Study of the theta dependence of vacuum energy density in chiral effective Lagrangian models.

Dipartimento di Fisica E. Fermi Corso di Laurea Magistrale in Fisica

Curriculum Fisica Teorica

Study of the theta dependence of

vacuum energy density in chiral

eective Lagrangian models

Candidate: Supervisor:

Francesco Luciano Prof. Enrico Meggiolaro

Abstract

Quantum Chromodynamics (QCD) is the theory that describes the strong interaction within the Standard Model of the elementary particles. It is well known that, if we consider the so-called chiral limit, in which L quark masses (the physical relevant case being L = 2 and L = 3) are sent to zero, the symmetry group of QCD becomes larger: the Lagrangian turns out to be invariant under the chiral group G = U(1)L⊗ U (1)R⊗ SU (L)L⊗ SU (L)R.

However, it was noted that the SU(L)L⊗ SU (L)Rchiral symmetry is

spon-taneously broken due to the condensation of quark-antiquark pairs, while the U(1)A axial symmetry is aected by anomaly: the action is not

invari-ant under U(1)A rotations but rather acquires a contribution proportional

to the topological charge density Q, which (despite being a total divergence) contributes to the path integral because of the non-trivial topological struc-ture of QCD. In particular, this is due to the existence of a class of Euclidean solutions of the classical equation of motion with nite Euclidean action and integer topological charge, known as instantons.

The discovery of instantons raised an important issue: if one introduces an additional Lagrangian term Lθ = θQ (known as topological term or

θ-term), its contribution is non-vanishing; moreover, it explicitly breaks the CP invariance of the theory. So far no violation of the CP symmetry in strong interactions has been observed experimentally, and the most recent experimental measurements of the (CP-breaking) neutron electric dipole mo-ment set the upper bound |θ| < 10−10_{. However, the most important eect}

of the insertion of the topological term in QCD Lagrangian for the work of this thesis is the resulting non-trivial dependence on θ of the vacuum energy density vac(θ), which is of particular interest. Indeed, being θ very small, it

makes sense to consider the Taylor expansion of the vacuum energy density around θ = 0: the rst coecients of such an expansion turn out to be im-portant physical quantities. More in detail, the coecient of the quadratic term is equal to the topological susceptibility χ, whose pure-gauge part was related by Witten and Veneziano to the mass of the meson η0 _{in the chiral}

limit, while that of the fourth order term is equal to the second cumulant c4 of the probability distribution of the topological charge density Q, which

also enters in the η0_{− η}0 _{elastic scattering amplitude.}

One of the most important and useful tools to analyse the low-energy i

ii regime of QCD, paying particular attention to its chiral symmetries, is the eective Lagrangian formulation, which considers the eective mesonic de-grees of freedom as the fundamental elds of the theory (often gathered in a matrix eld U). Very important examples are: i) the Chiral Eective La-grangian, which describes the dynamics of the L2_{− 1(non-singlet)}

pseudo-Goldstone bosons coming from the breaking of the chiral SU(L)L⊗ SU (L)R

symmetry; ii) the Extended Linear σ model, that describes both scalar and pseudoscalar mesonic degrees of freedom (including also the singlets); iii) the model of Witten, Di Vecchia, Veneziano et al., which, in the frame-work of an expansion for large number of colours, implements the U(1) axial anomaly by properly introducing the topological charge density operator Q as an auxiliary eld in the eective Lagrangian.

Moreover, a fourth example of eective Lagrangian is also present in the literature, in which the U(1) axial symmetry is spontaneously broken independently of the chiral SU(L)L⊗ SU (L)R one because of the existence

of a new order parameter related exclusively to the U(1)A symmetry and

independent of the chiral condensate. The eects of such axial condensate on the dynamics of the mesonic eective degrees of freedom is described by including in the eective Lagrangian a new (exotic) mesonic eld X, in addition to the usual (q¯q) mesonic elds U. In this thesis, we shall refer to this model as the Interpolating model (since, in a sense, it interpolates between the Extended Linear σ model and the model of Witten, Di Vecchia, Veneziano et al.).

It is also known (mainly by lattice simulations) that, at temperatures above Tc≈ 150 MeV, the chiral SU(L)L⊗ SU (L)R symmetry gets restored

(in the chiral limit). For what concerns the U(1) axial symmetry, its restora-tion or not at Tc is still an important open question in hadronic physics.

The main goal of the work of this thesis is the systematic study of the modications brought to the vacuum energy density by a small, but non-zero, value of the coecient θ, both in the zero temperature case and in the nite temperature one (above Tc), using the four eective Lagrangians described

above. Our aim is to derive the expressions for the topological susceptibility χand for the second cumulant c4 starting from the θ dependence of vac(θ):

this method will allow to obtain important results in a rather simple way. First, we have considered the case T = 0. The guideline of the compu-tational method used here is to exploit the fact that θ  1: as the vacuum energy density is given by the minimum of the potential of the eective Lagrangian, we looked for the eld conguration which minimizes the lat-ter, expanding all the eld variables in powers of θ. This way, we are left with an expression for vac(θ) which is in the form of a Taylor expansion

around θ = 0: this allows to directly read the expressions for the topological susceptibility χ and for the second cumulant c4. In the case of the Chiral

litera-iii ture, using the same method. Also the nal results that we have found for the topological susceptibility in the three other cases conrm some relations already known in literature (which, however, were obtained by means of a dierent approach, by studying directly the two-point correlation function of the operator Q). Instead, the results that we have found for the second cumulant are all original. We have checked that all the expressions that we have derived for both χ and c4, while being dierent, behave as expected in

some proper theoretical limits.

After that, we have evaluated numerically our results, so as to compare them with each other and with the available lattice data.

At last, we have analysed the nite-temperature case, examining the Extended Linear σ model and the Interpolating one (which are well dened also above Tc). Here, we decided to base our study on the chiral limit,

i.e., on the expansion in the quark masses up to the rst non-trivial order: this allows to greatly simplify the computations, without any assumption on θ, which in this case is a real free parameter (i.e., it is not obliged to be very small). Also in this case, our results for the topological susceptibility reproduce some expressions already known in the literature, even though they were obtained with the mentioned dierent approach, while those for the second cumulant are original.

Contents

Introduction vi

1 QCD chiral symmetries 1

1.1 Quantum Chromodynamics Lagrangian . . . 1

1.2 QCD historical development and main properties . . . 3

1.3 Chiral symmetries of QCD . . . 5

1.4 Exact and spontaneously broken symmetries . . . 7

1.5 U(1) axial symmetry . . . 9

1.5.1 't Hooft solution to U(1) problem . . . 10

1.5.2 Witten-Veneziano solution to U(1) problem . . . 11

1.6 Chiral condensates . . . 14

1.6.1 The chiral condensate . . . 15

1.6.2 Restoration of chiral and U(1) axial symmetries at -nite temperature . . . 15

1.7 The topological theta term . . . 17

2 Chiral eective Lagrangian models 23 2.1 Eective degrees of freedom of QCD . . . 23

2.1.1 Linear and non-linear parametrizations . . . 25

2.2 The Chiral Eective Lagrangian to O(p2_{) . . . 28}

2.3 The Extended Linear σ model . . . 30

2.4 The eective Lagrangian model of Witten-Di Vecchia-Veneziano et al. . . 32

2.5 An Interpolating model with the inclusion of a U(1) axial condensate . . . 34

3 Topological susceptibility and second cumulant: results at T=0 39 3.1 General remarks on the procedure . . . 39

3.2 The Chiral Eective Lagrangian O(p2_{) . . . 41}

3.2.1 First case: L=2 . . . 42

3.2.2 Second case: L>2 . . . 43

3.2.3 Considerations on the results . . . 46 iv

CONTENTS v

3.3 The Extended Linear σ model . . . 47

3.3.1 Considerations on the results . . . 50

3.4 The model of Witten, Di Vecchia, Veneziano et al. . . 52

3.4.1 Consideration on the results . . . 54

3.5 The Interpolating model . . . 56

3.5.1 Consideration on the results . . . 60

3.6 Summary of the results . . . 61

3.6.1 Topological susceptibility . . . 62

3.6.2 Second cumulant . . . 62

3.6.3 Chiral limit . . . 63

3.6.4 Innite quark-mass limit . . . 63

3.7 Numerical results . . . 64

3.7.1 The Chiral Eective Lagrangian O(p2_{) . . . 66}

3.7.2 The Extended Linear σ model . . . 66

3.7.3 The model of Witten, Di Vecchia, Veneziano et al. . . 67

3.7.4 The Interpolating model . . . 67

3.7.5 Comparison of the results with the literature . . . 68

4 Topological susceptibility and second cumulant: results at nite temperature 70 4.1 Finite temperature analysis in the case L ≥ 3 . . . 71

