Chiral Perturbation Theory for Light Mesons

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Chiral Perturbation Theory for Light

Mesons

Author: Niels VERCAUTEREN Supervisor: Dr. W.J. WAALEWIJN 10677623 Second examiner: Prof. dr. P.J. MULDERS

A thesis submitted in fulfillment of the requirements for the degree ofBachelor of Science Bèta-Gamma

at the

Institute for Theoretical Physics Amsterdam (ITFA)

12 EC

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Universiteit van Amsterdam

Abstract

Faculteit der Natuurwetenschappen Wiskunde en Informatica (FNWI) Institute for Theoretical Physics Amsterdam (ITFA)

Bachelor of Science Bèta-Gamma Chiral Perturbation Theory for Light Mesons

by Niels VERCAUTEREN

A very brief introduction to Field Theory is given, along with an introduction to chiral perturbation theory. Chiral perturbation theory is an effective field theory of quantum chromodynamics at low energies. It describes the light mesons as (pseudo-) Goldstone bosons from spontaneous breaking of chiral symmetry. Chiral pertur-bation theory is applied to the description of light meson masses and their con-stituent quark mass ratios, finding M2

π± = B(mu+ md); M_K2± = B(mu+ md); M_K20 =

B(md+ ms); M_π20 = B(mu+ md); Mη2 = B3(mu+ md+ 4ms)for the meson masses

with B an experimentally determined constant and mu/md≈ 0.66; Ms/md≈ 22 for

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situation. I am also grateful for all the understanding and support I received from my family and friends during this project.

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1 Introduction

In the Standard Model (SM) quarks and their interactions through gluons are de-scribed using Quantum ChromoDynamics (QCD). At low energy, when this interac-tion binds quarks together into hadrons, it is more commonly known as the Strong Nuclear Force. The Lagrangian in QCD describes the interaction of the color charge1 of quarks and gluons. The strength of this interaction is depending on the coupling strength g. The strong coupling constant αs is defined as[1] αs(Q) ≡ g(Q)

2

4π . The Q

stands for the energy (momentum) exchange of the interaction. Higher interaction energy corresponds to to a lower coupling constant.

Usually an interaction can be described by the Feynman diagrams corresponding to the lower order terms of the expansion in αs, meaning only diagrams with few

vertices - and therefore few factors of αs- contribute to the cross-section. If the

en-ergy (momentum) scales approach the scale ΛQCDof confinement, αsbecomes large.

This in turn entails the contribution of any order term in the expansion, so infinitely many Feynman diagrams (with an arbitrary number of vertices) contribute to the the cross-section. So at low energies, when quarks bind into hadrons, perturbative calculations using QCD are no longer viable.

We are still able to describe this regime, but instead of using QCD directly, we will use an Effective Field Theory (EFT) of the SM, called Chiral Perturbation Theory (ChPT). This EFT uses a newly constructed Lagrangian tailored to the particles of interest, but which has the same chiral symmetries as the QCD Lagrangian.

Since for heavy B mesons αsis small, even at low energies, this paper will focus on

the description of light mesons. To this end, the paper is structured into two main chapters,2: Field Theory and3: Effective Field Theory for Light Mesons. The first chap-ter is an introduction to field theory, in which an example of a free field expression is looked at, global symmetries are discussed and Spontaneous Symmetry Break-ing (SSB) is introduced. The purpose of this chapter is to cover the bare minimum needed to be able to understand the third chapter, the core of this thesis. Readers fa-miliar with field theory and its symmetries can skip this chapter, readers for whom this is not enough introduction, see the much more thorough treatments of the sub-jects in [10] and [11]. The third chapter starts with the concept of chirality. Using this concept, the symmetries of the QCD-Lagrangian are exposed, so that the Gold-stone bosons (GBs) resulting from spontaneous breaking of chiral symmetry can be described. The structure of these GBs then finally allows us to show our EFT in ac-tion, through the construction of the leading effective Lagrangian. Now knowing the workings of this technique, we apply it to describe massive (but light) mesons and conclude with predictions about their own and their constituent quark masses.

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2 Field Theory

In Field Theory (FT), a different approach is used than what you may be used to from Quantum Mechanics (QM). As turning from Classical Mechanics (CM) to QM requires a new way of thinking, so does the transition from QM to FT, or more ac-curately, Quantum Field Theory (QFT). In FT instead of describing individual parti-cles by their individual properties (CM) or by the macro properties of the ensemble (Thermodynamics), properties of all identical particles are described by the prop-erties of their collective field. Particles are then viewed as being excitations in this field, see fig. 2.1b. Even when no particles are present, the field still is and we call this the vacuum field, see fig.2.1a.

2.1 Free field Lagrangian

This section is based on the material in paragraph 2.3 in Peskin and Schroeder[10], which has a more in-depth description of the subject.

In field theory instead of describing particles, we describe positions in space and their associated field values. So instead of having operators which describe the property of let’s say momentum pi of the particle i, we now look at the property

of π(~x), which is the momentum density of the field at position ~x. The traditional commutation relations [qi, pj] = iδij now thus become:

[φ(~x), π(~y)] = iδ3(~x − ~y). (2.1) The index overlap δij is replaced by the dirac delta function δ3(~x − ~y), indicating

whether the positions ~x and ~y overlap. As expected, the fields φ and π do commute with themselves, as did the position and momentum of different particles ([pi, pj] =

[qi, qj] = 0):

[φ(~x), φ(~y)] = [π(~x), π(~y)] = 0 (2.2) In QFT the position and momentum density fields φ and π behave as their respective QM equivalents q and p. Likewise, QFT uses Lagrangians to describe the state of a system, which might consist of multiple fields. Most often though, instead of the actual Lagrangian L the Lagrangian density L is used, defined by

L = Z

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4 Chapter 2. Field Theory

(A) Vacuum field, corresponding to no particles being present.

