

As follows from dimensional analysis, the interaction term is relevant for d < 4 and

irrelevant for d > 4. Hence, for 2 < d < 4, it is expected that the quartic interaction

generates a ow from the free UV xed point to an interacting IR xed point. In

d = 4 − ε this xed point can be studied perturbatively in the framework of the

Wilson-Fisher ε-expansion [1, 2]. Indeed, the one-loop beta function for the theory in

d = 4− ε reads

βλ = −ελ+ (N + 8)λ2

8π2, (1.5)

and there is a weakly coupled IR xed point at

λ∗ =8π2

N + 8ε . (1.6)

Higher order corrections in ε will change the value of the critical coupling, but not

its existence, at least in perturbation theory. The anomalous dimensions of the fun-

damental eld φi and the composite φiφi at the xed point can be computed to be,

to leading order in ε

γφ =N + 2

4(N + 8)2ε2 +O(ε3) , γφ2 =

N + 2

N + 8ε+O(ε2) , (1.7)

5

corresponding to the scaling dimensions

∆φ =d

2− 1 + γφ = 1− ε

2+

N + 2

4(N + 8)2ε2 +O(ε3) , (1.8)

∆φ2 = d− 2 + γφ2 = 2− 6

N + 8ε+O(ε2) . (1.9)

Note that the dimension of the φiφi operator is 2 + O(1/N) to leading order at

large N , a result that follows from the large N analysis reviewed below. Higher order

corrections in ε for general N may be derived by higher loop calculations in the theory

(1.4), and they are known up to order ε5 [21, 22].

For d > 4, the interaction is irrelevant and so the IR xed point is the free theory;

however, one may ask about the existence of interacting UV xed points. Working

in d = 4 + ε for small ε, one indeed nds a perturbative UV xed point at (see, for

example [23])

λ∗ = − 8π2

N + 8ε . (1.10)

The anomalous dimensions at this critical point are given by the same expressions

(1.7) with ε→ −ε. Note that, because γφ starts at order ε2, the dimension of φ stays

above the unitarity bound for all N , at least for suciently small ε. However, since

the xed point requires a negative coupling, one may worry about its stability, and it

is important to study this critical point by alternative methods.

1.3 Review of large-N expansion

A complementary approach to the ε expansion that can be developed at arbitrary

dimension d is the large N expansion. The standard technique to study the theory

(1.4) at large N is based on introducing a Hubbard-Stratonovich auxiliary eld σ as

S =

∫ddx

(1

2(∂φi)2 +

1

2σφiφi − σ2

4λ

). (1.11)

6

Integrating out σ via its equation of motion σ = λφiφi, one gets back to the original

lagrangian. The quartic interaction in (1.4) may in fact be viewed as a particular

example of the double trace deformations studied in [24]. One can then show that at

large N the dimension of φiφi goes from ∆ = d−2 at the free xed point to d−∆ = 2

at the interacting xed point. At the conformal point, the last term in (1.11) can

be dropped1, and the eld σ plays the role of the composite operator φiφi. One may

then study the critical theory using the action

Scrit =

∫ddx

(1

2(∂φi)2 +

1

2√Nσφiφi

)(1.12)

where we have rescaled σ by a factor of√N for reasons that will become clear

momentarily. The 1/N perturbation theory can be developed by integrating out the

fundamental elds φi. This generates an eective non-local kinetic term for σ

Z =

∫DφDσ e

−∫ddx

(12

(∂φi)2+ 1

2√Nσφiφi

)

=

∫Dσ e

18N

∫ddxddy σ(x)σ(y) 〈φiφi(x)φjφj(y)〉0+O(σ3)

(1.13)

where we have assumed large N and the subscript `0' denotes expectation values in

the free theory. We have

〈φiφi(x)φjφj(y)〉0 = 2N [G(x− y)]2 , G(x− y) =

∫ddp

(2π)deip(x−y)

p2. (1.14)

In momentum space, the square of the φ propagator reads

[G(x− y)]2 =

∫ddp

(2π)deip(x−y)G(p)

G(p) =

∫ddq

(2π)d1

q2(p− q)2= − (p2)d/2−2

2d(4π)d−32 Γ(d−1

2) sin(πd

2)

(1.15)

1This applies formally to both IR and UV xed points.

7

and so from (1.13) one nds the two-point function of σ in momentum space

〈σ(p)σ(−p)〉 = 2d+1(4π)d−32 Γ

(d− 1

2

)sin(

πd

2) (p2)2− d

2 ≡ Cσ(p2)2− d2 . (1.16)

The corresponding two-point function in coordinate space can be obtained by Fourier

transform and reads

〈σ(x)σ(y)〉 =2d+2Γ

(d−1

2

)sin(πd

2)

π32 Γ(d2− 2) 1

|x− y|4 ≡Cσ

|x− y|4 . (1.17)

Indeed, this is the two-point function of a conformal scalar operator of dimension

∆ = 2. Note also that the coecient Cσ is positive in the range 2 < d < 6.

The large N perturbation theory can then be developed using the propagator

(1.16)-(1.17) for σ, the canonical propagator for φ and the interaction term σφiφi in

(1.12). For instance, the 1/N term in the anomalous dimension of φi can be computed

from the one-loop correction to the φ-propagator

1

N

∫ddq

(2π)d1

(p− q)2

Cσ

(q2)d2−2+δ

, (1.18)

where we have introduced a small correction δ to the power of the σ propagator as a

regulator2. Doing the momentum integral by using (A.1), one obtains the result

CσN

(d− 4)

(4π)d2dΓ(d

2)(p2)1−δΓ(δ − 1) (1.19)

where we have set δ = 0 in the irrelevant factors. The 1/δ pole corresponds to a

logarithmic divergence and is cancelled as usual by the wave function renormalization

of φ. Dening the dimension of φ as

∆φ =d

2− 1 +

1

Nη1 +

1

N2η2 + . . . (1.20)

2One may perform the calculation using a momentum cuto, yielding the same nal result.

8

2 3 4 5 6d

0.1

0.0

0.1

0.2

0.3η1

-2 3 4 5 6

d0

5

10

15

20

Cσ

Figure 1.1: The 1/N anomalous dimension of φi and the coecient of the two-pointfunction of σ in the large N critical O(N) theory for 2 < d < 6.

the one-loop calculation above yields the result

η1 =Cσ(d− 4)

(4π)d2dΓ(d

2)

=2d−3(d− 4)Γ

(d−1

2

)sin(πd2

)

π32 Γ(d2

+ 1) . (1.21)

Setting d = 4 − ε and expanding for small ε, it is straightforward to check that this

agrees with the 1/N term of (1.8). The leading anomalous dimension of σ also takes

a simple form [4, 11, 12]

∆σ = 2 +1

N

4(d− 1)(d− 2)

d− 4η1 +O(

1

N2) . (1.22)

and can be seen to precisely agree with (1.9) in d = 4− ε.

Note that the anomalous dimension of φi (1.21) is positive for all 2 < d < 6. Thus,

while the focus of most of the exisiting literature is on the range 2 < d < 4, we see

no obvious problems with unitarity in continuing the large N critical O(N) theory to

the range 4 < d < 6.3 A plot of η1 and of the σ two-point function coecient Cσ is

given in Figure 1.1, showing that they are both positive for 2 < d < 6.

3Recall that the unitarity bound for a scalar operator is ∆ ≥ d/2 − 1. One can also check thatthe order 1/N term in the anomalous dimension of the O(N) invariant higher spin currents, givenin [10], is positive in the range 2 < d < 6, consistently with the unitarity bound ∆s ≥ s+ d− 2 forspin s operators (s > 1/2).

9

1.4 The Gross-Neveu Model and its UV completion

In order to motivate our proposed cubic model as the UV completion of φ4 theory, we

rst review the relation between the Gross-Neveu (GN) model and its UV completion,

the Gross-Neveu-Yukawa model.

The same ε and 1/N -expansion schemes can be applied to fermionic theories as

well. The GN model has a quartic interaction given by:

LGN = ψj 6 ∂ψj +g

2(ψjψ

j)2 . (1.23)

As discovered in [25], the theory is asymptotically free in d = 2 for N > 2. The

β-function for the renormalized coupling constant g in d = 2 + ε starts with:

β = εg − N − 2

2πg2 + . . . , (1.24)

and the anomalous dimension of ψ is given by[3]:

∆ψ =1 + ε

2+N − 1

8π2g2 . (1.25)

Therefore this theory possesses a non-trivial UV xed point given by g∗ = 2πN−2

ε in

d = 2 + ε. And the anomalous dimension at that xed point is:

∆ψ =1

2+

1

2ε+

N − 1

4(N − 2)2ε+ . . . . (1.26)

However, by simple power counting, this theory is not renormalizable in dimen-

sions greater than two, and therefore in the UV, the large momentum behavior of the

theory cannot be properly described via perturbation theory. It would be desirable

to describe this xed point through some other means, such as the IR xed point of

10

some other theory. Indeed such a theory exists, and it is the Gross-Neveu-Yukawa

theory, as we will demonstrate shortly.

First, we can still apply the Hubbard-Stratonovich analysis to the Gross-Neveu

model, although it requires a bit more care. Introducing the auxiliary eld σ, and

dropping the quadratic term in the critical limit, we have the action

Scrit ferm =

∫ddx(− ψ0i/∂ψ

i0 +

1√Nσ0ψ0iψ

i0

), (1.27)

where i = 1, . . . , N and N = NTr1. The propagator of the ψi0 eld reads

〈ψi0(p)ψ0j(−p)〉0 = δiji/p

p2. (1.28)

The σ eective propagator obtained after integrating over the fundamental elds ψi0

reads

〈σ0(p)σ0(−p)〉0 = Cσ0/(p2)

d2−1+∆ , (1.29)

where

Cσ0 ≡ −2d+1(4π)d−32 Γ(d− 1

2

)sin(πd

2

)(1.30)

and we have introduced the regulator ∆. Note that the power of p2 in the propagator

is d2− 1 + ∆ instead of d

2− 2 + ∆ found in the scalar case. In order to cancel the

divergences as ∆→ 0 we have to renormalize the bare elds ψ0 and σ0:

ψ = Z1/2ψ ψ0, σ = Z1/2

σ σ0 , (1.31)

11

where

Zψ = 1 +1

N

Zψ1

∆+O(1/N2), Zσ = 1 +

1

N

Zσ1

∆+O(1/N2) . (1.32)

The full propagators of the renormalized elds read

〈ψi(p)ψj(−p)〉 = δijCψi/p

(p2)d2−∆ψ+ 1

2

, 〈σ(p)σ(−p)〉 =Cσ

(p2)d2−∆σ

, (1.33)

where we introduced anomalous dimensions ∆ψ and ∆σ and two-point function nor-

malizations Cψ and Cσ in momentum space. In particular, we have

∆ψ =d

2− 1

2+ ηGN , (1.34)

where ηGN = ηGN1 /N + ηGN2 /N2 + . . ., and to rst order in 1/N , we have:

ηGN1 =Γ(d− 1)(d− 2)2

4Γ(2− d2)Γ(d

2+ 1)Γ(d

2)2

(1.35)

We can expand this expression in d = 2 + ε and see agreement with our ε-expansion

result.

Motivated by the Hubbard-Stratonovich transformation, the Lagrangian of the

GNY model [26, 27] contains a scalar eld with cubic interaction with the fermions

as well as a quartic self-interaction which is also marginal:

LGNY =1

2(∂µσ)2 + ψj 6 ∂ψj + g1σψjψ

j +1

24g2σ

4 . (1.36)

12

In d = 4− ε, the β-functions to one-loop are given by[3]:

βg2 = −εg2 +1

(4π)2

(3g2

2 + 2Ng21g2 − 12Ng4

1

),

βg1 = − ε2g1 +

N + 6

2(4π)2g3

1 , (1.37)

where N = Nf tr1 = 4Nf . The anomalous dimension for the ψ eld is given by:

∆ψ =3− ε

2+

1

2(4π)2g2

1 . (1.38)

The model possesses an IR stable xed point at the critical couplings g∗i given by

(g∗1)2

(4π)2=

1

N + 6ε ,

g∗2(4π)2

=−N + 6 +

√N2 + 132N + 36

6(N + 6)ε . (1.39)

The anomalous dimension of ψ at the xed point is equal to:

∆ψ =3

2− N + 5

2(N + 6)ε . (1.40)

Which again matches against the largeN result expanded near d = 4−ε. Therefore

the UV xed point of the GN model is described by the IR xed point of the GNY

model. The GNY model is said to be the UV-completion of the GN model.

Similarly, in this thesis, we are trying to show that the O(N) symmetric ((φi)2)2

theory with a UV xed point in d = 4 + ε has a UV completion given by a cubic

theory with N + 1 scalar elds in d = 6− ε:

L =1

2(∂µφ

i)2 +1

2(∂µσ)2 +

g1

2σφiφi +

g2

6σ3 . (1.41)

13

We will demonstrate this by comparing against large N results of various operator

dimensions.

1.5 Overview of chapters

This thesis is organized as a collection of papers. Chapters 2,3,4,6 are based on

work published in [28, 29, 30, 31] respectively, with co-authors Simone Giombi, Igor

Klebanov, and Grigory Tarnopolsky. Chapter 5 is based on unpublished work.

In chapter 2, we propose the alternate description of the O(N) symmetric scalar

eld theories in d dimensions with interaction (φiφi)2 in terms of a theory of N + 1

massless scalars with the cubic interactions σφiφi and σ3. Our one-loop calculation

in 6− ε dimensions shows that this theory has an IR stable xed point at real values

of the coupling constants for N > 1038. We show that the 1/N expansions of various

operator scaling dimensions match the known results for the critical O(N) theory

continued to d = 6− ε. These results suggest that, for suciently large N , there are

5-dimensional unitary O(N) symmetric interacting CFT's; they should be dual to the

Vasiliev higher-spin theory in AdS6 with alternate boundary conditions for the bulk

scalar. Using these CFT's we provide a new test of the 5-dimensional F -theorem,

and also nd a new counterexample for the CT theorem.

In chapter 3, we continue the study of the same theory to three loop order. We

calculate the 3-loop beta functions for the two couplings and use them to determine

certain operator scaling dimensions at the IR stable xed point up to order ε3. We

also use the beta functions to determine the corrections to the critical value of N

below which there is no xed point at real couplings. We also study the theory with

N = 1, which has a Z2 symmetry under φ → −φ. We show that it possesses an

IR stable xed point at imaginary couplings which can be reached by ow from a

nearby xed point describing a pair of N = 0 theories. We calculate certain operator

14

scaling dimensions at the IR xed point of the N = 1 theory and suggest that, upon

continuation to two dimensions, it describes a non-unitary conformal minimal model.

In chapter 4, we consider a variant of our model with Sp(N) symmetry composed

of N anti-commuting scalars and one commuting scalar. For any even N we nd an

IR stable xed point in 6 − ε dimensions at imaginary values of coupling constants.

Borrowing our three loop calculations in 3, we develop ε expansions for several op-

erator dimensions and for the sphere free energy F . The conjectured F -theorem is

obeyed in spite of the non-unitarity of the theory. Our results point to the exis-

tence of interacting non-unitary 5-dimensional CFTs with Sp(N) symmetry, where

operator dimensions are real. For N = 2 we show that the IR xed point possesses

an enhanced global symmetry given by the supergroup OSp(1|2). This suggests the

existence of OSp(1|2) symmetric CFTs in dimensions smaller than 6. We show that

the 6− ε expansions of the scaling dimensions and sphere free energy in our OSp(1|2)

model are the same as in the q → 0 limit of the q-state Potts model.

In chapter 5, we provide further evidence of the equivalence between the Sp(2)

model and the 0-state Potts model. Using existing results, we nd that to four-loop

order, the anomalous dimensions and beta functions of these two theories are equal

up to a sign alternating with successive loop order. Using a graph theoretic approach,

we prove this equality as well as the alternating sign. This argument holds for all

loop orders in perturbation theory. This strongly suggests that these two models are

indeed equivalent, giving an alternative description of the 0-state Potts model.

Finally, in chapter 6, we study conformal eld theories with Yukawa interactions

in dimensions between 2 and 4; they provide UV completions of the Nambu-Jona-

Lasinio and Gross-Neveu models which have four-fermion interactions. We compute

the sphere free energy and certain operator scaling dimensions using dimensional

continuation. We provide new evidence that the 4− ε expansion of the N = 1 Gross-

Neveu-Yukawa model respects the supersymmetry. Our extrapolations to d = 3

15

appear to be in good agreement with the available results obtained using the numer-

ical conformal bootstrap. We apply a similar approach to calculate the sphere free

energy and operator scaling dimensions in the Nambu-Jona-Lasinio-Yukawa model,

which has an additional U(1) global symmetry. For N = 2, which corresponds to

one 2-component Dirac fermion, this theory has an emergent supersymmetry with 4

supercharges, and we provide new evidence for this.

16

Chapter 2

Critical O(N) Models in 6− ε

Dimensions

2.1 Introduction and Summary

This chapter is based on the work published in [28], co-authored with Simone

Giombi, and Igor Klebanov. We also thank David Gross, Daniel Harlow, Igor

Herbut, Juan Maldacena, Giorgio Parisi, Silviu Pufu, Leonardo Rastelli, Dam Son,

Grigory Tarnopolsky and Edward Witten for useful discussions.

We would like to extend the results of the ε-expansion and large-N expansion to

interacting O(N) models in d > 4. At rst glance, such extensions seem impossible:

the (φiφi)2 interaction is irrelevant at the Gaussian xed point, since it has scaling

dimension 2d− 4. Therefore, the long-distance behavior of this QFT is described by

the free eld theory. However, at least for large N , the theory possesses a UV stable

xed point whose existence may be demonstrated using the Hubbard-Stratonovich

transformation. At the interacting xed point, the scaling dimension of the operator

φiφi is 2 +O(1/N) for any d. For d > 6, this dimension is below the unitarity bound

d/2 − 1. It follows that the interesting range, where the UV xed point may be

17

unitary, is [32, 33, 34, 35]

4 < d < 6 . (2.1)

We will examine the structure of the O(N) symmetric scalar eld theory in this range

from various points of view.

We rst note that the coecients in the 1/N expansions derived for various quan-

tities in [4, 5, 6, 7, 8, 9, 10, 11, 12] may be continued to the range (2.1) without any

obvious diculty. Thus, the UV xed points make sense at least in the 1/N expan-

sion. However, one should be concerned about the stability of the xed points in this

range of dimensions at nite N . In d = 4 + ε, where the UV xed point is weakly

coupled, it occurs for the negative quartic coupling λ∗ = − 8π2

N+8ε + O(ε2) [36, 23].

Thus, it seems that the UV xed point theory is unlikely to be completely stable,

although it may be metastable. In order to gain a better understanding of the xed

point theory, it would be helpful to describe it via RG ow from another theory. In

such a UV complete" description, the O(N) symmetric theory we are after should

appear as the conventional IR stable xed point.

Our main result is to demonstrate that such a UV completion indeed exists: it is

the O(N) symmetric theory of N + 1 scalar elds with the Lagrangian

L =1

2(∂µφ

i)2 +1

2(∂µσ)2 +

g1

2σφiφi +

g2

6σ3 . (2.2)

The cubic interaction terms are relevant for d < 6, so that the theory ows from the

Gaussian xed point to an interacting IR xed point. The latter is expected to be

weakly coupled for d = 6− ε. The idea to study a cubic scalar theory in d = 6− ε is

not new. Michael Fisher has explored such an ε-expansion in the theory of a single

scalar eld as a possible description of the Yang-Lee edge singularity in the Ising

model [37]. In that case, which corresponds to the N = 0 version of (2.2), the IR

18

xed point is at an imaginary value of g2.1 This is related to the lack of unitarity of

the xed point theory. Using the one-loop beta functions for g1 and g2, we will show

in Section 2.3 that for large N the IR stable xed point instead occurs for real values

of the couplings, thus removing conict with unitarity. Remarkably, such unitary

xed points exist only for N > 1038. As we show in Section 2.4, for N ≤ 1038 the

IR stable xed point becomes complex, and the theory is no longer unitary. Thus,

the breakdown of the large N expansion in d = 6 − ε occurs at a very large value,

Ncrit = 1038. We will provide evidence, however, that for the physically interesting

dimension d = 5, Ncrit is much smaller.

Besides its intrinsic interest, the O(N) invariant scalar CFT in d = 5 has inter-

esting applications to higher spin AdS/CFT dualities. There exists a class of Vasiliev

theories in AdSd+1 [41, 42, 43, 44, 45, 46] that is naturally conjectured to be dual to

the O(N) singlet sector of the d-dimensional CFT ofN free scalars [47, 23]. In order to

extend the duality to interacting CFT's, one adds the O(N) invariant term λ4(φiφi)2.

For d = 3 this leads to the well-known Wilson-Fisher xed points [1, 2]. In the dual

description of these large N interacting theories, it is necessary to change the r−∆

boundary conditions on the scalar eld in AdS4 from the ∆− = 1 to ∆+ = 2+O(1/N)

[47]. The situation is very similar for the d = 5 case, which should be dual to the

Vasiliev theory in AdS6 [23]. One can adopt the ∆+ = 3 boundary conditions on the

bulk scalar, which are necessary for the duality to the free O(N) theory. Alterna-

tively, the ∆− = 2 +O(1/N) are allowed as well [48, 49]. This suggests that the dual

interacting O(N) CFT should exist in d = 5, at least for large N [36, 23]. Our RG

calculations lend further support to the existence of this interacting d = 5 CFT.

In Sections 2.3 and 2.5, using one-loop calculations for the theory (2.2) in d = 6−ε,

we nd some IR operator dimensions to order ε, while keeping track of the dependence

on 1/N to any desired order. We will then match our results with the 1/N expansions

1Renormalization group calculations for similar cubic theories in d = 6 − ε were carried out in[38, 39, 40].

19

d

2 4 6

6-ϵFexpansioninFcubicFtheory

4+ϵFexpansioninF-ϕ4Ftheory

4-ϵFexpansioninF+ϕ4Ftheory

2+ϵFexpansioninFNLσM

3dFCriticalFO(N)FCFT 5dFCriticalFO(N)FCFT

Figure 2.1: Interacting unitary O(N) symmetric scalar CFT's exist for dimensions2 < d < 6, with d = 4 excluded. In 6−ε and 4−ε dimensions they may be described asweakly coupled IR xed points of the cubic and quartic scalar theories, respectively.In 4 + ε and 2 + ε dimensions they are weakly coupled UV xed points of the quartictheory and of the O(N) Non-linear σ Model, respectively.

derived for the (φiφi)2 theory in [4, 5, 6, 7, 8, 9, 10, 11, 12], evaluating them in d = 6−ε.

The perfect match of the coecients in these two 1/N expansions provides convincing

evidence that the IR xed point of the cubic O(N) theory (2.2) indeed describes the

same physics as the UV xed point of the (φiφi)2 theory. Our results thus provide

evidence that, at least for large N , the interacting unitary O(N) symmetric scalar

CFT's exist not only for 2 < d < 4, but also for 4 < d < 6. In Fig. 1 we sketch the

entire available range 2 < d < 6, pointing out the various perturbative descriptions

of the CFT's where ε expansions have been developed.

In Section 2.6 we discuss the large N results for CT , the coecient of the two-

point function of the stress-energy tensor [12]. We note that, as d approaches 6, CT

approaches that of the free theory of N + 1 scalar elds. This gives further evidence

for our proposal that the IR xed point of (2.2) describes the O(N) symmetric CFT.

We show that the RG ow from this interacting CFT to the free theory of N scalars

provides a counter example to the conjectured CT theorem. On the other hand, the

ve-dimensional version of the F -theorem [50] holds for this RG ow.

20

Our discussion of the (φiφi)2 scalar theory in the range 4 < d < 6 is analogous to

the much earlier results [26, 27, 3] about the Gross-Neveu model [25] in the range 2 <

d < 4. The latter is a U(N) invariant theory of N Dirac fermions with an irrelevant

quartic interaction (ψiψi)2. Using the Hubbard-Stratonovich transformation, it is

not hard to show that this model has a UV xed point, at least for large N [3].

This CFT was conjectured [51, 52] to be dual to the type B Vasiliev theory in AdS4

with appropriate boundary conditions. An alternative, UV complete description of

this CFT is via the Gross-Neveu-Yukawa (GNY) model [26, 27, 3], which is a theory

of N Dirac fermions coupled to a scalar eld σ with U(N) invariant interactions

g1σψiψi + g2σ

4/24. This description is weakly coupled in d = 4 − ε, and it is not

hard to show that the one-loop beta functions have an IR stable xed point for any

positive N . The operator dimensions at this xed point match the large N treatment

of the Gross-Neveu model [53]. These results will be reviewed in Section 2.7, where

we also discuss tests of the 3-d F-theorem [54, 50] provided by the GNY model.

2.2 Summary of large N results for the critical O(N)

CFT

In chapter 1, we derived the anomalous dimensions of φ and σ to rst order in 1/N .

However, at higher orders in the 1/N expansion, a straightforward diagrammatic

approach becomes rather cumbersome. However, the conformal bootstrap method

developed in papers by A. N. Vasiliev and collaborators [4, 5, 6] has allowed to

compute the anomalous dimension of φi to order 1/N3 and that of σ to order 1/N2.

These results have been successfully matched to all available orders in the d = 2 + ε

and d = 4 − ε expansions, providing a strong test of their correctness. The explicit

21

form of η2 in general dimensions, dened as in (1.20), reads [4, 5]:2

η2 = 2η21 (f1 + f2 + f3) ;

f1 = v′(µ) +µ2 + µ− 1

2µ(µ− 1), f2 =

µ

2− µv′(µ) +

µ(3− µ)

2(2− µ)2,

f3 =µ(2µ− 3)

2− µ v′(µ) +2µ(µ− 1)

2− µ ;

v′(µ) = ψ(2− µ) + ψ(2µ− 2)− ψ(µ− 2)− ψ(2) ,

ψ(x) =Γ′(x)

Γ(x), µ =

d

2. (2.3)

The expressions for the dimension of σ at order 1/N2 [5] and for the coecient η3

[6] in arbitrary dimensions are lengthy and we do not report them explicitly here.3

However, since they are useful to test our general picture, we write below the explicit

ε-expansions of ∆φ and ∆σ in d = 6 − ε, including all known terms in the 1/N

expansion:

∆φ = 2− ε

2+

(1

N+

44

N2+

1936

N3+ . . .

)ε−

(11

12N+

835

6N2+

16352

N3+ . . .

)ε2 +O(ε3)(2.4)

and

∆σ = 2 +

(40

N+

6800

N2+ . . .

)ε−

(104

3N+

34190

3N2+ . . .

)ε2 +O(ε3) . (2.5)

In the next section, we will show that the order ε terms precisely match the one-loop

anomalous dimensions at the IR xed point of the cubic theory (2.2) in d = 6 − ε.

Higher orders in ε should be compared to higher loop contributions in the d = 6− ε

cubic theory, and it would be interesting to match them as well.

2Ref. [5] contains an apparent typo: in the denition of the function v′(µ) given in Eq. (21),α = µ− 1 should be replaced by α = µ− 2. The correct formula for η2 may be found in the earlierpaper [4].

3Note that [5] derives a result for the critical exponent ν, which is related to the dimension of σby ∆σ = d− 1

ν . We also note that a misprint in eq. (22) of [6] has been later corrected in eq. (11)of [53].

22

Using the results in [4, 5, 6, 53] we may also compute the dimension of φi and

σ directly in the physical dimension d = 5. The term of order 1/N3 depends on a

non-trivial self-energy integral that was not evaluated for general dimension in [6].

An explicit derivation of this integral in general d was later obtained in [55]. Using

that result, we nd in d = 5

∆φ =3

2+

32

15π2N− 1427456

3375π4N2

+

(275255197696

759375π6− 89735168

2025π4+

32768 ln 4

9π4− 229376ζ(3)

3π6

)1

N3+ . . .

=3

2+

0.216152

N− 4.342

N2− 121.673

N3+ . . . (2.6)

and

∆σ = 2 +512

5π2N+

2048 (12625π2 − 113552)

1125π4N2+ . . . (2.7)

= 2 +10.3753

N+

206.542

N2+ . . . (2.8)

We note that the coecients of the 1/N expansion are considerably larger than

in the d = 3 case.4 Assuming the result (3.54) to order 1/N3, one nds that the

dimension of φi goes below unitarity at Ncrit = 35. This is much lower than the value

Ncrit = 1038 that we will nd in d = 6−ε. This is just a rough estimate, since the 1/N

expansion is only asymptotic and should be analyzed with care. Note for instance

that in the d = 3 case, a similar estimate to order 1/N3 would suggest a critical value

Ncrit = 3, while in fact there is no lower bound: at N = 1 we have the 3d Ising

model, where it is known [56, 17, 18, 57] that ∆φ ≈ 0.518 > 1/2. Nevertheless, the

reduction from a very large value Ncrit = 1038 in d = 6− ε to a much smaller critical

value in d = 5 is not unexpected. For instance, an analogous phenomenon is known

to occur in the Abelian Higgs model containing Nf complex scalars. A xed point in

4In d = 3, one nds [5, 4, 6] (see also [18]) ∆φ = 12 + 0.135095

N − 0.0973367N2 − 0.940617

N3 + O(1/N4)and ∆σ = 2− 1.08076

N − 3.0476N2 +O(1/N3).

23

d = 4− ε exists only for Nf ≥ 183 [58, 3], while non-perturbative studies directly in

d = 3 suggest a much lower critical value of Nf . 5 Some evidence for the reduction

in the critical value of Nf as d is decreased comes from calculations of higher order

corrections in ε [59]. It would be nice to study such corrections for the O(N) theory

in 6− ε dimensions. It would also be very interesting to explore the 5d xed point at

nite N by numerical bootstrap methods similar to what has been done in d = 3 in

[18]. For the non-unitary theory with N = 0 such bootstrap studies were carried out

very recently [60].6

The large N critical O(N) theory in general d was further studied in a series of

works by Lang and Ruhl [7, 8, 9, 10] and Petkou [11, 12]. Using conformal symmetry

and self-consistency of the OPE expansion, various results about the operator spec-

trum of the critical theory were derived. As an example of interest to us, [10] derived

an explicit formula for the anomalous dimension of the operator σk (the k-th power

of the auxiliary eld), which reads

∆(σk) = 2k − 2k(d− 1) ((k − 1)d2 + d+ 4− 3kd)

d− 4

η1

N+O(

1

N2) (2.9)

where η1 is the 1/N anomalous dimension of φi given in (1.21). In d = 6 − ε, this

gives

∆(σk) = 2k + (130k − 90k2)ε

N+O(ε2) . (2.10)

For k = 2, 3, we will be able to match this result with the one-loop operator mixing

calculations in the cubic theory (2.2).

5We thank D. T. Son for bringing this to our attention.6 After the original version of this paper appeared, a bootstrap study of the O(N) model in d = 5

was carried out in [61] with very encouraging results.

24

In [11, 12], explicit results for the 3-point function coecients gφφσ and gσ3 were

also derived. These are dened by the correlation functions

〈φi(x1)φj(x2)σ(x3)〉 =gφφσ

|x12|2∆φ−∆σ |x23|∆σ |x13|∆σδij .

〈σ(x1)σ(x2)σ(x3)〉 =gσ3

(|x12||x23||x13|)∆σ

(2.11)

The coecient gφφσ was given in [11, 12] to order 1/N2 and arbitrary d. Expanding

that result to leading order in ε = 6− d, we nd

g2φφσ =

6ε

N

(1 +

44

N+O(

1

N2)

). (2.12)

This result indeed matches the value of the coupling g21 in (2.2) at the IR xed point,

as will be shown in the next section. To leading order in 1/N , the 3-point function

coecient gσ3 is related to gφφσ by [11]

gσ3 = 2(d− 3)gφφσ , (2.13)

which, as we will see, is precisely consistent with the ratio g∗2g∗1

= 6 + O(1/N) of the

coupling constants at the d = 6− ε IR xed point.

2.3 The IR xed point of the cubic theory in d = 6−ε

In this section, we show that the interacting scalar theory with Lagrangian (2.2) has a

perturbative large N IR xed point in d = 6− ε, and compute the anomalous dimen-

sions of the fundamental elds φi and σ at the xed point. The theory (2.2) has an

obvious O(N) symmetry, with the N -component vector φi, i = 1, . . . , N transforming

in the fundamental representation. Note that g1 and g2 are classically marginal in

25

d = 6, and have dimension ε2in d = 6−ε. The Feynman rules of the theory are shown

in Figure 2.2.

φi j

δijp2 σ

1p2

−g1δij −g2

Figure 2.2: Feynman rules of the theory in Euclidean space.

