Study of the Axion Potential in Chiral Effective Models around the Chiral Phase Transition

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UNIVERSITÀ DI PISA

D  F

C  L M  F

T  L M

Study of the Axion Potential in

Chiral Eﬀective Models around the

Chiral Phase Transition

Candidato

Salvatore Boaro

Relatore

Prof. Enrico Meggiolaro

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Introduction

antum Chromo-Dynamics, or QCD, represents a conceptually simple, economic, yet rich and successful theory of the strong interactions. It describes the dynamics of hadrons, the strongly interacting particles such as protons and pions, within the framework of a

non-Abelian gauge theory based on the colour gauge group SU(3)c and in terms of the

fundamental spin 1

2 constituents of hadrons, the quarks, whose interactions are mediated

by massless spin 1 bosons, the gluons, which experience strong interactions as well, as a consequence of the non-Abelian structure of the gauge group.

e richness of QCD is essentially due to the striking features emerging from its rather simple structure:

• Asymptotic freedom. e coupling constant, giving the strength of the interactions, becomes smaller at high energies or, equivalently, at short distances. is allowed making consistent predictions, based on perturbation theory, concerning the high energy phenomenology of strong interactions, which have been conﬁrmed by

ex-periments at the colliders. However, as we lower the energy to a scale ΛQCD ≈

400MeV, asymptotic freedom rapidly turns into an infrared slavery, since the

cou-pling constant becomes large, making perturbation theory no longer reliable. • Colour conﬁnement. Because of the strong dynamics of QCD at low energy, quarks

and gluons gather to form a rich population of bound states, i.e. the hadrons. How-ever, the lack of observations of free quarks and gluons, and the fact that all hadrons carry a zero net colour charge forced the introduction of the Conﬁnement Postulate, which states that asymptotic states must be colour neutral.

• Spontaneous breaking of chiral symmetries. QCD, restricted to the lightest nlquarks,

is invariant under the global symmetry ⊗nl

f =1U (1)f, the physically relevant cases

being nl = 2, for the up and the down quarks, and nl = 3if we include the strange

quark, whose masses are smaller than ΛQCD. is symmetry group is enlarged at

the classical level to ˜G = U (nl)R⊗U(nl)L= SU (nl)R⊗SU(nl)L⊗U(1)R⊗U(1)L,

where R, L refer to the independent right and le chiral components of the quark ﬁelds, if these quarks are taken to be massless. e strong dynamics of QCD at

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low energies leads to the condensation of quark-antiquark pairs and thus to a

non-vanishing Vacuum Expectation Value (VEV) for the quark bilinear⟨¯qq⟩ ̸= 0, called

the chiral condensate. is condensate induces the spontaneous breaking of the symmetry group ˜Gdown to U(nl)V.

As a consequence of the spontaneous breaking induced by the chiral condensate, the Goldstone’s theorem predicts the existence of massless spin 0 particles, the Nambu-Goldstone bosons (NGBs), with the same quantum numbers of the conserved currents corresponding

to the broken generators of the symmetry group ˜G. Indeed, these NGBs can be identiﬁed

with the triplet of pions (nl = 2), which can be enlarged to the octet of the light

pseu-doscalar mesons (nl = 3). ough not massless, since ˜Gis also broken explicitly by the

quark masses, they form the lightest hadronic excitations of the QCD spectrum.

Actually, the fact that at most eight Goldstone bosons were observed represented a puzzle, since the chiral condensate breaks also the U(1)A, where A stands for axial,

sub-group of ˜G, corresponding to a phase redeﬁnition of the quark ﬁelds, where opposite

chiralities transform with opposite phases. However, the lightest candidate for the U(1)A

Goldstone is the η′ _{meson, which is too heavy if compared to the lightest mesons. is}

statement was made precise by Weinberg, who found an upper limit for the mass of the

lightest isosinglet (I = 0) pseudoscalar meson given by MI=0 <

√

3Mπ (Weinberg’s

limit), where Mπ is the pion mass, which is violated by both the η and the η′mesons. is

puzzle was dubbed the U(1)A problem, which led Weinberg to argue that QCD has no

U (1)Asymmetry so that no NGB should be expected. Indeed, it was soon realized that

the U(1)Asymmetry is broken at the quantum level by the Adler-Bell-Jackiw anomaly, so

that, even in the chiral limit, the corresponding current Jµ

5 is not conserved, but rather:

∂µJ5µ(x) = 2nlQ(x)

where Q(x) is the so-called topological charge density and one can show that it is a total

divergence. While this explained why the η′_{cannot be a NGB, the precise mechanism at}

the origin of its large mass is still debated. ere are essentially two kinds of solutions proposed so far:

• Solution by ’t Hoo. Despite the fact that Q(x) is a total divergence, there exist solutions to the classical equations of motion of the gauge ﬁelds, with the gauge

invariant boundary condition Fµν = 0at inﬁnity (Fµν is the ﬁeld strength tensor

of the gauge ﬁeld), which yield a non-vanishing topological charge q = ∫ d4_xQ(x)_.

ese gauge conﬁgurations, called instantons, induce an eﬀective 2nl-fermion

inter-action term among the light quarks which provides a mass term for the η′_.

• Solution by Wien and Veneziano. In the context of the large Ncexpansion, where

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ex-v Introduction

ponentially suppressed and the mass spliing between the η′ _{and the light}

pseu-doscalars arises from Feynman diagrams for quark-antiquark annihilation.

As we have seen, despite the simple structure of QCD, we are still far from a complete understanding of the theory. Besides the η′ _{mass problem, in fact, there is still no ﬁeld}

theoretical proof of the Conﬁning Postulate, nor it is totally clear why the chiral conden-sate forms. Here the problem lies entirely in the non-perturbative nature of QCD at low energies, which called for new theoretical tools, the most important being Laice QCD (LQCD) and Chiral Eﬀective Lagrangians.

LQCD is a powerful, non-perturbative tool which has been employed to study QCD from ﬁrst principles, that is by using directly the fundamental QCD Lagrangian, generalized to a discretized laice spacetime. In particular, laice studies have shown that at high-temperature QCD loses both its colour conﬁning and condensation properties through two in principle independent phase transitions. ite amazingly, these transitions seem

to happen at the same critical temperature Tc= 150÷ 170 MeV.

A completely different approach is at the base of the Chiral Effective Lagrangian formal-ism, which represents the main tool employed throughout this esis. In this case, the colour confinement and the quark condensation are assumed since the focus is on the low energy phenomenology arising from these two phenomena combined. e fundamental hypothesis is that we need not know the full spectrum of the theory if we wish to study only the dynamics of the lightest particles, like those forming the light pseudoscalar me-son octet, which can be considered as the sole, new, fundamental degrees of freedom. en, their dynamics can be described by means of an Effective Lagrangian, which is wrien in terms only of the relevant degrees of freedom and which must reflect the same symmetries of the underlying theory.

In addition to the diﬃculties mentioned above, the discovery of instantons led to another problem, still waiting to be solved. In fact, the QCD Lagrangian can be aug-mented with the addition of the so-called topological θ-term, where θ is a free param-eter, which, because of the existence of instantons, gives non-zero contributions to the path integral. It can be shown that the operator Q(x) is odd under simultaneous

Par-ity and Charge conjugation (CP) transformations, leading to a direct CP violation in the

strong interactions. In particular, it induces a non-vanishing electric dipole moment for

the neutron given by dN ≈ eθm

2 π M3

N . However, the experimental bounds are very strict:

dN ≤ 10−26e·cm→ |θ| ≤ 10−10. Why should θ be so small or even 0 is known as the

”Strong-CP problem”.