(B) Excitation in the field corresponding to a particle with momentum ~p = (px, py) ≈ (2, 3).

FIGURE2.1: Field representations in momentum space.

To get a feel for this, consider as an example the theory of a single field φ(x), gov-erned by the Lagrangian density

L = 1 2 ˙ φ2− 1 2( ~∇φ) 2₋ 1 2m 2_φ2 = 1 2(∂µφ) 2₋1 2m 2_φ2_, _(2.4)

where m is the mass of the particles described by this field. The different components written out in the first line all have a physical interpretation. Since they are terms in the Lagrangian, they can all be interpreted as "energy costs". The term with time derivative of the field ˙φis the energy cost of "moving" in time; the second term, with the gradient of the field ~∇φ, is the energy cost of shearing in space and the last term including the mass m is the energy cost of having the field around at all. This is called the Lagrangian1of a free field since there is no external potential in the Lagrangian, so no external forces are acting on/interacting with the field.

In order to get useful information out of our Lagrangian, with which we do predic-tions, we need the equations of motion for the system. You are probably familiar with the classical Euler-Lagrange equations of motion, and using exactly the same reasoning as in CM we will derive those for QFT in the next section.

2.2 Euler-Lagrange equations

Suppose we start with a set of scalar fields φa(x)and a general Lagrangian

(den-sity) L(x) = L(φa(x), ∂µφa(x)). We now make a infinitesimal change in each field:

φa(x) → φa(x) + δφa(x).This in turn changes the Lagrangian: L(x) → L(x) + δL(x).

The amount with which L(x) changes is δL(x) = ∂L ∂φa(x) δφa(x) + ∂L ∂(∂µφa(x)) ∂µδφa(x). (2.5)

1_{Since we almost always work with the Lagrangian density instead of the Lagrangian in QFT, the} Lagrangian density is often simply referred to as the Lagrangian. Here this is also the case.

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δφa(x)

= 0, (2.6) where _δφδ

a(x) is a functional derivative. The functional derivative and this form of

the minimal action principle are further explained in appendixA.

When measured, the field will collapse into a classical state, so this is an accurate description of our initial and final states, but not for the paths taken by the particles in between (during possible interactions). For our purpose now, this will suffice, but for more detail, see chapter 22 in Srednicki[11]. So for now assuming this (2.6) holds, we have δS δφa(x) = Z d4y δL(y) δφa(x) (2.7) = Z d4y ∂L(y) ∂φb(y) δφb(y) δφa(x) + ∂L(y) ∂(∂µφb(y)) δ∂µφb(y) δφa(x) = Z d4y ∂L(y) ∂φb(y) δbaδ4(y − x) + ∂L(y) ∂(∂µφb(y)) δba∂µδ4(y − x) = ∂L(x) ∂φa(x) + Z d4y ∂L(y) ∂(∂µφa(y)) ∂µδ4(y − x) = ∂L(x) ∂φa(x) + Z d4y ∂µ ∂L(y) ∂(∂µφa(y)) δ4(y − x) − ∂µ ∂L(y) ∂(∂µφa(y)) δ4(y − x) = ∂L(x) ∂φa(x) − ∂µ ∂L(x) ∂(∂µφa(x)) + Z d4y ∂µ ∂L(y) ∂(∂µφa(y)) δ4(y − x) (2.8) = 0

The last term in (2.8) can be turned into an integral over the boundary of the 4D space-time region of integration2. We assume the configuration of the field at t → ±∞ is unaltered, i.e. the perturbation does not affect the initial and final states. This integral is then equal to zero, so we become the EL equations (one for each field φa)

∂L(x) ∂φa(x) = ∂µ ∂L(x) ∂(∂µφa(x)) . (2.9) 2.2.1 Klein-Gordon equation

Filling the Lagrangian for a free field (2.4) into the newly derived EL equations (2.9), we get the so-called the Klein-Gordon (KG) equation

∂2

∂t2 − ~∇ 2_{+ m}2

φ(x) = 0 or (∂2+ m2)φ(x) = 0, (2.10) where x = (t, ~x) is a four-vector and ∂2 _{≡ ∂}µ_∂

µ. This equation is the FT equivalent

of the Energy-Momentum relation in relativity. This can be seen by writing out the KG equation and using the Schrödinger equation (SE). Note that this next part is somewhat non-rigorous, but can be insightful. We start by writing out the KG

2_{Stokes theorem:}R Ωdω =

H

∂Ωωor, in our case: R∞ −∞d 4_y∂ ...= H∞ −∞d 4_{y[...] = 0.}

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6 Chapter 2. Field Theory equation (2.10) (∂2+ m2)φ = 0 (∂µ∂µ+ m2)φ = 0 ∂2 ∂t2 + (|~p| 2_{+ m}2₎ φ = 0 SE : Hφ = i∂ ∂tφ. (2.11) Comparing this to the Energy-Momentum relation from relativity, we see they are strikingly similar:

E = |~p|2+ m2 (2.12)

2.3 Global Symmetries

In this section we will discuss what symmetries are, by returning to equation (2.5) as promised and by showing a simple example of interacting fields.

