Random perturbations of the O(2) vector model

108  10  Download (0)

Full text

(1)

FACOLT `A DI FISICA Corso di Laurea in Fisica

Tesi di Laurea Magistrale

Random perturbations

of the O(2) vector model

Candidato:

Maria Nocchi

Matricola 583327

Relatore:

Prof. Alessandro Vichi

(2)
(3)

Random perturbations

of the O(2) vector model

Candidate: Maria Nocchi

Supervisor: Prof. Alessandro Vichi

(4)

It is not unreasonable that we grapple with problems. But there are tens of thousands of years in the future. Our responsibility is to do what we can, learn what we can, improve the solutions, and pass them on.”

(5)

Acknowledgements

I would like to express my gratitude to my supervisor, professor A.Vichi, for giving me the opportunity to know and appreciate the framework of Confor-mal Field Theories: expanding one’s areas of expertise is always a chance for growth.

Special thanks should be given to my family for supporting me from near and far: thanks to my parents and my brother for never making me feel alone over the distance; thanks to my second family, the big family of Praticelli, for taking care of me as blood brothers. I wish to thank my friend Luciano for all the long discussions about theoretical physics. Thanks also to all the nice people with whom I shared these years in Pisa.

Finally, I would like to thank Giammarco for always being there, at every stage of my life.

(6)

Contents

Introduction v

1 Conformal Field Theory in d dimensions 1

1.1 Locality . . . 1

1.2 Scale Invariance . . . 2

1.3 Killing equation and solutions . . . 3

1.4 Conformal group algebra . . . 6

1.5 Two-point functions . . . 9

1.6 Three-point functions . . . 11

1.7 Four-point functions . . . 13

1.8 CFT-data and the Operator Product Expansion . . . 14

1.9 Radial quantization and OPE . . . 15

1.10 Conformal Blocks and crossing symmetry . . . 18

1.11 Conformal Bootstrap . . . 20

2 Conformal perturbation theory 23 2.1 CPT with one deformation . . . 25

2.1.1 Marginal case . . . 26

2.1.2 Anomalous dimensions in the marginal case . . . 29

2.1.3 Slightly relevant case . . . 30

2.2 CPT with two deformations . . . 34

3 Random Bond Ising Model 35 3.1 Quenched disorder in the Ising model and Harris criterion . . . 35

3.2 Replica trick . . . 37

3.3 Four dimensions . . . 39

3.3.1 OPE of energy operators . . . 40

3.3.2 Correlators in the replicated theory . . . 42

(7)

3.3.3 Beta-functions . . . 46

3.4 Three dimensions . . . 49

3.4.1 Infrared critical exponents . . . 50

4 Isotropic Random Bond O(2) Model 53 4.1 Replicated theory . . . 55

4.2 Four dimensions . . . 56

4.2.1 OPE of energy operators . . . 57

4.2.2 Correlators in the replicated theory . . . 59

4.3 Three dimensions . . . 61

5 3d Anisotropic Random Bond O(2) Model 64 5.1 OPE coefficients in the replicated theory . . . 68

5.2 Beta-functions . . . 70 Conclusions 74 A Casimir equation 78 B Unitarity 80 C CPT (two couplings) 82 D CPT (three couplings) 86

(8)

Introduction

One of the keywords of modern research in theoretical physics is

Renormaliza-tion Group (RG). It can be regarded as a tool to understand the behaviour of

a theory under change of the scale at which it is observed. Among all phys-ical systems, a special role is played by those that are invariant under scale transformations and are therefore fixed points of the RG. Moreover, it is be-lieved that most of the scale invariant physical theories are actually conformal invariant.1 In that case the theory is described by a Conformal Field Theory (CFT), which is the focus of this thesis.

The RG was born as a perturbative method to compute physical quantities in terms of scale-dependent parameters, that characterize a Quantum Field Theory (QFT) [2]. However, the connection to scale invariance is more evident in the beautiful Wilson formulation, used in the theory of critical phenomena [3]. The starting point is a microscopic Hamiltonian. In practise, one performs an integral over high-momentum degrees of freedom. This defines the steps of the trajectory or flow in the space of the theories, namely the Renormalization Group flow. The endpoints of such a flow are called fixed points because they do not change under an RG transformation. Each fixed point is described by a scale invariant theory and most of them, as already mentioned, actually are invariant under the whole conformal symmetry. So, we define a QFT as a trajectory starting from an ultraviolet CFT and flowing to the infrared limit (at large distances), that could correspond to another non-trivial CFT. We therefore see that CFTs have a special role in the space of the theories.

The RG procedure generates a scale-dependent effective action character-ized by a set of local operators. They can be classified according to their behaviour under an RG transformation: if d is the space-time dimension, we define an operator O to be irrelevant when its dimension is [O] > d and so the

1To know more about the differences between scale and conformal invariance see for

instance the review [1].

(9)

that it is important in the IR. Finally, if [O] = d, we talk about a marginal operator and we do not know a priori if it is important for the macroscopic description.

If we repeat the RG transformation several times, at the end there exists a finite number of a few surviving operators. If two theories, at some scale, differ only by the content of irrelevant operators, they are described by the same IR fixed point, corresponding to a CFT. Regardless of the dynamics, some properties (e.g. critical exponents) can be the same for physical systems that on a microscopic level would seem very different. As an example we report the well-known IR Ising point of the φ4-theory: it describes the IR behaviour

of ferromagnets and the vapour to liquid transition in water.

Figure 1: The phase diagram of a magnetic system (left) and of water (right). Figure taken from [4].

This field-theoretic context of particle physics is able to accurately predict the universal properties of critical phenomena, in good agreement with exper-iments and numerical studies. This results in the partition of the space of the theories into universality classes, related to the existence of IR fixed points.

In the study of statistical systems or condensed matter systems one usu-ally deals with Hamiltonians defined on a regular lattice. However, it is very interesting to go beyond the study of regular Hamiltonians to treat the im-portant case of disordered systems. As a matter of fact, real systems contain some impurities or defects that correspond to random inhomogeneities. For example, in nature we do not find translationally invariant perfect crystals.

How to extend the standard apparatus of RG and universality to the study of disordered systems? The random systems should indeed be described by

(10)

to the standard case. The couplings correspond to random functions of posi-tion with a fixed probability distribuposi-tion. In addiposi-tion to this, they may also fluctuate in time, leading to the two possible cases of quenched and annealed disorder. In this work we will always refer to quenched disorder, when the ran-dom variables describing the disorder fluctuate over large times with respect to (w.r.t.) some observable time scale. In this case it can be shown that the proper analysis of the disordered system requires averaging over log Z, where

Z is the partition function. When averaging over the physical observables, one

performs the so-called quenched average.

We would like to understand if the fixed point corresponding to the pure system is stable against disorder, that is, if the random system has a critical behaviour different from the pure one. Storically, first Harris [5] and then Chayes et al. [6] worked to understand if the inclusion of the disorder could modify the critical properties of the system. Now we know that this can happen only if the random perturbation is relevant or marginal. If this is the case, even a small concentration of impurities can be important. It then becomes very interesting to analyze the RG flow, searching for new possible fixed points with the related non-trivial universality classes. Indeed it appears that the critical exponents of the random systems sometimes are different from the ones obtained in absence of disorder. This can be qualitatively predicted with the so-called Harris criterion, that requires only to know the critical behaviour of the pure system.