4.1.1 The Extended Linear σ model . . . 71

4.1.2 The Interpolating model . . . 74

4.2 Finite temperature analysis in the case L = 2 . . . 78

4.2.1 The Extended Linear σ model . . . 78

4.2.2 The Interpolating model . . . 82

Conclusive remarks 86

Introduction

Quantum Chromodynamics (QCD) is the theory that describes the strong interaction, within the Standard Model of the elementary particles, as the re-sult of interactions between quarks, the fundamental constituents of hadrons, and gluons, massless vector bosons which mediates the interactions. In par-ticular, QCD is formulated as a non-abelian gauge theory related to the colour group SU(3)c, where the quark matter elds are minimally coupled

to the gauge gluon ones. After the formulation of the theory, QCD was successfully compared with high-energy experimental evidences: the rst theoretical result were obtained using the perturbation theory, based on the assumption that the coupling constant g is small and that all the physical quantities of interest can be expanded as a perturbative series in powers of g.

The reason why this perturbative approach describes very well high-energy phenomena lies in a peculiar property of QCD, usually called asymp-totic freedom: it takes its name from the fact that the interactions between quarks, mediated by the gluons, become weaker as the energy increases, so that QCD tends asymptotically to become a free theory. This property jus-ties the use of the perturbative approach in the high-energy regime, due to the fact that the coupling constant g is small; on the contrary, the low-energy regime is characterized by a value of g comparable with unity, if not larger, and is therefore dominated by non-perturbative eects. Nevertheless, such a regime is very interesting since, from the experimental point of view, many important phenomena involving strong interactions take place at low energies, as for example the formation of bound states of quarks (hadrons) or the so-called quark connement. In this case, perturbation theory fails and ceases to be predictive: a dierent approach to study the behaviour of the theory in the low-energy regime is thus needed.

To address the approach used in this thesis, a brief introduction to the chiral symmetries of QCD is necessary. The symmetry group of QCD La-grangian is ˜G = U (1)u ⊗ U (1)d⊗ . . ., which leads to the conservation of

the number of quarks of each avour in strong interactions. However, if we consider the so-called chiral limit, in which L quark masses (the physical rel-evant case being L = 2 and L = 3) are sent to zero, the symmetry group of

INTRODUCTION vii QCD becomes larger: the Lagrangian turns out to be invariant under inde-pendent rotations of the chiral components of the quark elds (usually called left and right). So, the new enlarged symmetry group, which is called chiral group, is G = U(1)L⊗ U (1)R⊗ SU (L)L⊗ SU (L)R, where the subscripts L

and R stand for left and right. However, studying the structure of the observed hadronic multiplets, it was noted that such a symmetry is not real-ized exactly (in the Wigner-Weyl way), but it is rather spontaneously broken down to its vectorial subgroup H = SU(L)V ⊗ U (1)V. It is well known that

the SU(L)L⊗ SU (L)R chiral symmetry gets spontaneously broken due to

the condensation of quark-antiquark pairs which gives to the so-called chiral condensate h¯qqi a value dierent from zero. For what concerns the U(1) axial symmetry, the situation is more complicated. As we do not observe any con-served quantum number related to it, the rst hypothesis argued was that the axial symmetry were spontaneously broken. However, the attempts to interpret it as a broken symmetry failed: in particular, Weinberg [12] found an upper limit for the mass of the would-be Goldstone coming from such a breaking (the best candidate was the η0 _{meson), which was m}(th)

η0 <

√ 3mπ0,

violated from the experimental value of the mass of the meson η0_.

Actually, the U(1) axial symmetry is aected by the Adler-Bell-Jackiw anomaly [13]: the action is not invariant under U(1)A rotations due to the

fact that the fermionic functional measure is not unchanged, but rather ac-quires a contribution proportional to the topological charge density Q, a quantity related to the topology of gluon eld congurations. Since the topological charge density Q can be expressed as a total divergence of the Chern-Simons current, however, it was initially believed that it did not con-tribute to the path integral and, so, that the axial anomaly would not have had any eect on the theory. This is not true, due to the non-trivial topolog-ical structure of QCD: indeed, it was found [14] that there exists a class of Euclidean solutions of the classical equation of motion with nite Euclidean action and integer topological charge, known as instantons, which makes the contribution of the anomaly in the quantum theory be non-vanishing. They are purely non-perturbative objects which cannot be studied with the usual technique of the expansion in the small coupling g (or similar ones) and, so, they do not arise in the perturbation theory approach.

The discovery of instantons made clear that the nature of the violation of the Weinberg's limit was topological, and that topology was a determinant aspect of the dynamics of the low-energy degrees of freedom in QCD. As an important example, Witten and Veneziano [16] related the theoretical value of the mass of the η0 _{meson in the chiral limit to the pure-gauge part of the}

topological susceptibility χ, a quantity which is intimately related to the two-point correlation function of the topological charge density operator Q. The existence of instantons also raised another important issue: if one introduces

INTRODUCTION viii in the QCD Lagrangian an additional term Lθ = θQ, its contribution in the

quantum theory would be non-zero thanks to the existence of congurations with non-trivial topology. This term, usually referred to as topological term or as θ-term (from the name of the coecient that appears in front of it), is particularly interesting since it introduces an explicit breaking of the CP symmetry in QCD , absent in the original theory, referred to as strong-CP violation. So far no violation of the CP symmetry in strong interactions has been observed experimentally, so that θ is believed to be zero (recent mea-surements of the CP-breaking neutron electric dipole moment [19] suggest an upper bound of |θ| < 10−10_{), despite the fact that the coecient θ could}

assume, in principle, whatever value in [0, 2π). This ne-tuning problem, despite many possible solutions have been proposed (one of the most impor-tant came from Peccei and Quinn [22] and hypothesized the existence of a new particle called axion), is still an open issue.

However, it is nonetheless interesting to study the dependence of QCD on nite θ, since the insertion of the topological term with θ 6= 0 in QCD Lagrangian causes the modication of the partition function of the theory: indeed, such a new Lagrangian term brings a change in the action, which acts as a statistical weight in the path integral formulation of the partition function. In particular, this modication results in a non-trivial dependence on θ of the vacuum energy density vac(θ), which is of particular interest.

Indeed, being θ very small, it makes sense to consider the Taylor expansion of the vacuum energy density around θ = 0: the rst coecients of such an expansion turn out to be important physical quantities. More in detail, the coecient of the quadratic term is equal to the topological susceptibility χ, while that of the fourth order term is equal to the second cumulant c4 of

the probability distribution of the topological charge density Q, which also enters in the η0_{− η}0 _{elastic scattering amplitude.}

One of the most important and useful tools to analyse the low-energy regime of QCD, paying particular attention to its chiral symmetries, is the eective Lagrangian formulation: it was rst proposed by Weinberg [26] and then rened and widely used to study dierent aspects of the dynamics of the eective hadronic degrees of freedom at low energy. Due to the colour connement, indeed, we do not observe free quarks or gluons, so that the eective low-energy degrees of freedom are actually described by multilinear operators carrying the quantum numbers of the mesons we observe in nature. The basis of the eective Lagrangian formulation lies in considering them as the fundamental elds of the theory (often gathered in a matrix eld called U). Starting from the low-energy limit of the expansion in the momentum p, the famous Chiral Eective Lagrangian O(p2_{) was built [1, 2, 3, 4, 27, 28, 29],}

which is the basis of all the other most used models: this Lagrangian de-scribes the dynamics of the pseudo-Goldstone bosons coming from the break-ing of the chiral SU(L)L⊗ SU (L)R symmetry, therefore neglecting the axial

