Low-energy processes with three pions in the nal state

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Low-energy processes with three pions in the nal

state

Bruno Strandberg

Master Thesis

Uppsala University

Department of Physics and Astronomy

Division of Nuclear Physics

Supervisor: Stefan Leupold

June 12, 2012

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Introduction

The theory that describes modern particle physics is called the Standard Model. In the Standard Model, quantum eld theories are used to describe the interactions between particles. To a large extent quantum eld theories have proven to be very successful tools in predicting and describing the behavior of particle interactions, for instance Quantum Electrodynamics (QED, the quantized theory of electromagnetism) and the Electroweak Theory (a theory unifying electromagnetism and weak interaction) have provided results in excellent agreement with the experiments. The quantum eld theory describing the strong interaction is called Quantum Chromodynamics (QCD). In these theories elementary particles (quarks and leptons) and the force mediating particles act as relevant degrees of freedom.

When it comes to calculating the observables of physical processes, e.g. decay widths and cross-sections of the particle reactions, the analytical results obtained from the quantum eld theories are often innite series. To solve this problem, one needs approximation methods. What one usually does in QED is to expand the result in powers of the coupling constant that determines the strength of the interaction; in that way a contribution at a higher order in the coupling constant becomes a correction to the lower order part and at some point one can cut o the expansion. This sort of an expansion can only be applied when the coupling constant is small, which is the case for QED. In QCD, however, the coupling constant strongly depends on the energy region where the process under investigation is taking place and the coupling constant is not necessarily small. A perturbative expansion is possible in the high-energy region, but not in the low-energy regime, where the reactions considered in this thesis take place.

In order to be able to describe QCD processes in the low-energy regime, eective

eld theories are introduced. These are approximate theories that are expected to hold only in a certain energy interval. In eective eld theories only relevant degrees of freedom are considered in the Lagrangian (i.e. only the physical states that can be excited in the specied energy region) and when approximation methods are applied, the expansion is done in powers of energies and momenta, rather than in powers of coupling constants. If the contributions at higher powers of momenta act as corrections to the parts with lower powers of momenta (a desirable condition), one has a systematic way to make the results more precise.

An eective eld theory of QCD where only the lightest pseudoscalar mesons are relevant is called Chiral Perturbation Theory (ChPT) [23,5,6] (a modern and extensive overview is provided in [21]). This holds at energies signicantly below 1 GeV and has been rather successful in describing the reactions between the lightest pseudoscalar states. In [14] an eective eld theory has been proposed where both the pseudoscalar mesons and the lightest vector mesons act as relevant degrees of freedom.

For various reasons one of the most precisely measured reaction is e⁺e⁻ → hadrons [17,8]. Here e⁺ and e⁻ denote the positron and the electron, respectively. The formal-

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ism developed in [14] can be tested against the available data. This has been carried out in [9] for the process e⁺e⁻ → 2πin the low-energy region up to ≈ 1 GeV. In the present thesis the reaction e⁺e⁻ → 3πwill be studied. An important contribution in the latter reaction is the excitation of an intermediate omega meson, therefore also the process ω → 3πwill be studied in detail. This decay has already been calculated at tree level in [11]. The new aspect of the present work is the incorporation of pion-pion rescattering along the lines of [9]. After the description of the process ω → 3π, calculations for the reaction e⁺e⁻→ 3π are performed. This has not been carried out in the framework of [14] before. The impact of rescattering of two nal-state pions on the cross-section of the second reaction is also studied. Calculated results of both processes are compared with experimental data.

Section 1 of this thesis is devoted to give a theoretical background of QCD and the eective eld theories that are used to perform the calculations in section 2. In subsection 1.1 the QCD Lagrangian is presented and subsection 1.2 deals with the symmetries of the QCD Lagrangian in the chiral limit. Subsection 1.3 is dedicated to symmetry breaking and to its implications on QCD, leading to the existence of Goldstone bosons. In subsection 1.4 the eective eld theories, which are relevant for the calculations, are presented. In section2 the calculations for the reactions ω → 3π and e⁺e⁻→ 3π are performed and comparison with experimental data is provided. In subsection2.1the formalism of [9] is introduced to incorporate rescattering. The decay width of the reaction ω → 3π with the rescattering of two nal-state pions is calculated in subsection2.2. In subsection2.3 the calculations concerning the reaction e⁺e⁻→ 3π are presented and comparison with experimental data is provided. A summary and an outlook follow the two main sections of the thesis. Appendices A-C address technical details, appendixD provides a popular scientic summary of the thesis.

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1 Theoretical background

The starting point of the theoretical background is the QCD Lagrangian, presented in subsection 1.1. Then, in subsection 1.2, the symmetries of the system are studied in the chiral limit, where the three heaviest quarks are ignored and the masses of the three lightest quarks are taken to be zero. By applying Noether's theorem, this leads to seventeen conserved currents and to corresponding charge operators. It is discussed how these charge operators, in the chiral limit, lead to energy-degenerate multiplets. After establishing symmetry properties in the chiral limit, symmetry breaking is studied in subsection1.3. Two general examples will be provided to demonstrate how spontaneous symmetry breaking produces massless Goldstone bosons of the theory and how through explicit symmetry breaking these bosons turn massive. Then, in the light of the two examples, symmetry breaking is applied to QCD and a discussion is presented leading to a pseudoscalar meson multiplet and other avor multiplets. In subsection 1.4 the concepts of an eective eld theory are introduced. The eective eld theory called Chiral Perturbation Theory is constructed. Finally, an extension to an eective eld theory is briey presented, where vector mesons along with the Goldstone bosons act as relevant degrees of freedom. The theoretical background presented in this thesis is mostly based on [21]. Other sources will be cited explicitly.