Among the diﬀerent solutions to the strong-CP problem, the most convincing one is represented by the introduction of the axion, which, since its very ﬁrst appearance, has been gaining an increasing interest and is the main object of this esis. e axion was

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originally introduced by Weinberg and Wilczek on the base of a model previously pro-posed by Peccei and inn (for this reason we speak of the PQWW axion). It is based on the introduction into the Standard Model Lagrangian of an additional pseudoscalar ﬁeld (which essentially replaces the free parameter θ) and is endowed with an additional

U (1)symmetry, known as U(1)P Q, which is both spontaneously broken and anomalous.

By virtue of this extra U(1) symmetry, CP comes out to be dynamically conserved in this model and, moreover, a new pseudo-Goldstone boson appear, the axion itself. e original PQWW axion was soon experimentally ruled out, but new fundamental models, namely the Kim-Shifman-Vainshtein-Zakharov (KSVZ) and the Dine-Fischler-Srednicki-Zhitnisky (DFSZ) invisible axion models, were proposed and are still compatible with cur-rent experimental bounds. e axion described by these models is both very light and very weakly interacting with ordinary maer.

ese aspects immediately lead to the second reason why axions are so appealing, es-pecially nowadays. In fact, it was soon recognized that they represent good candidates to explain the current abundance of Dark Maer in the Universe. e most important mecha-nism of axionic Dark Maer production is the so-called vacuum misalignment mechamecha-nism, which consists in the assumption that, in the early Universe, the axion classical field was not located at the minimum of its potential and started oscillating only when the Hubble constant, which represents a sort of friction term, dropped below the value of the axion mass, even if this requires that the value of the field should not be so far from the min-imum. Because the couplings of the axion are weak, the energy density stored in the oscillations of the classical field does not dissipate rapidly, so that, through the computa-tion of the relative axion abundance, this allows to put a lower bound on its mass, which is also constrained from above by other astrophysical bounds.

In general, the evolution of the axion ﬁeld and thus its current abundance may depend on the form of its potential, whose expression is obviously determined by studying the interaction of the axion with the QCD particle content. However, especially at low tem-peratures, this can be achieved only resorting to the Chiral Eﬀective Lagrangians which

must be extended in order to accommodate the axion. Since the U(1)P Qsymmetry suﬀers

from the same anomaly of the U(1)Ain QCD, the inclusion of the axion into the Eﬀective

Lagrangian requires a proper implementation of the anomaly. e ways in which this can

be done parallel the two diﬀerent solutions proposed for the U(1)Aproblem.

e Extended Linear σ (ELσ) model, which includes also the scalar partners of the η′ and

of the light pseudoscalar mesons, implements the anomaly through a term inspired by the

2nl-fermion interaction term which mimics the interactions among fermions through the

exchange of an instanton.

e Wien-Di Vecchia-Veneziano (WDV ) model, which is rigorously derived within the

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vii Introduction

ﬁelds (that is, those describing the η′_{and the axion) to the topological charge density Q(x),}

which serves as an external source.

In addition to these models, in this esis, we shall make use of a third example of Eﬀective Lagrangian, where it is postulated the existence of an additional condensate which breaks spontaneously U(1)Awhile preserving SU(nl)L⊗SU(nl)R. We refer to this model as the

Interpolating Model, since the introduction of this new condensate allows to implement

the anomaly as in the WDV model and at the same time to take into account the eﬀects of the instantons.

In this esis, the axion potential has been studied by employing these three models. In particular, the focus has been on the comparative study of its form around its minimum, and thus the evaluation of the axion mass, and in a neighbourhood of the endpoints of its periodicity interval. e laer case is particularly interesting since, in some regions of the parameter space spanned by the parameters of the Eﬀective Lagrangians, the large mixing of the axion with the standard QCD degrees of freedom makes the axion poten-tial no longer meaningful as we approach these endpoints. e analysis has been made

both at T = 0 and at ﬁnite temperature, below and above the critical temperature Tc, by

means of a mean ﬁeld approach, where all the coeﬃcients of the models are assumed to be temperature dependent.

e plan of the work is as follows.

• Chapter 1. We give a brief, conceptual introduction to QCD and to its chiral symme-tries, emerging when some quarks are exactly massless. Aer a concise discussion

on the U(1)A problem, we review its possible solutions and how they led to the

Strong CP problem in QCD. en, we conclude the Chapter with a sketch of the Peccei-inn mechanism, which ultimately brought to the introduction of the ax-ion, followed by the description of the most important aspects of axion phenomenol-ogy: its mass, decays, astrophysical bounds and relevance in the physics of the Dark Maer.

• Chapter 2. We discuss the relevant degrees of freedom of QCD at low temperatures and the Eﬀective Lagrangian paradigm. en we proceed with the description of

the eﬀective models used in the esis, that is the ELσ, the WDV and the

Interpo-lating model, and how the axion can be consistently implemented in them. Finally, we employ the Eﬀective Lagrangian approach to derive the known and model inde-pendent axion-mass formula, valid to the lowest order in the quark masses and in the limit of weakly interacting axion.

• Chapter 3. e Chapter is devoted to the computation of the axion mass, both above

and below the critical temperature, in the ELσ and Interpolating model, since the

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results to provide more general expression for the so-called topological susceptibility

χQCD, deﬁned as the Fourier transform of the two-point correlation function of the

topological density evaluated at zero momentum, by exploiting the proportional-ity relation m2

axion ∝ χQCD. e zero-temperature expressions for the topological

susceptibility are then evaluated numerically and compared with the laice deter-minations.

• Chapter 4. In the ﬁrst part of the Chapter, we study the QCD phase diagram near

θ = π, reviewing the results known for the WDV model and presenting the original

results for the ELσ and the Interpolating models. We show the existence of ﬁnite

regions of the parameter space where CP is spontaneously broken, separated by a CP-preserving region through a second order phase transition surface, along which the lightest hadronic excitation becomes exactly massless. In the second part of the Chapter, we discuss the consequences of the structure of the QCD phase diagram at θ = π on the axion potential near the boundary of its periodicity interval.

• Conclusions. Finally, a conclusive Chapter will summarize the results obtained through-out the esis and through-outline possible future applications and developments of the present work.

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1

From QCD to the Axion

1.1 QCD Fundamentals

QCD, or antum Chromo-Dynamics, is a non-abelian gauge theory, being based on the

SU (3)c colour gauge group, and is the most successful quantum ﬁeld theory describing

the strong interactions. Its ﬁeld content consists in eight spin 1 gauge bosons, the gluon ﬁelds Aa

µ(x), with a = 1,· · · , 8, and in six ﬂavours of spin 12 Dirac spinors, each of which

living in a fundamental representation of SU(3)c, the up, down, strange, charm, boom

and top quarks indicated as qi

f(x), where i = 1, 2, 3 and f = u, d, s, c, b, t are respectively

the colour and the ﬂavour indices.

