Study of the mass of the axion and of its interactions with mesons and photons using chiral eﬀective Lagrangian models

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

Curriculum Fisica Teorica

Study of the mass of the axion and of its

interactions with mesons and photons

using chiral effective Lagrangian models

Candidate: Supervisor:

Giacomo Landini Prof. Enrico Meggiolaro

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Ringraziamenti

La scrittura di questa tesi di laurea magistrale rappresenta la conclusione di un impor-tante capitolo della mia vita. Colgo questa occasione per ringraziare tutte le persone che mi sono state vicine in questi cinque anni.

In primo luogo ringrazio il mio relatore, il Professore Enrico Meggiolaro, per tutto il sostegno che mi ha dato nella realizzazione di questa tesi, sia dal punto di vista sci-entifico sia nella sua stesura, e per la costante disponibilita’.

Voglio ringraziare la mia famiglia: i miei genitori Andrea e Marina e mia sorella Irene. Li ringrazio per tutto il sostegno, morale, logistico ed economico, che mi hanno dato in questi anni, per avermi sempre incoraggiato e per avermi dato ottimi consigli. Li voglio ringraziare in modo particolare per aver creduto in me durante i primi mesi di Universita’ e per avermi aiutato ad superare le difficolta’ che ho trovato in questo periodo.

Voglio ringraziare immensamente tutti gli amici che mi hanno tenuto compagnia in questi anni.

Grazie a Guazza che per me e’ stato (ed e’) come un fratello, che piu’ di tutti mi e’ stato vicino tanto nelle belle che nelle brutte occasioni, e con cui ho condiviso tante belle esperienze sia a Spezia che a Pisa.

Grazie a Pietro per tutte le giornate passate insieme al polo e tutte serate che abbi-amo trascorso in compagnia dal primo anno fino all’ultimo. Sono sicuro che l’amiciza resistera’ alla distanza anche nel futuro. Lo ringrazio anche per avermi aiutato nella presentazione grafica di questa tesi.

Grazie a Francesk per gli anni passati insieme, per avermi tenuto compagnia tra una lezione e l’altra e nelle (dis)avventure in laboratorio.

Grazie a Elisa, Sara, Ale e Simon per le meravigliose serate pisane trascorse insieme (con Pietro e Francesk) in questi ultimi due anni, che ricordero’ per sempre con gioia. Grazie a Fabio, Nikolo’, Eugenio e Dema per le belle giornate che abbiamo passato in compagnia a Spezia e Pisa e che si aggiungono ai tantissimi bei ricordi del tempo del liceo.

Grazie a Luca, Matteo e Carlo per le giornate trascorse insieme (ancora con Pietro e Francesk!) durante la triennale e per le serate passate a vedere il Trono di spade. Grazie a Sorre, Simone, Misha, Davide per avermi tenuto compagnia a Spezia.

Grazie a Sara che e’ stata la coinquilina piu’ simpatica con cui ho condiviso la casa in questi anni.

Grazie a Giulia per gli anni passati insieme, per i bei ricordi e per aver condiviso i

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Grazie ai miei insegnanti di musica, Gianni ed Ilaria, per avermi aiutato a coltivare questa mia grandissima passione nonostante tutti gli impegni universitari.

Ci tengo infine a ringraziare il Professore Enrico Girardi, che per me e’ stato un impor-tante punto di riferimento dal punto di vista scolastico e grazie al quale ho coltivato questa passione per la matematica e la scienza.

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Introduction

Quantum chromodynamics (QCD ) is the theory which describes the strong interactions in the framework of the Standard Model. It is a gauge theory, based on the non-Abelian ”coulor” group SU (3)C. The degrees of freedom of the theory are quarks, the

funda-mental constituents of hadrons, and gluons, massless vector bosons which mediate the interactions. A fundamental property of QCD is asymptotic f reedom: the strenght of the interaction between quarks and gluons becomes weaker when the energy increases. As a consequence the high-energy regime of the theory is well described by perturbative methods. On the other side the low-energy regime is highly non-perturbative and a different approach is needed. This low-energy regime is an extremely interesting sector of the theory since it can investigate different phenomena, such as confinement, hadron spectrum and dynamics and many others, whose explanation is crucial for a full un-dersting of the theory itself.

Despite several successes, in the ’70s it became clear that QCD presented an unsolved question: after the introduction of instantons by t’Hooft, physicists realized that, due to these new kinds of topological configurations of gluon fields, an extra term could appear in the QCD Lagrangian, the so called θ term, given by Lθ = θQ, where Q is

the topological charge density and θ is a free parameter. This term violates explicitly CP symmetry and predicts the existence of a non-zero value for the electric dipole of the neutron. Many experiments have studied the question but none of these has observed CP violations in strong interactions. In particular the measurement of the electric dipole of the neutron has set the experimental bound |θ|< 10−10. QCD is not able to explain the smallness of θ without assuming a fine-tuning of the parameter. This is known in the literature as the strong-CP problem.

Among the several possible solutions to the strong-CP problem, the most appealing one is the axion model, proposed by Peccei, Quinn, Weinberg and Wilczek (PQWW) in 1977-1978. The key idea is the introduction of a new global symmetry of the theory, the so called U (1)P Q, which is both spontaneously broken at an energy scale fa and

af-fected by a quantum anomaly proportional to the topological charge Q. The Goldstone boson of U (1)P Q, mixing with the other degrees of freedom of the theory, gives rise to a

new neutral pseudoscalar particle, the axion. The non-zero vacuum expectation value (v.e.v.) of the axion field is able to cancel out the θ term, restoring automatically the CP-symmetry.

There are different specific models for the axion and its interactions with the Standard Model fields. A general feature, common to all, is that both the axion mass and its interactions are supressed by the energy scale fa. The original PQWW model

identi-fied this scale with the electroweak breaking scale vEW ∼ 250 GeV, but this hypothesis

has been ruled out by the experiments. Nowadays the most appealing models, the so v

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called ”invisible axion models” are based on the assumption that fa vEW. As a

consequence the general expectation for the axion is a very light particle with highly supressed interactions. Thanks to these characteristics, the axion is also indicated as a promising candidate for Dark Matter.

The main goal of this thesis is the description of axion properties, such as its mass and interactions. There are different approaches to accomplish this purpose. One of the most powerful, the one we choose for this thesis, is the use of an ef f ective field theory. The general idea is that of ”integrating out” the high-energy degrees of free-dom and build a model, based on symmetry principles, which describes the low-energy regime of the theory. In the context of QCD this paradigm has achieved important results in the so called chiral Lagrangian f ormulation.

The starting point is the symmetry group of the QCD Lagrangian. Since the three lightest quarks (up, down and strange) have small masses compared to the charac-teristic QCD mass scale ΛQCD, it is a good approximation to neglect their masses,

defining the so called chiral limit. In this limit QCD is invariant under a global sym-metry group acting on quark fields with independent rotations of their lef t and right chiral components. This chiral group is G = U (1)L⊗ U (1)R⊗ SU (3)L ⊗ SU (3)R.