In investigating the RG flow of disorder-averaged observables, we encounter some mathematical difficulties. This problem can be overcome with the Replica

Trick. It allows us to rewrite the averaged disorder free energy as the simpler

quenched average of Zn, where n is a positive integer that we send to 0 at the end:

F ∼ log Z = lim

n→0

Zn− 1

n . (1)

To sum up, in the context of field theory, we first perturb the pure action with a random coupling, say h(x), associated with the local operator O(x) which tunes the system across the transition. We thus consider the deformation

SCFT→ SCFT+

Z

ddx h(x)O(x) , (2)

(11)

disordered free energy is studied via the Replica Trick2 that generates the interaction g Z ddx X A,B OA(x)OB(x) , (3)

where the new coupling g no longer depends on spatial coordinates.

In this scenario we can use the tools of Conformal Perturbation Theory (CPT) to study the evolution of the coupling induced by the disorder. In particular, knowing only the CFT data, namely the correlators, of the pure theory, we can deduce the properties of the random system studying the RG flow. CPT teaches us how to expand around any fixed point, not only the Gaussian one, as one is used to do in standard perturbation theory. See for instance the beautiful work of Cardy [7] about the general theory of critical behaviour, with the analysis of the perturbative RG.

In this framework the article [8] has studied the Random Bond Ising Model between 2 and 4 dimensions, dwelling on the interesting computation in d = 3. We would like to extend their analysis to the O(2) vector model, where we have a chance to treat both a random perturbation that preserves the global symmetry (like in the Ising case) and an operator charged under the O(2) group, which we obviously do not have in the Ising case.

Before entering in details of our work, let us see how it is organized. In the first chapter we present the basic concepts of CFT. Specifically, we explain why two- and three-point functions are completely fixed by the conformal invari-ance, contrary to the four-point correlators. We then present two important tools: Operator Product Expansion (OPE) and Crossing equations, related to the fundamental concept of Conformal Bootstrap, which uses only the confor-mal invariance to study critical systems.

Next, in the second chapter we introduce the CPT formalism in the case we deform the fixed point action with one scalar operator or two scalar operators. We thus derive the general expressions of the related beta-functions. We refer to Appendix C for the details of the computations and Appendix D for the discussion about three scalar deformations.

The third chapter is instead dedicated to a review of the Random Bond

2We remind that with the Replica Trick we recover the translational invariance, so that

we can exploit the knowledge of the pure model.

(12)

analysis of random system with the tools of CPT.

Finally, in the fourth and fifth chapters we focus on the Random Bond O(2) Model: we add a quenched disorder with a Gaussian distribution and we study the beta-function for the related coupling. Except for the last section, our strategy is to study the system in d = 4 − ε to have an analytically known limit and then we attempt an approximate calculation in the three dimensional case, which is closer to experimental realizations. In particular, in the last chapter we present the original study of the 3d Anisotropic Random Bond XY Model: we deform the fixed point Hamiltonian with tij (the symmetrical and traceless part of φiφj) and we consider a Gaussian distributed tensorial coupling. We apply CPT using the results from Numerical Conformal Bootstrap, recently obtained in [9].

(13)

Chapter 1

Conformal Field Theory in d

dimensions

1.1

Locality

We define a local QFT a theory that possesses a local symmetric conserved Stress Tensor, usually called Tµν, which is the generator of the translations.1

We can visualize these assumptiones in the Path Integral formulation of the QFT, focusing on the role of symmetries in a quantum context.

From Noether’s theorem we know that the invariance of the action under a continuous symmetry implies the existence of a classically conserved current. At quantum level we expect some constraints on correlation functions. Indeed, in the hypothesis of no anomalies,2 adding a current Jµ

a within a correlator results in an infinitesimal transformation for the fields. This can be seen in the Ward Identities for multiple insertions, namely

∂µh0| ˆT {Jaµ(x) n Y i=1 Φi(yi)} |0i = −i n X k=1 δd(x − yk) h0| ˆT {δaΦk(x) Y l6=k Φl(yl)} |0i , (1.1) where ˆT is the time-ordering operator. In the simple case of just one field, we 1In general, the canonical Stress Tensor is not symmetrical under space-time indices

exchange, but in Poincaré invariant theories one can always build a symmetric Stress Tensor, according to the Belinfante prescription. Moreover, we stress that we talk about a conserved Stress Tensor meaning that the operator equation ∂µTµν = 0 is valid away from other

operator insertions, as it will be clear in (1.1).

2We assume the functional integration measure in the Path Integral to be invariant under

the continuous symmetry.

(14)

can consider the space integral version of (1.1), that is3

h[Qa(x0), Φk(y)]i = −ihδaΦk(y)i , (1.2) where

Qa(x0) ≡

Z

dd−1x Ja0(x) (1.3)

is the Noether charge. It can be interpret as the generator of the infinitesimal transformation associated with the symmetry. In the specific case of the Stress Tensor we have = Z dd−1x Tµ0 , [Pµ, Φ] = −i∂µΦ . (1.4)

It can be seen that the Noether charge is actually a topological operator, implying the locality of the symmetry transformations. See [10] to know more about this and see also [11] for a full discussion of all the topics presented in this chapter.

1.2

Scale Invariance

Now we introduce the hypothesis of scale invariance, that is to say the theory is a fixed point of the RG flow.

A scale transformation affects all the dimensionful quantities: the space-time coordinates change according to

xµ→ (1 + λ)xµ , (1.5)

which is true for all λ. We can also refer to the metric tensor in a d-dimensional space-time to write

gµν → (1 + λ)2gµν . (1.6)

The variation of the related action is

(15)

δS ∝ Z ddxgTµνδgµν , (1.7) where g ≡ det(gµν) , Tµν ≡ √2 g δS δgµν . (1.8)

We can indeed look at the Stress Tensor either as the classical Noether current associated with the Poincaré invariance or as the answer of the action to the perturbation of the background geometry. When δgµν ∝ gµν, we see that

δS ∝

Z

ddxgTµµ , (1.9)

so that in the case of scale invariance, δS = 0 and Tµ

µ = ∂ρKρ. Then, since [Tµ

µ] = d, Kµhas the dimension of a conserved current in a free theory, namely [Kµ] = d − 1. Generically it is very hard to find a candidate for Kµ that is not a conserved current. Hence, in most cases the request of scale invariance implies conformal invariance. This is actually always the case in 2d and in perturbation theory in 4d ([12],[13]). See instead [14] for an example of a scale invariant physical theory which is not conformally invariant.

We conclude that in the assumption of scale invariance, the tracelessness of the Stress Tensor implies the invariance of the theory under a bigger symmetry group, allowing us to rescale the metric with some coordinate-dependent factor. Precisely we have invariance under the infinitesimal Weyl transformations, that leads to the invariance under finite transformations in the case of no anomalies.

1.3

Killing equation and solutions

In general, an arbitrary rescaling of the metric (Weyl transformations) changes the space-time geometry. We therefore define the conformal transformations starting from the subgroup of those Weyl transformations that are also diffeo-morphisms. We talk about conformal transformations because of their prop-erty to locally preserve the angle between two arbitray crossing curves.