It is not hard to compute the one-loop beta functions β1, β2 for the couplings

g1, g2. The relevant one-loop diagrams needed to compute the counter terms δφ, δσ,

δg1 for the σφφ coupling, and δg2 for the σ3 coupling are given in Figure 2.3. Note

that Gm,n denote the Green's function with m φ elds and n σ elds. The one-loop

diagrams are labeled 1 through 7 for convenience.

Let us begin with the computation of diagram 1 in Figure 2.3 7

D1 = (−g1)2

∫ddk

(2π)d1

(p+ k)2

1

k2= −g2

1I1 = − p2

(4π)3

g21

6

Γ(3− d/2)

(M2)3−d/2 . (2.14)

Here we have used the renormalization condition p2 = M2. This has a 1/ε pole in

d = 6− ε which must be canceled by the counter term −p2δφ. So we get:

δφ = − 1

(4π)3

g21

6

Γ(3− d/2)

(M2)3−d/2 . (2.15)

For δσ, we have two one-loop diagrams (2 and 3). However, other than having

dierent coupling constant factors, the integrals are identical to diagram 1. Note that7We will state approximate expressions for the one-loop integrals I1 and I2 that are sucient for

extracting the logM2 terms in d = 6− ε. The more precise expressions are given in Appendix A.

26

G2,0 = + 1 +

−p2δφ

+ · · ·

G0,2 = +

2

3

+

−p2δσ

+ · · ·

G2,1 = +

4

5

+

−δg1

+ · · ·

G0,3 = +

6

7

+

−δg2

+ · · ·

Figure 2.3: Diagrams contributing to the 1-loop β-functions.

these two diagrams have a symmetry factor of 2, and there is a factor of N associated

with the φ loop in diagram 3. So, we have:

D2 +D3 =1

2(−g2)2I1 +

N

2(−g1)2I1 = −Ng

21 + g2

2

2I1 . (2.16)

We arrive at the following expression for δσ:

δσ = − 1

(4π)3

Ng21 + g2

2

12

Γ(3− d/2)

(M2)3−d/2 . (2.17)

27

Now let us compute corrections to the 3-point functions. Diagram 4 gives:

D4 = (−g1)2(−g2)

∫ddk

(2π)d1

(k − p)2

1

(k + q)2

1

k2

= −g21g2I2 =

−g21g2

2(4π)3

Γ(3− d/2)

(M2)3−d/2 , (2.18)

where I2 is computed in Appendix A. Diagram 5 is again exactly the same as diagram

4, except for the coupling factors:

D5 = (−g1)3

∫ddk

(2π)d1

(k − p)2

1

(k + q)2

1

k2

= −g31I2 =

−g31

2(4π)3

Γ(3− d/2)

(M2)3−d/2 . (2.19)

The divergences in D4 and D5 must be canceled by the −δg1 counterterm. So we get

δg1 = −g31 + g2

1g2

2(4π)3

Γ(3− d/2)

(M2)3−d/2 . (2.20)

Finally, we calculate δg2. The diagrams have the same topology, and just dier in

the coupling factors. Also, in diagram 6, we have a factor of N from the φ loop. So

we nd

D6 +D7 = N(−g1)3I2 + (−g2)3I2

= −(Ng31 + g3

2)I2 =−(Ng3

1 + g32)

2(4π)3

Γ(3− d/2)

(M2)3−d/2 . (2.21)

This term is canceled by −δg2, so we have

δg2 = −Ng31 + g3

2

2(4π)3

Γ(3− d/2)

(M2)3−d/2 . (2.22)

28

The Callan-Symanzik equation for the Green's function Gm,n is:

(M∂

∂M+ β1

∂

∂g1

+ β2∂

∂g2

+mγφ + nγσ)Gm,n = 0 (2.23)

Let's rst apply it to G2,0. Then, to leading order in perturbation theory, the Callan-

Symanzik equation simplies to

− 1

p2M

∂

∂Mδφ + 2γφ

1

p2= 0 , (2.24)

which gives the anomalous dimension of φi

γφ =1

2M

∂

∂Mδφ =

1

(4π)3

g21

6. (2.25)

Note that γ1 > 0 as long as the coupling constant g1 is real. Analogously, we obtain

the σ anomalous dimension

γσ =1

2M

∂

∂Mδσ =

1

(4π)3

Ng21 + g2

2

12. (2.26)

Now we can compute the β functions for g1 and g2. We have

β1 = − ε2g1 +M

∂

∂M(−δg1 +

1

2g1(2δφ + δσ)) , (2.27)

where the rst term accounts for the bare dimension of g1 in d = 6 − ε. After some

simple algebra, we obtain

β1 = − ε2g1 +

(N − 8)g31 − 12g2

1g2 + g1g22

12(4π)3. (2.28)

Notice that when N 1, β1 is positive.

29

Finally, let us compute β2. The Callan-Symanzik equation gives:

β2 = − ε2g2 +M

∂

∂M(−δg2 +

1

2g2(3δσ)) , (2.29)

and so

β2 = − ε2g2 +

−4Ng31 +Ng2

1g2 − 3g32

4(4π)3. (2.30)

As a check, let us note that for N = 0 the beta function for g2 in d = 6 reduces to

β2 = − 3g32

4(4π)3, (2.31)

which is the correct result for the single scalar cubic eld theory in d = 6.

The single scalar cubic eld theory in d = 6 − ε has no xed points at real

coupling, due to the negative sign of the beta function (2.31). It has a xed point at

imaginary coupling, which is conjectured to be related by dimensional continuation

to the Yang-Lee edge singularity [37]. However, as we now show, for suciently large

N , our model has a stable interacting IR xed point. Note that for large N the beta

functions simplify to

β1 = − ε2g1 +

Ng31

12(4π)3(2.32)

β2 = − ε2g2 +

−4Ng31 +Ng2

1g2

4(4π)3. (2.33)

This can be solved to get8

g∗1 =

√6ε(4π)3

N(2.34)

g∗2 = 6g∗1 . (2.35)

8There is also a physically equivalent solution with the opposite signs of g1, g2.

30

It is straightforward to compute the subleading corrections at large N by solving the

exact beta function equations (4.5), (3.15) in powers of 1/N . This yields

g∗1 =

√6ε(4π)3

N

(1 +

22

N+

726

N2− 326180

N3+ . . .

)(2.36)

g∗2 = 6

√6ε(4π)3

N

(1 +

162

N+

68766

N2+

41224420

N3+ . . .

). (2.37)

Note that the coecients in this expansion appear to increase quite rapidly. This

suggests that the large N expansion may break down at some nite N . Indeed, we

will see in Section 2.4 that this large N IR xed point disappears at N ≤ 1038 (the

coupling constants go o to the complex plane). For all values of N ≥ 1039, the

xed point has real couplings and is IR stable, see Section 2.4. At large N , the IR

stability of the xed point can be seen from the fact that the matrix ∂βi∂gj

evaluated

at the xed point has two positive eigenvalues. These eigenvalues are in fact related

to the dimensions of the two eigenstates coming from the operator mixing of σ3 and

σφiφi operators, as will be discussed in more detail in Section 2.5.2.

We can now use the values of the xed point couplings to compute the dimensions

of the elementary elds φi and σ in the IR. From (2.25) and (2.26) we obtain

∆φ =d

2− 1 + γφ = 2− ε

2+

1

(4π)3

(g∗1)2

6

= 2− ε

2+

ε

N+

44ε

N2+

1936ε

N3+ . . . (2.38)

and

∆σ =d

2− 1 + γσ = 2− ε

2+

1

(4π)3

N(g∗1)2 + (g∗2)2

12

= 2 +40ε

N+

6800ε

N2+ . . . (2.39)

31

Note that both dimensions are above the unitarity bound in d = 6 − ε, namely

∆ > d2− 1, since the anomalous dimensions are positive. Moreover, note that the

order ε term in the dimension of σ cancels out and we nd ∆σ = 2 +O(1/N). This is

in perfect agreement with the large N description of the critical O(N) CFT. In that

approach, the eld σ corresponds to the composite operator φiφi, whose dimension

goes from ∆ = d− 2 at the free xed point to d−∆ +O(1/N) = 2 +O(1/N) at the

interacting xed point. Furthermore, comparing (2.38), (2.39) with (2.4), (2.5), we

see that the coecients of the 1/N expansion precisely match the available results

for the large N critical O(N) theory [5, 4, 6] expanded at d = 6− ε. This is a strong

test that the IR xed point of the cubic theory in d = 6− ε is identical at large N to

the dimensional continuation of the critical point of the (φiφi)2 theory.

We may also match the values of the xed point couplings (2.36),(2.37) with the

large N results (2.12), (2.13) for the 3-point functions coecients in the critical O(N)

theory. At leading order in ε, the 3-point functions in our cubic model simply come

from a tree level calculation, and it is straightforward to verify the agreement of (2.36)

with (2.12),9 and that the ratio g∗1/g∗2 = 6 at leading order at large N agrees with

(2.13).

2.4 Analysis of xed points at nite N

In this section we analyze the one-loop xed points for general N . First, let us dene:

g1 ≡√

6ε(4π)3

Nx , (2.40)

g2 ≡√

6ε(4π)3

Ny . (2.41)

9An overall normalization factor comes from the normalization of the massless scalar propagatorsin d = 6.

32

After this, the vanishing of the β-functions (4.5), (3.15) gives

Nx = (N − 8)x3 − 12x2y + xy2 , (2.42)

Ny = −12Nx3 + 3Nx2y − 9y3 . (2.43)

These equations have 9 solutions if we allow x and y to be complex. One of

them is the trivial solution (0, 0). Two of them are purely imaginary; they occur at

(0,±√

N9i). There are no simple expressions for the remaining six solutions, but it is

straightforward to study them numerically.

Two of them are real solutions in the second and fourth quadrant: (−x1, y1) and

(x1,−y1), with x1, y1 > 0. They exist for any N , but they are not IR stable (they are

saddles, i.e. there is one positive and one negative direction in the coupling space).

They can be seen in both the bottom left and right graphs in Figure 2.4, corresponding

to N = 2000 and N = 500 respectively.

The behavior of the remaining four solutions changes depending on the value of N .

For N ≤ 1038, we nd that all four solutions are complex. The bottom right graph

of Figure 2.4 shows that for N = 500 there are indeed only the two real solutions

present for any N discussed above.

For N ≥ 1039, all four of these solutions become real, and they lie in the rst and

third quadrants: they have the form (x3, y3), (x4, y4), (−x3,−y3), (−x4,−y4), with

x3, y3, x4, y4 > 0. To display the typical behavior of the solutions with N ≥ 1039, in

the bottom left graph of Figure 2.4 we plot the zeroes of β1, β2 for N = 2000, with

regions of their signs labeled. Combining these, we can get the ow directions in each

of these regions in the bottom left graph. We nd that for all N ≥ 1039, we have two

stable IR xed points that are symmetric with respect to the origin, and are labelled

as red dots in the gure. These correspond to the large N solution (2.36)-(2.37), and

its equivalent solution with opposite signs of the couplings. For very large N , we see

33

that these solutions satisfy g∗2 = 6g∗1 as in (2.35). The origin is a UV stable xed

point, and from the direction of the renormalization group ow we can see that all

other xed points are saddle points: they have one stable direction and one unstable

direction.

It is interesting to treat N as a continuous parameter and solve for the value Ncrit

where the real IR stable xed points disappear. First, we notice that we can factor

out x in (2.42), eectively making it quadratic. Moreover, if we subtract (2.43) from

yxtimes (2.42), we will cancel the Ny term, making the second beta function equation

a homogeneous equation of order 3. After some more manipulation, we obtain:

N = (N − 44)x2 + (6x− y)2 (2.44)

Nx2(6x− y)− y(4x2 − 4y2 + (6x− y)y) = 0 . (2.45)

It is convenient to rewrite these equations in terms of the following variables

x = α , y = β + 6α . (2.46)

After some algebra, we get:

(N − 44)α2 + β2 = N (2.47)

840α3

β3+ (464−N)

α2

β2+ 84

α

β+ 5 = 0 . (2.48)

Ncrit occurs when the curves dened by the two equations are tangent to each other.

Notice that we have eectively decoupled the two equations, since if we write z = α/β,

we get

α2(N − 44 + z−2) = N (2.49)

840z3 + (464−N)z2 + 84z + 5 = 0 , (2.50)

34

We can just rst solve the second equation, then easily solve the rst. To determine

the critical N , we want the second equation to have exactly one real root, thus we

require its discriminant to be zero:

∆ = 18abcd− 4b3d+ b2c2 − 4ac3 − 27a2d2 = 0 , (2.51)

where a, b, c, d are the coecients of the cubic equation:

a = 840 , b = 464−N , c = 84 , d = 5 . (2.52)

We then arrive at a cubic equation in terms of N :

∆ = 20N3 − 20784N2 + 19392N + 1856 = 0 . (2.53)

We can easily write down an analytic expression for N , and we nd

Ncrit = 1038.2660492... . (2.54)

Our numerical solution of the equations is consistent with this value.

For completeness, we now discuss the large N behavior of the four xed points

which are not IR stable (two of them are present for any N , and two of them only

for N > Ncrit). They are obtained if we assume that, as N →∞, x is O(1) and y is

O(√N). Then, at the leading order in N , we get:

Nx = Nx3 + xy2 , Ny = 3Nx2y − 9y3 . (2.55)

This can be solved to get four solutions: (√

56,√

16

√N), (−

√56,√

16

√N), (

√56,−√

16

√N),

(−√

56,−√

16

√N). They correspond to the four real xed points at large N that are

35

Figure 2.4: (a) zeros of β1 at N = 2000. (b) zeros of β2 at N = 2000. (c) RG owdirections at N = 2000. (d) only two non-trivial real solutions at N = 500.

saddle points. Using the equation from β1, we can also check that:

∆σ = 2− ε

2+ε

2

(Nx2 + y2

N

)

= 2±√

5√Nε+O(

1

N) . (2.56)

The ε√Ncorrection does not correspond to the conventional large N behavior.

36

2.5 Operator mixing and anomalous dimensions

2.5.1 The mixing of σ2 and φiφi operators

Let us compute the anomalous dimension matrix for operatorsO1 = φiφi√NandO2 = σ2,

where the√N in the denominator is to ensure that the two-point functions of O1

and O2 are of the same order. They both have classical dimension 4− ε in d = 6− ε,

so we expect them to mix.

Consider the operators O1, and O2 renormalized according to the convention

shown in Figure 2.5.

= 〈φ(p)φ(q)φ2(p+ q)〉 = 1p2

1q2

= 〈σ(p)σ(q)σ2(p+ q)〉 = 1p2

1q2

Figure 2.5: Renormalization conditions for the operators O1 and O2.

Let OiM denote the operator renormalized at scale M , and Oi

0 denote the bare

operator. We are looking for expressions similar to (18.53) of [62], i.e. of the form:

O1M = O1

0 + δ11O10 + δ12O2

0 + δφO10 (2.57)

O2M = O2

0 + δ21O10 + δ22O2

0 + δσO20 . (2.58)

The δij counterterms in the above equations are obtained by extracting the loga-

rithmic divergence from the diagrams shown in Figure 2.6, while the δφO10 and δσO2

0

terms correspond to external leg corrections, which are omitted in Figure 2.6.

37

= +

+

δ11

+

+

δ12

= +

+

δ21

+

+

δ22

Figure 2.6: Matrix elements of operators O1 and O2 to 1-loop order.

Each of the terms δij is given by canceling the divergent pieces of two of the above

diagrams, as shown. For example, let us compute the rst diagram of the δ11 counter

term in Figure 2.6. We have:

D1 = (−g1)2

∫ddk

(2π)d1

(k − p)2

1

(k + q)2

1

k2

= (−g1)2I2 =g2

1

2(4π)3

Γ(3− d/2)

(M2)3−d/2 . (2.59)

Next, we compute the second diagram of the δ11 counter term. Notice that there is a

symmetry factor of 2, and also a factor of N from a closed φ loop.

D2 =N

2(−g1)2

∫ddk

(2π)d1

(p+ k)2

1

k2

1

p2

=Ng2

1

2I1

1

p2= − 1

(4π)3

Ng21

12

Γ(3− d/2)

(M2)3−d/2 . (2.60)

These two terms are canceled by δ11; therefore, to leading order in ε we have

δ11 =

(− g2

1

2(4π)3+

Ng21

12(4π)3

)Γ(3− d/2)

(M2)3−d/2 . (2.61)

Similarly, we can calculate the other three δij. The integrals are the same, only

the factors of coupling constants and N (due to closed φ loops) are dierent. With

our normalization convention of O1, we need to multiply δ12 by a factor of√N , and

38

divide δ21 by a factor of√N . Then the matrix is symmetric, δ21 = δ12, and we nd

δ12 =

(−√Ng2

1

2(4π)3+

√Ng1g2

1(4π)3

)Γ(3− d/2)

(M2)3−d/2 , (2.62)

δ22 =

(− g2

2

2(4π)3+

g22

12(4π)3

)Γ(3− d/2)

(M2)3−d/2 . (2.63)

Thus, the matrix δij is

δij =1

12(4π)3

Γ(3− d/2)

(M2)3−d/2

−6g21 +Ng2

1 −6√Ng2

1 +√Ng1g2

−6√Ng2

1 +√Ng1g2 −6g2

2 + g22

. (2.64)

The anomalous dimension matrix is given by

γij = M∂

∂M(−δij + δijz ) , (2.65)

where we have dened

δijz =

δφ 0

0 δσ

. (2.66)

Now, using expressions for δφ and δσ from (2.15), (2.17) , we get:

γij =−1

6(4π)3

4g21 −Ng2

1 6√Ng2

1 −√Ng1g2

6√Ng2

1 −√Ng1g2 4g2

2 −Ng21

. (2.67)

The eigenvalues of this matrix will give the dimensions of the two eigenstates arising

from the mixing of operators O1 and O2.

39

After plugging in the coupling constants at the IR xed point from (2.36) and

(2.37), and keeping its entries to order 1/N , the matrix elements of γij become:

γij =−1

6(4π)3

4g21 −Ng2

1 6√Ng2

1 −√Ng1g2

6√Ng2

1 −√Ng1g2 4g2

2 −Ng21

(2.68)

= ε

1 + 40N

840N3/2

840N3/2 1− 100

N

. (2.69)

To order 1/N , the o-diagonal terms do not aect the eigenvalues, and we get the

scaling dimensions

∆− = d− 2 + γ− = 4− 100ε

N+O

(1

N2

)

∆+ = d− 2 + γ+ = 4 +40ε

N+O

(1

N2

) (2.70)

with the explicit eigenstates given by

O+ =

√N

6O1 + O2 (2.71)

O− =−6√NO1 + O2 . (2.72)

Satisfyingly, we see that to this order the dimension ∆− precisely matches the dimen-

sion of the only primary eld of dimension near 4 in the large N UV xed point of

the quartic theory. It was determined using large N methods in [10], and corresponds

to k = 2 in (2.10). Since there are no other primaries of dimension 4 +O(1/N) in the

critical theory, we expect that the other dimension ∆+ corresponds to a descendant.

Comparing (4.26) with (2.39), we see that to this order ∆+ = 2 + ∆σ, so that the

eigenstate with eigenvalue γ+ is indeed a descendant of σ.

40

We can actually show that ∆+ = 2 + ∆σ to all orders in 1/N (and for all xed

points) just using the β-function equations. To start, we redene our variables as

(2.40), (2.41). With this denition, the condition satised by x and y at the IR xed

point is given in (2.42), (2.43), or

1 = x2 − 8

Nx2 − 12

Nxy +

1

Ny2 (2.73)

1 = −12x3

y+ 3x2 − 9

Ny2 . (2.74)

The anomalous dimension matrix in this notation becomes

γij = ε

x2 − 4Nx2 1√

N(xy − 6x2)

1√N

(xy − 6x2) x2 − 4Ny2

, (2.75)

which has eigenvalues

λ± =(2x2 − 4

Nx2 − 4

Ny2)±

√|γ|

2ε, (2.76)

where |γ| is the determinant, given by:

|γ| = 144

Nx4 − 48

Nx3y +

4

Nx2y2 +

16

N2x4 − 32

N2x2y2 +

16

N2y4 . (2.77)

Also, from the rst line of (2.39), we see that in terms of x and y the dimension ∆σ

becomes

∆σ = 2− ε

2+ε

2x2 +

ε

2Ny2 , (2.78)

Thus, we have to show that:

∆σ + 2 = 4− ε+ λ+ (2.79)

41

which, after some algebra, is seen to imply

1− x2 +4

Nx2 +

5

Ny2 =

√|γ| . (2.80)

Replacing (1− x2) with (2.73), we have

− 4

Nx2 − 12

Nxy +

6

Ny2 =

√|γ| . (2.81)

Squaring both sides, and plugging in the expression for |γ|, we get that the following

equation is to hold for (2.79) to be true

36x4 − 12x3y + x2y2 =1

N

(24x3y + 32x2y2 − 36xy3 + 5y4

). (2.82)

To prove this equality, we again go back to (2.73) and (2.74). If we multiply both of

these equations by 3xy − y2

2and subtract the former from the latter, we get exactly

the expression above. Thus, ∆+ = 2 + ∆σ holds to all orders in 1/N .

2.5.2 The mixing of σ3 and σφiφi operators

Next, we would like to calculate the mixed anomalous dimensions of the ∆ = 6

operators, and show that they are all slightly irrelevant in d = 6 − ε. There are six

operators with ∆ = 6 at d = 6, they are:

O1 =σ(φi)2

√N

, O2 = σ3 , (2.83)

O3 =(∂φi)2

√N

, O4 = (∂σ)2 , (2.84)

O5 =φiφi√N

, O6 = σσ. (2.85)

42

However, in d = 6− ε, O1 and O2 have bare dimensions 3d−22

= 6− 32ε, while O3, O4,

O5, O6 have bare dimensions 2 + 2d−22

= 6 − ε. Therefore, in d = 6 − ε, O1 and O2

mix with each other, and O3, O4, O5, O6 mix with themselves.

We will compute the mixed anomalous dimensions of O1 and O2. Using the

expressions for the β functions, (4.5), (3.15), we can very simply compute the mixed

anomalous dimension matrix by dierentiating them with respect to appropriately

normalized couplings (the rescaling is needed because two-point functions of O1 and

O2 are o by a factor of 3N). After some algebra, we get:

M ij =

− ε

2+

3Ng21−24g21−24g1g2+g2212(4π)3

−12g21+2g1g212(4π)3

√3N

−12g21+2g1g212(4π)3

√3N − ε

2+

Ng21−9g224(4π)3

. (2.86)

We plug in the xed point value of g1 and g2 from (2.36) and (2.37) to get:

M ij =

ε− 5040N2 ε+ ... 840

√3

N√Nε+ ...

840√

3N√Nε+ ... ε− 420

Nε− 154560

N2 ε+ ...

. (2.87)

Notice that at this IR xed point, both eigenvalues of M ij are positive, which implies

that the xed point is IR stable. This matrix has eigenvalues

λ1 = ε− 3780

N2ε+O(

1

N3) (2.88)

λ2 = ε− 420

Nε− 155820

N2ε+O(

1

N3) . (2.89)

The relation between scaling dimensions and the eigenvalues of the matrix is given

by10

∆ = d+ λ . (2.90)

10As a simple test of this formula, note that at the free UV xed point the eigenvalues of (2.86)are λ1 = λ2 = − ε

2 , giving dimensions ∆1 = ∆2 = 3(d/2− 1) as it should be.

43

Thus, the scaling dimensions of the mixture of the two operators are:

∆1 = d+ λ1 = 6 +O(1

N2) (2.91)

∆2 = d+ λ2 = 6− 420

Nε+O(

1

N2) . (2.92)

There are no O(ε) correction to the scaling dimensions, as we expected. As a further

check, we note that ∆2 agrees with the dimension of the k = 3 primary operator

given in (2.10). In the large N Hubbard-Stratonovich approach, this operator is σ3

[10]. Our 1-loop calculation demonstrates the presence of another primary operator

whose dimension is ∆1. Presumably, the corresponding operator in the Hubbard-

Stratonovich approach is (∂µσ)2.

We can also show that at the other xed points discussed in Section 2.4, M ij has

one positive and one negative eigenvalues. If we use (2.40), (2.41) and plug them into

(2.86), we get

M ij =

− ε

2+ ε

2N(3Nx2 − 24x2 − 24xy + y2) ε

2N(−12x2 + 2xy)

√3N

ε2N

(−12x2 + 2xy)√

3N − ε2

+ 3ε2N

(Nx2 − 9y2)

. (2.93)

We have shown from solving (2.55) that

x = ±√

5/6 , y = ±√

1/6√N . (2.94)

Plugging these in, we nd that the eigenvalues of M ij at these xed points are:

λ1 = ε , λ2 = −5

3ε , (2.95)

which conrms our graphical analysis that all of these xed points are saddle points.

44

2.6 Comments on CT and 5-d F theorem

A quantity of interest in a CFT is the coecient CT of the stress-tensor 2-point

function, which may be dened by

〈Tµν(x1)Tρσ(x2)〉 = CTIµν,ρσ(x12)

x2d12

, (2.96)

where Iµν,ρσ(x12) is a tensor structure uniquely xed by conformal symmetry, see e.g.

[63]. For N free real scalar elds in dimension d with canonical normalization, one

has CT = Nd(d−1)S2

d, where Sd is the volume of the d-dimensional round sphere. In the

critical O(N) theory, using large N methods one nds the result [11, 12]

CT =Nd

(d− 1)S2d

(1 +

1

NCT,1 +O(

1

N2)

)

CT,1 = −(

2C(µ)

µ+ 1+µ2 + 3µ− 2

µ(µ2 − 1)

)η1 , µ =

d

2

(2.97)

where C(µ) = ψ(3− µ) + ψ(2µ− 1)− ψ(1)− ψ(µ), and ψ(x) = Γ′(x)Γ(x)

. It is interesting

to analyze the behavior of CT,1 in d = 6 − ε. The anomalous dimension η1, given

in (1.21), is of order ε, but there is a pole in C(µ) from the term ψ(3 − µ) ∼ −2/ε.

Hence, one nds at d = 6− ε

CT,1 = 1− 7

4ε+O(ε2) (2.98)

and so

CT =d

(d− 1)S2d

(N + 1− 7

4ε

)+O(ε2) . (2.99)

Thus, as d → 6, we nd the CT coecient of N + 1 free real scalars. This provides

a nice check on our description of the critical O(N) theory via the IR xed point of

(2.2). The O(ε) correction may be calculated in this description as well, using the

1-loop xed point (2.36), (2.37), but we leave this for future work.

45

The appearance of an extra massless scalar eld as d → 6 is also suggested by

dimensional analysis. The dimension of σ is 2+O(1/N) in all d, so as d approaches 6,

it becomes the dimension of a free scalar eld. Then the two-derivative kinetic term

for σ (as well as the σ3 coupling) become classically marginal. Note also that the

negative sign of the order ε correction in (2.99) implies that CT decreases from the

UV xed point of N + 1 free elds to the interacting IR xed point, consistently with

the idea that CT may be a measure of degrees of freedom [12]. However, expanding

CT in d = 4 + ε, one nds

CT =d

(d− 1)S2d

(N − 5

12ε2)

+O(ε3) . (2.100)

According to our interpretation, the critical theory in d = 4 + ε should be indentied

with the UV xed point of the −φ4 theory, while the IR xed point corresponds to

N free scalars. Then, (2.100) is seen to violate CUVT > CIR

T . A plot of CT,1 in the

range 2 < d < 6 is given in Figure 2.7. Finally, let us quote the value of CT in the

interesting dimension d = 5

Cd=5T =

5

4S25

(N − 1408

1575π2

). (2.101)

Again, we observe that this value is consistent with CUVT > CIR

T for the ow from the

free UV theory of N+1 massless scalars to the interacting xed point, but not for the

ow from the interacting xed point to the free IR theory of N massless scalars. Thus,

the latter ow provides a non-supersymmetric counterexample against the possibility

of a CT theorem (for a supersymmetric counterexample, see [64]). In contrast, the

5-dimensional F-theorem holds for both ows, as we discuss below.

It was proposed in [50] (see also [65, 66]) that, for any odd dimensional Euclidean

CFT, the quantity

F = (−1)d+12 F = (−1)

d−12 logZSd , (2.102)

46

3 4 5 6d

1.0

0.5

0.5

1.0CT,1

-

-

Figure 2.7: The O(N0) correction CT,1 in the coecient CT of the stress tensor two-point function in the critical O(N) theory for 2 < d < 6; see (2.97). Note that CT,1is negative for 2 < d . 5.22, so that in this range of dimensions Ccrit

T < NC free sc.T

for large N . Therefore, the CT theorem is respected in 2 < d < 4, but violated for4 < d . 5.22.

should decrease under RG ows

FUV > FIR . (2.103)

Here F = − logZSd is the free energy of the CFT on the round sphere, which is a

nite number after the power law UV divergences are regulated away (for instance,

by ζ-function or dimensional regularization). Note that when we put the CFT on the

sphere, we have to add the conformal coupling of the scalar elds to the Sd curvature.

This eectively renders the vacuum metastable, both in the case of the −φ4 theory

and in the cubic theory. Similarly, the CFT is metastable on R × Sd−1, which is

relevant for calculating the scaling dimensions of operators.

For d = 3 (where F = F ), the F-theorem (2.103) was proved in [67]. However,

higher dimensional versions are less well established. Using the results of this chapter

we can provide a simple new test of the 5d version of the F-theorem (in this case

F = −F ). As we have argued, the 5d critical O(N) theory can be viewed as either

the IR xed point of the cubic theory (2.2) or the UV xed point of the quartic scalar

47

theory. This implies that F should satisfy

NFfree sc. < Fcrit. < (N + 1)Ffree sc. , (2.104)

where Ffree sc. is minus the free energy of a 5d free conformal scalar [50]

Ffree sc. =log 2

128+

ζ(3)

128π2− 15ζ(5)

256π4' 0.00574 . (2.105)

The value of F at the critical point can be calculated perturbatively in 1/N by

introducing the Hubbard-Stratonovich auxiliary eld as reviewed in Section 2. The

leading O(N) term is the same as in the free theory, while the O(N0) term arises from

the determinant of the non-local kinetic operator for the auxiliary eld [24, 68, 50].

The result is [23]

Fcrit. = NFfree sc. +3ζ(5) + π2ζ(3)

96π2+O

(1

N

). (2.106)

The same answer may be obtained by computing the ratio of determinants for the

bulk scalar eld in the AdS6 Vasiliev theory with alternate boundary conditions [23].

We see that the left inequality in (2.104) is satised, since the O(N0) correction

in (2.106) is positive (this check was already made in [23]). More non-trivially, we

observe that the right inequality also holds, because 3ζ(5)+π2ζ(3)96π2 ' 0.001601 is smaller

than the value of F for one free scalar, eq. (2.105).

48

2.7 Gross-Neveu-Yukawa model and a test of 3-d F-

theorem

The action of the Gross-Neveu (GN) model [25] is given by

S(ψ, ψ) = −∫ddx

[ψi/∂ψi +

1

2g(ψiψi)2

]. (2.107)

Here N = Ntr1, where tr1 is the trace of the identity in the Dirac matrix space, and

N is the number of Dirac fermion elds ψi (i = 1, . . . , N). The parameter N counts

the actual number of fermion components, and it is the natural parameter to write

down the 1/N expansion (factors of tr1 never appear in the expansion coecients if

N is used as the expansion parameter).