INTRODUCTION ix singlet meson η0_{, which is pretty massive with respect to the non-singlet}

mesons due to the anomalous contribution to its mass. Using the Chiral Eective Lagrangian O(p2_{), it was possible to obtain important information}

on the mass spectrum of the pseudo-Goldstone bosons. The results obtained were in good accordance with the experiments, except for the η meson mass: in this case, there was a small but relevant discrepancy among the result predicted by the theory and the experimental data. The simplest solution to this problem was found by adding the axial singlet eld to the Chiral Eective Lagrangian: two of the most used models which implement such an addition are the Extended Linear σ model [30], that describes both scalar and pseudoscalar degrees of freedom, including also the singlets ('t Hooft [31, 32] argued that this model is the correct description of the dynamics of the theory in presence of instantons), and the model of Witten, Di Vec-chia, Veneziano, et al. [35, 36, 37], which, in the framework of an expansion for large number of colours, implements the U(1) axial anomaly by prop-erly introducing the topological charge density operator Q as an auxiliary eld in the eective Lagrangian, and correctly reproduces the anomalous contribution to the mass of the η0 _{(given by the famous Witten-Veneziano}

formula). Moreover, another example of eective Lagrangian is also present in the literature, in which the U(1) axial symmetry is spontaneously broken independently of the chiral SU(L)L⊗ SU (L)R one because of the existence

of a new order parameter [39] related exclusively to the U(1)Asymmetry and

independent of the chiral condensate. The eects of such axial condensate on the dynamics of the eective degrees of freedom is described by including in the eective Lagrangian [39, 40, 41] a new (exotic) mesonic eld X, in addition to the usual mesonic elds U, which interact with each other in a non-trivial way. In this thesis we shall refer to this model as the Interpolat-ing model since, in a sense, it interpolates between the two previous models. Through this eective Lagrangian, the phenomenological eects coming from the possible presence of this new condensate, in particular the modication on the mass spectrum (both at T = 0 and at nite temperature) and on the meson radiative decay, have been studied.

All what has been described until now considers T = 0. However, it is known (mainly by lattice simulations [17]) that, at temperatures above a certain temperature Tc≈ 150MeV, thermal uctuations break up the chiral

condensate, causing the complete restoration of the chiral SU(L)L⊗SU (L)R

symmetry in the chiral limit: this leads to a phase transition called chiral transition. For what concerns the U(1) axial symmetry, its restoration or not at T = Tcis still an important open question in hadronic physics.

The main goal of the work of this thesis is the systematic study of the modications brought to the vacuum energy density by a small, but non-zero, value of the coecient θ, both in the zero temperature case and in the nite

INTRODUCTION x temperature one (above Tc), using the four eective Lagrangians described

above. Our aim is to derive the expressions for the topological susceptibility χand for the second cumulant c4 starting from the θ dependence of vac(θ):

this method will allow to obtain important results in a rather simple way. The thesis is organized as follows.

In Chapter 1 we present the historical development and the most im-portant properties of Quantum Chromodynamics, and describe its chiral symmetries in the limit of a certain number of massless quarks, paying par-ticular attention to their spontaneous breaking and to the presence of the axial anomaly; then, we briey resume, starting from lattice results, the role of the chiral condensate as an order parameter for the chiral symmetries and their restoration with temperature; at last, we analyse the modications brought to the theory by the insertion of the θ-term in the Lagrangian of QCD and their importance in the work of this thesis.

In Chapter 2 we present the chiral eective Lagrangian formulation and the four frameworks we have worked on in this thesis: starting from the anal-ysis of the eective degrees of freedom of QCD in the low-energy regime, we rst describe the Chiral Eective Lagrangian O(p2_{) proposed by Weinberg;}

then, we present the Extended Linear σ model, the model of Witten, Di Vecchia, Veneziano, et al. and the Interpolating model.

In Chapter 3 we present and critically analyse the results that we have obtained for the topological susceptibility χ and the second cumulant c4 in

the four dierent frameworks, considering T = 0, describing also the com-putational techniques used in our work; the last section of the Chapter is devoted to evaluate numerically, where possible, our results so as to compare them with each other and with the available lattice results.

In Chapter 4 we present the results that we have obtained in the nite temperature case: in particular, we have analysed the behaviour of the vac-uum energy density in the Extended Linear σ model and in the Interpolating model in the range of temperature Tc < T < TU (1), that is, in the situation

in which the chiral SU(L)R⊗ SU (L)L symmetry is restored while the U(1)

axial one is still broken.

In the last chapter we draw our conclusions, summarizing the results obtained in our work and discussing their possible future applications and development.

Chapter 1

QCD chiral symmetries

1.1 Quantum Chromodynamics Lagrangian

Quantum Chromodynamics (QCD) is the quantum eld theory that de-scribes strong interactions: it is a non-abelian gauge theory related to the colour group SU(3)c. The fundamental matter variables of the theory are the

quark elds, which appear in six avours (up,down,strange,charme,bottom,top). Each one of these avours exists in its turn in three states of colour:

qf =   q_f1 q_f2 q3 f   (1.1)

where the subscript f indicates the particular avour. The three colour components of the quark eld transform under the action of the colour group according to the fundamental representation of SU(3), that is:

qf −→ U (x) qf (1.2)

with

U (x) = eiθa(x)Ta (1.3) where θa_{(x) are the parameters of the transformation, while T}a _{are the}

generators of the fundamental representation of SU(3), related to the Gell-Mann matrices. They satisfy the commutation rule:

h

Ta, Tbi= ifabcTc (1.4)

where fabc are the structure constants of SU(3). Moreover, the generators

are normalized as:

Tr  TaTb  = 1 2δab (1.5) 1

1.1. QUANTUM CHROMODYNAMICS LAGRANGIAN 2 If we now dene q =      qu qd qs ...      (1.6)

where each component of the vector is given by (1.1), the (classical) La-grangian density of the theory is given by [6]:

L = −1

2Tr (FµνF

µν_{) + q (iγ}

µDµ− M ) q (1.7)

where M = diag (mu, md, ms, . . .)is the quarks mass matrix.

Dµ is the covariant derivative dened as:

Dµ= ∂µ+ igAµ (1.8)

where g is the QCD coupling constant, while Aµ is the matrix gluon gauge

eld dened as Aµ= 8 X a=1 Aa_µTa (1.9)

and the elds Aa

µ are the gluon gauge elds.

Fµν is the eld density tensor and it is dened, starting from (1.9), as:

Fµν = ∂µAν − ∂νAµ+ ig [Aµ, Aν] ≡ 8 X a=1 F_µνa Ta (1.10) where: F_µνa = ∂µAaν − ∂νAaµ− gfabcAbµAcν (1.11)

If we now request (1.7) to be invariant under the transformation (1.2), we nd the transformation rule of the gauge elds:

Aµ−→ U AµU†+

i

g(∂µU ) U

†

1.2. QCD HISTORICAL DEVELOPMENT AND MAIN PROPERTIES 3

1.2 QCD historical development and main

proper-ties

In the mid-20th _{century, experimental physicists started classifying the}

hadrons in multiplets, nding that these multiplets coincide with the octet and the decuplet of SU(3). This work leads to the basis of QCD, the quark model, proposed by Gell-Mann and Ne'eman [7] in the early 60's. In their works, they propose that QCD presents an SU(3) symmetry (nowadays called avour symmetry) which generalizes the isospin SU(2) symmetry proposed by Heisenberg in the 20's.

In the quark model, Gell-Mann asserted that the whole multiplets struc-ture of hadrons could be explained by the existence of particles of spin 1

2,

which he called quarks, present in three avours: up, down and strange. The observed hadrons would have been bound states of three quarks (baryons) or of a quark-antiquark pair (mesons).

The Gell-Mann's SU(3) symmetry, as the Heisenberg's SU(2), is not ex-act: it is explicitly broken by the fact that the quark mass matrix is not identity-like. Nevertheless, as we will study in more detail later, these sym-metries are a good approximation to study strong interactions in certain limits.

Turning back to the history, the hypothesis that the hadronic states ob-served in nature were composite particles were hinted also by the results of the experiments on deep inelastic scatterings: the study on high energy collisions between electrons and hadrons suggested that the latter had an internal structure, while up to that time they were considered as elementary. The quark model, though simplifying the description of strong interac-tions, left many problematic issues unresolved. The most important were: no free quark had been observed; there were some states (for instance ∆++_,

∆−) whose wave functions could not be arranged so as to be in accordance with the spin-statistics theorem.