The conformal transformations are indeed invertible maps that do not affect the metric up to an overall local rescaling, namely

(16)

Figure 1.1: Example of a conformal transformation of a part of a square lattice. Figure taken from [7].

gµν(x) → gµν0 (x

0

) = Λ(x)gµν(x) . (1.10)

Let us focus on the infinitesimal transformation by demanding that

xµ→ x0µ = xµ+ εµ(x) . (1.11)

Then the metric tensor transforms according to

gµν → gµν0 = ∂xα ∂x0µ ∂xβ ∂x0νgαβ = gµν− (∂νεµ+ ∂µεν) + O(ε 2) . (1.12)

If we expand the trasformation matrix in (1.10), we find the Killing equation

∂νεµ+ ∂µεν = 2 d∂ρε ρg µν ≡ 2 d ˆ f (x)gµν (1.13)

and after a few simple steps we get the second order differential equations

(2 − d)∂µ∂νf (x) = gˆ µν ˆf (x) , (1 − d) ˆf = 0 .

(1.14)

(17)

con-straints on ˆf : each smooth transformation is conformal in one dimension and

this is trivial since in this case there cannot be any notion of angle.

The case d = 2 admits a special description because of the infinite dimen-sion of the conformal group that consists of the conformal analytical trans-formations.4 For our work we do not need to go into detail on this and we

concentrate on d > 2. In this case we find

ˆ

f (x) = c0+ cµ1 , (1.15)

where c0 and cµ1 are constants. We conclude that5

εµ(x) = cµ+ aµνxν + bµνρxνxρ . (1.16)

We find that the conformal symmetry is a Lie group with (d + 1)(d + 2)/2 pa-rameters, isomorphic to the non-compact group SO(d+1, 1). The infinitesimal transformations are

→ xµ+ cµ (translations) ,

→ xµ+ ωµνx

ν (rotations) ,

→ xµ+ λxµ (dilatations) ,

→ xµ+ 2(b · x)xµ− x2 (special conformal transformations (SCTs)) , (1.17)

where bµ is an arbitrary constant vector in Rd.

The finite transformations are as usual obtained by exponentiation of the previous linear ones. Actually, the quadratic nature of the SCTs prevents us from passing directly to the finite form. We thus use the trick of writing the infinitesimal transformation as a composition of an inversion (I), a translation (P ) and again an inversion. Even if the inversion is a discrete transformation which is not connected to the identity, we use it twice, so there are no problems and we can write the finite SCT as a series of infinitesimal transformations, each one written as “IP I”. Finally we obtain the finite transformations

4It can be shown that in 2d the Killing equation corresponds to the Cauchy-Riemann

equations for the coordinate change on the complex plane.

5Actually it can be shown that the 3-index tensor in (1.16) has just one independent

(18)

xµ→ xµ+ Cµ (translations) , xµ→ xµ+ Λµ νxν (rotations) , xµ→ xµ+ λxµ (dilatations) , x µ− bµx2 1 − 2(b · x) + b2x2 (SCTs) . (1.18)

1.4

Conformal group algebra

We have seen that the conformal group contains the Poincaré group: it cor-responds to take Λ(x) = 1 in (1.10). We thus expect that it has the usual Poincaré algebra and then there are the other generators associated with the dilatations and the SCTs. The aim of this section is to understand the struc-ture of the representations of this algebra to classify the operators of a general CFT.

Implementing our transformations on some general function, we find the generators written as differential operators,

= −i∂µ (translations) ,

Mµν = i(xµ∂ν − xν∂µ) (rotations) ,

D = −ixµ∂µ (dilatations) ,

Kµ= −i(2xµxρ∂ρ− x2∂µ) (SCTs) ,

(1.19)

where Pµ and Mµν are the usual Poincaré generators. Then we can deduce the fundamental commutation rules, namely

[D, Pµ] = iPµ , [D, Kµ] = −iKµ , [Kµ, Pν] = 2i(ηµνD − Mµν) , [Mµν, Pρ] = −i(ηµρPν − ηνρPµ) , [Mµν, Kρ] = −i(ηµρKν − ηνρKµ) , [Mµν, Mρσ] = −i(Mµρηνσ − Mµσηνρ− Mνρηµσ + Mνσηµρ) , [Mµν, D] = [Pµ, Pν] = [Kµ, Kν] = [D, D] = 0 . (1.20)

(19)

Now, if we think about a general QFT with some symmetry, the operators are typically classified in terms of the irreducible representations (irreps) of the symmetry group. We want to understand which quantum numbers we need to do this in the framework of a quantum CFT.

The crucial point is the possibility of studying only the operators in the origin and this is good because we exactly know the commutators of the gener-ators of the translations with the other ones. Hence, let us study the genergener-ators of the subgroup which leaves x = 0 invariant and then translate them at x 6= 0. We denote the generators of this subgroup as G.

If the operators implementing the transformations are unitary we can write

O(x) = e−iPµxµO(0)eiPµxµ ,

[G, O(x)] = [G, e−iPµxµO(0)eiPµxµ] = e−iPµxµ[ ˜G, O(0)]eiPµxµ ,

(1.21) where ˜ G ≡ eiPµxµGe−iPµxµ =X n in n!xµ1. . . xµn[P µ1. . . [Pµn, G]] . (1.22)

After a finite number of commutators, if we find [Pµ, G] ∝ Pµ, the series stops. This is precisely our case since when G ∈ {D, Mµν} the series convergences at the first step, while if G = Kµ, we need to consider the second term. In particular we have ˜ D = D + xµPµ , ˜ Mµν = Mµν− xµPν+ xνPµ , ˜ Kµ= Kµ+ 2xµD + 2xρMρµ+ 2xµxρPρ− x2 . (1.23)

Consequently, we will focus on commutators of {Kµ, D, Mµν} with the operator O evaluated in zero. From the rotational invariance and Schur’s lemma we can write

[Mµν, Oa(0)] = i(Sµν)abOb(0) , [D, Oa(0)] = i∆Oa(0) ,

[Kµ, Oa(0)] = 0 ,

(20)

where Sµν is a matrix representation defined by the action of the Lorentz transformation on the field; in other words, it is the spin operator associated with the field.

In conclusion we translate (1.24) at x 6= 0:

[Mµν, Oa(x)] = i(Sµν)abOb(x) − i(xµ∂ν− xν∂µ)Oa(x) , [D, Oa(x)] = i∆Oa(x) + ixµ∂µOa(x) ,

[Kµ, O(0)] = −2ixµ∆Oa(x) + 2xρi(Sρµ)abOb(x) + i(2xµxρ∂ρOa(x) − x2∂µOa(x)) .

(1.25)

From the commutation rules in (1.20), we also see that [D, Kµ] ∝ −Kµ and [D, Pµ] ∝ Pµ, implying that Pµ and Kµ behave like ladder operators for D, building the so-called conformal multiplet of O. Given that ∆ is the energy dimension of the operator, we get

D(Pµ) |ψi = i(∆ + 1) |ψi ,

D(Kµ) |ψi = i(∆ − 1) |ψi .

(1.26)

We have to pay attention to the action of Kµbecause we could apply it unless we reach negative energy states and this is impossible in the framework of unitary theories (see Appendix B to know more about the implications from unitarity). We therefore restrict our analysis to KµO = 0 and by this choice we define O to be the primary operator of the conformal multiplet (or the

quasiprimary operator if we deal with two-dimensional CFTs). For each

repre-sentation there is a primary and then we talk about the descendants, obtained by application of Pµ.