The beta functions and anomalous dimensions of this model in d = 2 + ε can be

calculated to be (see, for instance, [3])

β(g) = εg − (N − 2)g2

2π+ (N − 2)

g3

4π2+O(g4) (2.108)

ηψ(g) =N − 1

8π2g2 − (N − 1)(N − 2)

32π3g3 +O(g4) (2.109)

ηM(g) =N − 1

2πg − N − 1

8π2g2 +O(g3), (2.110)

where ηM is related to the anomalous dimension of the composite eld σ = ψψ. We

can solve the beta function for the critical value of g at the xed point:

g∗ =2π

N − 2ε

(1− ε

N − 2

)+O(ε3) . (2.111)

49

Plugging this value of g∗, we can nd the dimensions of the fermion eld and the

composite eld:

∆ψ = d− 1 + ηψ(g∗) =1 + ε

2+

N − 1

4(N − 2)2ε2 +O(ε3) , (2.112)

∆σ = d− 1− ηM(g∗) = 1− ε

N − 2+O(ε2) . (2.113)

The UV xed point of the GN model in 2 < d < 4 dimensions is related to the IR

xed point of the Gross-Neveu-Yukawa (GNY) model, which has the following action

[26, 27, 3]

S(ψ, ψ, σ) =

∫ddx

[−ψi

(/∂ + g1σ

)ψi +

1

2(∂µσ)2 +

g2

24σ4

]. (2.114)

In d = 4− ε, the one-loop beta functions of the GNY model are given by:

βg21 = −εg21 +

N + 6

16π2g4

1 (2.115)

βg2 = −εg2 +1

8π2

(3

2g2

2 +Ng2g21 − 6Ng4

1

). (2.116)

For small ε, one nds that there is a stable IR xed point for any positive N :

(g∗1)2 =16π2ε

N + 6(2.117)

g∗2 = 16π2Rε, (2.118)

where

R =24N

(N + 6)[(N − 6) +√N2 + 132N + 36]

. (2.119)

The anomalous dimensions of the elementary elds are given by

γψ =1

32π2g2

1 , γσ =N

32π2g2

1. (2.120)

50

and plugging in the value of the xed point couplings, one obtains

∆ψ =d− 1

2+ γψ =

3

2− N + 5

2(N + 6)ε (2.121)

∆σ =d− 2

2+ γσ = 1− 3

N + 6ε . (2.122)

The large N expansion of the 3d critical fermion theory may be developed in

arbitrary dimension following similar lines as in the scalar case reviewed in Section 2:

one introduces an auxiliary eld σ to simplify the quartic interaction, and develops

the 1/N perturbation theory with the eective non-local propagator for σ and the

σψψ interaction. The anomalous dimensions of ψ and σ in the critical theory have

been calculated respectively to order 1/N3 and 1/N2, and in arbitrary dimension d

[53, 69]. To leading order in 1/N , the explicit expressions are given by

∆ψ =d− 1

2− Γ (d− 1) sin(πd

2)

πΓ(d2− 1)

Γ(d2

+ 1) 1

N+O

(1

N2

)(2.123)

∆σ = 1 +4(d− 1)Γ (d− 1) sin(πd

2)

π(d− 2)Γ(d2− 1)

Γ(d2

+ 1) 1

N+O

(1

N2

). (2.124)

Expanding these expressions in d = 2 + ε and d = 4− ε, one can verify the agreement

with (2.113) and (2.122) respectively. Note that both ∆ψ and ∆σ go below their

respective unitarity bounds for d > 4; so, unlike in the scalar case, there is no unitary

critical fermion theory above dimension four. One may check that the available

subleading large N results [53, 69] also precisely agree with the 1/N expansion of

(2.113) and (2.122), giving strong support to the fact that the 3d critical fermionic

CFT may be viewed as either the IR xed point of the GNY model, or the UV xed

point of the GN theory.

We may use the two alternative descriptions of the critical fermion theory to

provide a simple test of the 3d F-theorem, similar to the tests carried out in [50]. In

the UV, the GNY model (which has relevant interactions in d = 3) is a free theory

51

of N fermions and one conformal scalar, while the GN model is free in the IR. Thus,

the value of F for the critical fermion theory should satisfy

NFfree fermi < Fcrit. fermi < NFfree fermi + Ffree sc. . (2.125)

In d = 3, one nds

Ffree sc. =log 2

8− 3ζ(3)

16π2' 0.0638 . (2.126)

The value of F in the critical theory may be computed perturbatively in the 1/N

expansion, and one obtains the result [68, 50]

Fcrit. fermi = NFfree fermi +ζ(3)

8π2+O

(1

N

). (2.127)

Because ζ(3)8π2 ' 0.0152 < Ffree sc., we see that indeed (2.125) is satised in both

directions.

52

Chapter 3

Three-Loop Analysis of the Critical

O(N) Models in 6− ε Dimensions

3.1 Introduction and Summary

This chapter is based on the work published in [29], co-authored with Simone Giombi,

Igor Klebanov, and Grigory Tarnopolsky. We also thank J. Gracey and I. Herbut for

very useful correspondence, and to Y. Nakayama for valuable discussions.

We would like to extend the one-loop analysis was carried out in 2 to three loops.

We have the cubic O(N) symmetric theory of N + 1 scalar elds σ and φi in 6− ε

dimensions with the Lagrangian:

L =1

2(∂µφ

i)2 +1

2(∂µσ)2 +

1

2g1σ(φi)2 +

1

6g2σ

3 , (3.1)

The one-loop beta functions showed that for N > Ncrit there exists an IR stable

xed point with real values of the two couplings. We argued that this IR xed point

of the cubic O(N) theory is equivalent to the perturbatively unitary UV xed point

of the O(N) model with interaction (φiφi)2, which exists for large N in 4 < d < 6

[32, 33, 34, 35]. The 1/N expansions of various operator scaling dimensions we found

53

in chapter 2 agree with the corresponding results [4, 5, 6, 7, 8, 9, 10, 11, 12] in the

quartic O(N) model continued to 6− ε dimensions.

A surprising result of chapter 2 was that the one-loop value of Ncrit is very large:

if Ncrit is treated as a continuous real parameter, then it is ≈ 1038.266. Our interest

now is in continuing the d = 6 − ε xed point to ε = 1 in the hope of nding a

5-dimensional O(N) symmetric unitary CFT. In order to study the ε expansion of

Ncrit, in section 3.2 we calculate the three-loop β functions, following the earlier work

of [38, 39, 40].1 In section 3.4 we nd the following expansion for the critical value of

N :

Ncrit = 1038.266− 609.840ε− 364.173ε2 +O(ε3). (3.2)

Neglecting further corrections, this gives Ncrit(ε = 1) ≈ 64, but higher orders in ε can

obviously change this value signicantly. It is our hope that a conformal bootstrap

approach [13, 14, 15, 16], perhaps along the lines of [61], can help determine Ncrit

more precisely in d = 5. The bootstrap approach may also be applied in non-integer

dimensions close to 6, but one should keep in mind that such theories are not strictly

unitary [71].2

The major reduction of Ncrit as ε is increased from 0 to 1 is analogous to what is

known about the Abelian Higgs model in 4−ε dimensions.3 For the model containing

Nf complex scalars, the one-loop critical value of Nf is found to be large, Nf,crit ≈ 183

[58]. However, the O(ε) correction found from two-loop beta functions has a negative

coecient and almost exactly cancels the leading term when ε = 1, suggesting that

the Nf,crit is small in the physically interesting three-dimensional theory [59].

1These papers considered cubic eld theories of q − 1 scalar elds that were shown in [70] todescribe the q-state Potts model. These theories possess only discrete symmetries and generallydier from the O(N) symmetric theories that we study.

2 Another possible non-perturbative approach to the theory in 4 < d < 6 is the Exact Renormal-ization Group [72]. This approach does not seem to indicate the presence of a UV xed point in thetheory of N scalar elds, but a search for an IR xed point in the theory of N + 1 scalar elds hasnot been carried out yet.

3 We are grateful to Igor Herbut for pointing this out to us.

54

Another interesting property of the theories (4.2) is the existence of the lower crit-

ical value N ′crit such that for N < N ′crit there is an IR stable xed point at imaginary

values of g1 and g2. The simplest example of such a non-unitary theory is N = 0,

containing only the eld σ. Its 6 − ε expansion was originally studied by Michael

Fisher [37] and the continuation to ε = 4 provides an approach to the Yang-Lee

edge singularity in the two-dimensional Ising model (this is the (2, 5) minimal model

[73, 74] with central charge −22/5). From the three-loop β functions we nd the ε

expansion

N ′crit = 1.02145 + 0.03253ε− 0.00163ε2 +O(ε3) . (3.3)

The smallness of the coecients suggests that N ′crit > 1 for a range of dimensions

below 6. In section 3.5 we discuss some properties of the N = 1 theory. We show

that it possesses an unstable xed point with g∗1 = g∗2 where the lagrangian splits

into that of two decoupled N = 0 theories. There is also an IR stable xed point

where g∗2 = 6g∗1/5 + O(ε). A distinguishing feature of this non-unitary CFT is that

it has a discrete Z2 symmetry, and it would be interesting to search for it using the

conformal bootstrap methods developed in [60]. We suggest that, when continued to

two dimensions, it describes the (3, 8) non-unitary conformal minimal model.

In Section 3.4.1 we also discuss unstable unitary xed points that are present in

6−ε dimensions for all N . For N = 1 the xed point has g∗1 = −g∗2; it is Z3 symmetric

and describes the critical point of the 3-state Potts model in 6− ε dimensions [75].4

3.2 Three-loop β-functions in d = 6− ε

The action of the cubic theory is

S =

∫ddx(1

2(∂µφ

i0)2 +

1

2(∂µσ0)2 +

1

2g1,0σ0φ

i0φ

i0 +

1

6g2,0σ

30

), (3.4)

4We are grateful to Yu Nakayama for valuable discussions on this issue.

55

where φi0 and σ0 are bare elds and g1,0 and g2,0 are bare coupling constants.5 As

usual, we introduce renormalized elds and coupling constants by

σ0 = Z1/2σ σ, φi0 = Z

1/2φ φi,

g1,0 = µε2Zg1Z

−1/2σ Z−1

φ g1 , g2,0 = µε2Zg2Z

−3/2σ g2 .

(3.5)

Here g1, g2 are the dimensionless renormalized couplings, and µ is the renormalization

scale. We may write

Zσ = 1 + δσ , Zφ = 1 + δφ , Zg1 = 1 + δg1 , Zg2 = 1 + δg2 (3.6)

so that, in terms of renormalized quantities, the action reads

S =

∫ddx(1

2(∂µφ

i)2 +1

2(∂µσ)2 +

g1

2σφiφi +

g2

6σ3 (3.7)

+δφ2

(∂µφi)2 +

δσ2

(∂µσ)2 +δg1

2σφiφi +

δg2

6σ3).

To carry out the renormalization procedure, we will use dimensional regularization

[76] in d = 6− ε and employ the minimal subtraction scheme [77]. In this scheme, the

counterterms are xed by requiring cancellation of poles in the dimensional regulator,

and have the structure

δg1 =∞∑

n=1

an(g1, g2)

εn, δg2 =

∞∑

n=1

bn(g1, g2)

εn, δφ =

∞∑

n=1

zφn(g1, g2)

εn, δσ =

∞∑

n=1

zσn(g1, g2)

εn.

(3.8)

The anomalous dimensions and β-functions are determined by the coecients of the

simple 1/ε poles in the counterterms [77]. Specically, in our case we have that the

5We do not include mass terms as we are ultimately interested in the conformal theory. In thedimensional regularization that we will be using, mass terms are not generated if we set them tozero from the start.

56

anomalous dimensions are given by

γφ = −1

4

(g1

∂

∂g1

+ g2∂

∂g2

)zφ1 , (3.9)

γσ = −1

4

(g1

∂

∂g1

+ g2∂

∂g2

)zσ1 (3.10)

and the β-functions are

β1(g1, g2) = − ε2g1 +

1

2

(g1

∂

∂g1

+ g2∂

∂g2

− 1)(a1 −

1

2g1(2zφ1 + zσ1 )),

β2(g1, g2) = − ε2g2 +

1

2

(g1

∂

∂g1

+ g2∂

∂g2

− 1)(b1 −

3

2g2z

σ1 ). (3.11)

In other words, in order to determine the anomalous dimensions and β-functions, we

have to calculate the coecients of the 1/ε-divergencies in the loop diagrams, from

which we can read o the residues a1(g1, g2), b1(g2, g2), zφ1 (g1, g2), zσ1 (g1, g2).

Working in perturbation theory, we will denote by a1i the term of i-th order in

the coupling constants, and similarly for the other residue functions. Then, using the

results for the Feynman diagrams collected in the Appendix, we nd the anomalous

dimensions

γφ =− 1

2zφ12 − zφ14 −

3

2zφ16

=g2

1

6(4π)3− g2

1

432(4π)6

(g2

1(11N − 26)− 48g1g2 + 11g22

)

− g21

31104(4π)9

(g4

1(N(13N − 232) + 5184ζ(3)− 9064) + g31g26(441N − 544)

− 2g21g

22(193N − 2592ζ(3) + 5881) + 942g1g

32 + 327g4

2

),

(3.12)

57

γσ =− 1

2zσ12 − zσ14 −

3

2zσ16

=Ng2

1 + g22

12(4π)3+

1

432(4π)6

(2Ng4

1 + 48Ng31g2 − 11Ng2

1g22 + 13g4

2

)

+1

62208(4π)9

(96N(12N + 11)g5

1g2 − 1560Ng31g

32 + 952Ng2

1g42

− 2Ng61(1381N − 2592ζ(3) + 4280) + g6

2(2592ζ(3)− 5195)

+ 3Ng41g

22(N + 4320ζ(3)− 8882)

)

(3.13)

and the β-functions

β1 =− ε

2g1 + (a13 −

1

2g1(2zφ12 + zσ12)) + 2(a15 −

1

2g1(2zφ14 + zσ14)) + 3(a17 −

1

2g1(2zφ16 + zσ16))

=− ε

2g1 +

1

12(4π)3g1

((N − 8)g2

1 − 12g1g2 + g22

)

− 1

432(4π)6g1

((536 + 86N)g4

1 + 12(30− 11N)g31g2 + (628 + 11N)g2

1g22 + 24g1g

32 − 13g4

2

)

+1

62208(4π)9g1

(g6

2(5195− 2592ζ(3)) + 12g1g52(−2801 + 2592ζ(3))

− 8g21g

42(1245 + 119N + 7776ζ(3))

+ g41g

22(−358480 + 53990N − 3N2 − 2592(−16 + 5N)ζ(3))

+ 36g51g2(−500− 3464N +N2 + 864(5N − 6)ζ(3))

− 2g61(125680− 20344N + 1831N2 + 2592(25N + 4)ζ(3))

+ 48g31g

32(95N − 3(679 + 864ζ(3)))

),

(3.14)

58

β2 =− ε

2g2 + (b13 −

3

2g2z

σ12) + 2(b15 −

3

2g2z

σ14) + 3(b17 −

3

2g2z

σ16)

=− ε

2g2 +

1

4(4π)3

(−4Ng3

1 +Ng21g2 − 3g3

2

)

+1

144(4π)6

(−24Ng5

1 − 322Ng41g2 − 60Ng3

1g22 + 31Ng2

1g32 − 125g5

2

)

+1

20736(4π)9

(− 48N(713 + 577N)g7

1 + 6272Ng21g

52 + 48Ng3

1g42(181 + 432ζ(3))

− 5g72(6617 + 2592ζ(3))− 24Ng5

1g22(1054 + 471N + 2592ζ(3))

+ 2Ng61g2(19237N − 8(3713 + 324ζ(3))) + 3Ng4

1g32(263N − 6(7105 + 2448ζ(3))

).

(3.15)

In the case N = 0 (the single scalar cubic theory), our results are in agreement with

the three-loop calculation of [39].

3.3 The IR xed point

Let us introduce the notation

g1 ≡√

6ε(4π)3

Nx, g2 ≡

√6ε(4π)3

Ny. (3.16)

59

In terms of the new variables x and y, the condition that both β-functions be zero

reads

0 =1

2x(−8x2 +N(x2 − 1)− 12xy + y2)

− 1

12Nx((536 + 86N)x4 + 12(30− 11N)x3y + (628 + 11N)x2y2 + 24xy3 − 13y4

)ε

− 1

288N2x(12xy5(2801− 2592ζ(3)) + y6(−5195 + 2592ζ(3))

+ 48x3y3(2037− 95N + 2592ζ(3))

+ 8x2y4(1245 + 119N + 7776ζ(3)) + x4y2(358480− 53990N + 3N2 + 2592(5N − 16)ζ(3))

− 36x5y(−500− 3464N +N2 + 864(5N − 6)ζ(3))

+ 2x6(125680− 20344N + 1831N2 + 2592(25N + 4)ζ(3)))ε2 (3.17)

and

0 =− 1

2

(9y3 +N(12x3 + y − 3x2y)

)

− 1

4N

(125y5 +Nx2(24x3 + 322x2y + 60xy2 − 31y3)

)ε

− 1

96N2

(N2x4(27696x3 − 38474x2y + 11304xy2 − 789y3) + 5y7(6617 + 2592ζ(3))

+ 34224Nx7 − 6272Nx2y5 + 16Nx6y(3713 + 324ζ(3))− 48Nx3y4(181 + 432ζ(3))

+ 48Nx5y2(527 + 1296ζ(3)) + 18Nx4y3(7105 + 2448ζ(3)))ε2. (3.18)

These equations can be solved order by order in the ε expansion. Using also the 1/N

expansion, we nd the xed point values

x∗ = 1 +22

N+

726

N2− 326180

N3− 349658330

N4+ ...

+

(−155

6N− 1705

N2+

912545

N3+

3590574890

3N4+ ...

)ε

+

(1777

144N+

29093/36− 1170ζ(3)

N2+ ...

)ε2, (3.19)

60

y∗ = 6(1 +162

N+

68766

N2+

41224420

N3+

28762554870

N4+ ...

+

(−215

2N− 86335

N2− 75722265

N3− 69633402510

N4+ ...

)ε

+

(2781

48N+

270911− 157140ζ(3)

6N2+ ...

)ε2. (3.20)

This large N solution corresponds to an IR stable xed point and generalizes the

previous one-loop result. This xed point exists and is stable to all orders in the 1/N

expansion.

If results beyond the 1/N expansion are desired, one can determine the the ε

expansions of x∗, y∗ for nite N as follows. Plugging the expansions

x∗ = x0(N)+x1(N)ε+x2(N)ε2+. . . , y∗ = y0(N)+y1(N)ε+y2(N)ε2+. . . (3.21)

into (3.17)-(3.18), the leading order terms are found to be [28, 78]

x0(N) =

√N

(N − 44)z(N)2 + 1z(N) , y0(N) =

√N

(N − 44)z(N)2 + 1(1+6z(N)) ,

(3.22)

where z(N) is the solution to the cubic equation

840z3 − (N − 464)z2 + 84z + 5 = 0 (3.23)

with large N behavior z(N) = 840N + O(N0)6. This solution is real only if N >

1038.27, as can be seen from the discriminant of the above cubic equation. Once the

x0(N), y0(N) are known, one can then determine the higher order terms in (3.21) by

solving the equations (3.17)-(3.18) order by order in ε.

6The other two roots have large N behavior z(N) ∼ ±√

5N and they are unstable IR xed points[28].

61

For N 1038 the nite N exact results are close to (3.19)-(3.20), but for N ∼

1038 they deviate somewhat, indicating that, close to the critical N , the large N

expansion is not a good approximation (see also Figure 3.1 below).

3.3.1 Dimensions of φ and σ

In terms of the rescaled couplings x, y dened in (3.16), the anomalous dimensions

read

γφ =x2

Nε− x2

12N2

((26− 11N)x2 + 48xy − 11y2

)ε2

+x2

144N3

(6(544− 441N)x3y − 942xy3 − 327y4

+ x4(9064 + (232− 13N)N − 5184ζ(3))

+ 2x2y2(5881 + 193N − 2592ζ(3)))ε3, (3.24)

γσ =Nx2 + y2

2Nε− 1

12N2

(13y4 +Nx2(2x2 + 48xy − 11y2)

)ε2

+1

288N3

(N2x4(2762x2 − 1152xy − 3y2)

+ 2Nx2(−528x3y + 780xy3 − 476y4 + 3x2y2(4441− 2160ζ(3)) + 8x4(535− 324ζ(3))

)

+ y6(5195− 2592ζ(3))

)ε3. (3.25)

62

Plugging the xed point values (3.19)-(3.20) into these expressions, we get the con-

formal dimensions of σ and φ at the xed point

∆φ =d

2− 1 + γφ

= 2− ε

2+

(1

N+

44

N2+

1936

N3+ ...

)ε+

(− 11

12N− 835

6N2− 16352

N3+ ...

)ε2

+

(− 13

144N+

6865

72N2+

54367/2− 3672ζ(3)

N3+ ...

)ε3, (3.26)

∆σ =d

2− 1 + γσ

= 2 +

(40

N+

6800

N2+ ...

)ε+

(−104

3N− 34190

3N2+ ...

)ε2

+

(− 22

9N+

47695/18− 2808ζ(3)

N2+ ...

)ε3. (3.27)

One can verify that these results are in precise agreement with the large N calculation

of [4, 5, 6] for the critical O(N) model in general d, analytically continued to d = 6−ε.

This provides a strong check on our calculations and on our interpretation of the IR

xed point of the cubic O(N) scalar theory.

The 1/N expansions are expected to work well for N 1038. For any N larger

than the critical value, the ε expansions of the scaling dimensions may be determined

using (3.25) and the exact analytic solutions for the xed point location (x∗, y∗). For

example, in Figure 3.1 we plot the coecient of the O(ε3) term in ∆σ as a function

of N and compare it with the corresponding 1/N expansion.

3.3.2 Dimensions of quadratic and cubic operators

In chapter 2 the mixed anomalous dimensions of quadratic operators σ2 and φiφi

were calculated at one-loop order. These results were checked against the O(1/N)

term in the corresponding operator dimensions for the O(N) φ4 theory [10]. In this

chapter, we carry out an additional check, comparing with the O(1/N2) correction

63

exact result

up to 1 N8

up to 1 N4

1200 1400 1600 1800 2000N

-0.002

0.000

0.002

0.004

0.006

0.008gs at OHe3L

Figure 3.1: The O(ε3) in ∆σ as a function of N for N ≥ 1039. The 1/N expansionapproaches the exact result as we include more terms.

found in [79], but still working to the one-loop order in ε (it should be straightforward

to generalize the mixing calculation to higher loops, but we will not do it here).

In the quartic O(N) theory with interaction λ4(φiφi)2, the derivative of the beta

function at the xed point coupling

ω = β′(λ∗) = 4− d+ω1

N+ω2

N2+ . . . (3.28)

is related to the dimension of the operator (φiφi)2 by

∆φ4 = d+ ω . (3.29)

In [10, 79] the coecient ω1 was computed as a function of dimension d:

ω1 =2(d− 4)(d− 2)(d− 1)Γ (d)

dΓ(2− d

2

)Γ(d2

)3 . (3.30)

64

The coecient ω2 has a more complicated structure for general d which was rst

found in [79]. Using this result, we get that in d = 5,

∆φ4 = 4− 2048

15π2N−8192(67125π2 − 589472)

3375π4N2+. . . ≈ 4−13.8337

N−1819.66

N2+. . . (3.31)

Let us also quote the expansion of ω2 in d = 4− ε and d = 6− ε:

ω2 = 102ε2 +

(−259

2+ 120ζ(3)

)ε3 + . . . , d = 4− ε (3.32)

ω2 = −49760ε+237476

3ε2 +

(−92480

9+ 32616ζ(3)

)ε3 + . . . , d = 6− ε (3.33)

In d = 4− ε, one can check that the above results correctly reproduce the derivative

of the β-function [2]

β = −ελ+N + 8

8π2λ2 − 3(3N + 14)

64π4λ3

+33N2 + 480Nζ(3) + 922N + 2112ζ(3) + 2960

4096π6λ4 +O(λ5) (3.34)

at the IR xed point

λ∗ =8π2

N + 8ε+

24π2(3N + 14)

(N + 8)3ε2

− π2(33N3 − 110N2 + 96(N + 8)(5N + 22)ζ(3)− 1760N − 4544)

(N + 8)5ε3 +O(ε4) .

(3.35)

In d = 6− ε, the dimension of the (φiφi)2 operator in the quartic theory should be

matched to the primary operator arising from the mixing of the σ2 and φiφi operators

in our cubic theory. In chapter 2, the mixing matrix of σ2 and φiφi to one-loop order

was found to be

γij =−1

6(4π)3

4g21 −Ng2

1 6√Ng2

1 −√Ng1g2

6√Ng2

1 −√Ng1g2 4g2

2 −Ng21

. (3.36)

65

Computing the eigenvalues γ± of this matrix, and inserting the values of one-loop

xed point couplings

g1∗ =

√6ε(4π)3

N

(1 +

22

N+

726

N2− 326180

N3+ . . .

), (3.37)

g2∗ = 6

√6ε(4π)3

N

(1 +

162

N+

68766

N2+

41224420

N3+ . . .

)(3.38)

we nd the scaling dimensions of the quadratic operators to be

∆− = d− 2 + γ− = 4 +

(−100

N− 49760

N2− 27470080

N3+ . . .

)ε+O(ε2), (3.39)

∆+ = d− 2 + γ+ = 4 +

(40

N+

6800

N2+

2637760

N3+ . . .

)ε+O(ε2). (3.40)

The operator with dimension ∆+ is a descendant of σ. The operator with dimension

∆− is a primary, and comparing with (3.30), (3.33), we see that its dimension precisely

agrees with the results of [79] to order 1/N2. The higher order terms in ε can be

determined from mixed anomalous dimension calculations beyond one-loop, and we

leave this to future work.

We now calculate the mixed anomalous dimensions of the nearly marginal oper-

ators O1 = σφφ and O2 = σ3. Using the beta functions written in equations (4.5)-

(3.15), we can determine the anomalous dimensions of the nearly marginal operators

by computing the eigenvalues of the matrix

Mij =∂βi∂gj

. (3.41)

Strictly speaking, this matrix is not exactly equal to the anomalous dimension mix-

ing matrix, because it is not symmetric. However, we could make it symmetric by

dividing and multiplying the o-diagonal elements by a factor√

3N + O(ε), which

corresponds to an appropriate rescaling of the couplings. This clearly does not change

66

the eigenvalues of the matrix, and hence we can directly compute the eigenvalues λ±

of (3.41), and obtain the dimensions of the eigenstate operators as

∆± = d+ λ± . (3.42)

Plugging in the xed point values x∗ and y∗ from equations (3.19)-(3.20), we nd that

∆+ = 6 +

(155

3ε2 − 1777

36ε3 + ...

)1

N+O(

1

N2) ,

∆− = 6 +

(−420ε+ 499ε2 − 1051

12ε3 + ...

)1

N+O(

1

N2).

(3.43)

The dimension of the σk operator in the quartic O(N) model is known to order 1/N

as function of d [10], and may be written as

∆(σk) = 2k +k(d− 2)

((k − 1)d2 − d(3k − 1) + 4

)Γ (d)

NdΓ(2− d

2

)Γ(d2

)3 +O(1

N2) . (3.44)

Our result for ∆− agrees with the ε expansion of this formula for k = 3 in d = 6− ε.

3.4 Analysis of critical N as a function of ε

We now investigate the behavior of Ncrit above which the xed point exists at real

values of the couplings. This can be dened as the value of N (formally viewed as a

continuous parameter) at which two real solutions of the β-function equations merge,

and subsequently go o to the complex plane. Geometrically, this means that the

curves on the (g1, g2) plane dened by the zeroes of β1 and β2 are barely touching, i.e.

they are tangent to each other. Therefore the critical N , as well as the corresponding

67

critical value of the couplings, can be determined by solving the system of equations

β1 = 0 , β2 = 0,

∂β1/∂g1

∂β1/∂g2

=∂β2/∂g1

∂β2/∂g2

.(3.45)

Note that the condition in the second line is equivalent to requiring that the de-

terminant of the anomalous dimension mixing matrix of nearly marginal operators,

Mij = ∂βi∂gj

, vanishes. This means that one of the two eigenstates becomes marginal.

Working in terms of the rescaled coupling constants dened in (3.16), we can

solve the system of equations (3.45) order by order in ε. We assume a perturbative

expansion

x = x0 + x1ε+ x2ε2 +O(ε3),

y = y0 + y1ε+ y2ε2 +O(ε3),

N = N0 +N1ε+N2ε2 +O(ε3)

(3.46)

and plugging this into (3.45), we can solve for the undetermined coecients uniquely.

At the zeroth order, we get the equations

N0 + 8x20 −N0x

20 + 12x0y0 − y2

0 = 0,

12N0x30 +N0y0 − 3N0x

20y0 + 9y3

0 = 0,

6 +(N0 − 44)x0

6x0 − y0

=6N0x0(y0 − 6x0)

3N0x20 − 27y2

0 −N0

.

(3.47)

The above system of equations can be solved analytically, as was done in chapter 2.

We nd that, up to the signs of x0 and y0, there are three inequivalent solutions

x0 = 1.01804 , y0 = 8.90305 , N0 = 1038.26605, (3.48)

x′0 = 0.23185i , y′0 = 0.25582i , N ′0 = 1.02145, (3.49)

x′′0 = 0.13175 , y′′0 = −0.03277 , N ′′0 = −0.08750. (3.50)

68

IR

IR UV

x

y

-3 -2 -1 0 1 2 3

-40

-20

0

20

40

Figure 3.2: The zeroes of the one-loop β functions and the RG ow directions forN = 2000. The red dots correspond to the stable IR xed points, while the blackdots are unstable xed points. As N → Ncrit, the red dot merges with the nearbyblack dot, and the two xed points move into the complex plane.

The rst of these solutions, with Ncrit = 1038.26605 +O(ε), is of most interest to

us because it is related to the large N limit of the theory. For N > Ncrit, we nd

a stable IR xed point at real couplings g1 and g2.7 This xed point is shown with

the red dot in Figure 3.2 (there is a second stable IR xed point obtained by the

transformation (g1, g2) → (−g1,−g2), which is a symmetry of this theory). There is

also a nearby unstable xed point, shown with a black dot, which has one stable and

one unstable direction. As N approaches Ncrit from above, the nearby unstable xed

7It is stable with respect to ows of the nearly marginal couplings g1 and g2. As usual, there aresome O(N) invariant relevant operators that render this xed point not perfectly stable.

69

point approaches the IR stable xed point, and they merge at Ncrit. At N < Ncrit,

both xed points disappear into the complex plane. As discussed in [80], this is a

rather generic behavior at the lower edge of the conformal window: the conformality

is lost through the annihilation of a UV xed point and an IR xed point. In [80] this

was argued to happen at the lower (strongly coupled) edge of the conformal window

for 4-dimensional SU(Nc) gauge theory with Nf avors. It is interesting to observe

that the same type of behavior occurs at the lower edge of the conformal window of

the O(N) model in d = 6− ε, which extends from Ncrit to innity.

Let us identify the operator that causes the ow between the unstable xed point

and the IR stable xed point of our primary interest. It is one of the two nearly

marginal operators cubic in the elds that were studied in section 3.2. By studying

the behavior of the dimensions ∆1 and ∆2 as N → Ncrit we nd that ∆2 → 6 − ε.

Therefore, it is the operator corresponding to ∆2 that becomes exactly marginal for

N = Ncrit and causes the ow between the IR xed point and the nearby UV xed

point for N slightly above Ncrit. In bootstrap studies of the quartic O(N) model

this operator was denoted by σ3 [10], i.e. it can be thought of as the triple-trace

operator" (φiφi)3. The theory at the unstable xed point has an unconventional large

N behavior where x ∼ O(1) and y ∼ O(√N), so that corrections to scaling dimension

proceed in powers of N−1/2 as in chapter 2.

Let us now go back to nding the higher order corrections to Ncrit given by (3.48)

(the higher order corrections to the other critical values (3.49)-(3.50) will be discussed

in the next section). Once we have solved the leading order system (3.47), we can plug

the solution into (3.46) and expand (3.45) up to order ε2. From this we obtain simple

systems of linear equations from which we can determine x1, y1, N1 and x2, y2, N2. We

nd

x1 = −0.00940 , y1 = −0.21024 , N1 = −609.93980,

x2 = 0.00690 , y2 = 1.01680 , N2 = −364.17333.

(3.51)

70

Thus, to three-loop order, we conclude that

Ncrit = 1038.26605− 609.83980ε− 364.17333ε2 +O(ε3) . (3.52)

We have also checked these expansion coecients via a direct high-precision numerical

calculation of Ncrit for very small values of ε. The large and negative coecients

indicate that in the physically interesting case of d = 5, Ncrit is likely to be much

lower than the zeroth order value (this is analogous to the result [59] for the Abelian

Higgs model). If we just use the rst three terms and plug in ε = 1, we get:

Ncrit ≈ 64.253 . (3.53)

For N < Ncrit the anomalous dimensions, such as γφ, are no longer positive (in

fact, they become complex). This loss of positivity of γφ can also be seen as N is

reduced in the quartic O(N) model. For example, using the 1/N expansion of γφ in

d = 5 [6]

γφ =32

15π2N− 1427456

3375π4N2

+

(275255197696

759375π6− 89735168

2025π4+

32768 ln 4

9π4− 229376ζ(3)

3π6

)1

N3+ . . .

=3

2+

0.216152

N− 4.342

N2− 121.673

N3+ . . . (3.54)

we nd that it stops being positive for N < 35. This critical value is not too far from

(3.53).