Under a gauge transformation U(x)∈ SU(3)c, the ﬁelds transform according to:

U (x) ≡ eiαa(x)Ta :      qi_f(x)→ U(x)ijqjf(x) Aµ(x)→ U(x)Aµ(x)U†(x)− i g(∂µU (x))U †_(x) (1.1)

where Aµ(x)≡ Aaµ(x)Ta, g is the QCD coupling constant and Taare the SU(3) generators,

spanning the corresponding SU(3) algebra, and which satisfy:

[Ta, Tb] = ifabcTc (1.2)

where fabc _{are the SU(3) structure constants. In the following, by T}a _{we shall always}

denote the fundamental representation of the SU(3) algebra, deﬁned by:

    

fabcfully antisymmetric

Tr[Ta

Tb] = 1 2δ

ab (1.3)

e classical QCD Lagrangian, with the quarks colour indices understood, is given by:

LQCD =− 1 2Tr[FµνF µν_{] +}∑ f ¯ qf(i /D− mf)qf (1.4) where:

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• (Dµ)ij ≡ ∂µδij − igAaµ(x)(Ta)ij is the covariant derivative in the fundamental

rep-resentation;

• Fµν ≡ FµνaTa ≡ [Dµ, Dν]is the ﬁeld strenght tensor, whose explicit expression is

given by: Fµν = ∂µAν − ∂νAµ+ ig[Aµ, Aν] = (∂µAaν − ∂νAaµ− gf abc_Ab µA c ν)T a

Under gauge transformations, Fµν transforms covariantly:

Fµν(x)→ U(x)Fµν(x)U†(x)

QCD possesses very interesting features both in the high and in the low energy limit. At high energies, QCD phenomenology is dominated by the phenomenon of the

asymp-totic freedom [1, 2], that is the strong interactions become weaker as we go to smaller

distances. is can be proved perturbatively through the computation of the QCD β-function:

β(gR)≡ µ

∂gR

∂µ (1.5)

where gR is the renormalized coupling constant and µ the renormalization scale. e

β-function can be expandend in powers of gR:

β(gR) = β0g3R+ β1gR5 + β2g7R+· · · (1.6)

with only the first two coefficients being independent on the renormalization scheme. Asymptotic freedom comes from the observation that the flavour and the colour content

of QCD makes β0negative. In fact:

β0 =−

1 (4π)2

11Nc− 2Nf

3 (1.7)

so that for Nc= 3colours and Nf = 6ﬂavours, β0 < 0. is implies that, taking only the

ﬁrst term in (1.6), gRdecrease as we increase µ, making also this approximation sensible.

Solving (1.5), taking the ﬁrst non-vanishing order in gR, we ﬁnd:

g_R2(µ) =− 1 β0log ( µ2 Λ2 QCD ) (1.8)

where ΛQCD is the so-called QCD scale parameter, which is constant under the

renormal-ization group ﬂow. Its value in the MS scheme is ΛQCD = 332± 17 MeV [3].

On the other hand, the same result shows that, as we approach energies comparable to

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3 1.2 Chiral Symmetries and the U(1)AProblem

down. erefore, in order to study the low energy regime of QCD, new methods have been developed, that is Laice Simulations and Chiral Effective Lagrangians. e former, based on the laice formulation of gauge theories, is a very powerful tool which is em-ployed to study the QCD spectrum and in general its phenomenology especially at low temperature, where QCD is non-perturbative and shows some of its most striking features: the colour confinement and the spontaneous breaking of chiral symmetry. e postulate of colour confinement was introduced soon aer the quark model proposed by Gell-Mann and Ne’eman [4–6] in order to explain why the observed QCD spectrum consists only of colour neutral bound states of quarks, the hadrons, and thus the lack of observations of quarks as free asymptotic states. To date, contrary to the case of asymptotic freedom which is established as a theorem, only heuristic arguments [7] can be given in favour of colour confinement and no rigorous proof is still available.

e spontaneous breaking of chiral symmetries and the chiral eﬀective lagrangians in-troduced to study its phenomenological consequences will be discussed in greater detail respectively in the next Section and in the next Chapter.

1.2 Chiral Symmetries and the U(1)

A

Problem

e Lagrangian (1.4) has a number of classical global symmetries, in particular it is invari-ant under global phase redeﬁnition of the quark ﬁelds:

qf(x)→ eiαfqf(x), f = u, d, ...

ese transformations correspond to the group

G =⊗

f

Uf(1)

which, because of Noether’s theorem, implies the number conservation for each ﬂavour of quarks in strong processes (weak interactions violates ﬂavour conservation).

is symmetry group and the corresponding set of conserved currents become larger, at least at the classical level, once we take the massless limit for nl ﬂavours of quarks, also

called the chiral limit. In order to beer see this, we restrict to these nl ﬂavours and

introduce the quark multiplet:

q =     q1 ... qnl    

en, we explicit the right-handed and le-handed chiral components of the quark ﬁelds, deﬁned by: qL = 1 + γ5 2 q, qR = 1− γ5 2 q

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where γ5 ≡ −iγ0γ1γ2γ3.

e quark sector of the Lagrangian (1.4) is then given by:

LQ = iqLDq/ L+ iqRDq/ R− (qRM qL+ h.c.) (1.9)

where M = diag(m1,· · · , mnl). e opposite chiralities are coupled only through the

mass term, so that in the chiral limit the Lagrangian is invariant, at the classical level, under the following transformations:

qχ → ˜Vχqχ≡ eiαχVχqχ, Vχ = eiθ a

χTa ∈ SU(n l)χ

where χ = L, R. erefore, the classical symmetry group is enlarged to: ˜

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

Rather than making the two chiralities transform independently, we shall consider transformations in which both chiralities transform at the same time. ey are given by:

• Vectorial transformations ˜VL= ˜VR= ˜V ∈ U(nl)V

   qL→ ˜V qL qR→ ˜V qR , q→ eiαV_eiθaVTaq

• Axial transformations ˜VL = ˜VR† = ˜A∈ U(nl)A1

   qL→ ˜AqL qR→ ˜A†qR , q → eiαAγ5_eiθ_AaTaγ5_q

Indeed, it can be proved that any transformation belonging to U(nl)L⊗ U(nl)R can be

wrien as the composition of an axial and a vectorial transformation, so that the group ˜G

can be safely rewrien as: ˜

G = SU (nl)V ⊗ SU(nl)A⊗ U(1)V ⊗ U(1)A

Now we discuss whether these symmetries are or are not observed in Nature.

