El papel del quark charm en la regla Delta I = 1/2

UNIVERSIDAD AUTÓNOMA DE MADRID

El papel del quark charm en la regla

∆I = 1/2

(On the rôle of the charm quark in the ∆I = 1/2 rule)

Memoria de Tesis Doctoral

realizada por

Eric Endreß

dirigida por

Dr. Carlos Roberto Pena Ruano

Presentada ante el Departamento de Física Teórica de la Universidad Autónoma de Madrid

para la obtención del Título de Doctor en Ciencias Madrid, diciembre de 2013

Motivation and aim

Light hadron physics is one of the key sources allowing to shape the understanding of strong and weak interactions. The dynamics of hadrons at low energies is involved in many interest- ing fundamental problems. Examples are the determination of the fundamental parameters of the Standard Model such as quark masses and CKM matrix elements, or the investigation of hadron structure properties including baryonic and mesonic form factors. Further exam- ples comprise the understanding of CP violation in non-leptonic kaon decays, or dynamical phenomena such as the spontaneous breaking of chiral symmetry and the enhancement of the octet channel in kaon decays leading to the puzzling ∆I = 1/2 rule. Experimentally these phenomena are thoroughly explored. Although there exists a well-established, fundamental description of the strong interaction in terms of a quantum field theory, Quantum Chromo- dynamics (QCD), the theoretical understanding and calculations of these problems remain challenging tasks. This is mainly due to the peculiar non-perturbative character of the low- energy QCD regime which prohibits the application of perturbation theory.

The formulation of QCD on a discrete space-time lattice, lattice QCD, provides a suitable tool to tackle the aforementioned phenomena from first principles, i.e. without any assumptions nor model-dependencies. Lattice QCD is a non-perturbative approach to the gauge theory of the strong interaction. It amounts to compute mathematically well-defined regularized, Euclidean functional integrals by means of numerical simulations with controlled systematic errors. Nowadays, the lattice approach is the principal non-perturbative tool for detailed stud- ies of flavor physics in the quark sector of the Standard Model and for gaining insight into the physics that may lie beyond. In addition to rising computing power thanks to remarkable developments in the hardware and software industry, recent breakthroughs in the design of al- gorithms, engineered for lattice QCD applications, set the stage for performing lattice studies in the physical regime (near the continuum limit, with large volumes and at physical input quark mass values). These prerequisites allow that large-scale computer simulations of e.g.

hadronic matrix elements achieve accuracies which have decisive impact on current studies of Standard Model flavor physics. Besides, it is possible to set limitations for potential Standard Model extensions.

Kaon decays, which involve the interplay of strong and electroweak interactions, generally re- main in a special focus of active research. An important phenomenon that lacks a satisfactory theoretical explanation is the ∆I = 1/2 rule, which is associated with non-leptonic K → ππ decays. Parametrized through the ratio of transition amplitudes,∣A0∣/∣A2∣ ≈ 22, the rule refers to the experimental observation that the decay amplitude of an (strong) isospin-¹₂ kaon which decays into a two-pion state of total isospin I = 0 is about twenty times larger than the de- cay amplitude into a final two-pion state of I = 2. In other words, the strangeness-changing non-leptonic decays of kaons are observed to exhibit a selection rule: ∆I= 1/2 transitions are considerably enhanced compared to the ∆I = 3/2 decay channel. Within the Standard Model, short-distance QCD and electroweak effects yield only a moderate contribution to the ratio.

Therefore, the main part of the enhancement of A₀ is expected to come from long-distance, i.e. non-perturbative, QCD effects or, if this is not the case, from new physics.

In the low-energy regime of QCD various sources at different energy scales might be responsible for the large ∆I = 1/2 enhancement. This includes pionic final-state interactions at very low energies around 100 MeV, physics at the intrinsic QCD scale of a few hundred MeV, or physics at the scale of the charm quark mass, i.e. around 1.3 GeV. So far, it remains unclear whether the experimental observation is possibly the result of an accumulation of several effects, or mainly due to a single cause or mechanism.

In 2004, a theoretically well-defined strategy was proposed which aims at disentangling non- perturbative QCD contributions from its various potential origins. It is based on an approach that avoids the challenging direct lattice computation of K → ππ decay amplitudes. Instead, the low-energy constants (LECs) of the associated CP-conserving ∆S = 1 weak Hamiltonian in the chiral low-energy effective theory are computed, from which the amplitudes can be de- termined at some given order in the chiral expansion. Thus, the idea amounts to match lattice computations of suitable correlation functions to the corresponding counterparts calculated within the framework of chiral perturbation theory (χPT) which allows to extract the LECs.

The simplification of the proposed strategy comprises to leave aside final-state interactions of the two pions and the intricate computation of lattice four-point correlation functions in large volumes. Moreover, the computations are rendered simpler because the matching procedure does neither require physical kinematics nor physical quark masses as long as the valid regime of χPT is reached.

An indispensable, albeit computationally demanding, ingredient of studying kaon decays on the lattice is the use of Ginsparg-Wilson fermions, i.e. of a fermionic discretization which preserves chiral symmetry at non-zero lattice spacing and has an exact chiral symmetry in the limit of vanishing quark masses. Preserving chiral symmetry does not only enable a reliable matching to χPT predictions but, furthermore, simplifies decisively the renormalization pat- tern of the involved four-fermion operators. Mixing of composite operators mediating weak

∆S= 1 transitions is alleviated to the extent that four-fermion operators renormalize like their counterparts in the continuum.

The main focus of this strategy is to understand the rôle of the charm quark as a possible source behind the mechanism of the ∆I = 1/2 enhancement. Indeed, one of the standard proposed explanations for the phenomenon is related to the fact that the charm quark mass is much larger than the masses of light quarks. For this reason, the leading LECs of the chiral low-energy effective theory are proposed to be monitored as a function of the charm quark mass. In order to do so, the charm quark is kept as an active degree of freedom in the formu- lation of the effective theory, i.e. the charm quark is not integrated out, and thus remains a tunable input parameter in the simulations.