However, studying the structure of the observed hadronic multiplets, it became clear that this symmetry is not realized exactly (in the Wigner-Weyl way) but it is rather spontaneously broken to its vectorial subgroup H = U (1)V ⊗ SU (3)V. The broken

axial generators give rise to nine Goldstone bosons (eight from SU (3)A and one from

U (1)A). The introduction of quark masses explicitly breaks the chiral group. As a

consequence the Goldstone bosons acquire a mass term, becoming pseudo-Goldstone bosons. These light bosons have been identified with the QCD lightest pseudoscalar mesons (pions, kaons and etas). In addiction to that we have to remark that the U (1)A

subgroup is affected by a quantum anomaly proportional to the topological charge Q. As a consequence the corresponding Goldstone boson is massive even in the chiral limit. In order to describe the low energy dynamics of the pseudoscalar mesons, the chiral Lagrangian formulation was introduced: the key idea is building the most general La-grangian, whose degrees of freedom are the pseudo-Goldstone bosons of the chiral group G, so as it transforms under G exactly as the QCD Lagrangian. In particular, since we are interested in low-energy dynamics, we can expand the Lagrangian in powers of the derivatives (and so for the amplitudes in powers of the momentum p). In this thesis we have considered the leading-order Chiral Effective Lagrangian O(p2). This Lagrangian has achieved important results in the computation of the mass-spectrum and interac-tions of the pseudoscalar mesons. Anyway, it does not include the effects of the U (1)A

group and the corresponding Goldstone boson. The next step is therefore the inclusion of U (1)A. In doing that it is crucial the role of the axial anomaly, which has to be

implemented in the Lagrangian model. There exist different models in the literature: the main difference between them is exactly the description and implementation of the U (1)A anomaly.

As the axion is expected to be a very light particle, it is possible to include it in the description of the low-energy dynamics of the theory. Therefore we have consid-ered different chiral Lagrangian models and we have added to them the axion degree of freedom. To accomplish that, we have applied a ”minimal” procedure: following the previous works of Di Vecchia and Veneziano, the only terms we have added to our Lagrangians are a kinetic term for the axion field and an anomalous term, i.e a term

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vii Introduction

which can reproduce the U (1)P Qanomaly. The implementation of the anomalous term

depends on the specific model.

Let’s analyze briefly the four models we have considered in this thesis. The first one is the above-mentioned Chiral Effective Lagrangian O(p2). The second is a model proposed by Witten, Di Vecchia and Veneziano (WDV): the Lagrangian of this model includes the topological charge density operator Q as an external background field. The presence of Q enables a simple procedure to implement the U (1)Aanomaly. Even if this

Lagrangian is well defined in the large NC limit (NC being the number of colours), we

have used it to make predictions in the physical case NC = 3. The third model is the so

called ”extended non-linear σ-model (EN Lσ)”: the main difference with WDV is that

the U (1)Aanomaly is implemented without Q, but rather by mean of a determinantal

interaction, which mimics the interactions of quarks generated by istantons. The last model, the so called ”Interpolating model”, ”interpolates” between WDV and EN Lσ,

keeping both Q and the determinantal interaction and predicting the existence of a new exotic hadronic state. The model is based on the introduction of a U(1)-axial breaking condensate, in addiction to the usual quark-antiquark chiral condensate.

It is not difficult to include the axion field in all the four models. The form and prop-erties of the anomalous axion term depend explicitly on which one we are considering: as an example in WDV and Interpolating models it is a pure quadratic term, while in the others it produces also higher order interactions.

Let’s now discuss the original work done in this thesis. First of all, we have expanded the Lagrangians up to the second order in the pseudoscalar fields and derived the pseudoscalar mass matrices for all models. From these we have extracted both our predictions for the axion mass and the axion-mesons mixing angles. To accomplish this purpose we have followed a double procedure: we have introduced two perturbative parameters, mq/ΛQCD and b ∼ Fπ/fa, where mq is a scale with the same order of

magnitude of the mass of the three lightest quarks, while Fπ is the pion decay

con-stant. Since both the parameters are expected to be 1, we have performed first a perturbative expansion at the leading order in b, making no assumptions on the quark masses; then we have worked in the opposite limit. The results found with the two approaches have been compared at the end of the derivation.

Next we have introduced the well-known QCD chiral electromagnetic anomaly (which gives rise to π0, η and η0 decays in two photons) and, from the knowledge of the

previ-ously computed mixing angles, we have derived both vertices and decay widths for the electromagnetic decay of an axion in two photons, in the same perturbative schemes discussed above. The predictions of these observables are of a crucial importance, since almost all the experimental search for the axion is based on its interaction with elec-tromagnetic fields.

In the literature it is possible to find results for the Chiral Effective Lagrangian O(p2) in the case of two light flavours (both for axion mass and electromagnetic vertex) and of the WDV model for three light flavours (only axion mass). As a consequence the great part of the results found in this thesis (all of them are derived for three light flavours) are original. The goal of this section is not only to derive analytical (and numerical) expressions for our observables but also to make a critical comparison between them and analyze the model dependence of the predictions.

Finally we have analyzed mesons decays involving the axion: in particular we have focused on the decays of η and η0 in two pions (neutral or charged) plus an axion,

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η/η0 → ππa, and the decay K+ → π+_{a. The first ones are the lowest energy hadron}

decays with a single axion involved: we have first expanded the Lagrangians up to the fourth order in the pseudoscalar fields (third order terms vanish for symmetry prin-ciples); then we have derived the coupling constants; finally, we have computed the amplitudes and the decay widths for each process.

The charged kaon decay is interesting because of well-known experimental bounds on it and it was important to rule out the original PQWW model. This decay requires to take into account weak interactions between mesons; we have used some well-known ”tricks” found in the literature to perform our computations.

Since these hadronic channels are less studied than the electromagnetic one, our aim was to find some interesting and original results, testable by experiments. As a conse-quence we have limited ourselves to work in a double expansion at the leading order in both the pertubative parameters mq/ΛQCD and b.

All the results found in the thesis have been discussed and compared with each other and with the ones found in the literature. We have evaluated several numerical results and compared them with the experimental data, so as to test the goodness of our as-sumptions and also to give some bounds on the parameter b (i.e on the PQ breaking scale fa).

The thesis is organized as follows.

In Chapter 1 we present the most important properties of Quantum Chromodynamics, and describe its chiral symmetries in the limit of L massless quarks, paying particular attention to their spontaneous breaking and to the presence of the axial anomaly; then, we analyze the strong CP problem and the Peccei-Quinn solution, focusing on the gen-eral properties of axions. Finally we describe some of the most popular axion models. In Chapter 2 we present the chiral effective Lagrangian formulation and the four models we have worked on in this thesis: starting from the analysis of the effective degrees of freedom of QCD in the low-energy regime, we first describe the Chiral Effec-tive Lagrangian O(p2) proposed by Weinberg; then, we present the model of Witten, Di Vecchia and Veneziano, the extended non-linear σ-model and the Interpolating model. For each of them we discuss how to implement the axion degree of freedom and we show explicitly how the Peccei-Quinn mechanism cancels the θ term.

In Chapter 3 we present and critically analyze the results that we have obtained for the axion mass in the four different models, describing also the computational techniques used in our work; the last section of the chapter is devoted to evaluate nu-merically, where possible, our results so as to compare them with each other and with the experimental data.

In Chapter 4 we present the results that we have obtained for the axion decay in two photons and mesons decays involving the axion. For each of them we derive both the coupling constants and the decay widths, describing in detail the computational techniques adopted. Finally we evaluate numerically, where possible, our results so as to compare them with each other and with the experimental data and we also give some bounds on the parameters of the models.

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ix Introduction

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Chapter 1

The Strong CP problem and

Axions

1.1 Quantum Chromodynamics Lagrangian

Quantum Chromodynamics (QCD ) is the theory which describes the strong interac-tions sector of the Standard Model (SM). It is a Quantum Field Theory, more specifi-cally a non Abelian gauge theory, related to the coulor group SU (3)c. The fundamental

degrees of freedom of the theory are quarks and gluons: the quarks are the matter vari-ables (fermionic 1₂-spin fields) and appear in six different f lavours (up, down, strange, charm, bottom and top). Each one of these exists in turn in three states of colour:

qf =   q1_f q2_f q3_f   (1.1)

where f is a flavour index.