A symmetry is said exact or realized à la Wigner-Weyl if it is implemented on the Hilbert

space of the theory by a unitary operator ˆU whose generators annihilate the vacuum and

commute with the Hamiltonian H. is means that we can write ˆU = eiϵa_Qa

, where

the generators Qa _{coincide with the conserved charges and satisfy: Q}a|Ω⟩ = 0 and

1_{e notation is a bit misleading, since U(n}

l)Aand in particular the ”subgroup” SU(nl)Aare not proper

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5 1.2 Chiral Symmetries and the U(1)AProblem

[Qa_{, H] = 0 =}_⇒ dQa

dt = 0. In addition to this, the one-particle states gather into

ir-reducible representations of the symmetry group.

e simpler case is the U(1)V subgroup of the group ˜G. In fact, the corresponding

con-served current:

J_Vµ = ¯qγµq

leads to the conservation of the baryon number, which is experimentally well established. As for SU(nl)V ⊗ SU(nl)A, it was soon realized that this symmetry cannot be exact. In

fact, since the SU(nl)A generators Qa5 contain the γ5 matrix, they anticommute rather

than commute with the parity operator P. en, given any hadronic state|h⟩ and the state

|h′_{⟩ = Q}a

5|h⟩, they should be both degenerate in mass and have opposite parities:

   [Qa₅, H] = 0 : H|h′⟩ = HQa₅|h⟩ = Q₅aH|h⟩ = MhQa5|h⟩ = Mh|h′⟩ {Qa 5, P} = 0 : P |h′⟩ = P Q a 5|h⟩ = −Q a 5P|h⟩ = −ηhQa5|h⟩ = −ηh|h′⟩ (1.10) However, no degenerate partner with opposite parity has been observed for any of the known hadrons. Instead, the observed spectrum could be explained by assuming that the

subgroup SU(nl)V ⊗ SU(nl)A is spontaneously broken. A global symmetry group G is

spontaneously broken if not all its generators annihilate the vacuum, but only a subset which generates the subgroup H. In this case, we say that that group G is spontaneously broken to the group H. e Goldstone’s theorem guarantees that for each broken genera-tor there is a massless spin 0 boson with the same quantum numbers of the corresponding generator. At the same time, being the subgroup H still realized à la Wigner-Weyl, the spectrum organises into irreducible representations of the subgroup H.

is picture correctly ﬁts with the observations, even though no massless hadron is ob-served. In fact, the SU(nl)V ⊗ SU(nl)Agroup is not exact at the classical level but

ex-plicitly broken by the quark masses. However, if the quark masses are suﬃciently small,

i.e. mi ≪ ΛQCD, this picture is not entirely spoiled since we can still identify the states

corresponding to the Goldstone’s bosons which now acquire a small squared mass propor-tional to the quark masses so that they become Pseudo-Goldstone’s bosons. erefore the remnant of the spontaneous breaking of the chiral symmetries should be found among the lightest hadronic excitations. Indeed, these correspond to the octet of the parity-odd pseudoscalar mesons, which therefore possess the correct quantum numbers to be

consid-ered as the Pseudo-Goldstone’s bosons corresponding to the n2

l−1 generators of SU(nl)A,

ﬁxing, therefore, the number of light quarks to nl = 3, that is the up, down and strange

quarks. Moreover, they, together with the other hadrons, can be grouped to form

irre-ducible representations of SU(3)V. However, the masses of the particles within each

representation can diﬀer signiﬁcantly, while if we split these representations into proper

SU (2)subgroups, which correspond to the isospin subgroups, the mass spliing decreases

dramatically. At the same time, the triplet of pions is much lighter than the other parti-cles in the pseudoscalar mesons octet. is suggests that, if we restrict to two ﬂavours of

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quarks, we get a more accurate approximation of the chiral limit. is is consistent with having the up and the down quarks much lighter than the strange.

In principle, the same conclusions should hold for the U(1)Asubgroup once we replace

in (1.10) the corresponding generator. However, as said above, only eight light pseu-doscalar mesons have been found. In fact, the best and lightest candidate for the

Pseudo-Goldstone’s boson corresponding to the U(1)Agroup is the η′ meson, which is too heavy

for this solution to be feasible. is statement was made quantitative by Weinberg [8] who found, through an Eﬀective Lagrangian approach and neglecting the up-down mass spliing, that the mass of the lightest isosinglet (I = 0) meson is bounded from above by:

MI=0<

√

3Mπ0 (1.11)

which is the famous Weinberg’s limit. However, given the measured masses for π0, η′ and

η, that is Mπ0 ≈ 135 Mev, Mη ≈ 547 Mev and Mη′ ≈ 958 Mev, we can see that this limit

is not satisﬁed. is puzzle was dubbed by Weinberg himself as the ”U(1)Aproblem”: we

shall discuss the most accredited solutions in the next section.

1.3 Diﬀerent U(1)

A

Solutions, One CP Problem

e resolution of the U(1)Aproblem came with the ﬁnding that, even in the chiral limit,

U (1)Acannot be a symmetry of the Lagrangian at the quantum level Because of the

Adler-Bell-Jackiw anomaly [9, 10]. is can be seen both from a direct computation of the

Schwinger-Dyson equations2_{for the conservation of the axial current and from the}

non-invariance of the fermionic measure in the path integral formulation of QCD [11]. In both cases, the result is:

∂µJ µ 5 = 2i nl ∑ j=1 mjq¯jγ5qj+ 2nlQ (1.12)

where Q is the Topological charge densitiy and is given by:

Q = g 2 64π2ϵ µνρσ_Fa µνF a ρσ (1.13)

ough the presence of the anomaly is the accepted solution to the lack of a ninth Nambu-Goldstone’s boson, it is not ﬁrmly established the precise mechanism which leads to the

large mass spliing between the η′ _{and the PNGBs of the meson octet, since the sources}

for this spliing can be both classical (’t Hoo) and quantum (Wien and Veneziano).

2_{e Schwinger-Dyson equations describe how the classical field equations get modified at the quantum} level, where the classical fields are replaced by correlation functions involving the corresponding operators.

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7 1.3 Diﬀerent U(1)ASolutions, One CP Problem

1.3.1 e (Classical) Solution of ’t Hoo

It is easy to prove that the topological charge density is actually a total derivative:

Q = g 2 64π2∂µK µ _(1.14) where: Kµ= ϵµαβγAa_α ( F_βγa −g 3f abc_Ab βA c γ ) (1.15) Despite this, it should not be expected that its spacetime integral, the topological charge, which gives the non-conservation of the axial charge, vanishes. In fact, Belavin et al.

showed [12] that the correct boundary conditions are not Aµ = 0at spatial inﬁnity, but

instead Fµν = 0, that is Aµmust be pure gauge. Indeed, they found non-trivial solutions

to the classical equations of motions with these boundary conditions, later known as

in-stantons, whose associated topological charge is ﬁnite. In two seminal papers [13, 14], ’t

Hoo argued that instantons do have physical consequences which are clearly inherently non-perturbative in their nature. In particular, he provided an eﬀective term which mim-ics the interactions among fermions through the exchange of an instanton, in the form of a 2n-fermion interaction term (this will be relevant for the discussion of the next Chapter):

L2n ∼ Cdet

st (¯qs(1 + γ5)qt) +h.c. (1.16)

which has the right quantum numbers to provide a mass for the chiral singlet η′ _(this

will be clear in the calculations of the next Chapters). However, a direct computation of the coefficient has not been obtained because of the rather complicated dynamics of instantons, which, at low temperatures, are characterized by a large overlap, so that it is difficult to count separately the contribution of different instantons. is is instead possible at large temperature, where the overlap is negligible and the so-called Dilute Instanton Gas Approximation (DIGA) may be employed to carry perturbative calculations [15].

Nevertheless, though no precise quantitative expression can be found for the η′ _mass,

’t Hoo solution provides a reasonable qualitative framework to explain the large mass spliing. Indeed, a rough estimate of the instanton contribution to the coeﬃcient ofL2nl

is C ≈ exp(−8π2

g2

)

which at low temperatures, where g is not small, is not negligible.