We have abstractly seen how a Lagrangian describing a system can be dependent on multiple fields in the previous section2.2, let’s make this concrete by regarding a Lagrangian consisting of two fields:

L = 1 2∂ µ_φ†_∂ µφ − 1 2m 2_φ†_{φ −} 1 2λ(φ †_φ)2 _(2.13)

Several things about this equation are new concepts and we will go through them one by one. First we notice the introduction of φ†, the complex conjugate transpose of the field. So far we have worked with real fields only, but the extension to complex fields is straightforward and mostly a notational nicety. we simply write two real fields φ1and φ2as the real and imaginary components of a complex field φ,

φ = φ1+ iφ2. (2.14)

Instead of using the real fields φ1 and φ2, we use φ and φ†, since these are also

in-dependent and can always be recombined as φ1 = ½(φ − φ†) ; φ2 = −½i(φ + φ†)

to retrieve the real fields. The other new concept introduced here is the interaction between fields, expressed by the last term in (2.13). Because this term is of power order four, it describes the four-point interaction of the fields.

Now we know the meaning of the different parts of this Lagrangian, we get to the heart of this section, symmetries. A symmetry is the change of the field(s), under which the equations of motion remain invariant. We will see what this means through the fol-lowing example, in which we use the Lagrangian above (2.13) and see what happens to it when we transform the fields φ and φ†as

φ → e−iαφ and φ†→ e+iαφ†, (2.15) where α is an angle. This is a so-called U(1) transformation, a rotation in the complex plane. The "U" refers to unitarity of the transformation group, the "(1)" indicates that α is a 1×1 matrix, ergo a scalar. We will see other transformations in the next

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= L (2.16) The Lagrangian remains unchanged. This can be seen more directly by noticing that only pairs of φ†and φ are present in the Lagrangian and their combined transforma-tion results in the same pair, φ†φ → e+iαφ†e−iαφ = φ†φ. Looking at the EL equations (2.9), we see that if the Lagrangian is unchanged, so are the equations of motion. Therefore, we found a symmetry! Our Lagrangian L is symmetric under the U(1) transformation (2.15).

A final note on nomenclature; the reason that this section is named Global Symmetries has to do with the existence of a different kind of symmetries, local symmetries, for which the transformation of the field is explicitly dependent on the position in spacetime. Global symmetries however act the same way on the field at each point in spacetime and are all we will need in this thesis. Now we have had our first encounter with a symmetry, the next section will build from this, looking at what happens when existing symmetries are lost, or "broken".

2.4 Spontaneous Symmetry Breaking

Likewise to how the concept of a symmetry was explained by an example in the previous section 2.3, an example will be used in this section to explain Sponta-neous Symmetry Breaking (SSB); although we will use a different, more general La-grangian. Symmetries as in the previous section2.3 are exact, meaning that they are unconditionally valid[2]. Some symmetries are approximate, meaning they are valid under certain conditions and may become broken otherwise. A symmetry can be broken either explicitly or spontaneously. Explicit symmetry breaking refers to when the equations of motion are not invariant under the considered symmetry group, by a term in the Lagrangian explicitly breaking the symmetry. SSB refers to a situation in which, given a symmetry of the EL equations, one or more ground states - solutions of the EL equations - exist, which are not invariant under the action of this symmetry, without the introduction of any term explicitly breaking the symmetry[2]_.

We are interested in the SSB phenomena and will now illustrate it with an example. Consider the Lagrangian

L = 1 2∂

µ_φ†_∂

µφ − V (φ), (2.17)

where V (φ) is the so-called "Mexican Hat" potential, which looks like the potential in fig.2.2. This potential is such that the initial ground state is unstable, a small per-turbation and the ball in fig.2.2will roll down into a new (stable) ground state at (B). While on the peak of potential, the ball is in a state with rotational U(1) symmetry: no matter which way it faces, the potential looks the same in any direction. When it rolls down however, this symmetry is lost: the potential looks different in different directions.

The new situation still has two degrees of freedom, only where before they were in directions of the imaginary and real components of the field φ, now the degrees of freedom are in the azimuthal and radial direction. The excitation in the radial

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8 Chapter 2. Field Theory

FIGURE2.2: Mexican Hat potential.

On top (A) there is rotational U(1) symmetry. If the ball rolls down into the ground state at (B), this symmetry is broken. Edited from [4].

direction corresponds to a massive state, after all, the mass term we had in (2.2) is now absorbed3into the potential V(φ). In contrast, the other degree of freedom - the excitation in the azimuthal direction - corresponds to a massless state. This massless state is a direct result of the symmetry breaking and is interpreted as a massless particle. In general, the reduction (or rather the difference) in symmetry determines the group that describes the new particle(s) resulting from SSB. Here the reduction is from U(1) to no symmetry at all (denoted as 1), so the difference in symmetry describing the massless particle is itself a U(1) group and we denote it

B ∈U(1)/1 = U(1). (2.18) Massless particles resulting from SSB are called Goldstone Bosons (GBs). In this case, there is only one such boson (described by B), since the difference in symmetry can be represented by U(1). In the next chapter we will see a "larger" reduction in symmetry, resulting in eight GBs.