The three colour components of the quark fields transform under the action of SU (3)c

according to the fundamental representation of the group :

qf → U (x)qf (1.2)

with

U (x) = eiθa(x)Ta (1.3)

where θa are local parameters of the transformation and Ta are the generators of SU (3)cin the fundamental representation (the index a runs from 1 to 8). They satisfy

the algrebra commutation rules : h

Ta, Tb i

= ifabcTc (1.4)

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

Gluons are the gauge bosons of the theory, described by eight vector fields: 1

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Aµ= AaµTa (1.5) transforming under SU (3)c as Aµ→ U AµU†+ i g(∂µU ) U † _(1.6)

The field density tensor is defined as

Fµν = ∂µAν− ∂νAµ+ ig [Aµ, Aν] (1.7)

and transforms as

Fµν→ U FµνU† (1.8)

If we define the field

q =     qu qd qs ...    

and the Covariant derivative

Dµ= ∂µ+ igAµ (1.9)

the (classical) Lagrangian describing the theory, invariant under SU (3)c

transforma-tions, is given by [1]:

L_QCD= −1

2Tr [FµνF

µν_{] + ¯}_{q (iγ}

µDµ− M ) q (1.10)

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

1.2 Main properties of QCD : Confinement and

Asymp-totic freedom

The quark model was proposed at beginning of ’60s by Gell-Mann and Ne’eman [2]. The latters, by studying the structure of hadronic multiplets, hypothesized that the strong interactions present an SU (3) symmetry (nowadays it is called flavour symme-try), which generalizes the isospin SU (2) symmetry proposed by Heisenberg.

According to Gell-Mann, the whole multiplets structure of hadronic spectrum could be explained by the existence of 1₂ spin particles named quarks, present in three dif-ferent flavours (up, down and strange). The observed hadronic particles would have

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1.2 Main properties of QCD : Confinement and Asymptotic freedom 3

been bound states of three quarks (baryons) or a quark and an anti-quark (mesons). Gell Mann’s flavour symmetry is not exact. It is explicitly broken by the different values of quark masses of different flavours. Nevertheless, as we will see, it remains a good approximate symmetry, when two or three flavours are considered massless. The quark model, though simplifying the description of strong interactions, presented many issues. Among the most important:

• no free quarks observed in experiments

• existence of states (such as ∆++_{) 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 [3], proposed a new quantum number, which they called colour: according to them, each flavour state should exist in a number NC of coloured copies. To make experimental data (in

particular, e+e− → hadrons cross section and π₀ → γγ decay width) fit the theory, the proper number of colour turned out to be three. Starting from this assumption, these scientists hypothesized the link between the colour charge and a new non-Abelian gauge group: the colour SU (3)C gauge group of QCD.

The introduction of the colour quantum number enabled a simple solution to the spin-statistics problem, giving the correct properties of symmetry to the problematic wave functions.

For what concerns the absence of free quarks, it has been formulated the so-called colour confinement postulate: formally, it establishes that a generic observable hadronic state must be a singlet under transformations induced by the colour gauge group SU (3)C.

This explains why quarks (fundamental representation) and gluons (adjoint represen-tation) can never be observed while mesons and baryons, singlets of SU (3) have been observed for a long time. It still does not exist a demonstration of confinement, but the study of Lattice QCD, a version of the theory on an Euclidian discretized space-time, performed by numerical simulations, provides strong indications that this claim is correct.

Another fundamental property of strong interactions is asympotic freedom. QCD is a renormalizable theory : the U V divergences of loops diagrams can be absorbed in a finite number of counterterms in the Lagrangian, so as we can make finite and well defined predictions at any order in perturbation theory, in terms of renormalized pa-rameters. In 1973, studying the renormalization group equations, Gross and Wilczek [4], and Politzer [5], showed that the β function of QCD, defined as

β = µ∂gR

∂µ (1.11)

where gR is the renormalized coupling and µ is the renormalization scale, is negative

in the neighbourhood of gR = 0. Indeed, we can expand the β function in powers of

the renormalized coupling :

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

The first two coefficients are independent of the particular renormalization scheme used, and their values are :

β0 = 1 (4π)2 11NC− 2Nf 3 (1.13) β1= 1 (4π)4 34 3 N 2 C− 13 3 NC − 1 NC Nf (1.14)

where NC is the number of coulors and Nf is the number of flavours of quarks. For

NC = 3 the coefficient β0 is positive for Nf < 17, and the first term is the dominant

contribution in the perturbative regime, where gR 1. The consequence of this

negative sign is that while the energy scale µ increases, the strenght of the renormalized coupling decreases, leading to a free theory for µ → ∞.

In particular, by solving (1.11) at one loop, we find

αs,R= g_R2 4π = 1 4πβ0ln µ2 Λ2 QCD (1.15)

where ΛQCDis called QCD mass scale parameter and its magnitude order is ΛQCD≈ 0.4

GeV in the M S renormalization scheme [6].

Therefore for high-energy processes (µ ΛQCD) QCD ’s predictions can be achieved

using the (renormalized) perturbation thery.

On the contrary, when the energy scale µ → 0 the strenght of interactions (gR)

in-creases leading to a strongly interacting theory. Of course when gR reaches values of

order of unity we have to take in account also multi-loops diagrams, which modify the β function, but the qualitative picture remains the same. The physical consequence is that low-energy QCD (µ . ΛQCD) is a strongly interacting theory that cannot be

studied using the usual perturbative methods. This result is in qualitativeaccordance with the idea of confinement: strong interactions are required to keep quarks and glu-ons bounded.

The low-energy sector of QCD offers an incredibly rich variety of phenomena, whose comprehension is crucial for a complete understanding of the theory. Since perturba-tion theory fails in this regime, there is need for alternative techniques. Two among the most popular are :

• Lattice QCD, based on numerical simulation • Chiral Effective Lagrangians.

In this thesis we will make use of the second technique, as we will see in the next chapters.

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1.3 Chiral symmetries of QCD 5

1.3 Chiral symmetries of QCD

The QCD Lagrangian is invariant under transformations induced by the group U (1)u⊗

U (1)d⊗ U (1)s⊗ ..., which acts on quark fields as

qf → eiαfqf (1.16)

Noether’s theorem guarantees the existence of a conserved current Jµ, satisfying ∂µJµ=

0, for each generator of the symmetry group. Every conserved current is related to a conserved charge Q = R d3xJ0(x). In this particular case we have J_fµ = ¯qfγµqf and

Qf =R d3xq †

fqf, which is related to the conservation of the number of quarks-antiquarks

of flavour f .

However, if we make the assumption that L light flavours have zero mass (tipically L = 2, taking the up and the down quarks massless, or L = 3, including also the strange quark) the symmetry group of QCD Lagrangian becomes larger.

This assumption is, obviously, an approximation, but it works really well in many predictions of low-energy QCD. The reason is that the two (or three) lightest quarks’ masses are much smaller than the typical QCD low-energy scale (it can be taken as ΛQCD or the mass of the proton mP ≈ 1 GeV), so the error made by neglecting them

is quite small. This regime is known as the chiral limit.

To prove this, we introduce the chiral components of quark fields, usually called lef t and right, defined as :

qL= 1 + γ5 2 q ≡ PLq (1.17) and qR= 1 − γ5 2 q ≡ PRq (1.18)

where we have adopted the convention

γ5≡ −iγ0γ1γ2γ3 (1.19)

We focus only on the L massless flavours. In terms of these components, (1.10) can be re-written as LQCD= − 1 2Tr [FµνF µν_{] + ¯}_q LiγµDµqL+ ¯qRiγµDµqR− ¯qLM qR− ¯qRM qL (1.20)

Taking M = 0 the Lagrangian is invariant under independent transformations of qL

and qR

qL→ eVLqL (1.21)

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where eVL and eVRare elements of U (L) :

e

VL= eiαLVL= eiαLeiθ

a

LTa _(1.23)

e

VR= eiαRVR= eiαReiθ

a

RTa (1.24)

where Ta are the generators of SU (L).