1.3.2 e (antum) Solution of Witten and Veneziano

Another solution to the U(1)A problem can be given by studying QCD in the large Nc

limit. is framework was introduced by ’t Hoo [16] who generalized the SU(3)cgauge

symmetry of QCD to SU(Nc)and proposed to consider expansions in powers of _N1_c. e

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implies:

g ∼ √1 Nc

(1.17)

From (1.13) we can see that the U(1)A anomaly vanishes in this limit as _N1_c, and can

be therefore treated perturbatively. We expect in this case the emergence of a ninth (Pseudo)Nambu-Goldstone’s boson, which in this sense receives a mass from the anomaly.

is idea is the basis for the solution to the U(1)Aproblem by Wien [17] and Veneziano

[18]. Here we shall review the reasoning followed by Wien in [17].

e starting point is the possibility of adding to the QCD Lagrangian the so-called θ-term:

LQCD → LQCD+ θQ(x)

which, because of instantons, can give ﬁnite contributions to the action, despite being a total derivative. e following step is to study the dependence of the physics on the parameter θ in the large Nclimit, assumed non-trivial to leading order in _N1_c as suggested

by the study of simpler models, in particular the vacuum energy E(θ) deﬁned as:

E(θ) = i V T log Z[θ] (1.18) where Z[θ] = ∫ [dAµ][d¯q][dq]ei ∫ d4_x(L QCD+θQ(x))

and its second drivative at θ = 0: d2_E dθ2 θ=0 =−i lim k→0 ∫ d4_xeik·x_{⟨Ω|T Q(x)Q(0)|Ω⟩ ≡ −i lim} k→0U (k) (1.19)

e function U(k) can be expanded in powers of 1

Nc:

U (k) = U0(k) + U1(k) + U2(k) + ...

e leading term U0(k) gathers all the contributions coming from the diagram which

do not contain quark loops. From the large Nc expansion rules, in fact, any quark loop

induce a suppression of a factor of 1

Nc to the corresponding diagram. erefore this term

represents the pure Yang-Mills part of the expansion which is of orderO(N0

c).

e NLO term, instead, collects the contributions from the diagrams with a single quark loop coming from the propagation of mesonic type intermediate state. We can thus write:

U1(k) = ∑ mesons |⟨Ω|Q(0)|m⟩|2 k2− M2 m =O(N_c−1)

is, however, leads to a paradox once we consider the chiral limit, where any dependence of the physics on θ disappears. In this case, in fact, a U(1)Atransformation of parameter

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in the partition function Z[θ] produces a shi in the parameter θ → θ − 2nα, so that we

should have lim

k→0U (k) = 0. However, there seems to be no way in which the leading term

can be cancelled by the NLO one. is cancellation occurs only if there exists a meson

singlet|s⟩ such that ⟨Ω|Q(0)|s⟩ ̸= 0, whose mass M2

s =O(Nc−1)and such that:

A≡ lim k→0U0(k) = |⟨Ω|Q(0)|s⟩|2 M2 s (1.20) where A is the pure Yang-Mills Topological Susceptibility. We interpret this singlet as the

η′ meson. e matrix element ⟨Ω|Q(0)|s⟩ can be deduced from the anomaly equation

(1.12). In fact: ⟨Ω|Q(0)|η′_{⟩ =} 1 2nl ⟨Ω|∂µJ µ 5|η′⟩

And in momentum space:

⟨Ω|∂µJ5µ|η′⟩ = −ikµ⟨Ω|J5µ|η′⟩ By deﬁnition3_: ⟨Ω|Jµ 5|η′⟩ = i √ 2nlFη′kµ So that: ⟨Ω|Q(0)|η′_{⟩ =}√_2n lFη′Mη2′

which, once replaced in (1.20), gives the celebrated Wien-Veneziano formula for the singlet mass: M_η2′ = 2nl F2 η′ A = 2nl F2 π A (1.21) where Fη′ = Fπ = O( √ Nc)to leading order in _N1

c, with Fπ the pion decay constant. In

addition to this, large Nc rules imply that Fπ = O(

√

Nc), so that Mη2′ = O(Nc−1), as

expected: in the limit Nc→ ∞ the η′becomes an exact Nambu-Goldstone’s boson.

is formula can be used to give a prediction on the value of A, which is:

A ≈ (180MeV)4

Surprisingly, this value has been conﬁrmed by laice simulations [19].

is solution to the U(1)Aproblem is radically diﬀerent from the semiclassical approach

by ’t Hoo. Instanton processes, in fact, cannot reproduce the 1

Nc decay of the η ′ _mass,

being exponentially suppressed by e−8π2g2 ∼ e−Nc. However, this is only a naive scaling,

no rigorous proof is available that all instanton contributions are so strongly suppressed.

3_With√_2n

lfactored out, Fη′does not depend on nlto leading order in_N1

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1.3.3 e θ-term and the Strong CP Problem

e discovery of the U(1)A anomaly and of its non-zero physical eﬀect despite being

a total derivative, due to the existence of instantons, while leading to the resolution of

the U(1)A, brought to another problem connected to the structure of the QCD vacuum

induced by the non-trivial topology of instantons [20].

When studying the Euclidean Path Integral formulation of QCD, the correct boundary conditions to be imposed for the gauge ﬁelds are Fµν = 0 and thus Aµ = g−1(x)∂µg(x),

where g(x)∈ SU(3)c, at euclidean inﬁnity, that is the sphere S3. is deﬁnes a set of maps

of the group SU(3)cinto S3, which can be classiﬁed into homotopy classes corresponding

to the homotopy group Π3(SU (3)c). is homotopy group coincide withZ, so that each

class is characterized by an integer ν called winding number, which coincide with the integral of the topological charge density:

ν = ∫

d4_xQ(x)

erefore, depending on the boundary conditions, we have a discrete inﬁnity of vacua|ν⟩,

each deﬁned by its winding number.

e functional integration over a given homotopy class has a very interesting meaning, since it gives the tunnelling amplitude between two vacua:

⟨ν1| exp(−Hτ)|ν2⟩ →_τ_→∞ ∫ [dAµ]ν1−ν2· · · exp ( −∫ d4_x_L QCD )

which is non-zero because of the instantons. erefore, none of this topological vacua can be the true vacuum, which is instead a proper superposition of them. To ﬁnd the

true vacuum, we observe that, by deﬁnition, each state |ν⟩ is stable under local gauge

transformations and is connected to the other only through ”large” gauge transformations (LGTs), which do not die off at infinity. It is then possible to define a LGT G which changes the winding number by one unit:

G|ν⟩ = |ν + 1⟩

Since G commutes with the Hamiltonian, the energy eigenstates must be G eigenstates

and since G is unitary, its eigenvalues are phases eiθ _{and the corresponding eigenstate}

|θ⟩ = eiνθ_{|ν⟩. erefore, each state |θ⟩ deﬁnes a vacuum for a physically disconnected}

sector, since the transition amplitude vanishes.