This now concludes the necessary Field Theory and Symmetry concepts needed for the description of light mesons in the next chapter3; any additionally needed con-cepts will be treated as needed in that same chapter.

3_{To see that the excitation in this direction is a massive particle, approximate the potential by a} Taylor series in the radial direction around the ground state in the valley, at (B) in fig. 2.2. Although an explicit formula is not given for the shape of the potential, one can see that an expansion would at least need a quadratic term for a decent fit and that this would be the dominant term for not too large excitations. Realizing the quadratic term in the Lagrangian corresponds to the mass of the particle, as in our example2.13, this means that the rising potential gives mass to this state.

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3 Effective Field Theory for Light

Mesons

In this chapter we get to the heart of the thesis, in which we use a method called Chi-ral Perturbation Theory (ChPT) as an effective field theory to describe light mesons. Before applying ChPT, we will look into what chiralty means physically and math-ematically. In the sections following that,3.2 and3.3, chirality is used to study the symmetries of QCD and the SSB leading to Goldstone bosons respectively. Knowing how the symmetry is broken, we construct an effective Lagrangian Lef f (section3.4)

and move on to improve this Lagrangian in section3.5so that it describes massive particles. Finally we use the description of light mesons obtained in this manner to make predictions about the masses of their constituent quarks.

3.1 Chirality

Chiral Perturbation Theory (ChPT), as the name suggests, relies on the chirality of the particles it describes - in this case quarks. So before moving on to the description of symmetries in QCD, a description of chirality will be given.

Quarks have a distinguishing property called flavour, which determine the quark type e.g. up , down or top quarks. Quarks are not always in a flavour eigenstate[3]_{, but}

can be in a superposition of flavour eigenstates. Since here we are only concerned with light quarks, we can write the quarks as having components in the up, down and strange flavour space.

q = (u, d, s) (3.1) So one could imagine the flavour of the quark as being a vector in 3D flavour space. Since quarks are fermions, they are spin-½ particles and we can decompose the quark fields into the components which have their spin either aligned (right handed) or anti-aligned (left handed) with the vector indicating their velocity, as shown in fig.

3.1. This is called the helicity of the quark(field). Since for massive particles this de-pends on the frame of reference, we work with an almost equivalent principle called chirality. Chirality of a particle (or field) is equivalent to helicity for massless parti-cles (moving at the speed of light), but also invariant under Lorentz transformations for particles that do have mass. Therefore it is a useful property in the description of particles or fields. Mathematically, decomposing the quark fields into their chiral components is done as[8] q = 1 2(1 − γ 5_{)q +}1 2(1 + γ 5_)q = PLq + PRq = qL+ qR. (3.2)

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10 Chapter 3. Effective Field Theory for Light Mesons

(A) Left handed.

Spin anti-aligned with velocity vector.

(B) Right handed.

Spin aligned with velocity vector. FIGURE 3.1: Chiral components of light quarks. The red and blue arrows indicate the "rotation direction" corresponding to the spin, the

grey arrow indicates the velocity vector. Figure from [12].

Where q is a 4-component spinor, indicating the direction and orientation of the quark spin and γ5≡ iγ0_γ1_γ2_γ3_{is the product of the Dirac matrices (multiplied by i),}

γ5 =     0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0     . (3.3)

This expression for the chiral components of the quarks can now be used to analyze the symmetries of QCD.

3.2 Symmetries of L

QCD

The part of the QCD Lagrangian describing light quark fields (but not their masses) can be written in terms of the chiral components of the quark fields (qL, qR) as[8]

L0

QCD = i¯qLDq/ L+ i¯qRDq/ R. (3.4)

Where /D ≡ γµDµ , γµ are the Dirac matrices, Dµ is the covariant derivative and

¯

q ≡ q†γ0.

Now we see that if we perform the chiral U(3)Lx U(3)Rflavour transformations,

(qL, qR) → (LqL, RqR) where L, R ∈ U(3)L,R, (3.5)

we get the same L0_QCD back:

L0_QCD → i¯qLL†DLq/ L+ i¯qRR†DRq/ R

= i¯qLL†L /DqL+ i¯qRR†R /DqR

= L0_QCD (3.6) Since the fields are transformed, but the Lagrangian remains unchanged, this is a symmetry of the theory. Please note that this symmetry is only valid for the part of LQCD that we looked at, so for the up, down and strange flavours of massless

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To describe massive (but light) mesons, we will have to move away from the chiral limit by adding quark mass back perturbatively into our effective Lagrangian. This is done in section3.5. To do that however, we first need to construct such an effective Lagrangian (section3.4), for which in turn we need to know which particles we are actually describing.

3.3 Field U for Goldstone bosons

The chiral symmetry of the QCD Lagrangian in the chiral limit (L0_QCD) is sponta-neously broken, as was the U(1) symmetry in the Mexican hat example (see section

2.4). In this case however, the SSB is not caused by a term in the Lagrangian (as was the case with the potential), but by non-perturbative physics responsible for binding quarks in hadrons. So for the SSB of L0_QCD it is impossible to show an image of the process, as was done for the Mexican hat potential in fig. 2.2of the previous chap-ter2. The following description of the SSB will therefore be purely mathematical. Whenever possible I will compare results or effects to the previous example, in an attempt to keep it as intuitively as possible. For a more comprehensive description of this topic, see chapter 3.1 and 3.2 in Kubis[8].