1.1 The QCD Lagrangian

Quantum Chromodynamics is the gauge theory of the strong interaction with color SU(3) as the gauge group. This means that the corresponding Lagrangian is invariant under a local SU(3) transformation (also called a gauge transformation) in color space.

The Lagrangian reads

L_QCD =X

f

¯

q_f(iγ_µD^µ− m_f) q_f − 1

4G_µν,aG_a^µν, (1.1) where f denotes the avor (the six avors are up, down, strange, charm, bottom and top), qf is the color triplet of a quark avor (the three colors are red, green, blue), Dµ

is the covariant derivative, mf is the quark-mass matrix and Gµν,a is the eld strength tensor. More explicitly, the quark color triplets are

q_f =





q_f,red q_f,green

q_f,blue



 (1.2)

and the covariant derivative is built as D_µ= ∂_µ− ig

8

X

a=1

λ^C_a

2 A_µ,a, (1.3)

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where g is the universal QCD coupling constant. The presence of 8 matrices λ^C_a and gauge potentials Aµ,a in (1.3) can be understood after considering some properties of the SU(3) group. The superscript C on λ stresses the fact that the matrices act in color space. All elements of SU(3) acting on the fundamental representation can be expressed as an exponential

U (ϑ) = exp −i

8

X

a=1

ϑ_aλ_a 2

!

, (1.4)

where ϑa are real numbers and λa are eight linearly independent matrices, called the Gell-Mann matrices. From here on Einstein's summation convention will be used to shorten the notation. The Gell-Mann matrices read

λ₁ =





0 1 0 1 0 0 0 0 0



, λ₂ =





0 −i 0

i 0 0

0 0 0



, λ₃ =





1 0 0

0 −1 0

0 0 0



,

λ₄ =





0 0 1 0 0 0 1 0 0



, λ₅ =





0 0 −i 0 0 0

i 0 0



, λ₆ =





0 0 0 0 0 1 0 1 0



,

λ₇ =





0 0 0 0 0 −i 0 i 0



, λ₈ = 1

√3





1 0 0 0 1 0 0 0 −2



 (1.5)

and satisfy the commutation relation

λ_a 2 ,λ_b

2

= if_abcλ_c

2 . (1.6)

The additional properties of the Gell-Mann matrices are

λ_a = λ^†_a, (1.7)

Tr (λ_aλ_b) = 2δ_ab, (1.8)

Tr (λ_a) = 0. (1.9)

The non-vanishing values of the structure constants fabc(up to index permutations) are f₁₂₃ = 1, f₁₄₇= f₂₄₆ = f₂₅₇ = f₃₄₅ = 1

2, f₁₅₆ = f₃₆₇ = −1

2, f₄₅₈ = f₆₇₈ =

√3

2 . (1.10)

The structure constant fabc is completely antisymmetric in exchanging any pair of its indices. An independent gauge potential Aµ,a corresponds to each of the eight

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linearly independent matrices λa of the group SU(3). Finally, before turning to the transformation properties of the Lagrangian, the generalized eld strength tensor is presented:

G_µν,a = ∂_µA_ν,a− ∂_νA_µ,a+ gf_abcA_µ,bA_ν,c. (1.11) As mentioned above, the QCD Lagrangian (1.1) is invariant under the SU(3) gauge transformation in color space. This implies a transformation behavior of the gauge potential in the form

λ^C_a

2 A_µ,a → U (ϑ (x))λ^C_a

2 A_µ,aU (ϑ (x))^†− i

g [∂_µU (ϑ (x))] U (ϑ (x))^†. (1.12) Note that in equation (1.12) the transformation parameters ϑ = ϑ (x) depend on x, which means the transformation is local, i.e. it is a gauge transformation. To see how (1.12) guarantees the invariance of the Lagrangian, consider the calculation (below, the xdependence of U and Aµ,a is implicit)

U D_µU^† = U ∂_µ− ig

8

X

a=1

λ^C_a 2 A_µ,a

! U^†

= U ∂µU^† + UU^†∂µ− ig

8

X

a=1

U λ^C_a 2 Aµ,a

U^†

= ∂_µ+ ∂_µ U U^† − (∂_µU ) U^†− ig

8

X

a=1

U λ^C_a 2 A_µ,a

U^†

= ∂_µ− ig

8

X

a=1

U λ^C_a 2 A_µ,a

U^†− i

g (∂_µU ) U^†

!

= ∂_µ− ig

8

X

a=1

λ^C_a

2 A⁰_µ,a = D⁰_µ. (1.13)

The primes in the last line denote transformed objects. So if the gauge potential transforms as (1.12), then the covariant derivative transforms as

D_µ → U D_µU^†. (1.14)

The quark elds transform according to

q → U q, q → ¯¯ qU^† (1.15)

and thus the rst term of the Lagrangian (1.1) is indeed invariant. The invariance of the generalized eld strength tensor term of the Lagrangian is ensured by observing that

G_µν ≡ λ^C_a

2 G_µν,a → U (ϑ (x)) G_µνU (ϑ (x))^† (1.16)

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and then rewriting the gluonic part of the Lagrangian, with the help of (1.8), to the form

− 1

2Tr (G_µνG^µν) . (1.17)

Due to the cyclic property of the trace, term (1.17) of the Lagrangian is invariant under (1.16) [21,10].