The enlarged symmetry group is

G = U (L)L⊗ U (L)R= U (1)L⊗ U (1)R⊗ SU (L)L⊗ SU (L)R (1.25)

It is easy to prove that the subgroup SU (L)L⊗ SU (L)R can be written as the

com-position of a vectorial transfomation, i.e VL = VR ≡ V , and an axial one, in which

VL= V_R† ≡ A.

Equivalently the U (1)L⊗ U (1)R subgroup can be expressed as a composition of a

vec-torial transformation (αL= αR), and an axial one, (αL= −αR).

Finally we can write

G = U (1)V ⊗ U (1)A⊗ SU (L)V ⊗ SU (L)A (1.26)

The symmetry group acts on quark fields as            U (1)V : q → eiαVq U (1)A: q → eiγ5αAq SU (L)V : q → eiα a VTaq SU (L)A: q → eiγ5α a ATaq

The corresponding Noether’s currents are            U (1)V : Jµ= ¯qγµq U (1)A: J5µ= ¯qγµγ5q SU (L)V : Vaµ= ¯qγµTaq SU (L)A: Aµa = ¯qγµγ5Taq

All the currents are conserved in the chiral limit at a classical level. The quark mass term explicitly breaks the symmetry group G down to the U (1)V subgroup.

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1.4 Exact and spontaneously broken symmetries 7

1.4 Exact and spontaneously broken symmetries

The symmetries of the Lagrangian can be realized in two different ways, leading to different properties of the physical spectrum:

• Exact symmetry (a’ la W igner-W eyl): all the generators of the symmetry group annihilate the vacuum state which is, therefore, invariant under all the transfor-mations induced by the group. In this case, the particle spectrum is organized in multiplets corresponding to the irreducible representations of the symmetry group and we observe a conserved quantity for each generator.

• Spontaneously broken symmetry (a’ la N ambu-Goldstone): not all the genera-tors annihilate the vacuum state, but only a subset of them, which generates a subgroup H. The vacuum state, therefore, is invariant only under transformations 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 par-ticle with spin equal to zero (called Goldstone boson) exists and it is described by the same quantum numbers of the corresponding broken generator.

The U (1)V symmetry is realized a’ la Wigner-Weyl : the related conserved charge is

the baryon number.

Let’s focus on the chiral group SU (L)L⊗ SU (L)R. This symmetry can not be an

exact one. Indeed, if we take a generic hadronic state |hi and an axial charge, related to the SU (L)A symmetry,

QA_a = Z

d3xA0_a (1.27)

it’s easy to see that the new hadronic state |h0i ≡ QA

a|hi would be degenerate in mass

with |hi but with opposite parity. Indeed, calling H the Hamiltonian of the system, for a particle at rest we have H|hi=mh|hi, where mh is the mass of the state |hi; using

[H,QA_a]=0, which is satisfied for an exact symmetry, we find

Hh0 = HQA_a |hi = QA_aH |hi = m_hQA_a |hi = m_h h0

(1.28)

Analogously, calling P the parity operator and ηh the parity of the state |hi, we have

Ph0 = P QA_a |hi = P QA_aP†P |hi = −QA_aP |hi = −η_h h0

(1.29)

Therefore if the chiral group was realized a’ la Wigner-Weyl, for every hadron we would observe a corresponding particle degenerate in mass but with the opposit parity. Since these particles have never been observed, it was proposed that the chiral group has to be spontaneously broken. Nevertheless observations of the hadronic spectrum show that particles can be classified in approximate multiplets of SU (L)V, suggesting

the existence of a residual symmetry of the vacuum under SU (L)V subgroup.

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SU (L)L⊗ SU (L)R→ SU (L)V (1.30)

The basis of the symmetry-breaking hypothesis lies in the fact that in nature we ob-serve an hadronic multiplet composed by particles much lighter than the other hadrons: it is the JP = 0− pseudoscalar meson octet, composed by pions, kaons and eta (π0,π±,K±,K0,K

0

and η). Moreover, these hadronic states carry the same quantum number of the axial charge operators (pseudoscalars): these properties make these par-ticles good candidates to be the Goldstone bosons related to the SU (L)L⊗ SU (L)R

symmetry breaking.

Even though the pseudoscalar mesons are the lightest observed hadrons, they are not massless, as it would be expected if they were Goldstone boson. The explanation lies in the explicit chiral symmetry breaking brought, as said before, by the mass term in the QCD Lagrangian. Nevertheless, being the masses of the first two or three quarks light if compared with the QCD scale ΛQCD, the corresponding mass term can be considered

as a small perturbation in the Lagrangian, making the chiral group a good approximate symmetry of the theory: this leads to the existence of pseudo-Goldstone bosons related to the symmetry-breaking pattern. These bosons are no longer expected to be massless, and can now be identified with the lightest pseudoscalar mesons.

In addiction to that, another source of explicit symmetry breaking comes from electro-magnetic interactions of quarks, which give a contribution to charged pseudo-Goldstone bosons masses.

Finally, we discuss what is the proper number of light quark flavours to be consid-ered almost massless. As we have pointed out, the pseudoscalar meson octet is much lighter than the other hadronic multiplets: this observation leads to set the number of light flavours to L=3. Indeed, if we consider the symmetry-breaking pattern (1.30), we expect a number of Goldstone bosons equal to the number of broken generators, that is, L2− 1 = 8. 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 precise

if the number of light flavours is L=2 (Heisenberg’s isospin symmetry) rather than L=3 (Gell-Mann’s symmetry). This evidence is explained by the fact that, as we have already discussed, in nature three quark flavours 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, the case L=2 is more precise but the case L=3 can explain a larger range of phenomena (pions kaons and etas spectrum and interactions). In the following thesis we will always work in the L=3 case.

1.5 U(1) axial anomaly

We have now to discuss the role of the U (1)A symmetry. With the same reasoning

made in the previous section it’s easy to prove that this subgroup can not be realized a’ la Wigner-Weyl.

The natural idea is assuming that this U(1) symmetry is spontaneously broken, giving rise to a new pseudo-Goldstone boson, singlet of SU (L)L⊗ SU (L)R, pseudoscalar, with

a mass term due to quark mass matrix. The observed candidate is the η0.

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1.5 U(1) axial anomaly 9

describes the eight pseudoscalar mesons originated by (1.30) plus the singlet pseudo-Goldstone boson corresponding to the U (1)Abreaking. Neglecting the small parameter

∆ = mu− md, which quantifies the size of isospin breaking, he found a non-zero mixing

angle from the meson mass matrix between the singlet and the isoscalar element of the pseudoscalar octet. The lightest eingenvalue of the mixing matrix should satisfy the following inequality:

mlight<

√

3mπ0 (1.31)

However, the two candidates with the right quantum numbers, the η and the η0, does not obey the limit (1.31) [6]. (in particular the η0 mass is mysteriously larger than the octet mesons’ ones). This issue is known as the U (1) problem.

1.5.1 t’Hooft’s solution of the U(1) problem

The key observation is that U (1)Ais a symmetry of QCD (in the chiral limit) only at the

classical level. At the quantum level, it is affected by the so called Adler-Bell-J ackiw anomaly [8]: due to the non-invariance of the fermionic measure in the functional integral under U (1)Atransformations, the divergence of the axial current in the chiral

limit is classically null but non-zero because of quantum corrections. Therefore, the effect of the anomaly is that of explicitly break the symmetry, also in the chiral limit. The expression for the axial current divergence in the chiral limit is

∂µJ5µ= 2LQ (1.32) where Q ≡ g 2 64π2 µνρσ_Fa µνFρσa (1.33)

is the topological charge density operator.

This expression shows that the axial charge is not conserved even in the chiral limit :

∆Q5 = Z +∞ −∞ dtdQ5 dt = Z +∞ −∞ dt Z d3x∂0J50= 2L Z d4xQ(x) ≡ 2Lq (1.34)

where q is the topological charge operator.

