Axion from high-quality accidental Peccei-Quinn symmetry

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Universit`

a di Pisa

Dipartimento di Fisica “Enrico Fermi”

Tesi di Laurea Magistrale in Fisica

Axion from high-quality accidental

Peccei-Quinn symmetry

Candidato: Marco Ardu

Relatori : Prof. Alessandro Strumia, Dott. Daniele Teresi

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Abstract

We propose two models where a U(1) Peccei-Quinn (PQ) global symmetry arises accidentally and is respected up to high-dimensional operators, so that the axion solution to the strong CP problem is successful even in the presence of Planck-suppressed operators. One model is SU(_{N ) gauge interactions with fermions in the fundamental and a scalar in the symmetric.} The axion arises from spontaneous symmetry breaking to SO(_{N ), that confines at a lower} energy scale. Axion quality in the model needs _{N & 10. SO bound states and possibly} monopoles provide extra Dark Matter (DM) candidates beyond the axion. In the second model the scalar is in the anti-symmetric: SU(_{N ) broken to Sp(N ) needs even N & 20.} The cosmological DM abundance, consisting of axions and/or super-heavy relics, can be reproduced if the PQ symmetry is broken before inflation (Boltzmann-suppressed production of super-heavy relics) or after (super-heavy relics in thermal equilibrium get partially diluted by dark glue-ball decays).

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Ringraziamenti

Vorrei ringraziare i miei relatori, Prof. Alessandro Strumia e Dott. Daniele Teresi, per avermi guidato, con la loro disponibilit`a, conoscenza ed esperienza, nella stesura di questa tesi. Posso solo sperare di avvicinarmi alla vostra padronanza e comprensione della Fisica ad Alte Energie.

Ringrazio Luca di Luzio, Giacomo Landini e Jin-Wei Wang, con i quali ho condiviso la mia prima esperienza nella scrittura di un articolo scientifico, sempre disponibili nel chiarire i miei dubbi.

Un ringraziamento speciale alla mia famiglia, in particolare ai miei genitori, per l’immenso supporto psicologico ed economico di questi 5 anni. Ogni mio traguardo `e anche merito vostro.

Grazie ad amici e colleghi, per avermi accompagnato in questi anni. Poter condividere con voi le mete raggiunte `e ci`o che le rende veramente speciali.

Un grazie infinito ad Eleonora, per esser stata sempre al mio fianco. Non credo di meritare quanto mi fai sentire amato e il tuo supporto mi tiene sano nei momenti pi`u difficili.

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Chapter 1 Introduction

A deeper understanding of the vacuum structure of Quantum Chromodynamics, closely related to the discovery of instanton field configurations [1], has furnished a solution to the so-called ‘ U(1)A problem’ [2], while introducing a potential CP-violating operator in

the QCD Lagrangian. The value of the θ angle, the relevant parameter that quantifies the (strong) CP violation, is strictly bounded by the non-observation of the neutron dipole electric moment dn< 1.8× 10−26 e cm [3], which impliesθ < 10−10 [4, 5].

Given that the possible values of θ belong to the interval [0, 2π], such a small value is regarded as highly unnatural. One of the most theoretically appealing way to explain the smallness of the θ parameter, which is still not ruled out experimentally, is known as the Peccei-Quinn (PQ) mechanism [6,7].

A global U(1)PQ symmetry is postulated, anomalous under SU(3)c and spontaneously

broken at some energy scale fa. The corresponding pNGB goldstone in the spectrum, the

axion [8, 9], acquires a VEV which actively cancels out the CP-violating term, solving dynamically the strong CP problem (also providing a Dark Matter (DM) candidate)

However, global symmetries are not regarded as fundamental features of Quantum Field Theory, but they rather arise accidentally in gauge theories, famous examples being the baryon and lepton numbers.

This conflicts with a global U(1)PQsymmetry that must be realized at very high precision

to relax the topological θ-term. Adopting an Effective Field Theory (EFT) perspective, new physics can generate higher-dimensional operators suppressed by some energy scale ΛUV,

generally breaking the accidental U(1)PQ symmetry of the renormalizable Lagrangian. The

strong CP problem solution requires the UV explicit breaking contribution to the axion potential to be 10−10 smaller than the QCD energy density

fa ΛUV d−4 f4 a . 10−10Λ 4 QCD (1.1)

estimated for a d dimensional breaking operator. Assuming new physics at the Planck’s scale ΛUV ' MPl = 1.2× 1019 GeV and taking into account the phenomenological bound

fa & 109 GeV [10], d & 9 is needed to satisfy eq. (1.1). This is known as the PQ quality

problem [11, 12,13, 14].

This encourages us to seek axion models in which the PQ symmetry is an accidental symmetry of the Lagrangian, protected by early explicit breaking by some fundamental principle, for instance, gauge invariance. In the present work, we propose a gauge theory based on the SM gauge group GSMplus a SU(N ) [15], in which an accidental chiral U(1)PQ

symmetry it’s protected by higher dimensional operator up to _{N dependent dimension, in} principle arbitrary.

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Chapter 1

The thesis is outlined as follows: in Chapter 2 a brief review of the U(1)A problem is

laid out, along with some aspect of chiral perturbation theory.

The discovery of the instanton solutions and how their inclusion in the path integral results in the strong CP problem is discussed in Chapter 3.

In Chapter 4 the general approach of an axion solution is presented, with the original Peccei-Quinn model as well as the more popular invisible axion models. Furthermore, a more detailed discussion on the PQ quality problem is given.

In Chapter5we give a summary of Early Universe cosmology, discuss thermal production of Dark Matter and axion cosmology.

Finally, in Chapter 6 we present two model with a high quality PQ symmetry, studying their phenomenological implication.

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Chapter 2 The U(1)

_A

problem

2.1 Quantum Chromodynamics

Quantum Chromodynamics (QCD) has been established, through numerous experimental results, theoretical arguments and lattice simulations, as the description for the strong inter-action. At low energies, the strong force is the strongest of all fundamental interactions and it’s responsible for the binding of quarks inside hadrons, and for the interaction between the latters, thanks to which protons and neutrons are held together to form the nuclei of atoms. QCD is a quantum field theory that features Nf = 6 fermion fields, the quarks, that

carry Nc = 3 colors. The interactions between the quarks, mediated by the 8 gluons, arise

by the requirement of invariance under a local (position dependent) SU(Nc) transformation

of the quark fields.

2.1.1 Non-Abelian SU(

N

c

) gauge theory

Let us consider the lagrangian for Nf free fermions withNc colors

L0 F = Nf X f =1 Nc X i=1 ¯ ψf i(x)(iγµ∂µ− mf)ψf i(x) (2.1)

where the ψ-s are grassmann variables. The theory is invariant under a global U(Nc)

trans-formation

ψf i(x)→ ψ0f i(x) = Uijψf j(x) (2.2)

with U∗

jiUjk =δik. To lighten the notation, let us define

Ψf ≡    ψf 1 .. . ψf Nc    Ψ 0 f =U Ψf with U†U = U U† =1Nc×Nc (2.3)

If we focus on the SU(Nc) subgroup (i.e U ∈ U(Nc) and detU = 1), every element can be

expressed in terms of the Lie algebra generators

U (θ) = exp   N2 c−1 X a=1 iθa_Ta   Ta, Tb = ifabcTc Tr TaTb = 1 2δab (2.4)

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2.1. Quantum Chromodynamics Chapter 2

where Ta _{are traceless}_N

c× Nc hermitian matrices and fabc is completely anti-symmetric in

a, b, c. The Nc colors furnish the defining representation of SU(Nc).

Clearly, the local version of the group action is not a symmetry of the Lagrangian, because if we perform a position dependent transformation

Ψf → Ψ0f(x) = U (x)Ψf(x) (2.5)

the derivative ∂µΨ does not transform as Ψ. This is inherently linked to the definition of

the derivative, which involves the sum of fields defined in different space-time points (xµ _and

xµ₊_dxµ_{), that do not transform covariantly.}

Parallel Transport

Given two space-time points xµ _and _yµ_{, a path} _C

y←x:zµ(s) such that s∈ [0, 1], zµ(0) =xµ

and zµ_{(1) =}_yµ_{, we define the parallel transport} _{W (C}

y←x) as:

1. W (Cy←x)∈ SU(Nc)

2. W (_{∅) = 1, where ∅ represents any curve with zero measure} 3. If Cy←x =Cy←w◦ Cw←x → W (Cy←x) = W (Cy←w)W (Cw←x)

4. If we defineC−1

x←y :zµ(1− s) → W (Cx−1←y) = W (Cy←x)−1

5. Under a gauge trasformation W (Cy←x)→ W (Cy←x)0 =U (y)W (Cy←x)U−1(x)

It follows from the property 5) that ˜Ψ(y)≡ W (Cy←x)Ψ(x) transforms under a gauge

trans-formation as Ψ(y).

