Detemining low-energy constants in χPT from decays of decuplet baryons

Måns Holmberg

Master Thesis in Physics Department of Physics and Astronomy Uppsala University

Supervisor: Stefan Leupold Abstract

Since quantum chromodynamics is non-perturbative at low energies, we cannot use conventional perturbative methods. Instead, we turn to chiral perturbation theory to restore predictability at low energies. At next-to-leading order for baryons, the low-energy constants cM and cE, related to the magnetic and electric decuplet- to-octet transitions moments, are unknown. We determine the value of cM by fitting the calculated decay widths of radiative two-body decays of decuplet baryons to data, enabling us to predict the branching ratios of Σ∗0 → Σ0γ ∗0 0 − 0 − and Ξ → Ξ γ. The value of cE is similarly determined using the Ω → Ξ e ν¯e decay, with less precision, however, due to experimental uncertainty. Using our values of cM and cE, we predict the branching ratios of 20 radiative three-body decays of decuplet baryons. In the future, experimental facilities such as the Facility for Anti-proton and Ion Research in Germany will allow us to confirm our predictions and to pin down more accurately the low-energy constants for baryons at next-to-leading order, thus, helping us to further probe the non-perturbative regime of quantum chromodynamics at low energies. Populärvetenskaplig sammanfattning på svenska

Elementarpartiklar, så som elektroner och kvarkar, beskrivs av den så kallade standardmodellen – en kvantfältteori som kombinerar de tre starkaste fundamentala krafterna: elektromagnetism, den svaga växelverkan och den starka växelverkan. För att beskriva verkligheten med standardmodellen används störningsteori, ett kraftfullt matematiskt verktyg som bygger på antagandet att partiklar interagerar svagt med varandra. Men, vid låga energier bryter den starka växelverkan mot detta antagande, vilket gör att störningsteori blir oanvändbart. Problemet uppstår då kvarkar och gluoner binder samman för att bilda större partiklar, så kallade hadroner. Detta gör att vi inte kan använda kvantkromodynamik, den delen ar standardmodellen som hanterar den starka växelverkan, för att beskriva hadroner. Istället kan vi använda kiral störningsteori, en effektiv kvantfältteori som approximerar kvantkromodynamik vid låga energier på ett sätt som återställer vår förmåga att använda störningsteori. Kiral störningsteori konstrueras utefter kvantkromodymanikens symmetrier när de tre lättaste kvarkarnas massor går mott noll, där man skriver ner alla relevanta termer som uppfyller dessa symmetrier upp till en viss ordnings noggrannhet. Denna ordning beskriver hur viktiga alla termer är och gör att man i praktiken kan arbeta med ett ändligt antal termer. I detta arbete använder vi oss av kiral störningsteori för baryoner, vilket är partiklar som består av tre kvarkar, till andra ordningens noggrannhet. Termerna i kiral störningsteori är proportionella mot obestämda konstanter, så kallade lågenergikonstanter. För att beskriva fysiska fenomen måste vi därför först bestämma dessa lågenergikonstanter på något sätt. Vi gör detta genom att jämföra experimentell data med våra egna beräkningar av olika sönderfallsprocesser. Vid andra ordningens noggrannhet för baryoner finns två lågenergikonstanter som ännu inte är bestämda. Vårt mål är därför att göra just detta, med hjälp av att studera sönderfall av exciterade hyperoner. Hyperoner är en typ av baryoner som är uppbyggda av uppkvarkar, nerkvarkar och särkvarkar, med andra ord: de tre lättaste kvarkarna. Genom att jämföra två- och tre-kroppssönderfall av exciterade hyperoner med befintlig data kan vi bestämma de två tidigare okända lågenergikonstanterna, något som möjliggör oss att förutspå sönderfallshastigheterna av hela 22 okända sönderfallsprocesser. I framtiden kommer experiment, så som Facility for Anti-proton and Ion Research i Tyskland att kunna bekräfta våra förutspådda sönderfallsprocesser och därmed hjälpa oss att med bättre noggrannhet bestämma andra ordningens lågenergikonstanter för baryoner. Anledningen varför vi studerar hadronfysik är för att utöka vår förståelse av det problematiska lågenergiområdet inom kvantkromodynamik, vilket vi effektivt kan göra med hjälp av kiral störningsteori. Contents

1 Introduction 3 1.1 Outline ...... 5

2 Background 6 2.1 Symmetries – Lie groups and Lie algebras ...... 6 2.1.1 Lie groups ...... 6 2.1.2 Lie algebras ...... 8 2.1.3 Relating Lie groups and Lie algebras ...... 11 2.1.4 Making real Lie algebras complex ...... 12 2.1.5 Group actions and applications ...... 13 2.1.6 Representations ...... 16 2.2 The Lorentz group ...... 19 2.3 Representations of the Lorentz group ...... 22 2.4 Gamma matrices ...... 24 2.5 Spinors and Parity ...... 26 2.6 The special unitary group (under construction) ...... 26 2.7 Representations of SU(3) (under construction) ...... 28 2.8 Introduction to Quantum Chromodynamics ...... 28 2.8.1 Noether’s theorem ...... 32 2.8.2 Exact symmetries of QCD ...... 35 2.8.3 Approximate symmetries of QCD ...... 36 2.8.4 Chiral symmetry (under construction) ...... 38 2.9 Spontaneous symmetry breaking (under construction) ...... 39 2.10 Chiral perturbation theory (under construction) ...... 41

3 Methods and calculations 42 3.1 Decay processes and kinematics ...... 43

1 Måns Holmberg Master Thesis

3.2 NLO Chiral Lagrangian for baryons (under construction) ...... 47 3.3 Matrix elements ...... 50 3.3.1 Flavor factors ...... 51 3.3.2 B∗(J = 3/2) → Bπ ...... 53 3.3.3 B∗(J = 3/2) → Bγ ...... 55 3.3.4 B∗(J = 3/2) → Bπγ ...... 57 ∗ − 3.3.5 B (J = 3/2) → Be ν¯e ...... 62

4 Results and discussion 68 ∗ 4.1 Determining hA using B (J = 3/2) → Bπ at LO ...... 68 ∗ 4.2 Determining cM using B (J = 3/2) → Bγ at NLO ...... 69 ∗ − 4.3 Determining cE using B (J = 3/2) → Be ν¯e at NLO ...... 70 4.4 NLO predictions of BR(B∗(J = 3/2) → Bπγ)...... 72

4.5 The importance of cE at NLO ...... 76

5 Conclusion and future directions 78

Appendix A Properties of spin-3/2 fermions (under construction) 80

Appendix B Vertices (under construction) 81 B.1 TBΦ ...... 82 B.2 TBΦγ ...... 82 B.3 BBΦ ...... 83 B.4 TT Φ ...... 84 B.5 ΦΦγ ...... 84 B.6 BBγ ...... 85 B.7 T T γ ...... 87 B.8 T Bγ ...... 88

Appendix C Calculation of the matrix element of B∗(J = 3/2) → Bγ 89

Appendix D Electromagnetic current conservation of B∗(J = 3/2) → Bπγ 93

6 Bibliography 98

2 Chapter 1

Introduction

One of the main research challenges of modern particle physics [1] is to understand the confinement regime of quantum chromodynamics (QCD), where perturbation theory, the primary mathematical tool at hand, breaks down. This confinement regime occurs at low energies, where the expansion parameter, the QCD coupling constant, is no longer small. Confinement means that quarks do not exist as free entities but, instead, that they are confined to bound states, called hadrons. Studying hadrons is therefore crucial to understand this energy regime. Hadrons of specific interest, due to increasing experimental activities, are the hyperons – being baryons containing strange, up and down quarks. Studying the decays of (excited) hyperons will allow us to learn more about the intrinsic structure of hadrons and thus probe parts of the non-perturbative confinement regime. At very low energies, chiral perturbation theory (χPT) [2, 3, 4, 5, 6] provides a systematic improvable effective field theory (EFT) to describe and predict QCD phenomena. Since the three lightest quark masses are light (< 100 MeV) on a hadronic scale (∼ 1 GeV), the limit of vanishing (up, down,

strange) quark masses is a good approximation to QCD. In this limit, SU(3)L × SU(3)R×U(1)V chiral symmetry arises, in which the left- and right-handed parts of the Dirac fermion fields (describing quarks) transform independently. This chiral symmetry appears to be spontaneously broken, giving rise to the pseudoscalar meson octet (being pions, kaons, and the η meson), as Goldstone bosons1. The perturbative expansion is carried out in powers of (three-)momenta and masses of the Goldstone bosons, denoted p. The expansion breaks down if the Goldstone momenta become too large, which corresponds to an energy scale were not considered degrees of freedom become active, i.e., the effects of individual quarks become important. In χPT, being an EFT, one writes down all possible terms consistent with the symmetries of QCD in the limit of massless quarks. However, there are an infinite amount of terms that fulfills this

1Since quarks are only approximately massless, chiral symmetry (or more specifically, flavor symmetry) is not exact. This gives rise to Pseudo-Goldstone bosons, since the pseudoscalar meson octet states acquire a small mass proportional to the square root of the quark masses [5].

3 Måns Holmberg Master Thesis

requirement. We herefore construct a power-counting scheme to organize the large number of terms according to their importance. Terms which give rise to lower powers of p are more important, since we expand around small p. This power-counting scheme allows us to omit higher order corrections, giving us a finite amount of terms that we can deal with. Each interaction term of this perturbative expansion (in p) has an unknown coupling constant (from a pure χPT standpoint), that in principle could be determined from the full theory of QCD; but here we run out of luck, since this would require knowledge about the non-perturbative confinement regime of QCD which we wished to probe in the first place. These coupling constant encode information about resonances that come into play at higher energies; therefore, we call them low-energy constants (LEC:s). In practice, there are two approaches we can use to determine LEC:s. We either can determine them by fitting calculated decay widths (from χPT) to experimental data, or we can do it by direct numerical calculations; obtained by putting QCD on a discretized spacetime-lattice (so-called lattice QCD) [7]. we will adopt the first approach. In this work, we are going use the complete and minimal relativistic Lagrangian at next-to-leading order (NLO) for three-flavor χPT in the presence of baryon octet states and baryon decuplet states, constructed in [8]. Different representations of the flavor symmetry group (SU(3)) group give rise to different hadronic multiplets. In particular, the lowest-lying spin-1/2 and spin-3/2 hyperons are contained in the octet and decuplet representations, respectively. At leading order (LO) most LEC:s

are already well known, apart from the LEC called HA, that gives rise to the coupling of a Goldstone boson to the decuplet states, since we do not have simple observables to pin it down. Instead, one

can estimate the value of HA in the limit of a large number of colors (Nc) in QCD [9, 10, 11, 12]. Moreover, χPT deals with explicit symmetry breaking by the quark masses; however, at LO and NLO flavor symmetry is still exact. Since the strange quark is much heavier than that of the up and down quarks, we expect a more significant flavor symmetry breaking with increasing strangeness. Therefore, LEC:s at LO and NLO are expected to differ in value for a given process2, depending on the amount of strangeness involved. In order to quantify this flavor symmetry breaking at LO, we will determine ∗ 3 ∗ ∗ the LEC hA in the case of Ξ → Ξπ decays (one already know hA in the case of ∆ and Σ decays).

The hA interaction term gives rise to the coupling of a decuplet baryon to a pair of Goldstone boson and baryon octet state. At NLO, we know less about the various LEC:s, some of which are unknown. Because of this, we will determine the values of two unknown LEC:s; cM and cE, related to a magnetic transition moment and an axial vector electric transition moment of the decuplet-to-octet transitions, respectively. We ∗+ + ∗0 will determine cM by comparing the calculated decay widths of ∆ → Nγ, Σ → Σ γ, Σ → Λγ

2Flavor symmetry breaking occurs first at next-to-next-to-leading order (NNLO). Also, all LEC:s do have a precise value. The apparent splitting of LEC:s for different strangeness is caused by fitting to data, since nature does not stop at NLO as we do. 3Decuplet baryons, i.e., exited baryons, are denoted by ∗.

4 Måns Holmberg Master Thesis

with experimental results [13]. As in the case of hA, we expect to see a splitting of cM for different

strangeness (Deltas has S = 0 and Sigmas has S = 1), due to flavor breaking. To pin down cE, we will 0 − compare the calculated decay width of Ω → Ξ e ν¯e with data. Unfortunately, the measured branching 0 − ratio of Ω → Ξ e ν¯e have a large uncertainty, hence, limiting our knowledge of cE. Furthermore,

the authors of [14] do not include the cE interaction term in what they claim is a complete NLO Lagrangian; hence, they imply that this term should be included only at higher orders. In [8] we argue 0 − against this and in this work, we will show, by the example of Ω → Ξ e ν¯e, that the cE interaction term is important already at NLO. After determining the values of these LEC:s, we will be able to predict the decay widths of Σ∗0 → Σ0γ and Ξ∗0 → Ξ0γ, as well as of 20 radiative three-body decays. These three-body decays will

also allow us to determine cE more accurately, once experimental data is available. The future experiments at FAIR (Facility for Anti-proton and Ion Research) in Germany will help us study the two and three-body decays considered in this work; thus, promising a better understanding of hyperons, their interactions, and the corresponding LEC:s. The two experimental collaborations HADES (High Acceptance DiElectron Spectrometer) [15] and P¯ANDA (Antiproton ANnihilation at DArmstadt) [16] will both provide excellent opportunities to measure decays of excited hyperons. All of this, to further probe the confinement regime of QCD.

1.1 Outline

This thesis is structured as follows: chapter 2 contains the theoretical background and is divided into three main parts. The first part covers the mathematics of symmetries, the second part is an introduction to QCD, and the third part is about chiral perturbation theory for mesons and baryons. In chapter 3, we present the kinematics of decay processes and performs the calculations of the decay widths. The results and discussions are covered in chapter 4. In the end, chapter 5 summarizes the content of this thesis and provides suggestions for future directions. There are also four appendices which mostly cover explicit calculations too long for the main text. Appendix A contains the main properties of Rarita-Schwinger fields, i.e., spin-3/2 fermions. In appendix C we give the full calculation of the decay width of B∗(J = 3/2) → Bγ, i.e., the radiative two-body decay in section 3.3.3. Appendix B contains the derivations of most vertices used to calculate the decay widths. Finally, in appendix D we check the electromagnetic current conservation of B∗(J = 3/2) → Bπγ from section 3.3.4.

5 Chapter 2

Background

Here we cover a broad range of the theoretical background – from the mathematics of symmetries to chiral perturbation theory. In this chapter, we will introduce the most necessary information to understand topics such as continuous symmetries, spontaneous symmetry breaking, spinors, hadronic multiplets, the fundamentals of QCD and chiral perturbation theory for mesons and baryons. Section 2.1 about symmetries is a thorough introduction to Lie groups and Lie algebras and, as a consequence, is more mathematically oriented than the rest of the thesis.

2.1 Symmetries – Lie groups and Lie algebras

In general, the importance of symmetries in physics cannot be overstated. Symmetries are crucial in many fields of physics, especially when it comes to particle physics; where we mostly deal with continuous symmetries. The gauge symmetry of QCD and the approximate chiral symmetry are two examples of critical continuous symmetries. In general, continuous symmetries are described by Lie groups. Therefore, let us give an introduction to Lie groups and Lie algebras (used to understand Lie groups). We do this to finally understand chiral perturbation theory, which relies on the properties of the Lorentz group O(1, 3) and the SU(3) flavor group, as well as spontaneous symmetry breaking. The material covered in this section is not new, instead, it comes from several sources: The information about Lie groups, Lie algebras and their representations was mainly gathered from [17, 18], the information about group actions is from [18, 19, 20], and finally, the information about the mathematics of spontaneous symmetry breaking is from [21, 22].

2.1.1 Lie groups

A real Lie group G is a group and simultaneously a real smooth manifold, where the group product G × G → G and the inverse map G → G are both smooth. If G as a manifold is instead complex, and

6 Måns Holmberg Master Thesis

the group product and its inverse are complex analytic, then G is said to be a complex Lie group. We will restrict our attention to matrix Lie groups only, which are closed subgroups of the real or complex group of all n×n invertible matrices where the group product is given by matrix multiplication, i.e. the real or complex general linear group GL(n, R) or GL(n, C). Because of this, we are not going to write matrix Lie group explicitly. To begin with, let us define a few concepts of Lie groups that will prove to be useful: A Lie group G is said to be abelian if every two elements of the group commute, that is: [g, g0] = gg0 − g0g = 0 for all g, g0 ∈ G. Otherwise, the group is called non-abelian. A Lie group homomorphism between the Lie groups G and H is a continuous map Φ: G → H such that Φ(gg0) = Φ(g)Φ(g0) (2.1)

0 for all g, g ∈ G. Note that this implies that Φ(eG) = eH , where eG and eH are the identity elements of G and H (i.e., identity matrices), respectively. If Φ is a bijection it is called a Lie group isomorphism. If there exists a Lie group isomorphism between two Lie groups G and H, we call them isomorphic and write G =∼ H, which informally can be read as G is essentially the same as H. Isomorphic groups do not need to be distinguished as their group properties are the same. Two elements g, g0 of a Lie group G are said to be connected if there exists a continuous curve γ(t):[a, b] → G such that γ(a) = g and γ(b) = g0. A Lie group G itself is said to be connected (or path connected) if for every two elements g, g0 ∈ G there exists such a continuous curve. Moreover, a group is said to be simply connected if it is connected and if every closed path (loop) can be continuously shrunk down to a point (i.e. the manifold contains no holes). Let G and H be two Lie groups, then the Cartesian product G × H = {(g, h): g ∈ G and h ∈ H} together with the group multiplication (g, h) · (g0, h0) ≡ (gg0, hh0), for all g, g0 ∈ G and h, h0 ∈ H, is called the direct product of the two groups. The direct product of two Lie groups is a Lie group. Since G and H are two matrix Lie groups, i.e., being closed subgroups of GL(n, C) and GL(m, C), respectively; the Cartesian product G × H is a closes subgroup of GL(n + m, C). Dealing with global geometrical objects, such as Lie groups, can be hard, and it is often easier to deal with local (linearized) objects instead. This idea is key in the theory of Lie groups – rather than studying the group itself, one studies the linearized version of it, which is called the Lie algebra. In a sense, geometry is hard, and algebra is easy. Linearization of the Lie group about the identity produces a new set of elements that form the Lie algebra.

7 Måns Holmberg Master Thesis

2.1.2 Lie algebras

Consider a one-dimensional subgroup of a Lie group G, defined as a Lie group homomorphism γ : R → G (where R is a group under addition). From the requirement (2.1), which in this case translates to

γ(s + t) = γ(s)γ(t), it follows that γ(0) = eG = I. Think of a one-dimensional subgroup as an abstract curve in G where every two elements along the curve commute. Intuitively (and informally), this means that the curve is in some sense straight. Consider for instance, the group of rotations in 3 dimensions (i.e., SO(3)) and let R(θ) be the rotation of an angle θ around the vector nˆ, then, clearly R(θ + φ) = R(θ)R(φ); because rotating around nˆ with an angle θ and then by another angle φ is the same as doing the combined rotation at once (this is just saying that rotations in 2 dimensions commute). Therefore, R is the one-parameter subgroup of the rotations around nˆ. The key point here is to realize that all information about R is completely specified by nˆ; it sounds like R satisfies some differential equation with nˆ as the initial condition. Going back to the general case, we can construct a differential equation precisely like that. Let us solve γ(s + t) = γ(s)γ(t) by taking the derivative with respect to s on both sides and setting s = 0 afterwards. Doing this yields the following differential equation:

d d γ(s + t)| = γ(s)γ(t)| ⇔ γ0(t) = γ0(0)γ(t) . (2.2) ds s=0 ds s=0 with the unique solution (with γ(0) = I as initial condition) given by

γ(t) = exp γ0(0)t
, (2.3) where exp is the regular matrix exponential. Analogously to the previous example, γ is now fully determined by γ0(0) – the initial condition. We could say that γ0(0) generates all elements of γ(t), because for each initial condition γ0(0), there exists a unique solution in some interval of t (if γ and γ0 are continuous). Furthermore, notice that γ0(0) is not an element of the group1, but rather it is part of 0 0 the tangent space of G at e, given by all possible g (0), denoted TeG. We say that γ (0)t is an element

of the Lie algebra g which coincides with TeG as a vector space, hence, informally

Lie group = exp (Lie algebra) . (2.4)

a Every element x ∈ TeG = g can be written as x = α Xa, where a = 1, ..., N, and where {Xa} =

{X1, ..., XN } is some basis of g. The sum over repeated indices is implicit. An arbitrary element of a

1E.g., a tangent line to a circle at some point is not part of the circle itself.

8 Måns Holmberg Master Thesis

2 a Lie group G that is connected to the identity can thus be expressed as g(~α) = exp(α Xa) ∈ G, where the components of the basis are given by

∂ Xa = a g(α) . (2.5) ∂α α=0

These are called the (infinitesimal) generators of the group, because they are all the possible initial directions (in an abstract sense) at identity, whose one-parameter subgroups reach all elements that are connected to identity. In physics, one usually defines the generators with an extra i, such that an a element of the group is g(~α) = exp(−iα Ya), i.e., Ya = iXa. For now, we will not adopt this convention. So far so good, however, the linear structure of g alone is quite uninformative, except for encoding information about the dimension of the manifold (dim(G) = N). Moreover, to justify calling the Lie algebra an algebra, we need a bilinear form that defines the product of two elements. We want this product to give us information about the multiplication of group elements. Let g, g0 ∈ G be connected to the identity, hence, we can write g = eX and g0 = eY where X,Y ∈ g. Then, in general, gg0 = eZ 6= eX+Y ; instead, the argument Z is given by the Baker-Campbell-Hausdorff formula (described in section 5.6 in [17])

1 1 Z = X + Y + [X,Y ] + ([X, [X,Y ]] + [Y, [X,Y ]]) + ··· . (2.6) 2 12

The corresponding element of the Lie algebra of the product of two group elements is thus expressed in terms of nested commutators. All one need to express Z to an arbitrary precision is therefore knowledge about the commutation relation. Consequently, the commutator encodes structure about the group multiplication and is precisely the bilinear form we seek. Note that elements of the Lie algebra only commute if and only if elements of the Lie group commute. A Lie algebra is also called abelian if its elements commute with each other. Now for the more abstract definition of a Lie algebra: A Lie algebra g is a real or complex vector space equipped with an antisymmetric bilinear form [ · , · ]: g × g → g that satisfies the Jacobi identity (2.7). The map [ · , · ] is called the Lie bracket (or Lie product). A bilinear form on some vector space V is a map V × V → V that is linear in each argument separately. Moreover, the antisymmetric property ([X,Y ] = −[Y,X] ∀ X,Y ∈ g) of the Lie bracket directly follows from the definition of the commutator. Note that in general, the Lie bracket does not have to be defined as the commutator (because multiplication might not be defined); however, when restricting our attention to matrix Lie groups only, the Lie bracket is always given by the commutator

2If not, there is no way to reach this group element from the Lie algebra; hence, (2.4) does not always come with an equal sign.

9 Måns Holmberg Master Thesis

(Hall). The Jacobi identity is

[X, [Y,Z]] + [Z, [X,Y ]] + [Y, [Z,X]] = 0 , for all X,Y,Z ∈ g , (2.7) which holds even for Lie brackets of non-matrix Lie groups. Because the Lie bracket is closed (a

fun exercise) we are able to write the Lie bracket of two basis components Xa and Xb as a linear combination: c [Xa ,Xb] = fab Xc , (2.8)

c where fab are real or complex numbers called structure constants, that obviously are antisymmetric in a, b. Upper and lower indices, corresponding to the components of vectors and dual vectors (covectors), respectively, do not have to be distinguished if the metric tensor (of the corresponding manifold) is positive definite; e.g., as in the case of a Euclidian metric. A counterexample is the Lorentzian metric, which is not positive definite. Note that if we simultaneously multiply the structure constants and the generators by a constant the algebra itself does not change, hence there is a freedom of normalization3. Let us now give two examples of some important Lie algebras. Consider the group GL(n, C) = {g ∈ M(n, C) : det g 6= 0}, where M(n, C) is the set of all n × n complex invertible matrices. Every group X X tr(X) element can be written as g = e ∈ G, where X ∈ gl(n, C), with the condition 0 6= det(e ) = e . However, because the exponential function is always nonzero, this condition does not restrict X in any way, meaning that gl(n, C) = M(n, C) is the associated Lie algebra, with the commutator as bracket. Moreover, consider the special linear group SL(n, C) = {g ∈ M(n, C) : det g = 1}, then, using the same argument, it easy to see that sl(n, C) = {g ∈ M(n, C): tr (g) = 0}. Furthermore, here are some useful definitions regarding Lie algebras: Similar to Lie groups, we can have a continuous map between Lie algebras. A Lie algebra homomorphism between Lie algebras is a continuous map φ : g → h such that

φ ([X,Y ]g) = [φ(X) , φ(Y )]h (2.9)

for all X,Y ∈ g. Again, if φ is bijective with a continuous inverse, we call it a Lie algebra isomorphism, denoted g =∼ h. Let h and g be two Lie algebras, then h is called a Lie subalgebra if it is a vector subspace of g that is closed under the Lie bracket, i.e [X,Y ] ∈ h for all X,Y ∈ h. For example, all real traceless matrices 4 form a Lie subalgebra of GL(n, R) . Let U and V be two real or complex vector spaces, then the direct sum of U and V is defined as

3The normalization can include complex numbers without making the Lie algebra complex. The dimension has to change for it to be truly complex. 4Because tr ([X,Y ]) = tr (XY ) − tr (YX) = 0.