Subsequently, a two-step procedure is considered. In the first step, the charm quark is chosen to be degenerate with the three lightest quarks, i.e. mc= mu = md= ms. In the chiral limit the theory thus has an exact SU(4)_L× SU(4)_R symmetry and the results of four-flavor lattice QCD are matched to the corresponding predictions of SU(4) χPT. Computations in this setup reveal intrinsic QCD contributions. In the second step, the effects stemming from the heavier charm quark mass are exposed by monitoring the transition amplitudes as m_c increases from the mass-degenerate limit towards its physical value. However, as soon as mc≠ mu= md= ms

additional contributions to the correlation functions have to be evaluated. These substantially complicate lattice simulations due to the appearance of closed quark loops, which involve quark propagators that start and end at the same lattice site. Their computation by means of simple lattice techniques yields an ill-behaved numerical problem, which disallows to draw physical statements.

For that reason, this dissertation puts the focus on the introduction, development and test-

ing of variance reduction techniques with the aim to extract a signal from the contributions involving closed quark loops. This comprises the combination of two well-established lattice tools, namely, low-mode averaging and the technique of stochastic volume sources as well as the introduction of a promising technique that is new in the context of lattice QCD; referred to as probing.

A non-degenerate charm quark mass additionally complicates the renormalization pattern of the problem since more composite operators contribute to the transition amplitudes while a different computational setup for computing quark propagators is needed for some of the observables. In particular, mixing among the operators occurs. Different approaches of ex- tracting the required renormalization constants are discussed, proposed and tested.

Despite having been suggested about one decade ago, no conclusion regarding the rôle of the charm quark could be made due to the large statistical noise occurring when diagrams with closed quark loops are involved. In this work it is shown how the introduction of stochastic and probing techniques allows to obtain results for bare transition amplitudes at mc≠ mu for moderately large values of the charm quark mass.

Objetivo y motivación

La física de hadrones ligeros es una de las áreas clave para nuestra comprensión de las interac- ciones fuertes y débiles. Muchos problemas esenciales involucran la dinámica hadrónica a bajas energías: e.g. los valores de parámetros fundamentales del Modelo Estándar (como las masas de los quarks y los elementos de la matriz CKM); fenómenos dinámicos espectaculares como la rotura espontánea de la simetría quiral o el favorecimiento de las desintegraciones de kaones en el canal de octete (la famosa regla ∆I = 1/2); la violación de CP en las desintegraciones no leptónicas de kaones; o las propiedades estructurales de los hadrones, como factores de forma mesónicos y bariónicos, exploradas por muchos experimentos. A pesar de que poseemos una descripción fundamental de las interacciones fuertes en términos de una teoría cuántica de campos, la Cromodinámica Cuántica (QCD), sigue siendo muy difícil obtener a partir de la misma soluciones cuantitativas a los problemas mencionados, debido al carácter intrínseca- mente no perturbativo de su régimen de baja energía.

La formulación de las teorías cuánticas de campos en un retículo espacio-temporal proporciona una herramienta natural para tratar este problema con ayuda de simulaciones numéricas. La QCD en el retículo permite realizar cálculos no perturbativos que no asumen ninguna propiedad dependiente de un modelo particular para los fenómenos descritos, de modo que es posible con- trolar completamente los errores sistemáticos. Esta formulación constituye la principal fuente de información no perturbativa en el estudio de la dinámica del sabor en el sector quark del Modelo Estándar y en sus extensiones. En los últimos años ha tenido lugar una revolución en los algoritmos utilizados en QCD en el retículo, que abre la posibilidad de realizar cál- culos directamente en el régimen físico (cerca del límite continuo, en volúmenes grandes, y, sobre todo, a valores físicos de las masas de los quarks). En este contexto, resulta posible por primera vez plantear cálculos de elementos de matriz hadrónicos cuya incertidumbre total es lo suficientemente pequeña como para tener un impacto en el análisis desde primeros principios de la dinámica del sabor en el Modelo Estándar, así como en la obtención de límites para las diversas extensiones propuestas para el mismo.

Los desintegraciones de kaones, las cuales implican interacciones electrodébiles y fuertes, per- manecen en el centro de atención de la investigación en Física. Un fenómeno importante, que aún carece de una explicación teórica convincente, es la regla ∆I = 1/2 asociada a desinte- graciones no-leptónicos K → ππ. Parametrizada por el cociente de amplitudes de transición,

∣A0∣/∣A2∣ ≈ 22, la regla hace referencia a la observación experimental que la amplitud de desin- tegración de un kaon de isospin-¹₂ (isospin fuerte), el cual decae en un estado de dos piones con isospin total I= 0, es 20 veces mayor que la amplitud de desintegración en dos piones de isospin I = 2. En otras palabras, los desintegraciones no-leptónicos de kaones que cambian extrañeza exhiben una regla de selección: las transiciones con ∆I = 1/2 son considerablemente mayores comparados respecto a los canales de desintegración con ∆I = 3/2. En el Modelo Estándar, efectos electrodébiles y de corta distancia de QCD contribuyen moderadamente al cociente.

Por lo tanto, la mayor parte del aumento de A₀ se espera de efectos de larga distancia, o sea

efectos no-perturbativos de QCD, o, si no es el caso, de nueva física.

En el régimen de bajas energías de QCD existen varios orígenes potenciales, a diferentes escalas de energía, de la regla ∆I = 1/2: interacciones piónicas de estado final a energías muy bajas, alrededor de 100 MeV; física a la escala intrínseca de QCD de pocos cientos de MeV; o física a la escala de la masa del quark charm, es decir alrededor de 1.3 GeV. Actualmente sigue sin haber una respuesta clara a la pregunta de si la observación experimental es el resultado de la acumulación de varios efectos, o se debe primordialmente a una única causa o mecanismo.

En 2004 fue propuesta una estrategia teóricamente bien definida, con el objetivo de aislar las contribuciones de QCD no perturbativa con distinto origen. Se basa en la idea de evitar las dificultades del cálculo directo de las amplitudes de desintegración K → ππ, obteniendo en su lugar las constantes de baja energía (LECs) del hamiltoniano débil ∆S = 1 CP-simétrico asociado a las mismas en la descripción efectiva de baja energía, a un orden dado en la expan- sión quiral. Para ello, se igualan funciones de correlación adecuadas, calculadas en el retículo, con sus equivalentes calculados mediante la Teoría de Perturbaciones Quirales (χPT), lo que permite extraer las LECs. La simplificación que comporta esta estrategia implica ignorar las interacciones de estado final entre los dos piones, pero también evita el complicado cálculo de funciones de correlación a cuatro puntos en volúmenes grandes necesario para una deter- minación directa de las amplitudes. Además, el cálculo no tiene por qué ser realizado en la cinemática física en términos de masas de quark, siempre que las mismas permanezcan en el régimen de validez de χPT.