71

It is also instructive to study the theory using the 4−ε expansion. The anomalous

dimensions of φi is [2]

γφ =N + 2

4(N + 8)2ε2 +

N + 2

16(N + 8)4

(−N2 + 56N + 272

)ε3

+N + 2

64(N + 8)6

(−5N4 − 230N3 + 1124N2 + 17920N + 46144− 384ζ(3)(5N + 22)(N + 8)

)ε4

+O(ε5) (3.55)

For positive ε this expansion gives accurate information about the Wilson-Fisher IR

xed points [1]. For negative ε there exist formal UV xed points at negative quartic

coupling where we can apply this formula as well. In that case, γφ becomes negative

for suciently large |ε| and N < Ncrit, indicating that the operator φi violates the

unitarity bound. For example for d = 5, corresponding to ε = −1, we nd Ncrit ≈ 8.

Inclusion of the O(ε5) term (see [81]) raises this to Ncrit ≈ 14.

We see, therefore, that the estimates of Ncrit using the quartic O(N) theory in

d = 5 are even lower than the three-loop estimate (3.53). It seems safe to conclude

that the true value is much lower than the one-loop estimate of 1038. To determine

Ncrit in d = 5 more precisely, one needs a non-perturbative approach to the d = 5

theory, perhaps along the lines of the conformal bootstrap calculation in [61].

3.4.1 Unitary xed points for all positive N

Let us note that not all real xed points disappear for N < Ncrit. The unstable real

xed points that are located in the upper left and lower right corners of Figure 3.2

exist for all positive N , and we would like to nd their interpretation.

The xed point with N = 1 has a particularly simple property that g∗1 = −g∗2.

This property of the solution holds for the three-loop β functions, and we believe that

it is exact. Using this, we note that the action at the xed point is proportional to

(σ+ iφ)3 + (σ− iφ)3. Therefore, the theory at this xed point enjoys a Z3 symmetry

72

acting by the phase rotation on the complex combination σ+ iφ. This cubic classical

action appears in the Ginzburg-Landau theory for the 3-state Potts model (see, for

example, [75]).8 Therefore, we expect the Z3 symmetric xed point to describe the

3-state Potts model in d = 6− ε. The dimensions of operators at this xed point are

related by the Z3 symmetry. For example, we nd

∆φ = ∆σ = 2− 1

3ε+

2

3ε2 +

443

54ε3 +O(ε4) . (3.56)

This is in agreement with the result of [39]. By calculating the eigenvalues λ± of the

matrix Mij = ∂βi∂gj

, we also nd the dimensions (3.42) of the two cubic operators to

order ε3:

∆− = 6− 14

3ε− 158

9ε2 −

(17380

81+ 16ζ(3)

)ε3 = 6− 4.66667ε− 17.5556ε2 − 233.801ε3 ,

∆+ = 6− 83

18ε2 −

(38183

648+ 4ζ(3)

)ε3 = 6− 4.61111ε2 − 63.7326ε3 . (3.57)

The dimension ∆+ corresponds to the operator (σ+ iφ)3 + (σ− iφ)3 which preserves

the Z3 symmetry and is slightly irrelevant for small ε. The dimension ∆− corresponds

to the relevant operator σ(σ2 +φ2) which breaks the Z3. Thus, the Z3 symmetry helps

stabilize the xed point at small ε.

Unfortunately, the 6 − ε expansions (3.57) have growing coecients, and it is

not clear for what range of ε the xed point exists. Thus, one may not be able to

interpolate smoothly from the Z3 symmetric xed point in d = 6− ε to d = 2 where

the 3-state Potts model is described by the unitary (5, 6) minimal model [73].

For all N ≥ 2 we nd unstable xed points with O(N) symmetry. These xed

points always have a relevant cubic operator, corresponding to a negative eigenvalue of

the matrix ∂βi∂gj

. Also, they exhibit an unconventional large N behavior involving half-

integer powers of N , similarly to the unstable xed points that appear for N > Ncrit

8We are grateful to Yu Nakayama for pointing this out to us.

73

and are shown by the black dots in the upper right and lower left corners of Figure

3.2. We leave a discussion of these xed points for the future.

3.5 Non-unitary theories

In addition to the xed points studied so far, which are perturbatively unitary and

appear for N > Ncrit, there exist non-unitary xed points for N ′′crit < N < N ′crit. The

leading values of N ′crit and N′′crit are given in (3.49) and (3.50), respectively. Using the

method developed above for nding the higher order in ε corrections to Ncrit we get

N ′crit = 1.02145 + 0.03253ε− 0.00163ε2

x′ = i(0.23185 + 0.08887ε− 0.03956ε2

), y′ = i

(0.25582 + 0.11373ε− 0.04276ε2

)

(3.58)

and

N ′′crit = −0.08750 + 0.34726ε− 0.88274ε2

x′′ = 0.13175− 0.16716ε+ 0.12072ε2 , y′′ = −0.03277 + 0.13454ε− 0.35980ε2

(3.59)

Unfortunately, the latter expansion has growing coecients, and we cannot extract

any useful information from it. On the other hand, the higher order corrections to

N ′crit are very small, which suggests that N ′crit > 1 for range of dimensions below 6.

The theory with N = 0, which contains only the eld σ, was originally studied by

Michael Fisher as an approach to the Yang-Lee edge singularity in the Ising model [37].

Since the coupling is imaginary, it describes a non-unitary theory where some operator

dimensions (e.g. σ) are below the unitarity bounds. In d = 2, this CFT corresponds to

the (2, 5) minimal model [74], which has c = −22/5. A conformal bootstrap approach

to this model [60] has produced good results for a range of dimensions below 6.

74

The N = 1 theory, which has two elds and two coupling constants, has a more

intricate structure. This theory is distinguished from the N = 0 case by the presence

of a Z2 symmetry φ→ −φ. Examining the β functions atN = 1 and the eigenvalues of

the matrix ∂βi∂gj

, we observe that there exist a stable xed point with g∗2 = 6g∗1/5+O(ε),

and an unstable one with g∗1 = g∗2. Introducing the eld combinations

σ1 = σ + φ , σ2 = σ − φ , (3.60)

we note that for g∗1 = g∗2 the interactions of the N = 1 model decouple as ∼ σ31 +σ3

2, i.e.

at this xed point the theory is a sum of two Fisher's N = 0 theories. However, one

of the ow directions at this xed point is unstable, since the corresponding operator

has ∆O = 6− 10ε/9 +O(ε2) and is relevant (this value of the dimension corresponds

to the negative eigenvalue of the matrix Mij = ∂βi∂gj

at the g∗1 = g∗2 xed point). This

dimension has a simple explanation as follows. The ow away from the decoupled

xed point is generated by the operator O = σ1σ22 + σ2σ

21. This is allowed by the

original Z2 symmetry φ → −φ, which translates into the interchange of σ1 and σ2.

Thus,

∆O = ∆N=0σ + ∆N=0

σ2 = 2 + 2∆N=0σ , (3.61)

where we used the fact that in the N = 0 theory, ∆N=0σ2 = 2 + ∆N=0

σ because σ2 is a

descendant. Using (3.25) for N = 0, we nd

∆σ = 2−5

9ε− 43

1458ε2+

(8ζ(3)

243− 8375

472392

)ε3 = 2−0.555556ε−0.0294925ε2+0.021845ε3

(3.62)

Substituting this into (3.61) we nd the dimension of the relevant operator O, which

indeed precisely agrees with ∆O = d + λ−, where λ− is the negative eigenvalue of

Mij = ∂βi∂gj

at the g∗1 = g∗2 xed point. Using the ε expansion (3.62), we nd that O

continues to be relevant as ε is increased. For ε = 4, i.e. d = 2, we know the exact

75

result in the (2, 5) minimal model that ∆N=0σ = −2/5, which implies ∆O = 6/5. This

strongly suggests that O is relevant, and the decoupled xed point is unstable, for

the entire range 2 ≤ d < 6. To describe this CFT in d = 2 more precisely, we note the

existence of the modular invariant minimal model M(3, 10), which is closely related

to the product of two Yang-Lee (2, 5) minimal models [82, 83].

The ow away from the unstable xed point with g∗1 = g∗2 can lead the N = 1

theory to the IR stable xed point where g∗2 = 6g∗1/5 + O(ε). Using our results we

can deduce the ε expansion of various operator dimensions at this xed point. For

example,

∆φ = 2− 0.5501ε− 0.0234477ε2 + 0.0200649ε3 + . . .

∆σ = 2− 0.561122ε− 0.0358843ε2 + 0.0236057ε3 + . . . (3.63)

By calculating the eigenvalues λ± of the matrix Mij = ∂βi∂gj

, we nd the dimensions of

two operators that are slightly irrelevant in d = 6− ε

∆− = d+ λ− = 6− 0.88978ε+ 0.0437732ε2 − 0.039585ε3 ,

∆+ = d+ λ+ = 6− 0.773191ε2 + 1.59707ε3 . (3.64)

As ε is increased, these expansions suggest that the two operators become more

irrelevant. It would be interesting to study this Z2 symmetric xed point using a

conformal bootstrap approach along the lines of [60].

Assuming that the N = 1 IR xed point continues to be stable in d = 5, 4, 3, 2, it is

interesting to look for statistical mechanical interpretations of this non-unitary CFTs.

A distinguishing feature of the N = 1 CFT is that it has a discrete Z2 symmetry,

while the N = 0 theory has no symmetries at all. As we have noted, in d = 2 the

CFT can be obtained via deforming the (3, 10) minimal model by a Virasoro primary

eld of dimension 6/5 (this is the highest dimension relevant operator in that minimal

76

model). After analyzing the spectra of several candidate minimal models, we suggest

that the end point of this RG ow is described by the (3, 8) minimal model with

c = −21/4.9 Let us note that M(2, 5) and M(3, 8) are members of the series of

non-unitary minimal models M(k, 3k − 1).

In addition to the identity operator, theM(3, 8) model has three Virasoro primary

elds which are Z2 odd and three that are Z2 even. Comparing with the theory in

6 − ε dimensions, we can tentatively identify the leading Z2 odd operator as φ and

the leading Z2 even one as σ. Obviously, further work is needed to check if the stable

xed point in 6− ε dimensions with g∗2 = 6g∗1/5+O(ε) continued to ε = 4 is described

by the non-unitary minimal model M(3, 8).

3.6 Appendix A. Summary of three-loop results

The Feynman rules for our theory are depicted in Fig. B.1

= −dαβγ

α

β γ

α

γβ= −(δg)αβγ

= δαβ1p2

= −p2(δz)αβ

α β

α β

Figure 3.3: Feynman rules.

9Note that this value is greater than the central charge of the UV theoryM(3, 10), which is equalto −44/5. For ows between non-unitary theories the Zamolodchikov c-theorem does not hold, andit is possible that cUV < cIR.

77

where we introduced symmetric tensor coupling dαβγ and counterterms (δg)αβγ, (δz)αβ

with α, β, γ = 0, 1, .., N as

d000 = g2, dii0 = di0i = d0ii = g1,

(δg)000 = δg2, (δg)ii0 = (δg)i0i = (δg)0ii = δg1,

(δz)00 = δσ, (δz)ii = δφ, (3.65)

where i = 1, ..., N . The general form of a Feynman diagram in our theory could be

schematically represented as

Feynman diagram = Integral× Tensor structure factor. (3.66)

The Tensor structure factors are products of the tensors dαβγ and (δg)αβγ, (δz)αβ,

with summation over the dummy indices. Their values for dierent diagrams are

represented in Fig. D.1 and D.2 after parentheses 10. The Integrals already include

symmetry factors and are the same as in the usual ϕ3-theory; their values are listed

in Fig. D.1 and D.2 before the parentheses.

10To nd the Tensor structure factor we used the fact that it is a polynomial in N , so wecalculated sums of products of dαβγ , (δg)αβγ , (δz)αβ explicitly for N = 1, 2, 3, 4, ..., using WolframMathematica. Having answers for N = 1, 2, 3, 4, .. it's possible to restore the general N form.

78

3.6.1 Counterterms

zφ12 =− g21

3(4π)3, zσ12 = −Ng

21 + g2

2

6(4π)3, a13 = −g

21 (g1 + g2)

(4π)3, b13 = −Ng

31 + g3

2

(4π)3,

(3.67)

zφ14 =g2

1

432(4π)6

(g2

1(11N − 26)− 48g1g2 + 11g22

),

zσ14 =− 1

432(4π)6

(2Ng4

1 + 48Ng31g2 − 11Ng2

1g22 + 13g4

2

),

a15 =− 1

144(4π)6g2

1

(g3

1(11N + 98)− 2g21g2(7N − 38) + 101g1g

22 + 4g3

2

),

b15 =− 1

48(4π)6

(4Ng5

1 + 54Ng41g2 + 18Ng3

1g22 − 7Ng2

1g32 + 23g5

2

),

(3.68)

zφ16 =g2

1

46656(4π)9

(g4

1(N(13N − 232) + 5184ζ(3)− 9064) + g31g26(441N − 544)

− 2g21g

22(193N − 2592ζ(3) + 5881) + 942g1g

32 + 327g4

2

),

zσ16 =− 1

93312(4π)9

(2Ng6

1(1381N − 2592ζ(3) + 4280)− 96N(12N + 11)g51g2

− 3Ng41g

22(N + 4320ζ(3)− 8882) + 1560Ng3

1g32 − 952Ng2

1g42 − g6

2(2592ζ(3)− 5195)),

a17 =g2

1

15552(4π)9

(− g5

1(N(531N + 10368ζ(3)− 2600) + 23968)

+ g41g2(99N2 + 2592(5N − 6)ζ(3)− 9422N − 2588) + 2g3

1g22(1075N + 2592ζ(3)− 16897)

+ 2g21g

32(125N − 5184ζ(3)− 3917)− g1g

42(5184ζ(3) + 721) + g5

2(2592ζ(3)− 2801)),

b17 =− 1

2592(4π)9

(2g7

1N(577N + 713)− 48g61g2N(31N − 59)

+ g51g

22N(423N + 2592ζ(3) + 1010)− g4

1g32N(33N − 1296ζ(3)− 6439)

− 27g31g

42N(32ζ(3) + 11)− 301Ng2

1g52 + g7

2(432ζ(3) + 1595)). (3.69)

79

Chapter 4

Critical Sp(N) Models in 6− ε

Dimensions and Higher Spin dS/CFT

4.1 Introduction and Summary

This chapter is based on the work published in [30], co-authored with Simone Giombi,

Igor Klebanov, and Grigory Tarnopolsky. We also thank D. Anninos, D. Harlow and

J. Maldacena, and especially S. Caracciolo and G. Parisi, for useful discussions.

The O(N) invariant theories of N massless scalar elds φi, which interact via

the potential λ4(φiφi)2 possess interacting IR xed points in dimensions 2 < d <

4 for any positive N [1]. These theories contain O(N) singlet current operators

with all even spin, and when N is large the current anomalous dimensions are ∼

1/N [2]. Since all the higher spin currents are nearly conserved, the O(N) models

possess a weakly broken higher spin symmetry. In the Anti-de Sitter/Conformal Field

Theory (AdS/CFT) correspondence [84, 85, 86], each spin s conserved current in a d

dimensional CFT is mapped to a massless spin s gauged eld in d + 1 dimensional

AdS space. For these reasons, it was conjectured [47] that the singlet sector of the

critical O(N) models in d = 3 is dual to the interacting higher spin theory in d = 4

80

containing massless gauge elds of all even positive spin [41, 42, 43, 46]. This minimal

Vasiliev theory also contains a scalar eld withm2 = −2/`2AdS, and the two admissible

boundary conditions on this eld [48, 49] distinguish the interacting O(N) model from

the free one (in the latter all the higher spin currents are conserved exactly). A review

of the higher spin AdS/CFT dualities may be be found in [87].

A remarkable feature of the Vasiliev theories [41, 42, 43, 46] is that they are

consistent not only in Anti-de Sitter, but also in de Sitter space. On general grounds

one expects a CFT dual to quantum gravity in dS4 to be a non-unitary theory dened

in three dimensional Euclidean space [88]. In [89] it was proposed that the CFT dual

to the minimal higher spin theory in dS4 is the theory of an even number N of

anti-commuting scalar elds χi with the action

S =

∫d3x

(1

2Ωij∂µχ

i∂µχj +λ

4(Ωijχ

iχj)2

). (4.1)

This theory possesses Sp(N) symmetry, and Ωij is the invariant symplectic matrix.

This model was originally introduced and studied in [90, 91], where it was shown to

possess an IR xed point in 4 − ε dimensions. The beta function of this model is

related to that of the O(N) model via the replacement N → −N . According to the

proposal of [89], the free UV xed point of (4.1) is dual to the minimal higher spin

theory in dS4 with the Neumann future boundary conditions on the m2 = 2/`2dS bulk

scalar eld, and its interacting IR xed point to the same higher spin theory but with

the Dirichlet boundary conditions on the scalar eld. In the latter case, the higher

spin symmetry is slightly broken at large N , and the de Sitter higher spin gauge

elds are expected to acquire small masses through quantum eects. 1 A discussion

1A potential diculty with this picture is that unitarity in dSd+1 space requires that a massiveeld of spin s > 1 should satisfy m2 > (s− 1)(s+ d− 3)/`2dS [92, 93, 94]. In other words, there is anite gap between massive elds and massless ones (the latter are dual to exactly conserved currentsin the CFT). However, since the masses are generated by quantum eects and are parametricallysmall at large N , this is perhaps not fatal for bulk unitarity. It would be interesting to clarify thisfurther.

81

of the de Sitter boundary conditions from the point of view of the wave function of

the Universe was given in [95, 96].

In this chapter we consider an extension of the proposed higher spin dS/CFT

correspondence [89] to higher dimensional de Sitter spaces, and in particular to dS6.

Our construction mirrors our recent work [28, 29] on the higher dimensional extensions

of the higher spin AdS/CFT. It was observed long ago [32, 33, 34] that in d > 4 the

quartic O(N) models possess UV xed points which can be studied in the large N

expansion. The UV completion of the O(N) scalar theory in 4 < d < 6 was proposed

in [28]; it is the cubic O(N) symmetric theory of N + 1 scalar elds σ and φi:

S =

∫ddx

(1

2(∂µφ

i)2 +1

2(∂µσ)2 +

1

2g1σ(φi)2 +

1

6g2σ

3

). (4.2)

For suciently large N , this theory has an IR stable xed point in 6− ε dimensions

with real values of g1 and g2. The beta functions and anomalous dimensions were

calculated to three-loop order [28, 29],2 and the available results agree nicely with the

1/N expansion of the quartic O(N) model at its UV xed point [4, 6, 7, 10, 11] when

the quartic O(N) model is continued to 6 − ε dimensions. The conformal bootstrap

approach to the higher dimensional O(N) model was explored in [61, 20, 98].

To extend the idea of [89] to dSd+1 with d > 4, we may consider non-unitary

CFTs (4.1) which in d > 4 possess UV xed points for large N . The 1/N expansion

of operator scaling dimensions may be developed using the generalized Hubbard-

Stratonovich transformation, and one nds that it is related to the 1/N expansion in

the O(N) models via the replacement N → −N . In d = 5 this interacting xed point

should be dual to the higher spin theory in dS6 with Neumann boundary conditions

on the m2 = 6/`2dS scalar eld (corresponding to the conformal dimension ∆ =

2The one-loop beta functions of the model (4.2) in d = 6 were rst calculated in [97].

82

2+O(1/N) on the CFT side).3 In search of the UV completion of these quartic CFTs

in 4 < d < 6, we introduce the cubic theory of one commuting real scalar eld σ and

N anti-commuting scalar elds χi:

S =

∫ddx

(1

2Ωij∂µχ

i∂µχj +1

2(∂µσ)2 +

1

2g1Ωijχ

iχj σ +1

6g2σ

3

). (4.3)

Alternatively, we may combine the elds χi into N/2 complex anti-commuting scalars

θα, α = 1, . . . , N/2 [90, 91]; then the action assumes the form

S =

∫ddx

(∂µθ

α∂µθα +1

2(∂µσ)2 + g1σθ

αθα +1

6g2σ

3

). (4.4)

We study the beta functions for this theory in d = 6 − ε and show that they are

related to the beta functions of the theory (4.2) via the replacement N → −N . For

all N there exists an IR xed point of the theory (4.3) with imaginary values of g1

and g2. This is similar to the IR xed point of the single scalar cubic eld theory

(corresponding to the N = 0 case of our models), which was used by Fisher [37] as

an approach to the Lee-Yang edge singularity. The fact that the couplings are purely

imaginary makes the integrand of the path integral oscillate rapidly at large σ; this

should be contrasted with real couplings giving a potential unbounded from below.

Our results allow us to study the 6− ε expansion of the theory (4.3) with arbtrary

N , and we observe that at nite N there are qualitative dierences between the

Sp(N) and O(N) models which are not seen in the 1/N expansion. In fact, for the

Sp(N) model there is no analogue of the lower bound Ncrit that was found in the

O(N) case [28]. For the lowest value, N = 2, we observe some special phenomena.

In this theory, which contains two real anti-commuting scalars, it is impossible to

formulate the quartic interaction (4.1); thus, the cubic lagrangian (4.20) seems to be

3The d = 5 free Sp(N) model corresponds to Dirichlet scalar boundary conditions in the dS6 dual.It should be possible to extend this free Sp(N) model/dS higher spin duality to general dimensions,since the Vasiliev equations in (A)dSd+1 are known for all d [46].

83

the only possible description of the interacting theory with global Sp(2) symmetry.

Furthermore, it becomes enhanced to the supergroup OSp(1|2) because at the IR xed

point the two coupling constants are related via g∗2 = 2g∗1. The enhanced symmetry

implies that the scaling dimensions of σ and χi are equal, and we check this to order

ε3. An example of theory with OSp(1|2) symmetry is provided by the q → 0 limit

of the q-state Potts model [99].4 We show that the 6 − ε expansions of the scaling

dimensions in our OSp(1|2) symmetric theory are the same as in the q → 0 limit of

the q-state Potts model.5 This provides strong evidence that the OSp(1|2) symmetric

IR xed point of the cubic theory (4.20) describes the second order transitions in the

ferromagnetic q = 0 Potts model, which exist in 2 < d < 6 [100].

Using the results of [78], we also compute perturbatively the sphere free energies of

the models (4.3). The sphere free energy for the OSp(1|2) symmetric model is found to

be the same as for the q = 0 Potts model. In terms of the quantity F = sin(πd2

) logZSd ,

which was introduced in [78] as a natural way to generalize the F -theorem [66, 54, 50]

to continuous dimensions, we nd that the RG ow in the cubic Sp(N) models in

d = 6 − ε satises FUV > FIR for all N ≥ 2. We show that the same result holds in

the model (4.1) in d = 4 − ε. 6 This is somewhat surprising, since for non-unitary

CFTs the inequality FUV > FIR is not always satised. It would be interesting to

understand if this is related to the pseudo-unitary" structure discussed in [91], and

to the fact that these models are presumably dual to unitary higher spin gravity

theories in de Sitter space.

4We are grateful to Giorgio Parisi for informing us about this and for important discussions.5We thank Sergio Caracciolo for suggesting this comparison to us and for informing us about the

paper [100].6The models (4.1) and (4.3) also satisfy the F -theorem to leading order in the large N expansion

(for all d), since the leading order correction to FUV − FIR is of order N0, and was shown to satisfythe F -theorem in the corresponding unitary O(N) models [50, 23, 78].

84

4.2 The IR xed points of the cubic Sp(N) theory

The beta functions and anomalous dimensions for the Sp(N) symmetric model (4.3)

can be obtained by replacing N → −N in the corresponding results for the cubic

O(N) model (4.2), which were computed in [28, 29] to three-loop order. Indeed,

writing the action in the complex basis (4.4), we see that the Feynman rules and

propagators are identical to those of the O(N) theory written in the U(N/2) basis,

the only dierence being that the N/2 complex scalars are anticommuting. Hence, for

each closed loop of the θα we get an extra minus sign, thus explaining the replacement

N → −N .

Using the results in [28, 29], the beta functions for the Sp(N) model are then

found to be:

β1 =− ε

2g1 −

1

12(4π)3g1

((N + 8)g2

1 + 12g1g2 − g22

)

− 1

432(4π)6g1

((536− 86N)g4

1 + 12(30 + 11N)g31g2

+ (628− 11N)g21g

22 + 24g1g

32 − 13g4

2

)+ . . . ,

β2 =− ε

2g2 +

1

4(4π)3

(4Ng3

1 −Ng21g2 − 3g3

2

)

+1

144(4π)6

(24Ng5

1 + 322Ng41g2 + 60Ng3

1g22 − 31Ng2

1g32 − 125g5

2

)+ . . . . (4.5)

We have omitted the explicit three-loop terms, which can be obtained from [28, 29].

Similarly, the anomalous dimensions of the elds χi and σ take the form

γχ =g2

1

6(4π)3+

g21

432(4π)6

(g2

1(11N + 26) + 48g1g2 − 11g22

)+ . . . ,

γσ =− Ng21 − g2

2

12(4π)3− 1

432(4π)6

(2Ng4

1 + 48Ng31g2 − 11Ng2

1g22 − 13g4

2

)+ . . . .

(4.6)

With the beta functions at hand, we can look for non-trivial xed points of the RG

ow satisfying β1(g∗1, g∗2) = 0, β2(g∗1, g

∗2) = 0. For all positive N , we nd two physically

85

equivalent xed points with purely imaginary coupling constants, hence all operator

dimensions remain real. These xed points are IR stable for all N (the stability

matrix Mij = ∂βi∂gj

has positive eigenvalues). Note that this is dierent from the O(N)

versions of these models [28, 29], where one nds a critical N , whose one-loop value is

' 1038, below which the IR stable xed points with real coupling constants disappear.

However, to all orders in the 1/N expansion, the xed point couplings and conformal

dimensions in the Sp(N) models are related to the ones in the O(N) models by the

replacement N → −N .

Figure 4.1 shows the RG ow directions for N = 2. The arrows indicate how

the coupling constants ow towards the IR. The two IR xed points are physically

equivalent because they are related by gi → −gi. At higher values ofN , the qualitative

behavior of the RG ows and xed points remain the same. We still have a UV

Gaussian xed point, and two stable IR xed points.

A special structure emerges for N = 2. In this case we nd the xed point solution

g∗2 = 2g∗1 , g∗1 = i

√(4π)3ε

5

(1 +

67

180ε+O(ε2)

), (4.7)

and the conformal dimensions of the fundamental elds are equal (the three-loop term

given below can be obtained from the results in [28, 29])

∆σ = ∆χ = 2− 8

15ε− 7

450ε2 − 269− 702ζ(3)

33750ε3 + . . . . (4.8)

We show in the next section that the equality of dimensions is a consequence of a

symmetry enhancement from Sp(2) to the supergroup OSp(1|2).

It is natural to ask whether symmetry enhancement can occur at other values of

N . For instance, we can explicitly check for which N the dimensions of σ and χ are

equal. A direct calculation using the beta functions and anomalous dimensions up

to three-loops shows that this only happens for N = 2 and N = −1. The latter

86

x

y

UV

IR

IR

-3 -2 -1 0 1 2 3

-6

-4

-2

0

2

4

6

Figure 4.1: The zeroes of the one-loop β functions and the RG ow directions for

the OSp(1|2) model. The coordinates are dened via g1 = i√

(4π)3ε5x, g2 = i

√(4π)3ε

5y,

and the red dots correspond to the stable IR xed points.

case corresponds to the 3-state Potts model xed point of the the theory with two

commuting scalars [29]: it has g∗1 = −g∗2 and enhanced Z3 symmetry.7

The anomalous dimensions of some composite operators may be similarly obtained

from the results in [28, 29]. Let us quote the explicit result for the quadratic operators

arising from the mixture of σ2 and Ωijχiχj. These operators have the same classical

dimension, so we expect them to mix. The 2×2 anomalous dimension mixing matrix

γab was given in [28, 29] up to one-loop order. Extending those results to two-loop,

7For N = −1, there is also a (non-unitary) solution with g∗1 = g∗2 which corresponds to twodecoupled Fisher models [37], and hence the dimensions of the fundamental elds are trivially equal.

87

we nd the mixing matrix

−g21(N+4)

6(4π)3+

g21(g21(21N−134)−6g1g2(2N+5)+5g22)108(4π)6

Ng1(6g1−g2)6(4π)3

+Ng21(18g21+161g1g2+15g22)

108(4π)6

g1(g2−6g1)6(4π)3

− g1(6g31(11N+20)−g21g2(11N−324)+12g1g22−13g32)216(4π)6

−Ng21+4g226(4π)3

+Ng21(160g21+36g1g2−41g22)−181g42

216(4π)6

(4.9)

where index `1' corresponds to the operator Ωijχiχj, and index `2' corresponds to σ2.

The diagrams contributing to the calculation are listed in Fig. 4.2.

Figure 4.2: Diagrams contributing to the mixing of σ2 and Ωijχiχj operators to two-

loop order. Notice that in these diagrams, we have not distinguished the σ eld andχi elds. Therefore, each one of these diagrams represent several diagrams where theelds are dierent. The solid dots represent one-loop counter terms for either thepropagator or the vertex, and the crossed dots represent one-loop counter terms forthe operator mixing.

The two eigenvectors of γab give the two linear combinations of σ2 and Ωijχiχj,

and the eigenvalues γ± give their anomalous dimensions, so that ∆± = d − 2 + γ±.

We nd that one of these combinations is a conformal primary, and the other one

is a descendant of σ. Indeed, after plugging in the xed point couplings, we nd

∆− = ∆σ + 2. For instance, in the large N expansion we nd the results

∆− = ∆σ + 2 = 4 +−40ε+ 104

3ε2

N+

6800ε− 341903ε2

N2+ . . . ,

∆+ = 4 +100ε− 395

3ε2

N+−49760ε+ 237476

3ε2

N2+ . . . .

(4.10)

Upon sending N → −N , the dimension ∆+ can be checked to be in agreement with

the available large N results for the critical exponent ω in the quartic O(N) theory8

[10, 79].

8This critical exponent is related to the derivative of the beta function in the O(N) theory withquartic interaction; see [29] for more details on the comparison to the large N results.

88

Figure 4.3: Dierent Padé approximations of ∆χ for the N = 2 model.

4.2.1 Estimating operator dimensions with Padé approxi-

mants

In the previous section we calculated ε expansions for the operator dimensions of the

cubic Sp(N) theory in d = 6 − ε. We may use the following Padé approximant to

estimate the behavior of operator dimensions as we continue d:

Padém,n[d] =A0 + A1d+ A2d

2 + . . .+ Amdm

1 +B1d+B2d2 + . . .+Bndn. (4.11)

For N = 2, by demanding that the ε expansion have the same behavior as (4.8), we

can x four coecients in the Padé approximant. Here we use Padé1,2, and Padé2,1,

to obtain the following estimate for the operator dimension in d = 5:

∆N=2χ = ∆N=2

σ ≈

1.467 with Padé1,2

1.459 with Padé2,1 .

(4.12)

As expected, this is below the unitarity bound in d = 5, which is ∆ = 32. The plots

of dierent Padé approximants are shown in Fig. 4.3. For the next primary operator,

89

whose scaling dimension is given in (4.30), using the Padé2,1 approximant we estimate

∆+ ≈ 3.35 in d = 5. In four dimensions we estimate ∆+ ≈ 2.7; this suggests that the

N = 2 model does not have a free eld description near d = 4.

In the Sp(4) case, we may assume for 4 < d < 6 that the IR xed point of the

cubic theory (4.3) is equivalent to the UV xed point of the quartic theory (4.1).9 If

so, then not only do we know the 6− ε expansion of ∆χ to O(ε3), we can also use the

quartic theory result in d = 4 + ε, which is known to O(ε5):

∆χ =

2− 0.529827ε− 0.0126197ε2 + 0.0157244ε3 in d = 6− ε

1 + 0.5ε− 0.03125ε2 + 0.015625ε3 − 0.0730951ε4 + 0.0195503ε5 in d = 4 + ε

(4.13)

Together, these expansions allow us to determine ten coecients of the Padé ap-

proximant. Disregarding the approximants that have a pole, we obtain the estimate

∆N=4χ ≈ 1.478 in d = 5.

4.2.2 Sphere free energies

It is also interesting to compute the sphere free energy at the IR xed point in

d = 6 − ε. The leading order term in the ε expansion for the corresponding O(N)

models was computed in [78]. Sending N → −N , we nd the following result for

F = sin(πd2

) logZSd in the cubic Sp(N) models

FIR = FUV −π

17280

(g∗2)2 − 3N(g∗1)2

(4π)3ε+O(ε3), (4.14)

where FUV = (1 − N)Fs, and Fs the value corresponding to a free conformal scalar.

Plugging in the explicit solutions for the xed point couplings g∗1, g∗2, it is straight-

9This equivalence holds for Sp(N) models with suciently large N , but it is not completely clearif it applies for N = 4.