1.2 The chiral limit

The masses of the six quarks are approximately [17]





m_u ≈ (1.7 − 3.1) MeV md≈ (4.1 − 5.7) MeV m_s≈ (80 − 130) MeV



 1 GeV <





m_c≈ (1.18 − 1.34) GeV mb ≈ (4.13 − 4.37) GeV m_t ≈ (172.9 ± 1.5) GeV



.

One can see that the masses of the three lightest quarks are very small compared to the masses of the three heaviest quarks. In addition, the up, down and strange quarks are also light compared to the masses of the lightest hadronic states (e.g. mρ= 0.77 GeV) which are not Goldstone bosons of the theory (a discussion concerning the Goldstone bosons and spontaneous symmetry breaking is presented in subsection 1.3). As the goal is to develop an eective eld theory of QCD in an energy region below 1 GeV, symmetry properties of the QCD Lagrangian (1.1) are studied in the limit where the three heaviest quarks are ignored and the masses of the three lightest quarks are taken to be zero, mu = m_d = m_s= 0. This approximation is called the chiral limit. Ignoring the three heaviest quarks seems justied because in eective eld theories only the states that can be excited in the energy region should enter the Lagrangian.

In the literature, one can actually nd at least two distinct approaches to study the symmetry properties of QCD. One of the approaches is presented in [16], in which case the symmetry properties are studied in a step-by-step manner. First the transformation under group U(1) is considered; then the approximation mu = m_d is made and the behavior of the Lagrangian under SU(2) is investigated, leading to the isospin symmetry.

Finally one approximates mu = m_d = m_s and arrives at a pseudoscalar meson octet, a vector meson octet and a baryon octet. Although this approach might be more intuitive, the connection to Goldstone's theorem and spontaneous symmetry breaking is not so easily established in that case and therefore, in this thesis, the approach presented in [21] is followed.

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1.2.1 The QCD Lagrangian and its symmetry group in the chiral limit The QCD Lagrangian in the chiral limit is

L⁰_QCD = X

l=u,d,s

¯

q_liγ_µD^µq_l−1

4G_µν,aG_a^µν

= X

l=u,d,s

(¯q_R,liγ_µD^µq_R,l+ ¯q_L,liγ_µD^µq_L,l) −1

4G_µν,aG_a^µν, (1.18) where the projection operators PR = ¹₂(1 + γ₅)and PL= ¹₂(1 − γ₅) have been used to split the rst term of the Lagrangian into right-handed and left-handed counterparts.

The projection operators satisfy

PR+ PL = 1, (1.19)

P_R,L² = PR,L, (1.20)

P_RP_L = P_LP_R= 0. (1.21)

As the covariant derivative is avor independent (meaning it acts in the same way on u, d and s), the Lagrangian (1.18) is invariant under global avor transformations of the left-handed and the right-handed elds. The nontrivial transformations are



 u_L d_L s_L



 → U_L



 u_L d_L s_L



= exp −i

8

X

a=1

ϑ^L_aλ_a 2

! e^−iϑ^L



 u_L d_L s_L



, (1.22)



 uR

d_R s_R



 → U_R



 uR

d_R s_R



= exp −i

8

X

a=1

ϑ^R_aλ_a 2

! e^−iϑ^R



 uR

d_R s_R



. (1.23) Note that the exponential parts containing the Gell-Mann matrices (now acting in

avor space) are members of SU(3)L and SU(3)R (cf. (1.4)) while the exponents e^−iϑ^L,R ∈ U(1)_L,R, respectively. It is clear that the classical global symmetry group of the Lagrangian (1.18) is

U (3)_R× U (3)_L = U (1)_R× SU (3)_R× U (1)_L× SU (3)_L. (1.24) 1.2.2 Noether's theorem

According to Noether's theorem, a transformation which leaves the Lagrangian invariant leads to conserved currents from which one can extract conserved charges. As the theorem is well known, the details are not covered here and a classical approach can be found in e.g. [16, 7]. Briey, if one has a Lagrangian

L = L (Φ_i(x) , ∂_µΦ_i(x)) (1.25)

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and an innitesimal transformation

Φ_i(x) → Φ_i(x) − i_a(x) F_i^a[Φ_j(x)] , (1.26) where a(x) is a small parameter and Fi^a[Φ_j(x)] are the generators of innitesimal transformations, then one obtains conserved currents and the divergences of currents

J^µ,a = −i ∂L

∂∂_µΦ_iF_i^a= ∂ (δL)

∂∂_µ_a, (1.27)

∂µJ^µ,a = ∂ (δL)

∂_a . (1.28)

For a conserved current ∂µJ^µ,a= 0 and one obtains a conserved charge Q^a(t) =

ˆ

d³xJ₀^a(~x, t) , (1.29) which is a constant of motion. This, so far, has been an approach on a level of classical

eld theory. After quantizing the theory the elds Φi, the conjugate momenta Πi =

∂L/∂ (∂₀Φ_i), the currents J^µ,a(x)and the charges Q^a(t)become operators. It can then be shown that in case of an innitesimal transformation linear in the elds

Φ_i(x) → Φ_i(x) − i_a(x) tâ_ijΦ_j(x) , (1.30) where the generators of the transformation tâij = Tâ satisfy the Lie algebra

T^a, T^b = iCabcT^c, (1.31) the charge operators emerging from Noether's theorem also satisfy the Lie algebra

Q^a(t) , Q^b(t) = iC_abcQ^c(t) . (1.32) A derivation of this relation can be found in [21] in section 2.3.3. The reason why it is presented here is because there are some aspects worthy of being stressed at this point.