However it is possibile to write the topological charge density Q as the divergence of the Chern-Simons current, defined as

Q = ∂µKµ; Kµ= g2 8π2 µνρσ_Tr Aν(∂ρAσ+ 2 3igAρAσ) (1.35)

By substituting (1.35) in (1.34) one could argue that the axial charge is conserved, being the integral of a total divergence. This way, despite the axial anomaly, the sym-metry would be restored.

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thanks to the existence of topologically non trivial solutions named istantons : they are Euclidean gauge fields configurations, with topological charge different from zero, which minimizes the Euclidean gauge action [9].

t’Hooft’s work solves the U(1) problem : since the axial U(1) is no longer a symmetry we do not expect the existence of another pseudo-Goldstone boson in the spectrum. Despite solving the problem, this solution raised new questions. In particular: what is the effect of the anomaly on the spectrum? Is it related to the η0? Or, in other words, why is the η0 mass so large if compared with the masses of the octet’s mesons? To date, the best and easiest solution to the U(1) problem is that proposed by Witten: he 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 mechanism and η0 mass

Witten’s reasonement [11] is based on the assumption that the U (1)A is both broken

spontaneously on the QCD vacuum and explicitly by the quantum anomaly. This hy-potesis predicts the existence of a psuedo-Goldstone boson, singlet of SU (L)V, with a

non-zero mass also in the chiral limit, thanks to the anomaly contribution. This new state can be identified with the η0.

To study the anomaly contribution to the singlet mass the best idea is that of consider-ing an appropiate limit in which the anomaly is suppressed and the symmetry restored (always spontaneously broken); in such a limit (being also in the chiral limit) the singlet psuedo-Goldstone becomes a true Goldstone boson with zero mass. Introducing then the anomaly as a small effect, it will give rise to the singlet mass.

The anomaly is supressed in the large NC limit: we take the limit NC → ∞, where NC

is the number of coulors of the theory; in order to keep ΛQCD constant we can see from

(1.15) that we must take g2NC fixed [10], so g2 → 0 as 1/NC. Indeed, considering the

behaviour of g2 and (1.33) it’s obvious that the topological charge

Q ∼ 1 NC

→ 0 so that the anomalous term vanishes.

Following Witten we introduce the two point function of the topological charge density operator:

χ(k) ≡ −i hQQi (k) = −i Z

d4xeikxhT Q(x)Q(0)i (1.36)

where the angle brackets denote the expectation value on vacuum. For generic operators Oi it is defined as

hT O1O2i ≡

R [dΨ]O1O2eiS[Ψ]

R [dΨ]eiS[Ψ] (1.37)

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1.5 U(1) axial anomaly 11

measure.

The value of (1.36) for k =0 is known in the literature as the topological susceptibility: χ ≡ −i hQQi (0) = −i

Z

d4x hT Q(x)Q(0)i (1.38)

We notice first of all that the topological susceptibility is null in the chiral limit. Indeed we can express it in the form

χ = i V T 1 Z[0] d2Z[θ] dθ2 |θ= 0 (1.39) where Z[θ] = Z [dA][d¯qdq]eiR d4x LQCD+θQ _(1.40)

is the partiction function of the theory with a parameter θ which plays the role of an external current.

Now, it is easy to prove that, in the chiral limit (actually, it is sufficient that at least one quark is massless), the partition function turns out to be totally independent of θ. Indeed let us consider the following U (1)Atransformation:

q → eiαγ5_q _(1.41)

If M =0 the QCD Lagrangian is unchanged; the field Q(x) does not depend on fermionic variables, so the only variation is caused by the anomaly in the fermionic functional measure

[d¯qdq] → [d¯qdq]e−iR d4x2LαQ(x) (1.42)

Since we have only implemented a change of variables in the functional integral, the partiction function must be the same

Z[θ] = Z

[dA][d¯qdq]eiR d4x(LQCD+(θ−2Lα)Q) = Z[θ − 2Lα] (1.43)

Selecting α = θ/2L, we conclude Z[θ] = Z[0], that is the partiction function is indipen-det of θ and so, from (1.39), χ = 0.

We can now expand the two point function in powers of 1/NC

χ(k) = A0(k) + A1(k) + A2(k) + ... (1.44)

where the Ai represents the contribution to χ(k) of order N_C−i [10], coming from

dia-grams with i loops of quarks (every loop of quarks brings a factor 1/NC): A0 is a pure

gauge term, A1 includes all diagrams with one fermionic loop, etc.

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A ≡ lim

k→0A0(k) (1.45)

and we assume A 6= 0. This hypotesis is confirmed by lattice calculations, which predict A = (180 ± 5 MeV)4 [54].

How it possible that χ = 0 if A 6= 0 and the following terms are suppressed by powers of 1/NC? Let’s focus on the first two terms, recstricting our analysis at the first order

in 1/NC: the A1 term can be intepreted as the sum of diagrams with the exchange of

one meson (it is the propagation of a colour singlet ¯qq) between two Q fields. In this way (1.44) can be expressed in the form

χ(k) = A0(k) + X mesons | hΩ|Q(0)|ni |2 k2_{− M}2 n (1.46)

where |Ωi is the vacuum state while |ni is a generic meson state with mass Mn.

The only possibility to ensure that χ(0) = 0 is assuming the existence of a meson state S, with mass M_S2 ∼ 1/N_C. Moreover this state must be a pseudoscalar flavour singlet, in order to be coupled to the vacuum state by the field Q.

From (1.46) we find

χ(k = 0) = 0 = A − | hΩ|Q(0)|Si |

2

M_S2 (1.47)

We have finally to calculate the matrix element of Q between the vacuum and the singlet state.

By using equation (1.32), and parametrizing

hΩ|J₅µ|S(p)i = i√2LFSpµe−ipx (1.48)

where FS is the singlet decay constant, we find

hΩ|Q(0)|S(p)i = 1 2L∂µhΩ|J µ 5|S(p)i = 1 √ 2LFSM 2 Se −ipx _(1.49)

Finally, substituting (1.49) in (1.47) we find M_S2 = 2LA

F2 π

(1.50)

which is the famous Witten’s formula for the singlet mass. In the last step we have used that, to the lowest order in 1/NC,

FS ≈ Fπ (1.51)

where Fπ ≈ 92.2 MeV is the pion decay constant.

Let us remark that, being A = O(N_C0) and Fπ = O(

√

NC) [12], we have M_S2= O(1/NC),

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1.6 The topological θ term 13

singlet mass, which tends to zero in the large NC limit.

Obviously, introducing a non-null quark mass matrix M, there is a contribution to M_S2 proportional to M, which, together with the anomalous term, is able to explain the large mass of the η.

1.6 The topological θ term

In the partiction function of QCD, introduced in the previous chapter, we have con-sidered (see equation (1.40)) a term proportional to the topological charge density Q. We call it θ term, being the parameter θ a free constant. Is this term only an external source or it can be added to the QCD Lagrangian?

Let’s recap how to write the most general Lagrangian for a gauge theory. Given the degrees of freedom of the theory (quarks and gluons), the Lagrangian must include all the gauge-invariant (and Lorentz-invariant) terms. As an optional requirement we may include only the renomalizable terms.

Let’s focus on the θ term,

L_θ= θQ (1.52)

This term is Lorentz-invariant, gauge invariant and even renormalizable. As we have seen, despite Q is the divergence of a current, Lθ is non null because of instantons. We

can conclude that Lθ is a legitimate term in our Lagrangian.

This new term violates the CP-symmetry. In fact :

Q ∼ F_µνa F_aµν ∼ ~Ea· ~Ba (1.53)

being ~Ea( ~Ba) the chromoelectric(magnetic) field, is invariant under charge conjugation,

while it is a pseudoscalar under parity and time reversal. With θ different from zero QCD is no longer CP-invariant.