Any U(3) group can be written as a product of a SU(3) group and a phase U(1), since the only extra restriction of SU(3) compared to U(3) is that SU(3) must have its determinant equal to one instead of it lying somewhere on the unit circle. We can therefore rewrite the symmetry as

U(3)L× U(3)R=SU(3)L× SU(3)R× U(1)V × U(1)A, (3.7)

where V ≡ L + R and A ≡ L − R. U(1)V is a conserved current corresponding to

the baryon number and U(1)Ais broken by the U(1)Aanomaly, so we are left with

SU(3)L× SU(3)Rsymmetry pertaining to chirality. From experiments[6]it turns out

that this symmetry is spontaneously broken, pointing to the emergence of Goldstone bosons1. The reduction in symmetry is as follows

SU(3)L× SU(3)RSSB→ SU(3)V. (3.8)

In the Mexican hat example, the symmetry was reduced by U(1), giving rise to one Goldstone boson described by B ∈ U(1)/1 = U(1). Since here the symmetry is reduced by SU(3), we describe the emerging Goldstone bosons by

U ∈SU(3)L× SU(3)R/SU(3)V. (3.9)

Since U ∈ SU(3), we can write it as

U ≡ eiλaφaF a ∈ {1, 2, ..., 8}, (3.10) 1_{Some theoretical reasoning also points to this, see [}₇_]

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12 Chapter 3. Effective Field Theory for Light Mesons where φ1..8are the arising massless states called Goldstone bosons and λ1..8are the

Gell-Mann matrices. The Gell-Mann matrices form a basis for the SU(3) group and since there are eight of these, there are also eight corresponding Goldstone bosons. Now we know that there is SSB, that leads to eight Goldstone bosons, which adhere to the structure described by U . Now we can construct our own effective Lagrangian Lef f expressed in terms of U for us to work with, as we will do in the next section.

3.4 Constructing L

ef f

In constructing the effective Lagrangian Lef f for Goldstone bosons, there are several

restrictions. Lef f must be:

1. Lorentz invariant 2. A scalar

3. Invariant under SU(3)L× SU(3)R

The first two may seem trivial - since we want our theory to be valid in any inertial reference frame and a Lagrangian is always a scalar - but they do place important restrictions on the allowed expression for Lef f, as we will see. The third restriction

is less trivial. One might expect Lef f only needing to be invariant under U ∈ SU(3),

since that describes the Goldstone bosons, which we are interested in. However, we need the Lagrangian of our effective theory to behave in the same way as the QCD Lagrangian, since it is exactly from this behavior - the symmetries describing it - that these Goldstone bosons appear. Taking these restrictions into account, we can begin construction of Lef f consisting of U , U†and derivatives thereupon.

A first naive attempt might be to simply set Lef f =Tr[U ]. While the trace makes it

a scalar, it does not fulfill all requirements, since it is not invariant under SU(3)L×

SU(3)R:

L_{ef f} =Tr[U ] → Tr[LU R†]. (3.11) Adding U†to get Lef f =Tr[U U†] is a step in the right direction, since it now seems

to fulfill all requirements:

L_{ef f} =Tr[U U†] →Tr[U U†]. (3.12) However U is still a unitairy matrix, so this would set Lef f = 3, a constant and

not very useful in any description of the dynamics of nature. Any combination of only U and U† give something which is either a constant or not invariant under SU(3)L× SU(3)R. Therefore, we necessarily introduce derivatives, setting

Lef f =→Tr[∂µU U†] =Tr[∂µU U†]. (3.13)

This is however clearly not Lorentz invariant, since the expression has an uncon-tracted index. This finally brings us to the simplest working expression for Lef f:

L_{ef f} ≡ Tr[∂µU ∂µU†] →Tr[L∂µU R†R∂µU†L†] =Tr[∂µU ∂µU†]. (3.14)

Of course one could work with a different expression, which uses higher order derivatives of U . These higher order terms have small contributions, since looking at the expression of U (3.10) we see that each derivative gives a suppressing factor

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factor F and is the expression we will use from now on:

L(2)_{ef f} =¼F2Tr[∂µU ∂µU†] . (3.15)

The ¼ is there to ensure the standard normalization of the kinetic term.

Now we have an expression for the effective Lagrangian describing massless states (Goldstone bosons). The particles we want to describe, light mesons, are linear com-binations of emerging bosons, but not massless. So in order to give an accurate description of these mesons, we need to reintroduce the masses into our effective Lagrangian.

3.5 Including quark masses

In describing the Goldstone bosons, we only regarded the massless quark part of the QCD Lagrangian, L0

QCD. The full QCD Lagrangian does describe the mass of

the quarks.

L_QCD = L0_QCD − Lm

QCD+ ... (3.16)

The mass term includes the masses of the up, down and strange quarks by means of the diagonal matrix M =diag(mu, md, ms)[8]:

Lm

QCD = ¯qLMqR+ ¯qRM†qL, (3.17)

where ¯q ≡ q†γ0.

This mass term in the QCD Lagrangian breaks the the chiral symmetry explicitly, as we can see when we perform our previous chiral transformations (3.5):

Lm_QCD = ¯qLMqR+ ¯qRM†qL→ ¯qLL†MRqR+ ¯qRR†M†LqL6= LmQCD. (3.18)

The mass matrix M doesn’t transform and hence breaks the symmetry.