First of all, from (1.6) it is clear that the Gell-Mann matrices satisfy the Lie algebra.

If now one rewrites, for example, the SU (3)_R transformation of the Lagrangian (1.18) into an innitesimal form,

SU (3)_R = exp −i

8

X

a=1

ϑ^R_a λ_a 2

!

= 1 − i

8

X

a=1

ϑ^R_aλ_a

2 + O ϑ² , (1.33) where ϑ^Ra acts as a(x)and ^λ₂â acts as tâij in (1.30), then it is clear that the conserved charge operators QâR(t)that emerge from Noether's theorem also satisfy the Lie algebra.

In addition, if the symmetry currents are conserved, the charge operators are time independent and therefore they commute with the Hamiltonian. We will return to this point in subsection1.2.4.

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1.2.3 The symmetry currents of the QCD Lagrangian in the chiral limit The global symmetry currents of the QCD Lagrangian in the chiral limit are most easily obtained when one studies the behavior of the Lagrangian (1.18) under a local, innitesimal form of the transformations (1.22), (1.23). It might seem contradictory to obtain global symmetry currents by investigating the behavior under local transformations, but it can be done by nding an appropriate generating functional for the theory. The method is briey described in subsection 1.4.1 of this thesis. A more detailed review can be found in section 2.4.4 and appendix A of [21]. There exists a dierent approach to obtain the conserved currents that does not require considering local transformations (see e.g. [16,10]). However that approach is less straightforward and therefore here the considerations from [21] are presented.

The aim is to use (1.27) to nd the conserved currents. To that end one should calculate δL. Let us, for the moment, only consider the right-handed part of (1.18).

We should also not worry about the second term of the Lagrangian, given in (1.17), because it transforms trivially under avor transformations. A short calculation gives

δLR = L (q_R⁰ , ¯q_R⁰ , ∂µq_R⁰ , ∂µq¯⁰_R) − L (qR, ¯qR, ∂µqR, ∂µq¯R)

= q¯_RU_R^†iγ_µ∂^µ(U_Rq_R) − ¯q_RU_R^†iγ^µ ig

8

X

a=1

λ^C_a 2 A_µ,a

!

(U_Rq_R)

− q¯_Riγ_µ∂^µq_R− ¯q_Riγ^µ ig

8

X

a=1

λ^C_a 2 A_µ,a

! q_R

!

. (1.34)

The second terms with the gauge elds cancel each other because URacts in avor space whereas the Gell-Mann matrices which multiply the gauge elds act in color space - thus the dierent matrices commute and one can use UR^†U_R= 1. We are left with

δL_R = q¯_RU_R^†iγ_µ(∂^µU_R) q_R+ ¯q_RU_R^†iγ_µU_R(∂^µq_R) − ¯q_Riγ_µ∂^µq_R

= q¯_R

8

X

a=1

∂^µϑ^R_a λ_a

2 + ∂^µϑ^R

!

γ_µq_R. (1.35)

Above, the innitesimality of the transformations was used, which enables one to approximate

U_R^†(∂^µUR) ≈ −i

8

X

a=1

∂^µϑ^R_aλ_a

2 − i∂^µϑ^R

!

. (1.36)

After analogous calculations for the left-handed elds the total δL is

δL⁰_QCD = ¯qR 8

X

a=1

∂^µϑ^R_aλ_a

2 + ∂^µϑ^R

!

γµqR+ ¯qL 8

X

a=1

∂^µϑ^L_aλ_a

2 + ∂^µϑ^L

!

γµqL. (1.37)

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Applying (1.27) to the equation above, where ∂^µϑ^R,L_a play the role of ∂µ_a, results in 16 conserved currents

L^µ,a = q¯_Lγ^µλa

2 q_L, ∂_µL^µ,a = 0, (1.38) R^µ,a = q¯Rγ^µλ_a

2 qR, ∂µR^µ,a = 0. (1.39) The currents (1.38) transform as an (8,1) multiplet under the symmetry group

SU (3)_L× SU (3)_R,

i.e. they transform as an octet with respect to the left-handed transformations and as a singlet with respect to the right-handed transformations. The currents (1.39) transform as a (1,8) multiplet under the same symmetry group. Alternatively to equations (1.38) and (1.39) one can use linear combinations

V^µ,a = R^µ,a+ L^µ,a = ¯qγ^µλ_a

2 q, (1.40)

A^µ,a = R^µ,a− L^µ,a = ¯qγ^µγ₅λ_a

2 q, (1.41)

which transform as vector and axial-vector current densities under parity. In obtaining (1.40) and (1.41) the properties of the projection operators (1.19) - (1.21) have been used. In addition to the 16 currents (1.38) and (1.39) one can also extract a singlet vector current

V^µ= ¯q_Rγ^µq_R+ ¯q_Lγ^µq_L= ¯qγ^µq, ∂_µV^µ= 0, (1.42) which is obtained by taking ϑ^La = ϑ^R_a = 0 in transformations (1.22), (1.23) and using ϑ^L= ϑ^R in the U (1)_L,R parts. There is also a singlet axial-vector current that emerges from choosing ϑ^L= −ϑ^R,