Once that the true vacua are deﬁned, we must express the path integral in their basis:

⟨θ′_{| exp(−Hτ)|θ⟩ →} τ→∞ δ(θ ′_{− θ)Z[θ]} where: Z[θ] = ∑ ν eiνθ ∫ [dAµ]ν· · · exp ( −∫ d4_x_L QCD ) = ∫ [dAµ]· · · exp ( −∫ d4_x(_L QCD+ iθQ) )

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Which, once re-expressed in Minkowski coordinates, leads to the addition to the QCD Lagrangian of the famous θ-term:

LQCD → LQCD+ θQ

is additional term has the property of being odd under the discrete time reversal symme-try T or equivalent under parity P and charge conjugation C simultaneously, that is under

CP. However, while CP violating processes are well known in the weak interactions, no

sign is found in the strong interactions. In particular, the most famous CP-breaking

ob-servable is the neutron electric dipole moment dN, which approxiamtely given by [21,

22]: dN ≈ M_π2 M2 N e|θ| ≈ 10−16|θ|e · cm

Where Mπ is the pion mass, MN the neutron mass and e the electron charge. However,

experimentally only an upper bound is known dN < 10−26e· cm which leads to an upper

bound for θ:

|θ| < 10−10

e reason why θ is so small, or even 0, leading to a yet unobserved CP-violating phe-nomenology in the strong interaction is not known. is issue is what is called the strong

CP problem. Axions, the main object of this esis, represent the most compelling

solu-tion to this problem.

Before moving to the next section, an important remark. e coeﬃcient θ as introduced above is physical only if we assume the quark mass matrix to be real. In fact, because of the axial anomaly, the θ-term can be entirely moved to the mass term which acquires an additional imaginary part. To see this, we write the mass term assuming a more general, complex, diagonal mass matrix in addition to the

theta-term:

Lm+θ =−qR†MqL− qL†M†qR+ θQ

If we now perform a U(1)Atransformation of parameter α, that is:

qL→ eiαqL, qR → e−iαqR

en, because of the anomaly the coeﬃcient θ is shied as θ → θ − 2nlα, but the mass

matrix acquires an additional phase:

Lm+θ =−qR†e

2iα_Mq

L− qL†e−2iαM†qR+ (θ− 2nlα)Q =−q†RM′qL− q†LM′†qR+ θ′Q

While both θ and M have changed, we can see that the combination (arg det M + θ) is

constant:

arg detM′_{+ θ}′ ₌_{arg det(e}2iα_{M) + θ − 2α =}

arg(e2inlαdetM) + θ − 2n

lα =arg detM + θ ≡ ¯θ

It is to ¯θ which we shall refer as the physical θ. Clearly, when the mass matrix is real, we have ¯θ = θ.

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1.4 Axions: Models and Phenomenology

Many solutions have been proposed to aack the strong CP problem:

1. Vanishing quark up mass;

2. Spontaneously broken CP symmetry;

3. A new chiral symmetry exact at the classical level and anomalous.

As for the ﬁrst one, a vanishing quark mass makes the classical Lagrangian invariant un-der U(1)Atransformation limited to the single quarks. Choosing properly the phase, then,

this axial transformation can be used to rotate away the θ-term through the anomaly. is solution, however, has been ruled out by laice simulations, which are consistent with all quark masses being non-zero. In addition to this, recently it has been criticized that the CP problem can be solved by a single massless quark [23].

e second option [24–26] consists in starting from a CP-invariant Lagrangian with a zero bare value for θ, and then break spontaneously CP, inducing an eﬀective θ-term.

How-ever, θ can receive contributions from the loops and in order to get θ < 10−9 _{we must}

require that θ vanishes at the 1-loop level. In doing so, new problems emerge, for exam-ple enhanced Flavour Changing Neutral Currents because of the interplay with the CP violation coming from the weak sector of the Standard Model.

e third possibility is the more solid solution to the strong CP problem and leads di-rectly to the introduction of the axions. Originally, Peccei and inn [27, 28] proposed

the existence of a new U(1) symmetry, later on known as U(1)P Q, which acts by chiral

rotations on the quarks. is symmetry is thus anomalous, but if exact at the classical level it implies the vanishing of some quark masses. In order to avoid this, an additional Higgs doublet was introduced in the Standard Model (SM) Lagrangian that, together with the ﬁrst one, is coupled à la Yukawa to the quarks, which thus receive a mass once the Higgs doublets take a non-zero Vacuum Expectation Value (VEV), and is charged under

U (1)P Q in such a way to make the U(1)P Q exact at the classical level and, through the

anomaly, rotate away the θ angle. is can be immediately seen from the relevant terms of the Lagrangian:

LY ukawa = ΓuijQ¯LiΦ1uRj+ ΓdijQ¯LiΦ2dRj + ΓlijL¯LiΦ2lRj+ h.c. (1.22)

Where Φ1, Φ2 are the two Higgs doublets, QLi, LLi the arks and the Leptons doublets

and Γu

ij, Γdij, Γlij the Yukawa matrices. e U(1)P Qis precisely the new U(1) symmetry,

independent of the Hypercharge U(1)Y, that can be deﬁned thanks to the existence of the

second Higgs doublet. Under U(1)P Qthe ﬁelds transform as:

uRj → e−iαxuRj, dRj → e−i α xd Rj, lRj → e−i α xl Rj

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13 1.4 Axions: Models and Phenomenology

Φ1 → eiαxΦ1, Φ2 → ei

α xΦ₂

As we can see, only the right-handed quarks transform under U(1)P Q. Moreover, their

charges are such to make U(1)P Q anomalous under both SU(3)c and U(1)Y. Soon

af-ter, it was recognized by Weinberg [29] and Wilczek [30] that, because of the non-zero

VEVs acquired by the Higgs doublets, the U(1)P Q must be also spontaneously broken,

leading to the existence of a Nambu-Goldstone’s boson, the axion, which, because of the above-mentioned anomaly, acquires a ﬁnite mass, so that it is actually a

Pseudo-Nambu-Goldstone boson4_{. In fact, the axion is the common phase ﬁeld ϕ in Φ}

1and Φ2: Φ1 = v1 √ 2e i_vFxϕ ( 1 0 ) Φ2 = v2 √ 2e i_xvFϕ ( 0 1 ) where vF = √ v2

1 + v22, the coeﬃcient x now is x = vv21 and where ϕ transforms non-lineary

under U(1)P Q:

ϕ→ ϕ + αvF

as expected by a Nambu-Goldstone boson. For this reason, the axion of the original model of Peccei and inn is called the Peccei-inn-Weinberg-Wilczek or PQWW axion.

Nev-ertheless, the original PQ model was soon ruled out. In fact, being the scale Fϕ of the

spontaneous breaking of the PQ symmetry the weak scale, many processes involving

ax-ions should have been seen, like the decay K+ _{→ π}+ _{+ ϕ. Soon aer, however, new}

models of axions where proposed where Fϕ ≫ vF, the so-called Invisible axion Models,

which are still consistent with the present bounds. ere are basically two types of these models:

• the Kim-Shifman-Vainshtein-Zakharov (KSVZ) axion [31, 32]: the SM is augmented with a scalar σ, neutral under the SM gauge group and a quark Q charged only

un-der SU(3)c(but not living necessarily in a fundamental representation as ordinary

quarks), both charged under U(1)P Q. e ﬁeld σ acquires a VEV⟨σ⟩ = Fϕwhich

spontaneously breaks the PQ symmetry, whose associated PNGB is the axion, and gives a mass to Q (an explicit mass term is forbidden by an ad hoc discrete symmetry, which represents the drawback of this model). Both σ and Q are extremely heavy, while the axion mass and its coupling to ordinary maer are highly suppressed by the scale Fϕ.