Un elemento indispensable, aunque numéricamente costoso, en el estudio de las desintegra- ciones de kaones en el retículo es el uso de fermiones de Ginsparg-Wilson, es decir de una discretización fermiónica que preserva la simetría quiral a espaciado reticular distinto de cero y posee una simetría quiral exacta en el límite de masas de quark nulas. La preservación de la simetría quiral no sólo permite una comparación fiable con las predicciones de χPT, sino que además simplifica dramáticamente la renormalización de los operadores de cuatro fermiones contenidos en el hamiltoniano efectivo. De hecho, la mezcla de operadores que median transi- ciones débiles ∆S= 1 se reduce a la misma estructura que aparece en el continuo.

La característica más importante de esta estrategia es su énfasis en comprender el papel del quark charm como el posible responsable del mecanismo subyacente a la regla ∆I = 1/2. El hecho experimental de que su masa es mucho mayor que las típicas escalas de baja energía de QCD, de alrededor de pocos cientos de MeV, ha dado lugar a la conjetura de que su carácter pesado es el principal responsable del aumento de la amplitud A₀. Por este motivo, la estrate- gia implica estudiar las LECs del orden dominante de la teoría quiral efectiva como función de la masa del charm. Para ello, el quark charm se mantiene como un grado de libertad activo en la formulación de la teoría efectiva, es decir no se integra a través de su escala asociada, y por lo tanto se mantiene como un parámetro de input en las simulaciones.

A continuación, se desarrolla un procedimiento en dos pasos. Primero, la masa del quark charm se mantiene degenerada con la de los tres quarks ligeros, mc= mu= md= ms. La teoría resultante tiene una simetría SU(4)_L×SU(4)_R en el límite quiral, y se comparan los resultados de QCD en el retículo con cuatro sabores con las predicciones correspondientes de la χPT SU(4). En el segundo paso, se aísla el efecto de la masa física del charm estudiando las ampli- tudes de transición a medida que se aumenta m_c desde el límite de masas degeneradas hasta su valor físico. Sin embargo, n el momento en que mc> m^u, aparecen nuevas contribuciones a las funciones de correlación, que deben ser evaluadas. Esto complica de manera sustancial las simulaciones en el retículo, debido a la aparición de loops de quarks cerrados, es decir de propagadores que vuelven a su punto de partida. Su cálculo mediante técnicas simples resulta imposible debido a los elevados niveles de ruido estadístico, lo que había impedido hasta ahora extraer información física.

Por este motivo, la presente tesis se centra en la introducción, desarrollo y comprobación de

técnicas de reducción de varianza, con el objetivo de extraer una señal de las contribuciones que incluyen loops de quarks cerrados. En particular, se estudia la combinación de dos técnicas establecidas en QCD en el retículo: low-mode averaging y fuentes estocásticas volumétricas;

así como una tercera técnica, muy prometedora y nueva en este contexto, denominada “prob- ing”.

Además, una diferencia de masas no nula entre el charm y el up complica la renormalización de los operadores compuestos que intervienen en las amplitudes de transición, por lo que son necesarias nuevas técnicas de cálculo para los propagadores de quark en ciertos observables.

En particular, aparecen fenómenos de mezcla de operadores. Se introducirán, discutirán y estudiarán varios métodos para extraer las constantes de renormalización asociadas.

A pesar de haber sido sugerido hace una década, ninguna conclusión sobre el papel de la masa del quark charm ha sido formulada debido a los elevados niveles de ruido estadístico que ocurren cuando diagramas con loops de quarks cerrados son considerados. La presente tesis muestra cómo la introducción de las técnicas de fuentes estocásticas volumétricas y de “prob- ing” permite extraer una señal para las amplitudes de desintegración desnudas, con m_c≠ mu, para valores moderadamente grandes de la masa del charm.