In order to solve these problems, Han, Nambu, Greenberg and Gell-Mann [8] proposed a new quantum number, which they called colour: according to their theory, each avour state should exist in a certain number of coloured copies. To make experimental data (in particular, e+_e− _{→ hadrons cross}

section and π0 _{→ γγdecay widht) t the theory, the proper number of colour}

turned out to be three. Starting from this assumption, these scientists hy-pothesized the link between the colour charge and a new non-abelian gauge group: the colour SU(3)c gauge group of QCD.

1.2. QCD HISTORICAL DEVELOPMENT AND MAIN PROPERTIES 4 The introduction of the colour quantum number permitted to easily solve the spin-statistics problem, giving the correct properties of symmetry to the problematic wave functions. For what concerns the absence of free quarks, it had been formulated the so-called colour connement postulate: formally, it establishes that a generic hadronic state must be invariant under the action of SU(3)c, that is, it must be a singlet under transformations induced by the

colour gauge group.

Another important property of strong interactions is the so-called asymp-totic freedom. As a result of experiments, a precise behaviour of those inter-actions had been found out: they become weak at small distances (that is, at high energy), while they became stronger at large distances among quarks (that is, at low energy: this property of QCD is called infrared slavery). In 1973, Gross and Wilczek [9], and Politzer [10] showed that QCD's β function, dened as:

β = µ∂gR

∂µ (1.13)

where gRis the renormalized coupling constant while µ is the renormalization

scale, is negative in a neighbourhood of gR = 0. Indeed, it is possible to

expand the β function in powers of the renormalized coupling constant as: β = −β0g3R− β1gR5 − β2gR7 + . . . (1.14)

where the coecient βi gathers the contributions from i + 1-loops diagrams.

The values of the rst two coecients are: β0 = 1 (4π)2  11Nc− 2Nf 3  (1.15) β1 = 1 (4π)4  34 3 N 2 c −  13 3 Nc− 1 Nc  Nf  (1.16) where Ncis the number of colour of the theory while Nf denotes the number

of avours. We do not give the results of the successive coecients since a very important property demonstrated for the β function is that the two coecients β0 and β1 are totally independent of the particular

renormaliza-tion scheme chosen, while the successive ones are scheme-dependent. For Nc = 3, the coecient β0 is positive for Nf < 17, and the rst term is the

dominant contribution in the perturbative regime, where gR  1. Hence,

in this regime, due to (1.13), while the renormalization scale µ increases the renormalized coupling constant gR decreases, tending asymptotically to

zero (this is the asymptotic freedom property of QCD); on the contrary, if µ decreases, gR increases. However, the expression (1.15) is a one-loop

re-sult and, therefore, it is the dominant contribution only in the perturbative regime: when gRincreases, becoming comparable with unity, it is necessary

1.3. CHIRAL SYMMETRIES OF QCD 5 to take into account also the contributions due to the diagrams with two or more loops, which modify the β function.

More in detail, substituting β = −β0g_R3, with the expression (1.15), into

(1.13) and integrating, it is immediate to nd the expression for the strong coupling constant to one loop level:

αS(µ) ≡ g2_R(µ) 4π = 1 4πβ0ln  µ2 Λ2 QCD  (1.17)

The parameter ΛQCD is called QCD scale parameter, and it is of order

ΛQCD ≈ 0.4 GeV in the MS renormalization scheme [11].

From what has been said, it is evident that perturbation theory is not an adequate tool to be used while investigating QCD in the low-energy regime. However, this region oers many interesting phenomena: the desire of knowl-edge pushed to develop new techniques to study the low-energy limit, the most important of which for this thesis is the chiral eective Lagrangian formulation, which will be described in greater detail in the next chapter.

1.3 Chiral symmetries of QCD

The Lagrangian (1.7) is invariant under transformation induced by the group ˜G = U (1)u⊗ U (1)d⊗ U (1)s⊗ . . ., which acts on the quark elds as

follow:

qf → eiαfqf (1.18)

Noether's theorem guarantees the existence of a conserved current Jµ_for

each generator of the symmetry group, to which is related a conserved charge Q =R d3x J0(x). In this particular case, the conserved currents are given by J_fµ= q_fγµq_f, where the subscript f indicates the particular avour, while the conserved charge Q = R d3_{x q}

fγ0qf =R d3x q_f†qf is related to conservation

of the number of quark of avour f.

However, if we now assume L light avour masses to be zero (the most usual cases are L = 2, in which the up and down quarks are massless, and L = 3, with also the strange one with no mass), it is easy to see that the symmetry group become larger: this is the so-called chiral limit.

To get convinced of this, it is convenient to consider the chiral compo-nents of the quark elds, usually called left and right, dened as:1

qL≡

1

2(1 + γ5) q ≡ PLq ; qR≡ 1

2(1 − γ5) q ≡ PRq (1.19)

1_{Here and in the rest of the thesis, γ}

1.3. CHIRAL SYMMETRIES OF QCD 6 In terms of these components, (1.7) can be rewritten as:2

L = −1

2Tr (FµνF

µν_{) + iq}

LγµDµqL+ iqRγµDµqR− qLM qR− qRM qL (1.20)

It is easy to see that, if M = 0, the Lagrangian (1.20) is invariant under independent rotation of qL and qR:

qL→ ˜VLqL ; qR→ ˜VRqR (1.21)

where ˜VL and ˜VR are elements of U(L), that is, unitary L × L matrix,

independent of x: ˜ VL= eiαLeiθ a LTa = eiαL_V L ; V˜R= eiαReiθ a RTa = eiαR_V R (1.22)

So, the enlarged symmetry group, usually called chiral group, is:

G = U (L)L⊗ U (L)R= SU (L)L⊗ SU (L)R⊗ U (1)L⊗ U (1)R (1.23)

It can be demonstrated that a transformation related to the subgroup SU (L)L⊗ SU (L)R can be expressed as a composition of a vector

trans-formation (in which VL = VR ≡ V) and an axial one (in which VL =

V_R†≡ A). To prove this, it is sucient to demonstrate that, given a general SU (L)L⊗ SU (L)R transformation:

(

qL→ VLqL

qR→ VRqR

(1.24) it is always possible to nd two elements V and A such that (1.24) can be

rewritten as: ₍ qL→ AV qL qR→ A†V qR (1.25) Comparing (1.24) and (1.25), we nd:        A =VLV_R† 1₂ V =  VLV_R† 1 2 VR (1.26) We point out that, being VLand VRelements of SU(L), also the product

VLVR† is an element of the group and, as such, it can be expressed as eiθ

a_Ta

for certain parameters θa_{. Given that, the matrix}_V LV_R†

1₂

turns out to be simply eiθa_Ta_/2

.

1.4. EXACT AND SPONTANEOUSLY BROKEN SYMMETRIES 7 In an analogue but easier way, it can be demonstrated that a U(1)L⊗

U (1)R transformation can be expressed as a composition of a vector

trans-formation (in which αL= αR) and an axial one (in which αL= −αR). So,

it is possible to identify (1.23) with:

G = SU (L)V ⊗ SU (L)A⊗ U (1)V ⊗ U (1)A (1.27)

This symmetry group acts on the quark elds as:            U (1)V : q → eiαVq U (1)A: q → eiγ5αAq SU (L)V : q → eiθ a VTa_q SU (L)A: q → eiγ5θ a ATa_q (1.28)

According to Noether's theorem, for each generator of the symmetry group there exists a conserved current. In the case of the group G, they are:

         U (1)V : Jµ= qγµq U (1)A: J5µ= qγµγ5q SU (L)V : Vaµ= qγµTaq SU (L)A: Aµa = qγµγ5Taq (1.29)

These currents are all conserved, at least at classical level, in the chiral limit. The mass term in the Lagrangian (1.20) explicitly breaks up the symmetry of the Lagrangian under the action of the group G, preserving only its U(1)V part.