90

forward to verify that, for all N ≥ 2, we have

FUV > FIR . (4.15)

For instance, for N = 2 we nd

FIR = −Fs −π

43200ε2 +O(ε3) . (4.16)

Note that for the Fisher model [37] of the Lee-Yang edge singularity, corresponding to

N = 0 and imaginary g∗2, the inequality (4.15) does not hold. For general non-unitary

theories the inequality does not have to hold; remarkably, it does hold for the Sp(N)

models with positive N .

Similarly, using the results of [78] for the quartic O(N) theory in d = 4−ε, we can

compute the sphere free energy at the IR xed point of the model (4.1) in d = 4− ε.

We nd

Fquartic = Ffree −π

576

N(N − 2)

(N − 8)2ε3 +O(ε4) , (4.17)

where Ffree = −NFs. We observe that Ffree > Fquartic is satised for all N > 2. 10

The case N = 8 is special and needs to be treated separately. Here the one-loop

term in the beta function vanishes, and we have

βλ = −ελ+15

32π4λ3 +O(λ4). (4.18)

Therefore, at the IR xed point λ2∗ = 32π4

15ε+O(ε3/2), and we get

Fquartic = Ffree −π

360ε2 + . . . . (4.19)

10For ε < 0 the interacting theory has a UV xed point. So, in this case Ffree < Fquartic, again in

agreement with the conjectured F theorem.

91

4.3 Symmetry enhancement for N = 2

Let us write the cubic Sp(2) model in terms of a real scalar σ and a single complex

anti-commuting fermion θ:

S =

∫ddx

(∂µθ∂

µθ +1

2(∂µσ)2 + g1σθθ +

1

6g2σ

3

). (4.20)

For g2 = 2g1, i.e. the xed point relation (4.7), this action possesses a fermionic

symmetry with a complex anti-commuting scalar parameter α

δθ = σα , δθ = σα , δσ = −αθ + αθ . (4.21)

As a consequence of this symmetry, the scaling dimensions of σ and θ are equal, as

seen explicitly in eq. (4.8). This complex fermionic symmetry enhances the Sp(2)

to OSp(1|2), which is the smallest supergroup. The full set of supergroup generators

can be given in the form

Q+ =1

2

(σ∂

∂θ+ θ

∂

∂σ

), Q− =

1

2

(σ∂

∂θ− θ ∂

∂σ

),

J+ = θ∂

∂θ, J− = θ

∂

∂θ, J3 =

1

2

(θ∂

∂θ− θ ∂

∂θ

),

(4.22)

and it is not hard to check that they satisfy the algebra of OSp(1|2):

[J3, J±

]= ±J±,

[J+, J−

]= 2J3 (4.23)

[J3, Q±

]= ±1

2Q±,

[J±, Q∓

]= −Q± (4.24)

Q±, Q± = ±1

2J±, Q+, Q− =

1

2J3 (4.25)

We expect the operators of the theory to form representations of OSp(1|2). For

example, let us study the mixing of the Sp(2) singlet operators σ2 and θθ. Setting

92

N = 2 in (4.9), we nd the scaling dimensions of the two eigenstates:

∆+ = 4− 2

3ε+

1

30ε2 +O(ε3) , ∆− = 4− 8

15ε− 7

450ε2 +O(ε3). (4.26)

The rst of these dimensions corresponds to the conformal primary operator

O+ = σ2 + 2θθ, (4.27)

which is invariant under all the OSp(1|2) generators. The second corresponds to

O− = σ2 + θθ, which is a conformal descendant because at the xed point it is

proportional to ∂µ∂µσ by equations of motion. Indeed, we nd ∆− = ∆σ + 2.

Continuation of the results for the cubic model with OSp(1|2) global symmetry to

nite ε points to the existence of such interacting CFTs in integer dimensions below

6. In [99] it was argued that the q → 0 limit of the q-state Potts model is described

by the OSp(1|2) sigma model [101, 102]. There is evidence that the upper critical

dimension of the spanning forest model is 6 [100], and the 6 − ε expansions of the

critical indices are [38, 100]

η = − 1

15ε− 7

225ε2 − 269− 702ζ(3)

16875ε3 +O(ε4) , (4.28)

ν−1 = 2− 1

3ε− 1

30ε2 − 173− 864ζ(3)

27000ε3 +O(ε4) . (4.29)

Remarkably, the operator dimensions we have calculated (4.8) and (4.26) agree with

these expansions upon the standard identications

∆χ =d

2− 1 +

η

2, ∆+ = d− ν−1 . (4.30)

93

Similarly, we can match the 6 − ε expansions of the sphere free energies. For the

q-state Potts model it is not hard to show that

Fq = (q − 1)Fs +π(q − 1)(q − 2)

8640(3q − 10)ε2 +O(ε3) , (4.31)

and for q = 0 this matches the F of our OSp(1|2) model, (4.16). These results provide

strong evidence that the OSp(1|2) symmetric IR xed point of the cubic theory (4.20)

describes the q → 0 limit of the q-state Potts model.

Numerical simulations of the spanning-forest model [100], which is equivalent to

the q → 0 limit of the ferromagnetic q-state Potts model, indicate the existence

of second-order phase transitions in dimensions 3, 4 and 5. The critical exponents

found in [100] are in good agreement with the Pade extrapolations of 6−ε expansions

exhibited in section 2.1. This provides additional evidence for the existence of the

critical theories with OSp(1|2) symmetry. It would be of further interest to nd the

critical statistical models that are described by the Sp(N) invariant theories with

N > 2.

94

Chapter 5

Equivalence between the Sp(2) model

and the 0-state Potts model

5.1 Preliminaries

In this chapter, we will show that the 0-state Potts model and the Sp(2) models

are indeed equivalent at the level of tensor structures. The N + 1-state Potts model

contains a cubic interaction dened as [38]:

1

6gdijkφiφjφk . (5.1)

Where,

dijk =N+1∑

α=1

eiαejαe

kα . (5.2)

95

Where the eiα satises:

N+1∑

α=1

eiα = 0 , (5.3)

N+1∑

α=1

eiαejα = (N + 1)δij . (5.4)

N∑

i=1

eiαeiβ = (N + 1)δαβ − 1 . (5.5)

Whereas in the O(N) model, the coupling constants are:

d000 = 2, d0ii = di0i = dii0 = 1 , (5.6)

where the index i runs from 1 to N .

First, we note that the 4-loop results for (N + 1)-state Potts model as well as

the O(N) cubic model have been computed in [103]. For the O(N) cubic model, the

Sp(2) model corresponds to setting N = −2, and g2 = 2g1 = 2g, the beta function

and anomalous dimensions are:

βSp2 = −g2ε+

5

4g3 − 335

72g5 +

(144889

5184+

53ζ(3)

4

)g7

+

(−7502393

31104+

53π4

144− 5297ζ(3)

18− 125ζ(5)

3

)g9 , (5.7)

γφSp2 = −1

6g2 +

25

108g4 +

(−14213

7776+

5ζ(3)

6

)g6

+

(78667

5184+

8π4

135+

209ζ(3)

108− 160ζ(5)

9

)g8 . (5.8)

The 0-state Potts model corresponds to N = −1, but the coupling and other

quantities are all divergent. To extract meaningful results, we set N = −1 + ∆. We

96

get that the beta function and anomalous dimensions are:

βPotts = −g2ε−

(5

4+O(∆)

)∆2g3 +

(−335

72+O(∆)

)∆4g5

−(

144889

5184+

53ζ(3)

4+O(∆)

)∆6g7

+

(−7502393

31104+

53π4

144− 5297ζ(3)

18− 125ζ(5)

3+O(∆)

)∆8g9 , (5.9)

γφPotts =

(1

6+O(∆)

)∆2g2 +

(25

108+O(∆)

)∆4g4 +

(14213

7776− 5ζ(3)

6+O(∆)

)∆6g6

+

(78667

5184+

8π4

135+

209ζ(3)

108− 160ζ(5)

9+O(∆)

)∆8g8 . (5.10)

Notice that the terms are very similar to the Sp(2) model in the ∆→ 0 limit, if we

rescale g → g/∆. In fact, the only dierence is that the terms have alternating signs in

successive loop orders. Namely, one loop contributions have opposite signs, two loop

contributions have the same sign, three loop contributions have the opposite sign, etc.

However, these are completely equivalent if we make the substitution g2 → −g2, as

one could easily verify. Therefore, they will give exactly the same answer for physical

quantities such as the anomalous dimension at the xed point.

This strongly suggests that the two models might be equivalent. We will show that

the tensor structures are indeed equal for general zero, two and three-point functions,

up to the alternating sign at successive loop order.

5.2 Review of Potts model tensor calculation

This section reviews the diagrammatic technique used in [38] by considering the fol-

lowing 3 point function example 5.1, which we will refer to as T5: We have:

97

Figure 5.1: Diagram for what Gracey calls T5

T5dijk =

∑

l1,...,l6

dil1l2djl3l4dkl5l6dl1l3l5dl2l4l6 (5.11)

=∑

α1,...,α5

∑

l1,...,l6

(eiα1el1α1

el2α1)(ejα2

el3α2el4α2

) . . . (el2α5el4α5

el6α5) . (5.12)

Now we perform the sum over the l's, using 5.5, for example:

∑

l1

el1α1el1α4

= (N + 1)δα1α4 − 1 . (5.13)

We will get 6 such factors, then we have:

T5dijk =

∑

α1,...,α5

eiα1ejα2

ekα3[(N + 1)δα1α4 − 1][(N + 1)δα1α5 − 1][(N + 1)δα2α4 − 1]

(5.14)

[(N + 1)δα3α5 − 1][(N + 1)δα2α5 − 1][(N + 1)δα3α4 − 1] . (5.15)

If we multiply the 6 factors out, we will get 64 terms, carrying various powers of

(N + 1) and −1. Each of these terms can be thought of as a coloring of the original

6 propagators in the graph. Each solid line corresponds to a factor of (N + 1), while

98

each dashed line corresponds to a factor −1. We will have one diagram where all 6

lines are solid, 6 diagrams where 5 lines are solid..., until one diagram where all lines

are dashed. Notice also that from 5.3, a three-point diagram will only be non-zero

if all its vertices are connected by solid lines, since an isolated vertex will give zero.

Also, when there are free clusters of isolated solid lines in the middle of the diagram,

each would contribute another factor of (N + 1) from having a free index in the sum.

Likewise, a vertex with three dashed lines can be considered as a trivial cluster, and

also contribute a factor of (N + 1) due to having a free index under the sum.

Let's consider our example T5. There are:

One diagram with no dashed lines: (N + 1)6.

6 diagrams with one dashed line: −6(N + 1)5.

15 diagrams with two dashed lines, but three of them don't contribute since they

have isolated points. 12(N + 1)4.

2 diagrams with three dashed lines. Each of them contain an isolated trivial

cluster, giving an additional factor of (N + 1), we get: −2(N + 1)4.

No diagrams with four or more dashed lines since the outer vertices would no

longer be connected.

In total, we have T5 = (N + 1)6 − 6(N + 1)5 + 10(N + 1)4.

In the N → −1 limit, only the lowest power contributes (in this case 10(N + 1)4),

since other terms are higher order in ∆. So we only need to nd the lowest term for

each tensor structure.

This has a natural interpretation. It can be easily argued using induction [38],

that all terms contributing to the lowest power of (N + 1) have no solid line cycles,

i.e. it consists of only (possibly disconnected) trees, also known as a forest.

Thus, in order to compute the tensor structures in the 0-state Potts model, we

need to enumerate the forests, while remembering that each one will carry +1 or −1

depending on how many dashed lines there are.

99

Our proof will be roughly as follows. The majority of the work will be to rst

show that all two-point tensor structures are the same in the Potts model and the

Sp(2) model. Then we can use this to deduce that it holds for three-point functions

and zero point functions as well.

5.3 Prove the equivalence for two-point functions

5.3.1 Some general observations about the diagrams

First, let's consider the Potts model. For any given two-point tensor structure, it will

only be non-zero if there's a solid line going through the middle and connecting the

two outer points. There must be one and only one such line for each forest, otherwise

there will be a cycle. For each forest, we call this line the principal line. There can

be trees that are connected to the principal line, which we call rooted trees, and

there can be trees not connected to the principal line, which we call isolated trees.

In the Sp(N) model, we only consider the two-point function of θθ. For the same

topology, the tensor value for σσ must be the same due to Osp(1|2) invariance that

was proved in chapter 4. To motivate the analogy to the Potts model case, we will

denote lines of θ by solids, and σ by dashes. It can be easily reasoned that because all

vertices in Sp(N) model are either dash-dash-dash or dash-solid-solid, for a Feynman

diagram to be valid, it must be that there is again one and only one principal line

connecting the two outer points. If this is not the case, then a solid line must either

terminate (resulting in an invalid vertex of solid-dash-dash), or self-intersect (resulting

in an invalid vertex of solid-solid-solid). In this case, there can be no rooted trees or

isolated trees in a valid Feynman diagram. However, there can be other cycles not

connected to the principal line. Further. the cycles do not share any edges.

5.2 shows an example of possible structures of a given topology in the Potts case

and the Sp(2) case, with the same principal line.

100

Figure 5.2: Example illustrating various terminologies.

In the proof we will also need the following fact. We start with a given topology

and a principal line. Suppose there are P vertices on the principal line. V vertices

outside of the principal line, and E edges not on the principal line. We have the

following relation:

2E = 3V + P . (5.16)

This can be reasoned as follows. Suppose we rst add in the vertices without any

edges (except those on the principal line). Since the interaction is cubic, every single

vertex must have 3 edges. Each of the P vertex on the principal line already has

2 edges, so they need 1 more edge. Each of the additional V vertices not on the

principal line needs 3 edges each. Whenever we add an edge not on the principal line,

we saturate two edges. Hence the above relation.

Therefore, we have:

E − V =V + P

2=

total number of vertices2

= number of loops in the diagram .

(5.17)

We will use this fact later on.

101

We will prove a stronger statement than the original claim. We will show that for

a given topology and an arbitrary principal line, the tensor factors calculated from the

0-state Potts model agree with the Sp(2) model, with a factor of (−1)loop order. Since

this equivalence holds for any principal line, the equivalence extends to all tensor

structures of that topology, and our original claim follows.

5.3.2 Dening three sets of graphs: GC, GF , GA

The proof relies on the relations between three quantities, which we will dene now.

Given a topology and a xed principal line, with V vertices and E edges outside of

the principle line, where each edge can be either dashed or solid, we dene three sets

of graphs.

1. The set of cycle graphs, GC, consisting of all graphs on the topology

where there can be zero or more solid line cycles. These cycles cannot intersect

with the principal line, or themselves. These clearly have a 1-to-1 correspondence

with Feynman diagrams in the Sp(2) model. Each vertex is either dash-dash-dash or

dash-solid-solid. For each individual graph gC ∈ GC , suppose it has L total vertices

contained in n solid line cycles (hence V −L vertices not contained in them) we dene

its value to be V (gc) = 2V−L× (−2)n. This corresponds to the value of the Feynman

diagram in the Sp(2) model. Since there will be V −L dash-dash-dash vertices, each

carrying a factor of 2, and n closed fermion loops, each carrying a factor of −N ,

where the number of species N = 2 in our case. Therefore the sum of all graphs in

GC equals to the sum of all Feynman diagrams in the Sp(2) model with the given

topology and principal line.

2. The set of forest graphs, GF, consisting of all graphs on the topology where

there are no solid lines cycles, including those formed with the principal line. These

are just the set of all forest diagrams on the given topology, which have a 1-to-1

correspondence with the diagrammatic method of calculating the tensor factor of the

102

0-state Potts model. for each individual graph gF ∈ GF , suppose it has k solid edges

outside of the principal line (hence it has E−k dashed lines) we dene its value to be

(−1)E−k. Therefore the sum of the value of all graphs in GF gives exactly the tensor

factor in the 0-state Potts model with the given topology and principal line.

Therefore our goal is to show the sum of value of all graphs in GC is equal to

GF . To do so we need to dene the following quantity which, unlike the other two

quantities, does not have an obvious interpretation in terms of Feynman diagrams of

either theory.

3. The set of arrow graphs, GA, consists of all graphs on the topology where

we draw arrows on each edge with the following rules. Imagine initially all edges not

on the principal line are dashes. For each of the V vertex not on the principle line, we

make one of four choices: to color one of the three lines connected to this vertex to be

solids (each of these choice carry a factor of 1, we say that such a vertex is owing),

or not coloring any of the lines and leaving them as dashed (this carries a factor of

−1, we call such a vertex empty). The value of an individual graph gA ∈ GA is just

the product of all the value on its vertices, which is V (gA) = (−1)# of empty vertices.

Let's look at the properties of GA. We can make each of the 4 choices indepen-

dently for each vertex, getting a total of 4V possibilities. If we sum the value of all

such 4V diagrams, we will get (3− 1)V = 2V , since the choice made on each vertex is

independent.

Notice that we allow a line to be colored twice by its two adjacent vertices. We

will keep track of which vertex colored which line with arrows. If a line connecting

vertices A and B is colored by A, then an arrow is drawn on the line pointing away

from A, likewise for B. If a line is colored by both vertices, we draw a double arrow.

This is illustrated in 5.3. Since each vertex can color up to 1 of its adjacent lines,

there can never be more than one arrow coming out of any vertex.

Let us now look at the properties of the diagrams in GA.

103

Figure 5.3: Line 1 is colored by vertex A, line 2 is double-colored by vertices B andC. All 3 vertices are owing.

Figure 5.4: Illustrating some properties of GA

1. It is impossible to create a cycle with part of the principal line. Since in order

to form a cycle with the principle line, you would need to color (k+1) lines with only

k vertices, which is impossible. See 5.4 (a).

2. While it is possible to create a cycle disconnected from the principal line, it

must be a simple cycle. For example, if you want to form a double cycle, with k

vertices, you need at least (k+1) colored lines. See 5.4 (b). Thus, all cycles created

must be disjoint.

Our main proof proceeds as follows. First we will show that the sum of all diagrams

in GC corresponds to the sum of all diagrams in GA that does not result in any cycles,

ie. all forest diagrams in GA.

104

Then we will show that sum of all the forest diagrams in GA equals to the sum of

diagrams in GF times the factor (−1)E−V , which we've shown to be (−1)number of loops.

This explains the observation we made in the beginning of this chapter that the

successive loops have alternating signs.

5.3.3 Relation between GA and GC

In this section we will show that the sum of all diagrams in GC corresponds to the

sum of all forest diagrams in GA.

First we look at the diagram in GC where all lines outside the principal line are

dashes, there is only one such diagram. That means all of the V vertices outside the

principal line are dash-dash-dash, with each contributing a factor of 2. This diagram

has value 2V . On the other hand, as we've already noted, the sum of all diagrams in

GA is exactly 2V .

Conclusion 1: the only diagram with no cycle in GC is equal to the sum

of all diagrams in GA

Now we look at a diagram in GC where there is exactly one solid line cycle.

Suppose the cycle has length L. This diagram has V − L triple dash vertices, each

contributing a factor of 2, and L solid-solid-dash vertices, each contributing a factor

of 1. Finally, it has a solid line cycle, which gives an additional factor of N = −2. In

total, it has value −2× 2V−L.

Now consider all the diagrams in GA where that particular cycle is solid. We don't

care about the choice of other vertices (in particular, they can form additional cycles).

So the other V −L vertices contribute a factor of 2V−L. The remaining L vertices in

a loop must all be owing. Further they either all ow clockwise or counterclockwise,

giving 2 possibilities, and each owing vertex gives a factor of 1. In total the sum of

all such diagrams is 2× 2V−L.

The above argument can be easily generalized to any other cycle. So we have:

105

Conclusion 2: the sum of diagrams with exactly one cycle in GC is equal

to the negative of the sum of all diagrams with at least one cycle in GA

Now we look at a diagram in GC with exactly 2 solid line cycles. As we argued

before the two cycles must be disjoint. Suppose the length of the two cycles are L1 and

L2. Then the value of this diagram is the product of: 2V−L1−L2 (for the triple-dash

vertices), (1)L1+L2 (for the solid-solid-dash vertices), and N2 = 4.

in GA, it corresponds to all diagrams with these two particular cycles, regardless

of the choices of the other V −L1−L2 vertices. However, each of these cycles have 2

ways to be connected (clockwise or counterclockwise), therefore there is an additional

factor of 4.

Conclusion 3: the sum of diagrams with exactly two cycles in GC is

equal to the sum of all diagrams with at least two cycle in GA

We can see how this generalizes. Let GC,n denote the sum of all diagrams in GC

with exactly n cycles. When we sum up all possible diagrams in GC , we get,:

∑

all diagrams

GC = GC,0 +GC,1 +GC2 + . . . . (5.18)

Then let GA,n+ be the sum of all diagrams in GA with at least n cycles, we have:

∑

all diagrams

GC = GC,0 +GC,1 +GC,2 +GC,3 . . . , (5.19)

= GA,0+ −GA,1+ +GA,2+ −GA,3+ + . . . . (5.20)

In other words, we have the sum of: all diagrams in GA, minus all diagrams with

at least one cycle in GA, plus all diagrams with at least two cycles in GA, etc. By

the principle of inclusion and exclusion, the sum is exactly equal to the sum of all

diagrams with no cycles in GA, leaving only the forest diagrams in GA.

106

Figure 5.5: There is only one possible coloring for a rooted tree: every arrow pointstowards the root.

Therefore, we nally conclude that, the sum of all diagrams in GC is equal to the

sum of all forest diagrams in GA.

5.3.4 Relation between GA and GF

Now we will argue that each diagram in GF (which is a forest) is counted with

exactly weight 1 in GA, up to an overall factor depending on the number of loops in

the topology. Therefore if the sum of all diagrams in GF is also equal to the sum of

all forest diagrams in GA, establishing our original claim.

Case 1: a single rooted tree If the forest diagram contains a single rooted

tree with k vertices and nothing else, there are k vertices which need to color exactly

k lines. All k vertices must be owing and the direction of the arrow is xed, as

illustrated in 5.5. Thus, there is exactly one conguration which colors each rooted

tree. Suppose there are V vertices and E edges in total. in GA, k of the vertices are

owing, (V − k) of them are empty, and hence it has a factor of (−1)V−k. In GF ,

there are E − k dashed lines, it carries a factor (−1)E−k. Thus, to match the overall

factor, we must multiply the Potts result by (−1)E−V = (−1)number of loops.

107

Figure 5.6: There are (k+1) choices for the empty vertex, and k choice for the double-colored line. But they have opposite signs.

Case 2: a single isolated tree For a single isolated tree in GA, there are k + 1

vertices which need to color k lines. This means that either one of the vertices doesn't

color any lines, or one of the lines is double colored. There are k+1 choices for where

the empty vertex is, and k choices for where the double-colored line is. However,

there is a sign dierence due to the fact that if a vertex doesn't color any lines, it carries

a factor −1. Thus, the total factor is (−1)V−k(k+ 1) + (−1)V−k−1k = (−1)V−k. This

is illustrated in 5.6 Notice that the same graph in GF will carry a factor of (−1)E−k.

Again, in order to match the Potts model, we need an overall factor of (−1)E−V . This

is consistent with case 1.

Case 3: multiple rooted tree and isolated trees This is a simple generaliza-

tion of the previous two cases, indeed the GA value is again consistent with the GF

value, up to the same overall factor.

Combining the results from the previous section, we see that, for a given topology

and a given principal line:

108

∑

all diagrams

GF = (−1)#ofloops∑

all forest diagrams

GA (5.21)

= (−1)#ofloops∑

all diagrams

GC . (5.22)

This proves our claim, that for a given two-point function topology and principal

line, the value of the tensor structure in the Sp(2) model is equal to that in the 0-

state Potts model. If we sum over all possible principal lines, the equality still holds.

Therefore all two-point function tensor structures in the Sp(2) model are equal to

that of the 0-state Potts model.

5.4 three-point functions and zero point functions

The generalization to 3-point functions is rather simple. Suppose we have a three-

point function T3dijk in the Potts model. We would like to show that it's equal to a

three-point function in the Sp(2) model. Again, there are two choices of three-point

functions in the Sp(2) case: σ3 or σθθ, we can choose either one due to supersymmetry

(except σ3 is bigger by a factor of 2), so for convenience, we choose the latter.

We can convert the 3-point function to a 2-point function by merging two of its

lines into one (In the Sp(2) case, we must merge the σ line with one of the two θ

lines). This eectively creates a new two-point function tensor structure T2new related

to T3 via:

T2new =∑

j,k

T3dijkdijl = T3T2δ

il , (5.23)

where T2 is the simplest one loop tensor structure dijkdjkl.

Thus, to show that T3 is equal, we just need T2 and T2new being equal, which is

true.

109

The generalization to zero point functions is also simple. Notice that relation 5.20

still holds when we ignore the existence of the principal line. The consideration for

diagrams in GF is also simpler because there will only be isolated trees, and no rooted

trees. The equivalence will continue to hold without any issue.

Notice that the equality between all zero point functions in the two models imply

that their sphere free energy F will be the same.

110

Chapter 6

Yukawa CFTs and Emergent

Supersymmetry

6.1 Introduction and Summary

This chapter is based on the work published in [31], co-authored with Simone Giombi,

Igor Klebanov, and Grigory Tarnopolsky. We also thank David Gross, David Poland,

Silviu Pufu and David Simmons-Dun for useful discussions.

Physical applications of relativistic quantum eld theories with four-fermion in-

teractions date back to Fermi's theory of beta decay. The rst application to strong

interactions was the seminal Nambu and Jona-Lasinio (NJL) model. In the original

paper [104] they considered the model in 3 + 1 dimensions with a single 4-component

Dirac fermion and Lagrangian

LNJL = ψ 6 ∂ψ +g

2

((ψψ)2 − (ψγ5ψ)2

). (6.1)

In addition to the U(1) symmetry ψ → eiβψ, this Lagrangian possesses the U(1)

chiral symmetry under ψ → eiαγ5ψ. Using the gap equation it was shown that the

chiral U(1) can be broken spontaneously, giving rise to the massless Nambu-Goldstone

111

boson. This was the discovery of the crucial role of chiral symmetry breaking in the

physics of strong interactions.

One of the goals of this chapter is to study a generalization of (6.1) to Nf 4-

component Dirac fermions ψj, j = 1, . . . Nf :

LNJL = ψj 6 ∂ψj +g

2

((ψjψ

j)2 − (ψjγ5ψj)2), (6.2)

and its continuation to dimensions below 4. We dene N = 4Nf , so that N is the

number of 2-component Majorana fermions in d = 3. In addition to the chiral U(1)

symmetry ψj → eiαγ5ψj this multi-avor NJL model possesses a U(Nf ) symmetry.1

When considered in 2 < d < 4 this model gives rise to an interacting conformal eld

theory which describes the second-order phase transition separating the phases where

the U(1) chiral symmetry is broken and restored.

In addition to studying the NJL model with the U(1) chiral symmetry, we will

present new results for the Gross-Neveu (GN) model [25], which has a simpler quartic

interaction

LGN = ψj 6 ∂ψj +g

2(ψjψ

j)2 . (6.3)

Instead of the continuous chiral symmetry it possesses the discrete chiral symmetry

ψj → γ5ψj. As discovered in [25], in d = 2 this theory is asymptotically free forN > 2.

When considered in 2 < d < 4 this is believed to be an interacting conformal eld

theory which describes the second-order phase transition where the discrete chiral

symmetry is broken.

In d > 2 the four-fermion interactions (6.2) and (6.3) are non-renormalizable.

While they are renormalizable in the sense of the 1/N expansion [107], at nite N it

1 It is also often called the chiral Gross-Neveu model [25]; in d = 2 it is equivalent to the SU(2Nf )Thirring model [105, 106].

112

is important to know the UV completion of these theories. In [26, 27] it was suggested

that the UV completion in 2 < d < 4 is provided by the appropriate Yukawa theories.

The UV completion of the GN model (6.3) contains a real scalar eld σ and Nf

4-component Dirac fermions ψj, j = 1, . . . Nf :

LGNY =1

2(∂µσ)2 + ψj 6 ∂ψj + g1σψjψ

j +1

24g2σ

4 . (6.4)

This theory, known as the Gross-Neveu-Yukawa (GNY) model, will be discussed in

section 6.2.

The UV completion of the U(1) symmetric NJL model (6.2), which contains a

complex scalar eld φ = φ1 + iφ2, was introduced in [27]:

LNJLY =1

2(∂µφ1)2 +

1

2(∂µφ2)2 + ψj /∂ψ

j + g1ψj(φ1 + iγ5φ2)ψj +1

24g2(φφ)2 . (6.5)

It has a continuous U(1) chiral symmetry under2

ψj → eiαγ5ψj , φ→ e−2iαφ . (6.6)

This theory, which we will call the Nambu-Jona-Lasinio-Yukawa model (NJLY), will

be discussed in section 6.3.

There is a large body of literature on the GN and NJL CFTs in d = 3 and their

applications; see, for example, [109, 26, 27, 110, 111, 112, 113, 114, 115, 3, 116,

117, 118, 119]. We will carry out further studies of these CFTs using the 4 − ε and

2 + ε expansions followed by Padé extrapolations. In addition to studying the scaling

dimensions of some low-lying operators, we will calculate the sphere free energy F .

The latter determines the universal entanglement entropy across a circle [66], and is

2In d = 3, one may express the Lagrangian (6.2) in terms of 2Nf 2-component Dirac spinorsχi, χi+Nf by writing ψi = (χi, χi+Nf ), i = 1, . . . , Nf . See for instance [108] for the explicit relationbetween 4-component and 2-component notations in 3d. The 2-component spinors χi± = (χi ±χi+Nf )/

√2 have charge ±1 under the U(1) symmetry (6.6).

113

the quantity that enters the F -theorem [120, 54, 50, 67]. We will also discuss CT ,

the normalization of the correlation function of two stress-energy tensors. For the

GN model, the 1/N and ε expansions of CT were studied in [119]; in this chapter we

extend these results to the NJL model and also provide the numerical estimates in

d = 3 for various values of N .

When considered for Nf = 1/2, i.e. the single 4-component Majorana fermion

(which is equivalent to one Dirac fermion in d = 3), the NJLY model is expected to

ow to the well-known supersymmetric Wess-Zumino model with 4 supercharges. In

d < 4 this theory denes a CFT with emergent supersymmetry" [121, 122], in the

sense that the RG ow drives the interactions to a supersymmetric IR stable xed

point, where the global U(1) symmetry becomes the U(1)R symmetry (see gure 6.1).

We will provide additional evidence for this using the 4 − ε expansion of the NJLY

model with Nf = 1/2 to two loops, both for certain scaling dimensions and for the

sphere free energy. A three-loop calculation of scaling dimensions, which supports

the emergent supersymmetry, was carried out recently [123].

Even more intriguingly, when the GNY model is continued to Nf = 1/4, which

corresponds to a single 2-component Majorana fermion in d = 3, it appears to ow

to a CFT with 2 supercharges [121, 124, 125, 118, 126]. We will show that the O(ε2)

corrections to scaling dimensions of operators continue to respect this emergent super-

symmetry.3 This provides new support for the existence of an N = 1 supersymmetric

CFT in d = 3. Our Padé extrapolations of operator dimensions including the ε2 cor-

rections are in good agreement with the conformal bootstrap approach to the N = 1

supersymmetric CFT in d = 3 [118]. We also estimate CT and F for this theory.

These results will be presented in section 6.4.

3This is reminiscent of the symmetry enhancement from Sp(2) to the supergroup OSp(1|2) atthe IR stable xed point of a cubic theory in 6− ε dimensions in chapter 4.

114

UVUV

=1 SUSY=1 SUSY

IsingIsing

??

-1 0 1 2 3 4 5

-4

-2

0

2

4

6

8

(g1)2

g2

GNY model with N=1

UVUV

=2 SUSY=2 SUSY

O(2)O(2)

??

-1 0 1 2 3 4 5

-5

0

5

10

(g1)2

g2

NJLY model with N=2

Figure 6.1: RG ow and xed point structure for the GNY model with N = 1 (oneMajorana fermion in d = 3) and the NJLY model with N = 2 (one Dirac fermion ind = 3), obtained from the one-loop β-functions (6.7) and (6.68) in d = 4 − ε. Theattractive IR xed points have emergent" supersymmetry with 2 and 4 superchargesrespectively. The red triangles denote unstable xed points with negative quarticpotential which can be seen in the one-loop analysis in d = 4− ε; their fate in d = 3is unclear.