A^µ= ¯q_Rγ^µq_R− ¯q_Lγ^µq_L= ¯qγ^µγ₅q. (1.43) The singlet axial-vector current, however, is only conserved on a classical level and not preserved in the quantized theory. This means that due to quantization, the classical symmetry group (1.24) of the chiral Lagrangian (1.18) reduces to a symmetry group

SU (3)_R× SU (3)_L× U (1)_V= SU (3)_V× SU (3)_A× U (1)_V. (1.44)

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1.2.4 The conserved charges of the QCD Lagrangian in the chiral limit We are now ready to discuss the charge operators that correspond to the currents (1.38), (1.39) and (1.42). They read

Q^a_L(t) = ˆ

d³xq_L^†(~x, t)λ_a

2 q_L(~x, t) , a = 1, ..., 8 , (1.45) Q^a_R(t) =

ˆ

d³xq_R^† (~x, t)λ_a

2 q_R(~x, t) , a = 1, ..., 8 , (1.46) Q_V (t) =

ˆ

d³x

q^†_L(~x, t) q_L(~x, t) + q_R^† (~x, t) q_R(~x, t)

. (1.47)

The commutation relations between the charge operators are

Q^a_L, Q^b_L

= if_abcQ^c_L, (1.48)

Q^a_R, Q^b_R

= if_abcQ^c_R, (1.49)

Q^a_R, Q^b_L

= 0, (1.50)

[Q_V, Q^a_L] = [Q_V, Q^a_R] = 0. (1.51) We now return to the discussion at the end of subsection1.2.2, where it was already mentioned that for conserved symmetry currents the charge operators are time independent and commute with the Hamilton operator. To establish a better understanding what are the consequences of the charge operators and their commutation relations, we turn to ordinary quantum mechanics (see e.g. [20]). Recall that whenever two operators A and B commute, the operators share a common eigenket |a⁰, b⁰i that species the quantum mechanical state. In the ket, a⁰ and b⁰ are eigenvalues of the operators A and B, such that

H |E, C_Vi = E |E, C_Vi , (1.54)

Q_V |E, C_Vi = C_V |E, C_Vi . (1.55) At this point it is appropriate to mention that the conserved charge operator QV leads to baryon number conservation, classifying the hadrons into mesons (baryon number 0) and baryons (baryon number 1). Baryon number is always conserved, which means that the current from which the operator emerged, (1.42), is always conserved. This is not the case for (1.40) and (1.41) - the rst one is realized only approximately and leads

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to avor multiplets, e.g. a vector meson octet, while the second one is spontaneously broken and produces the Goldstone bosons of the theory (see subsection 1.3).

Let us now restrict ourselves to the mesonic sector, meaning we x the eigenvalue of the charge operator QV to be 0. By virtue of equation (1.40), we can use a combination Qâ_V = Qâ_R+ Qâ_L instead of QâR - this is benecial because when the symmetry breaking is discussed in subsection 1.3, the symmetry groups SU (3)_V and SU (3)_A, rather than SU (3)_R and SU (3)_L are studied. As the charge operators Qâ_V commute with H (and Q_V), they can be used to construct energy-degenerate multiplets (neglecting for now the fact that the symmetry current (1.40) is only approximately realized). This is done much in the same way as in ordinary quantum mechanics the angular momentum operators Jx,y,z are used to construct multiplets. Recall that in ordinary quantum mechanics, the eigenvalues of the operators

J² ≡ J_xJ_x+ J_yJ_y+ J_zJ_z, J_z

with the properties

J², J_z

= 0,

J², H

= [J_z, H] = 0

specify the quantum mechanical state |l, mi. For a given eigenvalue l of the operator J², the eigenvalue m of the operator Jz may have a value from the interval [−l, l]. For example, this means that for l = 1, one has a three-fold energy-degenerate multiplet with m = 0, ±1. The combinations

J± ≡ J_x± iJ_y

are the ladder operators that, when acting on the ket |l, mi, increase or decrease the eigenvalue m of the operator Jz. This means that the ladder operators can be used to

navigate between dierent states inside the multiplet. As in the present case we deal with an SU(3), rather than an SU(2) group, the process of constructing the multiplets is more cumbersome. However, the principle is the same - one uses the operators ~Q_V2

, Q³_V and Q⁸V to serve the purpose of J² and Jz and constructs ladder operators from the rest, which enable to navigate between states inside the multiplet.

If the axial-vector symmetry were not broken, the operators QâA= Qâ_R−Qâ_Lcould be used in a similar manner and one would expect to observe parity doublets of multiplets.

This means that e.g. for each vector multiplet there would be an axial-vector multiplet with similar masses. These parity doublets are not observed, which is interpreted as a strong indication that the axial-vector symmetry is spontaneously broken.

As a nal remark for this section - the charge operators Q^aV, Q^a_A and QV should be considered as generators of innitesimal transformations of the Hilbert space associated

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with H_QCD⁰ . Again, one can draw a parallel with the angular momentum operators from ordinary quantum mechanics. Angular momentum operators satisfy the Lie algebra and they are generators of rotational transformations.

1.3 Symmetry breaking

In this section, the concepts of spontaneous symmetry breaking and explicit symmetry breaking are discussed. A symmetry is said to be spontaneously broken if the ground state of the system is not invariant under the full symmetry group of the Lagrangian.

Explicit symmetry breaking is induced by adding a small perturbation to the Lagrangian which is not invariant under the full symmetry group of the Lagrangian. A simple example of spontaneous symmetry breaking is studied in subsection1.3.1and an example of explicit symmetry breaking is presented in subsection 1.3.2. In subsection 1.3.3 the results from subsections 1.3.1 and 1.3.2 are translated to the framework of QCD.