1.4 Exact and spontaneously broken symmetries

In a physical system, a continuous global symmetry can be realised in two dierent ways, which lead to dierent properties of the spectrum of the theory:

• Exact symmetry (á la Wigner-Weyl): all the generators of the sym-metry group annihilate the vacuum state which is, therefore, invariant under all the transformations induced by the group. In this case, the particle spectrum is organized in multiplets corresponding to the irre-ducible representations of the symmetry group;

• Spontaneously broken symmetry (á la Nambu-Goldstone): in this case, not all the generators of the symmetry group G annihilate the vac-uum state, but only a subset of them, which generates a subgroup H. The vacuum state, therefore, is invariant only under transformations

1.4. EXACT AND SPONTANEOUSLY BROKEN SYMMETRIES 8 induced by H. Goldstone's theorem guarantees that, for each broken generator (that is, a generator which does not annihilate the vacuum state), a massless particle with spin equal to zero (called Goldstone boson) exists and it is described by the same quantum numbers of that broken generator.

It is possible to demonstrate that the U(1)V symmetry is realised in the

Wigner-Weyl way: this leads to the conservation of the baryon number. The discussion on the U(1)Asymmetry will be postponed to a dedicated section.

For what concerns the SU(L)V ⊗ SU (L)A chiral symmetry, it cannot be

exact. Indeed, calling |hi a generic hadronic state and: QA_a =

Z

d3_{x A}0

a(x) (1.30)

an axial charge related to the SU(L)A symmetry, it is easy to see that the

state |h0_{i ≡ Q}A

a |hi would be degenerate in mass with |hi but would have

opposite parity. Indeed, recalling that, for a particle at rest, H |hi = mh|hi,

where H is the Hamiltonian of the system and mh is the mass of the state

|hi, and using the fact that H, QA

a = 0, we nd:

H |h0i = HQA_a |hi = QA_aH |hi = m_h|h0i (1.31) Analogously, calling P the parity operator and ηh the parity of the state |hi,

we nd:

P |h0i = P QA_a |hi = P QA_aP†P |hi = −QA_aP |hi = −ηh|h0i (1.32)

demonstrating that |h0_{i has the same mass as |hi but opposite parity.}

Therefore, if the SU(L)V ⊗ SU (L)A were realised in the Wigner-Weyl

way, each hadron present in nature should be coupled to a mirror particle degenerate in mass but with opposite parity. Given that these particles are not seen in nature (for example, no scalar pions had ever been found), it had been proposed that SU(L)V ⊗ SU (L)A were spontaneously broken.

Never-theless, the presence of approximate hadronic multiplets hints the existence of a residual symmetry of the vacuum under the SU(L)V subgroup. So, it

has been argued that the chiral symmetry-breaking pattern is: ˜

G = SU (L)V ⊗ SU (L)A −→ ˜H = SU (L)V (1.33)

The basis of the symmetry-breaking hypothesis lies in the fact that in nature we observe the existence of a hadronic multiplet composed of particles much lighter than the other ones: it is the JP _{= 0}− _{pseudoscalar mesonic}

octet. Moreover, these hadronic states carry the same quantum number of the axial charge operators (1.30): these properties make these particles good candidates to be the Goldstone bosons related to the SU(L)V ⊗ SU (L)A

1.5. U(1) AXIAL SYMMETRY 9 Even though the pseudoscalar mesons are light, they are not massless, as it would be expected if they were Goldstone boson. The explanation is in the explicit chiral symmetry breaking brought, as said before, by the mass term in the Lagrangian (1.7). Nevertheless, being the masses of some quarks light if compared with the QCD scale ΛQCD ∼ 0.4 GeV, the mass term related to

these light avours can be considered as a perturbation, making SU(L)V an

approximate symmetry of the theory: this leads to the existence of pseudo-Goldstone bosons related to the symmetry-breaking pattern (1.33). These bosons, so, are no longer expected to be massless, and can now be identied with the light pseudoscalar mesons.

Finally, we discuss which is the proper number of quark avours to be considered light. Considering the hadronic spectrum, we see that the pseu-doscalar mesonic octet is much lighter than the other hadronic multiplets: this observation leads to set the number of light avours to L = 3. Indeed, if we consider the symmetry-breaking pattern (1.33), we expect a number of Goldstone bosons equal to the number of broken generators, that is, L2_{− 1.}

Furthermore, within the octet, the pions are much lighter than the other mesons: this is a sign that the approximate SU(L)V symmetry is more

pre-cise if the number of light avour is L = 2 (Heisenberg's isospin symmetry) rather than L = 3 (Gell-Mann's symmetry). This evidence is explained by the fact that in nature three quark avours exist which have mass lighter than ΛQCD: the up, the down and the strange; among them, the up and the

down are much lighter than the strange.

Therefore, in the following sections of the thesis, we will consider only L = 2or L = 3 as the number of light avours.

1.5 U(1) axial symmetry

The U(1)Aaxial symmetry requires an in-depth look. With a reasoning

analogous to that of the previous section, it is possible to demonstrate that this symmetry cannot be realised in the Wigner-Weyl way: indeed, this would entail, as before, the existence for each hadron of a mirror particle carrying the same mass but opposite parity.

Therefore, it is natural to think that the U(1)Aaxial symmetry is

sponta-neously broken. Due to the symmetry-breaking, we expect the existence of a new Goldstone boson, light but not exactly massless because of the presence of the explicit symmetry-breaking brought by the mass term.

Starting from the hypothesis that the U(1)A axial symmetry were

spon-taneously broken and that the complete symmetry-breaking pattern were: U (L)L⊗ U (L)R→ U (L)V

S. Weinberg [12] analysed a chiral eective Lagrangian model (in the next chapter this method will be described in greater details) which describes,

1.5. U(1) AXIAL SYMMETRY 10 setting L = 3, the eight pseudo-Goldstone bosons πi(i = 1, . . . , 8) originated

from (1.33), plus the singlet pseudo-Goldstone boson S originated from the U (1)L⊗ U (1)R→ U (1)V breaking.

Neglecting eects proportional to the small dierence  = md− mu (it

is the parameter which quanties the breaking of the SU(2) isospin which, as said, is a well approximated symmetry), he found that the mass matrix has only one mixing term, between S and π8. Weinberg calculated that the

smallest eigenvalue of the mixing matrix obeys the inequality: mlight<

√

3 mπ0 (1.34)

However, in nature, the closest candidates are the η and η0 _{mesons but,}

being mπ0 = 140 MeV, m_η = 548 MeV and m_η0 = 958 MeV, none of them

obeys (1.34). This issue took the name of U(1) problem. 1.5.1 't Hooft solution to U(1) problem

The crux of the problem lies in the fact that the current Jµ

5 related to

U (1)Asymmetry in (1.29) is conserved (in the chiral limit) only at classical

level. Indeed, at quantum level the U(1)Aaxial symmetry is aected by the

so-called Adler-Bell-Jackiw anomaly [13]: due to the non-invariance of the fermionic measure in the functional integral under U(1)A transformations,

the divergence of the axial current in the chiral limit is classically null but non-zero because of quantum correction.

Explicitly, the expression for the divergence of the axial current is: ∂µJ5µ= 2i L X f =1 mfqfγ5qf+ 2LQ (1.35) where Q ≡ g 2 64π2 µνρσ_Fa µνFρσa (1.36)

is called topological charge density operator3_{, while its integral over the}

space-time q = R d4_{x Q(x)is called topological charge. The expression (1.35) shows}

that the axial charge Q5 ≡R d3x J50 is not conserved even in the chiral limit

because of the presence of the topological term. Indeed: ∆Q5 = Z +∞ −∞ dtdQ5 dt = Z +∞ −∞ dt Z d3x ∂0J50= 2L Z d4x Q = 2Lq (1.37) However, it is known that the topological charge density operator Q can be rewritten as the divergence of the Chern-Simons current, dened as:

Q = ∂µKµ with Kµ= g2 8π2 µνρσ_Tr  Aν  ∂ρAσ+ 2 3igAρAσ  (1.38)

3_{In the work of this thesis, we set }0123

1.5. U(1) AXIAL SYMMETRY 11 The form (1.38) could suggest that ∆Q5 is null, being the integral of

a total divergence. This way, despite the anomalous nature of the U(1)A

symmetry, the axial charge would be eectively conserved.

Actually, the integral in (1.37) is not null due to the existence of topolog-ically non-trivial solutions, the most relevant of which are called instantons: they are Euclidean gauge eld congurations of topological charge q = 1 with nite, and minimal, Euclidean action. From the fact that they possess nite action, it follows that their contribution to the integral (1.37) is not null even though Q is a total divergence [14].