From an experimental point of view, we know that QCD does not violate CP, with a very high accuracy. In particular we can find a relationship between the θ parameter and an experimental quantity, which is the neutron electric dipole moment dN [13]:

dN ≈

M_π2

M_N3 e|θ|≈ 10

−16_{|θ|e cm} _(1.54)

where MN is the neutron mass and Mπ the pion mass.

From the experimental data [14] we know that dN < 10−26e cm, which leads to the

upper buond

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The relation (1.54) is based on dimensional ground but considers also some more re-fined properties of the neutron electric dipole moment: in particular, dN is expected to

be linear in the light quark masses (and, so, proportional to M_π2) since, 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 M_N3 (which in this case roughly plays the role of the typical nucleon mass scale).

More refined relations among the neutron electric dipole moment and the θ angle were derived by Baluni [15], in the framework of the so-called bag model, by Di Vec-chia, Veneziano et al. [16], using the Chiral Perturbation Theory, and by many other physicists using different approaches. The aim was to translate the above mentioned experimental bound into a constraint on θ computing the quantity cN, defined as:

dN = cN|θ|10−13e cm. There is a substantial global accordance on the magnitude of

this coefficient (0.001 6 |cN|6 0.01) but not on its sign (see Sec. 7.1 of Ref. [12] for a

more detailed discussion).

These considerations suggest that either θ = 0, and this will be a fine-tuning prob-lem, or there must be a mechanism responsible for suppressing the value of θ in QCD. This discussion is known as the strong CP-problem.

Before discussing the possible solutions, we have to study in detail the properties of the θ term. First of all, it is strongly related to the quark mass term. Indeed let’s write the mass term in a more generic form

LM= −¯qRMqL− ¯qLM†qR (1.56)

where M is a general complex mass matrix. In terms of chirality projectors, it becomes

LM= −¯qPLMPLq − ¯qPRM†PRq = −¯qMPLq − ¯qM†PRq = −¯q M + M † 2 + M − M † 2 γ5 q ≡ −¯q (A + iγ5B) q (1.57)

where A and B are hermitian operators. The term including the γ5 violates CP exactly

as the topological θ term. This suggests a connection between the two terms.

To understand this connection, let us consider the change of variables induced by the SU (L)L⊗ SU (L)R⊗ U (1)A transformation ( qL→ qL0 = eVLqL= eiαVLqL qR→ qR0 = eVRqR= e−iαVRqR (1.58) to be implemented on Z = Z [dA][d¯qdq]eiR d4xLQCD (1.59)

where in LQCD we have included the topological θ term and the generic mass term

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1.6 The topological θ term 15

Under the chiral transformation (1.58) both the fermionic functional measure (anomaly) and the mass term are non-invariant :

[d¯qdq] → [d¯qdq]e−iR d4x2LαQ(x) (1.60) LM → L_M0 = −¯q_RM 0 qL− ¯qLM 0_† qR (1.61) where M0 = eV_R†M eVL.

Therefore the partiction function becomes Z = Z [dA][d¯qdq]eiR d 4_xL0 QCD _(1.62) in which L0_QCD= −1₂Tr[FµνFµν] + ¯qiγµDµq − ¯qRM 0 qL− ¯qLM 0_† qR+ (θ − 2Lα)Q.

But, since we have implemented only a change of variables, (1.59) and (1.62) describe exactly the same physics, i.e they are completely equivalent. They formally differ only by the changes

(

M → M0 = eV_R†M eVL

θ → θ0 = θ − 2Lα (1.63)

We notice that, if M is invertible, that is, if det M 6= 0, we have:

det M0 = (det eVR)∗det M det eVL= e2iLαdet M (1.64)

or in other words

arg(det M0) = arg(det M) + 2Lα (1.65)

Looking at equations (1.63) and (1.65), we notice that, under the transformation (1.58) the quantity

θphys ≡ θ + arg(det M) (1.66)

is invariant. This quantity is therefore a physical parameter.

The θ and the (CP violating part of the) mass term can be transformed one in each other by chiral rotations , but the physical observable that quantifies the strenght of CP violation is θphys. Its presence, that is, θphys 6= 0 , is a sign of the strong CP

violation. Indeed, (1.66) tells us that, in that case, if the mass matrix is invertible, it is impossible to cancel the topological term by a chiral rotation. Indeed, if we implement a transformation setting

α = θ

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which implies θ0 = 0, the whole CP violation is moved on the mass term; on the other hand, to diagonalize the mass term, leading to M =diag(mu, md, ...), we must

imple-ment a chiral rotation which makes θ0 6= 0, and the whole violation is carried by the topological term.

The relation (1.55), which gives an upper bound on the magnitude of the θ coefficient, is therefore referred to the combination (1.66). If we want QCD to be a CP-conserving theory we must find some mechanism for which θphys is suppressed (taking directly

θphys = 0 would be a fine-tuning problem).

1.6.1 Massless quark up

The strong CP-problem can be solved if the QCD Lagrangian is invariant under an appropriate U (1) chiral rotation. In particular this transformation must act as the identity on the quark mass matrix and must be broken explicitly by a quantum anomaly proportional to the topological charge density Q, so as to shift θ to zero.

The simplest way to accomplish that within the Standard Model is considering QCD with the up quark massless, mu = 0. The Lagrangian is therefore invariant under the

chiral rotation

qu → eiαγ5qu (1.68)

which is the usual U (1)A transformation acting only on the up quark. The symmetry

is broken by the quantum anomaly (1.42) with L=1, while the quark mass term is obviously invariant. Therefore we find      M → M θ → θ0 = θ − 2α θphys→ θ 0 phys= θphys− 2α (1.69)

We can choose α = θ₂ and the mass matrix to be real and diagonal, and consequently θphys = 0, solving the strong CP problem.

This solution has the advantage of not requiring neither Standard Model’s extensions nor new symmetries apart from usual chiral transformations. It also does not intro-duce new particles. On the other side it is still unexplained why the up quark must be massless, leading to a new fine-tuning problem.

Anyway the experimental data do not support this hypotesis : in particular we have [6]

mu

md

≈ 0.56 (1.70)

which is inconsistent with the assumption mu = 0. Therefore it is necessary to find

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1.7 Axion solution to the strong CP problem and Peccei-Quinn mechanism 17

1.7 Axion solution to the strong CP problem and

Peccei-Quinn mechanism

As we have discussed in the previous section, the strong CP problem can be solved if our Lagrangian is invariant under an appropriate U(1) chiral rotation. The idea behind the Peccei-Quinn (PQ) mechanism [17] is that of extending the Standard Model with a new scalar field so as to define a new U(1) symmetry group, named U (1)P Q. It is

characterized by two basic properties : first of all, it is spontaneously broken at an energy scale fa, giving rise to a new Goldstone boson; then, it is explicitly broken

by a quantum anomaly proportional to the topological charge Q. The axion is the Goldstone boson associated to U (1)P Q(the physical axion mixes with the pseudoscalar

mesons so it is a linear superposition between the U (1)P Q Goldstone boson and them)

and under a generic U (1)P Q transformation with a constant parameter γ, the axion

field a translates to

a → a0 = a + γfa (1.71)

The Lagrangian must be invariant under this translation (apart from the anomaly term).

The most general QCD Lagrangian including the axion is the following : L_QCD+axion= LQCD+ θQ + 1 2∂µa∂ µ_{a + a} P Q a fa Q + Lint[∂µa, Ψ] (1.72)

where aP Q is a free parameter.

Let’s analyze this Lagrangian: we have a kinetic term for the axion field, a term propor-tional to Q, which reproduces the quantum anomaly, and interactions with fermions, which must contain only derivatives of the axion field so as the Lagrangian is invariant under (1.71). These interaction are of a kind

Lint[∂µa, Ψ] =

1 fa

¯

Ψγµγ5∂µaΨ (1.73)

It is really interesting that all axion interactions (and we will see also its mass) are suppressed by the scale fa.