1.3.1 Spontaneous symmetry breaking of a global, continuous symmetry The goal is to give an explicit example how a symmetry of the Lagrangian can be spontaneously broken. We consider a Lagrangian invariant under SO(3):

L~Φ, ∂_µΦ~

= 1

2∂_µΦ_i∂^µΦ_i −m²

2 Φ_iΦ_i− λ

4 (Φ_iΦ_i)². (1.56) Here λ > 0 is just a parameter, m² < 0 and the three elds Φi are Hermitian. The Hamiltonian of the system is

H = ∂L

∂₀Φ_i∂₀Φ_i− L

= 1

2∂₀Φ_i∂⁰Φ_i−1

2∂_kΦ_i∂^kΦ_i+ V (Φ_i) , (1.57) where

V~Φ = m²

2 Φ_iΦ_i+λ

4 (Φ_iΦ_i)². (1.58)

The Lagrangian (1.56) is invariant with respect to a rotation Φ_i → e^−iα^k^T^k

ijΦ_j, (1.59)

where the Tkconstitute a representation of the rotation group SO(3) and satisfy the Lie algebra [Ti, T_j] = i_ijkT_k. The matrix representation is chosen to be T^k

ij = −i_kij, such that, for example, matrix T¹ is

T¹ = −i (_1ij) = −i





111 112 123

₁₂₁ ₁₂₂ ₁₂₃

₁₃₁ ₁₃₂ ₁₃₃



= −i





0 0 0

0 0 1

0 −1 0



. (1.60)

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The conserved charge operators that generate the rotational transformations can be extracted from Noether's theorem. They read

Q_a= −i ˆ

d³x∂₀Φ_it^a_ijΦ_j, a = 1, 2, 3. (1.61) The aim is to investigate the properties of the ground state. The ground state ~Φmin

should be constant in space and time, so that the terms with derivatives in the Hamilto- nian (1.57) would go to 0. The ground state should also minimize the potential, which yields

∂V~Φ

∂Φ_j = Φ_j m²− λΦ_iΦ_i = 0 → (1.62)

~Φ_min =

r−m²

λ ≡ v,

~Φ =

q

Φ²₁+ Φ²₂+ Φ²₃. (1.63) The ground state ~Φmin can point in any direction, which means that the number of vacua is innite. Choosing a specic direction for the vacuum can be achieved by disturbing the ground state by an innitesimal perturbation not invariant under SO(3).

The perturbation can be selected to be such that the ground state becomes (see section 3.1 of [21] for details)

Φ~_min = v~e₃ = v



 0 0 1



. (1.64)

Now the vacuum is not invariant with respect to rotations about axes 1 and 2:

e^−iα¹^T¹Φ~_min = 1 − iα₁T₁+ O α²₁~Φ_min

≈ v



 0 0 1



− iα₁



−i





0 0 0

0 0 1

0 −1 0



v



 0 0 1









= v



 0 0 1



− α₁v



 0 1 0



6= ~Φ_min; (1.65)

analogously

e^−iα²^T²~Φ_min 6= ~Φ_min. (1.66) However, as T3Φ~_min = 0, the rotation about the 3. axis leaves the vacuum invariant:

e^−iα³^T³~Φ_min = ~Φ_min. (1.67)

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As a next step one wants to study the excitations on top of the vacuum. To that end the elds are parametrized such that they incorporate the vacuum into the potential and hence into the Lagrangian. The eld parametrization can be chosen to be

Φ3(x) = v + η (x) ,

∂_µΦ₃(x) = ∂_µη (x) . (1.68)

Finally the potential (1.58) has to be rewritten in terms of the new eld η (x), yielding V =˜ 1

2 −2m² η²+ λvη Φ²₁ + Φ²₂+ η² + λ

4 Φ²₁+ Φ²₂+ η²2

− λ

4v⁴. (1.69) Concentrating on the terms quadratic in the elds, one nds that there are two massless Goldstone bosons and one massive boson

m²_Φ

1 = m²_Φ

2 = 0, (1.70)

m²_η = −2m². (1.71)

Note that m² < 0, such that m²η > 0. The crucial point is that for each generator of transformations that does not leave the vacuum state invariant (or, equivalently, does not annihilate the vacuum), here Q1 and Q2 from (1.61), one obtains a massless Goldstone boson. This feature is independent of the system which is studied.

Next we extend the above statement to a more general case. Suppose one has a system with nG generators of transformations that leave the Lagrangian invariant (in the example above these were the three generators Q1, Q2 and Q3). The corresponding symmetry group is called G. After nding a ground state of the system, there are supposedly nH generators of transformations which leave the ground state invariant (Q3 in the above example) and the symmetry group is called H. The transformations generated by H are associated with a subgroup of the group that is associated with the transformations generated by G, i.e. H is a subgroup of G. A massless Goldstone boson corresponds to every generator nGwhich produces a transformation that does not leave the vacuum state invariant. This means that the number of massless Goldstone bosons is nG−n_H. In the example above, this would yield 3−1 = 2 massless Goldstone bosons.