Since W belongs to the gauge group, we can write it as the exponential of a matrix in the Lie algebra W (Cy←x) = exp(iA(x, y)). If we consider a straight infinitesimal path from

xµ _to_xµ₊_dxµ_{, we expect the parallel transport to be infinitesimally close to the identity}

W (Cx+dx←x)' 1 + igsAµ(x)dxµ ' eigsAµ(x)dx

µ

(2.6)

where all equalities are true at first order in dx and the factorization of gs is a matter of

convention. Aµ is known as the connection and lives in the vector space spanned by the

generators, it can therefore be expanded as Aµ(x)≡ Aaµ(x)Ta. TheAaµ are the Nc2− 1 gluon

fields.

From the transformation rule of the parallel transport

W0(Cx+dx←x) =U (x + dx)W (Cx+dx←x)U−1(x), (2.7)

it is straightforward to obtain the gauge transformation of the connection, expanding eq. (2.7) at first order indx

A0µ(x) = U (x)Aµ(x)U−1(x)−

i gs

(∂µU (x))U−1(x). (2.8)

We can now define a covariant differential

DΨ(x) _{≡ W (C}_x−1_←x+dx)Ψ(x + dx)_{− Ψ(x)} (2.9)

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and, consequently, a covariant derivative Dµ such that DΨ(x) = DµΨ(x)dxµ

Dµ=∂µ− igsAµ=∂µ− igsAaµT a

. (2.10)

By definition, DµΨ(x) transforms as Ψ(x), therefore

LF = Nf X f =1 ¯ Ψf(iγµDµ− mf)Ψf =LF0+ Nf X f =1 gsAaµΨ¯fTaγµΨf (2.11) is gauge invariant.

Pure gauge lagrangian

We were able to extend the global symmetry to a local one introducing N2

c − 1 new spin-1

fields that interacts with the quarks. We need to introduce the kinetic terms of the gluons, without spoiling the local symmetry. Actually, since we are using gauge invariance as our guiding principle to construct the theory, we should add every term allowed by the gauge symmetry, stopping at the renormalizable level if we want the theory to be predictive.

The following identity

[Dµ, Dν]Ψ = [∂µ− igsAµ, ∂ν − igsAν]Ψ = −igs(∂µAν − ∂νAµ− igs[Aµ, Aν])Ψ (2.12)

holds for every Ψ, such that

[Dµ, Dν] =−igs(∂µAν − ∂νAµ− igs[Aµ, Aν]) =−igsGµν. (2.13)

The anti-symmetric tensor Gµν is known as the curvature and its transformation under the

gauge group is easily found by observing that D0

µ=U DµU−1 G0µν = i gs D0 µ, D0ν = i gs U [Dµ, Dν]U−1 =U GµνU−1. (2.14)

The curvature transforms in the adjoint of the gauge group and it can also be expanded in terms of the generators as

Gµν =GaµνTa = (∂µAaν − ∂νAaµ+gsfabcAbµAcν)Ta. (2.15)

We can now construct a gauge invariant Lorentz scalar

− 1₂Tr(GµνGµν) =− 1 4G a µνG aµν _(2.16)

where the coefficient in front has been chosen to canonically normalized the kinetic term. The gauge symmetry allows an analogous term, built using the field tensor and its dual

µνρσTr(GρσGµν) = 2 Tr ˜GµνGµν

. (2.17)

We can further expand it in terms of the connection

µνρσTr(GρσGµν) =4µνρσTr((∂µAν − igAµAν)(∂ρAσ − igAρAσ))

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having used that the term containing four connections is zero, because any cyclic permutation of the 4 indices leaves the trace invariant but gets a _{− from the tensor.}

From eq. (2.18) it’s easy to see that

µνρσTr(GρσGµν) =∂µKµ with Kµ= 4µνρσTr Aν_∂ρ_Aσ ₋2igs 3 A ν_Aρ_Aσ (2.19)

where Kµ is known as the Chern-Simons current. Naively, one can think that since this

term is a total derivative, its integral over all space-time vanishes. However, as it will be discussed in the next sections, there exist topologically non trivial field configurations, known as instantons, with a non vanishing R ∂µKµd4x and finite action [1].

In the original QCD Lagrangian, this CP odd term1 _{was not considered, so that P and} CP were automatically conserved as was experimentally verified in strong processes, scoring a feature that was considered one of the major success of QCD. What will be explored in this and the next chapter is that simply disregarding this term is inconsistent with the existence of instanton solutions and with the heaviness of the η mesons, a puzzle dubbed the ‘ U(1)A

problem’ [2].

Explaining the absence (or a very tiny presence) of CP violation in the strong interaction is what ultimately motivates the introduction of the axion.

Finally, the full QCD Lagrangian density reads

LQCD =− 1 4G a µνG aµν₊ Nf X f =1 ¯ Ψf(i /D− mf)Ψf +θ g2 s 32π2G˜ aµν_Ga µν (2.20)

where the choice of normalization of the last term foreshadows the upcoming discussions.

2.1.2 Discrete symmetries

In this section we discuss the properties of the QCD lagrangian density under discrete trans-formation: ˆC, ˆP and ˆT .

Parity

Parity corresponds to the change of sign of the spatial components of the 4-position

xµ = (x0, ~x)−→ xPˆ µ0

= (x0,−~x) xµ −→ gPˆ µµxµ. (2.21)

Summation convention is suspended in writing the second expression. In the Hilbert space of a quantum theory, parity acts as a unitary (and hermitian) operator U ( ˆP ). On the quark and gluon fields 2

U ( ˆP ) ˆΨ(x0_{, ~x)U}†_{( ˆ}_{P ) = γ}0_Ψ(x_ˆ 0_, −~x) U ( ˆP )Aaµ(x 0 , ~x)U†( ˆP ) = gµµAˆaµ(x 0 ,−~x). (2.22)

We remind that we are working in the Weyl representation for the γ matrices

1_{Discrete symmetries of the lagrangian will be discussed in the next section}

2_{Fermion transformations under discrete transformation can be derived imposing the invariance of the}

free Dirac action

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Charge conjugation

The charge conjugation effectively switches a particle with its antiparticle. On fermions acts as

U ( ˆC) ˆΨ(x0_{, ~x)U}†_{( ˆ}_{C) = iγ}2_Ψ_ˆ†_(x0_{, ~x).} _(2.23)

In the previous section we have derived the expression for the covariant derivative using the parallel transport, defined to act on fermion fields that transforms in the fundamental rep-resentation of SU(Nc). Clearly, the same procedure can be applied in general, the difference

being that the generators have to be considered in the desired representation.

If under charge conjugation fermions transform as stated in eq. (2.23), the covariant derivative should now act on a field that transforms in the anti-fundamental

Dµ ˆ C

−→ Dµ∗. (2.24)

The generators of the fundamental of SU(Nc) are hermitian, tracelessNc×Ncmatrices. Some

are real and symmetricTa

real, while the others are purely imaginary and anti-symmetricT b imag.

Since in the anti-fundamental the generators are _−Ta∗ 3_{, the conjugation effectively acts on} the corresponding gluon fields as

U ( ˆC) ˆAaµ_real(x0_{, ~x)U}†_{( ˆ}_{C)) =}_{− ˆ}_Aaµ real

U ( ˆC) ˆAaµ_imag(x0_{, ~x)U}†_{( ˆ}_{C) = ˆ}_Aaµ

imag. (2.25)

Time Reversal

Time reversal corresponds to changing the sign of time

x0 Tˆ

−→ −x0 _xµ Tˆ

−→ −(gµµ)xµ. (2.26)

In this case, on quantum states, time reversal acts as an anti-unitary operator. On quark and gluon fields

U ( ˆT ) ˆΨ(x0_{, ~x)U}†_{( ˆ}_{T ) = γ}1_γ3_Ψ(_ˆ

−x0_{, ~x)}

U ( ˆT ) ˆAa

µ(x0, ~x)U†( ˆT ) = gµµAˆaµ(−x0, ~x). (2.27)

QCD discrete symmetries

We can now determine the symmetry properties of the QCD lagrangian under discrete trans-formations. Faµν_Fa µν ˆ P , ˆT −−→ g2 µµg 2 ννF aµν_Fa µν =F aµν_Fa µν (2.28) µνρσ_Fa µνF a ρσ ˆ P , ˆT −−→ µνρσ_g µµgννgρρgσσFµνa F a ρσ =− µνρσ_Fa µνF a ρσ (2.29)

where space-time coordinates are understood to be properly transformed. In the last equality of eq. (2.29) has been used that the four indicesµ, ν, ρ, σ are all different due to the contrac-tion with the tensor. For every local, Lorentz invariant and with a hermitian hamiltonian QFT, ˆC ˆP ˆT is a symmetry of the theory [16, 17].

Consequently, the two terms in eq.s (2.28) and (2.29) are, respectively, CP (CT) even and CP odd.

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2.1. Quantum Chromodynamics Chapter 2 Bilinear Cˆ Pˆ Tˆ ¯ ΨΨ + + + ¯ Ψγ5_Ψ ₊ _- -¯ Ψγµ_Ψ _- _g µµ −gµµ ¯ ΨTa realγ µ_Ψ _- _g µµ −gµµ ¯ ΨTa imagγµΨ + gµµ −gµµ

Table 2.1: Transformation properties of fermion bilinear

For fermions bilinears the symmetries properties are summarized in Table2.1.