What has been said so far appears as the solution of the U(1) problem: since the axial symmetry does not hold anymore at quantum level, we do not expect the existence of another (pseudo-)Goldstone boson. One can object that, starting from (1.35) and (1.38), it is possible to dene a current

˜

J₅µ≡ J₅µ− 2LKµ

whose divergence is zero: it will lead to the existence of a conserved charge ˜

Q5. But this reasoning is not correct: indeed, the Chern-Simons current

Kµ is not gauge-invariant and, so, the application of ˜Q5 on vacuum cannot

create any physical state.

Despite solving the problem of the U(1)A symmetry, 't Hooft solution

raised new questions. In particular: what happens to the related pseudo-Goldstone boson? Or, in other words, why is the η0 _{mass so large if compared}

with the masses of the octet mesons? To date, the best and easiest solution to the U(1) problem is that proposed by Witten and Veneziano: they explained the fact that the η0 _{mass is much larger than expected starting from the}

anomalous nature of the U(1)A symmetry.

1.5.2 Witten-Veneziano solution to U(1) problem

In general, a broken symmetry is best understood by studying the limit which leads to a theory where the symmetry is conserved. To nd such a limit, Witten decided to follow the hint given by 't Hooft: the 1/Ncexpansion

in the Nc→ ∞limit, where Ncindicates the number of colours of the theory.

As it can be easily seen from (1.15) and (1.17), in this limit, in order to keep ΛQCD constant and have a smooth large Nclimit [15], we must impose that

the product g2_N

c is xed.

Witten realised that in the innite number of colours limit all the contri-butions coming from the anomaly get suppressed. Indeed, being g2 _{∼ 1/N}

c,

from (1.36) we see that:

Q ∼ 1 Nc

1.5. U(1) AXIAL SYMMETRY 12 so that the anomalous term in (1.35) vanishes.

Assuming that the axial U(1) symmetry is spontaneously broken, Witten supposed that the Goldstone boson related to it (that is, for L = 3, the η0

meson) is massive even in the chiral limit due to an anomalous contribution. Let us resume the reasoning carried out by Witten [16] to explain the η0

mass. He started from considering the two-point function of the topological charge density operator:

χ(k) ≡ −i hQQi (k) = −i Z

d4x eikxhT Q(x)Q(0)i (1.39) where the angle brackets denote the expectation value on vacuum. For some generic operators Oi, it is dened as:

hT O1O2. . .i ≡

R [dΨ] O1O2. . . eiS[Ψ]

R [dΨ] eiS[Ψ] (1.40)

where [Ψ] denotes the set of all the elds in the theory, while [dΨ] is the related functional measure.

The value of (1.39) for k = 0 is called topological susceptibility and it is usually indicated with χ:

χ ≡ −i hQQi (k = 0) = −i Z

d4x hT Q(x)Q(0)i (1.41) We notice that, in the chiral limit, the topological susceptibility is null. Indeed, we can express it in the form:4

χ = −i Z d4x hT Q(x)Q(0)i = i V T 1 Z[0] d2Z[θ] dθ2     θ=0 (1.42) where Z[θ] = Z [dA][dq][dq]eiR d4x(LQCD+θQ(x)) (1.43)

is the partition function of the theory with the insertion of a topological contribution, the so-called theta term. Now, it is easy to demonstrate that, in the chiral limit (actually, it is sucient that at least one quark is massless), the partition function turns out to be totally independent of θ. Let us consider the following U(1)Atransformation on the fermionic variables:

(

q → q0 = eiαγ5 _q

q → q0 = q eiαγ5 (1.44) 4_{V T =R d}4_{x is an innite space-time volume to be factorised out from correlation}

1.5. U(1) AXIAL SYMMETRY 13 where α is the parameter of the transformation. If M = 0, LQCD is

un-changed; moreover, Q(x) is independent of the fermionic elds. So, the only variation in (1.43) is that caused by the anomaly in the fermionic functional measure, that is:

[dq][dq] → [dq0][dq0] = [dq][dq]e−iR d4x 2LαQ(x) (1.45) So,

Z0[θ] = Z

[dA][dq][dq]eiR d4x{LQCD+(θ−2Lα)Q(x)} = Z[θ − 2Lα] (1.46)

However, we have only implemented a change of variables, so that the value of the integral is exactly the same. Therefore, selecting α = θ

2L, we conclude

that Z[θ] = Z[0], that is, the partition function is independent of θ and, so, from (1.42), χ = 0 in the chiral limit.

This remark leads to an apparent paradox. To make it clear, we start from expressing χ(k) as a sum

χ(k) = A0(k) + A1(k) + A2(k) + . . . (1.47)

where each Aj gathers the contributions to χ(k) coming from diagrams with

j loops of quarks: so, A0(k)is a pure gluonic term, while A1(k)includes all

diagrams with one fermionic loop, etc. This expression turns out to be an expansion in powers of 1/Nc [15]: A0(k) is of order g4Nc2 ∼ Nc0, while each

following term is suppressed by 1/Ncwith respect to the previous one.

Let us now dene:

A ≡ lim

k→0A0(k) (1.48)

It is the topological susceptibility in the pure Yang-Mills theory (and in the Nc → ∞ limit): we will meet it again frequently in the following

chapters.

We now assume A 6= 0 (assertion conrmed by lattice results [5]). But then, how is it possible that χ(k = 0) = 0 if A 6= 0 and the following terms are suppressed by powers of 1/Nc? To understand Witten's solution, we restrict

(1.47) to the rst order in 1/Nc, that is, only to A0(k) and A1(k). Then,

considering that the term A1(k)corresponds to an exchange of a meson (it

is the propagation of a colour singlet qq state) between two Q elds , we can express it in the form:

χ(k) = A0(k) + X mesons |hΩ| Q(0) |ni|2 k2_{− M}2 n + . . . (1.49) where |Ωi represents the vacuum state while |ni is a generic mesonic state having squared mass M2

1.6. CHIRAL CONDENSATES 14 In order to ensure that χ(k = 0) = 0, the only possible solution is to suppose the existence of a mesonic state S whose mass is M2

S ∼ N1c. This

could sound strange, but actually the squared mass of an approximate Gold-stone boson is in general linear in the symmetry breaking parameter, which in this context is represented exactly by 1/Nc. Moreover, to let the state S

be coupled to the vacuum state by the eld Q, it must be a pseudoscalar avour singlet.

From (1.49), we see that:

χ(k = 0) = 0 = A − |hΩ| Q(0) |Si| 2 M2 S ⇒ A = |hΩ| Q(0) |Si| 2 M2 S (1.50) So, the last step consists in the calculation of the matrix element hΩ| Q(0) |Si. We remind that, in the chiral limit, ∂µJ₅µ= 2LQ(x), and that hΩ| J₅µ(x) |S(~p)i =

i√2L FSpµe−ip·x, where FS is the singlet decay constant. So:

hΩ| Q(x) |S(~p)i = 1 2L∂µhΩ| J µ 5(x) |S(~p)i = 1 √ 2LFSM 2 Se −ip·x _(1.51) Finally, substituting (1.51) in (1.50), we nd: M_S2 = 2LA F2 π (1.52) which is the famous Witten-Veneziano formula. Here, we have used that, to the lowest order in 1/Nc,

FS ' Fπ

Let us remark that, being A = O(N0

c) and Fπ = O(

√

Nc) [5], we have

M_S2 = O_N1

c as expected. Witten-Veneziano formula xes the anomalous

contribution to the SU(3) singlet mass, which tends to zero in the limit of large number of colours.

1.6 Chiral condensates

The realisation of a certain symmetry in the Wigner-Weyl way is related to the vanishing of an order parameter: it is null if the symmetry is exact while it takes values dierent from zero if there is a spontaneous symmetry breaking.

As said in the previous sections, in the chiral limit vector symmetries are exact, while it is believed that the axial ones present a spontaneous breaking. Therefore, it is important for the study of the chiral structure of QCD to establish the order parameters related to the axial symmetries.