To sum up, in an ordinary QFT an irriducible representation is identified by the rotation quantum number (l) and the stabilizer group6 of x = 0 is simply

{Mµν}. In a CFT we have instead {Mµν, D, Kµ} with the related quantum numbers {l, ∆, 0}.7 The eigenvalue ∆ is real and it assumes the minimum value

6The stabilizer is the set of the elements of a group that have no effect on x. Later we

will introduce the concept of conformal frame; in that case we can also define the stabilizer group as the set of conformal transformations that does not affect the conformal frame configuration.

7In general we could assume the third quantum number to be different from 0; however

(21)

for the primary state, while there is no reason to expect a maximum value to exist. We can thus visualize the typical representation of the conformal group as an infinite tower based on the primary. We have already seen that the action of Pµ on a state is to raise up its scaling dimension ∆, but it is worth to underline that it also changes the spin of the state and hence we expect that in each level of the tower there can be states in different Lorentz group representations, depending on the way we contracted the indices.

In general we can study finite conformal transformations, knowing that

xµ→ x0µ , ∂x

∂xρ = Ω(x)Λ µ

ρ , (1.27)

where the Jacobian consists of Ω(x), a scale factor depending on the specifc transformation and Λµ

ρ, related to the rotation group. This means that con-formal transformations locally look like a rotation combined with a dilatation. We are interested in this object because it enters the transformation rules of the operator. We define the matrix R(Λ)AB to represent the finite rotation Λ in the representation of spin l, where A is a collective index. The general transformation law is thus

OA ∆,l(x) → O 0A ∆,l(x 0 ) = 1 Ω(x)R(Λ) A BO B ∆,l(x) . (1.28)

For example, under dilatations, x → λx, we have

O → λ−∆O (1.29)

and that is why we identify ∆ as the energy dimension of the operator.

1.5

Two-point functions

In the framework of CFTs, we do not discuss about the S-matrix like we use to do for general QFTs, because a CFT does not really admits a particle inter-pretation. In fact, we are interested in the general structure of the correlation functions and this approach is powerful because then we have just to verify if a specific theory is scale invariant to know its correlators.8

we can directly study an irriducible representation as the state defined by the set {l, ∆}, meaning the vanishing of [Kµ, O(0)].

8In a scale but not conformal invariant theory the correlators are different from what we

(22)

Now, since hO(x)i = 0 because of the dilatational and translational sym-metries, we directly treat the two-point function of two scalar primary fields. Using only the symmetries of a CFT, we easily find that9

hO1(x1)O2(x2)i =

C1

|x12|∆1+∆2

, (1.30)

where x12≡ x1− x2.

Until now we have not used the SCTs yet; thery are complicated but re-lated to the translations via the inversion in a simple way. Even if it is not rigorous, we could impose the invariance under the inversion (which actually is descrete!) to therefore require the invariance under SCTs. Under inversion the coordinates change according to xµ→ xµ/x2 and the Jacobian is

∂x0µ ∂xν = 1 x2 δ µ ν2xµx ν x2 ! . (1.31)

Imposing the covariance under inversion, we can see that two primary fields are correlated if and only if ∆1 = ∆2, that is to say they have the same scaling

dimension. The conclusion is that in a scalar CFT the most general two-point function is hO1(x1)O2(x2)i =      C1 |x12|2∆1, if ∆1 = ∆2 0, otherwise . (1.32)

An important result is that we can diagonalize the space of operators with the same dimension if the theory is unitary, so that

hO1(x1)O2(x2)i =

δ12

|x12|2∆1

, (1.33)

where we have absorbed the constant into the definition of the operators.10

If the operators have spin, they have non-trivial two-point functions only if they belong to the same Lorentz representation. Moreover, regardless of the type of operators, they always must have the same dimension, while the prefactor can change. For instance, in the case of two vector operators we have

9For example we can get this result taking advange of the scale invariance of the

ground-state, hO1(x1)O2(x2)i = h(U O1(x1)U−1)(U O2(x2)U−1)i. Developing it for an infinitesimal

dilatation, we find h[D, O1(x1)]O2(x2)i + hO1(x1)[D, O2(x2)]i = 0 that leads to a differential

equation, giving (1.30).

10There exist some non-unitary CFTs, like logarithmic CFTs, where this diagonalization

(23)

hV1µ(x1)V2ν(x2)i = Iµν |x12|2∆1 δ∆1,∆2 , (1.34) where Iµν ≡ ηµν− 2x µ 1212 x2 12 . (1.35)

It is usually called “tensor structure” because it has the same spin structure of the l.h.s. of (1.34). Finally we can generalize for the case of symmetric traceless tensors with an arbitrary number of indices, that is

hTµ1µ2...µn(x 1)Tν1ν2...νn(x2)i = 1ν1. . . Iµnνn+ perm − traces |x12|2∆1 δ∆1,∆2 . (1.36)

1.6

Three-point functions

We have seen that the conformal symmetry completely fixes the form of the two-point functions, contrary to the ordinary QFTs where we have much more arbitrariness. Now we would like to explain why the same thing happens also with the three-point functions, using an argument about the number of degrees of freedom contained in a correlator.

Let us take three different space-time points {x1, x2, x3} in the hypothesis

to be in the same time.11 We act on this set of points with the conformal

transformations to impose x1 ≡ (0, . . . , 0), x2 ≡ (0, 1, 0, · · · , 0) and finally

x3 ≡ (0, ∞, 0, . . . , 0).

We conclude that if we have only three points we cannot build any con-formal invariants: we guess the three-point functions to be fixed from the conformal symmetry.

In the case of scalar primary operators the most general ansatz one can do is

hO1(x1)O2(x2)O3(x3)i =

λ123

|x12|a|x13|b|x23|c

, (1.37)

where λ123 is a constant but physical coefficient, that we cannot scale away

since we have already used such a freedom to diagonalize the two-point

corre-11In the Minkowsky space we have to consider three space-like separated points, while

we should pay attention if we were dealing with time-like separated points, since the usual Feynamn ε-presciption should be taken into account.

(24)

lators. This is an important point: different coefficients correspond to different theories. We also underline that in the case of scalar fields we can exchage any pair of them without any effect, so that λ123 is completely symmetric. Then

we can fix {a, b, c} as functions of the scaling dimensions of the operators using the dilatations and the SCTs. We find that

a = ∆1+ ∆2− ∆3 ,

b = ∆1+ ∆3− ∆2 ,

c = ∆2+ ∆3 − ∆1 .

(1.38)

It is common to define the kinematic factor

K3(∆i, xi) ≡

1

|x12|∆1+∆2−∆3|x13|∆1+∆3−∆2|x23|∆2+∆3−∆1

, (1.39)

where the index i labels the operators appearing in the three-point functions and the related points in which they are inserted.

As a consequence, the functional form of the three-point function is com-pletely fixed by the conformal symmetry as it happened for the two-point function, but this time we have no constraints on {∆i}.

We can also consider the three-point function of two scalar operators with a traceless symmetric tensor operator, namely

hO1(x1)O2(x2)Tµ1...µn(x3)i = λ123(Z123µ1 . . . Z µn 123− traces) |x12|∆1+∆2−∆3+n|x13|∆1+∆3−∆2−n|x23|∆2+∆3−∆1−n , (1.40) where Zijkµx µ ik x2 ikx µ ij x2 ij . (1.41)

In conclusion, each three-point function of a CFT has a fixed structure with a finite number of constant coefficients depending on the representations of the operators.