TABLE OF CONTENTS

Table of contents

1 Kaon Physics in the Standard Model 1

1.1 Standard Model . . . 1

1.1.1 Symmetries and field representations . . . 1

1.1.2 Lagrange formalism . . . 3

1.1.3 Quark flavor mixing . . . 4

1.1.4 Strong Isospin . . . 6

1.1.5 GIM mechanism . . . 6

1.2 Non-leptonic kaon decay phenomenology . . . 7

1.2.1 CP violation in neutral kaon decays . . . 7

1.2.2 ∆I = 1/2 rule . . . 9

1.3 Low-energy effective theory of kaon decays . . . 11

1.3.1 Effective weak Hamiltonian . . . 11

1.3.2 Classification of operators . . . 14

1.3.3 Three-flavor versus four-flavor theory . . . 17

2 The lattice QCD approach 19 2.1 Lattice regularization . . . 20

2.1.1 The lattice . . . 20

2.1.2 Gauge fields . . . 21

2.1.3 Matter fields . . . 23

2.2 Simulating observables . . . 25

2.2.1 Monte Carlo integration . . . 26

2.2.2 Correlation functions . . . 28

2.3 Parameters and reach of lattice simulations . . . 31

2.3.1 Input parameters . . . 31

2.3.2 Reach of lattice simulations . . . 33

2.4 Neuberger fermions . . . 35

2.4.1 Operator . . . 37

2.4.2 Properties . . . 37

2.4.3 Computational aspects . . . 39

2.5 Chiral perturbation theory (χPT) . . . 42

2.5.1 Quenched χPT . . . 43

2.5.2 Kinematical regions: p-regime and -regime . . . 44

2.6 Non-leptonic kaon decays . . . 46

2.6.1 Direct computation . . . 47

2.6.2 Indirect computation . . . 48

3 Contribution of the charm quark to the ∆I= 1/2 rule 49 3.1 Introduction . . . 49

3.2 Strategy . . . 50

3.3 Step 1: Degenerate charm quark . . . 53

3.3.1 Lattice formulation . . . 53

3.3.2 χPT formulation . . . 54

3.3.3 Renormalization factors and Wilson coefficients . . . 56

3.3.4 Matching condition . . . 57

3.3.5 K → ππ amplitudes . . . 58

3.4 Step 2: Non-degenerate charm quark . . . 59

3.4.1 Lattice formulation . . . 59

TABLE OF CONTENTS

3.4.2 χPT formulation . . . 62

3.4.3 Matching condition . . . 63

3.4.4 Renormalization factors for four-quark operators . . . 65

4 Propagator techniques 71 4.1 Introduction . . . 71

4.2 Low-mode averaging (LMA) . . . 73

4.3 Hybrid approach . . . 77

4.4 Stochastic volume sources (SVS) . . . 78

4.4.1 Dilution . . . 80

4.4.2 Application to Eight-diagrams . . . 82

4.4.3 Application to Eye-diagrams . . . 84

4.5 Probing . . . 90

4.5.1 Application to flavor singlet two-point functions . . . 93

4.5.2 Application to Eye-diagrams . . . 96

5 Numerical results for a non-degenerate charm quark mass 102 5.1 K→ π transitions . . . 102

5.1.1 Results in the p-regime . . . 105

5.1.2 Results in the -regime . . . 112

5.1.3 Dependence on charm quark mass . . . 116

5.2 Subtraction coefficients . . . 117

5.2.1 Non-perturbative determination . . . 117

5.3 Implications for physical amplitudes . . . 118

6 Summary and conclusions 121 7 Resumen y conclusiones 124 Appendix 127 A Theoretical remarks 128 A.1 Euclidean space . . . 128

A.2 Wolfenstein parametrization . . . 129

A.3 Current interaction terms . . . 130

B Technical details 131 B.1 Data analysis . . . 131

B.1.1 Jackknife error . . . 131

B.1.2 χ²-minimization . . . 132

B.2 Algorithmic aspects . . . 133

B.2.1 Conjugate gradient algorithm . . . 133

B.2.2 Generalized minimal residual algorithm . . . 134

B.2.3 Low-mode preconditioning . . . 135

C K → π correlation functions 137 C.1 Eight-diagram . . . 140

C.1.1 Color-connected . . . 140

C.1.2 Color-disconnected . . . 144

C.1.3 Combined techniques . . . 146

C.2 Eye-diagram . . . 152

C.2.1 Color-connected . . . 152

TABLE OF CONTENTS

C.2.2 Color-disconnected . . . 156

C.2.3 Combined techniques . . . 158

C.3 Closed Eye-diagram . . . 164

D Renormalization 166 D.1 Some comments on renormalization . . . 166

D.1.1 Rôle of Ginsparg-Wilson fermions . . . 166

D.1.2 Three versus four active flavors . . . 166

D.2 K→ vacuum matrix elements . . . 168

D.2.1 K → vacuum transition via Q^±₁ . . . 168

D.2.2 K → vacuum transition via Q^±₂ . . . 172

E Chiral perturbation theory formulas 173 E.1 Corrections to chiral weak Hamiltonians . . . 173

E.2 NLO corrections to K→ π matrix elements of Q^±₂ . . . 175

F Miscellaneous 176 F.1 Variance of stochastic volume sources . . . 176

F.1.1 Noise types . . . 177

F.1.2 Dilution . . . 179

F.2 Propagator on a unit gauge field . . . 182

F.3 Pseudoscalar density approach . . . 183

G Plots 185 G.1 Probing method . . . 186

G.1.1 Goodness . . . 186

G.1.2 Single closed Loops . . . 187

G.1.3 Two-point correlation functions . . . 189

G.1.4 Eye-diagram . . . 191

G.2 K→ π transitions . . . 196

G.2.1 Fit range dependence in -regime . . . 196

Acknowledgement 198

References 199

TABLE OF CONTENTS

Chapter 1

Kaon Physics in the Standard Model

1.1 Standard Model

The Standard Model (SM) is a gauge theory of the elementary particles. Disregarding gravity it describes the strong, weak and electromagnetic interaction of spin-¹₂ fermions, that is, of quarks and leptons. The fundamental interactions, which are formulated in terms of abelian and non-abelian gauge theories, are mediated by the exchange of spin-1 particles, the gauge bosons.

Within the SM formulation the elementary particles obtain their masses via the Brout-Englert- Higgs mechanism [1, 2, 3, 4]. The associated Higgs particle, a spin-0 boson, has been missing experimentally for many years. In 2012, however, the ATLAS and CMS experiment at the LHC, CERN[5], both reported on the discovery of a Higgs-like particle [6, 7, 8, 9]. An unam- biguous identification as the SM Higgs boson is a central part of current research activities at the LHC.

The basic symmetries and fields of the SM are summarized in the initial section 1.1.1, followed by the presentation of its Lagrangian 1.1.2, and quark flavor mixing 1.1.3. Of particular im- portance throughout this work will be the concept of strong isospin and the GIM mechanism which are briefly reviewed in section 1.1.4 and 1.1.5 respectively.

1.1.1 Symmetries and field representations Gauge fields

The gauge group of the SM is the direct product

SU(3)_C× SU(2)_L× U(1)_Y, (1.1)

which has the associated quantum numbers color C, weak isospin T and hypercharge Y , respectively¹ .

The postulated color gauge group SU(3)_C [10, 11, 12, 13] forms the basis of the theory of the strong interaction, Quantum Chromodynamics (QCD), and its associated particles are the eight color-carrying gluons, described by the gauge fields

SU(3)_C∶ G^a_µ, a= 1, . . . , 8. (1.2)

1The group SU(2) does not carry the subscript corresponding to weak isospin. As reviewed in the next section, the used letter accounts for constructing doublets of weak isospin made up of left-handed (L) quarks and leptons.

1.1 Standard Model Kaon Physics

As all gauge fields, they transform in the adjoint representation of their group.