We conclude by observing that every term in eq. (2.20), with the exception of theθ term, is separately invariant under all discrete transformations. However, if the mass matrix is complex, the mass term contains a CP odd operator. Indeed, in such case

− Lmass= ¯ΨLMΨR+ ¯ΨRM†ΨL (2.30) that is equal to −Lmass = ¯Ψ(A + iγ5B)Ψ (2.31) A = A† = M + M † 2 (2.32) B = B†=_−iM − M † 2 . (2.33)

The term involving _{B is CP odd and we will see that can be rotated away and absorbed in} the physical ¯θ definition. In the next sections we ignore every source of ˆC ˆP violation, hence we work with the QCD lagrangian with θ = 0 and with a real, diagonal, mass matrix in flavour space.

2.1.3 Perturbative QCD and running coupling

As in any interacting quantum field theory, vacuum-to-vacuum amplitude of time ordered products of gluon and quark fields can be computed perturbatively through Feynman dia-grams, considering a formal expansion in powers of the coupling gs.

UV divergences arise in computing loop diagrams, and a regularization procedure is nec-essary to deal with the divergent integrals. A UV cut-off Λ or a renormalization scale µ are introduced in the usual regularization schemes, and the parameters of the theory are renormalized, which means redefined in such a way that all correlation functions are finite when the regulator is removed. Many different renormalization schemes are possible.

The Lagrangian parameters and field normalizations are infinite and known as bare gs → gsB Aµ→ A

µ

B Ψ→ ΨB mf → mf B. (2.34)

They can be related to the renormalized fields and parameters, that are finite and appear in physical observables, through infinite scheme dependent multiplicative factor

gsB =ZggsR AµB =Z 1/2 A A µ R ΨB =Z 1/2 ψ ΨR mf B =Zmmf R. (2.35) 3_{U (θ)}∗ _{= exp(iθ}a₍

−Ta∗_{)), by conjugating the commutation relations for the T}a _{we get}_−Ta∗_,

−Tb∗₌

ifabc₍

−Tc∗₎

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2.2. Continuous chiral symmetries Chapter 2

Clearly, the underlying theory should not depend by the arbitrary regulator introduced. This statement, applied to the bare correlation functions, results in the Renormalization Group Equation (RGE), that relates renormalized correlation functions at different energy scales. One important consequence of the RGE solution is that, effectively, the renormalized coupling of the theory depends on the energy scale through the differential equation

µdgR(µ)

dµ =β(gR(µ)). (2.36)

β(gR(µ)) is accessible through perturbative calculations and admits an expansion in powers

of the renormalized coupling

β(gsR) =−β0gsR3 − β1gsR5 − β2g7sR+O g 9

sR. (2.37)

The coefficients of the expansion are in general scheme depedent, but it can be shown that β0, β1 are not. A two loop calculation yields [18]

β0 = 1 (4π)2 11Nc− 2Nf 3 β1 = 1 (4π)4 34 3 N 2 c − 13 3 NcNf − Nf Nc . (2.38)

For Nc = 3 and Nf = 6, β0 > 0 which implies that the theory is asymptotically free. To

lowest order

µdgR(µ)

dµ =−β0g

3

sR. (2.39)

Integrating it, we find

µe−12β0g2sR ₌cost. = Λ_QCD → g2 sR(µ) = 1 2β0ln µ ΛQCD . (2.40)

From eq. (2.40) asymptotic freedom is apparent and we identify a physical mass scale, ΛQCD,

at which the theory is no longer perturbative.

2.2 Continuous chiral symmetries

In order to explore the continuous global symmetry of QCD, let us rewrite the fermion sector of LQCD in terms of chiral eigenstates Ψ = ΨL+ ΨR, whereγ5ΨL =−ΨL and γ5ΨR= ΨR.

LF = ¯ΨRDΨ/ R+ ¯ΨLDΨ/ L− ¯ΨRMΨL− ¯ΨLMΨR. (2.41)

The mass matrix is _Mf f0 = mfδf f0 and flavour indices are implied. In general, the only

classical continuous symmetry is

Nf

Y

f

U(1)f, under which

Ψf U(1)f

−−−→ eiαf_Ψ

f. (2.42)

The combination of all U(1)f such that all flavours have the same charges U(1)V, corresponds

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Note that if _{M ∝ 1 (m}f =m for every f ) or M = 0 , the global symmetry group G is

much larger G = U(Nf)L⊗ U(Nf)R. ForUL ∈ U(Nf) andUR∈ U(Nf), the action of G on

the quark fields is

ΨL → Ψ0L =ULΨL Ψ¯L → ¯Ψ0L = ¯ΨLU_L† (2.43)

ΨR → Ψ0R =URΨR Ψ¯R → ¯Ψ0R = ¯ΨRUR† (2.44)

G can be decomposed as G = U(Nf)L⊗ U(Nf)R= SU(Nf)L⊗ SU(Nf)R⊗ U(1)L⊗ U(1)R=

SU(Nf)V ⊗ SU(Nf)A⊗ U(1)V ⊗ U(1)A, in which

SU(Nf)V : ( ΨL→ V ΨL ΨR→ V ΨR SU(Nf)A: ( ΨL→ AΨL ΨR→ A†ΨR (2.45) U(1)V : ( ΨL→ eiαΨL ΨR→ eiαV ΨR U(1)A : ( ΨL→ eiβΨL ΨR→ e−iβΨR (2.46)

where V, A ∈ SU(Nf). Writing V and A as exponential, we shorten the notation using the

full Dirac spinor

SU(Nf)V : Ψ→ eiα

a_Ta

Ψ SU(Nf)A : Ψ→ eiβ

a_Ta_γ5

Ψ (2.47)

U(1)V : Ψ→ eiαΨ U(1)A: Ψ→ eiβγ

5

Ψ. (2.48)

Many experimental and theoretical results, some of which will be mentioned in the upcoming discussion, suggest that the quark masses are neither equal nor zero, soG is not a symmetry. However, the masses of the lighter flavours (up, down and strange4_{) can be considered small} compare to the physical scale ΛQCD of the theory, allowing us to treat the chiral symmetry

as an approximate symmetry, that is explicitly broken by the quark masses.

Even if we consider their masses negligible (m _{→ 0, know as the chiral limit), we have} to discuss how the continuous symmetry is realized at the quantum level.

2.2.1 Quantum Symmetries and Anomalies

Classically, Noether’s theorem states that if the Lagrangian is invariant under a global trans-formation

Φ_{→ Ψ + f}aδaΦ. (2.49)

we can find a set of conserved currents

Jµ a =−

∂L ∂(∂µΦ)

δaΦ ∂µJ_bµ = 0 (2.50)

and define the corresponding constant charges Qb

Qb(x0) = Qb =

Z

d3_xJ(x0_{, ~x).} _(2.51)

A quantum field theory is defined through the partition function

Z = Z [DΦ] exp(iS(Φ, ∂Φ)) = Z [DΦ] exp i Z d4_{xL (Φ, ∂Φ)} (2.52)

4_{This assumption for the strange quark is not very accurate, indeed the SU(3) flavour symmetry is more}

badly realized than the SU(2) of Isospin.

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where Φ represents all fields. If we perform a change of variables

Φ(x)_{→ Φ(x) + f}a_(x)δ

aΦ(x) (2.53)

with fa_{(x) arbitrary infinitesimal functions, the Lagrangian is replaced by}

L (Φ, ∂Φ) → L (Φ, ∂Φ) +δL δΦf a_δ aΦ + δL δ(∂µΦ) ∂µ(faδaΦ). (2.54)

If we assume that fa_{(x) goes to zero sufficiently fast at infinity for every a, we can partial}

integrate to find Z d4_x δL δ(∂µΦ) ∂µ(faδaΦ) = Z d4_x δL δ(∂µΦ) ∂µ(δaΦ)− ∂µ δL δ(∂µΦ) δaΦ fa_. _(2.55) Defining δaL ≡ δL δΦδaΦ + δL δ(∂µΦ) ∂µ(δaΦ) (2.56)

and recognizing the classical Noether’s current in eq. (2.55), we see that Z d4_x_{L (Φ, ∂Φ) →} Z d4_{x [}_{L (Φ, ∂Φ) + (δ} aL + ∂µJaµ)f a_]_. _(2.57)

Under a local change of variables, the measure in the path integral can also transform non trivially, which can be parameterize introducing a function_Aa(x) known as anomaly, defined

such that [DΦ]_{→ [DΦ] exp} i Z d4_x Aa(x)fa(x) . (2.58)

The partition function, which transforms as

Z _→ Z [DΦ] exp i Z d4_{x (δ}a_{L + ∂} µJaµ+Aa)fa exp(iS(Φ, ∂Φ)) (2.59)

has to be invariant under a change of dummy variables, so that expanding the exponential at first order we find

Z [DΦ]

Z

d4_{x (δ}a_{L + ∂}

µJaµ+Aa)faexp(iS(Φ, ∂Φ)) = 0. (2.60)

This has to be true for every choice of the arbitrary functions fa_{, leading to the condition}