115

6.2 The Gross-Neveu-Yukawa Model

The β-functions for the GNY model with action (6.4) in d = 4 − ε, up to two-loop

order, are [114]4

βg2 = −εg2 +1

(4π)2

(3g2

2 + 2Ng21g2 − 12Ng4

1

)+

1

(4π)4

(96Ng6

1 + 7Ng41g2 − 3Ng2

1g22 −

17g32

3

),

βg1 = − ε2g1 +

N + 6

2(4π)2g3

1 +1

(4π)4

(− 3

4(4N + 3)g5

1 − 2g31g2 +

g1g22

12

), (6.7)

where N = Nf tr1 = 4Nf . The model possesses an IR stable xed point at the critical

couplings g∗i given by

(g∗1)2

(4π)2=

1

N + 6ε+

(N + 66)√N2 + 132N + 36−N2 + 516N + 882

108(N + 6)3ε2 +O(ε3) ,

g∗2(4π)2

=−N + 6 +

√N2 + 132N + 36

6(N + 6)ε+

1

54(N + 6)3√N2 + 132N + 36

×(

3N4 + 155N3 + 2745N2 − 2538N + 7344

− (3N3 − 43N2 − 1545N − 1224)√N2 + 132N + 36

)ε2 +O(ε3) .

(6.8)

Of course, there is also a xed point g∗1 = 0, g∗2 = gIsing2 = 16π2ε

3+ O(ε2) which

corresponds to the decoupled product of the single-scalar Wilson-Fisher xed point

and Nf free fermions. By looking at the derivative of the beta functions at the xed

points, one can verify that (6.8) is attractive for all Nf , so one can ow to it from the

Ising" xed point along a relevant direction. Let us mention that there is formally

also a third xed point obtained from (6.8) by changing the sign of√N2 + 132N + 36.

This xed point is unstable in d = 4− ε due to the negative quartic coupling, g∗2 < 0,

but its dimensional continuation may produce a CFT in d = 3.

4We have reproduced them using the general two-loop results [127, 128, 129, 130, 131] for theYukawa theories, which are reviewed in the Appendix.

116

The scaling dimensions of σ and ψ are found to be [114]

∆σ = 1− ε

2+

1

2(4π)2Ng2

1 +1

(4π)4

(1

12g2

2 −5

4Ng4

1

),

∆ψ =3− ε

2+

1

2(4π)2g2

1 −1

8(4π)4(3N + 1)g4

1 . (6.9)

At the IR stable xed point (6.8), one gets5

∆σ = 1− 3

N + 6ε+

52N2 − 57N + 36 + (11N + 6)√N2 + 132N + 36

36(N + 6)3ε2 +O(ε3) ,

∆ψ =3

2− N + 5

2(N + 6)ε+−82N2 + 3N + 720 + (N + 66)

√N2 + 132N + 36

216(N + 6)3ε2 +O(ε3) .

(6.10)

These dimensions agree with [112] after correcting some typos in eq. (11) of that

paper (in particular, the coecient 33 should be changed to 3). Our O(ε) term in ∆ψ

corrects a typo in eq. (4.44) of [3].

Setting g∗1 = 0, g∗2 = 16π2ε3

+O(ε2) in (6.9), we may also recover the result at the

Ising xed point ∆Isingσ = 1 − ε

2+ ε2

108+ O(ε3). One can then see that the Yukawa

operator σψψ is relevant at this decoupled xed point, and can trigger a ow to the

IR stable xed point (6.8). We expect this to be true in d = 3 as well, since it is

known that ∆3d Isingσ ≈ 0.518 [57], and so ∆3d Ising

σψψ≈ 2.518 < 3.

The anomalous dimension of the operator σ2, which determines the critical expo-

nent ν−1 = 2− γσ2 , may be read o from eq. (18) of [114]

γσ2 =g2

(4π)2− 1

(4π)4(g2

2 +Ng2g21 − 2Ng4

1) + 2γσ . (6.11)

5Throughout the chapter, one can obtain the corresponding ε expansions at the unstable xedpoint by changing the sign of the square root.

117

At the xed point (6.8) we nd

∆σ2 = d− 2 + γσ2 = 2 +

√N2 + 132N + 36−N − 30

6(N + 6)ε+

1

54(N + 6)3√N2 + 132N + 36

×(

(3N3 + 109N2 + 510N + 684)√N2 + 132N + 36

− 3N4 − 658N3 − 333N2 − 15174N + 4104)ε2 +O(ε3) . (6.12)

We have also calculated the one-loop anomalous dimensions of the operators ψψ and

σ3:

∆ψψ =(N + 2)g2

1

(4π)2+ 2∆ψ , ∆σ3 =

3g2

(4π)2+ 3∆σ . (6.13)

At higher orders these operator will mix, and one has to nd the eigenvalues of their

mixing matrix. At the xed point (6.8), we nd

∆ψψ = 3− 3

N + 6ε+O(ε2) , ∆σ3 = 3 +

√N2 + 132N + 36−N − 12

2(N + 6)ε+O(ε2) .

(6.14)

The rst of these dimensions corresponds to a descendant of σ, as can be seen from

the fact that it equals 2 + ∆σ.

Let us also review the known result for the 4 − ε expansion of CT in the GNY

model, which was discussed in [119]; the diagrams contributing to the term ∼ g21 are

shown in g. 4.7 there. Evaluation of these diagrams yields

CT = NCT,f + CT,s −5N(g∗1)2

12(4π)2=

d

S2d

(N

2+

1

d− 1− 5Nε

12(N + 6)

), (6.15)

118

where Sd = 2πd/2/Γ(d/2), and we used the values of CT for free scalar and fermion

theories:

CT,s =d

(d− 1)S2d

, CT,f =d

2S2d

. (6.16)

6.2.1 Free energy on S4−ε

In order to renormalize the theory on curved space, one should add to the action all

the relevant curvature couplings that are marginal in d = 4− ε [132, 133]

SGNY =

∫ddx√g(ψi 6 ∂ψi +

1

2(∂µσ)2 +

d− 2

8(d− 1)Rσ2 + g1,0σψiψ

i +g2,0

24σ4

+η0

2Rσ2 + a0W

2 + b0E + c0R2),

(6.17)

where R is the scalar curvature, W 2 is the square of the Weyl tensor, and E the Euler

density

E = RµνλρRµνλρ − 4RµνRµν +R2 . (6.18)

The parameters η0, a0, b0, c0 are bare curvature couplings whose renormalization can

be xed order by order in perturbation theory. On a sphere, the Weyl square term

drops out, and to the order we work below we will only need the renormalization of

the Euler coupling b0 (the R2 term and the renormalization of conformal coupling

are expected to play a role at higher orders [132, 133, 130]). The corresponding beta

function can be extracted from the results of [130, 134], and we nd

βb = εb− 11N/4 + 1

360(4π)2− 1

(4π)8

1

36

(9

32Ng6

1 +3

8N2g6

1 +1

4Ng4

1g2 −1

96Ng2

1g22

)+ . . . ,

(6.19)

119

where b is the renormalized coupling, and its the relation to the bare one b0 = µ−ε(b−11N/4+1360(4π)2ε

+. . .) can be inferred from the above beta function. The coupling independent

term is related to the a-anomaly of the free fermions and scalar.

The calculation of the sphere free energy now proceeds as in [135, 136]. Keeping

terms that contribute up to order ε2, we have

F = NFf + Fs −1

2g2

1,0

∫ddxddy

√gx√gy〈σψψ(x)σψψ(y)〉0

− 1

2!(4!)2g2

2,0

∫ddxddy

√gx√gy〈σ4(x)σ4(y)〉0

− 1

4!g4

1,0

∫ddxddyddzddw

√gx√gy√gz√gw〈σψψ(x)σψψ(y)σψψ(z)σψψ(w)〉0 + δFb ,

(6.20)

where δFb = b0

∫ddx√gE is the contribution of the curvature term, and Ff , Fs are

the sphere free energies of free fermion and scalar, which can be found in [78]. Starting

from the at space propagators

〈σ(x)σ(y)〉 = Cφ1

|x− y|d−2, 〈ψi(x)ψj(y)〉 = δjiCψ

γµ(x− y)µ|x− y|d , (6.21)

where Cφ = Γ(d2− 1)/(4π

d2 ) and Cψ = Γ(d

2)/(2π

d2 ), and then Weyl transforming to

the sphere, we nd

∫ddxddy

√gx√gy〈σψψ(x)σψψ(y)〉0 = NCφC

2ψ

∫ddxddy

√gx√gy

s(x, y)3d−4= NCφC

2ψI2(

3d

2− 2) ,

∫ddxddy

√gx√gy〈σ4(x)σ4(y)〉0 = 4!C4

φ

∫ddxddy

√gx√gy

s(x, y)4(d−2)= 4!C4

φI2(2d− 4) .

(6.22)

120

Here I2(∆) denotes the integrated 2-point function of an operator of dimension ∆,

which is given by [137, 138, 50]

I2(∆) =

∫ddxddy

√gx√gy

s(x, y)2∆= (2R)2(d−∆) 21−dπd+ 1

2 Γ(d2−∆

)

Γ(d+1

2

)Γ(d−∆)

. (6.23)

For the 4-point function, we nd

〈σψψ(x)σψψ(y)σψψ(z)σψψ(w)〉0 = 6N2C2φC

4ψ

1

s2d−2xy s2d−2

zw sd−2xw sd−2

yz

− 3NC2φC

4ψ

(s2xws

2yz − s2

xzs2yw + s2

xys2zw

(sxysyzszwsxw)d

)×(

2

(sxyszw)d−2+

1

(sxzsyw)d−2

),

(6.24)

where we used a shorthand notation for the chordal distance sxy ≡ s(x, y). The

integral of this 4-point function over the sphere cannot be calculated explicitly, but

one can nd it as a series in d = 4− ε. For this we used the Mellin-Barnes approach,

which is described in [135, 136]. The result for the integral reads

∫ddxddyddzddw

√gx√gy√gz√gw〈σψψ(x)σψψ(y)σψψ(z)σψψ(w)〉0 =

= −N(N + 6)

2(4π)4ε− N

(N − 6 + 6(N + 6)(3 + γ + log(4πR2))

)

12(4π)4+O(ε).

(6.25)

Putting everything together, we nd for the free energy in d = 4− ε

F = NFf + Fs −1

2g2

1,0NCφC2ψI2

(3

2d− 2

)− 1

2 · 4!g2

2,0C4φI2(2d− 4)

+1

4!g4

1,0

(N(N + 6)

2(4π)4ε+N(N − 6 + 6(N + 6)(3 + γ + log(4πR2))

)

12(4π)4+O(ε)

)+ δFb .

(6.26)

121

Now replacing the bare couplings with the renormalized ones

g1,0 = µε2

(g1 +

N + 6

32π2

g31

ε+ . . .

), g2,0 = µε

(g2 + . . .

), b0 = µ−ε

(b− 11N/4 + 1

360(4π)2ε+ . . .

)

(6.27)

we nd that all pole cancels, and the free energy is a nite function of the renormalized

couplings g1, g2, b.6

As explained in [135], in order to calculate the free energy at the critical point we

should now tune all couplings, including b, to their xed point values. Using (6.8),

we get

F = NFf + Fs −Nε

48(N + 6)

−((N2 + 99N + 18)

√N2 + 132N + 36 + (80N2 + 2103N + 6381)N + 108

)ε2

7776(N + 6)3

+ δFb +O(ε3) . (6.28)

From the curvature beta function (6.19), we nd at the critical point

b∗ =11N/4 + 1

360(4π)2ε+

1

(4π)8

1

36

(9

32N(g∗1)6 +

3

8N2(g∗1)6 +

1

4N(g∗1)4g∗2 −

1

96N(g∗1)2(g∗2)2

)1

ε

(6.29)

and, using (6.8) and∫ddx√gE = 64π2+O(ε), we nd that the Euler term contributes

δFb =N(882 + 66

√N2 + 132N + 36 +N(516−N +

√N2 + 132N + 36))

15552(N + 6)3ε2 +O(ε3) .

(6.30)

6Note that the coupling dependent part in the renormalization of b is necessary to cancel polescoming from diagrams at the next order. However, we still have to carefully include the Euler term,as in [135, 136], as it aects the free energy at the xed point to order ε2.

122

Substituting this into (6.28), and writing the result in terms of F = − sin(πd/2)F ,

we nd

F = NFf + Fs −Nπε2

96(N + 6)− 1

31104(N + 6)3

(161N3 + 3690N2 + 11880N + 216

+(N2 + 132N + 36

)√N2 + 132N + 36

)πε3 +O

(ε4). (6.31)

6.2.2 2 + ε expansions

In this section we review the known results for operator dimensions at the UV xed

point of the Gross-Neveu model in d = 2 + ε, and then compute its sphere free energy

to order ε3.

The action for the Gross-Neveu model [25] in Euclidean space in terms of bare

elds and coupling reads

SGN = −∫ddx√g(ψi 6 ∂ψi +

1

2g0(ψiψ

i)2)

+ b0

∫ddx√gR , (6.32)

where i = 1, . . . 2Nf , and we have included the Euler term which is needed for the

calculation of the sphere free energy below.

The β-function for the renormalized coupling constant g in d = 2 + ε is known to

be [139, 3, 140]

β = εg − N − 2

2πg2 +

N − 2

4π2g3 +

(N − 2)(N − 7)

32π3g4 +O(g5) . (6.33)

Therefore, one can see that there is a perturbative UV xed point at a critical coupling

g∗ given by7

g∗ =2π

N − 2ε+

2π

(N − 2)2ε2 +

π(N + 1)

2(N − 2)3ε3 +O(ε4) . (6.34)

7This corrects a typo in g∗ in [3] on page 59 (there it is denoted by uc).

123

The scaling dimensions are found to be [140]

∆ψ =1 + ε

2+N − 1

8π2g2 − (N − 1)(N − 2)

32π3g3 +O(g4) ,

∆σ = 1 + ε− N − 1

2πg +

N − 1

8π2g2 +

(N − 1)(2N − 3)

32π3g3 +O(g4) , (6.35)

where σ ∼ ψψ. At the UV xed point this gives

∆ψ =1

2+

1

2ε+

N − 1

4(N − 2)2ε2 − (N − 1)(N − 6)

8(N − 2)3ε3 +O(ε4) , (6.36)

∆σ = 1− 1

N − 2ε− N − 1

2(N − 2)2ε2 +

N(N − 1)

4(N − 2)3ε3 +O(ε4) , (6.37)

It is also not hard to determine the dimension

∆σ2 = d+ β′(g∗) = 2 +1

N − 2ε2 +

N − 3

2(N − 2)2ε3 +O(ε4) . (6.38)

Let us now turn to the calculation of the free energy on S2+ε. To order g4, we

have

F =NFf −1

2!

(g0

2

)2

S2 −1

3!

(g0

2

)3

S3 −1

4!

(g0

2

)4

S4 + b0

∫ddx√gR , (6.39)

where Ff is the free fermion contribution, derived as a function of d in [78], and

Sn =

∫ n∏

i=1

dxi√gxi〈ψ4(x1)...ψ4(xn)〉conn0 , (6.40)

with ψ4 ≡ (ψiψi)2. Using the at space fermion propagator in (6.21), and then

performing a Weyl transformation to the sphere, we nd

124

S2 = 2N(N − 1)C4ψI2(2d− 2) ,

S3 = 8N(N − 1)(N − 2)C6ψI3(2d− 2) , (6.41)

where the integral I2(∆) is given in (A.12), and I3(∆) denotes the integrated 3-point

function [137, 138, 50]

I3(∆) =

∫ddxddyddz

√gx√gy√gz

[s(x, y)s(y, z)s(z, x)]∆= R3(d−∆) 8π

3(1+d)2 Γ(d− 3∆

2)

Γ(d)Γ(1+d−∆2

)3. (6.42)

For the integrated 4-point function we nd

S4 = 24N(N − 1)C8ψ

∫ 4∏

i=1

dxi√gxi

(2(N2 − 3N + 4)

(s13s14s23s24)2d−2+

2N

s2d−412 s2d−4

34 sd13sd14s

d23s

d24

+2s4

13s424s

d12s

d34 + sd+4

12 sd+434 + 4s2

12s234s

d+213 sd+2

24 − 4s412s

434s

d14s

d23 − 4s2

14s223s

d+212 sd+2

34

(s12s213s

214s

223s

224s34)d

− 4(N − 1)s2−2d

12 s2d13s

2−2d34 s2d

24 + s213s

224 + s2

14s223 − s2

12s234

s3d−213 sd14s

d23s

3d−224

), (6.43)

where smn ≡ s(xm, xn) is a chordal distance on a sphere. Using the methods described

in [135, 136], we nd

S4 =C8ψ(2R)4(2−d) 21−dπ

d+12

Γ(d+12

)

e32γ(2−d)

(πd/2)3 192N(N − 1)

((N − 2)2

(− 4

ε2+

10

ε−(25 +

7π2

12

))

− 2(N − 2)( 3

5ε− 11

5

)+ 1 +O(ε)

). (6.44)

After expressing the bare coupling g0 in terms of the renormalized one

g0 = µ−ε(g − N − 2

2π

g2

ε+

((N − 2)2

4π2ε2− N − 2

8π2ε

)g3 + ...

), (6.45)

125

we nd a surviving pole in F of order g4/ε. This pole can be cancelled provided we

renormalize the Euler density parameter as

b0 = µε(b− N

48πε+N(N − 1)(N − 2)

30(4π)5

g4

ε

),

βb = −εb+N

48π− N(N − 1)(N − 2)

6(4π)5g4 , (6.46)

where the coupling independent term is due to the trace anomaly of the free fermion

eld, and we used∫ddx√gR = 8π +O(ε). In order to obtain the correct expression

for F at the UV xed point in d = 2 + ε, we should now set g = g∗ and b = b∗ =

N48πε− N(N−1)(N−2)

6(4π)5g4∗ε. The contribution of the Euler term δFb = b0

∫ddx√gR at the

xed point (6.34) is then

δFb = − N(N − 1)

48(N − 2)3ε3 +O(ε4) , (6.47)

and putting this together with the contributions of S2 and S3,8 we nd

F = NFf +N(N − 1)

24(N − 2)2ε2 − N(N − 1)(N − 3)

16(N − 2)3ε3 +O(ε4) , (6.48)

or, in terms of F = − sin(πd/2)F :

F = NFf +N(N − 1)πε3

48(N − 2)2− N(N − 1)(N − 3)πε4

32(N − 2)3+O(ε5) . (6.49)

This result agrees with the one obtained in [135] using a conformal perturbation

theory approach; this provides a non-trivial check on the procedure used here which

involves the curvature terms.8Note that even though we had to compute S4 to x the renormalization of b, we cannot obtain

F to order ε4. That would require xing βb to order g5.

126

6.2.3 Large N expansions

In this section we test the 4 − ε and 2 + ε expansions by comparing them with the

known 1/N expansions [109, 69, 141]. The general form of the large N expansions of

scaling dimensions in the GN model is9

∆ψ =d− 1

2+

1

Nγψ,1 +

1

N2γψ,2 +O(N−3) ,

∆σ = 1 +1

Nγσ,1 +

1

N2γσ,2 +O(N−3) ,

∆σ2 = 2 +1

Nγσ2,1 +

1

N2γσ2,2 +O(N−3) , (6.50)

where

γψ,1 = − Γ(d− 1)

Γ(d2− 1)Γ(1− d

2)Γ(d

2+ 1)Γ(d

2), γψ,2 = 4γ2

ψ,1

((d− 1)Ψ(d)

d− 2+

4d2 − 6d+ 2

(d− 2)2d

),

(6.51)

γσ,1 = −4d− 1

d− 2γψ,1 ,

γσ,2 = −γ2ψ,1

(6d2Θ(d)

d− 2+

16(d− 1)2Ψ(d)

(d− 2)2− 4(d− 1) (d4 − 2d3 − 12d2 + 20d− 8)

(d− 2)3d

)

(6.52)

and

γσ2,1 = 4(d− 1)γψ,1 ,

γσ2,2 = −16dγ2

ψ,1

d− 2

(( 1

d− 4− d2

2+

4

(d− 4)2− 6

d− 2− 1

d− 3

2

)Ψ(d) +

4

(d− 4)2γψ,1

+3dΘ(d)

8

(9− d+

12

d− 4

)− (d− 3)d

d− 4(Φ(d) + Ψ(d)2)− 5d

2+

1

2(d− 4)− 2

(d− 4)2

− 7

d− 2− 4

(d− 2)2+

5

2d− 3 +

d2

2− 1

d2

). (6.53)

9The 1/N3 term in ∆ψ may be found in [142, 143].

127

In these equations we dened

Ψ(d) = ψ(d− 1)− ψ(1) + ψ

(2− d

2

)− ψ

(d

2

),

Θ(d) = ψ′(d

2

)− ψ′(1) ,

Φ(d) = ψ′(d− 1)− ψ′(

2− d

2

)− ψ′

(d

2

)+ ψ′(1) , (6.54)

where ψ(x) = Γ′(x)/Γ(x) denotes the digamma function.

In d = 3, the above large N expansion of the scaling dimensions read

∆ψ = 1 +4

3π2N+

896

27π4N2= 1 +

0.1351

N+

0.3407

N2, (6.55)

∆σ = 1− 32

3π2N+

9728− 864π2

27π4N2= 1− 1.0808

N+

0.4565

N2, (6.56)

∆σ2 = 2 +32

3π2N− 64(632 + 27π2)

27π4N2= 2 +

1.0808

N− 21.864

N2. (6.57)

For F , the O(N0) result can be obtained from the general formula [68, 78] for the

change in F under a double trace" deformation δS = g∫O2

∆, where O∆ is a scalar

primary operator of dimension ∆. In terms of F = − sin(πd/2)F , the result is

δF =1

Γ (d+ 1)

∫ ∆− d2

0

du u sin(πu)Γ

(d

2+ u

)Γ

(d

2− u). (6.58)

In the present case O∆ = ψψ, and so ∆ = d− 1. Therefore,

F = NFf +1

Γ (d+ 1)

∫ d2−1

0

du u sin(πu)Γ

(d

2+ u

)Γ

(d

2− u)

+O(1/N) . (6.59)

The 4− ε expansion of this agrees with the large N expansion of (6.31), i.e.

F = NFf + Fs −(πε2

96+πε3

192

)+

(πε2

16− πε3

32

)1

N−(

3πε2

8− 17πε3

32

)1

N2+O(1/N3),

(6.60)

128

Similarly, the 2 + ε expansion of (6.59) agrees with the large N expansion of (6.49),

i.e.

F = NFf +

(πε3

48− πε4

32

)+

1

N

(πε3

16− πε4

16

)+

1

N2

(πε3

6− 3πε4

32

)+O(1/N3) .

(6.61)

For CT , the relative O( 1N

) correction to the answer for free feermions is given in

[119]:

CT = NCT,f

(1 +

CT1

N+O(1/N2)

),

CT1 = −4γψ,1

(Ψ(d)

d+ 2+

d− 2

(d− 1)d(d+ 2)

). (6.62)

Its expansion in d = 4− ε can be seen to match (6.15), in particular reproducing the

extra propagating scalar present in the GNY description.

6.2.4 Padé approximants

To obtain estimates for the CFT observables in d = 3, we will use two-sided" Padé

approximants that combine information from the 4− ε and 2+ ε expansions. Namely,

we consider the rational approximant Pade[m,n] =∑mi=0 aid

i

1+∑nj=1 bjd

j , where 2 < d < 4 is the

spacetime dimension, and we x the coecients ai, bj so that its Taylor expansion

near d = 4 and d = 2 agrees with the available perturbative results. Clearly, the

degree" n + m of the approximant is bound by how many terms in the ε-expansion

are known. Such approximants may be derived for any nite N , and it is useful to

compare their large N behavior as a function of d to the 1/N -expansion results listed

in the previous section. When several approximants Pade[m,n] are possible for the

same quantity, we use such comparisons to large N results to choose the one which

appears to work best.

129

For ∆ψ, ∆σ and ∆σ2 we know the 4 − ε expansion to order ε2, and the 2 + ε

expansion to order ε3. This allows to use Padé approximants with m+n = 6. For ∆ψ

and ∆σ, we nd that Pade[4,2] has no poles in 2 < d < 4 and for large N is in good

agreement with the results (6.51) and (6.52). For ∆σ2 , we perform the Padé on the

critical exponent ν−1 = d−∆σ2 , and then translate to ∆σ2 at the end. In this case,

we nd that Pade[1,5] is the only approximant with no poles, and it matches well to

the large N result. The resulting estimates in d = 3 for these scaling dimensions are

given in Table 6.1. In gure 6.2, we plot our 3d estimates for ∆ψ and ∆σ, compared to

the large N curve obtained from (6.57) by eliminating N to express ∆σ as a function

of ∆ψ. The N = 1 values, which correspond to the N = 1 SUSY xed point, are

obtained in Section 6.4.

N 3 4 5 6 8 20 100∆ψ (Pade[4,2]) 1.066 1.048 1.037 1.029 1.021 1.007 1.0013∆σ (Pade[4,2]) 0.688 0.753 0.798 0.829 0.87 0.946 0.989∆σ2 (Pade[1,5]) 2.285 2.148 2.099 2.075 2.052 2.025 2.008

F/(NFf ) (Pade[4,4]) 1.091 1.060 1.044 1.034 1.024 1.008 1.0014

Table 6.1: Estimates of scaling dimensions and sphere free energy at the d = 3interacting xed point of the GN model.

For the sphere free energy of the interacting CFT, F , we nd it convenient to

perform the Padé approximation on the quantity

f(d) = F −NFf , (6.63)

which is essentially the interacting part of the free energy in the GN description, but

it includes the contribution of a free scalar from the GNY point of view. Using the

results (6.31) and (6.49), we can use Padé approximants with n + m = 8, and we

nd that Pade[4,4] has no poles and agrees well with the large N result (6.58). The

resulting d = 3 estimates for F , normalized by the free fermion contribution NFf ,

130

N=1N=1

Large N

SUSY

1.02 1.04 1.06 1.08 1.10Δψ

0.6

0.7

0.8

0.9

1.0

Δσ

Figure 6.2: Padé estimates in d = 3 of ∆σ versus ∆ψ for N = 1, 2, 3, 4, 5, 6, 8, 20,compared to the large N results (6.57). The N = 1 value corresponds to the SUSYxed point discussed in Section 6.4. The black dotted line is the SUSY relation∆σ = ∆ψ − 1/2.

are given in Table 6.1 for a few values of N . In Figure 6.3, we also plot the result of

the constrained Padé approximants for F −NFf as a function of 2 < d < 4 for a few

values of N , showing that they approach well the analytical large N formula (6.58).

N=4

N=6

N=20

N=100

Large N

2.5 3.0 3.5 4.0d

0.005

0.010

0.015

0.020

0.025

0.030

F˜-N F

˜f

Figure 6.3: Padé estimates of F −NFf in 2 < d < 4 compared to the large N result(6.58).

We can use our estimates to make some tests of the F -theorem. In the GNY

description, we can ow to the critical theory from the free UV xed point of N

fermions plus a scalar, while in the GN description one can ow to the critical theory

131

to the free fermions. Then, the F -theorem implies the inequalities

NFf + Fs > F > NFf . (6.64)

We veried that our estimates satisfy these inequalities for all values of N . As an

example, for N = 4 we get FGN/(NFf + Fs) ≈ 0.93 and FGN/(NFf ) ≈ 1.06. In

the GNY description, we also see that we can ow to the critical GN point from the

decoupled product of the Ising CFT and N free fermions. This implies

NFf + FIsing > F . (6.65)

Using the estimates for FIsing derived in [135], we have checked that this inequality

indeed holds. Using the Padé approximants as a function of 2 < d < 4, we can also

verify that both (6.64) and (6.65) are satised, in terms of F , in the whole range of

d. This is in agreement with the generalized F -theorem" [135].

Finally, we discuss N = 2, which is a special case where the β-function in d = 2

vanishes exactly; therefore, the theory has a line of xed points. For N = 2 we

cannot apply the strategy described above because the 2+ε expansions (6.37) become

singular. Directly in d = 2, the GN model is equivalent to the Thirring model for a

single 2-component Dirac fermion which can be solved via bosonization and has a line

of xed points; the dimensions of ψ and σ = ψψ depend on the interaction strength.

Therefore, for these operators we can only perform the one-sided" Pade[1,1] on the

4−ε expansions. In d = 2 it is known that ∆σ2 = 2 (the operator is exactly marginal)

and c = 1; we impose these boundary conditions on the Padé approximants. Then

the results in d = 3 are

∆N=2ψ ≈ 1.076 , ∆N=2

σ ≈ 0.656 , ∆N=2σ2 ≈ 1.75 , (6.66)

132

and for the sphere free energy

FN=2 ≈ 0.254 ≈ 1.16(2Ff ) ≈ 0.9(2Ff + Fs) , (6.67)

in agreement with the F -theorem.

6.3 The Nambu-Jona-Lasinio-Yukawa Model

Using the results of [127, 128, 129, 130, 131], we have found the following β-functions

for the NJLY model (6.5) up to two loops:

β1 = − ε2g1 +

1

(4π)2

(N

2+ 2

)g3

1 +1

(4π)4

(−8

3g3

1g2 +1

9g1g

22 + (7− 3N)g5

1

),

β2 = −εg2 +1

(4π)2

(10

3g2

2 + 2Ng2g21 − 12Ng4

1

)− 1

(4π)4

(20

3g3

2 − 96Ng61 − 2Ng2g

41 +

10

3Ng2

1g22

).

(6.68)

The one-loop terms above agree with the Nb = 1 case of the results in [113]. Solving

(6.68), we nd the following xed point:

(g∗1)2

(4π)2=

ε

N + 4+−N2 + 448N − 1096 + (N + 76)

√N2 + 152N + 16

100(N + 4)3ε2 +O(ε3) ,

g∗2(4π)2

=3(√N2 + 152N + 16−N + 4)

20(N + 4)ε+

9

500(N + 4)3√N2 + 152N + 16

×(

3N4 + 114N3 + 764N2 − 26192N + 1280

− (3N3 − 114N2 − 1932N − 320)√N2 + 152N + 16

)ε2 +O(ε3) .

(6.69)

To get a positive solution for g∗2 we have picked the + sign for the square root.

The other choice of sign gives another xed point which is presumably unstable. In

addition, there is clearly a xed point with g∗1 = 0 and g∗2 corresponding to the O(2)

133

Wilson-Fisher xed point. The solution (6.69) yields an IR stable xed point for all

values of N .

The scaling dimensions of the elds are found to be

∆ψ =3− ε

2+

1

(4π)2g2

1 −1

4(4π)4(3N + 2)g4

1 ,

∆φ = 1− ε

2+

1

2(4π)2Ng2

1 +1

(4π)4

(1

9g2

2 −3

2Ng4

1

). (6.70)

At the xed point (6.69) these give:

∆ψ =3

2− N + 2

2(N + 4)ε+−76N2 + 98N − 1296 + (N + 76)

√N2 + 152N + 16

100(N + 4)3ε2 +O(ε3) ,

∆φ = 1− 2

N + 4ε+

56N2 − 498N + 16 + (19N + 4)√N2 + 152N + 16

50(N + 4)3ε2 +O(ε3) .

(6.71)

The NJLY has two types of operators quadratic in the scalar elds: the U(1)

invariant operator φφ, and the charged operators φ2 and φ2. The one-loop scaling

dimension of φφ was determined in [113]. Using [131], we nd up to two loops

∆φφ = d− 2 +4

3(4π)2g2 −

4

3(4π)4

(g2

2 +Ng2g21

)+ 2γφ ,

∆φ2 = d− 2 +2

3(4π)2g2 −

2

3(4π)4

(4

3g2

2 +Ng2g21 − 6Ng4

1

)+ 2γφ . (6.72)

At the xed point this gives

∆φφ =2 +

√N2 + 152N + 16−N − 16

5(N + 4)ε+

1

250(N + 4)3√N2 + 152N + 16

×(

(17N3 + 104N2 + 1252N + 1120)√N2 + 152N + 16

− 17N4 − 4646N3 + 2304N2 − 187712N + 4480)ε2 +O(ε3) (6.73)

134

and

∆φ2 =2 +

√N2 + 152N + 16−N − 36

10(N + 4)ε+

1

125(N + 4)3√N2 + 152N + 16

×(

(3N3 + 571N2 + 688N + 240)√N2 + 152N + 16

− 3N4 − 924N3 + 7806N2 − 47688N + 960)ε2 +O(ε3) . (6.74)

The 4− ε expansion of CT in the NJLY model proceeds similarly to that for the

GNY model presented in [119] and reviewed in section 6.2. In the NJLY model there

are two real scalar elds, and we need to replace Tσ by Tσ1 + Tσ2 . It is not hard

to check that each diagram contributing to the term ∼ g21 picks up a factor of 2

compared to the GNY model, since each of the internal scalar lines can be either φ1

or φ2. Thus, we nd

CT = NCT,f + 2CT,s −5N(g∗1)2

6(4π)2=

d

S2d

(N

2+

2

d− 1− 5Nε

6(N + 4)

). (6.75)

6.3.1 Free energy on S4−ε

The calculation of F for the NJLY model follows the same steps as the one for the

GNY model discussed earlier. The integrals are also nearly identical, except for

combinatorics factors due to the fact that we have two scalar elds, one of which has

iγ5 coupling.