1.6. CHIRAL CONDENSATES 15 1.6.1 The chiral condensate

Using the anticommutation rules for the quark elds, it is possible to demonstrate that:

QA

a(0) , qγ5Tbq(0) = −

1

Lδabqq(0) − dabcq Tcq(0) (1.53) Taking its expectation value on vacuum and assuming that the SU(L)V

symmetry is realised in the Wigner-Weyl way, we nd: QA

a(0) , qγ5Tbq(0) = −

1

Lδabhqqi (1.54) If hqqi = 0, we conclude that QA

a |Ωi = 0 ∀a, that is the condition

for the axial symmetry to be exact; on the other hand, if hqqi 6= 0 we have QA

a |Ωi 6= 0, which is a sign of the spontaneous breaking of SU(L)L⊗

SU (L)R symmetry. It is natural, so, to choose as order parameter for the

SU (L)L⊗ SU (L)Rsymmetry the quantity hqqi, known as chiral condensate.

Moreover, it is possible to demonstrate that the chiral condensate is a sort of order parameter also for the U(1)A symmetry: as long as it is non-null,

that symmetry is spontaneously broken.

Obviously, in the presence of the mass term, there is an explicit breaking of all chiral symmetries, so that the chiral condensate of the entire theory will always be dierent from zero. We are considering, however, the theory in the chiral limit: there will be spontaneous breaking of the chiral symmetries if the chiral condensate remains dierent from zero even if the quarks are massless.

1.6.2 Restoration of chiral and U(1) axial symmetries at -nite temperature

In the previous paragraph, we have dened the order parameter for the chiral symmetry assuming the temperature T to be zero. The eect of the temperature on the calculation of expectation values of quantum operators is to restrict the set of eld congurations on which compute the functional integral to those periodic within an Euclidean time interval tE = β = _kT1 .

In particular, the expectation value of a generic operator O at a certain temperature T is:

hOi_T = TrOe

−βH

Tr [e−βH] (1.55)

where H is the hamiltonian of the considered system. The expression (1.55) is nothing but the quantum average over Gibbs' ensemble.

For what concerns the chiral condensate, its expectation value varies with the temperature: in particular, it is dierent from zero at T = 0 but,

1.6. CHIRAL CONDENSATES 16 at a certain temperature Tc, it becomes null and keeps being zero when

temperature increases. Tc is the chiral phase transition temperature, and it

is dened as the temperature at which the chiral condensate becomes zero in the chiral limit: if temperature is higher than Tc, the thermal energy

breaks up the qq condensate, restoring the chiral symmetry; conversely, if the temperature is lower than Tc, the chiral condensate is dierent from

zero and the chiral symmetry is broken. From lattice determinations [17], it is known that Tc ∼ 150 ÷ 165 MeV, very similar to the deconnement

temperature Td which separates the conned (or hadronic) phase form the

deconned (or quark-gluon plasma) one.

On the other hand, the role of the U(1)Asymmetry for the nite

temper-ature phase structure is yet unclear. One expects that at high tempertemper-atures also this symmetry will be eectively restored since, at least for T  Tc, the

density of instanton congurations, responsible for the U(1)A breaking, is

strongly suppressed due to a Debye-type screening [18]. However, it is not yet known whether the fate of the U(1)Asymmetry has or has not anything

to do with the fate of the SU(L)L⊗ SU (L)R chiral symmetry.

From the theoretical point of view, this question can be investigated by comparing (setting L = 2) the behaviour of the two-point correlation func-tions of the mesonic channels presented in Table 1.1.

Meson Interpolating operator I JP

σ (f0) Oσ = qq 0 0+ ~ δ (~a0) O~δ = q~τ₂q 1 0+ η Oη = iqγ5q 0 0− ~ π O~π = iqγ5~τ₂q 1 0−

Table 1.1: Meson channels for L = 2. ~τ denotes the Pauli matrices Under SU(2)Atransformations, σ is mixed with ~π, while η is mixed with

~_{δ. Therefore, the restoration of this symmetry at T = Tc requires identical} correlators for these two channels, which implies, in particular, identical chi-ral susceptibilities5 _{(χσ = χπ and χη = χδ) and identical screening masses}

(Mσ = Mπ and Mη = Mδ). On the other hand, under U(1)A

transforma-tions, σ is mixed with η while ~δ is mixed with ~π: an eective restoration of the U(1)Asymmetry should imply, therefore, that these channels become

degenerate with identical correlators.

5_{We remind that a generic chiral susceptibility is dened as}

χf ≡

Z

d4x DT Of(x)O†_f(0)

1.7. THE TOPOLOGICAL THETA TERM 17 Therefore, it is possible to dene two new order parameters, one for each axial symmetry:

χ_{SU (2)} ≡ χ_σ− χ_π ; χ_{U (1)}≡ χ_δ− χ_π (1.56) If the temperature is below Tc, the chiral condensate makes these two

parameters be dierent from zero. Though, lattice simulation results for the correlators, in the case of L = 2 and T > Tc, seem to indicate that χU (1)6= 0

up to a temperature TU (1) ∼ 1.3 Tc [17]: this would suggest that the U(1)A

symmetry remains broken also above the chiral phase transition.

But, if below Tc the chiral condensate guarantees that the order

pa-rameters are dierent from zero, above Tc, where the chiral condensate is

broken up, what will let χU (1)be dierent from zero? A possible way to keep

χ_{U (1)} 6= 0is to suppose the existence of another chiral condensate, connect-ing 2L fermionic elds: this condensate, representconnect-ing an order parameter for the U(1)Asymmetry, must be non-null above Tcup to TU (1). The discussion

on this new condensate is postponed to the next chapter.

1.7 The topological theta term

This section of the chapter is devoted to analyse in more depth the topo-logical term. An intriguing question is: is the theta term, present in (1.43), only an external source term or is it a possible term which can be added to QCD Lagrangian? If the second hypothesis were correct, we would add a term

Lθ= θQ (1.57)

which would be renormalizable but which would violate CP-symmetry and T-symmetry. Indeed:

Q ∼ TrhFµνF˜µν

i

∼ ~Ea· ~Ba (1.58)

is invariant under charge conjugation while it is pseudoscalar under parity and time reversal. The discussion on the presence or absence of the topolog-ical term is known as strong CP-violation problem.

From the experimental point of view, it is known that strong interactions do not violate CP-symmetry with a very high accuracy. For what concerns the relation among the topological term and CP-breaking physical quantities, in particular, one can nd a relation between the magnitude of the coecient θand the neutron electric dipole moment dN [1]:

dN '

M2 π

M_N3 e |θ| ' 10

−16_{|θ| e · cm (1.59)}

where MN is the neutron mass, while Mπ is the pion mass. From the

1.7. THE TOPOLOGICAL THETA TERM 18 bound:

|θ| < 10−10 (1.60) The relation (1.59) is based on dimensional ground but considers also some more rened properties of the neutron electric dipole moment: in particular, dN is expected to be linear in the light quark masses (and, so, proportional

to M2

π) since, as it will be demonstrated later, if one of these masses goes

to zero, no CP-violation occurs and, so, the neutron electric dipole moment must be null; moreover, based on dimensional ground, dN carries a dimension

of the inverse of a mass, which is obtained by dividing for M3

N (which in this

case roughly plays the role of the typical nucleon mass scale).

More rened relations among the neutron electric dipole moment and the θ angle were derived by Baluni [20], in the framework of the so-called bag model, by Di Vecchia, Veneziano et al. [21], using the Chiral Perturbation Theory, and by many other physicists using dierent approaches. The aim was to translate the above mentioned experimental bound into a constraint on θ computing the quantity cN, dened as: dN = cN|θ| × 10−13e · cm.

There is a substantial global agreement on the magnitude of this coecient (0.001 ≤ |cN| ≤ 0.01) but not on its sign (see Sec. 7.1 of Ref. [5] for a more

detailed discussion).

These considerations suggest that either θ = 0 or there must be a mecha-nism responsible for suppressing the value of θ in QCD. A number of possible explanations have been proposed for the origin of this mechanism, as for ex-ample the axion-model by Peccei and Quinn [22] in which, adding a new particle (called axion), it is possible to dynamically rotate away the θ depen-dence from the theory; however, an in-depth analysis of this issue is beyond the scope of this thesis.