The electromagnetic and weak interaction are combined into the electroweak Glashow-Weinberg- Salam theory [14, 15, 16, 17], which obeys the symmetry group SU(2)_L×U(1)_Y. The associated gauge fields can be labeled by

SU(2)_L∶ W_µⁱ, i= 1, 2, 3, (1.3)

U(1)_Y∶ B_µ. (1.4)

After the spontaneous symmetry breaking (SSB) of the electroweak symmetry into the elec- tromagnetic U(1)_EM gauge symmetry, linear combinations of these four fields describe the massless photon γ and the massive W^±and Z⁰ bosons. The charges associated with the gauge symmetries are related via

Y = 2(Q − T₃), (1.5)

where Q denotes the electric charge and T₃ the third component of the weak isospin. Under the symmetry group of the SM, SU(3)_C× SU(2)_L× U(1)_Y, the gauge fields transform in the following irreducible representations

G^a_µ∶ (8, 1, 0), (1.6)

W_µⁱ ∶ (1, 3, 0), (1.7)

B_µ∶ (1, 1, 0). (1.8)

Thus, the gluonic fields, for instance, transform as an octet under SU(3)_C transformations, as a singlet under SU(2)_L transformations and, like gauge bosons in general, do not have hypercharge.

Matter fields

The building blocks of a gauge theory are massless left-handed (L) and right-handed (R) fermions

ψ_L≡ P₋ψ=1

2(1 − γ5) ψ, ψ_R≡ P₊ψ= 1

2(1 + γ5) ψ. (1.9) They are distinguished by their weak isospin while there are two different types of fermions:

quarks and leptons. Both exist in three generations or flavors with in total six members each.

The six leptons are the electron, muon and tau and their corresponding neutrinos. The quarks are additionally classified in up-type quark (up, charm, top) and down-type quark (down, strange, bottom). Up-type quarks are connected with down-type quarks as well as charged leptons with neutrinos by the exchange of W^± bosons. Since W^± bosons couple only to left- handed fermions, the latter form SU(2)_L doublets, whereas right-handed fermions transform as singlets. For each generation this leads to the following multiplets

Q^I_L= (u^I_L

d^I_L) , u^I_R, d^I_R, L^I_L= (ν_L^I

e^I_L) , e^I_R, (1.10) where u and d refer to the up-type and down-type quark of each generation and the superscript I marks (weak) interaction eigenstates. Neutrinos are denoted by ν². Unlike all other known elementary fermions, neutrinos have so far only been observed as left-handed particles. The existence of right-handed neutrinos, i.e. the possibility that neutrinos are Dirac particles, can currently however not be excluded.

While leptons take part in the weak and electromagnetic interaction only, quarks additionally

2In experiments neutrino oscillation [18] has been verified [19]. The implied mixing of neutrino flavor states with neutrino mass states evidences non-zero neutrinos masses.

Kaon Physics 1.1 Standard Model

interact strongly. Consequently, quarks are triplets of the color group SU(3)_C, i.e. each field carries a color index, α= 1, 2, 3, which is omitted in the notation.

Regarding the symmetry group of the SM the matter fields are classified as follows

Q^I_L∶ (3, 2, 1/3), (1.11)

u^I_R∶ (3, 1, 4/3), (1.12)

d^I_R∶ (3, 1, −2/3), (1.13)

L^I_L∶ (1, 2, −1), (1.14)

e^I_R∶ (1, 1, −2). (1.15)

Scalar field

The symmetry of the SM allows a complex scalar field Φ which helps to spontaneously break the SU(2)_L×U(1)_Yinto the U(1)_EM gauge symmetry. This so-called Higgs field is introduced as a doublet of weak isospin and has neither spin nor electric or color charge. It transforms according to

Φ∶ (1, 2, −1). (1.16)

1.1.2 Lagrange formalism

The Lagrangian of the massless Standard Model L_SM is invariant under its symmetry group SU(3)_C× SU(2)_L× U(1)_Y and can be divided into four parts

L_SM= L_Fermion+ L_Gauge+ L_Higgs+ L_Yukawa. (1.17) Arranging the matter fields in the vector

ψ^T = {(u^I_L, d^I_L), u^I_R, d^I_R,(ν_L^I, e^I_L), e^I_R} , (1.18) the first term can be written

L_Fermion = ¯ψiγ^µD_µψ (1.19)

= ¯ψiγ^µ[∂µ+ ig_sG^a_µ¯λ_a

2 + igWµⁱ

¯σ_i

2 + ig^′B_µ¯y

2] ψ, (1.20)

where g_s, g and g^′ are the dimensionless coupling constants of the corresponding symmetry group. The matrices of generators, which are expressed in terms of the Gell-Mann matrices λ_a and the Pauli matrices σ_i, are given by

¯λ_a= diag {λa, λ_a, λ_a, λ_a,0, 0, 0} , (1.21)

¯σ_i= diag {σi,0, 0, σ_i,0} , (1.22)

¯y= diag {1/3, 1/3, 4/3, −2/3, −1, −1, −2} . (1.23) The covariant derivative Dµensures gauge invariance by treating the object Dµψunder gauge transformation on the same footing as the original field ψ. In doing so, Dµcouples matter and gauge fields.

The dynamics of the gauge particles is contained in the Yang-Mills part L_Gauge= −1

4G^a_µνG^µνa−1

4W_µνⁱ W^µνi−1

4B_µνB^µν, (1.24)

where the field strength tensors are given by

SU(3)_C∶ G^a_µν = ∂µG^a_ν− ∂νG^a_µ− g_sf_abcG^b_µG^c_ν, (1.25)

1.1 Standard Model Kaon Physics

SU(2)_L∶ W_µνⁱ = ∂µW_νⁱ− ∂νW_µⁱ − gijkW_µ^jW_ν^k, (1.26)

U(1)_Y∶ B_µν = ∂µB_ν− ∂νB_µ. (1.27)

Due to the non-abelian structure of the special unitary SU(N) groups, their associated gauge bosons acquire three-point and four-point self-interactions. This manifests in the terms pro- portional to the structure constants f_abc and _ijk which are quadratic in the gauge fields.

As in L_Fermion, mass terms are explicitly forbidden because they violate gauge invariance. For this reason, boson and fermion masses are generated via the Higgs mechanism³. Boson masses emerge from the Higgs-Gauge interaction term

L_Higgs= DµΦ^†D^µΦ+ V (Φ), V(Φ) = µ²Φ^†Φ− λ(Φ^†Φ)², (λ > 0). (1.28) Its potential V consists of a φ⁴-theory with a negative mass term, i.e. µ² < 0. The square of the covariant derivative leads to three-point and four-point interactions between the gauge and scalar fields.