• the Dine-Fischler-Srednicki-Zhitnisky (DFSZ) axion [33, 34]: the original PQ model

is augmented with a scalar ﬁeld σ carrying PQ charge and whose VEV⟨σ⟩ = Fϕ

breaks spontaneously U(1)P Q.

4_{Actually, the axion is the true (pseudo-)Nambu-Goldstone boson corresponding to a proper,} non-anomalous combination of U(1)P Qand U(1)A.

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In all the cases so far considered, the axion can be implemented eﬀectively into the ordi-nary SM and in particular into the ordiordi-nary QCD Lagrangian as:

L = LQCD + 1 2∂µϕ∂ µ_{ϕ +}_L int[∂µϕ, Ψ] + aP Q Fϕ ϕQ (1.23)

where Ψ gathers all the QCD ﬁelds. e last term is necessary to ensure that the U(1)P Q

symmetry has a chiral anomaly under PQ transformation ϕ → ϕ + αFϕ:

∂µJP Qµ = aP QQ

where aP Qis the anomaly coeﬃcient, in particular:

aP Q=      Nf 2 ( x + 1 x ) , PQWW 1, KSVZ, DFSZ

From the form taken into the Lagrangian, it is evident that the axion represents a dynam-ical θ angle. is observation has important consequences and allows to deduce many general properties of the axion. In fact, because of the large value of Fϕ, as we shall see

explicitly in the next Chapters, the axion is by far the lightest particle in the QCD spec-trum, so that it is usually correct to integrate out all the heavier ﬁeld and get an eﬀective Lagrangian which describes the dynamics of the axion alone:

Lϕ=

1 2∂µϕ∂

µ_ϕ_{− V}

1(ϕ)− V2(ϕ, ∂µϕ) (1.24)

e term V1(ϕ), collecting all the non-derivative coupling of the axion, resembles the

vacuum energy in the presence of a θ-term deﬁned in (1.18), that is V1(aP Qϕ/Fϕ) = E(θ).

is implies that:

m2_axion = d 2 dϕ2V1(aP Qϕ/Fϕ) ϕ=0 = a 2 P Q F2 ϕ d2 dθ2E(θ) θ=0 ≡ a2P Q F2 ϕ χQCD (1.25)

where χQCD is the full QCD topological susceptibility. From this relation, we can see that

in the limit of one quark geing massless, since the physics is supposed to be independent of θ, E(θ) is constant (solving the strong CP problem in the absence of axions) and so is

V1(aP Qϕ/Fϕ), which implies that, in this limit, the axion is massless. is result can be

understood also in terms of the symmetries of the Lagrangian (1.23), which now include at the classical level the U(1)Arestricred to the massless quark. In fact, performing at the

same time a U(1)P Qtransormation with parameter α and a U(1)Aone with parameter β,

if we impose:

aP Qα + β = 0

no anomaly is produced. erefore this combination of U(1)Aand U(1)P Qis realized at

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the Goldstone’s theorem requires the existence of a massless Goldstone’s boson, which in this case is the physical axion.

A more precise version of the above argument is reviewed in [35], where the eﬀective potential for the axion is computed integrating out quarks and gluons in the path integral formulation of QCD, leading immediately to (1.25).

A more formal proof can also be given and is based on a diagrammatic calculation. In fact from the Lagrangian in (1.23) we can read the Feynman rules for the axion:

• Propagator: i

p2;

• Interaction term: -aP Q Fϕ .

e axion physical mass is deﬁned as the pole of the dressed propagator, which is given by the following sum of diagrams:

which corresponds to the following geometric series:

i p2 ∞ ∑ k=0 ( − i p2 a2 P Q F_ϕ2 ⟨QQ⟩F.T . )k = i p2_{+ i}a2P Q F2 ϕ ⟨QQ⟩F.T.

We can expand⟨QQ⟩F.T.in powers of the momenta:

⟨QQ⟩F.T. = iχQCD+ χ′p2+· · ·

It is shown in [36] that χ′m2η′ χQCD

≲ 0.15, so that the momentum dependent terms can be

neglected once we replace the η′_{mass with the axion mass. is means that:}

0 = m2_axion+ i a 2 P Q F2 ϕ ⟨QQ⟩F.T. p2_=m2 axion =⇒ m2_axion ≈ a 2 P Q F2 ϕ χQCD

is method makes evident which are the assumptions necessary for the validity of this relation, namely the large Fϕ limit. In fact, this limit allows to neglect the inevitable

mixing of the axion with all the ﬁelds that couple to the anomaly, the η′ _{in the ﬁrst place.}

is mixing comes from a non-zero value for 2-point functions like⟨ϕ(x)η′_(y)_{⟩. Clearly,}

this assumption is also implicitly used in the other ”classical” derivation of (1.25).

e important relation m2

axion = a2

P Q

Fϕ χQCD can be considered as an equivalent of the

Wien-Veneziano formula which relates the η′_{mass to the pure Yang-Mills susceptibility.}

It can be used in both ways: to compute χQCD, once that the axion mass is known, and

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A famous formula exists for the axion mass [37], valid to lowest order in the quark masses (mi ≪ ΛQCD ≪ Fϕ): maxion = aP Q Fϕ FπMπ 2 √ mumd mu+ md (1.26) In the next Chapter we are going to prove this formula, which, apart from the factor aP Q

Fϕ ,

is model independent and relates the axion mass directly to the scale Fϕ

aP Q ≡ fϕ: maxion ≈ 6.3 eV ( 106 GeV fϕ ) (1.27)

e scale fϕ is constrained both from above and from below. e laer is basically an

astrophysical bound, since it is related to the stellar evolution. In fact, all axion models are also characterized by how axions interact with the photons. is interaction is evident

in the PQWW and KSVZ models, where also the leptons are charged under U(1)P Q, which

is thus anomalus also under the QED gauge group. e corresponding term is given by:

Lϕγγ = α 8πKϕγγ ϕ fϕ ϵµνσρFµνFσρ (1.28)

where Kϕγγ is model dependent, in particular [38]:

Kϕγγ =                    Nf 2 ( x + 1 x ) mu mu+ md , PQWW 3e2_Q− 4md+ mu 3(mu+ md) , KSVZ 4 3 − 4md+ mu 3(mu+ md) , DFSZ (1.29)

where eQis the charge of the heavy quark introduced in the KSVZ Model. is coupling

is particularly important, since the process ϕ → γγ is the dominant decay channel of

the axion. Into the stars, this term causes energy loss through Compton scaering γe →

ϕe and Primakoff effect, that is a photon which inside strong electromagnetic fields is

converted into an axion. Above a certain value of maxion, the energy lost through axion

emission would imply an enhanced consuption of nuclear fuel and thus a too fast stellar evolution. Besides this, other observations (Supernova 1987A gamma ray burst duration,

white dwarfs cooling, etc…) are consistent with a lower bound on Fϕgiven by:

fϕ> 109GeV =⇒ maxion < 10meV

Being the scale Fϕso high compared to the SM scales and since the couplings of the axion

to SM ﬁelds are suppressed by Fϕ, we have that Astrophysics requires the axion to be

very light and also very weakly interacting. For this reason, axions were soon adressed as a promising source of Dark Maer [39–41]. e most important mechanism of axionic