10 Måns Holmberg Master Thesis

the set (given by the Cartesian product)

U ⊕ V = {(X,Y ): X ∈ U,Y ∈ V } , (2.10)

equipped with component-wise addition and scalar multiplication. Note that dim(U ⊕ V ) = dim(U) +

dim(V ). The same can be done with Lie algebras as vector spaces, i.e. g1 ⊕ g2 is defined as (2.10), but with the additional Lie bracket

0 0 0 0 [(X,Y ) , (X ,Y )]g1⊕g2 ≡ ([X,X ]g1 , [Y,Y ]g2 ) (2.11)

0 0 for all X,X ∈ g1 and all Y,Y ∈ g2. The direct sum of two Lie algebras is again a Lie algebra. Elements

of g1 and g2 (i.e., (X, 0) and (0,Y )) have intersection (0, 0) and commute with each other according to (2.11). Conversely, if we have two subalgebras a and b of some Lie algebra g that span g as a vector space, have intersection 0 and where [X,Y ] = 0, for all X ∈ a and Y ∈ b; then (X,Y ) → X + Y is an isomorphism of a ⊕ b to g.

2.1.3 Relating Lie groups and Lie algebras

For completeness, we ask ourselves if every (finite-dimensional and real) Lie algebra is the corresponding Lie algebra of some matrix Lie group. Lie’s third theorem (theorem 5.25 in [17]) says that this is the case! We will not use this fact in this thesis, but it is nice to know that whenever you find yourself with a closed commutation relation; there is a corresponding symmetry described by a matrix Lie group. Furthermore, how do Lie group homomorphisms relate to Lie algebra homomorphisms? That is, if we have a Lie group homomorphism Φ: G → H, is there a corresponding Lie algebra homomorphism between g and h (the Lie algebras of G and H)? The answer is yes (theorem 3.28 in [17]). There exists a unique linear map φ : g → h such that Φ(eX ) = eφ(X), where φ satisfies (2.9). Naturally, φ is the

derivative of Φ at identity, since φ is the map TIG G → TIH H = TΦ(IG)H, that is

d tX φ(X) = dΦIG (X) = Φ(e ) , (2.12) dt t=0

for all X ∈ g. On the contrary, the reverse (harder) question is more interesting, as we would like to study Lie algebras instead of Lie groups. Do all Lie algebra homomorphism give rise to a Lie group homomorphism? Unfortunately, not in general, but under the additional requirement that G is simply connected, the converse is true and is also given by Φ(eX ) = eφ(X) (theorem 5.6 in [17]). An essential corollary of this theorem is that if G and H are two simply connected Lie groups with Lie algebras g and h, respectively; then if g =∼ h, we necessarily have G =∼ H (corollary 5.7 in [17]). Also, let G ⊂ GL(n, C) and H ⊂ GL(m, C) be two matrix Lie groups with Lie algebras g and h,

11 Måns Holmberg Master Thesis

respectively. Then, it is not hard to show that the Lie algebra of G × H ⊂ GL(n + m, C) is isomorphic to g ⊕ h (exercise 3.9 in section 5 in [17]).

2.1.4 Making real Lie algebras complex

Let us consider changing the field of a Lie algebra from R to C. This is motivated by the fact that it is often easier to classify the representations of some real Lie algebra by first making it complex. In particular, we will see this first hand when dealing with representations of the Lorentz group. From a mathematical point of view, this is further motivated by the fact that C is algebraically closed, whereas R is not. The complexification or complex extension of a real vector space V is

VC = {X + iY : X,Y ∈ V } , with i(X + iY ) = −Y + iX , (2.13) which is isomorphic to V ⊕ iV (but the notation (X,Y ) is a bit clumsy compared to X + iY ). We would like to stress that this is nothing more complicated than when one constructs the complex plane

from a pair of real numbers, that is, when V = R, we obtain VC = C. Given a Lie algebra g, the

complexification gC defined as above (as a vector space), has the unique Lie bracket extension, given by

[X + iX ,Y + iY ] = ([X ,Y ] − [X ,Y ]) + i ([X ,Y ] + [X ,Y ]) . (2.14) 1 2 1 2 C 1 1 2 2 1 2 2 1

This Lie bracket is again bilinear, antisymmetric and satisfies the Jacobi identity (Proposition 3.37 ∼ ∼ Hall). For example, one can easily check that gl(n, R)C = gl(n, C) and sl(n, R)C = sl(n, C). Also, consider the unitary group U(n), consisting of all unitary n × n complex matrices; whose Lie algebra is † u(n) = {X ∈ M(n, C): X = −X}, since

† gg† = eX eX = I, (2.15)

for all g ∈ U(n) connected to the identity (i.e., with positive determinant) and for all X ∈ u(n). To see that this Lie algebra (and group) is real, we note that (iX)† = iX is hermitian (and not antihermitian)

and thus not part of the algebra. Next, let us investigate the complexification u(n)C, which consists of all elements of the form X + iY , for all X,Y ∈ u(n). We saw above that iY is hermitian, which means that X + iY is a sum of hermitian and antihermitian matrices. This is a well-known way of ∼ decomposing any complex matrix; hence, we conclude that u(n)C = gl(n, C). Likewise, we consider the special unitary group SU(n), which consists of all unitary n×n complex matrices with unit determinant. We saw earlier that group elements with unit determinant translate to traceless elements of the Lie ∼ algebra, thus, since the trace is linear, we conclude that su(n)C = sl(n, C), similar to above.

12 Måns Holmberg Master Thesis

Let us now address why this is useful. It trivially turns out that there is a one-to-one correspondence between the representations of a real Lie algebra and its complexification; which come in handy in situations where, compared to the real case, it is easier to study the complexification of a real Lie algebra. We will later see that a Lie algebra representation is a special case of a Lie algebra homomorphism; therefore, the above statement follows from (proposition 3.37 in [17]):

Let g be a real Lie algebra with complexification gC and let h be some complex Lie algebra. Then, every (real) Lie algebra homomorphism φ : g → h can be uniquely extended to a (complex) Lie algebra

homomorphism of gC to h, given by

φ(X + iY )C = φ(X) + iφ(Y ) (2.16)

for all X,Y ∈ g.

2.1.5 Group actions and applications

3 Given an element R of the group of rotations in 3 dimensions, i.e., SO(3), we rotate a vector ~x ∈ R 3 using matrix multiplication, to obtain R~x. We say that this is a group action of an element of R . In more general terms, a (left) group action from a group H to a set M is a mapping ψ : H × M → M such that ψ(I, m) = m ∀ m ∈ M,I identity of H, (2.17) ψ(h , ψ(h0, m)) = ψ(hh0, m) ∀ h , h0 ∈ H , m ∈ M.

The group action is called transitive if and only if for any two elements m, n ∈ M, there exists an element g ∈ G such that n = ψ(g, m); hence, giving a sense of connectedness. 0 0 0 −1 Consider the example of a group action of G on itself, defined by (g , g) 7→ Cg(g ) = gg g , for 0 g, g ∈ G. This is called the conjugate action. When G = GL(n, C), the conjugate action is what we know from linear algebra to be a similarity transformation. The group action defines an equivalence relation on M; we say that

m ∼ n if and only if n = ψ(h, m) for some h ∈ H. (2.18)

We call the equivalence class of m ∈ M the orbit of m:

ψ(H, m) = {ψ(h, m): h ∈ H} . (2.19)

2 ∼ ∼ To see why this is called an orbit imagine M = R = C and H = SO(2) = U(1), with matrix multiplication as group action. We find that all vectors of equal length belong to the same orbit, thus,

13 Måns Holmberg Master Thesis

justifying the name. Furthermore, if the group action is transitive, only one orbit exists – M itself. More on this later. In the case of the conjugate action, the orbit of g ∈ G is called the conjugacy class of g. The set of all orbits is called the quotient of the action ψ, denoted

M/H = {ψ(H, m): m ∈ M} . (2.20)

2 ∼ + Hence, M/H is obtained from M by identifying every orbit with a single element, e.g., R /SO(2) = R is the set of all possible radii (which cannot be negative). Let H ⊆ G be a subgroup and M = G, with the natural group action ψ(h, g) = gh. Given this, the orbit of g is called the left coset and denoted gH = {gh : h ∈ H}. Similarly, the set of all left cosets is the quotient G/H, which is not necessarily a group!. Given a group action ψ : G × M → M of a group G and a set M, we define the stability group (or as we physicists like to call it – the little group) at m ∈ M by

Gm = {g ∈ G : ψ(g, m) = m} . (2.21)

This is a subgroup of G (Gallier, Warner). Given this definition, symmetry should immediately come to mind, since the stability group is defined by the set of group elements that, given some action, leaves some point m ∈ M invariant. This is precisely where we are going. In fact, the stability group is a key component behind the mathematics of spontaneous symmetry breaking – which in turn is a key concept of chiral perturbation theory.

Let us now show that the choice of m ∈ M for the stability group Gm is arbitrary for a transitive group action, since −1 Gψ(g,m) = gGmg , (2.22) which is easy to show (excuse the slight abuse of notation):

0 0 Gψ(g,m) = {g ∈ G : ψ(g , ψ(g, m)) = ψ(g, m)}

= {g0 ∈ G : ψ(g0g, m) = ψ(g, m)} (2.23) = {g0 ∈ G : ψ(g−1g0g, m) = m}

−1 = gGmg .

Hence, the stability group of m and ψ(g, m) belong to the same conjugacy class, which means that the stability groups of any two elements are necessarily isomorphic, since subgroups of the same conjugacy class are isomorphic. Let us now come back to the statement that for a transitive group action there is only one orbit (or

14 Måns Holmberg Master Thesis

left coset), which is the whole set itself. It might seem intuitive, but we prove it regardless (Gallier, Scherer): Let G be a group and ψ : G × M → M be a transitive group action on the set M. Then, we define the new map ψm : G → M, for a fixed m ∈ M, defined by ψm(g) = ψ(g, m), for all g ∈ G.

Consider the set of ψm acting on the left coset gGm:

0 0 ψm(gGm) = {ψ(gg , m): g ∈ Gm}

0 0 = {ψ(g, ψ(g , m)) : g ∈ Gm} (2.24)

= {ψm(g)}

Consequentially, ψm induces a map ψ˜m : G/Gm → M, defined by ψ˜m(gGm) = ψm(g). We conclude 0 that ψ˜m is injective, since for all g, g ∈ G,

0 0 −1 0 −1 0 0 ψm(g ) = ψm(g) ⇔ ψ(g , m) = ψ(g, m) ⇔ ψ(g g , m) = m ⇔ g g ∈ Gm ⇔ gGm = g Gm . (2.25)

The map ψ˜m is also surjective, since the group action is transitive on M, which make it such that for all ∼ g ∈ G there is an ψ˜m(gGm) = ψ(g, m) = n ∈ M. As a result, ψ˜m is a bijection, such that G/Gm = M. You might wonder what the nature of such quotient sets is. So far, we have not used the fact that we are dealing with Lie groups and, furthermore, we have not yet said anything about the sets which the groups are acting on. The reason why we have not covered this is that it is a bit technical in general; therefore, see [21, 19, 20] if you are curious about the details of quotient spaces. In the special case when we have a Lie group G and a closed Lie subgroup H ⊆ G; the quotient set G/H of a smooth group action forms what is called a homogeneous manifold (or space). A homogeneous manifold is a smooth manifold endowed with a smooth Lie group action. Moreover, if H is normal5, G/H is actually another Lie group (theorem 3.64 in [19]). Now, with all of this machinery at hand, we can describe spontaneous symmetry breaking of some physical system in terms of group theory. Let different physical states of the system be given by m ∈ M, where M is a smooth manifold; and let the system as a whole, described by some scalar function S(m) (e.g., an action), be invariant under the transformations of the Lie group G, given some smooth group action ψ : G × M → M. Then, the minimum of this scalar function, corresponding to a

ground state m0 of the system, might not be invariant under the group action of some elements g ∈ G,

i.e., ψ(g, m0) 6= m0 with the minimum S(ψ(g, m0)) = S(m0). That is, the ground state is not unique! On the other hand, there might be a proper6 closed Lie subgroup of G whose elements do leave the

ground state itself invariant. This subgroup is the stability group Gm0 . But, since the ground state is

5A subgroup H of a group G is normal in G if it is invariant under conjugation by any element of G, i.e. if and only if ghg−1 ∈ H for all g ∈ G and h ∈ H. 6If the subgroup is not proper, it could be the trivial subgroup containing only the identity element, which does not lead to any symmetry breaking. Note that a subgroup H of a group G is proper if and only if H ⊂ G, that is, H 6= G.

15 Måns Holmberg Master Thesis

degenerate, the stability group is not unique either; in fact, all different ground states which can be reached by the group action7 give rise to different stability groups, that are conjugate to each other (as 0 shown in (2.22)). For a specific ground state m0, two group elements g, g ∈ G are equivalent if they 0 0 0 are related by g = g h for some h ∈ Gm0 , following ψ(g, m0) = ψ(g h, m0) = ψ(g , m0). This forms an

equivalence class, given by the left coset gGm0 . Finally, the quotient G/Gm0 is the set of all such cosets,

i.e., representing all group elements which transform m0 into different ground states. It follows that

G/Gm0 is isomorphic to the set of all ground states, which is a homogeneous space that we denote by

M0. The dimension of the space of all ground states is dim(M0) = dim(G/Gm) = dim(G) − dim(Gm) (corollary 3.7.26 in [21]). In quantum field theory, it is in this space where the so-called Goldstone bosons live. More on this and its application to QCD is described in section 2.9.

2.1.6 Representations

A finite-dimensional representation of a Lie group G is a Lie group homomorphism

Π: G → GL(V ) , (2.26) where V is a real or complex vector space, and where GL(V ) denotes all invertible linear transformations of V . Similarly, a finite-dimensional representation of a Lie algebra g is a Lie algebra homomorphism

π : g → gl(V ) , (2.27) where gl(V ) = End(V ) is the set of all linear transformations of V (also called an endomorphism of V ). By choosing a basis of GL(V ) and gl(V ), we can identify them with GL(n, C) and M(n, C), respectively. If two representations are isomorphic we do not distinguish between them, for example if two representations only differ by a choice of basis (thus are related by a similarity transformation). Representations are invertible linear transformations on some vector space V , which means that we can construct two actions Π: G × V → V and π : g × V → V , defined as (g, v) → Π(g)v and (X, v) → π(X)v, respectively, for g ∈ G, X ∈ g and v ∈ V . There are a few ways one can go about combining representations to generate new ones. Here we cover direct sums and tensor products of group representations. We start with direct sums:

Let Π1 , Π2 ,..., Πn be n representations of the matrix Lie group G acting on the vector spaces

V1 ,V2 ,...,Vn. The direct sum of these representations is a new representation Π1 ⊕ Π2 ⊕ · · · ⊕ Πn of

G that acts on V1 ⊕ V2 ⊕ · · · ⊕ Vn, defined by (similar to (2.10))

(Π1 ⊕ Π2 ⊕ · · · ⊕ Πn(g)) (v1 , v2 , . . . , vn) = (Π1(g)v1 , Π2(g)v2 ,..., Πn(g)vn) (2.28)

7 I.e., the group action is transitive when acting on the subset of all ground states (M0).

16 Måns Holmberg Master Thesis

for all g ∈ G. The definition is the same for Lie algebras. In the language of matrices, we can write the representation of direct sums as a block diagonal matrix, where the elements of the representations

make up the blocks. In this representation of representations, the vectors v1 , v2 , . . . , vn on which the representations are acting becomes a single combined column vector. Direct sums of Lie algebra representations work in an analogous way. See (section 4.3.1 Hall) for more information. Next up is tensor product representations. We first need to know the tensor product acting on vector spaces. Given two vector spaces, we can build a new vector space by introducing the tensor product. Let U and V be two finite-dimensional real or complex vector spaces, then, the tensor product of U and V , denoted U ⊗ V , is a vector space W , endowed with a bilinear map φ : U × V → W . One can show that a tensor product of U and V always exist and that it is unique, since any two tensor

products (W1, φ1) and (W2, φ2) of U and V are isomorphic. Therefore, we do not need to specify φ as long as U and V are two finite-dimensional real or complex vector spaces; thus, we denote φ(u, v) as

u ⊗ v for all u ∈ U and v ∈ V . Moreover, suppose that e1, e2, ··· , en is a basis of U and f1, f2, ··· , fm is a basis of V , then

{ei ⊗ fj : 1 ≤ i ≤ n , 1 ≤ j ≤ n} , (2.29)

is a basis for W . From this we see that dim(U ⊗ V ) = dim(U) dim(V ). We can think of the tensor product of vector spaces as a generalization of the outer product, since the outer product is the special case (u ⊗ v = uT v). The uniqueness of the bilinear map φ and (2.29) is described in more detail in (theorem 4.14 in [17]). To obtain tensor product representations, we need to define the tensor product of linear maps. Again, let U and V be two finite-dimensional real or complex vector spaces and let A : U → U and B : V → V be two linear maps. Then, the tensor product A ⊗ B is the unique linear map

(A ⊗ B): U ⊗ V → U ⊗ V, (2.30)

defined by (A ⊗ B)(u ⊗ v) = (Au) ⊗ (Bv) , (2.31)

for all u ∈ U and v ∈ V . The tensor product is given by the Kronecker product when representing the two linear maps as matrices.

Finally, let Π1 be a representation of the Lie group G and let Π2 be a representation of the Lie group H, acting on the vector spaces U and V , respectively. Then, according to (2.31), the tensor

product representation Π1 ⊗ Π2 acting on U ⊗ V , is a representation of G × H, given by

(Π1 ⊗ Π2)(g, h)(u ⊗ v) = (Π1(g)u) ⊗ (Π2(h)v) , (2.32)

17 Måns Holmberg Master Thesis

for all g ∈ G,h ∈ H and u ∈ U, v ∈ V . It is important to note that this notation is, unfortunately,

ambiguous in the special case when H = G, since Π1 and Π2 would be representations of the same

group G, meaning that we can either regard Π1 ⊗ Π2 as a representation of G or G × G. Hence, we

need to be careful to specify in what way we use Π1 ⊗ Π2. Now for the case of Lie algebras. Let G and H be two matrix Lie groups with algebras g and h.

Again, let Π1 and Π2 be representations of the Lie groups G and H, acting on the spaces U and V

respectively. Then, consider the tensor product representation Π1 ⊗ Π2 of G × H. The associated Lie

algebra representation, denoted π1 ⊗ π2, is then (remember (2.12))

d tX tY (π1 ⊗ π2)(X,Y ) = Π1(e ) ⊗ Π2(e ) dt t=0     d tX d tY (2.33) = Π1(e ) ⊗ IV + IU ⊗ Π2(e ) dt t=0 dt t=0

= π1(X) ⊗ IV + IU ⊗ π2(Y ) ,

tX tY with g = e ∈ G, h = e ∈ H and X ∈ g, Y ∈ h. Here, π1 and π2 are the corresponding Lie algebra representations of g and h. Since the associated Lie algebra of G × H is g ⊕ h, in the case of matrix Lie

groups, we have that π1 ⊗ π2 is a representation of g ⊕ h, that is acting on U ⊗ V . Explicitly, π1 ⊗ π2 acts on U ⊗ V as

(π1 ⊗ π2)(X,Y )(u ⊗ v) = (π1(X)u) ⊗ v + u ⊗ (π2(Y )v) , (2.34) where u ∈ U and v ∈ V . Moreover, let Π be a representation of a Lie group G acting on a vector space U. A vector subspace V ⊂ U is invariant if Π(g)v ∈ V for all v ∈ V and g ∈ G. A representation is called irreducible (or simple) if it has no invariant subspaces except the trivial ones: {0} and V itself. In addition, a representation is completely reducible (or semisimple) if V is isomorphic to a direct sum of irreducible representations. The tensor product of two irreducible representations of a group G will, in general, not be irreducible when viewed as a representation of G8. The process of decomposing the tensor product of two representations (irreducible or not) as a direct sum of irreducible representations is called Clebsch-Gordan decomposition. Let us now give an example of one of the most important representations – the adjoint representation. For a matrix Lie group G, with Lie algebra g, the adjoint representations is the map Ad: G → GL(g), −1 9 given by g 7→ Adg(X) = gXg for a fixed X ∈ g . This has its origin as the derivative of the conjugation action at identity. The corresponding representation of the Lie algebra is obtained from

8The same tensor product representation is, however, irreducible when instead viewed as a representation of G × G. 9 gXg−1 X −1 −1 n You can check that Adg(X) is indeed part of g by showing that e = ge g ∈ G, using the hint: (gXg ) = gXng−1.

18 Måns Holmberg Master Thesis

(2.12):

d tX −tX adX (Y ) = d(AdeX )e(Y ) = e Y e = XY − YX = [X,Y ] , (2.35) dt t=0 for all X,Y ∈ g. In physics, we often consider the explicit matrix elements of the adjoint representation (of the Lie algebra), which turns out to be given by the structure constants. To see this, we choose a

basis {Xa} of g and consider what these become in the adjoint representation. Doing this, we get

c adXa (Xb) = [Xa ,Yb] = fab Tc , (2.36)

c c which written in components are (adXa (Xb)) = fab . Instead of this cumbersome notation, we usually c c write [Xa]b = fab . This is in accordance with [23].

2.2 The Lorentz group

The Lorentz group O(1, 3) is a Lie group that is given by the set of all transformations that leave the T µ ν µ 4 symmetric bilinear form hx, gyi = x gy = gµνx y = x yµ invariant, where x, y ∈ R are four-vectors and g = diag(1, −1, −1, −1) is the (flat) metric of spacetime. That is, the Lorentz group consists of all T 10 Λ ∈ M(4, R) such that hΛx, gΛyi = hx, gyi or equivalently g = Λ gΛ, i.e. ,

 T O(1 , 3) = Λ ∈ M(4, R): g = Λ gΛ . (2.37)

We will not discuss the origin of the above bilinear form, and why the metric has this specific signature, however, one should know that all of this follows as a direct consequence of the postulates of special relativity. The Lorentz group is not connected, in fact, it consists of four disjoint components, corresponding 0 0 11 to det Λ = ±1 and Λ 0 ≥ 1 or Λ 0 ≤ −1 . None of these components are simply connected. The Lie algebra of a Lie group only deals with group elements which are connected to the identity; hence, the Lie algebra can only reach the elements of the Lorentz group that are in the component that includes 0 the identity – the det Λ = 1, Λ 0 > 1 component. We call this restriction (which is the only component that is also a subgroup) of the Lorentz group the proper orthochronous Lorentz group12, which we will denote by SO(1, 3)+. We find the Lie algebra of SO(1, 3)+ by

T T −1 g = ΛT gΛ ⇐⇒ I = ΛT gΛg−1 ⇐⇒ I = etX getX g−1 ⇐⇒ I = etX etgXg , (2.38)

10This group is in general called the indefinite orthogonal group. 11We find the first condition by taking the determinant of g = ΛT gΛ and the second condition by looking at the µν σρ µ ν µ = ν = 0 component of the same relation in index notation, i.e. g = g Λ σΛ ρ. 12Proper refers to det (Λ) = 1, which means that spatial orientation is preserved, i.e., improper rotations are not 0 allowed. Furthermore, orthochronous refers to Λ 0 ≥ 1, which means that the direction of time is preserved.

19 Måns Holmberg Master Thesis

and by then differentiating both sides and setting t = 0,

d tXT tgXg−1 T −1 T 0 = e e ⇐⇒ 0 = X + gXg ⇐⇒ X g = −gX . (2.39) dt t=0

As a result, the Lie algebra of SO(1, 3)+ is given by

T so(1, 3) = {X ∈ M(4, R): X g = −gX} . (2.40)

We can now count the degrees of freedom to find the number of generators, which we can do directly by plugging in a generic 4 × 4 matrix into (2.39), or by

T µ µ ρµ µ ρ µν νµ (X g) ν = −(gX) ν ⇐⇒ X gρν = −g ρX ν ⇐⇒ X = −X . (2.41)

In the last step we contracted the ρ index and raised the lower index by the metric on both sides. Thus, we have found that elements of the Lie algebra are anti-symmetric 4 × 4 matrices (with both indices up) and these we know have 6 degrees of freedom, meaning that there are 6 generators. Next, let us choose a set of generators that have a nice physical meaning and use these to figure out the commutation relations; these are13:       0 0 0 0 0 0 0 0 0 0 0 0       0 0 0 0  0 0 0 i 0 0 −i 0       J1 =   ,J2 =   ,J3 =   , (2.42) 0 0 0 −i 0 0 0 0 0 i 0 0       0 0 i 0 0 −i 0 0 0 0 0 0

and       0 i 0 0 0 0 i 0 0 0 0 i       i 0 0 0 0 0 0 0 0 0 0 0       K1 =   ,K2 =   ,K3 =   . (2.43) 0 0 0 0 i 0 0 0 0 0 0 0       0 0 0 0 0 0 0 0 i 0 0 0

The set Ji should be familiar as the generators of rotations in 3 dimensions and the set Ki are the generators of boosts. Using these generators, a general Lorentz transformation of SO(1, 3)+ can now

be written as Λ = exp(iθiJi + iφiKi), where the extra i in the exponent exists to cancel the i in the generators, something that is purely a convention. The corresponding commutation relations of the Lie algebra in this basis are

[Ji ,Jj] = iijkJk , [Ji ,Kj] = iijkKk , [Ki ,Kj] = −iijkJk . (2.44)

C By going to the complexification of the Lie algebra, i.e., so(1, 3) −→ so(1, 3)C, we can create a new

13 µ µν These are not anti-symmetric since they are of the form (Ji) ν and not (Ji) .