The perturbative expansion of the free energy is given by

F = NFf + 2Fs −1

2g2

1,0

∫ddxddy

√gx√gy〈O1(x)O1(y)〉0

− 1

2!(4!)2g2

2,0

∫ddxddy

√gx√gy〈O2(x)O2〉0

− 1

4!g4

1,0

∫ddxddyddzddw

√gx√gy√gz√gw〈O1(x)O1(y)O1(z)O1(w)〉0 + δFb ,

(6.76)

135

where we dened the operators O1 = ψj(φ1 + iγ5φ2)ψj and O2 = (φ21 + φ2

2)2, and

δFb = b0

∫ddxE is the Euler term. Evaluating the correlation functions above, we

nd

F = NFf+2Fs−1

22g2

1,0NCφC2ψI2

(3

2d−2

)+

1

2 · (4!)264g2

2,0C4φI2(2d−4)+

1

4!g4

1,0I4+δFb ,

(6.77)

where I2 is given in (A.12) and I4 corresponds to the integrated 4-point function of

O1, for which we nd

I4 =N(N + 4)

(4π)4ε+N (19N + 72 + 6(N + 4)(γ + log(4πR2)))

6(4π)4+O(ε) . (6.78)

Now replacing the bare couplings with the renormalized ones

g1,0 = µε2

(g1 +

N + 4

32π2

g31

ε+ . . .

), g2,0 = µε

(g2 + . . .

)(6.79)

we nd that all poles cancel as they should. As explained in the GNY and GN

calculations, in order to obtain the correct expression for F at the IR xed point in

d = 4− ε we need to include the eect of the Euler term. Using an improved version

of the result from [130, 134], adapted to the presence of γ5 in the vertices, we get:

βb = εb− 11N/4 + 2

360(4π)2− 1

(4π)8

1

1296

(Ng2

1(24g21g2 − g2

2 + 9g41(3N − 7))

). (6.80)

From this we can solve for the xed point value b∗ with the couplings given in (6.69),

and we nd that the Euler term contributes

δFb =N(−1096 + 76

√N2 + 152N + 16 +N(448−N +

√N2 + 152N + 16))

7200(N + 4)3ε2 .

(6.81)

136

Putting this together with the integrals in (6.77), we nally nd in terms of F :

F = NFf + 2Fs −Nπε2

48(N + 4)− 1

14400(N + 4)3

(149N3 + 2372N2 + 1312N + 64

+(N2 + 152N + 16

)√N2 + 152N + 16

)πε3 +O(ε4) . (6.82)

6.3.2 2 + ε expansions

In this section we consider the theory in 2 + ε dimensions with 4-fermion interactions

which respect the U(1) chiral symmetry. We begin with the action in Euclidean space

of the form [106]

SSPV = −∫ddx(ψi/∂ψ

i +1

2gS(ψiψ

i)2 +1

2gP (ψiγ5ψ

i)2 +1

2gV (ψiγµψ

i)2), (6.83)

where i = 1, . . . 2Nf is the number of two-component spinors, and γ0 = σ1, γ1 = σ2

and γ5 = −iγ0γ1 = σ3. For gV = 0 and gS = −gP this reduces to the well-known chiral

Gross-Neveu model [25] in d = 2. However, as we will see below, for our purposes it

is not consistent to set gV = 0 the corresponding operator respects the U(1) chiral

symmetry and gets induced.

The one-loop beta-functions and anomalous dimension of the ψ eld were found

in [106] using MS scheme and read

βS = εgS −1

π

(1

2(N − 2)g2

S − gPgS − 2gV (gS + gP ))

+ . . . ,

βP = εgP +1

π

(1

2(N − 2)g2

P − gPgS + 2gV (gS + gP ))

+ . . . , (6.84)

βV = εgV +1

πgPgS + . . . ,

137

and

∆ψ =1

2+

1

2ε+

N

4(2π)2

(g2P + g2

S + 2g2V

)− 1

4(2π)2

(4gV (gS − gP ) + (gS + gP )2

).

(6.85)

We note that at the leading order in the 2 + ε expansion the evanescent operators do

not appear [106, 144, 145]. One of the UV xed points of (6.84) is10

g∗S = −g∗P =N

2g∗V =

2πε

N, (6.86)

which corresponds to the SU(2Nf ) Thirring model [106, 146]. Indeed using the

relation for the SU(2Nf ) generators (T a)ij(Ta)kl = 1

2(δilδ

kj − 2

Nδijδ

kl ) and the Fierz

identity in d = 2 (γµ)αβ(γµ)γδ = δαδδβγ − (γ5)αδ(γ5)βγ one nds

(ψψ)2 − (ψγ5ψ)2 +2

N(ψγµψ)2 = −2(ψγµT

aψ)2 . (6.87)

So the action for this model is

SSU(2Nf )Thirring = −∫ddx(ψi/∂ψ

i +1

2g((ψiψ

i)2 − (ψiγ5ψi)2 +

2

N(ψiγµψ

i)2)),

(6.88)

with the beta-function and anomalous dimension

βg = εg − N

2πg2 +O(g3), ∆ψ =

1

2+

1

2ε+

N2 − 4

8π2Ng2 +O(g3) . (6.89)

It is plausible that this model is the continuation of the NJLY model (6.5) to d = 2+ε.

One nds that the UV xed point is g∗ = 2πε/N , and the critical anomalous dimension

10Equations (6.84) have four dierent non-trivial xed points. Two of them correspond to the GNand SU(2Nf ) Thirring models.

138

reads

∆ψ =1

2+

1

2ε+

1

2

( 1

N− 4

N3

)ε2 +O(ε3) . (6.90)

Also one nds that the dimension of the quartic operator is

∆φφ = 2 + ε+ β′(g∗) = 2 +O(ε2) . (6.91)

We can also calculate the free energy of this model. To order g2, we have:

F = NFf −1

2!

(g0

2

)2

S2 +O(g30) , (6.92)

where Ff is the free fermion contribution and

S2 =

∫dx1dx2

√gx1√gx2〈O(x1)O(x2)〉conn0 , (6.93)

where, under the Thirring description (6.87), O = −2(ψγµTaψ)2. Going through the

combinatorics, we nd

S2 = 4(N2 − 4)C4ψI2(2d− 2) . (6.94)

Evaluating this in d = 2 + ε and plugging in the xed point value (6.86) we nally

nd

F = NFf +(N2 − 4)πε3

24N2+O(ε4) . (6.95)

139

6.3.3 Large N expansions

For the NJL model, the 1/N expansions of operator dimensions again assume the

general form (6.50), where now [110, 147]

γψ,1 = − 2Γ(d− 1)

Γ(d2− 1)Γ(1− d

2)Γ(d

2+ 1)Γ(d

2), γψ,2 = 2γ2

ψ,1

(Ψ(d) +

4

d− 2+

1

d

),

(6.96)

γφ,1 = −2γψ,1, γφ,2 = −2γψ,2 +4d2(d2 − 5d+ 7)

(d− 2)3γ2ψ,1 (6.97)

and

γφφ,1 = 4(d− 1)γψ,1 , (6.98)

γφφ,2 = −8γ2ψ,1

(3d2(3d− 8)Θ(d)

4(d− 4)(d− 2)− (d− 3)d2 (Φ(d) + Ψ2(d))

(d− 4)(d− 2)+

4(d− 2)d

(d− 4)2γψ,1+

4

d− 2+

1

d

+(d− 4)d2

d− 2− (d− 3)2d2

(d− 4)2(d− 2)+ Ψ(d)

((d− 3)d2 (d2 − 8d+ 20)

(d− 4)2(d− 2)2− d2 + 1

)).

(6.99)

There is a considerable similarity between these results and the corresponding results

for the GN model. For example, γψ,1 in the NJL model is 2 times that in the GN

model. This is because the NJL lagrangian (6.2) contains two separate double-trace

operators, and hence there are two auxiliary elds, φ1 and φ2, that couple to the

fermions. A similar factor of 2 appears in various other quantities.

140

When (6.97) and (6.99) are expanded in d = 4−ε and d = 2+ε, they are consistent

with our ε-expansion results.11 Setting d = 3 in these equations gives [110, 147]

∆ψ = 1 +8

3π2N+

1280

27π4N2= 1 +

0.2702

N+

0.4867

N2,

∆φ = 1− 16

3π2N+

4352

27π4N2= 1− 0.5404

N+

1.6547

N2,

∆φφ = 2 +64

3π2N− 128(364 + 27π2)

27π4N2= 2 +

2.1615

N− 30.684

N2.

(6.100)

Turning to the sphere free energy, we can obtain the O(N0) result for F by simply

doubling the δF from eq. (3.8) of [78]. Therefore, we have

F = NFf +2

Γ (d+ 1)

∫ d2−1

0

du u sin(πu)Γ

(d

2+ u

)Γ

(d

2− u)

+O(1/N) . (6.101)

The 4− ε expansion of this agrees with the large N expansion of (6.82), i.e.

F = NFf + 2Fs −(πε2

48+πε3

96

)+

(πε2

12− πε3

18

)1

N−(πε2

3− 17πε3

36

)1

N2+O(1/N3),

(6.102)

and its 2 + ε expansion agrees with the large N expansion of (6.95), which yields

F = NFf +πε3

24− πε3

6N2+O(1/N4) . (6.103)

For CT , the presense of the extra scalar eld compared to the GN case [119] again

poses no diculty. After some simple excercise commuting γ5, we conclude that all

the diagrams in [78] contributing to CT in the GN model should receive a factor of 2

11 The form of (6.91) agrees with the large N result (6.99), but we haven't compared the coecientof ε2.

141

due to the presence of two scalar elds. Hence, we nd for the NJL model

CT = NCT,f

(1 +

CT1

N+O(1/N2)

),

CT1 = −4γψ,1

(Ψ(d)

d+ 2+

d− 2

(d− 1)d(d+ 2)

), (6.104)

i.e. the correction CT1 is just twice the corresponding term in the GN model (note

that γψ,1 in the NJL model is twice that of the GN model). This result is, in particular,

consistent with the fact that in the limit d → 4 we expect CT1 to reproduce the

contribution of two free scalar elds. Its 4 − ε expansion can be also seen to agree

with (6.75).

6.3.4 Padé approximants

Following the same methods as described in Section 6.2.4, we now use the 4− ε and

2 + ε expansions12 derived in the previous sections to obtain rational approximants

of scaling dimensions and sphere free energy at the NJL xed point in 2 < d < 4,

for N > 2 (the case N = 2, which displays the emergent supersymmetry, is treated

separately in Section 6.4). The results in d = 3 are given in Table 6.2, indicating which

approximants was chosen in each case. In gure 6.4, we also plot our 3d estimated for

∆ψ and ∆φ, compared to the large N results. From the expression for ∆φ in (6.100),

it appears that the expansion does not converge well already at O(1/N2), therefore

we have just used (6.100) to order 1/N to produce the plot below.

For the sphere free energy F , we again nd it convenient to perform the Padé

approximation on the quantity f(d) = F −NFf , which corresponds to the interacting

part of the NJLY free energy plus the contribution of two free scalars. In Figure 6.3,

we plot the resulting Padé approximants as a function of 2 < d < 4 for a few values

of N , showing that they approach well the analytical large N formula (6.101).

12For ∆φ, we use the boundary condition ∆φ = 1 +O(ε) in d = 2 + ε.

142

N 4 6 8 10 12 20 100∆ψ (Pade[3,2]) 1.074 1.054 1.041 1.033 1.027 1.016 1.0029∆φ (Pade[2,1]) 0.807 0.870 0.903 0.923 0.937 0.962 0.992∆φφ (Pade[3,1]) 2.018 2.041 2.055 2.062 2.064 2.06 2.022

F/(NFf ) (Pade[5,2]) 1.109 1.064 1.045 1.034 1.028 1.016 1.0029

Table 6.2: Estimates of scaling dimensions and sphere free energy at the d = 3interacting xed point of the NJL model.

Large N

1.00 1.02 1.04 1.06 1.08Δψ0.70

0.75

0.80

0.85

0.90

0.95

1.00

Δϕ

Figure 6.4: Padé estimates in d = 3 of ∆σ versus ∆ψ for N = 4, 6, 8, 10, 12, 20, 100,compared to the large N results (6.100).

N=4

N=6

N=20

N=100

Large N

2.5 3.0 3.5 4.0d

0.01

0.02

0.03

0.04

0.05

F˜-N F

˜f

Figure 6.5: Padé estimates of F −NFf in 2 < d < 4 compared to the large N result(6.101).

We can again use our estimates to test the d = 3 F -theorem and its proposed

generalization in 2 < d < 4 in terms of F . We nd that in the whole range of

143

dimensions, the inequalities

NFf + 2Fs > F > NFf (6.105)

hold, in accordance with the conjectured generalized F -theorem [78, 135]. Using the

results in [135] for the free energy of the O(2) Wilson-Fisher model, we have also

checked that

NFf + FO(2) > F , (6.106)

which is consistent with the fact that the theory can ow from the xed point con-

sisting of free fermions decoupled from the O(2) model, to the IR stable NJL xed

point.

6.4 Models with Emergent Supersymmetry

6.4.1 Theory with 4 supercharges

A well-known supersymmetric theory with 4 supercharges is the Wess-Zumino model

of a single chiral supereld Φ with superpotential W ∼ λΦ3. In d = 4 the model

is classically conformally invariant, but it has a non-vanishing beta function and is

expected to be trivial in the IR. Continuation of this model to lower dimensions

(dened such that the number of supercharges is xed in 2 ≤ d ≤ 4) was discussed

in [78, 148, 149]. In d = 3, one nds the N = 2 theory of a single chiral supereld (a

complex scalar and a 2-component Dirac fermion) with cubic superpotential, which

ows to a non-trivial CFT in the IR [150]. In d = 2, the model matches onto the (2, 2)

supersymmetric CFT with c = 1, which is the k = 1 member of the superconformal

discrete series with c = 3kk+2

.

144

The component Lagrangian of the WZ model in 4d reads

LWZ =1

2(∂µφ1)2 +

1

2(∂µφ2)2 +

1

2ψ /∂ψ +

λ

2√

2ψ(φ1 + iγ5φ2)ψ +

λ2

16(φφ)2 , (6.107)

where ψ is a 4-component Majorana fermion and φ = φ1 + iφ2. The beta function of

the model in d = 4− ε is [151]

βWZ = − ε2λ+

3

2

λ3

(4π)2− 3

2

λ5

(4π)4+

15 + 36ζ(3)

8

λ7

(4π)6+O(λ9) , (6.108)

from which one nds an IR xed point with λ2∗ = 16π2ε

3+ O(ε2). The dimension of

the chiral operator φ at the xed point is determined by its R-charge to be

∆φ =d− 1

2Rφ =

d− 1

3. (6.109)

One also has the exact result ∆φ2 = ∆φ + 1, since the operator φ2 is obtained from φ

by acting twice with the supercharges (this is because, due to cubic superpotential,

one has the relation Φ2 = 0 in the chiral ring). It also follows from supersymmetry

[121, 123] that the dimension of the operator φφ at the xed point is given by

∆φφ = d− 2 + β′(λ∗) = 2− 1

3ε2 +

1 + 12ζ(3)

18ε3 +O(ε4) , (6.110)

which agrees with the explicit three-loop calculation of [123]. In performing Padé

extrapolation of this result to d = 3, we have found that Padé[1,2] and Padé[2,1] give

answers close to each other. Their average is ≈ 1.909, which is very close to the

value ≈ 1.91 reported using numerical bootstrap studies [148, 149]. We can also take

into account the fact that in d = 2 the dimension of φφ should approach 2. Since

it also approaches 2 in d = 4, it is not a monotonic function of d, which makes

Padé extrapolation dicult. If we instead perform a two-sided" extrapolation of

145

ν−1 = d − ∆φφ, and then return to ∆φφ in d = 3, then we nd ≈ 1.94. This is

somewhat further from the numerical bootstrap estimate.

Since in d = 4 the WZ model includes a complex scalar and a 4-component Majo-

rana fermion (i.e. one half of a Dirac fermion), one would expect it to correspond to

Nf = 1/2 and N = 2 in the NJLY model [121, 122]. Note that the NJLY Lagrangian

(6.5), specialized to the case of a single Majorana fermion, coincides with the WZ

Lagrangian (6.107) provided13

3g21 = g2 =

3

2λ2 . (6.111)

Indeed, setting N = 2 in the result for the xed point couplings (6.8), we nd:

(g∗1)2

(4π)2=

1

6ε+

1

18ε2 +O(ε3) ,

g∗2(4π)2

=1

2ε+

1

6ε2 +O(ε3) .

(6.112)

This is precisely consistent with the relation (6.111), and gives evidence of the emer-

gent supersymmetry in the N = 2 NJLY model. Note that for this value of N , the

chiral U(1) symmetry of the NJLY model becomes the U(1) R-symmetry of the WZ

model.14 Further evidence can be found by setting N = 2 in the 4− ε expansions of

the operator dimensions (6.70) and (6.74), which give

∆ψ =3

2− ε

3, ∆φ = 1− ε

3, ∆φ2 = 2− ε

3, (6.113)

in agreement with the supersymmetry. In particular, the fact that the O(ε2) terms

vanish is consistent with the exact result (6.109), and we also see ∆φ2 = ∆φ + 1 as

13One should rescale ψ → ψ/√

2 in (6.5) to get a canonical kinetic term when the fermion isMajorana.

14For a gauge theory in 1 + 1 dimensions which exhibits emergent supersymmetry, see [152]. Inthat case a global U(1) symmetry turned into the U(1)R-symmetry of the (2, 2) supersymmetric IRCFT.

146

discussed above. Furthermore, setting N = 2 in (6.73), we nd ∆φφ = 2−ε2/3+O(ε3),

in agreement with the result (6.110) in the WZ model.

It is also interesting to look at the sphere free energy. Setting N = 2 in the 4− ε

expansion (6.82) of F , we get

FN=2 = 2Fs + 2Ff −πε2

144− πε3

162+O(ε4) . (6.114)

This precisely agrees with the expansion of (5.23) in [78], which was derived using a

proposal for supersymmetric localization in continuous dimension. We note that the

curvature term (6.81) contributes atO(ε3) order to F : δFb = − πε3

1296. This contribution

is crucial for agreement with [78]. Thus, (6.114) provides a nice perturbative test of

the exact formula for F as a function of d proposed in [78] for the Wess-Zumino

model.

In d = 3, the result obtained from localization [153] yields FW=Φ3 ≈ 0.290791. In

[135], the value of F for the O(2) Wilson-Fisher xed point in d = 3 was estimated

to be FO(2) ≈ 0.124. Using also the value Ff = 18

log(2) + 3ζ(3)16π2 in d = 3 [50], we see

that

2Ff + FO(2) ≈ 0.343 > FW=Φ3 , (6.115)

in agreement with the RG ow depicted in Figure 6.1.

Finally, we discuss CT of the N = 2 SCFT with superpotential W ∼ λΦ3. Its

exact value in d = 3 has been determined using the supersymmetric localization

[64, 154]:CTCUVT

=16(16π − 9

√3)

243π≈ 0.7268 , (6.116)

where CUVT = 4CT,s is the value for the free UV theory of two scalars and one two-

component Dirac fermion. Let us compare this with a Padé extrapolation of the ratio

147

CT/CT,s using the boundary conditions

CTCT,s

=

1 in d = 2 ,

5− 116ε+O(ε2) in d = 4− ε ,

(6.117)

where the 4 − ε expansion was obtained by setting N = 2 in (6.75). The Pade[1,1]

approximant with these boundary conditions is

CTCT,s

=49d− 76

d+ 20, (6.118)

which in d = 3 gives CT/CT,s = 71/23 ≈ 3.087. Comparing to the free UV CFT,

this result implies CIRT /C

UVT ≈ 0.77. This is not far from the exact result (6.116),

demonstrating that the Padé approach works quite well. It would be useful to know

the next order in the 4− ε expansion, which may improve the agreement.

6.4.2 Theory with 2 supercharges

It has been suggested that in d = 3 there exists a minimal N = 1 superconformal

theory containing a 2-component Majorana fermion ψ [121, 124, 125, 118, 126]. This

theory must also contain a pseudoscalar operator σ, whose scaling dimension is related

to that of ψ by the supersymmetry,

∆σ = ∆ψ −1

2. (6.119)

Some evidence for the existence of this N = 1 supersymmetric CFT was found using

the conformal bootstrap [118].

To describe the theory in d = 3, one can write down the Lagrangian [121, 124,

125, 118]

LN=1 =1

2(∂µσ)2 +

1

2ψ 6 ∂ψ +

λ

2σψψ +

λ2

8σ4 . (6.120)

148

This model has N = 1 supersymmetry in d = 3; the eld content can be packaged

in the real supereld Σ = σ + θψ + 12θθf , and the interactions follow from the cubic

superpotential W ∼ λΣ3.15 It is natural to expect that this model ows to a non-

trivial N = 1 SCFT in the IR. Note that the theory cannot be described as the UV

xed point of a lagrangian for a Majorana fermion with quartic interaction, because

the term (ψψ)2 vanishes for a 2-component spinor.

The theory (6.120) is super-renormalizable in d = 3, and one may attempt its 4−ε

expansion [121]. To formulate a Yukawa theory in d = 4, one strictly speaking needs

a 4-component Majorana fermion, which corresponds to the GNY model with N = 2.

However, the GNY description may be formally continued to N = 1. A sign of the

simplication that occurs for this value is that√N2 + 132N + 36, which appears in

the 4− ε expansions (6.10), (6.12), equals 13 for N = 1. For this value of N , we nd

that the xed point couplings in (6.8) become

(g∗1)2

(4π)2=

1

7ε+

3

49ε2 +O(ε3) ,

g∗2(4π)2

=3

7ε+

9

49ε2 +O(ε3) .

(6.121)

This is consistent with the exact relation 3g21 = g2 in the SUSY model. Indeed the

GNY Lagrangian (6.4), formally applied to the case of a single 2-component Majorana

fermion, coincides with (6.120) when 3g21 = g2 = 3λ2. The result (6.121) gives a two-

loop evidence that the non-supersymmetric GNY model with N = 1 ows at low

energies to a supersymmetric xed point.

In [121] it was found using one-loop calculations that ∆σ = 1− 3ε7

= ∆ψ − 12. Let

us check that the supersymmetry relation (6.119) continues to hold at order ε2. Using

15To obtain the component Lagrangian (6.120), one should eliminate the auxiliary eld f usingits equation of motion f ∼ λσ2.

149

(6.10) with N = 1, we indeed nd

∆σ = 1− 3

7ε+

1

49ε2 +O(ε3) , (6.122)

∆ψ =3

2− 3

7ε+

1

49ε2 +O(ε3) . (6.123)

The dimensions of operators σ2 and σψ should also be related by the supersymmetry,

∆σ2 = ∆σψ − 12. Since σψ is a descendant, we also have ∆σψ = ∆ψ + 1. Thus, the

supersymmetry relation assumes the form [125]

∆σ2 = ∆ψ +1

2= ∆σ + 1 . (6.124)

Substituting N = 1 into (6.12) we nd

∆σ2 = 2− 3

7ε+

1

49ε2 +O(ε3) , (6.125)

so that the supersymmetry relation (6.124) holds to order ε2. These non-trivial checks

provide strong evidence that the continuation of the GNY model to N = 1 ows to a

superconformal theory to all orders in the 4− ε expansion.

If we apply the standard Padé[1,1] extrapolation, we nd

∆σ =8d− 11

25− d , (6.126)

which in d = 3 gives ∆σ = 1322≈ 0.59. 16 This is close to the estimate of ∆σ obtained

using the numerical bootstrap [118]. It is the value ∆σ ≈ 0.582 where the boundary

of the excluded region touches the SUSY line ∆σ = ∆ψ − 1/2.

16We note that our estimate ∆σ2 ≈ 1.59 in the N = 1 theory is below 2, just like the estimate(6.66) in the N = 2 theory. This is in contrast with the large N behavior (6.57) where ∆σ2 > 2.Thus, perhaps not surprisingly, the theories with N = 1, 2 are, in some respects, rather far from thelarge N limit.

150

It is also important to know how the theory behaves when continued to d = 2.

It is plausible that the d = 2 theory has N = 1 superconformal symmetry, and

the obvious candidate is the tri-critical Ising model [124, 126], which is the simplest

supersymmetric minimal model [73, 155]. The Padé extrapolation (6.126) gives ∆σ ≈

0.217, which is quite close to the dimension 1/5 of the energy operator in the tri-

critical Ising model. This provides new evidence that the GNY model with N = 1

extrapolates to the tri-critical Ising model in d = 2; in gure 6.6 we show how the

operator spectrum matches with the exact results in d = 2. Imposing the boundary

condition that ∆σ = 1/5 in d = 2 enables us to perform a two-sided" Padé estimate.

The resulting value in d = 3 is ≈ 0.588, which is very close to that following from

(6.126). The agreement with the bootstrap result ∆σ ≈ 0.582 is excellent.

3

∆

d

σ

σ2

ψψ, σ3

ψ, ψ

432

Φ( 110

, 110

)

Ψ( 35, 110

), Ψ( 110

, 35)

Φ( 35, 35)

L−1L−1Φ

SS = X( 32, 32)

1

2

σψ, σψ

L−1Ψ, L−1Ψ

≈ 0.59

≈ 1.09

≈ 1.59

≈ 2.09

≈ 2.59

≈ 2.79

M4,5 N = 1 SCFT

Figure 6.6: Qualitative picture of the interpolation of operator dimensions from thed = 4 free theory to the tri-critical Ising model M(4, 5) in d = 2, indicating ourestimated values for the N = 1 SCFT in d = 3. In 2 ≤ d < 4, the operators σψ, anda linear combination of σ3 and ψψ, are expected to be conformal descendants of ψand σ respectively.

From the β-functions (6.7) for the GNY model we may also deduce the dimensions

of two primary operators that are mixtures of σ4 and σψψ. They are determined by

151

the eigenvalues λ1, λ2 of the matrix ∂βi/∂gj. At the xed point for N = 1 we nd

∆1 = d+ λ1 = 4− 3

7ε2 +O(ε3) ,

∆2 = d+ λ2 = 4 +6

7ε− 95

49ε2 +O(ε3) , (6.127)

Since ∆1 corresponds to the θθ component of the supereld Σ3, we know that

∆σ3 = ∆1 − 1 = 3− 3

7ε2 +O(ε3) . (6.128)

As a check, we may set N = 1 in (6.14) and see that the order ε term in ∆σ3 indeed

vanishes. It is further possible to argue that this primary operator, when continued

to d = 2, matches onto the dimension 3 operator in the tri-critical Ising model, i.e.

the product of left and right supercurrents. This implies that ∆σ3 is not monotonic

as a function of d and is likely to be somewhat smaller than 3 in d = 3.17 Simply

setting ε = 1 in (6.128) gives ∆σ3 ≈ 2.57, but this probably underestimates it. One

way to implement the exact boundary condition ∆σ3 = 3 in d = 2 is to extrapolate

the quantity ∆ = (∆σ3 − 3)/(d− 2), instead of ∆σ3 itself [22]. This approach would

yield the estimate ∆σ3 ≈ 2.79 in d = 3. This slightly relevant operator is parity odd

in d = 3, while the dimension ∆1 corresponds to an irrelevant parity even operator.

Similarly, we can perform extrapolation of F and check if the d = 2 value is close

to πc/6 where c = 7/10 is the central charge of the tri-critical Ising model. Setting

N = 1 in (6.31), we have

F = Fs + Ff −π

6

(ε2

112+ε3

98+O(ε4)

). (6.129)

Performing a Padé approximation of the quantity f(d) = F − Ff , we nd that the

average of the standard Padé[2,1] and Padé[1,2] approximants yields F /Fs ≈ 0.68,

17This is in line with a statement in [118] that a kink lies on the SUSY line when ∆σ3 is slightlysmaller than 3.

152

which is quite close to c = 7/10. Therefore, in order to get a better estimate in d = 3,

it makes sense to impose it as an exact boundary condition in d = 2. Following this

procedure, and taking an average of the Padé approximants with n+m = 4, we nd

the d = 3 estimate

F ≈ 0.158 . (6.130)

In the UV, we have the free CFT of one scalar and one Majorana fermion, which

has FUV = 14

log 2 ≈ 0.173, and therefore we nd F IR/FUV ≈ 0.91. This is a check

of the F -theorem for the ow from the free to the interacting N = 1 SCFT. It is

also interesting to compare the value of F at the SUSY xed point to the decoupled

Ising xed point in Figure 6.1. A plot comparing F − Ff with FIsing, which was

obtained in [135], is given in Figure 6.7. It shows that F < Ff + FIsing in the whole

range 2 < d < 4, in agreement with the generalized F -theorem [78, 135] and the

expectation that the SUSY xed point is IR stable.

F˜Ising

F˜N=1 - F

˜fer

2.0 2.5 3.0 3.5 4.0 d

0.05

0.10

0.15

0.20

0.25

Figure 6.7: Comparison of F − Ff at the SUSY xed point and FIsing (from [135]) in2 ≤ d ≤ 4.

Finally we consider CT . Its 4 − ε-expansion for general N is given in eq. (6.15),

and we can use it to estimate the CT value at the N = 1 SUSY xed point in d = 3.

We can perform a Padé approximation on the ratio CT/CT,s, where CT,s is the free

153

scalar value, using the boundary conditions

CTCT,s

=

710

in d = 2 ,

52− 19

28ε+O(ε2) in d = 4− ε ,

(6.131)

where we have imposed the d = 2 value corresponding to the tri-critical Ising model.

A Pade[1,1] approximant with these boundary conditions is

CTCT,s

=497d− 728

62d+ 256, (6.132)

which in d = 3 gives CT/CT,s ≈ 1.73. This value lies well within the region allowed

by the bootstrap in g. 8 of [118] (note that ∆ψ ≈ 1.09). Comparing to the free UV

CFT of one scalar and one Majorana fermion, we have found CIRT /C

UVT ≈ 0.86. It

would be useful to know the next order in the 4 − ε expansion in order to obtain a

more precise estimate. Based on the comparison of the Padé for the N = 2 model

with the exact result, we may expect the ε2 correction to reduce CT somewhat in

d = 3.

Let us also include a brief discussion of the non-supersymmetric xed point of the

N = 1 GNY model, marked by the red triangle in gure 6.1. While it has g∗2 < 0

in d = 4− ε, it may become stable for suciently small d. Changing the sign of the

square root in (6.31), we nd the 4 − ε expansion of the sphere free energy at this

xed point:

Fnon−SUSY = Fs + Ff −π

6

(ε2

112+

6875ε3

889056+O(ε4)

). (6.133)

Extrapolating this to d = 2, we estimate cnon−SUSY ≈ 0.78. This is close to central

charge 4/5 of the (5, 6) minimal model. This model includes primary elds of con-

formal weight 1/40 and 21/40, so that one may form a spin 1/2 eld with weights

154

(h, h) = (1/40, 21/40) that could correspond to ψ in the Yukawa theory (such a eld

is not present in the standard modular invariants that retain elds of integer spin

only).18 In d = 3 our extrapolation gives Fnon−SUSY ≈ 0.16. The latter quantity is

bigger than (6.130); therefore, the non-SUSY xed point, if it is stable, can ow to

the SUSY one in d = 3.

It is also interesting to look at the scaling dimensions of σ, ψ and σ2 at the non-

supersymmetric xed point. Changing the sign of the square roots in (6.10) and

(6.11), we nd for N = 1:

∆non−SUSYσ = 1− 3

7ε− 95

6174ε2 +O(ε3) ,

∆non−SUSYψ =

3

2− 3

7ε− 115

37044ε2 +O(ε3) ,

∆non−SUSYσ2 = 2− 22

21ε+

1117

9261ε2 +O(ε3)

(6.134)

Using Pade[1,1] extrapolations to d = 2 we get

∆non−SUSYσ ≈ 0.077 , ∆non−SUSY

ψ ≈ 0.63 , ∆non−SUSYσ2 ≈ 0.297 . (6.135)

These numbers are not far from the corresponding operator dimensions in either the

(5, 6) or the (6, 7) minimal models. For example, in the (5, 6) interpretation the exact

scaling dimension of ψ in d = 2 should be 11/20, while in the (6, 7) it should be 19/28.

The Padé value we nd lies between the two, but the accuracy of the extrapolation

all the way to d = 2 is hard to assess.