Z [DΦ] (δaL + ∂µJaµ+Aa) exp(iS(Φ, ∂Φ)) = 0 D ∂µJˆaµ E =₋DδaL + ˆˆ Aa E . (2.61)

Even if the classical theory is invariant under the corresponding global transformation with constant fa

Φ(x)→ Φ(x) + fa

δaΦ(x) (2.62)

which means thatδaL = 0, in the quantum theory the Noether’s currents are not necessarily

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Chiral Anomaly in a gauge theory

Let us consider a transformation on fermionic fields that fill a representation _{R under a} gauge group _G

Ψm(x)→ Umn(x)Ψn(x) (2.63)

wherem, n are collective indices that runs over all relevant spaces, for example in QCD they represents flavour, color and spinor indices. The transformation is equivalently written as

Ψm(x)→

Z

d4_{y δ}4_(x_{− y)U}

mn(y)Ψn(y)≡ Uxm,ynΨyn (2.64)

where in defining _Uxm,yn we formally extended the summation convention to the space-time

coordinates. Correspondingly ¯

Ψxm → ¯Ψyn U¯yn,xm where U¯yn,xm ≡ (γ0U†(x)γ0)nmδ4(x− y). (2.65)

In changing variables in Berezin integration, we have to multiply for the inverse of the Jacobian

[D ¯ΨDΨ]→ (det ¯U det(U))−1_{[D ¯}_ΨDΨ]. _(2.66)

Let us consider a unitary non-chiral transformation U , meaning

U (x) = exp(iα(x)t) (2.67)

whereα(x) is an arbitrary function and t is an hermitian matrix which is equal to the identity in Dirac space. In this case

¯

UU = 1 (2.68)

hence the change of variables amount to a trivial transformation of the Berezin measure. On the other hand, for a chiral unitary transformation

U (x) = exp iα(x)γ5_t (2.69) we find that ¯ U = U (2.70) therefore [D ¯ΨDΨ]_{→ (det U)}−2[D ¯ΨDΨ]. (2.71)

The Dirac delta represents the identity matrix for the space-time coordinatesδ4_(x_{−y) ≡ 1} xy

and the following formal equality holds

exp(A)_{⊗ 1}xy = exp(A⊗ 1xy). (2.72)

Using it in writing _{U and employing the identity det ≡ exp Tr ln , we can write} det_{U = exp Tr ln exp iα(x)γ}5t⊗ 1xy = exp iα(x) Tr tγ5δ4(x− y)

(2.73)

where the trace is extend over all indices. In eq. (2.73) we identify the anomaly as defined in the previous section

A(x) = −2 Tr tγ5_δ4_(x_{− x).} _(2.74)

It is not very surprising that, as a consequence of the hand-waving manipulations that we have performed, the final result is ill-defined in dealing with continuum space-time coor-dinates. We need to give meaning to eq. (2.74) introducing a regularization for the delta

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function. There is hope that the final result will be finite when removing the regulator, indeed while it’s true that the delta function is infinite, the trace of γ5 _{vanishes. The delta}

function diverges in the UV, so we introduce a cut off Λ and a smooth function f (s) that satisfies the following conditions

f (0) = 1 lim s→∞f (s) = 0 (2.75) lim s→∞sf 0_{(s) = lim} s→0sf 0_{(s) = 0.} _(2.76)

We regulate the anomaly as

A(x) = −2hTrntγ5_{f ( /}_D2 x/Λ2)

o

δ4_(x_{− y)}i

y→x (2.77)

where /Dx is the covariant derivative in the representation of the fermion fields Ψ and we

sub-stitute y→ x only after the action of f(− /D2x/Λ2) on the right. This choice of regularization

is manifestly gauge invariant. In momentum space

A(x) = −2 Z

d4_k

(2π)4Trtγ 5

f (−(/∂x− ig /A(x))2/Λ2) eik(x−y)

y→x (2.78) =₋₂ Z _d4_k (2π)4 e

ik(x−y)_Trtγ5_{f ((i/}_{k + /}_∂

x− ig /A(x))2/Λ2) y→x (2.79) =₋₂ Z _d4_k (2π)4 Trtγ 5_{f ((i/}_{k + /}_D x)2/Λ2) . (2.80)

After performing the change of variables k = k0Λ, we find

A(x) = −2 Z d4_k0 (2π)4 Λ 4_Trtγ5_{f ((i/}_k0_{+ /}_D x/Λ)2) (2.81)

and for notation’s sake we relabel k0 _{→ k. Observing that, for two generic operators T} µ Tν0, the product (γµ_T µ)(γνTν0) can be written as (γµ_T µ)(γνTν0) = 1 4([Tµ, T 0 ν][γ µ_{, γ}ν_{] +} {Tµ, Tν0}{γ µ_{, γ}ν }) = 1 4 [Tµ, T 0 ν][γ µ_{, γ}ν_{] + 2T}µ_{, T}0 µ (2.82) we work out the argument of the function f

i/k + D/x Λ 2 =_−k2+ 2ik· Dx Λ + / D2_x Λ2 (2.83) =_−k2+ 2ik· Dx Λ + D2 x Λ2 − ig 4Λ2Gµν[γ µ , γν] (2.84) ≡ −k2 _{+ ∆.} _(2.85)

Since the function f is smooth, it admits a Taylor expansion

f (_−k2_{+ ∆) =}_{f (}

−k2_{) +}_f0₍_−k2_{)∆ +}f00(−k 2₎

2 ∆

2 ₊_{. . .} _(2.86)

Higher orders either give a trivial contribution when taking the limit Λ _{→ ∞ or do not} contain the minimum number (4) of gamma matrices to have a non zero trace with the γ5_.

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The last condition is never satisfied at first order, and we therefore disregard it. At the end, the only non zero contribution is given by the square of the last term in eq. (2.84)

A(x) = g 2 16TrD(γ 5_[γµ_{, γ}ν_][γρ_{, γ}σ_{]) Tr}_{tF µνFρσ} Z d4_k (2π)4f 00₍_−k2₎ _(2.87)

where TrDrepresents the trace over Dirac space. The proper way to evaluate the integral is to

work from the beginning in the euclidean path integral and perform an analytic continuation when going back to Minkowski space. Effectively, we can consider the rotation k0 → ikE4

where kE4 runs from −∞ to +∞5, whileki = (~kE)i and k2E =kE42 +|~kE|2 =−k2

Z d4_kf00₍_−k2_{) =}_i Z d4_k Ef00(kE2) = i Z ∞ 0 2π2_k3 Ef00(kE2)dkE =iπ2 Z ∞ 0 sf00(s)ds (2.88) =iπ2_sf0_(s) ∞ 0 − iπ2 Z ∞ 0 f0(s)ds (2.89) =iπ2 _(2.90)

where we have employed the properties of f written in eq. (2.76). Considering that

TrD(γ5[γµ, γν][γρ, γσ]) = 16iµνρσ (2.91)

and that the fermion fields fill the representation _{R under the gauge group}

Gµν =GaµνTaR (2.92) we finally find A(x) = − g 2 16π2 µνρσ GaµνG b ρσTrtTaRTbR . (2.93)

Let us now consider a gauge theory involving only chiral fermions, for instance left-handed, in a representation _{R of the gauge group}

L ∼ ¯ψLi /DψL. (2.94)

The anomaly of a symmetry transformation on ψL can be computed introducing a dummy

right handed component that do not couple with the gauge fields

Ldum ∼ ¯ψLi /DψL+ ¯ψRi/∂ψR (2.95)

and considering as the path integral measure [D ¯ψDψ] where ψ = (ψR, ψL)t. This does not

affect the theory, since the correlation functions do not involve the fictitious and decou-pled right handed field, but allows us to compute the anomaly using the previous results, considering a transformation

U (x) = exp(iα(x)PLt) = exp(iα(x)t/2) exp −iα(x)tγ5/2. (2.96)

Only the chiral component effectively changes the measure and in making the substitution α(x)_{→ −α(x)/2 we find} AL (x) = g 2 32π2 µνρσ GaµνG b ρσTrtTaRTbR . (2.97)

5_{In other words, the analytic continuation from the Euclidean is straightforward because we do not}

encounter any singularity thanks to the analyticity of f .

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The difference in the case of a right-handed fermion would have been replacing PL ↔ PR,

amounting in a sign difference

AL_{(x) =}_−AR_(x). _(2.98)

This is important when considering gauge theories that involve fermions of both chiralities, such as the Standard Model (SM).