20 Måns Holmberg Master Thesis

basis 1 1 N = (J + iK ) ,N † = (J − iK ) (2.45) i 2 i i i 2 i i which are closed under the commutator and commute with each other. We find

† † † † [Ni ,Nj] = iijkNk , [Ni ,Nj ] = iijkNk , [Ni ,Nj ] = 0 . (2.46)

Therefore there are two sets of commuting Lie algebras which have the same form of angular momentum operators as su(2), but over the field C. Hence, in the basis (2.45), we form two Lie subalgebras of

su(2)C. Furthermore, these subalgebras span so(1, 3)C, only intersect at the origin and commute with

each other. This means that so(1, 3)C is isomorphic to su(2)C ⊕ su(2)C. This is good, since we know how su(2) representations work.

A finite-dimensional irreducible representation of su(2) (or equivalently su(2)C, since one can always restrict the complex representation to the real part only) are specified by a positive integer or half-integer n = 0, 1/2, 1, 3/2,... , representing spin. Therefore, we label these su(2) representations by

πn, which, can be shown to be 2n + 1 dimensional [23].

From section 2.1.6 we know that a representation of the direct sum su(2)C ⊕ su(2)C is given 0 by πn ⊗ πn0 , where n, n specifies the two su(2) representations; let us use the following shorthand 0 notation π(n,n0) = πn ⊗ πn0 . We can now conclude that a (n , n ) representation (of the Lie algebra) is (2n + 1)(2n0 + 1) dimensional. There is a subtlety here, coming from the fact that the Lorentz group is not simply connected, thus, a representation of the Lie algebra of the Lorentz group might not be a representation of the group, see section 2.1.3. It turns out that a representation of the Lie algebra corresponds to a representation of the group only if n + n0 is an integer (see section 5.6 in [24]). If instead n + n0 is a half-integer, this is not true; however, in this case, we get a so-called projective representation of the Lorentz group. It is these projective representations that contain half-integer angular momenta, meaning that they are of great interest in physics. One usually extends the group to also include projective representations, to what is called its universal cover (or universal covering group). We can quickly figure out that the universal cover of SO(1, 3) is SL(2, C), since

C ∼ so(1, 3) −→ so(1, 3)C = su(2)C ⊕ su(2)C ∼ = sl(2, C) ⊕ sl(2, C) (2.47) ∼ = sl(2, C) ⊕ i sl(2, C) ∼ C = sl(2, C)C ←− sl(2, C) ,

21 Måns Holmberg Master Thesis

shows that the complexification of both SO(1, 3) and SL(2, C) have the same Lie algebra (up to 14 isomorphism) . Because SL(2, C) is simply connected, it follows that SL(2, C) is the universal cover of SO(1, 3) (which can be shown to be unique; see section 5.8 in [17]). You can read more about projective representations and the topology of the Lorentz group in section 2.7 in [24]. From now on, we consider the only universal cover.

We can find the spin content of a representation π(n,n0) by decomposing it as a direct sum of irreducible representations of su(2), i.e., Clebsch-Gordan decomposition. This is described in more detail in Appendix C in [17]. Here is the result15:

π(n,n0) = πn ⊗ πn0 = πn+n0 ⊕ πn+n0−1 ⊕ · · · ⊕ πn−n0+1 ⊕ πn−n0 . (2.48)

With this in mind, let us briefly discuss the representations of most interest in this thesis: The scalar representation, the left- and right-handed Weyl spinor representations, the vector representation; and finally, the Dirac spinor representation to describe spin-1/2 particles (with both left- and right-handed components) and the Rarita-Schwinger representation to describe spin-3/2 particles.

2.3 Representations of the Lorentz group

We start with the irreducible representation (0, 0), a 1 dimensional representation with angular momenta † j = 0. We see that the only 1 × 1 dimensional ”matrices” π(0,0)(Ni) and π(0,0)(Ni ) that satisfy the † commutation relations (2.46) are π(0,0)(Ni) = π(0,0)(Ni ) = 0. Meaning that this representation concerns objects which do not change under Lorentz transformations, i.e., these are (Lorentz) scalars. Things get a bit more interesting when considering the two 2-dimensional irreducible representations

(1/2, 0) and (0, 1/2) with angular momenta j = 1/2, where we can choose π(1/2,0)(Ni) = σi/2 and † † π(1/2,0)(Ni ) = 0 and π(0,1/2)(Ni) = 0 and π(0,1/2)(Ni ) = σi/2, respectively. Here, σi are the three Pauli matrices       0 1 0 −i 1 0 σ1 =   , σ2 =   , σ3 =   . (2.49) 1 0 i 0 0 −1

Writing the transformations of the two representations explicitly (at the group level), we have16

σi σi
σi σi
iθi +φi iθi −φi Π(1/2,0)(Λ) = e 2 2 , Π(0,1/2)(Λ) = e 2 2 . (2.50)

14This is an explicit example showing that Lie algebras, being intrinsically local, contain less information than Lie groups, with global properties; which is the reason we chose to study Lie algebras instead of Lie groups in the first place. 15This shows that the tensor product of two irreducible representations is not necessarily irreducible. Also, note that π(n,n0) is irreducible when viewed as a tensor product representation of the direct sum of two su(2), whereas it is reducible when viewed as a tensor product representation of su(2). 16 † −1 † −1 We note that Π(1/2,0)(Λ) = Π(0,1/2)(Λ) and Π(0,1/2)(Λ) = Π(1/2,0)(Λ) .

22 Måns Holmberg Master Thesis

Hopefully, you recognize the Pauli matrices as the angular momentum operators for spin 1/2 particles. The two-component objects these representations act on are called Weyl spinors. We say that a spinor in the (1/2,0) representation is left-handed and a spinor in the (0,1/2) is right-handed, which we denote by ψL and ψR (ignoring indices). If we consider a rotation in this representation, one finds that spinors have to rotate by an angle of 4π to return to their original configuration (an effect which is closely linked to the notion of projective representations), e.g., a rotation by 2π about the x3 axis is e(iπσ3) = −1. Clearly, this is not how vectors behave under rotation; spinors are fundamentally different. Furthermore, consider the 4-dimensional representation (1/2, 1/2) with angular momenta j = 0 and 1 (see (2.48)). The spin 0 part has 1 component and the spin 1 component has 3 components, hence, this describes a four-vector, with 1 time component and 3 spatial components. This representation is the so called standard representation, in which π(X) = X, where π is a Lie algebra representation † † and X is an element of a Lie algebra; i.e., π(1/2,1/2)(Ni) = Ni and π(1/2,1/2)(Ni ) = Ni . For more information about this representation and its connection to spinors, see chapter 34, 35 and 36 in [25]. So far we have only dealt with representations of the proper orthochronous Lorentz group, but let us now extend our consideration to include two discrete Lorentz transformations: parity transformation P and time-reversal transformations T , given by17

    +1 0 0 0 −1 0 0 0      0 −1 0 0   0 +1 0 0      P =   , T =   . (2.51)  0 0 −1 0   0 0 +1 0      0 0 0 −1 0 0 0 +1

In particular, consider how the generators of rotations (2.42) and boosts (2.43) transform under parity transformation, which flips all spatial coordinates. Rotations are obviously unchanged, since they only treat spatial coordinates, which all transform in the same way under a parity transformation,

i.e., Ji → PJiP = Ji. Boosts on the other hand do change under parity, since they mix spatial and † time coordinates. One finds Ki → PKiP = −Ki. As a result, Ni → PNiP = Ni , meaning that a left-handed Weyl spinor transforms into a right-handed Weyl spinor under parity transformation,

and vice versa, i.e. ψL ↔ ψR. In some sense, this justifies the left-right naming convention, because something left-handed is changed into something right-handed under parity transformation, just like normal hands in a mirror. We learn that if we want to describe a physical system with (massive) spinors that is parity trans- formation invariant, we will always need both right-handed and left-handed Weyl spinors. Therefore,

17These discrete transformations can be used to go between the four disjoint components of the Lorentz group.

23 Måns Holmberg Master Thesis we introduce the four-component Dirac spinor (also called bispinor):

  ψL ψ =   , (2.52) ψR

that transform under the direct sum representation (1/2, 0) ⊕ (0, 1/2), explicitly given by18:

  Π(1/2,0)(Λ) 0 S(Λ) =   . (2.53) 0 Π(0,1/2)(Λ)

Next up, the Rarita-Schwinger representation of spin-3/2 particles. This is obtained by the tensor product representation of the vector representation (1/2 , 1/2) and the Dirac spinor representation (1/2 , 0) ⊕ (0 , 1/2) (Weinberg 5.6):

1 1 1   1 1  1   1  1 , ⊗ , 0 ⊕ 0 , = , 1 ⊕ , 0 ⊕ 1 , ⊕ 0 , . (2.54) 2 2 2 2 2 2 2 2

The dimension of the vector space we act on is therefore 4×4 = 16, thus, we write the object on which this representation is acting as ψµ. Because of this, ψµ is sometimes called a vector-spinor. We can convince ourselves that (2.54) is correct by looking at the spin content of both sides; e.g., looking at either the left- or right-handed component of the Dirac representation, we find (1⊕0)⊗(1/2⊗0) = 3/2⊕1/2⊕1/2 and (1/2 ⊗ 1) ⊕ (1/2 ⊗ 0) = 3/2 ⊕ 1/2 ⊕ 1/2 (see (2.48)). However, since we only want do describe spin-3/2 particles, we have to somehow get rid of these extra spin-1/2 components. We do this in appendix A, after introducing gamma matrices.

2.4 Gamma matrices

We now introduce a set of matrices that allows us to combine four-vectors and Dirac spinors in a meaningful way – Gamma matrices. The full motivation behind introducing these matrices is a bit technical, but outside the scope of this thesis. We note, however, that these matrices have its origin in the fact that a four-vector can be expressed as an object with two left-handed spinor components and two right-handed components; thus, one can act with a four-vector on a left- or right-handed Weyl spinor as if it were a matrix. The gamma matrices provide a compact way of doing precisely this, for Dirac spinors. Again, see (Srednicki chapter 34, 35, 36) for an in-depth introduction19.

18 We name this representation S, in accordance with [25], instead of Π(1/2,0)⊕(0,1/2), because of convenience. 19There is an even more general way of introducing gamma matrices than what is covered by Srednicki; using the Clifford algebra, an algebra of the form (2.56) (or its Euclidian analog), of the Lorentz group (in any dimension). See (Hamilton chapter 6.2) for a deep dive into this topic.

24 Måns Holmberg Master Thesis

The four 4 × 4 gamma matrices γµ, µ = 0, 1, 2, 3, are defined as (in the Weyl or chiral basis):

    i 0 I2×2 0 σ γ0 = , γi = i = 1, 2, 3 , (2.55)    i  I2×2 0 −σ 0

and satisfy the anticommutation relation

µ ν µν {γ , γ } = 2g I4×4 . (2.56)

Covariant gamma matrices (i.e., with a lower index) are defined just like covariant four-vectors. We µ µ can then act with a four-vector A on a Dirac spinor ψ by the product A γµψ. Naturally, we want to be able to construct Lorentz scalars from Dirac spinors; in order to eventually build a Lorentz invariant Lagrangian. One candidate is ψ†ψ, but this is not Lorentz invariant, since S(Λ) is not unitary20 Instead, as     Π (Λ)† 0 Π (Λ)−1 0 S(Λ)† = (1/2,0) = (0,1/2) = γ0S(Λ)−1γ0 , (2.57)  †  −1 0 Π(0,1/2)(Λ) 0 Π(1/2,0)(Λ) we define ψ¯ ≡ ψ†γ0, such that ψψ¯ is Lorentz invariant. Note that (γ0)2 = I.

Moreover, it is useful to define a new matrix γ5, which is a product of the four gamma matrices, as follows:   5 0 1 2 3 −I2×2 0 γ ≡ iγ γ γ γ =   . (2.58) 0 I2×2

Using (2.56), one can show that

2 {γµ γ5} = 0 , (γ5) = I. (2.59)

We will use a lot of identities of gamma matrices when calculating the decay width. Most often, these are identities of traces of gamma matrices, that we will prove as we go along. Now, let us use gamma matrices to study how spinors transform under the discrete parity transformation.

20For rotations (θ 6= 0, φ = 0) we see from (2.50) that S(Λ) is unitary; however, for boosts (θ = 0, φ 6= 0), S(Λ) is clearly not unitary (because of the missing i in the exponent). In fact, there are no finite-dimensional unitary representations of the Lorentz group, since the same argument repeats for all finite-dimensional representations. This leads to: ψ†ψ → (S(Λ)ψ)†S(Λ)ψ = ψ†S(Λ)†S(Λ)ψ 6= ψ†ψ.

25 Måns Holmberg Master Thesis

2.5 Spinors and Parity

Under the parity transformation, we know from before that a left-handed Weyl spinor transforms into a right-handed Weyl spinor and vice versa. For Dirac spinors, this means

P ψ −→ γ0ψ . (2.60)

As a consequence, ψ¯ transforms as follows

P † ψ¯ −→ (γ0ψ) γ0 = ψγ¯ 0 . (2.61)

We then consider how Lorentz scalars of the form ψAψ¯ transform under parity, where A is the identity or some combination of gamma matrices. We call these Lorentz scalars Dirac spinor bilinears (or fermion bilinears) since they can be regarded as bilinear forms (ψ, ψ) → ψAψ¯ . From (2.60) and (2.61) we conclude that P ψAψ¯ −→ ψγ¯ 0Aγ0ψ . (2.62)

In table 2.1 we show the behavior under parity transformation for various A.

Bilinear Transformation under parity Name ψψ¯ +ψψ¯ Scalar

ψγ¯ 5ψ −ψγ¯ 5ψ Pseudocalar µ µ ν ψγ¯ ψ +P νψγ¯ ψ Vector µ µ ν ψγ¯ γ5ψ −P νψγ¯ γ5ψ Axial-vector

Table 2.1: Behavior under parity transformation of Dirac spinor bilinears.

We now need to describe one more Lie group before constructing the QCD Lagrangian – the special unitary group SU(N).

2.6 The special unitary group (under construction)

The special unitary group of degree n, denoted SU(n), is a Lie group defined by the set of all n × n † unitary matrices with determinant 1, i.e. U U = I and det (U) = 1. Topologically, the group is simply connected, meaning that every element of the group can be continuously transformed into the identity, and compact, meaning that the parameters of the group vary over a closed interval.

26 Måns Holmberg Master Thesis

Any element U ∈ SU(n) can be expressed as

U = eiH , (2.63) where H is an hermitian and traceless n × n matrix. Unitarity implies that H is hermitian:

† −iH† iH −iH iH U U = e e = e e = I , (2.64)

and det (U) = 1 implies that H is traceless:

det (U) = det eiH
= eitr(H) = 1 . (2.65)

The dimension of the group is not equal to the degree, but is instead n2 − 1. This is easily seen by counting the number of linearly independent hermitian and traceless matrices: In general, an arbitrary complex n × n matrix has 2n2 independent parameters. Hermiticity imply real diagonal entries (n parameters) and that the lower off-diagonal entries are fixed by the upper off-diagonal entries. There are (n2 − n)/2 upper off-diagonal complex entries, giving n2 − n parameters, counting both the real and imaginary parts. In total, a hermitian matrix has n2 independent parameters.

n2 − n n + 2 = n2 . (2.66) 2

Tracelessness reduces this by one, giving n2 − 1.

It is convenient to express H in terms of a basis set of hermitian and traceless matrices Ta, where a = 1 , ..., n2 − 1. Doing this, we can write (2.63) as

U = eiωaTa , (2.67)

2 with n − 1 parameters ωa ∈ R. The matrices Ta are called the generators of the group, and are chosen to obey the normalization condition 1 tr (T T ) = δ . (2.68) a b 2 ab For every Lie group there is an associated Lie algebra, which in the case of SU(n), is denoted su(n). The Lie algebra is used to study infinitesimal group transformations (connected to the identity). Every Lie algebra has a Lie bracket that specifies the algebra, a binary operation, which is bilinear, skew symmetric and satisfies the Jacobi identity. For su(n), the Lie bracket is

[Ta ,Tb] = ifabcTc . (2.69)

27 Måns Holmberg Master Thesis

Here, fabc are the structure constants of the group. If fabc = 0, the group is said to be abelian, meaning that every pair of group elements commute. Otherwise, the group is said to be non-abelian. For example, any group with only one generator must be abelian, because the generator clearly commutes with itself. It is easy to show that fabc must be completely antisymmetric. Multiplying (2.69) by Td on the right, and taking the trace (using the normalization (2.68)), we find

i f = − tr ([T ,T ] T ) . (2.70) abd 2 a b d

Thus fabc is completely antisymmetric, due to the cyclic property of the trace.

2.7 Representations of SU(3) (under construction)

2.8 Introduction to Quantum Chromodynamics

Quantum chromodynamics, or QCD for short, is the theory describing the strong nuclear interaction of quarks and gluons. It is a non-abelian gauge theory (also called Yang-Mills theory) with symmetry group SU(3), that is, a type of quantum field theory that is invariant under local SU(3) transformations. The fact that the theory is non-abelian, which refers to the non-commutative property of two arbitrary elements of the symmetry group, makes QCD a strikingly different theory compared to quantum electrodynamics (QED), as we will find out shortly. Analogously to electric charge in QED, QCD also comes with conserved charges, namely three color charges: red, green and blue. Note, however, that these colors have nothing to do with the real colors of light. Color is a "new" quantum number that was introduced shortly after the discovery of quarks – to explain the apparent violation of the Pauli exclusion principle of quarks inside some hadrons that otherwise would have identical quantum states (REF). Specifically, the problem occurred when baryons had quarks of all the same flavor, e.g., as in the case of the ∆++ baryon that consists of three up quarks. To fix this, physicists introduced color: A quantum number with three possible values (not counting anticolor) to make each of the quarks inside a baryon unique. This realization, together with the discovery of asymptotic freedom, among other theoretical developments, naturally lead to the conclusion that SU(3) was the correct gauge group of QCD (REF). The gauge group of QCD is also referred to as the color group. The basic building blocks of QCD are the quarks q, being matter fields that live in the fundamental

representation (3) of the color group; and gluons Aµ, being gauge fields that belong to the adjoint representation (8). Antiquarks, denoted q¯, with opposite color-charge (anticolor) belong to the complex conjugate representation (3¯). Hence, quarks and antiquark are color triplets, and gluons are color octets. The quarks are massive spin-1/2 Dirac fermions that come in six different flavors, named: "up", "down", "strange", "charm", "bottom" and "top". A more descriptive name instead of flavor would be

28 Måns Holmberg Master Thesis

type, as different flavors are different types of particles, but we will stick to the convention. Let us denote these flavors with an index f = 1, ..., 6, where 1 = up, 2 = down, etc. Each quark field is an

object with three indices: qcfs, with color index c = 1, 2, 3, flavor index f = 1, ..., 6 and spinor index s = 1, 2, 3, 4 (because they are fermions), i.e.

  qred,fs   qcfs =  q  . (2.71)  green,fs  qblue,fs

Let us now construct the QCD Lagrangian by means of gauging the free quark Lagrangian. This refers to the process of localizing the symmetry of a non-interacting field theory which, to begin with, is symmetric with respect to a continuous global symmetry. A global symmetry is one which is coordinate independent, and by contrast, a local symmetry is one which is coordinate dependent. For the global symmetry to become local one is forced to include additional (gauge) fields in the action, which ends up mediating interactions between the previously free matter fields. This method of gauging is called the gauge principle – a principle that underlies most, if not all, modern theories of elementary particles; from the Standard Model to string theories. We start with the free quark Lagrangian that is invariant under global color SU(3) transformations. The free quark Lagrangian density is given by

µ

Lfree =q ¯cfs iδcc0 δff 0 (γµ)ss0 ∂ − δcc0 δss0 Mff 0 qc0f 0s0 =q ¯ i∂/ − M q , (2.72) where all repeated indices are summed over. We will from now on suppress these indices in favor of readability. The mass matrix M is a diagonal matrix in flavor space, that encodes the masses of each quark flavor, or if you want, the amount of chiral symmetry breaking (see 2.8.4), i.e.

M = diag(mu , md , ms , mc , mb , mt) . (2.73)

µ µ Note the Feynman slash notation: ∂/ = γ ∂µ, where γ are the four Gamma matrices (see section 2.4) that we choose to be in the Weyl basis because of its usefulness when dealing with massless fermions. Consider the global gauge transformation

q → Uq (2.74) q¯ → (Uq)†γ0 =qU ¯ † , where U ∈ SU(3) acts only on color. It will prove useful to express this transformation as U =

exp (−iθata), where θa is a set of 8 real parameters and ta = λa/2 are the generators of the corresponding

29 Måns Holmberg Master Thesis

Lie algebra in the fundamental representation (given by the 3 × 3 Gell-Mann matrices λa). Think of the exponential map above as the regular matrix exponential, given by the usual power series expansion21. This transformation leaves the free Lagrangian density22 unchanged, or invariant,

†
†
Lfree → qU¯ i∂/ − M Uq =qU ¯ U i∂/ − M q = Lfree , (2.75)

just as we promised. This works because U is constant. However, in accordance with the gauge principle, we now enforce local gauge invariance, i.e. U → U(x). The coordinate dependence of U(x) is

embedded in θa, thus θa → θa(x), where x 7→ θa(x) are smooth maps from Minkowski space to the real numbers. Clearly, this does not leave our free Lagrangian invariant, as the derivative of U(x) no longer vanishes, hence giving an extra term in (2.75):

†
Lfree → Lfree + iq¯ U (x) ∂U/ (x) q . (2.76)

Because the problem occurs when the derivative acts on U(x) we define a new operator to fix

this: the covariant derivative Dµ(x), which will replace the ordinary partial derivative. The covariant derivative should have the following transformation property to solve our problem:

† Dµ(x) → U(x)Dµ(x)U (x) ⇔ Dµ(x)U(x) → U(x)Dµ(x) . (2.77)

To achieve this, we define the covariant derivative as

Dµ ≡ ∂µ − igAµ , (2.78)

where Aµ(x) is a vector field that transforms as

 i  A (x) → A0 (x) = U(x) A (x) + ∂ U †(x) . (2.79) µ µ µ g µ

By replacing the ordinary partial derivative ∂µ with the covariant derivative Dµ in (2.72), we remove the extra term in (2.76), making the Lagrangian locally gauge invariant. In the process, we had to

introduce the vector field Aµ(x), which, as can be seen by rewriting the last term of (2.79),

i 1 U(x) ∂ U †(x) = − ∂ θa(x)ta , (2.80) g µ g µ

is a linear combination of the generators ta. Hence, there are actually eight new vector fields, given by

21There is a more abstract definition of the exponential map that does not involve matrices, but we do not need this in the case of SU(3) (because it is a matrix group). 22Throughout this thesis, when you read Lagrangian, add the word density afterwords in you mind.

30 Måns Holmberg Master Thesis

a a Aµ(x) = Aµ(x)t , which means that Aµ(x) is an element of the Lie algebra (i.e Aµ(x) belongs to the adjoint representation). These fields describe the exchange bosons of the strong interaction, that is, the eight gluons. Next we construct the gluon field strength in order to give dynamics to the gluon fields, defined as

i   F ≡ [D ,D ] = ∂ A − ∂ A − ig [A ,A ] = ∂ Aa − ∂ Aa + gf abcAb Ac ta . (2.81) µν g µ ν µ ν ν µ µ ν µ ν ν µ µ ν

Here, f abc are the structure constants of the color group. Because the field strength tensor is constructed as the commutator of two covariant derivatives, it inherits the transformation property in (2.77), that is 0 † Fµν → Fµν = U(x)FµνU (x) . (2.82)

Due to this, the contraction of two field strength tensors cannot enter the Lagrangian alone, because the term would not be gauge invariant. This is in contrast to QED (with U(1) gauge symmetry) where this term would be gauge invariant. Instead, we may use the cyclic property of the (color) trace µν to construct a suitable kinetic term, i.e. tr (FµνF ). Note that this term includes both three- and four-point interactions of gluons! This remarkable fact is due to the non-Abelian property of SU(3),

responsible for [Aµ ,Aν] 6= 0. One can show that these gluon-gluon interaction terms cause the effect of asymptotic freedom [1, 26]. Together with the appropriate normalization, we have finally reached the full QCD Lagrangian: 1 L = − tr (F F µν) +q ¯iD/ − M
q . (2.83) QCD 2 µν We note that gauge invariance forbids gluon mass terms, making gluons massless; like photons. In principle, there is one more gauge invariant term with dimensionless coupling constant23 we could add, namely the θ-term: g2θ µνρσtr (F F ) , (2.84) 32π2 µν ρσ which can be written as a derivative in the following way

g2θ  2ig  ∂ Kµ ,Kµ = µνρσtr A ∂ A − A A A . (2.85) 8π2 µ ν ρ σ 3 ν ρ σ

One might conclude that the θ-term is irrelevant to physics because it contributes only with a surface term to the action. This is true on a classical level, but when taking into account non-perturbative quantum effects, this term does, in fact, contribute (see section 26.3 in [22]). Therefore, if we take the gauge-principle seriously, we are obligated to include this term. But, it turns out that the θ-term explicitly violates P and CP-symmetry, in contrast to (2.83), which is something that is not observed

23We have not touched on this here, but this is an important condition not to be overlooked, that is, interaction terms must have dimensionless coupling constants in order for the theory to be renormalizable. This is described in [1, 25].