(25)

1.7

Four-point functions

From the argument presented at the beginning of the previous section, we can infer that we need at least four points to build conformal invariants. Hence, let us consider a set of four points, namely {x1, x2, x3, x4}.

If we assume d ≥ 2, after fixing three points, we can rotate around the axis to bring x4 to a specific plane. We would like to see how such an

in-tuitive freedom translates to some arbitrariness in the structure of four-point correlators.

From translational invariance we have to consider differences of points; moreover, from rotational invariance we have to build scalar quantities like (xij)2 and finally from dilatational invariance we can restrict our attention to ratios of the mentioned quantities. We therefore define the two conformal

cross-ratios or anharmonic ratios,

u ≡ x 2 12x234 x2 13x224 , v ≡ x 2 14x223 x2 13x224 . (1.42)

We expect there are more of them in the case of n-point correlators with

n > 4; indeed it can be proved that if we have n distinct points, we can build n(n − 3)/2 independent cross-ratios.

Alternatively to the previous notation, we can also choose the parametriza-tion x1 = (0, 0, ¯0) , x2 = (0, 1, ¯0) , x3 = z − ¯z 2i , z + ¯z 2i , ¯0 ! , x4 = (0, ∞, ¯0) , (1.43)

where ¯0 stands for the remaining (d−2) null components in Euclidean signature and in this basis we have

u = z ¯z , v = (1 − z)(1 − ¯z) . (1.44)

A particular choice of the set {xi} is called conformal frame and it is a kind of gauge fixing that allows us to study the correlation functions directly in this conformal frame without losing generality, because of the covariance.

Effectively, we see that we cannot fix four points just with the conformal symmetry, that is why we expect the four-point functions to have a functional dependence on variables not fixed a priori. For instance, in the case of scalar

(26)

operators we have

hO1(x1)O2(x2)O3(x3)O4(x4)i = K4(∆i, xi)g(u, v) . (1.45) In the simple case of four identical operators we get

hφ(x1)φ(x2)φ(x3)φ(x4)i =

1

|x12|2∆φ|x34|2∆φ

g(u, v) , (1.46)

where g(u, v) is invariant and all the transformation properties are in the pref-actor of the last expression. In this context we actually meet the first arbi-trariness of our analysis.

1.8

CFT-data and the Operator Product

Ex-pansion

So far we have found that a CFT is characterized by

• irreducible representations: listing the operators present in the theory is equivalent to give the two-point functions of the CFT that are entirely determined by the quantum numbers of the operators;

• three-point functions coefficients: the physical constants λ123 correspond

to the vertices of the theory.

Now we would like to understand if we also have to introduce in the above list the two-variable function g(u, v) appearing in (1.45). The fundamental question is whether we can reduce the n-point correlation functions to the (n − 1)-point correlation functions, so that we can link g(u, v) (from four-point functions) to λ123 (from three-point functions).

All we have to deal with is the product of local operators in different points inside the correlation functions. Given two operators in two irreps, their prod-uct results in an operator which is not a priori in an irrep of the symmetry group. In fact, we can write

O1× O2 ∼

X

i

C12iOi , (1.47)

(27)

In the framework of ordinary QFTs this sum is asymptotic, while in the particular case of conformal theories it has a finite radius of convergence, cor-responding to the distance to the next operator insertion. This is precisely the assertion we need to reduce an n-point correlator to an infinite sum of (n − 1)-point correlators. Schematically (1.47) implies

hO1O2O3i ∼

X

i

C12ihOiO3i ∝ C123 , (1.48)

that is to say the three-point function is different from zero if and only if O1× O2 ⊃ O3. From (1.48) we can also deduce that there should be a relation

between the coefficients of the operators in the expansion and the three-point functions coefficients λ123; we will clarify this correspondence later.

From this argument we conclude that indeed g(u, v) is uniquely determined by applying this Operator Product Expansion (OPE).

1.9

Radial quantization and OPE

Usually in the quantization of a field theory we choose a direction of evolution called time and we consider U (t) ≡ eiH(t), where H is the Hamiltonian of the system. If we define a set of configurations over a temporal slice, say t = t1,

then H evolves it to some t = t2.

This is not a suitable choice for a CFT since we have defined the irreps starting from the primary state. The standard time evolution operator H would correspond to P0and we have seen that it does not commute with D. As

a consequence, it would evolve the primary state into a mixed combination of primaries and descendants. This argument leads us to choose a new evolution operator, namely D, with a related new foliation of the space-time.

In the standard quantization, the Hilbert space is a section of the space-time with t = const. Now we rather have a sort of “constant dilatation surface”: we think in terms of concentric spheres because the new evolution direction is taken along the radius, that is why we use to talk about Radial Quantization. The evolution operator in the Euclidean space is thus

U = e−D(r1/r0) , (1.49)

(28)

state, a concentric sphere of radius r1 we call |φfi. Evolving a state forward or backward in time in the standard view now corresponds to contract or expand the sphere with the parameter (r1/r0).

In this context, we assume that a Path Integral exists and is formally defined by the action SCFT, so that we can schematically write the probability

of transition from |φii to |φfi as hφf| e−D(r1/r0)|φii =

Z

CFT e−SCFT/~ , (1.50)

with the boundary conditions φ(|x| = r0) = φi and φ(|x| = r1) = φf.

In this formalism we define a state as a choice of the initial condition

φ(|x| = r0) = φi, while φf is free.

We can also extend this approach to write the vacuum state. In ordinary QFTs, the unnormalized vacuum can be written as the Path Integral over the field configurations with the limit t0 → −∞, namely12

|0iun= lim t0→−∞

Z

φ(t0)=φi

Dφ e−S/~ . (1.51)

If our theory is conformal, the previous definition translates into sending the sphere to a point (r0 → 0).

Next we consider the possibility of having an operator in the origin and define the state

|O(0)i ≡

Z

Dφ O(0)e−S/~ , (1.52)

where O(0) has the usual quantum numbers {l, ∆}. The Path Integral defines a state over the sphere with radius r1 that has the same quantum number of the

operator: this is the so-called Operator-State Duality which is the key concept of the OPE. In general the insertions of operators in the sphere generate a state that we can expand in a basis of the eigenstates of D: for example, we can interpret the vacuum of a CFT as the insertion of the identity operator inside the sphere.

If we now complicate things a little requiring that the operator is inserted in a point different from the origin, we have

12If we divide this expression by its norm we get the normalized vacuum which depends

(29)

|O(x)i ≡

Z

Dφ O(x)e−S/~ , (1.53)

with

|O(x)i = O(x) |0i = e−ixµP

µO(0)eixµPµ|0i = |O(0)i − ix · P |O(0)i + . . . .