In order to have a consistent gauge theory, all anomalies that correspond to the action of the gauge group must vanish. In QCD, the SU(Nc) anomaly is evidently zero because the

gauge transformation is vector-like (non-chiral) on the Dirac fermion. Employing the symmetry ofµνρσ_Fa

µνFρσb under the exchangea ↔ b, we write the anomaly

in eq. (2.93) as A(x) = − g 2 16π2 µνρσ GaµνG b ρσTr tTaRTbR = − g2 32π2 µνρσ GaµνG b ρσTr tTaR, TbR . (2.99)

It is understood that the calculations have been computed in a background of fixed gauge fields, in the sense that the Noether’s current operator satisfies

D ∂µJˆµ E A=− D δ ˆLE A+ D ˆ_AE A (2.100) where hOiA ≡ Z [D ¯ψDψ]OeiS[A, ¯ψ,ψ] Z [D ¯ψDψ]eiS[A, ¯ψ,ψ] (2.101) Chiral anomaly in QCD

The axial SU(L)A and U(1)A of the chiral group in QCD, which correspond respectively to

the generators t = αa_(x)Ta _and _{t =}_{1 acting on flavour space, are potentially anomalous.}

The SU(L)A anomaly vanishes because the αa(x)Ta is traceless, while

A(x)U(1)A =− g2 s 32π2 µνρσ_Ga µνG b ρσTrc({Tac, T c b}) Trf(1) (2.102) =_−L g 2 s 32π2 µνρσ_Ga µνGaρσ (2.103) ≡ −2LQ(x) (2.104)

where we have used Trc({Tac, Tbc}) = δab and defined the topological charge density

Q(x) = g 2 s 64π2 µνρσ_Ga µνG a ρσ. (2.105)

The topological charge density is proportional to the divergence of the Chern Simons current defined in eq. (2.19), hence the change of the QCD action under an axial U(1)A rotation is

different from zero only if we take into account the instanton solutions.

2.2.2 Spontaneous symmetry breaking and the effective chiral

la-grangian

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1. The symmetry is not anomalous;

2. The transformation acts as the unitary operator ˆU = eiωbQˆb _{on the Hilbert space and}

ˆ

Qb_{|Ωi = 0, which means that the vacuum |Ωi is invariant ˆ}_U_{|Ωi = |Ωi.}

From the second property we see that the conserved charges are also the generators of the infinitesimal transformation, meaning that for ωa ₁

e−iωaQˆa_ΨeiωaQˆa

' (1 − iωa_Q_ˆa_{)Ψ(1 +}_iωa_Q_ˆa₎

' Ψ − iωah ˆ_Qa_{, Ψ}i _{= Ψ +}_ωa_δa_Ψ. _(2.106)

If 1. is satisfied but 2. is not for some ˆQ˙a_{, the vacuum is not invariant under the group}

symmetry and Spontaneous Symmetry Breaking (SSB) occurs. For every broken generators ˆ

Q˙a _{a spin 0 mass-less particle appears in the spectrum, with the same quantum numbers of}

ˆ

Q˙a_{|Ωi [}₁₉_].

Isospin and Gell-Mann flavour symmetries, that corresponds to the SU(2)V and SU(3)V

symmetries that emerge in taking respectively the limit mu, md → 0 and mu, md, ms →

0, were known to be approximate symmetries for the strong interaction even before the establishment of QCD.

From now on, we will assume that SU(L)V is Weigner-Weyl realized in the chiral limit.

If SU(L)Ais an exact symmetry, for every hadron state|hi with mass Mh and intrisic parity

ph, we can find a partner with the same mass and baryon number, but opposite parity. In

the algebra of the axial charges h ˆ_Qa A, ˆQ b A i =ifabc_Q_ˆj A a, b, c = 1, . . . , L 2 − 1 (2.107)

we can identify a SU(2) sub-algebra h ˆ_Qi A, ˆQ j A i =iijk_Q_ˆj A (2.108)

and define the ladder operators ˆQ±_A = ˆQ1

A± ˆQ2A. Considering the state

h0_± ≡ ˆQ±_A|hi, we find that

1. If the symmetry is realized h ˆQ±_A, ˆHi= 0, so that h0_± and |hi are degenerate

2. h ˆQa A, ˆQV

i

= 0, where ˆQV is the generator of the U(1)V transformation, that

cor-responds to the baryon number. This can be verified observing that the conserved current of the U(1)V symmetry, JVµ = ¯ΨγµΨ, is invariant under an axial SU(L)A

transformation. 3. The ˆQa

A are pseudo-scalars, therefore

h0_± has intrisic parity equal to −ph.

These hadronic degenerate partners with opposite parity have never been observed, ruling out the possibility that SU(L)A is an exact symmetry. An analogous parity doubling would

be present if the U(1)A was exact. Furthermore, through lattice simulations, it has been

found that [20]

hΩ| ¯ΨΨ_{|Ωi 6= 0.} (2.109)

The expectation value in eq. (2.109) is known as the chiral condensate. Note that

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2.2. Continuous chiral symmetries Chapter 2 but since h ˆQa A, ¯Ψ i =i(δa AΨ) =¯ − ¯ΨTaγ5 and h ˆQaA, Ψ i =i(δa AΨ) =−Taγ5Ψ, we find hΩ|h ˆQa A, ¯Ψγ5TbΨ i |Ωi = − hΩ| ¯ΨTa_{, T}b_Ψ|Ωi _(2.111) =₋1 Lδ ab hΩ| ¯ΨΨ_{|Ωi − d}abc_{hΩ| ¯}ΨTc_Ψ |Ωi (2.112)

where in the last equality we have used that Ta_{, T}b_{is a L × L hermitian matrix and}

admits an expansion in the basis_{{1, T}a_{}. The coefficient of the identity matrix is determined}

imposing the trace normalization Tr Ta, Tb_{= δ}ab_{. The term involving the symmetric}

tensor dabc _{is zero if the SU(L)}

V symmetry is Wigner-Weyl realized6

hΩ|h ˆQa A, ¯Ψγ5TbΨ i |Ωi = −1 Lδ ab_{hΩ| ¯}_ΨΨ_{|Ωi .} _(2.113)

Clearly, if SU(L)A was exact, the Left-Hand-Side would have been zero for everya, b.

SU(L)A breaking and effective chiral lagrangian

We conclude that SU(L)A undergoes complete SSB and we fall in the assumptions of the

Nambu-Goldstone Theorem. In the following breaking pattern

G0 = SU(L)V × SU(L)A → H = SU(L)V. (2.114)

L2 _{− 1 pseudo-scalar Nambu-Goldstone bosons (NGB) emerge, associated with the broken}

generators of SU(L)A. Because of the explicitH breaking, due to the non zero quark masses

and electromagnetic interaction, the NGBs acquire masses and we refer to them as pseudo-NGBs.

They fill the same representation under the unbrokenH; in the relevant cases L = 2, 3 we identify them respectively as the iso-triplet of the pions (π−, π0_{, π}+_{) and the pseudoscalars}

octet (π−_{, π}0_{, π}+_{, η, K}+_{, K}−_{, ¯}_K0_{, K}0_{) of Gell-Mann SU(3). In this framework, their lightness}

compared to the other hadrons is naturally accounted for.

We can explore the low energy behaviour of the pNGBs with an effective field theory, arising from topological arguments [21,22].

In general, let us consider a symmetry groupG0 _{and a vacuum manifold that is invariant}

under a subgroup H ⊂ G. The πa_{(x) (a = 1, . . . , N and N = dim(G}0₎_{− dim(H)) NGBs will}

transform under the symmetry group with some mapping

~π _{−→ ~π}G0 0 = ~f (g, ~π) (2.115)

where ~π = (π1_{, . . . , π}N_{). Being a representation of the group, if} _{e is the unit element and}

g1, g2 ∈ G0, the mapping satisfies

~

f (e, ~π) = ~π f (g~ 2, ~f (g1, ~π)) = ~f (g2g1, ~π). (2.116)

The NGBs fields corresponds to quantum excitations along the vacuum manifold, so if we choose a vacuum configuration ~0, we can write the NGBs fields

~π = ~f (˜g,~0) = ~f (˜gh,~0) (2.117)

6_{We define C}

ij= hΩ| ¯ΨiΨj|Ωi with i and j flavour space indices. An exact symmetry is implemented by

a unitary operator U that leaves the vacuum invariant, so that Cji= hΩ|U†Ψ¯iU U†ΨjU|Ωi = hΩ| ¯Ψ0iΨ0j|Ωi.

Since Ψ0

i = VijΨj, Cji = V_ki†VjlhΩ| ¯ΨkΨl|Ωi = (V CV†)ji. Due to Schur’s Lemma Cji ∝ δij and

hΩ| ¯ΨTc_Ψ

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where in the last equality we’ve used that, by definition, h∈ H leaves the vacuum invariant. The transformation needs to be non-linear to get a non trivial results. If the mapping of eq. (2.117) is invertible, the NGBs fields can be uniquely identified with the elements of the coset space G/H. The coset space is divided into equivalence classes

g0 _{∼ g} if g0 =gh for some h_{∈ H} (2.118)

and for every field configuration ~π we choose a representative ˜g(~π). Let us focus on the specific case at hand, G0 _{= SU(L)}

L× SU(L)R and H = SU(L)V.

We write ˜g = ( ˜VL, ˜VR) = ( ˜VLV˜R†,1)( ˜VR, ˜VR) = nh, h ∈ SU(L)V. We have chosen the

representative as a SU(L)L transformation.