31 Måns Holmberg Master Thesis

experimentally in the strong interaction (Weinberg 26.3); forcing the θ parameter to be unnaturally small. This leads to the "fine-tuning" problem known as the strong CP problem, which is an area of active research. Here, an improved understanding of QCD at low energies could help reveal signs of new physics. Because gluons are massless, we expect the strong force to be long-ranged just like electromagnetism. In reality, this is not what we observe, and the reason is something called (color) confinement; which makes it impossible to isolate color charged particles. However, there is currently no analytical proof that confinement exists in QCD or any other Yang-Mills theory, but analytical and numerical hints suggest it does [27, 28]. For now, we can understand confinement as the growth of the QCD coupling constant with decreasing energies (i.e., the opposite of asymptotic freedom). This is related to the fact that gluons interact with each other already at tree-level, because of their non-zero color charge, as compared to photons, which are neutral under the electromagnetic force. Interactions of photons are therefore suppressed but do occur at higher orders in the perturbative expansion (due to loop corrections). As we saw above, gluon interactions come from the non-abelian property of the gauge group. Also important is the fact that only gauge invariant objects are observable, which naturally explains why quarks are grouped in quark-antiquark states (mesons) and three-quark states (baryons). Surely, for arbitrary flavor and spin structures, these combinations of quarks are gauge invariant:

q¯cfs(x) qcf 0s0 (x) , cde qcfs(x) qdf 0s0 (x) qef 00s00 (x) . (2.86)

We have now constructed the QCD Lagrangian and seen how it is determined by imposing color SU(3) gauge symmetry. It turns out the Lagrangian possesses several global symmetries as well, but in order to discuss these further we first need an understanding of Noether’s theorem; which establishes a relation between symmetries and conservation laws. With this, let us further investigate the symmetries of QCD.

2.8.1 Noether’s theorem

Noether’s theorem tells us that every continuous symmetry of the action leads to a conserved quantity, which highlights another role of symmetries in physics; by being explanatory – explaining how conservation laws come about. It is easy to prove Noether’s theorem in the case of classical scalar fields, which is what we will do next. A generalization to fermions is made similarly. Suppose we have a set R 4 of scalar fields φi(x) and the Lagrangian density L(φi , ∂µφi). We say that the action S = d x L has

32 Måns Holmberg Master Thesis

0 a continuous symmetry if there is a continuous transformation φi → φi that leaves the action invariant

0 S[φi] = S[φi] . (2.87)

0 Consider an infinitesimal global transformation of the fields, i.e φi = φi + δφi, from which we get that the variation δL (coming from L → L + δL) must be either zero or a total derivative, in order for µ (2.87) to be satisfied, i.e., δL = ∂µK . Using the chain rule, we can write δL as

∂L ∂L δL = δφi + ∂µ(δφi) ∂φi ∂(∂µφi) (2.88)  ∂L ∂L   ∂L  = − ∂µ δφi + ∂µ δφi . ∂φi ∂(∂µφi) ∂(∂µφi)

Next, because the Euler-Lagrange equations of motion are satisfied, the first term vanishes, resulting in:   ∂L µ ∂µ δφi − K = 0 , (2.89) ∂(∂µφi) This implies that there is a conserved quantity

µ ∂L µ µ ∂ 0 j ≡ δφi − K , ∂µj = j (x) + ∇~ · ~j(x) = 0 , (2.90) ∂(∂µφi) ∂t whenever there is a continuous transformation that leaves the Lagrangian invariant, up to a total µ µ derivative ∂µK . This quantity j is called a Noether current density. We can also define the Noether charge by integrating over the time component of the Noether current:

Z Z Z 3 0 3 ∂L(x) 3 Q ≡ d x j (x) = d x δφi = d x Πi(x)δφi , (2.91) ∂φ˙i where Πi(x) is the conjugate momentum of the field φi(x) (that satisfies the usual canonical commutation relations). See chapter 3 in [25]. Naturally, we find this quantity to be conserved as well

d Z ∂ Z Q = d3x j0(x) = − d3x ∇~ · ~j(x) = 0 , (2.92) dt ∂t where the last equality comes from Gauss’ law, assuming ~j(x) → 0 at infinity. Furthermore, consider a transformation of some symmetry group G that mixes different fields, a a 2 a φi → Uijφi = φi − iθ (X )ijφj + O(θ ), where Xa are the generators of G and where θ are some a constants. That is, compared to before: δφi = −iθ (Xa)ijφj. This means that there are actually a conserved charges and currents, since the normalizations of these are arbitrary (what is important is

33 Måns Holmberg Master Thesis

that they are conserved), i.e. (ignoring the total derivative)

Z µ ∂L 3 ja = (Xa)ijφj ,Qa = d x Πi(x)(Xa)ijφj . (2.93) ∂(∂µφi)

With this, it is not hard to see that Qa generates the same group as the original generators themselves. Let us consider the example of translation invariance, both in time and space. This leads to the infinitesimal transformation

ν ν ν µ x → x + δx , φi(x) → φi(x) + δx ∂µφi(x) . (2.94)

The Lagrangian density transforms in the same way as the fields, and is hence left invariant up to a total derivative ν L(x) → L(x) + δx ∂νL(x) . (2.95)

Now, Noether’s theorem tells us that there is a conserved current associated with the transformation, which according to (2.90) is given by

  µ ∂L ν µν j = δxν ∂ φi − g L . (2.96) ∂(∂µφi)

There are really four conserved currents, as δxν are four arbitrary constants

µ ν ∂L ν µν (j ) = ∂ φi − g L , (2.97) ∂(∂µφi)

µν µ ν µν that defines the so called stress-energy tensor T ≡ (j ) , which is conserved ∂µT = 0. We first identify that the T 00 component is the Hamiltonian density H, since

00 ∂L T = φ˙i − L = H . (2.98) ∂φ˙i

The four conserved Noether charges are then

Z Z H = d3x T 00(x) ,P i = d3x T 0i(x) , (2.99) where H is the Hamiltonian operator and P i is the three-momentum operator. Consequently, we have found that spacetime-translation invariance is responsible for energy and momentum conservation. Moreover, recall from quantum mechanics that the unitary time evolution operator is defined as

U(t) = exp (−iHt) . (2.100)

34 Måns Holmberg Master Thesis

Therefore, since a conserved charge Qa is time-independent, we have

† U(t) Qa U(t) = Qa . (2.101)

By then taking the time derivative of both sides and setting t = 0 in (2.101), we conclude that all conserved charges commute with the Hamiltonian24

[Qa,H] = 0 . (2.102)

Before applying this knowledge to QCD, let us end this general discussion of symmetries with a word of caution. There might be symmetries of the action that do not leave the integration measure in the quantum mechanical path integral invariant; hence, a symmetry of the action is not always a symmetry of the full quantum theory. When this happens, it is referred to as an anomaly. We already saw a related phenomenon above, where the θ-term (2.84) in the QCD Lagrangian still gave rise to physical effects, even though its effects should vanish classically. We will see this effect again in section 2.8.4.

2.8.2 Exact symmetries of QCD

Before going into chiral symmetry, let us first discuss two exact (global) symmetries of the QCD Lagrangian. If we apply a common global phase to each of the quark fields, i.e., a U(1) transformation, the Lagrangian remains invariant. This is similar to the global gauge transformation in (2.74), but acting on all spaces, as an overall factor, not just on color. Then, according to Noether’s theorem, the conserved current (defined up to an overall factor) is

V µ =qγ ¯ µq . (2.103)

We call the current V , since it transforms as a vector under parity (see table 2.1). The conserved charge, called the baryon number operator, is then

Z Z 3 0 3 † QV = d x j = d x q q , (2.104) which counts the number of quarks minus the number of antiquarks [1, 25]. The charge transforms as the zeroth component of a vector under parity, i.e., remain unchanged. If instead we allow for different phases for the different flavors, but without mixing, we get six

24One can also derive the same relation using Hamiltonian mechanics (see section 7.6 in [24]), where the Poisson bracket of Qa and H vanishes, which, in quantum mechanics, translates to (2.102).

35 Måns Holmberg Master Thesis

conserved charges; one for each flavor:

Z 3 † QV f = d x qf qf , (2.105) which is the flavor number operator. The operator counts the number of quarks of a specific flavor minus the number of antiquarks of that flavor. Here, the index f is not summed over. We have thus found that both baryon and flavor number is exactly conserved in QCD. Note that the U(1) symmetry responsible for baryon number conservation is a special case of the symmetry responsible for flavor number conservation, where all phases are set equal.

2.8.3 Approximate symmetries of QCD

The two exact symmetries found above, baryon and flavor number conservation, are hinting at a larger symmetry group. The next logical step would be to allow for mixing between quark flavors. But this is not allowed, due to the different masses of the quark flavors. However, by looking at the mass spectrum of the quarks, we notice a hierarchy:

Quark Up Down Strange Charm Bottom Top Mass (PDG) 1-3 MeV 4-6 MeV 90-100 MeV 1.3 GeV 4.7 GeV 170 GeV

There are two lighter quarks, up and down, with similar masses md ≈ mu ≈ few MeV, a somewhat

heavier quark, the strange quark, with ms ≈ 100 MeV, and three much heavier quarks in the GeV

range and above. It is, therefore, reasonable to approximate the QCD Lagrangian by setting md = mu, since the mass difference is small. An excellent follow-up question is then: Small compared to what? The answer is: Compared to the typical scale of hadronic masses25 ∼ 1 GeV. Doing this, the QCD Lagrangian is invariant under SU(2) isospin transformations

  † u q → Uq , q¯ → qU¯ , where q =   , (2.106) d

thus U ∈ SU(2). As before, the number of obtained conserved charges equals the number of symmetry transformation, which is the same as the number of generators of the symmetry group, as we saw explicitly in (2.93). Thus, resulting in 22 − 1 = 3 conserved vector currents of SU(2) isospin symmetry,

µ † µ jV a = q γ taq . (2.107)

25Pions are hadrons with a mass of ∼ 140 MeV, but, as we will see, these are Goldstone bosons in the chiral limit.

36 Måns Holmberg Master Thesis

The corresponding conserved charges are called the isospin operators, given by:

Z 3 † QV a = d x q taq , (2.108)

where ta = σa/2 are the generators of SU(2), hence a = 1, 2, 3. Here, σa are the three Pauli matrices that we saw before (REF). We can extend the symmetry further by including the strange quark in this approximation, such

that md = mu = ms. This approximation is not as good as that of isospin symmetry, as the mass of the strange quarks is considerably larger than that of the up and down quarks. However, it is still justified, because the mass difference is still smaller than ∼ 1 GeV, but we are reaching the limit of applicability. Including the charm quark in this scheme is, however, not justified (Georgi section 16.4)! Also, this symmetry provides further insights into the ordering of hadrons into multiplets, as will become apparent later on. We obtain a Lagrangian invariant under

  u †   q → Uq , q¯ → qU¯ , where q =  d  , (2.109)   s where U ∈ SU(3). We obtain 32 − 1 = 8 conserved currents and charges of the form (2.111) and (2.107),

but with the generators of su(3) instead, given by the eight Gell-Mann matrices λa, i.e. ta = λa/2. This approximate symmetry is naturally called flavor symmetry. The charges satisfy the same commutation relation of su(3):

[QV a ,QV b] = ifabcQV c . (2.110)

µ Since jV a transforms as a vector under parity, we label flavor symmetry by SU(3)V. One natural question arises: Does the QCD Lagrangian in (2.83) also exhibit an axial-vector SU(3) symmetry, where the conserved charges transform as an axial-vector instead? The answer is no, for two reasons; equal quark masses is not a sufficient condition in this case and, moreover, axial-vector charges do not form a Lie algebra on their own(!). The corresponding axial-vector currents would,

according to table 2.1, come with an extra γ5:

Z 3 † QAa = d x q γ5taq , (2.111) with the commutation relation

[QAa ,QAb] = ifabcQV c . (2.112)

Here, we get the vector charge because the square of a γ5-matrix is unity. Thus, the axial-vector

37 Måns Holmberg Master Thesis

charges do not close under commutation, meaning that they do not form a Lie algebra. Instead, if we consider left- and right-handed charges, which can be obtained by a simple linear combination of vector and axial-vector charges, we get two mutually commuting Lie algebras. However, for the left- and right-handed SU(3):s to be a symmetry group of the theory, equal quark masses is no longer a sufficient condition; instead, completely massless quarks are required. In this limit of massless quarks, we obtain chiral symmetry.

2.8.4 Chiral symmetry (under construction)

To recap, we have seen that in the approximation of equal quark masses, i.e., md = mu = ms, the

QCD Lagrangian is invariant under SU(3)V × U(1)V transformations (where U(1)V corresponds to an exact symmetry, namely baryon number conservation). Furthermore, we saw that axial-vector transformations do not form a group, since the charges (i.e.,generators) do not close under the Lie bracket. Consider splitting the quark fields into their left- and right-handed Weyl components. We do this by the following projection operators

1 1 P = (1 + γ ) ,P = (1 − γ ) , (2.113) R 2 5 L 2 5

that projects a Dirac spinor onto its left- or right-handed component, such that q = PRq + PLq. We

define qR ≡ PRq and qL ≡ PLq (compare with section 2.3). Using these projection operators, we can decompose the QCD Lagrangian into its left- or right- handed component, resulting in

1 L = − tr (F F µν) +q ¯ iDq/ +q ¯ iDq/ − q¯ Mq − q¯ Mq . (2.114) QCD 2 µν R R L L R L L R

From this we conclude that in the limit of massless quarks, that is, md = mu = ms → 0, the left- and right-handed components of the quark fields transform independently. Naturally, this limit is called

the chiral limit. In the chiral limit, we obtain the chiral symmetry group U(3)L × U(3)R, such that

† † qL → ULqL , q¯R → q¯RU , qL → UqL , q¯L → q¯LU , (2.115)

38 Måns Holmberg Master Thesis

with U ∈ U(3)L × U(3)R. From this, we obtain 18 conserved currents:

λ Lµ =qγ ¯ µP a q a L 2

µ µ λa Ra =qγ ¯ PR q 2 (2.116) µ µ L =qγ ¯ PLq

µ µ R =qγ ¯ PRq .

µ µ µ µ such that ∂µLa = ∂µRa = 0 and ∂µL = ∂µR = 0 in the chiral limit. Instead of these chiral currents, we could use the linear combination of vector and axial-vector currents:

λ V µ = Rµ + Lµ =qγ ¯ µ a q a a a 2

µ µ µ µ λa Aa = Ra − La =qγ ¯ γ5 q 2 (2.117) V µ = Rµ + Lµ =qγ ¯ µq

µ µ µ µ A = R − L =qγ ¯ γ5q .

µ µ µ ∂µVa and ∂µAa scales linear with M, ∂µV = 0 and

3g2θ ∂ Aµ = µνρσtr (F F ) + O(M) (2.118) µ 32π2 µν ρσ

It turns out that we can justify such an approximation; of setting the up, down and strange quark masses to zero.

In the limit of massless (light) quarks, i.e. md = mu = ms → 0, the QCD Lagrangian is invariant

under global U(3)L × U(3)R transformations, where the subscripts stand for left and right, respectively.

SU(3)L × SU(3)R × U(1)V (2.119)

2.9 Spontaneous symmetry breaking (under construction)

Let the potential of a scalar (or pseudoscalar) field φ(x) be invariant under a continuous unitary transformation, given by an element of a unitary representation of a Lie group G. That is, for any g ∈ G: V (φ) = V (U(g)φ) . (2.120)

39 Måns Holmberg Master Thesis

Consider the vacuum, where the expectation value of φ(x), denoted φ0, is such that it minimizes the potential energy, i.e.

∂V = 0 . (2.121) ∂φ φ=φ0

However, this vacuum is not unique if for some g ∈ G we have U(g)φ0 6= φ0; which means that both

U(g)φ0 and φ0 minimizes the potential. In this case, the symmetry G is said to be spontaneously broken. It might still be the case that G is not completely broken, as there may exist some g0 ∈ G 0 which do in fact leave the vacuum invariant, i.e. U(g )φ0 = φ0. These elements form the stability 26 group Gφ0 ⊂ G, with the group action defined by ψ(g, φ) = U(g)ψ:

Gφ0 = {g ∈ G : U(g)φ0 = φ0} . (2.122)

This means that each g ∈ G that satisfies U(g)φ0 6= φ0 is not uniquely determined, instead, all elements 0 gg of the coset gGφ0 have the same property, i.e.

0 0 0 U(gg )φ0 = U(g)U(g )φ0 = U(g)φ0 6= φ0 , g ∈ Gφ0 . (2.123)

Furthermore, using the assumption that the group action is transitive, that is, one can reach all possible φ by a transformation of the kind U(g)φ, it follows that there is a bijection between the set of all cosets

gGφ0 , denoted G/Gφ0 , and the set of all scalar fields φ (OLD CHAPTER). This bijection is defined by

gGφ0 7→ U(g)φ0.

Since, Gφ0 is a Lie subgroup, its generators Ta form a closed Lie subalgebra. Then, if G is assumed

to be compact we can always find a basis of G which include Ta, together with some other independent

generators Xb, such that the full set of generators is

{T1 ,T2 ,...,Tn ,X1 ,X2 ,...Xm} , (2.124)

where n = dim(Gφ0 ) and m = dim(G) − dim(Gφ0 ) (Weinberg volume 2, 19.6). We say that Ta are the unbroken generators and Xb are the broken generators. Thus, considering that Ta and Xb span the Lie algebra of G, any element connected to the identity can be expressed as

a b g = exp(θ Xa) exp(ξ Tb) , (2.125) where θa, ξb are a set of real parameters. b By construction, exp(ξ Tb) ∈ Gφ0 ; which means that our scalar field φ can be written only in terms

26Which is a Lie subgroup, since G is a Lie group.

40 Måns Holmberg Master Thesis

of the generators Xa:

a b a φ = U(g)φ0 = U(exp(θ Xa))U(exp(ξ Tb))φ0 = U(exp(θ Xa))φ0 , (2.126) thus, demonstrating the same redundancy as shown in (2.123). We are however forgetting that φ(x) is a function of Minkowski space; to fix this, we impose an x-dependence on θa. Also, since

a ∼ φ(x) = exp(θ (x)Xa)φ0 ∈ M = G/Gφ0 . (2.127)

2.10 Chiral perturbation theory (under construction)

41 Chapter 3

Methods and calculations

The main goal of this thesis is to determine three previously unknown LEC:s – hA at LOand cM and

cE at NLO; by comparing the calculated partial decay widths of four types of radiative and weak decays of decuplet baryons: B∗(J = 3/2) → Bπ, B∗(J = 3/2) → Bγ, B∗(J = 3/2) → Bπγ and ∗ − B (J = 3/2) → Be ν¯e with experimental data (once available). The LEC hA has been determined

before in the case of Delta and Sigma baryons [29], but not in the case of cascades. Furthermore, hA

has also been estimated from large-Nc considerations [30, 9, 10, 11, 12]. In this chapter, we will cover the methods and calculations used to achieve this goal; starting with a short description of general two- and three-body decays and the relevant χPT Lagrangian at LO and NLO. We will then proceed with the calculations of the relevant matrix elements. We are first going to calculate the partial decay widths of decays of the form B∗(J = 3/2) → Bπ,

to determine the value of hA in the case of cascades, a LEC present in the LO χPT Lagrangian. This will also serve as a consistency check (mostly on my part), by comparing with previous calculations. Next, we will calculate the partial decay widths of the radiative two-body decays B∗(J = 3/2) → Bγ, which depend only on cM ; which we should be able to determine, since the partial decay widths of Σ∗+ → Σ+γ, Σ∗0 → Λγ and ∆ → Nγ have been measured [13].

Once the value of cM is known, we can determine value of cE using the partial decay widths of two ∗ ∗ − three-body decays: B (J = 3/2) → Bπγ and B (J = 3/2) → Be ν¯e; however, experimental data only − 0 − exists in the case of Ω → Ξ ν¯ee (unfortunately with large uncertainty). Nevertheless, we should be able to make predictions of the partial decay widths of B∗(J = 3/2) → Bπγ decays, even with this

limited knowledge of cE. Once more experimental data is available, this situation will improve.

42 Måns Holmberg Master Thesis

3.1 Decay processes and kinematics

To calculate the decay width Γ of a decay process, i.e., the inverse lifetime, we have to perform two

calculations. First, we must find the Lorentz invariant Feynman matrix element M1→n, that encodes information about the interactions of the involved particles. This quantity can be obtained from a given Lagrangian using perturbation theory, hence, by finding all contributing Feynman diagrams up to some order. Second, we must perform an integral over the allowed phase space volume, which in 2 general depends on |M1→n| and the kinematics of the decay. If one does not perform this integral, one obtains the differential decay rate dΓ, which is interesting to study if the matrix element depends on some kinematic parameters. Consider a decay process for a particle at rest with mass M and

momentum P decaying into n particles with mass mj and momenta ~pj, thus j = 1, ..., n, with the q 2 2 corresponding energy Ej = mj + ~pj . Then, the differential decay rate is given by [25, 1]

1 dΓ = |M |2dLIPS , (3.1) 2M 1→n n where dLIPSn is the Lorentz-invariant phase-space measure

  n n 3 X Y d pj dLIPS = (2π)4δ(4) P − p . (3.2) n  j (2π)32E j j j

As the name suggests, this measure is Lorentz-invariant, meaning that we are free to select any convenient reference frame to perform the integration. One usually selects the center-of-mass frame (CM frame), where the initial particle is at rest, i.e. P = (M,~0). Often, a particle can decay into several different final states or decay modes, each with their own

decay rate, called partial decay rates. Together, the sum of each partial decay width Γk gives the total decay width X Γtotal = Γk . (3.3) k Furthermore, the branching ratio is given by the ratio of a particular partial decay rate and the total decay rate, i.e. Γk BRk = . (3.4) Γtotal A particle’s lifetime τ is given by the inverse of the decay rate, that is

1 τ = . (3.5) Γtotal

Since the dimension of the decay rate is an inverse time, it has the same dimension as mass (or energy)

43 Måns Holmberg Master Thesis

using natural units. In fact, the total decay rate is associated with the width of the peak of a particle’s irreducible mass distribution. By irreducible, we mean as measured by a particle accelerator with infinite precision. Therefore, one usually refers to the decay rate as decay width, which is what we will do. When conducting experiments, it is common to not measure the spin orientation of the initial and final states. In this case, the spin information of |M|2 is unnecessary to drag along in calculations. Instead, one considers the spin-averaged matrix elements squared h|M|2i, obtained by averaging over the initial spin and summing over the final spin orientations. In our case, the spin information could be of benefit, since it can be measured from the decays of octet hyperons [31]; however, this is outside the scope of this thesis.