The Noether’s current associated to U (1)P Qis not conserved because of the anomaly :

∂µJ_{P Q}µ = aP QQ (1.74)

The whole effect of the new symmetry is therefore to replace the static θ with a dy-namical degree of freedom, that is the axion field.

Now we can act a U (1)P Q transformation (1.71) with γ = _a_{P Q}θ and cancel the θ term.

We notice that our transformation has no effects on the quark mass matrix so we are always free to choose it real and diagonal. As a result U (1)P Q sets to zero θphys solving

the strong CP problem. This is known as Peccei-Quinn mechanism.

Obviously, before the transformation, the axion field in the Lagrangian must satisfy hai = −_aθ

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non-zero v.e.v. of the axion field dinamically restores CP, i.eDθ + aP Q_fa_a

E = 0. Let’s analyze some general properties of the axion. First of all we notice that there is a spontaneously broken but anomaly-free U (1) symmetry, which is a subgroup of U (1)A⊗ U (1)P Q : if we denote with β the U (1)A parameter and with γ the U (1)P Q

one, it’s easy to see that if we choose

βL + γaP Q= 0 (1.75)

the resulting U(1) subgroup is anomaly-free. In fact, thanks to the mixing with the neutral pseudoscalar mesons, the physical axion can be seen as the Goldstone boson associated to this anomaly-free subgroup [18]. There are two important consequences : first of all, in the chiral limit the axion is massless (no contributes from the anomaly to its mass); secondly, when we introduce quark masses we break explicitly U (1)A and so

also its subgroup. It follows that out of the chiral limit the axion has a non-zero mass and it is a pseudo-Goldstone boson.

Summing up, the Peccei-Quinn mechanism solves the strong CP problem introduct-ing a new pseudoscalar boson, elettrically neutral, sintroduct-inglet of the chiral group and (as we will discuss) with a very small mass, which, together with its interactions, is con-trolled by the scale fa.

1.8 Axion’s models

1.8.1 PQWW model

The Peccei-Quinn-Wilczeck-Weinberg (PQWW) [17,19,20,21] model introduces an ad-ditional complex scalar field as a second Higgs doublet. One Higgs (Hu) gives mass to

the up-type quarks and the other one (Hd) gives mass to the down-type quarks. A

free-dom of the model is the choice of which dobulet gives mass to leptons. The Lagrangian must be invariant under the gauge group of the Standard Model SU (2)L⊗ U (1)Y and

the global symmetry U (1)P Q, which acts with chiral rotations (it distinguishes between

left and right fermions). There is a certain freedom in the choice of how U (1)P Q acts

on fermionic fields; one convenient choice is

uR→ e−iγxuR (1.76)

dR→ e−i

γ

xd_R (1.77)

where u are the up-type quarks and d the down ones, while x is a fixed parameter. The left fermions are invariant in this scheme.

The two Higgs transform as

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1.8 Axion’s models 19

Hd→ ei

γ

xH_d (1.79)

so as the Yukawa interactions are invariant :

LY ukawa,quarks = YuQ¯LHuuR+ YdQ¯LHddR+ h.c. (1.80)

The gauge group, together with U (1)P Q are spontaneously broken by the Higgs

po-tential at the electroweak scale v ∼ 250 GeV, which in this model coincides with fa.

The axion field a is introduced as the common phase field in Hu and Hd, which is

ortogonal to the weak hypercharge. On the vacuum : Hu = v1 √ 2e iax_v 1 0 (1.81) and Hd= v2 √ 2e i_xva 0 1 (1.82) with v =pv2 1+ v22.

It is easy to see that under U (1)P Q the axion transforms as

a → a + γv (1.83)

which is of the form (1.71).

From the transformation laws (1.76) and (1.77) it is clear that (some of) the Stan-dard Model fields are charged under U (1)P Q. The axion couples to them through the

Yukawa term. Focusing only on the quark sector, the symmetry current for U (1)P Q is

J_{P Q}µ = v∂µa + x Nf am X i=1 ¯ uiRγµuiR+ 1 x Nf am X i=1 ¯ diRγµdiR (1.84)

where Nfam is the number of families of up and down quarks in the Standard Model

(Nf am=3 ). The current is affected by a quantum anomaly of a kind (1.73), with

aP Q = Nfam 2 x + 1 x (1.85)

This model has been ruled out by experiments: as expected, all axion interactions are suppressed by the scale fa; the problem is that in PQWW model this scale is

taken as the electroweak scale v, leading to axion couplings and mass too large and not compatible with experimental data (see beam-dump experiments [22] and collider experiments [23]).

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A way to overcome this problem is the introduction of the so called invisible axion models: they introduce scalar fields which carry PQ charge but are singlets of the Standard Model gauge group. As a result, it is possible that these fields, through some potential, spontaneously break U (1)P Q at a scale fa which is now a free parameter of

the theory. We can assume fa v leading to very suppressed axion couplings and

mass, still compatible with experiments.

There are two important classes of models in the literature: the KSVZ model and the DFSZ model.

1.8.2 KSVZ model

The KSVZ model [24,25,26] introduces a complex scalar field φ, which is a singlet of SU (2)L⊗ U (1)Y and carries U (1)P Q charge. The only field which interacts with it is

an hypothetical heavy quark qh. The Lagrangian describing these fields is

L = ¯qhiγµDµqh+ (∂µφ)†(∂µφ) − g(φ¯qhRqhL+ h.c.) − V (φφ†) (1.86)

The potential is built so as the v.e.v. of the scalar field is different from zero and U (1)P Q is spontaneously broken at a scale fa

hφi ∼ fa (1.87)

Neglecting the quantum fluctations of the modulus of φ we have φ = √fa

2e

ia

fa _(1.88)

where a is the axion field.

From the interaction term we derive the mass of the heavy quark, mqh =

fa

√

2g, which

(taking fa v) can be integrated out at low energy. The ”complex mass-interaction”

term mqh(e

i_faa

¯

qhRqhL+ h.c.) is equivalent to an anomaly term proportional to the QCD

topological charge density, of the kind of equation (1.72) (the equivalence between the anomaly and the ”complex mass term” is the same we have used in section 1.6 to define θphys ). Once the heavy quark has been integrated out, the anomaly term is the only

modification to the Standard Model Lagrangian. In particular the KSVZ axion has not tree-level couplings to the Standard Model matter fields.

1.8.3 DFSZ model

DFSZ model [27,28,26] is a variant of the PQWW one. It introduces two Higgs doublets, exactly as in PQWW, plus a complex scalar field φ, singlet of the Standard Model gauge group. Both φ and the two Higgs carry the appropriate P Q charges. Fermions are charged as in PQWW model so as the Yukawa term (1.80) is invariant. The scalar potential is built as the most general combination of φ and the two Higgs which is invariant under both the gauge group and U (1)P Q and leads to the spontaneous

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1.8 Axion’s models 21

breaking of both the symmetries. Thanks to the introduction of the new scalar singlet field, the gauge group and U (1)P Q are broken at two independent scales. It is always

possible to choose the parameters of the theory so as the electroweak breaking scale is set to v ∼ 250 GeV while the P Q scale is the free parameter fa.

The scalar fields are parametrized as in equations (1.81),(1.82) and (1.88), with three different phase fields. The axion emerges from the mixing between these phase fields and its tree level interactions with fermions come from the Yukawa term.

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Chapter 2

Chiral effective Lagrangians with

the axion

2.1 Effective degrees of freedom of QCD

One of the characteristic properties of QCD is the confinement: all the observable hadronic states must be singlets of the gauge coulor group SU (3)C. As a consequence,

the fundamental degrees of freedom of the theory, quarks and gluons, can never be directly observed in experiments. The physical particles which we observe are always bound states of quarks, antiquarks and gluons, combined in such a way that the re-sulting state is gauge-invariant. Mesons and baryons are the most familiar examples of these states.