We define ˜VR=V , U = ˜VLV˜R† and we choose for U the canonical parametrization

U (~π) = exp L2−1 X a=1 iπa_τa α ! where τa = 2Ta. (2.119) Under a G0 _{transformation} ˜ g0 =g˜g = (VL, VR)(U V, V ) = (VLU V_R†,1)(VRV, VRV ). (2.120)

In eq. (2.120) we read the trasformation forU

U (~π)_{−→ U(~π}G0 0) =VLU (~π)VR†. (2.121)

Low energy effective lagrangian

UsingU and its transformation under the chiral group, we can write a G0 _{invariant effective}

low energy Lagrangian

L(0)

χPT=f0(U ) + f2(∂µU, ∂µU ) +O p4

(2.122)

where f0 and f2 are hermitian operator that involves respectively 0 or 2 derivatives. No

term with one derivative of U is allowed by Lorentz invariance, while f0(U ) can only be a

function of the constant Tr U†_U_{. We define the hermitian operator}

∆µ=−iU†∂µU (2.123)

and work out its transformation under the G0 _action

∆µ G0

−→ VR∆µVR†. (2.124)

It is possible to rewritef2(∂µU, ∂µU ) as an appropriate G0 invariant combinations of (∆µ)ij×

(∆µ₎

kl. The only two possibilities are

Tr(∆µ) Tr(∆µ) Tr(∆µ∆µ). (2.125)

U , being unitary, can be written as exp(M ) with M traceless and anti-hermitian. Using the following identity ∂µeM = Z 1 0 dxexM∂µM e(1−x)M (2.126) 22

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2.2. Continuous chiral symmetries Chapter 2 we see that Tr(∆µ)∝ Tr e−M∂µeM = Tr e−M Z 1 0 dxexM_∂ µM e(1−x)M (2.127) = Tr(∂µM ) = ∂µTr(M ) = 0 (2.128)

where the ciclycity of the trace has been used in writing the second equality. We conclude that the effective Lagrangian takes the form

L(0)

χPT=C Tr(∆µ∆µ) = C Tr ∂µU†∂µU. (2.129)

There are 2 free parameters that need to be fixed: C and α in the definition of U . They can be determined by imposing a canonical normalization for the kinetic terms and identifying the pion decay constant fπ ' 130.4 MeV in this framework. Expanding U in powers of the

fields U (~π)_{' 1 + i}π a_τa α (2.130) C Tr∂µU†∂µU ' C α2 Tr τ a_τb_∂ µπa∂µπb = 2C α2∂µπ a_∂µ_πa _(2.131)

we obtain the condition

2C α2 =

1

2. (2.132)

Under an infinitesimal axial transformation

U −−−−→ AUA ' (1 + iωSU(L)A a

τa/2)U (1 + iωaτa/2)' U + iωa

/2{τa

, U}. (2.133) The corresponding Noether’s currents are

Aµ a =− δLef f δ(∂µU )_ij δaUij − δLef f δ(∂µU†)_ij δaUij† =_{−iC/2 Tr ∂}µU†{τa_{, U}_}_{+ h.c} =_{−iC/2 Tr ∂}µU†U τa₊_∂µ_U†_τa_U_{+ h.c} =_{−iC/2 Tr τ}a∂µU†, U + h.c. (2.134) Since ∂µ_(U†_{U ) = 0, we find} ∂µ U†, U = −∂µ U, U† (2.135) so that using it in eq. (2.134), the leading order of the axial currents reads

Aµ

a =−iC Tr τ

a_∂µ_U†_{, U}_{' −iC Tr τ}a_−i(∂µ_πb_)τb_,_{1 = −}4C

α ∂

µ_πa_. _(2.136)

The matrix elements of Aa

µ between a one-pion particle state and the vacuum is related to

fπ Ω Aa µ πb_{(p) =} 4C α iδ ab_p µ. (2.137)

Indeed, in the weak decay of the charged pion π− → µ− _{+ ¯}_{ν, we define the pion decay}

constant as

Ω Aµ

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where Aµ _{is the axial hadronic currents involved in the V-A weak interaction. Considering}

the case L = 2, in which Ψ = (u, d)T

Aµ_{= ¯}_uγµ_γ5_{d = ¯}_Ψγµ_γ50 1 0 0 Ψ = ¯Ψγµ_γ5(τ1+iτ2) 2 Ψ≡ A µ 1 +iA µ 2. (2.139)

In terms of the neutral components πa_{, the negative pion state is written as}

π− = |

π1_{i − i |π}2_i

√

2 (2.140)

and the matching of the amplitudes leads to 4C

α =

fπ

√

2 ≡ Fπ ' 92 MeV. (2.141)

Finally, from eqs. (2.132) and (2.141), the effective Lagrangian reads

L(0) χP T = F2 π 4 Tr ∂µU †_∂µ_U_{+ O p}4 with U (~π) = exp i~π· ~τ Fπ . (2.142)

Chiral pertubation theory The Lagrangian mass term

− Lmass= ¯ΨLMΨR+ ¯ΨRM†ΨL (2.143)

explicitly breaks the chiral symmetry G0_{. If we replace the constant mass matrix with a}

dynamical field (spurion) that under the G0 action transforms as M → M0 ₌_V

LMVR† (2.144)

the symmetry is restored. We want the pertubation to the effective chiral Lagrangian, induced by the massive quarks, to share the spurionic property of the mass term, fixing the form of the explicit breaking to

LχPT=L_χPT(0) +δLmass (2.145) with δLmass = F2 π 2 Tr BMU + B ∗_M†_U†_. _(2.146)

Since the pions are pseudoscalars, U transforms under parity as

U (x0_{, ~x)} Pˆ

−→ U†(x0_,

−~x) (2.147)

so that we take real B to avoid parity violation

δLmass =

F2 π

2 B Tr MU + M

†_U†_. _(2.148)

In the basis with a real and diagonal mass matrix 7

hΩ| ¯ΨfΨf|Ωi = − hΩ|

δLQCD

δmf |Ωi .

(2.149)

7_{Summation convention is suspend in the next expressions.}

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The effective chiral Lagrangian must reproduce the correlation functions of the full theory, so we require hΩ|δLQCD δmf |Ωi ≡ hΩ| δLχPT δmf |Ωi . (2.150)

Taking into account that the vacuum expectation value of U (~π) is equal to the identity (because for the physical pion fields _hΩ|πa_{|Ωi = 0), we are led to the following relation}

between the value of B and the chiral condensate

hΩ| ¯ΨfΨf|Ωi = −Fπ2B, (2.151)

accessible through numerical simulations.

2.2.3 The U(1)

_A

problem

The absence of the hadronic parity doubling historically led to believe that the U(1)A

sym-metry underwent SSB, adding a pNGB in the spectrum. Moreover, following a similar approach to the one adopted in section 2.2.2, it can be shown that

hΩ|h ˆQA, ¯Ψγ5Ψ

i

|Ωi = −2 hΩ| ¯ΨΨ_|Ωi (2.152)

so the presence of the chiral condensate undermines the quantum invariance under U(1)A

of the QCD vacuum.

This picture immediately ran into troubles. Considering the chiral limit for the up and down quarks (L = 2), no signs of a light meson, with a mass ∼ mπ, were present. The

lightest isospin singlet meson known is the η, but its mass is significantly larger than the pion mass.

The effective Lagrangian approach of the last section teaches us that η has to regarded as the isospin singlet in the pNGB octet of the broken SU(L)A, which is heavier than the

pions due to the presence of the heavier strange quark. So assuming the U(1)A SSB in the

L = 3 case, the obvious candidate to be the SU (3)V singlet is theη0 meson, that once again

is much heavier than the Kaons and η.

This qualitative picture can be made more quantitative considering an effective La-grangian for the Goldstone modes in the presence of the breaking pattern

SU(L)V ⊗ SU(L)A⊗ U(1)A→ SU(L)V (2.153)

constructed following the very same steps of the previous section. As shown by Weinberg [2], this leads to an iso-singlet light meson _{|li with a mass}

Ml<

√

3mπ (2.154)

that is not observed.

The resolution of this puzzle relies on the presence of the quantum anomaly in eq. (2.104)

A(x)U (1)A =

g2 sL

16π2∂µK

µ _(2.155)

whose effect is non trivial because of the existence of the instanton field configurations. It is tempting to argue that since

∂µJ µ U(1)A+ g2 sL 16π2∂µK µ A =D∂µK˜µ E A= 0 (2.156)

we can apply the Goldstone theorem to the conserved current ˜Kµ_{. This is technically true,}

but the resulting Goldstone mode will result in un-physical excitation, because the conserved current ˜Kµ _{is not gauge invariant (because} _Kµ _{is not).}

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Chapter 3 Instantons and the strong CP

Problem

3.1 Classical instantons

Let us consider the Euclidean version of the pure gauge action for a non-Abelian SU(Nc)

gauge theory SE[A] = 1 4 Z d4_x EGaEµνGaEµν (3.1) where Ga

E0i = −iGa0i and GEija = Gaij, evaluated at the Euclidean space time point xEµ,

defined after perfoming a π/2 rotation in the complex plane of Minkowski time

xµ

→ ˜xµ_{= (˜}_x0_{, ˜}_xi_{) = (x}0_eiπ/2_{, x}i₎

≡ (−ixE4, xEi). (3.2)

We drop the E index for simplicity, but it’s understood that from now on we are working in Euclidean spacetime. A field configuration contributes to the path integral if the action is finite, therefore the field tensor Gµν must vanish in the limit |x| → ∞ faster than 1/|x|2.