1.3.2 Explicit symmetry breaking

In order to study the mechanism of explicit symmetry breaking, the Lagrangian (1.56) is modied by adding a small perturbation to the potential (1.58):

V (Φ_i) = m²

2 Φ_iΦ_i+λ

4(Φ_iΦ_i)²+ aΦ₃. (1.72) The modied potential is invariant under SO(2) rather than SO(3). The values Φi that minimize the potential are found by using ~∇ΦV = 0and read

Φ₁ = Φ₂ = 0, λΦ³₃+ m²Φ₃ + a = 0. (1.73)

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Using the perturbative ansatz

hΦ3i = Φ⁽⁰⁾₃ + aΦ⁽¹⁾₃ + O a²

(1.74) to solve for Φ3 yields the minimum conditions

Φ⁽⁰⁾₃ = − r

−m²

λ , Φ⁽¹⁾₃ = 1

2m². (1.75)

After parametrizing the potential by Φ3 = hΦ₃i + η one obtains the masses

m²_Φ₁ = m²_Φ₂ = a

r λ

−m², (1.76)

m²_η = −2m²+ 3a

r λ

−m². (1.77)

The crucial point here is that the two Goldstone bosons which were massless according to (1.70) now have a mass (1.76). The squares of the masses are proportional to the symmetry breaking parameter a.

1.3.3 Symmetry breaking in QCD

We are now ready to discuss the consequences of symmetry breaking in QCD. As was concluded in subsection 1.2.3, in the chiral limit the QCD Lagrangian has a symmetry group (1.44). The expectation for spontaneous symmetry breaking to occur is motivated by experimental data. Both vector and axial-vector charge operators QâV = Qâ_R+Qâ_Land Qâ_A = Qâ_R− Qâ_L, corresponding to the symmetry currents (1.40) and (1.41), commute with the Hamiltonian of the theory HQCD⁰ . As was discussed in subsection1.2.4, without symmetry breaking both sets of operators could be used to construct approximately energy-degenerate multiplets and one would expect that for each state of positive parity one would also have a state of negative parity with a similar mass. For baryons, however, such parity doublets are not observed. Moreover, at low energies (.1 GeV) in the mesonic sector (baryon number 0), one observes a vector meson octet and a pseudoscalar meson octet, but not an axial-vector meson octet that could be a parity doublet of the vector meson multiplet. Also the lowest-lying scalar states are signicantly heavier than the pseudoscalar states. The members of the pseudoscalar meson octet transform under parity and avor transformations in the same manner as QâA- they are good candidates for Goldstone bosons. Therefore, motivated by the absence of parity doublets and the properties of the mesonic pseudoscalar states, one concludes that the generators of transformations QV and QâV leave the ground state invariant,

Q_V |0i = Q^a_V |0i = 0, (1.78)

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whereas the generators Q^a_A do not annihilate the ground state,

Q^a_A|0i 6= 0. (1.79)

This leads to spontaneous symmetry breaking of the SU (3)_Asymmetry. We now apply the line of reasoning presented in subsection1.3.1. Restricting ourselves to the mesonic sector, there are altogether sixteen generators of transformations that leave the La- grangian invariant, nG = 16, and the corresponding group is G = SU (3)_V× SU (3)_A. There are eight generators of transformations that leave the ground state invariant, nH = 8, and the subgroup is H = SU (3)_V. The number of Goldstone bosons is thus n_G− n_H = 8.

So far we have exclusively worked in the chiral limit, meaning that the quark masses have been taken to be zero. This means that after spontaneous symmetry breaking of SU (3)_A, the eight emerging Goldstone bosons are massless and the avor multiplets obtained by using the operators Q^a_V are degenerate in energy, i.e. the members of the multiplets have exactly the same masses. However, the chiral limit is only an approximation and for more realistic results, the quark masses have to be taken into account.

The mass term of the Lagrangian (1.1) with non-vanishing quark masses mixes the left-handed and the right-handed elds,

L_M = −¯qM q = − (¯q_RM q_L+ ¯q_LM q_R) (1.80) with M = diag (mu, m_d, m_s). After conducting calculations similar to the ones presented in subsection1.2.3, one obtains

δL_M = −i

" ₈ X

a=1

ϑ^R_a

¯ q_Rλ_a

2 M q_L− ¯q_LMλ_a 2 q_R

+ ϑ^R(¯q_RM q_L− ¯q_LM q_R)

+

8

X

a=1

ϑ^L_a

¯ q_Lλ_a

2 M q_R− ¯q_RMλ_a 2 q_L

+ ϑ^L(¯q_LM q_R− ¯q_RM q_L)

#

. (1.81) With the help of equation (1.28), the divergences of symmetry currents read

∂_µL^µ,a = ∂δL_M

∂ϑ^L_a = −i

¯ q_Lλ_a

2 M q_R− ¯q_RMλ_a 2 q_L

, (1.82)

∂µR^µ,a = ∂δL_M

∂ϑ^R_a = −i

¯ qR

λ_a

2 M qL− ¯qLMλ_a 2 qR

, (1.83)

∂_µL^µ = ∂δL_M

∂ϑ^L = −i (¯q_LM q_R− ¯q_RM q_L) , (1.84)

∂_µR^µ = ∂δL_M

∂ϑ^R = −i (¯q_RM q_L− ¯q_LM q_R) . (1.85)

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Combining them into the divergences of vector and axial-vector currents gives

∂_µV^µ,a = i¯q

M,λ_a

2

q, (1.86)

∂_µA^µ,a = i¯q

M,λ_a

2

γ₅q, (1.87)

∂_µV^µ = 0, (1.88)

∂_µA^µ = 2i¯qM γ₅q + 3g²

32π²_µνρσG_a^µνG_a^ρσ. (1.89) Note that the anomaly present in equation (1.89) was not written out in equations (1.84) and (1.85) - as the calculations to obtain the anomaly are rather lengthy, we here refer to section 2.3.6 in [21] for details. As one can see, the current (1.42), which leads to baryon number conservation, is also conserved with non-vanishing quark masses, i.e. it is always conserved. In the limit mu = md = ms 6= 0 the currents (1.40) associated with the group SU (3)_V are conserved (1,^λ₂^a = 0), while the currents (1.41) associated with the group SU (3)_Aare not. This means that in addition to spontaneous symmetry breaking, the SU(3)Asymmetry is also explicitly broken by the mass term. The singlet axial-vector current has, in addition to the anomaly, an explicit divergence due to quark masses.