(1.54)

The first term is the primary with eingenvalue ∆O but the second one has

eigenvalue (∆O+1) and so forth for the other terms. We remind that Pµ|O(0)i is not a primary because

Kν(Pµ|O(0)i) = ([Kν, Pµ] + PµKν) |O(0)i 6= 0 . (1.55) Finally we can also consider a pair of operators, one in the origin and the other one in a different point:

|ψi ≡

Z

Dφ O(x)O(0)e−S/~ . (1.56)

We can decompose this state in the complete basis of the dilatations and the rotations to write

|ψi =X

i

c(x, P ) |ii = (c0(x) + cµ1(x)Pµ+ cµν2 PµPν + . . . ) |ii , (1.57)

where |ii corresponds to a local operator in the origin, namely |ii = Oi(0) |0i. Even if the sum is over all the states of the theory, we expect the conformal symmetry to be able to group the states in a non-trivial way: we can restrict the sum only to primary states. Each of them comes with its descendants. In conclusion we can expand a product of two operators inside a correlation function writing it as a sum over all the primaries, namely

Oi(x) × Oj(0) =

X

primary Ok

COiOjOk(x, P )Ok(0) . (1.58)

Thanks to the translational invariance, we can generalize last expression for a pair of operators in any two points,

(30)

Oi(x) × Oj(y) =

X

primary Ok

COiOjOk(x − y, P )Ok(y) (1.59)

and this is true inside the correlation functions, whenever the other operators {On(xn)} have |x2n| > |x12|. Then the scale invariance allows us to fix the

coefficients of the expansion up to a number; we can write

Oi(x) × Oj(y) = X primary Ok cOiOjOk |x − y|i+∆j−∆k Ok(y) + descendants ! , (1.60)

where {cOiOjOk} is a set of pure numbers usually called OPE coefficients or structure constants.

1.10

Conformal Blocks and crossing

symme-try

Having understood how to characterize a CFT, we now proceed to the study of the four-point functions to reach an important consistency set of equations. For simplicity, let us consider the four-point function of scalar primary fields, hO1(x1)O2(x2)O3(x3)O4(x4)i, where we choose a pair of operators and

expand it via its OPE. Before entering in details of the computation, we un-derline that there is no counterpart of Wick’s theorem: we do not have to sum over the possible permutations but just consider a specific OPE-channel. For example, if ∆1 = ∆2 and ∆3 = ∆4, we can write

hO1(x1)O2(x2)O3(x3)O4(x4)i =

X

i

C12i(x12, ∂1)C34i(x34, ∂3)hOi(x1)Oi(x3)i

=X i λ12iλ34i |x12|2∆1|x34|2∆3 gi,li(u, v) , (1.61)

where as usual the prefactor reproduces the right properties of the four-point functions under the conformal transformations. This implies that the other factor must be an invariant, even if it depends on the exchanged operators.

The quantity gi,li(u, v) is called Conformal Block and it is the contribution

(31)

resummed in just one function.13

In the previous paragraph, we used the OPEs in the channels |x12| → 0

and |x34| → 0, but it is clear that we could also consider for instance |x14| → 0

and |x23| → 0. In the case of all scaling dimensions of the operators be equal

to ∆, therefore we have hO1(x1)O2(x2)O3(x3)O4(x4)i = X i λ12iλ34i |x12|2∆|x34|2∆ gi,li(u, v) =X i λ14iλ32i |x14|2∆|x32|2∆ gi,li(v, u) , (1.62)

where the index i selects those operators Oi contained in the proper OPE. In the second equality the arguments of the Conformal Block are swapped because when we make the replacement (2 ↔ 4) we automatically also have that (u ↔ v) (1.42).

Since we have three ways to make pairs out of four operators, typically we talk about the s-channel, corresponding to (12)(34), the u-channel for (13)(24) and finally the t-channel, that is (14)(23). In diagramatic language this corre-sponds to the symbolic equivalence

2 1 P k0 k 0 4 3 3 1 = P k00 k 00 4 2 1 = P k000 k000 2 4 3

The Crossing Equation in (1.62) links consistently the quantum numbers of the operators lying in the spectrum of the CFT with the interactions, that is to say the OPE coefficients.

Starting from any four operators, we can write down these consistency equations: we generally have an infinite number of them, leading us to define a CFT as a set of quantum numbers and OPE coefficients that satisfy all the possible crossing equations.14

13Often the quantities W

Oi ≡ C12i(x12, ∂1)C34i(x34, ∂3)hOi(x1)Oi(x3)i are called Confor-mal Partial Waves (CPWs) and we have WOi= K4gi,li(u, v), even if the distintion between CPWs and Conformal Blocks is a matter of convention.

14In the case of four identical scalar primary operators, the crossing equation acquires a

(32)

We finally remark that only certain operators can contribute to the four-point function and therefore just some of them can appear in the Conformal Blocks; for example, it can be seen that the OPE of two scalar operators contains only traceless symmetric tensors.

For further informations on Conformal Blocks and in particular for their possible derivation from the Casimir differential equation we refer to Appendix A.

1.11

Conformal Bootstrap

We conclude this chapter presenting the fascinating Conformal Bootstrap ap-proach to the study of CFTs.

We have seen that most of the physically important critical points are described by conformal invariant theories. It is therefore interesting trying to solve them using only the conformal invariance: this results in a non-perturbative approach.

In the 70s the fundamental works of Ferrara et al. [16] and Polyakov [17] presented the conjecture of conformal invariance which resulted in an alge-braic approach with profound consequences on OPE.15 In particular, starting

from the enhanced conformal symmetry of the critical points, one introduces the assumption of an associative operator algebra. This leads to important constraints on the observables defined in the CFTs.

In the modern sense of the term, the concept of Conformal Bootstrap was first introduced by Belavin et al. [18] in the investigation of the 2d CFTs. They proposed for the first time the study of the critical 2d Ising model interpreted as a CFT. Their procedure constrained the parameters of the theory to determine its spectrum, with no need of a Lagrangian formulation.

We indeed know that the anomalous dimensions of the operators are di-rectly related to the critical exponents measured in experiments. One could obtain them starting from the Crossing Equation: the knowledge of the Con-formal Blocks leads to a system of equations giving the OPE coefficients and the scaling dimensions.

This apparatus worked very-well for the the 2d CFTs and gave rise to their classification in terms of Minimal Models. Later the Conformal Bootstrap

15In this framework one of the assumptiones is that the product of local operators is

(33)

Figure 1.2: Ising critical exponents. The Conformal Bootstrap determinations (blue region) vs Monte Carlo studies (dashed rectangle). Figure taken from [20].

idea has been improved numerically to the d > 2 case [19], with a particular interest in 3d systems. A very important result has been the world record determination of the leading scaling dimensions in the 3d Ising model [20], see figure 1.2.

We can see that this method leads to a small isolated allowed region for the scaling dimensions of the Ising model. Moreover, it is actually applicable to the general 3d O(N ) vector models, although the result is not as accurate as the Ising study, as it’s clear in figure 1.3.

The general aim is to isolate each CFT to classify the space of CFTs.

We’ve indeed seen that we can completely characterize a CFT by its CFT data, namely {∆O, lO, Cijk}, since they imply the knowledge of any correlation function. Determining this set for physical systems can be very difficult. The Conformal Bootstrap introduces the interesting new possibility of focusing only on those values of dimensions and coefficients that are actually physically consistent. One therefore obtains universal bounds over the most interesting quantities, without needing the whole infinite set of CFT data.

In this thesis we will use the Numerical Conformal Bootstrap results for the scaling dimensions and OPE coefficients of the 3d O(N ) models, for N = 1 and N = 2.

(34)
(35)

Chapter 2

Conformal perturbation theory

We want to study the short-distance behaviour of the correlation functions of a QFT that lies at a fixed point of the RG flow and so it exhibits scale invariance and typically conformal invariance. We present the formalism for a general d-dimensional theory.