This can certainly be accomplished if the fieldsAa

µ → 0 sufficiently fast, but it’s also satisfied

if the connection approaches a pure gauge configuration

igsAaµ(x)T

a |x|→∞

−−−−→ ∂µU (ˆx)U−1(ˆx) (3.3)

where the gauge group elements are direction dependent. Note that the asymptotic config-uration is invariant if we replace U (ˆx)→ U(ˆx)U0, where U0 is a fixed SU (Nc) element. We

can choose U0 such that in any desired direction ˆx0, U ( ˆx0) = 1. This defines a mapping

from points ˆx of the S3 sphere S3 ={x ∈ R4| |x| = 1} to the group manifold

S3 → SU(Nc) (3.4)

with the point ˆx0 mapped into the unit element of the group1. A 3-sphere can be mapped

to the interior of a 3-cube I3 = [0, 1]3, with the boundary identified as a single point,

conveniently chosen to be ˆx0. The mapping in eq. (3.4) is therefore equivalent to

I3 → SU(Nc) (3.5)

1_{For the logic of the discussion, the base point doesn’t need to be the identity, but it could be any fixed}

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3.1. Classical instantons Chapter 3

with the boundary of I3 mapped to the identity of SU(Nc). Two different mappings

U1(z1, z2, z3), U1(z1, z2, z3) are said to be homotopically equivalent if it exists a continuous

function f (t, z1, z2, z3), with t∈ [0, 1] such that

f (0, z1, z2, z3) =U1(z1, z2, z3) f (1, z1, z2, z3) = U2(z1, z2, z3) (3.6)

f (t, p) =1 for every p in the boundary. (3.7) This is an equivalence relation, that can be employed to divide the set of all mappings into equivalence classes. For every element of the quotient space, i.e an equivalence class c, we can choose a representative U (z1, z2, z3;c) and define the product c1× c2 as the equivalence

class that contains the mapping

U (z1, z2, z3;c1× c2) =

(

U (2z1, z2, z3;c1) z1 ≤ 1/2

U (2z1− 1, z2, z3;c2) z1 ≥ 1/2

. (3.8)

It can be shown that this product gives to the quotient space, known as third homotopy group Π3( SU(Nc)), a group structure.

The perk of identifying topological non-equivalent configurations is that if we find a local minimum of the action in an certain equivalence class, we know that a finite perturbation cannot change the topology type, hence we find a local minimum for configurations in all equivalence classes.

3.1.1 The Cartan-Maurer integral

It is often convenient to characterize topological non-equivalent configurations in terms of topologically invariant quantities that distinguish from one another. Let us consider a map-ping from ad = 2n+1 dimensional manifold M , with local coordinates θ1, . . . , θd, to elements

U (θ1, . . . , θd) in a representation of a Lie group G. We define the Cartan Maurer integral as

C[U] = i1...id Z dd_{θ Tr} ∂U (θ) ∂θi1 U−1(θ) . . .∂U (θ) ∂θid U−1(θ) . (3.9)

This is independent of the choice of local coordinates2_{and it’s invariant under a small change} of U → U + δU, indeed δC[U] = di1...id Z ddθ Tr ∂U (θ) ∂θi1 U−1(θ) . . . δ ∂U (θ) ∂θid U−1(θ) . (3.10)

having used that we can push every δ∂U (θ)_∂θ

ij U

−1_(θ) _{to the right using the cyclic property}

of the trace and adjusting the indices as written in eq. (3.10) is an even permutation of the indices (since d_{− 1 is even).}

δ ∂U (θ) ∂θid U−1(θ) = ∂(δU (θ)) ∂θid U−1(θ)−∂U (θ) ∂θid U−1(θ)δU (θ)U−1(θ) (3.11) =U (θ) ∂ ∂θid U−1(θ)δU (θ) U−1_(θ) _(3.12)

2_{Considering a change of chart θ}0

i(θ) and using the chain rule for the derivatives, we find that the

con-traction with the tensor amount to multiply the trace for the Jacobian of the change of variables.

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When we partial integrate we only act on the remainingd− 1 U−1_{(θ) terms, because, thanks}

to the anti-symmetry of the tensor, second order partial derivatives vanish. The resulting d− 1 = 2n terms come in pairs of opposite sign and their sum is hence zero. This shows that every element of an equivalence class c, that can be continuously deformed in one another, shares the same value of _{C[c]. From the definition of multiplication between elements of the} homotopy group, it is straightforward to see that

C[c1 × c2] =C[c1] +C[c2]. (3.13)

This tells us that if we find a non zero Cartan-Maurer invariant _{C[c] for a given equivalence} class c, we find inequivalent classes cn _{for every} _{n integer with} _C[cn_{] =}_n_{C[c], which means}

that Z ⊂ Π3(G).3

In our case of interest Π3(SU (Nc)) = Z, so this procedure completely characterizes the

homotopy group. Let us discuss some general features that allow us to compute the invariant more easily. If we consider two elements of the Lie group U (θ), U (ψ) their product is

U (θ)U (ψ) = U (θ0(θ, ψ)) (3.14)

given that U is a representation. Differentiating with respect of θ0

j, keeping ψ fixed ∂U (θ) ∂θi ∂θi ∂θ0 j U (ψ) = ∂U (θ 0₎ ∂θ0 j (3.15)

and multiplying on the right of both sides by the inverse of U (θ0_{), we get}

∂θi ∂θ0 j ∂U (θ) ∂θi U−1(θ) = ∂U (θ0) ∂θ0 j U−1(θ0). (3.16)

Employing the previous identity in the definition of the Cartan-Maurer invariant

C = i1...id Z dd_θ0_Tr ∂U (θ0) ∂θ0 i1 U−1(θ0). . .∂U (θ0) ∂θ0 id U−1(θ0) (3.17) =i1...id_Tr ∂U (θ) ∂θi1 U−1(θ) . . .∂U (θ) ∂θid U−1(θ) Z dd_θ0_det ∂θ ∂θ0 (3.18)

where θ and θ0 are regarded as independent variables considering that the element ψ is arbitrary. We define the metric

γij(θ0) =− 1 2Tr ∂U (θ0₎ ∂θ0 i U−1(θ0)∂U (θ 0₎ ∂θ0 j U−1(θ0) (3.19)

and observe that using eq. (3.16), the following holds

γij(θ0) = ∂θk ∂θ0 i ∂θl ∂θ0 j γkl(θ). (3.20)

Taking the determinant of both side

det(γ(θ0)) = det ∂θ ∂θ0

2

det(γ(θ)) (3.21)

3_{Substituting U}

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3.1. Classical instantons Chapter 3 we find C = i1...id_Tr ∂U (θ) ∂θi1 U−1(θ) . . .∂U (θ) ∂θid U−1(θ) s 1 det(γ(θ)) Z ddθ0pdet(γ(θ0_)). _(3.22)

Let us employ what we’ve found to calculate the Cartan-Maurer integral for a mapping S3 → SU(2), with an element in the defining representation of SU(2).

Bott [23] has shown that every mapping S3 → SU(Nc) can be continuously deformed to

a mapping S3 → SU(2) where SU(2) ⊂ SU(Nc) is the subgroup that acts on the first two

components of the fundamental representation of SU(Nc). Since the Cartan-Maurer integral

only depends on the equivalence class, we need to compute it for topologically non-equivalent configurations of the SU(2) subgroup.

An element of SU(2) in the defining representation can be written as

U (θ) = θ0× 1 + 2i~t · ~θ (3.23)

where ~t = ~σ/2 and θ2

0 = 1− |~θ|2. Employing the algebraic properties of the pauli matrices

titj =δij/4 + iijktk/2, it can be shown that the following identities hold

Tr(titj) =δij/2 Tr(titjtk) = i 4ijk (3.24) Tr(titjtktl) = 1 8(δijδkl+δilδjk − δikδjl) (3.25) γij(θ) =− 1 2Tr θi θ0 + 2iti (θ0− 2itkθk) θj θ0 − 2it j (θ0+ 2itkθk) (3.26) =δij + θiθj θ2 0 =δij + θiθj 1_{− |~θ|}2. (3.27)

The determinant of the metric is hence equal to

p

det(γ(θ)) = q 1 1_{− |~θ|}2

. (3.28)

Substituting in eq. (3.22) and taking the limit θ → 0, the invariant reads C = −8iijk_Tr(t itjtk) Z d3_θ0 q 1_{− |~θ}0_|2 . (3.29)

The integral is easily computed in polar coordinates, multiplying by a factor 2 to account for the two possible values of θ0

0 for fixed |~θ0|2

C[c] = 24π2_. _(3.30)

This holds for every mapping homotopic to U (θ) in the equivalence class c and we know that topologically non-equivalent configurations for every integer n _C[cn_{] = 24nπ}2 _exist.4

4_{A configuration with a negative Cartan Maurer integral}

−24π2_{is found considering the anti-fundamental}

of SU(2).