Since for energies . ΛQCD QCD is a non-perturbative theory, it would be

impossi-ble to describe the low-energy dynamics directly using the QCD Lagrangian. The solution to this problem is using some effective degrees of freedom of the theory, which can be considered as composite operators, multilinear in the fundamental fields, which carry the quantum numbers of the physical hadronic states. Their dynamics can be described by an effective Lagrangian, which must be endowed with the same transfor-mation properties of the QCD Lagrangian.

In this thesis we will focus on the low-energy dynamics of the pseudoscalar mesons composed by the three lightest quarks. In particular, since the axion is expected to be a very light particle, we can include it in this description. Our aim is to study the axion mass and its interactions with photons and mesons, making use of this low-energy formulation of the theory.

• First we will describe a general procedure to derive an effective Lagrangian for the lightest pseudoscalar mesons, which, as we have already pointed out, can be identified with the pseudo-Goldstone bosons of the chiral group (1.30) and U (1)A.

In particular we will describe four different models, which will be specified in the following.

• Then we will introduce the axion field and describe how it can remove explicitly the θ term (and so, CP-violation).

• Finally we will focus on axion properties analyzing the different effective La-23

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grangian models we have introduced.

The field variables of our low-energy Lagrangian are the pseudo-Goldstone bosons com-ing from the spontaneous breakcom-ing of axial symmetries. So as to satisfy the correct symmetry transformation properties, the effective Lagrangian L must be invariant, in the chiral limit, under the group

U (1)V ⊗ SU (L)L⊗ SU (L)R (2.1)

while under U (1)Atransformations (1.41), it must transform, in the chiral limit, as

L → L − 2LαQ (2.2)

so as to reproduce the anomalous fermionic measure transformation (1.42).

Notice that (2.1) invariance must realized in such a way that the symmetry-breaking pattern (1.30) is reproduced.

Over the years, many different chiral effective Lagrangian models have been developed and used to study the low-energy dynamics of the pseudoscalar mesons. In this thesis, we have focused on four of these models, which have been critically compared one with each other.

In these models, the meson fields are represented by a L × L complex matrix Uij,

which can be written, in terms of quark fields, as Uij ∼ ¯qj

1 + γ5

2

qi = ¯qjRqiL (2.3)

up to a multiplicative constant. Under a parity transformation, we have (

qiL(x0, ~x) → qiR(x0, −~x)

qiR(x0, ~x) → qiL(x0, −~x)

(2.4)

from which the U field transforms as

Uij(x0, ~x) → U_ij†(x0, −~x) (2.5)

From (2.5) we can see that the hermitian part of the matrix U describes the scalar degrees of freedom, while the antihermitian part describes the pseudoscalar ones. Under a chiral transformation (1.58) the field U transforms as

U → eVLU eV_R† (2.6)

The most simple model, symmetric under the whole symmetry group (1.26) which can be built is the so-called Linear σ-model, originally proposed in [29] but later elab-orated on and extended (see [30]). The related Lagrangian, in the chiral limit (M=0 ), is the following :

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2.1 Effective degrees of freedom of QCD 25 L0 = F_π2 4 Tr h ∂µU ∂µU† i − V0(U, U†) (2.7) V0(U, U†) = 1 4λ 2 πTr " F2 π 2 U U †_{− ρ} πI 2# +1 4λ 02 π Tr F 2 π 2 U †_U 2 (2.8)

where Fπ ≈ 92 MeV is the pion decay constant [6], the first term is ”kinetic-like”,

while the second one is a ”Mexican hat ” potential term. Here we notice the importance of the coefficient ρπ: its sign defines the symmetry properties of the vacuum state,

determining whether the minimum of the potential (this is nothing but the vacuum expectation value < U > of the meson field U ) is invariant under the whole symmetry group (1.26) or only under one of its subgroups.

In particular, if ρπ > 0, (1.26) is spontaneously broken by a non-zero value of hU i,

which turns out to be [31]:

hU i = I (2.9)

and which is invariant only under the U (L)V = U (1)V ⊗ SU (L)V subgroup of (1.26).

Therefore, in this hypotesis the symmetry group G is spontaneously broken to its vec-torial subgroup.

On the other hand, if ρπ < 0, we have hU i = 0, which is invariant under the whole

group, signalling the restoration of the chiral symmetry in the Wigner-Weyl way. Since we want to study the dynamics at T = 0, and in this regime, as we have seen, the chiral symmetry is spontaneously broken, we have to consider the case in which ρπ > 0.

The considerations on the behaviour of the symmetry properties of the vacuum call us to focus on how to choose the parametrization of the field U.

As described in section 1.4, Goldstone’s theorem states that, in the presence of a spon-taneously broken symmetry, there exists, in the spectrum of the theory, a spin zero massless boson corresponding to each broken generator. In addiction to the Goldstone bosons, the spectrum contains also some ”heavy” states, with mass comparable to ΛQCD. Nevertheless, in the low-energy regime, these heavy degrees of freedom can be

considerd as static fields, decoupled from the light ones: therefore the dynamics of the theory will be dominated by the Goldstone bosons.

As we have seen, in our case the chiral group SU (L)L⊗ SU (L)Ris spontaneously

bro-ken to its vectorial subgroup: the σ-model can reproduce this scenario if we choose to express the field U in the following non-linear representation, also called polar decom-position:

U (x) = H(x)Γ(x) =I + ˜H(x)eiΦ(x) (2.10)

where:

• H(x) = I + ˜H(x) is a generic hermitian L × L matrix

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• ˜H(x) is an hermitian matrix which takes the null value in the minimum of the potential: so it describes the fluctuations of meson scalar fields

• Γ(x) = eiΦ(x) _{is a generic U(L) element}

• The field Φ(x) is defined as:

Φ(x) = 1 Fπ πa(x)λa+ r 2 LS(x) ! (2.11)

and describes the pseudoscalar excitations of the spectrum, i.e the Goldstone bosons of the axial symmetries.

The matrices λa are the generators of SU(L), normalized as

Trhλaλbi= 2δab (2.12)

(in the case L=3 they are the Gell-Mann matrices ), πa are the Goldstone bosons of SU (L)L⊗ SU (L)R, while S, usually called the ”singlet ”, is the Goldstone boson of

U (1)A.

Substituting (2.11) into (2.7-2.8) we find that ([31]) the scalar fluctuations described by ˜H have non-zero squared masses M_scalars2 ∼ λ2

πFπ2, while the fields πa and S are

massless: this confirms they are Goldstone bosons.

Therefore, since we are interested in describing the low-energy dynamics of the ef-fective degrees of freedom (that is the Goldstone fields πa _{and S), we can decouple}

the scalar massive fields by letting λπ → ∞: this way we are implementing the static

limit ˜H → 0, giving infinite mass to all the scalar fields. In this limit, looking at the potential term in (2.8), we are forcing the constraint Fπ2

2 U U † _{= ρ}

πI, which implies

TrU†_U_{= constant: therefore the term proportional to λ}02

π is just an irrilevant

con-stant term, which can be dropped by setting λ02_π = 0.

Finally, neglecting the scalar degrees of freedom, and setting L = 3 our field is written as U = e i Fπ πa_λa₊q2 3S (2.13)

and so the σ-model, in the chiral limit, becomes L = F 2 π 4 Tr h ∂µU†∂µU i (2.14)

which is usually called Non Linear σ-model. In fact (2.14) turns out to be the first term in the so called Chiral Effective Lagrangian expansion, which will be discussed in greater detail in the following section. Notice that so far we have not include neither quark mass matrix M nor the effect of the U (1)Aanomaly. In the next sections we will

first recover (2.14) with the effective Lagrangian approach then we will include also the above mentioned contributions.

Study of the mass of the axion and of its interactions with mesons and photons using chiral eﬀective Lagrangian models