In order to still be able to use our effective Lagrangian in the description of massive particles, we will have to incorporate this symmetry breaking by the quark masses into it. We must do this in such a way that our effective Lagrangian behaves the same as the QCD Lagrangian with the quark masses included. So our effective Lagrangian must display symmetry breaking in the same way the QCD Lagrangian does, when introducing quark masses to it. The trick to ensure this is to imagine the mass matrix M as not being a static 3 × 3 matrix, but as a field2 which transforms in a certain way. If we write M as transforming as

M → LMR†, (3.19)

2_{This may seem nonphysical, the quark masses are set values after all. It may help to think of the} masses as being coupled to the Higgs field and therefore transforming, but in essence this trick is no more than just that, a mathematical trick.

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14 Chapter 3. Effective Field Theory for Light Mesons it will conserve symmetry in the massive QCD Lagrangian:

Lm_QCD → ¯qLL†LMR†RqR+ ¯qRR†RM†L†LqL= LmQCD. (3.20)

If we now use the same transformation in the construction of our new massive ef-fective Lagrangian Lm

ef f , it will ensure Lmef f breaks in the same way by the

non-transforming actual M as the massive QCD Lagrangian.

3.5.1 Construction Lm_{ef f}

As we did in section3.4, we can now construct our effective Lagrangian, Lm_{ef f}. The procedure is the same, but now we have a new building block M, with its transfor-mation M → LMR†. Since the masses are assumed small (compared to for instance the invariant mass of a collision), higher powers of M give a suppresion as do higher order derivatives of U . Keeping to lowest possible order, but still adhering to the re-strictions3.4, we get[8]

Lm

ef f =¼F2Tr[∂µU ∂µU†+ 2B(MU†+ M†U )], (3.21)

where B is another so called Low Energy Constant (LEC), to be determined by ex-periment.

This remarkably simple expression is an effective Lagrangian in which the masses for mesons have been added perturbatively. This means we can use this expression instead of the full QCD Lagrangian when describing these mesons. Even though this theory describes mesons as a whole, it also enables us to make predictions about their constituent quarks by looking at the ratio of the meson masses. This is per-formed in detail in the next subsection.

3.5.2 Pion and quark masses

Starting with our effective massive Lagrangian Lm

ef f, we can expand U = e

iλaφa F ≡

eiφF in powers of φ. Now writing <..> instead of Tr[..] to denote the trace for brevity,

L(m)_{ef f} = F 2 4 < ∂µU ∂ µ_U† + 2B(MU†+ M†U ) > ≈ F 2 4 < ∂µ(1 + iλaφa/F )∂ µ_{(1 − iλ} bφ†_b/F )+ 2B(M(1 − iλaφ†a/F − λaφaλbφb/F2) + M†(1 + iλaφa/F − λaφaλbφb/F2)) > = F 2 4 < λaλb F2 ∂µφa∂ µ_φ† b+ 4B(M(1 − λaφaλbφb/F 2_{) >} =< λaλb 4 ∂µφa∂ µ_φ† b+ F 2_{BM(1 − λ} aφaλbφb/F2) > ≈< λaλb 4 ∂µφa∂ µ_φ† b− BMλaφaλbφb> . (3.22)

Going to the last line, we threw away the constant term F2_BM_{since any constant}

added to the Lagrangian would not affect the equations of motion. Since < λaλb >=

2δab , we recognize the first term as the kinetic term, similar to the kinetic term in

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< BMλaφaλbφb > = < B × 0 m_d 0 0 0 ms  ×     0 1 0 1 0 0 0 0 0  φ₁+   0 −i 0 i 0 0 0 0 0  φ₂+ ... +   1/√3 0 0 0 1/√3 0 0 0 −2/√3  φ₈   2 > = B md(φ21+ φ22+ φ23+ φ24+ φ25− 2 √ 3φ3φ8+ 1 3φ 2 8) + mu(φ21+ φ22+ φ23+ φ24+ φ25+ 2 √ 3φ3φ8+ 1 3φ 2 8) + ms(φ24+ φ25+ φ26+ φ27+ 4 3φ 2 8) . (3.23) This is not very informative in this form, so we rewrite this as φiMijφj. Note: Mij

are elements of the meson mass matrix M , not to be confused with the quark mass matrix M. The result is an almost diagonal matrix:

B               md+ mu 0 0 0 0 0 0 0 0 md+ mu 0 0 0 0 0 0 0 0 md+ mu 0 0 0 0 1₂ 2m_√u 3 − 2m_√d 3 0 0 0 ms+ mu 0 0 0 0 0 0 0 0 ms+ mu 0 0 0 0 0 0 0 0 md+ ms 0 0 0 0 0 0 0 0 md+ ms 0 0 0 1₂ 2m_√u 3 − 2m_√d 3 0 0 0 0 1₃(md+ 4ms+ mu)               (3.24)

This allows us to directly read of some of the meson’s masses (squared). For exam-ple, we know that the pions π±have constituent quarks up and down - π+_{= u ¯}_d_and

π− = ¯ud- so we associate the φ1φ1and φ2φ2entries in the matrix with the squared

masses of the π±pions. Noting that the mass of the K0and the ¯K0are the same, we can read of six squared meson masses :

M_π2±= B(mu+ md), MK2± = B(mu+ md), M_K2¯0 = M_K20 = B(md+ ms).