At the beginning we have the so-called unperturbed CFT, that is our initial theory with its operators and its correlation functions. We deform it by adding a set of operators that could change its IR behaviour. This will start a new RG flow that will end at a possible new fixed point, changing for example the scaling dimensions of the operators or equivalently the critical exponents. To verify this, we consider the initial fixed point Hamiltonian H∗, so that the perturbed partition function is1

Z = Tr e−H∗+Pigi R

ddx O

i(x) , (2.1)

that we can expand in powers of the couplings. The exponential of the per-turbed theory can be defined as usual by its power series

1Actually, the standard form of the perturbed partition function has usually a minus in

front of the perturbation. However, in studying random systems, couplings depend on space and so we can use CPT only in the replicated theory, as mentioned in the introduction. In this framework it is a common feature that in the leading perturbation due to the disorder the sign of the coupling is negative. It can be seen that this does not lead to problems regarding whether the Hamiltonian is bounded from below: the partition function is still well-defined. See [7].

(36)

eRddx PigiOi(x)=1 + Z ddx X i giOi(x) + 1 2 X i,j gigj Z ddx ddx1 Oi(x)Oj(x1) + 1 3! X i,j,k gigjgk Z ddx ddx1 ddx2 Oi(x)Oj(x1)Ok(x2) + . . . . (2.2)

We would like to extract the beta-functions of the couplings from this expres-sion and verify if there are new fixed points.

Following [8], we compute the beta-functions studying the overlaps hOm|0ig,V with the definitions

hOm| ≡ h0| Om(∞) , Om(∞) ≡ lim x→∞x 2∆OmOm(x) , |0ig,V ≡ ePigi R Vd dx O i(x)|0i , (2.3)

where g stands for the set of the couplings {gi} and V is a volume around the origin in which we deform the CFT. So we have2

hOm|0ig,V =hOm(∞)i +

X i gi Z V ddx hOi(x)Om(∞)i +1 2 X i,j gigj Z V ddx ddx1 hOi(x)Oj(x1)Om(∞)i +1 6 X i,j,k gigjgk Z V ddx ddx1 ddx2 hOi(x)Oj(x1)Ok(x2)Om(∞)i + . . . , (2.4)

where the correlation functions have to be evaluated w.r.t. the fixed point Hamiltonian. In order to perform such calculations, we can use the OPE of pairs of operators, namely

2Since we are doing perturbation theory around a critical point, we expect the partition

function to be singular and we can see it from the divergence of the integrals in the limit of infinite volume [7].

(37)

Oi(x) × Oj(0) ∼ δij |x|2∆O + X k Cijk |x|OkOk(0) + . . . , (2.5)

where the values of the OPE coefficients depend on the normalization of the operators of the CFT and they are computable by doing Wick contractions with free fields.

In (2.5) we see that the contribution of the identity comes only when we consider two equal operators which have a non-vanishing two-point function, as predicted by (1.32) and then we consider the contribution of other primaries. In the previous chapter we have also understood that there is no analogue of Wick’s theorem when we choose which pair of operators expand via the OPE; however now we deal with integrated correlators, so we should pay attention to which OPE converges in the various regions. Obviously we would have also all those terms from the outside of the convergence areas of the OPE-channels, but we are searching for the UV divergent terms.

Focusing on the short-distance singularities, we set x = 0 in (2.4) and

R

V ddx becomes a volume factor.3 We obtain

hOm|0ig,V ∼V gm+ 1 2V X i,j gigj Z

ddx hOi(0)Oj(x)Om(∞)i

+1 6V X i,j,k gigjgk Z

ddx ddx1 hOi(0)Oj(x)Ok(x1)Om(∞)i + . . . ,

(2.6)

where we have to regularize the d-dimensional integrals.

We will use this general expression to explore the case of one coupling or two couplings associated with marginal or slightly relevant deformations, when ∆Oi = d − δi. We will consider δi = 0 or δi  1 respectively.

2.1

CPT with one deformation

Let us first study the case of deforming the partition function with one cou-pling, meaning that i = j = m = 1 in (2.6), namely4

3We are actually ignoring x-dependent boundary effects on the other integrals.

4We streamline the notation removing the index from the coupling and the operator,

(38)

hO|0ig,V ∼V g + 1 2V g 2Z V ddx hO(0)O(x)O(∞)i + 1 6V g 3Z V ddx ddx1hO(0)O(x)O(x1)O(∞)i + . . . . (2.7)

We want to compute it for the two cases of a marginal operator or a slightly relevant one, using two different regulators.

2.1.1

Marginal case

Let us start with the case of ∆O = d. We use a cutoff regulator at the scale

µ to study the integrated three-point function via the OPE in the channel

|x| → 0; we thus have O(x) × O(0) ∼ 1 |x|2d + COOO |x|d O(0) + . . . , hO(0)O(x)O(∞)i = COOO |x|d + . . . , (2.8)

where we have used the vanishing of the one-point function and the definition of O(∞) (2.3). From now on we will refer to the OPE coefficients with the capital letter, contrary to what has been done in (1.60), but obviously they are always pure numbers.

In conclusion we have

Z

V

ddx hO(0)O(x)O(∞)i ∼ COOO Sd−1log µ , (2.9)

where

Sd−1

2πd/2

Γ(d/2) (2.10)

is the volume of the unit (d − 1) sphere. We streamline the notation defining

A ≡ COOO Sd−1, so that

Z

V

ddx hO(0)O(x)O(∞)i ∼ A log µ . (2.11)

Then we have to consider the integrated four-point function, where we have three regions of divergences corresponding to the three possible OPE-channels,

(39)

namely |x| → 0 or |x1| → 0 or |x − x1| → 0. First of all we have hO(0)O(x)O(x1)O(∞)i ⊃ 1 |x|2dhO(x)O(∞)i + COOO |x|d hO(0)O(x)O(∞)i , (2.12) from which Z V ddx ddx1 hO(0)O(x)O(x1)O(∞)i ∼3 Z Rd d x ddx1 ( 1 x2dhO(x1)O(∞)i +COOO xd hO(0)O(x1)O(∞)i ) , (2.13)

where the integrals are restricted to the region R defined by |x| < |x1| and

|x| < |x − x1|. It can be seen that (2.13) has a double-log contribution and

a single-log term: the first one enters the one-loop beta-function and we can easily see where it comes from. In fact, using the OPE in the channel |x1| → 0

in the three-point function, we obtain

3 Z R ddx ddx1 C2 OOO xd xd 1 = 3 2A 2log2µ ≡ B log2µ . (2.14)

To sum up, the logarithmic contributions to the integrated three- and four-point functions are

Z

V

ddx hO(0)O(x)O(∞)i ∼ A log µ ,

Z

V

ddx ddx1 hO(0)O(x)O(x1)O(∞)i ∼ B log2µ + C log µ .

(2.15)

We replace these developments into (2.7) and we demand the overlap, which is a physical observable, to be UV cutoff independent, i.e.