The spontaneous breaking of gauge invariance effectively redistributes several degrees of free- dom of the involved fields and eventually three massive, physical vector fields can be identified:

the W^± and Z⁰ boson.

Fermion masses are generated starting from the Higgs-Fermion interaction Lagrangian L_Yukawa= Yij^(u)Q¯^I_L,iu^I_R,jΦ+ Yij^(d)Q¯^I_L,id^I_R,j˜Φ + Y_ij^(e)¯L^I_L,ie^I_R,j˜Φ + h.c. , (1.29) which couples left-handed and right-handed fermions via the Higgs field. The abbreviation h.c. refers to the hermitian conjugate counterpart and ˜Φ ≡ −iσ₂Φ^∗. The introduced Yukawa couplings Y are parametrized by 3× 3 matrices in generation space, i.e. here i, j = 1, 2, 3 refer to the first, second and third generation of fermions, respectively. The superscripts (u) and (d) again denote up-type and down-type quark.

Under the spontaneous breaking of the electroweak symmetry, the Yukawa couplings give rise to mass terms. Denoting the vacuum expectation value of the Higgs field by v and relating the Yukawa couplings to the quark mass matrices M via

M^(u)= v

√2Y^(u), M^(d)= v

√2Y^(d), M^(e)= v

√2Y^(e), (1.30) the Lagrangian of fermion mass terms results in

L_Yukawa= Mij^(u)¯u^I_L,iu^I_R,j+ Mij^(d)d¯^I_L,id^I_R,j+ Mij^(e)¯e^I_L,ie^I_R,j+ h.c. . (1.31) In this notation the SU(2)_L doublets are decomposed into their components.

1.1.3 Quark flavor mixing

By definition the quark mass eigenstates correspond to the basis in which the quark mass matrix is diagonal. The mass matrices derived in eq. (1.31) refer to the interaction eigenstates and can be diagonalized by means of four 3-dimensional unitary matrices

V_χ^(u)V_χ^(u)†= 1, V_χ^(d)V_χ^(d)†= 1, χ= L,R, (1.32) such that in the quark sector

V_L^(u)M^(u)V_R^(u)†= diag(mu, m_c, m_t), (1.33)

3While a Higgs-like boson has now been found, it remains an open problem under active study whether electroweak symmetry breaking takes strictly place as proposed in the SM (via a potential for a fundamental scalar field etc.), or via some other mechanism (extended Higgs sectors, dynamical symmetry breaking due to strong dynamics, etc.).

Kaon Physics 1.1 Standard Model

V_L^(d)M^(d)V_R^(d)†= diag(md, m_s, m_b). (1.34) The diagonalization results in a change of basis relating the mass eigenstates, denoted without superscript I, to the weak interaction eigenstates via

u_χ= Vχ^(u)u^I_χ, d_χ= Vχ^(d)d^I_χ, χ= L,R. (1.35) The electroweak, boson-matter interaction content of eq. (1.20) splits into a charged current and neutral current interaction term, i.e. L^int_EW= L^CC_EW+ L^NC_EW. The quark part of the charged current can be written as

L^CC_EW= − g

√2¯u^I_L,iγ^µd^I_L,iW_µ⁺+ h.c. , (1.36) where the charged physical W bosons are related to the SU(2)_L gauge fields via

W_µ^±= √1

2(Wµ¹∓ iWµ²) . (1.37)

In the mass basis the Lagrangian turns into L^CC_EW= − g

√2¯u_L,iγ^µ(V_L^(u)V_L^(d)†) d_L,iW_µ⁺+ h.c. . (1.38) That is, the charged weak interaction links the three up-quarks with a unitary rotation to the triplet of down-quarks and, thus, couples quark (mass) eigenstates of different generations.

The rotation is given by the Cabibbo-Kobayashi-Maskawa (CKM) matrix [20, 21]

V_CKM≡ V_L^(u)V_L^(d)†=⎛

⎜⎝

V_ud V_us V_ub V_cd V_cs V_cb V_td V_ts V_tb

⎞⎟

⎠, V_CKMV_CKM^† = 1, (1.39) whose elements satisfy

∑

i=d,s,b∣Vji∣²= 1, j = u, c, t, and ∑

j=u,c,t∣Vji∣² = 1, i = d, s, b. (1.40) In summary, the CKM matrix relates the eigenstates of the weak interaction to the mass eigenstates. Per convention the mixing occurs in the down-type quark sector, i.e.

⎛⎜

⎝ d^I s^I b^I

⎞⎟

⎠= V_CKM⎛

⎜⎝ d s b

⎞⎟

⎠. (1.41)

A unitary 3×3 matrix depends on nine real parameters. After removing five of them by suitable phase rotations of the quark fields, the CKM matrix is parametrized by four remaining param- eters: three rotation angles plus a complex phase. One of the two standard representations adopted by the Particle Data Group [22] is given by [23]

V_CKM=⎛

⎜⎝

c₁₂c₁₃ s₁₂c₁₃ s₁₃e^−iδ13

−s₁₂c₂₃− c₁₂s₂₃s₁₃e^iδ13 c₁₂c₂₃− s₁₂s₂₃s₁₃e^iδ13 s₂₃c₁₃ s₁₂s₂₃− c₁₂c₂₃s₁₃e^iδ13 −c₁₂s₂₃− s₁₂c₂₃s₁₃e^iδ13 c₂₃c₁₃

⎞⎟

⎠, (1.42) where c_ij ≡ cos θij and s_ij ≡ sin θij. The three angles θ_ij are the three real mixing parameters and the phase δ₁₃ is chosen to describe the relation between the 1^st and 3^rd family. The other common representation is the Wolfenstein parametrization [24], which is summarized in Appendix A.2.

1.1 Standard Model Kaon Physics

Within the electroweak quark sector of the SM, the complex phase e^iδ13 is the exclusive source of CP violation (CPV)⁴, where CP is the combined symmetry of charge conjugation C and parity inversion P. Charge conjugation interchanges particles and antiparticles by conjugating all internal quantum numbers. Under P the handedness of space is reversed. The violation of CP is closely connected to the number of generations. A world with only two flavors would not exhibit any CP violation, whereas the existence of an additional generation would yield three complex phases and thus provide three sources of CP violation.