As was discussed in subsection 1.3.2, a term that explicitly breaks the symmetry of the Lagrangian leads to massive Goldstone bosons. This is exactly what happens in QCD - the SU (3)_A symmetry of the chiral Lagrangian (1.18) is spontaneously broken, leading to eight Goldstone bosons. The mass term in the QCD Lagrangian behaves as a perturbation which breaks SU (3)_A explicitly and yields masses for the eight Goldstone bosons.

Finally, in reality the three lowest quarks have dierent masses, such that the SU (3)_V symmetry becomes approximate and the multiplets constructed through the charge operators Q^a_V are approximately degenerate in masses. The non-equality of the masses of the three lightest quarks also causes the structure of the multiplet containing the Goldstone bosons. The construction procedure of the multiplets can be found in [16]. The pseudoscalar meson octet and the vector meson octet, which contain the mesons that enter the eective eld theories, are depicted in gure 1.1.

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Figure 1.1: The pseudoscalar meson octet and the vector meson octet. Image taken from [22].

1.4 Eective eld theory

So far, symmetries of the QCD Lagrangian in the chiral limit have been studied. With the help of Noether's theorem, it has been shown that in the chiral limit one would expect parity-doubled energy-degenerate multiplets to emerge. Then, after considering spontaneous and explicit symmetry breaking of the group SU (3)_A, it has been discussed how one arrives at a pseudoscalar meson octet (consisting of the Goldstone bosons of the theory) and other avor multiplets - the sets of states actually observed in nature.

As was mentioned in the introduction, a perturbative expansion in the coupling constant does not work for QCD in the low-energy regime. To obtain a method to describe the reactions taking place in that energy region, an eective eld theory is constructed. The eective eld theory at hand is called Chiral Perturbation Theory (ChPT), where the Goldstone bosons act as relevant degrees of freedom. In this section, the goal is to give an overview how the eective theory is constructed. To that end, rst the formalism of the generating functional is introduced in subsection 1.4.1. Then the lowest order eective theory is built in subsection 1.4.2, followed by a short overview of the Wess-Zumino-Witten action in subsection 1.4.3. Finally in subsection 1.4.4 an eective eld theory will be presented where also the vector mesons act as relevant degrees of freedom.

1.4.1 The generating functional and external elds

In quantum eld theories, the objects of interest are n-point functions or Green functions, which are vacuum expectation values of time-ordered products. These functions lead to Feynman matrix elements and therefore are crucial for calculating physical observables of the theory - the decay widths and cross-sections. There exists a formalism to incorporate all Green functions that could possibly emerge into a single formula, which is called the generating functional. To do that, one has to modify the Lagrangian

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of the theory by introducing couplings with external elds:

L = L⁰_QCD+ L_ext = L⁰_QCD+ ¯qγ_µ

v^µ+ 1

3v^µ_(s)+ γ₅a^µ

q − ¯q (s − iγ₅p) q. (1.90) The external elds are color-neutral, Hermitian 3 × 3 matrices that act in avor space.

They read

v^µ =

8

X

a=1

λ_a

2 v^µ_a, a^µ=

8

X

a=1

λ_a

2 a^µ_a, s =

8

X

a=0

λ_as_a, p =

8

X

a=0

λ_ap_a. (1.91)

In the equations above we have introduced λ0 = p2/3 diag (1, 1, 1). Notice that the ordinary QCD Lagrangian is obtained by choosing v^µ = v_(s)^µ = a^µ = p = 0 and s = diag (m_u, m_d, m_s). The next step is to dene the generating functional in the form

exp (iZ [v, a, s, p]) =

0

T exp

i

ˆ

d⁴xL_ext(x)

0

. (1.92)

Now any Green function consisting of Hermitian bilinear quark forms can be obtained by taking functional derivatives of the generating functional with respect to the external

elds. It should be stressed that as the generating functional produces all possible Green functions, it is a fundamental object of the theory.

The extended Lagrangian has to meet certain requirements under a local SU (3)_V× SU (3)_A× U (1)_V

symmetry. As a rst step, the extended Lagrangian has to be written in terms of left-handed and right-handed elds. This is achieved by using v^µ = ¹₂(r^µ+ l^µ) and a^µ= ¹₂(r^µ− l^µ) along with the properties of projection operators. The result is

L = L⁰_QCD+ ¯q_Lγ_µ

l^µ+ 1

3v^µ_(s)

q_L+ ¯q_Rγ_µ

r^µ+1

3v_(s)^µ

q_R

− ¯q_R(s + ip) q_L− ¯q_L(s − ip) q_R. (1.93) Next we consider local transformations

q_L = exp

−iϑ (x) 3

V_L(x) q_L, (1.94)

q_R = exp

−iϑ (x) 3

V_R(x) q_R, (1.95)

Low-energy processes with three pions in the nal state