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Dark Maer production is the so-called ”Misalignment Mechanism”, which consists in the assumption that in the early Universe the axion ﬁeld was misaligned, that is not located at the minimum of its potential⟨ϕ⟩ = 0. If ⟨ϕ⟩

fϕ is not too large, the evolution of the axionic

ﬁeld is described by the equation: ¨

⟨ϕ⟩ + 3H(t) ˙⟨ϕ⟩ + m2

axion(t)⟨ϕ⟩ = 0 (1.30)

where H(t) is the Hubble constant and m2

axion(t)the axion mass at a given cosmological

time. is represents the equation for a damped harmonic oscillator: at early times, the

friction term dominates and⟨ϕ⟩ is frozen, while once 3H(t) ≈ m2

axion(t), the axion ﬁeld

starts oscillating, giving rise to the production of non-relativistic axions, which form the so-called Cold Dark Maer (CDM). Computations of the predicted present-day axionic Dark Maer abundance based on this mechanism give the following result [42]:

Ωϕh2 = 0.5 ( fϕ 1012GeV )7 6 (θ2_i + σ_θ2)γ (1.31)

where θi = ⟨ϕ⟩_f_ϕ is the misalignment angle, σθ its mean-squared ﬂuctuation possibly due

to inﬂation, γ a dilution factor due to the entropy released once the axion ﬁeld starts oscillating. From the WMAP bound on CDM abundance:

Ωϕh2 < 0.12

assuming no dilution (γ = 0), an average misalignment angle θ2

i = π

2

3 and negligible

ﬂuctuations, we obtain:

fϕ < 3× 1011GeV =⇒ maxion > 2× 10−5eV

Cosmological axions are particularly relevant because they can be directly detected through their coupling to photons (1.28):

Lϕγγ =− α 2πKϕγγ ϕ fϕ ⃗ E· ⃗B

In fact, Sikivie [43] ﬁrst suggested that axions passing through an electromagnetic cavity in the presence of a strong magnetic ﬁeld can resonantly convert into photons once the cavity resonance ω is centered on the axion mass. Being non-relativistic (the predicted

axion velocity is β ≈ 10−3_{), the axion mass directly determines the range of resonant}

frequencies required:

2× 10−5 eV < maxion< 10meV =⇒ 240 MHz < ω < 240 GHz

such wide range requires the development of tunable cavities. Currently, the Axion Dark Maer eXperiment (ADMX) is reaching the sensibility required to detect cosmological axions.

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2

Chiral Eﬀective Lagrangians with the

inclusion of the Axion

2.1 e Order Parameter for Chiral Symmetries

In the previous Chapter, we have shown that the chiral group ˜G = U (nl)R⊗U(nl)Lmust

be spontaneously broken down to the vectorial subgroup U(nl)V and how this justiﬁes

the existence of eight PNGBs, the ninth being only a would-be Nambu-Goldstone boson because of the anomaly.

In the proof of the Goldstone’s theorem, the spontaneous breakdown of the global sym-metry is studied through the introduction of an order parameter:

δa≡ ⟨Ω|[Q, A]|Ω⟩ (2.1)

where A is a local interpolating operator with the correct quantum numbers to allow

δa ̸= 0, so that Q(t)|Ω⟩ ̸= 0 and thus the symmetry that should be generated by Q is

spontaneously broken. In the case of the chiral group, we choose as interpolating opera-tors for the axial charges Qa

5 the pseudoscalar bilinears Pa= ¯q(0)γ5Taq(0). It is possible

to prove that: ⟨Ω|[Qa 5, P b_]_{|Ω⟩ = −}1 nl δab⟨Ω|¯q(0)q(0)|Ω⟩ (2.2)

e quantity ⟨Ω|¯q(0)q(0)|Ω⟩ ≡ ⟨¯qq⟩ is called chiral condensate and represents the

or-der parameter for chiral symmetries: when vanishing, the chiral group ˜G, modulo the

anomaly, is realized à la Wigner-Weyl.

e chiral condensate clearly represents the key quantity to study the chiral properties of QCD, especially at ﬁnite temperature. In particular, laice studies have proved the existence of a critical temperature Tcabove which the chiral condensate has a drop,

sig-nalling the occurrence of a phase transition, the Chiral Phase Transition. e value of the

critical temperature is Tc ≈ 150 − 165 MeV [44]; laice studies have also shown that in

the proximity of the same temperature, QCD loses its conﬁning properties and enters the

ark-Gluon Plasma phase, where quarks and gluons are no longer conﬁned in colour

neutral hadrons. In general, we should not expect the chiral condensate to vanish, even above Tc, since the chiral group is explicitly broken by the quark masses. In fact, both Tc

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2.2 Eﬀective Degrees of Freedom

As discussed at the beginning of the previous Chapter, the non-perturbative regime which characterizes QCD at low energies requires the development of new tools. We have al-ready seen how Laice Simulations allow to deduce and recover many properties of QCD starting from first principles, that is from the fundamental QCD Lagrangian itself. A com-pletely different approach is at the base of the Chiral Effective Lagrangian formalism. It consists in the fundamental hypothesis that we do not need to know the high energy the-ory governing the short distance physics to study the dynamics of the low-lying region of the observed spectrum. In this case, though the fundamental theory is known, that is QCD, perturbation theory breaks down before we can make any reasonable prediction on the physics of the hadron spectrum and in particular on the dynamics of its lightest meson components. en, the low-energy physics we wish to study can be described by an Effective Lagrangian, wrien in terms of the relevant degrees of freedom alone, which must display the correct symmetry properties necessary to match with the observations. In our case, we want to study the dynamics of the light mesons, which is consistent with

a U(nl)L ⊗ U(nl)R symmetry spontaneously broken by a non-zero chiral condensate

⟨¯qq⟩ = ⟨¯qRqL⟩ + ⟨¯qLqR⟩ to an approximate U(nl)V, explicitly broken by the quark mass

term, with the U(1)A subgroup further broken by the anomaly. Since mesons are

antiquark bound states and the chiral condensate is the expectation value of a quark-antiquark bilinear, it is natural to represent the eﬀective degrees of freedom correspond-ing to the mesonic ﬁeld in terms of a nl× nlcolour neutral complex matrix Uij:

Uij ∝ ¯qj (_{1 + γ} 5 2 ) qi = ¯qjRqiL (2.3)

so that a non-zero chiral condensate corresponds to a non-zero VEV⟨U⟩, in particular (in

the chiral limit):

⟨U + U†_{⟩ ≡} _√Fπ

2Inl×nl ∝

⟨¯qq⟩ nl

Inl×nl =⟨¯uu⟩Inl×nl (2.4)

e Eﬀective Lagrangian wrien in terms of the ﬁeld U must reproduce the same discrete and global symmetry ofLQCDat the classical level, that is parity and the symmetry under

the chiral group ˜G, and provide a mechanism for the spontaneous breaking of the group

˜

G itself. ese symmetries, however, are actually explicitly broken respectively by the

topological θ-term and by the quark masses: we shall see later how to implement the symmetry breaking terms. Under parity, the quark ﬁelds transform according to:

P :    qiL(x0, ⃗x)→ qiR(x0,−⃗x) qiR(x0, ⃗x)→ qiL(x0,−⃗x) (2.5) so that: ¯ qjRqiL(x0, ⃗x)→ ¯qjLqiR(x0,−⃗x) =⇒ Uij(x0, ⃗x)→ Uij†(x0,−⃗x) (2.6)