An important property of the topological term is that it is related to the quark mass term. Indeed, let us write the mass term in a more generic form: LM= −qRMqL− qLM†qR (1.61)

where M is a general complex mass matrix. The relation (1.61) can be rewritten as: LM= −qPLMPLq − qPRM†PRq = −qMPLq − qM†PRq = − q M + M † 2  + M − M † 2  γ5  q ≡ −q [A + iBγ5] q (1.62) where A and B are hermitian operators. The term including γ5 violates

CP-symmetry exactly as the topological term. So, it is not senseless to believe that the two terms are related. To understand the connection between them,

1.7. THE TOPOLOGICAL THETA TERM 19 let us consider the change of variable brought by a SU(L)L⊗SU (L)R⊗U (1)A

transformation6 ( qL→ qL0 = ˜VLqL= eiαVLqL qR→ qR0 = ˜VRqR= e−iαVRqR (1.63) to be implemented on: Z = Z [dA][dq][dq]eiR d4xLtot (1.64)

where Ltot = −1₂Tr [FµνFµν] + iqγµDµq − qRMqL− qLM†qR+ θQ.

Un-der the transformation (1.63), we have non-invariance both in the fermionic functional measure and in the mass term:

• [dq][dq] → [dq0][dq0] = [dq][dq]e−iR d4x 2LαQ(x) • LM→LM0 = −q0RMqL0 − qL0 M†qR0 = −qRM0qL− qLM† 0 qR where M0 _{= ˜V}† RM ˜VL.

Therefore, the modied partition function is: Z = Z [dA][dq][dq]eiR d4xLtot0 (1.65) where L0 tot= −12Tr [FµνFµν]+iqγµDµq−qRM 0_q L−qLM† 0 qR+(θ − 2Lα) Q.

But, having implemented only a change of variables, the values of (1.64) and (1.65) are exactly the same. So, there is equivalence among two expres-sions which dier only by the changes:

(

M → M0 = ˜V_R†M ˜VL

θ → θ0= θ − 2Lα (1.66) We notice that, if M is invertible, that is, if det M 6= 0, we have:

det M0 = (det ˜VR)∗det M det ˜VL= e2iLαdet M (1.67)

or, in other words:

arg(det M0) = arg(det M) + 2Lα (1.68) So, from the relations (1.66) and (1.68), we see that, under the transfor-mation (1.63), the quantity:

θphys≡ θ + arg(det M) (1.69)

6_{We do not consider the U(1)}

V part of the chiral group since it is an exact symmetry

1.7. THE TOPOLOGICAL THETA TERM 20 is unchanged. Its presence, that is, θphys 6= 0, is a sign of the strong

CP-violation. Indeed, (1.69) tells us that, if the mass matrix is invertible, it is impossible to cancel the topological term by a chiral rotation. Indeed, we can implement a transformation setting

α = θ 2L

which implies θ0 _{= 0: the whole violation carried by θ}

phys is moved on the

mass term; on the other hand, if we consider M = diag(m1, . . . mL), with

mi ∈ R ∀i, we have arg(det M) = 0, and the violation is totally carried by

the topological term. At last, we notice that, if at least one quark is massless, we have det M = 0 and, so, it is possible to rotate away all the dependence on θphys from the theory.

The relation (1.60), which gives an upper bound on the magnitude of the θcoecient, is referred to the combination (1.69).

Let us now explain the importance of the θ-term in the work of this thesis. As it can be seen from (1.43), the presence of the topological term modies the partition function of the theory. In particular, we are interested in the θ-dependence of the vacuum energy density vac(θ), dened as:

Z[θ] ≡ 1 Ne −iΩvac(θ) _{⇒ } vac(θ) = i Ωlog Z[θ] + const. (1.70) where N is a normalizing constant while Ω = V T is the four-volume consid-ered7_.

Being θ very small, it makes sense to Taylor-expand the vacuum energy density around θ = 0: vac(θ) = vac(0) + 1 2 ∂2_ vac(θ) ∂θ2     θ=0 θ2+ 1 24 ∂4_ vac(θ) ∂θ4     θ=0 θ4+ . . . (1.71) It is possible to demonstrate that only even powers of θ appear in (1.71) since the coecients of the odd-powered terms vanish at θ = 0. What is interesting is that the coecients of the expansion are related to very important quantities: they can be determined from appropriate correlation functions of the topological charge density at θ = 0. As an example, starting

7_{The expression (1.70) is given in the Minkowski space-time. Later, we will make use}

of the expression of the vacuum energy density in the Euclidean space-time, that is: vac(θ) = −

1 ΩE

log ZE[θ] + const.

1.7. THE TOPOLOGICAL THETA TERM 21 from the denition (1.70) of the partition function, let us make more explicit the expression (1.42). We have:

∂Z[θ] ∂θ = −iΩ ∂vac(θ) ∂θ Z[θ] (1.72) and ∂2Z[θ] ∂θ2 = −iΩ " ∂2vac(θ) ∂θ2 − iΩ  ∂vac(θ) ∂θ 2# Z[θ] (1.73) However, we notice that:

1 Z[θ] ∂Z[θ] ∂θ     θ=0 = iV T hΩ| Q(0) |Ωi = 0 =⇒ ∂vac(θ) ∂θ     θ=0 = 0 (1.74) since the expectation value of the topological charge density on the vacuum is null. Therefore, we are left with:

1 Z[θ] ∂2Z[θ] ∂θ2     θ=0 = −iΩ ∂ 2_ vac(θ) ∂θ2     θ=0 (1.75) so that, looking at (1.42), we can conclude that:

χ = ∂ 2_ vac(θ) ∂θ2     θ=0 (1.76) So, the coecient of the θ2 _{term in (1.71) coincides with the topological}

susceptibility. We see here that, according to Witten's mechanism, the most plausible solution to the U(1) problem requires a non-trivial dependence on θ of the vacuum energy density: the numerical evidences for it, obtained through Monte Carlo simulations of the lattice formulation, appear quite robust.

Another important quantity is the second cumulant c4 of the probability

distribution of the topological charge density operator Q, dened as [5]: c4≡

i Ω hQ

4_{i − 3hQ}2_i2

(1.77) which is related to the η0 _{− η}0 _{elastic scattering amplitude [23] and to the}

non-gaussianity of the topological charge distribution [5]. Given that: hQni = i n Z[θ] ∂nZ[θ] ∂θn     θ=0 (1.78) with calculations analogous to those carried on for the topological suscepti-bility, it is possible to nd the relation:

c4 = ∂4vac(θ) ∂θ4     θ=0 (1.79)

1.7. THE TOPOLOGICAL THETA TERM 22 Therefore, the expansion (1.71) can be rewritten as:

vac(θ) = vac(0) + 1 2χθ 2₊ 1 24c4θ 4_{+ . . . (1.80)}

The work of this thesis consists mainly in computing the dependence on θof the vacuum energy density in some relevant eective models (which will be described in the next chapter) to exploit the relations (1.76) and (1.79) so as to obtain the expressions of the topological susceptibility and of the second cumulant in terms of the fundamental parameters of the theories.

Chapter 2

Chiral eective Lagrangian

models

2.1 Eective degrees of freedom of QCD

One of the peculiar property of Quantum Chromodynamics is the colour connement, according to which all observable hadronic states must be colour singlets, that is, invariant under the action of the gauge group SU(3)c.

As a consequence, it is impossible to isolate and observe the fundamental degrees of freedom of QCD, that is, quarks and gluons, since they are not colour singlets: the physical states that we observe (hadrons) are bound states of quarks, antiquarks and gluons, combined in a way such that the resulting state is gauge-invariant.

Therefore, the eective degrees of freedom of QCD can be considered as composite operators, multilinear in the fundamental elds, which carry the quantum numbers of the physical hadronic states. Their dynamics can be described by an eective Lagrangian which must be endowed with the same property of transformation of the fundamental Lagrangian (1.7).

In our case, we are interested in the description of the low-energy regime of QCD with light quarks, paying particular attention to the chiral sym-metries described in the previous chapter. To address this issue, a chiral eective Lagrangian formulation has been developed, in which the eld vari-ables describe the pseudo-Goldstone bosons coming from the breaking of the axial symmetries. So as to satisfy the correct properties of symmetry, the eective Lagrangian L must be invariant, in the chiral limit, under the group

G = SU (L)A⊗ SU (L)V ⊗ U (1)V (2.1)

while under U(1)A transformations (1.63), it must transform, in the chiral

limit, as (see (1.46)):

L → L − 2LαQ (2.2)