In contrast to the charged current interaction part, the neutral current interaction term L^NC_EW remains diagonal in flavor space even after the transformation into the mass basis. Hence, at tree level the Standard Model lacks flavor changing neutral currents (FCNC). For completeness the expression of L^NC_EW is given in Appendix A.3.

1.1.4 Strong Isospin

Flavor quantum numbers are assigned to quarks. Except for charged weak processes, these quantum numbers are conserved by strong, electromagnetic, and neutral weak interactions and are therefore useful to describe the quark content of hadrons. Whereas strangeness, charm, bottomness and topness are associated with their corresponding quarks, the quantum number of strong isospin I combines up and down quarks.

Historically, the concept of isospin has been introduced because the nucleon can be viewed as having an internal degree of freedom with two allowed states: the proton and the neutron.

These have similar masses and are not distinguished by strong interactions. Due to the as- sociated internal SU(2) symmetry, proton and neutron form an isospin doublet with I = 1/2.

In the quark model language this is attributed to the fact that up and down quark form an isospin doublet and their interchange allows to transform neutron and proton into each other (isospin symmetry). The up and down quark have different isospin projections in z-direction, I_z = ±1/2, while the isospin projection is related to the quark content of particles via

I_z =¹₂[(nu− n_¯u) − (nd− nd¯)] , (1.43) where nu and n_d are the numbers of up and down quarks, and n_¯u and n_d¯are the numbers of up and down antiquarks, respectively. Several examples of particles and their isospin content are

π⁻[¯ud] ∶ 1(−1), π⁺[u ¯d] ∶ 1(+1), π⁰[¯uu, ¯dd] ∶ 1(0), (1.44) K⁺[u¯s] ∶ ¹₂(+¹₂), K⁰[d¯s] ∶ ¹₂(−¹₂), K¯⁰[s ¯d] ∶ ¹₂(+¹₂), K⁻[s¯u] ∶ ¹₂(−¹₂), (1.45) where the notation “particle[quark content] ∶ I(Iz)” is used.

1.1.5 GIM mechanism

In 1970, the Glashow-Iliopoulos-Maiani (GIM) mechanism [29] was introduced to explain the suppression of FCNC ∆S = 1 and ∆S = 2 processes, i.e. transitions where the strangeness quantum number changes by one or two units. Its formulation involved the prediction of the charm quark.

As an example, experiments reveal a strong suppression of the decay K_L⁰ → µ⁺µ⁻ whose branching ratio is measured to be [22]

B_µ+µ⁻= Γ(K_L⁰→ µ⁺µ⁻)

Γ(K_L⁰ → all) = (6.84 ± 0.11) × 10⁻⁹. (1.46)

4 The non-trivial structure of the QCD vacuum [25], which allows to resolve the axial U(1)A anomaly, predicts a CP-violating term proportional to the vacuum angle ¯θ [26] (strong CP violation). To agree with the measurements of the neutron electric dipole moment dn≃_M^e

n(_M^mq

n

θ¯) [27, 28], the parameter is bounded to θ¯≤ (10⁻⁹− 10⁻¹⁰) [22].

Kaon Physics 1.2 Non-leptonic kaon decay phenomenology

¯s

d

µ⁺

µ⁻ W⁺

W⁻ V_us^∗

V_ud

K_L⁰ u ν_µ

-

¯s

d

µ⁺

µ⁻ W⁺

W⁻ V_cs^∗

V_cd K_L⁰ c ν_µ

Figure 1.1: Box diagrams generating the K_L⁰ → µ⁺µ⁻decay.

The process is generated by the box diagrams of Figure 1.1. In the second diagram, the up quark is replaced by a charm quark while its contribution to the amplitude occurs with the opposite sign. So, in the limit of exact flavor symmetry, i.e. mc = mu, the two diagrams cancel exactly. However, the flavor symmetry is broken by the mass difference of the quarks (m_c≫ mu) such that the sum of the two diagrams is of the order

g⁴(m²c− m²u) M_W² ∼ m²_c

M_W² , (1.47)

The amplitude does not vanish but is strongly suppressed with respect to the case where the is no charm quark (first diagram only). This compensation of the up and charm quark contribution is referred to as GIM mechanism. The suppression mechanism works because both mu and mcare very light when measured in units of M_W; the mass of the W boson. The fact that their masses are very different in terms of the QCD scale is of little relevance here.

1.2 Non-leptonic kaon decay phenomenology

Non-leptonic K→ ππ decays have been playing an important rôle in the study of weak interac- tions and the shaping of the Standard Model. In 1964, they were the source for the discovery of indirect CP violation [30] and, in 1999, they additionally provided first experimental evidence of direct CP violation [31, 32].

Kaon decays, in general, involve the interplay of electromagnetic, strong and weak interac- tions. Thereby, especially the low-energy contributions of the strong interaction pose a huge theoretical challenge. Due to their intrinsic non-perturbative character, the calculation of the involved hadronic matrix elements is a complex task.

Kaon decays remain in the focus of current research since advances in their theoretical under- standing have the potential to reveal glimpses of new physics. In particular, the ∆I= 1/2 rule is still an unsolved phenomenon even after several decades of intensive research. To get an understanding of the underlying mechanism is the key motivation of this work.

In the next two sections, CP violation (section 1.2.1) and the ∆I = 1/2 rule (section 1.2.2) are reviewed. An introduction to these topics can for instance be found in Ref. [33]. All stated experimental values are taken from the PDG book 2012 [22] or are derived therefrom.