In principle, the matrix element M1→n depends on the four-momenta of each particle in the decay (both initial and final), giving us 3 × number of particles = 3(n + 1) kinematic variables, since the energy of each particle is fixed by its three-momenta (explaining the 3 above). Furthermore, the

matrix element M1→n must be a Lorentz scalar, meaning that it can only depend on Lorentz invariant quantities, built from four-momenta and spinors; but, as we do not care about the spin-orientation (since we averaged over spin), we only get Lorentz invariant combinations of four-momenta as free variables. Lorentz invariance further reduces the number of free variables to 3(n+1)−10 = 3n−7, since

M1→n should not depend on the frame of reference. We get 10 constraints from the three directions of boosts and rotations, as well as the conservation of energy and three-momenta. In the case of two-body decays, i.e. n = 2, we therefore have no free kinematic variables. The only combination of four-momenta that we can build is the scalar product of the two four-momenta of the outgoing particles, which is easily shown to be constant. Therefore, the matrix element is constant in the case of two-body decays; consequentially, the decay width is easily obtained by integrating over the constant Lorentz-invariant phase-space in (3.1). Performing this integral in the CM frame gives us

p Γ = cm h|M |2i Θ(M − m − m ) , (3.6) 8πM 2 1→2 1 2 where M is the mass of the initial particle and where pcm = |~p1| = |~p2| is the so called center-of-mass

momentum. Here, ~p1 and ~p2 are the three-momenta of the corresponding four-momenta of the two outgoing particles. One can also rewrite center-of-mass momentum using the Källén function

λ(a, b, c) = a2 + b2 + c2 − 2(ab + bc + ac) , (3.7)

in the following way: 1 p = λ1/2(M 2, m2, m2) . (3.8) cm 2M 1 2

44 Måns Holmberg Master Thesis

Note that the step function Θ in (3.6) makes sure that the decay width vanishes when M ≤ m1 + m2. In the case of three-body decays, i.e. n = 3, we instead have 2 free kinematic parameters. Let M

and P be the mass and four-momenta of the decaying particle and let m1, m2, m3 and p1, p2, p3 be the respective mass and four-momenta of the three outgoing particles. Consider the following two Lorentz scalars 2 2 2 2 m12 = (p1 + p2) , m23 = (p2 + p3) . (3.9)

These variables are the invariant masses (squared) of the (1, 2) and (2, 3) particle systems, respectively. A three-body decay can be considered as two sequential two-body decays1. It is not hard to show that 2 2 one can write all possible four-vector combinations using the masses, m12 and m23; confirming that 2 these two invariant masses (squared) span all kinematics. Naturally, we ask if arbitrary values of m12 2 2 and m23 are allowed. The answer is no. Instead, the kinematical boundaries for m12 are given by

2 2 2 (m1 + m2) ≤ m12 ≤ (M − m3) . (3.10)

2 The kinematical boundaries for m23 are given by

2 2 2 (m23)min ≤ m23 ≤ (m23)max , (3.11) with q q 2 2 ∗ ∗ 2 ∗2 2 ∗2 2 (m23)min/max = (E2 + E3 ) − E2 − m2 ± E3 − m3 , (3.12) where 2 2 2 2 2 2 ∗ m12 − m1 − m2 ∗ M − m12 − m3 E2 = ,E3 = , (3.13) 2m12 2m12

are the energies of particles 2 and 3, respectively, in the frame where ~p1 +~p2 = 0 (called the rest frame of

m12). Using these two invariant masses as kinematic variables, one can compute the double-differential decay width: 1 1 dΓ = h|M |2i dm2 dm2 Θ(M − m − m − m ) , (3.14) (2π)3 32M 3 1→3 12 23 1 2 3

2 2 which is obtained by differentiating the decay width (3.1) with respect to m12 and m23 and then integrating over the Lorentz-invariant phase-space. 2 2 The first thing to note is that the only dependence of m12 and m23 can come from the matrix element itself, meaning that the double-differential decay width would be constant in the case of a constant matrix element. This might seem trivial, and it is, but it is important to realize; especially when illustrating the double-differential decay width. Illustrations of the double-differential decay width

1This generalizes to n-body decays, which can be considered as a chain of sequential two-body decays.

45 Måns Holmberg Master Thesis

are usually done by a so-called Dalitz plot2, which is a two-dimensional plot of (3.14) as a function of 2 2 m12 and m23. We obtain the integrated decay width by integration over the invariant masses. This integral is usually performed numerically, due to the complicated boundary and the often complicated matrix element. 2 Consider again the ~p1 + ~p2 = 0 frame. Here we can write m23 as

2 ∗ ∗ 2 ∗ 2 ∗ 2 ∗ ∗ m23 = (E2 + E3 ) − (~p2) − (~p3) − 2|~p2||~p3| cos θ , (3.15)

∗ ∗ where ~p2 and ~p3 are the respective three-momenta of particles labeled 2 and 3, and where the angle theta is the angle between these two three-momenta, given by (compare with (3.8))

∗ ∗ 1 1/2 2 2 2 ∗ ~ ∗ 1 1/2 2 2 2 |~p1| = |~p2| = λ (m12 , m1 , m2) , |~p3| = |P | = λ (m12 ,M , m3) . (3.16) 2m12 2m12

2 We can then use cos θ as integration variable in the double-differential decay width instead of m23. Doing this, we obtain (ignoring the step function):

1 1 dΓ = |~p∗||~p∗|h |M |2i dm2 d(cos θ) . (3.17) (2π)3 16M 3 2 3 1→3 12

We dropped the minus sign coming from (3.15), which is justified if the integration over cos θ is later performed from −1 to 1 (and not in the opposite order)3. Both (3.14) and (3.17) contains the same 2 information, however, since the integral domain of the latter is a square (i.e., dm12 and d(cos θ) are independent), it is easier to work with. For more information about the kinematics of two- and three-body decays, see section 47 in [13].

2The value of the double-differential decay width is usually illustrated as different colors, brightness-levels, or as the height of a surface. 3 2 2 Since m23,min → cos(0) = 1 and m23,max → cos(π) = −1.

46 Måns Holmberg Master Thesis

3.2 NLO Chiral Lagrangian for baryons (under construction)

The LO chiral Lagrangian including the spin-3/2 decuplet states is given by (REF)

(1) ¯
Lbaryon = tr B (iD/ − m(8)) B ¯µ α ν abc ν abc + Tabc (iγµνα(D T ) − γµν m(10) (T ) ) D F + tr(B¯ γµ γ {u ,B}) + tr(B¯ γµ γ [u ,B]) 2 5 µ 2 5 µ

hA  ade µ b c e µ d abc + √  T¯ (uµ) B + ade B¯ (u ) T 2 2 abc d e c b µ H − A T¯µ γ γ (uν)c T abd (3.18) 2 abc ν 5 d µ with tr denoting a flavor trace. We have introduced the totally antisymmetrized products of two and three gamma matrices

1 γµν ≡ [γµ, γν] = −iσµν (3.19) 2

and

1 γµνα ≡ (γµγνγα + γνγαγµ + γαγµγν − γµγαγν − γαγνγµ − γνγµγα) 6 1 = {γµν, γα} = +iµναβγ γ , (3.20) 2 β 5

respectively. a The spin-1/2 octet baryons are collected in (Bb is the entry in the ath row, bth column)   √1 Σ0 + √1 ΛΣ+ p  2 6   − 1 0 1  B =  Σ − √ Σ + √ Λ n  . (3.21)  2 6  Ξ− Ξ0 − √2 Λ 6

47 Måns Holmberg Master Thesis

The decuplet is expressed by a totally symmetric flavor tensor T abc with

1 T 111 = ∆++,T 112 = √ ∆+, 3 1 T 122 = √ ∆0,T 222 = ∆−, 3 1 1 1 T 113 = √ Σ∗+,T 123 = √ Σ∗0,T 223 = √ Σ∗−, 3 6 3 1 1 T 133 = √ Ξ∗0,T 233 = √ Ξ∗−,T 333 = Ω . (3.22) 3 3

The Goldstone bosons are encoded in  √ √  π0 + √1 η 2 π+ 2 K+  √ 3 √   − 0 1 0  Φ =  2 π −π + √ η 2 K  ,  √ √ 3  2 K− 2 K¯ 0 − √2 η 3 2 † † † u ≡ U ≡ exp(iΦ/Fπ) , uµ ≡ i u (∇µU) u = uµ . (3.23)

The fields have the following transformation properties with respect to chiral transformations [32, 6]

U → LUR† , u → L u h† = h u R† ,

† † uµ → h uµ h ,B → h B h , (3.24) abc a b c def ¯µ † d † e † f ¯µ Tµ → hd he hf Tµ , Tabc → (h )a (h )b (h )c Tdef .

The chirally covariant derivative for a (baryon) octet is defined by

DµB := ∂µB + [Γµ,B] , (3.25)

for a decuplet T by

µ abc µ abc µ a a0bc µ b ab0c µ c abc0 (D T ) ≡ ∂ T + (Γ )a0 T + (Γ )b0 T + (Γ )c0 T , (3.26)

for an anti-decuplet by

µ ¯ µ ¯ µ a0 ¯ µ b0 ¯ µ c0 ¯ (D T )abc ≡ ∂ Tabc − (Γ )a Ta0bc − (Γ )b Tab0c − (Γ )c Tabc0 , (3.27)

and for the Goldstone boson fields by

∇µU ≡ ∂µU − i(vµ + aµ) U + iU (vµ − aµ) (3.28)

48 Måns Holmberg Master Thesis with

1   Γ ≡ u† (∂ − i(v + a )) u + u (∂ − i(v − a )) u† , (3.29) µ 2 µ µ µ µ µ µ where v and a denote external sources. The NLO Lagrangian contains all independent terms of order p2 where p denotes a small momentum or Goldstone boson mass. The pertinent building blocks of O(p2) are

µν µν µ ν µ ν ν µ χ± , and O2 = f± , u u ,D u + D u (3.30)

† † † with χ± = u χu ± uχ u and χ = 2B0 (s + ip) obtained from the scalar source s and the pseudoscalar source p. The scalar source contains the quark mass matrix. he field strengths are given by

µν µν † † µν f± ≡ u FL u ± u FR u (3.31) with µν µ ν ν ν µ µ µ µ ν ν FR,L ≡ ∂ (v ± a ) − ∂ (v ± a ) − i [v ± a , v ± a ] . (3.32)

The field strengths constitute flavor octets. Interactions with electromagnetism can be studied by the replacement

 2  3 0 0  1  vµ → eAµQ,Q =  0 − 0  , (3.33)  3  1 0 0 − 3 with the photon field Aµ and the proton charge e. Transition sector

¯e µν d abc ade ¯ µν b c
icM ade Bc γµγ5(f+ )b Tν −  (Tν)abc γµγ5(f+ )d Be

¯e µν d abc ade ¯ µν b c
+ i cE ade Bc γµ(f− )b Tν −  (Tν)abc γµ(f− )d Be (3.34)

Decuplet sector:

¯ µν c abd ¯ µν ¯ µν dM i (Tµ)abc (f+ )d Tν + bM,D tr(B{f+ , σµνB}) + bM,F tr(B[f+ , σµνB]) (3.35)

To do: organize stuff, specify which terms correspond to which interactions, include the values of

49 Måns Holmberg Master Thesis

known LEC:s and references.

3.3 Matrix elements

In order to obtain the decay widths of B∗(J = 3/2) → Bπ, B∗(J = 3/2) → Bγ, B∗(J = 3/2) → Bπγ ∗ − and B (J = 3/2) → Be ν¯e, using the relevant interaction terms of the LO and NLO Chiral Lagrangian ((3.18), (3.34) and (3.35)), we need first to calculate the corresponding matrix elements. Matrix elements can, in general, not be obtained as closed-form expressions, which is why we turn to perturbation theory. In perturbation theory, one approximates the matrix elements by a series expansion of some small parameter, usually a coupling constant (if sufficiently small, which is not the case for QCD at low energies), or as in the case of χPT, small momentum transfer. The standard recipe of calculating matrix elements using perturbation theory involves Feynman diagrams – an ingenious way of intuitively visualizing terms in the perturbative series expansion. After identifying which interaction terms of a given Lagrangian give rise to the process of interest, one proceeds by finding the rules that govern the Feynman diagrams; given by vertices, propagators, and external lines. The different interaction term of the Lagrangian gives rise to different vertices, as we said above. Our main task will be to find these different vertices; given that the propagators and external lines are determined not by the interaction terms themselves but rather by the type of fields in the theory. The Feynman rules for propagators and external lines can be found in any textbook on quantum field theory, e.g., in [25]. However, since these rules for spin 3/2 fermions are usually not covered by textbooks, we provide them in Appendix A. Moreover, all vertices used (except for the weak-decay) are conveniently found in Appendix B. After finding the Feynman rules of a given process, i.e., the vertices, propagators, and external lines, we are left with the task of calculating the (approximate) matrix element. When dealing with spin average quantities, much of this work ends up involving taking traces of (sometimes a large number of) gamma matrices – a not so trivial task4. We will resolve to computers to perform this calculation when the number of gamma matrices in the trace is larger than 4. We will use Mathematica [33] for this purpose, which is a program designed to do symbolic computations, together with a package called FeynCalc [34, 35]. Before calculating the matrix elements (and decay widths when possible by hand), let us define some useful formalism to treat flavor factors more systematically.

4When calculating the matrix element of the three-body decay B∗(J = 3/2) → Bπγ we will at some point take the trace of 16 gamma matrices, which gives an expression containing 135135 different terms; hence, making it infeasible to do by hand.

50 Måns Holmberg Master Thesis

3.3.1 Flavor factors

To render the concept of flavor factors clear, we first start with an example. Consider the decay of ∗ a decuplet baryon into an octet baryon and a pion, i.e., B (J = 3/2) → Bπ, given by the hA term of (3.18). Not all combinations of hadrons are allowed in this decay, due to flavor symmetry and charge conservation5; therefore, the natural question arises of what types of hadrons are allowed. By

expanding uµ up to linear terms of Φ, we obtain the interaction term we seek:

hA ¯e µ d abc − √ adeBc (∂ Φ)b Tµ . (3.36) 2 2Fπ

We could simply sum over all flavor indices (that is: a, b, c, d, e) explicitly and get all allowed such decays, but that is a cumbersome task, especially for more complicated decays. Imagine instead that we could write the fields as

1 T abc = T n(tn)abc ,Ba = √ Bi(si)a , Φa = Φj(sj)a , (3.37) µ µ b 2 b b b

thereby splitting the fields themselves from the flavor structure. Here, the indices n, i, j now specifies the types of hadrons in the decay, e.g. we might choose n = 1 as Σ∗+, i = 8 as Λ and j = 1 as π+, in order to represent the decay of Σ∗+ → π+Λ. Then, we could rewrite (3.36) as

hA 1 T e d abc ¯i µ j n − √ √ ade(si )c(sj)b (tn) B ∂ Φ Tµ , (3.38) 2 2Fπ 2

and define the flavor factor to be

(1) 1 T e T d abc c ≡ √ ade(s ) (s ) (tn) . (3.39) nij 2 i c j b

The number (1) is just a way of numbering the different flavor factors that we will encounter. Also, notice

that sj appears transposed (3.39) and not in (3.38). This makes sure that the meson is an outgoing particle and not an incoming particle, e.g., since the interaction term responsible for Σ∗+ → π+Λ has ¯ µ − ∗+ ¯ µ + ∗+ the form Λ ∂ π Σµ and not Λ ∂ π Σµ . Therefore, (3.39) is what ends up in the vertex. To accomplish (3.37), we define a new basis for the octet and decuplet states, respectively. We first

define the basis si, where i = 1, ..., 8, using the Gell-Mann matrices, for traceless 3 × 3 matrices, given

5The decay must also be energetically possible, i.e., the sum of masses of the final state particles must be less than the mass of the decaying mother particle. Again, this is the job of the step functions in (3.6) and (3.14).

51 Måns Holmberg Master Thesis

by

1 1 s1 = √ (λ1 + iλ2) , s2 = √ (λ1 − iλ2) , s3 = λ3 , 2 2 1 1 s4 = √ (λ4 + iλ5) , s5 = √ (λ5 − iλ5) , 2 2 1 1 s6 = √ (λ6 + iλ7) , s7 = √ (λ6 − iλ7) , s8 = λ8 . 2 2

This new basis ensures that we can write the Goldstone bosons in (3.23) and the lowest lying baryon octet in (3.21) as (3.37), with

(Bi) = Σ+ , Σ− , Σ0 , p , Ξ− , n , Ξ0 , Λ
, (3.40) (Φj) = π+ , π− , π0 ,K+ ,K− ,K0 , K¯ 0 , η
.

When combining flavor factors from different vertices, it is useful to know how elements of this new basis combine; therefore, we express the completeness relation of the Gell-Mann matrices (REF):

2 (λ )a(λ )c = 2δaδc − δaδc , (3.41) i b i d d b 3 b d

in terms of the new basis to see what happens. The sum over only i = 1, 2 yields

2 X 1 1 (λ )a(λ )c = (s + s )a(s + s )c − (−s + s )a(−s + s )c i b i d 2 1 2 b 1 2 d 2 1 2 b 1 2 d i=1 a c a c (3.42) = (s1)b (s2)d + (s2)b (s1)d

a T c a T c = (s1)b (s1 )d + (s2)b (s2 )d ,

T where we used that s1 = s2. Since (3.42) also holds for the pairs s4, s5 and s6, s7; and, since s3 and s8 are diagonal, we conclude that

2 (s )a(sT )c = (λ )a(λ )c = 2δaδc − δaδc . (3.43) i b i d i b i d d b 3 b d

This relation will prove much useful later on.

Next, we define the basis tn, where n = 1, ..., 10, in order to write the decuplet baryons (3.22)

as (3.37). First, let eabc be the 3 × 3 × 3 tensor with all its elements equal to zero, except for the

abc-element, which is equal to one. The basis tn spans all symmetric traceless 3 × 3 × 3 tensors and is

52 Måns Holmberg Master Thesis

given by 1 1 t1 = √ (e113 + e131 + e311) , t2 = √ (e223 + e232 + e322) , 3 3 1 t3 = √ (e123 + e132 + e213 + e231 + e312 + e321) , 6 1 1 t4 = √ (e112 + e121 + e211) , t5 = √ (e233 + e323 + e332) , (3.44) 3 3 1 1 t6 = √ (e122 + e212 + e221) , t7 = √ (e133 + e313 + e331) , 3 3

t8 = e111 , t9 = e222 , t10 = e333 .

Similar to the octet case, we can now write the decuplet baryons as

(T n) = Σ∗+ , Σ∗− , Σ∗0 , ∆+ , Ξ∗− , ∆0 , Ξ∗0 , ∆++ , ∆− , Ω
. (3.45)

With explicit forms of the basis s and t at hand, we can evaluate (3.39) in the case of Σ∗+ → π+Λ (1) (nij = 181), which turns out to be c181 = −1. If we were not to take the transpose of sj in (3.39), we would instead find a vanishing flavor factor, since Σ∗+ decaying into π− and Λ is forbidden. We will now begin to calculate the matrix elements with this convenient way of handling flavor factors.

3.3.2 B∗(J = 3/2) → Bπ

We start off with the two-body decay of a decuplet baryon decaying into an octet baryon and a pion:

πj

Tn

Bi

This decay occurs already at LO and depends only on the LEC hA, as we saw in the example used to explain flavor factors. The relevant interaction term at tree-level, found in (3.21), is:

hA e µ d abc √ adeB¯ (u ) T . (3.46) 2 2 c b µ

By expanding uµ to linear order in the Goldstone bosons Φ we end up with (3.38), which gives us the

53 Måns Holmberg Master Thesis

following vertex: Φj

P q µ hA (1) µ Tn, µ ≡ (VTBΦ)nij(q) = √ cnijq , (3.47) 2 2Fπ p

Bi where the Goldstone boson momenta q comes from the derivative in the interaction term (3.38). The flavor factor is again: (1) 1 T e T d abc c ≡ √ ade(s ) (s ) (tn) . (3.48) nij 2 i c j b We then follow the general rules of Feynman diagrams, see (Srednicki), to obtain the matrix element

hA (1) µ Mnij = i √ uc¯ nij q uµ , (3.49) 2 2Fπ

and the matrix element squared

2 2 hA (1) 2 µ ν ∗ |Mnij| = 2 (cnij) q q uu¯ µ (¯uuν) 8Fπ h2 A (1) 2 µ ν (3.50) = 2 (cnij) q q uµu¯νuu¯ 8Fπ 2 hA (1) 2 µ ν = 2 (cnij) q q tr (uµu¯νuu¯) . 8Fπ

Here, uµ denotes a (spin-3/2) Rarita-Schwinger spinor, i.e., an incoming decuplet state. This computa- tion can be done by hand without too much struggle. We average (and sum) over the spin degrees of freedom to obtain: 1 X |M |2 = |M |2 nij 4 nij spin 2 h (1) X = A (c )2qµqν tr (u u¯ uu¯) 32F 2 nij µ ν π spin h2   A (1) 2 µ ν / 3/2 = 2 (cnij) q q tr −(P + M)Pµν (p/ + mB) 32Fπ (3.51) h2   1 1   A (1) 2 µ ν / / / = 2 (cnij) q q tr (P + M) −gµν + γµγν + 2 (P γµPν + Pµγν P ) (p/ + mB) 32Fπ 3 3M h2   P · q  A (1) 2 / 2 / / = 2 (cnij) tr (p/P + MmB + terms linear in γ) −3mΦ + /q/q + 2 (P /q + /qP ) 96Fπ M h2   2(P · q)2  A (1) 2 / 2 = 2 (cnij) tr (p/P + MmB) −2mΦ + 2 , 96Fπ M

54 Måns Holmberg Master Thesis where we used that6

2 a/a/ = a I4 and more generally a//b + /ba/ = 2a · b I4 , (3.52)

and the very helpful fact that the trace of an odd number of gamma matrices vanishes [25]. Furthermore, 3/2 Pµν (p) is the so called spin-3/2 projector, defined in appendix A. Then, choosing the rest frame of the decaying particle, i.e., P = (M,~0), the scalar product P · q in the last term of (3.51) becomes

(P · q)2 P · q = ME ⇐⇒ = E2 , (3.53) Φ M 2 Φ which together with

q 1 p = |~q| = |~p| = E2 − m2 = λ1/2(M 2, m2 , m2 ) , (3.54) cm Φ Φ 2M Φ B

results in h2 2 A (1) 2 2 /
|Mnij| = 2 (cnij) pcm tr p/P + MmB 48Fπ h2 A (1) 2 2 (3.55) = 2 (cnij) pcm (P · p + MmB) 12Fπ 2 hA (1) 2 2 = 2 (cnij) pcmM (EB + mB) , 12Fπ where we used P · p E = . (3.56) B M Finally, we find the partial decay width

2 pcm 2 hA (1) 2 3 EB + mB Γnij = 2 |Mnij| = 2 (cnij) pcm , (3.57) 8πM 96πFπ M with q 2 2 EB = pcm + mB . (3.58)

3.3.3 B∗(J = 3/2) → Bγ

Next, we move on to the radiative two-body decay where a decuplet baryon is decaying into an octet baryon and a photon:

6 µ ν µ ν Proof: a//b + /ba/ = a b {γµ, γν } = 2a b gµν I4 = 2a · b I4.

55 Måns Holmberg Master Thesis

γ

Tn

Bi

∗ Just as in the case of B (J = 3/2) → Bπ, which only depends on one LEC (at LO), namely hA, this

decay only depends on cM , a LEC found in the NLO Lagrangian. The interaction term responsible for this decay at tree-level can be found in (3.34) and is:

¯e µν d abc icM adeBc γµγ5(f+ )b Tν , (3.59)

µν where the field strength f+ , in (3.31), is given by (up to linear order in Φ)

µν µν † † µν µν µν f+ = uFL u + u FR u = FL + FR + O(Φ) . (3.60)

As we saw in (3.33) that we obtain the interaction with photons by replacing vµ → eAµQ (and setting the axial vector source to zero). From which we obtain (see (3.32))

µν µν FR,L = eF Q, (3.61)

since [Aµ,Aν] = 0 and where F µν = ∂µAν − ∂νAµ. Using this, we rewrite the interaction term and again, separate the flavor structure:

e µν d abc 2 T e d abc i µν n icM adeB¯ γµγ5(f ) T = √ icM e ade(s ) Q (tn) B¯ γµγ5F T + O(Φ) , (3.62) c + b ν 2 i c b ν where we define7 (9) 1 T e d abc c ≡ √ ade(s ) Q (tn) . (3.63) ni 2 i c b The corresponding vertex is

7The number (9) is arbitrary. It only tells in what order we did the calculations of this thesis.

56 Måns Holmberg Master Thesis

γ, α

P k µα (9) µα µ α Tn, µ ≡ (VT Bγ)ni (k) = −2icM e cni (kg/ − k γ )γ5 , (3.64)

p

Bi which gives the matrix element

(9) µ ∗ν ν ∗µ Mni = −2icM ecni (k ε − k ε )uγ ¯ µγ5uν , (3.65)

and the matrix element squared

2 2 2 (9) 2  α β β α µ ∗ν ν ∗µ |Mni| = 4cM e (cni ) k ε − k ε (k ε − k ε ) tr (uνu¯βγαγ5uuγ¯ µγ5) . (3.66)

Evaluating the spin-averaged expression requires a few more steps than that needed to calculate the spin-averaged matrix element squared of B∗(J = 3/2) → Bπ, therefore, we have collected this calculation in appendix C. After performing this calculation, we find

16c2 e2 |M |2 = M (c(9))2p2 M (E + M) , (3.67) ni 3 ni cm B which results in the decay width

p 2c2 e2 E + M Γ = cm |M |2 = M (c(9))2p3 B , (3.68) ni 8πM 2 ni 3π ni cm M with q 2 2 EB = pcm + mB , (3.69)

and 1 p = E = λ1/2(M 2, m2 , 0) . (3.70) cm γ 2M B

3.3.4 B∗(J = 3/2) → Bπγ

Now for the radiative three-body decay where a decuplet baryon decays into an octet baryon, a pion and a photon:

57 Måns Holmberg Master Thesis

γ

Tn πj

Bi

Compared to before, we now have 12 interaction terms(!). At NLO, these are (coming from (3.18), (3.34), (3.35) and (LO Lagrangian Goldstone Bosons)):

F 2   L = π tr (∇ U)†∇µU + i tr B¯DB/
+ iT¯µ γ (DαT ν)abc int 4 µ abc µνα D F h + tr Bγ¯ µγ {u ,B}
+ tr Bγ¯ µγ [u ,B]
+ √A  B¯e(uµ)dT abc 2 5 µ 2 5 µ ade c b µ 2 2 (3.71) H − A T¯µ γ γ (uν)c T abd + ic  B¯eγ γ (f µν)dT abc + ic  B¯eγ (f µν)dT abc 2 abc ν 5 d µ M ade c µ 5 + b ν E ade c µ − b ν ¯ abc µν c abd ¯ µν
¯ µν
+ idM (Tµ) (f+ )d(Tν) + bM,D tr B{f+ , σµνB} + bM,F tr B[f+ , σµνB] .