The Pade[1,1] extrapolations of (6.134) to d = 3 yield ∆non−SUSYσ ≈ 0.555 and

∆non−SUSYψ ≈ 1.068. Interestingly, these values are not far from the feature at

(∆σ,∆ψ) ≈ (0.565, 1.078), which lies below the supersymmetry line in gure 7 of

18 Another possibility is that the continuation of the non-supersymmetric xed point to d = 2gives the (6, 7) minimal model (we are grateful to the referee for suggesting this). Its central charge6/7 ≈ 0.857 is also not far from the extrapolation of (6.133) to d = 2. The spectrum of the (6, 7)model contains a spin 1/2 operator with weights (h, h) = (5/56, 33/56).

155

[118]. Changing the sign of the square root in (6.14), we also nd

∆non−SUSYσ3 = 3− 13

7ε+O(ε2) , (6.136)

which suggests that in d = 3 the dimension of this parity-odd operator is less than 3.

A higher loop analysis of the operator dimensions is, of course, desirable.

156

Appendix A

Useful integrals for one-loop

calculation

Let us evaluate the following useful one-loop integral in dimensional regularization

I(α, β) =

∫ddq

(2π)d1

q2α(p+ q)2β

=

∫ 1

0

dxxα−1(1− x)β−1 Γ (α + β)

Γ(α)Γ(β)

∫ddq

(2π)d1

[q2 + x(1− x)p2]α+β

=

∫ 1

0

dxxα−1(1− x)β−1

Γ(α)Γ(β)

∫ ∞

0

dttα+β−1

∫ddq

(2π)de−t(q

2+x(1−x)p2)

=1

(4π)d2

Γ(d2− α

)Γ(d2− β

)Γ(α + β − d

2

)

Γ (α) Γ (β) Γ (d− α− β)

(1

p2

)α+β− d2

(A.1)

In the calculations in d = 6− ε, we often encounter the case α = 1, β = 1, which gives

I1 ≡ I(1, 1) =1

(4π)d2

Γ(3− d

2

)Γ(d2− 1)2

(2− d2)Γ (d− 2)

(1

p2

)2− d2

(A.2)

For the purpose of extracting the logarithmic terms in d = 6− ε, it is sucient to use

the approximation

I1 → −p2

6(4π)3

Γ(3− d/2)

(p2)3−d/2 . (A.3)

157

We will also need the following three-propagator integral

I2 =

∫ddk

(2π)d1

(k − p)2

1

(k + q)2

1

k2(A.4)

= 2

∫ 1

0

dxdydzδ(1− x− y − z)

∫ddk

(2π)d1

[x(k − p)2 + y(k + q)2 + zk2]3(A.5)

= 2

∫ 1

0

dxdydzδ(1− x− y − z)

∫ddk

(2π)d1

[k2 − 2xkp+ 2ykq + xp2 + yq2]3(A.6)

Dening l ≡ k − xp+ yq, we get:

I2 = 2

∫ 1

0

dxdydzδ(1− x− y − z)

∫ddl

(2π)d1

[l2 + ∆]3(A.7)

with ∆ = x(1 − x)p2 + y(1 − y)q2 + 2xypq. Evaluating the standard momentum

integral, we obtain

I2 =

∫ 1

0

dxdydzδ(1− x− y − z)1

(4π)d/2Γ(3− d/2)

∆3−d/2 . (A.8)

In the RG calculations of Section 3, we use the renormalization conditions [62] p2 =

q2 = (p+ q)2 = M2, which imply 2p · q = −M2, and hence ∆ = M2(x(1− x) + y(1−

y)− xy). So we can write

I2 =1

(4π)d/2Γ(3− d/2)

(M2)3−d/2

∫ 1

0

dxdydzδ(1− x− y − z)1

(x(1− x) + y(1− y)− xy)3−d/2 .

(A.9)

In d = 6− ε the Feynman parameter integral becomes

∫ 1

0

dxdydzδ(1−x−y−z)(

1− ε

2log(x(1− x) + y(1− y)− xy) +O(ε2)

)=

1

2+O(ε) .

(A.10)

Thus,

I2 =1

2(4π)3

(2

ε− logM2 + A+O(ε)

), (A.11)

158

where A is an unimportant constant. For the purpose of extracting the logM2 terms

in d = 6− ε, it is sucient to use the approximation

I2 →1

2(4π)3

Γ(3− d/2)

(M2)3−d/2 . (A.12)

159

Appendix B

Summary of three-loop results

The Feynman rules for our theory are depicted in Fig. B.1

= −dαβγ

α

β γ

α

γβ= −(δg)αβγ

= δαβ1p2

= −p2(δz)αβ

α β

α β

Figure B.1: Feynman rules.

where we introduced symmetric tensor coupling dαβγ and counterterms (δg)αβγ, (δz)αβ

with α, β, γ = 0, 1, .., N as

d000 = g2, dii0 = di0i = d0ii = g1,

(δg)000 = δg2, (δg)ii0 = (δg)i0i = (δg)0ii = δg1,

(δz)00 = δσ, (δz)ii = δφ, (B.1)

160

where i = 1, ..., N . The general form of a Feynman diagram in our theory could be

schematically represented as

Feynman diagram = Integral× Tensor structure factor. (B.2)

The Tensor structure factors are products of the tensors dαβγ and (δg)αβγ, (δz)αβ,

with summation over the dummy indices. Their values for dierent diagrams are

represented in Fig. D.1 and D.2 after parentheses 1. The Integrals already include

symmetry factors and are the same as in the usual ϕ3-theory; their values are listed

in Fig. D.1 and D.2 before the parentheses.

1To nd the Tensor structure factor we used the fact that it is a polynomial in N , so wecalculated sums of products of dαβγ , (δg)αβγ , (δz)αβ explicitly for N = 1, 2, 3, 4, ..., using WolframMathematica. Having answers for N = 1, 2, 3, 4, .. it's possible to restore the general N form.

161

B.1 Counterterms

zφ12 =− g21

3(4π)3, zσ12 = −Ng

21 + g2

2

6(4π)3, a13 = −g

21 (g1 + g2)

(4π)3, b13 = −Ng

31 + g3

2

(4π)3,

(B.3)

zφ14 =g2

1

432(4π)6

(g2

1(11N − 26)− 48g1g2 + 11g22

),

zσ14 =− 1

432(4π)6

(2Ng4

1 + 48Ng31g2 − 11Ng2

1g22 + 13g4

2

),

a15 =− 1

144(4π)6g2

1

(g3

1(11N + 98)− 2g21g2(7N − 38) + 101g1g

22 + 4g3

2

),

b15 =− 1

48(4π)6

(4Ng5

1 + 54Ng41g2 + 18Ng3

1g22 − 7Ng2

1g32 + 23g5

2

),

(B.4)

zφ16 =g2

1

46656(4π)9

(g4

1(N(13N − 232) + 5184ζ(3)− 9064) + g31g26(441N − 544)

− 2g21g

22(193N − 2592ζ(3) + 5881) + 942g1g

32 + 327g4

2

),

zσ16 =− 1

93312(4π)9

(2Ng6

1(1381N − 2592ζ(3) + 4280)− 96N(12N + 11)g51g2

− 3Ng41g

22(N + 4320ζ(3)− 8882) + 1560Ng3

1g32 − 952Ng2

1g42 − g6

2(2592ζ(3)− 5195)),

a17 =g2

1

15552(4π)9

(− g5

1(N(531N + 10368ζ(3)− 2600) + 23968)

+ g41g2(99N2 + 2592(5N − 6)ζ(3)− 9422N − 2588) + 2g3

1g22(1075N + 2592ζ(3)− 16897)

+ 2g21g

32(125N − 5184ζ(3)− 3917)− g1g

42(5184ζ(3) + 721) + g5

2(2592ζ(3)− 2801)),

b17 =− 1

2592(4π)9

(2g7

1N(577N + 713)− 48g61g2N(31N − 59)

+ g51g

22N(423N + 2592ζ(3) + 1010)− g4

1g32N(33N − 1296ζ(3)− 6439)

− 27g31g

42N(32ζ(3) + 11)− 301Ng2

1g52 + g7

2(432ζ(3) + 1595)). (B.5)

162

Appendix C

Sample diagram calculations

C.1 Some useful integrals

Many of the diagrams listed in gure D.1 are recursively primitive, so they can be

easily evaluated using the integral:

I(α, β) =

∫ddp

(2π)d1

p2α(p− k)2β=

Ld(α, β)

(k2)α+β−d/2 , (C.1)

where

Ld(α, β) =1

(4π)d/2Γ(d

2− α)Γ(d

2− β)Γ(α + β − d

2)

Γ(α)Γ(β)Γ(d− α− β). (C.2)

For the more complicated integrals, we use the mathematica program FIRE [156],

which uses integration-by-parts (IBP) relations to turn them into simpler master

integrals, which we then evaluate by hand.

There are two categories of diagrams which show up quite frequently as subdia-

grams, the special KITE diagrams and the ChT diagrams shown in Figure C.1.

163

ChT (α, β) =

α 1

β 1

1SK(α) =

1 1

1 1

α

Figure C.1: The Special KITE and ChT diagrams, the numbers labeling each prop-agator denote its index.

The special KITE diagram is a two-loop diagram corresponding to the following

integral:

SK(α) =

∫ddp

(2π)dddq

(2π)d1

p2(p+ k)2q2(q + k)2(p− q)2α. (C.3)

Notice that the power of the middle propagator is arbitrary. Via the Gegenbauer

Polynomial technique as described in [157], this integral can be expressed as an innite

sum of gamma functions.

SK(α) = − 2

(4π)d1

(k2)4+α−dΓ2(λ)Γ(λ− α)Γ(α + 1− 2γ)

Γ(2λ)Γ(3λ− α− 1)

×(Γ2(1/2)Γ(3λ− α− 1)Γ(2λ− α)Γ(α + 1− 2λ)

Γ(λ)Γ(2λ+ 1/2− α)Γ(1/2− 2λ+ α)

+∞∑

n=0

Γ(n+ 2λ)

Γ(n+ α + 1)

1

(n+ 1− λ+ α)

), (C.4)

where λ = d/2− 1. In the case of d = 6− ε we have found that, for example:

SK(2−d2

) =1

(4π)d1

(k2)6−3d/2

(−54

ε2+−71 + 24γ

1296ε+−14641 + 8520γ − 1440γ2 + 120π2

15520+. . .

)

(C.5)

The ε-expansion of the above result can also be veried indirectly with the mathe-

matica packages MBTools implementing the Mellin-Barnes representation [158].

The ChT diagram is another variation of the KITE diagram. It correspond to

the integral:

ChT (α, β) =

∫ddp

(2π)dddq

(2π)d1

p2α(p+ k)2βq2(q + k)2(p− q)2. (C.6)

164

In this diagram, one triangle of the KITE diagram all have indices 1, and the other

two lines have arbitrary indices α and β. This diagram was evaluated in position

space by Vasiliev et. al. in [5]. Their answer is:

ChT (α, β) =πdv(d− 2)

Γ(d2− 1)

1

(x2)d/2−3+α+β

×(

v(α)v(2− α)

(1− β)(α + β − 2)+

v(β)v(2− β)

(1− α)(α + β − 2)+v(α + β − 1)v(3− α− β)

(α− 1)(β − 1)

),

(C.7)

where v(α) = Γ(d/2−α)Γ(α)

. For our purpose, we just need to fourier transform this

expression to momentum space.

We also need variations of the SK and ChT diagrams, with a particular index

raised by 1, for example. However, we can use FIRE to relate them to the original

version of these diagrams.

C.2 Example of a two point function diagram

We will evaluate the three-loop ladder diagram which is the rst diagram in Figure

D.1(e). It corresponds to the integral:

LADDER =

∫ddpddqddr

(2π)3d

1

p2(p+ k)2(p− r)2r2(r + k)2(r − q)2q2(q + k)2, (C.8)

where the loop momenta are p, q, and r. The external momentum is k. Using FIRE,

it can be reduced to a sum of ve master integrals, denoted as MA, ... , ME:

LADDER =4(2d− 5)(3d− 8)(9d2 − 65d+ 118)MA

(d− 4)4k8− 12(d− 3)(3d− 10)(3d− 8)MC

(d− 4)3k6

+32(d− 3)2(2d− 7)MB

(d− 4)3k6+

4(d− 3)2ME

(d− 4)2k4+

3(d− 3)(3d− 10)MD

(d− 4)2k4.

(C.9)

165

k k

p

r

q

p − rr − q

p + k

r + k

q + k

MA MB MC

MD ME

Figure C.2: The LADDER diagram can be reduced to ve master integrals

The diagrams corresponding to the master integrals are listed in Figure C.2. Among

these master integrals, only MD is non-primitive, the rest can be calculated easily.

However, if we integrate over the middle loop, we see that MD is in fact related to

the special KITE diagram SK(2− d/2). We have:

MA =Ld(1, 1)Ld(1, 2− d/2)Ld(1, 3− d)

(k2)4−3d/2, MB =

(Ld(1, 1))2Ld(1, 4− d)

(k2)5−3d/2

MC =(Ld(1, 1))2Ld(1, 2− d/2)

(k2)5−3d/2, (C.10)

MD = Ld(1, 1)SK(2− d/2), ME =(Ld(1, 1))3

(k2)6−3d/2.

Plugging in d = 6− ε and expanding in ε, we nd that:

LADDER =k2

(4π)3d/2

(− 2

9ε3+−115 + 36γ + log k2

108ε2

+−4043 + 18(115− 18γ)γ + 18π2 − 18 log k2(−115 + 36γ + 18 log k2)

1296ε+ . . .

).

(C.11)

C.3 Example of a three point function diagram

We will evaluate the three-loop diagram found in Figure D.2(f). In order to employ

the same techniques used for the two-point functions, we impose that the momentum

running through the three points are p, −p, and 0, as a three-point function with

three arbitrary momenta are much more dicult to compute.

166

p p

0

(a)

p p p p

0 0

(b) (c)

Figure C.3: The three orientations of the same diagram topology correspond to dif-ferent integrals.

However, since the momenta are asymmetric, it is necessary to consider all three

orientations of each topology of the diagram. Notice that the tensor factors men-

tioned in the previous section will also be dierent. As an illustration, let's denote the

three orientations of the diagram we are considering by I1, I2, and I3. After taking

into account that one of the external momenta is zero, they are each equivalent to a

two-point function as shown in Figure C.3. All lines have indices 1, except those with

black dots, which have indices 2.

I1 contains a subdiagram that is equivalent to ChT (1, 2), which can be evaluated

easily using our formula before; after that, the diagram is primitive. The other two

diagrams be reduced via FIRE into the master integrals MA, MB, MC , and MD as in

the LADDER diagram. Again, in d = 6− ε, we nd that:

167

I1 =1

(4π)3d/2

(1

6ε3+

5− 2γ − 2 log k2

8ε2

+173 + 18(γ − 5)γ − π2 + 18 log k2(−5 + 2γ + log k2)

96ε+ . . .

)(C.12)

I2 =1

(4π)3d/2

(1

6ε3+

5− 2γ − 2 log k2

8ε2

+125 + 18(γ − 5)γ − π2 + 18 log k2(−5 + 2γ + log k2)

96ε+ . . .

)(C.13)

I3 =1

(4π)3d/2

(1

6ε3+

5− 2γ − 2 log k2

8ε2

+125 + 18(γ − 5)γ − π2 + 18 log k2(−5 + 2γ + log k2)

96ε+ . . .

). (C.14)

168

Appendix D

Beta functions and anomalous

dimensions for general Yukawa

theories

In this appendix we list known general results for β- and γ-functions for general

Yukawa theories in d = 4 with the Lagrangian

L =1

2(∂µφi)

2 + ψ /∂ψ + ψΓiψφi +1

4!gijklφiφjφkφl , (D.1)

where φi, with i = 1, . . . , Nb are real scalar elds, ψα, with α = 1, . . . , Nf are four-

component Dirac spinors, and the matrices Γi have the following general form

Γi = Si ⊗ 1 + iPi ⊗ γ5, Γ†i = S†i ⊗ 1− iP †i ⊗ γ5 (D.2)

and act in the avor and spinor spaces and are not necessarily Hermitian. We also

assume that γ25 = 1.

Using the papers [127, 128, 129] and [130] one can nd the β- and γ-functions of

the general Yukawa theoryfootnoteWe note that one can use results of [127, 128, 129]

169

for the four-component spinors, see sec. 4 in [128]. For the γ-functions the result

reads (see formulas (3.6), (4.4) in [127] and (7.2) in [130])

γψ =1

(4π)2

1

2ΓiΓ

†i −

1

(4π)4

(1

8ΓiΓ

†jΓjΓ

†i +

3

8tr(ΓiΓ

†j)Γ†iΓj

),

γφ,ij =1

2(4π)2tr(ΓiΓ

†j) +

1

(4π)4

( 1

12giklmgjklm −

3

4tr(ΓjΓ

†iΓkΓ

†k)−

1

2tr(ΓjΓ

†kΓiΓ

†k)).

(D.3)

For the anomalous mixing matrix of the Nb(Nb+ 1)/2 quadratic operators Oij = φiφj

with i 6 j we have [131]

γij,kl = γφ,mkδij,ml + γφ,mlδij,km +

Mij,kl +Mij,lk , k 6= l ,

Mij,kk , k = l ,

(i 6 j, k 6 l) ,

(D.4)

where δij,kl is the Kronecker delta (e.g. δ11,11 = 1, δ11,12 = 0, δ12,12 = 1, δ22,22 = 1,. . . )

and

Mij,kl =1

(4π)2gijkl −

1

(4π)4

(gikmngjlmn + tr(ΓlΓ

†m)gijkm − 2tr(ΓiΓ

†kΓjΓ

†l )). (D.5)

For the β-functions we have (see (3.3) in [128] and (7.2) in [130])

βi =1

(4π)2

(1

2(Γ2†Γi + ΓiΓ

2) + 2ΓjΓ†iΓj +

1

2Γjtr(Γ

†jΓi)

)+

1

(4π)4

(2ΓkΓ

†jΓi(Γ

†kΓj − Γ†jΓk)

− Γj(Γ2Γ†i + Γ†iΓ

2†)Γj −1

8(ΓjΓ

2Γ†jΓi + ΓiΓ†jΓ

2†Γj)− tr(Γ†iΓk)ΓjΓ†kΓj

− 3

8tr(Γ†jΓk)(ΓjΓ

†kΓi + ΓiΓ

†kΓj)− Γjtr

(3

8(Γ2Γ†j + Γ†jΓ

2†)Γi +1

2Γ†jΓkΓ

†iΓk)

− 2gijklΓjΓ†kΓl +

1

12giklmgjklmΓj

)(D.6)

170

and (see (4.3) in [129])

βijkl =1

(4π)2

(1

8

∑

perm

gijmngmnkl −1

2

∑

perm

tr(ΓiΓ†jΓkΓ

†l ) +

1

2

∑

a=i,j,k,l

tr(Γ†aΓa)gijkl)

+1

(4π)4

( 1

12gijkl

∑

a=ijkl

gamnpgamnp −1

4

∑

perm

gijmngkmprglnpr −1

8

∑

perm

Tr(Γ†nΓp)gijmngklmp

+1

2

∑

perm

gijmntr(ΓkΓ†mΓlΓ

†n)− gijkl

∑

a=ijkl

(3

4tr(ΓaΓ

†aΓmΓ†m) +

1

2tr(ΓaΓ

†mΓaΓ

†m))

+∑

perm

(tr(Γ†mΓmΓ†iΓjΓ

†kΓl) + 2tr(ΓmΓ†iΓmΓ†jΓkΓ

†l ) + tr(ΓiΓ

†jΓmΓ†kΓlΓ

†m))),

(D.7)

where∑

perm denotes the sum over all permutation of the indices i, j, k, l and Γ2 = Γ†iΓi

and Γ2† = ΓiΓ†i , also

∑a=ijkl f(a) ≡ f(i) + f(j) + f(k) + f(l). Traces tr(Γi . . . ) are

over the avor and spinor indices and tr1 = 4, trγ5 = 0.

Looking at the previous expressions for the β- and γ-functions, one can easily

generalize the result of [130] for βb, when the matrices Γi are not Hermitian:

βb = − 1

(4π)8

1

144

(1

8tr(Γ†iΓ

2ΓiΓ2) + tr(Γ†iΓjΓ

†iΓjΓ

2) + tr(ΓiΓ†jΓiΓ

†kΓjΓ

†k)− tr(ΓiΓ†jΓkΓ†iΓjΓ†k)

+3

4tr(Γ†iΓj)tr(Γ

†iΓjΓ

2 + Γ†iΓkΓ†jΓk) + gijkltr(ΓiΓ

†jΓkΓ

†l )−

1

24giklmgjklmtr(Γ

†iΓj)

).

(D.8)

Now to use these formulas for the GNY model one simply takes

Γ1 = g11Nf×Nf ⊗ 1, g1111 = g2 . (D.9)

171

and nds the results (6.7), (6.9), (6.11) and (6.19). In the case of the NJL model, we

have

Γ1 = g11Nf×Nf ⊗ 1, Γ2 = ig11Nf×Nf ⊗ γ5 ,

g1111 = g2222 = g2, g1122 = g1221 = · · · = 1

3g2 , (D.10)

and we obtain the results (6.68), (6.70), (6.72) and (6.80).

172

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

×

Ng

3 1(g

3 1+

3g2 1g 2

+2g

1g

2 2+

g3 2)+

g6 2

g3 1((

N+

2)g

3 1+

3g2 1g 2

+g 1

g2 2+

g3 2)

×

Ng

3 1((

N+

1)g

3 1+

2g2 1g 2

+g 1

g2 2+

2g3 2)+

g6 2

2g4 1(g

1+

g 2)2

×

Ng

2 1(N

g4 1+

4g3 1g 2

+g

2 1g

2 2+

g4 2)+

g6 2

2g3 1(2

g3 1+

Ng

2 1g 2

+g

3 2)

×

Ng

2 1(2

g4 1+

(N+

2)g

3 1g 2

+g 1

g3 2+

g4 2)+

g6 2

g3 1(g

1+

g 2)(

(N+

2)g

2 1+

g2 2)

×

Ng

3 1(2

g3 1+

2g2 1g 2

+g 1

g2 2+

2g3 2)+

g6 2

g2 1((

N+

2)g

4 1+

2(N

+1)

g3 1g 2

+g

4 2)

×

Ng

4 1(2

g2 1+

5g2 2)+

g6 2

4g4 1(g

2 1+

g2 2)

×

Ng

2 1(4

g4 1+

Ng

2 1g

2 2+

2g4 2)+

g6 2

4g4 1(N

g2 1+

g2 2)

×

Ng

2 1((

N+

2)g

4 1+

3g2 1g

2 2+

g4 2)+

g6 2

g2 1((

3N+

2)g

4 1+

(N+

1)g

2 1g

2 2+

g4 2)

×

Ng

2 1(4

g4 1+

Ng

2 1g

2 2+

2g4 2)+

g6 2

g2 1((

N2+

4)g

4 1+

2Ng

2 1g

2 2+

g4 2)

(c)

(b)

(a)

×

2Ng

4 1+

Ng

2 1g

2 2+

g4 2

g2 1((

N+

2)g

2 1+

g2 2)

×

Ng

3 1(g

1+

2g2)+

g4 2

2g3 1(g

1+

g 2)

×

Ng

2 1+

g2 2

2g2 1

Val

ue

of∂

∂p2

(sum

ofgr

aphs)

−1

(4π)9

1 9ǫ3

( 1−

5ǫ 6+

7ǫ2

144

)

−1

(4π)9

1 9ǫ3

( 1−

7ǫ

12

+71ǫ2

288

)

1(4

π)9

1108ǫ3

( 1−

3ǫ 2+

103ǫ2

144

)

1(4

π)9

118ǫ3

( 1−

11ǫ

12

+2ǫ2 9

)

1(4

π)9

154ǫ3

( 1−

19ǫ

12

+121ǫ2

144

)

1(4

π)9

118ǫ2

( 1+

ǫζ(3

)2

−7ǫ 8

)

−1

(4π)9

1648ǫ3

( 1−

11ǫ

12

+23ǫ2

144

)

−1

(4π)9

1324ǫ3

( 1−

11ǫ

4+

103ǫ2

36

)

−1

(4π)9

1324ǫ3

( 1−

11ǫ

12

−13ǫ2

144

)

−1

(4π)3

1 6ǫ

1(4

π)6

1 6ǫ2

( 1−

ǫ 3

)

−1

(4π)6

136ǫ2

( 1−

11ǫ

12

)

FigureD.1:Valuesof

derivativesof

two-pointdiagrams.

The

upperrowin

parenthesisisfor〈σσ〉a

ndthelower

isfor〈φφ〉.

173

FigureD.2:Valuesof

three-pointdiagrams.

The

upperrowin

parenthesisisfor〈σσσ〉a

ndthelower

isfor〈σφφ〉.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

×

Ng

3 1+

g3 2

g2 1(g

1+

g 2)

×

3(N

g3 1(g

2 1+

g 1g 2

+g

2 2)+

g5 2)

g2 1((

N+

3)g

3 1+

5g2 1g 2

+2g

1g

2 2+

g3 2)

×

3(N

g2 1(2

g3 1+

g3 2)+

g5 2)

g2 1((

N+

4)g

3 1+

2(N

+1)

g2 1g 2

+g 1

g2 2+

2g3 2)

×

g 2(3

Ng

4 1+

g4 2)

2g3 1(g

2 1+

g2 2)

×

3(N

(N+

1)g

7 1+

Ng

6 1g 2

+N

g5 1g

2 2+

2Ng

4 1g

3 2+

Ng

3 1g

4 2+

g7 2)

g2 1((

2N+

3)g

5 1+

(N+

7)g

4 1g 2

+(N

+6)

g3 1g

2 2+

3g2 1g

3 2+

g5 2)

×

3(N

g7 1+

2Ng

6 1g 2

+2N

g5 1g

2 2+

Ng

4 1g

3 2+

Ng

3 1g

4 2+

g7 2)

g2 1((

2N+

3)g

5 1+

(2N

+7)

g4 1g 2

+(N

+5)

g3 1g

2 2+

2g2 1g

3 2+

g 1g

4 2+

g5 2)

×

3(N

g7 1+

N(N

+2)

g6 1g 2

+N

g5 1g

2 2+

2Ng

3 1g

4 2+

g7 2)

g3 1((

2N+

3)g

4 1+

(2N

+7)

g3 1g 2

+5g

2 1g

2 2+

3g1g

3 2+

2g4 2)

×

3(2

Ng

7 1+

2Ng

6 1g 2

+N

2g

5 1g

2 2+

Ng

3 1g

4 2+

Ng

2 1g

5 2+

g7 2)

g3 1((

3N+

4)g

4 1+

(3N

+4)

g3 1g 2

+(2

N+

1)g

2 1g

2 2+

5g1g

3 2+

2g4 2)

×

3(2

Ng

7 1+

2Ng

6 1g 2

+N

2g

5 1g

2 2+

Ng

3 1g

4 2+

Ng

2 1g

5 2+

g7 2)

g2 1((

N(N

+1)

+4)

g5 1+

(2N

+6)

g4 1g 2

+(2

N+

3)g

3 1g

2 2+

(N+

2)g

2 1g

3 2+

g 1g

4 2+

g5 2)

×

3(2

Ng

7 1+

2Ng

6 1g 2

+N

g4 1g

3 2+

2Ng

3 1g

4 2+

g7 2)

g2 1((

N+

4)g

5 1+

(4N

+6)

g4 1g 2

+(4

N+

2)g

3 1g

2 2+

g 1g

4 2+

2g5 2)

−1

(4π)3

1 ǫ

1(4

π)6

1 2ǫ2

( 1−

ǫ 4

)

−1

(4π)6

112ǫ2

( 1−

7ǫ

12

)

−1

(4π)6

1 4ǫ

−1

(4π)9

1 6ǫ3

( 1−

3ǫ 4+

3ǫ2 4

)

−1

(4π)9

1 3ǫ3

( 1−

3ǫ 4−

ǫ2 4

)

−1

(4π)9

1 3ǫ3

( 1−

ǫ 4−

5ǫ2 16

)

1(4

π)9

118ǫ3

( 1−

5ǫ

12

−47ǫ2

144

)

1(4

π)9

1 9ǫ3

( 1−

5ǫ

12

−47ǫ2

144

)

1(4

π)9

118ǫ3

( 1−

5ǫ 4+

23ǫ2

48

)

174

(k)

(l)

(m)

(n)

(o)

(p)

(q)

(r)

(s)

(t)

(u)

×

3(2

Ng

7 1+

N2g

6 1g 2

+2N

g5 1g

2 2+

Ng

4 1g

3 2+

Ng

2 1g

5 2+

g7 2)

g2 1((

3N+

4)g

5 1+

(3N

+4)

g4 1g 2

+(N

+3)

g3 1g

2 2+

(N+

3)g

2 1g

3 2+

g 1g

4 2+

g5 2)

×

3(N

2g

7 1+

2Ng

6 1g 2

+3N

g5 1g

2 2+

Ng

2 1g

5 2+

g7 2)

g2 1((

3N+

4)g

5 1+

(2N

+6)

g4 1g 2

+(2

N+

1)g

3 1g

2 2+

(N+

2)g

2 1g

3 2+

2g1g

4 2+

g5 2)

×

3(N

(N+

2)g

7 1+

Ng

5 1g

2 2+

2Ng

4 1g

3 2+

Ng

2 1g

5 2+

g7 2)

g2 1((

4N+

4)g

5 1+

(5N

+2)

g4 1g 2

+(N

+2)

g3 1g

2 2+

(2N

+1)

g2 1g

3 2+

g 1g

4 2+

2g5 2)

×

3(4

Ng

7 1+

N2g

4 1g

3 2+

2Ng

2 1g

5 2+

g7 2)

g2 1((

N2+

8)g

5 1+

(2N

2+

4)g

4 1g 2

+2N

g3 1g

2 2+

4Ng

2 1g

3 2+

g 1g

4 2+

2g5 2)

×

3(4

Ng

7 1+

N2g

4 1g

3 2+

2Ng

2 1g

5 2+

g7 2)

g2 1((

4N+

4)g

5 1+

N(N

+4)

g4 1g 2

+4g

3 1g

2 2+

(2N

+4)

g2 1g

3 2+

g5 2)

×

3(N

(N+

4)g

6 1g 2

+N

g4 1g

3 2+

Ng

2 1g

5 2+

g7 2)

2g3 1((

N+

4)g

4 1+

(2N

+3)

g2 1g

2 2+

2g4 2)

×

3(N

2g

7 1+

2Ng

6 1g 2

+2N

g5 1g

2 2+

Ng

4 1g

3 2+

Ng

3 1g

4 2+

g7 2)

2g3 1(3

g4 1+

(N+

3)g

3 1g 2

+2g

2 1g

2 2+

2g1g

3 2+

g4 2)

×

(3N

g6 1g 2

+3N

g5 1g

2 2+

Ng

3 1g

4 2+

g7 2)

g3 1(2

g4 1+

(N+

2)g

3 1g 2

+g

2 1g

2 2+

g 1g

3 2+

g4 2)

×

3(2

Ng

7 1+

2Ng

5 1g

2 2+

3Ng

4 1g

3 2+

g7 2)

g2 1(6

g5 1+

(3N

+4)

g4 1g 2

+6g

3 1g

2 2+

4g2 1g

3 2+

g5 2)

×

4Ng

6 1g 2

+3N

g4 1g

3 2+

g7 2

(N+

2)g

7 1+

4g5 1g

2 2+

g3 1g

4 2

×

3(N

g7 1+

Ng

6 1g 2

+3N

g5 1g

2 2+

2Ng

4 1g

3 2+

g7 2)

g3 1((

2N+

5)g

4 1+

7g3 1g 2

+3g

2 1g

2 2+

5g1g

3 2+

2g4 2)

1(4

π)9

118ǫ3

( 1−

17ǫ

12

+10ǫ2 9

)

1(4

π)9

136ǫ3

( 1−

17ǫ

12

+11ǫ2

18

)

−1

(4π)9

1108ǫ3

( 1−

29ǫ

12

+19ǫ2 9

)

−1

(4π)9

1108ǫ3

( 1−

7ǫ

12

−11ǫ2

48

)

−1

(4π)9

1108ǫ3

( 1−

7ǫ

12

−11ǫ2

48

)

−1

(4π)9

136ǫ2

( 1−

11ǫ

12

)

1(4

π)9

112ǫ2

( 1−

ǫ 4

)

1(4

π)9

1 6ǫ2

( 1−

23ǫ

12

+2ǫ

ζ(3

))

1(4

π)9

1 6ǫ2

( 1−

29ǫ

24

+ǫζ

(3))

−1

(4π)9

1 3ǫ

−1

(4π)9

1 3ǫ

( ζ(3

)−

1 3

)

FigureD.3

175

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