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3.1.2 Bogomol’ny inequality

We rewrite the pure gauge action as

S[A] = 1 2 Z d4_{x Tr(G} µνGµν) = 1 4 Z d4_{x Tr}h_(G µν∓ ˜Gµν)2 i ±1₂ Z d4_{x Tr}_G µνG˜µν (3.31)

where ˜Gµν = 1/2µνρσGρσ is the dual tensor. We’ve used that, in space-time with the metric

δµν,

µνρσµναβ = 2(δραδσβ− δρβδσα) (3.32)

to obtain that ˜GµνG˜µν =GµνGµν. From eq. (3.31) we identify a lower bound for the action

S[A]_≥ 1 2 Z d4_{x Tr}_G µνG˜µν (3.33)

which is reached when the curvature is dual G = ˜G or anti-dual G =− ˜G. Observe that the RHS of the above inequality is proportional to the divergence of the Chern Simons current, or more precisely the Euclidean version of it:

1 2Tr GµνG˜µν = ∂µKµ 4 whereKµ = 4µνρσTr Aν∂ρAσ− 2igs 3 AνAρAσ . (3.34)

Exploiting the antisymmetry of , the Chern-Simmons current is re-written as

Kµ= 4µνρσTr Aν∂ρAσ− 2igs 3 A ν AρAσ = 4µνρσTr AνGρσ 2 − igs 3 AνAρAσ . (3.35)

and the integral is evaluated using Gauss-Green theorem 1 2 Z d4x TrGµνG˜µν = lim r→∞ Z S₃r dσµµνρσT r(AνGρσ/2− igs 3 AνAρAσ) (3.36) =₋igs 3 rlim→∞ Z Sr 3 dσµµνρσT r(AνAρAσ) (3.37)

whereσµis a 4-vector normal to the hypersurface and with modulus equal to the infinitesimal

surface element on the 3-sphere. In the second row we have neglected theAνGρσterm because

the curvature must vanish in the limit r _{→ ∞. This can be achieved, as we have previously} discussed, if the connection approaches a pure gauge configuration at infinity

igsAµ(x) |x|→∞

−−−−→ ∂µU (ˆx)U−1(ˆx). (3.38)

We substitute the asymptotic behaviour inside the integral 1 2 Z d4x TrGµνG˜µν = 1 3g2 s lim r→∞ Z Sr 3 dσµµνρσT r((∂νU )U−1(∂ρU )U−1(∂σU )U−1). (3.39)

The infinitesimal surface element on the sphere in cartesian coordinates is d4_xδ(_{|x| − r) and}

the normal direction is given by the 4-versor ˆxµ_{. Defining}_θ

4 ≡ |x|, the latter is expressed as

ˆ xµ ₌_∂

µθ4. If θi, with i = 1, 2, 3, identifies a point of S3

1 2 Z d4x TrGµνG˜µν = 1 3g2 s lim r→∞ Z S₃r d4xδ(θ4− r)µνρσ ∂θ4 ∂xµ ∂θi ∂xν ∂θj ∂xρ ∂θk ∂xσ× Tr ∂U (θ) ∂θi U−1(θ)∂U (θ) ∂θj U−1(θ)∂U (θ) ∂θk U−1(θ) (3.40)

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3.2. The Strong CP problem Chapter 3

where we’ve used that U is direction dependent. Noting that

µνρσ ∂θ4 ∂xµ ∂θi ∂xν ∂θj ∂xρ ∂θk ∂xσ =4ijkdet ∂θ ∂x (3.41)

we recognize the Jacobian of the change of variables_{xµ_{} → {θ}4, θi}, finding that the integral

is proportional to the Cartan-Maurer invariant of the asymptotic configuration 1 2 Z d4x TrGµνG˜µν = = 1 3g2 s ijk Z S3 d3_{θ Tr} ∂U (θ) ∂θi U−1(θ)∂U (θ) ∂θj U−1(θ)∂U (θ) ∂θk U−1(θ) = 1 3g2 s C[U]. (3.42) Using the results of the previous section _{C = 24π}2ν, where ν is an integer known as the Chern-Pontryagrin index or winding number, we conclude that

Z d4x TrGµνG˜µν = 16π 2_ν g2 s (3.43)

which yields the following lower bound for the action

S[A]≥ 8π

2_|ν|

g2 s

. (3.44)

Belavin et al. showed in [1] that the configurations

igsAµ =

(r− r0)2

(r_{− r}0)2+ρ2

∂µU1(ˆx)U1−1(ˆx) (3.45)

have dual curvature (G = ˜G), hence are solutions for the classical equation of motion. U1

belongs to the SU(2) subgroup of SU(Nc) that acts on the first two components of the

defining representation U1(ˆx) = x4+i2~t· ~x r (3.46)

for which we computed theν = 1 winding number. The parameters r0 andρ are respectively

known as the instanton center and size.

Solutions with anyν _{∈ Z exist, and can be explicitly constructed considering |ν|} configu-rations as in eq. (3.45) (or with U1−1 if ν is negative) with centers r0i sufficiently distant that

non linearities of the field equation become negligible.

3.2 The Strong CP problem

3.2.1 Topological term

We can discuss how the instanton solutions enter in the path integral considering a generic weight factor f (ν) for configurations with winding number ν. If we have a local operator_O, its vacuum expectation value is in general

hOi = X ν f (ν) Z [DΦν]O exp − Z d4_x _{L [Φ]} X ν f (ν) Z [DΦν] exp − Z d4_x _{L [Φ]} (3.47) 31

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3.2. The Strong CP problem Chapter 3

where Φ represents all fields of the theory and the measure [DΦν] is restricted to gauge fields

configurations with winding number ν. Suppose that we can decompose R4 _{= Ω}

1 ∪ Ω2 in

two non overlapping large volumes and that the operator _{O has support contained in Ω}1.

Approximately, we can divide the path integral in configurations with winding numberν1 in

Ω1 and ν2 in Ω2, such thatν = ν1+ν2.5

hOi = X ν1,ν2 f (ν1+ν2) Z [DΦν1]O exp − Z Ω1 d4_x _{L [Φ]} Z [DΦν2] exp − Z Ω2 d4_x _{L [Φ]} X ν1,ν2 f (ν1+ν2) Z [DΦν1] exp − Z Ω1 d4_x _{L [Φ]} Z [DΦν2] exp − Z Ω2 d4_x _{L [Φ]} . (3.48) The cluster decomposition principle (Chap. 4 of [24], Chap. 23.6 of [25]) states that the expectation value should be the same if we omit the volume Ω2, but this is true only if the

weight factor satisfies

f (ν1+ν2) =f (ν1)f (ν2). (3.49)

This fixes the form of f (ν) to

f (ν) = exp(iνθ). (3.50)

If the Poyntragin index is expressed in terms of gauge fields, the inclusion of the instanton in the path integral effectively amount to adding to the Euclidean Lagrangian density

LE → LE − iθ g2 s 32π2G a µνG˜ a µν. (3.51)

Going back to Minkowski space-time, substituting d4_x

E → id4x and GE34 → −iG30, we get

LQCD → LQCD +θ g2 s 32π2G a µνG˜ aµν_. _(3.52)

Starting with the QCD Lagrangian with no topological term, the solution of the U(1)A

prob-lem furnished by existence of instantons requires us to include the θ term in the Lagrangian density.

ForNf flavours QCD, the partition function then reads

ZQCD = Z [DA][D ¯ΨDΨ] exp i Z d4_xL0 QCD[A, ¯Ψ, Ψ] +LM+θQ(x) (3.53)

where L_QCD0 is the QCD Lagrangian density with massless quarks and no topological term, while

− LM = ¯ΨLMΨR+ ¯ΨRM†ΨL. (3.54)

Let us perform the change of variables in flavour space

ΨL→ ULΨL ΨR→ URΨR (3.55)

with UL, UR∈ U(Nf). Defining the determinants

detUL,R=eiϕL,R (3.56)

5_{This is approximately true if the volumes are large compared to the instanton size, since the integral of}

the Poyntragin density g2 s/32π

2_Ga µνG˜

a

Axion from high-quality accidental Peccei-Quinn symmetry

Universit`

a di Pisa

Dipartimento di Fisica “Enrico Fermi”

Tesi di Laurea Magistrale in Fisica

Axion from high-quality accidental

Peccei-Quinn symmetry

Candidato: Marco Ardu

Relatori : Prof. Alessandro Strumia, Dott. Daniele Teresi

Abstract

Ringraziamenti

Contents

Chapter 1

Introduction

Chapter 2

The U(1)

A

problem

2.1

Quantum Chromodynamics

2.1.1

Non-Abelian SU(

N

) gauge theory

Parallel Transport

Pure gauge lagrangian

2.1.2

Discrete symmetries

2.1.3

Perturbative QCD and running coupling

2.2

Continuous chiral symmetries

2.2.1

Quantum Symmetries and Anomalies

2.2.2

Spontaneous symmetry breaking and the effective chiral

la-grangian

2.2.3

The U(1)

problem

Chapter 3

Instantons and the strong CP

Problem

3.1

Classical instantons

3.1.1

The Cartan-Maurer integral

3.1.2

Bogomol’ny inequality

3.2

The Strong CP problem

3.2.1

Topological term

_A