As before, we obtain interactions with photons by vµ → eAµQ (see (3.33)). An extensive derivation of all vertices that (3.71) gives rise to is found in appendix B. Moreover, let us list all relevant propagators. A Dirac spin-3/2 fermion has the propagator (see appendix A):

p (T ) µ ν = Pµν (p, m) (3.72) i(p/ + m)  1 2p p pµγν − pνγµ  = − g − γ γ − µ ν + , p2 − m2 + i µν 3 µ ν 3m2 3m

likewise, a Dirac spin-1/2 fermion has the propagator (Srednicki):

p i(p/ + m) = P (B)(p, m) = , (3.73) p2 − m2 + i

and finally, a charged or neutral scalar boson with mass m has the propagator (Srednicki):

p i = P (Φ)(p, m) = . (3.74) p2 − m2 + i

After obtaining the Feynman rules associated with (3.71), we find that the following six types of diagrams are allowed:

58 Måns Holmberg Master Thesis

γ Bi Φj k p q P q P P Tn Φj Tn γ Tn γ Φl k Bl k p k + q k + p q p Bi

Φj Bi

γ Φj γ k q k P P P Tn Φj Tn γ Tn Φj Bl q Tm k Tm q q + p k + p q + p p p p

Bi Bi Bi

From left to right and from top to bottom, we label these diagrams 1 to 6. The momentum P of the decaying particle is incoming, while the other three momenta are outgoing. The corresponding matrix elements are:

(1) µν ∗ Mnij =u ¯(VTBΦγ)nij(k)uµν

(2) X µ (Φ) l ν ∗ Mnij = u¯(VTBΦ)nil(k + q)P (k + q, mΦ)(VΦΦγ)lj(k + q , q)uµν , l

(3) X ν (B) l µ ∗ Mnij = u¯(VBBγ)il(k)P (k + p, mB)(VTBΦ)nlj(q)uµν , l (3.75) (4) X (B) l µν ∗ Mnij = u¯(VBBΦ)lij(q)P (q + p, mB)(VT Bγ)nl (k)uµν , l

(5) X αν (T ) µβ ∗ Mnij = u¯(VT Bγ)mi(k)Pαβ (k + p, Mm)(VTT Φ)nmj(q)uµν , m

(6) X α (T ) βµν ∗ Mnij = u¯(VTBΦ)mij(q)Pαβ (P − k, Mm)(VT T γ)nm (k)uµν , m

l l where mΦ, mB and Mm are the Goldstone boson masses, octet baryon masses and decuplet masses, respectively. The sum over l and m ensures that we include all possible propagators (allowed by flavor symmetry). We see that a general matrix element could be written as (ignoring flavor indices)

µν M(i) =uA ¯ i uµεν , (3.76)

µν where i = 1,..., 6 indicates the diagram and where Ai can be read off from (3.75). Using this compact

59 Måns Holmberg Master Thesis

notation, the spin-averaged matrix element squared is

6 6 2 X X ∗ h|M |i = hM(i)M(j)i , (3.77) i=1 j=1 where ∗ 1  µν 3/2 ¯ρσ hM M i = Tr (p/ + mB)A (P/ + M)P A gνσ , (3.78) (i) (j) 4 i µρ j since ∗  µν ¯ρσ ∗ M(i)M(j) = Tr uuA¯ i uµu¯ρAj ενεσ . (3.79)

The spin-averaged matrix element squared is finally calculated using Mathematica. The double differential decay width is then given by (3.14). Furthermore, we can also determine which decays are allowed by flavor symmetry by checking for which flavor indices the matrix elements (3.75) are nonvanishing. By doing this and checking if the decays are energetically possible, we obtain the information in table 3.1.

60 Måns Holmberg Master Thesis

Decay (n, i, j) Contributing diagrams (propagators) Σ∗+ → Σ+ π0 γ (1, 1, 3) 3 (Σ+), 4 (Σ+), 5 (Σ∗+), 6 (Σ∗+) Σ∗+ → Σ0 π+ γ (1, 3, 1) 1, 2 (π+), 3 (Σ0, Λ), 4 (Σ+), 5 (Σ∗0), 6 (Σ∗+) Σ∗+ → Λ π+ γ (1, 8, 1) 1, 2 (π+), 3 (Σ0, Λ), 4 (Σ+), 5 (Σ∗0), 6 (Σ∗+) Σ∗− → Σ− π0 γ (2, 2, 3) 3 (Σ−), 6 (Σ∗−) Σ∗− → Σ0 π− γ (2, 3, 2) 1, 2 (π−), 3 (Σ0, Λ), 5 (Σ∗0), 6 (Σ∗−) Σ∗− → Λ π− γ (2, 8, 2) 1, 2 (π−), 3 (Σ0, Λ), 5 (Σ∗0), 6 (Σ∗−) Σ∗0 → Σ+ π− γ (3, 1, 2) 1, 2 (π−), 3 (Σ+), 4 (Σ0, Λ), 5 (Σ∗+) Σ∗0 → Σ− π+ γ (3, 2, 1) 1, 2 (π+), 3 (Σ−), 4 (Σ0, Λ) Σ∗0 → Σ0 π0 γ (3, 3, 3) 3 (Λ), 4 (Λ) Σ∗0 → Λ π0 γ (3, 8, 3) 3 (Λ), 4 (Σ0) Ξ∗− → Ξ− π0 γ (5, 5, 3) 3 (Ξ−), 6 (Ξ∗−) Ξ∗− → Ξ0 π− γ (5, 7, 2) 1, 2 (π−), 3 (Ξ0), 5 (Ξ∗0), 6 (Ξ∗−) Ξ∗0 → Ξ− π+ γ (7, 5, 1) 1, 2 (π+), 3 (Ξ−), 4 (Ξ0) Ξ∗0 → Ξ0 π0 γ (7, 7, 3) 3 (Ξ0), 4 (Ξ0), 5 (Ξ∗0) ∆++ → p π+ γ (8, 4, 1) 1, 2 (π+), 3 (p), 5 (∆+), 6 (∆++) ∆+ → p π0 γ (4, 4, 3) 3 (p), 4 (p), 5 (∆+), 6 (∆+) ∆+ → n π+ γ (4, 6, 1) 1, 2 (π+), 3 (n), 4 (p), 5 (∆0), 6 (∆+) ∆0 → p π− γ (6, 4, 2) 1, 2 (π−), 3 (p), 4 (n), 5 (∆+) ∆0 → n π0 γ (6, 6, 3) 3 (n), 4 (n), 5 (∆0) ∆− → n π− γ (9, 6, 2) 1, 2 (π−), 3 (n), 5 (∆0), 6 (∆−)

Table 3.1: All energetically possible B∗(J = 3/2) → Bπγ decays allowed by flavor symmetry.

Finally, the matrix elements (3.75) should conserve the electromagnetic current corresponding to the Noether current of a U(1) global transformation. One can show that if the matrix element conserves the electromagnetic current, then, the sum over all (3.76), i.e. the total matrix element, will vanish if one replaces εν by the photon momenta kν (see section 67 in [25]). Therefore, we define (l)µ ρµ Mnij ≡ u¯(Anij)l uρ and, indeed, we find that

X (l)µ Mnij kµ = 0 . (3.80) l

This serves as a good check of the validity of the matrix elements in (3.75). The calculation of (3.80) is covered in detail in appendix D.

61 Måns Holmberg Master Thesis

∗ − 3.3.5 B (J = 3/2) → Be ν¯e

Let us now cover the weak three-body decay where a decuplet baryon decays into an octet baryon, an electron and an antineutrino:

ν¯e

− Tn e

Bi

The relevant interaction terms are (coming from (3.18), (3.34), (LO Lagrangian Goldstone Bosons) and (charged current part of the Lagrangian of weak interactions))

gw µ − + hA e µ d abc Lint = √ ν¯eγ PLe W + √ adeB¯ (u ) T 2 µ 2 2 c b µ

¯e µν d abc ¯e µν d abc (3.81) + icM adeBc γµγ5(f+ )b Tν + icEadeBc γµ(f− )b Tν + h.c. F 2 + π Tr(u uµ) . 4 µ where h.c. denotes the hermitian conjugate of the preceding four terms. We obtain interactions with the W-bosons via (Scherer)

gw +
lµ = vµ − aµ → −√ W T+ + h.c. , (3.82) 2 µ with   0 Vud Vus   T+ =  0 0 0  . (3.83)   0 0 0

Here, Vud and Vus denote two elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix (REF). We note that (3.83) is constructed such that

gw µ + − √ q¯Lγ W T+qL + h.c. , (3.84) 2 µ

becomes the standard charged-current weak interaction of light quarks. That is, the vanishing

components of T+ is due to electromagnetic charge conservation. Moreover, the weak gauge coupling

62 Måns Holmberg Master Thesis

gw in (3.82) relates to Fermi’s constant GF and the W mass mW (at leading order) by (Scherer): √ 2 2gw −5 −2 GF = 2 = 1.16638 × 10 GeV . (3.85) 8mw

As for the Feynman diagrams; the two contributing diagrams at tree-level, according to (3.81), are:

i i p1 B p1 B P P n n T ν¯e T ν¯e k p2 k p2 k W − Φj W − p3 p3 e− e−

Note that k = p2 + p3. We will refer to the left and right diagrams as diagram 1 and 2, respectively. The vertices regarding the weak interaction are not covered in appendix B, therefore, let us cover them − − here, starting with the vertex W → e ν¯e. This interaction comes from the hermitian conjugate of the first term of (3.81), i.e.

gw µ − +
† gw − † 0 µ 0 0 − √ ν¯eγ PLe W = √ (e ) PLγ γ γ γ νeW 2 µ 2 µ (3.86) gw + µ − = √ e γ PLνeW , 2 µ which leads to ν¯e

− µ gw µ W ≡ (VW −ν¯ e− ) = i√ γ PL . (3.87) µ e 2

e−

Next, the Φ → W − vertex, coming from (omitting irrelevant interactions)

F 2 F 2  1   1  π Tr(u uµ) = π Tr − ∂ Φ − l − ∂µΦ − lµ 4 µ 4 F µ µ F π π (3.88) F = π Tr ({∂ Φ , lµ}) , 4 µ where we used 1 uµ = i∇µU + O(Φ2) = − ∂µΦ − lµ + O((Φ + l)2) . (3.89) Fπ

63 Måns Holmberg Master Thesis

Next, we insert (3.82):

Fπ µ gwFπ +µ Tr ({∂µΦ , l }) = − √ Tr ({∂µΦ ,T+}) W + h.c. 4 4 2 (3.90) gwFπ i +µ = − √ Tr ({si ,T+}) ∂µΦ W + h.c. , 4 2

from which we find k gwFπ − µ µ j W ≡ (VΦW− ) = − √ cjk , (3.91) Φ µ j 4 2 where we define the flavor factor

(10) √ cj ≡ Tr ({sj ,T+}) = 2Tr (sj T+) = 2 2(0 ,Vud , 0 , 0 ,Vus , 0 , 0 , 0) . (3.92)

Next up, the T → W −B vertex. Starting with the LO contribution (again, omitting irrelevant interactions):

hA e µ d abc hA e µ d abc √ adeB¯ (u ) T = √ adeB¯ (−l ) T 2 2 c b µ 2 2 c b µ

hA gw e d abc +µ = √ √ adeB¯ (T+) T W (3.93) 2 2 2 c b µ

hA gw 1 T e d abc i n µν + = √ √ √ ade(s ) (T+) (tn) B¯ T g W , 2 2 2 2 i c b µ ν where we define the "flavor" factor:

  0 0 0 0 0 0 0 0    0 0 − V√ud 0 0 − V√us 0 V√ud   6 3 2     V√ud 0 0 − V√us 0 0 0 0   6 6     0 0 0 0 0 0 0 0      1  0 0 − V√us 0 0 0 V√ud V√us  (11) √ T e d abc  6 3 2  cni ≡ ade(si )c(T+)b (tn) =   . (3.94) 2  0 0 0 − V√ud 0 0 0 0   3     V√us 0 0 0 0 0 0 0   3     0 0 0 0 0 0 0 0       0 0 0 0 0 −Vud 0 0    0 0 0 0 0 0 Vus 0

64 Måns Holmberg Master Thesis

Then, we have the NLO contribution

¯e µν d abc ¯e µν d abc icM adeBc γµγ5(f+ )b Tν + icEadeBc γµ(f− )b Tν ¯e µ ν ν µ d abc = iadeBc (cE − cM γ5)γµ(∂ l − ∂ l )b Tν

gw e d µ +ν ν +µ abc (3.95) = −i√ adeB¯ (cE − cM γ5)γµ(T+) (∂ W − ∂ W )T 2 c b ν

gw 1 T e d abc i µ +ν ν +µ n = −i√ √ ade(s ) (T+) (tn) B¯ (cE − cM γ5)γµ(∂ W − ∂ W )T , 2 2 i c b ν where we used (see (3.31)) µν µ ν ν µ µ ν f± = ∂ l − ∂ l − i[l , l ] + O(Φ) , (3.96)

and µν µ ν ν µ µ ν µν FL = ∂ l − ∂ l − i[l , l ] ,FR = 0 . (3.97)

(11) Note that the flavor factor is the same, i.e. cni . We obtain the vertex:

− Wν

  n k µν gw (11) hA µν µν µ ν T ≡ (VTW −B) = i√ c √ g + (cE − cM γ5)(kg/ − γ k ) . (3.98) µ ni 2 ni 2 2

Bi

(1) Finally, we need the T → BΦ vertex, which has the flavor factor cnij. This vertex can be found in appendix B. Before constructing the diagrams, we need the propagators. For small momentum transfers 2 2 2 (k  mW ≈ (80GeV) ), the W-boson propagator reduces to   −i kµkν i kµkν igµν 2 2 gµν − 2 + 2 2 → 2 , (3.99) k − mW + i k mW k mW

giving us Fermi’s effective theory, while the usual scalar propagator is

i 2 2 . (3.100) k − mΦ + i

2 2 2 2 2 The  can be set to zero, since k = (p2 + p3) = m23 will not hit the pole at k = mΦ. Moreover, since the propagators have a trivial spinor structure (i.e. no gamma matrices), we can

65 Måns Holmberg Master Thesis

contract the indices (of the metric) of the W-boson propagator and write both diagrams of the form

µν ∗ ¯ρσ ∗ ¯ M(i) =u ¯3Aiµv2Piu¯1Bi uν , M(j) =u ¯σBj u1Pj v¯2Ajρu3 , (3.101)

µ µν where Pi and Ai , Bi are defined as:

i 1 1 P1 = 2 ,P2 = − 2 2 2 , (3.102) mW mW (k − mΦ)

gw A1µ = A2µ = (VW −ν¯ e− )µ = i√ γµPL , (3.103) e 2

  µν µν gw (11) hA µν µν µ ν (B )ni = (VTW −B) = i√ c √ g + (cE − cM γ5)(kg/ − γ k ) , (3.104) 1 ni 2 ni 2 2 and (sum over j)8

µν µ ν gw hA 1 (1) (10) µ ν gw hA (11) µ ν (B )ni = (VΦW − ) (VTBΦ) = −√ √ c c k k = −√ √ c k k . (3.105) 2 j nij 2 2 2 4 nij j 2 2 2 ni

In (3.105), we used 1 j 1 T e T d abc g f cnijc = √ ade(s ) (s ) (tn) (sj) (T+) 4 2 2 i c j b f g 1  2  = √  (sT )e(t )abc(T )f 2δgδd − δgδd ade i c n + g b f 3 f b 2 2 (3.106) 1 T e d abc = √ ade(s ) (T+) (tn) 2 i c b

= cni , which holds true since the s-matrices satisfy (3.43) and since T+ is traceless.

The matrix element square is then

2 2 2 X X ∗ h|M |i = hM(i)M(j)i , (3.107) i=1 j=1

8The sum over j means that we sum over all Goldstone boson propagators. We should be careful, however, since if more than one Goldstone boson gives a contribution, we would have to use different masses mΦ in P2 (see (3.102)). Luckily, it turns out that only one Goldstone boson is propagating for a given initial and final state. If this was not the (11) (11) (1) (10) case, then cni , in (3.94), would have elements that mixed Vud and Vus, given that 4cni = cnij cj (shown in (3.106)), which is not what we find.

66 Måns Holmberg Master Thesis with ∗ 1 ∗  ¯  hM M i = PiP Tr (p/ + m3)Aiµ(p/ − m2)Ajρ (i) (j) 4 j 3 2 (3.108)   × Tr −(P/ + M)P 3/2(P )B¯ρσ(p + m )Bµν , νσ j /1 1 i since ∗ ∗ ¯
 ¯ρσ µν M(i)M(j) = PiPj Tr u3u¯3Aiµv2v¯2Ajρ Tr uνu¯σBj u1u¯1Bi . (3.109)

The calculation of (3.107) is performed using Mathematica.

67 Chapter 4

Results and discussion

In this section, we will cover the results of the calculations of all considered decay widths. By comparing

to experimental values, we obtain the values of hA, cM and cE. Using the obtained values of these LEC:s, we predict the decay widths of Σ∗0 → Σ0γ and Ξ∗0 → Ξ0γ, as well as of 20 radiative three-body − 0 − decays. Furthermore, we explicitly check, using the decay rate of Ω → Ξ e ν¯e, that the cM and cE interaction terms do, in fact, contribute already at NLO.

∗ 4.1 Determining hA using B (J = 3/2) → Bπ at LO

We found in section 3.3.2 that the decay width of B∗(J = 3/2) → Bπ at tree-level is determined by

one interaction term at LO – the hA term in (3.18). After finding the Feynman rules, we calculated the matrix element and found the (partial) decay width to be (3.57):

2 hA (1) 2 3 EB + mB Γnij = 2 (cnij) pcm , (4.1) 96πFπ M with 1 q p = λ1/2(M 2, m2 , m2 ) ,E = p2 + m2 , (4.2) cm 2M Φ B B cm B (1) and where the flavor factor cnij is defined in (3.48). 3 Before finding hA, we first give some intuition why the decay width is proportional to pcm, and not

some other power of pcm. By dimensional analysis, we know that we must have three powers of energy −2 3 (or momenta) to compensate the dimension of Fπ ; but why must these come from pcm and not some other combination of masses? 3 The pcm dependence is what we expect from the non-relativistic limit of a resonance decays widths

68 Måns Holmberg Master Thesis

(for two-body decays). In this case, the decays widths is (see section 48 in [13])

2l+1 Γ ∼ (pcm) , (4.3) where l is orbital angular momentum of the two final-state particles. Then, given the positive parity of P 3 + l the initial-state (i.e., J = 2 ) and the parity (+1)(−1)(−1) of the final-state (coming from the octet P 1 + P 1 − baryon J = 2 and pion J = 2 ), we find that l must be odd. To conserve angular momentum, 3 only l = 1 is possible, hence Γ ∼ (pcm) . One can also do pure power counting of pcm, similar to the 3 analysis in section 4.5. Doing this, we also find Γ ∼ (pcm) .

Finally, we obtain the value of hA by fitting to experimental decay widths [13]. The results are summarized in table 4.1 below. With, e.g., Σ∗+ → (πΣ)+, we mean both Σ∗+ → π+Σ0 and Σ∗+ → π0Σ+, that, is the overall charge of the final state.

Decay BR [%] hA Σ∗+ → π+Λ 87.0 ± 1.5 2.39 ± 0.03 Σ∗+ → (πΣ)+ 11.7 ± 1.5 2.2 ± 0.1 Ξ∗− → (πΞ)− 100 2.0 ± 0.2 Ξ∗0 → (πΞ)0 100 2.00 ± 0.06

Table 4.1: LO determination of hA (in the last column) and the experimental values [13] for the branching ratios. The errors comes from the experimental uncertainty.

Given that hA in the case of the delta baryons (S = 0) is around 2.88 [29]; we observe a downwards

trend of the value of hA as the strangeness increases, since we found hA ≈ 2.3 in the case of sigma

baryons (S = 1) and hA ≈ 2.0 in the case of cascades (S = 2). Moreover, considering the large-Nc

limit, one either finds hA ≈ 2.67 [30, 9, 10] or hA ≈ 2.26 [11, 12]. This discrepancy of around 20% due to flavor breaking appears completely acceptable for an LO calculation of the decay widths.

∗ 4.2 Determining cM using B (J = 3/2) → Bγ at NLO

The partial decay width of B∗(J = 3/2) → Bγ is given by:

2c2 e2 E + M Γ = M (c(9))2p3 B , (4.4) ni 3π ni cm M

as found in (3.68), with 1 p = λ1/2(M 2, m2 , 0) . (4.5) cm 2M B

69 Måns Holmberg Master Thesis

(9) The flavor factor cni is defined in (3.63). The decay widths of the various radiative decays are provided in table 4.2.

(9) −1 Decay 2cni BR [%] |cM | [GeV ] √ ∆ → Nγ −2/ 3 0.60±0.05 2.00±0.03 √ Σ∗+ → Σ+γ 2/ 3 0.70±0.17 1.89±0.08 Σ∗− → Σ−γ 0 < 0.024 — √ Σ∗0 → Σ0γ −1/ 3 0.18±0.01 — Σ∗0 → Λγ 1 1.25±0.13 1.89±0.05 √ Ξ∗0 → Ξ0γ 2/ 3 4.0±0.3 — Ξ∗− → Ξ−γ 0 < 4 —

Table 4.2: NLO predictions (in bold) and experimental values [13] for the branching ratios of the radiative decuplet decays (third column). The second column shows the calculated flavor factors and the last column contains cM as determined from the respective experimental value.

−1 Matching to the measured decay widths results in the average |cM | = (1.92 ± 0.08) GeV . Note 2 that we cannot determine the sign since the decay width depends on cM . With this input, we could make a NLO prediction for the unknown decays rates Σ∗0 → Σ0γ and Ξ∗0 → Ξ0γ, which are provided in table 4.2 as the bold entries. Unfortunately, the decays of Σ∗− → Σ−γ and Ξ∗− → Ξ−γ vanishes at NLO, since these decays break flavor symmetry, or more specifically U-spin symmetry1. At NLO, flavor symmetry is not explicitly broken, rendering U-spin symmetry to be exact. If we were to consider higher orders of the chiral expansion, we would at some point break flavor symmetry. In the case of the radiative decays, this happens at NNLO.

Moreover, as in the case of hA at LO, we again see the effects of unbroken flavor symmetry in the value of cM for different strangeness, since |cM | ∼ 2 for the Delta baryon decay and |cM | ∼ 1.9 for the Sigma decays. Compared to the B∗(J = 3/2) → Bπ decay, where parity and angular momentum conservation demanded l = 1 (the final-state orbital angular momentum), we now have both l = 1 and l = 3 as 7 possibilities. However, the (pcm) contribution is not seen at NLO since it is heavily suppressed. Again, 3 if we do a power counting of pcm, as in section 4.5, we find Γ ∼ (pcm) at NLO.

∗ − 4.3 Determining cE using B (J = 3/2) → Be ν¯e at NLO

The three-body decay of a decuplet baryon decaying into an octet baryon, an electron, and an electron antineutrino is a bit more complicated to calculate, compared to the previous two-body decays. The

1Like isospin concerns up and down quarks, U-spin concerns down and strange quarks.

70 Måns Holmberg Master Thesis

main source of complication comes from taking the trace of gamma matrices – a repetitive task that computers do very well. Therefore, we calculate the decay width with the matrix element squared (3.107) using FeynCalc in Mathematica. The general form of the decay width is

2 2 2 Γ = αhA + βcM + γcE + δhAcE , (4.6) where α, β, γ and δ are constants that only depend on the particle masses. ∗ − The only decay width of B (J = 3/2) → Be ν¯e that has been determined experimentally is − 0 − − 0 − Ω → Ξ e ν¯e, unfortunately, with large uncertainty. The branching ratio of Ω → Ξ e ν¯e at

LO and NLO, depending on cE, is shown in figure 4.1. Here we used hA = 2.0 as we determined ∗ −1 by B (J = 3/2) → Bπ (in the case of cascades) and |cM | = 1.92 GeV as we determined by B∗(J = 3/2) → Bγ.

− 0 − Figure 4.1: NLO branching ratio of Ω → Ξ e ν¯e as a function of cE, together with the LO result and the experimental value (PDG). The gray box is showing the uncertainty of the measurement.