µ d

hO|0ig,V = 0 . (2.16)

(40)

β +A 2 2gβ log µ + g 2 ! +1 6 " 3g2β B log2µ + C log µ ! +g 3 6 2B log µ + C !# = 0 . (2.17)

We have to expand the beta-function up to the third order in the coupling,

β(g(µ)) = β1g2+ β2g3+ . . . , (2.18)

to bring out the relations between its coefficients and the coefficients of the logarithmic divergences from the integrated correlators. We easily get

β1 = − A 2 , β2 = − C 6 , A 2 = 2 3B , (2.19)

so that the one-loop beta-function is

β(g) = −1

2 Sd−1COOOg

2+ . . . . (2.20)

Finally we show how to find the next term studying the four-point correlator. We have seen that if we have four different points we can build the conformal cross-ratios u and v defined in (1.42), that in our case become

u = |x2| 2 |x1|2 , v = |x1− x2| 2 |x1|2 . (2.21)

We can generally write

hO(0)O(x)O(x1)O(∞)i = 1 |x|2d F |x1| |x| , |x − x1| |x| ! . (2.22)

Performing a simple change of variables we get

Z V ddx ddx1 hO(0)O(x)O(x1)O(∞)i ∼ Sd−1 Z 1 1/µ d|x| |x| Z ddx1F (x1) = Sd−12 Z 1 1/µ d|x| |x| Z |x| 1/µ

d|x1| hO(0)O(e1)O(x1)O(∞)i ,

(2.23)

where e1 = (1, 0, . . . , 0) is a unit vector in the 1-direction.

As already said, this expression has a double-log divergence and we have to subtract it to compute C in (2.15). To do this, the authors of [8] use the

(41)

trick of considering a generalized free field theory5 deformed by a double trace

operator and at the end they obtain the two-loop beta-function

β(g) = − 1 2Sd−1COOOg 2 1 6 Z ddx F (x) − 1 2C 2 OOO 1 xd(x − e 1)d + 1 xd + 1 (x − e1)d !! g3+ . . . . (2.24)

See their work for the details.

2.1.2

Anomalous dimensions in the marginal case

In addition to the perturbation with O, let us now deform the action also by

λR

ddx Φ(x). Similarly to what done in the previous section, we define the

state hΦ| given by an insertion of

Φ(∞) ≡ lim x→∞x

2∆ΦΦ(x) (2.25)

and the state

|0ig,λ,V ≡ egRVd dx O(x)

RVd dx Φ(x)

|0i . (2.26)

In expanding this perturbed groundstate we remind that we can compute the anomalous dimension taking a derivative of the beta-function w.r.t. λ and then imposing λ = 0. In this spirit we can consider directly only linear terms in λ in the expansion in (2.26). We find the overlap

hΦ|0iλ,g,V ∼V λ + V λg Z V ddx hΦ(0)O(x)Φ(∞)i +1 2V λg 2S d−1 Z ρ Z ddx hΦ(0)O(e1)O(x)Φ(∞)i + . . . . (2.27)

We suppose O to be marginal and we extract the beta-function for λ using the OPE

5One talks about a generalized free field because of its dimension being greater than the

free field one, that is (d − 2)/2. The correlation functions of this theory are computed by doing Wick contractions with the elementar definition of two-point functions.

(42)

O(x) × Φ(0) ∼ CΦΦO

|x|d Φ(0) + . . . . (2.28)

In particular, the doule-log contribution from the integrated four-point func-tion is

Sd−1

Z

ρ

Z

ddx hΦ(0)O(e1)O(x)Φ(∞)i ∼ Sd−12 log

2µ ( CΦΦO2 +1 2COOOCΦΦO ) (2.29)

and then as usual there is also a single-log term. At the end we obtain

βλ = −λ Sd−1CΦΦOg + 1 2C 0 g2+ O(g3) ! , (2.30)

where C0 is the coefficient of the single-log divergent term in the integrated four-point function wich requires a careful analysis. Taking a derivative of the beta-function w.r.t. λ we get the anomalous dimension of the related operator, that is

∆Φ(g) = ∆Φ+ γΦ,1g + γΦ,2g2+ . . . . (2.31)

The coefficients of this expansion are

γΦ,1= −Sd−1CΦΦO , γΦ,2 = −

1 2C

0

. (2.32)

2.1.3

Slightly relevant case

Next we analyze the possibility of deforming the pure theory with a slightly relevant operator. In this scenario, we actually assume the existence of a family of CFTs where δ = d − ∆O is a continuous parameter and we would like to see

the differences w.r.t. the previous case.

This time, instead of a cutoff regulator, we use dimensional regularization. Notice that a finite δ works as UV regulator for the theory, hence we expect poles in 1/δ. We can choose the bare coupling to reabsorb these divergences. In (2.7) we consider

(43)

and in the MS-scheme we have to determine the renormalization constant that cancels poles in 1/δ. We compute the beta-function starting from

dgB d log µ = 0 , (2.34) since β(g) = −δ " d dglog(Zgg) #−1 . (2.35)

We therefore define suitable counterterms, namely

gB = µδZgg = µδ 1 + g a δ + g 2 b δ2 + c δ ! + . . . ! g , (2.36)

implying that the beta-function is

β(g) = −δg + ag2+ 2cg3+ . . . , (2.37)

where we have to choose {a, b, c} to cancel the UV-divergences. Assuming

µδ ' 1, we have hO|0ig B,V ∼V gZg + 1 2V g 2 Zg2 Z V ddx hO(0)O(x)O(∞)i + 1 6V g 3Z3 g Z V ddx ddx1hO(0)O(x)O(x1)O(∞)i + . . . , (2.38)

from which at g2-order we find

a = β1 = − 1 2 Z V ddx hO(0)O(x)O(∞)i δ−1 , (2.39)

while at g3-order we have

c = β2 2 = − 1 6 Z V ddx ddx1 hO(0)O(x)O(x1)O(∞)i δ−1 , (2.40)

where we consider the term proportional to δ−1 in the Laurent expansion. First we explicity compute the one-loop beta-function, using the leading-OPE inside the three-point function, obtaining

β1 = − 1 2 Z V ddx hO(0)O(x)O(∞)i δ−1 = − 1 2COOOSd−1 , (2.41)

(44)

in agreement with (2.20). We underline that in this scenario the OPE coef-ficient is always evaluated in δ = 0 because all the CFT-data depend on δ smoothly by hyphotesis. So, the one-loop beta-function is

β(g) = −δg − 1

2COOOSd−1g

2+ . . . . (2.42)

The coefficient β2 is instead related to the term proportial to δ−1 of the

inte-grated four-point function. From the conformal invariance we can write

Z V ddx ddrhO(0)O(x)O(re1)O(∞)i δ−1 = Sd−1 Z V ddx Z 1 0 dr rd−1hO(0)O(x)O(re1)O(∞)i δ−1 , (2.43)

which can be further simplified using dilatation invariance to fix x1 → rx1 and

also using a Ward identity for the four-point function under rescaling. Finally we obtain Sd−1 Z V ddx Z 1 0 dr r 2d−1 r2∆OhO(0)O(x)O(e1)O(∞)i δ−1 = 3 Sd−1 Z Rd d xhO(0)O(x)O(e1)O(∞)i δ0 , (2.44)

where R : |x| < |x − e1|, |x| < |e1| = 1. So we see that

β2 = − 1 2 Sd−1 Z Rd dxhO(0)O(x)O(e 1)O(∞)i δ0 , (2.45)

which is divergent as |x| → 0. We renormalize it using the OPE in the channel |x| → 0, hO(0)O(x)O(e1)O(∞)i ⊃ 1 |x|2∆O + COOO2 |x|∆O , (2.46)

where it is now clear that the last term, after integration, must be subtracted to renormalize the theory. At the end we can write

β2 = − 1 2Sd−1 " Z Rd dxhO(0)O(x)O(e 1)O(∞)i − Sd−1COOO2 δ # . (2.47)

Figure

Updating...

References

Related subjects :