(3.25) This is a remarkable result. The masses of the pions do not - as might be naively expected- scale linearly with the masses of their constituent quarks, but with the square root of the sum of these masses. Somewhat unfortunate is the remainder of the constant B, so that we can’t do any theoretical predictions of the actual quark masses. We can however make predictions of the ratio of these masses, we find[8]

mu md = M 2 K+ − M_K20 + M_π2+ M2 K0 − M_K2++ M_π2+ ≈ 0.66 (= 0.42(4)), (3.26) ms md = M 2 K0 + M_K2+− M_π2+ M_K20 − M_K2++ M_π2+ ≈ 22 (= 19.3(7)). (3.27)

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16 Chapter 3. Effective Field Theory for Light Mesons Indicated behind the approximate results by this ChPT calculation, are the results obtained by lattice-QCD calculations[5], which are more accurate. The ChPT results can be improved in accuracy by e.g. including electromagnetic corrections or by retaining higher order derivatives of U in the expansion of L(m)_{ef f}.

For the non-diagonal part of the matrix above (3.24), we can regard these entries seperately in a different basis. In this new basis, φ3and φ8do not directly correspond

to the flavor eigenstates of mesons, but linear combinations of them do. The problem we are left with is:

B φ3 φ8 md+ mu m_√u 3 − md √ 3 m_√u 3 − md √ 3 1 3(md+ 4ms+ mu) ! φ3 φ8 , (3.28) which can be diagonalized by the rotation[8]

π0 η =cos − sin sin cos φ3 φ8 , = 1 2arctan √ 3 md− mu 2ms− mu+ md . (3.29) So this gives M_π20 = B(mu+ md) and Mη2 = B 3(mu+ md+ 4ms). (3.30) Combining this result with the squared masses we already found in (3.25), we can deduce the Gell-Mann-Okubo mass formula:

4M_K2 = 4B(mu+ ms)

≈ 4B(mu+ md 2 + ms)

= 3M_η2+ M_π2, (3.31) which is accurate to within 7% compared to what is found in nature[8].

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4 Conclusions

The goal of this thesis was to give the reader an insight into the workings and appli-cability of chiral perturbation theory. Mostly this was done by the usage of examples preceded by their necessary theoretical foundation. These examples accumulated into a discussion of the symmetry properties of the QCD Lagrangian and the con-struction of an effective Lagrangian adhering to these same properties.

The constructed effective Lagrangian was then shown to have predictive power in the last subsection 3.5.2of the previous chapter 3. There the meson masses were predicted (up to an overall experimentally determined factor B), finding M_π2± =

B(mu+md); M_K2±= B(mu+md); M_K20 = B(md+ms); M_π20 = B(mu+md)and Mη2 = B

3(mu + md+ 4ms). This is a curious result, since it shows that the masses of the

mesons scale with the square root of the sum of their constituent quarks.

The fact that ChPT can lead to such a profound physical insight shows that this is a successful method to describe nature. Perhaps even more compelling to this end, is that this theory was shown to be able to make predictions about the mass rela-tions of light mesons and about the mass ratios of the quark constituents of these mesons; without the need for an experimentally determined constant. The famous mass relation of light mesons is also called the Gell-Mann-Okubo mass formula; 4M_K2 = 3M_η2+ M_π2. The mass ratio predictions found in this paper are mu/md≈ 0.66

and Ms/md≈ 22, but one could make even better predictions by taking

electromag-netic effects into account. Although it falls beyond the scope of this paper, this would entail adding a term to the Lagrangian describing the interaction between the quarks and photons, see e.g. section 3.7 in [8].

Besides increasing the accuracy of mass (ratio) predictions, this theory has many more applications in the realm where the Standard Model is unusable, so when αs

becomes too large for regular perturbative methods. One such example of a differ-ent application is the description of scattering between light mesons in φ4 _theory,

producing predictions for the invariant amplitudes of these processes.

When taking into account higher order terms, chiral perturbation theory can even be extended to include baryons, making it a thoroughly extensive framework to describe mesons and their interactions.

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A

Functional Derivative and Minimal

Action Principle

Let’s start with a more detailed look into the functional derivative: δφb(y)

δφa(x)

= δbaδ4(y − x) (A.1)

This describes the derivative to a specific φa(x)with a fixed in a specific point x.

If we are interested in the variation of φb(y)however, δφb(y), we get something you

might not expect:

δφb(y) = X a Z d4x δφb(y) δφa(x) δφa(x) =X a Z d4x δbaδ4(y − x)δφa(x) (A.2)

Since φb (b fixed) may depend on any φa, we sum over these1(Pa). This is not

suf-ficient however, because each field φa has infinitely many points x in which it can

vary. Since this variation in φa in turn might lead to a variation in φb (if it is

de-pendent on φa), we integrate over all space-time (H d4x). This way we get the full

expression for the variation of φb(y).