−1 Matching to the measured decay width results in two solutions of cE: (0.5 ± 2.0) GeV and (−5.2 ± 2.0) GeV−1, where these error come only from the experimental uncertainty (the total error

is larger). We cannot distinguish between the two solutions of cE by only considering the integrated decay width. However, by looking at the distribution of the double differential decay width, we

obtain more information. The double differential decay width for the two solutions of cE and cM is illustrated in figure 4.2. We used the rest frame of the Cascade baryon and the electron and the kinetic variables m2(Ξ0e−) and cos(θ) (where θ is the angle between the three-momenta of the cascade and

the antineutrino). In this frame, we can distinguish between the two solutions of cE and pin down

the sign of cM , since all plots in figure 4.2 are different. The relative sign of cM and cE would be the easiest thing to distinguish since this determines if the distribution of the double differential decay

71 Måns Holmberg Master Thesis width is concentrated towards lower or higher values of m2(Ξ0e−).

0.0125 0.008

0.0100

0.006

0.0075

0.004

0.0050

0.002 0.0025

0 0

−1 −1 −1 −1 (a) cM = 1.92 GeV , cE = 0.5 GeV (b) cM = 1.92 GeV , cE = −5.2 GeV

0.0125 0.008

0.0100

0.006

0.0075

0.004

0.0050

0.002 0.0025

0 0

−1 −1 −1 −1 (c) cM = −1.92 GeV , cE = 0.5 GeV (d) cM = −1.92 GeV , cE = −5.2 GeV

− 0 − Figure 4.2: The double differential decay width of Ω → Ξ ν¯ee for the two solutions of cE and for different signs of cM . The unit of the double differential decay width is arbitrary.

.

4.4 NLO predictions of BR(B∗(J = 3/2) → Bπγ)

∗ ∗ We have now determined the value of hA (in the case of both Σ and Ξ ), the absolute value of cM

and the range of possible values of cE. Using these findings, as well as all previously known LEC:s (see section 3.2), we can predict the branching ratios of all B∗(J = 3/2) → Bπγ decays at NLO. This is

72 Måns Holmberg Master Thesis

achieved by calculating the decay width with (3.77) using the FeynCalc2. We summarize the results in table 4.3 below. These are new findings since none of these branching ratios have been measured or predicted before.

Decay LEC dependence at NLO BR at LO BR at NLO ∗+ + 0 −7 −7 Σ → Σ π γ hA,HA, D, cM , dM , bM,D, bM,F 7.17 × 10 9.45 × 10 ∗+ 0 + −5 −5 Σ → Σ π γ hA,HA, F, cM , cE, dM , bM,D 2.86 × 10 2.97 × 10 ∗+ + Σ → Λ π γ hA,HA, D, cM , cE, dM , bM,D -- ∗− − 0 −7 −7 Σ → Σ π γ hA, dM , bM,D, bM,F 5.08 × 10 5.26 × 10 ∗− 0 − −5 −5 Σ → Σ π γ hA,HA, cM , cE, dM , bM,D 3.32 × 10 3.59 × 10 ∗− − Σ → Λ π γ hA,HA, cM , cE, dM , bM,D -- ∗0 + − −5 −5 Σ → Σ π γ hA,HA, D, F, cM , cE, bM,D, bM,F 4.76 × 10 4.78 × 10 ∗0 − + −5 −5 Σ → Σ π γ hA, D, F, cM , cE, bM,D, bM,F 2.68 × 10 2.71 × 10 ∗0 0 0 −8 Σ → Σ π γ hA, D, cM , bM,D 0 2.31 × 10 ∗0 0 −6 Σ → Λ π γ hA, D, cM , bM,D 0 3.24 × 10 ∗− − 0 −6 −6 Ξ → Ξ π γ hA, dM , bM,D, bM,F 6.55 × 10 6.87 × 10 ∗− 0 − −3 −3 Ξ → Ξ π γ hA,HA, cM , cE, dM , bM,D 1.09 × 10 1.12 × 10 ∗0 − + −4 −4 Ξ → Ξ π γ hA, D, F, cM , cE, bM,D, bM,F 9.34 × 10 9.48 × 10 ∗0 0 0 −6 Ξ → Ξ π γ hA,HA, D, F, cM , bM,D 0 1.62 × 10 ++ + −3 −3 ∆ → p π γ hA,HA, cM , cE, dM , bM,D, bM,F 1.34 × 10 1.42 × 10 + 0 −5 −5 ∆ → p π γ hA,HA, D, F, cM , dM , bM,D, bM,F 4.48 × 10 5.81 × 10 + + −4 −4 ∆ → n π γ hA,HA, D, F, cM , cE, dM , bM,D 6.04 × 10 6.08 × 10 0 − −4 −4 ∆ → p π γ hA,HA, D, F, cM , dM , bM,D, bM,F 8.35 × 10 8.38 × 10 0 0 −6 ∆ → n π γ hA,HA, D, F, cM , bM,D 0 6.55 × 10 − − −3 −3 ∆ → n π γ hA,HA, cM , cE, dM , bM,D 1.81 × 10 1.87 × 10

Table 4.3: Branching ratio at LO and NLO of all energetically possible B∗(J = 3/2) → Bπγ decays allowed by flavor symmetry. The second column is showing the LEC dependence at NLO. Here we used −1 −1 cM = +1.92 GeV and cE = 0.52 GeV . We used a photon energy cutoff at 25 MeV for decays with an infrared divergence at low photon energies. The decays Σ∗± → Λ π± γ run over the pole of Σ0 and cannot be integrated directly.

The infrared divergence is a result of final-state radiation of photons that, in the limit of vanishing photon energy, can make a propagating particle on-shell. Consider writing the invariant mass m2(BΦ) as 2 2 m (BΦ) = M − 2MEγ . (4.7)

2This computation took several hours using a modern laptop (even when the task was running in parallel on different CPU cores).

73 Måns Holmberg Master Thesis

Then, we see that a vanishing photon energy corresponds to the maximum of m2(BΦ) (i.e., M 2). We see precisely this infrared divergence in figure 4.4 at large m2(BΦ). One deals with this problem by enforcing a cut-off of the photon energy when integrating the differential decay rate. This cut-off can be arbitrarily chosen to match the photon energy resolution of any experiment. In this work, we chose the cut-off at 25 MeV, corresponding to a relative photon energy resolution of around 5 − 10% at the upcoming PANDA¯ experiment [36].

0.006

0.004

0.002

0

Figure 4.3: Double differential decay width of Ξ∗0 → Ξ−π+γ in the frame where the pion and octet baryon goes back-to-back, and where the angle θ is the angle between the three-momenta of the octet baryon and photon.

Looking at table 4.3 we note that decay widths with neutral final states vanish at LO, even though LEC:s of the LO Lagrangian contribute at NLO. From table 3.1 we see that the only diagram 3, 4 and 5 (see section 3.3.4) give a contribution to these decays. Diagram 4 and 5 vanish at LO since they are

proportional to cM , which makes the D, F and HA dependence trivially go away. Then, looking at the structure of the third diagram (coming from the BBγ-vertex):

(3) X (1) h (6)  (7) (6) i Mnij ∼ hAcnlj cil (··· ) + bM,Dcil − bM,F cil (··· ) (··· ) , (4.8) l

(6) we see that we could lose the hA dependence at LO, but not at NLO, in the case when cil = 0 and (7) cil 6= 0. This is precisely what happens; as we see explicitly in table 4.3, since hA always comes

together with bM,D and not bM,F , in the case of neutral final states. This makes the BBγ-vertex

74 Måns Holmberg Master Thesis

different from the ΦΦγ and the T T γ-vertices, since the BBγ-vertex couples to neutral particles (via

bM,D), which does not occur for the ΦΦγ and T T γ vertices. Furthermore, it is reassuring that the branching ratio of these neutral decays are small at NLO, since they vanish at LO; indicating that the NLO contribution is in fact a small correction. Moreover, it is only natural that decays involving only neutral particles and photons should be small, since such decays would have been even more suppressed if hadrons did not contain charged quarks. That is, neutral radiative decays probe the inner structure of hadrons. We can also explain why some decays that have a nonvanishing branching ratio at LO are very small, especially for Σ∗+ → Σ+ π0 γ and Σ∗− → Σ− π0 γ. This is the case since they are missing diagram 2, which has the largest contribution of all diagrams, because of the pion propagator.

Given the large uncertainty of HA (from (3.18)), we varied its value around HA = 2.0 (by a normal distribution with a standard deviation of 1) to estimate the sensitive of the decay widths on the value

of HA. This changed the decay widths insignificantly (much less than 1%), except in the case of Ξ∗0 → Ξ0 π0 γ and ∆0 → n π0 γ, which changed by 10% and 2%, respectively.

We also considered a negative cM , but this changes the decay widths by only a few percents (often much less) in the case of the charged final states decays. Also, the decays with neutral final-states (and vanishing decay width at LO) changed significantly (up to 80%). But this is to be expected since their decay width comes only from NLO.

Furthermore, we can also use these radiative three-body decays to reduce the uncertainty of cE; once data are available. For this purpose, the decays Ξ∗− → Ξ0 π− γ and Ξ∗0 → Ξ− π+ γ are of most interest, since they both have a relatively large branching ratios ∼ 10−3 and because of the small (total) decay width of Cascade baryons (as compared to broad Delta baryons). As compared to table 4.3, we find that the decay widths of the two cascade decays decreases by around 20% by changing

to the negative solution of cE. In figure 4.4 we consider how the single differential decay width of ∗0 − + Ξ → Ξ π γ changes when comparing the solution of cE.

75 Måns Holmberg Master Thesis

∗0 Figure 4.4: Showing how the two solutions of cE impact the single differential decay width of Ξ → − + −1 Ξ π γ using cM = +1.92 GeV . Here we used a photon energy cutoff at 50 MeV (otherwise the cE-dependence is hard to see). The gray interval comes from the uncertainty of the two solutions of cE.

We see that at high photon energies, where the single differential decay width is diverging, both 2 solutions of cE converges. Instead, it is at higher photon energies, i.e., lower m (BΦ), where we could

distinguish between the two possible cE. Also, since the change of the single differential decay width of

the two possible cE is not drastic, one would need a good resolution to determine the correct value, especially in the lower m2(BΦ) range.

Let us now explicitly show that hA is an LO term and that cM and cE belong to NLO.

4.5 The importance of cE at NLO

The authors of [14] do not include the cE interaction term (of the decuplet-octet transition sector

(3.34)) in what they claim is a complete NLO Lagrangian. Therefore, they imply that the cE term − 0 − only contributes at higher orders. Here, we argue, by the example of the Ω → Ξ e ν¯e decay width, why such a term does, in fact, contribute already at NLO. To do this, we explicitly check the scaling

behavior of the cE interaction term (in (3.34)), compared to the hA and cM terms, when the Goldstone boson mass and the decuplet-octet mass difference go to zero3. A small decuplet-octet mass difference is as important as a small Goldstone boson mass to make, e.g., the center-of-mass momenta (4.2) of the B∗(J = 3/2) → Bπ decay small, since

1 1 1/2 p = λ1/2(M 2, m2 , m2 ) = m2 − (M − m )2
m2 − (M + m )2
 . (4.9) cm 2M Φ B 2M Φ B Φ B

3 The decuplet-octet mass splitting vanishes in the large Nc limit, thus, making the decuplet and octet baryons degenerate. They become part of the same (extended) representation [37, 38].

76 Måns Holmberg Master Thesis

Therefore, let us consider mΦ → amΦ and M → mB + a(M − mB), where a ∈ [0, 1]. Hence, a = 0

implies pcm = 0. Doing this, we expect the different terms of (4.6) to scale differently with small a, which tells us how important each term is. Moreover, we set mν, me → 0 in order for a not to be bounded from below. From the kinematics alone, each term of (4.6) picks up an O(a3) dependence. We can see this ∗ by looking at the general double differential decay width in (3.17), where each momentum (|~pν| and ∗ 2 |~pe|) contributes with O(a) in the same way as (4.9). Lastly, the integral over dm12 contributes with another O(a), since 2 2 2 dm12 ∼ M − mB ∼ O(a) . (4.10)

As for the matrix element itself, we see that the lepton trace (i.e. the first trace) in (3.108) contains 2 2 two soft momenta: p2 and p3; hence, contributing with O(a ). Consequentially, the hA term in (4.6) 5 scales as O(a ). Note also that (3.105) contains two k = pν + pe, one from the TBΦ vertex and one from the ΦW − vertex, which contribute with higher order of a. 2 2 2 As for the NLO terms, both the cM and cE terms pick up another O(a ), since these terms

contribute with the momentum k, as seen in (3.104). This should make the cM and cE terms in (4.6) 7 6 scale as O(a ). The hAcE term, being an interference term, should instead scale as O(a ). The arguments above tell us how each term of the decay width scales with a at minimum, but, there

could be unforeseen kinematic effects that, for instance, make the cE contribution less important (i.e., scale with higher powers of a). Therefore, we also checked the importance of each term numerically. Doing this yields precisely the predicted results, that is: α ∼ O(a5), δ ∼ O(a6) and β ∼ γ ∼ O(a7). We conclude that the LO contribution is more important than the NLO contribution in precisely the way we would expect and that the cE interaction term of the decuplet-octet transition sector is correctly classified to be NLO.

77 Chapter 5

Conclusion and future directions

To conclude, we went from the mathematics of Lie groups and Lie algebras to the Lagrangian of QCD, and finally, to χPT for mesons at LO and baryons at NLO. We have shown that χPT, being an effective field theory where we treat momenta as the expansion parameter, offers a systematic approach to deal with the low-energy problems of QCD. All of this has rendered us able to fulfill the aim of this thesis,

that is, to determine the values of the LEC:s hA, cM and cE, by studying decays of decuplet baryons using the χPT at LO and NLO.

The value of hA, being the LEC concerning the decay of a decuplet baryon into an octet baryon and a pion, was determined to be around 2.3 for Sigma baryons and around 2.0 for Cascade baryons.

The latter of which had not previously been obtained. Next, we determined the value of cM from three radiative two-body decays, as shown in table 4.2. We found the modulus to be around 2.0 GeV−1 for −1 −1 Delta baryons and around 1.9 GeV for Sigma baryons, and on average |cM | = (1.92 ± 0.08) GeV .

We could not determine the sign since the decay width depended on the square of cM . The discrepancy

of the value of hA and cM for different baryons (with different strangeness) is to be expected since flavor symmetry is not exact. As stated before, the breaking of flavor symmetry is systematically treated in χPT; however, these effects come into play only at NNLO and above. Note again that all LEC:s do have a precise value and that the apparent splitting only comes about when fitting to data, that is, the discrepancy is a result from the comparison of the calculated NLO decay widths and nature (who does − 0 − not stop at NLO). We then studied the decay of Ω → Ξ e ν¯e, since its branching ratio had been − 0 − measured, to determine cE. However, as in the case of cM , the decay width of Ω → Ξ e ν¯e depends 2 −1 −1 on cE, giving us two possible values: cE: (0.5 ± 2.0) GeV and (−5.2 ± 2.0) GeV . The large errors

come from the large experimental uncertainty. We also explicitly showed that cE interaction term contributes at NLO and should thus be included in a complete NLO Lagrangian for baryons, contrary

to [14]. Note that the values of cM and cE has never been determined before.

We also found that we one could in principle determine the solutions of cM and cE by studying full

78 Måns Holmberg Master Thesis

− 0 − double differential decay rates of Ω → Ξ e ν¯e, since this is for the different possible values of cM and

cE. This is illustrated in figure 4.2.

Using these value of cM we could determine the branching ratio of the unknown decays rates Σ∗0 → Σ0γ and Ξ∗0 → Ξ0γ to be 0.18±0.01 and 4.0±0.3, respectively. These branching ratios and the

determined value of cM are published in [8]. We also studied the radiative three-body decays of the form B∗(J = 3/2) → Bπγ, which, using the values of the various LEC:s we studied, made it possible to predict the branching ratios of 20

unobserved decays, summarized in table 4.3. Moreover, we can better pin down the value of cE once data is available for these decays. Studying the decays of Cascades would be of most interest in order ∗− 0 − to obtain as much information about cE as possible, specifically, the decays of Ξ → Ξ π γ and Ξ∗0 → Ξ− π+ γ. We want to stress that most of these decays (apart from the ones with a tiny branching ratio) will be within reach of experiments in a few years when facilities such as FAIR in Germany come online. Once this happens, we will be able to study more hyperon decays, like the ones considered in this thesis, to learn more about the low-energy behavior of QCD. From here, there are several future directions that one could take. Maybe one of the most exciting possibilities is to study the polarization of the considered decays, as this would allow a more detailed test of the χPT predictions and/or access to interaction structures that appear only beyond NLO. Another possibility would be to study loop corrections of the simpler (two-body) decays, to better determine the LEC:s and to obtain better predictions. Loop correction contribute to higher orders in χPT [5], meaning that NNLO order terms and beyond have to be considered at some point; however, the amount of terms at higher orders is huge. This is what intrinsically limits the applicability of any EFT, but at low enough energies, this is not a problem. In the work presented here, LEC:s have been obtained from fits to data and experiments have been identified that can provide an even better access to these LEC:s. On the pure theory side, it is not possible at the moment to obtain all LECs from lattice QCD. Yet, in the future it will be interesting to test and improve our understanding of the strong interaction by comparing QCD lattice predictions and experimental results for the LECs.

79 Appendix A

Properties of spin-3/2 fermions (under construction)

To do: write some explanations. A Dirac spin-3/2 (vector) fermion has the propagator

p (T ) µ ν = Pµν (p, m)

i(p/ + m) 2 pµpν = − P 3/2(p) + i(p/ + m) p2 − m2 + i µν 3m2 p2 (A.1) i p pαγ + γ pαp − µ αν µα ν 3m p2 i(p/ + m)  1 2p p pµγν − pνγµ  = − g − γ γ − µ ν + , p2 − m2 + i µν 3 µ ν 3m2 3m

3/2 where Pµν (p) is the projector on spin 3/2, and is defined by

3/2 1 1
P (p) ≡ gµν − γµγν − pγ/ µpν + pµγνp/ . (A.2) µν 3 3p2

The spin-1/2 spinor satisfies (p/ − m)u(p, m) = 0 . (A.3)

The spin-3/2 spinor satisfies γ · u(p, m) = 0 ,

p · u(p, m) = 0 , (A.4)

(p/ − m)uµ(p, m) = 0 .

80 Appendix B

Vertices (under construction)

To do: write some text to accompany the derivations.

Let us expand some useful quantities using this replacement, starting with uµ expanded to linear terms in Φ:

† † uµ = iu (∇µU)u

† † = iu (∂µU + i[U, vµ]) u         (B.1) iΦ iΦ e iΦ 2 = i 1 − ∂µ 1 + − Aµ [Φ,Q] 1 − + O(Φ ) 2Fπ Fπ Fπ 2Fπ

∂µΦ ie 2 = − − Aµ [Φ,Q] + O(Φ ) , Fπ Fπ where we used i 2 [U, vµ] = [Φ, vµ] + O(Φ ) Fπ (B.2) ie 2 = Aµ [Φ,Q] + O(Φ ) . Fπ µν Next, the field strengths f± :

µν µν † † µν f± = uFL u ± u FR u         iΦ µν iΦ iΦ µν iΦ 2 = 1 + FL 1 − ± 1 − FR 1 + + O(Φ ) (B.3) 2Fπ 2Fπ 2Fπ 2Fπ   µν i  µν µν i  µν 2 = FL + Φ,FL ± FR − Φ,FR + O(Φ ) , 2Fπ 2Fπ where µν µ ν ν µ µν FR,L = e (∂ A − ∂ A ) Q = eF Q, (B.4)

81 Måns Holmberg Master Thesis

results in µν µν 2 µν ie µν 2 f+ = 2eF Q + O(Φ ) , f− = F [Φ,Q] + O(Φ ) . (B.5) Fπ

B.1 TBΦ

Inserting (B.1) into the hA term of (3.71) gives

hA ¯e µ d abc hA  ¯e µ d abc ¯e µ d abc √ adeBc (u )b Tµ = − √ adeBc (∂ Φ)b Tµ + ieadeBc A [Φ,Q]b Tµ , (B.6) 2 2 2 2Fπ where the first term gives the B(J = 3/2) → A(J = 1/2) Φ interaction

hA ¯e µ d abc hA T e d abc ¯i µ j n − √ adeBc (∂ Φ)b Tµ = − ade(si )c(sj)b (tn) B ∂ Φ Tµ , (B.7) 2 2Fπ 4Fπ

corresponding to the vertex (correct)

Φj

pT pΦ µ hA (1) µ Tn, µ = (VTBΦ)nij(pΦ) = √ cnij pΦ , (B.8) 2 2Fπ

pB

Bi with the constant (1) 1 T e T d abc c ≡ √ ade(s ) (s ) (tn) . (B.9) nij 2 i c j b

(1) Note that cnij is antisymmetric in i, j.

B.2 TBΦγ

There are two interaction terms giving rise to the B(J = 3/2) → A(J = 1/2) Φ γ interaction. Starting with the second term of (B.6)

hAe ¯e µ d abc hAe 1 c d abc ¯i j µ n −i √ adeBc A [Φ,Q]b Tµ = −i √ √ ade(si)e [sj,Q]b (tn) B Φ A Tµ . (B.10) 2 2Fπ 2 2Fπ 2

82 Måns Holmberg Master Thesis

The second one is the cE term of (3.71)

¯e µν d abc cEe ¯e µν d abc icEadeBc γµ(f− )b Tν = − adeBc γµF [Φ,Q]b Tν Fπ x (B.11) cEe 1 T e d abc ¯i j µν n = − √ ade(si )c [sj,Q]b (tn) B Φ γµF Tν , Fπ 2 where we inserted (B.5). The vertex is

γ, ν pγ

pT pΦ e  h   T , µ Φ = (V )µν (p ) = c(2) √A gµν + c p gµν − γνpµ , (B.12) n j TBΦγ nij γ nij E /γ γ Fπ 2 2

pB

Bi where (2) 1 T e  T d abc c ≡ √ ade(s ) s ,Q (tn) (B.13) nij 2 i c j b

B.3 BBΦ

D ¯ µ
D 1 T
¯i µ j k tr Bγ γ5{uµ ,B} = − tr si {sj , sk} B γ γ5∂µΦ B . (B.14) 2 2Fπ 2

F ¯ µ
F 1 T
¯i µ j k tr Bγ γ5[uµ ,B] = − tr si [sj , sk] B γ γ5∂µΦ B . (B.15) 2 2Fπ 2

Φj

pB pΦ 1   B = (V ) (p ) = Dc(3) + F c(4) p γ , (B.16) k BBΦ kij Φ kij kij /Φ 5 2Fπ

pB0

Bi where 1 1 c(3) ≡ tr sT {sT , s }
, c(4) ≡ tr sT [sT , s ]
. (B.17) kij 2 i j k kij 2 i j k

83 Måns Holmberg Master Thesis

B.4 TT Φ

HA ¯µ ν c abd HA ¯µ ν c abd − Tabcγνγ5(u )dTµ = Tabcγνγ5∂ ΦdTµ 2 2Fπ (B.18) HA c abd ¯µ ν m = (tn)abc(si)d(tm) Tn γνγ5∂ ΦiTµ , 2Fπ

Φi

pT pΦ H T , µ = (V )µν (p ) = − A c(5) p γ gµν (B.19) m TT Φ mni Φ mni /Φ 5 2Fπ

pT 0

Tn, ν

(5) T c abd cmni = (tn)abc(si )d(tm) . (B.20)

B.5 ΦΦγ

F 2   F 2   π tr (∇ U)†∇µU = π tr (∂ U † + i[U †, v† ]) (∂µU + i[U, vµ]) 4 µ 4 µ µ 1   = tr (−i∂ Φ + [Φ, v† ]) (i∂µΦ − [Φ, vµ]) + O(Φ3) (B.21) 4 µ µ 1 = tr (∂ Φ∂µΦ + ie {∂ Φ, [Φ,Q]} Aµ) + O(Φ3) + O(A Aµ) 4 µ µ µ ie ie tr ({∂ Φ, [Φ,Q]}) Aµ = tr ({s , [s ,Q]})(∂ Φi)ΦjAµ (B.22) 4 µ 4 i j µ The derivative can act both on the incoming and outgoing particle, hence we add both cases to avoid

ambiguities. Consider the flavor factor where Φi is the outgoing particle

(6)  T
cij = tr si , [sj,Q]

T
= 2 tr si [sj,Q]

T T
= 2 tr si sjQ − si Qsj) (B.23)

T T
= 2 tr si sjQ − sjsi Q)

T
= 2 tr [si , sj]Q .

In this case, the derivative acts on Φi. If instead the derivative acts on the incoming particle Φj, the corresponding flavor factor is T
(6) 2 tr [sj, si ]Q = −cij . (B.24)

84 Måns Holmberg Master Thesis

The vertex is then

γ, µ

p Φ p γ µ ie (6) µ µ
Φ = (V ) (p , p 0 ) = − c p + p . (B.25) i ΦΦγ ij Φ Φ 4 ij Φ Φ0

pΦ0

Φj

It is easy to show that this flavor factor is symmetric

(6) T
cij = 2 tr [si , sj]Q

T T
= 2 tr ([si , sj]Q) (B.26) T
= 2 tr [sj , si]Q

(6) = cji ,

thus the vertex is also symmetric with respect to i, j. It is also diagonal (proof by Mathematica).