1.2.1 CP violation in neutral kaon decays

Kaons are bound states of the strangeness conserving strong interaction and are therefore naturally described in terms of eigenstates of strangeness. Arranged in doublets of strong isospin, in the quark-model language the strong interaction eigenstates are

K= (K⁺

K⁰) = (u¯s

d¯s), K¯ = (K¯⁰

K⁻) = (s ¯d

s¯u). (1.48)

1.2 Non-leptonic kaon decay phenomenology Kaon Physics

Individual neutral kaon states are not eigenstates of CP transformations. Choosing the phase convention

CP∣K⁰⟩ = −∣ ¯K⁰⟩, CP∣ ¯K⁰⟩ = −∣K⁰⟩, (1.49) CP eigenstates are constructed as follows

∣K_e⟩ = √1

2[∣K⁰⟩ − ∣ ¯K⁰⟩] , (CP-even), (1.50)

∣K_o⟩ = 1

√2[∣K⁰⟩ + ∣ ¯K⁰⟩] , (CP-odd). (1.51) Whereas produced via strong interactions, neutral kaons mix and decay due to weak inter- actions which do not conserve strangeness. The two physical neutral kaon states, i.e. the eigenstates of the weak interaction, are denoted K_S⁰ and K_L⁰. They have a well-defined mass and decay width and are linear combinations of the CP eigenstates∣K_e⟩ and ∣K_o⟩, viz.

∣K_L/S⁰ ⟩ = √ 1

1+ ∣˜∣²(∣K_o/e⟩ + ˜∣K_e/o⟩) , (1.52)

= 1

√2√

1+ ∣˜∣²((1 + ˜) ∣K⁰⟩ ± (1 − ˜) ∣ ¯K⁰⟩) , (1.53) where ˜ is a complex parameter. In the hadronic channel the state K_S⁰ exhibits mostly decays into two pions, whereas K_L⁰ predominantly decays into three pions. The decay into three pions has much less phase space available and, therefore, its observed lifetime is about two orders of magnitude longer than that of K_S⁰. Hence, the subscripts refer to long-lived (L) and short-lived (S) states respectively.

Two pions, π⁰π⁰ or π⁺π⁻, in a state with relative angular momentum l= 0 must be CP-even.

If CP were a conserved symmetry of the weak interaction, K_L⁰ would be equivalent to K_o and thus CP-odd. Hence, the decay K_L⁰ → ππ would be forbidden. However, in nature this decay exists and CP violation (CPV) occurs in a twofold way

K_L⁰ ∼ K_o + ˜K_e

?

direct CPV←→ ^′ ππ

? indirect CPV←→  ππ

Due to K⁰− ¯K⁰ mixing, the physical eigenstate K_L⁰ comprises a small CP-even component

∝ ˜K_e which can decay into two pions without violating CP. This small admixture of opposite CP, decaying in a CP-conserving way, is referred to as indirect CPV and parametrized through

= T[K_L⁰ → (ππ)₀]

T[K_S⁰→ (ππ)₀], ∣∣ = (2.228 ± 0.011) × 10⁻³. (1.54) Additionally, the CP-odd component of K_L, i.e. K_o, can directly decay into a final, CP-even two-pion state. This direct CPV is expressed by

^′=√1 2

⎛⎜⎜

⎜⎜⎜

⎝

T[K_L⁰ → (ππ)₂]

T[K_S⁰→ (ππ)₂]− T[K_S⁰ → (ππ)₂] T[K_S⁰ → (ππ)₀]

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶_≡w

⎞⎟⎟

⎟⎟⎟

⎠

, (1.55)

Kaon Physics 1.2 Non-leptonic kaon decay phenomenology

Meson I(J^P) Mass[MeV] Lifetime[s] Decay mode Ratio[%]

π⁰ 1(0⁻) 134.9766(6) 8.52(18) × 10⁻¹⁷ π^± 1(0⁻) 139.5702(4) 2.6033(5) × 10⁻⁸

K^{+ 1}₂(0⁻) 493.677(16) 1.2380(21)×10⁻⁸ K⁺→ π⁺π⁰ 20.66(8) K_S^{0 1}₂(0⁻) 497.614(24) 0.8954(4)×10⁻¹⁰ K_S⁰→ π⁰π⁰ 30.69(5) K_S⁰→ π⁺π⁻ 69.20(5)

Table 1.1: Experimental values used for the calculation of the kaon decay amplitudes.

and is three magnitudes smaller than the indirect one. More precisely Re^′

 = (1.65 ± 0.26) × 10⁻³. (1.56)

The experimentally measured quantities are the ratios of decay amplitudes

η₀₀≡ T[K_L⁰ → π⁰π⁰]

T[K_S⁰→ π⁰π⁰] = (2.221 ± 0.011) × 10⁻³, (1.57) and

η₊₋≡ T[K_L⁰ → π⁺π⁻]

T[K_S⁰→ π⁺π⁻]= (2.232 ± 0.011) × 10⁻³. (1.58) The quantities are related via

η₊₋=  + ^′ 1+ w/√

2, η₀₀=  − 2^′ 1− w√

2. (1.59)

1.2.2 ∆I = 1/2 rule

The ∆I = 1/2 rule refers to the experimental observation that the amplitude of a kaon decaying into two pions which are in a state of total strong isospin I = 0 is about twenty times larger than the decay amplitude into a final two-pion state of I = 2.

Pions have strong isospin I = 1. For this reason, the final two-pion state occurring in the process K→ ππ can have strong isospin I = 0 or I = 2. A state with I = 1 is ruled out by Bose symmetry since the spinless final pions in this decay have zero relative angular momentum l= 0. Considering that kaons have strong isospin I = 1/2, the weak decay changes the isospin content of initial and final state by either ∆I = 3/2 or ∆I = 1/2. Experiments evidence that the transition via the channel ∆I = 1/2 is substantially enhanced.

The partial widths of the different decay modes can be derived from the experimental values listed in Table 1.1, where J denotes the total angular momentum. It follows

Γ_π⁺_π⁻ = Γ (K_S⁰→ π⁺π⁻) = 5.087(1) × 10⁻¹² MeV, ∆I= 1 2,3

2, (1.60) Γ_π⁰_π⁰ = Γ (K_S⁰→ π⁰π⁰) = 2.256(5) × 10⁻¹² MeV, ∆I= 1

2,3

2, (1.61) Γ_π+π⁰ = Γ (K⁺→ π⁺π⁰) = 1.098(6) × 10⁻¹⁴ MeV, ∆I =3

2. (1.62) The corresponding isospin changes between the final two-pion and initial kaon states are indicated on the right-hand side. In the last row only ∆I= 3/2 transitions are possible since