Minimal Action Principle

Since the the minimization of the action demands

δS = 0, (A.3) this - by the equations (A.1) and (A.2) above - means that

δS δφa(x) = 0, (A.4) as is used in section2.2. 1_{c.f. df =}P i ∂f ∂xidxi

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B Popular Scientific Summary

Alle materialen in de wereld om ons heen en wij zelf ook bestaan uit moleculen, die weer uit verbonden atomen bestaan. Atomen hebben doorgaans een afmeting van rond de 50 picometer (pm), ook wel 0.5 ångström (Å) genoemd. Ter vergelijking, de doorsnede van een haar is een miljoen keer groter. Het atoom op zijn beurt bestaat weer uit een nucleus (kern) met daar omheen een electronen"wolk". De straal van de nucleus is op zijn beurt weer 10.000 keer kleiner dan de straal van het hele atoom.

FIGURE B.1: Artistieke impressie van het proton, met daarin twee up en één down quark die bij elkaar gehouden worden door de gluonen

(gele kronkellijnen). Uit [9].

Ook de nucleus bestaat weer uit kleinere on-derdelen, protonen en neutronen, samen nucle-onen genoemd.

In de figuur hiernaastB.1is een artistieke inter-pretatie van één proton weergegeven. Zoals je ziet heeft ook het proton (net als het neutron) weer een substructuur met nog kleinere deeltjes. Dit keer zijn we echter bij de allerkleinste bouw-stenen van de natuur aangekomen: quarks1_.

De quarks worden bij elkaar gehouden door glu-onen, "virtuele" deeltjes (ze hebben geen massa) aangeduid als kronkellijnen in de figuur. Het bij elkaar houden van de quarks middels gluonen wordt ook wel de sterke kernkracht genoemd, of met een mooi woord KwantumChromoDy-namica (QCD2). QCD is onderdeel van het

Stan-daard Model, een model in de natuurkunde dat drie van de vier fundamentele krachten bevat, de Sterke Kernkracht, de Zwakke Kernkracht en Electromagnetisme. Zwaartekracht is een lastig verhaal in dit opzicht, om dat het moeilijk verenigbaar is met de andere krachten in één model, zelfs met de ontdekking van het Higgs deeltje. De normale aanpak met behulp van storingsrekening in het standaarmodel is echter in sommige situaties niet erg praktisch (lees:onbruikbaar) om grotere deeltjes dan quarks te beschrijven. Lichte mesonen zijn hier een voorbeeld van. Mesonen zijn deeltjes die uit twee quarks bestaan in plaats van drie zoals protonen en neutro-nen. Mesonen bestaan maar heel kort nadat ze onstaan (ongeveer 10 miljoen keer korter dan hoe lang het duurt om met je ogen te knipperen), maar we kunnen ze wel direct detecteren met onze apparatuur, in tegenstelling tot losse quarks. Omdat we deze deeltjes direct kunnen meten is het interessant om beter te weten wat hun eigenschappen zijn en hoe ze zich gedragen.

In dit paper is naar een manier gekeken om deze meson deeltjes te beschrijven, onder specifieke omstandigheden (bij lage totale energie van het systeem). Het resultaat is een voorspelling van de massa van deze lichte mesonen en er kan zelfs iets gezegd worden over de massa verhoudingen van de quarks waaruit ze bestaan, hoewel quarks niet de basis vormen voor deze theorie!

1_{We laten snaar theorie buiten beschouwing omdat er nog andere theorieën zijn die daar mee} con-curreren.

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Bibliography

[1] Bugra Borasoy. “Introduction to chiral perturbation theory”. In: The Standard Model and Beyond (2008), pp. 1–26.

[2] Elena Castellani. “On the meaning of symmetry breaking”. In: Symmetries in physics: Philosophical reflections (2003), pp. 321–334.

[3] Augusto Ceccucci. “The CKM quark-mixing matrix”. In: Lawrence Berkeley Na-tional Laboratory (2008).

[4] Exploring the Cosmos. What the Higgs is going on? Digital Image. [Online; ac-cessed August 8th, 2017]. URL:http://exploringthecosmos.tumblr. com/post/17698353435/what-the-higgs-is-going-on.

[5] CTH Davies et al. “Precise charm to strange mass ratio and light quark masses from full lattice QCD”. In: Physical Review Letters 104.13 (2010), p. 132003. [6] Robert L. Jaffe, Dan Pirjol, and Antolello Scardicchio. “Parity doubling among

the baryons”. In: Physics reports 435.6 (2006), pp. 157–182.

[7] Robert L. Jaffe, Dan Pirjol, and Antonello Scardicchio. “Parity doubling and SU (2) L× SU (2) R restoration in the hadron spectrum”. In: Physical Review Letters 96.12 (2006), p. 121601.

[8] Bastian Kubis. “An introduction to chiral perturbation theory”. In: arXiv preprint hep-ph/0703274 (2007).

[9] Brookhaven National Laboratory. Electron-Ion Collider. Digital Image. [Online; accessed August 10th, 2017]. URL: https : / / www . bnl . gov / rhic / eic . asp.

[10] M.E. Peskin and D.V. Schroeder. An Introduction to Quantum Field Theory. Ad-vanced book classics. Avalon Publishing, 1995.ISBN: 9780201503975.URL:https: //books.google.nl/books?id=i35LALN0GosC.

[11] Mark Srednicki. Quantum field theory. Cambridge University Press, 2007. [12] Flip Tenado. Helicity, Chirality, Mass, and the Higgslity-mass-and-the-higgs.

Digi-tal Image. [Online; accessed July 16th, 2017].URL:http://www.quantumdiaries. org/2011/06/19/helicity-chirality-mass-and-the-higgs/.

Chiral Perturbation Theory for Light Mesons