B.6 BBγ

There are three interaction terms giving rise to this vertex.

i tr B¯DB/
= i tr B¯(∂B/ + [Γ/,B])
(B.27) = i tr B¯(∂B/ − ieA/[Q, B])

1   Γµ = u†(∂µ − ivµ)u + u(∂µ − ivµ)u† 2 (B.28) = −ivµ + O(Φ)

[Γµ,B] = −i[vµ,B] + O(Φ) = −ieAµ[Q, B] + O(Φ) (B.29)

e eA/ tr B¯[Q, B]
= tr sT [Q, s ]
A/B¯iBj 2 i j e = − tr [sT , s ]Q
A/B¯iBj (B.30) 2 i j e = − c(6)A/B¯iBj , 4 ij (6) where we used the existing flavor factor cij from (B.23).

85 Måns Holmberg Master Thesis

¯ µν
T
µν ¯i j bM,D tr B{f+ , σµνB} = bM,De tr si {Q, sj} F σµνB B (B.31) b e = M,D c(7)F µνσ B¯iBj , 2 ij µν (7) (6) where we define cij similar to cij

(7) T
T
cij = 2 tr si {Q, sj} = 2 tr {si , sj}Q} . (B.32)

This flavor factor is also symmetric

(7) T
cij = 2 tr {si , sj}Q

T T
= 2 tr ({si , sj}Q) (B.33) T
= 2 tr {sj , si}Q

(7) = cji ,

but not diagonal (proof by Mathematica).

¯ µν
T
µν ¯i j bM,F tr B[f+ , σµνB] = bM,F e tr si [Q, sj] F σµνB B (B.34) b e = − M,F c(6)F µνσ B¯iBj , 2 ij µν µν The structure F σµν will correspond to the vertex

µν µ ν ν µ F σµν → i(pγ gα − pγgα )σµν , (B.35) where the free Lorentz index α of the two metrics is to be contracted with the polarization vector ∗α  (pγ, λ) later on. The vertex can be simplified using the antisymmetric property of σµν

µν µ ν ν µ
F σµν → i pγ gα σµν − pγgα σµν

µ ν ν µ
= i pγ gα σµν + pγgα σνµ

µ = 2ipγ σµα (B.36)

µ = −pγ [γµ , γα]   = − p γ − γ p . /γ α α/γ

86 Måns Holmberg Master Thesis

γ, α

p B p γ α e (6) α  (7) (6)  α α  Bi = (VBBγ) (pγ) = −i c γ − e bM,Dc − bM,F c p/ γ − γ p/ (B.37) ij 4 ij ij ij γ γ

pB0

Bj

(6) (7) The vertex is symmetric (but not diagonal) in i, j as both cij and cij are symmetric.

B.7 T T γ

There are two interaction terms giving rise to this vertex.

α ν abc α ν abc α a ν a0bc α b ν ab0c α c ν abc0 (D T ) = ∂ (T ) − (Γ )a0 (T ) − (Γ )b0 (T ) − (Γ )c0 (T ) (B.38) α ν abc α  a ν a0bc b ν ab0c c ν abc0  = ∂ (T ) − ieA Qa0 (T ) + Qb0 (T ) + Qc0 (T )

¯µ α ν abc ¯µ α ν abc iTabcγµνα(D T ) = iTabcγµνα∂ (T ) (B.39)  a a0bc b ab0c c abc0  ¯µ α ν + e(tn)abc Qa0 (tm) + Qb0 (tm) + Qc0 (tm) Tn γµναA Tm .

We can write  a a0bc b ab0c c abc0  (tn)abc Qa0 (tm) + Qb0 (tm) + Qc0 (tm)

 a bca0 b acb0 c abc0  = (tn)bcaQa0 (tm) + (tn)acbQb0 (tm) + (tn)abcQc0 (tm) (B.40)

c abd = 3(tn)abcQd(tm) , which is true because the basis t is totally symmetric. We have the interaction

c abd ¯µ α ν (8) ¯µ α ν 3e(tn)abcQd(tm) Tn γµναA Tm = 3ecij Tn γµναA Tm , (B.41) with (8) c abd cnm = (tn)abcQd(tm) , (B.42)

(8) which is symmetric in n, m as Q is diagonal. cnm is also diagonal (proof by Mathematica).

87 Måns Holmberg Master Thesis

¯ µν c abd ¯ µν c abd idM (Tµ)abc(f+ )d(Tν) = 2idM e(Tµ)abcF Qd(Tν) c abd ¯n µν m = 2idM e (tn)abcQd(tm) Tµ F Tν (B.43) (8) ¯n µν m = 2idM e cnmTµ F Tν ,

γ, α

p T p γ νµα (8) νµα µ να ν µα
Tn, µ = (VT T γ)nm (pγ) = ie cnm 3γ − 2dM (pγ g − pγg ) . (B.44)

pT 0

Tm, ν

B.8 T Bγ

¯e µν d abc 2 T e d abc ¯i µν n icM adeBc γµγ5(f+ )b Tν = √ icM e ade(si )cQb (tn) B γµγ5F Tν 2 (B.45) (9) ¯i µν n = 2icM e cni B γµγ5F Tν

(9) 1 T e d abc c = √ ade(s ) Q (tn) (B.46) ni 2 i c b

γ, α

pT pγ T , µ = (V )µα(p ) = −2ic e c(9)(p gµα − pµγα)γ . (B.47) n T Bγ ni γ M ni /γ γ 5

pB

Bi

88 Appendix C

Calculation of the matrix element of B∗(J = 3/2) → Bγ

Starting from (3.66), the spin average of the matrix element squared is

1 X |M |2 = |M |2 ni 4 ni spin

2 2 (9) 2 X  α β β α µ ∗ν ν ∗µ = cM e (cni ) k ε − k ε (k ε − k ε ) tr (uνu¯βγαγ5uuγ¯ µγ5) spin

2 2 (9) 2  β ν αµ β µ αν α ν βµ α µ βν = cM e (cni ) −k k g + k k g + k k g − k k g (C.1)

 3/2 × tr −γαγ5(p/ + mB)γµγ5(P/ + M)Pνβ

2 2 (9) 2  β ν αµ β µ αν α ν βµ α µ βν = cM e (cni ) −k k g + k k g + k k g − k k g


3/2 × tr − γαpγ/ µP/ − mBMγαγµ + terms cubic in γ Pνβ ,

2 where we used {γµ, γ5} = 0 and (γ5) = I4. First note that the sum over spin also includes the 3/2 polarization of the photon. Also note that the terms which are cubic in γ vanish, since Pαβ (see appendix A) contains only terms with either zero or an even number of γ-matrices.

89 Måns Holmberg Master Thesis

Let us continue by evaluating one term at a time, starting with the first term of (C.1):

β ν αµ 
3/2 − k k g tr − γαpγ/ µP/ − mBMγαγµ Pνβ    β ν αµ
1 1 = −k k g tr γαpγ/ µP/ − mBMγαγµ −gνβ + γνγβ + (P/ γνPβ + PνγβP/ ) 3 3M 2    1 µ
P · k = − tr γ pγ/ µP/ − 4mBM k/k/ + (P/ k/ + k/P/ ) 3 M 2 2 2(P · k) µ

= − tr γ pγ/ µP/ − 16mBM (C.2) 3M 2 16(P · k)2 P · p  = m + 2m 3M 2 B M B 16E2 P · p  = γ M + 2m 3 M B 16E2M = γ (E + 2m ) , 3 B B where we used fact that photons are massless and

µ
µ µ
tr γ pγ/ µP/ = tr −γµγ p/P/ + γ p/{γµ, P/ } = tr (−4P · p + 2P · p) = −8P · p , (C.3)

since ν ν {γµ, P/ } = P {γµ, γν} = 2P gµν = 2Pµ . (C.4)

The second term of (C.1) becomes:

β µ αν 
3/2 k k g tr − γαpγ/ µP/ − mBMγαγµ Pνβ    β µ αν
1 1 = k k g tr γαpγ/ µP/ − mBMγαγµ −gνβ + γνγβ + (P/ γνPβ + PνγβP/ ) 3 3M 2    1 ν
1 = tr γ p/k/P/ − mBMk/ −3kν + γνk/ + ((P · k)P/ γν + Pνk/P/ ) 3 M 2

4
2

= tr p/k/P/ − mBMk/ −M k/ + (P · k)P/ (C.5) 3M 2 4 2 3
= tr −m p/k/P/ k/ + (P · k)p/k/P/ P/ + mBM k/k/ − mBM(P · k)k/P/ 3M 2 B 16 = −2M 2(k · p)(P · k) + M 2(p · k)(k · p) − m M(P · k)2
3M 2 B 16E2M = − γ (M + m ) , 3 B

90 Måns Holmberg Master Thesis where we used1

µ µ ν σ ρ µν σρ µρ νσ µσ νρ γ aγ/ µ = −2a/ , tr (γ γ γ γ ) = 4 (g g + g g − g g ) , (C.6)

and

k · p = k · (P − k) = k · P = MEγ . (C.7)

The third term of (C.1) becomes:

α ν βµ 
3/2 k k g tr − γαpγ/ µP/ − mBMγαγµ Pνβ    α ν βµ
1 1 = k k g tr γαpγ/ µP/ − mBMγαγµ −gνβ + γνγβ + (P/ γνPβ + PνγβP/ ) 3 3M 2

1  β β 2 2
 = tr k/pγ/ P/ − mBMkγ/ −3M kβ + M kγ/ β + (P/ kP/ β + P · kγβP/ ) 3M 2

1  2 2 β β = tr −3M k/p/k/P/ + M k/pγ/ P/ kγ/ β + k/pγ/ P/ (P/ kP/ β + P · kγβP/ ) 3M 2 (C.8) 3 3 β β  +3mBM k/k/ − mBM kγ/ kγ/ β − mBMkγ/ (P/ kP/ β + P · kγβP/ )

1 2 2
= tr −3M k/p/k/P/ + 2M (P · k)k/p/ − 4mBM(P · k)k/P/ 3M 2 4 = tr −6M 2(k · p)(P · k) + 2M 2(P · k)(k · p) − 4m M(P · k)2
3M 2 B 16E2M = − γ (M + m ) , 3 B where we used µ ν ρ µ γ a//bγµ = a b γ γνγργµ

ν ρ µ = a b γ γν(−γµγρ + 2gµρ)

ν ρ µ = a b γ (γµγνγρ − 2gµνγρ + 2gµργν) (C.9) ν ρ = a b (4γνγρ − 2γνγρ + 2γργν)

ν ρ = 2 a b {γν, γρ}

= 4 a · b .

1 µ ν µ ν µ µ Proof: γ aγ/ µ = a γ γν γµ = a γ (−γµγν + 2gµν ) = −4a/ + 2a/ = −2a/, since γ γµ = 4I4.

91 Måns Holmberg Master Thesis

Finally, the last term of (C.1) becomes:

α µ βν 
3/2 − k k g tr − γαpγ/ µP/ − mBMγαγµ Pνβ    α µ βν
1 1 = −k k g tr γαpγ/ µP/ − mBMγαγµ −gνβ + γνγβ + (P/ γνPβ + PνγβP/ ) 3 3M 2   
4 2 = −tr k/p/k/P/ − mBMk/k/ −4 + + 3 3 (C.10) 6
= tr k/p/k/P/ 3 48 = (P · k)2 3 48E2M 2 = γ . 3

Putting everything back together yields the (surprisingly compact) expression of the spin-averaged matrix element squared

E2M |M |2 = c2 e2(c(9))2 γ (16(E + 2m ) − 32(M + m ) + 48M) ni M ni 3 B B B (C.11) 16c2 e2 = M (c(9))2p2 M (E + M) . 3 ni cm B

92 Appendix D

Electromagnetic current conservation of B∗(J = 3/2) → Bπγ

Our goal is to show that: X (l)µ Mnij kµ = 0 . (D.1) l This should hold even if the mass of the photon is nonzero, therefore, we set k2 > 0. The structure of µν ∗ the vertices that are coming from the field strength F is trivially vanishing when replacing α → kα, since µν µ να ν µα ∗ µ να ν µα F → i (k g − k g ) α(k) → i (k g − k g ) kα = 0 . (D.2)

Using this, the first diagram reduces (see section 3.3.4) to

(1)µ µν Mnij kµ =u ¯(VTBΦγ)nij(k)uµkν (D.3) (2) hAe µ = cnij √ k uu¯ µ . 2 2Fπ

Next, the second diagram becomes (note that we do not sum over j)

(2)µ X ν (Φ) l µ Mnij kµ = u¯(VTBΦ)nil(k + q)uνS (k + q, mΦ)(VΦΦγ)lj(k + q , q)kµ l

hA (1) ν ν i (−i)e (6) µ µ = √ c (k + q )¯uuν c (k + 2q )kµ 2 2F nij (k + q)2 − m2 4 jj π Φ (D.4) h e 1 1 √A (1) ν ν (6) 2 = cnij(k + q )¯uuν 2 cjj (k + 2(k · q)) 2 2Fπ k + 2(k · q) 4

hAe (2) ν ν = − √ cnij(k + q )¯uuν , 2 2Fπ

93 Måns Holmberg Master Thesis

(6) where we used that cij , from (B.26), is diagonal and

1 (1) (6) 1 1 T e T d abc T
c c = √ ade(s ) (s ) (tn) (−2) tr sj[s ,Q] 4 nij jj 4 2 i c j b j

1 1 T e T d f T g abc = − √ ade(s ) (s ) (sj) [s ,Q] (tn) 2 2 i c j b g j f   1 T e d f 1 d f T g abc (D.5) = −√ ade(s ) δ δ − δ δ [s ,Q] (tn) 2 i c g b 3 b g j f

1 T e T d abc = −√ ade(s ) [s ,Q] (tn) 2 i c j b (2) = −cnij .

T In (D.5) we used the identity (3.43) and the fact that [sj ,Q] is traceless (as it is antisymmetric). Similarly, the third diagram becomes (where we do not sum over i)

(3)µ X µ (B) l ν Mnij kµ = u¯(VBBγ)il(k)kµS (k + p, mB)(VTBΦ)nlj(q)uν l

e i(k/ + p/ + mB) h (6) / √ A (1) ν = −i cii u¯k 2 2 cnijq uν 4 (k + p) − mB 2 2Fπ 2 e ((−p/ + mB)k/ + k + 2(k · p)) h (6) √ A (1) ν (D.6) = cii u¯ 2 cnijq uν 4 k + 2(k · p) 2 2Fπ

1 (1) (6) hAe ν = cnijcii √ q uu¯ ν 4 2 2Fπ

(2) hAe ν = cnji √ q uu¯ ν , 2 2Fπ where we used 1 1 c(1) c(6) = − c(1) c(6) = c(2) , (D.7) 4 nij ii 4 nji ii nji (1) which holds since cnij (defined in (3.48))) is antisymmetric in i, j. The fourth and fifth diagram vanishes trivially, as they are proportional to the structure from F µν, i.e.

(4)µ (5)µ Mnij kµ = Mnij kµ = 0 . (D.8)

The last diagram is given by

(6)ν X α (T ) βµν Mnij kν = u¯(VTBΦ)mij(q)Sαβ (P − k, Mm)(VT T γ)nm (k)uµkν . (D.9) m

94 Måns Holmberg Master Thesis

βµν µ We first simplify (VT T γ)nm (k)uµ using γ uµ = 0, by

βµν βµν (VT T γ)nm (k)uµ ∝ γ uµ

βνµ = −γ uµ 1 = − {γβν , γµ}u 2 µ 1 = − γµγβνu 2 µ 1   (D.10) = − γµγβγν − γµγνγβ u 4 µ 1   = − −γβγµγν + {γµ, γβ}γν + γνγµγβ − {γµ, γν}γβ u 4 µ 1   = − −γβ{γµ, γν} + {γµ, γβ}γν + γν{γµ, γβ} − {γµ, γν}γβ u 4 µ  β µν ν µβ = γ g − γ g uµ

Using this, we can simplify the structure coming from the spin-3/2 propagator in (D.9):

 1 2(P − k) (P − k) (P − k)αγβ − (P − k)βγα  g − γ γ − α β + γβµνu k . (D.11) αβ 3 α β 3M 2 3M µ ν

Let us do this term by term: βµν µ µ gαβγ uµkν = (γαk − kg/ α) uµ , (D.12) 1 1 − γ γ γβµνu k = − (4γ gµν − γ γµγν) u k 3 α β µ ν 3 α α µ ν 1 = − (4γ gµν − γ {γν, γµ}) u k 3 α α µ ν (D.13) 1 = − (4γ gµν − 2γ gµν) u k 3 α α µ ν 2 = − γ kµu , 3 α µ µ where we used that P uµ = 0 and (P/ − M)uµ = 0,

2 2   − (P − k) (P − k) γβµνu k = − (P − k) (P − k) γβgµν − γνgµβ u k 3M 2 α β µ ν 3M 2 α β µ ν 2 = − (P − k) (P/ − k/)kµ − (P − k) (P − k)µk/
u 3M 2 α α µ (D.14) 2 = − (P − k) P/ kµ − (P − k) P µk/
u 3M 2 α α µ 2 = − (P − k) kµu , 3M α µ

95 Måns Holmberg Master Thesis

and finally

((P − k) γ − (P − k) γ ) 1   α β β α γβµνu k = ((P − k) γ − (P − k) γ ) γβkµ − kg/ µβ u 3M µ ν 3M α β β α µ 1  = (P − k) γ γβkµ − (P − k) γ kg/ µβ 3M α β α β β µ µβ − (P − k)βγαγ k + (P − k)βγαkg/ uµ 1  = 4(P − k) kµ − (P − k) {k,/ γ }gµβ (D.15) 3M α α β µ µ  − γα(P/ − k/)k + (P − k) γαk/ uµ 1 = 2(P − k) kµ − γ P/ kµ
u 3M α α µ 2 1 = (P − k) kµu − γ kµu . 3M α µ 3 α µ

µ As a result, almost every term of (D.11) cancels and we are left with only −kg/ αuµ, hence

i(P/ − k/ + M) S(T )(P − k, M)(V )βµν(k)u = 3iec(8) kg/ µ u αβ T T γ nm µ nn (P − k)2 − M 2 α µ k2 + 2(P · k) − k/(P/ − M)
= 3ec(8) gµ u (D.16) nn k2 + 2(P · k) α µ

(8) µ = 3ecnn g αuµ .

Here we also used that the mass in the propagator is the same as the mass of the incoming decuplet (8) state, since cnm, in (B.42), is diagonal; and

1 P · k = − (P − k)2 − M 2
. (D.17) 2

By combining (D.16) and the rest of (D.9), we obtain

(6)µ (8) (1) hAe µ Mnij kµ = 3cnn cnij √ q uµ 2 2Fπ (D.18) (2) (2) hAe µ = (cnij − cnji) √ q uµ , 2 2Fπ where the last equality holds, since one can show that (proof by Mathematica)

(8) (1) (2) (2) 3cnn cnij = cnij − cnji . (D.19)

96 Måns Holmberg Master Thesis

Finally, summing over all diagrams yields the wanted result

X (l)µ hAe  (2) µ (2) µ µ (2) µ (2) (2) µ M kµ = √ c k − c (k + q ) + c q + (c − c )q uu¯ µ = 0 . (D.20) nij 2 2F nij nij nji nij nji l π

97 Chapter 6

Bibliography

[1] Michael E. Peskin and Daniel V. Schroeder. An Introduction to Quantum Field Theory. Westview Press, 1995.

[2] Steven Weinberg. Phenomenological Lagrangians. Physica, A96:327, 1979.

[3] J. Gasser and H. Leutwyler. Chiral Perturbation Theory to One Loop. Annals Phys., 158:142, 1984.

[4] J. Gasser and H. Leutwyler. Chiral Perturbation Theory: Expansions in the Mass of the Strange Quark. Nucl. Phys., B250:465, 1985.

[5] Stefan Scherer. Introduction to chiral perturbation theory. Adv. Nucl. Phys., 27:277, 2003.

[6] Stefan Scherer and Matthias R. Schindler. A Primer for Chiral Perturbation Theory. Lect. Notes Phys., 830, 2012.

[7] Gilberto Colangelo et al. Review of lattice results concerning low energy particle physics. Eur. Phys. J., C71:1695, 2011.

[8] Måns Holmberg and Stefan Leupold. The relativistic chiral Lagrangian for decuplet and octet baryons at next-to-leading order. Eur. Phys. J., A54(6):103, 2018.

[9] Vladimir Pascalutsa, Marc Vanderhaeghen, and Shin Nan Yang. Electromagnetic excitation of the Delta(1232)-resonance. Phys. Rept., 437:125–232, 2007.

[10] T. Ledwig, J. Martin-Camalich, V. Pascalutsa, and M. Vanderhaeghen. The Nucleon and ∆(1232) form factors at low momentum-transfer and small pion masses. Phys. Rev., D85:034013, 2012.

[11] Roger F. Dashen and Aneesh V. Manohar. Baryon - pion couplings from large N(c) QCD. Phys. Lett., B315:425–430, 1993.

98 Måns Holmberg Master Thesis

[12] A. Semke and M. F. M. Lutz. Baryon self energies in the chiral loop expansion. Nucl. Phys., A778:153–180, 2006.

[13] C. Patrignani et al. Review of Particle Physics. Chin. Phys., C40(10):100001, 2016.

[14] Shao-Zhou Jiang, Yan-Rui Liu, Hong-Qian Wang, and Qin-He Yang. Chiral Lagrangians with decuplet baryons to one loop. Phys. Rev., D97(5):054031, 2018.

[15] M. Lorenz et al. Strange hadron production at SIS energies: an update from HADES. J. Phys. Conf. Ser., 668(1):012022, 2016.

[16] M. F. M. Lutz et al. Physics Performance Report for PANDA: Strong Interaction Studies with Antiprotons. 2009.

[17] B.C. Hall. Lie Groups, Lie Algebras, and Representations: An Elementary Introduction. Graduate Texts in Mathematics. Springer, 2003.

[18] Y. Kosmann-Schwarzbach and S.F. Singer. Groups and Symmetries: From Finite Groups to Lie Groups. Universitext. Springer New York, 2009.

[19] F.W. Warner. Foundations of Differentiable Manifolds and Lie Groups. Graduate Texts in Mathematics. Springer New York, 2013.

[20] Jean Gallier. Notes on group actions manifolds, lie groups and lie algebras, April 2005.

[21] M.J.D. Hamilton. Mathematical Gauge Theory: With Applications to the Standard Model of Particle Physics. Universitext. Springer International Publishing, 2017.

[22] S. Weinberg. The Quantum Theory of Fields. Number v. 2 in The Quantum Theory of Fields 3 Volume Hardback Set. Cambridge University Press, 1995.

[23] H. Georgi. Lie Algebras In Particle Physics: from Isospin To Unified Theories. Frontiers in Physics. Avalon Publishing, 1999.

[24] S. Weinberg. The Quantum Theory of Fields. Number v. 1 in Quantum Theory of Fields, Vol. 2: Modern Applications. Cambridge University Press, 1995.

[25] M. Srednicki. Quantum Field Theory. Cambridge University Press, 2007.

[26] David J. Gross and Frank Wilczek. Ultraviolet behavior of non-abelian gauge theories. Phys. Rev. Lett., 30:1343–1346, Jun 1973.

[27] Kenneth G. Wilson. Confinement of quarks. Phys. Rev. D, 10:2445–2459, Oct 1974.

99 Måns Holmberg Master Thesis

[28] Richard W. Haymaker. Confinement studies in lattice qcd. Physics Reports, 315(1):153 – 173, 1999.

[29] Carlos Granados, Stefan Leupold, and Elisabetta Perotti. The electromagnetic Sigma-to-Lambda hyperon transition form factors at low energies. Eur. Phys. J., A53(6):117, 2017.

[30] Vladimir Pascalutsa and Marc Vanderhaeghen. The Nucleon and Delta-resonance masses in relativistic chiral effective-field theory. Phys. Lett., B636:31–39, 2006.

[31] Elisabetta Perotti, Göran Fäldt, Andrzej Kupsc, Stefan Leupold, and Jiao Jiao Song. Polarization observables in e+e− annihilation to a baryon-antibaryon pair. Phys. Rev. D, 99:056008, Mar 2019.

[32] Elizabeth Ellen Jenkins and Aneesh V. Manohar. Chiral corrections to the baryon axial currents. Phys. Lett., B259:353–358, 1991.

[33] Wolfram Research, Inc. Mathematica, Version 11.0. Champaign, IL, 2019.

[34] Vladyslav Shtabovenko, Rolf Mertig, and Frederik Orellana. New Developments in FeynCalc 9.0. Comput. Phys. Commun., 207:432–444, 2016.

[35] R. Mertig, M. Böhm, and A. Denner. Feyn calc - computer-algebraic calculation of feynman amplitudes. Computer Physics Communications, 64(3):345 – 359, 1991.

[36] Klas Marcks von Würtemberg. Response of the PANDA electromagnetic calorimeter to photons with energies below 100 MeV. PhD thesis, Stockholm University, 2011.

[37] Edward Witten. Baryons in the 1n expansion. Nuclear Physics B, 160(1):57 – 115, 1979.

[38] Roger Dashen and Aneesh V. Manohar. Baryon-pion couplings from large-nc qcd. Physics Letters B, 315(3):425 – 430, 1993.

100